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MATHEMATICS 
 LIBRARY 
 
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MATHEMATICAL TABLES: 
 
 CONTAINING THE 
 
 COMMON, HYPERBOLIC, AND LOGISTIC 
 
 LOGARITHMS. 
 
 SINES, TANGENTS, SECANTS, § VERSED SINES 
 BOTH NATURAL AND LOGARITHMIC. 
 
 TOGETHER WITH 
 SEVERAL OTHER TABLES 
 
 ki . USEFUL IN 
 t 
 
 MATHEMATICAL CALCULATIONS. 
 
 i To which is prefixed, 
 + Large and Original History of the Discoveries and Writings 
 
 relating to those Subjects ; 
 
 WITH THE 
 
 COMPLETE DESCRIPTION AND USE OF THE TABLES. 
 
 THE FOURTH EDITION. 
 
 BY CHARLES HUTTON, 
 ANE LL.D. F.R.S. &e. 
 N, : AND PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY, WOOLWICH. 
 i { 
 | 
 
 LONDON: 
 
 PRINTED FOR G. AND J. ROBINSON, AND R. BALDWIN, 
 PATERNOSTER-ROW, 
 RY S. HAMILTON, SHOE-LANE, FLEET-STREET. 
 
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 REV. NEVIL MASKELYNE, D.D. F.R.S. 
 
 ASTRONOMER ROYAL, &c. 
 
 IN 
 
 TESTIMONY OF RESPECT 
 
 FOR HIS EMINENT LEARNING AND ABILITIES, 
 
 AND 
 
 FOR HIS LAUDABLE AND DISINTERESTED PROMOTION 
 
 AND ENCOURAGEMENT OF THESE, 
 
 AS WELL AS ALL OTHER USEFUL SCIENCES, 
 
 AND OF GRATITUDE 
 ) FOR HIS GENEROUS ADVICE AND ASSISTANCE IN THIS WORK, 
 
 AND ON VARIOUS OTHER OCCASIONS, 
 
 | THESE SHEETS 
 
 ARE HUMBLY INSCRIBED, 
 
 BY HIS MOST OBEDIENT SERVANT; 
 
 THE AUTHOR. 
 
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PREFACE. 
 
 Tu Every ample introduction prefixed to the follow- 
 ing collection of Mathematical Tables, has superseded 
 the necessity of using many words here by way of pre- 
 face, and has left me little more to mention than the ne- 
 cessity and occasion of this work, with some account of 
 the contents and mode of execution. 
 
 The undertaking was occasioned by the great incor- 
 rectness of all the editions of Sherwin’s or Gardiner’s 
 tables, and more especially of the bad arrangement in the 
 fifth or last edition. Finding, as well from the report 
 _ of others, as from my own experience, that those edi- 
 tions (to say nothing of the very improper alteration 
 in the form of the table of sines, tangents, and se- 
 cants in the last of them) were so very incorrectly 
 printed, the errors being multiplied beyond all tolera- 
 ble bounds, and no dependence to be placed on them 
 for any thing of real practice, I was led to undertake 
 the painful office of preparing a correct edition of 
 another similar work. And I was lucky enough to 
 meet with a bookseller of sufficient spirit to be at the 
 great expense of printing the book, as well as to al- 
 low me what I demanded for my trouble in prepar- 
 ing it; which demand, however, was nothing ade- 
 quate to the great labour attending it, as I was well 
 
 A - gware 
 
vi PREFACE. 
 
 aware that the profits of the book would not enable 
 him fully to reward my pains. | 
 
 I have in the first place, therefore, used all the 
 means in my pow er to render the work correct. I be- 
 gan by collating the third or best edition of Sherwin’s 
 tables, with some others of the most perfect works of 
 the same kind, as Briggs’s, Vlacq’s, Gardiner’s quarto 
 book, &c.; by which means I detected many errors 
 in each of them, which had not before been disco- 
 vered; and of these, between twenty and thirty were 
 in the two editions of Gardiner’s quarto work, printed 
 at London in 1742, and at Avignon in 1770; the 
 errata of which two books are here printed at the end 
 of the tables in this work. But, besides detecting 
 many unknown errors in my copy of the said third edi- 
 tion of Sherwin, which was no more than what I ex- 
 pected, I discovered, with no small surprise, that the 
 last figures in the table of logarithms were not uni- 
 formly true to the nearest unit, except in a very few 
 pages at the beginning and end ‘of the table; al- 
 though Mr. Gardiner,*the editor of that edition, had 
 made the table correct in that respect in his own 
 quarto work before mentioned, which was also printed 
 in the same year 1742, with the said third edition of 
 Sherwin! The errors from this cause, in that third edi- 
 tion, amounted to several thousands ; and they have 
 continued to run through all the editions of Sherwin 
 ever since that time! But I have here corrected them. 
 Nor have I employed less attention in correcting the 
 ecm than in previously correcting the copy ; every 
 
 proof 
 
PREFACE. vi 
 
 proof having been several times read over, and com- 
 pared with the best of the books hitherto printed, by 
 several persons attending to the reading of every 
 proof-sheet. 
 
 But in giving this edition to the world, I was not 
 satisfied with barely making it correct. I was aware 
 that the materials themselves might be much im- 
 proved; and I have accordingly enlarged, or other- 
 wise greatly amended them,: in various respects. 
 Among the improvements of the old materials may 
 be reckoned the following :—namely, in the large ta- 
 ble of logarithms, the proportional parts, near the 
 beginning,.are more conveniently arranged, being now 
 all placed in the same opening of the book where their 
 corresponding differences occur; the logarithms to six- 
 ty-one figures are brought to their proper place in the 
 book, and more conveniently disposed all in one page; 
 the large table of sines, tangents, and secants, is more 
 commodiously arranged, and rendered more distinct 
 and convenient for use; the natural sines, tangents, 
 secants, and versed sines, being all separated from 
 the others, and placed all together on the left-hand 
 pages, and the logarithmic ones facing them on the 
 right-hand pages; the common differences, in both, 
 set between the two columns to which each of them 
 answers; and the versed sines here introduced into 
 their proper place in the same pages with the sines, 
 tangents, and secants. Besides these, there are some 
 other alterations in the new tables here given, and the 
 
 reader will find a number of very important mmprove- 
 AQ ments 
 
vill PREFACE. 
 
 ments in the description and use of the whole; espe- 
 cially in the arithmetic of logarithms, and in the resolu- 
 tion of plane and spherical triangles, according to the 
 present improved methods of calculation used by the 
 Astronomer ‘Royal, and other persons si most ex- 
 perienced in these matters. 
 
 The improvements in the tables, by the introduc- 
 tion of new matter, are both great and numerous. 
 The tables numbered 2, 3, and 4, are here added, be- 
 ing an entire new set, with their differences, for finding 
 numbers and logarithms to twenty places. ‘The co- 
 lumns of common differences, in the pages of natural 
 sines, &c., are now first introduced: As are also the 
 tables of hyperbolic and logistic logarithms; the loga- 
 rithmic sines and tangents for every second, in the 
 first two degrees of the quadrant; together with a 
 table of the length of arcs, a table to change com- 
 mon. and Lorperbelic ales from the one to the 
 other, &c.—the uses and exemplifications of the 
 whole being very amply detailed. 
 
 But the greatest alteration of all is the very exten- 
 sive and new introduction here given, instead of the 
 former inadequate and heterogeneous one, consisting 
 of about 180 pages of new matter, on a methodical 
 plan, containing the historical account and description 
 of all trigonometrical writings, and the tables relating 
 to that subject, both natural and logarithmic; besides 
 the complete use of the tables in this work. Inven- 
 tions are here ascribed to the proper authors, and 
 their methods and improvements described and com- 
 
 pared. 
 
PREFACE. ix 
 
 pared. ‘This historical description will evidently ap- 
 pear to be the result of immense labour and reading. 
 And, indeed, I have painfully gone over all the books 
 which are here so minutely described ; and that de- 
 scription with a detail in some degree adequate to 
 their great merits; especially the works of Napier, 
 Briggs, Kepler, &c.; which was the more necessary, 
 as the writings and methods of those great masters 
 had not been anywhere properly described and dis- 
 criminated, although they are in themselves highly 
 curious and important. 
 
 ‘These readings and commentaries haveubeen car- 
 ried on to an extent far beyond what was at first in- 
 tended. But the tables having been in the press for 
 the space of seven or eight years, I had thereby an 
 opportunity of collecting and examining a still greater 
 number of books; so that I was gradually led on, and 
 my views and plans rendered still more extensive and 
 complete. This delay, therefore, though in many re- 
 spects it proved very inconvenient and disagreeable, 
 has at length been the occasion of rendering these 
 commentaries more perfect and satisfactory. 
 
 Besides what immediately relates to trigonometrical 
 subjects, the reader will here find many other cu- 
 rious and uncommon articles, relating to their several 
 authors and their discoveries, which have occurred in 
 the course of my reading, and which appeared of too 
 much consequence to be passed over unnoticed, in the 
 analysis of their several compositions. Among these 
 is the discovery of the first author of the binomial 
 
 theorem, 
 
oe PREFACE, 
 
 theorem, and the differential method,, which are due 
 to Mr. Henry Briggs, whose writings are replete with 
 ingenious and original matter, and are well deserving 
 to be more generally known and. studied than they 
 have been for some time past. 
 
 This long course of examination and description, 
 however, having been carried on for so many years, at 
 different intervals, and interrupted by various avoca- 
 tions, and by business of different kinds, it will be no 
 wonder if this circumstance may have occasioned 
 some inequalities in the style and composition of this 
 history ; and for which, therefore, should any such 
 appear, it is hoped the occasion will plead an apology. 
 
 * ,* In the large table of common logarithms, when the first of the last 
 four figures in any logarithm changes from a 9 toa 0, ina line, in which 
 case the first three or constant figures are prefixed to the next following line, 
 instead of these three, it often happens that young beginners by mistake 
 take out the three constant figures next above the said line. To guard 
 against this error, the figures in this edition are so contrived, that where 
 the said change happens, a bar is placed over the cipher, thus 6, in 
 order to catch the eye, and remind the learner that the change there takes 
 place. In this edition, too, the black rules formerly drawn across the 
 pages, at the intervals of every five, or six, or ten lines, have been taken 
 out, leaving thin white spaces across the pages instead of them. These 
 improvements, besides that of new and better formed figures here now in- 
 troduced, and other attentions, contribute to render this edition of the 
 tables more convenient and correct than either of the former ones. 
 
 Dec. 1800. 
 
A short Abstract of the principal Contents, may be as follows: 
 
 1. In the Introduction. 
 
 History of trigonometrical tables 
 before the invention of loga- 
 
 rithms, with the various me- 
 
 thods of construction - - 1 
 On the word sinus - = - 17 
 History of logarithms .- - 20 
 
 Nature of logarithms - - - 22 
 Invention of logarithms - - 24 
 Different sorts of logarithms - 25 
 Construction of logarithms - 42 
 By Napier - - - - - = 42 
 
 Kepler - - - --- = 49 
 
 Briggs - - - - - - 61 
 
 Briggs’s 'Trigonometria Britan. 7 
 Relation between logarithms and 
 certain curves - - - - 84 
 Gregory’s construction - - 87 
 Mercator’s Logarithmo-technia 87 
 Gregory’s Excercit. Geometricz 97 
 
 SirI. Newton’s methods - - 102 
 Halleys - - - - + = 107 
 Sharp’s - - - - - - - 110 
 By Fluxions - - - - = 11) 
 Cotes’s Logometria - - - 112 
 Taylor’s construction - - - 116 
 Long’s method - - - - 118 
 Jones's - - - - = = = 119 
 Reid, &c. - - - - = = 4322 
 
 Page 
 
 , Page. 
 Dodson’s Anti-log. canon - 122 
 Description and use of logarith- 
 
 mic tables - - - = - 125 
 Definition and notation - - 125 
 Properties of logarithms - - 127 
 Construction of logarithms - 127 
 
 Description and use of our tables 129 
 
 Of our large table - - - 129 
 Logarithmical arithmetic - 134 
 Of the table to 20 places - 137 
 Of the table to 61 places - 142 
 Of the hyperbolic logarithms 146 
 Of the logistic logarithms - 147 
 
 Of the log. sines and pe ae to 
 every second - - - 
 
 Of the general tableof log. sines, 
 tangents, &c. - - - - 150 
 
 Trigonometrical rules - - 156 
 
 The cases of plane triangles resolved 
 
 148 
 
 by logarithms - - - - 160 
 The cases of spherical triangles 
 
 resolved by logarithms - 162 
 Use of the versed sines - - 170 
 Of the traverse table - - - 172 
 Of Mercator’s sailing - - - 175 
 Of the length of circular arcs 178 
 
 Of comparing the common and 
 
 hyp. logs). - - - - - 179 
 
 2. In the Tables themselves. 
 
 ry 
 
 ab. Page. 
 
 1. Logarithms from 1 to 100000 1 
 2. Logarithms, &c. to 20 places 187 
 3. Id. with differences - - 107 
 4, Numbers to spepeitis to 20 
 places - - - - 200 
 5. Logarithms to 61 placés - 203 
 6. Id. with differences - - 207 
 7. Hyperbolic logarithms - 208 
 8. Logistic logarithms - - 212 
 g. Sines and tangents to seconds 217 
 
 Tab. Page, 
 10. Natural and logarithmic 
 sines, tangents, secants, and 
 
 versed sines - - - - 248 
 11. Traverse table - - - 338 
 12. Length of arcs - - - 340 
 
 13. Table to change common 
 and hyperbolic logarithms 
 
 from one to the other - 34) 
 14. Points of the compass - 341 
 15. Errata in Gardiner’s and 
 
 other logarithms - - - 342 
 
] f 
 
 ’ LBits o 
 
 o™ 
 
INTRODUCTION. 
 
 I. OF TRIGONOMETRICAL TABLES, &c. 
 
 N ECESSITY, the fruitful mother of moft ufeful inventions, 
 gave birth to the various numeral tables which compofe the following 
 work. Aftronomy has been cultivated from the earlieft ages. The 
 ‘progrefs of that {cience, requiring numerous arithmetical computa- 
 tions of the fides and angles of triangles, both plane and fpherical, 
 gave rife to trigonometry ; for thofe frequent calculations fuggefted the 
 neceflity of performing them by the property of fimilar triangles; and 
 for the ready application of this property, it was neceflary that certain 
 lines defcribed in and about circles, to a determinate radius, fhould be 
 computed, and difpofed in tables. Navigation, and the continually 
 improving accuracy of aftronomy, have alfo occafioned as perpetual 
 an increafe in the accuracy and extent of thofe tables. And this itis 
 evident muft ever be the cafe, the improvement of trigonometry 
 uniformly following the improvement of thofe other ufeful fciences, 
 for the fake of which it is more efpecially cultivated. 
 
 . The ancients performed their trigonometry by means of the chords 
 of arcs, which, withthe chords of their fupplemental arcs, and the con- 
 {tant diameter, formed all {pecies of right-angled triangles. Beginning 
 
 _ with the radius, and the are whofe chord 1s equal to the radius, they 
 divided them both into 60 equal parts, and eftimated all other arcs and 
 chords. by thofe parts, namely all arcs by oOths of that arc, and all 
 chords by 60ths of its chord or oftheradius. At leaftthis method is as 
 old as the writings of Ptolemy, who ufed the fexagenary arithmetic for 
 this divifion of chords and arcs, and for aftronomical purpofes.—And 
 this, by-the-bye, may be the reafon why the whole circumference is 
 divided into 300, or 6 times GO, equal parts or degrees, the whole cir- 
 cumference being equal to G times the firft arc, whofe chord is equal to 
 the radius: unlets perhaps we are rather to feek for the divifion of the 
 circleinthe number ofdaysintheyear; for thus, theancient year confift- 
 ing of 400days, thefun or earch in each day defcribed the sv0th part of 
 the orbit ; and thence n.ightarife the methodof dividing every circleinto 
 360 parts; and radius being equal to the chord of Gv of thofe parts, 
 the fexagefimal divifion, both of the radius and of the parts, might thence 
 arife. ‘Lrigonometry however muit have been cultivated long before 
 the time of Ptolemy; and indeed Theon, in his commentary on 
 
 Ptolemy’s 
 
wu HISTORY OF 
 
 Ptolemy’s Almagett, 1. i. ch. 9. mentions a work of the philofopher 
 Hipparchus, written about a century and a half before Chrift, confift- 
 ing of 12 books on the chords of circular arcs; which muft have been 
 a treatife on trigonometry. And Menelaus alfo, in the firft century 
 of Chrift, wrote 6 books concerning fubtenfes or chords of arcs. He 
 ufed the word nadir (of an arc), which he defined to be the right line 
 fubtending the double of the arc; fo that his nadir of an arc was the 
 double of our fine of the fame arc, or the chord of the double arc; and 
 therefore whatever he proves of the former, may be applied to the 
 latter, fubftituting the double fine for the nadir. 
 
 ‘The radius has been fince decimally divided ; but the fexagefimal 
 divifions of the arc have continued in ufe to this day. Indeed our 
 countrymen, Briggs and Gellibrand, having a general diflike to all fex- 
 agefimal divifions, made an attempt at fome reformation of this cuitom, 
 by dividing the degrees of the arcs, in their tables, into centefms or 
 hundredth parts, inftead of minutesor 60th parts. The fame was alfo 
 recommended by Vieta, and- others; anda decimal divifion of the 
 whole quadrant* might perhaps foon have followed, had it not been 
 for the tables of Vilacq, which came outa little after, toevery 10 fe- 
 conds, or 6th parts of a minute.—But the complete reformation would 
 be, to exprefs all arcs by their real lengths, namely in equal parts of 
 the radius decimally divided: according to which method I have 
 nearly completed a table of fines and tangents. 
 
 It is not to be doubted that many of the ancients wrote on the fub- 
 jett of trigonometry, asbeing a neceffary part of aftronomy; although 
 few of their labours on that branch have come to our knowledge, and 
 {till fewer of the writings themfelves have been handed down to us. 
 
 _ We are in pofieffionofthe three books of Menelaus, on fpherical tri- 
 gonometry; but the fix books are loft which he wrote upon chords, 
 being probably a treatife on the conftru€tion of trigonometrical tables, 
 
 The trigonometry of Menelaus was much improved by Ptolemy 
 (Claudius Ptolemzus) the celebrated philofopher and mathematician. 
 He was born at Pelufium, taught aftronomy at Alexandria in Egypt, 
 and died in the year of Chrift 147, being the 78th year of his age. 
 In the firft book of his Almageft, Ptolemy delivers a table of arcs and . 
 chords, with the method of conftru€tion. This table contains 3 
 columns ; in the 1{t are the arcs to every half degree or 30 minutes; 
 in the 2d are their chords, exprefied in degrees, minutes and feconds, 
 of which degrees the radius contains 603; and in the 3d column are 
 the differences of the chords anfwering to | minute of the arcs, or the 
 30th part of the differences between the chords in the 2d column. In 
 the conftruction of this table, among others, Ptolemy fhows, for 
 the firft time that we know of; this property of any quadrilateral 
 infcribed inacircle, namely, that the rectangle under the two diagonals, 
 is equal to the fum of the two rectangles under the oppofite fides. 
 
 This method of computation, by the chords, continued in ufe till 
 about the middle centuries after Chrift ; when it was changed for that 
 of the fines, which were about that time introduced into trigonometry 
 
 * This has lately been'done by the French mathematicians, in-their new logarithmic tabies, 
 
 by 
 
TRIGONOMETRICAL TABLES, &c. 3 
 
 by the Arabians, who in other refpe€ts much improved this fcience, 
 which they had received from the Greeks, introducing, among other 
 things, the three or four theorems, or axioms, which we ufe at pre- 
 fent as the fotindation of our modern trigonometry. 
 
 The other great improvements that have been made in this branch, 
 are due to the Europeans. ‘Thefe improvements they have gradually 
 introduced fince they received this ference from the Arabians. And 
 although thefe latter people had Jong ufed the Indian or decimal fcale 
 of arithmetic, it does not appear that they varied from the Greek or 
 fexagefimal divifion of the radius, by which the chords and fines 
 were exptefled. _ . 
 
 This alteration, it is faid, was firft made by George Purbach, who 
 was fo called from his being a native of a place of that name between 
 Auftria and Bavaria. He was born in 1423, ftudied mathematics 
 and aftronomy at the univerfity of Vienna, where he was afterwards 
 
 _ profeffor of thofe fciences, though but for a fhort time, the learned 
 world quickly fuffering a great lofs by his immature death, which 
 happened in 1462, at the age of 39 years only. Purbach, befides 
 enriching trigonometry and aftronomy with feveral new tables, theo- 
 rems, and obfervations, fuppofed the radius to be divided into 600,000 
 equal parts, and computed the fines of the arcs, for every 10 minutes, 
 in fuch equal parts of the radius, by the decimal] notation. 
 
 This project of Purbach was completed by his difciple, companion, 
 and fucceflor, John Muller, or Regiomontanus, who was fo called from 
 the place of his nativity, the little town of Mons Regius, or Konings- 
 berg, in Franconia, where he was born in the year 1436. Regiomon- 
 tanus not only extended the fines to evety minute, the radius being 
 600,000, as defigned by Purbach, but afterwards, difliking that {cheme 
 as evidently imperfect, he computed them likewife to the radius 
 1,000,000, for every minute of the quadrant. He alfo introduced the 
 tangents into trigonometry, the canon of which he called fecundus, 
 becaufe of the many and great advantages arifing from them. Befides 
 thefe, he enriched trigonometry with many theorems and precepts. 
 Through the benefit of all thefe improvements, except for the ufe of 
 logarithms, the trigonometry of Regiomontanus is but little inferior 
 to that of our own time. His treatife on both plane and fpherical 
 trigonometry, is in 5 books; it was written about the year 1464, and 
 printed in folio at Nuremburg, in 1533. And in the fifth book are alfo 
 various problems concerning rectilinear triangles, fome of which are 
 refolved by means of algebra: a proof that this {cience was not wholly 
 unknown in Europe before the treatife of Lucas de Burgo. Regio- 
 montanus died in 1476, at the age of 40 years only; being then at 
 Rome, ‘whither he had been invited by the Pope, to affift in the re- 
 formation of the Calendar, and where it was fufpected he was poi- 
 foned by the fons of George ‘Trebizonde, in revenge for the death of 
 their father, which was faid to have been caufed by the grief he felt 
 on account of the criticifms made by Regiomontanus on his tranfla- 
 tion of Ptolemy’s Almageft, . 
 
 Soon after this, feveral other mathematicians contributed to the im-=. 
 
 provement of trigonometry, by extending and enlarging the tables, 
 though 
 5 
 
P | ~ HISTORY OF 
 
 though few of their works have been printed ; and particularly John 
 ‘Werner of Nuremburg, who was born in 1468, and died in i528, 
 and who it feems wrote five books on triangles. 
 
 ~About the year 1500, Nicholas Copernicus, the celebrated modern 
 reftorer of the true folar fyftem, wrote a brief treatife on trigonometry, 
 both plane and fpherical, with the defcription and conftruction of the 
 canon of chords, or their halves, nearly in the manner of Ptolemy; to 
 which is fubjoined a canon of fines, with their differences, for every 
 10 minutes of the quadrant, to the radius 100,000. ‘This traét is in- 
 
 _ferted in the firft book of his Revolutiones Orbium Celeftium, firtt printed 
 in folio at Nuremburg, 1543. It is remarkable that he does not call 
 thefe lines fines, but femiffes Jubtenfarum, namely of the double arcs.— 
 Copernicus was born at Thorn in 1473, and died in 1543. 
 
 In 1553 was publithed the Canon Fecundus, or table of tangents, 
 of Erafmus Reinhold, profeffor of mathematics in the academy of 
 Wurtemburgh. He was born at Salfieldt in Upper Saxony, in the 
 year 1511, and died in 1553. 
 
 To Francis Maurolyc, abbot of Meffina in Sicily, we owe the in- 
 troduction of the Labula Benefica, or canon of fecants, which came 
 out about the fame time, or a little before. But Lanfbergius errone- 
 oufly afcribes this to Rheticus. And the tangents and fecants are 
 both afcribed to Reinhold, by Briggs, in his A@athematica ab antiquss 
 minus cognita, (p. 30. Appendix to Ward’s Lives of the Profeffors of 
 Gretham College.) 
 
 Francis Vieta was born in 1540 at Fontenai, or Fontenai-le-Comte, 
 in Lower Poitou, a province of France. He was matter of requefts at 
 Paris, where he died in 1603, being the 63d year of his age. Among 
 other branches of learning in which he excelled, he was one of the 
 moft refpectable mathematicians of the 16th century, or indeed of any 
 age. His writings abound with marks of great.originality, and the 
 fineft genius, as well as intenfe application. Among them are feve- 
 ral pieces relating to trigonometry, which may be found in the col- 
 leétion of his works publithed at Leyden in 1646, by Francis Schooten, 
 befides another large and feparate volume in folio, publifhed in the 
 author’s lifetime at Paris in 1579, containing trigonometrical tables, 
 with their conftruCtion and ufe ; very elegantly printed, by the king’s 
 mathematical printer, with beautiful types and rules; the differences 
 of the fines, tangents, and fecants, and fome other parts, being printed 
 with red ink, for the better diftin€tion ; but inaccurately executed, as 
 he himfelf teftifies in page $23 of his other works above mentioned. 
 ‘The firft part of this curious volume is entitled Canon Aathematicus,- 
 feu ad Triangula, cum Appendicibus, and contains a great variety of 
 tables ufefulin trigonometry. ‘The firft of thefe is what he more pe- 
 culiarly calls Canon Mathematicus, feu ad Triangula, which contains, 
 all the fines, tangents, and fecants for every minute of the quadrant, 
 to the radius 100,000, with all their differences; and towards the end 
 of the quadrant the tangents and fecants are extended to 8 or 9 places 
 of figures. ‘They are arranged like our tables at prefent, increafing on 
 the left hand fide to 45- degrees, and then returning upwards by the 
 right hand fideto90.degrees ; fo thateach number and its complement 
 
 {land 
 
TRIGONOMETRICAL TABLES, &c: 5 
 
 ‘fkand on the fame liné. But here the canon of what we now call 
 
 ' tangents is denominated fecundus, and that of the fecants fecundiffi~ 
 mus. For the general idea prevailing in the form of thefe tables, is, 
 not that the lines reprefented by the numbers are thofe which are 
 drawn inand about a circle, as fines, tangents, and fecants, but the 
 three fides of right-argled triangles ; this being the way in which 
 thofe lines had always been confidered, and which ftill continued for 
 ~ fome time longer. And therefore he confiders the canonas a feries of 
 plane right-angled triangles, one fide being conftantly 100,000; or 
 rather as three feries of fuch triangles, for he makes a diftin€ feries 
 for each of the three varieties, namely, according as the hypotenufe, or 
 the bafe, or the perpendicular, is reprefented by the conitant number 
 100,000, which is fimilar to the radius. Making each fide conftantly 
 100,000, the other two fides are computed to every magnitude of the 
 acute angle at the bafe, from 1 minute up to 90degrees, or the whole 
 quadrant. Each of the three feries therefore confifts of two parts, as 
 reprefenting the two variable fides of the triangle. When the hypo- 
 tenufe is made the conftant number 100,000, thetwo variable fides 
 of the triangle are the perpendicular and bafe, or our fine and cofine ; 
 when the bafe is 100,000, the perpendicular and hypotenufe are the 
 variable parts, forming the canon feecundus et fecundifimus, or our 
 tangent and fecant; and when the perpendicular is made the conftant 
 100,000, the feries contains the variable bafe and hypotenufe, or alfo 
 canon feecundus et feecundifimus, or our cotangent and cofecant. Of 
 courfe, therefore, the table confifts of 6 columns, 2 for each of the 3 
 feries, befides the two columns on the right and left for minutes, 
 from 0 to 60 in each degree. 
 
 The fecond of thefe tables is fimilar to the firft, but all in rational 
 numbers, confifting, like it, of 3 feries of 2 columns each ; the radius, 
 or conftant fideof the triangle, in each feries, being 100,000, as before; 
 and the other two fides accurately exprefled in integers and rational 
 vulgar fractions. So that we have here the canon of accurate fines, 
 tangents, and fecants; or a feries of about 400 rational right-angled 
 triangles. But then the feveral correfponding arcs of the quadrant, 
 or angles of thofe triangles, are not exprefled. Inftead of them, are 
 inferted, in the firft column next the margin, a feries of numbers de- 
 creafing from the beginning to the end of the quadrant, which are 
 called numeri primt bafeos. -It-is from thefe numbers that Vieta con- 
 ftrudts the fides of the 3 feries of right-angled triangles, one fide in 
 each feries being the conftant number 100,000, as before. ‘The theo- 
 rems by which thefe feries of rational triangles are computed from the 
 numeri primi bafeos, or marginal numbers, are inferted all in one page 
 at the end of this fecond table, and in the modern notation they may 
 be briefly expreffed thus. Let p be the primary or marginal number 
 on any line, and r the conftant radius or number 100,000; then if r 
 denote the hypotenufe of the right-angled triangle, the perpendicular 
 and bafe, or the fine and cofine, will be refpedtively, . 
 
 dp? MS Ty (which laft we may reduce to tp oe Lait 
 
 When 
 
6 | HISTORY Of 
 
 When r denotes the bafe of the right-angled triangle, then the pers 
 pendicular and hypotenufe, or the tangentand fecant, are exprefled by 
 
 AY: A gt 
 rp] and are Me Ey (which laft we may reduce to aa r) 4 
 
 and when r denotes the perpendicular of the right-angled triangle, 
 the bafe and hypotenufe, or the cotangent and cofecant, are then 
 exprefled by 
 
 ; Ipt_-] Ty2 
 £ pr— ee (or 5 r), and pr 9 al (ameter r) 
 
 So that Vieta’s general values will be as we have here collected them 
 together in the following expreffions, immediately under the words 
 fine, cofine, &c; and juft below Vieta’s forms I have here placed 
 the others to which they reduce and are equivalent, which are more 
 contracted, though not fo well adapted to the expeditious computation 
 as Vieta’s forms. 
 
 Sine Cotine ‘Langent Secant Cotangent aes 
 yh ata BG ox: r sata spr— ms | tor-+ al 
 as ad a Ped 
 nao par! Frert spor’ npr ae 7H - ‘ 
 
 All thefe expreflions, it is evident, are rational; and by affuming p of 
 different values, from the firft theorems Vieta computed the corre- 
 {ponding fides of the triangles, and fo exprefled them all in integers 
 and rational fractions. 
 
 To the foregoing principal tables are fubjoined feveral other {maller 
 tables, or fhort {pecimens of large ones: as, a table of the fines, tan- 
 gents and fecants for every fingle degree of the quadrant, with the 
 correfponding lengths of the arcs, the radius being 100,000,000 + 
 another table of the fines, tangents, and fecants, for each degree alfo, 
 exprefled in fexagefimal patts of the radius, as far asthe 3d order of 
 parts ; alfo two other tables for the multiplication and reduction of 
 fexagefimal quantities. 
 
 The fecond part of this volume is entitled Univerfalium In/pettio- 
 num ad Canonem Mathematicum Liber fingularis. St contains the 
 conftruction of the tables, a compendious treatife on plane and 
 {pherical trigonometry, with the application of them to a great 
 variety of curious fubjects in geometry and menfuration, treated in 
 a very learned manner ; as allo many curious obfervations concern- 
 ing the quadrature of the circle, the duplication of the cube, &c. 
 Computations are here given of the ratio of the diameter of 2 
 circle to the circumference, and of the length of the fine of 1 
 minute, both to many places of figures; by which he found that 
 the fine of 1 minute is between 2,903,8$1,959 and 2,908,882,056 
 alfo, the diameter of a circle being 1000 &c, that the perimeter of 
 ae infcribed and circum{cribed polygon of 393,216 fides, will be as 
 ollows ; 
 
 perim, 
 
TRIGONOMETRICAL TABLES, &c, q 
 
 perim. of the infcrib. polygon 314,159,265,35 
 
 perim. of the circum. polygon 314,159,265,37 
 and that therefore the circumference of the circle lies between thofe 
 two numbers. 
 
 ‘Although no author’s name appears to the volume I have been de- 
 {cribing, there can be no doubt of its being the performance of Vieta ; 
 for, belides bearing evident marks ofhis mafterly hand, it is mentioned 
 by himfelf in feveral parts of his other works collected by Schooten, 
 and in the preface to thofe works by Elzevir the printer of them: as 
 alfo in M, Montucla’s Hiffcire des Mathematiques, which are the only 
 notices I have ever feen or heard of concerning this book, the copies 
 of which are fo rare, that Inever faw one befides that which is in 
 my own poffeffion, nor ever met with any other perfon at all ace 
 quainted with fuch a beok. 
 
 In the other works of Vieta, publifhed at Leyden in 1646, by 
 Schooten, as mentioned above, there are feveral other pieces relating 
 to trigonometry; fome of which, on account of their originality and 
 importance, are very deferving of particular notice in this place. 
 And firft, the very excellent theorems, here firft of all given by our 
 author, relating to angular fections, the geometrical demonftrations 
 of which are fupplied by that ingenious geometrician, Alexander 
 Anderfon, then profeflor of mathematics at Paris, but a native of 
 Aberdeen, and coufin-german to Mr. David Anderfon, of Finzaugh, 
 whofe daughter was the mother of the celebrated Mr. James Gregory, 
 inventor of the Gregorian telefcope. We find here, theorems for 
 the chords (and confequently fines) of the fums and differences of 
 arcs; and for the chords of arcs that are in arithmetical progref- 
 fion, namely, that the firft or leaft chord is to the 2d, as any one after 
 the 1{t, is to the fum of the two next lefs and greater : for example, 
 as the 2d to the fum of the 1ftand 3d, and as the 3d to the fum of 
 the zd and 4th, and as the 4th tothe fum of the 3d and 5th, &c3 
 fo that, the 1ft and 2d being given, all the reft are found from them 
 by one fubtraction and one proportion for each, in which the ! ft and 
 2d terms are conftantly the fame. Next are given theorems for the 
 
 chords of any multiples of a given arc or angle, as alfo the chords of 
 their fupplements to a femicircle, which are fimilar to the fines and 
 cofines of the muitiples of given angles; and the conclufions from 
 them are exprefled in this manner: ift, thatifc be the chord of the 
 fupplement of a given arc @, tothe radius 1, then the chords of the 
 fupplements of the multiple arcs will be as in the annexed table: 
 where the author obferves that the 
 figns are alternately + and—; that | Arcs| Chords of the Supplements. 
 the vertical columns of numeral co- | qq | ¢ 
 efficients to the terms of the chords, | 9g | ¢2-_92 . 
 ‘are the feveral orders of figurate | 3a | c} — 3c 
 numbers, which he calls triangular, | 4@ | c+ — 4c? + 2 
 pyramidal, triangulo-triangular, trie | 5@ | © —— 5c? + 5e 
 angulo-pyramidal, &c, generated in | 4 rc bet ie ofl 
 the ordinary way by continual additions; \ 74 | ¢1— Teo + 140 — (eo | 
 not indeed from unity, as IN THE oe: soa 
 
 GENE 
 
s 3 HISTORY OF 
 
 GENERATION OF POWERS, but beginning with the number 2; and 
 that the powers obferve always the fame progreflion: fecondly, that 
 if the chord of an arc abe called J, and d the chord of the double 
 arc 2a, then the chords of the feries 
 of multiple arcs will be as in this 
 table; where the author remarks as 
 before on the law of the powers, 
 figns, and coefficients; thefe being 
 - the orders of figurate numbers, 
 raifed from unity by continual ad- | 5a | 4#— 3d? 4 1 
 ditions, after the manner of the genefis | 9a | ds’ —4BP 4+ 3d 
 of powers, which generation in that 74 | a — 5d+-+ 6d?—1 
 way he fpeaks of as a thing géne- | 54 | ABN oh 100" 48 
 rally known, but without giving any POG sili) fx Sey yee es 
 hint how the coefhcients of the terms of any power may be found 
 from one another only, and independent of thofe of any other power, 
 as it was afterwards, and firft of all, I believe, done by Henry Briggs, 
 about the year 1600: and 3dly, that if C be the chord of any are a, 
 to the radius 1, then the feries of the chords and fupplemental chords 
 of the multiple arcs 
 
 will be thus; where [Arcs| — “Chords and Chords of Sup. 
 the values are al- | 1¢|Chord = +C 
 
 ternately chords, and | 2¢ |Sup.ch.= —C?+ 2 
 
 chords of the fup- | 3u|Chord = --C? + 3C 
 
 plements of the arcs | 4¢ |Sup.ch.= +C*—-4C? + @ 
 
 on the fame line,and | 5¢|}Chord = +C?—5C*+ 5C 
 
 the law of the pow- 6a.) Sup, ch. —C® 4 6C! — oC" ft 2 
 ers and coefficients | 7% [Chord = —C' + 7C°— 140° + 7C 
 as before, but every [&c: 1 &c- 
 
 alternate couplet of lines haying their figns changed. 
 
 Another curious theorem is added to the above, for finding the fum 
 of all thefe chords drawn in a femicircle, from one end of the diameter 
 to every point in the circumference, thofe points dividing the circum- 
 ference into any number of equal parts; namely, as the leaft chord is 
 tothe diameter, fo isthe fum of the faid leaft chord and diameter 
 and greateft chord, to double the fum of allthe chords including the 
 diameter as one of them. ? 
 
 As the above theorems are chiefly adapted for the chords of multipie 
 angles, a few problems and remarks are thenadded (whether by Vieta 
 or Anderfon does not clearly appear, but1 think bythe latter) concern- 
 ing the application of them, to the fection of angles into fubmultiples, 
 and thence to the computation of the chords or fines, or a canon of 
 triangles. The general precept for the angular fections 1s this: felect 
 one of theabove equations adapted to the propernumberof the fection, 
 in which will be concerned the. powers of the unknown or required 
 quantity, as high as the index of the fe€tion; and from this equation 
 find that quantity, by the known methods for the refolutien of equa- 
 
 ram ROE ee ee mart 
 Ure SOT EE ae tae A ceo 
 70 — 4074+ C5 —C'.,..: = ¢ 
 
 where 
 
+a 
 
 TRIGONOMETRICAL TABLES, &e. 9 
 
 where g is the chord of the given are or angle, and € the required 
 chord of the 3d, 5th, or ith part of it. And it is fhewn, geometri- 
 cally, that the firft of thefe equations has 2 real pofitive roots, the {e- 
 cond 8, and the laft 4; alfo from the fame principles the relations of 
 thefe roots are pointed out.” | 
 
 - ‘The method then annexed for conftru€ting the canon of fines, from 
 the foregoing theorems, is thus: By dividing the radius in extreme- 
 and-mean ratio, is obtained the fine of 18 degrees ; this quinquifecied, 
 gives the fme of 3° 36’. Again, by trifeGing the arc of 60°, there 
 is obtained the fine of 20°; this again trife€ted gives that of 6° 40’; 
 and this bifeCted gives that of 3° 20’: Then, by the theorem for the dif- 
 ference of two arcs, there will be found the fine of 16’, the difference 
 between 3° 36’ and 3° 20’: Laftly, by four fucceflive bifeCtions, will 
 at length be found the fines of 8”, 4’, v’, and 1”. This laft being found, 
 the fines of its multiples, and again of the multiples of thefe muilti- 
 ples, &c, throughout the quadrant, are to be taken by the proper 
 theorems before laid down. | 
 
 And the fame fubject is ftill farther purfued and explained, in the 
 tract containing the anfwer given by Vieta, to the problem propofed 
 to the whole world by Adrianus Romanus. 
 
 -In the fame collection of Vieta’s works, from page 400 to 432, is 
 given a complete treatife on practical trigonemetry, containing rules 
 for refolving all the cafes of plane and {pherical triangles, by the Canon 
 Mathematicus, or table of fines, tangents and fecants. 
 
 The next authors whofe labours in this way have been printed, are 
 Rketicus, Otho, and Pitifcus: to all of whom we owe very great | 
 improvements in trigonometry. 
 © George Joachim Kheticus, profeflor of mathematics in the univer- 
  fity of Wittemburg, and fometime pupil to Copernicus, died in £576; 
 in the 60th year of his age. He conceived, and executed, the great 
 defign of computing the triangular canon for every 10 feconds of the 
 quadrant to the radius (1000000000000000) confifting of 1, followed 
 by 15 ciphers. ‘The feries of fines which Rheticus computed to this 
 radius, for every 10 feconds, and for every fingle fecond in the firft 
 and laft degree of the quadrant, was publifhedsin folio at Francfort, 
 
 1613, by Pitifcus, who himfelf added a few of the firft fines com- 
 
 puted to the radius | 0000000000000000000000. 
 But the large work, or whole trigonometrical canon, computed by 
 Rheticus, was publifhed in 1556 by Valentine Otho, mathematician 
 to the Electoral Prince Palatine. This vaft work contains all the three 
 feries for the whole canon of right-angled triangles (being fimilar 
 to the fines, tangents, and fecants, by which names I fhall call them), 
 with all the differences of the numbers, to the radius !OO0Q00C0000. 
 Prefixed to thefe tables, are feveral books on their conftru¢ction and. 
 ufe, in plane and f{pherical trigonometry, &c. Of thefe, the firit three 
 are by Rheticus himfelf; namely, book the firft, containing the demon- 
 {trations of 9 lemmas, concerning the properties of certain lines drawn 
 in and about circles: the 4d book contains !0 propofitions, relating to 
 the fines and cofines of arcs, together with thofe of their fums and dif- 
 ferences, their halves and doubles, &c. The 3d book teaches, in 13 pro 
 
 C pofitia 
 
10 | HISTORY OF 
 Wee 
 
 pofitions, the conftruction of thecanontotheradius 1000000000000000. 
 By fome of thecommonpropertiesof geometry, having determined the 
 fines of a few principal arcs, as 30°, 46°, &c, in the firft propofition, 
 by continual bifections he finds the fines of various other arcs, down 
 to #5 minutes. Then in the 2d propofition, by the theorems for 
 the fums and differences of arcs, he finds all the fines and cofines, 
 up to 90 degrees, ina feries of arcs differing by 1° 30’. And, in the 
 sd propolition, by the continual addition of 45’, he obtains all the 
 fines and cofines in the feries whofe common difference is 45’. In 
 the th propofition, beginning with 45’, and continually bifecting, © 
 he finds the fines and cofines of the feries of half arcs, till he arrives 
 at the arc of 14°" 19, the fine of which is found to be J, and its 
 cofine 999999999999999. In the fifth propotition are computed the 
 fine and cofine of 30”, or half a minute. In the 6th and 7th pro- 
 pofitions are computed the fines and cofines for every minute, from 
 1’ to 45’, as well as of many larger arcs. "The 8th propofition ex- 
 tends the computation for fingle minutes much farther. In propofi- 
 tions 9 and 10 are: computed the tangents and fecants for all arcs in 
 the feries whofe’'common difference is 45’; and thefe are deduced 
 from the fines of the fame arcs by one proportion for each. In 
 the remaining three propofitions, 11, 12, 13, are computed the 
 tangents and fecants for feveral {mall angles. And from all thefe 
 primary fines, tangents, and fecants, the whole canon is deduced 
 and completed. 
 
 The remaining books in this work are by the editor Otho ; namely, 
 a treatife, in one book, on right-angled plane triangles, the cafes of 
 which are refolved bY the tables : then right-angled f{pherical trigono- 
 metry, in four books; next oblique {pherical trigonometry, in five 
 books; and laftly, feveral other books, containing various {pherical 
 problems. 
 
 Next after the above are placed the tables themfelves, containing 
 the fines, tangents, and fecants, for every 10 feconds in the quadrant, 
 with all the differences annexed to each, in a fmaller chara@ter. “he 
 numbers however are not called fines, tangents, and fecants, but, like 
 Vieta’s before defcribed, they are confidered as reprefenting the fides 
 of right-angled triangles, and titled accordingly. ‘They are alfo, in 
 like manner, divided into three feries, namely, according as the radius, 
 or con{tant fide of the triangle, is made the hypotenufe, or the greater 
 leg, or the Jefs leg of the triangle. When the hypoterufe is made 
 the conftant radius 100V0000000, the two columns of this cafe, or 
 feries, are called the perpendicular and bafe, which are our fine and 
 cofine ;> when the greater leg is the conftant radius, the two columns 
 of this feries are titled hypotenufe and perpendicular, which are our 
 fecant and tangent; and when the lefs legis conftant, the two co- 
 lumns in this cafe are called hypotenufe and bafe; which are our 
 cofecant and: cotangent. After this large canon, is printed another 
 fmaller table, which is faid to be the two columns of the third feries, 
 or cofecants and cotangents, with their differences, but to 3 places of 
 figures leis, or to the radius 10000000. But I cannot difcover the 
 reafon for adding this lefs table, even if it were correct, which is very 
 
 fay . 
 
TRIGONOMETRICAL TABLES, &e. i 
 
 far from being the cafe, the numbers being uniformly erroneous, and 
 different from the former through the greateft part of the table. 
 
 Towards the clofe of the 16th century, many perfons wrote on the 
 fubject of trigonometry, and the conftruction of the triangular canon. 
 But, their writings being feldom printed till many years afterwards, it 
 is not eafy to aflign their order in refpe& of time. I fhall therefore 
 mention but a few of the principal authors, andthat without pretend- 
 ing to any great precifion on the {core ef chronological precedence. 
 
 In 1591 Philip Lanfbergius firft publifhed his Geometria Triangul- 
 rum, in four books, with the canon of fines, tangents, and fecants ; a 
 brief, but very elegant work ; the whole being clearly explained : and 
 it is perhaps the firft fet of tables titled with thofe words. ‘he fines, 
 tangents, and fecants of the arcs to 45 degrees, with thofe of their 
 complements, are each placed in adjacent columns, in a very commo- 
 dious manner, continued forwards and downwards to +5 degrees, and 
 then returning backwards and upwards to 90 degrees: the radius is 
 10060000, and a fpecimen of the firft page of the table is as follows: 
 
 O Sinus | -  'Tangens Secans ite 
 6) 0/10000000;) O} infinitum. 10000000] infinitum. {00 
 1 |; 2909 9999999) 2909 34377466738 10000000 34377468193)'59 
 
 2 || 5818) 9999998) 5818 17188731915 |1000000217188734924 58 _ 
 
 — -——_—— --— 
 
 “3 || 8727| 9999996) 8727|11459152994|10000004,11459157357'57 
 4 |11626} 9999993}111636] 8594363048 ,|10000007! 8594368860) 56 
 5 j|t4o44 ties 14544) 6875488693' 
 
 10000011} 6875495966) 55 
 ee se 
 
 <9 
 
 Of this work, the firft book treats of the magnitude and relations 
 of fuch lines as are confidered in and about the circle, as the chords, 
 fines, tangents, and fecants. In the fecond book is delivered the 
 conftruétion of the trigonometrical canon, by means of the properties 
 ‘Jaid down in the firft book. After which follows the canon itfelf. 
 And in the third and fourth books is fhown the application of the 
 table, in the refolution of plane and {pherical triangles Lanfhberg, 
 who was born in Zealand 1561, was many years a minifter of the 
 gofpel, and died at-Middleburg in 1632. 
 
 ‘Lhe trigonometry of Bartholomew Pitifcus was firft publifhed at 
 Francfort in the year 1599. This is a very complete work; contain- 
 ing, befides the triangular canon, with its conttruction and ufein . 
 refolving triangles, the application of trigonometry to problems of 
 furveying, altimetry, architecture, geography, dialling, and aftronomy. 
 The conftru€tion of the canon is very clearly defcribed: And, in the 
 third edition of the book, in the year 1612, he boafts to have added in 
 this part, arithmetical rules for finding the chords of the 3d, 5th, and 
 other uneven parts of an arc, from the chord of that_arc being given ; 
 faying, that it had been heretofore thought impoflible to give fuch 
 rules; But, after all, thofe boafted methods are only the application of 
 
 al 
 x 
 
1s HISTORY OF 
 
 the double rule of Falfe-Pofition to the then known rules for finding 
 the chords of multiple arcs ; namely, making the fuppofition of fome 
 number for the required chord of a fubmultiple of any given arc, then 
 - from this aflumed number, computing what will be the chord of its 
 multipie arc, which is to be compared with that of the given arc ; 
 then the fame cperation is performed with another fuppofition; and fo 
 on, as in the double rule of pofition. The canon contains the fine, 
 tangent, and fecant for every minute of the quadrant, in fome parts 
 €o 7 places of figures, in others to 8 ; as alfo the differences for every 
 10 feconds. The fines, tangents, and fecants, are alfo given for every 
 10 feconds in the firft and laft degree of the quadrant, for every 2 
 feconds in the firft and Jaft 10 minutes, and for every fingle fecond in 
 _ the firft and laft minute. In this table, the fines, tangents, and fe- 
 ' cants, are continued downwards on the left-hand pages as far as to 
 45 degrees, and then returned upwards on the right-hand pages, fo 
 that the complements are always.on the fame line in the oppofite or 
 facing: pages. : | 
 
 ‘The mathematical works of Chriftopher Clavius (a German jefuit, 
 who was born at Bamberg in 1587) in five large folio volumes, were 
 printed at Moguntia, or Mentz, in 1612, the year in which the au- 
 thor died, at the age of 75. Inthe firft volume we find a very ample 
 and circumftantial treatife on trigonometry, with Kegiomontanus’s 
 canon of fines, for every minute, as alfo canons of tangents and fe- 
 cants, each in a feparate table, to the radius 10000000, and in a 
 form continued forwards all the way up to 90 degrees. The expla- 
 nation of the confiruCtion of the tables is very complete, and is 
 chiefly extracted from Ptolemy, Purbach, and Regiomontanus. The 
 fines have the differences fet down for each fecond, that is, the quo- 
 tients arifing from the differences of the fines divided by 60. 
 
 About the year 1600, Ludolph van Collen, or a Ceulen, a refpect- 
 
 able Dutch mathematician, wrote his book De Circulo et Adfcriptis, . 
 
 in which he treats fully and ably of the properties of lines drawn in 
 and about the circle, and efpecially of chords or fubtenfes, with the 
 conftruciion of the canon of fines. The geometrical properties from 
 which thefe lines are computed, are the fame as tlHofe ufed by former 
 writers ; but his mode of computing and exprefling them is different 
 from theirs; for they actually extrafted all the roots, &c, at every 
 
 ftep, or fingle operation, in decimal numbers; but he retained the © 
 
 radical expreffions to the laft, making them however always as fimple 
 as poflible: thus, forinftance, he determines the fides of the polygons 
 of 4, 8, 16, 32, &c, fides, infcribed in the circle whofe radius is 1, to 
 ‘be as in the table here annexed: . 
 
 where the point before any figure 
 (as /.2) fignifies the root of all 
 that follows it; fo the laft line 
 is in our notation the fame as 
 
 Jot VI-WV2. And as } 99 Pf 24 fsd-m/2 
 the perfect management of fuch | Ke, Bc. 
 
 futds’: was then’ (not * generally rrr eae 
 knownly 
 
 Length of each fide. 
 
 4 | /2 
 eC: 
 
 16 | /.2—-/.2+4,/4 
 
 ~ 
 —————————— eo 
 
% 
 
 TRIGONOMETRICAL TABLES, kc. is 
 
 known, he added'a very neat tra&t on that fubjeét,’ to facilitate the 
 computations. Thefe, together with other diflertations on fimilar ge- 
 ometrical matters, were tranflated fromthe Dutch language, into Latin, 
 by Willebrord Snell, and publifhed at (Lugd. Batav.) Leyden in 
 1619. It was in this work that Ludolph determimed the ratio of the 
 diameter to the circumference of the circle, to 46 figures, fhowing 
 that, if the diameter be 1, the circumference will be 
 greater than 3-14159 26535 89793 23846 26433 83279 50288, 
 but lefs than 3:14159 26535 89793 23846 26433 $3279 50289 3 
 which ratio was, by his order, in imitation of Archimedes, engraven 
 on his tomb-ftone, as is witnedied by the faid Snell, pa. 54, 55, Cyclom 
 metricus, publifhed at Leyden two years after, in which he treats the 
 fame fubje&t in a fimilar manner, recomputing and verifying Lu- 
 dolph’s numbers. And, in the fame book, he alfo gives a variety of 
 geometrical approximations, or mechanical folutions, to determine 
 very nearly the lengths of arcs, and the areas of fectore and fegments 
 of circles. . ue . r 
 Befides the Cyclometricus, and another geometrical-work (Apollonius 
 Battavus ) publifhed in 1608, the fame Snellius wrote alfo four others 
 dottrine triangulorum cansnice, in which is contained the canon of 
 fecants, and in which’the conftruction of fines, tangents, and fe= 
 cants, together with the dimenfion or calculation of triangles, both 
 plane and fpherical, are bricily and clearly treated. After the author’s 
 death, this work was publifhedin 8vo, at Leyden 1627, by Martinus 
 Hortenfius, who added to it a tract on furveying and fpherical prob= 
 lems. ‘Wulebrord Snell was born in 1591 at Koyen, and died in 
 1626, being only 35 years of age. He was profeflor of mathematics 
 in the univerfity of Leyden, as was alfo his father Rodolph Snell. 
 Alfo in 1627, Francis van Schooten publifhed, at*Amfterdam, in 
 a {mail neat form, tables ‘of fines; tangents, and fecants, for every 
 minute of the quadrant, to 7 places of figures, the radius being 
 10000000 ; together with their ufe in plane trigonometry. Thefe 
 tables have a great character for their accuracy, being declared by the 
 author to be without.one fingle error. This however muft not be 
 ‘underftood of the laft figure of the numbers, which I find are very 
 often erroneous, fometimes in excels and fometimes in defect, by not 
 being always fet down to the neareft unit. Schooten died in 16595 
 while the fecond volume of his fecond edition of Defcartes’ geometry 
 was in the prefs: He was alfo author of feveral other valuable works 
 in geometry and other branches of the mathematics. 
 
 The foregoing are the principal writers on the tables of fines, tan- 
 gents, and fecaifts, before theinvention of logarithms, which happened 
 about this time; namely, foon after the year 1600. ‘Tables of the 
 natural nurabers were now all completed, and the methods of com- 
 puting them nearly perfected : And therefore, before entering on the 
 difcovery and conitruction of logarithms, I fhall flop here a little, to 
 give a fummary of the manner in which the faid natural fines, tans _ 
 ‘gents, and fecants, were actually computed, after having been gra- 
 dually improved fromm Hipparchus; Menelaus, and Ptolemy; who ufed 
 “f nh only 
 
14 HISTORY OF 
 
 only the chords, down to the beginning of the 17th century, whem 
 fines, tangents, fecants, and verfed fines were in ufe, and when the 
 method hitherto employed had received its utmoft improvement. 
 
 In this explanation, I fhall here firft enumerate the theorems by 
 which the calculations were made, and then defcribe the application 
 of them to the computation itfelf. 
 
 Theorem 1. ‘The fquare of the diameter of a circle, is equal to the 
 fam of the fquares of the chord of an arc, and of the chord of its 
 fupplement to a femicircle. 
 
 2. The reGtangle under the two diagonals of any quadrilateral 
 infcribed in a circle, is equal to the fum of the two rectangles under 
 the oppofite fides. 
 
 3. Lhe fum of the fquares of the fine and cofine (hitherto called 
 the fine of the complement), is equal to the fquare of the radius. 
 
 4. ‘The difference between the fines ef two arcs that are equally 
 diftant from 60 degrees, or 7 of the whole circumference, the one 
 as much greater as “the other is lefs, is equal to the fine of half the 
 difference of thofe arcs, or of the difference between either arc and 
 the faid arc of 60 degrees. 
 
 5. The fum of the cofine and verfed fine, is equal to the radius. 
 
 6. The fum of the fquares of the fine and verfed fine, is equal to 
 the {quare of the chord, or to the Sods of double the fine of half the 
 arc. 
 
 7. The fine is 2 mean proportional between half the radius and the 
 verfed fine of double the arc. 
 
 g. A mean proportional between the verfed fine and half the radius, 
 is equal to the fine of half the arc. 
 
 9. As radius is to the fine, fo is twice the cofine to the fine of 
 twice the arc. | 
 
 10. As the chord of an arc, is to the fum of the chords of the 
 fingle and double arc, fo is the difference of thofe chords, to the 
 chord of thrice the arc. 
 
 11. As the chord of an arc, is to the fum of the chords of twice 
 and thrice the arc, fo is the difference of thofe chords, to the chord 
 of five times the arc. : 
 
 12. Andin general, as the chord of anarc, is to the fum of the 
 chords of z times and #-+1 times the arc, fo is the difference of 
 thofe chords, to the chord of 27+1 times the are. 
 
 13. The fine of the fum of two arcs, is equal to the fum of the 
 products of the fine of each multiplied by the cofine of the other, and 
 divided by the radius. 
 
 14. The fine of the difference of two arcs, is equal to the difference 
 of the faid two products divided by radius. 
 
 15. The cofine of the fum of two arcs, is equal to the difference 
 en the produéts of their fines and of their fetch divided by 
 radius 
 
 16. The cofine of the difference of two arcs, is equal to a Hh ae of 
 the faid products divided by radius. 
 
 Be A {mall arc is equal to its chord or fine, nearly. 
 
 8. As cofine is to fine, fo isradius to tangent. i 
 19. Radius 
 
TRIGONOMETRICAL TABLES, &e. | 15 
 I 
 
 19, Radius is a mean proportional between the tangent and co- 
 tangent. — 7 
 
 20. Half the difference between the tangent and cotangent of an 
 arc is equal to the tangent of the difference between the arc and its 
 complement, Or, the fum arifing from the addition of double the 
 tangent of an arc with the tangent of half its complement, is equal to 
 the tangent of the fum of that arc and the faid half complement. 
 
 21. The fquare of the fecant of an arc, is equal to the fum of the 
 
 {quares of the radius and tangent. 
 
 42. Radius is a mean proportional between the fecant and cofine. 
 Or, as cofine is to radius, fo is radius to fecant. 
 
 23. Radius is a mean proportional between the fine and cofecant. 
 
 24. The fecant of an arc, is equal to the fum of its tangent and 
 the tangent of half its complement. Or, the fecant of the difference 
 between an arc and its complement, is equal to the tangent of the 
 faid difference added to the tangent of the lefs arc. 
 
 25. The fecant of an arc, isequal to the difference between the 
 tangent of that arc and the tangent of the arc added to half its com- 
 plement. Or the fecant of the difference between an arc and its 
 complement, is equal to the difference between the tangent of the faid 
 difference and the tangent of the greater arc. 
 
 From fome of thefe 25 theorems, extracted from the writers before 
 mentioned, and afew propofitions of Euclid’s elements, they com- 
 piled the whole table of fines, tangents, and fecants, nearly in the 
 following manner. 
 
 By the elements were computed the fides of afew of the regular 
 figures infcribed in a circle, which were the chords of fuch parts of 
 the whole circumference as are exprefled by the number of fides, 
 and therefore the halves of thofe chords the fines of the halves of the 
 arcs, So, if the radius be 10000:00, the fides of the following fi- 
 gures will give the annexed chords and fines. | 
 
 a 
 
 Arc {ub-| Its chord, |Halt] Its fine, 
 
 The fignre tended or fide | arc jor £ chord 
 Triangle OS 17320508 | 60°| 8060254’ 
 Square gO 14142136 | 45 | 7071068 
 Pentagon 72 11755705 |. 36 | 5877853 
 Hexagon GO 10000000 | 30 | 5000000 
 Decagon 36 6180540 | 18 | 3090170 
 Quindecagon| 24 4158234 | 12 | 2079117 
 
 _ Of fome, or all of thefe, the fines of the halves were continually 
 taken by theorem the 6th, 7th, or 8th, and of their complements 
 by the 3d; then the fines of the halves of thefe, and of their com- 
 plements, by the fame theorems ; and fo on, alternately of the halves 
 and complements, till they arrived at an arc which is nearly equal to 
 its fine. ‘Thus, beginning with the above arc of 12 degrees, and its 
 fine, the halves were obtained as follows; 
 
 The 
 
’ 
 
 ten’ 
 
 ‘6 HISTORY OF 
 
 ‘Fhe halves |Pheir nnés {TheCompJ” Sines The halves{ Sines. 
 
 P62 2 7} 1045285 of thefe 33°’ 154.4639) - 
 (3 + §238 48°  % 17431448 IO 30 240153 
 7: 30 261769 69 0335804: $ 15 11434926) 
 45 | 130896 79 30 19832549 27 45 |4656145 
 oy nei Nie 84 45 0958049 rote te ep 
 The comp. 46 30 7253744 57. fags670 
 | of thefe su is oe 73 30 19588197 
 84 9945218 Se ne 1 Hate 
 . £84 45 k 2 15 |884987 
 
 fez | 9980208} crihele Hale Te 
 Vedanta eee 4067366, | 28 30 |4771588) 
 89-29, 9999149 1h 34 30 [5664062] | 14 * 15 (2461533, 
 cae. 17-15 |2965416) | 36 - 45-|5983246) 
 The halves /39 45 |6394390|  |~ Comps. ‘5p 
 
 of thefe Jef 23. 15 |3947439 61 30 |8788171 
 | 42 6691306 ct bec 6 45 19692309 
 . i 5 OWS * 
 A 3583079 , The comp. 53 1S 8012538 
 (10 =. 30: |: 1822355 66 oy Re voe: 40% Maman Comet eifeee neta) urea 
 | & «315 | 915016 | | 55° 30 |241262 Flatt. 
 43 30 | 6883545 72 45 |9550199,. | 99 #5 _ SI 12931 
 24 45 | 3705574 50 «15 (7688418 Comp. |, | 
 144 15 | 6977905 66 45 |91879121 | 59 158594064: 
 
 The fines of fmall arcs are then deduced in this manner. From 
 the fine of 46’, above determined, are found the halves; which will 
 be thus: 
 
 PGB QM R 3) 6a BOTAN Set PEGG OS 
 
 ROMAINE 6 DE eS 65449,4 
 
 BPP Tare: Seeing haat 32724,8 
 Now thefe laft two fines being evidently i inthelfaine. SHG Bs their 
 arcs, the fines of all the lefs fingle minutes will be found by fingle 
 proportion. So the 45th part of the fine of 45’, gives 2909 for the 
 fine of 1’; which ‘may be doubled, tripled, &c, for the fines of 2’, 
 3’, &c, up to 45’. 
 
 Then, from all the foregoing primary fines, by the theorems for 
 halving, doubling, or tripling, and by thofe for the fums and dif- 
 ferences, the reft of the fines are deduced, to complete the quadrant. 
 
 But having thus determined the fines and cofines of the firft 30° 
 
 of the quadrant, that is, the fines of the firft and laft 20°, thofe of 
 the intermediate 30° are, by theor. 4, found by one fingle fubtrac-. 
 tion for each fine. 
 The fines of tha whole quadrant being thus completed, the tan- 
 gents are found by theor. 18, i9, 20, namely, for one half of the 
 quadrant by the 1éth and 19th, and the other half, by one fingle 
 addition or fubtraction for each, by the 20th theorem. 
 
 And laftly, by theor. 24 and 25, ‘the fecants are deduced from the 
 tangents, by addition and fubtra€tion only. % 
 
 Among the various means ufed for conitructing the canon of fines, 
 tangents, and fecants, the writers above enumerated feem not to teen 
 
 een, 
 
“ 
 
 TRIGONOMETRICAL TABLES,’ &c. U7 
 
 been poffeffed of the method of differences, fo profitably ufed finces 
 and firft of all I believe by Briggs, in computing his trigonometrical 
 canon and his logarithms, as we fhall fee hereafter, when we come to 
 defcribe thofe works. ‘They took however the fucceflive ditterences 
 of the numbers after they were computed, to verify or prove the truth 
 of them ; and if found erroneous, by any irregularity in the laft differ- 
 ences, from thence they had a method of correéting the original 
 numbers themfelves. At leaft, this method is ufed by Pitifcus, Trig. 
 lib. 2, where the differences are extended to the third order. In 
 page 44 of the fame book alfo is defcribed, for the firft time that I 
 know of, the common notation of decimal fractions, as now ufed. 
 And this fame notation was afterwards defcribed and ufed by baron 
 Napier, in po/itio 4 and 5 of his pofthumous works, on the conitruction 
 of logarithms, publifhed by his fon in the year 1619. But the decimal 
 fractions themielves may be confidered as having been introduced 
 by Regiomontanus, by his decimal divifion of the radius, &c, of the 
 circle; and from that time gradually brought into ufe : but continued 
 long to be denoted after the manner of vulgar fractions, by a line 
 drawn between the numerator and denominator, which laft however 
 was {oon orhitted, and only the numerator fet down, with the line 
 below it; thus it was firft 31-34, then 312%; afterwards, omitting 
 the linep it became 31°°, and laftly 31,;, or 31.33, or 31°35: as’ 
 may be traced in the works of Vieta, and others fince his time, gra- 
 dually into the prefent century. . 
 
 Having often heard it remarked, that the word /ize, or in Latin and 
 French finus, is of doubtful origin ; and as the various accounts which 
 Ihave feen of its derivation are very different from one another, it 
 may not be amifs here to employ a few lines on this matter. Some 
 authors fay, this isan Arabic word, others that it is the fingle Latin 
 word fmus; and in Montucla’s Hi/fare des Mathematiques it is conjec- 
 tured to be an abbreviation of two Latin words*. The conjecture is 
 thus exprefled by the ingenious and learned author of that excellent 
 hiftory, at pa. xxxili, among the additions and corre€tions of the firft 
 volume: ‘ A Voccafton des finus dont on parle dans cette page, comme 
 d’une invention des Arabés, voici un étymologie de ce nom, tout-a-fait 
 heureufe et vraifemblable. Je la dois a M. Godin, de Académie 
 Royale des Sciences, Direteur de Ecole de Marine de Cadix. Les 
 finus font, comme l’on fcait, des moitiés de cords; et les cordes en 
 Latin fe nomment infcript@. Les finus;font donc femiffes infcriptarum, 
 ce que probablement on écrivit ainfi pour abréger, S. Ins. Dela en- 
 fuite s’eft fait par abus le mot de finus.” Now, ingenious as this 
 conjecture is, there appears to be little or no probability forthe truth 
 ofit. For, in the firft place, it is not in the leaft fupported by quo- 
 tations from any of the more early books, to fhow that it ever was.the _ 
 practice to write or print the wordsthus, S. Js. upon which the con- 
 ieCture is founded. . Again, it is faid the chords are called in Latin 
 infcripta ; and it is true that they fometimes are fo: but I think they 
 are more frequently called /ubtenfe, and the fines /emiffes /ubtenfarum 
 
 * That is, in the firft e.:tion o: his book, But he has omitted this improbable conje@ture in 
 the new edition of 1799, 
 D . of 
 
18 . HISTORY OF. 
 
 of thedouble arcs, which will not abbreviate into the word fmus. 
 But it may be faid, what reafon have we to fuppofe that this word is 
 either a Latin word, or the abbreviation of any Latin words what- 
 ever ? and that it feems but proper to feck for the etymology of 
 words in the language of the inventors of the things. For which 
 reafon it is, that we find the two other words, tangens and fecans, are 
 Latin, as they were invented and ufed by authors who wrote in that 
 language. Hut the fines are acknowledged to have been invented 
 and introdueed by the Arabians, and thence by analogy it would 
 feem probable that this is a word of their language, and from them 
 adopted, together with the ufe of it, by the Europeans. And indeed 
 Lanfberg, in the fecond page of his trigonometry above mentioned, 
 exprefsly fays that its Arabic: His words are, Vox finus Arabica ef}, 
 et proinde barbara; fed cum longo ufu approbata fit, et commodior non 
 fuppetat, nequaquam repudianda eft: faciles enim in verbis nos effe oportet, 
 cum de rebus convenit. And Vieta fays fomething to the fame purport, 
 in page 9 of his Univerfalium Infpectionum ad Canonem Mathematicum 
 Liter : His words are, Breve finus vocabulum, cum fit artis, Saracenis 
 prafertim quam familiare, non eft ab artificibus explodendum, ad laterum 
 femiffium infcriptorum denotatimnem, 8c. 
 
 Guarinus alfo is of the fame opinion: in his Euclides Adaudtus, Sc. 
 tract. xx. pa. 307, he fays, Sinus vero eff nomen Arabicum ufurpatum in 
 hance fignificationem a mathematicis ; although he was aware that a Latin 
 origin was afcribed to it by Vitalis, for he immediately adds, Licet V1- 
 talis in fuo Lexico Mathematico ex €0 velit finum appellatum, quod claudat 
 curvitatem arcus. » 
 
 Long before I either faw or heard of any conjecture, or obfervation 
 concerning the etymology of the word finus, I remember that I ima- 
 gined it to be taken from the fame Latin word, fignifying breaft or bo- 
 fom, and that our fine was fo called allegorically. Ihad obferved, 
 that feveral of the terms in trigonometry were derived from a bow to 
 fhoot with, and its appendages; as arcus the bow, chorda, the ftring, 
 and /agitta the arrow, by which name the verfed fine, which repre- 
 fents it, was fometimes called ; alfo, that the tangens was fo called 
 from its office, being a line souching the circle, and /ecans from its cut- 
 ting the fame: I therefore imagined that the fizus was {fo called, either 
 from its refemblance to the breaft or bofom, or from its being a line 
 drawn within the bofom (/inus) of the arc, or from its being that part 
 of the ftring (chorda) of a bow (arcus) which is drawn near the 
 breaft (/inus ) in the at of fhooting. And perhaps Vitalis’s defini- 
 tion, above-quoted, has fome allufion to the fame fimilitude. 
 
 Alfo Vieta feems to allude to the fame thing, in calling /aus an 
 allegorical word, in page 417 of his works, as publifhed by Schooten, 
 where, with his ufual judgment and precifion, he treats of the pro- 
 pricty of the terms wed in trigonometry for certain lines drawn in 
 and about the circle; of which, as it very well. deferves, I fhall here 
 extraét the principal part, to fhow the opinion and arguments of fo 
 great a man on thofe names. ‘ Arabes autem femifles infcriptas du- 
 plo, numeris prefertim 2zftimatas, vocaverunt allegoricé Sinus, atque 
 ideo ipfam femi-diametrum, que maxima eft femiflium infcriptarum, 
 Sinum Torum. Et de iis fua methodo canones exiyerunt qui circum- 
 
 feruntur, 
 
TRIGONOMETRICAL TABLES, &c. 19 
 
 feruntur, fupputante prefertim Regiomontano bené jufté et accuraté, 
 in iis etiam particulis qualium femidiameter adfumitur 10,000,000. 4 
 
 “¢ Fx canonibus deinde finuum derivaverunt recentiores canonem 
 femiffium circumfcriptarum, quem dixére Fecundum ; et canonem 
 eductarum € centro, quem dixére Facundifimum et Beneficum, hypo- 
 tenulis addiftum. Atque aded femiffes circumfcriptas, numeris pre- 
 fertim eftimatas, vocaverunt Fecundss, Sinus numerofve videlicet 
 quanquam nihil vetat Fecundi nomen fubftantivé accipi. Hypote- 
 nufas autem Beneficas, vel etiam fimpliciter Hypotenufas : quoniam 
 hypotenufa in prima ferie finds totius nomen retinet. Itaque ne no- 
 vitate verborum res adumbreter, et alioqui fua artificibus, eo nomine 
 debita, preripiatur gloria, prepofita in Canone Mathematico canon- 
 icis numeris in{criptio, candidé admonet primam feriem effe Canonem 
 Sinuum. In fecunda vero, partem canonis foecundi, partem canonis 
 foecundiflimi, contineri. In tertia, reliquam. 
 
 ‘Sane preter infcriptas et circumfcriptas, circulum etiam adficiunt 
 alive linese recte, velut Incidentes, Tangentes, et Secantes. Vertm 
 illz voces fubftantive funt, non peripheriarum relative. Ac{fecare qui- 
 dem circulum linea recta tuncintelligitur, cum in duobus punctis fecat. 
 Ttaque non loquuntur bené geometricé, qui eductas é centro ad metas 
 circum{criptarum vocant fecantes improprié, cum fecantes et tangen- 
 tes ad certos angulos vel peripherias referunt. Immo vero artem con- 
 fundunt, cum his vocibus neceffe habeat uti geometra abs relatione. 
 
 * Quare fi quibus arrideat Arabum metaphora; que quidem aut 
 omnino retinenda videtur, aut omnind explodenda; ut femifles in- 
 {criptas, Arabes vocant finus; fic femifles circumfcripte, vocentur 
 Profinus Amfinufve ; et educte écentro Transfinuofe. Sin allego- 
 ria difpliceat, geometrica fane infcriptarum et circum{criptarum no 
 mina retineantur. Et cum eductzé centro ad metas circum{criptarum, 
 non habeant hactenus nomen certum neque elegans, voceantur fané 
 profemidiametri, quafi protenize femidiametri, fe habentes ad fuas 
 - circumfcriptas, ficut femidiametri ad infcriptas.”’ 
 
 Again{t the Arabic origin however of this word (/mus) may be 
 urged its being varied according to the fourth declenfion of Latin 
 nouns, like manus ; and that if it were an Arabic word latinized, it 
 would have beenranked under either the firft, fecond, or third dee 
 clenfion, as is ufual in fuch adopted words. 
 
 So that, upon the whole, it will perhaps rather f{eem probable, that 
 the term /inus is the Latin word anfwering to the name by which the 
 Saracens called that line, and not their word itfelf. And this conjeCture 
 feemis to be rendered {till more probable by fome expreflions in pa. 4 
 and 5 of Otho’s Preface to Rheticus’s Canon, where it is not only faid, « 
 that the Saracenscalled the half-chord of double the arc finus, but alfo 
 that they called the part of the radius lying between the fine and thearc 
 finus verfus, vel fagitta, which are evidently Latin words, and feem to” 
 be intended for the Latin tranflations of the names by which the Ara- 
 bians called thefe lines, or the numbersexprefling the lengths of them. 
 
 And this conje€turehas been confirmed and realifed, by a reference 
 to Golius’s Lexicon ofthe Arabic and Latin languages. Inconfequence 
 I find that the Arabic and Latin writers on trigonometry do both of 
 them ufe thofe words in the fame allegorical fenfe, the latter being 
 the Latin tranflations of the former, and not the Arabic words ae 
 
 : rupted. 
 
Baits HISTORY OF : 
 
 ~ 
 
 rupted. ‘Thus the true Arabic word to denote the trigonometrical fine, 
 is WAS) pronounced 7eid, (reading the vowels in the French man- 
 ner), meaning finus indufit, veffisque, the bofom part of the garment ; 
 the verled fine is Rey Sebim, which is fagitta, the arrow 5 the arc 18 
 On > which ‘is arcus, the arc; andthe chord is y a) Vitr, that. 
 is, chorda, the chord. 
 
 OF LOGARITHMS. 
 
 Tae trigonometrical canon of etd fines, tangents and fecants, 
 being now brought to a confiderable degree of perfection, the great 
 length and accuracy of the numbers, together with the increafing 
 delicacy and.number of aftronomical problems and fpherical triangles, 
 ‘to the refolution of which the canon was applied, urged many perfons, 
 converfant in thofe matters, to endeavour to difcover fome means of 
 diminifhing the great labour and time, requifite for fo many multi- 
 plications and divifions, in fuch large numbers as the tables then con- 
 fitted of. And their chief aim was, to reduce the multiplications and 
 
 divifions to additions and fubtractions, as much as poflible. 
 For this purpofe, Nicholas Raymer Urfus Dithmarfus invented an 
 ingenious method, which ferves for'one cafe in the fines, namely, 
 when radius is the frit term in the proportion, and the fines of two 
 arcs are the fecond and third terms; for he fhowed, that the fourth 
 term, or fie, would be found byonly taking halfthe fum or difference 
 of the fines of two other arcs, which fhould be the fum and difference 
 of the lefs of the two former given arcs, and the compiement of the 
 greater. ‘This is no more, in effect, than the following well-known 
 theorem in trigonometry : as half pen is to the fine ons arc, {fo is 
 _ the fine of another arc, to the cofine of the difference minus the co- 
 fine of the fum of the faid arcs. ‘The author publithed this ingenious 
 device in 1588, inhis Pundamentum Afironome. And three or four 
 years afterwards it was greatly improved by Clavius, who adapted it 
 to all- proportions in the refolution of f{pherical triangles, for all 
 fines, tangents, fecants, verfed fines, &c; and that whether radius 
 ee in the proportion or not. All which he explains very fully in dem. 
 53, did. 1. of his treatife on the Afrolabe. See more on this fubjec 
 in in Longomont. Aftron. Danica. pa. 7, et feq. This method, although 
 ingenious, depends not on any abftract property of numbers, but 
 only on the relations of certain lines, drawn in and about the circle ; 
 and it was therefore rather limited, and fometimes attended with 
 
 trouble in the application. 
 
 After perhaps various other contrivances, iNceffant endeavours at 
 length produced the happy inventionoflogarithms, whichare of direct 
 ‘and univerfal application toall numbers abftractedly confidered, being 
 derived froma property inherent in numbers themfelves. ‘This pro- 
 perty may be confidered, either as the relation between a geometrical 
 feries of terms and a correfponding arithmetical one, or as the relation » 
 between ratios and the meafures of ratios, which comes to much the 
 fame thing, having been conceived in oneof thefe ways by fome of the 
 writers 
 
y 4% be . 
 
 LOGARITHMS. ta 2% 
 
 writers on this fubje€t, and in the other by the reft of them, as well as 
 in both ways at different times by the fame writer. A fummary idea 
 of this property, and of the probable, refleCtions made on it by the 
 firft writers on logarithms, may be to the following effect : 
 
 The learned calculators, about the clofe of the iéth, and begin- 
 ning of the i7th century, finding the operations of multiplication and 
 divifion by very long numbers, of feven or eight places of figures, which 
 they had frequently occafion to perform, in refolving problems relating 
 to geography and aitronomy, to be exceedingly troublefome, fet them- 
 felves to confider whether it was not poflible to find fome method of 
 leffening this labour, by fubftituting other eafier operations in their 
 fiead in purfuit of this obje€t, they reflected, that, fince in every 
 multiplication by a whole number, the ratio, or proportion, of the 
 product to the multiplicand, is the fame as the ratio of the multiplier 
 tounity, it will follow that the ratio of the product to unity (which, ac- 
 cording to Euclid’s definition of compound ratios, is compounded 
 of the ratios of the tzid product to the multiplicand and of the mul- 
 tiplicand to unity), muft be equal to the fum of the two ratios of the 
 multiplier to unity and of the multiplicand to unity. Confequently, if 
 they could find a fet of artificial numbers that fhould be the reprefenta- 
 tives of, or fhould be proportional to, the ratios of all forts of numbers 
 to unity, the addition of the two artificial numbers that fhould repre- 
 fent the ratios of any multiplier and multiplicand to unity, would an- 
 fwer to the multiplication of the faid multiplicand by the faid multi- 
 plier, or the fum arifing from the addition of the faid reprefentative 
 numbers would be the reprefentative number of the ratio of the pro- 
 duct to unity;-and confequently, the natural number to which it 
 fhouid be found, in the table of the faid artificial or reprefentative 
 numbers, that the faid fum belonged, would be the product of the faid 
 multiplicand and multiplier. Having fettled this principle, as the: 
 foundationsof their wifhed-for method of abridging the labour of cal~ 
 culations, they refolved to compofe a table of fuch artificial numbers, 
 or numbers that fhould be reprefentatives of, or proportional to, tie 
 xatios of all the common or natural numbers to unity. 
 
 The firft obfervation that naturally occurred to them in the purfuit 
 of this {cheme was, that whatever artificial numbers fhould be chofen 
 
 _to reprefent the ratios of other whole numbers to unity, the ratio of 
 
 equality, or of unity to unity, muft be reprefented by 0; becaufe that 
 
 ratio has properly no magnitude, fince, when it is added to, or fub- 
 
 tracted from, any other ratio, it neither increafes nor diminifhes it. 
 The fecond obfervation that occurred to them was, that any num- 
 
 Der whatever might be chofen at pleafure for the reprefentative of the 
 . Yatio of any given natural number to unity; but that, when once fuch 
 choice was made, all the other reprefentative numbers would be 
 
 thereby determined, becaufe tney mult be greater or lefs than that firft 
 reprefentative number, in the fame proportions in which the ratios re- 
 prefented by them, or the ratios of the correfponding natural numbers. 
 to unity, were greater or Jefs than the ratio of the faid given natural 
 number to unity. ‘Thus, either 1, or 2, or 3, &c, might be chofen 
 for the reprefentative of the ratio of 10 tol. But, if 1 be chofen for it, 
 
 the reprefentatives of the ratios of 1Q0 to 1 and 1000 to 1, whichare 
 
 . 
 
 double 
 
*, Saeea 7 HISTORY OF 
 
 double and triple of the ratio of 10 to 1, muft be 2and 3, and cannot 
 be any other numbers ; and if 2 be chofen for it, then the reprefenta- 
 tives of theratios of lOUto 1 and 1000 to 1, will be 4 and 6, and cannot 
 be any other numbers ; and, if 3 be chofen for it, then the reprefenta- 
 tives of the ratios of 100 to 1 and 1000 to 4, will be 6 and 9, and 
 cannot be any other numbers; and fo on. . 
 
 The third obfervation that occurred tothem was, that, as thefe arti- 
 ficial numbers were reprefentatives of, or proportional to, ratios of the 
 natural numbers to unity, they muft be -expreflions of the numbers of 
 fome fmaller equal ratios that are contained in the faid ratios. Thus, 
 ‘if 1 be taken for the reprefentative of the ratio of 10 to 1, then 3, 
 which is the reprefentative of the ratio of 1000 to 1, will exprefs the 
 number of ratios of 10 to | that arecontained intheratioof 1000 to 1. 
 Andif, inftead of 1, we make 10,000,000, or ten millions, the reprefen- 
 tative of the ratio of 10 to 1, (in which cafe 1 will be the reprefentative 
 of a very {mall ratio, orratiuncula, whichis only theten-millionth part 
 of the ratio of 10 to 1, or willbethe reprefentative ofthe 10,000,000th 
 root of 10, or of the firft or fmalleft of 9,999,999 mean proportionals 
 interpofed between 1 and 10), the reprefentative of theratio of 1000 to 
 1, which will in this cafe be $0,000,000, will exprefs the number of 
 thofe ratiuncula, or {mall ratios of the 10,000,000th root of 10 to 1, 
 whichare contained in the faid ratio of 1000 tol. And the like may 
 ‘be fhown of the reprefentative of the ratio of any other number to 
 unity. And therefore they thought thefe artificial numbers, which 
 thus reprefent, or are proportional to, the magnitudes of the ratios 
 of the natural numbers to unity, might not improperly be called the 
 LoearitHns ‘of thofe ratios, fince they exprefs the numbers of 
 {maller ratios of which they are compofed. And, then, for the fake 
 of brevity, they called them the Logarithms of the faid natural numbers 
 themfelves, which are the antecedents of the faid ratios to unity, of 
 which they are in truth the reprefentatives. ? 
 
 The foregoing method of confidering this property leads to much 
 the fame conclufions as the other way, in which the relations between | 
 a geometrical feries of terms, and their exponents, or the terms of an 
 arithmetical feries, are contemplated. Inthis latter way, it readily 
 occurred that the addition of the terms of the arithmetical feries cor- - 
 refponded to the multiplication of the terms of the geometrical feries; 
 and that the arithmeticals would therefore form a fet. of artificial 
 _ numbers, which, when arranged im tables, with their geometricals, 
 would anfwer the purpofes defired, as has been explained above. 
 
 From this property, byaffuming four quantities, two of them as two 
 terms in a geometrical feries, and the others asthetwocorrefponding 
 terms of the arithmeticals, or artificials, or logarithms, it is evident 
 that all the other terms of both thetwoferiesmay thence be generated. 
 And therefore there may be as many {ets or {cales of logarithms as we 
 pleafe, fince they depend entirely on the arbitrary affumption of the 
 firft two arithmeticals. And all poflible natural numbers may be 
 fuppofed to coincide with fome of the terms of any geometrical pro- 
 greflion whatever, the logarithms or arithmeticals determining which 
 of the terms in that progreflion they are. 3 
 ~ It was proper however that the arithmetical feries fhould be fo af- 
 
 fumed, 
 
LOGARITHMS. 23 
 fumed, as that the term 0 in it might anfwer to the term ] in the geo- 
 metricals; otherwife the fum of the logarithms of any two numbers 
 would be always to be diminifhed by the logarithm of 1, to give the 
 logarithm of the product of thofe numbers : for which reafon, making 
 © thelogarithm of 1, andafluming any quantity whatever for the value 
 of the logarithm of any one number, the logarithms of all other num- 
 bers were thence to be derived. Andhence, like as the multiplication 
 of two numbers is effeCted by barely adding their logarithms, fo divi- 
 fion is performed by fubtracting the logarithm of the one from that 
 of the other, raifing of powers by multiplying the logarithm of the 
 given number by the index of the power, and extraction of roots by 
 dividing the logarithm by the index of the root. It is alfo evident, that 
 in all icales or fy{tems of logarithms, the logarithm of 0 will beinfinite ; 
 namely, infinitely negative if the logarithms increafe with the natural 
 numbers, but infinitely pofitive if the contrary; becaufe that while the 
 geonietrical feries mult decreafe through infinite divifions by the ratio 
 of the progreflion, before the quotient come to or nothing; the loga- 
 rithms, or aritmeticals, will in like manner undergo the correfpond- 
 ing infinite fubtra€tions or additions of the common equal difference ; 
 which equal increafe or decreafe, thus indefinitely continued, mutt 
 needs tend to an infinite refult. | 
 
 This however was no newly-difcovered property of numbers, but 
 what was always well known to all mathematicians, being treated of 
 in the writings of Euclid, as alfo by Archimedes, who made great 
 ufe of it in his Arenarius, or treatife on the number of the fands, 
 namely, in affigning the rank or place of thofe terms, ofa geometrical 
 feries produced from the multiplication together of any of the fore- 
 going terms, by the addition of the correfponding terms of the arith- 
 metical feries, which ferved as the indices or exponents of the former. | 
 Stifelius alfo treats very fully of this property at folio 35. et feq. and. 
 there explains all its principal ufes, as relating to the logarithms of 
 numbers, only without the name; fuch as, that addition anfwers to: 
 multiplication, fubtraCtion to divifion, multiplication of exponents to 
 involution, and dividing of exponents to evolution ; all which he ex- 
 emplifies in the rule-of-three, and in finding fevetal mean propor- 
 tionals, &c, exactly as is done in logarithms. So that he feems to have 
 been in the full poffeffion of the idea of logarithms, but without the 
 neceffity of making a table of fuch numbers. For the reafon why 
 tables of thefe numbers were not fooner compofed, was, that the ac- 
 curacy and trouble of trigonometrical computations had not fooner 
 rendered them neceflary. It is therefore not to be doubted, that 
 about the clofe of the fixteenth and beginning of the feventeenth 
 century, many perfons had thoughts of fuch a table of numbers, be. 
 fides the few who are faid to have attempted it. | 
 
 It has been faid by fome, that Longomontanus invented logarithms : 
 but this cannot well be fuppofed to have been any more than in idea, 
 {ince he never publifhed any thing of the kind, nor ever laid claim to 
 the invention, though he lived thirty-three years after they were firit 
 publifhed by baron Napier, as he died only in 1647, when ge: 
 
 cey 
 
 “ 
 
24 HISTORY OF 
 
 been long known and received all over Europe. Nay more, Lorn 
 gomontanus himfelf afcribes the invention to Napier: vid. Aftron. 
 Danica, p. 7, &c. Some circumftances of this matter are indeed re- 
 lated by Wood in his Athena Oxonienfes, under the article Briggs, on 
 the authority of Oughtred and Wingate, viz. “That one Dr. Craig, 
 a Scotchman, coming out of Denmark into his own country, called 
 upon Joh. Neper baron of Marchefton near Edenburg, and told him, 
 among other difcourfes, of a new invention in Denmark (by Longo- 
 montanus as ’tis faid) to fave the tedius multiplication and divifion in 
 aftronomical calculations. Neper being folicitous to know farther of 
 him concerning this matter, he could give no other account of it,’ 
 than that it was by proportionable numbers. Which hint Neper 
 taking, he defired him at his return to call upon him again. Craig, 
 after fome weeks had paffed, did fo, and Neper then fhowed him a 
 rude draught of that he called Canon mirabilis Logarithmorum. Which 
 draught, with fome alterations, he printing in 1614, it came forth. 
 with into the hands of our author Briggs, and into thofe of Will. 
 Oughtred, from whom the relation of this matter came.” 
 Kepler alfo fays, that one Jufte Byrge, afliltant aftronomer to the 
 landgrave of Heile, invented or projected logarithmslong before Neper 
 
 did; but that they had never come abroad, on account of the great 
 ~ reférvednefs of their author with regardto his own compofitions. It is 
 alfo faid that Byrge computed a table of natural fines for every two 
 feconds of the quadrant. : 
 
 But whatever may have been faid, or conjectured, concerning an 
 thing that may have been done by others, it is certain that the world 
 is indebted, for the Abel hema ae of logarithms, to John Napier, or 
 Nepair*, or in Latin, Neper, baron of Merchifton, or Markinfton, 
 
 in 
 
 i} 
 
 * The origin of which name, Crawfurd informs us, was from a (lefs) peer/e/s action of 
 ene of his ancefiors, viz. Donald, fecond fon of the earl 6f Lenox, in the time of David 
 the Second. ‘* Some Englifh writers, miftaking the import of the term baroz, having 
 _called this celebrated perfon lord Napier, a Scotch nobleman. He was not indeed a pecr 
 of Scotland: but the peerage of Scotland informs us, that he was of a very ancient, ho- 
 nourable, and illu(trious family ; that his aneeftors, for many generations, had been poffeffed 
 ef fundry baronies, and, amongfi others, of the barony of Merchiftoun, which de(cended to. 
 ‘him by the death of his futher in 1608. Mr. Briggs, thereiore, very properly ftyles him Baro 
 Mercheftonii. Now, according to Skene, de verborum fiynificatione, ‘1n this realm (of Scotland) 
 he is called a Barronne, quha haldis his landes immediatelie in chiefe of the king, and hes 
 power of Pit and Gallows ; Fof/a et Furca; qubilk was firft inffitute and granted be king 
 Malcome, quha gave power to the Barrones to have ane Pit, quhairin wemen condemmed for 
 thieft fuld be drowned, and ane Gallows, whereupon men thieves and trefpaffowres fuld be hang- * 
 ed, conforme to the doome given in the Barron Court thereanent.’ So that a Scotch baron, 
 though no peer. was neverthelefs a very confiderable petionage, both in dignity and power ” 
 Reid’s Essay on Logarithms... The name of the illuftrious inventor oflogarithms, and his family, 
 has been varioufly written at different times, and on different occafions., In his own Latin 
 works, and in (perhaps) all other books in Latin, it is Mefer, or Neferus Baro Mercheftonii s 
 By Briggs, in a letter to Archbifhop Uther, he is called Nafer, lord of Markinfion: In Wright’s 
 tranflation of the logarithms, which was revifed by the author himfelf, and pub- 
 lithed in 1616, he is called Nepair, baron of Marchifion; and the fame by Crawfurd 
 and fome others: But M‘Kenzie and others write it Majier, baron of Merchifton 3 - 
 which being alfo the orthography now ufed by the family, I fhall adopt in this. 
 work, I obferve alfo, that the Scotch Compendium of Honour fays he was only 
 ~ Sir 
 
LOGARITHMS. 25 
 
 in Scotland, who died the 3d of April 1618, et 67 years of age. Ba- 
 ron Napier added confiderable improvements to trigonometry, andthe 
 frequent numeral computations he performed in this branch gave oc- 
 cafion to hisinvention of logarithms, in order to fave part of the 
 trouble attending thofe calculations ; and for this reafon he adapted 
 his tables peculiarly to trigonometrical ufes. 
 
 This difcovery he publifhed in 1614, in his book intituled Mirificz 
 Logarithmorum Canonis Defcriptio, referving the conftruction of the 
 numbers till the fenfe of the learned concerning his invention fhould 
 be known. And, excepting the conftrudtion, this is a perfeét work 
 on this kind of logarithms, containing in effect the logarithms of all 
 numbers, and the logarithmic fines, tangents, and fecants, for every 
 minute of the quadrant, together with the defcription and ufes of the 
 tables, as alfo his definition and idea of logarithms. 
 
 Napier explains-his notion of logarithms by lines defcribed or ge-- 
 nerated by the motion of points, in this manner: He firft conceives a 
 lineto be generated by the equable motion of a point, which paffes over 
 equal portions of it in equal {mall moments or portions of time: he 
 then confiders another line as generated by the unequal motion of a 
 point, in fuch manner that, in the aforefaid equal moments or por- 
 tions of time, there may be defcribed or cut off, froma given line, parts 
 which fhall be continually in the fame proportion with the refpective 
 remainders, of that line, which had before been left: then are the fe- 
 veral lengths of the firft linc, the logarithms of the correfponding parts 
 of the latter. Which defcription of then: is fimilar to this, that the 
 logarithms are a feries of quantities or numbers in arithmetical pro- 
 greflion, adapted to another feriesin geometrical progreflion. The firft 
 or whole length of the line, which is diminifhed in geometrical pro- 
 greflion, he makes the radius of a circle, and its logarithm 0 or no- 
 thing, reprefenting the beginning of the firft or arithmetical line ; and 
 the feveral proportional remainders of the geometrical line, are 
 the natural fines of all theother parts of the quadrant, decreafing down 
 to nothing, while the fucceflive increafing values of the arithmetical 
 line, are the correfponding logarithms of thofe decreafing fines: fo that, 
 while the natural lines decreafe from radius to nothing, their loga- 
 rithms increafe from nothing to infinite. Napier made the logarithm 
 of radius to be 0, that he might fave the trouble of adding and fub- 
 tracting it, in trigonometrical proportions, in which it fo frequently 
 occurred ; and he made the Jogarithms of the fines, from the entire 
 quadrant down to O, to increafe, that they might be pofitive, and fo 
 in his opinion the eafier to manage, the fines being of more frequent 
 ufe than the tangents and feeants, of which the whole of the latter 
 and half the former would, in his way, be of a different affetion from 
 the fines ; for it is evident that the logarithms of all the fecants in the 
 quadrant, and of all the tangents above 45°, or the half quadrant, 
 would be negative, being the logarithms of numbers greate: than the 
 radius, whofe logarithm is made equal to O or nething. 
 
 Sir John Napier, and that his fon and heir Archibald, was the firft lord, being tailed to that 
 dignity in 1626. Be this however gs it may, I fhall conform to the common modes of exprei+ 
 fion, and call him indifferently, Buren Negi or Lerd Nupicr. x 
 
 : $ 
 
4 Is HISTORY OF 
 As to the contents of Napier’s table; it confifts of the natural fines 
 
 and their logarithms, for every minute of the quadrant. Like moft 
 other tables, the arcs are continued to 45 degrees from top to bottom 
 on the left-hand fide of the pages, and then returned backwards from 
 bottom to top on the right-hand fide of the pages: fo that the arcs 
 and their complements, with the fines, natural and logarithmic, ftand 
 on the fame line of the page, in fix columns; and in another column, 
 in the middle of the page, are placed the differences between the lo- 
 garithmic fines and cofines on the fame lines, and in the adjacent co- 
 Jumns on the right and left; thus making in all feven columns in 
 
 each page. Ofthefe columns, the firft and feventh contain the are 
 and its complement, in degrees and minutes; the fecond and fixth, 
 the natural fine and cofine of each arc; the third and fifth, the loga- 
 rithmic fine and cofine ; and the fourth, or middle column, the dif- 
 ference between the logarithmic fine and cofine which are in the third 
 and fifth columns. 
 
 To elucidate the defeription, the firft page of the table is here in- 
 ferted, as follows: 
 
 (PL a i re a cer arn ne eR RU RE BER 
 Gr. OU Tari 
 
 POY Wen wes RY: ‘ . 
 min. Sinus. Logarithmi. Diftcrentie. Logarithmi.|, Sinus. 
 O bat O || Infinitum. | Infinitum. O | 10000000 | 60 
 a | 2900 81425681 | 81425680 ] 10000000 | 59 
 2 | 5818 CAS94213 - FASOS211 | ; 2 99999098 | 58 
 3/8 87' we | “70436 9504 70430: 560 4) 9999906 | 57 
 4. 11636 67562740 | 67562739 7 || 9999903 | 56 
 5 |14544 | 65331315 | 65331: 1304 Seti 99 99989 _ a 
 } 
 
 6 117453 || 63508099 | 63508083 
 7 }20362 || 6!966595 | 61900573 
 8 (23271 60631284 | 60651256 
 
 “9 [26180 "59453453 | 50453418 
 
 16 | ggggus6 | 54 
 22} 9999980 | 53 
 28 | g99Qy74 | 52 
 35 |) 9999907 
 
 51 
 1029088 || 58399857 | 58399814 43 || g999959 | 50 
 11 (81997 || 57446759 | 57440707 | 52 | 9999950 | 49 
 12 34900 50576646 - 56576584 02 ~ 9999940 48 
 
 13 137815 || 55776222 | 557760149 
 4 40724 || 55035148 55035004 
 
 Pa Rein) is. irks ot 
 
 73 || 9999928 | 47 
 48 | 55035004 | 84 |) 9999917 a, 
 43632 || 64345225 | 54345129 96 |}. 9999905 | 45 
 46541 || 53699843 53099734 ~ 109 goggsg2 | 44 
 17 |49450 |) 53093600 | 53093577 123 | 9999878 | 43 
 52359 || 52522019 | 52521881 
 19 155268 || 51981350 } 51981202 
 58177 || 51468431 | 51468301 
 4 = ees i AS STIRS ESS 
 | \Ot086 || 50980537 | 50980450 
 22 163995 || 505 15342 50315137 £05 Q999795 | 38 
 23 j00904 50070827 500706 13 | 224 e) 9999776 37 
 24 (69813 || 49645239 | 49644995 244 | 9999756 | 30 
 257 L721 a 49237 30 | 49236765. 205 9999736 | 35 
 26 175630 | 48844826 48844539 _ 287 | __ 9999714 34 
 27 178539 |, 48407431 | 48467122 309 | 9999692 | 33 
 
 138. ~ 9999893 42 
 154 || 9999847 | 41 
 
 170 ||_g999831 | 40 
 187 QO9Q090813 | 39 
 
 28 |81448 |) 48103763 | 48103431 332 | 9999008 | 32 
 | 29 184357 || 47752889 ee 356 | 9999044" } 31 
 | 30 [87205 | 47413852 | 47413471 | 381 | 9999619 “30 
 
 89 Befides: 
 
LOGARITHMS. 27 
 
 Befides the columns which are aétually contained in this table, as: 
 above exhibited and defcribed, namely, the natural and logarithmic 
 fines and their differences, the fame table is made to ferve ailfo for the 
 logarithmic tangents and fecants of the whole quadrant, and for the 
 logarithms of common. numbers. For, the fourth or middle column 
 contains the logarithmic tangents, being equal to the differences be- 
 tween the logarithmic fines and cofines, when the logarithm of radius’ 
 is O, becaufe cofine : fine: : radius : tangent, that is, in logarithms, 
 tangent = fine — cofine. Alfo the logarithmic fines, made negative, 
 become the logarithmic cofecants, and the logarithmic cofines made ~ 
 negative, are the logarithmic fecants ; becaufe fine : radius : : radius : 
 cofecant, and cofine : radius: : radius : fecant; that is, in logarithms, 
 cofecant = 0 — fine = —fine, and fecant = 0 — cofine = — co- 
 fine. And to make it anfwer the purpofe of a table of logarithms of 
 common numbers, the author direéts to proceed thus:. A number being 
 given, find that number in any table of natural fines, or tangents, or 
 fecants, and note the degrees and minutes in its arc; then in his table 
 find the correfponding logarithmic fine, or tangent, or fecant, to the 
 fame number of degrees and minutes; and it will be the required lo- 
 garithm of the given number. 
 
 After his definitions and defcriptions of logarithms, Napier explains 
 his table, and illuftrates the precepts with examples, fhowing how to 
 take out the logarithms of fines, tangents, fecants, and of common 
 numbers ; asalfo how to add and fubtract logarithms. He then pro- 
 ceeds to teach the ufes of thofe numbers; and firft, in finding any 
 of the terms of three or four proportionals, fhowing how to multiply 
 and divide, and to find powers and roots, by logarithms : 2dly, in tri- 
 gonometry, both plane and {pherical, but efpecially the latter, inwhich 
 he is very explicit, turning all the theorems for every cafe into loga- 
 rithms, computing examples toeach innumbers, and then enumerating 
 a fet of aftronomical problems of the {phere which properly belong to 
 each cafe. Napier here teaches alfo fome new theorems in_ {pherical 
 trigonometry, particularly, that the tangent of half the bafe: tang. * 
 fum legs; : tang. } dif. legs: tang. i the alternate bafe; and the ge- 
 neral theorem for what are called his five circular parts, by which he 
 condenfes into one rule, in two parts, the theorems for all the cafes 
 of right-angled {pherical triangles, which had been feparately demon- 
 itrated by Pitifcus, Lanfbergius, Copernicus, Regiomontanus, and 
 others. 
 
 he defcription and ufe of Napier’s canon being in the Latin lan- 
 guage, they were tranflated into Englith by Mr. Edward Wright, 
 an ingenious mathematician, and inventor of the, principles of what 
 has commonly, though erroneouily, been called Mercator’s Sailing. 
 _ He fent the tranflation to the author, at Edinburgh, to be revifed by 
 him before publication; who having carefully perufed it, returned 
 it with his approbation, and a few lines introduced befides into the 
 tranflation. But, Mr. Wright dying foon after he received it back, 
 it. was after his death publithed, together with the tables, but os 
 
 number 
 
28 HISTORY OF 
 
 number to one figure lefs, in the year 1616, by his fon Samuel 
 Wright, accompanied with a dedication to the Eaft-India Company, 
 as alfo a preface by Henry Briggs, of whom we fhall prefently have 
 occafion to {peak more at large, on account of the great fhare he bore 
 in perfecting the logarithms. In this tranflation, Mr. Briggs gave 
 alfo the defcription and draught of a fcale that had been invented by 
 Mr. Wright, and feveral other methods ofhisown, for finding the pro- 
 portional parts to intermediate numbers, the logarithms having been 
 on'y printed for fuch numbers as were the natural fines of each mi- 
 nute. And the note which Baron Napier inferted in this Englifh edi- 
 tion, and which was not in the original, was as follows: ‘ But be- 
 “ canfe the addition and fubtraétion of thefe former numbets may 
 “< feem fomewhat painfull, I intend (if it fhall pleafe God)in a fecond 
 “‘ edition, to fet out fuch logarithms as fhall make thofe, numbers 
 “* above written to fall upon decimal numbers, fuch as 100,000,000, 
 *€ 200,000,000, 300,900,000 &c, which are eafie to be added or 
 ¢ abated to or from any other number.” ‘This note had reference to 
 the alteration of the fcale of logarithms, in fuch manner, that 1 fhould 
 become the logarithm of the ratio of 10 to 1, inftead of the number 
 
 2°3025851, which Napier had made that logarithm in his table, and. 
 
 which alteration had before been recommended to him by Briggs, as 
 we fhall fee prefently. Napier alfo inferted a fimilar remark in his 
 
 Rabdologia, which he printed at Edinburgh in 1617. | 
 The following is the preface to Wright’s* book, which, as far as 
 where 
 
 * Of this ingenious man I fhall here infert in a note the following memoirs, as 
 they have been tranflated from a Latin piece taken out of the annals of Gonvile and Caius 
 College at Cambridge, viz. ‘* This year (1615) died at London, Edward Wright of Gar- 
 vefton in Norfolk, formerly a fellow of this college; a man refpected by all for the integrity 
 and fimplicity of his manners, and alfo famous for his fkill in the mathematical fciences : infos 
 much that he was defervedly ftyled a moft excellent mathematician by Richard Hackluyt, the 
 author of an origina! treatife of our Englifh navigations. What knowledge he had acquired 
 in the fcience of mechanics, and how ufefully he emploved that knowledge to the public as 
 well as private advantage, abundantly appear both from the writings he publifhed, and from 
 the many mechanical operations ftill extant, which are ftanding monuments of his great in- 
 duftry andingenuity. He was the firft undertaker of that dificult but ufeful work, by 
 which a little river is brought from the town of Ware in a new canal, to fupply the city of 
 London with water ;. but by the tricks of others he was hindered from completing the work 
 he had begun. He was excellent both in contrivance and execution ; nor was he inferior to 
 the moft ingenious mechanic in the making of inflruments, either of brafs, or any other 
 matter. To his invention is owing whatever advantage Hondius’s geographical charts have 
 above others ; for it was our Wright that taught Jodocus Hondius the method of conttructing 
 them, which was till then unknown : but the ungrateful Hondius concealed the name of the 
 true author, and arrogated the glory of the invention to himfelf. Of this fraudulent praétice 
 the good man could not help complaining, and juftly enough, in the preface to his 
 Treatife of the Corretion of Errors in the Art of Navigation ; which he compofed with ex- 
 cellent judgment, and after long experience, tothe great advancement of naval affairs. For 
 the improvement of this art he was appointed mathematical lefturer by the Eaft- 
 India Company, and read leétures in the houfe of that worthy knight Sir Thoraas 
 Smith, for which he had a yearly falary of 50 pounds. This office he difcharged 
 with great reputation, and much to the fatisfaction of his hearers. He publithed in 
 Englifh, a book on the doétrine of the fphere, and another concerning the con- 
 firuction of funedials. He alfo prefixed an ingenious preface to the learned Gil- 
 
 , _ _ bert’s 
 
 » 
 
LOGARITHMS. 25). 
 
 where it mentions the change from the Latin into Englifh, is a literal 
 tranflation of the preface to Napier’s original ; but what follows that, 
 is added by Napier himfelf. And I willingly infert it here, as it con- 
 tains a declaration of the motives which led to this difcovery, and as 
 the book itfelf is very fcarce. ‘‘ Seeing there is nothing (right well 
 beloved ftudents,in the mathematics) that is fo troublefome to Mathe- 
 maticall practife, nor that doth more moleft and hinder Calculators, 
 than the Multiplications, Divifions, fquare and cubical Extractions 
 of great numbers, which, befides the tedious expence of time, are for 
 the moft part fubject to many flippery errors: I began therefore to 
 confider in my minde, by what certaine and ready Art I might remove 
 thofe hindrances. And having thought upon many things to this pur- 
 pofe, I found at length fome excellent briefe rules to be treated of 
 (perhaps) hereafter. But amongft all, none more profitable than this, 
 which together with the hard and tedious Multiplications, Divifions, 
 and Extractions of rootes, doth alfo caft away from the worke it felfe, 
 even the very numbers themfelves that are to be multiplied, divided, 
 and refolved into rootes, and putteth other numbers in their place, 
 which performe as much as they can do, onely by Addition and Sub- 
 traction, Divifion by two, or Divifion by three; which fecret inven- 
 tion, being (as all other good things are) fo much the better as it fhall 
 be the more common; I thought good heretofore to fet forth in La- 
 tine for the publique ufe of Mathematicians. But now fome of our 
 Countrymen in this Ifland well affeCted to thefe ftudies, and the more 
 publique good, procured a moft learned Mathematician to tranflate 
 the fame into our vulgar Englifh tongue, who after he had finithed 
 it fent the Coppy of it to me, to be feene and confidered on by my- 
 felf. I having moft willingly and gladly done the fame, finde it to be 
 moft exact and precifely conformable to my minde and the originall. 
 Therefore it may pleafe you who are inclined to thefe ftudies, to re- 
 ceive it from me and the Tranflator, with as much good will as we 
 recommend it unto you. Fare yee well.” 
 
 There are alfo extant copies of Wright’s tranflation with the date 
 1618 in the title: but this is not properly a new edition, but only 
 
 bert’s book on the load-ftone. By thefe and other his writings, he has tranfmitted 
 his fame to late{t pofterity. While he was yet a fellow of this college, he could not 
 be concealed in his private ftudy, but was called forth to the public bufinefs of the 
 kingdom, by the queen’s majefty, about the year 1593. He was ordered to attend 
 . the earl of Cumberland in fome maritime expeditions, One of thefe he has given a 
 faithful account of, in the way of a journal or ephemeris, to which ‘he has prefixed 
 an elegant hydrographical chart of his own contrivance. A little before his death, he 
 employed himfelf about an Englifh tranflation of the book of logarithms then _ lately 
 found out by the honourable Baron Napier, a Scotchman, who had a great afte&tion for 
 him, This pofthumous work of his was publifhed foon after, by his only fon Samuel 
 Wright, who was alfo a fcholar of this college. He had formed many other ufeful defigns, 
 but was hindered by death from bringing them to perfeétion. Of him it may be truly faid, 
 that he ftudied more to ferve the public than Kimfelf; and though he was rich in fame, and 
 in the promifes of the great, yet he died poor, to the fcandal of an ungrateful age.” 
 
 Other anecdotes ofhim, as well as many other mathematical authors, may be found in 
 the curious hiftory of navigation by Dr. James Wilfon, prefixed to Mr. Robertfon’s excellent 
 treatife on that fubject. . . ah 
 
 & 
 
30 . HISTORY OF 
 
 the old work with a new title-page adapted to it (the old one being 
 cancelled), together with the addition of fixteefi pages of new matter 
 called “ An Appendix to the Logarithms, fhowing the practice of 
 the calculation of triangles, and alfo a new and ready way for the 
 exact finding out of fuch lines and logarithms as are not precifely to 
 _be found in the canons.” But we are not told by what author: pro- 
 bably it was by Briggs. } 
 . Befides the trouble attending Napier’s canon, in finding the pro- 
 portional parts, when ufed as a table of the logarithms of common 
 numbers, and whichwas in part remedied by the fore-mentioned con- 
 trivances of Wright and Briggs, it was alfo accompanied with another 
 inconvenience, which arofe from the logarithms being fometimes +. 
 or additive, and fometimes — or negative, and which required there- 
 fore the knowledge of algebraical addition and fubtra@tion. And this 
 inconvenience was occafioned, partly by making the logarithm of 
 radius to be 0, and the fines to decreafe, and partly by the compen- 
 dious manner in which the author had formed the table; making the 
 three columns of fines, cofines, and tangents, to ferve alfo for-the 
 other three of cofecants, fecants, and cotangents. | 
 
 But this latter inconvenience was well remedied by John Speidell, 
 in his New Logarithms, firft publifhed in 1619, which contained all 
 the fix columns, and in this order; fines, cofines, tangents, cotan- 
 gents, fecants, cofecants: and they were befides made all pofitive, by be~ 
 ing taken the arithmetical complements of Napier’s, that is, they were 
 the remainders leftby fubtraGting each of thefe latter from, 1000000. 
 And the former inconvenience was more effectually removed by the 
 faid Speidel!, in an additional table, given in the fixth impreflion of 
 the former work, in the year 1624. This was a table of Napier’s 
 logarithms for the round or integer numbers 1, 2, 3, 4, 5, &c, to 1000, 
 together with their differences and arithmetical complements 3 as alfo 
 the halves of the faid logarithms, with their differences and arithme- 
 tical complements; which halves confequentiy were the logarithms 
 of the fquare roots of the faid numbers. Thete logarithms are-how- 
 ever a little varied in their form from Napier’s,namely, foas toancreale 
 fron: 1, whofe logarithm is 0, inftead of decreafing fo 1, or radius, 
 whofe logarithm Napier made 0 likewife ; that is, Speidell’s logarithm 
 of any number #, is equal to Napier’s logarithm of its reciprocal ;: 
 fo that in this laft table of Speidell’s, the logarithm of 1 being 0, the 
 logarithm of 10 is 2302584, the logarithm of 100 is twice as much, 
 or 4605168, and that of 1000 thrice as much, or 6907758. 
 
 This table is now commonly called Ayperbolic logarithms, becaufe 
 the numbers exprefs the areas between the afymptote and curve of the 
 hyperbola, thofe areas being limited by ordinates parallel te the other 
 
 -afymptote, and the ordinates decreafing in geometrical progreflion. 
 But this is not a very proper method of denominating them, as fuch 
 areas may be made to denote any fyftem of logarithms whatever, as 
 w efhall fhow more at large in the proper place. idee 
 
 In the year 1619, Robert Napier, fon of the inventor of logarithms, 
 
 . | publithed 
 
LOGARITHMS. | 31 
 
 publithed a new edition of his late father’s Logarithmorum Canenis 
 Diferiptio, together with the promifed Legarithmorum Canonis Cona 
 firuétio, and other mifcellaneous pieces, written by his father and by 
 Mr. Briggs.—Alfo one Bartholomew Vincent, a bookfeller at Lugdu- 
 num, or Lyons, in France, printed there an exact copy of the fame 
 two works in one volume, in the year 16203; which was four years 
 before the logarithms were carried'to France by Wingate, who was 
 therefore erroneoufly faidto have firft introduced them into that country. 
 But I {hall treat more particularly of the contents of this work, after 
 I have enumerated the other writers on this kind of logarithms. 
 
 In 1618 or 1619, Benjamin Urfinus, mathematician to the Elector 
 of Brandenburg, publifhed, at Cologn, his Cur/us Mathematicus, in 
 which is contained a copy of Napier’s logarithms, with the addition 
 of fome tables of proportional parts. And in 1024, he printed at the 
 fame place, his Trigonometria, with a table of natural fines and. their | 
 _ Jogarithms, of the Napierian kind and form, to every ten feconds in 
 the quadrant; which he had been-at much painsin computing. 
 
 In the fame year 1624, logarithms, of nearly the fame kind, were 
 alfo publifhed, at Marpurg, by the celebrated John Kepler, mathema- 
 tician to the Emperor Ferdinand the Second, under the title of Cdr“as 
 Logarithmorum ad Totidem Numeros Rotundos, premifa Demonftratione 
 fegitima Ortus Logarithmorum eorumque U/usy &c; and the year follow- 
 ing, a fupplement to the fame; being applied to round or integer 
 numbers, and to fuch natural fines as nearly coincide with then» 
 ‘Thefe are exa€tly the fame kind of logarithms as Napier’s, being the 
 fame logarithms of the natural fines of arcs, beginning from the quad- 
 rant, whofe fine or radius is 10,000,000, the logarithm of which is 
 made ©, andfrom thence the fines decreafing by equal differences, 
 down to 0, or the beginning of the quadrant, whilft their logarithms 
 increafe to infinity. So that the difference between this table and 
 Napier’s, confifts only in this, namely, that in Napier’s table the are 
 of the quadrant is divided into equal parts, differing by one minute 
 each, and confequently their fines, to which the logarithms are adapted, 
 are irrational or interminate numbers, and only exprefied by approx~ 
 imate decimals ; whereas in Kepler’s table, the'radins is divided into 
 equal parts, which are confidered as perfect and terminate fines, having 
 equal differences, and .to ‘which terminate fines the logarithms are 
 lrere adapted. By this means indeed the proportions for intermediate 
 numbers and logarithms are eafier made, but then the correfponding 
 arcs are not terminate, but irrational, and only fet down to an ap- 
 proximate degree. So that Kepler’s\table is more convenient as a 
 table of the logarithms of common numbers, and Napier’s as the lo- 
 garithmic fines of the arcs of the quadrant. In both tables, the loga- 
 rithm of the ratio of 10 to 1, is the fame quantity, namely 23025852 ; 
 and as the radius, or greateft fine, 1s 10,000,000, whofe logarithm is 
 made 0, the logarithms of the decuple parts of it will be found by 
 adding 23025852 continually, or multiplying this logarithm by 4, ¥, 
 4, &c 3 and hence the logarithm of 1, the farft number, or {malleft 
 fine, in the table, is 161180959, or 7 times 2302 &c. 
 
32 
 
 HISTORY OF 
 
 Befides the two columns, ofthe natural fines and their logarithms, 
 with the differences of the logarithms, this table of Kepler’s confifts 
 alfo of three other columns ; the firft of which contains the neareft 
 ares, belonging to thofe fines, exprefled in degrees, minutes and fe- 
 conds; and the other two exprefs what parts of the radius each fine 
 is equal to, namely, the one of them in 24th parts of the radius, and 
 minutes and feconds of them; and the other in 60th parts of the ra- 
 dius, and minutes of them. 
 
 As a fpecimen I have here extraéted the latt page of the table. 
 printed exactly as in the work : 
 
 siaces SINUS. -| Partesvice- {| LoGarirumM! Partes 
 Circulicum | feu numeri 6 t differentiis.Af i 
 differentiis. abfoluti. 1mz quar ew. jCUM dliferentils. exagenarigz, 
 1 ene 101.58 
 80. 3. 461 98500.00 |23. 38. 24) 1511.36+ | 59. 6 
 2 0. Lg 101.47 
 80. 23. 58 | 98600.00 |23. 39. .50; 1409.890+ | 59. 16 
 20. 53 |————_| ——_>$ —__— —_- 101.37 —_————- 
 80. 44. 51 | 98700.00 23. 41. 17] 1308.524+ | 59. 13 
 | 21. 2 101.26 r 
 81. 6. 33 | 98800.00 |23. 42, 43! 1207.26 5G he, 
 me VO Da Seana 
 81. 29. .26 | 98900.00 |23. 44. 10) 1106.09+ | 59. 20 
 24. 6 101.06 | 
 81. 53. 32 | 99000.00 |23. 45. 36; 1005.03+ | 59. 24 
 D5, 6 meee meee fee ence 100.96 
 82. 18. $38 | 99100.00 |23. ~ 47. 2 904.07+ | 59. 28 
 26. 28 100.85 
 82. 45. 6 | 99200.00 |23. .48. 29 803.224 SQ 81 
 07.554 |-——- | +. 100.76 —_—— 
 83. 13. O | 99300.00 |23. 49. 55 702.46 A Dis Pe 
 30. 20 100.65 
 83. 43. 20] 99400.00 |23, 51. 22 601.81 59. 38 
 82. 40 PAIS SURAT VDE 100.56 eee ia 
 84. 16. O | 99500.00 |23. 52. 48 501.254 | 59. 42 
 3:6..180 100.45 
 84. 52. 30 | 99600.00 }23. 54. 14 400.80 50,. . 46 
 PS Magee Dag Pa aN ot NES Blk panes 100.35 i se a ool 
 85. 33. 39 | 99700.00 |23. 55. 41 300.45 59. 49 
 48. 54 100.25 
 86. 22. 33 | 99800.00 |23. 57. 7 200 20 59... ~53 
 Dee ie VEL, (yn Pema Romane, Peal ROUT ie PATA POST Fo Be ee Eee Sua 
 87. 26. 15 | 99900.00 j23. 58. 34 100.05 50.; 30 
 Sen” Maia: Bt ONL 100.05 
 90. O. QO. |100000.00 /24, O. Q| 000000.00 60. oO 
 
 To 
 
LOGARITHMS. __ 33 
 
 To the table, Kepler prefixes a pretty confiderable tract, containing 
 ‘the conftruction of the logarithms, and a demonttration of their pro- 
 perties and {tructure, in which he confiders logarithms, in the true 
 and legitimate way, as the meafures of ratios, as fhall be fhown 
 more particularly hereafter in the next part, where we {hall treat of 
 the conftruction of logarithms. 
 
 Kepler alfo introduced the logarithmic calculus into his Rudolphine 
 tables, publifhed in 1627 ; and inferted inthat work feveral logarithmic 
 tables; as, firft, a table fimilar to that above defcribed, except that 
 the fecond, or column of fines, or of abfolute numbers, is omitted, 
 and, inftead of it, another column is added, fhowing what part of the 
 quadrant each arc is equal to, namely the quotient, exprefied in inte- 
 gers and fexagefimal parts, arifing from the dividing the whole qua- 
 drant by each given are; 2dly, Napier’s table of logarithmic fines to 
 every minute of the quadrant; alfo two other fmaller tables, adapted 
 for the purpofes of eclipfes and the latitudes of the planets. In this 
 work alfo Kepler gives a fummary account of logarithms, with the 
 def{cription and ufe of thofe that are contained in thefe tables. And 
 here it is that he mentions Juftus Byrgius, as having had logarithms 
 before Napier publifhed them. 
 
 Befides the above, fome few others publifhed logarithms of the fame 
 kind about this time. But let us now return to treat of the hiftory 
 of the common or Briggs’s logarithms, fo called becaufe he firft com- 
 puted them, and firft mentioned them, and recommended them to 
 Napier, inftead of the firft kind by him invented. | 
 
 Mr. Henry Briggs, not lefs efteemed for his great probity, and 
 other eminent virtues, than for his excellent fkill in mathematics, was 
 at the time of the publication of Napier’s logarithms, in 1614, pro- 
 feflor of geometry in Grefham college in London, having been ap- 
 pointed the firft profefior after its inftitution: which appointment he 
 held till January 1620, when he was chofen, alfo the firft, Savilian 
 profeflor of Geometry at Oxford, where he died January the 26th, 
 1687, aged about 74 years. 
 
 On the publication of Napier’s logarithms, Briggs immediately ap- 
 plied himfelf to the ftudy and improvement ofthem. In a letter to 
 Mr. (afterwards Archbifhop) Ufher, dated the 10th of March 1615, 
 he writes, “ that he was wholly taken up and employed about the 
 noble invention of logarithms, lately difcovered.”’ And again, ‘* Napier 
 lord of Markinfton hath fet my head and hands at work with his new 
 and admirable logarithms : I hope to fee him this fummer, if it pleafe 
 God ; for I never faw a book which pleafed me better, and made me 
 more wonder.” ‘Thus we find that Briggs began very early to com- 
 pute logarithms: but thefe were not of the fame kind with Napier’s, 
 in which the logarithm of the ratio of 10 to 1 was 2°3025851 &c; 
 _for, in Briggs’s firft attempt he made 1 the logarithm of that ratio; 
 and, from the evidence we have, it appears he was the firft perfon 
 who formed the idea of this change in the fcale, which he prefently 
 -and.generoufly communicated, both to the public in his lectures, and 
 to lord Napier himfelf, who afterwards faid that he alfo had thought 
 
 . of the fame thing; as appears by the following extract, tranflated iin 
 F the’ 
 
have been as in the firft column ; but after they 
 
 54 HISTORY OF 
 
 the preface to Briggs’s drithmetica Loggrithmica : “ Wonder not (fays 
 he) that thefe logarithms are different trom thofe which the excellent 
 baron of Marchifton publithed in his Admirable Canon. For when I 
 explained the doctrine of them to my auditors at Grefham college in 
 London, I remarked that it would be much more convenient, the lo- 
 garithm of the fine total or radius being 0 (as in the Canon Mirificus), 
 
 if the logarithm of the 10th part of the faid radius, namely, of 5° 44’ 
 
 21”, were 100000 &c3 and. concerning this I prefently wrote to the 
 author; alfo, as foon as the feafon of the year and my public teaching 
 would permit, I went to Edinburgh, where being kindly received by 
 him, I ftaida whole month. But when we began to converfe about 
 the alteration of them, he faid that he had formerly thought of it, and 
 wifhed it; but that he chofe to publith thofe that were already done, 
 till fuch time as his leifure and health would permit him to make others 
 more convenient. And as to the nature of the change, he thought 
 it more expedient that O fhould be made the logarithm of 1: and 
 100000 &c the logarithm of radius; which I could not but acknow- 
 ledge was much better. ‘Therefore, rejeCting thofe which I had be- 
 fore prepared, I proceeded, at his exhortation, to calculate thefe: and 
 the next f{ummer I went again to Edinburgh, to fhow him the prin- 
 ciple of them; and fhould have been glad to do the fame the third 
 
 fummer, if it had pleafed God to {pare him fo long.” 
 
 So that it is plain that Briggs was the inventor of the prefent fcale 
 of logarithms, in which 1 is the logarithm of the ratio of 10 to 1, and 
 2 that of 100to 1, &c3 and that the fhare which Napier hadin them, 
 was only advifing Briggs to begin at the loweft number 1, and make 
 the logarithms, or artificial numbers, as Napier had alfo called them, 
 
 ‘to increafe withthe natural numbers, inftead of decreafing ; which made 
 
 no alteration in the figures that exprefled Briggs’s logarithms, but only 
 in their affection or figns, changing them from negative to pofitive; 
 fo that Briggs’s firft logarithms to the numbers in 
 the fecond column of the annexed tablet, would 
 
 were changed, as they are here in the third co- 
 
 Jumn ; which isa change of no eflential difference, ae fala fs 
 _as the logarithm of the ratio of 10 to 1, the radix 9 \ Ot? 
 _of the natural fyftem of numbers, continues the 1 sn a 
 fame, a change in the logarithm of that ratio being en | O 
 _the only circumftance that can effentially alterthe |—1] 10 1 
 
 fyfem of logarithms, the logarithm of | being 0, [-—2 |100 | 2 
 
 And the reafon why Briggs, after that interview, ae NO” 3 
 
 —- a 
 
 rejected what he hadbefore done, and began anew, 
 was probably becaufe he had adapted his new 
 logarithms to the approximate fines of arcs inftead 
 
 of the round or integer numbers, and not from their being logarithms 
 
 of another fyftem, as were thofe of Napier. 
 
 On Briggs’s return from Edinburgh to London the fecond time, 
 namely, in 1617, he printed the firft thoufand logarithms, to eight 
 places of figures, befides the index, under the title of Logarithmorum 
 
 ' Chilias Prima. But thefe feem not to have been publifhed till after 
 
 the 
 
 a a 
 
-LOGARITHMS. 35 
 
 the death of Napier, which lIrappened on the 3d of April 1618, as 
 before faid; for, in the preface to them, Briggs fays, “ Why thefe 
 logarithms differ from thofe fet forth by their moft illuftrious inventor, | 
 of ever refpectful memory, in his Cavion Mirificus, ty 1s TO BE HOPED 
 his pofthumous work will fhortly make appear.” And as Napier, 
 after communication had with Briggs, on the fubje& of altering the 
 {cale of logarithms, had given notice, both in Wright’s tranflation, 
 and in his own Rabdologia, printedin 1617, of his intention to alter 
 the fcale (though it appears very plainly that he never intended to 
 compute any more), without making any mention of the fhare which. 
 Briggs had in the alteration, this gentleman modeftly gave the above 
 hint. But not finding any regard paid to it in the faid pofthumous 
 work, publifhed by lord Napier’s fon in 1619, where the alteration 
 is again adverted to, but {till without any mention of Briggs; this 
 gentleman thought he could not do lefs than ftate the grounds of that 
 alteration himfelf, as they are above extracted from his work pub- 
 lifhed in 1624. 
 
 Thus, upon the whole matter, it feems evident that Briggs, whe- 
 ther he had thought of this improvement in the conftru@tion of loga- 
 rithms, of making | the logarithm of the ratio of 10 to 1, before lord 
 Napier, or not, (which is a fecret that could be known only to Napier 
 himfelf), was the firft perfon who communicated the idea of fuch an 
 improvement to the world ; and that he did this in his le€tures to his 
 auditors at Grefham college in the year 1615, very foon after his per- 
 ufal of Napier’s Canon Mirificus Logarithmorum in the year 1614. He 
 alfo mentioned it to Napier, both by letter in the fame year, and on 
 his firft vifit to him in Scotland in the fummer of the year 1616, when 
 Napier approved the idea, and faid it had already occurred to himfelf, 
 and that he had determined to adopt it. It would therefore have been 
 more candid in lord Napier to have told the world, in the fecond edi- 
 tion of this book, that Mr. Briggs had mentioned this improvement to 
 him, and that he had thereby been confirmed in the refolution he had 
 already taken, before Mr. Briggs’s communication with him, to adopt 
 it in that his fecond edition, as being better fitted tothedecimal notation. 
 of arithmetic which was in general ufe. Such a declaration would 
 have been but an act of juftice to Mr. Briggs; and the not having 
 made it, cannot but incline us to fufpect that lord Napier was de- 
 firous that the world fhould afcribe to him alone the merit of this 
 very ufeful improvement of the logarithms, as well as that of having 
 originally invented them ; though, if the having firft communicated 
 an invention to the world be fufficient to entitle a man to the honour 
 of having firft invented it, Mr. Briggs had the better title to be called 
 the firft inventor of this happy improvement of logarithms. 
 
 In 1620, two years after the Chilias Prima of Briggs came out, Mr. | 
 Edmund Gunter publifhed his Canon of Triangles, which contains* 
 the artificial or logarithmic fines and tangents, for every minute, to 
 feven places of figures, befides the index, the logarithm of radius be- 
 ing 10-0 &c. Thefe logarithms are of the kind laft agreed upon by 
 Napier and Briggs, and they were the firft tables of logarithmic fines 
 and tangents that were publithed of this fort. Gunter, alfo, in 1623, 
 
 . HO 2 reprinted 
 
36 - HISTORY OF 
 
 reprinted the fame in his book De Seéfore et Radio, together with the 
 Chilias Prima of his old colleague Mr. Briggs, he being profeffor of 
 aftronomy at Grefham college when Briggs was profefior of geometry 
 there, Gunter having been elected to that office the 6th of March 
 1619, and enjoyed it till his death, which happened on the 10th of 
 December 1626, about the forty-fifth year of his age. In 1623 alfo, 
 Gunter applied thefe logarithms of numbers, fines, and tangents, to 
 ftraight lines drawn on a ruler; with which, proportions in common 
 numbers and trigonometry were refolved by the mere application of a 
 " pair of compaffes ; a method founded on this property, that the loga- 
 rithms of the terms of equal ratios are equidifferent. ‘This inftrument, 
 in the form of a two-foot {cale, is now in common ufe for navigation, 
 - and other purpofes, and is commonly called the Gunter. He alfo 
 greatly improved the fector for the fame ufes. Gunter was the firft who 
 ufed the word co-/ine for the fine of the complement of an arc. He alfo 
 introduced the ufe of arithmetical complements into the logarithmieal 
 arithmetic, as is» witnefled by Briggs, chap XV. Arith. Log. And it 
 has been faid, that he flarted the idea of the logarithmic curve, which 
 was fo called becaufe the fegments of its axis are the logarithms of 
 the correfponding ordinates. 
 The logarithmic lines were afterwards drawn in various other ways. 
 In 1627, they were drawn by Wingate on two feparate rulers fliding 
 againft each other, to fave the ufe of compafles in refolving propor- 
 tions. They were alfo,in 1627, applied to concentric circles, by 
 - Oughtred. Then in a fpiral form by a Mr. Milburne of Yorkshire 
 about the year 1650. And, laftly, in 1657, on the prefent fliding rule, 
 by Seth Partridge. | 
 The: difcoveries relating to logarithms were carrie¢ to France by 
 Mr. Edmund Wingate, but not firft of all, as he erroneoufly fays in 
 the preface to his book. He publifhed at Paris, in 1624, two {mall 
 tracts in the French language : and afterwards at London, in 1626, 
 an Englith edition of the fame, with improvements. In the firft of 
 thefe, he teaches the ufe of Gunter’s ruler; andin the other, that of 
 Briggs’s logarithms, and the artificial fines and tangents. Here are 
 contained alfo, tables of thofe logarithms, fines, and tangents, copied 
 from Gunter. ‘The edition of thefe logarithms printed at London in 
 1635, and the former editions alfo I fuppofe, has the units figures 
 difpofed along the tops of the columns, andthe tens down the mar- 
 gins, like our tables at prefent; with the whole logarithm, which was 
 only to fix places of figures, in the angle of meeting: which is the 
 _ firft inftance that I have feen of this mode of arrangement. 
 But proceed we now to the larger ftruCture of logarithms. 
 Briggs had continued from the beginning to labour with great in- 
 » duftry at the computation of thofe logarithms of which he before 
 *publifhed a fhort {pecimen in {mall numbers. And, in 1624, he pro- 
 duced his 4rithmetica Logarithmica—a ftupendous work for fo fhort a 
 time !—-containing the logarithms of 300V0 natural numbers, to four- 
 teen places of figures befidés the index, namely from 1 to 20000, 
 and from 90000 to 100000; together with the differences of the lo- 
 garithms. Some writers fay that there was another chiliad, namely, 
 
 from © | 
 
LOGARITHMS. 3T 
 
 from 100000 to 101000; but none of the copies that I have feen have 
 more than the 30000 above mentioned, and they were all regularly 
 terminated in the ufual way with the word Finis. The preface to 
 thefe logarithms contains, among other things, an account of .the 
 alteration made in the fcale by Napier and himfelf, from which we 
 before gave an extra&t; and an earneft folicitation to others to under- 
 take the computation for the intermediate numbers, offering to give 
 inftructions, and paper ready ruled for that purpofe, to any.perfons 
 fo inclined to contribute to the completion of fo valuable a work. In 
 the introdudtion, he gives alfo an ample treatife on the conftru@tion © 
 and ufes of thefe logarithms, which will be particularly defcribed 
 hereafter. By this invitation, and other means, he had hopes of 
 colleCting materials for the logarithms of the intermediate 70000 
 numbers, whilft he fhould employ his own labour more immediately 
 on the canon of logarithmic {nes and tangents, and fo carry on both 
 works at’ once ; as indeed they were both equally neceflary, and he 
 himfelf was now pretty far advanced in years. 
 
 Soon after this, Adrian Vlacq, or Flack, of Gouda_in Holland,” 
 completed the intermediate feventy chiliads, and republifhed the Arith- 
 metica Logarithmica at that place, in 1627 and 1628, with thofe in+ 
 termediate numbers, making in the whole the logarithms of all num- 
 bers to 100000, but only to ten places of figures. To thefe was 
 added a table of artificial fines, tangents, and fecants, to every minute 
 of the quadrant. ‘ : ‘ 
 
 Briggs himfelf lived alfo to: complete a table of logarithmic fines — 
 and tangents for the hundredth part of every degree, to fourteen places 
 of figures befides the index ; together with a table of natural fines for 
 the fame parts to fifteen places, and the tangents and fecants for the 
 fame to ten places; with the conftruction of the whole. Thefe tables 
 were printed at Gouda, under the care of Adrian Vlacq, and moftly 
 finifhed of before 1631, though not publithed till 1633. But his death, 
 which then happened, prevented him from completing the application 
 and‘ufes of them. FElowever, the performing of this office, when 
 dying, he recommended to his friend Henry Gellibrand, who was then 
 profeflor of aftronomy in Grefham college, having fucceeded Mr. 
 Gunter in that appointment. Gellibrand accordingly added a preface, 
 and the application of the logarithms to plane and {pherical trigono- 
 metry, &c; andthe whole was printed at Gouda by the fame printer, 
 and brought out in the fame year, 1633, as the Lrigonometria Artifi- 
 cialis of Viacq, who had the care of the prefs as above faid. » This 
 work was called. Lrigonometria Britannica; and befides the-arcs in 
 degrees and centefms of degrees, it has another column, containing 
 the minutes and feconds anfwering to the feveral centefms inthe firft 
 column. | . 
 
 In 1633, as mentioned above, Vlacq printed at Gouda, in Holland, 
 his Trigonometria Artificialis; five Magnus Canon Triangulorum Loga- 
 rithmicus ad Decadas Secundorum Scrupulorum.confirudius. ‘This work 
 contains the logarithmic fines and tangents to ten places of figures, 
 with their differences, for every*ten feconds in the quadrant. To 
 them is alfo added Briggs’s table of the firft 20000 logarithms, but 
 carricd only to ten places of figures befides the index, with their dif- 
 
 ferences. 
 
BEN a iy HISTORY OF 
 
 ferences. The whole is preceded by a defcription of the tables, and 
 the application of them to plane and fpherical trigonometry, chiefly 
 extracted from Briggs’s Trigonometria Britannica, above mentioned. 
 
 Gellibrand publifhed alfo, in 1635, 4n Inftitution Trigonometricall, 
 containing the logarithms of the firft 10000 numbers, with the natural 
 fines, tangents, and fecants, and the logarithmic fines and tangents, 
 _ for degrees and minutes, all to feven places of figures, befides the 
 index ; as alfo other tables proper for navigation; with the ufes of 
 the whole. Gellibrand died the 9th of February 1636, in the 40th 
 year of his age, to the great lofs of the mathematical world. 
 
 Befides the perfons hitherto mentioned, who were moftly computers 
 of logarithms, many others have alfo publifhed tables of thofe artifi- 
 cial numbers, more or lefs complete, and fometimes improved and 
 varied in the manner and form of them.. I fhall here juft advert to 
 a few of the principal of thefe. . 
 
 In 1626, D. Henrion publifhed, at Paris, a treatife concerning 
 Briggs’s logarithms of common numbers from | to 20000, to eleven 
 places of figures; with the fines andtangents to eight places only. 
 
 In 1631, was printed, at London, by one George Miller, a book 
 containing Briggs’s logarithms, with their differences, to ten places 
 of figures befides the index, for all numbers to 100000; as alfo the 
 logarithmic fines, tangents, and fecants, for every minute of the qua- _ 
 drant ; with the explanation and ufes in Englifh. 
 
 The fame year, 1631, Richard Norwood publifhed his Trigonome- 
 trie ; in which we find Briggs’s logarithms for all numbers to 10000, 
 and for thefines, tangents, and fecants, to every minute, both to feven 
 places befides the index.—In the conclufion of the trigonometry, he 
 complains of the unfair practices of printing Vlacq’s book in 1627 or 
 1628, and the book mentioned in the laft article. His words are, 
 “© Now whereas I have here, and in fundry places in this book, cited 
 Mr. Briggs his Arithmetica Logarithmica, (left 1 may feem to abufe the 
 reader,) you are to underftand not the book put forth about a month 
 fince in Englifh, as a tranflation of his, and with the fame title; being 
 nothing like his, nor worthy his name; but the book which himfelf 
 put forth with this title in Latin, being printed at London anno 1624. 
 And here I have juft occafion to blame the ill dealing of thefe men, 
 both in the matter before mentioned, and in printing a fecond edition 
 of his 4rithmetica Logarithmica in Latin, whilft he lived, againft his ~ 
 mind and liking ; and brought them over to fell, when the firft were 
 unfold ; fo fruftrating thofe additions which Mr. Briggs intended in 
 his fecond edition, and moreover leaving out fome things that were in 
 the firft edition, of {pecial moment :a practice of very ill confequence, 
 and tending to the great difparagement of fuch as take pains in this 
 kind.” i 
 
 Francis Bonaventure Cavalerius publithed at Bologna, in 1632, his 
 Direétorium Generale Uranometricum, in which are tables of Briggs’s 
 logarithms of fines, tangents, fecants, and verfed fines, each to eight 
 places, for every fecond of the firft five minutes, for every five feconds 
 trom five to ten minutes, for every ten feconds from ten to twenty 
 minutes, for every twenty feconds from twenty to thirty minutes, for 
 
 _ every thirty feconds from 30’ to 1° 30’, and for every minute in the 
 reft 
 
LOGARITEMS. | 39 
 
 reft of the quadrant: which is the firft table of logarithmic verfed . 
 fines that I know of. In this book are contained alfo the logarithms 
 of the firft ten chiliads of natural numbers, namely, from 1 to 10000, 
 difpofed in this manner: all the twenties at top, and from 1 to 19 on 
 the fide, the logarithm of the fum being in the {quare of meeting. In 
 this work, alfo, I think Cavalerius gave the method of finding the 
 area or {pherical furface contained by various arcs defcribed on the 
 {furface of a fphere; which had before been given by Albert Girard, 
 in his Algebra, printed in the year 1629. 
 
 _ Alfo, in the Trigonometria of the fame author, Cavalerius, printed in 
 1643, befides the logarithms of numbers from | to 1000, to eight 
 places, with their differences, we find both natural and logarithmic 
 fines, tangents, and fecants, the former to feven, and the latter to 
 eight places; namely, to every 10” of the firft 30 minutes, to every 
 30” from 30’ to 1°; and the fame for their complements, or back- 
 wards through the laft degree of the quadrant ; the intermediate 83° 
 being to every minute only. 
 
 Mr. Nathaniel Roe, ‘* Paftor of Benacre in Suffolke,” alfo reduced 
 the logarithmic tables to a contracted form, in his Tabule Logarith- 
 mice, printed at London in 1633. Here we have Briggs’s logarithms 
 of numbers from 1 to 100000, to eight places; the fifties placed at 
 top, and from | to 50 on the fide; alfo the firft four figures of the 
 logarithms at top, and the other four down the columns. They con- 
 tain alfo the logarithmic fines and tangents to every 100th part of de- 
 grees, to ten places. . 
 
 Ludovicus Frobenius publifhed at Hamburg, in 1634, his Clavis 
 Univerfa Trigonometrie, containing tables of Briggs’s logarithms of 
 numbers, from I to 2000; and of fines, tangents, and fecants, for 
 every minute; both to feven places. | 
 
 But the tables of logarithms of common numbers was reduced to 
 its moft convenient form by John Newton, in his Zrigonometria Bri- 
 tannica, printed at London in 1658, having availed himfelf of both the 
 improvements of Wingate and Roe, namely, uniting Wingate’s dif- 
 pofition of the natural numbers with Roe’s contracted arrangement 
 of the logarithms, the numbers being all difpofed as in our bett tables 
 at prefent, namely, the units along the top of the page,. and the tens 
 down the left-hand fide, alfo the firft three figures of each logarithm 
 in the firft column, and the remaining five figures in the other co- 
 Jumns, the logarithms being to eight places. This work contains alfo 
 the logarithmic fines and tangents, to eight figures befides the index, for 
 every 100th part of a degree, with their differences, and for 1000th 
 parts in the firft three degrees.—In the preface to this work, Newton 
 takes occafion, as Wingate and Norwood had done before, as well as 
 Briggs himfelf, to cenfure the unfair practices of fome other publifhers 
 of logarithms. He fays, “In the fecond part of this inftitution, thou 
 art prefented with Mr. Gellibrand’s Trigonometrie, faithfully tranflated 
 from the Latin copy, that which the author himfelf publifhed under 
 the title of ‘Lrigonometria Britannica, and not that which Vlacq the 
 Dutchman ftyles Zrigonometria Artificialis, from whofe corrupt and 
 imperfect copy that feems to be tranflated which is amongft us gene- 
 
40 , HISTORY OF 
 
 rally known by the name of Gellibrand’s Trigonsmetry ; but thofe whe 
 either knew him, or have perufed his writings, can teftify that he was 
 mo admirer of the old fexagenary way of working ; ‘nay that he did 
 
 preferre the decimal way before it, ashe hath abundantly teftified in 
 all the examples of this his trigonometry, which differs from that 
 other which Vlacq hath publithed, and that which hath hitherto borne 
 his name in Englifh, as inthe form, fo likewife in the matter of it; 
 for in the two laft-mentioned editions, there is fomething left out in 
 the fecond chapter of plain triangles, the third chapter wholly omitted, 
 und a part of the third in the {pherical ; but in this edition nothing : 
 fomething we have added to both, by way of explanation and demon- 
 {tration.” : 
 
 In 1670, John Caramuel publifhed his Jfathefis Nova, in which are 
 eontained 1000 logarithms both of Napier’s and Briggs’s form, as alfo 
 1000 of what he calls the Perfect Logarithms, namely, the fame as 
 thofe which Briggs firft thought of, which differ from the laft only 
 in this, that the one increafes while the other decreafes, the radix.or 
 logarithm of the ratio of 10 to 1 being the fame in both. 
 
 The books of logarithms have fince become very numerous, but 
 the logarithms are moitly of that kind invented by Briggs, and which are 
 now in common ufe. Of thefe, the moft noted for their accuracy or 
 ufefulnefs, befides the works above mentioned, are Vlacq’s {mall vo- 
 lume of tables, particularly that edition printed at Lyons in 16703 
 alfo tables printed at the fame place in-i760 3 but moft efpecially the 
 tables of Sherwin and Gardiner. Of thete, Sherwin’s Mathematical 
 _ Lables, in 8vo, formed the moft complete colleGtion of any, contain- 
 ing, befides the logarithms of all numbers to 101000, the fines, tan- 
 gents, fecants, and verfed fines, both natural and logarithmic, to every 
 minute of the quadrant. ‘Che firft edition was in 1706 ; but the third 
 edition, in 1742, which was revifed by Gardiner, is efteemed the moft 
 correct of any, though containing many thoufands of errors in the 
 final figures: as to the laft or. fifth edition, in 1771, itis fo errone- 
 oufly printed that no dependance can be placed in it, being the moft 
 inaccurate book of tables I ever knew; I-have a lift of feveral thou- 
 fand errors which I have corre¢ted in it, as well as in Gardiner’s oc- 
 tavo edition. , 
 
 Gardiner alfo printed at London, in 1742, a quarto volume of 
 “Tables of Logarithms, for all: numbers from 1 to 102100, and for 
 the fines and tangents to every ten feconds of each degree in the qua- 
 drant 5 as alfo, for the fines of the firft 72 minutes to every fingle 
 fecond: with other ufeful and neceflary tables ;” namely, a table of 
 Logiftical Logarithms, and three fmaller tables to be ufed for finding 
 the Jogarithms of numbers to twenty places of figures. Of thefe 
 tables of Gardiner, only a {mall number was printed, and that by 
 fubfcription ; and they have always been held in great eftimation for 
 their accuracy and ufefulnefs. 7 
 
 _ An edition of Gardiner’s colleCtion was alfo elegantly printed at 
 | Avignon in France, in 1770, with fome additions, namely, the fines 
 and tangents for every fingle fecond in the firft four degrees, and a 
 {mall table of hyperbolic logarithms, copied from a treatife on Fluxions 
 
 by 
 
LOGARITHMS. | A 
 
 by the late ingenious Mr. Thomas Simpfon : but this is not quite fo ~ 
 correét as Gardiner’s own editiow. ‘The tables in ail thefe books are 
 to feven places of figures. 
 
 There have alfo lately appeared the following accurate and elegant 
 books of logarithms ; viz. 
 
 1. Logarithmic Tables, by the Jate Mr. Michael Taylor, a pupil 
 of mine, and author of The Sexagefimal Table. His work confi{ts 
 of three tables ; 1 {t, The logarithms of Common Numbers from | to 
 1260, each to 8 places of figures ; 2dly, The Logarithms of all Num- 
 bers from 1 to 101000, each to 7 places; 3dly, The Logarithmic 
 Sines and Tangents to every Second of the Quadrant, alfo to 7 
 ‘places of figures: a work that muft prove highly ufeful to fuch per- 
 fons as may be employed in very nice and accurate calculations, fuch 
 as aftronomical tables, &c. ‘The author dying when the tables were 
 nearly all printed off, the Rev. Dr. Mafkelyne, Aftronomer Royal, 
 has fupplied a preface, containing an account of the work, with ex- 
 cellent precepts for the explanation and ufe of the tables: the whole 
 very accurately and elegantly printed on large 4to. 1792. 
 
 2. “Tables Portatives de Logarithmes, publiees 4 Londres par Gar- 
 diner,” &e. This work is moft beautifully printed in a neat portable 
 8vo volume, and contains all the tables in Gardiner’s 4to volume, 
 ‘with fome additions and improvements, and a confiderable degree of 
 accuracy. On this, as well as feveral other occafions, it is but juftice 
 to remark the extraordinary {pirit and elegance with which the learned 
 amen and the artifans of the French nation undertake and execute 
 works of ‘merit. Printed at Paris, by Didot, 1793. 
 
 3. Afe¢ond edition of the ** Tables Portatives de Logarithmes,” &e. 
 printed at Paris with the Stero types, of folid pages, in 8vo, 1795, by 
 Didot. ‘This edition is greatly enlarged, by an extenfion of the old 
 tables, and many new ones; among which are the log. fines and 
 tangents to every ten thoufandth part of the quadrant, viz..in which 
 the quadrant is firft divided into 100 equal parts, and each of thefe 
 into 100 parts again. 
 
 4, Other more extenfive tables, not yet quite completed, ordered 
 ‘by the Board of Longitude in France, and under the dire€tion of M. 
 Prony, in which the quadrant is decimaily divided into 10000 equal 
 
 arts. 
 
 ee The logarithmic canon ferves to find readily the logarithm of any 
 affigned number; and we are told by Dr. Wallis, in the fecond volume 
 of his Mathematical Works, that an antilogarithmic canon, or one to 
 find as readily the number correfponding to every logarithm, was be- 
 gun, he thinks, by Harriot the algebraift (who died in 1621), and 
 completed by Walter Warner, the editor of Harriot’s works, before 
 1640; which ingenious performance, it feems, was loft, for want of 
 ‘encouragement to publith it.”” 
 
 ** A {mall fpecimen of fuch numbers was publifhed in the Philofo- 
 ‘phical Tranfactions for the year 1714, by Mr. Long of Oxford ; but 
 it was not till 1742 that a complete antilogarithmic canon was pub- 
 Jithed by Mr. James Dodfon, wherein he has computed the numbers 
 yi ita to every logarithm from 1 to 100000, for 11 places of 
 
 gures.”” | | 
 
 G THE 
 
42 ! CONSTRUCTION OF 
 
 THE CONSTRUCTION OF LOGARITHMS, &e. 
 Havine defcribed the feveral kinds of logarithms, their rife and 
 
 invention, their nature and properties, and given fome account of the 
 principal early cultivators of them, with the chief colleCtions that have 
 been publifhed of fuch tables; I proceed now to deliver a more par~ 
 ticular account of the ideas and methods employed by each author, 
 and the peculiar modes of conftruction made ufe of by them. 
 
 And firft, of the great inventor himfelf, Lord Napier. 
 
 NAPIERS CONSTRUCTION OF LOGARITHMS. 
 
 The inventor of logarithms did not adapt them to the feries of na- 
 tural numbers 1, 2, 3, 4, 5, &c, as it was not his principal idea to 
 extend them to all arithmetical operations in general ; but he confined 
 his labours to that cireumftance which firft fuggefted the necedlity of 
 the invention, and adapted his logarithms to the approximate numbers 
 which exprefs the natural fines of every minute in the quadrant, as 
 they had been fet down by former writers on trigonometry. 
 
 ‘Vhe fame reftricted idea was purfued through his method of con- 
 flructing the logarithms. As the lines of the fines of all arcs are parts 
 of the radius, or fine of the quadrant, which was therefore called the 
 Jfinus totus, or whole fine, he conceived the line of the radius to be 
 defcribed or run over, by a point moving along it in fuch manner, 
 that in.equal portions of time it generated, or cut off, parts in a de-~ 
 creafing geometrical progreflion, leaving the feveral remainders, or 
 {ines in geometrical progreflion alfo ; whilft, another poimt, in an in- 
 definite line, defcribed equal parts of z¢ in the fame equal portions of 
 time ; fo that the re{pective fums of thefe, or the whole line generated, 
 were always the arithmeticals or logarithms of thefe fines. 
 
 Thus, az is the given radius, on which all the fines are Sines Log. 
 to be taken, and A&c the indefinite line containing the °° 4]? 
 
 logarithms ; thefe lines being each generated by themotion -|; -|: 
 
 of points, beginning at A, a. Now, at the end of the |}, 
 ft, 2d, 3d, &c,. moments, or equal fmall portions of = | 
 time, the moving points being found at the places marked 
 1, 2, 3, &c, then za, z1, 22, 23, &c, will be the feries -4 J. 
 of natural fines, and AO (or0), Al, A2, A3, &c, will be  -)5 
 their logarithms; fuppofing the point which generates az “|x | 
 to move every where with a velocity decreafing in propor- — -|&e -6 
 tion to its diftance from z, namely, its velocity in the points | 
 0, 1, 2, 3, &c, to be refpectively as the diftances 20, zl, 2 | 
 22, 23, &c, whilft the velocity of the point generating the ve 
 
 logarithmic line A&c remains conftantly the fame as at 
 
 firft in the point A or 0.. 
 
 Hitherto the author had not fully limited his fyftem or feale of lo- 
 garithms, having only fuppofed one condition or limitation, namely, 
 that the logarithm of the radius az fhould be 0. Whereas two inde- 
 pendent conditions, no matter what, are neceflary to limit the feale 
 or fyftem of logarithms. It did not occur to him that it was proper 
 
 te form the other limit, by affixing fome particular value to an afligned 
 number, 
 
* 
 
 LOGARITHMS. | 43 
 
 number, or part of the radius : but, as another condition was neceflary, 
 he affumed ¢his for it, namely, that the two generating points fhould 
 begin to move at a and A with equal velocities; or that the incre- 
 ments al and Al, defcribed in the firft moments, fhould be equal ; as 
 he thought this circumftance would be attended with fome little eafe 
 inthecomputation, Andthis is the reafon that, inhistable, the natural 
 fines and their logarithms, at the complete quadrant, have equal dif- 
 ferences ; and this is alfo the reafon why his fcale of logarithms hap- 
 ae accidentally to agree with what have fince been called the hyper- 
 olic logarithms, which have numeral differences equal to: thofe of 
 their natural numbers at the beginning ; except only that thefe latter 
 increafe with the natural numbers, and his on the contrary decreafe; 
 the logarithm of the ratio of 10 to 1 being the fame in both, namely 
 2°30258509. 
 
 And here, by the way, it may be obferved, that Napier’s manner of 
 conceiving the generation of the lines of the natural numbers, and their 
 logarithms, by the motion of points, is very fimilar to the manner in 
 which Newton afterwards confidered the generation of magnitudes in 
 his do€trine of fluxions; and it is alfo remarkable, that, in art. 2. of 
 the Habitudines Logarithmorum et fuorum naturalium numerorum invi= 
 cem, in the appendix tothe Con/iruétio Logarithmorum, Napier {peaks of 
 the velocities of the increments or decrements of the logarithms, in the 
 fame way as Newton does of his fluxions, namely, where he fhows 
 that thofe velocities, or fluxions, are inverfely as the fines or natural 
 numbers of the logarithms; which is a necefflary confequence of the 
 nature of the generation of thofe lines as defcribed above; with this 
 alteration however, that now the radius az muft be confidered as 
 generated by an equable motion of the point, and the indefinite line 
 A&c by a motion increafing in the fame ratio as the other before de- 
 creafed ; which is a fuppofition that Napier muft have had in view 
 when he ftated that relation of the fluxions. 
 
 Having thus limited his fyftem, Napier proceeds, in the pofthumous 
 work of 1619, to explain his conftruction of the logarithmic canon ; 
 and this he effects in various ways, but chiefly by generating, in a 
 very eafy manner, a feries of proportional numbers, and their arith- 
 meticals or logarithms; and then finding, by proportion, the loga- 
 rithms to the natural fines, from thofe of the neare{t numbers among 
 the original proportionals. 
 
 After defcribing the neceflary cautions he made ufe of, to preferve a 
 {ufficient degree of accuracy, in fo long and complex a procefs of cal- 
 culation ; fuch as annexing feveral ciphers, as decimals feparated by 
 a point, to his primitive numbers, and rejecting the decimals thence 
 relulting after the operations were completed; fetting the numbers. 
 downto the neareft unitin the laft figure; and teaching the arithme- 
 tical proceffes of adding, fubtraCting, multiplying, and dividing the 
 limits between which certain unknown numbers mutt lie, fo as to ob- 
 tain the limits between which the refults muft alfo fall; I fay, after 
 _ defcribing fuch particulars, in order to clear and fmooth the way, he. 
 enters onthe great field of calculation itfelf. Beginning at radius 
 10000000, he firft conftrudts feveral defcending geometrical feriesy 
 but of fuch a nature, that they are all quickly formed by an eafy nee 
 
 finua! 
 
44 CONSTRUCTION OF 
 
 tintal fubtraction, and a divifion by 2, or by 10, or 100, &c, which 
 is done by only removing the decimal point fo many places towards 
 the left hand, as there are ciphers in the divifor. He conftruéts three 
 tables of fuch feries: “Fhe firft of thefe confifts of 100 numbers, in 
 _ the proportion of radius to radius minus 1, or of 10000000 to 
 9999999; all which are found by only fubtracting from each its 
 10000000th part, which part is alfo found by only removing each 
 figure feven places lower: the laft of thefe 100 proportionals is 
 found to be 9999900:0004950. | 
 
 The 2d table contains 
 50 numbers, which are 
 
 : f (No.| First TaBLe. Seconp TaBLe. i 
 inthe continual propor- | 1” |19900000.000000011 0000000.000000 
 tion of the firft to the laft 2 | 9999G99.0000G00! 9999900.000000 
 in the firft table, namely, | 3 | g999998.0000001! 9999800.001000 
 of 10000000-0000000,' | 4 | 9999997.0000003} 9999700.003000 
 to 9999900'0004.950, or &c. j&c tul the 100th term,| &c to the 50th term 
 nearly the proportion of | °0 Race miher eI e220 27, 
 100000 to 999993; thefe [190 _1 9999900.0004950. 
 
 therefore are found by 
 only removing the figures of each number 5 places: lower, and fub- 
 tracting them from the fame number : the laft of thefe he finds to be 
 9995001°222927. Anda fpecimen of thefetwo tables is here annexed. 
 The third table confifts of 69 columns, and each column of twenty- 
 one numbers or terms, which terms, sn every column, are in the con- 
 tinual proportion of 10000 to 9995, that is, nearly as the. firft is to 
 the Jaft in the 2d table; and as 10000 exceeds 9995 by the 2000th 
 part, the terms in every column will be conftructed by dividing each 
 upper number by 2, removing the figures of the quotient 3 places 
 lower, and then fubtracting them; and in this way it is proper to 
 conftruct only the firft column of 21 numbers, the laft of which wilk 
 be 9900473°5780: but the 1ft, 2d, 3d, &c, numbers in all the co- 
 Jumns are in the continual proportion of 100 to 99, er nearly the 
 proportion of the firft to the laft in the firft column; and therefore 
 thefe will be found by removing the figures of each preceding number . 
 two places lower, and fubtracting them, for the like number in the 
 next column. <A fpecimen of this 3d table is as here below. 
 
 THe Tuirp Taste. 
 
 Terms: Ist Column, a Column. ud Column. oC Uk Lue o>tii Coiumn. 
 1 |10000000.0006)G4YO00000.C000|Y801000.0000| &c tor }5048858 8900 
 9995000.0006/9$95050.0000|97 96099.5 the Ath |5046334.4605 
 3 | 9990002 5006\9890102.4750/9791201.4503}) 5th, O.h, 9043811.2932 
 A | 99085007 .4987|9885 157.4237|9780305.8495 7th, &c 5041289.3879 
 5 | 9980014.995019880214.8451|9781412.6gG67| col. till '5038768.7435 
 &c é&e ull &C &c the laft , &c, 
 21 | G900473.5780|9801468.8423|9703454 1539 
 
 bo 
 
 Thus he had, in this 3d table, interpofed between the radius and 
 its half, 68 numbers in the continual proportion of 100 to 99 3 and 
 interpofed Letween every two of thefe 20 numbers in the proportion 
 
 of 
 
LOGARITHMS. 45 
 
 16000 to 9995: and again, in the 2d table, between 10000000 and 
 9995000, the two firft of the third table, hehad 50 numbers in the pro- 
 portion of 100000 to 99999; and laftly, in the 1f{t table between 
 10000000 and 9999900, or the two firft in the 2d table, 100 numbers 
 in the proportion of 10000000 to 99999993 that is, in all, about 
 1600 praportionals ; all found in the moft fimple manner, by little 
 more than eafy fubtractions ; which proportionals nearly coincide 
 with all the natural fines from 90° down to 30°. 
 
 To obtain the logarithms of all thofe proportionals, he demonftrates 
 feveral properties and relations of the numbers and logarithms, and 
 illuftrates the manner of applying them. ‘The principal of thefe pro- 
 perties are as follow: ift, that the logarithm of any fine is greater 
 than the difference between that fine and the radius, but lefs than the 
 faid difference when increafed in the proportion of the fine to radius* 3 
 and 2dly, that the difference between the logarithms of two fines is 
 lefs than the difference of the fines increafed in the proportion of the 
 lefs fine to radius, but greater than the faid difference of the fines 
 increafed in the proportion of the greater fine to radius.+ 
 
 Hence, by the 1ft theorem, the logarithm of 10000000, the radius 
 or firft term in the firft table being 0, the logarithm of 9999999, 
 the 2d term, will be between 1 and 1°0000001, and will therefore 
 be equal to 1°00000005 very nearly: and this will be alfo the com- 
 mon difference of all the terms or proportionals in the firft table: 
 therefore, by the continual addition of this logarithm, there will be 
 obtained the logarithms of all thefe 100 proportionals ; confequently 
 100 times the ie firft logarithm, or the laft of the above fums, will 
 - give 100°000005, for the logarithm of 9999900°0004950, the laft of 
 the faid 100 proportionals. 
 
 Then, by the 2d theorem, it eafily appears, that -0004950 is the 
 difference between the logarithms of 999900'0004950 and 9999900, 
 the laft term of the firft table, and the 2d term of the fecond ehtes 
 
 * By this firft theorem, r being radius, the logarithm of the fine s is between 
 
 ys and the ; and therefore, ~when s differs but little from r, the logarithm 
 J . 
 
 of s will be nearly equal to———— Sain - ot oe the arithmetical mean between the li- 
 
 mits r—s andr 5 but ftill nearer to (—s)V— — or are rs the geometrical 
 
 mean between fee {aid limits. 
 
 4 By this fecond theor em, the difference beers the Jeger of the two 
 
 Ss— 
 fines S and s, lying between the limits S rand = 
 
 Fr will, when thofe 
 
 S 
 9 S S—s) : 3 
 fines differ but little, be nearly equal to a or aay caine r, their arith- 
 S—s 
 metical mean ; or nearly = = ety the geometrical mean ; or nearly = car 2rs 
 
 by | fubfituting i in the latt denominator, 4 (S+ts) for 4/ Ss, to which it is nearly 
 
 equal ‘ 
 5 this 
 
as ts CONSTRUCTION OF 
 
 this then being added to the laft logarithm, gives 1000005000 for the 
 logarithm of the faid 2d term, as alfo the common difference of the 
 logarithms of all the proportions in the 2d table; and therefore, 
 by continually adding it, there will be generated the logarithms of all 
 thefe proportionals in the fecond table; the laft of which is 5000°025, 
 anfwering to 9995001°222927, the lait term of that table. 
 
 Again, by the 2d theorem, the difference between the logarithms 
 of this laft proportional of the fecond table, and the 2d term in the 
 firft column of the third table, is found to be 1:22353873; which being 
 added to the laft logarithm, gives 5001°2485387 for the logarithm of 
 9995000, the faid ud term of the third table, as alfo the common 
 
 difference of the logarithms of all the proportionals in the firft column 
 
 of that table; and this therefore, being continually added, gives all 
 the logarithms of that firft column, the laft of which is 100024°97077, 
 the logarithm of 9900473°5780, the laft term of the faid column. 
 
 Finally, by the 2d theorem again, the difference between the loga- 
 rithms of this laft number and 9900000, the 1{t term in the fecond 
 column, is 4'78°3502 5 which being added to the laft logarithm, gives 
 100503°3210 for the logarithm of the faid 1{t term in the fecond 
 column, as well as the common difference of the logarithms of all 
 the numbers on the fame line in every line of the table, namely, of all 
 the I{t terms, of all the 2d, of all the 3d, of all the 4th, &c, terms in 
 all the columns ; and which therefore, being continually added to 
 the logarithms in the firft column, will give the correfponding loga- 
 rithms in all the other columns. — | 
 
 And thus is completed what the author calls the radical table, in 
 which he retains only one decimal place in the logarithms (or artifi- 
 
 _ tals, as he always calls them in his tract on the conftruétion), and 
 four in the naturals. A fpecimen of the table is as here follows; 
 
 RADICAL TABLE. 
 
 Terms 1% Column. _ 2d Column. 69th Column. 
 Naturals: {Artificials|| Naturals. | Artificials Naturals. | Artificial 
 1 $10000000.0000 01} 9900000.0000; 100508.3}} 5048858.8900' 6834225,8 
 2 9995C00.0000 5001.2}] 9895050.0000} 105504.6|| 5046333.4605|6839227.1 
 3 9990002.5000}  10002.5}} 9890102.4750! 110505,8]} 5043811.2932'6844228.3 
 A 0985007.4987! —15003.7]} 988515'7.4297! 115507.1|| 5041289.3879!6849229.6 
 a 99800149950}  20005.0}] 9880214.8451} 120508.3]} 5038768.7435)6854.230.8 
 &c &c till &c &e. &c &e &e |. 
 91 9900475.57801. 100025.0}} 9801468.84293) 200528.2}) 4998609.4054)6934250.8 
 
 Having thus, in the moft cafy manner, completed the radical table, 
 by little more than mere addition and fubtraCtion, both for the natural, 
 numbers and logarithms ; the logarithmic fines were eafily deduced 
 from it by means of the 2d theorem, namely, taking the fum and dif, 
 ference of each tabular fine and the neareft number in the radical table, 
 annexing 7 ciphers to the difference, dividing the refult by the fum, 
 then hali the quotient gives the difference between the logarithms of the 
 
 pis ‘ . faid 
 
LOGARITHMS. 4% 
 
 faid numbers, namely, between the tabular fine and radical number ; 
 confequently adding or fubtracting this difference, to or from the 
 given logarithm of the radical number, there is obtained the loga- 
 rithmic fine required. And thus the logarithms of all the fines, from 
 radius to the half of it, or from 90° to 30°, were perfected. 
 
 Next, for determining the fines of the remaining 30 degrees, he 
 deliverstwo methods. In the firft of thefe he‘proceeds in this’ man- 
 ner: Obferving that the logarithm of the ratio of 2 to 1, or of half 
 the radius, is 6931469°22, of 4 to 1 is the double of this, of 8 to I 
 is triple of it, &c; that of 10 to 1 is 23025842°34, of 20 to 1 is the 
 fum of the logarithms of 2and 10; and fo on, by compofition for 
 the logarithms of the ratios between 1 and 40, 80, 100, 200, &e, to 
 10000000 ; he multiplies any given fine, for an arc lefs than 30 de- 
 grees, by fome of thefe numbers, till he finds the produét nearly equal 
 to one of the tabular numbers ; then by means of this and the fecond 
 theorem, the logarithm of this product isfound ; to which adding the 
 logarithm that anfwers to the multiple above mentioned, the fum is 
 the logarithm fought. But the other method is ftill much eafier, 
 and is derived froin this property, which he demonftrates, namely, 
 as half radius is to the fine of half an arc, fo is the cofine of the faid 
 half arc to the fine of the whole arc; or as 3. radius : fine of an arc :: co~ 
 fine of the arc: fine of double arc; hence the logarithmic fine of an 
 arc is found, by adding together the logarithms of half radius and of 
 the fine of the double arc, and then fubtracting the logarithmic co- 
 fine from the fum. 
 
 And thus the remainder of the fines, from 30° down to 0, are eae 
 fily obtained. But in this latter way, the logarithmic fines for full one 
 half of the quadrant, or from 0 to 45 degrees, he obferves, may be 
 derived ; the other half having already been made by the general 
 method of the radical table, by one eafy divifion and addition or fub- 
 traction for each. 
 
 I have dwelt the longer on this work of the inventor of logarithms, 
 becaufe I have not feen, in any author, an account of his method of con 
 itructing his table, although it is perfectly different from any other 
 method ufed by the later co.nputers, and indeed, almoft peculiar to his 
 {pecies of logarithms. ‘The whole of this work manifelts great inge- 
 nuity in the defigner, as well as much accuracy. But notwithftanding 
 the caution he took to obtain his logarithms true to the neareft unit in 
 the lait figure fet down in the tables, by extending the numbers in the 
 coniputations to feveral decimals, and other means, he had been dif- 
 appointed of that end, either by the inaccuracy of his affitant com- 
 puters or tranferibers, or through fome other caufe ; as the logarithms 
 in the table are commonly very inaccurate. It is remarkable too, that 
 in this traét on the conftruction of the logarithms, Lord Napier never 
 calls them logarithms, but every where arfificials, as oppofed in idea 
 to the naturalnumbers : and this notion, of natural and artificial nums 
 bers, [take to have been his firft idea of this matter, and that he altered 
 the word artificials to logarithms in his firft book, on the defcription of 
 them when he printed it, in the year 1614, and that he would ae 
 
 : ave 
 
sg CONSTRUCTION OF 
 
 have altered the word every where in this pofthumous work if he had 
 lived to print it: for in the two or three pages of appendix, annexed 
 to the work by his fon, from Napier’s papers, he again always calls 
 them logarithms. | ‘This appendix relates tothe change of the loga- 
 rithms to that {cale in which 1 is the logarithm of the ratio of 10 to 1, 
 the logarithin of 1, with or without ciphers, being 0 ; and it appears to 
 have been written after Briggs communicated to him his idea of that 
 change. | 
 
 Napier here in this appendix alfo briefly defcribes fome methods 
 by which this new {pecies of logarithms may be conftru@ted. Having 
 fuppofed 0 to be the logarithm of J, and 1, with any number of ciphers, 
 as 10000000000, the logarithm of 10, he direéts to divide this loga- 
 rithm of 10, and the fucceflive quotients, ten times by 53 by which 
 divifions there will be obtained thefe other ten logarithms, namely, 
 2000000000, 400000000, 80000000, 16000000, 4200000, 640000, 
 128000, 25600, 5120, 1024: then this laft logarithm, and its quo- 
 tients, being divided ten times by 2, will give thefe other ten loga- 
 rithms, 512, 256, 128, 64, 32, 16, 8, 4,2, 1. Andthe numbers 
 anfwering to thefe twenty logarithms we are directed to find in this 
 manner; namely, extract the 5th root of 10 (with ciphers), then the 
 5th root of that root, and fo on, for ten continual extractions of the 
 5th root; fo fhall thefe ten roots be the natural numbers belonging to 
 the firit ten logarithms, above found in continually dividing by 5: next, 
 out of the laft 5th root we are to extract the fquare root, then the 
 {quare root of this laft root, and fo on, for ten fucceflive extraCtions 
 of the fquare root; fo fhall thefe laft ten roots be the natural numbers 
 correfponding to the logarithms or quotients arifing from the laft ten 
 divifions by the number 2. And from thefe twenty logarithms, 1, 2, 4, 
 8, 16, &c, and thejr natural numbers, the author obferves that other 
 logarithms and their numbers may be formed, namely, by adding the 
 logarithms, and multiplying their correfponding numbers. 
 
 It is evident that this procefs would generate rather an antiloga- 
 rithmic canon, fuch as Dodfon’s, than the table of Briggs; and that - 
 the method would alfo be very laborious, fince, befides the very 
 troublefome original extractions of the 5th roots, allthe numbers would 
 be very large, by the multiplication of which the fucceflive fecondary 
 natural numbers are to be found. 
 
 Our author next mentions another method of deriving a few of the 
 primitive numbers and their Jogarithms, namely, by taking continu- 
 ally geometrical means, firft between 10 and 1, then between 10 and 
 this mean, and again between 10 and the laft mean, and fo on; and 
 taking the arithmetical means between theircorrefponding logarithms. 
 He then lays down various relations between numbers und their loga- 
 aithms; fuch as, that the produéts and quotients of numbers anfwer 
 to the fums and differences of their logarithms, and that the powers 
 .and roots of numbers anfwer to the products and quotients of the 
 logarithms by the index of the power or root, &c3 as alfo that, of 
 any two numbers whofe logarithms are given, if each number be raifed 
 .to the power denoted by the logarithm of the other, the two refults 
 
 fi 2% will 
 
LOGARITHMS. 49 
 
 will be equal, He then delivers another method of making the loga- 
 rithms to a few of the prime integer numbers, which is well adapted 
 for conftructing the common table of logarithms. ‘This method ea- 
 fily follows from what has been faid above; and it depends on this 
 property, that the logarithm of any number in this fcale, is 1 lefs than 
 the number of places or figures contained in that power of the given 
 number whofe exponent is 10000000000, or the logarithm of 10, at 
 leaft as to integer numbers, for:they really differ by a fraction, as is 
 fhown by Mr. Briggs in his illu{trations of thefe properties, printed at 
 the end of this appendixto the conftruction of logarithms. I fhall here 
 fet down one more of thefe relations, as the manner in which it is ex- 
 prefled is exactly fimilar to that of fluxions and fluents, and it is this: 
 Of any two numbers, as the greater is to the lefs, fo is the velocity 
 of the increment or decrement of the logarithms at the lefs, to the ve- 
 locity of the increment or decrement of the logarithms at the greater: 
 that is, in our modern notation, as _X:Y:: § to x, where x and # are 
 the fluxions of the logarithms of X and Y. 
 
 Kepler’s Conftruction of Logarithms. 
 
 The logarithms of Briggs and Kepler were both printed the fame 
 year, 16243 but as the latter are of the fame kind as Napier’s, we 
 fhall here give this author’s conftru€tion of them, before proceeding 
 to that of Briggs’s. 
 
 We have already (pa. 31 & /eg.) defcribed the nature and form of 
 Kepler’s logarithms, fhowing that theyare of the fame kind as Napier’s, 
 but only a little varied in the form of the table. It may alfo be added, 
 that, in general, the ideas which thefe two mafters had on this fubje&, 
 were of the fame nature: only it was more fully and methodically laid 
 down by Kepler, who expanded, and delivered in a regular fcience, 
 the hints that were given by the illuftrious inventor. The foundation 
 and nature of their methods of conftruction are alfo the fame, but 
 . only a little varied in their modes of applyingthem. Kepler here, firft 
 of any, treats of logarithms in the true and genuine way of the mea- 
 {ures of ratios, or proportions*, as he calls them, and that in a very 
 full and fcientific manner: and this method of his was afterwards fol- 
 lowed and abridged by Mercator, Halley, Cotes, and others, as we 
 fhall fee in the proper places. Kepler firft ereéts a regular and purely 
 mathematical fy{tem of proportions, and the meafures of proportions, 
 treated at confiderable length in a number of propofitions, which are 
 fully and chaftely demonftrated by genuine mathematical reafoning, 
 and illuftrated by examples in numbers. This part contains and de- 
 
 * Kepler almoft always ufes the term proportion inftead of ratio, which I alfo 
 fhall do in my account of his work, as well as conform in expreffions and nota: 
 tions to his other peculiarities. It may alfo be here remarked, that I obfervethe 
 fame practice in defcribing the works of other authors, the better to convey the 
 idea of their feveral methods and ftyle. And this may ferve to account for fome 
 feeming inequalities in the language of this hiftory. 
 
 H monttrates 
 
56 CONSTRUCTION OF 
 
 monftrates both the nature and the principles of the ftructure of lo- 
 garithms. And in the fecond part he applies thofe principles in the 
 actual conftruction of his table, which contains only 1000 numbers, and 
 their logarithms, in the form as we before defcribed : and in this part 
 he indicates the various contrivances made ufe of in deducing the lo- 
 garithms of proportions one from another, after a few of the leading 
 ones had been firft formed, by the general and more remote princi- 
 ples. Heufes the name /ogarithms, -given them by the inventor, 
 being the moft proper, as exprefling the very nature and effence of 
 thofe artificial numbers, and containing as it were a definition in the 
 very name of them; but without taking any notice of the inventor, 
 or of the origin of thofe ufeful numbers. eo 
 As this tract is very curious and important in itfelf, and is befides 
 very rare and little known, inftead of a particular defcription only, | 
 I fhall here give a brief tranflation of both the parts, omitting only 
 the demonftrations of the propofitions, and fome rather long illuftra- 
 tions of them. 
 The book is dedicated to Philip, landgrave of Heffe, but is without 
 either preface or introduction, and commences immediately with the 
 fubjeét of the firft part, which is entitled Zhe Demonftration of the 
 Structure of Logarithms; and the contents of it are as follow : ? 
 Poftulate 1. ‘That all proportions equal among themfelves, by what- 
 ever variety of couplets of terms they may be denoted, are meafured 
 _or expreffed by the fame quantity. ; 
 Axiom |. Iithere be any number of quantities of the fame kind, 
 the proportion of the extremes is under{tood to.be compofed of all the 
 proportions of every adjacent couplet of terms, from the firft to the laft. 
 1 Propsition. 'The mean proportional between two terms, divides 
 the proportion of thofe terms into two equal proportions. 
 Axiom 2. Of any number of quantities regularly increafing, the 
 means divide the proportion of the extremes into one proportion 
 more than the number of the means. 
 Poftulate 2. That the proportion between any two terms is divi- 
 fible into any number of parts, until thofe parts become lefs than any 
 propofed quantity. Gaye 
 An example of this fection is then inferted in a fmall table, in dividing the pro- 
 portion which is between 10 and 7 into 1073741824 equal parts, by as many 
 jean proportionals wanting one, namely, by taking the mean proportional be- 
 tween 10 and 7, then the mean between 10 and this mean, and the mean between 
 10 and the laft, and fo on for 30 means, or 30 extractions of the {quare root, the 
 laft or 30th of which roots is 99999999966782056900; and the 30th power of 2, 
 “which is 10737141824, fhows into how many parts the proportion between 10 and 
 4, or between 1000&c, and 700&c, is divided by 1073741824 means, each of 
 which parts is equal to the proportion between 1000&c, and the 30th mean 
 999 &c, that is, the proportion between 1000&c, and 999&c, is the 1073741824th 
 part of the proportion between 10 and 7. Then by aflumingthe fmall differ- 
 ence 00000000033217943100, forthe meafure of the very fmall element of the 
 proportion of 10 to 7, or for the meafure of the proportion of 1000&c, to 999&c, 
 or for the logarithm of this laft term, and multiplying it by 1073741824, the 
 number of parts, the product gives 35667.49481.37222,14400, for the logarithm 
 of the lefs term 7 or 700&c. 
 
 Poftulate 3. That the extremely fmall quantity or element of a pro- 
 pertion 
 
LOGARITHMS. — 51 
 
 portion may be meafured or denoted by any quantity whatever; as, 
 for inftance, by the difference of the terms of that element. 
 
 2 Propofition. Of three continued proportionals, the difference of 
 the two firft. has to the difference of the two latter, the fame propor- 
 tion which the firft term has to the 2d, or the 2d to the 3d. } 
 
 3. Prop. Of any continued proportionals, the greateft terms have 
 the greateft difference, and the leaft terms the leait. 
 
 4 Prop. In any continued proportionals, if the difference of the 
 greateft terms be made the meafure of the proportion between them, 
 the difference of any other couplet will be lefs than the true meafure 
 of their proportion. 
 
 5 Prop. In continued proportionals, if the difference of the greateft 
 terms be made the meafure of their proportion, then the meafure of 
 the proportion of the greateft to any other term will be greater than 
 their difference. , 
 
 6 Prop. In continued proportionals, if the difference of the greateft 
 term and any one of the lefs, taken not immediately next to it, be 
 made the meafure of their proportion, then the proportion which is 
 between the greateft and any other term greater than the one before 
 taken, will be lefs than-the difference of thofe terms; but the pro- 
 portion which is between the greateft term, and any one leis than that 
 firft taken, will be greater than their difference. 
 
 7 Prop. Of any quantities placed according to the order of their 
 magnitudes, if any two fucceflive proportions be equal, the three 
 fucceflive terms which conftitute them will be continued propor- 
 fionals. 
 
 8. Prop. Of any quantities placed in the order of their magnitudes, 
 if the intermediates lying between any two terms be not among the 
 mean proportionals which can be interpofed between the faid two 
 terms, then fuch intermediates do not divide the proportion of thofe 
 two terms into commentfurable proportions. , 
 
 Befides the demonftrations, as ufual, feveral definitions. are here given; as 
 of commenfurable proportions, &c. 
 
 9 Prop. When two exprefible lengths are not to one another as 
 two figurate numbers of the fame fpecies, fuch as two {quares, or 
 two cubes, there cannot fall between them other expreflible lengths, 
 avhich fhall be mean proportionals,-and as: many in number as that 
 fpecies requires, namely, one in the {quares, two in the cubes, three 
 in the biquadrats, &c. 
 
 10 Prop. OF any expreffible quantities, following in the order of 
 their magnitudes, if the two extremes be not in the proportion of two 
 {quare numbers, or two cubes, or two other powers of the fame kind, 
 none of the intermediates divide the proportion into commenturables. 
 
 11 Prop. All the proportions, taken in order, which are between 
 expreflible terms that are in arithmetical proportion, are incommen- 
 furable to one another. As between ®, 13, 18. . ie 
 
 12 Prop. Of any quantities placed in the order of their magnitude, ie 
 
 the. 
 
 LIBRARY 
 LINIVERSITY OF TLLINOIS 
 
52 CONSTRUCTION OF 
 
 the difference of the greateft terms be made the meafure of their pro- 
 portion, then the difference between any two others will be lefs than 
 the meafure of ¢heir proportion ; and if the difference of the two leaft 
 terms be made the meafure of their proportion, then the differences of 
 the reft will be greater than the meafure of the proportion betwee 
 their terms. . 
 
 Corel. If the meafure of the proportion between the greateft ex- 
 ceed their difference, then the proportion of this meafure to the faid 
 difference, will be lefs than that of a following meafure to the difer- 
 ence of its terms. Becaufe proportionals have the fame ratio. 
 
 1% Prop. If three quantities follow one another in the order of mag- 
 nitude, the proportion of the two laft will be contained in the pro- 
 portion of the extremes, alefs number.of times than the difference 
 of the two leaft is contained in the difference of the extremes : And, 
 on the contrary, the proportion of the two greateft will be contained 
 in the proportion of the extremes, oftener than the difference of the 
 former is contained in that of the latter. } 
 
 Corol. Hence, if the difference of the two greater be equal to the 
 difference of the two lefs terms, the proportion between the two 
 greater will be lefs than the proportion between the two leis. 
 
 14 Prop. Of three equidifferent quantities, taken in order, the pro~ 
 portion between the extremes is more than double the proportion be- 
 tween the two greater terms. 
 
 Corol. Hence it follows, that half the proportion of the extremes 
 is greater than the proportion of the two greateft terms, but lefs than 
 the proportion of the two leatt. 
 
 15 Prop. If two quantities conftitute a proportion, and each quan- 
 tity be leflened by half the greater, the remainders will conftitute a 
 proportion greater than double the former. 
 
 16 Prop. The aliquot parts of incommenfurable proportions are 
 incommenturable to each other. 
 
 17 Prop. If one thoufand numbers follow one another in the na- 
 tural order, beginning at 1000, and differing all by unity, viz. 1000, 
 999,998,997, &c3 and the proportion between the two greateft 1000, 
 999, by continual bifecétion, be cut into parts that are fmaller than 
 the excefs of the proportion between the next two 999, 998, over the 
 faid proportion between the two greateft 1000, 9993; and then for 
 the meafure of that {mall element of the proportion between 1000 and 
 299, there be taken the difference of 1000 and that mean proportional 
 which is the other term of the element. Again, if the proportion be-~ 
 tweetr 1000 and 998 be likewife cut into double the number of parts 
 which the former proportion, between 1000 and 999, was cut into: 
 and then for the meafure of the {mall element in this divifion, be taken 
 the difference of its terms, of which the greater is 1000. And inthe 
 fame manner, if.the proportion of 1000 to the following numbers, * 
 as 99, &c, by continual bifection, be cut into particles of fuch 
 magnitude, as may be between 3 and } of the element arifing from 
 
 the fection of the firft proportion between 1000 and 999, the meafure 
 | of 
 
LOGARITHMS. 58 
 
 of each element will be given from the difference of its terms. “Then, 
 
 this being done, the meafure of any one of the 1000 proportions will’ 
 be compofed of as many meafures of its element as there are of thofe 
 
 elements in the faid divided proportion. And all thefe meafures, 
 
 for all the proportions, will be fufficiently exaét for the niceft calcu» 
 
 lations. 
 
 All thefe fe@tions and meafures of proportions are performed in the manner of 
 that defcribed at poftulate 2, and the operation is abundantly explained by 
 numerical calculations. | 
 
 18. Prop. ‘The proportion of any number, to the firft term 1009, 
 being known ; there will alfo be known the proportion of the reft of 
 the numbers in the fame continued proportion, to the faid farft term. 
 
 So from the known proportion between 1000 and 900, 
 there is alfo known the proportion of 1000 to 810, and to 7295 
 
 | And from 1000 to 800, alfo 1000 to 640, and to 5123 
 
 And from 1000 to 700, alfo 1000 to 490, and to $433 
 And from 1000 to 600, alfo 1000 to 360, and to 2163 
 And from 1000 to 500, alfo 1000 to 250, and to 125. 
 
 Corol. Hence arifes the precept for {quaring, cubing, &c3 as alfo 
 for extracting the {quare root, cube root, &c, out of the firft figures 
 of numbers. For it willbe, as the greateft number of the chilhad, 
 as a denominator, is to the number propofed as a numerator, fo is 
 this to the fquare of the fraction, and {fo is this to the cube. 
 
 19 Prop. The proportion of a number to the firlt, or 1000, being 
 known; if there be two other numbers in the fame proportion to each 
 other, then the proportion of one of thefe to 1000 being known, there 
 ‘will alfo be known the proportion of the other to the fame 1000. 
 
 Corol. 1. Hence from the 15 proportions mentioned in prop. 18, 
 will be known 120 others below 1000, to the fame 1000. 
 
 For fo many are the proportions, equal to fome one or other of the faid 153, 
 that are among the other integer numbers which are lefs than 1000. 
 
 Corol. 2. Hence arifes the method of treating the Rule-of-Three, 
 when 1000 is one of the given terms. 
 
 / 
 
 _ For this is effected by adding to, orfubtracting from, each other, the meafures 
 of the two proportions of 1000 to‘each of the other two given numbers, at- 
 cording as 1000 Is, or isnot, the firft term in the Rule-of-Three. 
 
 20 Prop. When four numbers are proportional, the firft to the fe- 
 eond as the third to the fourth, and the proportions of 1000 to each 
 of the three former are known, there will alio be known the propor- 
 tion of 1000 tothe fourth number. 
 
 Corel. 1. By this means other chiliads are added to the former. 
 
 Coro/, 2. Hence arifes the method of performing the Rule-of-Three, 
 when 1000 is not one of the terms. Namely aa the fum of the 
 meafures of the proportions of 1000 to the fecond and third, take that 
 of 1000 to the firft, and the remainder is the meafure of the propor- 
 tion of 1000 to the fourth term, 
 
 Definition. 
 
54 CONSTRUCTION OF 
 
 Definition. . The meafure of the proportion between 1000 and any 
 lefs number as before defcribed, and expreffed by a number, is fet 
 oppofite to that lefs number in the chiliad, and is called its LoGa- 
 RITHM, that 1s, the number (api§uos) indicating the proportion 
 (Avvo) which 1000 bears to that number, to which the logarithm is 
 annexed. — 
 
 21 Prop. Tf the firft or greateft number be made the radius of a 
 circle, or finus totus; every lefs number, confidered as the cofine of 
 fome arc, has a logarithm greater than the verfed fine of that arc, but 
 Jefs than the difference between the radius and fecant of the arc; ex- 
 cept only in the term next after the radius, or greateft term, the loga- 
 rithm of which, by the hypothefis, is made equal to the verfed fine. 
 
 That is, if CD be made the logarithm of AC, or the mea- EE 
 fure of the proportion of AC to AD; then the meafure of 
 _the proportion of ABto AD, that is the logarithm of AB, 
 will be greater than BD, but lefs than EF. And this is the 
 fame as Napier’s firft rule in page 45. . 
 
 A BCD 
 
 22. Prop. The fame things being fuppofed; the fum of the verfed 
 fine and excefs of the fecant over the radius, is greater than double 
 the logarithm of the cofine of an arc. 
 
 Corel. The log. cofine is lefs than the arithmetical mean between 
 the verfed fine and the excefs of the fecant. 
 
 Precept 1. Any fine being found in the canon of fines, and its de- 
 fe& below radius to the excefs of the fecant above radius, then fhall 
 the logarithm of the fine be lefs than half that fum, but greater than 
 the faid defect or coverfed fine. 
 
 Let there be the fine 90970.1490 of an arc: bpd is 
 Its defect below radius is 29.8510 the coverf, and lefs than the log. fine; 
 Add the excefs of the fecant 29.8599 
 
 Sum 59.7109 
 its half or 29.8555 greater than the logarithm. 
 ‘ "Therefore the log. is between 29.8510 . 
 and 29.8555 
 
 Precept 2. The logarithm of the fine being found, you’ will alfo 
 find nearly the logarithm of the round or integer number, which is 
 next lefs than the fine with a fraGtion, by adding that fractional ex- 
 efs to the logarithm of the faid fine. 
 
 Thus, the logarithm of the fine 99970.149 is found to be about 29.854 ; ifnow 
 the logarithm of the round number 99970.000 be required, add 149, the frac- 
 tional part of the fine, to its logarithm, obferving the point, thus, 
 
 29.854 
 149 
 
 the fum 30.008 is the log. of the round number 99970.000 nearly. 
 
 23 Prop. Of three equidifferent quantities, the meafure of the 
 proportion between the two greater terms, with the meafure of the 
 proportion 
 
LOGARITHMS. 5s 
 
 proportion between the two lefs terms, . will conftitute a proportion, 
 which will be greater than the proportion of the two greater terms, 
 but lefs than the proportion of the two leaft. 
 
 1 Vina 
 Thus if AB, AC, AD "be three quantities, having AB CG D 
 the equal differences BC,CD; and ifthe meafure ofthe 
 proportion of AD, AC be cd, and that of AC, AB be 1 1‘ 
 be ; then the proportion of cd to cb will be greater than 
 the proportion of AC to AD, but lefs than the propor- 
 tion of AB to AC. 
 
 24.’ Prop. The faid proportion between the two meafures is lefs 
 than half the proportion between the extreme terms. ‘That is, the 
 proportion between bc, cd, is lefs than half the proportion between 
 AB, AD. 
 
 Corel. Since therefore the arithmetical mean divides the propor- 
 tion into unequal parts, of which the one is greater, and the other lefs, 
 than half the whole; if itbe inquired what proportion is between 
 rate proportions, the anfwer is, that it is a little lefs than the faid 
 
 alf. | | 
 
 2 ® 
 
 + 
 
 4n Example of finding nearly the limits, greater and lefi, to the meafuve 
 of any propofed proportion. 
 
 Tt being known that the meafure of the proportion between 1000 and 900 is 
 10536.05, required the meafure of the proportion 900 to 800, where the terms 
 1000, 900, 800, have equal differences. Therefore as 9 to 10, fo 10536.05 to 
 11706.72, which is lefs than 11778.30 the meafure of the proportion 9 to 8. 
 Again, as the mean proportional between 8 and 10 (which is 8.9442719) is to 
 10, fo 10536.05 to 11779.66, which is greater than the meafure of the propor- 
 tion between 9 and 8. - 
 
 Axiom. Every number denotes an expreflible quantity. 
 
 25 Prop. If the 1000 numbers differing by 1, follow one another 
 in the natural order; and there be taken any two adjacent numbers, 
 as the terms of fome proportion; the meafure of this proportion will 
 be to the meafure of the proportion between the two greateft terms 
 of the chiliad, ina proportion greater than that which the greateft 
 term 1000 bears to the greater of the two terms firft taken, but lefs 
 than the proportion of 1000 to the lefs of the faid two feleCted terms. 
 
 So, of the 1000 numbers, taking any two fucceffive terms, as 501 and 500, 
 _the logarithm of the former being 69114.92, and of the latter 69314.72, the 
 difference of which is 199.80. ‘Therefore, by the definition, the meafure of the 
 proportion between 501 and 500 is 199.80. In like manner, becaufe the logarithm 
 of the greateft term 1000 is 0, and of the next 999 is 100.05, the difference of 
 thefe logarithms, and the meafure of the proportion between 1000 and 999, is 
 100.05. Couple now the greateft term 1000 with each of the felected terms 501 
 and 5003; couple alfo the meafure 199.80 with the meafure 100,05 ; fo fhall the 
 proportion between 199.80 and 100.05, be greater than the proportion between 
 1000 and 501, but lefs than the proportion between 1000 and 500. 
 
 Corol. 1. Any number below the firft 1000 being propofed, as alfo 
 its logarithm, the differences of any logarithms antecedent to that 
 propofed, 
 
56 CONSTRUCTION OF 
 
 propofed, towards the beginning of the chiliad, are to the firft loga- 
 rithm (viz. that which is afligned to 999) in a greater proportion than 
 1000: to the number propofed; but of thofe which follow towards the 
 laft logarithm, they are to the fame in a lefs proportion. 
 
 Corol. 2. By this means, the places of the chiliad may eafily be fill- 
 ed up, which have not yet had logarithms adapted to them by the 
 former propofitions. 
 
 26. Prap. The difference of two logarithms, adapted to two adjacent 
 numbers, is to the difference of thefe numbers, in a proportion greater 
 than 1000 bears to the greater of thofe numbers, but lefs than that 
 af 1000 to the lefs of the two numbers. 
 
 This 26th prop. is the fame as Napier’s fecond rule, at page 45. 
 
 271 Prop. Having giventwo adjacent numbers, of the 1000 natural 
 numbers, with their logarithmic indices, or the meafures of the pro-— 
 portions whichthofe abfolute orround numbers conftitute with 1000, 
 the greateft ;, the increments, or differences, of thefe logarithms, will 
 be tothe logarithm of the {mall element of the proportions, as the fe- 
 cants of the arcs whofe cofines are the two abfolute numbers, is to the 
 greate{t number, or the radius of the circle; fo that, however, of 
 the fatd two fecants, the lefs will have to the radius a lefs proportion 
 than the propofed difference has to the firft of all, but the greater 
 will have a greater proportion, and foalfo will the mean proportional 
 between the faid fecants have a greater proportion. 
 
 Thus if BC, CD be equal, alfo bd the logarithm of A B, 
 and cd the logarithm of AC; then the proportion of bc 
 to. c d will be greater than the proportion of AG to AD, 
 but lefs than that of A F to AD, and alfo lefs than that of 
 the mean proportional between AF and AG to AD. ° 
 
 Corol. 1. The fame obtains alfo when the two terms differ, not 
 only by the unit of the fmall element, but by another unit, which 
 may be ten fold, a hundred fold, or a thoufand fold of that. 3 
 
 Coral. z. Hence the differences will be obtained. fufliciently exact, - 
 efpecially when the abfolute numbers are pretty large, by taking the 
 arithmetical mean between two fmall fecants, or (if you will be at 
 the labour) by taking the geometrical mean between two larger fe- 
 cants, and then by continually adding the differences, the logarithms 
 will be produced. . 
 
 Coral. 3. Precept. Divide the radius by each term of the afligned 
 proportion, and the arithmetical mean (or ftill nearer the geometrical 
 mean) between the quotients, will be the required increment ; which 
 being added to the logarithm of the greater term, will give the lo- 
 garithm of the lefs term. | 
 
 : Example 
 
LOGARITHMS. ~ 57 
 
 Example. 
 
 Let there be given the logarithm of. 700, viz. $5667.4948, to find the log ta 699. 
 Here radius divided by, 700 gives 1428371 &c. 
 and divided by 699 gives 1430672 &c. 
 the arithmetic mean is 142.962 
 which added to 35667.4948 
 
 - gives the logarithm to 699 $5810.4568 
 
 Corel. 4. Precept for the logarithms of fines... 
 
 The increment between the logarithms of two fines, is thus bad: 
 find the geometrical mean between the cofecants, and divide. it by 
 the difference of the fines, the quotient will be the difference of the 
 logarithms. 
 
 Example. 
 
 0° 1’ fine 2909 cofec. 343774682 The quotient 80000 exceeds the re- 
 ‘0 2 fine 5818 cofec. 171887319 | quired increment of the logarithms, 
 becaufe the fecants are here fo largev 
 
 dif. 2909 geom.mean 2428 nearly. 
 
 Appendix. Nearly in the fame manner it may be fhown, that the 
 fecond differences are in the duplicate proportion of the firft, and the 
 third in the duplicate of the fecond. Thus, for inftance, in the be- 
 ginning of the logarithms, the firft difference is 100.0000, viz. equal 
 to the difference of the numbers 100000.C0000 and’ 99900.000U0.; 
 the fecond or difference of the differences, 10000; the third 20. 
 Again, after arriving at the number 50000.00000, the logarithms 
 have for a difference 200.00000, which is to the frit difference, as 
 the number 160000.00000 to 50000.00000; but the fecond differ- 
 ence 1s 40000, in which 10000 is Ponianed four times ; and the 
 third 328, in which 20 is contained 16 times. But fince in treating 
 of new matters we labour under the want of proper words, where- 
 fore left we fhould become too obfcure, the demonftration is omitted 
 untried. 
 
 28. Prop. No number exprefles exa@tly the meafure of the propor- 
 tion, between two of the 1000 numbers, conftituted by the foregoing 
 method. 
 
 29. Prop. Ifthe meafures of all proportions beexprefied by numbers 
 or logarithms ; ; all proportions. will not have afligned to them their 
 due portion of meafure, to the utmoft accuracy. 
 
 40 Prop. If to the number 1000, the greateft of the chiliad, be 
 ied others that are greater ee it, and the logarithm of 1000 be 
 made 0, the logarithms belonging to thofe greater. numbers will be 
 negative. 
 
 This concludes the firft or fcientific part of the work, the principles 
 of which Kepler applies, in the fecond part, to the meat conftruction 
 of the firft 1000 logarithms, which conftruétion is pretty minutely 
 defcribed. This part is intituled 4 very compendious Method of ¢con- 
 Jtructing the Chiliad of Logariims ; and it is not improperly fo called, 
 the method being very conc'‘e and eafy. The fundamental principles 
 
 are briefly thefe: That at the beginning of the logarithms, their m- 
 I  crements 
 
58 CONSTRUCTION OF 
 
 crements or differences are equal to thofe of the natural numbers : that 
 the natural numbers may be confidered as the decreafing cofines of in- 
 creafing arcs; and that the fecants of thofe arcs at the beginning have 
 the fame differences as the cofines, and therefore the fame differences 
 as the logarithms. ‘Then, fince the fecants are the reciprocals of the 
 cofines, by thefe principles and the third corollary to the 27th propo- 
 fition, he eftablifhes the following method of conftituting the 100 firft 
 or {malleft logarithms to the 100 largeft numbers, 1000, 999, 998, 
 997, &c, to 900. viz. Divide the radius 1000, increafed with feven 
 ciphers, by each of thefe numbers feparately, difpofing the quotients 
 in a table, and they will be the fecants of thofe arcs which have the 
 divifors for their cofines; continuing the divifion to the 8th figure, as 
 }t is in that place only that the arithmetical and geometrical means 
 differ. Then by adding fucceffively the arithmetical means between 
 every two fucceffive fecants, the fums will be the feries of logarithms. 
 Or 4 adding continually every two fecants, the fucceflive fums will 
 be the feries of the double logarithms. 
 
 Befides thefe 100 logarithms, thus conftructed, he conftitutes two 
 others by continual bifeCtion, or extractions of the {quare root, after 
 the manner defcribed in the fecond poftulate. And firft he finds the 
 logarithm which meafures the proportion between 100000.00 and 
 97656-25, which latter term is the third proportional to 1024 and 
 1000, each with two ciphers ; and this is effected by means of twenty- 
 four continual extractions of the fquare root, determining the greatest 
 term of each of twenty-four claffes of mean proportionals; then the 
 difference between the greateft of thefe means and the firft or whole 
 number 1000, with ciphers, being as often doubled, there arifes 
 2371.6526 for the logarithm fought, which made negative is the loga- 
 rithm of 1024. Secondly, the like procefs is repeated for the propor- 
 tion between thenumbers ! 000 and 500, from which arifes69314.7193 
 for the logarithm of 500; which he alfo ¢alls the logarithm of du- 
 plication, being the meafure of the proportion of 2 to 1. | 
 Then from the foregoing he derives all the other logarithms in the 
 chiliad, beginning with thofe of the prime numbers 1, 2, 3, 5; 7, &c, 
 in the firft 100. And firft, fince 1024, 512, 256, 128, 64, 32, 16, 8, 
 4, 2, 1, are all in the continued proportion of 1000 to 500, therefore 
 the proportion of 1024 to 1 is decuple of the proportion of 1000 to 
 500, and confequently the logarithm of 1 would be decuple of the 
 
 Jogarithm, of 500, if O were taken as the logarithm of 1024 but fince _ 
 
 the logarithm of 1024 is applied negatively, the logarithm of 1 muft 
 be diminifhed by as much: diminifhing therefore 10 times the loga- 
 rithm of 500, which is 693147.1923, by 2371.6526, the remainder 
 
 690775.5422 is the logarithm of 1, or of 100.00, what is fet down in 
 
 the table. 
 And becaufe 1, 10, 100, 1000, arecontinued pro- | Nos. |Logarithms. 
 
 100| 230258.5141 _ 
 
 portionals, therefore the proportion of 1000 to | is 101 460517.0282 
 
 triple of the proportion of 1000 to 100, and confe- 1} 690775.5422 
 ently Mos the lovar, : .1] 921034.0563 
 quently = of the logarithm of I is to be put for the .01]1151292.5703 
 
 logarithm of 100, viz. 230258.5141, and this is | goiliagis51.0944 
 
 alfo the logarithm of decuplication, or of the pro- !.0001[1611809.5935 
 portion.of 10 tol, And hence, multiplying this 
 Wise logarithm 
 
 7 
 
LOGARITHMS. 5 
 
 S 
 
 logarithm of 100 fucceffively by 2, 3, 4, 5, 6, and 7, there arife the 
 logarithms to the numbers in the decuple proportion, as in the 
 margin. 
 
 Alfo if the logarithm of duplication, or of the (Log. of 1690775.5422 
 proportion of 2 to 1, be taken from the logarithm | of 2 to 1) 69314.7195 
 of 1, there will remain the logarithm of %¥; and | log. of 2}621460.S229 
 
 5 y 3 log. of 10 460517.0281 
 from the logarithm of 2 taking the logarithm of pegOe yaa cacti 
 10, there remains the logarithm of the proportion ; iio SpORSTTTATE 
 of 5 to 13 which taken from the logarithm of 1, ' “*' 
 there remains the logarithm of 5. See the margin. 
 
 __ For the logarithms of other prime numbers, he has recourfe to 
 
 thofe of fome of the firft or greateft century of numbers, before found, 
 viz. of 999, 998, 997, &c. And firft, taking 960, whofe logarithm is 
 4082.2001; then by adding to this logarithm the logarithm of dupli- 
 cation, there will arife the feveral logarithms of all thefe numbers, 
 which are in duplicate proportion continued from 960, namely 430, 
 240, 120, 60, 30, 15. Hence the logarithm of 30, taken from the 
 logarithm of 10, leaves the logarithm of the proportion of 3 to 1; 
 which taken from the logarithm of 1, leaves the logarithm of 3, viz. 
 580914.3106. And the double of this diminifhed by the logarithm 
 of 1,-gives 471053.0790 for the logarithm of 9. 
 
 Next, from the logarithm of 990, or 9 X 10 X 11, which is 
 ¥005.0331, he finds the logarithm of 11, namely, fubtra&t the fum 
 of the logarithms of 9 and 10 from the fum of the logarithm of 990 
 and double the logarithm of 1, there remains 450986.0106 the loga- 
 rithm of 11. | 
 
 Again, from the logarithm of 980, or 2 X 10 X 7 X 1, which is 
 2020.2711, he finds 496184.5228 for the logarithm of 7. 
 
 And from.5129.3303 the logarithm of 950, or 5 x 10 x 19, he 
 finds 396331.6392 for the logarithm of 19. 
 
 In like manner the logarithm 
 
 to 998 or 4 X 13 X 19, gives the logarithm of 13; 
 to 969 or 3 X 17 X 19, gives the logarithm of 17; 
 
 _ to 986 or2 X 17 X 29, gives the logarithm of 29; 
 to 966 or6 X 17 x 23, gives the logarithm of 23; 
 
 _ to 930 or 3 X 10 x 31, gives the logarithm of 31. 
 
 And fo on for all the primes below 100, and for many of the 
 primes in the other centuries up to 900. ° After which, he directs 
 to find the logarithms of all numbers compofed of thefe, by the pro- 
 per addition and fubtraCtion of their logarithms, namely, in finding 
 the logarithm of the product of two numbers, from the fum of the 
 ee of the two factors take the logarithm of 1, the remainder 
 is the logarithm of the produ€t. In this way he fhows that the loga- 
 rithms of all numbers under 500-may be derived, except thofe of the 
 following 36 numbers, namely, 127, 149, 167, 173, 179, 211, 223, 
 251, 257, 263, 269, 271, 277, 281, 283, 293, 337, 347, 349, 355, 
 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 
 439, 443, 449. Alfo, befides the compofite numbers between 500 
 and 900, made up of the produdts of fome numbers whofe logarithms 
 have been before determined, there will be 59 primes not compofed. 
 
 of 
 
sy 
 
 60 CONSTRUCTION OF 
 
 of them ; which, with the 36 above mentioned, make 95 numbers in. 
 -all not compofed of the produts of any before them, and the loga- 
 rithms of which he directs to be derived inthis manner; namely, by 
 confidering the differences of the logarithms of the numbers inter- 
 fperfed among them: then by that method by which wereconftituted 
 the differences of the logarithms of the fmalleft 100 numbers in’a con- 
 tinued feries, we are to proceed here in the difcontinued feries, that 
 is, by prop. 27, corol. 3, and efpecially by the appendix to it, if it 
 be rightly ufed, from whence thofe differences will be very eafily 
 fupplied. | | 
 Vhis clofes the fecond part, or the aétual conftruction of the loga- 
 rithms; after which follows the table itfelf, which has been before 
 _ defcribed, pa. 32. Before difmifling Kepler’s work, however, it may 
 not be improper in this place to take notice of an erroneous property 
 laid down by him in the appendix to the 27th prop. juft now referred 
 to; both becaufe it is an error in principle, tending to vitiate the 
 ‘practice, and becaufe it ferves to fhow that Kepler was unacquainted 
 with the true nature of the orders of differences of the logarithms, 
 notwithftanding what he fays above with refpect to the conftruction 
 of them by means of their feveral orders of differences, and that con- 
 _ fequently he lias no legal claim to any fhare in the difcovery of the dif- 
 ferential method, known at that time to Briggs, and it would feem to 
 him alone, it being publifhed in his logarithms ‘in the fame year, 1624, 
 as Kepler’s book, together with the true nature of the logarithmic 
 orders of differences, as we fhall prefently fee in the following ac- 
 count of his works. Now this error of Kepler’s here alluded to, is 
 .in that expreflion where he fays the third differences are inthe duplicate 
 ratio of the fecond differences, like as the fecond differences are in the 
 -duplicate ratio of the firft; or, in other words, that the third differences 
 are as the /guares of the {econd differences, as well as the fecond dif- 
 ferences as the fquares of the firft; or that the third differences 
 are as the fourth powers of the firft differences. Whereas in truth 
 the third differences are only as the cubes of the firft differences. 
 Kepler feems to have been led into this error by a miftake in his num- 
 bers, viz. when he fays in that appendix, that the ¢hird difference is 
 328, in which 20 is contained 16 times ; for when the numbers are ac- 
 curately computed, the third difference comes out only 161, in which 
 therefore 20 is contained only 8 times, which is the cube of 2, the 
 number of times the one firft difference contains the other. It would 
 -hence feem that Kepler had haftily drawn the above erroneous prin- 
 ciple from this one numerical example, or little more, falfe as it is : 
 for had he made the trial in many inftances, although erroncoufly 
 -computed, they could not eafily have been fo uniformly fo, as to af- 
 ford the fame falfe conclufion. And therefore from hence, and what 
 he fays at the conclufion of that appendix, it may be inferred, that he 
 either never attempted the demonttration of the property in queftion, 
 or elfe that he found himfelf embarraffed with it, and unable to ac- 
 complifh it, and therefore difpatched it in the ambiguous manner in 
 which it appears. 
 But it may eafily be fhown, not only that the third differences of the 
 logarithms 
 
/ 
 
 LOGARITHMS. 6L 
 
 ‘logarithms at different places, are as the cubes of the firft differences; 
 but, in general, that the numbers in any one and the fame order of 
 differences, at different places, are as that power of the numbers in the 
 firft differences, whofe index is the fame as that of the order: or that 
 the fecond, third, fourth, &c, differences, will be as the fecond, third, 
 fourth, &c, powers of the firft differences. For the feveral orders of 
 ‘differences, when the abfolute numbers differ by indefinitely fmall 
 parts, are as the feveral orders of fluxions of the logarithms ; but if 
 
 j m 4 e 1 . e ‘ 
 x be any number, then — is the fluxion of the logarithm of x, to the 
 
 v 
 modulus m, and the fecond fluxion, or the fluxion of this fluxion, 
 
 z m x? “i d ‘ 
 i3 —- ——., fince x is conftant: and the third, fourth, &c, fluxions, 
 - 
 Q2m2x2>  2.3mat : . 
 Noten ane meat a &c; that is, the firft, fecond, third, fourth, fifth, 
 xv be 
 
 fixth, &c, orders of fluxions, are equal to the modulus m multiplied 
 into each of thefe terms, 
 a Li iy Se TIS hia, va Meh TCO aa 
 
 isa DOCiis 
 
 —— — —— — ——_-——-, ee 
 
 ? > 2 
 
 a Xx 2 ae 3 4 v 5 “es 
 
 where it is evident, that the fluxionof any order is as that power of the 
 firft fluxion, whofe index is the fame as the number of the order. 
 And thefe quantities would actually be the feveralterms of the differ- 
 -ences themfelves, if the differences of the numbers were indefinitely 
 {mall. But they vary the more from them, as the differences of the 
 abfolute numbers differ from x, or asthe faid conftant numerical dif- 
 ference 1 approaches towards the value of « the number itfelf. How- 
 ever, upon the whole, the feveral orders vary proportionably, fo as 
 {till fenfibly to preferve the fame analogy, namely, that two ath dif- 
 ferences are in proportion as the 7th powers of their refpelive firft 
 differences. : 
 
 Of Briggs’s Confiruction of his Logarithms. 
 
 Nearly according to the methods defcribed in page 48, Mr. Briggs 
 con{tructed the logarithms of the prime numbers, as appears from his 
 relation of this bufinefs in the Arithmetica Logarithmica, printed in 
 1624, where he details, in an ample manner, the whole conftruction 
 ‘and ufe of his logarithms. The work is divided into thirty-two chap- 
 ters or fections. In the firft of thefe, logarithms ina general fenfe are 
 defined, and fome properties of them illuftrated. In the fecond chap- 
 ter he remarks, that it is moft convenient to make O the logarithm of 
 13; and on that fuppofition he exemplifies thefe following properties, 
 namely, that the logarithms of all numbers are either the indices of 
 powers, or proportional to them; that the tum of the logarithms of 
 two or more factors, is the logarithm of their product; and that the 
 difference of the logarithms of two numbers, is the logarithm of their 
 quotient. In the third fection he ftates the other aflumption which is 
 neceflary to limit his fyftem of logarithms, namely, making 1 the 
 logarithm of 10, as that which produces the moft convenient form of 
 
 logarithms ; 
 
62 CONSTRUCTION OF 
 
 fogarithms: He hence alfo takes occafion to fhow that the powers of 
 10, namely, 100, 1000, &c, are the only numbers which can have 
 rational logarithms. The fourth fection treats of the characteriftic ; 
 by which name he diftinguifhes the integral, or firft part, of a loga- 
 rithm towards the left hand, which expreffes one lefs than the num- 
 ber of integer places or figures in the number belonging to that loga- 
 rithm, or how far the firft figure of this number is removed from the 
 place of units; namely, that O is the characteriftic of the logarithms 
 of all numbers from 1 to 10; and1 the characteriftic of all thofe from 
 10 to 1003; and 2 that of thofe from 100 to 1000; and fo on. 
 
 He begins the fifth chapter with remarking, that his logarithms 
 may chiefly be conftructed by the two methods which were mentioned 
 by Napier, as above related, and for the fake of which he here pre- 
 mifes feveral /emmata, concerning the powers of numbers and their 
 indices, and how many places of figures are in the products of num- 
 bers, obferving that the product of two numbers will confift of as 
 many figures as there are in both factors, unlefs perhaps the product 
 of the firft figures in each factor be exprefled by one figure only, 
 which often happens, and then commonly there will be one figure in 
 the product lefs than in the two fa€tors ; as alfo that, of any two of 
 the termsin a feries of geometricals, the refults will be equal by 
 raifing each term to the power denoted by the index of the other; 
 or any number raifed to the power denoted by the logarithm of the 
 other, will be equalto this latter number raifed to the power denoted 
 by the logarithm of the former; and confequently if the one number 
 be 10, whofe logarithm is one with any number of ciphers, then any 
 number raifed to the power whofe index is 1000 &c, or the logarithm 
 of 10, will be equal to 10 raifed to the power whofe index is the lo- 
 garithm of that number; that is, the logarithm of any number in this 
 {cale, where 1 is the logarithm of 10, is the index of that power of 
 10 which is equal to the given number. But the index of any in- 
 tegral power of 10, is one lefs than the number of places in that 
 power, confequently the logarithmofany other number, which is no 
 integral power of 10, is not quite one lefs than the number of places 
 in that power of the given number whofe index is 1000 &c, or the 
 logarithm of 10. 
 
 Find therefore the 10th, or 100th, or 1000th, &c, power of any 
 number, as fuppofe 2, with the number of figures in fuch power; 
 then fhall that number of figures always exceed the logarithm of 2, 
 although the excefs will be conftantly lefs than 1. 
 
LOGARITHMS. 63 
 
 No. of places 
 
 or logs. 
 
 se 
 
 An example of this procefs is | 
 here given in the margin; where 
 the 1ft column contains the fe- 
 veral powers of 2, the 2d their 
 correfponding indices, and the 
 3d contains the number of places 
 in the powers in the firft co- 
 lumn ; and of thefe numbers in 
 the third column, fuch as are 
 on the lines of thofe indices that | 12089 _ 
 confift of 1 with ciphers are 
 continual approximations to the 
 logarithm of 2, being always 
 too great by lefsthan 1 in the 
 laft figure, that logarithm being 
 30102999566398 &c. 
 
 And here, fince the exact 
 powers of 2 are not required, 
 but only the number of figures 
 they. confift of, as fhown by 
 the third column, only a few 
 of the firft figures of the powers 
 in the firft column are retained, 
 thofe being fufficient to deter- 
 mine the number of places in 
 them; and the multiplications 
 in raifing thefe powers are per- 
 formed in a contracted way, 
 fo as to have the fifth or laft 
 figure in them true to the near- 
 eft unit. Indeed thefe multi- 
 plications might be performed 
 in the fame manner, retaining 
 only the firft three figures, and 
 thofe to the neareft unit in the 
 third place; which would make | 1 433 
 this a very eafy way indeed of | 33977 
 finding the logarithms of a few 46129 
 prime numbers. 
 
 It may alfo be remarked, that thofe feveral powers, whofe indices 
 are 1 with ciphers, are raifed by thrice fquaring from the former 
 powers, and multiplying the firft by the third of thefe {quares ; mak- 
 ing alfo the correfponding doublings and additions of their indices : 
 thus, the fquare of 2 is 4, and the {quare of 4 is 16, the {quare of 16 is 
 256; and 256 mutiplied by 4 1s 10243 in like manner, the double of 
 1 is 2, the double of 2 is 4, the double of 41s 8, and 8 added to 2 
 makes 10. And the fame for all the following powers and indices. 
 The numbers in the third column, which fhow how many placeg 
 are in the correfponding powers in the firft column, are produced in 
 the yery fame way as thofe in the fecond column, namely, by three 
 
 dupli- 
 
 13 log. of 16 
 25 log. of 256 
 
 100 31 log..of 2 
 200 61 log. of 4 
 400 121 log. 16 
 800 241 log. 256 
 1000 302 log. 2 
 2000 603 log. 4 
 4000 j 1205 log. 16 
 8000 2409 |. 256 
 10000 3011 log. 2 | 
 6021 log. 4 
 40000 12042 log.16 
 25099 | 80000 - 24083 |. 256 
 100000 30103 log. 2 
 200000 60206 log. 4 
 400000 120412 & 
 99204 | so0000 240824 ° 
 1000000 301030 
 2000000 602060 
 4000000 1204120 
 8000000 2408240 
 
 10000000 | 3010300 
 
 20000000 | 602@600 
 40000000 12041200 
 80000000 | 24082400 
 100000000 | 30103000 
 200000000 | 60206000 
 400000000 | 120411999 
 800000000 | 240823997 
 
 1000000000! 301029996 
 
64 CONSTRUCTIGN OF 
 
 duplications and one addition; only obferving to fubtract 1 when the 
 product of the firft figures are exprefled by one figure, or when the 
 firft figures exceed thofe of the number or power next above them. 
 It may farther be obferved, that, like as the firft number in each qua- 
 ternion, or {pace of four lines or numbers, in the third column, ap- 
 proximates to the logarithm of 2, the firft number in the firft qua- 
 ternion of the firft column; fothe fecond, third, and fourth terms of 
 each quaternion in the third column, approximate to the logarithm of 
 4, 16, and 256, the fecond, third, and fourth numbers in the firft 
 quaternion in the firft column. And moreover, by cutting off one, 
 two, three, &c, figures, as the index or integral part, from the faid 
 logarithms of 2, 4, 16, and 256, the firft, fecond, third, and fourth 
 numbers in the firft quaternion of the firft column, the remaining fi- 
 gures will be the decimal part of the logarithms of the correfponding 
 firft, fecond, third, and fourth numbers in the following fecond, third, 
 fourth, &c, quaternions: the reafon of which is, that any number of 
 any quaternion in the firft column, is the tenth power of the corre- 
 {ponding term in the next preceding quaternion. So that the third 
 column contains the logarithms of all the numbers in the firft column: 
 a property which, if Dr. Newton had been aware of, he could not 
 well have committed fuch grofs miftakes as are found ina table of his 
 fimilar to that above given, in which moft of the numbers in the latter 
 quaternions are totally erroneous; and his confufed and imperfect 
 account of this method would induce one to believe that he did not 
 well underftand it. 
 
 In the 6th chapter our illuftrious author begins to treat of the 
 other general method of finding the logarithms of prime numbers, 
 which he thinks is an eafier way than tie former, at leaft when the 
 logarithm is required to a great many places of figures. ‘This method 
 confifts in taking a great number of continued geometrical means 
 between land the given number, whofe logarithm is required; that 
 is, firft extracting the fquare root of the given number, then the 
 root of the firft root, the root of the fecond root, the root of the 
 third root, and fo on till the laft root fhall exceed 1 by a very {mall 
 decimal, greater or lefs according to the intended number of places 
 to be in the logarithm fought: then finding the logarithm of this 
 {mall number, by methods defcribed below, he doubles it as often as 
 he made extractions of the {quare root, or, which is the fame thing, 
 he multiplies it by fuch power of 2 as is denoted by the faid number 
 of extractions, and the refult is the required logarithm of the given 
 number ; as is evident from the nature of logarithms. ‘The rule to 
 know how far to continue this extraction of roots is, that the number 
 of decimal places in the laft root be double the number of true places 
 required to be found in the logarithm, and that the firft half of them 
 be ciphers; the integer being one: the reafon of which is, that then 
 the fignificant figures in the decimal, after the ciphers, are direCtly 
 proportional to thofe in the correfponding logarithms ; fuch figures 
 in the natural number being the half of thofe in the next preceding 
 number, like as the logarithm of the laft number is the half of the 
 preceding logarithm. ‘Therefore any one fuch {mall number, with 
 
 its 
 
LOGARITHMS. es 
 
 its logarithm, being once found by the continual extractions of 
 fquare roots out ofa given number, as 10, and correfponding bifec- 
 tions of its given logarithm 1; the logarithm for any other fuch {mall 
 number, derived by like continual extractions from another given 
 number, whofe logarithm is fought, will be found by one fingle pro- 
 portion: which logarithm is then to be doubled according to the num- 
 ber of extractions, or multiplied at once 
 
 by the like power of 2, for the loga- 10,given no. _|1, itslog.f 
 
 rithm of the number propofed. To find | } 3162277 &e | O'S 
 the firft {mall number and its logarithm, | 3 ra sia nah : 
 our author begins with the number 10 | 4 | 4454781 6°0625 
 and its logarithm 1, and extracts con~ | 5 | 1:074607 003125 
 
 tinually the root of the laft number &e. &c. 
 and bifects its logarithm, as here re- , 
 giftered in the margin, but to far more places of figures, till he 
 
 arrives at the 53d and 54th roots, with their annexed logarithms, 
 as here below : . ; . 
 
 Numbers. Logarithms. 
 
 53 | 1+00000,00000,00: 06,25563,82986,40064,70 | 0:00000,00000,00000, 11 102,23024,59515,45404 
 54 | 1-00000,00000,00000,12781 ,91493,20052,35 | 0-00000,00000,00000,05551,,11512,31257,82702 
 where the decimals in the natural numbers are to each other in the 
 ratio of the logarithms, namely in the ratio of 2 to 1: and therefore 
 any other fuch imall number being found, by continual extration or 
 otherwife, it will then be as 12781 &c,is to 5551 &c, fo is that other 
 {mall decimal, to the correfponding fignificant figures of its loga- 
 rithm. But as every repetition of this proportion requires both a very 
 long multiplication and divifion, he reduces this conftant ratio to 
 another equivalent ratio whofe antecedent is 1, by which all the di- 
 yifions are faved : thus, | 
 
 as 12781 &c 35551 &es+,1000 &c: 43429448 1903251804, 
 
 that is, the logarithm of 1:00000,00000,06000,1 
 
 is 000000,00000,00000,04342,94481,90325, 18045 
 and therefore this laf{t number being multiplied by any fuch fmall de- 
 cimal, found as above by continual extraction, the prody@ will be 
 the correfponding logarithm of fuch laft root, : 
 
 But as the extraction of fo many roots is a very troublefome ope- 
 ration, our author devifes fome ingenious contrivances to abridge that 
 Jabour. And firft, in the 7th chapter, by the following device, to, 
 have fewer and eafier extractions to perform: namely, raifing the 
 powers from any given prime number, whofe logarithm is fought, 
 till a power of it be found fuch that its firft figure on the left hand is 
 1, and the next to it either one or more ciphers; then, having di- 
 vided this power by 1 with as many ciphers as it has figures after the 
 firft, or fuppofing all after the firft to be decimals, the continual roots 
 from this power are extracted till the decimal become fufticiently 
 {mall, as when the firft, fifteen places are ciphers; and then by mul- 
 tiplying the decimal by 43429 &c, he has the logarithm of this laft 
 soot; which logarithm muleelice PE the like power of the number. 2, 
 
 Ss giveg 
 
66 CONSTRUCTION OF 
 
 gives the logarithm of the firft number from which the extrace 
 tion was begun: to this logarithm prefixing al, or 2, or 3, &c, ace 
 cording as this number was found by dividing the power of the given 
 prime number by 10, or 100, or 1000, &c3 and iaftly, dividing the 
 refult by the index of that power, the quotient will be the required 
 logarithm of the given prime number. Thus, to find the 
 
 logarithm of 2; it is firit raifed to the 10th power, as in [217 
 the margin, before the firft figures come to be 103; then, 4 2) 
 dividing by 1000, or cutting off for decimals all the figures $4 
 after the firft or 1, the root is continually extrafted out of bo ; 
 the quotient 1,024, till the 47th extraction, which gives 64| 6 
 1,00000,00000,00000, 1 6851,60570,53949,71; the de= | 42a f 
 
 cimal part of which multiplied by 43429 &c, gives pee ; 
 0,00000,00000,00000,07318,55936,90623,9368 for its |,°!% 
 
 0 lav 
 logarithm; and this being continually doubled for 47 times, yeaa 
 gives the logarithms of all the roots up to the firft 
 number: or being at once multiplied by the 
 47th power of 2, viz. 140737488355328, PREGA LOS 
 which is raifed as in the margin, it gives 4 | 2 
 0;01029,99566,39811,95265,27744 for the ree bea 
 logarithm of the number 1,024, true to 17 or 32 | 5! 
 18 decimals: to this prefix 3, fo fhall 3,0102 64 | 6 
 &c be the logarithm of 1024: and laftly, be- 128 | 7 
 caufe 2is the tenth root of 1024, divide by oop aa Dog 
 10, fo fhall ©,30102,99956,63981,1952 be the Pieabeh bye 
 logarithm required to the given number 2. 1048576 {20} 
 
 The logarithms of {, 2, and 10 being now 1073741824 180} 
 known; it is remarked that the logarithm | 1099511627776 ie 
 140737488355328 {47 
 
 of 5 becomes known; for fince 10 + 2 is 
 == 5, therefore Jog. 10 —log. 2 = log. 5, 
 which is 0,69897,00043,36018,8058 ; and that from the multi- 
 plications and divifions of thefe three 2, 5,10, with the correfpond- 
 ing additions and fubtra€tions of their logarithms, a multitude of 
 other numbers and their logarithms are produced; io, from the 
 powers of 2 are obtained 4, 8, 16, 32, 64, &c; from the powers of 3, 
 thefe, 25, 125, 625, 3125, &c’; alfo the powers of 5-by thofe of 
 10 give 250, 1250, 6250, &c3 and the powers of 2 by thofe of 
 10, give 20, 200, 2000, &cs; 40, 400, 80, 800, &c3 likewile by 
 divifion are obtained 24, 14, 123, 63, 12, 3%, 62, &c. . 
 Briggs then obferves, that the logarithm of 3, the next prime num- 
 ber, will be beft derived from that of 6, in this manner: 6 raifed to 
 the 9th power becomes 10077696, which divided by 10000000, gives 
 1,0077696, and the root from this continually extracted till the 46th, 
 is 1,00000,00000,00000,10998,59 34.5,88155,71866 5 
 the decimal part of which multiplied by 43429 &c, gives 3 
 0,00000,00000,00000,04776,62844,78608,0304 for its logarithm ; 
 and this 46 times doubled, or multiplied by the 46th power of 2, gives 
 0,00336,12534,52792,69 for the logarithm of 1,0077696; to which 
 adding 7, the logarithm of the divifor 10000000, and dividing by ¥, 
 the index of the power of 6, there refults 0,77815,12603,8 Bye iee 
 ‘ or 
 
LOGARITHMS. 67 
 
 for the logarithm of 6; from which fubtraCting the logarithm of 2, 
 there remains 0,47712,12547,19662,44 for tae logarithm of 3. 
 
 In the eighth chapter our ingenious author defcribes an original and 
 eafy method of conftru€ting, by means of differences, the continual 
 mean proportionals which were before found by the extraction of roots. 
 And this, with the other methods of generating logarithms by dif- 
 ferences, in this book as well as in his Zrigonometria Britannica, are 
 
 J believe the firft inftances that are to be found of making fuch ufe of 
 differences, and fhow that he was the inventor of what may be called 
 the Differential Method. He feems to have difcovered this method in 
 the following manner : having obferved that thefe continual means be- 
 tween 1 and any number propofed, found by the continual extraction 
 of roots, approach always nearer and nearer to the halves of each pre- 
 ceding root, as is vifible when they are placed together under each 
 other; and indeed it is found that as many of the fignificant figures 
 of each decimal part, as there are ciphers between them and the in- 
 teger 1, agree with the half of thofe above them; I fay, haying ob- 
 ferved this evident approximation, he fubtracted each of thefe decimal 
 parts, which he called A or the firft differences, from half the next 
 preceding one, and by comparing together the remainders or fecond 
 differences, called B, he found that the fucceeding were always 
 nearly equal to 4 of the next preceding ones; then taking the dif- 
 ference between each fecond difference and 4 of the precedingone, he 
 found that thefe third differences, called C, were nearly in the continual 
 ratio of 8 to 1; again taking the difference between each C and 4 of 
 the next preceding, he found that thefe fourth differences, called 
 D, were nearly inthe continual ratio of 16 to 1; and fo on, the 5th 
 (£), 6th (F), &c, differences, being nearly in the continual ratio of 
 82 to l, of 64 to 1; ec, 
 
 Thefe 
 
63 
 
 Thefe plain obfervations 
 being made, they very na- 
 turally and clearly fuggett- 
 ed to him the notion and 
 method of conftructing 
 all the remaining numbers 
 ‘from the differences of a 
 few of the firft, found by 
 extracting the roots in the 
 ufual way. This will evi- 
 dently appear from the 
 annexed fpecimen of a few 
 _ of the firft numbers in the 
 laft example for finding 
 the logarithm of 6 3 where, 
 after the 9th number the 
 reft are fuppofed to be 
 conftructed from the pre- 
 - ceding differences of each, 
 as here fhown in the 10th 
 and lith. And it is evi- 
 dent, that in proceeding, 
 the trouble .will’ become 
 always lefs and lefs, the 
 differences gradually va- 
 nifhing, till at laft only 
 the firft differences re- 
 main; and that generally 
 
 each lefs difference is’ 
 fhorter than the next 
 
 greater, by as many places 
 as there are ciphers at 
 the beginning of the de- 
 cimal in the number to be 
 venerated from the dif- 
 ferences. 
 
 He then concludes this 
 chapter with an ingenious, 
 but not obvious, method 
 ef finding the differences 
 B,C,D,E, &c, belonging 
 to any ‘number, as fuppofe 
 the 9th, from that number 
 itfelf, independent of any 
 of the preceding sth, 7th, 
 6th, 5th, &c, and itis this: 
 Sth, &c powers ; 
 
 CONSTRUCTION OF 
 
 1,00776 ,96 
 1} 1,0038'7}72833,36962,45663,84655, 1 
 9) 1,00193,67661,36946,61675,87022,9 
 3] 1,00096,79146,39099,01798,89072.0 
 4! 1,00048,33402,68846,62985,49953,5 
 
 5} 1,00024,18908,78824,68563,80872,7 
 24,19201,34423,314:92 ,74626,7 
 292,55598 62928, 93'754,0 
 
 “@{-1,00012,09381,263597, (5459,49919,4 Al 
 12,09454,39412 ,34281,90436, 3 
 73,13015,! 2U822,46516,9 
 73,13899 ,65732,23438 5 
 884 44909, 1692! c 
 <7 1,00006,04672,35055,30968,01600,5 
 
 6;04690,63195,56729,71959,7 
 18,28143,25761,703 3592 
 18,28253,80205,61629,2 
 110,54443,91270,0 
 
 110,55613,72115,2 
 
 : 1169,30845,2 
 8] 1,00003,02331 ,60505,65775,96479,4 
 5,02336, 17527, 65484,00800, 2 
 4..51021,99708,04320,8. 
 457035 ,81440 942539,8 
 13,81732,58269,0 
 
 13,81805, 48908,7 
 
 _ 73,10639,7 
 
 73,11302,8 
 
 663, 1 
 
 9| 1,00001,51164,65999,05672,95046,8 
 
 1,51165,80252,82887,98239,7 
 1,14253,77215,03190,9 
 
 Hithertothe  1,14255,49927,01080,2 
 (maller differences 1,72711 ,971889,3. 
 are found by fub- 1,72716,54783,6. 
 traéting the larger from 456894, 5 
 the parts of the like pre- — 4,56915,0 
 
 ceding ones. 20,7 
 20,7 
 
 Here the greater differences : 65 
 remain after fubtra@ting 28555,89 
 the fmaller from the parts 28555 ,24 
 of the difference of 21588,99736,16 
 the next preceding 21588,71180,92 
 number. 23563,44303,7579'7,72 
 28563,22715,04616,80. 
 715582,32999,52836,47524,40 
 1,00060,75582,04436,80121,42907,60 
 
 ti ) 
 Plow pb) > 
 
 1K 
 
 RlH DbiH 
 Soon > P| OW PP. 
 
 Bh 
 
 Col 
 
 Sol tia wn 
 wi >» 
 
 aie 
 OF Pw ele) yao 
 
 be NH 
 ss) 
 
 os “Neko 
 
 ual 
 
 we! 
 
 IK tint = iS - 
 pr mwNovta's 
 
 IC 
 a 
 
 ro 
 
 | 2 
 
 1784,70 | 
 
 1784,68 | 
 
 2698,58897,@2 |: 
 
 2698 5711294 
 
 7140,80678,76 154,20 
 
 sa 1140,7'7980, 1904126 
 
 8779 1,02218, 15060 11453, 80 | 
 11,1,00000,37 790, 9504", 53'1080,5241 2,54 
 
 im 
 
 >> wwOOUU 
 
 tole 
 
 : raife the decimal A to the 2d, 3d, 4th, 
 
 then will the 2d (B), 3d(C), 4th (D), Be difs 
 
 ferences, be as here below, viz. 
 
 B= 
 
- 
 
 LOGARITHMS. 69 
 
 ei Az, ae 
 
 SB . rAS41A4, 
 
 =. ZAMIR EACH re le Ce RP plied “beat 
 
 Se QeAS 4 7A® + 1015. A7 4 19.6 ASE LEAS + br125 Aro, 
 
 . 139,A°4 -813A7+ 296.27, AS 4 834.43 A+ 1953253.A '°&e. 
 = : . 192 F ATL SION AS +1IA75 2 AP + 65372,2°,A!°8& Ce 
 ae ; ‘ MD teat a MP aaa ge 7068452423. At °&c. 
 | ire : : : .  §4902.8,2. DY eee ee ee 
 mm . 6 8 SB np to a ; 9805527 At °&e, 
 Cc. 
 
 Thus in the gth number of the foregoing example, omitting the 
 ciphers at the beginning of the decimals, we have 
 
 A = 1,51164,65999,05672,95048,8 
 A*= -  2,28507,54430,06381,6726 
 Ak’= = + + 3,45422,65230,48546,2 
 
 Ata =~ 2 -- = $,22156 97802;288 
 ASS Sig ee i = 7.80316, 8208° & 
 EPS Gare ae re oe PT, OSES) 1 
 &C. 
 
 Confequently, 
 EA? = 1,14253,77215,03190,8363 =B 
 
 ZA32-  § 1,72711,32619,74273 
 gAt-  - -. 65209,62225 
 tA3+4 4A 1,72711,97889,36498 = C 
 
 2A* 4,56887,35577 
 
 AS - - 6.90652 
 
 vs - - = +5 
 TAFLZIASLWZA® 4,50894,20234 =D 
 
 Ea ob - 20,71957 
 7A° - - - 83 
 2RASE7AL + 1 =) = 20,72040 = E 
 
 which agree with the like differences in the foregoing fpecimen. 
 
 > 
 
 {n the ninth chapter, after obferving that from the logarithms of 
 1, 2, 3, 5, and 10, before found, are to be determined, by addition 
 tnd fubtraction, the logarithms of all other numbers sphioh can be 
 
 4 produced: from thefe by multiplication and divifion; for finding the 
 Jogarithms of other ptime numbers, inftead of that in’ the feventh 
 chapter, our author then fhows another ingenious method of ob- 
 taining numbers beginning with 1 and ciphers, and fuch as to bear 
 a certain relation to {ome prime number by means of which its lo- 
 garithm may be found. ‘The method is this: Find three products 
 having the common difference 1, and fuch that two of them are pro- 
 duced from factors having given logarithms, and the third produced 
 from the prime number, ‘whofe logarithm i is required, either multi- 
 plied by itfelf, or by fome other number whofe logarithm is given : 
 
 ad 
 
#0 CONSTRUCTION OF 
 
 then the greateft and leaft of thefe three products being multiplied to- . 
 gether, and the mean by itfelf, there arife two other produéts alfo dif- 
 _4ering by 1, of which the greater, divided by the lefs, gives fora quo- 
 ‘tient 1 witha {mall decimal, having feveral ciphers at the beginning. 
 Then the logarithm of this quotient being found as before, from 
 thence will be deduced the required logarithm of the given prime 
 number. ‘Thus, if it be propofed to find the logarithm of the prime 
 mumber 7; here 6 X 8=48, 7X 7=49,and 5 X 10= 50, will be 
 the three products, of which the logarithms of 48 and 50, the 1ft 
 and 3d, will be given from thofe of their faCtors 6, 8, 5, 10: alfo 
 48 X 50 = 2400, and 49 X 49 = 2401, are the two new produdts, 
 and 2401 + 2400=1,000412 their quotient: then the leaft of 44 
 means between | and this quotient is , 
 _ 1,00000,00000,00000,02367,9824.9,04333,6405, which multiplied by 
 43429 & c,produces0,00000,00000,00000,01028,40172,88337,29715 
 for its logarithm 5 which being 44 times doubled, or multiplied by 
 17592186044416, produces 0,00018,09183,45421,30 for the loga- 
 rithm of the quotient 1,000413; which being added to the logarithm 
 of the divifor 2400, gives the logarithm of the dividend 2401 ; then 
 the half of this logarithm is the logarithm of 49 the root of 2401, 
 and the half of this again gives 0,84509,80400,14256,82 for the lo- 
 garithm of 7, which is the root of 49. The author adds another 
 example to illuftrate this method ; and then fets down the requifite 
 factors, products, and quotients for finding the logarithms of all other 
 prime numbers up to 100. 
 
 The 10th chapter is employed in teaching how to find the loga- 
 rithms of fraCtions, namely by fubtracting the logarithm of the de- 
 nominator from that of the numerator, then the logarithm of the 
 fraCtion is the remainder: which therefore is either abundant or de- 
 fective, that is pofitive or negative, as the fraction is greater or lefs 
 than |. 
 
 In the 11th chapter is fhown an ingenious contrivance for very ac- 
 curately finding intermediate numbers to given logarithms, by the 
 proportional parts. On this occafion, it is remarked, that while the 
 abfolute numbers increafe uniformly, the logarithmsincreafe unequally, 
 with-a déecreafing increment ; for which reafon it happens, that either 
 logarithms or numbers corrected by means of the proportional parts, 
 will not be quite accurate, the logarithms fo found being always too 
 {mall, and the abfolute numbers fo found too great; but yet fo how- 
 ever as that they approach much nearer to accuracy towards the 
 end of the table, where the increments or differences become much 
 nearer to equality, than in the former parts of the table. And from 
 this property our author, ever fruitful in happy expedients to obviate 
 natural difficulties, contrives a device to throw the proportional part, 
 to be found from the numbers and logarithms, always near the end of 
 the table, in whatever part they may happen naturally to fall. And 
 it is this: Rejeéting the charateriftic of any given logarithm, whofe 
 number is propofed to be found, take the arithmetical complement of 
 the decimal part, by fubtra€ting it from 1,000 &c, the logarithm. of 
 
 103; then find in the table the logarithm next lefs than this Hane: 
 metic 
 
LOGARITHMS. qb 
 
 metical complement, together with its abfolute numbers to this ta- 
 bular logarithm add the logarithm that was given, and the fum will 
 be a logarithm neceflarily falling among thofe near the end of the 
 table: find then its abfolute number, corrected by means of the pro- 
 portional part, which will not be very inaccurate, as falling near the 
 end of the table; this being divided by the abfolute number, before 
 found for the logarithm next lefs than the arithmetical complement, 
 the quotient will be the required number anfwering to the given lo- 
 garithm ; which will be much more correcét than if it had been found 
 from the proportional part of the difference where it naturally hap- 
 pened to fall: and the reafon of this operation is evident from the 
 nature of logarithms. But as this divifor, when taken as the num- 
 ber anfwering to the logarithm next lefs than the arithmetical com- 
 plement, may happen to be a large prime number ; it is farther re- 
 marked, that inftead of this number and its logarithm, we may ufe 
 the next lefs compofite number which has {mall factors, and its lo- 
 garithm; becaufe the divifion by thofe {mall factors, inftead of by the 
 number itfelf, will be performed by the fhort and eafy way of divi- 
 fion in one line. And for the more eafy finding proper compofite 
 numbers and their factors, our author here fubjoins an abacus or lift 
 . of all fuch numbers, with their logarithms and component factors, 
 from 1000 to 10000 ; from which the proper logarithms and factors 
 are immediately obtained by infpeCtion. ‘Thus, for example, to find 
 the root of 10800, or the mean proportional between 1 and 10800: 
 The logarithm of 10860 is 4,03342,37554,8695, the half of which is 
 2,01671,18777,4347 the logarithm of the number fought, the arith- 
 metical complement of which logarithm is 0,98328,81222,56535 now 
 the neareft logarithm to this in the abacus is 0,98227,12330,3957, 
 and its annexed number is 9600, the factors of which are 2, 6, 3; 
 to this laft logarithm adding the logarithm of the number fought, the 
 fum is 0,99898,31107,8204, whofe abfolute number, corrected by 
 the proportional part, is 99766,12651,6521, which being divided 
 continually by 2, 6, 8, the factors of 96, the laft quotient is 
 103,923048454713 which is pretty correct, the true number being 
 103,923048454133 = /10800. 
 
 We now arrive at the 1th and 13th chapters, in which our inge- 
 nious author firft of all teaches the rules of the Ditferential Method, 
 in con{tructing logarithms by interpolation from differences. ‘This 
 is the fame method which has fince been more largely treated of by 
 later authors, and particularly by the ingenious Mr. Cotes, in his 
 Canonotechnia. How Mr. Briggs came by it does not well appear, as 
 he only delivers the rules, without laying down the principles or in- 
 veftigation of them. He divides the method into two cafes, namel 
 when the fecond differences are equal or nearly equal, and when the 
 differences run out to any length whatever. ‘The former of thefe is 
 treated in the 12th chapter; and he particularly adapts it to the in- 
 terpolating 9 equidiftant means between two given terms, evidently 
 for this reafon, that then the powers of 10 become the principal mul- 
 tipli¢rs or divifors, and fo the operations performed mentally. The 
 fubftance of his procefs is this ; Having given two abfolute ae 
 
 . } wit 
 
"9 CONSTRUCTION OF 
 
 with their logarithms, to find the logarithms of 9 arithmetical means 
 between the given numbers: Between the given logarithms take the 
 “ft difference, as well as between each of them 
 and their next or equidiftant greater and lefs lo- 
 
 r equ ai 1] 48 94 
 garithms;-and likewile the fecond differences, or | 2| 35 £3 
 the two differences of thefe three firft differences; | 2] 22 33 
 
 j 3 pei : a 26 Se 
 then if thefe fecond differences be equal, multiply | 5 | "s <4 
 one of them feverally by the numbers 45, 35, |@|—5-3 7; 
 &c, asin the annexed tablet, dividing each pro- | 7] 15 84 
 duct by 1000, that is cutting off three figures | 8 | 25 33 
 from each; laftly, to +5 of the 1f difference ic 30 Be. 
 
 ef the given logarithms add feverally the firft 
 five quotients, and {ubtract the other five, fo 
 fhall the.ten refults be the refpeCtive firft differences to be continually 
 added, to compofe the required feries of logarithms, Now this 
 amounts to the fame thing as what is at this day taught in the like 
 cafe: We know that if 4 be any term of an equidiftant feries of terms, 
 and a, b,c, &c, the firft of the 1ft, 2d, 3d, &c, order of differences; 
 then the term 2, whofe diftance from A is exprefled by x, will be 
 thus, <= A+ wat ex. coe et ies Mah c+ &c, And if 
 : a 2 
 now, with our author, we make the 2d differences equal, then c, d, 
 é, &c, will all vanith, or be equalto 0, and 2 will become barely 
 
 = 4 + x2 rt ag 
 ; er" 
 
 in Series of terms. The differences. 
 Therefore ir we take x |4 oe } 
 fucceflively equally to A+ ott x30b Pott 2t00 Hott r3h0b 
 Portosre Fs? &c, we fhall AtTo9+ x08 ot t xp qb ott rgs0b 
 have the annexed feries At yott zoo! 15? ta500 Sot + 1360! 
 of terms with their dif- [4 tase t abe! raat ro0b S104 traa0! 
 ferences. Whereit istobe Bey 14 ile ; 14 Yee ities Tress 
 obferved, that our author Paki Hird j ret Ee ; 
 had reduced the difer- A4 Se a4 fob pas of RAM it : 
 ences from the Iftto the PROTO MO REO yo ea ery Saree 
 A+ SoA +5309 | 154 —xb5b= 751-1550} 
 2d form, as he thought A+ 
 it eafierto multiply by 5 * 
 
 than todivide by 2. Alfoall the laft terms (x. *~1 5) are fet down po 
 ° | 
 
 r 9 — I 45 
 a tot—zo05 = hI — +3356 
 
 fitive, becaufe in the logarithms dis negative.—If the two 2d differ- 
 ences be only nearly equal, take an arithmetical mean between them, 
 and proceed with it the fame as above with one of the equal 2d dif- 
 ferences.—He alfo fhows how to find any one fingle term, independ- 
 ent of the reft; and concludes the chapter with pointing out a mez 
 thod of finding the proportional part more accurately than before. _ 
 In the 13th chapter our author remarks, that the beft way of filling 
 up the intermediate chiliads of his table, namely from 20000 to 90000, 
 is by quinquifeCtion, or interpofing four equidiftant means between 
 two given terms; the method of performing which he thus particu- 
 larly defcribes. Of the given terms, or logarithms, and two or thre¢ 
 others on each fide of them, take the lft, 2d, 3d, &c, aeepore 
 
LOGARITHMS. 43 
 
 till the laft differences come out equal, which fuppofe to be the 5th 
 
 differences: divide the firft differences by 5, the 2d by 25, the 3d by 
 
 125, the 4th by 625, and the 5th by 3125, and call the refpective 
 
 quotients the 1ft, 2d, 3d, 4th, 5th mean differences; or, inftead of 
 
 dividing by thefe powers of 5, multiply by their reciprocals +%5 13% 
 
 rou» Tooss, rosso; that is, multiply by 2, 4, 8, 16, 32, cutting 
 off refpectively one, two, three, four, five figures from the end of the 
 
 products, for the feveral mean differences: then the 4th and 5th of 
 
 thefe mean differences are fufliciently accurate; but the lit, 2d, and 
 
 $d are to be corre€ted in this manner; from the mean third differ- 
 
 ences fubtract three times the 5th difference, and the remainders aré 
 
 the corre 3d differences; fromthe mean 2d differences fubtract double 
 
 the 4th differences, and the remainders are the correct 2d differences; 
 
 laftly, from the mean 1ft differences take the correét 3d differences, 
 
 and + of the 5th difference, and the remainders will be the correct 
 firft differences. Such are the corrections when the differences extend 
 as far as the 5th. However, in completing thofe chiliads in this 
 way, there will be only 3 orders of differences, as neither the 4th nor 
 5th will enter the calculation, but will vanifh through their {mallnefs: 
 therefore the mean 2d and 3d differences will need no correction, and 
 the mean firft differences will be corrected by barely fubtraCting the 
 3d from thea. Thefe preparatory numbers being thus found, all the 
 2d differences of the logarithms required, will be generated by adding 
 continually, from the lefs to the greater, the conftant 3d diiference; 
 and the feries of 1{t differences will be found by adding the feveral 
 2d differences; and laftly, by adding continually thefe 1{t differerices 
 to the 1ft given logarithm &c, the required logarithmic terms are 
 generated. 
 
 Thefe eafy rules being laid down, Mr. Briggs next teaches how by 
 them the remaining chiliads may beft be completed: namely, having 
 here the logarithm for all numbers up to 20000, find the logarithm to 
 every 5 beyond this, or of 20005, 20010, 20015, &c, in this manner 3 
 to the logarithms of the 5th part of each of thefe, namely 4001, 4002, 
 4003, &c, add the conftant logarithm of 5, and the fums will be the 
 logarithms of all the terms of the feries 20005, 20010, 20015, &c: 
 and thefe logarithms will have the very fame differences as thofe of 
 the feries 4001, 4002, 400%, &c; by means of which ‘therefore in- 
 terpofe 4 equidiftant terms by the rules above ; and thus the whole 
 canon will be <afily completed. | 
 
 Briggs here extends the rules for correCting the mean differences in 
 quinguifection, as far as the 20th difference ; he alfo lays down fimilar 
 rules for tri{eftion, and {peaks of general rules for any other fection, 
 but omitted as being lefs eafy. So that‘he appears to have been pof- 
 fefled of all that Cotes afterwards delivered in his Canonotechnia five 
 Conftructio Tabularum per Differentias, drawn from the Differential 
 Method, as their general rules exactly agree, Briggs’s mean and cor- 
 rect differences being by Cotes called round and quadrat differences, 
 becaufe he expreffes them by the numbers 1, 2, 3,° &c, written re- 
 fpectively within a {mall circle and fquare. 
 
 Briggs 
 
44 CONSTRUCTION OF 
 
 Briggs alfo obferves, that the fame rules equally apply to the corie 
 ftruction of equidiftant terms of any other kind, fuch as fines, tan- 
 gents, fecants, the powers of numbers, &c: and farther remarks, that, 
 of the fines of three equidifferent arcs, all the remote differences may 
 be found by the rule of proportion, becaufe the fines and their 2d, 
 4th, 6th, 8th, &c, differences, are continued proportionals, as are 
 alfo the 1ft, 3d, 5th, 7th, &c differences, among themfelves; and, like 
 as the 2d, 4th, 6th, &c differences are proportional to the fines of the 
 mean arcs, fo alfo are the 1ft, 3d, 5th, &c differences proportional to 
 the cofines of the fame arcs. Moreover, with regard to the powers of 
 numbers, he remarks the following curious properties; I{t, that they 
 will each have as many orders of differences as are denoted by the 
 index of the power, the{quares having two orders of differences, the 
 cubes three, the 4th powers four, &c; fecond, that the laft differences 
 will be all equal, and each equal to the common difference of the fides 
 or roots raifed to the given power and multiplied by 1x 2X3x4 &c, 
 continued to as many terms as there are units in the index; fo if the 
 roots differ by 1, the fecond differences of the {quares will be each 
 1 X 2 or 2, the 3d differences of the cubes each | X 2X 3 or 6, the 4th 
 differences of the 4th powers each 1X 2X3X4 or 24, and fo on; 
 and if the common difference of the roots be any other number #, 
 then the laft differences of the f{quares, cubes, 4th powers; 5th powers, 
 é&c, will be refpectively 27° 6°, 24n', 120n° &c. 
 
 Befides what was fhown in the eleventh chapter, concerning the 
 taking out the logarithms of large numbers by means of proportional 
 parts, Briggs employs the next or 14th chapter in teaching how, from 
 the firft ten chiliads only, and a {mall table of one page, here given, 
 to find the number anfwering to any logarithm, and the logarithm to 
 any number, confifting of fourteen places of figures*. 
 
 Having thus fully fhown the con{tru€tion and chief properties of 
 his logarithms, our ingenious author, in the remaining eighteen 
 chapters, exemplifies their ufes in many curious and important - 
 fubjects; fuch as The Rule-of-Three, or rule of proportion; finding 
 
 the roots of given numbers; finding any number of mean propor- 
 
 * Jtis no more than a large exemplification of this method of Briggs’s that 
 has been printed fo late as 1771, in a 4to tract by Mr. Robert Flower, under the 
 title of The Radix, A New Way of making Logarithms. Although Briggs’s work 
 might not be known to this writer.—Since this was written, I have been fa- 
 voured with the following anecdote, concerning Mr. Flower and his work, by 
 the Rev. Dr. Horfley, now bifhop of St. Afaph, the learned editor of the works 
 of Sir I. Newton. ‘* This Robert Flower wasa very obicure, and probably an 
 illiterate man. He was mafter of a writing fchool in the town of Bifhop Stort- 
 ford, in Hertfordfhire. He communicated his Radix, before he publifhed it, 
 to my late learned friend Math. Raper, Efq. of Thorley Hall. I was at Thorley 
 at the time upon a vifit to my father, who was rector.of the parifh; and I 
 well remember that Mr. Raper told me with great furprife, that Flower (who 
 was known to us both by name as the writing-mafter, of the neighbouring 
 market town) had fallen upon Briggs’s way of finding alllogarithms from the 
 firft ten chiliads. And he was fo well perfuaded that Flower had made the dif- 
 covery for himfelf, without any light from Briggs, that with his accuftomed 
 munificence he rewarded the man’s ingenuity with a prefent of ten guineas; 
 informing him, I believe, that his work had been done before, and diffuading 
 the publication. 
 
 tionals 
 
LOGARITHMS. 715 
 
 tionals between two given terms ; with other arithmetical rules: alfo 
 various geometrical fubjects, as 1ft, Having given the fides of any 
 plane triangle, to find the area, perpendicular, angles, and diameters 
 of the infcribed and circumfcribed circles; 2d, In a right-angled 
 triangle, having given any two of thefe, to find the reft, viz. one leg 
 and the hypotenufe, one leg and the fum or difference of the hypo- 
 tenufe and the other leg, the two legs, one leg and the area, the area 
 and the fum or difference of the legs, the hypotenufe and fum or 
 difference of the legs, the hypotenufe and area, and the perimeter 
 and area; 3d, Upon a given bafe to defcribe a triangle equal and 
 ifoperimetrical to another triangle given; 4th, To defcribe the cir- 
 cumference of a circle fo, thatthe three diftances from any point in 
 it to the three angles of a given plane triangle, fhall be to one an- 
 other in a giyenratio; 5th, Having given the bafe, the area, and the 
 ratio of the two fides, “of a plane triangle, to find the fides; 6th, 
 Given the bafe, difference of the fides, and area of a triangle, to find 
 the fides; 7th, To find a triangle whofe area and perimeter fhall be 
 expreffed by the fame number; 8th, Of four given lines, of which 
 the fum of any three is greater than the fourth, to form a quadrila- 
 teral figure about which a circle may be defcribed; 9th, Of the dia- 
 meter, circumference, and area of a circle, and the furface and fo- 
 lidity of the {phere generated by it, having any one given, to find any 
 of the reft; 10th, Concerning the elipfe, fpheroid, and gauging ; 
 11th, To cut a line or a numberin extreme and mean ratio; 18th, 
 Given the diameter of acircle, to find the fides and areas of the in- 
 fcribed and circumfcribed regular figures of 3, 4, 5, 6, 8,10, 12, and 
 16 fides; 13th, Concerning the regular figures of 7, 9, 15, 24, and 
 $0 fides; 14th, Of ifoperimetrical regular figures; 15th, Of equal re~ 
 gular figures; and 16th, Of the fphere and the five regular bodies; 
 which clofes this introdu€tion. Such of thefe problems as can admit 
 of it, are determined by elegant geometrical conftructions, and they 
 are all illuftrated by accurate arithmetical calculations, performed by 
 logarithms ; for the exemplification of which they are purpofely 
 iven. | 
 is At the end he remarks, that the chief and moft neceffary ufe of 
 logarithms, is in the doctrine of fpherical trigonometry, which he 
 here promifes to give in a future work, and which was accomplifhed 
 in his Trigonometria Britannica, to the defcription of which we now 
 proceed. : 
 
 Of BRIGGS’s Lrigonometria Britannica, 
 
 At the clofe of the account of writings on the natural fines, tan- 
 gents, and fecants, I omitted the defcription of this work of our 
 learned author, although it is perhaps the greateft of this kind, all 
 things confidered, that ever was executed by one perfon; purpofely 
 referving my account of it to this place, not only as it is connected 
 with the invention and con{truCtion of logarithms, but thinking it 
 
 ae a -deferved 
 
76 _ CONSTRUCTION OF 
 
 deferved mote peculiar and diftinguifhed notice, on account of the 
 importance and originality of its contents. The diyifion of the qua- 
 drant, and the mode of conf{tru€tion, are both new; andthe numbers 
 are for more accurate, and are extended to more places, than they had 
 ever been before. ‘The circular arcs had always been divided in a 
 {exagefimal proportion ; but here the quadrant is divided into degrees 
 and decimals, as thisis a much eafier mode of computation than by 
 60ths; the divifion being completed only to 100ths of degrees, though 
 his defign was to have exténded it to 1000ths of degrees. And, be- 
 fides his own private opinion, he was induced to adopt this method of 
 decimal divifions, partly at the requett of other perfons, and partly 
 perhaps from the authority of Vieta, pa. 29 Calendarii Gregoriant. 
 And it is probable that computations by this decimal divifion “would 
 have come into general ufe, had it not been for the publication of 
 Vlacq’s tables, which were extended to every10 feconds, or 6th parts 
 of minutes. But befides this method by a decimal divifion of the 
 degrees, of which the whole circle contains 360, or the quadrant 90, 
 in the 14th chapter he remarks that fome other perfons were inclined 
 rather to adopt a complete decimal divifion of the whole circle, firft 
 into 100 parts, and each of thefe into 1000 parts; and for their fakes 
 he fubjoins a fmall table of the fines of every 40th part of the qua- 
 drant, and remarks, that from thefe few the whole may be made out 
 by continual quinquifections; namely, 5 times thefe 40 make 200, 
 then 5 times thefe give 1000, thirdly 5 times thefe give 5000, and 
 laftly, 5 times thefe give 25000 for the whole quadrant, or 100000 
 for the whole circumference. 
 _ Butto return. Our author’s large table confifts of natural fines to 
 15 places, natural tangents and fecants each to 10 places, logarithmic 
 fines to 14 places, and logarithmic tangents to 10 places, each befide 
 the chara€teriftic. A moft ftupendous performance! The table is 
 preceded by an introduction, divided into two books, the one contain- 
 ing an account of the truly ingenious conftruction of the table, by 
 the author himfelf; and the other, its ufes in trigonometry, &c, by 
 Henry Gellibrand, profeffor of aftronomy in Grefham College, who 
 remarks in the preface, that the work was compofed by the author 
 about the year 1600; though it was only publifhed by the direction 
 of Gellibrand in 1633, it having been printed at Gouda under the care 
 of Vlacq, and by the ‘prittes nf his Lrigonometria Artificialis, which 
 came out the fame year. 
 
 After briefly mentioning the common methods of dividing the 
 quadrant, and conftructing the tables of fines, &c, from the ancients 
 down to his own time, he haftens to the defcription of his own pe- 
 culiar and truly ingenious method, which is briefly this: having firft 
 divided the quadrant into a {mall number of parts, as 72, he finds the 
 
 ‘ fine of one of thofe parts, then from it the fines of the double, triple, 
 quadruple, &c, up to the quadrant or 72 parts. He next quinqui- 
 feéts each of thefe parts; by interpofing four equidiftant means, by dif- 
 fer ences ; he then quinquifects each of thefe ; 7 and finally each of thefe 
 
 agaims 
 
LOGARITHMS. ri! 
 
 again; which completes the divifion as far as degrees and centefms. 
 ‘The rules for performing all thefe things he inveftigates and illu 
 trates in a very ample manner. In treating of multiple and fubmul- 
 tiple arcs, he gives general algebraical expreffions for the fine or 
 chord of any multiple whatever of a given arc, which he deduced 
 from a geometrical figure, by finding the law for the feries of fuccef- 
 five multiple chords or fines, after the manner of Vieta, whowas the 
 firft perfon that I know of, who laid down general rules for the 
 chords of multiples and fubmultiples of arcs or angles: and the fame 
 was afterwards improved by Sir 1. Newton, to fuch form, that radius, 
 and double the cofine of the firft given angle, are the firft and fecond 
 terms of all the proportions for finding the fines and cofines of the 
 multiple angles. For afligning the coefficients of the terms in the’ 
 multiple expreffions, our author here delivers the con{truCtion of 
 figurate or polygonal numbers, inferts a large table of them, and 
 teaches their feveral ufes; one of whichis, that every other number, 
 taken in the diagonal lines, fyrnifhes the coeflicients of the terms of 
 the general equation by which the fines and chords of multiple arcs 
 are exprefled, which he amply illuftrates; and another, that the fame 
 diagonal numbers conftitute the coefficients of the terms of any 
 power of a binomial; which property was alfo mentioned by Vietain 
 his Angulares Seétiones, theor. 6, 1; and before him, pretty fully 
 treated of by Stifelius, in his Arithmetica Integra, fol. 44 & feq.; 
 where he inferts and makes the like ufe of fucha table of figurate 
 numbers, in extracting the roots ofall powers whatever. But it was 
 perhaps known much earlier, as appears by the treatife on figurate 
 numbers by Nichomachus, (fee Malcolm’s Hittory, p. xviii). Though 
 indeed Cardan feems to afcribe this difcovery to Stifelius. See his 
 Opus Novum de Proportionibus Numerorum, where he quotes it, 
 and extracts the table and its ufe from Stifel’s book. Cardan, in p. 135, 
 &c, of the fame work, makes ufe of a like table to find the number 
 of variations, or conjugations as he calls them. Stevinus.too makes 
 ufe of the fame coefhcients and method of roots as Stifelius. See 
 his Arith. page 25. And even Lucas de Burgo extraéts the cube 
 root by the fame coefficients, about the year 1470. But he doesnot go 
 to any/higher roots. And this is the firft mention I have feen made 
 of this law of the coefficients of the powers of a binomial, common- 
 ly called Sir I. Newton’s binomial theorem, although it is very evi- 
 dent that Sir Ifaac was not the firft inventor of it: the part of it pro- 
 perly belonging to him feems to be only the extending it to fractional 
 indices, which was indeed an immediate effet of the general method 
 of denoting all roots like powers with fraCtional exponents, the theorem 
 being not at all altered. However, it appears that our author Briggs 
 was the firft who taught the rule for generating the coefficients of the 
 terms fucceflively one from another, of any power of a binomial, in- 
 dependentof thofe of any other power. Forhavingfhown, inhis Abacus 
 Hayyeysos (which he fo calls on account of its frequent and excellent 
 pie, aud of which a {mall {pecimen is here annexed), that the numbers 
 
 ‘ in 
 
Wa cere CONSTRUCTION OF 
 
 Bacus FIATXPHETOS. 
 
 H G 
 
 F E D C 
 —®|—@|+©| +@|—G|-@| +@| © 
 1 1 1 a, 1 1 i 
 
 9 8 7 6 Bi 4 
 
 36 28 21 15 10 
 
 1 
 3 2 
 
 abe? 3 
 
 84: 56 35 20 10 4 
 126 70 35 15 5 
 
 126 56 21 6 
 
 84 28 a 
 
 36 8 
 
 9 
 
 in the diagonal directions, afcending from right to left, are the coeffi- 
 cients of the powers of binomials,the indices being the figures in the - 
 firft perpendicular column A, which are alfo the coefficients of the 2d 
 terms of each power (thofe of the firft terms being 1, are here omit- 
 ted); and that any one of thefe diagonal numbers is in proportion to 
 the next higher in the diagonal, as the vertical of the former is to the 
 marginal of the latter, that is, as the uppermoft number in the co- 
 Jumn of the former is to the firft or right-hand number in the line 
 of the latter ; having fhown thefe things, I fay, he thereby teaches the 
 generation of the coeflicients of any power, independently of all other 
 powers, by the very fame law or rule which we now ufe in the bi- 
 nomial theorem. Thus, for the 9th power ; 9 being the coefficient of 
 the 2d term, and 1 always that of the Mt, to find the 3d coeflicient 
 we have 2:8::9:36; for the 4th term, 3:7::36:84; for the 5th 
 term, 4:6::84:126; and fo on for the reft. That is to fay, the co- 
 efficients of the termsin any power m, are inverfely as the vertical num- 
 bers or firft line 1, 2, 3,4,....m, and dire€tly as the afcending numbers 
 m,m—1, m—2, m—3,..+.1; in the f§yft column A; and that confe- | 
 quently thofe coefficients are found by the continual multiplication of 
 
 thefe fractions— ae ure hee bth , which is the very 
 1 2 3 4, m 
 
 theorem as it flands at this day, and as applied by Newton to roots 
 or fraCtiona] exponents, as it had before been ufed for integral powers. 
 This theorem then being thus plainly taught by Briggs about the 
 year 1600, it is furprifing how a man of fuch general reading as Dr. 
 Wallis was, could be quite ignorant of it, as he plainly appears to 
 be by the 85th chapter of his algebra, where he fully afcribes the in- 
 vention to Newton, and adds, that he himfelf had formerly fought 
 for fuch a rule, but without fuccefs: Or how Mr. John Bernouilli, 
 not half a century fince, could himfelf firft difpute the invention of 
 this theorem with Newton, and then give the dilcovery of it to Pafcal, 
 
 | who 
 
2 
 
 LOGARITHMS, - 79 
 
 who was not born till long after it had been taught by Briggs. See 
 Bernouilli’s Works, vol. 4. pa. 173. But I do not wonder that Briggs’s 
 remark was unknown to Newton, who owed almoft every thing to 
 genius and deep meditation, but very little to reading: and I have 
 no doubt that he made the difcovery himfelf, without any light from 
 Briggs, and that he thought it was new for all powers in general, as 
 it was indeed for roots and quantities with fractional and irrational 
 exponents. | 
 
 When the above table of the fums of figurate. numbers is ufed 
 by our author in determining the coefficients of the terms of the 
 equation, whofe root is the chord of any fubmultiple of an arc, as 
 when the fection isexprefled by any uneven number, he remarks, that 
 the powers of that chord or root will be the I1ft, 3d, 5th, 7th, &c, 
 in the alternate uneven columns, A,C, E, G, &c, with their figns 
 +- or — as marked to the powers, continued till the higheft power 
 be equal to the index of the fe€tion; and that the coefficients of 
 thofe powers are the fums of two continuous numbers in the fame 
 column with the powers, beginning with 1 at the higheft power, 
 and gradually defcending one line obliquely to the right at each 
 lower power: fo fora trifeCtion, the numbers are | in C, and 142 | 
 =3 in A; and therefore the terms are —1@)+ 3(1): for a quinqgui- 
 fection, the numbers are 1 in E, 14+4—5 in C, 243=—5 in A; 
 fo that the terms are 1(5—5@) +5): for afeptifeCtion, the num- 
 bers are 1 in G, 1+6=7 in E,4+10=14inC, and 34+4= 
 7 in A; and fo the terms are—1 Q+tigQ- 14(3) +17 ()): and fo 
 
 on; the fum of all thefe terms being always equal to the chord of the 
 whole or multiple arc. But when the fection is denominated by an 
 even number, the fquares of the chords enter the equation, inftead of 
 the firft powers as before, and the dimenfions of all the powers are 
 doubled, the coefficients being found as before, and therefore the 
 powers and numbers will be thofe in the 2d, 4th, 6th, &c, columns: 
 and the uneven fections may alfo be expreffed the fame way: hence, 
 
 for a bifeCtion the terms will be —1Q+43 for a trifection 
 1@)—6@ + 9); forthe quadrifetion —1 9) +8 G@—20 4) 4 16@); 
 for the quinquifection 1 @G—10 (8) +35 —50@ +25@; and fo 
 on. : ‘ 
 Our author fubjoins 
 another table, a {mall 
 ipecimen of which. is 
 here annexed, in which 
 the firft column confifts 
 of the uneven numbers 
 14.35..55 Se, | ‘the..reft 
 being found by addition 
 as before, and the alter- 
 nate diagonal numbers themfelves are the coeflicients:. 
 
 The 
 
so” CONSTRUCTION OF 
 
 The method is quite different from 
 
 | 1ft Vieta’s Table. 
 
 that of Vieta, who gives another ta- | 9 | __. 
 
 ble for the like purpofe, afmall part | 3 | 2d 
 
 of which is here annexed; which is | 4 | % | — 
 
 formed by adding, from the number | 7 : BF 
 
 2, downwards obliquely towards the | 7 |a4]}7 [au 
 
 tight; and the coefficients of the {8 | 20 lie | 9 
 
 terms ftand upon the horizontal line. | 9 | 27 80 | 9 | 5th}___ 
 10 35 150 25 ° 6th 
 
 Thefeangular feGtions were afterwards farther difcuffed by Oughtred 
 and Wallis. And the fame theorems of Vieta and Briggs have been 
 fince given in a different form by Herman and the Bernouillis, in 
 the Lerpfic Acts, and the Memoirs of the Royal Academy of Sciences. 
 Thefe theorems they expreiled by the alternate terms of the power of 
 a binomial, whofe exponent is that of the multiple angle or fection. 
 And De Lagny in the fame Memoirs, firft fhowed, that the tangents 
 and fecants of multiple angles are alfo exprefled by the terms of a 
 binomial, in the form of a fraction, of which fome of thofe terms form 
 the numerator, and others the denominator. ‘Thus if r exprefs the 
 radius, s the fine, cthe cofine, ¢ the tangent, and / the fecant, of the 
 angle 4; then the fine, cofine, tangent, and fecant of z times the 
 angle, are exprefled thus, viz. 
 
 . 1 n a—l nn—l.n—2n—3 nn—1.n—2.1—3.n—4 nj 
 Sit. 0.4 - ae fey tall Ale a) es elena ised ee noes Hite decile 5, @ 
 pe Taomliy aca, La, 3, Jen 1,905. a5 vor ae 
 1 2 nn—! n—2,2 PN—1 N—Q.romeB n—4 
 Cofine 2 A— ——~— x: ¢ — ¢ $ mene 4 &o. 
 pees 1.2 | 1.2.3.4 ae 
 n nmel © n.n—I1.n—2 423 n.n—1.n—Unme3.n—4 n—5 
 
 * eee ba eed ia “3 
 pio Song ae a 
 Tang. n d= 0S 
 
 ——$ dD ¥ 
 1.2.3.4.5 t° &c 
 
 SE 
 nnmetl n-2, . mim inmQn—3B n—4 j \ 
 r 2 r 
 
 r At PERLE t on Li GEC. 
 gg? 1.2. 3.4. 
 ° 9 
 {2 orr 2 to 
 Sec.nd=rxX wee — | —______. , 
 n Amel em ten meee tee ca 
 gt Gd - 1" 7 + gC 
 
 —_ _——-—Fr 
 Pes Pa: . 
 
 where it is evident, that the feries in the fine of x 4, confifts of the 
 even terms of the power of the binomial (c + 5)’, and the feries in the 
 cofine of the uneven terms of the fame power; alfo the feries in the 
 numerator of the tangent, confifts of the even terms of the power 
 (r +1)’, and the denominator, both of the tangent and fecant, confifts 
 of the uneven terms of the fame power (r+ 7)”. And if the diameter, 
 chord, and chord of the fupplement, be fubftituted for the radius, 
 fine and cofine, in the expreflions for the multiple fine and cofine, 
 
 the 
 
LOGARITHMS. 81 
 
 ¢he refult will give the chord and chord of the fupplement of » times 
 the arc or angle 4 ‘Thefe, and various other expreflions, for multi- 
 ple and fubmultiple arcs, with other improvements in trigonometry, 
 have alfo been given by Euler, and other eminent writers on the fame 
 fubjec 
 The before mentioned De Lagny offered a project for fubftituting, 
 
 inftead of the common logarithms, a binary arithmetic, which lie 
 called the natural hearithms, and which he and Leibnitz feem to have 
 both invented about the fame time, independently of each other: 
 but the project came to nothing. De Lagny alfo publifhed, in fe- 
 veral Memoirs of the Royal Academy, a new method of determining 
 the angles of figures, which he called Goniometry. It confifts in 
 meafuring, with a pair of compafles, the arc which fubtends the angle 
 in queftion: however, this arc is not meafured by applying its extent 
 to any preconf{tructed fcale, but by examining what part it is of half 
 the circumference of the fame circle, in this manner: from the 
 propofed angular point as a centre, with a fufficiently large radius, 
 a femicircle being defcribed, apart of which is the arc intercepted 
 by the fides of the propofed angle, the extent of this are is taken 
 with a pair of fine compafles, and applied continually upon the arc of 
 the femicircle, by which he finds how often it is contained in the fe- 
 micircle, with ufually a {mall arc remaining; in the fame manner he 
 meafures how often this remaining arc is contained in the firft arc, 
 and what remains again is applied continually to the firft remainder, 
 and fo the 3d remainder to the 2d, the 4th to the 3d, and fo on till 
 there be no remainder, or elfe till it become infenfibly fmall. By this 
 procefs he obtains a feries of quotients, or fractional parts, one of 
 another, which being properly reduced into one, give the ratio of 
 the firft arc to the femi-circumference, or of the propofed angle, to 
 two right angles or 180 degrees, and coniequently that angle in de- 
 grees, minutes, &C, if required, and that commonly, he fays, to a 
 degree of accuracy far exceeding the calculation of the fame by means 
 of any tables of fines, tangents or fecants, notwithftanding the apparent 
 paradox in this expreflion at firft fight. Thus, if the 1f are be 4 
 times contained in the femicircle, the remainder once contained in 
 the firft arc, the next five times in the fecond, and nally the fourth 
 two times in the third: Here the quotients are 4, 1, 5, 23; confequent- 
 
 ly the fourth or laft arc was j the 3d; therefore the 3d was Lor ve 
 : 5x 
 
 I 
 ‘ of the 2d, and the 2d was we 
 
 Fh 9 
 
 or 11 of the ~ and the firft or arc 
 
 ] 
 fought, was a or 43 of the femicircle; and confequently it con- 
 Sih 
 tains 377 degrees, or 37° 8° 34”3. Hence it is evident that this me- 
 thod is in fact nothing more dann an example of continued fraétions;, 
 the firft inftance of which was given by lord Brouncker. 
 But to return from this long digreflion; My, Briggs next treats of 
 
 M jnterpo- 
 
 Qa, 
 
SO). CONSTRUCTION OF 
 
 interpolation by differences, and chiefly of quinquifeCtion, after the 
 manner ufed in the 1th chapter of his conftru€tion of logarithms, 
 before defcribed. He here proves that curious property of the fines 
 and their feveral orders of differences, before mentioned, namely, 
 that of equidifferent arcs, the fines, with the 2d, 4th, 6th, &c dit- 
 ferences, are continued proportionals; as ‘alfo the cofines of the 
 means between thofe arcs, and the ift, 3d, 5th, &c differences. And 
 to this treatife on interpolation,by differences, he adds. a. marginal 
 note, complaining that this 13th chapter of his Arithmetica Logarith- 
 nica had been omitted by Vlaeq in his edition of it; as if he were 
 afraid of an intention to deprive him of the honour of the invention 
 of interpolation by fucceflive differences. The note is this: Modus 
 correétionis & me traditus eff Arithmetice Logarithmice capite 13, in 
 editione Londinenfi: Iftud autem caput und cum fequenti in. editione Bax 
 tava me inconfulto et tafcio omiffum fut: nec in omnibus, editionis illus 
 author (vir aliogui indufirius et won indocius), meam mentem’ videtur 
 affequutus: Ideoque, ne quicquam. defit cuiquam, qui integrum canonem 
 conficere cupiat, quedam maxime neceffaria illine hue transferenda 
 CE71/ Ul. 
 
 A large fpecimen of quinquifection by differences is then given, 
 and he fhows how it is to be applied to the conftru@tion of the whole 
 canon of fines, both for 100th and 1000th parts of degrees 5 namely, 
 for centefms, divide the quadrant firft into 72 equal parts, and find 
 their fines by the primary methods; then thefe quinquifected give 
 360 parts, a fecond quinquifection gives 1800 parts, and a third 
 gives 9000 parts, or centefms of degrees: but for millefms, divide the 
 quadrant into 144 equal parts; then one quinquifection gives 720, a 
 {econd gives 3600, a third 18000, and a fourth gives 90000 parts, or 
 millefms. . 3 
 
 He next proceeds to the natural tangents and fecants, which he 
 directs to be raifed in the fame manner, by interpolations from a few 
 primary ones, conftructed from the known proportions between fines, 
 tangents, and fecants; excepting that half the tangents and fecants 
 are to be formed by addition and fubtraction only, by means of fome 
 fuch theorems as thefe, namely, | {t, the fecant of an arc is equal to 
 the fum of the tangent of the fame arc, and the tangent of half its 
 complement, which will find every other fecant ; 2d, dauble the tan- 
 gent of an arc, added to the tangent of half its complement, is equal 
 io the tangent of the {um of that arc and the faid half complement, 
 by which rule half the tangents will be found, &c. _ 
 
 "In the two remaining chapters of this book are treated the con- 
 ftruction of the logarithmic fines, tangents, and fecants. This is pre- 
 ceded by fome remarks on the origin and invention of them. Our 
 author here obferves, that logarithms may be of various kinds; that 
 others had followed the plan of Baron Napier the firft inventor, 
 among whom Benjamin Urfinus is efpecially commended, who ap- 
 plied Napier’s logarithms to every ten feconds of the quadrant; but. 
 that he himfelf, encouraged by the neble inventor, devifed other lo- 
 fue ih OS garithms 
 
LOGARITHMS. . 83 
 
 _parithms that were much eafier and more excellent*. He fays he 
 put 10, with ciphers, for the logarithm of radius; 9 for the logarithm 
 fine of 5° 44’, whofe natural fine is one 10th of the radius; 8 for 
 that of 34’, whofe natural fine is one 100th of the radius, and fo on; 
 thereby making 1 the logarithm of the ratio of 10 to 1, which is the 
 characteriftic of his fpecies of logarithms. | 
 
 To conftruct the logarithmic fines, he directs firft to divide the 
 quadrant into 72 equal parts as before, and to find the logarithms of 
 their natural fines as in the 14th chapter of his Ar:thmetica Logarith- 
 mica; after which this number will bé increafed by quinquifection, 
 firft to 360, then to 1800, and laftly to 9000, or centefms of degrees. 
 But if millefms of degrees be required, divide the quadrant firft into 
 144 equal parts, and then by four quinquifections thefe will be ex 
 tended-to the following parts, 720, 3600, 18000, and 90000, or 
 millefms of degrees. He remarks, however, that the logarithmic fines 
 of only half the quadrant need be found in this manner, as the other » 
 half may be found by mere addition, or fubtra€tion, by means of this 
 theotem, as ‘the fine of half an are is to half radius, fo is the fine of 
 the whole arc to the cofine of the faid halfarc. ‘This theorem he il+ 
 luftrates with examples, and then adds a table of the logarithmic fines 
 of the primary 72 parts of the quadrant, from which the reft are to 
 be made out by quinquwifection. , 
 
 In the next chapter our author fhows the conftruCtion of the natural 
 tangents and fecants more. fully than he had done before, demonftrat- 
 ing and illuftrating feveral curious theorems for the eafy finding of 
 them. He then concludes this chapter, and the book, with pointing 
 out the very eafy conftruction of the logarithmic tangents and fecants 
 by means of thefe three theorems : 
 
 lft, As cofine : fire :: radius ;: tangent, 
 2d, As tangent : radius : : radius : cotangent, 
 8d, Ascofine : radius :: radius : fecant. 
 
 So that in logarithms, the tangents are found by fubtraCting the cofines 
 from the fines, adding always 10 or the radius; the cotangents are 
 found by fubtra€ting always the tangents from 20 or double the ra- 
 dius; and the fecants are found by fubtracting the cofines from 20 
 the double radius. | 
 
 The 2d book, by Gellibrand, contains the ufe of the canon in plane 
 and {pherical trigonometry. - 
 
 Befides Briggs’s methods of conftructing logarithms, above de- 
 {cribed, no others were given about that time. For as to the calcu- 
 lations made by Vlacq, his numbers being carried to comparatively 
 but few places of figures, they were performed by the eafieft of 
 
 Pp § ) bs Pp Y 
 Briggs’s methods, and in the manner which this ingenious man had 
 
 8§ & 
 
 pointed out in his two volumes. Thus, the 70 chiliads of logarithms, 
 
 * His words are ; “ Ezo vero ipfius inventoris primi cohortatione adjutus, ali- 
 es logarithmos applicandos cenfui, qui multo faciliorem ufum habent, preftan- 
 tiorem. Logarithmus radii circularis vel firnus totius, a me ponitur 10 &c.”” 
 
 ) from 
 
st CONSTRUCTION OF 
 
 from 20000 te 90000, computed by Vlacq, and publifhed in 1628, 
 being extended only to 10 places, yield no more than two orders of 
 mean differences, which are alfo the correct differences, in quinqui- 
 {e€tion, and therefore will be made out thus, namely, one-fifth of 
 them by the mere addition of the conftant logarithm of 5; and the 
 other four-fifths of them by two eafy additions of very fmall numbers, 
 namely, of the 1ft and 2d differences, according to the directions 
 given in Briggs’s Arith. Log. c. 13, p. 31. And as to Vlacq’s loga- 
 rithmic fines and tangents to every 10 feconds, they were eafily com- 
 puted thus; the fines for half the quadrant were found by taking 
 the logarithms to the natural fines in Rheticus’s canon; and then 
 from thefe the logarithmic fines to the other half quadrant were found 
 by mere addition and fubtraGtion; and from thefe all the tangents by 
 one fingle fubtraction. So that all thefe operations might eafily be 
 performed by one perfon, as quickly as a printer could fet up the 
 types; and thus the computation and printing might both be car- 
 ried on together. And hence it appears that there is no reafon for 
 admiration at the expedition with which thefe tables were faid to have 
 been brought out. | ; 
 
 Of certain Curves related to Logarithms. 
 
 About this time the mathematicians ef Europe began to confider 
 fome curves which have properties analogous to logarithms. Edmund 
 Gunter, it has been faid, firft gave the idea of.a curve, whofe ab- 
 {ciffes are in arithmetical progreflion, while the correfponding or- 
 dinates are in geometrical progreflion, or whofe abfciffes are the lo- 
 garithms of their ordinates; but I cannot find it noticed in any part 
 of his writings. The fame curve was afterwards confidered by othersg, 
 and named the Logarithmic or Logiftic curve by Huygens, in his Di/- 
 Jertatio de Caufa Gravitatis, where he enumerates all the principal 
 properties of this curve, fhowing its analogy to logarithms. Many 
 other learned men have alfo treated of its properties; particularly Le 
 Seur and Jacquier, in their commentary on Newton’s Principia; Dr. 
 john Keill, in the elegant little traét on logarithms fubjoined to his 
 edition of Euclid’s Elements; and Francis Maferes, Efq. Curfitor 
 Baron of the Exchequer, in his ingenious treatife on ‘Trigonometry ; 
 in which books the do€trine of logarithms is copioufly and learnedly 
 treated, and their analogy to the logarithmic curve &c fully difplayed- 
 —It is indeed rather extraordinary that this curve was not fooner 
 announced to the public; fince it refults immediately from baron 
 Napier’s manner of conceiving the generation of logarithms, by only 
 {uppofing the lines which reprefent the natural numbers to be placed 
 at right angles to that on which the logarithms are taken. ‘This curve 
 greatly facilitates the conception of logarithms to the imagination, and 
 affords an almoft intuitive proof of the very important property of 
 their fluxions, or very fmall increments, to wit, that the fluxion 
 of the number is to the fluxion of the logarithm, as the number 
 is to the fubtangent; as alfo of this property, that, if three ea 
 
 e 
 
LOGARITHMS. 85 
 
 be taken very nearly equal, fo that their ratios to each other may dif- 
 fer but a little from a ratio of equality, as for example, the three 
 numbers 10000000, 10000001, 10000002, their differences will be 
 very nearly proportional to the logarithms of the ratios of thofe 
 numbers to each other: all which follows from the logarithmic arcs 
 being very little ditterent from their chords, when they are taken very 
 {mall. And the conftant fubtangent of this curve is what was after- 
 wards by Cotes called the Modulus of the fyftem of logarithms: and 
 fince, by the former of the two properties abovementioned, this fub- 
 tangent is a 4th proportional to the fluxion of the number, the fluxion 
 of the logarithm, and the number itfelf; this property atlorded oc- 
 cafion to Mr. Baron Maferes to give the following definition of the 
 modulus, which is the fame in elfeét as Cotes’s, but more clearly ex- 
 prefled, namely, that it is the limit of the magnitude of a 4th propor- 
 tional to thefe three quantities, to wit, the diflerence of any two na- 
 tural numbers that are nearly equal to each other, either of the faid 
 numbers, and the logarithm or meafure of the ratio they have ta 
 each other. Or we may define the modulus to be the natural number 
 at that part of the fy{tem of logarithms, where the fluxion of the num- 
 ber is equal tothe fluxion of the logarithm, or where the numbers. 
 and logarithms have equal differences. And hence it follows, that 
 the logarithms of equal numbers or of equal ratios, in different fyftems, 
 » are to one another as the moduli of thofe fy{tems. Moreover, the ra- 
 tio whofe meafure or logarithm is equal to the modulus, and thence 
 by Cotes called the ratio modularis, is by calculation found to be the 
 ratio of 2°718281828459 Xc to 1, or of | to *367879441171 &e3 
 the calculation of which number may be feen at full length in Mr. 
 Baron Maferes’s treatife on the Principles of Life-annuities, pa. 274 
 and 275. 
 
 The hyperbolic curve alfo afforded another fource for developing 
 and illuftrating the properties and conftruction of logarithms. For 
 the hyperbolic areas lying between the curve and one afymptote, 
 when they are bounded by ordinates parallel to the other afymptote, 
 are analogous to the logarithms of their abfcifles or parts of the 
 afymptote. And fo alfo are the hyperbolic fectors; any fector bound- 
 ed by anarc of the hyperbola and two radii, being equal to the qua- 
 drilateral fpace bounded by the fame arc, the two ordinates to either 
 afymptote from the extremities of the arc, and the part of the 
 afymptote intercepted between them. And although Napier’s loga= 
 rithms are commonly faid to be the fame as hyperbolic logarithms, it 
 - is not to be underftood that hyperbolas exhibit Napier’s logarithms 
 only, but indeed all other poflible fyftems of logarithms whatever. 
 For, like as the right-angled hyperbola, the fide of whofe fquare in- 
 {cribed at the vertex is 1, gives Napier’s logarithms; fo any other 
 fy{tem of logarithms is expreffed by the hyperbola whofe afymptotes 
 form a certain oblique angle, the fide of the rhombus infcribed at the 
 vertex of the hyperbola in this cafe alfo being ftill 1, the fame as the 
 
 fide of the {quare in the right-angled hyperbola. But the areas of the ~ 
 . {quare 
 
ad CONSTRUCTION OF 
 
 fquare and rhombus, and confequently the logarithms of any one and 
 _the fame number or ratio, differing according to the fine of the an- 
 gle of the afymptotes. And the area of the {quare or rhombus, or any 
 infcribed parallelogram, is alfo the fame thing as what was by Cotes 
 ealled the modulus of the fy{tem of logarithms; which modulus wilt 
 therefore be expreffed by the numerical meafure of the fine of the 
 angle formed by the afymptotes, to the radius i; as that is the fame 
 with the number exprefling the area of'the faid {quare or rhombus, 
 the fide being 1: which is another definition of the modulus, to be 
 added to thoie we before remarked above, in treating of the logarith- 
 mic curve. And the evident reafon of this is, that in the beginning 
 of the generation of thefe areas from the vertex of the hyperbola, the 
 nafcent incremerit of the abfciffe drawn into the altitude 1, is to the 
 increment of the area, as radius is to the fine of the angle of the or- 
 dinate and abfciflé, or of the afymptotes; and at the beginning of the 
 logarithms, the nafcent increment of the natural numbers is to the in= 
 crement of the logarithms, as | is to the modulus of the fyftem. 
 Hence we eafily difcover that the angle formed by the afymptotes of 
 the hyperbola exhibiting Briggs’s fyftem of logarithms, will be 25 
 deg. 44 min. 25+ fec. this betng the angle whofe fine is 0°4342944819 
 &c, the modulus of this fyftem. 
 
 Or indeed any one hyperbola, as has been remarked by Mr. Baron 
 Maferes, will exprefs all poffible fyftems of logarithms whatever, 
 namely, if the {quare or rhombus infcribed at the vertex, or, which is 
 the fame thing, any parallelogram infcribed between the afymptotes 
 and the curve at any other point, be expounded by the modulus of 
 the fyftem; or, which is the fame, by expounding the area, intercepted 
 between two ordinates which are to each other in the ratio of rO tol, 
 by the logarithm of that ratio in the propofed fyftem. 
 
 As to the firft remarks on the analogy between logarithms and 
 the hyperbolic fpaces; it having been fhown by Gregory St. Vincent, . 
 —ainhis Quadratura Circuli 8 Seétionum Coni, publifhed at Antwerp 
 in 1647, that if one afymptote be divided into parts in geometrical 
 progreffion, and from the points of divifion ordinates be drawn parallel 
 to the other afymptote, they will divide the {pace between the afymp= 
 tote and curve into equal portions; from hence it was fhown by 
 Merfenne, that by taking the continual furns of thofe parts, there 
 would be obtained areas in arithmetical progreflion, adapted to 
 abf{cifles in geometrical progreflion, and which therefore were ana- 
 logous to a fyftem of logarithms. And the fame analogy was re- 
 marked and illuftrated foon after by Huygens and many others, who 
 
 _ fhow how to fquare the hyperbolic fpaces by means of the loga~ 
 rithms. 
 
 oe 
 
: LOGARITHMS. 87 
 Of * Gregory’s Computation of Logarithms. 
 
 On the other hand, Mr. James Gregory, in his Vera Circuli et Hye 
 erbole Quadratura, firft printed at Patavi, dr Padua, in the year 1667, 
 wee approximated to the hyperbolic afymptotic fpaces, by means of 
 a feries of infcribed and circumfcribed polygons, from thence fhows 
 how to compute the logarithms, which are analogous to thofe areas: 
 and thus the quadrature of the hyperbolic fpaces became the fame 
 thing as the computation of the logarithms. He here alfo lays down 
 various methods to abridge the computation, with the afliftance of 
 fome properties of numbers themfelves, by which we are enabied to 
 compofe the logarithms of all prime numbers under 1000, each by one 
 multiplication, two divifions, and the extraction of the fquare root. 
 And the fame fubject is farther purfued in his Evercitationes. Geome- 
 tric, to be defcribed hereafter. | 
 
 There are alfo innumerable other geometrical figures having pro- 
 perties analogous to logarithms: fuch as the equiangular fpiral, the 
 figures of the tangents and fecants, &c; which it is not to our pur- 
 pofe to diftinguifly more particularly. 
 
 Of + Mercator’s Logarithmotechnia. 
 
 In 1668, Nicholas Mercator publifhed his Logarithmotechnia, five 
 methodus confiruendi Logarithmos nova accurata, & facilis ; in which 
 he delivers a new and ingenious method for computing the logarithms 
 on principles purely arithmetical; which being curious and very ac- 
 curately performed, I fhall here give a rather full and particular ac- 
 count of that little tra€t, as well as of the {mall {pecimen of the qua- 
 drature of curves by infinite feries, fubjoined to it; and more efpe- 
 pecially as this work gave occafion to the public communication of 
 fome of Sir Ifaac Newton’s earlieft pieces, to evince that he had not 
 borrowed them from this publication. So it appears that thefe two 
 ingenious men had, independent of each other, in fome inftances 
 fallen upon the fame things. : 
 Mercator begins this work with remarking that the word Loga- 
 vithm is compoted of the words ratio and number, being as much as to 
 fay the number of ratios ; which he obferves is quite agreeable to the 
 nature of them, for that a logarithm is nothing elfe but the number 
 of ratiuncule contained in the ratio which any number bears to unity. 
 He then makes a learned and critical differtation on the nature of 
 
 * James Gregory was born at Aberdeen in Scotland 1638, where he was edu- 
 cated. He was profeflor of mathematics in the college of St. Andrews, and 
 afterwards in that of Edinburgh. He died ofa fever in December 1675, being 
 only 36 years of age. ; 
 
 + Nicholas Mercator, a learned mathematician, and an ingenious member of 
 the Royal Society, was a native of Holftein in Germany, but {pent moft of his 
 time in England, where he died inthe year 1690, at about 50 years of age. He © 
 was the author of many other works in Geometry, Geography, Aftronomy, 
 Aftrology, &¢. Ay ; 
 
 Fatlos, 
 
85. CONSTRUCTION OF 
 
 ratios, their magnitude and meafure, conveying a clearer idea of the 
 nature of logarithms than had been given by either Napier or 
 Briggs, or any other writer except Kepler, in his work before de- 
 fcribed; although thofe other writers feem indeed to have had in their 
 own minds the fame ideas on the fubject as Kepler and Mercator, 
 but without having expreffed them fo clearly. Our author indeed 
 pretty clofely follows Keplerin his modes of thinking and expreffion, 
 and after him, in plain and exprefs terms, calls logarithms the mea- 
 fures of ratios; and, in order to the right underftanding that defini- 
 tion of them, he explains what he means by the magnitude of a ratio. 
 ‘This he does pretty fully, but not too fully, confidering the nicety and 
 fubtlety of the fubjeét of ratios; and their magnitude, with their addi- 
 tion to, and fubtraction from, each other, which have been mifconceived 
 by very learned mathematicians, who have thence been led into con- 
 confiderable miftakes. Witnefs the overfight of Gregory St. Vincent, 
 which Huygens animadverted on in the Eferacss Cyclometrie Gregerii & 
 Sanéto Vincentio, and which arofe from not underftanding, or not ad- 
 verting to, the nature of ratios, and their proportions one to another. 
 And many other fimilar miftakes might here be adduced of otheremi- 
 nent writers. From all which we muft commend the propriety of our 
 author’s attention, in fo judicioufly difcriminating between the mag- 
 
 nitude of a ratio, as of a to J, and the fraction = ) or quotient arifing 
 
 from the divifion of one term of the ratio by the other; which latter 
 method of confideration is always attended with danger of errors and 
 confufion on. the fubject; though in the 5th definition of the 6th 
 book of Euclid this quotient is accounted the quantity of the ratio 3 
 but this definition is probably not genuine, and therefore very pro- 
 perly omitted by profeflor Simfon in his edition of the Elements. 
 And in thofe ideas on the fubject of logarithms, Kepler and Merca- 
 tor have been followed by Halley, Cotes, and moft of the other emi- 
 nent writers fince that time. 
 
 Purely from the above idea of logarithms, namely, as being the 
 meatures of ratios, and as exprefling the number of ratiuneule con- 
 tained in any ratio, or into which it may be divided, the number of 
 the like equal ratiuncule contained in fome one ratio, as of 10 to I, 
 being fuppofed given, our author fhows how the logarithm or meafure 
 of any other ratio may be found. But this however only by-the-by, 
 as not being the principal method he intends to teach, as his laft and 
 beft, and which we arrive not at till near the end of the book, as we 
 fhall fee below. Having fhown then, that thefe logarithms, or 
 numbers of {mall ratios, or meafures of ratios, may be all properly re- 
 prefented by numbers, and that of 1, or the ratio of equality, the 
 Jogarithm or meafure being always 0, the logarithm of 10, or the 
 meafure of the ratio 10 to 1, is moft. conveniently reprefented by 1 
 with any number of ciphers; he then proceeds to fhow how the _ 
 meafures of all other ratios may be found from this laft fuppofition. 
 And he explains the principles by the two following roca 
 
 IFT ky 
 
LOGARITHMS. $3 
 
 Firft, to find the logarithm of 100°5*, or to find how many raft- 
 
 wncule are contained in the ratio of 100°5 to 1, the number of ra/i- 
 uncule in the decuple ratio, or ratio of 10 to 1, being 1,0000000. 
 _ ‘The given ratio 100°5 to 1, he firft divides into its parts, namely, 
 100'5 to 100, 100 to 10, and 10 to 1; the laft two of which being 
 decuples, it follows that the charaCteriftic will be 2, and it only re- 
 mains to find how many parts of the next decuple belong to the firit 
 ratio of 100°5 to 100. Now if each term of this ratio be multiplied 
 by itfelf, the products will be in the duplicate ratio of the firft terms, 
 or this laft ratio will contain a double number of parts; and if thefe 
 be multiplied by the firft terms again, the ratio of the laft products 
 will contain three times the number of parts; and fo on, the number 
 of times of the firft parts contained in the ratio of any like powers of 
 the firft terms, being always denoted by the exponent of the power. 
 If, therefore, the firft terms, 100°5 and 100, be continually multiplied 
 till the fame powers of them have to each other a ratio whofe mea 
 fure is known, as fuppofe the decuple ratio 10 to 1, whofe meafure is 
 1,0000000; then the exponent of that power fhows what multiple 
 this meafure 1,0000000, of the decuple ratio, is of the required 
 meafure of the firft ratio 100°5 to 100; and confequenily dividing 
 {,0000000 by that exponent, the quotient is the meafure of the ratio 
 100°5 to 100 fought. ‘The operation for finding this, he fets down 
 as here follows; where the feveral multiplications are all performed 
 in the contracted way, by inverting the figures of the multiplier, and 
 retaining only the firft number of decimals in each product. 
 
 * Mercator diftinguifhes his decimals from integers thus 100[5, or 100 (5. 
 
 power 
 
 N J 
 
CONSTRUCTION OF 
 
 a 
 
 eed 
 1020150 - + 
 O510201 = «& 
 
 ~ 1020158 
 
 £0403 
 
 102 
 
 51 
 {O40706 - = 
 6070401 - = 
 1083068 - = 
 $603801 - - 
 1173035 - = 
 5303711 - = 
 1376011 .-. = 
 
 PPOGTSL <5) = 
 1393400 - = 
 6043981 = = 
 
 3534935 =, © 
 * 5894853 =. = 
 
 12852116 ~ - 
 
 Since 
 the 
 
 | power | | | | 
 100°5000 - - - 1} This power being] This being again too 
 6001 - - + Iigreaterthanthe decuplemuch, inftead of the 
 1005000 of the like powerof100,|16th, draw it into the 
 
 i 5025 which muft always be 1/8th or next preceding, 
 
 7010025 - «- - g{with ciphers, refumejthus | 
 
 BI00I01 - - githerefore the 256th 9340180 = - 448 
 7010025 power, and multiply it, 6070401 - - 8 
 10100 not by itfelf, but by the) 9720329 - - 456 
 
 00 next before, viz, by| 0520201 - - A 
 
 ie the 128th, thus , OODLE OS4  o eekOO 
 
 Ato puMousy tii. (oi geo ueeQOOLOlyo = i 2 
 ~ 41 6043981: = ~—  198|!0015603. -) = 462 
 67T8783L - = 38 Which power again 
 1106731 - - 64 exceedsthe limit; there- 
 Goanten fore draw the 460th 
 
 9340130 - - 448. 
 S208TIT. a gojinto the Lit, thus 
 
 Wa ““1 9916193. - - 460 
 - 8 : parca 
 
 ti, GSN TT 4s <2 Se GB 
 ~ ) 
 
 yrs Us This ROWS? Again ga Since therefore the 
 ceeding thefame power], . ips 
 _ 39 ne 462d power of 100°5 is 
 of 100 more than 10! 
 Pee a igreater, and the 461{t 
 
 ~ 0“times, I therefore draw, Pea b 
 ts 3 4 } ft: Q h . power 1S vat Sy than the 
 64\the tame 448th, not in hiathla tor the tame 
 - 64ito the 32d, but the next mit Ch agent é 
 receding, thus ore cath canta ne - 
 
 128)P BO BTS that the ratio of 100°5 
 
 APP ito 100 is contained in 
 256) 9340130 - = 448the decuple more than 
 256| 8603801 - = 161461 times, but lefs than 
 
 51210115994 - = 464)462 times. Again, 
 
 461 4 POS 9965774 
 462 2 10015603 
 
 / re) oO : 
 fact ? ower § 9916193 and the differences 
 
 49581 2 nearly 
 49329 
 
 equal ; 
 
 therefore the proportional part which the exaG power, or 10000000, 
 exceeds the next lefs 9965774, will be eafily and accurately found by 
 the Golden Rule, thus: 
 
 The juft power = - - 10000000 
 andthenextlefs = - - 9965774 
 
 the difference - - 34226; then 
 
 As 49829 the dif. between the next lefs and greater, 
 : To 44226 the dif. between the next lefs and juft, 
 2: So 1s 10000 ; to 6868, the decimal parts; and therefore the ra- 
 tio of 100°5 to 100, is 461°6868 times contained in the decuple or 
 
 ratio of 10 to 1. 
 
 Dividing now 1,0900000, the meafure of the de- 
 
 cuple ratio, by 461°6868, the quotient 00216597 is the meafure of 
 the ratio of 100°5 to 100; which being added to 2 the meafure of 
 
 100 
 
LOGARITHMS. | 91 
 
 100 to 1, the fum 2,00216597 is the meafure of the ratio of 100°5 to 
 1, that is, the log. of 100°5 is 2,00216597. 
 
 In the fame manner he next inveftigates the log. of 99:5, and finds 
 it to be 1,99782307. 
 
 A few obfervations are then added, calculated to generalize the 
 confideration of ratios, their magnitude and their affeQiions. It is here 
 remarked, that he confiders the magnitude of the ratio between two 
 quantities as the fame, whether the antecedent be the greater or the 
 lefs of the two terms: fo, the magnitude of the ratio of 8 to 5,is the 
 fame as of 5 to 8; that is, by the magnitude of the ratio of either 
 to the other, is meant the number of ratiuncule between them, which 
 will evidently be the fame, whether the greater or lefs term be the 
 antecedent. And, he farther remarks that, of different ratios, when 
 we divide the greater term of each ratio by the lefs, that ratio is of 
 the greater mafs or magnitude, which produces the greater quotient, 
 et vice verfa; although thofe quotients are not proportional to the 
 mafles or magnitudes of the ratios. But when he confiders the ratio 
 of a greater term to a lefs, or of a lefs to a greater, that is to fay, the 
 ratio of greater or lefs inequality, as abftracted from the magnitude of 
 the ratio, he diftinguifhes it by the word affection, as much as to fay, 
 greater or lefs affection, fomething in the manner of pofitive and ne- 
 gative quantities, or fuch as are affected with the figns + and—..... 
 The remainder of this work he delivers in feveral propofitions, as 
 follows. — | 
 
 Prop. 1. In fubtra€ting from each other, two quantities of the fame 
 affe€tion, to wit, both pofitive, or both negative ; if the remainder 
 be of the fame affection with the two given, then is the quantity fub- 
 tracted the lefs of the two, or exprefled by the lefs number; but if 
 the contrary, it is the greater. | 
 
 . f : a ath \a+2b 
 
 Prop. 2. In any continued ratios, as EYL PT EET &ce, (by 
 _ which is meant the ratios of @ to a+, a+b to a+2b, a+2b to 
 ‘a+b, &c,) of equidifferent terms, the antecedent of each ratio being 
 equal to the confequent of the next preceding one, and proceeding 
 from lefs terms to greater; the meafure of each ratio will be expreffed 
 by a greater quantity than that of the next following; and the fame 
 through all their orders of differences, namely, the 1ft, 2d, 3d, &c, 
 differences; but the contrary, when the terms of the ratios decreafe 
 from greater to lefs. 
 
 Prop. 3. Inany continued ratios of equidifferent terms, if the 1{t or 
 leaft be a, the difference between the 1ft and 2d 4, and ¢, d,e, &c, 
 ‘the refpeCtive firft term of their 2d, 3d, 4th, &c, differences ; then 
 fhall*the feveral quantities themfelves be as in the annexed f{cheme ; 
 
 a Ses roe iQ where 
 
 . 
 
$2 CONSTRUCTION OF 
 
 where each term is compofed lift term .- a4 
 
 of the firft term, together with 2d term - a+ 35 
 
 as many of the differences as od term - a+ 2b+4 ¢ 
 
 it is diftant from the firft term, [4thterm - a+ 3b+ 24+ d , 
 and to thofe differences join- 5thterm - a+46+ 6¢+ 4d+e 
 ing, for coefficients, the num-| &c. &c. 
 
 bers in the floping or oblique —,-~ 
 lines contained in the annexed // 
 table of figurate numbers, in | 
 the fame manner, he obferves, 
 
 as the fame figurate numbers 
 
 complete the powers raifed 
 
 ' from a binomial root, as had 
 
 long before been taught by 
 
 others. He alfo remarks, that ; 
 this rule not only gives any 
 one-term, but alfo the fum of 
 any number of fuccefliveterms | 
 from the beginning, making 
 the 2d coefficient the firft, the 
 ¢d the 2d, and fo on; thus, 
 the fum of the firft 5 terms is 
 5a + 106+ 10e+ 5d+ e. 
 
 In the 4th prop. itisfhown, |Ift term a 
 that if the termsdecreafe, pro-|2d term -a—4 
 ceeding from the greater to|jd term a@—2b +c 
 the leis, the fame theorems |4thterm a—3)+ 3c—d 
 hold good, by only changing |5thterm a—4b+ 6c —4d+e 
 the fign of every other term,|. &c. sate eso 
 as in the margin. 
 
 Prop, 6 and 7 treat of the approximate multiplication and divifion 
 of ratios, or, which is the fame thing, the finding nearly any powers 
 or any roots of a given fraction, in an eafy manner. The theorem 
 for railing any power, when reduced to a fimpler form, is this, 
 
 m eS 
 | the m power of, or, Gee yes ue eo nearly, where sis=a +, bs 
 andd=a ~4A, the fum and difference of the two numbers, and 
 
 é . 4 &:. 
 the upper or under figns taking place according chad Gd ob ssi Mian 
 
 improper fraction, that is, according as a is lefs or greater than b. 
 
 = a AGT Bre 
 And the theorem for extrafting the mth root of 30 gy ig 
 , : » 
 By. ms rity tk ‘ ¢ 
 (=e ve nearly; which latter rule is alfo the fame as the for- 
 b msid 
 
 mer, as will be evident by fubftituting -. inftead of m in the firit 
 
 theorem, 
 
LOGARITHMS. 93 
 
 m nsxmd 
 theorem. So that univerfally (Sr is = ickingy nearly. Thefe theo- 
 
 rems Ritter are nearly true, iat in fome certain cafes, namely 
 
 when s and — do not differ greatly from unity. Andin the 7th 
 n 
 
 prop. the author fhows how to find nearly the error of the theo- 
 rems. 
 
 In the 8th prop. it is fhown, that the meafures of ratios of equidif- 
 ferent terms, are nearly reciprocally as the arithmetical means be- 
 
 1 Gri aS 2 
 tween the terms of each ratio. So of the ratios RSE Nae: ae 
 3: jz 
 the mean between the terms of the firft ratio is 17, of the 2d 34, of 
 1 I 
 
 the 3d 51, and the meafure of the ratios are nearly as ee SOseerm 
 
 From this property he proceeds, inthe 9th prop. to find the mea- 
 
 : 9U*5 : : : 
 fure of any ratio lefs than 1008? which has an equal difference (1) 
 5 
 
 of terms. In the two examples, mentioned near the beginning, our 
 
 99/5 
 100° to be 
 
 author found the wen: or meafure of the ratio, of 
 
 00 
 
 £1769;%, and that of - Taaenen be 2165943 therefore the fum 43429 
 tS beh 99°5 100 99° 
 
 w , or — Seid -; or the logarithm of sd 
 
 100°5’ 100 100°5 1005 
 
 is nearer 43430, as found by other more accurate computations.—~ 
 
 isthe logarithm of - 
 
 the fame difference of terms 
 
 (1) with the formers it will Bes by prop. 8, as 100°5 (the mean 
 between 101 and 100) : 100 (the mee between 99°5 and 100.5) 
 
 OO 
 > : 43430.: 43213 the logarithm of or the difference between 
 
 the logarithms of 100 and 101. But the log. of 100 is 2; therefore 
 the logarithm of 101 is 2,0043213,—-Again, to find the loga- 
 
 rithm of 102, we muft firft find the logarithm of root the mean be- 
 
 tween its terms being 101°5, therefore as 101°5 : 100.: : 43430 
 
 101 
 42788 the logarithm of Lg ON the difference of the logarithms of 
 
 301 and 102. But the logarithm of 101 was found above to be 
 2,004321%3 3 therefore the logarithm of 102 is 2,0086001.—So that, 
 dividing continually 868596 (the double of 434298 the logarithm of 
 99°5 199 
 100°5 OF 901 
 then add 2 to the 1{t quotient, to the fum add the 2d quotient, and 
 fo on, adding always the next quotient to the laft fum, the feveral 
 fums will be the refpective logarithms of the numbers in this ferics 
 101, 102, 103, 104, &c. 
 
 The 
 
 ~~) by each number of the ferigs 201, 203, 205, 207, &c, 
 
94 - CONSTRUCTION OF 
 
 The next, or prop. 10, fhows that, of two pair of continued ratios 
 whofe terms have equal differences, the difference of the meafures of 
 the firft two ratios, is to the difference of the meafures of the other 
 two, as the fquare of the common term in the two latter, 1s to that 
 in the former, nearly. Thus, in the four ratios 
 
 b a+3b a+4b ; nee i 
 Oy ee St ot eet ie theatare eh eee (the difference 
 
 ath? a+ub a+4b° a+5b3 (a +2) 
 
 of the firft two, or the quotient of the two fractions) : is to the. 
 aa+8ab+15bb . 
 
 meafure of Set ‘tt fo (a+4d)* zis to (a+5)*, nearly. 
 
 In prop. 11, the author fhows, that fimilar properties take place 
 among two fets of ratios confifting each of 3 or 4 &c, continued 
 numbers. 
 
 Prop. 12 fhows that, ofthe powers of numbers in arithmetical 
 progreffion, the orders of diflerences which become equal, are the 2d 
 differences in the {quares, the 3d differences in the cubes, the 4th 
 differences in- the 4th powers, &c. And from hence it is fhown, 
 how to conf{truct all thofe powers by the continual addition of their 
 differences ; as had bees long before more fully explained by Briggs. 
 
 In the next, or 13th prop. our author explains his compendious 
 method of raifing the tables of logarithms, fhowing how to conftruct 
 the logarithms by addition only, from the properties contained in the 
 Sth, 9th, and 12th propofitions. For this purpofe, he makes ufe of 
 
 the quantity samp which by divifion he refolves into this infinite fe- 
 
 a ac aes ac 
 
 ties eit y He ai 3s + Ga &c (ia infin.). Putting thena = 100, the 
 
 : : preabie 5” ba 
 arithmetical mean between the terms of the ratio eT es 100000, 
 
 and ¢ fucceflively equal to 0°5, 1°5, 2°5, &c; that fo b—c may be re- 
 fpectively equal to 99999+5, 94998°5, 99997°5, &c, the correfponding 
 99999... 99998 99907 
 
 ig re a sae 9 [eres 
 100000 99999 99998 4 
 
 means between the terms of the ratios 
 
 a 
 it is evident that Pe will be the quotient of the 2d term divided by 
 
 the 1ft, in the proportions mentioned in the 8th and 9th propofitions; 
 aud when all of thefe quotients, are found, it rernains then only to 
 multiply them by the conftant 3d term 43429, orrather 43429°8, of 
 the proportion, to produce the logarithms of. the ratios 
 9999 99998 99997 .,, 1000 , iy 
 0000" 99999 99998" &e, till 7556, 3 then adding thefe conti- 
 nually to 4, the logarithm of 10000, the leaft number, or fubtracting 
 them from 5, the logarithm of the higheft term 100000, there will. 
 refult the logarithms of all the abfolute numbers from 10000 to 
 100000. Now when ¢ = 0°5; then , 
 
 ‘ a 
 
 d — 
 
LOGARITHMS. 95 
 
 * 
 
 , 3 : 
 os ‘001, ae “000000005, ~ = -000000000000025, o = -000000000090000000128, 
 &c. ; therefore = ie — +o p Bes: “> ~ &, is == +001000005000025000125, 
 
 In like mannerr if c= 15, then = will be == -001000015000225003375:; 
 
 ——€ 
 
 and if e == 2-5, then —— willbe = -0010000250006250156255 
 
 b—c 
 ac. But inftead of aiteratitie all the Values of -— po in the ufual 
 
 way of raifing the powers, he directs how they may be found by ad- 
 dition only, as in the tals propofition. Having thus 
 
 found all the values of se , the author then fhows, | 1 , 4342: 
 that they may be drawn into the conftant logarithm 
 43429 by addition only, by the help of the annexed 
 table of the firft 9 products of it. 
 
 The author then diftinguifhes which of the loga- 
 rithms it may be proper to find in this way, and which 
 from their component parts, Of thefe, the logarithms 
 of all even numbers need not be thus computed, being 
 compofed from the number 23; which cuts off one-half 
 of the numbers: neither are thofe numbers to he computed which 
 end in 5, becaufe 5 is one of their factors ; thefe laft are - of the 
 numbers; and the two together } -++ ; make 4 of the whole: and 
 of the other 2, the Ey of them, or ¥;, of the whole, are compofed.of - 
 3; and hence 4 + 4%, or 7 of the iabeys, are made up of fuch as 
 - are compofed of ’, 3,and 5. As to the other numbers which may 
 be compofed of 7, of 11, &c; he recommends to find zheir logarithms 
 in the general way, the fame as if they were incompolites, as it is not 
 worth while to feparate them in fo eafy a mode of calculation. So 
 that, of the 90 chiliads of numbers, .from 10000 to 100000, only 24 _ 
 chiliads are to be computed. Neither indeed are all of thefe.to be 
 
 1 
 
 pith i 
 
 A i a 
 calculated from the foregoing feries for ~— -—, but only a few of them 
 
 in that way, and the reft by the proportion in the sth propofition. 
 Thus, having computed the logayittinas of 10003 and 10013, omit- 
 ting 10023, as being divifible by 3, eftimate the logarithms of 10033 
 and 10043, which are the 30th numbers from 10003 and 410013; 
 and.again omitting 10053, a multiple of 3, find the logaxithins of 
 10063 and 10073. Then by prop. 4, fay, 
 
 As 10048, the arithmetical mean between 10033 and 10063, 
 
 to 10018, the arithmetical mean between 10003 and 10033, 
 
 fo 13006, the difference between the logarithms of 10003 and 100335 
 to 12967, the difference bet ween the logarithms of 10033 and 10063. 
 
 | 10048 “12967 
 That is, 1ft - - As} 10078 > ; 10018 :; 13006: 4 &c. 
 10108 ‘lta 
 
$6 CONSTRUCTION OF 
 
 (10058) ~ 12953 
 Again, As2 10088 > : 10028 : : 12993: 4 &c. 
 
 10118 | 
 10068 12940 
 And sdly, As} 10008 | sok OOBS ¢) REDS TOME ee CCS 
 &c. 
 
 And with this our author concludes his compendium for con- 
 ftructing the tables of logarithms. 
 
 He afterwards fhows fome applications and relations of the doc- 
 trine of logarithms to geometrical figures: in order to which, in_ 
 prop. 14, he proves algebraically that, in the right- 
 angled hyperbola, if from the vertex, and from any 
 other point, there be drawn BI, FH perpendicular 
 to the afymptote AH, or parallel to the other afympe 
 tote; then will AH: AI:: BI: FH. And, . 
 
 In prop.15, if AL=BI=1, andHI =a; then will 
 
 ‘ER 
 
 =1—a+a*—a* +a*—a> &c, inine 
 l+a 
 finitum, by a continued algebraic divifion, the procefs of which he 
 defcribes, {ftep by ftep, as athing that was new or uncommon. But 
 that method of divifion had been taught before, by Dr. Wallis, in 
 his Opus Arithmeticum. | 
 
 Prop. 16 is this: Any given number being fuppofed to be divided 
 into innumerable fmall equal parts, it is required to affign the fum 
 of any powers of the continual fums of thofe innumerable parts. 
 For which purpofe he lays down this rule; if the next higher power 
 of the given number, above that power whofe fum is fought, be 
 divided by its exponent, the quotient will be the fum of the powers 
 fought. ‘That is, if Nbe the given number, and a one of its innu- 
 merable equal parts, then will 
 
 a” -+(2a)" + (3a)" + (4a)? &c.....N* be = noe which theorem 
 he demonftrates by a method of indu€tion. And this, it is evident, 
 is the finding the {um of any powers of an infinite number of arith- 
 _ meticals, of which the greateft term is a given quantity, and the leaft 
 indefinitely fmall. It is alfo remarkable, that the above expreflion is 
 fimilar to the rule for finding the fluent to the given fluxion of a 
 power, as afterwards taught by Sir I. Newton. | 
 
 Mercator then applies this rule, in prop. 17, to the quadrature of 
 the hyperbola. ‘Thus, putting AI=1, conceive the afymptote to be 
 divided from J into innumerable equal parts, namely Ip=pq = qr 
 =a; then, by the 14th and 15th, 
 
 —) ena 2 3 
 ps = 1 eae fs, x cet But the area Blru is =the fum 
 
 =li— z7— ga3 kee 
 qt - 2a + 4a 8a" &c ps + qt tru, whichis = 
 
 rus 1l~—3a+ 9a? —27a3 &c 
 3— 64 +14a?— 36a3 &c, that is, equal to the number of terms 
 contained in the line Ir, minus the fum of thofe terms, plus the 
 fum 
 
- LOGARITHMS: 97 
 
 fum of the fquares of the fame, minus the fim of their cubes, plus 
 the fum of the 4th powers, &c. Putting now IA=1, as before, and 
 Ip=o-l the number of terms, to find the area Blips ; by prop. 16 the 
 
 a) seen 3# ‘ ; ; “12 ; 
 {um of the terms will be “y =-005, the fum of their fquares = 
 
 *000333333, the fum of their cubes :000025, the fum of the 4th 
 powers ‘000002, the fum of the 5th powers ‘000000166, the fum 
 of the 6th powers "000000014, &c. Therefore the area BIps is = 
 . *L — 005 + °000333333 — 000025 + *O00002 — ‘000000165 + 
 *000000014 &e = *100335347 — °005025166 = *095310i81 &c, 
 
 Again, putting Iq =°21 the number of terms, he finds in like 
 tanner the area BIqt = -2i1 —*02205 + °0030%7 —-000456202 + 
 ~ *00008 1682 — *000014294 + *000002572 — *000600:72 + 
 “000000088 &c = -213171345 — ‘022550954 = *190620361 &c. 
 
 He then adds, hence it appears that, as the ratio of Af to Ap, or 
 1 to 1°!, is half or fubduplicate of the ratio of Ai to Aq, or i to 
 1°21, fo the area BIps is here found to be half of the area Biqt. 
 Thefe areas he computes to 44 places of figures, and finds then: itill 
 in the ratio of 2 to 1. 
 
 The foregoing do€trine amounts to this, that if the reCtangle BI 
 x Ir, which in this cafe is -expreffed by Ir only, be put = 4, AL 
 being = 1 as before; then the ar¢a Blru, or the hyperbolic logarithm 
 of 1 + 4, or of the ratio of 1 to 1 + A, will be equal to the infinite 
 feries 1— $A? + $43 — TA* 4-44 &cs; and which therefore may 
 be confidered as Mercator’s quadrature of the hyperbola, or his gene- 
 ral expreffion of an hyperbolic logarithm in an infinite feries. And 
 this method was further improved by Dr. Wallis in the Philot. Tranf. 
 for the year 1668. eae . . | 
 
 In prop. 18 Mercator compares the hyperbolic areo/e with the ra- 
 tiuncule of equidifferent numbers, and obferves that, 
 the areola BIps is the meafure of the ratiuntula of AI to Ap, 
 the areola {pqt is the meafure of the ratiuncula of Ap te Aq, 
 the areola tqru is the meafure of the ratiuncula of Aq to Ar, &c. 
 
 Finally, in the 19th prop. he fhows how the fums of logarithms 
 may be taken, after the manner of the fums of the areole. And from 
 hence infers as a corollary, how the continual produ& of any given 
 numbers in arithmetical progreflion may be obtained; for the fum of 
 the logarithms is the logarithm of the continual product. He then 
 remarks, that from the premifes it appears, ininwhat manner Mer- 
 fennus’s problem may be refolved, if not geometrically, at ieaft in 
 figures toany number of places. And thus clofes this ingenious tract. 
 
 In the Philof. Tranf. for 1668 are alfo given fome further illuithias 
 tions of this work, by the author himfeli.. And in various places aifo 
 in a fimilar manner are logarithms and hyperbolic areas treated of by 
 Lord Brouncker, Dr. Wallis, Sir I. Newton, and many other learn- 
 ed perfons. 
 
 Of Gregory's Exercitationes Geometrice. 
 In the fame year 1608 came out Mr. James Gregory’s Exercita- 
 tiones Geometric, in which aie contdined the following pieces): 
 
 1, Appen- 
 
98 CONSTRUCTION OF 
 
 1, Appendicula ad veram circuli et hyperbole quadraturam ? 
 
 2, N. Mercatoris quadratura hyperbole geometricé demonftrata z, 
 
 3, Analogia inter lineam meridianam planifpherit nautici et tans 
 gentes artificiales geometricé demonftrata; feu quod fecantium natu- 
 ralium additio efheiat tangentes artificiales: 
 
 4, Item, quot tangentium naturalium additio efficiat fecantes. arti- 
 ficiales: 
 
 5, Quadratura conchoidis : 
 
 €, Quadratura ciffoidis: & 
 
 l, Methodus facilis et accurata componendi fecantes ect tangentes 
 artificiales. 
 
 The firft of thefe pieces, or the 4ppendicula, contains fome farther — 
 extenfion and illuftration of his Vera circuli et hyperbole quadratura, 
 occafioned by the animadverfions made on that work by the celebrated 
 mathematician and philofopher Huygens. 
 
 In the, 2d is demontftrated geometrically, the quadrature of the hy- 
 perbola; by which he finds a feries fimilar to Mercator’s for the logae 
 rithm, or the hyperbolic {pace beyond the firft ordinate (BI, fig. pa. 
 96.) In like manner he finds another feries for the fpace at an equal 
 diftance within that ordinate. ‘Thefe two feries having all their terms 
 alike, but all the figns of the one plus, and thofe of the other alter- 
 nately plus and minus, by adding the two together, every other term 
 is. cancelled, and the double of the reft denotes the fum of both fpaces. 
 Gregory then applies thefe properties to the logarithms; the conclu 
 fion from all which may be thus briefly exprefled : : 
 
 1+f4 
 fince A— $4’ 444i —4A* &c. = the log. of A 
 
 and 4 + 14% + 143 4+ 1A* &e = the log of > 
 
 therefore 24 + 2.4?4+2 45 +347 &c = the log. of £ of 
 or of the ratio of 1 —.4 tol1+ 4. Which may be accounted Gree 
 gory’s method of making logarithms. 
 
 ‘The remainder of this little volume is chiefly employed about the 
 nautical meridian, and the logarithmic tangents and fecants. It 
 does not appear by whom, nor by what accident, was difcovered the 
 analogy between a fcale of logarithmic tangents and Wright’s pro« 
 traction of the nautical, meridian line; which confifted of the fums 
 of the fecants.. It appears however to have been firft publifhed, 
 and introduced into the practice of navigation, by Henry Bond, 
 who mentions this property in an edition of Norwood’s Epitome of 
 Navigation, printed about 16453 and he again treats of it more fully 
 in an edition of Gunter’s works, printed in 1653, where he teaches, 
 from this property, how to refolve all the cafes of Mercator’s failing 
 by the logarithmic tangents, independent of the table of meri- 
 dional parts. This analogy had only been found to be nearly true 
 by trials, but not demonftrated to be a mathematical property. 
 Such demonftration feems to have been firft difcovered by Nicholas 
 
 Mercator, who, defirous ef making the moft advantage of this 
 nti bs, and 
 
LOGARITHMS. 99 
 
 and another concealed invention of his in navigation, by a paper 
 in the Philof. Tranf. for June 4, 1666, invites the public to enter 
 into a wager with him, on his ability to prove the truth or falfehood 
 of the fuppofed analogy. But this mercenary propofal it feems was 
 not taken up by any one, and Mercator referved his demonitration. 
 The propofal however excited the attention of mathematicians to the 
 fubject itfelf, and a demonftration was not long wanting. The firft 
 was publifhed about two years after by Gregory, in the track now 
 under confideration, and from thence and other fimilar properties, 
 here demonftrated, he fhows, in the laft article, how the tables of 
 logarithmic tangents and fecants may eafily be computed, from the 
 natural tangents and fecants. ‘The fubftance of which is as follows; 
 Let AI be the arc of a quadrant 4, 
 
 extended in aright line, and let 
 the figure AHI be compofed of 
 the natural tangents of every arc Q 
 from the point A, erected per- 
 pendicularto AI at their re{pect- 
 we points: let AP, PO, ON, 
 NM, &c, be the very fmall equal 
 parts into which the quadrant is 
 divided, namely, each 4, or +a0 
 of a degree: draw PB, OC,ND; p< M 
 ME, &c, perpendicular to AI. 
 
 Then it is manifeft, from what had been demonftrated, that the’ fi- 
 gures ABP, ACO, &c, are the artificial fecants of the ares AP, 
 AO, &c, putting o for the artificial radius. It is alfo manifeft, that the 
 rectangles BO, CN, DM, &c, willbe found from the multiplication 
 of the fmall part AP of the quadrant by each natural tangent. But, 
 he proceeds, there is a little more difficulty in meafuring the figures 
 ABP, BCX, CDV, &c; for if the firft differences of the tangents 
 be equal, AB; BC, CD, &c, will not differ from right lines, and 
 then the figures ABP, BCX, CDV, &c, will be right-angled trian- 
 gles, and therefore any one, as HQG, will be = 3 QH x QG: but 
 if the fecond differences be equal, the faid figures will be portions of 
 trilineal quadratrices ; for example HQG will bea portion of a trilineal 
 quadratrix, whofe axis is parallel to QH; and each of the laft diiler- 
 ences being Z, it will be QHG = + QH x QG—,42Z x QG: and 
 if the third differences be equal, the faid figures will be portions of 
 trilineal cubices, and then thall QHG be =i QH x QG— Vv (7% 
 QHxZxQG’*— 775, Z* x QG’): when the 4th differences are 
 equal, the faid figures are portions of trilineal quadrato-quadra~ 
 trices, and the 4th diflerences are equal to 24 times the 4th power 
 of QG divided by the cube of the latus reCtum; alfo when the 
 5th differences are equal the faid figures are portions of trilineal furfo- 
 lids, and the 5th differences are equal to 120 times the furfolid of QG 
 divided by the 4th power of the latus retum; and fo on in infinitum. 
 What has been here faid of the compofition of artificial fecants 
 from the natural tangents, it is remarked, may in like manner 
 
100 CONSTRUCTION OF 
 
 be underftood of the compofition of artificial tangents, from the 
 natural {ecants, according to what was before demonftrated. It is 
 alfo obferved that the artificial tangents and fecants are com- 
 puted, as above, on the fuppofition that 0 is the logarithm of 1, 
 and 10 OOCCOOUU0" the radius, and 23025505 2994145624017870 
 the logarithm of 1 5 but that they may be more eatily computed, 
 namely, by addition only, by putting 3 of adegree = QG=AP=1, 
 and the jegarithm of 10==°7915704467897819; for by this means 
 30H xQis is = {QH=QHG, and {QHxQG —+,Z x QG= 
 Z0H — +5 PN Qki 9 alfo 
 
 3QH x QG — v(7,QH x Z x QG*— yh? x QG*= 
 4QH — VW(-4QH x Z—+434,Z7) = QUG: and finally, by one 
 divifion only are found the artificial targents and fecants to 
 10000000. 00C0000, the logarithm of 10, putting ftill 1 for radius, 
 which are the differences of the artificial tangents and fecants, in the 
 table from that artificial radius 5 and to make the operations eafier in 
 multiplying by the number 7°157044678s781%, or logarithm of 10, 
 a table is fet down of its products by the firft 9 figures. But if AP 
 or QG be = zie of a degree, the artificial tangents and fecants will 
 anfwer to 13192840779829703 as the logarithm of 10, the firft 9 
 multiples of which are alfo placed in the table. But to reprefent the 
 numbers by the artificial radius, rather than by the logarithm of 10, 
 the author directs to add ciphers, &c.—And io much tor Gregory’s 
 Etxercitationes Geometrica. 
 
 The fame analogy between the logarithmic tangents ard the me- 
 ridian line, as alfo other fimilar properties, were afterwards more 
 elegantly demonftrated by Dr. Hailey in the Philof. Tranf. for Feb. 
 1696, and various methods given for computing the fame, by exa- 
 mining the nature of the fpirals into which the rhumbs are tranf- 
 formed in the ftereographical projeétion of the {phere on the plane 
 of the equator: the doctrine of which was rendered {till more eafy 
 and elegant by the ingenious Mr. Cotes, in his Legometria, firft 
 printed in the Philof. Tranf. for 1714, and afterwards in the collec- 
 tion of his works publifhedin 1732 by his coufin Dr. Robert Smith, 
 who fucceeded him in the Plumtan profeflorfhip of philofophy in the 
 Univerfity of Cambridge. | 
 
 The learned Dr. lfaac Barrow alfo in his Leéfiones Geometrica, 
 Lc, XI. Append. firft publifhed in 1672, delivers a fimilar property, 
 namely, that the fum of all the fecants of any arc is analogous to 
 the logarithm of the ratio of r +s tor —s, or radius plus fine to 
 radius minus fine; or, which isthe fame thing, that the meridional 
 parts anfwering to any degree of latitude are as the logarithms of 
 the ratios of the verfed fines of the diftances from the two poles. 
 
 Mr. Gregory’s method for making logarithms was farther exem- 
 
 lified in numbers, in a {mall traét on this fubje€t, printed in 1688, 
 i one Euclid Speidell, a fimple and illiterate perfon, and fon of 
 John Speidell, before mentioned among the firft writers on loga- 
 rithms, | | ! | is * ere 
 
 Gregorp 
 
LOGARITHMS. ° 101 
 
 Gregory alfoinvented many other infinite feries, and among them 
 thefe here following, viz. a being anarc, ¢ its tangent, and s the fe- 
 cani, to the radius r; then is 
 
 , {3 ¢5 ¢7 #9? & 
 pot fe 4 YO, 
 a ; or* oh oe Y hoe + or* ‘ 
 s a3 2a5 1747 62a° bec 
 ea ah or*. i5r* 1 315r° F. 2835r* ; 
 7 ® 4 ,5 7 
 0 5a Ola 2774 ee 
 
 s=7t oO, + aap + z20rs + “GoGar? 
 And if 7 and ~ denote the artificial or logarithmic tangent and fecant 
 
 of the fame arc a, the. whole quadrant being g, and e= 2a—gs 
 then is 
 
 Ses 73 7 6177 277979 3 
 ee ~ = &e, 
 6; r 24r* = 5040r° + 72.576r* 
 e3 e 6le7 27769 
 Toe 4+ —— 2 — 4 — 4 Xe. 
 t 62 + gars + Gosor® + 72576r% 
 ar at a® 17a® 622° 
 
 = ort jot 45R + o520r7 + 2835079 
 Alfo if / denote the artificial fecant of 45°, and / + / the artificial fe« 
 cant of any arc a, the artificial radius being 0; then is 
 azhe: sat sy ster (& 7l+ 1415 45215 
 ai Oke T f 3r2 sis Br + 3r4-  45r5 or 
 The inveftigation of all which feries may be feen at pa. 298 eft /eg. 
 Vol i. Dr. Horfley’s learned and elegant commentary on Sir I, New- 
 ton’s works, as they were given in the Commercium LEpiftolicum 
 No xx. without demonftration, and where the number 2% is alfo 
 wanting in the denominator of the firft term of the feries exprefling 
 the value of o. | 
 Such then were the ways in which Mercator and Gregory ap- 
 plied thefe their very fimple feries A— 7A” + 3A? — {A+ &c, and 
 A + TA* +3A* + 7A% &c, for the purpofe of computing logarithms. 
 But they might, as I apprehend, have applied them to this purpofe 
 in a f{horter and more direct manner, by computing, by their means, 
 only a few !ogarithims of {mall ratios, in which the terms of the feries 
 would have decreafed by the powers of 10 or fome greater number, 
 the numerators of all the terms being unity, and their denominators 
 the powers of 10 or fome greater number, and then employing thefe 
 few logarithms, fo computed, to the finding of the logarithms of other 
 and greater ratios, by the eafy operations of mere addition and fub- 
 traction. This night have been done for the logarithms of the ratios 
 of the firft ten numbers, 2, 3, 4, 5, 6) 7, 8, 9, 10, and 11, to 1, 
 in the following manner, communicated by Mr. Baron Maferes. 
 In the firft place, the logarithm of the ratio of !0 to 9, or of 1 to +% 
 or of 1 to 1 — x5, is equal to the feries 
 J 1 1 1 1 
 Tx 10 + 2x100 + 3xic00 + 4x 10000 F Sxi00000  &* 
 In like manner are eafily found the logarithms of the ratios of 
 11 to 
 
102 CONSTRUCTION OF 
 
 11 to 10; and then, by the fame feries, thofe of 121 to 120, and of 
 81 to 80, and of 2401 to 2400; in all which cafes the feries would 
 coriverge flill fafter than in the firft two cafes. We may then pro- 
 ceed by mere addition and fubtraCtion of logarithms, as follows : 
 
 Log. 43 = .L. 1° + L. S33L. ¥e di fe 2 furs = L.r5 —Ls 
 P27 eos — — 
 - I as key ou + Tt she! 2 44 a hee {, 9 nt 5 me ri 
 nf : Bet Ris L. ae Lies : et Sots: bey rs ti of ey \¥ 5 
 ey BR oe <G O! he ar Se Oa Ee meme alee see Feo TT. Mex 
 
 Having thus got the logarithm of the ratio of 2 to 1, or, in com- 
 mon language, the logarithm of 9, the logarithms of all kinds of 
 even numbers may be derived from thofe of the odd numbers, which 
 are their coefficients, with 2 or its powers, We may then proceed 
 as follows ; 
 
 L. 4= 21.2, L:- 100 = 2L:10, L.2401 =L.3403 + L. 2400, 
 A014" 40.4 )0. os 6h, Ja.) Fase LAO 
 
 LO L2 +) 1.4, G5 $24 = Le 1.85 00L. ib, 
 ba 410, L.2400 = L.100 +L.24,;L. 6=L.2+4 L.3. 
 
 ‘Thus we have got the logarithms of 2, 3, 4, 5, 6,7, 8,9, 10, and 11, 
 And tbis is, upon the whole, perhaps the beft method of computing 
 logarithms that can be taken. ‘There have been indeed fome methods 
 difcovered by Dr. Halley, and other mathematicians, for computing 
 the logarithms of the ratios of prime numbers to the next adjacent 
 even numbers, which are ftill fhorter than the application of the 
 foregoing feries. But thofe methods are lefs fimple and eéafy to un- 
 derftand and apply, than thefe feries; and the computation of loga- 
 rithms by thefe feries, when the terms of them decreafe by the 
 powers of 10, or of fome greater number, is fo very fhort and eafy 
 _ (as we have feen in the foregoing computations of the logarithms of 
 the ratios of 10 to 9, 11 to 10, $1 to 80, 121 to 120, &c,) that it is 
 not worth while to feek for any fhorter methods of computing them, 
 And this method of computing logarithms is very nearly the fame 
 with that of Sir Ifaac Newton, in his fecond letter to Mr. Olden- 
 burg, dated O€tober i676, as will be feen in the following article. 
 
 Of Sir Ifaac Newton's Methods. 
 
 The excellent Sir I. Newton greatly improved the quadrature of 
 the hyperbolical-afymptotic fpaces by infinite feries, derived from the 
 general quadrature of curves by his method of fluxions ; or rather 
 indeed he invented that method himfelf, and the conftruction of lo- 
 garithms derived from it, in the year 1665 or 1666, before the pub-~ 
 lication of either Mercator’s or Gregory’s books, as appears by his 
 Jetter to Mr. Oldenburg, dated OG. 24, 1676, printed in pa. 634 e 
 eq. vol. 3, of Wallis’s works, and elfewhere. ‘Uhe 
 guadrature of the hyperbola, thence tranflated, is to 
 this effect. Let dD be an hyperbola, whofe cen- 
 tre is C, vertex F, and interpofed fquare CAFE 
 = 1. In CA take AB and Ab oneach fide = 3 
 ot O'l : And, erecting the perpendiculars BD, bd; 
 half the fum of the fpaces AD and Ad will be 
 
LOGARITHMS. 103 
 
 OB ON gp CO at SENN tec 
 atid ‘the half dig. — 20! uy 00001 4 _— 4. 0:00000001 He. 
 2 4 8 
 
 Which reduced will ftand thus, | 
 1°0000000000000,0'0050000000000 The fum of thefe 0'1053605156577 is Ad, 
 
 3333333333 250000000 and _ the differ. 0°0953101798043 is AD, 
 20000000 1666666 In like manner, putting ABand Ab 
 142857 12500 each = 0'2, there is obiained 
 11il 100 Ad = 0:2231435513142, and 
 9 1 AD =0'18232155679309. 
 
 0°1003353477310,0°0050251079207 
 
 Having thus the hyperbolic logarithms of the four decimal num- 
 bers O'S, 0°9, 1°1, and 1°25 and fince a x = 2, and 0°8 and 
 0-9 are lefs than unity; adding their logarithms to double the loga- 
 rithm of 1°2, we have 0°6931471505597, the hyperbolic logarithm of 
 2. To the triple of this adding the logarithm of 0-8, becaufe 
 ZXxXx2éxK2 - . 
 Piemonte: 10, we have 2°3025850929933, the logarithm of 10. 
 Hence by one addition are found the logarithms of 9 and 11: And 
 thus the logarithms of all thefe prime numbers, 2, 3, 5, 11, are pres 
 pared. Moreover, by only deprefling the numbers, above computed, 
 lower in the decimal places, and adding, are obtained the logarithms 
 of the decimals 0°98, 0°99, 1:01, 1°02; as alfo of thefe 0:998, 0-999. 
 1-001, 1:002. And hence, by addition and fubtraction, will arife 
 the logarithms of the primes 7, 13, 11, 37, &c. All which logarithms 
 being divided by the above logarithm of 10, give the common loga- 
 rithms to be inferted in the table. , 
 
 And again, a few pages farther on, in the fame letter, he refumes 
 the conftruGiion of the logarithms, thus: Having found, as above, 
 the hyperbolic logarithms of 10, 0°98, 0°99, 1°01,1°02, which may 
 be effected in an hour or two, dividing the laft four logarithms by the 
 logarithm of 10, and adding the index 2, we have the tabular loga- 
 rithms of 98, 99, 100, 101, 102. ‘Then by- interpolating nine means 
 between each of thefe, will be obtained the logarithms of all num- 
 bers between 980 and 1020; and again interpolating 9 means be- 
 tween every two numbers from 980 to 1000, the table will be fo far 
 conftruéted. Then from thefe will be colleéted the logarithms of 
 all the primes under 100, together with thofe of their multiples ; 
 all which will require only addition and fubtraGtion ; for 
 
 1°9984 X 1020 10 98 99 1001 102 
 9945 Ping Te LiMely Ray sgl Fe tps A gi = 173 
 on = 71; er aig bf ata 79; 88 sons 893 eae at Le 
 
104 CONSTRUCTION OF 
 
 This quadrature of the hyperbola, and its application to the con- 
 ftru€tion of logarithms, are {till farther explained by our celebrated 
 author in his treatife on Fluxions, publifhed by Colfon in 1736, 
 where he givesall the three feries for the areas AD, Ad, Bd, in ge- 
 neral terms, the former the fame as that publifhed by Mercator, and 
 the latter by Gregory; and he explains the manner of deriving the 
 latter feries from the former, namely by uniting together the two 
 feries for the {paces on each fide of an ordinate, bounded by other 
 ordinates at equal diftatices, every 2d term of each feries is can- 
 celled, and the refult is a feries converging much quicker than either 
 of the former. And, in this treatife on fluxions, as well as in the 
 letter before quoted, he recommends this as the moft convenient way 
 of raifing a canon of logarithms, computing by the feries the hyper- 
 bolic {paces anfwering to the prime numbers 2, 3, 5; 7,11, &c, and 
 dividing them by 2°3025850929940457, which is the area corre- 
 fponding to the number 10, or elfe multiplying them by its reciprocal 
 0°4542944819032518, for the common logarithms. ‘ ‘Then the 
 logarithms of all the numbers in the canori which are made by the 
 multiplication of thefe, are to be found by the addition of their los 
 garithms, as is ufual. And the void places are to be interpolated 
 afterwards by the help of this theorem: Let 2 be a number to whichi 
 a logarithm is to be adapted, x the difference between that and the 
 two neareft numbers equally diftant on each fide, whofe logarithms 
 are already found, and let d be half the difference of the logarithms ; 
 then the required logarithm of the number x will be obtained by add- 
 
 3 
 ing d + si a? ipa &c to the logarithm of the lefs number.” This 
 theorem he demonftrates by the hyperbolic areas, and then proceeds 
 thus; “ The two firft terms d + = of this feries I think to be accu- 
 
 rate enough for the conftruction of a canon of logarithms, even 
 though they were to be produced to 14 or 15 figures; provided the 
 number whofe logarithm is to be found be not lefs than 1000. And 
 this can give little trouble in the calculation, becaufe x is generally 
 an unit, or the number 2. Yet it is not neceflary to interpolate all 
 the places by the help of this rule. For the fogarithms of numbers 
 which are produced by the multiplication or divifion of the number 
 laft found, may be obtained by the numbers whofe logarithms were 
 had before, by the addition or fubtraction of their logarithms. 
 Moreover, by the differences of the logarithms; and by their 2d and 
 3d differences, if there be occafion, the void places may be more ex- 
 peditioufly fupplied; the foregoing rule being to be applied only. 
 when the continuation of fome full places is wanted, in order to ob- 
 tain thofe differences, &c.” So that Sir I. Newton of himfelf dif- 
 covered all the feries for the above quadrature which were found out, 
 and afterwards publifhed, partly by Mercator and partly by Gregory 3 
 and thefe we may here exhibit in one view all together and that in 
 a general manner for any hyperbola, namely putting CA = a, AF 
 
 —a 
 
LOGARITHMS. 105 
 
 — 5, and-AB = Ab = *; then will BD = , and bd as 
 whence the areas are as below, viz. — 
 bx bx3 ‘ bx4 bxs 
 AD = bx — i hae enor an Ke 
 bx? bx3 bx* bx5 
 Ad bx + ha | + 3a? Lt 4a? + or &c, 
 yA Qbx$ Shien ers GAGE | EN 
 Bia naeats Aga haat teresa aa te 
 
 In the fame letter alfo, above quoted, to Mr. Oldenburg, our il- 
 luftrious author teaches a method of conftructing the trigonometrical 
 canon of fines, by an eafier method of multiple angles than that be- 
 fore delivered by Briggs for the fame purpofe, becaufe that in Sir 
 Ifaac’s way radius or | is the firft term, and double the fine or cofine 
 of the firft given angle is the 2d term of all the proportions by which 
 the feveral fucceflive multiple fines or cofines are found. ‘he fub- 
 ftance of the method is thus: The beft foundation for the conftruc- 
 tion of the table of fines, is the continual addition of a given angle to 
 itfelf or to another given angle, As, if the angle A be to be added; 
 
 A E C GQ I Sic anda ilo UN 
 infcribe HI, IX, KL, LM, MN, NO, OP, &c, each equal to the 
 radius AB; and to the oppofite fides draw the perpendiculars BE, 
 HQ, IR, KS, LT, MV, NX, OY, &c; fo fhall the angle A be the - 
 common difference of the angles HIQ, IKH, KLI, LMK, &c; their 
 fines HQ, IR, KS, &c; and their colinesIQ, KR, LS, &c. Now » 
 let any one. of them, LMK, be given, then the reft wiil be thus found: 
 Draw_'Ta and Kb perpendicular to SV and MV ; now becaufe of the 
 equiangular triangles ABE, TLa, KMb, ALT, AMV, &c,, it will 
 bea AB. sD Sa SLM TUS} saih ee Past ee ee 
 2K5,j and AB sBE is. 2L0 + La (2 bS = SEV) 310 2 RM) 
 -2Mb (=iMV — 3KS.) Hence are given the fines and cofines KS, 
 MV,LS, LV. And the method of continuing the progreffions is 
 evident. Namely, 
 Mae SaCe SER ae >MT + MX::MX:NV + NY &c 
 ; “ty GUM A NOR: Bs Boosie Nix OFF “as ME Vinge 
 LV: NX — LT ep MxXOY.— MV. &c 
 of AB: 2BEs > 3 MV :MT-— MX ::N& :NV/—NY.. &c. 
 And on the other hand, AB: 2AE:: LS : KT + KR, &e. 
 Therefore put AB = 1, and make BE x LT = La, AE x KT = 
 Sa, Sa — La = LV, 2AE x LV — TM = MX, &e. 
 The fenfe of thefe general theorems is this, that if P be any one 
 i among 
 
106 | CONSTRUCTION OF 
 
 among 2 feries of angles in arithmetical progreffion, the angle d being 
 their common difference, then as radius or 
 col! Patcof Pi! -b yr een Pa 
 | iene erate fn, tn Pune 
 ar . §cof.P: fin. P+ d— fin. P—d 
 1:2 fin. d:: sep “cot Ppa — cof Pe a 
 where the 4th terms of thefe proportions are the fums or differences 
 of the fines or cofines of the two angles next lefs and greater than 
 any angle P in the feries; and therefore, fubtraCting the lefs extreme 
 from the fum, or adding it to the difference, the refult will be the 
 greater extreme, or the next fine or cofine beyond that of the term P. 
 And in the fame..manner are all the reft to be found. ‘This method, 
 it is evident, is equally applicable, whether the common difference d, 
 or angle A, be equal to one term of the feries or not: when it zs one 
 of the terms, then the whole feries of fines and cofines becomes 
 thusj;iviz, asd! 2icolid ty: 
 
 fin. d: fin. 2d 2: fin. 2d: fin. d+ fin. 34: : fin. Sd: fin. 27+ fin. 4d:: fin. 4d: fin. 3d+ fin.5d &c, 
 col. dil cof 2d: : cof. 2d: cof. d-+- cof. 3d: : cof. 3d: cof. 2d+- cof. 4d :: cof, 4d: cof. 3d—+-cof.5d &c. 
 
 which is the very method contained in the directions given by Abra- - 
 ham Sharp, for conftructing the canon of fines. 
 
 Sir I. Newton remarks, that it only remains to find the fine and 
 cofine of a firft angle A, by fome other method; and for this purpofe, 
 he directs us to make ufe of fome of-his own infinite feries: thus, by 
 them will be found 1°57079 &c for the quadrantal arc, the fquare 
 of which is 2°4694 &c3 divide this fquare by the fquare of the num- 
 ber exprefling the ratio of 90 degrees to the angle A, calling the quo- 
 
 pd 3 4 
 tient z ;then 3 or 4termsofthis feries 1 mie T s — a Hees SiC, 
 will give the cofine of that angle A. Thus we may firft find an 
 angle of 5 degrees, and thence the table may be computed to the fe- 
 ries of every 5 degrees ; then thefe interpolated to degrees or half de- 
 grees by the fame method, and thefe interpolated again ; and fo on as 
 far asneceflary. But two-thirds of the table being computed in this 
 manner, the remaining third will be found by addition or fubtraction 
 only, as is well known. 
 
 Various other improvements in logarithms and trigonometry are 
 owing to the fame excellent perfonage; fuch as, the feries for ex- 
 prefling the relation between circular arcs and their fines, cofines, 
 verfed fines, tangents, &c; namely, the arc being a, the fine s, the 
 verfed fine v, cofine c, tangent ¢, radius 1, then is | ) 
 
 Gees SSS ee cb tie ld erat agres) &C 
 amet + fol 4 aot 4. riot + pte | aSdgv® &e. 
 Om, ls Ms aL Ze at? + oh me dott Oy ORs 
 $a == 003 tyie@® —. sort? + oxarsyo®? —. soprero0e"* &e. 
 sal = fe?. 4+ et opie?! + qotee® — orerrpost"® &c. 
 v = hat—_hat+ 700°. — goezet® + zerpgo0t ~~ aza0sTs00%"* &C 
 tmatie@ + pea tose? + oaegse? tb regpasat® &ee 
 
LOGARITHMS. 107 
 
 Of Dr. Halley's Method. 
 
 Many other improvements in the con{tru€tion of logarithms are 
 aifo derived from the fame doétrine of fluxions, as we fhall fhow here- 
 after. In the mean time proceed we to the ingenious method of the 
 learned Dr. Edmund Halley, Secretary to the Royal Society, and the 
 fecond’Aftronomer Royal, having fucceeded Mr. Flamfteed in that 
 honourable office in the year 1719, at the Royal Obfervatory at 
 Greenwich, where he died the 14th of January 1742, in the 86th 
 year of his age. His method was firft printed in the Philofophical 
 _'TranfaCtions for the year 1695, and it is entitled * A moft compen- 
 dious and facile method for conftructing the logarithms, exemplified 
 and demonftrated from the nature of numbers, without any regard to 
 the hyperbola, with a {peedy method for finding the number from 
 the given logarithm.” 
 
 Inftead of the more ordinary definition of logarithms, as numerorum 
 proportionalinm equidiferentes comites, in this tract our learned author 
 adopts this other, umeri rationem exponentes, as being better adapted 
 to the principle on which logarithms are here conftruéted, where 
 thofe quantities are not confidered as the logarithnis of the numbers, 
 for example, of 2, or of 3, or of 10, but asthe logarithms of the ra- 
 tios of 1 to2, or 1 to 4, or 1 to 10. In this confideration he firft 
 purfues the idea of Kepler and Mercator, remarking that any fuch 
 ratio is proportional to, and is meafured by, the number of equal 
 ratiunculz.contained in each; which ratiuncule are to be underftoad 
 as in a continued fcale of proportionals, infinite in number, between 
 the two terms of the ratio; which infinite number of mean propor- 
 tionals is to that infinite number of the like and equal ratiunculz be- 
 tween any other two terms, as the logarithm of the one ratio is to 
 the logarithm of the other: thus, if there be fuppofed between 1 
 and 10 an infinite fcale of mean proportionals, whofe number is 
 100000 &c in infinitum; then between 1 and 2 there will be 30102 
 &c of fuch proportionals; and between 1 and $ there willbe 47712 
 &c of them; which numbers therefore are the logarithms of the ratios 
 of 1 to 10, 1 to 2, and 1 to ¥. But for the fake of Ais mode of 
 conftru€ting logarithms, he changes this idea of equal ratiuncule, 
 for that of other ratiunculz, fo conftituted, as that the /ame in- 
 finite number of them fhall be contained in the ratio of 1 to every 
 other number whatever; and that therefore thefe latter ratiunculze 
 will be of unequal or different magnitudes in all the different ratios, 
 and in fuch fort, thatin any one ratio, the magnitude of each of the 
 ratiuncule in this latter cafe, will be as the number of them in the for- 
 mer. And therefore if between 1 and any number propofed, there 
 be takenany infinity of mean proportionals, the infinitely {mall augment 
 or decrement of the ficft of thofe means from the firft term 1, will be 
 a ratiuncula of the ratio of 1 to the faid number; and as the number 
 of all the ratiuncule in thefe continued proportionals is the fame, 
 
 their 
 
108 CONSTRUCTION OF 
 
 their fum, or the whole ratio, will be dire€tly proportional to the 
 magnitude of one of the faid ratiuncule in each ratio. But it is alfo 
 evident that the firft of any number of means, between i and any 
 number, is always equal to fuch root of that number, whofe index is 
 expreffed by the number of thofe proportionals from 1 ; fo, if 7 denote 
 the number of proportionals from 1, then the firft term after 1 will 
 be the mth root of that number. Hence, the indefinite root of any 
 number being extracted, the differentiola of the faid root from unity, 
 fhall be as the logarithm of that number. So if there be required the 
 logarithm of the ratio of 1 to 1 + g3 the firft term after 1 will be 
 
 (i+ 9)", and therefore the required logarithm will be as (1 + q\m —1. 
 1—m 1 1—n: - 1—2m 
 
 a 1 1 : 
 Buty {1 ueeap ed bee gick oro ed he Fem sed ORS 
 
 or by omitting the 1 in the compound numerators, as infinitely {mall 
 
 in refpect of the infinite number m, the fame feries will become 
 1 1 —m | 1 am om 
 Ag ee aa Pk Bae em 
 | 1 1 1 1 ; 
 Sock gi gt) he aa aed Bees and hence, finding,-the 
 
 q’ &c, or by abbreviation it is 
 
 differentiola by fubtraQting 1, the logarithm of the ratio of 1 tol +4 
 is as — x (g— 29 + 3¢ —ag' +4¢5—tq> &c.) Now the index m 
 may be taken equal to any infinite number, and thus all the varieties 
 of {cales of logarithms may be produced: fo, if m be taken 1000000 
 &c, the theorem will give Napier’s logarithms ; but ifm be takenequal 
 to 230258 &c, there will arife Briggs’s logarithms. 
 
 This theorem being for the increafing ratio of 1 to 1 + 43; if that 
 for the decreafing ratio of 1 to | — g be alfo fought, it will be ob- 
 tained by a proper change of the figns, by which the decrement of 
 the firft of the infinite number of proportionals will be found to be 
 
 Lai PRO RES ; é F 
 — into g + 39° + 39° + ag° &c, which therefore is as the logarithm 
 
 of the ratio of 1 to 1 — ¢. 
 
 _ Hence the terms of any ratio being a and 4, g becomes a or the 
 difference divided by thelefs term, when it is an increafing ratio ; or 
 g foe when the ratio is decreafing, or as ) toa. Therefore the 
 
 logarithm of the fame ratio may be doubly exprefied; for, putting x 
 for the difference b — a of the terms, it will be 
 
 1 ite x xe? x? x* 
 - m a 24? 3a° Aa’ 
 Bik x xe x? x* 
 
 OP ADtO oe ae ae i ee Oe 
 Mm b 26? ne 35° Abt 
 
 But if the ratio of ato 4 be fuppofed divided into two parts, namely, 
 into 
 
LOGARITHMS. 108 
 
 into the ratio of a to 4a + 4) or iz, and the ratio of tz to 4, then 
 will the fum of the logarithms of thofe two ratios, be the logarithms 
 
 of the ratio ofa to 6. Now by fubftituting in the foregoing feries, 
 the logarithms of thofe two ratios will 
 
 5 a x 2 3 4 5 
 
 iy on es see MRT te i dB EA 
 m z Qx? 32° Ac! 52? 
 iy x x? Fo x4 Eo 
 
 and— into —.— —. + —_ —___4 "_. &c; and hence the fum, 
 m z Da XB ix® 4iz* 52? 
 1 Bie 9x + 9x3 ox? Qx7 2x9 
 
 or —_ into ae oe ULE Net, Sit, 
 mm f-: ES Y i Lei Oz? 
 
 will be the log. of the ratio of a to J. 
 
 Moreover, if from the logarithm of the ratio of a to iz be taken 
 that of tz to 4, we fhall have the logarithm of the ratio of ab to 427; 
 and the half of this gives that of Wad to tz, or of the geometrical 
 mean to the arithmetical mean. And confequently the logarithm of 
 this ratio will be equal to half the difference of that of the above two 
 
 2 4 6 3 
 ratios, and will therefore be . into 3 + i + 5 +- — &c. 
 
 The above feries are fimilar to fome that were before given by 
 Newton and Gregory, for the fame purpofe, deduced from the con- 
 fideration of the hyperbola. But the rule which is properly our au- 
 thor’s own, is that which follows,-and is derived from the feries 
 above given for the logarithm of the fum of two ratios. For the 
 ratio of ab to jz’ or 4a’ + 4ab + 30’, having the difference of its 
 terms ta°— iab + 10° or (44—4a)? or <x’, which in the cafe of 
 finding the logarithms of prime numbers is always |, if we call the 
 fum of the terms {2° + ab = y’, the logarithm of the ratio of /ad to 
 1a + 4b or 4z will be found to be 
 
 Lento etl : Bee Yeo ae 
 IR Teg a OP) 87) Ae ORR Ran OPI 
 
 And thefe rules our learned authorexemplifies by fome cafes in 
 numbers, to fhow the eafieft mode of application in practice. 7 
 
 Again, by means of the fame binomial theorem he refolves with 
 equal facility the reverfe of the problem, namely, from the logarithm 
 given, to find its number or ratio: For, as the logarithm of the 
 
 ratio of 1 to 1 + g was proved to be (1 + g)"—1, and that 
 
 of the ratio of 1 to 1 —gto be+ * + 1—(1—gq)”5 hence, 
 calling the given logarithm L, in the former 
 t 
 
 cafe it will be (1+ ¢)"= 141, 
 
 and in the latter (1 —q)” = 1—Ls; 
 
 and therefore lg = (1+ mh f that is, by the binomial theorem, 
 and 1 —g=(1—L) 
 
 1 
 
a 
 
 110 CONSTRUCTION OF 
 
 l+grit4+mL+4+ in’ L? 4+ tm3 L3 4 mt Lt 4+ yi5m5 LS &c, 
 and l—g=1—mL+ jm’ L’— $m? L3 4+ Amt L4* — yh gms LS &c. 
 am being any infinite index whatever, differing according to the fale 
 of logarithms, being 1000 &c in Napier’s or the hyperbolic loga- 
 rithms, and 2302585 &c in Briggs’s. 
 
 If one term of the ratio, of which L is the logarithm, be given, the 
 other term will be eafily obtained by the fame rule: For if L be Na- 
 ‘pier’s logarithm of the ratio of a the lefs term, to J the greater, then, 
 ‘according as a or 4 is given, we fhall have, 
 
 6=aintol +L + 40’ + tL’ + AL &c, 
 a—éint0ol—L+ 4L’ — ¢L’ + 41 &e. 
 Hence, by help of the logarithms contained in the tables, may eafily 
 be found the number to any given logarithm to a great extent. For 
 if the fmall difference between the given logarithm L, and the neareft 
 tabular logarithm, either greater or lefs, be called /, and the number 
 anfwering to the tabular logarithm a, when it is lefs than the given 
 Jogarithm, but 4 when greater; it will follow, that the number an- 
 {wering to the logarithm L, will be 
 either aintol + 2+ 324+ 2P + 2H + 1,15 &c, 
 or into l —2 4+ 22 — 22 4+ lt — pts &e. 
 which feries converge fo quick, / being always very {mall, that the 
 firft two terms 1 + / are generally fuflicient to find the number to 
 10 places of figures. . 
 
 Dr. Halley fubjoins alfo an eafy approximation for thefe feties, by 
 which it appears, that the number anfwering to the log. is nearly 
 1+7 Cay 1—1/ c ah Napier’s}a +3 WP Sree ee fm Briggs’s 
 1—3/ I oOgs. and Sn—2] ail logs. 5 
 
 ri 
 
 where n is = 434294481903 &c = a 
 
 Of Mr. Sharp's Methods. 
 The labours of Mr. Abraham Sharp, of Little-Horton, near Brad- 
 
 _ ford in Yorkfhire, in this branch of mathematics, were very great and 
 
 meritorious. His merit however confifted rather in the improvement 
 and illuftration of the methods of former writers, than in the inven- 
 tion of any new ones of his own. In this way he greatly extended 
 and improved Dr. Halley’s method, above defcribed, as. alfo thofe of 
 Mercator and Wallis; illuftrating thefe improvements by extenfive 
 calculations, and by them computing table 5 of this book, confifting 
 of the logarithms of all numbers to 100, and of all prime numbers to 
 1100, each to 61 places. He alfo compofed a neat compendium of 
 the beft methods for computing the natural fines, tangents, and fe- 
 cants, chiefly from the rules before given by Newton; and by New- 
 ton’s or Gregory’s feries a= ¢ — 4 + H° —7t’ &c, for the arc in 
 terms of the tangent, he computed the circumference of the circle to 
 12 places, namely from the arc of 30 degrees, whofe tangent ¢ is = 
 
 4/3 to the radius 1. Other aftonifhing inftances of his ena 
 | | labour 
 
LOGARITHMS. lik. 
 
 labour appear in his Geometry Improv’d, printed in 1717, and figned 
 A. 8. Philomath, from whence the 5th table of logarithms above- 
 mentioned was extracted. This ingenious man was fometime aflift- 
 ant at the Royal Obfervatory to Mr. Flamfteed the firft Aftronomer 
 Royal; and being one of the moft accurate and indefatigable com- 
 puters that ever exifted, he was for many years the common refource 
 for Mr. Flamfteed, Sir Jonas Moore, Dr. Halley, &c, in all intri- 
 cate and troublefome calculations. He afterwards retired to his na- 
 tive place at Little Horton; where, after a life {pent in intenfe ftudy 
 and calculations, he died the 18th of July 1742, in the 91ft year of 
 his age. | 
 
 / 
 Of the Conftruétion of Logarithms by Fluxions. 
 
 It appears by the very definition and defcription given by Napier 
 of his logarithms, as ftated in page 42 of this IntroduCtion, that the 
 fluxion of his, or the hyperbolic logarithm, of any number, is a fourth 
 proportional to that number, its logarithm, and unity; or, which is 
 the fame, that it is equal to the fluxion of the number divided by the 
 number : For the defcription fhows that z1 : za or 1: : 21 the flux- 
 
 % mit 
 a3 but za is alfo equal to the 
 
 fluxion of the logarithm A &c, by the defcription ; therefore the flux- 
 
 ion of z1 : za, which therefore is = 
 
 ; 1 
 ion of the logarithm is equal to =, the fluxion of the quantity di- 
 
 vided by the quantity itfelf. ‘The fame thing appears again at art. 2 
 of that little piece in the appendix to his Con/fructio Logarithmorum, 
 entitled Habitudines Logarithmorum 8 fuorum naturalium numerorum 
 invicem, where he obferves that, as any greater quantity is to a lefs, 
 fo is the velocity of the increment or decrement of the logarithms 
 at the place of the lefs quantity, to that at the greater. Now this 
 velocity of the increment or decrement of the logarithms being the 
 fame thing as their fluxions, that proportion is this, x : a: : flux. log. 
 a: flux. log. x; hence if abe = J, as at the beginning of ‘the table 
 of numbers, where the fluxion of the logs. is the index or character- 
 iftic c, which is alfo 1 in Napier’s or the hyperbolic logarithms, and 
 43429 &c in Briggs’s, the fame proportion becomes « : 1 ::c¢: flux. 
 log. x; but the conftant fluxion of the numbers is alfo 1, and there- 
 
 i r . : cx r eae 
 fore that proportion is alfo this, x: *::¢:— = the fluxion of the 
 
 | logarithm of x; and in the hyperbolic logarithms, where cis = I, it 
 
 _ becomes = = the fluxion of Napier’s or the hyperbolic logarithm of 
 
 x. This fame property has alfo been noticed by many other authors 
 fince Napier’s time. And the fame or a fimilar property is evidently 
 true in all the fyftems of logarithms whatever, namely, that the mo- 
 dulus of the fyftem is to any number, as the fluxion of its logarithm 
 is to the fluxion of the number. 
 eat Now 
 
12 CONSTRUCTION OF 
 
 q 
 
 Now from th’s property, by means of the dodtrine of fluxions, 
 are derived other ways for making logarithms, which have been il 
 luftrated by many writers on this branch, as Craig, john Bernouilli, 
 and almoft. all thé writers on fluxions. And this method chiefly 
 confifts in expanding the reciprocal of the given quantity in an in- 
 finite feries, then multiplying each term by the fluxion of the faid 
 quantity, and laftly taking the fluents of the terms; by whic!) there 
 grifes an infinite feries of terms for the logarithm fought. 5o, to 
 find the logarithm of any number N; put any compound quantity 
 
 n+. 
 for N, as fuppofe —— , 
 BT fg sap di taey Mam ant, ara RAE at SB 
 . bein er ne Wier Cat eos sea ee aes 
 then the flux of the log. or N 5 pe ee a na ae” 
 i 4 x ee x* 
 ive log. OFO, Ober mei a Bb Seo &c. 
 the fluents give log. of N 2 i eevee yer Me a 
 j hie x re x3 at 
 And writing — » for x gives log. wees, et weet teneem ia ie i eee 
 n Ty ceo 3n3 44 
 ‘ oN yi 
 n nx n nx 
 Alfo, becaufe sor] te a5 y or log. = 0 — log. =", 
 ax % 4 ux 73 
 n x by x3 x* 
 theref. log. rena abt lg a 8 Map N28) FP 
 : n+ % n Qn? 3n3 An* 
 
 n a 
 i ROE: pace ia og Bak TEGaR aes 
 And by adding and fubtracting any of thefe feries, to or from one 
 
 another, and multiplying or dividing their correfponding numbers, 
 various other feries for logarithms may be found, converging much 
 quicker than thefe do. - 
 
 “In like manner by affuming quantities otherwife compounded for 
 the value of N, various other forms of logarithmic feries may be 
 found by the fame means. 
 
 Of Mr. Cotes’s Logometria. 
 
 Mr. Roger Cotes was elected the firft Plumian profeffor of aftro~ 
 homy and experimental philofophy in the univerfity of Cambridge, 
 January 1706, which appointment he filled with the greateft credit, 
 till he died the 5th of June 1716, in the prime of life, having not 
 quite completed the 34th year of his age. His early death was a 
 great lofs to the mathematical world, as his genius and abilities were 
 of the brighteft order, as is manifeit by the fpecimens of his per- 
 formance given to the public. Among thefe are his Logometria, firft 
 printed in number 338 of the Philofophical Tranfa€tions, and after- 
 wards in his Harmonia Menfuarum, publifhed:in 1722 with his other 
 works, by his relation and fucceffor, in the Plumian profeflorthip, 
 Dr. Robert Smith. In this piece-he firft treats in a general way of 
 
 ? | meafures 
 
. LOGARITHMS. 118 
 
 meafures of ratios, which meafures, he obferves, are quantities of any 
 Kind whofe magnitudes are analogous to the magnitudes of the ratios, 
 thefe magnitudes mutually increafing and decreafing together in the 
 famie proportion. He remarks, that the ratio of equality has no mag- 
 nitude, becaufe it produces no change by adding and fubtracting 5 
 that the ratios of greater and lefs inequality, are of different affections; 
 and therefore if the meafure of the one of thefe be confidered as po- 
 fitive, that of the other will be negative; and the meafure of the ra-.° 
 tio of equality nothing: That there are endlefs fy{tems of thefe, 
 
 which have all their meafures of the fame ratios proportional to cer- 
 
 tain given quantities, called moduli, which he defines afterwards, and 
 
 the ratio of which they are the meafures, each in its peculiar fyftem, 
 
 is called the modular ratio, ratio modularis, which ratio is the fame in 
 
 all fyftems. He then adverts to logarithms, which he confiders as 
 
 the numerical meafures of ratios, and he defcribes the method of ars 
 rar.ging them in tables, with their ufes in multiplication and divition, 
 
 faifing of powers and extracting of roots, by means of the corre« 
 {ponding operations of addition and fubtraction, multiplication and 
 
 divifion. 
 
 After this introduction, which is only a flight abridgment of the 
 doctrine long before very amply treated of by others, and particular- 
 ly by Kepler and Mercator, we arrive at the firft propofition, which 
 has juftly been cenfured as obf{cure and imperfe&t, feemingly through 
 an affectation of brevity, intricacy, and originality, without fufficient 
 room for a difplay of this qualification. The reafoning in this pro- 
 pofition, fuch as it is, feems to be fomething between that of Kepler 
 and the principles of fluxions, to which the quantities and expreflions 
 are nearly allied. However, as it is my duty rather to narrate than. 
 explain, I fhall here exhibit it exaCtly as it ftands. This propofition 
 is to determine the meafure of any ratio, as for inftance that of AC to 
 AB, and which is effected in this manner: Conceive the difference 
 BCto be dividedinto innumerable : hae 1 
 very {mall particles, as PQ, and A B PQ Cc 
 the ratio between AC and ABinto as many fuch very fmall ratios, as 
 , between AQ and AP: then if the magnitude of the ratio between AQ 
 and AP be given, by dividing there will alfo be given that of PQ to 
 AP; and therefore, this being given, the magnitude of the ratio be- 
 tween AQ and AP may be expounded by the given quantity at for, 
 AP remaining conftant, conceive the particle PQ to be augmented or, 
 diminifhed in any proportion, and in the fame proportion will the 
 magnitude of the ratio between AO and AP be augmented or dimi- 
 nifhed: Alfo, taking any determinate quantity M, the fame may be 
 expounded by M x a and therefore the quantity M x a will be 
 
 the meafure of the ratio between AQ and AP. And this meafure will 
 have divers magnitudes, and be accommodated to divers fy{tems, ac- 
 
 ee cording 
 
114 CONSTRUCTION OF 
 
 cording to the divers magnitudes of the affumed quantity M, which 
 therefore is called the modulus of the fyftem. Now, like as the fum 
 of all the ratios AQ to AP is equal to the propofed ratio AC to AB, 
 
 fo the fum of all the meafures M x — found by the known methods, 
 
 will be equal to the required meafure-of the faid propofed ratio. 
 
 _ The general folution being thus difpatched, from the general ex- 
 preflion, Cotes next deduces other forms of the meafure, in feveral 
 corollaries and fcholia: as 1ft, the terms AP, AQ, approach the 
 nearer to equality as the {mall difference PQ is lefs ; fo that either 
 
 me P ‘ n 
 M xX = or M x — will be the meafure of the ratio between AQ_ 
 
 and AP, to the modulus M. 2d, That hence the modulus M is to the 
 meafure of the ratio between AQ and AP, as either AP or AQ is to. 
 their difference PQ. 3d, The ratio between AC and AB being 
 
 given, the fum of all the will be given; and the fum of all the 
 
 P e e . e 
 Mx te is as M: therefore the meafure of any given ratio, is as the 
 
 modulus of the fyftem from which it is taken. 4th, Therefore, i? 
 every fyftem of meafures, the modulus will always be equal to th® 
 meafure of a certain determinate and immutable ratio ; which there- 
 fore he calls the niodular ratio. 5th, To illuftrate the folution by an 
 example: let z be any determinate amd permanent quantity, ¥ a va- 
 riable or indeterminate quantity, and x its fluxion ; then, to find the 
 meafure of the ratio between z-++ x and z—», put this ratio equal 
 to the ratio between y and 1, expounding the number y by AP, its 
 fluxiony by PQ, and 1 by AB: then the fluxion of the required 
 
 . . J. ¢ es 
 meafure of the ratio between y and 1 is M x ~~. Now, for y, reftore its 
 
 ~ 
 
 atx “aie ED | 
 val. —— » and for , the flux. of that value, ———va> fo fhall the fix-of 
 3 z—7 (3—~x) , 
 Ze | 2 ¥ xv? at 
 the meafure become 2M X IE AE: gM into: ee ere 
 
 * 5 
 
 : : x 5 ay 
 
 and therefore that meafure will be 2M into oo 358 +3 &e. 
 v 
 
 In like manner the meafure of the ratio between 1 + v and 1 will 
 be found to be - - s+ = M intov — 'v* + tuv3 — iy! &c. 
 And hence, to find the number from the logarithm given, he reverts 
 the feries in this manner: If the laft meafure be called m, we 
 
 4 7 
 hall have — or @ = u-— ju + jv? — iy’ + tv’ &e, 
 M 
 
 therefore Q?= - ov — v3 + {iv'— iv’ &cy 
 P 3 cy Pen ; Fah 
 and Q?= - = = -¥v gu! + yA &c, 
 andQ'= s= «© «+ - = = vi-—-2v &C, 
 andQ?= 2 f= - es ee ee wv &;5 
 then, 
 
LOGARITHMS. 115 
 
 then, by adding continually, we fhall have, 
 Q + iQ? = v— Iv 3;0' — 13 v' &e, 
 Q+ 304+ 1@V@=v—Ar+4+ 5 vw &, 
 Q+1Q°4 7@ 4 4Qi=v — Lv &e, 
 Q + 1Q?+ 1Q° + 1 Q'4+ 71,0) =vkc, 
 that isu = Q + 4Q’?+ 12° 4 4 Q'+4+ +44, Q' &c. And therefore 
 the required ratio of 1 + vto 1, is equal to the ratio of 1+ Q + 
 3Q: &c to 1. Putnow m =M, or Q=1, and the above will be- 
 come theratio of 1 +2+i+2+4+.4++, kcto 1, for the con- 
 ftant modular ratio. In like manner, if the ratio between 1 and 
 1 —v be propofed, the meafure of this ratio will come out M into 
 
 m 
 v + jv? + iv? + tu! &c; which being called m, and y= & 
 
 that ratio will be the ratio of 1 to 1 -Q44Q’—103 +40! &c. 
 And hence, taking m = M, or Q = 1, the faid modular ratio will 
 alfo be the modular ratio of lto 1—i 4+ 4—2+4 4.—,4), &e. 
 And the former of thefe exprefions, for the modular ratio, comes 
 out the ratio of 2,718281828459 &c to 1, and the latter the ratio of 
 ] to 0367879441171 &c, which number is the reciprocal of the 
 former. — 
 
 In the 2d prop. the learned author gives direCtions for conftru&- 
 ing Briggs’s canon of logarithms, namely, firft by the general feries 
 x3 x5 
 
 i; + soy &c, finding the logarithms of a few fuch 
 
 x 
 2M into a -|- rv 
 ratios as that of 126 to 125, 225 to 224, 2401 to 2400, 4375 to 
 4374, &c, from whence the logarithm of 10 will be found to be 
 2,302585032994 &c, when M is 1; but fince Briggs’s log. of 10 is 
 1, therefore as 2,302585 &c is to the modulus 1, fo 1s 1 ( Briggs’s log. 
 of 10) to 0434294481903 &c, which therefore is the modulus of 
 Briggs’s logarithms. Hence he deduces the logarithms of 7, 5, 3, 
 and 2. In like manner are the logarithms of other prime numbers 
 to be found, and from them the logarithms of compofite numbers by 
 addition and fubtraction only. 
 
 Cotes then remarks, that the firft term of the general feries 2M into 
 Ey) x3 sc? 
 ree © aa) eae &c, will be fufficient for the logarithms of inter- 
 
 mediate numbers between thofe in the table, or even for numbers be- 
 yond the limits of the table. “Thus, to find the logarithm anfwering 
 to any intermediate number; let aande be two numbers, the one 
 the given number, and the other the neareft tabular number, a being 
 the greater, and e the lefs of them; put z =a + e¢ theirfum, x = 
 a —e their difference, 7 = the logarithm of the ratio of atoe, 
 that is the excefs of the logarithm of a above that of e: fo fhall the 
 
 faid difference of their logarithms be A = 2M x = very nearly.— 
 
 And, if there be required the number anfwering to any given inter- 
 mediate logarithm, becaufe A is = 
 2Mx 2Mx 2Mx Aa Ae 
 
 E> Daw 8 Depa theref x = PTA Oh Symi Very nearly. 
 In 
 
Ais CONSTRUCTION OF 
 
 In the 3d prop. the ingenious author teaches how to convert the 
 canon of logarithms into logarithms of any other fy{tem, by means of 
 their moduli. And, in feveral more propofitions, he exemplifies the 
 canon of logarithms in the folution of various important problems 
 in geometry and phyfics; fuch asthe quadrature of the hyperbola, 
 the defcription of the logiftica, the equi-angular fpiral, the nautical 
 meridian, &c; the deicent of bodies in refifting mediums, the 
 denfity of the atmofphere at any altitude, &c, &c. 
 
 Of Dr. Taylor's Conftruétion of Logarithms, 
 
 Dr. Brook Taylor (a very learned mathematician, and fecretary to 
 the Royal Society, who died at Somerfet-houfe, Nov. 1731) gave 
 the following method of conftructing. logarithms, in number 352 of 
 the Philofophical Tranfa€tions. His method is founded on thefe 
 
 three confiderations: 1{t, that the fum of the logarithms of any two . 
 
 numbers is the logarithm of the produ&t of thofe numbers; 2d, that 
 the logarithm of 1 is nothing, and confequently that the nearer any 
 number isto 1, the nearer will its logarithm be to 0; 3d, that the 
 product of two numbers or factors, of which the one is greater and 
 the other lefs than 1, is nearer to 1 than that factor is which is on 
 the fame fide of 1 with itlelf; fo of the two numbers 2 and 4, the 
 product 5 is lefs than 1, but yet nearer to it than 3 is, which is alfo 
 Jef: than 1. On thefe principles he founds the prefent approxima~- 
 tion, which he explains by the following example. To find the 
 relation between the logarithms of 2-and 10: In order to this, he 
 
 128 8 Qi 3 
 aflumes two fra€tions, as 100 and io’ 10" and 755 whofe nume- 
 
 rators are powers. of 2, and their denominators powers of 10, the one 
 fraction being greater and the other lefs than unity or 1. Having 
 fet thefe two down, in the form of decimal fraétions, below each 
 other, in the firft column of the following table, and in the fecond 
 column A and B for their logarithms, exprefling by an equation how 
 
 comp, ar.235313 
 
 1,280c0000C0000;A= .. = 712, — 2/10 {12-7 0,28 
 
 0 soocccoco000/; B= .. = 3/22 — AiO} 20,33 
 1,024000000000/;C =A -+ B= 10/2 — 3/10] 7 0300 
 0.990352031429;D =B + 9C= 93/2 — 28/10] Z 0,30107 
 1004380277004). B/G aD — 169/2 — 59/10] 7 0,301020 ' 
 0,908959530107/F =D+ 2E= 485/2 — 146/10} Z 0,3010309 © 
 1,000162894105|G =~ E4 4F= 2136/2 — 643/10} 7 0,30102996 
 0,099936281874|H =F + 6G= 13301/22— 4004/10} Z 0,301029907 
 1,000035441215}1 =G+ 2H= 28738/2—  °8651/10] 7 0,3010299951 
 0,09097 1720830] K =H + = 42039/2— 12055/10} Z 0,3010299959 
 1,000007161046]L =I + K= 70777:.2— 21306/10} 7 0,30102999562 
 0,999993203514|M=K + 3L = 254370/2— 7€573/10| 4 0,30102999507 
 1,000000364511;N=L + M= 325147/22— 97879/10| 7 0,30 0299956635 
 0,9999997604087;O = M + 18N => 6107016/2 —1838335/10] Z 0,3010299950040 
 
 0 = 364511 O + 235313 N = 2302585825187 12 — 693147400972 /10 
 
 7 0,301029995663987| 
 they 
 
‘LOGARITHMS. 117 
 
 they are compofed of the logarithms of 2 and 10, the numbers in 
 gueftion, thofe logarithms being denoted thus, /2 and /10. Then 
 multiplying the two numbers in the firft column together, there is 
 produced a third number 1,024, again{ft which is written C, for its 
 logarithm, exprefling likewife by an equation in what manner C igs 
 formed of the foregoing logarithms A and B. And in the fame man- 
 ner the calculation is continued throughout; only obferving this 
 compendium, that before multiplying the two laft numbers already 
 entered in the table, to confider what power of one of them muit 
 be ufed to bring the product the neareft that can be to unity. Now 
 after having continyed the table a little way, this is found by only 
 dividing the differences of the numbers from unity one by the other, | 
 and taking the neareft quotient for the index of the power fought. 
 Thus the fecond and third numbers in the table being 0,8 and 1,024, 
 their differences from unity are 0,200 and 0,024; hence 0,200 + 
 0,024 gives 9 for the index; and therefore multiplying the 9th 
 power of 1,024 by 0,8 produces the next number 0,990352031429, 
 whofe logarithm is D= B + 9C, 
 
 When the calculation is continued in this manner till the numbers 
 become {mall enough, or near enough to 1, the laft logarithm is 
 fuppofed equal to nothing, which gives an equation exprefling the 
 relation of the logarithms, and from thence the required logarithm is 
 determined. ‘hus, fuppofing G = 0, we have 
 2136/2 — 643/10 = 0, and hence, becaufe the logarithm of 10 is 1, 
 
 we obtain /2 = = 0,30102996, too fmall in the laft figure 
 
 2136 
 only; which fo happens, becaufe the number correfponding to G is 
 greater than 1. ‘Andin this manner are all the numbers in the third 
 or laft column obtained, which are continual approximations to the 
 logarithm of 2, 
 
 There is another expedient, which renders this calculation ftill 
 fhorter, and it is founded on this confideration ; that when x is fmall, 
 
 (1 +x) is nearly = 1 + x. Hence if 1 + » and 1—<zbe the two 
 Jaft numbers already found in the firft column of the table, the 
 
 produét of their powers (I+) X (1—2) will be nearly = 1; and 
 
 hence the relation of m and n may be thus found, (1 + x) x (1— z) 
 is nearly =(1-++mx) X (1—nz) =1 + mx—nz— mnxz = 1 + mx—nz 
 nearly, which being alfo = 1 nearly, therefore min: :2:6:2 
 /, (1—2x) : 4.(1 + x); whence w/. (1 —z) + 2/.(1 +») =0. For exam- 
 ple, let 1,024 and 0,990352 be the laft numbers in the table, their 
 logarithms being C and D: here we have 1,024 = 1 + x, and 
 
 0,990352 = 1 — 23; confequently + = 0,024, and z — 0,009648, 
 
 ; 201. 
 and hence the ratio = in {mall numbers is can So that, for finding 
 the logarithms propofed, we may take 500D + 201C = 48510/2 
 
115. 
 Bl 
 
 CONSTRUCTION OF 
 
 *~ 
 
 —~11602/10 = 03 which gives /2 = 0,3010307. And in this mane 
 
 ner are found the numbers in the laft line of the table. 
 
 Of Mr, Long’s Method. 
 
 In number 339 of the Philofophical TranfaGtions, are given a 
 brief table and method for finding the logarithm to any number, and 
 the number to any logarithm, by Mr. John Long, B.D. Fellow of 
 ‘This table and method are fimilar to thofe defcribed 
 in chap. 14, of Briggs’s /rith. Logar. differing only in this, that in 
 this table, by Mr. Long, the logarithms, in each clafs, are in arith- 
 metical progreffion, the common difference being 1 ; but in Briggs’s 
 little table, the column of natural numbers has the like common dif- 
 ference. The table confifts of eight clafles of logarithms, and their 
 correfponding numbers, as follow : 
 
 C.. CC) @xon, 
 
 Lo!Nat. Numb. 
 
 29 '7 5043292347 
 38 [6,309573445 
 »7 |5,011872336 
 6 13,981071706 
 5 |3,162277660 
 »4 12511886432 
 93 11,9952693 15 
 92 11584893193 
 yi 11,258925412 
 509 1,230268771 
 8 1,202264435 
 7 t,174897555 
 61,148153621 
 5 1,122018454 
 4,1,096478196 
 3\1,071519305 
 21,047 128548 
 - 41,0238292992 
 
 mB s°>pwohr ou On & 
 
 1 mere 
 
 '0009|1,002074475|| 
 
 1,009 41,020939484 
 
 1,018591388 
 1,016248694 
 1,013911386 
 1,011579454 
 1,009252886 
 1,006931669 
 1,004615794 
 1,002305238 
 
 8|1,001843766 
 7|1,001613109 
 6|1,001382506 
 5|1,001151956 
 4|1,000921459 
 3]1,000691015 
 211,000460623 
 t|1,000230285 
 
 motoh OOnN © 
 - 
 © 
 © 
 © 
 joak 
 fied 
 Cr 
 ak 
 ise) 
 (=) 
 
 5000009} 1,000020724 
 
 1 11,000002302 
 
 Log.'Nat. Numb.||. Log. {Nat. Numb, 
 00009 |1,000207254 
 
 1,000184224. 
 1,000161194 
 
 |1,000023026 
 
 8/1,000018421 
 7\1,000016118 
 6|1,000013816 
 5 1,000011513 
 4| 1000009210 
 3)1,000006908 
 
 2)1,000004605 
 
 Nat. Numb, 
 
 1,000002072 
 1,000001842 
 000001611 
 000001381 
 000001151 
 000000924 
 1,000000690 
 1,000000460 
 1,000000230 
 
 =r 
 
 —_ 
 
 ary 
 
 500000009} ,000000207 
 
 8]1,000000184 
 7|1,000000161 
 6|1,000000138 
 511,000000115 
 4|1,000000092 
 3}1,000000069 
 2|1,000000046 
 1/1,000000023 
 
 where, becaufe the logarithms in each clafs are the continual mul- 
 tiples 1, 2,3, &c, of the loweft, it is evident that the natural num- 
 bers are fo many {cales of geometrical proportionals, the loweft be- 
 ing the common ratio,’ or the afcending numbers are the 1, 2, 3, &c 
 powers of the loweft, as expreffed by the figures 1, 2, 3, &c of their 
 correfponding logarithms. 
 third, &c clafs, is the 10th, 100th, 1000th, &c root of 10; and any 
 number in any clafs is the 10th power of the correfponding number 
 in the next following clafs. 
 
 'To find the logarithm of any number, as fuppofe of 2000, by this 
 table, Lock in the firft clafs for the number next lefs than the 
 firft figure 2, and it is 1,995262315, againft which is 3 for the 
 fir{t figure of the logarithm fought. Again, dividing 2, the number 
 
 4 
 
 Alfo the laft number in the firft, fecond, 
 
 propofed, 
 
LOGARITHMS. - 119 
 
 propofed, by 1,995262315, the number found inthe table, the quo- 
 tient is 1,002374467; which being looked for in the fecond clafs 
 of the table, and finding neither its equal nor a lefs, O is therefore © 
 to be taken for the fecond figure of the logarithm; and the fame 
 quotient 1,002374467 being looked for in the third clafs, the next 
 lefs is there found to be 1,002305238, againft which is 1 for the third 
 figure of the logarithm; and dividing the quotient 1,002374467 by the 
 faid next lefs number | 002305238, the new quotient is 1 ,000069070; 
 which being fought in the fourth clafs gives 0, but fought in the 
 fifth clafs gives 2, which are the fourth and fifth figures of the lo- 
 garithm fought : again, dividing the -laft quotient by 1,000046053, 
 the next lefs number in the table, the quotient is 1,000023015, 
 which gives 9 in the 6th clafs for the 6th figure of the logarithm 
 fought: and again dividing the laft quotient by 1,000020724, the 
 next lefs number, the quotient is 1,000002291, the next lefs than 
 which, in the 7th clafs, gives 9 for the 7th figure of the logarithm: 
 and dividing the laft quotient by 1,000002072, the quotient is 
 1,000000219, which gives 9 in the 8th clafs for the 8th figure of 
 _ the logarithm: and again the laft quotient 1,000000219 being di- 
 vided by 1,000000207, the next lefs, the quotient 1,000000012 gives 
 5 in the fame 8th clafs, when one figure is cut off, for the 9th figure 
 of the logarithm fought. All which figures collected together give 
 3,301029995 for Briggs’s logarithm of 2000, the index 3 being fi 
 plied ; which logarithm is true in the laft figure. 
 
 To find the number anfwering to any given loga- {3{1,995962315 
 rithm, as fuppofe to 3,3010300: omitting the cha- |o\0 
 racteriftic, againft the other figures 3,0,1,0,3,0,0, {1/1,002305238 
 as in the firft column in the margin, are the feveral {9/0 
 numbers as in the 2d column, found from their re- _|9,1,000069080 
 {pective Ift, 2d, 3d, &c claffes; the effective num- 0,0 
 bers of which multiplied continually together, the nit 
 laft product is 2,000000019966, which, becaufe 
 the characteriftic is 3, gives 2000,000019966, or 2000 only, for the 
 required number, anfwering to the given logarithm. 
 
 ~ Of Mr. Fones’s Method. 
 
 In the 61f volume of the Philofophical Tranfa€tions, is a fmall 
 paper on logarithms, which had been drawn up, and left unpublifhed, 
 by the learned and ingenious William Jones, Efq. The method con- 
 tained in this memoir, depends on an application of the doctrine of 
 fluxions, to fome properties drawn from the nature of the exponents 
 of powers. Here all numbers are confidered as fome certain powers 
 of a conftant determinate root : fo, any number x may be confidered 
 
 as the z power of any root, or that x = ris a general expreflion 
 
 for all numbers, in terms of the conftant root 7, and a variable expo- 
 
 nentz. Now the index z being the logarithm of the number », 
 
 therefore, to find this logarithm, is the fame thing, as to find what 
 
 power of the radical ris equa! to the number x. 
 ) From 
 
136 CONSTRUCTION OF 
 
 From this principle, the relation between the fluxions of any 
 number, +, and its logarithm z, is thus determined: Put r = 1 + 23 
 
 : 2 S : z+ Zz 
 then is x =r = (L+2), and# + «¢=(1 +2) “=(14n) xX 
 {t+ n) =m x (1+ n) , which by expanding (1 + n)'s omitting the 
 a 
 
 2d, 3d, &c powers of z, and writing ¢ for rey becomes 
 eben Xs gtigtig tig kes | 
 
 therefore + = axz, putting a for the feries g + 29? + ty° &e;y 
 
 a% . ° I 
 or fa = «#2, plitting f = i, 
 
 Now when r = 1 + 2 = 10, as in the common logarithms of 
 Briggs’s form; then »=9,q = ,9, and the feries g + 49’ + 44% 
 &c, gives @ = 2,302585 &c, and therefore its reciprocal f = 
 
 9434294 &c. Butifea=1 =f, the form will be that of Napier’s 
 logarithms. | 
 
 RIDE MO 
 From the above form rz = fx, or z =~» are then deduced many 
 
 curious and general properties of logarithms, with the feveral feries 
 heretofore given by Gregory, Mercator, Wallis, Newton, and 
 Halley. But of all thefe feries, that one which our author feleéts 
 f-—p 
 
 j » the 
 r . Cass 
 
 logarithm of is = 2fx: N+iN° + 5N? + 5N’ &c, in the 
 
 for conftructing the logarithms, is this, putting N = 
 
 cafe in which r— p is = 1, and confequently in that cafe 
 3 
 
 N= oho Spa : 
 
 Elence, having given any numbers, 9”, g, r, &c, and as many 
 ratios’ a, 5, c, &c, compofed of them, the difference between the 
 two terms of each ratio being 1; as alfo the logarithms 4, B, C, &c 
 of thofe ratios given: to find the logarithms P, 9, R, &c of thofe 
 numbers ; fuppofing f= 1. For inftance, if p= 2, g=3, r= 53 
 
 ; which feries will then converge very faft. 
 
 d . a | eed 7 : h ] 
 — > b — ; 3 — pee ae N W = 
 
 rithms 4, B, C, of thefe ratios a, 4, c, being found by the above 
 feries, from the nature of powers we have thefe three equations, 
 A= 22 —3P which equa- (P = 34 + 48 + 2C = log. of 2. 
 Bx=4P— Q—- Rt tions re- = 5A + 6B + 3C = log. of 3. 
 C=2R— Q—3P) duced give (R=7A+ OB + 5C = log. of 5. 
 And hence P+ R=104+4 13B + 7C is = the logarithm of 
 2X 5or 10. 
 An elegant tract on logarithms, as a comment on Dr. Halley’s 
 method, was alfo given by Mr. Jones, in his Syzopfs Palmariorum 
 Mathefeos, publifhed in the year 1706. And, in the Philofophicak 
 Tranfactions, he communicated various improvements in goniome- 
 trical 
 
LOGARITHMS. 121 
 
 trical properties, and the feries relating to the circle and to trigono- 
 metry. | | 
 
 The memoir above defcribed was delivered to the Royal Society 
 by their then librarian, Mr. John Robertfon, a worthy, ingenious, 
 and induftrious man; who alfo communicated to the Society feveral 
 little tracts of his own relating to logarithmical fubje€ts; he was 
 alfo the author of an excellent Treatife on the Elements of Naviga- 
 tion in two volumes; and he was fucceilively mathematical mafter 
 to Chrift’s hofpital in London; head mafter to the royal naval aca- 
 demy at Portfmouth ; and librarian, clerk, and houfekeeper to the 
 Royal Society; at whofe houfe, in Crane-Court, Fleet-Street, he died 
 in 1776, aged 64 years. 
 
 And among the papers of Mr. Robertfon, Ihave, fince his death, 
 found one containing the following particulars relating to Mr. Jones, 
 which I here infert, as 1 know of no other account of his life, &c, 
 and as any true anecdotes of fuch extraordinary men mutt always be 
 acceptable to the learned. ‘This paper is not in Mr. Robertfon’s hand 
 writing, but in a kind of running law-hand, and is figned R. M. 
 12 Sept. 1771. 
 
 “ William Jones, Efq. F.R.S. was born at the foot of Bodavon 
 mountain [Mynydd Bodafon], in the parifh of Llanfihangel tre’r 
 Bardd, in the ifle of Anglefey, North Wales, in the year 1675. 
 His father John George * was a farmer of a good family, being 
 defcended from Hwfa ap Cynddelw, one of the 15 tribes of North 
 Wales. He gave his two fons the common fchool education of the 
 country, reading, writing, and accounts, in Englifh, and the Latin 
 grammar. Harry his fecond fon took to the farming bufinefs; but 
 William the eldeft, having an extraordinary turn for mathematical 
 ftudies, determined to try his fortune abroad from a place where the 
 fame was but of little fervice to him; he accordingly came to Lon- 
 _don, accompanied by a young man, Rowland Williams, afterwards 
 “an eminent perfumer in Wych-Street. ‘The report in the country 
 is, that Mr. Jones foon got into a merchant’s counting-houfe, and 
 fo gained the efteem of his mafter, that he gave him the command 
 of a fhip for a Weft-India voyage; and that upon his return he fet 
 up a mehematical fchool, and publithed his book of navigation + ; 
 and that upon the death of the merchant he married his widow: that 
 Lord Macclesfield’s fon being his pupil, he was made fecretary to the 
 chancellor, and one of the D. tellers of the exchequer—and_ they 
 have a ftory of an Italian wedding which caufed great difturbance in 
 
 Lord Macclesfield’s family, but compromifed by Mr. Jones; which 
 
 * Tt is the cuftom in feveral parts of Wales for the name of the father to be- 
 come the furname of his children. John George the father was commonly call- 
 ed Sion Siors of Llambabo, to which parifh he moved, and where his children 
 were brought up.” 
 
 + This traét on navigation, intitled, ‘‘ A new Compendium of the whole Art 
 of Practical Navigation,” was publifhed in 1702, and dedicated “ to the reverend 
 and learned Mr. John Harris, M.A. and F.R.S.”’ the author, I apprehend, of 
 the “ Uniyerfal Di@ionary of Arts and Sciences,’ under whole roor Mr. Jones 
 jays he compefed the faid treatife on Navigation. 
 
 : R gave 
 
122 CONSTRUCTION OF 
 
 gave rife to a faying, that Macclesfield was the making of Jones, and 
 Jones the making cf Macclesfield.” | 
 Mr. Jones died July 3, 1749, being vice-prefident of the Royal 
 
 Society; and left one daughter, and his widow with child, which 
 proved a fon, who was the late Sir William Jones, one of the judges 
 in India, and highly efteemed for his great abilities and extenfive 
 learning. ' 
 
 - Euler’s method given in his Introd. in Anal. Infinit. is much the 
 fame, in manner and effect, as that of Mr. Jones, given above. 
 
 Of Mr. Andrew Reid and Others. 
 
 Andrew Reid, Efg. publifhed in 1767 a quarto tract, under the 
 title of An Effay on Logarithms, in which he alfo fhows the compu- 
 tation of logarithms from principles depending on the binomial theo- 
 rem andthe nature of the exponents of powers, the logarithms of 
 numbers being here confidered as the exponents of the powers of 10. 
 He hence brings out the ufual feries for logarithms, and largely 
 exemplifies Dr. Halley’s moft fimple conftruction. 
 
 Befides the authors whofe methods have been here particularly 
 defcribed, many others have treated on the fubje&t of logarithms, and 
 of the fines, tangents, fecants, &c ; among the principal of whom 
 are Leibnitz, Euler, Maclaurin, Wolfius, and prefeffor Simfon in 
 an elegant geometrical tract on logarithms, contained in his pofthu- 
 mous works, elegantly printed in 4to. at Glafgow, inthe year 1776, | 
 at the expence of the very learned Earl Stanhope, and by his Lord- 
 fhip difpofed of in prefents among gentlemen moft eminent for ma- 
 thematical learning. 
 
 Of Mr. Dodfon’s Anti-logarithmic Canon. 
 
 ‘The only remaining confiderable work of this kind publifhed, that 
 I know of, is the Anti-logarithmic Canon of Mr. James Dodfon, an 
 ingenious mathematician, which work he publifhed in folio in the — 
 year 1742: a very great performance, containing all logarithms 
 under 100000, and their correfponding natural numbers to 1! places 
 of figures, with alltheir differences and the proportional parts; the 
 _ whole arranged in the order contrary to that ufed in the common 
 tables of numbers and logarithms, the exact logarithms being here 
 placed firft, and increafing continually by 1, from 1 to 100000, with 
 their correfponding neareft numbers in the columns oppofite to 
 them; and by means of the differences and proportional parts, the 
 Jogarithm to any number, or the number to any logarithm, each to 
 11 places of figures, is readily found. ‘This work contains alfo, be- 
 fides the conftruction of the natural numbers to the given logarithms, 
 “* precepts and examples, fhowing fome of the ufes of logarithms, 
 in facilitating the moft dificult operations in common arithmetic, 
 cafes of intereft, annuities, menfuration, &c; to which is prefixed 
 an introduction, containing a fhort account of logarithms, and of 
 the moft confiderable improvements made, fince their invention, in 
 the manner of conftruCting them.” | 
 
 The manner in which thefe numbers were conftruCted, confifts 
 
 chiefly 
 
4 
 
 LOGARITHMS. 123 
 
 chiefly in imitations of fome of the methods before ‘defcribed. by 
 Briggs, and is nothing more than generatinga fcale of 100000 geo- 
 metrical proportionals, from 1 the leaft term to 10 the greateft, each 
 
 continued to 11 places of figures; and the means of effeCting this, 
 
 are fuch as eafily flow from the nature of a feries of proportionals, 
 and are briefly as follow. Firft, between 1 and 10, are interpofed 9 
 mean proportionals; then between each of thefe 11 terms there are 
 interpofed 9 other means, making in all 101 terms; then between 
 each of thefe a 3d fet ot 9 means, making in all 1001 terms; again 
 between each of thefe a 4th fet of 9 means, making in all 10001 
 terms ; and laftly, between each two of thefe terms, a 5th fet of 9 
 means, making in all 100001 terms, including both the 1 and the 
 10. The firft four of thefe 5 fets of means, are found each by one 
 extraction of the 10th root of the greater of the two given terms, 
 which root is the leaft mean, and then multiplying it continually by 
 itfelf according to the number of terms in the fection or fet ; and the 
 5th or laft feGiion is made by interpofing each of the 9 means by 
 help of the method of differences before taught. Namely, putting 
 10 the greateft term 
 
 I x i ae | rt 
 a= ASGHAR ri, (BE? ox © ORF = DD) * ono Eeand Er Snr ke now 
 extracting the 10th root of A or 10, it. gives 1,25892541138 =—B 
 fe ses ie 
 — A T° for the leaft of the 1ft fet of means; and then multiplying 
 it continually by itfelf, we have B, B?, B3, B+, &ctoBt° =A 
 for all the 10 terms: 2dly, the 10th root of 1,2589254118 gives 
 i ays i 
 1,0232929923 = C = Bt® — A‘ for the leaft of the 2d clafs of 
 means, which being continually multiplied gives C, C*, C’, &c, to 
 
 Creo = Bt? = A for all the 2d clafsof 100terms: 3dly, the 10th root 
 
 Zz I Lt 
 pi l,02 39920922 aiwes 1002 105038 tien) CTS —— BI Aree 
 for the leaft of the 3d clafs of means, which being continually 
 mitiltipliedsepives.D,'D%, Ds 8rcito .D ieoh = Cr2? Book for 
 the 3d clafs of 1000 terms: 4thly, the 10th root of. 1,0023052381. 
 
 ir I 
 gives 1,0002302850 = E = Dro — CTs — Bross = ATCO foy 
 the leaft of the 4th clafs of means, which being continually mul- 
 Ciplicd, dives Deis oc cD. bore ee ee eo Bre 
 for the 4th clafs of 10000 terms. Now thefe 4 claffes of terms, thus 
 produced, require no lefs than 11110 multiplications of the leaft 
 means by themfelves: which however are much facilitated by making 
 a fmall table of the firft 10 or even 100 produ€ts of the conftant 
 multiplier, and from thence. only taking out the proper lines and 
 adding them together: and thefe 4 clafies of numbers always prove 
 themfelves at every 10th term, which mutt always agree with the 
 correfponding fucceffive terms of the preceding clafs. The remain- 
 ing 5th clafs is conftructed by means of differences, being much 
 eafier than the method of continual multiplication, the 1ft and 2d 
 differences only being ufed, as the 3d difference 1s too {mall to enter 
 the computation of the fets of 9 means between each two terms of 
 the 4th clafs. And the feveral 2d differences for each of thefe fets 
 of 9 means, are found from the properties of a fet of proportionals 
 
 ly 7, 
 
124 CONSTRUCTION OF LOGARITHMS. 
 
 l,r, 7°, ry &c, as [Terms] itdif. | 2ddif. | 3ddif. | &e 
 difpofed in the lft co- 1x |(r—1) x|(7—1)P X|(r—1)3 X] 
 lumn of the annexed 1 1 aa | 1 
 
 table, and their fever 
 ral orders of differences 
 as in the other columns 
 of the table ; where it 
 
 is evident that each 
 column, both that of the given terms of the progreffion, and thofe 
 
 of their orders of differences, forms a f{cale of proportionals, having 
 the fame common ratio r, and that each horizontal line, or row, forms 
 a.geometrical progreflion, having all the fame common ratio r— 1, 
 which is alfo the 1{t difference of each fet of means; fo (r—1)* is 
 the 1ft of the gd differences, and which is conftantly the fame, as 
 the 3d differences become too {mall in the required terms of our pro- 
 ’ greffion to be regarded, as leaft near.the beginning of the table: hence, 
 like as 1, y-—1, and (r-—1)’ are the firft term, with its 1ftand 2d differ- 
 
 ences; for’, r° (r—1), and r” (r—1)’ are any other term with its 
 1ft and 2d differences. And by this rule the 1{t and 2d differences 
 are to be found for every fet of 9 means, viz, multiplying the 1 
 term of any clafs (which will be the feveral terms of the feries E, E’, 
 E°, &c, or every 10th term of the feries F, F’, Fy &c) by r—1 or 
 F—1 for the 1ft difference, and this multiplied by F—1 again for 
 the true 2d difference at the beginning of that clafs. Thus, the 
 10th root of 1,0002202:50 or E gives 1,000023026116 for F, or 
 
 the 1ft mean of the loweft clafs, therefore F —1 =r—1 = 
 3000023026116 is its 1{t difference, and the fquare of it is (r—1)? 
 
 F 2 f 10 
 
 = ,0000000005302 its 2d difference; then is ,000023026116F ” 
 ; 20 
 
 or ,00002302611G6E* the 1ft difference, and ,0000000005302F " 
 
 or ,0000000005 302K” is the 2d difference at the beginning of the 
 
 mth clais of decades. And this 2d difference ts ufed as the conftant 
 ad difference through all the 10 terms, except towards the end of 
 the table, where the ditterences increafe faft enough to require a 
 {mall corre€tion of the Yd difference, which Mr. Dodfon effects 
 by taking a mean 2d difference among all the 2d differences, in this 
 
 manner; having found the feries of 1ft differences (F—1) E™ 
 
 n—t-1 ‘ . ene 
 (F—1)E + »(F—1) E Bie &c, take the differences of thefe, and 
 gs of them will be the mean 2d differences to be ufed, namely, 
 
 F—1 pr! es E)’, me l (Er ta: ET) 
 
 > &c, are the mean 
 
 \ 
 
 2d diiferences. And this is not only the more exa@, but alfo the ea- 
 fier way. ‘The common 2d difference, and the fucceflive 1 ft differ- 
 ences, are then continually added, through the whole decade, to give 
 the fucceilive terms of the required progreilion. 
 
 DESCRIPTION. 
 
i C4 ae ase 
 
 DESCRIPTION AND USE 
 
 OF 
 
 LOGARITHMIC TABLES, 
 
 Auruoucu the nature and conftruction of logarithms have 
 been pretty fully treated in the preceding hiftory of fuch numbers, 
 ‘where the more learned and curious reader will find abundant fatif- 
 faction, I fhall here give a brief, eafy, and familiar idea of thefe 
 - matters, for the practical ufe of young ftudents in this fubject. 
 
 Lhe Definition and Notation of Logarithms. : 
 
 Logarithms are the indices or arithmetical feries of numbers, adapt- 
 ed to the terms of a geometrical feries, in fuch fort that 0 correfponds 
 to 1, oris the index of it, in the geometricals. 
 
 Thus iW wl YP 3 4... 5, &c. indices or logarithms, 
 
 is ee 8 16 32, &c. geometric progreflion. 
 
 ae ee bv 2 aes ‘i 5, &c. indices or logarithms, 
 ae eae a = 243, &c. geometric feries. 
 
 a2 3 + 5,. &c. indices or logarithms, 
 
 Where the fame indices ferve equally for any geometric feries; and 
 from which it is evident, that there may be an endlefs variety of 
 fyftems of logarithms to the fame common numbers, by varying the 
 2d term, 2, or 3, or 10, &c, of the geometric feries; as this will 
 change the original feries of terms, whofe indices are the integer 
 numbers, 1, 2, 3, &c3 then by interpolation the whole fyftem of 
 numbers may be made to enter the geometrical feries, and receive 
 their proportional logarithms, whether integers or decimals, 
 
 Or, the logarithm of any number is the index of that power of 
 fome other number, which is equal to the given number. So, if N 
 
 0 7 
 vi } 1, 10, 100, 1000, 10000, 100000, &c. geometric feries. 
 
 be =r’, then the logarithm of N is z, which may be either pofitive 
 or negative, andr any number whatever, according to the different 
 fyftems of logarithms. When N is 1, then =0, whatever the va- 
 jue of ris; and confequently the logarithm of 1 is.always 0 in every 
 fyftem of logarithms. When »# is = 1, then- N is = +3; confe- 
 quently 7 is always the number whofe logarithm is 1, in every fyftem. 
 When 7 is = 2°718281828459 &c, the indices are the hyper- 
 bolic logarithms, fuch as in our 7th table; fo that is the hyperbo- 
 
 lic logarithm of (2°718 &c)’. But in the common logarithms, r 
 is 
 
F 
 126 DESCRIPTION AND USE 
 
 is = 103 fo that the common logarithm of any number ( 10°) is 
 (x) the index of that power of 10 which is equal to the faid num- 
 ber. So 1000, being the 3d power of 10, has 3 for its logarithm; 
 
 1-69897 ; 
 and if 50 be = 10 , then is 1°69397 the common logarithm of 
 50. And hence it follows, that this decupal feries of terms 
 
 4 
 
 10% , 103 , 107,10", 10°, 10. 10. 10 ;, 10 "ts 
 
 or 10000, 1000, 100, 10, 1, ‘l ,*Ol ,*00L 4, -0001 , 
 have 4°. §)...38) gp ID 5 Beg endr sabes | 2s GS Ses = 
 re{pectively for their logarithms. 
 
 The logarithm of a number comprehended between any two terms 
 of the firft feries, 1s included between the two correfponding terms 
 of the latter, and therefore that logarithm will confift of the fame 
 index (whether pofitive or negative) as the lefs of thofe two terms, 
 together with a decimal fraCtion, which will always be pofitive. So 
 the number 50, falling between 10 and 100, its logarithm will fall 
 between 1 and 2, and is=1°69897, the index of the lefs term, toge- 
 ther with the fame decimal *69897 as before: aifo the number.°05, 
 falling between the terms*l and ‘01, its logarithm will fall between 
 —— 1 and — 2, and. is indeed = — 2 + °69897, the index of the 
 lefs term together with ftill the fame decimal:69897. Uhe index is alfo 
 called the characteriftic of the logarithms, and is always an integer, 
 either pofitive or negative, or elfe = 0; and it fhows what place is 
 occupied by the, firft fignificant figure of the given number, either 
 above or below the place of units, being in the former cafe + or po- 
 fitive, in the latter — or negative. es Re 2 , 
 
 When the characteriftic of a logarithm is negative, the fign — is 
 commonly fet over it, to diftinguith it from the decimal part, which 
 being the logarithm found in the tables, is always pofitive: fo 
 
 — 2+ °69897, or the logarithm of :05, is written thus 2°69897. 
 But on fome occafions it is convenient to reduce the whole expreflion 
 to a negative form ; which is done by making the charaCteriftic fi- 
 gure lefs by 1, and taking the arithmetical complement of the deci- 
 mal, thatis, beginning at the left hand, fubtract each figure from 9, 
 except the laft fignificant figure, which fubtra&t from 103; fo fhall the 
 
 remainders form the logarithm entirely negative. ‘Thus the logarithm 
 of -05, which is 2°69897, or — 2 + 69897, is alfo exprefled by 
 — 1*3010%, which is wholly negative. It is alfo fometimes thought 
 more convenient to exprefs fuch logarithms wholly as pofitive, 
 namely, by only joining to the tabular decimal the complement of 
 the index to 10: in which way the above logarithm is exprefled by 
 8*69897 5 which is only increafing the indices in the fcale by 10, 
 [tis allo convenient, in many operations with logarithms,to take 
 their arithmetical complements, which is done by ‘beginning at the 
 left hand, and fubtra€ting every figure from 9, but the lait figure 
 from 10: fo the arithmetical complement : 
 
 of 1°69897 Cand of 2°698972 where the index — 2, being nega- 
 is 8°30103 @° itis 11°30103 § tive, is added to 9, and makes 11. 
 ag The 
 
OF LOGARITHMIC TABLES. 127 
 
 The Properties of Logarithms. 
 
 From the definition of logarithms, either as being the indices of 
 a feries of geometricals, or as the indices of the powers of the fame 
 root, it follows, that the multiplication of the numbers will anfwer 
 to the addition of their logarithms; the-divifion of numbers to the 
 {ubtraction of their logarithms; the raifing of powers, to the multi- 
 plying the logarithm of the root by the index of the power; and the 
 extracting of roots, to the dividing the logarithm of the given number 
 by the index of the root required to be extracted. So. 
 
 1B Lae! ot? @ Xe si Ls a@i-b Deb 
 Tici8 ors Gis L. 3 4d. 6 
 B65, KO 74a Ais La eee. 0 4 Lg 
 2d. Li. a 5 1S) Die oe a 
 L. 18~+6 is=L. is—L. 6 
 . L. 7ox5+—9is = L. FO+L 5—bL. 9 
 L. fdorl+2is=bL.1—-L2=-0-LiI—L2 
 torn isc: — lex 
 z ~ 
 ™ 
 
 ¥ on , Bich 
 Ris n ° 1} Ne wt 
 Sd Lar ist 8le rs Lr On la ria re Lr is ~ Lr. 
 nN ‘ 
 
 I 3 3 
 Ls 20s = 01,2; D.-25 or Ly Bis fF 23 25 is ee ge 
 So that any number and its reciprocal have the fame logarithm, 
 
 but with contrary figns; and the fum of the logarithms of any num- 
 ber and its complement, is equal to 0. ; 
 
 Lo conftrud Logarithms. 
 It has been fhewn, in the foregoing hiftorical part, that the loga- 
 af 
 
 “hy ¢ db. 2g x x3 ¥? x 7 ty ‘ 
 rithm of — is = — rey Fee ag eas —— &c, where 
 ae a pr ieos tere ta? P “re 
 
 the sum, and » the difference of a and 4; alfo m= 2°202585092994 
 &c, the hyp. logarithm of 10. Therefore if a and 4 be any two 
 numbers differing only by unity, fo that x or 6— a may be = 1; 
 2 1 1 1 
 
 then fhall the logarithm of bbe = L. a + noe hana t Bye oe 
 Which gives this rule in words at length: call z the fum of any 
 number (whofe logarithm is fought) and the number next lefs by 
 unity ; divide °8685889638 &c (or 2 + 2°23025 &c) by z, and re- 
 ferve the quotient: divide the referved quotient by the fquare of z, 
 and referve this quotient: divide this laft quotient alfo by the fquare 
 of z, and again referve this quotient: and thus proceed, continually, 
 dividing the laft quotient by the {quare of z, as long as divifion can 
 be made. - Then write thefe quotients orderly under one another, the 
 firft uppermoft, and divide them refpectively by the uneven num- 
 bers 1, 3, 5, 7) 9, 11, &c, as long as divifion can be gs : 
 
 that 
 
123 “DESCRIPTION AND USE 
 
 that is, divide the firft referved quotient by 1, the 2d by 3, the 3d 
 by 5, the 4th by 7, &c. Add all thefe laft quotients together, then ~ 
 the fum will be the logarithm of 6 + a; and therefore to this loga- 
 rithm adding alfo the logarithm of @ the next lefs number, the 
 {um will be the required logarithm of d the number propofed. | 
 
 Er.1. To find the Log. of 2. » Ex,2. To find the Log. of 3. ° 
 Here the next lefs number is1, & 2+ 1/Herethe next lefs number is 2, and 2+ 3 
 =— 3 = 2, whofe fquare is 9. ‘Then = 5 = 2, whofe fquare is 25, to divide 
 
 3)'868588964| 1)-289529054(:289529654| by which always multiply by ‘04. Then 
 ©)°289529654| 3) -32169962( 1072332115 ):868588964] 1)'173717703(°173717793 
 ©) 32169962; 5) 3574440( 714888|25)-173717793] 3) 6948712( 2316237 
 ©) 3574440 7)  397160( 56737|25) 60948712) 5)  277948( 55590 
 
 9) 397:60' 9) 44129( 4903/25) 277948] 7) 11118( 1588 
 ©) 4412911) A903¢ ' 446/25) 11318} 9) 4A8( 50 
 Q) 4903 13) 545 ( 42|25) 445\i1) 18( 2 
 9) 545 15) O1¢ a TS - « °176091260 
 
 A annette ee y, i PF 2 
 9) | 61) Log. = *301020995 L. 2add - -301029905 
 Add L.1 = *000000000 Li3 0 ~ 4771291255 
 
 Log. of 2 = | *301020995 
 
 Then becaufe the fum of the logarithms of numbers gives the lo- 
 garithm of their product, and the difference of the logarithms gives 
 the logarithm of the quotient of the numbers, from the above two 
 
 ‘logarithms, and the logarithm of 10 which 1s 1, we may raife a 
 great many other logarithms, thus: 
 
 Ex. 3% Becaufe2 x 2— 4, therefore| Ex.6, Becaufe 32 = 9, therefore 
 
 tol.2 - - - *3010299952 L.3° - + = 477121254, 
 add b.2-+ - - °301029995% mult. by2 = + y Ee 
 fumisL.4 - - 6020599914 gives L.Q ~- °054242509 © 
 _Ex. 4. Becaufe2 x 3 —6, therefore! Ex. 7, Becaufe 42 = 5, therefore 
 toL.2 - = ~- *301020095 ‘from L. 10 - 1°000000000 
 add L.3  - = °477121255 takeL.2 - -. ‘*3010299952 
 ‘ fumisL,6 - - °778151250 leaves L.5 = °698Q700044 
 Ex.5, Becaufe 23 = 8, therefore |Ex. 8. Becaufe 12 =3 x 4, therefore 
 L,2- - - = 3010299952 toL.3 - - = °477121255 
 mult. by3 - - 3 addL.4 - + ‘602059901 . 
 ‘ gives]L.8 - - ‘908089987 |. givesL.12 - 1°079181246 
 
 And thus, computing, by the general rule, the logarithms of the 
 other prime numbers 7, 11, 13, 17, 19, 23, &c3 and then ufing 
 compofition and divifion, we may eafily find as many logarithms as 
 we pleaie, or may ipeedily examine any logarithm in the table. 
 
 THE 
 
( 199 ) 
 
 THE DESCRIPTION AND USE OF THE TABLES, 
 
 Tue following colle€tion .confifts of various tables, in the fol- 
 lowing order, viz. 1, A large table of logarithms to 7 places of 
 figures; 2, A table for finding logarithms and numbers to 20 places; 
 3, Logarithms to 20 places, with their 1f{t, 2d, and 3d differences 5 
 4, Another table of logarithms to 20 places, with their 1ft, 2d, and 
 3d differences 5 5, Logarithms to 61 places ; 6, Another table of !o- 
 garithms to 61 places, with their 1{t, 2d, 3d, and 4th differences 3 
 71, Hyperbolic logarithms ; 8, Logiftic logarithms, 9, Logarithmic fines 
 and tangents to every fecond of the firft ¥ degrees; 10, Natural and 
 logarithmic fines, tangents, fecants, and verfed fines, with their differ- 
 ences to every minute of the quadrant. After which follow feveral 
 {maller tables ; as a table of the lengths of circular arcs; a traverfe 
 table, or table of difference of latitude and departure, to every degree 
 and quarter point of the compafs; atable for changing the common 
 logarithms into hyperbolic logarithms; and a table of the names and 
 number of degrees &cin every point of the compafs ; as alfo lifts of 
 errata in various works of this fort. Of each of which in their order. 
 
 Of the large Table of Logarithms. 
 
 The firft is the large table of logarithms, to all numbers from 1 to 
 100000; by which may be found the logarithm to any number, and 
 the number to any logarithm, to 7 places of figures. ‘This table con- 
 fifts of two parts; the firft contains, in 4 pages, the firft 1000 num- 
 bers with their correfponding logarithms in adjacent columns ; the fe- 
 cond contains all the 100000 numbers and their logarithms, with the 
 differences and proportional parts, difpofed as follows: in the 1{t co- 
 lumn of each page are the firft 4 figures of the numbers, and along 
 the top and bottom of the columns is the 5th figure, in which columns 
 are placed all the logarithms, the firft 3 figures of each logarithm be- 
 ing at the beginning of the lines in the firft column of logarithms, 
 figned O at the top and bottom, and the other 4 figures in the remain- 
 ing columns. Sometimes the firft three figures of the logarithms are 
 found in the line next below the number, viz. when the fourth fi- 
 gures have changed from 9’s to 0’s, in which cafe, a baris placed 
 over the firft cipher, to catch the eye, thus 0. After the 10 columns 
 of logarithms, ftands their column of differences, figned D ; and laftly, 
 after that, the column of proportional parts, figned pro. pts. fhowing 
 what proportional part of each difference correfponds to 1, 2, 3, 
 &c, the whole difference anfwering to 10; or fhowing the +5, +s, 
 +35, &c, of the differences. , 
 
 Note, The logarithms in thefe columns are all fuppofed to be deci- 
 mals, and their correfponding natural numbers may be either inte- 
 gers or decimals or mixt numbers; for the fame figures, whatever be 
 their denomination, have the fame decimal logarithm, and_thefe 
 differ only in the index or characteriftic, which is the integer num- 
 
 9 ber 
 
130. DESCRIPTION AND USE 
 
 ber to be prefixed to the decimal part of the logarithm ; and this is 
 always the number which exprefles the diftance of the higheft deno-. 
 mination, or left-hand figure, of the natural number from the units 
 place. So that if the natural number confift of only one place of 
 integers, the index of its log. will be 0: if of 2, 3, 4, 5, &c, the 
 index of its logarithm will be refpeétively 1; 2, 3, 4, &c, being 
 1 lefs than the number of integer places: and the fame figures made 
 negative will give the index of the logarithm of a decimal, viz. if 
 the. natural number be a decimal, and its firft fignificant figure be 
 in the place of primes, 2ds, 3ds, 4ths, &c, the index of its logarithm 
 will be refpectively 1, 2, 3, 4, &c, or the figure which ex- 
 prefles the diftance of the firft place of the natural number from 
 the units place, but with a negative fign, as the number is below the 
 place of units, the fign being written above the index inftead of 
 before it, as that part only of the logarithms is to be confidered as 
 negative, the decimal part of it bemg always affirmative. And in 
 the arithmetical operations of addition and fubtra€tion with loga- 
 rithms, the negative indexes will have the contrary effet to that of 
 the decimal part of the logarithm, viz, when the logarithm is to be 
 added, the figure of the negative index muft be fubtracted, & vice 
 verfa. Hence if 4234097 be the tabular 
 or decimal part of the logarithm belong- 
 ing to the figures 2651, without any re- 
 
 4 
 
 Number Logar. 
 2651 3°4234097 
 
 gard to their particular denominations ; 265") 2+4.234097 
 then according as they are varied with 26°51 1°4234.097 
 re{fpect to the number of decimals, as in 2°651 0°4234097 
 the 1 ft annexed column, the index of “O61 114934097 
 their logarithm, and the complete loga- nbs bhhes4b04 
 s é, ‘ ue ig Oe Lee a)* x 
 rithm, will vary as in the 2d column here rrte: TNS BROIL 
 70026 51)3°423 L097 
 
 annexed. And hence, like as when the 
 natural number is given, we find the in- ot 
 dex of.its logarithm by counting how far its firft figure on the left 
 hand is from the units place; fo when:a logarithm is given, the de- 
 nominations of the figures in its natural number will be found by 
 placing the decimal point fo, that the number of integer places may be 
 { more than that of the index when pofitive, or by fetting the firft 
 fignificant figure in that decimal place, which is exprefied by the 
 number of the index when negative. : 
 
 Of “finding the Logarithm of a given Number, or the Number to a given 
 ° Logari thm. f 
 
 1. To find the Logarithm, of a Number conjfifting of 3 figures. 
 Find the number in the column of numbers in one of the firft 4 
 pages of the table, and immediately on the right of it is its logarithm _ 
 fought. So the logarithm of 72 is 1°8573525, and the logarithm of 
 
 3°33) is 075221442, when the proper index is fupphed, 
 ® Boh 
 
\ 
 
 OF THE TABLES. : 131. 
 
 2. To find the Logarithm of a Number confifiing of 4 Places. 
 
 In the firft column (figned N) in fome one of the pages of the table 
 after the firft four, find the given number, then againft it in the 2d 
 column (figned 0) is the logarithm fought. So the logarithm of 2254 
 is 3°3529539, and that of 31°32 is 1°4598218. 
 
 3. Lo find the Logarithm of a Number confifting of 5 Places. 
 
 / 
 
 Find the firft 4 figures of the given number in the firft column as 
 before, and the 5th figure at the top or bottom; then the 7 figures 
 of the logarithm are found in two columns on the line of the firft 4 
 figures of the given number, viz, the firft 3 figures of the logarithm 
 are the firft 3 common figures of the 2d column (figned 0), and the . 
 laft 4 figures are on the fame line, but in the column figned with 
 the Sth figure of the given number. So the logarithm of 23204 is 
 4*3655629, and that of 746°40 is 2°8729716, and that of °083178 
 is 2°9200085. : | 
 
 Note, When the laft four figures of the logarithm begin with a 
 cipher, or any figure lefs than the laft four in the 2d column begins 
 with, then the firft $ common figures are thofe in the next lower line: 
 fo in the laft example the firft 3 common figurés ate 920, and not 
 So. 
 
 4. To find the Logarithm of a Number of 6 Places. 
 
 Find the logarithm of the firft 5 figures by the laft article, and 
 take the difference between that logarithm and the next following 
 logarithm, or (which is the fame thing) find the difference neareft 
 oppofite in the laft column but one, figned D; then under that dif- 
 ference in the laft column (of proportional parts) and againft the 6th 
 figure of the given number, is the part to be added to the logarithm 
 before found for the firft 5 figures, the fum being the logarithm 
 fought. Soto find the logarithm of 3409°26: the logarithm. of 
 34092, the firft 5 figures, being 5326525, and the common difler- 
 ence 127, under which and againit 6 in the laft column is 76, 
 which being added to the former logarithm, and the proper index 
 prefixed, we have ¥*5326601 for the whole logarithm required. 
 
 5. Lo find the Légarithm of a Number of 7 Places. 
 
 Find the logarithm, of the firft ¥ figures by the 9d article, and 
 of the fixth figure by the 4th article; then for the logarithm of the 
 7th figure, divide its proportional part by 10, that is, fet it one place 
 further to the right hand than the laft figure of the logarithm reaches; 
 add all the three together, and their fum will be the logarithm re- 
 quired, , o 
 
 Thus 
 
132 DESCRIPTION AND USE 
 
 Thus, to find the logarithm of 3°409264. 
 
 The feveral parts beingtaken out according Numb. Logars’ 
 
 to the rule, and placed as in the margin, the 34092 - - 5326525 
 
 fum gives the whole logarithm fought. 6 - 76 
 Note, In the fame way we might take out 4 - 5b 
 
 the proportional part of an oth figure, divid- 3*40926 4-0'°5326606 
 ing its tabular part by 100, or fetting it two 
 
 places further to the right hand than the firft logarithm. Or the 
 whole proportional part for any number of figures above five, may 
 be found at once, by multiplying the common tabular 
 
 difference of the logarithms, found as before, by all 127 
 the figures after the 5th, cutting off from the product 54 
 as many figures as we multiply by, and adding the oU8 
 re{t to the logarithm of the firft 5 figures before 162 
 
 found. So in the laft example above, having found 81,28 
 
 the common difference 127, multiplying it by 64 the so0¢505 
 laft two figures, cutting off two, add the refit to the .=G5caq- 
 ; : : ; 0°5326606 
 logarithm of the firft 5, as in the margin. 
 For another example, fuppofe we wanted the Jogarithm of the fol- 
 lowing 8 figures 34092648. The operation by both methods will be 
 as below. | 
 
 127 
 94092 = = - = 5326525 ; 648 
 6mice other F6ix:i| wildest: 9 at01es 
 ch POE Ae saa Sil 508 
 Be hes me je 1,02 7162 
 34092648 - - ° 75326607 “82,296. 
 B326525 
 7°54326607 the fame as the other. 
 
 6. To find the Logarithm of a Vulgar Fraétion, or of a Mixt Number. 
 
 Either reduce the vulgar fraction to a decimal, and find its loga- 
 rithm as above. Or elfe (having reduced the mixt number to an 
 improper fra€tion), fubtract the logarithm of the denominator from 
 the logarithm of the numerator, and the remainder will be the loga- 
 rithm of the fraction fought. | ; 
 Ex.1.To find the log. of 3. or 0:1875| Ex.2. To find the log. of 13jor %. 
 From log,of 3 - - 04771213 |Fromlog.of 55.9 -..-  1°7403027 
 Take log. of 16 - - 1°2041200 |Take log. of 4 - - 0'6020600 
 Rem, log. of 3.or ‘1875 1°2730013 (Leaves log. of 58 or 13°75 11393027 
 
 1. Lo find the Natural Number anfiering to,any given Logarithm, 
 
 Find the firft 2 figures, next after the index of the given logarithm, 
 in the fecond column, figned 0, and the other 4 figures on the fame 
 line in one of the nine following columns if the figures of the loga- 
 
 rithm 
 
OF THE TABLES. 133 
 
 rithm be thus found exactly, then on the fame line in the firft column 
 are the firft four figures of the natural number, and the 5th is at the 
 top or bottom of that column in which the laft four figures of the log. 
 were found. So to find the number anfwering to the logarithm 
 2°5890108. In pa. 64 I find the firft three figures 539, and in co- 
 lumn 6 of the line above are found the other four *0108, becaufe the 
 firft three common figures are fuppofed to begin at that part of the 
 line above where they are placed): then on the fame line in the co-' 
 Jumn of numbers ftand the firft four figures 388°1, and 6 at the top 
 of the column, making in all 388-16 for the number fought ; having’ 
 placed the decimal point fo as to make three integers, being 1 more 
 than 2 the index of the given logarithm. 
 
 But if the given logarithm be not found exaétly in the table, fub- 
 tract the next lefs tabular logarithm from it, and look for the re- 
 mainder in the proportional parts under the difference between the 
 two tabular logarithms next lefs and greater than the given loga- 
 rithm, and againft it, or the part next lefs, isa 6th figure to be an- 
 nexed to the five figures before found. And if the remainder be not 
 found exaétly in the proportional parts, fubtract the next lefs part 
 from it, and annex a cipher to this 2d remainder, then againft the 
 neareft proportional part (either greater or lefs) is a 7th figure to be 
 annexed to the fix before found. And that figure will be the neareft 
 to the truth in that place, either too much or too little. , 
 
 Ex. To find the number an{wering to the logarithm 1°2335678. 
 
 The next lefs tab. log. is the log. of 17122 viz. 2335545 
 1ft rem. 133 
 The difference is 254 i - 5 for hp al —", 
 _ 2d rem. 
 and the table of pro. pts. gives¢_ _ sg tattle tmart oa) cues 
 
 So that the number fought is 17°12252, making two integers for 
 the index |. 
 
 Or the 6th and 7th figures may be found without the table of pro- 
 portional parts, by dividing the firft remaindér by the tabular differ- 
 ence, annexing one cipher to the dividend for each | 
 figure to be found. So, in the laft example, the re- 254)133,00(52 
 
 mainder 133, with two ciphers annexed, being di- 127,0 
 vided by the tabular difference 254, as in the margin, “600 
 the quotient gives 52 for the 6th and 7th figures, 508 
 
 the fame as before. - 
 
 {n like manner may be found the numbers to the following loga- 
 _ rithms. 
 Loar. 1°2345678 
 Numb,17+10200 
 
 3°7343003] 1 0921406)2°3720468/4:'6123004 3-2946809 
 5*423758 |'1236348 /-02355303 40954'39 |1970°974 
 
 OF 
 
134 DESCRIPTION AND USE 
 
 OF LOGARITHMICAL ARITHMETIC. 
 I. Multiplication by Logarithms. 
 
 Add together the logarithms of all the faCtors; then the fum isa 
 logarithm, the natural number correfponding to which, being found 
 in the table, will be the product required. | ) 
 
 Obferving to add, tothe fum of the afhrmative indices, what is 
 carried from the fum of the decimal parts of the logarithms. 
 
 And that the difference between the affirmative and negative in- 
 _ dices, is to be taken for the index to the logarithm of the product. 
 
 Ex. 1. To multiply 23:14 by 5:062. (2x. 2. To mul. 2581926 by 3'457291. 
 23°14 its log. is 1°3643634 2°581920 iis log. is 04119438 
 
 5062 its log. is 0.7013221 3°457291 - - - 0°5387359 
 
 — Product 117'1347  - 2°0080855) Prod. 8.92047 - - 0'9506797 
 ———— pe aad 
 
 Ex. 3. To mult. 3-902, and 597°16,|Ex. 4. To mult. 3°586, and 2°1046, 
 and ‘0314728 all together. and 0°8372, and 0'0294 all together, 
 
 3°902 its log. is 0°5912875 3°586 its log. is 0°5546103 
 50716 + = 2°7760907 21046 ~ - 03231606 
 “0314728 - ~ 2:4979353 1°9228292 
 Prod. 73 33533 - + 1°8653133 0'0294 - -. 2'4683473 
 
 Prod. °1857618- - 1'2689564 
 Here the 2 to carry cancels the 2, and 
 |there remains the 1 to fet down. 
 
 The 2 cancels the2, and the 1 to 
 carry from the decimals is fet down. 
 
 Il. Divifion by Logarithms. 
 
 ~ From the logarithm of the dividend, fubtra& the logarithm of 
 the divifor; the remainder is a logarithm, whofe correfponding 
 number will be the quotient required. 
 
 But firft obferve to change the fign of the index of the logarithm 
 of the divifor, viz. from negative to affirmative, or from afhirmative’ 
 to negative ; then take the fum of the indices if they be of the fame 
 kind, or their difierence when of different kinds, with the fign 
 of the greater, for the index to the logarithm of the quotient. 
 
 And when | is borrowed in the left-hand place of the decimal 
 part of the logarithm, add it to the index of the logarithm of the 
 divifor when that index is afhrmative, but fubtraét it when nega- 
 tive; then let the index thus found be changed, and worked with 
 
 as before. 
 
 Feel. 
 
OF THE 
 
 Ev, 1. To divide 24163 by 4507. 
 
 Divid. 24163 its log. 4°3831509 
 
 TABLES. 135 
 
 Ex. 2. To divide 37'149 by 52376. 
 Divid. 37'149 its log. 1.5699472 
 
 Divif. 4567 - - 36590310 Divif. 523°76 - - 2°7191323 
 Quot. 5°290782. - 0°7235109'  Quot. 07002752 - 28508148 
 
 Ex. 3. To divide 06314 by *007241. 
 
 Divid. ‘06314 its log. 2 8003046 
 3°8597985 
 0°9405061 
 Here 1 carried from the decimals to 
 
 Divif. 007241 
 Quot. 8°719792 - 
 
 the 3 makes it become 4%, whiclstaken 
 from the other 2, leaves O remaining. 
 
 Ex. 4. To divide ‘7438 by 12°9476. 
 Divid. 7438 its log. 1°8714562 
 
 Divif. 12°9476 - =—-:1'1121893 
 Quot. ‘05744094 - 2°7502669 
 
 Here the 1 taken from the 1 makes 
 it become 2 to fet down. 
 
 Ill. Lhe Rule of Three, or Proportion. 
 
 Add the logarithms of the 2d and 3d terms together, and from 
 their fum fubtract the logarithm of the Ht, by the foregoing rules ; 
 the remainder will be the logarithm of the 4th term required, 
 
 Or in any compound proportion whatever, add together the loga- 
 rithms of all the terms that are to be multiplied, and from that fum 
 take the fum of the others; the remainder will be the logarithm of 
 
 the term fought. 
 
 But inftead of fubtracting any logarithm, we may add its comple- 
 
 ment, and the refult will be the fame. 
 
 the logarithm of the reciprocal 
 mainder by taking the given logari 
 
 By the complement is meant 
 of the given number, or the re- 
 thm from 0 or from 10, changing 
 
 the radix from O to 103 the eafieft method of doing wich) ig to 
 begin at the left-hand, and fubtra&t each figure from 9, except the 
 laft fignifcant figure on the right-hand, which mutt be fubtradted 
 
 from 10. But when the index is 
 the reft as before. | And for every 
 
 negative, add it to 9, and fubtract 
 complement that is added, fubtra& 
 
 10 from the laft fum: of the indices. 
 
 Ex. 1. To finda 
 934, and 2°519, and357'4862. 
 
 Ath proportional to{Ex. 2. To finda 3d proportional te to 
 
 12°796 and 3°24718. 
 
 A 72°34 - comp. log. 8°1406215|As 12-796 - comp. ae 8 8929258 
 To2519 .- - = => 0:4014282'To 3- 24718 ~ 0'5115064 
 So 357'4802 = + = 2°5532502|/S0 3-24718 - = - 075115064 
 To 1244827 - - = 1:0951089)7'9 +g940216 = - - 19159380 
 
 Ex. 3. To fiod a number ia propor- 
 tion to 379145 as ‘85132 is to 
 0649. 
 
 Ex. 4. If the intereft of 100I. tor a 
 year or 365 days be 4°51]. what will 
 be the intereft of 279°25!. for 274 
 
 piv days ? 
 As 0649 - comp. log. 11° Beale 100 pales 8'0000000 
 To 85132 - - - = 1:9300928/45} g¢5 ( COMP: (OB } 7-4377071 
 So +379145 - =. = 1°5788054 279'25 2+ = = 2°4459032 
 HWaradet ool. enobaae| (0 4 274 2 =i at. > 2°4377506 
 “td eres (SQA ih a cme aialer ies O 6532125 
 To Q433296 - - - U'y740034 
 
 LV. Invo/u- 
 
136 DESCRIPTION AND USE 
 
 IV. Involution, or Raifing of Powers. 
 
 Multiply the logarithm of the number given by the propofed in- 
 dex of the power, and the product will be the logarithm of the power 
 fought. : 
 
 Note, In multiplying a logarithm with a negative index by any 
 affirmative number, the product will be negative.—But what is to be 
 carried from the decimal part of the logarithm will be affirmative.— 
 ‘Therefore the difference will be the index of the product; and it is 
 to be accounted of the fame kind with the greater. 
 
 Ex.1. To find ‘the 2d power offE x, 2. To find the cube of 3:07146. 
 
 2°57). 
 Root 2'57Q} its log. 0°4114682' Root 3°07146 its log. 0°487344g9 
 index - - - - 2 index ---- 3 
 
 Power 6°651756 ~ 08229364 Power 28°97575 - 14020347 
 (Ex. 4, To find the 365th power of 
 
 Ex.3. To find the 4th power of 1°0045. 
 709103, Root 1°0045 its log. 0°0019499 
 Root 09163 its log. 2°9620377 index - - - 305 
 index ----4 07405 
 Power ‘0000704938 = 5°8481508 116004 
 
 58497 
 
 Here4 tim ative ind é | ——— 
 RT iseutan een echt Power 5°148888 - 0°7117135 
 
 ing 8, and 3 to carry, the difference 
 5 is the index of the produét. l 
 
 hl 
 
 V. Evolution, or Extraétion of Roots. 
 
 Divide the logarithm of the power, or given number, by its index, 
 and the quotient will be the logarithm of the root required. 
 
 Note, When the index of the. logarithm is negative, and the di- 
 vifor is not exa€tly contained in it without a remainder, increafe it 
 by fuch a number as will make it exadtly divifible ; and carry the 
 units borrowed, as fo many tens, to the left-hand place of the de- 
 
 -cimal part of the logarithm ; then divide the refults by the index of 
 the-root. | : 
 
 Ex. 
 
. OF THE TABLES. 
 
 Ex.1. To find the fquare root of 
 305. 
 Power 365 +. - 2) 2°5622929 
 
 Root 1910498 = 1°2811405 
 
 Ex.3. To find the 10th root of 2. 
 Power 2 - - 10) 0°3010300 
 Root 1°071773 - 0'0301030 
 
 Ex. 5. To find the fquare root of 
 “093. 
 Power ‘093 - - 2) 2°9684829 
 Root ‘304959 - 1°4842415 
 Here the divifor 2 is contained ex- 
 actly once in 2 the negative index, 
 therefore the index of the quotient 
 
 137 
 
 To find the cube root of 
 12345. 
 
 Power 12345 - 3) 40914911 
 
 Root 23°11162 - 1°3638304 
 
 Ex. 2 
 
 Ex. 4. To find the 365th root of 
 1°045. 
 
 Power 1°045 365) 0'0191163 
 
 Root 1;00012L - 0°:0000524 
 
 Ex.6. To find the cube root of 
 ‘00048. 
 
 Power - - - $)4°6812412 
 
 Root ‘07820735 - 2'8937471 
 
 Here the divifor 3 not being exaétly 
 
 contained in 4, augment it by 2, to 
 
 make it become 6, in which the di- 
 
 is 1. vifor is contained juft 2 times; and 
 the 2 borrowed being carried to the 
 other figures 6 &c, makes 2°6812412, 
 
 which divided by 3 gives *8937471, 
 
 OF THE TABLES FOR LOGARITHMS TO TWENTY 
 ‘ PLACES. 
 
 ‘Turse are tables 2d, 3d, and 4th, beginning at page 187. Of 
 thefe, table 2 contains all numbers from | to 1000, and all uneven 
 numbers from 1000 to 11613; with their logarithms to twenty places : 
 table 3 contains all numbers from 101000 to 101139, with their lo- 
 garitbms to twenty places, and the 1ft, 2d, and 3d differences of thofe 
 logarithms : and table 4 contains all logarithms regularly from 00001. 
 to 00139, with their corref{ponding natural numbers to twenty places, 
 as alfo the 1ft, 2d, and 3d differences of thofe numbers. And by 
 means of them may be found the logarithm to any other number, and 
 the number to any other logarithm, to twenty places of figures. 
 
 (I.) To find the Logarithms to given Numbers. 
 
 Casz 1. If the given number 4 be found in any of thefe three 
 tables ; then its logarithm B is in the line even with it. 
 
 Casz 2. If Jis known to be the product or quotient of numbers 
 found in thefe tables; then B is the fum or difference of the logas 
 rithms of thofe numbers. 
 
 T Case 
 
138 DESCRIPTION AND USE 
 
 Case 3, Ifa’, the firft fix fignificant figures of a given number 
 b', be found in table 3; let @ be an integer, A’ its logarithm ; 
 the remaining figures of 4’; x the complement of 5 to d’ or 1; 
 D’, D’, D”, the 1ft, 2d, 3d differences of the logarithms in the fame 
 line with A’; f= + D” x «41 + D”: Then B' the logarithm of 
 the number J' will be 
 ; Ry tO At ee te Aid 
 tyD'+D' xd+A’ - - tol places of figures nearly. 
 ey UK oF Ale) eto 20 
 Ex.1. Given the number 4: = 0°01010,26227,6351, to find B’ 
 its logarithm nearly to twelve places. | 
 
 Here a’ = 101026 A’ = 00443,31579,747 
 § = 0:2276351 0D - --- + 9785,618— 
 “ D’ = 429881746 B’ = 2°00443,41365,365— 
 Exr.2. Given b' = 0:01010,26227,63509,626, to find B’ its log. 
 nearly to 17 places. - Here a = 101026. 
 
 5 == 0:22763,509626 ; x = 0'772365; D’ = 42988,174579; D” 
 = 425510. 
 
 we Ue cal mnths Gime ecm Lo ue ee a real Ot aa ee 
 De ee nie al or GABE LAST 8G 
 
 txD'+D' - - - - - - = + 42988,33890,31 
 axD'+.D' x3 - -« - «= ~ = 9785,65466,42 
 
 AU - = = 0:0443,31579,74695,33 
 
 AndixD’+D'xd4 A,orB’ = 2:004.43,.41365,40161,75 
 Ex. 3. Given b = 0°01010,26227,63509,62573,17345, to find B’ 
 its log. nearly to 20 places. a’ = 101086. 
 § = 0°22763,50962,5731733 7 =0°7172365490374; r+ 1 = 1°7723653 
 D'= 42988,17457,46301; D’= 42550,96343 ;D” = 84236. 
 
 Now 7 DUX WHEL oe ne ee a 4766 
 Du - - - + - = = 42550,96343 
 
 fi. et eet te ee ee = 4955146109 
 EU f men ee ee a G45 9,62757 
 Il’ ae Le = a ‘ad ‘ai - Gr 4.2988,17457,86301 
 taf+D' - - - - - - = - 42988,33890,49058> 
 yaft+D xd = - - - - -  9785,65466,45604 
 
 - = = 00443,31579,74695,3279 | 
 
 And Bo = 6-2-2 (6 ee + 2,00443,41365,40161,78395 
 
 Case 4. Ifthe number d do not come under one of the preceding 
 cafes: put for the firft five figures of 6; » for 101, the leaft, or 
 
 ; a oon y 
 fome one, of the numbers in table 3; then 7 Or— = a is to be had 
 XQ «a 
 
 b 
 in table 2, with A its logarithm; let 4° =-— or da, and a’ the 
 
 a 
 firft fix fignificant figures of 3’ (found in table 3) be an integer, 
 | and 
 
OF THE TABLES. 139 
 
 and A’ its logarithm ; put 3 for the remaining figures of 3’; x the” 
 complement of 6 to ad’; D', D’, D”, the lft, 2d, 3d, differ. 
 ences of the logarithms in fie fame ine with A’; Waar el Ulge,: 
 *+1+D”. Then B the logarithm of the number 4 will be 
 
 D' x4 A‘ + A = B+ A to 19 
 cee of 
 
 pxD’'+D' x3 +A +A=—B + Ato17 figuresinearly 
 
 Ss Behe ee eee ee nee 
 Ex. Given 6 = 3'14159,26535,89793,25846,26434, to find B to 
 twenty places. 
 
 Here a = 31415 Letiaics. + S831, 
 
 RI 
 
 aes 
 Then J’ = Ge COO; 15840,95144,02970,573 a’ = 101015. 
 
 5 ==0°84095 14402,971057; * = 0°15904,8559754 -F P= 1°15905; 
 
 . D'= 42992,85574,06337; D’= 42560,23099; DY = 84265. 
 BRING Wie ee EE Tlie ose em lee) RS) OB SE 
 DF ee, a) [SREY OO Aa 6O,83099 
 
 Ff da) ic ee pettiness We PS ahs 4.2560,55654 
 DONUTS nN a wi tas Mn hs Vina OO ROTEL 
 
 Di n= me ew te = (429999,85574,06337 
 
 a*f+D) fof Pa OU ge %72999,88968,66008 
 ye a RS Be ee, ENE Pare 36154,93242,03919 
 
 Woke ph aan tA 00438 ,58681,74054,30961 
 
 A - = 49276,038905,26837,50555 
 
 AndB .- - = = = «@ = 0°49714,98726,94133,85435 
 
 144 ji 
 Or, let a Tae: $°216"—=.0°536 x6, 
 
 Then 4’ = ba = 10°10336,19739447175,0549 ; a = 101033. 
 3 =0°6197%,94477,50549 3 x= 0°38026,0552253 nti) 1=1°38036; 
 
 D'’ = 42985 ,19618,80760; D’ — 42545,06747 ; how = 84219. 
 Now; D" xX *+1 Se ae a eT a ere iin, ta 
 
 | . Par ke hos aol ig SN da es A ET 
 
 SPER edit nite sie e ey im fe etn ODED OSS 
 
 APs hm itd mt) =. vm pin 4 eh mee iy BOBOILIOLO 
 
 dD - = = - == = 42985,19618,30760 
 
 ERD iar is ei ee eyes etn 42985,27107, 93670 
 eX OGG r ne) =; Sek $96ES9:67 187,886 11 
 
 A’ - = = 00446,32488,03359,61854 
 
 BP - = = = = = =. 1:00446,59127,70547,50665 
 Aoi im 4s, BOR 2 = 0°50731,60400,76413,65230. 
 
 =B—A «- « = = 0°49714,93726,94133,85435 
 
140 DESCRIPTION AND USE 
 
 \IL.) Lo find the Numbers to given Logarithms. # 
 
 Case 1. When the logarithm B is found in any of thefe three 
 tables: then its number 4 is in the line even with it. 
 
 Case 2. If the firft five figures (omitting the index) of a given 
 logarithm Bb’, be between 00432 and 00492: take them as an inte- 
 ger, and put A/ and C for the logarithms, in table 3, next lefs and 
 greater than B’, a’ and ¢’ their numbers; let D’ (= C’ — A’) and 
 D" be the 1ft and 2d differences in the line with A’; A = B’— A; 
 di = (c —\a’ =) 1: Sig = See then 2°. = a’ 
 3 D! D'+ 3 pb Wee 
 
 + 3, nearly true to 17 places of figures. 
 
 ix. Given the logarithm B’  - = 5,00446,59127,70547,507 
 to find J its number. A’ = 5,00446,32488,03359,619 
 a’ = 101033 | A = 0'26639,67187,888 
 O- - = 0°61973,944776 D’ = 0°42985,19618,808 
 6 = 101033:61973,944776 D’ — A = 0°16345,52430,920 
 X = 0'38026 
 D’ = 0:00000,42545 
 % X D” = 0°00000,08089,1 
 
 D’ + =: X D” = 0°42985,27707,9 
 But when any other logarithm B is given, fubdu€t -004321 from 
 the firft fix figuresof B: call the remainder R, and let A be the 
 logarithm in table 2, next lefs than R, or next greater than the 
 complement of R, and a its number: then B = B— A, or B = 
 B + A, will be within the limits of table 3, and 4 will be found 
 as in the preceding example ; and if B’ = B— A, then’ = al’; 
 
 orif BD = B+ 4, then d=, 
 
 Case 3. If A’, the firft five figures (omitting the index) of a 
 given logarithm B’, be found in table 4: let a’ be its number; and 
 put A’ asaninteger, and A the remaining figures of B’, and X the 
 complement of A to D’; a’, d’, d", the lft, 2d, 3d differences of 
 the numbers in the fame line with a’; f = d’— 3d" x K+41; 
 then the number 4’, whofe logarithm is B’, will be . 
 
 dxAQ+t+a = = to 12) 
 
 @—iXd’xAt+d -. - to17 places of figures nearly. 
 
 ad—iXf xA+ad - # to 20 , 
 
 Ex. Given the logarithm B’ = 0:00006,93311,37711,69929, ta 
 find 4 its number to 20 places. Here A’ = 00006, 
 
 A= 0'93311,37711,69929; X—0'°06688,622883; X + 1 =1:066886; 
 a! = 23029,29742,212933 d’= 53027,52746 3d”. = 1-22100. 
 
 Now 
 
OF THE TABLES. 141 
 
 Nowfd”" Xx X+1 - - - 2 - = = = = 43422 
 
 ee ais wie a OF LESS A eae SSOTT SOG 
 Fim 8 meg ee tt -63027,09324 
 
 1Xf Pde APOE TERS INAS gh 1773,39115 
 d’ = iti! oh inn, tek e poms @9089, 20 $2 O90 S 
 
 a—Z7Xf - - = - = = & = = 23029,27968,82178 
 a— z Xf x A - 2 - - CA ae - ae’ 21488,93801,72000 
 a - - = = 10001,38 164,64943,57474 
 
 Ande vem Safle le. iim) «eo 1100015,96535,81462,9474 
 
 Case 4. If the logarithm B do not come under one of the pre- 
 ceding cafes. Put A for the logarithm in table 2, next lefs than B, 
 or next greater than the complement of B, and a its number; let 
 B' = B — A, or B = B + A; and A’, the firit five figures of B’ 
 may be had in table 4, with a’ its number; put A’ as an integer, 
 and let A be the remaining figures of B'; X the complement of A 
 to D’; d, d’, d", the 1ft, 2d, 3d differences of the numbers in the 
 fame line with a’; f= d'— id” x X + 1: then the number 6, 
 whofe logarithm is B’, will be 
 
 d xA+a@ xa=al' to 0} 
 
 places of figures 
 
 d—141XavxXxA+a Xa al to 16 
 2 nearly. 
 
 @—iXf xA+4 x a=al' to 19 
 Ex, Given B = 4°46272,61172,07184,15204, to find 4 its number. 
 
 Let A = 1°46239,79978,98956,08733. a = 29. 
 B/ = B— A =5-00132,81193,08228,06471. A’ = 00132 
 
 A=0°81193,08228,06471; X=0°18806,917723; X + 1=1°18807; 
 d= 23096,20835,345893 d’ = 53181,59733;d" = 1°22457. 
 
 Nowid” x X+1 - - - = + - = - = 48496 
 Bien ie er Deals ms ue lg os ddgn = cial Le ie SOAS LEA TO 
 eae te im, se eerie ee OSS PROT Sa 
 
 Se wert ties a Pa BA ee a het 0 OOD, Sa Ud 
 
 DP Nee a hee ee ae im | ite et SAI G JOSS A. tt 5 80 
 @—-3 Xf nee we oe wm 23096,15534,48187 
 PE Xf XA - = = = = © 18752,48284,85771 
 , a = = = 10030,44036,01963,96855 
 be = 2 = & = = &  10030,62788,50248,82626 
 6 mal = «© = =~ 0'90029,08882,08665,72159,6154 
 
142 DESCRIPTION AND USE, &c. 
 
 Or, given B = 4°46372,61172,07184,15204, to find 5. 
 
 Let A = 215365 5584425,71530,11 208. ata 344. 
 B=B + A = 100028 ,45597,78714,26409. A’ = 00028. 
 A = 0°45597,78714,264093 X= 0°54402,21986; X+ 1 Cal 54402 
 @ = _23040,96629,915215 d”== ~ 53054396394; d"” = = 122163. 
 NS eens I ep Mey Oh ry er aN al ge eee 
 DEE Set we me I?! RR tra i AO S054 7590634. 
 Fmt fe ee eR ee OS e660 
 ee ie ea et a NS BL ae To 
 Be ig lites DN! 83040 0669091503 
 ee NT be iy ee ye ee 23040,82198,70342 
 d—-i*#XfXA - - - = - ~ J0506,10496,55627 
 ~ @ = = = 10006,44931,70511,67281 
 BaP oh sor V euimeym yim yep OGS. 155437,81008,22908 
 sg i - = = = = 0'00029,08882,08665,72159,616 
 
 on THE TABLES FOR LOGARITHMS TO SIXTY-ONE 
 PLACES. 
 
 Thefe are tables 5 and 6, from page 203 to page 207; the former 
 containing the natural numbers in regular order from 1 to 100, and 
 after that all the primes up to 1100, with their correfponding loga- 
 rithms, to fixty-one places of figures 5 5 and the latter in page 207 
 contains all numbers in order from 999980 to 1000020, with their 
 logarithms, to fixty-one places, as alfo the Ift, 2d, 3d, and 4th dif- 
 ferences of thefe logarithms. And the ufe of thefe tables; in finding 
 the logarithm to any number, or the number to any logarithm, each 
 to fixty-one places of figures, will be as follows. 
 
 ee, 
 
 1 Any 
 
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146 DESCRIPTION AND USE 
 
 OF THE TABLE OF HYPERBOLIC LOGARITHMS. 
 
 This is table 7, in pages 208 - -.- 211, which contain the feries 
 of numbers 1°01, 1°02, 1°03 &c to 10°00, with their hyperbolic 
 logarithms to feven places of figures. They are-fo called becaufe 
 they fquare the afymptotic {paces of the right-angled hyperbola ; 
 and they are very ufeful in finding fluents, and the fums of infinite 
 feries. The table, as well as the following rules, were firft given at 
 the end of Simpfon’s fluxions, but they were rendered much more 
 correct.in the French edition of Gardiner’s tables, printed at Avi- 
 gnon in 1770, being very incorreét in the laft figure in Simpfon’s 
 book. But both thofe books aré very erroneous in the Pregelee for 
 finding logarithms by the table. 
 
 1. When the given Number is between 1 and \0. 
 
 From the given number fubtraét the next lefs tabular number, 
 _divide the remainder by the faid tabular number increafed by half 
 the remainder; add the quotient to the logarithm of the faid tabu- 
 lar number, and the fum will be the logarithm of the number 
 propofed. 
 Ex. To find the hyperbolic loga- ; 
 rithm of 3:4.5678. Here the next lefs 3°45239) °00678 (0019633 
 
 nue is 3°45, and its logarithm. — 1°2483742 
 °2383742, the remainder or dividend log. 1°24033715 
 
 cae 718,4its half 339, which joined to 
 
 the tabular number 3°45, gives the divifor ;. the quotient 0019633 
 added to the tabular logarithin 12335742, gives 1°2403375 the re- 
 quired logarithm of 3 45678: 
 
 2, When the given Number exceeds 10. 
 
 Find the logarithm of the number as above, fuppofing all the 
 figures after the firft to be decimals, then to that logarithm add 
 2 13095 851, or 4°6051702, or 6°9077553, &c, according as the given 
 number Panta Oo Or 3, or 4, &c, places of integers. ‘Thatas, 
 add 2*302585092994 multiplied by the index of the power of 10, 
 by which the given number was divided to bring it to one integer, 
 or within the limits of the table. 
 
 Ex. To find the hyperbolic logarithm of 345+678. 
 
 This number divided by 100 or 10°, to bring it within . 1°2403375 
 the limits of the table, or removing the decimal point 4 6051702 
 two places, gives 3:45678, the logarithm of which as 5+8450077 
 ‘above found is 1°2103375, to which adding 4 6051702 
 
 the hyperbolic logarithm of 100, the fum is 5°8455077 the hyper- 
 bolic logarithm te a of 345°678. 
 
 " Note; 
 
OF THE TABLES. 147 
 
 Note, The hyperbolic logarithm of any number may be alfo found 
 from Briggs’s logarithms, viz. multiplying Briggs’s logarithm of the 
 fame number by the hyperbolic jopea a of 10, viz. 
 
 _ Multiplying by - = 2*30258,50929,94045 ,68401,79914, 
 
 Or dividing by its reciprocal °43429,44819,03251,82765,11289. 
 
 OF THE LOGISTIC LOGARITHMS. 
 
 Thefe are in table 8, pages 212 - = - 216, which contain the 
 logiftic logarithm of every fecond as far as the firft $0’ or 4800”. 
 
 The logiftic logarithm of any number of feconds is the difference 
 between the logarithm of 3600” and the logarithm of that number 
 of feconds. 
 
 The chief ufe of the table of logiftic logarithms, i is for the ready 
 computing a proportional part in minutes and feconds, when two 
 terms of the proportion are minutes and OS hours and minutes, 
 or other numbers. 
 
 When two terms of the proportion are common numbers, their 
 common logarithms may be ufed inftead of their logiftic logarithms, 
 puiting the logarithm | where its complement fhould be, and the 
 contrary. 
 
 1. To find the Logiffic Logarithm of any Number of Minutes and 
 Seconds, within the Limits of the Table. 
 
 At the top of the table find the minutes, and in the fame co- 
 lumn, even with the feconds on the left-hand fide, is the logiftic - 
 logarithm. 
 
 Note, When hours are made any terms of the proportion, they 
 are to be taken as if they were minutes, and the minutes-of an hour 
 as if they were feconds. 
 
 2. To find the Logiftic Logarithm of any Number not exceeding 4800. 
 
 In the 2d row, next the top of the table, find the number next lefs 
 than that given; then in the fame column, even with the difference 
 on the left-hand fide, is found the logiftic logarithm. 
 
 When two given terms of the proportion are common numbers, 
 one or both greater than 4800, take their halves, thirds, &c, in- 
 ftead of them. But when only one of .the given terms is a common 
 number, and that greater than 4800, take its half, third, &c, and 
 multiply the 4th term by 2, 3, &c. 
 
 The logiftic logarithms in this table are all affirmative, as well 
 above as below 60’; but'the index of thofe above 60’ is — 13 be- 
 low 60’ down to ey the index is 0; and below 6’, the indices (being 
 either 1, 2, or 3) are expreffed in the table. } 
 . ‘ EXAMPLES. 
 
148 DESCRIPTION AND USE 
 
 EXAMPLES. 
 
 A:60° .- - Jo.log.[As60’.- + lo. log.jJAs 60’ - ~~ lo. log. 
 To4612" - O1135!T9 78’ 27”. 1-8836/To 1531 Pincay oe 
 So 8 7 - 0'8088/So 13 53. = 0 6357/50 40 12”. - (01135 
 To 615 - 09823'To18 9g - 05198 3 To 1179.-'' -\ 04848 
 As 46°12" co, 1°8865}As 78’ 27” co, 0.1164 As 40°12” co. 1°8865 
 To 60:'0 -' o0000/To60 0 - 0-0000'T0 1179 - - 04848 
 So 615  -  0'9823/S0.18 g -  05193'So60'0"  - - 0:0600 
 ‘To 8:7 = 0°8688/To13 53 .- 0°6357/To 1531 - - 03713 
 As60'. - co. 00000; As 24"  - co. 1°6021 As 24" - co. 1'6021 
 To 4721 - 1°8823}T046° 11" - 0°'1137\ To 76 34” ~~ 18941 
 So 37'28” - 0:2045/S0 8" 7° - _0'8088/§, 13553” - * 0:6357 
 To 2048 - - oad or VOR Nir ey “05846 To. 445 ee S10 
 As 4721 = c0.0'1177|As 46°11” co. 1°8863| As 76’ 34” co. 0'1059 
 To 600" - 9'0000)/To 24" - _- 0'3979'To 24" - .- 03979 
 So 2948 + - 00808!So 15°37” - 0°5840 50 44°17" - 071319 
 To 37/28". - -0°2045/To 8° 7’ = 0-8688/T0 13" 53° =. 06357 
 
 The logiftic logarithms may conveniently be ufed in trigonome- 
 trical operations, when two of the terms are fmall arcs, with the 
 logarithmic fines or tangents of other arcs; obferving, that inftead 
 of the logarithmic fine or tangent, to take the complement of their 
 logifiic logarithm ; ; and the contrary. 
 
 But this may be as readily and more naturally done by the loga- 
 rithmic fines and tangents themfelves of fuch fmall arcs, as taken 
 from the next following table of fines and tangents for r every fecond 
 of the firft 2° or 120’, 
 
 OF THE LOGARITHMIC SINES AND TAN GENTS +0 
 EVERY SECOND, 
 
 Table 9, pages 218 - - - 247, contains the log. fines and tan- 
 gents for every fingle fecond of the firft 2 degrees of the quadrant ; 
 he fines being placed on the left-hand pages, and thé tangents on the 
 right. ‘The degrees and minutes are placed at the top of the co- 
 Jumns, and the feconds on the left-hand fide, of each page, the 
 logarithmic fine or tangent being found, in the common angle of 
 meeting. So of 1° 52” 54” the log. fine is 8° 5163: 420, and the log. 
 tangent $°5165762, 
 
 ‘Uhe fame numbers are alfo the coho and cotangents of the laft 
 2 degrees of the quadrant, thofe degrees with their minutes being 
 placed at the bottom of the columns, and their feconds afcending | 
 
 on 
 
 | 
 
OF THE TABLES. | AAI 
 on the right-hand fide of the pages. So the cofine of 88° 7’ 6” is 
 8°5163420, and its cotangent 8°5165762. 
 
 _ When it is required to find the fine or tangent &c to 3ds &c, or 
 
 any other fractional part. of a fecond, fubtraGt the tabular fine or- 
 tangent of the complete feconds from the next to it, in the table, 
 and take the like proportional part ofthe difference ; which part 
 added to, or taken from, the faid tabular fine or tangent, accord~ 
 ing as it is increafing or decreafing, will give the fine or tangent 
 required. 
 
 fix. 'To find the log. fine of 1°52’ 54” 25’ or 1° 52’ 54° 23 or #y. 
 
 Here the fine of 1° 52’ 54” 19 52’ 54” fine 8°5163420 
 taken from the next Jeaves 1 52 55  - 8'51040061 
 641, which multiplied by 5 dit. O41 
 and divided by 12, or multi- 5 
 plied by 25 and divided by 3 12)3205 
 60, gives 267 the pro. part; pro. part. 2607 
 this added to the firft fine gives 1°52’ 54” =~ 825163420 
 that which was required. | 1° 52’ 54°" 25'"8°5 163087 
 
 On the contrary, if a fine or tangent be given, to find the cor- 
 refponding arc ; take the difference between it and the next lefs ta- 
 bular number, and the difference between the next lefs and greater. 
 tabular numbers, fo fhall the lefs difference be the numerator, and 
 the greater the denominator, of the fraCtional part to be added to. 
 the arc of the lefs tabular number; which fraction may alfo, if re- 
 quired, be either turned into a decimal, or into 3ds &c, by multi- 
 plying the numerator by 60, and dividing by the denominator. 
 
 Ex. To find the arc whofe fine is 85163900, 
 
 Finding the number is between the 
 fines. of 19 52’ 55" and 1° 52° 54’, 
 take the differences between the fines 
 as in the margin, and the differences oo 
 give £77 for the fraction of a fecond, sits ra ed 
 or #% nearly, which abbreviates to Ht rae 
 2" = 45"; and therefore the arc fought is 1° 52’ 54” 45”, 
 
 Where the 1ft differences of the fines and tangents alter much, 
 as near the beginning of the table, the 2d, 3d, &c, differences may 
 be taken in, and then the logarithmic fine or tangent will’be exprefled 
 by this feries, viz. . 
 
 19 52'55” = 85164061 
 1 52 54 - 85163420 
 l 52 54 45” 8°5163900 
 
 x—] x«—2 die 8 
 
 x—1 
 Q—=A+xD’'-+x, Z D+. nau’ jz Dk; ornearlyA+D’/—iD", x; 
 
 where A is the next lefs tabular logarithm, D’, D”, D’’, &c, the 
 1ft, 2d, 3d, &c differences of the tabular logarithms, and x the 
 fractional part of the arc over the complete feconds. 
 
 SLs, ; Lx. 
 
150 DESCRIPTION AND USE © 
 
 £x. To find the log, tangent of 5’ 1” 12°” 24!" of 511” nh OFS) 1) 206, 
 Here A=7:1641417,; x —-52,;.D’ = 14404; 
 
 / 
 
 Tang. Dp’ Be 
 
 ~ 
 
 §.0%.« 4+1626964! nue, and the mean 2d diff. D’” = —48. Hence 
 5 + 7164141 ed A 1) 
 “ ; “Pavciane 174404 9 | ay 1641417 
 43-7 10 5821) 4350-47 SD cee on eee 
 5 3° - 7-1670178 | Bry 
 ¥ ¥, casein 2 Vi “= F a A 
 2 
 Therefore the tangent of.5’ 17 12/” 24°" .. 4. =) 7 1644398 
 
 ‘ And on the other hand, when the fine or tangent is given, and 
 falls near the beginning of the table, from the fame feries we may 
 find « the frational part of a fecond. - For fuppofe it be required 
 to find the arc whofe tangent is 7°1644398. This falling between 
 the tangents-of 5’ 1" and 5’ 2”, take the differences, &c, as above 
 P| u 
 rT ee 
 or 2981 = 14404 *— 24. «*— x, or — 24 x* 4 14428 x = 2981; 
 which gives x = +2067” nearly = 12" 24’. Therefore the arc 
 ‘required is 51” 12” 94’, Or rather the approximate value A + 
 agate AEE: ye QA ae aan 2981 
 D’/ — iD’ ox = Q, gives x = D—ib’ > 
 
 and the feries gives 7°1644398 = T1641417 +x#D'+ x. 
 
 se 
 
 14404424 — 14498 — 
 
 *2067, the fame as before. 
 ) £3. . 
 OF THE LARGE TABLE OF NATURAL AND LOGA- 
 ‘RITHMIC SINES, TANGENTS, SECANTS, AND VERSED 
 SINES. ; 
 
 Table 10, page 248 - = - - 337, contains all the fines, tangents, 
 fecants, and verfed fines, both natural and logarithmic, to every 
 minute of the quadrant, the degrees at top, and minutes defcending 
 down the left-hand fide as far as 45°, or the middle of the quadrant, 
 and-from thence returning with the degrees .at the bottom, and the 
 minutes afcending by the right-hand fide to 90°, or the other half 
 of the quadrant, in fuch fort, that,any arc on the one fide is on the 
 fame line with its complement on the other fide; the refpective 
 fines, cofines, tangents, cotangents, &c, being on the fame line 
 with the minutes, and in the columns figned with their refpe@tive 
 names, at top when the degrees are at top, but at the bottom when 
 the degrees are: at the bottom. ‘Tlie. natural fines, tangents, &c, 
 are placed all together on the left-hand pages, and the logarithmic 
 ones all together, facing them, on the right-hand pages. Alfo in - 
 the naturals there are two columns of the common differences, and 
 in the logarithmic 3’columns of common differences, each column 
 of differences being placed between the two columns of numbers 
 having the fame differences; fo that thefe differences ferve. for 
 both their right-hand and left-hand adjacent columns: alfo each 
 differential number is fet oppofite the {pace between the numbers 
 whofe difference it is. ‘Thé numbers on the fame line ‘in thofe co- 
 lumns having fuch common diserences, are mutually complements 
 
 : of 
 
OF THE TABLES? 7 ts¥ 
 
 of each other; fo that the fum of the decimal figures of any two 
 {uch numbers, is always 1 integer, with O in each place of deci-. 
 mals. } 
 
 All this will be evident by infpe€&ting one page of each fort, as 
 well as the method of taking out the fine, &c, to any degrees and 
 complete minutes. It is however to be obferved, that in’ all the 
 log. fines, tangents, &c, and in fuch of the natural as have any 
 fignificant figure for their index or characteriftic, the indices are 
 exprefied in the table, and the feparating point is placed berween_ 
 the index and the decimal part of the number; but in feveral co- 
 lumns of the natural fines, &c, having 0 for their integer or-index, 
 both the index and decimal feparating point are omitted ; and where- 
 ever this is the cafe, it-is to be underitood that all the figures in fuch 
 columns are decimals, wanting before them. only the feparating point 
 and index 0. , 
 
 The fine, tangent, or fecant of any arc, has the fame value, or 
 is expreffed by the fame number, as the fine, tangent, or fecant of 
 the fupplement of that arc; for which reafon the tables are carried 
 only to a quadrant or 90,degrees. So that when an arc is greater 
 than 90°, fubtraét it from 180°, and take the fine, tang. or fecant of 
 the remainder, for that of the arc given. But this property does not 
 take place between the verfed fines of arcs- and their fupplements : 
 and to find the verfed fine of an arc greater than 90°, proceed thus: 
 in the natural verfed fines, to’radius add the natural cofine, the fum 
 will be the natural verfed fine; and in the log. verfed fines, add 
 0°3010300 to twice the log. fine of half the arc, the fum, abating 
 radius 10:0000000, will be the log. verfed fine required. 
 
 1. Given any Are ; to find its Sine, Cofine, Tangent, &c. 
 
 Seek the degrees at the top or bottom, and the minutes refpect- 
 ively on the left or right; then on the fame line with thefe is the 
 fine, &c. cach in its proper column, the title being at the top or’ 
 bottom, according as the degrees are. , 
 
 * But when the given arc contains any parts of a minute, interme- 
 diate to thofe found in the table: take the difference between the 
 tabular fines, &c, of the given degrees and minutes, and of the mi- 
 nute next greater; then take the proportional part of that difference 
 for the parts of the minute, and add it to the fine, tangent, fecant, 
 and verfed fine, or fubtra&t it from the cofine, cotangent, cofecant, 
 or coveried fine, of the given degreesand minutes; fo fhall the fum 
 or remainder be the fine, &c, required. 
 
 Note, ‘The proportional part is found thus, as 1’ is to the given 
 intermediate part of a minute, fo is the whole difference to the pro- 
 portional part required ; which therefore is found by multiplying 
 the difference by the {aid intermediate part. Alfo that intermediate 
 part may be exprefied either by a vulgar fra€tion, or a decimal, or a 
 fexagefimal in feconds, thirds, &c, and the fraction or fexagefimal 
 
 may 
 
152 | DESCRIPTION AND USE 
 
 may be firft reduced to a devimal, if it Be thought better fo to do, by 
 dividing the numerator of the fra€tion by the denominator, or by di- 
 viding the fexagefimal by 60. 
 
 BE, XOA MoPobh E:S: 
 1. To find the natural fine of |2. To find the natural tangent of 
 
 ye 4.9/ 938” 190°": | §° Q’ 10” 24". 
 Inthe columnof difference between 8° 10’ tang. - = 1435084 
 - the natural fines of 1°48’ and 1° 4g) ~ BH £ = 1432115 
 is the difference 2907 ; and 28” 12'” diff. 2909 
 
 being —= 28°22” — 47’: therefore as Bl este oe 
 a ( 11 : 2060 ::: (10" 24'"=) +7'1- 
 1: 2907 :: -47: the pro. part +1306 9 os . ) Re eee 
 to which add fin. 1948’ - 0314108 8° 9 10 24” ade ad 
 makes fin. of 1° 48’ 28” 12’” 0315474. ‘ 1432630 
 
 3. To find the nat. coverfed fine of| 4:10 find the logarithmic cofine of 
 4° 6" 5” 40”, 0° 8-42" 
 
 17’ ; ; , 
 L = 2902 (tab. dif.) 735 =| —274/1:156(tab. dif.) ::°'7’=42”: pr.pt. --95 
 5” 40”: pro. part 2 6° 8’ cofine - 9°9975009 
 4° @’coverf. - - 9285026 6° 8'42” - 5 99974974 
 
 49.6’ 5” 40°" *. ~ 9984752 
 
 6. To find the] i 
 .To find the log. fec. of 7919 50"| > go ar an” mayne Teen OF 
 
 . 5’ , . : Py 
 1:160tab.dif.:: 2.50": pr. pt. +133! ? 2581 tab. dif. :: 203 = | 
 6 set 12” 20”; pro. part ads 
 79 12’ fecant = 10°0034381 39° 4’ cotan. - 10°0905978 
 79 19’ 50” s 10'0034514 39° A’ 12” 90” 10°0905447 
 
 The foregoing method of finding the proportional part of the ta- 
 bular difference, to be added or fubtra¢ted, by one fingle propor- 
 tion, is only true when thofe differences are nearly equal, and may 
 do for all except for the tangents and fecants of large arcs near 
 the end of the quadrant in the natural fines, &c, and in the log. 
 fines, &c, except the fines and verfed fines of {mall arcs, the tan- 
 gents of both large and fmall arcs, and the fecants of large arcs. 
 And when much accuracy is required, thefe excepted parts may be 
 found by the feries ufed in the laft article, viz. Q= A+ x* D' + 
 
 sear? See aps Sr lew eee ae StURETE NE 
 Wa prc AL) A! abet haste t— D" &c.or = A + D— ¢ DD. ~x near- 
 
 2 2 3 
 ly; where A is the tabular number for the degrees and minutes 
 D’, D’, D", &c. the I ft, 2d, 3d, &c tabular differences, and rv the 
 fractional part ever the complete minutes, &c3 at leaft it may be 
 
 proper to find the tangents and fecants of very large arcs from this - 
 feries; but as to the log. fines, verfed fines, and tangents of {malt 
 arcs, they may alfo be found, perhaps eafier, from their correfpond- 
 
 ing matural ones, viz. find the natural fine, veried fine, or eae 
 : ry 
 
 ~ 
 
 -” 
 
OF THE TABLES. 
 
 153 
 
 of the given fmall arc, and then find the log. of fuch natural number 
 by the 1ft or large table of logarithms, which will be the log. fine, 
 &c, required. And the log. tangent and fecant of large arcs will 
 be alfo found by taking the difference between 20 and their log. 
 cotangent and cofine refpeétively. And laftly, the natural tangents 
 and fecants of large arcs may alfo be found by firft finding their log. 
 tangent and fecant, and then finding the correfponding number. 
 
 EXAMPLES. 
 
 1. To find the log. fine of 1° 48’ 28" 12”, 
 
 The natural fine, found in Ex. 1. above is’ 
 
 "03154745 and the log. of thisis$-4.989636 
 which is the log. fine required. L 
 
 3. To find the log. tang. of 2° 23’ 33” 36”. 
 2° 23’its nat. tan. = -. 0416210 
 1:2914 tab. dif.::°56’ = 33” 36”: + 1632 
 . 2° 23’ 33” 36” nat. tan. - 0417842 
 Itslog.2 23 33 36 log. tang. 86210121 
 
 5. To find the log. fec. of 88° 11’ 31” 48”, 
 Its complement is - -- 1 48 28 12 
 
 Its log. fineinEx. l.is - ~- 8°4989636 
 Which taken from - - - 20-:0000000 
 
 Leaves 1. fec. 88° 11'31”" 48"" 11°5010364 
 
 In the 6th example, the natural fecant 
 is found by the differential feries to be 
 31°698339.. But by taking the number to 
 the Jogarithm of it, as found in the 5th 
 example, it is 31°698333 ; which feems to 
 be the more accurate, as well as the eafier 
 
 way ; and indeed this method by the feries | - 
 
 feems to be, in fome inftances, more trou- 
 blefome, and lefs accurate, than finding 
 the fecant by dividing 1 by the cofine. 
 
 2. To find the log. verf. of 1° 43’ gg7 19" 
 1° 48/ nat. verf. - - - 0004934 
 1:92 tab. dif.::-47' = 28"12":+ 43 
 1°48/ 98” 39" nat. verf. - *0004977 
 Itslog.1 45 28 12 log. verf. 6°6969676 
 
 4. To find the log. tang. of 87°36’ 26” 24", 
 Its complement is - - 2 23 33 36 
 
 Whofe log. tang. in Ex. 3. is 8°6210121 
 Takenfrom - - -+ - - 20:0000000 
 
 Leaves log.tan.87° 36’ 26” 24" 11°3789879 
 
 6. To find the nat. fec. of 88° 11’ 31” 48". 
 
 20976074 
 ‘ ow to ' 
 88 10131°257577/7 Oe 
 88 11 /31°544246 
 $8 12/31°836225 
 88 13132133663 
 
 850 9 | D” 
 
 dD” 
 seananld 16 
 286669). 4/144 
 291979)2 1149 
 297438) 
 
 Hence A = 31°544246; D' = 291979; 
 D” = 5310; the mean D” = 146; 
 
 45S 168) o> 31-48""5)00, Pane Od 
 K—1 x—2Z 
 
 te = 06188. 
 
 Then A - = = = - = -31°544246 
 
 xD’ om” we - - ad - = 
 
 1)’ - hd = = bad 
 
 154748 
 
 ——664 
 
 ~~ 31°698339 
 
 2. Given 
 
‘154 DESCRIPTION AND USE 
 
 2. Given any Sine, Tangent, Sc. to find its Are. 
 
 Take the difference between the next Jefs and greater tabular 
 numbers of the fame kind, and the difference between the given 
 number and the faid next lefs or next greater tabular number, ac- 
 cording as the: given number is a fine, tangent, &c, or a cofine, 
 cotangent, &c, noting its degrees and minutes ; then the two dif- 
 ferences will be the terms of a vulgar fra€tion of a minute, to be 
 added to thofe minutes, to give the arc required. | 
 
 And this vulgar fraction may alfo, if required, be reduced to a 
 decimal by dividing the lefs or numerator by the denominator, or 
 brought to fexigefimals, by multiplying by 60, &c. Alfo, where 
 the tabular differences are printed, the fubtraCtion of the lefs tabular 
 number from the greater is faved. 
 
 EXAMPLES. 
 
 1. To find thearcto the natural fine|3. To find the arctologarithm cofine 
 
 03154714. 9°9974974. 
 Anf. 1° 48’ 28” 12” 0315474 6° 8’ - 9:9975069 
 Subtr. 1 48’ next lefs0314108 Anfwer 6° 8/ 42 9°9974974 
 1366 95 
 60 60 
 Tab. diff. ~ 2907)81960(28"| Tab. difference 136)5700 
 5814 544 
 
 23820 260 
 23256 Veet st 
 
 564 
 
 60 
 
 2907) 33840(12” 
 
 4. To find the arc to logarithm cot. 
 10°0905447. 
 29° 4! 100905978 
 Anf.39°4'12"20"10:0905447 
 
 2, To find the arc to natural tang. 
 *1432630 , 
 Next greater 1435084 
 
  Anf. 8° 9/10” 24” 1432630 531 
 Nextlefs,fubt.fr.each1 432115 | 60 
 7 515 Tab. difference 2581)31860(12" 
 60° 2581 
 Tab. difference 2969)30900(10" 6050 
 29690 5162 
 1210 888 
 60 GO scan 
 712600(24"" 2581)53230(20 
 5938 5162 
 13220 1660. 
 
 The 
 
OF THE TABLES. | 155 
 
 The above method of proportioning by the firft difference alone, 
 can only be true when the other differences are nothing, or very 
 {mall 5 brut other means muft be ufed when they are large, viz. for 
 the natural tangents and fecants of very large arcs; and for the lo- 
 garithmic fines, and verfed fines of {mall arcs, alfo the log. fecants 
 of large arcs, with the log. tangents and cotangents both of {mall 
 and large arcs. When the log. fine, verfed fine, or tangent of a 
 {mall arc is given, by means of the table of logarithms find the cor- 
 xefponding natural number, and then the arc anfwering to it in the 
 table of natural fines, &c. But when the log. tangent or fecant of 
 a large arc is propofed, fubtract it from 20, the remainder is the 
 
 . log. cotangent or cofine, which will be the log. tangent or fine of a 
 {mall arc which is the complement of that required, which comple- 
 ment will be found as in the laft remark, by taking the correfpond- 
 ing natural number, and finding it in the natural tangents or fines ; 
 then fubtracting that complemental are from 909, leaves the required 
 large arc anfwering to the propofed log. tangent or fecant. And 
 when the natural tangent or fecant of a large arc is propofed, change 
 it into the log. tangent or fecant of the fame, by taking the log. of 
 the propofed natural number ; then proceed with it as above in the 
 laft remark.—Or, what relates to the log. fines and tangents of {mall 
 arcs, or cofines and cotangents of large ones, will be beft performed 
 by the foregoing table for every fecond of the firft 2 degrees. | 
 
 EXAMPLES. 
 
 ) 
 
 t. To find the arc to natural tangent, 2. To find the arc to natural fecant 
 
 50°0000000. 31 6983333. 
 20'0000000 200000000 
 Given 50°0000000 itslog.1 16984700, Given 31°608% its log. 11°5010365 
 ‘02 - - - - 8 3010300 "0315474 - - 8 4989635 
 "0197830 nat. tan. of L° 8’ 0314108 nat. fine of 1° 48’ 
 2370 1366 
 60 | 60 
 2910) 130200 (44” . 2907) 81960 (28” 
 1164 5814 
 1380 93820 
 1164 23256 
 210 — «04 
 60 60 
 12960 (44”” 33840 (12/" 
 1164 2907 
 1320 : 4770 
 Hence from = = 90° 0' 0” 0 Hence from - - 90° 0' 0” O%. 
 Take thecomp. + - 1 844 44! Takethe comp. - | 48 28 12 
 
 Leaves arc required. 68 51 15 16| Leaves arc required $5 11 31 48 
 
 TRIGONO- 
 
(sb Sy 
 
 TRIGONOMETRICAL RULES. 
 
 / 
 
 1. Ty a tight-lined triangle, whofe fides are a, B, C, and their oppofite 
 angles a, 4, c; having given any three of thefe, of which one is a Bas: ; to 
 
 find the reit. 
 
 > 
 
 Put s for the fine, s’ the cofine, t the tangent, and t’ the cotangent, of an arch or 
 angle, to the radius r; alfo x for a logarithm, and u’ its arithmetical comple- 
 
 Then 
 
 ment. 
 
 When three fides a, B, c, are 
 given, 
 
 eat Po = A+B +c or femiperimeter. 
 a hens, 8 poi ay ayo Se P55 AS eae), 
 
 Cafe 1. 
 
 A Se cr 
 Ae 3 ae Abo), 
 AXB 
 
 L. s.dc= Z(t. p—a-L. -u, Pow a! hats B). 
 u's.c= S(t. P+. pet wl abt! B). 
 
 Note, When a = B, then 
 Ae — Ft Gh: 
 
 See x, Ands’ CL A 
 A 2 } A® , 
 
 Cafe 2. Given two fides a, B, and their 
 included angle c. 
 
 —Le,andt.d=AT® Ai 8s 
 
 Put s = 90° 
 A-++B 
 
 | thena=s +d; andb=s—d, And 
 
 thes Wie enamee 2S ef plien 
 rr 
 
 Or in logarithms, putting L. a = ) 
 2L. (A —B), and L. R= 1L.2a + 1, 2B 
 + Dees $. 5 c — 20; 
 we fhall have L, C = 3 OL. (@+R). 
 
 - one value, 
 
 If the angle ¢ be right, or = 90°; 
 
 then t. a Arzstbm 4; 
 
 os A 
 a plese 
 cm 7a, or = — 8B, or = a? + 8%, 
 S.a s.6 
 If a = B; we fhall have 
 a ath = 90°— 1'¢,and 
 
 Cafe 3. When a fide aed its oppofite 
 angle are atnong the terms given. 
 
 A Bits ‘ 
 Chen 2 vies 2 Tront which 
 Sia 2 pied) sie 
 
 equations any term wanted may be found. 
 
 When an angle, as @, is 90°, and a. and 
 c are given, then 
 
 BW a® —c* = / (a+c) X (A—C). 
 Andiu.Bm=i(l.at+eo+L.a4—C). 
 
 Note, When two fides a, B, and an 
 angle @ oppofite to one of them, are 
 given; if a be lefs than 3, then 4, c, 
 c have each two values; otherwife, only 
 
 I. In 
 
eal 
 
 7 
 
 TRIGONOMETRICAL RULES. 
 
 angles a, 4, 
 reft. 
 
 Ca/e 1, Given the three fides a, B, c. 
 Calling 2p the perim. orp = 3 (a+z+c). 
 
 hen a tiecamee Ay & (FB) 
 
 S.AXS.B 
 S.P XS. (P—C 
 And eee res lho). 
 §; A XSi B 
 
 Pek. 8. 4 c= (x. S. P—A-L. $.P—g-br!s. a-L1’s. B) 
 
 ‘hes c= (tu. 8. pe. s. p—c-br’s. a-Ev's. 8), 
 And the fame for the other angles. 
 
 Cafe 2. Given the three angles. 
 Put 22 -a +6 -+¢. Then 
 
 . gicary oP ** (e). And 
 
 { 
 
 Sa KS; 
 Wiemr yl PT X § (p~4) 
 s.a@ Xs. 6 
 
 8. 7 C= F(t. 8/p-f tes! p—ce+rv’ s, atr/s. b) 
 
 Ok. S304 (1. 8!p—atr, s'p—b-ex! s. at-1's.b) 
 And the fame for the-other fides. 
 
 Note, The fign 7 fignifies greater than, 
 and Z lefs than; alfo w the difference. 
 
 e 
 Cafe 3. Given a, 8, and included an- 
 
 gle c, 
 
 To find an angle a oppofite the fide a, 
 
 let ris’c::t.a:t:M, like or unlike ys 
 
 ascis 7 or Z 90°; allon= Bom: 
 
 then s. N:S.M%2:t.¢:t. a, like or-un- 
 likecas Mis 7 or Z pB. 
 
 Orilets’2.a+B:s Lawme:: Lh datine, 
 
 which is 7 or Z 9O°as a+ Bis 7orZ 180». 
 ands.ia+B:S.AKB::t/le:t. N,7 90°. 
 thena=mM+wn;andé—=™M—wN. 
 
 Again let r:s’c::t.a:t. ™m, like or 
 unlike 4 as¢ is 7 or Z 90°; and N = 8B 
 
 nM, 
 
 2'c—20; then L. s. $ 
 
 157 
 
 * 
 
 eel {pheric triangle, whofe three fides are A, B, C; and their oppofite 
 ff. In a fph Oe Aa heen of thefe fix terms being given, to find the 
 
 Then s’M:s’n::s' a: 3’ ¢, like or un- 
 like Nasc¢cis 7 or Z Q0o. Or, 
 
 s.AXS.BxXs*1i¢, ,,-—-— 
 
 8.36 te 4/ pe ae & on 
 
 rr ar 
 
 In logarithms, putt.a = 2b. s.2a meg; 
 andy. W S= Li ~S.'A" 42 02 6) ew Des, 
 
 C=L. (ate), 
 
 Cafe 4. Given a, 4, and included fide c, 
 
 Firft, letr:s’c::t.a:t’ m, like orun- 
 like aas cis 7orZ 90° ; allouxsw-m, 
 Then s/n": $m: 3.t..G if. A,. like or un- 
 like xas ais 7 or Z 90°. 
 
 On lets 2469 4 acs hoes cet: M, 
 7 of £ Q0easa+Sis 7 or Z 180°; 
 ands.ja+6:3.am6::t.tc:t.n,790°3 
 thna=mM+tN;andgao=uMay un. 
 Again, letr:s'c::t. a: t’ m, like or un- 
 like a@as cis 7 or Z 900; 
 
 anduz= bw m: ‘ 
 8.a::s'a:8 ¢, like or unlike 
 a@asmis 7-or-Z b,» 
 
 then s, m : 
 
 Cafe 5. Given a, 3, and an oppofite ans 
 gle a. 
 Mftisca + S'ar:; B's. 6, 2 or Z g0°. 
 and. Let r:s'B::t.a:t’m, like orunlike 
 Basais 7 or £ 90°; 
 and t.a:t.B::9s'm:s' 7, like or unlike 
 Aasdais 7 or Z 90°; 
 then ¢= m2, two values alfo. 
 Sdly. Letris’a:: t. B+: t.m, like or 
 
 unlike Bas dis 7 or Z 90°; . 
 and s'2: 8A ;:s'M:3' wn, like or unlike 4 as 
 ais 7 or Z 90°; 
 then c <= mM = n, two values alfo, 
 
 But if a be equal to 8, or to its fup- 
 plement, or between B and its fupple- 
 ment; then. is @ like to » : alfo cig — 
 mn, and ¢ =M => NH, as Bis like or 
 unlike a, 
 
 C. afe 
 
158 
 Cafe 6. 
 
 Given a, 3, 
 fide a. 
 
 Sty 8a 28, a’: 2 8.098. By 7 Or 2, 00%, 
 
 and. Letr:s'4::t.a:t.M, like or un- 
 like das Ais 7 or Z QO°; 
 andt.a:t.6::8.M:s.N, 7. or Z 90°: 
 
 then c = mM + n, as wis like or unlike 4, 
 
 Bdly. Letr:s'a::t.4: t! m, like or un- 
 likebasa 7 or Z QO°; 
 ands’6:s'a::s.m:s8.u, 7 or Z QO: 
 
 then c — m = 2, asais like or unlike 4. 
 
 But if a be equal to xz, or to its fupple- 
 ment, or between B and its fupplement ; 
 then B is unlike 4, and only the lefs values 
 of n, 2, are poflible. 
 
 Note, When two fides a, B, and their 
 oppofite angles a, 4, are known ; the third 
 fide c, and its oppofite angle c, are readily 
 _ found thus: 
 s-lanmb:s.tatb:i:tEamse:t Zc. 
 s.2.00B8:S2.a+Bi: than b:5e. 
 
 and an oppofite 
 
 TRIGONOMETRICAL RULES. 
 
 Ill. In a right-angled fpheric trian- 
 gle, where u is the hypotenufe, or fide op- 
 pofite the right angle, sp, Pp the other two 
 fides, and 4, p their oppofite angles ; any 
 two of thefe five terms being given, to find 
 the reft ; the cafes, with their folutions, are 
 as in the following table. | 
 
 The fame table will alfo ferve for the 
 quadrantal triangle, or that which has one 
 fide = 90°, » being the angle oppofite to 
 that fide, B, P the other two angles, and 
 6, p their oppofite fides ; obferving, inftead 
 of # to take its fupplement: or elfe mu~ 
 tually changing the terms /ike and unlike fot 
 each other where u is concerned, and its real 
 value is taken. | 
 
 ' 
 
 Cafe |Given Reg‘ 
 
 6 \|s.H: r :: 8.B : 8.0, and is like B 
 A P A eae es LB nay ty 7 or Z 90° as w is like or unlike 3B} 
 Poul SOB Ps 2 Se Sie Pll . 
 B yr :8.H:: 8.6 : s.B, liked 
 2 °y fi oe ee tah 7 7 or Z 90° as uw is like or unlike 
 i p PRM sy SCl.OF Fue 
 1 SO er es bee eee > a 
 3 ‘ P r 2 tp:2 £b «:teee >, each 7 or Z 90's bot values true | 
 pss: 7 tt 80 sh 4 | 
 H r :tB:: 8p :tH, 7 or Z 90° as Bz is like or unlike p 
 Bed oy al ce y 3'sBis, sp 3 80, likes 
 P P r $8.B22 tp. ¢ te, golikep 
 | H r ¢8'B:: se : s'H, Z or 7. 90° as B is like or unlike P 
 B ‘ j ' 
 Hy ce b r ¢s.P:: te : tb, likes 
 p T(E SDs h. BR ak Pye Cee ee, 
 H r :tb:: tp : su, 7 or Z 90° as bis like or unlike p 
 6 ‘i Be Sea (ee i i i sb .: sB, hiked 
 PL Epis ot BD tas BR, MKD 
 serliipeoateligatetralpennne termysreattnsinleae eect hea GAN DIT CE TE OM ne te 
 
 $-OuL WUT. TO UN WSs 
 
 ee ee ee an ene 
 
 Thi 
 
TRIGONOMETRICAL RULES. _ 159 
 
 ? ‘ 
 
 The following Propofitions and Remarks, concerning Spherical Tri- 
 angles, (fele&ted and communicated by the Reverend Nevil Mafke- 
 lyne, p. D. Aftronomer Royal, F.R. s.) will alfo render the Calcu- 
 lation of them perfpicuous, and free from Ambiguity. " 
 
 «¢ 1, A fpherical triangle is equi- 
 lateral, ifofcelar, or fcalene, accord- 
 ing as it has its three angles all equal, 
 or two of them equal, or all three 
 unequal ; and wice ver/a. 
 
 2. The greateft fide is always op- 
 pofite the greateft angle, and the 
 {malleft fide oppofite the fmalleft 
 angle. 
 
 3. Any two fides taken together, 
 are greater than the third. 
 
 4. If the three angles are all acute, 
 or all right, or all obtufe; the three 
 fides will be, accordingly, all lefs than 
 90°, or equal to 90°, or greater than 
 g0°; and vice vera. 
 
 5. If from 
 the three an- 
 gles aA, B, C, 
 of a triangle 
 Asc,aspoles, /..., A 
 
 there be de- ; Ke 
 
 {cribed, upon eee ek 
 - the furface of 
 the fphere, three arches of a great 
 circle DE, DF, FE, forming by their 
 interfections a new {pherical triangle 
 peF; each fide of the new triangle 
 will be the fupplement of the angle 
 
 at its pole; and each angle of the 
 fame triangle, will be the fupplement 
 of the fide oppofite to it in the trian~ 
 glea Be. 
 6. In any tri- C 
 
 angle a BC, or 
 A ’c, rightan- 
 gled in A, itt, 
 Theanglesat the 
 hypotenufe are 
 always of the 
 famekindas their 
 oppofite fides ; 2dly, The hypotenufe 
 is lefs or greater than a quadrant ac- 
 cording as the fides including the 
 right angle are of the fame or diffe- 
 rent kinds; that is to fay according 
 as thefe fame fides are either both 
 acute or both obtufe, or as one is 
 acute and the other obtufe. And, 
 vice verfa, 1{t, The fides including 
 the right angle, are always of the 
 fame kind as their oppofite angles: 
 2dly, The fides including the right 
 angle will be of the fame or different 
 kinds, according as the hypotenufe is 
 lefs or more than 90°: but one at 
 leaft of them will be of 90°, if the 
 hypotenufe is fo.” 
 
 A 
 
 THE 
 
i60 ) 
 
 THE CASES OF PLANE TRIANGLES RESOLVED BY 
 LOGARITHMS. 
 
 J N this and the following folutions 
 
 of fpherical triangles, it is to be ob- 
 
 ferved, that when we fay the fine, 
 tangent, &c, we mean the logarithmic 
 fine, tangent, &c. as found by the 
 table, 
 
 C 
 
 A 
 
 Prop.I, Having the angles, and one 
 - fide ; to find either of the other fides, 
 
 Add the logarithm of the given fide 
 to the fine of the angle oppofite to the 
 fide required, and from the fum fub- 
 tract the fine of the angle oppofed to 
 the given fide ; the remainder will be 
 the logarithm of the fide required. 
 
 Example. In the triangle pcn, hav- 
 ing the angle cpB 90°, cBD 51° 56, 
 BcD 38° 4, and the fide Bb 197°3; 
 to find the fide cn. 
 
 2'2951271 log. of 197°3 
 
 G°8961569 fin. of 51° 50’ 
 12°1912040 tum 
 
 9°78909880 fin. of 38°4 
 
 2'4012700 log.251°9278 cp req, 
 
 Or you may add the complement 
 of the fine of the angle oppofed to 
 the given fide, to the two other loga- 
 rithms, the fum (abating radius) is 
 the logarithm of the fide required ; 
 as fhown in art. 3. of Log. Arith. 
 And it is to be obferved that the com- 
 plements of the fines in the table are 
 to be found in the columns of the 
 cofecants; for (pafling over the firft 
 
 unit) the cofecants of the fame arcs 
 are the complements of the fame 
 fines. Alfothe complements of the 
 tangents, are the cotangents. 
 Example, The fine of 38° 4’ being 
 9°7899880, the cofecant of 38° 4’ is 
 10°2100120, which (omitting the firft 
 unit) is the complement of the faid 
 fine. 
 
 ' 
 
 0'2100120 co. of fin. 38° 4 
 2°2951271 log. of 197°3 
 9'8961369 fin. of 51° 56’ 
 2'4012700log.251'9278,as before, 
 
 But if one fide and the angles; of a 
 right-angled triangle, be known, and 
 you would have the other fide, as in 
 the former example, the operation 
 will be eafier thus : 
 
 Add the tangent of the angle op- 
 pofite to the fide required, to the lo- 
 garithm of the given fide, the fum 
 (abating radius) is the logarithm ef 
 the fide required. | | 
 
 10°1061489 tan. 51° 56 
 
 22951271 log. of 197°3 
 
 2 4012700 log. 251'9278 asbefare, 
 
 Prop. II. Having tewo fides, and an 
 angle oppofite to one of them ; to find 
 the other tavo angles, and the third 
 
 fide, 
 
 Add the fine of the angle given, to 
 the logarithm of the fide adjoining 
 that angle, and from the fum fubtract 
 the logarithm of the fide oppofite to 
 that angle, or add its arithmetical 
 comp. the remainder or fum will be 
 the fine of the angle oppofite to the 
 adjoining fide. _ 
 
 Example. In the triangle az €, 
 having the fide ac 800, Bc 320, it 
 
 the 
 
OF RIGHT-LINED TRIANGLES. 
 
 the angle anc 128° 4 
 angles BAC, ACB, and the fide av. 
 
 7°0969100 ar. com, log. 800. 
 2°5051500 log. of 320. 
 9°8961360 fin. 128° 4’, 
 0°4981909 fin. 18 21 Bac. 
 
 Having Bac and asc, the angle 
 AcB is their fupplement to 1809, viz. 
 33° 35’; and you may find the fide as 
 by the firft propofition, 
 
 Prop. II]. Having tavo fides, and the 
 angle between them ; to find the other 
 two angles, and the third fide. 
 
 If the angle included be a right 
 
 angle, and the radius to the logarithm > 
 
 of the lefs fide, and from the fum 
 fubtra&t the logarithm of the greater 
 fide, or add. its. arith. comp. the re- 
 mainder or fum will be the tangent of 
 the angle oppofed to the lefs fide, 
 
 Example, Jn the triangle gcn, hav- 
 ing the fide BE 197°3, and cp 251°9; 
 to find the angles BCD, cep, and the 
 fide cr, 
 
 7°59877 28 ar. com, log. 251°9 
 12°2051271 rad. + log. 197°3 
 9°8958989 tan, 389 4” RED. 
 
 But if the angle included be ob- 
 lique, add the logarithm of the dif- 
 ference of the given fides to the tan- 
 gent of half the fum of the unknown 
 angles, and from the fum fubtra& the 
 logarithm of the fum of the given 
 fides, or add its complement ; the re- 
 mainder or fum will be the tangent of 
 half their difference. 
 
 Example. In the triangle abc, hav- 
 ing the fide aB 502, BC 320, and the 
 angle AEG 128° 4’; to find the angles 
 BAC, ACB, and the fide AC. 
 
 td 
 
 ’; to find the 
 
 161 
 
 The fum of the given fides is 882, 
 and the difference 242, the half fum 
 of the unknown angles is 25° 58’. 
 
 7°0545314 com, log. 882 
 2°3838154 log. of 242 
 9°6875402 tang. 25° 58’ 
 
 9° 1258870 tang. 7 37 
 
 ; 25 1 SR 
 Angle acgp. - 33 35 fum, 
 Angle caB = 18 21 dif. 
 
 Thefe 7° 37’ being added to 25° 
 58’ the half fum of the angles un- 
 known, the fum is 33° 35’ for the 
 greater angle acB; and the fame 7° 
 37’ being fubtrated from 25° 58’, the . 
 remainder is 18° 21’ for the leffer an- 
 gle cag. Lafily, knowing the angles, 
 and two fides, the third ee may be 
 found by the firft propofition, 
 
 Prop. FV, Having the three fides ; to 
 Jind any angle. 
 
 Add the three fides together, and 
 take half the fum, and the differences 
 betwixt the half fum and each fide: 
 then add the complements of the lo- 
 garithms of the half fum, and of the 
 difference between the half-fum and 
 the fide oppofite to the angle fought, 
 to the logarithms of the differences of 
 the half-fum, and the other fides ; half 
 their fum will be the tangent of half 
 the angle required. 
 
 Example. In the triangle a Bc, 
 having the fide aB 562, ac 800, and 
 BC 320; to find the angle aBc. 
 Ac=800|u=841 - co. 7'0752040 
 AB 502/H—ac=41 co. 838721601 
 BC—=320|/H—aB=279 - 2°44560042 
 fum 1682)H--Bo=521 - 2°7168377 
 Jium 841H fum 20°62450-0 
 
 ‘Lang. of 64° 2’= tum 1073124310 
 W hofe double 128°4 is the angle aBc. 
 
 Y THE ° 
 
(\ ¥6o"") 
 
 THE CASES OF SPHERICAL TRIANGLES RESOLVED 
 BY LOGARITHMS, 
 
 ‘Tue refolution of fpherical tri- 
 angles is to be performed by the table 
 of fines, tangents, and fecants ; which 
 we fhall fhow by the 28 propofitions 
 following : whereof 16 are of right- 
 angled, and 12 are of oblique trian- 
 gles; and firft - 
 
 Of right-angled Triangles. 
 B 
 
 h a 
 Prop. I. Having the legs; to jind the 
 bypotenu/e. 
 
 Add the cofine of one leg, to the 
 cofine of the other leg; the fum, (a- 
 bating radius) is the cofine of the hy- 
 potenufe required. 
 
 Example. In the right-angled tri- 
 angle aBc, having ac 27° 54’, and 
 BC 11° 30’; to find as the hypote- 
 nufe. 
 
 99911927 cofin. 11° 30! 
 9'0463371 cofin, 27 54 
 -9'9375298 cofin. $0 AB. req. 
 
 Prop. II. Having the two legs ; to find 
 either of the angles. 
 
 Add the fine of theleg next the an- 
 
 gie fought, tothe cotangent of the 
 
 a ther leg : the fum, (abating radius) is 
 the cotangent of the angle required. 
 
 Example. Inthe right- -angled tri. 
 angle ABC, ae ac 27° 54, and 
 nc 11° 30°; to find the angle Bac. 
 
 Q° 6701807 fin, next leg. 27° 54’ 
 
 10°6915374 cot. opp. leg. 11 30 
 10°3617181 cotan, Bac 23 30 
 pate sisinbee Bi. tn 
 
 Prop. V. 
 
 Prop. III. Having the hypotenufe, and 
 one of the angles ; to find the other 
 angle, . 
 Add the cofine of the hypotenufe 
 
 to the tangent of the angle given ; the 
 
 fum, (abating radius) is the cotangent 
 of the angle required. 
 
 Example. In the right-angled tri- 
 anole agc, having the hypotenufe 
 AB 30°, and the angle aBc 69° 22’; 
 to find the angle Bac. 
 
 9°9375306 cofin. hyp. aB 30° 00’ 
 10°4241806 tang. aBc - 69 22. 
 10°3617202 cotan. BAC - 23 30. 
 
 Prop. 1V. Having the hypotenu/e, and 
 one of the angles ; to find the leg next 
 the given angle, 
 
 Add the tangent of the hypotenufe 
 to the cofine of the angle given ; the 
 fum, (abating radius) is the tangent 
 of the leg required. 
 
 Example. In the right-angled tri- 
 angle asc, having the hypotenufe 
 AB 30°, and the angle aBc 69° 22’; 
 to find the leg Be. 
 
 9'7614393 tang. hyp. aB. 30°00 
 9'5470188 cofin. ABc 86-69 22 
 9°3084581 tang.Bc - - 11 30 
 
 Having the hypotenufe, and 
 one of the angles; to find the eg ops 
 pofed to the given angle, ~ 
 Add the fine of the hypotenufe to 
 the fine of the angle given; the fum, 
 
 -(abating radius) is the fine of the leg 
 
 required. 
 
 Example. In the right-angled tri- 
 angle a BC, having the hypotenufe 
 AB 30°, and the angle Bac 23° 30’; 
 to find the leg Bc, 
 
 9'6989700 fin. hyp. aB 30° 00’ 
 
 9'6006997 fin. BAC - 23 30 
 
 Q 2990007 fin.’BC = 11 30 
 
 Prop, 
 
RESOLUTION OF SPHERICAL TRIANGLES. 
 
 Prop. VI. Having one of the legs and the 
 angle next it; to find the hypotenu/e. 
 
 Add the cotangent of the given leg, 
 to the cofine of the given angle; the 
 fum, (abating radius) is the cotangent 
 of the hypotenufe required. 
 
 Example. In the right-angled trian- 
 gle anc, having the leg ac 27° 54’, 
 and the angle pac 23° 30’ ; to find the 
 hypotenufe as. 
 
 10°2761563 cot. ac - 27° 54’ 
 _9'9623977 cof. Bac - 23 30 
 .10°2385540 cot. hyp. aB 30 00 
 
 Prop. VII. Having one of the legs, 
 and the angle next it; to find the 
 other leg. 
 
 Add the fine of the leg given to the 
 tangent of the angle given; the fum, 
 (abating radius) is the tangent of the 
 leg required. 
 
 Example. In the right-angled trian- 
 gle anc, having the leg ac 27° 54’, 
 and the angle Bac 23° 30’; to find the 
 leg BC. 
 
 9'6701807 fin. ac 27° 54’ 
 9'6383019 tan. BAC 23 30 
 0°3084826 tan. Ec 11 30 
 
 Prop. VIII. Having one of the legs, 
 and the angle next to it ; to find the 
 other angle, 
 
 Add the cofine of the given leg to 
 the fine of the givenangle; the fum, 
 (abating radius) is the cofine of the 
 angle required. . 
 
 Example. In the right-angled trian- 
 gle anc, having the leg gc 11° 30, 
 and the angle arc 69° 22’; to find the 
 angle Bac. 
 
 9°9911927 cof. Ec 11° 30’ 
 
 9°9712084 fin. aBcOQ 22 
 
 9:9624011 cof. Bac 23 30 
 
 Prop. 1X. Having one of the legs, and 
 the angle oppofed unto it ; to find the 
 hypotenufe. ' 
 
 Add the radius to the fine of the 
 
 _ given leg, and from the fum fubtract 
 
 163 
 
 the fine of the given angle, or add its | 
 cofecant ; the remainder or fum is the 
 fine of the hypotenufe required. 
 
 Example. \n the right-angled tri- 
 angle anc, having the leg Bc 11°30’, 
 and the angle Bac 23° 80 ; to find the 
 hypotenufe AB. 
 
 9°2996553 fin. Bc 11°30’ - 
 
 O 3993003 cof, BAC 23 30 
 
 976980556 fin, aB 30 reqd. 
 
 Prop. X. Having one of the lees, and 
 the angle oppofed unto it ; to find ihe 
 other leg. 
 
 Add the tangent of the given leg, 
 tothe cotangent of the given angle; 
 the fum, (abating radius) is the fine of 
 the leg required. 
 
 Example. In the right-angled tri- 
 angle asc, having the leg 5c 11°30’, 
 and the angle Bac 23° 30’; to find 
 the leg ac. 
 
 e 
 
 9°3084626 tang. Be 11° 30’ 
 10°3616981 cot. Bac 23. 30 
 
 poe es ee Se 
 
 9°6701607 fin. ac 27 54 
 
 Prop. XI. Having one of the legs, and 
 the angle oppofed unto it; to find the 
 other angle. 
 
 Add the radius to the cofine of the 
 given angle, and from the fum fubtraét 
 the cofine of the given leg, or add the 
 fecant ; the remainder or fum is the 
 fine of the angle required. 
 
 Example. In the right-angled trian- 
 gle apc, having the leg pc 11° 30’, 
 and the angle Bac 23° 30’; to find 
 the angle aBc. 
 
 9°96230977 cof. BAc 23° 30° 
 0088073 fec. Bc 11 30 
 
 9'9712050 fin. asc 69 22 
 
 Prop. XII. Having one of the legs, and 
 the hypotenufe; ta find the angle next 
 the given leg. } 
 
 Add the tangent of the given leg, 
 to the cotangent of the hypotenule, 
 the fum, (abating radius) }s the co- 
 fine of the angle required. 
 
 Example, 
 
164 
 
 Example, Yn the right-angled tri- 
 angle anc, having the leg ac 27° 54’, 
 and the hypotenufe ax 30° ; to find 
 the angle pac, 
 
 9°7238436 tan. ac 27° 54’ 
 10 2385606 cot. aB. 30 OO 
 99624042 cofi, Bac 23 30 
 
 Prop. XIII. ‘Having one of the legs, 
 and the hypotenufe ; to find the angle 
 oppofed to the given leg. 
 
 Add the radius to the fine of the 
 given leg, and from the fum fubtract 
 the fine mE the hypotenufe, or add its 
 cofecant ; the remainder or fum will 
 be the firie of the angle required. 
 
 Example. Yn the right-angled trian- 
 gle aBc, having the leg Be 11° 307, 
 
 and the hy potenufe AB 30°} to find | 
 
 the angle Bac. 
 9'°2096553 fin. leg. Be 11° 30 
 0°3010300 cofec, hyp. AB 30 CO 
 9 6000853 fine of Bac 23 30 
 Prop. XIV. Having one of the legs, and 
 the bypotenuje ; to. ae, the other leg. 
 Add the radius to the cofine of the 
 
 hypotenufe, and from the fum fub-- 
 
 tract the cofine of the given leg, or 
 add its fecant ; the remainder or -fum 
 is the cofine of the leg. required. 
 Example, In the right-angled tri- 
 anele anc, having the leg rc 11° 30’, 
 and the hypotenute AB 30° ; 3 to find 
 the leg ac. 
 9:9375306 cofin. AB 30° 00 
 0'0088073 fec. Bcll 30 
 99403379 cofin. ac 27 54 
 
 ’ 
 
 Prop. XV. Having the angles: to find 
 the bypotenufe, 
 
 Add the cotangent of one oblique 
 angle to the cotangent of the other 
 oblique angle; the fum, (abating ra- 
 dius) 1s the cofine of the hypotenute 
 required, 
 
 Example. In the right-angled tri- 
 angle A BC, having the angle pac 
 
 - pofite to the angle fought, 
 
 THE RESOLUTION OF 
 
 23° 30’, and the angle ane 69° 22’; 
 to find the hypotenufe as. 
 0°3616981 cot. BAc 23° 30° 
 9°5758104 cot. aBc 09 22 
 
 9°9375085 cof. hyp. AB30 OO 
 
 Prop. XVI. Having the angles; to find 
 either of the legs. 
 
 Add the radius to the cofine of ei- 
 ther oblique angle, and from the fum 
 fubtra&t the fine of the other oblique 
 angle, or add its cofecant; the remain- 
 der or furh will be the cofine of the 
 leg oppofite to the angle, whofe cofine 
 was taken. 
 
 Example, In the right-angled tri- 
 angle ABC, having the angle BAC 
 3° 30’, and the angle ABC 69° 22’; 
 
 te find the leg Bc. 
 
 0°9623977 cofin. BAC 23° 30’ 
 
 0'0287916 cofec. aBc 6Q 22 
 
 0°9911893 cofin. Bc 11 30 
 
 Of Oblique Triangles, 
 
 ri oS 
 
 Prop. XVII. Having the three fides, 
 to find any of the angles, 
 
 Add the three fides together, and 
 take half the fum; alfo the difference 
 between the half fum and the fide op- 
 ‘Then add 
 the cofecants, or the complements of 
 the fines, of the other fides, to the 
 fines of the half fum and of the faid 
 difference; half the fem of thefe four 
 logarithms is the cofine ot half the 
 angle required, 
 
 Example, In the triangle szp, hav- 
 ing the fide zs 40°, rs 70°, and pz 
 38° 30°; to find the angle ars. 
 
 Ps 
 
SPHERICAL TRIANGLES. 
 
 Ps=70° O'lcofec. 0'0270142 
 PZ—38 36 cofec. 0°2058505 
 zs=40 Ofin. }fum  0°9833805 
 
 y Sum 148 30 jin. dif. 9'7503579 
 1fum 74 15 2) 19°9660031 
 zs—40 0O cof, 15° 47’ 9°9833015 
 Diff. 34 15} zes 31 34 required. 
 
 Prop. XVII. Having the three angles; 
 te find any of the fides, 
 
 ‘ Let the angles be changed into fides, 
 taking the fupplement of one of them ; 
 then the operation will be the fame as 
 in the-former propofition. 
 
 _ Prop. XIX. Having tawo angles, and a 
 fide oppofed to one of them; to find the 
 fide oppofed to the other angle, 
 
 Add the fine of the fide given to the 
 fine of the angle oppofite to the fide 
 required, and from the fum fubtract 
 the fine of the angle oppofite to the 
 fide given, or add its cofecant; the 
 remainder or fum will be the fine of 
 the fide required. 
 
 Example, In the triangle szp, hav- 
 ing the angle, szp 130° 3’ 12”, 
 sPz 31° 34’ 26”, and the fide zs 40°; 
 to find the fide ps. 
 9'8080675 fin. zs 40° 0’ 0” 
 9 paisa fila: i 49 56 
 
 0'2808858 1165 § } 
 cof, spz 
 
 A8 
 
 31 35 
 34 
 
 70;, OY O 
 
 1165 
 Q: 9729842, fin. ps reqd. 
 See pa. 171 following. 
 
 Prop. XX. aoe two angles, anda 
 fide oppofed to one of them ; to find the 
 fide between the angles given. 
 
 Let a perpendicular fall from the 
 angle unknown, on its oppofite fide : 
 then add the cofine of the given angle 
 next the given fide, to the tangent of 
 the given fide; the fum, (abating ra- 
 dius) is the tangent of the firft arc, 
 comprehended between the given an- 
 gle next the given fide, and the feg- 
 ment of the fide where the: perpendi- 
 cular falls, 
 
 163 
 
 And the fecond arc, comprehended 
 between the fame fegment and the 
 other angle, is to be found thus: 
 add the fine of the arc found, to the 
 tangent of the given angle next the 
 given ‘fide, and from the fum fub- 
 tract the tangent of the other angle. 
 given, or add. its cotangent ; the re- 
 mainder or fum will be the fine of the 
 fecond arc. 
 
 The fum or difference of thefe two 
 arcs will be the fide required. 
 
 Example. In the triangle sz Py 
 having the angle zps 31° 34’ 26”, 
 zsP 30° 28’ 12”, and the fide rz 38° 
 
 30’ ; to find the fide sp. 
 9°9303781 S188 ie 
 440 cof. zPs Saga 
 Q'9006052 = tan. Pz 38 30 0 
 9°8310273 tan.PRIiftarc34 7 30 
 9 7488608 aaa 
 og cee Ve EdD 
 9°7884529 Me SA x 
 1227 tan. ZPs 26 
 02301404), . 30290 . 
 agg y cot tee f 20 8 
 
 9°7079103 fin.sr2darc 35 52 30 
 
 add pr Iftarc 34. 7 30 
 fumissP 70 0 O 
 See page 171 following. 
 But when the perpendicular falls 
 
 out of the triangle, the difference of 
 the two arcs wilh be the fide required, 
 
 Prop. XXI. Huving two angles, and a 
 ‘fide oppoftte to one of them; to find ihe 
 third angle, 
 
 Let a perpendicular fall from the 
 angle unknown, on its oppofite fide : 
 then add the cofine of the given fide 
 to the tangent of the adjacer nt angle; 
 the fum, (abating radius) is the, co- 
 tangent ‘of the firit angle to be found, 
 comprehended by the given fide and 
 the perpendicular. 
 
 And the fecond angle, comprehend- 
 ed by the perpendicular and the fide 
 unknown; 1s to be found thus: add the 
 fine of the angle found, to the cofine 
 of the given angle oppofite to the given 
 
 fide, 
 
166 
 
 fide, and from the fum fubtraé the 
 cofine of the other angle given, or add 
 its fecant; the remainder or fum -will 
 be the fine of the fecond angle. 
 
 The fum or difference of thefe two 
 angles will be the angle required. 
 
 Example. In the triangle sz P, 
 having the angle vs 31° 34’ 26”, 
 zsp 30° 28' 12”, and the fide pz 38° 
 
 30’ ; to find the angle szr. 
 
 98935444 cofin. Pz 38°30’ 0” 
 
 0°7884529 $1 34 
 Wears ea 086 
 
 9°6821200 cot. 1ft ZpzrR64 18 50 
 
 9'9547619 64 18 
 deh fin. PZR } 
 
 WES ¢ 
 oe cof, zsP ay Pere 
 0 0695443 31 34 
 
 336 fec. ZPS } ; 26 
 
 9° 9508447 fin. 2d Z szr 65 44 21 
 thenadd 1fiZpzr 64 18 50 
 
 the fumis szP 130 3 11 
 See page 171 following. 
 
 But when the perpendicular falls 
 out of the triangle, the difference of 
 the two angles will be the angle re- 
 quired, 
 
 Prop. XXH. Having tawo fides, and 
 the angle between them ; te find either 
 ah the other angles, ‘ 
 et a perpendicular fall from the 
 
 unknown angle, which is not required, 
 on its oppofite fide: then add the co- 
 fine of the given angle to the tangent 
 of the given fide oppofite to the angle 
 required ; the fum, (abating radius) 
 is the tangent of the firft arc, compre- 
 hended between the given angle and 
 the fegment of the given fide where 
 the perpendicular falls. 
 
 And the fecond arc is the difference 
 of that fide and the firft arc, being 
 comprehended between the fame feg- 
 ment and the angle required. 
 
 THE RESOLUTION OF 
 
 Now add the fine of the firft are, 
 to the tangent of the given angle, and 
 from the {um fubtraét the fine of the 
 fecond arc, or add its cofecant; the 
 remainder or fum will be the tangent 
 of the angle required. 
 
 Example. In the triangle sz p, 
 having the fide pz 38° 30’, Ps 70°, 
 and the angle zPs 31°34’ 26” to find 
 the angle zsp. 
 
 ° _ y 0 / we 
 9 oan cofin, ZPs ?) a4 26 
 99006052 tang.rz 38 Oo 
 
 9°8310273 tan.pR,lftarc34 7 30 
 taken from ps70 O O 
 2 
 
 leaves sr, 2d arc 35 5¢ 
 
 iss mingare sa Hi. 
 
 932. spat Ge 
 9'7884529 Pash G3 Bee COP 
 1227 tang. ZPs : _ 26 
 02320011) Vc. gp $35 53 - 
 873 . - .—30 
 9°7090270 tan.zPps req. 30 28 12 
 
 See page 171 following. 
 
 To find both the unknown angles. 
 
 Add together the cofecant, or the 
 complement of the fine, of half the 
 fum of the given fides, the fine of half 
 their difference, and the cotangent of 
 half the angle given; the fum, (abat- 
 ing radjus). is the tangent of half the 
 difference of the angles required. 
 
 Add.alfo together the fecant, or the 
 complement of the cofine, of half the 
 fum of the given fides, the cofine of 
 half their difference, and the cotan- 
 gent of half the angle given ; the fum, 
 (abating radius) is the tangent of half 
 the fum of the angles required. 
 
 Then add the half difference of the 
 
 angles required, to their halffum,and ~ 
 
 you will have the greater angle; and 
 fubtract the half-difference from the 
 half-fum, and you will have the leffer 
 angle required, the fame as in the 
 former operation, 
 
 PS 
 
SPHERICAL TRIANGLES. 
 
 167 
 
 PS io 70° O' ~—_ |Cofec. 3 fum 0°09060719|Sec. $ fum 0'2334015 
 by 38 30 Sin. 2 diff. 9°4336746\Cofin. + diff. 9-9833805 
 Sum 108 30 Cot. 4 zps 10'5486352\Cot. 4 zes 10 5486352 
 Diff. 31 30 T.49°47'30"10°07290817|T.80°15'42” 10°7654172 
 1 Sum "5A 15 Halt {um of angles required is . 80° 15’ 42” 
 2 Diff. 15 45 Half the differenceis . . 4 .° 49 47 30 
 “—"F zps— 31 34 26 |The greater angle szpis . . 130. 3 12 
 1 £2ps=15 47 13 The lefler angle zsp is, as before, 30 28 12 
 
 Prop. XXIII. Having tawo fides, and 
 the angle between them; to find the 
 third fide. 
 
 Let a perpendicular fall from either 
 of the angles unknown, on its op- 
 pofite fide : then add the cofine of the 
 given angle, to the tangent of the fide 
 from whofe end the perpendicular is 
 let fall; the fum (abating radius) is-the 
 tangent of the firft arc, comprehended 
 between the given angle and the feg- 
 ment of the fide where the perpendi- 
 cular falls. 
 
 And the fecond arc is the difference 
 of that fide and the firft arc, being 
 comprehended between the fame feg- 
 ment and the end of the fide required. 
 
 Now add the cofine of the fec6nd 
 arc, to the cofine of the fide from 
 whofe end the perpendicular falls, 
 and from the fum fubtra¢t the cofine 
 of the firft arc found, or add its fe- 
 cant; the remainder or fum will be 
 the cofine of the fide required. 
 
 Example. Inthe triangle szp, hav- 
 ing the fide pz 38° 30", Ps 70°, and 
 the angle zps 31° 34’ 26”; to find the 
 fide zs. 
 
 9'9303781 o1c 
 . 440 cofin. ZPs 
 
 9°9006052 tang,Pz . 38 
 9'8310273 tan.pR,1ftarc 34 
 
 taken from es 70 O O 
 leaves sr, 2d arc 35 52 30 
 
 ages Be Feni De ts ee 
 
 30 
 9'8935444 
 
 35’ sa? 
 —34 
 30 O 
 
 7 30 
 
 cofin. Pz 38 30 0 
 
 0'0820236. 8469 
 ey) fec, PR 30 
 
 1°8842553 cofinzsreq.40 O O 
 See page 171 following. 
 
 Prop, XXIV. Having two fides, and 
 the angle oppofite to one of them; to 
 jind the angle oppofed to the other fide. 
 
 Add the fine of the angle given, to 
 the fine of the fide oppofite to the an- 
 gle required, and from the fum fub- 
 tract the fine of the fide oppofite to 
 the angle given, or add its cofecant ; 
 the remainder or fum will be the fine 
 of the angle required, 
 
 Example, In the triangle szp, hav- 
 ing the fide ps 70°. zs 40°, and the 
 angle szP 130°3' 12"; to find the an- 
 gle zPs. 
 
 9'8838294 49° 56’ .” 
 eK fin. fup. gis ree, 
 Q'8080675 fin. zs 40 0 QO 
 
 00270142 cofec. rs . 70 O O 
 Q'7189961 fin. zpsreq. 31 34 26 
 See page 171 following. 
 
 Prop. XXV. Having two fides, and 
 the angle oppofite to one of them; to 
 find the third fide. . 
 
 Let a perpendicular fall from the 
 angle between the fides given, on its 
 oppofite fide ; then add the cofine of 
 the angle given, to the tangent of the 
 given fide next that angle; the fum, 
 (abating radius) is the tangent of the 
 tirft arc, comprehended between the 
 given angle and the fegment of the 
 fide where the perpendicular falls. 
 
 Now the 2d arc, comprehended be- 
 tween the fame fegment, and the end 
 of the fide required, is to be found 
 thus: add the cofine of the firft arc, 
 to the cofine of the given fide oppo- 
 fite to the angle given, and from the 
 
 fum 
 
168 _ 
 
 fum fubtraét the cofine of the other 
 given fide, or add its fecant; the re- 
 mainder er fum will be the cofine of 
 the fecond arc. 
 
 The fum or difference of thefe two 
 arcs will be the fide required. 
 
 Example. In the: triangle sz p, 
 having the fide pz 38° 30’, sz 40°, 
 and the angle srz 31° 34’ 26”; to 
 find the fide ps. 
 
 99303781 
 fast cof. sPz os ial 
 9'9006052 tan.pz 38 30 O 
 9°8310273tang.Priftarc34 7 30 
 . - r / 
 9 Sea cofin. pr 9°4 oA 
 
 9°8842540 cofin.sz. 40 0 O 
 071064556 fec, Pz 38 
 
 9'9086432 cofin.sn2dare 35 52 30 
 add pr, iftarc 34 7 30 
 gives psreq. 70 O O 
 
 See page 171 following. 
 
 But when the perpendicular falls out 
 of the triangle, the difference of the 
 two arcs will be the fide required, 
 
 Pron. XXVI. Having two fides, and 
 the angle oppsfed to one of them 3 to 
 Jind the angle between them. 
 
 Let a perpendicular fall from the 
 angle between the fides given, on its 
 oppofite fide: then add the cofine of 
 _ the given fide next the ave angle, 
 
 to the tangent of that angle ; the fum, 
 (abating radius) is the cotangent of 
 the firft angle to be found, compre- 
 hended by the given fide next the an- 
 gle given, and. by the perpendicular. 
 
 Now the fecandangle, comprehend- 
 ed by the perpendicular and the other 
 given fide, is to be found thus: add 
 the cofine of the firft angle found, to 
 the tangent of the giver tide next the 
 angle given, and from the fum fub- 
 tract the tangent of the other given 
 fide, or add its cotangent; the re- 
 mainder or fum will be the cofine of 
 the fecond angle to be found, 
 
 3 Ls ah Ae 
 
 38° 30’; to find the fide sz. 
 
 THE RESOLUTION OF 
 
 The fum or, the difference of the 
 firft and fecond angles, will be the 
 angle required. 
 
 Example. In the triangle $2 P, _ 
 having the fide pz 38° 30’, sz 40°, 
 and the angle spz 31° 34/ 26”; to 
 find the angle szp. ; | 
 
 9'8935444 cofin. Pz 38° 30° 0” 
 
 9 7884520 31 34 . 
 1927 tang. sZzP ' 96 
 
 9'6821200cotan.pzr,1ft 264 18 50 
 
 rea io a ee id big eel 
 
 437 ; .—10 
 99006052 tang. pz . 38 30 O 
 0°0701865 cotan.sz . 40 0 O 
 
 Q°6137213 cofin.szr,2d 265 44 22 
 add pzx, lit 264 18 50 
 gives szp, req. 130 3 12 
 
 See page 171 following. 
 
 Prop. XXVII. Having tavo angles, 
 and the fide between them; to find 
 either of the other jivies. 
 
 Let a perpendicular fall from the 
 given angle, which is next the fide re- 
 quired, upon its oppofite fide: then 
 add the cofine of the given fide to the 
 tangent of the given angle oppofite to 
 the fide required; the {um (abating 
 radius) is the cotangent of the firft 
 angle to be found, comprehended by 
 the given fide and the perpendicular. 
 
 And the fecond angle is the differ- 
 ence between the firft and the given 
 angle next the required fide, being 
 comprehended by the perpendicular 
 and that fide. 
 
 Now add the cofine of the firft an- 
 gle found, to the tangent of the fide 
 given, and from the jum fubtract the 
 cofine of the fecond angle, or add its 
 fecant; the remainder or fum will be 
 the tangent of the. fide required. 
 
 Example. In the triangle szpr, 
 having the angle spz 31° 34’ 26”, 
 szP 130° 3’ 12’, and the fide pz 
 
 0'8935444 
 
SPHERICAL TRIANGLES. 
 
 9°8035444 cofin. Pz - 38° 30’ 0” 
 9'7884529 $1 34 
 1297 | tang. SPZ 56 
 
 9°6821200cot. PzR, Ift Z, 64 18 50 
 taken from szP 130. 3 12 
 
 leaves SZR, Qd Li 65 44 22, 
 
 ‘96368859 | ‘ { 64 19° 
 ya aeealeh PZR “ethane 
 9 9000052 tang. pz - 38 30 O 
 
 0°3861750 (aoe 65 44 
 1028 b is ae pater. 22 
 
 9'9238126 40.0 0 
 See page 171. following. 
 
 To find both th: unknown fils. 
 
 Add together the cofecant, or the 
 -complement of the fine, of half the fum 
 of the angles given, the fine of half 
 
 tan. sz req. 
 
 169 
 
 their difference, and the tangent of 
 
 half the given fide; the fum (abating 
 radius) is the tangent of half the dif- 
 ference of the fides required. 
 
 Add alfo together the fecant, or the 
 complement of the cofine, of half the 
 fum of the given angles, the cofine ef 
 half their difference, and the tangent 
 of half» the given fide; the fum 
 (abating radius) is the tangent of half 
 the fum of the fides required. 
 
 Then add half the difference of the 
 fides required, to their half fum, and 
 you will have the greater fide; and 
 fubtraét the half-ditference from the 
 half-fum, and you will have the leffer 
 fide required, the fame as in the 
 former operation, 
 
 SZP 130° 3'12” |Cofec. I fum 0:0056062! Sec. 3 fum  0'796836 
 SPZ 31 34°20 |Sin, i diff. 9°870352 i Cofin. 4 diff. Lapis 
 Sum 161 37 38 |Fang.£Pz 9°5430936|Tang. i rz 9 5430936 
 Diff. 98 28 46 |'Tang. of 15° g: "4980535 | Tang. of 55° 10°1547733 
 tSum 80 48 49 Half fam of the fides required'ig.\ += <5 -5," 55° 
 Diff. 49 14 23. |Half their differenceis - -° - = - = 15 
 PZ. 38 30,0, ,| The greater fide sp'is)) 9-504 5 
 1 pz 19 15 oO |Leffer fide sz is, as before Me ee pe li iy a0 y | 
 
 Prop. XXVIII. Having two angles 
 and the fide beiween them 3 ta find 
 the third angle. 
 
 Let a perpendicular fall from either 
 of the angles given, upon its. oppofite 
 fide: then add the cofine of the fide 
 given to the tangent of the given an- 
 gle, from which the perpendicular does 
 not’ fall; the fum {abating radius) is 
 the cotangent of the firft angle, com- 
 prehended by the given fide and the 
 perpendicular. 
 
 And the fecond angle is the differ- 
 ence between the firt and the given 
 angle that the perpendicular fell from, 
 being comprehended by the perpen- 
 dicular and the fide oppofite to the 
 other angle given. | 
 
 Now add the fine of the fecond 
 angle to the cofine of that given angle 
 from which the perpendicular did not 
 full, and from the fum fubtra& the fine 
 
 —_—_ 
 
 of the firft angle found, or add its co- 
 fecant ;. the remainder or {um will be 
 the cofine of the angle required. 
 
 Example. In the triangle szp, hay- 
 ing the angle $ZP 130° "3" 1207) $P4 
 31° 34’ 26", and the fide pz 38> 30’ ; 
 to find the angle psz. 
 
 98935444 cofin. Pz - 88 30 O 
 9°7884529 Sida’ S 
 iad ARB MORE Lo oe gig 
 
 96821200 cotan-pzR,1ft Z,64 18 50 
 taken from szP 130 3 12 
 leaves $ZR, QdZ, 05 44 22 
 
 0°0451773 } K 6419": 
 On Gso PZR ; OME 15 
 *0303781 : 
 9°9 Ae cofin. sPz ee cy 
 sais ban m9 is 44. 
 <O0Q } oe OE 
 
 9°9354550 cofin. psz req. 30 28 O 
 See page 171 following. — 
 FOR . 
 
179 
 
 THE RESOLUTION OF 
 
 FOR THE USE OF THE VERSED SINES MAY BE ALSO ADDED 
 THE FOLLOWING PROPOSITIONS. 
 
 Prop. I. Having iwo fides of a /pheric 
 triangle, with the angle between 
 
 them ; to find the third fide. 
 
 y 1 ha together the log. verfed fine 
 
 of. the contained angle, and the log. 
 / fines of the two fides ; the fum (abating 
 
 twice the radius) is the logarithm of 
 a number to be found, which added 
 to the natural verfed fine of the dif- 
 ference of the two given fides, the 
 fum will be the natural verfed fine of 
 the third fide fought. 
 
 ~Or when the contained angle is 
 above gO°, add the log. verfed fine of 
 its {upplement, ‘and the log. fines of 
 the two fides together; the fum (a- 
 bating twice the radius) is the loga- 
 rithm of a number to be found, and 
 fubtracted from the natural verfed fine 
 of the fum of the two given fides, the 
 remainder will be the natural verfed 
 fine of the third fide fought. 
 
 Example 1. In the triangle SZP, 
 having’ the fide pz 38° 30’, ps 70°, 
 and the angle zPs 31° 34’ 26"; to find 
 the fide zs. 
 9°1703625 log.ver. finezsP 31° 34'26” 
 9'7941496 log. fine of Pz 38 30 O 
 99729858 log. fine of ps70 O' O 
 8: 9374979 log; of the numb. 865960 
 Nat. vert. diff. fides 31° 30’ 1473598 
 Nat.verf. zs40° - - - 2339558 
 
 Example 2. In the triangle sez Pp, 
 
 “having the fide rz 38° 30, zs 40°, 
 
 and the angle szp 130° 3’ 12”; to 
 find the fide ps. 
 
 The angle vzp is the fupplement 
 of sze. 
 9°5520590 log. verf. vzp 49° 56° 48” 
 9:7941496 log. fin. Pz 38 30 O 
 9'8080675 log. fin.zs 40.0 O 
 9'1542761 log. of the number1426514 
 Nat. verf, fum fides 78° 30’ 8006321 
 Nat. verf.rs 70° = - 6579807 
 
 This propofition may be very ufeful 
 in finding the difiances of places on 
 the earth, whofe longitudes and lati- 
 tudes are known; the diftances of 
 ftars, whofe declinations and right 
 afcenfions, or longitudes and latitudes, 
 are known; and confequently the al- 
 titudes, or common altitude of two 
 ftars, or two altitudes of the fun, and 
 time between the obfervations, or 
 difference of azimuth, being taken, 
 the latitude of the place may readily 
 be found, 
 
 Prop. If, Hawing tavo angles of a 
 Jpheric triangle, and the fi le between 
 them; to find the third angle, 
 
 .. Let the angles be changed into fides, 
 
 and the fide into an angle ;. then pro- 
 ceed as in the former propofition, and 
 the refult will be the fupplement of the 
 third angle. Butif one of the given 
 anglesexceed gO*, take itsfupplement, 
 and the refult will be the third angle. 
 
 The 
 
SPHERICAL TRIANGLES. 171 
 
 “ 
 
 The following remarks and directions, for rendering the proportional 
 part of a logarithm always additive, and for ufing ¢ + t,c—?, &e, 
 for sorc &c, in the foregoing propofitions, 20, 21, 22, 23, 25, 
 26, 27, 28, were communicated by the Rev. Nevil Maskelyne, 
 p. D. aftronomer royal, and F. Rr. s. the fourth cafe having been 
 invented by him many years fince, and delivered to the computors 
 of the Nautical Ephemeris, as precepts neceflary in computing the 
 moon’s diftances from the ftars in fome cafes, and the reft he has 
 now added on this occafion. 
 
 “ The refult of trigonometrical calculations will be fometimes 
 inaccurate, owing to the logarithms not being carried to a greater 
 number of places in the table, as will fuficiently appear from the 
 logarithmic differences being fmall.. This will happen where the 
 anfwer comes out in the cofine of a very {mall angle, or the fine of 
 an angle near 90°. ‘The greatnefs of the differences of the log. 
 fines of {mall arcs, or cofines of large ones, will fometimes affect the 
 accuracy cf the refult of the fecond part of the’ operation, unlefs 
 the firft arc be found to a fmall part of a minute or fecond: ‘To pre- 
 vent fuch error, and render the computation eafier, putting t, t’, 
 s, c for the tangent, cotangent, fine, and cofine of the 1ft arc or 
 angle, then in the 2d part of the work, . 
 
 In Prop. 20, if the firft arc is very fmall, for s ufec+t 
 
 21 - - angleis very {mall, for s ufec—t 
 22 - - - are is very {mall, for s ufec+?# 
 93-- - = are isneargo®, for—c ule t—s 
 25 - » -= are isneargo®, for c ufes—t 
 26 - - - angleisneargo®, for c ufes +4 t’ 
 27 - - + angleis neargo*, for co ufes4t’ 
 28 - - = angleis very {mall, for—s ufet’ —c 
 
 This obviates the neceflity of finding the firft arc to a very minute 
 exactnefs, which otherwife would be neceflary in taking out the fine 
 or cofine of the fame arc in the fecond part of the work. 
 
 Where the foregoing precepts direct to fubtra& a fine or cofine, 
 it will be readier in practice to add a cofecant or fecant; and where 
 they direct to fubtract a tangent (which is done in prop. 26) it will 
 be readier to add a cotangent. ‘This method being ufed if it be 
 required to find the logarithmic fines, &c, to the exattnefs of a fe¢ 
 cond, and the logarithm is increafing (as in the fines, tangents, and 
 fecants), write down the: logarithm for the degree and minute with-' - 
 out the feconds ; and alfo write down the proportional part for the 
 feconds; but, if the logarithm is decreafing (as in the cofines, co- 
 tangents, and cofecants) write down the logarithm for the next 
 greater minute, and alfo write down the proportional part for the 
 complement of the feconds to 60; and proceed in like manner 
 . with 
 
172 USE OF THE TRAVERSE TABLE. 
 
 with every logarithmic fine, Benne &e, ufed in the work; the fum 
 ofall the logarithms (abating one or two’radii or tens in the index, 
 according: as 2 or 3 logarithmic fines, &c are ufed in the part of the 
 work in quetftion) ori be the: dogarithmic fine, cofine, tangent, Or Co» 
 tangent required. 
 
 EBx.1. To find the log. fine of | 34° 17’ 24” 
 Here log. fine of 34° 17° - - 9°7507287 
 - And as 60: 24 or as 10:4::1853:  - au 
 ' O77 50802 
 Ex.2, To find the log. cof, of 55° 42° 36” 
 Loy, cof. of 552% 4387 .0-)) =» -) 'O'7 507287 
 60: 24 \0O—36), or 10: 4::1853: - 741 
 
 9'7 508028 
 
 Ex. 3. Inthe triangle PLS, given y 
 
 ee ae eee 30! ie to find Ls by prop, 23 5 ae 
 
 PE 8b 16 0 sp being perp, PL. of bate . 
 
 L i 
 
 Bi /20° Bie. fone e Diae 1S2 0H ; R 
 
 aera eH i Ba ace, we art @ Oe Me 
 
 ek MMs Yo esos fee tan. 11°062435 co!, foun akin alates 
 Ce es) LUD Ses ate lk: atte 9814 ¢ tang. eae A 879319686 
 rp 8443 43 tan. 11°03496603 - - - - - = ~ 11°0349663 
 PL $8910 0 cofec, PD 84° 44" 2” - = 100018374 
 LD 2 4) 26 17, — 17 - 83 
 pA NA cofin. LD 4 27-.  - , 9:9986888 
 Ae ute 70 
 cofins Ls 20°53°24 -- = OU7017:14 
 
 Here, to avoid the trouble of finding the proportional part for the 
 large logarithm difference of the cofine of ps, that cofine is found by 
 fubtraGing the tangent of it (already found) from the fine, which is 
 eafily Pan becaufe the differences are fmall: And, for the fame 
 reafon, the tain of the tangent and cofecant of pp, are ufed inftead 
 of its fecant. 
 
 N. B. The perpendicular fhould. always be let fall from the end of 
 the fide, Ps or PL, which diters moft from 90°, over or under.” 
 
 OF THE TRAVERSE ‘TABLE. 
 
 3! / 
 HIS traverfe table, or table’ of 
 difference of latitude and departure, 
 
 in page 338 and 339, is fo contrived,. 
 
 as to have the whole in one view, and 
 is fo plainly titled as to want little or 
 no explanation, 
 
 The diftances 1, 2, 3, &c, at the 
 top and bottom, may be accounted 
 10, 20, 30, &c, and the 10 as 100 
 
 if the minutes of latitude andaert- 
 ure anfweringy to the courfe be in- 
 creafed in the fame proportion; fo 
 ‘that if the diftance confitts of two 
 fignificant figures, the difference of 
 latitude, and the departure, is each 
 to be taken out at twice; and if of 
 three figures, at thrice. 
 
 | The 
 
USE OF THE TRAVERSE TABLE. 
 
 The chief defign of this tables i is for 
 the ready and exadt working of. tra- 
 verfes ; but it may alfo be applied to 
 the folution of the feveral cafes of 
 plain failing, and to fome other ufes. 
 
 Prop. I. Having the courfe and di- 
 Stance, to find the difference of latitude 
 and departure, 
 
 Seek the courfe on the left-hand 
 ‘of both pages downwards, if lefs than 
 four points, or 45 degrees; or if 
 greater, on the right-hand upwards ; 
 aud even with it in the double column, 
 figned at the top and bottom with the 
 diftance, is found both the difference 
 of latitude and the departure. 
 
 Example1. A fhip fails ssw 3 w 
 37 niles; the difference of latitude, 
 and the departure are required, 
 
 ’ Find the courfe 23 points on the 
 left-hand fide of each’ page, and even 
 with it in the double columns figned 
 3, and 7, the two figures of the di- 
 ftance, the difference of latitude for 
 30 is 25'732, and for 7 is 6:004, 
 the fum is 31°736 for the whole dif- 
 ference of latitude; and the depart- 
 ure for 30 is 15°423, and for 7 is 
 3°599, the fum is 19°022 for the 
 whole departure. 
 
 Thus, Dif. Diff. Lat. Dep. 
 30 ~ = 25°732 - 15°493 
 
 7 = = 6-004 - 3°599 
 
 _ 37 miles 31°736 - 19°022 
 
 Example 2. A fhip fails sz 49° 
 148 miles; the difference of latitude 
 and the departure are required. 
 
 Find the courfe 49 degrees on the 
 
 right-hand fide of each page, and’ 
 
 even with it in the double columns 
 figned 10, 4, and 8, the difference of 
 latitude at 100 miles is 65°606, at 40 
 is 26°242, and at 8 is 5:248; the fum 
 is Q7 OGO for the whole difference of 
 latitude. And the departure at 100 
 miles is 75°471, at 40 is 30°188, and 
 
 173 
 
 at 8 is 6:038; the fum is 111°697 for 
 
 ‘the whole departure, Thus, 
 iaDift: Diff. Lat. Depart. 
 100° «+= 65°06 woe saya. 
 AQ - - 20242 - - 30188 
 8 - =) 5248 = (+) °6:038 | 
 148 miles 97°096 - - 111°'697 
 
 Prop. II. Having feveral courfes and 
 diftances; to find the difference of 
 latitude and the departure. 
 
 Make a table in the following man- 
 ner, and put therein each courfe and 
 diftance ; then find the difference of 
 latitude and departure to each courfe 
 by the preceding, and place them, in 
 the proper column; the difference of 
 the {ums of the northings and fouth- 
 ings, is the whole difference of lati- 
 tude ; and the difference of the fums 
 of the eaftings and weftings, is the 
 whole departure. 
 
 Example, A thip from the latitude 
 of 50° north, fails according to the 
 courfes and diftances fet in the tra- 
 verfe table ; the differences of latitude, 
 and the departure, are found at the 
 bottom. 
 
 Boh Bt et 
 Sa? sen FH) fs 
 4gsury F 
 = th @ 
 aS) rj 
 orimaosoHoraSs C}s 
 ROT DDO = =} s 
 ) ¢ 
 = | 
 ol a © 2 a 
 oe S| Ris 
 GS i S14 
 =f. OH eB db OB els 
 Ma rwle  wonoanls FS) 
 QOS Bora |] +] y 
 BLE Go tw aT Ss 
 — 
 o S OVO} ey wo 
 g| S ets 1 yl G 
 24 Bee Pe Breopp | Y | ei 
 = De) at s 
 . SS ae 2 
 be = 
 Blo Rian nats aS 
 Bi SESSeo aks 
 HI} ~T Gd} OTD S 
 0) BO SH 
 wloO clp bp oop 
 This 
 
174 
 
 This propofition may be applied in 
 the furveying of large tracts of land, 
 as a county, &c, and was made ufe of>* 
 by Mr. Norwood in meafuring the 
 diftance from York to London, as the 
 road led him, obferving- the feveral 
 bearings by his circumferentor, and 
 finding by fuch a table his feveral dif-. 
 “ferences of latitude, and departure, 
 by which he obtained the diftance be- 
 tween the parallels, of London and 
 York, pretty near the truth, fo long 
 ago as the year 1635; as may be feen 
 in his Seaman’s Practice. 
 
 Alfo in plotting the furvey of a 
 county thus taken, the circuit ftation- 
 lines, though confifting of many hun- 
 dreds, may be reduced to a few for 
 the firft clofing, and the like for the 
 intermediates of each line firft plot- 
 ted, by which every ftation may per- 
 haps be more truly placed than by any 
 other method: the diftances in the 
 table may be chains of 66, or 100 
 feet, as well as miles, or any other 
 ymeafure that the differences of latitude 
 and departure would be had in. 
 
 Prop. U1. Having the difference of 
 latitude, and the departure ; to find 
 the courfe and diftance. : 
 
 Seek the given difference of latitude 
 and departure, taken together, in their 
 columns, or the neareft numbers to 
 them ; and the courfe is even there- 
 with at the fide, and the difiance at 
 the top and bottom: but if the given 
 difference of latitude and departure 
 cannot be found nearly, take 4, 2, 
 &c. part, or any equal multiple of 
 them that can be found; then the 
 courfe is even with them at the fide, 
 and fuch a part of the dittance, as was 
 taken of the difference of latitude and 
 departure, at the top and bottom, 
 
 Example 1. Given the difference of 
 Jatitude 59 miles s, and the departure 
 68 miles w; the courfe and diftance 
 are required. 
 
 In the double column over g, even 
 with 49° at the right-hand fide, is 
 
 USE OF THE TRAVERSE TABLE. 
 
 found together the given difference of 
 latitude and departure ; therefore the 
 courfe is 49° sw, and the diftance 90 
 miles. 
 
 Example 2. Given the difference 
 of latitude 30 miles nw,’ and the de- 
 parture 18 miles; the courfe and 
 diftance are required. 
 
 Here the given difference of lati- 
 tude and departure, or any numbers 
 near them, are not to be found to- 
 gether in the table; therefore taking 
 ~ or the double of each, the courfe is 
 found to be 31° n E£, and the diftance. 
 35 miles. 
 
 _ Note. A table computed to every 
 mile in the diftance up to 100 miles 
 would more readily folve this example. 
 
 Prop. IV. Hawing the departure and 
 middle latitude: to find the difference 
 of longitude, according to the method 
 ufed by W. Jones, Eiq. F. R.S. 
 
 Seek the given departure, or the 
 next le{s number in the columns figned 
 lat. even with the middle given lati- 
 tude found among the courfes, and 
 at the top and bottom (figned dift.) 
 is the difference of longitude fought ; 
 which if not found dire€tly at once, 
 may be taken out at twice or thrice. 
 
 Exemple1. Being yefterday noon 
 in the latitude of 37° 17’ N, and this 
 day noon in 38° 43’ n, and by the 
 table the departure is found 70°921 §; 
 the difference of longitude is required. 
 
 Jn the column figned lat. under 9, 
 even with 38°, the middle latitude 1s 
 found 7°0921!; therefore 9O miles is 
 the difference of longitude fought. 
 
 Example 2. Being yefterday noon 
 in latitude 46° 25’ n, and this day at 
 noon in 47° 35’ n, fo that the middle 
 latitude is 47° n,.and the departure 
 is found 112°53 miles w; required 
 the difference of longitude ? 
 
 In the column figned lat, over 10 
 at the bottom, even with 47 at the 
 
 tights 
 
OF MERCATOR’S SAILING. 
 
 Yight-hand fide, is 6°8200; therefore 
 fubduting 68200 from 112°53, the 
 remainder is 44°33; then over 6 is 
 4°0920, and 40°92 fubdudted from 
 44°33 leaves 3°41, which is found 
 over 5; therefore the difference of 
 longitude is 165 miles-wett. 
 
 If the middle latitude be not an 
 even degree, but have odd minutes ; 
 find the difference of longitude, for 
 the even degrees next lefs and great- 
 er, and add a proportional part of the 
 difference between the two refults to 
 the lefier ; the fum will be the differ- 
 ence of longitude fought. 
 
 Suppofe the middle latitude in the 
 laft example had been-47° 20’ n, then, 
 after finding the difference’ of longi- 
 tude as before for 47°, find- it -alfo 
 for 48°, which is 168 miles; then x 
 of the difference being added to the 
 former, gives the difference of longi- 
 tude 166 miles weft. 
 
 Note. Though this method is not 
 in all cafes near the truth, yet when 
 the miles are geographical, it is fufh- 
 ciently near fer daily pra¢tice in any 
 voyage, as well as eafy, and very ex- 
 peditious. 
 
 175 
 
 Prop. V. Having the latitudes, and 
 the longitudes of two places, to find 
 the bearing and diftance, 
 
 Seek the complement of the middle 
 latitude among the degrees, and the 
 difference of longitude in minutes 
 among the diftances, the departure 
 anfwering is found in its proper co- 
 lumn ; then with the difference of Ja- 
 titude and departure, find their bear- 
 ing or cour{e and diftance by the third. 
 
 Example. Wet the Lizard be given 
 in the latitude of 49° 50’. N, and 5° 
 21° w longitude, and Cape Ortegal 
 in the latitude. of 44° 10’ n, and 70° 
 43’ w longitude; to find the bearing 
 and diftance. 
 
 The difference of longitude is 142’ 3 
 and in the columns figned dep. under 
 10, 4, and 2, even with 43° the co- 
 middle latitude, are found 6°8200, 
 27280, and 1°3640; then increafing 
 the two former as before fhown, their 
 fum is 96°844 miles w, for the de~ 
 parture ; and the bearing, or courfe, 
 anfwering to 340 miles difference of 
 latitude, with Q6°844 departure, is 
 found about 16° sw: and the diftance 
 about 354 miles. 
 
 OF MERCATOR’S SAILING. 
 
 a HE ufes of the table of meridional parts are fully fupplied by 
 the table of logarithmic tangents, as is demonftrated in N° 219 
 of the Philofophical Tranfaftions. It is there proved, 1ft, That the 
 meridional line, or fcale of Mercator’s Chart, is a fcale of the log. 
 tangents of the half complements of the latitude. 2dly, That fuch 
 log. tangents of Mr. Briggs’s form, are a fcale of the differences of 
 longitude, on the rumb which makes an angle of 51° 38’ 9” with 
 the meridian. And 3dly, That the differences of longitude, on dif- 
 ferent rumbs, are to one another as the tangents of the angles of 
 thofe rumbs with the meridian. itl 
 
 Hence it follows, that the difference of the log. tangents of the half 
 _ complements of the latitudes, is to the difference.of longitude a fhip 
 makes in failing on any rumb from the one latitude to the other, as 
 the tangent of 51° 38’ 9” (whofe logarithm is 10°1015104) to the tan- 
 gent of the angle of the rumb or courfe with the meridian ; fo that: 
 
 I. If two latitudes, and the difference of longitude be given, the 
 courte and diftance are readily determined by this rule. 
 Take, 
 
176 OF MERCATOR’S SAILING. 
 
 Take, by help of the tables, the difference of the log. tangents of 
 the half complements of the latitudes, efteeming the la{t three figures 
 to be a decimal fraCtion ; and add the complement of its logarithm to 
 the logarithm of the difference of longitude reduced to minutes, and 
 the conftant log. 10°1015104; the fum (abating radius) fhall be the 
 log. tangent of the courfe. And to the log. fecant of the courfe, add 
 the logarithm of the difference of latitude reduced to minutes, the fum 
 (abating radius) fhall be the logarithm of the diitance in minutes. 
 
 Example, Given the Lizard to be in latitude 49° 55’ n, Barbadoes 
 in 13° 10’ n, and their difference of longitude 53°00’, or 3180° w3; 
 to find the courfe and diftance. 
 
 i Co. lat ra 38° 25’ |. tan. 9°39930821.3180'= 3°5024271 
 zee" Lizard = =20 221, tan. 95620477 conft. log.10°1015104 
 
 diff. 3372°G05 its co. log. 64720346 _ 
 
 Log. tang. of the courfe 49° 59’ 10” sw - = - - - 10:0759721 
 Log. fec. of the courfle 49 59 10 - = - = = = 1071918067 
 Log. of 2205’ diff. of the; latitudes’: oie Sb 58 fate eos de BB 4840816 
 
 Log. of 3429°373 diftance of Barbadoes from the Lizard — 3°5352153 
 
 If. If two latitudes and the courfe be given, the difference of lon- 
 gitude is obtained with the fame eafe: for as the tangent of 51° 38° 
 g” is to the tangent of the courfe, fo is the difference of the log. 
 tangents of the half complements of the latitudes, to the difference. 
 of longitude fought. ‘Therefore, to the complement of the conftant 
 log. 1071015104, add the log. of the difference of the log. tangents 
 of the half complements of the latitudes, and the log. tangent of the 
 courfe, the fum (abating radius) will be the log. of the difference of 
 longitude in minutes. 
 
 Example. Given the latitudes 49° 55’ and 19° 10’, and courfe 
 49° 59’ 10"; to find the difference of longitude. 
 
 Lat. 13° 10’, its } co lat. 38° 25’ 1. tan. 9°8993082 
 
 Lat:49 55 - - - 20 231.tan.9°5620i77co.conft.log.9°3984896 
 
 diff. 3372°605 - its log.3°5279654 
 Log. tang. of the courfe 49° 59°10" - - - -- - - 10°0759721 
 Log. of 3180’ = 53° for diff. of longitude - - - - 3°5024271 
 
 By this rule, having two good obfervations of the latitude, and the 
 courfe duly fteered, the reckoning of a fhip’s way is beit afcertained, 
 efpecially if you fail near the meridian. | 
 
 Ill. if the latitude departed from, the courfe fteered, and diftance 
 failed, be given; to find the fhip’s latitude, and difference of longitude. 
 
 Firft, the latitude is obtained from the confideration that the diftance | 
 is to the difference of latitude, as radius to the cofine of the courfe, 
 which is common to plain failing. Therefore to the log. of the diftance 
 add the log. cofine of the courle, the fum (abating radius) is the log. 
 of the difference of latitudes; which ditierence added to the lefier 
 latitude, or fubtraCted from the greater, the fum or remainder is the 
 prefent latitude: then having the two latitudes, and the courfe, the 
 difference of longitude is found by the fecond. 
 
 . : Example 
 
OF MERCATOR’S SAILING. 177 
 
 Example. Waving failed from the Lizard, in lat. 49° 55’ N, ona 
 
 courfe 49° 59' 10” fouth-wefterly 3429-378 miles: required what lon- 
 gitude and latitude the fhip is found in. 
 Log. of 3429°478 the diftance failed . . . . . . 3°5352153 
 Log. cofine of 49°59 10" thecourfe . . . «°° « 998081933 
 Log. of 2205’, or 36° 45’ diff. of the latitudes . . . 3°3434086 
 Now fubtraGting 36°45’ from 49° 55, the remainder 13° 10’ N, is the 
 latitude the fhip is found in. 
 
 By which latitude, now known, the difference of log. tangents 
 will be found 3372:605, and the further procefs in nothing differing 
 from the fecond rule, by which the difference of longitude will be 
 found 53° 00’. 
 
 Thus the dead reckoning by the log. line, and daily account of a 
 fhip’s way, are duly kept, and the trouble very little more than by 
 plain failing. 
 
 Thefe are all the cafes that occur in practice ; the reft, which are 
 moftly fpeculative, are either eafily reducible to thefe, or elfe not to 
 be performed by logarithms, and therefore come not at prefent under 
 our cognizance, | 
 
 But it is to be noted, that both the complements of the latitudes 
 are to be eftimated from the fame pole of the world ; which may be 
 from either ; and therefore if one latitude be n, and the other s, to 
 have their complements, you muft add 90° to one of them, and fub- 
 tract the other from 90, and then the operation will be the fame as 
 in the preceding cafes. ihe 
 
 Example. Given St. Jago, one of the Cape-de-Verd iflands, in the 
 latitude of 14° 56’ Nn; and the ifland St. Helena, in latitude 15» 
 45’ s, and their difference of longitude 30° 12’ £; to find the courfe 
 and diftance. Bi | 
 Colat 13e Jago 52° 2s’ , 1. tan, 10°1144965 1. 1812’ 3°2581582 
 
 " (St.Helena 37 73.1. tan. 9°8790845 conft.log.10°1015104 
 2354°120 itsco. log. 6°6281714 
 
 Log. tang. of the courfe 44° 11'53” sE . . ~ 99875400 
 Log, fec. of the, courfey 4411-583) oo. ee 101445200 
 Log. of 1841' diff. of the latitudes . ... . . .  3°2650538 
 Log. of 2567°875 diftance of St. Helena from St. Jago — $*4.095738 
 
 Or if it be thought eafier, when one latitude is N, and the other s, 
 you may add 90‘ to each of them, the fum of the log, tangents of | 
 their halves (abating twice the radius) will be the fame as the diffe- 
 rence of the log. tangents of the former. For an example, take the 
 fame latitudes as in the precedin ; 
 
 14° 56'=104° 56’ 2. 52° 28']. tan. 10°1144965 
 Rhea 20% a 45 =105 45 ? BR ae: 5221, tan. 10°1209155 
 The fum (abating twice the radius) equal to the former 
 
 AUCANCeDe etry Trea iee Ss Vaile | te are Mee AGS tel 20 
 
 Alfo, when both latitudes are of the fame name, that is both n or 
 both s, you may add 90° to each of them, the difference of the log. 
 tangents, of half thefe fums, will be the fame as of the log. tangents 
 
 of half the complements of thofe latitudes. 
 24 TABLE 
 
178 
 
 CIRCULAR ARCS. 
 
 ‘TABLE FOR THE LENGTHS OF CIRCULAR ARCS. 
 
 ‘TL us is table 12, and conftitutes page 340. It contains the 
 lengths of every fingle degree up to 180, and of every minute, fecond, 
 and third, each up to 60. The form of it is obvious; the length of 
 each degree, minute, fecond or third, immediately following it on the 
 fame line in the next column. And the two following examples will. 
 
 fhow the ufe of the table. 
 
 Ex. 1. To find the length of an arc 
 of 57° 17 447 48". 
 
 Take out from their refpective 
 columns the lengths anfwering to 
 each of thefe numbers fingly, and 
 add them all together, thus: 
 
 57° 0°9948377 
 Ly " 49451 
 4.4” : 2133 
 4.9/7" 39 
 the fum or 1°O0000000isthe 
 
 whole length, and is equal to the 
 radius ; that is, the length of an arc 
 of 57° 17 44” 48” is juft equal 
 to the radius of the circle. 
 
 Ex. 2. To find the degrees, mi- 
 nutes, &c in the arc 1, which is 
 equal to the radius. 
 
 Subtract from it the next lefs ta- 
 bular arc, and from the remainder 
 the next lefs again, and fo on till 
 nothing remain; andoppofite tothe 
 feveral numbers fubtracted, will be 
 the degrees, minutes, &c ; thus: 
 
 Given length 1°0000000 
 BI AS ten, «5 SBR LT 
 
 51623 
 aca se 49451 
 
 2172 
 dA ik, ares a Wel ee 
 wie ie, emgage 1 
 
 _ So that the arc which 1s equal to 
 theradius contains 57° 17/4448". 
 
COMMON AND HYP. LOGS. 
 
 179 
 
 TABLE FOR COMPARING HYP. AND COMMON LOGS. 
 
 Tus is table 13, and is the upper part of page 341. It contains 
 the hyperbolic logs. anfwering to the firft 100 common logs. and is 
 
 very ufeful for fpeedily changing the one into the other. 
 
 Ex. 1. To find the hyp. log. an- 
 fwering to the common log. 
 0°9542425. 
 
 Beginning at the left hand, and 
 dividing the given number into 
 periods of two figures each, in- 
 cluding the index, take out the hyp. 
 log. to each period, omitting two 
 figures at the 2d period, four at 
 the third, and fix at the 4th; then 
 add them ‘all together, thus: 
 
 Ex. 2. To find the common log. 
 an{wering to the hyp. log. 
 2°19712246. 
 
 Subtra& continually each next 
 lefs tabular hyp. log. from the 
 given number, and from the re- 
 mainders ; and the feveral com- 
 mon logarithms anfwering to thefe 
 tabular hyp. logs, joined toge- 
 ther, will be the com. log. re- 
 quired, thus : 
 
 com. log. hyp. log. hyp. log. 
 09 o. 01 Iso 206 given 2°1972246 
 54 ‘ 1243396 09 2°0723226 
 ye 5526 1248980 
 25. 58 54 1243396 
 0°9542425 2°1972246 anf, 5584 
 Sirs 24, 5526 
 ae. 
 25 58 
 
 0°9542425 anfwer. 
 
 The remaining pages contain the {mall table of the names and de- 
 grees, &c, in the points of the compafs ; which needs no illuftration ; 
 and a copious lift of fuch errors, with their eorrections, as have been 
 difcovered in the principal books of logarithms; among which are 
 many that have been detected by myfelf, both in the Avignon edition 
 of Gardiner, and in Gardiner’s own quarto edition, as well as in 
 the French tables by Callet, and by Didot: which renders this lift 
 more complete than any former one 3 and it will be found very ufeful 
 in correcting thofe books of tables which are already in the pofleffion 
 of the public. As to all the editions of Sherwin’s and Gardiner’s 
 tables in o€tavo, the errors in them are far too numerous to be printed 
 in this or any other work, as they amount to many thoufands, even 
 in the edition of 1742, publifhed by Gardiner, in which the laft figures 
 of the logarithms are ufually not correct to the neareft unit, except in 
 a very few pages at the beginning, and at the end of the table, {fo that 
 it cannot be depended on for nice calculations, 
 
 TABLE 
 
ite 7 
 | i." 
 
 vib cab 
 eM 
 
 petals pty” Poa. it a bys a) ey 
 
 4, py. pit God, arty > athe 
 ‘age Gia ok HAAS wha? Ret, ‘sitio ath 
 
 Uga¢ vadcel igi, tiie 5 eras as a Pree 
 Nee red at shee ok Cath ag a hi & 
 An iyibas: 22109 Nf. Me A alg: oe: we 
 “an : Boone) cob pniaeleak cian gi ‘sna poaihaci a! 
 
 o 
 
 € aaah moldy eB ep ‘etyehee Sele) ed Serie 2 . nt 
 
 peeks Det: hhh 8 Bigg ip eek 
 iii sgl ey a ‘3 fyaeoe Bldies | 
 Sunen fea a “a Mabie tas Re te : 
 ie OMe see 
 ma write. 7 
 Sy Ai Pi in 6 eit a 
 Rif, Pisa esthy 
 
TABLE I. 
 
 CONTAINING 
 
 THE LOGARITHMS 
 
 OF 
 
 ALL NUMBERS, 
 From 1 to 100000. 
 
 2B 
 
1|0’0000000 
 2|9°3010300 
 319°4771213 
 4'0:6020600 
 5|0'6989700 
 610°7781513 
 7|0°'8450980 
 8|0°9030900 
 9|0°9542425 
 10}1°O0000000 
 11|1°0413927 
 12/1°0701812 
 13}1°1139434 
 14|1°1461280 
 15}1°1760913 
 16]1°2041200 
 17}1°2304489 
 18/1°2552725 
 19|1°2787530 
 20|1°3010300 
 2111°3222193 
 22)1°3424227 
 23|1°3617278 
 24]1°3802112 
 25|1°3979400 
 26|1°4149733 
 27\1°4313638 
 28|1°4471580 
 29]1°4623980 
 30}1°4771213 
 31|1°4913617 
 32}1°5051500 
 3311°5185139g 
 34/1°5314789 
 |35}1°5440680 
 36|1°5563025 
 37|1°5682017 
 38}1'°5797836 
 3911°5910646 
 40}1'6020600 
 41}1°6127839 
 42|1°6232403 
 4311°6334685 
 4411°6434527 
 45|1°6532125 
 46|1°6027578 
 47|1°6720979 
 45|1°6812412 
 40\1°6901961 
 5C|1°09897 
 
 (2) Numb. 1to100,and|_LoGcaRiTHMs —N.100 4.00 
 their Log. with Indices.||N.| Log. ||N.| Log. |[N.| Log. 
 
 51]1°7075702||100|0000000 
 52\1°71600331/101|0043214 
 53]1'7242759||102'0086002 
 5411°7323938]|10310128372 
 55|1°7403627|110410170333 
 
 50|1°7481880]/105|0211893 
 57|1°75587490||100}0253059 
 58]1°7634280]107|0293838 
 50]1°7708520]|108/0334238 
 600}1°7781513||109|0374205 
 
 61}1°7853298)]1 1010413927 
 62)1°7923917||111]0453230 
 63}1°7993405)|112/0492180 
 64|1°8061800!|113}0530784 
 65}1°81291341|11410569049 
 
 66}1°8195439]}115|0606978 
 67|1°8260748]|1 16}0644580 
 68|1°8325089}|117|0681859 
 60|1°8388491||118|0718820 
 70|1°8450980)|119|0755470 
 71\1°8512583}/12010791812 
 72\1°8573325||12110827854 
 73|1°8633229]|122|0803598 
 74\1°86923 17||/123|0899051 
 75\1°8750613}|124\09034217 
 76)|1°8808136)|125}0969100 
 77\1°8864907}|126| 1003705 
 78|1°8920946)}127|1038037 
 79\|1°897627 1)|128]1072100 
 80}1*GO30900)/129]1105897 
 81}1°9084850)|130]1139434 
 §2}1°9138139]1131]1172713 
 83]1°9190781]|132}1205739 
 8411°92427931|133)1238516 
 85|1°9294189)134)1271048 
 
 86}1°9344985}/135]1303338 
 87|1°9395193}|136]1335389 
 88]1°944482711137|1367206 
 89]1°9403900}|138)1398791 
 90}1°954242511139]1430148 
 Q1}1°95904.14}}140]1461280 
 Q2]1°903787 811411492191 
 93}1°9684829]|142|1522883 
 Q4}1°G73.127911143|1553360 
 95|1°97772301114411583625 
 Q611°98227 1211145}1613680 
 
 .97|1'°9867717|1146}1643529 
 
 98]1°991226111147|1673173 
 99]1°9950352/1148!1 702617 
 
 100|2°0000000)]149|173 1863 
 
 N.100 L.00 
 
 150|1760913 ||205|3010300 
 151}1789769||201|3031901 
 152]1818436}}202)3053514 
 153}1846914/!203|3074960 
 154|1875207||204|3096302 
 155|19033 17||205|3 117539 
 156|193 1246}|/206)3 138672 
 157|1958997||207|3 159703 
 
 158|1986571 
 15912013971 
 
 160|204.1200 
 161|2068259 
 162|2095150 
 163|2121876 
 164|2148438 
 
 165|2174839 
 166|2201081 
 107|2227165 
 168|2253093 
 164|2278867 
 170|2304489 
 171|2329961 
 172|2355284 
 173|2380461 
 174|2405492 
 175|2430380 
 176|2455127 
 177|2479733 
 178|2504200 
 179|2528530 
 180|2552725 
 181|2576786 
 182|26007 14 
 183|2624511 
 184/2648178 
 18512671717 
 186|2695129 
 187/27 18416 
 1882741578 
 189|2704618 
 190|2787536 
 191|2810334 
 192/2833012 
 193|2855573 
 19412878017 
 195|2900346 
 196|2922501 
 197|2044662 
 
 198|2966652!| 
 
 208|3180633 
 209|3201463 
 210|3222193 
 211|3242825 
 212/3263359 
 213|3283796 
 214|3304138 
 
 215|3324385 
 21613344538} 
 21713364597 
 21913384565 
 219|3404441 
 
 22013424227 
 22113443923 
 22213463530 
 223|3483049 
 2.24'3502480 
 
 225|3521825 
 22613541084 
 227|3560259 
 228|3579348 
 229|3598355 
 230|3017278 
 23113636120 
 232'!3654880 
 233|3073559 
 234|3092159 
 235/37 10679 
 23613729120 
 23713747483 
 
N. 250 L. 39 
 
 351}5453071 
 352|5465427 
 135315477747 
 1135415490033 
 305/4842998 
 306/4857214 
 13071487 1384 
 308/4885507 
 309|4899585 
 310/44913617 
 311}4927004 
 311312'4941546 
 
 355|5502284 
 35615514500 
 357|5526682 
 358|5538830 
 
 32915171959 
 330)5185139 
 33115198280 
 332)5211381 
 33315224449 
 334|5237465 
 335|5250448 
 336/5263393 
 337|5276299 
 
 OF NUMBERS. 
 
 40016020600 
 401|603 1444 
 402|/6042261 
 403|6053050 
 40410063 814 
 40510074550 
 406|}6085260 
 407|6095944 
 408/6106602 
 409|0117233 
 
 450/6532125 
 45110541765 
 45210551384 
 45310560982 
 454/0570559 
 45510580114 
 45016589046 
 457'0599162 
 458/6008055 
 459/0618127 
 460/0627578 
 461|0637000 
 462/0046420 
 463|06655816 
 46416605180 
 
 468|6702454 
 
 46610711728}, 
 
 470/07 20976 
 4711673020 
 472|0739420 
 47310748611 
 
 47410757783 
 47510766936 
 
 )||476107 70070 
 
 47710785184 
 478|079427¢ 
 4709|0803355 
 480|6812412 
 481|0821451 
 482|0830470 
 483|083947 1 
 7|\|484|0848454 
 485|0857417 
 486|0806363 
 487|0875290 
 488/0884.198 
 48910893089 
 71|490|0901961 
 11/49116910815 
 3}1492|09190651 
 49310928469 
 494|09037200 
 495|0946052 
 496|6954817 
 
~~ th 
 
 (4) LOGARITHMS N.500 L. 69 
 N.| Log. ||N.| Log ||N.| Log. ||N.| Log. ||N.| Log. 
 
 500/6989700]|550|74.03 627 ||60017781513||650|8 129134) |700|8450980 
 50116998377 ||551\741 1516]160117788745]|65 1|8135810)|701|8457180 
 50217007037 115521741939 1||602|7795965||652|8142476)|702|8463371 
 503|7015680}1553|7427251||603|7803 173||653/8149132)|703 
 504}7024305||55417435098]|604|7810369]|65418155777||704|8475727 
 
 505|70329141|555|7442930]|605|7 817554]|655|8162413]|705|8481891 
 506/704.150511556|7450748||606]7824726]|656|8 169038} |706|8488047 
 507|7050080}|557|7458552)|607|7 83 1887]|65718175654!|707|84941 94 
 508|7058637||558|7466342]|608)783903 6||058/8 182259) |708|8500333 
 50917007 178]|559|74741 18||609|7846173]|650|8 188854) |709|8506462 
 
 510/7075702 
 511|7084209 
 512/7092700 
 
 560/748 1880}|610}7853298]|660/8 195439) |7 10/8512583 
 561|7489629]|61 11786041 2]|661|820201 5}|711}8518696 
 562|7497363||612|48675 14]|662|8208580)|7 12/8524800 
 
 51317101174]|563|75050841|613!7874605||063|8215135)|7 13|8530895 
 
 51417109631 
 
 515]7118072 
 516|7126497 
 517|7134905 
 518/7143208 
 
 564/7512791}/014|7881684]|664|8221681)|7 14/8536982 
 
 505|7520484||615|7888751||665|8228216 |715|8543060 
 566|/7528164||616|7895807}||066'8234742)17 16|8549130 
 5671753583 1||617|7902852]|667|8241258)17 1718555192 
 568|7543483||618]7909885||668|8247765||718|8561244 
 
 519|7151674||569'75511231/619|7916906]|669|8254261)|7 1918567289 
 
 5207160033 
 52117168377 
 52217176705 
 523'7185017 
 52417193313 
 
 525/7201593 
 52617209857 
 
 570}7558749]|620|7923917||670|8260748]|720|8573325 
 57117566361 ||621|}7930916||67 1|8267225]|721|8579353 
 572|7573960]| 62217937904! |672|8273693||722|8585372 
 573|7581546]| 623!7944880}|073/8280151||723/8591383 
 574|7580119]|62417951846]|074/8280590||724|8597386 
 
 5751759667 8]|625|7958800]|6075|8293038]|725|8003380 
 576'7604225||626|7965743]|676|8299467]|726|/8609366 
 
 527|7218106||577|7611758]|627|7972675||677|8305887||727|8015344 
 
 52817226339 
 52917234557 
 530|7242759 
 531|7250045 
 53217259116 
 
 {53317267272 
 
 534|7275413 
 
 535|7283538 
 53617291648 
 537|7299743 
 538|7307823 
 530|7315888 
 
 540/7323938 
 54117331973 
 54217339903 
 543|73470908 
 54417355989 
 54517363965 
 546|7371926 
 54717379873 
 54817387806 
 54917395723 
 
 N.| Log. |jN.} Log. ||N. 
 
 578|7619278||62817979596]|678/83 12297||728|8621314 
 579|7626786||62917986506]|67 9/83 18098] |720|8027275 
 580!7634280]|630]7993405]|680/83 25089} |730/8033229 
 58117641761 163 1]8000294}|681|833 147 1||73 118039174 
 582!7649230]|632|/8007 17 1||68218337844]|732|8645111 
 583|7656686]}633|8014037||68318344207]|7331805 1040 
 584|7664128]/634|8020893]|684/8350561)|734|8050961 
 
 5851757 1559||635|8027 737||6085|83 56906) |735|8062873 
 586|7678976||636|803457 1||686|8363241 ||736|8668778 
 587|7686381||637|8041394]1687|8360567 |1737|8074075 
 588/7693773||638|8048207||688|837 5884 ||738|8080564 
 589/7701153}|639|8055009]|689|8382 192||739|8080444 
 
 590/7708520]|640) 8061 800} |690/83 88491 ||740/86923 17 
 59117715875||641|80685 80||691|8394780)|741/8698182 
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 (6) LOGARITHMS N. 10000 L.000 
 
 03|0013009|3442/3875/4308]474115 17415607 160391647 216905 alrvaltyslirg 
 733717770|8202|8635|9067|19499199321036410796|1228 5 ainloitleie 
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 07|9030205 0720]1157/1588 2019}/245 1|2882)/33 131374414174 9139113901389 
 4605|5036|5467|5 898/03 28]1675917 190|7620/8051|8481 /431/430/429 
 
 1| 43} 43| 43 
 1010]00432.14|364414074/4504/4933'5363|5793|6223|0652/7082 : Be: ane Be 
 4 ” 0088/05 1 7|0947|1376})3,129 129|129 
 7512179411837 1|8800|92291/9659|0088)05 1] 7/0947]137 47972 its 
 
 J y 5'216)215/215 
 6094|052316952173 807 809||823 81866619094/952319951 612591058 257 
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 4660|5088|55 16159441037 2||67 9017 227|7655|8082|8510 013881587 he 
 
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 6002|6427|68531727917704118130|85561898119407|9832 a 717 170 
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 21}009025710683}1 108]1533]19059]2384|2800/3234|3059/4084 elasrloselese 
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 7239|7662|8086|85 10|8933}19357|9780}0204/0627|1050)} |425/424/425 
 26|0111474]1897|2320)/2743|3 166]3590/4013/4436/4850|5282)'1| 45] 49) 42 
 
 27|  5704|6127/0550/6973173961'7 818/824 1 |8664\9086 9509}|° a at i | 
 
 28} _ 9931|035410776|1 198}1621}}2043|2465}2887/33 10)3732I),1 ool cdlicg 
 29/0124 154|4576|4998|5420]5842]16264|6685|7 107175291795 1||s\o1 sla 1901019 
 8372187941921 51963710059l104801090111323]1744121651]6|2551254|254 
 
 71298129'7/296 
 8}3401339|338 
 
 4.29142 1[420 
 1] 49} 42) 42 
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 36 0153598/4017 4436/4855|5274 5693 6112/0531 6950 7 3091/3 12°7|126)126 
 
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 6}253}253)252 
 
 39} 615516573|6991|7400|7 827|18245|8663|908019498/9916))!o9519951204 
 1040101 70333)0751\1168|1586|2003|12421|2838|325613673|4090 ; an oe -. 
 41} 4507/40241534215759161 76|16509317010)7427|7844|8260)/- ere 
 42) . 8677\90941951119927|0344110761[117711594/2010|2427 7m _ no 
 43|0182843|3259|3676/4092|4508]4925|5341/5757|6173|0589 A ie 
 
 4510191163]1578:1994|241012825]|3240|3056/407 1/4486/4902!/4)1 68/1 67/167 
 
 46} 5317|5732 6147|6562|6977|17392|7 807|8222|8037 9052 nie ih Ne 
 47| 9467|9882.0296/0711]1126)}1540}1955/2369/2784/3 108) -\ oo alooaloq9 
 
 48|0203013/4027 4442/485615270 5684 0099 6513 0927 7341\\9133519341934. 
 775518169 8583|8997|941 1||9824|023810652|1066|147 oljo|s77's76|375 
 
 —— | i i | I ‘cid 
 
 9 Dif.&Pro.Pts. 
 
IN. 10500 L. 021 OF NUMBERS. (7) 
 2 3 4 5 6 0: 8 9 Dif. &Pro.Pts. 
 
 ————$ | —_———__] ——_—__| |_-__- -——_ ] - 
 
 1050|0211893|2307|2720|3 134135471|3061|43741478715201/5614|| | 416/415 |414) 
 6027|6440)6854|7267)7680)|8093)/8506}891919332/9745]|1) 42) 42) 41 
 521022015710570|0983} 1396]1808}|222 112634/3046134.50|387 1] : ce ihe hig 
 4284)4696)|5 109|5521}5933/16345|6758)717017582|7904 Aiastiee py 
 8406}8818|9230\9642/0054/0466|0878]1289]1 7012113} slooglaoglao: 
 5510232525|2936)3348|37591417 1\4582|4994|5405|5817|6228] 6 250)049}248 
 6639|7050/7462!7 873|8284)|8695|9 1061951 719928|0339 ‘ poalaatiant 
 57|0240750|1161|1572}1982)2393||2804/3214|3625/4036|4446} glacial sn ilar 
 4857|5267|567 3}6088/6498/}6900/73 19|772018 1309/8549 aTSTTSTraT 
 8960|9370/97 80/01 90/0600}|1010}1419]1829/2239]2640 nerey yi 
 1060/0253059|3468/3878)4288}4697 ||5107|5510|5926/0335}6744/2) 33] 89] 82 
 7154'7563|797 2|8382!8791119200|9609/001 8/0427/0836)|3) 1241124 125 
 6210261245|1654|2063|2472128811328913698|4107/4515|4924 - cri ee heh 
 5333|5741|6150)0558|6967 |7375|7783)/8192|8600\9008} 410 lo ario4n 
 
 9416|9824/0233|0041)1049)1457)1805|2273|2680)3088)/7logclossjogs, 
 6510273496|3904/43 12147 19|5127 |5535,5942|6350)6757)7 165]/§}930|35u)320 
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 6710281644|205 112458|2865|3272'13679|408614492/4899|5306)_|#10|409 
 
 68} 5713/6119|/6526|603217339]17745|815218558 8904/9371 4 ai 
 69} —9777|0183}0590|0996}1402]|180s}2214|2620)3026)3432]/°) 82) 82 82 
 
 1070|0203838]4244/4640|5055]5461115867|6272|667 8|7084|7480)|4|164|164)163 
 
 71| —7895|8300|8706/911119516/|9922/0327/0732/1138]1543 : one be ras 
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 4173|4576/4970'5382/5785||2| 81] 81] 81 
 
 6188|6590|6993|7396 7799) 8201|860419007/9409/9812 " a bis ae 
 79|0330214/0617|1010|1422}1824/2226/2629/303 1|3433/3835 5120412031005 
 
 8257|8059|900]9462/9864 |0265 0667/1008) 1470 1871})7/280 284 9 
 8210342273|20741307 5/347 71387 8||4270|4080)508 1|5482/5884 9136613651365 
 
 83) 6285)/0086/7087)7487|7888||8289|8090/909 1/949 1/9892 anootiGs 
 84103 50293}0093|1094}1495|1895]|22906|2696/30906'3407/3 897 x grieniwn: 
 85} —4297|4698|5098)5498|5898)}0298/0698/7098'7498)7898!/9) 81} 81} 80 
 86] 8298/8608)9098|9498|9898!|0297|0697|1097| 14.961 8g6)3}121/121}121 
 
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 87 |0302295|2695|3094/3404)/3 893 |4293|4092/509 1/549 1|5890 soneaalont 
 
 id ||6|242)24.2}241 
 89|0370279|0678]1076)1475|18741'2272]267 113070:3468|3867]!mlogs|ogology 
 ~ 4265|4663|5062|5460|5858]10257|6655|7053)745 11784g}|8}323)322/322 
 
 ‘ or a te ; 4119}364| 363/362 
 8248/8046|90441944219839|10237|0635|1033|1431]182 OT GnESs 
 02/0382226|2624/3022/34109|38171\4214/4612|5009|5407|5 804})_|#01|400)/599 
 
 6202/6599|6996!7393|77911/8188!8585|8982|937919776 . = a a 
 
 8106}8502|8898|/929419690)/0086|04.82 087 8) 1274) 1670)'5|201/200)200 
 
 eS . 24.1|/240/23 
 97|0402066'2462|2858/325413650114045|4441 4837/5232/5628 6 Ca ilcatene 
 
 98} 6023/6419/0814/7210)7605/'8001/8390/879119187/9582)' also 1\3901319 
 99] 9977|0372|0767)1162)1557||1952|2347|2742)3 1373532) 9|361|360/359 
 
 O l 9g) 3 4 5 6 TAs 9 |Dif. Pro. Pts. 
 
(8) LOGARITHMS N.11000 L.041 
 N. O | l 2 3 4, 3 6 7 8 9 |!Dif.ePro.Pts. 
 
 1100)0413927/4322)47 10|551 1/5506] 5900|6295|069017084|7479]| |398|397|396 
 Ol 7873)|8208/8662|9050/945 1119845 |0239/0633} 1028/1422}/1] 40} 40] 40 
 
 0210421816/2210|2604|2998/3392I|3786|4180/4574 4908,5361) 71 Pol alte 
 
 des 4 92971 411591159]158 
 64] _9691/0084)0477/087 11264] 1657|2050)2444!283713230} |; 991901108 
 
 05|0433023|4016/4409/4802)5 1951|5587|5980|0373]|6766)7 159}19|299|258)238 
 
 an a 7|279|2718|277 
 06} 7 551)7944/8337)8729/9 1221951419907 |0299|0692/1084]} 2h. 1 olos alg47 
 
 1} 40} 39} 39 
 1110|0453230|362 1/401 2/4403/4795]]5 1186/5577 |5968|6359|6750||2| 79} 79} 79 
 
 13] 4952/534215732/0122)65 12|/0902/7292)7682) 8072/8462). 23710361236 
 14] 8852/9242/9632/002 1/041 1/,0801]1190}1580)1970/23590|l"|o77/o761075 
 
 3 8/316|315|314 
 15]0472749/3138)3528 391714306]}4096/5085|5474| 5864/6253 olaselanslons 
 17|0480532\0921|1309|1698/2087]|2475|286413253|3641/4030]|_[992[991|390 
 
 18] 4418]4806|5195/5583|59721|636016748]7136)7525 7913|/) ie a 3° 
 
 3]118}117]117 
 1120/0492180)]2568|2956/3343|373 1/41194500/4894]5281|5660]/4|157|156|156 
 21) -6056|6444|6831/721817600]7993|8380)876719154)9541}9/196/1961195 
 22} 9929/03 16|0703|10g90)1477}1863}2250|2637/3024/341 11, oraloralors 
 2305037 98]41 84/457 1}4958/5344)5731 61 17/6504 68g0 7277 8131413131319 
 
 24| 7603|8049|8436/8822|920819595|9981/0367/0753]1139]/91353|3521351 
 
 i] 39] 39] 38 
 35}  9959|034110724]1106]1489]|187 112254)2636'301913401 lo] 74) 77) 79 
 
 4115411541154 
 
 5}193}193}192 
 38|0561423!1804/2186/2567|2949/|3330/3712|4093/4475/4856)-).colo43 930 
 
 39 5237 5619 6000/03 81|6762)1'7 14317524 7905|8287 8668 7127012701269 
 
 1140] 9049|9429/9810/0191/0572/10953|1334)1714|2095|2476 oat aie nt 
 41|0572856|3237|3618/3998|4379|14759]5140|5520,5900|6281)) "I 
 42|  6661|7041174221780218182|18562/8942|932219702|0082)|_|P83|$82/381 
 43 |0580462/0842]1222/1 602|1982/12362/27411312113501|3881)|1} 38} 38} 38 
 
 44| — 4260/464015019|5399|5778]6158|0537\0917|7290|7676 |), 77), 75,76 
 
 45| 8055|8434/8813/9193|9572|19951|0330|0709] 1088] 1467)14|153|1531152 
 46|0591846|2225|2604|2983|3362/|3741|4119|4498/4877|5256 eae ie os 
 47| 5634|6013/6391|0770\7148)|7527|7905|S284|8662\9041 
 
 48} 941919797/0175|/0554|0932)|1310|1688|2066|2444)2822|!eleoglsogl305 
 49/0603 200/357 8|3956,4334)47 1 2/\5090|5468/5845|6223|6601)|9/34.5/34.4/343 
 
 | N 9 Dif.&Pro.Pts, 
 
{1160 
 
 1180 
 
 N.11500 L. 060 OF NUMBERS, 
 
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 2g 7 8 19 
 
 SSS eee ee Ee FPA peeQeg aes ae STE OREN] FANE "RSS 
 
 §110610753]1131]1508}1885 2262 2630301 7|3394/3771 A148||1 
 
 52|  4525/4902)52709|5050/6032)/04090/6786 7103/7540 7916 
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 55| 582016196|6572/6948/7324117690|8075/8451 
 56] —957819954|0330/0705}1081}114.56]183 2|2207|258312058 
 57|0033334'3709)4084}4460/483 5/152 10|5585}5960,6335|67 11 
 58} — 7086)7461/7836|821 1|8585]/8960|9335|97 10008510460 
 §9|0640834]1209/1584)1958/2333]|2708|3082)3457' 7/3831 4205] 
 
 4580|495415320|5703|607 7/645 116826|7200\7574 7948}| 9 
 8322|8696|9070]9444/98 18]/01 9210566|0940}1314}1688}}3 
 5050 5424 
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 62'0652061/)2435|2800)|3 182|3556)3930}4303|4677 
 63} 5797|6171|0544/6917)7291)|7664|8037|8410; 
 64|  9530/9903/0276|0649]1022)/1395]1768/2141 
 * 65|0663250|3632/4.005/437 7147 50115 12315495|5868 
 66 
 
 72\  9276|9647/0017|0388|0758}]1 cm ih 1869/2240 2610} 
 73\0092980|33 50)372 1/409 1}44611483 1]52011557 1/5941 631114 
 744  668117051/7421|7791|8160]|8530|8900\9270 9639 0009} 9|3371336|: 
 
 75|0700379|0748]1118]1487|1857 mops 2596|2965'3335/3704]|_[371/370}: 
 760} 4073/4442/4812/518115550/1591916288|0658/7027|7396)| 1 
 77\  7765|8134|85031|8871 9240 9609|9978|034-7 07 15/1084 
 78107 11453|1822|2190/2559|292713296|366414033/4401/4770 
 79|  5138/5506|5875|6243|661 11}6970|7348)77 16|8084/8452 
 
 8820/9188/9556|9924|0292||0660]1028]1396)1763/2131 
 8110722499|2867/323413 60213970]'433714705|5072|5440|5807 
 82}  617516542/0010\7277176441|801 1/837 9/874619113\9480 
 83} 98471021 5|0582|0940]1316]|1683|2050/241 6/2783/3150 
 84\07335 17|38841425114617|4984]|535 1157 17|6084|0450}6817 
 
 85] 7184/7550/7916|8283|8649|19016|9382/9748'0114/0481 
 86|0740847/1213}1579]1945123 1111267713043|3400|3775|4141 
 
 87} 4507/4873/5230|560515970]|6336|0702|7008|7433|7799 
 88] . 8164/8530/8895|9261|9626||9902/0357|0723/1088}1453 
 89|0751819|2184/2540]2914|327011364414010/4375/4740/5105 
 1190} . 5470/5835|6199|0564|6920)|7294|765g|8024|83 88/8753 
 Ql} 9118/948219847/021 110576]}0940}1305]1669|203.4)2398 
 
 92|0762763'3127|349113855|4220]14584/4948153 1 2|/5676|6040 
 93} 6404/6768|713217496|786011822418588|8952/93 16|\9680 
 94|0770043/0407 (077 111134]1498}]1862]2225/2589|2952/3316 
 95} 3679|4042)4406/47601513315496]58509|6222/6585\6949 
 96] 7312/767518038|8401|8764]|912719490}9853 (021610579 
 Q7|0780942/1304]1667|2030]2393]|2755|3 118|/3480/3843/4206 
 98} 4568/4093 115293]5656]6018}|}03 80|6074317 105|7467|7830 
 
 99} 8192)855418916/927819640]}0003|0365|0727|108911451 
 
 SSS ee ee ee eet 
 
 913311330|329 
 
 (9) 
 
 Dif. &Pro. Pts 
 
 41152}159)151 
 5]190]190]189 
 6|228|227|297 
 7]1266|265}265] 
 8130413031302 
 9/342(341|340 
 
 “137713761375 
 
 | 38] 38] 38 
 75) 15) 75 
 3]113 113/113 
 41151}150/150 
 5}J891188)188 
 6]226] 226/225 
 11264) 263/263 
 81302/301{300 
 9)339}3381338 
 
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 61224 294 223 
 
 [568/307 366 
 
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 3]1101110]110 
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 5]184)184]183 
 612211220|220 
 71258}257|256 
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 3]110]109}109 
 
 6}219)218)218 
 7)256)255}254 
 8}292)29 11290 
 913291328 va 
 
1210 
 
 1220 
 
 1230 
 
 LOGARITHMS 
 3445 1 6 9 
 
 1200|0791812|2174|2536]2898|3260)|362213983|4345|4707|5008 
 Ol] © 5430/5792/6153]051516876}|7238|7599|7961|8322|8683 
 02!  9045]9406|976710128/0490)]085 1|1212)1573|193412295 
 03/0802656|3017|3378|3739)4.100||4.461 482215 183|5543|5904 
 04} 6265|06620|698617347|7707|/8068|8429187 80|9150/9510 
 05} 9870/0231|0591|0952}1312)11672/203212393|2753 3113 
 0608 13473/3833/4193}4553|4913|5273|5633|5993|6353|6713 
 07| 7073|7432)7 792815285 12/887 11923 119591|9950103.10 
 08|0820660]1020]1388]1748|2107]/2467|2826|3 185|3545|3904 
 O9 
 7854|82 13/857 118930,9289]|9648|0007|03 65/07 2411083 
 11/083 144.1|1800)2159]2517/2876||3234135093|3951|4309|4668 
 12} 5026/5385|5743 6101/6459 6817171701753417892|8250 
 13| — 8608/8966)/9324 9682/0040 0398|0756)1114)1471|1829 
 1410842187|2545|2902|3260 3618)|3975|4333/4600|5048|5405 
 15 57| 63|6120)6478)6835)7 192)17550|7907|8264|8621|8979 
 16} —9336|9693|0050|0407 07641|1 121|1478]1835|2192|2549 
 17/0852906|3263|3619/3976,43331/4690|5046|5403/5700/011 
 
 18} | 6473|0820\7186)|7542'7890]/82.55|8612|8908|93 2419681 
 19|0860037|0393|0750}1106\14.62/11818/2174|2530/2886/3242 
 
 3598|3954/43 10/4666)5022|15378|5734|6089'0445|6801 
 21 7157|7512)7868]8224'8570||893519290|9046|0001 |0357 
 22|08707 12}1067|1423}1778|2133]12480|2844|3 190|3554|3909 
 23)  4205/46204075|5330)5685|}6040|6395|0750)7 10417459 
 241  7814|8160)85241887819233119588|9943|0297|065211006 
 
 2510881361]1715|2070|2424/2770)13 133|3488|3842/4196/4550 
 261 © 4905|5250|5013|5967/6321||0676|7030\7384|7738|8092 
 27 
 
 5510)5872'0226|6579|693 2117 28517630|7992/8345|8608 
 905 119404975701 10|0463],0816}1169]1522)1875|2228 
 3 110902581|2933'3286|3639|3991|14344|4697|5049|5402|5755 
 32!  6107|6400,0812|7164|75171|'7869/822218574|8926|9279 
 
 063 119983|0335|0687|1039]]1392|1744|2096|2448/2800 
 34)09013152|350413855|4207|4550|1491 11520315614|5966|0318 
 
 29 
 
 TE BALO 
 
 4263|4622)4981|534115700/|0050|641816777|7 136|7495]|— 
 
 N. 12000 'L.079 
 
 Dif.&Pro.Pts. 
 362|361|360 
 
 1] 36} 36, 36 
 2) 72) 72) 72 
 3]109}108}108}} 
 4l145114411 44 
 5118141817180 
 6|21'7/21'7/216 
 11 253}2531252 
 8} 290}289/288 
 9)326}325/324 
 
 359}358/357 
 
 4)1444143)143 
 5}180}1'79}1'79 
 6/215)215/214 
 71251|2511250 
 8/28'7|286|286 
 9/323|322)321 
 
 356 355 354 
 
 6}214) 213/212 
 1124.9} 249/248 
 8}285] 284/283 
 9}3201320|319 
 
 353)352|351 
 
 8445 8800/9153 9507|9861|10215|0569|0923|1276]1630)| < 
 
 6)212/211/211 
 7|24'7)24.6|246 
 8}282/282/981 
 9/318}317]316 
 
 35013491348 
 
 1] 35] 35) 35 
 2).70) 70F 70 
 3}105}105}104 
 
 ~ 116 210/209}209 
 11245 |24.4)244 
 8/280}279]278 
 
 3}1044104)104) - 
 41139]138}138 
 5}1741173)173 
 6/208}208|207 
 7)243}24.2124.2 
 8/278127 71276} 
 9}312]3114311 
 
 Dif. &Pro.Pts. 
 
N.12500 L.096 OF NUMBERS. 
 8 | 9 
 
 + - | ——- -- ——— ——— | ———__— 
 — —_—— 
 
 ——— 
 ee a= 
 —_— 
 
 100|9448]9795/0142/0490/10837 11 184]1531]1879|2220 
 "250)090910 ‘ook 32.67|3614|3962/14309/4656/5003/5349|5696 
 6043/0390|0737|7084/743 1 1777 81241847 118817|9164 
 
 63} 4034/4377|472115005|5409||5752|0096|0440/67 8417 127 
 64, 7471|/7814)8158/8501 8845}1918819532/9875}0219 0502 
 65|1020905|1249]1592)|193.5/2278]1262 1 |296513308|305 113994 
 66 ene 7\4080|5023|5366|5700]|0052|6395|6738}708 117423 
 
 Ra 66|8109 8452 poe 9137/9480 spn 0165 009% O850]|_ 
 
 7g|  5309|5648|5988|6328|6668|7007|734717687|8026|83606 
 79| 8705|904519385|9724'0063||0403 [07421082] 1421/1760 
 1280}1072100]243912778/31171345713796|4135|4474|4813/5152 
 81} 5491|5830\6169|650816847]|7 186|7525!7864|8203/8541 
 82| 888019219|955819896/02351057410912/1251/1590)1928 
 
 85}. 9031]9300|9707|0045|0383 }0721)1059|1396}1734|2072 
 80/1092410|2747/3085|3423|3760)40098/443 5/47 731511115448 
 87| 5785|0123|6460/0798)7135||7472|7810|814718484/8821 
 88 
 
 ay 
 
 (11) 
 Differ. 
 3441343 
 
 1} 34] 34 
 
 2} 69) 69 
 311031103 
 
 1411381137 
 
 5{172/172 
 6}206|206 
 
 m1 712411240 
 
 8}275)274 
 91310}309 
 
 1342)341 
 
 1] 34] 34 
 2) 68] 684 
 
 | 3)103]102 
 
 41137/136 
 51171]171 
 6}205]205 
 7|2391239 
 8/2'74/273 
 9'308/307 | 
 
 340|339 
 1] 34! 34 
 2| 68] 68 
 3}102]102 
 44136]136 
 51170}170 
 6}204]203 | 
 712381237 
 8/272/271 
 91306]305 
 
 338|337 
 
 1} 34) 34 
 2) 68) 67 
 
 1) 3}101;101 
 
 4/135)135 
 5} 169}169 
 6) 203/202 
 12371236 
 8}270/270 
 9|304)303 
 
 336)335 
 1} 34} 34 
 2] 67) 67 
 S}1011101 
 411341134 
 
 1/5}168)168 
 
 6/202/201 
 112351235: 
 8}269}268 
 913021302 
 
 3341333 
 1 33 “33 
 2) 67| 67 
 
 131100] £00 
 1411341133 | 
 
 5)167 167 
 6 200200 
 
 <1} 7/234|233 
 
 / || 8.267|266 
 913011300 
 Differ. 
 
(12) LOGARITHMS N. 13000 -L.113 
 O Pepe Qu SW ae Be Ge 1 T= 8 i Differ, 
 1300|1139434|9768|0102|0436|0770}{1 104]1437]1771|2105|2439]] }334)393 
 01|1142773|3107|3441|3774]4108114442]4775|5 109|5443|5776)1| 93] 33 
 
 6110)644310777)71101744417777)811118444/8777(9111)?| 67} 67 
 
 411341133 
 04}1152776/3 109|3442/3775|4108]1444.11477415107|5439157721 511 67l1 6% 
 
 6105|6438/677 117 103|7436}|7 76918 101|8434)/8767|9099 3 it ia 
 0432/976410097|042.910762]|1094|1427|1759|2091|2424 Blo chiaae 
 07|1162756|3088|3420|3753/4085]|44.1 7147491508 1|5413}5745 [91491139 
 
 6077|0409|074.1|707317405]|7 737 |8069|8401)8733|9005 }339)531 
 Ly 98!.38 
 
 6027|6358|6689|702 1|7352||7683}801 4|8345|867619007}|3/100| 99 
 
 ed | oh, 41133/132 
 9338|96690000|033 1|0662}|0993|1324] 1655|1986)23 16})*)'29) °° 
 
 6}199}199 
 
 8|266|265 
 9|299|298 
 
 3301329 
 
 5739|6068|6397 |6726/7055}|7384)77 13|8042/837 1/8099)}4|132)132 
 9028/9357|9686|001 410343||0672!1000|1329]1657|1986 : qr oe 
 
 22|12123 15}2643}2972)3300)3628|/3957 [4285/40 14/4942/52701) 05 11029 
 | 5598|5927|6255|0583|691 1|\7239|7568)/7896|8224/8552} slogalogs |. 
 
 8880|9208/9536|986410192||0520|084§]1 175}1503/1831|19]297|296 
 
 25|12221509/2487/281413142|3470113707|4125/4453/4780)5108} |328}327 
 5435|5763|6090164.1 8|674.5}|7073|7400\7 727 |8055|8382}11 35} 33 
 87091903 6/93 64969 1 |0018}103-45|067 2|000)1327/1654 : st 8 
 28]1231981|2308|2635|2962|3280||361 6|3942/4269/4596/4923 aah 
 
 AILS1I131 } 
 5250|5577|5903|6230|0557||0883|7210)7537|7 8603/8 19011511 64l164 
 
 8516}8843|9169|9496|9822|/014¢  )-' 75|0802)1128)1454 , te se 
 31}1241781/2107|2433|2759|3086}/341 2,375 3; 1064/4390/47 16 slaselo6a 
 5042)/5368|5694|6020|63 46||667 2|6905, 75 24/7050/797 Ollologsloo4 
 8301|8627/89531927919005)9930/0250)0582)0907/1233 i396 305 
 34/125 1558]1884|2209}2535|2800}}3 1 86)35 1 1/3837}4162/4487})-\— 
 4813|5138|5463|5788|61 141|10439|0764|7080)741417739}/2) 65) 65 
 8065|8390)8715|904019365}/9690(001 5|03390064)0989}3] 98) 98 
 37|1201314]16309]1964/2288/2613)|2038}3263|3587|3912}423711,), 163 
 450114886}5210)5535|5859},6184|6508/0833/7 157|748116|1 96/195 
 7806}8130|8454|877 919103] 9427|9751]0076}0400)0724|Irl228\008 
 
 1340|1271048|1372]1696|2020|2344||2668|2992)33 16|3640)3964||9/261|260 
 
 9'293}293 
 41) —4288|4612/4935|5259(5583| 5907|6230|6554|0878|7202|—-— 
 42) 7525|7849}8172|8496)8819}/9143 9466)97 90/01 130437} -|—— Se 
 43 |1280760!1083|1407|1730|2053 p87? 2700'3023)3346)3070},) --} o. 
 44| 3993|431614639/49625285|'5608|593 1|6254]0577/0900I5) on] ox | 
 
 7 22317546/7869/8191|8514}|8837|9160|9483|9805|0128 ' i a | 
 46/129045110773}1096}1418]1741]/2064)2386]2709/303 1/3354) | OF 
 -3076/3998}432114643|4965| 5288/56101593 2/6255|5577 |n\oonloo0g 
 689017 221|7543|7865|8187]|8510]8832/9154|947619798lisio59|258 
 40|1300119/044110763}1085]1407]/1720}205 11237212694|301 6})/9|292|291 | 
 
 —_——_— - } }— -—____ --—— 
 J | 
 ———— |__| 
 
 21314151617 | 8] 9 If Dife. 
 
.13500 L.130 | OF NUMBERS. (13) 
 N. 3 | 4 ny i PATS 
 
 Te coal ee tebe hone i ne ta A Tope are <ct OF ne Staraperind (> -prereare 
 
 6553/0875)7196|7518 7830 8161|848218803|9124|9446}]1| 32} 32 
 9797|0088|0409|0730/1052)}1373 1094 2015 2336 2657 : | e 
 
 4)129]128 
 6187 6507 6828 7149 7469 7790}8111}843 118752|9072I15|1 611161 
 
 55] ~~ 9393/97 13|003.4}03 54/067 5|10995}13 16} 1636)1956|2277 eb ue 
 50}1322597|2917|3237|3558)3878|4198]4518|4838|5158|5478||.|,-21)- 
 
 5798|6119|643016758 707817398)77 18]8038 8358 807 8!l9le90]289 
 
 3 1] 32 32 
 1360} 53809)5708|6028|6347|6666|16985|7305|7624}7943|8262)|2| 64] 64 
 61} 8581/8900)9219}9538/9857 101 76/0495|08 1411 133}1452)/5| 96) 96 
 
 62]1341771|2090|2400|2728|3046|3365|3684)4003 |4321/4640) 28 1% 
 
 63}  4950/5277|5596|5914/0233}16551|6870)7 188]7507|7825}|6|; 99|194 
 64) 8144)8462)8780)9099/941711973 5|0054|0372|0600}1 008}!7|294/293 
 65]1351327|1645|19631228 11259911291 7|3235|3553/387 114189 i ae si 
 66] 4507/4825|51431546115779|6096|04.1416732|7050|7367|—— 
 67|  7685|8003|8320|8638|8956]192731959119908|0226|0543}|_|?18/517 
 - 68}1360861}1178]1496)1813/213 112448/2765|3083/3400)3717]|11 32) 32 
 69] 4034/43521466914986153031562015937/6255|6572|0889 : gehts 
 1370} 7206/7523/784018157|847311879019107194241974.1|0058]/4]127/197 
 — 71|1370375/0691|1008} 1325]1641|1195812275}259 11290813 225]|5|159]159 
 72) 3541/8858417414491/4807]/512415440)57566073|0389 "159. de 
 73) 6705|7022)7338}7654|7970]828718603|89 19192351955 1||s\a54lo54 
 
 Poe 0183 0490 0815}113141447}1763 2079 2395|271 1hi9l286l285 |. 
 
 - 81]1401937/2251|2566 irioh 3500/3 823|4.138]4452 4766 7/221)221 
 
 5080|5395|5709|6023|0337|665 1}6966|7280|7594|7908||3|09° = 
 
 -87|1420765 ‘ine 1391 net 20171|2330|2643/205613260|3582)|4|!26)125 
 
 88}  3895/4208|/4520)483315146||545915772|6084|6397|67 10 : real e. 
 
 ' 7022)7335|7648!7 960/827 3||8586|8898}921 119523|9836)\n\o091019 
 1390 143014810460|0773|1085|1398}|1710|202212335|2647|2950}|8|29 1250 
 '/3271135841389614208|452011483215 14415456|5768|608o}|(299(282 | 
 6392167041701 6|7328|7640]|7952|8264|8576|88ssig1go}|_|2!2/51! } 
 
 . 951119823 |0135|044610758}|1070|138 1|1693|2005|23 16]|!| 31] 31 
 - 94}144262812930|3251 3562/3874 [4185 4497 4808|5119|5431||2| 62} 6? 
 
 8854191 65|9476|9787|0098]}040910720) 103:1]1342|1653}}5]155/156 
 97|145 196412275 |2586|2897|3207113518|3820/4 1401445014761])197)° 87 
 
 507 2}8882}9699}6004/0814}6625|6939(7240)7 55017867 : 25 bho 
 
 _—_ 
 ) ff | | sf | ES ES TT SF 
 
 3 4. 5 | 8 Differ 
 
(14) LOGARITHMS N: 14.000 1.146 
 N.] O We) 2 h8s ti 46 a WE: [To hs Differ.’ - 
 
 1400]1461280|1591|1901/2213 |2521}2831|3141|3451|3761/4071|| }310|309 
 Ol] 4381/4691)5001)5311|562111593 116241/655116861|7170)|1| 31) 31 
 02] 7480|7790|8100]8409)87 19]|9029|933819648|9958|0267 : Ke os 
 03|1470577|0886/1196/1505}1815/2124)2434|2743)3052/3362) 53} 54 
 OA 307 113980/4290}4599/4908]15217|5527|583016145|0454 511551155 
 05| © 6763|707217381|7690|7999}||8308|86171892619235|9544 f a Ae | 
 06]. 9853|0162)047 110780}1089|]1397|1700}2015 2324/2639} I. ar 
 07|1482941|3250)3558 3867 4175 4484 4793|5101|5410)5718 9/27910"8 
 08}  6027|0335|6643|0952|7200)7 5009/7877 3185/8493/8802))—\so8)i597 
 09 Q110|9418|9726|003 5/0343)/005 1}0950}120711575|1883 Tail 31 
 1410]1492191|2499|2807|3115|3423//3731 4039/4347 (405 5/4902)12) 62) 61] | 
 11} 5270)5578|5886|6193|0501|/6809)7 116)7424/7732)8039)|9) 92} 92} | 
 12] 8$347|8055|89621927019577||9885|0192/0499/0807|1114 
 13}15014221172912036|2344|265 1|/2958/3265|3573)3 880/4187 
 14] 4494,4801|5108|5415|5722||0030)|0337|0044|695 117257 
 15] 7564\7871|8178|8485|8792)\9099|9406|97 12/0019|0326 7 
 17|  3699/4005/4311}4618/4924/1523 1]5537|5843/01 50/6456] | 2/2 | | 
 18] 6762\7069|7375|7681\7987||8293|8600}8906|92 12/9518} ,)on | <1 | 
 19] 9824/0130|0436)0742|1048])1354|1660}1966|2272)2578})5] ol gal | 
 1420]1522883'3189|3495|3801|4107||441 2/471 8|5024|532915035 
 21]  5941|6240|0552/0858/7 103|'7469|7774|8080|8385|869] 
 
 5{t544154 | | 
 6|185}184 
 712161215 
 
 51153]153 
 f . me, FOL 841183 | | 
 22 8996/9301 960719912|0217),0523}/0828}1133 1430|1744 "12141014 | | 
 
 23|1532040|2354|2659|2964|3270)/3575|3880}4185}4490}4795]is|2451244.| 
 94) §100'540515710/6015|6320)|6625|69209|723.417530)7844) 9275/2775 | | 
 
 25] 8149|8453|8758|9063|9368||9672]9977|02810586)0891}), |304)505 | 
 26|1541195)1500/1804)2109}2413)|27 18|3022/3327 363 1|3935}||1| 30) 30] | 
 27| 4240|4544|484815153|5457]|5761|0005|0370|0074|6978)2) 61) 61 
 28|  7282|7586|7890|8194{8498]|8802|9106}9410)97 1410018)|,5ol15, 
 
  2911550322/0626|0930|1234|1538||1842|2145|2449|2753|3057 5{159]152 
 
 1480] | 3360|3664/3968/4271|4575||4870|5 182|5486/5789|6093|/ : i ie 
 311 — 6396|6700|7003|7307|7610)|7914]8217|8520|882419127] a\- <1 
 32} 9430197330037 0340}0643|10946}1240/1553|1856)2159 lolerafors | 
 3311562462/2765 [3068/1337 11367411397 7|428014.583/488615 180} 4,907 |. 
 34] 5492\5794|6097|6400/0703)|7006|7308 701 1/7914/8216 
 
 35} 8519|8822/9124|9427|9729]/0032|0334]0637 0939) 1242)|2} 60} 60 | | 
 36/157 1544]1847 |2149)/2452)2754|3056)3359/3661 [3963/4205 : ie 3° | 
 37| 4568|4870/5172)5474|5776|1007916381 0083 }0985/7287)| -\ ricrne 
 
 — 381 = 7589|7891|8193|849518797|,9099/9401 |9702|0004/0306)'611 8 11131 
 39|1580608|0910}1212]1513}1815|/2117/2418/2720)3022/3323}'7\211\211 
 
 1440| 362513927|422814530/483 1115 133|5434|573616037|6338 3 a eh 
 
 41|  6640|6941|7243|7544|7845|'8146)8448|8749|9050)9351) —— 
 42| 9653/9954/0255/0556|0857]/1 158|1459|1760|2061|2362 ; sae 
 43|1592663|2964'3205|3506|3867|4 168|4469/4770|5070}5371))1| 30) 20 
 44} 5672/597310273105741087 5\\7 175|747 6017777807 7|8378 
 
 3| 90} 90 
 451 °8678|8979|9280/9580|9881}/0181 0481|0782}1082)1383 
 
 41120)/120 
 
 46|1601683}1983 |228412584128841/3 1 84|3485|3785|4085|43 85 Z a ao 
 - 47)  4685|4985|5286|5586|5886]|6186\0486|6786|7080)73 86}, 1 ola99 
 48|  768617986|8285|8585|8885]|9185|9485|9785|0084]03 84]| glo401039 
 40/1610684/0984/1283|1583{1883]|2182|2482/2781|3081133801/9|270|269 
 
 N.f-o }il2isl+isilel7 [39 bo 
 
 4/123}193 } | 
 
 8/ 2461246 |. 
 92771276 | 
 
 4}1291192} | 
 
 81243/242 | 
 
TN. 14500 DIGG] 3) 40R NUMBERS. _ 
 |N. | a A ih © 9. || Ditter. 
 
 ey bey Powe nase, | saa Ree we 
 ——| | -_———— | | | 
 
 1450}1613680/3980/4270|4578]4878]15 177154771577 61607 5|0375}| |298|297 
 6674|60973)7 27375721787 1\18170|847018769|9068/9367}|1) 30f 30 
 9666}9965|0264/0563|0862}}1161]1460}1750|205812357||2), 60}. 99 
 53|1622656)2055/32541355313852|/41 501444014748 15047/5345)1, 89). 89 
 
 4]119]119 
 5644|5943|6241/6540|6830}1713717430|7734'8023|833 115114901149 
 
 8630)8928]922719525|9824||0122/042010719)1017]1315 lhe a 
 eh it 8)238)/23 
 
 4596/4894|5 19215490|5788||6086|038416082'6979/7277] slosslogy 
 
 7575|7873|817 1|8406G|8767|19064|9362|906019958|0255||——_ 
 
 50|1640553/0851]1148!1446]1743}|204112339|2036!293413231 
 
 a Hh (2961295 
 
 6502|6799|7007|7394|760 1 7988 8285|8582|8880/9177||-|——_ 
 
 04741977 110068|0365|0662 0959 12.56]1553}1850|2146 
 
 63|1652443|274013037|3334|3631 
 
 8376|8673|8969|9265|9562/\9858|0155|0451|0747|1043 2) 28/125 
 - 66}1661340)1636]1932)2228/2525]2821/3117|3413/3709/4005|/zJo97lo97 
 812371236 
 9}266/266 
 69|1670218/0514|0809}]1105}1400 10901991 2287|2582|2878 
 ~ 317313460|3764/4060)43 55/4650 4946|5241/5536/5831 
 6127|6422)67 17\7012|7308)/7003)7898)8193|8488/8783], |294|293 
 . 907 8|9373|9668|99063 /0258||0553/0848}1143}1438|1733}|1| 29} 29 
 73|1682027|2322)2617|2912/3207113501/3796)4091/4386/4680 
 4975|5260|5564|5859'0153|16448 6742'703 71733 117626 
 7920|8215|8509|8803/90981!/9392'9686|998 11027 5|0569 
 76|1690864}1158]1452)1746|2040}/233 5|26209|2923}3217/3511)/ ie 176 
 3805|4099|4393|4687|4981115275|5560|5863|615716450 35 205 
 
 6744|7038|73321762617920|8213'8507|8801/9094/0388)||, ae ss 
 
 9682|9975|0269|0563 /0856}1 150)1443]1737|2030/2324 
 
 5551)584416137|0430)07 23 |/7017!7310|7003|7890/8189}} jogo1901 
 8482)/8775|9068/9361/9654119047|0240|0533|0826]1119 near 
 83|1711412]1704]1997|2290|2583|/2876)/3 168|3461)3754/4046]lol sal 53 
 4330|4632/4924|5217|5509||5802|6095|6387|G680|6972)|3| 8s] 87 
 72,65|\7557|7849|8142|8434|18727 9019/93 1 1|9604|9806 i ae ue 
 86|1720188|0480|0773)1065|1357||1649]1941/2233|2526/2818] [4175 I1-72 
 3110|3402|3694|3986/4278]!45701480215154|5446/5737||7le041004 
 6029|632 1|6613/6905/7197||748817780|807 283 64/865 5]/8|234)233 
 8947|9230|9530|9822/01 13/10405|0697|0988}1280}157 1||9!2621262 
 
 93 16|960619897|01 8710477 |(0767|1057|1348/1638]1928 ° ar pa 
 5118|5408|5698|5988|6278]656716857|714717437177271\ losolos 1 
 
 8016|8306|8596/8885]91 75119405197 54|00441033310623)}9 2611260 
 
 Se Oe | | | | || J | S| LL Se 
 
 11/2}/3/4451617 18 | 9 | Diter 
 
Sy LocARirums N.15000 L.176 
 
 1500 17600913 1202]1492]1781}207 1}2360|2649|2939|3228/35 18}! |290|289: 
 3807|4096)/43 86)467 5}4.9041152.53)5543|583216121|6410)| 1] 29} 29 
 6699|0988|727 8|7567|7856/|8145|8434|87231901 2/9301}|2| 58) 58 
 
 9590|9879|016810457|0745}}1034]1323|1612|1901/21 9071, 7} 87 
 04/1772478/2767|3056/3345|3033//3922/421 11449014788]5076)]5|1a5l14.5° 
 5365|5054|5942)623 1105 19}6808/7096|7385|7673|796]1 ||9|!'74|173 
 8250)8538)8826191 15|9403|19691|998010268 0550/0844 Gries | 
 
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 46]1892095|2376/2657|2938|3218|3499|3780}4061143.4214622}|5}141]141 
 
 47| — 4903|5184)5465/5745|6026|6307|0587|6868)7 148]7420]]>| 6°11 60 
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 52} 8917|9197|9477|9757|003G|103 16|0596/0876)1155 435] _ i 
 
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 54) 4510/4790|5060|5348|5628]}5907/618716466|6745|70251l 5|1401140 
 55} 7304|7583|7862|8 1421842 1118700/897919259/0538 pet7 i i . ‘ 
 56}1920096)037 5|0054/0933} 1212|/1491)1770|2049/2328)2607]| 3\55 :1o95 
 57| 2886|3165|3444/372314002]1428 1145591483815 117|5396||glo50lo51 
 
 1560|1931246}1524/1803/208 1/2359/|2038/29 16/3 194/3473/3751 278277 
 61} 4020)4307/4585/4804)5 142)5420|5698|5976/0254/0532)|7|-—>. ee 
 62} _ 6810)7088|7360|7644|7922)/8200)8478|875619034/9312)15) 56) 55 
 63) —9590}9868/0145/0423/0701]10970|1257}153.4]1812/2090]/3| 83] 83 
 
 64/1042367|264.5|29023/3200/3478)|3750|4033/43 1 1/4588|4866 : se oe 
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 67|1950090|0967]1244}1521)1798]/2075/2353|2630}2907/3184 sleep ps 
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28) LoGARtrHms N.16000 L.204 
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 1600|204.1200]147 1]1743|2014|2285]]2557|2828|3090|337 1/3042 (|272)271, 
 
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 5178|5443|5709|597 4|62401/6505|677 1|7037|7302)7 508}. bee ia 
 
 7833 8098 8304 8620|88951|91 60/9425 9091 9956 022161160 159 
 
 38} 31309/3404'36060/3934/4199/14464 47 30 4905 5260 5525|8/213}212 
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 45|2161659|1923|2187/245 127 1511297913243 |3507|377 1|40341/4|106/105 
 
 46] 42098}4502)4826/5090)5354115617/588 1 |6145|6409|6672)/9|192|152 
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IN. 16500 L.217 OF NUMBERS. (19) 
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 1650|2174830]5 103|5366|5629 
 
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 3] 79) 79 |) 
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 3225|3487|37401401 114273114535]4797|5050/532 115583 lolo3gio57 | 
 
 58} 5845/6107|6369|063 1/0893 |\7 155|7417|7678|7940/8202|-— 
 
 59| 8464187261898719249/9511119773|0034}0296|0558|08 19 
 1660|2201081|1342]1604|1866|2127||23809|2650)291213173|3435}| ogaiogy 
 
 61]  3696/3958)4219/448114742))5003/5265|5526|5788|6049)|7}-Gel-o¢ 
 
 62| 6310/6571/6833|7094\7355]7617|787 818139|8400)|8661 9] 59] 59 
 
 63] 8922/9184'9445|9706|9967 ||022810489|0750]101 1|1272)|3| 79] 78 
 
 64|2211533]1794|2055|23 1612577|]2838/3099|3360|3621!3882)/4|105|104 
 
 67| 9356|9617|9877\0138|0398||0058/0919]1179)1440/1700 : eoae 
 68/2221960]2221|248112741|3002|3262/3522137831404314303||21236.295 
 
 69| 4563/4824/5084|5344|5604!|5864/0124163 84|6645|6905 
 
 1670) -7165|7425|7685|794.5|8205||8465|8725|8985|924519505 
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 73| -4959|5219154701573815998||62.57|051716776|7036)7295||-| 22 
 
 5130}130 
 
 76) 2740|2999|3258/3517)3777|/4030|4295|4554/4813|5072)/5|198 a 
 77| — 5331|5590|5849/6107 |6306)}6625|0884)7 143|7402/7661 Ilslogglooy 
 78| 7920/8178|843718696/8955|19213|947 21073 11999010248 jole341233 
 79|2250507 |0766}102411283|154.1||1 800|2050|23 17|2576)2834 
 
 1680} 3093/3351/3610/3868/4127}}4385|464414902)5 160/5419 
 81} 5677|5935|6194|6452)67 10/6960|7 227 |7485)774318002)| losslos4 
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 24. a sea 5876/612 230380 6632 
 
 25 7891/8143 /8394/8040/8898 (9150 
 26/2370408|0660,091 1}1163]1414 1666 
 
 27|  2923/3175|3426|3678/3929/4181 (4432 4683|4935|5186 oe 
 28}  5437/5089'5940/6191/6443| 6040945 71 7196\7448/7699}19) 50} 50 
 29} = 7950,8201/8452/8703 89559200, 9457/9708/0959 021013] 76] 75 
 
 07 12/0963}1214}1465 |17 16|1967|2218)2409)2720) 5 |: o6l106 
 31 2071 3222/3472 3723139744225 ,44764727149077|5228]\ 611511151 
 32 5479)5730,5980 623 1|6482|6732|6983/7234|7484|773 5||7|176|176 
 33} 7986 823618487 873718988 19238|9489'9739|9990|/0240}|8 202/201 
 
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 35} = 2995|3245/349513746/3996)42 
 
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 36] °549715747|599816248|6498 6748 69087 248|7498|7748 
 
 37] 7998 8248'8498|8748 8998 9248|9498'9748/9998|0248 DahIA 
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 44} 54651571415903|621216461 | 67 10|695017208/7457|7705 slogol199 
 45] 7954|8203/8452/870118950/919919447|9696|9945|01 941912251224 
 46/2420442/069 1/0940}1 1890/1437 /|1686|1935|2183|2432/2680 
 47|  2029/3178/342613675|392314172/4420'4669/4917|5166 
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IN.17500 L.243. OF NUMBERS. _ (21) 
 
 24.8]247 
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 7|169)169 
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aa LOGARITHMS N. 18000 1.255 
 
 4772)}5013|5253|54941573415975 6215 456|6696|6037 
 7177\7418/7658|7899/|8139}8380 8620)8860/9101 9341 
 
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 7464!|7697|7930)8164|8397||8630|8864|9097 |9330|9504 70 
 9797 |0030|0263|0496|0730}10963)1 196|14.29]1662)1895 | 9:34 
 63|27021 29|2362/2595|2828|3061)3294|3527|3760|3993 4226)" 1” 
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 6788|7021|7254|7487|7720)|7953|8185|84.18|8651|8884} —_|8|186 
 9116193491958219815|0047|1028010513|0745\0g78|1211) = 21228 
 67|2711443|1676]1908)}2141|2374}|2606|2839|307 1/3304/3536 232 
 3769/4001|4234|4466|4699]/493 115 163|5396/5028/5801 1) 23; 
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 84|2750809|1039}1270)1500]1731||1961|2192|2422/2653/2883 eo : 
 31141334413574/3805/4035|!4265/4496147264956015187]] fal ag 
 5417150471587 716108)/6338||6568]6798|7028|72509|7489 3} 69] 
 
 | 7719|7949|8179|8409|8640]|887019100193301956019790|2304) 92} 
 88|2760020/0250|0480|07 10|0940]|1170|1400]1630|1860|2090} 2}! 1>) 
 
 6092/63 21|6550|6780)7000]|7238'7467|7006 7925|8154)1229)4|. 92 
 838318612|8841}/9070|9299|19528 9757 9980/0 215|0444)|  jpii15 
 
-LOGARITHMS N.19000 L.278 
 
 03| 4388 1616 eae 5072|5301 5520 seer 5985|6213|6441 
 - 04] 6660/G898|7126]7354175821|7810|8038|8266|8494|872211228] 51) 14 
 05| 8950}9178}9406|9634|9862]}0090|03 17/0545|0773/1001 6137 
 
 07 3507 3735|3962/4190}4418]}4645}4873 5101/5328 5556 91205 
 
 og} — 8059|8287|8514|8742/s909 919719424 one 9879|0106|| | 227 
 
 i, 9419 sea 9873|010010327 10554 0781}1007/1234 ns ml159 
 17| 6221 6448 6674/6901|7 127]|7354|7580|7807|8033|8260 2206 
 
 1920} 3012/3238 3465 3691139171/14143|4369 ssoal4 4821|5048}}226}4) 90 
 21}  5274|5500)5726|5952|6178'|640416630|6856|708217308 5i113 
 92] 7534/77601798618212/8438|S663|88Sql911519341|9567)| — ||136 
 23| 9793|001910245|0470|0696|]0922]1 148]1373/1599]1825 gl181 
 
 25 4307|4533]4759/4984152 1015435 |5661|5886)|61 12/6337 225 
 26 6563]0788|7014|7239|74.65||7690|7916|814.118366)8592 1} 23 
 27;  8817|9043|0208}9493|97 19/1994.4101 60|0394|0620|0845 2} 45: 
 
 29| . 3322|354713773|3998|42231/444814673/4898)5123/5348 
 1930] 557315798|6023/6248|6473||6698|6923|7148|7373/7598]| _ |6{135 
 31) _7823|8048|8273)8497|8722|/894719172|939719622|9846], 1/118 
 
 1 
 35| 681017034|7250174831770711793218156|8381|8605|8829]| {2 45 
 36] 9054192781950219726|9951|10175|0390|0624|0848}1072}} 5} 67 
 
 88| 3538/3762/3986/4210/443.41l4658/4882|510015330)5554224 eh 3) 
 39} 5778|6002|6226|6450|60741||6898]7 122173461757017793 M157 
 Lpalh 8017|8241|8465]8689|8913 ]9136|9360|9584)9808|0032 8/179 
 
 49) 2492|27 16|293013 16313387|1361018834 4057 7 4981|4504 223 
 43) 4728|49521517515399|5622||584516069|6292/6516|6730]| . |1| 22 
 44} 6963|7186|7409|7633|7856]8079|8303|0526|8740|8973|| — Iol_ gx 
 45| — 9196|9419|9643|9866|0089||03 12/0536|07 59|0982|1205}),,,. || 8° 
 
 47| 3660|3883|4106}4329,455214775 4998|5221|544415667}] © [elis. 
 48] 5890|6112)/6335|6558/6781)|7004:7227)7450|7673|7890 8l178 
 49]  8118|834118564|8787\9010 9282/9455 967819901|0123]| —_|9{201 
 
 No oc lalelsla fate lz | si] of pipes. 
 
N.19500 L.290 OF NUMBERS. _ | (25) 
 ; 3 | 4 5 PhO fF 
 
 1950|2900346)0569}07 92|1014)1237}1460]1682)1905 
 
 51]. 2573|2795|3018|3240/3463}3686|3908]413 114353/4576 
 
 4798|502115243|5466|5688!1591016133|6355|6578|6800 
 7022\724517467|7690|7912118134|8356|8579|8801|9023 
 
 56} 3689/3911|4133|4355/4577]4799|5020)5242/5404|5680]- 7716193 
 57} —5908/6130|6352|6574|6796]/701817240)7461|7683]7905) — {1112 
 58]  8127/8349|8570|879219014|19236|0458/96079/9901|0123 9/200 
 59|2920344/0566/0788}1 009}1 23 1]}1453!1167411896|2118|2339 
 1960} 2561/2782/3004)3225|3447/3668)/3 890/411 1/4333/4554 
 61} 477614997|5219|5440|5662/5883|6105|6326|0547|6769 221 
 62} 69090'721117433|7654|7875||8097|83 18|8539|87 60/8982 lf 22 
 63} 9203|942419645|9867/0088/}0309/0530/075 1|0973}1 194) Q) 44 
 
 64/293 1415,1636]1857|2078]2299] 2520/2741 2062/3 183|3405 
 
 6866|7087|7307|7527|7748]79068|8 1 88|8408/8629|8849 
 9069/9289/95 10}973019950/101 70|0390/0610/083 11051 
 
 74| 3471|3691/3911/4131)4351]4571}4791|501 11523115451 3] 66 
 75|  5671|5891\6111|633116550|6770|6990|7210/7430\7650} ‘51,98 
 76} 7809/8080|8309]8520|8748]|8968/9188/9408/9627|9847 61132 
 77|2960097|0286|0506|0726|0945}|1 165|1385]1604|1824|2043 154 
 78| 2263|2482|2702]29221314113361|3580|3800/4019/4238 8/176 
 79| 4458/4677|480715116|5336115555|5774|59941621316433)) —_—|9!198 
 1980} 665216871|7091173.10\7520117748!7968|8187|8406|8626 
 81} 8845/9064/9283/950219722)|094110160/0379/0598\0817 
 
 82|2071037|1256)1475|1694119131/2132/2351|2570)27 89/3008 iiion 
 3227|344613665|388414103|14322/4541|4760/497915 198||7191,] 7, 
 
 91129907 13/093 1)1 14.09|1367|1585}|1 803/202 112230/2457|2675 
 2893/3 111|332913547|3765113983/4201|4419/4637|4855|/218| 215 
 5073|5291|5500|5727|5945||6162|6380|6598|6816|7034|| |) 77 
 3] 65 
 4) 87 
 5}109 
 6131 
 W153 
 81174 
 91196 
 
 9429|9647|986410082|0300105 17|0735|0953}|1170]1388 
 96/3001 605}1823/204112258|2476]/2693|291 1/3 128]3346|3563 
 
(26) LOGARITHMS .20000 L.301 
 
 i ime | ——_— | > | OO OO NS OO OO OO nhn iO  _—_— Os Eee el ee 
 
 2000/3010300/05 1710734095 1]1168]]/1386}1603]1820/2037 2254 
 247 112688|2905|3 122/3339113556|3773|3990/4207 442411217 217 
 4641|4858]5075|5291|5508]1572.5159421615916376 6593 1} 22 
 
 06} 330913526/3742|3950/4175]|4392/4608/48251504.1|5257 
 07}  547415690|5906|6123|/0330/16556|0772|6988|7204'7421 
 08} 7063717853 sais 8286|8502)]|87 18}8935/0151 sais 9583 
 
 2010'3031961|2177 se 2609 28251|304.113257/3473 36893905 
 3714553|476914.9841|520015416|563215848 6064] - 
 
 16 re 5121 Bane 5552 Aa 5982/61 ee slo4i 6628/6844 
 17} 7059|7274|7490)7705|7920||813.5|835 1/856618781/8996 
 18] 9212|9427|9642|9857|0072!10288'0503107 18}0933|1148 
 
 2020|  3514|37291394414150/437411458q|48031501 8|5233/5448 
 211 - 566315878|6093|6308|6523/16737|6952\7 1671738217597 
 22 
 
 27)  8537|8752|8966\9180/939411960919823/003 7/025 1|0465 
 
 28)3070080}0804}1 108}1322/1536)1750|1964/2178|2392/2606)214)) aE 
 
 29}  2820)/303513240|3463|367 71/389 1|4105|431914532/4746) |—— 
 2030} 49060)517415388/5602|5816)16030|6244|6458|667216885 
 
 31 
 
 99100123 /033710550|0763||0976)1 He 1402}1616|1829 
 38|3092042)2255|2468]268 1|28941I3 107|3320|3533|3746/3959 
 
 43|  2084|28096)3 109/3321|3534113746/3959|417 114384|4596 ; fs 
 44} 4800)5021|523415446|5659|1587 1|6084|6296/6508|6721 3] 64 
 45| 698317145 (73581757017783}7005/8207|8419)8032)8844} | 8 
 4 vay q 
 
 re | ae ee | ———_—______ } —, —_______ | 
 ——-f{ 
 
N, 20500 L.311 OF NUMBERS. 
 
 ee | | 
 TT 
 
 73|  59093\6203)|64.1 2/662 1/683 1117040'7250|745G intone 7878 
 74|  8088/8297|8506|8716|8925/9134/9344'19553|9762'9972 
 75|3170181/0390|0600|0809}1 01 8} 1227|1437|1646)1855|2064 
 70} = 2273/2483/2692/2901/31 1033 19)3528/3738)3047|4156 
 
 9767|9974/0182|0389 
 95|3211840/2048/2255|2462 
 
 0596]}0804]101 1]1218}1426/1633 
 2669 
 
 25 
 7}146 
 
 ——| 
 
 TT | | a | = ——— [( ————————eeiyrorrnrnrne es 
 — [——$—$—$—$— << | ———_—_—_. 
 
(28) LOGARITHMS N.21000 1.392 
 N.| Oo Li? } 3. pe Sa 647 
 
 pe) eS Se, ee ee ee | a ee Se eS ee are | Meese ee 
 
 2100132221 93|2400|2607|2813|3020113227/3434|3640|3847/4054 | 
 01] 4261/4467/467414881|5087|15294|550115707|5914/6121 207. 
 02} 6327|05341674016047)7 153}\736017567|7773|7980|8 186 1] 21 
 03] — 839318599|880619012192191\94251963219838|0045\0251)) {2 4! 
 04|3230457|0664|0870|1077|1283]|1489|1696]1902|2108|2315]| {5} gs 
 
 05| 25211272712934|3140/3346113552137509139651417114377|| 5/104 
 06]  4584/4790/4996|5202|5408]|5615|5821|6027|6233/6430]| | 185, 
 07|  6645/685117058|7264|7470|7676!7882|8088|8294/8500 
 6s}  870618912/91181932419530l9736199421014810354'0560 2001 c° 
 09|3240766)0972|1178]1384|1589|11795|2001|2207|2413/2619]| |—— 
 
 2110} 2825/3030)3236|3442|3648]/3854\4059|4265|447 1|4677 ! 
 11} 4882/5088/529415499|5705||5911|0117|6322/6528|6734 206 
 12 6030)7 145|7350|7556)7 762)|7967/817318578|8584!8789 1) 21 
 13] . 8995|9201|9406|9612/9817)|0023|0228|0433|0639'0844 2} 41 
 14|3251050)1255|146111666]1872||2077|228212488]2693|2898 3} 62 
 
 15]  3104/3300|3514|3720/3925)|4130/4336/454114746,4951 51103 
 16] 5157/5362/556715772|5978||0183|6388|6593|6798|7003 6124 
 17| 7209|7414/7619|7824|8029||8234|8439/864418849/9055 7 sis 
 18} = 9260|9465|9670/987 5 |0080)|0285|04g0|/0695|0g00}1 105]lo05 oft85 18 
 19|3261310):515|1719|1924)2120]2334/2539|2744|2904913154 
 
 2120) - 3350)3563|3768/3973|4178]4383)4588]4792|4997/5202 
 21} 5407/5611|5816|6021|6226/}6430\663.5|0840)|7044|7249)| ° 
 22| 7454!7058/7803|8068/8272||8477|8682|8886/G00 1|92905 1) 9] 
 23}  9500/9705|9909/01 14/03 18]}|0523'0727|0932|1136)1341 a} ad 
 241327 1545]1750)1954'2158/2363||2567|277212976)3 18113385 3} 62 
 
 25]  3589/3794'3998|4202/4407]4611/4815|502015224/5428]] [31,55 
 26|  50633|5837|6041\6245|6450]10054|6858/7062)7 267|747 1 61193 
 27| 7075\7870|80:33|82871|8492)|8690/Sg00!9104/9308)95 12 71144 
 28] 9716\99201012.4/032810533/1073710941|1145/1349]1553/1204)8|164 
 29|3281757|1961)2165|2369|2572|12776|2980)3184|3388|3592\| [2/185 
 
 2130|  37096/4000/4204'4408/46121}4815/5019|5223|54271563 1 
 31]  5834|6038\6242|0446|6650)}0853|7057|7261|7465|7608 
 321’ 7872'8076\8279,8483|8687 ||8890|90941g298/9501|9705 
 33] 9G09|0112/0316|0519|0723)10926|1 130)133411537|1741 ol 41 
 34|3201044/2148]2351|2555|275812962)3 165|3300|357 213775 31 61 
 
 35} 3979|4182/4386/4589/4792|4996|5199|5402|5606|580g]| {4} 82 
 36] 6012)02166419|6622/0826|17029|7232|7436)7039|7842 61129 
 87|  8045|8248/8452)805518858/1906119264|04681967 119874 "143 
 38/3300077|0280/0483|0686|0889}|1093|1296]1499}1702)1905}}593|8]163 
 
 Peet SE See ieee Se a ee | ee ee 
 
 Tletstalstelta7{st9 Iplips. 
 
N.21500 L.332 OF NUMBERS. 
 
LOGARITHMS + N.22000 L.342 
 
 2200/342422714424/4622148 19|501615214154.4.1|5608|5806|0003 
 6200 6398 6595 6792 6990 7187|7384/7581 7779 797 
 
 2116/23 13)2510 2707 bad 3101/3298)3495|3692/388Q)|19713 i ee 
 
 3300)3495)3691 07 
 525215447 5642 5837 
 
N.22500 L.352 -OF NUMBERS. 
 
 7239|7432\7 624|7 816}8009)|8201/8393 ibn 8/8970 
 9162|9355|9547|9739\993 1110123|03 16|0508:07 00/0892 
 
 plop 8381|8572 
 st “56 9148/9340 953 1972319 915 0107\0299|0490 
 
 25901279 112982)3174 3366 355713749 39404132 4324 
 4515 4707 4898 5090 5281 5473) si 5856,6048 0239 
 
 210192,0382|0572 
 86/3590762 0952 1142 1332 1522111712}1902 2092/2282 2472 
 2662|2852/3041|3231 3421 vie 3801 3991/4181 4370 
 
LOGARITHMS _N. 23000 L. 361 
 
 17|7 056)7 845 |803.4118222184 1 1]8600|8780|8977 
 9166|9355|954419732/99211:0110|029810487|067610865 
 
 02|3621053}1242]1430}1619/1808}1996|2185]/237412562|2751 
 2939/3 128|3317|3505|369413882)/407 1 4259 4448/4036 
 
 9260)9448}9035|9823|0010/0197 0385, 
 1813651134]1322}1509|1696]18841207 112258, 2446126332820 
 39441413 1/4318/4505|4693 
 
 4880150671/5254|5441 apacehe 5816|6003'6190 G7 6564 
 
OF NUMBERS. 
 
 N.23500 L.371 | 
 aa 718° 19 | D[Pro.| 
 
 gost 84902 sera 8854|9035 g216 9307 hat pany 99400121 
 99 3800302|048410665|0846]1027 
 
 ————_—— | -———_. | ———— | —_——_- - |} | ————_ | ———_—_] ——__ | ——_ |] —||“—|—~— 
 
(34) LOGARITHMS N.24000 L380 
 
 Nop O yrs) 2013.5] Falibe [P6cf i718; | 9-2 Bee. 
 
 See EEE ee es ee ees) ee a a ns hed 
 
 2400 38021 12|2293|2474}2655|283613017|3 198|3379|35600/374111181 
 Ol] 3922/4102/4283/4464}4645114826|5007|5 188|5368|5549 
 02} 5730/591116092|627 2|6453]16634/681516995|7 1 76/7357 
 03] 7538|7718|7899|8080}8261)1844 118622/880318983|9164 okie 
 04]  9345]952519706|9887|0067|1024.810428|0600|0790|0970 3] 54 
 05|381115111331|1512}1693]1873|1205412234)2415|2595|2776 4) 712 
 O06}  2956/3137/3317|3498|3678]3859|4039/422014400/4580 
 07| 4761}4941|5122|5302/5483|15663|5843|602416204/6384|| © |nl1om 
 O08} 6565|674.5|6926)7 106|7286]7467|7647|7827|8007|8188 81145 
 O09] 8368/8548'8729|/8909/9089}|9269|94.50|9630/98 10/9990 9163 
 24.1C}3820170/035 1/053 11071 1}0891]]107 111252}1432}1612)1792 
 VP ti 1972|2152/2332/2512/2693||2873'3053/3233|341 3/3593 
 (12) =3773|3953|4133|43 13/4493114673|4853|5033/5213|5393 
 13}  557315753|5933/6113|6293]16473 |665316833|7013 7193 
 14 7373|7553|7732|7912|8092||8272|8452|8632/8812/8992 
 
 15} = 9171|9351|953 1197 11}9891||(0070|0250/0430|0610/0790 180 
 16/3830969}1 14.9|1329]1500|1688}] 1868|2048|2227|2407|2587 1) 18 
 17} 2767'2046,3126)3306|3485113665|3844/4024|4204/4383 
 18} 4563 17.43)4922 5102'528111546115640/5820,6000/6179 Al 70 
 19] 6359 sie lees 6897|70771'7256|7436|761517795|7974 5] 90 
 
 2420} 815418333/8513/8692|8871|9051/923019410958919769] |° 
 
 21]- 9948}0127\0307|0486|0665]}084.5]1024|1203]1383|1562 gltad 
 22|3841741]1921|2100}2279|24.509]/2638)/28 171299613 176|3355 O}163F 
 23] - 353413713|3893|407 2/425 1114430|4609/47 89/4968 |5 147 ce Py 
 24}  5326)5505|5684|5864|6043 6222/6401 |0580)/6759|6938 
 25 7117\7297 7470 7655|7834|8013 8192|837 1|8550|8729 
 
 178 
 
a 24500 L.389 
 
 OF NUMBERS. 
 
 4185 sa1909 6 4533/4707 4881 3083 5229) 5403 5577/5 | 
 
‘| 35) , ~LOGARITHMS N.25000 L.397 
 Nediquy ia 2 
 
 pees Rags fa wae! ST eS 5 A SS | ———| >| —_] ——. | ——-] —— |, ———— || -—_] — 
 
 OS 3275 3448 Abee nak 3968}\4141 re 14 4487 1660 4834 
 Og 5007/5 180/5353 5520 5699||5872|6045]6218}6391|0564 
 
 171 — 8832'9005|9177/9350/9522\|9605 9867 6040|0212,0385 
 18 #010 0730 Bye 1075 a7 1420 1892 1764] 1937|210¢ 
 
 31 292113093'3205|3436|3608]/37 791395 1|4122/42904\4465 
 32| 4037|4809/4980)5 1525323115495 |5666/5838]6009/61 80 
 33| 63521/6523/6695|6866|70381'72091738117552|7723|7895 
 34| 8006)8237/8409|8580)|8752|18923|9094|9266|9437\9608 
 35} §780/99511012210294}0465|10636/0807|0970}1150)1321 
 30404 1492/1664/1835|2000|2177|12349|2520|200 1|2862/3033 
 37| 3205)33763547/37 18|3889]]406 1/423 2|4403)4574'4745 
 38} 4916 5087/5258 5429|5601115772|59043/6114|6285|6456 
 390] 6027 6798) 696917 140)73 11}'7482)7653|7824|7995|8100 
 2540] 8337/8508|8670|8850/9021||9192|9363|9534|9705|9876 
 4114050047 02180388|0559|0730}}0901)1072|1243)1414)1585 
 42} 1755\1920)200712268|2430}2610]2780]295 1|3.122/3293 
 43} 3464)3034/3805|3976/4147|143 1714488|465g|4830|5000 
 44 5171|5342!5512)568315854||6025|0195|0366|6537|6707 
 45] 6878/7049'721017390|7560/773117902|8072|8243184.13 
 46] 8584/8755'8925|9096|92061\9437|9607|9778|9948|0119 
 47|4060289|046010630 0801}097 1}]1142)13.12)1483]1053}1824 
 
FN-25500 L.406. |. OF NUMBERS. 
 
 et a eee 
 
 5572|574215913 6083 6253/0424'6594|0704 634 
 7105 7275 7445 7015|7786) Teel 8126'3 8296 8406 8037, 
 
 77647 7817 ra 
 79/3 08 15 9084 
 
 | 
 
 laiienat 1190 135911527 1606 1865|2034|2203|2372/2541 
 72; 2710/2878)3047/3216'3385113554|3723|3891|4060/422g 
 731  43098|4507/4735|490415073115242/54101557015748|5917 
 74,  6085|5254}6423)059216760)6920|7098|7 260! 7435|7 004 
 75 777 2\7941)81 10,8278)8447 1861 0|8784/8953/91 2119290) 
 76| 9459|9627|9796|9964/0133/0301|0470|063910807|0976 
 77,4111144}1313]148111650/1818]1987|2155/2324' 2492/2661 
 2820|2998/3 166|3334|35031367 113840/4008|4177/4345 
 
 7964|8132'8300,8468|8636|8804/897 1/91391930719475} 
 96439811 9978)0140 03 14110482|/0649'0817\0985}1153 
 89/4131321|1488]1656)1824]199112159|2327|2495|2062/2830 
 2998/3 165'3333/3501|366813836|4004!4 17 11433G ason| 
 4074|4842'5009|5177 5345 5512 5080 5847 0015 01 ga! 
 
 6391|6559|672616893]7060)|7 227|739417501|7729|7896 
 8063|8230/8397|8564|873 1|$898}9005|9232|9390|9506 
 
(38) / LOGARITHMS N.26000 L.414 
 
 CS ee ee eee ee 
 
 14/ 3056/3222'3388|355413720}3886|4053/4219/4385}4551 
 —15|°  4717/4883}5049/5215|538115547|5713|58709|0045|621 1 
 16} 6377|6543\6700|6875|7041 7207|7373 acid 705|7871 
 
 17 
 18 
 
 21 
 22 
 23 
 
 3114201208]1374}1539}1704|1869]/203.4|2190/2364/2520|2694 
 32} . 2859|3024/3189/3354)35109}3684|3849|/4014/4179/4344 
 33}. 4509/4674!4838'5003/5168]}5333/5498|5663/5828|5993 
 34] 6158)/6323/6487|6652;68 17 6982/7 147/73 12|7477|7641 
 35 7806797 118136|830118465|18630/8795|8960)9 1 25|9289 
 36] 9454|961919784/9948/01 13]10278'0442|0607|07 72/0937 
 3714211101|1266)143 1|1595|1760) 1925|2089/2254)/24 19/2583 
 38? 2748/2913'3077|32421340611357 1|3736/3900}4005/4229 
 39] 4394/4558/4723|4888|5052}|5217|5381|5546|57 10/5875 
 2640! 6039/6204|6368/0533|6697|/6862|7026)7 19117355|7520 
 41} | 7684/7848|8013/817718342)/8500|867 1|8835|8999|9164 
 42} 9328/9493|96571982 119986)|01 50/03 14104709|0643|0807 
 4314220972) 1136|1300}1465|1629]]1793|1057/2122/2286/2450 
 44} 2615/2779\294313107/327 1}3436/3600)3764/3928|4093 
 45| 4257/4421 4585 749|4913 5078 5242 ar 5570/5734 
 
 tnd 
 
 OPP RPT aT ara 
 
N.26500 L. 423 “ OF NUMBERS. 
 
 ————} ~~ -- 
 I eo ee 
 
 5550|5713 877 G040}6203 a oat 693 6857 7020 | 
 
 7837 ||8000|8 163/8327|8490,8653 
 8816|8980/914319306/946919633197 961995910122'0286 
 61}4250449|0612)0775|0938)1 102}1265|1428'1591/1754]1917 
 
 62| 2081|2244/2407/2570)2733]|2896/3059/322213385/3549 
 63]. 3712'3875|4038|4201 4364114527|46904853/5016/5179 
 64]  5342)5505|5608)583 1159940157163 20|0483/0646,0809 
 
 6972|7135|7298|7461)\7624|7787|7950'81 13|8276/8439 
 
40) | LOGARITHMS N. 27000 
 NG} 0 36g Wl hk | Do ch Cd cl 
 2700 43 13638'3798|3950|41 20142811 444214603147 03/4924 5085! 
 01! 5246/5407/556715728|5880 6050162101637 7110532\0003| 
 02| 685317014171751733617406 7657 7818|7978|8139|830C' 
 03 8460)8621|8782/894219 103 19264 '9424/9585|9740/9900, 
 041432006710227|0388]054910700| 0870/1080] 119111352)1512 
 
 05| 1673/1833 1994}2154}2315}247 5!2636]2796120957/31171 
 
 O6| 3278/343813599/37591392014080 424 1/4401|456214722 
 
 07|  4883150431520315364|5524] 5085/5845 0051616616326} 
 
 08|  6487\6647|6807\6968|7 128 7288) '7449|7 60017769 7930 
 09} . 8090/8250)84111857 1/8731 8892 9052921 2193721953 
 2710} — 9693}9853/0013'0174|0334| 0494065410815 0975}1135 
 11 4331295 1455|16161776|1986] 2096/2250 2416012577|2737 
 12) - 2897/3057|3217/337713537/3697'3858 401814178 4338 
 13}. 4498/4658}4481814978)5 138!5298'5458/561815778|5038 
 14] 6098'6258|6418)6578|6738|6898'7058/7218/7378|7538 
 15} — 7698|7858|8018/8178|8338]8498'8658|8818|8978|9138 
 16} © 929810458196171977719937|10097|025710417/0577|0737' 
 
 17|4340896|1056|1216)1376]1536]1696)1855/2015/2175/2335 | 
 
 18]  2405/2654/2814'297413 1343293'3453|3613/3773|3932| 
 19} 4092'4252/4412 45711473114891/5050/52105370|5529. 
 2720| — 568015849|6008'61 6816328 ]/6487|6647|6807/696617126 
 21 7285|7445|7605|7 70417 924/|8083/|824318403/8502 8722) 
 22} . 88811904119200,9360/95 1919679|9838/999810157|03 17 
 23 4350476 0636|0705'0055 1114111274/1433]1593|1752 fens 
 24} . 2071/2230 poo *9 2709 2868/3028/3187|3346|3506 
 
 25|  3665'382413984'4143/4303 liasali6a1 4781\4040|5099| 
 
 26] § 5259|5418)557 7\5730 5896 16055'6214|637416533 6692 
 27| ° 68511701117170:7329 7488 7648)7897|796618125|8284' 
 28]  8444'8603/8762'892 1|G080 |024010390|9558)0717|9876 
 
 | 29 436003 5 0194/0354,05 131067 3 
 
 2730| 16261178611 945 /2104}2268 [2422/2581 2740 2899 3058 
 | 31) 321733763535 3694/3853. 4012/4 Tt 43304489 4048 
 32 pe 1496615 12.515284 SAAS [5 5602'5761159201607816237 7 
 33 6906555 6714'687317032'1719117350) 750917667 7826 
 sear §144'8303'846218620 87791893 89097 925619415. 
 9573197321989110050 isda! 0367|0526,0685'0843!1002 
 
 2580) 
 
 0831 gga} 149}1308|1467} 
 
 in| 
 
 1]0199}0358/0516 | 
 
 L..431} 
 
 iéi 
 
|N.27500 1.439 OF NUMBERS. wh) 
 
 Nitlioee te? | Sl 47s 1 61.7.) 8) 94 Dip 
 2750|4393327|3485|3643/3801|3959]/4116)4274|443214590/4748] | 
 5 4906/506415222/5379|55371|5695|5853/601 11616916326 158 
 52]  648416642|6800/6958]7 11572731743 117589|774717904 1] 16 
 53] 806218220|837818535|8693}/885 1|9009|9166/932419482 : 32 
 54] — 963919797\99550112/0270]10428]0585|0743|0901/1058]] fF 
 55|4401 216}1374]1531|1689]1847112004|2 162)23 19|2477|26035 5} 79 
 56| 2792)/2950/3107|3265|3422135801373813895|4053/4210 6) 95 
 57|  4368/4525/4683|4840/4998]5 155/5313|5470\5628/5785]) |. 
 58] ~— 5943/6100/6258164.15|6572116730|6887|7045|7202|7360} — {ols 40 
 1 50) 7517|7674|7832|7980|81471/8304|8461|8619|8776|8033 
 12760, 9091|924819406|9563/9720]19878|003 5|0192|0349|0507 
 | 6114410664|082110979}1136]1293]|1450|1608|1765|1922/2080 
 62} 223712394/2551|2708|2866]3023|3 180|3337|3494|3052 
 63}  3809/39660/4123/4280)4438}}4595/4752|4900|5006/5223 
 64) 5380'5538'5695|5852|6009]16 166/63 23|0480/6637 167094: 
 65} — 6951)7108|7205|7423|7580}|7737|7894|805 1|8208|8305}| , 57 
 66| 8522/8679'8836/8993|9150]1930719464|962 119778|9935 
 67|44200921024.9|/0405|0562|07 19/}0876|1033]1190)1347|1504 
 68]  1661]1818}1975|2132/2288|12445|2602|2759|2016|3073 
 69|  3230/3386/3543/3700|3857||4014|417114327|4484/4641 
 27701 + 4798'49541511115268|5425|15582|5738)5895|/6052/6209 
 71} ©306516522|6670|0835|0992)/7 140|7306|7462)761 9/7776 157 
 72| — 7932|8089|8246|8402|8559|87 16|/8872/9029/9185|9342 1) 16 
 73} 9499|0655|0812/9960|0125||0282/0438/0595|07 51/0908 2} 3) 
 74|443 1005|1221]1378]1534]16911|1847|2004|2160|23 17/2473 3} 47 
 -75| .2630)2786|2943|3099/3256]341213569/372513882/4038} fs! oo 
 76] 4195|435114507/4664|4820]1497 7/5 133|5290|5446/5602 6} 94 
 7/|  +5759|5915|6072|6228/03 84116541 |0697|0853|7010)7 166 71110 
 78| = 732217470|7635|7791|7948||8104|8260|8417 8573/8729 8}126 
 
 | 79] 8$885|9042/9198/93541951 119667|9823|9979 0136/0292 
 2780}4440448'0604)07 60/091 7/1073}}1229|1385]1541/1698]1854 
 81 2010)|2106/2322/2478/263 511279 112947/3 103/3259|3415 
 
 82} 35711372713883/4040/4196)/4352/4508|4664/4 820/4976 
 83] = §132)5288]5444|5600|5756||59 1 2|(0068|6224163 80/0536 
 84] 6692/6848)700417 160/73 16747 2|7628|7784|7940|8096 
 85| 8252'8408/8564|872018876]|903219188]9343/9499|9055" 
 86] 9811|996710123/0279|0435|10590|0746 0902} 1058}]1214 
 
 156 
 
 89 44954641 4797|4952|5108}15264|541915575/573115886 
 2790} 6042619816353 6509|6065||6820/6976)7132)7287|7443 
 
LOGARITHMS N. 28000 L. 447 
 
.28500 L.454 OF NUMBERS. 
 
 2850}4548449|8001|8753|8900|9058}\92 10|9363/95 15|9008 9820 
 
 9972|0125|0277|0429 0581 |0734}0886|1038]1191'1343) 
 
 52/4551495|1647]1800|1952/2104|/2257/2409|2561 2713 2865! 
 3018/3170)3322/3474'3627 
 
‘<a. LOR ARITEMS) “N.29000 1.462 | 
 
 ai 4623980|413014270|44201457911472014878|5028|5 17815328 
 5477 5627 5777 5926|6076/}62261637516525 6075 6824. 
 
.29500 L.469 OF NUMBERS. (45) 
 N. O ZA D \Pro.} 
 
 a fy a a ff | | 
 mr mr J _—_ 
 
 14 
 5214701 164/13 11|1458|1605 1752 SAN ea Nts ; A 
 53] — 2634|2782)2929|3076)3223113370)3517|3004|381 1/3958), 4 -|2} 29} 
 54) —4105/4252/4399/4546)4693|14840/4987)5134|5281|5428)) “| 44 
 55|  5575|572215869|6016(61631631016457|6604|6750/6897|) {31 =, 
 56} 7044/719117338|74851763217770\7926|8073 6l 88 
 
 71 
 
 79\4730488|063410780|0926|1073 1219}1365]1511|1657/1803 aie 
 741  34101355613702|3848|3994|41404286/44321457814724 14g i 
 75| 4870|5016|5162|5308|545415600'5746|589 11603716183] [4] 58 
 76} 6329|6475|662116767\6913117059'7205 Wash 7497|7642)| : he 
 
 78) —9247|9393/9539|9684|9830]19976/0122 0068 0413/0559 : re 
 
 79\4740705|085 11099711 14211288}}143411580)1725}187 1/2017 O}1S1 
 
 poe ok: Of ce Se Rec perce ik» all Sec Yon 
 
N.30000 1.477 
 
 LOGARITHMS 
 
 ——| |__| |__| | | ——_—_ |__| | | eS 
 
 02} 4107/4252/43961454114686|4830/49751511015264|5409 
 03} 5553/5698|584315987 6132| 62761642 116566167 10/6855 
 
 04, 690017144|728817433175781772217867|801 118156|8300 
 05 
 
 06 
 
 08 
 
 09 
 
 ll] = 7108/725217396|7540\76841782017973|8117|8261|8405 
 12}  8550)8694/8838/8982/91261927 7 7 
 13] 9991/0135]0280}0424/0568 
 144791432)1577|172111865|2009 
 
 15} §2873'3017|3161/3305|344013593|3737|388 11402514169 
 16, 4313)4457/4601/4745}4889/5033|5 177/532 1154655609 
 17| =5753'589716041|6185}6329|16473'6617|676116905}7048 
 
 18} 7192/733617480|7624|77681'7912|8056|820018343|8487 
 19} 863118775189191906319207 193501040419638/9782|9926 
 
N.30500 L.484 OF NUMBERS. 
 
 = | +] - J] | ———- —_ } —- | —————— | —_—-— | -—--—— 
 8 | eee | er een ee ee 
 
 3050 4842998)3 141/3283/3426|3568|137 10|3853/3995|4137|4280 
 51|  4422'4564/4707/4840!499 1115 1341527615418|5561/5703 
 52 584515988 6130'627216414116557|6699|684.1|6984\7 126 
 53 Sone 7553\7695|783717979|8121|8264|8406 8548 
 54 poooe 8833/8975|9117|9259}9401)9543 
 
- LOGARITHMS N.31000 L.491 
 
 a a ee ee ed fae! | Go, es | 
 
 9573 97 11|9850!19988|0126'0265|0403|0541 
 
 2062/2200'2338'2476|261 5||2753|289113020/316713306 
 344413589] 
 
N.31500 L.498 OF NUMBERS. | (49)f 
 
 ee ee) ee ee ee ee 
 — J 
 
 AAS Ow 
 WT 
 C 
 
 ome 2) 
 — 
 |S 
 © 
 
 137 
 
LOGARITHMS _ N. 32000 L.505 | 
 3{ 41516 E 
 
 1635}177 1|1907|2043}/2178}23 14|2450|2585|2721 
 28571299213 12813264133 901135351367 113806|3942|4078 136 
 4213}4349|4485]4620)47 5611489 11502715 163|5298|5434 1] 14] 
 5509|5705}5841|5976|0112116247|6383|0518|6654|6790) 51 41) 
 6925|70611719617332|7467||760317738|7 8741800918145 al 54y 
 8280|84.1 6/855 1|8687|8822|1895819093}9229|9364}9500) + — 5} 68) 
 96351977 119906|0042101771103 12}044'810583107 1910854 2 ped | 
 
 M 
 
 2603|2738|2873)3007/3 142132771341 1135403681816)  |49>, 
 3950}4085|4220143 541448911462.4/475814893|502815 163 al on 
 52971543 21556715701(5836|1597016105|6240|6374|6500]] = 8} 41 
 6644|6778]6913|7047|7182117317|745117586\7720/7855)— | ne 
 7990}812418250/8393|8528}/8663|8797|8932|9066)9201 él 81h 
 9335]9470|96041973910873|10008|0142/0277/041 110546 "| 95 
 
 29|5090680)08 1 5}094G}1084}1218}11353}1487)1622}1756}1891 8/108 
 2025|2160|2294|2429|2563\/269712832|2960/310113235) = 
 3370|350413638|3773|3907114042/4 1 76)43 104445 579} 
 
 ' .4714/4848H4982/5117|525 1115385|5520|5654157 8815923 
 
 6057|0191|6326|6460}6594116729|6863|6997]7 13217266 
 74007 53417 669178031793 7||807 2|8206|8340|8474|860g 
 8743|8877|901 1|9146|g28011941419548|9682|981719951 
 
 36/5 100085/0219|0354|0488|0622]107 56/0890] 1024}1159]1293 
 1427|1561}1695|1820}1964]/2098}223212366]2500|2634 
 2768}2903|303713 17 113305113430|3573|3707|3841 13975 
 4100|424414378]45 12/46461147804914|5048|/5182|5316 
 5450|5584157 1815852/50861|61 20|625416388]6522/6656} 134) \ 
 0790|602417058)7 1921732611746017 59417 7 28|7862|7996 
 8130|8264/8398/8532|8666||8800|8934|9068|9202|9336 134 
 9469|9603|9737|987 1{0005}}0139|0273|0407|0541/0675]| || 3° 
 
 4415110808|0942}1076]1210]13441}1478]1612}1745|1876]2013 31 40 
 214712281}2415}254812682|12816|295013084|3218|3351} —|4] 54 
 3485|361013753|3887|40201141 5414288/442214555|468 / a 
 482314957|5090|522415358115492|5625|5759|5893|6026 nt 4 
 6160)6294|6428)6561|6695}]5820|6962|7096|7 2301736. 8H107 
 7497/7 93 1|7764}7898|8032}8 1 65|8299|8433|S566}8700} —_|94121 
 O 11,2193 | £00 3 7 9 | D|Pts 
 
4140|4282/4415/4548|4681 te 5079152125345 
 5478|561015743|5876|6009||0142 
 
 6274'040710540|6672 
 
MAD iF ge Pie Pe RR GoRErsEre ko Reed Mineesun feta eo —_—_————— | —_» | —_____—__] ———_——__-_ ] - ————_ | | ——___.__ |... 
 
 6968|7099|7230|736117493)7624\7755|7886|8019|8149 
 8280|841 118542/8674|8805|18936|906719198/932019461 
 959219723|985419985|01161024810379|/0510|0641 pi led 
 12|5200903} 1034}1166|1297|1428] 1559]1690}1821)1952|2083) 
 2214|2345/247 7|2608|2739]|2870/3001)3 132|3263/3394 
 3525|3656,3787|3918/4049]4180/43 1 1/4442/4573/4704 
 4835'4966|5097|5228|5350|5490|562115752|5883|6014 
 6145|6270'6407/653 8|6669||6800/693 1|7062)7 193|7324: 
 7455\7586177 171784717978||8 109|82401837 1/8502|8633! 
 8764|8895|9026|9156|9287|19418|9549|9080/08 1 119942 
 
 5250 5389|5519|564915779|5908|6038|6168|6298|0427 
 6957 0687 6817/6946 7076 7206/7336 7405 759517725 
 
N.33500 L.525 OF NUMBERS. 
 
 N.;cO eta ais al sfie 
 
 65| 985119980|0109|0238|0367|10496|062.510754|0883 1012 
 
 3390| —1997|2125|2253|238112509||2637|2766]28941302213 150 
 
 Q8} 2234/2362)/2480|2617|2745]1287 3|3001)3 12813256/3384 
 QO] ~ 3512)3639 37 707 '3895|4023 4150}4278}4406/4534/460) 
 
(54) LOGARITHMS N.34000 L.531 
 ct O 1 pat 1) 30 1t4 a 
 
 -_ |---| - ————_ | | |  —  ——— ) | | | 
 
 Ol 60661619416322 944.9|6577|16705|6832 6017088 7215 
 02| 7343 para 75981772617 8541'7081) pio 8237 $364|8492 
 
 37] 3795|1922/2048/2174/2301 2427/2553 |2680|2806|2932 
 38}  3059/3185/3311|3438|3564]]369013 817|3943/4069/4. 195 
 30] 4322/444814574147011482711405315079|5206|5332/5458 
 
 440] §584)57.1115837|5963|0089/6216)/634216468/6594/6721 
 41] 6847|697317090|72251735211747 81700417 730)7856]7982 
 421  8109/8235|8361|8487|8613/18739}/8866/8992/91 18|9244 
 48| 9370|9496|9622'9740/9875}]0001|0127|0253|037910505 
 441537063 110758|0884|1010}1136]1262)1388]1514/1640]1766)). 
 45, 1892/2018)/2144/2270/2396}12523|2649|2775|/2901 13027 
 46]  3153/3279|3405|353.1|0057113783]3900]/403 5/4101 |4287 } 
 47| 4413/4530/4665|4791/491715043}5 100|5295|5421|5547 
 48} 567315790|59024|6050/0176]}630216428|055416680|0806 
 40} 6932)7058|7 184173 1017436117561|768717813|7939/8065 
 
 JN.) OCf rl 2] 3s [4d 5 
 
N.34500 L537 OF NUMBERS. 
 
 9450}9575|970119827|9953||0079|02051033010456|0582 
 52/5380708|0834|0959]1085|12111/1337|1463}1 588}17 1411 S40}} 
 
4401]4525]4649|4773/4897||5021)5145|526915393|5517|1 2416 
 5641|576515889|601316137|16201|6385 sete 6632)|6756 
 
N.35500 L.550 OF NUMBERS. 
 
LOGARITHMS N. 3 6000 L. 556 
 
.86500 L562 OF NUMBERS. 
 
 3650|56229209)3048)3 167|3286|3405||3524|3642)3761/3880)3 999) 
 51] 4118}4237}4350}4475/4594))47 13|4832/495115070)5189 
 52] 5308}5427|5546}566415783!159021602116140|0259/0378 
 53] 6497|6616|673410853}6972||70911721017329)7448|7567 
 54] = 7085/7804}7923]8042)8161]18280)8398|85 17|8630)/8755 
 
 8874|8993|9111]9230\93491194681958719705|982419943 
 5615630062018 1|0299]04-18]0537/}0656)0775|0893}1012}1131 
 1250}1368]1487}1606|1725]|1843]1962!208 1|2200]2318 
 2437|2556]2674)27 9329 1 2303 1}3 149/3268/3387|3505]| 
 3624|374313861|3980/4009]]421814336]4455]4574\4692 
 
 4811/4930|5048]5107|528511540415523|56411576015879 
 5997|0116|6235|63.53|6472||6590|6709|68281694 6/7065 
 7183|73021742117539|7 058177 7 O|7 895|8013]8132/8251 
 8369|8488/8606/87 25|8843]|89621908 1|9199193.18|9436 
 955519973]9792|9910|0029}/014710206|038410503|0621 
 
 SODBAREHONO KY 
 es S00. te DD ne 
 
 ‘| 
 
 4293/44.1214530]464814767]14885|5004]5 1221524015359 
 547715595|5714 5832 5051 6069} 618716306 O44 6542 
 
 6117 71023510353 ae 6590 rie 082.6|6944|7062|7 180 
 7298 7416|753417652\7770||7888|8006|8 12418242 Bae ] 18 
 
 08 
 
(60) _ LOGARUTHMS N.37000 LL. 568 
 
 ts Pf is Paterna Tt i es oe 
 
 03 5537 
 
 06} 9054/91 71}9289 940019523 964019757 hey 9992/0109 
 
 22} 77064|7880/7997)81 14|8230]|8347|8464|8580|8697|8814 
 
 25| 1263/1379|1496]1613)1720}]11846)1962|2079|2195|2312 
 26| 2420/2545|2062 2778|2895/|301 1/3 128|3244)3361/3477 
 27| 3594|371013827/3943/4000)/41 77|4293|4410)4526 4643 
 28] 4759|4876/4992|5109|5225||534115458|5574|5691|5807 
 29| 592416040/6157|6273|6390/16506 6623|673916855|6972 
 3730] )7088!7205|73211743817554||7670|7787|7903|8020)8136 
 31|  8252/8369!8485/8602|8718]18834 895 1|9067/9184|9300 
 32} 941619533\9649/9765|9882}9998)01 15|023 1 0347/0464 
 33|5720580}0696(0813|0929|1045]}1 162)1278/1394]1511)1627 
 34] 1743 1859 1976}2092|2208]}2325|244 1|2557|2674)2790 
 
 35| 2906)30221313913255/337113487|3604|372013836|3952 
 36} 4009/4185/4301/4417/453414650/4766)48824900|5115 
 37| — 5231)5347|5463/5580|5696||5812|5928|6044 [61 61/6277 
 38] 639316509|66251674216858 6974|7090)|7206|7322|7438 
 39} 755517671/7787|7903|8019}9135|8252|8368|8484| 8600 
 
 3740} 8716/8832 $948 9064/9 1 80||9297/9413|9529/9645/9701 
 41] 987719993 0109 0225103410457 574 O0690/0806/0922 
 
 45| 4518 4634 4750/4866 es 5098 521415330 546 5562 
 46| 5678 5794|591016026|614 116257|637316489|/6605|6721 
 47| 683716953|706917185|730111741617532 7648 7764|7880 
 
 pouemeeeneen? Leenmemeseme=en seed eee 
 [$$$ J —— } |__| | | 
 
N.37500 L. 574 OF NUMBERS. 
 
LOGARITHMS N.388000 ‘L. 
 91D 
 114 
 
 113 
 
 579, 
 
 A A tr pc 
 
OF NUMBERS. 
 
 N.38500 L. 585 
 
 385015854007/472014833]404 
 51} 5735 84815961|6073 61 86/]6299 6412 0525 ar 3750 
 
 2496|2609|27221283.4|294711805913 17213285|3397]35 10 
 3622)3735|3847|3 960/407 2}}4185)4298/4410/4523 463 5h. 
 
(64) | | LOGARITHMS N.39000 L591} 
 
 a ee ee es ee | ee ee a a ee ee 
 
 07|  8434|854518656|876818879|18990\9101|9212193231943.4]| 
 08} 9546/9657|97681987919990]101 01/021210323|0434|0546 
 
 “Se[ 39] —3800)3970/4080)4191)4301)/441 114521}4632/4742/4852 
 43940} 4962/5072/5183/5203|540315513]5624|5734|5844|5954 
 
 43] 8208/8378/8488/8598/8708/|881 8/8929 |9039|9149/9259 
 44} —930919479|958919699|98 10}/9920}0030]0140)0250/0360 
 
 | | | | | | J | | 
 
1N.39500 L. 596 OF NUMBERS. (65) 
 N.| 0 «AD SAA a nl Soleo ad shal a 
 
 * ————e| 5am od Dae Rares epee Popped & ooesers Geena be espe ae 
 
 51] 707017180)7290}7400|7510)7620 #730 7840|7950|805g 110 
 52} 8169|8279/8389|8496|8600|/87 19/8829]8939/9048]9158 Uli 
 53 9268 9378}94838}9598)9708|198 1719927 |903 710147 "ib ps we 
 
 57| 366113770)3880/3990/4090)/4209)/43 19|4429|4538 4048 gles 
 58 ied 48608 4977|5087|5 197 5300)5416)5526 5030 ith 9199 
 
 (~>) 
 — 
 =o 
 nse 
 (8) 
 QO 
 — 
 oO 
 .D 
 CO 
 bo 
 ©) 
 6.2) 
 CO 
 Co 
 N 
 bs I 
 e8) 
 oe 
 6 om. 
 SI 
 io) 
 or 
 © 
 Oe | 
 @ 
 “Ty 
 3 
 (=>) 
 io) 
 @ 
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 yS 
 es) 
 Ss. 
 bo 
 cr 6 
 af 
 On 
 
 07 pares ron 4841|4950|50600)|5 169|527 9 5388 has 5607 
 68} 5717|5826|5936|6045|6154||6264 pe vk 0592 shies 
 
 92) 1905/2014/212312232/23401124491255812667|2776|2884 109 
 93] 2993/3102/3211/3319/3428]|3537|364613754|3863|3972 111 
 94] ~4081/4189|4208'4407/4516]14624147331842/4905015059 = ab 
 
 95 516815277/5385|5494'5603]|5711|582015929|6037|6146 444 
 96|  6255|636316472|658116690||6798|6907|70161712417233 5155 
 Q7| 734117450\7550|7607|77 7 O17 8851799318 102/82 1 1/8319 aln6 
 98} 8428/8537|8645|8754|8862)1897 1|Q080/9188!90297|9405 gig 
 99} 951419623/973119840\9948}/0057|01 660274103 83/0491 9/98 
 
 9 | Di Pts. 
 
(66) LOGARITHMS N. 40000 L. 602 
 N.{ O Lj 214.3 
 
 er | nn rr 
 
 0036|014410252 
 122711335 
 
 20 1973|2080,2188|2296|24041!25 1 2'2610|272712835|2043 
 
 4030}  3050/3158|3266|3374|3482)|3580|3697/3 805/39 12/4020 
 31} 4128/4236/4343/4451|455914607|4774/4882/4990|50908 
 32) 5205/5313/542115528)563615744)5851|5959|0067/6175 
 33} ~ 6282|6390|}6498\6605]67 13]}682 1|6928)7036)7 14417251 
 34 7350174671757 417 082\7 790\|7 897|8005}8 1 12)8220/8328 
 35] 843518543/8651|8758|8866)|897 4908 11918019290\9404 
 36] 9512196191972719834|9942!}0050|0157|0265|037210480. 
 3716060587|0695|0803|0910]1018]}1125)1233]1340)1448 1556| 
 38] 1663/1771/1878}1986]/2093}|2201}2308|24 16)2523)2631} 
 
 4040} 3814)/3921/4029/4136142441]43511445091456614674/4781) | 
 41] 4889/490615103/521 1/53 18}15426|55331504.1|5748|5856} 
 A2| 59631607 1/617816285|6393]}6500|6608)|67 15|6823|6930 
 43) = 7037)7145|72.52|7360)74671|7574\7082\77 80|7897|8004 
 44| 8111)8219|8326)8434|85411}8048|87 56|8863|897 19078 
 45| 9185/9203|940019507/961 5|10722|9829|9937|0044)0151 
 46|6070259|0366|0473|0581|0688|(0795,0903] 1010}1117|1225} 
 
 47} 1332}1430]1547]1654]1761} 1809] 1976}2083|2190}2298} 
 
 A8| 2405|2512/2620/2727/28341/2941|3040|3156|3263 837) 
 
 40] 347813585 3692 3800)39071|4014)4121}4229 4336/4 
 
N.40500 L.607 OF NUMBERS. 
 
(68) LOGARITHMS N.41000 L.612 
 O Py 2vis | 4. 647 
 
 4100|6127830|7944]8050181 5618262 _ 8474|8580|8686|8792|| 
 8898/G004|9109|921 5|932.1|19427|9533|9630|9745|9851 
 9957|0062/01 68/0274 0380) 0486|0592/0698|0803|0909 
 
 6304 Apres Bae 6621 pide 
 736117467|7573|7678|7784 
 8418/852418630/8735|8841 
 
 ri 6930|7044]7 150|7256 
 
 8947/9052 9158|9263|9369 
 
 1587 wr 1798 1904'2009}}2115|2221|2326|2432 
 
 2043/2748/2854 2960 3065]/3 17 1)3276|3382)3487|3593 
 3698)3804/3900 4015 4121]14226/4332/4437|454314648 
 4754148509 490515070 5176) 5281|5387|5492|5598|5703 
 |4|6020,6 125'6231116336 0442 0547 
 
 7918]8023|8129|8234'8340|8445|8550'865618761|8867 
 897 2|9078)01 83 ie 9499|9605|97 10)9815|9921 
 
 Si 8554 8659|87 64'8879]6 8C se 9080/9185|9290)9395 
 9501 9606 9711 98169921 0026)013 11023 7/0342)0447 
 7'\007.2)|1078|1183{1288]1393}1498 
 20241|2120'223.4|2339|2444/2549 
 307413.170|3284|3390|3495|3600 
 14125 4230|4335|4440/4545|4050 
 
 §175|.5280|5385|5490|5595 5700 
 en 6330|6435|0540|0645|0750 
 
 1603}1708|1813 1918 
 265412759|2864|2969 
 3705|3810)3915|4020 
 4755\4860\4965 5070 
 Baie 5910)6015}6120 
 
 4197/4302/4407|451214617114721|4826/493 1|5036|5141 
 
 5245|5350|5455155601\5664 5769 587415079|6083/6188 
 6293/6398 6502/6607 0712} '0817|0921 702617 13117236 
 
 [eee 
 ——_—___ |__| ———______ | 
 
N. 41500 L.618 OF NUMBERS, (69'} 
 
 | —_———_. } ——___— | ——__—— |! -—_ ——_—_- | ——______| 
 
 2) eta rat 
 
(70) LOGARITHMS N.42000 L.623 
 
 -_—— ff -__ ] |, - ——- J — 
 
 0727 9830/9933 
 
 17 6250036 0139;0242/0345 0448 055 110654,0757|0860|0963 | 
 
 2095 ee 2404!2507/|261-0|27 13/2816/2919|3022 
 7 
 
 42201  3125|3227/3330/3433/3536]3630/3742|384.513948/4051 
 : 4154/42564359}4462)4565)/4008/477 1/4874|4977|5079 
 
 | 
 
 | 
 
 2377 ae 212685 811704 2890|2993/3096'3 1198/3301 
 3404)3506,3609/37 r 1213814|/8917|4020)4122/4225/4328 
 
N. 42500. L. 628 OF NUMBERS. 
 
 4911 5013151151521815320154221552415626 5728/5830 
 5933 6035 6137162391634.11|644316545|6647167 5016852, 
 
(72) LOGARITHMS N.43000 L.633 
 
 4300|633468514786]4887|4988|5089 51905291 53911549215593 
 5795|58960|5997 |6098}|6199/6300|640116502)6608 101 
 
 4773/4873 4974|5075151761527615377154781557915679 
 5780|5881|598216082|6183||6284|6385|6485|6586|6687 
 
 rm ff fl a a | ae ee | | — fe  - 
 
N.43500 L.638 OF NUMBERS. 
 
 4350|6384893|4992 5092|519215292115392|5492|5591|5691|5791 
 5891|5991|6090}6190}6290}|6390 04 S289 6689 6789) 
 
 31938319483 epee 9782 
 380 0480}05800679|0779 
 
 43 
 
 31958219681 
 
 -§634|5733|5832'1503 116030)|6129/6228/6327 vee 6524 
 6623/6722|6821\6920/7019 73 1817217|7315|741417513 
 7612)771117810}7900|8007}'8 106/8205|8304/8403|8502 
 
/ 
 
 ~ 
 
 LOGARITHMS N.44000 L.643 
 
N. 44500 L.648 OF NUMBERS. | 
 
 Niels OO MPR 1.3.4.3 1-4 6 | 7 | 8 | 9 | D [Pro 
 4450|6483600|3698|3795|3893|3990)'4088141 86 4283/4381 478 
 51] 4576/4674)4771|4860|49606)|5064/5101/5259|5356|5454 
 52}  §552)5049|57471584415942)|6039/6137|623410332'642¢ 
 53}  0527|002416722168201691 7/1701 5!7112)7210|730717405 
 
 9452|9549 “sh 9744|9842||9939 0037 0134/023 110329 
 ~ 57|6490426|05241062 107 19/08 16]]0914]101 1/1108/1206/1303 
 
 58} 1401/1498]1595}1693|1790}|1888}1985\2083 218012277 
 59} 2375|24721/2570|2607|2764!|2862/2959|3056/3154 3251 
 4460! 33409/3446/3543|3041|3738|/3835/3933/4030/4128'4225 
 61} 4322/442014517/461414712)/4809|4906/5004/5 10115198 
 62} 5296/5393/5490/5588/5685}|/5782|5880/5977|0074/6172 
 63} 6269/6366/6463/6561|00581|6755|6853 ee 7047/7145 
 
 68}  1132/1220]1326]1423/1520)1618]1715|1812/1909/2006 
 GQ} 2104/2201/2298]2395|2492)/2589/2687|2784/2881|2978 
 4470} 3075/3172/327013367/3464/13561/3658|3755|3852'3950 
 71| 4047/4144/4241/4338/443 5/4532/4029/4727/482414021 
 72| 5018|5115|5212153009|5406)5503/5601|5698/5795|5892 
 73| 5989|6086/6183|6280/6377116474|657 11666967 60/0863 
 74, 609601705717 154}7251|7348]|7445|7542|7 039 (a0 Rae 
 75| 7930|8027/8124)8222/83 19/1841 6/85 13|8610)/8707 8804 
 76} - 8901|8998!9095|9192/9289||9386|9483|9580\9077|9774 
 77| —9871|9968|0065|0162|02.59||0356|045310550|0047 0744 
 
 78|651084.1|0938]1035]1132|1220||1326)1423|1520/1617|1714]| _ 
 
 1811|1908}2005|2102|2198}|2295|2392|/2480|2586|2083 
 
 27 80/287 7|2074|307 113 168)|3265|3362/3459|3556/3053 
 37409|384613943|4040/4.137||4234|433 1|4428|4525 4622 
 4719|4815}4912|5009/5 106)|5203|5300/5397|5494'5591 
 5784|5881|597 81607 5/161 72|6269|6365|0462 6559 
 6656|6753|6850|0947|7043)|7 140|7237|7334|743 1/7528 
 7624'7721}7818|7915|8012)8109|8205|8302|8399'8496 
 8593|8690|87 86|8883|8980}9077|9174|9270|9307 (9464 
 _ 9561 19657 97 54/985 119948}|004.5|0141|0238/0335 0432 
 
 89 saooribes 1690]1786|1883}|1980|2076|2173/2270/2367 
 4400} 24063'2560)2657/2754/2850)|2947/3044/3140/3237/3334 
 gl 343 1\3527|3624|372113817|\3914/401 1|4107/4204/4301 
 92} 4397\4494)4591|}4688/4784!488 1|4978|5074/517 115268 
 
 536415461155581565415751||5847|5944|0041 1613716234 
 
LOGARITHMS N. 45000 L. 653 
 
 . re 
 
N.45500 L.658 OF NUMBERS. | (77) 
 
 N.} O°] 1) 2/13 | 41.5 | 6:1 7-] 8 1:9 | D]Pro 
 
 4550105801 141020910305 '0400\0496)|059 110687|0782|0877,0973 
 51 1068|1164]1259/1355/1450]11545}1641)1736|1832}1927 Q5 
 §2|  2023)2118|2213/2309|2404}|2500)2595|2690)2780/2881 : OG 
 53| 2977|3072|3167/3263/3358)|3453|3549|36441374038351 °}.a)9 
 54| 3930/4020/4121/4216/43 12/1440714502|4598/4693/4788 4138 
 55|  488414970|507415170)5265||5361|5456)5551|5647|5742 oe : 
 56| 5837|5932|6028|6123|6218]|63 14|6409|6504|6600|6695 , 4 
 57| 6790|6886)698 117076717 1)17267|7362\7457|7553|7648 876 
 58} 774317838/7934|8020|8 1 24||8220)83 15|8410;8505|8601 9186 
 59} 8696/8791|8886|8982)9077}19172/9267 9363 eas nee 
 
 4560} 9648/9744'9839/99034|0029}'012.5|0220,03 15|0410,0506 
 
 61}659060110696|079 1 (0886|0982)|1077|1172)1267)136211458 
 
 62| 1553|1648|1743]1838/1934|12020/2124/2210/2314/2410 
 63} 2505}/2600!2695|2790/2885]i2981|3076)/3 171 32003361 
 64, 3456|3552|3647/3742)/3837113932\4027/4122/42184313]) . 
 65} 4408/4503/4598/4693/4788]4883|4970|5074)5160|5264 
 66} 53509|545415549|504415740)|5835|5930/0025|/012016215 
 67} 6310|6405|6500|0595|6690)|6786|6881|6976|707 117 166 
 68 726117350\745 1|75461764.1117736)7 83 11792.6|8021|8117 
 69|  8$212)/8307|8402|849718592)|8687|87 82/887 7|8072|9067 
 4570|  91621925719352|0447|9542)1963 719732|982719922'0017 
 
 7 1|66001 12|0207|0302/0397|04.92|10587|0682)07 7 7108720967 Q5 
 
 72) = 1062}1157}1252}134711442)11537|1632)1727/1822|1917 
 73|  2012|2107/2202|2297|2392)12487/2582)267 7|2772|2867 
 7A] ~~ 2962/3057/3151|32461334 11134361353 113626/3721/3816 
 75} +3911|4006/410114196/4291114386/448 1/4575/4670)4765 
 76} 4860}4955|5050)5145|5240/15335)5430)5524/501915714 
 77\ —58009|5904|5999|6094|6189||6284|6378|6473|0568'16663 
 781 675816853|0948|7042!7137)|\7232|7327|7422'751717612 
 79| 7706/780117896'799 118086)'8181|8275|8370|8405/8560 
 4580] 8655/8750|8844/8939|90341/9129|9224193 18'9413/9508 
 
 2446|2541|2636|2730|2825]|2920/3015|3 109 
 3393|348813583|3678/3772|'386713962/4056 
 4341|4435|4530/4625]47 10|4814|4909|5003 
 5287|5382|547715571|566615761|5855|5050 
 
 3204/3299 
 415114246 
 5098/5193 
 6045/6139 
 
 91] 9073/9168|9262/935719451|19546|9640|9735/9830\9924 
 
 92|662001 9/01 13}0208|0303/0397|10492|0586|068 1|0775|0870 Q4 
 93| 0964]1059/1154]1248]1343/1143711532|162611721|1815 1/9 
 94| — 1910|2004|2099|21942288||2383|247712572'2666)2761 aoe 
 95| 2855]2950/3044)3 139/3233/13328|34221351 71361113706 4138 
 96| 3800)3895|3989|4084/41 78]|427314367|4462'455614651 S47 
 97| 4745|4840/493.4/5028|5 123/15217]5312/5406'5501/5595] {5/6 
 98} 5690/5784]587015073 6067 6162|6256/6351/6445|0540 875 
 
 ek See, ee. ae 
 —_—<—$__— jf ——_..__.— ——_ -|_— | | I eee! ee 
 
(prcerientipinde | Ciequeaeatemectne | <teeemeer am | eager || werinenmaninian |amntntmenatvabisie i Teliadenniaigensent> loleicetinnanngtens | uabecicnentonse? | cinckmasiemasionss-l tssitpnsneninasinee! 
 
 1353 1447|1542]1636/1730)11825}1919/2013/2108/2202 
 
 2296|239 112485!2579/2674|127 68|2862/2950/305 113145 
 3239/3334/3428/3522|3616)/3711/3805|3899|3994|4088 
 4182}42761437 114405/4550]/465314748/48421493615030 
 5125|5219|531315407|5502)}5596/5690)/57 841587015973 
 6067|61611625616350|64441|6538|6032'6727/682 116915 
 7009|7 103'7 198!7202|7386)17480|757417660|776317857 
 7951|8045|8 140)}8234|8328]18422)/8516 861018705 8799 
 889318987|9081/9175|92701|/9304|9458/9552\9046/9740 
 -9835]9929'0023/01 17/021 1}10305/0399'0494|0588|0682 
 
 ~ 1717|1811]1905]1999|2093]|2188!2282/2376|247012564 
 2658|2752/2846!204013034]|/3 128/3222'33 171341113505 
 3590|3693|/3787/3881|3975||}4009/4163 /4257|435114445 
 4530|4633/4727/482114915)}5009'5 104'5198|5292|5386 
 
 548015574|5668)|5 762)5856)|59506044 OIa8 623216326) 
 . 642016514|6608/670216796||6890'6 7172\7266 
 8111|8205 
 
 7360|745417 548|764217736)17830\7924/8018 
 8290/8393 8487|8581|8675||8769|8863 8057|9051|9145 
 9239/9333/9427/952119615||9709/0803 98969990 008 
 241606501 78/0272|0366,0460\0554||0648'0742/0836.0930) 1023}; 
 1117}121111305}1399|1493)|1587|1681|1775|1869|1962 
 2056/2150}2244/2338|2432/12526|2020)27 13|2807|2901 
 3183/3277|3370113464/3558|3652'374613840 
 3934|4027141211421514309||4403/4497 4590/4084 478 
 4872|4906|5059|5153|5247)|5341|5435 552915622 5716) 
 
 5810]5904/5998|6091|6185||6270|6373 6466 
 
 1434]1528]1622 we 1809|}1903]1996/2090/2184 2977 
 237 1|2465|2558|2652|2746'|2839|2933|3027|3 120/321 
 33071340113495|3588'3682 13776/3869|3963 405614150 
 42441433 7|443 1145251461 8)|47 12/4805|4899|4993|5086 
 §180|5273|5367|5461/5554||564815741|5835|5929|6022 
 6116}6209|6303|6396|6490)|6584|667 71677 1|6864|6058 
 705 1\714517238/7332|74261|75109|761317706|7800|7893 
 7987|8080|8174|826718361)||845418548|8642/8735|8829 
 8922/9016'9109|9203|9296 9390/9483 |9577|967019764 
 98571995 1|6044 0138 023 1|10325|04.18|0512|0605|0699 
 
 LOGARITHMS N.46000 L. 662 
 
 9186 
 
N.46500 L. 667 OF jth: ce 
 N.| 0 18) Sh | 8 
 
 4650 6674530 4623/47 16|4810 nas 4990)50 90|5183|5277 5370 
 51| 5463/555715050|574415837|15930.60241611716210)0304 
 52] 6397|6490/6584)|5677|0770)|6804/0957|705 1\7 144|7 237 
 53) 733117424)7517|7611/7704)|7797'|7 89179841807 7|8170 
 54] 8264/8357/8450/8544/8637118730/882418917|G010|9104 
 55} 919719290\93831947719570 9663 9757|9850\9943|0036 
 56|6680130)0223|03 16/04.10|0503)|0596|0680|0783|0876|0960 
 1062}1156)1249}134211435]/1529}1622]1715|1808}1902 
 1995|2088]21 8112275/2368]|2461|2554|2647|274112834 
 2902713020)3 1 1413207/3300)'3393|3486)|3 5803 673'37606 
 
 3859|3952/4046|4139/4232)|4325]4418]45 1 1|4605|4698 
 479114884|49771507 1 c 164]|5257|5350)5443'5536|5630 
 5723158 16|5909|6002 '6095]|6188|6282)6375|0468/6561 
 6054|6747|6840/0034|7027)|7 120/72.13|7306]7390|7492 
 7585|7670)77 7217 805\7958||805 1181441823 7|8330/8423 
 85 16/8610)8703|8796|8889)|8982!907 5|9168/9261/9354 
 9447|95401963319727 9820199 13|0006 0099 019210285 
 
 1308]1402)1495}1588/1681}|1774) 1867|1960|205312 146 
 2239/2332|2425125 18/261 11|2704/279712890|2983/3076 
 
 3160|3262'3355|3448/354.11/3634|3727|3820|3913|4006 
 4009|4192/4285}4378/447 1}/4504146056/4740|/4842/4035 
 5028/5 121|521415307|5400}}549315586|56079|577 215865 
 5958|605 161441623 7|6330)||6422/65 1 5|6608/6701|67094 
 6887|60980|7073)7 16617 2.50/17352|7445|7537|7630|7723 
 7816|7909|8002/8095\8 1 88]/8281|8373/8466/8559|8652 
 8745|8838|803 11902491 171|0200|9302/90395|9488/9581 
 9674|9767|9859|9952/0045]|0138/023 1|0324|0416|0509 
 
 1530|1623]1716|1809|1902]|1995|2087|218012273|2366 
 
 2450|2551|264412737|2830)/2922/301513108]320113294 
 3386|3479|3572|3665|3758}|3850)3943 4036/41209/4221 
 43 14|4407/4500|4592/4085 4778 487 114963|5056)5 149 
 5242|5334/542715520|5613||5705|5798|5891|5983|6076 
 6160/6262|63 5416447|0540]10032/6725|6818)691 1|7003 
 7006|7180|7281}7374174671||7 5507 65217 745|7 837|7930 
 8023/8116/8208/8301/8394]|8480618570|8672|8764|8857 
 8950|9042'9135|9228/9320)9413190505195981969 119783 
 9876/9969/0061 |01 54|0247||0339/0432/0524|0617|07 10 
 
 s5lél7tslo |plpet 
 
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 2283|237412466|2558!2640}/274 1|2833|2925/3016)/3 108 
 3200|3292|3383/3475|3567113058|37 50/3 842|3934/4025 
 4117|4200|4300)/4392/4484114575|4667|475914850)4942 
 5034]5126]5217/5309|5401 |1549215584|5676|5767|5859 
 595 1|6042/613416226)63 17||6409|050116592|0084/6775 
 6867|6959|7050)7 14217234)17325|741717500|7 600|7692 
 7783\7875|7967|8058]8 1 50}!8242|/8333|8425|85 16|8608 
 8700879 1|8883|8974|9066}/9158|/924919341|9432\9524 
 961519707|9799|9890|9982)|/007310165|0257|0348/0440 
 
 43|676053 11062307 14|0806|0897||0980}1081|1172]1264]1355 
 
 1447]1538]1630}1721}1813]}1905|1996/2088|2170|227 1 
 
 2362|2454|2545|2637|2728|12820]291 113003|3094|3 186 
 327713360|346013552|3643113735|3826/3918|4000|4101 
 
 ee |_| I) | qcj-— q— i um \—q_ 
 
.47500 L.676 OF NUMBERS. (81) 
 
 Oo 71/2;3)}4175 16 ]7 7 8] 9 ID IPro. 
 
 > | - —— ——— | ——— | — | | ———_— | ————— ——— | | 
 —_—_ | a 
 
 4750|6760930|7028)7 119|7210|7302!7393|7485|7576|760717759 
 
 51] 7850/7942|8033|8125|8216|/8307|8399|8490|8582|8673 Ql 
 52)  8764/8856|8947/903819130]922 1193 13!940419495|9587 1) 9 
 53} 96781977019861|9952|0044}}0135|0226]0318|0409|0500] 2) 
 5416770502/0083|0774|0866|0957]| 104.9]1 140}123111323]1414 4136 
 55} 1505)1597}1688]1779}1871}1962|205312145|2236]/2327 5/46 
 56] 2418/2510)/2601/2692|2784]|2875|2966|3058)3 1490/3240 : 4s 
 57|  3332'3423/351413605|3697]|3788|3879|307 114002/4153 gig 
 58} 424414336144271451814600]|4701|4792|48831497 5/5066 9182 
 50| = 5157|5248]5340/543 11552211561315705|5796|5887|5978 roe 
 4760| 6070)6161/6252/6343|6434]16526/6617|6708|67909|6801 
 61}  60982/7073/7164/7255173471743817529|7620177 12/7803 
 
 62} 780417985|S076,8168/82501183.50/844 1/853 2|/8623/8715 
 63 8806|8897 898819079 917 1110262/93531044419535|9626' 
 64} —9718|9809;9900/999 110082|101 7310264103 56/0447 |0538 
 65|6780629/07 20/08 1 1|0902/0994|]1085]1 176]1267]1358 1449, 
 66] 1540|1632}1723|1814|19051]/1996|2087|2178|2269|2360, 
 67|  2452/2543/263412725)|28 16112007|2998/3080)3 1 80/327 1 
 68}  3362)3454|3545|3636/372711381 8|3900|4000/4001 4182) 
 69]  4273/4364/4455|4546/4637|1472914820/491 1|5002|5093 | 
 477 5184/5275|5366|5457|5548)56309|5730|582115912/6003. 
 
 71 6094/61 85|6276|636716458}16549|6640/673 11682216913 | 
 72) = 700417095|7 186|7277|7308)\7459|75501764.1|7732|7823, {91 
 73| 7914|8005|8096|8187|8278||8369/84.60|855 1}8642/8733 
 74, $824)891519006|909719188]}9279\90370|046119055219643 
 
 9734|9825|9916/0007|0098)|01 88/027 9/03 70)046 1 |0552 
 76|67906431073.410825 |0916]1007]| 1098]1 189|1280}1371|1461 
 
 77| 1552|1643]1734|1825|1916]12007/2098/2189|2280)237 1 
 78| 2461/2552/26431273412825}2916|3007|3098|3 1809/3279) 
 79|  3370/346113552|3643|3734)/3825/3916|4006|4097/4.188 
 
 4780| 4270\4370/4461/4552/4642'14733148241491515006|5097 
 81] 51871527815369|5460 5551] 5642|5732|5823/591416005 
 82} 6096/6187|6277|6368'6459'16550/064.11673 116822)6913 
 83! 7004|709517185|727617367)|7458|754017630|7730\7821 
 84]  7912/8002/8093/8184|8275||8366|8456|8547|863 8/87 29g 
 85} —8819|8910/9001/9092/9 182/19273|9364/9455|9545|9036 
 86} 9727/981819908\9999|0090}101 81|027 1103621045310544 
 87|6800634|0725/08 16|0906/0997/11088!1179|1269|1360)1451 
 88} 1541 1632 1723}1814}1904/11995|2086/2176|2267/2358 
 89} 2448/2530/2630/2720)281 1/12902|2092|3083)3 17413264 
 
 4790} § 3355|3446'3536|3627/37 18/13 808/3899/3990/408014 17 1 
 Ql] . 4262/4352/4443/4534146241/47 1 5/4806|4896|498715077 
 92} 5§168/5250|5340|54401553 11/562 1157 12|5802|589315084 gO 
 93]  607416165|6256/63-461643 71165271661 8|6700|6799|68900 1| 9 
 04]  698017071\7161)7a5917345|743317524 7614170517796] [olan 
 95 78801797 7|8007|8 1 58}8248/18339/8430|8520/861 1|8701 4\36- 
 90} —8792/8882)8973|9063|91 5411924419335 0426/95 16|9607 9/45" 
 97} 9697|9788|9878]9969|0059/1015010240|033 1]042110512) {9/23 
 
~ LOGARITHMS N.48000 L. 681 
 
 a a ms Pe ay fs fe ff 
 
 4150|4249|4330|4420/4520114610}4700/47 90)4880/407 1 
 500115151)5241|5331|5422|155 12|5602|56092|57 83/5873 
 
 3173|3263|3353|3443|3533 [3623/37 13|3803|3803|3983 
 4073|4103/4253/4343|4433||4523/4013/4703/4793/4883 
 
 7673/77 63\7 853|7942|803 2118 122/82 12/8302|8392/8482 
 8572|8062|8752|/8842 
 
 A863/4953|5043/5132/52221153 1 2|5402/5492/558115671]| 
 5761585 115940|6030)6120)|62 10|6300/6389|647 9/6569 
 
 6659|6748/6838|6928|7018}|710717197|7287|737 7|7460 
 755617646)7 736\782.5|7915||8005|8005|818418274/8304 
 8454|8543/8633/8723/88 13]18902/8992|9082)917 1/9261 
 93511944.1|953019620/97 10/197 9919889|9979|0068/0158 
 4216850248/0338|0427|05 1710607 |10696|07 86/087 6|0965| 1055 
 43} = 114.5]1234)1324]1414]1503]}1593]1683]1772)1862)1952 
 2041213 1]2221/23 10/2400}|2490)2570|2060)|2759/2848 
 
 2038|302713 11713207|3296||3386|347613505|3655|3744 
 3834|30241401314103|4193]|4282/4372/44611455 1/4041 
 4730}4820!4909/4990|5080)|5 178]5268|5357|5447|/5537 
 5626|57 16|5805|5895|5984||607 4|6164|625316343/6432 
 6522/661 116701/6791|6880}|6970)7050)7 149|7238)7328 
 
 On he lars | ateie | 74 sa oD! 
 
N.48500 L.685 OF NUMBERS. (83) 
 213141516171 819 ND pro. 
 
 7596|7686)7776 |7805|7955|8044/8134/8223 
 
 51}  8313|8402/8492|8581|867 1 ||8760}8850)8939|9029)9118 89 
 52} 9208|929710387|9476 9506, 965519745|9834|992410013) | 1 9 
 53|6860103|0192|0282|037 1|0461|0550|0640|0729|08 1 90908 ans 
 
 7 5414843/4933)5022)5111/5201|5290/5380 9180 
 59] 5469|5558|5648|5737/582615916|6005|6095|6184/6273}) ~~ -—— 
 4860} 6363/6452165411663 110720)0809|0890|6988!707 8!7 167 
 
 61 72,501734617435|7 5241761417 70317 792!7882;797 1|8060 
 
 62]  8$150/8239|8328/841818507 }8596|8685|8775|8864|8053 
 
 66} 1721|1810]1900]1989/2078]2167|2257|2346|243 5/2524 
 67}  2613]2703|2792)/288 1|2970)|3060)3 149/323 8/332713416 
 68] 3506)3595/368413773|3863]|3052/404114130/42190/4309 
 69] 4398/448714576/4665/4755|14844|4933/5022)511115200 
 
 14870] 5200/5379|5468|555715646]5735|5825|59 1 4|0003/6092 
 71| 6181|6270|6360/64409|0538)|6627/67 16|6805|0895|0984 
 72} 707317162|725 1173407420117 5 18|7608|7697/77 86/7875 
 731  796418053|8142/823 1/832 118410184909|8588|867 7/8766 
 74{  8855|8944|9033|9123/92129301193901947919568/9057 
 75|  9746/9835|9924\0013/0103/10192/0281|0370|0459/0548 
 
 v7) 1528/1617|170611795|1884]]1973(2062|215 1/224012329 
 78]  2.418|2507|259612685|2774|2863/29521304113 1303219 
 79] — 3308|3397/3486)3575|3664|3753/3842/393 11402014 109]| gg 
 
 44880] 41098]4287/4376/4465|4554|14643/4732'4821\4910/4999 
 81} 5088/5177|5266/5355|5444|5533|5622'57 11|5800 5889 
 82] 5978'6067|6156|6245|633.4|16423/65 1 1|6600|6689,6778 
 83 6867|6956|7045|7 134|7 223173 12)7401|7490\7579'7068 
 84, 7757)\784.5!17934'8023|8112//8201|/8290'8379|8468/8557 
 85] 86468735|8823|8912|9001|}9090/9179/9268/9357|9446 
 86|  9535!962419712'980119890|19979\0008'01 571024610335 
 87 6890423 05 12|0601|/0690/07 79]10868|0957/104.5}1134}1223 
 88] 1312.1401]1490/1570|1667]}1756}1845]1934|2023/2112}) 
 89} 2200,2289/237812467|2556|12645|2733/2822)291 113000 
 
 4890} 3089'3177|326613355|3444/13533|3621/3710'3799|3888 
 gl 3077|4005|4154!4243}4332]1442 114509/4598/408714776 
 92} 4864/4053/5042/5131|5220]5308|5397/5486/5575|5663 
 93} 5752'584115930'6018/0107|019616285|6373|0462/6551 ol18 
 Q4| 6640}6728|6817|6906)69951\7083)7 1 72!726117350|7438 3/26 |; 
 95| 752717616|7704|7793|78821|797 118059|8148|8237|8325]| |4/°> |: 
 96} 841418503|8591|8680]876G]|8858/8946|9085|912419212 6153 |: 
 97} 930119390|9478/9567|9056}1974419833|9922|0010|0099 162 | 
 98|69001 88|0276|0365|0454|0542}1063 110720|/0808|0897|0986 8|70\1 
 
 —$— |§ —— | | ———___ | | ——_| | | ES SES —e 
 
LOGARITHMS N.49000 L. 690 
 
 1699}1788]187611965|2053]12141|2230!23 18|2407|2495 
 258412072)2760]2849/29371302613 114|3202/3291|3379 
 3468}3556/3644/3733|3 8211139 10|3998/4086)4175|4263 
 4352|4440]4528/4617|47051|4793'4882/4970|5058|5147 
 5235|5324154.12)5500'55891|}507715705|585415942|0030 
 6119|6207|6295|0384'6472||0560|6649|6737/0825|69 14 
 7002|7090)7 179|7207|7355||7444|7 53217 620\7 700\7 797 
 7885|7974|8062|8150'8238]'8327/84.15|8503|8592|8680 
 8768]/8857|8945|9033/9121||9210|9298|9386/9474|9563 
 9651|9739|9828/9916/0004 ||0092/01 8110269|0357|0445 
 
 1416]1504]1593|1681|1760||1857/1945|2034/212212210 
 2.298|2387|2475|2563/265 1||2739|2828/29 16|3004|3092 
 3180|3269/3357|3445|3533]|3621137 10|3798|3880|3974 
 
 4062|4151|4239/4327/4415]}4503/4591|4680/4768/4856 
 4944|503215120|5200|5297]15385|5473|5561|5049/5737 
 5826]591416002/6090)6178]/6266|635.4|6443|653 110619 
 6707|6795|0883|607 117059]|7 148|7236|7324|7412)7500 
 75881707 6\7 764!7853|7941 8029/8 1 17/8205|8293|/8381 
 8460/8557|8645!8733|8822||/8910|8998|9086/9174|9262 
 
 935019438|952619614'970211979019878|9967|0055|0143 
 
 —_——, | ———_____ ] ___ |__| ——___}] —______ J | —_____ | —_ -_—_—__ |- -———__ | | ff 
 
N.49500 L.694 
 
 2550 6946052|6140|6227 |63 15|6403|}649 116578|666616754|6842 
 
 51} 6929|7017}7105|7192|7280))7368]7456|7543|763 117719 87 
 52} 78006 1) 9 
 53 
 54 
 551695043 710524061 2/0700|07 87/10875|0962! 1050)1138}1225 
 56} 13813}1401/1488|1576|1663]|1751|1839|1926|2014|2102 si 
 57) 2180)2277|2364|2452/2540/12627127 15|2802|2890/2978 8l70 
 58}  3065|3153/3240/332813416]13503|359113678|3766)3854 alg 
 59 3941|4020/41 16/4204142091|\4370914407/4554|4642/4729 3 Daa 
 4960}  4817/4904/4992/5070]5 1671|5255)5342'5430)5517|5605 
 61} 569215780|5867|5955|6042)|613016217|6305|6393|6480 
 62}  6568}0055|6743'6830)6918}|/7005|7093!7 180|7268|7355 
 63 7443|7530!7618|7705|7793]|7 880|7 968'8055|8143/8230 
 64|  8318|8405|8493|8580/8668)|/8755|8843/8Q30|9018/9105 
 65| 9193/9280\9367/9455}9542119630 9717\9805 9892\9980 
 66|6960067}0155|0242/0330|04.17/10504}0592'0679|0767|0854 
 67} 00942)1029)1116)1204)1291}113709)1466)1554]1641|1728 
 68 1816}1903}1991/2078/2166]12253|2340|2428/2515|2603}| 
 69} 26090)2777|2805|2052|3040)/3 127/32 14/3302|3389|3477 
 4970}  3564!365113739|3826|3913]/4001/4088)4.176|4263/4350 
 71) 4438/4525/4612)4700)4787|/487414962!5049)5137|5224 
 721 5311153909|5486|5573/5661115748|5835|5923|6010/6007 
 73} 6185|6272/6359|/6447/6534||6621|6709'6796|6883|6970 
 74, 7058)7145|7232\7320|74071|749417582'7660|7756|7844 
 75} .7931|8018]8105|8193|8280}18367|8455'8.54.2|8620|87 16 
 76] 8804/8891|8978|9066/91 53/}924019327|9415|9502/9589 
 77|_ 967 
 78|6970549|0636 
 7Q\ °1421]1508)1596]1683]1770)|1857|1945|2032)2119|2206 
 AQ80| 22093/2381/2468)/2555|2642)|2729|2817|2904/2991/3078 
 81] 3165|3253'3340/3427|35 14|360113680|3776/3803|3950 
 82] 4037/4124/4212/4290|4386)/4473|4560/4647/4735/4822 
 83 4909/4.996|5083|5170)5257||5345|543 2155 19|56060|5693 . 
 84} 5780)}5867|5955|6042/6120)|62 16|/6303|6390\6477|6505 j 
 85| 6652|6739|\6826|6913}7000)\7087)7 174|7261|7349|7436 
 86} 75231761017697177841787 1117958|804.518132/8220/8307 
 87|  8394|848118568|8055|87421'882918916|9003|9090/9177 
 88} 9264/9352/9439195261G013}19700/9787198741996 110048 
 89|698013 510222/0309|0396|0483||057010657/07441083 110918 
 4990} 1005}1002/1180}1267}1354))1441}152811615]1702|1780]| 87 
 gl 1876]1963/2050)2137/22241|23 1 1]2398/2485|257212659 
 92} 2746}2833/2920/3007|3004113.18 113268/335513442)3529 86 
 93| 3616|3703/3790/3877|3964||405 114138}4224]4311/4398 7S 
 Q4| 4485/4572/465914746/4833||4920|5007|5094|5181/5268 | 214 
 05} 5355|5442)552015616|5703/1579015877|5964|6050/6137 4] 34 
 96|  6224/6311/639816485|6572)|16650|6746'6833|6920|7007 B43 
 97}  7093|7180\7267|735417441\|7528|7615|7702|7789|7876] — |“ 
 7963|8040|8136|8223|83 10]|8397|/8484|857 118058|8744 3169 
 
 ef 
 
 OF NUMBERS. 
 
 ST ee eee ee 
 
 883 1]8918|9005|9092/9179]9266|9353/9439|9526)9613 
 
LOGARITHMS N.50000 LL. 698 
 
 —_—_—_ |———————|[(——“——_— [orl 4 |e hE I I 
 
 0977|1064|1150)1237/1324111410|1497]1583| 167011757 
 1843]1930|2017|2103|2190||2276|2363|2450|2536|2623 
 
 27009|2796|2883|2969|3056]'3 142'3229|33 16|3402/348q 
 3575|3662|3748)383 5|3922||4008|4095/41 8 1/4268|4354 
 4441|4528|461414701/47871|487414960|504715 133|5220 
 5307|5393|5480)5566|5653||5730|5826|59 121599910085 
 6172|6258|0345|6432|6518)|6605|6691|0778)6864\695 1 
 703717 124|7210\7297|73831|7470|7556|7043|7720|7816 
 7902\7989|807 5|8162|8248)|833.5|8421|8508/8594|8681 
 8767|8854|8940|9027/9 1 13}19 199|928619037 2|9459|9545 
 9632/97 18|9805|9891\9978]]0064/01 5 1|0237|0323/0410 
 
 1361}1447|1534|1620|1706)|1793|1879}1966)2052/2138 
 2225|23 11|2398]2484)2570|12057|2743/2830/29 1 6/3002 
 30803 175|3262)3348|34341|352 113607|3694/3780|3866 
 3953|4030/4 125|4212/4298)}|4385|447 114557|4644/4730 
 4816/4903|4980)5075/5 162)||5248/5334|5421|5507|5594 
 5680}5766|5853|5939/6025||61 12/6198|6284|637 116457 
 654316620167 1616802\688s}|6975|70611714717234|7320 
 740617493|7579\76605|7752)|7838|7924|8010!8097|8183 
 8269|8356|8442)8528|86014//8701|8787|8873|8960\9046 
 9132/9218]9305|939119477||9563:96050|9730|9822|9908 
 99951008 1|0167|025410340 (eae 05 1210598|0085|077 1 
 
N.50500 L.703 OF NUMBERS. 
 
 — | J] ff —  —  — ] 
 
 8071 ioe $2491530818414|s500l8ss0l8672187581684 
 
 5704|5880/5966/6052 ae 
 6652 ks 6823 0909 6005 
 seus 
 
 6293 6300|6395|6480|6566 
 7080)7 166)7252/7338|7423 
 ae 8023/8109 8195 8280 
 
(88) LOGARITHMS N.51000 L.707 
 
 Coed SEE PEST Viren | pamemall =yameed been po REE le<oeee— see 
 pe fp 
 
 06}7080808|0893 097 8] 1063]1148 1939 1318/1403}1488}1574 
 2084. zL09 2254|2339|2424 
 
 18] 1003/1088|1173/1257]1342 he 1512|1597|1682|1766 
 19} 1851]1936/2021/2106|2191||2275|2360/2445/2530|2615 
 
 5120]  2700|2784|2869|2954/3030)/3 12413 200/3293|3378|3463 
 21 3548/3633'37 1713802|3887|13072|4057|414114226/4311 
 22) 43096/4481/4565/4650)4735]4820/4904\4980|5074|5159 
 
 - 23) §244/532815413|5498/5583)|5667|5752|5837|5922|6006 
 24}  6091/6176|6261|6345|6430)|65 15|6600|668416760/0854 
 25| 6939|7023]7108|7193|7278)|7362|7447)|7532\7617)7701 
 
 26] ~=—- 77861787 117955'8040|8125||82 10/8294/8370|846418548 
 27 8633|87 18/8803 |8887|8972|9057|9141|9226)93 1119395 
 
 9480|9565/9050/9734198 19]19904/9988 0073/0158|0242 
 
N.51500 L.711 ‘OF NUMBERS. 
 
 o_o eee a a he 
 
 77\7140782,0866|0949|1033}1117 
 
 2459|2543|2627127 1 1|2795)/2878|2962|3046/3 13013214 
 
LOGARITHMS N.52000 L.716 
 
 PEPE ELE, ERE VE PIS tasty) iene SRR prs Fine mune ek, ¢ | powered & Lineal.) Venn © sae 
 
 0869|0952|1036]1119}12031]128611370|1453]1537|1620 
 1703|1787|1870|1954}20371/2121/2204122881237 1/2455 
 
 4970|5053/5136|5210|5302)|53 84|5467|5550|5633|571 
 5799|588 1|5964|604 7161 30}|6213/6296)6378)646 110544 
 
 ee fy 5 | -- J | — | ——— ——_} —— | —-————_ | — 
 
N.52500 L.720 
 
 OF NUMBERS. 
 525017201593|1070|175811841119241/2007|2089/2172)/2255/2337), | 
 2420}2503|2586|2668]27 511283429 1 6|2990/3082|3 164 82 
 3247|3330/3413|349513578||366113743|3826|3909/39091 1) 8 
 40741415714239/432214405|1448714570/4653147354818] | 2/16 
 4901/4983|5066)5 149|523 11153 14/5397 |547915562)/5045 4133 
 572715810|5892)5975|0058]16140|6223|6306|6388/6471 5/4 
 6554|6636)67 19|6801|6884]|6967|704917 132|7215|7297]| | ®/#9 | 
 7380|740217545|7 028/77 10)|7793)7 87517958804 18123 3le6 
 8206|8288|837 1|8454|8536)|8619|8701 |8784/8867|8949 ol74. | 
 9032191 1419197|9279|9362||9445|9527|9610\9692\9775 1pm 
 985719940|0023/0105'0188)(0270/03.53|043.5|05 18/0600 
 
 1508]1591|1674|1756)1839)1921/2004|2086|2 169/225 1 
 2334|241 6'2499/2581|2664||2746|2829/291 1/2994|3076 
 3159|324.1'332413406'3489]13 57 1|3654|3736|3819/3901 
 
 3984 4066/4149 423 1143 14114396/4470/4561|4644|4726 
 4800/1489 1'4973|5056)5138}|522 115303153 861546815551 
 5633157 16,5798 588115963||6045|6 128/62 10|6293'6375 
 6458/0540 6623|6705|6787)|6870/095217035|7117|7200 
 7282 73647447 7529\7012)\769417777|7859|7941|8024 
 8106|8189/827 118353|8436)185 18|8601|8683/8765|8848 
 8930/9013|9095/9177|9260)19342\942419507|9589\9072 
 9754 9836|9919|000100841101 66102481033 11041310495 
 
 1401/1484]1566|1648|173111181311895|1978|2060'2142 
 2225|2307/2380|247212554]12636/27 19/2801 2esa|2g00 
 
 551773717820 
 7984|8066|8 148]823 1/83 13}|8395|8477185 5918642 
 8806|8888|897 119053|9135)|9217192991938219464 
 9628197 10197921987 51095 7|10039|012 110203|0286 
 
 7902 
 8724 
 9546 
 0368 
 
 ——___ ] — = |] | ——————__ ] —_——_—__ | ————__— 
 
(92) LOGARITHMS N.53000 L.724 
 Name S|, 2) SU eS | 
 
 — |) —_—_—_—_— | —_—_- |__| ———_ |__| |_| MI 
 
 5310}  00945}1027}1109)1191]1272/11354}1436]}1518]1599]1681 
 11 1763]1845|1927|/2008]2090)|21 7212254|233.5|241 7/2499 
 12]  2581/2662|2744/2826/2908]|2980|307 1/3 153|3235/3316 
 13}  33098|3480!3562)3643/3725||3807|3889|3970)4052/4134 
 14] 4216/4207|4370/4401|4542|\462414706)|4788|4860/405 1 
 
 15} 5033)/511415190|5278/5360/5441|5523|5005|5686|5768 
 16} 5850)5931|/6013/6095/6176)}6258|03401042210503/0585 
 17| 6667|6748/6830/691216993)|707 5|7 157|7238|7320)7402 
 18} 748317565/7647'7728/7810)7892!7973|8055|8137/8218 
 19] 8300/8382)8463/8545/8626]/8708|8790|887 1|8953|9035 
 5320|  9116190198/928019361|9443119524|9606/9688|97 60/985 1 
 21) . 993310014/0096|0177|0259||034.110422/0504|0585|0667 
 
 23|  1565|1646|1728/1809]1891]]1973|2054/2136|2217/2299 
 24| 2380/2462/25441262512707|12788|2870|2951/3033/3115 
 
 25}  31096|3278|33509|344.1/3522)/|3604|3685|3767|3849|3930 
 26| 4012/4003/4175)4256/4338)|44.10|450114582|4664/4745 
 27|  4827|4908/4990'507 215 153)1523515316|5398|5470|5501 
 28| 564215724!5805|5887|5968]|6050)613 1/6213|6294|6376 
 29} 6457/0530\6020/6702'07 83|\0865|6946|7028|7 100|7191 
 5330| = 727217354|7435|7517|7598||767017761|7842|7924|8005 
 31} 8087|8168/8250|833 1|84.13]849418576|8657/87309|8820 
 32} 890118983/9064\914619227||9300|9390/9472/955319634 
 33] — 9716|979719879|9960|0042)10123|0204'0286|0367|0449 
 
 35} — 1344}1426)1507)1588}1670]1751]1833]1914]1995|2077 
 36} 2158)2240/2321)/2402/2484/12565|2647|272812800|2891 
 37}  2972|/3053/3135/3216|3298/3379|3460|3542|3623|3704 
 38] 3786|3807|3948/4030/4111)4192/427414355|4437|4518 
 39} 4590)4681|4762/4843}4925|5006|5087|5 169|5250)5331 
 
 5340| 54.13]5494|5575|5057|5738]15819|5901/5982'6063|6144 
 41} 6220/0307|0388/0470/65511|6632/67 1416795|6876|6958 
 42} 7039)7120|7201|7283|7364|7445|752717608|7680|77 70 
 43} 7852\7033|8014|8096]8177118258|/8339|842 1|8502/8583 
 44] 8604/8746)8827/8G08|8Q90]|907 1191 52|9233/90315|90396 
 45)  9477|9558|/9640'9721|9802||9883|9965|0046/01 27\0208 
 46|7280290/037 1104520533 |0614|}0696|0777 |0858/0930}102 1 
 47}  1102/1183/1204)1346]14271|11508]1580/1670)1752]1833 ad 
 48} 1914/1995|2076/2158/2239]2320/2401/2482/2504|2645 
 
 | 8l65 
 49| _ 2726/2807|2888|2970/305 1/13 132/3213|3294/3375|3457 ol 
 
 No ;rf2}3}4fofe]7! 38 fo [pps 
 
 81 
 1] 8 
 2116 
 3/24 
 4/32 
 {5/41 
 6/49 
 
N.53500 L.728 “OF NUMBERS. ee ee 
 
 DEMONS ere So (oe Po Oat 8) Sr DiIPro 
 5350 7283538 3619|370013781|3863|13944/4025|4106/4187/4208], | 
 51 43501443 1145 12/4593}46741/4755148360/4918)49909/5080 81 
 
 52 5161}5242/532315404|5486 5667 5648/5720|581.0/5891 : . 
 
 59 0838/0919]1000]108 1} 1 162]|1243/1324]1405]1486)1567 
 
 5360} — 1648]1720]1810}1891]1972}/2053|2134)/2215/2296|2377 
 61] 2458/2539/2620|2701/27 82/2863 |2944|3025]3 100/3 187|| 81 
 62} 3268/3349/3430)351113592||3673/3754|3835|3916)3997 
 63| 4078)4159/4240/4321|4402)/'4483/4564/464514726/4807 
 64]. 4888/4969'5050/513 1/5212/|52902/5373|5454/5535|5016 
 
 65| 5697|5778|5859|5940|602 1} 6102\6183/6264|6345|6426 
 66| 6507/6588/6669/6749|6830}'691 1|6992/707317 1154/7235 
 67| 73161739717478|7559|7640]'7721|7801|7882|7963|8044 
 68} 8125/8206/8287|8368|8440|'8530/8610/8691/8772)8853 
 69} 893419015|9096/9177/9258]'9338 '9419/9500)958 1/9662 
 5370} 9743|9824'9905|9985 0066]'0147/0228/0300|0390/047 1 
 7 117300552|0632)07 131079410875]10956)1037/1118}1198|1279 
 721 = 1360/144.1|1522}1603|1683}|1764)184.5|1926|2007/2088 
 73| 216812240/2330)241 1/249212573|2053|2734)2815|2806 
 74| . 2977|3057|3138/3219|3300]3381'3461|3542/3623|3704 
 
 75| ° 3785|3865|3946|4027|4108]/41894269|4350/443 1/4512 
 76| — 459314673/475414835|4916]|4997|507715 15815230|5320 
 77|  5400|548115562|564315723]|5804|5885|59606|6046|6127 
 78| 6208/6289|6369|64.50|653 1||16612|6692'677316854|6935 
 79| = 7015|7006)7 17717258]7338]|7419|7500,758117661|7742 
 5380] 782317903/7984|8065|8146]|8226/8307|8388]8468|8549 
 81] 8630)8711|879118872)8953]|9033/91 14\9195|927619356 
 82] - 943719518]9598|967919760]/9840/992 1|0002/0082'0163 
 83173 10244|032410405|0486/0567||0647|0728 O809|0880|0970 
 84, =: 1051]1131]1212}1292|1373))1454 153411615 1696|1776 
 
 85 1857|1938/2018]2099|21 80}}2260/2341|2422|2502/2583 
 86} — 2063/274412825|2905|2986]1306713 147|322813309|3389 
 87| 3470/3550/363 1137 12/379211387313953/4034/41 15/4195 
 88] 4276/4356|4437|451814598}|4679/4759/4840)492 1/5001 
 89} 5082)5162/5243/532415404115485|5565|5646|5727|5807 
 5390} 5888]5968/0049/612916210]}6291/637 1/6452/6532/6613 
 
 92} 7499/7579|7660)7740|78211|7902|7982|8063|8 143|8224 80 
 93|  8304|8385|8465|8546|8626]/870718787|8868|8948|9029 Bo 
 94} 91091919019270\9351/9431119512|959219673|9753|9834|| | 215° 
 95! 9914|9995|0075101 5610236103 171039710478|0558|0639 4/32 
 96|73207 1 g|0800/0880/0961 | 1041 }]1 122}1202/1283}1363]1444 5/40 
 97|  1524|1605|1685]1760|1846]|1927|2007|2087|216s|2248| | 8 
 Q8| = 232G]2400|24.90/2570)265 11273 1|2812|289212972/3053 ales 
 99] __3133/3214)3294)3375)3455]3535|3616)3696|3777|3857 9172 
 
 9 D/Pes4 
 
4 
 
 LOGARITHMS N.54000 L.732 
 
 1170/125011330)1411]1491/11571/165211732|1812|1892 
 1973/2053|2 133/22 13]2204}12374|2454/2535|2615|26905 
 277 5|2856|29361301 6}3096]/3 17713257|3337|341 713498 
 3578/3658/3738|38 1 9|3899}|3979|4059|4140/4220/4300 
 4380/4461145411462114701]1478114862)4942/5022'5 102 
 5183}5263|5343)5423|5503||5584|5004|5744|5824|5904 
 5985|0065|6145|622516305||63 8610466/6546|062 6/6706 
 6787|6867|6947|7027|7 107]|7187|7208'7348|7428|7508 
 7588/76691774017 820|7 90G1|7.98g}800G)8 1 50}8230/83 10 
 8390|8470|8550'8030)87 1 1||879 11887 1|895 11903 119111 
 919219272/9352/9432195 1 2/19592|96072/9752'9833|9913 
 9993/0073/0153'0233|03 13}(03931047410554|0034107 14 
 
 713037|3117 
 
 9598|9078|97 58|9837|9917|19997|007 7\0157 0237 317 
 
 1196]1276}1356|1436|1516]}1596|1676|1756|1836}1916 
 
 19Q5|207 5|2155)2235}23 15112395|2475|2555|2035/2715 
 _ 2794|2874|2954|303413 114413 194|327413354|3434|3513 
 3593|3673]37 53|3833/3913|13993/4073|4152)423 2/4312 
 43924472145 52\4632/47 11114791/487 1/495 1/503 1/5111 
 5191|52701535015430155 10}15590|5670!5749|5829|5g09 
 5989|6069|6140|6228}6308}|63 88|6468|6548|6628|6707 
 6787|6867|6947|702717 107117 186|7266|7346'7426|7506 
 7585|706517 74517 825|7 9O5|17 G84|8064/8 1441822418304 
 8383]|8403|8543|8623187 021/87 82/8862/8942|9022/9101 
 Q181|926119341|9420|95001195 80}9660|97 401981 919899 
 
 ee ee Oe | Se J | J ——— | | 
 
N.54500 L.736. OF NUMBERS. 
 
Deer lok ab OS 
 
 HE Mrs, Hone wa rwpeay ommend baa sometad Fg tmmnen [os wires fire wath they Smee A Re OR RRR BRANES Lory ace 
 
 5510| —1516}1595}1674|1752/18311|1910|1980|2068/2146|2225 
 11| 2304|2383/2462/2541/2619|12698/2777|2856|293513013 
 12| 309213171|32501332813407113486/3565|364413722/3801 
 13] 3880/3959/4037/4116)4195|4274/4353|443 11451014589 
 14 466814746|4825 4904 4983}|5061|514015219|5298|5376 
 
 15] 5455|5534|5613,5091|5770)|5849|5928|6006|6085|6164 
 16} 6243/6321 |6400 6479|0557||\6636)67 15|0794|0872/6951 
 17| 7030)7100'7187\72660|7345)|742317502)7581|7660\7738 
 18| 7817|7896)7974|8053|8132||\8210/8280|8368|844 7/8525 
 19] 8604|8683|8761|/8840|8919]18997/9076|9155/9233|93 12 
 5520| 9391104609'9548 :9627|9705]|9784/9863 
 2117420177|0256|0335 0413|0492/|057 110640|0728|/0807|0885 
 22|  0964/1043}1121|1200|12709)]1357/143611515|1593|1672 
 23) 1750|1829]1908!1986|2065||2144'2222/2301|2370/2458 
 24|  2537|2615|2694,2773/285112930/3008|3087/3 166|3244 
 
 25|  3323/3401134803550|3637/'3716|3794|3873|3052|4030 
 26| 4109/4187 fod dan 442314502/4580/4059/473714816 
 27|  4895|4973|5052'5130|5209}5288|5366|5445|5523|5602 
 28] 5680)5759|5837|5916|5995|16073/615216230|6300|6387 
 20]  6466/6544\6623/670210780/}68509|6937|7016)7094/7173 
 5530| 7251)7330)7408|7487|7 505]7644)7 722|780117880\7058 
 31] 8037/8115/8194'8272]835111842918508|8586|8665|8743 
 32}  8822/8900;8979 9057|9136)921419293/937 1/9450,9528 
 33] 9007/9685|0764'9842/9921 {94 Q9|0078|0156|0235/0313 
 34(7430392/0470,0540'0027|0705|07 84|0862|094111019|1098 
 35} =: 1176)1255]1333)1412}1490}|1569|1647|1725|1804|1882 
 36] | 1961/2039/2118/2196/2275]/2353|243 1|2510|2588/2667 
 37| — 2745|2824/2902'2981|3059113137|3216/320413373/3451 
 38]  353013608/3086'37 651384313922|4000!4078]4157/4235 
 39]  4314/4392144701454914627|14706/478448021494115019 
 5540| 5098)5176|5254|5333}541 1115490|5568|5646|5725|5803 
 41 §882}5960|603 8/61 17|0195||6273/63.52!6430/6508|0587 
 42} 6065|0744|6822|0900/697 911705717 135|721417292|7370 
 A3 7449|7527|7605|7 68417 7 621'7 84.1|7910|7997|8076|8154 
 44]  $232/8311|83809|8467|8546]18624|87021878 1|88509|8937 
 45] 9016\9094|9172/9250/9329119407 |9485|9564|9642/9720 
 46] —9799|9877\9955 0034 112/01 go 0268/0347 1042510503 )). 
 
 ee eed | keene’! ieee ee ened 
 
OF NUMBERS. 
 
 N.55500 L.744 
 
 —_— SE 
 OE) a ne in_s-«_~«— | —— ——_ | —_— 
 
 0854/0932 
 163.1}1709 
 
 85|7470232103 10103 87|0465|0543}/062 11069810776 
 86}  1000]1087|116511243]1320)11398)1476]1554 
 
 88}  2564/264212719|2797|2875}}2953|303013 108/3 
 80] 3341|3419|3497|3574|3652)13730'3807|3885|3903|4040 
 5590| 4118/4196)4273/4351|4420114507/4584|4662/4740 
 gl 4805|4973|5050)5 128)}5206/15283/536 115430155 16|5594 
 
N.56500 Dre OF NUMBERS. 
 
 PASE APR ITEN | kMer I PEE DRE, ha ATE 
 
 2,4758/4835}491 1/4988 
 5141|5218}5294|5371 iat a 5601|5677|5754 
 
 583115907|5984|6060|6137||6214'6290|63671644316520 
 
 [| J | S| — —  - | I OO 
 7 me cf ef ee | 
 
LOGARITHMS N.57000 L.755 
 
N 
 
 N.57500 L.759 OF NUMBERS. 
 , O BZ i\-3) 4 5 
 575017590078|6754|0830}0905|0981\\7056/7 132|7207|7283|7358 
 
 - 4016|40091|4166)4241|43 16)14392/4467/45421461 7/4603 
 
 — 6035|6111/6186|6261|6336)64.1 1/6486|6501|6636)0711 
 
 ——_—_—- 
 
 1208]1283]1359]1434]15101585]1661]1736|1811|1887 
 1962|2038]2113}21809]2264]2339)24 1 5|2490|2566)2641 
 27 17|2792|2867|2943/3018)13094/3 169]3245|3320/3395 
 347 11354613622|/3697|3772/|38481392313999/4074/4149 
 4225|4300)4376|445 1|4526/4602,4677|47 531482814903 
 4979|5054|5 130|5205|5280]5356'543 1|5506|5582/5657 
 5733|5808|5883|5959|00341\6109/6185|6260\6336/64.10 
 6486|6562|6637|67 12|6788||}6863 |6938/7014|7089/7 164 
 7240\7315|7390|7466|7541 AarOy0o2 7767\7842|7918 
 7993|8068}8 144/82 19|8204/|83 70'8445|8520)8596/867 1 
 8746|8822/8897|897 2|9048119 123|9198|92741034910424 
 9500/9575|96501972.5|9801|19876 995 1/0027|0102/0177 
 
 1005}1081]1156]1231|1307||1382}1457|1532|1608|1683 
 1758|1833]1909|1984|2059]/2134|2210|2285|2360|2435 
 2511\2586)2661/2737|28 1 2||2887|2962/3037|3113|3188 
 3263|3338|/3414|3489/3564113 639/37 15|3790/3865|3940 
 
 4768|4843/4918|}4993|5069)|5 144|5219|5294|5369|5445 
 
 §520|5595|5670|5745|582 1||5896/597 116046|6121|6197 
 6272|6347|642216497|6573||0648|672316798|6873|0948 
 70247 090|7 17417 24.9|7324117400|747 5|7550|7625!7700 
 77'75\785117926|8001 |8076)|8 15 1|8226|830118377|8452 
 8527|8002|867 7|8752|8828!|8903|897 819053 |9128|9203 
 
 92781/9354|942919050419579]19054|9720|9804/9879|0055 
 
 0781|0856|093 1;1006}1081)|1156])1232)1307|1382]1457 
 153211607|1682)1757|1832)/1907|1982!2058/2133|2208 
 2283|2358|2433|2508|2583)|2058|2733|2808|2883/2959 
 
 30343 100|3 184|3250|3334)|3400|3484|3559|3634|3709 
 3784|3850|3934|4000/4085|/4 160/423 5/43 10/4385|4460 
 4535|4610|4685|4760)483 5|4910/4985|506015135}5210 
 5285153601543 5/55 10)5585||506015735|5810/5885|5Q060}| - 
 
 6786|0861|0936|701 1170861|7 16117236|7311|7386|7461 
 75361701 1\7686)776117836)|791 117986|8061|8136/8211 
 8286/8361|8435|85 10|/8585||8660|8735|88 10/8885|8960 
 903.5|9110/9185|9260|933 5||0410|94 85|9560/963.5|97 10 
 9785|9860)9935|0010)0085}\0160}0235}03 10103850459 
 
— J J J —  —  — sJXq ——— ——— } ———_ Gobet, gs ie Ree pg? 
 
 (102) LOGARITHMS N. 58000 
 
 1014}1080}1 163]1238]1313)|1388) 1462)1537/161211687 
 
 1761}1836}1911]1986}2060}|2 135/22 10)2285|2350|2434 
 2500|2583/2658}2733|2808]|2882/2957|303 21310713181 
 3256/1333 113406|3480}3555]|36303704137 791385413929 
 40031407 8/41 53}4227/4302)14377/445 114526|460114676 
 4750|4825|4900/4974|5049)/5 124'5198|5273/534815423 
 54971557 2|5647157211579611587 1'5945}6020|/6095161 69 
 6244|63 19|6393|6468|6543)|6617|6692|6767|684.116916 
 6991|7065|7 140}7215|7280]17364|743017513|7588|7663 
 
 18} 7737|7812|7886|7961/8036]|8110.8185|8260/833.4|8409 
 19} 8484)8558/8633|8707|8782)|/8857!893 1|9006/908 119155 
 9230)9304/9370|9454|9528 9003/9678 975219827|9901 
 
 9976)005 110125|0200|0274/|0349,0424|0498/05 73|0647 
 1468/1542/1617|1692}1766)|1841/1915|1990/2065|2139 
 2214/2288/2363]243 7/25 12/|2586/2061|2736/281012885 
 
 2.959|3034/3 108]3 183|32.58]|3332'3407/348 1 135563630 
 3705|3779|385413928/4003]|4078!4152/4227/4301|4376 
 4450)4525|4599/467414748|14823|4897/4972|5046/5121 
 5195|52701534415419|54031|556815643/57 171579215866 
 59411601 5|6090/6164/6239)|63 13/6388|6462|653 7/6611 
 
 6686|6760/6835|6909|69841|7058}7 13217207|728 117356 
 
 31] 7430/7505|7570|765417 7 28)17803|7877|7952|8026|8101 
 32} 8175|825018324/8399|8473)1854 718622|8696|877 1/8845 
 33]  8920)8994/9069/9143/9218)19292/9366)944.11951519590 
 34| 96641973919813|9888/9962)1003 6/01 1 110185}0260|0334 
 
 L. 763} 
 
N.58500 L.767 OF NUMBERS. 
 
 |] |] + | ————_- 5 —-———. 
 
 1199)1273]1347|1421 14g 1569|1643 es 1791]1866 
 1940)2014/2088}2 162)}2236]/2310|2384/2458]2532|2606 
 
 65}  2680)2754/2828/2902)/2076||3050)3 1124/3 198|3273/3347 
 66}  342113495|35009|3643)37 17113791 |3805|3939/4013/4087 
 67| 4161)4235|4309)4383)4457)145314005|/4679|47 53/4827 
 68} 4901/4975|5049/512315197)1527115345|5419|5493|5567 
 60) 864115715|5780|58631593711601 1|6085|61 591623316307 
 
 5870|  638116455/6520|6603|667711675 116825|689916973|7047 
 71 712117195|7200\7343|7417||\749117565|7630|77 1317787 
 72\  7860)7934|8008/8082/8 15 1/8230\8304'/8378|8452|8520 
 
 7A 9330\9413|9487|9561/9635 970907 83|9857 9931 0005 
 75\7690070|0153|022710300,03741|0448'052210596|0670,0744 
 76} 0818/0892'0966/1040)1114/}1187/1261}1335]1409)1483 
 77 15571163 1|1705|1779/1852}|1926|2000/2074/2148/2222 
 78} 2206/2370)244412517/2591||2065 2739 9|2813}2887/2061 
 79|  3035/3108)3182/3256)3330)3404 3478/3552/3626)3099 
 
 5880} 377313847/392113995|4069 4143) 216'4290}4364)/4438 
 81] . 4512)4586)4650/4733/4807 4881/4955 5020|5103'5176 
 82} 5250/5324|539815472/55460/1561 9008 5767|584115915 
 83] 5988/6062|6136|6210|6284116358 6431 6505|0579,6653 
 84| 6727|6800)6874/6948|7022)\7096; 7169! 7243|7317|7391 
 85| 7465!7538|7612|7686)7760 Paaaloor| 7981|8055|8129 
 86}  8203/8276)8350|8424|8498]/857 118645187 19|8793|8867 
 87} 80940)0014/9088)9162/9235/19309/90383'9457|0530'0604 
 88} 967819752|982619899/9973]|0047 |012110194/0268/034.2, 
 
 5890} —:1153}1227|1300)1374)1448}11 522/1595|1669]1743)1817 
 Ql} 1890]1964/2038/2111]2185|/2259/2333/2406|2480|2554 
 92} 262712701|2775|2849|20221/299613070)3 143/3217/3291 73 
 03| 3364/343813512|/3585|36501373313807/3880|3954/4028 1 7 
 94) 4101/4175|4249/432214396/447014543/4617/469114764 alse 
 
 95]  4838]4912/4985|5059|5 133]|5206|5280|53541542715501 4129 
 06} 5575|5648)5722|579615869]|5943|601 7|/6090/616416238 5|37 1 
 97| 631116385|6459|6532|6606||667 9/67 53|6827|6900|6974. 5) 
 © Q8} 7048/712117195|726917342)174.1017489|756317637/77 10 alsa f 
 99] __778417858/793 1|8005|807 8}}8152/8226}8299|8373|8447 166 
 
 18: Tow, D | Pts. 
 
LOGARITHMS N.59000 L. 710 
 
 ce RM aS A 0 AE I RSA 
 5900|7708520 85948067 8741 8815) 888818962/9035/9109|9183 
 01] —9256}9330,9403/9477}95519624|9098/977 119845 |9918 74. 
 02} 9992 0066101 39|02 13|028610360!043410507/0581 |0054 1} 7 
 03|77 10728|080110875|0949}1022/1096]1 169}1243]1316]1390 aoe 
 O04} 1463]1537|1611)1684]1758]183 1)190511978|205212125 4130 
 05| 2199|2273|2346|24'20|2493112567|2640|27 14|2787|2861 5/37 
 06] — 2934/3008|3081|3 155/3229]/3302|3376)344913523|3596) | O14 
 07} 3670)3743/3817|3890|3964/14037/41 11/4184)4258/4331 gi59 
 O8| 4405|4478!4552/4625|46901147 72/4846/49 19/4993 15000 9]67 
 09} 5140)5213]5287|5360|5434)|5507|558115054|5728|5801 a 
 5910| 5875|594816022|6095]6169]}6242163 16|6380|6463|6536 
 11] 6610|6683/6757|6830|6903)}6977|7050)7124|719717271 
 12} 7344'7418/749117565|7638)|77 12|7785|7858|7932|8005 
 13] 8079|8152'|8226/8290|8373118446|85 1 9|8503|8066|8740 
 14| 8813/8887|8960|903419107]/918019254193271940119474 
 15} 95471962119694/9768|984.111991519988|0061}0135|0208 
 1617720282|0355|0428/0502/0575]]064910722/0795|0869 0942 
 17{ 1016}1089}1162}1236}1309]}1383]1456}1520|1603|1676 
 18] 1750/1823}1896)1970|2043}!21 17/2190)/2263/233712410 
 19] 2483/25571/2030|2704|2777|/2850|2924|2997/3070)3 144 
 5920| 3217/32900)3364/3437|351013584|3657/373 1|3804|3877 
 211 - 3951\4024)4007/417 114244]14317|4301/4464)4537/4611 
 22} 4684/4757/\483 1/4904|497 711505 115124/5 197/527 115344 
 23} 5417|5491|5564)563 7/57 11115784|5857|593 116004 |6077 
 24- 6150/0224/6297|6370|04441165 17|6590|/6004|6737 |6810 
 25| 68841695717030)7 103|71771'725017323|73907|7470|7543 
 - 26) — 7616\7690\7763\7836|7910117983|8056|8 1290/8203 8276 
 271  83409|8423|8496|8560|8642)/87 16/8780|8862|8935|9009 
 28]  90821915519228|9302/93751\9448|952119505|9668|9741 
 29} 9815/9888/9961|003410107]10181|025410327|0400/0474 
 5930|7730547 |0620|069310767 |0840]/09 13|0986}1060}1133|1206 
 31 1270|1352)1426)1499|1572||1645}17109]1792)1865|1938 
 _ 32) 2011/2085|2158)223 1/2304]123771245 112524|2597|2670 
 33} 2743)/2817)/2890|29063|303 6/3 100|/3 183|3256/33290|3402 
 34]  3475)3540|3022|3695|3768]/384.113915|3988]4001 4134 
 35} 4207|4280/4354/442714500]4573/4646147 190|4793 |4866 
 30] 4939)5012/5085/5158/5232/15305|5378|545 11552415597 
 37; . 5070|/5744|5817|5890|5963]}6036/6109/6183/6256/6329 
 38} 6402/6475|0548|6621|6094116768/684.1 1691416987 |7060 
 30]  7133}720617280!7353|7426||7490|7572|7645|771817701 
 5940}  786417938/801118084/8157|18230|8303|8376|84409/8522 
 41| — 8596/8669|8742|8815|8888]/8961/9034/9107/9180/9253 
 42] © §326/94001947319546/0619||9602|9765|9838)/991 1 |9g984 73 
 43|7740057 |0130|0203|0277|03 50||0423|0496|0560|0042/07 15 1) 7 
 44] 0788/0861|0934]1007|1080}|1153/1226]1299| 1372/1446 Ele 
 45}  1519]1592|1665}1738)1811]}1884]1957|2030}2103|2176 4129 
 AO} = 2240)2322/2395|2468|2541|12614|2687|2760)2833|2906 9/37 
 47}  2979|3052|3125/319813271113345|3418/3491|3564|3637 1 
 48} 3710/3783|3856|3920|4002||4075|4148/422 11429414367 alss 
 40} 4440)4513}4580/4650/4732'14805|48781495 1|5024|5097 9166 
 NiO fie fio. | 3°} 4 yi 52] 6° [17 1:8h 119 Pp [pes 
 
N.59500 L.7'74 OF NUMBERS. 
 
.60500 L.781 OF NUMBERS. 
 
1(108) : LOGARITHMS N. 61000 L.785 
 
 |] ——————_ jf —_——_—_ ] ———_ - 
 
 15] 3965/4036|4107/4178}4249]/4320/4391|4462/4533|4604]| 7) 
 16| 4675}4746/4817/4888/4959]|503015 101|517215243|5314]| |. 
 
 25 1061/1132}1203]1274 realy iets 1486]1557|1628 60) 
 26} ~§1770\1841/1912/1983/2053}|2124/2195|2266|2337|2408 
 27| 2470|2550/2621|2691|2762]|2833|2904|2975|3046)3 117 
 28}  3188/3258/3329)3400/347 1|\3542/3613|3084|3754/3825 
 20}  3896)3967\4038/4109}4180)/4250)432 1|4392/4463/4534 
 
 6130} 4605|4676/4746)4817/4888]1495915030)5 101/517 1/5242 
 31}  5313|5384|5455|5526|5506||5667|5738|5809|5880|595 1] 
 32} 6021/6092/6163|623416305|163 761644605 1 7|0588|/605 
 33 6730|6800)687 11604.2}7013}|7084\7 1551722517 206|7367 
 34| 743817500\7579|7650177211177 9217 803|7933|8004|8075 
 35| 8146/8216/8287|8358]8420]18500/8570|864 1187 12/8783 
 36] —8854|892.418995|9066|9137|192071927 8|9349|942019490 
 37)" 956119632|9703|077419844 9915 9986/0057 701 27/0198 
 
 46} 5926 5990;6067 6138|6208||6279|6350/6420}649 1/6561 5/56 
 47| 6032/6703 6773|6844|6015|I6985|7056)7 127|7197|7268 tio 
 
 —.aoeaIeEeEG0Q9Qee__ee ie | LN I I I IY —|————— 
 
N.61500 L.788 OF NUMBERS. 
 
 52\7890163}0234]0304}0375 
 ~ 086910940} 1010]10811151]/122211293]1363]1434]1504 
 
(110) LOGARITHMS N.62000 L.792 
 
 AES eC eenroe) reereenee ary FS PERE! | NBR NE {oR ea —— | —_———_ 
 ————- |] ——_]— 
 
 6018 Bee 0158|6228|6298}|6368|64381650816578}6648 
 6718 6788 6858|6928|6998||7068'7138 7208|7278|7348 
 
 6210 re 986]1056)1126}1196]|1266)1336]1406)1475|1545 
 1615}1685|1755|1825|1895||] 965|2035|2105/217512245 
 23 14|2384|245412524|25941/2664|2734|2804'2874/2944 
 3014/3083|315313223/3293!/3363/3433|3503135 73/3643 
 37 121378213852/3922/3992|'4062'413214202/4279/4341 
 
 4411144811455 114621|46911'4761/483 114900,4970|5040 
 5110|5180)5250!5320/5390)|5459|5520 559915060 5739 
 5809|5879'5948'601 8}6088 6158 0228|6298'0367|6437 
 6507|6577|0647/67 17/6787116856 6926 6996 7060 7136 
 720617 275|7345\741517485||7555|7625|7694'776417834 
 6220| 790417974|8043/81 13/8183 8253/832318393|8462/8532 
 8602/8672 8742881 1/8881 895 1|9021}9091/9160/9230 
 9300}937019440:9509'9579]|9649)97 1919789'9858}9928 
 9998|0068|0138/0207/0277|(0347/0417 ted og 0626 
 
 24|7940696|07 66|0835|0905|0975}|1045}111411184)1254}1324 
 
 1394|1463]1533|1603]1673|/1742|1812|1882!1952|2021 
 209112161223 1/2300|2370)'2440!25 102579 2649127 10 
 
 3486135561362613695/3765|383.513904|3974404414114 
 4183}4253 in 4462)|4532}4002/467 11474 1/4811 
 4880/4950|5020 5090)5 159||5229|5299}5368'5438/5508 
 5578|564715717 OF e/ pee 5926|5996|6065|6135|6205 
 
 $2) 62741634410414/6484/0553||6623|6609316762'6832/6902 
 33} 69711704.1)7111/7180|7250)|7320|73891745017 52917598 
 34) 7608]773817807)\7877'|7947||8016|8086)8 1 56|8225|8295 
 
 8365|8434|8504/8574/8643)187 1318782|8852|8922/89g 1 
 90611913 119200|9270/9340)|9409'947 919549 9601 8/9688 
 9757|982719897|9966/0036}10106101 75|0245/03 14|0384. 
 
 39] 1150}1219]1289}1359]1428}11498}1567|1637|170711776 
 6240}  1846}1915]1985/2055|212412194/2263]2333|/2403|2472 
 Al] 2542/2611/268112751/2820||2890 2959|302913098/3 168 
 ‘A2| 3238/3307|3377/3446135 16118586 3655|3725|3794|3864 
 43] 3033/4003/4072/414214212)1428 11435 114420!4490/4559 
 44, 4629|4698)4768)4838]4907|14077|5046/511615185|5255 
 45] 5324/539415464/5533|5603115072'5742/5811)5881|5950 
 AG} . 6020}6089|61509|6228|6298)16367|643716500|657610646 
 47} 6715|0785|0854|692416993)|7063|7 13217202727 117341 
 48} 741017480)754017610\7688117758!7827|7807|7 900|8036 
 
 mm mre ff ee a ff fe —— | — | —————___] | —_____] —___——— 
 
N.62500,L.795 . OF NUMBERS. (111) 
 
 70 
 52|7960190}0259 32910398 68110537|0606|067 607 4.5|0815 Ae 
 0884/0954]1023/1093}1162}/1232/1301)1370}1440]1509 aila 
 
 1579|1648]1718|1787|1857||1926]1995|2065|2134|2204 ites 
 
 2273|2343/241 21248 1/255 1112620\2690|2759|2829|2898 5135 
 2.967/303713 106|3176|32451133 14/338413453/3523|3592 642 
 2662)373 11380013870|3939]4.009|4078/41471421 714286 pa 
 4356|44251449414564/4633114703|4772/484 1/491 1/4980 9163 
 5050|51109|5188|525815327115390|5466|5535|5605|5074|| = |—— 
 5743|5813]5882|595 1 |602 1||G090|6160|6229|6298|6368 
 6437|6506|6576|6645|07 141167 84|68.53/6923|6992|7061 
 713117200|72691733917408]174771754717616|7085|7755 
 
 1290]1359|1428|1498|15671|1636}1706|1775}1844|1913, 
 1983|2052)2121|2191|2260)3329|2398|2468|2537|2606 
 
 2675|2745|2814!2883|2952||3022/3091/3 160|3220/3299 
 3308/3437|3507|3576|30451137 1413784/3853|3922\3091 
 4060/4.130/4199]4268/4337 |440714476/4545|4614/4684 
 4753|4822|489 1/4961 |5030/15099/5 168|5237|5307|5376 
 5445)551415584|5653|57 22}|579115860}5930/5999|6068 
 
 6137|6207|6276|6345|6414||6483/|6553|6622/669 1/6760 
 6829|6899|6968}703717 1 06117 17.517245|73 14|7383|7452 
 752117590\7660)7 720/77 98)\7867|7936|8006|807 5/8 144 
 8213|8282|835 118421|8490)8559|8628|8697|8766|8836 
 8905/8974/9043/91 12/9181/1925119320\9389'9458/9527 
 9596|9606|9735|9804/9873)19942'001 110080'0.1 50/0219 
 
 2302}243.1|2500|2569|2638 |2707127 76|2846/291 5/2984 
 
 3053/3.122/3191}3260|3329)|3398|3467|3536'3006,3675 
 3744|381313882/395 1|4020 4089/4 158/4227|4296/4366 
 443 5|4504|4.57314642/47 1 1/}4780/4849/4918149087|5050 
 512515 194|5263|5333|5402)1547 11554015609|56078|5747 
 §816}5885|5954|6023|6092!|61611623016299/6368|6437 
 
 6506/657 5|6645|67 14/67 83)|6852|692 1|6990)7059)7 128 
 7107\726617335|7404|7 4731175421761 117680!7749|7818 
 788717956|8025|8094|8 163]/8232'8301|83 70/8430|8508 
 8577|8646/87 15|8784|8853||8922!899 1 |Q060/9120101908 
 92.6710336/9405|947419543||961 2|9681|9750/90819/0888 
 
 po? 26}0095|016410233||0302/037 1 elie 0578 
 
| (112) LOGARITHMS N.63000 L.799 
 
IN.635 L.802  ‘ OF NUMBERS. 
 
 6250|802773717 8001787 4|7942|801 111807918 1481821618284|8353 
 
 5418030472/054.0/0609|067 710745||08 14/0882j0951/1019}1087 
 
 83/8050248|03 16103851045310521||0589|0057 072510793 0861 
 
 68/804003 1/0099'0167/023 510303||037210440'0508)0576|0044 
 
 OA Nei ee oe ae SS | Gee 8 
 
 9789/9857|9925|9994|0062)|0130|019910267|033 50404 
 
 1150/1224/1292)1361]1429))1497|15660]1034/1702}1771 
 -1830}1907|1976)2044|21 12/1218 112240/2317|2385|2454 
 2.522/2590|2659|27 27127 95]|2864 |2932|3000!3060)3 137 
 320513274|3342134.1013478]13547|3015|3683/3752/3820 
 3888/3957|402514093/4161)14230/4298/4300)4435 4503 
 457 1|4639|4708)4776|4844)/4913/4981|504915117|5186 
 5254|5322|539 115459|5527||5595|5004. 372|5800 5868 
 5937|6005|6073/014.1102 10}1627 816346/6414/6483/6551 
 ‘0019|6687|6756|682.416892||0960|70291709717 105 
 7302|7370|743 817 50617 575)1704317 74 11777917 848|7916 
 7984'8052/8121/8189|8257}|8325|8393|8462'8530/8598 
 8066/873 5'8803/887 1|8939||90071907 6,9 14419212/9280 
 934819417194851955319621|9690|9758'9826,9894|9962 
 
 4802'4870/4938)5006/5074)|5143/5211 
 54831555 1|5619|5087|5750)|5824|5892/5960 0028/6096 
 61 04/0232 6300/0368|6437||0505/0573|004 11070916777 
 6845'6913/6981|7049)7 118)|7186|7254)7322'73G0|7458) 
 7526'759417662!7730)7798)|7 8667 934|8003)/807 1/8139 
 8207|8275/83431841 1|/8479/18547|801 5|8083/87 51/8819 
 8887 |8950|9024|9092/9 1001|9228/9296/9304'943219500 
 9568/9603 6'9704|9772'9840|19908/9976\0044 0112/0180 
 
 0929/0997/1065]1133/1201||1269]1337/1405,1473]1541 
 1609] 1677/1745]1813]1881)]1949/20171208512153/2221 
 2289|23571242.5/240312501112029/2097127652833|2901 
 29691303713 10513173(324113309183773445 351313581 
 3040|37 17/3785|3853/3921]/3989|4057 4125/4193 4261 
 4329|439714465/4533/400111400914737|/4805 4873/4041 
 5009|5077/5145|5212/5280)|534815416 548415552 5620 
 5688|5756|5824|5892|5960!|6028|6096/6164/0232/6300 
 6368|643616504|657 116639|16707|6775|0843)69 1 116979 
 70471711517 183|7251|731917387|7455 7523'7590 7658 
 7726|7794\7 862179307 998||8066]8 13418202 8270|8338 
 8405|847318541/8609|867 7||8745|8813 s8silso4g 9017 
 9085|9152/922019288/9356)|942.4|9492/9560 962819606 
 97641983 1198G919967|003.5/01031017 11023y'030710374 
 
(114) 
 
 N. 
 
 O 1 
 
 640C] 30618001868 
 
 13 8070612|0680, 
 
 14 
 
 247812546 
 3157 
 
 2 
 
 LOGARITHMS 
 
 3°| 4& 
 
 N.640 L.so6| 
 8 | 9 | D|Pro. 
 
 SS ee | ee nen, vee yen 
 
 oN Sh BA 
 
 1935|2003|207 112139|2207|2275|2343|2410 
 2614|2682|27 501128 17|2885|2953|302 1/3089 
 
 32.25|32.92|3360|3428|349613564|3632|3699|3767 
 
 3835|3903|397 1|4038|4 106/41 74|4242143 10/4378/4445 
 40409|47 17|4784114852}4920149088/5056|5 124 
 5§327|5395|5463|1553015508|5066|5734|5802 
 6005|6073|614.1}6208]627 6|0344164 12/6479 
 6683|0751|681816886|6954]7022!7080)7 157 
 7361|7428|7496117564|703217600|7767|7 835 
 8038]/8106/81741/8242|8300|837 71/8445|8513 
 8716|878418851]8919|8987|9055/912219190 
 939319461 |9529]9596|9664|9732|9800|9867 
 0071 01380206 
 
 4513|4581 
 519115259 
 586015937 
 6547 | 
 72.2.5|7293 
 7903 soi 
 
 8580|8648 
 9258 9326 
 9.935|0003 
 
 0274|0342104091047710545 
 
 0748/08 16,0883 }095 1)1019}1086)1154/1222 
 1290|1357 '142511493|1560]1628)1696|1764]183 111899 
 1907|2034|2102}2170'2237)12305/2373/2440/2508/2576 
 2644|2711)2770/2847/29142082/3050)311713185|3253 
 
 3320/3388 
 3997/4065! 
 407414741 
 5350 5418) 
 
 3456/352313591/3059137 2613 794|3862/3929 
 4132|4200:4268)1433 5|4403]/447 1|4538/4606 
 4809'4877/4944||5012'5080|5147|5215|5283 
 
 5486) 
 
 5553 5021 
 
 5680|5756|582415891/5959 
 6365|6432|6500|6568|6635 
 
 602716004'6162'6230 6297 
 6703/6771 6838 6906 6974)7041/7 100\7176|7244/7312 
 7379|7447'7514/7582'7650/77 17\7785|7853|7 92017988 
 8055/8123/8191/8258 8326|8393/8461)|8520|8596|8664 
 873 1|8799 8867/8934 9002 |9069,9137|9204|9272'9340 
 9407/9475 9542/9610 0678|9745|9813|9880|9948 (001 5 
 
 27|/8080083|015 1 0218|0286.0353)|0421|0488|0556|0624'0691 
 0750 082608940961 1029]]1096)1 164]1232]1299|1367 
 1434 1502 1869)1037 1704}1772)1840}1907|1975|2042 
 2110/2177 2245|2312'2380|2447/25 1 512582/2650|2718 
 278512853 2920 2988 3055 3123/319013258/3325/3393 
 3460/3528 359513063 3730/3798'3805|3933/4000/4068 
 A136|4203/4271 433814406 4473'4541|4608|467614743 
 481114878/4946|5013/508115 148/5216|5283|5351|5418 
 
 5486]5553|5620/5688'5755||5823)5890|5958|6025|6093 
 0160|6228/6295/63 63'6430|6498'6565|0633|6700\6768 
 6835|0903/6970]703 7|7 1057 172\72401730717375|7442 
 7510|7577|7045177 121778017847|7914|798218049|8117 
 8184|8252|8319/8387|84541852 118589|8056/872418791 
 8859|8926|/8994|9061|9128]19196,92631033 119398|9466 
 9533|9600|96681973519803 1987 0!993 8|0005|007 2/0140 
 
 088 1|0949|1016]1084]1151]1218|1286):353]1421|1488 
 1555|1623!1690|1757|1825||1892|1960|2027|2094|2162 
 
 222G|2207|2364)/2431 2490) 2506|2634|2701|2768 
 
 324013307|33751344213509 
 3914|3981/4048]4116/4183 
 4587|4654|4722147809|4856 
 5260|532815305|5462|5530 
 
 men fe 
 
 2.903|2970)3038|3105 ieee: 
 3577|3644)37 111377013846 
 42.50\4318)4385|4452 4520 
 
 4924/4099 1|5058)5126 5193 
 
 o [1/2138] 4 
 
OF NUMBERS. 
 By A PSY G7 
 
 6450 8095597 5064|573215790|5860115934|6001 |6008/6136)6203 
 62701633 8|64.05|6472'6540)|5607|6674|0742|68090|6876 
 6944|701 117078)7 146|7213]|72801734717415|7482/7549 
 7617\7684)775117810\7 886||7953/8020/8088) 8155/8222 
 8290|8357|8424|849118559||8626|8693/876118828|/8S95 
 8962|9030/9097 |9 1641923 2119290|90366/943319501\9508 
 
 963 5|9702/97 7019837 |9904/1997 2|0039/0106)01 73/0241 
 
 57 810030803 75|044210510'0577110644|07 1110770|0846|0913 
 0980!1048}1115}1182]124011317}1384]1451/1518]1586 
 1653)1720/1787|1855/1922}]1 98g]/2056|2123)2191/2258 
 
 2325|2392|246012527|2594|12661|2720|2796|2863|2930 
 2997|3065)3 132'3199/3266]13333|3401|3468|3535|3602 
 3670|3737138041387 113938]4006|4073/4 1401420714274 
 434214409/4476/4543|4610|4678|4745/4812|487914946 
 5013/5081|5148/5215|528211534915417/5484'555 115018 
 
 5085}5752|582015887|595411602 1 {6088/61 56|6223/6290 
 6357|6424|6491|6558|6626||6693|67 60|0827|6894'6961 
 7920|7096)7 163|7230)7297||736417432|7400|7560|7633 
 7700!7767'7834'7902'7909||803 6/8 103|8170|8237/8304 
 8372|8439/8506'8573|8640!18707|8774|8841/8900|8976 
 9043191 101917719244'93 1119378194461951319580'9647 
 71| 9714/978119848\9915\9982 0050)0117/0184 0251 0318 
 72'8110385]0452105 19|0586|0053)|07 2 1|0788/0855 0922/0989 
 1056]1123|119011257|1324111392|1459|1526|1593|1660 
 1727)1794|1861|1928/1995112062)2120|2197|2264/233 1 
 2398/2465|2532|2599|2600)}2733|2800|2807|2034/3001 
 3068/3 135|320313270/3337)|3404|347 113538'1360513672 
 3730|3806/3873 3940/4007 4074/4 141|420814275/4349! 
 4400|4476)4544/4611|4078]|4745|4812/4870/4946|5013 
 5080)5 1475214528 115348)/5415|548215549'5616|5683 
 5750|5817|5884'595 1/601 8!16085|6152/6219|628616353 
 642016487\655416621|66388116755|0822\68s0 695617023 
 790017 157|722417291 |7358)|742517492'7559|702617603 
 7700|7827'|7 89417 961|8028)|8095|8 162/8229 8290/8363 
 
 2447/25 1412581|2048)27 15|12782|2848/2915|2982 3040 
 3116}3183'3250)33 17|33841/3451|351813584|365113718 
 
(116) LOGARITHMS N.650 L. 812 | 
 
 —_——_— | -————_—_————.- | — — | — J | | | | | I I | 
 
 6540} 5777|5844|591015977|6043)16109|6176|624216300|6375 
 
 IO «OW ~] 
 
 O MIR & OMe 
 Doe OWW— 
 
 Cy 
 — 
 
 7144 
 7811|787717944|801 1|8077|/3144|821 118278|8344 
 84788544861 11867818744||8811/8878]8944|901 1 
 
 8411 
 9078 
 
 15} 9144/90211|9278/034419411 947719544 9611|9677/0744 
 16{ 981119877|G944/001 1|0077 0144021 1/027 7|0344,0411 
 17|8140477/0544|0610|0077|07441/0810,0877|0944!1010;1077 
 
 1144/12101127711343114101|1477|1543]1610]1677|1743 
 
 19] 1810/1876|1943!2010|2076)|2143'221012276|2343/2409 
 2 2476 '2543|2609/2676|2742)|2809|2876|2942|3009|3075 
 21}  3142,3209/327513342|3408]|3475'3542|300813675'3741 
 22}  $8083875|394114008/4074/414 1 4207/4274/434 1/4407 
 
 4474|4540|4607/4674|4740 4807/4873 4940'5006 5073 
 5140'5206|5273|5339|5406)|5472/5539|5005|5672'5739 
 
 9132/9198/9265/933 119398 946419531 9597\9664|9730 
 9797 |9863/9930/9996|0063)10129 0196)/0262/0329,0395 
 32'81504.62/0528/05095|006110728)|0794:0861|0927\0994| 1060} 
 33| 1127/1 193]1260)1326)1392}/1459)1525]1592)1658]1725 
 34] —1791]1858|1924)1991]2057}|2124 219012257|2323/23 86 
 35| 2450/2522)25809/265512722)|2788/2855|2921|2988/3054 | 
 36} 3120/3187|3253/3320)3386)/3453'351013586|3652/3718 
 37|  3785|3851|3918|398414051]14117/4183|4250}4316/4383 
 38| 4440/451614582/4648|4715|4781/4848]4914/408115047 
 39] 5113/5180|5246/5313153791|544515512|5578|5645|5711 
 
 46} 9760)\9826)9893|9959|0025||10092/0158|/0224|0291|0357 
 47 |816042310490|0556|0622|06891/0755|082 1|0888/0954) 1020, 
 
 Migoule ie. 3 
 
IN.655 L.816 OF NUMBERS. 
 
 N.| O Med FA Ba Ee 
 6550|8162413]2479|2546|2012|2078]|2745|281112877/2943 3010), | 
 
 51) 3076)3142/3200|3275|33.4.1||3407|347413540|3 6063673 
 521  373913805|387113938]4004]/40701413714203]426914335 17 
 53|  4402/4468]4534|4600/4667||473314790|4866)4.932/4998 a 
 54) 50041513115197|52631532011539015462|5528|5594/5001 Me 
 55) —: 572.715793|585091592.6]5992]/0058/0124/019116257 6323 5133 
 56| 63809|0456/6522/6588|0654|107 2 1167871|G853|6919|69086 640 
 571 7052/7118]7184]7251|7317117383|744917515|7582|7648 aye 
 58} 771417780)784717913/7979}|8045]8 1 11/8178]8244'83 10 9}59 
 59|  83706|8443/8500|8575|8641}'8707|8774|8840/8906/8972 bs 
 656 9038/9 105/917 1/923719303]|9369|9436'9502/9568/9634 
 it On 9700/9767|9833!9899|0965| 003 1|0098|01 6410230'0296 
 62)8170362'0428|0495|0561|0627||0093|07 59|0826|/0892\0958 
 63|  1024/1090]1156/1223]1289]}1355|1421)1487|/1553)1620 
 64  1686)1752]1818]1884}1950}/2017/2083/2149|2215/2281 
 65| 2347|241312480|2546126121|\2673!2744/2810)2876]2943 
 66] 3009/3075|3 141]3207|3273]13330|3406|3472/353813004 
 67|  3670/3736|3802|3869|3935]/4001/4067/4.133|4199/4265 
 68|  433114398/4464|4530/4596]4002!4728/470414800/,4927 
 69}  4993/5059/5125|5191|5257|,5323/5380|5455|5521/5588 
 057 5054|5720/5786|5852|5918},5984|6050/61 16/6182'6249 
 a 6315|638116447165 13|0579||6045|67 1 11677716843|6909 
 7; 697 6)7042)7 108|7 17417240]7306)7372)7438|7504|7570 
 73 7036)770217768!7835|79011'7907|8033/8099|8105|823 1 
 74|  8207|8363|8420/8495|8561)/8027|8693'8759|8825/8892 
 75|  8958|9024|9090/9156|9222|9288/9354'9420 9486,9552 
 4 9618|968419750|9816)9882]|9948 0014'0080/0146,0212 
 >71818027810344l04 101047710543] 00091007 5\074110807/0873 
 zs} — 0939{1005|1071|113711203|112691133511401]146711533 
 79 1599|1605]173111797|1863 192011905 206112127/21G3 66 
 6580] 2250}2325123911245712523]|2580|20551272112787|2853 
 81] 2010129851305 11311713 183/13240/3315/338113447/3513 
 82)  3579/3645|371113777/3843113900130751404.1/4107/4173 
 83]  42391430514370/4436/4502/14508|403.4|4700/476614832 
 84]  4898]496415030/5096|5 162)|5228|5204|5360/5426|5492 
 85}  555815624|5690|5756|5822/|5888)5053|6019)608516151 
 86|  6217|6283/634916415|6481]10547|6613/667016745/6811 
 87| 6877|6943|7008|7074)7 140]17206/7272|7338|7404|7470 
 88] — 7536|7602/7668|7734\7800}|7866|703 1|799718063|8129 
 89] 8195|8201|832718393|8459|18525|859 1|86561872218788 
 6590}  8854|8920/8986/9052/9118]1918419250193 15|938119447 
 Ql] 9513}9579106451971119777119843|9908/9974|0040|0106 
 92)8190172/0238/0304\0370\0436)10501\0567 |0633|0G9910765 65. 
 93| — 0831/0897/0962|1028]1094]]1 160} 1226]1292]1358]1424 1} 7 
 Q4|-  1489]1555]1621]1687|1753]|1819)1885]1950/2016|2082 a 
 Q5| 2148)2214/2280)23461241 1})2477|2543/2600|2675|2741 4|26 
 96} 2806/2872)2938/3004|3070)3 136)3202/326713333|3399 5)33 
 97} 3465/3531|3597/3662|3728113794'3860 3926 3991/4057) [Phe 
 Q8| 4123/4189/425514321/4386)/4452/4518)4584|4650/4715 8h50 
 99) 4781/4847 4913/4979 5045)/51 10/5176)5242|5308|5374 9159 
 O LET a3 4s D | Pts. 
 
 SiGe wel. 8 [PS 
 
(118) LOGARITHMS N.660 L: 819 
 N. O Poy 2 Sa Be OU TR F 8 al Oe ie. 
 
 ——_—_——— _ | ——— | —_____ pe 
 
 6600|8195430|5505|557 1|5637|5703115708|5 83.415 900|5 90010032 
 O01} 6097|6163|6229|629516360}|642.6|64921655816624/6689 
 , 02} 6755)6821|6887|6953}701 81170847 150\7216|7281]7347 
 03} 7413/7470|7545|7610|7676)17742)|7808|7873|7039}8005 
 04}  8071/8136)8202/8268]/8334118309|8405/853 118597|8062 
 05|  8728)8794|8860|892.5}8991]|9005719 12319 1881925419320 
 06}  93801945110517|9583}964G)197 14/97 80|9846|99 12/9977 
 
 08} 07000766\0832/0898|0963}}1029]1005}1 100} 1226}1292! 
 O09} 1358/1423|1489]1555]1620)|1686!1752}181711883]1949 
 
 6610}  2015|2080/2146/2212|2277/12343|2400/247412540/2606} . 
 11] 2072/2737|2803|2860|2934||3000/3066)3 13 1/3 197/3263 
 12| 3328/339413460)3525135911|3057/37 23/37 881385413920 
 13} 30985|4051/4117/4182)4248)143 14/4379,4445/45 1114576 
 14{ 464214708/4773/4839|49005||4970|5036)5 102/5 167|5233 
 
 15|  5208/5364'5430)549515561||5627|5602|5758/582415889 
 16} 50955|6021'6086/61 52)6218/16283|6349|6414/6480/6546)}, 
 17| 661116677\6743|60808|6874)|6939!7005|707 1|7 1136/7202 
 18} 7208/7333'7399|7404|7530)|7596\7061|7727|7793|7858 
 19| 792417989'8055'8121|818618252/8317/8383|844918514 
 6620| 8580/8645'8711|8777|18842)|8908|/8973|9039\9105\9170 
 2} 9230/9301 9367|9433|9498)|9564|9629 969519761 9826)! 
 221° 9892/9957 0023'00890|0154||022010285/035 110416j0482 
 23|8210548|0613 0679|0744/0810)|0875|0941|1007|1072)1138 
 24} 120311269/1334}1400}1465][/153 1|1597|1662)1728]1793 
 
 25}  1859}1924'1990/2055|2121//2187/2252)23 18|2383/2449 
 26} 2514/2580.264512711|2776|2842|2908 2973|3039|3 104 
 27} 31703235 330)13366|3432/13497|3503|3628/3694|/3759 
 28|  3825:3891/39564022/4087)|4153/4218/4284/4349/4415 
 
 291  4480.4546.461114677/4742|4808/4873/4939|5004|5070 
 
 6630 51351590) 5266|533215397||5403'5528)5594/505915725 
 31] 5790'5856.59211598716052)/61 18|0183|6249'63 14/6380 
 32| 6445'6511'6576\064216707)0773|6838|6904'6969|7034 
 33] 7100171651723 1|7296|7362'17427|7493|7558|7024'7689 
 34] 7755/7820'788617951|801 7//8082|8147|8213|8278|8344 
 35| 8409 $475 8540 86061867 1118737|8802'8867|8933|8998 
 36} 9004/9129'10195|9260/9326)19039119450/0522/0587|9653 
 37] 9718|9784|9849|991419980)004510111|0176\024210307 
 
 39} 1027|1092)1158]1223}1288)|1354]1'419/1485|1550]16015 
 
 6640} - 1681/1746/1812]1877}1942/|2008/2073|2139|2204|2269 
 41} 2335)/2400|/2466)/253 112596112662|2727|2793/2858/2923 
 42} 2080|/3054/3119/3185/3250/133 16/3381 |3446)/3512|3577)| 
 43}  3643/3708|3773|3839|3904||3969/403 5/41 00/41 66)423 1)’ 
 441 4209614362/4427|/4492/4558||4623/4688/4754/481914884 
 
 45} 4950)5015|5081|5146/52111|/5277|5342|5407 [5473/5538 
 46} 5003!5660|5734|5790|5865||5930|5905 |0061|6126|6191 
 47| 6257|632210387|6453/6518||6583|6640|07 14167 70/6845 ane 
 48} 6910|6975|7041}7106]71711|7237|7302|73671743317408 
 
 8152 
 49| _7563|7629|7694|775917825||7890|7955|8021|8086|8151 9159 
 
 N.[-0 [il2ls{4hslel718 19 |p \[pts. 
 
 65 
 1147 
 12113 
 3/20 
 4126 
 5(33 
 6/39 
 
N.665 L.822 - OF NUMBERS. (119){ 
 
 ——_— Cee ee ee, | ee ene, Sannin ane 
 —_—_|—_—_—_———" 
 
 8002 Aci 8132'8197|8202]|8327 ofsoaeas 852318588 
 8914||/8979 sie 9100|9174/9239 
 
 9030|9005'9761|9826 9891 
 
 7 \0412)0477'0542 
 1063|1128/1193 
 
 9|17141177911844' 
 9 2300)2365 243024951 
 2560|26251269 1|2756|282) 28861295 1/3016|308113 146, 
 321118276'334113406/3472)13537 idee pel 373213797 
 4383|4448 
 
er | | LOGARITHMS N.670 L.826 
 
 peo, Papen sl paar saed ERE Heide) FAL ma. “retpets 2} hima qed lemegmaneagpag MRRP! AREY i i og 
 
 28 8860/8924|8989'9053]9118]}0183/9247193 12 937010441 
 29 9505|9576|9034,9699|97 63}19828/98931995 710022\0086 
 
 75|543915503|5568]|5032|5097|576115826|5890 
 
 | 39]  5955|6010/0083|6148/6212/16277/0341|6400|6470/6535 
 6740] 6599|6663|572816792|0857||\6921|6986)7050)7114)7179 
 41]  7243|7308|737217437175011|7505|76301769417750|7823 
 
N.675 L.829 OF NUMBERS. (12Y) 
 
 Prag Peco ae te sl eve DL Opie | feeaarrene Caen & SORRENE (PURE: DSR 
 
 75 9093)91571922119285|9349|1941319478'9542/9006|9670 
 70} 973419798|9862|9926!9990||0054|01 1910183/024710311 
 77|83 1037 5|0439|0503|0567|063 10695107 59|0823|0887|0952 
 78} 1016)1080}1144}1208}1272])1336)1400|1464)1528]1592 
 
 1656}1720}1784]1849/1913||1977/2041/2105/2169|2233 
 
 93|832061610680|074410808|0872 
 
 1394|1458]152211587|1051]11715|1779|1843|1908]1972 
 2.036|2100]2164/222912293112357/2421|2485|2550/2014 
 
 2678)2742/2806|287 112935|2990/3063|3 127|3 19213256 
 3320/33 8413448135 12|3577||304 113705137 691383313898 
 3962/4026/4090)4154}4218)4283/4347/441 114475|4539 
 4004|4668]4732/4796|4860]492414988'5053/5117|5181 
 5245!5300|5373|5438/5502)'5506)/3030/5094/5758|5823 
 
 5887|5951}6015|6079/6143 ||0207|/6272/6336|0400/6464 
 6528)/0592|0656|6721|6785|'6849|6913/6977|704117105 
 7 100)7234|7208|7362|7426|'7490\7 55417018/768317 747 
 7811}7875|7930|8003/8067|\8 13 1/8195|8260/8324/8388 
 8452/85 10|8580|8644/8708)|8772)8837/8901/8905 19029 
 
 2489/2553 2617 2681|2745|2809|2873 
 
 2937|3001|306613 130)3 194|3258|3322/3386)3450)3514 
 3578|36421370613770|383413898|3962'402614090/4154 
 A218|4282/43.46)44 1014474|4538/4602|4660]473014794 
 4858]492214986/5050\5114!'5178|5242/530615371(5435 
 5490|5563|5627/5691|5755 Is19/5 5883)/509047|6011 9075 
 6139}6203|6267}633 1 6395|6450|0523)05870051 | 
 0778|6842|6906}6970 704) | 
 
 7|2361|2425 
 
 8058/8122/818618250 8314 
 8098/8762|8826)8890|8954 
 9337/9401 |9405|9529|9593|9657 
 997710041 10105|0169|0233| 0296 
 
 yy 09 03 tO ee 
 So Somer) 
 
 1255]1319]138311447|1511 | 
 1895}1958|2022!2086]2150) 
 2534|2598}2062/2725|27 80): 
 3173 3237 3300|3364|3428 
 
 Od 20 
 Ors 
 
 ~_ 
 SO we Cd 
 
 200-1 Sr 
 
 A) 
 a 
 
 2R 
 
(122) LOGARITHMS N. 680 7 832 
 N.p 0 [2 | slap a] e[/7 | 8 | 9:) DyPa. 
 
 680018325080! 5 153|5217}528115345|1540815472)55360|5600|5004 
 Ol 572815792 5855)5910/5983 60471011116175|0230)5302 64 
 O2 630016490)0 6494|0558|0022/16686|6740|0813|0877|5941 1) 6 
 03| 7005}7069|71321719617260||7324]7388|7452|7515|7576 as 
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 5237|5298153601542 1154821 554315005|50061572715788 
 5850/59 1 115972|6034|6095]'6156|6217/6279|0340,0401 
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 7075\7 130|7 19717250|7320)|73 8 117442/7504|7565|7020 
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 8300/8361|8422)8483/8545||8600|8667/87 28/87 89|885 1 
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 GO}85 10130/0197/0258|03 20/03 8 1||0442/0503|0564|0626|0687 
 
 97 
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 0748/080910870|0932/0993}!1054]1115/1176]1238]1299 
 1360]1421!1482|1544|1605]|1666]1727]1788]1849]1911 
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(128) LOGARITHMS. N.710 L.851| 
 reg) 1 8 | 9 || D iPro. 
 7100/85 12583|2645|2706|276712828]12880|295013012/307313 134 
 Ol 3195]3256133 17|3370|3440)|35011356213623|368413746 
 G2}  3807/3868)3929)3900/4051}141121417414235|4296/435 
 03} 4418]4479/4540|4602/4603)/4724/4785|4846)4907|4968 
 04}  5030)5091515215213]5274115335|5396)5457155 19/5580 
 05| 5641/5702}5763/5824|5885||5946|6008|6069/6130/6191 
 06} 6252/6313|/6374|643 516496||65581661 9|6680/074 1 |6802 
 07; 6863/6924/6985 7230/7291\7352\7413 
 08} 7474|753517590|7657)7 7 10||7780\7841|7902'7963|8024 
 09} 8085/8146|/§207|8208)83 29/1839 1|8452/8513/8574/8635 
 47110} 8696)8757|8818|8879|8940)i9001|906219124/9185|9246 
 11 9307|9368/9429/94901955 1119012|0673|9734/9795|9856 
 12] 9917!9979|0040/0101)0162/102231028410345\04060467 
 13|8520528/0589|0050)/07 1 110772|10833|0894|0955|1017|1078 
 14 1139)1200)1261/1322)1383}|1444|1505]1566|/1627|1688 
 13 1749)1810)187111932]1993/12054/2115/2176'2237|2298 
 16| 23509|2420)2481|2542|/2604|120605|2726|2787|2848 2909 
 17|  2970/303 113092'3153!3214)327513330|3307/3458'3519 
 18] 3580)36411/3702|3763!382413885|3946|4007 406814129 
 
 41090/42.5 1/43 12/4373/4434114405|4550.4617/4078 4739 
 4800)4861/4922/4983|5044)5105/5166/5227 528815349 
 §410)547 115532)5593|5054|'57 15|5776|5837'5898 505g 
 
 22|  6020,0081|6142 6203 6264) 6325|6380(6447 6508|6568 
 231  6629|66901675116812'6873|6934169957056 7117/7178 
 24|  723091730017361/742.217483 17544|7005 7666'7727/7788. 
 
 25 7849|7910)797 1|8032/8092 81 53/8214/827 5|8336/8397 
 26} 8458/85 19}8580 8641|8702||8763|8824/8885 8946|9007 
 27} 90608/9129,9189/9250/93 11)190372/9433'9494,95559616 
 
 9677/9738'9799|9860'9921|19982|0042 0103016 6410225 
 2.9'853028610347|0408/046910530||(059110052/07 13'0773|0834 
 
 0895|0950/1017|1078)}1130}/1200)1261 1399/1383 1443 
 1504}1505 
 
 2113/2174 
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 1626|1687]1748]|1800|1870]1931 
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 1992/2052 
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 35| 30940!14001'4062/412214183/14244/4305|4366|442714488 
 36| 4548)4009|4670/473 1/4792)4853/4914/4974|5035|509 
 37| = §1571|5218|5270'5340)5400)1546115522/5583/564415705. 
 38} | 5765|5826|5887|5948|0009)10070/6130)0191|6252/63 13 
 39] 6374/0435|6495|0550|60017)||6078|0730|/0800\6860)/6921 
 7140] 6982|7043}/7104|7 105|7225|728617347|7408!7460|7530 
 41} 7590}70511771217773|7 834||780417955|8016|8077|8138 
 42} 8198]8250|8320/8381]844 2/18502|8563/8624|8685|8746 
 
 a ey ee eS ee eens —— J ——_—_—_——_ J ~ | | | 
 
 Deh Ae eae Iai: 
 
N.715 L.854 OF NUMBERS. 
 
 78|8560035|0095|0156|021610277|1033710398|0458 0519/0579 
 
 (155) 
 
 ON Pe a. 18 Se de Se 5 Oo Ve Fe Be 1 D> VD PPro: 
 
 ————— | ———_——— | — —— | |] | — | | | | 
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 4882|4943|500415064|5 125/15 1861524715307|5368/542G 
 54809|55501561 11567 1/57321157031585415914|5975|0036 
 6096/6157|621816278|6330||6400|646110521|6582/6643 
 6703|5764|6825|6885|5946]17007}7067|7 128|7 189|7249 
 
 73101737 1|7432)749217553\|7614|7674|7735|7790|7856)|_ 
 
 79171|7978|8038/8099/8 160)|8220/828 118342|8402)8463 
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 9737 |0797|985819091919979||0040|0101/0161/0222/0283 
 
 0950|1010}1071)1131/11921|1253]1313}1374|1435}1495 
 1556|1616|167711738]1798)11859]191g9]1929'204.1/2101 
 2162/2223122831234412404||2465/2526/2586'2647/2707 
 2768!2829|2889|2950|3010)|307 1/3132)3192/3253/3313 
 3374|3435|3495|3556|301611367713738|3798/3859/3919 
 3980}404114101/4162/4222/|4283|4343/4404/4465/4525 
 4586/4646]4707|476814828||4889!4949/5010:5070/5131 
 51921525215313|5373|54341|549415555|561 615676 5737 
 5797|5858|5918|59079|0039}|0100/6161|6221/6282'6342 
 6403|6463|6524/0584|664.5||6706|6766|6827|6887/6948 
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 8219/8280|8340/8401|8461||/8522/8582|8643'8703'8764 
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 3059|3119/3180/324013301|13361)3421134821354213603' 
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 4268/4328'4389|4440/4500/|4570/4630)469 1/475 1/4812 
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 547615537|5597|5058157 18115779|5839|589915960|6020 
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 6585|6745|6806|6866|6926)6987|704717 1087 168|7229 
 728917349|741017470|753117591|7651|771217772|7832 
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 910119161|9221|9282|9342l9402/9463|9523/9584|9644 
 
 97049765|982.5198851994.6)}0000|0067|01271018710248! 
 
 091210972} 103211093]1 1531]1213]1274]133411394]1455 
 1515}1575|1636}1696]1756|1817]1877|193711998|2058 
 2118]2179|223G12290|2360]|2420|2480}254112601|2661 
 27221278212842|2003|2963|13023|3084]3 144132043265] 
 
| ee LOGARITHMS N.720 L.857t 
 Pte richie) pou raat i } 
 BNE Le Qi] Seen i ID Pro. 
 
 oe eS ee | SS I ED 
 
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 O4{ 5737|5797'5858)5918}597 8|603 8|6000|61 59|62.19}6280 
 05} 6340}6400'0460/652116581|/664.116701|6762/6822 6882 
 
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 O09} 8750)8810,887 1/893 118991}/905 1191 12|9L72/9232'9292 
 7210) 9353|941319473'953319594/19654197 14190774 gs3slosgs 
 11} —9955|0015|0075|0136101960256'03 161037710437 0497 
 
 12 8580557 0617|067810738 
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 1761|1822]1882]1942 
 
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 0918/097G|1039110909 
 1521}1581|1041)1701 
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 236312424/2484/2544. 2.004, 2064/27 2.4|2785|2845|2905 
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 3567|362713087 3748|3808 386 
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 2073 0034 0094 0154 0214)\¢ 
 
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 9581|9641 9701 9761/98 21{19881|904.1{0001|0061/0121} 
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 0782|0842'0902/0962/10231| 1083 
 
 1383|144311503|1563!16231683]1743N 803]1863]1924 
 1984|2044/2 104|2164'22242284/2344|2404|2464:2524 
 2584|2644 2704/27 64 2824||2884)2944)3005/3005 3125 
 318513245/3305|33651342.5|349513.54.5|3605|3605/3725 
 378513845 |3G0513965|4025|4085/4145}4205/4265/4325 
 43851444 5|4505|4505/4625114685/4746|4806 pan hen 
 4986|504615 106)5 1665226 |5286|5340|5406/5460'5526 
 5586156461570015766 '5826115886|5946/6006|0060/6126 
 6186)0246 1300/6360 \6426]6486|6546|4600|6666|6726 
 6786158-46'6906)6 96617 0261708617 146|72061726617326 
 7386|74461750617566\76267686|7746]780617866\7925 
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 Se ne a a la Ato avs AAA Richy ere Ate eae CE 
 
 51 9/6 4039 4099/4159 1219 {4279u988 asyel 4458/4518 
 52| 4578/4638/4698/4758/4817/1487714937|4997|5057|5117 
 53| = §177|523715290715356154.1615476|5536155901565615716 
 541 5776|5835|589515955|0015!1007516135}61951025410314 
 55! °6374\6434|649416554,6614|16673|6733|679310853|6013 
 56| 60973|7033)7092)7 152'7212)'727217332)730217452)7511 
 
 9426,9486|9546/9005|196651972519785|9845|9905 
 61} ggd4 602410084 0144/0204/10263/0323/0383/0443/0503 
 62'8610562}0622'0682)/07 42;0802/|086 11092 11098 1}1041/1101 
 63} .1160)1220)1280 134011400 1459]1519|1579]1639]1699 
 04 1758 eu ate 1938]1997|2057|2117/217712237 (2206 
 65| 2356/2410/2476/2536\2505||2655]2715|277 512834 12894 
 
 06 2954 301418073 3133|3193]3253/3313/3372)3432'3492 
 
 67| 3582/3611/367113731137901 3850 39103970 4030/4089 
 68} 41409/4200)4200/4328/43 88/4448)4508 4507|4627 4687 
 69] 4747/4806)4800/4926/4986].5045/5105 5165/5225 |5284 
 
 7270|  5344|5404|5464|5523)5583|/564315 703, 5762 580015860 
 71|  5941/6001|6061/6121/6180!'6240 0}6300)0860)0 641916479 
 
 72)  6539/05¢ 98)6058/67 18'6778||6837|6897 69571701 67076 
 73} 7136|7196\7255|73151737 5\'743417404175541761417673 
 74|  773317793\7852'7912 a7 2118031 S091 8151 8211/8270 
 +5} . 8330183908449 '8509'85G69!/8628'8688'874818808 8867 
 
 76| 8927|8987 
 77 9524 9583 (9643'97 0319702 2 i982 9}/9882'994 1/0001, ee 
 78|8020121/0180 024903000350 041910476 05380598 0058 
 76 07 17|0777|0837 089610956 1016]1075)1135|1194)1254 
 7280 1314113 873)1433 14981552 1612]1072|1731)179111851 
 81 1910 1970|2030,2089/2149}!2 200|2208 /2328'2387|12447 
 82}  2507/256612626'2086|2745)|/2805/2805 2024 2984) 3043 
 83} . 3103}3163|3222'3282 3342 3401134611352035803640 
 84| 3699 376913819138783938 3997 4057/4117 417614236| 
 85| 42061435514415!44741453411459414053/47 13 sy70l4 832 
 86] 4892/4951/501 1/507015130)15 190}5249|5309/5 530815428 
 87| 5488]5547|5607|5066'5726]15786|5845|5905|5904 6024 
 88]  6084\6143/6203/6262/6322 6389 644.1|0501|6560 6620) 
 89| . 6680}0739\6799|/6858'6918]10977|7037|7097|7 156/7216 
 7290| _ 7275)7335|73904'7454'751417573|7633|7692|7752/7811 
 gl 787 117931}7990'8050,8109)|81069)8228/8288 ears ica 
 92}  846718526/8586'8545|8705]/8764|8824/8883 
 93 9062 9122/9181}924 1|9300}19360/9419)947g 0539 0598 
 
 o4 jose 971710777 esd 9896 9958 0015 odie Ore4 on 
 
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 9040}91 001100) i922 25 928519345 94049464 ; 
 
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(132) LOGARITHMS N.730 L.863 
 Nao a RE, 
 
 — aeares a Ioeiws paar liseaeme cued | bs) aR = SOLEIL haayr sea DED (MLTR | 
 Pee peso VP pane NRC 
 
 01 ae 3883|3942|4002 4061 412114180|4240 42.90 4359 60 
 02]  4418/4478/4537|4597|465014716]4775|48351489414954 ae 
 03] 5013|507215132|5191|525115310|5370|5420915489|5548 3118 
 04!  560815607/5727|5780|584515905|5964|6024|6083|6143 4124 
 05| — 6202|6262|6321|63816440]}6499|6559|6618|6678|6737) ~—|>|3° 
 06| 6797|6856|6916|6975|7034||709417 153|721317272|7332 49 
 07 739117451175 10|7500|7020'|7088|7748|7807|7807|7926 8l48 
 O08} 7985|/8045|8104}8164|822318283/8342)/8401|8461|8520 9154 
 09| 8580|8639|8698]8758}8817|18877|8930\8996|9055\9114 
 
 7310| 9174|9233|929319352/94 1111947 1/053019590|964919708 
 11} 9768/9827|9887|9946}0005/}0065/0124'0184|0243|0302 
 12/8640362]/042 11048 1|0540}0590||0659|07 18'0778|0837|0896 
 13}  0956]1015}1075]1134}1193]}1253/1312}1371|143 111490 
 14} ~— 1550|1609/1668]1728]1787||1846]1906)1965|202512084 
 15|  2143}/2203]2262)2321|23811}2440)2500)\2559/2618|2678 
 16] 273712796/2856|2915|2974113034|3093/3 152/3212/3271 
 17| 3331|3390)3449/3500|3508!13627|3087|3746|3805|3865 
 18|  3924|3983/4043/4102/41611/4221|4280|4339/4399]4458 
 19} 4517/4577/4636/4095/47551148 14|4873/4933]4992]505 1 
 
 7320) 5111|5170'522915280|5348)|5407|5407|5526|5585|5645 
 21 5704157 63|5823}5882|5941]}6001|6060)61 19/6179|6238 
 22}  6297|0357|/6416|6475|0534)0594|0053/6712|67 72/083 1 
 23 6890/6950!7000|7068\7 1287 187|7246'7305|7365|7424 
 24, 7483/7543'7602!7601|77211|7780 7839\7898 7958/8017 
 25|  8076]8136/8195|8254/83 13|18373|8432|8491/8551|8610 
 26| 806609|8728/8788|8847|8906|/8966|9025'1908419143|9203 
 271  92621932119380|9440/94.99119558|9618]9677197360/9705 
 28]  98551991419973|0032|00921/01 5 1|0210\0269|032g|0388 
 2.9|8650447 |0500|0566|0625/0684|10743|0803 0862\092 1|0980 
 
 7330)  1040|1099)1158]1217]1277/1336)1395/1454]1514]1573 
 31] © 1632}1691]1751|1810]1 869||1928)]1988/2047|2106]2165 
 $2| 2225)2284/2343|2402/2461112521|2580/2639|2698]2758 
 
 Bl QBlg. 2876)2035 2995|3054!/3113|3172/3231|329113350 
 
 34} 34009/3468)3527|3587|364613705|3704/3 82413 883|3942 
 35| 4001}1060/4120/4170/4238]14297|4350)441 61447514534 
 36]  4503|4052/47 12/477 114830)4889]4948/5008|5067|5 126 
 37| 5185)5244|5304|536315422/15481|5540|560015659)/5718 
 38] 5777|5830|5895|59551601 416073|0132)/61911625 1|63 101] | 
 39| 6360910428|648710546|6606||6005|07 24/6783|6842)|6901 
 
 7340| 6961|7020\7070|7138)719717256|7316)7375|7434|74093 
 43) 7552|761117671|7730|7780)|7848|7 907|7960|802.5|8085 
 42}  $144/8203]82621832118380)8440|8490/8558|8617|8676 | 99 
 43] 8735)8794|8854|8913/8972|}903 1/9090/914919208/9268 a8 
 44| ~ 932719386|9445|9504|9563|\9622|968119741lgs00|9ssq] | 3113 
 45} 9918/9977|0036|0095|0155|1021410273/033 21039110450 424 
 46|8660509|0568|0627|0687|0746]10805|0864|092310982|1041 9190 
 47| ..1100}1160|1219|1278]1337]}1396]1455|1514]1573|1632) 8? 
 48| 1691]1751}1810)1869]1928111987}2046]2105/2164|2223 8)" 
 40} 2282/2342/2401)2400 2519}12578 2037 2696|2755 2814 9153 
 
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N.735. L. 866 OF NUMBERS. (133) 
 
 Ne Oo eee oor Pe a SP Gh PO be A pipe 
 735018662873|293 2|29921305 113 1 10)|3 169|3228|3287|3346|3405|| | 
 51| 346413523|358213641|3701||3760|3819/387813937 |3996 59 
 52| 4055'411414173|42321429 114350/4400|4468]452814587 1] 6 
 53|  4646/4705|4764|4823|4882]14941 |50001505915 11815177 aie 
 54, 5230152.95|5354|5413|5472115532|559115650|5709|5768 | % 
 55| 582715886|59045\600416063)/6 12216181 |6240}6299|6358 5130, 
 56| 6417|G476|0535|059416053|167 12|677 116830168890|6949 Spe 
 57|  7008\7067|7126)7185|7244||7303|7362|7421)7480\7539|| | 31° 
 58| 7598|76571771617775|7834|1789317952/801 1/8070|8 129 053 
 59| 8188|8247|8306|8305|8424|18483|8542|8601|8660)87 19 one 
 7300} 8778'8837|8896|8955 901 4]|9073|913219191\9250/9309}| 59 
 61}  9368/9427/9486|9545|9604)|9663 9722/97 8 1\9840\9899 - 
 62} 9958/0017|0076 01350194 0253/03 12/037 1/0430|048G) 
 63|867054810607|0666|07 25 07 841|0843 |0902|096111020)1079 
 64| 1138]1197|1256)1315}1374]11433|149211551|1610|1669 
 65| 1728]1786|1845|1904!1963}/202212081|2140}2199|2258 
 66| 231712376|2435|24904/2553)|261 21267 1|2730|2789|2848 
 67|  2907/2966|3025|3084/3 142!/3201/3260/33 1913378'3437 
 68}  34096/3555|3614\3673/3732)|379113850|}3909|3968!4027 
 69|  4086|4145|4203|/4262/4321]14380}4430|44908|4557/4616, 
 7370| 4075|4734|4793/4852/4011|'4970}5028!508715 14615205) 
 71| §264/532315382/5441|5500!15559|5618|5677|573515704| 
 72) . 585315912/5971|6030'6089]|614816207|6266)63 2516383) 
 73| 6442/6501'6560|6619|6678)|6737 |6796|6855|6914|6972 
 741 .7031]7090\7149|7208|7267/17326|7385'7444|7502|7561 
 75| 7620!7670|7738|779717856||7915|7974|8032'8091 8150 
 76|  8209|8268|8327/|83 868445118503 |8562/862 118680 8739 
 77\  8798|8857|8916/8974|9033/|9092|9151|9210,9269 9328 
 78|  9387194451950419563|9622\190811974019799|9857 0916 
 79| ~ 9975|003410093/0152\0211|10269|0328|038710446'0505 
 7380|8680564|0622|068 1|0740|0799)||0858|09 17\0976)1034)1093 
 81] 1152|1211]1270)13209|1387/}1446]1505|1564| 1623/1682 
 82]  1740|1799|1858]1917/1976]12035|2093|2152/2211|2270 
 83|  2320/2388/2446|2505|2564]12623 |2682/2740|2799!2858 
 841 20917/2976|3035|309313 152/13211|3270/3320|338713446 
 85}  3505|3564|3623'3681|3740137909|3858/3917/39754034 
 86]  40093/4152/421114269/4328]/4387|4446]4505|4563/4622 
 87| 4681/4740]479014857/4916)14975|5034150931515 1/5210 
 88]  5260|5328)5386)5445|550415563|5622|5680|5739|5798 
 89} 5857|5915|597416033|6092)|615116200|6268/63 27/6386 
 7390} 6444/0503/0562'6621|6679|}6738}5797|6856|691510973)| 
 91) 7032|7091|7150'7208|7267/|7326|7385|7443|75021756] || 
 92) 7620)7078|7737!7706|7855/17 9131797 2/803 1|8090/8 148 58 
 93]  8207/8266]832518383|8442/18501|8560|8618|867718736 ne 
 94] 8794|8853)8912/897 1/9029||9088]9147/9206|9264l9323|}  |212 
 95} 9382194411/949919558]9617)|19075197341079310852\9910 4125 
 96} 9G69|0028/0086/0145|0204]10263|0321|0380|043910497 5129 
 97|8690556|061 5|0674107321079 1||0850/0908|0967] 1026] 1085 . et 
 98} = 1143}1202/1261}1319}1378]]1143711495]1554]1613]1672 lis 
 99] __1730}1789]1848}1906}1965)]2024/2082]2141)2200|2259 9152 
 2131415161718 19 I pD\pts 
 
 O ] 
 
(134 LOGARITHMS N.740 L.869 
 1 Dt Sas 5 16 : 7 
 7400|86923 17|2376|243512493|2552||261 1126692728 
 
 2787|2845 
 
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 bel eerchoceae nh, selnseahnemonaatiores 7479\7537), | 9|33 
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 08| 7010 hes dong ‘ s is eel 7880 7948|8006|8065|8123 
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 711] 8768188278889|6044\9000 1906119120191 781923719200 
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 12} 9354/041319471195 | 400|0467 
 7410233|0292,0350|0400|0467 
 13) 9940/9999 a me ieee 0819 0877/0936|0994|1053 
 14|8700526|0584 wei wid 1404114631152211580 1639 
 15} 1112}1170/1229)1287 2.166|2224 
 He 7 1990/2049|2107]21 
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 LAME cst hited 85/3044|3 10213 1611322032781333713395 
 pleas tetas ea 3629|3688|3746,38053863)3922(3981 
 19| 345413512 il iN 41073 4332/439014449|450714566 
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 22) 5210/5268/5327/5385)5 3146'62041626316321 
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 23) 5795|5853)5912/59700 “216731167 891684816906 
 Sei UMR acres visit ie Wein aba yen ae ie 7432 mene 
 An 7 140!719917257|7310,7374|7432 
 25 0905/7023 “pte bis re ssl 900.7059 8017/8076 
 zane iv 8310|8368}}8427/8485/8544|8602|8660 
 2$|  871918777/8890/880418953 901 119070.9128|91 87/9240 
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 ie 923019289193479405|9464|19522195801963010097 ee \ 6195 
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 49| _0980}1038!1G96}1155]1213)|127 : | 
 
 7 ee 
 
N.745 1.872 OF NUMBERS. 
 
 Not cGy ee 2S Se Ge ls SEL) 9 hp iPe6 
 
 745018721503|1621|1670|173811796]]1854|19121197 11202012087 aa 
 51] 2146/2204)22.62}23 20/23 70112437|2495/25541261212670 
 52| . 2728/2787/2845|2903}2962||3020/307 8|3 13613 195|3253 
 53|  3311|/8369|3428/3486/3544||3603|3661137 191377713836 
 54| — 389413952|4010/4069/-4127||/4185!424314302/436014418 
 55] 44761453514593/465 1147001147 68/482614884|4042|5001 
 ».56} = 5059|5117)51751523415292/153.50/540815467/5 5525/5583 
 57| 5641)5700)5758/5816|5874115933|599 1|604910107|0166 
 58] 6224/6282/634016398164571165 15/6573/663 1166g0|6748 
 
 913419193/925 119309|9367119425'94841954219600/9658 
 971619774|9833|989 1|9949 |0007 (0065/0124/0182|0240 
 
 66} 0880)0938/0996)1054/1113/}1171)1229)1287]1345]/1403 
 67| 1462/1520)1578)1636}1604)}1752 18101869 1927/1985 
 68} 2043/2101/2150/2218/2276)2334 2392/2450 2508/2566 
 O69}  2625/2683)/2741|279912857/12915|2073/3032/3090)3 148 
 7470}  3206)3264|3322'3380/3430)/34097/3555'3613)/367 113729 
 71 3787|3845|3904|3962|4020)/4078'4136.4194/4252'4311 
 72\| 4360)442714485|454314001/14659/4717/4775 4834/4892 
 73) 4950/5008|5006)5 124/5182'15240 520815357 5415)5473 
 74 553115589 5647/5705 5763 5821 58805938 59060054. 
 75|  6112'6170\6228|6286}0344'16402/646116519 657716635 
 76}  6693)0751\6800|6867 6925 0983|704.1!7 100 71587216 
 77|  727417332|7390|744817506|7564|7622|7680 77387797 
 78} = 7855/7913/797 1|8029|8087 8 145)8203 8261183198377 
 79 8435/84G3|855 118610|8668'18726|8784!8842/8900 8058 
 7480]| 9016,9074/9132/9190 9248 9306/936419422 94809538 
 
 95979055197 13)977 1|9829}19887 190450003 0061/0119 
 
 _ 82'8740177/0235|0293/03 51 0409, 0467|0525|0583|0041|0699 
 83} 075710815;087410932/0990'}1048/1106)1164}1222/1280 
 
 133|139611454|151211570)1628|16861174411802|1860 
 
 1918|197612034|2092)21 50 2.208!2266/232412382/2440 
 2498|2.556|26 14267 2]273012788!284612904|2962'3020 
 
 3948)|4000/4064|4122/4180 
 
 89 4700! 
 7490 5108!5166)5224|5282/5340 
 gl 5920 
 92] 50978/6036)0004/6152\621016268163 25}63 83|04411649G 
 93 8471600516963 |702 117070 
 0A 7058 
 05 7716|7774|7 83217 890}7 948 |/8006}8064)8122}8180/8238 
 QO} 82096/835418412/8470|8528]/8585|86431870118750|8817 
 
(136) LOGARITHMS N.750 L.875 
 
 1192}1250/1307|1365}1423]]1481|153911597|1655|1713 
 1771|1828|1886]1944}2002||2060}21 1812176|2234|22092 
 234912407|2465|2523|25811!2639|2697|2755|2813|2870 
 2.928|2986|3044)3 10213 160}}3218|3275|33331339 1/3449 
 
 3507|3565|3623|3681|3738}13796|3854/391 2/3970/4028 
 
 046]  4086/4143}4201|4259143 17||4375|4433|449 11454814606 
 07| 466414722/4780/4838|4890]4953|501 115069|5127|5185 
 os} 5243/530015358|5416|54741]5532/55901564815705|5763 
 
 5821|5879|5937|5995|6052\|61 10|6168|622616284|63.42 
 
 6399|6457|6515|0573|663 1116689|6746|6804|6862/6920 
 6978|703.5|7093)7 15 1|7209||7267|7325|7382|7440|7408 
 755017614|767 117720\7787||7845|790317 960!801 88076 
 8134]8192|8240|8307|8305]/8423/848 1|8530|859018654 
 87 12/8770|8828/8885|8943]|9001|9059/91 161917419232 
 ~ 929019348]9405/94631052 1|9579|9037|969419752\9810 
 9868!992.519983|004.1|0099)|0157|0214|027210330\0388 
 
IN. 755 L.877 OF NUMBERS. 
 
 — | —____. ] /-—_—_—___ ] -————____ ] ————_.- | ———— | —___. 
 
 ee er eeneeneneeeans| eens I ene 
 
 2000/2057/2.115|217212230)2287 
 257 5|2632|209012747|2805|2862 
 3149/13207|320413322|337 913437 
 37241137 82)383913890/3954/4011 
 4299|14350|44.14)447 1/452014586 
 48731493 114988]50460/5103}516) 
 
 1770}1827}1885|1942 
 | 2460|2517 
 2919|2907713034|3092 
 34944355213600|3 667 
 4069/4 126]4184/4241 
 4643/470114758|4816 
 
 3425|3482/3540)3597|3054137 1213769 
 
 ss2gossab 3941 13998/4056|4113/4170/4228 4285/4342 
 4400\4457}4514}4572 4029)20 4086)4744|4801 si te 
 
 841 1|846818525|8582/8640)8697|8754/88 1 1/8869 
 898319041 9098 9155|9212/19270|9327/9384|9441 
 
 8926 
 9499 
 
 1216 
 1788 
 
 070110758]0815|0873/0930/}0987| 1044111021159 
 1273]1330|1388|1445|1502/]1559|1617|1674|1731 
 184611 903]1960|2017|20741|2132|2189|2246|2303'2361 
 
 2418/247512532|2580|2647|12704|2761|2818|2875|2933 
 2990'3047 |3 104/3 162'3219113276|3333|3390/3448 3505 
 3562/3610|3676/3734/3791 ||3848/3905|3902|4020.4077 
 4134/4191|4248|4306|4363||4420)447 714534 4592/4649 
 4706)47631482014877|4935||4992|50491510615 163/522) 
 
 5278}5335}5392|5440|5507|5564|5621|567815735|5792 
 5850!5907 |59041602 1 |6078}|6135|6193|625016307\6364 
 64211647 815536|6593|6650]|67 07|0764|682116879|6936 
 6993}7050|7 107|7 164|7222))7270|7330|7393|7450\7507 
 90) 7504|7622)7679|7736|7793)|7850 7907|7964|8022/8070 
 
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2 3 
 
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 5615|507215720)5786//5844|5901|5958|001 5/6072 
 6186|624316300)6357||0414|047 116528/0585/6642 
 
 6756|6813|6870|6927||6984|7041|7098)7 1155/7212 
 7326|7383|7440\7497||7554|7611|76609|7720|7783)| 
 7897|7954|801 1|8008}/812.5|8 182/823Gg|8296/8353 
 8467|8524|8581|8038)|8695|87 52|8809|/8806/8923 
 (9037190941915 1|/9208)/9265|9322'93701943 619493 
 fee 966419721197 78]'9835|9892'9949|0006,0063 
 77|0234|0291)0348| (040510402105 19|0575\0032 
 One 0803!0860|0917||0974|103 1/1088]1 1451202 
 1259)1316]1373]1430)1487||15441100111658}1715|1772 
 1820}1886)1943/2000/2057|/2114]217112228]/2285/2342 
 2398|2455|2512)2569|2626]||2683]2740/2797|2854/2911 
 2968|302.5|3082)3 139|3 196)|3253/33 10|3367|3424|3481 
 3537/3594|305 113708|3705/|3822/3870/3936|3003|4050 
 4107|4164)/4221|4278)433 5) 4392)4448/4505|4502/4019 
 4676)4733/4790/4847 4G04||4961|5018)/5075|5132/5188 
 §24515302)53509|5416|5473]/5530|5587|5644|570115758 
 5815|587 1|5928|5985|0042||0099|0156|02 13|6270|6327 
 6384)044.1,0497|6554/601 1||6008/6725/6782/683916896 
 6953|7010|70006)7 1237 180)|7237|7 2941735 1|7408|7465 
 7522\7578/7035|7692/7 749 7 806}7 863|792017977|8034 
 8090/8147/8204|8261 8318 837 5|8432|/8489|8545|/8602 
 8059/87 16|8773|8830|8887||89441900019057/9114/9171 
 922819285|9342/9399/9455/105 12/9500/9626/968319740 
 9797|9853|99 1019907 |0024|008 1}0138/0195|025 1}0308 
 2'0479|0536|0593 ||064.9|0706|0763|0820/0877 
 0934/0990) 1047}1104/1161/}1218]1275]1331/1388]1445 
 1502|1559 1616}1673/1729)|1786]1843]1900)1957|2014 
 2070)2127/21841224 1/2298 /12354}241 1/2468]2525/2582 
 2639|2695|2752|2800|2866)|2923|2080!3036/3093)|3 150 
 
 6047'0103/0100|0217 0274 
 
 3207 |3264/332013377|3434 
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 5479|5536)5592 5049 5700, 
 
 DOLh ts: | & ba 
 
 3491|3548|3604|3061|3718 
 
 4059/4116/4173|4229|4286 
 4627|4084|47411479714854 
 5195|5252|5308|5305|5422 
 5763|58191587015933/59g0 
 6330|0387|6444|6501/6558 
 
N.765 L.883 OF NUMBERS. 
 O l 2 
 
 Pia ae | Hi 
 
 Rann CHRIS IRS OT cIRS a aiRe AD Ne ne ee 
 
 0586|0643/0700]0757/0813|10870|0927|0983|1040]1097 
 1154|1210]1267}1324]1380]11437]1494/1551/1607|1664 
 172111777]1834]189 1/1948] 2004]2061/2118|2174|2231 
 
 2288]/2344/2401/2458/25 1411257 1|2628|2085/274 1/2798 
 2855/29 1 1]2968|3025/308 1113 138]3 195/325 1|3308|3365 
 342 113478/3535|3592/3648]13705|3762/3818/3875|3932 
 3988/4045/4102/4 158/42 15||4272|4328/4385|444214498 
 4555/4612/4668 Bane 4838]4895|4952|5008|5005 
 
 §122)5178)5235|5292,534815405}5462/5518/5575|563 1 
 5688/5745/5801 5858/5915 597 1|6028/6085|6141|6198 
 6255163 1 116368|6425'6481110538|6594'605 11670816764 
 682110687 8/693410991!7048]7 10417 16117217|/7274|7331 
 7387 |7444\750117557|70141707 117727177 84|7 840|7 897 
 7954|8010/8067 8124/81 80]|823718293|8350'8407|8463 
 - 8520/8576,8633|8690'8746|18803|8860 891618973 gO029 
 9086/9143/9199 925619312 9369|9426/9482'9539|9595 
 9652 hes 9705|9822'9878]|9935 9992 0048/0105/0161 
 
A603 oes 4772\4828||488414940/4006 505215 108 
 5165} Deane 2.77\5333|5380||54451§501|5558|5014|5070 
 
 5726|5782 
 6287|6343/6 
 6848]6905|6961|7017|70731|7120|7185|7241|7207|7353 
 7410|7406:7522\7578 
 
 5838 5894)5950}|9007|6063/0119 6175/0623) 
 
 8532 8588/9044 8700|8756||881218868|8924|8980|9037 
 
 g093 914919205 9261193 17|193731942019485|954119597]| 
 
 9053/97 109706 9822/0987 8]10934|9990|0046|0102/0158 
 45 8890214 Ki des 0382 0439 Ope 0551 0607 066310719 
 
4210)|4266|4322}4378 4434114490/4546 4601 465714713 
 
(142) LOGARITHMS N.780 L.892 
 Of 1] 24. Sef 4d Slt 6 ]i'7 [: 8! 1:9 Pay tape 
 
 —- — — | — J —————————__ | ————_ —] | -———— | | | |] 
 ee 
 
 Ol 1503|1558]1614}1670}1725/11781/1837]1892]1948|2004 
 02} 2059/2115/2171|2226/2282)|2338/2393/244012505|2560}). 
 03} 2616/2672|2727|2783|2830|/2894|2950)\3006|306113117 
 04} 3173)3228/3284/334013395||345 113500|3562|361 8/3673 
 05} 3720/3785|3840/3896|3952)|4007|4063)4.1 10/41 7414230 
 O06} 4285/4341/4397|4452)4508]14564|4619|4675|473 1/4786 
 07| 4842/4897/4953|5000|5064115 12015 176/523 11528715342 
 O8}| 5398|5454|5500|5505|5621)|5676|5732|57871584315899 
 09! 5954/6010/6065/6121/6177||6232/6288/634416399/6455 
 7810| 6510}6566|/6622|6677|6733|16788|6844|6900|6955/701 1 
 11{ | 7066)7122)7178!7233)728017344|7400)7456175 11/7567 
 12} 7622)7078/7734/7780|7845||7900/7956\801 118067/|8123 
 13} 8178/8234/8280|8345]84011|18456'85 12/8567|8623/8678 
 14] 8734/8790|8845|8901/895619012'9068|91 23/9170\9234 
 
 9290|9345|9401 |9457/9512 ga6s 9623 9€79|973419790 
 9846|9901 9957001 210068 012301 79:0234|0290/0346 
 17|8930401|0457|05 12 0568/0623110679,0734/0790|0846|0g01 
 
 18} 0957|1012}1068)1123}1179)1234)1290)/1345|1401|1457 
 19 1512}1568|1623!1079 ha 1790/1845) 1901}1956|2012 
 7820|  2068/2123/2179|2234|2290)|2345|2401/2456/25 12/2567 
 21 2623|2678|2734/27 80/284 5)|2G00)/2956)3012/3067/3 123 
 
 3178/3234|32809'3345|3400, 3456351 113567 3622/3678 
 23)  3733)/3780|3844'3900)3955/1401 1/4006/41 22/4177/4233 
 4455 4510/4566 4621/4677 4732|4788 
 
 24} 428814344/43909 
 251  4843/4890|4954|5010 5063 5121/51 7615032 528715343 
 26} 5398!5454|5500|5565|5620)/5676'573 115787|5842|58g8. 
 27;  5953|6009|6064/6120)6175)1623 1|0286'6342|6397|0453 
 28 6508|6564/6619/6675|6730 O7BO)054: 0897|6952|7007 
 29} 7063}7118|7174|7220|7285]7340 (S008?) 7507|7562 
 7830|  7618/7673'7720\7784|7830)17805 7950)8006 8061}8117 
 31 8172/8228!8283/8330|8394|/8450/8505/8560/861 6/8671 
 32| $727|8782'8838/8893/894919004|9059\9115|917019226 
 33 9281/933719392|9448/9503/19558!9614\9660|9725|97 SO 
 34| 9836/9891|0947/0002/0057/|01 13/0168|022410270|0335 
 
 35|8940390|0445|0501 055610612'|0667/0723|0778|0833|0889 
 
 36} 0944]1000)1055]1111 1160 1221/1277|1332}1388}]1443 
 37] 149811554)1609)1605]1720)1776)183111886]1942}19907 
 38] 2053)/2108/2163)/2210)2274112330|2385/2440/2406/2551 
 30] 2007/2062|2717/2773/2828]2884|2939|2994|3050)3 105 
 7840] 3161/3216)3271/3327|/3382]3438|3493|3548]3604136059 
 All 3715/3770/3825|388 113936300 1|404714102}4158|4213 
 42) 4268143241437914435|4490]4545|4601/4656)47 1 114767 
 43} 4822/4878]4933/4988/5044115090]/5 154|5210|5205|5320 
 44} 5376)5431}54871554215507115053|5708/5763|5819|5874 
 45} 5920|598516040/0096}6151}|6206|6262)/63 171637216428 
 46|  6483/6538]65941064910704116760,68 1 5|6870|6926|608 1 
 
 7037|7092\7 147|7203]7258]73 13|7360|7424|7470|7535 
 7590|7645|7701|7756|7811||7867|7922|7977|8033|8088 
 8143/3190|825418300}8365]|8420|8475|853 1|8580|864 1 
 
 SS ee eS ed ee ee ne ee ee ee ee ee eee 
 
 O ;/14,2)3 1,4] 516,718 19 | D}Pts. 
 
N. 785 L.894 OF NUMBERS. 
 
 oe ——— 
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 2421|2476)253 1/2586|204112696|275 1/2806) 2861/2916 
 a 3301|3356.341 113466 
 
ee, Ce eS ee nnn een | a a, Fe 
 
 3960/4015|40701412.5|418014235|4290/4345 
 
 15|  45009/4564/461 914674/4720]4784/483 8/4893/4948/5003 
 16} 5058)5113/5168)5222/5277)5332)5387|5442 5497/5552 
 17| 5606/5061/5716|5771|5826|15881|5936|5090,6045 6100 
 18]  6155'6210:6265,6320'6374||6429|/6484'0539'0594 0649 
 | 19]  6703\6758'6813/6868\6923||0978|7032'7087|7 142'7197 
 $7920) 725217307;\7361/7416\747 1||75260|7581\7636 7090 7745 
 23} — 7800\7855|7910'7965'8019||8074'8129181 1848239 8294 
 22;  $8348/8403'8458/8513 8568 8622|8677|87 32,8787 8842 
 23; 8807 895119006 906 119116)917 1/9225 9280 9335'9390 
 24; 0445949090554 9009 9664)\07 19|90774|9828 ‘9883 9938 
 25} 9093 Gods 
 26/89090541 05985, 50 0705 0760)0815|0869 
 27|  1089)1143)1198)12 
 28} 16361691 174611801 1856)|1910|1905/2020: 
 29] 2184/2239 safle 2403 |2458)2513 sce es 2677 
 
 7930, 2732127872841 28961295 1|3006/3060131 15'317013.225 
 31 3279133343389 344413499) 3553/3608 3603 37183772 
 32|  382713882/3937;: 
 33| 4375/4420|4484 
 
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 37) 6564 6619 6073,0728 67 83||6837|6892/69471/700217056 
 38} 7111|7160)7220:7275|7330)73841743917494754917608 
 
N.795 L.900 OF NUMBERS. 
 
 |] | | SO |  — | EO I EI OO” 
 
 4218 ies 4327|4381144361449114545|4600/405414709 
 4764|4818]4873]4928]4982)|5037|5091|5146|5201/5255 
 §310|5364|5419154745528}|5583|5637150902|5747|5801 
 §856]5910|5965|6020|607 41/61 20|6183|6238|0293|0347 
 
 6402]6456165 1 116566|6620|1667 5|67 20/07 84/6839/6893 
 6948}7002|705717 11217 166]]7221|727517330|7384|7430 
 7494|754817603|7057|77 12117 7060|782117876)7930)7985 
 803 9]809418148/8203 |8258]183 12]83671842 1|8476)8530 
 8585|8640|8694|8749|8803]18858|89 1 2|8967|9022|9076 
 9131/9185|9240|929419340||/9403/0458/951319567|9022 
 9676|973 119785|9840|989411994.9|0004|0058)01 13|0167 
 0}|04.94|054.9|0604|0658|07 13 
 0767|0822/0876|003 110985}|1040}1094]1 149]1203}1258 
 1313}1367|1422)1476]153 1||1585]1640|1694|17409)1803 
 1858]1912|1967|2021/2076]2130)2185/2239|2294/2349 
 2403|2458|25 1 2|25671262 1|/2676|2730)2785|2839|2894 
 2948|3003|3057)3 112/31 66/1322113275|3330)33 84/3439 
 3493|3548|3602/3657137 1 1||3766|3820|3875|3929/3984 
 4038|4093|4147/4202/42.56]|43 ] 1/4365|4420/4474/4529 
 
 4583|463 814692'4747|4801)|4856|49 10/4965|501 9/5074 
 512815183]5237|529215346|540115455|5509/5504/5018 
 5673|5727|57 82158361589 1115945|6000|0054/0109|0163 
 6218/6272|63271638 1\6436)|0490|6544|0599|6053/0708 
 6762|6817|687 116926|6980)|703 517 089)7 144|7 198\7252 
 7307|7361|74161747017525117579\763417 688|774317 797 
 785 1|7906|7960)|801 5|8069)}8124|8178/8233|8287)|8341 
 8396|8450|/850518550|861 4||8068/872318777|883 1|8886 
 8940|8995|9049/9 104/91 58||921219267|9032 1193769430 
 9485/953919594|964819702!|/97 57/981 119866\9920\9974 
 
 0573|0628|0682/0736|079 1||0845|0900|0954|1008]1003 
 1117/1172|1226}1280]1335]|1389]1444)1498}1552)1607 
 1061]1716)1770|1824]1879]|1933|1988|2042/2096/2151 
 2205|2260/23 14|2368124231|247712532|2586|2640/2605 
 
 2740|2804/2858}29 12|2967||302 1/3076|3 130/3 1843239 
 320313347|3402/3456135 1 11/3565|3619|367413728/3782 
 3837|3891|3946|4000)4.054/14109/4163/4217/4272/4326 
 438 1)/4435/4489/4544/4598||405214707/47011481 5/4870 
 4924/4070|5033|5087|5 142115 196|5250|5305/5359/5413 
 
 5468)5522'5577|563 1|5685||5740|5794|5848|5903/5957 
 601 1|6066 6120)617416229|16283|0337|0392/6446|6500 
 6555|6609/6663|67 18|6772)|6826|688 11693 5|6980|7044 
 7098|7 152)7207|72611731517370|7424|7478|7533|7 587 
 7941|7696)77 50/7 80417859/7913|7967|8022/8076|8130 
 8185|8239/8293/8348/8402||8456)85 1 1|8565|8619/8674 
 
 rn | eee | 
 
 O 112 
 
 —— ff | Pe —_————— J —— —__| -—_—__ —_- ] —__-___- = SS) oe 
 
LOGARITHMS _ N.800 L.903 
 4 
 
 4450|4505/4559|4013 4607 |4721|4775|48294883\4037 
 4992|5046,510015154|5208]5262153 1615370\542415479 
 5533 5587/5641 5695|57491580315857|592 115965|6020 
 6074 6128161 82'6236|629016344|6398|6452|650610560 
 661 5|6669'0723 67771683 1|16885|6939|6993|7047/7101 
 7155|7210'7264173 18|7372'7426|7480)753417588|7642 
 7750'7804|785817913|7967|8021|8075|8129|8183 
 
 9967002 110075|10129|0183|0237|029 1/0345 
 
 9859/9913 
 0507|0561 061 5||(0600|07 2407 78/0832 0886 
 
 _30|/9050399|0453 
 
 2560|2615|2669|2723|2777|283 112885|2939|2993|3047 
 
2U 2 
 
 JN.805 L.905 OF NUMBERS. (147) 
 N.| O Pete: te ae Ww Se Ge 1 8.4). 9 ot D iPro, 
 805019057950|8013|8007|812118175||8229|828218336|8390|8444 
 51| 8$498]855218606|8660187 14]/8768|8822|8876]8930|8984 54 
 52] 9038!9092/9146|919919253]]9307|9361|941 5|9469/9523 k . 
 53] 9577|9631|9685}9739|9793||9847 19901 |9954|0008|0062 3116 
 54|90601 16|0170|0224/0278|0332/10386|0440|0494|0548|0002 4l22 
 55| 0655|0709|0763|0817/087 1|0925|0979|1033]1087/1141 : oH 
 56} 1195|1248}1302}1356/1410]1464]1518|1572)1626)168 Hoe 
 57| —-1734{1788]1841/1895|1949||2003|2057}2111/2165}2219 8h43 
 58| 227312327/2380!2434|2488!12542|2596/2050)2704/2758 9149 
 59| 2812/2865!2919/2973/3027//308 1/3 135|318913243/3297 
 8060} 33501340413458'351213566]3620/367413728/378113835 
 61}  3889|3943)3997|405 1|4105||4159/4212/4260/43 20/4374 
 62} 4428)448214536.45090\4643)14697/475 1|4805/4 8590/4913 
 63] 4067/5020 50745 128/5182||5236|5290'5344 5397/5451 
 64} 550515559)5013|5607|57211|577415828|5882/5936|5990 
 65| 6044|6098/015 1|6205'6259)|63 13|0367|642 1164 74/0528 
 66] 6582|6636|6690\0744'6798)|685 1 |6905|6959'7013|7067 
 67| 712117174|7228'7282'73361|73901744417497/7551|7005 
 68] 76509|771317767|7820|7874|17928|7982|8036)8090|8143 
 69} . 819718251|8305'8359/8412 Th Newt 8574,8628|8082 
 8070! 8735/8789|8843'8897/8951119004,9058/91 1219166 9220 
 71| 9273193271938 11943519489)|0543 | 
 72| 9812\9865|9919'9973|0027||008 1 Sie 01880242 02906 
 73190703 50|0403 0457/05 11105650018 0672/0726 0780/0834 
 74| 088710941/0995|10409/1103//1156)121011264'1318]1372 
 75|  142511470|1533/1587|1640]11694'1748 1802) 1856/1909}|° 
 76} = 1903)/2017/207 1)2124/2178)/2232 2286/2340 2393 |2447 
 77|  2501|2555|2608'2662!27 162770 2823/2877:293 1 (2985 
 78| 303813092 31463200 3254)/3307|3361 3415/3409 3522 
 79|  3576136303684|37371379 11138453809 3952/4000 4000 
 8080} 4114]4167\4221|4275]4320|14382'4436144904544/4597 
 81| © 4651/4705|4759|4812/4866 4920'4974 5027/5081/5135 
 82| 5188]5242/5296/5350)5403115457/551 115565 56185072 
 83} 5726|5780,583315887|5041/5994\6048|0102 '6156'6209 
 84|  6263163171637016424|047810532'6595|0039'669316747 
 85| 6800|6854/6908'6961)7015]|7069'7 123|7176.7230'7284 
 86| 73371739117445|7498|7552l1760017660 771377 67/7821 
 87| 787417928|7982|8036|8089]/8143'8197/8250'8304'8358 
 88 841 1/8465|/8519|8573|8626)/8680)/8734 8787/8841 8895 
 89} 8948!9002'|9056|91090|9163|1921719270'032419378 9432 
 8090} 94851953919593|9646|9700)|9754/9807|986119915|9968 
 91|9080022}0076/0129/0183|0237||0290\03 4410398045 1'0505 
 Q2| 0559}0612/0660,0720]07731/08271088 1109340988 1042 53 
 Q3| 1095/1149)1203/1256]1310]]1364]1417|1471/1525 1578 i] 5 
 94}  1032|1680)1739/1793]1847]]}1900|1954/2008|2061 | bi : ie 
 Q5| —- 2109|2222122761232912383}|2437|2490|2544/2598 2651 421 
 96} = 2705|2759]2812|2866] 2920|'2973|3027|3080 313413188 5\Q7 
 Q7|  3241|3295|3349|3402|3450 3510 3563(3617|36703724 oe 
 98] 3778)3831|3885/3936|39921|4046|4099/4153|4207/42600)] 155 
 __ 99 A314/4368|442 1/447 5145261|458214636/468G]4743'4797 lh 
 MCORP LEA Sa Spek eh s ) op tpe: 
 a aaa iain en i ee 
 
LOGARITHMS ~ N.810 L.908 
 N. O Ley De Sa ee Fee 1 13. tO ai) Pro. 
 
 SET een Ga ae LE oe Lee ERNE FURL RCOS BER SEE Gertie 
 
 Ol] 5386/5440|5494|5547|5601|5654|5708|5762)5815|5809 54 
 
 02| — 5922|5976}6030/6083]6137110190|624416298|635 116405 : . 
 
 04) 6994|7048)7102)7 155|7200/|7262)73 16|7360|7423|7477 aise 
 05| 7530/758417637176011774517798|7852|7905|7950|8012 5127 
 06} 8066|8120)8173|8227/8280]8334|8387|8441|8495|8548) | S22 
 07 8602/8655|8700}87 62/88 1 6|/8870|8923/8977|9030|9084 8143 
 08}  9137/919119245|9298|9352)/9405]9459/95 12}9566/9619 9}49 
 
 11] — 074410798|0851\0905|0958}|1012}1065}1110]1172)1226 
 12] 1279]1333|1386]1440}1494)|1547|1601|1654|1708|1761 
 13] 1815]1868}1922}1975|2029)|2082}2 136}2 1 80|2243|2297 
 14, 2350)2404/2457/251112564)|261 8]267 1|272.5|2778|2832 
 15|  2885]2939|/2992/3046|3099]/3 153|3206/3 260)33 13|3367 
 16] 3420/347413527|3581|3634)/3688/374113705|3848|3902 
 17} 3955]4009/4062/4116/4169/4223/4270/4330/4383/4437 
 18} 4400/454414597|465 1|4704)/4758]/48 1 114865|4918/49072 
 19] 502515079|5132|5186)5239]5203|5340|5400|5453|5507 
 8120}  5560/5614|5667/572115774/15828)588 1|503515988|6042 
 21}  6095/6149|6202|62506|6309||6362)041 616469|6523|6576 
 22| 66301668316737|6790|0844)/080971595 1|7004|705817111 
 
 716517218|727 117325|737811743217485|7530)7592|7 646 
 
 24| 76991775317806/7860!7913||7966|8020/8073|8 12718180 
 25| 8$23418287/8341|8304/8447/1850118554/8008|8661/87 15 
 26|  87681882218875|8929|89821903519089|914219196|9249 
 271  930319356)409/0463 9516 9570|9623196771973010784. 
 
 9837|98901994419997 esi 010410158/021 11026410318 
 291910037 110425|0478|0532|05850638|06921074510799|0852 
 
 0905|0959]L012}1066 111g 1173|1226)1279)1333]1386)|  ~ 
 
 8130 
 31} — 1440]1493]1546)1600|1653||1707|1760}1813]1867/1920 
 32| 1974/202712081/2134|2187)|2241|2209412348]2401|2454 
 33] 2508]2561|2615|2668|272112775|2828|288212935|2988 
 
 34] 3042/3095|3148|/3202|325513300|336213415]3469|3522 
 35|  3576/3620)3682|3736/3789!1384213896|3949|4003 [4056 
 36} 4100/4163/4216)4270/432311437614430/448314536/4590 
 37| 464314697/4750/480314857||4910/4963/5017|5070/5123 
 38] 5177|5230/528415337|5390)544415497|5550|5604|5057 
 39| 5710)576415817/587 115924159771603 1/6084|6137/6191 
 8140] 6244/6297/635116404/6457)165 1 116564|6618|667 1/6724 
 A} 6778/0683 1|6884/6938}699 1|/7944|7098]7 15117204|7258 
 42) 7311)7304174181747 117524)/ 5781703 1|7684|7738|7791 53 
 43} 78441/7898|7951|8004/8058//8111/8164/8218/827 118324 I] 5 
 44] 8378|8431184841853818591)18644|86981875 118804|8858 3 + 
 45] 8911}896419018}9071|9124119177|923119284|9337|9391 4121 
 46]  944419497/955 1960419657 197 11|9764/9817|/987 119924 {27 
 47| —_997710030/0084/0137/0190/024410297/035010404|0457 : ty 
 48/9110510|0564|061 7|0670|0723//077710830/088310937 |0990 al4o 
 49] 1043}1096}1 150}1203}1256}1310/1363|1416/1470]1523 948 
 Nelo OL TLS) 34 bevel 7a (oun ipa. 
 
N.815 L.911 OF NUMBERS. (149) 
 
 a a ee, el ieee’ ieee | od ee ee ee 
 
 8150}9111576]1629}1683 1736|1780||1843}1896]1949|2002|2056 
 
 51} 2109}2162}22.15|2260|2322)|237512429|2482|2535|2588 54 
 52] | 2642/2695|2748|2802/2855]!2908]2961 |301 5|3068/3121 15 
 53| 317413228/328113334|3387113441 134941354713601|3054 ps 
 54]  3707|3760|3814|3867|3920}3973/402714080/4133/4186]] | 15 
 55] 4240/4293|4346/4390|4453||4506/4550|461214666/4719 5/27 
 56] 4772)}482514870|493214985||5038|509215 145|5198|5251 6/32 
 57] 5305|5358|5411|5464|5518||5571|5624|5677/5731|5784|| | ahs 
 58] 5837|5890|5943|5997|6050)16103|6156|62 1016263163 16 9l49 
 59| 6369|6423|6476|6529165821|6635|6089|6742|6795|6848||  |-— 
 8160} 6902/6955|7008!7061)7 114)|/7168|7221|727417327|7381 
 61} 74341748717540175031764711770017753|7806|7859|7913 
 62] 7966|8019)807218126)/8179//8232|8285|8338|8392|8445 
 63| 8498/855118604|8658/87 111187641881 7|8870|892418977 
 64) 9030/9083/9136)9190|9243|19296|9349|9402/945619509 
 65| 09562'9615|9668/9721|9775||0828|9881|9934|9987|0041 
 66|9120094/0147/0200!025310306/}0360/041310466|05 1910572 
 67 0626|0670|0732/0785|083 8/1089 1109450998} 105 1}1104 
 68} 1157|1210)1264}1317/1370)|1423]1476]1529]1583|1636 
 69 1689|1742)1795]1848]1902}/1955|2008/2061|2114/2167 
 817 2221|2274|2327123 80/2433}|248612530|2593|2646|2609 
 71 2752|2805|2858)29 12}2965||301 81307 113 124/3177|3230 
 72}  3284'3337|339013443/3496]|3549|3602|3656/3700|3762 
 73|. 3815|3868!3921|3974/4028}|408 1|4134'4187/4240/4293 
 74| 434614399/4453/4506)4550}|4612/4065147181477 11482 
 75|  4878|493114984|5037|5090)5 14315 196|524915303|535 
 76)  5400|5462/5515/5568)562 1115674/57281578 11583415887 
 77\|  5940}5993|6046/6090}6 1 52||6206|6259/63 12|6365/6418 
 78|  6471|6524'6577|6630/6683)|'673716790\6843|6896|6949 
 79} 7002)7055|7 1087 161)72141/7268)7321173741742717480 
 8180]. 7533|7586\7630\7692|7745||779817852'7905|7958|801 1 
 81]  8064/8117/8170/8223|8276]'83 2018382'1843618489|8542 
 82}  8595|8648/8701/8754|8807/|8860/8913|896619019|9072 
 83 9126/9170\9232'90285]9338}193911944419049719550|9603 
 84] 9656/9700|0762/981519868|19922|9975|0028|008 1/0134 
 85|913018710240!0203/0346|03901104521050510558|061 110664 
 86] 0717|0770|0824/087710930)|0983]1036]1089|1142}1195 
 87} 1248]1301}1354/1407/1460/11513]1566)1610)1672)1725 
 88]  1778|1831]1884/1937]1990)|2044|2097/2150|2203]2256 
 89] 2300123621241 5/2468)]25211|25741262712680|2733/2786 
 S190]  28309]/2892!2945/2998|/305 11/3 10413 157|3210)3263/3316 
 Ql] 3369|3422/3475/352813581||3634|3687|374013793|3846 
 92}  3890|3052/4005/4058]41 11)/4165]4218!427 11432414377 53 
 93|  4430/4483/4536/4580]4642/}4695|4748]4801/4854|4907 fo 
 94 4960|5013/5066/511915172/15225|5278/533 115384/54371| 53 ||" - 
 95]  5490/5543|5596|5649|5702)|5755|5808|5861|591415967 At 1 
 96} 6019/6072/6125|/6178)623 1]}6284|6337|6390/}6443|6496 5j27 
 97|  6549|6602|6655|6708|6761116814|6867\692016973|7020]] [2/3 
 98] = 7079|7132)7185|7238]7291||734417307|7450)7503|7550 8lag 
 00] 7609|7662|7715|7768)782111787417927|7980/8033|8086 9148 
 
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 8920|897 1/902 1|907219123)|9173|9224|9274|9325|9375 
 04261947 71952719578|9628}|9079/9729197 80/983 119881 
 9932|9982|0033|008310134}|0184|0235|0286|033610387 
 0740|079110842/0892 
 1246)1296}1347|1398 
 
 0943 |0993|1044/1094]1145]11195 
 1448}1499]1549}1600}1650)}1701/1751)1802)1852]1903 
 1953|20042055|2 105}2156/12200'225 7/2307 1235812408 
 24509}2509|2560|2610/206 1/27 1 1/2762/2812/2863|2914 
 2964|3015|3065/3 1160/3160 Aaa? 33 18)/3308/3419 
 3460|3520/3570/362 11367 1113722/377213823|387 313924 
 3974|4025|4075)4 120/41 76/14227/4277143281437 8}442G 
 4783|4833|4884]4034 
 
 2037||2088|2139/2189/2240!22911) 
 
 —_—_—_— 
 
a LOGARITHMS  N.860 L. 934 
 2 aA Rew ae de : 
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 2049|2099|2.150]2200/22.501123011235 112402'245212502 
 2.553|2603|2054|2704'2754|12805|2855|2906129563006 
 3057/3 107(3 15813208 325g 13300133591341034603511 
 3561/3611 11360213712'3763113813|386313914'396414015 
 4005/41 15/41 66|42 1614266] 431 7|4367/4418'4468/4518 
 456914619|407014720 4770)4821|4871149221497215022 
 
 50731512315173|5224'527411532515375|5425 54765520 
 5576\5927|507 7157 28)57 7 8||5828/5879 592915979, 6030 
 60801613 116181|623.1'6282!16332'638216433'6483'6533 
 6584|663.4166851673516785| 683616886 6936698717 037 
 7087|7 138|7 188|7239;7289 733017 7390|7440,7490 7541 
 75911704117692|7742!77921'7843|789317943 799418044 
 8095|8145/8195|8240'8206)'83461839718447/8497/8548 
 8598|8648|86991874¢ 98709) '8850)8900|89509001 9051 
 G10119152|920219252'9303 9353 9403|945419504'9554 
 9605|965519705|97 50 9806) 9856|9907|9957|0007|0058 
 8630 9360108 0158}0209}0259 0309) 0360'0410/0460'0511/0561 
 
 061110601 Vea 07620812! 0863/0913 0963)1014}1004 
 11809 1416|1466)1517|1567 
 
 3126|317713227|3277 3327| 
 3629|3670|3730'3780.3830]'388 11393 1139811403 114082 
 413214182)/423314283/4333]) 
 4635|4685|4735|4786|483 6} 
 
 5137151881523 815288/5338 
 5640|5690]5 7411579115841} 
 6143|019316243/6293|6344] 6 
 6645|6695|6746'6796,6846 
 714817 198|7248}729817340 
 7650|7700}7750|7801|7851 
 8152/8203|8253/8303/83 53 
 865518705|8755|8805/8855 
 
IN.865 L.937 OF NUMBERS. 
 
 3050 9370] 6110211 }0261103 12103 62|'0412'0462105 121056310613 
 51 0663)07 13|0704/08 1410864}1091 4/0964}1015}1065]1115 
 §2)  1165]/1215]1265|1316]1366)|14.16}1406/1516)1567]1617 
 53|  1667|1717}1767|1818/1868}|1918)1968!2018}2069|2119 
 54| 2160/2219/2269|23 19|2376)|2420/2470/252012570)2621 
 55| = 2671|2721N2771|2821|287 1112922|2972|3022)307 213 122 
 56} 3172|3223|327313323|3373|13423|347413524)3574/3624 
 57| 3674137241377513825|387511392513975|4025|407 514126 
 58| 4176|4226/4276)4326/4376)4427/4477145271457714627 
 59} 4077|4728]4778,4828/48781/492814978/5028/507915 12g 
 
 8660} 5179|522915279'5329|5380]|5430 5480/5530 5580/5630 
 61]  5680/573115781|583 1|5881]/593 1|5G8 1,603 116082/6132 
 
 62)  6182/6232/0282/633210382]'043216483/05331/0583|6033 
 
 63} 6683/6733|6783'6834|0884]|6934|6984'7034)708417 134 
 64° 718417235 7285/7335)7 73851:7435174851753517585|7036 
 
 65| 7686773617786. 783617886] 793617986 8037|S087|8137 
 
 66] 8187|8237|8287\8337|8387| 8437 7|8488)5538 8588)8638 
 
 07 8688|8738|/8788'8838/88ss 8939 8989) 9039 9089/9139 
 
 68} —9189|923919289'9339|9389 9440 949019 954019590)! (0640 
 09} 9690\9740/9790,9840/9890;:9941|9991 poe 0091/0141 
 8670/9380191/024,1/0291 034110391 |(0441|0492,054210592/0642! 
 
 71 0692/0742/0792/0842|0892 0942/0992) 1042 1093}1143 
 72) —-1193|1243}1293/1343}1393}|1443/149311543]1593|1643 
 73 1693|174411794)1844)1894})194411994'2044|2094'2144 
 741 2194)2244/2294/2344|2394//2445 2495/2549 2595/2645 
 75 2095|2745|2795|2845/2895)/2945|2995 3045|3095/3 145 
 76| 3195|3245 329613340 3396||3446134¢ G6 354013596,3646 
 21 369613746|3796.38461389613046 3990 4040 40904146} 
 781  4196/4247|4297/434714397|/4447/4407.45471459714647 | 
 79) 4097/474714797/4847}4897]4047 4997 5047}509715147] 
 8680) 5197.5247 5297|5347|5397||5447 5497 5547/5598 5648} 
 8] 5098'5748157 98|/5848|5898)|\5G48 5998 0048/6098) '6148|| 
 82} 6198'6248|0298|0348|039s!/6448\0498 '6548|0598'6648 
 684810898 0948/6998 704s 70987 148} 
 84|°  7198'7248|7298/7348/7398)|7448 7498\7548\" 7598\7048 | 
 85 7008'774817798'7 84817 898)|7948 7998 
 86] 8198|8248/8298/8348/8398)\S448/8498,8548/8598/8648 
 87} 8698}8748]/8798)8848|8898]\89048!8998' 9048}9098/9148 
 88] 9198/9248/9298/9348/9398)'G44819498: 9548 9598|0048 
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 g1 0697 |07471|0797|0847)|0897||0947 |09097) 1047|1097|1147 
 92} =: 1197|1247}1297|1347|1397||1447|1497|154711597|1647 
 - 93) 1697}1747|1797|1847}18971||19471199712046|2006)/2 146 
 Q4| =: 2196/22.46/22:96/2346/2396)|2440/2496/254612596/2646 
 95 2696|2746|2796|2846)2895)|2946|2096/304.5|3095|3 145 
 96} 3195)324513205|334513395||3445|3495|3545|350513045 
 97| 3695|3745|3795|3845|38941|394413994|40441409414 144 
 Q8| 4194|4244}42941434414304]1444414404/4544|14593/4643 
 99] 4693/4743}4793)4843)4893)1494314993 5043 |5093)5 143 
 
 83} 6698 674816798 
 
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 (160) LOGARITHMS N. 870 L.939 
 
 N. O ] 24S 4A Wed cI BRUTE 8 9 || D |Pro 
 870019395 193|5242|520215342|5392)15442/5492155421559215042| | 
 01] — 5692}5742]5792/5841/589111594115991|6041\609 1/6141 50 
 
 02} 6191/6241}0291|0341/6390}|}0440/6490/6540/0590|6640 I] 5 
 
 03 6690}0740}6790|0840/088G}||69390|0980|7039|7089|7 139 si15 
 04] — 7180|7239)7280)7339/7388)|7438]7488|7538|7588/7 035 4120 
 05} = 7088/7738)7788|7837|7 887||7937|7987 |8037 |8087 |8137 5/25 
 o6| 8187 $937 8286/8336|8386]184361848618536/8586/8636 6/30] 
 07} 8685]873518785|8835|8885||893 5|8985|9035|9084\9134 gl4o 
 OS} . 9184]9234|9284]9334/9384/1943.4/9483 1953319583 9033 914.5 
 { 09]  968310733|9783|9833|9882]19932|9982}0032/0082/0132 1 ae 
 87 10}0400182/023 11028 1/033 11038 1]||043 1]0481|053 11058010630 
 11 0680]0730/07 80|0830,0880 te 0979 1029}1079]1 ae 
 
 19) 4007|4717|4706)4816/4866}/49 1 6/4960/5015'5005/5115 
 8720| 5165|5215]5264/5314|5364]154.14|5464|5513/5563/5613 
 af 5663157 13|5762]58 12/5 862)|59 1 2|/5962/601 1\|606116111 
 22| 6161|6211|6260/63 10'6360]|6410/646016509 6559/6000 
 23 6659|0709|07 58|0808/6858]|0908|6957|7007|7057|7 107 
 241 7157/72001725017306/7356||7405|7455|7505 cy 605 
 25|  765417704|7754|7 804|7853]|790317953|8003'8053 8102 
 96  8152/8202/8252/8301/8351]18401|845 118500 85508600 
 27|  8050|8700}8749/8799'8849]|8899|8948|8998 9048:9098 
 28}  9147191971924719297|93401|9396|9446/9496|9545|9505 
 29] 9645|9095107441979419844||9894|9943 |\9993 0043 (o0gs 
 8730 9410142/0192/024210292|034 1/039 1|04-41 049110540 0590) 
 3} 0640/!06g0/0739/0789|0839]|088G)093 8|0988|1038 1088 
 32} 1137}1187]}1237|1286)1336]]/1386]1436}1485}1535|1585 
 33 1635}1684]1734}1784]1834]|1883]1933}1983/2032/2082 
 
 MS IEG Se RE a cet 
 
N.875 L.942 OF NUMBERS. 
 
 es Fle ee nn) SE Sn (ena 
 
 62} 6032|0082/6132'6181|623 1/6280|6330)637016429|6479 
 63| 652810578|6627|6677|0726)6776|6825|6875|09025|6974 
 
LOGARITHMS N.880 L.944 
 
 OWf 1; 2] ope hs fe] F | 8S | 9 Die 
 3800 944482714870] 1925]4975|5024|15073151231/5172|522215271) | 
 5320|5370|5419|5468155 1 8}1556715616|5666|57 1515764 50 
 5814|5863|5912|5962|601 1]|6060/61 10|6159|6208/6258 1 5 
 630716356|6406164.5516504116554|6603|66521670216751 A 
 
 6800|6850/689916948]6098}|704-7|7096)7 146|7 195|7244 4190 | 
 7294|734317392|74421749 1||7540|7590|763G|768817 737 5{25 
 7787\783017 885|793.5|7984||8033|808318 132/81 81/8231 
 8280/8320|8370|842818477|/8527|8576|8025|867418724 8140 
 8773|8822|887 21892 118970}|9020|9069)91 18}9 16719217 914.5 
 9266193 15|9365|9414\9463}/95 13}9562/96 1 11|9660\97 10 | 
 9759|9808|9858]9907190956}|0006|0055|0104}01 53/0203 
 
 0745|079410843|0893|0942}|099 11104.1|1090}1 139]1188 
 1238]1287|1336]1386}143.5||1484]1533]1583|1632|1681 
 1730]1780|1820|1878|1928||1977|2026|2075}2125|2174 
 
 2223/227212322)237 112420}12460|25 19|2568}261 7|2667 
 27 16|2765|2814|2864)2913||/2962)/301 1/3061|3 1110/3159 
 3208/3258|330713356|3405113455|350413553|3602/3052 
 3701|3750|3790|3 840|3 89811394713996/4046/4005/4.144 
 4193/424314292/434114390/|4440/4480|4538]458714637 
 4680)4735|4784/483414883)|493 2/498 1/503 1/5080!5129 
 5178'5227|5277|5326|5375||5424|5474|552315572|56021 
 507 1|572015769|5818}5867|5917|5966|001 5|6064/61 14 
 6 163/021 2\626 1163 10|6360||6400|6458|6507|0557 |6006 
 6655\670416753|6803/6852}|6901|}6950}7000}7049/7008 
 7147\7196|7246|7295|7344117393|7442|74921754 117590 
 703917688)7738|7787|7836||7 885|7 934|7 984|8033|8082 
 8131 8180/8230|8270|8328||837718426|847 6|8525|8574 
 8623|8672|87221877 1|8820I18869|89 1 8|8968}9017|9066 
 9115/9164/921 41926393 12/19361190410|9459}9509/90558 
 _ 9007 905619705 19755|9804)|9853 9902 9951 0000 0050 
 
 0591|0640/0689 0738 07871\0836\0886 0935 0984/1033 
 1082/1131)1181/1230}1270)|1328}1377|}1426}1476]1525 
 1574|1623}1672}1721)177 1}11820}1869}1918}1967/2016 
 
 2066/21 1512164221 3|2262}/23 1 1|/2360)2410}2450/2508 
 2557|2606|2655|2705|27 54128031285 212901/2950/2099 
 3049|3098)3 147|3 19613245113 204/3343133931344213491 
 3540)3589|3638'3687|3737||3786/383.5|3884|3933|3982 
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 5014|5063]|5 1 12/5161|52105260|5300|5358}5407|5450]| 
 5505|555415603|505215702/1575 1|5800|5849|5898|5947 49 
 5996|6045|609416144161 93||6242|6291|6340/6389|6438 1] 5 
 64871653616586|663.5|6684||67 33167821683 1|6880|6929 3l15 
 6978|7027|7077\7 12617 175||722417273|7322|737 1|7420 4)20 
 7460\7518|7568|7617|76661|77 1517764178 13|7862|7911 5/25 
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 845 1/8500|8549|8598|8647||8697|8746|879518844|8893 8/39 
 8942/899 119040)9089/9138 9187|923619285|9335 9384) 9144 
 
 OTTE als te ele 
 
N.885 L.946 OF NUMBERS. Oe) 
 N. O Bre yS eS) Sys 
 
 adh eS | ee | et | ne | es | | | — a | ens | ne 
 
 885019469433|94821953 119580] 962011967 8197271977 61982519874: 
 
 51 99231997 2|0022)007 110120/0169/0218/0267/03 10)0365 
 
 52|9470414|0463|0512)0561|0610}}0659|0708|0757|0807|0856 1) 5 
 
 53} 0905/0954/1003]1052] 1101/1 150}1199]1248}1297| 13.46) 3 \ 
 
 54 1395|1444}1493]1542}1591111640}1680}1739)1788]1837 sf20 
 
 55| 1886]1935]1984]2033}2082}/213.1/2180/2220/227 812327! 5)25 
 
 56 2376)/2425|247412523]25721|2621/2670/2719/2768}28) 7| 6/29 
 
 57] 2866]2915|2965/3014|3063]3112/3161|3210/3259|3308) | 7134 
 
 58|  3357/3406|3455|3504/3553||3602/365 113700|3740|3798 9144 
 
 59] 3847/3896|3945|3994/4043}/4092/414114190/4239/4288 15 2a 
 8860] 4337/4386/4435}4484/4533|4582/463 1|4680/4720/4778 
 
 61 48271487 6|4925|4974|5023}]507 2/512 1|5170|521915268 
 
 62| 5317'5366)5415|5464|551315502'561 1/5600|5709|5758)] 49 
 
 63} 5807)5856|5905|5954|6003 pas2 0x01 6150|0199/0248 
 
 6297|634616395|6444|0493]|6542 6591|664016680|6738: 
 
 65|  6787\6836)6885|693.4|6983|7032!7081\71301717917228 
 66] 7277/7326:7375|7424|747375 221757 117620|766917718 
 67| 7767\7816786517914|7963}8012'8061'81 101815418208 
 68] 8257\8306'8355|8404|845318502'855 1\s600/8649 8698 
 69] _8747/879618844|8893]8942|8991 19040 gosqi9138|9187 
 8870 9236192851933419383|9432|0481 953095791062819677 
 71) 9726 .977519824|9873|9922|997 1 0020.0063|0117|0166 
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 73} —0705|0754|0803|085 2109010950 0998'1047|10961 1145 
 74} 1194/1243/12921134111390]}143911488'1537|158611635 
 
 7a 1684|1733/1781}1830)1879}1928 1977\2026 2075/2124 
 76| 2173/2222!227 11232012369 2418 2407 2515|2564|2613 
 77\|  2662'271 1/2760} 2800)2858||2907 | 2956 3005|3054/3 102 
 78}  3151/3200'3249/3298|3347 3396 3445 3.494/3543/3592 
 79|  3041|3089/3738|3787/3836}}3885 3934 3983/4032/408 1 
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 81| 4619|4668/471714765|4814]}4863 49124961 5010|5059 
 82} §10815157/520515254|5303/15352 5401154501549915548 
 83| 5597/5646)5694|57431579215841 '5890'5939/598816037 
 84}  608516134|6183|0232|6281110330 6379'642816477|6525 
 85 657416623|667 21072 116770/16819 6868169 16|6965|7014 
 86} 7063/71 12!7161|7210|725917307'7356'7405|7454)7503 
 87| 7552'760117650!7698|7747 7796)7845 789417943|7992 
 88}  $040|8089/8138)8187/8236]8285'833.4'83 82/843 1/8480 
 89] $529/8578|/8627/8070/8724118773 'S822'887 1|8920/8969 
 8890} 9018/9066/9115|9164)9213}}9202'93 1 1/9360/9408]9457 
 91 9506/9555|9604}905319701 1975019799\984819897|9946 
 
 92} 9995}0043|0092 pes 0190}0239/0288'0336)0385|0434 48 
 
 5 
 
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 94} 09711102011 069 aie 1167]]1215|1264|1313]1362l1411 Bly 
 95} 1460}1508|1557|1606}1655]11704|1752}1801'1850|1899 4l19 
 96] =1948]1997|2045}2094|2143]|2192/2241|2289'2338]2387 9124 
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 98|  29242973/3022|3070)3 1 19/|3.168|3217/32663314/3363|| | alg 
 99| 3412/346113510|3558|36071|365613705|3754|380213851 943 
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POO ATE TET 22 So ae ie os 6.) Te 8 1 Sa ipeS 
 
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 02}  4876|492514973/5022/507 1115 120151691521 7|526015315 
 03} 5364}5413|5461|5510|5559||5608|5656|5705|5754|5803 
 
 6339|6388|6437|6486|653.4||6583|6632|/6681|6729|6778 
 6827|6876|692.4|6973|7022\|707 117119'7168|7217|7266 
 
 07| 7315|7363)7412'7461|7510117558|7607|7656|7705|7753 
 08} 7802/785117900\7948]7997||8046|8095/8143/8192/8241 
 0g| 829018338|8387/843018485|\8533/85 82/803 1|8680|8728 
 8910|  8777|8820|8875|8923/8972)/19021 9069 91 1819167|9210 
 1] 9264193 13|9362\941 119450}19508|9557,9606/065419703 
 
 9752|980119849|9898|9947 19995 00440093 0142/0190 
 13/9500239|0288/0337|03 85|0434/10483/053 10580 0629|0678 
 0726|07 7 5|08241087 21092 1/0970) 1019 1067|1116]1105 
 
 15] 1213/1262113111360|1408||1457|1506 1554|1603|1052 
 16 1701|1749|1798|1847]1895||1944!1993'2042/2090|2139 
 17} 218812236'2285|2334|2382|/2431 24802529 2577)2626 
 18|  267512723|277212821|2869|2918129673016/3064/3113 
 19h 2 162)8210 3259|330813356|3405/3454'3502(355 1/3600 
 
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 21 4139]41 84/4233/428114330 4379/4427 4470 4525|4574 
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 23|  510915158\5200'525516304115352 5401/5450\5498)5547 
 24 whined 9 5093|5742/5790||5 8309/5888 5930/5985 0034 
 25 6082)013 1/}6180'6228|6277 63266374'6423 6472/6520. 
 26| — 6569\6617|6666)67 1 5|6763|\6812|6861 |6g09|6958\7007 
 27}  705517104/7153(720117250l7299'7347 73901744517493 
 
 28| 754217590|763917688|7736|7785'7834.7882|7931\7980 
 29 8028)8077 8126)8174|8223|/827 1/8320 8369|8417|8466 
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 311 9001/9050\909819147/9195 19244'9293'93.411939019439. 
 32| — 9487|9536|958419633|9682/|97309779 9827 987619925 
 33| .9973|0022100711011910163)0216 0265'0314|036210411. 
 
 35| 0946,0994]1043]1091}1140)|1189|1237/1286]1334)1383, 
 36| =1432/1480)1529/1577|1626)|1675|1723/1772|1820)1869, 
 37} 1918}1966)201 5|/2063/2112)/2161'2209'2258/2306|2355, 
 38} 2404/24521250112549|2598)|2646'2095'2744|2792)/2841 
 39} 2880|2938]/2987/3035|3084/3 1132/3 181|3229|3278)3327| 
 8940| 3375)342413472'3521|3569||361 8\3667/3715|376413812' 
 41} 386113910)3958/400714055/14104!41 52/4201 |4250/4208 
 42} 43471/4395|4444'449214541|\4580|4638)4687 4735/4784 
 43|  4832/4881|49209|4978]5027||\5075|5124|517215221|5269 
 44, 531815300/5415|540415512)|[5561|5609|50658|5706|5755 
 45} . 5803)585215901|5949150908)|0046|6005|6143 1619216240)! 
 46} 62809/6337|6386|043 5|6483 ||6532|6580|6629|6677|0726 
 47 6774|0823|687 11|6920/6969!|701 7|7006)7 1 1417163|7211 
 48} 7200/7308)7357|7405|7454 |750217551|7590|7648|7607 
 49} 774517794|784217891|7939||7988|8036|8085|8133}8182 
 NeOe Tae? PSP anaes Web 4 PAS AOU. D | Pts 
 
N.895 L.951 OF NUMBERS. 
 
 i a ee seemed —enammeeeed eee)  , 
 
 —_—_— |—_——___ 
 
 7440;\7480|7537 7580/7634 
 792417973|8021|8070,8118 
 
 3731/3779|382713876'39241|3972|402 1 |4060)4.1 17/4166 
 4214'4262/43 1114350|4407114456/4504/4552/400114649 
 4007|4746|4794|4842/480 1||14939|/4987/5036)5084'5 132 
 §181|522015277|5326|53741/5422/547 115519)}556715616 
 5664/57 12|5761|5800|5857||5906|5954|6002|605 1 |G6099 
 6147/|6196|6244|6292'6341110380|6437 |04801653416582 
 663 1\6679|07 27167 7 6|682.4)|6872\692 1169691701 7|7065 
 
 7597\7645|7694|7742177901|7838|788717935|7983|8032 
 
— |§ ——————_ | ————— | ————————]} | J fF | | 
 
 9050 9560486 6534{6582|6630)667 8|16726|6774|6822|6870|6918 
 51 6966)7014)7062)7 1 1017 158]|7206|7254 7302 7349 7397 
 
 0803/0851 |0898/0946|0994]|104.2/1090}1138]1186]1234 
 
 1282]1330/1378]1426]14741/1522|1570)1618}1665|1713 
 1761]1809|1857|1905|1953||200112049|2097|2145|2193 
 224.1|2289|2336|23 84/243 2||2480|2528|2576|2624|2672 
 2720|2768/28 16!2864|291 1]129509|3007|3055|3 1103/3151 
 319913247/3295|3343/3301113439|348613 53413 582/3030 
 
 3678)/3726|3774|3822)3 8701139 18|3966|4013/4061|4109 
 4157/4205 /4253|4301/4349]|4397|4445|4402/454014588 
 4636|4684/4732)4780/4828]14876/49241497 1 |5010|5067 
 5115|5163)5211|5259|5307||5355|5402|5450|5498/5546 
 5594|5042|5690|573815786)/5833/588115929|5977|0025 
 6073|612116169|6217|62641163 12/6360|64.08/64.56|0504 
 655216600 6647|6695|674311679 11683 9|6887|603 5|6983 
 703017078)7 126)7 174|7222)|\7270\7318'7366|7413|7461 
 750017557|7005|7053|7 70111774817 796|7844/7892|7040 
 7988}8036|8083/813 1/8179]/8227|8275/8323|837 1|8418 
 8466/85 14/8562/8610'8658)|8706|8753'8801|8849|8807 
 8045|8993/904119088/9136]19184/923219280/932819376 
 94231947 119519|9567|9015||9663|97 10197 58/G806}9854 
 
 9902/9950|9997|0045/0093/10141|0189|0237/028410332 
 
 1337]1385|1432|1480|1528||1576|1624|1672|1719|1767 
 1815]1863|1911|1958|2006]12054]2102|2150/2198]2245 
 22:93|234112389|2437|2484)|2532|2580|2628/2676|2723 
 277 1|2819|2867|2915|2962\|301013058)3106'3154|3202 
 
 3249/3297|3345|3393/3441113488)353613584/363213680 
 3727|3775|3823|387 11391 9113960/401414062/41 1014157 
 420514253/4301|434914396)|4444|44921454014588|/4635 
 46831473 114779|482714874)14922/4970|5018|5065|5113 
 516115209|5257|5304153521154001544815495|5543/5591 
 
 5639|5687|5734)5782)/5830!|/5878|5925|5973|602 110069 
 6117)|6164|6212/6260/0308)|6355|6403/645 11649910547 
 659416042|6690)673 8/67 85||6833/088 1169201697 6|7024 
 7072\7120\7 167|7215|7263)|73 1 1|7358|}7406|7454|7502 
 7549|7597|7045|7 093/774 111778817 836|7884|793217979 
 8027|8075|8123/8170|82 1 8/18266/83 1 4/836118400|8457 
 8505|8552/8600/8648/8695||8743/879 1|8839|8886|8934. 
 8982|9030/907719125|9173}1922119268/93 16|9304|9412 
 945919507 |9555/9603/9050}19698/9746/97931984 119889 
 
 9937/9984 0032 0080)0128}10175}0223/027 1/03 18/0366 
 
CiGS)anenny, 1 ebOGARTEENES N.910. L.959 
 
 089 11093G}0987]10341082]|1130]1177]1225|1273|1321 48 
 1368] 1416]1464}1511)1559}|1607|1655}170211750}1798 dy 5 
 
 03} — 1845/1893}1941/1989}2036}2084}2132)2179|2227/2275) {219 | 
 04} — 2322/2370]2418]2466|25 13]|2561}2609|2656|2704/2752)| | 1,4 
 
 2800|2847|2895|2943|2990||3038/308613 133|3181|3229 5} 24 
 3276|3324|33721342013467113515|3563|30101365813706 6}29 
 3753/3801138401389613944|1399214039/4087|4135|4193]| | 7154 
 
 4230|4278|4326]4373|4421114469/45 1614564|461214659 ol43 
 4707|4755|4802|4850/4898]/4945/4993|5041/5088/5136 
 
 5184/523 1152701532715374||54221547015517/5565|5613 
 5660)5708}5750|5803|585 1||5890|5946|5994|0042\6089 
 6137|6185|623 216280163 28)/63 75|6423|047 11651816566 
 6614|6061|6709|6757|6804||6852'6900/694.7|6995|7043 
 7090)7 138|7 180|7233\7281||7328|7376|7424|747 1\7519 
 7507|7614|7662 77107757 7 805|785317 900)7948|7996 
 8043)809 1]8138/8 186,8234|/8281|8320|8377|8424/8472 
 8520|8567|8615|8662/87 10]18758!18805|8853|890118948 
 8996|90441909 119139191 86]19234|9282/93 291937 7|9425 
 947 2/9520|9507|9615|9663)|97 10|9758|9806,9853\9901 
 9948}9996|0044/009 110139||0186|0234|0282/0329 0377 
 21196004251047 2105 20/0567|0615||0663|07 10|0758 0805\0853 
 0901}0948}0996}1044!10911}1139/1186]1234]1282)1329 
 1377|1424}1472)1520)1567||1615}1662}1710)1758'1805 
 1853}1900}1948]1996|2043}|2091|2138|2186/2234'2281 
 
 2329|2376|2424|2472)25 19|'2567|261 4|266212709|2757 
 
 26|  2805/2852|2900!2947/2995|'3043/3090|3 138/13 185|3233 
 27| 3281|3328)3376)3423'347 1|3518/3566|3614/366113709 
 28} 375613804|385113890/3947|'39904|4042|4089/4.137/4184 
 29| 4232|4280)432714375|4422)447014517/4565|4613/4660 
 g13 4708}4755|4803}4850,4898/49046)4993|504115088/5 136] * 
 31 5183}523 115279|5326|5374| 542 115409|5516|5564/561 1 
 32} 5650)5707|5754|5802|5840]'5897|5944|5992|6039\6087 
 33]  6135|0182/0230\6277\6325| 6372/6420/6467/6515|6563 
 34| 6610)0058]070516753/6800 Seng 689516943|0990\7038 
 35} 7086|7133|7181)7228)|7276)7323|737 1|7418|7406)7513 
 36| 756117008/7056|770417751 7799 7846)7 8941794117989 
 37| 8036/8084|8131)8170|8226)8274/832 118309/8416/8464 
 38} 8512!8559|/8007!8654|8702| 8749/8797|8844|8892/8939 
 39} 8987/903419082/912919177| 9224|9272/03 191036719414 
 9140} 9462/9509)9557|9605|9052)190700|9747 |9795|9842/9890 
 
 9937|9985 0032 0080 0127}10175 0222 0270|03 1710365 
 
IN.915 L.961 OF 
 
 83 
 
 85 
 86 
 
 ¢ 
 
 84/06303 191036610413 ad 
 
 3220|3207133 14/3362 
 3693|3741|3788/3835 
 4167|4214|4262/4309 
 4040/4688|4735|4783 
 511415161)5209!5256 
 5587|5635|5682!15729 
 
 6061}0108/6155'6203 
 
 7007|7055|7 10217149 
 7481\752817575/7622 
 7954/8001 '8048|8096 
 8427|8474'852118569 
 
 8900/8947 '8994 9042/9089 9136 
 
 937319420/946719515 
 9846/9893 99409988 
 
 0792|0839'0886:0933 
 1264}1312|}1359|1406 
 
 5045/5092 
 
 NUMBERS. 
 
 7197117244 
 797017717 
 8143||8190 
 
 é 
 
 729117330|7386|7433 
 7764\7812/7859|7906 
 8238'8285183321838 
 
 8616]8663'87 11|8758/8805/8853 
 
 0184/9231|9278/9326 
 
 9502}19009|9057 97041975 1/9799 
 0035 /0082,01300177/0224|027 1 
 
 0508|10555'0602/0650 
 09811/1028/1075}1123 
 
 0697|0744 
 1170/1217 
 
 '5139|5187|5234|15281|5328/53761542315470 
 
 5517 5565/5612 5659|5706]|5753|5801|5848/5895|5042 
 5990|6037|60841613 1/6170}16226/627316320|6367106415 
 6462/6509|6556|/66041605 1|16698|6745|0792/6840|/6887 
 6934|698 1\7028]707 617 1231\7 170|721717265|7312|7359 
 7400|7453|7501|7548/7595 “642 7680|7737\7784|7 831 
 
 ef emt ef ff | | SS 
 
 445 |.61 7 g 19 |p [Pts. 
 
 ARoPoOwr KR 
 202 DW 
 (Ee SSekcrouw” 
 
 ;oo 
 
LOGARITHMS 
 
 9719 
 
 2596/2643 
 11] 3068/3115 
 
 259 1|2738|27 85||2832!2874|2926)2974|/3021 
 3102/3209 3250) 3304/33 5 1133G8|3445134G2 
 12| 3530|3586'3034|368 1|37 28137 7513822|3869/391 6/3964 
 13} 4011/4058'4105/4152/4199'14246]4294/4341|438814435 
 14} 4499145294576/4623/467 1147 18147651481 21485014906 
 15} 4953/5001/5048/5095|5 142)|5 189|5236|5283|5330|5378 
 16} 542515472'5519|5566|5613)5660|5707|5755|5802|5849 
 17}  5896)5943 5990|5037|6084'/613 1/61 79,0226 627310320 
 18| 6367/6414/6401|6508|5555||6603|6650 6097107440791 
 19| 6838|0885|0932|6979/7027)|7074)7 121|7 108)}7215|7262 
 9220) 7300|7350'7403/745 1174981754517 592'7630|7086\7733 
 
 21| 7780\7827|7874|7922)7969||8016 8003/81 10|8157|8204 
 22}  8251/8298)8345]8392}8440]8487)8534 8581 8628|8675 
 8816/8863 sted 8958}9005'9052/9099/9146 
 
 989919946.9993/0040|0087 
 0370|0417/04641051110558' 
 
 Gl0746 079 
 1076}1123}1170)1217)1264)1311]13.58)1405)1452|1490} 
 1546)1594|1641|1688|1735]|1782|1829'1876}1923]1970 
 2017|2064}211112158}/2205||2252 2299 2346 2393/2440 
 
 17 01/2723 2770.28 17|2804}29 11 
 3240,3287}3334)3381 
 3604|371113758/3805|3852 
 4134]4181|4228|427514322 
 4604465 1/4098]4745/4792 
 
 7472'7510|7500|7013 
 780117848]]7 89517 942!7 98G/803 6/8083 }} 
 
 _ 953y19586|963319680|07 271197741982 1/98681991 519962 
 19660009|0056/010301 49101 96}0243}0290}033 7103 841043 I 
 
 N.920 L.963 
 
 OMOA1R GO 099 
 woWUOr OH ON 
 
 i C2 G00 ht 
 
N.925 L.966 
 
 7048|7095|7142 
 
 7985/8032 
 
 65 
 
 66 
 
 67| 9392190438/9485]9532 
 
 68} _ 9860'9907|9954|0001 
 
 6919670329 0370|042310469 
 9270) 0797/0844/089 1/0938 
 
 22 
 
 nt ae 
 
 6416/0402! 
 
 1092}1139|1185|1232 
 
 ns. fetes fem ff 
 
 O TEP yee os 
 
 75\| 3130913186 
 76| 36073654 
 771 4076'4122'4.16914' 316) 
 78|  4544/4590'463714684 
 79} 5012/5059 
 
 #9280 5480) 552715573 5020) 
 
 5048 phe pean 6088 6 
 050910550'66 
 6884/6930/6977!7024)7 
 
 1559|1606}1053!1700, 
 2027/2073|2120)2107/2214)|/226 
 
 4362 ripe 1450) 4503 
 
 OF NUMBERS. 
 
 9250 906017 1464]151111558]16051|1652 1699 1746 1793] 1840 
 1887|1934}1981|2028}207 51212212 168]2215}2262/2309 
 2356|24.03}2450|249712544|125091|2638|2085|2732)2779 
 282.6|2873129 1912966/3013]13060)3 107|3 154/320113248 
 3295|33421338013436|3483)|353013577|3623|307013717 
 3764/1381 113858|3905|395 2113 999|4046|4093/4140/4187 
 
 /4233\428014327143741442 1114468145 154562|4000|4656} - 
 4703/47 50)4796|4843/4890]1493 7149841503 1|5078/5125 
 §172\5219/5260|53.1215359!|5406|5453|5500|554715594 
 5641}5688/5735|57 82|5828||5875|5922|5969|001 0/0063 
 611016157|620416251162971163441639 110438|048516532 
 6579|0626|6673|07 20|67 60/68 13|6860|6907|0954|7001 
 
 7 188\7235})7282\7320|7370|7423|7470 
 
 7517\7504\7010170571\7 7041775117 798|784517 89217939 
 
 8079|8126'8173]822018267|83 14/8300|8407 
 
 8454 3501/8548 8595|8042 8689 873518782 pire 8876 
 
 Haba. 1112858 
 323313280'3326|337 
 3701137483795 384113888 3935|3982 
 
 | 
 
 5105/5152'5199]1524615293|53391538615433 
 
 6 
 
 841 735117398/7445|749217538 76 7 720\7772 
 85] 7819'7866/791317959|8006 
 
 86] . 8287'8334/8380)842718474 
 
 87]  8754'8801/8848|8895'80- 
 
 88}  9222\9209/9316 93029409 9456/9503 
 80 ee 9730 9783 se 9877 978 9970 0017 eis 0110 
 
 9579}9026/9073 97209707 9813 
 0048 ‘0095 0141|0188/0235|0282 
 0516/0563 
 
 0985 
 
 42.031143 10|435614403/4450\4497 
 473 1]4778]4825'487 1/4918'4965 
 
 5667||57 145761 |5807|5854)5901 
 
 141721117 258\7305 
 
 1279||132611372]141911400|1513 
 1740]|1793|1840]1886]1933]1980 
 
 A5AQ 4506 4643 
 
 —— |] -———. | ———_—_—__ | -————__ | -————_-- | —__—_ 
 
LOGARITHMS N.930 L.968 
 
 3227 78274 982013307 
 3693 374037 86 vapeuewast 
 4159 4206,4252/4299, 4346). 
 4625 46724718 4765 ( 
 5091 51385184 5231 
 5557|5603'5650 MA 
 6023 6069 ,6116)6162 
 648816535 6582/6628'6 
 7047'7094 
 7513/7559 
 7978 8025 807316 8118/8105 nfs 1 8258 8304. 
 8584|8630)/8677|87 23/8770 
 
 8816/8863)/89 10/8956'9003)|904.9/9096'9 1421918919235 
 9282|932819375|9422'9468!195 15/9501 (9608 905419701 
 9747)|9794/9840)9887 | ) 
 
.935 L.970... 
 
 OF NUMBERS. 
 
 me fe | | Sf SS SS] 
 _—_—_— |, 
 
 3686'3733)3779|38201387 2)13918)3905/401 114057|4104 
 4150,4197|424314289]4336114382/4429/447 5/452 1/4568 
 4614'4660}4707/47 5314800114846)4892/4039/4985'503 1 
 507815124|517 1/521 7|5263||53 10\5350)5402/54409'5495 
 5542|5588|5034'5081|5727 15773/5820|5860)5912'5959 
 6005|6052'6098'6144/6191}0237|6283|0330 6376 0422 
 646905 15|0502/6008|665410701)/0747/0793 Sen GonP 
 693216979 7025'707 1171187 104'72111725717303'7350 
 7396\7442 748017535 7581176281707 417720\7767'7 813 
 7859|7906|7952'7998|8045 8091)8137/8184 8230 8276 
 8323 8309/84 15|8402|8508 18554 8001|8647\80y4 8740 
 8786,883318879)8925|3972 yoo 911119157 |9203 
 9249'929061934:2'9388/9435 5271957419620 ‘0666 
 07 131975 980519852 9898 Qg44 9991 0037 0083 0130 
 76197201 7, 0222/0269\03 15/0361 | 0408'0454|0500 5470593 
 
 9481 
 
 a9 0639 0085 0732'0778108241087 1'0917|0963|101011056 
 78] 1102/11. 1241 1288 1334|1380|142611473!1519 
 79 1505|1012 1658)1704}1751 17971843 1889 1936 1982 
 
 9380} 2028'2075)2121|2167 
 81 2491/2538 25841263012677\|2723/2769|2815|2862 2908 
 82|  29543001|3047/3093/3 139/13 186/323 2/327 8133251337 1 
 83] 3417/3463)3510)3556/3602 |3649|3695|37411378713834 
 841 3880'392613973'40) 9140651/4111/4158142041425014206 
 85|  4343'4380|4435/4482/4528)14574/4620/4007/47 1314759 
 86]  4805'4852/4898/4944140991]15037|5083/5 129|5 17615222 
 871  5268)5314)5361/5407|5453115500|5546/559215638)5085 
 88} 57311577715823'5870159165962|6008'60551610116147 
 89| 6193/0240/6286/6332|6378)/6425|647 116517|0563/6610 
 9390| 6656)6702\0748/6795|0841|10887|6933|6980}7026|7072 
 gl 7118'7 1651721 1|725717303)17350\7390174421748817535 
 Q2 758117 027|7073|7720)7 7 00)|7 81 217 858)7905}7951|7997 
 93|  8043/8089}813618182/8228/8274/832 1{8367|8413|845¢G 
 04]  8500/8552|8598|8644)8600)/87 37|87 83|S820/887 5/8922 
 95 8968/9014/906019107/9 15311919919245|9201|9338/9384 
 96} 9430\9476|9523|9569/961 5/9061}9707|9754|9800,9846 
 97} 9892 9938 9985 003 1 aor 0123|0170/0216|0262 0308 
 
$203 slg240 8295|834218388||8434|8480'8526]857 218618 
 
 912019172'9218/926493 10)/935610402/944919495/9541)| 
 9587 |903319679|07 251077 111981 719864|991 019956|0002 
 -1919740048|0094'0140/0186|02321\027910325|037 11041710463 
 0509\055510001|0647 |0693||0740/07 86|0832/0878'0924 
 0970}1016|1062!1 108|1154}/1201]1247/1293/1339|1385 
 143 }|1477/1523|1569|1615||1661|1708|1754]1800|1 846 
 1892|1938}]1984|2030/207 6]'2122/2168!2215|2261)2307 
 2353/23 90|244.5|249 11253 7112583|2629'267 51/272 112768 
 2.814'2860|290612952129981|304.4|309013 13613 182'3228 
 3274|3320|3367|3413/34591|3505|3551/3597|3643 368g 
 37351378 11382713873139] OlI3965/401 11405814104'4150 
 4196/4242|4288]4334|13801/4426/4472/4518 4504 4610 
 4650/4702/474814795148411'4887 4933|4979|5025 507 
 
 5117}5163}520915255}5301].5347}5393)5439 548510591 
 5577 |5023|5070157 16}5702}5808/585. 4'590015946'5992 
 6038|0084|0130)0176|6222) 626s}63 i4l63606406.6452 
 6498|654410590.6036 ier 07 291677 5|082116867\6913 
 
 8800/8846]/8892/8938/8984]|9030|9076|9122/9168 9214 
 
ee, [enn (ne ee | ee ee ee eee eed | eee 
 
 4778}4824/4870]49 15|4961||5007|5053|5099|5 14.5|5191 
 5237|5283|5320|5375|542 1115467155 1315559|5005|505 1 
 §697|5743|5788|5834|5880}159261597 21001 8|6064/6110 
 61506|6202/6248|6294|6340)|63 86/643 21647 8/05 23/0569 
 661 5|6661|6707|67 5316799|1684.5|689 1|693710983|7029 
 7075\712117166)7212}7258)17304|735017396|7442|7488 
 7534\7580!7626\7 07 2)77 18|17 7603/7 800|7855|790117 947 
 7993|8039|8085|813 118177||8223/8269/83 15|8360|8400 
 8452|/8498/8544|8590|8636]|8682|8728/8774|8820|8805 
 8911/8957|9003/9049|9095|/9 14.11918719233|9279|9325 
 9370)9416 9462|950819554]|9600/964.019692|9738/9784 
 62| —982919875'99211996710013|1005910105/015 1019710243 
 6319760288|0334'0380\0426/04721105 1810564|061010656|0701 
 64| 0747|0793'0839|0885/093 1|10977}1023}1069]1114|1160 
 1206)1252!1298|1344/1390]|1436)1481|1527}1573}|1619 
 1065|1711)1757|1803}1849}|1894|1940}1986|2032|2078 
 2124|2170'2216)2261/2307/12353|2390|2445|249 1/2537 
 
 3041/3087 /3133/3179'3225||3270)3316/3362'3408/3454 
 3500/3546 3592 363713683 3720|3775/3821'3867/3913 
 3958}4004/4050/409614142/]4188/4233/4279/4325/4371 
 4417|4463!4500|4554|4600)|4646|4692|4738|4784|4830 
 4875|4921/4967|5013/5059]15 1051515015 196'5242|5288 
 
 5792|5838|5884|593015976||602 16067161 13|6159|6205 
 625 116296/6342|6388!043.4||6480|/0525'057 1|661 716063 
 6709|0755|6800/6840}6892)||0938/0984|7030)7075)7 121 
 716717 213|7250|7305|73501\73960|744217488 1753417579 
 76251707 1177 1717763'7808}|785417900!7946/7992|8038 
 8083/8 120/81 75|822 1182671183 121835818404'84 50/8496 
 854118587|8033|8070/87 25|187 70/88 16|/8862|/8908|8954 
 9000|9045/909 1 /9137/9183/192291927419320'9300|9412 
 9458)9503|9549|9595'004.11\9080/9732|9778/9824|9870 
 9915996110007 Don cogo 0144|0190|0236/0282/0328 
 85|9770373104.19|0465|0511 (0556 0602|0648!0694/0740|0785 
 
 083 110877|0923|0969g'1014/|1 060}1 106)1152/1197|1243 
 
(176) LOGARITHMS N.950 L.977) 
 
 | 
 
 975019795}9841}0887 19932 
 9978|0024}0069)01 15/0161 0207 10252|0298|0344/03 89 
 
 363113677 3793 3768 vats 3860 Sbace 3051 3997/4042 
 4088/4134)4179|4225)4270 43 1614362 fe 4453/4499 
 
 8194 
 8650 
 
 8230|8285|833 1|8376/18422'8467/8513|8559/8604 
 8695'8741|8787|8832||8878'8923|8969|901 5|Q060 
 9106.9151/9197|9243|9288)|0334:9379|942519470/9516 
 9562/9607|9653|9098/9744 979019835 988119926/9972 
 28/979001 70003 0109|0154/0200)10245 wea 033710382|0428 
 
 0473 (05 19,0504|0610/0656)|0701'074710792|0838/0883 
 0929/097 5|1020)1066]1111/]1157}1202)1248]129411339 
 1385/1430)1476)152111567}|1613}1658/1704|1749)1795 
 1840/1886}193 1|1977/2023]|2068'21 14/2159}2205|2250 
 33 2296/2341 |2387|2433)|2478]|2524|2569/2015|2660|2706 
 275 112797|2843|2888!2934]|2970/3025/3070/3 1116/3161 
 
 320713253|3298133441338911343.5|348013526/357 113617 
 
 ne ee ee ee es | ee a a Sonny | Ca eee 
 
59 4125]4170)4215'426114306)|435 2/4397 4443 |4488/4533 "sae 
 
 19560} 45709|/4624 467014715 4761||4806)485 1/4897/4942/4988 
 61} 5033/5079)512415169)5215||/5260|5306)/5351|5397|5442 
 62}  5487|5533|5578|562415069||57 14|5760/5805|585 115896 
 63] | 594215987|6032/6078|/6123)|6169|6214\6259/6305|0350 
 64| 6396|644116486/6532/6577)||6623|0668)67 1416759/6804 
 
 65| 6850)6895|6941|6986)703 1)\7077|7 122|7168/7213|7258 
 66} 7304!734017395|7440\7485/1753 11757 6|7622|7667|7712 
 67| 7758|7803|7849/78094|79390||7985|8030|807 5/8 12118166 
 68] 8212/8257/8302|/8348/8393)/8439/8484/8520/8575|8620 
 69| 86066)871118756)8802|/8847||8892/893 8/8983 |9029|9074 
 9570 9119|9165/9210!9256'9301]19346|9392/9437|948219528 
 9573 9019 96640709 p78 9800/9845|989 119936|9982 
 
 78}. 2748 cece 2839 28842920 297 5|3020/3006'3 1113156 
 79|  3202/3247/3292133383383)|3428/3474/3519|3564|3610 
 9580| 3655|3700/3746 3701|3836 3882)/3927|3972|4018)4063 
 81} 4108/4154)4190|4244 4290)/4335|4380/4426/447 1/4516 
 
LOGARITHMS N.960 L.982 
 
 3165|3210)3255|3300|3346)/339113436|348 11352713572 
 3017|3662)/3707|37 53137 98]13 843|3888]39341397 9/4024 
 4069|41 15/4 100)4205/4250]142951434.1}43861443 114476 
 452214567146 12)4657|470211474814793|4838|4883|4928 
 4974|5019|5064)/5109)5 155||/520015245|5200|5335|5381 
 54261547 1/55 10]5561|56071|5052150971574215787|5833 
 5878|5923)5908/6014/6059]/6104|60149/6194|6240|6285 
 6330/6375 '0420|6406)/65 1 1|}6556|6601|664616092/6737 
 07 82|6827|087 2/69 1 $}0963}|7008|7053|7098)7 143|7189 
 7 234/727 9'7324\7300\74151|746017 505|7550|7595|7 641 
 7080)7 731177 76)7 82117 867||791217957|8002|8047 8092 
 $138/8183|8228/8273/83 1 8!18364|8409|8454|8499'8544 
 8589/863 5|8080|87 25/87 70)|88 1 5|8860/8906|895 1/8996: 
 904-119086/9132/917719222||9267193 12/9357|9403|9448 
 9493/9538/0583|9028/9674)197 1919764|9809|9854|9899 
 9945|9990,0035|0080)01 25)/0170/02.16|0261|0306/035 1 
 
 0848/0893/093 8/0983] 1028)/1073}1119]1164}1209}1254]| 
 1299}1344|1390}1435}1480)/1525]1570}1615|1660|1706 
 175111796|1841}1886]193 1)|1976|2022/2067/21 12/2157 
 2202/2247|2292|2338}2383}|2428/2473/25 18|2503|2008 
 2054|20099|274412789128341/2879|2924|296g/301 5|3000 
 3105/3 150)3 195|324013285]|333 113376|342 1/3466/3511 
 3556|36011364.013092/3737113 782'38271387 2/391 713962 
 4007/4053/4098/4 143/41 88/'4233/4278/4323|4368/4413, 
 4459|4504|454914594|4039]4684/472091477414819/4805 
 4910/4955|5000|504.5|5090)/5 135/51 8015225|527 115316 
 5301/5406|545 1|5490|554 1115586|563.115077|5722|5707 
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 6263|6308|6353/6398/6443]16488/6533|0579|6624/0609 
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 761617661|770017 75 1177906)|7841 |788617 93 11797 6|8021 
 8066/81 11|8157/8202/8247||8292|8337|8382!8427/8472 
 8517|8562|8607|8652|8097||/8743|87 88|8833|8878!8923 
 38908)9013/9058)9103|9148}19193|9238)9283/9328|9374 
 9419/9464|9500|9554|9590||/964.419689197 34197 79/9824 
 9869/99 14|9959|0004/0049/|0095|0140/0185|0230/0275 
 
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 3923/3968/401 3/405 8/4 103/|4148]41 93|4238/4283/4328) » 
 
 4373|4418/446314508/4553}|4598|4643|4088]4733/4778 
 4823/4868 5183]5228 
 
N.965 L.984. OF NUMBERS. 
 N. aS Eee A 
 
 Core hs gre Pgnanps Eamrumers [ipaimres Fave’ pacer b ge: 2 aug Uren 
 a ed 
 
 ys i 5813/5858 Gos eae 5993 dose bos 6128 
 6173|0218|620316308| 6353/03 98|64431648816533|0578 
 6623|6608]67 13|6758|0803||6848/6893|693 8|6983|7028 
 7073\7118|7 163)7208)7253||7298|7343|7388|7433|7478 
 7523\7508|7613)7658)77031|7 74817 793|783 817 88317928 
 7973|801 8|8063/|8 107|8 1 52/8 197|824218287|833 2/8377 
 
 842218467|85 1 218557|8602||8647|8092'8737|8782'8827 
 887 2|8917|8962|9007| 90529097 \9142191871923219277 
 93221936794 12|9457|9502/195469591196361968119726 
 977119816|9861'|9906|9951 9990/0041 0086|013 1/0176 
 611985022 110266103 1 110356|04.0110446'049 11053 5|0580|0625 
 0670107 15|0760}0805}0850}0895 |0940,0985|1030)1075 
 
 63] —1120/1165}1210)1255!1300)1345)1389/1434|147911524 
 64} 1560}1614/1059}1704! 1749/1 794'1839)1884)1920)1974 
 65|  2019|2064/2108/2153/2198||2243|2288'2333|2378/2423 
 66]  2468/2513/2558|2603|2648]12693 |2737|2782/282712872 
 67|  20917/2962/3007|3052)3097 1/3 142/3 187|3232)3277|3321 
 68! 3366'3411)8456)350113546]3591 eet ele 3726/3771 
 
 691  3816386113905/3950|3995/4040'4085/4 1301417514220 
 
 9670, =4205|4310,4355)4390)4444144894534!45 701462414669 
 71] 4714,4759|4804/4840|4893/}4933/4983/5028|5073/5118 
 72| 5163'5208/5253|5298|534215387 |54321547715522|5507 
 73| 5612'5057|5702)5747|5791 58365881 5926|597 1|6016 
 741  6061/6106'6151\6196/6240 ‘ea oth 6375|0420\6465 
 75\|  6510'6555|6600|6644/0689 aerate 682416800|6914 
 76 
 
 6959!7003|7048|7093|7 138 BN bei 7273\731817363 
 
 77 7407|7452|740717 542175877 032'7677|772217766/7811 
 78 7850/7 9O117946|799 1|803 6808 1 | 8125|8170|8215|8200 
 79 8305|8350 §395|8440|8484 8529 8574'8019|8664|8709 
 
 8754 8798)/8843|8888/8933|/8978 9023 9068|911219157 
 920219247|929219337|9382|\9426 947 1195 161956119606 
 965 119696|0740)97 85/9830 9875 99209905 0010|\0054 
 83'9860099'014410189|0234|027 9/03 3240308 041310458'0503 
 84) —054810593|0637|068210727/0772/0817/086210907/0051 
 
 85} 0996|1041}1086|1131]1176}}122011265|1310]1355|1400 
 86, 1445|1489}1534]1570}1624111669)17 14 heals Lert 1848 
 
(180), ~ LOGARITHMS N.970 L.9sef 
 pues gti Ns 1) 21) 8) age GE 117 | 8 119 ee 
 
 8165|8210|8255 obs 8344]|8389]84341847 8|8523|8508 A5 
 
 8613]8657|8702'874718792118837 |8881|8926|897 1/9016 I 
 9060}9105/9150\9195|9239)||928419329|937419418|9463 f 
 9508]955319597|9642|06871|9732|97 70/982 1\9866)\g9 11 a 
 9955|0000/0045|0090|0134/}01 7910224 |0269/03 13/0358 5/23 
 
 07} 0850!0895/094.0\0985)1029}/107411119]1163}1208]1253 8136 
 
 os 1298]1342]1387|1432)1477/|152111566|1011|1656}1700 9l41 
 
 09 1745]1790}1834|1879]1924]|1969|2013}2058|2103/2148 oe a 
 9710} 2192}223712282)/2326)237 1]|2416/2461/2505|2550/2595 
 
 11 2640/2684]2729|277-4|28 1 8}/2863|2908]2953|2997 |3042 
 
 12} 3087/3131|3176|3221|3266]|33 1013355|3400)3444'3489 
 
 13]  3534/3570|3623/3068/37 13||3757|3802|3847 3892/3930 
 
 14 3981}4026/4070)4115|4160)|4205/4249/4294|4339/4383 
 
 15| 4428/4473/4517|4562|4607]||4652\4696/474114786,4830 
 
 16] 4875|4920)4964|5009/5054]|5099/5 143|5188)5233|5277 
 
 17| 5322)53671541115450/55011|5545|5590|5035|5080'57 24 
 
 18} 5760/5814}5858|5G03|5948||5992|6037|6082/6126\6171 
 
 19} 6216)626116305|6350\6395||0439|6484165209|6573|0018 
 9720| 6663/6707|6752/6797|684 .1||6886/693 1|6975|7020/7005 
 
 21 7100|7 15417 199)|7 24317288]17333/7377|7422|7467/7511 
 
 22} 7550/7601|7646|7690)7735||7780\7824|786917914|7958 
 
 23 8003]8048|/8092/8137|8182|'8226'827 1183 16/8300/8405 
 
 24,  8450/8494|8539'8583|/8628]18673/87 17|8762|8807|885 1 
 
 25 8890)894 1|8985/9030/907 5}19119. 9164 9209|9253|92908 
 
 26} 9343/9387|9432'9477|9521||9566, 9611 9655|9700|9745 
 
 9789|9834987 8/9923/9968 00120057 0102/0146/0191 
 28|9880236|0280/0325|0370|04.14|10459 0503|0548/0593|0037 
 
 29| 0682/0727|0771|0816|0861||0905 10950 0994|1039)1084 
 9730} =1128}1173]1218/1262/1307 135211306 144114851530 
 31} 1575}1619|1064}1709)1753}|/1798}1842)1887|1932|1976 
 32| 2021/2066/2110/2155|2200/|2244 2289 2333|2378|2423 
 33} 2467/2512/2556|2601|2646 269027 35|2780|2824|2869 
 34| 20913]2958/3003|3047/3002!/3 136'3 181|3226|3270)3315 
 35}  3360)3404|3449|3403/3538]13583 |3627|3672/37 16/3761 
 36]  38006)3850|3895/3939|13984||4029 4073/41 18/4162|4207 
 37 4252}4296/434 1|4386|4430]1447514519|4564|4609/4653 
 38! 4698/4742/4787|4831|4876]1492 1 14965|5010|5054|5099 
 39} —5144|5188]5233/527715322/15367|5411|5456|550015545 
 9740| 5590|5634|5670|5723|5768]|5813|5857|5902|5946/5991 
 | 41) 6035/6080/6125/6169|6214]|6258|0303|6348|6392|6437 
 42} 6481|652616570)6615|6660||6704|6749'6793/6838|6882 44 
 43} 6927|0972'7016}706117105||715017194|7239 7284 7328 1) 4 
 441 737317417|746217500|7551\\7596|7040|7685|7729|7774 aye 
 - 45}  7818!7863)7908|7952|7097||8041|S086|8130|8175|8220)] _  |4|18 
 46}  8264/83009|8353|8398|8442|18487|853 1|8576|8621|S665 9/22 
 47| 871018754|8790|8843|8888||8932|8977|9022\9066|91 11 y ee 
 48} — 9155)9200|9244}9289|9333}|9378|9423|9467 (95129556 gl35 
 
 49} 9601196451969¢19734|9779|19823|9868|9913|9957|0002|| __|o|40 
 6/71819 |\D IPts. 
 
“ LOGARITHMS N. 980 L.991 
 
 9800 0912261}2305|234912394|2438 9482|2527|257 1|2615|2000 
 2704|2748|279312837!288 1]12925!297013014|3058)3103}} 
 314713 19113236|3280 3324113360|3413|345713501/3540 
 3590|3634|3670|3723'3767||381 2|3856/3900|3944|3 980} 
 40331407 7|4122|4166)42 10/14255|429014343/438714432 
 4476|4520/4565|4609|4653]|4697/4742/4786|483014875 
 4910|4963/5007|5052'5096)15 140)5185|52209|5273|5317 
 5362|5406}5450/5495'5539]15583|5627|567 2157 161576 
 5805|5849|589315937|59821|6026|6070)61 15161596203 
 6247/629216336/63 80 6424||6460105 13|6557|6602/6646 
 6690;673.416779|6823'6867)1691 1|6956)7000!7044|7088 
 
 8903/8947|8992}9036'9080)/9124|9160|9213|9257/9301 
 9345|9390/943410478)9522||9507/961 1/9655|9699|0744 
 9788|98321987619921|9965||0009|0053|0098/01 42/0186 
 18 9920230027 5|0319 0303)0407 0451|0496/0540/058410628} 
 | 067310717\076110805,0850||0894|0938)0982!1026}107 1 
 1115|1159}1203 124811299 1336|1380)1424}1469}1513 
 1557|1601|164 16901734 1778|1822]1867|1911}1955 
 1999|2044|2088]2 132 2176|/2220!2265/2309/23 53/2397 
 2441|2486|2530|2574'261 8]|2662/2707|27 5 1|2795|2839 
 
 3768)3812|3856|39003944|3989 
 4210|4254)4298|/4342'43 86/4431 4475 451014563|4607 
 465 114696|4740)47 84|4828]|4872)- )11: 
 
 5093/5 138|5182|5226|5270)153 1415358 5403|5447/5491 
 
 4033)}4077|41211/4165 
 
 597 7|6021|606516109/61541'61¢ 98|6242 6286/633 6375 
 641916463|65071655116595||66406684|67 2816772 6810 
 6860,6905|6949|6993|703711708 117 125]7170\7214 7258) 
 7302|7346\7390/7435|747917523|7567/761 117655176909 
 7744177 88|7 832|787617920)|7964,8009|8053!8007|8 14 1 
 8185|8229|8274/83 1 8'8362|8406/8450/849418538)8583) 
 8627|867 1/87 15|8759)}8803)|8847|8892|8936/8980 9024) 
 9Q068}91 12|9156|920119245|10289|9333|937 71942119465 
 9510|9554|9598|9642\96861|97 30|977 41981] 9/9863|9907 
 -995119995|0039/0083/01 2810) 72/021 6|02601/0304|0348}| 
 
iN. o- | rl2l3}4h5 164171819 [Dip 
 9850|9934302/4400/445014.495/4530]14583|4627/467 1147 15/4759 
 51)  4803/4847]4891|4935|4980}15024|500815 1 12|5156|5200 
 52\ 524415288)/533215376/5420)|546415500|55531559715041 
 53)  5685|57201577315817|5861)|59005|594.015993 1603710082 
 54)  6126/0170/6214/6258'6302)/6346)6390|043.4|647 8/0522 
 55} 6566}0010|6654|6698)|6743]167 87/683 11687 5|69 1 9169063 
 50} 7007}7051|7095]7 139)7 183}|7 2271727 1|731517359|7404 
 57) 7448]749217536|7580|76241|7 668/77 12|7750|7800|7844 
 58} 7888]7932!7976|8020|8064)|8 108/81 52/8 197/824 118285 
 59} 8320/8373]8417|8461|8505||8549'8503|803 7/868 118725 
 9860} 87609/8813|/8857|8901)8945]/8989|\9033|9077|9 12219166 
 61 9Q210}9254192098]/9342'9386]|9430/9474195 18195629606 
 
 62} 9650|9694/973 819782/9826]19870'991419958|0002/0046 
 6319940090|0134/0178|0222/0266}'03 10103 55|0399|/0443|0487 
 64| ©53110575|0619|0663'0707]1075 1\0795|0839|0883|0927 
 65) 0971]1015}1059)1103/1147]|1191]1235]1279/1323]1367 
 66} 141111455]1499/1543|1587]|163 1|1675|17 19}1763/1807 
 607 1851|1895/1939}1983/2027|/207 1/21 15|2150|2203/2247 
 68} 229112335/2379|2423|/2467]/25 1 1|2555|2500'2643|2087 
 
 69} 2731/2775/2820|2864!2908]|2952!2996|3040/3084)3 128 
 9870)  3172/3216)3260/3304/3348113392'3436'3480'3524|3568 
 71\ 3612/36056'3700|3744/3783]'383 1|3875 391913963 4007 
 72\ 4051|4095/4130|4183/4227/14271 paeniesouisa Oe A447 
 73| 4401|4535|4570|4623|4607|/471114755 47994843 4887 
 74; 4931|4975|5019|5003|5107/|5151)5195 ped ita 327 
 75\ 5371)5415|5450|5503'5547|15591|5635|5679'57 2315767 
 
 5811|5855|5899|5943|5987|}003 1|6075|61 19|616316207 
 
 77|  6251|6295 6338|6382'6426) 6470/65 14,6558,0602|6646 
 73|  6690)0784,6778 6822'6866)69101695-4'6998'7042|7086 
 79\ 7130/71 747 218/7262:7306||7350/7394 743817482 7525 
 g8so| = 7569/7613, 7657 7\770117745\\7 7 8017 8337 877'7921\7965 
 81} 8009/8053 8097 8141/8185]|8229/8273183 17/836118405 
 82} 8448 8492/8536 8580'8624)|8668]87 12/87506:8800/8844 
 83! 8888!8932 8976|9020|9064||9108/9152|9196'9239/9283 
 
 C 950206 0250 O: ore 0338 0382 0426 0470/05 sha 0601 
 
 1963 2007205 1|2095|2130}!|2182|2220/2270/2314 2358 
 
 gl 240212446 24 90|25341257 8}12622/2605|2700|27 53|2797 
 Q2| 2841 2885) 929|2973|301 71/306 1)3 10413 14813 19213236 
 93} 3280)3324/3308/341213450}13500}3 543135871363 113675 
 94}  3710)3703|3807/385 11389513 930|3082/4026/4070}4114 
 95} 4158)4202/4246/4290/43341143 771442 11\446514500]4553 
 90} 4597|/4641/4085|4729/477 211481 614860)/4904)494814992 
 97} 5036/508015 12315 1671521 11525515290|5343153871543 1 
 - Q8} 5474)5518)/5562|5606|5650}/5004|5738/5782|5825|5869 
 96) 5913/5957 6001 6045 pep 0133 6170/6220 626416308 
 
N.990 L.995] 
 
 LOGARITHMS 
 516 
 
 RATA Fa) \GP.AG LCS SRREEE PELE] ELL PILE GPE N IE “VE PFT eee PERE > “nersrer 
 
 a3 oe es atts 71736117405 re 7492 ee 7580\7624 
 03 7668!7712)7755|7799|7 84317887 |793 1|7975|8019|8062 
 04} 8106/8150/8194/8238/8282/18326|8360184.13|8457|8501 
 05} 8545/8589|8632|8676|8720/18764'8808|8852|8896|8939 
 06} 8983]9027|907119115|9159]9202/9246|9290|9334|9378 
 07| 9422/9465/9509)/9553/9597||9041 (9685/97 2819772/9816 
 08 86019904|9948/999 1/003 51007 9|0123/0167|021 110254 
 09|9960298/0342'03 86|0430/0474 105 1710501|0605|0649/0693 
 9910) 0737|0780!0824/0868/0912/}0956|09990]1043}1087}1131 
 11} = 1175|1219|1262)1306|1350)1394]1439}1481]1525|1569 
 12}  1613]1657|1701]1744|1788]1832]1876]}1920]1963|2007 
 13] = 2051/2095|2139/2182|2226)2270)23 14/235812402|2445 
 14|  248912533/2577/2621|2664)2708|2752|2796|2840|2883 
 15]  2927/2971|3015/3059\3 102 13 146)3 190|3234|3278|3321 
 16|  3365|3409|3453|3497/3540|3584|3628/3672|37 16/3759 
 17| 3803)3847|389113935|39784022/4066]4110/4153/4197 
 18} 4241/4285|4320/4372/441614460\450414548|459 1/4635 
 19] 4679|4723)|4766/4810|4854/4898|4942)4985|5020|5073 
 9920} 5117|5161|5204|5248/5292)|5336|537015423/5407|5511 
 21} 5554|5598|5642|5086|5730)577315817|5861|5905|5948 
 22} 5992/6036/6080/6124|6167)/621 116255|6299|634216386 
 23; 6430\647416517|0501|0605)|6649|6693/6736|67 80|6824 
 24!  6868/6911|6955|0999\7043)|7086)7 130/7174|7218|7261 
 25|  7305|73401\7303|743617480)7524/7508!761 117055|7699 
 26}  774317786|7830'7874'7918)7961|8005|8049|8093/8136 
 27\ 818018224182688311'8355|8399|8443|8486/853018574 
 28] 8618/8661|8705|8749'8793||8836'8880)8924|8968\901 1 
 29} 9055/9099/9143'9186,9230)/9274/93 181936119405/9449 
 9930!  9492/953619580'9624|9667)|97 1119755|9799|9842\9886 
 31] 9930\9974|0017/0061 |0105||0148,0192/0236|0280}0323 
 
 33 04/0848/0892|0936:0979]|1023/1067}1110]1154/1198 
 34] 1242)1285]1329)1373]1416)/1460)1504)1548]1591/1635 
 35| 1679)1722)1766|1810)1854}|1897)194111985/2028/2072 
 36}  2116/2160/2203/2247|2291112334/2378)2422/2405/2509 
 371 2553/2597|2640|2684/2728]1277 1/281 5|2850|2903|29046 
 38]  20990)3034|3077/3 12113 165]|3208|3252/3296|3340/3383 
 39} 3427/347113514|3558|3602||3645|3680|3733/37 763820 
 9940; 3864/3908/395113995/4039}|4082)4 126/41 70)4213|4257 
 41} 4301/4344/4388/4432|4475||45 19/4563|46071405014694 
 42} 4738/478114825|4869/4912]14956|5000/5043/5087|5131 
 43] 5174/5218)5262)5305|5340/15393/5436|5480]5524|5567 
 - 44) §611/50655|5699|5742|5786)|583015873|5917|5901|6004 
 45| 6048)/6092/6135|6170|6223|16266/63 10|}6354|0397|0441 
 46} 6485/6528/6572|0616|6650)}0703|6747|6790/6834/6878 
 47 692 1|6965|7000]7052\7 006117 130)7 183|7227|7270|7314 
 48} 7358]740117445|7480|7532)17576|7620\7603|7707|775) 
 49} 7794|7838|7882!7025|7960]|8013)8056|8100]8 14418187 
 N.| 0 12d Se Pah Oh. Welw a D1 Pe 
 
77\9990000,0043|0087|0130|0174||0217/026110304|0348/0391 
 
 667|87 1 1|8755|8798|8842||8885|8920|8973|9016|9060 
 
 9540\9584|96271967 1197 15|19758}9802|9845}9880| 9933 
 9976|0020|0064|01071015 1/]0195|0238]028210325)0369 
 
 7823\7 866|7910!7053|7997|/8040|8084/8 281817118215 
 8258|8302/8345|83 80|84321847 6185 1 9\8563/8607|8650 
 869418737 878118824|8868]/89 1 1/8955|8998!9042/9086 
 9129'917319216/9260|9303|19347|9390 0434/9047 7|9521 
 9564196081965 1|9695|9739|9782982619869|9913|9956 
 
 0435|0479|052210566|0609||0653|0696 0740\0783|0827 
 0870)091410957|1001}1044/}1088)113111175]1218}1262 
 
 1305}1349]1392)1436)1470}11523)1567|1610)1654|1697 
 
 2611/265412698/2741|2785|2828 2872/2915 2 
 3040/3080)3 13313 176|322013263/3307'3350|3394|3437 
 
 2 a a es | a eel ied ieee | 
 ee ee 
 
 99501997823 118274|83 18]8362|8405|18449| 8493/8530 
 . 
 
(186) 
 
 N 
 
 CONAGAONE| 
 
 Logarithms. 
 
 00000 00000 C0000 00000 
 30102 99956 63981 19521 
 47712 12547 19662 43730 
 60205 99913 27962 39043 
 69897 00043 36018 80479 
 77815 12503 83643 63251 
 84509 80400 14256 83071 
 90308 99869 91943 58564 
 905424 25094 39324 87459 
 00000 00000 00000 00000 
 04139 26851 58225 04075 
 07918 12460 47624 82772 
 11394 33523 06836 76921 
 
 14612 80356 78238 02593 | 
 
 17609 12590 55681 24208 
 20411 99826 55924 78085 
 23044 89213 78273 92854 
 25527 25051 03306 06980 
 27875 36009 52828 96154 
 30102 99956 63981 19521 
 32221 92047 33919 26801 
 34242 26808 22206 23506 
 36172 78360 17592 87887 
 38021 12417 11606 02294 
 39794 00086 72037 60957 
 41497 33479 70817 96442 
 431360 37641 58987 31189 
 44715 80313 42219 22114 
 46239 79978 98956 08733 
 47712 12547 19662 43730 
 49136 16938 34272 67067 
 50514 99783 19905 97607 
 51851 39398 77887 47805 
 53147 89170 42255 12375 
 54406 80443 50275 63550 
 55630 25007 67287 26502 
 56820 17240 66994 99681 
 57978 35966 16810 15675 
 59106 46070 26499 20050 
 60205 99913. 27962 39043 
 61278 38507 19735 49451 
 62324 92903 97900 46322 
 63346 84555 79586 52641 
 64345 20764 86187 43118 
 65321 25137 75343 67938. 
 
 66275 78316 81574 07408 
 
 67209 78579 35717 46441 
 
 68124 12373 75587 21815 
 69019 60800 28513 66142 
 69897 00043 36018 80479 
 
 N. 
 
 | 75587 48556 72491 39883 
 
 06378 78273 45555 26930 
 | 96848 29485 53935 11696 
 
 | 98677 17342 66244 85178} 
 
 6 MOTs Ss PF 
 For finding Logarithms and Numbers to 20 Places of Figures: 
 
 Logarithms. 
 
 70757 01760 97936 36584 
 71600 33436 34799 15963 
 72427 58696 00789 04563 
 73239 37598 22068 50710 
 74036 26894 94243 84554 
 74818 80270 06200 41635 
 
 76342 79935 62937 28255 
 77085 20116 42144 19026 
 77815 12503 83643 63251 
 
 78532 98350 10767 03389 
 79239 16894 98253 87488 
 79934 05494 53581 70530 
 80617 99739 83887 17128 
 81291 33566 42855 57399 
 
 81954 30355 41808 67326 
 82607 48027 00826 43415 
 83250 89127 06236 31897 
 83884 90907 37255 31616 
 84509 80400 14256 83071 
 
 85125 83487 19075 28609 
 85733 249064 31268 46023 
 86332 28601 20455 90107 
 86923 17197 30976 19202 
 87506 12633 91700 04687 
 88081 35922 807901 35196 
 88649 07251 72481 87146 
 89209 46026 90480 40172 
 89762 70912 90441 427909 
 90308 GQ86g 91943 58564 
 90848 50188 78649 74918 
 91381 38523 83716 68972 
 91907 $0923 76073 90383 
 92427 92860 61881 65843 
 92941 89257 14292 73333 
 93449 $4512 43567 72162 
 93951 92526 18618 52463 
 94448 26721 50168-62639 
 94939 00066 44912 78472 
 95424 25004 39324 87459 
 95904 13923 21093 59992 
 
 97312 78535 99698 65963 
 97772 36052 88847 76032 
 
 98227 12330 39568 41336 
 
 99122 60756 92494 85664 
 99563 51945 97549 91534 
 00000 00000 00000 00000 
 
Tab. 2. 
 
 Logarithms. 
 
 00432 13737 82642 57428 
 00860 01717 61917 56105 
 01283 72247 05172 20517 
 01703 33392 98780 35485 
 02118 92990 69938 07279 
 02530 58652 64770 24085 
 02938 37776 85209 64083 
 03342 37554 86949 70231 
 03742 64979 40623 63520 
 04139 26851 58225 04075 
 
 04532 29787 86657 43410 
 04921 80226 70181 61157 
 05307 84434 83419 72280 
 05690 48513 36472 59405 
 06069 78403 53611 68365 
 06445 79892 26918 47776 
 06818 58617 46161 64380 
 07188 20073 06125 38547 
 07554 69613 92530 750925 
 07918 12460 47624 82772 
 08278 53703 16450 08150 
 08635 98306 74748 22910 
 08990 51114 39397 93180 
 09342 16851 62235 07009 
 09691 00130 08056 41436 
 
 10037 05451 17562 90052 
 10380 37209 55956 86425 
 10720 990G6 47868 30050 
 11058 97102 99248 96370 
 11394 33523 06836 76921 
 11727 12956 55764 26081 
 12057 39312 05849 80847 
 12385 16409 67085 79225 
 12710 47983 64807 62036 
 13033 27684 95006 11667 
 13353 89083 70217 51418 
 13672 05671 56406 76856 
 13987 90864 01236 51138 
 14301 48002 54095 08046 
 14612 80356 78238 02503 
 
 14921 91126 55379 90171 
 15228 83443 83056 48131 
 15533 60374 65061 80996 
 15836 24920 95249 65545 
 16136 80022 34974 89212 
 
 16435 28557 84437 09629 
 16731 73347 48170 09872 
 17026 17153 94957 38724 
 17318 62684 12274 03826 
 17609 12590 55681 24208 
 
 LOGARITHMS tO 20 PLACES. 
 
 N. 
 
 151 
 152 
 153 
 154 
 155 
 
 156 
 
 .22530 92817 25862 853605 
 
 | 23552 84469 07548 91683 
 
 (187) 
 Logarithms. 
 
 17897 09472 93109 43087 
 18184 35879 44772 54718 
 18469 14308 17598 80313 
 18752.07208 36463 06668 
 19033 16981 70291 48445 
 19312 45983 54461 59693 
 19589 96524 09233 73676 
 19805 70869 54422 62321 
 20139 71243 20451 48293 
 20411 99826 55924 78085 
 
 200682 58760 31849 70958 
 20951 50145 42030 94439 
 21218 76044 03957 80704 
 21484 38480 47697 88404 
 21748 39442 13906 28283 
 
 22010 80880 40055 09905 
 22271 64711 47583 27998 
 
 22788 67046 13673 53841 
 23044 89213 78273 92854 
 23299 61103 92153 83613 
 
 23804 61031 28795 41456 
 24054 92482 82599 71984 
 24303 80480 86294 44028 
 
 24551 26078 14149 82161 
 24797 32003 61806 62756 
 25042 00023 08893 97904 
 25285 30309 79893 16057 
 25527 25051 03306 06980 
 25707 85748 69184 51029 
 20007 13879 85074 79513 
 26245 10897 30429 47118 
 26481 78230 09536 46451 
 26717 17284 03013 80159 
 26951 20442 17916 31218 
 27184 10065 36498 96929 
 27415 784092 63679 85484 
 27646 18041 73244 14260 
 27875 30009 52828 96154 
 28103 33672 47727 53764 
 28330 12287 03549 60858 
 28555 73090 07773 76060 
 28780 17299 30226 0470: 
 29003 46113 62518 01129 
 29225 60713 56476 05185 
 290446 62261 61592 92737 
 29666 51902 61531 11055 
 29885 30764 09706 65010 
 30102 99956 639081 19521 
 
LOGARITHMS 
 Logarithms. N. Logarithms. 
 
 30319 00574 20488 87144 251 | 39907 37214 81038 13934 
 30535 13694 46623 76949 252 | 40140 05407 81544 00573 
 30749 60379 13212 91805 253 | 40312 05211 75817 91962 
 30963 01674 25898 75626 254 | 40483 37166 19938 05946 
 31175 38610 55754 29930 255 | 40654 01804 33955 17062 
 31386 72203 69153 40038 256 | 40823 99053 11849 56171 
 31507 03454 56917 75346 257 | 40993 31233 31204 53716 
 31806 33349 62761 55006 258 | 41161 97059 63230 15891 
 32014 62861 11054 00229 259 | 41329 97640 81251 82752 
 32221 9209047 33919 26801 260 | 41497 33479 70817 96442 
 32428 24552 97692 66508 261 | 41664 05073 38280 96192 
 32633 58609 28751 43606 262 | 41830 12913 19745 45602 
 32837 96034 38737 72339 263 | 41995 57484 89757 86897 
 33041 37733 49190 83605 264 | 42160 39268 69831 06369 
 33243 84599 15605 33119 265 | 42324 58739 36807 85042 
 33445 37511 50930 89753 266 | 42488 16366 31066 98746 
 33645 97338 48529 51038 267 | 42651 12013 64575 22202 
 33845 04936 04004 83041 268 | 42813 47940 28788 82458 
 34044 41148 40118 33837 269 | 42075 22800 02407 98009 
 34242 26808 22206 23506 270 | 43136 37641 58987 31189 
 34439 22736 85110 60775 271 | 43296 92908 74405 72052 
 34635 29744 50638 62932 272, | 43456 89040 34198 70940 
 34830 48630 48160 67348 273 | 43616 26470 40756 03721 
 35024 80183 34162 80678 274 | 43775 05628 20387 y6378} 
 35218 25181 11362 48416 275 | 43933 26938 30262 05032 
 35410 84391 47400 91801 276 | 44090 90820 65217 70659 
 35602 58571 93122 72010 277 | 44247 97000 64448 55378 
 35793 48470 00453 78926 278 | 44404 47959 18070 27507 
 35983 54823 39887 90413 279 | 44500 42032 73597 55426 
 36172 78360 17592 87887 280 | 44715 80313 42219 22114 
 36361 19798 92144 30876 281 | 44870 63199 05079 89286 
 36548 79848 90899 67297 282 | 45024 91083 19301 09692 
 36735 59210 26018 97219 283 | 45178 04355 24290 23550 
 36921 58574 10142 83901 284 | 45331 83400 47037 07052 
 37106 78622 71736 26920 285 | 45484 48000 08510 20362 
 37291 20029 70106 58069 286 | 45036 60331 29043 00517 
 37474 83460 10103 86529 287 | 45788 18967 33992 32522 
 37057 69570 56511 95447 288 | 45939 24877 59230 85066 
 37839 79009 48137 68500 289 | 40089 78427 50547 85708 
 38021 12417 11606 02294 290 | 46239 79978 98950 08733 
 38201 70425 74868 38408 291 | 46389 29889 85907 28908 
 38381 53659 80431 27671 292 | 46538 28514 48418 29150 
 38560 62735 98312 18648 293 | 46086 70203 54109 45624 
 38738 98263 38729 42431 294 | 46834 73304 12157 29393 
 38916 60843 04532 46621 295 | 46982 20159 78162 99505 
 39093 51071 03379 12702 206 | 47129 17110 58938 58245 
 39269 69532 59005 73074 207 147275 64493 17212 35204 
 39445 16808 26216 26531 208 |47421 62640 76255 23347 
 39619 93470 95736 34113 209 | 47567 11883 24429 64807 
 300 | 47712 12547 19662 43730 
 
 | 39794 00086 72037 60057 
 
 Tab. 2./ 
 
Tab. 2. © 
 
 Logarithms. 
 
 47856 64955 93843 35712 
 48000 69429 57150 63208 
 48144 20285 02305 01157 
 
 | 48287 35836 08753 74239 
 
 48429 98393 46785 83867 
 48572 14264 81579 99834 
 48713 83754 77186 48475 
 48855 07165 00444 26189 
 48095 84704 24834 64247 
 49136 10938 34272 67967 
 49276 03890 20837 50555 
 49415 45940 18442 79214 
 
 49554 43375 46448 48481 
 
 49692 96480 73214 93198 
 49831 05537 89000 51009 
 49968 70826 18403 81842 
 50105 92622 17751 49455 
 50242 71199 84432 67814 
 50379 06830 57181 12808 
 
 50514 99783 19905 97607 |‘ 
 
 50650 50324 04872 07813 
 50785 58716 95830 90479 
 50920 25223 31102 89008 
 51054 50102 00612 13961 
 51188 33609 78874 37878 
 51321 76000 67939 00285 
 51454 77526 60286 07250 
 51587 38437 11679 08015 
 51719 58979 49974 29513 
 51851 39398 77887 47805 
 51982 79937 75718 73861 
 52113 80837 040360 29426 
 52244 42335 00319 87140 
 52374 64668 11564 47520 
 52504 48070 36845 23894 
 
 52633 92773 89844 04886 
 52762 99008 71338 62619 
 52891 67002 77054 73363 
 53019 96982 03082 16009 
 53147 89170 42255 12375 
 53275 43789 924907 72042 
 53402 61060 56135 03134 
 53529 41200 42770 49214 
 53055 84425 71530 11205 
 53781 90950 73274 12095 
 
 53907 60987 92776 60977 | 
 54032 94747 90873 71854 }. 
 
 54157,92439 46580 91506 
 54282 54269 59179 89654 
 54406 80443 50275 63550 
 
 TO 20 PLACES. _ 
 
 (189) 
 
 Logarithms. 
 
 54530 71164 05824 08109 
 54654 26634 78131 01682 
 54777 47053 87822 56550 
 54900 32620 25787 82277 
 55022 83530 55094 09088 
 
 55144 99979 72875 17515 
 55266 82161 12193 19655 
 55388 30266 43874 36478 
 55509 44485 78319 14782 
 55630 25007 07287 26502 
 
 55750 72019 05057 92307 
 55870 85705 33165 70550 
 55990 66250 36112 51880 
 56110 13836 49055 99035 
 56229 28644 50474 70586 
 
 506348 10853 94410 66639 
 56466 60642 52089 33799 
 56584 78186 73517 65972 
 56702 63661 59060 36910 
 56820 17240 66994 99081 |. 
 56937 39096 15045 87635° 
 
 57054 29398 81897 50739 
 57170 88318 08687 60551 
 57287 16022 00480 16450 
 57403 12677 27718 85165. 
 57518 78449 27661 05006 
 570634 13502 05792 85054 
 57749 17998 37225 33781 
 57863 92099 08072 34193 
 57978 35906 16810 15675 
 58092 49756 75019 30154 
 58200 33629 11708 73285 
 58319 87739 08022 74038 
 58433 12243 07530 80379 
 58546 07295 08500 67625 
 58058 73046 71754 95581 
 58771 09650 18911 40100 
 58883 17255 94207 24221 
 58994 96013 25707 73624 
 59100 46070 26499 20650 
 59217 67573 95860 80741 
 50328 60670 20457 24707 
 59439 25503 75420 69811 
 59549 62218 25574 12259 
 59659 70956 204600 23278 
 
 59769 51859 25512 30577 
 
 | 59879 05067 63115 06588 
 
 50988 30720 73687 84531 
 60097 28956 860748 22954 }. 
 60205 99913 27962 39043 
 
(190) 
 
 N. 
 
 05321 25137 75343 67938 
 
 LOGARITHMS Tab. 2. 
 Logarithms. N. Logarithms. 
 60314. 43726 2018% 30054 451 | 05417 65418 77960 53526 
 60422 60530 84470 06666 452,| 65513 84348 11382 11322 
 60530 50461 41109 44887 453 | 65609 82020 12831 87416 
 60638 13651 10604 96470 454105705 58528 57103 91532 
 60745 50232 14668 55397 455 | 65801 13966 57112 40470 
 60852 60335 77194 11326 456 | 65806 48426 64434 98447 | 
 60059 44092 25220 03756 457 | 65991 62000 69850 22235 
 61066 01630 89879 95148 458 | 66086 54780 03869 18934 
 61172 33080 07341 80361 459 | 66181 26855 37261 24043 
 61278 38507 19735 49451 460 | 66275 78316 81574 07408 
 61384 18218 76069 20586 461 | 66370 09253 89648 14507 
 61489 72160 33134 59560 462 | 66464 19755 56125 50397 } 
 61595 00516 56401 02097 463 | 66558 09910 17953 13567 
 61700 03411 20898 94867 464 | 66651 79805 54880 86819 
 061804 80967 12092 70862 465 | 60745 29528 89953 92175 
 61909 33300 26742 74528 466 | 66838 59166 90000 16740 
 62013 60549 73757 51775 467 | 60931 68805 66112 16309 
 62117 62817 75035 19750 468 | 67024 58530 74124 03422 
 62221 40229 66295 30085 469 | 67117 28427 15083 26486 
 62324 92908 97900 46322 470 | 67209 78579 35717 46441 
 62428 20958 35668 30744 471 | 67302 09071 28896 17406 
 62531 24509 61673 86030 472 | 67394 19986 34087 77590 
 62634 03673 75042 33900 473 | 67480 11407 37811 56716 
 62736 58565 92732 63127 474 | 67577 83416 74085 06050 
 62838 89300 50311 53811 475 | 67669 36096 24866 57111 
 62940 95991 02718 91860 476 | 67760 69527 20493 14968 
 63042 78750 25023 86460 477 | 67851 83700 40113 92022 
 63144 37690 13172 03126 478 | 67942 78906 12118 88022 
 63245 72921 84724 24725 479 | 68033 55134 14503 22010 
 03346 84555 795860 52641 480 | 68124 12373 75587 21815 
 63447 72701 60731 60075 481 | 68214 50763 73831 76601 
 63548 37468 14912 09274 482 | 08304 70382 38849 57929 
 63648 78963 53365 44270 483 | 68394 71307 51512 14088 
 63748 97205 12510 70559 484 | 68484 53616 44412 47193 
 63848 925069 54637 32941 485 | 68574 17386 02263 65057 
 63948 64892 68586 02563 486 | 68663 62692 62293 38169 
 64048 14369 70421 84040 487 | 68452 890612 14634 33246 
 64147 41105 04099 53358 488 | 68841 98220 02710 61953 
 64246 45202 42121 37003 489 | 68930. 88591 23620 24494 
 64345 26764 86187 43118 490 | 69019 60800 28513 66142 
 64443 85894 67838 53601 491 | 69108 14921 22968 47275 
 64542 22693 49091 89206 492 | 69196 51027 67360 32233 
 64640 37262 23069 50023 493 | 69284 69192 77230 01587 
 64738 29701 14619 82453 494 | 69372 69489 23646 92596 
 64836 00109 80931 58951 495 |69460 51989 33568 72013 
 } | 64933 48587 12141 86869 496 |69548 16764 90197 46052 
 65030 75231 31936 47555 497 |69635 63887 33332 11081 
 65127 80139 98144 00199 AQ8 | 69722 93427 59717 53034 
 65224 63410 03323 17492 499 | 69810 05456 23389 91417 
 500 
 
 69897 00043 36018 80479 
 
To 20 PLACES. 
 
 ee ee 
 
 69983 77258 67245 71728 
 70070 37171 45019 33455 
 70156 79850 55927 39710 
 70243 05364 45525-29094 
 70329 13781-18661 37906 
 
 70415 05168 39799 11483 
 
 | 70500 79593 33335 97571 
 
 70586 37122 83919 25467 
 70671 77823 36758 74657 
 
 170757 01760 97936 36584 
 
 70842 09001 34712 73179 
 79926 99609 75830 75692 
 71011 73651 11816 27342 
 
 71096 31189 95275 73238: 
 
 71180 72290 41191 00906 
 71264 97016 27211 35413 
 71349 05430 93942 50516 
 71432 97597 45233 02273 
 71516 73578 48457 85186 
 
 )|71600 33436 34799 15963 
 
 71683 77232 99524 47424 
 71767 05030 02262 15714 
 71850 16888 67274 23926 
 71933 12869 83726 65124 
 72015 93034 05956 87758 
 72098 57441 53739 00419 
 72181 06152 12546 60821 
 72263 39225 33812 25800 
 72345 56720 35185 75774 
 72427 58696 GO789 04563 
 72509 45210 81469 06485 
 72591 16322 95048 18268 
 72672 72000 26572 26372 
 72754 12570 28556 41723 
 72835 37820 21228 44562 
 72916 47896 92770 01979 
 72997 42856 99555 60687 
 73078 22756 66389 17530 
 73158 87651 86738 70217 
 73239 375908 22968 50710 
 73319 72651 06569 43688 
 73399 92865 38386 92473 
 73479 98295 88846 94758 
 73559 88996 98179 90461 
 73039 65022 76642 43999 
 73719 20427 04737 23243 
 73798 73263 33430 77381 
 73878 05584 84369 15899 
 73957 23444 50091 90848 
 74036 26894 94243 84554 
 
 Logarithms. 
 
 74115 15988 51785 04887 
 74193 90777 29198 90180 
 74272 51313 046908 25871 
 74350 97647 28429 748909 
 74429 20831 22670 23889 
 74507 47015 82057 47088 
 74585 51951 73728 90044 
 74663 41989.37578 74947 
 74741 18078 86423 20561 
 74818 80270 06200 41635 
 
 74896 28612 56161 40659 
 74973 63155 69061 08808 
 75050 83948 51346 22909 
 75127 91039 83342 29214 
 75204 844783 19438 52758 | 
 
 } | 75281 64311 88271 43077 | 
 
 75358 30588 92906 57989 
 75434 83357 11018 87173 
 75511 22663 95071 17229 
 75587 48556 72491 39883 
 75663 61082 45848 05004. 
 75739 60287 93024 20038 
 75815 46219 67389 97403 
 75891 18923 97973 52044 
 75966 78446 80630 48844 
 
 76042 24834 23212 04587 
 70117 58131 55731 42849 
 76102 78384 20529 05229. 
 76267 85637 27436 19789 
 76342 70935 62037 28255 
 76417 61323 90330 73454 
 76492 20846 49888 48429 
 76500 85547 59014 08638 ; 
 76641 28471 12309 48072 
 76715 58660 82180 44858 
 76789 76160 18090 65146 
 76863 81012 47614 47606 
 76937 73260 76138 48915 
 77011 52947 87101 64120 
 77085 20116 42144 10026 
 
 77158 74808 81255 36467 
 77939, 17067 22919 77766 | 
 77305 46933 64262 60640 
 77378 64449 81193 54785 
 77451 69057 28549 56404 
 77524 62597 40236 42868 
 77597 43311 29369 08740 
 77670 11839 88410 84329 
 
 | 77742 68223 89311 37983 
 
 77815 12503 83643 63251 
 
(192) LOGARITHMS Tab. 2. 
 IN; Logarithms. N. Logarithms. 
 
 601 | 77887 44720 02739 52089 651 | 81358 09885 68191 94767 
 602177959 64912 57824 55233 652 | 81424 75957 31920 19807 
 603 | 78031 73121 40151 30874 653 | 81491 31812 75073 92143 
 604} 78103 69386 21131 82730 054 | 81557 77483 24267 26771 
 605 | 78175 53740 52468 88629 655 | 81624 12999 91783 06560 
 606 | 78247 26241 66286 20678 656 | 81690 38393 75660 27536 
 607 | 78318 86910 75257 58096 657 | 81756 53695 59780 77566 
 608 | 78390 35792 72734 93761 658 | 81822 58936 13955 49034 
 609 | 78461 72926 32875 35534 659 | 81888 54145 94009 86128 
 610 | 78532 98350 10767 03389 660 | 81954 39355 41868 67326 
 611 | 78604 12102 42554 23362 661 | 82020 14594 85640 23665 
 612 | 78075. 14221 45561 19356 062 | 82085 79894 396909 93382 
 613 |78746 04745 18415 03774 663 | 82151 35284 04773 13504 
 614. | 78816 83711 41107 67997 664. | 82216 80793 68017 48947 
 615 | 78887 51157 75416 73059 665 | 82282 16453 03104 59703 
 616 | 78958 07121 64425 45710 666 | 82347 42291 70301 06661 
 617 | 79028 51640 33241 68205 667 | 82412 58339 16548 96620 
 018 | 79098 84750 88815 83768 668 | 82477 64624 75545 67041 
 619 | 79169 06490 20117 97680 669 | 82542 61177 67823 11077 
 620 | 79239 16804. 98253-87488 670 | 82607 48027 00826 43415 
 621 | 79309 16001 76580 19075 671 | 82072 25201 68992 07464 
 622 | 79379 03846 90818 70077 672 | 82736 92730 53825 24408 
 623 | 79448 80460 59169 01544 673 | 82801 50642 23976 84648 
 624 | 79518 45896 82423 98736 674 | 82865 98965 35319 82140 
 625 | 79588 00173 44075 21915 675 | 82930 37728 31024 92146 
 626 | 79657 43332 10429 68002 676 | 829904 66959 41635 02884 
 627 | 79726 75408 30716 43958 677 | 83058 86686 85144 31601 
 628 | 79795 96437 37196 12719 678 | 83122 96938 67063 35530 
 629 | 79805 00454 45208 92535 679 | 83186 97742 80501 68250 
 630 | 79934 05494 53581 70530 680 | 83250 89127 06236 31897} . 
 631 | 80002 93592 44134 31302 681 | 83314 7111G 12785 15740 
 632 | 80071 70782 82385 01364 682 | 83378 43746 56478 91563 
 633 | 80140 37100 17355 10238 683 | 83442 07036 81532 56340 
 634 | 80208 92578 81732 68977 684 |} 83505 61017 20116 22655 
 635 | 80277 37252 91975 66903 685 | 83569 05714 92425 57335 
 636 | 80345 71156 48413 87336 686 | 83632 41157 06751 68735 
 637 | 80413 94323 35350 43063 687 | 83695 67370 59550 43142 
 638 | 80482 06787 21162 32330 688 | 83758 84382'35511 30726 
 639 | 80550 08581 58400 16068 689 | 83821 92219 07625 81484 
 640 | 80617 99739 83887 17128 690 | 83884 90907 37255 31616 
 641 | 80085 80205 18817 42225 691 | 83947 80473 74198 40758 
 642 | 80753 50280 68853 27334 692 | 84010 60944 56757 80499 | . 
 643 | 80821 09729 24222 07249 693 | 84073 32346 11806 74605 
 644 | 80888 58073 59812 10001 694 | 84135 94704 54854 91375 
 645 | 80955 97146 35207 76849 695 | 84198 48045 90113 88524 
 646 | 81023 25179 95084 08529 696 | 84260 92396 10562 11027 
 647 | 81090 42806 68700 38446 697 | 84323 27780 98009 42305 
 648 | 81157 50058 70593 33482 698 | 84385 54226 23161 09175 
 649 | 81224 40968 00369 23101 699 | 84447 71757 45681 40948] 
 650 | 81291 33566 42855 57399 700 | 84509 80400 14256 83071} 
 
Logarithms. 
 
 84571 80179 66658 65700 
 84633 71121 29805 27631 
 84695 53250 19823 95834 
 84757 26591 42112 21203 
 84818 91169 91398 70650 
 84880 47010 51803 76071 
 84941 94137 96899 40499 
 85003 32576 89769 017908 
 85064 62351 83066 54285 
 85125 83487 19075 28609 
 
 85186 96007 20766 30258 
 85247 99936 36856 37036 
 ‘85308 95298 51865 55853 
 14 | 85369 82117'76174 30176 
 5 | 85430 60418.01080 61474 
 } }:854.91 30223 07855 56000 
 |85551 01556 67800 12230 
 | 85012 44442 42300 34303 
 ‘85072 88903 82882 60777 
 85733 24964 31268 46023 
 85793 52647 19429 03588 
 85853 71075 69639 11829 
 “| 85913 820972 94530 82137 
 4 | 85073 85061 97146 90071 
 86033 80065 70993 69601 
 (86003 66207 00093 71401 
 80153. 44108 59037 83621 
 86213 13793 13037 18556 
 Q | 80272 75283 17074 62377 
 80332 28601 20455 90107 
 86301 73769 57860 45405 
 86451 10810 58391 86161 
 86510 39746 41127 94317 
 86569 60599 16070 53320 
 806628 73390 84194 90351 
 } | 86687 78143 37408 854094 
 89740 74878 50051 47490 
 86805 63618 23041 56431 
 
 809023. 17197 30976 19202 
 86981 82079 79328 16804 
 87040 39052 79027 07156 
 87098 88137 60575 29242 
 ‘| 87157 20355 45878 70260 
 87215 02727 48202 84304 
 87273 88274 72668 80072 
 87332 06018 15308 77842 
 87390 15078 64461 35972 
 87448 18176 99466 47155 
 87500 12033.91700 04687 
 
 To 20 PLACES. 
 
 80804 44383 94825 73669 
 
 (193) 
 
 Logarithms. 
 
 87503 99370 04108 38075 
 87621 78405 91642 24527 
 87079 49762 00700 57664 
 $7737 13458 00774 05175 
 87794 69516 29188 241606 
 )| 87852 17955 01206 53302 
 ‘87909 58795 00072 75709 
 87966 92056 32053 53715 
 88024 17758 95480: 35691 
 ‘| 88081 35922 80791 35190 
 88138 46567 70572 82637 
 88195 49713 39600 49675 
 ‘88252 45379 54880 46591) 
 88309 33585 75689 92806 
 88366 14351 53017 60792 
 ‘88422 87606 32603 93559 
 88479 53639 48980 95947 
 88536 12200 31511 99900 
 88592 63398 01431 03960 
 88640 07251 72481 87146 
 88705 43780 50956 97446 
 88761 73003 35736 15102 
 88817 94939 18324 90897 
 88874 09606 82802 59621 
 88930 17025 06310 28924 
 
 88986 17212 58188 43743 
 89042 10188 009014 26482 
 89097 95969 80688 93146 
 89153 74576 72504 45605 
 89209 46026 90480 40172 
 
 89265 10338 77300 32684 
 89320 67530 50848 00262 
 89370 17620 57943 3009022 
 890431 60626 84438 44228 
 80486 90567 45252 54155 
 89542 25460 39407 89332 
 80597 47323. 59004 55847 
 89652 62174 89555 31780 
 89707 70032 09420 30627 
 89762 70912 90441 4279u 
 89817 64834 97676 5535) 
 80872 51815 890493 50098 
 89027 31873 17603 80309 
 89982 05024 27006 26100 
 *Q0036 71286 56470 28771 
 g0091 30677 37669 04053 
 90145 83213 96112 34727 
 90200 28913 50729 42476 
 90254 67703 13991 39205 
 90308 99869 91943 58564 
 
92941 89257 14292 73333 
 
 LOGARITHMS ‘Fab: 23 
 Logarithms. N. Logarithms. 
 
 90363 25100 84237 65931 851 | 92992 95600 84587 87568 
 90417 43682 84163 50176 852 | 93043 95947 66700 11382 
 Q0471 55452 78680 94182 853 | 93094 90311 67523 03000 
 90525 60487 48451 26187 854 | 93145 78706 89005 05981 
 90579 58803 67808 51437 855 | 93196 61147 28172 64091 
 90633. 50418 05090 64409 856 | 93247 37646 77153 22648 
 Q0687 35347 22070 41738 857 | 93298 08219 23198 16429 
 90741 13607 74586 15992 858 | 93348 72878 48705 44247 
 90794 85216 12272 30432 859 | 93399 31638 31242 30263 
 90848 50188 78649 74918 860 | 93449 84512 43567 72162 
 90902 08542 11156 03069 861 | 93500 31514 53654 76252 
 9095500292 41175 30847 862 | 93550 72658 24712 79506 
 91009 05455 94068 16682 863 | 93601 07957 15209 59266 
 |91062 44048 89201 23277 864.| 9365137424 78803 28705 
 91115 76087 39976 61243 865 | 93701 01074 04814 21935 
 91169 01587 53861 14669 866 | 93751 78920 17346 63791 
 91222 20565 32415 487094 867 | 93801 90974 76210 29438 
 91275 33036 71322 99882 868} 93851 97251 76491 90081 
 91328 39017 60418 47451 869 | 93901 97764 486066 46875 
 91381 38523 83716 689072 870 | 93951 92526 18618 52463 
 91434 31571 19440 77180 871 | 94001 81550 07663 20336 
 91487 18175 40050 40107 872 | 94051 64849 32567 22084 
 91539 98352 12269 83977 873 | 94101 42437 05569 72637 
 91592 72116 97115 79081 874 | 94151 14326 34403 03562 
 91645 39485 49925 08762 875 | 94200 80530 22313 24507 
 91698 00473.20882 21619 876| 94250 41061 68080 72880 
 91750 55095 52546 67071 877 | 94299 95933 66040 51823 
 91803 03367 84880 14389 878 | 94349 45159 06102 56585 
 91855 45305 50273 55312 879} 94398 88750 73771 89354 
 91907 80923 76073 90383 880 | 94448 26721 50168 62639 
 91960 10237 84110 99107 881 | 94497 59084 12047 91274 
 92012 33262 90723 94049 882} 94546 85851 31819 73123 
 92064 50014 00787 58096 883 | 04596 07035 77568 58562 
 92116 60506 37738 71297 884.1 04645 22050 13073 08817 
 92168 64754 83002 08477 885 | 94094 32706 97825 43234 
 92220 62774 39016 39271 886 | 94743 37218 87050 75544 
 92272 54579 93259 99155 887 | 94792 30198 31726 39220 
 902324: 40186 30276 50506 888 | 94841 290657 78601 01974 
 92376 19608 28700 27500 889 | 94890 17609 70213 09496 
 92427 92860 61881 65843 890 | 94939 00066 44912 78472 
 92479 59957 97912 17467 891 | 94987 77040 30874 78993 
 92531 20914 99649 50266 892 | 95036 48543 76123 06390 
 92582 75746 24742 33016 893 | 95085 14588 88546 42595 
 92634 24466-25655 05551 894 |95133 75187 95917 67077 
 92085 67089 49692 34320 895 | 95182 30353 15911 97430 
 } | 92737 03630 39023 53422 896 195230 80096 62125 19721 
 92788 34103 30700 91221 897 | 95279 24430 44092 08537 
 92839 58522 56713 82649 898 | 95327 63366 67304 37013 
 92890 70902 43952 67285 899 | 95375 90017 33228 76700 
 
 95424 25094 30324 87459 
 
te I I RE A RR A A 
 Tab. 2. TO 20 PLACES. (195) 
 
 Logarithms. N. Logarithms. 
 951 | 97818 05169 37413 93185 
 952 | 97863 69483 84474 34489 
 953 |.97909 29006 38326 40853 
 954 | 97954 83747 04095 115 
 955 | 98000 33715 83746 34242 
 956 | 98045 78922 76100 07543 
 957 {98091 19377 76843 50538 
 958 | 98136 55090 78544 41531 
 959 |} 98181 86071 70663 59928 
 9600 | 98227 12330 39568 41336 
 961 | 98272 33876 68545 35933 
 962 | 98317 50720 37812 90123} 
 963 | 98362 62871 24534 51542 
 964 | 98407 70339 02830 77450 
 965 | 98452 73133 43792 50533 
 960 | 98497 71264 15493 34209 
 907 | 98542 64740 83001 67360} 
 908 | 98587 53573 08393 66714 
 969 | 98632 37770 50705 32737 
 970 | 98677 17342 66244 85178 
 971 | 98721 92299 08004 86280}: 
 972 | 98766 62649 26274 57690}. 
 Q73 | 98811 28402 68351 91117] 
 74 | 98855 89568 78615 52768} 
 975 | 98900 46156 98536 81607: 
 976 | 98944 98176 66691 81474] 
 977 | 98989 45037 18773 07091 
 Q78 | 99033 88547 87001 44015 
 979 | 99078 20018 03137 82547} 
 980 | 99122 60756 92494 85664 
 981 | 99166 90073 79048 50979 
 982 | 99211 14877 86949 66707 
 Q83 | 99255 35178 32135 62275 
 984 | 992090 50984 31341 51745 
 985 | 99343 62304 97611 73216 
 986 | 90387 69149 41211 21109 
 Q87 | 99431 71526 69636 73242 
 988 | 99475 69445 87028 12117 
 989 |} 99519 62915 97179 40527 
 ~990 | 99503 51945 97549 91534 
 991 | 90607 30544 85275 32836 
 g92.| 99051 10721 54178 05574 
 903 | 99604 92484 95381 175900 
 994 | 99738 03843 97313 31202 
 095 | 990782 30807 45725 45489 
 090 | 99825 93384 23008 73156 
 947 197634 99790 03273 41875 997 | 99869 51583 11655 719088 
 948 | 97080 83373 38066 25572 9Q8 | 99013 05412 87371 10938 
 949 197726 02124 27202 67028 999 | 99956 54882 25982 30864 
 97772 36052 88847 76632 1001 | 00043 40774 709318 64067 
 
 901 | 95472 47909 79002 97417 
 902 | 95520 65375 41941 73047 
 903 | 95568 77503 13505 79441 
 904 | 95616 84304 75303 30844 
 905 | 95604 85792 05203 31508 
 906 | 95712 81976 76813 06938 
 907 |} 95760 72870 60095 25585 
 908 | 95808 58485 21085 11053 
 909 | 95856 38832 21967 44887 
 910 | 95904 13923 21093 59902 |. 
 911 |95951 83769 729008 24763 
 912 |}9590909 48388 28416 17069 
 913 | 96047 07775 34208 94458 
 914 | 96094 61957 33831 41757 
 915 | 96142 10940 66448 27507 
 916 |96189 54736 67850 38456 
 917 |96236 93356 70021 09152 
 918 |96284 26812 01242 43564 
 919 |96331 55113 86111 26520 
 920 |96378 78273 45555 20930 
 921 |96425 96301 96848 92205 
 922 |96473 09210 53629 34029 
 923 |96520 17010 25912 05530 
 924 |96567 19712 20106 69918 
 925 {90614 17327 39032 60638 
 1926 | 96661 09866 81934 33089 
 927 |96707 97341 44497 07970 
 928 | 96754 70762 18862 06340 
 {929 |96801 57139 93641 76318 
 930 |96848 29485 53935 11696 
 931 | 96894 96809 81342 62206 
 932 | 969041 59123 53981 36262 
 933 |96988 10437 46499 94285 
 934 | 97034 68762 30093 35830 
 935 |}97081 16108 72517 77408 
 936 197127 58487 38105 22944 
 937 |97173 95908 87778 26303 
 938 | 97220 28383 79064 406008 
 939 | 97266 55922 66110 92210 
 940 | 97312 78535 99698 65963 
 941 197358 96234 27256 90834 
 942 197405 09027 92877 36927 
 943 197451 10927 37328 37338 
 944 197407 19942 Q8068 97112 
 945 | 97543 18085 09262 94738 
 940 | 97589 11364 01792 76237 
 
 3C2 
 
re rem 
 
 (196) Dono diss DOG ARE TEMES Tab. 2, 
 
 N. Logarithms. 
 
 N. Logarithms. | 
 
 1083 | 03462 84566 25320 36037 
 
 1085 | 03542 97381 85148 31517 
 1087 | 03622 95440 86294 53993 
 
 1089 | 03702 78797 55774 95610) 
 1091 | 03782 47505 88341 87761) 
 1093 | 03862 01619 49702 79227 
 1095 |03941 41191 76137 14316 
 1097 |04020 66275 74711 13222 
 1099 | 04099 76924 23490 56747 
 1101 | 04178 73189 71751 77529} 
 1103 |04257 55124 40190 59866)’ 
 1105 | 04336 22780 21729 50254}. 
 1107 |04414 76208 78722 80639 
 1109 |04493 15461 49160 06471) 
 1111 |04571 40589 40867 61503) 
 1113 |04649 51643 34708 31364 
 
 1115 |04727 48673 84779 47827) 
 1117 |04805 31731 15609 05702 
 1119 | 04883 00865 28350 04281 
 1121 | 04960 56125 94973 15180 
 
 1123 |05037 97562 61457 78469 
 1125 |05115 25224 47981 28895 
 1127 |05192 39160 46106 54029} 
 1129 | 05269 39419 24967 86114 
 1131 }05346 26049 25455 29384 
 1133 | 05422 99098 63397 24592 
 1135 |05499 58615 29741 5248 
 1137 |05576 04646 87734 77923 
 1139 | 05652 37240 79100 36209 
 1141 }05728 56444 18214 63835 
 
 1143 |05804 62303 95281 73884 
 1145 |05880 54866 759060 79892 
 1147 |05956 34179 01267 67648 
 1149 |06032 00286 88285 17768 
 1151 |06107 53236 29791 80185 
 115g | 06182 93072 94699 02164 
 1155 |06258 19842 28163. 11355 
 1157 | 06333 33589 51749 55393 
 1159 | 06408 34359 63595 90543 
 1161 ]|06483 22197 38573 83830 
 
 1003 | 00130 09330 20418 11880 
 1005 | 00216 60617 56507 67623 
 1007 | 00302 94705 53618 00717 
 1009 | 00389 11662 36910 52172 
 1011 |00475 11555 91001 06349 
 
 1013 | 00560 94453 60280 42845 
 1015 | 00646 60422 49231 72283 
 1017 | 00732 09529 22744 59739 
 1019 | 00817 41840 06426 39490 
 1021 | 00902 57420 86910 24725 
 1023 | 00987 56337 12160 15771 
 1025 | 01072 38653 91773 10408 
 1027 | 01157 04435 97278 19720}: 
 1029 | 01241 53747 62432 92943 
 1031 | 01325 86052 83516 54691 
 
 1033 | 01410 03215 19620 57904 
 1035 | 01494 03497 92936 55824 
 1037 | 01577 87563 89049 96243 
 1039 | 01601 55475.57177 41240 
 1041 | 01745 07295 10536 15583 
 1043 |.01828 43084 26530 86897 
 1045 | 01911 62904 47072 80707 
 1047 | 019904 66816 78842 33384 
 1049 | 02077 54881 93557 85901 
 1051 | 02160 27160 28242 22008 
 1053 | 02242 83711 85486 51839 
 1055 | 02325 245906 33711 46987 
 1057 | 02407 49873 07426 26758 | 
 1059 | 02489 59601 07485 00279 
 1061 | 02571 53839 01340 66012 
 1063 | 02653 32645 23296 75697 
 1005 | 02734 960077 74756 52817 
 1067 | 02816 44194 24469 89253 
 1069 | 02897 77052 08778 01749 
 1071 | 02978 94608 31855 63385 
 1073 | 03059 97219 65951 08414 
 1075 | 03140 84642 51624 13598! 
 1077 | 03221 57032 97981 58511} 
 1079 | 03302 14446,.82910 67304 } 
 1081 |.03382 56939 53310 84328 
 ee a 
 
ro 20 PLACES. 
 
 97) 
 dec 2. HDS; : 
 
 101022'00441 59622 79337 39356|42989 87670 14079|42554 33310184245 
 101023|00442 02612 67007 53435/42989 45115 80760|42553 49005|84244 
 101024'00442 45602 12123 34204142989 02562 31704/42552 64821|84239 
 
 101025|00442 88591 14685 6590842988 60009 66883/42551 80582/84239 
 10102600443 31579 74605 32791|42988 17457 86301|42550 96343/84236 
 101027'00443 74567 92153 19092/42987 74906 89958/42550 12107|84233 
 101028\00444 17555 67060 09050/42987 32356 77851142549 27874|84230 
 10102900444 60542 99416 86901/42986 89807 49977|42548 43644|84228 
 
 101030,00445 03529 89224 36878/42986 47250 06333/42547 59416|84225 
 10103 1/00445 46516 36483 43211}42086 04711 40917|42546 75191|84225 
 101032/00445 89502 41194 90128/42085 62164 71726)42545 90966|84219 
 101033|00446 32488 03359 61854142985 19618 80760/42545 06747|84219 
 101034/00446 75473 22978 42614|42084 77073 74013|42544 22528/84214 
 
 101035|00447 18458 00052 16627/42084 34529 51485/42543 38314|84215 
 101036|00447 61442 34581 68112/42983 91986 13171142542 54099/84209 
 101037|00448 04426 26567 81283142083 49443 59072|42541 69890/84209 
 101038/00448 47409 76011 40355|42983 06901 89182|42540 85681/84205 
 101039/00448 90392 82913 29537142982 64361 03501/42540 01476|84204 
 
 101040/004409 33375 47274 33038]42982 21821 02025/42539 17272|84199 
 101041|00449 76357 69005 350031429081 79281 84753|42538 33073|841 99} 
 101042/00450 19339 48377 19816/42981 30743 51080/42537 48874|84196} 
 101043/00450 62320 85120 71496}42980 94206 02806/42536 64678/84193 
 110104400451 05301 79320 74302}42980 51669 38128)42535 80485/84189} 
 
 101045|00441 48282 30990 12430/42980 00133 57643/42534 96296|84180} 
 10104600451 91262 40129 70073142979 66598 01347|42534 12107|84187} ° 
 101047100452 34242 06728 31420}42979 24064 49240/42533. 27920/84181} 
 101048/00452 77221 30792 80660142978 81531 21320/42532 43739]84181 
 ; 01049|00453. 20200 12324 Oig80 42978 38998 77581/42531 59558)/8-41 78 
 
(198) “LOGARITHMS ~ ‘Fab. 3. 
 Num. Logarithms. Diff. 1. Diff. 2. {D. 3. 
 
 101050|00453 63178 51322 79561|42977 96467 18023|42530 75380|84177 
 101051100454 06156 47789 97584|42077 53936 42643|42529 91203|84172 
 101052|00454 49134 01726 40227142977 11406 51440/42529 07031184170 
 101053|00454 92111 13132 91667|42976 68877 44400|42526722861|84168 
 101054|00455 35087 82010 36076/42976 26349 21548|42527 38693|84167 
 
 101055|00455 78064 08350 57624/42975 83821 82855|42526 54520/84162 
 101056|00456 21039 92181 40479/42975 41295 28320/42525 70304/84161 
 101057|00456 64015 33476 68808|42974 98769 57965|42524 86203/84158 
 101058|00457 06990 32246 26773|42974 56244 71762/42524 02045|84155 
 101059|00457 49964 88490 98535|42074 13720 69717|42523 17890/84154 
 10106000457 92939 02211 68252/429073 71197 51827|42522 33736|84150 
 101061100458 35912 73409 20079}42973 28675 18091/42521 40586/84148 
 101062/00458 78886 02084 38170/42972 86153 68505/42520 65438/84146 
 10106300459 21858 88238 06675 42972 43033 03067/42519 81292/84143 
 101064|00459 64831 31871 09742142072 01113 21775/42518 97149184140 
 
 101065|00460 07803 32984 31517/42971 58594 24626/42518 13009/84138 
 101066|00460 50774 91578 56143|42971 16076 11617|42517 28871|84137 
 101067|00460 93746 07654 67760/42070 73558 82746|42510 44734|84132 
 101068'00461 36716 81213 50506'42970 31042 38012/42515 60602/84130 
 101069100461 79687 12255 88518/42969 88526 77410/42514 76472|84130 
 
 101070|00462 22657 00782 6592842969 46012 0093842513 92342/84124 
 10107100462 65626 46794 66866/42969 03498 0859642513 08218/84123 
 101072100463 08595 50292 75462/42968 60085 00378'42512 24005|84122 
 101073|00463 51564 11277 75840|42968 18472 70283 142511 390973|84117 
 10107400463 94532 29750 52123'42967 75061 3631042510 55856|84116 
 
 10107500464 37500 05711 88433/42967 33450 80454/42509 71740!84113 
 10107600464 80467 39162 68887|42966 90941 08714)42508 87627|84111 
 101077100465 23434 30103 77601|42966 48432 21087/42508 035160)84107 
 101078/00465 66400 78535 98688!42906 05924 17571\42507 19400/84107 
 101079|00466 09366 84460 16259/42905 63416 98162)42506 35302/84102 
 101080/00466 52332°47877 14421142965 20010 62860/42505 51200/84102 
 101081|00466 95297 608787 77281|42964 78405 11060!42504 67098|84007 
 101082|00467 38262 47192 88941|42964 35900 44502/42503 8300184095 
 101083|00467 81226 83093 33503/42903 93306 61561|42502 98g06|84095 
 101084|00468 24190 76489 95064|42963 50803 62055|42502 14811/84090 
 101085100468 67154 27383 57719|42963 08391 47844142501 30721)84087 
 101086|00469.10117 35775 05563\42902 65800 17123|42500 46034|84087 
 101087 (00469 53080 01605 22686/42962 23389, 70480|42499 62547/84083 
 101088|00469 96042 25054 93175/42061 80890 07942/42408 78464/84080 
 101089|00470 39004 05045 01117/42961 38391 20478|42497 94384/84078 
 101090|00470 81965 44336 30595|42960 95893 35094/42497 103006184077 
 101091/00471 24920 40229 65680}42960 53306 24788/42406 20220|84073 
 101092/0047 1 67886 93625 90477142960 10899 98559142495 42150|84069 
 101093|00472 10847 04525 89030/429590 68404 56403/42494 58087|84070 
 101094}00472 53806 72030 45430|42959 25909 983 16/424G3 74017|84005 
 101095|00472 96765 98840 43755/429058 83416 24290/42492 89952|84063 
 101090|00473 30724 82256 6805442058 40023 34347142492 0588G|84060 
 101097 |00473 82083 23180 02401|42957 98431 28458]42491 21829/84061 
 101098|00474 25641 21611 30850|42957 55040 06629)424090 37768|84052 
 10100000474 68598 77551 37488]42057 13449 68861|42489 53710/84056 
 
Tab. 3. to 20 PLACES. (199) 
 Diff. 1. Diff.-2. 
 
 101100100475 11555 91001 06349/42956 70960 15145|42488 69660|84050 
 10110100475 54512 61961 21494/42056 28471 45485]42487 85610/84048 
 101102|00475 97468 90432 66979/42055 85083 50875]42487 01562|84046 
 101103|00476 40424 76416 26854/429055 43496 58313]42486 17516|84043 
 10110400476 83380 19912 85167|42955 01010 40797/42485 33473]84040 
 
 101105100477 26335 20923 25964/42954 58525 07324/42484 49433/84040 
 101106100477 69289 79448 33288/42054 16040 57801|42483 65393/84034} 
 1101107|00478 12243 95488 91179/42053 73556 92498/42482 81350}84035 
 101108|00478 55197 69045 83677/42953 31074 11139|42481 97324/84030 
 101109|00478 98151 00119 9481642952 88592 13815/42481 13294|84028 
 
 10111000479 41103 88712 08631|42952 46111 00521/42480 29266/84025 
 101111}00479 84056 34823 09152/42952 03630 71255|42479 45241/84026 
 101112|00480 27008 38453 80407/42951 61151 26014/42478 61215/84018 
 101113|00480 69959 99605 06421/42951 18672 64790/42477 77197|84020 
 101114/00481 12911 18277 71220)42950 76194 87602/42476 93177|84015 
 
 101115|00481 55861 94472 58822/42950 33717 94425/42476 09162/84013 
 1101116|00481 98812 28190 53247|42049 91241 85263/42475 25140184012 
 101117|00482 41762 19432 3851042949 48766 60114/42474 41137|84008 
 10111800482 84711 68198 98624/42049 06292 18977|42473 57120184006 
 10111900483 27660 74491 17601/42048 63818 61848/42472 73123184002 
 
 10112000483 70609 38309 79440|42048 21345 88725|42471 89121184003 
 101121|00484 13557 59055 68174'42947 78873 99004|42471 05118/83996 
 101122/00484 56505 38529 6777842047 36402 94486/42470 21122/83998 
 101123|00484 99452 74932 62264/429046 93032 73364142469 37124/839003 
 10112400485 42399 68865 35628/42046 51463 36240/42468 53131/83990 
 
 101125100485 85346 20328 71868|42946 08994 83109/42467 69141|83988 
 101126|00486 28292 29323 54977/42945 66527 13968/42466 85153183988 
 101127|00486 71237 95850 68945/42045 24060 28815/42466 01165|83982 
 101128|00487 14183 19910 97760/42944 81594 27650/42465 1718383981 
 101129|00487 57128 01505 25410,42944 39129 10467/42464 33202/83978 
 
 10113000488 00072 40034 35877/42043 96664 77205|42463 49224|83978 
 101131100488 43016 37209 13142|42943 54201 28041/42462 65246/83971 
 101132/00488 85959 91500 41183/42943 11738 62795|42461 81275183972 
 101133|00489 28903 03239 03978/429042 69276 81520/42460 97303|/83969 
 10113400489 71845 72515 85408/42942 26815 84217/42460 13334|83965} 
 
 101135|00490 14787 99331 69715/42941 84355 70883/42459 29369/83964 
 101136|00490 57729 83687 40598/42941 41896 41514/42458 45405/83962 
 101137|00491 00671 25583 82112/42940 99437 96100/42457 61443|83957 
 101138|00491 43612 25021 78221)/42940 50980 34660/42450 7748683956 
 101139100491 86552 82002 12887]42040 14523 57180/42455 03530}83954 
 
 Num. Logarithms. 
 
Ter! apg . A BL Ev. ne a 
 
 (ZOO) «, : LOGARITHMS AND Tab. 4, 
 
 00007|10001 61193 04083 sa 23029 82769 74039|53028 74846]1 22106 
 Joo00s 10001 84223 77455 52806}23030 35798 4 48885|53029 06082 1 22104 
 
 10002 53316 43942 12421123031 94892 06063|53033 63282]1 22115 
 1000276348 38834 18484/23032 47925 69345153034 85397|1 22120 
 
 11 
 12 
 
 0038 10005 52773 14511 54360)23038 84424 54566|53049 51008|1 22151 
 
 00925}10005 75811 98936 08926|23039 37474 05574|53050 73159|1 22156 
 026,10005 98851 36410 14500|2303Q9 90524 78733/53051 95315/1 22158 
 00027) 10006 21891, 26934 93233/23040 43576 74048|53053 174731 22161 
 2810006 44931 70511 67281/23040 96629 91521/53054 39634/1 22163 
 00029} 10006 67972 67141. 58802/23041 49684 31155|53055 61797|1 22167 
 
 00030'10006 91014 16825 89957|23042 02739 92952/53056 83964]! 22170 
 00031/10007 14056 19565 82909|28042 55796 76916153058 06134|1 22170 
 032/10007 37098 75362 59825|23043 08854 83050|53059 28304|1 22177 
 00033|10007 60141 84217 42875|23043 61914 11354153060 50481|1 22177 
 00034|10007 83185 46131 54220123044 14974 61835153061 72658]1 22180 
 
 00035)/10008 05229 61 106 1606423044 68036 34493|53062 94838}1 22185 
 
 0038 10008 75365 24405 26242)/23046 27228 855611|530606 61397|1 22192 
 
 00030}10008 98411 51634 11803/23046 80295 46958|53067 83580}1 22195 
 00040)10009 21458 31929 58761}23047 333603 30547/53069 05784}1 22196 
 100041}10009 44505 65292 89308|23047 86432 36331153070 27980}1 22202 
 00042)10009 67553 51725 25630}/23048 39502 64311153071 50182}1 22202 
 00043}10009 90601 91227 89950)23048 92574 14493153072 72384|1 22200 
 00044/10010 13650 83802 04443]23049 45646 86877153073 945g0}1 22208 
 00045}10010 36700 29448 91320]23049 98720 81467|53075 10795}| 22213 
 00046}10010 59750 28169 72787}23050 51795 98205]53076 39011}1 22213 
 00047|10010 82800 79965 71052] 23051 04872 37276|53077 61224}1 22218 
 09048]10011 05851 84838 0832£}23051 57949 98500}53078 83442]1 22219 
 190040}10011 28903 42788 00828 25052 11028 819421/53080 056611 22224 
 
‘Tab. 4. NUMBERS to 20 PLACES. — ... (201 ) 
 Differ. 1. its: 2.) D3 
 
 00050/10011 51¢ 51955 55 53810 8 88770 23052 64108 ‘08 87603 53081 2 27885 1 22225 
 00051/10011 75008 17925 706373/23053 17190 15488|/53082 50110}1 22228 
 00052/10011 98061 35115 91861|23053 70272 05598]53083 72338]1 22232 
 00053}10012 21115 05388 57459/23054 23356 37930|53084 94570}1 22233 
 00054}10012 44169 28744 95395|23054 76441 32500/53086 10803]1 22238 
 
 00055|10012 67224. 05186 27901/23055 29527 49309|53087 39041]1 22238 
 0005610012. 90279 34713 77210|23055 82614 88350/53088 61279]1 22244 
 00057/10013 13335 17328 65560|23056 35703 49629|53089 835231 22245 
 100058|10013 36391 53032 1518023056 88793 33152153091 05768|1 22247 
 00059|10013 59448 41825 4834123057 41884 38920/53002 28015|1 22252 
 
 00060)10013 82505 83709 87261|23057 94976 66935|53093 50267]1 22254 
 00061}10014 05563 78686 54196/23058 48070 17202153004 72521|1 22255 
 00062|10014 28622 26756 71398/23059 01164 89723|53095 94776|1 22261 
 00063/10014 51681 27921 61121/23059 54260 84499/53097 17037|1 22262 
 100064} 10014 74740 82182 45620)23060 07358 01536/530g8 392Q90|1 22263 
 
 00065|10014 97800 89540 47156/23060 60456 40835|53099 61562]1 22270 
 00066/10015 20861 49996 87991/23001 13556 02397|53100 83832]1 22270 
 00067/10015 43922 63552 90388/23001 66656 86229|53102 06102|1 22273 
 00068]10015 66984 30209 76617|23062 19758 92331|53103 28375|1 22276 
 00069/10015 90046 49968 68948/23002 72862 20706/53104 50651|1 22280 
 00070|10016 13109 22830 89054/23063 25966 71357|53105 72931|1 22282 
 0007 1}10016 36172 48797 61011]23063 79072 44288/53106 95213]1 22284 
 00072|10016 59236 27870 05299/23064 32179 39501/53108 17497|1 22287 
 00073}10016 82300 60049 44800/23064 85287 56998/53 109 39784|1 22290 
 00074/10017 05365 45337 01798/23005 38396 96782)53110 62074|1 22295 
 00075/10017 28430 83733 98580/23065 91507 5885653111 84369]1 22295 
 1(00076}10017 51496 75241 574306,23006 44619 43225/53113 06664]1 22299 
 '00077|10017 74563 19861 00061|23006 97732 40889|53114 28963]1 22301 
 (00078}10017 97630 17593 50550/23067 50846 78852/531 15 51264]1 22305 
 00079)10018 20697 68440 290402/23068 03962 3011653116 73559|1 22306 
 00080)10018 43765 72402 59518/23068 57079 0368553117 95875|1 22312 
 0008 110018 66834 29481 63203!23069 10196 990560'53119 18187]1 22312 
 00082)10018 89903 39678 62763/23069 63316 17747|53120 40499|1 22315 
 00083}10019 12973 02994 80510!23070 16436 5824653121 62814]1 22318 
 00084}10019 36043 19431 38756)23070 69558 21060153122 85132I1 22324} 
 
 00085]10019 59113 88989 59816/23071 22681 06192/53124 07456|1 22321 
 00086]10019 82185 11670 66008|23071 75805 1364853125 29777|1 223209 
 00087} 10020 05256 87475 7965623072 28930 43425|53126 52106]1 22320 
 00088] 10020 28329 16406 23081/23072 82056 95531153127 74435|1 22332 
 00089]10020 51401 98463 18612/23073 35184 69966/53128 96767|1 22336 
 
 90} 10020 74475 33647 88578/23073 88313 66733|53130 19103]1 22339 
 0009 1}10020 97549 21961 55311,23074 41443 85836/53131 41442/1 22339 
 00092)10021 20623 63405 41147|23074 94575 27278|53132 63781]1 22346 
 00093]10021 43698 57980 68425123075 47707 91059/53133 8612711 22345 
 0009410021 66774 05688 59484)23076 00841 77186/53135 08472]1 22351 
 
 00095}10021 89850 06530 36670|23076 53976 85058/53136 30823]1 22350 
 00096}10022 12926 60507 22328/23077 07113 16481153137 5317311 22358 
 00097110022 36003 67620 38800|23077 60250 69054153138 75531\1 22355 
 00098}10022 59081 27871 08463|23078 13389 45185|53139 97886)1 22363 
 00099]10022 82159 41260 53648|23078 66529 43071153141 2024011 22362. 
 
 3D 
 
 Log. Number. _ 
 
LOGARITHMS and NUMBERS to 20 PLACES. 
 Differ. 1. Diff. 2. 
 
 }00100}10023 05238 07789 96710}23079 19670 63320|53142 42611]1 
 00101|10023 28317 27460 60030}23079 72813 05931153143 64979]1 
 00102110023 51397 00273 65970)|23080 25956 70910153144 87346]1 
 00103|10023 74477 26230 36880|23080 79101 58256153146 09720]1 
 00104|10023 97558 05331 95136|23081 32247 67976|53147 32094] 
 
 00105}10024 20639 37579 63112/23081 85395 00070/53148 54472}1 
 00106}10024 43721 22974 63182/23082 38543 54542/53149 76852]1 
 00107|10024 66803 61518 17724|23082 91693 31394|53150 99235}1 
 00108]10024 89886 53211 49118]23083 44844 3062953152 21621}1 
 00109/10025 12969 98055 70747|23083 97906 52250)53153 44011]1 
 
 00110)10025 36053 96052 31997/23084 51149 96261|53154 66403}1 
 00111/10025 59138 47202 28258/23085 04304 hier aoe 88796}1 
 }00112)10025 82223 51506 90922/23085 57460 51460'53157 11194}1 
 06113}10026 05309 08967 42382/23086 10617 62054/53158 335941 
 res 28395 19585 05036|23086 63775 96248/53159 55997/1 
 
 00115/10026 51481 83361 01284|23087 16935 52245/53160 78404|1 
 0011610026 74569 00296 53529|23087 70096 30649/53162 00811}1 
 00117|10026 97656 70392 84178!23088 23258 3140053163 23223}1 
 0011810027 20744 93651 1563823088 76421 54683/53164 45638}1 
 001 19|10027 43833 70072 70321/23089 20586 00321/53165 680541 
 0012010027 66922 90658 70642/23089 82751 68375153166 90475/1 
 00121/10027 90012 82410 39017|23090 35918 58850]53168 12897]1 
 00122)10028 13103 18328 97867|23090 89086 71747/53169 3532211 
 00123|10028 36194 07415 69614)|23091 42256 07069|53170 57752}1 
 00124'10028 59285 49671 76683/23091 95426 64821153171 80182}1 
 00125110028 82377 45098 41504|23092 48598 45003!53173 02616)1 
 00126)10029 05469 93696 86507/23093 01771 47619|53174 25054]1 
 00127|10029 28562 95468 34126/23093 54945 72673/53175 47492)1 
 soleil 51656 50414 06799|23004 08121 20105|53176 69935/1 
 00129 
 00130;10029 97845 19833 17004|23095 14475 82481153179 14829]1 
 00131|10030 20940 34308 99545|23095 67654 97310|53180 37279)1 
 00132)10030 44036 01963 96855|23006 20835 34589/53181 59733]1 
 00133}10030 67132 22799 31444|23096 74016 04322|53182 82190}1 
 00134)10030 90228 96816 25706/23007 27199 76512/53184 04649Q)1 
 00135)10031 13326 24016 02278/23097 80383 81161/53185 2711211 
 00136/10031 36424 04399 83439|23098 33569 08273|53186 495761 
 00137|10031 59522 37968 91712|23098 86755 57840|53187 72045]1 
 [00138 10031 82021 24724 49561|23099 39943 29894|53188 94514]1 
 
 (00139|10032 05720 64667 7945523009 93132 24408153190 1698G]1 
 Siiieneretrinana esas ca gig Map ESS G88 Say gD al eid wo 2 
 
 (202) 
 
 Log. Number. 
 
 Tab. 4. 
 D3: 
 
 10029 74750 58535 26964|23094 61297 90100/53177 923811 22 
 
 22450}; 
 22454 
 
 22475} 
 22476 
 
N Tab. 5. 
 
 410°60205 
 510°69897 
 6|0°77815 
 710°84509 
 810:90308 
 
 10|1 -00000 
 11/1°04139 
 
 15}1-17609 
 
 16}1-20411 
 17}1°23044 
 18)1°25527 
 19}1-27875 
 20}1 30102 
 
 21)1°32221 
 22)1°34242 
 23)1°36172 
 24/1 +38021 
 
 1°44.715 
 9}1°46239 
 30}1°47712 
 
 ]42)1-62324 
 43/1-63346 
 4411-64345 
 45/1+65321 
 46) 166275 
 47/1-67209 
 48|1-68124 
 49)1-69019 
 ail 69897 
 
 so 0157 
 ath: 71600 
 
 60/1°77815 
 
 99913 
 00043 
 12503 
 80400 
 99869 
 25094 
 00000 
 
 26851 
 12460 
 33523 
 80356 
 12590 
 
 99826 
 89213 
 25051 
 36009 
 99956 
 
 92947 
 26808 
 78360 
 12417 
 00086 
 
 33479 
 37641 
 80313 
 79978 
 12547 
 
 16938 
 99783 
 39398 
 89170 
 80443 
 
 25007 
 17240 
 35966 
 46070 
 99913 
 
 385677. 
 92903. 
 
 84555 
 26764 
 25137 
 
 78316 
 78579 
 12573 
 60800 
 00043 
 
 01760 
 33436 
 
 58696 
 
 37598 
 26894 
 8027() 
 48556 
 79935 
 20116 
 12503 
 
 63981 
 19662 
 27962 
 36018 
 83643 
 
 14256 
 91943 
 39324 
 00000 
 58225 
 4.7264 
 06836 
 78238 
 55681 
 
 55924 
 18273 
 03306 
 52828 
 63981 
 33919 
 22206 
 17592 
 11606 
 72037 
 710817 
 58987 
 42219 
 98936 
 19662 
 
 34272 
 19905 
 171887 
 42255 
 50275 
 67287 
 66994 
 16810 
 26499 
 27962 
 19735 
 
 97900, 
 
 79586 
 86187 
 75343 
 
 81574 
 35717 
 75587 
 28513 
 36018 
 97936 
 34799 
 00789 
 22968 
 94243 
 06200 
 71249] 
 62937 
 421 4d 
 83643 
 
 TABLE V_ 
 
 Bricas’s Logarithms of all Numbers to 100, and of Primes 
 under 1100, to Sixty-one Places. 
 
 LOGARITHMS To 61 
 
 ee 
 "1/0°00000 00000 00000 00000 
 2|0°30102 99956 
 310-47712 125417 
 
 195421 
 43729 
 39042 
 80478 
 63250 
 83071 
 58564 
 874.59 
 00000 
 
 04075 
 82772 
 716920 
 02592 
 24208 
 78085 
 92854 
 06980 
 96153 
 19521 
 
 26800 
 23.596 
 87886 
 02293 
 60957 
 9644.2 
 31188 
 22113 
 08733 
 43729 
 
 67966 
 97606 
 47804. 
 12375 
 63549 
 26501 
 99680 
 15675 
 20650 
 39042 
 
 49450 
 46322 
 5264.0 
 43117 
 67937 
 07408 
 4644 1 
 21814 
 
 66142 4 
 
 80478 
 
 36583 
 15963 
 04563 
 50709 
 84553 
 41635 
 39883 
 28254 
 19026 
 
 63250 
 
 00000 
 
 57388 
 50279 
 FATT 
 62611 
 
 87667 
 22162 
 12166 
 00558 
 00000 
 
 01999 
 25056 
 65051 
 59551 
 12890 
 
 49555 
 01698 
 37947 
 63334 
 37388 
 
 712441 
 39388 
 77771 
 62445 
 25222 
 02440 
 50837 
 96940 
 28467 
 50279 
 67041 
 86944 
 52278 
 39087 
 84773 
 75335 
 84506 
 00723 
 15330 
 74777 
 
 94118 
 09330 
 50881 
 76177 
 63169 
 15160 
 42193 
 99834 
 ASZ5 
 62611 
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 39829 
 29922 
 88226 
 64610 
 
 34329, 
 
 13613 
 65856 
 06563 
 87667 
 
 00000 00000 
 
 94724 
 03255 
 89448 
 05275 
 97979 
 
 58592 
 84173 
 06510 
 00000 
 11243 
 92704 
 57942 
 53317 
 08530 
 
 78897 
 94328 
 01234 
 75156 
 94.724. 
 
 61847 
 65967 
 12251 
 87428 
 10551 
 
 52666 
 09765 
 48041 
 62969 
 03255 
 00118 
 13622 
 714498 
 89052 
 63868 
 95959 
 89539 
 70481 
 61197 
 8944.8 
 49968 
 56572 
 53229 
 60692 
 11785 
 06915 
 9GA49 
 82153 
 17185 
 05275 
 7 97583 
 4739 
 
 sted 
 04489 
 76518 
 42766 
 79012 
 57693 
 84535 
 97979 
 
 49302 
 11530 
 98605 
 50697 
 
 60833 
 63619 
 47908 
 23061 
 00000 
 02424 
 10136 
 32843 
 
 62228 
 
 92931 
 
 98605 
 
 68257 
 20064 
 08741 
 27238 
 
 25659 
 83895 
 53121 
 
 11527 
 04462 
 74801 
 14423 
 60833 
 
 3D2 
 
 12922 
 
 97210 
 33703 
 72364 
 
 49302 
 75150 
 51726 
 18954 
 59438 
 01394 
 
 82145 
 34592 
 62224. 
 25499 
 11530 
 41572 
 46513 
 13955 
 83005 
 14316 
 21667 
 12944 
 42934 
 44374 
 
 18079 
 244.52 
 22215 
 01029 
 73759 
 
 50697 
 
 45233 
 1 31448 
 
 PLACES. 
 
 00000 
 67681 
 92001 
 35363 
 32318 
 59683 
 34835 
 03045 
 84002 
 00000 
 17067 
 27365 
 08297 
 OQ4517 
 24319 
 
 7072" 
 00075 
 51684 
 T9511 
 67681 
 26837 
 84748 
 96975 
 95046 
 64636 
 15979 
 716003 
 710199 
 12542 
 92001 
 23037 
 38409 
 09068 
 6TT57 
 67153 
 
 19366 
 79829 
 47193 
 00298 
 35363 
 
 95305 
 94518 
 88087 
 52430 
 16320 
 64657 
 01598 
 62728 
 69671 
 32318 
 92076 
 43661 
 26955 
 43685 
 49385 
 
 37881 
 71512 
 80224 
 89267 
 59683 
 
 00000 
 89881 
 28864 
 19762 
 10118 
 18745 
 12396 
 69644 
 57728 
 00000 
 
 (2190 
 08627 
 297838 
 62277 
 38982 
 59525 
 67378 
 47609 
 29337 
 89881 
 01260 
 92071 
 11034 
 98508 
 20237 
 19069 
 86592 
 52159 
 94.417 
 28864 
 
 01558 
 49407 
 31054 
 57259 
 82514 
 37491 
 72690 
 19218 
 5805 
 "9762 
 13633 
 O1141 
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 81953 
 67846 
 00915 
 03098 
 88390 
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 10118 53 
 
 96242 
 08951 
 (2401 
 16474 
 12309 
 4.2040 
 58201 
 84299 
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 18745 
 
 2 S776 
 
 00000 00000 
 46210 
 19069 
 92421] 
 53789 
 65280 
 32396 
 38632 
 38139 
 00000 
 
 46645 
 11491 
 38706 
 "8607 
 2858 
 84843 
 4.2504 
 84350 
 39449 
 46210 
 51466 
 52856 
 33609 
 57702 
 07578 
 84917 
 57208 
 24818 
 88715 
 19069 
 
 30418 
 31054 
 65714 
 88715 
 $6185 
 30560 
 16631 
 8.5660 
 
 85413 
 58648 
 710826 
 14586 
 
 4.4.06 1 
 54065 
 56239 
 17296 
 00000 
 30945 
 29474, 
 82718 
 39478 
 73235 
 41652 
 63973 
 02709 
 75989 
 85413 
 
 12713 
 16359 
 61882 
 14887 
 29173 
 
 68131 
 T5944 
 24891 
 38410 
 58648 
 
 4.6559 
 27065 
 89594 
 49386 
 68651 
 88122 
 25466 
 61402 
 41366 
 70826 
 
 70890 
 98126 
 34145 
 01772 
 31883 
 4NQ95 
 78270 
 00300 
 08130 
 14586 
 22622 
 53544 
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 61358 
 45532 
 
 10304 
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 34926 23823 
 93076 52155 2 
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 924.2] 
 
 83368 
 97676 
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 39067 
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 79820 
 4.2994, 
 03913 
 64793 
 789 
 61574 
 31128 
 29493 
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 71029 
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 104.275 
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 104275)20 
 
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 756055\23 
 611480|24 
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 116184126 
 895969)27 
 244899128 
 653969|29 
 298656)30 
 
 383498|31 
 521373|32 
 264047|33 
 
 932075135 
 
 805862/36 
 176799)37 
 172463)38 
 
 20854.9/40 
 
 073567 41 
 43928 1/42 
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 173939)/44. 
 4.93038}45 
 860329/46 
 873294147 
 T1STS5|A8 
 072699149 
 895725150 
 102341151 
 2204.59/52 
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 758244)58 
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N (204) LOGARITHMS TO 61 PLACES. Tab. SN 
 
 61}1°78532 98350 10767 03388 57485 13757 32134 92633 78757 11340 42120 "703489 
 62}1°79239 16894 98253 87488 04429 94842 90874 90718 91439 76629 31972 487773 
 63}1°79934 05494 53581 70530 22720 65102 86681 18838 30124 70535 71361 633662 
 64]1°80617 99739 83887 17128 24333 68346 95816 06091 39288 77265 12478 625648 
 6511°81291 33566 42855 57399 27662 63217 83540 40615 39306 92495 97304. 907635 
 
 66]1°81954 39355 41868 67325 89667 69222 63257 76750 20936 11925 5007 368321 
 §7|1°82607 48027 00826 43414 91316 29226 06858 09496 26080 56861 38691 179160 
 68]1°83250 89127 06236 31896 76476 83777 32308 35439 4'7141 34926 34800 012234 
 69}1°83884 90907 37255 31616 28050 15506 30485 88976 39898 52679 20531 054711 
 70}1°84509 80400 14256 83071 22162 58592 63619 34835 72396 32396 54065 036350 
 
 71}1°85125 83487 19075 28609 28294 35035 42913 52704 19901 60039 19762 766499 
 72|1°85733 24964 31268 46023 12724 90683 70969 87048 27372 76771 73535 910137 
 73|1°86332 28601 20455 90107 43869 00470 30853. 44528 68255 31165 74851 100020 
 74|1°86923 17197 30976 19202 21895 84263 62247 47511 62571 62842 10879 281074 
 75|1°87506 12633 91700 04686 75501 13806 12925 56637 49101 26647 87822 090107 
 
 76|1°88081 35922 80791 35196 38112 65205 91537 14875 09100 31871 46815 276738 
 77\1°88649 07251 '72481 87146 24162 29835 66043 51902 74586 79041 85011 001740 
 78|1°89209 46026 90480 40171 52719 55921 93676 67980 47934 03987 26779 414841 
 7911°89762 70912 90441 42799 48213 86478 24968 64828 62019 02515 03156 163513 
 80190308 99869 91943 58564 12166 84173 47908 03045 69644 38632 56239 312824 
 
 81)1°90848 50188 78649 74918 01116 13020 46123 68005 15456 76278 34593 194626 
 82)1°91381 38523 83716 68972 31507 44692 67382 62987 03515 29579 56503 177842 
 83|1°91907 80923 6073 90383 27603 52027 26124 70016 37658 08063 04535 293708 
 8411°92427 92860 61881 65843 47219 51296 13755 62200 81023 43887 83539 543555 
 85{1°92941 89257 14292 73332 64309 99603 84400 32393 77496 96293 78560 699410 
 
 86|1°93449 84512 43567 72161 88270 47953 71518 55769 64765 84220 19558 351768 
 87)1°93951 92526 18618 52462 78746 66224 37030 04544 23282 07784 97058 952625 
 88/1°94448 26721 50168 62639 14166 55416 50332 20112 71834 85277 87185 278214 
 89}1°94939 00066 44912 78472 35433 69702 44112 46651 61858 10024 45836: 328694 
 90}1:95424 25094 39324 87459 00558 06510 23061 84002 57728 38139 17296 597313 
 
 91|1°95904 13923 21093 59991 87214 16534 96462 43133 01584 71103 36783 048259 
 9211°96378 78273 45555 26929 52549 01700 17560 32338 90797 26031 32708 964604 
 93/1°96848 29485 53935 11696 17320 03373 53103 15038 30422 49488 05207 682155 
 9411°97312 78535 99698 65962 79582 94173 69366 69279 92979 89205 63683 477569 
 95/1°97772 36052 88847 76632 25945 81032 43629 11829 39455 93238 90575 963914 
 
 96/1-98227 12330 39568 41336 37223 76877 58044 30410 78271 50123 85713 820029 
 9'7)1°98677 17342 66244 85178 43618 11665 57744 94258 41584 63886 69747 187207 
 98)1°99122 60756 92494 85663 81714 11909 76541 37353 34674 11003 93543 176974 
 realli 51945 97549 91534 02557 77753 25486 01069 59918 84784 48242 562703 
 Adee iit 13737 $2642 57427 51881 78222 93791 32192 89355 20645 25914 058186)101 
 
 103/2-01283 72247 05172 20517 10711 94580 23942. 43905 23496 97603 05647 528079|103 
 107!2-02938 37776 85209 64083 45412 39461 43564 61268 16891 63401 93519 816620|107 
 109|2°03742 64979 40623 63520 05133 07613 87528 66422 04522 82798 36821 104005|109 
 113/2°05307 $4434 83419 72279 52270 28609 44818 47783 83623 62209 73395 157054113 
 12'7/2°10380 37209 55956 86424 69874 21827 28625 85765 63239 79239 38677 68782Q|127 
 
 131)2:11727 12956 55764 26081 00542 70697 73859 47801 63117 12162 69689 '770335|131 
 1371213672 05671 56406 76856 29266 27114 78973 36782 29707 46423 50456 632444]137 
 139]}2-14301 48002 54095 08045 64332 02319 84731 44797 32967 91785 93396 574308|139 
 149/2°17318 62684 12274 03825 73635 42628 33705 39346 71326 37222 11012 048653}149 
 151|2°17897 69472 93169 43686 90730 55337 30278 84460 93428 77687 74510 971401]151 
 
 11571219589 96524 09233 73676 14811 29897 28370 50651 90992 78552 95873 594477/157 
 163;2°21218 76044 03957 80764 00914 35925 99475 49930 97247 35985 06185 303704163 
 167/2°22271 64711 47583 27998 40759 09920 46753 44613 38401 33125 82289 069635167 
 173/2°23804 61031 28795 41456 05302 58758 46588 77816 83269 13492 66453 988743/173 
 179}2°25285 30309 79893 16957 03826 91773 05861 94310 72090 67852 86239 4'77285|179 
 
 181/2°25767 85748 69184 51028 97436 76412 29249 22479 59232 72291 88769 574799}18) 
 191/2°28103 33672 47727 53763 50435 98270 61031 84957 36134 17824 30405 891262}191 
 193)2°28555 73090 07773 76059 72386 46353 31082 10979 21601 94604 88412 889733}193 
 19712-29446 62261 61592 92737 17443 17717 15501 75120 64672 00453 36906 180720)197 
 199|2*29885 30764 09706 65010 00217 84419 80284 14948 88771 49827 32431 907065]199 
 
 21112:32428 24552 97692 66508 1558) 29927 88565 15502 58502 90193 86869 014730)211 
 223|2°34830 48630 48160 67347 51762 16240 35284 44534 24237 98021 08177 231582223 
 22'7'2°35602 58571 93122 72010 50489 64753 67294 74838 78261 56058 48416 494656|227 
 229)2°35983 54823 39887 99412 79298 65526 65887 03358 93242 54328 14002 593934)229 
 233}2°36735 59210 26018 97215 91388 35476 85936 08884 54098 32289 45750. 381402/233 
 239]2°37839 79009 48137 68500 16611 60147 89212 27092 22421 69429 85262 599734)239 
 
N Tab. 5. LOGARITHMS TO 6] PLACES. (205) N 
 
 241|2°38201 70425 74868 38407 68839 66454 63294 43845 75422 87941 37116 090780/241 
 25112°39967 37214 81038 13934 05493 16706 90408 18574 66685 39315 23086 557977|\251 
 25712°40993 31233 31294 53716 28954 65919 63183 09299 89891 62261 22190 657085|257 
 263}2°41995 57484 89757 86897 22335 83870 11811 42207 55733 87652 55581 84'7682|263 
 269|2°42975 22800 02407 98008 72285 15871 27175 37709 54680 -10337 16358 202492)269 
 
 2711243296 92908 74405 72952 11801 94875 18026 90280 28099 71147 47196 959683)271 
 27'7\2°44247 97690 64448 55377 77563, 19599 75831 09223 84739 72572 00838 275546|277 
 28112:44870 63199 05079 89286 39179 16275 08871 55000 84994 87733 11091 225526)281 
 283]2°45178 64355 24290 23555 89519 10570 23772 98828 25398 13326 05411 834686}283 
 293|2:46686 76203 54109 45624 37585 12602 18133 14970 80293 87633 91801 387293)293 
 
 307|2:48713 83754 77186 48475 46084 36539 33504 93281 89817 26663 11352 567959|307 
 311]2:49276 03890 26837 50555 30231 83253 64155 85949 18519 90441 42567 782324)311 
 313}2°49554 43375 46448 48480 81265 04861 24315 15792 98693 98571 52993 196813313 
 317/2°50105 92622 17751 49455 32290 16378 22488 04877 22158 71549 07278 111979|317 
 331]2°51982 79937 75718 73860 81406 07340 85663 50827 13549 69614 46087 295510/331 
 
 337|2°52762 99008 71338 62619 00147 90194 51019 87041 58106 86338 94145° 590771|337 
 347/2°54032 94747 90873 71853 53573 03206 97397 86865 56176 91243 65052 250367|347 
 349)/2°54282 54269 59179 89654 01719 77159 63066 31783 00866 75487 04181 990296/349 
 35312°54777 47053 87822 56549 70693 15968 56119 79362 71500 87293°47356 1717651353 
 359|2°55509 44485 78319 14781 65293 94413 89970 02357 64461 12862 45018 194841|359 
 
 367|2°56466 60642 52089 33798 75290 93006 90914 75947 52157 57773 73388 529180|367 
 373}2°57170 88318 08687 60550 68969 38701 43991 49308 33032 45651. 82236 8284'75/373 
 37912°57863 92099 68072 34193 14620 59454 44405 29413 87210 96923 21381 081258/379 
 383/2°58319 87739 68622 74037 90461 29502 11234 47857 39787 51936 81090 658346/383 
 389}2°58994.96013 25707 73624 49469 11731 95270 14076 41221 24688 95645 0643841389 
 
 3972-59879 05067 63115 06587 68482 40668 63112 25522 37562 91876 18078 588386/397 
 401/2°60314 43726 20182 30654 46411 48149 42549 75189 88963 37359 82761 562011)/401 
 409]2°61172 33080 07341 80360 95027 17736 46679 00320 51595 65255 67279 407052)/409 
 41912°62221 40229 66295 30985 07395 99373 73621 25514 08166 99180 26223 814'797/419 
 42112°62428 20958 35668 30744 40669 23421 44371 09437 88488 01681 56998 058298/421 
 
 43112°63447 72701 60731 60075 02803 26184 67878 49873 63233 16232 39160 168424/431 
 433|2°63648 78963 53365 44269 80664 49685 26766 08604 17833 53839 54652 633209)433 
 439/2°64246 45202 42121 37063 37411 50613 31363 46233 64482 93197 78492 698498)/439 
 (443]/2°64640 37262 23069 56023 01044 89864 53902 83230 69450 39547 31960 218878/443 
 44912°65224 63410 03323 17491 90263 53743 43105 35027 59942 01108 72112 409383/449 
 
 45712°65991 62000 69850 22235 35461 45220 47714 05940 16155 52489 85626 587883|/457 
 461|2°66370 09253 89648 14507 46818 18487 42153 71937 47244 04839 02463 622776|461 
 463/2°66558 09910 17953 13567 41931 08438 70855 40157 65450 46974 53874 838090|463 
 467/2°66931 68805 66112 16308 80510 89779 99674 10010 61401 55968 77553 654228/467 
 4'7912°68033 55134 14563 22009 69639 66962 31078 27266 76340 01805 94696 676822/479 
 
 4872-68752 89612 14634 33246 32050 64435 '75372 38433 54413 59009 69060 272887/487 
 491)2°69108 14921 22968 47275 36909 83546 39435 54324 95219 43164 65484 935064/491 
 499/2°69810 05456 23389 91416 59050 36033 38846 ‘73162 68889 76585 04407 216866|499 
 415031270156 79850 55927 39709 82240 90279 52805 50061 79311 53264 13100 626989|503 
 
 40912-70671 77823 36758 74656 80767 11564 25501 75116 31022 82795 59327 732505)509 
 
 521/2°71683 17232 99524 4'7423 63411 86589 82340 55592 48804 35659 10389 037518|521 
 523/2°71850 16888 67274 23926 01265 78891 07882 05229 27624 54022 80340 618542|523 
 541)2°73319 72651 06569 43687 93482 43895 35766 02744 51126 54918 07249 958843/541 
 
 547 2°73798 73263 33430 717381 26473 72542 06411 41123 32573 38734 83672 5442941547 
 557}2°74585 51951 73728 90044 34334 98899 38696 26667 22982 65562 88916 047639|557 
 
 563/2°75050 83948 51346 22909 45827 07761 08389 89309 27510 02997 46276 527041|563 
 569/2°75511 22663 95071 17228 70555 24030 20058 87808 40566 56954 49337 6621641569 
 571|2°75663 61082 45848 05004 02841 30031 39578 08074 83371 59899 19622 253745/571 
 S77/2°T6117 58131 55731 42848 88336 67563 87165 18349 94631 00807 86067 506949577 
 537/2°76863 81012 47614 47606 35592 98596 71376 19981 12599 05673 24995 7158554587 
 
 593}2°77305 46933 64262 60639 66715 59821 78133 09249 84055 79640 65224 216122)593 
 599)2°77742 68223 89311 37982 81725 69101 74684 25198 87827 14494 37552 485037|/599] 
 601/2°77887 44720 02739 52088 58506 99987 83983 48917 52297 24032 80181 145090/601 
 607|2°78318 86910 75257 58096 01956 30455 95072 14062 42317 98498 79486 868541|607 
 613)2°78746 04745 18415 03774 22662 81456 45078 29528 38564 77870 60511 887769]613 
 
 617)2°79028 51640 33241 68204 54661 67275 45331 98845 73431 10231 76836 317560|617 
 619)2°79169 06490 20117 97679 79674 34394 50849 41105 79264 06695 48606 1340851619 
 631)}2°80002 93592 44134 31301 69298 49975 36836 15526 21483 45926 22618 8194061631 
 641/2°80685 80295 18817 42224 83770 09638 02810 30784 64091 37064 08860 016375641 
 643/2°80821 09729 24222 07249 19385 05465 83232 48443 16034 72535 33279 475692)643 
 _ (647]2°81090 42806 68700 38445 84305 62795 35772 33374 52752 88620 55534 785384|647 
 
N (206) LOGARITHMS TO 61 PLACES. Tab. 5 N 
 
 653/2°81491 31812 75073 92142 93105 65465 57968 44420 93073 59911 14836 790768) 653 
 659/2°81888 54145 94009 86128 04846 07065 03884 71245 58914 63114 16630 487450} 659 
 661}2°82020 14594 85640 23664 65718 97680 09240 24475 29556 41077 27411 001763} 661 
 673}2°82801 50642 23976 84647 61709 94824 66587 84392 73852 95699 07219 527629) 673}. 
 677}2°83058 86686 85144 31600 60170 60287 15791 96987 21869 42085 75219 422855} 677 
 
 683}2°83442 07036 81532 56339 98239 41016 94314 12519 92074 22395 15101 356100} 683 
 691}2°83947 80473 74198 40758 33677 24326 62643 33706 67025 71535 20888 200815} 691 
 70112°84571 80179 66658 65706 40223 37250 30440 16828 60606 06710 99378 642626] 701 
 709|2°85064 62351 83066 54285 38844 79778 89914 12079 23464 57372 91344 715434} 709 
 719/2°85672 88903 82882 60776 76506 51400 88113 55319 50785 66409 97910 273675} 719 
 
 727\2°86153 44108 59037 83621 34642 48678 39613 39988 10242 96505 05660 709999] 727 
 733/2°86510 39746 41127 94317 28131 02559 86776 12051 12268 36141 01539 967269) 733 
 739|2°86864 44383 94825 73669 35855 14263 03827 78685 62960 06015 93030 162646} 739 
 743)2°387098 88137 60575 29242 26723 41225 78659 86402 35201 25826 22906 426196} 743 
 751|2°87563 99370 04168 38974 59851 09251 08913 79777 69486 72300 09449 287788} 751 
 
 75712°87909 58795 00072 75709 02275 46289 28831 29598 55610 77568 18424 909661} 757 
 761\2°88138 46567 70572 82636 87243 35559 42944 66262 26115 19329 16113 770466| 761 
 769}2°88592 63398 01431 03960 42922 39990 68928 55438 24266 73676 32539 540297} 769 
 77312°88817 94939 18324 90897 46881 27193 74602 82128 27448 51788 65363 250475} 773 
 787|2°89597 47323 59064 55847 49105 93093 84403 00557 33235 30892 05759 509372) 187 
 
 797)/2°90145 83213 96112 34726 66008 27220 37150 60763 80048 04080 90214 871170] 797 
 809/2°90794 85216 12272 30432 36285 45880 42151 46893 16537 70803 38111 022662} 809 
 811)2°90902 08542 11156 03069 03305 483522 97484 96977 10258 36812 36616 489430} 811 
 821}2°91434 31571 19440 77180 40593 41703 71406 12897 21030 05294 12843 731072} 821 
 82212°91539 98352 12269 88976 77077 56599 55165 51291 17341 03959 46095 528115] 823 
 
 827/2°91750 55095 52546 67071 16671 84496 53756 13593 71081 63043 50219 579982} 827 
 829|2°91855 45305 50273 55311 51367 88077 88199 00092 68851 27047 81176 310395} 829 
 839}2°92376 19608 28700 27499 86012 26886 40032 82838 28125 42235 16955 539741} 839 
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 103|0°0295588]1°53|0'4252677|2'03|0°7080358)2'53|0°9282 193|3°03|1° 1085626 
 1°04/0'0392207]|1°54/0'4317824/2'04|0'7 129498|2'54|0'9321641|3'04)1°'1118575 
 1‘05]0°0487902)1°55|0'4382549|2°05|0°7 178398)/2'55|0'9360934|3°05]1'1151416 
 1'06|0'0582689}1°56/0°4446858]2:06|0'7227060|2°56|0'9400073}3 061° 1184149 
 1°07|0°0676586]1°57|0°45 10756|2°07|0'7275486)|2°57|0'9439050|3'07|1°'1216776 
 1-08/0'07696 10} 1°58/0°4574248/2'08|0'7323679|2°58|0'9477894|3 08]1° 1249206 
 1-09|0:0861777|1'59|0'4637340|2'09|0:737 164.1]2°59|0°9516579|3°09|1'1281711 
 1°10|0'0953 102}1:60|0°4700036)2:1010'74.19373|2°60|0'9555114/3°10/1°1314021 
 1°11]0°1043600}1'61|0°'4762342/2:11|0°74668709)2°6110°90593502)|3°11}1°1346227 
 1°12/0°1133287]1°62)0°4824261|2°12/0°7514161|2°62|0'963 1743/3°12/1°1378330 
 1°13|0°1222176]1°'63|0°'4885800|2:13/0°7561220!2°63/0'9669838)3'13}]1°1410330 
 1°14/0°1310283]1°64|0'4946962)2°14|0°7608058/2°64|0°9707789|3'14|1°1442228 
 1°15|0°'1397619]1°65|0'5007753|2'15|0°7654678|2°65|0°'9745596)3'°15]1° 1474025 
 1°16|0°1484200}1 660°50681 76|2°16|0°7701082|2°66|0'9783261)3°16}1°1505720 
 1°17|0°1570037 1-67/0°5128236 2°17|0°7747272 2°67 0°9820785|3°17|1°1537316 
 1°18}0°1655144|1 ‘68'0°5187938 2'18|0'7793240|2°68|0'9858168/3°18]1'1568812 
 1°19]0°1739533|1'69(0°5247285|2°19|0°783901 5|2°69|0'98954.12/3°19}1° 1600209 
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 1°21|0°1906204}1°71 (0°5364934 2'21|0°7929925)2°7 1|0°9969486|3°21|1°1662709 
 1°22/0°1988500]1°72,0°5423243/2°22 Pear eOraey 1°00063 19|3°22}1°1693814 
 1:23/0°'2070142]1'73'0'5481214/2'2310°8020016'2°73|1°0043016)3'23)1°1724821 
 1:24|0°2151114]1‘740'553885 1/2°2410'8064759)2'74|1'0079579)3 '24]1°1755733 
 1:25|0°223 1436}1°75|0°5596158|2°25|0'8109302,2°75|1°01 16009|3°25]1°1786550} 
 1°26|0°2311117}1°76,0°5653138/2:26 0'8153648,2°76 1:0152307/3°26]1°1817272 
 1°27|0°2390169}1'77|0°5700795|2'27 0°8197798)2'77 1'0188473/3‘27|1° 1847900 
 1°28|0'2468601/1°78|0'5766134|2°2810'8241 754,278 1:0224.509|3°28]1°1878434 
 1°29|0°2546422)1°79|0°5822156)2°29|0°8285518 2°79|1'0260416}3'29}1° 1908876 
 1°30|0°2623643 pies kana 2°30 0'8329091|2'80 1°0296194|3°30}1°1939225 
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 1:3210°2776317|1°82/0°5988365|2'32|0°8415672!2°8211°0367369|3°32|1' 1990648 
 1°3310'28517809]1'83/0°0043 160|2°33|0°8458683/2'°83]1°0402767/3°33|1'2029723 
 1'34/0'2926696}1°84'0'6097656|2°34/0°8501 500|2°84]1°0438041}3°34|1'2059708 
 1°35]0°3001046]1°85|0'6151856/2°35|0'85441 53!2°85]1°0473 190|3°35}1°2089003 
 1°36|0°3074847|1'86,0°6205765|2'36|0°8586616 2°86}1°0508217/3°36}1°2119410 
 1°37/0°3 148107|1'87|0'625093 84|2'37|0'8628899|2'87}1°0543 121/3°37/1'2149127 
 1°38/0°3220835!1:88/0°63 127 18]2°38|0°867 1005|2°88] 1°0577903)3°38)1'2178757 
 1°39|0°3293037)|1'89|0'6365768|2'39/0'87 12933)2°89}1°0612565]3°30}1°2208299 
 1°40|0'3304722}1-GO|0°6418530|2°40|0°8754687|2°90|1°0647 107|3°40)1°2237754 
 1°41|0°3435897|1‘9110'647 1032|2°41|0°8796267|2'91|1°0681531]3°41]1°2267 123 
 1°4210°3506500] 1-92/0'6523252|2:42|0°8837675|2'92]1°0715836|3°42]1'2296406 
 1:4310:3576745]1°9310'6575200}2°43|0'8878913|2'93]1°0750024|3°43]1°2325603 
 1°4410°364643 1|1°94|0'6626880|2°44|0°8019980)2'94]1'0784096)/3'44/1°2354715 
 1°45|0°3715636)1'95|0°66782094|2°45|0°8960880)2°95] 1°08 18052)}3°45|1'2383742 
 1:46|0'3784364)1-96|0'6729445|2'46|0-9001613]2°96]1°085 1893|3°46}1°24.12686 
 1:47|0°3852624}1'97|0°6780335|2°47|0'9042182]2°97|1°0885610}3°47}1'2441546 
 1'48|0°3920421]|1'98|0°6830968!2'48|0'9082586)|2°98]1°0919233)3°48)1°2470323 
 41-40|0'°3987761]1-9Q|0'6881346)2°4g|0'9122827|2°90}1°0952734|3°49}1'2499017 
 1°5€|0°405465 1|200|0°693 1472)2°50|0°9162907|3 00} 1-°0986123|3°50|1'2527030 
 
ts _ HYPERBOLIC LOGARITHMS. , (209 ) 
 
 N.| Logar. | N.} Logar. | N.| Logar. 
 3°51]1°2550100/4'01|1°388791 2)4"51]1°506297 1|5°01/1°6114359|5°51|1-7005646 
 3°52|1-2584610/4'02]1°3912810]4°52|1°508512015°02}1°6134290|5°52]1-7083779 
 3'5311°2612979|4'03|1°3937664]4'53]1°5107219|5°03]1°6154200)5°531-7101878 
 3°54|1°2641267|4 04|1°3962447/4°54)1°5129270|5'04/1°6174061|5°54]1-7119945 
 3°55|1°2669476)4°05|1°3987 169]4°55]1°515127215°05|1°6193883/5°55/1-7137970 
 3°56)1°2697605|4'06]1°4011820/4°56]1°5173226|5-06/1'6213605|5°50}1°7155981 
 3'57|1'2725656)4'07|1°4036429/4'°57|1°5195132|5°07|1°6233408|5°57|1°7173951 
 3°58|1°2753628/4-08]1:4060970|4'58| 1-521 6990|5°08}1°6253113/5°58]1-7191888 
 3°5Q|1°2781522/4-00|1°4085450)4"59}1°5238800]5°00}1°6272778|5°50|1-72007093 
 3°60, 1*2809338)4:10}1°4109870]4°60}1:5260563)5°10}1°6292405|5°60}1 -7227666 
 3°61/1'2837078)4"11|1°4134230)4°61]1°5282278]5°11)1°6311994|5°61!1-7245507 
 3°62} 1°2864740)/4"12)1°415853214°62)1°530394715°12}1°633154415°62)1-7263317 
 3°63] 1°2892326)4-13]1°4182774|4°63] 1°5325569)/5°13]1°6351057|5°63]1-7281094 
 3°64, 1°2919837/4"14]1°4206958]4'64]1°534714415°14]1°6370531|5'04]1-72g8841 
 3°05|1°2947272 4°15|1°423 1083]4"65]1°5368672)5°15]1°0389967|5°65]1°7316555 
 3°66,1°2974631 4*16]1°4255151/4°66]1°5390154)5°16)1°6409366|5°66)1-7334239 
 3°67|1°3001917/4°17|1°4279160)4°67|1°541159115°17|1°6428727/5°67|1°7351891 
 3°68 1'3029128/4'18]1°4303 112/4°68}1°5432981|5°18]1°6448051|5°68]1-7369512 
 369 1:3056265/4"19|1-4327007|4-69|1°5454326)5°19|1-6467337|5-69|1:7387102 
 3°70,1'3083328)4°20|1°4350845/4°70)1°547562515°20) 1°6486586)5°70)1°7404662 
 -7il1-3110819 4°21]1°4374626)/4-71|1°5496870]5°21)1°6505799|5°71|1°7422190 
 3°72 1°3137237|4-22|1 439835 114-72) 1°55 1 8088|5°22|1°6524974|5°72)1 ‘7439089 
 3°73) 1'3104082)4°23]1-4422020]4-73]1°5539252|5°23/1°6544113]5°73]1°7457155 
 3°74'1°3190856)|4"24|1°4445633|4'74]1°550037 1|5°24|1°0503215/5°-74]1°7474593 
 3791 3217558 4'25|1°4469190)4°75|1'°5581446|5°25|1°658228115°7511°7491998 
 3'76,1°3244190/4'26]1-4492692]/4°76]1°5602476|5:26]1°6001310|5°76]1°7509375 
 3°77|1°3270750)|4°27|1°4516138|4°77|1°5623463]5-27|1°6620304/5°77]1-7526721 
 3°78) 1°3297240)4'28]1°4539530/4°78| 1°5644405|5°28)1°6639261)/5°78\1°7544037 
 3°79) 1°3323660/4'20]1°4562868)/4-79]1'5605303]5°20|1°6058182|5°79}1-7561323 
 3°80) 1°335001 1/4°30}1°4586150/4'80)1°5686159|5°30|1°6077068|5°80)1°7578579 
 3°81)1°3376292|4°3 1|1-46093709/4'81]1'570697 1|5°31|1°6695918)5°81}1:7595806 
 3'82/1°3402504/4°32]1°463255414°82/1°572773915°32|1°6714733/5°8211-7613003 
 3°83] 1°3428048/4°33]1°4655075/4'83|1°5748465|5°33|1°0733512'5'83|1°7630170 
 3°84) 1°3454724/4°34|1°4678743/4°84]1°576914715°34|1°0752257/5'84|1°7647308 
 3°85|1°348073 1/4°35|1°4701758]4°8511°5789787|5°35|1°67.70966)5°85}1-7664416 
 3°86) 1°3506672/4'36]1:472472114:86]1°581038415-36)1°6789640)5°86]1-7681496 
 3°87|1°3532545|4°37|1-4747630)4°87]1°5830930|5°37|!°6808270|5°87|1-7698546 
 3°88) 1°3558352/4°38]1°4770487|4°88]1°585145215°38|1°682088415°88]1-°7715568 
 3°80) 1°3584092/4°30]1°4793202|/4:89]1°5871923|5°30|1°684545415°80|1-7732560 
 3°90} 1°3609766)4"40)1°4816045}4-90|1-5892352|5-40|1°6863990|5-90 1°7749524 
 3°91/1°3035374|4-41}1-4838747)/4-9111-5912730|5°41]1°6882491|5'91|1-7766458 
 3°92|1°3000917|4"42/1-4861397|4'92]1°5933085|5°42]1 ‘6900958/5°92}1-7783304 
 3°93|1°3086394/4"43]1-4883996] 1-93|1°5953390|5°43 1°6919391)5°93}1-7800242 
 3°94)1°3711807|4°44]1-4906544|4-9411-5973053|5°44|1°693779115'94|1-7817091 
 3°95)1°3737150}4°45]1-4929041] 4-95] i-5993876|5°45]1°6956156|5°95|1-7833912 
 3°90}1°3702440)4-46}1 4953486] 4-96] 1-6014057|5-46|1:6974488|5°96]1-7850705 
 3°97) 1°3787661)|4"47}1°4973884] -97|1-603419815-4711-699278615°97 1°7807469 
 3°98) 1°3812818)4:48} i -499623C} -9&|1-605429¢15-48]1°701105 15°98] 7884200 
 3°99| 1°3837912)4"40]1-5018527} -9¢|1-60743.5¢ 15-4g]1°7029283|5 -99|1-7900914 
 { #°00} 1°3862944]4-5C]1°5040774|5-0C]}1:60094370|5-50|1°7047481|6-00 1°7917505 
 
 ak 
 
(210) HYPERBOLIC LOGARITHMS. et yeaa. 54 
 
 N. arta N.| Logar. |N.| Logar. | N.| Logar. | N.} Logar. 
 
 Rs be eee 
 aN Shel AEBS Hegel ap Panes a 
 
 6 O2I1° O50 6-52/1° oe *. O2H1 487659 5h reps Les 20816384 
 6-03]1°7967470|5-53]1°8764069|7-0311-9501867|7-5312°018895018'03|2°083 1845 
 6-04]1°7984040]6-54|1°8779372I7-04|1-9516082|7°54!2°0202222|8-04|2°0844291 
 6°05] 1-8000583|6-55|1°8794650)7-05]1-9530276|7-55|20215476|8-05|2°0856721 
 
 6-06] 1°8017098}6°56]1°8809906)7 -06}1°0544451|7-56|2°02287 12/8-06|2°0869136 
 6°07}1°80335806)0°57}1°8825138)/7°07}1"955860517-57|2°024193 1|8°07|2°0881535 
 6-08]1°8050047)6'58]1°8840347/7 08] 1°9572730|7°58|2°0255132|8'08|2°0893919 
 6°09] 1°806648 1]6°59]1°8855533|7 00] 1°9586853]7°59|2°02683 1 6|8-09|2:0906287 
 6° 10}1°8082888|6-60)1°8870696)7°10}1-9600948)7 :60|2'0281482)8"10|2°0918641 
 6-11]1°8099268|6-61]1°8885836)7"11]1-901502217-61!2°0294632|8-11|2:0930979 
 6-12]1°8115621|6-62]1:8900954]7-12|1-9629077|7 -62|2°0307764|8-12|2:0943301 
 6°13]1°8131947|6°63}1'8916048)7:13]1:9643 1 12/7:63|2°0320878)8:13/2°0955609 
 6°14]1°8148247|6°64|1+8931120)7°14}1-9657128/7-64|2°0333976|8'14|2°0967901 
 6:15]1°8164521|6°65|1°8946160)7°15]1°967112417°65|2°0347056|8: 15|2°0980179 
 6:16]1°8180768|6:66|1°8961195/7'16}1°96851 
 6-17}1'8190988)6'67!1'8976108'7°17|1°9699056|7-67|2°0373 166|8:17|2° 1004689 
 6°18}1°8213183|6:68]1:8991 180)7°18]1°9712994|7-68|2'0386195|8°18/2° 1016922 
 6° 19}1:822935 1]0:69]1:9006130|7°19}1-972691 217 -69|2°0399208)8" 19|2° 1029139 
 6°20}1°8245493|6:70)1°9021075)7°20)1:974081017:70|2'0412203|8:20)2°1041342 
 6:21}1'8261600|6°71]1 "9035990)7" 21}1:9754690)7°7 1'2°0425182)/8°21/2°1053529 
 6°22|1°8277699)}6°72)1:9050882)7°22)1°9768550\7°72/2° @438144/8°22/2° 1065702 
 6:23}1:829376310°73|1°9005751|7°23]1°9782390)7 °73|2°0451089/8°23/2° 1077800 
 6°24]1:8309802/6'74|1 ‘908059017 -24]1°9796212)7°74|2'0464017|8-24|2° 1090003 
 16°2.5|1°8325815|0°75|1°9095425)/7'25|1°081001 5|7°75|2°0476928/8°25|2° 1102125 
 
 6:26}1°8341802)6'76)1°9110229)7-26)1°9823798|7'76,2'0489823|8-26|2' 1114246 
 6:2711°8357764|6'77|1°9125011/7°27|1°9837563)7° 772 0502702)8°27/2°1126345 
 6°28]1:8373700]6'78]1°913977117°28)1°9851300|7-78'2°0515563|8°28/2° 1138430 
 6°29}1°838961 1|6°79}1°9154509|7°29}1 “986503517 7'°79\2'0528409/8'29|2° 1150500 
 6°30}1°84054.96)0'80|1°9169226|7°30|1°9878743]7°80|2°0541237|8°30|2°1162555 
 
 6°31|1°8421357|6°8111°9183921/7°3 1]1°9892433|7°81|2°0554050)8°31|2°1174596 
 16°3211:843719216°82|1°9198595|7°32|1°9906103|7-82|2°0566846|8-32'2°1186623 
 6°33|1-8453002/6°83]1:9213247|7°33]1°9919755|7°83|2°0579625|8'33|2° 1198034 
 6°34|1°8468788|6:84|1°9227877|7:34|1:9933388]7-84!2°0592388|8°34|2° 1210032 
 6°35|1°8484548|6°85]1°9242487|7°35]1 9947003|7°85|2°0605 135|8°35|2°1222015 
 
 6-36|!°8500284|6-86]1-9257074)7:36]1:9960599|7°86|2'0617866)|8-36/2' 1234584 
 6°3711°8515995|6'87|1°927 164117°37|1:9074177|7'87|2°063058 1|8°37|2°1246539 
 6:38|1°853 1681)6-88]1°928618717°38]1°998773617'88|2°0043270|8-3 8/2 1258479 
 6:39|!°8547343|6:80]1°9300711)/7°39|2°0001 277|7°80|2°0655901|8°39|2°1270405 
 6-40}1°:8562980)6°90}1°93152.14|7-40|2°001 480017 G0|2°0668628)/8-40|2°12823 17 
 6°41|1°8578593|6'91|1°9329696)7°41|2'00283041791|2°0681278)}8'41/2° 1294215 
 6°42}1:859418116-°92]1°9344158|7:42|2°0041791|7°92|2°06939 1 2|8'42|2° 1306008 
 6°43]1:8609745|6-93]1°9358598!7-43|2°0055259|7'93|2°0706530)8°43|2°1317968 
 6°44|1°8625285|6-94]1°9373018}7°44]2°0068708}7 94|2°07 19133|8"44)]2° 1329823 
 6-45]L° 8640801 6°95|1°938741717-45|2°0082140|7°95|2°0731710|8'45|2°1341004 
 
 6°46)1° 8656293 6°96]1°9401795}7 *46|2°0095554)796|2'0744290/8'46)2° 1353492 
 
 6°47|1°8671761|6°97|1°9416152|7°47|2°0108950]7 971207 5684518°47|2° 1365305 
 -|6°48)1°8687205/6°98}1°9430480!|7'°48]2°0122325]7 -98}2°0769384|8°48/2°1377104 
 6:49|1°8702625|0°90}1°9444805|7-4912°0135688)7 -Q012°0781907|8"°40|2° 1388890 
 6°50|1°87 18022|7°O0|1'9459101]7°50|2°014903018-00|2 079441 5|8°50}2° 1400662 | 
 
pA AE RS EES, REELS CR SAE 
 —— | —_— | -——— 
 
 8°52 . 1494168 8° G2 2° pete § 9°32)2° Badd Gab 
 8°5312°1435894|8°93|2°18941649°33|2°2332350 
 8°54}2°1447610)8°94)2°1905356|9°34/2°2343062 
 8°55|2°14593 13)/8'95/2°1916535|9°35|2’2353763 
 8°56}2°147 1002|8'Q6/2*1927702|0°36/2°2364453 
 8°57|2°1482677|8°97|2°1938857|9°3712'2375131 
 8°58|2*1494340)8'98/2°1949999/9'°38|2°2385797 
 8°5G}2°1505988|8'g9Q|2° 1901 128/9°30|2°2396453 
 8°60}2°1517622\9'00|2"1972246/9°40|2°2407007 
 8°61)2'1529244/9°01/2°198335119°41|2°2417729 
 8°62|2°154085 119'02'2'1994443/9'42|2°2428351 
 8°63}2°1552445|9'03!2°2005524|9'43/2°2438961 
 8°64}2°1564026|9'04'2:2016592/9°44|2'2449560 
 8°65}2°1575593/9° 05/2" *2027648/0°45|2°2460147 
 
 8°66|2°1587147|9'06'2°203869 119°46|2°2470724 
 8°67|2°1598688|9°07|2°2049723|9'47|2°2481289 
 8°08}2°1610215|9°08|2°206074219‘48/2°249 1843 
 8°60|2°1621729|9°09|2'207 1740|9'49|2'2502386 
 8°70}2°1633230)9° 10/2°2082744/9°50|2°2512918 
 8°71|2°1644718]9"11/2‘200372719'5112'2523439 
 |8°72|2°1656192)/9°12|2°2104698)9'52|2°2533947 
 8°73]2°166765419°13/2'2115657|90°53|2°2544446 
 8°74)2'1679102/9'14/2'2126604|90°54/2°2554935 
 8°75|2° 169053719 15|2'213753910'55|2°2505411 
 8°70/2°1701950|9'16)|2°2148462|9°56/2°2575877 
 8°77|2°1713368]9"17/2°2159373|9'57|2'2586332 
 8°78|2°1724764|9"18|2°2170272|9'58/2°25960775 
 8°79|2°1736147|9"19|2°2181159|9°59|2°2607209 
 8°80}2°1747517|9'20/2'2192035|9°00|2°2617631 
 8°81|2°1758874)9'21|2°2202898|9'61|2°2628042 
 8°82)2°1770219|9°22|2°2213750|9'62|2'2638443 
 8°83]2°1781550\9'23|2'2224590/9'°63|2°2648832 
 
 22731563 
 22741856 
 22752139 
 2:2762411 
 22772673 
 22782924 
 2'2793105 
 22803395 
 2:2813615 
 22823824 
 
 2°2834023 
 2'2844211 
 2'2854389 
 2'2864557 
 2°2874715 
 2°2884862 
 2‘2894999 
 2°2905125 
 2°2915241 
 2°2925348 
 2°2935444 
 2°2045529 
 2°2055605 
 2°2905070 
 2'2075726 
 2°2085771 
 2°2995806 
 2°3005831 
 2*3015846 
 2°3025851 
 4°6051702 
 6:9077553 
 9°2103404 
 
 8°84)/2°1792809}9'24|2"2235419|9-64)2°265921 1]100000}1 1°51292546 
 
 8°85/2°180417519°25|2'224623 6/9°65|2°2669579 
 
 8°8612°1815468|9'26|2°2257040\9'66|2°2679936 
 8°87|2"1826748|9°27|2°2267834\9'67|2'2690283 
 
 8°88/2"1838010)9°28)/2°2278615\|g°68|2°2700619}. 
 
 18°8]2"1849270|9°29|2:2289385|9°69|2°27 10944 
 Is: 90}2°18605 13}9°30/2'2300144|9°70|2°2721259 
 eae A ERY SP a ee 
 
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(212) | LOGISTIC LOGARITHMS. Tab. 8: 
 
 QoS} 4-5 p67.7 , 8 [9 (10411412) 134 14 
 
 “| 0 | 60 | 120} 180] 240 | 300 | 360 | 420 | 480} 540 | 600 | 660 | 720 | 780 | 840 
 0} [1°7'782]1-4771]1-3010]1 +176 1]1 -0792|0000/9331|875 118239]778217368]699(]6642)6320 
 1)3+5563)1°7710)1-4735}1+2986|1 +1743] 1 -0777/9988|9320]87421825 1/7774] 736 1|6984 166376315 
 2)3-2553]1-7639]1 46991 -2969| 1 -1'725]1-0763]9976|9310|8733|8223|7767|7354|6978|6631]6310 
 3/3 :0792|1-°7570}1 46641-2939} I +1707] 1-0749}9964|9300|8724|8215|7760|7348]6972|6625|6305 
 412-954.9]1-17501)1-4629}1 +2915]1+1689}1-0734}9952|928918715|8207]7'753|7341 |6966|6620|6300 
 5|2°8573|1-7434]1-4594/1-2891|1-1671]1-0720|9940/9279|8706]81 99}774.5|7335}6960]6614|6294 
 
 6|2+7789]1+'7368]1-4559]1-2868]1°1654|1+0706}9928|9269|8697|8191/7738)/732816954|6609|6289 
 7/2°7112|1°7302)}1-4525}1 *284.5]1°1636]1°0692)9916|9259)8688)8 1 83/773 1)7322|694816603|6284 
 8}2°6532|1°7238]1-44-91]1+2821]1°1619]1°0678/99051924.9/8679]8 175/77 241731 5}694.216598)6279 
 9/2°6021}1°'71'75}1°44.5'7|1°2798}1 1601] 1 -0663/9893/9238)8670/8 167/77 17|7309|6936)6592)6274 
 10]2°5563}1°'7112|1°442411-2775|1°1584|1-0649/988 1/9228] 866118159)7710)7302)6930}658 7/6269 
 
 11}2°5149)1°'7050}1-4390/1 -2753}1°1566]1°0635)9869}9218/8652|8152)7'703}7296/69241658 116264 
 12/2°4°771 {169901 4357/1 °2730)1°154.9]1°0621/9858)9208|/8643|8 144|7696]7289|69 1 8165 76}6259 
 13}2°44.24/1*6930)1°4325]1 *2707|1°1532|1°0608)9846|9 1 98/8635]8136)7688/7283/69 1 2/6570|6254, 
 1412°4102/1°68'71|1-4299/1 -2685]1°1515]1°0594)9834)9 1 88/8626/8 128/768 117276 6906}6565|/6248 
 15}2°38092|1°6812/1°4260]1 °2663}1-1498}1°0580|9823/91 78/861 '7}8 120/76'74)7270|6900}6559|6243 
 
 16|2°3522|1°6755}1°4228)1 -264.0)1°1481]1 °0566/9811/9168}8608]8 1 12/7667/7264/6894|6554)6238 
 17/2°3259)1 *6698}1+4.196/1'2618)1°1464/1°0552)9800)9158/8599|8104/7660/7257'6888/6548 6233 
 18/2°3010)1 °664.2|1°4165}] -2596)1-1447|1°0539/9'788)9 148/859 1|8097|7653/725 1/6882 6543}6228 
 19/2°27'75|1°658'7)|1°4133}1°2574/1°1430}1°0525|977'7/9 138/858 2|8089)764.6]7244/6877 6538/6223 
 20/2°2553)|1°6532}1°4102/1°2553)1°1413}1°0512/9765)9 1 28/8573/808 1 |7639|7238|687 1/6532/6218 
 
 Q1]2°2341|1°64'78] 1-407 1]1+2531\1°1397]1-0498|9754191 19 8565|8073!7632]7232 6865 ed Sdn 
 22)2°2139}1°6425}1 -404.0/1*2510)1°1380}1°0484)9742/9109/8556|8066|7625|7225|/6859 6521 6208 
 23)}2°1946|1°63'72}1°4010}1°2488) 1°1363]1°0471}973119099\8547/8058/76 1817219, 
 2412°1'761}1°6320}1°3979}1 +2467) 1°1347]1°045819720/9089/8539|8050/761 1|7212)684-7/6510'6198 
 25|2°1584)1°6269}1°3949]1+244.5/1°1331]1°044419'708|9079!8530|804.3/7604)7206/684 116505/6195 
 
 26/2°1413]1°6218/1°391 9}1°2429411-131411°04311969719070|8522 3035!75977 7200,6836|6500)6188 
 27/2°1249)1-6168}1 -3890)1 +2403 1-1298|1-041 819686|9060 8513|8027)7590}71 93, 6830|6494, ‘6183 
 28/2°1091/1°6118}1°3860}1°2382/1-128911 °0404196'7519050'850418020/7583/7187 
 29|2+0939]1-6069|1°3831}1 23691-1266} 0391196649041 |84.96|8012/7577/7181 
 30|2°0799}1°6021|1+3802}1-2341/1+124.9}1 -03781965219031|8487|8004'7570/7175 
 
 312-0649} 1-5975|1 37731 -2320)1-1253|1-0365}964119021/84'79|7997|7563}7 1 68 6807]64736163 
 $2]2-051 911 -5925(1°3745'1-2300|1-1217]1035219630|9012'847017989/7556|71 62\6801 6467/6158] 
 
 33|2°0578|1-5878|1-3716|1-2279 1-1201|1 °0339/96191900218462|7981|754917156|6795|6462'6153 
 
 34(2-0248)1-5839|1-3688|I- 2259/1+1186]1°0326/9608|8992|8453/7974)7542 naar sata 
 
 16818 set ah 
 6812/6478 6168 
 
 35|2°0122|1°5786|1°3660)1-2239 1+1170|1-031319597|8983}8445]7966]75357143|6784|645 1/6143 
 
 36/2°0000/1°5740|1°363Q)1- erelt *115411°0300)958618973/843'717959|7528] 7137/6778 6446) 6138 
 37}1°988111°5695/1°3604 Peake °1138]1°0287)9575|8964/8428/7951|75221713116772)644.1 | 6133 
 38} 1°9'765}1°565 1}1°35'76{1°21'78)1°1193]1°027419564)8954/8420/794.4175 1 51'7124'6766|6435)6128 
 39) 1°9652)1°5607|1+3549)1+2159)1°1107}1°0261)9553/894.5|841 1/7936}7508}71 18 6761/6430 6123 
 
 15563 1°3522|1°2139 1°1091}1°0248 9542)8935|8403/7929/7501 hist Set 6425, ‘61 18 
 
 40}1°954,2) 
 
 4.1}1+943511°5520]1°3495]1°2119]1-1076]1°0235/9539/8926|8395}7921/7494)7106 6749 6490161 13 
 4,2}1°933 1|1°54'77|1 +3468) 1°2099)1-1061] 1 +0223)9521|8917|8386]79 14)7488]7100/6743)6414/6108 
 43} 1+9228)1°5435]1'344.111-2080/1-104.5}1°0210/9510|8907|8378/7906)748 1|7093|6738|6409 6103 
 44}1°912811°5393)1*3415|1-2061)1-1030)1°0197|9499|8898}8370|7899/74'74) 708'7/6732)64004,6099 
 
 45|1°9031}1°535 1}1°3388]1°2041/1°1015}1°0185|9488/8888)836 1|789 1|7467)| 708 1|6726 ssa] 
 
 4.6] 1+8935}1°5310)1°3362)1-2022)1-0999]1:0172/94'78|8879|8353|7884)746 1|7075|6721/6393|6089 
 471188421 *5269/1°3336)1+2003;1°0984}10160/94.67|8870|8345178'77|/7454| 7069/67 I 5|6388 6084. 
 4.8]1°8751]1'5229]1°3310]1-1984!1-0969] 1 -0147/9456|8861|8337]7869|744'7|7063|6709}6383|6079 
 49) 1°8661)1°5189)1°3284)1°1965)1°0954]1°0135}9446|885 1|832817862|744 1)7057|6704/6377|6074 
 50}1°8573)1°5149)1°325911-1946)1-0939]1 70122194 35|8842)8320/7855}7434) 7050/6698)/6372|6069 
 
 51)1°8487}1°5110 *3233}1+192'7/1-0924)1°0110)9425|8833)83 1 2)784.71742'71704.416692/6367/6064 
 52/1 °8403)1°5071)1+3208) 11908) 1-0909]1 -0098}941 4|8824)8304|7840)742 1|'7038/6687|/6362)6059 
 I ‘BoeG l me 1°3183 11889/1-0894 1 -0085}94.04/88 1 4|8296|7832/74 14]7032|668116357/6055 
 
 *8239)1°4994)1°3158}1°1871!1-0880}1-0073/939318805|8288|7825174.07| 7026] 6676)635 11/6050 
 8159) 1°4956)1°3133)1°1852)1-0865}1°0061/9383|8796|8279178 18}74.01)7020}5670)\6346)6045 
 
 *8081]1+4918}1°3108]1°1834}1-0850}1-004.9}9372|878'7/82711781 1]7394|'70 14] 6664|6341)6040 
 8004} 1°4881/1°3083}1-1816)1+0835]1°0036/9362/8778|8263/780317387|7008|6659|6336|6035 
 7929} 1°4844/1°3059}1°1797)1 -0821}1 00241935 118769/8255/7796]738 117002] 6653|633 116030 
 ->> 1+4808}1*3034)1°17'79}1 -0806]1-0012/934.118'760|824'7/7789}737416996| 6648) 6325|6025 
 
 T7821 °4771'1°301 011-176 111 °079211 -0000)93311875 118239}'7782}7368)69901664.216320]602 1 f 
 abe tert uta eave tt eerie hs sch eS dei RAY Ne p07 Ws aha NRL 
 
 FW Pe ats 
 
den 8 8s LOGISTIC LOGARITHMS, | (218) 
 22 23 24 25 | 26 | 27 28 29 30 
 
 26|5897|562415368]512514896|4678/4471 |4273]40831390213727|3560|3399|3243'3093 
 27|5892|562015364/5 1 2214892)4675144.67|4269]/4080138991372513557|3396|324 11309 1 |2946 
 
 4 j 3896|3722(3555|9393}3238)3088 2943/2803 
 29)5883/9611)9355)511414885}4668/4460/4263}4074)3893/371913552/339 1/3236/5086/294 1/2801 
 ross nor her sige 
 
 31158'74|5602|534'7/5 106/487'7|4660/44.54|4256/40681588'7|37 13|354.6|3386|323 1|3081/2936 
 
 2948/2808 
 
 ra93t 
 
 soselissolieasisisrl4oaolio5el3e7 cpus rm 
 4 41423'71404913869}< a ape pl Seg ite! akon 
 
EZER) yh, LOGISTIC LOGARITHMS. Tab. 8. 
 er nora RR SRR a en a RRR PS SE ETE 
 | 32) 33 1 34) 35 | 36 137) 58 |39, 40,41 , 42) 431 44) 45 (46 47 
 11920] 1980}204.0]2 1 00/2 160}2220]22801234.01240012460}2.520|2580}264.0]2700}2760] 2820 
 
 —_—j——_—__. | —————— ] ——— fF —————_— |} ———_ | ————_ } —]|—— | —~— ] —~ |_ ] — — | ——-_ | ———__ | —_- | ———- 
 
 0/2'730}2596}2467|234 1/221 8}/2099|1984)1871}1761)1654)1549}144'7)1347}1249]1154) 1061 
 | 112728125941246512339]22 1 6]2098) 198211 869}1'759}1 65211547) 144.5)154.5]1248)1152)1059) 
 212'72.5}2592}24.62) 2337] 221 4]2096}1980}1 867) 1757}165011546) 1443) 1344)1246)1151)1057 
 3/2723}2590)]2460) 2335] 22 12}20941978}1865}1755}1648} 154411 44.2)/134.2}1 24511149] 1056 
 412°72112588]24.5812353}22 1 0}2092)1976]1 863[175411 647}154211440}1340]1243)1148]1054 
 5}2719}2585]/2456)}233 1}2208/2090] 1974] 1862) 1752]164.5}1540}1438)1339]1241)1146]1053}- 
 
 6/271 612583]24.54/2328) 2206/2088} 1972}1 860} 1750} 164.3}1539]143'7)1537}1240)1145}1051 
 7/27 14|258 1}245 2/2326) 220412086] 1 970}1858]1'748) 16411537] 1435)1335]1238}1 143}1050 
 8,271 2}2579}2450)/2324/2202/2084)1968)1856}1746) 1640] 1535)1453)133411237}1141)1048 
 9/2710) 2577/2448) 2322) 2200}2082|1967/1854)1745}1638]1534}1 432/1332]1235)1140)1047 
 10,2°707}2574|2445}2320) 2198/2080} 1 965}1 852} 1'743]1636]1532}1430/1331}1233}1138]1045 
 
 1 12705}2572/2443}2318| 21 96|2078|1 963]1850]1741]1694|1530|1428|1 59911 259]1197|1044 
 12/2703/2570)244 1/23 1 6}2194)20'76}1961}1849)1739}1633/1528]1427)/1327}1230}1 135)1042 
 13,270112568]2439)231 412 192/2074)1959]184'7}1 73'7]1631}1527]1425)1326}1229]1134/1041 
 14,2698|2566|2437|23 12121902072} 1957|1845|1736]162911525]1423|1324]1227 
 
 15,2696} 2564/2455/25 1 0}2188/2070}1955}1843)1734)162'7}1523)14.22/1322)1225/1130)1057 
 
 16 2694}2561|2433}2308\2186}2068]1953}1 841}1'732}1626]1522)|14.20)1321]1224)1129}1036 
 17/2692}2559|243 112306]21 84) 2066]1951]1839}1730/1624]1520]1418)13 19}122211127|1034 
 18 2689]2557|2429]2304/2182}2064| 1950]1838]1728}1622}1518]1417/1317]1221}1126|1033 
 19 268'7/2555|24.2612302/21 80/2062}1 948]1836]1 72711620|1516|1415]131 6]1219]1124/1031 
 
 (1 26B5)2555}2426/2900)2 178) 2061194618341 725)1619}1525 VAT SISTA Skt RG} 1O8D 
 21 2683]255112429|2298|21'76|2059}1944 1832]1723]1617}1513}1412/1313]1216]1121|1028 
 
 22 268 1]2548]2420]2296]21 74) 2057|1942}1830]1721{1615|151 1]1410)1311}1214]1119|1027 
 23,2678|2546}24 18/2294) 2172) 2055|1940)1828)1719|1613}1510}1408)1309]1213/1118) 1025 
 in ikdaa baie | great se | eng Fab ts bi lp tn heap alae AT NO So 
 
 25: 2674)254-2) 24 1412289}2169)2051)1936]1825]1716)1610}1506]1405)1 3061 209)1 115)1022 
 
 26267212540) 241912987)21 67}2049]1994|1 s25|1714[1 608|1504|1403|1304|1208|1115|1021 
 27|266912538/2410)/2285)/2165)2047}1933/182111712)1606)1503}1402)1303]1206)1112)1019 
 28 2667)2535)2408/2283)/2163)/2045|1931)1819})711)1605}1501}1400)1301}1205}1110)1018 
 29/2665}2533)2405)2281/2161)2043/1929/1817]1709}1603/1499/1398)1500}1203)1109/1016 
 i spi gmt gen i. Sea Sd MAL ks Waa Dd dE IR Hein. Reba Habe IL) <8 
 
 31,2660}2529) 2401) 227'7]/2157}2039}1 925/181 4]1705|1599|1496]1395|1296}1200]1105)1013} 
 32,2658}252'7/2399)/ 2275/21 55[203'7}1923/181 2)1703]1598)1494)1393/1295]1198]1104/1012 
 33 2656}2525}2397/2273)2153}2035}1921)18 1 O1'702}1596}1493}1392/1295]1197]1102)1010} 
 34,2654)2522) 2395/2271 /2151/2033}1919}1808)1700)1594)1491}1390)1291}1195]1101}1008 
 35\2652/2520)2393)2269/2 | 4912032/1918/1806}1698}1592)1489]1388 1290}1193)109911007 
 36,2649)2518|2391 2267/21 471203011916|1805{169611591/1487|1387|198811 991109811005 
 37264-7125 1 6}2389) 2265)2145)2028/191411803}1694}1589/1486]1385)1287}1190)1096)1004 
 38 2645)25 14/2387/2263/2 143/2026/1912)1801)1693/1587/1484}1 383)1285}1189]1095|1002 
 39 2645}25 1 2/2384'2261/2141/2024)1910)1'799]1691)1585]1482)1382)1283]1187)1093/1001 
 
 40}2640}2510)2382/2259}21 39/2022) 1908]1797|1689)1584)148 1]1380/1282]1186]1091|0999 
 
 41}2638}2507|2380/2257/2137)2020) 1906}1'79511 687}1582|1479]1378)1280}1184)1090\0998 
 42:2636}2505]2378!2255|2135}2018]1904}1794}1 686]1580]1477]1377)1278}1 182}1088)0996 
 43'2634|2503]2376|2253|2133}2016|1903{1'799|1 6841578] 1476|1375|1277|1 181]1087|0995 
 44/2652|2501]2374 225 1|2131|201 4|1901|1790|1689|157711474)13°73|1275|1179|1085}0993 
 4.5)262912499] 2372) 2249)21 29/20 12/1899]1788}1 680]1575|1472|1372|127411 178|1084|0992 
 i 
 
 4.642627|24.97]23'70)224'712 127/201 0}189'7]1'786]1678]1573}1470}13'70)1272)1 1'76|1082|0990 
 4°7}262.5]2494)2568}224 5/21 25/2009/1895}1785}1677)1571)1469]1368)1270)1 174110810989 
 48/2623} 24.92}2366 2243/21 25/2007] 1893} 783}167511570}1467]1367|1269}1 173}1079}0987 
 
 4.9/2621/2490/2364/224 1/212 1/2005|1891]1781]1673)1568)1465}1365}1 26711171 }1078}0986 
 5 ag fae‘ imeeed sbnh Task elles uM hla Mal pened vase easy! sah ada | 
 51]2616]2486]2959/9997 21 17/2001 |1888]1"777|1670|1565|1469|1 369|1 26411681074 |0983 
 52)/2614}2484/2357 2235/2115}1 999) 1886}1775|1668]1 563} 1460}1360)}1262)1167}1073)098 1 
 53/26 12} 2482)2355,2233121 13]1997]1884/177411666]1561114.59]1359}1261|1 165]1071/0980 
 54126 10}2480/23.53/223 1/2111]1995| 18821177211 664)1 559}14.57]1357/1 25911 163)1070]0978 
 sid Nj sc epg |S DCI iGo” Pak see Nk sled tak ela! swe nth 
 
 56/2605) 24°75)2349/2227/2107)1991)1878)1768)1661]1556)1454)1354)1256)1160}1067)0975 
 5'72603}2473)234'7|2225/2105]1989) 1876]1766]1659}1 55411 452]1352)1 25411 159}1 06510974 
 58}2601 1247 1}234.5,2223}2 1 03/1987} 1875] 1765}1657/1552|1450}1350}1 253] 1 1571064 }09724- 
 59/2599/2469}2343)/2220)2101} 1986) 1873]1763]655}1551)1449}1349}1251)115651062,0971 
 
 601259612467 |234 11221 8WO9911 984) 187111761 JL 654115491144 7/134 711 249] 1544106 1}0969 
 
 2 
 er 
 
Tab. 8. LOGISTIC LOGARITHMS. (215) 
 150151 152) 53) 54) 55 | 56) 57 1 58 | 59 160) 61 | 62 | 63 
 7/19880}2940}3000|3060}31 203 1 80}3240]3300}3360|3420|348013540/3600|3660|3720|3780 
 ~ 010969]08801079210706|0621(0539|045810378/0300102231014.7|0073 99281985819788 
 110968}0878|0790]0704|0620]0537|045610377/02981022 110146|007219999|9927|9856|9787 
 21096610877|0789]0703|061 9}0536|045510375|0297|022010145]0071|999819926}9855/9786 
 310965}0875|07871070210617/0535]04541037410296|021910143}0069]9996|9925}9854|9785 
 410963]0874|0786|0700}06 1 6|0533}04.52|0373|02941021 8101 42/0068|9995|9923}985319784 
 510962]08721078510699|06 15105321045 11037 1]0293|0216|0141}0067|99941992919852/9789 
 
 6]0960}0871)0783/069'7}0613}053 1}04.50}0370/0292/0215)0140/0066}9993)992 11985119781 
 71095910869|0782/0696]0612/0529}04.48/0369)029 110214101391006419992/99201984919780 
 8]0957|0868}0780|0694406 1 010528}044:7}0367/02891021310137|0063)9990)99 191984.8}97'79 
 910956|0866)0779|069310609}0526}044.6|0366)0288/02 1 1/0136|0062)9989)991 81984719778 
 10]095410865|077'7/0692}0608]0525/044.4)0365/028 7}0210/0135}006 119988199 16|984.619'7'77 
 
 11]0953)086310776|0690|0606|05241044.3}0363|0285|020910134/0060|998719915|984519775 
 121095 1]086210774]068910605|0522}044.2|0362'028410208|0132|005819986/99141984.4197"74 
 13}09350|08601077310687|0603|0521104401036110283|0206|01310057|998419913/984919773 
 1410948]085910772|0686|06021052010439|0359102821020510130/0056]9983/991 2/984 1/9770 
 15]094'710857/0770]0685|060110518|0438|035810280]020410129|0055|9982)991019840\9771 
 
 16|094.51085610769|068310599|0517/0436|0357}02-79|020210127|0053/9981|990919839|97"70 
 .11710944]0855|076710682|0598]0516|0435/03561027810201101261005219980/990819838|9769 
 1810942}0853|0766]068010596|0514]0434]0354|02'76|020010125)005 119978 |9907}9837|9767 
 19]09411085210764|067910595|05 13|04321035310275|019910124|0050/9977/9906|9835!9766 
 2010939/085010763|0678|0594105 12/043 11035210274|019710122|004.91997619905|983419765 
 
 2.110938108491076210676105921051010430103501027310196/0121/004719975|\990381983319764 
 221093.6]0847|0760|06751059 1{0509}042810349/027 1]0195|0120,0046|9974/9909/983219763 
 2310935}0846 0759|0673}0590|0507|0427|034810270]0194/0119|0045|997219901]983 119762 
 2410933}0844 |0757|0672|0588|0506]042610346)0269|01 99/01 17,0044)9971|9900)980'9761 
 2510952}0843)0756)067010587/0505]0424)0345)0267/0191/0116 0042/9970 9899] 982019759 
 
 26|0930]0841|0754|0669|0585|0503/042310344 0266/01 90)0115,0041)9969 9897/9827}9758 
 2'710929]084.010753}0668|0584]0502|0422|034210265|0189|01 14/0040|9968|9896|9826.9757 
 981099710838 10775 110666|0583|050 11042010341 |02641018710112/0039|99669895198a5/9756 
 $2910996/0837/0750106651058 1104991041 91034.010262|0186|011 110038|996519894|982419755 
 a a i va ete sth bs ss Ret ib sl Be ie 
 
 31}0995|0824,0747|0662|0579|049704 1 6/0337|0260}0184)0109 0035/9963 /9892|982219753 
 321092110833 |0746)0661]0577}0495}04 1 3|0336]0258/0182)0107/0034}9962)9890}9820,9751 
 33109201083 1107441065910.576|0494104141033510257/018110106|0033/9960/98891981919750 
 3410918|083010743|06581057410493|041 210333|0256|0180,0105|0031]9959:9888]981 819749 
 3510917/082g1074110656105'73|049 1/041 1/033210255|01'7910104|0030/9958/988719817/9748 
 
 361091 5]082'7|0'74.0|06551057210490]04.1 0/033 1|0253)0177/0 103/0029|9957\9886|98 1619747 
 371091 410825'07391065410570}0489/0408|0329/025 2/0 1'76|0101/0028/9956/9885|98 1519746 
 38/09 1 2)0824./073-71065210569|048'7/0407/0328/025 110175/0100|0027 7954)9883 981319745 
 39/091 11082210736}065 10568}048610406}0327|0250)0174/0099/0025/9953 9882198 12:9744 
 40}090910821107341064.910566|0484)04.0410326)024.8)0172)/0098/002419952:988 11981119742 
 
 4110908108 19|0733|0648|0565|0483]04.03}03241024-7)0 t'71/0096/0023/995 1 9880)9810/9'741 
 42/0906}0818|073 11064'7]0563}048210402/0323/0246)01'70)/009.5|0022/9950'9879 9809/9740 
 4310905|08 1 6|0730|064510562}0480/0400/03221024410 1 69/00941002 1/9948 '987'71980819739 
 4410903/0815|07291064440561]047910399}0320/0243/0 16'7/0093/00191994'7'9876|980719738 
 45}0902/08 1410727|0642}0559|04-78}0398]03 | 91024210 1 66)009 1/0018/994.6/9875]980519737 
 
 46|0900/081210726|0641/0558}0476]03960318|0241/0165|0090/0017|994.5198'741980419736 
 47/0899/08 1 1|0724]0640]0557}0475]0995]0316(025910163]0089)0016}9944 9873}9803 9734 
 {48|0897}080910723|0638/0555]0474|039410315}0258101 62/0088)0015}9942/987219802\9735 
 4910896|0808|0721/0637105541047210392103141023'710161|0087|0013|994.1|98'701980119732 
 50/0894/0806/0720|0635|055210471]03911031310235/0160}0085|001219940/986919800/9731 
 
 51/0893}0805/0719}0634)055 1104.70/0390/03 1 1/0234)0158}0084)001 1/9939)9868/979819730 
 521089 110803/0717}063310550(0468/0388)0310\0233/015710083)00 1019938/9867|9797|/9729 
 53}0890}0802)07161063110548|04671038 7/0309 10232) 0156]0082/0008/9937|9866)9796|9728 
 594/0888)0801)0714)06501054'7}046610386/0307/0230)015510080}0007|9935/9865]979519727 
 551088 710799071 3}0628105461046410384/0306/0229|0153}0079|0006}993419863]9794)9725 
 
 56}088510798)071110627}054-4)046310383/0305|0228/0152100781000519933|9862)9793|9794. 
 571088440796107 10/06261054.3104.6210382\030410227101 5 1100'7'7/0004)9932/986 1197929723 
 581088310795|0'709)06241054110460/038 140302/0225101501/007510002/9931|986019790)9'722 
 39088 110'79310'707| 0623/054.01045 9105791030 11/0224)0 148]0074|0001}9929]9859]97891979 } 
 450 0880'0792(07061062 110.539104.58}03'7810300'0223101 4710073000019 92898581978 8)97 206 
 
C216)40 LOGISTIC LOGARITHMS. Tab. 8. 
 
 12/9706'9639/9573|9508/9444/9380)93 18}9257|9 1 96/9136|9977/901918962|890518849|8'794 
 1319705)9638|9572/9507}9443)9379)93 17/9256)9195)9135)907619018/896118904/8849)8793 
 | 1419704)9637/9571/9506)9442/9978)95 161925519 194191541907619017/896018903|8848)8792 
 
 15}9703'9636|9570)9505|944.0/9377/93 J 5|9254/9 193/19 133)9075|9016/895918903/884'7|8'792 
 
 16'9°702'963519569|9504'9439|9376]93 1 4|9253/9192}913219074190 15!8958|8902|8846 oni 
 179701|9633/956719502,9438/9375|9313|9252/9 1 91]9131|9073|9015|8957|8901|8845/879 
 
 1819699 9632)9566)9501 94.37/9374{93 121925 1/919019130|9072|901418956|890018844 ae 
 1919G98'9631/9565|9500/9436193731931 1192501918919 1 29190711901 3/3955188991884318788 
 20'96971963019564194991943519372/951019249/9188/91 28/9070 901218954/8898/8842|8787 
 
 21/9696, 9629]0565|9498'9454]937 1]9309}924810187/9128}90691901 |/s95a}ss97|8841/8786 
 28/0695 962818962)949719533}9970)99089247/9185)912719068/90 1 0895 9ias90)B840) 8785 
 
 3'9694.962719561|949619432/936919307 9246|9185)9 1 26/9067/9009|8952)8895|883918784 
 24/9695 962619560)9495|943 |9369}9306(0245 9184|9125|9066)9008 895 |s804l8898)87 785 
 25'9692'962319559}9494/943019367|93051924419183/9 1 24190651900718950/88931883718782 
 
 26 9690 96241985s/9499|9420/9865}9504|9043/9182l0123]o06sI0006|5949]s¢99\8857/8781 
 27|9689 9622)9557|9492}9428|9365/9303|924 1}91819122/9063}9005 8948|889 1|8856|878 | 
 28 9688, 9621]9555)949 1}9427/9564 193021924019 18019121 /9062/9004|8947/8890/8835/8780 
 2919687 9620|9554}9490/9426|9363/930 1|9239/9 17919 1 20}906 1|9003|8946|8889|8834/8'7'79 
 3019686 9619|9553/9488/94-25|9362/9300|9238'91'78/91 1919060|9009'894.518888|8833|8'778 
 
 31]9685,96181955219487/9424/9361}929919237)9177191 18|9059|9001|s944|8sseiss32|8777 
 32/9684 96171955 119486|942219360/9298|9236/91'76)9 1 17/9058|9000|8943 8887/8831 18776 
 
 33/9683 96 16)9550)9485}9421|955919297/9235/9175|9 1 16/9057/8999|8942)8886|8830|8775 
 34/968 1 9615/9549)9484/9420/9358)9296}923 34/917419 11 5|9056|8998/8941/8885|8829/8774 
 
 3519680 9614)9548|94.83/941 9/9356|9294/9 2353/9 173/91 14/9055|8997|8940|\8884|8828/8773 
 
 3619679'9612/9547194821941819355}9293/923919 1 79191 13]9054|8996|8939|8s83\982718772 
 
 $7/9678.961 11954619481 /9417193549292/923 19171911 219053/8995|8938|saselss26(8771 
 38/9677, 9610|954.5|94801941 6'9353]9291|9230)9170]9111]9052|899418937/8881|8825|8771 
 39/9676, 960919544/9479194 15 9352}9290/92 291916919110/9051)8993/8956|8880/8825|8770 
 4019675|9608/954219478/941 41935 1/9289/922819 168/91 0919050/899918933/88'79|8824/8'769 
 
 4119674, 9607|9541/947'7/9413/935019288)9227/9 167 9108/9049)8992/8935|8878/8823/8768 
 42/9672'9606|9540|94'76|9412\934919287|9226|91 66/9107 9048}8991|89341887'7|8822|8'767 
 43|967119605|9539194'75|9411/93 4.8|9286]9225/9 1 65/9 106|904'7/8990|8933/88761882 118766 
 44/9670/9 960419538194.73194 1 0/934719285/922419 16419 105|9046/8989/8932)887518820|8765 
 45|9669/9603/953'7}94'72}94.1191934.6|928419223/9 163/91 04)9045/8988)/8 93 118375188 19/8764]; 
 
 46|9663/960 119536}947 1/9408]9345}9283}9222/9 1 6219 103/9044/8987|8930/8874188 18/8763}, 
 47/9667 /9600}9535}9470)9407/93449282/922 1/916 119 102}9043/8986/8929/887 73|8817|/8762 
 4819666/959919534194.69/94.06/9343/928 11922019 1 60/9 101/904-2/8985|8928/8872|88 16/8761 
 4919665/9598/9533/94.68]94.05/9342/9280192 191915919 1 00}/9042/8984/8927/8871/8815|/S761 
 501966419597195321946 71940419341 /9279}92 1 8)9158}9099}9041|8983)8926/8870}88 14/8760 
 
 5119662)959619530194.66|9402/934.019278192 1719 157}9098|904.0/8 982/8925}8869}88 1318759 
 521966 11959519529194.65194.01193391927'7/9216)9 156|9097/9039|898 1 |892418868/881318758 
 53|9660|}9594)/9528)/9464194.0019338192 76/921 5/9 155}9096|9038|8 980)8923!18867|88 12/8757 
 541965919593)952719463}9399|933'7192751921 419 15419095}9037/8979]8922) 8866/88 1 1/8756 
 5519658/9592/9526/9462/93981933619274192 1 3)9 | 53}9094}9036/8978]8921)8865/8810)8755 
 
 561965719590195251946 1 |9397|933.5}927 3192 I 2)9152}9093|9035|897'718920/8864)8809/8 754 
 5'7/965619589}9524194.6019396|933419272192 1 11915 1)9092}9034/8976189 1 9/8863|8808]8753 
 581965519588}9523}945919395 93331937 119210/9150)909 1|9033/8975189 1 8|8862|8807|8752 
 45996531952 719.5221945 71939419332192701920919 1 4919090}905218974189 1 81886 11880618752 
 6096529586195 211945619393 1933 1]926919208!9 1 48]9089/903 1|8973189 171886 1'!8805]8751 
 Bt AEE LL SST SS ST TE TR 
 
TABEE IX, 
 LOGARITHMIC 
 
 SINES AND TANGENTS 
 
 TO 
 EVERY SECOND 
 
 IN THE FIRST TWO DEGREES, 
 
 3 F 
 
(218) O Deg. LOG. SINES. | Tab. 9. 
 
 6°4637261|6°764.7561)6 -94084:75/70657860/7*1626960)7-2418771)7°3088239160 
 1/4°685574.916 ‘470904716 '7683602)6 '9432534)7°0675918}7° 164141 2)7 24308 18)}7°3098567|59 
 214*986604.9|6°4'77966516°771934716 '9456462)7 0693901 17°165581717°2442832|7°3108870158 
 315°1626961|6'484915416°7754800]6 -948025917°0711810]7-°1670173)7°2454813)]7°3119149)57 
 415°2876349|6°4917548]6°7789965]6 -9503926|7°072964.6)7 °1684483]7 °2466760)7°3129404156 
 5|5°3845449)6°4984882|6 °782484916 °9527465)7°0747408]7°1698745]7°24786'75|7°3139635155 
 615 °4637261|6°5051188|6°78594.5416 ‘9550878)7- 0765099 7°171296117°2490557|'7°3149842/54 
 
 15°5306729|6°511649715°7893786|6 9574.1 64/7- 0789717 7°1727131)7:2502407)7°3160024|53 
 815°5886649]6°5 1808385 °7927848)6 95973277 0800264|7°1741254)]7°251422517°3170183)52 
 9)5°639817416°5244239/6°796 164516 °9620366/7 081 774117°1755332/7°2526010)7°3180318)51 
 10)5°685574916°5306729|6°7995182/6°964.3284)7 0835 148/7°1769364)7°253776417°3 190430150 
 1115°7269676|6°5368332|6°802846116°9666082'7 '0852485]7°178335117°2549485]7°320051 8/49 
 12)5°764756116°5429074/6°806 1488/6 °968876017-0869753)7°1797295)7 +2561 176)7°3210583/48: 
 
 13]5°7995182)6°548897716°8094265|6°971132117-088695317°181 1190|7°2572835|/7°322062414'7 
 1415°8317029|6'5548066|6°8126796|6°973376517:0904.08517'1825043]7 258446217 *3230643/46 
 1515°861666116°5606361|6°815908616°9756094)7 0921 14.917°1838853]7°259605917 3240638145 
 16|5°8896948/6°5663884/6°8191137|6°9778309/7 0938 14717185261 8}7°260762517°325061 0/44 
 1715°9 1602386 572065616 822295416 9800410)7'0955079|7'1866340]7°2619160)]7°3260560/43 
 18|5°94.084'74|6°5776695|6°8254539|6'98224.00!7-097194.5|7:1880018]7'263066417°3270487/42 
 19}5°964328516°5832019]6°8285896)|6 984427917 -0988745|7'189565417°264213817°328039 1/41 
 2015'986604.9|6°5886648|6 83170296 -986604817°10054.8117°190724-7]7 265358217 °3290272|40: 
 2116°007794.2|6 594059916 834793916 988770917 -1022153|7°1920797|7°2664996]7°330013 1/39 
 22/6 0279975|6°5993887|6 837863216 °9909262)7°1038760|7°1954306)7 -2676380)7°3309968)38 
 2316 °04'7302'7|6°6046529|6°84.09 109]6°993070817"105530517°194.777217 268773417 °3319783/37 
 2416°0657861|6°6098541|6°8439373/6°9952050/7:107178717°1961197|7°269905817 *3329575|36 
 
 2516°083514.9|6°6149938|6 °8469428)/6°997328717:1088206|7°1974.58017°2710353}7'°3339345|35 
 26|6°100548216 °6200733|6°8499277|/6°9994.4.20/7-1104564|7°1987923]7°272161917°3349094134. 
 2716°1169386|6°6250941|6°8528922;7°00154.51)7°1 12086017 °200 1 224.17 °2732856|7°3358821133 
 2816°1327329]6°6300575|6°8558365|7 003638 1/7°1137095|7°2014485|7°2744.06317 °3368525|32 
 2916°14'79'729/6 °634.964.916 858761 117°005721117°1153270|7°2027706)17 °275524.2|7°3378209131 
 30]6°1626961|6°63981'74\6°861666117°007794117'°116938517°2040886]7°2766392|7°3387870|30 
 
 3116°1769366/6 644.6 162)/6 864.55 18/7°0098579]7'1185440]7°2054027)7°277751417°33975 1) 129 
 3216 °1907248|6 6493627 |6 86741 84|7'0119107|7-12014.36|7°20671 28}7°2788607|7°3407130|28 
 33|6*2040888|6°6540578/6°8702663)7°013954.4/7°1217374/7°20801 89)7°279967917°3416727|27 
 3416°217053816°6587027|6'873095517°0159886|7°123325317°209391 117°2810708]7 342630426 
 35|6°22964.29]6°6632985|6'8759065)7°018013217°124.90'74|7°21061 95)7°282171 717343585925 
 36|6°2418774|6 6678461 |6°8786994)7°0200285/7°1264838)7°2119140)7'283269817°3445394/24 
 
 37|6*2537766|6°67234.66|6°8814745|7°022034.5]7°1280545|7°2135204.6)7°284.3651/7'3454907123 
 38}6°2653585)/6°6768009|6 88423 19/7°0240313]7°1296195|7°2144.914)7°2854.57'7/7°3464400/22 
 39/6 °2766395|6 68.1 2100/6°8869719!7'0260189}7°1311789]7'215774.4)7 '2865475|7°3473872/21 
 40]6°28'76349/6 °6855748]6°8896948/7°0279975/7 °-1327328]7°2170536|7°287634.6]7°3483323|20 
 41/6 °2983587|6°6898962/6 °8924007/7 0299671 |7 "134281 1}7°2183290|7°2887190)7°3492754119 
 4216°3088242/6°6941750|6°8950898]7 03192787 '1358238|7°2196008]7°2898006|7 °3502165]18 
 
 4316+3190433|6°6984121|6°8977624/'7-0338'796\7°13 73612)7:2208688|7°2908796|7°3511555|17 
 4.4}6°3290275|6°702608216°9004187)7°0358228/7°1388931|7°2221331]7°2919560]7°3520925/16 
 4.5|6°338'78'74|6 °7067641|6*9030588]7-0377573!7 14041 967 *2233938}7°2930296/7°3530275]15 
 46 6°3483327|6°7108807)|6 *9056829]7 -0396832!7 °1419408}7°224650817*2941006/7°3539604/1 4 
 44716*35'767276-7149586|6°9082913)7-04.16006|7'1434.566/7°2259041]7°2951690/7°35489 1413 
 
 4.8}6°3668161)/6°7189986)6°9108841)7°0435096)7°1449672)7°2271539]7°296234'7'T-3558203}12 
 
 4.916 °3'75770916 °7230013/6°9134615|'7°0454103)'7*1464.726]7°2284.001]7°2972979)7 356747311 
 50|6°3845449]6 °7269675|6°9160237|7°0473026)7° 1497977 7229642717 °298358417*3576723)10 
 51/6 °3931450|6°7308978|6°9 18570970491 868|7°14.94677|7°23088 181729941 64)7°3585954. 
 5216 -4015782)/6°734'7929/6°921 1033)7°0510628)7-1509576)7°2321 1'75|'7°3004.718)}7°3595165 
 53/6 40985076 °7386533/6 92362097 -0529307/7*15244.23)7 °2333494)7°3015246|7°3604356 
 5416°417968616 °7424.79'7/6°926 1 24.117 °0547906)7 "1539221 17°234.5779)7'3025749)'7°3613528 
 
 55|6°425937616°746272'7|6°92861 29)7 -05664.26)7°1553967|7-2358030)7°503622'7/7 "3622681 
 56/6°4337629/6°7500328/6°9310875/7°0584868)7*1568664|7°23'T024.6]7°304667917°3631814 
 5'7)6*4.41449'716°753'7607/6°9335481!7-0603231/7°15833 1 2/7°2382429/7°3057106/7°3640929 
 58)6*44.9002916°757456916°935994817°0621517/7°1597910|7°2394577|7 306750917 3650024. 
 59}6°4.564269/6°7611218|6°9384278/7 06397277 °16124.59|'7°2406691|7°3077886}7 3659100 
 6016 °463726 1/6 764756 1|6°9408473)7-0657860)7°1626960]7°2418771)|7°3088239)7 "3668157 
 
 Cor ww ko D-A1D 0 
 
 LOG, COSINES. 89 Deg. 
 
O Deg. -LOG.TANGENTS. (219) 
 “a OQ’ 1 py a3 4! 5S rp Te Ww 
 
 0 6°4637261|6°764-7562)6 940847517 0657863)7*1626964)7°2418778)7°3088248)60 
 1/4°685574916-470904'7|6 °7683603}6 -9432536|7 067592 1)\7°1641417|'7°2430825)7°3098576)59 
 2\4.°986604.916°4'779666/6°771934.7|6 945646417 0693904! 7165582 117°2442839)7°3108879/58 
 3/5°162696116°484915416°7754800|6 948026 1!7-0711813|7-16'701'78)7°2454819)7°3119158)57 
 415+28'7634.916°49 1754916 °7789966|6-9503928)7-0729649|7°1684488]7°2466767)7°3 129413156 
 5|5+384544.9|6 498488216 °782484.9/6°9527467)7 074-741 2)7° 169875017 °24.78682)7 313964455 
 6|5°4.637261|6°5051188)6°7859455|6°9550879}7 0765 102)7°1712966}7 "24905647 314985 1/54 
 
 115*5306729/6°5116497|6°7893786|6'9574166|7°0782720/7° 17271367 °25024 1 4)7°3160034153 
 815588664916 °5180838]6 °792'7849]6 95973287 080026817 °174.1259]7°2514231)7°3170193}52 
 915°63981'74|6°524424.016°7961646|6°9620368]7°081774417°1755337/7°2526017)7°3180328)51 
 1015°6855'74.9|6°5306729|6-7995 183]6-9643286|7-083515117°1769369]7°253'7771)7°3190440)50 
 11|5°7269676|6°5368332|6-8028462|6 -9666084|7°08524.88)7 *1783356]7°2549492)7°3200528)49 
 12|5°164'756116°5429074|6 806 1489|6°9688762|7°0869756|7°1797298|7 :2561183]7°3210592/48 
 
 -113}5°7995182}6°5488977|6 -8094.266|6 971 1323|7-0886956]7°1811195|7°257284.2)7°3220634/47 
 14|5°8317029|6'5548066]6°8126797|6°9733767}'7 090408817" 1825049]7°2584469|7°3230652)46 
 15}5°8616661)|6°5606361/6'8159087/6°9756096/7°0921153]7'1838858)7'2596066]7°3240648)/45 
 16}5°8896948]6°566388516°8191 138|6°977831 1]7°0938151)7°1852623}7°2607632)7°3250620)44 
 17/5°9160238|6'5720656|6°8222955|6-980041217°0955082)7°186634.5|7°2619167|7°3260570/43 
 18|5°94084'74|6°57'76695|6:825454.016°9822402)7-09'71948)7°1880023)'7°2630672|7°3270496)42 
 
 19]5°9643285}6°5832020/6 82858976 984428 117°098874.9]7°1893659|7°264.214617°3280400)4 1 
 2015 *986604.9|6°588664.9|6°8317030|6°9866050]7'10054.8.4)7°1907252)7°2653590)'7°3290282)40 
 2116°007'794.216°594059916 :834.794.0)6 988771117 °1022156|7°192080217°2665003}7°3300141}39 
 22|6:0279975|6°599388'716 83'78633|6 99092647 '1038'764|7*193431 1]7°2676387/7°3309978/38 
 23/6 °04'73027|6°6046530/6°84.091 10)6°9930710}7'1055309]7'1947'777|7°2687741]7°3319793/37 
 2416 065786116 °609854.2|6 8439374)6°9952052|7°1071790|7°1961202/7°2699066/7°3329585|36 
 
 2516°083514.916°6149938]6 °84694.29|6°9973289]7 '1088210)7°1974.586]7°2710361|7°3339356|35} 
 26|6°10054.82}6°6200733/6°84.99278]6°99944.29)7°1104.567|7°1987928]7°2721627|7'°3349 104)34 
 27/6°1169386|6°6250941|68528923)7°00154.54)7°1120864)7°20012307 *2732863)7°3358831}33 
 28/6 -1327329)6°6300576/6 °8558367|7°0036383)7°1137099|7°201449117°27440'7 1]7°3368536|32 
 29/6°14'79'729|6 °634.9649/6°85876 1 217°0057213]7°1153274)7° 202771 117°2755250]7°3378219/31 
 30/6°162696116°63981'74)6 -86 1 6662)7°0077943]7"1169389}7°2040892/7°2766400]7 338788 1|90 
 
 31}6°1'769366)6°6446163)6°864.55 19)7°0098575)7°118544.417°2054.032)7°277752117°339752 1/29 
 32|6°190724.8]6°6493627|6°8674185|/7°011910917°12014.40]7°2067133]7°278861517°3407140)28 
 33}6*2040888]6°654.0578)6 °8702664)7'013954.6/7°1217378]7°2080195)7°2799679|7°34.16738|27 
 3416 °21'70538]6°658'7027/6 8730957) 7°0159888]7 1233257)7 209321 7|7°2810716}7 "34.263 1 4126 
 3516°22964.29/6 *663298516°8759066]7°0180135]7°1249078)7°2106201)'7°2821725]7°3435870|25 
 36 6°2418774 6667846 1|6°8786995]7:0200288]7°126484.2)7°211914.5]'7°2832706]7'34454.04|24 
 
 37|6°2537'766|6°6723466|6°88 14746]7°0220348]7-128054.917°2132052/7°2843659/7 34549 18123 
 38|6°2653585)6°6768010}6 88423207 °02403 1 5]'7°1296199|7°2144.920)7°2854585|7°3464.41 1/22 
 39|6°2166395]6°681210116°8869721)7°0260191}7°1311793}]7°2157750)7°2865483|7°3473883/21 
 4.0}6°287634.9]6°685574.9|6 °8896949|7°0279977|7°1327339)7°217054.9)7 287635417 °3483334|20 
 4.116+2983587|6 68989636 °8924.008]7'0299673)7-1342815|7°2183296]7°288'71 98]7°3492765]19 
 4.2|6°308824.216°694.1751|/6°8950900|7°0319280]7°135824.2)7°2196014|7°289801 5]7°35021'76|18 
 
 $45/6°31904.33)6°698412116-8977626]7-0338799]7-1373616|7°2208694]'7°2908805)7°3511566|17 
 4416*3290275|6''7026082|/6°9004188|7°0358231|7+1388935]7°222133'7|7°29 19568)7°3520936| 16 
 4516 °3387874|6°'7106764.2|6°9030589}7°037'75'76)7 +] 404.200]7°223394.47°2930304|7°3530286)15 
 46)/6°3483327)6 *7108808|6°9056830|7°0396835]7°14.1941 2]'7°2246514)7°2941015/7°353961 5/14 
 4°7/6°35'76727|6 *714958'7|6°90829 1 4)7°0416009]'7*1434570)7'225904.8)7°2951698)7°3548925]13 
 48)/6°3668161/6°7189987/6-9108842/7:0435099|7 °1449676/7°227154.5|7°2962356|7°3558215/12 
 
 4916°3'757'7109|6°723001416°913461'7|'7°04.541 05)'7°1464730)7°2284.007|7°2972987/7'3567485]1 1 
 50)6°3845449)6 *7269676|6°9160239)7°0475029/7-14'79732|'7°22964.33)7 2983593/7°3576735|10 
 451]6°3931450|6°7308979/6-918571 1)7°04.91870)'7°149468 1]7°2308824)7-2994.173)7°3585965 
 5216 *4015'782)6°734'7929|6-9211034)'7-0510630)'7°1509580)7°2321180/7°3004'727|'7'°3595176 
 53}6°4098507|6°7386534|6-923621 1|'7°05293 1 0)'7-15244.28]7'2333500)'7°3015255|'7°3604368 
 5416 °4179686)6*7424'798)6926124.2)'7:054'7909]7°1539225]7°2345786|7 302575817 °3613540 
 
 55)6°42593'76]6*74.62728/6 9286130} -0566429)7 155397217 °2358036|'7T°3036235|7'3622692 
 56/6 °4337629|6*7500329|6-9310876)7°05848'71|'7°1568669]'7°2370253 7°304.6688)7°3631826 
 57|6°44144.97|6°'753'7608|6 9335489) 0603234)7°1583316]7°2382435/7°30571 15/7°3640940 
 58/6 *449002916°7574570/6'9359950)7'0621520)7° 159791417 °2394583|7'3067517)7°3650035 
 59}6°456426916°'7611219/6-9384280|'7-0639730)'7"16124.64|'7°24.06698]7°3077895)|'7°3659112 
 60 6463726 1/6 °764'7562|6°94084751'7 0657863] 162696417 °2418'77817 -3088248)'7°3668 169) 
 
 A SOae Ui oahe: fs 578 50” |). Sage 5a’ 6a” 159" 
 LOG. CONTAGENTS. 3F2 89 Deg. 
 
 Mm wo 
 
 Slanwune a-r 
 
(220) 0 Deg. — LOG. SINES, Tab. 9. 
 8 ry 10’ 1k 12’ 
 
 0)'7°366815'7}'7°41'7968117°4637255}7°505118117°5429065|7°5776684|7 6098530)7°6398160)\60 
 1)7-3677195|'7°4187716|'7°4644487)7°5057756|7°543.5092)7-5782249)7 6 103697|7 640298359 
 2)'7*3686215]'7°419573'7|'7 465170717 °506432 1/7 °544111217-5787806|7 6 108858}7°6407800/58 
 3/7-°3695216|'7°4203'74.2|7 46589 1 6)7°50708'76)7°544.71 2317579335676 114012)7°6412612157 
 4)'7-3704.198}7:4.211733]'7°466611217°50774.29)7°5453125]7°5798899]7°6119161)'7'6417419|56 
 5}'7'3'713162|7-4219709}'7'4673296)7 5083958} 7°5459120)7°5804.43.5)7°6 1 243047 6422221155 
 6}'7°3'7221071'7°4227670!7 °4680469|7°5090483}7°5465106)'7°5809964)7°61294.40)7 -642701 7/54 
 
 1\'7°3'731034)'7°42356 1'7|7°4687629]7 5096999175471 08417 °5815485)]7°613457117 6431808)/53 
 8}'7°3'139943|'7°4.243549]7°4694.7'78)7°5103506)7'54'77053]7°5821000)7°6139695)|'7°6436593/52 
 9}'7-3'748832]7°4251467|7°4'701915}7°5110002)7°5483015}7°5826508}7°6144813)7°6441373/51 
 1017°375'7705]7°42593'70}7°4'709041)7°5116489]7°5488968]7-5832009]7°6149926)7°6446149|50 
 11]7°3'766559]'7°4267259)7°4716154)7°5 122966)7°54.949 1 3)7°5837503)'7 6155032)7°64509 18/49 
 12)'7°37'75396|7°42'75134)7°4723257/7°512943417°5500850)7 58429 90)7 61 60132)7°6455683)48 
 
 13]'7°3'784.21 41'7°4282995}7-473034'717 513589917 °5506779)7°5848470)7 61 65227|7 6460449147 
 1417°3793014)'7°429084.117°473'7426|7°514234.0]/7°5512700}7'°585394317°6170315)'7°6465196/46] | 
 15}'7°380179617°4298673}7°4744493)7°5148779]7°551861317°5859409]7°617539'7|'7 6469945145 
 1617°381056117°4306491]'7°4'75154.917 515520817 °552451817°'586486917 618047417 °6474689|44 
 17/7°38 1930817 431429517 °4758594)7°5 161628)7°553041417°58'70321|7°618554.4)7'64.794.28)43 
 18}'7°3828038]7°4322085!'7°476562717°516803817°5536303/7°5875767|7°6190609}7 6484.16 1/42 
 
 19!7°3836'750|7°4329861|'7°4772649)7 5174439) 7°5542184)7°588 1 206!7°6195668)7°6488839/41 
 20)'7-384.544.417°433762417°4779659|7°5180830]7°5548057|7 °588663817°620072117°6493613/40 
 91)'7°3854129)7°434.5372|7°4786658/7°5 18721 2|7°5553921|7°5892063]7 6205768) 7 6498331 |39 
 22)'7°3862'789|7°4353106}7'47936467°5 1 93585]7°5559778|7 58974817 °6210809]7*6503043/38 
 23|'7°38714.24]7°436082'7|7°4800623/7 51 99948)7°5565627|7°5902893]7 °6215844|7°6507751/37 
 24/7 °3880050]7°4368534]7 4807588]7°5206302)7°5571 4.69|7°5908298)7 -6220873]7 6512454136 
 
 25|'7°3888658]7°43'76228)7°48 1 4542/7 °521 2646)]7°5577302|7°5913696]7°6225897|7°651 7151/35 
 26|'7'389724.9|'7°4383908)7°4821485]7°5218982)7°5583127|7°5919088]7°6230915)7°6521844)54 
 2'11'7°3905824|7°439 157417 °4898417/7°5225308)7'°558894.517°59244.7317 62359277 6526531133 
 28/'7°3914.3811'7°4399227|%'4835338)7°5231625!7°559475517'°592985 117 '°624.0933/7 6531214132 
 29)'7°3922922|7°4406866]7°4842248)7 °5237933)7°560055717°5935223]7 624.5934!7 6535891131 
 30|7°393] 446/'7°441 44.92/7°484914'7)7°5244.23117°5606352|7°5940588)7 °6250928)7'6540563/30 
 
 3117°3939953/7°442210417°4856035]7°525052117°5612138)7°5945946]7°6255917)7°654.523 1129 
 32/'7°394.84.4.41'7°44.29703]7°486291 3)7°525680117°561791717°5951298]7°6260901}7°6549893/28 
 33)'7°5956918!7°4437289|7°4869779)7 5263073) 7°5623689/7°5956643|7°6265878]7°6554550|27 
 34)7°39653'75|7°444486217°48 7663417 °5269335] 7°56294.52)7°5961981|7'6270850)7'6559203|26 
 35|7°397738 1 6|'7°44.524.21)7°48834.79|7°527558817°5635208)7°59675 1317627581 6|7°6563850|25 
 36/7°3982241)7°4459968]7°48903 13!7°5281833]7°5640957|7'5972639)7°6280777) 7 6568492/24 
 
 3'7\'7°3990650|7'44.67501|7°48971 36|7°5288068]7°564.6698)7'°597795817 6285732) 7 6573130)23 
 3817 °399904.217 44.7502 1]7°490394.9)7 529429517 565243 117°598327017°6290681]7 657776222 
 397400741 8177448252917 °4910750)7°530051 217°5658157|7°5988576|7 629562417 -6582390/2 1 
 4.0)'7°4015'7'78}7°449002317°49 1754.1 |'7°530672117'5663875)7°5993876)7°6300562|7°6587012/20 
 41]'7°40241211'7°449'7504d7°4924.32217°5312920)7°5669585]7°5999 1 6917 °63054.95]7°6591630}19 
 4217403244917 °4504973!7°4931092|7'5319111|7°567528917°60044.5517°631042117°6596243/18 
 
 43]7-4040761]7°45124.28|7°493'785 1|7°5325294)7 568098417 6009735|7°631534.2|7°6600850]17 
 441'7°4.04.905'7)'7°45 19871 |'7°4944600)7°5331467|7°5686672)7°6015009}7°6320258)7 °66054.53/16 
 451'7°405733'7)'7°4527302|7°4951339|7°533763 1]7°5692353)7°602027717 632516817 °66 10052)15 
 4.617°4065601|7°4534'719}7°4958067|7°5343787|7°5698026)7°602553817°6330073|7 -6614645}14 
 AT *4.0'738501'7 4542 1 24)7°4964.784)7°534993417°5703692)7*6030792)7 633497117 °6619233113 
 48]7°4082083)7°4549516/7°4971492/7°5356073]7°5709351|7°603604.0)7°6339865|7°66238 17/12 
 
 491'7°4.09030117°4556896|7°49781 88/7 °536220217°5715002|7 6041 282)7°6344.753]7°6628395)1 1 
 50)7°4.098503]7*4.564263)7°498487517°5368324|7°572064617 604651817 °6349635|7°6632969|10 
 5117°4106689]7°45'71618}7°499155 117°53'74436]7°5726282)7 605174717 6354.5 12)7-6637538 
 5217°41 14860)7°45'78960)7°499821 7/7 °5380540)7'5731912)T'6056970]7°6359384)7°6642103 
 5317°41230 16]7°4586290)7°5004873/7°5386635]7°5737533]7 6062187|7°6364250)'7 6646662 
 5417°4131156)7°4593607)7°5011519]7-5392722)7°5743148]7°6067397|7°6369 1 10)7°6651217 
 
 55}'T°4139282)7 -46009 1 2/7 °5018154)7°539880017'5748755)7°6072602\7°6373965)7 6655767 
 56}'7*414.7392)7-4608205|'7°5024'780)'7°54.048'70)'7°57543.56)7 *6077800]7 63788 15]7°6660312 
 5'1)'7°415548'7|7°4615486)7°5031395}7°541093 1 7°575994.9]7 608299 117 °6383659]7 666485 
 
 LOG. COSINES. 89 Deg. 
 
0 Deg. LOG. TANGENTS. (221) 
 10’ ; 13; 13’ 
 
 1/7 36 77207)7° 4187731 7 “4644506 ‘| 503077 18|'7" 5435119 7 5782280 (as 6103733 7 6403024159 
 2!7 368622717 °4195752)7°4651726)7°506434.3)7°5441138)7'578783'7\7 6 108894)7 -640784.2/58 
 3|7+3695228|'7°4203757/'7°4658934]7°5070899]7°54.4'714.917°5793387)7°611404.9/7 6412654157 
 4l'7*3'704210}7°421 1748)7°4666 1 30)7°50774.4.4)7 5453 1521'7'5798930/7 °6119197/7°6417461156 
 51'7°3'1131741'7°421972417°467331 5)'7°5083980)'7°5459 14'717°58044.6617 °6 1 2434.0)7 6422262/55 
 6|'7°3'7221 19]7°4227685]7 °468048 717 °5090506/7°5465133)'7°5809995]7 61294777 6427059154 
 
 | '1'T*3'7310461'7°4235632]7°4687648]7°5097022|7°54'71111]7:5815517/7°613460717°6431850153 
 81'7°3'7399551'7°424356417°469479717°5 10352817 °5477080)7 582103217 °6139732!'7 -6436635/52 
 9|7+374884.5|'7°425 1482]'7°4'701934/7°511002517°5483049]7 582654017 °614485017°6441416/51 
 10/7:3'75'7'7181'7°425938617°4'7090601'7°5 11651217 °548899517°5832041|7°6149963!7 6446191150 
 11]7°3'7665'721'7°42672°'75|'7°4716173}7 512298917 °5494.94.117'°583753517 615506917 645096149 
 121'7°37'754.0817°42'75150)7°4723276|7°51294.5'7|7°5500878)7°5843022)7:616016917°6455725/48 
 
 13}7°3784.226]7°4283010)7°4730366|7°5135915|7°5506807|7°5848502)7°6165264|7 6460485147 
 1417°3793026|7°4290857|7°473'74.4.517 5 142363]7°551272817°5853975|7 6 1'70352|7 6465239146 
 15|7°3801809|7°4298689]7 4744.5 13)7°5148802)7°551864.0]7°585944117°6 17543517 6469988145 
 16|'7°3810574)7°4306507|7°475 1569)7°5155231|7°552454.517°586490 117-61 8051 1176474732144 
 17/7°3819321|'7°4314311)7°4758613]7°5161651|7°553044.2)7°5870353/7 618558217 °64794'7 1143 
 18]'7°3828051|7°4322101|7°4765646!7'5168061|7°553633117'°5875799)7°6190647|7 6484204142 
 
 1917°38367631'7°432987'7|'7'4772668|7'°51744.62/7°554.291 2)'7°588 1238]7°6195705|7°6488933)/41 
 20]7°384.54.5'7|7°433'764.01'7°4'7796'79]7°5 1808.5417°5548084)7°5886670]7°620075817 6493656/40 
 2117°385413417°4345388|7°4'786678|7°518723617°5553949]7°5892096)7 62058057 6498374139 
 29)7°386279417°4353123)'7°4'793666|7 5 193608]7°5559806]7°58975 1417 °6210847|7-6503087/38 
 23/7°387143'7|7°4360843]7°4800642]7'519997917°556565617 9902926)7 621588217 6507795137 
 24)7°3880063]7°436855 1|7°4807608|7°5206326|7'°55714.9717°590833117°622091 117-65 12497136 
 
 25|]7°3888671|7°43'7624.4|'7°48 14.562]7°5212670|7°5577330)7'5913730)7°622593517°65171 95135 
 26]'7°3897263]7:4383924)7°4821 50517°5219006|7°5583156|7°5919121]7°623095317°6521888134 
 2417°390583717°439 1 590|7°4828437]7'°522.533Q|7°5588974|7°5924.50617 623596517 6526575133 
 2817°391439517°439924317°483535917°523 1649/7 °5594'784|7°592988417 -624.097217 6531258139 
 2917°392293517°44.0688217*484226917'°523'7957|7 5600586] 1 °593525617 624597217 6535935131 
 30|7°393145917°4414.50817°4849 1 68]7°5244256|7°5606380)7°5940621)7°6250967/7-6540608130 
 
 311'7°3939964|7°44221 2117°485605617°5 250545 156121647 7°53945980]'7°625595617 6545275129 
 
 36/7°3982955|'7'4459985|'7'4890334|7°5281858)7:5640986|7'5972673]7°62808 1 6|7°656853'7/24 
 
 3'7|7°3990663]7°4467518)7°489'715717°5288093)7 56467277 °5977992)7 6285771|7- 65731 74/935. 
 38]7°3999055|7°4475038]7 -4903969]'7°529431 9)'7°5652460|7°598330417°629072017°6577807|29 
 39|7°4007431|7°4482546]7°4910771]7-5300537|7°56581 86|7°598861 1|'7°629566417°6582435|21 
 40|'7°4015791|7°449004.0)7°4917562|7°530674.6|7°5663904)7°59939 10/7 °6300602|7°6587057120 
 411'7°40241 3517449752117 °4924.343]7°53129467°566961517°5999203/7°6305534/7°659 1675119 
 42)7°4032463]7°4504990]7°4931113)7°531913'7/7°5675318/7°60044.90)7°6310461]7"°6596288]18 
 
 431'7°4.04.0775]7°45 12446]7°493787217°5325319)7°5681014)'7°60097'70|7°6315382)7-6600896|17 
 4,4)7°404.907117°45 1988917494462 117°5331492)7°5686702)7°6015044)7°6320298|7°6605499]16 
 4517 °405'7351|7°4.527319)7°495136017°533'765717°569238317 602031 1)'7°6325208)7°6610097)15 
 46|7°4065616]7°453473'7|'7°4958088]7°53438 1 3!7°5698056}7°6025572)7°6330113)7-66 146901 4 
 4°7|'7°4.073864)7-4542141]7°4964806]7°5349960)7°5'703'722)7°6030827|7 633501 2/7 *6619279/13 
 48)7°4082097|7°4549534)7°49'71513]'7°5356098}7°570938 1|7°6036075}7°6339905|7°6623863)12 
 
 49]7°4090315)7°45569 13]7°4978210)7°5362228)7°5715032)7°604131'7|'7°6344.793!7 662844111 1 
 50|'7°4098517|7°456428 117 °498489 7/7 °5368349|7°57206767 °6046553)7 °6349676|7 °6633015110 
 51]'7°4106'703]7°4571635}'7°4991573}7°5374462)7°57263 1 3)/'7°605 178217 °6354553)T 6637585 
 
 517°45'789'78]'7°499823917 °5380566|7°573 1 94.2''7°6057005|7°6359424)7°6642149 
 53)'7:4123030}7°4586308]7°5004895]7°5386661)7°573756417°6062222)'7°6364290)7 6646709 
 54)7°41311'71)7°459362517°501154117°5392748/7°57431'79)7°6067433)7*6369151)7°6651263 
 
 55}'7°4139296/7°4600930)7-5018176/'7°5398826)7°5748786)7°6072637)7 *6374006/7 6655813 
 56)'7'414'74.06}'7'4608223)7-5024802)'7°5404896)7°5754386]7 *6077835)7 °6378856|7 6660359 
 5'1'7°415550117°4615504)'7°5031417)7°5410958)'7°5759979|7-608302717 *6383'700!7 6664899 
 58)'7°4163581)'7°4622773)7-5038022/7°5417011)7°5765565)7 °608821 3)7°6388539|7 6669435 
 59)'7°41'71 646)'7°4630030)'7°504.46 1 8)'7°5423055)7°5771 14417 °6093392)7 °63933'73|7°6673966 
 60 7°4179696 7°46372731'7°5051203)7°542909 1|7°5776715)|'7°6098566!7°639820 176678492 
 
 Orepw WO & Or on-10 0 
 
 LOG. COTANGENTS. 89 Deg. 
 
(027) 0 Wer. LOG. SINES. Tab. 9. 
 
 NS ee, a, en) ee eet Ce 
 
 0}7°66784.4.5]'7°6941733|7+'7189966|7°7424.775}7 76475377 °7859427|7-8061458]7 8254507160 
 1)7-66829671'7 694.5988)" "71939867 °7428583)7'7651154)7°'7862872)7 -806474'7|7°8257653159 
 217°668'74.84}7 °695024.0/7°71 98001)7°'7432388)7 °7654.769!'7 786631 5|'7-8068033]7 8260797158 
 37-6691 996}7°695448'7/7°7202013)7°7436189]7*7658380)7°7869755|7-8071317)7°8263938|57 
 41'7:6696503]7°6958730|7°7206021|7°7439987|7 °766 1 989]7°7873 192)'7-8074599]7°8267077156 
 5}'7-6701006|'7°6962969\'7 "721002617 "744.378 1|7°7665594|7°78'76627/7-8077878]7 8270214155 
 6|'7°6'705504]7°6967204|7°7214027/7 744°7573)7°7669197)7-7880058)7°8081154)7°8273348)54 
 
 71'T-6'70999817-69'714.3517°721802417°74.51360]7°7672797/7 °7883488]7 808442817 8276481153 
 8|'7-6'7144.86]'7°69'75662!7°7222017|7°7455145]/7°7676393}'7-78869 1417808769917 827961 1/52 
 9|'7-6'7189'70|'7-69'79884|'7 "722600717 "745892617 76799877 -789033'7|7°8090968]7 828273851 
 10}'7-6'72345017°6984.103}7°722999317°'746270517°768357'7|\'7°7893758)7T 80942357 828586450 
 13}7°672792.5|7-698831 7/7 °7233976)7°7466479|7°7687 1 65|'7*'789717'7|7°8097499|7'8288987/49 
 1 2)'7°673239517-6992528)7°723'1955)7°74'7025 1)7°7690750)7 +7900592)7°8 10076117 °8292108)/48 
 
 13!7-6736861|'7-6996734)7°7241930|7°'4'7401 9]7°'7694.3321'7-'7904.005|7-8104020)'7'8295227147 
 1417-6'74.132917-7000936|7°724590217°74'7'7784|7°'76979 1 017-790741.5/7°810727'7|7'8298343/46 
 15}'7-674.5779|7-7005134|'7°7249869}7°7481546]7°'7'701486]7-7910823]7-8110531|7°8301458/45 
 16|'7-6'750231]7°'7009398|'7''7253834]7°'7485304|7°'7705059|7-791 4228/7-81 13'783]7°8304570|44 
 17)7-6754.678|'7-'7013518]'7-°725'7794)7°7489059|7°7708629|77917630|7-3117032|7'8307680/43 
 18}'7-6'759121|'7-7017'7047°7261'752|7°74.9281 117°7712196|7-7921029)7°8120279]7°8310787|42 
 
 19]'7°6763559]'7°'7021886|7°7265705)7°7496560)7°7715'760)7 °7924.426/7°8 1 23524!7 8313893141 
 20}7°6767993]7°'7026064|7°7269555)7°7500306/7°771 932217 °7927820)7'8 1 26766] 7°8316996/40 
 21)7°6772422'7°7030238|7°7273601|7°7504048)7°7722880!7°7931212)7°8130006/7°8320097/59 
 22)'7°6'7'7684'1|'7°7034.4.07/7°727'754.417°750778'7|'7 772643517 7934.60 1|7°8 135243] 7°8323195/38 
 ‘4 23)7°6781267|7°'703857317°7281483]7°7511523!'7°772998817°7937987|7 8 136478) 7°8326292137 
 2417 °6785683]7°7%04273517°7285419)7°7515255!7°7733537|7°7941371|7° 813971 1|7°8329386|36 
 
 2517°6'7900941'7 °704.6893}7°728935 17°75 1898517-773'7084|'7 794475217 °814.294.1|7°8332478/35 
 26|7°679450117°705 104717 °'7293279)7 "7.522 } 1/'7°'1'74.06281'7°'7948130)7°8146168]7°8335568|34 
 27117°67989041'7°105519717°7297 20417 °752643417°774.4.169/7°7951506)7°814.9394 7 °8338656|/33 
 28]'7°6803302)7-'705934.3|7°7301 1 25]7°'753015417-774'770'7|'1"7954.8°'791'1°8152617|7 8341741132 
 29/7 °6807695]'7°'10634.85]7°7305043)7°75338'71/7°775 1 24.9|'7°795825017'°815583'7]7'°8344825/31 
 30/'7°68120841'7°'7067623|7°7308957\'7°753'758417°7'7547741'7°'7961617|7°8159055| 18347906130 
 
 31)'7°68 16469)7°'7071'75'7|7°7312868)7 754129417 -775830317°7964.983|7°8 1 622'71|7°8350985|29 
 32\'7°682084.9)7°'7075887|7°7316776)7°754500117°'7761830|7°7968345]7°81 654841 7°8354062/28 
 33)'7°682522417 708001 4)7°7320679|7°7548705)7°776535417°7971705!'7°8168695|7°8357136)27 
 34)'7°6829596!7°7084.136]7°7324579)7°7552406|7°7768874|7°7975063|7°S1'71 904|7°8360209/|26 
 35|7°6833963]7°7088254|7°7328476)7°755610417°7'7723921'7°7978418|7°8175110]7°8363279|25 
 36|7°6838325)7:7092369)7°7332369)7°7559798)7 +7775907|7°798177017°81783 14178366347 |24 
 
 3'1|'7-6842683!7-70964.80)7°7336259)7°7563490)7-7'7794.20)'7°'7985120}7°8181516|7°8369413/23 
 38/7°684703'717 °7100586}7°7340145]7°75671'78)'7°7'782929]7-7988467|7 8184715] 7°83724.7'7|22 
 39)7°6851387|'7°7104.689]7°734402817°7570863)7-7786436|7°7991811|7°8187919|7°8375538/21 
 40/7°6855732)7-7108788}7°734:7908}]7°7574545)7-7789939|7-7995153)/7°8191 106|7°8378598|20 
 41)7°6860072/7°71 12883/7°7351783)'7°7578224)7 °7'793440|7°7998493|7°S194298!7°838 1655/19 
 42)7°6864409/7°'7116975|7°%355656/7°758 1 900)'7°7'796938/7-8001830]7°81974.88]7°8384710}18 
 
 43/'T*6868'74 117-7 121062)7°7359525]7°7585572|7-7800434)7°8005 164|7°8200676|7°8387763|17 
 4417 °6873069]7 7125 146]7°7363390)7°758924.2)7-7803926|7°80084.96|7°820386117°8390814\16 
 45|'7°68'7'7392)7 712929517 '°736'7252)'7°7592908)'7°7807416|7°801182517°820704.3)7'°8393863]15 
 46/7°6881 71 1)'7°7133301|7°737111117°759657217°7810903|7-8015151|7°821022417 8396909]14 
 47|'7*688602617°71373'13|7°7374966!7°7600232)7 +78 1438'7|7-80184'7517°82134.02|7°8399954)13 
 4817°6890337|'7:7141442|7°73'78818!7''7603889/7°7817868)'7°802179'7|7°821657817 840299612 
 
 49)7°6894.643)7-714.5506)7°7%382666|7°7607543/'7 +7821 3471780251 16]7°8219751|7°8406036|11 
 50/7°689894517-714.9567|7°738651 1|7°761119417+7824.822|7-8028432]'7°822292291'7 840907410 
 51]7°6903243)'7*'7153624)7°7390353|7°7614.842)7-7828295|7-80317461'7 822609 1}7 8412110 
 5217 °69075361'7 °715'7677|7°73941 9117 °761 8487) °7831765|'7°80350581'7°8229258)'7 8415144 
 53]7°6911826)7°'7161726)7°7398026]7°7622 12917 -7835233|7 803836717 *82324.22|'7°8418176 
 5417°69161 1117-7 1657'72|'7°'74.0185'7|'7°7625768)7-7838697|7 8041673!7°8235584|7 8421205 
 
 5517°692039217°71698 1417 °74.05685|'7°7629403!7 +7842 159)'78044.97'7|7°8238'743)7°84.24.233 
 56}7°6924668]7-7173852|7°74095 10|'7*7633036|7°784.5618!7+8048278]7°8241901|7°8427258 
 5'7/7*6928941)'7°'71'77886/'7 -74.13331)7°%636666|7-7849075]7 *8051577|7°8245056)7°843028 1 
 58)7°6933209)7°'718191'7)7°74.17149}7°'7640292)7+78.52.528)7 805487317 "824820917 8433302 
 59)7°693'7473)7°7185943)7°'742096417-'164391 6)7-7855979/7 80581 6717 °8251359)7'8436321 
 60 7°6941'733)'7-'7189966|7°7424-1'75|'7°764153'1|7°78594.2'1|'7 8061 458]7°825450717 8439338 
 
 4x 1 ai’ | a0) 3g | 38’ | 37 (| a 
 
 LOG, COSINES. 89 Deg. 
 
 ie ee D-10 
 
0 Deg. LOG. TANGENTS. (223) 
 
 ) 19’ 10’ 
 
 ~017-667849317 -6941786|7-7190026|7-742484.1|7°7647610]7°7859508|7°8061547|7-8254604/60 
 117-66830141'7°6946042)7°719404.5]7°7428649]7-7651228]7-786295417 8064836] 8257750159 
 2)7-6687531)7°6950293]7°7198061)7°74324.54)7 "765484317 *7866396)7°8068 1 23/7 8260894158] 
 317 -6692043|7°6954541]'7-7202073]'7°7436255]7°765845417'7869836|7°8071407|7°8264036157 
 41'7°6696551/7°6958784|7°720608 117 °744.0053]/7°7662063)7°7873274)7 807468817 8267175156 
 517°6701053/7°6963023)7°7210086|7°7443848]7-7665669]7°7876708)7 80779677 8270312155 
 61'7°6'705552|7°6967258]7 721408717 °744.764.0)7°' 766927 1)'7 "7880140178081 24417 8273.44.65 41 
 
 "17+6910045|7°697148917°721808417°745142817 °76728'71|7°788356917°8084.518)7°8276579153 
 81'7-6'714.5341'7°697571617°72220781'7 745521 217°%67646817 78869967 °8087789]7 8279709152 
 9|'7:6719018|7°697993817°722606817°74.5899417°768006 117-78904.20|7'°809105917°8282837|51 
 1017°67234.98!7°698415717°7230054/7°746277217°7683652|7°7893841|7°8094325|7 828596250 
 11}7°6727973]7°6988371 |7°7234037|\7°746654717- 768724017 7897259] 7 °8097590)7 8289086149 
 121'7°67324.43]7°699258217°723801617° 714°7031 917 *769082517°79006'75|7'810085 117°8292907}48 
 
 1317°6736909|7°69967881'7°724.199117°747408 717 °7694407)7°790408817°8 10411 117°8295326]47 
 1417°67413'71|7°7000990]7-7245963|7°7477852|'7°%697986]7-79074.9817'°810736817°829844.3146 
 1517°6'74.582°717°7005189)7°7249931)7°748161417°770156217°7910906|7°8110622]7°8301557]4: 
 2 76750279 7°'1009383}7°7253895}7°'7485372/7°7705135)7°79 14.31 1]7°81138'74)7°8304669144 
 7°675479'7/'7'°%013573/7-7257856|7 7489 12817°7708'705)'7°79177131'7°811712417°8307779143 
 
 ial *67591'70|7°7017759}7°7261813]7°7492880) 7°77 1227217792 1113]7°8120371|7°8310887/42 
 
 1917°6763608|7°702194.117°7265176'717°7496629]7'°7715836]7°7924.510]7°8125615]7°8313999]41 
 2017 °676804.217°7026119)7°7269717|7°75003.74)7°7719398]7°792790417°81 2685817 °8317096|40 
 2117°67724'7117°7030293]7 727366317 °75041 1717°7722956)7°'7931296]7°813009817 °8320197|39 
 2217677689617 '°7034463]7 °727760617'°750785617'°77265 1 21'7°7934685|7'°813333517°8323296)38 
 23]'7°678131'7°7038629/7 728154517775 1159217 '°T730064 a 7938071]7°8136570)7°8326399137 
 2417 °6785733/7 704.2791 17°728548117°751532517°773361417 7941455) 7°8139803]7°832948 7136 
 
 25]7°6'79014417°704694917°72894.1317°751905417°773'71 6117°794.4836| 7°8143033]7°8332579135 
 26]7°679455117°7051 10317°729334.217°752278017°7740705]7°794.821 51 7°814626117-8335669134 
 on iT" pipes 7: 1055253}7 7 72972677 r He 6504)17°774.4.24617°7951590|7°8 14948617 8338757133 
 
 3917'68514.38)7°7104746}7°7344092]7°7570934|7°7786514|7°7991898]7°8188006]7°8375641421 
 40)7°6855783)7°710884.6]7°7347972|7°7574616|7°77900 1 8]7°799524.017 819120117 °837870 1/20 
 41)7°6860124)7°7112941)7°7351848]7°7578295|7°77935 19]7°7998579]7°8194393}7°838 1758119 
 421'7°6864460)7°7117032)7°7355720|7°758197117°'77970 1'7|7°S001916]7°8197583]7°8384813}18 
 
 43)7°6868792)7-7121120/7°7359589|7°7585644)7°78005 1317800525 1)7°8200770|7 838786717 
 144)7°6873120)7°7125203)7°73634.55|7°7589313]7°780400517 800858217 °820395617°839091 8116 
 45/7°687744.4)'7 °'7129283]7°736731717°7592980|7°78074.95)7°801191217°8207139]7°8393966]15 
 46]7°6881763)7°7133359]7°737117617°7596643}7°78 10989)7°8015238/7°821031917°8397013}14 
 47)7°6886078)7°7137432)7°7375031|7°760030417°78 144.66]7°8018563}7'82134.9717°8400058]13 
 487 °6890389|7°7141500|7-7378883]7 *7603961|7°78 17948} 7802188417 '8216673}7°8403100}12 
 
 4917°6894.695)7°7145565]7°7382731)7°7607615]7°7821426}7°8025203|78219847|7°8406140}11 
 5017°6898997)7°7149625/7°7386577|7°76 11 266}7°782490217 °8028520|7°822301817°8409179]10 
 5117°6903295)7°7153682'7°7390418)7°7614.915]7°7828375}7°S031834)7°8226187)7°8412215 
 5217°6907589)7°7157736}7°7394257)7°7618560]7°7831845}7 8035 146]7 822935417 8415249 
 53/7°6911878)7°7161785]7°739809 1|7°7622202)7°783531 317 °8038455|7°82325 18178418280 
 9417°6916163]7°7165831]7°740 192317 -7625840]7°7838778/7 -8041761}7°8235680)7°8421310 
 
 55|7°692044.417°716987317 *'740575 1)7'76294.76]'7 °784.224.017 *804.5065|7°823884017 8424338 
 5617°692472117°7173911)7°'7409576|7°7633109)7°7845699)7 -8048366|7°8241997)|7°8427363 
 5717°6928993!7°7177945|7°741339717°7636739|7'7849 155}'7°8051665|7°8245153}7°8430387 
 58/7°6933262/7°7181976]7 "741721517 °7640366]7°7852609}7 80549627 °8248305]7 8433408 
 59)7°6937526|7°7186003/7°7421030/7°764398917 *7856060)7 *8058256|7°8251456)7 8436427 
 60 7°694.178617°7190026}7°74.2484.1|7 °764761017°7%859508)7: ia al 7825460417 °84394.44 
 
 Or wOho D-A1M 0 
 
 LOG. COTANGENTS. ~ 89 Deg. 
 
(224) O Deg. LOG. SINES. Tab. 9. 
 
 1)7-844.23.531'7°8619517|'7°8789736/7 8953534791 11378|7°9263685|7°941083117°9553153)/59 
 21'7°844536617°8622410/7°8792517/7 895621 2/791 13960792661 7917941324 117°9555486|58 
 31'7°844.83'77|'7°8625300/7°8795297|7 8958889|7 9] 1654217°9268671)7°941565117°9557818/57 
 41'7°84513851'7°8628189|7°8798075/7°896 156417 °91191 2117-9271 162/7°941805917 9560 14.9}56 
 517°845439217°8631075|7-8800850]7'896423'7\7 9121 69917 92736517 -9420465}7°9562478)|55 
 6|'7'845739617°863396017-°8803625|7*8966909)7 91 24.276]7°9276139]7 942287 117°9564806)54 
 
 "1'7*84.60398]'7°8636843]7°880639717°8969579)7-9 1 2685 11'7°9278626)7 °94.25275|7 956713353 
 8|'7°8463399]7°8639723/7-°8809167|7°8972248]7 912942517 -9281 1111794276777 °9569458)52 
 9}7°84.66397|T°8642602/7°8811936]/7°8974.914]7°9131997]'7°9283595]7°943007917-9571782|51 
 10|'7°8469393]'7'864.5479|7°8814703]7°8977580|'7 913456717 9286077|7 °94324.'79|7°9574105|50 
 11|'7°84'72387|7°864.835417°881'74.69]7°8980243|7-9137136]7°9288558|7 °9434877|7°957642°7/49 
 12|'7°84'753'7917°865 1228|7°882023217°8982905|7°9139704|7°929 103717 943727517 -957874.7/48 
 
 13}7°847836917-°8654.099]7°882299417°898556517°914.2269]7°9293516|7°943967117°9581067)47 
 1417°8481357/7°8656968(7 °8825754)7°898822417°914483417°929599917 944206617 °9583385146 
 15}7°8484.34317°8659836]7°882851217°8990881|7°914.739717°92984.67)7°94444.5917°9585 702/45 
 16|'7°8487326|7°8662702|'7°883 1269/7 '°899353617-914995817°930094117 944685 1]7°9588017)44 
 1'717°84.90308|7°8665565|7°8834.023/7°8996190/7°915251817°9303414)'7'944924917°9590331143 
 18]'7°84.93288]|7'86684.2'717°883677617 899884217 °91550761'7°9305885)7'°945163117°959264.5|42 
 
 19]'7°84.96265|7°867128717°883952817'90014.93)7°915763317°930835417 945401 917°9594956/41 
 20|'7°84.9924.117°86'74.14.5|'7°884227'717°90041 411'7°9160189]7°9310823]7°9456406|7°959726714.0 
 2117°8502215|7°8677001|7'884.5025|7'9006789]7 916274317 °931328917 945879217 9599576139 
 2917°850518617°867985617°884777 1|7°9009434)'7°916529517°931575517°9461176|7°9601885138 
 23/7°85081561'7°868270817'°88505 1517 °9012078/7°916784617°9318219]7°9463559]7'9604199/37 
 2417 °8511123]'7°8685559|7°885325817 °9014.72117°9170395|7'932068217 -9465940|7°9606497136 
 
 2517'851408817°86884.08]7°8855999}7°9017362/7°917294.37°932314.3]7°9468321|7°9608802|35 
 2617°85170521'7°869125417°8858738)7°9020001!7°9175489]7°9325603]7'9470700]7 9611105134 
 2°71'7*852001317°869409917°88614'75|7°9022639|7°91 7803417 °9328061]7'°9473077|7 9613407133 
 28]'7°8522973!7°869694.2]7 886421 1!7°9025275|7°918057817°9330518]7 947545417 9615708152, 
 291'7°852593017°869978417 '886694.5|7°9027909|7°9 1831 2017°9332974I7 947789917 -9618008)31 
 30]7°8528885|7°8'702623]7°8869677)7 °903054.2/7°9185660|7°9335428]7 9480203]7'9620306/30 
 
 311'7°853183917°8'70546117°88724.0'7|7 9033 173|7°918819917:933788117°9482575|7°9622603/29 
 32/7 °8534790|'7°8'708296|7°8875136|7°9035803|7°9190736|7:9340332|7°9484.94.6]7°9624899/28 
 3317°8537'739]'7°8'71113017'°8877863/7 "903843 1}7°9 19327917 -9342783]7 948731617 °9627194)27 
 3417 °854068'7\'7°8'71396217°8880589]7 '9041057/7'919580717°934.5231|7'9489685|7 9629487126 
 35]7°854363217°871679217°888331 2|7°9043682/7°919834017°934.7679]7 9492052|7°9631780|25 
 36}7°85465'75|7°8'71962117°88860341'7 904630517 °9200871/7°93501 2517 9494.41 817963407124 
 
 3°7\'7°854.95 1'7|7°8'7224.4'717°8888'75417 °904892'717°9203401]7°93.52569]7 °9496783|7°963636 1/23 
 3817°85524.56|7°87252'7217°889 14'73]7°905154'7|7°920593017°9355019/7 9499146]7°963864.9/22 
 39}7°8555393/7°872809517°88941 90/7 90541667 °92084.57/7 935745417 °9501508]7°964093'7/21 
 40)'7°8558329/7°8730916|7°8896905)7°9056783/7 921098317 :935989517°9503869|7°9643223/20 
 411'7°8561262/7°8733'735|7°8899618)7'9059398)7°921350717 -9362334|7°9506229|7'9645508/19 
 4,2|'7°856419317°8736552|7°8902330)7 906201 217 °921603017'°9364'772|7°950858'7|7 9647799118 
 
 43]7°856712317'8'739367|7°8905040|79064624)7°921855117°9367208]7°951094.417°9650075|1'7 
 4.417 °85'70050]7°8'74218 1|'7°890774.9'7 -9067235|7°9221071|7°9369643]7°9513300]7°9652356}16 
 4517°85729767°8'744.993}7°89 104.55/'7 -9069844)7°922358917°93%2077|7 951565417 °9654637|15 
 46]'7°8575899}7°874'7803|7°8913160)7-90'724.5 1|7°9226106}'7°9374509]7°95 18008}7°96.569 1 6 14. 
 47]7°85'78821]7°875061 1}7°8915864)7-9075057|7°9228621]7°93'76940]7°9520360]7 "965919413 
 48]7°858174.0)7°87534.1'7|7°8918565)7°9077662]7 9231 135}7°937936917-9522710]7°9661470|12 
 4.91'7°8584.658}7°8756222)7°892 1 265/7'908026517°9233648]7°9381798]7°9525060]7°9663746]1 1 
 
 50}'7°858'75'7417°8759025|7°8923963)7 908286617 9236 159]7°9384.224|7°95274.08)7°9666020}10 
 51}7°859048'7}7-8'76 182617-8926660)7*90854.66]7 °9238668}7°9386650]7°9529755]7 9668293 
 
 co tO 
 
 ——— | | | | 
 
 LOG. COSINES. 89 Deg. 
 
0 Deg. LOG. TANGENTS. (225) 
 , 1) 24 25’ 26 ayn 28’ 29’ 30’ 
 "0(7°8439444|7-8616738/7°8787077|7°8950988)7 91089387926 1344|7°9408584|7-9550996160 
 117*8442459]7'8619632|7°8789861|7°8953668]7°91 1 1522/7-9263840]7-94.10999}7-9553330]59 
 217 +844.54712|7°8622525/7 879264217 8956347) 7°91 1410517°9266333]7 -9413407]7-9555663158 
 31'7+8448493]7°8625415|7°879542917°8959023|7-91 16686]7'9268826]7°9415817|7°955 7995157 
 4|7°8451499]7'8628304|7°8798199]7°8961699|7°9119266]7°9271317|7°9418225|7°9560326/56 
 517+8454498}7°8631191/7-S800975]7°8964372|7°9 12184.4|7°9273807|7'94.20632)7'9562655155 
 6|7*8457503}'7°8634076|7'S80375017'8967044)7°91244.2117°9276295|7 942303717 9564984154 
 
 "17*84.60505|7°8636958/7°8806522)7 8969714179 126996)]7°927878217°942544117°9567310}53 
 8}7°8463506|7°8639839|7°8809293]7°897238317°91295'70/7°928 1 26717 °94.2784.417"°9569636|52 
 917°8466504|7°864.271 9|7-88 1 2062/7°8975050]7'°913214.2)7°928375 1|7°94.3024617°9571961/51 
 10}7°8469500|7'8645596/7°8814829/7°8977715|7°9134713/7°9286233)7°9432646]7°9574284150 
 11]7°8472494)7°864847117°8817594]7°8980379)7°9137282|7 928871417 9435045] 7 *9576606|49 
 19/7°8475487)7°865134417°8820358]7°8983041)7°9139856/7°9291194)'7°943'744.2|7°9578926148 
 
 13]7°8478477|7°865421 617°8823120/7°8985701 17914241 6}'7°9293672|7 943983917°9581 946147 
 14)7°8481465|7°865'708517°882588017 °8988360|7 914498017 '9296149]7 944223317 9583564146 
 15|7°84844.51)7°8659953|7'°8828639]7°899101-7/7°9147543]7°929862517 °9444627|7°958588 114.5 
 16|7°848743517°8662819|7°8831395|7°8993673'7°915010517°9301099|7°9447019]7°9588 197/44 
 177°8490416}7°8665683]7°8834150)7°899632'7|7°9152665|7°9303571|7°944.9410)7'959051 1143 
 18}7°849339617°866854.5|7°8836903]7'8998979)7°9 15522417 °930604317 945 1800)7°9592825/42 
 
 19]7°8496374|7°8671405)7°8839655|7°9001 630)7°915778117°9308512)7°94.5418817-9595 157) 41 
 _120}7°849935017'8674.2631'7°884.24.0417°9004.27917 9 16033617°931098 117945657517 959744714 
 
 21]7°8502323}7°86771 20)/7°8845152|7°9006926|7'9162890/7'93134.48/7°9458961|7°9599757/39 
 2217°8505295)7°867997417°884.7899]7°9009572)\7°9165443/7°931591317°9461345|7'9602065138 
 23}7°8508265]7°868289717 8850643}7°9012216)7°9167994|7'931 837817 °9463728]7 9604373137 
 9417°8511232)/7°8685677|7°8853336|7°9014859/7°9170543)'7°932084017°94661 1017 9606678136 
 
 25)7°851419817°8688526|7°8856127/7°9017500)7°9 17309 1}7°9323302|7-94684.91|7-9608983135 
 26|7°8517161)7°8691373|7°88.58866|7'°90201 39/7 °9175638|7:932576217'947087017'9611287134 
 2'7\'7°8520123|7°8694218]7°8861604|7°9022777)|7°9178183|7°9328220]7 °947324817°9613589133 
 28|7°8523083]7°8697062!7°8864339]7 °9025413/7°9180727|7:9330678/7 9475624 |7°9615890)32 
 29]7°852604.0/7°8699903]7°886707417°9028048|7°9 183269)7°9333133]7'°9478000|7'°9618190|31 
 30|7°8528996)7°8'702'743]|7°8869806|7°9030681|7°9185809|7°9335588)7 948037417 9620488130 
 
 31}7°853194.917°870558017°8872537|7°9033312/'7°9188348)7'933804117'9482746]7°9622786129 
 32/7°8534.900!7°8708416|7°88'75266/7°9035942|7°9190886/7°934.049317°948511817'°9625089/28 
 33|7°853'785017°871 1250]7°8877993|T°9038570]7°9 1934.22)7°934.294.3/7°9487488/7 9627377127 
 3417°854079'7/7°8'714.082|7°888071817°9041197/7°9195957'7°934539917 948985617 9629670) 26 
 35]7°8543'743/7°8'716913]7°888344.2|7°9043829/7°91984.90)7°934.'7839]7°949229417 9631 963)25 
 36|7°8546686/7°871974117°888616417°90464.45 |'7°9201029)7:9350286|7 °9494.590|7'9634254/24 
 
 37|7'8549628)7° 8722568] 7°8888885|7*904.906'7|7°9203552)7'9352730]7'9496955|7'963654.4]23} 
 38]7°8552567|7°8725393|7°389 1 603|7°9051687|7'°9206081]7'93551 747949931 9|7-9638833)22 
 39}7°8555505]/7°8728215|7°8894320|7°9054306)7'9208608|7°9357616|7'9501681}7°9641 121/21 
 40|7°855844.0/7°8731037|7°889'703617°9056923/7°9211134|7-9360057|7 "950404217 *964.34.08!20 
 41/7°8561374)7°8733856]7'8899749]7'9059539/7°921365817°9362496]7 9506402|7 9645693] 19 
 42)7°8564305)7°8736673)7'890246 1 |7°9062153|7'9216181|7°9364934|7°9508760}7 9647977118 
 
 45)7°8567235|7'8'739489|7°890517117°9064765|7°9218702|7°9367370|7°9511118]7'9650260}17 
 44'7°8570163)7*874.2303|7°8907880|7:9067376|7°922 1 222/7°9369805}7°951347417°9652541| 16 
 45)7°85'73088)7°87451 15]7°8910587]7'9069985/7°9223'74.1|'7-9372939]7'951582817°9654829]15 
 46}7°8576012)7°8'747925]7°891329217-9072593|7-9226258)7'9374672/7'9518 18217-9657 10114 
 
 LOG. COTANGENTS. 3G 89 Deg. 
 
(226) O Deg. LOG. SINES: ~ - | i= 9. 
 ely Bah Vn 3a! 
 0} 7*9688695}7-9822334}7-995 1980/8-0077867|8 020020718 031919518 04350098 0547814 60 
 1} 79690960 }7-9824.527|7°9954108)8-0079934)8 02022 17/8 0321150)8 0436913)8 0549670159 
 2/7°9693220}7 982671817 -9956235/8 00820018 02042268 032310518 043881618 0551524158 
 3]7°9695475]7°9828909]7 995836 1|8 00840668 *0206234|8 032505918 0440719|8 0553378157 
 4|7°9697736}7'9831098)7 99604878 ‘0086 131)/8°0208242)8 03270 1 2\8 0442621 |8°0555231156 
 
 5}7°9699993}7-9833287}7°996261 1/8 °0088194)8 02102488 °0328965}8 -0444522|8°055'7084155 
 6|7°9'702248/7 °98354'74]7 -9964734]8 °0090257|8 021 2253/8 "03309 16|8-0446422/8 05589351544 
 
 7|7°970450317 -9837660}7°9966856/8 "00923 18]8-0214258/8-0332866|8 044832 1/8 °0560786)53 
 8|7°9706756/7 -9839845]7°99689 77/8 °00943'7918 021626118 03348 16/8 04502208 °0562636/52 
 _9}7:9709008/7-984.2029/7:997109'7|8 0096439|8-0218264|8 033676518°0452 1 17/8 °0564485151 
 10)'7-9711258}7-9$44.212|7-997321 68 -0098497|8 -0220266|8:0338713|8 04540148 0366333/50 
 11]7°9713508)7-9846394/7°9975334)8°0100555|8°0222267/8 0340660)8 "04559 10/8 056818 1/49 
 412)7°9715756)7-9848574)7°9977451)8 -0102612)/8-0224267|8 °0342606|8'0457805|8°0570028)48 
 
 13|7-971800417-985075417°9979566|8 0 104668|8°0226266|8-0344.551|8:045970018 0571874147 
 147972025017 -9852933}7 998168 1/8°0106722|8 -0228264)8°0346495/8°0461593)/8 057371 9/46 
 15|7°9'72249517-9855110]7°9983795/8 -01087'76|8-023026 118034843918 04634868 °0575563/45 
 16|7°9724738]7-9857286|7°9985908)8 01 10829)/8-0232257/8 03503828 04653788 0577407144 
 1'7/'7°972698 117°9859461)7°9988020/8 01 12881)8"0234.252|8-0352323|8:0467269/8'0579250/43 
 18|7°972922217 -9861636/7°9990130|8°0114932|8°023624.7|8 035426418 °0469159)/80581092/42 
 
 19]'7-°97314631'7-9863809|7'9992240/8-01 16982|8:023824018°0356204|8°04'71 048)/8 0582933141 
 20/7°9733'702\7°9865981|7°999434.9]8-01 1903 118 -024.0233|8°0358 1 4.3/8 047293718 0584774140) 
 2117°9735940]7°986815117°99964.56|8°0121079|8 "0242299418 03600828 -04-74825|8 "0586614139 
 29/7-9738177|7-987032117°9998563]8°0123 1 96/8 -024.4.21 518036201 9]8°04-76'71218"05884.53|38 
 2317°974.04.121'7°9872490|8-0000669]8"01 25172/8:0246205|8°036395618 -0478598|8 0590291137). 
 2417-9794264'7|7 987465818 0002773|8-012721 7/8 °024.81 9418°036589218°048048318°0592128/361 
 
 25/7-9744880/7-9876824}8-0004877)8"01 29261 |80250182|8-0367826)80482366)8°0593965)35} 
 26|'7-9'747113|7-9878989|8 0006979|80131304|8-02521 69|8-0369760|8-0484251|8°0595801|34 | 
 27/7-974934417-9881154|8°0009081|8 °013334-7|8°02541 5518-0371 693]8-0486134/8°0597636|33 
 
 98|7-9751574|7°9883317|8"001 1 181]8-0135388|/8"02561 40}8"0373626]8 048801 6/8°05994.70|324 
 29/7+9753802/7°98854'79|8 0013281]8013'7428}8-025812518°0575557/8°0489897/8"0601304)3 14 
 30/7-9756030]7-988'7641)8*00153'79|80139468|8°0260108|8 0377488180491 778/8°06031371308 
 
 311'7:9'758257|7°9889801|8 -001 7477/8014] 5506/8 "026209118 °03794.1'7/8°04.93657|8*0604969|29} 
 32/7°9760482|7°989 1 960|8°0019573/8°014554.3|8-0264.0'79|8-0381346|8°0495536/8°0606800)28) 
 33)7-9762'706]7°98941 1'7|8°0021669]80145580|8°0266053/8 03832748 °049'7414|8°0608630/271 
 3417976492917 989627418 °0023'763/8°0147615|8°0268033|8 038520118 0499991 |8°0610460/26 
 35|7:9'6715117°9898430|8°0025856|8°014.9650/8°02'7001 9180387 128/8-050! 167/8°0612289]25}: 
 361'7°976937217°9900585|8°002794.9|8 °0151684/8-0271990|8°0389053/8-0513043/8"061 41 17/24 
 
 37|7-9771592|7°9902738|8'0030040|8-0153716|8°0273967 8+0390975|8-0504918|8°0615944123 
 38)7°97'7381017°990489118°0032131/8°0155748|8°02'75943|8°039290118°0506799/8°06177711228 
 39|7°977602817°990'704.3]8"0034220|8°0157779|8-02'7791 918 °0394.82418-050866519°0619597121: 
 4.0!'7-9778244.1'7°990919318°0036308/8'015980818°027989318"0396746|8"051053'719 0621429120); 
 4.1|7°9780459]7-99 1 1342/8°0038396|8"0161837|8"0281867)8'0398667|8°0512408/8'0623246/19}, 
 4,217-9782673|7°9913491|8004.0482]8°016386518°028383918"040058818°05142'79|8"0625070 18} 
 
 43}7+9784886|7-9915638|8-0042568|8°0165899|8-028581 1/8-040250718-0516149|8°0626899|1"74. 
 4.417°9787098|7-9917784|8°004465218 016791 8|8*0287789|8 -0404426|8-0518018|S°0628714|16)) 
 45|7-97893097-991999918"0046735|8-01 6994.3|8-0289752|8-040634318+0519886|8°0630536|15}, 
 46}79791518]7-9922073|8-004881 8/8-0171967|8-0291791|8-0408260|8-052175418-0632356|141- 
 4'7\7-9793726|7°992421 6|8-0050899|8-0173991 |8-0293689|8-0410176|8°0523620/3-06341'76|13}, 
 48|7-9795934|7-9926358|8°00529%9|8-01'76013|8-0295656/8°041 209918 -0525486)8°0635995|12), 
 
 4.9|7+9798140|7-99284.99]8 005505918 01 78034/8°0297623 8-0414006|8-0527351 8° 063781311} 
 50\7° mieceto a 9930659 8: OOSTAIT 8 0180055 8° 0299588 8 OH1S920 8° tein 8°0639630 10} 
 
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 53|7* 9817945 ue 9947720 8 “0073729 8. 0196184)8- 0915280 8° 0491198) -0544101 8 0634143 
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 LOG. COSINES.,. 89 De. 
 
0 Deg. LOG. TANGENTS. (227) 
 
 32’ ag 1 ad 1 35" 360" 37’ 38’ 30’ 
 0}7 ‘9688886 THO822598/7 9952192 B0078Q92 8 MAWS45 a: 0319 4.6/8 Mane 8 VS ane 1160: 
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 217° 9693408 7:9826919|7°9956448)8° 0082226 8° “0204465 8° 0393357 8: “0439089 8: “0551804 38 
 3}'7+9695667}7 98291 10)7'9958574|8 008429218 °02064.73/8 1032531 1/8°0440985)8°0553655)57 
 417°9697925!7°9831299/7°9960700|8 0086357}8 020848 1/8 :0327265)8 0442887) 80555512156 
 5}7+9'700182/7°9833488]7°9962824|5 008842018 °0210487/8°032921 7/8 °0444788) 8°05 57364155 
 6)'7°9702438}7°9835675|7°9964.94'718 -0090483/8.021 2495/8 °0331169)8 0446689805592 16154 
 
 + 7/'7°9704692)7°9837862|7:996'707018 '0092.54.5/8°02144.98|8°03331 20!8 0448588! 8°0561067153 
 8/'7:970694.517°984.004'7|7°996919118-0094606/8°0216501/8°0335069|8-0450487'8 0562917152 
 9)7°9709198)'7°984.2931|7°997131 1|8°0096666/8°0218504/8°033701 8/8 0452385 8056476751 
 
 '10}'7°971 14.4.9]7°984.4.4,1417°9973430]8 -0098725}8-0220506|80338967|8 0454282 8°0566615150 
 
 11/'7°9713698]7°984.6596|7°9975548|8°0100783}8:0222507/8'034.0914|/8-0456178'8°0568463/49 
 12/7°9715947)7 98487-7717 °997'7666|8 '010284.0)8°0224.507|8°0342860|8°0458074'8°0570310148} 
 
 413/7-97181941'7°9850957|7°9979782|8-0104896|8°0226507|8'0344806|8 0459968 8°0572156 4 
 14}7°9'72044.1]7°9853135}7°9981 897/801 0695 118°0228505|8:0346750|8 046 1862 8°0574009/46 
 15]'1°9722686|'7:9855313]7°9984011/8:010900518'0230502/8-0348694|8° 0463755 8°0575846}45 
 
 -$16}7°9'724°3017-985'74.9017 9986 1 2418°0111058}8°0252499/8°035063'7|8" 0465647, 8°0577690|44 
 17}7°972°71'73}'7°9859665]7'°9988236|8'0113110)8°0234494|8°035257918 046 7538) 8: 0579534143 
 18)7°972941417°9861839]7°999034.6)8°0115161/3°0236439|8- eee 8: 0469429 8:0581376|42 
 19/7-9'731655|'7°9864.013}7°9992456|9'011721 1/8 °0238483]/8°03.36460)8° 0471318/8° 0588217/41 
 20)'7°9'733894|7-9866185|7°9994565|8'0119260/8°02404.75/8° akeiee 8° 047320718: ‘0585058/40 
 21179736 13217+9868356|7°9996673|8°0121308/8"0242467/8'0360338/8-0475095|8°0586898139 
 22)7°9738369]7°987052617°9998 78018 0 1 233.56/8 02444588 °0362276)' 
 
 30/7+9756224|7-9887847|8"0015598)8"013969918-0260353|8:037 '1'146|8°0492050|8- 0603423|70 
 
 3] TON5S45] 7*9890007/8°001'7696|8°0141738]8 0262336|8:03796'76/8°0493930|8 060525 5)2¢ 
 $32|'7°9760676]7°98921 66|3'0019799|8°0143'77518°0264.318|8:0381605|8"0495809|8:0607087128 
 5317°97629011'7°9894.324|8°0021888}8 014.581 218 :0266299|80383533/8°0497687|8:0608918|27 
 3417-97651 24}7-989648 1|8°0023983|8:0147848/8-0268279|8 038.546 1|3'0499564|8°061074.8126 
 35\7°9767346)7°989863'7|8°0026076|8°014.9883}8-027025818:0387387|8°050144118-0612577/25 
 36|7°9769567|7°9900792)8 0028 1 69}8-015191 6|8°0272236/8 0389313|8:0503317|8°0614405}24 
 
 377 -9771787|'7+9902946]8 '0030260]8°0153949]S°0274.213}8 039 1238/8 -0505192/8-061 6233} 23 
 3817°9'7'74.006)7°9905099]8 003235 118015598 118°0276190/8:0393162/8°0507066)|8° 0618060 22 
 39}7°97'7622417-990725 1/8 0034441 {8 ‘015801 218:0278166/8°039508518°0508939/8 0619886121 
 40}7°977844.0)7-99094.01/8°0036529]8 °0 16004218 -0280140/8°0397007/8°0510842}8 0621711/20 
 - {4117°9780655]7°991 155 1/8°0038617|8°0 1620711802821 14/8-039899818-0512683/8-0623536119 
 +217 °97828'7(0}7°9913699|8°004.0703|8 01 64.099]8-0284.087/8 040084918 051455418062 29359118 
 
 4317°97850831'7-99 158478 0042789]8 01 66127|8°0286059|8-0402768]8°05 1 6424}8-0627 1.89] 17 
 44)7°9787295|7-9917993|8 °0044.8'74/8°0168153|8°0288030/8-0404687|8°0518294/8-0629005/16 
 4.5}7°9789506|'7-9920138|/8:0046957|8'0170178|8-0290000}8 -04.06605|8 0520162|8-0630826|L5 
 146]7°9791715}7°992228318-004904.0)8'0172203/8 029 1 969/8-0408522|8-0522030|8 0632647114 
 4/7/'7°979392417 +9924.4.26|8°0051121/8°01'74.226)8-0293938/8-0410459|8-0523897|8 0634467113 
 481'7:9796131|7°9926568|80053209/8:017624.8/8-0295905|8-0412554/8°0525763|8° 063628612 
 
 491'7-9't98338}7°992870918-005528218°01'78270|8:0297872)8 041 4269/8 :0527628/8 063810411 1 
 50}'7°9800543)7°993084.9]8 :0057360|8'0180291|8*0299838)8 041 6183/8 °0529493|8 0639922110) 
 51/'7+980274'7}7-9932988]8 '0059438]8 01823 10/8°0301802|8-0418096/8°0531356/8-0641739) 9, 
 $52/7°9804.950)'7°99351 26/8 0061514)/8°0184329}8 -0303766)8-0420008/8-0533219]8°0643555} 8 
 531798071 52}'7°9937263|8°0063590|8"01 8634:7/8-0305729/8-0421919)8°0535081|8°0645371} 7 
 D417 19809353}7-9939399|8-0065665|8°0188364|8-0307692/8°0423829|8-0536943|8-0647185] 6 
 
 551%*9811552!'7-9941534/8-0067738]8 :0190397|8-0309653/8-0425739|8°0538803|80648999 
 . [56)'7°9813'751|7-994.366718 00698 1 1/8:0192394|/8-031 1613/8 -0427648|8°0540663|8 0650812 
 
 5717°981 594817 -994.5800|8°0071883]8 01 94408|8°031 35'73|8°0429555|8 054252218 0652625 
 58!7+981814.5}'7-9947932|8-0073953/8 01 96422|8-0315531/8-043 1462/8 -0544380/8 0654436] ° 
 59/7-982034.0)7-9950062|8-0076023|8 01984348 -0317489]8-04.33369|8 05462318 -06,56247 
 60}7+9822534]7 99521 92)8-0078092|8 0200445803 19446|8-04.35274|80548094)8 0658057 
 
 "ap | a6 toe" Tae | aa" | ae" 
 vee LOG. COTANGENTS. 3G2 89 Deg. 
 
(228) O Deg. LOG. SINES. | Tab. 9. 
 
 218°0661381|8-0768526/8°087309 1/8 0975 198/8°1074958/3"1 172478 8:1267856 8*1361183/58 
 318°0663188]8:0770290|8:0874813/8:0976879|8°1076601 8°11'74085 81269428 8°1362722|57 
 418°0664995/8°0772052/8-0876534/8 °0978560|8*1078244/8°1 17.5691)/8'1270999 8°1364260)56 
 518*0666801 807738 15|8°087825418°0980240/8*1079886!8. 1177297 8-1272570|8°1365797/55 
 6|8*0668606|8:0775576|8 ‘087997418 0981920/8-108152818°1178902/8-1274140)8°1367334\54 
 
 713067041 1|8°077733'7|8 088 1692|8-098359918-1083169/8°1180507|8°1275710|8'1368871|53 
 815°0672215|/8:0779097|8 088341 1/8°098.5277|8" 108480918: 11824 11/8°127727918°1370407/52 
 9/8-06'74.018/8°0780856|8°08851 28/8 °0986955|8°108644.9/8-1183714 8°127884.8/8°1371942)51 
 10|8°0675820|8°0782614|8°088684.5|8°0988632|8*108808818-1 185317 8°] 280416|8°1373477|50 
 1118:067762218:07843'72|8°088856118:0990309|8°1089726|8'1186919,8*1281983|8°137501 1/49 
 12|8-06'79423/8:0786129|8:0890277|8-099 19848‘1091364/8°1188520 8'1283550/8°1376545/48 
 
 13|8°0681223/8:0787886|8:0891991|8°0993659]8"1093001/8°1190121/8*1285117/8°1378078|47 
 14/8+0683022|8-078964 1 |8°0893706/8 09953348" 1094638)8°1191722/8'1286682 8°1379610|46 
 15]8°068482118°0791396|8"089541 918:0997008|8"10962'74\8°1 193322)8°1288248/8°1381 143/45 
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 17|8°06884.16/8:0794.904/8-0898844|8-1000353)/8°1099544/8°1196519,8°1291376|8'1384205/43 
 18|8:0690212/8:0796657|8'090055518*100202518°11011'78/8°1198118 8°1292940|8°1385736/42 
 
 19)8-0692008/807984.09|8-0902266/8*1003697/8-1102812)8:1199715,8°1294503)8 1387265 )/41 
 20|8-0693803/8-0800161/8-0903976/8*1005367|8"110444.5|8°1201312,8°1296065)8"1388795/40 
 21)8-0695597|8-0801912/3:0905685)/8*1007037|8"1106077|8"1202908 8+1297627/81390324)39 
 22|8-0697390|8-0803662/8:0907394|8"1008706|8"1107709|8*1204.504'8: 1299188 /8°1391852|38 
 23|8-0699183)8'080541 1\80909102|8-10103'7518°1109340 8-1206099|8"1500749 8°1393380|37 
 24)8°0'700$75|8 08071 60)8'0910810)8°101204.3/8"1110970/8.1207693 8*1302309|8"1394907)|36 
 
 2518°0702'766|8:0808908|8:091251618°1013'71018'1112600/8-1209287,8'1303869|8°1396434)35]_ 
 2618:070455'718°0810655/8:091422218°101537718°1114229 8-1210881/8°13054298 8°1397960|34 
 2718°0706346|8°08124.01|8:0915928/8-1017043/8°1115858/8°121 24.74 §: 1506986 /3:k00048% 33 
 28]8:0708135|8°0814147)8'0917632/8°1018709/8°1117486 8°1214066)8°1508544 8°1401011)/52 
 2913-0709923)8 -0815892)8°0919336]8-1020374|8"11191 13]8"1215657)8°1310101|8'1402535)/31 
 30}3°0711'711|8°081'7637|8092104.0/8°102203818"1 12074.018°1217248/8°131 1658/8" 1404059/30 
 
 3113°0'7934.98|8-0819380|8:0929'74.3]8+102370118'1129366/8'1218839,8°1313215)|8°1405583/29 
 3213°0715284|8°0821 123|809244.4.5/8*102536418°1123999|8-12204.29/8'1314770|8"1407105/28 
 33|8°0717069/8-0822866/8'09261 46]8° 102702718" 1 125617/8°122201 8)8°1316325|8*1408628/27 
 34|18+0'71885418°0824607|8 092784718 -1028688/8*1127241/8*1223607/8°1317880|8'1410150|26 
 35|8:0720637|8 0826348 /8:092954.7|8"1030349|8°1128865|8°1225195)8°1319434)/8 "1411671 
 36|8°072242 1|8:0828088|8-0931246|8°103201018"1130488/8"1226782/8:1320987|8'1413192/24 
 37/8:0724.203/80829828)8-093294.5/8-1033669)8"1 1321 10)8-1228369|8-1322540|8°141471 2/25 
 3818°0725985|8°0831567|8°0934.643|8°1035328|8"1133732|8°1229956|8°1324.093|8"1416232)22 
 39|8°0727'765|8°0833305|8°093634.018"1036987/8+113535418") 2391541/8°1325644|8°1417751/21 
 4.018°0729546|8°083504.2\8°0938037|8:1038645|8*11369'74/8°1233127|8-132'7196|8°1419270|20 
 41}80731325|8-0836779|8 09397338" 1040302|8°113859518" 1234'71 1/8°1528'746|8*1420788|19}. 
 4.218 073310418 08385 15|8°094.142818*1041959|8°1140214|8'1236295|8.1330296|8'1422306|18 
 
 4.318°0'734882/8 08402518 °09431235/8'1043615|8°1141833|8°1237879|8*1331846|8*1423823)17 
 4418-073665918°084198518°09448 17|8-104.5270|8"114345 1|8°12394.62|8"1333395/8"1425339|16 
 4518-073843618'0843'719|8°0946510]8-1046925|8"114.5069|$°124104.4)8°1334.943/8°1426855|15 
 46|8°0740211|8-084545218°0948203|8- 10485'79|8-1146686|8°1242626|8° 1336491 |8 142837114 
 4718°0741986|8°0847185|8°0949895|8*1050239|8'1148302/8"1244207|8"1338039|8-1429886|13 
 4818°0'%43'761|8:084891'7/8°0951587|8"105188518"1 14991 818°124.578718-1339586|8°1431400|12 
 
 4918-074.553418°0850648/8°0953277|8*1053537|3°115153418°1247567|8 1341 13218"°143291 411 
 50|8°074730718°08523'79|8 :0954.968]8*105518818°115314818°124894'7|8°1342678]8"1434427|10 
 51]8-°0749080]8-0854.109|8:0956657|8°1056839]8*1154762]3°1250526|8°1344.223|/8"1435940 
 $2]3-075085118+0855838|8-0958346|3-1058490|8-115637618*125210418°134576718° 1497453 
 53{8+075262218 08575668 096003418" 1060139]8+1157989]8°1253682|8*1347311|8°1438964 
 54{80754392|8085929418 096172 1|8°1061788|8"1 159601]8"1255259|8° 134885518 ° 1440476 
 
 5518°0756161/8°0861021|8°0963408/3" 10634378 °1161213|8'1256836]8:1350398)8°1441987 
 56}8°0757930/8°086274'7/8 -0965094|8°1065085|8°116282418-125841 2]8°1351940]8°1443497 
 57|8°0759698}8 -08644.73|8 09667808" 10667328: 116443418 1259987/8*1353482)8*1445006 
 58}8°0761465|8'0866198/8-0968465]3'1068378]8-1 1660448 1261562/8°1355023|8* 1446516 
 59]8:0'763231|8 086792218 097014918 °10'70024|8°1167654|8"1263136]8°1356564|8"1448024 
 
 Slo nmrmareu aArAML© 
 
 —_—————— | SS | |__| | 
 
 LOG. COSINES. 89 Deg. 
 
0 Deg. «LOG. TANGENTS. | (229) 
 Al’ A2/ 
 
 “08: 8-0658057 8-0765306 By 08699 70 8° nbesane va cg 8° "1169634 8° 1265099 8° spin deh 60 
 1|8:0659866 8:0767071 
 2/8 -0661675,8:°0768835 
 3/8 -0663483|8°0770599 
 418-0665290 8:0772362 8:0876859|8:0978901 |8°1078601|8°1176064/8°1271389|8°1364667 
 5|8°0667096 8°0774125'8°0878.579|8-0980582|8'1080243|8"1 177670|8:1272960|8 1366205 55: 
 6|8-0668902'8:0775886 8-0880299|8:0982261|8"1081885|8°1179276 8 "1274531 /8 1367742) 54: 
 
 713-067070718'077764'7|8°0882018/$70983941|8-1083526|8°1180881|/8"1276101/8°1369279153 
 818-067251118°077940718-088373718:098.5619]8"1085167|8'1182485|8"127767018-1370815)52 
 918-067431418:0781167!8-08854.55|/8-0987297|8" 1086807|8"1184088/8°1279239|5-1372350)51! 
 1018°0676117|8:0782926|8-088'71'72|8°0988975|8°108844.6]8"] 185691)8"1280807|8°1373886] 50; 
 1118°067791.9|8-0784.684'8-088888818°099065 1/8:1090085]8'1187294)/8'1282375)/8°1375420149 
 1218-0679720|8'0786441'8-0890604/8:0992327|8"1091723}8"1188896)8128394.2'8' 137695448 
 
 1318°0681520|8°0788198/8-0892319|8:0994003|8"1193361]8"1190497)8"1285509/8"1378488]47) 
 14/8:0683320|8°0789954/8:0894.033|8°0995677}8"1194998|8-1192098|8*12870'75)8°1380020)46 
 1518°068.5118/8°0791709|8-089574'7|8:099735118"1 19663418" 1193698)8-1288641/8'°13815535/45 
 1618:0686911|8°07934.6418-0897460|8'0999025|8'1198269]8:1195297/8"1290206/8°1383085|44 
 1'718°0688'71418-0795218'8°08991'7218"100069818'°1199904|8"1 196896|8"1291770|8 138461 6]43 
 18|8-0690511|8'0796971|8-0900884|8"1002370|8"1101539|8+1198495)8"1293334/$°1386 147149 
 
 1918-0692306|8:079872318-090259518" 1004041 [871 103173]/8"1200092)8*1294897/8°13876774.1: 
 20|8°0694102|8°08004'75|8-0904.305]8" 100571 2/8°1 104806/8"1201689)8"1296460)8 "138920744 
 21|8°0695896|8°0802226/8-0906015/8'1007382)8"11064.38]8"1203286/8° 1298022)8°1390736139 
 221806976908 °0803976|8-090772418*1009052/8"1108070|8"1204882)/8*1299583/8-1392264135 
 23|8-0699483|8°0805726/80909432|8:1010721|8°1109702/8-12064-'77/8"130114.4/8°1393792134 
 2418°07019'75|8°08074'7518°0911 140/8'1012389|8"1111332)8°1208072/8°1302705|/8'1395320136 
 
 2518°070306618°0809223|8:091234.7|8-1014057/8°111296218*1209666|8" 130426518" 1396847135 
 2618-0704.85'7|8°081097018-09 14.55318°101572418°11 1459218°1211260)8°1305824)8°1398373/34. 
 27|8-070664:7/8°081271'7|8-0916259|8°1017390|8"1116221|8'1212853|8-1307383|8°1399899138 
 28|8°07084.36|8°0814463/8°09 1'7964/8°1019056/8°1117849]8"121444.6|8 "130894118 °1401425|39 
 29|8°0710225|8'0816208/8°0919668/8- 1020721 |8°1119477)8°1216037 8:1310498 8°1402949)31 
 3018-0712012/8°0817953/8°09213'72|8:1022386|8°1 12110413 °1217629|8'1312056)8° 1404474130 
 
 31}8:0'713'799|8°081969'7|8-09230'75|8°1024049|8°1122730|8°1219219}3°1313612\8-°1405997\29 
 32]8°071558618°082144.0|8°0924.77'7|8'1025'713/8'1124356/8°1220810/8°1315168)8:1407521/28 
 33|8°07173'71|8 08231 83}8°09264'79|8° 102757518 +1 1259818 +1222399/8°1316723)8-1409043/27 
 34|8°0719156|8°0824.925|8:0928180/8°1029037|8"1127606|8"1223988|8"1318278/8:1410566/26 
 35|8:0720940|8°0826666|8°0929880/8" 1 030698)8"1129230|8°1225577|8'°1319833/8:1412087/25 
 36|8:0722723/8°08284.06|8-0931579|8°1032359]8-°1130853|8°1227164)8°1321386/8°1413608|24 
 
 37|8°072450618°0830146|8:0933278]8-1034.019|8°11324.76|/8°1228752)8'1322940)8°1415129/23 
 38|8°072628818-0831885|8'0934977|8°1035678/8'°1134.098/8-1230338]8°13244.92)8-1416649|29! 
 39}8°0728069|8-0833624|8°093667418-1037337/8'1135720/8*1231924|8*132604.4|8°1418168)/21 
 4018 -0729850|8°0835361|8'0938371|8"1038995/8"1137341/8°123351018'1327596|8-141968%20 
 4118:0731629|8°0837098|8-0940068|8"104.0653/8°1138961|8°1235095|8'°132914.7/8-1421206)19} . 
 4218°07334.08|8°0838835|8°0941763|8"1042309|8"1 14.0581|8"1236679]8°1330697/8'14.22724)18 
 
 43/8°0735186|8'08405'70)8:09434.5818"1043966|8"1142200|8" 1238263/8*1332247/8 1424941117 
 44)8-0736964|8°0842305|8°094515318°1045621|8'1143819]8°1239846|8'1333796|8°1425758)16 
 145|8°073874118°0844039|8-094.6846|8°104'72'76/8°114.543'7|8"12414.29/8'°133534.518°14272 74115 
 46|8°074.0517|8°084.5773}8°0948539|8"10489318'114'7054/8° 124301 148°1336893]8°1428790)14 
 14718 -074229918-084'7506}8:0950232|8°1050584/8" 114867 1|8°1244.592|8°1338441|8-1430305/13 
 48}8°0744.06'7|8 :0849233|8-0951993/8*105223'718"115028718°124617318'°1339988/8°1431809I12 
 
 4918-0'74.584113°0850969|8:095361418"1053890|8"1151903)8°1247753|8'1341535)8° 143333411 
 50}8°074:761 418°0852700}8°0955305|8°1055542)/8'1153518|8'1249333]8 134308 1|8° 1434848 10 
 51|8°0749386|8°085443018 0956994/8°1057193)8°1155132)8"1250912)8°1344626)8°1436361| 9 
 52/8°0751158/8-0856160]8°0958683}8°1058843/8°1156746/8°1252491|8- 134617 1|8 1437874 
 53)8°075292918°0857888|8°0960372|8° 10604938 "1 158359|8"1254069|8°134'7715|8-1439386 
 5418075469918 °0859616/8°0962060/8*106214.2|8°1159972|8°1255646|8* 134925918" 1440897 
 
 55|8°0756469)8 -086 1344/8 :0963'747|8 06379 1/8°1161584,8°1257223)8'1350802)8-1442408 
 5613°0758238)8 0863070|8-0965433|8+1065439)8°1163195)8"1258'799)8"1352345)/8 1443919 
 57|8:0760006]8 -0864'796|8 09671 19|8:1067087/3"1164806|8"1260375|8*1353887/8 * 144.5429 
 58]8°076177318°0866522|8°096880418: 1068733)" 1166416)8*1261950)8°1355429|8:1446938 
 5913 °0'763540]8 '0868246|8°09'70488]8"1070380|8°1168025)8°1263525/8°1356970|8°1448447| | 
 60 3°0765306|8°0869970|8°09721'7218°1072025|8"1169634|8°1265099]8*1358510|8:1449956). 
 
 pr | | es | | | 
 
 LOG. COTANGENTS. 89 Deg. 
 
(230) O Deg. _ LOG. SINES. Babs, 
 
 1)8°1451040/8°1540552/8°1628255|8:"1714.223/8°179852118°1881213|8°196236018-2042019159 
 2/8°14.52.54'7/8°1542028/8°1629702|8" 17156418 1799912}8-1889578/8"1963'700|8 2043334158 
 3}8°1454054|8°1543504|8°163 1 $49/8°1717059)|8°1801303/8-1883943/8"1965039|8-2044649]57 
 - 418°14.55560|8°1544979|8°1632594)8°17184'77/8°1802693/8: 1885307/8°1966378)8°204.5963}56 
 | 5]8°1457065|8°154.6454/8°163404.0)8°1'719894/8-1804083/8-1886670)/8°196771'%|8°2047277155 
 } 618-14585'70|8°1547928/8°1635485)8°1721310/8*1805472/8*1888034|8°1969055|8 "2048591154 
 
 718°14600'7518* 15494028: 1636929|8'1722726|8° 1806861|8°1889397|8°1970399) 8°2049905153 
 8}8°1461579)8°1550876/8"1638373|8°1724142}8°1808250|8°1890759/8°1971729/8°2051218152 
 9/8°1463082/8"1552348|8° 16398 1'7/8°1725557|8'1809638}8°1892121/8'1973066]8°2052530}51 
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 28/8°1491549|8° pele 3: 1667151 g. 17.52359|8°183592'7|8°191791 7/8" 19983878 °2077593}32 
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 31/8°149602'7|8°1584625|8°167145218°1'756576|8°1840064/8'1921976|S’20023'79| 8-208 1306] 29} 
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 34/8°1500500/8'1589003|8<167574.8/8°1'760789|8"1844196/8 1926032|8*2006353)8 "208521 6|26 
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 3'7|8*1504968/8"1593386|8*168004.0|8 * 1764998)8°1848325/8"1930083|8°2010330)8 2089 121/23 
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 42|8+1512406|8'16006'74|8'168'7183/8°177:2003|8"1855197/8"1936827|8°201 695 118°2095623]18 
 
 43|8°1513891|8°1602130/8"168861 1|/8°17734.03|8°1856570|8" 19381 '75|8°2018274/8°2096929/17 
 44/8°151537'718° 1603585|8-1690038|8"1774802/8°185794.3|8°1939529|8°2019597|8°2098201/16 
 4.5]8°15 16862|8°1605040/8°1691464/8*1776201|8°1859315|8"1940869|8"202091 9|8-2099520) L5 
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 48/8°1521314)8°1609403/8°1695740/8*1780394|8'1863430|8-1944907|8:2024885/8 210341 9/12 
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 "1525761 |8°1613761|8°170001 2/8°1784584|8°1867540|8'°194894] [8°202884.3/8-2107309} 9 
 
 ‘LOG. COSENES, - 89 Deg. 
 
0 Deg. LOG. TANGENTS. 
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 rie a dn SRA Gs1 Hib ob alb 1 HONEDAs? RAND EAI TOA TERR Onk Te bolda 
 
 I He 14514648 11580993 8: 11698715 8°1714'701|8°1799018 3: 1881730|8*1962896|8°2042575|59: 
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 318°1454.47818"1543946|8°163160918°1717538)8° 1801800|8"1884460}8°1965576|8°2045206|57 
 418°145508418°15454.299|8°1633055|8*1718956|8" 1803191 |8°1885824|8° 196691 5|8°2046521/56 
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 818°1469004|8°1551319|8°1638835|8"1'724625]8"1808749)8°1 89 1278|8°1972268|8°2051776)52 
 918+146850818"1559'799|8-1 64097918-1726038)8°18101 37/8" 1892640)8"1973605|8°2053089)51 
 30/8-146501 1|8°1554265|8* 1 64172218" 1727453] 8°1811525|8°1894002/8'1974.949)8°2054401}50 
 11]8+1466514)8°1555737|8 16431 65|S-°172886g|8" 181 2913/8" 1895363|8°1976278)8°2055714)49 
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 16}8°14'74020|8°1563090|8" 16503'72/8-1735934|8°181984.4|8" 19021 64|8°1982953)8°2062268)44 
 17|8*4475519|8 1564559|8°1651819/8"1787346|8° 182 1229/8" 1903523)8"1984287|/8°2063578143 
 ] 1818°1477018]8- 1566028/8-1653251|8*1738757|8" 18226 15/8" 1904881|8°1985621|8°2064887,42 
 
 19}8°14'78517|8°1567496|8 1654690|8* 17401 68/8" 182399718" 1906239|8*1986954)8-2066196)41 
 20|8-148001 5|8°1568964)8° 1656 128]8" 1'74.15'79|8" 182.598 1)/8"1907597/8°1988286/8-2067505)40 
 2718°1481519|8°1570431|8" 1 657566}8"1742989|8" 1 82676.4)8° 190895418: 19896 19)8°2068813)39 
 99|8-1483009|8°15'71 898]8°1659004{8*1744.398]8"182814.6)8° 191031 1/8°1990950/8°20701 20/38: 
 93|8°1484.506|8°157336418° 166044118" 174.5807|8°1829529)8" 19 11667}8" 199928 2/8*2071428137 
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 95|8*1487497|8°1576295|8° 16633 1418-1 '74862418-183229918-°1914379|8" 199494318 -207404 1135 
 96|8°1488992|8°1577'759|8°1664-74.918-1750039|8*183367318"1915734|8:1996273/812075348)34 
 
 9718*14.9048'7|8°1579224|8-1666185|8°1'751430|8-1835053|8-191'7088!8"1997603|8*2076653|33 
 
 98|8-1491980|8°158068'7/8°166761 9}8°175284.6|8°1836433|8°1918442/8°1998933/8-207795 9132 
 20|8°14934'74}8°1582151|/8-1669054/8*1'754252|8-1837813|8*1919796/8°2000269/8°2079264)31 
 30|8°1494967|8°1583613/8°167048'7/8°1755658}8° 18391 92/8 ° 1921 150/8*2001590)8"2080568/30 
 
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 48|8*1521751|8'1609858|8- 169621 4/8-1'780887|8°186394218"1945439|8-2025435|8*2103985|12 
 
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 52|8*1527681)8°1615669/8-1701911|8°17864'74|8°1869423|8-1950818|8-2030716|8"2109171 
 
 53/8°1529162|8° 16171 21|8*1703334|8'1787870|8"1870799|8-1952161|8*2032035|8°21 10467 " : 
 454/8-1530643/8°1618572)8-1704756/8-1789265/8-18721 61|8"1953505|8°203335418-2111762 6 . 
 
 55|8°1532123/8-1620029|8-1706178|8°1790659|8°1873529|8-195484818°203467218°2113057| 5 
 56|8°1533603|8*1621472|8-1707600|8 1792054|8°1874897|8-1956190|8"2035990|8°2114351| 4 
 
 57\8°1535082|8"1622929/8- 170902118: 17934.4'7|/8°1876264)8-195753218:2037308|/8°2115646| 3) 
 
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 LOG. COTANGENTS, 89 Deg. 
 
(232) O Deg. toe. stnzs. | 1 Deg. Tab. 9. 
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 5/8*21254.07/8°2202155,8:2277570}8°235 16978 °2424.58018°2496260)8 2566775) 8°2636164|55 
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 13]8°2135719|8°22129286|8°228752718°236 1 48618°2434.206|8'250572818 257609118 '264.5332147 
 14|8°213'7006|8'2213551/8°228877018°2362'708|8 °24354.08/8°250691 1|8°257725518°26464.7 7146 
 15/8°2138293|8'2214815/8°2290013/8°2363930]8*2436609/8°2508092|8'25784.17/8*2647621/45 
 16|8°2139579|8°2216079)]8°2291255]8°2365151|8°243'78 10)8°25092'7418°25795 8018-2648 766)44 
 17|8°214086518°221734.3}8°2292497| 8° 236637218 °243901 118:2510455|8°2580742/8°2649909]|43 
 18|8°214.215118*221 8606/8°2293739|8°2367593/8°244.091 2/8°2511636/8°2581904/8°2651053)42 
 
 “[19]8°2143436}8°2219869)8'2294980/8 23688 1 3/8 °244141 2/8°2512816|8°2583065|/8°2652196)41 
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 2218 °214.7290|8'2223656)8°2298'701/8°23724'72/8 -244.501018°25163.56/8°2586548| 8° 2655624138 
 2318214857418 °222949 1'7|8°2299941/8°2375691 |8°244.6209)8° 2517535]8°258'7709| 8°2656766|37 
 2418°2149857/8°22261'78/8'2301 181/8°23749 10/8 °244.74.0'7|8 25187148 °2588869! 8°2657908)36 
 
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 3418°2162671|8°2238769|8°2313556|8°238707718'2459373'8' 2530485 |8°2600452|8°2669308|26 
 35}8°2163950|8*224.0026|8 23 14792)/8-2388292|8°2460568/8°2531661|8°2601608/8*26704.4.6/25 
 36{8°2165229|8°2241283|8°2316027/8°2389506|8°24.6 1762/8 °2532836|8-2602764|8°267 1585124 
 
 37|8°2166508}8°2242539/8 231726218 °2390720/8 24629578 2534.01 1|8*2603920|8°2672729/23 
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 4.0|8:217034.1|8°2246306/8" 23209658 °2394361 8-2466597/8°2597533 8°2607387/|8 2676134120 
 41\8°21'71618]8°224.7561 |8°2322198}8°2395574 8°-2467730)8-2538706 8°260854.1/8°2677271]/19 
 4218 ° 2172895822468 1 5|8*232343 118239678 6|882468922 8°2539880|8"2609696|8*2678407|18 
 
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 4618°21'7799818°2253830]8°2328360|8 °2401633/8°2473689/8 2544569 |8°261431 1/8°2682949] 14 
 47}8°21792'73|8°2255083|8°2329592)8-240284.4|8 24748 80/8 °2545 74d (8°26 15463|8°2684084/13 
 4.818°2180547|8°2256335]8°2330823)8°2404054/8-247607118°3.5469 1 2/8-2616616|8°2685219}12 
 
 49}8*2181821)8°2257587|S*2332053]8 -2405264|8 *247726 118 +2.548083|8°251 7768/8 268635311 
 50)8°2183095)/8°2258839|8 233328418 -24064.7418 *24'784.5 118 -2549254]8°26189290|8°268748'7| 10 
 . 151}8°2184368/8°2260090|8°233.45 1413 °2407683/8°247964.0/8 255042418 °2620072|8:2688620 
 52/8°2185641/3°2261341)8'2335743/8°2408892|8°2480829/8-2551594|8*262 1223/8 -2689754. 
 53|8*2186913)/8°2262591|8°2336973/8°2410101/8°249201 8)8°2552'764|8°2622375|8*2690887 
 548-2188 186/8-226384 1|8°2338202/8 -241 1310/8'248320 7/8 2553933 |8-2623525|8 "2692020 
 
 §5|8°21894.5'7/8° 226509 1|8°2339430/8°2412518/8°2484395/8-2555102|8°26246'76|8°2693152 
 56|8°2190729|8°2266341|8°2340659|8 °2413725|8°2485583/8 2556271 |8°2625826]8 2694284. 
 57182192000) 8°2267590|8°2341 886/824 14933)8 *248677118°25574.3913-262697618°2695416 
 58}8:2193270)8°2268839|8°234.31 14)8°241614018°2487958/8°2558607|$°2628 1 25/8 -2696548 
 5918°2394.54.1/8*2270087\8°2344341|8*241734'7/8*248914.518°255977518°2629275| 8 *269 7679 
 60}8°219581 118-2271335/8°2345568/8 °2418553)|8°249033218°2560943/8:2630424] 8°2698810 
 
 -_ |) | |] | | | — I 
 
 3 Bee 8 Oy 50/ 58’ 57’ 50’ 
 LOG. COSINES. 89 Deg. | 88 Deg. 
 
 SoKWwOWHL Or D100 
 
 z 
 
0 Deg LOG. TANGENTS. f 1 Deg. (253) 
 
 “tr 56’ : 57’ 59! 59’ f O’ 1 9/ 3’ 
 “018-21 19526|8*21964.08]8 ‘2271953/8°2346208)8- 82419215 8° 2491015 8*2561649|8°2631 153/60 
 118-2120818/8'2197678/8°2273201/8 °234'74.35|8°24204.21|8°24.92902)8 °2562817|8-2632302|59 
 2/8-21221 10|8°219894'7|8°2274.4.4.9/8 °234866 118° 242 1627/8 °2493388'8°2563984 18° 2633451158 
 3/8 +2123402/8°2200216|8°2275696)8°234.988'7|8° 242283318 °2494.574/8°2565 15118 *2634.599157 
 418°2194694(8°2201485/8°22'7694.3/8°2351113}8°2424038/8*2495760|8°256631'7|/8°263574'7/56 
 518+2125985|8°2202'75418 *2278 1 90/8 235233918 242524.4|/8°2496946)8 '2567484/8 2636895155} 
 6|8°2127275]8°2204.022)8 °22794.36|8°2353.564/8°24264.48]8 2498 131/8°2568650/82638043/54 
 
 718 +212856618°2205289|8°2280682|8 2354.789]8 *24.2'7653]8 °24.993 1 5|8°2569815/8°2639190)53 
 8/8 °2129855|8°220655'7|8°228 192718235601 3)8*242885'7|8 °2500500/8°2570981|8°2640337|/52 
 9/8 -213114.5/8°2207864)8 °2283173/8°235723'7/8° 2430061 |8°2501684)/8°2572146|8° 26414835 1 
 10/8:21324348 -2209090]8 228441 7/8 °2358461|8°243 1 264)8*2502868/8°25733 10}8°2642630/50 
 11}8°2133723]8*2210356|8°2285662/8 23596848 °243246'7/8 250405 1|8°25744'75|8°2643776]49 
 12)8°2135011]8°2211622/8°2286906|8°2360908|8"2433670|8 250523418 *25 756398264492 1/48 
 
 13/8-2136299|8-2919888 8°2288150/8°2362130)/8°2434872)8 *2506417|8°2576803/8°264.6067/4-7 
 1418°2137587|8°2214.15318°228939318°2363353|8°2436075|8°2507600|8'2.577966|8"°264721 2146 
 15(8°21388'74]8'221541818°2290636|8°2364575|8 243727618 -2508782|8°2579 1 29/8: 2648357145 
 16/8°214016 118221668218 °2291879|8°2365796|8*24384.'78/8 250996418" 2580299|8°2649501144 
 17|8°2141447/8°221'794.6]8°2293 1 21/8°236701 8/8 °2439679/8 251114518 °2581455|/8°2650645/43 
 18|8°2142733|8°2219210]8°2294363|8 °2368239|8°2440880/8 25 12326/8°2582617|8°2651789|42 
 
 19|8°2144.01918*22204.73]8°229560518°236946018 '244.2080|8°251350718°2583779|8°2652933141 
 2018°214.5304)8° 2221 73618 °2296846/8°23'70680|8 °2443280/8 °2514.688]8°2584941|8°2654076/40 
 2118°214658918°222299818-2298087|8°237 1 90018 °244.4.48018 25 1586818°2586102|8°2655219/39 
 29|8+214.78'74/8°222496018°2299327|8°23731 208° 244568018 °251704.818°2587263|8°2656362138 
 23]8°214.915818°229552218°2300568|8°23'74339]8 "24468798 °25 18227|/8° 258842418 -2657504/37 
 24|8°215044918°222678418°2301807|8°2375558/8°2448077/8°25194.07|8°2589584|8 2658646136 
 
 25|8°2151725|8°2298045/8°2303047|8 237677618 244927618 25205 86|8 "25907448 2659788135 
 96|8-2153008 8:2299305 8*2304.286 8°237'7995|8°24.504.7418 °2521764)/8°2.59 1 90418" 26609291394 
 278215429118 *2230566|8°2305525|8 "237921 3/8" 24.5 16'72)8 252294318 2593063|8°2662071)/33 
 28|8°2155573|8°2231826/8°2306763|8"23804.30]8" 245286918 °2524 1 21/8°2594.293)8 266321 213 
 
 2918 *2156855|8°2233085|8°2308001|8 238 1 64.8|8°2454.066|8°2525298|8°2595381/8° 266435231 
 30|8°2158137|8°223434.5|8°2309239|8 23828658 '2455263|5 25264-7618 °259654.0/8°2665492/30 
 
 31|8°21594.1 818°2235604|8°2310476|8° 2384.08 118 °24.564.6018 °252'765318°2597698)/8 °2666632|29 
 32|8°2160699|8"2236862|8°2311713)8°238529718 °24.57656|8*252882918 °2598856|8 *2667772|28 
 33|8°2161979|8*2238 1 20|8°2312950/8°2386513]8*2458852)/8*2530006|8°260001 4/8 *26689 1 1127 
 34/8°2163259/8°22393'78]8 °2314186|8°2387729/8°24.6004.7/8 °253 1 182|8°2601 171}8°267005 1/26 
 35]8°2164.539|8*224.0635|8*23 15429|8°238894.418 246 1 24.2|8°253235818' 260239818 +2671 189/25 
 
 6|8°2165818/8°2241892)8-2316658/8"25901 59/8°246243'7/8°2533533/8° 2605485 8°2672328)24. 
 
 37|8°2.167097|8°22431 4.918:231'7893|8°2391373]8°2463632|8°2534708|8° 2604641 8*2673466|23} 
 38/8 '21683'75|8°22444.05/8°2319128|/8°239258818 246482618 °2535883|8°2605797|/8°2674.604/220- 
 39|8°2169653]8°224.5661|8°2320363/8°2393802|8°24.66020]8°2537058|8°2606953)8°267574.2121 
 40/8 -21'70931|8 22469 17/8 *2321597/8°239501 5]8°24672 13/8 °2538232/8°2608 108/8°2676879}20) 
 4.1/8 *2172209}8°2248 1'72|8°232283 118 °2396228/8 °24684.07|8°25394.06|8°2609263)8°26780 16/19 
 4,218 °21'734.86]8°22494.9'7/8°2324.064|8°2397441]8 246959918 254057918 '26 1 041 8/8°2679153]18 
 
 4,3]8°21'74'762]8*22.50682|8°2325299718 *239865418°2470799|8-2541'752|8°261 157318*26802989]17 
 4.4/8°2176038]8°225 1936]8'2326530|8°239986618 +2471 9848 °254.2925|/8°2612797/8-268 14.25/16 
 4.5|8°2177314|8-2253190|8-2327763|8"240 107818 °24731'7618°2544.09818 "261 388 118°2682561115 
 46|8°2178590]8°2254443|8°232899518 240228918 -24'74368|8°2545270|9°2615034/8*2683696|14 
 
(234) 1 Deg. - LOG. $INES. Tab. 9. 
 td 4’ 5’ 6’ ZN 8/ ' Q’ 
 
 0/8 -2698810|8°2766136|8°28324.34)8-289773418 29620678 °3025460|8 °308794118°3149536/60 
 1]8°2699941|8 -2'767249]8 °2833530|8°28988 1 4/8 2963131 |8-3026509|8°3088975|8°3150555|59 
 218-270107118-2768362|8°283462618 289989418 296419518 302755818 °3090009/8 3151574158) 
 3}8°2702201|8°2769475]8*283.5722|8° 29009748 296525918 °3028606/8°309 1042/8 °3152593}57 
 418°2'70333118°2770587|8°2836818]8°2902053)8 °2966322|8 302965418 °3092075|8°3 15361 1/56 
 5|8-2'704.4.6 1|8°27'71700|8°2837913)8°2903132/8-2967385]8 °3030702/8-3093108/8°3154630}55 
 6|8:2705590/8 "27728 1 118°2839008]8 2904.21 118°2968448|8 303174918 -3094140/8°3155648)54 
 
 718-2706719|8-2773925|8*2840103/8°2905289/8 :296951 1/8 °3032796|8 30951 73/8 °3156665|53 
 8)8°2707847/8'2'775034]8 2841 19'7/8-290636'718-2970573/8 :3033843|8 -3096205|8°3157683)52 
 918 -27089'76|8°27'76 14.5|8°284.2292)8 290744518 +297 1635/8 °3034890)8 °3097237/8°3158700)51 
 10]8°2710104/8°2777256|8 °2843386)8 °2908523/8 -2972697|8 °303593'7|8°3098268/8°3159717|50 
 11]8°2711232|8°2778367|8°2844.479|8 *2909600)8 -2973759|8 30369838 -3099299)8 3 160734)49 
 12/8°2712359|8 27794778 °284.55 7318 °29 1067718 297482018 30380298 °3100330|8°3161751/48 
 
 13]8-2713486|8°2'78058'7|8*2846666|8°291 1754/8 °297588 1|8°3039075|8°3101361|8°3162767/47 
 14)8-2714613]3°278 1696/8 °2847759)8 291 283 1 18°2976942/8 30401 20/83 102392|8°3163783)/46 
 15}8°2'715'74.0}8°2782806|8 284885 1 [8°29 1390718*2978002)8°304 1165/8 '3103422!8°3164799/45 
 16|8°2'716866|8°2783915|8°2849943|8 -2914983/8 -2979063|8 304221 0|8°31044.52/8'3165815|44 
 1'7/8:271'7992|8 -2785023]8°285103.5]8°2916059|8 :29801 23/8 °3043255|8°3105482/8°3166830/43 
 18|8°2719118|8°2786132/8°28521 27/8 °2917134|8°298 1183/8 304429918 °3106512|8°3167845/42 
 
 1918°272024.3|8°2787240]8°2853219|8°291821 0/8 °298224.9/8°304534418°3107541/3'°3168860/41 
 20/8 -2721368]8°2788348}8°2854310|8°291928518:2983301|8°3046388/8°310857018°3169875140 
 2118 -27224.9318:27894.56|8°28554.01(8°292035918 298436018 °30474.3118°3109599|8°3170889/39 
 12918 -272361818-2790563|8°28564.91|8°292143418-2985419!8°30484.7518°3110628]8°3171903/38 
 4 23|8°2724'74.2|8-279 1670]8°2857582/8°2922508/8*2986477|5 30495 1818°3111656|8°317291 7/37 
 2418 +2725866|8°279277'718 "28.586 72/8 29235 82|8°2987536|8'3050561(8°31 12684/8°3173931136 
 
 2518°2726990]8"2793883|8"28.5976218°292465618°2988594'3°305160418°3113719/8°3174.945135 
 as §°2728113}8*2794989|8°286085 1|8°292572918-2989651/8°3052646|8'311474.0|8'3175958134 
 
 7|8°2729236|8-2796095|8°286 1941 |8°2926802)8-2990709|8-3053688/8"3115767|8'°3176971|33 
 33 8°2730359]8°279720118°2863030]8 °2927875|8°299 1'766/8°3054730|8'°31 16794|8°3177984132 
 2918-2731481]8°2798306|8 28641 1818°2928948|8 29928238 °3055772|8'°3117821/8'3178996)/31 
 30]82732604|8°2799411|8°286520'718-293002018 2993879850568 13/831 18848|8°3180008/30 
 
 311/8°2733725|8*28005 16|8:2866295]/8°2931092!18°2994936|8*3057855|8'3119874/8°3181021|29 
 32/8 °273484'7/8*280162118°2867383/8 °2932164)8*299599218°3058896/8°3120901/8°3182032/28 
 33/8 °2735968]8°2802725|8 *28684'71 |8°2933235]8 °299704.818°3059936|8°3121927/8°3183044/27 
 348 *273'7089|8°2803829/8°286955818°2934.306/8°2998 10418 °3060977/8°3.122959|8°3184055/26 
 35]8°2738210)8°2804933]8 28706458 *2935378)8 *2999 1 59/8°3062017|8°3123978|8°3185067/25 
 36/8*2739331)8°2806036|8°287 1732/8 °2936448]8°300021 418°3063057/8°3125003|8°3186077|24 
 
 37|8*274045 1|8°2807 139|8°287281 8/8°2937519|8 3001 269|8°3064097|8°3126028/8°3187088|23 
 38|8+2741571|8°2808242/8-2873905|8 -2938589|8°3002324|8°3065136|8°3127053|8°3188098|22 
 15918 *274.2690)8°280934.5/8°287499 I 8 -2939659]8 3003378|8°3066175)8°3128077|8°3189109/21 
 4.0}8+2'74381018"28 1044'7|8"876076|8 -2940729|8 300443218 306721 4|8°3129101|8°3190119/20 
 41)8-2744929|8°281 1549|8-28771 6218 -2941798|8-3005486|8°3068253|8'3130125|8°3191128)19 
 42/8 +2746048]8 281 2650|8°28'7824'7|8-2942867/8 -3006539|8 306929 1|8°3131149|83192138/18 
 
 43]8 +2747] 66]8°2813'752|8°287933218 :294393618-3007593|8'3070330]8°3132173/8°3193147/17 
 4.4)8°2°748284)8°28 1485318 288041 7/8 °2945005|8°3008646]8°3071368|8°3153196)8°3194156]16 
 44518°2749409)8 ° 2815954 8*2881501/8°294.6073|8 °3009699|8°3072405/8°3134219|8 319516515 
 46|/8-2750520)8 °281705518°2882585|8 2947141 18-3010751|8-307344.3|8°313524218°3196173/14 
 47/8275 1637|8°2818155]8°2883669|8 294820918301 180418 -3074480|8'3136264)8°3197182/13 
 4818 °2752754|8*2819255|8*2884.75218°294.9277|8 301 2856/8°3075517|8°3137287|8°3198190)12 
 
 49)8 *2753871|8°2820355}8 -2885836]8 29503448 °3013907|8°3076554|8°3138309]8°3199198}11 
 50]8'2754987|8°282145418°2886919]8°295 1411|8*301495918°3077590|8°3139331]8°3200205)10 
 51]8*2756105)8"°2822553)8+288800218 -29524.78]8°3016010/8°3078626|8°3 1 40352/8°3201213 
 52|8*2757219|8°2823652)8°288908418 °295354.4|8 30170618 °307966218'314.1374|8°3202220 
 53/8*2758335|8°2824-75 118°28901 66/8 295461 1]8*3018112/8°3080698]8°3142395|8°3203227 
 54]8*°2759450)8°282584.9]8 +2891 248)8 -2955677|8°3019163/8°3081734|8°3143416)8°3204233 
 
 55|8°2760565)8°282694:7)8 -2892330/8-295674.218 -30202 1 318 *3082769|8°3144436|8 3205240 
 56]8°2'761680)8°2828045/8 °289341 1/8 2957808]8*3021263/8°3083804)8°3145457/8°3206246 
 57/8 *2762794|8 2829 1 4.318 *2894.4.9218 295887318 -30223 1318 °3084839|8°31464.77|8°3207252 
 58|8°2763909|8 *283024.0/8 °2895573|8 295993818 -3023362|8°3085873|8°31 4'7497|8:3208258 
 59}8°2765022)8 *283133'7|8 -2896654/8 -296 1003/8 302441 1/8°3086907]8'3148516}8°3209263 
 60 8°2'766136]8°283943418 °2897734/8 :2962067|8 -3025460]8°3087941|8°3149536|8-3210269 
 
 LLU PURO ie SOTA net ocean | CO Cr > 
 
 CE ie i, a ioe ed : 
 LOG, COSINES. 88 Deg. 
 
 Soe wooro MH -10 © 
 
1 Deg. LOG. TANGENTS. (205) 
 
 tw A’ 5’ 6° 5 ye 8’ oy 
 0|8-2699563|8-276691 2/8-2833234)8-2898559|8-29629 1 7/8-3026335|/8°308884213- 3150462160} | 
 1/8°2'70069418°2768026/8*2834531|8°2899640)8 296398 1/8 °3027385|8°3089876|8 3151489159 
 218°2'701825]8°2769 1139/8 *2835428/8°2900720]8 :2965046/8 -3028433/8 309091018 °3152501158 
 3}8°2'70295518 277025318 283652418 °290 1800/8 °2966 1 10)3°302948918 309194418 -3153520157 
 418 -2704085)8°2771365|8 °283'7620/8 °29028 79/8 2967.1 '74|8 °303053 1}8°309297'7/8 3154539156 
 
 | 5|8°2705215|/8°27724'78/8*2838'71 618°2903959)8°296823'713°3031579]8°3094010|8 3155558155 
 6|8°270634.5|8°2773590]8 °28398 1 118°2905038}8°2969300}8 30326278 °309504.318 31565 76/54 
 
 7|8°27074'74)8°2'774.70218 -284.0906|8°29061 1 7|8'2970363/8 -30336'74/8 309607618 °3157595153 
 8/8°2708603'8°2'775814/8 -2842001/8'2907195/8°29'7 £426|8 °3034'729|8 °3097109}8 3158613152 
 9/8 °2'709732)8°27'76925|8 28430968 °29082'7418 °297 248918 °303576918°309814118°3159630151 
 10}8°2'710860/8°27'78036}8 ‘284.41 90|8°2909352|8*297355118 '30368 1 6|8°3099173}8°3160648150 
 11]8°2711989/8°27'7914.7|8 -284.528418°2910430|8 2974.61 318 °303'7862)/8°310020518'3161665/49 
 12}8°2713116|8°2'780258/8:284637818°29 1 1507|8°297567518°3038909|8°3101236|8°3162682/48 
 
 13}8°2714.24.4/8°278 136818 -284'74'7]|8 29 1258418 °2976736}8 °303995518°3 1022678 3163699147 
 1448°2'7153'71|8°27824'78]8 -284.856518°29 13661|8°2977797|8°304.100118°310329818°3164715146 
 1518°2716498]8°2783.58818 1284965818 °2914.73818°2978858)8 °304204.6/8 °310432918 3165739145 
 16|8°2'71'7625/8 278469718 *2850750|8°2915815|8°297991918 304309218 °3105360|8'°316674.8144 
 17/8°2718'751|8°2785806]8 °285 1843/8 °29 1689 1)8°2980980]8 °3044.137/8 "310639018 3167764143} - 
 18|8°2'71987'7|8°278691518 285293518 °291796718°298204.0/8 304518218 °31074.20|8 3168779142] - 
 
 19/8°2721003|8 278802418 -2854.02'7|8 291 904.2/8°2983100/8°304622618°31084.5018°3169'795]41 
 20]8°2722129)8°2789 13218-2855 1 1818°2920118]8 298415918 °30472'71|8°31094.79/8°3170810140 
 21}8+27723254/8*279024.018:2856210|8°29211 93/8 °298521 9)8°30483 151831 10508}8°3271825139} 
 22|8*2724379|8 279 1348]8-285730)118°2922268]8°2986278]8°304935918-31 1 153818°3172839|38] . 
 23)8°2725504/8°27924.5518 285839918 °292334.9]8°298733718 °3050403]8°3112566|8'°3173854137 
 2418 -2726628]8°2793.563|8 -2859489/8°2924.41718'298839518°30514.46]8°3113595|8-31 748681361 
 
 25|8°27271752|8 279467018 *28605'7218°29254.9 118 °298945 418 305248918 °31 14623/3°3175889/35] 
 26|8°27288'76|8°2795776|8°2861662|8°2926565|8°299051 218 °305353918°311565118°31768951341 | 
 2718 °2729999|8 :2796882/8°2862752|8°292763818°299 1570}8°3054.5'75/8°31 16679|8°3177909133 | - 
 28/8°2731129)/8°279798818:286384.1/8°292871 1|8°2992627|8 305561 718-31 1770718 3178929132 
 2918 °2'73224.5|8*279909418 +2864.93 1|8°292978418°2993685|8 °3056659]8°311873418°3179935131 
 30|8*2'733368}8 *2800200|8 -286601 9]8"2930857|8*2994'74.2)8°3057701|8°311976118°3180948)30 
 
 31|8°27344.90)8*2801305)8°286'7108]8°2931930|8°2995799|8-3058'743]8°3 1 20788]8:3181960}29 
 32|8°273561 2/8 280241 0/8°2868 1 96|8°2933009)8°299685518 305978418 3121815183 18297528 
 3318 °2736734/8°2803515|8°2869284|8°2934.0'74|8 299791 118306082518 °312284.118:3183985127| > 
 3418*273'7856|8°28046 1 918-28703'7918-2935 14.5|8°2998967|8°3061866|8'312386718°3184997126 
 3518 °27389'7'718*280572318°287 1 4.60|/8°293621 718 °3000023]8°3062907|8°31 2489383186008 25 
 36|8°2'740098)|8°2806827|8°28'7954.7|8'293728818°300107918°306394.718°3125919|8°318701 9194. 
 
 3718 °2741 21818 *2807930|8°2873634/8°2938359]8 30021 34/8°3064.98'7/8°3 12694418 °318803 123 
 38|8°2742338]8-280903418°2874720|8°29394.29]8 °3003189]8°3066027/8°312796918°3189041/22 
 
 9/8°2'74.34.5818 °2810136|8°28'75807|8°2940500]8°3004.24.418 °3067067]8°312899418°3190052191 
 4.0/8 °2'74.4.5'7818-28 1 1239/8°28'76893/8°2941570]8°3005298]8'3068106]8'°3130019/8-3191062120 
 4,1|8°2'74.5698]8°281234.2/8-28'77979|8°294.264.0]8 °3006353|/8'°306914.518°3131043]/8°3192073]19 
 42/8 *2'74.68 1718°2813444/8°28'7906518°294.3709|8°30074.07|8°30701 84/8 °3132068|8°3193083118 
 
 4.318 °274.7936/8 28 14.54.5/8°28801 50|8°294.4'7'79|8 °30084.60]8 3071 223]8 °3183092/8°3194099}17 
 4.418°274.9054/8 +28 1564'7|8*288 123518°294584818°30095 1 418307226 1/8°3134115/8°3195 102116 
 "|45)8°2'750173/8-28 16'748)8-2882320/8 294.69 1 6/8 °3010567|8°3073299/8°3135139/8:3196111]15 
 4,6/8°2751291/8°2817849|8 -28834.04|8°294'T985|8°301 1620/8°307433'7|8°3136162/8°3197120/144. 
 47]8°2'7524.0818*28 1895018:2884488]8°294.905318'301 267318°30753'7518°3137185/8°31981 29113 
 48/8°27753526|8*282005 118 :2885572|8°2950121/8°3013725|8°307641 218 °3138208/8°3199137/19 
 
 4918 °2754.64.3}8°2821151/8-2886656|8°2951 189/8°3014-778/8°30774.4.9/8°S 139230|8°3200145]11 
 50/8 °2755'760|8 1282225 118-2887'740}8*2952256]8 301 5830/8 °30784.86)/8 *3140253/8°3201154/10 
 51/8 °27568'76]8 +2823350)8 °2888823)8+2953324/8°3016881/8°3079523/8°3141 275|/8°3202161 
 52|8°275'799 218 *2824.4.5018*2889906|8°295439 1/8 °3017933|8°3080559|8 3142296/8 3203169 
 53}8*2759108}8:2825549]8 -2890988]8°2955457/8"30 1 8984/8308 1596/8 °3 14331 8/8°5204176 
 54)8°2760224/8+2826647|8°2892071|8 *2956524/8:3020035|8*308263 1/8 *3144339|8 3205183 
 
 55|8°2761340|8°282774.6]8 +2893 1 53/8°2957590|8°3021086|8°3083667/8°3145360/8°3206190 
 56/8 °27624.55]8°282884.4)8-2894.23.5/8°2958656/8°3022136)3'3084'703|8°314638 1|8°3207197 
 57/8 °2763570]8°2829942|8-2895316|8°295972118:3023 1 86|8°3085738/3°3147402/8-3208203 
 58/8 *2764.684]8 283 1040/8-2896397|8°296078'7}8 30242368 :3086773]8 °3148422/8 3209210 
 5918 °2765'798]8°2832137|8-28974'78|8°296 185218 °3025286|8 308780718 '314944.2/8 3210215 
 60 8*276691918-283323418-2898559|8-296291 7|8°3026335/8-3088842)8°3150462/8-321 1291 
 
 rm a a | ff 
 
 Tea ee io To) 
 
 3H 2 LOG. COTANGENTS. — 88 Deg. 
 
(236) 1 Deg. LOG. SINES._ | Tab. 9. 
 
 en nn ed! (eee  , ee 
 
 018 °3210269]8 °3270163]8°3329243|8 -338'752918 *344504.318°3501805|8°3557835|8 3613150160 
 1]8°32112'7418-3271155/8 ‘333022118 °33884.94)8 *344.5995/8 °3502745|8 355876218 3614066159 
 218321227818 °327214.6/8-333119918°3389459|8 344694718 35036858 35596908 3614982158} | 
 3]8°3213283|8°327313'7|/8°3332176)8 339042318 °344'789918*350462418 °3560617|8 361589757} | 
 418°3214.287/8°3274127/8°3333153/8°339 1387/8 °3448851/8°3505563/8°3561544)/8°36168 13156 
 5}8°3215292)8°32751 18/8 33341308 °3392351 |8°344.9802/8 °3.506502)|8°35624'71|8°3617728155} 
 6]8°3216295}8 °3276108/8°3335107/8°33933.15)/8°34.50753|8 35074418 *3563398)8 361864354 
 
 718+3217299|8°32'7709818'3336084|8°3394.2'79|8°34.51'704/8°35083'7918°356432418'3619558153 
 818'3218303]8°3278087|8°333'7060|8°339524.2|/8°345265518 350931 818356525 118°36204-72|59 
 9/8 °3219306|8°3279077|8°3338036|8°3396205|8°34.5360518°3510256|8°3566177|8 3621387151 
 10|8*3220309|8°3280066/8 333901 2/8 -339'7168|8°34.54.55518°351119418°3567103|8°3622301150 
 11]8°3221311|8°3281055/8°3339988|8°3398131|8°34.55505/8°351213918°3568029|8 "3623215149 
 12|8°322231 4/8 "328204418 -3340963]8°3399093|8°34.564.55|8°3513069|8"°3568954|8 3624129148 
 
 13]8°3223316]8°3283032|8°3341938)8°3400055/8 3457405535 14006|8 356988018 3625042147 
 14/8 °3224318/8°3284021/8°3342913/8°3401018|8'3458354/8°351494.418°3570805|/8°3625956/46 
 15}8°3225320|8°3285009]8°3343888|8 °340197918 °345930418°3515881|8°3571730)/8"3626869/4.5 
 16|8°3226322)8°3285997/8°3344863)8 340294 1|8°3460253|8°351681'7/8°3572654|8°3627782)4.4. 
 17}8°3227323]8°3286984/8 °334,5837|8 °3403902|8 3461 20118°351775418°3573579|8 3628695148 
 18|8°3228324|8°3287972|8°334681 118 °3404864|8 °3462150|8°3518690}8'3574503|8'°3629608)/4.9 
 
 19]8°322932518°3288959|8°334.7785|8 "340582518 °346309818°3519626/8°357542718°3630520141 
 20/8 °3230326|8°328994.6|8°3348759|8 °3406785|8°3464.04'7|8'352056218'°357635 118°363 1433/40 
 2118°3231326|8°3290933|8°3349732/8°340774.6|8°3464.995|8°352149818°3577275|8°3632345/39 
 29/8 °3232326|8°3291919|8°3350706|8°3408706/8"°346594.2|8°3.5224.3318°35781 9983633257138 
 23/8°3233326|8°3292906|8°335 167918 °34.09666|8°3466890/8°3.523369/8'°3579 1 29/8°3634169137 
 2418 *3234326|8°3293892)/8°335265118'°341062618°346783'718 352430418 °3580045/8°3635080/36] 
 
 2518 °323.532.5|8°3294878|8 335362418341 1586|8°346878418°352523918 358096818 3635991135 
 26|8°3236325|8°3295863|8°3354.597|8°341 2546/8 °346973118°3526173/8°3581891|8°3636903/34 
 2°7/8°3237324)8°329684.9|8'°335556918 341 3505|/8°3470678|8°3527 108/8°35828.14/8°36378 14133} 
 28/8°3238322|8°3297834|8°3356541 (8°34 1 44.64|8°3471 625/8°352804.2|8 3583736|8°3638724|32 
 29]8°3239321|8°3298819|8°33575 1218°34154.23|8°347257118°3528976|8°3584658|/8 3639635131 
 30|8°324.031918°329980418°3358484)8 341 638218°34.735 17/8 °3529910|8'3585580]8°3640545130 
 
 31]8°324131'7|8°3300788|8°3359455|8'341734018'°3474.463]8°353034.4/8°3586502/8'°3641456/29 
 32|8°324.2315|8°3301773/8°33604.26|8 °3418298)8 '34.754.09|8'3531778|8'°35874.24/8'°3642366/25 
 33}8°3243313/8°3302'75'7|8°3361397|8 341 9256/8°34'76354|8°3532'71 118°3588345|8 3645275127 
 3418°324.43 10|8°3303740]8°3362368|8'°34.202 1 418 °34'77300|8'353364.4/8'°3589266|8 364.41 85/26 
 35|8°3245308|8°3304'72.4/8'3363338/8°3421 1'72|8°347824.5|8°353457718°35901 87/8 °3645095/25 
 36|8°3246305|8°3305708|8'°3364309]8°34.22 1 29/8°34.79189|8°353551018°3591 10818 °3646004/24. 
 
 37/8°324.'7301|8°3306691|8°3365279/8°3423086|8 °3480134/8°35364.4.2|8 °3592029|/8°3646913/23 
 3818°3248298|8°3307674|8°3366248]8 °3424.043|/8°348107918°35373'74|8°359294.9]8 °3647829|99 
 39}8°3249294|/8-3308656/8°3367218/8°3425000|8°3482023|8°353830618'°359387018'°3648730/21} 
 4.018 °3250290|8°3309639|8°336818'7/8°342595718°348296718 353923818 °3594'79018°3649639|20 
 4118*3251286|8°331062118°3369 156|8 ‘34.2691 3|8°348391 118°3540170]8°3595709|3°3650547119 
 4.218 *3252289|8°3311603/8°33'70125|8'°3427869|8 °348485418°354110218°3596629|8 3651455118 
 
 43]8°325327718°331258518°33'71 09418 °342882518°348579818 354203318 359754918 3652363117 
 4418*3254.2'79|8°331356718'°3372063|8°342978 1|8°3486741 18°354296418 5598468]8°3653271116 
 4.518°3255267|8°331454818°3373031 |8°3430736|8 348768418 °354389518°3599387|8 "36541 79115] 
 4618°3256262|8°331552918°3373999/8 3431 691|8°34886 2718354482618 °3600306|8°3655086|14. 
 4°18 *3257256|8°3316510|8°3374.967|8°3432646|8°34895 7018354575618 °3601225|8°3655993]13 
 4813°3258250|8°33174.9118°3375934|8 3433601 |8°34905 1218 °3546686|8°360214.3|8'°3656900}12 
 
 4.9]8°3259244]8 -33184'72|8°337690218°3434.556|8°34.9 14.5418°354'761'718°3603061|/8°3657807] 11 
 50]8°3260238|8°331945218°337786918°34355 10|8°3492396|8°3.548546|9°3603979|8 3658713110) 
 51|8°3261232|8-3320432/8°3378836|8 343646518 "349333818 -354.94'7618 360489718 3659620 
 52/8 °3262225|8°332141 2/8°3379803|8 343741 9/8°349428018°3550406]8°3605815|8'°3660526 
 53]8°3263218]8-3322399|8°3380769|8°3438372)8 "34.9522 1/8°3551335|8°3606733|8 366 1432 
 5418 °326421118°33233'71|8°338 1736/8 '°3439326|8'34.96 1 6218°3552264|8°3607650|8 3662338 
 
 55{8°3265204)8°332435018°3382702)8°344027918 34971 03]8°3553193}8°3608567|8°3663244 
 56}8 °3266196|8*3325329|8 33836688 344 1233/8 °3498044)8 °355412218*3609484|8 "3664149 
 5'713°3267188|8*3326308]8 *3384633/8°3442 1 86]8 34989858 °3555050/8°36 104018 °3665054 
 58/8 *3268180|8°332728'7|8°338559918 34.431 38)8°349992518-355597918 361 1317/8°3665959 
 59/8 °3269 172|/8°332826518'3386564|8 344.409 1 |8°3500865]8 355690718 361 2234/8 °3666864 
 §0[8°32701 63/8 °332924.3/8 :3387529/8°344.504318°350180518‘3557835|8°36131 50|8°3667769 
 
 — _ | —— —o Ooo SS eee OO 
 
 LY AR hae 45’ 44’ 43’. V 49 41’ 40’ |” } 
 LOG. COSINES. ~ 88 Deg. 
 
 SOrFwoora A-100 
 
1 sch aa came TANGENTS. (237) 
 
 13-921 2997/8-807219818-9951298|8-398952518-344705718+ 3303839)8-985988 16 '3615219)59 
 913 -3213232|8°3273126|8-3332206|8'3390493]8 -344801018-350477518-3560809|8-3616129|58 
 318-3914237|8-32741 17]8-3333184|8-3391458|8-3448962|8-3505715|8-356173718°3617045)577). 
 
 F 4l,:2215249/8-3275108]3-3334161|8°3392423]8 344991 418-3506655|8-3562664|8°3617961/56}. 
 
 — 
 
 '5[8-3216246|3°32'76099|8-3335139|8 "33933878 -3450866/8-350759418-356359218°3618877155): 
 6|3-3217251]8°3277090]8-33361 1 6|8°339435 118+3451817/$°3508533|8°35645 1918°3619793154) 
 7}8-3218255|8°3278080]8 3337093/8'3395316]8-3452769|8-350947218-3565446|8°3620708153}. 
 8]3-3219259|8°3279070|8°3338070|8°3396279|8 -3453720|8°35 1041 1/8 -3566373)8 3621 623/52 
 918-32.20262|8°3280060]8-3339046)8°3397243]8-34.54671|8-3511350/8-3567299]8 3622538151) 
 
 1018-3221 266|8°328105018+3340023)8 '3398206|8-3455621|8-3512288]8-3568226|8 3623453150 
 
 1 1]8+3222269|8-3282039|8-3340999|3°3399 1 69|8 345657218 °35 13226183569 15218362436 7149 
 
 1218 -322327918°3283028|8°3341975/8.340013218 -34.57522/8 351416418 °3570078|8 362528 1148 
 
 15|3 322497418 -328401'713°3342950|8°3401095|8 -34.58479|8°351510218'3571 00418-36261 96147 
 
 141$ 322597718 -3285006|8°3343926|8 "340205818 -345942918-351604018°357192918-36271 10146]. 
 
 15]8*3226279|8°328599518°3344.901|8°3403020]8 -3460372/8°351697718 357285518 °3628023]45 
 
 15|8-3227981|8-3286983|3°334.5876|9°3403982|8-346132118°3517914|8°3573780|8°3628937| 44 
 
 17|8°3228983]8'328'797 1|8'3346851|8'°340494.4)8 346227 118°351885118°3574705|8'°3629850143 
 
 13]8-3229285|8°3288959]8'334'7826|8 34059068 -3463220|8-3519788]8°35'75630|8 3630763] 42 
 
 1913°3230286|8"3289947|8'33483800|8 3406867|8 34.641 69|8°3520725|8 357655583631 67641. 
 
 2018 -3231298718°3290934|8°334.9774|8 340782818 3465117/8°352166118'3577479|8°3632589]40 
 
 2118 °323298318°3291991|8'335074818 '3408789/8 -3466066|8°3522597|8 357840318 °3633502139 
 
 2218 323328818 °3292908|8°3351729|8°3409 75018 3467014)$°352353318°357932'7|8°3634414438 
 
 2318 +3234.28918°3293895|9°3352695|9'341071 118-3467962|8-°3524469]8°358025 1183635327137 
 
 24)8°3235289/8 -3294882)8°3353669|8 341 1671)8.3468910/8°3525405/8 3581 173|8°3636239)36) 
 
 2518 °323628918°3295868/8°335464.2/8'°341263 118 °346985718°3.52634.018'°358209818 3637150135 
 26|8°3237289|8*3296854|8°3355615|8°341359 118 -3470805|8°3527275|8 358302218 36380621344 - 
 2718 -3238288|8-3297840]8-°335658'7|9'3414551]8 347 1'752|8°3528210/8°358394.5|8 3638974133} 
 2818-3239287/8°3298826|8°3357560]8 34155 11)8°3472699|8°3529 1 45/8 °3584868|8°3639885132 
 29/8 °324.028618°329981118°3358532/9 341647018 347364618 353008018 °358579018 "36407961314 
 3018 °324.1285|8°3300796|8°3359504|9' 341742918 347459218 °3531014/8°3586713/8°3641707/30 
 
 $118°324.2284|8°330178118°33604'76|9'°3418388]8 -34.75539|8°3531 948/18 °358763518°364261 7129) 
 32/8 °324328918 330276618 °336144'7|8'341934'718 347648518 °3532882|8 358855718 °3643528]28 
 3318 °3244.280|8°3303'751|8°336241 9/8 °3420305]8 34.7743 118 '3533816|8°358947918 3644438127 
 3418 °324.527818 330473518 '°3363390|8°3421263|8-34783'77|8 °3534'750|8°359040 1 |[8°364.5348126 
 3518°3246276|8°3305719|8°3364361|8'°342222 118 -34.7932918°3535683/8°359 1322|8°3646258125 
 3618°324.72'73|8-3306703|8°336533118'342317918 -3480268/8 353661 6/8 '°3592243/8°3647168194 
 
 3'7/8°324827018°330768'7|8 °3366302|8' 34.241 37/8 °348 121 3/8°3537549/8 359316518 3648078123 
 
 38]8°324.9267|8°3308670|8'°3367272|8°342509418 34821 5818°353848218°359408618 "36489871298 
 3918325026418 °3309653|8°336824218'3426052|8°34831 03/8°353941418°359500618°3649896)/21 
 40/8 *3251260|8°3310636|8°336921 2|8°3427009]8 °3484.04'718°354.034'7|8 359592718 -3650805|20 
 4118°325225'718°3311619|8°3370181|8°342796518°3484.99 118°3541279|8 359684118 °3651 714119 
 4.218°3253253|8°3312601|8°33'71 151|8°3428922|8 °3485936|8°354221 1|8°359776'718 3652623119 
 
 43)8°3254.249|8°331358418°33'721 2018 °34.29878]8 3486879|8 °3543143/8 359868718 3653591117 
 4418°3255244|8°3314.566|8°33'73089|8°343083518 348782318 °3.54407418 359960718 3654439116 
 4.518°3256240|8°331554818°337405818°3431791|8°3488'767|8°3545006]8 360052713 3655347115 h 
 4.6]8°3257235|8°331652918°33'75026|8 °3432746|8 348971018 °354.593'7/8°3601446]8 *3656255]14 
 4'718°3258230|8°331751118°33'75994)8°3433'702|8°34.90653|8°3546868)8 °3602365|8°3657163113 
 4818°3259224|8°33184.9218°33'76963|8°34.3465'718'°349 159618 °3.54'7799|8°360328418 3658070112 
 
 49|8°3260219|8°33194'73|8°337'7930/8°3435612|8°349253918 °3548729)3 -3604.203/8°3658978]1 1 
 50|8°3261213)8°33204.54|8°3378898|3-3436567|8°349348 1 [8 °3549660/8°3605121)8 3659885110 
 51|8°3262207|8°3321434|8°33'79866|8°343'7522|8°3494.4.2318 3550590|8°3606040|8 3660792 
 52/8 *3263201|8°3322415/8°3380833|8°3438476|8 °3495365|8°3551520/8 3606958/8 3661698 
 53|8°32641 94|8°3323395|8 °3381800|8°343943 1/8 *34.96307|8°35524.50|8°36078'76|8 3662605 
 54|8°3265188|8°3324375|8°3382767|8°344.0385|8°349724.9|8 °355337918 "360879418 3663511 
 
 55|8°3266181|8°332535418-3383733|8°34.41339]8 349819 118°3554.309|8 -360971 183664417 
 56|8°32671'73|8°3326334|8 °3384'700|8 °344229218 3499 152/18 °3555238]8 °3610629|8°3665323 
 5'7/8°3268166|8 332731318 -3385666|8 °3443246/8 35000738 °3556167/8°361 1546/8°3666229 
 58/8°3269 158/8°332829218 -3386632|8 344419918 °3501014|8°35570968 361 2463)8°3667135 
 5918 °3270151|8°3329271|8 -338759718°3445 1 5218°35019548°3558024)8 3613380|8°3668040 
 60 $°3271 143]8-333024918+3388563]8°34.46 1 05|8°350289518°3558953|8 °3614297/8 3668945 
 
 rr | ne ST Fe 
 
 OF wouk& D-100 
 
 LOG, COTANGENTS, — $8 Deg. 
 
(238) 1 Deg. LOG. SINES. Tab. 9. 
 fe 21" 22/ 23’ | 24’ 25° | 20° 
 
 ——_eeeeee_e=_s=_s_=_=s ES SO | | i OO 
 
 0|8°366776918°3721710]8°3'774.988|8 382762018 ‘387962218 °3931008)8 398 179318403 1990|60 
 118 '3668674|8°372260318°3775870|8°38284.9218-38304.83|8°3931859}8°3982634)8 4032822159 
 218366957818 °3723496|8 °3'7'76753|8°3829364|8°388 13451839327 10/8 -39834'75|8°4033653/58I 
 3]8°36'704.82}8°372438918 37776358 °383023.518°3882206|8 393356 1 |8°3984316|8°4034485157 
 418 367 138618°3725282|8°37785 1718°383110618°3883067|8'39344 1 218-3985 157|8°4035316|56 
 5}8°3672290|8°3726 1748 °3779398]8°383197818°3883927|8°3935263|8°3985998/8 4036147155] | 
 618 °3673193/8°3727067|8:3780280]8°383284.818°3884788|8 39361 13/8 -3986839|8 4036978154 
 
 118°3674.097|8°3727959|8 °3'78116118°3833'71 918°388564.8]8 393696418 °3987679|8 4037809153 
 
 818 °5675000|8°372885118°378204.2|8°3834.590/8°3886509|8°39378 1418 398851918 °4038639]5 
 
 9/8 -3675903/8°3729743/8 378292418 -383546018°3887369|8°3938664|8 398935918 4039470151 
 10]8°3676806]8°3730635|8 -3783804|8 °383633018°3888229]8 39395 13/8 '39901 99}8 -4040300/50 
 11|8°3677'798|8°3731526|8°3784685|8°3837201|8°3889088|8 °3940363]/8 399 1039/8 *4041 130/49 
 1218367861 1]8°3732418]8-378556618°38380'70|8°3889948]/8°3941213}8°3991879]8-4041960/48 
 
 13/8 -3679513|8°3733309|8-37864.46|8°383894.018°3890807|8 3942062/8°399271 8)8 -4042790/47 
 14/8 -36804.15)8°3734200]8 -3787326|8 38398 1 0/8389 1 666|8°39429 1 1]8°3993557|8 4043620146 
 15)/8°3681317|8°373509 1 |8°3788206|8°384.0679|8°3892526|8 °3943760|8 39943978 -4044449|45 
 16/8-3682219|8°3735981|8°3789086|8°384 1 548]8°3893384|8 °3944609]8 -3995256|8 4045279144 
 17/8-36831 20}8°3736872}8-3789965|8 384241 7/8°3894243]8 39454578 399607418 40461 08/43 
 18/8 °3684022|/8°373'7762)8 379084518 °3843286]/8 3895 102)/8°3946306/8 "39969 13|8°4046937/42 
 
 1918°3684.923/8°373865218°3791724)8°384.4155]8°3895960}8'3947154|8°399775 1184047766141 
 20)8°368582418°373954.218°3'79260318°3845023]8 °3896818/8 394800218 °3998590)8 4048594140 
 21|8-3686725/8°374.04.3 1|8°3'7934.8218 '384.5892/8 389767618 39488508 399942818 -4049423)/39 
 2218 °368762518°3741321|8°3794361(8°384676018°389853418'394969818 '4000266]8 *405025 1138 
 2318 °3688526/8'°374221 (0[8°3795239|8°384-'762818 °3899399|8°395054618°4001104/8°4051080/37 
 248 *36894.26|8°3743099|8°37961 17|8°38484.96|8°390024.918'395139318°4001941]8-4051908136 
 
 2518°3690326/8°3743988|8'3'796996/8°384.936318°3901 107|8'395224.0/8°4002779|8 4052736135 
 26/8°3691226|8°3744877/8 -3797874/8°385023 1|8°3901 96418 °3953088|8 40036 1 6}8°4053563/34 
 27/8 °36921 2518 °3'74.5766|8°3798'75 1|8°3851 09818 °390282118'°3953935]8 "400445318 405439 1/33 
 28/8°3693025/8'°374.6654/3°3799629|8°385 1965/8 °3903678/8°395478 118 '400529018°40552 18/32 
 2918°3693924|8°374-754.2|8°3800507|8'°3852832/8 3904534|8'°3955628/8 4006 1 2718 -4.056046)31 
 30|8°3694823/8°3748430/8°3801984|8°385369918 390539 1/8 395647518 4006964/8°4056873|30 
 
 31|8°3695722|8°374931818*3802261 |8°3854.565|8°390624.718°3957321|8 400780 1/8 405770029 
 32|8 369662 1|8°3'750206|8°38031381/8°3855432|8°3907103/8°3958167|8°4008637|8 4058527128 
 33/8 °3697519|8°375109418-3804.015|8°385629818°3907959|8'3959013)]8 -40094.73/8 °4059353|27 
 34/8 °3698418]8°3'751981}8*3804891 18°38571 6418°39088 1 5|8'°3959859|8°4010309}8 4060 180/26 
 35/8°36993 1 6|8°3752868]8°3805768/8°3858030|38°3909671/8°3960705|8 "401 1 14.518°4061006|25 
 36|8-3'700214|8°375375518°980664.4|8°385889618:3910526|8'°3961550]8°401198118-4061832124 
 
 3718°3701111|8-3'754642|8°3807520|8°3859761/8°391 1382/8°3962395|8 401 2816/8 4062658)23 
 38/8°3'702009]8°3'755528|8°3808396|8°3860627/8°39 1223°'7/8 39632418 °4013652|8"4063484/22 
 39/8°3702907|8 375641 518°38092'71|8°3861492/8°3913099/8°3964.086]8 °4014487/8 -4064310]21 
 40|8°3703804|8°3757301|8°3810 1 4.7/8°3862357/8 °39 1394'7|8 396493018 -4.015322)8 4065135]20 
 4118 °3'704'701|8°375818'7}8°381 102218°386322218-3914801|8°3965775|8°4016157/8°4065961}19 
 42)8°3705598|8°3759073}8°381 1897/8°3864.08718°3915656/8°3966620}8-40 1 6992)8 -4066786)18 
 
 4318 *3'106494|8 °3759959]8°3812772|8°386495 118°39 1651 0|8'396'746418 401 7826|8°4067611]1 7}. 
 4418°3'707391|8 '3'76084.4|8 -381364'7|8°38658 1 6|8°391736418'3968308}8 401 8661/8°4068436)16 
 45/8 °3'708287/8°376 1 729|8°38 14.522|8°3866680]8°39 1821 8|8 "3969 1 52/8 °4019495|8-4069261]15 
 4.6/8 °3709183)8°3762615)/8°3815396|8°386754418°39 190'7218°396999618 -4020329|8-4070085}1 4 
 47/8 -3'710079]8°3'763500|8°3816271|8°3868408]8°3919926|8°397084.018 -4021163|8-4070910}13 
 48/8°3'710975|8°3'76438418 38 1714.5|8°3869271|8°3920779|8°3971683}8°402199718 407173412 
 
 4918 *3'7] 18'70|8°3'765269|8°381 801 9|8°387013518 3921 633|8°39'72527}8°4022831/8-407255811 1 
 50|8°3'71 2'166|8 °3766153|8°381889218°387099818 -39224.86|8°39733'70]8 *4023664/8°4073382110 
 51/8°3'713661|8°3767038/8-3819766|8-38'71 86118 -3923339|8°397421 3}8°4024497|8-4074206 
 52)8°3'714556|8°3'767922|8°3820639|8 387272418 3924.1 91|8°397505618 402533118 4075030 
 53)8°3'715451/8°3768806/8°3821 513|8°3873587|8°392504.4|8°3975898/8 4.0261 64/8 4075853 
 5418°3'716346|8°376968918-3822386|8 387445018 °3925897|8 397674118 -4026996/8 4076677 
 
 55|8°371'724.0)8°3'7'705'73|8°382325818 387531218 °3926749]8 397758318 402782918 4077500 
 56|8°3718134/8-37'714.56|8°3824 1318-3876 1'74|8°3927601}8°3978425)8 -4028662|8 4078323 
 57{8°3'719028) 8 °3772339/8 -3825004|8-387'7037|8 392845318 °3979268]8 -4029494}8 4079146 
 58|8°3719922|8 377322218 382587 6|8:3877898)8*3929305|8°3980 10918 -4030326|8°4079969 
 59/8 °3720816)8°37'74105|8*3826748)/8°3878'76018°39301 56]8.398095 1/8 °4031158)8-4080791 
 60 8°3721710)8°37'7498818 -3827620)8 -3879622)8-393 1008/8 °3981793|8-4031990)8 4081614 
 
 e——_—  | |  / |S | 
 
 ae Geek sa eee Oe, Bae 
 LOG. COSINES. 88 Deg. 
 
 "OR NOWES DA10W 0 
 
 4 
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] Deg LOG. TANGENTS. , (239) 
 
 ef Ys J | | | 
 — | —-—_—_—————————_ | —_—_——_——_ 
 
 118-3669850}8°3723809]8 3777 106]8 382975818 "388 1780183933 187|/8°3983994/3 -4034.213}59 
 2\8°36'70755|8°3724.70318 ‘377798918 °383063 1 |8°3882642)3 °393403918°3984835)8 4035045158 
 3|8°3671660)8°3725596|8°377887 218-383 1503]8 -3883.50418°393489 1/8°398567718°4035877)157 
 418°367256448'372648918 377975418 °38323'7418 -3884365|8°393.5742|/8°3986519)84036709)56 
 5|8°3673468]8°3727383|8°3780636]8 °3833246/8°3885227|8 °3936593/8'°398736013"°4037541155 
 6|8°367437218°3728275|8°378151918*3834117|8°3886088|8 -393'744.4/8 398820118 4038372) 54 
 
 98 °3675276|8°3729 168/8 378240018 °3834989/8 -3886949|8°3938295|8°398904.2)3 °4039203)53 
 
 8|8:3676180/8°373006 1/8 "378328218 °3835860]8 -3887809/8°3939145|8°3989883/8°4040035/52 
 
 9|8*3677083/8°3730953|8°3784164|8 383673 118°38886 70/8 °3939996/8*3990723|8-4040866)5 1} 
 10|8°3677987/8°373184.5|8°378504518°3837601)8°3889530|8 39408468 °3991564)8°4041696]50 
 11|8°3678890|8°3732737|8°3785926|8°38384'72|8°389039 1/8 -3941 696'8 399240418 -4042527)49 
 12/8*3679793]8°3733629|8°3786807|8°3839342)8°389 1251)|8°394.2546/8 39932448 4043358148 
 13/8*3680696|8°3734521|8°3787688/8°3840213}8 3892111 |8-3943396|8 39940848 40441 88147 
 14|8°3681598|8°3735412|8°3788569|8-3841083]8°3892970/8-3944246)8'3994924|8 404.501 8146 
 15|8°3682501|8-3736304|8°3789449|8 384195318 -3893830)8-394.5095|8°3995764|3 °4045848}45 
 16|8°3683403]8°3737 1 95|8:3790329]8 38428 22/8 -3894689|8 °3945945|8 °3996603|8 4046678 
 178 °3684.305!8°3738086]8°379 1 209]8 -3843692)8 3895548 )8°3946794|8°399 744.218 4047508 
 18|8°3685207/8°3738976|8°3792089]8 38445618 '38964.08|8°394.'7643/8°399828913°4048337/42 
 
 1918°3686108'8°3939867)8°3792969|8 °384543018 -389726618°3948492'8°399912118 4049167141 
 20|8-3687010|8-3740757|$'3793849|8-3846299|8-3898125|8-3949340|8°3999959]8-4049996)40 
 21|8°368791 1\8°374164718'°3794728|8 384.7 16818°389898418°3950189/8°400079813 4050825139 
 2918368881 9'3°374.2538|8°3795607|8°3848037|8°389984.2/8°395 1037/8 400163718 4051654138 
 23|8°3689713|8°37434.2'7|3°3796486]8 384890518 °3900700|8°3951885|8°40024'75/8°4052483/37 
 2418°3690614'8°374431'718°379736518 384977418 °390155818:3952733)8 400331318 405331 1136 
 
 2518°3691514/8°3'74.5206|8'379824.418 ‘38506428 °3902416|8°3953581/8°4004151|8°40541 40195 
 2618 '369241418°3'746096|8'3799 12218°385151018°39032'74/8+3954.4.29|8°4004989]8°4054.968134. 
 2718'3693315|8°3746985|8°3800001|8°385237818-3904131|8°3955276|8°4005897]8 4055796133 
 28|8°3694.21518°3'74787418°380087918 °3853245|8°390498918°39561 2418 °400666448°4056624 
 
 29|8°3695114|/8°3748762|8°3801757|8°38541 1318°390584618-39569'7 118°400750218-40574.5213 1} 
 3018°3696014)8°3'74965 118°380263413°38.54980|8°3906703|8°3957818)/8°4008339]8 4058280130 
 
 3118-36969 13/8°3'750539|8°3803512}3°3855847|8 39075608 °3958665|8 400917618 4059107129 
 32|8-3697812/8°375 142818 °3804390|8°38567 1418 °39084.1'7/8°395951 1/8°4010013}8°4059935}28 
 33/8 369871 1|8°3752316|8°3805267|8°385758 1)8°3909273/8°3960358|8°4010850}8 "4060762127 
 34|8°3699610|8°3753203/8°38061 44|8°3858448/8°3910129|8°3961204/8°4011686}8 4061589126 
 3518°3700509|8°3754091|8°3807021|8°385931418 39 10986'8°3962050|8°401252318°4062416125 
 36|8 3701407|8°375497918-3807898|8°38601 80/8391 1842/5-3962897/8°4013359|8 4063242124 
 
 37|8°3'702306|8°3755866|8'3808774|8°3861046/8°3912697|8°3963'74.2|8°4014195)/8 4064069 
 
 3818 °370320418°3756753|8°3809650|8°38619 12/8 °3913553|8°3964.588|8°40 1503 118 °4064895]22 
 3918 °3704.102|/8°375'7640|8°3810527|8°3862778}8 °3914409/8°3965434|8 °4015867/8°4065722121 
 4018 °3704999|8°375852'7/8°38 1 14.03]8°3863643]8 -3915264/8°3966279|8 °4016702|8 4066548120 
 4.118°370589718°3759413/8°3812278|8°3864.509]8°391611918°39671 2418 °4017538/8°4067374119 
 4218°3706794)8 °3760299|8 °381315418'386537418°39169'74|8°3967969|8°40 1837318 4068199118 
 
 4.318°3'7107692|8°376 1 186|8°3814030|8°3866239|8:39 1'7829|8 39688 1 4|8-4019208/8-4069025)17 
 4418 °3'708589/8°3'762072|8°381 4.905|8°3867 1041839 18684/8 '3969659|8°4020043/8-4069850]16 
 4518°370948518°3'762958|8-381 5780|8'3867969]8 39 19538|8-3970503|8°4020878|8 40 70676]15 
 46|8-3710382|8°3763843]8'38 1 6655|8°3868833]8 -3920393|8-3971348)8°4021713|8 4071501114 
 4718371 1278/8 °3'76472918 °3817530|8°3869698|8 -392124'7|8°3972192/8°402254.7/8:4072526)13 
 48]8°37121'7518°3765614|8-38 184.04/8°3870562|8 3922101 |8-3973036|8°4023381|8 4073151112 
 
 4918 °3713071|8°3766499|8°38 1 9279|8°3871426/8°3922955/8°3973880|8 40242 16|8°4073975/1 1 
 50|8°3713967|8 316738418 °3820153/8°3872290|8°3923808)8°3974.724|8 *4025050}8°4074800110 
 51]8°3714862/8°3'768269|8 '3821027|8°387315318°3924.66218°3975567|8°4025884(8 4075624 
 5218°3'715758|8°3769153)/8°3821901 |8°3874017/8°392551518°397641 118°4026717|8 4076449 
 5318°3716653]8°37'70038|8°3822775|8°38 7488018 °3926368/8°397725418°402755 1/8°4077273 
 5418 °3717548]8°3'770922|8°382364.8|8°3875743|8°392722 1 18°3978097|8 -4028384|8°4078097 
 
 es | ns | 
 
(240) 1 Deg. — ' LOG. SINES. es fee. Se 
 
 0)8°408 1614)8°4130676]8 -4179190/8*42271 68/8 °42'7462 118 °4321561/8°4367999)8 "4413944160 
 118°4082436]8°4131489]8*41 7999418 °4227963]8°42754.0818°4322339|8°4368768|8°4414706}59 
 2/8-408325818°4132302|8°4180798)/8°4228758]/8 "42761 941843231 17|8°436953818°441 5468/58 
 3|8-4.084080|8°413311518°4181602)8-4.229553/8°4276980|8°432389518°437030718 441 6229157 
 418°4084.902/8 °413392'7|8°4182405/8°4230348/8°4277766|8°4324672|8°437 107718 °44.16990|56 
 518° 4085723 8°4134'74.0|8-4183209|8-423 1 14.2/8°4278552/8°43254.5018 4371 84.6/8°441775 1155 
 6|8°4.086.54.5|8°4135552|8°4184012/8°423 19378 °427933818 432629718 437261 518°441851 2/54 
 
 '%18-4087366|8°4136364/8°41 8481 5|8°4232731/8 4280 1 2418432700418 °4373384/8 441927353] | 
 8/8°4088187|8°413'7176]8°41856 18/8 °4233525]8*4280909]8 432778 1|8°43741 53/8 4420034152 
 918°4089008|8°4137988/8 418642118 °42343 19/8 °4281694)8°432855818°4374.991 |8°4420795151 
 
 10]8°4089829|8°4138800]8 °4187223/8 4.2351 13/8°4282480/8°432933518°4375 69018 °4421555150} 
 
 11]8°4090650|8*413961 1]8°4188026/8°4235907|8°4283265|8°43301 12/8 437645818 "44223 15149 
 
 12]8°4091471|8°41404.22/8°4183828/8°4236700|8°4284050|8*4330888]8 437722718 "4423076148 
 
 13]8°409299 1 |8°414123418°4189630|8 4237494]8 428483518433 166518°437799518°4423836147) 
 1418°4093111/8°414.204.5|8°4190432)/8-4238287|8°428561 9/8 433244 1 |/8°4378763/8"4424.596|46 
 15}8°4093931|8°4142856|8'419 123418°4239080|8 428640418 °4333291718°4379531/8°44295355145 
 16|8°4094:'75 118°414366618'4192036|8°4239873|8°42871 8818°4333993/8°4380298|8 "4426115144 
 17|8°4.095571|8°41444-77|8 °4192838)/8-4.24.0666|8°428'79'7218°4334.769|8 438 1066/8 4426875 1/43 
 18}8°409639 1 |8°414.528'718°4193639/8°4.2414.5818°4288756|8°433554418° 4381833|8°4427634142 
 
 1918°4097210]8°414.6098|8°41 94.441 /8°424.925 118°428954.0/8°433632018°4382601/8°4428393)4 1 | 
 20/8 40980298 °4146908|8°419524.9/8°424304.3]8°429032418*433'7095|8°4383368/8 4429152140} 
 2118°409884.918°414'7718|8°4196043/8°4243836|/8°4291 108/8°433787118°4384.135|9°442991 1139 
 
 29|/8°4099662'8 41 48.528|8°41 9684418 4244623/8°429189118°4338646|8°4384.90918 4430670138) 
 23|8°410048618°414933'7|8°41976448°424.54.20|8°429267518°43394.2 1 18°4385669]8°4431429 137) 
 24|8°4101305|8°415014'7/8°4198445|8°424621 1/8-42934.58/8-4340196|8-4386435)8" 443218 '7|36) 
 
 49518-41021 2418°4150956|8°419924.518°4247003/8 4294241 |[8*434.0970|8°438720918°4432946|35) 
 26|8°410294.9|8 :4151765|8°4200046|8 4.247'795|8 4.29 5024|8 434 174518 4387968|8 4433704134! 
 27/8°4103'760|8 415257518 °420084.6|8 °4248586|8*4295807|8°434.25 1 918°438873418 4434462133 
 2818 °4104.578|8°4153383|8°420 1 64.6)8°4249577|8°4296590/8°4343294|8 438950] |8°4435221 |32 
 29|8°4105396|8°4154.192/8°42024.4.6|8°4250168)/8°42973'72|8 434406818 °4390266|8 4435978 /31 
 30|8°4106214|8°4155001 |8°420324.5|8°4250959|/8°4298 1 54/8°434484.9|8°439 1032/8 °4436736/30] | 
 
 31|8°4107032|8 4155809 |8°4.204.04.5|8 425 1750|8°429893'7|8 434561 6|8°439 1798 |8°4437494|29 
 32|8°410784.918°4156618/8°4204.84.418 425254118°429971 9/5 *4346389|8'4392564|8 4438251128 
 33|8°4108667|8°415'7426|8°4.205644|8 425333 1/8 °4300501|8'434'71 6318 °4393329|8 "4439009 27 
 3418°4109484/8°41 582348 °420644.3/8°4254 1929/8 -4301283|8-434'7937|8°4394094|8°4439766|26 
 3518°41 10301|8°415904.2|8°420724.2)8°42549 1 2)8-4302064|8°434.8'71 0/8 *4394859|8°4440523|25 
 36|8°4111118|8°4159850|8°4208040|8 °4255'702|8°4302846|8 434948318 439562484441 280/24 
 
 37|8°411 1934|8°4160657|8°4208839|842564.99|8 43036278 435025618 "45963898 444203723 
 38|8°411275118°4161465|8°4209638'8 "42572898 °43044.09|8 435102918 °43971 54/8 °4442794/229 
 39|8°411356718°4162279|8 42 104.36|8°425807118°4305190|8°435180218°43979 [918 °444.3551 21 
 40|8°4114383)8°41630'79|8 421 1234/8 °42.5886118°4305971 |8°43525'7418°4398683/8°444.4307|20 
 4,1]8°41 15200/8°4163886|8°4212052)8 -4259650|8°430675 1/8 435334'7|8°4399447|8 °444.5063}19 
 4.918 °4.116015|8°41 64693 /8°4212830/8°42604.39|8°430753218°43541 1918°440021 2/8°4445820]18 
 
 43|8°411683118°41654.99/8°4213628|8 -4.261229|8-430831 3)8'4354892|8°4400976|8°4446576|17 
 448 °4117647|8 41 66306|8°42144.26|8 -4.262018/8°4309093|8°4355664|8°440174.0|8 4447339116 
 4518°4118469/8°41671 12|8°4215223/8°4262806|8°430987318°435643618°4402503/8°444808 7/15 
 4618°4.11927818°4167919|8°4216020/8°4.263595|8°431065418°435720718°4403267|8°4448843]14 
 4'7|8°4.120093/8°4168725|8°4216818|8°4264.383|/8-431143418°43579'7918 "44.0403 1|8°4449599]13 
 48]8°4120908]8°416953 1/8 °421'7615|8-4265172|8-4312213|8°435875 1|8°4404794|8 445035412 
 
 498-41 21793]8*41 70336 |8-421841 218-4265960|8-431299318-4359522|8°4405557|8-°4451 109] 1 
 50|8°4129537/8°4171142/8°4219208|8°4266748/8°4.31377318°4360293|8 °4406321|8°445 1865/10 
 51]8*41 2335218 °4171948|8 -4220005|8'4267536]8°431 455218 °436 1 064|8°4407083]8 4452620 
 5218°4124166/8 °4172753/8 4220801 18 °4.26832418 -43 1535218 °436183518°440784.6|8 4453375 
 53|8°4124.98 118 °4173558/8°4221598/8-42691 1 1/8°431611 1/8°4362606]8°4408609)8°44541 29] 
 5418 °4125795]8 °41'74.563/8 422239 4)8 426989918 431689018 °4363377|8°4409372/8 4454884 
 
 55|8°4126609)8 °4175168|8*42231 90/8 -4270686|8-431'7669|/8°4364.148]8°4410134)3 4455638 
 56|8°412'74.22)8 41 75973/8 *4223986]8 42'7 1474/8 431844718 43649 1818 °4410896/8 4456393 
 57|8°4128236/8 °4.1'76'7'7'7|8°4.224'782|8°427226 1 18°43 1922618 436568818 441 165918 4457147 
 58}8°4129050/8 °41'77582/8°4.22557718 427304818 432000418 °436645918°441242118°4457901 
 59]8°4.129863/8°4178386|8°4226373/8-4273834|8-4320783]8 °4367229|8 44131838 °4458655 
 60/8- 4,1306'76]8 °41'79190|8-4.227168|8 4274621 |8-432 1561/8 -4367999|8 441394418" 4459409 
 
 Yom iD we D~WMOwo 
 
 LOG, COSINES, 88 Deg. 
 
Dod 65 dp MOE ELS Ses TANGENTS, (241). 
 
 Re ee ee a ee a ee, | eee ee oe eee Oe ee 
 
 018*4.083037|8°4132132)8°4180679|8 -4.228690|8-4.2761 76|8°4323150)84369622|8-4415603]/60} 
 1]8°4083859/8°4132945/8*41 8 1483/8 -4229485)/8°42'76963/8 432392918 -4370393|8 4416365159 
 2/8 40846828 '4133759/8°4182288/8 423028 1/8-4277750/8°4324707|8 4371 163/8°4417127158 
 3}8°4085505|8*4134572/8°4183092)8°4231076/8°427853'7|8°4325486|8°4371933/8*441 7889) 574 
 4|8°4086327|8°4135385|8 °4183896|8°4231872|8°4279324|8°4326264|8°4372703/8 44.1865 1156 
 5|8°4087149|8°4136198|8°4184'700/8°4232667/8°42801 10/8 °4327042/8 °4373473)8 4419413155 
 6|/8°4087971|8°4137011/8°4185504|8°42334.62)8°4280897/8°4327820/8 °43'74.24.2|8 4420 174/54 
 
 7|84088793/8°4137823|8-4186307|8:4234257|8.428 1 683/8°4328598|8-4375012/8-4420936|53 
 8]8°4089615)8°4138636/8°41871 1 1/8°4235051]8°4282469|8°4329375|8°437578 1|8°4421697 52 
 
 10}8°4091258)8°4140261|8-4188717)|8'423664.0|8°4284.041/8-4330930|8'°4377320}8- 4423219150 
 11]8*4092079|8°4141073)8°4189520|/8°4237434)8°4284826|8°4331707|8'4378089]8 °44.23980|49} 
 12}8*4092900|8°414.1885/8°4190323/8-4238229|8 428561 2/8 °4332484)8°4378857|8 °44.2474.1148 | 
 
 13}8*4093'721|8°41 4.2696|8°4191126/8°4239023/8-4286397|8 433326 1|8 °4379626|8 "4425502147 
 14]8°4094.542/8°4143508/8°4191929/8°42398 16|8°4.287 1 82/8 -4334038]8'4380395|8-4426262146 
 15|8°4095362/8°414.4.3 1 918°4192731/8°4240610/8'428'7968/8°4.3348 15|8°4381163/8°4427023)/45] 
 16|8°4096183}8°4145131|8*4193533/8°4241404|8°428875 2/8 4.33559 1|8°4381931/8°4427785/44 
 17|8°4097003/8 "41459428 °4194336|8°4242197|84289537|8°4336368]8°4382700|8°4428543)4 
 18}8°4097823|8°4146753/8°4195138/8-4242990]8-4290322)8 -433'7144)8°4383468)8 °4429303/42] 
 
 1918°409864.3|8°4147564)8°419594.0|8'°4243783|8°429 1 106|8°433'7920|8°4384235|8°4430063]/41 
 20]8°40994638°41483'74]8 4196741 |8°4244.576|8°429189 1|8°4338696|8°4385003|8°4430829140 
 21]8°4100283]8°4149185|/8°4197543/8°424.5369|8°4.29967518°43394.7918°438577 1/8443 1589139 
 22}8°4101103]8°4149995|8°419834.4|8 -4246162|8°42934.5918°4340248|8°4386538]8 4432341138] ° 
 2318°4101929]8°415080518°4199146]8°424695418°4294.24.318°434.1023/8'4387306|8°44331 01137] 
 2418°4102741)8°4151616|8°419994'718°424774'7|8°4295027|8°4341799|8°4388073/8 -4433860/36] 
 
 25]8°4103560|8°4152425]8+420074818°4.24853918°429581 118°43425'74|8°438884.0]8 4434619135 
 
 26]8°4104379|8°4153235]8°4201549]8 4249331 |8°429659418°434334.9|8°438960718 4435378)34) 
 27/8°4105198|8°4154.045]8-4202349|8-4250123|8°4297377|8°434412418°439037418:4436 137133} 
 28]8°4106017)8°4154854|8°4203150/8°4250915|8°4298 16 1|8°4344899|8°4391140/8°4436895/32} 
 29)8°4106835/8°4155664)8°4203950/8-4251706|8°429894.418°4345674|8°439 1907/8 4437654131} 
 30|8°4107653)|8°4156473/8°4204'750|8°42524.98/8°4299727|8 °434.64.4.8]8°4392673/8 443841 2/30} 
 
 31/8°410847218*4157282|8°4205550|8°4253289]8°4300510|8°434.7223|8°43934.4.0}8°44.39171/29} 
 32/8 °4109290|8*415809 1|8°42063.50|8°4254.080]8°430 1299/8 -434.799'7/8°4394206|8 °4439929/28} 
 33|8°411010'7|8°4158900)8°4207150]8°4254872|8°4302075|8°434877 1|8°4394.97218°444068 7127} 
 3418°4110925/8°4159708|8°420795018°425566218'°4302857|8°434954.5|8°439573818°44.41 44.4) 26 
 35|8*4111743/8°416051'7|8°4208'749|8 °4256453|8°4303639)8 43503 1 9/8°4396503)/8-4442209/25 
 36|8°4112560/8°4161325|8°4209549|8°425724418°43044.2218°4351093/8-4397269|8°44.42960}24 
 
 37/8-41133'77|8°4162133]8-4210348|8°425803418°430520418°4351867|8:4398034/8-444371 7123 
 38}8°4114194|8°4162941/8°421114'718°425882518°4305985/8°4352640/8°4398800|8°44.4.4.4.75]22 
 39|8°4115011|8°4163'749)8°4211946/8°425961518°4306767/8°435341 3/8 °4399565/8°444.525Q/21 
 40/8°4115828]8°4164556|/8°4212'74518°42604.05|8°430754.9|8°435418'7|8°4400330)8°444.5989/20 
 41|8°4116645|8°4165364)/8°4213543/8°426119518°4308330]8°4354960|8°44.01095)8°4446746] 19 
 42'8°411'7461|8°4166171 |8°4.214.34.218°426198518°43091 11/8°4355733/8°4401860/8°4447503]18]. 
 
 43/8°4118278 |8°4166979|8-4215 140|8-4262'774|8-4309892|8°4356506|8°4402624)8 °444825 9117 
 44'8°4119094)|8-4167786|8°4215938|8-4263564|8°4310673/8°4357278)8°44.03389/8-444.90 1 6]16 
 45}8°4119910)/8°4168593|8°421 6736|8°4.264353|8°43114.54)8°4358051|8°4404153/8°4449772115 
 4.6|8°4120726|8°4169399]8 -4.217534/8°4265142|8°43 1223518 '4358823|8 °-4404.918/8°44.50529]14. 
 7/8°4.121541 18°417020618*4218332)8°42659329/8°4313016|8°4359595)8°4405682/8°4451285]13 
 sls 22357/8°4171012/8:4219130|8°4.26672018°43 13796|8°4360367|8°44.0644.6]8°4452041]12 
 
 49/8 °4123172/8°417181918°4219997|8°4967509|8°4314.5'76/8°4361139]8°4407209)8°44.52797111 
 50)8°4123988/8-417262518°4220725|8°4268298|8°4315356|9°436191 1|8°4407973}8°4453552| LO 
 
 55|8°4128062[8-4176654|8-4224'709|8 427223918 4319255 8-4.365'768]8°4411790 8°44.5'7329 
 56/8°4128876)8 -417'74.5918 -4.225505|8°4273027|8°4320034|8:4366540/8-4412553/8-4458084 4 
 
 59|8° 4131318 8: “AVT9874 8 499789418 4.275389)8° eet 8: 4368852 8: “4414841 8° “4460348 
 60 8-413213218-4180679]8-4228690)8-4976176|8-4323150|8-4369622|8 44d 5603]8°4461103 
 
 COrwnwoko D-AIWwo 
 
 4 
 
 ee sa LOG. COTANGENTS, _ 88 Deg. 
 
(242). 1 Deg. _. LOG. SLNES. Tab. 9. 
 Pek waa. 37’ 38’ 30/ AO’ 
 0/8°44594.09]8°4504.4.02/8°4548934|8°459301 318 -463664.9|8°46798.50|8°4722626|8 4764984160 
 1]8°4460163]8-4505148|8-4549672)8-4595744)8-463'7372|8-4680567|8°4725535|8°4765686|59 
 2)8-4.460916|8-4505894)8°455041 0|8-45944'74|8-4638096|8-468 1 289|8-4724044)8 476638858 
 3|8°44.616'70]8°4.506640|8°4551148|8°4595205|8 46388 1918°468 1999)8 °4724.753|8°476709 1157 
 418 °44.62423/8°4.507385|8°455 1886/8 °4595936|8 463954218 °46827 15/8 °4725462|8°4767793156 
 5|8°44.63176|8°4508131|8°4552624|8 -4596666|8 46402658 °4683431|8°4'726 17118 °4768495)55 
 6|8+4463929|8-4508876|8-4553362|8-4597396|8-4640988|8-4684 1 47|8-4726880|8°4769 197154 
 
 7/8 °4.4.64.682)8°4509621|8°4554.099|8 4598 1 26|8 464171 1/8°4684862|8°4727589/8 -4769899)53 
 8|8°4465435)8°4510366|8'°4554837|8°4598856|8 464 2434/8 46855788 472829718 47 70600|52 
 9)8°44.66188]/8°4.511111]8°4555574|8°4599586|8°4643 1 56|8*4686293|8"4729006/8 4771302151 
 10|8°446694.0)/8°4511856]8°4556311]|8°46003 16/8 -4643879/5°4687009|8 472971 4/8 °4772003)50 
 11}8°4467693/8°4512601/8°4557048|8 "460 1046/8 :4644.60 1|8°4687724|8 473042918 °4772705)/49 
 12 haneas actea tas rhe bey eens 8°4645523 evieae itil ih as 8°4773406/48 
 
 21]8-4475210/8°452004.0)8 456441 2/8 -4608335|8 46518 1 8/8°4694870)8 -4737498)8 4779712139 
 22/8-4475961)8°4520784)8 °4565148/8-4609064|8 -4.652539]8 -4695583)8 -4738205|8 "478041 2/38 
 25|8-44°76712)8°4521527|8-4565884|8°4609'792)8 -4653260)8 469629 7/8 47389 19/8 "478 1112/37] 
 2418 -4.4'77462)8°4522270)8°456661 9/8 -4610520)8 465398 1|8°469701 1|8-473961 8/8 °4781812/36] | 
 
 25/8-4.47821 3/8°4.523013)8°456735418°461 124.818°4654'702|8°4697725|8°474032518 4782511135 
 26/8 °44'78963|8°4523755/8°4568090|8°461 1976|8°4655422|8°4698438]8 474103218 °478321 1134 
 27718-44797 1 418°45244.98|8°4.56882518°461 2703/8°465614318°4699 15118 °474173818°47839 1 1133 
 28/8°448046418°452524018°4.569560/8°461 343 118°4656863/8°4699865|8 474244418 4.7846 10|32 
 29/8 °4.48121418°4525983/8°4.5'7029518°4614158]8°46.57583]8°4700578)8 474315013 °478.5309131 
 3018448 196418°4526725|8'4.5'71029|8°4614886]8 465830318 "4701 291|8°4743856|8°4786009130 
 3118448271 4/8°4527467/8°4571764/8°4615613|8'4659023|8°4'702003/8 °-4'74.4.562|8 °4786708|29 
 3218 °44834.63|8°4528209|8'45724.98|8°4616340)8465974318°4702'7 16/8 °474.5268|8°47874.07128 
 33|8°4484213|8°4528951/8°4575233|8-46 170678 °466046318°47034.29|8 °474.5974|8°4788 105/27 
 34|8- 4484962/8° 4529693/8°4573967|8 461779418 4661 182|8°4704141 |8°4'746679|8°4788804)26 
 3518°448.5712/8°4530434|8°4.57470 118°4618520]8°4661 90218°47048.5418°4747385|8°4789503125 
 3618448646 118°45311'76|8°457543518°46 1924718 °466262118°4705566|8 474809018 °4790201 24 
 37/8 °44.87210/8°4531917|8°4576169|8°4619973|8-°4663340|8°4706278)|8°4748795|8 "479090023 
 38|8°4487959/8-4532659|8 457690218 462070018 -466405918°470699018°474.9500|8 4791598122 
 39}8*4488708|8°45334.00|8°4577636|8 462142618 '4664778]8°470770218 °4750205|8°4792296]21 
 4.0/8 °44.89456|8°4.534.141|8°4578369|8°4622152)8°46654.9718°47084 1418 °4'7509 10/8 °4792994/20 
 41]8°44.90205/8°4.53488 1|8°4.579103|8°4622878|8 466621 6|8°4709 196|8°4751615|8 4995692119 
 4.2}8°44.90953/8°4535622|8°457983618°4623604|8-4666935|8°4709837|8 °4752320|8°4794390118 
 
 4318°44.91701|/8°4536363|8°4580569|8-4624330|8 -4667653|8°4710549|8°4753024|8°4795088117 
 4418°4492450)8°453'7103|8°458 1302|8°4625055|/8°4668372|8°471 1 260|8 4753729 |8 4795785] 16 
 45|8°4493198)8°4537844)8 4582035/8°462578 118'4669090|8°471 19'71|8°47544.33|8°4796483}15 
 4.6|8°4493945|8°4538584/8'4582768|8°4626506|8-4669803|8"4712682|8°4755137|8°4797180}14 
 4718°44.9469318°4539324|8°4.583500|8 462723 118467052618 47 13393|8 4755841 (8 "479787815 
 48/8 -4495441/8°4540064/8°4584233|8 462795 7|8-4671 24.4|8'471 410418 475654518 °4798575/12 
 
 49]8°44.96188)8°4.540804|8°4.584965|8°4628682|8°4671962|8'4714815|8 475724918 4799272111 
 50}8°4496936/8*4.541543/8°4585697/8 -4629406/8°4672680)8 47 15526|8°4757953/8 4799969110 F 
 91/8 °44.97683|8°4542283/8°4.586429|8 463013118 -4673397|8°4'716236|8°4758656]8 4800666 
 52)8°4498430/8°4.543023/8°4.587 161 |8°4630856|8 46741 15|8°47 16947|8°4759360/8°4801362 
 53/8*44.99 1'7'7|8°4.543762|8°4587893|8°4631580|8°4674832/8°4717657|8°4'760063|8°4802059 
 5418 '44.99924/8°4.544.501)8*4588625|8°4632305/8°4675549|8°4718367|8°4760766|8°4802755 
 
 95{8°4500671)8°4.54.5240|8°4589357|8*4633029|8-4676266|8°4'719077|8°4'761470|8 4803452 
 56}8°4.501417/8°4545979|8°4590088)8°4633753]8°4676983|8°4719787|8.47621 73/8 4804148 
 
 718 °4.502164|8 °454671 8/8°4.5908 19/8 °4634477/8 -467770018'4720497|8.4762876]8 4804844 
 38/8 °4.5029 10/8°4.54745'718 °4.591551|8°4635201/8°46784 1'7/8°4721207|8.4763578)8 °4805540 
 9918 °4.503656/8 °4548195|8°4592282/8 463592518 4679 1 34|8°4721916)8.4'764.28 1/8 °4806236 
 50/8 °4.504402)8 -454.8934/8°4593013|8-4636649)8 -4679850]8 °4722626|8.4764984/8 4806932 
 
 te dae & capt 6s 10’ 
 
 LOG. COSINES. ~ 88 Deg. 
 
 Se WOO DA-A1M 0 
 
1 Deg. | -. LOG. TANGENTS... .. (243) | 
 
 1}8: “4461857 8° “4306878 8: 4551498 8° “4505545 8: 4639211 8: “4682440 8: a 25248 8° ‘167636 59 
 218°446261118°4.507624|8°4552176|8°4596277|8 46399358 °4683159)8°4725957/8°47683391538 
 3|8°44.63365|8°4508371|8°4552915/8°4597008|8 °4640659|8°46838 75/8 °4726667|8 °476904.2157 
 418°4464119]8°45091 1'7/8°4553654|8°4597739]8 4641382|8°4684592|8°472737718 476974.5|56 
 518°44.64873]8°4509863/8°4554392/8 “4.598470|8 4.64.2 106/8°4685309/8 °4'728086|8°4770448)55 
 6|8°44.6562'718°4.5 1060918 -4555130/8°459920 118 °4642830/8°4686025/8°4728796/8°4771150154 
 
 718°4466380/8 451135418 45558688 °4599939|8 -4643553|8 4686741 |8°4.729505|8°4771 853153 
 
 818°4467133/8°4512100|8°4556607|8 '4600662/8 -4644276]8 °4687458)8°473021418°4772555)152 | 
 
 9/8 °44.6788'7|8°45 12846|8°455 734418 °460 1393/8 °4645000|8 46881 74/8 °4730923/8°477325715 1 
 10]874468640)8°4513.591)|8°4558082/8°46021 23/8 °-4645723/8°4688890/8°475 1632/8 °4773959]50 
 11]8°4469393/8°45 14336/8°4558820|8 °4602853]8 464644618 °4689605|8°47323-41|8 4774661149 
 1218°44.7014.6|8°451508 118°4559558|8°4603584|8 4647 168)8°4690321|8°4733050|8°4775363/48 
 
 13|8°44.70898/8 45 15826/8'°4.560295|8 46043 1 4|8 464789 1|8°4691037|8°4733758|8 4776065147 
 14|8°4471651/8°4516571|8°4561032|8*4605043]8 464861 4/8°469 1752/8 °4734467/8 4776766146 
 15|8°4472404|8°4517316|8°4561769|8°4605773|8 4649336]8 °4692468]8°4735 17518 °4777468145 
 16|8°4473156|8°4518061|8°4562506|8°4606503)8 *4650059)8 -4693 183|8°4735884)/8°4778 169/44 
 17|8°4473908/8°4.518805|8°4563243/8°460723213°465078 1 |8°4693898/8°4736592)8 477887 1143 
 18|8°4474.660|8°451954918 °4563980}]8°4607962|8 °4.65 1503}8°46946 13/8 °4737300|8°4779572|49 
 
 19]8°44'75412)8°4520294|8 456471 7|8°460869 1/8 °4652225/8°4695328/8°4738008)/8°4780273/41 
 
 20)8°44'76164)8°4521038/8°45654.53|8 *46094.20|8 -465294'7/8 46960438 °4738'71 5/8 °4780974140 
 21)8°44'7691 6|8°4521782/8'°4566190)8°461014.9/8-4653669|8°4696757|8°4739423)8°4781675)39 
 22/8 °44'77667|8°4522526|8°4566926/8°4610878/8 4654390/8°46974'72|8 °4'74.0131/8°4782375)/38 
 23/8°44'784.1 918 °4523269|8°4.567662/8°4611607|8°465511218°4698186)/8°4740838)8 °4783076)37 
 24[8°4479 1'70]8°4.524013]8°4568398/8 °46 12336/3 "465583318 -4698900|8°4741545|8 4783776136 
 
 60 8: ‘4506131 8° “4.350699 8° 4594814 8° Pies 8° 4681 72518° ‘194538 3: 4.766933)8°" "4808920 
 
 31 i LOG. COTANGENTS. 38. Deg. 
 
(244) 1Deg. _ LOG. SINES. as: 
 
 8°480693218 -484847918 -4889632/8°4930398]8 -4970784/8 501079818 -5050447|8 -5089736/60 
 1/8°4807628|8°4849168]8°489031418°493107418°497 145418 -5011462|8°5051105|8°5090388)59 
 2/8 -4808323]8-484985718°4890997|8°4931'750|8°49'72 1 24/8 -5012126|8°5051762|8°509 104.0158 
 3/8°4809019|8°4850546]8°4891679]8 °4932426|/8°497279418-5012790/8 505242018 °5091691157 
 418 48097148 °4851235/8-489236 1 |8°4933102|8°4973463)8 5013453/8*5053077|8 -5092343)56 
 5}8°4810410)/8°4851923)/8-489304318°4933778|8°497413318°5014116/8°5053735|8*°5092994155 
 -6|8°4811105/8°4852612/8°4893726|8°49344.53|8°4974802)8 -5014'780/8°5054392|8°5093646)54 
 
 18°4811800|8°4853300|8°48944.0718°4935129|8°49754'7218 501544318 °505504.9|8°5094297153 
 $18°48124.95|8°4853989]8 489508918 °4935804|8'°497614118:°5016106|8'°5055706/8-5094.948152 
 9/8-4813190]8°4854677|8-48957'71|8°4936480|8°4976810|8°5016769|8°5056363/8-5095599)5 1 
 10/8°481388418°4855365|8°48964.5218°493'715518°497747918°501743218°5057020|8°5096250)50 
 11}8°4814.57918°485605318°489713418°4937830|8°497814818-501809518-505767'718°509690 1149 
 12/8-4815273|8 4856741 |8-48978 1 6|8-4938505|8°4978817/8°5018757|8'5058333/8-5097552148 
 
 13]8-4815968)/8:4857429|8-4898497|/8-4939180|8-497948518-5019420|8°5058990|8-5098209|47 
 14}8°48 1 6662/8°4858116|8°48991'78/8°4939855|8°498015418°5020082|8°5059646/8 5098853146 
 15|8°4817356]8°485880418°4899859|8°4940530/8°49808 2318 °502074.518 °5060303/8°5099503/45 
 16|8°4818050]8 *4859491|8°4900540|8°494.1204|8°4981491/8-5021407|8°5060959/8°5100154/44 
 17|8°4818744/8°48601'79/8°490122118°494.1879|8°498215913°5022069/8°5061615)|8°5100804)43 
 18/8°4819438/8°4860866|8°4901 902/8°4.94.2553|8°4982827/8 °502273 1/8 °5062271/8-5101454/42 
 
 19]8°4820132/8°4861553)8-4902582)8-4943228|8 -49834.95/8°5023393/8°5062927|8°5102104/41 
 20|8°4820825|8°486224.0/8-4903263]8°4943902/8°4984163/8 °5024055|8°5063583)8°5 102754140 
 21/8:4821519/8° 4862927/8- 4903943)8°4944576 8° “4984831)8 -5024717/8"5064239)8-5103404)39 
 
 38 34833991 §-4874583 Has ranae $-4956020 Bagge eo 8-5033951 8°50753'71|8°5114437)}22 |, 
 39|8°4833983]8°4875273|8°4916173]8*4956692/8°499683518 -503661 1/8°5076025}8°5115085)}21 
 40|8°4834.67418°487595'7|8 49 16852|8°495736418 4997501 |8 5037271 |8-5076679]8°5115733|20 
 41|8°4835365 8°4876642 8°4.91'7530/8°4958036]8 4998 167/8°5037931|8°5077333)/8'°5116381}19 
 4.2|8°4836057|8°487732718 491 8208]8°4958708}8*4998833|8°5038590|8°5077987|8°5117029]18} 
 
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 4418 °4837439)8 °4878696)8 491956418 -4960051 |8°5000164)8°5039909|8°5079294)8°5118324)16 
 45 8°4838129|8°4879380]8°492024.2/8 49607238 °5000829|8 -5040569|8°5079947|8°5118972]15 
 46|8°4838820|8°4880064|8 -4920920|8-4961394|8°500149518-5041228/8-5080601]8°5119619}14 
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 4.818 °484.0201|8°488 1432/8 -4922275/8 -4962737|8 '5002825|8°5042546|8 508 1907/8°512091412} 
 49]/8*4840892/8°4882116|8°4922953)8-4963408|8°5003490)8 -5043205|8°5082560/8°5121561)11 
 50|8°4841582/8°4882800/8°4923630/8 4964079 |8 500415518 504386418 °5083213/8 5122208110} 
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 52|8°4842962/8 °4884167|8°4924984/8 4965421 18°5005485]8 50451818 °5084518|8°5123502 
 53|8*4843652/8°4884.85 1/8 °4925661|8°4966092|8°5006149]8 50458408 °50851'71/8°5124148 
 5418°484434.9/8 -4885534/8°4926338|8°4966763|8 50068148 :504.6498|8°5085823|8 5124795 
 
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 55|8°484.5032/8°4886217|8°4927015|8-4967433|8°50074'7818 5047 157|8°508647618°5125442) 5}: 
 56|8°484.572118°4886900]8 °4927692/8 4968 104/8°5008142)8 -5047815/8°5087128|8 5126088) 47 
 
 5'1/8°484641] 1]8°4887583)8°4928368|8-4968774|8°5008806|8 50484738 °5087780|8'°5126735| 3 
 
 58|8°484'7100]8°4888266/8°492904.5/8 496944418 °50094'7118°5049131 |8°5088432/8°5127381) 2 
 
 5918°484.7790|8°488894.9/8 492972118 -49'7011418°5010135|8°5049789|8-5089084|8 5128027] 1 
 
 60]8°48484'79|8 °488963218 -4930398/8 -4970784|8°501079818 50504478 °5089736)8°5128673} 0 
 
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 LOG. COSINES. 88 Deg. 
 
1 Deg. “LOG, TANGENTS. (245) 
 50’ | 51 
 
 1]8°4809616|8°48.51195]8°4892380]8 493517918 °4973598)8 °5013646|8 °5053329|8 509265359 
 218 °431031218°48.51884|8 '4893063)8 °4933855|8°4974269)8 °50 1431118 *5053987/8°5093305|58 
 3]8-4811008]8°4852.574)8 °489374618 °4934.532]8°4974939/8 °5014975)8*5054646/8 5093958157 
 418°481170418°4853263|8 489442918 -4935208)/8°4975610/8:5015639]8 *5055304|8 5094610156 
 518°4812400|8 °4853953]g "4895 1 12/8 °4935885/8°49'76280)8 °501 6303/8 505596218 °5095262/55 
 618 °481309618°485464218°4895794)8 °4936561|8°4976950)8°5016967/8 -5056620/8 "5095914154 
 
 718 °4813'792|8 °4855331|8 489647718 493723718 497762018 -5017631}8°5057277|8°5096566/53 
 8/8 °481448718°4856020|8 °4897159|8:493'79 1 4)8 °4978290/8-5018295)|8°5057935]8 5097218152 
 918 -4815183|8°485670918 489784213 °4938590]8 497895918 °501895818 50585938 °5097870)51 
 10]8°4815878|8 °48.5739'7|8 °48985 2418 °4939266/8°49796 2918 -5019622)8 *5059250|8°5098521/50 
 11]8°48165'74|8 °4858086]8°4899206)8 493994118 °4980299)8 -5020285)8°5059908/8'5099173)49 
 12/8°4817269]8 °48587'75|8°4899888/8 49406 17/8°4980968|8 502094918 -5060565)8°5099824./48 
 
 13]8°481'796418 -4859463/8°4900570/8°4941293)8°498 1638/8 -5021612/8-506122218*51004'75/47 
 14/8 °4818659|8°4860151|8*4901252/8:4941968/8°4982307/8 °5022275|8-5061879|8'5101127|46 
 15]8°48 19353/8°4860839]8 °490193418 *4942643|8°49829 76/8 502293818 5062536/8 5101778145 
 16]8°4820048]8°486152818°4902615)|8°49433 1918 °498364.5)8°502360 1/8°5063 193|8°51024.29)/44 
 17|8°4820743|8*4862216]8°4903297|8 °4943994)8 °49843 14/8 °5024264/8°5063850)8°5103080)43 
 18|8°4821437|8:4862903/8 °4903978|8°4944669]8 *4984.983)8°5024927/8 °5064507|8'5103731)42 
 
 19}8°4822131|8°4863.591/8°4904.66018 494534418 498565218 °5025589|8°5065164|8°5104381/41 
 20/8 *4822826]8 486427918 °490534118 "494601 9]8°4986320/8 5026252)8 -5065820|8°5105032/40 
 2118°4823520]8 -4864.966|8 "490602918 °4946694|8 "498698918 50269 14/8 -506647'7|8°5105683)|39 
 2918 -4824.2.1 413486565418 °4906703|8 °494736818 498765 7/8 °5027576|8°506713318'5106333!38 
 2318 482490818 486634118 °490738418*4948043]8°498832518°5028239/8 506778918 5106983137 
 2418 +482,5602|8°4867028/8 490806518 °4.948'7 1 7/8 °4988994)8 -5028901|8°5068445|8'°5107634/364 
 
 2518482629518 48677168 °4908'74.5]8 *494939218 498966 2)8 -5029563|8 50691018 °5108284|35 
 26/8 -4826989]8 *4868403|8"°4909426]8 -4950066]8 4990330)8 -5030225|8-5069757|8°5 108934134 
 2718 °4827682]8 °4869089|8 4910106|8°4950740)8 °4990998/8 -5030887)/8°5070413)8'5109584133 
 28/8 °4828376]8 48697768 °491078'7/8 49514 1418°4991666|8°5031548/8°5071069/8"5 110234139 
 2918 48290698 48 70463]8 °49 1 146'7/8°4952088/8 °4992333)/8°5032210/8 -507172418°5110883/31 
 3018 *4829'762/8 °48'71149]8°49121 47/8 °-4952762|8 4993001 |8 °5032871/8°5072380/8'5111533/30} 
 
 31|8°48304.55|8 °48'71836/8 49 1289'7/8 -4953435|8'4993668)8 '5033533/8 -5073035|8°5112183/29} 
 
 4.1|8°4837379]8 °4878695|8°4919621|8°4960167|8°5000338/8°5040142/8 -5079584|8°5118673]19 
 4.2/8 483807118 °4879380|8°4920300)8 *4.960839)8 °5001.004)8'5040802)8-5080239)8-5119322118 
 
 43/8 -4838'763/8 *4880065/8:4920979/8 -4961512}8°5001671|8-5041462/8-5080893|8:°5119970)1 74. 
 4418 483945418 -4880750}8°492 1658/8 °49621 8418 -5002337}8°5042122)/8°5081547|8°512061 8116 
 45|8°4840146]8 °488 143518 *4922336/8°4962856|8 '5003005/8 504278 2/8 5082201 |8°5121267 
 
 46|3°4840837]8 *48382120|8°4923015]3 '4963.529/8°5003669/8*5043442/8-5082855/8°5121915|1 4) 
 4.7/8 484152818 '4882805|8 -4923693)8 '4964201|8°5004335]8"50441 02/8 -5083509|8°5122563113 
 48/8 °484222018 -4883489]8 4924371 |8 *49648'73|8°5005000/8°504476218 50841 6318 °512321 1112 
 49)8 484291118 °48841'74|8 *4925049/8 496554418 °5005666/8 504542 118-5084817|8°5123859]1 1 
 50/8 °484.3602/8 488485818 °4925727|8 49662 16)8"5006332}8 '504608 1/8 °5085470/8°5 124506]10} 
 51/8 °4844.292)8 °4885543/8°4926405/8°4966888|8 °5006997|8°5046740/8 508612418 5125154 
 92/8 '4844983)8 -4886227/8 °4927083|8°4967559|8 *5007663)/8 °5047400)8:5086777/8 5125801 
 93/8 48456748 48869 11|8°492776118°496823 1|8°5008328|8°5048059]8 -508743018 5126449 
 54/8 4846364/8 48875958 °4928438)8 '4968902/8 -5008993|8°5048718)8 -5088084|8°5127096 
 
 5518 °484'7055|8°4888279|8°4929116)8 496957318 °5009658|8-504937'718°5088797|8 5127743 
 56/8 °484774.5|8 *4888962|8 °4929793)8°4970244)8 °5010323}8°5050036/8:508939018 "5128391 
 57|8 484843518 '4889646|8 493047 1/8 49709 1 5|8:5010988|8-505069518-509004.218°5 129038 
 98}8°4849 125)8-4890330]8 4931 14818 °497158618°5011653|8-5051353}8-5090695|8 5129685 
 
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(246) 1 Deg. LOG. SINES. | Tape, 
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 “0\8-512867518- 5167264|8°520551418°524343018*528101 718531828 1|8°5355228/8°539 1 863/60 
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 2/8°5129965/8-5168544|8°5206783)8 52446888 °5282264/8 53195 18)/8°-5356455|8 5393079158 
 3}8°513061118°5169184)8°52074 1718 524.53 17/8 °52828 88/8 °5320136/8°53.57068/8 5393687157 
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 518°5131902/8°5170464/8 °5208686/8 °52465 7418 5284 13.5]8 53213728 -5358293/8-5394902)551 
 6/8 °5132548/8°5171104)8°5209320/8 -5247203/8°528475818'°5321990/8 -5358906|8°5395510/54 
 
 718°5133193/8°5171'743}8°5209954)8°524.7832)8 °528538 118°5322608/8 535951 8/8 -539611 7/55 
 8/8°5133838/8*51712383/8°5210588)8°52484.60/8°5286004|8 532322618 536013118 *5396725/52 
 9)8°5134484/8°5173023/8°521 1222/8 °5249088|8 528662718 '532384.4|8 °5360743)/8 539733215 1 
 10/8°5135129/8°51'73662/8°5211856|8°524971 7|8'°5287250|8°532446 1/8 °5361356|8°5397939150 
 11)8°513577418°5174301/8°52124.90]8°5950345|8°52878 7318532507918 536 1968/8 °5398546|49] * 
 1218°5136419/8°51'74941/8°5213123/8°5250973/8°52884.95/8°5325696|8°5362580)8°5399 155/48 
 
 1318 -5137064]8°5175580|8°5213'75'718°525 160 1|8°52891 18/8°53263 13|8°5363 192/8°5399760|47 
 14/8 +5 137708]8'°5 17621 9|8°5214390/8°5252229|8°528974.1 8532693 118 53638048 °5400367|46 
 15/8°513835318°51'76858|8°521502418°5252857/8°5290363|8°532754818 536441 6/8 °5400974145 
 16)8°5138997/8°5177497/8°521565718 525348518 °5290985/8 °5328 16518°5365028|8 54015811444. 
 17/8°513964218'5178135|8°5216290|8°5254112|8°529 160818'532878218'°536564018°540218714.3 
 18}8°5140286|8'5178774|8°5216923)/8°5254'740|8°5292230/8°5329399|8 536625 1/8 °5402794/42 
 
 1918°5140931/8 5 1'79413|8°5217556/8°5255367|8:529285 2/8 533001518 5366863|8 540340041 
 2018°5141575/8°518005 118°52181 89]8°5255995/8°529347418°5330632)8 536747418 5404007140 
 2118°514.2219|8°5 180689]8"°521882219-525662218°5294096|8'°5331 24918 °536808618"5404613/39 
 22)8°5142863]8°5 18 1328|8°52194.55/8°52.5724.918'°5294-7 1 818 "533 1 86518 *5568697)8 5405219138 
 2318°5 143507/8°51 8 1966}8°5220087/8°5257877|8'5295339|8 533248918 *5369308/8 5405825137 
 
 2418-3 144.150}8°5182604)8*5220720/8 -5258504|8°529596 1 18°5333098]8 *5369920/8°540643 1136 
 
 2518°3144'79418 5 18394.2/8'°52213.5218°525913118°5296583/8 533371 418°5370531|8 54070371351 | 
 26)8°514545818°5183880]8°5221985/8°5259757|8°529720418 '5334330|8°537 1 14.218°5407643}34 
 278514608 1/8°5184518/8°5222617/8'5260384|8°529 7826/8 °5334.94.6|8°537 175218°540824.9133 
 28/8 °514672518°5185156!18°522324 918 526101 118°529844.718 533556218 537236318 54038541929 
 2918 °514.7368|8°5185793|8°522388 118-5261 637(8°5299068|8°533617818'°5372974|/8 54094601318 - 
 30/8°514801 118°5186431/8°5224.5 1318 -5262264/8°5299689]8°5336794/8 53735858 "5400066130 
 
 3118°514865418°35187068|8°522514518°5262890/8°5300310}8 533741 01853741 95/8 5410671129 
 3218°514.9297/8'518'7706|8'°522577 7/8 5263.51 7/8°530093 11853380268 °5374806|8 541127628 
 33/8 °514994.0|8°5188343/8 "522640818 -3264 1 43/8°5301552/8 °5338641/8°5375416|8 "5411882127 
 3418 °5150583}8°5188980/8°522704.0)8 -5 264.769]8°5302173)8 533925718 537602618541 2487196 
 35|8 °5151226/8°5189617]8°5227672|8°526539518°5302795|8 °533987218-5376636|8 541309295 
 36}8°5151869}8°5190254/8°522830318°5266021|8°530341 4[8°5340487|8°537724-718°5413697104. 
 
 3718°5152511[8°519089118°5228934/8 °526664'7/8°5304034/8°5341 103/8°5377857/8 -5414302|03 
 38]8-515315418°5191598|8°5229566|8°5267273|8°5304655|8 53417 18|8°53'78466|8°5414907129 
 3918 °5153796|8°51921 64(8°5230197/8-5267898|8 530527518 °5342333)8°537907618 "5415511191 
 4018 °5154438/8-51928018°5230828|8 *526852418°530589518 534294818 °537968618'°5416116/20| 
 4118°5155080/8'51934.98}2 °5231459/8°5269149|8°5306516|8°5343563/8°5380296|/8°541 6721/19 
 4.218 -5155799|8°519407418°-3232090|8-5269775|8°53071 36|8°5344177|8°5380905|8°5417325|18 
 
 4318 +5156364/8°5194'71 0/8°5232720|8-52704.00|8°5307756|8°5344799/8 538 1515|8°5417929]17 
 $4418-5157006|8-519534'7/8 523335 118-5271 025|8°530837518°5345407|8°53821 2418 °5418534]16 
 45/8°515'7648/8°5195983/8°5233982/8-5271651]8°5308995|8°5346021|8°5382734|8 5419138|15 
 46/8°5158290/8°5196619/8°52346 1 2/8°5272276|8 530961 518534663618 °5383343|8 541974214 
 4718 °5158931|8°5197955|8°5235243/8°527290 1 (8531 0235|8'°534-7250]|8 538395218 5420346|13 
 4818 -5159573|8°5197891|8°5235873|8°5273525|8°5310854/8°5347864|8 5384.56 118 5420950)12 
 
 4918°5160214/8:5198526|8°523650318°5274150|8°531 1473/8 °53484-78|8°5385170|8 5421554111 
 50/8°5160856|8*5199162/8'5237133|8-5274775|8°5312093!8°5349092)8 °5385779|8°54221 58/10 
 
 1)8°5161497/8°5199798/8°5237763/8 -5275400)8 53 1 2712/8 °5349706|8 -5386388)/8 5422762 
 52/8°5162138/8°5200433/8°5238393)8-5276024|8 °5313331{8'5350320]8 -5386997|8 5423365 
 53/8 °51627'79|8°5201069/8°5239023)8 -5276648)8°5313950|8°5350934)8 -5387605|8°5423969 
 5418-5 163420/8-5201704|8°5239653/8°5277273/8°5314569/8°5351548/8-5388214|8 5424572 
 
 55 8+5164061/8°5202339|8*5240283|8-527789718-3315188/8°535216118 4388822 85425176 
 56 8°5164701/8*5202974/8-52409 1 2/8-527852118-5315807|8°5352775/8°538943 1/8 -5425779 
 
 718 °5165342/8°5203609|8°524.1542)8 5279 145/8°5316426/8 *5353389]8 5390039 /8 5426382 
 58 8°5165983/8°520424.4/ 8 °5242171|8°5279769|8*531'704.4/8 °5354002/8 539064-:7/8 5426986 
 5918°5166623|8-5204879|8*5242800/8 -5280393'8:5317663|8 53546 15|8°53912955/8°5427589 
 60/3 °5 167264/8°5205514/8°5243430/8 +528 1017/8 °5318281|8°5355228)/8 5391863 8°5428 192 
 
 7) | r, TR) Oar 77 aes alae” as 3/ 2/ 1’ | 0’ 
 LOG. COSINES. ~~ 88 Deg, 
 
 ae Amo 
 
1 Deg. LOG. TANGENTS, 247} 
 * 52’ 53’ 5A! 55’ 50’ 
 
 318*5132918/8°5171533|8*°5209808)/8 5247749) 8'5285363/8°5322654)8 °5359629/8 °5396292)57 
 4)8°5133564)8°5172173|8°52 10443 hipeperdied seer eget 8°5323273 8°5360242/8°5396900 56} 
 5}8°5134211)8°51728 14/8 °5211078|8°5249008)8°5286611/8 5323892 8°5360856/8 5397509 55 
 6(8°5134857|8°5173455/8°5211713)8°5249638 8°5287235|8°5324510/8°5361469,8°5598117): 
 
 718°5135503|8°5174095|8°521234818°5250267/8°528785918 5325 129'8-5362082|8 5398725153 
 818:5136149|8°5174735|8°5212982|8 5250896 8°5288483/8°532574.7/8°536269618°5399333/52 
 9/8-5136795|8°5175375|8°521 3617/8525 1525/8°5289106|8°5326366|8°5363309)8 5399941)51 
 10/8-5137441)|8'5176016)8°5214.251|8°5252154/8°5289730/8°5326984)/8°5363922)8 5400549|50 
 11/8°5138087|8°5176656|8°5214886|8°5252783|8°5290353/8°332760218°5364.535,8'5401157/49 
 12/8°5138732'8°5177296|8°5215520/8°5253412/8°5290977|8°5328220/8 5365 14818°5401765/48 
 13)8-°513937818°5177935|8°5216154|8'5254041/8°5291600|8°5328838]8°5365761 |8°5402372/47 
 1418 °514.0023/8°5178575|8'°5216789|8'°5254669|8'52922293)8°53294.56)8°5366373'8 5402980146 
 15]8°5140668/8'5179215/8"521742318°525529818°529284.7|8°5330074|8 536698618 '°5405587145 
 1618°514131418°5179854)8°5218057|8°5255926)8 '529547018°5330692|8°5367599|8 "5404 195/44 
 1718°5141959]8°51804.9418°5218690|8'°5256555|8'°3294093|8°5331310]8°536821 118°5404802143 
 1818 +514.260418°5181133/8°521932418'°5257183/8°5294.716|8°5331927/8°536882318 540540942 
 19]8-514324.9]8°5181772|8-5219958|8°525781 118°5295338/8°5332545/8°5369436/8'5406017/41 
 ‘1 20/8-5143894)8°518241218 522059 118'52584.39/8 °529596 1/8 °5333 1 62/8°5370048|8°5406624140 
 2118-5144.53918°5183051|8°5221295|8°5259067|8°529658418°5333779|8°5370660/8°540723 1139 
 2218 -514518318°5183690/8-5221 8.58]/8°52.59695|8°529720618'°533439718°537197918-°5407838138 
 23/8 -514.582818°5184.32918-5229499/8°526032318°5297829/8°5335014|8°53'7 1884/8 "5408445137. 
 2418 °51464.72|8"5 1849678 "522312518 "526095 1/8°529845 1/8 °533563118°33724.96|8'°540905 1136 
 2518-51471 1'718°5185606|8°5223758|8°5261579|8°5299073/8°533624818°537310818°5409658|35 
 2618514776 18°51 8624.5|8°5224.39 1 |8°526220618 5299696|8 °-533686518'°5373719|8 5410264134 
 2718-514840518°5186883]8°5225024|8°5262834/8'5300318]8 533748218 '°5374331|8°5410871|/33 
 28/8 -514.9049]8°5187522/8°5225657|8'526346 1/8 °530094.018 533809818 °5374942|8 541147732 
 2918 -5149693)}8°5188160|8°5226290|8 526408818 530156218-533871 5|8°5375554/9°541 9084/31. 
 15018 °515033'7/8°5188798/8°522692918 526471 6|8°530218318°533933118°5376165/8'5412690|30) 
 
 31)8°5150981|8°5189436|8°5227555|8'5265343|8°5302805)/8°533994818 537677718 °5413296|29 
 
 36/8 °3154199)/8°5192626|8°5230717|8°5268477|8°53059 1 2)8°5343029|8°53798398-°541 6326, 
 
 718 5154842|8°5193263|8°523134.9}8°5269103/8'5306534|8°5343645|8°5380442|8°5416931 
 38/8°5155485|8°5193901|8°5231980/8°5269730 
 
 4318°515869918'519708718'5235139/8°527286118°5310259|8°534733918°5384105/8°5420563)17 
 4418-5 15934.218°519772418 523577018 327348718 531088018 °534795418°5384-71 5/8°5421 168/16 
 4518 :515998418°5198361/8°52364.0 11852741 1318 °531 150018 °5348569/8°538.532518 5421773115 
 4(18*5160627|8°5198997/8°523703318'°5274.759]8°531219118°534.918418°5385935(8°542293781 14 
 4718°5161269|8°5199634/8°5237664/8'5275364/8 53127441 18534979918 538654518 °3422983/13 
 4818+5161911|8°520027118°523829518°5275990|8°531336118°5350414/8°5387155|8°5423588)12 
 4918 +516255318°520090718'°5238926|8°527661518°531398 1/8°5351029|8°5387765/8°54241935]1 1 
 5018°5163195|8*°5201543/8°5239557|8°5277 24.118 °5314601 185351 644|8°538837418 542479710 
 91/8-5163837|8°5202180/8°5240187|8°5277866]8 °5315221|8°535225918 53889848 5425402 
 5218-5 1644'79|8°52028 | 6|8°5240818/8°527849 118 °531584118°535287318 538959318 '°5426006 
 5318°5165121/8°5203452/8°5241449/8 52791 16|8°5316461/8°5353488]8°5390203|/8 5426610 
 5418°5165762|8°5204088|8'5242079/8'°527974118°531708118°535410218°53908 12/8 °5427214, 
 9518 °5 16640418 520472418 °5242709,8°528036618:5317700]8°535471'18°539142118°5427819 
 5618 °516704.518°5205360|8°5243340/8 528099 1/8°531832018°535533118°5392030]8 5428425 
 5718 *516768'718°520599519°52439'70|8'°528161618°531893918'°535594519°5392639]8 5429027 
 5818°516832818°5206631|8°5244600/8°528224118-531955918°5356559|8°5393248]8 5429631 
 59/8°5168969|8°5207267|8'5245230/8°528286518:5320178/8 535717318 °539385718 5450234 
 60/8-5169610|8-5207902|8°5245860|8 5283490)3 532079718 °5357787|8 539446618 5430838 
 
 AES ORE See 3 a lake 
 - LOG, COTANGENTS, 88 Deg. 
 
 lomo mee Gy. <I 92.0 
 
 ~ 
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(248) ODeg. © NATURAL stnES,&c. 
 Sine |Dif.|Covers}| Cosec. | Tang.|Cotang.| Secant | Vers. 
 
 0/0000000 2909 1000000} Infinite. |0000000] Infinite. {1-0000000|0000000 
 1}0002909 2909 99970913437 *7468]0002909|3437 +7467] 1 0000000/0000000 
 2\0005818 2909 9994182]1718°8735|0005818}1718°8732)1 -0000002)0000002 
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 4:0011636 2908 99883641859 °43689/001 1636|859°43630}1 -0000007/0000007 
 5}0014544 2909 9985456|68'7°54.960|0014544/687°54887|1°00000111000001 1 
 6|0017453 9909 99825471572°95809|001 7453/5 72°95721}1°000001510000015 
 0020362 9909 9979638}491+10702}0020362)491 -10600}1 -0000021/0000021 
 
 271 
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 39|0113444| qo | 
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 4210122170] gq (08778308 1°853150/01221 79/81 8470410000746 0000746 
 
 43]0125079 2908 9874921179 *949684/0125088/79 :943430)1-0000782,0000782 
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 4510130896 2909 9869104)76°S96554/0130907|/76-390009/1-0000857 
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 50)0145439 2909 9854.56 1/68 °757360/0145454/68 °750087]1 0001058/0001058 
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 52/0151 256|gon,919348744/66° 1 13036/0151273)66°105473}1 -0001 14410001144. 
 53}0154165 9(9045835164°865716 
 
 54\0157073 2909 9842927165 °66459510157093|63 656741 |1*0001234)0001234 46 
 
 55]0159982 9840018]62°507153}0160002|62°499154/1'000128010001280 
 
 5610162890|so9¢ : 
 5710165799 2908 9834201/60°314110}0165821/60*305820}1 :0001375}0001375 
 58}0168707 9909 9831293}59+274308)0168731159 *265872|1 :0001423/0001423 
 
 5910171616 2908 9828384158 °269755|01'71641158 +261 174}1-0001473/00014.73 
 
 60]0174524) 98254.76]5'7°298688]01'74551157-289962]1 -0001523|0001523 
 Secant |Cotan.| Tang. ! Cosec, [Covers|D| Sine_ 
 
 a CosinelDif. Vers. 
 ’ NATURAL SINES, &c. 
 
 0000857), 
 
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 37 
 38 
 
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 40/9999065113 
 
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 47|9998720 5] 
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 9998695} 3 
 3519998577 2 
 2519998527] 1 
 
 9998477 0 
 
 Deg. 89 
 
 » 
 
O Deg. LOG. SINES, &C. (249) 
 
 — . | Diff. | Cotang. | Covers.| Secant |D| Cosine | _ 
 i OT i a cl A Tea 10-000000}10-0000000}  9|10-000000)6¢ 
 “Ojinf. Neg.| 7. | Infinite [Inf. - Neg.|Inf. Seeds Infin. |15.5369739/9-999873"1|10-0000000 ;{10-000000}5$ 
 1/6-4637261), 298D- |15-536973912°6264229 P1860 726415010307 13-2352438|9-9997473}10-0000001} ,|9-9999999)58 
 9f6- 7647561}, 1h hgi of! 3:235243919'228482216-764756Il ocoo1 5 13-0591525|9-9996208|10-0000009} 1/9°9999998)57 
 316+94.08473 1760912 13°0591527/3°580664'716:94084'75 1249388 a eaaeian 9+9994944110-0000003 5 9-9999997156 
 4|7-065'7860 1249387 12°934.214.0/3°8305422 706571863 969101 acedied 99993679 10:0000005 919:9999995 55 
 5]7-1626960) 75, gy |12°8979040/4-0245622}7-1626964 791814] 19.758199019-9999414|10-0000007| 2199999993154 
 418771] 1218111 9-7581999|4°1827246|7-2418778 669470| 12" 981222 ae PE 
 6/7241 8974) eoa Gs « "119-691175919°9991148]10-0090009 3|9°999999 15 
 7) 1°3088239 Beppe ig ells hi dod Pay 0 12-6331831|9-9989882|10-0000019} 319-9999988)52 
 of7-4179681] 21 1224119-5820319 4°5349070/7 4179696) a sm 5c0 12-5362727|9-9987548}10-0000018} 5 |9°9999982150 
 10|7-4637255] 927 4119-5369745|4-626421917-4637273 413930)) 9.3.94.979719-9986081110-0000029 4(9°9999978]49 
 1117°5031181 413926 12°4.9488 19)4°709207217°5051203 3778881 19.4.5 9-9984814110'0000026] -/9°9999974148 
 5429065] 2178841 19.4570935/4-7847843]7-5429091 847604) 2 4970909 5 MTA 
 pi We ts US t3 ba ‘ 3195 1|12°422328619°9983546|10-0000031] .|9-9999969|47 
 13/7 3776684) 30 9 1g) 12422331 6/4°8543084/7-5776715) so 954 12+3901434]9°9982278|10-0000036 5 {9999996446 
 14/7°6098530) 649639)! 2°3901470)4°9186777/7 6098566) 999635 12-3601799|9°9981009|10-0000041] 219°9999959/45 
 16|7°6678445] O53500|12:9521555]5-0346614]7-6678492 263294] 19.395891419-9978471|10-0000053) °/9-9999947|43 
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 ae 2482331 9.2810034/5"1369663/7"7190026 2348 1 5|12°2809974 2 6 sh 
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 12-21404.99|9-9973389110-0000081 
 202039 12°1938453/9°9972118110 0000089 
 193057 12°174.5396|9°9970846|10:0000097 
 
 184840 12°156055619°9969574)10 0000106 
 Fac 72ll Seanadonih-oadeadelOipanit 
 
 170339 12+1212993/9°9967029|10-00001 94. 
 163911 12°104901919:9965756|10°0000134 
 157950/10-0891069|9-9964483|10-0000144 11|2'9999856)32 
 15240611 9.973865619°9963209{10-0000155 Lol? 9999845 )31 
 14724011 9.0959 141619°9961935|10-0000165 19} '2999835)30 
 
 142412 m7|, ,|9°9999823|29 
 12-0449004}9°9960660}10-000017], meal! | 
 
 13789011 9:931111419-9959385|10-0000188 19)? '9999812)28 
 13364811 9.917746619-9958110|10-0000200 1912 '9999800/27 
 12965811 9-0047808|9'9956834|10-0000219 1312 2999788)26 
 12590011 1 .992190919-99555581!0-00002e5 19|2'2999775}25 
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 1G19|11'9235003}5'8519848]8-0765306| 14464) 11-9130030/9-9946615}10-0000324),219-9999676]18 
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 211890 12°214.0573]5°2708595|7°7859508 
 202031 12°1938542|5°311266117°8061547 
 193049 12°1745493|5°3498763]7°8254604 
 184831 12°1560662|5°3868430)7°843944.4 
 
 28 l19-1383377|5-4223003)7-8616738 
 170330 12°121304'7|5°4563669]'7 8787077 
 163901 12°1049146|5°48914'7517°8950988 
 157939 12-0891207|5°52073.5917°9108938 
 152397 12°073881015'5512157/7°9261344 
 14.7229 12°059158115°5806620|7°9408584 
 
 1091 9-0449181|5-6091427]7-9550996 
 137879 12°0311302|5°6367191|7:9688886 
 133636 12°0177666|5 6634468|7°98229534, 
 129646 12:0048020|5'6893'765|7°9952192 
 125887 11°9922133]5°7145546|8 0078099 
 122340 11-9799793)5°7390233 80200445 
 
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 11-7959297|6°107138448 2041259 
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 Cosine | Diff. | Secant | Covers. Cotang. 
 
(250) 1 Deg. NATURAL SINES, &c. 
 
 0)0174524 999919825476 |57298688}01 7455 1|57'289962|1 -0001523|0001523 
 10177432} 9|9822568}56°359462/0177460}56°350590I1 -0001574/0001 574 
 2/01 80341] 921981 9659]55°450534)0180370155 4415 17/1 00016270001 626 
 310185249] 56 g1981675 1]54°570464)0183280}54°561 3001 °0001679/0001679 
 410186158|59,).|9813842|53-71 7896|0186190}53-708587|1 °0001733|0001733 
 s A 999 8|9810934,92°89 1564/0189 100}52:882109]1-0001788]0001787 
 5JO19I9 TAC o ng 
 
 7|019.4883 bi 9805117)51°312902}0194920)51 3031571 0001900|0001899 
 
 810197791) 99.g]9802209}50°558396)0197830}50°548506)1 0001957}0001956 
 
 91020069559 )9]9799301}49°825762)0200740)49 8 15726]1 0002015 0002014 
 
 1010203608} 59 y9)9796592)/49"114062)0203650]49°103881 1-0002075 0002075 
 
 111020651 6).,99,4/9793484|48 42241 1/0206560 48°412084 1-0002133 0002133 
 
 1210209424) 49. 3197905'16|47 7499 74,0209470]47°739501]1 ‘0002194/0002193 
 
 13]021 2332}, 991978766847 095961021 2380|47-085345]1 °0002255}0002255 
 
 [410215241] 559919 784759}46°459625/021529 1]46-448862/1 -00023 170002317 
 
 15}0218149},9 491978185 1/45 °840260)0218201)45°82935 11 0002380/0002380 
 
 16/0221057|,99g19778943)45 °237195|0221111]45°226141]1 000244.4/0002444 
 
 1710225963] 9999|9776035|44"649795}0224021 |44°638596] 1 -0002509}0002508 
 
 180226873 2908 9773127|44°07745810226932/44°066113}1°0002575}0002574 
 
 19/0229781 a 9770219/43°519612)0229842}43°508 1 22}1°000264110002640 
 
 2010232690) 5999197673 10/42°975713)0232753|42°964077)1 0002708|0002708 
 
 2110235598 |g 99,919 764402|42°445 24.5|0235663]42°433464 1 -0002776|0002776 
 
 22/0258506)99 991976149441 °92771 70238574141 915790}1 000284510002844 
 
 251024141 4/59 19758586 /4 1 422660]0241 484/41 -410588/1°0002915]0002914 
 
 9410944322 2908 9755678/40°929630)0244.395/40°917412}1°0002986|0002985 
 
 25102472305 99219752770 /40-4482010247305/40-43583"7|1 -0003058|0003057 
 26}0250138)599919749862:39°977969;02502 1 6|39-965460|1 °0003130|0003129 
 27102530461. 9746954395 1854910253127139°505895|1°0003203/0003202 
 28}02559541599,919744046 39 069577 110256038|39-05677 1/1 -0003277|0003276 
 291025886 2|5 99/9741 138 38°630683/0258948]38-617738}1 -0003352|0003351 
 
 : 38°201550|0261859|38°1884.59|1 °0003428/0003427 
 
 3110264677 990919735323 3'7°%81849'0264770137'768613|1°0003505|0003503 
 3210267585 2908 97324.15'37°371 2731026768 1|37°35789Q]1°000358210003581 
 3310270493 99919729507 36°969528)0270592)36-956001/1°0003660)0003659 
 3410273401) 949919726599 36576332 0273503]36-562659|1°0003739|0003738 
 35|0276309|9 6 5x|9723691 3619141 4(027641 4|36°177596|1°000382010003818 
 
 36|027921 6) 5991972078435 °814517/0279325)35 800553] 1*0003900|0003899 
 
 371028212419 9.19717876|35°44539 1|0282236|35 43128911 -0003982|0003980 
 38}0285032) 54019714968 |35-083800}0285148|35°069546)1 0004065}0004063 
 3910287940150 41971 2060/34"7295 1 5}0288059]34"715115]1 :0004148|0004146 
 401029084715 99.9|9709 15334-38231 6|0290970|34°36777 11 0004232}0004230 
 41]0293'15 5159 n1970624.5|34°041 994|0293882}34'027303|1 *000431'7/00043 16 
 
 4.910296669),,49 19703338|33-708345|0296793|33-693509|1-0004403(0004401 
 
 2008 97004.30|33°38117610299705|33°366194|1°00044.90|0004488 
 ‘14410302478 290% 9697522133 0603001030261 6|33°04.51'73|1°0004.578/0004.576 
 
 45}0305381,54/9694615(32°74.553710305528132°730264|1*0004666|0004664 
 9691407|32°4367 1310308439]22°421295]1 0004756|0004753 
 9688800|32°133663]031 13.51132°118099]1°0004846|0004843 
 9685892)/31 '836225}0314263|31°820516]1°000493'7}0004934 
 
 9682985/31°544.24.6103171'74/31°528392)1 :0005029|0005026 
 9680078|31°257577103820086]31°241577/1°0005121|0005119 
 5110322830 290% 9677170|30°97607410322998|50°959928]1 °0005215|0005212 
 5210325737 090N 9674263|30°699598]0325910130°683307]1 °0005309}0005307 
 53]0328644 9908 9671356|30°42801'7}0325822130°411580]1 -0005405|0005402 
 5410331552 290% 9668448)30°16120110331734)30"144619}1°0005501)0005498 
 
 + 
 
 55)03344.59 2904 9665541}29°899026|0334646]29*88229911 -0005598}0005595 
 56|0337366 9908 966263429 °641373}0337558]29°62449911 *0005696}0005692 
 5'710340274)- 9659726)/29°388 1 240340471 |29°371106}1°0005794)0005791 
 96568 19}29°139169}/0343383]/29°122005}1'0005894/0005890 
 2904 96539 12/28 °894.398}0546295)|28°877089 1°0005994)0005991 
 | 60}0348995 9651005]28°653708/034.9208]28 *636253} 1 °0006095}0006092 
 
 Cosine]Dif.} Vers. | Secant |Cotan. 
 
 ~. 
 
 49}0317015 
 
 5010319922}, 5 5.4 
 
 9808026}52°090279)0192010)52°080673]1°0001843)0001843 56 
 
 ‘88 
 
 101 
 101 
 
 Tang. | Cosec. |Covers| D} Sine 
 
 Tab. 10. 
 
 Sine |Iif|Covers| Cosec. | Tang. | Cotang.} Secant | Vers. | D [Cosine 
 
 511999847760} 
 5919998426159 | 
 5319998374 ]58 
 5 419998321157 
 5 419998267|56 
 BGP 998213}55 
 9998157|54] 
 51999810153] 
 7 g{9998044) 526 
 591999798651 
 B19 997927}50 
 6019997867149 
 69[9997807)48 
 Gol999TT45]47 
 651999 7683]46 
 64(9997620]45 
 64 (9997556/44 
 BG (999749243 
 GaI0997426)42 
 63 (9997360 +1 
 6319997292] 40 
 6819997224) 39 
 (9997156138 
 7 1{9997086|37 
 791999701536 
 17919996943]35 
 751999687 1/34 
 74 |9996798|33 
 175|9996724|32 
 519996649)31 
 7G |0996573}30 
 7g|9996497|29 
 9996419|28 
 7919996341]27 
 $0 19996262|26 
 3 1|9996182]25 
 31 |9996101)24 
 
 g3i9996020/25 
 g3|9995937}22 
 g4,9 999854421 
 86 9995'T70}20 
 85 9995684|19 
 8h 9995599}18 
 
 9995512117 
 9995424116 
 9995336|15 
 999524'7}14 
 999515715 
 
 9 9995066] 12° 
 
 93 9994974111 
 9994881}10. 
 
 9994598 
 
 9919994110 
 
 9994009]. 
 9993908 
 
 (Yo) 
 to 
 ie) 
 - 
 a 
 oO 
 Or 
 SO Mook Or HD-10 0 
 
 ~ 
 
 D CS. 88. 
 
1 Deg. - LOG. SINES, &c. : (251) 
 
 9-9999358|60 
 5219999931659 
 14 9+9920964|10-0000706|2219-9999294158 
 117368847 5, (9:9999271/57 
 62° |9-9999224155 
 2419-9999200154 
 I9-9999175 
 
 2519-9999150 
 25 
 
 67326] 5.» 
 66298}, 1 
 65300 
 
 64333 
 63393 
 
 6|8°2832434, 
 718 °2897 734. 
 
 53 
 52 
 
 i 
 
 aq rl 9°99991 95151 
 11-6974540|6-3041058|8-3026335|°. OH acadionls0 
 6248111 1 -6912059]6-3166 2619 .9999974149 
 
 1218-3210269 dee 11-6789731|6-3410714/8-3211221 
 
 59080 
 
 27 
 
 Sactibe 
 
 “111°6728857|9°9906'792]10-0000979 | 
 2910611 1 -6669751]9°9905501]10-0001006 dele mason 
 58286 2831411 1.661 143%719-9904210 10-0001034 anil 2998966 
 S514 IT542 11°655389519°9902919]10-0001061 9 9:9998939 
 
 : [56790 7|10-0001089)7"/9°9992911 
 56030) 1°6498195|6°3993855|8°3502895]2 245 « 180le oedasas 
 56030 6°4105928|8°3558953]>, 4) 2 eoscaig Nien 
 55315 9-9998853 
 2919-099882.4 
 ae 9-9998794 
 
 45 
 hd 
 43 
 42 
 al 
 40 
 39 
 
 2154648 
 11°633223116-432.3826|8-366894.5 Bonk 
 11-6278290|6-4433720|8'3722915)25, 
 
 11°6225012|6°4.540294|8°3776223 
 
 “19:9998577|52 
 
 9319 +999854.4131 
 39 
 
 ae 47978) 
 31 4297168 aris? 
 
 146940 
 
 [45945 
 45465 
 
 45500 
 45028 
 
 19°9998971|23 
 3619-9998935/29 
 3 (9°9998199/21 
 Dl 
 27/9 +9998125|19 
 re 9-999808g]18 
 nal 9 899805017 
 ~919-9998012|16 
 "319-99979%4. 
 
 “19-9997935114 
 
 44115 
 43672 
 43239 
 42813 
 
 #2995) 1 .593306719-9867910110-0001950 
 41585 11°514949519°9865306]10:0002026 
 #11911 1.5 10830419-9864003|10-0002065 me 
 no |40806 11°506749819-9862700|10-00021041?~ |9°8997896|13 
 
 40054 : 
 
 4 s 
 
 TL 1 *4'75414019-9852262|10-0002430], .|9-9997570 
 flict 11-4'71651019-9850955|10-0002473 i 9-9997527 
 hed me ® 137 7 peters 97484 
 11-4681719|6-7627520|8-5520797| safe 909788: 
 
 37264 
 36947 
 
 ~ianieaae St Saas re 
 Sete J ee 
 OOOO SS 
 
 Diff} Secant |Covers.|Cotang.| Diff] Tang. Verseds.! 7 iis 
 3K 2 Deg. 88. 
 
 Cosine 
 
ct mt 
 
 9993390155 
 
 1064390 
 107 9993284154 
 
 2906 
 3 2907), 
 2907 
 
 12/038387 
 
 113 9992517}47 
 114 9992404 46 
 114 9992296 40 
 116 99921'76)44. 
 
 9992060)43 
 
 ll. 6isaqtaaatan 
 117/9991944}42 
 
 2564183211 -0007602|0007596 
 25°451700|1°0007716|0007710 
 
 125 
 127 
 
 22-602015|1-0009783|0009773 pe 
 
 22°4'76353|0445353/22'4.54096]1°000991 2\0009902 130 
 22°3304.99|0448268/22°308097)1-001004.210010032 131 
 9549276|22°18652810451 183/22°163980}1°0010173\00101 
 22°02171011°0010305100102 
 
 954346412 1°904090]045'7012/21°881251]1 0010438100104 
 
 63 
 131 
 94, 133 
 
 $3919514408|20-593409|0486 1 66]20-569115]1-001181 10011797 
 4810488498|-> 0 9195 11502120-4'70926|0489082 
 4910491403]... .{9508597}20°349893]0491997|20-325308|1-001 209610012081 
 
 2905 
 5010494308 9906 9505692|20°23028410494.913)/20-205553}1 001 223910012295 
 eJUO ; : A.QM 
 4 is 
 
 949988 1}19°99524110500746|19-9'7021911 0012529 0012514 
 
 » {2905 
 
 1520986095 
 
 ——— OOOO |] OO OS | | [I sd] 
 
 or S or 
 
 » 
 
ODeg. Lt LOG. SINES, &c. | (253) 
 
 0|8-5428192 11*4569162|9-9845725|10-0002646 = peau 
 1 55461216 sen “458309 1|9-984441'7|10-0002691|;» 2-999750959 
 918-5499948)5> 3. 4497317|9-9843 108|10-0002735], .|9°9997265 58 
 (9)8°5535386) 50755 4461834/9-9841799|10-0002780},,4|9°9997220157 
 4]8°5570536}5, 02. 4496 638]9-9840490|10-0002826), .|9°9997174)56 
 15]8-560540415 7-04 “439 1'724|9°9839180|10-0002872\ 199997 126)55 
 |6)8'5639994) 3455 ‘435'71088|9-9837869|10-0002918}, .|9°999 el 
 17)8°5674510)5.4 047 3] 1-4322725}9-9856559|10-00029641,,,|9-9997036153 
 |8)8:5708357|35-.05 114988632|9°9835248|10-000301 1], ]9°9996989|52 
 
 gls-a742139/05-° 
 (0)8-5775660|355 05 
 ‘q/8-5808923 
 $+5841933}> 0010 
 2 32761 
 
 gl8-5874694).,, 
 
 4/8-5907209 i. 
 5]8°5939483).5,.04 
 glg-5971517I0~ 
 
 *4.254.80319°9833936|10-0003058 ” |9*999694215 1 
 ‘4221 23419 9832624) 10-0003106];,9|9°9996894/50 
 ‘4187923/9-983131 21100003154] ;,¢19°9996846)49 
 -4154864|9-9830000|10-0003209|,,)9-9996758/48 
 11+4122055|9-9828687|10-000325 1] ;,919° 9996749 |47 
 411 1-408949 119°9827373|10-0003300 Rolo 9996700 46 
 4057168|9°9826060}10-0003350)4.99°9996550)4< 
 4025083|9°9824745|10-0003399) 5 5 |9 9996601 44 
 *3993233|9°9823431|10-0003450]5 919°9996550/43 
 *3961614/9°9822116|10-0003500}. 1 |9 9996500 42 
 11+3930293/9°9820801|10-0003551]5 ;|9°9996449141 
 11+3899057|9-9819485|10-0003602}5,9|9°9996398|40 
 11*3868111/9°9818169}10-0003654},)9'9996346/59 
 11+3837384|9°98 16853}10-0003706]5 9|9 999629433 
 11+38068'73|9°9815536|10-0003758}..4)9°999624. 2/37 
 11°3776573|9°9814219|10 000381 1}~.|9°99961 89/56 
 
 AK 
 Te 
 
 31569 
 31340 
 31115 
 30894. 
 30675 
 30459 
 {30247 
 
 30037 
 
 11-3780584|6-9430837)8-6223497|5 500 
 11-3750347|6-949093918-6253518 
 
 } ‘ 29032 
 0|8°6396796 28838 
 
 29086|1 4. abe 
 9ga94{_!'3599069)9-9806308 
 
 56 “ie 
 10-0004303}y¢|9°S 
 
 11-3432983/7-012596918-6571490 herd 
 11-3405259/7-0181461|8-6599279|,6 4 
 
 27440 
 27267 
 27099 
 
 11+3345669}9-9794415|10-000464'7|5¢ 
 11+3318402|9-9793092|10-0004705}5, 919° 
 11+3291303/9-9791768]10-0004764|¢9|9° 
 11°3264372|9-9790444|10-00048241 ¢ 019° 
 
 11-3237607)|9-97891-19}10°00048841¢ 09° 
 11°3211004|9:9787795}10°000494.416 0/9" 
 11+3184563|9-9786469]10-0005004]¢ |S 
 11+3158281|9-9785144|10-0005065}¢ )|9° 
 11°3132156|9°9783818} 0-0005126|g0)9° 
 11+3106187|9-9782491110-0005188i¢5 
 
 25663}! 1 3080371)9°9781164}10-0005250) 55 ; 
 O55 Aft t'3054708)9 97798371 0-0005312) ¢|9°9994685110 
 2536 6| | 1'3029194|9-9778510)10-0005375|g.419'999462.5 
 7219593 g|11°3003828|9-9777182]10-0005438)¢ 4 
 250%15|1 1'2978610|9°9775853}10-0005502| ¢4)9" 
 24930]! 1'2953535/9°9774525)10-0005565}¢,5,)9 9S 
 11-2928605|9-9773195}10-00056301¢ ,\9°9994370) : 
 ~1.g| + 1*2903815/9°97'71866]10-0005694¢ .|9°9994306 
 111 -2879166|9-9770536}10-0005759|¢ 5 
 11+2854655|9+9769206|10-0005824),4)9° 
 "283028 1]9°9'167875}10-0005890\¢ @|9°9994110 
 11+2806042\9-9766544}10-0005956) |9°9994044 
 
 ——_— | ————_— ——] | | I 
 
 0|8°6676893 
 '1|8°6703932 
 
 26765 
 
 26603 
 26441 
 26282 
 26125 
 25969 
 25816 
 
 _—_——— | | eS | 
 
(254) 3 Deg. NATURAL SINES, &€¢. Tab. 10. 
 
 Vers, |D.\Cosine 
 
 2 a ee 
 
 210529169 mich 9470831]18°89754.510529912)18°8'71068]1 °0014030/001401 1], ¥ , |9985989}58 
 3/0532074|< >"? 194.67926|18*794377|0532829|18°76775411-0014185/0014165 *19985835157 
 410534979 290: 9465021118 °69233010535746| 18 °665562\1°001434.1}0014320 9985680156 
 510537883} 9 9 *194.62117|18°591357|0538663| 18564473] 1°0014497/00 14476 Lomo 24/95 
 610540788 390 5 94.5921 2)18-491530|0541581}18-4644'71}1 -0014655)0014633} ,; 9985367|54 
 
 710543693 ety 94.5630] 18°39274.210544498118°365537|1°0014813/0014791 159 9985209153 
 810546597 9453403|18°295005]034 741 6|18-267654}1-0014972|0014950| | 54 998505052 
 
 9105495091290 194.504.98118"198303510550333|18+1708071-001513210015109|,>. 1998489 1151 
 
 10/0552406|9941o4.47594|18-102619|0553951|18-0749771 -0015295|001 5269] 2) [9984731]50 
 111055531 1\790"19444689]18-00793'710556169|17-980150]1 -0015454/00154301+04|9984570149 
 
 1210558215 con 9441785|17-91424310559087|17-886510]1-0015617|0015599|, 25 9984408148 
 
 1310561119|54,,|945888 1]17°82152010562003|17-793442|1-0015780/0015 75511 64 998424.5)47 
 1410564024. 99n4|9439976|1'7°729753)0564925 17:701529}1°0015944/0015919], ., (998408 1146 
 1510566928], 6), |9433072|1'7°638928|0567841]17°610559|10016109)0016085}, ¢. 9983917}45 
 1610569832|54 9 ,|9430168|17°549030]0570759|17'5205 16/1 -0016275|001 6249), --\998375 1/44 
 1710572736|5— 419427264 17 °4.6004.6]0573678|1'7°431385]1 :0016442/0016415|, ..9983585}43 
 
 18|0575640|,,,4|9424360|17-37196010576596\17-345155|1-0016609)0016589|  ¢ .0983418}42 
 
 19]05785441,,. 04{9 $214.56] 1728476 1]0579515]17-255809]1 00167781001 67501 6g 9983250}41 
 994.18559}17°198434|0582434}1'7°169337]1 °0016947|0016918],_,|9983082/40 
 
 20/0581 448 
 2110584352 sel 9415649] 1'7°112966|0535359|17-083724|1.0017117]001 7088]... [9982912139 
 22105872565. 4|941 2744) 1'7°028346|0588271|16-993957|1 -0017288/0017258} ;.,|9982742)/38 | 
 
 23105901 60}54, , [9409840] 16:944.559|0591 190]16+915025]1-0017460)0017430 “a 9982570)37 
 240593064] 4419406936) 16-861594}0594109| 16°851915|1 00176351001 76091 | .|9982998)36 
 
 251059596719 9,)4|9404033|16-779439]0597029|16-74961 441 0017806(0017775} tng 9982225135 
 2610598871 9401129]16-698082|0599948|16+668112|1-0017981|0017948} ;.|9982052134 
 27|0601'7'74 9398225|16+61751 2\0602867|16-587396|10018156|00181 23], -7|9981877}33 
 28/0604678| 5 9:4|9995329|1 6°5377717|0605787|16-507456|1 °0018332/0018299 melon ol 82 
 29}0607582) 95 ,4)959241 8] 1 6°458686|0608706|16-428279I (001 850410018475} | 772981525131 
 30|0610483|5 99 ;|9989515)16°380408/061 1626|16-349855|! 0018687001865] 1. 9981348}30 
 
 3110613389} 49.4{938661 1|16-302873|0614546|16-272174|1-0018866|0018830 9981176}29 
 3210616292 2004 9383708}16 +226069|0617466]16-195225]1-0019045|0019009} ; 44 998099128 
 3310619196]. ,<{9380804|16- 1499571062056] 1 6-118998|1 -€019225}0019189), ¢ 9280811127 
 34{0622099|55,,19377901|16 0746 17|0623306|16-043489]1 0019407100 19369} ; ¢ {9980631126 
 3510625002 9374998] 15+99994810626226|15-968667]1 °001958910019550), g|9950450)/25 
 360627905154 94|93720935|15°925971 (062914715 -894545]1:0019772|0019735} |g. 9980267/24 
 
 7] 
 3740630808}. 9,|9369192}15°852676|0632067]15+821105}1 -0019956|0019916} 1, 9980084193 
 38/063371 1/56 )3|9366289| 1 5:780054{0634988}1 5 -748397]1 °0020140]00201001 997990022 
 3910636614] 9 9|9363386|15-708096|0637908}15 -676233} 1 -0020326|0020284) | ¢¢ 9979716)21 
 40106395 17}, 4019360483] 15°636793|0640829]1 5 -604784 | 0920512100204701, a. 9979530}20 
 4110642420 935'7580|15°566135|0643'750|15°533981]1:0020699|0020657], a. (99 79 943}19 
 
 42|0645303},4019354677|15-4961 14|0646671|15-4638 14]! 0020887]0020844) 5 ¢|0979156]18 
 
 4310648226], 4.9{9351'774]15°42672 1]0649599! 15-394276|1 0021076|0021032 9978968)17 
 4410651129 9348871]15+357949/0652513}15-325358]1 ‘0021 26610021221), 4,|99 78779116 
 4510654031]>" “1934396911 5°289788|0655435|15-257052]1°002145710021411], 5 [9978989115 
 
 2903 
 
 47|0659836|9 494 ; 
 480662739], 4.4{933726 I|15-088896|0664 199]15-055723]1 -0022034}0021985), - 9978015/12 
 
 49106656419 99.9]9934359| 15 023103106671 21114989784 1 -0022228/0022179), 4, 9977821111 
 50}0668544}5 54919331456] 14-95788210670043} 14-9244 17|1 -0022423|0022373), 94 9977627110 
 51]0671446]50 )4]9328554 1 4895226|0672965) 14-8596 16|1 “00226 19|00225671 56 9977433] 9 
 5210674349174 451932565 1] 148291 2810675887) 14-795572)1 00228 15}0022763}, 4. 9977237 
 530677251] 5 )919222749| 14°765580|0678809| 14-731679] 1 -0023013}0022960} , 4, |99 77040 
 Re 0680153}, 9 99{9319847]14*702576|0681 732] 14°668529/1 002321 1]0023157] ga|99 76545 
 
 53}06830351o9991931694.514-640109|0684654} 14-6059 16}1 002341 010023355 |oy¢ 99716645 
 56|068595 715 4.4193 14043] 14+578172|0687577) 14-543835}|1 002361010" *3555| ogg 9976445 
 _{5710688859}55 9919311 141]14°5 1675710690499 14-482273}1 -0023811 40.3755 599 9976245 
 580691761155, 99|9308239]14-455859|0695422 14421 250)1-002401310023955}540 9976045 
 5910694663|5> \°|930533'7|14°3954'7 110696345] 14560696} 1 0024.21 6|0024157],,,|99 19845 
 
 60I0697565|- 9 O- 
 
 1930243.5|14-335587|0699268}14-300666)1-002419|0024359) 9975641 
 
 ~|Cosine|Dif.| Vers. | Secant |Cotan.| Tang. | Cosec. |Covers|D. Sine |” 
 
 “JOR WOAH D-AID 
 
—E——— 
 
 LOG. SINES, &c, (255) 
 
 Tang. | Diff. Cotang. | Covers.} Secant D| Cosine 
 8°7193958}. 5 ys|11+2806042/9-9766544'10-0005956 66/9'9994044]60 
 11-2787960)7"1416791)8-7218065|5.9509/11-2781937 99765213 10-0006022127 9-9993978159 
 11-2764054|7-1464636|8-7 242035], 90 |11-2757965 9-9763881)10-0006089) 57 9-999391 | 58 
 23645]! 1°2740279/7°1512219)8-7265877]2~ 15]11-273412319-9762549 10-0006156 59/9 9993844157 
 36fg sof 411 -271663417"155954218-7280580(22712|1 1-o710411 9-9761216 10-0006224)?5'9-9993776)56 
 5391611 1 .9693118|7-160660918°7313174{229 >| 11 -o6g6806 9:9759883)10-0006299/°S/9-9993708155 
 
 :2669728|7°165342218-7336631 ee 11-2663369/9-9758550| 10-0006360|°519-9993640154 
 
 11-2646465/7-1699984)8-73.59964| 9. 4|11-2640036]9-9757216|10-0006428 g9|9°9993572153 
 1 1'2625325)7-1746297/8 75831721530. -|11-2616828/9-9755882'10-0006497 ol" 9993503|52 
 1 1*2600509)7°1792365}8-7406258155 94, |11-2593742/9-9754547, 10-0006567 6919 9993433}51 
 SOL A14/71858189/8-7429299} 99. 1 5|11-2570778/9-97532 12] 10-0006636|r 5|9°9993364150 
 22655]! 1'2554640)7°1883773/8-7452067[5 405 9|11-2547933)9-9751877 10-0006707 7019'9993293}49 
 ; goes 11*2551985)7°1929118/$-7474799) 96 63111 2525208/9-9750541/10-0006777 71|9' 999322348 
 22490]! 1'250944'7)7-1974228)8-7497400]99 ; 4 [1 1-250260019-9749205|10-0006848 7119°999315 9147 
 1afg-7512973|,5 00 99377) 11°2480106|9-9747868|10-00069 19) 919-499508 1/46 
 15]8-7535278 Lost 11 +2464729|7°206375018-754226915 39 6111 -2457731 9-9746552\10-0006991|- 9-9995009/45 
 Mo 29150) 1 1°2435469]9°974519410-0007062 7g|9 9992938 )44 
 
 ee 9203911 1°2413319/9-9743857/10-0007135}0'}9-9992865143 
 sion) 11+2391281)9-9742519}10-0007207}..419-9999795/42 
 
 2999790141 
 9-99995"0/30} 
 99999498198 
 9-9999404137 
 9+9999349|36 
 9-9999074135 
 9:9999198134 
 9-9999199133 
 
 Diff.| Cosec. |Verseds. 
 
 ~|23906 
 
 a 0 
 
 112376634) 7°2240071|8°7630647 
 11*2354889|7°2283597|8°765246 
 
 11+2347535]9-973984 1|10-000795414 419° 
 11-2325825/9-9738509|10-0007498|. 
 21497{11°230422319-9737169|10-0007509 
 21391 |11:2982726]9°9735829]10-0007576l, 
 11+2268986|7°245555118°7738665 21984 11°2261335]9°9734489 10-0007651 n 
 11-2247774)7-2498019}8-7759959}9 | ,|11-2240048/9-9753141|10-0007796 
 21089|11'2218864|9-9751800|10-00078021r6 
 20981|!! 2197789|9-9730458|10-0007878\40 
 20889]! 1°2176801|9-9729117|10-00079541 
 207gglt1°2155921/9-9727774|10-000803) 
 20689|11'213513919-9726431|10-0008 
 
 11°2114456/9°9725088 10°0008185 
 11+2093870]9:9723745|10-0008263 
 11+2073380|9°9722401|10-0008341 nol?" 
 11-2052986|9°9721057/10-0008420 79)?" 
 11°2032687|9:9'719719|10-0008499|,~ 
 
 9 
 11-2012481|9-9718367|10-00085"8) 
 
 3 
 11-1992368/9-9'717021]10-0008658 
 11°1972347/9°9715675}10-0008738 
 
 "121636 
 
 Gr Gr & Bb 
 
 76 
 
 11°2163952/7°2665810|8°78440'79 
 11-2143247|7°270725818°7864861 
 
 20606 
 
 20508 
 20411 
 
 |! 1:1921808}7°3150572/8-8087179| 5 (25 
 11-1902298]7-3189773/8-810683 mi 
 
 4119573 
 11+1882736|'7-3228797|8°8126407 
 11+1863332)7°3267646|8'8145894. 
 11°1844015]7°3306329|8'8165294, 
 
 11°1824.783]7°3344827/8°8184608 
 
 11°1873593 
 11°1854106 
 11+1834706]9°9706243|10-0009309 
 11+1815392]9-9704894]10-0009399|°: 
 11°1796162)/9-9703545|10-00094'75|~: 
 11°1777016}9-9'702195]10-0009559 
 A{11°175795419-9700845 100009643 
 11°17389'74 
 
 19404 
 19317 
 19239 
 19146 
 
 11+168252219-9695440110-0009989 
 11+1663866|9-9694088|10-0010069 
 
 8-8435845)!9! 041, 1.1 564155/7-386668319-8446497 
 ‘| Cosine | Diff} Secant 
 
 9°9989408 
 
 Sine 
 
(256) 4 Deg. NATURAL SINES, &c. Tab. 10. 
 
 “| Sine [Dif-|Covers| Cosec. | Tang. |Cotang.| Secant | Vers. | D. = 50 
 “0[0697565| 59019302495 |14-935587|0699268]14-300666| | -0024419|0024359,5) 
 0 0697565 930292435 14°335587 0699268 ; 7 :002.462310024.563 9975437 59 
 2902 33114-97620010'705 14+241134)1+0024623 563loQ4 
 smo") -19299533|14°276200]07021 91|14-24 M04 )onaeoe 
 slovosserl99llogogengll4-21'70410705115|14-182099I1 opeanagiosaroNiia seg nee nS 
 : 6 PBC 88 090: 9293750)14"158894|0708038|14°123536]1-0025035|0024972|5q2 if ates Ba 
 mand UE meth le By 071096 1}14065459| 1002524100251 78}. SE ip s 
 ional 31220 oggrg07)14-043504)0713885)14-00785611-0023449/0025385 207/92. ihalag 
 ; hiner 54 19285026|15-986514)071 680913-950719| | -0025658|0025592|59 419974408 : 
 lov instal... |9282124|13-92998510719735113-894045|1 -002586710025801 209 Rodan i 
 Jor207 ye 290! |o979903113-873913)072965r{13-837827|1-0026078 pt-pt CAO i on 
 sloveseval2°! looveaa9i13-818291l0795561|19-782060| l-02628910026920 an eden 
 me Seat 290219973420)13-763 1 15|0'728505|13-726738] 1 0026501 Ai hist 212 ahieete aah 
 481222192705 19/13°70837910731430|13°671856|1-002671 410026643 Q1Q}-= | 
 M 270519|13°708379]0731430 : Zloar 4a} 
 ip passes 2901) oogrgte|t3-654077 073435413 6174091 (0026928/0026855}9 419973145/48 
 15]0735283 y9q_|9264717|13-600203]0737279]13-563991|1-0027149 nausea eererinse 
 10738184 290 19961 916|13-54675810740004|19-309799 1:0027358]0027283!9) 5 re 
 14 07381845 9261816 13-4933) 0743195 13*456625|1°002757410027498 216 99'72502/45 
 1510741085!99,) 19258915 Pe Ub tc Hives 0746059|13-403867l1-002779110097'714 91719972286) 44 
 16 074598699 epee ee omabens 13-351518|1-002800910027931|9 1 4|9972069|43 
 (70746887 oq 99/92 Bite -2995°74)1 -0028228]0028149]5, 91997185 1]42 
 id 074978 son, 9250213]13°337116|075 190413299574 218 
 
 Ra ee 9971633|41 
 ide 47312|13-285719|07549829|13-248031]1 0028448|0028367] 999/99" 
 
 se he ga eer ee eT con ede Ee pamald pnb fr gn 299|9971413/40 
 a1]078389 2° 0“l9241 51 1/13+184106/07G0euo|13-146127|1-0028890|0028807 iia TAPS} 
 pe Mepbae st 9238610|13°133882|0763606|13-09575%7|1-00291 1210029028log9 Sette 
 25)0764290.290C 99557 10l13-080404|0766839]13-085769)1-0029936 Seniesa 
 ooh I O*E 0 C900 Ean 345 92 ‘9 1-0029560}0294721, 
 oA 07671909994 9252810]13-0345'76|07694.58]12°996160 224 
 
 ‘ 4|12+ : 0029696}99419970304)35 
 i 9229909|12-985486)0772384|12-946924]1 0029785 224 
 2¢)07712991, 29° o2o70p9l12-256765|077591 1}12-808038]1 -009001010029990 296|99700804 
 2 0775801| 22 00l9004.1 09|12-888410/0778957112-849557|1 -003023710030146 26|2069854 a9 
 Bel toes 5200'9991 209|12-84041 610781 164|12-801417{1-0030464/0030372 2.97|2969628/32} 
 2910781691 220991 ss0|12-792779)07840a0|19-753654|1 -0030693}0030599 no9i2960401/31 
 2900199 2745495107870! 7]! 270620511 003092210030827I55° | 
 3010784591 ogppi9212409|1 2745495107870 17]1 2706205 ice 
 
 00} pa y 12°65912.5}1 003115210031055199(199689435129 
 [310787491 ogg p/9212509 iaeene pha ie 12390 1-0031383 0031285 rt 9968715}28 
 321079039119 g99/9209609 aoe slo7ostogl12-36599711 003161 510031515 931|9968485|27 
 3310793290 99 9}9206710 ieee oat 0798796|12°5 19949) 1-003184'7/0031'746 9391996825426} 
 3410796190 og, rae 12-514240]0801655|1 2-474991|1-003208110031978 233|9968022|25) 
 Sel0801 9891272 "lo19a7 1119 sesggs\os0a5e1|19-4eseg cannon ay IGM 
 361080 9 2900)" Aya E *383°'76811°0039.55 110032445 19967555123 
 3 , 9195114)|12°424078}0807509]12°383768 neds 234 
 Solos eer e898 gtecse 5|12-385210)0813969|12-294609]1 -003s024l0032015 236 en 
 apd Aha Festa PRES top 293]12+250505|1-0033261(0033151}. 66849]2 
 ibsteae nts eanecperean eaters ostm mosses 
 10: 5 ret OTN NPR Biased ; ‘ 663741; 
 49108193851 2099 9180615{12°204274|08221 5012163236) 1 -003374010033626)93q 
 
 12899 Baie nots rel]. 211°003398010033865 996613517 
 143/08 2228 4 ogog/? 177716 ee ibac de eerien 1-0034221/0034105 re 9965895}16 
 a Has je 2899 elie 12-076098|0830936112-034629|1 -003446310034345 94] Hee “" 
 ; 2899 12°033970/0835865}11°992549]1 *0034.706|0034586|94 01996: 
 46}0830981\5.099|9169019}12°0 94|11+95037011700349501003482815 4 4199651 79113 
 ; " 4{11+¢ : 243 
 4810836778 eae 9163229)11°95059510839723}11*90868211 003519510035071 Q4d. 
 
 ale :003544.01003531519, :|9964685/L1 
 9160323|11+9093401084265311 1 86728911 :0035440 945/996 
 a mse 9157424}1 1-868370}0845583]1 1-826167)1 “0035687 |00355601945 gsoi440 10 
 1]0845474|500510154526|11 -827685]0848519|1 1-785333|1-0035934|0035805]o4 7/9964 
 + iad 2899191516971 178727410851 44911 1-744"77911 poses a tee O44 aoe 
 Ralogsyony{2o ‘ 721 1+704.500|1 00364 299948 
 4 ; 9148729|11-74'714110854379I1 1-7 
 n ee 9145851 1 1"707282)0857302[1 1-664495|1 0086681 /0036547}q40)9963453 
 5510857067 i 9142933|1 1-66'7693|0860233}11-624761 1-oa6952 ibaa 250 teh fe 
 56|0859966|~~ ~~ |9140034)11°6283'72|08631 6311 1°58529411-003'718: ‘ 30 9967078 
 9962452 
 
 898 -550523/0869025]1 150715411 -0037689|0037548 | 
 fk id 2 DR ace 1956|1 1-4684'74| 1 -0037943]0037800| > 4|9962200 
 591086866015. -19131340]11-511990|087 7800|955|9962200 
 BOIOSTISS 79 [912844311 147371 3108748811 1-430051-00381981003 
 
 — | — . 
 (a eee 
 
 SHWYHE DAML 
 
 _ eS 
 —_————$—. fy ——____.__ 
 
 os 
 
4 Deg. LOG. SINES, &c. (257) 
 
 , 
 
 Sine 
 048°8435845 
 1)8°84538 74 
 218 °84.71827 
 3]8°8489707 
 418°8507512 
 518 °8 525245 
 618°8542905 
 
 718°8560493 
 8/8°8.578010 
 918°8595457 
 10/8°8612833 
 11/8*8630139 
 12|8°8647376 
 
 13/8-8664545 
 14/8°8681646 
 15|8-8698680 
 116|8°3715646 
 
 417/8-8732546 
 18/8°8749381 
 
 918-8766150 
 2018-8782854 
 2118:8799493 
 2918-8816069 
 25|8-8832581 ae 11: 
 2418-8849031] 5 u|11-1150969]7"4694166|8-8861850 
 
 - {5|8-a865418}, ae 
 
 26|8°8881743 16264112" 
 2818°8914209 16149 ite 
 29/8 8930351], 60382 af 
 
 '30|8°3894.6433 16022 
 
 3118°8962455 1596 11°103754517°4921359|8°8975963 
 
 32/8 8978418 ides 11-1021589}7°4953335]8 8992026 
 
 33|8°8994329 11°1005678]7°4985193|8-9008030 
 15846 
 
 3418°9010168 15787 
 
 3718-9057358 
 3818-9072975 ed 
 39|8-9088595|;?>°0 
 40/8-9104039] 12>" 
 
 : 115448 
 41[8-9119487] 7008 
 4918-9134881| 7005 
 
 43/8-9150219|,-... 
 44418-9165504 sd 
 45/8°9180734)7 570 é 15282 
 46|8-9195911], 21 4.|11-0804089|7-5389038)8-9210957| |? 
 4718-9211034 11-0788966(7-5419338|8-9226136 
 48|8-9296105|1>)1 
 
 4918*9241123 14966 
 
 50/8°9256089 
 5118°9271003 14863 11+0728997)7°5539489]8°9286581 
 452/8-9285866], , 11°0714134]7°5569268}8°9301552 
 14812 
 53/8°9300678 14761 
 
 5418°93 15439 14.711 
 
 155|8-95301501,- 
 5618-9344811 Hee 
 57[8-9359422|, #011 
 58|8-9373985|) 490. 
 59/8-9388496| 4) [11° 
 60|8-9402960| 4464111. 
 
 —_ 
 
 D.) Cosine 
 aogotl 
 
 80291, 1 -1546126}7-390278518-S464554 
 11:1528173}7-393873618°8482597 
 
 11°1474'755}7:4045706 
 11°145'7095)7-4081071)}8°8554034 
 
 11°1439507)7°4116293/8°8571713 
 *1421990/7°4151372'8°8589321 
 
 8°8536283 
 
 9-9988780/53 
 9°9988689)52 
 9-9988598)51 
 9°9988506)50 
 
 11*1387167|7°4221 id whan peep dl 
 11+136986 1|7°4255767|8°8641725 
 1°135262417°4290288 |8°8659055 
 11°1355455)}7°4324673/8 8676317 
 
 POL 11: 88693511 
 17034|11:1318354)7-4958921 5 
 
 16966) 5. 
 16900], 5. 
 16835} 1). 
 
 16769 
 *1233850|7°4528163|8°8778487 
 16704154. 
 
 16639}1 1. 
 16576114. 
 
 17127 
 
 17061} 1.1.9°79901|9-9664240|10-0012053 
 1699511 1 .1955306/9-9662879|10-0012147 
 
 16929}, 5. 9*9661517|10°0012242 
 16864|1 11238377 ol7l0: 
 
 *19-998794"7|44 
 
 ' 15,8°8727699 
 1284354)7°44270 | 9°9987853)/43 
 
 12674.5417°4460862)8°8744694 
 
 9-9987663)41 
 
 —_— 
 —s 
 
 9-99874'7 1139 
 9-9987375138 
 97 9°9987278)37 
 
 16547 9°9987181136 
 
 16484 
 
 9-9986790132 
 9-9986691131 
 9°9986591|3 
 
 9°9986492)29 
 9°9986392/28 
 
 1 
 11*1040158)9°9645146/10-0013409 a 
 
 11-1024037|9-9643779|10-0013508}, 9, 
 11-1007974}9°9642412}10-0013608}; 54 
 15647 11-0991970)9°9641044/10-0013708}, 01 9-9986292 27 
 15 ae 11-0976023)9°9639676|10-0013809 101 9-9986191 2 . 
 13633 11-0960134|9-9638308}10-0013910}, 02 9-9986090/25}. 
 
 11-0944303/9°9636939|10:0014019 109 9°9985988/24 
 
 1577 Pie 
 "21, % 11-0928528|9-9635570|10-00141141 ; 99°9985886)23 
 11-0927095|7-5149751|8-9087190|;94 pol 1+0912810 9-9634200}1 eon 4216] 09 g9ys5784 20 
 11-0911465)7"5173923)8-9109853), 5¢q_|11-080714719 9652830|10-0014318}, g9|9"9985682 a 
 1oal? 298557912 
 
 4 -5204.98218°9118460 
 11+0895961)7°5204982 460115550 1903/9 9935475}19 
 
 1108805 13}'7°5235931(8-913401¢ 
 15497), 1.9850491/9-962871811 0-00 14.628 101(9 9985372|18 
 
 11+0865119}7-5266769|8-9149509| 247 4 
 11-083504819°9627346|10°0014.732 105 9°9985268)17 
 11-081 9660|9-9625974|10-0014837], 19-9985 163}16 
 11-0804325|9-9624602110-0014949|, 7|9-9985058|15 
 7H oei97998495314 
 2291) 1-077381419°9621856|10-0015159- 0 19-9984848113 
 9-9984749| 19 
 
 1°0974.04.5]7°5048560)8°9039866 
 1°0958315]7°5080071/8°9055697 
 
 mS Bd | cg eae | cnet Deg. 85. 
 
(258) 5 Deg. NATURAL SINES, &c. Tab. 10. 
 
 _118|0923706 
 
 Jt 
 ° a 
 
 12910955562\o.9¢ 
 
 43910984514 
 
 ‘1! Sine |Dif |{Covers 
 
 —} —_— | —— _ -| ——— ---~— ]} ———— -- —_ [| ———__-} -- —— 
 
 210877353 
 3}0880251 
 410883148 
 
 9960926|56 
 
 257 
 258 996066955 
 
 710891840 
 8|0894.738 
 . 910897635 
 10}0900532 
 11]0903429 
 12|0906326 
 
 13|0909223 
 140912119 
 1510915016 
 16|091791315 
 17|0920809 
 
 9959107|49 
 
 263 
 264 995884448 
 
 ii 9958315146 
 sro |9958049/45 
 9957783|44 
 
 268) g9% , 
 268 9957515|43 
 
 269 
 sy |9956708|40 
 9067605|10°72.5070|0936474]10°678348] 1 -004375310043563 9 9956437|39 
 
 9064709) 10°691859)0939409]10°644992) 1 -0044028/0043835 th ) 
 
 273 
 
 275 
 
 1910926602}. x1 
 golog2a499|-05. 
 210932395 
 2210935 
 
 ~|2896 
 
 a7l094977 logos 
 2810952666|ogq¢ spied (| Pest 
 9044438] 10°46.5046|0959955}10-417158| 1 -0045970|0045760|97|9954240)31 
 
 9041542|10°43343110962890]10°385397|1 -004625 110046038 979 9953962/30} 
 
 3110961353999 5|9038647}10-402007/0965826]10-353827|1 -00465331004631'T}94)|9953683|29 
 32|0964248] 999419035752] 10°370772/0968763}10-322447]1 -0046815}0046597|o¢ {9953403 |28 
 33|0967144)q.g0 x|9032856| 10°339726|0971699}10-291255}1-0047099|0046878]404|9993122/27 
 34/0970039]9gq [902996 1}10:308866/0974635|10-260249]1 -0047383 00471 60|9g419952840]26 
 351097293419 99 5|9027066| 10-278 190/0977572]10-229428} 1 004-7669 |0047443}g¢4|9952557/25 
 
 9024171}10°247697|0980509}10°198789}1 -0047955|0047726 mie 9952274\24 
 
 9021276|10°217386|0983446]10°168332 1-0048242)0048010 
 901838 1}10°187254|0986383)10+138054|1-0048530 0048295 
 
 30|0958458logq 5 
 
 28519951705|22 
 cae 9951419/21 
 3g q{9951132}20 
 
 995084419 
 
 288 
 9909(9950556|18 
 
 38|0981619|ogo. 
 4010987408] 550. 
 4110990303150. 
 il099319"Ieo ot 
 dade 2895 
 43[0996092l 5.0, 
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 43[1001881|5954 
 4811010563 
 49]/10134.57 
 3011016351 
 51|1019243] 5005 
 5911022138 
 
 rg ¢ 
 
 8995225|9-9524787|1009886)9-9021 125] -0050864}0050607|,4|9949393|14 
 899233 1|9°9238943}1012824/9-8733823}1 0051 160/0050899)49419949101]13 
 89894.37)9°8954744)101576319-8448166|1 -00514.56|0051193 
 
 94 oon aanm 
 094 9948807} 12 
 
 8980755)9-811 1880)1024.580/9-7600927|1 -005235110052079 99 9947921 
 897786 2/9 +7834 124)1027520)9-7321713}1°0052651/0052375)5,.4] 9947625 
 
 9 
 
 2894 & 
 5311025039), 5 4.4|8974968|9-7557944| 1030460|9-7044075|I -0052952|0052673)4y9| 9947397] ‘7 
 54{1027925 | 4 ,[8972075|9-7283327]| 1033399|9-6768000} I -0055254|0052972, 4419947028] 6 
 55|1030819}.9..4|8969181]9-7010260]1036340/9:6493475|1-0053557|0053271|_,,|9946729| 5 
 56}1033712}44.|8966288|9-6738730]1059280/9+6220486|1 -0053860|0053572}5, [9946428] 4 
 57|1036605|9 5 :|8963395|9°6468724] 1042220/9-5949029| I -0054164|0053875|4, [9946127] 5 
 58|1039499],,. .|8960501|9-6200229| 1045 161|9-5679068|1 -0054470]0054175] 5 -|9945825| 2 
 59}1049392}, 9 3|8957608/9-5933233} 1048 10149-5410613|1 -0054776|0054477l4,,,{9945523| 1 
 60}104.5285|-”” “189547¥5|9°5667729| 105104219-5143645] 1 -0055083|0054781| "(99452191 0 
 
 ee fee ff 
 
 Tang. | Cosec. |Covers D.| Sine 
 
 Deg. 84, 
 
 ~ 
 
 Cosine! Dif.| Vers. 
 
) | | . 7 | 259) 
 - LOG. SINES, &c. ) ( 
 - 248 Secant D. epete,| 
 
 9-9983442/60 
 11°0580489|9-9609967|10-0016559), | [9-998344 
 11-0565956/9-9602588|10-0016668} | ./9-9983332159 
 47911 1-0551477|9-9601209}10-0016780}, ; [99983220158 
 1451/1 1 -0537046|9-9599829|10-0016891}; ,|9°9983109'57 
 1438411 1.052266 9-9598449|10-0017003} ,4|9°9982997156 
 149381) 1-0508324|9°9597069]10:00171 151 15 9-9982889155 
 Ene 11°0494033/9-9595688}10°0017228}; | .|9°9982772 a 
 
 ) aes 5|10- 9-9982660/53 
 
 “6004105|8*9520211} ; j go|11°0479789|9°9594506]10-0017340) 1 419° 
 11-0497129/7-6004103}8'952021 11, gol11-0479789/9‘9594300]10-0017340 ee ees 
 
 11°0483043]7°6032331|8°9534410 14154 S Negrete ool yned 
 11-046900417°6060468/8°9548564 14108 11°0451436)9°9591545 i 
 
 11 -0455009}7-6088515|8°9562672) 1 ye, 7110-0017796 
 : BH se 115 
 11-044 1060}7°6116468|8°9576735}, 40 14 -19*9587394110-0017911\14 21S 
 11°0427157/7°6144333|8°9590754 13974) | 1°0409246)9°9587394 ( a Sidhe roi 
 : nl.6 8-9604728],.... 95860 L15|r 
 A ek et te PG $-9618659 ett 11-0381341|9-9584626]1 ri rade 116p 27 81859)46 
 Py 5 9°958324210- 7119)" 
 11038571 2)7°6227395)/8 9632545}, 3¢ 4 4}11°0367455/9°95 00183 
 Bly 4. 219°9581857}10 0018374 1 1 ¢ 
 1103719867 °6254906)/8 964.0388) 549)! 1°0353612 47 111000184901; 3 
 Regn 8-9660188 11-033981219°9580471 4901117 
 {1-03446e0/7-6909668/8-9673044 i324 {tt-0326056]9-2579086]10-0018607]3 35 Bat 
 -6386920|8-9687658}, ano! 1°0312349|9-9577699]10-0018725|; 17|9-9981275]41 
 11-033 1066}7°6336920|8-9687658] gan. Nos 1000188421; 7 419-9981 158140 
 110317513 76364086 : ara 13629'1 1 .ogs41/9-9574926110-0018960 119|9°9981040|39 
 11-0304001]7° (ai »" 113588 4 9-9573539|10-0019079 
 . C ()¢ S 119 
 11-0290532)7°6418 164/8°97285471 45 4 .111°02714535)9" 2151110°00191 t 
 11-0277105|7°6445078/8 9742092) 3.4, rppneidets pened ied. soataait #8 ; 
 11°9263720/7°6471908)8°97555971 4) 65 nie 
 
 134.23 
 13382 
 13341 
 13301 
 13269 
 13229 
 13182 
 13144. 
 13104 
 13066 
 13027 
 12989 
 
 14526 
 
 : 14178 
 6/8-9488739|) 4148 
 713-9509871 
 818-9516957 nh 
 913-9530996 
 5 13995 
 10f8-95a4991} 3972 
 11]8-9558940}, 5309 
 12}8-9579843) 500° 
 
 1318-9586703 
 
 14/8-9600517 ae 
 15}8-9614288|) 3/95 
 16|8-96280141 3000 
 17]8-9641697] 5045 
 18]8-9655337| 13g 
 
 19/3-9668934 
 
 1 
 20|8-9682487 ite 
 22/8-9709468| 3500 
 
 13385 
 13344 
 
 1 
 26|8-9762996 eect 
 
 27|8-9776183), 
 
 28|8-9789408 eee 
 29/8-9802589|13 145 
 30/8-9815729|) 5150 
 
 129 
 39)8-9841889 9-9559643]10-0020284) | 939° 
 11-0194683]9°9558251}10-0020407} | 9319°9979593}27} 
 11-0111579}9 9556899}10-0020530}; 9319-29794 70}26} 
 11-0098513}9°9555466]10-0020653} ; 93|9t9979347]25} 
 
 11°0085486|9°9554073}10°0020777 12 
 
 12827 
 12788 
 12751 
 12713 
 12675 
 12638 
 12601 
 12565 
 12527 
 
 4719-0033179 ina 
 
 48190045634. 12419 
 
 2383 
 2348 
 2312 
 2278 
 12249 
 12207 
 
 “112173 
 12139 
 12104 
 12070 
 12037 
 
 11-0046633|9°9549892110-0021 150} 19519" 
 1287611 1 .0033757|9°9548497]10-0021973 jo¢!: 
 
 . 005 -|9-9978347]17 
 109995353|9°9544311)10°0021 653) 94/9°997834717 
 Throng |10-9982625|9-954291.4|10-0021 780]; o- ron cheer: 
 .006 18]10-0021907]; 5-|9°9S 5 
 10-9991840}7°7016959|9 0030066) 5°25 oi ies ee? faced Pith hal igal99977966| 141 
 7°7067124)9 54 12584 10-9932076|9°9537325|10-0022290] ; 9,19: 99777LO}12 
 10°9954566 7°7092098 9 0067924 12547 P , wool, 
 10-9919529]9:9535927}10-00224 18} ; 9g/9°9977582| 11 | 
 12513 10-990701619'9534528110:0022954.-7 130 9°997 7453} 10 
 12477 10+9894.539 9*9533199110:00229677 129 9-9977323} 9 
 1240771 0:9869690|9-9530329]10-0022936) 13 nie RIL 
 se 10:9857318|9'9528929]10-0023067)  4,.|9°9976933 
 a2 
 
 9-9527528}10100231977] :, ,|9-9976805 
 12904; )-9859675}9-9596127|10-0025528 {39)2°9976672 
 Lpas7|10°9820406|9-9524725]10-0023460} | 5,]9-99785.40 
 12257110-980816919:9523825}10°0023599| , 5, 
 12909 
 7*1386303]9-0216202 10: 
 
 Covers. }Cotang.| Diff.| Tang. \Verseds. 
 
 $L2 | Deg: 84. | 
 
 5019°0070436 1 
 51/9 0082784. 1 
 52|9-0095096 1 
 53}9°01073'74 
 
. (260) 6 Deg. NATURAL SINES,- &c. 
 
 ‘Tab. 10.. 
 
 ‘| Sine |Dit.|Covers| Cosec. | Tang. |Cotang.| Secant | Vers. |D.[Cosine 
 
 “011045285, 
 1}1048178 dae 
 
 — 
 aiiieaidall ft: —— | ———— 
 
 211051070 aes 8948930|9-5141 110|1056925|9-46144 16|] -0055699|0055391|;,|9944609}58 
 3{1053963|569918946037|9-4879984I 1059866|9°43515311] -0056009}0055697}.9.|9944503)57 
 
 4110568561509 
 5}1059748 hte 
 6|1062641 9899 
 
 8943 144]9-4620296|1062808/9-4090384| { -005651 900560041" |9943996|56 
 8940252|9-4362033]1065750|9°3830663]10056631|005631 2); ,19943688}99 
 893735919 -4105184]106869219°3572355] 1 -0056945|0056621]- 419943379154 
 
 7|1065533]9¢ 99[8934467|9 3849739] 1071 634|9°3515450}1 -005725610056930|, ; »|9943070)2S 
 8/1068425| 5495{8931575)9 3595689 1074576|9°3059936]1 0057570)0057240)°,  ,|9942760/02 
 9}1071318)5949[8928682/9 3343006) 10775 19]9°2805802|1 -0057885}0057552)3 ; 919942448) 1 
 10]1074210|5655/8925790)9 309 1699] 1080462|9°2553035]1-0058200]00578641, ; 3|9942136)00 
 11]1077102}9 49518922898 /9-284 1749|1083405|9°2501627| 1 -0058517/0058177], 1 4|9941823}49 
 12]1079994|56 9 [8920006|9°2593145]1086348/9-205 15641 0058834|0058490). | -|9941510/48 
 8917115|9-2345877|1 089291 |9-1802838|1-0059153}00588035},, .|9941195)47 
 
 13}1082885 
 14110857 77\seq, 
 15]1088669|o¢ 
 16}1091560}og99 
 17}1094452}5 
 18}1097343|9994 
 
 19]110023415<, 
 
 2011103126 aot 
 23/1111799)56 
 
 2411114689|999, 
 
 Q5/1117580\5 
 26)1120471 tue 
 
 8905548/9°1369949]1101066/9°0821074) 1 +0060435)0060072 3 
 §902657|9" 1129200) 1104010/9°0578867)1°0060757|0060390 300 
 
 8899766/9:088979511106955|9°0337933)1-0061081}0060710 301 
 8896874|9°0651512}1109899|9'0098261/1:0061405}0061031),; 
 §893983|9°0414553]1112844/8°9859843]1-0061731/0061352 9 
 8891092/9 ‘0178837}1115789|8°9622668}1 -0062057/0061674 393 
 8888201|8 9944354] 1118734|8°9386726}1 +0062384|0061997]}4., 
 
 ‘ Py ba: 1 
 8882420|8°94'7905111124625'8°89 18505] 1:0063040|0062645 306 
 8879529/8 924821 1}1127571|8°8686206]1 :00633'70|0062971 
 
 27|1123361 89013.876639|8-901856711 13051'7|8°8455103|1 -0063701/0063297)55. 9936703 33 
 28]1126259l9¢4|8875748)8 87901 09| 1 133463 /8-8225 186] 1 -0064032|0063625)4,.19956375/52 
 29]1129142l9 9 9/8870858/8 8562828) 1564.10)8"7996446]1 0064364006593), 9936047}31 
 3011 132032|5499/9867968/8 833671 5|1139356 8"77688744 | 0064697|006428 1], 9935719)50}- 
 31]1134922|,)..4,{8865078|8°8111761/1 142303|8-7542461]10065031]006461 554 2993369 ies 
 52/1137812|99 5, |8862188)8 78879571 145250 8-7317198|1 °0065366|0064949}, 9935058)28 
 33/1140702)5 69, 8859298|8°7665295|11481 97/8-7099077 1°0065702|0065273 9934727 if 
 34]1143592]9 0 4|8856408)8-7443766)1151144/8-6870088]1 0066039]0065605]4.56 9934595 *y 
 3511146482|9 0.4, [8853518|8-7223361]1154099'8-6648223} 1 0066376|0065938), 9934062 29, 
 36]1149372], 2 |8850628/8"7004071)1157039/8-6427475]1 00667 140066272}, 9933728|24 
 37|1152261 884'7739|8°6785889| 1 159987|8°6207833|10067054|0066607] 4 |9933393}23, 
 
 38]1155151 Heise 884484.918-6568805|1162936 
 39}1 158040), 09718841960)8"6352819|| 165884.8-5771838|1°0067735|0067279}, 
 40]1160929]509 5 [8839071 |8°6137901]1168852 8-5555468|10068077}006761614. 
 41]1163818)5, 00 |8836182[8-5924063]117178 1/8-5340179]1 -00684190067953|; 
 
 42]1166707), 
 
 2889 
 43]1169596 
 
 4411172485 
 4541175374 
 4611178263 
 411181151 
 4811184040 
 
 4911186928 
 50}1189816 
 ST}1192704 
 5211195593 
 53}1198481 
 54)1201368 
 55]1 204256 
 5611207144 
 597/1210031 
 5811212919 
 59}1215806 
 60}1218693 
 
 Cosine|Dif. 
 
 2889 88304048 54995841 177679, 8-491 2772/1 -0069108}0068633 341 
 8827515/8*5288923]1180628 8°4700651|1 :0069453/0068974 
 
 el 8824626|3°5079304|1183.578 8°4489573|1°0069799|00693 3); 
 8821737 
 
 8°4870721}1186528, 
 
 2889 
 
 2888 
 881307218 °-4251105}1195378'8°3655536} °0071 193/0070690 348 
 
 sane[881018418-4046586) 1198329|8-3419558|1-0071544|0071 035}, 
 
 2889 
 2888 
 2887 
 2888 
 2888 
 2887 
 2888 
 2887 
 2887 
 
 88044078 3640534] 1204230/8 3040586) 1°0072248)/0071729 
 88015198 °34.38986) 12071828 -2837579|1-0072601,0072078 3 
 
 2 
 
 8792856}8°284.0171)1216036/8 2234384) 1-0073666 0073127), 0 
 
 8'78'7081|8-244.5'748] 1 22194 1/8-1837041]1°0074380}0073831 353 
 8784194)8:224995 2) 122489318 1639786] | -0074739}0074184 554 
 87813078 *2055090) 1227846]8* 144346411 0075098}0074538 
 
 Vers, | Secant Cotan. Tang. Cosec. 
 
 8798632/8°3238415)1210133/8 2635547) 1°0072955|00724.27 349 9927573 
 8'795744|8°3038812)1213085/8-243448511 0073310/0072776 351 9927224 
 
 8789969|8-2642485] 1218988}8-2035239}1 -0074023100734-79} 43.5 |9926521 
 
 —— | | ane 
 
 Covers| D.| Sine 
 
 8914223}9-2099934)1092934]9-1555436|1+0059472|00591 20}. i. 9940880}46 
 #891 1351]9-1855505]1095178|9:1309348]1 0059792|0059437]. 1 |9940563]45 
 890844019-161 1980 109812219-10645641-0060113|0059754|~,./9940246|44 
 
 9939928]43 
 9939610|42 
 
 9939290|41 
 9938969|40 
 9938648|39 
 9938326/38 
 9938003/37) 
 
 888531 18-971 1095|1121680|8-9152009}1 -0062712(0062921 °° 419937679|96. 
 
 9937355/35 
 9937029134 
 
 8*5989990I1 -0067394|0066943}--0|9933057)22]. 
 
 9932'721|21 
 
 9932384|20} 
 a 9932045|19 
 883329318 °5711295}1174730'8-5125943]1 -0068'763/0068294 a 9931706]18 
 
 993136717 
 9931026]16 
 
 19930685|15, 
 
 ones 8°427953111°0070146|0069658}5 ; 4|9930342)14 
 8818849)8-4663165]1189478 8-4070515{1 0070494007000 1] ; 5 |9929999 13} 
 
 881596018 4456629] 1 192428 '8°3862519]1 :0070843|0070345 34 9929655}12 
 
 9929310}11 
 9928065]10 
 
 8807296/8°3843065}1201279'8-3244.577|1:0071895|0071382 ce 9928618} 9 
 
 9928271 
 9927922 
 
 9926873 
 9926169 
 
 9925816 
 9925462 
 
 “low wobko na 
 
 Deg. 83. 
 
Deg. LOG: SINES, &c. goat. (REE i, 
 
 D.) Cosine 
 
 9-9976143160 
 9+997601 1159 
 9-9975877158 
 iat 99975743157 
 1 34{079975609|56 
 a 9-0975475155 
 132 19°9975340)54 
 99975205153 
 9+9975069|52 
 9-9974933)51 
 
 Tang. | Diff.) Cotang. | Covers.) Secant. 
 
 LT TTT 
 
 0192346}. 10-9783798|9-9520518]10-0023857| , 4, 
 Balsa, 10-9771662}9-9519115]10-0025989 
 -0216318 opge|10°9759559/9-95 17711] 10-0024193 
 0228254. it 10°974'749019-9516307|10-0024257 
 0240157 12038}, 9.97354.5219-9514902|10°0024391 
 0252027 10°974'7975|7-750599919-0276552 
 
 0263865 Hae 10°973613517°752974219-0288524 
 
 136 
 
 9°9507874|10°0025067 ie 
 
 11909 
 118 
 
 0310890}, 5 4nn|10°96891 10}7:7624064}9-0336093 
 0522567], 5 24; |10-9677433]7-7647485|9-0347906 
 0334219], ; a 10-9665'78817°767084319-0359688 
 0345825 
 
 03574071; 25% 
 0368958] 1119 
 0391966]; 4550 
 0403424114 159 
 
 10°9558'70 119949386] 10-0026446 9°9973554141 
 11537 ate Gok rh 9-9499375|10-0026586). 7° 997341 4140 
 1150711 9-953565'1/9°9490963|10-0026797 1410 29 13273/39 
 11478) 1 0-95241'7919'9489551|10-0026868 [Allo 3288 
 11449 10°951273019°9488139|10-0027009 9°997299 1/37 
 
 feu 10-9501311|9°9486726|10-0027150 oe 9199 12850|36 
 
 10°94.89999/9°9485313]10-0027299 14g}? 297270835 
 
 1136111 0-9478561/9-9483899 1000274341, gl? 9972566)54 
 
 ibe 10-9467229/9-9482486 0-00275771 75 pg er 
 7°8036246|9°0544074 10-9455926)9°9481071}10-0027720}, 4 -19°9972280}5 
 10-9483646|7°8036246 11275119.944465 |9°9479656|10-0027863 i 9:997215%)31 
 
 1124611 9.943340519°9478941110-0028007 pagfe 29 11993)50 
 
 11218 
 10-9429187|9-9476825110-0028151 c 
 
 11189} 19.941999819-°9475409|10-0028296 ae 9:9971704{28 
 
 11162} 19.939985619-9473993]10-0028441 [9-9971559|27 
 
 11133]19.9388703/9°9472576110-0028586 Lael 9971414)26 
 
 111061} 9.93977597%|9-9471159}10-0028739 Lael 9971 268|25 
 
 rine 10-9366518|9°9469741|10-0028878} | ; 219°9971 122/24 
 
 5a| 10*9355467/9°9468323]10-0029094) | , -19-9970976 
 
 1102311 9.934444419°9466904|10-00991"71 ae 9-9970829|22 
 10997711 9-935344'719°9465486|10-0029318 14r7{29970682|21 
 10969) 1 9-9392478|9-9464066|10-0029465 1Agl? 9970535|20 
 1094311 9.0311535|9°9462646|10-0029613 14g{0 997038719 
 toate 10+9300619|9°9461226]10-002976 1] ; 7 |9-9970239)18 
 
 "| 10-9289730)9-9459806]10-0029910)  , |9+997009017 
 10863}; 9.9973867/9-9458385|10-0030059 149(97996994 1116 
 Ips; o|10-9268031|99456963|10-0030208} * 719-9969 79915 
 10810110-9957921|9-9455541}10-0050358|1>19-9969649|14 
 1078411 9-9946457|9°9454119|10-0030508]; 219-9969499|13 
 
 1075811 ().993567919-9452696|10-0030658 a 9°9969342112 
 9-9969191|11 
 
 9 2 
 Wine 10-9224947/9-9451273}10-0030809}, , 
 1070'119-9914240|9-9449850|10-0030960 1 3g|9'9969040}10 
 Ice | t0-9203559]9"9448426] 10-0031119}) 2319-99688 
 1069511 9-9199904/99447001}10-0031264|!2219-9968736 
 152l0. 
 1 ea|9"9968584 
 5 29|9°9968431 
 
 +0482786 
 
 -0494005|; 1130 
 0505194]; +169 
 0516354] 415. 
 “052748511 1 195 
 "053858811 j 45 
 
 10988 
 10961 
 10932 
 10905 
 
 10877 
 10849 
 10822 
 10795 
 10767 
 10741 
 
 10714 
 10687 
 10660 
 10634 
 10608 
 10581 
 
 10555 
 
 0582711 
 0593672 
 “0604604. 
 
 0615509 
 “0626386 
 "0637235 
 "0648057 
 "0658852 
 "0669619 
 
 “0680360 
 "0691074 
 “0701761 
 "0'7124.21 
 0723055 
 0733663 
 
 0744244 
 ‘0754 799 10530 
 0765329 
 10503 
 0775832|, 53x 
 0786310}. .* 
 
 10452 
 0796762 10407 
 
 10°94.1'7289)7°8 1693929 0611297 
 10°9406328/7°8191386/9.0622403 
 10°9395396|7'8213323|9 0633482 
 
 10-9384491|7°8235205|9 °0644.533 
 10°9373614|7°8257032|9 0655556 
 10°9362765/7°8278804/9 0666553 
 10°9351943/7°830052219°067'7522 
 10°9341148/7°832218519 0688465 
 10°9330381)}7°834379419 0699381 
 
 10:9319640/7°836534919 -0710270 
 10°9308926)7°8386851/9 0721133 
 10°9298239/7°8408299}9 0731969 
 10°9287579)7°8429695/9 0742779 
 10°9276945)7°8451037|9 0753563 
 10°9266337)7 *847232719°0764321 
 
 10°9255756}7°849356519 0775053] 
 10°924.520117°8514'75 1190785760 
 10°9234671/7°853588519 0796441 
 10-9224168/7°8556968/9 0807096 
 10°9213690}7°8577999|9 0817726 
 10°9203238|7°8598980]/9 0828331 
 
 — 
 — 
 
 a 
 S 
 
 1063011 9.91 8997419-944.55'77|10-0031416 
 tape 10°9171669|9°9444151|10-0031569 
 
 S 
 
 0817590 
 
 >9110:9129499/9-9438446|10-0032183 
 1048011 9.91 1901919*943'7019]10-0032338 
 1045711 9-9108562/9°9435591|10-0032493 
 
 Se wOOko A190 C 
 
 > 
 
(262) 7 Deg. NATURAL SINES, &€. ‘Tab. 10. 
 
 ") Sine |Dif.|Covers} Cosec. | Tang. | Cotang.{ Secant | Vers. |D.|Cosine 
 
 } ({1218693},. ,.18781507]8-2055090}1227846/8" 1443464] | 0075098}0074538}., , 9925462160 
 1221581}50¢,18778419|8"1861 157|1230798|8-1248071|1 -0075459|0074893),, 1992510759 
 1224468] 6 9-18775532|8" 1668 145]1233752|8- 1053599} 1 -0075820]0075249) 45 19924751}58 
 51122735 5|oq0 |8772645]8" 1476048] 1236705 |8-0860049| t -0076182/0075606],, 9924594157 
 4]1250241}9 6 2 -|8769759|8"1284860}1239658|8 0667394] | 0076545|0075963/5,..|9924097 
 5/1233128], 9 ¢,|8766872|8"1094573} | 2426 1 9|8-0475647|1 -0076908|0076321|5¢,|9923679}55 
 6]1256015]o 46 «|8763985|8-0905 182} 1 245566 |8-0284796} | “0077273|007668 1/4 419923319154 
 
 71238901], ,1876109918-071 6681|1248520}8 0094835} 1 0077639}0077041 |q ¢]9922958)53 
 8]1241788) 55 5 -|875821 218°0529062]125 147479905756] 1 -0078005|0077401)3 ¢ 99922599452 
 0}1244674 99g6|8792326|8°0342321]1254429 79717555} 1:0078372|0077763/5.-; 9922237)}51 
 10]1247560}9 96 -|8752440|8-0156450)1257384 79530224 | 007874 1}00781 26)... 9921874)50 
 11]1250446]0 6 2|8749554)7-9971445]1260339]7-9343758} 1 0079 110}0078489). 47 /9921511}49 
 19]1253332 2886 874.6668/7°9787298] 1 263294/7°9158151|1-0079480/0078853 36 9921147}48 
 
 13}1256218|9. .|8743782|7"9604003}1 266249|7'8973396| | 007985 1}007921 8]... .}9920782)47 
 14]125910415 6 -|8740896)7-942 1 556|1269205]7-8789489} | -0080222|00795844 5 ¢4|9920416/46 
 15|1261990]o¢ ,|87530 10)7-9239950}1 272 161]7-8606425|10080595}007995 1}. .,/9920049)45 
 16]1264875/9.¢02|8735125]7-90591 79] 12751 17/7°84241 9 111 0080968|008051 8}; 41991 9682/44 
 17}126776 11994 518732239) 7°8879238} 1 278073}7°8242790}1 -0081343/0080686)4, 
 18}1270646 
 
 939 5|8729354/7°8700 120] 1 28 1030)7-806221 21-008 1'718/0081056 nti 9918944/42 
 19|1275531 
 
 agg 5|8726469)7°8521821}1283986)7-78824.59] 1 -0082094)0081426) 4199918574] +L 
 20]1276416]og 0 4|8793584)7°8344335|1286943/7'7703506]1 "0082471 0081796 |4179/9918204 40 
 21]1279302\9 96 5|8720698)7°8167656]1289900}7°7525366|1 °0082849/00821 68/4.4)9917832)39 
 22}1282186],,00 .|8717814)7-7991778) 1 292858\7-7348028) 1 -0083228|0082541|5,49917459}38 
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 9984(8709159)7°746890 1] 1501731 /7-68207691 I -0084369|0083663}qn4)99 16337|35 
 26|1293725}9g¢ 4|8706275) 77296 176}1304690 /7-6646584)1 0084752|0084059]5,7/9915961)34 
 27|1296609]o 4.870339 1|7°7124227}1307648)7 6473 1741 °0085135/00844 1 6|g7g19915584/33 
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 9994|9094738)7°6612976}1516525/7°5957541|110086290|0085551\4g9|9914449)90 
 
 2884 8691854) 7°6444075]1319484/7°5787179}1 *0086676/0085931 14g 5 9914069|29 
 2883 8688970/7-6275923}1322444)7°5617567|1-0087064|0086312)305 9913688|28 
 2884 $686087/7°6108516}1325404)7°5448699}1 :0087452|0086694/404 9913306|27 
 2884 $683203)7°594.1849]132836417°5280571}1°008784210087077 385 9912923 
 2883 8680319)7-5775916|1331324/7°51 13178] 1°0088232/0087460 385 9912540 
 
 99.3|907 7436) 7°5610713}1334285|7:49465 14] 1-0088623}008784.5|4¢% 9912155|24 
 
 977] 132544°7 9999190 74553/7°5446236]1337246|7°4780576| 1-0089015|0088230 386 9911'770}23 
 
 2888 
 2887 
 2887 
 2886 
 
 -138]1328330 2883 8671670)7°5282478}1340207\7°4615357}1°0089408|0088616 384 9911384)22 
 $911931213 8668787\'7°5119457|13431 68)'7°4450855110089802|0089003 387 991099721 
 44011334096 8665904|'7°495'710611346129}7°4287064!1°0090196|0089390 9910610)20 
 
 a 991022119 
 39919909832 18 
 43}1349744) 9 9 ¢ 38657256) 7°4474335}1355015!7°3799909] 1 -0091586|0090558 391 9909442)17 
 1357978|7°36389 16]! -0091784|0090949],.4|9909051 16 
 
 8651491}7°4155959}1360940}7 34786 10}1:0092183}0091341)49.4|9908659|15 
 8648608]7°3997798}1363903|7°3318989}1°0092583}0091734 9908266}14 
 3939907843113 
 
 395 
 395 99074'78}12 
 
 46]1351399 
 4nf1354974] 55a 
 
 4911360038 8639962)7°3527377|1372793/7°28441 841} -009378810092917 396 9907083)11 
 50}1362919}59¢9|8637081)7-3371909}1375757|7'2687255| 1 0094192|0093313 304 9906687}10 
 5141365801 86341 99]7°3217102113'7872117°25309871 1 -0094596|/0093710 397 9906290} 9 
 52|1368683 863131'7|7°3062954]1381685|7°2375378]1 °0095001|0094107 309 9905893} 8 
 3]13'71564 8628436]7°2909460}1384650]7 222042211 -009540810094.506 399 9905494] 7 
 441374445 862555517°27566161138761517°2066116}1°009581510094905 9905095] 6) 
 55|1377327 8622673]'7*260441 7] 1390580]7° 191245611 °0096223/0095306 401 9904694] 5 
 36}1380208 8619719217 °24.52859]139354517° 175943711 *009663110095707 402 9904293} 4 
 5111383089 8616911}7°2301940}139651017°1607056}1°009'704110096109 409 9903891} 3 
 58}1385970 8614030}7°2151653]13994.7617°1455308]1:0097452/009651 1 04 9903489} 2 
 59]1888850 8611150]7°2001996]1 40244.2]7°1304.190}1-0097863}0096915 404 9903085} 1 
 6011391731 8608269}7°1852965]14054.08]7°115369711-0098276|0097319 9902681} 0 
 4 / 
 
 Deg. 82. 
 
 81991931443}. 
 
» er LOG. SINES, &c. 
 
 Cosec. |Verseds.| Tang. | Diff.) Cotang. |Covers.| Secant 
 
 10°9141055]7°8723806/9°089 1438 10431 10°9108562/9°9435591}10°0032493 
 10°9130779]7°8744436]9 0901869 10408 10°9098131/9°9434163}10-0032648 
 10°9120527|/7°8765017|9°09 12277 10383 10°9087723'9*9432735)10°0032804 
 10°9110300}7°8785550]9 0922660 10360 10°9077340/9 '9431306)10°0032960 
 10°9100097}7°8806033}9 0933020 10335 
 10°9089918/7°8826469]9 0943355 10312 
 0920237 10130 10°9079763'1°8846856]9 0953667 10288 
 
 09503671} 191 97|10-9069635]7-8867196]9-09639531 p94, 
 0940474]; y9gq|10-9059526]7-3887487]9-0974219}) 0541 
 0950556) (9 59|10°9049444|7-890773219-0984460} 991 
 0960615]; 09 36|10:9039385|7-8927928|9-0994678} | 41.94 
 0970651} 99} ;|10°9029349}7-8948078|9- 10048721, 75 
 |-0980662| 49g 9|109019338|7-8968181]9-1015044} 4145 
 
 0990651) 996 5|10°9009349}7-8988238)9-10251921, 195 
 1000616] 9949}10°8999384/7-9008248)9-1035317} 1 91 95 
 1010558} 99 g}10°8989442)7-9028212/9°1045420); yg) 
 1020477] ggo¢}10°8979523/7°9048130/9°1055500}) y95-|10°8944500/99412691]10-0035023}, 6, 
 1030373} 9g7,5)10-8969627)7-9068002|9-106555711 993 4|10°8934443/9 941 1256|10-0035184}) 65 
 9850|19°8959754|7'908782919-1075591] 1 oq; 3|10°3924409|9-9409821}10-0035345}) go” 
 
 9399|10-8949904]7-9107610|9'108560 10-8914396|9'9408385}10-003550711 4 
 9805|10'8940076|7-9127346|9-1095594 10-8904406}99406949}10-0035670) 64 
 9783) 10°8930271|7-9147038]91105562| gg4 ¢|10°8894438)9°9405513|10-0035893, 6.19: 
 9760|10°8920488|7°9166684/9°11 15508 10°88844.99}9°9404076}10°0035996), ¢5 
 9738|10°8910728]7-9186286|9°1125431 10°8874569)9°9402638|10 0036159), ¢ 119°9S 
 9694|10°889127417-922535819- 1145213 10°8854787|9°9399762}10-00364871, ¢ 5 
 
 9679) 10°8881580]7-9244827/9°1155072 10+884492819°9398324)10-0036652}, ¢5|9° 
 9650|!0°8871908/7°9264253/9-1164909 10°883509 1}9°9396885}10°0036817|) ¢; 
 
 0858945), yon¢ 
 08692211; y955 
 0879473) 997 
 08897001 9995 
 0899903 
 
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 id 10155 
 
 156\7. 
 
 10-9066980|9-9429876|10-0033116 ae . 
 
 10°9056645|9°9428447|10°0033273 15719" 
 10°9046333)9°9427016|10-0033430 158\) 
 
 10°9036045|9°9425586/10°0033588 158\” 
 10°9025781|9°9424155|10°0033746 158 
 10°901554.0|9°9422723]10-0033904 159\?" 
 10-9005322/9°9421291/10°0034063 159 
 10°8995128|9°9419859/10°0034229 159\” 
 10°8984.956|9°94184.26|10°0034381 160\"" 
 
 10+8964683|9-9415560|10-0034701); 6 
 10°8954.580|9°9414126]10-0034862I, «|S 
 
 9990 
 9968 
 9946 
 9923 
 9902 
 9880 
 9859 
 9837 
 9815 
 
 11059924 
 1069729 
 1079512 
 
 "1108726 
 11118420 
 1*1128092 
 
 "1137742} 9g9g|10°8862258/7-9283636)9°1174724} 9,194 |10:8825276|9°9395445]10-0036982} 6719" 
 111147370} 9¢Q7|10°8852630/7-9302975]9°1184518] 9,-,4{10°8815489|9-9394905]10-00371481) 6619" 
 11156977] 95g5|10°8843023/7-9322071|9°1194291] 5, .|10-8805709|9-9392565|10-00373141  ¢n19° 
 1166562) 9565/10°8833438]7-9341523/9-1204043) 9,45 
 
 11176125] 9549|10-8823875|7-9360734]9 1213773) 759 
 
 151185667] 9591|10°8814333|7-9379901)9°1293482 
 
 1195188] 9599|10-880481217-9399027|9-1233171 
 
 1204688] 9479|10-879531217-9418110]9'1242839 168 
 11214167] g457|10-8785833)7-9437151|9°1252486 10°87475 14]9°9383914}10-0058319}; 64 
 
 11223624, 
 
 9437|10°8776376|7-9456150]9'1262112 
 "1233061 
 
 941 6|10°8766939/7:°9475107/9°1271718 
 9395|10°8757523!7-9494023)9-1281303 
 9374|10°8748 128/7-951289819-1290868 
 9354|10°8738754|7-9531732/9-1300413 
 9334|10°8'729400/7-9550525]9'1309937 
 
 9313|10°8720066|7:9569276|9°1319442 
 9299|10-8710753|7-9587988]9"1328926 
 9973]! 0°8701461/7-96066599°1338391] ; 
 9259|7 0°8692188/7-9625290|9-1347835 
 9933] 10°8682936]7:9643880/9-1357260 
 9219|!0°8673703/7-9662431|9-1366665 
 
 9193|10°8664491/7-968094919-1376051 
 
 *1251872 
 °1261246 
 *12°70600 
 
 10°8680558/9°9373802/10-0039508 17] 
 10°867107419°9372356|10°0039679 179 
 10°866160919°9370909/10-0039851 179 ‘ 
 10°8652165/9°9369462}10-0040023 173 
 10°864274019°9368015}10°0040196 173 
 10°8633335|9°9366567/|10°0040369 173 
 
 "1344702 9173] 10°8655298)7-969941419°1385417 
 "1353875 9153|10°8646125)7°9717846|9°1394.764 
 1363028) 91 39|10°8636972|7°9736239)9°1404092 71}10°0041064 15 
 "1372161 9114) 19°8627839)7°9754593)9° 1413400 
 
 10*8586600]9-9359321|10-0041239}, 719° 
 1381275 10+8577311|9°9357870)10-0041414| 21s 
 61390370 
 
 "1399445 
 
 9095|!0°8618725)/7-9772908]9-1422689 
 
 9075|10°8609630)7-979 1 184]9:1431959 
 “ 9056|10°8600555)7-980942219-1441210 
 9036|10°8591499/7-982762119-1450442 
 
 9018|10°8582463/7-984.578219-1459655 
 
 3998|10°8573445|7-9863905|9-1468849 
 
 10-8564447/7-9881990|9-1478025 
 
(264) 8 Deg. © NATURAL sINEs, &c. Tab. 10. 
 
 Sine |Dit.|Covers}| Cosec, | Vang. 
 011391731 2881 8608269]7°1852965]1404085}7 +1 153697} 1 -0098276)0097319 
 11139461 2lo 30 
 
 213974925860 
 
 /13}1400372 96 6 
 411403252log¢9 
 
 5114061325906 
 
 9902681\60 
 He: 9902275159 
 +n {9901869158 
 7" s 9901462157 
 h ng 9901055156 
 as 9900646155 
 
 9900237154 
 
 411 
 9899826)53 
 
 41 ligg99415152 
 412/9899003151 
 
 aig 8.588108]77082694111426179)7:011'7441}1°0101187/01001'74 
 
 81141477219 9-74/8585228| 706827771 42914716 9971806) 1 °0101607}0100585 
 
 J 91141765 1]9g69[8582349]7°0539205]1439 1 15]6-9826781}1-010202710100997}, 5, 
 | (01142053 199x0]8579469]7°0396220|1 435084)6-9682335} 1 -0102449)0101410) ; 319898990150 
 
 29779|8576590| 7°0253820|1 43805316 -9538473|1-010287110101823], 1, }9898177/49} 
 1211426289]og 791857371 1|7°0112001]144 1022|6-9395199|1-010329410102938], | ./9897762/48 
 
 13]1429168]9g~79|8570832|6-9970760}1443991}6-9252489|1-0103718]0102653}, ; ¢|9897347147 
 14]1432047]9919|8.567953|6-9830092I1 44696 1]69110359]1-0104143}0103069) , | 9896991146 
 15|1434926]9g -9|8565074|6-9689994| 144993 1|6°8968799] 1 -010456810103486), ; .|989614/45 
 16114378035|9g79|856219.5|6-9550464|1452901|6-°8827807|1 -0104995|0103904], | 59896096144 
 17|14406841990|855931 6|6-941 1496|1455872|6°8687379| | -0105422/0104323), | 5/9895677/43 
 18/1443562]99%7¢|8556438|6"9273089|1458842|6°8547508|1-0105851/0104749}, 989 2258/42). 
 
 19]1446440]ogx9/8553560|6'9135239|1461813|6°84081 96]1°01 0628010105162}, 00 
 20/1443 19}9979|855068 1|6"89979491 1464784|6-826943"7] 1 -010671010105584) 4, 
 2211455075|o9~ 9(8544925]6°8724995|1470727|6-7993565]1 0107573|0106428) {9893572138 
 23]14.5'7953|g2n|85420476°8589338|1473699|6-7856446]1 010800601 068521 ; 4 .|9895148)37 
 24|146083019376/85391 70}68454229| 1476672|6-7919867|1 0108440]0107277], 4,|9892725)36 
 
 25]1463708]997n|8536299)6+8319642|147964416°7583826]1 -0108875]0107702), 9 
 26|14665951997g|853341 5|6°8185597|1482617]6-74453 18]! -0109310/0108128}, 9. 
 27|1469463]99~71853053"7|6-8052082|1485590)6°7313341|1-0109747}0 108555} 45 
 428|14723401997718527660|6°7919095|1488563)6"7 17889 1)1+0110184]0108983), 54 
 29|147521 Tl9qur18524783)6-7786632|1491536]6°7044966] | -0110622|01094 12}; 5719890588)31. 
 30|1478094|9g-77|8521906|6°765469 1|14945 10|6-691 1569}1 °0111061|0109841), 55 
 
 311480971 lo g7n718519029]6°7523268|1497484)66778677|1 0111501101 10279}, .., 
 32/1483848]9 37 6|85161526°7392360|1500458|6-6646307|1 011 1942/01 10703}; 54} 
 }33[14867241997118513276|6-7261965|1503433]6-6514449|t-0112384]01 11135], 44 
 34148960 1log76|851039916-7132079|1506408]6-63831 00}1 °0112827]0111568], 5). 9888432196 
 95|1492477logr 6|8507523|6-7002699|1509383]6-6252258}1 “01152701011 2009] 5 9887993/05} 
 36|1495353}997n|8504647|6°6873822|1512358|6°61219191-0113715|0112436), 5 .|988 7564) 24. 
 37|1498230|9gx76|8501770|6°6745446]1515335|6-5992080|1 +01 14160]01 12879}, 4. 
 38]1501106|9g75|8498894|6"6617568|1518309|6°5862739} I 0114606|01 13308], 4, 
 39}150398 119g176|84960 1 9]6-6490184]1521285|6-5733892|1°0115054|01 13745}; 46 
 40/1506857]9 97618493 143]6°6363293]1524262|6 5605538) t-0115502101 14183) 54 
 41}1509733] 5 975|8490267|6-6236890|1527238)6-5477672|1°01 1595110114622), 55 
 $42115 12608log76|84873996-61 109'73|1530215|6°5350293} 1 0116400]0115061), 4 5 9884939}18 
 
 9g178|848451 6/6°5985540|153319216-5223396|t-011685110115502), 5 reas bi 1". 
 44/15 18359}9 9x75|8481 64 1/6°5860587|1536170]6-5096981|1-0117303]01 15943}, 4 [0884057116 
 45}152123419¢175|8478766|6°57361 1 2|1539147]6-4971043)1-0117755)0116385) 4.44 asta 15 
 46]1524109| 9¢u7518475891!5-5612113]1542125|6-4845581/1 0118209}01 16828]; 9885172114 
 47}1526984}9 9+) 1847501 6|6-5488586|154510316-4720591)1°0118663]01172791, , ,p992-25i13 
 
 4 A , % J , 
 48}152985819g-7518470142|6:5365526|1548082|6°4596070]1 01191181011 7716]; ; Jose 228412 
 
 49]1532733|9 gx 4|8467267|6°5242938|1551061|6-4472017|t-0119575/0118162}, , 99818 
 50}1535607]99475|8464393}6-51208 1 2]1554040}6-4348428}1 -0120032)0118608), 4. 988139211 0; 
 51]1538482|9 973 /846151 816-4999 148|155701916-4225301]1 -01 2048910119055]; 1 o|2880945) 
 J52[1541356]9 97 4{8458644|6-487794.4]1559998]6-4102633]1 -0120948101 19503], 1 [9880497 
 53]154423019 9x 418455770|6°47577 1 95]1562978|6-3980429|1 -0121408|0119959}, 
 541154710415 9x 1|8452896/6°4636901|1565958|6-3858663}1 -0121869/0120401 
 
 9886255]91 
 988581799 
 9885378]19 
 
 9973| 944427516 °42787 1 9}15749006 °3496092)1 -0123256/01 21755 
 58}1558598 9874 844.1402}6-4160216}157788 116°3376126}1 -0123720]01 22208 
 591561472 2873 $43852816°404.215411580863}6°3256601)1-012418510122662 
 
 Ne ed ed Ce 
 
 Deg. $l. 
 
8 Deg. LoG. SINES, &c. (265) 
 ‘) Sine {Dif 
 019°14355551,-- 
 119-1444582 
 9191453493 
 319-1462435 
 4|9-1471358 
 
 Cosec. |Verseds.| ‘T'ang. |Dit| Cotang. ‘Covers. | Secant |D.] Cosine 
 
 | a. | ——— | ———- -——_—_————_] -——_ 
 i | RD 
 
 10-8564447/7-988199019-1478025|. ..|10°8521975|9-9349158/10-0042472 
 91577] 0.851981819-9347'705|10-0042650|- -919-9957350159 
 
 10°850367919-934625 11100042828 Ag 9+9957172158 
 
 8942 
 
 8923 Si tone 9102 
 390,{108528642}7°9953955}9-1514543]9 97 
 
 {|10-8519738}7-997185319-1523627}, \a5 
 36r%{10°8510852)7-9989713]9-1532602\5 9° 
 $19: 1506864}045110-8495136|3-0025 125|9-1550769 
 9}9+1515694}5 45 3|10°8484306)8 004307619-1559780 
 10}9°1524507]3~705|10-8475493]8-0060790|9' 15687 
 11/91533301}o.- -|10-8466699|8-0078468/9-15777 
 19/9-1542076]o4 00 
 9:1550834len., 
 13|9: 4 
 14|9-1559574 ant 
 15|9°1568296|57 
 16/9°1577000}3 <3 6 
 17/9°1585686}, 006 
 18]9°15943541o 235 |10-8405646/8°0201213/9-1640083 
 19|9-1603005 
 20/9-1611639}ecs4 
 
 9011 
 gg93 
 T3)gq%75 
 
 10*8457994(8-009611 3 1saeroe838 
 08457994 0 ohne 
 
 8923 
 8904 
 
 519°9955552149 
 
 10°8413294/9°9331688}10-0044630 oh 9°9955370/48 } 
 
 10-8404354}9°9350230|10-0044819] |, .|9°9955188|47 
 10-8395431|9-93287711|10-0044995] | °3|9°9955005}46 
 10-8386527|9°9327311]10-00451 73}, 35 19°9954822/45 
 10-8377639|9'9325851|10-0045361] | 5° |9°9954639|44 
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 (266) 9 Deg. NATURAL sings, &c. Tab. 10.° 
 
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 838970316-2100359 tatorefetinoas 1-013223010130504/40%|9869496/441 | 
 
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9 Deg. LOG. SINES, &c. 2 (267) 
 
 Tang. |Dif| Cotang. Covers. | Secant [13.| Cosine 
 
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 10°7476271}8°2072293/9 2594285 114 
 
 10-'745546813-211424119-2615779 
 69121 0-744855619°212817919-2622921 
 69001 0-744165619-214.209419-2630053 
 
 ccc PASE 
 68°77 10°%434.767 
 
 7120 
 9°2637173 1110 
 
 682i 
 6811 
 
 68001 9.7380059|8-226632919-2693749 
 
 9-2626729|-"98l10-7373973|8-2280023 
 
 9*2647030 
 9°265377 
 
 9°2667232 
 
 6713 
 6702 
 6691 
 6681|— 
 667011 (799931 118-242923519-2777343 
 6649 
 6638 
 
 9°2680647 
 
 9-2727263 
 
 9°2773366 
 92779911 
 9-2°78644.5) 
 9-2792970 
 
 —_— OO} I | SS 
 
(27) 211 Der NATURAL SINES, &c. Tab. 10. 
 1’] Sine Dif [Covers i | 
 
 559 9813486)55 
 8074780]5°1942125}1961922)5°09704 ‘ 56] 9812927/54 
 
 80'7192615+1865228] 196494315 °0892061)1" 56 
 
 ,18069072)5°178856 7 0191805}0188195 56 
 8066218]5°1712198}1970986 01923891018875 7), .- 
 
 806336415'°16359241197400815 0658352! 1°019297310189320 
 
 8037686)5 "696024812001 22914: 0198279|01944.24 
 51803483415'088628412004.248}4 0198873}0194995 
 51803198915°0812539 1+0199468}0195567 
 
 802627815 -0665701]2013327}4"9669037]1- 
 "9594474/1- 
 25/1979425]o9 x ,/8020575}5°0519726}2019581}4"9520125)1- 
 26]1982276logs 1 |8017724|5 044'7060}2022409]4°9445990} | 
 27}1985127|og5 {8014873} 0374607 *9372068} | ° 
 28]1987978]59 5 4{801 202215 -0302367 
 
 29]1990829}9¢ 5 9)8009171|5+0230337 4°9224859 
 30}199367915¢>5 {8006321}5°0158517/205452 49151570} 1° 
 3111996530 2850|800347015 0086907203755 4°90'78491]1- 
 32)1999380 08 5918000620}5-0015505 4°9005620)1°02 
 33|2002230 2950/9977 
 
 3412005080 
 
 281 
 Sane 79550394°8900700 
 
LOG. SINES, &c. : (271) 
 Tang. Dif | Cotang. Covers. | Secant | D.| Cosine 
 9-2886523] x4 0) 10°7113477]9-9080510 10-0080534},, , .|9:9919466)30 
 10711067371 9:90789'78}10-0080780|- , 219991922059 
 10°7100007199077445}10:0081026)<. 219-99 18974|58 
 10+7093287|9"9075911|10-0081273}5 ; .|9-9918727157 
 Ph 10°71086576|9:9074377\10°0081520\-7,.19:9918480|56 
 hss 10°7161641|8*270710919 29201 26} g ag 1|10°70798 74199072842 10008176715, -|9°9918233]55 
 As 10:'7155197]8+27201 19]9-292681 7 6.¢2.4]10°7073183|9°9071307 10-0082014|5 ; 4)9°9917986)54 
 5104 1077148763 8:2733111|9*2933500! ¢aro|10°706650019-9069772 10-0082263), , .|9°9917737)53 
 iy 5{10°7142339 §-27146082|9-2940179) aa «,|10°7059828/9-9068236 100082511), ; 4/9991 7489]52 
 e401 L077135924 $-2759035|9:2946856] ¢anq|10°7053164|9-9066699 10°0082760}5 ; |9:9917240)51 
 350 | 10°7129520)8-2771967 9-2953489|a01 5|10°704651 1]9°9065163 10-0083009|<+719-9916991}50 
 2 911.0°'7123125|8°2784880)9 2960134) cage 
 
 6385 ly : lke Me . ~ e 
 316 10-7116740|8°2797774|9°2966769) Geog 
 
 Sine 
 9:9805988 
 9-2812483 
 9-2818967 
 9°2825441 
 92831905 
 » 9*2838359 
 | 9-2844803 
 
 549 5{LO"7194012)8°2641757 
 
 aigy {0° 71875 17]8:2654867)9 “2895263} 675 
 
 Ot eg|10-7181033|8-2667957}9 2899995} 55, 
 10°7174559]/8°2681028]9-2906713 
 
 64641 9.71.68095{5+2694078|9-2913424 or y> 
 
 10:7026605'9-9060549]10-0083759!, 5 |/9°9916241/47 
 11-0-7019989/9-905901 1]10-0084010\4 5 1|9°9915990146 
 10:7097643|$-2836341|9°2986618! ¢599|10°7015382)9-905 7471 10-0084261),5 1|9°9915739145 
 10:7091296|8-2849158)/9°299321 6] es a« 10-7006784|9°9055932}10°0084512)5519°9915488)44 
 10:17084960|8°2861956/9°2999804. gs 1)! 0°7000196 9°9054392 10-00847641,) - 9|9°9915236)43: 
 10:7078633|8°2874735|9°3006383; @x 4) |10°6993617|9°9052851 10-0085016|9; 19°99 14984/42 | 
 10*6987046'9°9051310|10-0085269) 9 54/9'°991473 141 
 10*698048619°9049769|10°0085522 2% 9:99144.78140 
 33 10°6973934/9-9048227 10-0085775|9 5 4}9°9914225)39) 
 3 10°6967391|9°9046685|10°0086029}9« 4|9°9913971)38 
 39143} 55 ,|10°6960857|9°9045 142 10°0086283!95 5|9°9913717|37, 
 9°41 1-6954333 9°9043599|10-008652319 5 5 |9°9913462136 
 10°694'7817}9°9042055}10-0086793})9,,5|9°9913207/35 
 10°6941311)9'9040511]10°0087048}55 619799 12952)34 
 10°6934813/9-9038966|10-008730 4195 6\9°9912696)33; 
 10+6928525/9-9037421 10-0087560)g 5 6|9'99 12440 39: 
 106921845 )9°9035876|10-008781 69 57|9°9912184)3 1. 
 10°69153'74'9°9034330|10-0088073|95~ 
 
 10°6908912/9°9032783|10-0088330} 959 
 10°69024.59)9-9031236 10°0088588)956 
 10°6896015,9°9029689]10-0088846)950 
 10*6889579,9°9028141]10-0089 1041959 
 10°6883152/9 902659310 -0089363 959 
 10°6876734/9°9025044110-0089622'9 59|4" 
 10°6870325|9-9023495|10-008988 l}o¢q|9°9910119)23; 
 10+6863924)9°9021945110°0090141 564 |9°9909859)22 
 10°6857532)9°9020395]1 00090402156 ]9°9909598)21 
 10°6851149|9°9018845|10°0090662\0¢6, 9*9909338!20 
 10°6844'77419°9017294,10°0090923 9 g9|9° 990907719 
 10°6838408|9°9015742|10-0091185)9¢9/9°9908815}18: 
 10°683205019:9014190|10-0091447}o go/9°9908553}17 
 10682570 1199012638 |10-009 1'709]o ¢0|9°990829 1}16 
 ine 10°6819360|9:9011085}10-0091971]5¢4)9°9908029)15 
 061 {1076905268 8+322504'7(9°3186972! 639 4|!0°6813028)/9:9009531 10-0092234|,¢,|9°9907766|14 
 6051 10-6899202|8°3237298]9 °3193295} 631 |10°6806705)9:9007978 100092498] -9|9°9907502)13 
 
 10°6893151|$°5249532|9°3199611| gan]! 0°6800389/9'9006425 10-0092761]o¢5|9°9907239)12 
 
 8 9°3106849| 494.5 
 
 919-3112892 6034 10:6887108|8°3261748|9-3205918] goqq]10°6794082 9+9004869}10-0093026|9¢,|9°9906974 1] 
 
 0193118926] 4595 10°6881074|8°327394'79°3212216| oq q|10°6787784 9+9003313|10-0093290!9 ¢5|9°9906710}10 
 
 1931249511651 10°687504.9|8-3286 128193218506] gogo) !0°6781494 9+9001758}10:00935551o¢5|9°9906445) 9 
 
 | 29°3130968|gq¢ 10°6869032|8°329829219°3224783| go7|10°6775212 9-9000202]10:0093820], ~-|9°9906180} 8 
 
 '3/9°3136976}. 499 10-°6863024|8°3310439]9°323 106 Ll go¢,|10°6768939 9+899864.5|10-0094086],¢.19°9905914} 7 
 10°6857025|8°3322569|9°323732'Tl ao 51|10°6762673 9*8997088]10-0094352} 5 ¢-19°9905648 
 
 (14193142975 5990 
 '5|9°3148965 5989 10°6851035|8°3334682)9°3243584| 654.0 10°6756416|9:3995531|10°0094618},--|9°9905382 
 16)9°3154947 5O4 10°6845053|8 334677819 °3249832) 6944 10°6750168|9'8993973}10-0094885}5 ~ 9°9905115 
 9719°3160921 5964 10°6839079)|8°3358857|9 3256073) co 10°6743927/9 899241 4}10-0095 152/56. 9°9904348 
 5956 10°6833115|8°33'70918)9°3262305| coo 10°6737695}9°8990855 10°0095420 968 9°9904580 
 10°682'7159|8 33829639 °3268529| 651 6 10°67314'7119°8989296}10-0095688), 9°9904312 
 10°6821211|8°339499119°3274745 10°672.5255|9°8987736|10-0095956 9°9904.044 
 
 Covers. |Cotang. Difl Tang. |Verseds. |D.|~ Sine 
 
 6327 
 6318 
 
 10:7034610|8°2963660|9°3052183l¢ 596 
 4519-29771 64 1] 651,91! 0°7028359)8-2976289}9 5053689 6498 
 119-2977883|e599|10°7022117|8-2988899/9 3065187 Gage 
 319-2984116 5993 10:701588418°300149119°3071675 6430 
 /3/9-2990339| 6543 10°7009661|8°3014064)9°3078155! 6 4, 
 19-2996553) 25 2|10°700344'7|8-3026619 9:3084626! 61.40 
 i19-3002758),<19|10°6997242)8:3039 156|9-3091088|¢4 53 
 |2]9-3008953} 6 1 gr-|10°6991047/8°3051675/9 S09TS41 64.44 
 /3}9°3015140\ jr,|106984860]8-3064175)9°3103985 64.96 
 (§]9-3021317]4 j <a|10°6978683)8 3076657 9°3110421] 649% 
 A ipo !0°6972515|8°3089 122/93 116848! ¢4 45 
 7|9:3089794 ce 
 mig. QA 
 3193045934. ae 
 /'1]9°3064303|6 104 
 /3/9-3076503|,., 
 “alo-3082590|°0°. 
 
 9°9911154 
 9°9910896)26] | 
 
 10-6960206|8*3113996)9°3129675' ¢.404 
 10-6954066|8°3126406]9°3136076 g396 
 10°6947934)8-3138798|9°S 142468! 659.4 
 10°6941811/8:3151172|9°3148851) 65775 
 10*6935697|8°3163529}9°3155226| gg¢66 
 10°6929593|8'3175868|9°3161592 ¢35. 
 10°6923497/8°3188189|9°3167950! gg49 
 07g | 10769174 10]8°3200493}9°31 74299} 63.4 
 
 10-6911332|8°3212779}9°3180640!6339 
 
 n aD 
 Tannnm 
 
 slonwuare fon 
 
 Cosec. | D. 
 
(272) 12Deg. NATURAL SINES, Xc. _ Tab. 10. 
 
 ‘! Sine 1Dit}Covers| Cosec. Tang. =o Secant | Vers. 
 ] 
 
 7990883|48097343\21 25 56014-7046301|1-0223406|0218524 6l60 
 284512978059)4:8031613/2128606/4*69'79100) 1 -0224039|0219129]2/2197808771|59 
 284517415 19514°19660661213164'74-6912083] 1 *0224672'021973512.,19780265158 
 2845} 79 79348)4-7900702|213468814-6845248 I -0225307/02203421.00 
 2845}79(99503]4+7835520|2137730|4-67785951 1 022594210220950|- 59779050156 
 S615 17906659) 4+77705 1912140772] 4-67 121 241 40226578 0221558] 519 
 2845/79038 1414-7705699121 438 14/ 46645832 1022721 6[02221 681.5 (9777832154 
 
 2844 ) : 
 a 79009'70|4%641058]2146857|4°657972 111 10227854'0222778 61] 9777222153 
 
 212084807 
 312087652 
 412090497 
 1 512093341 
 612096186 
 112099030 
 8121018774] 94 117898 126/4°7576596)2 149900) 46518788} | 0228493 )0225389) 5 
 92104718 a tArrs95082|4-75123 1 9|2152944|4-6448034)1 -0229133}0224001/5 5 
 L0}210756) 49 +317 399439)4+7448206]2155989|4°6382457| I 0229774022461 3|., 719775387150 
 1112110405 nye 71889595|4-7384277]2159032] 46317056] 1 023041 61022520716 
 Jigl21 1324s); 0° °/7866752|4-7320524]2162077|4-6251892|1-02310590225841 
 15/2116091|,.,.{7383909|4+795694.5]21 6512946186783} 1 -0231703/0226456), 41977394447 
 14{2118934/297917881066|4-7193549)2168167/4°6121903|1-0232348,0227072\¢, - | 
 1512121777 78225|47130513121'71213]4+6057207) 1-02329941022 7689] 5, 41977231 1]45: 
 16|2124619)<°°'-)7875381|4°7067256|2174259/4°5992680|1 023364.110228307] 4, 4} 
 619 
 
 1712127462 
 “7869696)4*6941660}21 80353) 45864141) 1 -0234937}0229544) .45 9770456142 
 
 18}2130304 
 19}2133146], . , ,|7866854|4-68791 19}2183400/4"5800129}1 -0235587/0230164|¢ 9 ;|9769856/41 
 2012135988 553i 786401 2|4*681674812186448|4°5736237| 1 *0236237|0230785] .5519769215140 
 Q1{2138B20}, ° | [7861 171 [46754548] 2189496) 4°56726 15} | -0236889)0231407)¢ 9319708993139 
 2212.141671)5 OF 7858329|4+6692516]2192544/4°56091 1 1]1 -023754110232030],54|97679 70138. 
 2312144519}, 27 | 7855488}4-663065 2/21 95593/4°5545776)1 0238 195}0232653) 65 |9767347137 
 24/2147353), ie 7852647|4*6568956|21 98643) 4'5482608} 10238849 /0233277| 4, .[9766723|36 
 25}2150194),,. , ,|7849806/4-6507427|2201 692/454 19608) 1-02395040233902 ¢9419766098)35 
 26]2153035)5 7 ||7846965|4-6446064 22047421 45356773|1024016110234528|¢5 2197654712134 
 2712155876) Saco 7844 124/46384867|2207795|4°5294105) 1 02408 18/0235155|_5-19764845/33, 
 9312158716 9840 784128414°63238351221 0844)4°523 1601) 1°0241476|0235782 6 9764218132 
 {29}2161556), °° | 7838444 ]4-6262967]2213895|4"5 1692611 -024213510236411|¢ 1976358931 
 30|21 64396), 0 9|78556044-6202263]2216947)4'5107085}10242795)0237040 
 12167236 a 4011932764146 141722}2219995 F'5045072|1 “0243456 0237670164 }9762330|29 
 2170076) 96 4/7829924|46081345|2223051/4°4985221}1-02441 1810238501 ¢5 11976 1699}28 
 2172915); 039) 7827085}4-6021 12612226 104)4"4921532|1 -0244781/0238932|55. | 
 2175754 9 959] 2424645961070 22291 57)4°4860004 1 0245445]0239565| 545 
 85,21 78593) 05 4) 7821407/4-5901 174122522 1 1/4-4798636)1-02461 100240198} 55 
 36|2181432), 0-0) 7818568 |4-5841439|2235265| 44737428) 10246776 024033216 5.1¢ 
 99949) 815729}4 5781862122383 19)/4°4676379) 1 -0247442)0241 467) 6 1975: 
 438|21871 10/5 06 217812890 [457224441224 1574) 44615489) 1-02481 10/0242103).,,19757897}22 
 439}2189948 7810052\|4°5663183 2244499|4-4554756 1+0248'779| 0242740 
 
 4312201300 7798'70014-5427'7091225 665 44°43 13392) 1 095 1463|0245294 734706)17 
 14412204197 ne 
 [45]22069744, ° i 779302614°5310903 
 
 46|2209811|, 5 ae 7790189\4°525273012265827)/4°413399611"0259486102472 19 
 447}221 2648), 0 WISTS 5245194711 64a\2 12” 
 148]2215485 9656 17845 1514°5136844122'7194.414°4015164)1-0254839|0248506|~ , 19751494112 
 44912218321}... 7731679|4°50'79129]2275003)4°995597711 025551 80249151... 19750849] L 
 $50|2221158)5 05 1778842|4-5021565|2278063|4*3896940]1 0256197|0249797} - ie 
 512223994), 64 2|7776006|4-49641 521228 11 23)4°3838054)1-0256877}0250444.- 
 
 52}222683015 0 7773170|4-4906889|22841 84)4°377931711-025'7558/025 10911. 
 453}2229666)5 05 97'10334|4-4849'7'75|228'724.414°372073 111-025 824010951739 
 454]2252501]0 032 7'76'74:99| 4.°4.7928.1012290306|4°3662293)1 -025899310252388 
 
 455]2235337 985% 7764663|4-4735993|2293367|4"3604003}1 °025960710253038} . 
 156|2238172)5 oo 7'7161828|4°46'79324.122964.29]4'3545861'110260292/0253689] ~ 
 1572241007], 0% ‘ 
 45812243842 2834 
 15912246676 9835 
 160}2249511 
 
 | Cosine|Dif 
 
 = | | | | 
 
 2 
 < 
 : 
 9 
 
5, 
 12 Deg. | «LOG. SINES, &c. At et GaT 
 
 Covers.| Secant {D.} Cosine 
 
 rc | ef | + | ——— | 
 — | —_—_—_——_———— |} ———_ | ———_ —_—_—_—_ 
 
 019°31'78'789 5939 10°682121118-3394991|9°3274745 
 1/9°3184728 593] 10°681527218-340700219-3280953 
 2/9-3190659 nase 
 3|9°3196581 5914 10°6803419]8°3430975]9°3293345 
 419°3202495 5905 10°6797505}S°3442936]9-3299528) 4, 6 
 5}9°3208400F oo. 10°679 1600|8-3454880}9-3305 704), | 68 
 6)9°321429 7} 3 39 10°6785703|8-3466808/9°3311872},, 59 
 7/9°3220186] 544, 10°6779814/8°3478'71919°3318031 6152 
 8/9°3226066): ono 10:67739348°3490614}9°3324183]_, Ag 
 919°323 1938/9 6) 10°6768062|8°350249219 3330327 6136 
 10/9-3237802)5 4. 5 10°6762198]8°351435419°3336463 6198 
 11]9°3243657]; 4 48 10°6756343]8-3526200|9 334259 1], 120 
 1219°3249505 5839 10*6750495|8-3538029/9-354871 1}, 1 19 
 13|9°3255344 5830 10°674465618°354984.219°3354823 6104 
 1419°3261174 5823 10°673882618°3561639/9°3360927 609" 
 15|9°3266997 5814 10:6733003|8°357341919°3367024 6089 10°6632976]9 °8964.283]10°0100027 
 1619°3272811 seine 10°672'7189]8°358518419'3373113 6081 
 17|9°3278617 5799 10°6721383|8°3596932/9 3379194 6073 
 18|9°3284416 is 10°671558418°3608664|9'3385267 6 66 
 
 19}9°3290206 ana g|10°6709794|8-362038 1/9°339 1333]. .|10-6608667|9°895801 1)10-0101127 
 20193295988); -|10°6704012/8-363208 1}9°339739 1] -0 55 
 2119'3301761}<,, .-|10°6698239|8-364376519°340344 1 
 
 ; ~| 766 
 22/9°3307527}.- 56 
 23]9°3313285)505 
 24|9°3319035),, 
 
 269]7. 
 6183 
 
 271 
 
 273 
 
 2 
 
 275, 
 
 10°659655919°8954872110-0101680|2.," 
 
 10-6686715}8-3667086|9°3415519|°0°> 
 
 ong 
 . 5 . A1Q-34 5 
 5734{10°6675225|8-369034419-3427566 
 719°3336237 Ss 
 28193341955 
 
 5710 10°6658045|8°3725114|9°344.5580 
 
 10°6652335|8°3736672)9°3451570 P 
 
 260 9-9903775159 
 10°680934118°34189971/9-° 3287153 deol? 2909906158 
 269 . Q083rFl45K 
 Qn 9+9903237157 
 “110°67004'7219°8981492110°0097033 ay 99902967156 
 10:6694296/9°8979930}10°0097303 27] 9°9902697155 
 10°6688128|9 °8978367|10°0097574 9°99024.26154 
 
 10°6681969|9°8976804}10-°0097845 279 9°9902155)53 
 10°6675817|9°8975241|10°0098117 O71 9°9901883)/52 
 10°6669673)9'8973677|10-0098388 273 9°9901612)51 
 10°6663537|9°8972112|10-0098661 Qn0 9-9901339]50 
 10°6657409)9°8970547|10-:0098933 273 9°9901067|49 
 10°6651289/9°8968982)10°0099206 9-9900794]48 
 
 10-6645177/98967416}10-0099479},,,.,|9-9900521]47 
 ~|10-6639073]9°8965850]10-0099753|-°*19-9900247146 
 tap 9°9899973145 
 10-6626887}98962716|10-0100302|5,2|9-9899698|44 
 10°6620806|9°8961 148]10-0100577 ae 9°9899423143 
 10-6614733|9°8959580|10-01008359|~,/9-9899148|42 
 
 ae 9-9898873]41 
 10°660260919°8956442]10-0101403 9°9898597140 
 apis a 99898320159 
 10°6692473|8°365543419 3409484)... ~~ |10°6590516/9°8953309]10-0101957 on" 9-9898043|38 
 4 10°658443 1198951732) 10-0102234 aie 9-9897766|37 
 10°668096518°3678723/9-3421546 sel 10°65784.54/98950161110°0102511|~! '19-9897489136 
 
 6019 10°6572434|9°8948589]10-0102789 279 9°9897211135 
 26|. 0 '6669489|8°3701950/9°3433578 6005 10°65664.22\9°8947017}10-0103068 or7g|° 9896932|34 
 10°6663763)8°3713539|9°3439583 5997|40 0960417 9°894.5445}10°0103346 agol? 2896654)33 
 990|! 0°6554420 9°8943872] 100103626 ang|?'9896374|32 
 99|L0°6348430|9°894229910-0103905}, 47 |9°9896095) 31 
 
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 5 280 
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 34|9°3376099 
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 35|9°338 1762 5656 
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 ie 10°6524.546|9°8936000110-0105027 “We 9+9894973/9% 
 1) 10°662390118°3794232 9-3481407 tn 10°651859319°8934425]10-0105308 989 9-9894.692126 
 10°6618238]/8°380569819°3487359 an 10°6512643/9°8932849]10-0105590 289 9°98944.10}25 
 5647 10°6612582|8'3817149|9°3493290 O46 10°6506710|9'8931272110:0105872 9-9894.128]24. 
 
 pears agg|? 9893845|23 
 10660129418 -384000419°3505143 reaps 10°6494857|9°8928117}10 0106438 med 9°9893562|22 
 10-6595662|8°3851409|9:3511059|2~ +9110-6488941|9-8926539|10-0106721|-°9|9-9893279121 
 
 40}9°3409963 oe 10°659003'7|8°3862799]9-3516968 5909) 1 9-6483032|9-8924961|10-0107005 284 9°8892995/20 
 
 41]9-3415580}> oh 10°6584420|8-°38741 
 4919-34211 90}20 ane 285 
 10-6573208|8-389687819°3534650 
 
 7419+3529869 is : 10-647'7131|9°8923382|10-0107289 nh 9:9899711/19 
 jo|!0-6578810|8-388553319:3528763)? 694|10-6471237/9°8921802|10-0107573|,049°9892407] 18 
 
 10*64.65350|9°8920222|10°0107858 9°9892149117 
 
 10-6567614|8-3908207|9-3540530}? 97 |10°6459470]9°8918642]10-01081 44), 2°19-9891856|16 
 10-6562027}8-391 959919 +3546409|>572|106453598/9°8917061|10-010842915°°19-9891571]15 
 10-6556448|8-393082219-3552267)>° >| 1 0-6447733]9'8915480|10-0108715}-00|9-9891285}14 
 
 19 -3.44.9124 10°6550876|8-394210719-3558 1 26{- 8" |10°644187419°8913898|10-010900a|~°" 
 
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 5837 10°6430179}9°8910733}10-0109576 987 
 5899 10°6424342/9°8909150 100109863 288 
 5893 10°6418513]9°8907566)]10°0110151 989 9*9889849 
 
 10°6523130}8°39983 10}9°3587310 5816 10°6412690}9°8905982)10°01 10440 9°9889560 
 10°6517603}8 *4009506|9 3593126 10°64.0687419°8904397)10°0110729 9°9889271 
 
 9|10°6512083}8-4020688|9-3598935 sath 10-6401065]9-8902812}10-0111018]--419-9888982 
 
 : 5505 10°6506571|8:4031855}9°3604736 jqn9 988869. 
 
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 10°6490078}8-4065270/9°3622 1 OOF. ‘ 79119 Poo 7822 
 
 9°9890424/11 
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 CHU ORY A-106 
 
(274) 13 Deg. NATURAL SINES, &c. 
 
 012249511 
 1/2259348 
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 15/2286341 444 
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 16|2294835|,9% [7705165 |4°35761 13]2357758|4°24 131771 0274192|0266875 
 17|2297666| 50 3 | |7702334/4-35224 1 912360829|4-2358009] | “0274897 10267549 
 118]2300497|5¢5 | |7699503|4-3468861)2563900}1"2302977]1 “0275603026821 1 
 19|2303328],,....17696672/4°341 543812366971 42248080]! -0276310]0268881 
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 768818 1|4°3255977|2376189]4°2084196]1°0278437|0270895 
 35]1°027914810271568 
 
 0280573|0272916 
 2399|7075862}4°5045225)2388485 4° 1867546)1 0281287027359) 
 2890| 1674033 |4+2992867}239 1 560/4°1813713}1 -028200210274267 
 {28}2328796| 5059/7671 204|4-2940640|2594635}4"1760011}1-0282717/0274944 
 29}2331625]595|7668975]/4'2888543|239771 1}4"1 706440} 1 °0283454|0275622 
 30|23344541 5650 |7665546]4'2836576|2400788] 4" 1652998] 1 028415210276301 
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 5 1}2393808}5 95 ~|7606192)4°1774438)/246549 1/4°05598771 °029944.910290742 
 5212396633}. mn 7603367/4°1725210/24.68577|4°0509 17441 °030018810291439 
 53/2399457 7600543}4°1676 102)24'71663}4°0458590]1 °0300928)0292137 
 54}2402280 
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 56}2407927 
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 58/2413574 
 $59)2416396 
 
 7594896]4°1578243}2477837 4035777911 030241 110293534 
 7592073}4:15294.9 1}2480925/4-0307550 
 7589249}4°1480856]2484013]4°025 7440] 1°050389810294935 
 7.5986426}4° 1432339/2487102}4°020744.6]1°0304643/029563 
 
 7583604|4° 1383939]249019 114°015'75'70) 1'0305389}0296339 
 60|2419219 {758078 1{4°1335655)/2493280]4°0107809}1°0306136}0297043 
 
 ~ |Cosine|Dif| Vers. | Secant |Cotan. 
 
 2823 
 2822 
 2823 
 
 1655 
 
 771649114°3792257|234.5479]4°2655218}1 027138 11026421 1} - 
 
 1975606}1°0279860|0272241 67 
 
 766271 8|4°278473812403864/4-1599685]1 028487 1/0276980 
 7657062/4*268 1449]241 0019}4"149344.6|1-028631 110278349] 2° 197 
 425786711241 61'76/4°1387719] 1 -0287755|0279706 
 4°25 274'74124.19255/4'1335046|1°028847910280390 
 4°24°764.0212429334|4°1282499]10289203/0281074 
 4 4°1230079]1-0289929|0281760)- 
 4°2374637|242849414-1177784]1-0290655|0282446} 2 
 
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 NOOV70S € Ly) 
 1°0297257|0288657 694 
 
 75917 20|4" 1627 1141247475 0}4°0408 125]1 -0301669]0292835 69 
 
 “1702 
 1702 
 
 Tang. | Cosec. Covers|D.}.. Sine | 
 
 Tab: 10; 
 
 7 Sine (Dit Covers} Cosec. Tang. Cotang. Secant | Vers. |D.| Cosine 
 a 
 
 2 Jo741077156 
 oe 5(7736320|4'4175859)2324007 430291 36]1 -0266499|0259581]°2 0197404 19}55 
 2 34|7783487 441 20637}2327075|4*297 24401 -026719410260240} 19739760) 54 
 G6 1 (97139100153 
 7'72%39114°401061612333207|4°28.594'72|1°026858610261 561122 "19738439|52| 
 779.4988l4°3955817/23362'74 428031 99]1 026928310262292/°longanmgl51 | - 
 
 66 
 668 
 667 
 669 
 
 672 
 6720, 
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 73|9728432)97 
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 972708435 
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 675 
 676 
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 678 
 679 
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 681 
 
 68 
 684 
 
 686 
 686 
 68% 
 687 
 689 
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 690 
 69] 
 692 
 693 
 693 
 
 696 
 695 
 697 
 698 
 698 
 
 9709258 
 9708561 
 9707863 
 
 9 9707165 
 
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 1°030315410294.234 01 9705766 
 
 9705065 
 9704363 
 10219703661 
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 66501391 24/47 
 7710828!4°368391012351617/4°2523923) 1 -0272785|0265541)- g|9734459)46 
 
 9733793/45 
 9733125|44 
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 9716180|19 
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 9714809|17 
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 9710649/11 
 9709953)10 
 
 ‘Jouwunra A-100 
 
13 Dege ~  LoG. sINEs, &c. é | (275) 
 
 Tang. |Dit| Cotang. | Covers. | Secant |D.}) C 
 10°647912013°4087475|9-3633641 10°6366359'9°8893291/10°0112761 
 -110'6386059919°8891703110°0113053 
 10°646819018°4109622)/9-3645155 5746 10635484519 °8890114)10°0113345 
 9*3650901 5740 10°6349099]9°S888525110°0113637 
 
 5 10°634335919°8886935)10°0113930 
 5/9°3548150}2 
 
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 119°3526349 ee 
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 319°3537264/2 55 
 
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 994 (99885 188)53 
 10°6320468}9-8880571]10-0115106]5,,.|9°9884894)52 
 10)-6314762)98878978|10-01 15401 |54219°9884599)5 | 
 106309063|9°8877386) 10-0115697)5 .|9°9884503150 
 10°6303371|9*8875792I 10-01 15999}5~°|9-9884008|49 
 
 10°645 1850)8*4142736)9°3662374|~ 
 9 10°6331900|9°8883754] 100114518 
 
 ,{ 10°644099318 416474 119°3673819 5413 10°6326181)9°8582162}10°0114812 
 5410 10°64355 7418 °4175723|/9°3679532 5% : 
 
 9)9°3569836 5404 10°643016418*4186690)9°3685258 5699 
 10|9 3575240 5 10°6424'760|8°4197644/9 5690937 5699 
 
 1119°3580637 ean 10°6419363]8-420858319-3696629}” 
 
 207 OOR 
 
 12/9-3586027}> 55.,|10°6415973]5-4219508|9-3702515]2 26, 10-6297685/9°S874199|10-0116288 a 9-9883712148 
 13}9°3591409 5316 10°6408591}8°4230420)9°3707994 uy 10°6292006|9°8872605)10°0116585 goq\2 9885415 47 
 1419°3596785 53 10°64.03215}$'424131819°3713667 S664 10-6286333}9°8871010)10°0116882}5~ {|9°9883118]46 
 
 2979.9880891145 
 
 15|9°3602154 Hae 10°6397846)8-4252201|9-3719533]> 2 9|10°6280667|9-8869415]10-0117179|544 
 16}9°3607515|5 5. .|10-6392485]8 -4263072/9-372499}- 2 -110-6275008/9°8867819|10-0117477}5q./9°9882525 44 
 17|9°3612870f207~ 561g) 1076269355|9°$866223|10-0117775]54 019988222543 
 18|9°3618217 eau 5q|10°6263709/9-8864627] 10°01 18073]55,0|9°988 1927142} 
 19|9°3623558 10°6376442|8°4295599/9-3741930| « ..|10-6258070|9°8863030}10-0118372}59 (9988162341 
 20/9°3628892 10°6371108|3°4306414|9°3747563]5 °5-|10-6252437/9 886 1432|10-0118671 BOY 9881329140 
 21|9-3634219 10°6365781|8°4317216|9°3753190\. 27 S0p{? 2881029159 
 29019 ..9880'729138 
 
 9 
 29193639539 10°636046113°4328004/9°3758810 9620 300 
 my 301 9*9880429137 
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 10°6387130|8°4273928|9°3730645 
 
 5334 
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 10°635514919-433877819-3764423 Ran 
 10.6349842|3-434953919-3770030)> a5 
 
 25]9°3655458} «5 99|10°6344542]8'4360286)9'3775631]5 x0, 39% 987982735 
 26|9°3660750\2 -" ~ 5a, |9°9879525/94 
 27|9-3666036)0 799 hi a 9-9879223133 
 2819°3671315 pth 10°6528685}84992447|9°3799394)5 3° mia 79678921152). 
 29193675537), --110°6323413|8-4403141|9-3797969 BRAG 3030 2878618)5 1 
 30|9°3681853 5958 10°631814.7}8°4413821|9°3803537 3563 303 9°9878315130 
 31|9°36871111,., . .|10°6312889|8-4424488|9°3809100), « .. 21: 
 
 39|9°3699363 an 5|10°6307637]8 4435 14219°3814655}5 557 |10-6185345|9°8342225|10-0 122299} 
 
 33|9°3697608}5 550) 10°6302399}8 4445783 |9°3820205|5 5% 4 
 
 34|9°3702847)-555|10-6297153]8-4456410|9°9825748)5 5 3 
 35|9°3708079}> 55 «|10°6291921|8-4467024)9°3831285}5 55 
 
 36]9°3'713304)25 + 2|10:6286696|8-4477625|9 3836816 » 
 
 a 9-9877099|26 
 o fs 9+9876794|25 
 7110°0123512 S65 9°9876488124 
 sor(o 9876183)23 
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 2049-9875 263/12 
 3098 875263 
 
 5512 
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 2 F gel 0-6245 139|8 -4561964|9-3880837)> oe 
 9°3760034)2 | <|10-6239966]8-4572446]9°388631 2/2 7° 0 : 
 9°3765194)2,)°|10-6234806)8"4582920|9-389 178112 a: 
 9°37 703477fs 17 £|10°6229653]8-4593378|9-389 72441. 
 9°3775493} 55 7 | t0*6224507]8"46038249-3902700|> Fs 
 
 140 
 
 /10-6214935|8-4624677|9-3913595)2 144 
 
 10°6209106]8"4635085)9°3919034)5 137, 
 52/9°3796015}., 5 ,|10°6203935]8-4645480|9 3924466]. 7 ou 
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 56|9°3816434, 02° jy aygsTo2es| 4 
 57/9-3821593 0c" 38 3149.9g6998it 3 
 
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 10°6165243)8-4728139|9-396771 IP oo? 99869041] 0 
 
 Covers. 
 
(276) 14 Deg. NATURAL SINES, &c¢. Tab. 10. 
 
 Cotang.| Secant | Vers. D.|Cosine 
 9702957|60! 
 
 704). ad Ort 
 705] 102255 59 
 
 Sine |D:i1}Covers} Cosec. Tang. 
 0|2419319], . 59] 7580781|4-1995655|2499280]4-0107809I I -0306156)0297045 
 “112422041 |7922)15:7°7959|4-1287487]2496377014-00581 65|1 -0306884|0297747 
 92424863 see 7575137|4-193943.5|2499460|4-0008636} 1 0307633|0298452|, .°10701548/58] 
 3[2497685|500|7572315]4-1 1914981250255 1|3-9959225|1 -0308383]02991 58}, }9700849157 
 §12430507),075|7569495|4"1 143675|2505649}5-99099241 I 0309134|0299864) .) ° 1970015656 
 
 2822175 666'7114-109596712508734|3-9860739I 1 -0309886103005791 -°°l9699406155 
 
 5/2433329 9821 708 
 6|2436150 282] 7563850)|4*1048374/25 1 1826|3°9811669|1-0310639/030128 09 9698720|54 
 712438971 039] 7.561029|4°1000893/2514919|3°97627 12/1 -0311393|/0301989 969801 1}53 
 
 g|2441792\00°1!7558208]4-0953526|2518019]5-9713868|1 -0312147103026991" '1o697301|52 
 | 91245.4613(002417555397/4-090627212521 106|3-9665137|! -0312903|0303409) 11969659151 
 10]2447433|900017552567|4-085913012524200|3-9616518]  -0313660}0304121|" 1519695875150} 
 11]24502541282175.4.0746|4-0812100|2527994|3-956801 1)1-0314418|0304833)" 11969516" |49 
 19}24530714|50, 017546926 |4-076518 1/2530389|5-9519615|1-0315177/0305547|,, -|9694450|48 
 
 13/2455894 19 1544106|4°0'7183'74/2533484)3°94'7133111°0315956|0306260 15 969374047 
 14/2458713 2820 754.128'714°0671677/253658013°9423157/1 -0316697/0306975 “16 9693025146 
 15}2461533 2819 75384.67/4°0625091/2539676|3°937509411-0317459|0307691 9692309 45 
 41612464352 2819 153564814°0578615)/2542773|3°93271 41/1°0318222/0308407 718 9691593]44 
 17/2467171 2819 7532829)4°053224.9)2545870|3°9279297/1 -0318985}0309125 "18 9690875 43 
 
 254896813°9231563)]1°0319750/0309848 . 969015" 
 
 18)2469990 9819 7530010/4°0485992 
 19/24'72809 2818 7527191|4°0439844/2552066/3°918393'7| 1 -0320516/0310562 
 
 2012475627) 0 1 (7524373 4°0393804/2555165|3°9136420}1 °0221282)0311281 
 2112478445 2818 7521555 4°0347872)2558264 3'908901 1|1°0322050}03 12002 
 2912481263 2318 75) £75'7|4-0302048 2561363 3:9041710}1°0322818/0312723 
 2312484081 7515919\4°0256332!2564463/3°89945 161 -0323588|0313445 
 2412486899 OBI" 7515101/4-0210722 2567364 38947429} 1 -0324359|0314168 m9 9685832|36 
 
 25|2489716 2817 7510284'4°0165219'2570664)3°8900448}1 -0325130)0314892 "95 9685108}35 
 26/2492533 28] 7507467/4°0119823)\2573766|3 88535741 °0325903)03515617 nO 9684383]34 
 2'7|2495350 2817 7504650/4°0074532)2576868)/5°8806805/1 -0326676/0316342 nO4 9683658}33 
 2812498167 28)" 7501 833)4-002934'7)2579970 3°8760142)1-052745110317069 Qn 9682931|32 
 29 25009841981 ¢ 7499016,3°99842672583073 3°8'713584!1-0328227/0317796 798 9682204/31 
 30}2503800 2816 7496200'3 9939292 2586176|3°8667131)1-0329003/0318524 708 9681476/50 
 312506616 2816 749338413 °9894421/2589280/3 86207821 -032978 110319252 3 
 32/2509432 2816 7490568/3°9849654) 2592384|3°8574537|1°0330559|0319982 730 
 33}2512248 815 7487752\3°980499 1 2595488/3°8528396}1 -033 1339|0520712 ”3] 
 34/2515063 O816 748493'7|/3°9760431/2598593]3°8482358]1 °0332119|0321443 739 
 35}2517879 28})5 74821 21/5°9715975)2601699)3 84364241 -0332901|0322175 133 
 
 2520694 0814 7479306 digas fen wahagis 3°8390591}1°0333683'0322908 "34 
 2815 
 
 74'76492)3°9627369 26079 1 1|3°8344861/) -03344.67|/0323642 134 
 814 74'7367'7|3 95832191261 1018}35°8299233}1 -0335251|0324376 "36 
 
 74.70863|3 95391 '71|2614126|3°8253707)1 0336037/0325112 
 2815 74 736 
 2814 
 
 72] logenonn|3g 
 
 9686555)37 
 
 9677825 
 9677092 
 
 68048)3°9495224)2617234|3°S20828 1/1 °0336823)0325848 731 
 746523413°9451379|2620342/3 816295711 033761 1/0326585 "3N 
 
 74596073 9363988 2626560/3°8072609|1 -0339188/0328061 " 
 74.56794)3 °9320443|2629670|3°8027585|1-0339979'0328800 "AY 
 74.5398 1|3°927699'7/2632780]3°7982661\1 °0340770}0329541 
 9813 7451168)3°9233651/263589 113°7937835|1-034.1563/0330282 "41 
 744835513 °9190403)2639002/3°789310911 -034.2356 0331025 "43 
 9 714.4.554215°9 14°7254)26421 1413°784848 111 -0343151|0331766 "4A 
 
 6 74399 1813-9061 250|2648339|3-7759519|1-0344743|0333254 "145 
 743'7106|3°9018395}265145213°771518511-0345540|0333999 746° 
 
 71434295|3°89'75637|265456613 °7670947|1 -0346338|0334.745 man 
 7431483|/3°89329'76|2657680|3°7626807|1034713810335499], m 
 
 7423050|3-8805570|2667025}3-7494963]1 0349542033773, a 
 7420240|5-8763293}2670141|3-7451207|1 -0350346|0358487|,.. 
 13.°74.0754611-0351 15010339238 on 
 14.14.61 9)3-8679025|2676374/3°73639801 1 -035195510339989 
 
 “1741 1810)3°863'70331267949 213 °7320508}1-0352762/0340742 1 
 
 (72588190 
 Cosine |Dif] V ers. 
 
: Deg. LOG. SINES, &c.” nit (2777) 
 
 Sine Cotang. Covers. |} Secant |D.] Cosine 
 3836752 10-6032289}9-8797140]10-0130959),, | i 9-986904 1/60 
 9841815 33'74{ 10602691 1]9-8795522110-01312741, | 319-9868726|59 
 3846873/3 5367) 10°6021537|9°8793905]10-0131 590}, °}9-986841 0158 
 3851924 10°6148076]8 -4'759000|9°3983830122° ‘110-60161'70/9°3'792286] 1020131906}? °19-9868094/57 
 
 -3856969|>0 +} 10-6 143031|8-476924619-3989191f°2°110-6010809|9-8790668]10-0132299)°! “ly -ag67778156 
 
 3862008] 71) 9110-6137999]8-4779480|9 -3994547}> 4° 6|10-6005453|9-8789049|10-0152539}91 “|9-9867461155 
 “3867040|?-| 10-6132960|8-4789701|9 39998961? 5 |10-60001049-8787429|10-01328 56), | 19-9867 144)54 
 ‘9872087|, 0 49| 10°6127983}8-479991019 4005240}... .|10°5994760|9°8785809|10-0133173],,, |9-9866827/55 
 38770872) 1 4|10°6122913)8-4810107]9-4010578}25°5|1 0-5989420/9-8784188|10-0133491]. | 8I9-9866509|52 
 $882101|/,5|10°6117899}8-482029 1|9-4015910}29>~|10-5984090|9-8782567|10-0133809|> | $l0-986619151 
 “3887109]3",/,5|10-6112891|8-4830464|9-4021237)207110-5978763|9 -8780946|10-0134198|> |°19-9865872150 
 ‘98921 11]7495|10-6107889|8-4840625|9-4026558}2. >: |10-5973440)9-8779324]10-01944471- °19-9865553}49 
 9897106}; 95] 10°6102894/8-4850773}9 4031873]? \?|10-5968127]9-877770 1100134767] s) 019°9865253148 
 
 10°5962818]9°87'76078] 100135087), |9°9864.915147 
 
 3902096 F ; 
 391079 He, : ae 10°595151419°87744.54. 10-0135407}35 9°9864593/46 
 3912057 509 9{t0°9952216/9°8772830|10-0135727).5 |9°9864273/45 
 3917028 ee - “110°594692419 8771 206] 10-0136048/% -*19-9863952/44 
 
 28711 0.594163719-876958 1|10°0136370\>=2 
 
 3921993 4950 10-°607800718°4901336!9 4058363 5081 36 9:9863630/43 
 "99269591 , oxe 10°6073048/8°491 141219-4063644 5078 10°593635619-8767955|10-0136692 309 9*9863308|42 
 39319035 10°5931081|9-8766329]10°0137014 9-9862986|41 
 
 494.7 
 4949 
 4935 
 
 2 o40)t0:592581 1|9-8764703|10-0137397]52519-9862663) 46 
 59510 2920547/9 8763076]10-0157660)55519-9862340|39 
 »{10°5915288)/9°8761449/10-0137983/>. 
 
 3936852 
 3941794 
 
 394.6729 9-9862017/38 
 9951658|fo55 ope |10-5910035|9-8759821|10-0138307 Bo fio 9361693137 
 3956581|1 576 ,{10:5904788}9-8758 1 92]10°0138631\5" 19-986 1369136 
 3961499 5956 10°5899546/9-8756563|10-0138955 395 9°9861045/35 
 ‘3966410 3037 [1 075894310/9-8754934 10-0139280i5 °° 9-9860720134 
 3971315 10-6028685/8°5001573}9-41 10921}; 55 ,|10°5889079}9-8753304)10-0139606 ae 9-986039.4133 
 3976215}; 59, |10-6023785|8-501153219-4116146]2559|10°5883854/9-8751674|10-0139931), 57 }9-9860069)32 
 3981109 10-6018891/8°502148019°4121366). >. 11 0-587863419-8750043|10°01 40958|>= "199859749131 
 
 35pp| 05875419 |9-8748419]10-01405841°0 
 
 10-5868211|9-8746780}!0-0140911/,,, |9-9859089|29 
 10:5863007}9 8745 147}10-0141258'2 | /9°9858762/26F 
 10°385750919-8743515]10-0141566')~°|9-9858434]27 
 
 Bosses 5192}, 1. 9." 10° {328 9: 5 
 
 4005489 5 1gr7|L0985261 79874188 1}10°0141 894/50 }9-9858 106) 26 
 ‘4010348 519Q\t 0'9847430/9 °8740248|10°0142223', 0 9°9857777|25 
 4015201 3 7g| 0984224819 8758613|10-01425511, 5 19-98574491 24 
 4020048), 5, 517]/10°583707219-8736978|10-01428811, ,|0-98571 19)23 
 4024889), 24. 5 166 {10728319019 8735343/10-0143210), 2° |9-9856790/22 
 4029724) 4 530) 10°9970276)8°5120326]9°41 73265), | —|10°5826735|9 87337071 0-0143540)27 | 19°9856460) 21 
 4034554), > ,|10°5965446/8°5130148]9-4178425]>, 0-/10°5821573/9°8732071|10-0143871|-" | 19°9856129]20 
 
 10°5816420 9°8730434/10-0144209'"* °19-9855798/19 
 10°58112'71/9°8728797 10-0144533!. 5 9°9855467/18 
 10-5806 126/9°8727159}10°0144865/,. ,19°9855135)17 
 10°5800987/9°8725521/10-0145197/9" *|9-9854803]16 
 10-579585419°8723883/10-014552917"~|9- 9854471115 
 10°5790725|9°8'722243]10-0145862!"" |9'9854198]14 
 10°5785602'9-8720604|10°0146195/""'"19-9853805|13 
 9-9853471/19 
 
 (4039378 
 
 4044196|7918 
 
 4813 
 
 shed 10°5941383]8°5 17908919 4204146 5129 
 Nines 10°593658718°518884-)19°4209975 5193 
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 4784145 oe j , on 5117 
 4779 10°5927013]8*5208320}9°4219515 5113 
 
 4063413 
 4068203 
 4072987 
 
 4077766 510% 355 
 4082539). bs 10°591746118°522775219-4229795/?. ~! 333 19°9852803} 10 
 4087306 i ty 10°5912694}8°523745 1/9°4234838)” 155 51979852468). 9 
 4092068 . 56 L09907932/8 52471 40194239935 a8 9-98521335! 8 
 4096824. ie % 1 1075903 176]8-5256817]9 424.5026], 00 -110-5754974)9°8710755]10-0148202}.2°19°9851798) 7 
 4101575 j745|10°5898425|8 5266485 ]9-42501 13/5 4 110-5749887/9°8709112/10-0148538]>- 19-985 1462] 6 
 4106320 BONN agg 9851125). 5 
 4111059 en 10-5739729/9-8705824 10014921 1}5°-19-9850789| 4 
 4115793 10-5734658/9°87041'79]10-0149548 ary: 9°9850452! 3) 
 ”) 10:5729592)9-8702534)10-0149886],7|9-9850114] 2 
 4195945 33g(9 9849774) 1 
 4129962 ?9°19-984.9458] 0 
 D.l Sine |’ 
 
(278) 15 Deg. NATURAL ‘SINES, &c. Tab. 10. 
 
 / 
 
 Sine |Dit|Covers] Cosec. |'Tang. |Cotang.| Secant | Vers. |D.|Cosine 
 } [2588190], 5 | 741 1810|3-8637033}2679499|5 7320508] -0352762]0340742I,, . .|9659258|60 
 2591000]5° | -17409000}3°8595 1351268261 0|3-7277131]1 -0353569|0341495|.2" 19658505159 
 2593810) 939] 7406190)3°8553332|2685728|3-°7233847] | -0354378|0342249 
 2596619) ,90 jq|740338 13 °8511622|2688847 [3 °7190658| 1 °0355187|0343004]~719656996)57 
 25994298} 9 ¢ jol'74005'72/3°84700061269 1 967|3:7147561}1 0355998|0343760|1? *19656240156 
 2602237] 9 ya|7397763)3'84 2848212695087 [37104558] -0356809/0344516]/2 1965548455 
 2605045] 95 4173949553 '8387052|2698207|5°7061648]1 0357621 0345274 te 9654726154 
 2607853], y9/7392147/58345713}2701328|5:7018830|! -0358435]0346053), ./9655968)53 
 812610662) °°" |7389338|3-8304467|2'70444.9|3-6976 104) 1 0359249|034679 1] /9"19653209152 
 9126 13469}-°> 738653113°826331 3]270757113-6933469] 1 0360065|0347551| (519652449151 
 10}2616277|,¢9|7583723|3'822225 1 |27 1 0694{3-6890927|1-0360881}0348311] 1901965 1689}50 
 11]2619083|- 2" °|7380915|3°8181280|2713817|3°6848475] 1 036 169910349073 Ee 9650927149 
 1212621899}, 5 4.17378 108|3°8140399|2716940)5-6806115}1 "03625 17|0349835| "19650165 /48 
 
 13}2624699),, .0..|7375301}3°8099610)2720061|3 6763845 1+036333'7}0350598 : ” 964.9402)47 
 1412627506) ~~» {'73'79494/3°805891112723188'3°6721665|1 :0364157/0351362) /°*|9648638|46 
 1512630312) <2 2 °l'736968813-80183011272631313°6679575|1°0364979|0352127 765 9647873|45 
 16]2633118) 59 e|7366882/5"7977782|2729438|3 6637575|1-0365801]0352892 pe 964'7108|44 
 17|2635925}, : 7364075|3°793735212732564/3°6595665|1°0366625|0353659 "67 9646341 |4 
 
 18)2658730|9¢ 06 7361270|3°7897011|2%35690|3'6553844| 1 036'74.4.9|0354426 nee 964557442 
 1912641536 7358464/3°785676012738817\3'651211111°0368275|0355194 769 9644806)41 
 20|2644342)0 0 5 217355658/3°781 6596/2741 945|3°6470467|1°0369101)0355963 iss 9644.03'7|40 
 21/264714'7|5 9 5917352853|3°7776522'2745072'3 64289 1 1] 1°0369929|0356732 tes 9643268|39 
 291264995915 54 5/7350048 3°9136535 274320 1|5-6387444 1-0370757|0357503 a 964.2497|38 
 23/2652 7571" > 1734724313 °7696636/275 1330 .3°6346064)1°0371587/03582'74. 9641726|37 
 
 24)2655561 athe 734.4439|9°7656824 2754459 365047711 “037241 7|0359046|,1,.°}9640954}36 
 
 95|2658366 7341634)3°7617100 2757589'3 "6263566 1-0375249/0359819 way 9640181135 
 
 7338830|3°7577462/2760719)3°622244.711 0374082 0360593 ne 9639407|34 
 Q 7336027|3°7537911)2763850)/3°618141 5|1°0374915|0361367 Sf 9638633|33 
 2812666777 28041 1223223)3 “7498447 276698 1|3°614.04.69|1°0375750|0362142 ey 9637858|32 
 29/2669581 aang) ssosl9 3°7459068 2770 113)3'6099609} 1 :0376585}0362919 goo 081/51 
 30|2672384 9 803 7327616|3 7419775 2773245 '3°6058835|1°0377422|0363695 aH 9636305|30 
 
 31|2675187 73248 13}3°'7380568'2776378'3°6018146)1°0378260|0364473 9635527 29 
 
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 32}2677989),.- .4/7322011|3-7341446 2779519 35977543} -0379098,0365252| 70 9654745|28 
 35}2680792)5 6 19/7319208|3'75024092782646 3-5937024|1 "0379938 0366051] 1°\9633969|27 
 34|2683594) 5°) 217316406|9°1263457/2785780 3°5896590|1 “058077903668 1 1|,/996331 89126 
 
 83|2686396)940,-17313604]3"7224589 2788915 3°5856241)1 °0381621|0367592),"9 119639408125 
 
 36|2689198)5° i 7310809|3°7185805 2792050 3-5815975|1 0382463/0368374 Oo, 9631626|24. 
 ; 7308000|3°7147105'2795186 3°5775794\1:0383307|03691571_ . ,|9630843|23 
 
 7305199|3-7 108489 2798322 3-5735696|1+0384152 0369940), °|9630060)29 
 39) 2697602) 5 «  4|'7302398)/3°7069956 2801459 3°5695681]1 °0384998,0370725 hae 9629275/21 
 40}2700403)9 29 {7299997)9 703 15062304597 35655749) 1 0385844103715 10) --9628490/20 
 41|2703204. 7296796|3°6993139 28077353 5615900|1-0386692)/0372296 ; 9627704/19 
 
 42/2706004)0500/7293996|3-6954854}28 10873 355761331 -0387541|0373083 757962691 7|18 
 45|2708805} .,4,|7291195|3°691 6652/28140123:5536449|1 0388991 /03738701,_. .|9626150|17 
 44)271 1605) 9917988395|3°6878532 2817152 35496846] 03892420374658|. 0 9|9625549| 16 
 45}2714404|7/99}7983596|3-6840493|2820299/3-5457325|1+0390094|0375448) 2919624552115 
 46|2717204|.-99/7282796|3 '6802536|2823432'3-5417886|! -0390947)0376238|01962762|14 
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 48}2722802), 17°/7277198|3 6726865 |28297115{5:5339251| -0392655|0577820\0 196221 8019 
 
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 49]2725601}9,,9|7274599/9-6689 15 1|2852857 3-5300054]1-0399511/0378613},_, |9621387)11 
 50/2728400|* "17271 600/3-665151812835999'3-5260938]1 -039436810379406| 19620594110 
 51/2731198|*."9}726880213-66139641283914.3'3-522190911-03952296'0380200 ae 9619800 
 
 2}2733997|,,..,|7266003|3-657649 1 2842286|3-5182946| | 0396085 10380995]... .|9619005 
 53]2736794]5,, /17263206|3-6539097|2845430/3-5144070|1-0396945|0381790), 19618210 
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 ST 27T47984 ie 7252016}3°6390315)28.580 1 2/3'4989356|1-0400396 00 
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 HOSTS3S TT an9 72464,23|3°6316395/2864306|3 49 12470} 1°0402127/0386582 9613418 
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 7:243626|3°6279553|2867454 13-4874 144]1-040299410387383|°° 19612617 
 
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 Cosine|Dif} Vers. | Secant |Cotan.} Tang. ‘ 
 
351 
 
 | 4318788 
 } 4325285 
 4327777 
 
 Deg. - ‘Eoo. srnzs, &c. : (279) 
 Dif) Cosec. |Verseds.) Tang. |Dif| Cotang. | Covers.] Secant |D.) Cosine 
 471 9|10°5870038)8-5324253|9-4280525],, 1. |L0°5719475|9°8699243]10-0150562), , .}9°9849438/50 
 4707 |10°9865326|8-5333844)9 4285575); 7 -|10°5714425|9-8697596]10°0150901)2" 0 ]9 9849099} 59 
 1/701 [L079860619]8 53434239 -429062 11; 9; 0|L0°570937919-8695949]10°0151 24C],% A [99848760158 
 4696] 107985591 8]8-5352992)9 4295661}; -110-5704339|9-S694501110-0151580)5,. |9°9848420)57 
 ! 4690|L072851222)8 536255 1|9 4300697]; \110°5699303}9°8692653)10-0151919)5" 1 |9-984808 1156 
 | 4153468), °° 5 |10-5846532|8-5372098/9-4305727f; 05 -|10°5694273]9-8691004]10-(15 2260]. | -|9-9847740)55 
 4158152]; 60 |10°5841848|8-5381635)9-4310753}5 15 5|10°5689247|9°8689355]10-0152600),7 | |9-9847400) 54 
 4162832] , -,. ,10°58377168|8-5391161/9 4315773]. <|10°5684227|9:'8687706]10-0152941}., , .19°9847059153 
 105832494 }8-5400677|9:4320789}; 4 | 0|10-5679211/9°8686056]10-0153285}. ie 9:9846717|52 
 466 3|1072827826)8 -5410182/9-4525799); 1) .|10-5674201/9-8684405}10-0153625]5; 5}9°9846375)51 
 4176837); ¢5.4|10°5823 163]8-5419676|9 4330804); 01 |10:5669196|9°8682754}10-0153967)> , ,19°9846033)50 
 4181495), 54] 10-5818505|8-5429 160|9 4335805}; 99 «|10°5664195/9°8681102}10-0154310 34219 -9845690/49 
 . 4186148}; 1 -|10°5813852|8-5438633|9-4340800); 99 |10°5659200|98679450/10-0154653 ath 99845347148 
 | 9g5|!0°5654209/9°S677798}10-0154996],, ,|9-9845004147] 
 498 1 |10°5649224)9 °8676145}10-0155340} 199844660) 46 
 10°5799927|85466990|9 °4355757|, 9. -|10°5644243/9°8674491/10-0155684/; -}9°9844316)45 
 4606 |1079795296)8 5476422/9 4360733}, 9. 1|10-5639267|9°S672837|10-0156029), A 9°9843971|44 
 | 4209330} ; 659]! 0°5790670)8 -5485843)9-4365704] , 5 ¢ .|10°5634296|9°8671182|10-0156374)5 719°9843626)45 
 | 4213950) 67 g|10°5786050)8-5495253)9-4370670), 6. |10°5629330|9°8669527|10-0156719), iy 9-9843281|49 
 } 4218566) 5 6 1 y|10°5781434|8-5504654/9-4375631 10°5624369}9°8667872|10-015 7065), , |9°9842935)41 
 ) 4223176) | ¢,|10°5776824)8-5514044|9°4380587] 42 7/10°5619413 9-8605216|10°015741 13 [9 °9S42589)40 
 | 4227780); 699] 10°5772220/8-5523423/9-4585538) 45) 0 10°561446219°8664559}10-0157758|2*.19-9842249139 
 ) 10°5767620|8°5532793|9°4390485 rene 10°5609515|9:8662902|10-0158105],.,.|9°9841895/38 
 | 4236974) 1 5 49|10°5763026)8-5542152)9-4395426) ; 9.44|10°5604574/9°8661244/10°0158452 ys 9°9841548)37 
 4|10°5758457/8-5551500)9-4400363) , 034 10-5599637)9-8659586 10-0158800}, , 4|9°984] 200/36 
 499"7|10°5594705 9°8657928) 10-0159148),, ./9°9840852135 
 54|105589778|98656268|10-0159497} /9°9840503)34 
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 13]2959859 gum o| 1040 141/3'378539 1/3098705)3°2271 546) | 0469096)0448077). 6 119551923/47 
 1412962638|5...|7037362)3°3753707|3 10 1893}3°2238375]1 -04'7004.0/0448938}3 6.41955 1062/46} 
 2965416 anne 7034584)3°3722084/5105083}3°2205263}1-0470986)04498011, ..19550199/45 
 2968 194lon--17031806|3°3690524|5 108272/3°217221 5{1°0471932/04.50664] . 6.|9549336144) 
 17le97097] yet 70290299|3°365902613111462'3°2139228}1 047287910451527 865 95484'73]43 
 
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 2973749 a 702625 1/3°3627589)3114653}3°210630411-0475828)/0452392}. 519547608 |42 
 19}2976526 lon | 1025474 |3*3596214|5117845}3°2073440 10474777 10453257] 3 ¢nf9546745 41 
 20/2979303|5« x -|7020697|3-356490013 1 21036}3 2040638} 04757 28/0454 12415 6719545876 |40 
 21 2982079|oor 7047921/3°35336471312429513°2007897}1°0476679|0454991 868 9545009)|39 
 2210984856 ohne 701514413°35024.5513127499)5°1975217|1-0 63210455859 868 9544141}38 
 23}2987632\50, -|7012368|3'3471324)313061 6/3" 19425981 -0478586|0456727]gn)|9945273/37 
 2412990408 ante 7009592|3'344.0254/3133810}3°1910039}1+0479540/0457597]g17,19542403|36] 
 2512993184 gtr 5|70068 16/3°3409944)5137005 3°1877540}1°0480496)0458467] en 1|9541533}35 
 262995959 ants 700404 1|3°3378294/3 14.0200}3*1845102)1°048145310459338) 9x7 9/9540662)34 
 
 2998734|,,..,..|7001266]3'3347405]3143396|3° 181272411 -04824 1 110460210]gr4/9539790|393 
 28/3001509|9-,/91699849] [3-33 16575|3146593|3 1780406) 1-0485570/0461085|¢q5195389 17}32 
 
 91300428415 - 71699571 6|3°3285805|3149790}3"1748147]1°0484330/0461956]n3|9538044)5 | 
 50|3007058|o, ~ ,|6992942|3°3255093/3152988|3-171 5948} 1 :0485201/0462830}g. 695371 70/30 
 
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 3020926|5 1,\{6979074|3°3102432|3 168986]3"1555840|1 04901 13|0467214lg79/9532786)2 
 302369915; 516976301 |3°3072076/31721 87|3°1523994| 1-049 1080|0468093/g99/9531967 
 3026471),.-.|6973529|3°3041778/3175389]3"1492207]1 -0492049|0468973}g¢ /9531027/23 
 6970756)3'301 153913178591 |3*1460478] 1049301 9/0469854)g ¢5/9530146|22 
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 46)505 14131). [694858715 -2771700/3204252/5 +1208729]1-0500815/0476929}g¢ 419523071|14 
 4713054183 ONG 6945817|3°274.1977|3 20744 389 
 48]5056953|5. - 6943047 3-271 231 1/3210649)3" 1146353] 1-050277410478706] 44952129412 
 49'3059725}.5,. -.]6940277|3°2682702|3213858)9"1115254]1 0503756 0479596), 491952040411 
 501306249215, 69{0997508|3 2653 149}3217067|3-10842 10} 1-0504758}0480486)g 4 |9519514]10 
 51)306.5261 [9.4 |693473913 2623652/3220278|3 1055225} | 0505722|0481377]¢q9)9518623 
 52}3068030), - °6{6931970|3°259421 1|5223489}3-102229 1] -0506706}0482269]49,,19517731 
 53}3070798},~ 7 .1929202|5*2564825}3226700)5 099 141 6}! 0507692/04831 621o9 319516838 
 454/30735 665. ¢.16926434|5 2535496/39299 12/5 -0960596] I *0508679|0484056| gq 19515944 
 55}3076334) 5, -.6923666)3-2506222/3233125}3-092985 1} | :050966710484950 4 9 .19515050 
 56}3079102I,~ .-16920898|3-2477003|3236338)3 08991 22] 1 -0510656|0485846| 45 -19514154 
 57|3081869],, °.16918151|5*2447840)3239552/3-0868468] I -0511646|0486749|.4-19513258 
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 5813084636 an 
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 17 Des. LOG. SINES, &c. 
 E Sine |Dif] Cosec. |Verseds.| Tang. |Dif 
 
 Cotang. Covers. Secant. |D. 
 
 \9°4659353 10°5340647|8°64.04342/9*4853390 
 10°53365 17/8 °641279119°4857907 
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 10°5246729|8°6596569|9°49562981) 759 95 
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 47}9°484895 1|494r|10-5151049|8-6792611819-5061586 rere 10-4938414|9°8417233]10-0212635) 431 
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 10-5119665|8 685697319 -5096224 
 56]9°4884240]5.595|10-5115'760)8-6864984]9-5100539 
 57)9°4888 142|5 5 99|10-5111858/8-6872986|9-5104849 
 58}9:4892040\3 06 ,|10-5107960/8-688098 1]9°51091 561,451 
 59}9:489593415 09 4|10-5104066)8-6888969|9-51 13460] 5 ,9|10-4886540/9-8396412|10-0217526 
 60|9-4899824|"~”* |10-5100176|8-6896949|9:5117760| ~~” |10-4882240|9°8394674|10-0217937 
 
 Cosine |Dif| Secant. | Covers, Cotang. Dif Tang. Verseds. Cosec. {D. 
 
 A a 30 2 
 
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 10-4916739}9'8408567]10-0214666 ; 9. 
 10°4919414]9-8406832|10-0215073/; 54 
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 410 
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 10°5146610}9-8498052}10-0194037] .., .19°9805963)60 
 
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 10°4903776)9°8403361|10°0215889 4.09 99784111} 5 
 
 nlo*9183293 
 
 (283) 
 
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 9*9804415)56 
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 9-9§03250153 
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 9+9802471/51 
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(284) 18 Deg. “NATURAL SINES, &c. Tab. 10, 
 
 ‘) Sine |Dif|Covers| Cosec. |Tang.|Cotang.| Secant | Vers. |D.|Cosine 
 
 (|309017C ra 6909850 |5-2360680]3249197|3-0776835]1-05 146220489435). .|9510565]60 
 1]3092936] 5./2?16907064|3-2351736|325241 5|3-0746400]1 -0515617/0490334 ong 309966159 
 213095709] >,.615904.298]3-2302846|5255630/3-07160201 1051661 2/0491234|"0\9508766)58 
 313098468] 510°]5901539|3-227401 1|3258848]3-0685694] | 0517608|0492135 oa 9507865|57 
 413101254] 5. 0015898766|3-2245230)3262066]3-0655421]1 -0518606]0493037/2|9506963)56 
 5131039991 20° 16896001/3-221 6503]3265284|3-0625203} 1 -05196050493939/9"219506061155 
 6131067644 © 10 15895236]3-218783013268504|3-°0595038} 1 °0520604)0494843|904 9505157154 
 3+2159210/3271724/3-0564928]1'0521605|04957471 1950495353 
 °16887706|3-2130644|3274944|3 0534870] | 0522607}0496652)90919503348152 
 913115058],,.04|38849.12|3°2102152}3278 163]3-0504866|1-0529610]0497557/009 9502445151 
 10]3117829} 622158821 78|9°2073673|328 138 7|3 04749 1 51 -05246 1410498464 ‘3 {\9501536)50 
 1113190586 ee 687941 4]3*°204.5266|3284610]3°044.501 81052561 9]0499371|90 9500 629)49 
 1213123346] 5.2°15876651/3°2016913}3287833|3-0415173| 1-05 
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 16|31344001,5~-.|5865600 
 17}3137168 
 
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 2762 
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 19603651329428 113'0355641|1°0528641/0502098|9! 0lg4g7999146 
 
 -19521701329750513°032595411 7052965 10503009]! 119466901145 
 
 §857314[3°1819913]331041 1|3-020'7728]1°0533699)0506659 O15 9493341)41 
 og (0854552]3" 1791978}33 13639}3-0178301}1-0534714)05075 74), 919492426 )40 
 21}3148209]5~ °.|6851791)9'1764095}3316868)5-0148926/1°0535730)0508489/0 | }9491511)39 
 2913150960 nl 684903 1]3*1756264{5320097}3-0119603|1+0536747/05094035|, "19490595|38 
 2313153730 be _ 6846270(3°170848413323327|3 -0090330]1 1053'7'765 10510322 Bin 9489678137 
 2413156490) oa 684351013" 1680756|3326557|3°0061109 POSGBTS SOO L240 91 91° 738760)36 
 |2513159250}5,, <|6840750)3"1653078}3329788|3' 0031939} 1 0539805 (0512158|,,, 948 7842)55 
 26/3 162010] 51 ¢,15837990]3"1625452/3532020/3-0002820|1 0540826 0513078),> |9486929/34 
 2713164770 Sol §835230}3°15978'76|3336252'2°997375 1110541 8490513998 991 9486002)33 
 28/3167529| 5.0. «|6832471|3"157035 1/3339485|2-9944734)1°05428730514919]55 ,9485081|52 
 29/3170288|5,-.> |682971 9) 
 
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 30|5173047),.°> |5826953| f 
 31)3175805}9ny 6 0824195|5"1488079|3349188]2-9857989|1 0545950051 7687)... [9482313129 
 59131 78s6s|2 1 9|6821.45713+1460756193242 12-9829 167) 10546978 051861 12419481 389128 
 
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 33/3181321|o.2°|s81867 3-1493485 3355660|2-9800400|1 0548007 0519536 9251948046427 
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 3413184079 i 681592113°1406259|3358896|2°9771683 1-0549037/0520462|5°° 9479538126 
 3513186836 ade 681316413°1379086|3362134|2°9743016 1-0550068)0521388 ~ |947861 2125 
 ‘4126 194.0'713°135196213365372/2-9714399]1-0551101 
 
 -1324887|3368610/2°968583111°0552134/0523244 
 580480413°1297862/337185012°965731211-05531690524173 ae 94'7582'7/22 
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 5213235670 2750 6766330|3°0924620)3417267|2°9263 152) 1*°0567768|0537264 OA] 9462736 
 53|3236422 on%0 8763578|3°0898319|54205 1 6/2°9235358}1 05688 1910558205 941 9461795 
 —454)32391'74 9759 6760826|3°0872066|3423765|2°9207610}1°0569871/0539 146 94.3 9460854 
 5513241926 9750 675807415 0845 860/3427015/2°9179909]1 -0570924|0540089 94.3 9459911 
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 58}3250180 O75] 74982015:0767525|3436770|2°9097089| 1 0574090/054.2922 9046 9457078 
 59}3252931 9751 6747069|3°074.1507|344.0023/2 9069576 1 °0575148}0543868 946 9456132 
 60/3255682| ~ ‘?'16744318|3°0715535/3443276|2-904.2109]1°0576207/0544814 9455186 
 
 ~ (CosinelDif | Vers. | Secant |Cotan.| Tang. | Cosec. Covers!D.| Sine 
 
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Deg. LOG. SINES, &c. (285) 
 
 Dit| Cosec. |Verseds.| Tang. [Dif | Cotang. |Covers. | Secant {D.| Cosine 
 3886 10°510017618-6896949|9-°5117760 4297 10°4882240 9°§394674110°0217937 410 9-97 82063160 
 4903710 3839 10°5096290|8-690492119°5129057 494 10°487794319 -8392935|10°0218347 419 9+9781653}59 
 4907592 Neen 10°50924.08'8-691288619°5126351 1,290 10°487364919°8391195]10°0218759 All 9°9781241158 
 4911471 Bat 10°5088529\8°692084419°5130641 4086 10°486935919°8389455110°0219170 419 9°9780830157 
 4915345 3871 10°5084655|8°692879419 -5134997 935 10°4865073]9°8387714110-0219582 419 9:9780418/56 
 4919216 3987 10°5080784)8 °693673619 5139210 Ge 10°486079019°8385973}10:0219994 413 9°9780006155 
 49230835 9965 10°507691'7|8 69446729 °5143490 ae “110°485651019°8384231]10°0220407 413 9°9779593154 
 4996946 Gane 10-5073054/8°6952599|9-514.7766 40x35 10°485223419-8382489] 10-0220820 414 99779180153 
 4930806 3855 10°5069194/8°696052019°5152039 %Q70 10°4847961 9°8380746 10°0221234 Al: 99778766152 
 493466 1 3852 10°5065339/8-6968432)9-5156309 4266 10°484369119°8379003)10°0221647 ALS 9:9778353}51 
 4938515 3846 10°5061487/|8°6976338}9-5160575 4263 10°4839425/9°8377259]10-0222062 ree 9-9777938150 
 4942361 3844 10°5057639)8-698423619*5 164838 4.259 10°4835162)9°8375515|10°02224.77 ALS 9°97 77523149 
 4.946205 ae 10:5053795'3°6992127|9-516909% 4056 10°4880903/9'°$373770|10:0222892 41S 99777108 48 
 4950046 110°5049954'8°700001019°5173353! . 10°4826647|9°8372024)10°0223307 416 
 
 Sine 
 4899824 
 
 fa) i 
 ‘4953883 ried 10°504611'7187007886|9-5177606 ign 10°482239419°8370278110°0223725 417 9°9776277146 
 4997716)5.555 jo:501938) 8°7015755]9°5181853]494 ¢| 104818145 pea 10-0224 140 1 1g|9 9TTSB6O}45 
 4961545 3895 0°5038455'8°702361'719°5186101 4243 10°4813899 366 18. 100224596 418 9°97 1544 44, 
 496.53710)5.55|10-5034630)8 70314711915 190344]; 949 10°4809656/9°8365037|10-02249741, 1 -|9-9775026)43 
 4969192). 0, |10-5030808 8-7039318|9-3194585| ,9,34|10°4805417)9 8365289) 10-02253911, 7 319-9774609]49 
 4975010)... | ,|10-5026990)8-7047158/95198819] 4 o49| 10°4801181/9°8361540)10-0225809}, 1 019977419 1]41 
 , 4 |* “201 MES sKonenkol. ~% ry, >QA “Qe in -2262 “10-Q% mols 
 ACS rt ane aoe coin ee eel mere tect 
 “ 3807 30518 7062815)9 5207282 4.296 as Wo .ccs : peated 1. 1 fe dr 
 4984442/5 1) 5|10°5015558)8°7070633/9-5211508) , 999] 10°4788492 |S 96291/10-0227066)4 1 /9°9772934/38 
 4988245) 09 10°5011755)8-7078444|9-5215730] 499) 10°4784270}9 8354540) 10-0227485} 4 9)/9°9772515/3% 
 4992045 3479 5|£9°5007955/8°708624719-521 9950491 6|10°4780050 9°8352789}10-0227905) ; 1 |9'97720935|36 
 erecta tiered feperer et ersaire 
 633] on gq\t 0°50 67/8°7101833/9 522837914516 4771621 BV 200 228 Ke aot: (71253)34 
 5008421), 5 -|10:4996579|8-7109615|9-5232589| j o9¢|10°#767411]9°8347532110-0229168) ,.9519°9770832)33 
 5007206), | >10-4999794 8°7117390/9-5236795| 19,|10°4763205|9°8345778| 1 0-0229590| 49519-97704 10132 
 5010987 Sinn 10°4989013'8:712515'7/9°5240999 40001 £2°4759001 9°8344024110-0230012 4,99 9°9769988!31 
 9014764)". | |10°4985236 /8-7132918|9+5245 199] 4 o9|10-4754801]9°8342269|10-02304341 5.3|9°9769566)30 
 Gy32508 770) 19: 4977c00'6.T1aosialocatecy [Alot o.t7ae4 1 1/9-8926759) 10-oas 9903 oresradles 
 5026075|2/07 By farsoese aisciss Bae Hab Aantens dosgaaucnol le-aosinneteet i autedonion 
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 ig 3q{ 1074966403 /8-71716 14|9-5266150|j 19 1|10°4733850, 488]10-0232553| 95/9 9767447125 
 9037353). 2 | 10°4962647|8-7179332/9-5270331| 44 nf! 0°4729669)9 333173 1)10-0232978) 55/9 9767022124 
 p10 aug t0"4998895 8-7187044 9°5974508), 1 4|10°4725492 983299791 0-0233403},.9¢/9°9766597}29 
 (5044853)? /79 10-4955147/8-7194748/9-5278689l. , ,7/10°4721318|9°S328213)10-02338291, 5 -19°97661 71122 
 5048598 gr; |10°4951402 8°720244.5/9°5282853 ns 10°47 171 4'7/9°8326454 10-0234953| jon 99765 745/21 
 5052339 579g 104947661 8*721013519-5987021 A165 10°4'712979,9°8324694110-0234682 4or 9:9765318120 
 5056077|; 5 | 10°4943923/8-7217818]9-5291 186), , g1|10°47088 14 9°8322933) 10-0235109!) 9n/9-9764891|19 
 5059811|3.. ieee 8:7225494/9-5295347); 2 |10°4704653,9°8321172/10-0235536|; 94(9°9764464]18 
 9063542). o17110-49364358 8 -7233163|9-52995031, 1 x -|10-4700495|9°8319411/10-02359641 ; 9¢|9°9764036|117 
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 5067269 juga 10-4992731 8°71240825/9-5303661); 1x9) 104626539 9°8317649|10-0236399) 44 
 9070992),,_99| L0°4929008 8 -7248480/9-5307813],, 15 o| 10°4692187/9°8315686)10-02368211;9 
 10-4925288|8-7256129|9-5311961|;, 4 |10°4688039|9°8514125)10-0237250 
 
 5074'719)3,120 
 ae 4146 | 
 10-492157218-7263'770|9-5316107/¢ 4 £0|10°4683893)9°83 1 2359}10-0237679|49/9°9762321|13 
 
 9 
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 41 p(9'9761891]12 
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 5082141 ah 10-4917859/8-7271404/9-5320250},, 1 ga] 10°4679750}9°8310595}10.0238109|45 
 985850) .)¢)10°4914150)8-7279032/9-5324389],, 1 ._|10°4675611/9°8308830) 10-0238539) 44) |9°976 1461/11 
 5089556] o110-491044418-7286653|9-5398596), . 5] 10°4671474|9°S307064]10-0238970l45 1 |9°9'761030}10 
 5093258|, 1 9(10-4906742/8-7294267]0-5339659|7 5 | 10°466734 1|9°8305299|10-0239401] | 5919°9750599 
 
 5096956); |10-4903044}8-730187419 5556789], 1 4,|10°4663211/9°8503532110-0239835}45 9-976 
 5100651), 20 9|10-4899349]3-75094'74|9-53409 116], | .,|10-4659084|9 8301765] 10-0240264{ 419-975: 
 5104343}, °° 9|10°4895657|8-731706719-5345040] «|| 10°4654960]9 8299997} 10-0240697| /54)9-9759503 
 
 9 
 
 8 
 
 7 
 
 6 
 10°4891969)8°732465419°5349161 10°4650839)9 82982295) 10-0241 130 4.33 9°9758870] 5 
 
 + 
 
 3 
 
 2 
 
 1 
 
 5111716|,°9|10-4888284|8-7332233/9-5353073{¢ 1!" 10-4646722/9-8296461]10-0241563]15519-9758437 
 5115397}-7" _{10:4884603|8-7339806|9-5357395}_ | | 10°4642607|9-8294692|10-0241996) ,5419-9758004 
 51190743004 8:7947373|9-5361505|411°|10-4638495]9°8292929|10-0242430]43519-9757570| | 
 5122749]; °° 10°4877251|8-73.54932|9-5365615] +)» |10°4634387|9'8291 159110-0242865|15419-9757135 
 5126419}7° '"110-4873581)8-7362485/9-5369719| 7+ 99|10-4630281|9-8289381|10-0243299]"~ 19-9756701 
 
 +|—— | | -_-eooeoeoeoo os - | ———_——_ 
 | —— J —— |_—_—___—____...... | 
 
(286) 19 Deg. NATURAL SINES, &c. _ Tab. 10. 
 ‘! Sine [Dif |Covers} Cosec. |Tang.}Cotang. Secant | Vers. [D. [Cosine 
 "0|5255662|,-,|6744518}3-0715535|3443976|2-90421 0911 0576207]0544814), | ]9455186|60 
 
 113258432|29016741568)3-0689610}3.446590|2-9014688|1 -0577267]0545762|- *9}9454258)59 
 
 913261189) 2/9|673881813°066573 1/3449785|2-89873 1411 °0578328|054671 0)" 45}94.53290)58 
 
 313263959). /"016736068|3-0637898}3453040|2-8259986|1 -0579390|0547659|940 1945254157 
 
 4 ene h hl i449 6733319|3°0612111)3456296}2°893270411 -0580453}0548609 950 9451391156 
 
 | 
 
 513269430) ~,/#°la75057013-0586370|3459553|2-8905467|1 -0581517]05495591990194.5044 155 
 6|32721 79 th 6727821)3-0560675)34628 10|2°8878277|1-0582583|0550511 op 9449489|54 
 13274998]... 16725079|5-0535026|5466068|2'8851132]1-0583649|0551463).- 19448537153 
 §[327'7676|-- ¥9|672232413-0509425)3469327|2°8824033|1 0584717105524 1 6|99 194475 84/52 
 9159804.24|-/29{67195°76|3-0483864/3472586|2°8796979| 1 -0585786|0553370/99 49446630151 
 10139831 72|~,/456716828]3-04.5895213475846|2°8769970]1 :058685510554325|9991944.5675|50 
 1113285919|22 #1671408 113°043288.413479 107|2°38743007|10587926|0555280|99|9444'720149 
 
 3° * 167003473 030622 1|3495420|2°8608363]1 -0593298}0560069 ate 9439931|44| — 
 1713302398 pier 669760213 -028 1023/3498685|2°85821 6811 -0594376|0561029|7 01943897) |43 
 181530514410 791669485613 0255868}3501 950|2°8555517|1 0595454056 1990|"61 9488010 /42 
 
 19/3307880],,,, (069211 1]3°0230759|3505216/2°852891 1/1 -0596534)0562952 4 re 9437048|41 
 20133 106341-..*2|668936613°0205693}35084832°8502349]1 -0597615|0563915|"°3}9436085|40 
 31133133°7 Edbes 6686621}3-018067213511750|2-8475831\1-059869710564878|2 0719435199139 
 2919316193 ae 668387713°0155694/3515018|2°8449356|1 059978 110565843 
 93133 18867\~ 6166811 3313-0130760)3518287 
 
 2413321611 he 6678389|3-010587013521556 
 
 joni9434157)38 
 2+8422996]1-0600865/0566808 ae 9433199137 
 2839653911 -0601951|0567773 . 943999%|36 
 19513324355! |6675645|3-0081024/3524896'2°83'70196|1-060303710568'740 ogr( 043126035 
 26|3527098|2-416672902|3-0056221|3528096|2-8345896] I -06041 9510569707] o 94.30293|34 
 330984110 Ot 21667015913-00314691353136812°8317639]1 0605214 76|° 919490394133} 
 27)5329841|-.0171667015913-00314 '3531368)2°8317639]1-0 0570676|40-|9429324|35 
 33395 667416|3°0006746)3534640!2829 1 426]! 0606304 51999194.09355132 
 9813339584121 491666741 63-0006746)353464012°829 1426|1 0606304057164. socio 3 
 9913535526\-0*2)6664674 2°8 26525611 0607395 10572614] °" 19497386131 
 3013338060|~.' |6661931|2+995744313541 186|2°8239129|1-0608487 
 
 0573585/9 19426415130]. 
 )12°8213045}] -0609580/0574556]._ 1942544429 
 ae 2°8187003}1 -0610675|0575529{0,°194244'71|28 
 33133462931 -.* 16653707|2-988381 1/355 1010|2°8161 00441 0611770/0576502\~/19403498|27 
 34133490341. + /|6650966 2+98593591355428612-8135048}1 -0612867/0577475 ons 9422525}26 
 ~ 7141/6 648295)2-9834936\5557562)2°8109 1341 0613965|0578450 - 9421550125 
 6645484/2-98 10565 5560840)/2°8085263}1 -0615064/0579425}, .919420575|24) 
 5 ( 
 
 or sp COA 27442978623 1|35641 18]2-8057433]1 0616164 ]0580402),_ 19419598)23 
 3813359996|-." '6640004]2-9761942/356739712-803 1 64.6|1°0617265|0581379|- 
 
 3 
 
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 is 79418621|22] 
 3913362735 Hy (6097265 2+9737695}3570676|2-800590 111 °0618367|0582356 ithe 941764421 
 40133654°75 74016634525 2-97713490|3573956}2-7980198|1 0619471 0583335}q,,0|9416665}20 
 41/3368214|2,.29|6631786|2:9689327/3577237|2-7954.59'7|1 0620575 05843141, .°19415686)19 
 4213370933 ane 6629047} 2-9665205/3530518|2°792891 711 -0621681/0585295 9414705]18 
 
 431337369 t Fa 6626309/2:964112513583801]2-7903329]1 :0622788|0586276 81 941372417 
 4$4/33'76499!~ (9°|662357 112-96 17087|3587083|2-7877809l1 0623896058727 oggfot 2743/16). 
 4513379167 re 6620853]2°9593090|3590367|2-%852307|1°0625005|0588240 AE 941176015 
 {46]358 1905 |, |6618095|2-9369135|5599651/2-7826853|1 06261 15]0589223) [941 077714 
 47/9384642)o 3. 6615358}2°954522 11359693612 -7801440]1 -0627227|0590207 935 9409793}13 
 48|3387379|505. 6612621/2°9521348|/3600222|2-77'76069]1°0628339|0591192 94.08808}12 
 49133901 16l9,, -|6609884,2°9497516|3603508)2-77.50738}1 °0629453]05921 78} 9 ¢.9407822|1 1 
 50|3392859I,,.: ; 6G07148 2°94.737251360679512°7725448|1°06380568105931 65 98% 9406835)10 
 5113395589], 4 -|660441 1]2-9449975)/3610082)/2-77001 99]1 “06316840594 1529. 19405848 
 5213398325 Hi 6601675/2-9426265/3613371)/2-7674990}1 0632801 05951401, 19404860 
 53}3401060|5.6. 6598940)2°9402597|36 1 6660)2-7649822I1 0633919]0596129 g99{2 409871 
 54|3403796|,.3 6.59620412°93'78 968/36 1 994.9|2-7624695]1 0635038105971 19}¢¢ .|9402881 
 
 5513406531 9F54 6593469 2°9355380/3623240/2-7599608]1 °0636158}0598109 999 9401891} 
 56|34.09265 on35 6590735|2°9331833/362653112°7574561|1°0637280|0599101 999 9400899 
 57)341 2000 on34 6588000 2°9308326}3629823]2-7549554|1*0638403}0600093 993 9399907 
 58|3414734 on34 6585266 2°9284858)36331 1512-7524588]1 -063952710601086 993 9398914 
 59}3417468 9x53 §582532}2°92614311363640812-74.9966111°0640652|0602079 995 9397921 
 60}3420201]" '°"1657979912-92380441363970212-7474.774| 1 :0641'77810603074" ~~ 9396926 
 
 ~ ICosine|Dif| Vers. Secant |Cotan. Tang. Cosec. 
 
 Ore 1000 & Or D «1 CO 
 
 Covers|D.\ Sine |’ 
 
 - Dee. 70. 
 
Verseds.| ‘T'ang. |Dif| Cotang. | Covers.| Secant |D.| Cosine 
 
 — |__| | | ome 
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 10°4873581/8°7362485/9°5369719 4102 10°463028 1/9*8289381}10°0243299 4.36 9°975670 1160 
 10°4869914]8°737003019 °5373821 4099 10°4626179]9°S287609]10°0243735 435 9°9756265159 
 
 5133750 3660 10°4866250}8°7377570]9 °5377920 4097 10°4622080/9°8285837|10°0244170 436 9°9755830158 
 10°4862590|8*7385102/9°5382017 4093 10°4617983/9°8284065]10:0244606 4.37 9°9755394157 
 4 10°4858933]8 °739262819 °5386110 4090 10°4613890/9°8282292}10-0245043 43 9°9754957156 
 10°4855279]8°740014'719°5390200 10°4609800}/9°8280518}10°02454.79 438 9°9754521155 
 9°9754083)54 
 
 50 sAQK ¢ Q. - nC 4087 AR ¢ -QOMe h AOAS 
 5148371 36 10°4851629)8°7407659]9°5394287 4084 10°4605713/9°8278744]10°0245917 437 
 
 5155660 eas 10°4844340|8°742266419-54.02453 ise 10-459T547|9-82751 94] 100246 792/45 4}9-9753208]59 
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 5166569 ns 10°4833431]8-744512119-5414678}10 2] 10-4585322|9°8269866]10-0248109 de 
 5170198 : 
 5173824 : 
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 10°480449018-7504'716|9-5447148), ., |10°4552852|9°8255631/10-0251639] “19°97 
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(288) 20 Deg. NATURAL SINES, &c. ~ rab. 10. 
 
 Sine [Dif |\Covers| Cosec. 
 
 Tang.|Cotang.| Secant. | Vers. |Dif Cosine 
 3420201 | ong 4 |6979799/2°923804413639702/2°7474774) 1 064177810603074) oo. 9396926|60] 
 3422935) 9n4/6577065|2°9214697)3642997/2° 74499271 °0642905)0604069] 4461939593 1]59] 
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 3428400) 94739]5971600]2°9168121}3649588/2-7400352)1°0645163/0606062) 05,19393938|57 
 413431133] 91391/6968867)2°9 144892/3652885|2°7375623)] 0646 29410607060) 99 4)9392940|56 
 5}3433865| 97 39|6966135)2'9121703)3656182)2°7350934|1°0647425)0608058) 99419591942] 95 
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 713439329]. 79 {056067 1|2°9075443}3662779]2°7301 674 1-064969310610058 100019389942|93 
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 9344479 11995 0]6555209|2°9029339}3669379)2-7252569]1 065196410612060 938794091 
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 Cosine |Difl Vers. | Secant !Cotan. 
 
20 Deg. LOG. sINEs, &c. (289) 
 
 ‘534051 i} 4 gq] 10°4659485|5-7803705|9-56 1065S 
 5343986], «-110°4656014)8°'78 10866]9-°5614588 3994 
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 3P | Deg. 69. 
 
 pele 10°4486444]8°8161390|9°5807691 
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 Dif] Tang. |Verseds. Cosec. |D. 
 
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(290) 21 Deg. 
 
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 *7085139138972746|2°5171507| 1 °0760237;/0706525 
 7065323139761 1412°5150183]1°0761481107 
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 2702578413982853 
 700606 1/3986224 
 
 2°6966709|3992968 
 *6947079)3996341 
 
 2°6868867/4009841|2°4938645|1°077S988)0718386 
 684939 114015218 
 *682994.5)4016596)/2°4896706} 1 
 *68 1053014019974 
 
 9 . 
 
 Secant Cotan. 
 
 NATURAL SINES, &XC. Tab. 10. 
 
 790426 1|9838640/2-6050891)1-07 1145010664196} |, , .]9935804)60 
 7883153|5841978]2-6028258|1-0712647|0665234I 6 1'519334761)59 
 7862059|5845317|2°6005659| [ 0713844|0666289| 9, 5|9338718)58 
 7840999|3848656/2°5983095|1 -0'715043|0667327] 1 91 
 7819973}3851996/2-5960564|1-07 1624410668372} 1 9 19931625)56 
 798982/3855337/2°5938068|1 °0717445|0669418} , 19350582159 
 777802-4|3858679|2-59 1 5606| {-0718647/0670465), 4 19929535) 54 
 7757100|3862021}2-5893 177] -0719851,0671519} | y4 9328488153 
 77362 11/3865364|2-5870782| | -0721 05606725611 | 44.o]9327439152 
 
 105019326390)9 1 
 103019325340150 
 7652988|3878744)2-578 1 539] 1-0725887 0676762} 95 019323238]48 
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 2°5516999|1°0 495 0689442} 19 ¢19310558)36 
 2-54.95 16()|1 0741720 06205041 | 419309496135 
 2°5473359) 10742946 0691566) 1 9 4|9308434|34 
 7345630)3929027}2-5451591}1 0744173 0692630) qe 4|9307370}53 
 7325400'3932386122542985511 0745402 0693694 1065 9306306132 
 730520313935 74512°5408151 1-0746631,0694759 1063 930524 1/31 
 728.5038/3939105)2°5386479) | 0747862 06958241 4 19304176150 
 
 7264905|3942465|2-5364839|1°0749095 0696891] 4.193031 09129 
 72448 (04'3945827|2-5545231}1 0750328 06979581; y¢ g19302042/28 
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 67139()6}4036879}2°47'7 1612) 1 -0'784.080|0727072 1089 9272928 
 6694.67 21404.026 212 4750869] 1°078534710728 161 "192713839 
 
 Tang. t Cosec. [Covers Dif! Sine 
 
 : Deg. 68. 
 
21 Deg. LOG. SINES, &Cs (291) 
 mr er TTI ES DE OLE TIER IE OIE LEDER nae oa 
 ‘) Sine |Dif| Cosec. |Verseds.| Tang. |Dif| Cotang. | Covers.| Secant 
 U}9°5543:292} 49.4. 0{10°4456708}8 822296 1195841774] 0 5{10°4158226 9:8072860}10-0298483 
 1/9°5546581)3547|10-44534 1918 -8229774|9 584554913, |10°415445 1/9-8071022]10 0298968 
 219-5549868)35 6 ,|10°4450132/8-823658219-5849521/3-,-0|10°4150679}9 8069183] 10 0299453 
 3]9°5553152)39.4)|10-4446848]8 -8245385]9-585309 1]. 54/10-4146909)9 8067344) 10-0299939 
 419°5556433}3 54 5|10°44435678-8250 182)9-5856859|5. 5 5|10°4145141/9-8065503)10-0300426 
 5}9°5559711)552,6|10°4440289/8-8256975|9 586062415, | 10-4139376|9 8063665) 100300913), 
 6195562987) 35 75|10°44370 15/8 826375919 °5864386)3. .)|10°4135614/9°8061821)10-0301400); .419°9698600)54 
 7195566259} gon)|10°4433741]8 8270539195868 147]4-5110°4131853/9 8059980] 10-0301 888), . 29°9698112/53 
 8/9-5569529}3 9 .-|10-443047 1)8-8277314|9-587190414-7 a{10°4128096)9-8058137]10 0302376), 44|9°9697624) 52 
 9}9-5572796!55 6 4|10°4427204)8 8284084]9-5875660|4. 5 4)10°4124340/9-S056294]10 -05U2864], 3419°96971360151 
 0/9-5576060)3.5 6 )|10°4423940/8-8290848]9-58794 1315-,50)110-4120587)9 805445 1]10-0303353) , 40|9-9696647150 
 1/9-5579321!3 9 ¢.4|10-4420679/88297606/9'5883163]5,% 4|10°4116837/9-S052606)10-0303842)j | 19°9696158)49 
 21955825 79}4 55 -|10°4417421/8°8304360]9-58869 12/5. 1 5)10°4113088)9-8050762]10 0304332) 70 1|9°9695668/48 
 3/9°5585835|505. 10°4414165/8-891 1107|9°5890657l4~, ,|10°4109343/9-804891610-0304825|, 9)9°9695177]47 
 4/9°558908810 95 5)10°4410912)8-8317850/9-5894401]5. 4 | |10°4105599)9-8047070}10°0305313), 0 1|9°9694687/46 
 5}9°5592338!45 1 -|10-4407662]88324587|9 5898 142/449] L0°4101858)9-8045224]10-0305804l) 99)9°9694196/45 
 6 9°55955851394 4 1044044 13)8*8331318]9-5901881]47-110-40931 19/9-8043377|10-0306296]; 99199693 704144 
 7)9° 559882913. 1 9|10°4401 171|8°833804419°5905617 37734, 10°4094385 9-8041529] 100306788), 4.9:9°9693212143 
 819560207115.) 3q]10°4397929|8-°8344765/9°590935 115, 7|10°4090649/9 803968 1]10 0307250), 94/9°9692720/42 
 9'9°5605310)499,}10°4394690/8 835 1486/9 -5913082)479,))10°4086918/9°8037832|10-0307773] 4 9419°9692227)4 1 
 
 0}9°5608546)5534{10°4391454/8 8358190959168] 2l5npn|10°4083188/9°8035985]10-0308266) 
 
 D.]. Cosine 
 99701517166 
 9-9701032159 
 19970054758 
 9:9700061157 
 nan 9+96995 74156 
 ish 9+9699087155 
 
 485 
 485 
 
 A 
 
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 “19°9691 734140 
 10°438822118°8364595 9:5920539|gr9, 10°4079461 9°8034133 10'0308759 Fosi9 9691241 39 
 10°4384990'8°8371594 9°5924263!gn09 10°4075737|9°8032233110-0309254), 9 -2690746 38 
 10°438176318°8378288 95927985 lyn5|10-4072015 9°8030432/10°0309 7481, 9 -19°9690252'97 
 10°4375315:8'839166019°5935423 371 110°4064577 9°§026728110:0310738 4 9°9689262135 
 10°4572096)|8°8398337|9°5939138 saigiyo auaueee 98024875] 10°0311234 9°9638766134 
 10°436887918°8405010/9-594285 1 cone 10°4057149!9°$023021/10°0311730 49 9°9688270133 
 10°436566518°8411677/9°5946561 379 giL0 4093439 9°8021167}10-0312227], o.19°9687773132 
 10°43624548°841833919-5950269|5,..~,10°404973 1]9-°8019313]10-0312724)*5..19°9687276)31 
 10°4359246/8°8424.99619*5953975 3104 10°404602519°8017458/10°0313221 498 9°9686779139 
 10°4356040/8°843164719 5957679 37 10°4.042321)9°8015602/10°0313719 4 9°968628 1129 
 10°435283'7|8*843829419°5961380]9 gg 10°4038620/9°8013746)}10°0314217 4.99 9°9685'783)/28 
 10°4349637|8°8444934/9°5965079 3697|10°4054921 9:8011889]10°0314716 4.99 9:9685284)07 
 3195 10°4346439/8°845 1570/9 °5968776l.- ,/10°4051224. 9*8010031410°0815215 9'9684785)/26]. 
 5|9°5656756 3199 10°434324418 845820019°59724.70 3699 10°4027530)9°8008173;)10:'0315714 500 9°9684286)25 
 619°5659948 3189 10°434005218°8464826/9°5976162 3690 10°4023838!9°8006315110°0316214 501 9:9685786!24 
 7\9°5663137 318% 10°4336863|8°84'7144.5}9°5979859 3688 10°4020148}9°80044.56110°0316715 501 9-9683985!25 
 8\9°5666324 Sida 10°433367618'847806019°5983540 3685 10°4016460}9°8002596/10-0317216 50] 9-968278409 
 9 9°5669508 318] 10°43304.92)/8'848467019°5987225 683 10°401277519°8000735}10°0317717)~ 5 -19°9682283/2] 
 0 9°5672689\4 1 10°432731118°849127419-5990908 36 v4 10°4009092)9-7998875}10°0318219 : |9°968178 120 
 1 9°567586813 116 10°432413918°8497873/9°5994588 3679 10°4005-412/9°7997015}10-0318721)- 9 99681279119 
 219°5679044 3173 10°432095618°8504467/9 5998267 3676 10°4001733)9°7995151)10-0319223 ee 9°9680777\18 
 319°5682217 3170 10°4317783/8°8511055/9°6001943 367 10°3998057/9°7993288}10°0519726 503 99680274117 
 449°5685387 3168 10°4314613}8°8517639|9°6005617 me 10°5994383/9°7991425110°0320229}* -< 199679771116 
 519°5688555 3166 10°4311445|8°852421719-60092896 ae 10°3990711/9°7989561/10:0320733;), 3 9-9679267}15 
 619°5691721 3162 10°4308279]8°853079019-6012958 3664 10°3987042)9°7987697|10°0521237 fd 9°S67T8 769114 
 719.5694883 3160 10°4505117|8°8537358|9 6016625), ~- 110'398337519°7985832]10-0321742 9:°9678258/13 
 819°5698043 3157 10°430195'7]8°854392 1196020290 36 ye 10°3979 71019 7983966}10°0322247 506 996777532 
 919°5701200 10°429880018°855047919 6023953 3660 10°3976047/9 °7982100}10°0322753 9:96'7794.711] 
 019°5704355 10°4295645/8°8557082)9 6027613 , 10°3972387|9°7980233]10°0323259 99676741110 
 1/9°57075C6 10°4.292494/8°8563579/9 603127115 ~~ -110°3968729|9-7978366]10-0323765)< ~ 7199676235) 9 
 19°5710656 10-4289344)8-8570121/9-6054927|; °° 7|10-3965073}9-7976498|10-0324272 2 19°9675728) 8 
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 10-4276774|8 8596240]9-6049529|- 2 .410°3950471)9-7969020}10-0326303/23)9- 9673697) 4} - 
 10°4273638|8°8602757)9-60531741,, 2, o110°3946826|9-7967 149110-03268121?719-967318Et 3 
 
 819-5729495 ita 10°427050518°860926819-6056817 65M 10°3943183]9-7965278|10-0327321\2;"|9-9672679] 2 
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 ‘1 Cosine [Dif Secant” [Covers. Cotang. {Dif Tang. |Verseds.| Cosec. |D.| Sine |’ | 
 
(292) 22 Deg. = naturan Sines, &c. Tab. 10. 
 
 ‘| Sine [Dif |Covers Vers. |Dif |Cosine 
 013746066), -...16253934|2°6694672/404026212-4750869|1 0785341072816] 109 1|9271839]60 
 113748763 6254 297|2°6675467|4043646|2-4750145|1-07866 16107292591 | 04, 
 91975 145010 2 21624854) 12°66562921404703112 4709470 1 078'788510730342 
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 4137568525050 
 
 Cosec. |'Tang.|Cotang.| Secant 
 
 9266380)55 
 4 926528 6}54 
 
 9264192/53 
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 11377571 4lg 694 |6224286]2 6485054]4077531]2-4524.649]1 °0799364]0740195 
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 1413783794 oe 
 15}3786486)o 690 
 16)3789178}5 90 
 17/3791870)g 690 
 18)3794562}5 693 
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 6218899/2-6447323|4084.318}2°448389 1] 1 :0801928/07423941, | 4 AIS 
 6216206|2°6428502/4087713}2°4463559] 1030321 2/0743494 
 412°64.0971 0/4091 10812 °4443256]1-0804497/0744595 
 6210822]2°639094614094.504}2°44.22982]1°0805784|/0745697 
 6208150]2°637221 11409790 1]2°4402736]1-0807071]07467 
 2°6353506|4101299/2°43825 19] 1 -0808S60/074-7905 
 2+6334828}410469712°4362331{1-08096501074.9007 
 2°6316180/4108097|2°43421'7211°0810942}0750112 
 56126297560] 41 11497]2°4322041]1 081 223410751218 
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 2°6260406]4118300}2-4281864]1°0814823}0753432, 19. 
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 6186607|2°6223366]4125 1 06]2°4241801]1°0817417/075 
 2-6204888]41285 10/2-4221819]1 081871 50756758117 | | 
 6181230)/2°6186459/413191512-4201851}10820015]0757869}, | 5 | 
 61'7854.1]2°6168018]413532 1]2°4181918}1°082131610758980 1110 
 {6175853|2"614962.4]4138728|2-4162013|1-0822618|0760099}, ; ,519239908|31 
 2-6131259}4142136]2-4142136|1 -082392910761205], ; 1 5|9238795180 
 6170478}2-6112929}4145544)2-4122286]1 0825297076231 8], 1 ./9237682)29 
 6167791}2°6094613]4148953]2+4102465]1°0826533)0763433), | 1 519236567]28 
 2°6076332)4152363}2-4082672|1°0827840/0764548) | 1 619255452)27 
 2°6058078]415577412°4062906|1°082914910765664 1116 
 6159732)2"6039852/4159 186]2°4043168}1°0830458/0766780) | 18 
 2°6021654/4162598}2°402345711°0831769|0767898 1118 
 26003484 2/2-40037'74}1 -083508 1107690161, 14 0 
 96841015 1676]2°5985341]4169426)2-39841 18} 1°0834595}07701351 104 
 59}3851008]o gq .|6148992}2-5967225]4172841]2-3964490]1 0835 70907712551 15} 
 40}3853693]5 604 7/2°5949137}41'762572°3944889) 10837025107 72376) 591 
 6143623/2-5931077/4179673|2-39253 1 6} 1-0838342107734971) 190 
 J614094012-5915043]4183091}2:3905769]1 083966 1]0774619], 4 54 
 2°38862.50}1°0840980|0775'742 
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 6132890]2°5859107]4195348]2°384.'7293} 10843623 
 6130208]2°58411821419676912°3827855|1°0844947 
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 51}5883199) 56 j|5116801]2-5751963}4215885|2-373 1 063|10851582/0784754], 1519215246 
 52}5885880)523 161141 20|2-5734 1991421731 1]2-371179 1|1-0852913}0785884], 5 > 9214116 
 53}3888560].46 (61 11440]2-57 1 6462}4220738|2-3699540] 1 0854245]0787014|) 1 0/9212986 
 54138912401; om (6 108760]2-5698752}42241 6512-36733 16|1 08555 7810788146), | °5(921 1854 
 
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 Covers|Dif! Sine 
 
LOG. SINES, &c. 
 
 Dif| Cosec. |Verseds.| T'ang. Dif| Cotang. Covers. ; Secant Dz. Cosine 
 10-4264246|8-8622277|9-6064096) 5... -]10°3935904|9-7961533}10-0328341 9:9671659|5\° 
 | 10-426 1120]8-8628774|9°6067732 mated 103932268 9:71959660 A Hawn 
 5742003], 5 {10425799 7|8-8635265|9-607 1566), 10-3928634]9-7957786|10-0529363 
 5T45125)5 5 1, |10-4254877|8 86417521960 74997} 6 10:3925003|9-7955912 10-0329875}3 17/9 
 5748240] 5 g|10°4251760]8-8648233}9 6078627/> «0, 10-3921973}9°71954034 10-0330386)5 3/9" 
 *B751356]3 | 1 |10°4248644|8-8654710|9°60822541,. 696 10°3917746)9°7952161 10-0330899)55 5)9° 
 “5754468)> 1} o|10°4245532|8-8661181|9-6085880)5 0. 10*3914420|9%950285]10-0331412)5 15 
 ‘SISTSTS| 5 y-|10°4242422)8-8667648)9" 6089503 }5 50, 10°3910497|9°7948408]10-0331925).5, 4 
 *5760685]5 10-4259315|8-8674109]9-6093124) 5.7 .|10°3906876/9 7946531 10-0352438)5 1 
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 364 9{2805479|2"3840625|4620649|2"1641983]1°1015916]0922295}, 5, 19077775} 12 
 
 0637 577909 1}2°3691578/465599612°1477685}1°1030789}0934465 1998 
 3637 577 6454|2°367678714659536)2°146 1366} 1+103228310935693 1299 9064307 
 i 57738 17|2°3662016]4663077|2°144.5069} 1 °103377910936922 9063078) — 
 
 seqv{853068)2-41 1421014.557263|2°1942997] |-098947910900387}, 9619099613190. 
 
 2645)581 604412-39008281460653712"1708283|1°101000410917351|-~! “I9082649]16 
 
 3619 2900199]2-3810650]4627710]2-1608958}1-1018879]0924667 12219975933]10 
 3 3 
 
24 Deg. 
 Sine 
 ~019°6093153 
 119-6095969 
 219-6098808 
 3196101635 
 419°6 104465 
 519°6107293 
 
 LOG. SINES, XC. 
 
 ‘ 
 
 — 
 
 10°3906367/8°9367878 
 9834 10°390403 1/8°9373819 
 
 9°6485831 3399|10°3914169)9° 
 9-6489230) 3553 |10-3510770|9- 
 2°91 9-350737219- 
 
 <994110°3901197|8-937975619 6492628): 
 pte 3300 | [0:3503977]9° 
 oe” 2110°350058319° 
 
 9830 10°3898365|8°9385689/9 6496023}. 
 2898 10°3895535)8°939161819°64994.17 3399 
 3 »|10°3497191)9° 
 
 2895 10°3892707|8'9397542/9 6502809 300 
 8*940346219°6506199 3338 10°3493801)9° 
 
 6)9°6 110118) 54,,4}10-3889882 
 "719°6112941 999 1|!0°3887059)8-9409378]9°6509587 39gr|10°3490413 Os 
 819°6115762 2818 10°3884238/8-9415290|9 6512974 3385 10°348'7026|9° 
 9/9°6118580}5¢ ,-|10-388 14.20,8-9421197|9°6516359 3382 10°3483641]9: 
 10)9°6 121397] 50, ,|10°3878603)8 9427 101]9°6519742 339 1|10°3480258|9° 
 11)/9°6124211 2819 10°3875789}8°9433000|9°6523123 3380 10°34'76877|9 
 12'9°6127023} 99) 9|10°3872977/8'9438895|9°6526503 3378 10°3473497|9° 
 13}9°6129835} 5.45 .}10°3870167/8°9444785/9 6529831 3g76| 1073470119 Qs 
 14/9°6132641]5,;|10°3867359 hy 10°3466743}9° 
 1519°6135446) 10°3864554) 631 as 10°3463369|9° 
 16/9°6138250|595 ;|10°3861750/8-9462433 96540004155 ‘ti 10°3459996|9° 
 117 9°6141051)5~g9|10°3858949/8-9468307|9°6543375 39 69 10°34.56625|9° 
 18]9°6143850 3797|'0°3856150|8:9474177/9 6546744 33 3 10°3453256|9° 
 19]9°6146647|. 10-3853353)8 9480042 9*6550112 3365 10°3449888|9° 
 20)9°6 149441] 579 9] 10°3850559|8-9485904 96553477 /33 4) |10°3446523 9° 
 21/9°61522344 5.74 ,|10°3847766)8°949 176 1)9 6556841 346 10°3443159|9° 
 2219°6 155024. 2798 10°3844976|8°9497615 963602043364 10°3439796)9° 
 2319-6157812 an3% 10+3842188}8-9503464/9°6563564 3359 10°3436436|9° 
 9783 10-9859401)8 9509508 96566923 335% 10°3433077|9° 
 217g 9|!9°3836618)8-9515150 9°6570280|44 . 6 10°3429720)9° 
 24790 |19°3833836|8°9520987|9 6573636). 355 10+3426364|9° 
 27771 0°3831056)8-9526820|9°6576989 3359 10°34.23011|9° 
 27'75| 10°9828279)8-9532648|9 6580341 )45.5|10°3419659}9° 
 2774|10°382550418-9538473)9°6583692'5 1,9|10°3416308 9: 
 37H |10°3822730}8 954429419 6587041 394,¢|10°3412959)9 
 376|!0°3819959|8-95501 10]9°6590387 346 10°3409613}9° 
 2767 10°3817191|8°9555922/9 6593733 3343 10°3406267)9° 
 2176 5| 10°3814424)8-9561731]9°6597076 334,9| 1073402924 O 
 2769|10°3811659)8-9567533|9°660041 8)3.4 , 5|10°3399582)9° 
 376] |L0°3808897)/8 9573335|9 6503758 3339|1 0°3396242)9" 
 2759|/0°3806136/8-9579131|9°6607097 333% 10°3392903/9° 
 37196196622 2756|10°3803378|8-9584923/9-6610434 10°3389566)9° 
 38/9-6199378 2754) 19°3800622)8-959071 1/9°6613769 10°3386231|9° 
 39196202132 2759 10°3797868|8°9596 10:3382897|9° 
 40/9°62048841 10°3379566)9° 
 4119°6207634 10°33%6235|9° 
 42}9-°6210382 10°3372907|9° 
 * 143/9°6213127 10°3369580|9° 
 144196215871 10°3366255|9° 
 45|9°6218612 10°3362931|9° 
 46/9+6221351 27377|10°3778649|8-9636870)9 6640391 33901 1073359609 9: 
 47/9 -6224088 9736|10°37759 12)8 -964.2622/9°6643711 3339 10°3356289)9° 
 48/9-6226824 2'739|10°3773176|8-9648370|9 6647030 33 16|L0°3392970 9: 
 
 10°377044.3)8 96541 14/9°6650346 10°3349654/9° 
 10°3346338|9° 
 10°33543025|9° 
 10°3339712|9° 
 10°3336402)9° 
 10°3333093)9° 
 
 10°3329786/9° 
 
 2836 
 
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 24!9°6160599 
 2519°6163382 
 26)9°6166164 
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 2819°6179'721 
 2919614496 
 30|9°6177270 
 3119-6180041 
 3219-618%809 
 33/9°6185576 
 13419°6188341 
 35196191103 
 36|9°6193864 
 
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 3335 
 495}9°6617103\5044 
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 2748|£0°3792366)/8-960805 1|9-6623765]555 4 
 37 4.5|10°3789618|8-9613823|9-6627093], 
 91444|10°3786873 8-96 1959 119°6630420 
 374 |! 0°3784129|8-9625355|9°6653745 
 1739|10°3781388]8-96311149°6637069/355, 
 
 Dry 
 
 3325 
 3324 
 
 3316 
 5313 
 
 279 6 
 ahs 10+3759532|8-967705019"6663598 
 
 279] 10°3756810 8-9682774 9°6666907 ; 
 O118 10°3754089/8-9688494|9 6670214 
 Onl" 10°3751371/8°9694210)9 6673519 
 O14, 10°3748654|8 9699929 9°6676823 
 2719 10°3745940 8°9'705630)9 °66380 126 
 a1] 10°374322818°971133519°6683426 
 
 10°3740517\8-9717035}9°6686725 
 
 310 
 309 
 307 
 
 3305 
 3304. 
 3303 
 3300 
 3299 
 
 Q 
 
 - 15819-6254060 
 59196256772 
 60|9-6259483 
 
 Dif} Cosec. |Verseds| Tang. |Dif| Cotang. | Covers.} Secant. 
 
 *7711039}10°0398912 
 
 °7673835|10°040977 1 
 
 10°332648 119° 
 10°3323177\9° 
 10°3319874)9° 
 10)°33165 749° 
 10°3313275|9° 
 
 Cosine [Dif] Secant. | Covers. Cotang. Dif. Tang. |Verseds.| Cosec. 
 
 (297) 
 
 W.}] Cosine 
 ~ -.|9*9607302 
 2°19 -9606739 
 
 96319.9606176 
 564 
 
 7732475] 10-0392698 
 1730530|10-0393261 
 7'728583]10-0393824 
 1726636|10-0394388 
 172.4689] 10°0394952 
 7722741|10-0595516 
 7'720792|10-039608 | 
 7718843] 100396646 
 1716893] 10-0397212 
 7714942] 10-0397778 
 1712991|10-0398345 
 
 564, 
 965 
 
 9°9603354 
 
 7709087} 10°0399480 
 770'7134)10°0400048 
 7705180}10°0400616 
 7703225|10°0401185 
 7701271{10°0401754 
 7699315}10°0402324 510 
 7697359|10°0402894 5 
 76954.02)10°0403465 
 7693444) 10°0404036 
 7691486:10:0404607 
 7689528110°0405179 
 76387568] 10°0405752 
 7635608}10°0406325 
 7683648] 10°0406898 
 7681687|10°0407472 
 7679725|10°0408046 
 7677762|10°0408620 
 7675'799}10°0409195 
 
 9°9599952) 
 
 28919 .959881 5 
 
 56919.9598046) 
 
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 ie 9°9595393. 
 5 99)979994821 
 ie 9-9594.248 
 33{9°9593675 
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 2 (419 -9592598 
 ae 9-9591954 
 ar 9+9591380 
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 9 '919.9590299 
 
 9-9589653 
 27819-9589077, 
 21119.9588500 
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 599 986767 
 w19|9°9586188 
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 3451979585030 
 597 (9 9084450 
 59 1[9'9983869 
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 7669906|10°0410923 
 7667940|10°0411500 
 7665974)10°0412077 
 7664007|10°0412655 
 7662040) 10°0413233 
 7660072110°0413812 
 7658103|10°0414391 
 7656134|10°0414970 
 7654164110°0415550 
 7652193!10-0416131 
 7650222110°0416712 
 7648250|10°0417293 
 %646277|10-0417875 
 7644304)10°0418457 
 7642330|10°0419039 
 7640356] 10°0419622 
 7638381110°0420206 
 7636405)|10°0420790 
 16344.29110°04213'74 585 9 
 7632452110°0421959 585 9: 
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 7618595] 100426066}, 9 .|9°9573934 
 
 7616613}10°0426654 530 9°9573346 
 
 7614630} 10°0427243/" ~~ |9*9572757: 
 DI 
 
 Sine 
 
 60 
 59 
 58 
 
 564 9°9605612)57 
 
 9°9605048)56 
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 310 9-9597676 43 
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 9-9595964/40 
 
 39 
 38 
 37 
 36 
 
 35 
 34 
 38 
 32 
 31 
 30 
 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 29 
 21 
 20 
 19 
 18 
 17 
 16 
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 Deg. 65. 
 
(298) 25 Deg. NATURAL sinzs, &c. 
 
 Sine |Dit |Covers| Cusec. |‘Tang.|Cotang.| Secant | Vers. Dit |Cosine hii 
 577381 7i]2°3662016|4663077|2 1445069}] *1033779|0936922} | 4..9|9063078}60 
 a clO7T118112°3647265|46666 18]2°1428793]1 *1035277|09381521) 549/9061848/59 
 5168545|2°3632535|4670161]2°1412537)1°1036775j0939382} 5919060618158 
 2°36 178264675705 |2- 139630 1}1*1038275}0940614}, 545/9059386|57 
 3603136|4677250/2°1380085]1°1039777/0941846|, 4535190581 54/56 
 5760640}2°3588467/4680796)2°1363890|1 -1041279)0943078} | 51905692255 
 3 |9758006|2°3573818/4684542/2- 19477 14/1 *1042783|0944312), 55 419055688 54 
 
 ‘Tab. Qe 
 
 212-3559189|/4687890!2°1331559|1°1044.28910945546 1935 9054454153 
 
 521, 2+3544.5811469143912°1315493]1°104579510946781 193619053219 52 
 9/424.9895 263 3529999214 694.985|2°1290508]1°104730310948017 193719051983 5} 
 10}4252528}- oe 5 747472|2°35154.24|4698539)2° 1 283213)1°1048813}09492541, 93719050746 50 
 2635 235008754 702090/2 12671371 °1050324]0950491 193819049509 4,9 
 
 s an of 34.8634714705643!2°1251082) 1-105 183610951729 1939|9048271 48 
 13]4260425 , ag A 512°34.71838147091 9612°123504611°1053349/0952968 194919047032 47 
 
 2*34.5734.0/4.71275 1 12°1219030)|1"1054864)0954208 
 2°344.288 1|4:'716306]2°1203034/1-1056380/0955449 
 
 14442650565... |: P Ad 
 1249/9042068 43 
 194919040825 42 
 
 5731682}2°3428432/4719863)/2°1187057)|1°1057898/0956690 
 
 926 157.9005 110-341400914'723420)2°117110111°105941710957932 
 9]2639| 57264.2112-3399593}4726978] 21155 164|1°1060937/0959175 
 
 5723792}2+3385203)4730538 2:1 139246|1*1062458 0960418}, , ,/9059582)41 
 
 21123348]! 106398 10961662), , .19038338|40 
 
 2°33.56482/47376.59 21107470) | 1065506|0962907) | 4 4 -9037093)39 
 
 5}2°3342152)4741222/2-1091611}1*1067031)0964153} 9; -|9035847}38 
 
 4744785 )2°1075771]1 1068558]0965400} | 9 .|9034600|37 
 
 4748349 )2°1059951)1°1070087/09666471 5 5 6/9033353|36 
 
 3209276|4751914/2:1044150/1°107161 61096789514 9, 4190382105135 
 mea 5394/2°3285023)475548 1 /2-1028369/1+10753147109691441; 9. 419030856|34 
 seo.{5702767]2+3270%90|4759048 2-101 260711 -1074680/09703941, 5° 9029606193 
 9° 625]5700141]2-3256575|4762616 oss! 6 10762141097 16441) 52 1/9028356)52 
 
 2626156075 1510-3049381 14766185 |2°0981 140/1°107774910979805 1959(9027105)31 
 2 3488912-322890.5 14769755 2096543611 "1079285109741 471 919025855130 
 9296412-32140491477332612:0949751 
 
 Jao (2089639]2°3199912/4776899, 209340851 +1082363/09766 
 2625/568701412-3185794|4780472|2-09 1 84371 1083905 195% 
 mas 568439023171 695|4784046|2-09028091 *1085445}0979162|, 92 -19020838|26 
 
 5681'766|2°3157615]478762 1 |2-0887200}1 “1086989|09804181 57 -|901 9582123 
 2°3143554,4791197|2-0871610}1 10885331098 1675| 1, >|9018525]24) 
 2°3129513)4794774!2-0856039]1 *10900790982932), 4. .|9017068/23 
 2+311549014-79835212 -084.048711°109162710984190 1259 9015810)22 
 26|5 g99|2071274|2°3101486/4801932/2-0824953)1"109317610285449) 55019014551 
 5699| 00865 2|23087501 480551 2)2-0809438]1"1094726]0986708) 5 4° 
 ae 5666030}2-3075536|4809093]2-0793949|1 °1096277/0987969| ; 5p {1901 20: 
 ; a 5663409|2-3059588)/4812675}2-0778465|1109783010989236] | 5 .,|9010770|18 
 
 211 a¢9(9009508)1%7 
 
 2623 
 
 265 
 1265 
 
 1267 
 1268 
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 4}2+2934906|4844.959/2064.0008]1-1111869|10006 14. 
 
 56437325 Ho ae 2 | 
 57437586615 11 
 
 194 1|904579246] 
 
 4 
 
25 Deg. | LOG. SINES, &Xc. (299) 
 ‘1 Sine [Dif, Cosec. Verseds.| ‘Tang. |Dif] Cotang. | Covers. | Secant |D. 
 
 0/9-6259483] 9. )[10°3740517]8°9717035}9 66867251549 | 10°3313275}9-76 14630 10-0427 243}, 19-9572757160 
 119-62621911),-6 <{10°3737809|8-972275 119°6690023]555 <|10-3309977|9 7612647 10-0427832):. 5 9-9572168|59 
 2}9-6264897] 5.4 {10°3735103}8-9728424]9 669531945 5° 110-3306681/9 7610663)10-042842215019-9571578)58 
 319-6267601) 5 - y9|10°3752399]8°97341 13]9°6696613).5 ,|10-3303387|9-7608679|10-042901 2155 | 9-95 70988/57 
 4/9-6270303} 9... )10°3729697|8-9739797}9 6699906] 5 |10-3500094)9° 7606693 10-0429603155 5 |9-9570397|56 
 5|9°6273003}5 44 4|10°3726997}8-9745478)9°6703197]3 54] 10°3296803)9°7C04707/10-0430194).5 | |9-9569806|55 
 6/9°6275701] 9g ¢{10°9724299}8°975 1155/9 6706486]5 62 |10°S2935 14)9-76027 21 10-0430785)5 5, /9-95692 15154 
 7}9°62783971| 9 ¢.q|10°S721603|8-975682819-6709 7741.59 6] 10°3290226/9 7600734 /10-04313T7H 94/9°9568623]53 
 8/9628 1090} 5 .44|10°3718910}8-9762497/9-67130GU]55 6 2|10°3286940]9-7598746 100431970}; 53/9-9568030|52 
 91962837825 54 |10°3716218}8:97681 6319-671 6345 )30.45| 10-3283655]9°7596758'10-0432565]55°.19°9567437]5 1 
 10}9°6286472} 9 63 4}10-3713528|8-9773824)9 6719628}. 05] 10°528037219 7594769|10-0433156); 45 |9-9566844|50 
 11]9°62891 6019 60 5|10°3710840}8-9779482|9"67229 10] 35 ./.J10°S277090)9-7592779/10-0433750)5 4 4 /9*9566250|49 
 12)9°629 1845] ¢65|10°3708 155)8°9785 13519-6726 196]35~.4|10°32738 1019°7590789/10-043.4344)5 4 -19-9565656|48 
 13}9°6294529| 94g 9/10°3705471|8-9790785]9 6729468} 3974|10°3270532)9 7588798 |10-0434939), 
 1419°629721 19-7 9]10°3702789/8-9796431/9°6752745]3 5. 5|10°3267255)9 7586806] 10-0455534}5 06 
 15}9°6299890)9 4 4|10°37001 108980207319 6756020|59~3|10-3263980)9°75848 14] 10-0436 13015 9. 
 16 9*6302568 9675 ; 
 17/9-6305243| 96. 
 
 Cosine 
 
 o 
 
 10°3697432|8°98077] 1]9°6739294],., 79) 10°3260706)9°7582821'10-0456726 ae 
 10°369475718°9813346/9 6742566 390) 10929743419 °7980827 10043732215 on 19'S 
 10:3692083|8-9818976 9°6745836 3969 10°3254164]9°7578833|10-04579 19} 90 
 10°368941 1/8°982460319 6749105]. 5 ¢-)10°3250895 9°7570838 10-04385 1715 gn 
 10°3686742/8 983022619 6752372 3066 10°3247628)9°7574843]10°0459 114} 5 9 
 22/9°6318591 0664] 10'368 1409/8 9841460 9°6758903!3 99] 10°3241097]9°7570850 1004403 11 G00 
 23/9632 1255]5 4 1|10°3678745|8°9847072)9 6752165 326] 10 °323783519°7568852)10-044091 Ls g9 
 24196323916 9660] 1 0°3676084)8 °9852679}9 6765426155 «| 10°92345 74197566554 1004415 10) 9 9]9'9558490/36 
 2519°6326576 065710 3073424)8 9858283}9 6768686 40x | 10°3231314)9°1564856 10°0442110] 69 1]9°9557890)35 
 26,9°6329233 9656|L0°2670767)8 9863883 9°67 71944 la 0 50 10°3228056)9°7362856 10°044271 1] 69 1|9°9557289]34 
 27|9°6331889 2653 10°3668111!8°9869480 9677520113055 10°3224%79919°7560856 10°0443312 60 1/9 9996688)33 
 28/9-6334542 2659 10°3665458/8°9875072}9 6778456|39 59) 10°3221544,9°7558856 100443913! 65919°9556087/32 
 2919°633'7194. 2650 f0°3662806)8 9880661 
 
 30)9°6339844 264% 
 
 ip 
 
 9-6781 70912 a 10°3218291/9°7556895 10-0444515|q319°9555485)31| 
 10°5660156/8°9886246)9°678496 I 49.55) 10°3215039)9°7554853 10°0445 118] 6 9/9°9554882|30} | 
 10-3657509]8-9891827}9°67%88211)4.9 ; q(10°5211789)9°7552850'10-0445720] ¢919°9554980/29 | 
 10;3654863|8°9897404) 9-679 1460155 4; 9°9553676}28] | 
 10-3652220)8'9902978'9-6794708l49, x) 10°3205292'9°7548843'10-044692711gq),19°9553073|27] | 
 
 39.4.9) 10'S 208540/9°7550847 | 10-0446324! 6g 
 C 
 3245 9*9552469/26] | 
 
 10-3649578|8'9903548 9°6797953} 30; »|10°3202047/9°7546839 100447531] 695 
 '394 9| 10°3198802,9°7544833 10-0448136 ¢4/9°9951864 95] 
 36|9°6355699|9 55 6|10°3644501)8°9919676 9°6804440)4 9 j 5) 10°3195560)9'7542828 10-0448741 Gog gg tage a | 
 37}9°6358335|9 0. ,|10°3641665]8:9925235 9-680 7682145.9q|10°3192318)9°7540821 10-0449347]¢y¢|2°9550653 23] 
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(300) 26Deg. © NATURAL sINEsS, &c. Tab. 10. 
 
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 527 3{9328416)2-1406015]5283560) I -8926635|1-1309996|1158264| 13) 8841736 
 52791532584 1/2-13942381528798 1|1 -8919313]1-1311735]1159625] 425 8840377 
 7524 1|5323273|2-1382475}529 1 004|1 -8900006|1+1313475|1 160983); 55 |8839017 
 
 297115320702/2-1570726|5294'727]| I -8886713|1+1515217|1162344 36} |°92 1696 
 
 °971115318131}2-1358993|529845911 8873496|1-191696111163705 1569|9896295 
 15211531556 1|2-1347274|5502173|1-8860179{1+1318706|1 165067), 4, 8834933 
 
 =~ = 
 
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 2691530785312-1312205|5315364|1-882047011 -1323950}1 1691.59 
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 rr | 
 
 ‘osine!lDif| Vers. { Secant {Cotan. 
 
 . | Deg. 62. 
 
 ~ 
 
"Deg. tL _ LOG. SINES, &e. | (303) 
 Dif | Cosec. 
 
 10°34.2953219°0374005|9 *7071659 
 10°342705419°0579265/9°7074781 
 10°34.24.577|9°0384.522/9 "7077902 
 10°3422102/9°0389776/9-7081022 
 10°3419629}9°0395026)/9°'7084141 
 10°341715819°0400273)9 +7087258 
 10°3414688}9°0405517|9°'7090374 
 
 i).| Cosime 
 9+9498809160 
 
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 6572946 
 6575429 
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 658284215 410 
 6585312), 
 
 Verseds.| Tang. | Dif] Cotang. | Covers.| Secant 
 
 3199 10+2928541]9°7372002 10°0501191 : 
 3191 10°2925219 9°7369940 10:0501835 644 9*9498165|59 
 319) 10°2922098 9°71367878 10°0502479 645 9°9497521 98 
 3119 10°2918978|9*7365814 10°0503124 646 9°9496876 a7 
 3117 10°291585919‘7363750|10:0503770 645 9*9496250 56 
 3116 10°2912742|9°7361686|10°0504415 6417 9°9495585)55 
 <4, {107290962619 *7359621]10°0505062 646 9+94.94938)54 
 
 24.78 644 
 24.77 
 
 2475 
 
 2468 3114 
 
 6587780\ 94 66 10+2906512|9-7357555|10-0505 708) «, .|9°9494292)53 
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 6605005|o,,>4 971152545 1 45 |10°2884746]97343074]1 0-05 10248] ¢5 |9-9489752]46 
 66074.59|5 97118358}, p,5|10°2881642}9°734 1003)10-05108991 55 |9°9489101|45 
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 24 4,5|10°3382743]9-0473380/9-713076 1) 4 99|!0°2869239)9°7332711)10-0513505|¢55 9°9486495141 
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 6622145 O4AI 110*286304419°7328561)10°0514811 654 
 
 16624586) 5 4 40 300441 0°2859949|9°7326485/10-0515465} 65 4 
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 16634335|94.45|10°3365665|9-0509691|9 7152419] 59 4]10°2847581/9°7318174) 100518084 ¢ 5 ¢ 
 
 6612361 24.40 10*3387639)/9 -04629'76)9°7124.562 
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 6636768); 5 ;|10°3363232/9-0514865/9-7155508]5)qn|10°28444999°7316094|10-05 187401556 
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 16653749 : 
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 16639199 04.99 10°336080119-°0520036\9°7158595 303% 10°284140519°7314014)10°0519396 6517 
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 26 : 13084 Bi kero 
 
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 A 3081 [10°2829067]9-7305686|10-0522027 
 66513295, 5° |10'3348671|9-0545842/9° 71740141, 
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 76658586|q 11 5 Wl sn6 
 
 10°333899919-0566428|9'71863271g m6 10-2813673)9°7295262|10°0525326] 4, 99474674125 
 
 66634 etre 10+3336585/9-0571566|9'7189402)q 4. |10°2810598)9° 7293175] 10°0525987) ¢.¢5|9 9474013)22 
 528 2410 10°333417219-0576701|9°7192476)5.1.4)) 0°2807524 9+7291087}10+0526648} - ..4|9°9473352/21 
 6668238} 5 j yg]! 0°3331762}9-0581833 9°7195549| 34.5 |10°2804451 9°7288999| 10-05273 11]. .5|9'9472689) 20 
 16670647] 5 | | 10°3329353|9 -0586962)9°7198620 3040) "Ble a 9947202719 
 }°6673054| ~ 1. 110°332694.619-0592088|9°7201 690]... 4 910*2798310 664 9-9471364118 
 6675455 io, 10+3324541/9-0597210|9* 7204759149) ¢110°2795 24 9°7282729|10-0529300}¢ « ,|9°9470700 1% 
 16677863}. 40 10°332213'7|9+0602329|9'7207827 ae 10-2792173}9°7280638/10°05299641 , - |9°9470036] 16 
 16680265 10‘3319735}9-0607445|9°7210893}, y s|10-2789107 9°7278546|10-05506281, re 
 )*6682665 211 )978604.2/9°7276454110-0531293 G65|2 S 468707414 
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 )*6687461 ogo| 2779915 9°71272267|10-0532624} - - .|9°9467376}12 
 1-6689856|509, 10°3310144)9-0627877|9°722314714, nf 10+27'76853|9°7270172}10-0533290| ,19°9466710)11 
 96692250 ohio 10°3307750|9-0632977|9-7226207|,., 4{10°2773793 9#7268077|10°0533957] ¢¢.19°9466043/10 
 96694642 2390 10°3305358/9 -063807419°7229266 3058 10°277073419°726598 1]1 0°0534624. 668 9*9465376| 9 
 -6697032\ 5556 10°3302968|9*0643168}9°7232324 40 x | 10-276 7676 97263883}10-0535299) .019°9464708 
 9*6699420}5300 10°3300580}9-0648258/9-7235381)39 5 : 10°2764619\9°726 1787100535960) --4|9°9464040 
 6701807 anne 10°3298193}90653346/9°7238436|3 055 10-2761564]9 °7259689}1 005366291, .4/9°94633771 
 96704192] ,56,, 10°3295808|9-0658430]9-7241490},,.., 10-2758510)9°725759 1}10-0537298| ... .|9°9462702 
 9670657655 9.5) 10°3293424)9 06635 1 1]9-7244543]), “ 10+275545'9-725549 1}100537968] ..,|9°9462032 
 9°6708958) 53.6 9 10°329104219-0668589]9 7247595) 355 10°2752405|9°7253391}10-0538638] .119°9461362 
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 96713716 937711| 10°3286284|9-0678735|9 7253695 ]5 0 4 ¢ 10°2'746305/9°7249189}10:0539979] ., .|9°9460021 
 9-6716093|~? ‘ “110-3283907/9-0683803}9*7256744 4911 (-9743256/9°7247087|10-0540651 
 
 ‘Cosine |Difl Secant | Covers |Cotang. |Dif! Tang. Verseds.}| Cosec. 
 
 572199459349 
 (iinet tending tebe tiomrscnetilaion ti eitittAAL CCEA ONT ROLLA OOO ALLLCT AA 
 : a Deg. 62. 
 
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 52950 14)2°1254048/5332029}1: 
 
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 5251436!2°1058998)5395707|1- 
 52488'76}2°1047652}5399464| 1° 
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 520542 1|2°0856890)5463503]1° 
 5202869|2°084.57921546728 1]1° 
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 5195214/2°081258015478621|L° 
 5192663|2°0801536)5482404]1° 
 5190112)2-0790506|5486188]|1° 
 5187562|2°0779489/5489973]1° 
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 4814817537 Ae 518246312°075 7496154975471 1° 
 
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 517991 4)2°0746519]/5501335|1° 
 5177366|2°0735556|5505125)1° 
 51748 18/2:0724606)5508916|L* 
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 2546 51671 76)2°0691836/5520297)1° 
 
 5164630/2.0680940/5524093]1° 
 
 e\162084|2°0670056|5527890} 1 8090086] 1 *1426179}1248168)- 
 
 5159538|2°0659186|5531688]1- 
 5156993)2°0648323)5535488/1- 
 5154448}2°0637484|5559288] 1° 
 515190412°0626653/554309 11° 
 
 NATURAL SINES, &c. 
 
 “| Sine \Dit Covers| Cosec. 
 
 *8689065]1°1341527}1182845 
 
 *8649921/1'1346829|1186965 13 
 
 *8392184)1°1382529)1214606 
 
 *8315936]1°1393358}1222957 
 
 Tab. 10. 
 
 Tang. | Cotang. Secant. | Vers. [Dif |Cosine 
 
 &829476)60 
 Pye 8828110159 
 13¢7/8826743 58 
 13¢9|8825376 57] 
 1369/0824007 56 
 18822638155 
 i 3821269)54 
 8819898}53 
 (371)¢919597159 
 137 213917155151 
 1373 ° 
 isna 8815782|50 
 eh) 88144.09]49 
 Se 8813035|48 
 8636902|1+1548600}1188340) 3hB 8811660}47 
 862389611 -13.50372|1189716]  4,,/8810284 46 
 8610905}1°1352146|1191093 ison 8808907|45 
 8597928/1°1353921|1 192470 1378 8807530144 
 8584965)1+1355697/1 193848), vate 8806152143 
 8572015|1°1357476}1195296], 4 : 0|8804774) 42 
 8559080}1°1359255|1196606] , 380 880339441 
 8546159] 1°1361036|1197986 1381 880201440 
 853325211°1362819]1199367 1389 8800633|39 
 8520358]! -1364603}1200749} , 380 8799251/38 
 850'7479]1°1366389|1202131 1383 8797869)37 
 8494613)1°1368176}1203514 1384 8796486} 36 
 848176 1]1°136996511204898 1385 8795109}35 
 8468923) 1-1371755|1206283 1385 879371734 
 8456099}1-1573547|1207668}, 386 8792332}33 
 844.3289|1-1375341]1209054 139718790946] 32 
 8430499) 1°137713511210441), 2 
 
 8807265} 1*132570111170524 
 879407411 °1327453]1171890 
 8780898} 1°1329207)1173257 
 8767736}1°1330962|1 174624 
 8754588]1*1332719}11'75993 
 8741455}1°13344.78]1177362 
 8728336}1-1336238]1178731 
 87152311°1337999|1180102 
 8702141|1+1339762}1181473 
 
 8676003]1°1343293/1184218 
 8662955}1°1345060)1185591 
 
 lone 8789559|31 
 841770911°1378932}1211829 138 8788171}30 
 8404940} 1°1380730)1213217 8786783) 29 
 8785394) 28 
 8784.004|27 
 
 1389 
 1390 
 
 8379442}1°1384330}1215996 139] 
 
 8366713) 1°1386133]1217387 139] 8782613}26 
 8353999)1'1387937]1218778 1309 8781229125 
 8341297]1°1389742!1220170 1393 8779830) 24 
 8328610}1°1391550}1221563 1394 8778437|23 
 1394 8777043) 22 
 
 ~ 18775649121 
 
 8303275]1°1395169]1224351 1395 
 8774254120 
 
 8290623}1°1396980)1225746 1396 
 
 8277994) 1°1398794)1227142 1396 8772858|19 
 
 82653741°1400608}1228538 1398 8771462|18 
 877006417 
 
 825276711 14024.25)1229936 1398 
 
 8240 173} 1+1404243)1231334 1398 8768666] 16 
 
 82275931 °1406062)1232732 1400 8767268}15 
 
 $215026]1°1407883)1234132 1400 8765868} 14 
 
 8202473] 1°1409706]1235532 14011904468 13 
 
 §18993Q)1°1411530}1236933 14 876306712 
 8761665} 11 
 
 817740511 °1413356|1238335 1 
 8164892)1-1415183}1239737 1404 8760263}10 
 8152391} 1+1417012)1241141 1404 8758859 
 8139904)1°1418842)1242545 1404. 8757455 
 8127430|1°1420674)1243949 1406 8756051 
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 80404:78}1°1433541]1253803 
 
 CH DOORG ODAWO 
 
 Cosine|Dif| Vers. } Secant |Cotan. Tang. | Cosec. (Covers Dif! Sine |’ 
 
 Deg. 61. 
 
23Deg, Loc. sINES, &c. (305) 
 4 Dif) Cosec. |Verseds.| Tang. |Dif} Cotang. | Covers.{ Secant |D.| Cosine 
 
 Se 
 Se nee 
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 10°5283907]9-0683805|9°7256744| ,,, -|10°2743256|9-7247087|10-0540651|- ,|9-9459349|60 
 10-3281532|9-0688869}9-725979 1.1 *7/10-2740209|9-7244984|10-0541323]2719-9458677159 
 2379| 0°327915919-0693931]9-7262897}35 49 |L0-27371 63]9-7242880| 10-0541995]).7|9°9458005|58 
 3196725219} 470] 0°3276787}9"0698990}9-7265881|5 94 ,]10°2754119}9-7240776}10-054 2668] 7 |9-9457592)7 
 4196725583155 <o|10°3274417/9-0704046|9-7268925}20 *5|10-2731073/9°7238671|10-0543341|27 |9-945665: 
 
 5|9-6727959] 55 }10-3272048|9-070909919-7971 967], ,|10-2728033|9-7236565]10-0544015]>"-/9-9455985|55 
 
 ; 23671 19.3: ONS 2794 712344.59|10-0544690|2,/°19-9455310154 
 6|9-6730319| 9,.¢..|10°326968 1|9-07141 48/9-7275008),,5 |10-272499219-7234459|10-0544690) -7 
 
 7/9°6732684 10°3267316|/9°0719195|9°7278048 10°2721952)9°7232352}10°0545364 616 9 -9454636]53 
 8/9°6735047 2360 10°3264953/9°0724238|9°7281087 3037 10°2718913}9°7230244/10-0546040 67 9°9453960 52 
 9/9°6737409} 2 ~,|10°326259119 072927919 -7284124 3% 10:2715876|9°7228136 10-0546715 4 9°9453285/51 
 10|9°6739769 write 10°3260231|9°073431619°7287161 0 ‘110°2712839|9°7226027|10-054739 1 " 9+94.52609|50 
 11/9-6742128 10°3257872)9°0739350/9°7290196 ‘ 10*2709864)9°72239 17|10°0548068 "4 9°9451 932149 
 12/9°6744485 10°3255515}9°074438 1/9 7293230 3033 10°2706770|9'7221807)10°0548745 678 9*9451255/48 
 
 P19 lard . 1()- ’ : 47 
 
 13/9°6746840} ,,.. ,|10°3253160)/9°07494.09|9°7296263| 4,45|10°2703737/9°7219695|10-0549423 618 99450577 
 1419-6749194)229¢ 10°3250806}9-0754494 9+7299295 pid 10:270070519°7217584'10°0550101 $+944.9899]46 
 10°3248454'9°0759455 9°7302325)3 (91 0°2697675/9°7215471/10-0550780| 4.74/9°9449220)45 
 
 : 10°3246104/9 0764474 9°73053541399,|10°2694646|9°7213358|10-0551459)¢,,9|9°944854 1/44 
 3347|10°3249755|9-07694909-7308383 4,,~|10-2691617]9-7211244|10-0552138],4519°9447862143 
 4|10°3241408}9-0774502,9°7311410 10*2688590/9°7209129 10-0552818]-, 1 |\9°9447182/42 
 
 3026 
 
 344{10°3239063)9-07795119°7314456 49, 10-2685564}9-7207014/10-0553499 9°9446501|41 
 6 234411 03236719 |9-07845 18'9°7317460!00 2*|10-26825.4019-7204898'10-05541790>0 99445821140 
 20|9-676328115>74|10°3236719|9-0784518 460!aq0% 6 
 
 : 23)? | 10-393437719-0739521/9-7320484 20 -*|10-26795 1 6|9°7202781|10°0554861|00-19°9445139139 
 21/9°6765623' ¢-|10°3234377/9-0789521 4843050 | 3 445139 
 2219-6767963. o> #711 0-323903719-079452119°7323506 10°267649419°7200663:10-0555543|095/9°944445 7/38] 
 2219-6767963 0-3232037|9-0794521/9-7323506 35, 5 
 93 9°6770302\ 9 10°3229698 9:0799518)9°7326527 5 ol 0*2673473 9°7198545' 100556225 8 eee =p 
 24}9°6 772640, 5°, 5|10-3227360/9-0804512/9-°7329547 41 9|10°2670453)9°7196426 10-0556908| gq519°9443092136 
 25/967 749"5|y0. | 10-3225025/9-0809503 9°7332566 39 g|10°2667434)9*71 94307 /10-055759 116g 4 Ab Be me 
 26|9°6777309) 9.9.110°3222691/9-081449 1 |9-7335584)59 7] 10265441 6|9°7192186)10-0558275|6. tes 
 nig. 42) 2°? 3110-392035819-08194769-73386 10:2661399|9°7190066|10-0558959) <9 -/9°9441 041133 
 27|9°677964215.45|10°3220358/9-0819476|9-7338601|3q ; 4|10°266 4 i400 9-9440356|32 
 28/9:6781972l 93. 9|10°3218028)9-08244.58/9°7341616) 30; >|10°2658384)9 7187944 '10-0559644|¢05|9°9440356)3 
 
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 | 29/9°6784301155.5[10°3215699|9-0829437|9°7344631|5,_>|10°2655369/9°7185821/10-0560329 : 
 30|9-6786629|9 4|10-3213371|9-0834413/9°7347644'5. 9|10°2652356|9°7 183698 10-0561015|gq 4i9 9438985130 
 31)9°6788955}9g94|10°321 1045/9 -08399869-7350656)59 ; ;[10°2649344)9°7181575'10-0561701|4g719°9458299129 
 
 102643323|9°7177325 10-0563075|¢g7|9°9436925|27 
 10°264031519°7175199|10°0563'762 9°9436238}261 . 
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 02110-2622986|9°'7162429|10-0567898 691 |0 2 432102)20 
 
 32|9-6791279 , 10°2646333 
 33}9+6793602\949 |10°3206398|9-0849322/9°7356677 ang 
 34|9°6795993| 939 ))10°3204077|9 0854286 9°7359685 an. 
 35|9-6798243 95 1 |10-3201757|9-0859247|9-7362693 |) 4. 
 
 36 9°6800560!93 " 10°3199440/9°0864.204'9°'7365699 
 
 3'7|9°6802877 9 10°3197123 9°0869159)9°7368705, 
 38/9°6805191 : Deter eae IATL TGR 
 39)9°6807504 9319 10°3192496/9:0879059/9°7374712 
 40/9°6809816 310 10°3190184)9-0884.005/9°7377714 
 4.119°6812126)54,,.|10°318787419:0888948/9-7380715 ; 
 4219°6814434 se 10°3185566)9°089388'7|9-°7383714 9999 10°2616286}9°7158166)10°0569280 699 9*9430720118 
 
 45)9°68 16741] 44 )5110°3183259/9-0898824)/9 7386713) 9 q9,|10-2615287/9°7 156034) 10-0569972| gqq}9°9450028)17 
 44/9°6819046 i 10-°3180954|9-0903758/9°73897 10 rte ,|10°2610290}9°7153901}10-0570665| ¢q9/9'9420335 * 
 45]9°6821349|9° 5110-3178651/9-0908688|9°7392707], 49 .|10°2607293)9-715 1768}10-0571557]gq 4|9°9428645 8 
 46/9°6823651] 49 1|10°3176349}9 091361 6|9-7395702|,,49 3|10°2604298|9-7149633]10-0572051) 49 4|9°9427949) 14 
 47/9°6825952)9993110°3174048/9-0918541/9°7398696],9)4|10°260130419°7147498]10-0572745 Netter a 
 48/9°6828250} 9999) 10°3171750/9‘0923462)9°7401689} .405|10:2598311)9-7 145362)10-0573439]¢q59'9426561/) 
 49}9-6830548 10°3169452/9-0928381/9-7404681]4 99 1 |10°2595319|9-7145226|10-0574134) 9 5 masedee : ; 
 50|9°6832843 10-3167157/9-0933297|9-7407672| 144 9|10°2592328|9-7141089/10-0574829 aero 
 51|9°6835137 10°3164863/9-0938210/9*7410662| 595 4[10-2589338)9-713895 1}10-0575524| ¢94)9 9424476 
 52)9°6837430 10°3162570}9-0943120)9-7413650|99¢4|10°2586350/9-7136812|10-057622 1/495 operas 
 53|9-6839720 10°3160280)|9-0948027/9-7416638] 994 -|10°2583362)9-7134673/10-0576917 Sieber vA 
 54/9°6842010| 5564|10°3157990|9-095293 119-7419624 10-2580376|9°7132533}10-0577614| 694 
 55|9°684429"| oo 
 
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 57|9-6848868! 5 953110-31511529-0967625}9-7428577],, 40.9|10-2571423] 9°71 26 108] 10-0579 709] <4 /9" 
 58}9°6851151]993 j|10°3148849|9-0972517|9-7431559|5o97|10-256844 1|9-7123965]10-0580408| 29, 99419592 
 5219685345219 9 9) 10°51465689-0977406)9-74945401) 9 o|0°2565460]9°7121820] 10-0581 107} 9)? 9418895 
 60/9-6855719) “*{10-914498819-0982993/9-7497590) ~|10-256248¢|9-7119677] 10-0581807] | }9-941819¢ 
 Cosine |Dif} Secant | Covers. |Cotang.|Dif] Tang. |Verseds.j Cosec. {D.] Sine 
 
 3R Deg. 61. 
 
 “lOoH WOR BATIBO 
 
(306) 29 Deg. NATURAL SINES, &c. Pas 16s 
 
 Tang. Cotang.| Secant | Vers. [Dif {Cosine 
 902 2°0626653 5543091 1:804.04'78]1°143354111253805 141] 3746197160 
 5546894] 1°8028108}1°1435385]1255214 1411 8744786159 
 *6015751)1°1437293111256625 1419 8743375158 
 1°8003408]1°1439078]1 258037 1413 8741963157 
 2°05834601555831141°7991077/1°1440927)1 259450 1413 8740550156 
 2-0572695)55621 19]1°7978759}1-1442778)1260863}, , | .|8739137|55 
 2°056194.215565929)1°79664.54]1"144463011262278 1415 8737729154 
 795416211°144648411263693 1416 8736307 53 
 7941883/1°144833911265109 {A416 8734891152 
 7929616}1°1450196}1266525 1417 8733475151 
 791736211 -145205511267942 1418 8732058/50 
 55849041 1°7905121)1°1453915|1269360 1419 8730640)49 
 7892893]1°1455776|1270779 14.20 8729221148 
 
 5592629|1°7880678]1*1457639]1272199}, , 49|8727801/47 
 78684'75|1°14.59504]1273619 14.91/97 26381|46 
 7856285/1°1461371|1275040| |; 5|8724960/45 
 7844 107|1*1463238|12764621 3 55 
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 Cosine|Dif! Vers. | Secant |Cotan. Covers| Dif 
 
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 2245 
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 2217 
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 2204 
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 LOG. SINES, Xc. 
 
 9°744.0499 29 
 
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 2956 
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 10-3067692|9*1146705|9°7538203 
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 10°3063242)9°1156274|9°7544088 
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 10°3058797,9'1165831}9°7549969|o9.49 
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 10-2594363|9-699626 1|10-0622508|.,.|9°9377499 
 10-239 14.43]9-6994075|10:0623936|-5,19°S 
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(308) 30 Deg. NATURAL SINES, &c. Tab. 10. 
 Sine |Dif |Covers 
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 2181)19-9999378|9:1295764|9-7628969|_,.,, -|10°237103 1|9-6978750|10-0628347]_5-19-957 1653155 
 218011 9-299719819-129846449-7631881 sare 10°2368119|9-6976558}10-0629079},.-19-9370921)54 
 2179 991 9|10°2365208}9 6974365|10-0629811),,. sad eige 53 
 
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 2170 | 1.9-2347761}9-6961192{10-0694217 na@le 2265783147 
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 29041 1 0-234195319-6956795|10-0655689|,/~°19-936431 1145 
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(310): 31 Deg, NATURAL SINES, &C, Tab. 10. 
 
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32 Deg. LOG. SINES, &c. (313) 
 Sine |Dif| Cosec. |Verseds.| Tang. [Dif] Cotang. | Covers.|. Secant |D.| Cosine 
 
 ee EE TY ey a EL a EE Pe CS ah AT) PAU DD (OF nk CIOS A A SESS UE RR Ay ees SS A a pT Ee ee ee) ee 
 
 0}9°7 242097]... |10°2757903/9 181706 119°7957892) 10-2042108/9°6721725]10°0715795 9*9284205160 
 1)9°7244118 220 10°2755882/9°1821466)9°'7960703 10°203929719°6719445|10°0716585 : 9°9283415|59 
 Q)9°7246 138 2018 10°2753862|9°182586819°79635 13}. 10°2036487]9°6717165}10°0717375)_ 9°9282625/58 
 3)9°7248156 10°2751844}9°1830268|9°7966322 2808 10-2033678|9°6714884|10-0718166],,> 19°9281834'57 
 
 4|9°7250174]>)) °|10-2749826]9-1834665/9°7969130) 5. o| 1 0-2030870|9-6712602]10-0718957|.0,,|9-9281043156 
 5]9°7252189],,,9|10-2747811|9"1839060|9-7971938),4 4. 
 
 2015 10-2028062|96710319]10-0719749]; 919° 928025155 
 6|9°7254204 10°274579619°18434.52|9° 7974745 2806 10°2025255/9°6708036}10-0720541 793 9°92T94 591544 > 
 m9°T25621% 10°2743783]9°1847849/9°"7977551 2805 10°202244.9}9 °6'705752110°0721334), ,..|9°9278666)93 
 8}9°7258229)- 10°2741771}9°1852230|9°7980356 2804 10°2019644]9°6703467|10°0722127 4 9°927'7873)52 
 9|9°7260240 10°2739760|9° 185661 5|9°7983160 2804 10°2016840}9°6701181}10°0722921 194 9°9277079)51 
 
 10|9°7262249}; 10°2737751|9°1860998)|9°7985964 2803 10°2014036|9°6698895}10°0723715 195 9°9276285)50 
 11/9°7264257 2007 10°2735743|9'1865378|9°7988767 2909 10°2011233|9°6696607}10:0724510 195 9*9275490]49 
 12|9°7266264 2005 10:2733736|9°1869756|9°799 1569 10°2008431}9°6694319}10:0725305 196 9°9274695)/48 
 
 2801 
 
 13}9°7268269},,..,|10-2751731|9+1874132/9°79943'70}5.,|10°2005630)9°6692030]10-0726101],., |9°9273899|47 
 2800 7196|9.oo~ 
 
 nonl9'9273103/46 
 
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 1419-7270273 coe 10-272972719-1878505|9°799771'70 10-2002830|9-6689741|10:0726897 
 15}9°7272276) 9 y9|10-2727724]9-1 882876 |9°7999970] 5x 99] 10°2000050|9-6687450) 10-0727694}..4,19-9272306 45 
 16]9°7274278)..4,|10-2725729/9-1887245|9°8002769]...4|10°1997231|9-6685159|10-0728491],4 119-9271509\44 
 17|9°7276278] 7 90|10:2723722|9-1891611|9-8005367] 440.0] 10°1994433|9-6682867]10-0729289}.9 319-927071 1/43 
 
 18|9°7278277 1998 10°2721723/9°189597419 8008365 2706 10°1991635|9'6680574|10:0730087 799 9°9269913)42 
 19|9°72802'75 1996 10°2719725/9°1900336|9°8011161 2196 10°1988839|9 667828 1}10-0730886 9°9269114/41 
 20}9°7282271 10°2717729)9°1904695|9 8013957 2795 10°1986043}9°6675986]10-0731686 9°9268314)40 
 2119°7284.267 10-2715733]9*1909051/9°8016752|5,- , {10°198324819°6673691)10-0732486 9°9267514)39 
 
 800 
 80] 
 
 1993 
 
 1993 |9°9266714,38 
 
 9°9265913)|37 
 
 10:2713'74019-1913406|9°8019546|<,,~ *|10°19804.5419°6671395]10-0733286 
 
 2919°1286260 04 
 10-2711747|9°191775819°8022340]--* 2110-1977660|9-6669098]10-0734087 
 
 2319°7288253 
 
 24/9-7290244| 5° |10-2709756|9-1922107|9-8025133},,04|1 0° 1974867|9-6566801|10-0734888|5.,,9°9265112136 
 25|9°7292034], 4 4 .|10-2707766|9*1926454|9 8027925]. |10°1972075|9°6664502]10-0735690} 9 .|9°9264510)35 
 9619°'7294,223 10°2705177'7|9'193079919 8030716 ; 10196928419 °6662203]10°0736493]64°19'°9263507134 
 
 10196649419 :6659903]10-0737296},,.219°9262704133} 
 10°196370419°6657603|10:0738099|~ 7199261901132 
 10°1960915/9°6655301110:0738904-..°(9°9261096|31 
 10°1958127|9°6652999]10-0739708 805 9-9260292)30 
 
 27|9°7296211 ae 10-2703789]9"1935142/9'8033506|5.0, 
 28}9°7298197| 9 99|10-2701803/9-1939489|9-8036296|,.04 
 2919°7300189}, 529110-2699818/9-1943819|9-8039083) 5.00 
 30|9°73021 65} 534110-2697835|9-1948155|9°8041873),09 
 
 31)9°7304148} 1 9.4 1|10°2695852|9:1952488/9 8044661]... .|10°1955339|9 6650696} 10-0740513}.9,|9°9259487)29 
 32}9°713061291 | 9.4 9|10°2693871|9°1956819}9°8047447), ae 10°1952553/9*6648392110-0741319} 9, .|9°925868 1/28 
 33]9°7308109 10-2691891|9°1961147|9°8050233],,. 6110-1949'767/9°6646087| 100742125 B06{2 920 7875/27 
 34|9°7310087] ; 9.--|10°2689913;9°1965473)9 8053019 1361 10-1946981/9°6643781|10-0742931|>°19-9257069|26 
 35|9°7312064 10°2687936|9:1969797|9°8055803]5, 0 °110°1944197|9°6641475]10-0743739].0-19°925626 1/25 
 
 36|9-7314040 feee 10-2685960}9-1974118]9"8058587),,764|10-1941413|9°6639168|10-0744546], 199255454124 
 
 37|9°7316015 102683985|9°1978437|9°8061370]5,.05|10°1938630/9°6656860}10-0745354}.. 49 }9'9254646/25 
 38/9°7317989] >! *110-268201119°1982754|98064159| -..°-|10°193584819°6634.552|10-0746163}2, 5 9-9253837/22 
 39}9°7319961] | 9..5|10+2680039}9-1987068|9"8066933] 5... ,|10+1933067/9 '6632242}10-0746972}, 0/9°9253028)21 
 40}9°7321932 1640 10-2678068|9°1991380/9°80697 14, 048 10°1930286/9°6629932}10:0747'782 9-°9252218/20 
 41}9°7323909] 5 .5110-2676098|9-1995690]9-8072494 pA 10°1927506|9°6627621|10-0748592\>, 919°9251408/19} 
 42|9°7325870 1967| 10°2674130|9-1999997|9°8075273 ei ; 10-1924727|9°6625309}10-07494031, 1 5 9*9250597|18 
 4,319°7327837 1966 10+2672163/9:2004302)/9°8078052 otnn 10°1921948)9°6622996}10:0750214 9°9249786)17 
 10-26'7019'79°2008605/9°8080829 ahh 10°19191°'71|9°6620683]10°0751026 3 9-9248974/16 
 4519°7331768}, 4 .-|10:2668232|9°2012906/9°8083606], Hn 10°1916394}9°6618368}]10-0751839), ,5/9°9248161/15 
 46|9°7333731}, 9 5|10°2666269|9-2017204/9°8086383 on ' 10°1913617|9°6616053]10-0752651],, 7 |9°9247349) 14 
 4'1|9°7335693 1961 |10°2664307/9-2021499/9-8089158], 10°191084.2|9'661373'7|10°0753465 9:9246535|13 
 4819°733'7654 1960} 10° 2662346|9-2025793/9-809 1933 ones 10°1908067}9°6611421)10°0754279),, 4 9°9245721/12 
 
 491973396141, 5 10-2660386|9°2030084|9°8094.707 ang 10°1905293|9°6609103}10°0755093), , -|9°924490'7|11 
 50)9°7341572 1957 10°2658428/9°2034373)9°8097480]5-..110°1902520/9°6606785}10-0755908 9°9244.092)10 
 5119°7343529 {110:263647119°2038660|9:'8100253 Sa 5] 10° 189974'7]9°6604466] 100756723 I 9°9243277) 9 
 52)9°734.5485 1100-26545 15/9*204294.419°8 103025 dum |{L0°18969'7519°6602146]10°075 75391, 5 /9°9242461 
 453]9°7347440 10°2652560}9*2047226/9 8105796}, 10°1894204]9°6599825]10°0758356],, /9°9241 644 
 54)9°7349393 1959 10+2650607|9°2051506|9 8108566 OnN0 10°189143419°6597504}10°0759173 SL 9*9240827 
 
 55/9°735 1345), 9 «1|10-2648655]9-2055783|9'811 1536], .9{10°1888664/9-6595182)10-0759990}. 1 9/9°9240010 
 1950|10°2646704|9-2060058|9°8114105},,,-5|10°1885895]9-6592858|10-0760809}., 3/9°9239191 
 
 57)9°7355246) 19 1 o{10°2644754|9-2064331/9°8116873}, 20 
 
 58)9°7357195] 4 4n|10:2642805|9-2068602)9-8119641],, 
 
 59/9°7359149], 4, 4)10°264085819-207287019'8122408),,_ 0 | 300) 
 
 60]9°7361088 10-263891219-2077136|9-8125174 10°1874826|9-6583558|10-0764086] ~ 9*9235914 
 
 Cosine |Dif! Secant {| Covers. Cotang. Dif} T ang. IVerseds.! Cosec. D.| Sine |” 
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 10°188312719°6590535|10°0761627 9*9238373 
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 10°18'7'7592]9°6585884|10°07632661,, ,|9'9236754 
 
 TD Qo D108 
 
(314) 33 Deg. 
 
 NATURAL SINES, Xc. Tab. 10. 
 
 — }|§ ———j~— -—— ] 
 
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 Vers. | Secant ICotan.| Tang. | Cosec. 
 
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 *2138920|1'762035}, 6441823796528 
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 -214378411765334 1651 8234.666}26| ° 
 *2146218]1766985 1651 8233015125 
 *2148655|1768636 1652 8231364124 
 2151094 1770288 1653 8229712 93 
 *2153535)1771941 1654 8228059}22 
 *915597811773595 1654 8226405)21 
 *9158423)1775249 1655 8224751120 
 *2160870|1776904 1656 8223096} 19 
 *2163319|1778560 1656 8221440)18 
 
 °2165770]1780216 1654 821978417 
 *2168293}1781873 1658 8218127|16 
 *21'70678]1783531 1658 8216469}15 
 *2173135|1785189 1659 8214811}14 
 °2175594\1786848 1660 8213152]13 
 *21'78055|1788508 1660 8211492}12 
 
 *2180518|1790168] ; -.[8209839|11 
 *2182983}1791830], 7 /82081'70}10 
 *2185450|17934911 | --.4|8206509 
 *2187919|17951544 | 218204846 
 18903183 
 11664 
 *2192864|1798481] | {8201519 
 *2195339]1800146}, ..|8199834 
 -2197816}1801811 
 2900296] 1803477 
 -2202777|1805144 
 2205260}1806811 
 -2207'746}1808480 
 
 Covers|Dif!} Sine 
 
 ‘Deg. 55. 
 
 wo 
 — 
 fo) 
 fee) 
 — 
 fe s) 
 ito] 
 SlLoHn woke HA~Iwwo 
 
Deg. 
 ‘Sine 
 
 74715617 
 7477489 
 7479360 
 7481230 
 
 7483099 
 7484967 
 
 I OO | | 
 
 10°2524383)9°2329007|9 8289874. Qnon 
 10°252251119°2333138]9 8292599 ON 24, 
 10252064019 :233726'7|9°8295323}- 
 
 10°25187'70}9°2341393]9 8298047 2799 
 10°2516901|9°2345518/9°8300769 
 
 1870 
 
 10:2511302}9-2357879|9-8508934|59, 
 1 gg|10°2509438]9-2361995}9-831 1654), 
 10-2507575|9-2566109|9-8514374|>« 
 10250571319-237022119°831'7093 
 71496148 
 7498007] 
 m499866|,.. 
 7150179311897 
 1503579 
 
 10°2501993/9-2378438)9°832252919,, 1, 
 10-24.9827719-238664'719-8327963 
 1955|10°2496421]9-2390748|9 8330679 
 7505434) 05 3|10-2494566|9-2394847|9 8333394 
 75072871, -|10-2492713}92398944]9 8336109 
 7509140} ° ; 1 {L0°2490860/9-240303819-8338823 
 7510991), 557| 10248900919 -2407131/9-8341536 
 7512842 oe 10:248'715919°241129919-8344949 
 7514691), sie 10°2485309|9°2415310/9°8346961 
 9516538 (B44 10-248346219°241939619 8349673 
 7518385); 37 
 9520231|.°* 
 184 
 7522075): 
 
 2715 
 
 2712 
 2712 
 
 10°2481615}9°242348119°8352384 ate. 
 
 4|10°247976919°2427563)9-S35509415,, 
 1 g4g| 10°2477925|9°2431643)9-8357804 
 1523919 so] 10247608 1|9°2435721|9-8360513].. 4° 
 1525761), © , ;|10-2474239]9:2439797|9-8363221[,, 
 "527602 40 10+24'72398)9°244.38'7119°8365929 On0N 
 1529442), © « 4[10-2470558|9°2447942/9-8368636|5. 0 
 Besse 
 
 ] Bs 
 7534954) 396 
 7536790), 
 1538624 
 7540457 
 7542288 
 
 544119 
 7545949]! 
 "BATT 
 7549604! 
 1551431|! 
 7553256|!8 
 
 7555080 
 7956902), ¢ 
 7558724|- 
 7560544. 
 7562364 
 7564182), 
 
 10-2466882/9-2456079)9 837404915... 
 36|10'2465046|9°2460145|9 83767551, 
 34| 10° 246521019°2464208/9-8379460)5_ 7 
 3g 10°2461376|9-2468269]9-8582164), 
 3] |10°245954319°2472528)9 838486715, 1 
 1|10°2457712}9-2476385)9°838757110, 15 
 30| L0°2455881]92480440]9-83902731,,.., 
 9g] 10°245405 1]9-2484493)9 83999751) | 
 »[10+245299319°248854419°8395676|,. 
 110-2450396|9°2492593/9-8398377|~,, 
 27110+244856919°249664019-840107"\—. 
 102446744)9°2500684)9 8403776], on 
 10+2444.920|9°250472'719 8406475], 
 192211 0-2445098]9-2508767|9-8409174{2009 
 10-2441276|9-2512806|9-841 187115 24 4 
 90| 107 2439456]9-2516842/9-8414569|5 
 10-2437636|9-2520876|9-8417265|5 20,2 
 | 102435818]9°2524909]9-8419961]5 7° 
 
 10°24.34.00119°252893919°8429657 2694, 
 10°2432185]9°2532967)9°8425351 9695 
 1314 10:24303'70]9°2536993)9°8428046|; = * 
 7571444 1 10°24.28556]9°2541017|9°8430739)" 
 7573256 1812 10°24.2674419°254.5039}9°8433432\, 
 7575068 1810 10°24.24932)9°2549059}9 8436125 9 
 
 7576878 1809 10°242312919°2553077|9 8438817 
 7578687 1 10°2421313)9°255'7093}9°8441508 
 7580495 180" 10°2419505}9°2561107)9°8444199 
 7582302 1806 10°2417698/9°2565119}9°8446889 
 7584108 1805 10°2415892)9-2569128/9-8449579 
 7585913} ~*~ 110°2414087|9°2573136}9-8452268 
 
 2691 
 
 2690 
 2690 
 2689 
 
 Sosine |Dif| Secant | Covers Cotang.|Difl Tang. [Verseds.| Cosec. |D. 
 
 LOG. SINES, &c. 
 Dif} Cosec. {Verseds. Tang. Dif Cotang. Covers.| Secant {D.| Cosine 
 
 10:2515033|9-234964019-8303492 es 10°169650319-6430596]10-0818525 
 10-2513167]9+2355761)9-8906213|,, |10-1693787]9-6428215]10-0819380|a 6 
 
 72011 9-1 68562619°6421068110-0821949 : 
 191 9-168290719-6418685]10:0822806 
 10-250385219-237433019-8319811|>,19110-1680189|9°6416300110-0823664|.>9|9°91'76336/49 
 
 10°2500134}9 °2382543/9 8325246] 5, -|10°1674754)9 641 1528]10-0825381 
 ‘*110+167203'719°6409141|10:0826240 
 
 2715) 9-166389119°6401974110-0828821 86 
 *110°1661177/9°6399583 mt aoe 
 n{10°1658464/9°6397192]10-08305451 6 
 
 “|10°1655751)/9°6394800}10-0831407l¢ 6 
 
 '110°165032'7/9°6390012 10-0833134lg6 
 
 p| 1071644906 9°6385222|10-0834863'6 65|9°9165137 
 
 270911 0-1639487|9°6380428)10-0836594"g 
 °110°1636779|9°6378030|10-083746 liggal9*9162539 
 
 705} ()-1615133|9°6358814 
 
 9111 9.1601 623/9°6346776110-0848779'0.° 
 
 269311 0-156926119°6317799110-0859296G an 
 
 “eC SVT) 
 
 5 |9°9185742|60 
 g|9'9184890)59 
 944 (9°9184037)58 
 1 /9°9183183)57 
 (9°9182329|56 
 9-9181475|55 
 9+9180620|54 
 
 9°9179764)53 
 -|9°9178908152 
 9:9178051)51 
 9°917'7194)50 
 
 10°170740119°6440109}10°0815110 
 10°1'704677|9°6437732110°0815963 
 10°1701953)9°6435354110°0816817 
 10°1699231}9°6432975}10°0817671 
 
 10°1691066|9°6425834]10-0820236 
 10°1688346]9*6423452110-0821092 
 
 10°167'74'71|9°6413914 10°0824522 6 <0 9°9175478)48 
 
 9:9174619|47 
 9-9173760|46 
 9-9172900|45 
 6 119° 9172040/44 
 9/9°9171179}43 
 9:9170317|42 
 of9°9169455/41 
 3/9°9168593}40 
 1,|9°9167730|39 
 1|9°9166866|38 
 
 =19°9166002 
 
 10°1669321}9°6406753}10-0827100 8 
 10+1666606/9°6404364. 10-0827960 
 
 10°1653039|9°6392406]10-0832270) 4. 
 10°1647616,9°6387618|10-0833998 37 
 36 
 35 
 34 
 33 
 3|9°9161673/32 
 g19°9160805131 
 
 10°1642196)/9°6382825]10-0835728 866 9°9164272 
 
 gr {99163406 
 
 10°163407119°6375631110°0838327 86 
 10°1631364)9°6573931 10-0839195 9¢ 
 
 10:2468720|/9°245201219°8371343 ati 10°1628657|9°6370830)10-0840063 g¢¢\9'915993730 
 
 10°1625951|9°6368499110-0840931\, ..|9°9159069199 
 
 »|10+162324519°6366026/10-0841800 gn4|9°9158200)28 
 ~110+162054019°6363623 10°0842670 4 9°9157330/27 
 
 10°1617836|9°6361219)100843540-g. ,|9°9156460}26 
 10-0844411/9,,)|9°9155589)25 
 10+1612429/9°6356408|10-0845282 o,.9)9'9154718)24 
 10*16097279°6354001]10-0846 154'g4|9°9153846/23 
 10+1607025}9°6351594)10-0847026'g,.519°9152974)29 
 
 +11 0+1604324/9°634.9185110-084'7899!. (7 19-9152101121 
 
 ‘719-915 1298120 
 
 10°1598923|9°6344366|10-0849646 be 9+9150354119 
 10*1596224|9°6541955}10-0850521!,,9|9-914947 9115 
 
 10°1593525}9°6539543110:0851396 5 9°9148604)17 
 10*1590826|9°633'7131110°0852271 - 9°9147729116 
 10°1588129)9°6334'717}10-0853148]}o, -19°9146852115 
 10°1585431/9°6332303]10-0854024 9°9145976114 
 10°1582735]9°6329888]10-0854901 9*9145099}13 
 10°1580039)9°652'74'72110-0855'779 8'79 9°9144221119 
 10°1577343)9°6325055|10-0856658 9*9143342)1 1 
 10°1574649}9°6322637]10°0857536 ggol? 2142464110 
 10°1571954)9°6320218}10°0858416 88 9°9141584 
 0 9°9140704 
 10°1566568/9°6315378]10°0860176 1 9°9139824 
 10°1563875}9°6512957|10:0861057 889 9°9138943 
 
 10°1561183/9°6310533]10:0861939 ggql2 9188061 
 10°1558492/9°6308112}10-0862821 gggl? 2137179 
 10°1555801}9°6305688}10°0863'704. ggal? 9136296 
 10°155311 1/9°6303264)10'0864.587 3]9° 9135413 
 10°15504.21/9°6300838]10:08654'70 885 9-9134530 
 10°1547'732|9°6298412}10°0866355 9°9133645 
 
 Sine 
 
 Deg. 55: 
 
 “tO READ to Ur AIH 
 
(318) 35 Deg. 
 
 NATURAL SINES, &c. 
 
 Tab. 10. 
 
 ‘| Sine |Dif |Covers} Cosec. |‘Tang.|Cotang.| Secant , Vers. 
 
 4264236]1- 
 
 2383|;,. I: 
 25g 0(t261853 
 
 2382 
 2381 
 
 0/5'78.5764. 
 1)5738147 
 215740529 
 315742911 
 4)5745292 
 
 7|575243% oan 
 8575481119305 
 915757190 oan. 
 10)5759568\oan6 
 11/5761946193-~ 
 
 424281 0}1° 
 
 ~a|4219050|1° 
 
 ‘7149 1: 
 5 4216677 
 
 }- 
 fh 1420944.7 
 ‘93q9(t200077 
 |*799/4 ton708 
 4.195339 
 4192970 
 4190603 
 4188235] I 
 418586811 
 
 1 
 1 
 1 
 1 
 1 
 1 
 
 5823595\ga-1/4 
 3815895959|n07 
 
 4171677|1° 
 4169313|1° 
 4166950)1 
 "14164588)1° 
 
 4162296/1° 
 41598641 
 41 57503/1- 
 2360/4155143]1° 
 95600. ool, 
 oae, 415278311 
 
 415043]1° 
 
 2 
 oaneltt48064|t: 
 apn altl45706|1 
 35814 143348|1- 
 2358141 40990/1° 
 235714 13863311- 
 235714 1369761 
 
 nee Cn 
 oop HIS1D65|1- 
 sone eI 29210|1- 
 2355/4 106855I1° 
 
 4124501} 1° 
 
 7234568) 7124157 
 *7227534)7128543 
 "7220508 
 
 "7213489 
 7206477 
 "7199472 
 "7185484 
 
 "TPS 25 
 "7164556 
 
 *7143691/7181319]1°3925019|1°2311432]1877468 
 
 *7122890/7194.554|1°3899401]1°2319156}1889561 
 
 74,344.68]7002075|1°4281480|1°2207746|1808480 
 
 7427229) 700641 111 74272642}1 -2210233}1810148)) - 1 
 
 1670 
 1671 
 1672 
 1672 
 
 1680 
 1680 
 1681 
 
 4 
 
 — 
 . 
 
 7192475 
 
 os 
 - 
 
 7178501)7159297 ; 
 7163698]1°39592'79/1:2301161|1870686 
 7168100/1°3950698} 1°2303725]1872380 
 7157594171725045}1°394213 1/1°2306299)1874075 
 
 7150639)717691 1}1°3933571)1°2308861}187577 
 
 7136750)718572911°3916473]1° 
 71298 17/7190141]1°3907934/1+2316579}1880863} | 64, 
 
 7115970)7198970}1°3890876|L° 
 7109058/7203387]1°3882358]1° 596 
 7102152)7207806)1°387384.7|1-2326900}1887661 
 7095254)7212297)1'386534211 °2329486}1889362 
 
 7088362|7216650|1°3856844 1 -2332074]1891064 
 
 1701!o3, 
 1701)g44 
 1702 
 
 tT} 
 7067'730|7229950}1-385 1599112339850] 1896174] 4/81 0382 
 7060867|723436 {1 -3822929|1-2342446]1897878| |.) [8102122 
 518100416 
 
 1706 
 1706 
 1708 
 708|5093588 
 170913091879 
 170913990170 
 
 ICovers|Dif| Sine 
 
 8188182)58 
 8186512/57 
 
 812931423 
 
 20835}18 
 
 O99181 I ST40I15 
 1700131 14040|14 
 2339113 
 0638112 
 
Deg. LOG. SINES, &c. (319) 
 
 Sine |Dif| Cosec. |Verseds.| ‘Tang. Dif| Cotang. 1.) Cosine 
 1585913 1804 10°2414087/9°2573136/9 8452268 2688 10°1 54773219 6298412) 10°0866355 885 9:9133645160 
 1587717 1802 10°241228319°257714219°8454956 2688 10°154504419°6295985]10°0867240 385 9°9132760/59 
 1589519 1802 10°24.1048 119°2581145]9 8457644 2638 10°1542356 9°6293597 10:0868125 386 9°9131875|58 
 1591321 1800 10°2408679/9°2585147/9°8460332 2686 10°1539668 9°6291128 10-0869011 987 99130989 57 
 (593121 1799 10°240687919°2589147)9 8463018 2684 10°153698219'6288698}10-0869898 334 99130102 96 
 1594920 ~110°240508019'259314419°8465705 2685 10°153429519°6286267|10°0870785 387 9*9129215)5 
 10°153161019°6283836|10°0871672- 19-9128328154 
 
 1596718 fee 10-2403282|9-2597140|9'8468390|75° a 
 10-1528925|9-628140310-0872560),, .,|9-9127440|53 
 
 5089|10-1526240|9-6278970|10-0873449} 5 
 
 3084}10-1523556|9 6276536|10-0874338| 
 
 268311 -1520873|9-6274101{10°087522 
 
 5083 {10-1518190]9-6271665]10-08761 18 
 10°1515508|9°6269228|10-0877009 
 
 268% 892 
 oa 10-1512826|9-6266791|10-0877901 ,,|9°9122099}47 
 dog | [10°1510145|9-6264352}10-0878793},..19-912120746 
 segolLO: 150746449 -6261913]10-0879685] .0.5|9°9120315/45 
 speplt0:1504784|9-6259473| 100880578] 4 ,/9°91 1942214 
 
 10+1502104]9°6257031|10-0881472|5" , 9°9118528|43 
 
 2679119-14.99425|9-6254589|10-0882366| 7 #19°9117634|42 
 9-9116739|41 
 
 ra 10+1496747]9-6252147|10-0885261] 44, 
 Dery LO. 1494069]9 6249703 10-0884156) 9 6|9°9115844/40 
 2677711 0-149 139219°6247258] 100885052] .5-19°9114948/59 
 2677] .0-148871 5|9°6244813|10-U885949] 5 ./9°911405 1138 
 joa 10°1486039]9°6242367|10°0886845 898 9°9113155 37 
 3076l10-1483363|9°6239919 10-0887743}.,.9019°9) 12257)96 
 see 10+1480688]9-623747 1|10-0888641] qq 9|9°9111359}35 
 d6r74| LO" 1478013}9 6235022 10-0889540} 449|9°9110460/34 
 dan gltO’ 1475339}9-6232575}1 0-0890439} . 9 /9°910956 1198 
 doyg|t0" 1472665}9-6230 1 22}10-0891339|,,,)9°910866 1/32 
 “07311 0+1469992|9°6227670|10-0892239],,, | |9°9107761)31 
 coed 10:1467320]9 62252181 0-0893140), 5, 9°9106860)30 
 ae oan 10°1464643]9'6222765|10°0894041 909 9°9105959 29 
 1769] 10°2356920)9-2700321}9°8538023 yt 10°1461977|9°62203 1 1]10°0894943]9 ,.9|9°9105057|28 
 
 In| 0-2355151)9-270426219°8540604)) 2 1|10-1459306]9°6217855/10-0895845)qq 9-9104155}27 
 1646616|; ae¢|10°2353384]9-270820219°8549365) °° 10 1456635]9°6215400}10-0896749|,5,5)9°9103251 26 
 "71648382 10-235161819-271214019°8546034| -00|10-1453966|96212943]10°089765215,,; [9°9102348)25 
 
 #9650147]. 195110 -234985319-2716075/9°8548704) 20 2110-145 1296|9°62 10485|10-0898556 ae 9-9101444|24 
 
 1764 ; 
 "7651911 6% 10+2348089|9-2720009]9°8551372 co 10+1448628]9-6208026|10-0899461].,.,.|9-9100539|25 
 *7653674|1 705) 10-2346326|9-2725941]9°8554041], 2° |10°1445959/9 6205 567|10-0900366|4 46 /9'2099634)22 
 ¥1655436|) (0°|10-2344564|9°2727871|9°8550708), 221 |10-144329219 °6203107}10-0901 272] 7/9-9098728/21 
 #7657197|,/°1|10-9342803|9-2731799|9°8559376), °Fe|10-1440624)9-6200645}10-02021791,94/9°9097821 20 
 1658954 100110-2341043}9-2735725|9°8562042}, 27|10°1437958)9°6198183}10-0903085) 94199096915 19 
 #7660715}} 725|10-2339985]9-2739649)9-8564708] "2 9|10-1435292)9-6195720)10-0903995}4,),19'9096007]18 
 #7660473], ,.,|10-2397527/9-2749571|9-8567374), «,.|10-1492626)9-6193256|10-090490 I}q9/9-9095099)17 
 *1664229| 17911 0-2335771/9-274749 119 °85'70039| 700» |10°142996 19-6 F9079210-0905810]5,4|9°9094190}16 
 *1665985 2665} 1 9+142729619-6188526|10-0906719), 5 ,[9°9093281)15 
 "1667739 26641) 9.1 494632|9°6185860|10-0907529],,|9-9092571]14 
 "1669492 2663} 1 9-1421969]9-6183392|10-0908539|; 
 “1671244 
 
 % 
 
 7602106) ~99 10°2397894)9°2609 114]9°84.76444 90 
 ( 9°9124.772)50 
 
 890\9.9193889149 
 
 10-2394308]9'2617087/9°8481810 89] 
 9*9122991)48 
 
 7607483 ay 10*239251719-262107119°84844992 
 
 7623556 1781 10°237644419°265683219 8508608 
 9625337 [779 10°2374663]9°266079519°S511285 
 7627116 
 
 7% 10-236755319°2676629]9°8521987 
 16342221 ja 10:236577819°2680583/9 8524661 
 1635996] rong 10°2364.00419°2684534)9-8527335 
 1637769), - - 10°236223 119°268848419 8530008 
 "7639540 LTT 10°2360460]9°2692431|9°8532680 
 
 He 10-2330508|9-275923919-8578031 
 72 9|10°2328756]9°2763151]9-8580604 
 175o{10°2327004|9-2767062|9-8583357 
 i ge|02325254]9-2770970}9-8586019 
 i gglt0"2323506|9-2774876]9-8588680 
 
 10°2321'75819-277878 119°8591341 
 
 ae 10°232001119-278268319°8594002 
 
 110+ 141664319 6178455] 100910361 
 2662|1-1413981/9°6175985]10-0911273 
 2661110-1411520|9°61735 14|10-0912186 
 266111 9.140865919°6171042|10-0913099 
 266111 9-1405993|9°6168569|10-0914012 
 
 3698 )10*1403339)9-6166096|10-0914927 
 
 "7676494 
 *716'78 24.2 
 "1679989 
 "1681735 
 
 1683480 10°2316520}9°2790483/9 °8599321 2659 10°1400679)9°6163621|10°0915841 
 "7685223 10:231477'7|9°2794380}9°8601980 2658 10°1398020)9°6161146]10°0916757 O16\.. 
 "1686966 10-°2313034]9°2798274|9 8604638 10°1395362/9°6158669|10-0917673 916 9:9082327 
 
 noms 10°1392'10419-6156192110-0918589 
 26811 0-1390046)9°6153714]10-0919506 
 656] 9-1337390/9-6151235]10:0920424 
 
 Verseds. 
 
 Dif Paap: 
 
 "1688707 
 
(320) 36 Deg. NATURAL SINES, &c. Tab. 10. 
 
 4 
 
 Sine |Dif|Covers} Cosec. |'Tang. |Cotang. | Secant | Vers. |Dit| Cosine 
 
 0}5877859| 44 5.4|4122147]1°7013016|7265425| 3763819] 1-2360680}1909830 
 115880206|55 re 41197941 -7006208]726987111-3'755403}1°2363293}1911540 
 2}5882558|55 sof4117449)1-6999407]7274318]1-3746994] 1 -2365909}1915251 9158}: 
 31588491 0],,5> 94115090] -6992612}7278767|I 3738591]1-2368526]1914963}, | 51808503757 
 4/5387262},4> ,[4112738}1 6985825]728321 8|1-3730195|1-2371 146]191 6675] ,, + 4|8083325|56 
 5158896 15}555 j [41103871 -6979044] 728767 I|I 372 1806|1-2373768]1918388} ,, ,5|8081612155 
 6|5891964) 5° 4/4108036]1-6972271/7292195]1-3713423|1-2376393]1920101) 7 18079899)54 
 75894314], [4105686] -6965504}7296582I I -3705047}1-2379019]1921815],,, ,|8078185|53 
 15896663) 95; 9|4103337)1 “6958744/7301041]1 -3696678}1 -2381 64711923530] ,, 218076470|52 
 915899012 SaAc 4100988}1°6951990/7305501}13688315]12384278|1925246] /< | °)8074754)51 
 10}5901361}95 1 .|4098639|1 -6945244!7309963] 1 -3679959|1-2386911]1926962 8073038150 
 1115903-709|<27°140962911°6938504/7314428]1 367161 0|1-2389546] 1928679 8071321149 
 
 12)5906057 et: 4093943|1°693177117318894)1°3663267]1 -2392183|1930397 1718 8069603)48 
 
 1315908404 2346 4091596|1°692504.5]7323362]1°3654.931]1-2394823]1932115 ; 8067885|47 
 14}5910750!934 6 4089250|1°691832617327832|1 °364660211°2397464) 1933834 80661 66/46 
 1515913096], 40869041°6911613}7332303|1°3638279]1-240010811935554 1790 8064446|45 
 1615915442 34 & 408455811 °6904907|733617'77|1°3629963}1 -2402754| 1937274 1791 8062726) 44 
 
 SIL TTB 793 4 4082213|1°6898208]7341253}1°3621653|1°2405409]1938995 fwed 8061005|43 
 1815920132 2344 40'79868}1°6891516|7345730}1 361335011 -2408052}1940717 1 93 8059283 42 
 19]5922476} 54, .[4077524) 1 °6884830]735021 0]1-3605054|1°2410704| 1942440 1795 805756041 
 2015924819 i pe 4075181/1°687815117354691|1°3596764}1°2413359]1944163 14 805583'7|40 
 2115927163 She 4072837] 1°68'714:79]7359174|1°358848 1]1°2416016]1945887 104 8054119]39 
 2915929505], ie 4070495] 1°68648 14173636601 °3580204]1+2418675]1947611 1179 x[8052389)38 
 2315931847 2349 4068153}1°6858155]736814711°3571934)1-2421336]1949336 8050664137}. 
 2415934189} -0 2 ; 40658 1 1}1°6851503]7372636]1°3563670}1*2423999}1951062 1197 8048938/36 
 
 23 . 
 25)5936530 oe. 40634°'70]1°6844857|73'7'71 271 °3555413}1°2426665}1952789 1707 804.7211|35 
 2615938871 23.40 4061129}1°683821917381620]1'354'7162)1 +2429333)19545 16 1798 8045484)34 
 271594121115 405878911 °6831586]73861 15]1°3538918}]1°2432003}1956244 1798 8043756|33 
 28)5943550|50.09|4056450]1 -6824961]73906 1 1]1°3530680)1°2434675}1957972 29 804.2028]/32 
 2915945889 2330 40541 1111°681834217395110}1°352244.9}1+243'7349}1959701 8040299)31 
 30}5948228 2338 405 17'72|1°6811730}7399611|1°3514224)1°2440026}1961431 1731 8038569}30 
 3115950566], 8 40494.34]1°6805124174.04.113}1'3506006|1°2442704)1963162 1731 8036838/29 
 32)5952904 9: a ~(4047096| 1°6'798525|7408618]1°3497794]1 -2445385)1964893 8035107/28 
 3315955241 \- 5 4044759] 1°6791933}74131 2411 3489589] 1 -2448069|1966625 80333'75|27 
 S4IS957577 2336 404249311 67853471741 7633}1°348 13901 -245075441968358 8031 642/26 
 3515959913}545 -|4040087|1 °6778'768)7422 1 43}1°3473 198) 1+2453442/1970091 8029909}25 
 
 3615962249) 0 (4037751|1°6772195]7426655|1-346501 1|1-2456131|197 1825] 156180281 75)24 
 
 29, ‘ 
 37[5964584|o |4055416|1°6765629]7431 170}1 5456839] -2458823]1973560}, ..,|8026440|23 
 38|5966918},5. ,|4033089| | 6759070|7435686]] 3448658)1-2461518| 1975295}, 718024705|22 
 39}5969259) 555: 40307481 67525 17|7440204]1 3440499] | -2464214]1977031] "1802296921 
 40}5971586| 54 4|4028414|1 67459 70|7444724 I -3432331|1-2466913}1978768], 3 -/8021239|20 
 4115973919|5 555540260811 *6739430|7449246]1 3424177] °2469614| 1980505}, 5 418019495]19 
 
 42}5976251 9339 4023749] 1°673289717453770|1°3416029)1*24723 1'7/1982244 14 4 8017756}18 
 
 4315978583 40214.1'7|1°672637017458296]1 3407888} 1 °2475022]1983982], _ 
 44)5980915 933] 4019085]1°671985017462824]1 °3399753)1°24'77730| 1985722 
 4515983246], 4016754}1°6713336]7467354|1 °3391624/1°248044.0| 1987462 
 46|5985577 2399 40144.23)1°6706828}7471886}1°3383502)1°2483152)1989203 
 4715987906|545,|401 209-411 °6700328]74.764.20]1 *3375386) 1°248586611990944 
 48}5990236 Sbu0 4009764} 1°6692833}7480956]1°3367276| 1 -2488583]1992686 
 
 49)5992565 f 99}4007435}1°6687345| 748549411 °3359 172) 1-2491302)1994429 
 50}5994893 939 4005107|1°6680864)749003311 °3351075)1°2494023]1996173 
 5115997221 4002779} 1 *667438917494.575|1°334298411°249674.6119979 17 
 
 59|5999549|~>-14000451{1-666'7920]74991 191 -3334900|1 -249947111999662 
 53|6001876),5,,.(9998124| 1 -6661458|7503665|1 33268291 -2502199]2001407 
 
 546004202 0396 3995798]1°66550021750821 2) 1°3318750)1°2504929/2003153 1" 7996847 
 
 55|6006528| 9, ,|3993472 1 -6648553]7512762|1 -3310684|1-2507661|2004900}, ,, 47995100) 
 56|6008854]54, 0/3991 1461-66421 10|7517314|1 -3302624|1-2510396| 2006648), 157993352 
 57/6011179),3, 3988821[1 -6635673|7521 8671 -3294571|1 -2513133]2008396], 167991604 
 58/6013503}5'3, 4199864971 -6629249]7526493|1 -3986524]1-251 5879|2010145]1 /+0 7989855 
 5916015827] 29 2 *139841 73}1 662281 9175309811 327848311 -25 18613/2011895|2 > 7988105 
 2 1750996355 
 
 60]6018150}~°*7/3981850|1 +661 640117535541 11 °3270448}1 +2521 357/201 3645 
 
 —_—| —_—__ | | | [| | ] | CS - | -——]} 
 
 Cosiné|Dif | Vers. | Secant |Cotan. Tang. Cosec. |{Covers!Dif|} Sine | ’ 
 
 Deg. 53. | 
 
 — 
 
 ps 
 
 1741 
 1742 
 
 17 8007314112 
 
 800382'7|10 
 8002083 
 
 SOrRNwoOOfo D-10 0 
 
36 Deg. | LOG. SINES, &c.. (321) 
 i Dif} Cosec. |Verseds. Tang. Dif Cotang. Covers. | 
 faan 10°2307813|9-2809947/9-8612610 Li 
 
 Secant |D.} Cosine 
 10:1387390]9°6151235 10°09204.244 918 99079576150 
 2656 10°1384733]9°6148755]10-0921342 8 9*GOT8658}59 
 
 719°6146275)10-°092296.; on (9° 90771740158 
 ~|10°13794.2219 -6143795]10:0923180 9:9076820/57 
 »110°1376767|9°6141311]10-0924099 51 (9 GOTS9IOIISG 
 10°13'74113/9°6133827110-0925020]* 9 9°9074980/55 
 10°13'71459]9°6136343110-0995941 991 9°9074059]54 
 2Iy99/9°9073138153 
 10°1566152/9°6131372110-0927784. 993 9*9072216/52 
 10°1363500)9°6128885 10°0928707]- : '9°9071293 51 
 10°1360848]9°6126397 10:0929630 994, 9°9070370150 
 10°1358197/9'6123908 10-0930554 994/29 9069446 4.9 
 10°1355546/9°6121418 10°0931478 9959 9068522 48 
 
 10°1352895]9°6118928}10:0932403 99419 9067597147 
 10°1350245}9°6116436]10-0933329] *°'9-9066671 46 
 10°1347596/9'6113944110°0934255 ? 629065745 45 
 10°1344947|9°6111451}10:0935181 9499064819144, 
 10°1342298/9°6108956]10-0936108 go9 9°9063892/43 
 
 € 
 
 10°1339650]9°6106461|10-0937036 gg 9°9062964 42 
 
 1736 
 1734 
 
 2653 
 2652 
 2652 
 2651 
 2651 
 2651 
 
 2650 
 2649 
 2649 
 2649 
 2648 
 2647 
 
 iy94| 10°2290478)9-284873019-8639159 
 
 islo-7714-70 
 14|9-7716496|) 124 
 1519-7718150 i. 
 16/9°7719879 
 
 224 
 
 1718 10+2274967|9°28834.75|9°8662997 2647 10°133'7003}9*6103965]10-°0937964 9 pel abe edb 
 1717 10°2273249/9-2887326 9°8665644 264 10°1334356|9°6101469]10°0938893|"~* 9-906] 107/40 
 1717 10*2271532)/9°2891175|9°8668291 2646 10°1331709/9'6098971]10-0939893 9°9060177/39 
 1715 10°2269815/9°2895022/9-8670937 2646 10°1329063}9°60964'72]10°0940753 9 99059247138 
 1714 10°2268100|9°2898867|9°8673583 2645 10°1326417/9°6093972110:0941683)- 3 9°9058317/37 
 1713 10°2266386/9°290271 119'8676228 2645 10*1323772)9°6091472]10-0942614 9399 9057386 36} 
 
 10°1321127|9°6088971}10-0943546),..,. 9°9056454135 
 964,3| [9° 1318483|9°6086468]10-0944478 39 9°9055522)34 
 264.4) 10°1315840)9°6083965}10-09454111555 9°9054589/33 
 10-2259541/9°2918065 9°8686804)5 -4.9|10°1313196|9°6081461|10-0946344 34, 2°9053656'32 
 10°2257832|9°2921899|9 8689446 264.3] 10°1310554/9 6078956) 10-0947278 5 9°9052722:31 
 
 2649|10°130791 1/9°6076450]10-0948213 935 9°9051787/30 
 2641/10" 1305269/9-6073943]10-0949148],....9°9050852.99 
 26a, 1 |L0"1302628/9 °6071436}10-0950084 36 2°9049916/28 
 264011 0° 1299987|9 6068927] 10-0951020 37 9°9048980'27 
 264.0] 10°1297347)9 60664 17)10-0951957|0., 9°9048043)26 
 264)| 10" 1294707] 96063907] 10-0952894 938 9'9047106,25 
 9639|!0°1292067)9°6061396] 10-0953839\5°8 9-9046168/24 
 
 2638 10°1289428}9°6058883]10-0954.770 9 9 9°9045230 23 
 2638 10°1286790|9°6056370)10-0955709)- 40 9*9044291}29 
 2638 10°1284.152)9°6053856|10-095664.9 40 9°9043351)21 
 
 1712 
 1710 
 1710 
 1709 
 1708 
 1707 
 1705 
 1705 
 1704 
 1702 
 1702 
 
 063'7|10°1281514)9°6051341]10-0957589)5 7, 9°9042411)20! 
 9637|10'1278877/9 6048825|10-0958530|5, , 99041470119 
 
 9636) 10" 1276240}9-6046308| 10-0959471)5 ;, 9-9040529|18 
 
 2636 10°1273604)9°604379 1]10-0960413 Ae 9°9039587/17 
 2636 10+1270968)9°604.1 272!10-0961356 94.9.9 9038644116 
 mi 10+1268332/9°6038752]10°0962299)" 4429037701 15 
 10°1265698)}9°6036232/10-0963243 4 9-9036757|14 
 10°1263063}9°6033710}10:0964187|> - = 
 
 } 
 
 2634 AC yay ays . oe 
 9633 10°12604.29/9°6031188]10-0965132 945 
 2634. 10°1257'796}9 6028665] 10°0966077 4 
 263 10°125516219°6026141110-0967023 4 : 
 2632 10°1252530}9°6023616]10-0967969 Q4n 9°9032031 9 
 9639 10°1249898)9-6021090}10:0968916 Q4, 9°9031084 
 2631 10°1247266|9°6018563}10:0969864. 94, 99030136 
 9631 10°1244635)9°6016035]10°0970812 94,0 9°9029188 
 963 10°124.2004}9:6013506]10°0971761 950 9°9028239 
 9630 10°1239373/9-°6010977|10-0972711 9*9027289 
 9q|10°1236743]9 6008446] 10-0973661 950 A 
 10°1234114)9:6005914]10-0974611 05 Ge 
 10°1231485/9°6903382) 10-0975562)° 59 9°9024438 
 10°1228856/9-6000849/10-0976514!"~ “19-9023486 
 Cosec. }|D.| Sine 
 
 3°T Des. 53, 
 
 9°9024.868]12 
 
 9*9033923)11 
 9°9032977)10 
 
 9026339 
 9025389 
 
 OHM eo O10 
 
(322) 37 Deg. NATURAL SINES, &¢. _ Tab. 10. 
 ; Dif |Cosine 
 se (7986355 (60 
 1751 5/0 
 
 17 1 |7984604|59 
 
 7982853158 
 17153 i981 100157 
 
 Sine |Dif |Covers| Cosec. 
 
 ~0/6018150},.454|3981850|1 6616401 
 116020473), 
 216022795]. 
 
 Tang.|Cotang.| Secant | Vers. 
 753554111°3270448|1°2521357|2013645 
 
 3977205|1°6603586 
 3974883]1°6597187)75 
 416027439 2394 
 
 516029760|505,)39'70240} l-65844.09175583691 1 °3230368)1 *2535 108}2022406 
 £ 2320 
 6}6032080 
 
 9390 396792011 °6578030}756294 1] 1°3222570}1 *2537865}2024161 
 7/6034400)o34 9 3965600} 1°65'71657|756751 4}1°3214379}1*254.0625}20259 16 
 8|6036719 2319 396328 1}1°6565290}7572090}1 *3206393}1 -2543387|2027671 
 916039038 2518 396096211 *6558929/7576668} 1 '3198414)1°2546151/2029428 
 
 1016041356 2318 3958644) 1°6552575/758 124811 °3190441/1°2548917)203 1185 
 
 11/604.36'74] 54, ~|9956326]1 °6546227/7585829] 1318247411 °2551685|203294 
 
 3954009] 1°653988517590413}1°3174513}1 2554456) 2034701 1159 7965299}48 
 
 1216045991 cate 
 
 13]6048308]o41 6 395169211 -6533550|7594999]1 *3166559]1 °2557229]2036460 1760|7963540 47 
 
 14}6050624195 1 394.9376]1°6527221175995871°3158610)1°2560005}2038220 1760|1961780 46 
 
 15}6052940]5¢ ; 394'7060}1 *652089817604.1'77|1°3150668}1 *2562782/ 2039980 176 1 (1960020 45 
 
 16|6055255|o9 1 : 3944.74.5]1-°6514.58 117608769} 1°3142731|1°2565562|2041741 11769]71958259 44, 
 1394.24.30) 1°650827017613363} 1-3134801}1°2568345}2043503 17691926497 43 
 
 17/6057570 
 19/6059884154, , (09401 16|1 +6501 966] 7617959] -3126876|1 -2571 129}2045269| ,1°517954735|42} 
 
 19 6062198|54 14 3937802) 1+6495668}76225571°3118958}1°2573916|2047028 1764 7952979)41 
 20/6064511 lo313 3935489) 1 -6489376]7627157]1°3111046)/1°2576705|2048792 1764 7951208)40 
 21)/6066824 2319 3933176] 1°6483090}7631759]1°3103140}1°2579497|2050556 1766 794944.4)39 
 22\6069136 931] 3930864) 1 °6476811|7636363] 1 -3095239/1°258229 1/2052322 1765 794'7678|38 
 23,6071 44.7 3928553} 1 *647053'7| 7640969] 13087345) 1°2585087/2054087 116% 7945913|37 
 
 1167 79441 46}36 
 
 24)6073758 2311)3996949 1°6464270} 7645577} 1°3079457|1 *258788512055854 
 1168 7942379|35] 
 
 2311 
 25,6076069 392393 1]1-6458009] 7650188] 1°30715'75!1°2590686|2057621 
 1768 7940611/34 
 1769 7938843}33 
 
 26 6078379|5510 3921621)1 °64.51'754| 7654800} 1 -3063699}1°2593489}2059389 
 1770 7937074|32 
 
 79714084|53 
 iypo|7972329152 
 ipnf 7970572151 
 1'75m|7968815|50 
 1759{7967058)49 
 
 2716080589 Bik 3919311}1°6445506] 7659414) 1°3055828)1°2596294}2061157 
 
 28|6082998 2308 591'7002}1 -6439263}] 7664.03 1] 1°3047964|1°2599102/2062926 
 
 29|6085306 2308 391469411 -6433027|7668649]1°3040106)1°26019 1 212064696 1771 7935304|51 
 
 30|6087614 29308 3912386) 1+6426796| 7673270] 1°3032254)1 *2604724|2066467 1771 7933533}30 
 3910078) 1°6420572/7677893}1 °3024.4.07/1'2607539|2068238 1772 7931'762/29 
 
 32)}6092999 2306 390777 1}1°6414354|76825 17) 1°3016567/1°2610356|2070010 17 
 
 3316094535 2306 3905465] 1°6408 14217687144) 1 °3008733)1°26131'75}2071785 
 
 3416096841 2306 5903159} 1-6401936|7691773] 1-5000904)1 *2615997|2073555 1774 7926445 | 26 
 
 3516099147 13900853] 1°6395736|7696404|1*299308111 *2618820|2075329 1775 7924671}25 
 
 3616101452 2305 3898548]1°6389542}770103%71°2985265!1°262164712077104 1775 
 376103756 
 
 230 
 
 93 a 3896244} 1 -6383355]7705679|1 2977454) 1 +2624475| 2078879), .|799 
 386106060) 3 y4|2895940|1 +6377173]7710309} { 2969649 /1 -2627306| 2080655} 1, 9(7919945)22 
 39}6108363) 55 .5|3891637]1 6370997]7714948] | -2961850|1 26301 40]2082431), 91791 7569/21 
 40|6110666154|3889334|1 -6364828)771 9589] | 29540571 -2632975|2084208} 1 »1q|7915792|20 
 41}6112969| 9° 3887031]! -6358664|7724233] 1-2946270]1 -2635813]2085986| no[ 7914014) 19} 
 4216115270) 95 [9884730]! -6352507|7728878} | -2938488]1 2638655] 2087765]. 4|7912235/18 
 4316117572}, ,{3882428|1 6346355] 77595261 -2930713|1 +264 1496|2089544| |. 0|7910456)17 
 44}6119873155 92880127] +6340210|77381 76] | -2922943|1 264434 112091594] +, 9|7908676)16 
 45}6122173/95 [2877827|1 -6334070|7742827] | 2915 179]1 -26471 88] 2095 104] 179 {790689615 
 46/61 24473) 95 49 |9875527|1 6327937]774748 1] *290742 [1 2650038] 2094885] 1 7g|7905115|14 
 476126772] jq|2873228|I 6321 809}7752137|1*2899669} | -265289012096667| 1 7.|7903339|15 
 4816129071) 95, ,|9870929]1 -6315688]7756795] I 289192911 2655745] 2098450] 11.5.17901550112 
 49}6151369| 5 4,|9868631|1 -6309572|7761455|1 2884 182|1-2658601|2100235 7. ,{7899767|11 
 50}6133666)9594{2866334}1 6303469] 77661 18]1-2876447|1 -2661460]2102017] jg |7897983}10 
 51]6135964,99,q|2864056|1 -6297359|7770789|1 286871 8]1 2664322|2 103802] 79s{7896198 
 5216138260) 959 ¢{2851 7401-6291 261)7775448| |-2860995| 1-267 186]2105587| ng al 7894413 
 53]6140556|5y9¢|2999444 | “6285 1 69]7780 1 1'7]1 2853277] -267005212107373} ng 4|7892627 
 54]614285215 09 2|9857148|1 -6279085]7784788]1 °2845566|1-2672921|2109159] 73-|7890841 
 
 55|6145147 999% 3854853} 1+6273003/7789460]1 +283'7860]1°2675792121 10946 1789] 1889054 
 56|6 147442 2994 3852558} 1°6266929}779413511 +2830] 60|1°2678665121 129734 1789 7887266 
 57|6149736 2993 3850264] 1 *626086 | |77988 1 2}1-2822465]1-268 154112114523 1'789 7885477 
 586152029}, 3 384797 1}1°6254799|7803492}1 +28 14'7'76]1*2684419]2116312 7883688 
 59)6154322) 050 ,|3845678} 1 °6248743/7808 175} 1+2807094]1 -2687299]21 18102 
 
 3843385} 1°6242692/78 1 2856}1 -279941 6] 1+2690182/2119899} © 
 
 ~12293 
 ~ {Cosine|Dif! Vers. | Secant iCotan. Tang. | Cosec. ((Cavers|Dif| Sine 
 
 6016156615 
 Deg. 52. 
 
 slo m2 0 & D101 
 
37 Deg. LOG. SINES, Xc. | (323) 
 
 Cosec. |Verseds.| Tang. |Dit| Cotang. Covers. | Secant |D.| Cosine 
 10°2205370|9°3039829\9°8771144 0628 10°1228856|9°6000849]10°0976514 gsy 9-9023486160 | 
 10°220369419*3043604|9°8773772 9628 10°1226228/9-5998314]10°0977466), 5S 9+9022534|59 
 10-2202019|9°304.7376|9 *8776400 0621 10°1223600/9°5995779}10-0978419),,; 9-9021581155 
 10°2200345/9°3051148 9°8779027!5 pom 10+1220973|9°5993243)10°0979372 O54 9+9020628157 
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 10°2197000|9°3058684|9 8784281 9 ~ (11 0)+121571919°5988168)10°0981281 055 9:9018719155 
 10°1213093}/9°5985629 10°0982256| 5 9*9017764/54 
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 34/9°7851049|1 615 10°2148951{9°3167156|9 886026496) 5 10°1159736|9°5914162 10°1009216)9., 98990784196 
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 372 Deg. 52 
 
 | 
 Ee ay PS ARTE 
 
(324) 38 Deg. NATURAL SINES, &c. . Tab. 10. 
 
 ‘| Sine |Dit|Covers| Cosec. |'Tang. Cotang.| Secant Vers, Dif | Cosine 
 016156615 
 
 99go{2843385]1 -6242692]7812856)1-2799416] 1 -2690182/21 19892] ,,-,.|7880108/60 
 
 116158907] 959 1|384 10931 -6236648/78 175421 2791745] -2693067}2121 684), ,5|7878316|59 
 96161198], 55 ,{3838802)1 -6230609|7822229|1 -2784075] 1 -2695955]2123476} , 4, 5/7876524}58 
 316163489] 94 {383651 1]1 -6224576|78269 19] |-2776419] 1 -2698845]2125268|, -¢.4/7874752)57 
 4161657801595 19834220] | -6218549/783161 1|1-2768765|1 2701737212706] |, -4',/7872939)56 
 516168069|,.5.59|2831931(1 -6212528}7836305]1 :2761116|1 -2704632]2198855],-.,.|7871145|55 
 6}6170359]5 5 49|829641|1 “62065 15}7841 0021 -2753473|1 2707529/2130650} ; -¢.2|7862950)54 
 716172648]. .|3827352| I -6200504|7845700|1-2745835]1-2710429]2139445], ,. .|7867555)53 
 §/6174936} 4 3 5|98250641 | -6194500/7850400]1 -2738204}1 271335 1]2134241], 6 7/7865759|52 
 9]6177224 |, ¢-|3822776|I -6188502|7855 103|1 -2730578] 1 -2716235|2136037}, 0 2)7863965|51 
 10617951 1),5.,,|5820489] | -6182510/7859808|1-2722957|1-27 1914212137835), 4 .|7862165|50 
 11]6181798]595 4|3818209|1 6176524/7864515]1-2715349l1 -2722052}2139633} , 561786036749 
 126184084], 5 -|9815916|1-6170544|7869224| |-2707733]1-2724963)2141431] 9 4/7858569]48 
 
 2985 3813630}1°6164569]7873935} 1 '2'700130}1°2727877}/2143230 1800 7856 770|47 
 
 a 4 381134511 -6158600]7878649} 1 *269253211 °273079412145030 1801 7854970146 
 156190939 9285 8809061/1°6152637/7883364|1-2684940}1 -2733712}2146831 7853169|45 
 16}6193224 3806776} 1°6146680|7888082]1 -2677353}1°2736634. 
 
 2983 2148632 ade 1851368/44 
 17]6195507|59¢9|9804495} | 6140728}7892809]1 -26697'79] 1-2739557|21.50434| ,o05/7849566|43 
 19]6197790|,50 {38022101 -6134783]7897524|1 26621 96|1 -2742484|2159236|1 9) (7847764 /49 
 19}6200073|,,..9799997]1 -6128849}7902248]1 26546261 -2745419|2154059) _- ,,|7845961|41 
 20|6202353)5 06 ,|3797645|1 -6122908}7906975|1 -264706|1-2748343|2155843] v0 4 784415740 
 2116204636), 5°, [3795364)1-6116980\791 17031 -2639503]1-2751276|2157648|  S|7842952|39 
 221620691 7]9¢ [3799085]! -611 1057] 79164341 -2651950]1-275421212159453] ¢2|784054"7|38 
 2516209198). {9790809 “6105 140}7921 167]1 262440911 275715 1|2161259| 10) 017838741137 
 2416211478), -g)9788522| 1 6099228|7925909|1 -261 68601 276009 1|2168065| ,6|7836933|36 
 
 251621375 oon 3786243] 1-6093323}7930640]1 -2609323)1°276303412164873 1804 7835127135 
 2616216036 aoe (8396411 +608'74.2317935379|1-2601792]1:2765980/2166680 1809 7833320|34 
 2716218314 anny 3781686)1°6081528}794.012111°2594267|1°2768928/2168489 1809 7831511|33 
 28|62205921 9 5 4|3779408| 1 6075640)7944865]1 -2586747|1 -2771 878/21 70298), 5 41782970232 
 2916222870 9976 37771301 °6069757|794961 111°2579232| 1-2774831|2172108 1810 7827892131 
 30}6225146 oon 377485411 °606387917954959]1°2571723|1°277778712173918 18] 7826082)30 
 3116227423 9075 3772577| 1605800817959) 1011 °2564219|1°278074412175730 811 7824270129 
 32}6229698)45.-|9770302)1 -6052142)796386211 -2556721]1°2783705|21 77541) 1 4) 1822459)28 
 33|6231974 oH i 3768026|1°604628 11796861 7|1°2549229|1 2786667121793 
 
 3416234248 9074 3765752|1°6040426|/79733'74|1°2541749)1-2789632/2181167 
 35 6236522 on 3716347811 °6034577 
 36|6238796 9945 37612041 1°602873417982895]1°2526784\1°2795570 
 lovin 3758931) 1-°6022896|7987659|1°2519313}1°2798543/2186610 
 agHb 3756658|1°601706417992425}1°2511848}1°2801518 1gin 
 34; 3754386|1°6011237)7997193] 1 -2504388/1°2804495/2190243 1817 7809757\21 
 4016247885 oot 3752115}1°6005416|8001963|1°2496933]1-28074'75|2192060 7807940/20 
 41|6250156 9977) 3749844]1°5999600|8006736|1 -248948411°281045712193877 1819 
 4216252427 9966 3747573] 1°59937901801151111 2482040] 1°2813442|2195696 18 7804304118 
 
 4316254696 98"0 37453041°5987986|801 62881 -2474602)1°2816430)2197515 1890 780248517 
 4416256966 2966 3743034) 1*5982187|8021067/1*2467169}1°2819419)2199335 189 7800665)16 
 4516259235 2968 3740765) 1°59'76394|8025849]1 °24597421 *2822412 °] 7798845)15 
 46|6261505 0266 3738497)}1+59'70606/8030632! 1 *2452320)1°28254.07\2202976 1829 7797024) 14 
 47|626377]1 296", 3736229|1°596482418035418]1°2444903}1 *2828404;2204.798 1899 7795202\13 
 4816266035 996" 37339621 °5959048]8040206}1 *24374.92)1°2831404/2206620 1 ~ -17793380)12 
 
 823 
 4916268505 2266 3731695}1°5953276|8044997/1 -2430086) 1°28344.06|2208443 1824 TT9155 7/11 
 2266 372942911 *594'751118049790}1 °2422685}1-283741 1/2210267], 05 778973310 
 
 181 1(7818833)26 
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 5016270571 1894. 
 51|6272837 2965 37271 63)1°594175118054.584)1 2415290} 1°2840418)/2212091 1825 7787909 
 5216275102 2964 372489811 °5935996|8059382]1 °2407900} 1 °2843428)2213916 1896 7786084. 
 53|6277366 3722634] 1°5950247180641811°2400515]1°284644.0|2215742 7784258 
 
 ¢ ft hm 
 5416279631 ayes 3720369] 1°5924.504|8068985|1°23931 36) 1+28494.5512217569 tear 1782431 
 5516281894]. ¢-|3718106|1 -5918766|8073787]1 -2385'762| | -2852472|221 9396 7780604 
 56}628415776 [3715845 | -5913033|80785 93] -2378393] I -2855499]2221223], 05 
 576286420] 99613713580] 1-5907306|30834011 -2371030}1-2858514|2223051) | ¢4)7776949 
 58|6288682|;,,, 2 1|3711518|t-5901584/8088212]1-2363679| 1 -2861539}2224880) 04017775120 
 59]6290943]5.9¢5|3709057|1 -5895868|8093025]1 +2356319]1-2864566]2226710], 630 
 6()}6293204 370679 €|1+5890157|809784C|1 -2348972|1-286'7596|2228.540 7171460 
 
Deg. . LOG. SINES, &c. (325) 
 
 Sine [Dif] Cosec. |Verseds. Tang. |Dif| Cotane. | Covers. | Secant |Dif} Cosine 
 
 10°2106580}9-3263138}9-8928098} ,.., ,]10°107190219-584'7139]10-1034679 987 9°8965321160 
 7895036 1616 10°2104964]9-3266806|9-8930702 2604 10°1069298}9°5844549]10°1035666 988 9°8964334159 
 ,|10°210334819-+32704'73/9 8933306 5 10°1066694)9°584195'7}10°1036654. 98 9°8963346]58 
 .|10°210173419°327413719-8935909 2602 10°1064091]9°5839364]10°1037642 98 9*8962358}57 
 10°2100120)9°3277800|9-8938511 9603 10°1061489]9°5836771]10°1038631 99 9°8961369]36 
 10°2098507}9°3281461/9°8941114 10°1058886]9°5834176]10°1039621 9°8960379/55 
 
 10°2096896]9 °328512119-8943715 9602 10°1056285}9°5831581]10°1040611 od 9°8959386]54. 
 
 10°209528519:3288778|9°8946317 2601 10°1053683}]9'5828985]10-1041602 992 9°8958598]53 
 10°2093675]/9°3292434)9 8948918 260 10°1051082/9°5826387|10°1042594 999 9°8957406152 
 10°2092067}9-3296089|9°8951519]> 10°1048481]9°5825789]10-1043586 999 9°8956414151 
 10°20904.59]9 :3299'74.1/9°89541 19] 10°1045881}9°5821190/10°1044578 993 
 10°2088852|9*3303392/9-8956719 10°104328119°5818589}10-1045571 994, 
 10°2087246]9-330704119°8959319 2599 10°1040681/9°5815988}10:1046565 9°8953435/48 
 
 995 
 10°208564119°3310688]9°8961918}- g|9-1038082)9°5813386]10-1047560 995 9°895244014.7 
 3 10°2084.037/9-3314334)9°8964517 92599 10°1035483]9°5810783}10-1048555 995 9°8951445 46 
 7917566 1602 10°208243419-331797819°8967116 Q59¢ 10°1032884]9°5808179]10-1049550 00% 9°89504.50145 
 
 10°2080832/9-3321620/9°8969714. 959 10°1030286]9°5805574110-1050547| O5 -|9°8949453144, 
 10°20'7925119°3325261|9°897231¥ 2598 10°1027688}9°5802968)10°1051543 9°8948457143 
 
 10-2077631}9-3328900]9:897491 0153.4 -|10°1025090}9 580036 1/10-1052541| 940)9-8947459|42 
 10-2076032}9°3352537|9°8977507| 95 9n|10°1022499]9°5797753|10-1053539| ge (9°8946461|41 
 10-2074454}9-3336179|9-8980 104} 2110-101 9896|9'5795144|10- 1054557] ,99819-8945463|40 
 10+2072837}9°3339806|9-8982700152.5¢|10°1017300}9°5792534|10-1055537) 00|9-8044.463159 
 
 160 
 
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 10+2071240/9°3343438]9 8985296 10°1014'704]9°5789923}10-1056536), 7.7" |9°8943464138 
 10°206964.519+334'7068|9°8987892 595 10°1012108}]9°5787311|10'1057537 10 (2 8942463 37 
 4{10°2068051)9-3350697)9°8990487 9595|10°1009513]9°5784698)10-1058538 100 ]|2 994146236 
 
 10°2066457/9-3354323/98993089| 9. 10°1006918}9°5782085/10-1059539), ogi? '8940461)35 
 10°2064865|9°335794.9/9°8995677]- 10°1004323}9°57'79470110-1060542)5 9 49|9°8959458|54 
 10-2063273}9-33615729-899897 1|;,>4 10°1001729)9°577 6854 10-1061544}  494)9°8938456133 
 10°2061683/9°3365194/9 90008651959 ,|10-0999135|9°5774237|10- 1062548), /9°8937452132 
 10:2060093/9-3368814|9-9003459| 5 510°0996541/9°5771620)10°1063552I ; yy 1198936448131 
 7941496), 77|L0°205850419°337245219-9006052)5-54110-0993948]9°5769001 101064556 0035/2 899944450 
 7943083}, 5 ..|10-2056917/9°3376049|9 9008645), gq} 10°0991355}9°9766382}10-10655611; yp @/98934459|29 
 87110-205533019-33'79664)9-901 123" | 10°0988763|9°576376 11101066567), ,-,.19'8933433}28 
 10°2053744|9-3383278)9-901383015> 4910-09861 '70/9'5761139|10-1067574 {1007 9-s932496107 
 10°2052159|9-3386889|9-9016429) 53.0 7 10:0983578)9'5758517]10-1068581)  7'9°8931419)26 
 10°2050575}9*3390499/9°9019013|9 59 1|10-0980987|9°5755893|10-1069588), ,/, {19°8980412125 
 10°2048992/9°3594107/9°9021 6041954 j|10-0978596|9'5753269|10-1070596|  y4q/9°8929404/24 
 10:2047410}9°3397714)9-9024195!9 54 1|10-0975805|9'5750643}10-1071 605}, 9 9)9°8928395|23 
 10°2045829/9-3401319|9-9026786) 95 4|10-09732149'5748017]10-1079615 O10 99.2 1385}22 
 10+2044249]9-34.04922/9-9029376|5° ~110-097062419'5745390]10+1073625 19°8926375)21 
 10-2042670}9-340852419-9031966|,» 4.4] 10-096803419°5742761110+1074635 98925365120 
 10°2041091}9°3412124)9°9034555j9;; go} 10-096544.5]9°5740152|10-1075646) , 910°8924554119 
 6|10°20395 1419 °5415722/9-9037144) 9 5 ¢0}10-0962856|9'5737502/10-1076658} , 9°8923342118 
 10°2037938)9°341931919°9039733} 5» o.}10°0960267/9'5734870)10-1077671 32892232917 
 10°2036362)9°34229} 3)9 9042321155 ¢.4]10:0957679|9°5732238/10+1078684 9°8921316}16 
 10°2034788|9-3426507|9-9044910|5 7|19-0955090/9°5729605}10-1079697|, 7 {09920303115 
 10°2035214)9-3430098)9 904749755 4 4|10-0952503|9°5726970)10-108071 1], ,:/9°8919289114 
 10-2031 641}9-343368819-9050085/95<7110-0949915]9°5724535/10-1081726}, 9, 219°8918274115| 
 10°2030070)9+3437276}9 9052672! , ay 10°094'7328]9'5721699}10-1082742 L1G)? 22 t 7258)12 
 10°2028499)9 344086319 :9055259| oxo -110:094474119°5719062]10:1083758 oye 2916242] 11 
 10°2026929}9°3444448)9-9057843} 950 -|10-0942155]9°5716423]10- 1084774 O18. 09 19226410 
 10°2025360/9°3448031}9-906043 1195.46 10°0939569}9°5713784/10-1085792), 9°8914208] 9 
 10°2023792/9-3451612)9°9063017|5 5 -{10-0936983]9'571 1 144110-1086809 g{? 8913191 
 10°2022225/9-3455199}9-9065603},)5 45110-0924397|9°5708503|10-1087828| 01919-8912 172 
 10°2020659)9+345877049 9068188}, 54; 110-0931812/9°5705861110- 108884" 9-8911153 
 
 258 1020 
 10°201909419+346234719 9070773 2584. 10:0929227/9°5'703218110:1089867 1020 9°8910133 
 10°2017530)9°346592219 9073357 2594|10°0926643 9*5700573}10:1090887 1021 9°8909113 
 10°2015966)9°3469495}9:9075941 9534110°0924059 9°5697928]10-1091908 9°8908092 
 
 10-2014404|9-3473067|9-9078523|5>0',|10-0921475]9-5695282|10-1099999|!92119.go07071 
 
 7949495| 1984 
 
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 988716 10201 1282919-348020519-9083699 10-0916308]9:5689987|10-1094974]/9219-s905096 
 
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 Deg. 51. 
 
(326) 39 Deg. NATURAL SINES, &c. Tab. 10. 
 
 Sine {Dif |Covers| Cosec. |Tang.|Cotang.| Secant 
 360679611-5890157/809784.0]1 23489791 -286759612228540 
 
 2258 
 2257 
 2257 
 
 1834, 
 1835] ) 00464154 
 
 816311279) 517756794159 
 916313528 Baie vi “754957151 
 10l6315784/5020 1eg0|7753121 50 
 1116318039 7751283\49 
 
 22.54 
 
 1838 
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 1839}. 
 
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 ipiglT74208644 
 842) 7740044143 
 
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 1416324800|223|367 
 
 9951 (200994 1)1 5782680/8 1897641 °2210364)1-2925642/2263441 
 20)6338310 9(2061690)1 -57'77077|8 194625) -2203121)1*2928723 
 2949) Ooo 441) 1°5771479/8 199488] 1 °2195883)1° 
 *|3657192'1 -5'765887|8204354) 1 °2188650] 1 *2934892|22689 
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 1°5'754718/8214093]1°2174199}1> 
 
 224 
 is i 1-574914.1/8218965|1°2166989|1° 
 91 Gio048200 1 -57435710|8223840}1 21 59769] | -2947260 
 : nipi86459541 -5738004/82287 1 8) 1-2152569|1° 
 994 510043708) 1 -5732443)8233597]1°2145359) 1 
 994 5(0041463 15726887 |8258479]1 21381 62] 
 394 419099218 1 -5721337/824336411 °2150970]1 :2959670/2283754 
 99440096974 I -57157921824825 1]1 2123789} 2962779 |2285605 
 2945/0024 7130 1 571025218255140}1-2116601|1 2965890|2287456 
 9194,9(0092487]1°570471 7}8258031}12109424{1+2969004|2289308 
 56 | 4 9(200244 I 5699185/826292]| -2102259|1-2972191}2291 160 
 994910028002|1 *5693664|826782 I|1*2095085|1+2975240}229301 4 
 3625760)1-5688145|8272719|1 °2087924| 1+2978362|2294868} , o> 
 
 994 0(262919)1 568263 | [8277620]! 2080767|1 +298 148722967221, .17703278)93 
 D402 1 279|1 56771 23/8262523|1 207361 51 +2984614/2298577), .--/7701423)09 
 9940 | 2019029]! $671 619]8287429}1 2066468) 1+2987743}2300453}, 6317 
 9939(26167991 | -5666121/8292337|! -2059397|1 -2990876]2302290|, o>, 
 9.999(2614560}1 -5660628|8297247}1 '2052190]1+299401 12304147}, 62-17 
 
 [001232215655 141|8302160]] °2045058|1°2997148/2306004], °F 4 
 
 293 
 ate 361008411 -5649658|8307075]1°203793911 -3000288|2307863}, « 5 4 
 44|6392153),,° 15607847|1 -5644181|8311999|1-20308 101 +300343 12309729), °° 
 99362002610) -5638'708]83 169 1 9}1-2023693}1 +3006576)2311582), 4 
 993 6(0003974| t-5633241/8321 834] “201658 1]1°3009724/2313449), ¢ 2 
 
 “213601 138]1 -5627779|8326759| 1 -2009475|1°3012875|2315303 
 
 2235135989031 1-562232218331 6861 -2002373]1°301 602812317165 
 
 223 
 
 Wien 3596668} 1 -5616871{83366 1 5}1 1995276|1 +301 918412319027 
 5(}6405566) 5 9,,[9994434{1 561 1424183415471 1988 1 84/1 +3022343]2520890 
 5116407799} 9 54.71(959220 1/1 -5605982/8346481|1 +198 109741 -3025504|2322754 
 5216410032)-** i 3589968} 1+5600546|835 1418] 1°1974015}1°3028667|2524618 
 5346412264), 93199877361 -5595 115|8356357|1 °1966938}1+3031834}2326483 
 54|6414496), 55 35855041 -5589689/826 1298} | -1959866) | *5035003)/2328348 
 
 232 
 55/6416728]5 9. q[398327211 +5584268)8366 242] 1 +1952799}1°3038175]2350215 
 56|6418958 358104211 °5578852/8371 188}1°1945736)1°3041349|2332082 
 
 5716421189 vase 357881 1]1-5573441|8376136|1+1938679|1*3044526|2333949 
 
 59} 3425647], [3574355] -5562634|8386041]1 192457911 -505088812537686 
 5427876] "135721 2441 -5557238|8390996|1-1917536|1 *3054073)2559556 
 
 1849 
 1849 
 1850 
 1851 
 
 1851 
 1852 
 
 29|6358537 
 
 4816401097 
 4916403332 
 
Deg. LOG. SINES, &c. 
 
 Dif} Cosec. 
 10-2011282\9-348020519°9083692 
 1988718] |, go|10-2011289|9°348 
 
 1993394] 3 a7! 
 7994951] 13> 
 
 155411 9.999038419'3505137/9°9101766 
 1553}, ¢.199833119-350869219°9104347 
 
 1551 
 8005823 1549 
 
 8007372 
 
 1543)19-1981808|9°354'7691|9-9152714 
 154311 ¢-1980265|9°3551227|9°9135291 
 
 15371) 9-1972569|9-3568880|9°9148171 
 15371 9-197103219+3572406|9°9 150747 
 1536|19-196949619°35'75930)9°9153322 
 153411 9.1967962|9-3579453]9°9155896 
 8033579] /2 2110-1 966428|9°358297419° 9158471 
 1533}19-1964895|9°358649419°9161045 
 
 153211 9-1963363|9°3590011|9-9163618 
 1531]19-196183219-°3593528|9°9166192 
 
 "8027431 
 
 2575 
 
 QO. 
 
 80429757 
 8044284 ow A 
 “804 110°1954189)9°361108419°9179055 
 tars iy 10-1952664|9-36 1459 1}9°-9181627 
 1-8048861|1>90|10°1951139]9-3618096/99184198 
 38050385 1593 10°1949615|9°3621599 9°91 86769 
 }-8051908 1599 10°1948092}9-3625101/9°9189340 
 3°8053430 10°194657019°362860119°9191911 
 
 4951 |2 2410: 9-9194481 
 38054951 10+194.504919°3632100 
 3-8056479| 12> 1110°194352819-3635597/9°9 19705 | 
 9+8057991|1 2 19110-1942009|9-3639099|9°9199621 
 
 15191} 9.1 9404.9019-3642586/9°9202191 
 
 2569 
 
 1516119-1935940|9-3653058|9'9209898 
 1515}10-1934425|9-365654619°9212466 
 15141) 9.193291 1/9-3660032|9°9215034 
 1513!) 9.1931398|9+3663516|9°9217602 
 15121) 9-1929886|9:3666999|9°92201'70 
 151211 9.19283'74|9-3670480/9°9222737 
 
 151011 9-192686419-3673959|9°9225304 
 151011 9.192535419-3677437/9°9227871 
 150811 9-192384619-368091419-9230437 
 15081) 9-199933819-3684389|9-9233004 
 
 PAs he 
 
 9°8064060 
 9°8065575 
 '9°8067089 
 9°8068602 
 9°8070114 
 '9°8071626 
 
 9-8073136 
 9-8074646 
 9°8076154 
 98077662 
 98079169 
 
 2567 
 
 2566 
 2567 
 2566 
 
 25811) 9-9903397 
 
 258111 9-0895653 
 
 10-0867286 | 
 25771 19-0864.709|9°5636316|10-1115556 
 \2577 10°0862132)9°5634147|10°1116592 
 257611 ()-085955619°5631477 
 257611 (0-0856980 
 
 2574) 10-0844104 
 10*0841529 
 
 D574 1 
 
 257211 9-0893517 
 
 Dif| Cosine 
 
 (327) 
 
 ———— | | eee ft 
 
 10-091630819-5689987|10-1094974 
 2583} 0-0913725|9-5687338|10°1095997|, 9 |9°8904003}59 
 25831 19-(9911142|9°5684688)10°1097021 
 2582) 19.¢990856019:5682037|10°1098046 
 25821 19-090597819:5679385|10-1099071 
 } 9-5676732|10°110009% 
 155511 ¢.999193819-3501580/9°9099185]-2>-|10-0900815|9°5674078| 101101123 
 
 10-0898234)9°5671423]10-1102150}, og 
 
 9-5668766|10°11031 7811 998 
 2580) 10-0893073|9-5666109]10°1104206} Jog 
 25801 10-0890493]9°5663451]10°1105235], goq 
 2580} 10-0887913|9°5660792|10°1106264|, 430 
 2579) 1 0-0585334|9°5658132|10° 1107294 
 2579) 10-0882'755/9°5655471|10°1108325] 1 93 
 
 2809}10°1109356}) 939 
 9°5650146|10°1110388 
 
 10°1117628} 3% 
 2576110 -085440419°5626134|10°1119702 
 25751 10-0851829|9°5623461|10°1120740] 39 
 257 10°084.9253/9°562078 7101121779 1039 
 257511 0-0846678/9°5618112\10-1122818]) 40 
 
 9°5615436|10°1123858|1 946 
 9+5612759|10-1124898| 1 944 
 25714 10-0838955|9'°5610080}10°1125939 
 
 3110-0836389|9°5607401]10"1 126981} 194¢ 
 0-0833808/9"5604721|10 1128025}; 94.5 
 3110-0831235|9°5602040|10-1129066 944 
 25751 10-0828662|9'5599358|10°11301101 o44 
 2573) 10-0826089|9°3596675|10°11311541) 945 
 9-5593991|10"1132199 
 25721 1 9)-082094519°5591305}10°11332441) yy. 
 10-0818373|9°5588619}10°1134220|; g 45 
 10-081 5802/9°5585932|10°1135337 104. 
 10-0813231|9°5583244|10°1 13638411 94.6 
 25711) 9-9810660)9'5530553|10°1137432 
 257117 9-0808089]9°5577864 
 257017 9-0805519|9°5575173|10-1139530| 1959 
 257011 0-0802949|9°557 
 297011 0-080037919°5569787 : 
 257011 0-0797809|9°5367093|101142681| 950 
 25591 19-079524019-5564398110°1143733 
 10-0792671}9°3561701|10°1144785} 
 2569} | 9.0790102|9-5559004|101145838} 9 
 2568} 1 0-0787534|9°5556306|10°11468911; 9 
 256811 0.0784966|9°5553606|10°11479451  ¢ 
 2568] | 9-0782398|9-5550906|10°1149000|} 
 25681 10-0779830]9'5548204|10°1150055|; 956 
 10-0777263]9-5545502)10° 1151111) 
 2567] 1 9.07°74696|9°5542798|10°1152168] 1 )21/9°8847832 
 256711 9 -0771212919-55400941 10° 1153225}, 95 
 10-076956319-5537388|10-1154283} 1-5 4 
 
 10°1138481 
 
 10°1141630| 1954 
 
 1023 
 
 g 
 9 
 9 
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 9 
 9 
 9 
 
 1025) 
 1025 
 1026 
 1026 
 
 1027 
 
 1031 
 
 1036 
 
 1038 
 
 1042 
 
 1045 
 
 1049 
 
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 1052 
 
 Jo 
 
 10°0766996]9°5534681|10°1155341), o¢g 
 
 10:0764430)9°5531974 
 10°0'761865|9°5529265 
 
 Tang. Verseds. 
 
 10°1156401 
 10°1157460 
 
 Cosec. {Dif 
 
 1059 
 
 9°8879260135 
 9*8878221|34 
 9°8877182 
 9°8876142)32 
 9°8875102)31 
 9°8874.061|30 
 9°8873019}29 
 9°887197'7}28. 
 9°8870934)27 
 9°8869890/26 
 9°8868846|25 
 9°8867801|24 
 9*8866756)23 
 9°8865710)22 
 
 9°8863616'20 
 
 9°88604.70 
 
 3/9 8854162 11 
 7,\9°8853 109] 10 
 
 if \9°$848889 
 
 9°8905026)60 
 
 "8902979158 
 "8901954157 
 "8900929156 
 "8899903155 
 *8898877154 
 *8897850)53 
 *8896822|52 
 "889579415 1 
 
 33 
 
 9°8864.663|21 
 
 9°8862568)19 
 9°8861519)18 
 17 
 9-88594.20/16 
 9°8858370)15 
 9°8857319\14 
 9°8856267|13 
 9°8855215)12 
 
 9°8852055 
 9*8851000 
 9°8849945 
 
 98846775 
 '9°8845717 
 '9°884.4659 
 98843599 
 98842540 
 
 Sine 
 
 Deg. 50. 
 
 slonwure D100 
 
(328) 40 Deg. NATURAL SINES, &C. 
 “1 Sine |Dif|Covers 
 0154278 76],, 554/9572124|1-5557238]3390996| -1917536|1°3054073 
 11643010455 5°13569896|1"5551848]8395955]1°1910498]1°3057261|2 
 399410967665] 5546462]84009 1.5]1°1903465]1 306045 1 
 5941365441] +5541081]8405878]1 1896437] 1 306364412345 168 
 416436785] 5 499632 195]1-5535706(3410844]1"188941 4]1-3066839|2347040 
 5}5439011]55° -|3560989]1-5530335/8415819}11882395|1-3070038|2348913 
 
 6]6 4411236) 5595/3558764| | -5524970/8420789] |-1875382|1°3073239 
 
 1644346 | 3556539} 1 -551961018425755|1°1868375 
 
 $]6445685),,.5 ,|8554315]1-5514254|$430730]1 -1861369]1°3079649 
 9]6447909}), 5941955209 1]1 55089048435'705]1-1854370]1 3082858 5 
 10/6450132}505,5|3549868]1 -5503558|8440688| 1 1847376]! 3086069]2358286] ; ._ °|7641714)50 
 11/6452355|)959/3547645]1 54982 18)844.5670| | -1340387]1 30892842360 162] ; ¢, 017639838/49 
 [354542311 -5499889|8450655|1°1833402|1-3092501|2362040} ; 6 .|7637960/48 
 
 Cc tf 22 
 1216454577 999 1878 
 13|6456'798 390) 354320211 °548'7552|8455643] 1°18264.2911°309572012363918 
 1416459019}, 55 ,|3540981}1 -5482226|3460633|1-1819447 1*3098943]2365796 
 15|6461240),55 ‘5 3538760|1°5476906|8465625}1°181247'7|1°3102168|2367675 
 16|6463460], 6 | [3936540]! -5471590|8470620|1 -1805512 1°3105396|2369555 188] 
 17|/6465679|0~ | ~13534321]1-5466280|84'75617|1°1798551|1°3108626|2371436 188] 7628564/43 
 353210211 °5460974/8480617]1°1791595}1°3111860 7626683)\42 
 
 18|6467898 
 
 352988411 *5455673)84856 1 9]1 +1'784644)1°3115095|2375198 
 311 °5450378184906241°1777698]1°3118334 
 13525449]1+544.5087|8495631|1°1770756|1°3121575|2378964 
 901" 3523233]1-5439801|8500640}1*1763820|1°3124820)2380848 
 991 5}2221016]1 *5434520/8505653}1+1756388]1°3128066}238273 
 507 = {2 18801]1 5429244185 10667) 1°1749960]1°3131316]2 
 
 2352660 1815 764734053 
 
 92° , 
 
 pat 1875 7645465}52 
 356410 1876 7643590)/51 
 
 2914 35143721 °5418706|8520704| 1-1736 1201 °3137823/2388389 
 991 4f2? 12158]1 5413445/8525726|1-1729207 1°3141081 
 291 0f2909944|1 5408 189/8530750}1 °1722298}1°3 144341 
 091 Q/2207732]15402937|8535777} 1 °1715395|1°9147604 
 
 5 |3505520|1 -5397690|8540807|1-1708496|13150870|2395940}, 640 
 
 2912 
 
 991 1|2903308|1 -539244.9/85458391I -1701601]1°3154139]2397830}, gq 
 
 9211 (2201 097|1 53372 12)8550873}1 169471 211°3157410 
 12498886] 1 -5381980)/8555910)1 -1687827|1°3160684 
 
 sone{3496676|1-5376759|8560950|1 1680947|1°3163961 
 
 2209 
 2209 
 220% 
 2208 
 
 26|6485628 
 | 276487842 
 28)6490056 
 29}64.92268 
 30)6494480 
 31}6496692 
 32}6498903 
 
 2399163 mts 7607837|32 
 2394051]; ¢36|7605949/31 
 
 7604060) 30 
 
 7594606|25 
 3492258|1-5366313/8571037|1°1667200]1°3170523/2407287]; g44]7592713)24 
 
 3490049|1-5361100]85760841 -1660334]1°3173808}24091801 ; g9,)7990820/23 
 3487842] 1-5355892/8581133]1+1653479|1°3177 189 5|/298926|22} 
 22062 #89634]1 5350689}8586 1851-16466 15|1:3180386/241 2969}; g9;)7587031 es 
 226 ( 2493428]! 534549 1|859 1240]1-1639763]1 3183680)241 4864}, go 4|7585136/20 
 200 G{ 248122211 5340297]8596297] 1 16329 16|1*3186976|2416 760} go7|7583240]19 
 
 3479016}1°5335109|8601357|1-1626073)1°3190274/2418657]) go,/79815943/18 
 
 220) ; 
 90 mi 94768111 -5329925|/3606419)1+1619234]1°3193576)2420554] 1 399 1579446 17} 
 290; [474606]1 5524746]861 1484] 1-1612400]1°31968811|2492452}, 55417577548)16} 
 nog [24724021 -53 19572)8616551| 1160557 1|1°3200188|2424350], 3 49/7575650)15 
 Joya{3470199]1 -5314403}8621.621| 1598747]! -3203498|2426249|, 5)757375 1/14 
 3900(2467996|1 5509238|8626694| -1591927|1-3206810]24281491 49/7 
 
 34657941 -530407818631768}1°1535119|1°3210126 1901] 209991)12 
 
 2209 
 3463592|1-5298923/8636846|1 +1 578301|1-3213444]243 1950] go 756805011 
 3461391|1-5293773|864 1926] I -1571495|1°3216765]2433852| | 45|7566148}10 
 3459 190}1°5288627|8647009}1°1564693]1°3220089]2435754 1903 7564246 
 319913455990} I -5283487|8652094| I +1557896|1 "322341 6]243 7657} 99 [7562343 
 5316545209}5 651345479 |1-527835 1/8657181]1-1551104|1-3226745]2439561 4,)4|7560439 
 5416547408]. | 5[2452599]1 5275.21 918662279] 1-15443 1 6|1-3230078]244 1465); 9,,5|7558535]- 
 5516549607 -5268093|8667365]1-1537532|1-3233413}2443370} ; 4, 4(7556630 
 56}6551804),, ga[3448 1 96|1-5262971/8672460]1 -1530754|1 3236750)2445276| 4 ¢[1954724 
 57|6554002)5 5 [3445998] 5257854867558] [-1523979|1-3240091)2447182| 59017552818 
 on (2449802]1 525274 118682659] 1-1517210]1-3243435|2449089] | 554/755091 1 
 
 ¢ 3441605|1-5247634]3687769| 1 -1510445|1-3246781]2450996], 4ngi7549004 
 6016560590] > }3439410]1°524253 1186928671 1-1503634|1-3250130]2452904| 17547096 
 
 Cosine] Dif | Vers. | Secant }Cotan. Tang. | Cosec. 
 
 35|6505533 
 36/6507742 
 37/6509951 
 386512158 
 39|6514366 
 40|6516572 
 41|6518778 
 4216520984 
 
 4316523189 
 4d]6525394 
 45/6527598 
 4616529801 
 4716532004 
 4816534206 
 4916536408}, 
 5016538609 
 
 Xe) 
 
 oP Hs EG ee D106 
 
 Covers Dif| Sine 
 
 Deg. 49. 
 
40 Der. LOG. SINES, &¢. | 1 eA OD 
 Verseds. Tang. Dif 
 
 10°191932519+369133419-9938135 2566 
 10°1917820}9°3694804]9-9240701 25651007 Sand sasitecs 
 10°1916316]9-3698272|9 994.3966 2565 10°075673419 5523845 g 
 10°1914812)/9-370173919-9245831 2565 
 10°1913310]9°3705205]9-9248396 2564 
 10°1911808]9-370866919:9250960|~' 
 
 10°1910308/9-371213119-9253594. pret 
 10°1908808]9-3715599|9-9256088|. - 
 
 10*1907309]9-3719051/9-9958659|-2 0+ 
 10°1905811]9-3722508|9-9961915 Dna 
 10-1904314|9-372596519-9263"749 pian 
 10°1902818]9+372941919-9266341 
 
 10190132919-373287919-9268004, ite 
 
 Cotang. | Covers.] Secant [Dit] Cosine 
 
 10:0743912)9-5510276|10-1164896 
 10-0741348]9-5507559110°1165961 
 10-0738785/9-5504841]10-1167026 
 10-0736222|9-5502129110-1168099 
 10-0733659|9-54994.09]10-1169159 
 10°0731096|9-5496681|10-1170996 
 Blas go|10°0728534/9+5493959110-1171294 
 1898354}9-37397319-9974098|° oe 10-0725972|9-5491936110-1172360 
 10°1896841]9+3743221|9-99765909 10:0723410/9°5488511{10-1173439 
 10-072084819-5485786|10°117450] 
 10-0718287/9-5483060|10°1175579 
 ; 10-0715726/9-5480333|10°1176643 
 
 1499 
 1498 
 1497 
 
 1496 
 {1496 
 1494 
 
 1070 
 1069 
 1071 
 107] 
 
 10°1890879]9-3756999 9°9286835 2561 
 10°1889391]9-3760440 9°9289396 2560 
 10°188790419-3763879 9929195695 65 
 10:1886417}9-3767316 9°9294.516\~ 
 
 1488 
 a\1487 
 1487 
 1486 
 1485 
 14.84 
 
 1483 
 1482 
 1481 
 {29/9°8125965|/481 
 ‘}3019-819544 
 8125444. 1479 
 | 1478 
 3 14°77 
 3 . A 
 {33/9-8120878/0477 
 1475 
 14°74 
 1474 
 
 13819-8137950| 1473 
 
 10-070804419-347214.5]10-1179860 ieee 
 
 10-0705484195469413/10°1180939 ian 
 10°1884931}9°3770759/9-9297076|2° rn ites Pence ee part 1074 
 10°188344619-°3774186|9-9299636|-2°°110-0700364/9-5463947110-118308: 1076 
 10-1881962/9-3777619|9-9302195 seh 10-0697805 Hag tae fees 1076l9. 
 s , On * Ve 0:0 4 5 477 b 34 ~ 
 10-187899710-370de elo ae 2559100095245 9-5455740|10-118631 1/1077 
 10°1877516|9-3787908/9-9309879/-95!10-06901 28/9-5453002|10-1187388 1078 
 10+187603519°379133319-9310431 bya 10°0687569/9°5450264|10-1188466 1079 
 10°1874556/9°379476019-9514 98915 ?98|10-0685011/9-5447524)10°1189545 1079 
 10-187307719-379818419-9317547 gave 
 10-187159919-3801606|9-9320105(2 
 10-187012219-380502619-9590660 evs 
 10-1868646/9+380844.5/9-9325200\° he 
 10-1867171|9-3811863}9-9397777 2557{t0" Seated HAE 
 10°1865697/9°3815279 9:9330334 2556 00669666 
 10*1864293/9-381869319-9339890 Dis 
 10°1862750/9-3822106|9-9335446 oni 
 10-1861279|9-38955 1" 9-9538003195 94 
 10-1859808/9-382899719-9340559 ms 
 10°1858338]9-383233519-9343114 9% ‘A 
 10-18356869|9-3835'74,219-934.5670|2* 
 
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 10°0677338/9°5439298/10:1192785 
 10°0674780/9°5436554|10°1193866 
 
 10°066455419°54.25568]10°1198197 
 10°0661997/9'5422818/10-119998] 
 
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 Secant 
 
 Covers Cotang. Dif Tang. Verseds. _Cosec. if} Si 
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 10°0608369]9-5364839]10°122220 | 
 
Tang.|Cotang.|Secant.} Vers. |Dif [Cosine 
 343941615 242531186928671 11503684 1°3250130|2452904 
 
 016560590 
 116562785 
 216564980 
 316567174 
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 916580326]5 5, [34196741 -51968 1 5|8738935] I 144304 1|1-328039912470106| 1914 
 1016582516), 99341748411 -5191759)8744067] 1436526] | 3283776|24720201 9 15 
 11/6584706),5 9 4[3415294]1-5186708/8749201]1-142961 51 -32871 56|2473925] 942 
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 319, (2386881)! "5121459188 16186|1"1342773| 1-3331359|2498889 
 
 2195 
 
 219513 130633|1 5222! 66|8713316|1-1476687| | 326355412460543 1911 
 
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 2489279 1991 
 
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 2916621842 1997 74.9341 1/32 
 2916624022 1997 7491484131 
 } 9179 1999|7489557|30 
 3116628379 2178 3371621 |1+3086645)885244 0} k-2296321|1+3355362|2512371, 1998 7487629129 
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 '33|6632734)5, 6 3367266|1°5076739|8862829) 1 *1 28308811 °3362246125 162298 1930 7483772127} 
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 - (Cosine|Dif] Vers. Ree 
 
 Secant |Cotan. Tang. “Cosec. [Covers|Dif | Sine 
 
P20 
 
 41 Deg. LOG. sINES, &c. (331) 
 
 ‘| Sine {Dif} Cosec. 
 019°8 169429 
 119°8170882 
 219°8172334 
 
 Dit | Cosine 
 
 Verseds.| Tang. |Dif| Cotang. | Covers. 
 
 10°1830571/9°3896806/9:9591631)5.., 10°0608369|9-5364829|10°1222201}, yu 98777799160 
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 12!9'8186807 (443 10°1813193|9°3937245]9 9422233 |9 5 7. 
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 10°1805988|9°3954035 119 9454976]55 4 o)10-0565024 9°5317559|10°12409641 1 199 9°8759036/43 | 
 10°1 S0s A 780 1h 9:9437524 9 548 10-0562476|9°5314768}10-1242073|1 144 98757927142 
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 9+5286799|10°1253205), ‘ 9°8746795 92 
 29)9-S211 217) 499 3|10-0534461/9°5283997]10° 1254321] |; g|8"S745679/31 
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 10°1785927,9'4000837|9°9470630}|q 5 4 -|10°0529370 9°5278388)10°1256557111 1 9°8743445)29 
 10-1784500'9°40041 69/9-9473175 195s 1 5 |10°0526825,9"9275582 10°1257675} 1 po9|2 8742325 )/28 
 10°1783074.9°4007500]9°9475720|o > ;'»|10°052428019°5272774 {101258795 1) 1 99/9 8741205)27 
 10-1781649,9°4010829]9-9478265 15», »(10°0521735 95269266 |10-1259915} 4 94/9'8740085)26 
 10-1780225,9°4014157|9*94808 10)o» 1 3|10°0519190)9°5267157]10°1261035 4 193)9°8798965)25 
 10°1778802 9°4017484|9°9483355|o5 44 |10°0516645)9°5264346)10°1262156)) j 09 dos ay ata 
 10°1777379 94020809 9-94858991,, ~, ,|10°0514101/9'5261535]10°1263278) | | 94/9°8736722,28 
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 40/9-8226883} jj ql !0°1775117,9°4050775|9 949353 54 4/10 0506469}9°5255094)10-1266648)) 495 98709352)20 
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 45)9°82335971|1 45» 10-176602919-4047356]9-9506248|5» 1 |10°0493752)9°5239003/10°1272278 eB aot ie 15 
 146|9-8235386), | >|10°1764614.9°4050668)9-9508791|5> 5 .|10°0491209}9°5236182/10°1275406)1 ei" 372659414 
 147/9°8236800 10°1763200 9°4053978}9-951133-4),2 ;',)10°0488666)9°5235359]10°1274534), yoqi 
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 10°1758963}9°4063901]9-9518961|g> 5 5|10°0481039|9-5224885|10-1277924) | 1.5 /9°6720 
 
 10°1757552/9°4067206]9-9521503|5> 1 5|10-0478497/9 5222058) 10°1279055} | ,9]9 872094 
 
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 Foe LO0"1754733,9-4073811/9-9526587]5 555 )10°0473419,9-5216401)10°1281319)) 755, 98718681 
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 ? 10°1744891|9-4096883)9°9544374|~ 110:0455626)9°5196566 10+1289265 Ret 9BT1OT25 
 Cosine |Dif| Secant | Covers. |Cotang.|Dif| Tang. |Verseds.} Cosec. Haat pias. 2 
 ONTO AA LLL SE IE “ 
 Se Deg. 45. 
 
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 | 
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(3382) 42 Deg. NATURAL SINES, Xc. ‘Tab. 10. 
 
 ‘| Sine |Dit!Covers| Cosec. Lang. |Cotang.| Secant Se 
 ~ 016691306 3308694 1°494476519004.040] 1°1106125|1°34563271256 
 
 1/6693468 fe 3306532|1°493994.019009309} L*1099630}1°3459853)2! 
 216695628 2161 3304372) 1°4935118}9014580} 1:1093140)1+3463382 
 3}6697789 2159 330221 1]1°4930301}9019854)1°1086653 1°3466914 : 
 416699948 2160 3500052}1°4925488/9025131/1°1080171 1+34'70449 257634211 95) 
 516702108 2158 32978921 1-4920680|90304 1 1}1°1073693}1 3473987 /2578292 1950 
 
 6|6704.266 9158 3295734 1°4915876|9035693}1 -1067219}1°3477528 
 16706424 2158 3293576] 1 *4911076)90409'79) 11060750} 1°3481072 
 
 8|6'708582 
 9}6710739 
 
 3291418}1°4906280}9046267]1 °1054284)1 *3484619)2! 
 328926 111°4901489}9051557]1°104'7823] 1 °3488168 
 10]/6712895 2156 3287105|1°4896703/9056851)1°1041365]1°3491721 
 1116715051 2155 3284949] 1 °4891920)9062147|1°10349 12)1°3495277/2 1OK4 
 12)/5717206 Q155 32827941 °488'714.2}906744.6) 1 °1028463)1°3498836/259 1954 19541" 
 
 13}6719361 2154 
 
 2157 
 2156 
 
 3280639]1°4882369|9072748] I -1022019}1°3502398 
 14)6721515 2155 327848511 °4877599}9078053]1°1015578]1°3505963/2595863 1956 
 15}6725668},, 327633211 °4872834)9083360j1°1009141}1°350953112597819 19 
 16|§725821 2 5 32741791 °4868073}9088671]1°1002709} 1 -3513102/259977 
 17}6727973 0159 32720271 °4863317}/9093984!1 °0996281}1°3516677 
 
 fs 3269875]1°4858565|9099300) 10989857) 1°3520254 
 
 18/6730125|,,, > . 
 1916732276 ae 3267724|1-4853817|9104619] 1 -0983436|1-3523834 
 901673442717)? !13265573|1-484907319 109940]1 097702011 °352741712 
 2167365775 ,>p|3263423|1 -4844334/9 115265] -0970609]1-3531003 
 2967387275 1) [5261273] 4839599 9120599] -0964201|1 3534593261 1523] 1 44, 
 93167408765 | se (3259 1 24| 4834868 /9125929|1 -0957797|1°3538185|2613483| 1 590 
 2416745024155 Toh 
 Q5]6745172|q 1 4 .|3254828|1-4825420,9156591|1 -0945009|1 3545379 
 26|6747319| 5 | 44{925268 I |1-4820702|9141929|1 0938610] 1-3548980 
 Q716749466|—. * 1325053411 -4815988'9147270]1 0932993) 1-3552585 
 2816751612 Bre 3248388|1-481 1278! 
 29167537575 ¢2/3246243|1 4806573)9157969|1 -0919460|1-3559803 
 30167559025 1 ¢|9244098 1 -48018729163319|1 -0913085|1-3563417 
 316758046], , {924195411 -4797176|91 68665] 1-0906714)1-3567034|2629199| 964 
 3916760190], ; , 41925981011 4792483191 74020] -0900347| 13570654 
 3516762333}, 1 4/3237667|1 -4787795|9 179379] | -0893984|1 3574277 
 346764476], , ,9/9295524|1 47831 1 1]91 84740] 1 -0887624|1 -3577903|2635099| ong 
 3516766618), ; ,9|9299382|1 477843191 90104|1 0881 269|1-3581532 
 36|6768760|5 5 4 1 (3231240|1-4773755|9195471|1-08749 18]1-3585 164 
 s7l6770901\._ , 
 3816773041 bite 3.226959!1-4°764417192062 1411 -086222811°3592438 
 39677518 1! 20 [322481 911 -4759754|921 1590|1 -085588911 -3596080|2644939 
 40|6777320),, ,915222680|1 47550951921 69691 0849554) 13599725 |2 
 $1]6779459|5 1 49/3220541|1-4750440|9229350|1 “08492231 3603372 
 496781597], 1 44/9218403|1-4745790922773.4|1 -0836896|1 -3607023|2650854| or 
 43/67837341.. [991626611 -4°74114419235129l1-08305'7311°3610674 
 44167853771 a 3214120|1-4736502 
 4516788007] 21 2913211993|1-4731864 
 
 46|6790143|5 1 5° 
 
 2637060|1969 
 263902911 9¢9 
 3229099/1-4°769084,9200841|1 -0868571|1-3588800 
 
 92385 12}1°082425411°3614334 
 924.3905}1 0817939} 1°3617995|2656775 1975 
 3209857|1:4727230)9249301]1-0811628}1°3621658 1 
 4716792278 2135 320772211 #4722600/9254.700] 1 -0805321}1°3625324)2660725 19%6 
 48]6794413 2134 3205587|1°4717975|9260102)1-0799018}1°3628994 
 
 4.9)6 796547 Q194 3203453) 1°4713354)9265506! 1°079271 8] 1°3632667|2664678 197% 7335322 11 
 50}6798681 91992201319 1°4708736\92709 1 4|1°0786423}1°3636343 
 51)58003813 2133 
 
 52/6802946]5 155 
 
 31991871 -4704123/9276324]1 -07801 32|1-3640022|2668633}, 4-4 
 319705411 °46995 14/928 1738} 1-0773845|1+3643704 one 
 |3/6805078}5 ; 5; 3194922 
 54|6807209], 155 /3192791 
 |55|6809339},, 1.4 9/5190661 
 56|6811469). 
 
 7329388 
 7327409 
 {| TS 254.29 
 
 1*4.694910/928715411°0767561|1°3647389}2 
 1:4690309)9292573)1 -076128211°36510781267457 
 1°4685713)9297996] 10755006) 1 °3654770}2676551 1989 
 0130 3188531/1+4681120/9303421)}1 074873411 +365846412678533 1981 
 57 6813599\ 5109 3186401)}1°4676532/9308849| 1 0742467] 1 -3662162}26805 14 1983 
 5816815728 2128 3184272) 1°467 194819314280) 1 -0736203]1°3665865|26824.97 
 59158 17856 2198 318214411 °4667368/9319714)1:0729943]1-366956712684479 
 6016819984) ~ 3180016} 1°4662792}9325151]1-0723687] 1°5673275| 2686463 
 
 Covers|Dif 
 
 : | ? ~~ Deg AT. 
 
 — 
 
2 Deg. 
 
 8255109], _|10-1744891 
 68256519) 1403 
 *8257913 
 
 1401 
 8259314) 7), 
 9 
 
 ——— 
 
 9+5185207110°1293821 
 10-044.2925/9-5189365/10-1294.96] 
 10-04.40385/9°5179521|10-1296109 
 
 10°0437846]9°5176677]10°1297244. 
 
 2530|10°0435306/9-5173831 10-1298387 
 8267703 nea 10°043276719+5170984110-1999530 
 8269098 ona 10°043022819-5168136|10-1300674. 
 82704.93 
 
 10°0427689/9-5165287110-1301818 
 10-0425 150/9°5162436|10°1302963 
 
 10°0422611/9-5159585|10-1304109 
 10°0420073}9-5156733]10°1305256 
 
 139 10°172672119°4139546 9°957'7389 
 82746712"? 10°172532919-4142818 9°9579927 
 
 8277453 nea 10172254719-414935719-9535004 
 8278843] 135 
 
 *0414996/9-5151024/10-1307551|, 73 
 13gg|10°172115719-4152625]9-9587540 2394) 0°041245819-5148168|10-1508600 1149 
 8280231] °°°!10-171976919-415589) 9°9590080/9553|10-040992019-5 1453 11]10-1309848 1150 
 8281619] ,. 2537] 0°0407382}9-5142453|10+1310998} ||. |9- 
 32830061 1387 523 (10-0404845/9-5139504 10°1312149 tia 
 328.4393] /387 5208|10-040230719-5136%734110-1313300 1159 
 3285778]. door |10-0399770/9-5133879110-131 44.50 1139 
 3287163 ,{10-039723319-5131009|10:1315604 
 32885 ee 
 
 1154 
 1154 
 
 10°03946935/9°5128146 10°1316758 
 
 Sls ihe ec 
 “1ST. ; 2536 
 300966 | a 
 “12535 
 2536 
 
 306464 pide 
 
 307837 aa 
 
 309209}, 37 1|10°169079119-429415619-9643346 nye 
 
 310580} /4,75|10°1689420|9-429739919-964538) yee 1165 
 311950 10°1688050|9-423062619-9648416 2595], 0/0951584)9-5079289]10-1336466] 165 
 313320 10°1686680/9-423385819-965095 
 
 37631 
 
 1166 
 1167 
 1168 
 41168 
 
 121519] !36410. 1678481 9-4253295/9-9666157 
 
 1363|10°1677117|9-425644719-9668600 ai 
 24246) 3 24|10°167575419°4259660 9-9671995|~ 
 
 1358 10°1668950)9-4275756 9-9683893 2534, 
 1358 £0°166759219-4278969 9°9686427 2533 
 10°1666234/9+428918] 9°9688960 2533 
 10°166487819-4285399 9°9691493 9533 
 1355 10°166352219-4288601 99694026 2533 
 37833 10°166216719-4291809 9°9696559 
 
 10-0316107|9-5038806]10-1352844 
 10-0313573]9-5035906|10°1354010 
 
(334) 43 Deg. NATURAL SINES, &e. | Tab. 10. 
 
 -’! Sine |Dif Covers] Cosec. 
 
 Tang.|Cotang.| Secant | Vers. Dif |Cosine 
 
 “0/5819984 15 1 gn} 31800 16}1 466279219325 151]1-0723687|1-3675275| 2686463) | 4. ,17315537|60' 
 1168221115; o4(31'77889]1 -4658220]933059 i] -0717435}13676985|2688447 | 4. .|7311553)59 
 216824237]53 ¢|31 757631 4653652/9536034]1 -071 1 187}1-3680699]2690459| | 405 |7309568)98 
 3/6826363]q 1 9|3173637|1 -4649089/934.1479}1 0704945} I 3684416 
 4}6828489]o 1 9,(3171511]1-4644529|9346995| 1 -069870211 -3688 136]2694.403] 1 43~17305597156 
 
 - 5{683061315 1 93[5169387]146399'73|9352380]1 0692466 ]1 -3691859]2696390| 92 -|7303610)5. 
 6/6832738|5 105 
 
 3167262|1 -4655422}9357834]1 :0686233|1 -3695586|2698377|, 44 417301623)54 
 7683496 1 lo 1 95/3465139}14630875}9363299}1 -0680004/1 -369931 5|2700365} ; ggal7299635}53 
 8168369845 991316301 6]1 *4626331/9368753 
 916839107|5 5013160893]! -4621792 
 10]6841229},9 4 9 [8158771 ]1°4617257}9379683]1 0661341 }1-3710523 
 11]6843350\9 19 [3156650] °4612726|9985153}1 -0655128/1 -3714266]2708323} 1 gq | 129167749 
 12}684547119 1 99)3154529]1 4608 198}9390623}1 0648918/1 -371 801 1]2710314 1 991]7289586)48 
 13/6847591]9 1 99/3 152409}1 46036759396 101] °0642713}1-3721760]271 2305]  gqo)7287695)47 
 14]6849711]9 1 4.9/3 150289) 1+4599 1 56]9401579}1 063651 1}1-3725512)/27142971 1 g9q]7285703|46 
 15}6851830i9 1 1 g(3148170|1 459464 11940706 1]1-06803 151 -3729268]2716290} 1 94/7283710)49 
 16 6853948 '51 13/3 146052 1°4590130)941254.5}1°0624119!1°373302612718284. 1 7281716|44 
 17/6856066!93 ; g\3 143934] 45856251941 803311 061'7929|1 -3736788|2720278} 1 gq4|7279722143 
 18]6858184)o 1 1 g/3141816)1 -4581120/9423523|1 0611742 '1-3740553|27222791 1 ggg] 1277728) 42 
 19,6860300'9j 1 g/3159700 1 -4576621)9429017|] -0605560)1-3744521)2724268) 1 q9¢]!275752/41 
 20/6S62416 93 1 g/3157584) 145721 27/9494513] {-0599381 1-3748092|27262641 1 ggq{ 127373640 
 2116864532 91 1 5)3135468 1 -4567636|9440015]1 0593206) 1-3751867]2728260] j oqx|7271740)39 1 
 22/6866647 97 1 4|8133353)14563145}9445516)1°0587035|1-3'755645|2730257] 1 ggg] 7269743)/381 - 
 23}6868761 93 j 4/9151 259}1 4558666 /945 1021] 1-058086 741 :3759426)2732255} 1999) 1267745137 
 246870875 '9 ; 1 4/3129125)1-4554187}9456530) 10574704) 137632 10/2734253} 1 9qq/7265747) 96 
 95/6872988 o 1 1 31312701911 -454971219462049|1 -0568544 1-376699812756959logqq/7269748)35 
 26}6875101 94 j9)3124899|1 *454524119467556]1°0562388 1-37'70789]2738252|aqgp) 125! 748)34 
 27/6877213'9, 12/5 120787 14.5407'74194'73074) 10556235 1°37'74583|274.025 299 q1 | 7299 748/35 
 28}6879325'9 1 499120675] 1 453631 119478595}1 05500871: 
 29]6881459 9) ; 1|3118563|1-4531852,94841 19[1-0543942 1 -3782181|2744254loggql7295746)51 
 I 
 1 
 
 to 
 52) 
 
 30/6883346!9 1 99/31 164541 °4527397/9489646]1°0537801| 
 31]6885655 91 19/91 14345]] °4522946|9495176|1°0531664) 9792/2748259loqg3] 1291 741/29 
 32/6887765'9 1 gg/3112235/1°4518498|95007091 05255311 +3793602)2750262lagg4| (2#9738/25 
 33/6889873/9 1 ygi3110127)1 °4514055}9506245]1°0519401 |] -3°79'7416)2752266}o9 95 (247773427 
 3416891981!9; 9{3108019}! 4509616195 11784]1°0513275|1-3801233|275427 loggs| 1249729) 26 
 35/689.408915 | 4¢)310591 1]1-4505181|9517326| 1 °0507153'13805053|2756276loggs|1243724)25 
 36|6896195 91 y/3193805|1 -4500749)95228'71|1-0501034|1-3808877 2007) (24d 719/24 
 37}6898302' 9 1 95|3101698}1 -4496322}9528420) 1 0494920} 1 °3812'704]2760288] oq 9% 
 38)690040'7)9 1 95|3099593]1 *44.91898/9533971]1-0488809 1-3816534|2762295|o9 9x7] 1297705 22 
 391690251219 1 ¢3/3097488]1 4487478|9539526]1 0482702|1-3820367|2764302loggg|7235698)21 
 4.0/690461 719 1 9413095383}1 -4483063}9545083} 1 0476598} 1 *3824204/276631 0loqggl/233690)20 
 41]6906721]o | 93}3093279|1°44'7865 119550644 1-0470498]1 *3828044|2768319 7231681|19 
 421690882415 1 3/3091 1'76}1*4474.243/9556208] 1 04644021 °3831887/277032 7229671/18 
 431691092'1|9 1 .9/3089075}1 -4469839/9561774]1°0458310]1+3835734|2772339| 99 o| (227661) 17 
 4 44/691302919 1 y9)308697 11 °446543919567344]1-0452221]1-3839584 
 45}6915131)o 1 91 [30848691 446 104.3)9572917 
 46]6917232\9 | y9[3082768| 1 °44.56651/9578494]1 0440055] 3847294|2778372| 9015] /2 
 471691933219 1 ())/3080668}1°44.5226219584073| 104539771 3851153/2780385l99 1 4] 7219615/13 
 48}6921432)9)99)3078568/1 -4447878|9589655] 1 -0427904/1-3855017|2782398]99 1 5] (2! 7602/12 
 
 491692353 1}oy99}3076469} 1 -4449497|9595241|1 -0421833]1 -3858883 
 50}6925630| 999¢}9074370/ 1 -443912019600829|1-0415767]1-3862753|2786426| 991 5) (21957410 
 5116927728} 9 47|3072279} 1 -443474819606421]1 040970411 -3866626|2788441 
 5216929825}599-|3070175]1 *4450379}96 12016] 1 °0403645|1-3870503|2790456]o93 6 /209544F 
 531693 1922159 13068078] 1+4426013]961761 4|1-0397589]1+3874383 ; 
 
 5416934018 2096|3065982} 1 4421652196232 1 5)1°0391538]1 °3878266|2794489 
 
 55}6936114 2093 3063886|1 -4417295]9628819]1°0385489}1 -3882153/2796506 
 56|6938209 9995/2061 791] 1 441294 1}96344.27]1 -03'79445}1 +3886043/2798524 
 97}6940304}, Od. 3059696} 1 *44.08592|964.0037|1 °0373404]1-3889936/2800543 
 86942398 9093}2097602} 1 +44.0424.61964.565 1|1+0367367)1+3893832/2802562 2020}! 
 591694449 J 2093}2055509} 1 °439990 4/965 1268]1 -0361333}1-3897733 2020 
 6016946584 3053416]1°4395565/9656888} 1 -0355303}1 -3801636/2806602] © |7193398 
 
 Cosine|Dif} Vers. | Secant !Cotan.| Tang. | Cosec. [Covers!Dif| Sine | ” 
 
 © Deg. 46. 
 
 7203494 
 
 tw 
 i= 
 — 
 He 
 -~I 
 e7) 
 OF NULU D~WOHO 
 
3 Deg. LOG. SINES, &c. : (335) 
 Ir a A OE 
 Dit | Cosec. 
 
 10°1662167|9-4291809|9-9696559]o 540 
 10°166081219-4295015/9-9699091}554. 
 10°1659459|9°4298220|9-9701624))535 
 10°1658106|9°4301424/9-970415755 39 
 1016567549°4304626/9-9706689|5 30 
 101655403]9°4307827|9-9709221|5.35 
 10°1654052/9°4311027|9-9711754|5<30 
 9-8547297], «  o|10°1652703]9-4314295|9-9714286lo 540 
 9-8348646), 53 4|10°1651354 5-451 1499 GAG 9539 
 9°85499941." , 2110°1650006/9°452061 719° 2532 othe ey ay 
 9°8351341 ne 10-1648659}9-4323811]9-972188215531|10°0278118|9°4995182 10-13 10540 1186 cor ve 
 9-8352688 ia 5|£0:1647312/9-4327004/9 97244 13]y5,39[10°0275587)9-4992264 ‘eta ; his 1186\5 ce eorogglag 
 9°8354033} 5, 5110°1645967)9-4330196|9 9726945}95 49]10°0273055)9°4989345 : 1186 spt ae Fh 
 198352978) 4, 4|10°164462219-4333386)9-9729477959 ;|10°0270523/9-4986425/10°1374098| jgg\-' eo 2202 
 9-8356799 3 i‘ 10-1643278|9°4336574/9°9732008|y551|10°0267992/9-4983504}10-13 ae LI88iG cdossaclae 
 5 344 a ° len , in “0965 “4 “13 ef . 
 98359066], 54 9|10:164193419-4359762/9-97345391559 ODER DaRAIG do meratLON aereeerit to ueaeasalLl 
 pists ek tetnogettoso arte Hosmoslaerrs| o-traca tp aaie 
 o ‘ . i PO} ps 25 rd - zy : ¥ ’ 
 |9°$362091}, 37 -110-1637909|9-434931 6|9-9742133|.5 5 |10-0257867]9-4971808]10°1380042} ; 1 91 |9°8619958)42 
 9-8363431] |, .|10°1636569|9-435249819-9744664]9 x9 1| 10°0255336|9-4968881110-13812331 j 91 |9°8518767/41 
 9-8364771 10-1635229]9-4355678|9-97471 95|o59 ;| 10°0252805|9 4965953] 10-1582424) 1 94/9°8617576)40 
 9-8366109],355|10°163389 1|9-4358858|9-9749726|y 9 1| 10°0250274|9-4963024]10-1383617| 1193/9 °8616383/39 
 9-8367447] 995|10-1632553}9-4362036|9-9752257l9s 45] 10°0247743}9-4960095| 101384810}; 193/9°8615190)38 
 9-8368784| 9 /110-1631216]9-4365212/9-9754787 lor 4 {10024521 3]9-49571 62|10-13860031 ; 1 94|9°8613997|37 
 |9-8370121], 55 4|10-1629879|9-4368387|9-9757315 325 |10°024968219-4954229|10-1387197|) 195 /9°8612803)36 
 Sart 4 BAIQ. i 40l5 ean} 10°024015119°4951295| 101388391, 9°8611608/35, 
 Heater eerie cen os Racer eee eee SH 
 3113541 10-1625875|9*1977903|9-9764909|9%94 | 10°023509 1|9-4945424110-1390789| 1 97|9"8609215|33 
 10162454219 -4381075|9-9767440l95 39] 10-0232560)9°4942486]10°1591982) ; jg7/9°8608018)32 
 10°1623210)9-4384243}9-9769970195,50|10°0230030]9°4939547|10° 1395179] j99}9"8606821/31 
 10°1621878|9-438741119-9772500|95,30| 10-0227500|9-4936608}10-1394378} ; 1 g9|9°8605622130} 
 erretreenurg cron eeree eres Perea nate 
 *161921'7|/9°43937 "9771756012530 tna oe ‘al ahs 
 sBUiee rect dae LG eal odd 
 ; ; 59/9 4 319 °9782620)9599 ( ; vpheineck tS 
 tage 10°161523119+440322 719-9785 14919530] 10°0214851)9°4921891)10°1400381)] 99< 
 (526|10°1615904)9-4406386)9-9787679|05,30 
 10°1612578|9°440954-419-9790209}9599 
 10°1611253/9-441270019°9792738|95 59 
 >, [L0: 160992819-4415853)9°9795268ly 509 
 5 a{10*1608604!9-44-1900919-9797797l9 506 
 “110+1607281|9°44221 62/9-980032619 5, 
 5|10°160595919°4425313)9-9802856la50y 
 1991 |10°160463719-4428465}9-9805385)2 50 
 1 399{4 0° 16033 1 6/9-443161 119 9807914 
 isto 10+1601996)9°44347551/9°9810443 
 «2 {10*160087719-443790419-98 12972 
 1319 
 1317 
 
 Verseds.| Tang. |Dif| Cotang. | Covers.| Secant |Dit| Cosine | _ 
 
 94.99| 10°0303441 |9°5024294|10-1358725], 1 q|9°8641275160 
 10-0300909|9-5021388}10'1359904], 1 ,19°8640096)59 
 10-0298376|9-5018480|10-1361083}, ; 5 |9°8658917158 
 10+0295843]9-5015579|10°1362263|, ; 499863773757 
 10-0293311|9-5012663|10-1363443];  g 1 /9°8636557}56} 
 10-0290779}9-5009752}10+13646244 ; | g919°8635376)55 
 10°0288246|9-5006840}10-1365806] ; ) 94|9°86341 94154 
 10-0285'714)9-5003927|10-13669891; j g|9'8633011}53 
 10-0283182/9-5001013}10°1368172} 194 |9°8631828)52 
 10-028(0650}9-4998098|10°1369356| 7 ¢|9°8630644)51 
 
 Sine 
 
 9°6397895) 3. 
 
 9-8339188|, 52 
 9-8940541] 32° 
 9-8341894| 5? 
 9-8343246] 19> 5 
 98344597] 15°) 
 983459481) 59 
 
 1341 
 1340 
 1340 
 1338 
 1338 
 1337 
 1337 
 
 19°8380783 
 9°8382112 
 
 1o-8389%47 10°0207262)9-4913046}10°1403991)1 905 
 10°020§732/9°49 16096)10°1405196)1 995 
 10°0209203/9'4907144|10°1406401) 1996 
 10°0199674'9°4904191|10°1407607}1 997 
 10-0197 144|9°4901237}10-140881411 90g 
 10°0194615)9°4898282)10+1410022) 1098 
 
 *7140-0192086 9+4895326\10°1411230}] a99)9' 8988770) 15} 
 
 50} 10°01 89557/9°4892968|10°1412439} j 9 1 /9°8587561]15} 
 > 5.9{10:0187028]9-4889409/10-1413649) 93 yi9°8586351 14 
 10°1599358)9"4441049)9-9815501}9599)10°0184499/9-4886449 Behe repre ee eee es 
 13 7{t0°159804 1944441921998 1803019599] 10°0181970/9 4883488}10 eet Fact cee REED ie 
 13177]10°1596724]9-4447334)9-9520559l9599)10°0179441/9-4880525)10+1417282)j913) eo ae d 
 3 110+159540719-445047519 9823085 > 991 10017691 319°4877569110° 1418495] 1 94 919°8981505 
 1315 10'159540719 4450475 - r 11959G #499 119+487459'7110-1419708 198589299 
 131 5|10°1594092}9-44556 14/9-9825616})599|10°0174534)9-4874597 t0-1490999| 2) t19'8579078 
 LOs1 ge TTT) 4400 T3212 2828) 45g sogt! 0-011 1Bo 8 to Oa e022 150. oer agag 
 g|1071591463}9-4459889/9-983067319 59q|10°0169527|9-4868664 LAs eae SEAS tly arab 
 10°1590150|9-4463024)9-9833202) 50 4)10-0166798)9-4855696|10°1423352) 93 6 
 10°158883819-4466158}9-9835730)4 5.59|10-0164270|9°4862726)10°1424568) j 91779" 
 10°1587526|9-446929 119-9838259) 5,.52]10-0161741]9-4859755|10 1425780) 19171 
 10°1586215/9-4472422/9-9810787 
 B}9-84 15095) 0 -110-158490519-447555219°9843315 
 919°84164041 ', 3/10-1583596/9-44'7868 1|9-9845844 
 0|9-8417713 10°1582287|9-4481808|9-9848372 
 
 _¥ Cosine Dif] Secant | Covers. Cotang. Dift Tang. |Verseds.'! Cosec. Dif Sine 
 Apnea reget I I FT I RE ARR PO 
 
 deanna dnettiants canteen cinetat Se aE 
 | ee | Deg. 46. 
 
 3}9°8395363 
 /9°8396684 
 3]9°8398004 
 3/9°8399393 
 7/9 8400642 
 3}9°8401959 
 319°8403276 
 198404593 
 1/9-8405908 
 2)9°8407295 
 3/9°8408537 
 419°8409850 
 
 f 
 ; 
 . 
 
 A 
 9 
 i) 
 4 
 y 
 
 6|9°84124.74. 
 7\9°8413785 
 
 2529 
 2528 
 
 loolen) 
 Cr Gr 
 <= 
 wo > 
 © tO 
 Or 
 OO ur 
 “lomrwowrtir M-ATHO 
 ADE NE ee EE et 
 
 [oe] 
 Gr 
 fap) 
 co 
 ise) 
 > 
 
_ (836) 44 Deg. NATURAL SINES, &C. Tab. 10. 
 
 Sine |Dif |Covers|] Cosec.'|Tang. |Cotang.| Secant | Vers. 
 
 (15946584 2()99}30:534 16] | -4395565}9656888 1°0355303)1°3901636 2808602 
 145948676 909 18091324] 1°439I 231 9662511 1°0349277)1°3905543]2808623 
 ‘1995076 Tog j [3049233] 1 4386900 9668137 1°0343254 1°3909453 2810645 
 35952858 ]o 9). 1 [304714211 °4382.5'74|967376 71 0337235} +3913366]2812667 
 615954949 2090 304505 1]1°4378251]/9679399]1 -033 1 220} 1°3917283]2814690 
 6957039 2089 304.296 1]1°437393219685035}1 0325208) 1°3921203/2816713 
 
 1959128)99)49|3040872| 1 436961 6/9690674 1°0319199}1-3925127/2818737 2025 718126354 
 
 7/696 121 719938/3038783|1 4365305 9696316}1°0313195}1-392905412820762 2025 7179238153 
 816963300) 9 yg 5 |3036695} 1 °4360997/9 70 1962/1 °0307194)1 +3932985}2822787 2026 7177213/52 
 $1/6965392I9.)9-|3034608} 1 -4356693/9707610)1°0301196}1°3936918]2824813 2096 7175187|51 
 
 69674791998 6|303252 11 -4352393/9713262)1°0295203|1 -3940856/2826839 2027 717316150 
 11/6969565}49g6|3030435]1°4348097)97189 171028921 2)1-3944796|2828866 2098 7T171134)49 
 121697165 1}99g5|3028349]1 *434.3805|9724575)1 0283226) 1 *3948740)2830894. 2028 7169106/48 
 
 13}6973736|o9g 5130262641 43395 16)9730236}1 0277245} 1 +3952688}2832922o99/7167078/47 
 14}6975821}o93 4130241 79|1 °4335231/9735901]1 0271 263|1 -3956639}2834951/og4q|7165049)46 
 15}6977905}9935/3022095]1 -4330950)974 1569] -026.5287| 1 -3960593]283698 lloygp|7 16301914 
 16/6979988} 99331302001 2|1°4326672)9747240}1 0259315}1+396455 11283901 Iloj49|7160989|4 
 17|6982071}99g9/3017929}1'4322399|97529 14] 1 0253346}1-3968512]2341041loq3{7158959/43 
 18/6284153loqg 1 /3015847|1°4318129/975859 1] 024738111 -3972477|2843073| 9939/7 156927|42 
 
 19'6986234logg 11301397661 °4313863/9764279|1 -0241419]1-3976445]2845105]oq49{7154895|41 
 20/6988315 ea 3011685]! °4309600|9769956]1 °0235461]1-3980416|2847137 nie 7152863}40 
 216990396} 99g9]3009604]1 °4305342/9775643]1 °0229506}1-3984391]28491 70! 994 4/7150830|39 
 221699247619979'3007524|1 4301087/978 13331 °0223555]1-3988369|2851 204) 594417 148796)38 
 23/69945551997g|300544.5|1 *4296836)9787027|1 "021 7608} 1°3992351|2853238 7146762)/37 
 2416996633!9,)78|3003367]1 °4292588\97927 241 021 1664|1-3996336|2855273}o 
 
 25699871 1997313001 289]1 -4288345)979842411 0205723} 1 -4000325|2857309| 4946/7 142691)35 
 26} 7000789} 99777/299921 1|1°4284 105]98041 271 -0199786|1 40043 17}2859345| 944~|7140655|54 
 
 7|700286619976/2997134[] °4279868|9809833}1 0193853] 1 °4008313/2861382}594,|7138618)3 
 28} 7004942) 9762995058] 1°4275636/98 15543}1 0187925} 1 -401231212863419|o934|7136581)32 
 29/7007018]9975 2992989] -4271407/9821256|1 ‘01819971 -4016315|2865457loqgq/7134543)31 
 30/7009093} 297 4/2990907|1 4267 182)9826973]1 0176074} 1 -4020321|2867496| a 939|7 152504150 
 
 31/7011 167}9974/2988833}1 426296 1]9832699}1 01701 55}1°4024330]2869535]oq39|1130465|29 
 32/7013241 97312986759}! *4258743|98284 1 5]1 0164239} 1 +4028343}2871574| 994 1|7128426/28 
 33)70153141997312984686] 1 “4254529/9844141/1 0158326|14032360]28736 I Slog, 1 /7126385/27 
 34/7017387}a979 29826 13] 425031 9/9849871}1 0152418) 1-4036380]2875656}aq, 1|7124344/26 
 35}7019459}9979 29805411 42461 12}9855603}1 -0146512}1-4040403/2877697]994.4/7122303}25 
 36/70215311997,) 2978469} 1 4241 909/986 1339}1 -0140610}1-4044430)2879740\99,.9/7120260\24 
 
 207 1'2976399|1-4237710|9867079|1 013471 2|1-4048461|2881782 2044 “ 118278195 
 38|7025672l agg 2974328) 1 -42335 14]987282 1/1 -0128817}1-4052494|2883826|y9, ,{71 16174122 
 397027741 jay 79 2972259] 1 *4229323/9878567]1 0122925} 1 -4056532/2895870 
 40/7029811|9 465 2970189]1 °4225134/98843 1 6] 0117038} 1 -4060573|28879 14. 
 41]70318799., 632968121] *4220950/9890069]1 01 11153}1+4064617|2889959 
 42/70339 17139 ,57/2966053]1 42 16769/9895825|1 010527211 -4068665|2892005| 9541 
 43 703601419 :y412953986 1 +4212592)990158411 +0099394)1 407271 7]/2894052| 99 42|7105948)17 
 44)703808 1/9, 6)29619 19] 42084 18}9907346} 10093520} 4076772} 2896099] 5 4|7103901)16 
 45/704014719, 662959853]  -4204248/99 151 12)1-0087649}1 -4080831/2898146]99, 9|7101854)15 
 46) 7042219996 5[295778 7] -4200082/991888 1}1 008 17821 -4084893}2900194} 99; 47099806) 14 
 47) 7044278} 9()64|29.5.5722|1 °4195920|9924654 1 0075918] -408895812902243) 95, 4/7097 757/13 
 {48]70463 42\ 996 ,|2953658]1 -4191761|9930429) 1 °0070058|1°4093028}2904293}5 9, 4/7095707|12 
 
 49}7018406}o965/2951594]1 -4187605|9936208} | 0064201] -4097100}2906343| 9 ..{7093657I | 
 50}7050469]9563/2949531]1*4183454]994 1991] 1 -0058348}1-4101177]2908393| 95. 5 /7091607|10 
 5117052532194, 2947468] -4179306|9947777| I 00524971 -4105257]2910444| 59 2/7089556 
 52}705459 441996 1|2945406] 1 4175 161]9953566) 1 -0046651]1-4109340]2912496) 552 4|7087504 
 53]70566535}9 6 {| 2943345] | -4171020|9959358|1-0040807|1-4113427]2914549] 55, 4(7085451 
 54|7058716) 996, [2941284] | -4166883)9965154)1-0034968] I -4117517]2916602|5.4/7083398 
 537060776] 995912939224 ]1 -4162749]9970955}1 -0029151|1-4121612}2918655|9q5 4|7081545 
 56]7062835|,54 =9)2937165]1+41586 199976756] 1 °0023298|1 -4.125709|2920709 7079291 
 
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 59}7069011},,93-12930989| l -414625 1|9994184| | -0005819]1-4138024]2926876 
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 Cosine! Dit! Vers. | Secant |Cotan. Tang. | Cosec. [Covers Dif} Sine 
 
 Deg. 45. 
 
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44 Deg. LOG. SINES, &c. (337) 
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 1}9-8419021| 1398 
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 419+8492939 
 
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 1303 
 1302 
 1301 
 1300 
 1299 
 1299 
 1298 
 1297 
 
 2528 
 2527 
 2528 
 2528 
 2527 
 2528 
 2527 
 2528 
 2527 
 2528 
 2527 
 2528 
 2527 
 2597 
 2528 
 2527 
 2527 
 2527 
 2527 
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 2597 
 2527 
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 1226 
 
 819°8428154 1226 
 
 919-4994.56 
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 2019-3443725| 29311 0-15569'7519-4544084|9-9898006 
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 819-8554650148 
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 1231 
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 1233 
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 9°854.2329138 
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 123% 9°8541093/37 
 1237 
 1238 
 1239 
 1240 
 1240 
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 128815 ().. xanoto 4.565'75819°99 16616 
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 28/9-8454045 
 2919-8455339 
 
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 1283 
 1282 
 1282 
 1281 
 
 1280 
 1279 
 
 #919 8524959194 
 #6/9.8593713123 
 1247119.8599466 
 1248!9.g501918 
 tose |9°8919970]2 
 947248 16|10-1481279], 5 ¢19-8518721|19 
 252711 0-004.548019-4°721'789110+148259911 2 l9-g5 174% 1119 
 
 2527 10:0042953)9°4718761110°1483780 
 2526; 
 
 2527 
 
 10°1526733)9-4615065)9°995704'7 
 10°152545'7|9°4618134}9°9959573 
 
 9-851371 "115 
 125 219.8512465114 
 125419.9511011113 
 1254 3 
 1255 
 1256 
 
 pbAgnilocodRaaat -0035373|9-4'709669|10-1487535 
 tsi 1635 eid 90967154 2027 ined 9-4706636|10-148878¢ 
 pee 2526/7 ).0030320]9-4'703602110-1490043 
 
 LOO 
 
 10°1516550/9°4639593|9-9977260 10°0022740 
 
 ares Cnn vary, a 4.59110°1495067 
 R 10°1515280/9-4642654]9-9979787], 25, |10-0020213/9-4691452 ( 506 
 
 25261 oS801 5 
 2527 
 2526 
 2527 
 2597 
 
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 10°00126535}9°4682327|10°1498845 
 10:0010107)9°4679285)10'1500103 
 10°0007580}9°4676237 1960 
 1.0°0002527}9*4670142 10°1503887 12651: 
 ‘110-0000000}9*4667093|10+1505150 : 
 
 _ | J ist 
 i ee RES 
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 Cte WO & Or 
 
 — 
 
338 DIFFERENCE OF LAT. AND DEPARTURE. Tab. 11. 
 ee a Ne eel 
 Dist. 1. 
 Lat. Dep. 
 
 Course 
 Pts. }D. 
 
 0-9988)0°0491 
 0:9986|0:0523 
 C-997610°0698 
 0:9962)0:0872 
 0-9952)0°0980 
 
 0-989910:1467 
 0:987710-1564 
 0-9848)0°1736 
 (:9816|0-1908 
 0-9808|0-1951 
 0-978110-2079 
 0-974.410-2950 
 0-970310-2419 
 0-9700|0-2430 
 0-965910-2588! 
 0-9613}0-2756] 
 0-956910-2903 
 0-9563|0-2924| 
 0:951110-3090) 
 
 9|13°84.50)1°1025 
 3°8278]1°1611 
 3*82529)1°1695 
 3°8042)1°2361 
 3°782111°3023 
 3°7662)1°3476 
 3°7588|1°3681 
 3°7343]1°4335 
 3°7087|1°4984. 
 3°6955|1°5307 
 
 0:933610°3584 
 0°9272)\0°3746 
 0°923910'3827 
 
 *8672;0°7167 
 *8544)0°7492 
 *84:7810°'7654 
 
 2°8007|1°0751 
 2°7816|1°1238 
 2°7716}1°1481 
 
 0°9205\0°3907 2°7615}1°1'7221/3°6820)1°5629 
 4110°9135)0°4067)|1°8271)0°8135}|2°74.06]1 -22021|3°654211 6269 
 5}|0°9063\0-42261)1°812610°84.521/2°7189}1 -2679]|3°6252}1 6905 
 
 0:9040|0°4276 
 0°8988/0:4384 
 0:8910/0°4540 
 0*8829|0°4695 
 0°8819|0-4714 
 9}|0°9746|0°4848 
 0°8660)/0°5000 
 
 0°8577|0°5141 
 0*8572|0°5150 
 0°8480|0°5299 
 0°8387|0°5446 
 0°8315}0°5556 
 0+8290/0°5592 
 0°8192\0°5736 
 0:8090)0°5878 
 0°8032/0°5957 
 0-7986|0°6018 
 
 0°7880}0°6157 
 0°'7'77110°6293 
 0°773010°6344 
 0"'7660)0°6428 
 0:7547|0°6561 
 0°7431}0°6691 
 0°7410}0°6716 
 0:'7314}0°6820 
 10°7193)0°6947 
 0:7071)0°7071 
 
 “ll Dep. | Lat. 
 ners a ee 
 Dist. 1... 
 
 1°8080]0'8551 
 1°79'76|0°8767 
 1°7820|0°9080 
 1°765910°9389 
 1*7638|0°9428 
 1*74.9210°9696 
 173211170000 
 
 1+7155|1-0289 
 1-7143]1-0301 
 1°6961|1-0598 
 1-6773}1-0893 
 1+6629}1-1111 
 16581|1°1184 
 1°6383}1 +1472 
 1°6180}1+1756 
 1-606411-1914 
 1:5973]1 +2036 
 1-5760}1 +2313 
 1-5543]1°2586 
 1+5460}1-2688 
 1-5321]1°2856 
 1-5094{1°3121 
 1-4803]1+3333 
 1-481911-3431 
 
 2°7120}1°2827113°6160}1°7102 
 2°6964)1°3151)13°5952)1°7535 
 2°6730}1°3620}|3°5640) 18160 
 2°6488]1°4084)13°5318|1°8779 
 2°64.58)1°4142||3°52'77|1°8856 
 2°6239}1-4544||3°4985]1 -9392 
 
 2°5160}1+6339]|3°3547 
 2°4944) 1 -6667]|3°S259]2°2223 
 
 2-2641)1 9682) 3-0188 | 
 2-299412-00'74|'9-972612-6765||3°715713-3457/48]" 
 2-292912-0147)/2-9698|2°6862)[5-7048|3-3578|| 
 2°1941/2-0460'12-925412-72801|3-6568]3-4100/147) 
 2°158012-0840112-87'74|2-7786||3-596'713°4'733|(46} - 
 
 Dist. 3. || Dist.-4. 
 
Q 
 03 
 3 
 4 
 5 
 07 
 6 
 " 
 8 
 04 
 9 
 10 
 U1 
 i 
 12 
 13 
 14 
 14 
 15 
 16 
 13 
 17 
 18 
 19 
 14 
 20 
 21 
 29 
 2 
 23 
 24 
 125 
 24 
 26 
 27 
 2 
 
 5+9991|0-1047 
 
 19°9963/0°2094 
 
 5-992810-2944 
 5°991810:3140, 
 5*9854/0°4185 
 5°9772|0°5229 
 5-9711|0°5881 
 5+967110°6272 
 5*9553|0°7312 
 5941 6|0°8350 
 5+9351|0°8804 
 5+926110-9386 
 5*9088]1-0419 
 5*889811-°1449 
 §°8847|1-1705 
 5*8689]1-2475 
 5*84.62|1°3497 
 5*8218|1-4515 
 5°8202|1°4579 
 51956|1-5529 
 5“7676|1 6538 
 5741 6| 197417 
 5°13'78]1°7542 
 57063) 18541 
 5°6731|1°9534 
 5*6493]2-0213 
 5°6382|2-0521 
 5*601512'1502 
 5-5631|2°2476 
 5+5433/2°2961 
 5*523(]2°3444 
 
 1119°4813]2°4404 
 
 5°457812°5357 
 9°4239/2°5653 
 5°3928)2°6302 
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 5°29 15/2 °8284 
 5°247'7|2:9089 
 5°1962}3°0000 
 5°1464}3°0846 
 5*1430}3°0902 
 5°0883}3"1795 
 5°0320)3°2678 
 4°988813°3334 
 
 14°974.9}3-°3552 
 
 49149134415 
 4°8541|3°5267 
 4°819913°5742 
 4°791813°6109 
 4*7281|3-6940 
 46620137759 
 4°638 1]3-8064|| 
 4*5963}3°S5E7 
 4°5283)3°9363 
 
 ||4-458914-0148 
 
 4r44.5714-0294 
 4*3881|4-0920 
 4*3160)4-1680 
 4242614-2496 
 
 6-9989}0-1222 
 6:995710°2443 
 6-991610°3435 
 6:990410-3664 
 6*982910°4883 
 6:973410-6101 
 6*2663]0°6861 
 6:9617|0-7317 
 6°9478|0°8531 
 6*931910-9749 
 6+9249]1-0271 
 6-9138]1-0950 
 6°8937111-2155 
 6°8714|1°3357 
 6°865511°3656 
 6@si7041 4554 
 6*8206|1°5747 
 6°792111-6935 
 6-790211-7009 
 6°16 15|1°S117 
 6°7288|1-9295 
 6*698612-0320 
 6°6941]2-0466 
 6°6574|2°1631 
 6-6186|2-2790 
 6-5908]2-3582 
 6°577812-3941 
 6°5351|2°5086 
 6-4903]2-6229 
 6°467212-6738 
 §*443512°7351 
 *3948|2-8479 
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 2916}3-0686 
 -237013+1'7"79 
 1806|3-2863 
 ‘1734|3-2998 
 1223}3-3937 
 6°062913-5000 
 6-004113-5987 
 6°0002]3-6053 
 5*9363]3-7094 
 5*8707|3°8125 
 5°8203|3-8890 
 5*8033/3°9144 
 5°734114-0150 
 5*6631/4°1145 
 5+622514+1699 
 5+5904|4-2127 
 5+516114°3096 
 5-4400/4-4052 
 5°41 1114-4408 
 5*362314-4995 
 5*2830/4°5924, 
 5+202014°6839 
 5°1867|4-7009 
 5*1195)4-740 
 5*0354\4-8626 
 4°949714-94.97 
 
 DAAaAN dS 
 
 ADAH 
 
 zi Dep. | Lat. 
 enter | Res Gall Simaraaie 
 Dist. 7. 
 
 7-9988|0-1396 
 7-9951|0-2792 
 7-99040-3925! 
 7-9890|0-4187) 
 7-9805|0°5581 
 
 7*9696|0-°6972 
 7:9615|0°7841 
 
 7-9562|0°8362 
 7*9404|0°9750 
 7-9291]1°1134 
 
 7°9134/1°1738 
 7°9015]1 2515 
 7°8785]1°3892 
 7*8530}1-°5265 
 7°8463}1°5607 
 7°8252i1 6633 
 7°7950|1°7996 
 7°762411°9354 
 7°7602)1°9438 
 7°727412:0706 
 7°6901)2°2051 
 7°6555}2°3223 
 7°6504/2°3390 
 7°6085]2°4721 
 7°5642)2°6045 
 7°539412°6951 
 T'S 1V7512°7362 
 7°4686/2°8669 
 7°4175}2°9969 
 7°39 1013°0615 
 7°3640|3°1258 
 7'3084)3°2539 
 7°2505)|3*3809 
 7°231913-4204 
 7*190413*5070 
 7°1280}5°6319 
 7°0636|3°7558 
 7:055413°7712 
 6°997013°8785 
 6°9282)4 0000 
 
 6°8618}4°1128 
 6°8573}4°1203 
 6°784.4]4°2394. 
 6°709414°35'7) 
 
 6*5532\4°5886 
 6°4721)4°7023 
 6°425'714°'7656 
 6°389114°8145 
 6°304114°9253 
 6°217215°0346 
 6°1841]5°0751 
 6°1284]5°1423 
 6°037715°2485 
 59452153530 
 5°927615°3725 
 57850815 °4.560 
 5°T54715 5573 
 5*6509|5°6569 
 Dep. | Lat. 
 Dist. 8. 
 
 3X 2 
 
 6°6518)4°44461/7°4832}5:0001 
 6°6323}4°4735}17°46 13]5 0327 
 
 8-998610-15711'9-998510-1745 1189 
 
 8-978 1|0-6278]'9°9'75610°6976 186 
 8-9658|0°7844|!9-961910-8716|185 
 9°9518}0-9809! 
 9-945211-0453 184 
 9-92551-2187||85 
 9-9027]1-3917)182 
 
 8°9567|0°8822 
 8-950710:9408 
 8°9329]1 0968 
 8°912411 +2526 
 8:9026}1°3206 
 8°889Q11°4079 
 8°8633]1°5628 
 8°8346]1°7173 
 8°8271]1°7558 
 8°8033]1°8712 
 769312°0246 
 739712°1773 
 7303]2°1868 
 *6933]2°3294 
 8°651312°4807 
 8°612.5{2°6126 
 8°6067}2°6313 
 8°559512°7812 
 8°5097}2°9301 
 8°473913 0320 
 8°4572)3 0782 
 8 °402913 2253 
 8°344.713°3715 
 8°314.013°4442 
 §°284.5|3°5166 
 8:22] 9136606 
 8°1568|3°8036 
 8°1359]3'8480 
 8°089 1}3°9453 
 8°0191]4°0859 
 7°946514° 2252 
 7°9373}4°2426 
 7°8'716|4°3633 
 7°794.2)/4°5000 
 
 7-7196|4°6269 
 7°7145|4°6353 
 7-6324|4°7093 
 7:5480|4+°9018 
 
 8 
 8 
 8 
 8 
 
 7°372415°1622 
 7287 9/5-2901 
 7°22980|5°3613 
 7°187715°4163 
 092 145+5409 
 994315 *6639 
 ‘957115 "7095 
 *894.415-9851 
 6°792415-9045 
 6'6883|6°0222 
 6*6686)6:0440 
 6*582216'1380 
 6°474116°2519 
 613640|6"3640 
 Dep. 
 Dist. 9. 
 
 7 
 6 
 6 
 6 
 
 9°8918}1°4673 
 9°8769}1°5643 
 9°8481]1°7365 
 9°$163)1°9081 
 9-8079}1°9509 
 9°7815}2°0791 
 9*743712°2495 
 9°7030}2°4192 
 9°7003|2°4298 
 9°6593]2°5882 
 9-6120}2°7562 
 9°5694]2°9028 
 9°5630|2°9237 
 9*5106|3-0902 
 9°4.55213 2557 
 9°415413°3689 
 9°3969}3°4202 
 9°335813 5837 
 9°2718)3°7461 
 9°2388}3°8268 
 9°205013°9073 
 9°1355|4°0674 
 9°0631)4°2262 
 9°0599)4°2756 
 8°987914°3837 
 8*9101)4°5399 
 8°8295)4°6947 
 8°819214°7140 
 8°7462/4°8481 
 8°6603/5 0000 
 8°5717315°1410 
 8°571715°1504 
 
 _ Dist. 10. 
 
 1]. . FOR DEGREES AND QUARTER-POINTS. 839 
 Dist. 6. Dist. 7. Dist. 8. Dist. 92, Dist. 10. | Course 
 Pts./D|} Lat. ] Dep. || Lat. | Dep. || Lat. | Dep. || Lat. ‘Dep. || Lat. | Dep. |]D]Pts 
 
 >i 
 
 pln 
 
 dle 
 
 pM 
 
 plo 
 
 ES 
 
 [oo 
 
340 LENGTHS OF CIRCULAR ARCS. Tab. 12. 
 
 D} Arc Del Arc 
 
 °0174533]| 6111°0646508]}12112°1118484 
 ‘0349066 
 *0523599]}| 63}1°09955741/123 
 *0698132}) 64]1°1170107 ! 
 "0872665}| 65|1'1344640 
 *104'7198}| 66}1°1519173 
 *1221730]| 67|/1°1693'706 
 *1396263]| 68}1°1868239 
 “L570 7961). 69} 1°20427'791 1 
 *17453291 '70)1-2217305 
 19198621] 71)1-2391838 
 
 — | | —— | ——_—_ |-—__ | ——_ 
 
 ~SOAMDNSHE HOW eH 
 
 OODOAIDUE PS Oto o 
 
 12} +2094395]| 72|1-2566371 
 13] +2268928]| 73]1-2740904 
 14| +2443461]] '74]1-2915436 
 15} *2617994l! "7513089969 
 16] *2792527|| 76|1-3264.502 
 17] -2967060}| '77|1+3439035 
 18} °3141595}| '78]1-3613568 
 19} *3316126]| '79]1°3788101 
 20} 3490659] 80|1+3962634 
 21} *3665191]) $1]1-4137167 
 29} +3839724)| 82)1-4311700 
 
 401429571] 83}1°4486233 
 *4188790|| 8411°4660766 
 *4363323}| 85)1°4855299 
 "4537856)| 86]1°5009832 
 4712389]! $7/1°3184364 
 
 *4886922]| 8811°5358897 
 *5061455]} 89}1°5533430 
 *52355988]| 90/1°5707965 
 *5410521}) 91175882496 
 *55850541| 92)1°605'7029 
 *5759587)| 93}1°6231562 
 *5934119]| 94)1°6406095 
 *6108652)| 95)1°6580628 
 *62835185}} 96)1°6755161 
 *6457718)| 97|1°6929694 
 *6632251]} 98}1°7104227 
 *6806784}| 99}1°7273760 
 *6981317]/100|1°7453293 
 
 "7155850 101]1°7627825I1 
 
 413119264. 
 
 421 -73303831|10211-7809358 491199173 
 431 -7504916l10311-7976891 431125082 
 44] °76794491110441°81514.24 
 451 °785398911105/1-8325957 
 
 "8028515 
 *8203047 
 °8377580 
 "3992113 
 "8726646 
 *8901179 
 "9075712 
 9250245 
 “94.24.9778 
 "9599311 
 ‘9775844 
 "9948377 
 
 106}1°8500490 
 107} 1°8675023 
 108)1°8849556 
 09}1°9024089 
 
 114]1-9896753 
 115]2:0071286 
 116/2°0245819 
 117/2-0420352 
 118}2:0594885 
 119]2-0769418 
 12012-0943951 
 
 Del Arc 
 
 i 
 
 t 
 “ 
 
“Lab. 
 
 03 
 04 
 05 
 06. 
 07 
 08 
 09 
 10° 
 L 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 
 24 « 
 
 “25 
 
 A TABLE of Rumbs, shewing the Degrees, Minutes, and 
 Seconds, that every Point and Quarter-point of the Com- 
 
 CL IHYL. LO. 
 “O01 | 
 02'|- 
 
 13. 
 
 CL | HYP. LO. 
 
 02302585 || +26 
 04605170] +27 
 ‘06907755 |] +28 
 09210340] -29 
 "11512925 || +30 
 *13815511]] -31 
 "16118096 |} +32 
 "18420681 |} -33 
 :20723266 |] -34 
 "23025851 || +35 
 "25328436 |] +36 
 ‘27631021 || +37 
 -29933606 || +38 
 “32236191 || +39 
 "34538776 || -40 
 "36841361 |} +41 
 "39143947 |] +42 
 "41446532 || +43 
 ‘ASTAOILT || +44 
 ‘46051702 || +45 
 48354287 || 46 
 ‘50656872 || “47 
 "52959457 || 48 
 55262042 |} +49 
 ‘57564627 || +50 
 
 ~-59867212 
 
 °62169798 
 *64472383 
 °66774968 
 *69077553 
 *71380138 
 *73682723 
 *75985308 
 "78287893 
 *80590478 
 
 *82893063 
 °85195648 
 87498234, 
 *89800819 
 *92103404 
 "94405989 
 “96708574 
 *99011159 
 °01313744 
 °03616329 
 
 "05918914 
 "08221499 
 *10524084 
 *12826670 
 15129255 
 
 Ce ee ee ee ee eg 
 
 pass makes with the Meridian. 
 
 Bergtits North 
 
 NbE Nb W 
 NNE NNW 
 
 NEbWN Ne 
 NE NW 
 
 NEbE |NWbW 
 ENE WNW 
 EbN WbN 
 
 | East West 902 0° FQ Fast 
 
 Pts.qr. 
 
 0 
 
 CMTTTINAAMAMAMOGOGT EEE EOW WOW HWW MK HHHOO 
 
 0 
 
 OrOUWN KF OWNK THONWH CUNO rH OW IDR 
 
 SWwe CDW KE CH 
 
 4} 
 
 COMMON AND HYP. LOGARITHMS. 
 
 341 
 
 HYP. LO. 
 1°17431840 
 119734425 
 122037010 
 1°24339595 
 1'26642180 
 128944765 
 1*3124'7350 
 1°33549935 
 1*35852520 
 1°38155106 
 
 140457691 
 1°42760276 
 1°45062861 
 1°4°7365446 
 1°49668031 
 1°51970616 
 154273201 
 1°56575786 
 1°58878371 
 1°61180957 
 
 1°63483542 
 1:65786127 
 1°68088712 
 170391297 
 1-72693882 
 
 as) 
 a 
 — 
 
 Ce ee oe 
 SHOONWOWHOWUH ON 2 2s Shae 3 
 
 ET UDO DD Sr Gr Or Or oe EB OO 09 09 00 
 
 OUWKH STHODWH OW? 
 
 CL 
 
 76 
 1 
 78 
 79 
 80 
 81 
 82 
 83 
 84 
 85 
 86 
 87 
 88 
 “89 
 90 
 ‘91 
 99 
 93 
 94 
 95 
 96 
 ‘97 
 98 
 -99 
 1-00 
 
 SSE 
 
 . 
 
 SEbS | SWbsS 
 
 SEbE {SW bW 
 
 ESE 
 
 EbS 
 
 HYP. LO. 
 
 1°74996467 
 1°77299052 
 1°79601637 
 1°81904.229 
 1*84206807 
 1°86509393 
 
 2°02627488 
 2°04930073 
 2°07232658 
 2°09535243 
 2°1183'7829 
 2°14140414 
 2°16442999 
 2°18745584 
 2°21048169 
 2°23350754 
 2°25653339 
 2*271955924 
 2*30258509 
 
 Tab. 14. 
 South 
 
 SSW 
 
 WSW 
 WbS 
 
 West | 
 
+ 
 pena Sige 
 
 34.2 A LIST OF ERRORS DISCOVERED AND CORRECTED 
 
 I. In Gardiner’s Edition of 1742, in 4to. 
 
 In the Logarithms. In the Sines. 
 101213 3630 ay) OF: aes 8°0738436 
 14 4059 0:59 ae 8:2360264 
 15 4488 1 7 48 82949277 
 16 4917 by 2h 0 8°3879622 
 17 5346 2 4 0 8°5570536 
 1)... 94° *19 9°2960174 
 21308 5427 1S, BRL 0 9:3668676 
 26719 8202 _ 32 3. 20 9°724983T 
 29315 0899 37: 86...20 275 diff. 
 34259 7747 eo a OU 275 diff. 
 34728 6798 DP is OR a ae 162 diff. 
 35704 7534 55, . 48 10 9°9171322 
 511938 2106 65 4 20 97 diff. 
 59502 5316 65 4 30 98 diff. 
 60844. 2178 65 4 30 9°9575403 
 64445 1892 ED OU aD 9°9743838 . 
 65640 1686 15 SG BW 52 diff. 
 66607 5199 ys Sara's Foal 3 L 9°9893657 
 67329 2022 82 0 40 9°9957646 
 69519 1035 sh AY 0 9°9988962 
 
 71492 2574 
 73338 3291 
 
 13983 1319 In the Tangents. 
 
 M4294 | 9537 DT ee ake 8°7674175 
 74.742 5647 8 36 20 9-1799393 
 "5561 Q977 10) "TS" 50 9°2564.267 
 "6000 8136 1345 2SH 480 9°3756001 
 76041 0473 43 56 30 9-9839523 
 "6031 9907 44 12 20 9°9879549 
 "7316 2694 68 19 20 10°4006638 
 82958 8583 "1 21 0 10*4717147 
 
 73: 18 0 10°5228579 
 17 1 40 10°6375975 
 
 Absolute Numbers. 84 4 10 10°9871756 
 
 6462 6492 . 86 39 40 11°2340287 
 8668 8688 BT ii LDu s 20 11°3300317 
 9167 9157 88 20 30 11°5383295 
 G9 5 Oo a 12°8520268 
 
 L to 20 places, 916. 96189 
 
 Expl. & Use, p.11, 1.4 bot. mg, m1. 
 
 4 
 
 Note, that some very few places are omitted where a figure does not perfectly ap- 
 pear, as they are not real errors, and cannot mislead, but may easily be filled up by 
 the differences.—It is also to be observed, that some of these errors in both the books 
 are not in all copies of the same edition, as I have experienced by collating divers 
 copies: a circumstance probably occasioned by types sometimes working out at the 
 press, and carelessly supplied again; and sometimes by discovering and correcting 
 errors after the copies of some sheets ‘have been but partly worked off. And the same 
 in the French edition following.— All the real errors in both books are brought toge- 
 ther in these tables, both those I have seen printed elsewhere, and those received by 
 private communication, besides upwards of twenty detected by myself, in comparing 
 the proofs of my book with the like parts of the others. 
 
IN LOGARITHMIC TABLES. 
 
 Il. In the Avignon Edition of 1770. 
 
 In the Logarithms, 
 
 100288 4897 
 100499 6174 
 101213 3630 
 14 4059 
 15 4488 
 16 4917 
 17 5346 
 14151 "871 
 17740 9536 
 24626 3939 
 25803 6702 
 33071 4473 
 34259 TT47 
 34.728 6798 
 37268 3361 
 3'7696 2953 
 38119 1415 
 42431 6833 
 43284 3274 
 44218 5991 
 44781 0938 
 46309 6654 
 46559 0036 
 51193 2106 
 54681 8364 
 58987 1563 
 59502 5316 
 59889 3471 
 60844 2178 
 63064 71815 
 64149 1899 
 64347 5283 
 64445 1892 
 64881 1175 
 68128 3256 
 68761 3422 
 68859 9607 
 69339 9776 
 69519 1035 
 69533 1910 
 70076 5693 
 71021 3864 
 74.703 3380 
 81674 0838 
 84393 3068 
 $5328 0916 
 86486 9458 
 $9322 9584 
 89680 6956 
 94841 9961 
 93614 3408 
 Absolute Numbers. 
 4770 4670 
 3520 5520 
 71235 7135 
 1635 1535 
 
 1 
 Sheet ¢, last line, M;, m’. 
 
 Pro. Parts. 
 
 Sh, i. 
 
 9 
 
 Dif. 94 
 | 85 
 
 In the Sines. 
 
 OR GH ele 7+2852698 
 Qo! ee pao 7°6568492 
 On FT 3 8°0325059 
 O° 445i,7 30 8°1217248 
 Sih LOR Soe 8°5795094 
 Diy Sea ve 8°6535839 
 2° «Boe wee 8°6607629 
 Bi78, SQue 86660134 
 3) abled 8°7459722 
 Si RG Med 8°7789797 
 18 44 20 621 diff. 
 45 4 50 9°8500947 
 65 4 = 20 97 ditf. 
 30 98 diff. 
 30 9°9575403 
 Lon IS Ry ne ee 8 9°9649579 
 BS 9O8 ie: SO 52 diff. 
 SB) 583,10 99999154 
 In the Tangents. 
 Fo OR a Lie 7°4314311 
 Oslo Say 7°5349960 
 Oey Rae re8 7°6215882 
 0. S00, oo 78372579 
 O 2&4 16 7°8487435 
 Oo 24 p54 7°859933 
 ON TST ere 80348694 
 Lige TOTS 8°3628023 
 ) RNR De ake 8°3826268 
 3 0 7 8°7196777 
 3 6.28 8°734.7535 
 Rt 9 8°7633926 
 3 54°38 8°8347909 
 5 5 0 8°9491676 
 Ut) 39 40 9°5029635 
 23 R20 9°6297224 
 23. 22 0 9°6355321 
 bers 9 4 40 9°8464809 
 Shiney E33: SO 10°3770260 
 Wate 20 ad 10°5241600 
 88 Bie ae 11°4685399 
 Sheet S 1 79 deg. 
 T 4 12° 60 
 
 Table to 20 places. 
 
 ao 77085°20 &e 
 825 91645°39485 
 1083 03462-84566 
 
 Table ILI. of the same. 
 
 Diff. II. ib. 00127 
 Logist. - Log. 
 
 Hyp. L. 6 
 
 531 75:47 kes 
 acne 
 
 "15 >| 1°9095425 
 
 In the last Page. 
 
 line 
 
 20 | 0-0019633 
 22 1:24.03375 
 24 | 1-2403375 
 26 | 5-8455077 
 
 343 
 
: 
 
 344 A LIST OF ERRORS CORRECTED, &c. 
 
 III. The following List of Errors in Callet’s Tables Portatives have been 
 discovered in reading his book with the proof-sheets of the 2d edition of 
 my Tables. . 
 
 In Callet’s Tables Portatives, Paris 1783. 
 
 In the Introduction. In the Logarithms. 
 
 Page 9, line 1, read_t’/b 47891 2539 
 41, — 9, — compris c H 60844 2178 
 
 19, — 3¢ 64113 | 9461 
 
 44, —10, — L’/$c 64445 1892 
 46, — 28, — Ap- 64547 8761 
 
 70357 3073 
 76872 7682 
 77054 8515 
 78050 3729 
 99018 7142 
 
 IV. In Taylor's Tables, London 1792; besides those mentioned in the book 
 itself. 
 
 Page 56, line32, for +2, read -+ L. 2 In the Sines. 
 57, — 10and11,readonly one root 
 Pol in 4° 93' 38” | 43007 
 
 “Rs MOR ead | 4 93 39 | 43981 
 
 5 3 
 — 25, for — read = 
 
 — 27, for 4p read 4 p 
 
 V. In Callet’s Stereotype Edition, Paris 1795. 
 
 In the Logarithms. In the Differences. 
 
 1014 3795 214 — 7 150 
 
 24626 3939 185 — 1 19 
 27502 9406 185 — 3 56 
 33071 44.73 66 — 6 40 
 
 4.3130 7795 
 53919 7418 
 56246 0916 
 57319 2986 0? 909m 38%. 5 « 2599 
 64445 1892 O 24 54 9331 
 81674 0338 
 
 99018 7142 
 
 In Table I. to 20 Places. 
 
 965 56538 
 1071 94608 
 1085 85148 
 1105 | 21729 
 1115 84779 
 1125 47981 
 1135. [| 29741 
 
 Jn Table II]. of the same. 
 00132 Dif. 34589 
 
 In the Tangents. 
 
 Printed by 8. Hamilton, Shoe.Lane, Fleet-Street, 
 
mn I 
 Si Jf 
 
 i i se pe i 2 nay | 7 
 ; he i be a my Vad) ” ¥ 
 J 7 - = Age | 
 Thee ey py | 
 At 
 
| in 
 
 3 0112 016734995