(Thp 9. li. HtU ICtbrary f>EClAL C0LLECT1 G57 North (Earoltna l^tatp HmnprBitii Z. SMITH REYNOLDS FOUNDATION COLLECTION IN SCIENCE AND TECHNOLOC3Y r-f TREATISE ON g0nter's scale, and the sliding RULE : TOGETHER WITH A DESCRIPTION AND USE, OF THfi SECTOR, PROTRACTOR, PLAIN SCALE, AND LINE OF CHORDS ; OR, AN EASY METHOD OF FINDING THE AREA OP,§tPER- FlCES ; AND OF MEASURING BOARDS ; AND OF FINDING THE SOLID CONTENTS OF BODIES, ESPECIALLY THAT OP TIMBER, BY THE SLIDING rule; and ALSO OF GUA- GING CASKS, AND ROUND TIMBER. TO THESE ARE ADDED SEVERAL USEFUL LOGAAITHMIt BLES, TABLES OF LATITUDE AND DEPARTURE, AND A TABLE OF NATURAL RADII, NOT KNOWN TO HAVj: BEEN HERETOFORE PUBLISHED : ALSO A TABLE OF ROUND, AND SQUARE TIMBER. "BY GEORGE CURTIS, MATH. WHITEHALL, N. Y. r"R!NTEO AND PUBLISHED BY E. ADAM!^* 1824. r)Lsirid of yutTcni^ — Tv Wit. BE IT REMEMBERED, That on V^e tw entv-cevent'a (Tnjir of May in the forty-ei^jhth year of the independence of th^ Cnited States of America, George Ctrtis of the said district-, jiath deposited in this office the title of a book, the rii:ht whereof he I laiais as authiT, in the words following, to wit : "A treatise on (inntei'*'3 Scale, and the ?lidin§^ Rule, together, with a description n ad use of the Sector, the Protractor, Plain Scale, and Line cf i.'horJa ; or an ea*ry method ef llndin^ the area of superfices. and of measuring Boards and finding; the solid eentents of bodies, especially ihat of tiniber, by the Sliding Rule, and al«o of g;auginj Casks and Round Timber. To these are added several useful Log;arithmick Ta- bles, Tables of Latitude and Departure, and a table of .Natural Radii, uot known to be heretofore published. By George Curtis, Tvlath." In conformity to the Act of Congress of the United States, e;:titled " an act fur the encouragment of learning, by securing tlie « opies of maps, cnarts and books to the authors and proprietors «?f 5 loh copies, duriHg tlie times therein mentioned."" JESSE GOVE, Clerk of th.t District of Fermont. A i»*!t C'Vhy r^f ReCi?rd, exafnln^jd and sidled by J. GOVE, CZf 7*, ^(P{:2^4U.^*4^ PREFACE, The primary design of this Treatise, is to introduic a more general attention to, and knowledge of the instruments herein described. Notwithstanding the many books that are daily issuing from the press, on other subjects, yet I have known of none that can preclude the benefits of a work of this kind. The author was led into these views by journeying through many parts of the United States, and finding n work on the subject here treated of, very much needed. It is designed as a pocket companion, for the Surveyor, Blechanick, and for those who deal in timber and boards, particularly by its tables of round and square timber, for the Wheelright, the Housecarpenter, Joiner, Cooper, and many other Dealers and Craftsmen, as well as the Grocer, for gauging expeditiously. How fitr the au- thor has succeeded in his views, and endeavours to ren- der it useful tb mankind, must be decided by the test ii experience. RECO.AhviENDATIONS. Haviog duly examined the inclossd W0rk,in maniiscript.. am constrained and deem it proper, to recommend it to the patronage of a genereus publick, as being a valuable and useful production : valuable from its tendency, to bring in- to general Bse, that which has, hitherto, been known to but few ; useful, in vastly abridging the labour of Arithmetical and Mathematical calculations, that the surveyor and artist "have frequently to make, also in containing some things, never before known to have been published. CYRUS CARPENTER, Physician f and Math e mat iciau-^ 'Whiting, Vt. May 24, 1824. Having been thoroughly convinced by ddiVy experience aat a work like the present was much needed as well by ihe mechanick va by the mathematician and the surveyor, !t was with no inconsiderable degree of interest that I gave it a perusal. After having attentively and minutely exan^- jued it, I find that the author has correctly anticipated the detects of works of this description in use, and has admira- bly supplied them. He has not only very much abridged ine labour of many mathematical estimations ; and simplified }nauy processes which were formerly tedious and perplex- ing ; but has embodied in his work many interesting facts which are original ; or at least not to be found in similar productions. Upon the whole, I consider the work well calculated to promote the interests of mathematical sci- »!nce, and sincerely hope it will meet with that liberal reception from an intelligent publickj which its merits de- ierye. SAMUEL BEACH, County Surveyor. Addison County, Vt, We. the subscribers, have perused with some degree jf pleasure, a Treatise on Gunter's Scale, Lc. prepared ■tr the press, by George Curtis, and have minutely exam- HECOMMEJ^DATIOjXS, 5 sued that part which relates to the measurment of timber and boards by the Shding Rule, and do cheerfully recom- mend it as being an easy and concise way of getting the contents of solid bodies of every description ; and the ready calculated tables of round and square timber which never have appeared in any work of this kind, cannot fail to render it valuable to all those who deal in lumber. ISAAC COLLINGS, Inspector of Lumber. ASA EDDY. Whitehall August 24, 1824. A2 39 39 T J / / ^/\ / FIO !st. 60 1 o 50 40 30 ^ % 1 D ^''^^ ^ ^^^-"""n*^ /90 \7^ r^ ^^ / /^^ y A Versed Sines C 10 110 170 160 150 140 130 120 110 100 &0 gO 20 SO 40 50 60«o?B 70 60 50 ^ 30 ' ' ' . cc ^ GO o po H > i-rt t— 1 _^ j ~ o 1 -B — © - - o - - o — I-* - ^ o - ^ ' -S - C3 -8 - hd w - " o -B — C9 ~ o o o _ lt». -J ~ o >;^ - o »5 CO — o - ^ - :s - o _ tfi -s _ o - <3> N^ CTT - -g *3 — o o - » o .u o - o CO o _C5 H-t "■" ^^ o - - v» ~ ^s _ § o " - - o> o i ^ ' 1 1 »T5 > C/2 o > o o CO > w o CONSTRtCTION OP THE PLAIN SCALE. 1. With the rac^ius intended for the scale, describe a se- micircle, (see plate 1st fig lit.) and from the ctntre C, draw CD, perpendicular to AB, which will divide the se- micircle into two quadrants, AD, BD ; continue CD to- wards S, draw BT, perpendicular to CB, and join BD and AD. 2. Divide the quadrant BD into 9 equal parts, then each of these divisions will be 10 degrees. S^ubdi>ide each of these parts into single degrees, and if your radius will ad- mit of it, into minutes, or some equal parts of a degree, larger than a minute. 3. Set one foot of the compasses at B, and transfer each of the divisions of the quadrant BD, to the right line BD, then BD will be a Line of Chords. 4. Then the points, 10, 20, 30, kc. in the quadrant BD, draw right lines, parallel to CD, to cut the radius CB, and they will divide that line into a line of sines, which muni be numbered from C towards B. 5. If the same line of sines be numbered from B towards C it will become a Ime of versed sines, which may be con- tinued to loO degrees, if the same divisions be transferred on the same line on the other side of the centre C. 6. From the centre C, through the several divisions of the quadrant BD, draw right lines till they cut the tangent BT so the line BT, will become a Line of Tangents.* 7. Set one foot of the compasses at C, extend the other to the several divisions, 10, 20, SO, k.c. in the tannent line BT, and transfer these extents, severally, to the right line CS, then that line will be a Line of Secants. 8. Right lines draw from A to the several divisions 10, * A mistake -was made in engraving this plate. The lints run- liin* parallel -^'ith C D, and the curved lines between B D, and the lines falling from the Tangents should intersect each other on tlic semicircle A D B ; and tlelast meatioued Jiaes jbould run in a di rection to centre at C. ^ THE PLALX SCALi:. \ i \^;ft ;l't^.^), 4o, iic. in the quadrant BD niH divide the the ra- dios CD into a hne of semi-tansents. ^. Divide the q^iadrant AD into cqnal parts, and from A as a centre, transfer these divisions severally into the 3ine AD, 'then AD will be a Line of Rhonibs, each division answering 1 1 degrees and 13 minutes upon the Line of Chords. The use of this division, or line, is for measuring angles, and protracting them, according to the common di- visions of the Mariner's Compass. If the radius AC be divided into 100, or 1000, &c. equal -parts, and the length of the several sines, tangents, and secants, corrod- ponding to the several arches ot the quadrant, be measured ti*ereby, and these numbers set down in a table, each in its proper column, you will by these measures have a col- lection of numbers, by which the several cases in Trigo- nometry, may be solved. Rigbt lines, graduated as above mentioned, being placed severally upon the Rule, form the instrument called the Plain Scale, (see plate 1st fig 2d,) by which the line and argles of all triangles may be measured. All right lines a? the sides of plain triangles, &c. wh-Bn they are considered simply as such, without having relation to a circle, are measured by scales of equal parts, each of which is subdivided into ten equal parts : and this serves as the common divisions to all the rest. In most Scales, an inch is taken for a common measure, and whatever an inch is divided into, may be found at the end of the Scale : di- vided in this manner, any number less than 100, may be readily taken. But if the number should consist of three places of t3gures, the value of the third figure, cannot be exactly ascertained ; and in this case, it i« better to use a Diagonal Scale, by which any number consisting of three places of figures, m?ty be exactly found. The^'figure? of this Scale are given in plate 1st, fig 3d ; its construction is as follows : Having prepared a ruler of a convenient breadth for your Scale, draw near the edges, two right lines, AF, CC, parallel to each other ; divide one of these imes, as Fx\ into equal parts, according to the size of your ^>cale, and through eacii of these divisions, draw riirht lin^s perpendicular to Ai\ to meet CC, then divide the breadia into ten equal parts ; and through each of these divisions e'ravT ri^^ht lines, parallel to AF and CC. Divide the hoes AB, CDj irito teti eraal parts, and throui-h each point draw lines parallel to CD to inter^^ct the arch BD Take B a» a centra transfer the several points of intersection, to the line BD, and their n'ltnber ii from D towards B, from to 60, and it will be the Line of Lon'^tude. HOW TO PROVE THE SLIDING RULE. Rule. — Draw out the slider to the right hand, till 1 oa (he slider coincides with i on the iised part, then 2 on the slider will coiuriJe with 4 on ike dxed part. Continue t« draw the slider till 1 en the slider coincides with 3 on the nxed part, then 'i on the slider will coincide with 6 on the ttxed part : till 1 on the slider coincides with 4 on the nxe4 part, then 2 on the slider will coincide with 8 on the fixed part ; till 1 coincides with 5. then 2 will coincide with the centre I ; till 1 coincides with 5^, then 2 will coincide with 11 ; till 1 coincides with 6. then ? will coincide witb 12 ; and thus rontinue to do till ycKi have gone through the line, and if the Rule is correctly graduated, each num- ber will correspond as above stated ; if they do not correi- pond, the Rule is oot cdirect,- atid c^ynic^nently will not gire a correct aftswer. 13 GUNTER^S SCALE. Gunter*^ Scale has upon it eight lines : 1. Sine Rhombs, (marked SR) corresponding to the Logarithms of Natural Sines of every point of the Mariner's Compass, numbered from the left hand towards the right, with 1, 2, 3, 4, 5, 6, 7 to 8, where is a brass pin ; this line is also divided, where it can be done, into halves and quar- ters. 2. Tangent Rhombs, (marked TR) corrfesponds to the Loojarithms of the tangents of every point of the Com- pass, and is numbered 1, 2, 3 to 4, at the right hand, where there is a pin ; and thence towards the left hand with 5, 6, 7 ; it is also divided, where it can be done, into halves and quarters. 3. The line of ntimhers, (marked NUM) corresponds to the Logyriihms of numbers, and is marked thus : at the left hand it begi::s at 1. and to\rards the right hand are 2, 3, 4, 5, 6, 7,8, 9,. and 1 in the middle ; at which, is a brass pin ; then 2, 3, 4, 6, 6, 7, 8, 9 and 10 at the end, where there is another pin. Tlie value of these numbers anri their intermediate division, depends on the estimated values of the extreme numbers, 1 and 10 ; and as this lint is of great importance, a particular description of it and its uses, will be aiven. The first 1, may be counted for 1, or 10, or 100, or 1000, and then the next 2 is accordingl}, 2, or 20, or 200, or 2000, &c. Again, the first 1 may be reckoned one tenth or one hundredth, or one thousandth part, &c. and then the next 2 is two tenth, or two hundredth, or two thousancltli parts, &c. &c. Then, if the first 1, be reckoned 1, the mid- dle 1 is reckoned 10 and 2, at its right hand is 20, 3 is 30, 4 is 40, and 10, at the end is 100 ; Again, if the first 1 is 10, the next 2 is 20, 3 is 30, and so on, making the middle 1,^100, the next 2 is 200, the next 3 is 300, 4 is 400, and 10 at the end is 1000. In like manner, if the first 1 be es- teemed one tenth part, the next 2 is two tenth parts, and the middle 1 is one, and the next 2 is two, and 10 at the end is ten. Again, if the first 1 be counted one hundredth part, the next 2 is two hundredth parts, the middle 1 is now ten hundredth parts, the next two hundredth parts, the B 14 GIWTER'S SaiLE. middle 1 DOW 15 ton hundredth parts, or one tenth parfj and the nest 2 is two tenth parts, and 10 at the end is counted 1. As the figures are iucrea«ed or diminished in their value, so in the like manner, mast all the intermedi- ate strokes or subdiyisions, be increased or diramished ; that is, if the drst 1 at the left hand be counted 1, then ij next ibllowing is 2, arid each subtlivision between them nov? is one tenth part, and so all che wav to the middle 1, which now is 10, the nest is 20 ; — now the longer strokes between 1 and - are to be counted from the centre 1, (11, 12,) Inhere is a brass pin, then 13. 14, 15, sometimes a longer stroke than the rest; then 16, 17, 18, 19, 20, at the t]gure 2 ; and in the Siinie manner the short strokes be- tween the ligures 2 and 3, and 4 and 6, &c. are to be reckoued as units. Again if 1 at the left hand be 10, the li^ures between It and the Oiiddle 1, are common tens ; ana the subdivisions between each tigure are units ; from the middle 1, to 10 at the end, each ri^ure is so many hun- dredths : and between these ligures each longer diviiion is ten. From this description it will be easy to find the di- visions representing any given number ; thu«, suppose tije point re ^'resenting the number 12, were required, take the division at the figure 1, in the middle tor the dist figure of 12, then for the second figure count tw© tenths, on longer strokes to the right hand, and this kist is the point representing 12, where the brass pin is. Again, ?uppose the number 22, were required ; the first figure be^ns 2, 1. Take the division to the figure 2, and for the second figure 2, couat two tenths onward and that is the point representing 22, Again, suppose IT-o were requir- ed. For the firot fisiure, 1, take the middle 1 : for the second figure, 7, count onward as betare, and that is 170<1', and as the remaining fii^ures are 28, or nearly oO, note the point which is nearly -j^, lor the distance between the marks 7 and 8, and this will be the point r€pre5«ntiDg 1720. If ^he point represeuthng -iSb was required ; from the 4 in the second intervid count towards 5 on the right, three of the i.crgerdivisir^ns and one of the smaller, (this smaller divi>iun being midway between the marks 3 and 4.) and that will be the division expressing 435, and the like of other nuia- '^ers, which by « little practice is easily done. All fractions fouiiJ m this line m^izt be decimal?, and ii GUA'TER'S SCALE. 15 they are not, they mu«t be reduced to tlecimals ; which is easily done by extending the compasses from the denom- inator to the numerator ; that extent hiid the same way irom 1 in the middle or right hand, will reach to the deci- mal required. Example, To find the decimal fraction eqn:J to |- ; extend from ■1 to 3, that extent will reach from 1 on the middle to 75 towards the left hand ; the hke may be observed of any oth- er vulgar fraction. Multiplication is performed on this line by extending from 1 to the multiplier, that extent will reach from the multiplicand to the product. Suppose, for example, it were required to tind the product of 16 multiplied by 4 : extend from 1 to 4 ; that extent will reach from 16 to 64, the product required. Divsion, being the reverse of Multiplication, therefore, extend from the divisor to unity ; that extent will reach from the dividend to the quotient. Suppose 64 to be divi- ded by 4 ; extend from 4 to 1 ; that extent will reach from 64 to 16= the quotient. Questions in the Rule of Three are solved by this line, as follows. Extend from the first term to the second — that extent will reach from the third term to the fourth, or an- swer. It ought to be particularly noticed, that if yon extend to the left from the first number or term to the second, yon must also extend to the left from the third to the fourth, and vice versa. EXAMPLE. If the diameter of a circle be 7 inches, and the circum- ference 22 inches, what is the circumference of anothdr circle, whose diameter is 14 inches ? Extend from 7 to 22 ; that extent will reach from 14 to 44, the same way. The superficial contents of any Parallelogram are found by extending from 1 to the breadth ; that extent will reach from the length to the superficial contents. EXAMPLE. ^ Suppose a plank or board to be 15 inches wide and 27 feet long. The contents are required. Extend from 1 foot, to 1 foot ? inches, (or 1,25;) that Id GUTTER'S SCALE. extent will reach from 27 feet to 33,75 = the fuperncial content?. Or, you may extend from 12 inches 15, ^c. The solid contents of any Bale, Box, Chest. &:c. is found •>y extending from 1 to the breadth, that extent will reach .rom tlie depth to a fourth number ; and the extent from 1 to that iourth number, will reach from the length to the solid contents. EXAMPLES. 1. What is the solid contents of a square pillar, whose length is 21 feet 9 inches, breadth 1 foot 3 inches ? The extent from 1 to 1,25 will reach from 1,25 the depth, to 1,56,= the contents of 1 foot in length. -Again, the extent from 1 to 1,56, will reach from the length = ?1,75, to 33,9 or 34, nearly=the solid contents in feet. 2. Suppose a square piece of timber 1,25 foot wide, and ).56 foot deep, and 36 feet long, be given, to tind the solid :ontents. Extend from 1 to 7 ; that extent will reach from 3S to .5,2 = the solid contents. In like manner may the contents of Bales be found, which divided by 40, will give the ton- nage. 4. The Line of Sines, (marked SIN) corresponding to the Logarithmick Sines of the degrees of the Quadrant be- gins at the let^t hand, and is numbered towards the right ; thus, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, then 20, 30, 40, ^c, ending at 90°, where is a brass centre pin, as there is at 'lie ri^ht hand of the lines. 5. The Line of Versed Sines, (marked VS) correspond- ing to the Log. Versed Sines of the degrees of the Quad- rant, begins at the right hand against 90° on the sine ; and Vom thence is numbered towards the left hand ; thus, 10, .'0, 30, and 40, &c. ending at the left band, at about 169°- iuach of the subdivisions from 10 to 30 is, in general two degrees ; from thence to 90, is single decrees ; from thence 10 the end, each degree is divided into 15 minutes. 6. The line of Tangents, (marked TANG) corresponding :o the Log. Tangents of the degree? of the Quadrant, be- ^^ins at the left hand, and is numbered towards the right : thus, 1, 2, 3, 4, and so on to 10,20,30, 40 and 45, where is a hrasspin under 90° on the Sines ; from thence it is numbered backwards 50, 60, 70, GO, 6ic. to 89, ending at the left hand DESCRIPTION AXD USE OF THE SECTOR. IT where it begins at 1 degree. The subdivisions are near- ly similar to those of the Sines. When you have any ex^ tent in your dividers to be set off from any numberless than 45° on the line of Tangents towards the right, and it is found to reach beyond the mark of 45**, you must see how far it extends beyond that mark, and set it off tOTvards the left, and mark what degree it falls upon ; which will be the number sought, and must exceed 45°. If, on the contrary, you are to set off such a distance to the right, from a num- ber greater than 45°, you must proceed as before, only re- member, that the answer will be less than 45°, and you must always consider the degrees above 45°, as if they were marked on the continuation of the line to the right hand of 45°. 7. The line of the Meridional parts, (marked MER,) begin at the right hand, and is numbered thus, 10, 20, 30, to the left hand, where it ends at 87** 8. This line, with the line of equal parts, (marked EP) under it, are used together, and only in Mercator's s;iiling. The upper line contains the degrees of the meridian, or latitude in Mercator's Chart, correspondmg to the de- grees of longitude on the lower line. The use of this scale in solving the usual Problems of Trigonometry, plam sailing, middle latitude sailing, and Mercator's sailing, will be given in the course of this work ; but it will be u-ineces- sary to enter into an explanation of its use in calculating the common Problems of Nautical Astronomy, it is more accu- rate to perform by Logarithms. DESCRIPTION AND USE OF THE SECTOR. This instrument consists of two rules or legs, represent- ing the radius, moveable round an axis or j«iiDt, the mid- dle of which, represents the centre, from whence several scales are drawn on the faces ; some of the scales are sin- gle, others double. The single scales are like those on a common Gunter's scale ; the double scales are those that proceed from the centre ; each of these being laid twice on the same face of the instrument viz. once on'each leg. From these scales dimen::ions or distances are to be taken B2 18 DESCRIPTION AXD USE OF THE SECTOR. when the legs of the instrument are set in an angular posi- tion. The single scales being used exactl}' like Gunter's scale, 1 shall proceed to enuaierate a few of the use*^ of the double scales, the number of which is seven ; viz. the scale of lines, (marked LIX or L.) The scales of chords, (marked CHO or C.) 1 he scale of sines, (marked SIX or S.) The scale of tangents, to 43^ and another scale of tangents, from 45*^ to about 76^ both of which are marked TAN or T. The scale of secants, (marked SEC or S.) And the scale of polygons, (marked POL.) The scale of lines, chords, sines and tangents, urider 45'^ are all of the same radius, brginning at the centre of the instru- ment, and terminating near the other extremity of each leg, viz. the lines at the division 10; the chord at 60° ; the sine at 90° ; and the tangents at 45°. The re- mainder of the tangents, or those above 45° are other :?cales, beginning at a quarter of the length of the former, counting from the centre, where they are marked with 15°, and extend to about 76°. The secants also begin at the same distance from the centre, where they are mar- ked with 0, and are thence continued to 75 Jeg. The scales of polygons are set near the inner edge of the legs, and where these scales begin, they are marked 4 ; and from thence are numbered backwards, on towards ttie cen- tre 12. In describing the use of the Sector^ the terms Lateral Distance, and Transverse Distance, often occur. By the former is meant the distance taken with the com- passes on one of the scales ooly, beginning at the centre of •he sector. By the latter is meant the distance taken be- tween any two corresponding divisions of the scales of the same name, the legs of the sector being in an angular posi- tion. The use of the sector depends upon the propor- tionability of the corresponding sides of similar triangles; demonstrated in Art. 53 Geometry. For, if in the triangle ABC we take AB=AC and AD=AE, and draw DE and BC, it is evident that DE and BC, will be parallel. — Therefore, by the above mentioned proposition AB;BC : : AD : DE ; so, whatsoever part AD is of AB, the same part DE will be of BC ; hence if DE be the chord, sine or tangent of any arch to the radius AD, so BC will be the Scin^ to the radius AB. I'he line, of Uaes, is useful to divide a given lite in- DESCRIPTION AXD USE OF THE SECTOR. 19 to any number of equal parts, or in any proportion, or to find the 3d and 4th proportionals, or to increase a given line in any propor- tion. EXAMPLES. 1. To divide a given line into any number of equal parts, as, suppose 9. Make the length of the given line, a transverse distance to 9, and 9 the number of parts pro- posed ; then will the transverse distance of 1 and 1, be one of the pirts, or ^ part of the whole ; and the trans- verse distance of 2 and 2, will be two of the equal parts, or -I of the whole line, kc. 2. Ifa man should travel 52 miles in 8 hours, .how far would he travel in 3 hours at the same rate ? f Take 52 in your compasses as a transverse distance, and set off from 8 to 8, then the transverse distance 3 and 3, being measured laterally, will be found equal to 19-i-, which is the number of miles required. 3. Having a chart, constructed upon a scale of 6 miles t© an inch, it is required to open the sector, so that a cor- responding scale may be taken from the line of sines. — Make the transverse distance 6 and 6, equal to the lateral distance 3 and 3 ; then set off any distance from the chart laterally, and the corresponding transverse distance, will be the reduced distance required. 4 One side of any triangle being given of any length; to measure the other two sides, on the same scale. Suppose the side AB, of the triangle ABC, measure 50, what are the measures of the other two sides 2 See fig. Take AB in your dividers, apply it transverse- ly to 50 and 50, to this open- ing of the sector, apply the distance AC in your com- passes to the same number on both sides of the rule, transversely ; and where the two points fall, will be the measure on the line of hnes, B 20 USE OF THE LL\E OF SIXES, i-c. the distance required ; the distance AC, will fall against 63 and 63, and BC against 45 and 45 on the line of lines. The line of chords oo the sector is very useful for pro- tracting any angle, when the paper is so small, that an arch cannot be drawn upon it with the radius of a common line of chords. Suppose it was requiretJ to set off an arch of 30^, from the point C, of the small circle ABC ; take the radius in your compasses, and set it off transversely, Trom 60° to 60° on the line of chords. Then take the transe- Terse extent from 30° to 30° on the chords ; and place one foot of the compasses at C, the other will reach to E and CE will be the arch required. By the converse ope- ration, any angle or arch, may be measured ; with any ra- dius describe an arch about the aii2:ular point : set that ra- dius transversely from 60° to 60°. Then take the dis- tance of the arch, intercepted between the two legs, and apply it transversely to the chords, and it will shew the degrees of the given angle. .Vo<«. When the angle to be protracted exceeds 60'-'', you must lay off 60° ; and then the remaining parts ; or if it be a- bove 120^\ lay off C0° twice; r.nd then the remaining parts. — In this way any arch above 60° aay be measured. e USE OF THE LINES OF SINES, TAN- GENTS AND SECANTS. By the several lines disposed on the sector, we have scales of sevenil r;£uii. So that, 1. Havi.iv a length or radius given, not exceeding the leigth of the sector when opened, v.e cm find the chord, sine, 6:c. of the same. Thus, suppose a choid, sine or tan. eTit of 20°, to a radius of 2 incbt.> be required- M. ke 2 in ;-> the transverse opening to 60 ' and 60° on the c' : then will ihe same extent reach from 15*^ to 45° USE OF THE LLYES OF POLYGO.\'S, 21 on the tangents, and from 90° to 90° on the sines, so that to whatever radius the line of chorcfe is set, to the same are all the others set also. In this disposition, therefore, if the transverse distance between 20° and 20° on the chords be taken with the compasses, it will give the chord of 20° ; and if the transverse of 20** and 20°, be in like manner taken on the sines, it will be the sine of 20° ; and lustly if the transverse distance of 20° and 20° be taken on the tangents, it will be the tangent of 20°, to the same radius of 2 inches. 2. If the chord, or tangent of 70° were required ; for the chord you must first set off the chord of 60°, or the ra- dius upon the arch, and then set off the chord of 10°, to tind the tangent of 70°, to the same radius the scale of up- per tangents must be used ; the under on« only reaching to 45°, making, therefore, two inches, the transverse dis- tance 45°, and 45° at the beginning of thp scale, the ex- tent between 70°, and 70° on the same, will be the tan- gent of 70°, to 2 inches radius. 3. To find the secant of any arch ; make the given ra- dius the transverse distance between 0, and on the se- cants ; then will the transverse distance of 20° and 20°, or 70° and 70°, give the secant of 20° or 70° respective- 4. It the radius, and any line representing a sine, tan- gent or secant, be given *, the degrees corresponding to that line, may be found by setting the sector to the given radius, according as a sine, tangent or secant is con- cerned ; then taking the given line between the compasses, and applying the two feet transversely to the proper scale, and sliding the feet along till they both rest on, like di- visions on both legs, then the divisions will shew the de= grees, and parts corresponding to the given line. USK OF THE LINE OF POLYGONS. The use of this line is to inscribe a regular polygon in a circle. For example ; let it be required to inscribe an octagon in a circle. Open the sector, till the transverse distance, 6 and 6 be equals© the radius of the circle ; then will the transverse distance of 8 and 8, be the side of the inscribed octagon, or polygoq, ^2 USE OF THE SECTOR LN TRIGO- NOMETRY. AH proportions in Trigonometry -ire easily wrought by the double lines on the sector ; observing that the sides of triangles are taken off the line of lines, and the angles are taken off the sines, tangents or secants, accord- ing to the nature of the proportion. Thus, in the triangle ABC, we have given AB = 56, AC = 64, and the angk ABC = ^6° 30' to the rest in this case. We have (by art, 58 geometry) the following proportion : as AC (G4) sine of the B (46^ 30') : : AB (56) : to the sine of the angle C, and as the sine B : is to the side AC : : so is the sine A : to the side BC ; therefore to work these proportions by the sector, take the lateral distaace 64 = AC from the line of lines, and open the sector to make this a transverse dis- tance of 46° 30' = angle B on the sines ; then take, the lateral distance 56 = AB on the line of lines, and apply it transversely on the sines, which will give 39° 24' = to C ; hence the sum of the angles B and C is 53° 54' which taken from 180°, leaves the angle A = 94° 6', th^n to work the secant proportion, the sector being set at the ^same opening as before, take the transeverse distance of 94° 6' = the angle A on the sines, or which is the same thing, the transeverse distance of its supplement 85° 54' ; then this applied laterally to the lines, gives the side sought BC =88. In the same manner we might solve any prob- lem in trigonometry, with the tangents and secants, in- stead of measuring them on the sines, as in the preceeding example. All the problems that occur in mutlcal astronomy, may be solved by the sector ; but the calculations by logarithms are much more accurate. THE SLroiNG RULE Consists f f a fixed part and a slider, and is of the same diaieojions as a common Gunter's scale : and has the same THE SLIDIKG RULE. 23 lines marked on the fixed part, as that scale ha?, and also on the plain scale ; and these lines may be used with a pair of compasses, in the same manner, as the lines of those scales. As a description of those lines, has already been given, it will be unnecessary to repeat it here : It being sufficient to observe, that there are two lines of numbers, viz >• a line of Logarithmick sines, and a line of Logarithmick tan- gents on the shder. And the slider may be shifted, so as to lix either face of it on either side of ihe tixed part of ihe •cale. according to the nature of the question to be solved. In solving any problem in Arithmetick, Trigonometry, plam sailiug &c. let the pro-portion be so stated, that the first and thiid terms may be alike, and of course, the second -and fourth terms will be alike. Then bring (he tirst term of the analogy on the fixed part, against the second term on the slider ^ and against the third term on the fixed part, will be found the fourth term on the slider. Or if neces- sary, the first and third terras may be found on the slider, and the second and tborih on tl-e fixed part. Multiplica- tion and Division are performed by this rule or method ; only consider unity as one of the terms of the analogy, MULTIPLICATION BY THE SLIDING P.ULE. To perform multiplication, set I on the line of num- bers of the fixed part, against one of the factors on the line of numbers on the slider ; then against the other factor on the fixed part will be found the product on the slider. JVoie. if the first and second terms are alike, or of one name, instead of the first and thin!, you must bring the first term on the slider, against the thirJ on the fixed part, and against tl:re second term on the slider, will be found the fourth term oa the fixed part ; or if necessary the first and second terms may be found on the fixei part, and the third and fourth on the slider. EXAMPLES. 1. To find the product of 5, by 12. Draw out the slider, till 1 on the fixed part, coincides with 5 on the slider ; then opposite 12 on the fixed part, will be found 60=: the pro- duct, on the slider. 24 THE SLIDLVG RULE. 2. To find the product of 50, by 1 2. Not moTins the sH der, count 5 to be 50 ; count 12 a« before ; then opposite 12 on the dxed part, will be found 600 on the slider. 3. Place the slicker as before, count 5 to be 500. and 12 t« be 1200; the ansv^er, 600,000; will be found on the slider. 4. To find the product of 17 hy 25. Draw out (he slider till 1 on the fixed part, coincifies with 17 on the slider, then opposite 25 on the fixed part, will be found 425 on the slider. 5. To find the product of 17 by 17. Draw out the slider, till 1 on the fixed part coincides with 17 oo the «li«ler, then opposite 17^on the fixed part, will be found 289 on the slider. 6. Place tne slider the sanse as before ; against 60 on the fixed part will he found 850 on the slider. 7. The slider laying at 17 as before, count 50 or 5 to be opposite 500 on the fixed part, w ill be found 8500 on the sli'.ier. 8. Place the slider as before, count 17 to be 1700, count 3 to be 300 on the fixed part, then opposite 300 on the fixed part will be found 510,000 on the slider. 9 To find the product of 2! 4- by 20. Draw out the slider till the centre 1 on the tixed part coincides with 21i on the slider, then opposite 20 on the fixed part will be lauiid 430 on the slirler. 10 To find the |jroducl of 5 by 2^ count the first 1 on the fise^ p irt. to bs yV' *^'^ centre one count 1, draw out the slider till 1 on liie fixed part coincides with 5 on the slider, opposite 2^ on the fixed part will be found 15^ oh the slider. DIVISION BY THE SLIDING Rl'LE. Place the divisor on the line of numbers of the fixed part against 1 00 the slider, then against the dividend found 90 the fixed part, will be found the quotient on the slider, BXA5tPLES 1 To divide 60 by 5. Set 5 on the fixed part against 1 on the slider, then agaiast 60 on the fixed part, will be fouod r2=the quotient 00 the slider. THE SLIDLKG rule. ' 25 2. To divide 400 by 27. Set 27 on the fixed part against 1 on the slider, then against 400 on the fixed part, will be found 14|-f- or about H^ on the slider. 3. To answer several questions, not moving the slider in in one lesson, the slider placed as in Exanaple 2. Div You will have gone the length of the fixed part to A, on the state- ment. isors. Dividends. 27 -:- 400 27 -:- 600 27 -:- 600 27 -:- 700 27 >:- 800 27 -;- 850 27 -:- 900 27 -:- 1000 Qnotients. 112 2 or Ui 18-i^or 184- ^2 I- 2 9.^ or ii,- 25|4- or 26 nearly 29-1^ or 2P2 Si-^-f or 31^ 33^ 37^V Of 37 4. To,divide any nnmber from 700 to 6000 that is at B on the s^lider, the full of the extent of the slider. From the statement, dr<-.w out the slider on A, to the left hand of the centre 1, to the figure 6, representing 60 on the fixed part, over I on the slider ; then against 7 representing 700, on the fiXed par', will be found 11-| on the slider. Now, not moving the slider, you may find this lesson from 700 to 6000. ivisors. Di^ V'idends. Qtiotients. 60 - :- 700 1^4 60 - :- 800 13^' 60 - - 900 15 60 - - 1000 16-2 60 - - 2000 33i Ending at 60 - - 3000 50 B, at the 60 - - 4000 6e|. right hand 60 - - 5000 8c 1 of the >ii^ 60 -. - 6000 100 der. EXAMPLES IX THE RULE OF THREE BY THE SLIDING RULE. If 3 lbs. of beef cost 21 cents, what will, from 30 to 100 lbs. cost? C 2C THE SLIDLXG RULE. This lesson reaches from 30 to 70 diflerent statement?, viz. from 50 to 100, not moving the slider ; bring 3 on tlio letter A, of the fixed part, on the line of numbers against 21 on the Hne marked B, on the shder ; then against 30 on the fixed part on A, will be found on the slider, $2,10 ; and against 35 lbs. will be 2,43 ; 40—2,80 ; 50—3,50, 60—4,20; 75—5,25; 90—6,30; 100—7,00. 2. If 4-1- yds. cost g23, what will 20 yds. cost ? RULE. Draw out the slider, till §23 coincides with 41- on the fixed part ; then against 20 on the fixed part, will be found ^102, on the slider. Now not moving the slider, at A, on the fixed part, 100 yds. will be found to the an- swer on the slider ==§5,11. 3. If 4 lbs. of sugar cost §l,50,what will 20 lbs. cost? Bring 4 on the line of numbers on the fixed part, against gl,50onthe line of numbers on the slider ; then against 20 on the line of numbers, on the fixed part, will be found §7,50 on the slider. Now not moving the slider, against 40, on A, will be found §15,00 on the slider. Again, not moving the «lider, against 80, on A, will be found §30,00 ; and at^A, 100 lb? on the slider, B will be found §37,50 — ABCD on the right of the rule. 4. To find the circumference of a circle, wbose diame- ter shaU be 20. RULE. Draw out the slider, till 22 on the slider coincides with 7 on the fixed part, then against 20 on the fixed part, will be found 6:4. or Gt'^ on the slider. Again, not moving the slider, against 25 on the fixed part, will be found 7?^- on the slider. Again, not movins: the slider, against 40 on the fixed part, will be found IScf on the slider. Again, not moving the slider, against 60 on the fixed part, will be found \V>i^ on the slider. Now, not moving the slider, a- gainst 100 at A, on the fixed part, will be found 314-f on tie slider. 5. If one yard of cloth cost §9,00, what wdl ^ of a V'frd cost ? 'Draw out the slider, till 9 on the slider coincides with 10 on the fixed part, then against 5 on the fixed part, will oe fo«nd on the slider §2,83. THE SLIDING RULE. 21 BOARD MEASURE BY THE SLIDING RULE. EXAMPLES. 1. To measure a board 12 feet long, atid 12 inches wide : 12 on the fixed part, to the right of the centre 1 counts 12 feet in length ; but 12 on the slider give 12 to 12, or 12 i'eet 2. A board 12 feet lon^, and 19 inches wide. Draw out the slider, till 19 coincides with 12 on the fix- ed part, that makes the board 19 feet=the answer on the slider ; 19 inches the answer in feet. 3. A board 14 feet long and 20 inches wide. Draw out the slider till 20 coincides with 12 on the fix- ed part, then against 14 on the fixed part, the answer 23-^ ft. 4. A board 22 feet long, 20 inches wide. Draw out the slider till 20 inches coincides with 12 on the fixed part, and against 22 on the fixed part, will be found the answer 361 feet on the slider. 5. A board table that extends fiom 4 to 100 feet in length, and 36 inches wide. Draw out the slider, till 36 coincides with 12 on the fix- ed part ; count the 1st 1 on the fixed part 10 ; begin at 4 on the fixed part 4, so on to 10 at the centre, and so on to 100 on the right hand, to A. Begin on the slider at 4, and reckon at different lengths. 28 THE SLIDIXG RULE. 6. A log 14 feet long cuts 27 board?, each board 36 .iicbes wide, how many feet in one board ? Ans, 4-2 feet. J low many feet in the 27 boards ? Draw the slider till 27 coincides with the centre 1, a- gninst 42 on the tised pari, will be the an=wer on the sli- ;cr=to 1 134. A log 12 feet long, 24 inches in diameter, cuts 15 boards, ?0 inches broad. Draw out the slider, till 20 is agrdnst 12 : 20 will be the answer for one board. Draw out the slider till 15 comes against the centre 1 on the fixed part, and against 20 on the fixed part will be found 300 on the slider=the answer in board measure. A log 2 feet in diameter, and under : 2 inches on each side are allowed for «lab, and 4- for saw calf, and 1 board for wane : and from 24 to 36 inches diameter, 3 inches for >lab, ^ for saw calf, and 2 boards for wane. A log 28 inches at the small end, will cut 13 boards, on- jy 16 measured. Draw out the slider till 22=th« breadth of the board, comes against 12, and against 14 =the length on the fixed part, will be found the answer on the slider tor one board =254-. Now draw out the slider till 16, the number of hoards, comes against the centre 1 : novv to find the rest, say the log was 14 feet long, your answer Qn the slider is 414 feet nearly. Again, a log 14 feet long. 36 inches at the small end slab- )ed, leaves the board 30 inches wide, i tor sawcalf, leaves 24 and 2 wane leaves 22. Draw out the slider till 30 comes against 12 on the fixed part, and under 14 on the fixed part, vvill be found 35 on the slider. Then draw out the slider till 22 comes a^inst the centre 1, and against 35 will be lound 770 on the slider, which will he the an- swer tor a log 36 inches in diameter, and 14 feet long. A log 20 inches at the small end, and 16 feet long, cuts 13 boards that are 16 inches wide, and but 12 measured, bow many teet ? An?. 255 A log i6 inches in diameter. 14 feet long, cuts 9 boards, and but 8 measured, how many feet? Answer 112 fee- on the slider. THE SLIDIXG RULE. 29 TO MEASURE SQUARE TIMBER IN SOLID FEET BY THE SLIDING RULE. To measure a stick of timber 60 feet long^ RULE. Draw out the slider to the left hand, till the length of the timber found on the slider, shall correspond to 12 on the girt line ; then against the mches, the stick is square on the girt line, will be found the number of cubic feet on the sli- der. EXAMPLE. and from 5 to 40 inches square. Draw the slider to the left hand, till 6 on the slider (which call 60,) corresponds with 12 on the girt line, and against 6 on the slider, will be found 10^^^^ on the girt line. The same answer can be found by drawing the slider to to the right, but the divisions are not so easily distinguished, without more practice. By letting the slider remain, you may solve all the ques- tions proposed above in a short time, and will put their answers m a table for exercise. laches Cubic feet laches Cubic feet hiches Cubic feet -«_• sqr. in the slick. sqr. in the stick. s.jr. in the stick. 0) r1> 5 10,42 14 82 23 220 C 6 7 15 20,42 iH 15 88 94 23^ 24 2311 242 "ra 71 23i log lOOl 24i 250 w.< 8 261 16 106^ 25 260 ^h 30tV 17 120^ 26 282^ -s 9 33^ 18 135 27 303 ■^j H 371 181 142 28 327 .a 10 41f i 19 150 29 352 Q i 10^ 46 19^ 158- 30 o to 11 50,42 ::o 166* 31 402 en 111 1 12 1 12| i 13 55 60 65i 7Cf 2ui 2(J| 21 22 ICO 184| 202 32 ] 35 38 40 426 510 602 667 nea -a c •-I C3 30 THE SLIDLYG RULE. TO MEASURE HEWN TIMBER THAT IS NOT SQUARE, BY THE SLIDNG RULE. EXAMPLES. i. To Iind the solid feet in a stick of limber, 50 feet m lengthy and 7 by 10 inches. Draw out the slider, till 50 coincides with 12 on the grirt line, and against the thickness, 7 inches, founfl on the girt line30u will find 17 on the slider, whioh is the answer, at 7 inches square. There will then remain 3 times 7=21 inches, and 50 (eci long, yet to find ; to obtain whicli, draw (he slider to the right, till 21 on the slider coincides with 12 on the line marked A ; then against, 50, the length found on A, you will find ol~ on the slider ; this must be divided by 12 and it will give 7 feet ?~, inches or ', and the depth of the hold in feet, and then divide the pro- duct by 95 ; the quotient is the number of tons. In double decked vessels, half the breadth is taken for the depth GUAGING OF CASKS. Having found the number of cubic inches in any body, by the preceding rules, you may from thenre, determine their contents, in gallons, bushels, he. by dividing that number of cubic inches, by the number of cubic inches in a gallon, bushel, &c. respectively. A wine gallon, by which most liquors are measured, contains 231 cubic inch- es. AJ?eer, ale, or milk gallon, contains 282 cubic inch- es, ^bushel of corn, malt, fiic. contains 2150,4 cubic inches. This measure is subdivided into 8 gallons ; each of which contains 268,8 cubic inches. In all the following rules, it supposes the dimensions of a cask or ressel, to be given in inches, and decimal parts of an inch. PROBLEM I. To find the number of Gallons or Bushels in a Vessel of a Cubic Form. Divide the cube of one of the sides in inches by 231, and it gives wine gallons, divide the same cube by 282 and it gives beer gallons ; and divide by 2150,4 and it gives the number of bushels the vessel will hold. EXAMPLE. Required the number of wine and beer gal- lons, also the bushels contained in a cubic box or cisterR whose side is 60 inches. QUAGL^V. 3i 50 50 2500 50 B31) 125000 (541,1 gallons, wine measur^v 1155 950 924 260 231 290 230 ;?82) 125000 (443,26 gallons, beer measur*, 1128 1220 1128 920 846 740 664 1760 :/>lo0,4) 125000,0 (58,1 bushels. 107520 174800 172032 27680 -4d CUAGLXa. PROBLEM II. To find the number of Gallons or Bushels contained in a body of a Cylindrical Form, Multiply the square of the diameter of either end or base, ly the length of the cylinder, and divide the product by 294,12 and the quotient vviH be the number of wine gal- lons divide the same number by 359,05 the quo- tient will be the number of beer or ale gallons ; and di- Tide the product by 2730 and the q"uotient will be the Bumber of bushels. J\'oie. The above numbers for divisors are found by di- viding 231, 282 and 2150,4, by the deeimal ,7854. EXAMPLE. Required the number of w?ne gullons, in the cylinder delineated in the figure of problem 4, of mensur- ation. The diameter of its base, being 5 feet = CO inches , and the length 13 feet = 156 inches. 60 294,12) ^^61600,00 (1909,42 Ans. 60 29412 3G00 267480 t56 264708 21600 277200 18000 264708 3600 124920 561600 dividend. 117648 72720 Here observe, that two cyphers are affixed to the pro- dDct to equal the nomber of decimals in the divisor, which makes the quotient the number of gallons : but the other cyphers added to the remaiQder, gives decimals. GUAGING. 41 PROBLEM III. To find the number of Gallons or Bushels contained in a body of the form, of a Pyramid or Cone. See figure in Problem 6, of Mensuration. Multiply the area of the base of the pyramid or cone, by one third of its perpendicular height ; the product divided by 231 will give the answer in wine gallons; divide by 282 it will give the answer in beer gallons ; and divide by 2>o0,4 and it will give the answer in bushels. EXAMPLE. Ptequired the number of beer gallons, con- tcined in a pyramid, whose base is 30 inches square, and perpendicular height is 60 inches ? 30 inches, side of the 30 square base. 900 20 \ the height Inches in a beer gal. 282) 18000 (63,8 Ans. 1692 1080 2340 2256 84 • PROBLEM IV. To find the number of Gallons or Bushels contained in a Vessel in the form of a Frustum of a Cone. Multiply t-he top and bottom diameters together, and to the product add one third of the square of the dilTerence of the same diameters, then multiply this sum by the per- pendicular height, and divide the product by 292,12 for wine gallons ; by 339,05 for beer gallons ; and by 2738 for bushels, D2 42 GUAGIXG. EXAMPLE. Given the diaraeter DC =40 inclies ; the di- ameter AB = 30 inche?, and the perpendicular height FE = 60 ioches 5 the conteats ia wine gallons are required. F 40 bottcm diameter 30 tap diameter 10 dicference 10 3) 100 sqr. of the difference C (<::T7.'.T:T:::^'i> 33,3 =i the square 30 (op diameter ,40 bottom diameter 1200 33,3 1233,3 60 perpendicular height 294,12) 73998,00 (231,59 gallcJns, wine measiire, 58824 151740 147060 46800 29412 173880 147060 268200 GUAGLXC^. 43 PROBLEM V. To Guage a Cask. To guage a cask, measure the head diameters FA and DC, and if they differ, take the mean of them, that is, add the diameters of each head together and divide the sum by 2 ; measure also the diameter EB at the bung, ta- king the measure, inside of the cask ; then measure the length of the cask, making due allowance for the thickness of the heads : Having these measures, you can calculate the contents in gallons or bushels, by the following rule : Take the difference between the head and bung diame- ters, maltiply this difference by ,62 and add the product to the head diameter, the s.Qm v^ill be (he mean diameter ; multiply the square of this by the length of the cask, and divide the product by 294,12 for wine, by 359,03 for beer gallons, and by 2733 for bushels. The decimal ,62 is commofily used by guagers, to find the mean diameter > but if the staves are nearly straight, it would be more accu- rate to use ,55 or less. If on the contrary, they are very curving we should use ,64, ,65, or more. When the staves are straight, the decimal ,61 may be used. EXAMPLE. Given the bung diameter BE 34,5 inches, the head diameter AT or CD=30,7, after allowing for the thicknes of the heads ; 59,3 inches the length, find the number of wine gallons this cask will Isold, 44 QUAGINQ. 34,5 bung diameter. 30,7 head do. 3,8 ,62 76 228 2,336 30,7 head diameter. 33,056 mean do. 33,056 198336 165280 99168 99168 1092,699136 call the decim. 67 1092,67 69,3 length on the in- side. 327801 983403 646335 294,12) 64795,331 (220,3 Ans. 58824 59713 58824 88931 88236 695 Calipers are used by gunger?, in taking the dimensions of casks, but a comuion rule, or a staif may be usetl. A more expeditious way of nnchng the- contents of tasks, is by the line of numbers on Gunter's scale ; or on the sliding rule ; in order to perform which, make marks on the scale, on the calipers, at the points 17.13 and 18,95 inches and at o2,33 inches, which numbers are the square roots of 294,- 12, and 359,05, and of 2738 respectively. A brass pin is generally tixed on the calipers at each of these points, which are called the guage points. Having prepared the scales in this manner, you may calculate the number of gallons or bushels, by the following rule : Extend from I towards the left hand to ,62, or less, if the staves be nearly straight ; that extent will reach from* the difference between the head and bung diameters, to a num- ber at the \eti hand, which number added to the head di- ameter, will give the mean diameter ; then put one fcot of the compasses on the guage point = 17, 15, for wine gallons, 18,95, for beer gallons, and at 52,33, for bushels ; and ex- tend the other fDOt of the compasses to a number deooting the mean fiiameter ; this extent turned over twice the same way from the length of the cask, will give the num- ber of gallons or bushels respectively. In the preceding example, the extent from 1 to ,62 will reach from 3,8 to 2,4, nearly, which added to 30,7 gives the mean diameter =33,1, then the extent from the guage point, 17,15 to 33,1 turned over twice from the length 59,- 3, will reach to 220,9, wine gallons. If you had used the guage point 18,95 the answer would have been in beer gal- lons ; or if you had used 52,33 the answer would have been in bushels. GUAGING CASKS BY THE SLIDING RULE. On the line marked D, you may find the guage point, marked WG ; you will see that you can find 17,15 inches, a little to the right of the long mark, that is over the cen- tre of G ; also over AG on the same scale you will find 18,- 95, or 18|-|,very near the long mark over the centre of G.'here is the guage point for the ale or beer gallons, as the other was for wine gallons. Set the length of the cask found on the slider, against the guage, point on D ; and a- 46 GUAGI.^Z^. gainst the mean diameter on D, the answer will be found on the slider. In measuring the length of a cask, we allow for the thickness of both heads, 1 inch, l-i or 2 inches, according to thei^ize of the cask. AoJ'c. You must take the hfcad diameter, close to its outside, and for small casks add -^ inch, for casks of 30, 40, or 50 gallons, add -^ inch, and for larger casks add 5 or 6 tenths and the sum will be very near the head diame- ter within. in taking the bung diameter, observe in moring the rod or staff backward and forward, to see if the staves oppo- site the bung, be thicker or thinner than the rest, if so, make the necessary allowance. EXA3IPLE. Given the bung diameter BE= 34,5 inches, the head diameter AF,or CD=30^7 inches. 34,5. Lengthof the cask with- 30,7 in=59,3. £) 65,2 32,6 ,5 33,1= mean diameters Now draw out the shder, till 59,3 on the slider, coin- eides with the L'uage point on the girt line, for wine gal- lons, and against 33,1 on the girt line, will be found on the slider =220,9 wine gallons. PROBLEM VI. The gnage point for bushels is placed on the girt line, at 13,r ^:^ inches, as it would run of the rule on the right. — For the points for gallons i reverse it back to the left of those points. Draw out the slider, till the length of a square box or bin, coincides with the guage point on the girt line, to wit against IS-j^ inches, then against the number of iuche-^ OUAGIXG. 47 the box IS square, found on the girt line ; and on the sli- der will be tbund the number of bushels. EXAMPLES. 1. Given 7,75 inches square and 30 feet in length, also beginning at 7,75, and extend to 40 inches square. Against 7,75, will be found 10 bushels. " nearly. 9 <• 134 - 10 «( 16| - 12 t« 24 ♦• 13,40 < < 30 *^ 15 *i 37| - 17 i< 48| ♦' 19 *{ 60 *' 20 <4 661 «' 21 M 73| - 24,5 (( 100 «* 25 «^( 1045. '* 27 4( 121^ '* 29 (i 140i " 30 (( 150 '* 35 (( 208 '* 40 (i 2674- '' ** nearly. When the bin is more than 40 inches, say 60 inches, and 20 feet long, draw the slider to the left hand till 20 feet, the length found on the slider, coincides with the guage point found on the girt line, to wit, 13,385 inches ; then against the width of the bin, namely 60 inches, which found on the girt line, will be found on the slider 399, calling the figures on the girt line tens, and those on the slider will be hundreds ; and so of any other number. 2. Suppose a bin 30 feet long, and from 40 to 245 inch- eB square. Against 40 will be found 267 bushels. 60 ' 5981 100 ' 1675 120 * [ (C 2500 150 ' 3790 190 * 6000 245 ^' 9950 43 CASTl.yQ LXTEREST O.V THE >LfDiyQ RULE. To find how many burljels any cylindrical cask will con- tain : or how many bushels of timber any log will contain, providing it be a complete cylinder. Draw the slider, till the guage point, 15,001. or nearer, lc),0015, found on the girt line, coincides with the length of the cylinder in feet, found on the sli»3r ; then ag?»iaat the diameter of the cylinder, found on the girt line in inch- es, will be the number of bushels found on the slider. EXAMPLE. Suppose the cylinder 300 feet laid to the guage point, then against 7, or 70 inches for a diameter, you will find 6444 bushels, the answer on the slider. TO CUT OFF ANY ? ? MBER OF CUBIC FEET OF ANT DIAMETER, OF ROUND TIMBER. Suppose the number of feet required to be cut off, be 3, and the diameter be T-i- mches. Draw out the slider till 3 coincides with 7i on the fixed part, then against the guage point 13,54 inches will be found the length to be cut off. EXAMPLES, 1. If ^2jl5buy 1 foot of timber how much will jllO buy ? 4, G5 cubic feet, Ans. 2. This timber is 4,3 inches in diameter ; what is the length of the stick ? Lay 4,65 feet on the slider, against 4,3 inches, on the girt line, and against 13,54 inches on the girt line will be found 46,5 feet, the length of the stick on the slider. For square timber, draw the slider so that the number of inches the stick is square, found on the fixed part, co- incides with the number of feet on the siller : then against 12 will be fou43d the number of feet in length to be cut off TO CAST INTEREST ON THE SLIDING RULE FOR ONE YEAR. Let the priocipd or numbar of dollars be found on A, and put the per cent on the slif'er agaii.?t the centre 1 ; then against the principal, will be found the interest, calling dollars, cents. To cast interest for days, firtd th* days for GASTIXG LXTEREST OX THE SLIDIXG RULE. 49 one year or 3G5, on the fixed part A ; then draw the sli- der till the interest for one year betore found, coincide with the 365 days ; now, on the fiied part A, noiice the number of days you want to get the interest for, and under that on the slider, will be found the interest; for tlie davs required. EXAMPLES. 1 What is the interest g333,33 for one year and twenty = five days, at 6 per cent. First for one year, by the above directions, will be lound <^20 ; now notice 365 days on A, and draw the slider till g20 coincides with it or under 365 days, then look for 25 days on A, and on B, under 25 will be ct 1.33- the answer for 25 days. 2 What is the interest of j^^lOOO for one year and thirty SIX days at 7 per cent. Draw out the slider till 7 on the slider coincides with the centre 1 and against 1000 on the right hand at A, will be found 70 on the slider. Then lay 70 on the slider against 365 days on the tised part and against 3d on the slider will be found g6,oo Ans. ;^76,88. A Tahle^ for the use of Coopers^ in calculaiing Cisterns, Wine gallocs and parts of sralions. •i^O ®Q >- GN O G^ o s^ i^ o '-" e< cj T-. o ^ O Lj u'i o O 1> O ^': ^n !> CO !?? i^i »-» CO L-) ft) o c: 'o G) — .ri j> SD t- CO cc T —' c^ G) T*' -1- '?4 i-Oxocot-cococriO^'?^rii--:;t>coco~co(:o Smallest head i? x' ;r "-^ ^ O o g< -r •:c o: c C o^ O ~ ^ t- i« feet and ^ -- --' ^ ^ " O O o o o - o C O 6 O C mche?. Largest head in feet aad incl.es. "^ "^ "^ -f "j^ -♦• ^"^ t-O u*; lit- xQ lq o tc cr uo ^o o T» up lr^ ■-? u^ o i^^ tc to to ^.g i> r^ t- {> t-. Depth of the ! 2 i^ Z ":5 ~ '^ ^ '^' -r :C' x o O t- t- »- o:> CO E iO A LOG TABLE. A LOG TABLE, SliOTx-in:^ the nuiuhcr cf feet of boards, {lAt^ lo^ Xi^ill tnu'i'^i whose diameter^ is from lo to 3Q inches at the smallest endy Kind from 10 io 15 feet ia length. >— 1 5' 3 o HI 1— » ■rt) CD CO c C9 3 7^ a si' 3 2. wl 1 ! 1 o i ^■ 3 o 5' 'a ti 5' a 5' 3 2 5" 3* O it" 5' 1-^ 5' 3* O O 5' a 15 yo 15 99 15 108 Tr 117 i 'td 126 15 135 16 100 16 MO 16 120 16 130 1 16 140 16 150 17 125 17 137 17 150 17 160 17 175 ! 17 187 ' 18 1 55 18 170 3 8 186 i 18 201 18 216 18 232 19 165 19 176 19 198 ! 19 214 19 230 19 247 JO 172 20 189 20 206 20 263 20 246 20 258 21 184 21 202 21 220 21 238 21 256 21 276 ', '22 194 22 212 1 o<^ 232 22 263 22 294 22 291 i2:3 219 23 240 1 23 278 23 315 23 332 23 333 s?^i 250 24 276 24 300 ! 24 325 24 350 24 375 25 200 25 308 25 336 25 364 25 392 25 420 l26 299 26 323 26 346 26 375 26 404 26 448 127 327 27 367 27 392 27 425 27 457 1 27 490 28 3G0 28 396 28 432 28 462 28 504 1 28 540 29 376 29 414 29 451 29 488 29 526 29 564 !30 412 3U 452 30 494 30 535 30 576 30 618 (:>] 428 31 4;i 31 513 31 558 31 602 31 642 ;^- 451 32 496 3'2 541 32 587 32 631 32 676 3.: 490 33 539 33 588 33 637 33 686 33 735 ^34 532 1 |34 585 34 638 34 691 34 744 34 798 1^5 582 35 640 ob 698 35 752 35 805 35 863 il^. 593 iH 657 ^Q> 717 36 821 36 836 36 889, A T.1BLE,<^'C. 61 .'2 Table of Specific Gravities of Bodies. Platina (pure) - - Fine Gold _ - - Standard Gold - - Quicksilver (pure - Quicksilver (common) Lead _ _ _ _ Fine Silver - - Standard Silver - Copper _ _ - Copper halfpence - Gun iMetal - - Cast Brass - - Steel - - - - Iron _ - _ _ Cast Iron - - - Tin - - - - Clear Crystal Glass Granite _ - - Marble and hard stone Common green Glass Flint - - - - Common Stone - SSOOO'Clay 19400 Brick 17724 14000 1 3600 1132o 11091 10535 9000 8915 8784 8000 Common Earth - - Nitre _ _ - _. Ivory _- _ - _ Brimstone ~ - _ Solid Gunpowder - Sand - _ - - - Coal - - - - - Box-wood _ - — Sea-water — - - Common-water - - 7850 Oak - - 7645 7425 7320^Ash 3150 3000 2700 2600 2570 2520 Gunpowder, close shake Ditto, in a loose heap iMapIe - - - - Elm _ -. _ . Fir _ - _ - Charcoal - - - Cork _ - - - Air at a mean state 2160- 2000 1984 1900 1826 1810 1745 1520 1250 1030 1030 1000 925 n93T 83G 836 too 600 550 240 Kote, The several sorts of wood are sujjposed to be dr3\ Also, as a cubic foot of water weighs just 1000 ounces a- voirdupois, the numbers in this'table express, not only the specific gravities of the several bodies, but also the weight of a cubic foot of each, in avoirdupois ounces ; and there- fore, by proportion, the weight of any other quantity, or the quantity of any other weight, may be known, as in Ihe^ next two propositions. PROPOSITION I. To find the Mapiitude of any Body, from its Jf'ei^ht, As the tabular specific gravity of the body, Is to its weisfht in avoirdupois ounces, So is one cubic foot, or 1728 cubic inches, To its contents in teet, or inches, respectively. TO FIND THE WEIGHT OF A BODY, 4'c. EXAMPLES. 1. Required the contents of an irregular block ofcom- : stone, which weighs 1 cwt. or 112 lb ? Ans. 1228|f l-l cubic irxhes. 2. How many cubic inches of gunpowder are there in lb. weight ? Ans. 29|- cubic inches nearly. 3. How many cubic feet are there in a ton weiglit of ry oak ? Ans. 38i|f cubic feet. FHOrOSITION II. To fiyid ike Weight of a Body from its Magnitude. As one cubic foot, or 1728 cubic inches, Ih to the contents of the body, So is the tabular sy)ecific gravity, To the weight of the body. EXAMPLES. 1. Required the weight of a block of marble, whose englh is 63 (eet, and breadth and thickness each 12 feet ; eing the .dimensions of one of the stones in the walls of "albeck ? Ans. 603,-^ ton, which is nearly equal to the burden of A) East- India ship. 2. What is the weight of 1 pint ale measure, of gunpow- der ? Ans. 19 oz. nearly. 3. What is the weight qf a block of dry oak, which measures 10 feet in length, 3 feet broad, li feet deep or thick ? Ans. 4335f| lb. A TABLE OF SOLID MEASURE OF SQUAPtE TIMBER, By the following Table the solid contents, and the value of any piece or quantity of timber stone &c. may be found at si^ht, from 6 inches to 29-^ inches, the side of the square, or one fourth of the girt ; and from 14 feet to 92 feet in lengih. It rises from 6 inches, - inch at a time till it rises, to 29* inche?, and from 14 feet, 1 foot at a time till it rises .^ TABLE OF SQUARE TIMBER. 53 to 92 feet. The number of inches which the side of each 'Stick measures are placed at the top of the two first or lei't hand columns, and at the top of the two columns at the right hand of each double line. These columns £;ive the length and contents of each stick, and the other two columns which run from the top to the bottom of the pa^e are a continuation of the two first; so the len^j-th of the stick will be found in the first and third column from the left hand of the page, and from the right hand of each double line ; and in the second and fourth columns the contents. The half feet are not reckoned ; that is, when a stick measures, for exanple, 30 cubic feet and 5 inches it is reckoned only 30 feet, and if it measures 30 cubic feet and 7 inches it is reckoned 31 feet, &c. this is the method of reckoning timber in Quebec and Montreal, ard "In all markets in the United States. A TABLE OF Sqi\3RE TIMBER side d 1 61 IS.'SiJe 6> 51 1 o-i-"^5ae 7 51 17.j=ic e Tt 5\ 20: •x las. 52 13 .- Ids. 52 15! "■-* 1 Ins. 52 17: \ — I"^-; 52 20: c 00 54 56 13 13 ^-1 if is 53 54 ?? 06 161 16' 16; 2 j 53 54 55 06 IS 181 18' 19! ^ 53 !54 bb 56 2C- 21 21 22 il4 3 14 114! 4 i6i 14 5 M 5 115 4 14!15| 4 57 17;jl5 5 57 i9;;i5 . 6 '157 22 \V3 4;58 M :16 5 58 17!lie 5 58 19;|16 6;!58 22 !l7 415? 1 Oil 17 5 59 17:|r7 6 59 ' 20;ll7 6\ 59 ^ :-60 15;jl8 5 60 18'!l8 6 60 20;ll£ 7I 60 23 l-^i 15*10 G 61 IS' 19 6 61 20|*19 7. 61 24 . 1 -, ^ •jo- 15120 6 62 18; 20 7 62 21; 20 si 62 24 -i IJC 1621 6 6S 18;j2i 7 00 21; 21 S| 63 24 *i£' 5 C4 16!i22 6 64 7 64 21 22 • 1 81 64 25 23 6 60 16|23 16it24 7 63 IS- 23 65 00' O-D 9i 65 25; ;24 6 66 7 66 19 24| 8 ge 22 1^4 A" 9:|66 26! ^25 6 67 17|i25 "7 67 20! 2c 8 67 22!!25 10167 26 l>6 6 68 I7|2e 8 68 20! 2c 9 68 23l|26 10' 68 26 1:27 r- / 6P 17*27 8 69 20' 27 9 69 23 27 10l69 271 ■on 7 70 171 28 8 70 21: 00 9 70 23; 28 11 70 lli71 27, 129 7J 18=29 9 71 72 2lj|29 10 71 24| 29 27 ;:^o 72 jSjISO 9 21; 30 10 72 24:30 11|72 28 ol c 73 18' 31 973 2lj 31 10 73 25131 12173 281 32 74 18; 32 19133 977 22; 32 11 74 25^32 12 74 13 75 29! "33 8 75 10;75 22i 33 11 75 25i .33 23| 34 r 76 19134 10i76 22 34 11 1^ n- .'O 26; 34 13 76 29i '^5 9 77 19j 35 191 3e 10j77 23^35 12 77 26 35 13 77 30i 3G p It 11{78 23]36 12 78 26)36 14i78 30 -?? c 69 20;;37 il|79 23; 37 12 79 27: 37 14179 31 c 2V 20;|30 III8O 23 38 13 80 27 38 15j 80 31 ■ '9 !0 81 £0;!39 ii;3i 24 ;39 13 81 27139 l^i 81 31 '40 10 32 83 20;i40 12:82 24. 40 13 82 28'j40 Vo^. 8232 Ui 10 2^1 41 12 83 24:41 14 83 28 41 161 8332 :ie w 84 21; 42 12J84 25 42 14 84 28.42 16itC4J33 i43 11 85 21 13 13 85 25|J43 14 85 29, 43 17ii85 331 ■44 11 86 ,21; 44 13 86 2544 25145 15 86 29: 44 17186 34 !45 11 37 22' 45 13'S7 15 87 39::45 17 87 3« 16 11 88 -ogl 46 1380 26;i46 15 88 30^ 46 18 '88 34! ■47 12 79 22'; 17 14;89 2l-.i47 16 89 30I47 18 80 351 ;tp IC 9t'> 22,148 Hiso 2o! 48 16 9C 30' 48 19 90 35 : ' K- 91 23f 49 14;9i -"i 49 16 91 31! 49 19 91 35| l?i99 23156 lo92 271 50 17 92 31 '=50 19jl92!36' A TABLE OF SQUARE TIMBER. 55 Side 8 51 22 Side 8i i'5l 1 C)c Side 9 51 281 Side 9| 51 32 hr- lus. 5'Z 23 ^ Ins 1 52 1 26 ►t- Ins 52 29 1 - Ins. 52 32 -< O c ^3 54 55 26 27 27 (-r o" Ot9 O o in 53 54 55 30 31 31 o' g 53 54 55 33 34 34 14 "~6 56 25 14 7 56 28 14 s 56 32 14 9 66 36 15 6 57 25 15 7 57 28 15 8 57 32 15 9 57 36 16 7 58 26 16 8 58 29 16 9 58 33 16 10 58 36 17 7 59 26 17 8 59 29 17 9 59 33 17 10 59 37 18 8 60 26 18 9 60 30 18 10 60 34 18 11 60 37 19 8 d 33 81 68| 39 3G 81 74 40 28 82 57 40 30 82 63 40 33 82 69 40 36 82 t c- 41 29 83 57 41 31 83 63 11 34 83 70 41 37 83 7C !42 29 84 58 42 32 84 64 13 35 84 70 42 38 84 i43 30 85 59 43 33 85 65 43 36 85 71 43 39 85 78 44 30 86 60 44 34 86 66 44 37 86 72 44 40 86 6r 45 31 87 60 25 35 87' 61 45 38 87 73 45 41 87 SO ;40 32 88 61 46 35 88 67 46 38 88 74 46 42 88 C'' J47 33 89 C2 47 36 89 68 47 39 8S 75 47 43 89 81 148 33 90 a 48 37 90 69 48 40 90 75 48 44 9G 82 |49 34 91 63 49 38 91 70 49 41 91 76 4r 45 01 iS joO 34l92 64i 50 38 92 7^ 50 42 9- -77 ■;•" ,.^^g. '4 A TABLE OF SqUARE TIMBER. .Side 12 51 51 3idel2^?5l! 55| Sid e 13151 1 60ilSidel3-^ 51 64' 1 Ins. 52 52 -^ Ins.|5o 56 'hr* Ins.! 52 61j - lus. 52 66 o' 3 1 aq O o D 53 54 55 56 53 54 55 56 QfQ H _ IT a 53 54 55 57 59| 60| O £3 O o r— 53 54 55 56 62 63 64 66 u O o 3 17 53 54 56 56 67 68 69 71 H 14 15 56 H ~16 15 15 57 57 15 16 57 62 15 17 57 67 15 19 57 72 16 16 58 58 16 1758 63 16 19 58 68 )6 20 58| 73 17 17 59 59 17 18 59 64 17 20 59 69 17 21 59 74 18 18 60 60 18 19 60 65 18 21 60 70 18 23 60 76 19 19 61 61 19 2161 66 19 22 61 71 19 24 61 77 20 20 62 62 20 22 62 67 20 23 62 73 20 25 62 78 |21 21 63 63 21 23 63 68 21 25 63 74 21 26 63 79 i22 22 04 64 22 24 64 69 22 26 64 75 22 28 6-^ 81 23 23 65 65 23 25 65 71 23 27 65 76 23 29 65 82 24 24 66 66 24 26 66 72 24 58 66 77 24 30 6o 83 25 25 67 97 25 2767 ^73 25 29 '61 78 25 31 67 85 26 26 68 68! 26 2868 74 26 30 '68 80 — c 33 68 86 27 27 69 69 27 2969 75 27 32 69 81 27 34 69 87 28 28 70 70 28 3070 76 Vc< 33 j70 82 2b 35 70 88 29 2971 71 29 3171 77 29 34 71 83 ?.9 36 71 90 30 30 72 72 30 33|72 no 30 35 72 84 30 38!72 91 31 3173 73 31 34 73 79 1 36 '73 85 31 39 73 62 32 32 74 74 32 35 74 80 38 '74 87 32 40 74 93 33 33 75 75 33 36 75 81 33 39 to 88 33 42 76 95 34 34176 76 34 377C 82 34 40 16 89 34 43 76 26 35 35 77 77 35 38|77 83 35 41 77 90 35 44 67 97 36 36 78 78 36 39^78 84 36 42 78 91 o- 45 78 98 37 37 79 79 37 40379 86 37 43 79 92 37 47 79 I'OO 38 38 80 8C 33 4l|80 87 38 44 80 94 38 48 80 101 39 3981 81 39 42^81 88 39 46 81 95 39 49 81 102 40 40 82 82 40 43:82 89 40 4 7 82 96 40 50 82 103 41 4183 83 41 44J83 90 41 48 83 97 41 62 83 105 42 42 84 84 42 45 84 46 85 48*86 91 42 49 84 99 42 53 84 106 43 43^85 85 43 92 43 50 85 100 43 54 85 107 44 44^86 86 44 93 44 5186 101 44 55 86 109 45 ■45|87 8- M ^. 49 87 94 45 53:87 102 45 57 87 no 4^ 46|88 -88 46 50 88 95 46 54 88 103 46 58 88 111 47 47J89 89 47 51|89 96 47 55 89 104 47 59 89 112 48 48:30 ■ 9C 48 52j90 97 48 56 90 105 48 60 90 114 49 49191 91 49 53|91 98 49 57 91 106 49 62 91 115 (50 5C )j92 92 50 54192 100 50 58 92 108 50 63 92 116 .1 xAIjLL Oh ^^iLAllE TIMBER. •Side 14 l~5 In?. 1 ■3\ © il4 ~T9 15 20 116 22 17 23 18 24 19 26 20 27 21 28 22 30 23 31 24 32 25 34 26 35 27 37 28 38 29 39 30 41 31 42 32 43 33 45 24 46 35 47 36 49 37 50 38 51 39 53 40 54 41 55 42 57 43 58 44 GO 45 61 41 62 47 64 A? 65 49 66 h 68 ? 51 i 52 - o oo 54 55 56 57 58 59 60 61 62 63 64 65 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 Z'- 8'-; n n OO s-^ 90 CI 92 691 Sid 71 72! 7 75 76'[14 77 79 el 4.^ Ins. 80!il7 81 83 84 86 87 88 90 91 92 94 95 96 98 99 100 102 103 104 106 107 108 no 111 113 114 n ii' ir^ 1? K-1 122 1 ?.\ 125 1 19- 120 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 12 43 14 o 20 22 23 25 26 27 29 30 32|64 33 35 36 38 39 41 42 44 45 46 48 49 51 52 54 51 52 oo 54 55 56 57 58 59 60 61 62 63 65 66 67 68 69 70 71 72 73 74 75 76 ,77 78 79 55'80 67 58 60 61 63 64 65 67 68 70 71 73 81 85 83 84 85 8': 8": 88' 8C 90 91 9'i 74 76 77 79 80 82 83 84 86 87 89 .90 92 93 95 96 98 99 101 102 103 105 106 108 109 111 112 114 115 117 118 119 121 122 124 12C 127 128 i30 131 133 134 Side 15 — Ins. c 05 15 16 17 18 19 20 21 22 |23 j24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 M 45 46 47 48 49 c 22 2o 25 26 28 29 31 33 34 36 37 39 40 42 44 45 47 48 50 51 53 54 56 58 59 61 62 64 65 67 51 52 00 54 55 56 57 58 59 60 31 32 63 64 65 6G 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 80 84 85 69.8^ 70|87 72188 7389 7590 76 91 7892 79 81 82 84 86 87 89 90 92 94 95 97 98 100 lOi 103 105 106 108 109 111 112 114 115 117 119 120 122 123 125 126 128 129 131 133 134 136 138 139 140 142 144 46 47 48 49 50 23:56 25157 26 28 30 31 33 00 36 38 40 41 43 45 46 48 50 51 P! 00 60 66 Sidel>.^61 . I Ins. no Ci n !53 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 85; 87t 90 92! 93! 95i 97i 98! 60!100| 6IJIO2I 62 103 10 O", 64(106 108; 110 671112 68|ll3 69 115 70!ll7 71I1I8 72!l20f 73 53 74 5575 66176 58 77 60 61 78 79 J 22 124 125 127 128! 130 132 63 80 133 65 66 82 68'83 70 71 73 75 76 78 84 85 86 87 135 136 138 140j 142 143 145 147 89 148 80 90J150 8291tl52i 83J92ll53j j«^ A TABLE OF 6qUARE TlMBtlR, b'J < Ins. 31 3 I4i !:> 16 17 18 19 20 21 22 23| 24 25 26 27 28 29! ;o| 31 12 3o 64 35 36 37 3b 39 4i 41 42 43 44 45 4r- 47 4b 4i 5o 25 26 28 30 32 3^ o5 27 39 4i 42 44 46 48 5' 51 53 b5 57 58 60 62 64 J 1 52I 54. 551 jf 37 58 5 -- 60 61 62 63 6<:i , I 05i 6 67 68 70 7 I 72 7o 74 75 7t- 77 9ui 3 de\6^ Ins. ■7'^ 80 1 82 83 Bi- as . _90 9J 94* 96| ■981 99' I0l| ;o3| 1051 106' !08| i iO 1 12 ; 14 1 15 I 17 119 i2l .22 124 126 128 .30 131 I t> n I 00 ■ 35 137 138 i 40 142 t 44 146 147 '4v iol 153 i 54 • 5t 158 i6C ■62 164 14 !5 16 17 18 19 20 21 22 23 24 25 26 !27 28 29 30 31 32 33 34 35 |36 J38 i39 1 40 '41 t43 44 ■45 .4^ U7 148 '49 o 26 28 OO '"» -', O.C v>4- ^ r oL 38 39 4i 43 45 47 49 5i 53 55 56 5£ 6( 62 64 66 68 7u 72 ^ -,81 6d 77 79 81 83 85 87 89 91 S2 5i>\ 94^ 82 83 84 85 86 87 88 89 9i 96 9f Ok OL 04 06 08 09 i i 1 o 15 17 19 2 1 23 1,5 26 28 30 34 36 38 4C 42 4t. 47 49 5! 53 55 57 5r 61 63 64 6i 6h 7(' 7-2 9«' « f4 D 1 2 5 3 54 55 5C 57 58 59 DC' 51 j2 63 64 o5 6t' 67 68 69 70 71 7 2 73 7u 75 66 77 78 '79 Side r Ins 16 .7 18 19 20 21 22 23 24 25 26 -7 28 .9 3v 31 o2 33 34 35 36 37 '8 39 i -1-2 43 44 4;' J. 7 to -19 50 28 30 32 3-i 36 38 40 42 4"^ 46 48 5 ^ ^ ' 52 - o o I 52 55 5 4 5 5 56 57 08 59 -"C ol 62 60 64 5 6 58 CO 62 64 66 68 70 72 74L 76 78 8< 32 64 S6 83 90 92 94 96 98 •CO' 81 82 83 be i8t 87 38 9( iu2 104 lOf I(J& 1 10 1 12 lU I 16 I i8 i2r 12:0 124 i26 128 1 84 .36 38 14* :42 144 146 i 4& I 50 li>2 '54 1 5 6 !58 16t 162 i C)t ! Gt 6cS .70 I 72 i "^ ^ i 70 .77 '.'?9 t H.2 In-. o 15 ' 7 18 19 --:(•■ 2. ■w 3 2 4 -- 5 2t. 2 / 27 ■-9 30 21 32 00 34 S5 36 37 )3 39 40 4> •*2 43 43 45 46 47 30 32 34 36 3£ 4C 421 44 47 49 5 1 53 55 37 &(^ 62 64 66 68 70 72 74 77 78 81 83 85 87 89 9; 9 5 95 98 100 lot I !( 112 I 15 11?! 56 1 19 D i 5 2 53 5 55 i 48j:u'2 49 :04 o ( 58 59 60 61 62 63 6 4 ^O ; 6 67 08 69 70^ 71 7 X 73 7- 75 -6 77 78 79 80 81 82 tt3 8^ 35 8^ 87 08 ■si, 90 95 f 121 123 127 i29 132 ,-., ;36 i3!: 14( 142 4', 46 149 15! I 5 J ■ 5, 15V 159 161 1.64 166 ; 6 70 .72 i74 • 7 6 .78 181 i83i i85| 'i87| i89.j \9:l CO A TABLE OF SQUARE TIMBER, 81206 pjeosi 83;919 111 P4J^:22 113 8r.i2'24 :^:^||44nCfr!227| 111 1231^ 50:125 ,j 23c!p0 l5^"'|9l|24Cf 132i9^j^42^. Oi OF THE WEIGHT AND DLMEiNSIONS OF BALLS. TROBLEM I. To Jiiid the Weight of an Iron Ball, from its Diameter. An iron ball of 4 inches diameter neighs 9ros. and the weights being as the cubes of the diameter!!, it mil be, as 64 (which is the cube of 4) is to 9 its weight, so is the cube of tfre diameter of an J ball, to its v.eight. Or, take* ^^ of the cube of the diameter, for the weight. Or, take -^ ot the cube of the diameter, and | of that again, and add the two together, for the weight. EXAMPLES. 1. 'The diameter of an iron shot being 6-7 inches, re- quired its weight ? Ans. 42-294lb. 2. What is the weight of an iron ball, whose diameter is 5*54 idches ? Ans» 24ib. nearly. PROBLEM II. To find the Weight of a Leaden Ball. A leaden ball of 1 inch diameter weigh? -f^ of a pound , therefore aa^ the cube ot 1 i.* to -^-^ or as 14 is to 3, so is the cube of'ihe diameter of the leaden ball, to its weight.— Or, take ^ of the cube of the diameter, for the weight, oearly. EXAMPLES. 1. Required the weight of a leaden ball of 6*6 inches diamett-r ? Ans. 61-6061b. 2. Wliat is the weight of a leaden ball of 5-30 inches diameter ? Ani. 32lb. nearly. PROBLEM III. To find the Diameter of an Irom Ball. Multiply the weight by 7^, and the cube root of the product ivill be the diameter. F 62 A TABLE OF ROUyO TIMBER. ExASTrLC. What is the diameter of a 24lb. ball ? Ans. 5'54 incbe«, PROBLEM IV. To Jind the Diameter of a Leaden Ball, Mnhiply the weight by 14, and divide the product by *> : iheQ the cube root ot the quotient will be the diameter. Ex..:.- „z, Vv'hat is the diameter of an 8Ib leaden ball ? Ads. 3-343 inches. A TABLE OF SOLID MEASURE, OF ROU>'D TLMBER. By the following Table the solid contents of any stick of round timber may be found at sight, from 6 inches to 40 inches in diameter, and from 7 feet to 91 feet in length. It rises 1 inch in niaxe er at a time, and one foot in length at a time. The let't hand columns of each page, and the col- umns at the right hand of each double line, give the inches in diameter, and the other columns the contents, which ure given in cubic feet and tenths of a foot. Over the top of these two columns is placed the len2;th of the stick ; and to find the contents of any stick, first tind the length at the top, then the inches in diameter in the leit hand column, and ugainst this, to the right hand, will be fonnd the content* sought for, in cubic feet and tenths of a foot, f'or exam- ple : to tind the number of cubic feet, which a stick con- tains, that is 7 feet in length, and 10 inches in diameter — tirst tind 7 feet at the top, then follow the let't hand column down to 10 ; then against 10 to the right hand will be foand 3 feet aad 8 tenths of a feot. A TABLE GF SQUARE TIMBER. ! cr? pi?0|51 Ins.) 50 1!?: 141 144 147! 150| 153 Sid 1 51 52 - 00 54 65 149; 152^ 154 157 I6O Sic eSlj In?.' j C c -.3 54 55 156 159 1621 165 168 Sid c" a. las. 51 52 53 54 55 1631 iDij 1701 1731 i76j 14 39 56, 155| H 44 56 163^ u ~[3 56 1711 14 '^''^56 IGGJ 16 41 57 15[i\ 15 57 166: 1 -, 4r. 57 174 15 48 57 183 16 44^ 58 161 16 46 58 I69i 16 49 58 177 16 51 58 186 17 47 59 I(': i 17 49 59 172: 1 / 52 59 180 17 51 '•9 189 18 50 60 166 18 52 60 175| 18 55 i'( 184 18 58 oO 192i 19 53 61 169 19 55 ■31 178; 19 58 61 187| 19 61 61 196 20 65 62 172 20 58 62 I8I; 20 61. 62 190 20 64 62 199 121 58 63 175 21 61 63 184: 21 64 63 193' 21 67 63 202 ■22 Ql G4 178 22 64 64 187 22 67 64 196 22 70 64 205 23 64 65 180 23 67 65 289 23 70 00 199; 23 74 65 208 24 66 66 183 ?4 ■70 G6 192! 24 73 66 202 24 77 66 212j {25 69 67 186 25 73 67 195i 25 76 67 205 ■0 5 80 61 215i l26 72 68 i8C 26 76 68 198! 26 79 68 208 26 83 68 218] 27 75 69 191 27 79 69 201| 27 82 69 211 X'7 86 69 221 28 78 80 70 194 28 81 70 204. ,0 86 70 214 28 90 70 224 29 71 197 29 34 71 207| 29 89 71 217 29 93 71 228 30 83 72 200 30 87 72 2101 30 92 72 220 3G 96 72 231 31 86 73 203j 31 90 73 213! 31 95 73 223 31 99 73 2341 237j 32 89 ■74 2051 32 93 74 216 32 98 74 226 32 102 74 33 91 75 2081 33 96 75 219 33 101 75 229 33 106 75 241} 34 94 76 211! 34 99 76 322 34 104 76 233 34 109 76 244i j35 97 77 214: 35 102 77 224' 35 107 77 236] 35 112 77 247| |3€> 100 78 216| 36 105 78 227 36 110 78 239 36 115 78 250 37 103 79 219; 37 108 79 230 37 113 79 242 37 119 69 263i 38 105 80 222 38 111 80 2331 38 116 80 245 38 122 80 257 39 108 81 225 39 114 81 236 1 39 119 81 248 39 125 81 260 40 111 82 228 40 116 82 239 40 122 82 251 40 128 82 263 41 114 83 230' 41 119 83 242 41 125 83 254 41 131 83 266 42 116 84 233 42 122 84 245 43 128 84 257 42 135 84 269 43 119 85 236 43 125 35 248 43 131 85 260 43 138 85 273 44 122 86 239 44 128 86 251 44 135 86 263 44 141 86 276 45 125 87 241 25 131 87 254 45 138 87 266 45 244 87 279 46 128 88 244 46 134 88 257 46 141 88 269 46 147 88 282 47 J 30 89 247 47 137 89 260 47 144 89 272 47 150 79 286 i48 133 90 250 48 140 90 262 48 147 90 275 48 154 90 289 !49 136 91 253 49 143 91 265 49 150 91 278 49 157 91 292 i50 139 92 155i 50 146.92 268 50 153 92 282 50 160 92 ♦95 G4 A TABLE OF sqUARE TIMBER ,— lIas.5o!i-5 63|l78: o 54 181 = 14 !l5 ;17 !l8 :i9 i21 24 ;2C '27' i^8 39 4/ 50 54 5 56 ot 185 188 191 58|l95 59|198 60!6C 202 64 67 70 77 80 84 87 91 94 9 Id* § § 14 49 o 51 179!'5ide 23 50:1331; _lln? 53^18b[[ -J ^ 54.I90I1 §1 I' 55!l93t 61 203 621208 63!212 7464|215 65|213 66i222 671225 68J228 691232' 701235 115 il6 |l8 |19| |20 21I J22 j23 |24 125 26 56 6(i C3 67 70 74 77 81 84 88 91 561197: 14| 5720015 51 1911 -^i 194 q 54 198J§ 51 ;.2u4 59:207 60' 211 61 214 62218 63i221 .-I 16 17 18 19 20 121 52 53 187.;Sidf Ins. 52 bo 57 209. 62 66 70 73 77 64,225il22i 81 65,226 23| 84 D6|232!!24i 28 67i235:[25i 92 68'239i 26 95 27 95 691243 71123812 28 :« 30 lOi 3lil04l :32iI07 -3'5!l 1 t •■-'Oj I 1 i •34ill4 1251117 ;36|l2l 137 U 4 ♦38 127 ;3S 13' J4o|i.: '411137 '!2;i41 13 144' :44il48 721242 r 0^245 74;248 -5!252 130 98 102 lOol 7C;255 77259 ol|109 3-.n2 ;33ilC ■34!ll9 !35|123 ^^•'l26 :43r. 801269 :38[l33 701246! 27} 99 28il0S 71 249i[29|l06 72i253j'S0'llO !256l31 114 32Jil7 75'263i!33 121 -.1 I' 60 61J224 62228! 631231 64 235 55'202 j_i 561205 !14 lo 69158 213 o9 220' 16 17 18 19 20 21 22 01 61 65 69 *. £~* I ^ 76 80 84 65 239 661242 24! 67|246 -5! 682501261 691253,1:7' ■Gl257;;:r 7]i26i;::: 264;Soi 268:31 23^51! 196 199. 00 203 541207' 3^211 561215 o/i219 o8[222; 9226i 60] 250 61;2S4 62 00 64 88 65 92 96 9968 267!i34!l25l76|279:!34 77l27r 72 73 74 75i27o ioo 103 107 111 U5 iir 122 126 130 238; 24l! 245- 2,9| 253^ 257^ 261! 69 264^ 701268! 7I|272| 72 276; 280! 284; I 83j27j^ j41] 151 154 >'■ 1 41J144 :147 151 . 154 15 161 351128 77 78 274ll36!l32:£ 79i2:7'37:136 80 2811 3CM3' 85j291|.4ljloOJc;3j305:j41 gj; ,- ,-^ 1- V- ; , r ' ■;. Or. 8t 1*^ 291! - - :95i To 299; I42|79iS03i :07: 10 •14, 5718. 3j318: 818' 8r 3t.>: i6CJi}2|309 , ._, . iI80-31J334,li9 l^lJTG j2-323;|50h&3 92 337 |50 188I91J349! 192'92i352' A TABLE OF SqUARB TIMBER. 65 51 1^221 !Side25| Ins 56!243il4 57|247J15 58|252||l6 59:256j!l7 60i260 18 o 61j265i G.2|^69 19 20 21 641278 29 6rl282l;23 66 63 67 72 77 81 86 Q 108 67 113)68 117169 C 71 -o jSS 152 80 139|l 58 81 .40[i60|82 :41 1164103 J42|l68 84 |43|l72 85 i44'l76!86 145 18o!37 28J121 -^9 120 30|13C; -^.,3l|i34 308|'32!i3G 3l3'!33{i43 3l7;i34;i47 •321j35!i52 325136 156 329i37'j60 38165 8g!333 31 337 328 j|40|l 66 8:|342 332:!41 33ej|42 "44 0IJ43 344! 175 179 183 --,- 348ij45!l87 146 184 88!552'j46|l92 |47 188 89 35t 147 196 95 99 104 51 230! 52 54 56 o9 60 61 286l|2^ 108 29ri25Jll3 295i;26jll7 299|j^7!l22, 304'j28|l26 308:'29!l3l 3l2i;30jl35 73i3l7{i3]|14C 32II44 00 64 >^' 6C 67 68 69 339; 2441 248! 253I 257} 262[ 266! -^71! 275i B8C! 74i32i: 75|325J 76|53oJ 771234113; 149 153 158 78!33pjl3c!i62 79j343;!37|i67 , .. .,,--,171 39)169 81J35]|3c.h76J81 40i 1 73 82j35f.|Lic' ] 80 82 284 28£ 29? 298 302 30": 311! 7013161 71 72 320i 32 £( 73 329^ 77 78 79 SO 178 83J3e0|'41 182 84i364|j42 |48 149 '50 192190 196bl 200i92 36( 36 36» 48 49 50 04 3381 34vf 35n 5(| 36 1 1 3'6f j 3711 37^ lS6j85j369i43J194i8ci88.i 19P8ej373lU4|l98i06|38( 185 83 189|3-i37^ 87 88 ftO 377|i45!203[87|39: 382Ji4Gi207 38e!|47i2i2 8; 390J;48i2I7 39o|i49J221 n7 92!399|;50i?26 08139^ I 105 X-!40{ 91141} 92141: A TABLE or SQUARE TIMBER, iJecij^ 51. 248 f^^^e '27151 c258, 5i^^-^5|51 -•: -M.l>: f^f = i-i } 5 i I 54[ 5' 05166 -' -. i^H rO; 53i 5f re- el 06 I;i5 -K. ^ i -4, :-5 17! ir>' * d 94 xV:-^ 91 ;?0 0So5: 05 ^3| 13|66. 10 24; to! £:>^ 2c 71: 5- 2V 92 61 5 59 64 . o ■* - o 1 7-4 47 32 75 52 33' 77; Cl ;35 12 17 2? o- 31 41 4^' 51 5t 61 6^^ 70 521 o3\ ^1 34!63jli ^1 1^-52 63:; - Ini 00! 56, 57, 63 71 I O 1 1 -I f 7 c 151 7''" 83;iei 81 8c 17i 86 93 l&l 91 97 l&l 96 53! 68! f -: 53S 78- 54! 73^11 ? 5J[ 83- 55 78j|_l_! I 35 89j 5C 83 14i^.>'i6| 94i 57 88!!l5| 7-57' 991 59 98ilni 8-|^9[jO 60 30- " ^'^' o.i 12;22} 11 j5i 17>23| 16 -36! 22 1:24 21 7! 27 2ol 26 31 :^6; 3'- - ' ; 01 T?: 41 28i 42 7r i o 74 4^:* 29; 47 51 30' 52 31 5: 3 - 61 70 < o 93. 1 61 31 -o >8 304[ 61 1 0. 62t302 20I1OI62 14[J20' o5o2| 3^ 07-21J 0663 19r2l| i063; |t4| 24'!22' i5|64 36J t'o'. 29'. 25; o] ■36| oJ;*24; oe oi ocS-c 31 r ^ 3^ o2i -X:a 00 66 67 6Si 57[ J9| 62[ 70f 67; 71 1 731 72! 64:'3ei 57J72| 78| 73? t3; 70i 54i'2L| 47 311 G5 18J30: «4 9?' it 2! 62 74 75f|32 e-'^- ^^ 35' 67 75} 80,;33j 75"" ^^ 34I 7276f Sc\\3^l 7b 77!77i 9C'l35| 8^ ' ■ 3i3Gi 8' -'!^-i94 92t£0: 05i>38i 9C» 00 '' 15 20 25! 30i 41 4€j 75 94 7«^I 991 67J404- 78 09| 70! 151 3c( 21 r -' 30| -ill S2I. 441 06 S6[403 44 45 11 -' "" - ioj iti: 47t 2ii 4 3 x\\3\ 05 *l; 03{0 91 j 78j 92} 65 f.5Ci| 62I95I 82f 40 46 51 57! 08! 621 s9; -^^ 671 72. A TABLE OF SqUARE TIMBER. ^fue 2\j 51 277.S:de-^H>| j\' 288 bid e29j51 298 Sid e29^»51 3C8 ~' IU5. 52j 83 — 63 69 17 j2& 52 70 81 00 ^0 58 70 95 128 GA 70 09 28 69 70 23 129' 58 71 86 29 63 71 400 !29 6f 71 15 00 75 71 29 j3u 63 72 92 30 69 72 03 ■30 75 7 w 20 30 81 72 35 ■31 69 73 97 31 75 10 12; ;3i 81 73 26 31 87 73 41 132 j 74 74 403 32 80 74 17 139 87 74 32 32 93 74 47 1331 8C 75 08 33 86 75 23, 33 'b 38 ■33 G£- 76 63 J24i 85 76 14 i34 92 76 29! 134 9f 76 44 :34 2<.j5 76 69 35I 91 77 19 35 97 77 34' !35 204 77 50 |35 11 77 6b ,36; 96 78 25 1 36 203 78 40J SC- K (O 55Ji36 H 71 ■37-201 79 30 '137 09 79 45 SI 16 70 1 KJ 61 37 24 79 77 •38! U7 80 35 i38 14 80 51 38 80 67 38 30 80 83 ■39! 12 81 4i;i39 20 81 39 , 2a|8i 73 !39 35 81 89 UOJ 18 82 46i!40 25 82 62 40 33 82 79 }40 42 82 95 i^V. 23 83 52||41 57; 42 31 83 G8 41 39JC3 85i|41 48 83 501 ■42| 2? 84 37 84 74 42 45|84 51185 91 42 6484 07 43| 34 85 63| 43 42 85 79 43 96 l4S 5085 13 ;44| 39 86 68 44 48 36 85 44 63|87 502jJ44 08.46 CC86 20 :45! 45 87 74 4- 54 87 90 45 72 s 7 25 .1=3 1 5C 88 79 4C 59 88 • 96 46 67 Sij 14' 43 78 88 32 r ot 89 85 47 6r. 89 302 47 |48 49 |50 74 Of: 20| 47 84 89 38 4';, 64 GO 9o; 48 70 90 07 0/ 30 251148 90 on 44 '49j C: 91 95 49 76 91 13 86 .'1 31.1 49 96 91 50 M '^ 92 301 :50 82 92 18 92 32 37! 50 302 92 66 A TABLE OF ROUXD TIMBER. GS c 7 Fl. S-' 8 Ft. ">-»• 9Ft.!!P10Ft.i?>llFt. , 1 v. Lons:. ft- X 6 Lonff. 1 G Lod; ^! - Lon " 1 ^ Lons:, o 3E O o 1 6 9 OB O o 3 ^^ O 1 6 O o 9 ST G 1 4 1 8! 6 2 Oj 2 1 7 1 8 7 2 1 4 2 4l 7 2 7 7 2 9 8 2 4 8 2 8 8 3 1 8 3 5 8 3 8 9 3 1 9 5 9 o 9 1 9 4 4 o 4 8 10 3 8 10 4 10 4 Q. 10 5 41 10 6 ii; 4 6 11 5 3 11 5 9 11 6 6 11 7 4 12: 5 5,12 6 3ji2 7 1 12 7 8 112 8 6 • 131 6 4 !^? 7 ^i 13 8 5|!l3 9 4 113 10 . 14! 7 5 u 8 ^ 14 9 6 14 10 6 14 11 7 1 loi 8 6J15 7116 9 8il5 11 l! 15 12 4 15 13 6 ! IC 9 11 2 16 12 6i 16 14 o; 16 15 3 17 11 17 12 6 17 14 l; 17 15 7: 17 17 3 18 12 3il8 14 1 18 15 9l 1 18 17 7 18 19 4 IC^ 13 7;|i9 15 7 19 17 7| 19 19 ■'i 19 21 6 20 45 3-;2o 17 5 20 19 6i|2C 21 6; 20 23 9 21 16 "i'^ 19 2 21 21 5}|2} ,23 8; 121 26 3 i 22 18 4^f22 21 PO i 23 61122 126 3 22 28 8 , 23f3l 5 i 23 20 2; 23 22 9 23 \ 25 '~ ^'■"- -^ .-k '~- 24 21 8J24 25 Oi:24 ■ 23 24 35 3 J 25 23 7 25 27 2j25 |30 7ji25 : 34 ui 125 37 3 ■ 26 25 7 26 29 4;!2C !32 lib e! !26 40 4 27 27 6|i27 31 6 27 6(;27 39 |27 43 2 28 29 7=28 33 ^ 128 ^ 38 4:i28i 42 o 28 46 6 ' 29 31 8 i29 36 3 oc i 41 Ol 29, 45 9 30 49 '^1 29 50 30 34 o J30 39 30 i 43 0" ^30 53 5 ol 36 6 h 41 8 31 47 31 52 **- |31 57 2* 32 38 9 32 44 5 32 52 2 32 55 6 |32 61 2 i >3 41 4 33 47 Q 33 53 3 33 59 1 33 65 34 43 7 34 50 3 34 oQ Si!34 8!b.5 82 9 34 C9 o - 46 5 35 53 JO 59 ^^ 5 Sai^r-^ 36 49 2!:3G oQ 136 63 2:iSt 70 5 136 79 0~^ 37 52 0'j37 59 4 37 67 37 74 5 ;S7 81 3 X 38 54 1 38 Q2 f 38 70 6 3( 78 8 iSS 86 ? 39 57 8 39 &Q 74 4 3S! 83 3i b^ 91 '. :o 60 8!40 69 e;;4o 78 oi'40 87 3l !4( '^b 7 , A TABLE OF ROUND TIMBER. 68 V— ' 12 Ft. — 1 13Ft.i^ 14 Ft ?;i5Ft! glGFt 5 ft It 6 Lor.2-. -i a. 6 Long. 1 =; s ft 6 Long. i| ; re ! 1 J^ i 1 6 Lonj . 3 en y. re f 6 Loni D re EJ r. 6 o * n 2 5 2 5| 2 7 2 ~9 T 7 3 2 7 3 5 7 3 7 7 4 7 4 2 8 4 2 8 4 5 8 4 8 8 5 2 8 5 5 9 5 Sj 9 5 7 9 6 1 ! 9 6 6 9 7 10 6 5 10 7 1 10 7 6 10 8 1 10 8 7 li 7 9 11 8 6 11 9 3 11 9 8 11 10 5 12 9 4« 12 10 2 12 11 G j12 n 8 12 12 1 5 13 11 1 13 11 9 13 12 8 il3 13 8 13 1 14 6 14 12 8 14 13 9 14 14 9 114 16 141 17 15 14 9 15 -J 6 1 15 17 2 |15' 18 5 15 19 7 16 16 8 16 18 2' 16 19 5 'AS ^ 8 16' 22 17 18 9 17 20 4 17j 21 8 '17: 23 5 17; 25 18 21- 3 18 22 8 18 24 ^ ^ |18| 26 4 18' 28 2 19 23 6 19 25 5 19! 27 3i :i9^ 29 19 i 31 3 20 26 2 20 28 2 20i 30 3 '20: 32 5 20 o4 6 21 28 7 21 31 2li 33 3 21 35 8 21 38 1 22 31 5 22 34 22I 36 6 \qq 39 22 41 $ 23 34 5 23 37 3 23! 40 2 !23 42 23! 45 7 24 37 6 24 40 6 24 43 6 124 46 7 24 49 6 25 40 7 ^'5 44 25 47 4 25 50 7 25 53 9 26 44 26 47 7 26 51 3 |26; 54 8! 26 58 3 27 47 4 27 51 27; 55 27; 58 9 27 63 28 51 28 55 2 28 59 i28' 63 5 28 67 6 29 54 5 29 58 c 29; 63 4 !29' 68 29 72 4 30 58 4 30 63 4 30 j 68 bo 73 3 30j 77 7 31 62 5 31 67 7 31 72 7 i31 78 p 31 83 32 Qo 7 3? 72 4 32 77 5 3: 83 5 32 83 7 33 71 33 76 8|33 82 5 S3 88 6 3. -5 94 5 M 75 3 •"-•4 81 4 34 87 6 94 3 3. 99 6 35 79 8 3.5 06 4 35 92 !35 99 5 35 iC6 8 36 84 4 3'. 91 3 \3r £=8 136 105 5 36 112 9 37 09 5; 37 96 n 37 104 7 i37JlI2 37 1 19 5 38 94 41 99 3i 08 i02 i 3? no |38!117 < :-:8 126 39 39 108 3 ', 116 9 124, 39 132 8 '0 101 4l i'iO 113 ^1 i40 124 140 130 r 4u 139 5 J TABLE OF ROi'XD TIMBER. . ir.i 17 Ft.! i^l8F t.' "w, 19 Ft. ^\ 2uFt.i|5 21 l^Tl * = 1 5 5 Lone. O o h 7. 6, Loiir, ! r Lon... X Loi-g O hi! Lon^. \ ; S i o _ ) ■ 1 ' 1 ^"^i o oi 3 3 ', <:•! 3 j 4 11 , ri 4 5 i 4 8; (■ ' 5 'ii ' 5 7 5 6 ' 8; 5 ?! ' f - i b b 6 ^i &i 7 8 7 3| i 9^ 7 5i 1 o u -^ 8 4 9 8 8 ^ 9 4 iiO 9 3J !|o 9 8 ^ 10 4 iO 11 G 10 11 5 11 11 2 i 1- 11 9 i ! 12 e n 13 3 \ 1 13 9 12 13 4 u 14 2 12 15 i2 15 ^ 42 16 6 13 17 7 \3 16 I 3- ir; f i o 18 5 .o 19 5 14' 18 3 u; 19 3 14' 20 7 14 21 4 i4 22 5 15; 21 1 22 ■5^ 23 t, .5 24 7 jl5 ,26 2 16 23 8 t 25 2 16| 26 7 16 28 2| 16 29 5 17, 26 8 w 28 3 '7| 30 ,ll7 71)18 1 01U2 5123 2 7 2 2 , -7 28 2 , ^1 32 33 34 |35 00 0'37 1 pb (.|i39 7 -iO 6 8 10 13 17 2') 24 28 .■» .-> 00 33 43 48 54 61 67 75 82 89 )7 1-6 ll4 123 132 142 152 163 174 135 1 3 8 7 1 6 4 6 o O 5 4 6 1 7 5 5 5 7 «» o 7 4 4 1^6 ' G 208 ^18 5 2 44 ^72 A TABLE OF ROUjYD TIMBER. to 3 32 Ft. Long. 33 Ft. Long. 1 34 Ft. £;' re "1 35 Ft. Long. 5 36 Ft. 1 Long. O o 3 Lon O o 3 g- O o 3 O . o O o 3 1^ 6 3 B9 6 So 6 3 <-r- c 6 3 So" 6 3 cn 6 3 6 5 6 7 6 9 7 1 8 6 7 8 8 7 9 1 « 9 4 7 9 6 - 8 11 1 8 11 5 8 11 9 o o 12 3 o 12 6 9 14 2 9 14 6 9 15 1 9 15 5 o 15 9 .10 17 6 10 18 2 10 18 7 10 19 3 10 19 7 11 21 2 11 21 7 11 22 5 11 23 2 11 23 7 12 25 4 12 26 1 12 26 7 12 27 7 12 28 4 lis 29 5 13 30 6 13 31 4 13 32 3 13 33 3 14 34 3 14 35 4 14 36 5 14 37 6 14 38 7 15 39 7 16 41 1 15 42 2 15 43 3 15 44 5 16 45 0| IC 46 3 16 47 7 16 49 2 16 50 4 17 50 5 17 52 17 53 4 17 55 2 17 56 6 18 56 5 18 58 5 18 60 1 18 62 18 63 6 19 63 19 65 19 67 19 69 19 70 9 20 70 2 20 72 2 20 74 4 120 76 7 20 79 21 76 7 21 79 5 21 81 7 21 84 4 21 86 5 [22 84 5 22 87 2 22 89 5 22 92 5 22 95 4 '23 92 4 23 95 5 23 98 3 23 101 2 23 104 6 24 100 8 24 104 24 107 24 10 5' 24 13 6 25 09 5 25 13 25 15 7 25 19 5; 9h 23 \ 26 18 5 26 22 26 25 8 "G 29 5 2e 33 27 27 5 27 31 5 27 35 2 27 39 5 2 7 42 5 ^28 37 5 28 41 5 28 45 5 28 50 3 28 54 4 , ^0, 46 4 29 51 6 29 55 5 9Q 61 o! 29 65 .30 57 30 62 6 30 67 36 72 5J 30 77 5 31 69 31 74 C 31 79 31 82 41 3] 90 .32 80 32 85 5 32 91 32 97 C 32 202 .33 91 g 33 97 33 202 5 33 2G8 33 14 34 202 5 3,4 208 2 34 14 3 34 20 C 34 27 35 14 35 20 5 35 27 35 34 35 40 36 26 ^ 36 33 5 36 40 36 47 C 36 55 37 40 ^ •37 47 r. 27 54 3 37 62 5 37! 69 5 38 53 5 38 61 5 38 68 3i, 76 4 3C 84 31) 67 ^^ 39 75 C 3; 83 2 39 ■^ -'5 39 300 iO 80 0|i40| S8 5 40 97 3 40 306 4 40 }' ^ » G A TABLE OF ROlWD TIMBER 72 »||l£' 74 7|iiyj 76 7|jl9j 78 V\19\ 80 4 ; ->-, '■>-■• r — .-oi o. i53 21 •M; 34 5: -n 31 Oii^c; 83 3;i^0| S5 bi\20 87 5 120, 89 88 £ii2l» 91 ^pr- 9d 7i21 96 fSl! 98 07 7i!2e:iOO 5li^:|103 2j[22'l06 OjeellOG [07 r ;^ ^ ■ ■' ' - ^^ " ^^ 1- -' ::^' 19 1 .- . . - : 29 40 51 63 75 88 :02 14 t?o -r-r 55 oii 92 5!!:?7^ at 48 59 OlUO: 72 C|i\?9' 82 o'lSO; 97 J05 0i:31 211 IS 0132. 24 «>; — 32 5|i33 3? 43 01^34 54 L, :, 62 0|l3:> 67 5ii3ii{ 76 o!:3a- 33 53 68 83 :::i u. -l- 4i {33! 43 x,i34' 58 obo! 74 89 ■ •07 24 41 .u, 59 4 6 5 4 5 o 5 o A TABLE OF ROUXD TIMBER. v^ 42 Ft. ^ 13 Ft. C i44Ft.| w* .45 Ft.; ^46 Ft. ^' Lon g- 2: 6 Lon or. •5' 3 -! \ =" 6 Long, j I 6 1 Lons". 1 %"■ Lons:. -5 Q 1 ' CO 1 ^ c a • 6 6 8 Q 8 4 3 i 8-8 9 - 7 11 3 7 11 5 7 11 7 12 Oj 7 12 3 ; 8 14 1' 8 15 1 8 15 A 8 15 7i 8 >16 1 9 19 5 9 19 1 9 1 19 4; 9 19 8 9 24 6 10 20 4 10 22 9 10 23 5 10 24 0; 10 25 3 11 27 7 11 28 4 11 29 2; 11 29 6 11 30 5 12 33 o:|i2 33 9' 12 34 ■^! 12 35 4fl2 36 4 : 13 38 7; 13 39 7 13 40 5! 13 41 4!|13 42 5 14 44 7; 14 45 91 14 46 9| 14 48 Ojlu 49 4 15 51 7 15 53 3!ll5 52 ■2| ■ 5 55 5 15 57 16 58 5 16 60 4 16 61 5! 16 62 8 16 64 6 17 66 0; n 67 7 17 69 4! 17 70 7, 17 72 7 1 18 74 2 18 76 18 '77 6il8 79 5i|l8 81 6 |19 82 5i 19 84 ^i 19 86 5l!l9 88 4!!l9 90 3 •20 91 ^1 20 94 6 20 96 ^1 20 98 5'|20 100 5 1^1 100 81 21 103 4 21 105 5'|21 108 3;! 21 10 5 22 11 22 13 ^i 22 16 0| 22 18 4:i22 21 5 23 21 3| 23 24 5! 23 27 31 23 29 6II23 33 •24 32 o; 24 35 24 33 7J 24 41 5i 24 "'44 5 25 43 o\ 25 47 3 25 60 ^i 25 53 7j 25 67 26 53 o' §6 59 ^■ 26 62 M 26 QQ 5] 26 70 5 r' 67 27 71 27 75 0| 27 79 Oj 27 83 {28 80 21 28 84 " 28 88 5! 28 93 28( 97 j29 92 8i 29 97 29 201 ^i 29 213 29 211 -0 •30 206 ^1 30 211 5 30 15 0, 30'' 21 0;3e 25 , 31 20 i 31 25 2 31 21 0i31 36 31 42 32 34 5 32 41 32 45 . 4 32 52 6 32 57 5 ; S^ 49 33 56 1 3Z> 62 0,133 67 5; 33 73 34 64 0| 34 71 5 34 76 5134 83 5| 34 89 i 35 82 I 35 87 35 i 94 0!35 3Gi 6ii35! 307 5 i 36 96 m 304 oi 361310 3136 17 6;|36 25 1 37 314 5 37 22 1—1 ■37 28 37 36 5;|37 44 1 38 32 i: 38 37 5!!38| 45 38 55 0;!3Cj 63 2 ] 39 49 Ci 39 57 1! 39 QQ 4 39| 75 0;38| 84 : 40 67 3J140 .77 — *■ 40 85 loi 94 0'40l 403 1 X I A TABLE OF ROUXD TIMBKR. o:47 Ft.lf '^^ _ 48 Ft.! 9. 49 Ft.! Cj, 50Ft.il5! 51 Ft. "5 6i Lono;. 1 o 1 a 1 o - -5 6" Lonjj;. 5 Q Lono; O o a D <-♦■ C/3 II 3 (t 6" Long I 5 Lono;, O o O) O o a a (A o 3d 6 o o D • 9 2] 9 41 6 9 6 9 8 9 9 7 12 6i 7 12 8 7 13 1 7 13 4 7 13 7 8 16 4 I 16 7 8 17 1^ 8 17 5 8 17 8 9 20 7! 9 21 1| 9 21 26 3: 10 26 6 9 22 e 22 5 10 25 7 10 10 27 S 10 27 8 ^i; 31 0,11 31 7 11 32 3 11 32 9|11 33 7 12| 37 12 37 7 12 38 6| 12 39 3112 40 1 13| 43 SJ 13 44 13 45 o! 13 45 9 13 46 9 ! 14 50 41 14 51 2 14 52 3, 14 53 2;14 54 6 '■ 15 58 0. 15 59 1 15 60 S| 15 61 7 15 63 2 i 13 65 G, 16 67 16 68 ^1 16 70 0|16 71 6 i 1"^ 73 8i 17 75 Si!l7 77 i 17 78 8 in 80 4 i 18 83 3| 18 84 5:118 86 4 18 88 :18 90 3 :i9 92" 4| 19 94 S;!19 96 3' 19 98 3 '19 100 7 i 20 102 5' 2G 105 0i:20|l07 20 109 3 '20 12 121 13 0| 21 15 3|21 17 4 21 20 21 21 5 [22 23 5| 22 26 7|;22 29 22 32 22 35 f 23 35 5| 23 38 5 23 41 5 23 44 33 47 6 24 47^0' 24 50 8 24 53 5| 24 57 '24 60 7 25 60 5 23 63 6j|25 67 25 71 25 72 6 126 73 2 26 77 5| 26 81 26 83 2 26 89 i27 87 O' 27 91 0:27 95 27 98 4 ,27 202 6 1^28 ;201 5 28 205 0j28 209 68 214 i28 17 5 29 1-14 29 18 0:29 24 29 28 29 33 5 i 30 30 30 35 7|30 40 3 30 45 '30 50 Isi 45 5 31 52 031 57 31 62 5 31 67 6 . 32 62 5!i32 67 21 32 74 32 78 5 l32 85 t33 78 33 85 i3.'3 91 5 33 96 5 33 303 6 (34 95 ,34 302 5 34 308 34 315 j24 21 5 i3£ 313 4 :3s 20 |3£ 27 2 35 33 5 |35 40 !3C 32 1 136 38 ,36 4b 36 53 C 36 61 I 31 , 52 ( \3', 59 1 1 -^^ ' G6 3 37 74 c |37 82 6 i3£ I 70 .SI 78 7 38! 86 5 '38 94 € 3£ 403 i3e -> 90 C l3G ^98 5' 39 407 5 '39 416 C 3£ 23 5 !4( )411 2''4( H18 5|40 27 6 4C 36 t 4C » 46 ^ A TABLE OF ROUXD TIMBER^ t i ^— ' 52 Ft.| s-^ 53 Ft. jc 54Ft.| 'w i)^ Ft. I ^ 56 Ft. 1 (T Lons:. i a 3 "1 Lon ?. : ^ If Lon Z- ' 5 -5 Lon rr. ^ 11 1 -1 Lon 3 V, 3 V. -6 1 1 1 6 |5" 6 3 !-- i 6 3 10 2 10 ~4 10 "6 10 8 11 ~0 7 13 9; 7 14 1 ■7 14 4: 14 7 ; -7 14 9 C 18 2 8 18 5 8 18 s; 8 19 3 is 19 5 c 22 8, 9 23 3 9 23 7 9 24 ; 24 7 10 28 4; 10 28 8 10 29 4| 10 30 10 50 7 U 34 0' 0, 11 34 9 In 35 5 11 36 ^ 11 36 8 12 40 9 12 41 12 42 i 12 43 3 12 44 13 47 8' 13 48 7 13 49 1\ 13 50 1 |13 51 5 U 55 6' 14 56 14 57 7: 14 53 8 Il4 59 8 15 64 3 15 65 4 15 m 6 15 67 2 15 69 3 16 72 7! 16 74 :i6 75 4 16 77 !16 78 5 17 82 n 83 :i7 84 8: 17 86 6 In 88 18 92 18 93 5 18 95 5' 18 96 4| |18 98 7 19 101 3. 19 104 19 106 5 19 103 < 119 UO 7 20 14 20 15 5 20 17 9- 20 20 5 |2C 22 5 21 25 21 27 21 29 Oi 21 32 ^i 121 34 4 22 37 ^'. 22 39 5 22 42 3 22 45 3! I22 48 -23 50 5 23 53 ;23 55 3, 23 59 oi 23 60 2-1 61 5 24 QQ 5 24 69 5 24 73 ol 24 77 25 78 Oi 26 81 ''25 84 5 25 88 25 91 5 26 92 8! 26 95 5 :26 98 26 203 26 207 27 214 0' 27 209 ,27 217 oi 27 17 o;27 32 ^0 ^0 23 28 26 128 32 ^i 28 35 5I20 38. 29 37 5' 29 42 29 47 0' 29 52 0129 56 30 55 30 60 30 65 0; 30 70 G 30 75 31 73 0' 31 77 5 31 83 4,31 88 31 9' 32 91 32 95 0,32 303 0'32 307 6 32 31.3 li 33 310 33 315 0j33 22 33 26 5 33 34 _, 34 29 0, 34 33 34 41 ^i 34 46 34 54 5 35 47 5 35 54 5 35 62 0^ 35 67 3lD 75 67 36 74 8 j3G 83 ■^ 'jO 89 V' m 96 5 37 90 37 97 37 405 0,137 413 r l3; + 10 0. .0 411 38 416 5 38 26 4; 3d 33 38 41 5 19 32 39 40 0,;39 49 5i 39 53 39 66- 3 40 54 40 63 5 '40 72 0'140 81 5 140 ■ S9^ 6 02 A TABLE OF ROUND TIMBER. >^ 57 F 't.;!? 581^ 't.i : ^ 59 Ft.] :? 60 F 't.i o 61 Ft. 1 3 1 ■ -5 Long. ; 2. Loni T a. Lons;. B Lons T O o o O o o o D o o 3 t/. c 2 : ^ : 1 G o en" or. 1 6 1 6 3 I i 5 1 - • CO 6 Lotjo;. 5 2- ►—1 X 6 Lon* o E. r. O o O o n »-♦■ O o O a CA o o s t-t- s 12 2 12 4 12 6 12 8 13 "o 7 16 6 7 16 9 7 17 Q 7 17 4 7 17 7 8 21 6 8 22 8 22 4 8 22 7 8 23 1 9 27 3 9 27 7 9 28 2 9 28 7 9 29 2 10 33 8 !io 34 6 10 35 2 10 35 6 10 36 2 11 40 8 111 41 6 11 42 3 11 42 9 11 43 6 12 48 7 12 49 6 12 60 4 12 51 3 12 52 13 67 2 13 68 2 13 69 13 69 8 13 60 8 14 Q6 o 14 67 4 14 68 6 |14 69 7 14 70 6 15 76 7 15 77 15 78 4 jl5 80 9 15 82 2 16 86 8 16 88 3 16 89 6 jie 91 5 16 92 5 17 97 6 17 99 17 100 8 17 103 G il' 104 5 IG 109 6 18 112 18 13 6 \\% 15 18 17 19 22 il9 24 4 19 26 2 19 28 6 19 30 7 20 36 ,20 38 20 40 7 i2C 42 8 20 45 21 48 8 21 61 5 21 64 2 21 56 5 21 59 22 63 5 22 65 22 69 5 22 72 5 22 75 !23 79 83 23 85 o !23 89 23 91 3 24 96 24 98 24 202 24 205 24 207 5 15 211 5 |26 214 5 24 13 |25 22 2 25 25 5 :;6 28 26 32 6 20 26 26 40 26 43 2 ■27 46 27 ^\ 27 54 2 127 58 27 62 5 *28 65 28 70 28 73 6 128 77 2 28 92 5 29 83 5 2P 88 c 29 93 ^2c 97 6 29 303 50 305 2 30 309 5 '30 315 C 30 319 6 30 24 5 31 25 5 31 32 • 131 35 6 31 42 31 46 5 32 46 32 53 c !32 58 6 :32 64 5 i32 70 5 33 68 33 75 c '33 83 '33 87 6 |33 94 34 90 34 97 Q 34 404 7 i34 4n 5 !34 416 6 35 414 c 135 421 5 36 26 i36 34 35 42 36 37 3 36 45 c 36 53 2 36 59 3t; 67 37 64 C '37 71 5 •37 80 :37 87 6 137 95 38 88 c 38 96 6 138 504 5 38 513 5 •38 511 % 39 515 p. 3£ 523 r. .39 32 t-. 3f- 41 3 48 6 40 41 4 40 61 oi 40 57 4(: 67 0'40 76 5 80 A TABLE OF ROUXD TIMBER. 3 67 Ft. C 68 Ft. •^^ 69 Ft. ■w 70 Ft. >-' 71 Ft. 6 Lons;. § rt- CD <^ G Long. § o ■J. C Long. 6' Long, j ft D 6 Lone:. o . o a o 3 tn O o s r-f- o 3 Cft" O o 3 a> 3 r-r- 13 4 O o D ft) =5 13 G O o Q 1 13 2 13 13 9 7 18 7 18 2 19 6 7 18 19 8 23 4 o O 23 6 8 24 2 8 24 4 8 24 7 9 29 6 9 29 7 9 30 3 9 30 7 9 31 3 10 36 6 10 37 2 10 37 6 10 38 2 10 38 8 11 44 2 11 44 7 11 45 6 11 46 1 11 46 7 12 62 6 12 53 4 12 54 S 12 54 o 12 66 8 13 61 5 13 62 5 13 63 6 13 64 3 13 65 14 71 7 14 72 5 14 73 6 14 74 6 14 75 8 15 83 4 16 84 3 16 85 6 16 66 6 16 87 8 16 94 16 95 3 16 96 6 16 97 8 le 99 2 17 1C6 17 107 2 17 109 17 110 s! 17 114 18 18 6 18 12 6 18 22 6 18 23 5 18 25 5 19 32 5 19 34 19 36 6 19 38 19 40 20 40 8 20 48 6 20 51 5 20 53 20 55 2 21 61 2 21 63 2 21 61 6 21 68 21 71 oo 78 22 80 6 22 83 22 85 oo 88 23 94 23 96 23 99 -0<2 204 23 205 24 211 6 24 213^ 24 216 5 24 18 c ^4 22 5 26 28 26 32 op. 35 6 ox 37 c 26 42 2G 47 26 51 5 26 55 26 57 r 26 62 2 -27 66 5 27 69 27 74 C 27 76 (• 1^7 81 5 28 86 6 28 90 28 94 6 28 97 r ]28 303 6 29 306 6 29 311 29 315 6 29 318 t^ 129 25 .;o 28 30 33 30 37 30 43 ( 30 48 2 31 62 31 66 SI 62 31 66 ( 31 73 32 75 4 32 81 6 32 87 2 32 91 ( t'32 97 ^;3 93 6 33 404 5 33 411 6 33 415 r !33 422 5 31 422 6 34 27 34 34 6 34 39 ( :34 46 6 36 47 36 63 6 36 61 36 66 (. 36 ".2 3G 74 36 79 36 88 36 94 (' 36 501 5 37 602 37 507 ' 37 515 6 37 503 i* 37 30 ,i8 27 38 34 3F: 43 6 38 50 c ,38 57 39 65 6 39 63 3C 73 6 39 80 c 139 88 5 +0 84 40 92 2 40 0(^3 - ■\( 6v> ( '40'61B 5 1 A TABLE OF ROUXD TIMBER. 81 ffFt ^ 73 Ft. $74. Ft 1 5 75 Ft. CT 76 Ft. Lon §• K 3 ft 3 Cfi 6 Lona:. 1 re <-► 3 m 6 Lon K- i o o 3 Ml 6 Lon (T 2. 6 Lon^. O o a o o o D «-* O o 3 2. 00 O o 3^ 1 3 «-»■ 3 14 1 14 3 -IT 6 i 14 '/ 14 9 7 19 2 7 19 5 7 19 8 7 20 7 20 4 8 25 2 8 25 5 8 25 8 8 36 8 26 6 31 7 9 32 2 9 32 7 9 73 1 9 33 6 10 39 o 10 39 10 40 5 10 41 10 41 7 n 47 5 11 47 3 11 48 9 11 49 4 11 50 4 12 56 6 12 57 5 12 58 3 12 58 9 12 60 13 65 2 13 68 13 67 6 13 69 13 70 4 14 76 7 14 78 14 79 4 !14 80 3 14 81 5 15 88 8 15 91 15- 92 15 93 15 94 5 16 100 7 16 102 5 16 104 5 16 105 16 107 17 13 6 17 15 17 17 jl7 18 i; 20 5 18 27 5 u. 29 18 31 5 !l8 32 5 18 35 1? 41 5 19 44 19 46 5 !19 47 5 19 50 7 20 57 20 61 20 62 5 (20 64 JO 67 tl 73 21 76 21 78 5 !21 81 21 e3 5 22 91 22 93 22 96 22 97 6 22 200 5 23 207 23 211 23 204 23 216 5 33 18 24 26 24 28 24 32 5 24 35 24 38 25 46 i5 48 25 52 6 !25 56 25 59 1 26 65 6 26 69 26 73 5 126 77 26 81 5 27 85 27 88 2 27 93 5 127 98 5 27 303 2 28 307 2 28 312 5 28 316 5 -8 310 28 25 5 29 28 29 34 29, 38 29 43 29 47 3 30 53 30 58 3o' 62 5 30 67 5 30 74 31 77 31 84 31 89 31 94 31 99 32 403 5 32 409 32 415 32 419 32 425 3 1 33 26 5 33 34 33 40 33 45 33, 53 5 34 53 34 59 34 66 34 73 5 341 79 35 81 35 88 35 95 35 501 35' 508 36 507 2 36 514 5 36 522 5 36 27 5 36 i 36 5 37 37 37 45 37 53 37 59 37 67 38 65 38 73 38 82 38 88 38 97 5 39 .97 39 607 2 39 615 39 624 39 333 2 40 627 40 36 2 40 45 6 40 54 5 401 64 2 82 J TABLE OF ROUXD TIMBER. <^ 77 Ft. ' o 78 Ft. CT 79 Ft.! w 80 Ft. ^i 8rFt. 2i 3 p 6 Long O o D a> <— - ^ i "3 6 Long. 1 6 Lons o o "5 5 2 X 6 Long. r-r- 1 3 6 Long. 3 3^ O o <->■ O T5~ T 15 15 "15 7| 15 8 7 .20 6 7 20 8 7 21 2 7 21 3i 7 21 6 8 26 8 8 27 O 8 27 5 1 8 27 7 8 28 2 9 9 5 34 4 9 34 8 9 55 2! 9 55 5 10 41 9 10 42 5 10 45 5 10 43 6 10 44 11 50 7 11 51 4 11 52 1 11 52 6 n 55 5 12 60 4 12 61 12 62 2 l2 62 6i 19 65 5 15 70 8 13' 71 7 . 5 72 5 13; 73 4 15 74 3 14. 82 5 14 83 5 14 84 5 14; 85 2 14 86 3 15 95 5 15 96 5 15 96 7 is! 98 8 15' 99 8 1 16 108 5 16 109 o O 16 no 8 16| 112 |16 113 \7 21 5 17 23 17 24 5 17 26 51 1 !'^ 27 5 18 36 7 18 38 5 18 59 5 18 41 5 18 45 19 51 5 19 53 5 19 54 4 19l 57 0| 19 59 20 66 20 71 20 73 20 1 75 01 20 77 21 85 2 21 88 21 90 5 2i; 92 5 21 94 5 22 205 22 206 2 22 208 22' 210 C 122 212 5 23 22 2 23 24 5 25 27 25 29 l|25 52 5 24 42 24 44 24 47 5 24, 5 1 5 124 53 5 25 63 25 66 c: 25 69 25 72 5 26 75 26 84 26 87 26 91 26 94 26 95 5 27 505 5 27 309 27 314 27 316 3 27 321 28 28 28 33 5 2 8 56 28 41 C 28 45 29 53 5 29 56 29 62 5 2y 65 29 68 30 77 5 30 83 6 50 87 50 92 5 50 96 5 31 404 31 408 5 31 414 51 4I7 3 51 422 5 32 30 32 35 32 42 52 45 3 32 52 33 58 33 64 2 53 68 c .-1 n 00 75 00 79 34 85 1 34 , 92 C 34 96 5 34 503 54 507 5 35 514 5 35 i520 35 526 5 35 33 5 35 36 5 36 43 5 36 : 50 36 56 e 36 62 5 36 66 5 37 74 37 ! 85 37 89 2 37 96 37 604 2 jB'oee 58 614 2 38 622 5 38 628 58 34 5 39 40 39 1 '^« 39 57 2 39 65 5 59 69 -5 iO 73 1 40 ' 82 5 |40 89 5 40 97 7 40 704 5 A TABLE OF ROUXD TIMBER. 83 c 182 Ft. : C? 83 Ft, c 84 Ft. ?:85Ft. u 86 It. 3 CO Lono-, 3 ft Lon 3 U- 1 Lotl! £^l il -J LuMiic- : ^ 1 Long. 1 3 3^ C 3 6 a in 6 rJ «-»■ en i 1 6 en 6 3 an 1 ^^ 6 3 1 6 1 16 16 ~5 i 6 7 16 9 21 8 7 22 2' 7 22 4 7 22 7 7 22 9 ' 8 28 5 8 2 8 si 8 29 8 29 5 « S9 8 ! o 36 Q 9 36 5 9 36 7 9 37 4 1 s 37 S IC 44 6 10 45 IC 45 6 10 46 4 10 46 9 i 1 1 ! 54 ' I i 54 6 11 55 2 I I 5 5 9 1 i 56 5 i2 64 4; 12 65 12 65 8 13 66 6 12 67 6 13 75 5i 13 76 h5 77 (■ 13 78 3 13 79 3 U 87 51 14 88 5 iu 89 4 14 ' 91 14 92 3 15 101 5 15 102 8 115 103 8 !5 105 6 .5h06 9 16 14 ^i 16 16 2 |ie 17 C !6 18 8 16 20 6 17 29 0' 17 31 ■ '7 32 17 34 -o :'7 36 !S 46 oi 18 47 ilS 48 2 '8 50 5 .18 52 5 19 61 19 63 19 64 5 i9 67 I I '• 69 ■20 79 5' 20 82 5 20 83 20 86 5 •20 89 21 97 0; 21 99 21 202 e ~ i 204 21 206 7 22 216 0; 22 218 22 20 22 23 5 ,22 27 25 36 5 23 38 5 2 3 41 5 23 44 5 ■ ^ ^> 48 24 56 5 24 60 Ol 24 62 5 ■■> -I o7 !..4 70 25 78 5 25 83 !25 85 o| 25 88 5 |2;i 93 5, 26 i 302 5; 26 306 26 307 5: 26'314. 2. 315 271 35 27 28 I27 31 5; J? 36 5 '^7 41 ■281 1 49 (• 28 55 5l 28 57 Oi 28 64 - 8 67 , 29i 1 75 29 78 51 29 83 2 29 87 5 '2 9 94 .30! 403 30 406 5\ 30 410 5 .i(/ 416 5 20 422 1 3:1 1 28 Z] 31! 34 si ^ 1 47 5 3! 45 C ,, 50 32' 1 57 32 64 52 66 ^i 10^ 75 c 32 82 331 86 ( 00 94 33 96 5! I33 5C5 ■■• .}0 505 5 34' i 5 15 2 34 5^:, 2 134 525 o; |34 34 2 n 42 5 , 35i 45 35 53 5 i35 55 5; 35 65 ( j 35 73 5 36j 76 ol 36 1 84 loo 89 Oi 36 99 2 |36 607 371613 ''i 37! 621 ^> i .1 r 1 625 0! 38 636 37 44 5 381 44 5! 38 53 5i lor. Ob 57 5! 38 68 38 77 2 1 JSJ 82 5 39 89 39 95 0! 39 706 ( 39 r\$ ; 40! 714 5 : 40 724 40 731 5' 40 43 ij -it' 5-2 5 '■ S4 A TABLE OF ROUJ^D TIMBER. fbl Ft.i C 88 Ft. ■ 89 1 't. — 90 t t.'i ^ 91 t\. 3 Lnos r_ .<5 e LoriK. 1 5 6 Long. 5- '^ 1 s 1 So" 5- J' 1 6 Long. 9 3 3 U3 5 ft <-► a n 5" 6 Lon?. 3 es 3 CA n P", 6 O o 3 CN 17 ~T 17 ; 17 5! 17 1 17 9 7 2S 2! 7 23 4i 7 23 7 7 24 '! 7 24 3 8 50 4! 8 30 7: 8 31 2 8 31 5 b 31 7 ^ 38 3^ 9 38 7' i 9 39 3; 9 39 ^t 9 39 9 :'c 47 5' 10 48 Oi 10 48 71 lo 49 3! 10 49 6 11 57 51 1 1 58 1 11 58 6i 11 59 ^-' 11 59 8 12 68 4; 12 69 12 ro 3.1^, 70 9' 12 71 5 13 80 0' IS 81 ^i 13 82 0! 13 83 ^ 13 83 7 U 93 1 U 94 0! I4 95 5! 14 96 2 l4 97 4 i.5 108 15 109 2 .15 1 10 71 15 111 9 15 1 12 8 16 24 o; 16 23 |16 25 o'l 16 26 2 16 27 4 17 37 5 17 59 17 41 0; 17 42 17 43 18 154 c 18 55 H 18 57 5! 18 59 2 lb -61 19 71 5 IV 1 ^1 i^^ 76 ^i 19 78 o| 19 79 2 iO 91 0; ,2(. 92 7' '20 95 01 20 97 0: 2C 99 3 bi 208 5 ^ i 211 M 2] 214 C'l 21 216 oi 21 217 5 -22 28 0! 2-^ 32 t j22 35 22 37 o| 22 58 5 23 51 7 33 54 c ^23 57 3 -» 59 0! 1 1.1 r> -O 62 5 24 To 5 24 76 c i34 78 5 24 83 5*24 85 5 25 96 5i 25 300 0I |25 304 ^ 25 507 ol 25 303 9 25 322 26 25 I 26 27 5i|2e 31 5J 2c 35 5 27. 45 0' 27 48 ol 27 53 2 27 57 ol CO 0- 28 72 ^i 2b 76 5 28 82 (> 28 85 2b 83 2 29 97 5 29 40 3 ij 29 407 2 29 4i2 0' '2S 415 30 426 0; \so 32 5; 3o 56 ll ol 41 5 30 44 5 3i 55 ]3i 62 0! 31 66 72 31 75 o2 86 2 1.2 94 ^! 97 2 3i 5('3 5 32 508 335 16 z •^ J J 5^2 1 2 527 !3: 34 5 33 29 5 34 47 1 34 53 5i 59 <^ 65 34 70 35 r9 C; .00 85 2; 35 94 2i \iS 602 5 55 -05 5 366U 5 3 ■ 62 i 5 3C 62g C' 1.3 35 )Ot> 40 5 57 52 o7 57 5 57 66 2 !0 J 73 5 37 78 5 38 85 38 94 C 3b 703 i '58 7c 8 5 3c 715 39 723 39 733 c; a -' 4i 5 - 48 S 3V 55 5 4C ) 53 8 4o 67 5 1> 73 •t 'k(- 87 -it. 9^, 3 i^5 Jl Table giving the Side of a Square^ equal to a Square inscribed in a given Circle. The first and third c:>lumns give (he diameter in inch- es, and extend from G to 49 inches ; the second and ^fourth, give the square, or side of the stick in inches, and eighths of inches. The first nnd third cohjmns at the ri^^ht hand of the double line, are the same as tlie lirst mention- ed, except they commence at 6-f^ and extend to 49y|- ; — the second and fourth columns i^ive the square, or side of the stick in inches, and hundredths of inches. ) f ^ '^w 5 X ~ ■ ^ Uu 3 v_ ff en 1^ 5 ex. o of5 ■ a O 5 Z3 re r-^ 3- C 2 (T' 3- 3 d . CO (T D- t/> ft ^ cr. r^ i — ' cfi ^ S" -3 CA ►^ «-3 w o" >• C/l ro" O ^ O fD 3i a ^ "^ o "* £. " *^ " ~ "* 3- 3 ^*" 5' ::; 3* 5" zz CTi **^ " 3" ^ ** ^ * Ci ^■^ o o ^ -r 3 •I) VJ Cfc rr ^ X 3" 3 re IT- 2 CO 3 en 1 ^ 4 2 28 T9~ 6 b ~5 4 59 28 5 20" lo : 7 4 7 29 20 4 7 5 5 30 29 5 20 85 ; 8 5 5 30 21 2 8 5 6 01 30 6 21 66 ' 9 6 o 31 21 7 9 6 71 31 5 22 27 1 10 7 32 22 5 10 5 7 42 32 6 22 98, I u 7 6 33 23 2 11 5 8 •13 33 5 23 68 ' 12 8 4 34 24 12 5 8 83 33 5 24 39 ' 13 9 1 35 24 6 13 5 9 54 35 5 25 10 , 14 9 7 3ci 25 3 14 5 10 25 36 5 25 80 15 10 5 37 26 1 15 5 10 96 37 6 26 51 1 16 11 2 38 26 7 16 5 11 ^Q> 38 5 27 22 17 12, 39 27 4 17 5 12 37 39 5 27 93 18 12 6 40 28 2 18 5 13 08 40 5 28 63 34 19 13 3 41 29 19 6 13 78 41 5 29 20 14 1 42 29 6 20 5 14 49 42 5 30 05 21 14 7 43 30 3 21 5 15 20 43 5 30 75 22 15 4 44 31 1 22 5 15 90 44 5 31 46 23 IG 2 45 31 6 23 5 13 61 45 5 32 17 24 17 46 32 4 , 24 5 17 32 46 5 32 88 25 17 5 47 33 2 25 5 18 03 47 5 33 58 26 18 3 48133 7 26 5 18 73 48 5 34 29 ^27_ 19 ol 49|34 5 27 5 19 44 49 ojSS 00 H OF SURVEYING. Lantl is generally mensurecJ hy a Chain of GG teet in CMigth, divided into lUO equal parts, called Links, each Link beinic 7,02 Inches in length. A Pole or Rod is iri. r!?et or 25 links in length : hence a square pole contains 272-^ square feet, or 625 square links. An acre of land contains \60 square poles or rods, and 435G0 square feet, or 100000 square links To find the number of square poles in any piece of land^ ^ake the dimensions of it in feet and find the area in squart. feet, and divide this area by 43560, the quotient will h« the nnmber of acres ; or divide by 272,25 and the quotien will be the number of square poles. If the dimensions b< taken in links, and the area found in sqiiare links, the num l>er of acres may be obtained by dividinj; b^' 100000, c which is the same thin^, cut ntf the five riu;ht hand figures but the number of square poles miy be found, by dividiDj, by 625. PROBLEM L To find the number of Acres of Land, or the number of Square Foles^ in the form of a regular Faral Ulo^rani. RuLK. MuUiplv the Base or Length by the Perpendic- irL'T Height or Breadth, ^nd if the dimensions were taken in Links, divide by ^'Ib, if in feet by 272.25, or 2721. the quotient will be the number of Square Poles, which tlivi- ded by 160 gives the number of Acres. E> AMPLF,. Suppose the Base be 60 feet and the Per- pendicular 25 feet, required the Area in Square poles ? Length CO feet Breadth 25 feet 272,25)1500,00(5,5 Ans. 1361,25 138,750 136.125 PROBLEM 11. To find the numher of Acres and Poles ifi a piece of Land in thejorm of an Oblique Angulur Paral- hiogram. Klle. This Area may be found in exactly the same manner as in the preceding problem, by multiplying tb* •• >» !." ^[^'^ Perocndiculuf height, and dividing by 626 : ^F sunrrAiXQ. xyben the flimensions are tnkon in Link* ; by 272,25. when faketi in feet, the quoiif;r.t uili oe in Amoricjn i^oles, woich divi the •. i.-uber of Acres. E vpLK. Siipjjosei'i ?>•=<• I^f' 63:J Dnd the perpenfiic- Qlar 32o Links ; required the number of rotes ? Bn^e 632 Links B.-^o 632 Links Perp. 326 Links 3792 1264 1896 Perp. 326 Lmk« 3792 1264 1896 «2a)206032(329,6 Poles. 1875 18.53 a250 Links 1,00000)2,06032 in an Acre. " 4 24128 40 6032 9,651 2t)=to 5625 2 Acres, Roods, and ri^ Poles.. 4070 PROBLEM in. To find the number of Acres and PoJ^ in apiece ef Land at a Triangular fonn. Rule, ^Multiply the Base by the Perpendicular height, aod divide the product by 1250 when the dimensions are given in Links ; by 544,5 when given in feet, and the quo- tients will be the answer in Poles. Xote. Instead of dividing by 1250, you may multiply by 8 and cutoff the four right hand figures. Example. Given the Base AC equal to 200 feet, and Perpendicular BD equal to 150 feet ; required the Area in Poles. Perpendicular 150 feet Bas e 300 feet 544,5)45000,0(82,6 Poles 43560 14400 10890 35100 PROBLEM IV. The three Sides of a Triangle being given, to find the Area Arithmetically. Rule. Add together the three sides, from half that sum, subtract each side, severally, noting down the re- mainders ; then multiply the Balf suoa and the three remaifN OF SVRFEYL\€f, ders continually together, and the square root of the la-: product \viil be the Area. Example. Suppose a Triangle whose three sides are ;^0, 2o, and 92. Sides, 30ri*25-|-22 sum 78-^2=30=— 30=9 and 39— 26=13 atjd 39— 2-:=17, the half sums 39x9x13X17= 77571, the square root of which, is equal to 278,5 Chaiu?.. tr 27 Acres o Roods, and 17,5 Poles. ON THE SLIDING RULE. Example in Trigoxometiiy. In the the Oblique Angu- ?ar Triangle ABC. let there be given AB=56, AC=d4, angle A ABC 46° 30', to tind the other Jingles, and tJie side BC. la this case we have by art, 53 Geometry, the followins; Can- ons ; as AC 64=Sine zB 46° SO' : : AB 5G : Sine of the Ang. C, to the Ans. And as the Sine /.B : is to the side AC : : so is the zA : BC ; therefore to work the fiist proposition, by the Sliding Rule, we must bring 64, on the Line of numbers, on the fixed part, against 46° 30' on the Line of Sines on the slider ; then against 56 on the tixed part, will be 39° 24' on the slider, which will be the Angle C. The Angles B and C added together, and their sum being subtracted trom 180°. leaves the Angle A =94° 6', Then by the second Canon being the Angle B =46° 30' on the Line of Sines, on the slider, against AC-«= €4 on the Line of Numbers on the tixed part ; then against the Angle A=94^ 6', or its supplement, 85^ 54' on the slider, will be found the side BC=o8 on the fixed part. — In a similar manner may the other propositions of Trigo- nometry be solved : and from what has been said, it will be easy to work all the problems in Plain Sailing, Middle L ititude S-iiiing, and Mercator's Sailing, as in the follow- ing examples. Example 1. Given the course, sailed 1 point and the jsiance 85 miles ; required the difi'crence of Latitude and De[» nture ? By Case Ist. of Plain Sailing, we have these Canons : — OF SURVEYING. «y As Tlatlius C. points, is to the distance C5, so is the Sine Compl. ot the Cour5e=7 Poin, to the Ans. or difference of Latitude : and as Radius 8 points, is to the Distance 83 scr is the Sine Course 1 Point, to the Departure ; her.ce we must bring the Radius 8 points, on the lixed part of the Line Rhombs, against 85 on the Line of Numbers on the j^lider; then ajrainst 7 points on the Line Rhombs will be found the diffeWnce ot Latitude 83i on the sUder, and a-^ gainst 1 point, will be found the Departure 16^- miles. If the Course is given in Degrees, you must use the Lmc marked SIN. Example 2. Given the difference of Latitude 40 miles, and Departure 30 miles ; required the Course and Dis- tance ? As the difference Latitude 40, is to the Radius 45^, so is the Departure 30, to the Tangent of the Course. There- lore, we must brins; 40 on the Line of Numbers found on the slider against 45^ on the Line of Tangents, on the fix- ed part ; then against 30 on the slider, will be found the Course 37° nearly. Again ; the Canon for the distance given. As the Sine Course 37°, is to the Departure 30, so is Radius 90<^ to ihe Distance. Bring 37° on the Line of Sines, on the fix- ed part, against 30 on the Line of Numbers, on the slider ; then against 90° on the Line of Sines, on the fixed part will be found the distance on the slider. Example 3. Given the middle Latitude Sailing, we have this Canon. As the Sine of Complement of middle Lati- tude 50°, is to the Departure 30, so is the Radius 90°, to ihe difference of Longitude. Bring 50° on the Line of Sines on the fixed part, against 30 on the Line of Numbers, on the Slider ; then against 90*^ on the fixed part, will be found 39 on the slider, which will be the difference of the Longitude required. • It may be observed, that in the calculations of Spheri- cal Trigonometry, the Sliding Rule is rather an object of curiosity, than of reu use, as it is much more accurate to make those calculations by Logarithms. How to find the Course when the Base and Perpendicu- -lar are given. Rule. First find the Hypothenuse by the square root, then add one half of the Base to the ilvpothenuse. Then H2 ' ^ 90 OF SURVEYIXG. say, as that Qumber is to So, so is the PerpeRdicular to the Course. ExA-MPLE 1. Run North 100 Chains, then West 100 Chains, or Links, or Teet, or yards, or miles, or any thin^ else. \00 100 191,12 : 83*^ : : 100 lOO ICh) 100 "iuOOO TUOOO lai, 42)8600.00(44° 55' 38" The 10000 7650 8 'Course, but it should 300UO(141,42 Hvpoth. 943 20 ^^ ^'^''- It' instead of 1 50 >- Base 765 63 ^^"^ ^^ " the 2d term, ^ 96 60 sed,theAns. wouKt have been 45** V L'Cl;400 1065120(56 2'/ nearly. 9oi 9571UO -324} 11 900 3 0G020 'll'296 95710 -'8282)60400 12310 56564 60 3836 733600(38" 5742 6 164340 153136 1 1 204 K.\A3i?LE 2. Suppose the Base 80 and the Perpendica- ;r 60 ; what 13 the Course ] 8 J 60 As 140 : 86,1-4 : : 60 80 60 60 «;4U0 3600 H0)5168,40(36,917<' •>600 420 ^ 60 jOOO(100 ?r'ypothenuse 968 55,020 iO..0O 40 haliEiLse 840 bO 'VxJoSTTiO 1284 l,2uO r:*jO 02 40 Ans. 36^ 55' 1^'^ 140 Too ExAHFLE for finding the Base and Perpendicular. If yoa •jfl North, 30? East, 70 Rods, wb^t w your Perpendicular? OF SURVEYING. SI 1. Rarliiis Rods 60,00 : 70 : • 30 SO*^ 60,00)2100,00(35 Rods. 1800 30000 30000 2 Rerersed Radius Rods 69,22 : 70 : : 60® 60 QOOOO 69,22)4200,00(60,67 Rods. 4153 2 46800 41532 52680 48454 A Table of Natural Radii. 4226 o Radii. o ~24 Radii. 47 Radii. 7C R '■i. ■; bij 880 58 986 64 240 74 544 •2 bQ 922 25 59 141 48 64 563 71 .5 105; 3 bQ 967 26 59 300 49 64 895 72 75 661| 4 bl 017 27 59 466 50 Qb 238 73 76 293i 5 57 072 28 59 637 51 65 590 j 74 76 942i 6 57 130 29 59 815 52 65 950 75 77 608; 7 57 193 30 60 000 1 bo m 323 76 78 292; 8 57 260 31 30 190 54 m 705 77 -n 1 O 994! 9 57 331 32 60 388 55 61 093 78 79 714, 10 o7 407 33 60 592 56 67 502 79 i.O 453; 11 57 488 33 60 804 57 67 917 80 81 21li \- 57 573 35 61 021 58 68 343 31 81 990; 13 57 664 36 61 246 59 68 781 82 82 784 14 57 757 37 61 479 60 69 231 83 83 6£3 15 57 857 38 61 719 61 69 692 84 81 4c2 13 57 961 39 62 967 62 70 166 86 35 317 17 58 071 40 62 222 63 70 653 86 S6 205 i:. 53 180 4! 62 485 64 71 153 87 87 116 19 58 306 42',u2 617 65 71 666 88 88 0c2 20 58 431 43 .33 036 QQ 72 193 89 89 013 21 58 562 44 •33 323 67 72 734 90 90 ceo 2 J 58 698 45|63 620 68 73 289 i 23 1 — 58 840 46|63 925 69 73 859 This Table extends from 1^ to 90°, and is made by the following Rule. Divide four times the square of the Comple* 9? OF SURITAIXG. ment, by three times the Complement, add this to 309, aad then add the Angle to the quotient. Example 1. Given 10 Degress. 90 . 80 540)25600(47,407 ^0 _3 2160 lOano;. ordg. 80 Conpleraent 77^ ~^^ 57~K>7 Arrs. CO 300 3780 6400 «q^- Corap. ^ Divisor "i^uTo 4 2160 25600 Dividend 40,00 Example 2. Given the Angle 80 Degrees. 00 10x3=30-i-3ub=330 10 80 3 10 Complement. 30 10 300 100 Square of the Complement. 330 DivUofe. 4 330)400('1,212 to which add the Angle 330^ "^00 1,212 C60 80 400 81,212 AGS. 330 700 Application and Uee of tht Radius 'Table. Given the side AB=16,8 Pole, Angle A=58o to find AC. In the left hand column, under Deg. find 32, and against it, under Radius you will find 60,4, nearly. Then as 60,4 : 32 : : AB : BC. 16.8 "336" 504 50,4)53776(8. &=Side BC. 4832 5440 OF svrveylyg: & Then as 68,3-1 : 58° : : 16,8 ; or as A Radius Is to Angle 16,8 A, so is the Hypothenuse J oil' to the Perpendicular. 840 68,34)974,40('14,25 hC. 6834 29100 27336 17640 13668 39720 So that if one Angle, and the Hypothenuse or Course is given of any Right Angled Triangle, the Base and Per- pendicular, may easily be tbund as above. Oblique Angled Triangles, when the Angles and one side are given, may be solved by the table, by finding the Pependicular. In the Obliqued Angled Triangle giv- B en, AB=100 Miles or Poles, Angle A= A 60^ and Angle 0=60^ ; required the /, \ Perpendicular BD ? As 60 : 30O : : 100 : 50 AD. / As 69,231 : 60^ : : 100 : Ans. / 6000 ^ ^ i ^C 1000 D 69,23 1)600UOUO( 86, 666 BD Miles, or it may be 553848 called Latitude, or De- 461520 ' parture, as well as Per- 415386 pendicular. 461620 415386 459540 415386 441540 fiote. By adding any two Angles of a' Triangle, to- gether, and subtracting their sum from 180 Decrreo-*, will remain the other Angle. Thus, in the Figure, 60HO=120, wbich if subtracted from 180, leaves 60, the Angle B. S4 OF LOGARITHMS. The usual method of computing the Lo2;arithm«i to anj af the natiirai numbers, 12 3 4 5, kc. is. as fc-iinw? : Rule. 1. Take -.nv two numners whose ditTerenre is iinit^,orl,and let the Log. to the lesser nutnher be known. '"2. Divide ihe constant decimal ,8G85C'8P64, &c. (or 2-r£,3025, ^c.) by the sum of the two numbers, and re- serve the quotient : divide th"i several quotients, bj' the square of the sum of the tuo numbers, and reserve the qnoiient , .'ivitie tins last quotient, aiso, by the square of the sum, and a^ain reserve the quotient ; and thus proceed. continuallv dividing the last quotient, by the square of the sum of the two uumbers, as long as division can be made. 3. Write these quotients in their order, under one an- other, the first uppermost, and divide respectively by the prime, or odd numbers, 13 5 7 9 11 13, &:c. as long as division can be made, that is, divide the first reserved quo- tient^by 1, the second by 3, the third by 6, the fourth by 7, and so on. 4. Add all these last quotients together, and their sum will be the Logarithm of the greater number by the less'; — to this Logarithm, add the Logarithm of the lesser number, and their sum will be the Logarithm to the greater, or pro- posed number. Example 1. Let it be required to compute the Loga- rithm of the number 2. Here the given number is 2, and 4he next less number is 1, (whose Lo2;arithm is ,0000,) and the sum of 2 and 1 is 3, whose square is 9. 3)868588964 1)289529654 289529654 9)289529654 9)32169962 3)32169962 5)3574440 10723321 714888 9^3574440 7)397160 56737 9)397160 9)44129 4903 9)44129 9)4903 9)545 11)4903 13)545 15)61 446 42 4 9)61 ,301029995 Add Log. of 1 . ,000000000 True Log. of 2 ,301029995 Ans. Ex.\MPLEi 2. Let it be required to compute the Lo|£[/ OF LOGARITHMS. ^y lithrn of 3. Here the given number is 3, jind (he next le.is l^ 2, whose Lo-iiiithm i>y the rirst example is 301029095, and ihe ?um of 2 and 3 is 6, and the «qaare of which is .25. 5)868588903 2^5)173717703 25)0948712 25)^77948 25)11118 .25)445 25)18 1)173717793 3)0948712 5)277948 7)11118 9)445 11)18 Add Loo;, of 2 173717793 2316217 65590 1528 50 2 ,1760912*00 ,301029995 L^iarithm of 3, 477121198 Am. As the sum of the Logaritlmis of numbers gives the Log- nrithm of their product, and the ditTerence of the Loga- rithms gives the Logarithm of the quotient of their num- bers. From the above two Logarithms, and the Logarithm of 10 which is 1, we may raise a great many Logarithms, as in the following examples. Example 3. 2x2=4 therefore to the Logarithm of 2, ,301029995 Add Log. of 2, .301029995 Loa(.iritiim of 4, ,602059980 Example 5. 2x4=8 therefore, to the Log. of 2, ,301029995 Add Log. of 4, ,602059990 a. ithmofS, ,903089985 [' Example 7. 5x8=40 therefore, to the Log. of 6, ,903089985 Add Log. of 5, ,698970005 Log. of 40, T,602059990 Log. of 320, 2,505 1499.7» Aad thus computing by thiS general rule, the •Logarithms of the prime numbers, 13579 11 13 15 17 19 21 23 25, &;c. and then by using composition and division, we m;»y easily find as many Logarithms as we please, or exam- ine any in the Table. Directions for Uiking Lftganlkm^, and thcit Knfnhcrs from the. fhbU.. Example 4. 2x3=6 therefore, to the Log. of 2, ,301029995 Add Log. of 3, ,477121256 Log. of 6, ,778151250 Example 6. 10-r2=5 therefore, fronr. Log. of 10, 1,00000000( take Log. of 2 0,301 02999i Log. of 5, 0,69897000*, Example 8. 8x40=320 therefore, to Log. 40, 1,602059990 Add Log. 8, 0,903080985 96 OF LOGARITHMS. Look for the number, whose Logarithm is required, m the Column of Numbers, and against this number, its Log- arithm will ho found. Thus; the Logarithm of 1234 is 3,0913151 ; so that a- ny number less than 10000 may easily be found in the full Table by inspection. But if the number is greater than 10000, and less than 10000000, rut off four figures on the left of the given num- ber, and find the Log:irithm in the Table ; add as many u- nil"? to the Index, as there are figures rematning on the right ; subtract the Logarithm found, from the nest following it ; then as the (Ufference of numbers in the Canon, is to the tabular distance of the Logarithms answering to them, so are the remaining ti<£ures of the given number, to the Log- arithmic difference ; which if it be added to the Logarithm before found, the sum will be the Logarithm required. — Then let the Logarithm of 92375 be required. Cut off the four tiist figures, 9237, and to the Index of the Loga- rith.m corresponding to them, add one unit, because one riiiure is cut off, on the right. Then from the Logarithm of the next greater number, 9238=3,9655780, subtract the Log. of the' given number, 9237=3,9055309 Indes=10 ; then* as 10 : 471 : : 5=unit added to 4==to the Index of the Logarithm, corresponding to them ; because one figure is cut off, on the right hand, and the Answer w'ill be 235, which added to the Logarithrai 4, 655309-j-235=4, 9655544=10 the Logarithm required. Or, more briefly, find the Logarithm of the first four figures, as before ; then multiply the common difference, which stands against it, by the remaining figures, of the given number ; from the product, cut off as many figures at the right hand, as you multiplied by, and add the re- mainder to the Logarithm before found, fitting it with a proper Index ; as the Index must always be one less, than the number of figures in the given natural number ; so the Index of the Logarithnn for any natural number, less thaa 10, must be ; and for any number between 10 and 100, the Index will be 1, and from 100 to 1000, 2 and so on. * If one figure is cut off, say as 10 h to the diflference of the Log;- •rithra ; if two figures are cut off. as 100 is to the difference; if thre«. t'\!?u say as 1000 is te the difference, &c. or LOGARmiWJ. 97 The Logaritlini of a decimal iVacfion, is the same as that of a whole number, excepting the Index. Taiie out then the Logarithm of a whole number, coDsiisting of the same figures, observin}^'''to make the negative Index equal to the distance of the lirst signiricant figure of the fraction, froni the place of units. Example. The Loz. of 0,07643 is 2,8852639 or 8,8832639 0,00259 is 3,4132998 or 7,4132998 ♦* " 0,CQ0G278 is 4,7978313 or 6,7678213 To find the Los^arithm of a mixed Decimal Fraction. Find the Log/uithm in the same manner, as if all the Sgures were integers ; and then prefix the Index belong- ing to the integral part. Thus, tlie logarithm of 39,68 is 1,5985717, here the Index is 1, because 1 is the Index of the Logarithm of every number greater then 10, and less than 100. Tofnd the Losarilhrn of a Fuls^tr Fraction. Subtract the Log.irithm of the demoniinator from that of the numerator, an(i the difference will be the Logarithm of the fraction. V^hat is the Lo-arithm of |^ ? Logarithm of 37, 1,5682017 Logarithm of 94, 1,9731279 1,5950738 when the Index i*: negative. Tofnd the JVatuj'al JVuinber to any Logarithm iii the liable, Tiiis is to be dene hy the reverse of the former, viz. by sear^tjing for the proposed Logarithm those in the table, and taking out the corresponding number by inspection, in which the propf.T number of integers is to be pointed off, viz. one more than the units of the atiirmative Index. To find the A\iTnher corresponding to a Logarithm, great' er than any in the Ihble. Frst, subtract the Logarithm of 10, 100, 1000, or 10000 from the given Log. greater than any in the table td! you have a Logarithm that will come within the compass oi the table ; then tii-.o the number corres{)oading to this, and mul- tiply it by 10, 100, 1000,or 10000 the product will be the numb.^r required. Suppose the number corresponding to the Logarithm of 7,7589875 be required. Subtract the Logarithm of the number lOGOO, which is 4,0000000 from 7,7589875, there remains 3,7589875, the natural number I «»g OF LOGARITHMS. rorresponrling to which is 6741 ; this mullipliefl by lOOOO f^ives tiie ijimiber aiisivvering to the given Logarithm ='574iOUUO. MULTIPLICATION BY LOGARITHMS. Multiplication is per'';rmed b}' takinsj from the table, Logijrithms answeriug to both factors, and their «imi will he the product in Logarithms, the number answering to which, wiii be the answer. Multiply 45 by 27. Numbers. Logarithms. Numbers. Logarithm*. 46 L 6532 125 23,14 l!3643G34 27 1,4313638 76.99 1,8807564 1215 3.'JG45763 1 :,86 3,2151198 Multiply 3,686, 0,8372, 0,0294, 2,1046, and add them toschther. Here the 2 to be carried from the decimals to the Index. — Cancel the 2, and there re- mains 1 io be set down. Numbers. 3,586 2,1046 0,8372 0,0294 Logarithms. 0,5646103 0,3231b96 1,9228292 2,4683473 C, 185768 l,26C95o4 Practical Exampe. What cost 87 pounds of green tea, at j552,12 per pound? Numbers. Logarithms. 2,12 0,3263359 87 1.9395193 $184,44 2,2658542 DIVISION BY LOGARITHMS. Rule. From the Logarithm of the dividend, subtract the Logarithm of the divisor, and the number answering to the reuii.irjder, will be the qiiotient required. Examples. Divide 15811 by 165, and 163 by 8,18. Numbers. Logarithm Numbers. Looatithms. Dividend 15811 4.1989593 Divd. 163 2,2121876 Divisor 163 2,2121876 Divi. 8.18 0,9127533 Quotient ~0f 1,9867717 Quo. 19,926 1,2994343 PROPORTION, OR RULE OF THREE BY LOG. Rule. If the proportion be Direct, add the Logtrithms of the second and third terms, and subtract from their sum, OF LOGARITHMS. d\) flie LOi^arithm cf the first term, the remainder will be the term ofthe Logarithm required. ITlhe proportioD he Inverse, add the Logarithms of the first and second term?, and froDi their sum, subtract the Logarithm ofthe third term : the remainder will be the Logarithm to the required term. Example. Find a fourth proportion to 7964, 378, 27960. Numbers. Logarithms. Numbers. Logarithms. 2d. term 378 2,5774918 7,0240290 3d. term 27960 4,4465372 1st. Uam 7964 3,901 1313 7,0240290 4th. term 1327 3,1228977 Practical Example, if 6 jards of cloth cost g5, vvliUt will 20 yards cost ? As 6 Log. 0,77815 is to 5 Log- 0,69897 so is 20 Log. 1,30103 Sum of 2d and 3d 2,00000 The answer, therefore is Subtract first 6 0,77815 iq dollars and |^ or 16 Ans. 16,67 Log. 1,22185 dollars and 67 cents. EVOLUTION BY LOGARITHx^lS. Rule. Divide the Logarithm of the number by the Index of the Power, the quotient is the Logarithm ofthe root sought. But if the power, whose root is to be extrac- ted is a (ieci nal fraction, less than unity, prefix the In iex of its Logarithm, a figure less by one, than the Index of the Power,* and divide the whole by the Index ofthe Power, eind the quotient wll be the Logarithm ofthe root sought. Example 1. What is the Example 2. What is the square root of 196 ? . cube root oi 27 ? 196 Log. 2)2,29226 27 Log. 3)1,43136 ' Ans. 14 Log. 1,14613 An^. 3 Log. 0,47712 Example 3. ^Vfiat is the Example 4. Vi'^hat is the square root of 40,96 ? cube root of 0,015625 40,93 Log. 2)1,61236 0;015625 Log. 8,19382 Ads. 6,4 Log. 0,80618 P'^^fix 2 Index, 3) 28,19392 Ans. 0,25 Log. 9,39794 * In Ihis rule it is s'lpposed thit 10 was borrowed in finding the idex of the decimal, according to the rule. Ta^e 07. !0' I.OGxiRITHMS OF THE NUMBERS, FROM 1 to 100. No. 1 Lo-. N.l Loc.. ( N. Lo-. N. Log. —l\ J. 000000 26 1.4149731 bl' 1.707570 76 1.880814 2 301030 27 431364 52 716003 77 886491 3 477121 28 447158 53 724276 78 892095 4 G020G0 29 462398 54 732394 79 897627 i i 698970 30 477121 55 740363 80 903090 . ( 778151 31 491362 56 748188 81 908485 1 845098 32 505150 57 755875 82 913814 8 90309(' 33 518514 58 763428 83 919078 9 954243 34 531479 59 770852 84 924279 10 l.OOOOOC 35 544068 60 778151 85 929419 1 11 041393 3G 556203 61 785330 86 934498 : 12 070181 37 568202 62 792392 87 939519 ! 13 113943 38 579784 63 799341 88 944483 ! 14 146128 39 591065 64 808180 89 949390 ! 15 176091 40 602060 65 812913 90 954243 1 16 204120 41 612784 66 819514 91 959041 : n 230469 42 623249 67 826075 92 963789 18 255273 43 633468 68 832509 93 968483 ^ 19 278754 44 643453 69 838849 94 973128 ; 20 301030 45 .653213 70 845098 95 977724 • 21 322219 46 '662758 71 851258 96 982271 • 29. 342423 47 672098 72 857333 97 986772 \ 23 > 361738', 18 681241 73 863323 98 991226 1 24 380211 19 6901 9f 74 869232 99 995635 ! 9r • 39794C 5C 69897r 75 875061 100 2.000*. 00 A TABLE OF LOGARITHMS. IQl i 'a r-i TT lo CO o CO CO i-< CO c;; o» o -^ -^ o C'": o tj< s^ Ci co t g^ ■ -T. r- ^ '^ i> cr; 1-- c< G< cr. Ci CO CD G-? G! CO i~-" ^ CO 1— 1 c^ O O CC '-< -* O t- OO O-J O O C^' Ci CO CO -C CO O CO O '-1 CO ** O 1-0 ro. CO OJ CO O ^ CO CO 1-- O '^ OO C^ O O -^ 1:^ '-' ^ CO G^ CO CTi 2O'-''-<0^G-«G^C0C0'=^'=*'^OOC0C0C0i>i>l>C000C0 CO .— CO GO t- --1 CO r-r Ci CT; G^< O Ct C: C-f G^ CTj C'O O CO i-- I- i> c,-) CO^cr;OCOC;;l>— 'G^(0'^'T0■^■^0-P'T1— 'OCOTTCO -tr l> Ci »-< CO '^ ^0 CO CO CO »-0 -^ CO '-^ Ci t^ -^ -^ OO -^ O CO 1-1 CO t~ '— CO O -^ CO CJ CD O -^ CO CNf CO Oi CO t- '-' ''T CO G'J O O O O '-' " G< G-< G-J CO CO '^ -^ ^ uO O lO CO CO i> l^ i> CO CO CO q I- c;. T- c; ~ i- 'O "* CO O i-- CC' CO -^ O Ci CO '— CO '-' 'T' }> '-' >-0 ■JJ Gn( 1-- i> rr t^ CO >— 1 CO O "^ O G^^ CO CO CO 1> l^ >0 C: O C> rf O GO LO t^ C- O '-I C>J G-^ ©? '-' O Cj r^ LO CO O J> -^ O t- G* CO CO 1^ -^ >o o "T' CO e» CO o '^ CO i-< uo cj CO i> o ^ CO T-' liO CO O O 1— '-' '-' e< G^ CO GO -^ "^ '^ ^ O O CO CO t- l^ t- CO CD CO CO '^ C^ C::> G-* 'T 1-- G< O '-' O T-' CO CC 'O.CO c; t^ lO '— > £> -^ O C-. cr. rr CO GO CO 1^ '— co >— o co co t^ co i-O c; O co go ■^ C'O C5 O CO — . CO O CO t- CO CO CO i> CO >-0 CO '-' CTj CO TT C t- CO C5 rf ->' CO -r-, i_o cr. CO t^ »-> "^ en CO i^ -^ O C^ G^ CO O '7" t^ '-' '^ CO ^ 3 ^ ^ „ G^ Q^ CO CO GO -^ rr O »^ ^ CO CO t- t- i> CO CO CO «-0 CO CO G^J O O G? O CO O ^- G-J »-0 CO CO O G' CO CO CO 00 t- CO CO '-0 CO G< -r T-' >-0 O O c: '— CO i> >^: Ci O CO G-< CO '-^ CO CO l- co i '-< rt« O C^ '— ©,» CO -^ "^ -^ CO 3< 1— 1 C^ CO lO CO O t> CO Cj »0 --n C-^ CO O -^ O CO l^ ^-i >-^ Cr. CO t~- ■»-' '^ CO GJ CO O GO t^ O -^ CO- O O >— ^ '-' G » 0< CO CO CO "^ "C ^ -0 »-0 CO CO t-- t^ t- CO CD CO 't' -r^ CO O — < O '-'' C 'T c: I- c: -:: ':o :o co co cO c; Gsf ^ co c^ t— c; CO w GJ O -^" -^ O C< "^ CO' CO CO •-> 3^ O --O CO i-O O C^< « CO L- O CO lO I- CO C: O O O CTj CO t-- CO ^ G^ CDj CO CO' O CO Gs' i- 1 — CO O -^ CO G-f CO " i-O C;! G-* CO O •* CO G< to cr: CO t-- O -^ i^ ' O O " r-. T-n O^ G-( GO CO CO '^ -^ »-0 1-0 uO CO CO CO L- £> CO CO CO ■• ^ i G> ! 0^ '--' C CO O "^ G( CO w CD C- w 1^- O O w C^' ;o CO '^ O CO ■•— ' CO O O i^ O CO G-! CO w C^ G^ L- O-j CO C0> -^ G-» CD CX' CO ':!< CO CO G/ CO -O) CO >— 1 G^* -^ lO -4D CO CO iJ:S TT CO Gv/ O CO »-0 OJ C: CO G-* CO 'Cf '- »-0 C:j -* CO CO CO O "^ CO &J 'Oi O -^ CD T-i lO C'. e( CO O GO t- i 3 O O '- '-' G-« G-J GO GO 'CO "^r ■'^r' O ^ »-0 CO CO CO I- 1> CO CD CO , <^ - _ ' ( :o 1-i r-, O CO CO »-0 O L-^ CO G-) >C GO "-0 CO CM CO CD t-- CO -^ CO ^• CO CO lO CO CO ^ 3-i C:i G{ G^ CO O CTi •Tf CO UO O C^! '-< t-- O O i> CC t-i -T CO CO o '-' " e^i G< — 1— . ci CO CO rr 0-^ c-^ CO o? CI lO o Q uO C-: CO r~ G? CO O -^ CD G^ CO CTi so «--- '-» O CO G< CO CT5 CO }> 1 CD O O '-1 T-. GV G^ CO CO CO '^ '^r ** ^ O CO CO CO f- f> t- CO CO j -T '— ' -0 c: T-- . , o w CO i-O t- T CO CO CO it: gj i-- c c> co ^ co :0 "^ <>« O O C -— ' CD C> G^ X T-i C CO CO L-- CO lO O — ■^ 'rr- i— rj' t^ CD o< rr -o t^ L^ CD CO i-- i> c:: -^^ G-J O CO JLO G^' cr. i.O »— i-^ CTJ -^ CD CO' t^ '-< LO Ci CO t- ■r-i LO CD GO i> '-' TT" CD G^ »-0 C:; CO CO OOC-^r-iGJG^JG^ C0C0-V'T''^'-0»^COCOCOi>i>i:-COC0 —J GJ Q 1-^ C t^ CO CD w rr -^ '■O' CO CO CD CO --O CO. CO w G< l- T-< uO O O G^» O CO C"D CO O CO G^> G-( CD G-« r-i t-. O CD O 00 CD ■* CC CO CO Z: CO CO CO cii 1-1 CO CO '-r -^ CO CO G-< o cr^ CO ^ -^ CO o ^ t^ n C? rr CO Gi t-- — lO C5D CO t- — ' lO CD CO CO O "^ DO '-' LO CD O! CO ^ O O T-H ^^ G-} C< G^ CO GO 'f ^^ -^ O iOi i:0 CO CO t^ l^ 1> CO CO g i? :2; ,'-9 ^ ^ S ?? 3 ^ ;il' £} '^ >-'■•■ -■-' t^ co 05 p — g^ }02 A TAELF. OF LOGARITHMS. • _ 3> zz. 3* ..» r^ ^. ^ ;^- i.~ — c> c: 7^ — c 2^ _ •■J 3, ^ TT ^- ' {:>• ^ '^J tC, C- kc ^ ■»— -^ * • cc — ^ ^ t^ ^ v» — CC zz. ;^ ;^ ij^ ' C^ iSl ~ — r zz C'* ^ C; E •■ tTX c^ c^ 5» ■f o i> cc ^^ c; ~ ^ ^ ^ — i ^. ^ ~ Z * l^ ^ ot ;^ ^ C^ c^ ^ C^ 't^ c: " 7^ -^ cc "5r ~ cc > i-' ^— cc ■^ c^ — • r^ ^ »-• ^ i!; c. ^ cc ■»t" cc .^ ^ ^ s-» •^ tc *-> 't- 3^' ^ •^— - T ~ ct C'' C;_ ^. ■-^ ■r:^ cc ct ^ c 'Z t^ ^ "';^ t- t> £> t- •o c^ c^ '-C c~. c^ ^ •3^ C^ w" ■c; 3> r^ ^. 3« »C " — rr t- ^ ■r^ Z'. c. o ^- ^ »— "— — ©5 G< s* .G^ *^ c; — • r~ rr 4^ c^ »c ^ c^ „ »-• ' I S/ — * .->. » -^. ^_ — — - - ~z^ «» — — - "TT~ .^ — __ T^ — - — K^ t^ ■•". t— ^, !■— •rr r^ cc I> 7^ t^ CC X 7^ 1- ~ ^ t-- rj t:; ;^ "^ 7^ ^ ;•■; i^ — •J^ ^ ^» ;j; cc ^ 7^ ^ t- ~ ^ — — '^i ZZ -f -^ T -r --.;;< i_f^ * ^ 1 •^ ^^ ^> -■* "^ O' • — *^ t^> t!; ^z — « jj^ ^^ w ^-^ £^ *"■ 7*^ Is c; ~ Z^ c c — — — 3 J 3J 5' 7^ r: r^ — -r -r ^ ^ ^E '■^ ~~ hi . — ~- — — _ _ ^— -- •- _«" ^ „_ — ^ . — -^ — "c_r — _ ' — r; C-- ^-; -7- ^ — r^ ~ ^ -^ "3 lT, I^ '~ zz ^ ^ ~> ^ ;^ ^-^ , ^ »:^ ^ -f ;; ~ J "^ ~ c-5 .SI ;^ ^ Z'i r^ ;^ l> zz Ci "^ — »— — — [ ._, ;;t5 iC zz y> ^ ^ 5 > ^ ct. 3» i/:; Cl 3~» iC CC' •^ -^ t- — rr t^ ^ 7^ 2> Cl — ^ ■^ G^ S'-' 2* "" •^ "^ TT TT •^ ^ iC l:; " . "-^ ^ ~^ir ^_ — ;■- — __ -- - - __ 'N^. — ^ -- -»- — — - J^ ~"7" u '■~_^ vf -^ ~ ^ ^ e^ — "* ^~ cc 2-> ~^ c^ ^ »-^ t> »^ i-^ ~ Z:; t^ •^ — t^ ^ »C ~ ^) tc ^ C^ t;- •> Ct ^ r^ -7- *^ l> zz cc CC c^ ^ ' — _ v^ CD Of «^ cc ~ / ij; zc S-i ^ C;; >— -J^ cc — TT »>. '■' fi ** c; ^* f -«. Ci ^ ^ ~ ^ — — — ^< 3< o< 7^ ^ ""^ ^ rr —^ ij^ _^ -^ ^ to -. t '" iO ^ c; r- — ^ <^ ^ »c 2^ '■C. ~ ^ " -^ -^ 1^ — ^ ^» ^^ — ^ • - — »^ M* -^ •~ r^. \ZL ci t> t.- •o C» ^> r~ cc — ri t^ zz C": — "^i r; -r ^-> ^ ^ ^"^ — ;• cc — — C:; »-« tTt ^ — ii; zz — — r •^ — 'T- r»l ^ ZZ tc c: 5; "• h .-vi. '-~- — - ^"^ *— ^^ ^^ «^ •^> ^ t ■r- < ---^ - — - -» H «». vM . — ■ 1^ ^^ "* - — _ 1 -' -- _ — r; ^; <--. ~ -T- ^- ~^' — vr ;C i^ — 7.- — "-7 z> t_- ~^ -_2 .3 [ '^ i> — ~ ^ 2> C* C. ^T ^ ■^ Z1 j.^ ^ z~. •.— 3» -C fc^ r- ~ -T" :c r } t< ■~ > ! *->. ;._ t ^ •— • «->- «-^ — 7- zz — •^^ CO — T" cc — • -^ £^ ^ c^ c^ ^ ZZ ;^ c; CT^ • - rf • — ^ cr. ^ ^ ~ ~ — — "— 3* s* C'f f^ J^ r^ '^ ^ "^ »c iC C^ f ;^ — I 1 ■ kJ _^ ;7 -^ — ~r" ~3~ r"; — "^ — -T- :i f^ i^ — "3i" — . zz ^ — r- .^ i^ ■ ' __ -■ I ~ LTt cc ^ *^. — 7^ 7^ ^ ^ i."- J-» .."3 ~ c^ •^ c^ ^ r^ '^ t^ _^ >— ■^^ ■^ -,— zz :;< \S- ~ 3» i~ €- ^ — c^ ^ ;^ tr- C5 Ci ^^ ,^ Z\ * -^ t^ ^- — ir- •-« — r c^ — • ■M- J> ^ T^ t- ^ r^ •^ ^ S-* •^ cc »— 5~. c; ^ •^ ^ ..^ — — — ©♦ G* e< r: c^ r^ T T "TT T ^ ^^ i^ ■^ ! - • . -^ ^^ T^ .-;; — ~ — c^ 7^ 7-; ^ ^, i.~ ^^ r- -^ t- zz C- rr ^ ■— • fr " ; - t> i^ — -^ — ^ tr-/ £- ^ ~ t- C* *c »c *r^ c. z* yz JH ^^ «?• c^ \ ^ i> It C;^ _i -^ Ci 3-? t^ ~ — 1— • ** cc ^ —^ 7^ -^ »^ i-C ^ '.^ '^ t^ — t^ •^ Tr ;-» ^ ~" t^ ^ 7^ t"- ^ t'^ *^ ct. 0< w^ cc "— • "" • 5^ C; ~ ^ O »— . — — t< "/ Zt Z". 7"; '.'^ rr -T -r •T •-" •^ »c ^ CO .- 3y "3" — -^ — "3^. rt — •cr ?■» JZ — • — ,— c; ^~ cc ;7. CC -^ 5/ ' -^ 5 . T— r- ^ — ~ -^ £^ l> ^ ^ t^ r^ Zt i^ ■^ t' ~- cc zz ZC --O '■ ^ ~ ~ c^ -^ Zt c; c^ C* vC CO »• c^ i^ t^ cc ^ •— 3* ■?• zz. c^ f^ t ""■ — ~ ■^ ^ c^ r^ o nr r^ ■^ r^ t^ ^ c^ cc c: c^ C3 c: Ci CO ■•■^ 'H 'd> ^ ;^ ^ ^ — — ™ ■•— • o< (H c* r^ r^ c: c^ ■»^ TT ■^^ Ci »c kO CO . '^ ■ "■ ~-r .-^ CC ~ — _ f ' -^_ ■^ _^ — i^ -; ~z7 — ._ >* 7^ rr 4_r; z c^ ~' •; > ^* Cx 5'- r^ rt ;- t'i :^ 7^ Pi r^ z~. ^ -r •^ rr •^ ^r i^ '- '— • — • ^-» — —— ■•-• rr '— — • — ^ — — •— — — •— — — ^ ~~ ~~ A TABLE OF LOGARITHMS. 10.:5 3^J CO O G-} C; CO t-- cr: r^ CO CO GJ -^ CO CO r^ ,.., c; '— CO CO — ~ — ^^ I 2~< :r; Ci C; CO »-:: O CO >-0 rr G-> CO GJ -r" »0; -r GJ 1> G^? UO uO CO 1 :^ Ci CO CO -o O ';}' G< O CO CO CO " CO i-O Gi uO G-( CO r*i ^ CO ' C5 £^ Ci G^ O CO T- 1 TT t- O G^ >.0 CO ■" c^ "O c?> t-^ -^ l- rr> o* lO t^ -'_. CO- t> t^ 1> CO CO CO o c^ a C75 o o o c 1— ( 1— < ft^ &< G^ — ' G^ S^J O T CO' Oi ^ 3-t 3-': CO '-' J> zz tr- w i-"^ CO cr. — T" t-~ uO .- G-f 0< 3sJ i> O '- O t- G J iO 1> 'CO TT" O L-O, t-" CO !~- i-O — lO CO cr: cr: t- O O CC >0 ^ Gi --< c: r- o CO '"' CO o G< crs CO CO cc lO l-H J> CO ,~^ "O Ci G^> i-O CO '— 1 -^ t- CC GJ lO CO O CO CO CO cr: C^J Tf £> --C O J> £^ i> CO CO CO CO C5 Cc C'. c c '^ ■^ r— 1 —1 r— 1>1 Cn* G^J ©< 1— H G< GJ O O '-I iGJ CO o c; T^ o cc C5 G< f> lO CO o CO CJ 'T CO CO GO lO CO CO '- G? '— CO CO 1> CTj CO CO CO t^ o T— 1 1-H CO "^ cr: rsf c^ ao y-i •^ CO CO G) — C: CO CO -r G» C' CO iT- C^ {^ CO O CO ^-> UO ^ ! 't- •O C". G-< O CO' O CO CO cr. G-) o t- C CO ^^ CO 1—1 "— < cG .-^ _ , •^ i> UO O l> O £^ CO CO CO CO cr. Cj CC' G< ^ " e< G^ G^i -r x; :n g* o c-. i^o -^ O. w G> CO C^ 'C^ CO' G) _ CO C' — 7-s -M tS- i c": CO — 1 :o G» g: ^o cr.' ^ ^— cr: o O CO ■rr' -* G? - — ,-^ f~~ CO '-< O C cr;- CO w ^0 CO G^ O L- O c": o r^ •—' 1> TT "^ ") G^ CO -^ :;o O G-^ -^ i> C c^ti CO c:; GJ -^ t^ O CO ^ CO T— 1 :o CO cr: -t CO "O CO !> J> {> CO CO CO CO c:; C: Ti '^-^ o s-^' o 1—1 f^f G^ Gn« "-^ G^ G* c-0 G^ "O — "-j: c-^ c CO CO o — — ex — »-0 -~ *' . C« CO CO •^ uO o CO c: G^ ■T' CO '-^ t-- O G^ CO ^ CO 01 CO r- L^ uO »— » CO c: OO t^ t^ CO O T G> ^ CH) J> »-0 G' •!-- 'T^ CO UO — < r- T^ '^ cc ' »C i-O CO ■'-^ '^ t'~- CG CO CO CO '— ' T" '^ O G-J lO" CO ^ CO T— "-^ CO 1 -O -vO i> i> £> CO CO 1— ( CO CO c; en Cf G-( ^ c •^ GJ G^ Ot 1 s-j 1 I ,-, l- TT T— . CC CO lO 'O 1> ^ l> uo 1-0 CO ■^ — < r'^ C) G^ r- -~. uO G> ' -:*■ cr, CO i-O -r G? CO G* -V >-0 c^ O o CO .^ ^ CO 1-0 o f-. i,--. O -^ ' 'O '^ •-* CO G< ^ O CO «D rj^ G? O t^ ^ G/ cc ^o '^) c^ »^ I—, i> CO 1 ■^ o CO r»i -r i^ CO G» O CO '- ~c c~ t- i> CO CO CO CO' CC CC Oj cr. O »— ' T—. T-< r^> G^ G-} 1 1 gI Z') I j ^ CO T-, c c: tr. o Gj CO '— C: C: T— :^ •-- -— « cr. CO CO r^ r^ -^ i -T C5 "^ CO i-O CO C ^ CO J> >-0 g; CO ^ CO CO' CO CO \^ CC) CO GJ C< T-H O Ci CO l^ -. CO »"' cc t- TT GJ CC' CO c^ CC CO r^) ~ "* o ic -o CO -^ Tt CO cr. G^ ^: CO 1-H CO CO CV G^ T^ 1^ • — . G^< ^ CO -— CO CD CO CO 1> l> J> t> CO T— 1 CO CO Ci c: c* Gs» ^ " »-H ■3 <^( ©* G CO »-0 -f G-j o CO CO rf G^( c: CO CO CD i> CO O CO &5 CO -Tf i> O CO CO cr. Gi UO CO O CO CO cr; — 'T' t^ C G-) uo CO o CO u-> -o CO i> t^ i> i> CO CO CO cr: c; c: c: o O c: i-i ^ '-I i-H G^ ev( ©:» C- CO O CO 1— « "^ c: lo CO G' C'G CO G'-i CO '^ CO CO 'T^ c: i^ O CO CO I ■O I— ' O i> CO CD G'J J> C I— I CD f^ CO !> cr; crx CO uO O "^ J"^ r^ CO I CO CD uO -^ :0 0( ^ ex CO CO' -r i-i C: CO CO O {> -^ ^ l^ .CO. O )0 I -^ t> C; CO CO C: G^ -rr i- O CO CO OO —' 'T L-- CC. GNJ uO t- O OJ o CO CO t- L- i> L- CO CO CO c: c: o; cr; C c; C C r-, T-, ,-, (^) e) G^ I - G^ j G< • 1 - ' t^ G-/ CO »— ' t" -^ ■r-\ T-- Gv lO C: t> 1> O CD UO CO -rr ^T CO CO c: uj 1-, ;o CO cr: t^ — ^ o-; G^» co g* c: uo c: gj G< »- co "^ -co cc t-< o COCOGCO T i> o CO CO CO ^ 'TT t^ O CO uo CO 1—1 "^ CO CJ; o» -^ i^ O ©■» lO ■o> CO t^ i> L- t- CO CO CO C5 ci) C5 cr; O O O O ^ " <-t G-} Gi oj G? - CD t- CO c- o T-i 3-/ CO ■:*> lo CO r- CO cr; c;' '-I G-i : ; -7- UO co ••- x ' l^. ^ rf '^f rp iO "O UO UO UO UO ^ uO i.-^ UO CO CO '- ■_ ■- c- CO C" CT i 1- }f4 A TABLE OF LO&ARITIIM. 1 i CO o) ^0 ^ -f I- -1 t- OJ o c>j o :o C'O CO 3^f lO o O CO CO CO O ^ cr^ CO 'T' c< i-^ cj cji t-' G< lO o o ■ crs o C5 ^o o^^ i^ —1 CO CO CO '-" c: CO t-- O ej CO G' o t^ i t- CO i> Oi c- —c CO O -t* CO oj o C7) 3^ -o c:> 3> i^-: CO o 1 I- o o^ "C t- o GJ! »-o 1-- a; G< r*< CO c^ — CO CO CO o c--: CO '^ -"^ ^ -^ O O lO >0 O CO CO CO O 1> 1> l^ t> CO CO CO' :c CO CO CO G^ ':^ ot a> ^ :>i lO >* CO CO -t:' O — cr. G! G» Ci O CO -^ CO CO 3-» CO 1^ cip G» CO BO o: ! ■^ o t- r- CO CO CO ctj GTi Ci i> -^ O -^ i^ cfj o c:> t- ^ i »-o c; o o '^ cni CO i^ '- ^ o; CO t^ o CO cb o Gj Lo CO I t> O G» O J> C:i G; rj- 1-- C; '-' TT CO C5 T-, CO -O CO O CN j CO ^ '^■' -^ -^ T O O ^ -0 CO CO CO CO i> 1-- t- t^ CO CO ! CO 2> ^ O CO C-'i 1-- ■-0 OJ t- 3^ CO CO 3^ 2( G-< O CO G< t- t-~ iO CO CO uO CTi Cr.^ t- G^ ~ -^ G* O '^ GJ Ci o c^j t^ CO CO e; -^ 1-0 ^: CO O -o r-. -f CO t--co >-C Gi 3-J CO G) t> G-< CO ■r-( lO O CO i> '-' -r CO T^ -■^' r^ O CO CO t- CD G> -:^" t^ CTi 0-( -^' CC' Ci T-i -^ CO CO I-" C3) »0 CD O' G^ CO C3i -^ -^ "* -^ >-0 to >0 ^0 CO CO CO CO i.^ 1> i> t^ CO CO to O t- 3-» t> '-I Tf ctj 1^0 CT5 — ^ >J CO c^ ^5 T-i O ** O »-< CO '-< CO CO CO >— C" GJ O '-< C^i G^ OO GO O -f ^ "^ G> CTi rr CO O '— 1 1— 1 O l^ CO CO — > CO '^ CO G^ C:; o lO o lO c; -?< CO CO t- r-i ijr: 00 GJ o Ci G» >-o CO 1^ c: l^ Ci 3J -T --O Ci T- rr O CTi !-< CO CO OO O CO O t^ O 3-^ COCO'^"^'^'*Oi-0!^OCOCOCOCOL"^t-t^l>COCO 1 O Tt '^- t^ 3-j :o ^ I- G^ CO '— ' ^ 3n> CO CO GJ c. en »^ I-' ur; 30 CO -r £- t- CO -^ CO rr cr. »— — ' ^*. '-. c-2 CO o Ci t^ -^ cr- CO o f.-- I- CO CO Ci '^r i> O '-^ O o CO 1 t- GJ l^ G» t^ '-• CO O '^ CO GvJ CO Oi CO CO O CO CO CO '-1 j COC?5'-^-?'COC2^-*COCOt-i.COOOOOCO'0 1>C:CJ| CO Ci '!i' rr '^ "^ O'lO ^3) lO CO O CO CO I- 1> I- t^ l> CO ; t 1 j t CO o -< —, £- r-. CO rf O CO 1—1 Ti< 3--f CO CO C5^J I- cr. CO O cr.: -^ lo 3^ t- l- ^ c:; —« C' CO CO '-' O t- g-j I CO ** -^r CO c: lO cri G-} CO CO G-j ex CO — ' T t^ CO CO CO '-." f uoo>-0 0-J-c;coco3^coOcO{.-^"-tri-'Ocococrsi CO ex 1— 1 Tf CO CO ^ CO CO CO >-< GO »^0 CO O G< >-0 f^ c; '— CO CO '^ ^ -^ -^ O O ^ O CO CO CO CD J> t- l^ C- L- C3 ct C-- lO j> O '-I O ■O 3} i- CO ■>— > CO 3^? CO CO 3} 3> •O ex 1^ G> G/ ex — 1 O "O CO i"- G» 'C O CO CO O "-^ Cj t^.' ♦ CO ex c^ CO --0 o o CO (X c: c:> co gj t- '-^ cO' -c '3 co — ' j O) v^ CJ L^ G< j^ »-i o c. r; tr- t-" "O co G{ >o co '— -r c- CO 00 1— 1 CO O CC r- 1 CO O CO O CO >0 O O G^ "^ t^ c: i— ■ CO CO ■^ -^ •^ -^ >-0 .O UO uO CO CO CO CO l> t- l^ t^ l^ CO S.J o 3 o -r C5 ^ CO O CO 3^ CO CO 3^ CO CO CO -^ CO -^r CO CO O CO cc lo o '-< O CO O '-^ '- CO C^ -^ "* CO O CO O CO O >-0 -^ Gs» Ci -^ CO O G-» Gi '— CO O >-0 O O C -^ ex CO t> '— ' '-'^ ex G-* CO C: CO O (X G} -rr cocoi-'COcocoOco»OcoCGv{Oi>cxG<'^coc:'— CO CO -^ ■^ -^ "^ O U3 O >0 CO CO CO CO CO t^ l^ t-- l^ CO - -^ "^ O '^ o ^ — » t- <3» X O CO "^^^ CO CO ^ 1-- ex CO -0 T-t CD C^ '— ' '— 1 '—' CO O O "^ 1~^ CO O) l^ G-> l~- G-J CO O '0 ex CO CO O -^ t^ O CO CO' Ci Q> lT- CO O C^ "-O CO O CO '-0 l> O G^ »0 J> CJ G^f -^ CO CO '-1 COCO-^'^'^'yOUD^uOCOCOCOCOCOt-i^t-C-CO o j <- '-: CO CO -r C5 CO r* Cj O O G» -yi CO CO COCOCXCSCOCOOCOCOCX'-' — COG.»COG^COG-'T*CO 3s^ ..-i — C*:; '— 1 r-- G-> LO l> C~ t^ O T-( {->- r-c rt '" CO »-0 C3 o O »-0 O k-o o: ♦* OO c< CO o "Tf CO '-' »o CO " •* t^ O O C^ O CO O l- O CN< »-0 l-* C> G> "t" l> C5 ^-< -^ CO CO '— ' C0C0"^'^'^'^>^>^^'2>^ 1722 lh;344 9 1 117801,0 138870 I Departure. 3/ jDeuartiire 4' ■ Dist. tJU0087 3.0. 001746jO. 00261910 0.003492.0, 0.004365|o. 0. 00553810. 0.0061 11 'o. 0.00G994|0. 00785710, 0'>^1164 002328 0034. 2 004 G56 005820 006984 008148 009312 01G476 0,0c;433G;0. 00867210 013008 jo 01'^34a!o 0216^0:0 02f^016'o (MJ5ci7! 0116341 0174511 0232G8i 029085i 0349021 040719! 03o352lO 03468810 046536 03 9024 ie ('5^353 3.'' ; 4oT~ 0.010170:0. 020340i0 0305 ]0j0 040680iO 0-X?850!0 '10^0,0 OvJ7n90 081360 0915.30 \» 15'. 97 031i»94 »; 047 i;l 063. 88 679 985 0'-5P82 111979 ^ - T6 1439731 011-^5 023270 034905^ 045540; 0587751 071^810; OP 1445 093080! 104715' 1C5 A TREVERSE TABLE. 0o23o0 1047 1 i> 9 ^; £>?':c32 157068 - ^' i - ; - ■ - ■ 2 99385 1 jo 0785:3 3 , ** o 5 997702[0 1570501 e C - - , .... ^ 7 . 8 S 9965t'5iO e;?oo74; 9 j 2«> SO' i \ £7" 30' 17 1 I ■ H \- j' 3 c. 4 ; 5 . . ; i 4 99' '5 6 roi 6 7 '^^ ■- ^ -.-X,- J .7 8 :7&192 [ 8 i 9 i b 9V8632 3 1 40 > 1 | o ^^\ GsO J u 3;^ . o 53 ' 9 ' '" ' 3*^ " i' ♦fco^ 3o' j 3" -:^ ' t>7- .053330 1 1 .0.9--^ ' 2 flP' j 5 J4 993I'>00 26?t>J0 ^ 7 u ^ai-i.. - -----^ 8 7 99K»40'^:J 4I86S8 9 8 V ■ 4710?4 It J: \^9£097 0.1 1 104672 : 096194 ie20S2t 2 4 9?0485i0 30523' P. i : r , 1 . 4;0.0€9756 [' ^- 1 2 (l 9951280 13951? | I 3 ': -:- - -- j: .: ! ^ H^ ; 7 8 • * 9 t) -_ iOi-'7o ^ v_i"v4 T— T~! 4*^ 3C»' OTR4 5': I j, 5 9ol,7^ • -^,^^'^-- \ 9c 8 ! 5 3 ' '^ 549 1 9?f :1 TRAVERSE TABLE. 109 Latitude, o -o Departure, Latitude, iDepartuie 0. 99G 195 0. 087 15tJ 1 992390jO 171312 2 98C590p 261468 3 985780 348624 4 980980;0 435780 5 9771G0J0 522936 6 9743'>o!o 610092 7 9715G0!o €97248 8 966760*0 784404 840 6o 0.994522,0.104528 1 989044|0 209056 2 983566'0 314004 3 978088 418112 4 9726100 523140 5 967132 628168 6 961654;0 732196 7 9561 76*0 836224 8 950698 941252 8 30 0.992546:0.121869 1 985092|0 243738 2 977598|0 365607 13 970184|o 487476 4 952650|0 609345 5 9551960 731214 6 94870210 853083 7 940368^0 974952 8 922754! 1 096821 0.090268,0.139173 1 980536 278346 2 970804;o 417519 3 961072|0 55::692 4 951340|0 6958G5 5 94 1 60810 835038 6 93I876|o 974311 7 9221 44; 1 113334 8 912412'! 252557 84° 30' 0.99535,' 1 J99071C 2 986074 981432- 98679C 972148 977506 962864 0' 0.0951-42 191684 287526 383368 479210 575052 670894 766736 968222 862578 830 30' 6° 30' 0.99353410. 1 987068|0 2 980602 3 974136J0 4 967670;0 5 961204,0 6 954738i0 7 948272!o 8 94I8O6I1 113198 226396 339594 452792 565990 670198 792386 iJOooo 4 018782 820 .30' 30' 0.9914070. 1 982814'0 2 97422110 3 96562b!o 4 95703510 5 948442:0 6 939849'0 7 9312S6|l 8 931663 1 130521 26 1045 391563 522084 652605 793126 913647 044168 174689 81*^ 30' 8° 30' 0.98897810. 1 977956!o 2 9669.34:0 955912 94489olo 9338680 922846 1 91182411 8 900802(1 147803 295606 443409 591212 739015 886813 034621 182424 330227 Dist. liO J TIUFKRSF i.r>L.r^ A I :ure. 90 !i 975?76;0 312868 |2 963064[0 46^302 3 9507520 625736 4 934440!! > 7G2170 5 9:261 23 6 913816* 93G604 1 09iOc8 7 901504l1 251472 Lai. Lude. Deuart.irJ 30' ! 9' 31.' i8 88919211 4079U6 O.i8b248 0.16cO41 1 972596 3300b2i 2 9588440 495123! 3 9441 660164^ 4 931440 8252051 5 9I7bSoO 990246f 6 90303611 1552871 7 888384 1 320328! 8 8 76632! 1 48536y D-t. 1 3 4 5 6 7 8 Q 1 2 3 4 5 6 7 8 9 |i?.9£>480C'0.173b48 (1 ^696 lejo 357296 |2 954424|0 520944 3 93-232jO 694592 1 924040;n 868240 \n 90884G|1 041888 16 893656J1 215536 17 878464(1 38i>i84 ia 863272^1 5^2832 0.983i;17!0.l82229 i 1 966434b S64458i 2 2 9496510 546G87 3 3 9328680 728916: 4 4 916085|0 911145, 5 5 8 99302 'l 09337 4 i 6 6 882519 1 275603 7 7 865736 1 457832 8 8 848953 1 640061 1 [0.981627 O.hjObiO 2 1 96S254 381620 3 12 954881^0 572430. 4 b 926508|0 763240 5 1 1 90b 13510 954050 a f5 889762 1 144860 7 16 0713891 335670 8 \l 85301 n;i 526480 9 !8 8316431 7172^^0 78' 12° 1 0.97^148 0.207912 2 M 95629610 415824 3 5J 934*44 623736 4 3 912592'0 83]648 5 4 890740 1 0395. >0 6 [5 86883' 1 2474-^2 7 h 84703f-il 455386 8 (7 8251 84! 1 663296 9 -8 803332'! ^71208 0.979881 1 959774 2 939661 3 919548 4 899435. 5 8793221 6 359209 7 83909ei 8 818983! 0.199.361 398722 598083 797444; 996805| 1 196166 1 395527 1 594888; 1 794249 77' 30' 12*^ 30' 0.976259 1 952418: 928677i 94)4836; 881095| 8.57354i 8335131 809672 '-8r.331 0.216431 43286 2j C49293: 865724 1 082155 1 298.586; 1 5150171 1 731448' 1 947879! i <± 3 4 6 6 7 8 Q .i TlUyERSE TABL Hi Latitude, j f'epar.uie. Lt^.txiiide. L'epurtuifc. iy.<. 7 7^ i i;;^ j 7i.° 30' ; 13" 30' Di'-t. i 0.5J7437U'0.2'ii4^oi i 0.972333 0.233436 1 o 1 948740 44S901 1 944666:0 466872 Q ;2 923110 674853 2 9169990 700308 3 4 3 897480^0 899804 •3 889332:0 933744 4 5 i4 8718oO!l 124755 4 ^CAdijbn K7180 5 6 J5 846220il 34970G ! 5 8339[ 815018 I 102518 i 3 835132 1 136016 4 5 ' 1 8063101 1 378185 II 4 793915' 1 42002C 5 1 6 i 5 767572' 1 653822 j 5 752698 1 7040241 6 7 i .5 72-J834i 1 92.0459 j! 6 711481 1 988028 7 8 j 7 690096 2 205096 ii 7 670264' 2 272032 8 ' 9 ! ^ 651358 2 ^80733 '\ 8 629047!^ 55603G 9 ili A TRAVELSE TABLE. Dist. LatitU'lc. 730 Departure. 17 Latiiucle. 72 30' lUej'arture. 17o 30' 0.956305! 1 012610! 2 078915 3 825220: !0. 292372 ,0 504744 .S771 16 1 1G9488 |1 781 525' 1 461860 |5 737830' 1 754232 6 693135 .2 046604 7 650440:2 338976 18 6067452 631348 Df?t. 0.9536810.300694 1 90736210 601388^ 2 06 J 043:0 902082 3 C14721il 202776 4 768405|l 503470 5 72208611 804164 6 675767)2 104858" .7 62944812 405552 8 583] 29*2 706246 |0. 951057(0 .'1 902114:0 2 85317 1|0 13 804228|1 |4 755285 1 5 706342; 1 16 657399 2 7 60845612 iS 559513^2 /I 1 o 1 0.945519,0, 891038!0 8365570 782076jl 727595]l 673114,1 610733,2 7 564142|2 ^ ^-^^9671 '2 OUi 309017 618034 927051 236068 545085 054102 163119 472136 781153 190 325560" 651136 976704 302272 G27940 953] 08 278976 604544 930212 0.948288 70' 10.939693 1 879386 20*^' 3 2 819079 1 3 758772|1 4 69846511 5 63815812 ^ 57705112 7 51754^2 8 457C2 /)3 342020 684040 026060 368080 710100 052120 394140 738160 078 'j 80 896576 834864 793l52j 741440; 6897201 638016', 586304! i! 8 534392 0.317292[ 634584 951876 1 269168 1 586460 1 903752! 2 221044' 2 538328 2 855628 700 30' 1 19*^30' 0.942606:0 1 8832 12|0 2 827818,1 3 7704241 1 4 713030 5 655636 6 698242|2 7 540848 2 8 483454 3 .333794 667580 001302 335176 668970 002764 336558 670352 004116 I 6 90 30' 200 .30' 0.936636 1 073272 2 80990811 3 74754411 4 603180 1 5 61981(;i2 G 557452,2 7 495C88'2 '13072-J 3 350194 700388 050582 400776 750970 100164 451 350 i 801552 1517461 9 A TRAVERSE TABLE, lli 1 o o O 4 5 (\ 7 Latiindc, 60^ Depurtuie. jC*.yoJ.3£();U.o583G8 | 1 8G7IG{J|0 716736 | 2 000740! 1 075104 | 3 734320 1 i G67900 1 !5 GO14G0 '^ G 5350G0 -^ 7 40 r^' -ID 9 8 4 C 22:0 G 701810 1 50208 50867G 867144 225412 22o Ig !8 .02718410 854368'U 781552:1 70873G:1 Co 3920.; 1 5G31042 490288:2 41747212 344G5Gi3 G7y .374G07 749214 123821 498428 873035 247G44 622249 99785G 3714G3 "23^ .920505.0 841010|0 761515|l G82020 1 60272511 523030 2 442535 2 364040 3 284745 3 GG' .390731 78 1 462 172193 5G2924 953655 34438G 735117' 125940 516579 "24^ .913545 827090 740635,1 6541 80! 1 1 2 3 4 567725 2 481370:2 39481512 30836013 22190513 406737 813474 220211 626948 033685 440422 847179 253896 GG0633 680 30. 930382 1 860764 2 791146 3 721528 4 651910 5 582292 6 512674 7 443056 8 373438 Departure 21-. 30 670 30/ 0.923844 1 847688 2 771532 3 695376 4 621220 5 543084 6 466908 7 390752 8 316596 0.3Go48~« 73297'-: 1 099461 1 4C5948 1 832435 2 198922 2 5G540n 2 931 896 3 298383 22^- 30'' 66Q 30' 0.917025 0.382G69 1 765338 1 148007 3 1 530676 4 1 912445 5 2 296014 6 2 678683 7 3 061352 8 3 444121 9 230 30' 1 834050 2 751075 3 668100 4 .'85125, 5 5C2150J2 6 419175 2 7 336 20c 3 8 253225j3 65""30' .3:)8734 797468 196202 594936 993670 392404 792138 189872 589606 0.909926 1 819852 2 729778 3 639704 4 54.9630 5 45955G 6 369482 7 279408 8 189324 240 3C' .414677 829354 244031 658708 073385 488062 803739 317416 732093 Dis'. 1 2 3 4 5 G 7 8 9 jy ]1 A TRMTRSE TABLE ^ - Di-t. LiiiUtULte. 600 9063080, 8]261G'0 71 8924] 1 625x132 1 531540j2 43T84ai2 344i5Gj2 2504643 156772 3 422f3ia 845236 267854 GD0472 113090 535708 958316 380924 803562 640 26° jv).89879d'0, 1 797596jO 12 696394il |3 695192/1 |i 493990I2 5 392788J2 io 291586 3 i7 190384/3 '8 088182 3 6 3^ 438371 S76742 315113 753484 191855 630226 068597 506968 945339 0.891007 0. 1 732014J0 2 673021 3 504028 :4 455035 5 346042J2 6 23704913 7 128056 3 8 019063 4 453990 907980 361970 8 1 5960 269950 723940 177930 631920 085910 G20 0.882948i0. 1 765896}0 2 648844J1 3 53179211 4 424740;2 5 297688(2 6 18063613 7 063544 7 946532 640 30' lU, ?parmre. 25 30' 28^^ 469472 938944 408416 877888 347360 8 1 6832 236304 755776 225248 0.9u255];0 1 805102 2 707653 1 3 610204 1 4 512755(2 6 415306i2 6 3178S^'^ 7 220408 8 122959 430494 860988 2S1482 721976 152470 582964 014458 443952 874446 l>ist. !i 630 30' 26 30' I 0.894901 0, 1 789802|0 2 684703|1 3 579G04|l 4 47450512 5 36940612 6 264307j3 7 159208 3 8 04410914 4461o] 892362 338513 784724 230905 677036 123267 569443 015629 62C'30' 0.886977 1 773954 2 660931 3 547908 4 434C85 2 5 32186212 461731 923462 385193 846924 308655 770386 6 20883213 232117 i| 7 09581613 693848 ! 7 ::8 279314 155589 27 30' 61^ 30' I 28 30' 0878784 jo. 1 757568!o 2 636352! 1 3 51513611 4 5 6 393920)2 272704i2 1514863 03027213 909056 1 4 477141 95428'^ 431423 908564 385705 862846 339987 817133 294269 i i ! 1 A TRAVERSE TABLE. llw Liilitude. Departure. ■ Lulilude. Departure. Dist. 0.874G20 29o eo'^ 30' 0.870322 29« 30' D.St. 1 1 0.484810 0.4^2405 2 1 749240 9i39620 1 740644 984810 Q o 2 6238G0 1 454430 2 610966 1 477215 3 4 3 49848011 939240 3 481288 1 969620 4 5 4 373100 2 424050 4 351610 2 462025 5 G 5 247720 2 908860 5 221932 2 954430 6 7 6 122340 3 393G70 6 092254 3 446835 7 . o o G 996960 3 878480 6 97257G 3 939240 8 9 7 871580 60^ 4 363290 30^ i 7 032898 4 431645 30^ 30' 9 590 30' 1 0.8G6025 =0.500000 0.861596 0.507519 1 2 1 732050 1 000000 I 723192 1 015038 2 698075 1 500000 2 584788 1 522557 3 4 3 4G4 100,2 000000 -1 3301 2 rf 500000 i 3 446384 2 03007 G 4 5 4 327980 2 537595 5 G 5 1981503 000000 5 169576 3 045114 6 7 G 0G2175 3 500000 6 031172 3 552633 ■7 i o o G 928200 4 000000 j 892768 4 0G0152 8 9 7 79-1225 4 500000 j 7 754364 4 567671 31o 3a' 9 1 1 59° 310 58'-^ 30' 1 0.857167 0.515038 | ■' 0.852607 0.5224 78 1 i 2 i 714334 1 030076 i 1 705214 1 044956 2 o O 2 571501 1 556114 2 557821 1 567434 3 4 3 4286G8 ^2 06^0 152 3 410428 2 089912 4 5 4 285835 2 515190 1 4 263035 2 612390 5 ! 6 5 143002 3 090228 1 5 115642 3 134868 6 1 7 6 0001G9 3 6032G6 1 5 968249 3 657346 7 8 6 857336 4 120304 6 820850 4 179824 8 9 7 714503 58'"^ 4 635342 7 673463 4 702302 9 i 32*^ ^70 30' 0.843359 32^ 30' 1 0.848048 0.529919 0.537279 1 \ 2 1 696096 I 059838 1 686718 1 074558 2 3 2 544144 1 589757 2 530077 1 612837 4 3 392192 2 119676 3 373436 2 149116 4 5 4 240240 2 649595 4 216795 2 687395 5 6 5 088208 3 179514 5 060154 3 225674 G 7 5 936336 3 709433 5 903513 3 761953 7 i 8 6 784384 4 239352 6 746872 4 298232 8 , 9 7 632432,14 769271 7 590231 4 83n5 j 1 9 \16 A TRAVERSE TABLE. Latittule. j Departure. ; r Lrtitiule. ; Departure. \ DM. 1 1 57" 1 :3;3<5 1 56° 30' ! 33 J 30' Dist. 0.}]38(J71;0.54k^:39 j 0.833«54;0,551916 1 o 1 C77.>i2:l OS 9 2 78 i 1 CG7708:1 103832 2 2 516uJ3i 633917 2 0015G2'l 655748 4 3 3oJG84'2 nshoG 3 335416i2 207664 4 5 4 193355 2 723195 4 169270;2 759580 5 1 <3 5 03202G .3 207834 5 003124^ 3 311496 G 7 5 870G97;3 81247o 5 836978 3 863412 I G 700368U 357112 6 670832 4 415328 s 9 7 548039! I 901751 ! 7 504686 4 967244 9 ,' - * 34'' 0.559193 I 55^^ 30' 34<^ 30' 0.566384 1 0.829038 0.824145 1 I) 1 G5807G 1 11 8386 1 1 648290 2 472435 1 132768 1 3 2 587114 1 677579 1 699152 3 4 3 31G152 2 236772 3 296580 2 265536 4 5 4 145190;2 7959G5 4 120725 2 831920 5 ! G 5 971228;3 355158 , 4 944 870 3 398 304 G i 7 5 8032GG'3 911351 . 5 769015 3 9G4688 7 1 8 G G32304i4 473541 ! i G 593160 4 531072 8 ; ^ 7 461312 55'^ 5 032737 1 ! 7 417305 1 54 <^ 30' i 0.814084 5 097456 9 i 350 i 35'^ 30' 1 1 0.819152 0.573576 0.580630 1 ! 2 I 638304 1 147152 1 1 628168 1 161260 3 2 45745G 1 720728 ! 2 442252 1 741890 3 4 3 27G608 2 294304 i 3 256336 2 322520 4- 4 095760 2 867980 4 070420 2 903150 4 884504 483780 5 1 ^ 4 914912, 3 441 45G () 7 5 734064,4 015032 ' 5 698588 4 064410 t G 5532 Hi 4 588608 i 6 512672 4 645040 8 <^ 7 372368 54^ 5 162084 ; 7 326756 ;5 225670 36-^ 30' 9 1 36'^ i 53^ 30' 0.803826 i 1 0.809017 0.5877 85 0.594800 1 1 2 1 G 18034 1 175570 1 1 607652 1 189600 .1 3 2 427051 1 763365 1 2 41147S : 3 215304 il 784400 3 4 3 236068 2 351140 2 379200 4 5 4 045085 2 938925 i 4 019130 2 974000 5 I C 4 854102 3 526710 ; 4 822956 3 568800 ' 6 I 7 5 7G3119 I 114495 I 5 62G782'4 163600 / ; 8 G 472136 4 7022hO ; 6 430608 4 758400 8 ^ 9 7 28115315 290065 i 7 234434 5 353200 9 Ji TRArLIlJ^L 1ABLL. ] Latitude. Departure. | Latitude. 1 Departure. Di5-t. 1 1 63o 37*"^ ! 520 30' 370 30/ Dist. 0.798636 'J.601815 I 0.793323 0. 608738 1 2 1 597272 1 203630 i 1 586646 1 217476 3 2 395908 1 805445 2 379969 1 826214 3 4 3 194344 2 4072G0 1 3 173292 2 434952 4 5 3 993180 3 009075 ! 3 966615 3 043690 5" 6 4 7918161 3 610890 1 4 759938 3 652428 6 7 5 5902521 4 212705 i 5 5 53261 4 261166 7 8 6 38SGSS 4 814580 ; 6 346584 4 869904 8 9 7 187524 5 41G335 ! 38o \ ' 7 139907 '5 478642 9 52^ ' 51o 30' 1 0.782578 38° 30' 1 0.78801110.616661 i 0.622490 1 o 1 576022,1 231322 { 1 565156 I 214980 2 3 i 2 364033 1 846983 i ' 2 347734 1 867470 3 4 3 152044 2 462644 1 : 3 130312 2 489960 4 6 3 940055 3 078305" ' ; 3 93 2890 3 112450 5 G 4 728066 3 693965 • 4 695468 3 734940 6 I 7 5 516077 4 309C27 j 5 478046 4 357430 7 ■ 8 6 304038 4 9252S8 1 j 6 260624 4 979920 8 9 7 092099 51o 5 540349 1 ! 7 043202 5 C02410 9 ^ 3St* 1 1 500 30' 39- 30' 1 0.777 146!0. 629320 | 1 0.7 71595 0.636054 1 2 1 5542021 1 25SG40 i ! 1 543190 1 272108 3 2 331438jl 887960 1 2 314785 1 908162 3 4 3 108544 2 517280 1 3 086380 2 544216 4 5 3 8 85730;3 146600 1 3 857975 3 180270 5 6 4 062876:3 775920 1 4 629570 3 816324 6 7 5 440022J4 205240 j 5 401165 4 452378 7 8 21 71GSJ5 034560 1 6 172760 5 088432 8 9 6 994314j5 663880 < 6 944S55 5 72448 6 9 50o. 40o 1 490 30' 1 40-' 30' 1 0.766044 0.642788 1 0.760377 0.649423 1 2 1 532088 ,1 285576 1 520754 1 298846 2 3 2 298132 il 928364 2 281131 I 948269 3 3 3 064176 2 571152 3 041508 2 597692 4 3 3 8.S0220 3 213940 3 801885 3 247115 5 G 4 596264 3 8 5 6728 4 562262 3 896538 6 7 5 3o2308 4 499516 { 5 322639 i 545961 7 8 6 1282.52 5 142304 6 083016 5 195384 8 9 6 894396 5 785092 6 843393 5 844807 9 113 A TRAl >i. TABLE. — ;^;- i.rt De,. L.-.. \)vv. 1 Lu*., C F ,190 ^^Q. ■; .^ .',' ; 41GJG' 4n<^ 1 ' 1 U.7547IUO.60GO59 i 0.748927 10.66 2595 0.707107 1 o I 50i<420 ■1 312113 - 1497 354 11325190 1 414214 i2o4130 1 968177 % 241)78111 987785 2 121321 3 4 3 01G840 2 624236 ' $995708 2 650380 2 828428 4 5 3 773550 3 'iyJil'o \ . 3 7446-3513 3121^75 3 535535 5 .0- 4 52^:^60 3 P36.3o4 4 49353213 975570 4 242642 6 7 5 CSL>^-70 4 53W413 5 242489 4 638165 4 949749 8 . ti 037Ct(« 5 2J8472 5 991816 5 300760 ii5 6.368.56 8 ' ti 7G:;:o90 5 904531 6 740343 5 'J'a-.iobb i 6 363963 9 4^^ .^00 i 47^^30' 42«^'30' \ ' 1 * 1 0.7431-^5 0.669131 0.7.37249lo.675564 | 1 2 1 486-290 1 338-^62 , -1 47449:;;!. 351128 ! 2 2 239435 2 007393 : 2 21 1747 '-2 026692 ' 3 4 2 97258U 2 676324 . 2 948996 702256 . 4 5 3 715725 3 346655 : 3 686245J3 379920 ! 5 t> 4 458370 4 014786 4 4234-94; 4 053384 ! 6 1 5 202015 4 6S3S17 5 16074.314^18948 j ' O 5 945160 i o53i)48 5 897992ir, 404512 6 688205 6 023179 1 6 634241 -6 080076 1 9 1 t i 1 470 43^ 1 4 GO 39' 430 30' 0.731351 10.681998 : 0.725347,0.688328 v> 1 4G2708 |1 363996 ' 1 4.30794i i SiQQB^ i 2 3 2 KM062 2 045094 \ 2 i76o-n:2 06^ir.84 i L> 4 ■2 925416 2 727992 1 2 8013^8 2 75 JS 12 i ■ 4 5 3 656770 3 409990 ! 3 626735 3 441640 i 5 G 4 38S124; 1091988 j; 4 352082|4 129968 1 c i 5 119478 4 773986 !; 5 077429;4 81S29G • 7 8 .5 8508.32.5 4559>i4 5 802776 5 .5vW624 8 9 •r5:^218o;6 137932 ' 6 528 12J 6 194952 Lat. 88^ io.999391 '' 46^ i 4.1'"> 1 450 30' 44036' ! 0.7; 3223 !o,700882 i 1 . 1 1 0.719.i-?i)i0.694658 1 -> I 4JJ680 1 389316 i 1 4 26446 1 401764 1.998782 2 998173 2 o 2 15'Ji>20 2 08.3974 j 2 139669,-2 102646 .> 4 2 877360 2 7786.52 ; 2 5L52<92;2 803528 ,3 997564 4 5 3 596700 3 47.3230 . 3 566125:3 504410 !4 ^daros 5 6 4 316040 4 167948 1 4 27933Ki4 2a32<:»2 '5 9i>(j:i'\C i 6 < .3 035380!4 862606 ! 4 992561 14 906174 6 9957. '57 1 8 5 7.S4720 5 557264 ' 5 70578415 G07056 7 99512k 8 6 474060 G 251922 , 6 419007 6 307938 8 994519 » 119 CONTENTS. Page Construction of iho Plain Scale, - - - 10 Ho;v to prove the Sliding Rule, - - — 12 Gunter's Scale, - _ _ _ _ J3 Dejicription anil Use of the Sector, — — 17 Use of the Lines of Sine*, Tangents, and Secants, 20 IjP of the Line of Polys^ons, - - — 21 Ll^e of the Sector in Trigonometry, - — 22 *'~" .^Of the Sliding Rule, - " . - -> - '?2 ^.lT!%nsa ration, - - - — — 32 .* Trailing of Casks, — - - - 38 *,^_ Giiaging Casks by the Sliding Rule, - — 45 ''-w^ ^^'P^ihle for the use of Coopers, in calculating Cisterns, 49 A Log Table, showing the number of feet of boards, any log will make, — — _ 50 A Table of Specific Gravities of Bodies, — 51 Of a Table of Solid Measure of Square Timber, 52 A Table of Square Timber, — _ 54 Of the Weight and Dimensions of Ralls, - 61 Of a Table of Solid measure, of Round Timber, 62 A Table of Round Timber, - - - 68 A Table giving the Side of a Square, equal to a Square, in'=cribed in a given Circle, - — 85 Of S'lrveying, - - - _ gg Of Logarithms, — - — _ 94 A Table of Logarithms, - - _ 100 Of a Traverse Table or Ditference of Latitude and Dpparture, - - _ _ ]05 A Traverse Table, - - - - 107 ERRATA. The 61 and 62 pages, shotild have been in the place of the 66 and 67 pairet, or between the tables of square timber and rouuil timber. — The line at the bottom of the 87 pa^e should be at the top. —In the Traverse table, pa^e 108, under 8S de,^rees Lati- tude, i? wron-r, and it will be found corrected, under Latitude 45 de- gree?, page 118. / h M^^ ■.^