Class L_ A -545 Book —^-"iiiS GoK'right'N" - COPyRIGIlT DEPOSU^ / ELEMENTS OP SURVEYING- AND LEVELING. By CHARLES DAVIES, 'lL.D., AUTHOK OF A FULL COURSE OF MATHEMATICS. REVISED BY J. HOWARD VAN AMRINGE, A.M., Ph.D, PROFESSOR OP MATHEMATICS IN COLUMBIA COLLEGE. NEW YORK ■:• CINCINNATI •:. CHICAGO AMERICAN BOOK COMPANY a^^oi>__ DAVIESS MATHEMATICAL SERIES For Elementary Schools Davies's Primary Arithmetic Davies's Intellectual Arithmetic Davies's First Book in Arithmetic Davies's Standard Arithmetic Davies's Practical Arithmetic Davies's Complete Arithmetic For Secondary Schools Daviess University Arithmetic Davies's New Elementary Algebra Davies's Bourdon's Algebra Davies's Elementary Geometry and Tri2:onometry Davies's^ Legendre's Geometry and Trigonometry For Co.lleges and Advanced Students Davies's University Algebra Davies's Analvtical Geometry Davies's Analytical Geometry and Calculus Davies's Descriptive Geometry Davies's Elements of Surveying JUL -5 1898 J Twoirr IVED. ^^ C^ ^^ ^ ^^,^4 PREFACE. DAVIES' Elements of Surveying, first published in 1830, was designed as a text-book for the pupils of the U. S. Military Academy, and in its preparation little regard was had to the supposed wants of other institutions. The work was, however, received by the public with more favor than was anticipated, and soon became a leading text-book in Colleges, Academies, and the higher grade of Schools. For the purpose of adapting it more fully to the requirements of these institutions, the autlior made many changes in successive editions, and gave it his final revision in 1870. In the present edition, while the admirable features which have hitherto commended the work so highly to institutions of learning and to practical surveyors have been retained, some of the topics have been abridged in treatment and some enlarged, others have been added, and the whole has been arranged in the order of progressive development. It has been the intention to begin with the very elements of the subject and to combine those elements in the simplest man- ner, so as to render the higher branches of Plane Surveying com- paratively easy. The necessary principles of logarithms and plane trigonometry are given, and their mode of application shown. All the instruments needed for plotting have been carefully described ; and the uses of those required for the measurement of angles are fully explained. In the section on Magnetic Declination or Variation of the Needle, papers of the U. S. Coast and Geodetic Survey have been largely used. From them have been taken : — tables of annual changes in declination, and for computing the declination at any epoch, at various places in the United States, which will be found IV PREFACE. of especial value in re-running lines of old surreys ; also, new tables of the times and azimuths of Polaris when at elonsration, for use in determining the true meridian with compass or tran- sit, which with the rules given for interpolation are more accurate than any similar tables previously published. A full account is given of the system adopted in the survey of the public lands; and although the method is simple, it has, nevertheless, been productive of great results, by defining, with mathematical precision, the boundaries of lands in the new States, and thus settling their titles on an indisputable basis. In this connection official instructions and diagrams issued by the U. S. General Land Ofi&ce have been used, and, as the principal lines of a government survey must be run with reference to the true meridian, the Solar Compass and solar attachment to transit are described. A change made in the present edition, which must prove par- ticularly acceptable, is the transformation of the article on Mining Surveying into a complete treatise, in which the location of claims on the surface, the latest and best methods of underground travers- ing, etc., the calculation of ore-reserves, and all that pertains to the work of the Mining Surveyor, are fully explained, and illustrated by practical examples. This improvement is due, substantially, to John G. Murphy, Esq., E.M., at one time Territorial Geologist of Wyoming Territory, an expert Mining Engineer of large and varied practice. In addition to acknowledgments elsewhere made, the under- signed is indebted to Professor Eodney G. Kimball, of the Brook- lyn Polytechnic Institute, for valuable suggestions and labor, and to the Messrs. W. and L. E. Gurley, of Troy, N. Y., for their courtesy in furnishing him, for use, advance sheets of the last edition of their Instrument Manual and many cuts of their surveying instruments. J. H. VAN" AMRLNTGE, Editor of Davies' Cotjbsb op Mathekatics. CoLTjMBiA College, N. T., ) SepteTnber, 1883. S CONTENTS. BOOK I. INTRODUCTORY PRINCIPLES AND DEFINITIONS. SECTION PAGE I. — Logarithms 9-16 II.— Plane TRiGONOMETrvY 17-32 III. — Instruments for Plotting 33-39 IV. — General Definitions 40-41 BOOK II. CHAIN SURVEYING. I. — ^Measurement of Distances 43-58 Necessary instruments 43 To measure a horizontal line 44 Applications 48 Standard of measure 57 II. — Area or Contents of Ground 58-70 BOOK III. COMPASS SURVEYING. I. — Definitions 71-73 II. — Surveyor's Compass 74-78 III. — Work on the Field 79-90 Field notes 79 Necessary measurements on the field 80 Errors of compass 80 Measurements by off-set courses 82 General example 84 To correct local attraction 89 VI CONTENTS. SECTION PAGE IV. — Area or Contents of Ground 90-131 Traverse table and its uses 90 Balancing tlie work 94 Double meridian distances 97 Area 99 Plotting - 102 Examples 104 To supply omissions \n field notes Ill Anotlier mode of finaing areas 120 V. — Magnetic Declination or Variation of the Needle. . .131-152 Definitions 131 Daily variatior (with table) 132 Secular variation (with tables) 134 Method of finding declination (with tables) 140 To find true meridian with compass 145 Vernier compass and its use 150 Form of survey bill 152 BOOK IV. TRANSIT SUEVEYING. I. — Surveyor's Transit 153-165 II. — Measurement of Angles 166-183 Horizontal and vertical angles 166, 167 Azimuths and bearings 168, 169 To find true meridian with transit 172 Applications to heights and distances 174 III.— Ranging out Lines, Etc 184-189 To range out a line 184 Measurement of distances by transit 186 To survey a road, boundary of estate, etc 189 To survey streets of a town or city 189 IV. — Farm Surveying by Transit 190-195 BOOK V. LAYING OUT AND DIVIDING LAND. I.— Of Dividing Land 196-203 II. — Public Lands of the United States .... .203-211 CONTENTS. vii BOOK VI. TRIGONOMETRICAL SURVEYING. SECTION PAGE I.— Making the Survey 212-283 Definitions and general remarks 212 Base line 215 Signals 217 Heliotrope 218 Theodolite 219 Measurement of angles, and notes 221 Reduction to the centre 226 Survey of a harbor 228 II. — Filling up the Survey 233-240 By the compass 233 By the plane table 234 III. — Plotting the Triangulation 240-245 BOOK VII. LEVELING. I. — Definitions and Principles 246-248 II. — Instruments 249-257 Y Level 249 Leveling rods 255 Tests of adjustment 258 III. — Leveling in the Field 258-262 IV.— Section Leveling 263-276 Definitions and principles 263 Drawing the profile ... 270 Establishment of the grade 270 Examples 276 V. — Cross-Section Leveling 277-288 Slopes 277 Setting slope stakes 278 VI. — Computation of Earthwork 288-296 BOOK VIII. TOPOGRAPHICAL SURVEYING. Definition and principles 297 Examples and plotting 298 Vlll COK'TEiq'TS. 8KCTI0N P^ej. Shading and delineation 310 Topographical signs, U. S. Coast Survey 313 BOOK IX. RAILWAY CURVES. Definitions and principles 317 Location of curves 320 Laying oflf ordinates 327 Reversed and compound curves 329 BOOK X. MININ"G SURVEYING. I. — Definitions and General Principles 338-340 IL — Method of Locating Claims 340-345 III. — Underground Traversing, Etc 346-361 To make the traverse 346 To reduce the traverse 350 To plot traverse on the surface 353 To plot traverse on paper 360 IV. — Practical Applications 361-374 Problems 361 Example of a developed mine 366 Calculation of ore-reserves 370 Deposit mine with mill connections 372 APPENDICES. A. — Solar Compass and Solar Attachment to Transit 1-15 B.— Sextant 16-20 C. — Instructions to U. S. Mineral Surveyors 21-29 TABLES. Logarithms of Numbers 1-16 Logarithmic Sines and Tangents 17-62 Natural Sines 63-71 Traverse Table 72-161 BOOK I. INTRODUCTORY PRINCIPLES AND DEFINITIONS. SECTION I. LOGARITHMS. 1. The logarithm of a given number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number. The fixed mimher is called the base of the system. In the common system, to which alone reference is made in this section, the base is 10. Every number is, therefore, regarded as some power of 10, and the exponent of that power is the logarithm of the number. 2. If a number is an exact power of 10, its logarithm is a whole number. If a number is not an exact power of 10, its logarithm is composed of two parts, a whole number called the Characteristic, and a decimal part called the Mantissa. Thus, 225 being greater than 10^ and less than 10^, its logarithm is found to be 2.352183, of which 2 is the characteristic and .352183 is the mantissa. 3. In a table of logarithms, the mantissas only are necessarily given. The characteristic of the logarithm of a number is deter- mined by one of the two following rules : Rule I. — The characteristic of the logarithm of any whole number is positive, and numerically 1 less than the number of places of figures in the given number » iO ELEMENTS OF SURVEYING. [BOOK I. Rule II. — The characteristic of the logarithm of a decimal fraction is negative, and nuinerically 1 greater than the number of O's that immediately follow the decimal point. Note 1. —In the logarithm of a decimal fraction, the charac- teristic alone is negative, the mantissa being always positive. This fact is indicated by writing the negative sign over the characteristic: thus, 2. 371465, is equivalent to — 2 -f .371465. l^OTE 2. — It is to be observed, that the characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus, the characteristic of the logarithm of 725.4275 is the same as the characteristic of the logarithm of 725. 4. A Table of Logarithms is a table by means of which may be found the logarithm corresponding to any number, or the number corresponding to any logarithm. In the table appended, the mantissas alone are given; the characteristic may be found by one of the rules of Art. 3. The mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact power of 10. Hence, in finding the mantissa of the logarithm of a number, the position of the decimal point may be changed at pleasure. Thus, the mantissa of the logarithm of 456357, is the same as that of the number 4563.57 ; and the mantissa of the logarithm of 75, is the same as that of 7500. 6. To find the logarithm of a number between 1000 and 10,000. — Find the characteristic by the first rule of Art. 3. To determine the mantissa, find in the column headed ^'N" the left-hand three figures of the given number ; then pass along the horizontal line in which these figures are found, to the column headed by the fourth figure of the given number, and take out the four figures found there ; pass back again to the column SEC. I.] LOGARITHMS. 11 headed "0/' and there will be found in this column, either upon the horizontal line of the first three figures or a few hues above it, a number consisting of six figures, the left-hand two figures of which must be prefixed to the four already taken out. Thus, Log 8979 = 3.953228. If, however, any dots are found at the place of the four figures first taken out, or if in returning to the " " column any dots are passed, the two figures to be prefixed are the left-hand two of the six figures of the *' " column immediately below. Dots in the number taken out must be replaced by zeros. Thus, Log 3098 = 3.491081 Log 2188 = 3.340047 6. To find the logarithm of a number between 1 and 1000. — Find the characteristic by the first rule of Art. 3. To find the mantissa, fill out the given number to four places of figures (or conceive it to be so filled out) by annexing O's (see Art. 4), and find the mantissa corresponding to the resulting number, as in Art. 5. Thus, to find log. of 75 : characteristic is 1, by the rule ; the mantissa is the same as that corresponding to 7500, i. e., .875061; hence. Log 75 = 1.87506L In the same way, Log 2 = 0.301030. 7. To find the logarithm of a number greater than 10,000. — Find the characteristic by the first rule of Art. 3. To find the mantissa : set aside all of the given number except the left-hand four figures, aud find the mantissa corresponding to these four, as in Art. 5 ; multiply the corresponding tabular difference, found in column ^^D," by the part of the number set aside, and discard as many of the right-hand figures of the product as there are figures in the multiplier, and add the result thus obtained to 13 ELEMENTS OF SUKYETIi^G. [BOOK I. the mantissa already found. If the left-hand figure of those dis- carded is 5, or more, increase the number added by 1. Note. — It is to be observed that the tabular differ encey found in column ^^D," is millionths, and not a vhole number; and that, therefore, the result to be added *^to the mantissa already found " is millionths. Example. — To find the logarithm of 672887: the character- istic is 5 ; set aside 87, and the mantissa corresponding to 6728 is .827886 ; the corresponding tabular difference is 65, which mul- tiplied by 87, the part of the number set aside, gives 5655 ; as there are two figures in the multiplier, discard the right-hand two figures of this product, leaving 56 ; but as the left-hand figure of those discarded is 5, call the result 57 (which is millionths) ; adding this 57 to the mantissa already found, will give .827943 for the required mantissa ; hence, Log 672887 = 5.827943. In the same way. Log 3710053 = 6.569380. 8. To find the logarithm of a decimal. — Find the character- istic by the second rule of Art. 3. To find the mantissa, drop the decimal point and consider the decimal a whole number. Find the mantissa of the logarithm of this number as in preced- ing articles, and it will be the mantissa required. Thus, Log .0327 = 2.514548 Log .378024 = L 577520. IN'OTE. — To find the logarithm of a mixed number, find the characteristic by Note 2, Art. 3 ; then drop the decimal point and proceed as above. 9. To find the number corresponding to a given logarithm. — The rule is the reverse of those just given. Look in the table for the mantissa of the given logarithm. K it cannot be found, take SEC. I.] LOGARITHMS. 13 out the next less mantissa, and also the corresponding number, which set aside. Find the difference between the mantissa taken out and that of the given logarithm ; annex any number of O's, and divide this result by the corresponding number in the column "D." Annex the quotient to the number set aside, and then, if the characteristic is positive, point off, from the left hand, a num- ber of places of figures equal to the characteristic plus 1 ; the result will be the number required. If the characteristic is negative, prefix to the figures obtained a number of O's one less than the number of units in the nega- tive characteristic and to the whole prefix a decimal point ; the result, a pure decimal, will be the number required. Example. — Let it be required to find the number correspond- ing to the logarithm 5.233568. The next less mantissa in the table is 233504 ; the correspond- ing number is 1712, and the tabular difference is 253. OPEKATION. Given mantissa, 233568 Next less mantissa, . . . . 233504 . . 1712 253 ) 6400000 ( 25296 .-. The required number is 171225.296. The number corresponding to the logarithm 2.233568 is .0171225+. 10. Multiplication by Logarithms. — Rule. — Find the logarithms of the factors and take their sum; then find the number corresponding to the resulting logarithm, and it will he the product required. Example. — Find the continued product of 3.902, 5971.6, and .0314728. 14 ELEMENTS OF SUEVErii^G. [BOOK I. OPERATION. log 3.902 = 0.591287 log 5971.6 = 3.776091 log .0311728 = 2.197936 2.8653U .♦. 733.354, product Here, the 1 carried added to the 3 gives 4, which added to — 2 gives 2 as the characteristic of the logarithm of the product. 11. DiTision by Logarithms. — Rule. — Find the loga- rithms of til e dividend and divisor, and sitbtract the latter from the fomner; then find the number corresponding to the resulting logarithm, and it udll he the quotient re- quired, EXAMPLES. 1. Divide 24163 by 4567. log 24163 log 4567 OPEEATIOX, . 4.383151 . 3.659631 0.723520 .-. 5.29078, quotient. 2. Divide 0.7438 by 12.9476. log 0.7438. log 12.9476 . 0PEBATI02f . . T. 871456 . 1.112189 2.759267 0.057447, quotient Here, 1 taken from I, gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic sense. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of the SEC. I.] LOGAKITHMS. 15 12. Arithmetical Complement. — The aritlimetical comple- ment of a logarithm is the remainder obtained by subtracting it from 10. Thus, 8.130456 is the arithmetical complement of 1.869544. The arithmetical complement is denoted by the symbol (a. c). The following is the rule for the use of the arithmetical com- plement in division by logarithms: Rule. — Find the logaritlnn of the dividend, and the arithmetical complement of the logarithm of the divisor, add them together, and diminish the sum by 10 ; the nuwyher corresponding to the resulting logarithjn will be the quotient required. Examples.— 1. Divide 37.149 by 5^3.76. log 37.149 . . . 1.569947 (a. c.) log 523.76 . . . 7.2808 67 2.85081 4 .-. 0.0709273, quotient The operation of subtracting 10 is performed mentally. 2. Find x in the proportion, 602.647 : 2.29863 w x : .037293. log 602.647 . . 2.780063 (a. c.) log 2.29863 . . 9.638531 log .037293 . . 2.57162 7 log :c . . . . 0.99022 1 .-. a; = 9.7773 + . 3. Divide the product of 3.58884 and 5672, by the product of 89721 and 42.056. log 358884 . . . 5.554954 log 5672 .... 3.753736 (a. c.) log 89721 . . . 5.047106 (a. c.) log 42.056 . . . 8.376182 2. 731978 . • . 539.48, result. 20 is here subtracted, as (a. c.) has been twice used. 16 ELEMEI^TS OF SURVEYING. [BOOK I. Note. — If the logarithm, whose arithmetical complement is taken, exceeds 10, subtract it from 20, and reject 20 in the final operation. 13. Raising to Powers by Logarithms.— Eule.—2^7i^ the logarithm of the number, and inultiply it by the exponent of the power ; then find the number correspond- ing to the resulting logarithm, and it will he the power required. Example. — Find the 5th power of 9. log 9 0.954248 5 4.771215 .-. 59049, power. 14. Extracting Roots by Logarithms.— Rule.— i^iw,^ the logarithm of the number and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required. Example. — Find the cube root of 4096. The logarithm of 4096 is 3.612360, and one-third of this is 1.204120. The corresponding number is 16, which is the root sought. If the characteristic of the logarithm of the given number is negative and not exactly divisible by the index of the root, add to it such negative quantity as shall make it exactly divisible, and add also to the mantissa a numerically equal positive quantity. Thus, let the square root of .00863 be required. log .00863 = 3.936011 = 4 + 1.936011. 4 + 1.936011 log v. 00863 = "^ • = 2.968006. The number sought is therefore .09289 + . SEC. II.] PLANE TRIGONOMETRY. 17 SECTION II. PLAN E TRIGO NO M ETRY. 15. Plane Trigonometry is that branch of Mathematics which treats of the solution of plane triangles. In every plane triangle there are six parts : three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts, is called the solution of the triangle. 16. A plane angle is measured by the arc of a circle included between its sides, the centre of the circle being at the vertex, and its radius being equal to 1. Thus, if the vertex A be taken as a centre, and the radius AB hQ equal to 1, the intercepted arc BG will measure the angle A. Let ABCD represent a circle whose radius is equal to 1, and AC, BD, two diameters perpendicu- lar to each other. These diameters divide the circumference into four equal parts, called quadrants; and because each of the angles at the centre is a right angle, it follows that a right angle is measured by a quadrant. An acute a7igle is meas- ured by an arc less than a quadrant, and an obtuse angle, by an arc greater than a quadrant. 17. In Geometry, the unit of angular measure is a right angle ; so in Trigonometry, the primary unit is a quadrant, which is the measure of a right angle. Fig. 1. 18 ELEMEi^TTS OF SURVETIl^Q. [BOOK I. For convenience, the quadrant is divided into 90 equal parts, each of which is called a degree ; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are denoted by the symbols °, ', ". Thus, the expression 7° 22' 33", is read, 7 degrees, 22 minutes, and 33 seconds. Fractional parts of a second are expressed decimally. 18. The complement of an arc is the difference between that arc and 90°. The complement of an angle is the difference between that angle and a right angle. Thus, EB is the complement of AE, and FB is the complement of CF. In like manner, FOB is the complement of AGE, and FOB is the complement of COF. ^^^ In a right-angled triangle, the acute angles are complements of each other. 19. The supplement of an arc is the difference between that arc and 180°. The su2)plement of an angle is the difference between that angle and two right angles. Thus, EC (Fig. 3) is the supplement of AE, and FG the supplement of AF. In like manner, EOC is the supplement of A OF, and FOC the supplement of A OF. In any plane triangle, either angle is the supplement of the sum of the other two. 20. Instead of the arcs themselves, certain functions of the arcs, as explained below, are used. A function of a quantity is something which depends upon that quantity for its value. The following functions are the only ones needed for solving triangles ; SEC. II.] PLANE TRIGONOMETRY. 19 21. The sine of an arc is the distance of one extremity of the arc from the diameter through the other extremity. Thus, PM (Fig. 4) is the sine of AM, and P'M' is the sine of '^Z s t' AM'. 22. The cosine of an arc is the sine of the complement of the arc, ^^ complement sine" be- ing contracted into cosine. Thus, NM (Fig. 4) is the co- sine of AM, and NM' is the cosine of AM'. These lines are respectiyely equal to OP and OP'. 23. The tangent of an arc is the perpendicular to the radius at one extremity of the arc, limited by the prolongation of the diameter through the other extremity. Thus, AT (Fig. 4) is the tangent of the arc AM, and AT'" is the tangent of the arc AM'. 24. The cotangent of an arc is the tangent of its complement, " complement tangent " being contracted into cotangent. Thus, BT' (Fig. 4) is the cotangent of the arc AM, and BT" is the cotangent of the arc AM'. The sine, cosine, tangent, and cotangent of an arc, a, are, for convenience, written sin a, cos a, tan a, and cot a. These functions of an arc may also be considered as func- tions of the angle which the arc measures. Thus, (Fig. 4) PM, NM, AT, and BT', are respectively the sine, cosine, tangent, and cotangent of the angle A OM, as well as of the arc AM. 25. The sine of an arc is equal to the sine of its supplement ; and, in general, any function of an arc is equal to the corres- 20 ELEMENTS OF SUKVEYING. [BOOK I. ponding function of its supplement. Thus, if A is any arc or angle, sin A = sin (180° — A) ; cos A = cos (180° — A) ; tan A = tan (180° — A) ; cot A = cot (180° — A). Note. — These relations exist between the numerical values of the functions ; the algebraic signs, which they have in the different quadrants, are not considered. 26. A Natural Sine, Cosine, Tangent, or Cotangent, is the sine, cosine, tangent, or cotangent, of an arc whose radius is 1. A Table of Natural Sines is a table from which the natural sine, cosine, tangent, or cotangent of any arc may be found. The Table of Natural Sines, beginning at page 63 of the tables, gives the values of the sines and cosines only. If the tangent or cotangent of an arc. A, is desired, it may be found by the relation, sin ^ , . cos A tan A = J ; cot ^ = j* cos A sm A TABLE OF LOGARITHMIC SINES. 27. A Logarithmic Sine, Cosine, Tangent, or Cotan- gent is the logarithm of the sine, cosine, tangent, or cotangent of an arc whose radius is 10,000,000,000. A Table of Logarithmic Sines and Tangents is a table giving the logarithm of the sine and cosine, tangent and cotan- gent of any arc or angle. The logarithm of the tabular radius is 10. SEC. II.] PLANE TRIGONOMETRY. 21 Any logarithmic function of an arc or angle may be found by multiplying the corresponding natural function by 10,000,000,000, and then taking the logarithm of the result ; or more simply, by taking the logarithm of the corresponding natural function, and then adding 10 to the result. 28. In the table, beginning at page 18 of the tables, the logarithmic functions are given for every minute from 0° to 90°. In addition, their rates of change for each second, are given in the column headed " D." For the sine and cosine, there are separate columns of dif- ferences, which are written to the right of the respective columns; but for the tangent and cotangent, there is but a single column of differences, which is written between them. The angle obtained by taking the degrees from the top of the page, and the minutes from any line on the left hand of tlie page, is the complement of that obtained by taking the degrees from the lottom of the page, and the minutes from the same line on the right hand of the page. But, by definition, the cosine and the cotangent of an arc are, respectively, the sine and the tangent of the complement of that arc (Arts. 22 and 24) ; hence, the columns designated sine and tang, at the top of the page, are designated cosine and cotang at the bottom. 29. To find the logarithmic functions of an angle which is expressed in degrees and minutes. If the angle is less than 45°, look for the degrees at the top of the page, and the minutes in the left hand column; then follow the corresponding horizontal line to the column desig- nated at the top by sine, cosine, tang, or cotang, as the case may be ; the number there found is the logarithm required. Thus, 22 ELEMENTS OF SURVETIXG. [BOOK I. log sin 19° 55' . . . 9.532312 log tan 19° 55' . . . 9.559097 If the angle is greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right hand cohimn ; then follow the corresponding horizontal line back- wards to the column designated at the hottom by sine, cosine, tang, or cotang, as the case may be ; the number there found is the logarithm required. Thus, log cos 52° 18' . . . 9.786416 log tan 52° 18' . . . 10.111884 30. To find the logarithmic functions of an angle which is expressed in degrees, minutes, and seconds. Find the logarithm corresponding to the degrees and minutes as before ; then multiply the corresponding number taken from the column headed '"D" (which is millionths), by the number of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine or cotangent. EX A M P L E S . 1. Find the logarithmic sine of 40° 26' 26". OPERATION. log sin 40° 26' 9.811952 . Tabular difference 2.47 No. of seconds . 28 Product. . . . 69.16 to be added . ^ log sin 40° 26' 28" 9.812021 The same rule is followed for the figures discarded (in this case 16), as in Art. 7. SEC. il] plake tkigonometry. 23 2. Find the logarithmic cosine of 53° 40' 46". OPERATION. log cos 53° 40' 9.772675 Tabular difference 2.86 No. of seconds . 46 Product . . . 131.56 to be subtracted 132 log cos 53° 40' 46" 9.772543 If the angle is greater than 90°, we find the required function of its supplement (Art. 25). 3. Find the logarithmic tangent of 118° 18' 25". OPERATION. 180° Given arc . ... . . 118° 18' 25" Supplement 61° 41' 35" log tan 61° 41' ' . . . . 10.268556 Tabular difference 5.04 No. of seconds . 35 Product . . . 176.40 to be added . 176 log tan 118° 18' 25" 10.268732 31. To find the angle corresponding to any logarithmic function. This is done hy reversing the preceding rule : Look in the proper column of the table for the given logarithm ; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithm is not found in the table, then find the next less logarithm, and take from the table the correc- ponding degrees and minutes, and set them aside. Subtract the logarithm found in the table, from the given logarithm, annex 24 ELEMENTS OF SURVEYING. [BOOK i. two O's to the remainder, and divide this result by the corres- ponding tabular difference. The quotient will be seconds, which must be added to the degrees and minutes set aside, in thei^ase of a sine or tangent, and subtracted, in the case of a cosine or a cotangent. EXAMPLES. 1. Find the angle corresponding to the logarithmic sine 9.422248. OPEKATION. Given logarithm . . . 9.422248 Next less in table . . . 9.421857 ... 15° 19' Tabular difference 7.68 ) 391.00 ( 51", to be added. Hence, the required arc is 15° 19' 51". 2. Find the angle corresponding to the logarithmic cosine 9.427485. OPERATION. Given logarithm . . . 9.427485 Next less in table . . . 9.427354 . . . 74° 29' Tabular difference 7.58 ) 131.00 ( 17", to be subt. Hence, ihe required angle is 74° 28' 43". 32. Theorem I. — The sides of a plane triangle are propor^ tional to the sines of their opposite angles. In the triangle ABC let the large letters A, B, (7, designate the angles, and the corresponding small letters, a, h, c, the ^" Fig. 5. sides opposite ; then, « : 5 : : sin ^ : sin ^ ; a '. c : : sin ^ : sin C ; b : c : : sin ^ : sin C. SEC. II. J PLANE TKIGOKOMETKY. 25 33. Theorem II. — In any plane triangle, the sum of the two sides containing any angle, is to their difference, as the tangent of half the sum of the two other angles is to the tangent of half their difference. Thus, in the triangle ABG (Fig. 5), a + h : a—h :: imn^ (A-\-B) : tan|-(^— ^); a-\-c : a—c :: tan ^ {A-{-C) : tan ^ (A — C); Ij^c : h—c :: tani(^+C) : tan|(^— C). By solving any one of the above proportions, the first three terms being known, the tangent of half the difference of the two unknown angles is obtained, and from this tangent the half difference itself is found. The greater of the two unknown angles is equal to half their sum added to half their difference ; the smaller is equal to half their sum diminished by half their difference. 34. Theorem III. — In any plane triangle, if a line is drawn from the vertex of the vertical angle perpendicular to the base, dividing it into two segments : then, the sum of the two segments, or the whole hase, is to the sum of the two other sides, as the difference of those sides, to the difference of the segments. Thus, in the triangle ABG (Fig. 6), s-\-s' : b-i-c :: b—c: s—s', 35. Theorem IV. — In any right-angled plane triangle, radius is to the tangent of either of the acute angles, as the side adjacent to the side opposite. Let CAB (Fig. 7) be the proposed triangle, and denote the radius by R : then will ^6 ELEMENTS OF SURVETINa B* : tan C :: ACy or b : AB, or c ; Also, E* : tan B : : AB, or c : AC, or b, ^^ ^ 36. Theorem Y. — /?i an?/ right-angled triangle, radius is to the cosine of either of the acute angles, as the hypothenuse to the side adjacent. In the triangle CAB (Fig. 7), i^* : cos C : : BC, ov a '. AC, ov b', Also, i2* : cos B : : BC, or « : ^^, or c. 37. Solution of Triangles. — The relations between the sides and angles of plane triangles, stated in these five theorems, are sufficient to solve all the cases of Plane Trigonometry. Of the six parts, which make up a plane triangle, three must be given, and at least one of these must be a side, before the others can be determined. If the three angles only are given, it is plain that an indefinite number of similar triangles may be constructed, the angles of which may be respectively equal to the angles that are given and, therefore, the sides could not be determined. Assuming, with this restriction, any three parts of a triangle, one of the four following cases will always be presented: I. When two angles and a side are given. II. When two sides and an angle opposite one of them are given. III. When two sides and the included angle are given. IV. When the three sides are given. * If logaHthmic functions are used, R is equal to 10,000,000,000, and its logariihm is 10 ; otherwise, B is equal to 1. SEC. II.] PLANE TRIGONOMETRY. 27 CASE I. 38. When two angles and a side are given. In a plane triangle, ABC, there are given the angle A = 58° 07', the angle B = 22° 37', and the side AB, or c = 408 yards; to find C, a, and Z». (The sides lying opposite the angles A, B, and (7, are denoted by a, h, and c. Add the given angles, A and B, to- gether, and subtract their snm from 180° ; the remainder will be the other angle, C. Then from the proportion (Theorem I), Fig. 8. sin C : sin ^ : : c : a ; a may be fonnd ; and from the proportion, sin (7 : sin ^ : : (? : ^ ; h may be found. CASE II. 39. When two sides and an angle opposite one of them are given. In a plane triangle, ABC, there are given AC, or 5 = 216, CB, or a = 117, the angle A = 22° 37', to find the other parts. GEOMETRICALLY. Draw an indefinite right line A B'B ; from any point, as A, draw AC, making BAC = 22° 37' and make .4^=216. With (7 as a centre, and a radius equal to 117, the other given side, describe the arc B'B ; draw CB and CB' ; then will either of the triangles, ACB or ACB', answer all the conditions of the question. 28 ELEMENTS OP SURVEYING. [BOOK L TRIGONOMETRICALLY. From Theorem I, we have, a '. 1) : : sin ^ : sin ^, By applying logarithms, we have, (a. c.) log « (117) 7.931814 log Z> (216) 2.334454 log sin ^ (22° 37') .... 9.584968 log sin B, 45° 13' 55", or 134° 46' 05" . 9. 851236 The ambiguity in this and similar examples, arises in conse- quence of the first proportion being true for either of the angles ABC, or AB'C, which are supplements of each other and, there- fore, have the same sine (Art. 25). So long as the two triangles ^(75 and ACB' exist, the ambiguity will continue. But if the side CB, opposite the given angle, is greater than AG, the arc BB' , described from (7 as a centre and with a radius equal to the side a, will cut the line ABB', on the same side of the point A, in but one point, and then there will be only one triangle answering to the conditions. If the side CB is equal to the perpendicular Cd, the arc BB' will be tangent to ABB' , and in this case also, there will be but one triangle. When CB is less than the perpendicular Cd, the arc BB' will not intersect the base ABB' , and in that case no triangle can be formed, or it will be impossible to fulfill the con- ditions of the problem. In the example under consideration, there are two solutions, the first corresponding to ^ =: 45° 13' 55", and the second to ^^'C=: 134° 46' 05". FIRST CASE. A 22° 3r B 45° 13' 55" C 180—67° 50' 55" = 112° 09' 05". SEC. II.] PLANE TRIGONOMETRY. 29 Then, in the triangle A CB, sin ^ : sin (7 : : b : c, and applying logarithms, (a. c) log sin ^ (45° 13' 55") .... 0.148764 log sin C (112° 09' 05") .... 9.9G6700 log b (216) . 2.334454 log c 281.785 2.449918 SECOND CASE. A 22° 37' B 134° 46' 05" C 180° — 157° 23' 05" = 22° 36' 55". Then, in the triangle AC'B', m\B' : sin C : b : c', and applying logarithms, (a. c.) log sin B (134° 46' 05") .... 0.148764 log sin C (22° 36' 55") .... 9.584943 log b (216) . 2.334454 log c' 116.993 2.068161 2. Given two sides of a triangle, 50 and 40 respectively, and the angle opposite the latter, equal to 32° ; required the remaining parts of the triangle. Ans. If the angle opposite the side 50 is acute, it is equal to 41° 28' 59"; the third angle is then equal to 106° 31' 01", and the third side to 72.368. If the angle opposite the side 50 is obtuse, it is equal to 138° 31' 01", the thud angle to 9° 28' 59", and the remaining side to 12.436. Fig. 10. 30 ELEMENTS OF SUEVEYIN-Q. [bOOK I. CASE III. 40. When two sides and their included angle are given. Let ABC he a triangle ; AB and BC, a the given sides, and B the given angle. Since B is known, we can find the sum of the two other angles ; for, A + C =1S0''-B, and i{A^C) =i(lSO°-B). We next find half the difference of the angles A and Cy by Theorem II, viz., BC+BA : BG-BA :: tQXi\{A + C) : tan^(^-C), in which we consider BC greater than BA, and therefore A is greater than G ; since the greater angle must be opposite the greater side. Having found half the difference of A and G, by adding it to the half sum, |- (^ + C), we obtain the greater angle, and by subtracting it from half the sum, we obtain the less. That is, l(^A + G)+i{A-G)=A, and ^{A + G)-\(A-G) = a Having found the angles A and C, the third side A G may be found by the proportion, sin ^ : sin ^ : : a : h SEC. II.] PLAN^E TRIGONOMETRY. 31 CASE IV. 41. Having given the three sides of a plane triangle to find the angles. Let fall a perpendicular from the angle opposite the greatest side, dividing the given triangle into two right-angled triangles ; then find the difference of the segments of the base by Theorem III. Half this difference being added to half the base, gives the greater segment; and, being subtracted from half the base, gives the less segment ; the greater segment belongs to the right- angled triangle having the greater hypothenuse. We then have two sides and the right angle of each of two right-angled triangles, to find the acute angles. Example. — The sides of a plane triangle being given ; viz., BC =^ 40, AG =^ 34, and AB = 25 ; required the angles. BC : AC+AB :: AC-AB : CD-BD. That is. Then, And, 40 : 59 : : 9 : —^ = 13.275. 40 + 13.275 2 40-1-13.2 75 2 40 = 26.6375 = CD. = 13.3625 = BD. In the triangle DAC, to find the angle DAC. AG : DC :: sin D : sin DAG. Applying logarithms, we have, (a. c.) log ^(7(34) ....... 8.468521 log Z>(7 (26.6375) 1.425493 log sin D (90°) 10.000000 logsini).4C51°34'40" . , , 9.894014 '^2 ELKMlCxNTS OF SURVEYING. [BOOK L In the triangle BAD, to find the angle BAD. ,!/>' : /)7> :: sin D : sin BAD. ^\j)pl}ing logarithms, we have, (a. c.) h)gyiy? (25) 8.C020G0 log BI) (13.302:)) 1.125887 log sin /> (00°) lO.OQOQOO log sin y^y]/) 32° 18' 35" . . . 9.727947 Henoo, 00^-7)J = 00°— 51° 34' 40" = 38° 25' 20" = C, and, [)0'-BAJ) = 00°-32° 18' 35" = 57" 41' 25" == B, and, BAD + I)ACz=[)i° 34' 40" + 32° 18' 35" = 83° 63' 15" = A. 42. Solution of Right-angled Triangles. — The nn- know n parts of a right-angled triangle maybe found by one of tlio last four cases ; or, if two of the sides arc given, by means of tlie property that the square of the hypothennse is equal to the sum of the squares of the two other sides. Or, the parts may be found by Tlieorems IV. and V. EXAMPLES. 1. In a right-angled triangle BACy there are given tlie liypotliennse BC = 250, and the base .ir7 = 240; required ^" the other i^nrts. A?us. B = 73° 44' 23"; = 1(1° 15' 37"; AB = 70. 2. In a right-angli'd triangle BAC, there are given, J C = 384, and B = 53° 08' ; required tlie remaining parts. Am. A B =: 287.06 ; BC = 470.070 ; (7 = 3G° 52'. FiQ. 18. SEC. III.] INSTRUMENTS FOE PLOTTIXG. 33 SECTION III. INSTRUMENTS FOR PLOTTING. 43. The ordinary implements for making a diagram or plot of a survey are— drawing board ; T-square ; dividers ; ruler and triangle ; scale of equal parts ; semicircular protractor. Pig. 13. « 44. A Drawing Board (Fig. 13) is a rectangular Vjoard of about 24 by 30 inches, fths of an inch thick, made of several pieces of well seasoned white pine, fitted together with the grain running in different directions to prevent warping. It is important that its angles should be perfect right angles. O Pio. 14. 45. A T-square (Fig. 14) is a ruler about 2 feet in length let into a thicker piece of wood at right angles to it. One side of the cross-piece is even with the ruler and the other side projects somewhat, giving a shoulder on that side. It is a convenient instrument for drawing parallels and perpendiculars. 34 ELEMEi^TS OF SURVEYIljq^G. [book I. Fig. 15. 46. The Dividers (Fig. 15) consists of two legs la, he, which may be easily turned around a joint at b. One of the principal uses of this instrument is to lay off on a line, a distance equal to a given line. For example, to lay off on CD, a distance equal to AB. Open the points of the dividers to a a| IB greater distance than AB, place one point lightly upon A, and then slowly and con- tinuously close them till the point reaches B. Then raise the dividers, place one foot at C, and mark with the other the distance CE : this will evidently be equal to AB. E -D Fig. 16. Fig. 17. 47. Ruler and Triangle. — A Ruler of convenient size is about twenty inches in length, two inches wide, and a fifth of an inch in thickness. It should be made of a hard material, perfectly straight and smooth. The hypothenuse of the right-angled triangle, which is used in connection with it, should be about ten inches in length, and it is most conyenient to have one of the sides considerably longer than the other. The two following problems may be solved with the ruler and triangle. SEC. III.] INSTKUMENTS FOR PLOTTING. 35 I. To draw through a given point a line which shall be par- allel to a given line. 48. Let (7 be the given point, and AB the given line. Place the hypothenuse of the triangle against the edge of the ruler, and then ' — place the ruler and triangle on the paper, A B 60 that one of the sides of the triangle ^^°- ^^• shall coincide exactly with AB; the triangle being below the line. Then, placing the thumb and fingers of the left hand firmly on the ruler, slide the triangle, with the other hand, along the ruler, until the side which coincided with AB reaches the point C. Extend the first and second fingers of the left hand upon the triangle to hold it firmly in place, the thumb and remaining fingers steadying the ruler, and with the right hand, mark with a pen or pencil, a line through C: this line will be parallel to AB. II. To draw through a given point a line which shall be per- pendicular to a given line. 49. Let AB be the given line, and D the given point. Place the hypothenuse of the triangle against the edge of the ruler, as before. Then place the ruler and triangle so that one of the sides of the triangle shall coin- cide exactly with the line AB. Then slide the triangle along the ruler until the other side reaches the point D : then, draw through D, a right line, and it will be perpendicular to AB. The right angle of the triangle should be carefully tested by laying off a perpendicular through the same given point, D Fig. 19. 36 ELEMENTS OF SURYEYIi^^G. [book I. with the triangle in two positions— to the right of the per- pendicular, and then to the left of it. 50. A Scale of Equal Parts (Fig. 20) is formed by divid- ing a line of a given length, into equal portions. T..2 .3 .^ .S .C .7 .S .0 10 I I I I I I I I jzd a. Fig. 20. If, for example, the line ah^ of a given length, say one inch, be divided into any number of equal parts, as 10, the scale thus formed is called a scale of ten parts to the inch. The line ab, which is divided, is called the unit of the scale. This unit is laid off several times on the left of the divided line, and the points marked 1, 2, 3, &c. The unit of scales of equal parts is, in general, either an inch, or an exact part of an inch. If, for example, ah, the unit of the scale, were half an inch, the scale would be one of 10 parts to half an inch, or of 20 parts to the inch. If it were required to take from the scale a line equal to two inches and six-tenths, place one foot of the dividers at 2, on the left, and close the other to .6, which marks the sixth of the small divisions : the dividers will then embrace the required distance. df P \\ \ \\\\\ .09 \ \ \ \ \ W \ .08 w \ \\y\ \\\\ .07 M M M M M ■06 M M M 11 Ml .0 5 \ \ \eA \ \ \ .0 4 \ MM M .03 M M M M M .02 M M M M 1 .01 1 M M M ( 2 1 9 a. 12. 3 A .5.6.7 .8 .9 b Fig. 21. 51. A Diagonal Scale of Equal Parts (Fig. 21) is thus constructed Take ab for the unit of the scale, which may be SEC. III.] INSTRUMENTS FOR PLOTTING. 37 one inch, J, ^, or f of an incli, in length. On ab describe the square abed. Divide the sides ab and dc each into ten equal parts. Draw af, and the other nine parallels as in the figure. Produce ba, to the left, and lay off the unit of the scale any convenient number of times, and mark the points 1, 2, 3, &c. Then, divide the line ad into ten equal parts, and through the points of diWsion draw parallels to ab, as in the figure. Now, the small divisions of the line ab are each one-tenth (.1) of ab ; they are therefore . 1 of ad, or . 1 of ag or gh. If we consider the triangle adf, we see, that the base df is one-tenth of ad, the unit of the scale. Since the distance from a to the first horizontal line above ab is one-tenth of the distance ad, it follows that the distance measured on that line, between ad and af, is one-tenth of df: but since one-tenth of a tenth is a hundredth, it follows that this distance is one hundredth (.01) of the unit of the scale. A like distance, measured on the second line, is two hundredths (-02) of the unit of the scale ; on the third, .03; on the fourth, .04, &c. If it were required to take, in the dividers, the unit of the scale, and any number of tenths, place one foot of the dividers at 1, and close the other to that figure between a and b which designates the tenths. If two or more units are required, the dividers must be placed on a point of division further to the left. When units, tenths, and hundredths are required, place one foot of the dividers where the vertical line through the point which designates the units, intersects the line which designates the hundredths : then, close the dividers to that line between ad and be which designates the tenths: the distance so embraced will be the one required. For example, to take off the distance 2. 34, we place one foot of the dividers at I, and close the other to e : and to take off the 38 ELEMENTS OF SURVEYING. [book I. distance 2.58, we place one foot of the dividers at p and close the other to q. KoTE 1. — If a line is so long that the whole of it cannot be taken from the scale, it must be divided, and the parts of it taken from the scale in succession. Note 2. — If a line be given upon the paper, its length can be found by taking it in the dividers and applying it to the scale. >' Pig. 22. 62. A Semicircular Protractor (Fig. 22) is used to lay down, or protract angles. It may also be used to measure angles included between lines, already drawn upon paper. It consists of a brass semicircle, ABC. divided to haK degrees. The degrees are numbered from to 180, both ways ; that is, from A to B and from B to A. The divisions, in the figure, are made only to degrees. There is a small notch at the middle of the diameter AB, which indicates the centre of the protractor. SEC. III.] INSTRUMENTS FOR PLOTTING. 39 To lay off an angle with a Protractor. 53. Place the diameter AB on the Hne, so that the centre shall fall on the angular point. Then count the degrees con- tained in the given angle, from A toward B, or from B toward A, and mark the extremity of the arc with a pin. Eemove the protractor, and draw a line through the point so marked, and the angular point : this line will make with the given line the required angle. The ordinary brass or horn protractors are of but little value. Printed protractors of six and twelve inches diameter, upon heavy paper, or bristol board, and divided to quarter degrees, are very useful and reliable. By the following method an angle may be laid off with even greater accuracy than with a protractor. With a reliable scale of inches, divided to hundredths, set the dividers at five inches, and placing one point at the given vertex A (Fig. 23), describe the arc BO; take out from the y^ table the natural sine of half the given angle, y^ multiply it by 10, and call the product inches ; ^ b with the dividers take off this number of inches ^^^' ^' from the scale, and placing one point at B, describe an arc cutting BC 2J[> C ; CAB will be the required angle. It is evident that twice the sine of half the angle is the chord of the whole angle ; and since the radius used is 5 inches, we have 2 (Nat. sine Ja x 5) = 10 x ISTat. sine ^. 40 ELEMENTS OF SUEVEYING. [book I. SECTION IV. DEFI N ITIONS. 54. Surveying comprises all the oj^erartioDS necessary for finding the lengths and directions of the bounding lines of any portion of the earth's surface, the area of such portion, and for making on paper an accurate delineation, or map, of the surface and the boundaries. 55. Plane Surveying is that branch of surveying in which the curvature of the earth is neglected, as it may be when the survey is limited to small portions of the surface. 56. G-eodesy, or G-eodetic Surveying, is that branch in which the curvature of the earth is taken into account, as it must be in all extensive surveys. 57. A Horizontal Plane at any point is a plane perpen- dicular to the radius of the earth drawn to that point. 58. A Vertical Plane is a plane perpendicular to a hori- zontal plane. 59. A Horizontal Line is any line of a horizontal plane. 60. A Vertical Line is a line perpendicular to a hori- zontal plane. 61. An Oblique Line is a line inclined; i. e., neither par- allel nor perpendicular to a hori- zontal plane. Thus (Fig. 24), AB and DC fig. 24. are horizontal lines; BC and AB are vertical lines ; and A C and BD are oblique lines. SEC. IV.] DEFIKITIOKS. 41 62. The Horizontal Distance between two points is the horizontal line iutercepted between the two vertical lines passing through those points. Thus, DC or AB (Fig. 24), is the hori- zontal distance between the two points A and C, or between the points B and D. 63. A Horizontal Angle is an angle whose sides are hori- zontal ; the plane of its sides is also horizontal. 64. A Vertical Angle is an angle the plane of whose sides is vertical. 65. An Angle of Elevation is a vertical angle having one of its sides horizontal and the other oblique, the oblique side being above the horizontal side. Thus, BAC (Fig. 24) is the angle of elevation from A to G. 66. An Angle of Depression is a vertical angle having one of its sides horizontal and the other oblique, the oblique side being heloiu the horizontal side. Thus, DC A (Fig. 24) is the angle of depression from C to A. 67. An Oblique Angle is an angle the plane of whose sides is inclined to a horizontal plane. BOOK II. CHAIN SURVEYING. SECTION I. MEASUREMENT OF DISTANCES. 68. Any tape, rod, or chaiD, divided into equal parts, may be Uvsed as a measure for finding the distance between two points. Fig. 25. The measure in general use for land surveying is a chain of four rods, or sixty-six feet in length, called Gunter's chain (Fig. 25), from the name of the inventor. It is composed of 100 links, each joined to the other by two or three rings. Every tenth link from either end, is marked by a small attached brass pendant SEC. I.] MEASUEEMENT OF DISTANCES. 43 or tag, which is notched to designate its number from the end. The tag at the middle, or fifty-link, point is distinguished by being rounded, or by some other peculiarity of make. As the tags at equal distances from the two ends of the chain are marked the same, care must be taken not to mistake forty links for sixty, &c., and the reverse. To avoid such error, it would be better to have the tags marked in regular order from the begin- ning to the end of the chain, rather than from both ends to the middle. A link in measure includes a lar, with its connecting ring at each end ; when there are three connecting rings, a ring and a half at each end is included. The handles are of brass, and each forms part of the end link, to which it is connected by a nut, by which also the length of the chain is adjusted. To determine whether to measure from the inside of the brass handle, or from the outside, double back the last two or three links upon the preceding links and compare. The division of the chain into 100 equal parts is very con- venient, since the divisions, or links, are decimals of the whole chain, and in the calculations are treated as such. TABLE. 1 chain = 4 rods = ^^ feet = 792 inches = 100 links. 1 link = 7.92 inches. 80 chains — 320 rods = 5280 feet = 1 mile. An excellent chain for accurate measurements is Grum- man's patent "suspended chain," which is made of very light steel wire, is fitted with spring-balance, thermometer and spirit- level attachments, and is held above the surface when in use, the ends of the chain being marked upon the ground by the points of plummets let fall from the end notches. 44 ELEMENTS OF SUBVETING. [book II. Instead of a chain, which is liable to error because of the bars and rings becoming worn by frequent contacts, a steel ribbon or tape is often used. 69. Besides the chain or tape, the surveyor needs ten (or better, eleven) marking pins (Fig. 26), made of iron or steel wire, about an eighth of an inch in thickness and a foot long, sharpened at one end and bent into a ring at the other, for marking chain lengths on the ground ; a plumbob (Fig. 27) and line for referring, when necessary, points in the chain held horizontally to the inclined surface of the ground ; and a set of flag-poles, or ranging rods, for marking stations and ranging out lines. The marking pins should be strung upon an iron ring with a spring-catch, and this ring should be attached to a strap to be passed over the right shoulder suspending the pins at the left side ; or, better, the pins may be carried in a leather quiver strapped to the waist. The pins should be tagged with white cloth to enable the surveyor to find them again readily, when they have been left to mark a point. 70. To Measure a Horizontal Line. — The point where the measurement is to begin is located by a staff temporarily placed for the purpose, or by some one of the many permanent marks by which the angular points in a boundary are fixed. The other extremity of the line must be provided with a staff or flag which can be easily seen. Two chainmen are required, a fore-chain man, or leader, and Fig. 26. Fio. 27. SEC. I.] MEASUREMENT OF DISTANCES. 45 a hind-chainman, or follower. The more careful and expert of the two should be the follower. The leader, with the marking-pins and one handle of the chain in his right hand, starts off on the line, drawing out the chain to its full length. Both chainmen . now examine it to see that there are no inaccuracies in it, either from bent links or kinks in the rings joining the links. Having adjusted the chain for use, the leader resumes his place, to be directed by the follower, who stands behind the staff at the beginning, and sights to the staff at the end of the line, so that the measurement shall be made exactly along the established line. To facilitate this (on level ground) and to insure the correct alignment of the pin, at its proper distance, the chain and one pin should be held firmly in the right hand, as representcid in Figure 28. While the pin is being aligned, it should be held by the leader as far from the body as possible, so that the view of the flag be left unobstructed. To accomplish this, and at the same time draw the chain to the proper degree of tension, the right arm should be braced against the inside of the right knee. The follower directs, by the simple orders "right" or "left," according as the pin, held as described, is to be carried to the right or left to bring it into line with the flag. When the pin is truly in line, the chain at the same time being drawn straight and taut, the order " down " is given, when the leader bringing his left hand to bear on the top of the pin, forces it vertically into the ground, and resumes his course to the length of another chain. Fig. 28. 46 ELEMENTS OF SURVEYING. [BOOK IL If, for any reason, the pin can not be driven into the ground, the end of the chain length should be marked by driving the pin olliquelyy always at right angles to the chain ; if this cannot be done, a cross should be scratched on the ground at the exact point, and the pin laid down with its point at the mark. After one or two chains have been measured, on any line, the leader can, by glancing back to the station just left, place the pin nearly in' the right position ; the exact aligning should be left, however, to the follower. When the distance to be measured is more than ten chains, the pins, when exhausted, should be returned to the leader, the distance noted in a field-book provided for the purpose, and the chaining recommenced at the place of the tenth pin. If but ten pins are used, the follower has then but the hole made by the tenth pin to measure from. For this reason, eleven pins are often used, the eleventh being of different material from the others for distinction, and is used by the follower to measure from when the ten pins are returned to the leader. 71. All distances should be measured horizontally. Hence, when the ground slopes, one end of the chain must be elevated. Each chainman should be provided with a small plumb-line, so that the elevated end of the chain may be held directly over the proper point. . "When the raised end of the chain is only two, or even three feet above the ground, it will suffice, in many cases, to use a marking-pin, held lightly by the point, between the thumb and finger, instead of a plumb-line. When the chaining is on a steep inclination, other precautions should be observed. SEC. I.] MEASUREMENT OF DISTAKCES. 47 Suppose the chaining to be up hill. The leader draws the chain out to its full lengthj as in any other case, and then returns Fio. 30 to within such a distance of the follower, that when the chain is drawn out to that length horizontally, it shall not be too high to be held conyeniently. The follower holds his end of the chain carefully over the point or station, by means of the plumb-line, while he directs the leader in the usual manner. The point fixed in this manner, by the leader, must not be marked by a marking-pin, but by a small peg or nail. At the order, "Down," the leader does not go forward immediately, but waits until the follower comes up and takes the chain by the precise point held, the moment before, to the ground. This point is now held above the peg by the the follower, who uses the plumb, as before, and aligns the leader, who has taken hold of the chain a few links farther on, and is holding it to the ground. These short distances are not recorded. The end of a full chain is marked by a marking- pin. 48 ELEMENTS OF SURVEYIN"G. [book II. In chaining doivn-hill, the method is essentially the same. The leader uses the plumb, and determines by it where the peg is to be placed. At the end of a course, the part of a chain is measured by drawing the chain onli/ to the flag, where it is held by the leader, until the follower comes forward to the last pin, and counts the links. In measuring up the hill from A to C, or down the hill from C to A, the horizontal distances ahf c d, and / C, are measured and their sum is the horizontal dis- tance between A and C. Chaining doion hill gives more accurate results than chaining up ; for in chaining down, the follower holds the chain firmly upon the ground and no ordinary pull by the leader moves it. It is impossible to hold a chain perfectly steady over a point, by means of a long plumb-line, while the leader is pulling out. Fig. 31. 72. When the ends of a line can not be seen, each from the other, intermediate points between the two must be established. When a hill intervenes, such points may be established thus : let the surveyor and an assistant, each with a ranging-rod, place themselves as nearly in the line as possible, and in such position that each can see the other and the flag beyond him. The sur- veyor looking to the flag at the end of the line, directs the assistant into line with it ; the assistant then looks to the flag at the beginning of the line and directs the surveyor into line with it ; the surveyor from his new position redirects the assistant into line with the end flag-stafl ; the assistant then realigns the surveyor with the flag-staff at the beginning of the line ; the operation is repeated till both stand in the SEC. I.J MEASURExMENT OF DISTANCES. 49 desired Hue, when their positions are marked with the rang- ing-rods. 73. When a valley is to be chained across, intermediate points in the lower portions of it may be fixed, if necessary, by the surveyor holding a plumb-line so as to cover the flag-staffs at both ends of the line and directing an assistant to fix, between the two, ranging-rods which shall also be covered by the plumb- line. 74. When a wood intervenes between the two ends of a line, a trial line may be run out by ranging-rods placed at convenient distances in line with each other and with the staff at the beginning of the line, and as nearly as possible in the required line. Then draw on the ground a perpendicular (by a method to be shown presently) from the staff at the end of the required line to the trial line just run out, and measure the length of this perpendicular. The ranging-rods may then, by the property of similar triangles, be put in their true position in line. Thus, ^ G^ is to be measured ; the trial Ime AF \^ run out and the ranging-rods B, C, &c., fixed at known distances apart and as nearly in the line as possible ; the perpendicular 6^i^is measured ; then from the similar triangles, AFG and ABH, AF: AB : : FG : BH, The distance BH thus becomes known, and the ranging-rod at B is moved, on a perpendicular to AF, the required distance to its true position at H. The other ranging-rods are in like 50 ELEMENTS OF SURVEYING. [book II. Fig. 33. manner put in their true positions at /, K, &c., and the true line is marked out. 75. To trace on the ground the direction of a straight line, that shall be perpendicular, at a given point, to a given straight line. FIRST METHOD. Any three lines having the ratio 3, 4, and 5, form a right- angled triangle. Let AB (Fig. 33) be the given line and C the point at which the per- pendicular is to be drawn. Divide the number of links in the chain by 8, neglecting the remainder ; if we are using the 100-link chain the quotient would be 12 links. From the point C measure a distance towards A equal to four times this quotient (48 links) ; place one end of the chain at (7, and the end of the 96th link [(5 + 3) x 12 = 96] at A', then taking the end of the 36th link (3 x 12) pull out the chain so that the two portions EA and EG are taut, and E will be a point on the perpendicular required. This method supposes the chain to be correctly divided into links. SECOND METHOD. Let AD be the given right line, and D the point at which the perpen- dicular is to be drawn. Take the longest available distance on the tape or chain (the whole of it if possible), and place one extremity at D, and fasten the other at some point, as E, Fig. 34. SEC. I.] MEASUREMENT OF DISTANCES. 51 between the two lines which are to form the right angle. Place a staff at E. Then, having stationed a person at D, remove that extremity of the chain and carry it round until it ranges on the line DA, at A. Place a staff at A : then remove the end of the chain at A, and carry it round until it falls on the line AB, prolonged, at F. Then place a staff at F; ADF will be a right angle, being an angle in a semicircle. This method is independent of any errors of graduation of the chain ; it also gives the largest possible construction in the field, a matter of importance as insuring the most correct results. 76. There is a simple instrument for laying off right angles on the ground called the Surveyor's Cross. This instrument con- sists of two bars, AB and (72), Fig. 35, permanently fixed at right angles to each other, and firmly attached at F, to a pointed staff, which serves as a support. Four sights are screwed firmly to the bars, by means of the screws a, b, c, and d. As the only use of this in- strument is to lay off right angles, it is of the first im- portance that the lines of sight be truly at right angles. To ascertain if they are so, let the bar AB be turned until its sights mark some distinct object ; then look through the other sights, and place a staff on the line which they indicate ; let the cross be then turned until the sights of the bar AB come to this last line; if the other sights are Fio. 35. 5^ ELEMENTS OF SURVEYING. [bOOK II, directed to the first object, the lines of sight are exactly at right angles. The sights being at right angles, if one of them be turned in the direction of a given line, the other will mark the direc- tion of a line perpendicular to it, at the point where the instru- ment is placed. 77. From a given point without a straight line, to let fall a perpendicular on the line. — Let C be the given point, and AB the given line (Fig. 36). From C, measure a line, as CA, . to any point of the line AB. From o JSf'^ A, measure on AB any distance as i,,^'^ \ AF, and at F erect FE perpendicu- Fio. 36. lar to AB. Having stationed a person at A, measure along the perpen- dicular FE until the forward staff is aligned on the line AC: then measure the distance AE. From similar triangles, AE: AF :: AC: AD; in which all the terms are known except AD, which may, there- fore, be found. The distance AD being laid off from A, the point Df at which the perpendicular CD meets AB, becomes known. If the length of the perpendicular is desired, it may be found from the proportion, AE: EF :: AC: CD, in which all the terms are known except CD. It is always best to make AF nearly equal to AD when practicable ; for then AE will be nearly equal to AC, and the multiplication of errors will be avoided ; in the expression AC AC AD = -j^xAF, -j-p will be but little greater than 1. SEC. I.] MEASUREMENT OF DISTANCES. 53 Fig. 37. 78. To trace on the ground a straight line that shall pass through a given point and be parallel to a given straight line. Let ABhe the given line and P the given point. From F meas- ure any oblique line to AB, as PQf and mark its middle point, which call m in the figure. From any point of AB, as E, run a line through m, and prolong it till mS = Rm ; then PS will be the parallel required. 79. To determine the horizontal distance from a given point to an inaccessible object. Let A be an inaccessible object, and B the point from which the dis- tance is to be measured. FiEST Method. — At the point B, lay off BB perpendicular to the line BA, and measure along it any con- venient distance, as BB. At B lay off the right angle EBD, and measure any distance in the direction BD, Let a person at D align a staff on DA, while a second person at B aligns it on BB -. the staff will thus be fixed at C. Then measure the distance BC. The two triangles BOD and CAB being similar, we have, BG : BD \: CB : BA, in which all the terms are known, except the fourth, which is, therefore, found. 11 BC can be made equal to CB, then the measured distance BD is the distance required. The triangle BDC should be as 1^ B D Fig 3S. 54 ELEMENTS OF SUETEYING. [book II. nearly equal to ACE as practicable, and the angle BCD should not exceed 45°. Secoxd Method.— Let B be the given point, and A the inaccessible object ; it is required to find BA. Measure any horizontal base-line, as BC. Then, having placed staves at B and C, measure any conre- nient distances BD and CE, such that the points D, B, and A^ shall be in the same right line, as also, the points E, C, and A : then meas- ure the diagonal lines DC and EB. Kow, in the triangle BEC, the fig. 39. three sides are known, therefore, the angle ECB can be found. In the triangle CDB, the three sides are also known, therefore, the angle CBD can be determined. These angles being respec- tively subtracted from 180°, the two angles ACB and ABC become known; and hence, in the triangle ABC, we have two angles and the included side, to find the side BA. The lines BD and CE should each equal BC,if. possible, to facilitate computation, and the angle at A should not be less than 10°. n Thied Method. — Let AC he the distance required. Lay off the right angle CAB, and measure AB, any convenient distance. At B lay off the right angle CBD, and fix the point D, carefully, in line with AC. Measure AD. Then, AB^ AD : AB :: AB : AC..-.AC = AD (Legendre, Bk. IV. Prop. 23). KoTE. — When such problems occur Fig. 40. SEC. I.] MEASUREMENT OF DISTANCES. 55 in practice, the distance AC \s usually a portion of a longer line, so that the line CAD is well marked by stakes or pins, before AB is measured. 80. To prolong a line beyond an obstacle. — If the obstacle can be seen over, the surveyor should send an assistant, with a flag-pole or ranging-rod, to the further side of it, to a point approximately in line. At this point the assistant should hold the rod vertical and exposed to the surveyor, at one end of the line, who directs him to ^' right" or "left" till the ranging-rod covers or coincides with the flag-pole at the further end of the line, when it is inserted in the ground and marks a point in the desired line. Other points may be determined in like manner. This is called "ranging," or "ranging out" the line. When the obstacle cannot be seen over, the continuation may be effected as follows : First Method. — Let OA be the line to be prolonged. Lay off OAB = 120°, or CAB = 60". Measure AB, of such length as to permit BC to be measured without meeting the obstruction. Make ABC = 60°, and measure BC, equal to AB. If ^ be not in sight from C, make the angle BCP equal to 120°, and resume the survey of the line. AC is equal to AB or BC. Note. — This method may be employed in the absence of any angular instruments, by constructing an equilateral triangle with the chain. Measure half a chain from A towards C; then fasten 56 ELEMEISTS OF SURVEYIH^G. [book II. Fig. 42. the ends of the chain at A and the point so determined, and pull out the middle, thus forming an equilateral triangle. If there is not room to measure towards C, measure back towards 0, and construct the angle on the side of A away from B. The errors are cumulative in this method, but it is rapid, and will do for work not requiring great accuracy. Second Method. Take two points, A and B, at conyenient distance apart, one or two chains, and draw two offsets, AB and BF, at right angles to the direction of the line and of sufficient length to clear the obstacle ; draw^ BF through the extremities of the rectangular offsets and prolong it beyond the obstacle ; this last line ^vill be parallel to the original line of direction ; at G and H draw the lines at right angles to Bff, and make them equal in length to AB and BF; draw CD, it will be the pro- longation required. FG is equal in length to BC, provided the perpendiculars BF and GO are accurately laid out. This is a good method to use in passing trees or small obstacles, provided the distance between perpendiculars AB and BF, also CG and DH, is one chain or more. The perpendiculars are often laid off by- guess, which method will give a very fair prolongation, but the distance BC thus obtained will not be accurate. Third Method. — From any point A, measure AB; through its middle point x, run CD, making Dx = Cx ; then DB will be SEC. I.] MEASUREMEN^T OF DISTAI^CES. 57 parallel to A C. From D run a line, DE, through any point of the line AB, as y, making yE of such length that, By : Dy :: Ay : Ey. Then through z, the middle point of DE, run BH, making zH = Bz. HE will be the prolongation of A C. To find CH we have. By : BD : : Ay : AE, fi'om which AE is known ; AE—(AG-\^ HE) = CH, 81. To find the altitude of an object, when the distance to the vertical line passing through the top of it is known. — Let CD be the altitude required, and A C the known distance. From A, measure on the line A C, any conveni- ent distance AB, and place a staff vertically at B, Then placing the eye at A, sight to the object D, and let the point, at which the line AD cuts the staff BE, be marked. Measure the distance BE on the staff ; then, AB : BE :: AC: CD, whence CD becomes known. If the line ^(7 cannot be measured, on account of intervening objects, it may be determined by calculation, as in the preceding article, and then, having found the horizontal distance, the vertical line is readily determined, as before. 82. Standard. — As the chain varies in length from changes of temperature and from use, it should be daily compared with Fig. 44. 58 ELEMENTS OF SURVETIXG. [BOOK IL a standard kept for the purpose. A convenient standard for such comparison may be made by driving into a level and even piece of ground two stakes, sawed off even with the surface of the ground, distant from each other one chain, or CG feet, accurately measured, with nails driven into the heads of the stakes to mark the exact length of the standard. Marks made upon the coping of a wall, or a curb-stone, will answer the same purpose. If it is found that any line has been measured with either too long or too short a chain, the true distance may be found by the proportion ; The length of the standard : the length of the incorrect chain used : : the measured distance : the true distance. For areas the proportion would be ; The square of the length of the standard : the square of the length of the chain used :: the area found : the true area. SECTION II. AREA OR CONTENTS OF GROUND. 83. The surface of ground being, in general, broken and uneven, it is impossible, without great trouble and expense, to ascertain its exact area or contents. To avoid this incon- venience, it has been agreed to refer every surface to a hori- zontal plane; that is, to regard all its bouHding lines as hori- SBC. ll.j AREA OR CONTENTS OF GROUND. 59 ^A: % Fig. 45. zontal, and its area as measured by that portion of the horizontal plane which the boundary lines enclose. For example, if ABCD were a piece of ground, having an uneven surface, the whole would be referred to a horizontal plane, and that part of the plane which is included be- tween the bounding horizontal lines AB, BC, CD, DA, be taken for the measure of the area. In estimating land in this manner, the sum of the areas of all the parts, into which a tract may be divided, is equal to the area, estimating it as an entire piece ; but this would not be the case if the areas of the parts had reference to the actual surface, and the area of the whole were calculated from its bounding lines. 84. The unit of measure of any quantity is a quantity of the same kind, regarded as a standard. For lines, the unit is a right line of a known length, as 1 foot, 1 link, 1 chain, or any other fixed distance. In measuring land, the length of Gunter's chain is generally taken as the unit of linear measure. 85. The unit of measure for surfaces is a square described on the unit of linear measure. When, therefore, the linear measures are feet, yards, rods, or chains, the superficial measures, are square feet, square yards, square rods, or square chains ; and the numerical expression for the area, is the number of times which the unit of superficial measure is contained in the land measured. An Acre, which is the common unit of measure for land, is a surface equal in extent to 10 square chains ; that is, equal to a rectangle of which one side is ten chains and the other side one chain. 60 ELEMENTS OF SURVEYI2S"G. [book II. A Rood, is one quarter of an acre. Since the chain is four rods in length, 1 square chain con- tains 16 square rods ; and therefore, an acre, which is 10 square chains, contains 160 square rods, and a rood contains 40 square rods. A square rod is called a perch. 86. Land is generally computed in acres, roods, and perches, which are respectively designated by the letters A. R. P. When the linear dimensions of a survey are chains or links, the area will be expressed in square chains or square links, and it is necessary to form a rule for reducing such area to acres, roods, and perches. The reduction may be made by the following TABLE. Miles. Acres. Roods. Sq. Chains. Perches. Sq. Links. 1 640 1 2560 4 1 6400.0 10.0 2.5 1.0 102,400 16a 40 16 1 64,000,000 100,000 25,000 10,000 625 1 square mile = 6400 square chains = 640 acres. When the linear dimensions are links, the area will be ex- pressed in square links, and may be reduced to acres by dividing by 100,000, the number of square links in an acre ; that is, by pointing off five decimal places from the right hand. If the decimal part be then multiplied by 4, and five places of decimals pointed off, in the product, from the right hand, the figures to the left will express the roods. If the decimal part of this result be now multiplied by 40, and five places for decimals pointed off, as before, the figures to the left will express the perches. If one of the dimensions be in links, and the other in chains, sec: II.] AREA OR CONTENTS OF GROUND. 61 the chains may be reduced to links by annexing two ciphers ; or, the multiplication may be made without annexing the ciphers, and the product reduced to acres and decimals of an acre, by pointing off three decimal places from the right hand. When both the dimensions are in chains, the product is reduced to acres by dividing by 10, or pointing off one decimal place. From which it is evident that, 1st. If links he multiplied hy linhs, the product is reduced to acres by pointing off five decimal places from the right hand. 2d. If chains he multiplied hy links, the product is reduced to acres hy pointing off three decimal places from the right hand, 3d. If chains he multiplied hy chains, the product is reduced to acres hy pointing off one decimal place from the right hand, 87. Since there are 16.5 feet in a rod, a square rod is equal to 16.5 X 16.5 = 272.25 square feet. If the last number be multiplied by 160, there will result 272.25 X 160 = 43560 = the square feet in an acre. Since there are 9 square feet in a square yard, the last number divided by 9, will give 4840 = the number of square yards in an acre. 88. To find the area of a piece of ground in the form of ;\ square, rectangle, or parallelogram. Rule. — Multi-ply the base by the altitude, and the product will express the area (Geom., Bk. IV., Prop. IV.). 62 ELEMENTS OF SUBVEYING. [book IL Example. — To find the area of the rec- tangular field A BCD. Measure the two sides AB, BC ; suppose that AB = 14 chains 27 links, and BC=d chains 75 links. Then, Fig. 46. AB = 1427 links. BC = 975 links. ABxBC = 1391325 square links. = 13.91325 acres. 4 3.65300 roods. 40 26.12000 perches. Ans. 13 A. S R, 26 P. 89. To find the contents of a piece of land in the form of a triangle. FIRST METHOD. KuLE. — Measure either side of the triangle as BC, and from the opposite angle A, let fall a perpendicular AD, and measure this per^pendicular ; then, multiply the base and perpen- dicular together, and divide the product by 2 ; the result mill express the area of the triangle (Geom., Bk. IV., Prop. VI.). EXAMPLES. 1. What are the contents of a triangle whose base is 25 ch. 1 1., and perpendicular 18 ch. 14 1. ? Ans, 22 A. 2 R. 29 P. SEC. IL] AKEA or COl^TENTS OF GROUND. 63 2. What are ttie contents of a triangle whose base is 15.48 chains, and altitude 9.67 chains? Ans, K A. 1 E. 38 P. SBCONDMETHOD. Rule. — Measure the three sides of the triangle, Then^ add them together and tahe half their sum. From this half sum subtract each side separately. Then, multiply the half sum and the three remainders together, and extract the square root of the product ; the result will he the area (Geom. Mens., Art. 97). Or, after having obtained the three remainders, add together the logarithm of the half sum and the logarithm's of the respective remainders, and divide their sum by 2 ; the quotient will be the logarithm of the area. EXAMPLES. 1. Find the area of a triangular piece of ground whose sides are 20, 30, and 40 chains. BY FIRST BULE. 20 45 45 45 30 -20 —30 —40 40 25 1st rem. 15 2d rem. 5 3d rem. 2)90 45 = half sum. Then, 45 x 25 x 15 x 5 = 84375 ; and V^SSSTS = 290.4737 = the area. Ans. 29 A. R. 8 P. 2. What is the area of a triangle whose sides are 2569, 4900, and 5035 links? H ELEMENTS OF SURVEYING. [BOOK II. BY SECOND RULE. 2569 6252 6252 6252 4900 — 2569 —4900 —5035 5035 3683 Ist rem. 1352 2d rem. 1217 3d rem. 2 ) 12504 6252 = half sum. Then, log 6252 3.796019 log 3683 3.566202 log 1352 3.130977 log 1217 3.085291 2 )13.578489 Area in square links, 6155225 . . . 6.789244 Am. 61 A. 2 E. 8 P. 90. To find the area of a piece of land in the form of a trapezoid. Rule. — Measure the two parallel sides, and also the per- pendicular distance between them. Add the two parallel sides together, and take half the sum ; then multiply the half sum hy the perpendicular, and the product will be the area (Geom., Bk. IV., Prop. VII.). EXAM PLES. 1. What is the area of a trapezoid, of which the parallel sides are 30 and 49 chains, and the perpendicular distance between them 16 eh. 60 1., or 16.60 chains ? Fig. is. SEC. II,] AEEA OR CONTENTS OF GROUND. 65 30 + 49 = 79 ; dividiDg by 2, gives . . . 39.5 Multiply by .16.60 Area in square chains 655. 700 Ans. 65 A. 2 R. 11 P. 2. Eequired the contents, when the parallel sides are 20 and 32 ch., and the perpendicular distance between them 26 ch. Ans. 67 A. 2 R. 16 P. 91. To find the area of a piece of land in the form of a quadrilateral. Rule. — Measure the four sides of the quadrilateral, and also one of the diagonals ; the quadrilateral will thus he divided into two triangles, in both of ivhich all the sides will he hnown. Then, find the areas of the triangles separately, and their sum will he the area of the quad- rilateral. Example. — Suppose that the sides and diagonal AC, of the quadrilateral ABCD have been found, AB z=z 40.05 ch., CD = 29.87 ch., BC = 26.27 ch., AD = 37.07 ch.j and ^C = 55 ch.; required the area of the quadrilateral. Ans, 101 A. 1 B. 15 P. Note. — Instead of measuring the four sides of the quadri- lateral, the perpendiculars Bb, Dg, may be let fall on the diagonal AC The area of the triangle may then be determined by measuring these perpendiculars and the diagonal AC, The perpendiculars are Dg = 18.95 ch., and Bh = 17.92 ch. 66 ELEMENTS OF SURVEYING. [book II, 92. To find the contents of a field having any number of sides. Rule. — Measure the sides of the field and also the diagonals ; the three sides of each of the triangles into which the field mill he thus divided will then he known, and the areas of the triangles may then he calculated hy the preceding rules. Or, measure the diagonals, and from the angular points of the field draw perpendiculars to the diagonals and measure their lengths ; the hase and per- pendicular of each of the triangles will then he known. Example. — Let it be required to determine the contents of the field ABODE, having five sides. Suppose that the diagonals and per- pendiculars have been found, ^(7=36.21 ch., ^6'r= 39.11 ch., Bh=: 4.08 ch., Dd.zzz 7.26 ch., ^a = 4.19ch.; required the area of the field. Fig. 50. Area of triangle ABO = 73.8684 square chains, Area of " CZ>^= 141.9693 " Area of " AOE = 81.7399 " Area of " ABODE = 297.5776 '^ Ans, 29 A.3E,1 F. 93. To determine the area, when the diagonals and per- pendiculars cannot be measured; as in the case of swamp or submerged meadow. TiULE.— Measure the hounding lines of the area, and then determine the diagonals hy outside tie-lines. SEC. II.] AREA OK COl^TENTS OF GROUND. 67 Thus, let ^^CZ)J5' repre- sent the polygon including the area, and suppose its sides to be measured. Pro- long BA to X, making Ax any exact part of AB — say one third — and also prolong FA to y, Ay being one third of AE; then, because of similar triangles, the measured length of xy will be one third of BK In like manner CB may be determined. Having the perimeter and the diagonals proceed as in Art. 92. II 94. To find the contents when the boundary is an irregular line. It frequently happens that a plot to be surveyed is bounded partly by an irregu- lar line. In such a case, one or more straight lines are surveyed, and offsets measured from these lines, as often as may be required to afford data for the compu- tation of the area and a true delineation of the boundary. In the case represented in the figure, the stream from A to C is the boundary. The station B is selected for convenience, as it is evident, if the line were run direct from A to C, the labor of taking the offsets would be much greater. It will be observed that the offsets are so measured as to indicate the abrupt bends in the boundary; and furthermore, so 68 ELEMEIJ^TS OF SURVEYING. [book II. that the areas thus cut off may be considered as Veing bounded by straight lines, without sensible error. When the boundary is a crooked stream that is easily crossed, it is often convenient to survey a line across the bend, as in the figure, and locate by offsets upon both sides of the line. In any case, the small areas to be computed are only trapezoids and triangles. Fig. 53. 95. To find the area of a piece of ground in the form of s circle. Rule. — Measure the radius AG; then multiply the square of the radius by 3.I4I6 (Mens., Art. 105). Fig. 54. To find the area of a circular piece of land, of which the diameter is 25 ch. Ans. 49 A. R. 14 P. 96. To find the contents of a piece of ground in the form of an ellipse. Rule. — Measure the semi-axes AE, CE. Then multi-ply them together, and ^ their product by 3.I4I6, SEC. II. 1 AREA OR COKTEKTS OF GROUND. S19 To jSnd the area of an elliptical piece of ground, of which the transverse axis is 16.08 ch., and the conjugate axis 9.72 ch, Ans, 12 A.1R4: F, Note 1. — The following is the manner of tracing an ellipse on the ground, when the two axes are known. From Cy one of the extremities of the conjugate axis as a centre, and AU, half the transverse axis, as a radius, describe the arc of a circle cutting AJS in the two points F and G ; these poins are called the foci of the ellipse. Then, take a tape, the length of which is equal to AB, and fasten the two ends, one at the focus F, the other at the focus G. Place a pin against the tape and move it around, keeping the tape tightly stretched : the extremity of the pin will trace the curve of the ellipse. Note 2. — In determining the contents of ground, in the examples which have been given, the linear dimensions have been taken in chains and decimals of a chain. If the linear dimensions were taken in terms of any other unit, they may be readily reduced to chains. For, a chain is equal to 4 rods, equal to 22 yards, equal to 66 feet. Hence, 1st. Rods may be reduced to chains and the decimal of a chain, ly dividing hy 4. 2d. Yards may be reduced to chains and the decimal of a chain, by dividing by 22, 3d. Feet may be reduced to chains and the decimal of a chain, by dividing by 66. Note 3. — If it is thought best to calculate the area, with- out reducing the linear dimensions to chains, the result can be reduced to acres* 70 |:lements of surveyixg. [book ii. 1st. By dividing it by 160, when it is in square rods (Art. 85). 2d. By dividing it by J^Jfi, when it is in square yards (Art. 87). 3d. By dividing it by 43560, when it is in square feet (Art. 87.) BOOK III. COMPASS SURVEYING. SECTION I. DEFIN ITIONS. 97. The Axis of the earth is the immovable diameter about which it revolves ; and the poles are the points in which the axis meets the surface. 98. Any plane passing through the axis of the earth is called a meridian plane ; and its intersection with the surface is called a meridian line, or simply a meridian. 99. All the meridians converge towards the poles, but they vary so little from parallelism, within the narrow limits of surveys made with the compass, that they may, without sensible error, be regarded as parallel straight lines. 100. K a magnetic needle be suspended freely, and allowed to settle to a state of rest, a vertical plane passed through its axis is called the plane of the magnetic meridian ; and its inter- section with the surface of the earth is called the magnetic meridian, or sometimes, a North and South line. A line per- pendicular to a North and South line, is called an East and West line. 101. A line traced, or measured on the ground, is called a Course; and the angle which this line makes with the magnetic 72 ELEMENTS OF SURVETIKG. [BOOK III meridian, passing through the point of beginning, is called the Bearing. Thus, if we start from the point A, and measure in the direction AB, the line AB is the course, and the angle NAB is the bearing. When the course, like AB, falls between the north and east points, and makes an angle of 46° with the meridian, the bearing is read, north FioTse. 46° east, and is written, K 46° E. When the course, like ACy falls between the north and west points, and makes with the meridian an angle of 30°, the bearing is read, north 30° west, and is written, N. 30° W. When the course, like AD, falls between the south and west points, and makes an angle with the meridian of 70°, the bearing is read, south 70° w^est, and is written, S. 70° W. When the course, like AF, falls between the south and east points, and makes with the meridian an angle of 70°, the bearing is read, south 70° east, and is written, S. 70° E. A course which runs due north, or due south, is designated by the letter N, or S ; and one which runs due east, or due west, by the letter E, or W. 102. If, after having passed over a course, the bearing is taken to the back station, this bearing is called the lack sight, or reverse hearing. 103. The perpendicular distance between the east and west lines, drawn through the extremities of a course is called the northing or southing, according as the course is run towards the north or south. This distance is also called the difference of latitude, or simply the latitude, because it shows the distance which one end of the course is north or south of the other. SEC. I.J DEFINITIONS. 73 Thus, in running the course from A -^ to B, AC is the difference of latitude, c north. ^^ -re E S Fig. 57. w- 104. The perpendicular distance between the meridians passing through the extremi- ties of a course, is called the departure of that course, and is east or west, according as the course lies on the east or west side of the meridian passing through the point of beginning. Thus, in running the course AB, CB is the departure, east. 105. It is found convenient, in explaining the rules for sur- veying with the compass, to attribute to the latitudes and departures the algebraic signs, -|- and — . We shall, therefore, consider every northing as affected with the sign +, and every southing as affected with the sign — . We shall also consider every easting as affected with the sign +, and every westing as affected with the sign — . 106. The meridian distance of a point is its perpendicular distance from any assumed meridian. Thus, if the distance be estimated from the meridian NS, BC will be the meridian dis- tance of the point B. 107. The meridian distance of a line, is the meridian dis- tance of its middle point, and is east or west, according as this point lies on the east or west side of the assumed meridian. Thus, FG drawn through the middle point of AB, is the meridian distance of the line AB. The sign -f- will always be given to the meridian distance of a point or line, when it lies on the east of the assumed meridian, and the sign — , when it lies on the west. 74 ELEMENTS OF SURVEYIIi^G. [book III. SECTION II. SURVEYOR'S COMPASS. 108. The Surveyor's Compass consists of a compass-box, DCE\ a magnetic needle; a brass plate, AB, from twelve to fourteen inches long; two plana sights, ^i^and BG ; two spirit- FiG. 58. levels placed at right angles to each other ; a brass head, K, to fit the compass to a stand, which is sometimes a tripod and some- times a single staff, called Jacob-staff, pointed with iron at the lower end so that it may be placed firmly in the ground. 109. The compass-box DCE is circular, and generally about six inches in diameter. At the centre is a small pin, on which the magnetic needle is poised. This needle, if allowed to turn freely around the point of support, will settle to a state of rest ; SEC. II. J SUE VETO R'S COMPASS. 75 the direction which it then indicates, is that of the magnetic meridian. In the interior of the compass-box, there is a graduated circle divided to degrees, and half degrees ; the degrees are numbered from the extremities of the diameter NS, both ways to 90°. The length of the magnetic needle is a little less than the diameter of the graduated circle, so that the needle can move freely around its centre, within the circle, and its positions be noted on the graduated arc. 110. The open sights, AF and BG, are placed at right angles to the plate AB, and fastened to it firmly by screws. The sights have fine slits cut through nearly their whole length, interrupted at intervals by large circular apertures through which the object sighted upon is more readily found, as shown in the figure. 111. The spirit-level is a small glass tube, slightly curved toward the middle, nearly filled with alcohol, leaving a bubble of air in the tube, and closed at both ends. When the level is m a truly horizontal position, the bubble of air rests in the middle of the tube ; when the level is not horizontal, the bubble seeks the more elevated end. When the two levels have each the bubble at the middle, each is truly horizontal and their plane is hori- zontal ; therefore, the plane of the graduated circle and the magnetic needle, which is parallel to the plane of the levels, is, also, horizontal. 112. The brass-head, by which the compass is attached to the staff, is furnished with a ball-and-socket joint to give a universal motion for purpose of leveling. Sometimes a tripod is used as a support instead of a staff, in which case a plumb-bob is attached immediately under the centre of the graduated circle for the purpose of accurately placing the instrument over any 76 ELEMENTS OF SURVEYING. [book IIL desired point. When a tripod is used, a tripod-head with leyel- ing screws (to be described under the transit), instead of a ball- and-socket joint, is often used for leveUng. In the description which follows, a ball-and-socket joint will be assumed. Fio. 59. 113. To find the bearing of any course by the compass. Let PQhe the course whose bearing is desired. Place the compass exactly over the point P, by inserting the staff in the ground at that point, or by the plumb-bob if a tripod is used. It is usual for a sur- veyor to keep the south end of the compass towards him and to read the bearings from the north end, and this position of the compass will be assumed in description, unless otherwise stated. Turn the sights toward the staff or object at Q. Bring the bubbles to the middle of the spirit-levels by the pressure of the hand on different parts of the plate. Look through the smaU slit in the sight next the person, and turn the compass till the small slit in the sight opposite bisects the object at Q, being careful to keep the bubbles at the middle of the levels. Let the needle come to rest and take the reading indicated by the north end of the needle — it will be the angle NPQ, the bearing. When the needle is at rest, it lies in the magnetic meridian. The line of sights lies in the direction of the course ; when this line lies east of the meridian, the bearing is east ; when it lies west of the meridian, the bearing is west. When the bearing is east, the needle lies to the west of the line of sights, and the reverse ; hence, to facilitate the reading the E and W letters on the face of the compass are reversed from their natural position. In order to prevent a merely mechanical reading, the cover of SEC. II.] surveyor's compass. 77 the compass-box should be unscrewed and a circle of paper should be fitted into the bottom of the box so as to conceal the letters ; the student will then learn to read from the needle alone, the north end of which bears a distinguishing mark or color. It is only by such practice that blundering readings can be avoided. 114. To find by the compass the angle subtended at any point by two objectp, take the bearing of the course to each object, as explained in the last article. The question then is to find the angle between any two courses, when their bearings are known, which may be done as follows : Let NS be a meridian passing through A. Let AB, AC, AH, AD, and AF, be five courses running from A, We readily deduce the following PRINCIPLES, ^(7 is N 26° W AH'\^ ^ 65° W When the meridional letters are alike, and those of departure > also alike, fhe difference of the hearings is the angle lettveen the courses. AB is N 46° E ^C7is N26° W CAB = 72° When the meridional letters are alike, and those of departure unlike, the sum of the tearing s is the angle between the courses. 78 ELEMEKTS OF SURVEYING. [BOOK UL When the meridional letters ^C is N 26° W ') are unlike, and those of departure AD is S 66° W V alike, the angle letween the courses CAD = 180° -92° = 88° J is equal to 180^, minus the sum of the bearings. When the meridional letters are unlike, and those of departure also unlike, the angle hettveen the CAF= 180°— 40° = 140° J courses is equal to 180°, minus the difference of the bearings. ^CisN26° W AF is S 66° E Note. — The above principles are deduced, under the sup- position that the two courses are both run from the same angular point. Hence, if it be required to apply these rules to two courses run in the ordinary way, as we go around the field, the bearmg of one of them must be reversed before the calcu- lation for the angle is made. EXAMPLES. 1. The bearings of two courses, from the same point, are IS" 37° E, and S 85° W; what is the angle included between them ? Ans. 132°. 2. The bearings of two adjacent courses, in going round a piece of land, are ]^ 39° W, and S 48° W; what is the angle included between them ? A?is. 87°. 3. The bearings of two adjacent courses, in going round a piece of land, are S 85° W, and N 69° W ; what is the angle included between them? Ans. 154°. 4. The bearings of two adjacent courses, in going round a piece of land, are :N' 55° 30' E, and S 69° 20' E ; what is the anoxic included between them ? Ans, 124° 50'. SEC. III.] WOBK ON THE FIELD. 79 SECTION III. WORK ON THE FIELD. 115. When a piece of ground is to be surveyed, we begin at some prominent corner of the field and go entirely around the land, measuring the lengths of the bounding lines with the chain, and taking their bearings with the compass. It is not material whether the ground be kept on the right hand or on the left, and all the rules deduced for one of the cases, are equally applicable to the other. To preserve uniformity, how- ever, in the language of the rules, we shall suppose the land to be always kept on the right hand of the surveyor. Let A BCD be a piece of ground to be surveyed, A the point where the work is to be begun, and NS a meridian. On a sheet of paper, rule a single column, ^ inch wide, down the middle of the left hand page of the note looh, as in the example. I D C B A A 10.00 A 7.60 A 9.20 A 10.40 A (N. 45 E.) S. 45i W. (N. 35} W.) S. 36 E. (S. 62 W.) K 62 E. (S. 31iE.) N. 31i W. 80 ELEME>?^TS OF SURYEYII^G. [BOOK IIL Place the compass at A, and take the bearing to B, which is FAB ; suppose this angle has been found to be 31^°. The bearing from A to B is then N. 31J° W. Enter this bearing in the field notes at the right of station A. Then measure the distance from A to B, which we will suppose to be 10 ch. 40 1., and insert that distance in the column, above the station mark. We next take the bearing from B to C, 'N. 62° E., and then measure the distance BC = 9 ch. 20 1., both of which we insert in the notes as abore. At station C we take the bearing to D, S. 36° E., and then measure the distance CD = 7 ch. 60 1., and place them in the notes. At D we take the bearing to A, S. 45J° W., and measure the distance DA = 10 ch. We shall then have made all the measurements on the field which are necessary to determine the contents of the ground. 116. The reverse-bearing or back-sight, from B to A, is the angle ABH \ and since the meridians NS and EG are parallel, this angle is equal to the bearing NAB. The reverse-bearing is, therefore, S. 31 J° E, and should he entered in the notes in parenthesis opposite station B, as in the example. The reverse-bearing from C, is S. 62° W.; that is, it is the angle ICB = GBC. And generally, A reverse-hearing, or hack-sight, should always equal the for- ward-hearing, and differ from it only in hoth of the letters hy which it is designated. 117. In taking the bearings with the compass, there are two sources of error. 1st. The inaccuracy of the observations ; 2d. Local attractions, or the derangement which the needle expe- riences when brought into the vicinity of iron-ore beds, or any ferruginous substances. SEC. III.J WORK 01^ THE FIELD. 81 To guard against these sources of error, the reverse-bearing should be taken at every station ; if this and the forward- bearing are of the same value, the work is probably right ; but if they differ considerably, they should both be taken again. Electricity is a fruitful cause of annoyance in compass work. In cold, dry weather, any friction upon the glass cover of the needle- box, even that of a cold, dry wind, charges it with elec- tricity and the needle no longer traverses freely. To dissipate the charge touch the plate with a moistened finger, or breathe strongly upon it. If the surveyor uses a pocket lens to read the needle, he must see to it that no iron or steel screws are used in the casing; nickel-plated mountings are not admissible; hard rubber mountings are very troublesome, as they often become highly charged, and will drag the needle through 90°, or even 180°. Brass, or German silver, are the most satisfactory mountings. 118. In passing over the course AB, the northing is found to be HB, and the departure, which is west, is represented by AJff. Of the course BC, the northing is expressed by BG, and the departure, which is east, by ivrj GC. Of the course CD, the southing is expressed by CI, and the departure, which is east, by CF. Of the course DA, the southing is expressed by KA, and the departure, which is west, by DK. It is seen from the figure, that the sum of the northings is equal to HB + BG = EG ; and that the sum of the southings is equal to CI+EA = PA = EG ; hence, the sum of the northings is equal to the sum of the southings. If we consider the departures, it is apparent that the sum Gr- Ji 11 82 ELEMENTS OF SUEVEYIXG. [BOOK III of the eastings is equal to GC-\-CF =^ GF; and that the sum of the westings is equal to AH-\-DK ^ GF\ hence, the sum of the eastings is equal to the sum of the ivestings. We there- fore see, that when the survey is correct, the sum of the northings will he equal to the sum of the southvngs, and the sum of the eastings to the sum of the westings. It would, indeed, appear plain, even without a rigorous demonstration, that after having gone entirely round a piece of land, the distance passed over in the direction due north must be equal to that passed over in the direction due south ; and that the distance passed over in the direction due east must be equal to that passed over in the direction due west. 119. The boundaries of a field are generally occupied by fences, and frequently also by a border of shrubbery, so that chaining along the true boundary is impossible. In such cases, it becomes necessary to measure ^, ^^^^ an offset at each end of the course (and at right angles to it), and of suflBcient length to clear the obstructions ; the measurement is then made between these temporary stations. It is evident that the bearing and length of mn, the offset-course, are the same as those of MN. When such offset-courses are necessary for several successive courses, errors are likely to be committed, unless the surveyor is careful to make new offsets for each course. p^^ ^ To survey houndaries LM, MN, NO, &c., along which the chaining cannot he done, as in Fig. 64, proceed thus : Set the instrument at some point, A for example, far enough from LM to clear all obstacles, usually five or six feet; next SEC. III.] WOEK Oiq^ THE FIELD. 83 place a pole at By make Bl)=^Aa by measurement, laying off the perpendiculars by the eye alone ; now sight B and note the reading. While the compass is sighted on B, look across two notches cut in the rim of the compass-box in a line perpendicular to the line of the sights and passing through the needle pivot, and fix a plumb-line upon LM, as at « ; if the distance Aa, when re-measured, should differ much from that used in setting off Bi, the offsets must be corrected ; now measure La (a few links only), and, transferring the link held at a to A, continue the measurement along AB to B ; having set up the compass at B, sight to A for the back- sight, or reverse-bearing, and at the same time fix a plumb-line at & by "cross-sighting" as before ; transfer the link held at B to J), and continue to M, and enter the length of LM, thus obtained, in the notes. Without moving the instrument, measure the perpendicular Be, set a staff at D and make Dd, the perpendicular to MN pro- longed, equal to Be ; sight to D for the bearing of JfiV", cross- sight to fix c, and do the same at D to fix d', measure Mc, BD, dN; Mc-^BD—dN =z MJSf, which enter in the notes. In like manner, iVO = iVe + Z>il4-^0. 120. It has been customary, since the first settlement of this country, to use the compass in all land surveys, so that the description of lands, in purchase and sale, and by which they are recognized in the courts, involves the length and bearing of each straight line of the boundary. The method, therefore, is, at present, a necessary one. The errors to which the compass is liable are so numerous and so variable, even in the same instrument, that a change 84 ELEMENTS OF SURVEYING. [BOOK III. of practice is very desirable. Many surveyors, to insure a higher degree of accuracy, measure the angles of a field with the transit, and then, having determined the bearing of one side with sufficient accuracy, calculate the others by a method to be shown in a subsequent article. 121. In surveys of large areas, the surveying party should consist of at least four persons — viz., a compass-man, a flag- man, and two chain-men. In smaller areas, the work is gen- erally performed by the surveyor and one assistant ; the surveyor serving alternately as compass-man and hind-chainman, and the assistant as flag-man and fore-chainman. 122. In recording the notes of the survey, the advantage of beginning at the bottom of the page is this : that when standing on the line to be surveyed, and looking in the direction we propose to go, the column in the book lies before us just as the line does, and all measurements made to the right or left of the line are recorded at the right or left of the column. In surveys where many auxiliary notes are taken, a diagram is an important aid to a ready interpretation of the other notes. GENERAL EXAMPLE. . 123. To explain the method, in full, of making a compass survey and recording the notes, we will take an example of a farm, in which, in addition to the usual survey of the boundary, such other measurements are made as to enable us to make a correct map of the whole. Page 87 represents a farm to be surveyed, and page 86, the notes which are made, in the operations on the field. Beginning with the corner marked A, the bearing of the line AB is taken. In most cases, offsets from both A and B would be taken, in order that the survey may be clear of the fence, but such offsets are not recorded ; the surveyor must keep SEC. III.] WORK ON" THE FIELD. 85 in mind that it is the boundary of the field that is surveyed, and any device by which this is accomplished is no part of his notes. The record of the bearing of the first course is entered at the right of the column (page 86), while the letter designating the station, is placed to the left. The symbol A, which signifies station, is placed in the column, between the letter and the bearing, for each angle of the farm. In chaining the first course, the intersection of the line with any objects worthy of notice is recorded. The first record is of the road leading to the quarry. As it is an unimportant road, a single measurement of the distance on the course to its centre is sufficient to locate it. The distance is 4.30 chains. At 11.30 and 12.35 the sides of the turnpike are intersected. The bearing of the road, at this point, is also carefully taken and recorded. The intersections of the garden fence and of the brook are also noted (17.40) and (18.10) ; and these, with the entire length of the course (31.95), close the record of this line. At B, the back-sight upon A is first taken, and entered in the notes opposite the ^-station mark, as directed in Art. 116. The entry is omitted in the example to avoid croivding and confusing the notes. Then the bearing of BC is, taken. Next the bearing of the northernmost chimney of the farm-house is taken {N. 73° E.). Such bearings serve two purposes. They aid in the location of the objects observed, upon the map, and serve also, in case of errors, to aid in detecting their location. In general, in surveying large or small areas, some prominent point or points, within the boundary, should be selected, and their bearings, from different angles, carefully noted. K. I F JE D £ StaJttanA. WS7 A /7.S9 f/.3Q ^\ 6.45 S.40 A 7.65 A Id. SI A-\- ^.9 J J 3. TO /2.90 8.80 A'"" 8.S9 A 22.89 i9.50 J5.00 /3.00 40.00 7.70 6.00 A J. 40 A., 34.95' /3.70 /8.W 47.40 /^.3S //.30 4.30 A fo Stcdi/nvA. S30°W \FeTiceJT73''W SZ7°E S7/''£ SM^SO'E '■••-t%^ Fenc&S6d^4JE .^-'iS ,--' OAKTREE W£ASrS/0£ OFWHEATFIEUi 2.00 ZOO 7^30°35^ S87'>5'W Fenc&W^°E WS^E GarCUrvFeru^ TURNPIKE J^ySZS ir68''j5>r Fig. 65. -<^^ FARMHOUSE MAP OF FARM. Fig. 66. 88 ELEMEi^TS OF SURVEYIN"G. [BOOK III. The chimney of the farm-house and the oak-tree in the corner of the wheat-field, are thus employed in this survey. At C, the corner of the field, is in the centre of the brook, and from this point to D, the brook is the boundary. A straight line is run between the stations, and offsets are measured to each bend of the brook. It is necessary, in such a case, for the chainmen to exercise unusual care in keeping in the line between the stations, other- wise the lengths of the offsets cannot be correctly measured. At E, the bearing of the oak-tree is taken (IST. 73|-° E.). On the course between E and F, a marsh is encountered, which the chainmen pass, by an offset course. At F, another bearing is taken of the oak-tree (S. 44J° E.). At G, the bearing of the farm-house chimney is noted (S. 26° E.). At G and H the bearings of the division-fences are taken. On the course from H to /, the turnpike is again crossed ; the intersection of both sides, together with the bearing, are carefully noted. From / to K, the intersection and bearing of the fence between the potato and the wheat field, are recorded. The course from ^ to ^ closes the survey. To locate the buildings about the farm-house, a few measure- ments would be necessary ; but tbey may begin with the point already located by the bearings taken to the chimney nearest the north end of the house. The dimensions of the buildings, their distances apart, and the direction of one side of each afford sufficient data for locating them, correctly, upon the map. Note. — The advantage of the compass over other instruments with which angles are measured, lies chiefly in this : that the Bearing of a course may be measured at any point on the line. SEC. III.] WORK Ol^f THE FIELD. 89 When the angle between adjacent sides is taken with the Transit, the work can only be done at the corners of the field ; and when, as freqnently happens, a hill intervenes between two consecutive stations, it becomes necessary to locate a point on the hill, in the true line, and then return to the corner to measure the angle ; whereas, when the compass is employed, the establishment of the intermediate point on the hill affords the means of taking the proper bearing without going to the angle. Furthermore, the bearings may be measured with the compass, by placing it at the alternate stations only. The disadvantage of these rapid methods is that there is no check upon the needle readings, as there is in back-sighting. 124. To Correct Local Attraction.— Suppose that bearings and reverse-bearings have agreed for several stations, and that then a back-sight differs from the preceding fore-sight ; it may be concluded that local attraction affects the needle at the last station only. Suppose the fore-sight at P to have been N. 30 E., and the back-sight at S to be S. 28 W. ; suppose the fore-sight at S to be N. 28 W. ; what is the cor- rect fore-sight at iS ? From Fig. 67 we see that the needle at S, instead of being parallel to the needle at P (as indicated by the dotted needle) as it should be, has been deflected by local attraction (as shown by the full needle), so that its south end is moved 2° to the W., and its north end 2° to the E. of its proper position. Hence the fore-sight at S, being taken from the needle in a false position, is 2° too large, and should be only K 24 W. By similar reasoning, aided by a mental picture of the case, any bearing may be corrected. Fig. 67. 90 ELEMENTS OF SURVEYING. [book III. If 110 buck-sight and its fore-sight agree, then take the mean of the back-sight and fore-sight which differ least ; or seek some ground at a distance, free from local attraction, and then work into the plot to be surveyed, correcting each fore-sight on the way. SECTION IV. AREA OR CONTENTS OF GROUND. Having explained the necessary operations on the field, we shall now proceed to show the manner of computing the contents of ground. 125. The Traverse Table and its Uses. — This table shows the latitude and departure coiTesponding to bearings that are expressed in degrees and quarters of a degree, from to 90°, and for every course from 1 to 100, computed to two places of decimals. The following is the method of deducing the formulas for computing a traverse table ; by means of these formulas and a table of natural sines, the latitude and departure of a course may be computed to any desirable degree of accuracy. Let AD be any course, and NAD its bearing; then AE is the latitude, and ED the departure of the course AD, to the bear- ing NAD. From Art. 36, we have (using nat. sines) AD : AE w 1 : cos NAD, Fia. 68. Whence, lat. = course x cosine of bearing. SEC. lY.J AREA OR CONTENTS OF GROUND. 91 And from Art. 32, we have (since sine AED = 1), AD \ ED w I \ sine NAD. Whence, dep. = course x sine of bearing. We have then the following practical rule for computing the latitude and departure of any conrse. Look in a table of natural sines for the cosine and sine of the hearing. Multiply each hy the length of the course, and the first product will he the latitude, and the second will he the departure of the given course. EX A M^LES. 1. The bearing is 65° 39', the course 69.41 chains: what is the latitude, and what the departure ? Natural cosine of 65° 39' ...... .41231 Length of the course ....... 69.41 Product, which is the Dif. of Lat. . . 28.618437 1 Natural sine of 65° 39' 91104 Length of the course . 69.41 Product, which is the Departure . . . 63.2352864 2. The bearing is 75° 47', the course 89.75 chains ; what is the latitude, and what the departure? Natural cosine of 75° 47' 24559 Length of course 89.75 Product, which is the Dif. of Lat. . . 22.0417025 Natural sine of 75° 47 96937 Length of course 89.75 Product, which is the Departure . . . 87.0009575 In this manner, the traverse table given at the end of the book, has been computed. When the bearing is given in degrees 92 ELEMEN^TS OF SUEVEYIN'G. [BOOK III. and quarters of a degree, and the difference of latitude and departure are required to only two places of decimals, they may be taken directly from the trayerse table. When the bearing is less than 45°, the angle will be found at the top of the page ; when greater, at the bottom. When the distance is less than 50, it will be found in the column *' distance," on the left-hand page ; when greater than 50, in the corresponding column of the right-hand page. \T_ H .^ F E 126. The latitudes or departures of -^ courses of different lengths, but which have c the same bearing, are proportional to the G lengths of the courses. Thus, in the figure, ^^ the latitudes AG, AC, or the departures GF, CB, are to each other as the courses ^ AF, AB. Fig. 69. Therefore, when the distance is greater than 100, it may be divided by any number which will give an exact quotient less than 100 ; then the latitude and departure of the quotient being found and multiplied by the divisor, the products will be the latitude and departure of the whole course. It is also plain, that the latitude or departure of two or more courses, having the same bearing, is equal to the sum of the latitudes or depar- tures of the courses taken separately. It is always better to obtain the lat. and dep. by addition than by multiplication ; thus, if we sought the lat. and dep. of a course of 190 ft., bearing 36°, they would be found by multiplica- tion thus, lat. 19 X 10 = 15.37 x 10 = 153.70 dep. 19 X 10 = 11.17 X 10 = 111.70 giving results containing ten times the error of lat. and dep. of 19, as given in the table. By addition we should have I SEC. IV.] AREA OR CONTENTS OF GROUND. 93 lat. lOO + lat. 90 = 80.90 + 72.81 = 153.71 dep. 1^0 + dep. 90 = 58.78 + 52.90 = 111.68 which are closer approximations than the former. Hence, we should always make our multipliers as small as possible. EXAM PLE S. 1. To find the latitude and departure of 614, to bearing 29|-°. Latitude of 100 x 6 = 522.24 Latitude of _14 = 12.18 Latitude of 614 534.42 Departure of 100x6 = 295.44 Departure of 14 = 6.89 Departure of 614 = 302.33 2. To find the latitude and departure for the bearing 62J°, aud the course 7855 chains. Latitude for 7800 . 3602.00 Latitude for 55 . 25.40 Latitude for 7855 . 3627.40 Departure for 7800 . 6919.00 Departure for 55 . 48.79 Departure for 7855 . 6967.79 Note. — When the distances are expressed in whole numbers and decimals, the manner of finding the latitudes and departures is still the same, except in pointing off the places for decimals ; but this is not difficult, when it is remembered that the column of distances in the table may be regarded as decimals, by simply removing the decimal point to the left in the other columns. 3. To find the latitude and departure for the bearing 47f°, and the course 37.57. Latitude for 37.00 Latitude for .57 Latitude for 37.57 24.88 .38 25.26 Departure for 37.00 Departure for .57 Departure for 37.57 27.39 .42 27.81 94 ELEMENTS OF SURVEYIXG. [book III. 127. Balancing the "Work. — The field-notes having been completed, rule a new table, as below. Then find, from the traverse table, the latitude and departure of each course, and enter them in the proper columns opposite the station. Then add the column of northings, and also the column of southings; the two sums should be equal to each other. If they are not, subtract the less from the greater ; the remainder is called the error in latitude. Find the error in departure in the same way. This error for latitude or departure must be distributed among the latitudes or departures of all the courses, in propor- tion to the length of each course, observing to add the correction, when applied to the deficient column, and to subtract it, when applied to the other. This may be illustrated by the example of (Art. 115). stations. Bearings. Dis. Dif. Lat. Dep. Balance. :n". S. E. + W. Lat. Dep. A N. 311° w. 10.40 8.87 .... .... 5.43 + 8.86 -5.44 B K62° E. 9.20 4.32 8.13 + 4.31 + 8.12 C S. 36° E. 7.60 6.15 4.47 —6.15 + 4.46 D S. 451° w. 10. 7.01 7.13 -7.02 -7.14 Sum of courses, 37.20 13.19 13.16 12.60 12.56 13.16 12.56 Error in latitude .03 .04 Error in departure. The error in latitude, 3 links, is to be distributed among the northings and southings, in proportion to the lengths of the courses ; a part to be added to the southings, and the remaining part subtracted from the northings. The error in departure is similarly distributed among the eastings and westings. For this, SEC. IV.] AREA OR CONTEi^^TS OF GROUND. 95 two new columns are formed, called, the balanced latitudes and departures ; and to these columns the latitudes and departures are transferred, after the corrections have been made; the north latitudes being marked -f , and the south latitudes — , in order to distinguish them readily, and also, for convenience in the calculations which follow. The error of .03 in the latitudes is distributed among the latitudes, by subtracting 1 link from each of the northings of courses A and B, and adding 1 link to the southing of course D. This produces a balance. Of the error of 4 links in the departures, 1 link is added to each of the departures west, and 1 link subtracted from each of the departures east. This produces a balance. Note. — When a knowledge of the conditions under which the survey was made, enables us to determine that errors were more likely to occur at certain points, it is best to apply the corrections to those courses where it seems probable the errors were made. 128. The limit of error to be allowed depends, of course, upon the importance of the survey. In ordinary farming districts, the error should be as small as 1 link to 5 or 10 chains of perimeter. The "error of the survey" should be considered as the length of the line necessary to dose tlie boundary, and is equal to the square root of the sum of the squares of the errors of latitude and departure. Thus, in the above example, the error of the survey is 5 links. The perimeter being 37.20 chains, the error is about 1 link to 7.45 chains, or -^ of the perimeter. 129. It will be well to bear in mind the fact, that if the error in the perimeter has been made in one course only, and distributed, by the ordinary methods of balancing, among all the 96 ELEMENTS OF SURVEYING. [BOOK IIL courses, the error in area will be larger than the error in perimeter. 130. When the error is so large that a re-survey becomes necessary, the halancing should be carefully re-examined. In many cases, the location of the error may be determined by inspection of the computation, and a portion of the labor of a re-survey thereby saved. This refers more particularly to those cases where the error is one of chaining, and is mostly in one course. Errors of this kind occur sometimes with experienced chainmen, who draw the chain properly between the courses, but make occasionally an error in counting the fractional part of a chain at the end of a course. In such cases, the location of the error may be detected by observing, first what columns contain errors, and secondly the ratio of the errors of Latitude and Depari:ure. When the eiTor in the survey has been a single one, of dis- tance onlv, then the ratio between the errors of Latitude and Departure must be the same as the ratio between the Latitude and Departure of the course to be corrected. If the errors be in northings and westings, then the courses running either North and West, or South and East, should be examined. 131. The surveyor should take every possible precaution igainst en'ors in the bearings. This is accomplished by back- sighting, taking bearings of some one object from several sta- tions, and also by taking bearings of stations across the field. These precautions wiU give, in general,, sufficient data for the detection of an error in hearing ; for, by mapping the survey and drawing the lines to indicate the extra bearings, the error is revealed by the failure of the Hnes to meet at a common point. SEC. IV.] AREA OE CONTENTS OF GROUND. 97 132. One source of error, m large surveys with the compass, is frequently overlooked. This is the diurnal variation ; there is sometimes as much as 15 minutes variation during the day- light hours. Errors from this source can only be avoided by testing the compass, at intervals of two or three hours, by taking the bearing of the same line. 133. There is one kind of error frequently made in reading the compass when the bearing is nearly east or west. The error arises from reading North for South, or the reverse. If the survey is otherwise correct, the error in latitude is just twice the latitude of the course containing the error. DOUBLE MERIDIAN DISTANCES 134. After the work has been balanced, the next thing to be done is to calculate the double meridian distance of each course. For this purpose, any meridian line may be assumed. It is, however, most convenient to assume that meridian which passes through the most easterly or westerly station of the survey ; and these two stations are readily determined by inspecting the field-notes. Having chosen the meridian, let the station through which it passes be called the principal station, and the course which begins at this point, the first course. Care, however, must he taken not to confound this luith the cotcrse which begins at station 1, and which is the first course that is entered in the field-notes. It has already been remarked (Art. 105), that all departures in the direction east are considered as plus, and all departures in the direction west as minus. It 98 ELEMENTS OF SURVEYING. [book III M C Fio. 70. 135. To deduce a rule for finding the double meridian dis- tance of any course. Let SN be the assumed meridian. Let BG represent any course, and AB the preceding course ; also, let D and B be their middle points. Draw BH, BG, and CM, perpendicular to the assumed meridian NS. Draw also Al, EK, and BB, par- allel to N8. Then '2DG is the double meridian distance of the course BC, and "^EH = 2KG, is the double meridian distance of the course AB. Now, 2DG = 2GK+ 2KB + 2BB; but 2KB = IB is the departure of the course AB, and 2BD = MC is the de- parture of the course BC ; consequently, 2GD = 2GK + IB + MC\ hence, the double meridian distance of a course is equal to the double meridian distance of the preceding course, plus the departure of that course, plus the departure of the course itself : if there is no preceding course, the first two terms become zero. We therefore have the following EuLE. — L The double inej^idian distance of the first course is equal to its depaHure ; II. Tlve double meridian distance of the second course is equal to the double meridian distance of the first course, ■plus its departure, plus the departure of the second course ; III. Tixe double meridian distance of any course is equal to the double meridian distarice of the preceding course ; plus its departure, plus the departure of the course itself. Note.— It should be recollected that plus is here used in its algebraic sense, and that, when the double meridian distance SEC. IV.] AREA OR CONTEi^^TS OF GROUKD. 99 of a course, and the departure which is to be added to it, are of different names, that is, one east and the other west, they will liave contrary algebraic signs ; hence, their algebraic sum will be expressed by their numerical difference, with the sign of the greater prefixed. If the assumed meridian cuts the enclosure, the double meridian distances estimated to the east are plus, and those on the west must be taken with the minus sign. The double meridian distance of the last course should be equal to the departure of that course. A verification of the work is therefore obtained, by comparing this double meridian distance with that departure. AREA. 136. Let us resume the example of Art. 115. We will first write the differences of latitude and the double meridian distances of the courses, in the following table: Fig. 71. Stations. Dif. of Latitude. D. M. D. Area. + Area. A 4- cB + %l)a ^cAB B + Bs + ^p 2BsC C -Dy -h 2>iA 2ms CD D -Df + 2ed 2cmDA It is evident, that cB multiplied by 2ba, or cA, will give double the area of the triangle cAB. But cB and ba are both L.ofC. 100 ELEMENTS OF SURVEYIITG. [BOOK III. plus ; hence, the product will be plus, and must be put in the column of plus areas. Double the area of the triangle BsG, is equal to Bs multiplied by Hqp, wliich product is also plus. The area of the trapezoid msCD is equal to yD, or ms, multi- plied by nh (Geom., Bk. IV, Prop. VII) ; hence, double the area is equal to yD into 2nh, But since yD (being a southing) is minus, and 2nh plus, it follows that the product will be negative; hence, it must be placed in the column of negative areas. Double the area of the trapezoid cADm, is equal to Df, or mc, multiplied by 2de ; but, since Df is negative and 2de positive, the product will be negative. It is now evident that the difference between the two columns is equal to twice the contents of the figure ABOD ; and since the same may be shown for any other figure, we have, for finding the areas, the following general EuLE. — I. Multiply the double meridian distance of each course by its northing or southing, observing that like signs in the multiplicand and multiplier give plus in the product, and that unliJce signs give minus in the product. II. Place all the products luhich have a plus sign in one column, and all the products which have a minus sign in another. III. Add the columns separately and take the dif- ference of their sums ; this difference will be double the area of the laitd. Note. — When offsets are measured, the figures of which the offsets are bounding lines are, practically, triangles and trapezoids, and the areas of these are to be separately computed, and added to or subtracted from, as the case may be, the area obtained by the foregoing rule. SEC. IV.] AEEA OR CONTENTS OF GROUND. 101 137. We will now make the calculations of this example, in numbers, from the field-notes, which are the following : stations. Bearings. Distances. Diff. Lat. Dep. D. M. D. A N 31J° W 10.40 + 8.86 —5.44 + 18.02 — 7.14 -5.44 + 5.44 *B N62° E 9.20 + 4.31 + 8.12 + 8.12 C S 36° E 7.60 -6.15 + 4.46 + 8.12 + 4.46 + 20.70 D S 45i° W 10. — 7.02 — 7.14 4.46 -7.14 + 18.02 We see, from inspecting the notes, that B is the most westerly, and D the most easterly station. Either of them may, therefore, be taken for the principal station. Let us assume B for the principal station, through which the assumed meridian passes, and distinguish it by a star, thus *. Having done so, we enter the departure 8.12 in the column of double meridian distances, which is the double meridian distance of the course from B to C. The double meridian distances of the other courses are calculated according to the rule ; and as the last, which is that of the course from A to B, is equal to the departure of that course, the work is known to be right. Let us now form a new table, which will complete the arithmetical part of the work. 102 ELEMENTS OF SUEVETIXG. [book in. Sta. Bearings. Dist. Dif. Lat. D. M. D. Area. + Area. A X 31^° W 10.40 + 8.86 + 5.44 48.1984 *B N62° E 9.20 + 4.31 + 8.12 34.9972 C S 36° E 9.60 — 6.15 + 20.70 127.3050 D S 45i° W 10. — 7.02 + 18.02 126.5004 83.1956 253.8054 Area in sq. ch. A71S. SA. 2 B. 4.88 P. 83.1956 2 ) 170.6098 85.3049 PLOTTING. 138. To make a plot of the ground, draw any line, as WS, to represent the meridian passing through the principal station ; and on this line take any point, as B, to represent that station. Beginning at B, take the algebraic sum of the balanced latitudes and, also, of the balanced departures of the stations following. These sums are, respec- tiyely, the total latitudes and total departures (or longitudes, see Art. 148, Note) of the several courses with respect to the principal station, B. Tabulate the results (see example, p. 94) as shown in the margin. Through B, draw BX (Fig. 72) per- pendicular to NS. On XS lay off {upward since 4.31 is plus) Bs e;^ual to 4.31 units of the scale on which the Fig. 72. sta. *B C D A Total lat. from B. Total dep. from B. 0.00 4.31 -1.84 -8.86 0.00 8.12 12.58 5.44 SEC. IV.] AREA OR CONTENTS GROUND. 103 plot is to be made; and on BX lay off (to the right since 8.12 is plus) Bf equal to 8.12 units of the scale. From s, as a centre, with Bf, or 8.12, as a radius, describe an arc ; and from /, as a centre, with Bs, or 4.31, as a radius, describe an arc ; the intersection of these arcs will be the position on the paper of Station 0. On N8 lay off [downiuard since 1.84 is minus) Br equal to 1.84 units ; and on BX lay off to the right Bq equal to 12.58 units. From r, as a centre, with Bq, or 12.58, as a radius, describe an arc ; and from q, as a centre, with Br, or — 1.84, as a radius, describe an arc; the intersection of these arcs will be the position on the paper of Station D. Determine, in like manner, the position on the paper of each successive station ; connect the points so determined by straight lines, and a complete plot of the ground will be obtained. As a check, the distances BC, CD, &c., should be measured on the plot. 139. It is convenient, but not necessary, to take the most easterly or most westerly station as the principal station. It is simply to be remembered that north latitudes and east departures Sireplus, and south latitudes and west departures are minus ; that total latitudes and total departures are algebraic sums ; that plus latitudes are to be laid off on the meridian, from the principal station, upward, and minus latitudes dotumvard ; that plus departures are to be laid off on the east and west line, from the principal station, to the right, and minus depar- tures to the left. There are, of course, other metliods of plotting; but the one here given is, perhaps, the best m ease and accuracy of execution. If the latitudes and departures have been correctly balanced, and the drawing carefully made, the survey will 'certainly "close." 104 ELEMENTS OF SURVEYING. [book III. 140. -EXAMPLES. 1. It is required to determine the contents and plot of a piece of land, of which the following are the field-notes, viz.: Sta- tions. A Bearings. Dist. Dif. Lat. Dep. Balanced. D.M.D. + Area. + Area. + s £ + w Lat. Dep. N46rW 20.76 14.29 15.06 + 14.30 -15.04 15.04 215.0720 B N51f°E 13.80 8.54 10.84 + 8.55 + 10.86 10.86 92.8530 C E 21.35 21.35 + 21.37 43.09 D S56° E 27.60 15.44 22.88 —15.43 + 22.90 87.36 1347.9648 E S 331° W 18.80 15.72 10.31 —15.71 —10.29 99.97 1570.5287 F N 741° W, 30.95 8.27 29.83 + 8.29 -29.80 59.88 496.4062 31.10 31.16 55.07 55.20 804.3312 2918.4935 31.10 55.07 804.3312 Error . . .06 .13 Error. 2)2114.1623 Am. 105 ^. 2 i?. 33 P. 1057.08115 PLOT OF THE GROUND. Fig. 7a KoTE. — When the bearing is due East or due West, the error in latitude is nothing, and the corrections for latitude must be distributed among the other courses. So, when the bearing is due North or due South, the error in departure is nothing, and the error in departure must be distributed among the other courses. In the examples for practice, we have not been as careful to have as close balances as must be had in actual work on the field. SEC. IV.] AREA OR CONTENTS OF GROUND. 105 2. Eequired the contents and plot of a piece of land, of which the following are the field-notes. Stations. Bearings. Distances. A S 34° W 3.95 ch. B S 4.60 C S 36i° E 8.14 D N 59J° E 3.72 E ]Sr25° E 6.24 F N16° W 3.50 G N65° W 8.20 Ans. 10 A. E. 5 P. 3. Required the contents and plot of a piece of land, from the following field-notes. stations. Bearings. Distances. A S 40° W 70 rods. B ]Sr45° w 89 C N36°E 125 D N 54 E S 81° E 186 F S 8° W 137 G W 130 Ans. 207 A. 3 E. 33 P. 106 ELEMENTS OF SURVEYING. [book III. 4. Required the contents and plot of a piece of land, from the following notes. Stations. Bearings. Distances. A S 40i° E 31.80 ch. B N54° E 2.08 C N 29J° E 2.21 D N 28i° E 35.35 E N57° W 21.10 F S 47° W 31.30 Ans, 92 A. 3 R. 32 P. 5. Required the area of a survey, of which the following are the field-notes. Stations. Bearings. Distances. A N'42°E 5.00 ch. B East. 4.00 C N 9^E 4 00 D S 69° E 5.56 E S 36° E 7.00 F S 42° W 4.00 G S 75° W 10.00 H N39° W 7.50 If, in this example, we assume A as the principal station, the double meridian distances will all be plus, and the posi- tive area will exceed the negative. In balancing, we shall find the error in southing to be .28 ch., and in westing .22 ch. The area is 13 A. B. 11 F, It should, however, be remarked, that in all the examples the answers may be slightly varied by distributing the corrections. SEC. IV.] AREA OR CONTENTS OF GROUND. 107 6. What is the area of a survey of which the foUowiug are the field-notes ? Make the plot. Stations. Bearings. Distances. A N 75° 00' E 54.8 rods. B N 20° 30' E 41.2 C East. 64.8 D S 33° 30' W 141.2 E S 76° 00' W 64.0 F North. 36.0 G S 84° 00' W 46.4 H N 53° 15' W 46.4 I N 36° 45' E 76.8 J N 22° 30' E 56.0 K S 76°45'E 48.0 L S 15° 00' W 43.4 M S 16° 45' W 40.5 In this survey D is the most easterly and I the most westerly station. The area is equal to 110 A. 2 R. 23 P. It may yary a little, on account of the way in which the balancing is done. 7. What is the area of a survey of which the following are the notes? Make the plot. Stations. Bearings. Distances. A S 461° E 80 rods. B S 51}° W 55.20 C West. 85 D N56° W 110.40 E N 33i° E 75.20 F S 74i° E 123.80 Ans. 104 A. 1 R. 16 P. 108 ELEMENTS OF SURVEYING. [book III. 8. Eequired the area of the farm, of which the survey notes are given in Art. 123. stations. Bearings. Distances (in chains). A N" 68° 55' W 31.95 B N 8° E 1.40 C S 87° 05' W 22.89 D N 30° 35' E 8.39 E N 4^° 05' E 23.91 F N 87° 05' E 14.51 G S 64° 30' E 7.55 H S 71" E 12.21 I S 27° E 17.39 K S 30° W 16.97 Offsets between C and D to be added, 15.9450 chains. Offsets to be subtract- ed, 15.1885 chains. 9. To determine the bearing and distance from one point to another, when they are so situated that one cannot be seen from the other. Let A and C he the two points, and AB a, meridian passing through one of them. From either of them, as A, measure a course A 2, of a convenient length in the direction toward C, and take the bearing with the compass. At 2, take the bearing of a second course, and measure the distance to 3. At 3, take a third bearing and measure to 4. At 4, take the bearing to (7, and measure the distance from 4 to C. Then, the difference between the sum of the northings and Fig. 74. SEC. IV.] AEEA OR CONTENTS OF GROUND. 109 the sum of the southings will be denoted by AB ; and the difference between the sum of the eastings and the sum of the westings, by BC. The base AB, and the perpendicular BC of the right-angled triangle ABO, are then known. The angle at the base, BAC, is the bearing from ^ to C; or the equal alternate angle at C is the bearing from C to A, and the hypothenuse AC, is the distance. Having measured the bearings and courses on the field, form a table, and find the base and perpendicular of the right- angled triangle, in numbers ; after which, find the bearing and distance. stations. Bearings. Distances. N. s. E. w. A N 61° W 40 ch. 19.30 34.98 B N42° W 41. 30.47 27.43 C ^12° E 16.10 15.75 3.35 D N47°E 32.50 22.16 23.77 AB = .87.77 27.12 62.41 27.12 05 = 35.29 ch To find the angle BAC, or the bearing from A to C. Kadius : tan A : : AB : BC, or, AB : BC :: B : tan ^ ; that is, applying logarithms, (a. c.) log ^^ (87.77) 8.056654 log ^(7 (35.29) 1.547652 log i? 10. log tan J 21° 54' 12" 9.604306 110 ELEMEi^TS OF SURVEYING. [BOOK III. To find the distance AC. ^m A \ R '.: BC : AG) Applying logarithms, (a. c.) log sin A 21° 54' 12" 0.428242 logi^ 10. log BG (35.29) 1.547652 log ^C' 94.6 1.975894 Hence, the bearing and distance are both found. Note. — Had any of the courses run south, AB would have been equal to the sum of the northings, minus the sum of the southings. 141. The last problem affords an easy method of finding the bearing and length of one of the courses of a survey, when the bearings and lengths of all the others are known. It may be necessary to use this method when there are obstacles which prevent the measuring of a course, or when the bearing cannot be taken. Indeed, two omissions may in general be supplied by calculation. It is far better, if possible, to take all the notes on the field, and it is necessary to do so, if accuracy is required. Tor, when any of them are supplied by calculation, there are no tests by which the accuracy of the work can be ascertained, and all the errors of the notes affect also the parts which are supplied. If necessary, however, the following omissions in the field- notes may be supplied by calculation, viz : SEC. IV.] AREA OR COI^TEN-TS OF GROUN"D. Ill I. The bearing and the length of one course. 142. The following are the field-notes of a survey: stations. Bearings. Distances. A N60° W 9.72 ch. B N 17i° E 7.65 C N 15|° W 9.40 D N 63|° E 10.43 E S 49° E 8.12 F S 13^° E 8.45 G S 16f°E 6.44 H With the known bearings and distances, find the correspond- ing latitudes and departures, as in annexed table : stations. Bearings. Distances. N. s. E. w. A N60° W 9.72 4.86 8.41 B N 17i° E 7.65 7.31 2.27 C N 15f° W 9.40 9.05 2.55 D N 63f° E 10.43 4.61 9.36 E S 49° E 8.12 5.33 6.13 F S 131° E 8.45 8.22 1.98 G S 16|° E 6.44 6.17 1.86 H 25.83 19.72 21.60 10.96 19.72 10.96 Deficiency Denciencv /^ 1 1 Lat. South. 0«-Li- ^0 P,4- Deficiency XU. U* in Dep. West. The latitude of the course HA in the plot (Fig. 75), must. 12 ELEMENTS OF SUEYEYING. [book IIL therefore, be 6.11, and its departure 10.64; hence, in the right- angled triangle HP A, PA = 6.11, HP = 10.64; The angle PAH = angle QHA, or the bearing of the course HA. Find PAH, as in preceding ex- ample, thus, AP : PH :: R \ tan PAH; that is, by logarithms, (a.c.) logJP(6.11) = 9.213959 log P^ (10.64) = 1.026942 logi^ = 10. log tan PAH = 10.240901 Hence, PAH is 60° 8', and since the deficiencies were in south lat. and west dep., the bearing of HA is S. 60° 8' W. To find the length of the course, we have, sin PAH : R :: PH : HA ; that is, by logarithms, (a. c.) log sin PAH {60° 8') = 0.061887 \ogR = 10. log PH = 1.026942 log^^ = 1.088829 Hence, HA = 12.27 ch., which is the length of the required course. HA. SEC. lY.] AKEA OK COKTENTS OF GROUND. 1. In a survey we haye the following notes : 113 stations. BeariDgs. Distances. A N 31J° W 10 ch. B N 62}" E 9.25 C D S 45J° W 10.40 What is the bearing and distance from station C to D ? ( Bearing, S 38" 52' E. ^^' I Distance, 7.03 ch. 2. In a survey we have the following notes : stations. Bearings. Distances. A S 40i° E 31.80 ch. B ]Sr54° E 2.08 D N 28f E 35.35 E N57° W 21.10 F S 47° W 31.30 What is the bearing and distance from C to D ? , Bearing, JST 34° 47' E. Ans. i Distance, 2.19 ch. II. The bearing of one course and the length of another, when the courses are (1) contiguous; (2) separated. 143. First. — ^Example when the courses are contiguous. In Kg. 75, let the bearing of DE and the length of EF be required, the bearings and lengths of the other courses being known. 114 ELEMENTS OF SURVEYING. [book III. As before, find latitudes and departures for known bearings and distances, obtaining annexed table : stations. Bearings. Distances. N. S'. E. w. A N60° W 9.72 4.86 8.41 B N17i° E 7,65 7.31 2.27 N15f° W 9.40 9.05 2.55 D 10.43 E S 49° E F S 131° E 8.45 8.22 1.98 G S 16f° E 6.44 6.17 1.86 H S 60° 8' W 12.27 6.11 10.64 Draw a line DF and find the bearing and the length of DF in the plot ABCDFOH, as in Art. 142. The length will be found to be 15.506, and the bearing 87° 20' 19" ; the angle DFE is then equal to 87° 20' 19"— 49° (the bearing of EF) = 38° 20' 19" ; then in the triangle DBF, the sides DF (10.43), and FF (15.506), and the angle DFF (38° 20' 19") are knowii, and the angle DFF will be found from the proportion, DF : DF :: sin DFF : sin DFF to be equal to 112° 45' ; hence, the bearing of DF is 112° 45'— 49° = 63° 45'. The length of the course ^i^ will be found from the proportion, sin DFF{^^° 20' 19") : sin FDF (28° 54' 41") : : i)^ (10.43) : FF to be equal to 8.12. Note. — In finding the angle DFF doubt arises as to the proper angle, corresj)onding to the logarithmic sine, to be taken ; DFFmsij be either 67° 15', or 112° 45' (see Art. 39); and hence the bearing of DF may be either 18° 15, or 63° 45', without SEC. IV.j AREA OR CO:NtTEKTS OF GROUND. 115 affecting the length of the given course DE. If the smaller angle 67° 15' is taken as value of DBF, then angle FDE is 74° 24' 41", instead of 28° 54' 41", and the length of the required side EF, as determined from the proportion, sin DFE : sin FDE :: DE : EF will be 16.19, instead of 8.12, without affecting the given bearing. When the length of the side EF, whose bearing is required, is not greater than that of the auxiliary line DF, the method given is of no service, unless some independent means exist of determining which of the two possible solutions is the one to be taken. The following are the notes of a survey : Stations. Bearings. Distances. A North 12.84 ch. B N32°E 17.82 C N80°E 24. D S 48° E 27. E S 18° W F 46.21^ Eequired the length of the course EF, and the bearing of the course FA. Length of EF = 28.60. Bearing of FA = N 73° 28' 21" W. 144. Secon'D. — Example when the courses are separated. In Fig. 75 let the bearing of AB and the length of DE be unknown. From B, Fig. 76, one extremity of the course whose bearing is unknown, draw BP parallel and equal to DE, whose length is required. Draw then PQ and QE, equal and parallel to ^Cand CD, respectively; and draw PA. The lengths and bearings of 116 ELEMENTS OF SURVEYING. [lU)OK III. the courses PQ luul Ql' arc the Biimo as those of BC \\\\(\ ('/>, respectively; hence, in the li«,nne A r Q h' I'V H iha ^\ lengtlis and hearin<;s of :ill the courses except A /' are known ; and the length and boariiii; of A r may be found as in Art. 142. Then in (he triangle APB, the length of A P i\m\ A B are known, and (he angle .17V>' (whicii is the sum or di (Terence of the bearings of BPaml AP) ; hence, as in Art. 143, the length of BP or its equal DE, and the bearing of A B may bo found, ((lillespie's Land Surveying, j). 21)!)). The same doubt as to the proper solution exists here, as in the previous example. n 111. The bearings of two courses when they are (1) contiguous; (2) separated. 145. First. — Example when they are contiguous. In T'ig. 75, let the bearings of DE and EF be unknown. Draw a line from D io /' and find its length and bearing from the lati(udes and departures of the other courses as belbre. Then in the triangle DEF the three sides are known and the three anjilos may be found as in Ar(icle41. From the angle ^DjP and (lie b(^aring of DF, the bearing of DE may be found; and from llie angle EFD ami the bearing of DF, the bearing of EF may bo found. 146. Second. — Example when the courses are separated. Let AB and BE, 1^'ig. 75, be (lie courses whose bearings are un- known. Assume the ligure as in Eig. 7(). ^Phon (ho bearing and length of A r may be found as before. In the (riangle APB, all the sides are known and the angles may be found as in the SEC. IV.] AREA OR CONTENTS OF GROUND. 117 \l preceding article. Then the bearing of BP, or its parallel DE, may be found from the angle BPA and the bearing of FA ; and the bearing of AB, from the angle PAB and the bearing of AP (Gillespie's Land Surveying). IV. The lengths of two courses. 147. * Let A = bearing of the first unmeasured course, FG, of the survey. B = bearing of the second course, GIL = angle between the courses. L = total latitude, and Z> =: total departure of the two un- measured courses, obtained by comput- ing the total latitudes and total departures of the measured courses, UK to PF, as in Art. 142. Let X = length of the first course. y =z length of the second course. We shall then have, L = (lat. ic+lai y) =^ x cos A-\-y cos B, D = (dep. x-{-d(i]). y) = a; sin A-{-y sin B. Multiply (2) by cot B, D cot B ^^ X sin A cot B-\-y cos B» Subtract (3) from (1) and solve with regard to x, L—D cot B Fig 77. (1) (3) X cos ^— sin A cot B Multiply both numerator and denominator by sin B, • Prot H. 8. Manroe. E.M., Ph.D., and J. Woodbridge Davbs, C.E., Ph.D., In " School of Minee Quarterly," for November, 1881. Ifc 118 ELEMENTS OF SURVEYING. [book III. X = L siu B—D COS B sm {A—B) or, substituting for sin [A—B) its value, _ L sin B—D cos B sin C In like manner we have, D co^ A — L sin A (4) y = sin G (5) By taking into account the signs of the diSerent quantities, the above formulas (4) and (5) will solve any case that may arise, not only when the unmeasured sides are contiguous, as in the figure, but any two sides of the polygon, e,g., sides FG and ON, provided only that the courses are not parallel, or nearly so, as FG and MN. "'The following table gives the proper signs in a convenient form for reference : Signs of L. and D. Signs of functions of A. and B. Lat. N. ; L. is 4-- KE. bearing; sin 4-, cos +. Dep. E.; D. is +• S.E. bearing; sin +, cos — . Lat. S. ; L. is — . S.W. bearing; sin — , cos — . Dep. W.; D. is-. N.W. bearing; sin — , cos -f-. To find value of C. C = {A — B) if courses are in same or in opposite quadrants. C := {A-\-B) a the courses are in adjacent north or adjacent south quadrants. (7= 180° — (^+^) if courses are in adjacent east quadranti or adjacent west quadrants. SEC. IV.] AREA OK CONTEi^TS OF GEOUKD. 119 EXA M PL E Sta. A Bearings. Dist. Latitude. Departure. N. S. E. w. East. 48 cb. 48.00 B K. 71i° E. 5.18 1.64 4.91 C S. 13i° E. 34. 33.13 7.65 D South. 12. 12.00 E S. 89^ 10' W. (A) (x) F N. 80° 20' E. (B) (y) 1.64 45.13 60.56 L = +43.49 I) = —60.56 From the above table we find that, in order to balance the latitude and departure columns, the total latitude of the two unknown courses must be 43.49 North, and the total departure 60.56 West. Substituting in the formulas, 43. 49 X sin 80° 20' + 60. 56 x cos 80° 20' X = y = sin 8° 50' 60.56 X cos 89° 10' + 43. 49 x sin 89° 10' Solving, sin 8° 50' a: = 345.41; y = 288.91. As a check, the latitudes and departures of the two sides should be computed, and the survey closed as below. Latitude. Departure. Stations. Bearings. Dist. N. s. E. w. A East. 48. 48.00 B N. 711° E. 5.18 1.64 4.91 C S. 131° E. 34. 33.13 7.65 D South. 12. 12.00 E S. 89° 10' W. 345.41 5.02 345.37 F N. 80° 20' E. 288.91 48.51 284.81 50.15 50.15 345.37 345.37 120 ELEMEXTS OF SUEVEYING. [BOOK III. The following are the field-notes of a survey : Stations. Bearings. Distances. A North. 7.81 ch. B S. 76^° E. C S. 10° 47' W. 28.42 D N. 84i° W. 27.12 E N. 4^° W. F East. 16.68 Required the distance from B to C and from E to F. Bio G, 17.87; ^toi^, 21.82. 148. Another Method of Determining Areas.* The area of any right-line figure may be decomposed into a series of right trapezoids and triangles by letting fall from the vertices perpendiculars to any fixed straight line, called the Base. The triangles may be considered trapezoids having one parallel side reduced to zero. Some of the trapezoids may be suMradive ; but we may say that the algebraic sum of the trapezoids composing the series is the area of the figure. If the figure be bounded by a continuous or a broken curve, its area is, approximately, equal to that of a right-line figure whose vertices are contained in the curve. By properly locating these vertices. we may approximate the area as closely as may be required. Area of a Series of Right Trapezoids. Before deducing the necessary rules, a few preliminary defini- tions will be given. * J. Woodbridge Davis, C.E., Ph.D., 1879. 3EC. IV.] AREA OR CONTENTS OF GROUND. 121 The position of points in a plane may be fixed by referring them to two intersecting straight lines of the plane. The two intersecting lines are called Coordinate Axes. Thus, let XX' and YY' Y' R X A' Y Pig. 78. x intersect at A ; the position of the point P is known when RP, its distance from YY' on a line parallel to XX' , and QP, its distance from XX on a line parallel to YY', are known. RP, or its equal AQ, is called the Abscissa of the point P; and QP, or its equal AR, is called the Ordinate of P. AQ and QP together are called the Co- ordinates of P. The point A, in which XX' and YY' intersect, is called the Origin of- Coordinates. When XX' and YY' are perpendicular to each other, as is most conyenieni, the coordinates of points referred to them are called Rectangu- lar Coordinates. Consider any series of five right trapezoids lying consecutive, with their bases on one straight Hue. The area of the series, as found by ordinary method, the symbols of Fig. 79 used, is Di D, d y Ds 'D, D Fig. 79. i[Z),(o+J) + (A-^,)(J+c) + (A-A)(c+(i) + (A-A) {d + e) + {D-D,){e+f)-\. (1) This may be transformed into )^[D,{a-c)+D,{h-d)+D,{c-e)-\-D,{d-f)-^D(e-\-f)\ (2) From this is derived the following rule, which is true, at least, 122 ELEMEis^TS OF SURVEYING. [bOOK III. for any series of five right trapezoids arranged as in Fig. 79. For convenience let this be called Eule A. Rule A. — To find the area bounded partly by any broken line, determined from a base line by means of rectangular ordinates, and otherwise bounded by the base line and terminal ordinates : Multiply the distance of each interjyiediate ordinate from the first by the difference between the two adjacent ordinates, always subtracting the following from the pre- ceding in order along the brohen line. Also, multiply distance of last ordinate from first by the sum of last two ordinates. Divide the sum of these products by 2. If this rule apply to a series of n trapezoids, its area is i [A {a-c)^D, (&-.^)+etc. + i)„_, {i-l)J^B^ (i + z^O]- (3) Add another trapezoid to the latter end of this series. Its area is i(i)„+i-i>„)(^-hO- (4) Add this to (3). The result is the area of a series of n-{-\ trapezoids, and is expressed by the following : |[A(f^-c) + etc.+A_i(2:->^)+i)„O*-0 + A+i(^ + 0- Therefore, the rule applies to a series of n -f 1 trapezoids. If from the series of n trapezoids the last trapezoid, whose area is i(z>„-i>„.,)(y+^). be subtracted, the area of the remaining series of w — 1 trapezoids is the following : J [A (a— 6)^1), (J_6?)+etc. + Z)„_i (/+./)]. Therefore, the rule applies to a series oin—X trapezoids. SEC. IV.] AREA OR CONTENTS OF GROUND. 123 It is certain that the rule applies to a series of five trapezoids. Let n = 5. Then n-\-l = 6, and n — 1 = 4. Consequently, the rule applies to a series of six trapezoids, and to a series of four. For this reason it applies to series of three and seven trapezoids, and so on. Because a series containing any number of trape- Boids may be found by continually adding, or subtracting, one trapezoid to, or from, a series of five, and because the application of the rule is not affected by each such operation, it follows that Rule A applies to a series consisting of any number of trapezoids. If that portion of the broken line which forms the upper side of the last trapezoid be perpendicular to base, whether directed towards it or from it, the area of the last trapezoid is zero. But expression (4), which represents this area, is zero. There- fore the rule conforms to this case. If the last portion of broken line be retrogressive, as shown in Fig. 80, where the line termi- nates at ^3, or K^, the last trapezoid must be subtracted in order that the required area — htiween hroken line and ~base limited hy terminal ordinates — may be obtained. But in this case expression (4) is intrinsically negative, and therefore should, as before, be algebraically added to the preceding area. Hence the rule conforms to this case. The statements of last paragraph are true, whatever the number of trapezoids in the series; even should the trapezoid added compose, alone, the series. In consequence, since every possible broken line is some succession of progressive, retro- gressive, and perpendicular elements, and because the addition of each element and its accompanying trapezoid does not affect the rule, rule A, or formula (3), is perfectly general, whatever may be the complication of the broken line. Fig. 80. 124 ELEMENTS OF SURTEYiyG. [book III. Example. — Let it be required to determine the area of a tract of land whose description is as follows: Beginning at a poplar on the south bank of the Cumberland river; thence south 31 poles to a stone monument; thence east 180 poles to a beech ; thence north 40 poles to ii sycamore on the bank of said river ; thence, with the meanderiugs of the same, to the beginning. It was found convenient to determine the irregular boundary by means of offsets from the long straight side. The positions and lengths of these offsets are recorded in the first two columns below. To obtain the area by Eule A, a third column of alternate differences is obtained ; and in the fourth column are placed the products of corresponding quantities in first and third columns: half the algebraic sum is the area. Distances. Offsets. Differences. Products. 31 15 30 -8 -120 45 39 — 12 -540 77 42 86 39 -2 —172 113 44 • 3 339 161 36 4 644 180 40 76 13680 1 14663 -832 2 ) 13831 Square rods 6915J SEC. IV.] AREA OR COI^^TEI^TS OF GROUND. 125 The advantages of this method of calculation are readily per- ceived by inspection of the above example, where every product except the last, is found by mental work. To illustrate this more forcibly, let us suppose the base line to be moved, for instance, 2658 poles directly south. Now, all the products except the last remain as simple, in fact the same, as before, while the last only is increased. The old method of calculation would now require every product to be increased. If, to find the area of an irregular figure, parallel cross measurements be made at appropriate places, intersecting a given base line at known points, the figure may be considered a series of trapezoids. If the base line be perpendicular to the direction of cross-measurements, Eule A obviously applies. If the base line be inclined to the cross lines. Rule A may he applied, but the final result must he multiplied hy the sine of the angle of inclination. EXAMPLES. 1. Wallace Colyar agrees to sell to James Beckwith at ninety dollars per acre, a piece of unimproved bottom land, lying between the west bank of the Delaware river and the foot of the adjacent hill. The description of the survey is as follows : Beginning at a willow on the west bank of Delaware river, this being the northeast corner of Henry Gillespie's home place ; thence with the meanderings of the bank of the river — [numerous courses here omitted] — to a white oak, the corner of James Beck- with's land ; thence west with his south boundary to a spring at the foot of the hill ; thence meandering with the foot of the hill — [numerous courses omitted] — to a black walnut, Henry Gillespie's northwest corner ; thence, with his line east to the beginning. The east and west being sinuous but unchangeable natural bound- aries, it is only required to determine, as easily and accurately as 126 ELEMENTS OF SUKVEYING. [book III. possible, the area. Accordingly, a line is run due north through the land and cross- widths are taken at right angles. The distances on base line and the lengths of cross lines are recorded in the columns below. It is required to find the area in acres and the sum in dollars to be paid for the strips of land. Distances. Widths. 38 rods. 19 29 40 32 78 36 111 51 145 45 173 40 200 44 228 50 254 55 290 58 337 43 Ans. 94.05625 acres; 18465.06. 2. Caleb Hopkins employs John Bryant to cut and burn the underbrush in an irregular piece of hilly woodland, at one dollar and a half per acre, the agreement being that the chain shall be allowed to rest upon the surface through- out its length, and not be stretched horizontally. A base line was run N. 20° W. through the land, and cross- measurements were taken in directions due east and west. The distances in base line and the lengths of cross lines are as follows: SEC. IV.] AREA OR CON^TEKTS OF GROUND. 127 Distances. Widths. rods. 10 19 26 41 55 49 78 52 106 61 129 50 153 55 175 39 180 45 194 31 209 To find the Area of a Polygon. It has been proved that Kule A is true for any series of trapezoids, whatever may be the complication of the broken hne which forms the upper sides of the right trapezoids ; there- fore the rule is true when the broken line returns to its point of beginning. But in this case the area required as the rule is the area enclosed by the broken line itself, or is the area of any polygon. Now D, the distance between first and last ordinates, becomes zero, causing the only term, which depends upon a sum of two ordinates for one factor, to disappear. To treat this case in the most gen- eral way, suppose B, C, D, E, F (Fig. ^-''' 81), to be points anywhere situated in a plane, and referred in position by rectangular co-ordinates to any point A in same plane ; and suppose points to be con- FiG. 81. 128 ELEMENTS OF SURVEYING. [BOOK III. nected in the order uamed by a broken line, ending again at B. Let the ordinates of the points be denoted by a, h, c, d, e, f, and the abscissas by a, h', c\ cV , e' , f. Connect the origin by a straight line with any of the points, as B. Then the area of the polygon ABCDEFBA, which is equal to the area of the polygon BCDEFB, is by Eule A, i [h' {a-c) + c' {h-cl) + d' (c-e) + e' (d-f) +/ (e-i) + ^' (/-«)] (5) = i U>' (f-o) -{- c' {h-d) + d' (c - e) + e' (d - h) + /(^-^)] (6) = -\\P{f'- o') + c {h'-d') + d (c'-e') + e {d'-n + f(e'-h')]. (7) From expressions (6) (7) can be framed the following rule : KuLE B. — To find the area of any polygon whose vertices are fixed by rectangular co-ordinate measurements: Multiply the abscissa of each vertex by the difference between the ordinates of the two adjacent ve7i}ices ; or, ■multiply the ordinate of each vertex by the difference between the abscissas of the tico adjacent vertices ; always mahing the subtraction in the same direction around the polygon. Half the sum of these products is the area. If the origin be placed at a vertex, one term vanishes which- ever way the rule is used. If one of the co-ordinate axes be passed through two vertices, two terms vanish, when the rule is used one way. SEC. IV. J AREA OR CONTENTS OF GROUND. 129 The determination of the areas of polygons is a problem of * frequent occurrence in all branches of engineering and many other professions. Eule B is invariably simpler than the ordinary formula for this case, and is therefore presented above in the most general terms. If for words, abacissa, ordinatef vertex, in Eule B, be substituted longitude, latitude, station, that rule applies technically to the case of land surveys. Example. — Recently a survey of a tract of one hundred and fifty square miles of coal land in Tennessee was made on the co-ordinate system ; that is, every corner of the old grants, of the subsequent sales out of them, and of the older adverse claims within them, and every station of a road or other traverse, as well as all important points, as coal mines, outcrops, bridges, crossroads, springs, villages, etc., etc., was fixed in position with respect to an assumed point, by means of rectangular co-ordinates directed due north and east. Thus, any parcel of land could be easily plotted, when wanted, in any of the geological or other maps ; while merely a glance at the table served to indicate its position in the district. When the area of such a piece was required, it was only necessary to proceed as in the following example: station. Total latitude. Total longitude. Difference between alternate longitudes. Double areas. A B C D E 7087 ft. 10020 8181 5012 2873 94851 ft. 97403 101369 103155 98604 -T-1201 — 6518 -5752 + 2765 + 8304 8511487 — 65310360 —47057112 13858180 23857392 Area in sq. ft. = 66140413 2 130 ELEMEN^TS OF SURVEYING. [book III. If the latitudes were larger than longitudes, differences of latitudes should be made the factors with total longitudes. Note. — The total latitude of a station with respect to any assumed point, is the distance, measured on a nteridian through the assumed point, from the point to the foot of a perpendicular drawn from the station to the meridian ; and the total longitude is the dis- tance of the station from the meridian through the assumed point. To find the area of a polygon surveyed, from the field notes of the bearings and distances of its sides. After finding the latitude and departure of each course and balancing the survey, as already explained, make a column of total latitudes of the various stations referred to any one of them as origin. This selection of origin reduces one double area product to zero. Now, instead of calculating the total longitudes of all stations and finding the differences between alternate ones, according to Rule B in formula (7), we may obtain the same result in simpler manner by adding the departures of each pair of adjacent courses. Modified in this way the operation becomes governed by the following rule : Rule C—Jfioltiply the total latitude of each station bij the sum, of the departures of the two adjacent courses. The algebraic half sum of these products is the area. EXAMPLE. stations. Bearing. Distance. Latitude. Departure. Total Latitude. Adjacent Departures. Double Areas. N + s- E-i- W- A N 35° 00' E 2.70 2.21 1.55 B N 83° 30^ E 1.29 .15 1.28 2.21 2.83 6.2543 C S 57" 00' E 2.22 1.21 1.86 2.36 3.14 7.4104 D S 34° 15' W 3.55 2.93 2.00 1.15 -0.14 -0.1610 E N 56° SC W 3.23 1.78 2.69 -1.78 -4.69 8.3482 Square chains 2 )21.8519 . 10.9259 SEC. v.] MAGN'ETIC DECLINATIOIf. 131 To find the total latitude of each station, add to total latitude of preceding station the latitude of preceding course. If the latitude of last station, found in this way, be equal to the latitude of last course with reversed sign, the work is correct. However, the latitude of first station, or station taken as origin, is always zero ; of last station it is always the latitude of last course with reversed sign ; and of second station it is always the latitude of first course. To find the adjacent departures^ add the departures of the two courses, one on each side of the station. SECTION V. MAGNETIC DECLINATION OR VARIATION OF THE NEEDLE. The following articles, 149-164, are essentially, and so far as practicable verbatim, from papers of the U. S. Coast and Geodetic Survey. 149. The magnetic declination at any place is the angle which the compass-needle, when it is correctly constructed and freely suspended, makes with the true meridian. The true meridian is fixed, but the declination varies because the direc- tion in which the needle points is in a continuous state of change. Therefore, whenever a measure of the declination of the needle is taken, the exact time (year, day of month, and hour of the observations) should be recorded as well as the geographical position of the place, or its latitude and longitude expressed to the nearest minutes of arc. 150. The declination is called ^'West" when the north end of the needle points to the west of the true meridian, and it is 132 ELEMENTS OF SURVEYIN'G. [BOOK 111. called ^^East" when the north end of the needle points east of the true meridian. In order to give an idea of the amount of the declination at present observable within the limits of the United States we instance the followiDg places at or near which it reaches extreme value, which are given to the nearest whole degree (1878) : At Eastport, Me., decUnation 18° west. At the mouth of the Eio Grande, Texas, 8° east. At San Diego. Cal., 14° east. At Sitka, Alaska, 29° east. At Fort Yukon, Alaska, 36° east. 151. The accuracy with which the declination may be deter- i£iined depends chiefly upon the instrumental means, but also, and in a great measure, upon the care taken in the use of the instruments and the selection of the proper methods and times for observing. Omitting any detailed notice of the irregular variations to which the magnetic needle is subject, it becomes important for the purposes of the surveyor to refer particularly to the changes which have a special bearing upon his observations. These are the daily variation and the secular variation. 152. The Daily Variation. —It has been found that at about the time of sunrise the north end of the needle has a slow motion towards the east which soon ceases. The needle is then said to be at its eastern elongation ; its north end then begins a retrograde motion towards the west, and at about one o'clock in the afternoon reaches the point at which it is said to be at its western elongation, after which it again turns back towards the east. The times at which the needle reaches its eastern and western elongations vary with the seasons of the year (with BEC. v.] MAGI^'ETIC DECLIN^ATIOi?'. 133 the sun's declination), happening a little earlier in summer than in winter. The angular range between the eastern and western elonga- tions varies also with the season of the year. The average position of the needle for the day is called the mean magnetic meridian. At about six o'clock in the evening (and for about an hour before and after), throughout the year, the position of the needle coincides very nearly with the mean magnetic meridian, and this, therefore, is the time most favorable for making observations to obtain at once the mean declination. 153. For reducing the direction of the needle observed at other hours to the mean magnetic meridian the following table is furnished. It gives to the nearest minute the variations of the needle from its average position during the day, for each hour in the day for the four seasons of the year. Table for reducing the observed declination to the mean decli- nation of the day. The needle points east of the The needle points west of the mean magnetic mean magnetic meridian. meridian. A.M. A.M. A.M. A.M. A.M. A.M. a O o P.M. P.M. P.M. P.M. P.M. P.M. ^ h. h. h. h. h. h. § h. h. k. h. h. h. Bour 6 7 8 9 10 11 ^ 1 2 3 4 5 6 / / / 1 / / / / / / / / / Spring 3 4 4 3 1 1 4 5 5 4 3 2 1 Summer.... 4 5 5 4 1 2 4 6 5 4 3 2 1 Autumn .... 2 3 3 2 2 3 4 3 2 1 1 Winter 1 1 2 2 1 2 3 3 2 1 1 154. It appears from observations of the daily fluctuation of the declination that the mean of the extreme easterly and 134 ELEMENTS OF SURVEYING. [BOOK III. westerly positions m any one day approaches nearly (within half a minute) to the mean position of the day, as derived from hourly observations continued day and night. Since corrections to observed declinations to refer them to the mean of the day are generally very unsatisfactory, it is recommended to observe the declination for any one day at the epochs of the eastern magnetic elongation and of the western magnetic elongation and to take the mean position as representing the declination for that day. The epochs of extreme positions, as observed at Philadelphia, Washington, and Key West, apply, with comparatively small changes, to nearly all places within the United States and may be stated to be as follows : Referring to the north end of the magnet, the morning eastern elongation occurs, on the average, from May to September, inclusive, about 7-|- a.m.; in March, April, and October, about 8 a.m.; in November, about 8|- a.m., and in December, January, and February, about 9 a.m.; earliest time in August, about 7Ja.m. ; latest in January, about 9 a.m. These epochs, however, are subject to great fluctuations and cannot be depended upon in any one case within one hour and frequently they cannot be recognized at all, either on account of the small range of the daily fluctuation — the amount of which in winter is but one-half, nearly, of the amount in summer — which is easily disguised by small irregularities, or on account of disturbances, which reach their maxima in September and October, and generally are more predominant in winter than in summer. The afternoon loestern elongation occurs, on the average, about IJ p.m. from May to November, inclusive, and about 1| P.M. in the remaining months; also, earliest in Septem- ber — some minutes before 1 p.m. — and latest in January, about If- P.M. The afternoon epoch is subject to less fluctuation than the morning epoch. 155. The Secular Variation of the magnetic declination SEC. Y.] MAGNETIC DECLIITATIO]^". 135 is a subject of the greatest importance to surveyors. It mani- fests itself by a gradual change in one direction, which at first increases slowly, then more rapidly, diminishing again after- ward until the needle becomes stationary and subsequently returns by similar changes to its former position, the whole period extending over nearly two and a half centuries. Thus, at Philadelphia the declination was 8f° west in 1700, whence it diminished until in 1800 it reached a minimum 2°.l (2° 6'), and increased again to 6°.8 in 1880. At present all along the Atlantic and Gulf coasts the effect of the secular variation is to i7icrease west declinations or to decrease east declinations by from 2' to 5', but on the Pacific coast the effect is opposite in direction, viz., increasing east declinations by from 1' to 3'. In Alaska, however, there are indications of a decrease of east declination. 156. Lying between the region in which the variation of the needle is west and that in which such variation is east, is a line of no variation, on which the magnetic meridian coincides with the true meridian. The north (or seeking) end of the needle always inclines toward the line of no variation : hence, for all points east of this line, the variation is West; and for all points west of it, the variation is East. This line is not a fixed line, but changes its position from year to year. By referring to a map, such line for the United States may be traced. In 1870, it crossed Sault St. Marie at the lower end of Lake Superior, passed, very nearly, through Cleve- land in the State of Ohio, Ealeigh in North Carolina, and passed into the Atlantic Ocean near Wilmington, North Caro- lina. In 1875, it crossed Lake Superior and entered Michigan at White Fish Point, passed thence, very nearly, through Bay City Michigan, Oberlin Ohio, Parkersburgh West Virginia, 136 ELEMENTS OF SURVEYIl^^G. [book III. thence, with a slight curve to the southwest, to Fajetteville North Carolina, and thence into the Atlantic Ocean a little to the west of Cape Fear. The line of no variation is now moving quite rapidly to the south and west, and therefore it is that ivest declinations of the needle are increasing, and east declinations decreasing. 157. The U. S. Coast and Geodetic Survey has made a collection of magnetic declinations observed at various places in the United States and elsewhere. Mr. Charles A. Schott, chief of the Computing Division of the Survey, has tabulated the observed declinations, and deduced from them a Table of For- mulae. This table is here given so far as it relates to localities in the United States. The localities are arranged geographically, as far as practicable, and their positions are given by latitude and longitude (west of Greenwich). Formulae expressing the magnetic declination at various places in the United States. Locality. Latitude. Longitude. Expression for Magnetic Declination. Portland, Me o 43 r 38.8 o 70 16.6 O D^ +10.72 + 2.68 sin (1.33 ?w+ 24.1). / Burlington, Vt 44 28.2 73 12.3 D= +10.81 + 3.65 sin (1.30 w— 20.5) + 0.18 sin (7.0 m+132). j Rutland, Vt 43 36.5 72 55.5 D= f 10.03 + 3.82 sin (1.5 m— 24.3). Portsmouth, N. H 43 04.8 70 43.0 D= +10.63 + 3.17 sin (1.44 m— 4.7). 1 Newburyport, Mass.. 42 48.4 70 49.0 D= +10.07 + 3.10 sin (1.4 m+ 1.9). Salem, Mass 42 42 31.9 21.5 70 71 52.5 03.8 D=+ 9.80 + 3.61 sin (1.50 m— 1.0). D=+ 9.52 + 2.93 sin (1.30 7n+ 5.0). Boston, Mass ... Cambridge. Mass 42 22.9 71 07.7 D=+ 9.58 + 2.69 sin (1.3 m+ 7.0) + 0.18 sin (3.2 W2+ 44). Nantucket, Mass. . . . 41 17.0 70 06.0 D=+ 9.29 + 2.78 sin (1.35 OT+ 5.5). Providence, R. I 41 49.5 71 24.1 D=+ 9.10 + 2.99 sin (1.45 m— 3.4) + 0.19 sin (7.2 m + 116). Hartford, Conn 41 45.9 72 40.4 D=+ 8.06 + 2.90 sin (1.25 m— 26.4). Wew Haven, Conn . . . 41 18.5 72 55.7 D=+ 7.78 + 3.11 sin (1.40 m— 22.1). SEC. v.] MAGNETIC DECLIiq^ATIOJf. 13? Locality. Latitude. Longitude. Expression for Magnet.^ Oeclination. Albany, N.Y 42 39.2 73 45.8 D=+ 8.17 + 3.02 sin (1.4im— 8.3). Oxford, N. Y 42 42 26.5 52.8 75 78 40.5 53.5 D=+ 6.19 + 3.24 sin (1.35 7?i— 18.9). D=+ 3.66+3.47 sin (1.4 w— 27.8). Buffalo, N.Y Erie, Pa.. 42 07.8 80 05.4 D=+ 2.26 + 2.71 sin(l 55 m— 29.7). Cleveland, Ohio 41 30.3 81 42.0 D=+ 0.10 + 2 07 sin (1.40 7n— 6.2). Detroit, Mich 42 20.0 83 03.0 D=— 0.97 + 2.21 sin (1.50 m— 15.3). Saint Louis, Mo 38 38.0 90 12.2 D=— 7.15 + 2.33 sin (1.4 m— 20.1).* New York, N.Y 40 42.7 74 00.0 D=+ 6.40 + 2.29 sin (1.6 m— 5.5) + 0.14 sin (6.3 m+ 64). Hatborough, Pa 40 12 75 07 D=+ 5.23 + 3.28 sin (1.54 771^ 13.2) + 0.22 sin (4.1 wi+157). Philadelphia, Pa 39 56.9 75 09.0 D=+ 5.38 + 3.29 sin (1.55 m— 23.9) + 0.39 sin (4.0 m + 161). Harrisburg, Pa 40 15.9 76 52.9 D=+ 2.93 + 2.98 sin (1.50 m+ 0.2). Baltimore, Md 39 17.8 76 37.0 D=+ 3.20 + 2.57 sin (1.45 m— 21.2). Washington, D. C 38 53.3 77 00.6 D=+ 2.47 + 2.47 sin (1.40m— 14.6). Cape Henry, Va 36 55.5 76 00.5 D=+ 2.54 + 2.41 sin (1.50 m— 35.4). Charleston, S. C 32 46.6 79 55.8 D=- 2.14 + 2.74 sin (1.35 m 1.3). Savannah, Ga 32 04.9 81 05.5 D^- 2.54+2.32 sin (1.5 m- 28.6). Key West, Fla 24 33.5 81 48.5 D=— 3.90 + 2.93 sin (1.4 m— 33.5). Mobile, Ala 30 41.4 88 02 5 D=— 4.40 + 2.69 sin (1.45 m— 76.4). D=— 5.61 + 2.57 sin (1.4 m- 61.9). New Orleans, La 29 57.2 90 03.9 San Diego, Cal 32 42.1 117 14.3 D=— 12. 54 + 1. 64 sin (1.2 m— 180.0). Monterey, Cal 36 36.1 121 53.6 D=— 12. 82 + 3.54 sin (1.0 m-142.9). San Francisco, Cal. . . 37 47.5 122 27.2 D=— 13.34 + 3.23 sin (1.00 m— 130.3). Cape Disappointment, Wash. Ter 46 16.7 124 02.0 D=— 20. 72 + 2. 81 sin (1.2 m— 188.8). Sitka, Alaska 57 02.9 135 19.7 D=— 26.72 + 2.41 sin (1.6 m-107.1). Unalashka, Alaska. . . 53 52.6 166 31.5 D=— 18. 34 + 1.45 sin (1.4 m— 67.8). 158. The epoch to which the formulae refer is 1850 ; hence, in the "expression for magnetic decHnation/' m — t — 1850. To illustrate the use of the table. Let it be required to find the declination of the needle at Albany, N. Y., in August, 1879, or 1879.6. * Approximate expreBsion, 138 ELEMEi^TS or SUKVEYIls^G. [BOOK lil The tabular expression for magnetic declination at Albany is D = +8M7 + 3^02 sin (1.44° w-8°.3). m = 1879.6 — 1850 = 29.6. 1.44° m — 42°.624. .-. 1.44°m-8°.3 = 34°.324. Natural sin (34°.324) = .5638 ; hence, -Z> = + 8°.17 + 3°.02 x .5638 = + 9°.87 ; which gives the computed declination of the needle at Albany in 1879.6, as 9°.87 (west, since the result is phcs), which differs from the observed declination at that time but one-hundredth of a degree. If the time for which the declination is desired is prior to 1850, then m will be negative. For example, let the declination at Albany for 1836.8 be required ; then m = 1836.8 — 1850 = —13.2, and 1°.44 m = 1°.44 x ( — 13.2) = -19°.008, and 1°.44 ???- 8°.3 = — 27°.308, and sin (— 27°.308) = -.45877; .-. D = 8M7 + 3°.02 sin (— 27°.308) = 8°.17-1°.39 = +6°.78; which agrees exactly with the observed declination. The student is referred to Mr. Schott's paper. Appendix No. 9, IT. S. Coast and Geodetic Survey Report for 1879, for valuable tables of Magnetic Declinations. 159. From the same paper is taken the following table of computed annual changes in the declination of the magnetic needle for 1870, 1880, and 1885, expressed in minutes of arc, a + sign indicating north end of needle moving westward, a — sign indicating north end moving eastward: SEC. v.] MAGNETIC DECLINATION?". 139 TABLE. Locality. Annual change. In 1870. In 1880. In 1885. Portland, Me.. Burlington, Vt Eutland, Vt Portsmouth, N. H. Newburyport, Mass Salem, Mass Boston, Mass Cambridge, Mass. . . Nantucket, Mass. . . Providence, E. I Hartford, Conn. . . . New Haven, Conn. Albany, N.Y Oxford, N. Y Buffalo, N. Y Erie, Pa Cleveland, Ohio. . . . Detroit, Mich Saint Louis, Mo. . . . New York, N. Y. . . Hatborough, Pa. . . Philadelphia, Pa. . . Harrisburg, Pa Baltimore, Md Washington, D. C. Cape Henry, Va Ch.^rleston, S. C . . . + 2.4 + 5.0 + 6.0 + 4.4 + 3.9 + 5.0 + 3.4 + 2.9 + 3.3 + 3.8 + 3.8 + 4.6 + 4.3 + 4.5 + 5.1 + 4.4 + 2.8 + 3.4 + 3.4 •+2.4 + 4.6 + 4.9 + 4.1 + 3.9 + 3.5 + 3.8 + 3.5 + 1.6 + 6.0 + 5.6 + 3.7 + 3.3 + 4.1 + 2.9 + 2.1 + 2.7 + 3.7 + 4.3 + 3.7 + 4.3 + 5.0 + 4.2 + 2.5 + 3.0 + 3.2 + 2.5 + 4.5 -i-4.9 + 3.3 + 3.6 + 3.2 + 3.7 + 3.0 + 1.2 + 5.8 + 5.3 + 3.3 + 2.9 + 3.5 + 2.5 + 1.8 + 2.4 + 3.6 + 4.1 + 3.4 + 4.0 + 4.8 + 4.0 + 2.2 + 2.8 + 3.0 + 2.6 + 5.3 + 2.8 + 3.2 + 3.0 + 3.6 + 2.7 140 ELEMENTS OF SURVEYING. [book III. Locality. Annual change. In 1870. lu 1880. In 1885. Savannah, Ga Key West, Fla Mobile, Ala New Orleans, La San Diego, Oal Monterey, Gal San Francisco, Gal Gape Disappointment, W. Ter. Sitka, Alaska . . . Unalashka, Alaska + 3.6 + 4.3 + 2.8 + 3.1 — 1.9 -2.0 — 1.0 -3.4 + 1.0 + 1.6 + 3.5 + 4.2 + 3.4 + 3.5 -1.7 —1.5 —0.5 -3.1 + 2.1 + 1.9 + 3.3 + 4.1 + 3.7 + 3.7 -1.6 — 1.1 -0.3 -2.7 + 2.5 + 2.0 160. It will be observed that the amount of change is by no means the same, even in places not far remote from each other, as New York and Philadelphia. In grouping together a table of the present rate of change, much allowance must therefore be made for possible local pecu- liarities that have not been ascertained. A surveyor should, at the time and place of survey, deter- mine the true meridian and thence the magnetic declination for record with his survey. Method of ascertaining the Declination. 161. The best practical method of determining the true meridian of a place, is by observing the north star, Polaris. If this star were precisely at the point in which the axis of the earth prolonged pierces the heavens, then the intersection of the vertical plane passing through it and the place, with the surface of the earth, would be the true meridian. But the SEC. v.] MAGNETIC DECLINATION". 141 star being at a distance from the pole equal to 1° 30' nearly, it performs a revolution about the pole in a circle, the polar distance of which is 1° 30' nearly, and the time of revolution is 23 hours and 56 minutes. To the eye of an observer this star is continually in motion, and is due north but twice in 23 hours and 56 minutes, and is then said to be on the meridian. When it departs from the meridian, it apparently moves east or west for 5 hours and 59 minutes, and then returns to the meridian again. When at its greatest distance from the meridian, east or west, it is said to be at its greatest eastern or western elongation. 162. The western elongations from the beginning of April to the end of September, and the eastern from the beginning of October to the end of March occur in the day-time. If it be necessary to determine the meridian at that particular season of the year, and at night, let 5 hours and 59 minutes be added to or subtracted from the time of greatest eastern or western elongation, and the observation be made at night when the star is on the meridian. 163. The angle which the meridian plane makes with the vertical plane passing through the pole-star when at its greatest eastern or western elongation, is called the Azimuth of Polaris. The following extract, Art. 164, with the tables, is from the Report of the U. S. Coast and Geodetic Survey for 1881 : 164. The following tables of the times and azimuths of Po' laris ivhen at elongation have been prepared for the benefit of those surveyors and others who may prefer to make use of the pole-star for their determination of the true meridian, and whose instru- mental outfit for the measure of the declination may be limited to a compass -svith sights or to a small theodolite with compass-needle attached, and who may be without a chronometer. 143 ELEMENTS OF SURVEYIi^^G. [book III. The method was recommended to surveyors by Dr. Charles Davies in the revised edition of his work on surveying, and a description of it still forms part of the instructions of the Com- missioner of the General Land Office to the surveyors-general of public land of the United States (editions of 1855, 1871, and 1878). The tables given in these instructions have either become obsolete from lapse of time or are not sufficiently extended for future use. They were, therefore, recomputed, and in their present form and with the rules given for interpolation will be found to possess greater accuracy than any similar tables previously published. The tables include all elongations whether occurring by day or night. Polaris may be observed in day-time w^hen the sun is not too high even with moderately powerful telescopes ; besides, a complete table facilitates interpolation. Mean local time (astronomical, counting from noon) of the elongations of Polaris. [The table answers directly for the year 1885, for latitude +40° and for longitude 6 hours west of Greenwich.] Date. Eastern elongation. Western elongation. Jan. 1 h. m. 35.3 h. 12 m. 24.6 (< 15 23 36.1 11 29.3 Feb. 1 22 29.0 10 22.2 iC 15 21 33.7 9 27.0 Mar. 1 20 38.5 8 31.8 (< 15 19 43.4 7 36.6 Apr. 1 18 36.4 6 29.7 a 15 17 41.4 5 34.7 May 1 16 38.6 4 31.8 i< 15 15 43-7 3 36.9 June 1 14 37.1 2 30.3 iC 15 13 42.2 1 35.4 SEC. v.] MAGNETIC DECLIiq-ATIGlC. 143 Date. Eastern elongation. Western elongation. July 1 h. m. 39 6 h. m. 32.8 a 15 11 44.7 23 34.0 Aug. 1 10 38.2 22 27.5 i( 15 9 43.3 21 32.6 Sept. 1 8 36.7 20 26.0 a 15 7 41.7 19 31.1 Oct. 1 6 38.9 18 28.2 a 15 5 43.9 17 33.2 Nov. 1 4 37.0 16 26.4 i( 15 3 41.9 15 31.3 Dec. 1 2 38.9 14 28.2 i( 15 1 43.6 13 33.0 N. B. — To refer the tabular times to any year (limit about 10 years) subsequent to the epoch, add 0"\35 for every year. For years previous to epoch subtract 0°'.35 for every year. To refer the tabular times to any other latitude (between the hmits 25° and 50°), add 0'".14 for every degree south of 40° ; subtract 0°'.18 for every degree north of 40°. To refer the tabular times to any year in a quadriennium, For first year after a leap year the table is perfect. For second year after a leap year add . . . I'^.O For third year after a leap year add . . . 2"*. For a leap year and before March 1 add . . d'^.O And for remainder of the year subtract . . 1™.0 For any other than the tabular day subtract from the tabular time of elongation 3"'.94 for every day elapsed. It will be noticed that there occur two eastern elongations on January 9, and two western elongations on July 9. 144 ELEMENTS OF SUEVEYING. [book III. Eg f1 .0 «M m kO a> CO rH na ^ cii C4 GO 00 r-t CfQ ^ ce » pCj -M xo A + a> ti ^ c« rO A t--^_eoTHOoooi-i(Mint-T-iioooei3;t-«o«icx)OT}< W CO T); O O t- OC C: O rH « CO ■*' 10 t-^ 00 O ,-' (TO Tt" O O) C (N LO t-^ o«-r-iooo»CTf»OQOQOt-t-OOC50(NlOOse>300T)*'?ii-iooc>jiooo oi at Oi S5 1-1 *> CO 's* ■>a< «D 00 c ■^ Tj" JO S S 10 s «Dcooooom-*co OSO-r-lCiTl^lQOl— CS50 0a<^oseo t-'^i-IOSQO»»0»0!:C=COOO(NlftOi-VC;iOi-iOiCOOOOOO ot-incoi-iOCiC". oo(W-^«ooseoQOooo5«o-^ecc*eoiOQOeo Tje0T-'O ©» -^ 00 CO oi a (^ ct P3 n (Meor} move it to the east as the pole-star moves ^vesi, till Polaris and Delta both appear upon the plumb-line together; the line through the point of sight and the plumb-line will be, very nearly and with sufficient accu- racy, the true meridian. This method is practicable only when the star Delta is 'below the pole-star during the night ; when it passes the meridian above the pole, it is too near the zenith to be of service, in which case the star Zeta (^), the last star but one in the tail of the Great Bear, may be used instead. Delta Cassiopeiae is on the meridian below the pole-star at midnight about April 10, and is, therefore, the proper star to use at that date and for some two months before and after. Six months later, the star Zeta in the tail of the Great Bear will supply its place, and is to be used in precisely the same manner. Fig. 82 gives a representa- tion, drawn to scale, of N. Pole, Polaris, and the Constellations Cassiopeia and Great Beai ; and fw. 82. Polaris •IT.Tolt £• Casd •/3 a" -6* opeia li 148 ELEMENTS OF SUKVEYING. [book III. the line drawn through the star Delta (6) of Cassiopeia and Zeta (C) of the Great Bear, represents those stars on the meridian with the pole-star. The method given in this article for finding the true meridian cannot be used with advantage, on account of the haziness of the atmosphere near the horizon, at places below about 38° north latitude. 167. The variation of the needle should always be noted on every survey made with the compass, and then if the land be sur- veyed at a future time, the old lines can always be re-run. In re-running the lines of an old survey, a vernier is used for setting off the variation of the needle. 168. A Vernier is a contrivance for measuring snaaller divi- sions of a unit than those into which the line, to which it is applied, is divided. It is a graduated scale so arranged as to cover an exact number of equal spaces on the primary scale, or limb. It is divided into a number of equal parts greater by one than the number of equal spaces which it covers on the limb. The vernier may be applied to any limb or scale of equal parts. The modes of its application are extremely various ; the principle, howeyer, is the same in all, and may be illustrated by a simple diagram (Fig. 83). W // 12 IS 14- iJ 16 17 18 19 c 1 1 1 1 1 1 1 1 n B 6 6 Fig. 83. to Let ^^ be any lirrib or scale of equal parts, one of which let us suppose equal to ^ of an inch. Let CD be a vernier, equal in length, say to nine of these parts, and itself divided into ten equal spaces, each one of which is then equal to nine-tenths of ^^ or yfo of an inch. The difference between a space on the limb and a space on the vernier, is therefore equal to one- tenth of SEC. v.] MAGKETIC DECLINATION. 149 ^^ or y^ of an inch. This is the least space that can be measured by means of the vernier, and is called the least count j hence, The least count of a vernier is equal to one of the equal divisions of the limb divided by the number of spaces on the vernier. 169. The true reading of an instrument, for any position of the vernier, expresses the distance from the point where the graduation on the limb begins, marked 0, to the point of the vernier. In the diagram, that distance is expressed by nine units of the limb, or 9. If, now, the vernier be moved till the division 1 coincides with the division 10 of the limb, the point will have advanced along the limb a distance equal to -^ of one space, and, if we call each space J, the reading will become ^-\-^b. If we again move the vernier till the division 2 coincides with the division 11 of the limb, the point will have advanced an additional distance, equal to -j^h, and the reading becomes ^-{-^-^h ; when 3 coincides with division 12, the reading will become 9 + ^^, and so on, till finally, when the point 10 coincides with 19 of the limb, the distance 9 will have been increased by \^h, and will become 10, as it should, since, in that case, the point will have been moved a whole space, and will coincide with the division 10 of the limb. Hence, the following rule for reading an instru- ment which has a vernier : Read the scale, in the direction of the graduation, up to the line preceding the of the vernier ; this gives the number of whole units of the scale. Look along the vernier and find which of its lines coincides with a line of the scale ; this line of th© vernier gives the number of fractional parts of one unit of the scale to be added to the former reading. 150 ELEMENTS OF SUKVEYING. [book III. 170. In order to read a vernier correctly, especially if it be one with which we are not familiar, it is necessary to estimate, hy the eye alone, the fractional part of one unit to be added ; then if the vernier reading is nearly the same as the estimate, we may record the reading with confidence ; but if the estimate and the vernier reading disagree largely, then the cause of such disagreement will probably be a false reading of the vernier. H -6,5^; a^^^^ ^s. FiQ. 84. 171. In a vernier compass, the vernier is attached, as shown in the figure. A small arc HI is described on the bar AB, having its centre at the centre of the compass-box. This arc is divided to degrees, and sometimes to the parts of a degree. The vernier is permanently attached to the compass-box. When the point of this vernier coincides with the point of the graduated *arc HI, the north and south line of the com- pass-box lies in the plane of the sights. The compass-box is turned about its centre, without moving SEC. v.] . MAGNETIC DECLIi^ATION^. 151 the plate AB, by means of the milled screw L : and is fastened to the plate AB, by a clamping nut underneath the main plate. 172. To set off the variation of the needle on a yernier compass, turn the north end of the compass plate by the tangent screw L of the vernier (see Fig. 84), over the number of degrees in the variation, to the left (the observer is supposed to be standing at the south end of the plate and looking towards the north end), if the variation is westerly, to the right if it is easterly. 173. To run out lines of old deeds, set the compass upon one end of an ^^ original line," determined to be such by old "marks," or by testimony as to its having been undisputed for twenty years or more, and turn the vernier plate till the reading is the same as that given in the deed ; then run the other lines by the bearings and distances as given in the deed. 174. To make allowance for change in "varia'tion" in running out old lines, when only one "corner" can be identi- fied, the surveyor must see to it that the date of the deed from which he is working is the same as the date of the survey from which the description of property in the deed is taken ; very often descriptions are copied from former deeds, in which case the date of the survey must be discovered before running the lines. 175. If no *^ original" line of the property can be found, nor a corner and date of survey, then the surveyor must seek his data upon neighboring property and work back from there, through several deeds if necessary. 176. It is very desirable that the date of the survey from 152 ELEMENTS OF SURVEYIN^G. [BOOK lU. which a description is taken, and also the name of the surveyor, should be entered in every deed conveying lands. The form of a *^ Survey Bill '* may be somewhat as follows : Description of Property in Town of , County of , State of , surveyed on the sixteenth day of August, in the year eighteen hundred and eighty- three, and bounded as follows : Beginning at a point in the northwest corner of land owned by A. B. 0., and marked by a cross cut into the top of a stone sunk three feet into the ground, — from which point the east chimney of the house of A. B. 0. bore S 87^° E, distant 8.56 ch.— thence from the point of beginning N 18° E, 11.27 ch. to a stake ; — thence N 56° E, 15.26 ch. to a corner where two stone walls meet; — thence S 87° E, 5.56 ch. to the shore line of the bay ; — thence following the shore line to a point on the shore Hne bearing S 23° W, and distant in a straight line 13.47 ch. — from which point the middle of the eastern key-stone of the bridge across Beaver Dam Creek bears S 37|-° E ; — thence, fol- lowing the middle of the road to a point in the middle of the road bearing S 87° W, distant in a straight Hne 5 ch. — thence, &c., &c. ; the above described property containing, by cal- culation, — Acres, — Eoods, and — Perches, more or less. Erasmus Kuggles, Variation of the needle — ° — ' — " W (or E). Surveyor. BOOK IV. TRANSIT SURVEYING. SECTION I. SU RVEYOR'S TRAN SIT. 177. The transit is an instrument used for measuring hori- zontal and (when furnished with a vertical circle) vertical angles. It is placed on a tripod, TTT, to which it is screwed fast by means of a horizontal brass plate, DE, Upon the upper surface of the plate DE rest four screws with milled heads, called level- ing screws, which work through the second horizontal plate, FG, into cylindrical nuts, shown in the figure. The two plates DE and i^(r are called leveling plates. The lower leveling plate, DE, is made in two pieces — the upper piece, which is screwed fast to the top of the tripod, having a large opening in its centre, in which the smaller lower one is shifted from side to side, or turned completely around. By this arrangement, termed a *' shifting centre," the instru- ment is easily moved over the upper plate, and the plummet, which hangs from the centre, set precisely over a point without moving the tripod. The upper side of the plate DE terminates in a curved surface, which encloses a ball that is nearly a hemisphere with the plane of its base horizontal. This ball or hemispherical nut is scrjag5d fast to the smaller base of a solid conic spindle, that pas^ tlirough the curved part of the plate DE. To this sg||dle is firmly attached the second horizontal plate, FG. 154 ELEMENTS OF SURYEYIITG. [book IV. To this spindle, also, and above the plate FG, is fitted and fastened by spring-catch, a socket, called the main socket, to which is attached a third horizontal and circular plate, BB. Fig. 85. called the limb of the instniment. Fitted to the upper conical surface of the main socket is a second socket to which is united a thin circular plate, A A, called the Vernier Plate, SEC. I.J surveyor's transit. 155 which rests on the limb of the instrument and carries a compass- circle, standards, &c., as shown in Fig. 85. Fig. 86. 178. The sectional view, Fig. 86, shows the interior construc- tion of the sockets of the transit, the manner in which it is detached from the spindle, and the means by which it can be taken apart if desired. In the figure, the limb BB is attached to the main socket of LL, which is itself carefully fitted to the conical spindle P, and held in place by the spring catch C. The upper plate, A A, carrying the compass-circle, standards, &c., is fastened to the flanges of the socket, Z7Z7, which is. fitted to the upper conical surface of the main socket L ; the weight of all the parts being supported on the small bearings of the end of the socket, as shown, so as to turn with the least possible friction. A small conical centre, in which from below is inserted the strong screw, 8, is brought down firmly upon the upper end of the main socket LL, and thus holds the two plates of the instru- 156 ELEMEifTS OF SURVEYING. [BOOK IT. ment securely together, while at the same time allowing them to move freely around each other in use. A small disc above the conical centre contains the steel centre-pin upon which rests the needle, as shown ; the disc is fastened to the upper plate by two small screws, as represented. The main socket with all its parts is of the best bell-metal and is most carefully and thoroughly made, the long bearing of the sockets ensuring their firm and easy movement, while at the same time they are entirely out of the reach of dust, or other source of wear. When desired the whole upper part of the instrument can be taken off from the spindle by pulling out the head of the spring- catch at Cy and when replaced will be secured by the self-acting spring of the catch. The figure also shows the covers of the leveling screws, the shifting centre of the lower leveling plate, and the screw and loop for the attachment of the plummet. 179. On the upper surface of the plate FG, Fig. 85, rests a clamp which goes round the main socket and which, being com- pressed by the clamp-screw K, is made fast to it. This clamp is thus connected with the plate FG. Two small cylinders, aa, are fastened to the plate FG ; through these cyHnders thumb screws, //, called slow-motion screws or tangent screws, pass and abut against opposite sides of a piece projecting from the clamp- ring, thus preventing the clamp from moving in either direction. When the clamp is compressed against the main socket by the clamp-screw K, the limb of the instrument is prevented from revolving. One of the tangent screws /, being then loosened as the other is tightened, will slowly and steadily move the clamp- ring itself and with it, of course, the limb. The limb, BB, has a silvered circle near its outer edge, on which the graduation for horizontal angles is made. In the SEC. I.] SURVEYOR'S TRANSIT. 157 instrument described, the circle is divided into degrees and half degrees ; the degrees are figured in two rows, viz : from to 360, and from to 90 each way. The vernier plate, A A, has two openings, one of which is shown in the Figure 85, placed diametrically opposite each other, in which, attached to the plate A A and moving with it, are small silvered arcs called verniers, which serve to read the limb about which they revolve. The verniers are double, having on each side of the zero mark thirty equal divisions corresponding precisely with twenty- nine half-degrees of the limb ; they thus read to single minutes, and the number passed over is counted in the same direction in which the vernier is moved. The use of two opposite verniers gives the means of testing the correctness of the graduations, the perfection with which they are centered and the dependence which can be placed upon the accuracy of the angles indicated. Two spirit levels, at right angles to each other, are attached to the vernier plate by small adjusting screws, one being fixed to the standard which supports the telescope, so as not to obstruct the light which falls on the vernier opening beneath. The vernier plate turns freely around with the socket to which it is attached. It is made fast to the limb by the clamp- screw Q, Fig. 85, after which the smaller motions are made by the tangent-screw R. There is a compass on the vernier plate that is concentric with it, the uses of which have been explained under the head Compass. It also may be made to serve as a check upon the measurement of angles with the transit, as from the magnetic bearings of two courses may be found the angle between them, as explained in Art. 114. 180. The standards which support the horizontal axes of the 158 ELEMENTS OF SURVEYING. [book IV. B ^^|^~W^ = telescope with its attached level, rest on and are made fast to the Ternier plate. The vertical circle MM, Fig. 85, called the Vertical Limb, is securely fastened to the axis of the telescope; it is plated with silver, graduated to half degrees, and with its vernier enables the sur- veyor to obtain vertical angles to single minutes. There is a clamp-and-tangent arrangement connected with the tele- scope ; it consists of an arm at one eod encircliug the telescope axis, and at the other connected with the tangent-screw 0, Fig. 85. The clamp is fastened at will to the axis by a clamp-screw, and then by turning the tangent-screw the telescope is raised or lowered as desired. 181. The telescope is from ten to eleven inches long, firmly secured to an axis having its bearings nicely fitted in the standards, and thus enabling the telescope to be moved in either direction, or turned completely around if desired. The different parts of the telescope are shown in Fig. 87. The object-glass is composed of two lenses, so as to show objects without color or distortion, is placed at the end of a slide having two bearings, one at the end of the outer tube, the other in the ring CC, suspended within the tube by four screws, only two of which are shown in the cut. ^m. 8t. SEC. I.J SURVEYOR'S TRAI^SIT. 159 The object-glass is' carried out or in by a pinion working in a rack attached to the slide, and thus adjusted to objects either near or remote as desired. The eye-piece is made up of four plane convex lenses, which, beginning at the eye end, are called respectively the eye, the field, the amplifying, and the object lenses, the whole forming a compound microscope having its focus in the plane of the cross- wire ring BB. The eye-piece is brought to its proper focus usually by twisting its milled end, the spiral movement within carrying the eye-tube out or in as desired ; sometimes a pinion, like that which focuses the object-glass, is employed for the same purpose. 182. In order that the telescope may be directed to an object with precision, two spider's lines, or fine wires, are fixed at right angles to each other, and placed within the barrel of the tele- scope, and at the focus of the eye-glass. The cross-wire diaphragm is a small ring of metal suspended in the tube of the telescope by four capstan-head screws, ff, gg, Fig. 88. The ring can thus be moved in either direction by working the screws with an adjusting-pin. Across the flat surface of the ring two fine fibres of spider's web, or, better, very fine platinum-wire, are extended at right angles to each other, their ends being cemented into fine lines cut in the metal of the ring. The intersection of the wires forms a very minute point whicii, when they are adjusted, determines the optical axis of the telescope, and enables the surveyor to fix it upon an object with the greatest precision. 160 ELEMENTS OF SURVETIl^G. [BOOK IV. The openiugs in the telescope tube are made considerably larger than the screws, so that, when these are loosened, the ring may be moved a short distance in the direction of the length of either wire for purposes of adjustment as hereafter described, and may also be slightly turned for the same purpose. 183. To measure horizontal angles correctly, the limb of the instrument must be made truly horizontal. To level the instrument, extend the legs and place them so as to bring each bubble as nearly as possible to the middle of its tube. Turn the vernier plate till one of the levels is parallel to one pair of leveling screws, the other level will be parallel to the other pair of screws. If the bubble in either level is not at the middle, turn the leveling screws to which that level is parallel, in contrary directions, thus raising one side of the horizontal plate carrying the graduated circle and lowering the other till the bubble is brought to the middle of that tube ; with the other pair of leveling screws, bring the bubble in the other level to the middle of the tube ; when the bubbles in both levels are at the middle of their respective tubes, the plane of the levels, and, consequently, the graduated circle by which horizontal angles are measured, are truly horizontal. Two opposite leveling screws may be turned in contrary directions, by holding each screw with the thumb and forefinger, and turning so that both thumbs turn in, or both out. When the thumbs are turned outward, the left- hand side of the circle is raised ; when both are turned inward, the right-hand side is raised. The bubble always runs to the end of the level which is too high. If the leveling screws become jammed, so as to work hard, turn one of them only, forward or backward, till they are free again ; sometimes they work hard, from setting the other pair tight while at a large angle of inclination with the lower plate, in which case the other pair must be loosened again. I SEC. I.] SURVEYOR'S TRAIS'SIT. 161 ADJUSTMENTS OF THE TRANSIT. Before using the instrument, it must be adjusted; that is, the parts must be brought to their proper relative positions. There are four principal adjustments. First Adjustmeii^t. — To malce the axes of the levels, on the linib, perpendicular to the axis of the i7istrument. 184. Turn the horizontal limb until one of the levels is parallel to one pair of leveling screws ; the other will be parallel to the other pair. Bring the bubble in each tube to the middle with its parallel pair of leveling screws ; after which turn the horizontal limb half way round to reverse the levels, and if the bubbles remain in the middle of the tubes, the levels are properly adjusted. But if either bubble recedes from the centre, that level must be adjusted by raising the lower or depressing the higher end, one half by the leveling screws, and one half by the screws which fasten the level to the plate. Again reverse the levels, and if the bubble does not remain in the middle of the tube, correct the error as before, and repeat the operation till the bubble remains in the middle during an entire revolution of the plate. Each level must be adjusted separately. Second Adjustment. — To fix the intersection of the cross- ivires in the axis of the telescope, which is called the line of collimation. 185. Having screw^ed the tripod to the instrument, extend the legs, and place them firmly. Level carefully, then loosen the clamp-screw Q, of the vernier plate, and direct the telescope to a small, well-defined, and distant object. Slide the eye-glass of the telescope till the cross-wires are distinctly seen ; then with the 162 ELEMEi^TS OF SURVEYIi?-G. [BOOK IV. thumb-screw, which forces out and draws in the object-glass, adjust this glass to its proper focus, when the object, as well as the cross-wires, will be distinctly seen ; clamp the vernier plate and, by the tangent-screw R, bring the intersection of the cross- wires exactly upon a well-defined point of the object. Now move the eye from side to side across the eye-glass ; if there results any movement of the cross-wires away from the point sighted, it is evidence that the image of the point does not fall exactly upon the plane of the cross-wires, and the object-glass must be again focussed, till no such displacement can be detected. This displacement is called ** parallax." Having done this, sight some well-defined point, or the mid- dle of a pin, distant two or three hundred feet, and having clamped tbe horizontal motions, revolve the telescope about its horizontal axis and sight a pin in the opposite direction from the first and at the same distance from the plumb ; now loosen the lower clamp and, revolving the instrument about the vertical axis, sight the middle of the first pin again, and clamp ; reverse the telescope upon the horizontal axis, and if it now cuts the second pin, as before, the adjustment is correct; but if the intersection of the wires falls on one side of the second pin, bring it back over one quarter of the error, by the capstan screws on each side of the telescope over the cross- wire ring. As the eye-piece inverts the position of the wires, the operation of loos- ening one of the screws and tightening the other on the opposite side, must be conducted as if to increase the error observed. To insure the accuracy of the adjustment, repeat the whole operation as above described, until the reversals cut the pins exactly. 186. The reason for the above directions will be apparent upon examination of Fig. 89. SEC. I.] SURVEYOR'S TRANSIT. 163 Let AA' represent the line of collimation pro- longed both ways from the instrument at /, and sup- pose that the intersection falls at B, the cross- wires not being in adjustment; on reversing the telescope the intersection will fall at B', as far to the left of A' as B was to the right ot A; if now we turn the instrument 180°, by the horizontal limb, the intersection will fall at C, to the left of A, CA being equal to BA, and on reversal the inter- section will fall at C, CA' being equal to A'B' ; if the cross- wires be moved over G'A', the adjustment will be accomplished. But it is not well to risk introducing possible errors of gradua- tion of the horizontal limb, or errors of reading, into the opera- tion, and therefore instead of turning the instrument through 180°, the surveyor turns till he sights B the second time, thus passing over 180° ±(7^; this causes C to fall at C", C'G" being equal to GB, and therefore B'G" must be /owr times G'A', or four times the error of adjustment. 187. The adjustment of the horizontal wire, which is only an approximation to perfect adjustment, is accomplished thus : Set the zero of the vertical limb at the zero of its vernier, and clamp; sight some sharply defined point upon a wall or staff, distant about 200 feet ; unclamp and reverse the telescope upon its horizontal axis and set the zero of the vernier at 180° on the vertical limb ; now unclamp the lower clamp and revolve the instrument upon its vertical axis till the chosen point is again in the field of view ; if the point and the intersection of the cross-wires do not coincide, correct half the error by the screws, moving the cross-wire ring up or down, and the remaining half error by the leveling screws ; repeat the operations till the error is wholly corrected. 164 ELE3IEXTS OF SURVEYING. [BOOK IT. If the instrument should not be furnished with a vertical limb, the adjustment of the horizontal hair can only be guessed at, by bringing it to the centre of the field of the telescope. Thied Adjustment. — To make the axis of the attached level of the telescope parallel to the line of coUimation, 188. First level the instrument carefully, and with the clamp and tangent movement to the axis, make the telescope horizontal as near as may be with the eye ; then having the line of collima- tion previously adjusted, drive a stake at a convenient distance, say from one to two hundred feet, and note the height cut by the horizontal cross-wire upon a staff set on the top of the stake. Fix another stake in the opposite direction and at the same dis- tance from the instrument, and, without disturbing the telescope, turn the instrument upon its spindle, set the staff upon the stake, and drive in the ground until the same height is indicated as in the first observation. The top of the two stakes will then be in the same horizontal line, however much the telescope may be out of level. Now remove the instrument from fifty to one hundred feet to one side of either of the stakes, and in line with both ; again level the instrument, clamp the telescope as nearly horizontal as may be, and note the heights indicated upon the staff placed first upon the nearer and then upon the more distant stake. If both readings do not agree, correct nearly the whole error of the more distant reading by the tangent screw, and continue correcting and reading until the two readings agree ; now place a stake at a point in line T\ith the two already fixed and about one hundred feet beyond the more remote, and mark upon it the point cut by the cross-wires, and mark also the corresponding point upon the more distant of the first two stakes ; a line passing through these two marks will be a horizontal or level SEC. I.] SURVEYOE'S TRANSIT. 165 line with reference to the first position of the instrument, but not with reference to its present position. Set the instrument again at its first position, and by the tangent-screw make the readings, aboye or below the marks, the same ; the line of collimation will now be horizontal, and the bubble of the attached level must be brought, very carefully, to the middle of its run by the screws at the end of the level tube. Fourth Adjustment. — To make the axis of the vertical limb perpendimilar to the axis of the iiistrument. 189. Bring the intersection of the cross-wires of the tele- scope upon a plumb-line, or any well-defined vertical object, and move the telescope with the thumb-screw 0; if the inter- section of the cross-wires continues on the vertical line, the axis is horizontal. Or, the adjustment may be effected thus : Direct the inter- section of the cross-wires to a well-defined point that is consider- ably elevated ; then turn the vertical limb, until the cross- wires cut some other well-defined point, upon or near the ground; revolve the telescope on its axis, and turn the vernier plate 180° ; then, if in elevating and depressing the telescope the line of collimation passes through the two points before noted, the axis is horizontai If it be found, by either of the above methods, that the axis is not horizontal, it must be made so by adjusting the standards which support the telescope. One of these standards is made adjustable, and by raising the standard at the end towards which the deviation from the vertical occurs, or depressing the other, the adjustment is made. Note. — In making the third adjustment, the curvature of the earth is taken into account that the adjustment may be theoretically correct. 166 ELEMENTS OF SURVEYIifG. [BOOK IV. SECTION II. MEASUREMENT OF ANGLES. 190. To Measure with the Transit a Horizontal Angle subtended by two objects. — Place the axis of the instrument directly over the point at which the angle is to be measured. This is effected by means of a plumb, suspended from the centre of the plate which forms the upper end of the tripod. Haying made the limb truly level, place the of the vernier at any exact degree (merely to avoid minutes and seconds in the first reading), and fasten the clamp screw Q of the vernier plate. Then, facing in the direction between the lines which subtend the angle to be measured, loosen the lower clamp and sight one of the objects very nearly, without wasting time in trying to secure perfect bisection by the cross-wires ; tighten the lower clamp and make perfect bisection by the lower tangent screw. This being done, loosen the clamp-screw Q of the vernier plate, and direct the telescope to the other object ; the arc passed over by the point of the vernier, is the measure of the angle sought ; the difference of the two readings (if the 0° of the limb be not passed in turning to the second object), is the required angle ; if the 0° be passed over, then add 360° to the smaller reading and subtract the greater reading from this sum. Alivays le careful to tahe both readings from the same vernier. 1 Note 1. — In measuring horizontal angles, it does not matter whether the telescope has to be elevated or depressed to sight either or both of the objects, since the telescope revolves on its axis in a vertical plane, and the angles measured are always the horizontal projections of the angle. SEC. II.] MEASUREMENT OF ANGLES. 167 Note. 2. — When great accuracy is desired, a repetition, or several repetitions of the measure may be made, and a mean of the observations taken as the true measure. (See Arts. 254, 255.) In the measurement of vertical angles, it is necessary to understand, first: 191. The method of determining the index error of the vertical limb. Having leveled the horizontal limb, direct the telescope to some distinctly marked object, as the top of a chimney, and read the instrument. Eevolve the telescope on its axis and turn the vernier plate 180°, and having directed the telescope to the same object again, read the instrument. If the two readings are the same, the limb is adjusted ; that is, the of the limb coincides with the of its vernier, when the axis of the telescope is parallel to the horizontal limb. When the reading, found with the telescope in the first position, is greater than that obtained in the reversed position, the true elevation of the object, which is equal to a mean of the readings, may be obtained by subtracting half the difference from the first reading. If the first reading is less than the second, the half difference must be added to the first. Hence, To find the index error, take the reading of the limb when the telescope is directed to a fixed object, a)id then with the telescope and vernier plate both reversed. Take half the difference of these readings, and affect it with a minus sign if the first is the greater, or a plus sign if the second is the greater ; this is equal to the index error. Let the operation be repeated several times, using different objects, and a mean of the errors will be more correct than the result of a single observation. Then, second : 192. Having determined the index error, let the axis of the 168 ELEMENTS OF SURVEYIi?"G. [book IY. telescope be directed to any poiut either above or below the plane of the limb, and read the arc indicated by the of the vernier. To the arc so read apply the proper correction, if any, and the result will be the true angle of elevation or depression. The angle of elevation may be more correctly found by taking the elevation of the object, and repeating the observation with the telescope and vernier plate reversed, and then taking a mean of the readings for the angle required. 193. The true azimuth of a line or course is the angle which the vertical plane through it makes with the plane of the meridian. The azimuth of a line or course referred to some preceding course, or to any given line, is the angle made by the line or course with the prolongation of the line of reference or of a parallel to it through the angular point, the measurement being made around to the right. Thus, The azimuth of BC with AB, (Fig. 90), is the angle RBC ; the azimuth of CD with ^^ is the angle SOD (SQ being parallel to RA). ^P Fig. 90. 194. To find with the transit the azimuths of several successive courses with a given first course. — Place the transit at B (Fig. 91) and level it ; make the zero of the vernier coincide with the zero of the horizontal limb, and clamp the vernier plate ; direct the telescope to A, and clamp the limb; revolve the tele- scope on its horizontal axis, and it will then point in the direction of BE, the prolongation of AB ; un- R SEC. II.] MEASUREMENT OF ANGLES. 169 clamp the vernier plate and direct the telescope to C ; the reading will be the angle RBC, the azimuth of BC with AB. Clamp the vernier plate and remove the instrument to C ; reverse the telescope on its horizontal axis, loosen the lower clamp and sight B ; the horizontal limb now has its zero point in the direction of QPy or its parallel AR, as it had at B ; tighten the lower clamp and revolve the telescope on its axis ; unclamp the vernier plate, direct the telescope to D, and the reading will be the angle QCD, which CD makes with PQy or its parallel AR, and is the azimuth of CD with AB, Clamp the vernier plate and remove the transit to D ; reverse the telescope on its horizontal axis, loosen the lower clamp and sight C ; the limb will then have its zero point in the direction T8, or its parallel AR, as it had at C and B ; tighten the lower clamp and revolve the telescope on its axis ; unclamp the vernier plate and direct the telescope to E ; the reading will be the angle TDE, which DE makes with T8, or its parallel RA, and is the azimuth of DE with AB. Proceed in like manner with any number of successive courses. If the courses enclose a field, the reading at the last station, sighting to the first station occupied by the transit, should be 360°. The course AB, with respect to which the azimuths are taken, is called the Meridian of the Survey. 195. The magnetic bearing of a line or course, is the angle which it makes with the magnetic meridian, and its true bearing is the angle which it makes with the true meridian. In finding the area of a piece of ground, it is not necessary to have either the true or the magnetic bearing. It is sufficient to have the bearings of the several successive courses with respect to one of the courses taken as a meridian. These may be found from the azimuths as f oUows : 170 ELEMENTS OF SURVEYING. [book IV. First suppose a north and south line, and an east and west line to be drawn, and the graduation to be made from 0° to 360°, as represented in Fig. 92 ; let the course taken as the meridian be represented by the line JVS; then when the azimuth of the second course with the first, or meridian, is less than 90°, it is the bearing, and since the course lies between N and B, the bearing is JVU ; when the azimuth is 90°, the bearing is due east ; when the azimuth is between 90° and 180°, the course lies between S and F, the bearing with the first course, or meridian, will be SF, and may be obtained by subtracting the azimuth from 180° ; when the azimuth is 180°, the bearing is due south ; when the azimuth is between 180° and 270°, the course lies between S and W, the bearing is, therefore, SW, and may be obtained by subtracting 180° from the azimuth ; when the azimuth is 270°, the bearing is due west ; when the azimuth is between 270° and 360°, the course lies between iV and TF, the bear- ing is JVW, and may be obtained by subtracting the azimuth from 360°. For example: Let AB (Fig. 93) be taken as the meridian, and let the azimuths of the several courses with AB be as in the table ; then will the bearings of the several courses with AB, be as noted in the TABLE. station. Azimuth with AJB. Bearing with AB. A 0° North B 93° 30' S 86° 30' E C 175° 30' S 4°30'E D 257° S77° W SEC. II.l MEASUREMENT OF ANGLES. in 196. If it is desired to find the true or magnetic bearing of a course from its azimuth with a given course taken as a meridian, it may be obtained by finding the true or magnetic bearing of the course taken as meridian and subtracting it from, or adding it to, the azimuth of the given course, according as the bearing of the reference line "is NW or NE ; the result if less than 90° is the true, or magnetic bearing, and is NE ; if more than 90° and less than 180°, subtract it from 180°, and the result will be the bearing, 8E ; &c. For illustration, take the example in the last article, and let the magnetic bearing of the meridian course, or reference line, AB, be N 31| W. TABLE, Station. Azimuth with AB. Bearing with AB. Magnetic Bearing. A 0° North. N n\ w B 93° 30' S 86° 30' E ]S[62° E C 175° 30' S 4° 30' E S 36° E D 257° S 77° W S 45° 30' W If BCy of which the magnetic bearing is N 62° E, had been taken as the reference line, its bearing would have been added to the azimuths to obtain tha magnetic bearings of the successive courses, as follows : Station. Azimuth with B C. Bearing with BC, Magnetic Bearing. A 226° 30' S 86° 30' W N 31i W B 0° North. N 62° E C 82° N82° E S 36° E D 163° 30' S 16° 30' E S 45i W Note. — For 8E and SW in bearing of reference line, read iVPTand NE, respectively, in applying the above rule. 172 ELEMENTS OF SURVEYING. [BOOK IV. To find the True Meridian with the Transit. 197. Before making the observations it will be necessary to devise some means by which the cross- wires may be lighted, that they may be distinctly visible. To do this, take a board of about one foot square, paste white paper upon it and perforate it through the centre ; the diameter of the hole being somewhat larger than the diameter of the telescope of the transit. Let this board be so fixed to a vertical staff as to slide up and down freely; and let a small piece of board, about three inches square, be nailed to the lower edge of it for the purpose of holding a candle. About twenty-five minutes before the time of the greatest eastern or western elongation of the pole-star, as shown by the table of elongations (see Art. 164), let the transit be placed at a convenient point and leveled. Let the board be placed about one foot in front of the transit, a lamp or candle placed on the shelf at its lower edge, and let the board be slipped up or down until the pole-star can be seen through the hole. The light reflected from the paper will show the cross- wires in the telescope of the transit. Then let the vertical cross- wire be brought exactly upon the pole-star, and, if it is an eastern elongation that is to be observed and the star has not yet reached the most east- erly point, it will move from the line toward the east, and the reverse when the elongation is west. At the time the star attains its greatest elongation, it will appear to coincide with the vertical cross-wire for some time, and then leave it in the direction contrary to its former motion. As the star moves toward the point of greatest elongation, the telescope must be continually directed to it by means of the tangent-screw of the vernier plate ; and when the star has SEC. II.] MEASUREMEi^T OE AI^GLES. 173 attained its greatest elongation, great care should be taken that the instrument be not afterward moved. Next turn the telescope very carefully upon its horizontal axis and fix a peg in the ground, distant 150 or 200 feet ; to do this, let the light of the lantern shine through a small hole in a board, across the centre of which hole a plumb-line hangs, and by sighting the line thus seen the peg may be fixed. Also mark the point directly under the transit plumb ; the line passing through this point and the staff, makes an angle with the true meridian equal to the azimuth of the pole-star. From the table of azimuths (see Art. 164), take the azimuth corresponding to the year and nearest latitude. If the observed elongation was east, the true meridian lies on the west of the line which has been found, and makes with such line an angle equal to the azimuth. If the elongation was west, the true meridian lies on the east of the line found; and, in either case, laying off the azimuth angle with the transit, gives the true meridian. 198. Another method depends upon the fact that at the same angular distances east and west of the meridian the alti- tudes of any selected star are equal. Direct the transit to any bright star towards the south, and east of the meridian, and clamp the vertical limb ; carefully read the horizontal limb ; do not disturb the clamp and tangent of the vertical limb in any way, but loosen the clamp of the vernier plate, and after a suflBcient interval of time, two or three hours perhaps, as it is not good practice to set the altitude when the star is too near the meridian, turn the plate upon its spindle towards the west ; bring the star upon the cross-wire intersection as its altitude decreases, moving the vernier plate only ; it will now have the same altitude as before; read the horizontal limb ; take half the difference of the two readings, and set this half angle back towards the east ; the line thus determined is the meridian. 174 ELEilEXTS OF SURVETINTG. [book IV. Fig. 94. APPLICATIONS TO HEIGHTS AND DISTANCES. 199. To determine the horizontal distance to a point which is inaccessible by reason of an intervening river. — Let C be the point (Fig. 94). Measure along the bank of the river a hori- zontal base-line AB, and select the stations A and B, in such a manner that each can be seen from the other, and the point C from both of them. Then measure the horizontal angles CAB and CBA, with the transit. Let us suppose that we have measured AB =zmO yards ; CAB = A=bT 35", and CBA = B = 6^° 51'. Then, C= 180° - {A + B) = 57° 34'. To find the distance BC. sin C : sin A :: AB : BC. Applying logarithms, we have, (a. c.) log sin C (57° 34') .... 0.073649 log sin ^ (57° 35') .... 9.926431 logAB (600) 2.778151 log ^6' 600.11 2.778231 To find the distance AC. sin C : sm B :: AB : AC, and applying logarithms, we have, SEC. II.] MEASUKEMEKT OF ANGLES. (a. c.) log sin (7 (57° 34') .... 0.073649 log sin ^ (64° 51') .... 9.956744 logAB (600) 2.778151 log^C 643.94 2.808544 175 B< Fig. 95. To determine the altitude of an inaccessible object above a given horizontal plane. FIRST METHOD. 200. Suppose D to be an inac- cessible object, and BC the horizon- tal plane from which the altitude is to be measured ; then, if we suppose DC to be a vertical line, it will repre- sent the required distance. Measure any horizontal base-line, as BA ; and at the extremities B and A, measure the horizontal angles CBA and CAB. Measure, also, the angle of elevation DBC, ' Then, iu the triangle CBA, there will be known two angles and the side AB ; the side BC can therefore be found by calculation. Having found BC, we shall have, in the right- angled triangle J)BC, the base BC and the angle at the base, to find the perpendicular DC, which measures the altitude of the point D above the horizontal plane BC. Let us suppose that we have found, by measurement, BA = 780 yards. The horizontal angle CBA = B = 4.1° 24', the horizontal angle CAB = A = m° 28', and the angle of elevation VBC^ 10° 43'. l'^6 ELEMENTS OF SUEVEYING. [BOOK IT, To find, in the triangle BCA, the horizontal distance BC. The angle BCA = C = 180° — {A + B) = 42° 08'. Then, sin (7 : sin .4 :: AB \ BC', and applying logarithms, we have, (a. c.) log sin C (42° 08') .... 0.173369 log sin A (96° 28') .... 9.997228 log^^ (780) 2.8920 95 \ogBC 1155.29 yards . . . 3.062692 In the right-angled triangle DCB, to find DC. We have, from Theorem IV, R : tan BBC :: BC : DC. Applying logarithms, we have, (a. c.) logi? (90°) .... 0.000000 log tan i)^C (10° 43') . . . 9.277043 logBC (1155.29) . . . 3.0626 92 log DC 218.64 .... 2 .339735 Note 1. — It might, at first, appear, that the solution given requires that the points B and A should be in the same horizontal plane ; but it is entirely independent of such a supposition. For, the horizontal distance represented by BA is the same, whether the station A is on the same level with B, above it, or below it. The horizontal angles CAB and CBA are also the same, so long as the point C is in the vertical line DC. Therefore, if the horizontal line through A should cut the ver- tical line DC, at any point, as B, above or below C, AB would still be the horizontal distance between B and A, and AB would be the horizontal distance between A and C. If SEC. II.] [EASUKEMENT OF ANGLES. 177 If at Ay we measure the angle of elevation at the point D, we shall know in the right-angled triangle DAE, the base AU and the angle at the base ; from which the perpendicular DB can be determined. Let us suppose that we had measured the angle of elevation DAEy and found it equal to 20° 15'. First: In the triangle BAC, to find AC, or its equal AE. sin (7 : sm B :: AB : AC or AB. Applying logarithms, we have, (a. c.) log sin C (42° 08') .... 0.173369 log sin B (41° 24') .... 9.820406 log AB (780) 2.8920 95 log AB 768.9 ...... 2.885870 In the right-angled triangle DAE, to find DE. We have, from Theorem IV, E : t3inA : : AB (a. c.) logi^ (90°) . . log tan A (20° 15") . log^^ (768.9) . . logZ>^ 283.66 . . Now, since DC is less than DB, it follows that the station B is above the station A. That is, DB— DC = 283.66 — 218.64 = 65.02 = BC, which expresses the vertical distance that the station B is above the station A, DB : hence, 0.000000 9.566932 2.8 85870 2.452802 178 ELEMEIfTS OF SURVEYmG. [book IV. Note 2. — It should be remembered that the vertical distance, which is obtained by the calculation, is estimated from a hori- zontal line passing through the eye, at the time of observation. Hence, the height of the instrument is to be added, in order to obtain the true result. SECOND METHOD. 201. When the nature of the ground will admit of it, measure a base-line AB, in the direction of the object D, Then measure, with the instrument, the angles of elevation at A and B. Then, since the out- ward angle BBC is equal to the sum of the angles A and ADB, it follows that the angle ABB is equal to the difference of the angles of elevation at A and B. Hence, we can find all the parts of the triangle ABB. Having found BB, and knowing the angle BBC, we can find the altitude BC. This method supposes that the stations A and B are on the same horizontal plane; and therefore it can only be used when the line ^^ is nearly horizontal. Let ns suppose that we have measured the base-line and the two angles of elevation, and found, AB = 975 yards, A = 15" 36', and DBC= 27° 29'; required the altitude DC, Arts, i)a= 587.61 yards. SEC. II.] MEASUREMENT OF AN"GLES. 179 To determine the perpendicular distance of an object below a given horizontal plane. 202. Suppose G to be directly over the given object, and A the point through which the horizontal plane is supposed to pass. Measure a horizontal base-line AB, and at the stations A and B conceive the two horizontal lines AG, BG, to be drawn. The oblique fig. 98. lines from A and B, to the object, are the hypothenuses of two right-angled triangles, of which AG, BG, are the bases. The perpendiculars of these triangles are the distances from the horizontal lines AG, BG, to the object. If we turn the triangles about their bases AG, BG, until they become horizontal, the object, in the first case, will fall at G', and in the second at G". Measure the horizontal angles GAB, GBA, and also the angles of depression G'AG, G"BG. Suppose that we have measured, and found AB =: 672 yards ; BAG= 72° 29'; ^^(7= 39° 20'; angle of depression G'AG = 27° 49', and G"BG=ir 10'. First: In the triangle ABG, the horizontal angle AGB =: lSO°-{A-JrB) = 180°-111° 49' = 68° 11'. To find the horizontal distance AC. sin G : sin B :: AB : AG; hence, (a. c.) log sin G (68° 11') 0.032275 log sin B (39° 20') 9.801973 log^^ (672) 2.8273 69 \ogAG 458.79 2.661617 180 ELEMEJ^TS OF SURVEYING. [BOOK lY. To find the horizontal distance BC. sin C \ sin .1 :: AB : BC ; whence, (a. c.) log sin C (68° 11') 0.032275 log sin A (72° 29') 9.979380 log^^ (672) 2.8273 69 log^C 690.28 2.839024 In the right-angled triangle CAC, to find CC We have, Theorem IV., R : idJiA :: AC : CC; whence, (a. c.) log R (90°) 0.000000 log tan A (27° 49') 9.722315 log AC 458.79 2.661617 log CC 242.06 2.383932 In the triangle CBC", to find CC". We have, Theorem IV., R : tsinB :: BC : CC" j whence, (a. c) logR (90°) ...... 0.000000 log tan B (19° 10') 9.541061 log^C (690.28) 2.839024 log CC" 239.93 2.380085 Hence, also, CC — CC = 242.06 — 239.93 = 2.13 yards; which is the height of station A above station B, i i SEC. II.J MEASUREMENT OF ANGLES. 181 203. P R O B L E M S . 1. Wanting to know the distance between two inaccessible objects, which he in a direct level line from the bottom of a tower 120 feet in height, the angles of depression are measured from the top of the tower, and are found to be, of the nearer 57°, and of the more remote 25° 30' ; required the distance between the objects. Ans. 173.656 feet. 2. In order to find the distance between two trees, A and B, which could not be directly measured because of a pool which occupied the intermediate space, the dis- tances of a third point C from each of them were measured, and also the included angle A CB J it was found that. CB = 672 yards, CA = 588 yards. ACB = 55° 40'; required the distance AB. Ans. 592.967 yards. 3. Being on a horizontal plane, and wanting to ascertain the height of a tower standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51° ; then measuring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45' : required the height of the tower. Ans. 83.998. 4. Wanting to know the horizontal distance between two inaccessible objects JE and W, the following measurements were made; 182 ELEMENTS OF SURVEYIIS'G. [book IV. ^ Ei\ ' AB = 536 yards BAW= 40° 16' viz.: \ WAE = 57° 40' ABE = 42° 22' EBW = 71° 07'; required the distance EW, Ans. 939.617 yards. fig. loo. 5. Wanting to know the horizontal distance between two inaccessible objects A and B, and not finding any station from which both of them could be seen, two points C and D were chosen at a distance from each other equal to 200 yards ; from the former of these points A could be seen, and from the latter B, and at each of the points C and D a staff was set up. From C a distance CF was measured, not in the direction DC, equal to 200 yards, and from D a distance DE equal to 200 yards, and the following angles taken, ^ AFC = 83° 00', BI)E = 54° 30', viz.: \ ACD = 53° 30', BDC = 156° 25', ^AGF = 54° 31', BED = 88° 30'. A?is. AB = 345.459 yards. 6. From a station P there can be seen three objects. A, B, and (7, whose distances from each other are known : viz., AB = 800, AC= 600, and BC= 400 yards. Now, there are measured the horizontal angles, AFC = 33° 45', and BFC = 22° 30' ; it is required to find the three distances, PA, PC, and PB» SEC. II.] MEASUREMEN^T OF ANGLES. 183 GEOMETRICALLY. i With the three given sides construct the triangle ABC. Then, at A lay off the angle BAD z=z 22° 30', and at B the angle ABD = 33° 45', and note D, the point at which the two lines intersect. Through the points A, D, and B, de- scribe the circumference of a circle, and through G and D draw the line GDP \ the point P in which it intersects the circumference, will be the position of the station. By observing the equal angles in the figure, the trigono- metrical solution is not difficult. We find, Fig. 102. (PA = 710.198 yards. Ans. \PG — 1042.524 '' PB= 934.289 " Note. — This problem is much used in maritime surveying, for the purpose of locating buoys and sounding-boats. The trigonometrical solution is somewhat tedious, but the geomet- rical solution is very easy, as shown above. 184 ELEMENTS OF SURVEYING. [BOOK IV. SECTION III. RANGING OUT LINES, ETC. 204. To range out a line with the transit, place the instru- ment, carefully adjusted, over the first station ; direct the telescope to a distant and well-defined point in the desired line, and clamp both the vernier plate and the horizontal limb. The line of sight of the telescope is then in the vertical plane of the given line, so that points on the surface bisected by the intersection of the cross- wires will be in the required line; let an assistant, directed into the line by the observer at the transit, fix ranging-rods, or stakes conspicuously marked, as far as the power of the telescope extends. Eemove the transit to the third or fourth stake from the last set, and place it precisely over that position by plumb-bob, and adjust it for observation ; the telescope is ranged in the line by sighting backwards and forwards to the stakes already set. The line is then continued as before. If great accuracy be required, each operation must be repeated with telescope reversed, as only in this way can error in adjustment of cross-wires be eliminated. If the sighting with reversed telescope does not agree with the former sighting, ake a point midway between the two points sighted, as a point in the required line. 205. A line may be traced in forests or plantings, in which there are no general surface obstructions, by the aid of auxiliary parallel lines. In the illustration, Fig. 103, aa, Ih, cc, are the parallel lines, and aa, cc, the auxiliary lines. AB is a line in which h is a given point. The distances ab, he, in this line should be measured. SEC. III.] BANGII^G OUT LINES. 185 and also the angle which hh makes with AB. The line hb and the auxiliary lines should be traced on this angle, until the trace of one of the parallel lines be obstructed by a tree. Bs ^, B3 x.<^~- XJ. Fig. 103. such as hi at (1). Immediately on passing the obstruction on one of the auxiliary lines, a line should be traced on the measured angle or its supplement, as may be required, and traced to intersect the other auxiliary line. The angle made by these lines should be measured at the point of intersection to verify the trace of the intersecting lines. From these angular points in the auxiliary lines, distances to the point (1), equal to ab and cb, respectively, should be measured in the transverse line, and found to meet, but not overlie, one another. Then will the point of meeting in the transverse line be a forward point in the line bb. At a suitable distance forward from which the point (1) may be observed, a like determination of another point in bb should be made. The trace of the line bb should be taken upon these points and continued in connection with the auxiliary lines until the trace of one of the lines be obstructed, such as the line aa at (2). The trace of the obstructed line Bhould be taken up by measurements in the transverse line 186 ELEMENTS OE SURVEYING. [book IV. B^Ai, and in a forward parallel line, and the traces continued as described above. In like manner the obstructions at (3), (4), &c., may be passed, and the trace of the line continued for considerable distances with sufficient accuracy for most practical purposes. The continued parallelism of the lines at the meas- ured distance apart will be a sufficient verification (Smith's Treatise on Land Surveying.) 206. To measure distances by means of the transit. — The cross- wire ring in the telescope of the transit is often fitted with an arrangement called "Stadia," or *^ Micrometer." The Stadia, or Mi- crometer, is a compound cross- wire ring or dia- phragm, shown in Fig- ures 104 and 105, having three horizontal wires, of which the middle one is cemented to the ring as usual, while the others, bb and cc, are fastened to small slides, held apart by a slender brass spring hoop, and actuated by independent screws, dd, by which the distance between the two movable wires can be adjusted to include a given space upon a rod, held perpendicular to the optical axis in front of the object-glass, at a distance from it equal to its principal focal distance. When the wires are thus adjusted to include a certain space, as two feet for instance, upon a rod placed at a distance of 100 feet from the specified point on the optical axis, it is found that they will cover one foot at half that distance, or four feet at a distance of 200 feet ; thus the distance is proportional to the space intercepted upon the rod. By adding to the distance Fig. 105. SEC. III.] BANGING OUT LINES. 187 thus obtained, the principal focal distance of the object-glass, plus the distance of the object-glass from the middle of the horizontal axis, the distance of the rod from the station can be ascertained without the use of a chain. The focal distance of the object-glass can be readily obtained by sighting some very distant object, being careful to correct instrumental parallax (Art. 185), and then measuring the dis- tance from the object-glass to the capstan screws of the cross- wire ring, which call a ; now sight some object, distant about 100 feet, and measure the distance from the object-glass to the horizontal axis, which call h ; the sum of these, a-{-bf will be a constant, sufficiently exact, to be added to all distances obtained by readings on the rod. The spaces upon the rod used should be equal to that which the instrument intercepts at 100 feet from the point in front of the object-glass, and should be numbered from the bottom up ; each space should be subdivided into hundredths. The rod should have two movable targets, like those used upon leveling rods, and should also be furnished with an attached plumb, or level, to insure its vertical position. A distinct mark should be made upon it at the ordinary height of the horizontal axis of the instrument. In using the micrometer, sight the middle horizontal hair to the ^* height of instrument " mark^ and then direct the targets to be moved successively till they coincide with the micrometer wires, the rod being kept vertical. If the telescope has been level during this operation, the distance given by the rod plus the instrument constant, can be recorded ; but if the line of sight has been elevated or depressed, then from the distance given by the rod, including the instrument constant, must be subtracted the product of this distance and the square of the sine of the angle of deviation from the horizontal. If distances greater than 600 feet are to be measured, the 188 ELEMENTS OF SURYEYING. fBOOK IV. unit of the rod must be less than ^V of the standard distance, to avoid the use of a rod too long for practical management. If the distances are to be recorded in chains and links, set the stadia staff at 66 feet in order to obtain its unit, and then graduate it to this unit, and subdivide to hundredths. The stadia staff should be spiked, that it may be thrust into the ground to secure steadiness. The telescope should be a good one, giving a very sharp, clear definition of objects, and the micrometer wires should be very fine indeed in order to secure close readings. The degree of accuracy that may be attained is shown by the following table, deduced from that given in ^^Cours de Topo- graphic, par A. Lehagre, 1881," to which the student is referred for a very full description of the method : Focal length of object-glass in inches 6 8 10 12 14 Distance in feet which may be safely measured .... 600 800 1000 1200 1400 Eelative error in horizontal dis- tnnpp 1 1 1 1 1 UdilOC -g-^ -g-g-Q Yo^oO 1800 1400 With a telescope of ten inches focal length or over, and within the above limits, the stadia measurements are as reliable as chain measurements on fairly level ground, and are much more accurate than chain measures on bad, broken country. As the use of the micrometer requires care and consumes time, it is not recommended for short distances, except on bad ground, swamps, inaccessible distances, &c. 207. In connection with the chain or tape, the transit is used to obtain horizontal distances on sloping ground. The chaining is made on the surface of the sloping ground, and not by elevating the chain as described in Art. 71 ; and the angle of SEC. III.] RANGING OUT LINES. 189 the slope is taken with the transit, by marking on a rod the vertical distance from the horizontal axis of the telescope to the ground, and sighting to this mark on the rod held vertical at the end of the line measured ; the horizontal distance is equal to the measured distance multiplied by the cosine of the angle of the slope. 208. To survey a line, such as a road, boundary of an estate, &c., measure the angle of deviation which each line makes with the preceding line prolonged, or measure the azimuths which each line makes with the first line taken as a meridian (Art. 194), and measure, also, the length of each line and offsets to prominent objects. Care must be taken to centre the instrument exactly over each angular point, as any error in centreing will cause an error in the apparent direction of the object sighted, which will be the greater the nearer the object is to the instrument. 209. To survey the streets of a town or city, place the transit at the intersection of two or more of the principal streets, through which the longest lines of sigh t can be had ; find the angle which each of the streets diverging from this point makes with the principal street, and find, also, the angle of slope of each of the streets at this point ; measure, with the chain or stadia-rod, or both as checks one upon the other, the lengths of the lines of sight, and take offsets to the corners of all streets, to public buildings and prominent objects ; remove the transit to the next street and take the angles, angle of slope, measure- ments, and offsets as before, and so continue till the survey is complete. 190 ELEMENTS OF SURVEYING. [book IV. SECTION IV. FARM SURVEYING BY TRANSIT. 210. The figure and area of any piece of ground raay be found by beginning at any one of the angular points and going entirely around the boundary, measuring the length of the sides by chain or stadia-rod, and the angle which each side makes with the preceding side prolonged, called Angle of Deviation, or the azimuths of the several sides with a given first side as meridian. Let the farm to be surveyed be the one given in Article 123. Let the side AB be taken as the meridian of the survey. Measure with the transit the azimuths of the several successive sides with AB, as directed in Art. 194, and enter them in the notes at the left of the station mark. In the following illustration both the azimuth and the angle of deviation have been entered, though the surveyor would use but one, together with the bearings, which should always be entered as a check : Azimuths. Angles of Deviation, Bearings. 99° 30' 123° 30' / D A 22.89 N 30° 35' E (N 87° 5' E) 336° 00' 100° 55' \ A 1.40 S 87° 5' W (S 7° 50' W) 76° 55' 76° 55' / B A 31.95 N8° E (S 68° 55' E) A A N 68° 55' W From the azimuths determine the bearings of the several sides with AB, as directed in Art. 195, and let them be as noted on the following page ; complete the table and determine the area as in a compass survey : SEC. IV.] FARM SURVEYIXG BY TRANSIT. 191 m tzi SQ .a a "d .9 o 03 J3 , ^ .3 a o ao % X3 o 1-1 2 3 T-l 1-1 (U O ^ Z/ O « a a> ^ > 5S fi o 03 O H J ^ in o -^ o in S ^ 1 2 5^ c^ o 1*1 00 05 OS ti ^ 1 00 «5 in C3D eo' § 00 o (.- in eS eo C2 ^ 00 00 in ^ ^~^ Qt < + o i d ^' 1-1 18 o •^ , 1 50 «5 o 1 o £- s fi ;» JO «o ^ Tf< in O + o eo 05 1-1 eo tH l- l-C i 1-1 in g s in d ?o (TJ ^. .^ o 00 ■<5< eo CO ;£, eo CO eo C5 in 1* CO (N in f + ^ CO 0) Q + 1 + 91 + + 1 1-1 1 1-1 1 s U) o ©I e* 00 ^ in CO O CO ^ a cq ^J 05 CO o: eo Oi o (N 05 .^ s ■fH + O l-C 00 eo t^ ff* (N ai CD CO o* 1 1 T-l 1 1-1 tH + S + + 1 1 1 1 1 1 a 1-1 00 o* ?o & •■^ £ 1^ eo in «o 00 s d 1-1 T-l eo tH s PE) + K1 o ^ ^ IN CO o ^. S ^ S ■^ 00 ^ in ?§ CO eo 1 ^ s. o M S. ^ 35 ^ ^ ^ 1 ■f-< 00 eo t^ oi (N >2 in T-l r-l 1-1 in in • • H in ©« tH 00 TH t^ + 1-1 eo OS 00 bJC M in ^ C5 C5 1-1 tH m tH Ci l:- -w •saouB^sid C5 00 *^ 05 in in Oi CO 05 t-« o l-( 1-1 (N 00 eo' TiJ t-' ci t- ?o ^ eo (N c« 1-1 IH l-< .s txhq .5^ 5 t-. la ^ Eel o eo w E=l in H S in in 8 b4 o Sj o JJ f. o o o o 4, -k^ ^ to ^ s § s; -"^ (N ^ 1-1 00 ^ ^ GO 72 02 02 O! CZ2 z ^-1 In o eo ^ g S ^ o o o o o () o o ^ i s 1-1 S i ?2 •eaoijBjg < PQ O w fe o w HH W o CO -<*< , Oi o 192 ELEMENTS OF SURVEYING. [book IV. 211. If the surveyor does not record compass-bearings, as advised to do in last article, the notes may be kept in precisely the same manner as the field notes of the compass survey, as shown in Article 123 — substituting azimuths for the hearings of the seyeral courses, and angles instead of hearings for the prominent objects sighted to. 212. If the angles of deviation had been measured instead of the azimuths, the method of finding the area would have been substantially the same. The azimuths would have been determined from the angles measured, and the rest of the computation would have remained the same. The azimuth of any course with a course taken as meridian may be found from the angles of deviation, thus : The azimuth of the course taken as meridian is zero ; the azimuth of the second course is equal to its angle of deviation ; the azimuth of any succeeding course is equal to the azimuth of the preceding course increased by the angle of deviation of the course itself, as a simple diagram will show\ if the interior angles had been measured, the angles of deviation would have been obtained by subtracting each interior angle from 180° and the method would have been* stations. Interior Angles. Angles of Deviation. Azimuth with A B. A 98° 55' 81° 05' B 103° 05' 76° 55' 76° 55' C 280° 55' — 100° 55' (-24°) 336° D 56° 30' 123° 30' 99° 30' E 168° 30' 11° 30' 111° F 135° 45° 156° G 151° 35' 28° 25' 184° 25' H 186° 30' -6° 30' 177° 55' I 136° 44° 221° 55' K 123° 57° 278° 55' The other columns as before. SEC. IV.] FARM SURVEYING BY TRANSIT. 193 213. When an angle of deviation lies within the boundary of the survey, as at C, (see page 87), it must be called, negative ; when it lies without the boundary, it is an exterior angle of the polygon and is positive. The azimuth obtained, by the rule may be negative (as it is in the case of the course CD) ; but as the azimuth must be positive, 360^ must be added, to the negative result obtained to get the true azimuth. The algebraic sum of the angles of deviation should be equal to or differ but little from 360°, which fact serves to check the correctness of the angles recorded. 214. Where all the comers of a field may be seen from one of them, the area may be determined, as follows : Place the transit over the point A, Fig. 106, from which the other comers may be seen, and measure the angles BA C, CAD, and DAK Measure the lengths of the diagonals AC and AD, and of the sides AB and AB; then the areas of the triangles ABC, CAD, and DAB, may be separately found from the principle that the area of a triangle is equal to half the product of two sides and the sine of their included angle ; the sum of the areas of the triangles is equal to the area of the field. 215. If a description of the property is also required, the bearing of some one of the lines must be taken, and from this and the recorded angles and distances, the "Bearings and Courses" may be computed. 216. Where all the comers of the field may be seen from a point within, as A, Fig. 107. Place the transit over the point A ; measure the angles BAC, CAD, &c., at A, and the length 194 ELEMENTS OF SURVEYING. [book IV, of the lines AB, AC, &c., from A to the several corners of the field ; find the areas of the tri- angles BAG, CAD, &c., as in the last article, and add them together. If the boundary is irregular, as represented in the figure, measure offsets, calculate the contents of these smaller portions separately, and add or subtract them as may be necessary, to find the true area of the tract. Fig. 107. 217. — E X A M P L E S . 1. Eequired the contents and plot of a piece of land, of which the following are the field notes : stations. Azimuths with AB. Distances. A 0° 15.8 ch. B 34° 17.4 70° 30' 32.56 D 154° 30' 14.88 E 189° 24.96 F 230° 14. G 279° 32.8 A 360° 2. Eequired the contents and plot of a piece of land, of which the following are the field notes : SEC. IV.] FARM SURVEYING BY TRANSIT. 195 Stationp. Interior Angles. Distances, A 92° 30' 31.80 ch. B 94° 30' 2.08 C 155° 15' 2.21 D 179° 30' 35.35 E 94° 15' 21.10 F 104° 31.30 3. Required the area of a piece of land, of which the follow- ing measurements were made : AB = 20 chains AC = 22.57 AD = 28.64 AE = 40.80 AF = 30.95 angle BAC = 37° CAD = 46° 45' DAB = 42° 15' BAB = 26 Fig. 108. Offsets from the line AB were taken as tollows: At 3.00 ch. E offset 2.50 ch. « 6.50 " ' 0.00 " " 9.50 " L ' ' 1.60 " "13.00 " B ' 2.00 " "16.00 " B ' 1.40 " "20.00 " * 0.00 « BOOK V. LAYING OUT AND DIVIDING LAND, SECTION I. OF DIVIDING LAN D. 218. The surveyor is often required to lay off a given quantity of land, in such a way that its bounding lines shall form a particular figure, viz., a square, a rectangle, a triangle, &c. He is also often called upon to divide given pieces of land into parts containing given areas, or, into areas bearing certain relations to each other. The manner of making such divisions must always depend on a skilful and judicious application of the principles of geometry and trigonometry to the particular case. For example, if it were required to lay out an acre of ground, in a square form, it would be necessary to find, by calculation, the side of such a square, and then trace, on the ground, a figure bounded by four equal sides at right angles to each other. 219. To lay out a given quantity of land in a square form. Rule. — Reduce the given area to square chains or square rods ; then extract the square root, and the result will he the side of the required square. This square being described on the ground, will be the figure required. SEC. I.] OF DIVIDING LAND. 197 220. To lay out a given quantity of land in a rectangular form, when one of the sides of the rectangle is given. EuLE. — Divide the given area, reduced^ to square chains or square rods, by the given side of the required rectangle, and the quotient will he the other side. Then, trace the rectangle on the ground. 221. To run a line from the vertex of a triangular field which shall divide it into two parts, having to each other the ratio of m to 72. Let ABC hQ any triangular field. Divide the side BC into two parts, such that (Geom., Bk. IV, Prob. I) BD \ DC \\ m : n; and draw the line AD ; then will ABD : DAC :: m : n. For, the two triangles ABD, ADC, having the same alti- tude, are to each other as their bases (Geom., Bk. IV, P. 6, C.) ; hence, the triangle is divided into parts having the ratio of m to n. 222. To run a line parallel to one side of a triangular field, that shall form with the parts of the two other sides a m triangle equal to the - part of the field. Let CBA represent a triangu- lar field, and CA the side parallel to which the dividing line is to be drawn. On the side BC, take 55' equal Fig. no. 198 ELEMENTS OF SURVEYING. [book V. / 7W / fn to BC \ — , and on the side BA. take BF equal to BA \ — ; 11 ^ ^ n the line EF is the line required ; for, since it divides the sides ^Cand BA proportionally, it is parallel to the side CA (Geom., Bk. IV, P. XVI) ; and from the similar triangles, we have (Geom., Bk. IV, P. XXV), or BEF : BCA BEF ■ BCA BE' : BC 73 hence, 777 7)1 n ; BEF = - BCA. n Example. — Let it be required to divide the triangular field GAB, in which ^C=9ch., AB =z \1 q\\., and CB =. 7 ch., into two such parts that ADE shall be one-fourth of the whole field. In this case, we have, Fio. 111. n hence. wi = 1, AE=4: ch. 50 1. 4, and ^/^^^/\ = \. and AD — 5 ch. 50 1. 223. To run a line from a given point in the boundary of a piece of land, so as to cut off, on either side of the line, a given portion of the field. Make a complete survey of the field, by the rules already given. Let us take, as an example, the field whose area is computed in Ex. 1, Art. 140. That field contains 105 ^. 2 R. 33 P., and Fig. 112 is a plot of it. Let it now be required to run a line from station A in such a manner as to cut off, on the left, any part of the field ; say, 26 A. 2 R. 31 P, SEC. I.] OF DIVIDING LAND. 199 Fig. 112. It is seen, by exam- ining the field, that the division line will proba- bly terminate on the course CD. Therefore, draw a line from A to C, which we will call the first closing line. The bearings and lengths of the courses AB, BC, are always known, and in the present example are found in the table Art. 140, Ex. 1 ; hence, the bearing and distance from G to A can be calculated by Art. 142 ; they are, in this example, Bearing, S 9° 28' E; Course, 23.22 ch. Having calculated the bearing and length of the closing line, find, by the general method, the area which it cuts off; that area, in the present case, is 14 ^. OR. 26 P. It is now evident that the division line must fall on the right of the closing line AG, and must cut off an area AGH, equal to the difference between that already cut off, and the given area; that is, an area equal to 12 A. 2 R. b P, Since the bearing of the next course CD, and the bearing of the closing line A C are both known, the angle A CD which they form with each other, can be calculated, and is in this example, 79° 32'. Hence, knowing the hypothenuse AG, and the angle AGG at the base, the length AO, the perpen- dicular let fall on the course CD can be found, and is 22.82 chains. The base of a triangle is equal to its area divided by half the altitude. Therefore, if the area 12 y1. 2 i?. 5 P. be re- duced to square jhains, and divided by 11.41 chains, which 200 ELEMENTS OF SURVEYING. [BOOK V. is half the perpendicular AG, the quotient, which is 10.95 chains, will be the base CH. Hence, if we lay off from C, on CD, a distance CH, equal to 10.95 chains, and then run the line AH, it will cut off, from the land, the required area, viz., 26 A. 2 B, 31 F, Note 1. — If the part cut off by the first closing line should exceed the given area, the division line will fall on the left of AC. Note 2. — If the difference between the given area and the first area cut off, divided by half the perpendicular AG, gives a quotient larger that the course CD ; then, draw a line from A to D, and consider it as the first closing line, and let fall a perpendicular on D£J, Note 3. — When the point from which the division line is to be drawii falls between the extremities of a course, divide the course into two parts, at this point. Then consider one of the parts as an entire course, and the other as forming a new coarse, having the same bearing. The manner of making the calcula- tion will then be the same as before. 224. To cut off from a field, a given area, by a line running in a given direction. In this case, as in the previous one, a complete and correct survey is first necessary. Then, when the whole area is known, the position of the line may be approximately determined by the inspection of a correct map of the whole. We will take, for illustration, Example 4, Art. 140, of which Fig. 113 is the plot. SEC. I.] OF DIVIDING LAND. 201 Let it be required to cut off from this area, 50 acres, by a line whose bearing shall be S 60° E, or N 60° W. We will make a trial of a line starting at 25 chains from station 6, on the 6th course. We will call this station A, and the trial line AB. In order to determine if the area cut off is equal to the required area, we must first determine the length of ^^ and of B 6. These cannot be determined by the method of sup- plying lost notes. We must first calculate the length of a line, starting at the proposed point, and running to the station nearest to the other extremity of the closing line. In this example, from A to 5. This is easily found to be 36.406 chains, and its bearing N 81° 13' E. Fio. 118. 202 ELEMENTS OF SURVEYIN^G. [book V. Now, in the triangle ^^ 5 we have one side and the angles, to find the remaining parts. ^^ is found to be 28.88 chains and Bh to be 22.81 chains. We have now the complete field- notes of the area cut off. A S 60° E 28.88 ch. B N 28f E 22.81 5 N 57° W 21.10 6 S 47° W 25.00 The area is found to be 58.5029 acres. It now remains to move this line northerly, so that the area contained between its present position and the new one shall be equal to 8.5029 acres. Suppose the lines A 6 and ^ 5 be prolonged till they meet at some point, as F, Fig. 114. Calculate A V and BV, also the area ABV. ^F is found to be 92.19 chains and 5 F 88.18 chains. The area of the triangle ABV, is 127.29 acres. Let MN represent the line sought. Then, we have two similar triangles, with all the sides of the one, and the areas of each, known ; for, VMNmvi^t contain 8.5029 acres less than A VB. Then, AM and BN are easily determined. The complete notes of the area to be cut off, are FiQ. 114. M S 60° E 27.89 N N 28J° E 19,82 5 N 57° W 21.10 6 S 47° W 21.87 SEC. II.] PUBLIC LAI^^DS. 203 Note. — Fields are so variously shaped that it is difficult to give rules that will apply to all cases. It is by practice alone that facility is obtained in that branch of surveying relating to the division of estates. We have given only a few examples that may serve as general guides in the application of the principles of Plane Geometry to such cases as may arise. SECTION II. PUBLIC LANDS OF THE UNITED STATES. 225. Soon after the organization of the government, several of the States ceded to the United States large tracts of wild land, and these, together with the lands since acquired by treaty and purchase, constitute the public lands, or public domain. These lands were at first parceled out without reference to any general plan, in consequence of which the titles often conflicted with each other, and in many cases, several grants covered the same area. Through many years of labor and experiment, the present admirable system has been wrought out. J^26. This system is briefly that the territory to be surveyed shall be divided by true north and south, and east and west lines, into tracts, each of six miles square and containing as near as may be 23,040 acres, called Townships ; each township into thirty-six tracts, each of one mile square and containing as near as may be 640 acres, called Sections ; and each section into halves, quarters, and smaller portions, as may be deemed expedient. 227. In the survey, all primary lines north and south are 204 ELEMENTS OP SUKVEYIXG. [BOOK V. run on meridians of longitude, and all primary lines east and west are laid out on perpendiculars to the meridians. It is of the first importance, then, that meridians of longitude should be accurately determined, and that lines perpendicular to them should be determined with equal precision. The ordinary Surveyor's Compass is, for several reasons, not sufficiently accurate for the work of running out the standard and township lines, and such lines must be run by Burt^s Solar Compass, "or other instrument of equal utility." 228. The Solar Compass for determining a true meridian was invented by William A. Burt, of Michigan, and patented by him in 1836. It has been improved, by him and others, from time to time since, and in its present state is represented and described in Appendix A. 229. In commencing the division of the public lands in unsurveyed territory, an initial point is selected, with reference to its convenience in making the survey, perpetuated by a substantial monument suitably marked, and its true position in latitude and longitude determined. 230. From the initial point the principal base line is run out due east and west with the solar compass, and permanently marked at each 40 chains, or half mile, with a quarter-section corner, and at each 80 chains, or mile, with a section corner. 231. The principal meridian is then run out due north and south from the initial point, and also marked with monuments at intervals, like the base line. 232. As meridians of longitude converge toward the poles, the distance between two such meridians decreases as the surveyor goes north. To counteract the error that would other- wise result from the convergency of meridians, and also to arrest SEC. II.] PUBLIC LAIRDS. 205 error arisins^ from inaccuracies in measurements on meridian lines, standard parallels, or correction lines, are run east and west from the principal meridian and at stated intervals. On the north of the principal base-line, about latitude 35° north, these standard parallels are, in general, run at distances of e\erjfour townships, or twenty-four miles, and south of the principal base, at distances of every Jive townships, or thirty miles. Each of these standards is run out and marked in the same way a^ the principal base, and forms the base for laying out the townships north to the next standard parallel. The standards are num- bered according to their position with respect to the principal base-line, as 1st Standard Parallel South, 2d Standard Parallel South, 1st Standard Parallel North, &c. 233. The principal meridian, the base-line, and the standard parallels having been first run, measured, and marked, and the corner boundaries thereon established, the exterior lines, of townships are then run, measured, and marked. The townships, consisting of a series of townships lying along a parallel, are numbered north or south of the principal base ; the first series north of the base being Township 1 North, the second series north being Township 2 North, &c. ; and the first series south being Township 1 South, &c.; these are designated T. IN., T. 2N., T. 1 S., &c. The ranges, consisting of tiers of townships, are numbered from the principal meridian both ways; the first tier west of the meridian being Range 1 West, the first tier east being Eange 1 East, &c.; designated R. 1 W., R. 1 E., &c. 234. The accompanying map, from the U. S. General Land Office, representing a considerable portion of the State of Arkansas, will serve for illustration. The principal meridian in this survey is called the 5th 206 ELEME2STS OF SUKVEYING. [book y. S location f^ Jj and Office "L Boundary of LandBisiricts o + Lands offered for Sale i.W.UKWK^O^OH .»^.H.'^! Fig. 115. SEC. II.] PUBLIC LANDS. 207 meridian, and passes through the point of junction of the White river with the Mississippi. The principal base-line, running east and west, intersects this meridian a little to the east of White river; and from the meridian and base-line, reckoned from this point of intersection, all the ranges of townships are laid off. For example, 1 North, will apply to all the townships lying in the first row north of the base-line ; 1 South, will apply to all the townships in the first row south of the base line. Eange 1 East, will apply to all the townships lying in the first row, east of the 5th meridian ; and Range 1 West, will apply to all lying in the first row to the west of it. The small figures designate the rows of town- ships, reckoned north and south from the base-line, and the ranges reckoned east and west from the 5th meridian. Thus, Township 1 North, Range 4 West, has its exact place designated, and may be immediately located. 235. The diagram here given (Fig. 116), from the "Instruc- tions to the Surveyors-General of Public Lands of the United States," represents the required method of running out township lines. In the diagram, the upright figures (made thus, 1, 2, 3) commencing near the Principal Meridian and Base Line with No. 1, indicate the perambulations of the Surveyor in running the Townships and Correction lines. The Correction or Standard lines north of the base- are every four townships, and south of the base every five townships. The excess or deficiency of measurement on northern and southern boundaries is thrown on the westernmost half-mile. The measurements between meridian lines will, of course, always vary according to the latitude of the survey, besides being liable to be rendered inexact where the country is very hilly or broken. The convergency of the range lines as shown 208 ELEMENTS OF SUfiYEYING. [book V. by the measurements on this diagram, is according to calculation, as it exists between the parallels of 46° and 47° North Latitude. lHagram. r.x'ij ERIOKS OR i ( TOrvVT^SHIP LINES. 1 p 1 1 First correction or Standard Line ISTortli. 80| 80 |B0 1 80 1 80] 80 80|80 , 80, so 1 80 80 80, 80 , 80] 80 1 80 ,80 80, 30,80 ,80 ,80,80 22 K90 80'8O' sol SO'OO 769^ 80 '80'80' SD'Sa i(i9b} 80 '80' 80' 30 80 80 '80 'so 80 '80 § h'" 11 1" o oo ^^' 11 22 c © © o 3 e: © » © 1© ^ © o © o 09 © « © a © o o o © T.^-.jy.Z o L© © o o © o a n © © © © §21 "gio o o CO § ?wi^ 20 J7.7^J 9 O mi^ 10 njij 21 © 19 o 80 '80 80 80 '30 o 80 '80 '80 '80 '80 SO '80 '80 '80 'so 80 '80 '80 80 '80 ^ CO 8 © o 03 §9 8 ©20 19 © o o © ofi OD a © «§ © © o o © © -^1 © © a ^ o CO © CO T. S.JST i © a ^ o CO §18 J o a ,1 a 3 © W.-t7^ ir 7&n| 6© f^. % 18 ' © 16 o 80' 80 ' 80 ' 80 '80 o 80 ' 80 ' 80 ' 80 ' 80 80 ' 80 ' so ' SO '80 80 ' SO ' 80 80 '80 Q CO 5 © o u © 6 5 §17 16 © .1 © 1 o o s N 1 as o 03 © « o © © © © 'f 1 © © © 1 © ro © 1 O flO 115 1 o © o a o C3 i « CO J Q ra.2?/ ^9-2^, 3© ^23/' 4^ 79.23/ 15 a 13 o 80 '«0 ' 80 '80 '80 o SOJSO '80 ' 80 '80 © so > 80 ' 80 ' 80 ' 80 80 ' 80 ' 80 80 '80 09 2 © s «3 2 ©14 13 e © o o e CO © © © © © o o o oo © © CO © » o ca © TTZJV: § © © © o o CO « © © © n i lUIW. 1 jLznr o 1. RIB; 1 "0 © BJrSE 12 12 so 80 ,80 ,80 ,80,80 80 80 ,80 ,80 ,80|80 80 18O1 80,80,80 80 80 ,80 ,80 80 ,80 © Base Line Fig. 116. 236. The "Instructions to the Surveyors-General " for run- ning, measuring, and marking the exterior lines of townships, are as follows: SEC. II.] PUBLIC LANDS. 209 Townships situated North of the Base Line and West of the Principal Meridian. Commence at No. 1 (see figures on the diagram, Fig. 116), being the southwest corner of T. 1 N. — K. 1 W., as established on the base line ; thence north on a true meridian line, four hun- dred and eighty chains (6 miles), establishing the section and quarter-section corners thereon, as per instructions, to No. 2, whereat establish the corners of Tps. 1 and 2 N. — Es. 1 and 2 W.; thence east, on a random or trial line, setting temporary section and quarter-section stakes to No. 3, where measure and note the distance at which the line intersects the eastern boundary, north or south of the true or established corner. Run and measure westward, on the true line (taking care to note all the land and water crossings, &c., as per instructions), to No. 4, Avhich is identical with No. 2, establishing the section and quarter-section permanent corners on said line. Thence proceed in a similar manner from No. 4 to No. 5, No. 5 to No. 6, No. 6 to No. 7, and so on to No. 10, the southwest corner of T. 4 N. — E. 1 W. Thence north, still on a true meridian line, establishing the mile and half mile corners, until reaching the Standard Par- allel or correction line ; throwing the excess over, or deficiency under four hundred and eighty chains, on the last half mile, according to law, and at the intersection establishing the " Closing Corner," the distance of which from the standard corner must be measured and noted as required by the instruc- tions. But should it ever so happen that some impassable bar- rier will have prevented or delayed the extension of the standard parallel along and above the field of the present survey, then the deputy will plant, in place, the corner for the township, subject to correction thereafter, should such parallel be ex- tended. 210 ELEMEKTS OF SURVEY1N"G. [book V. North of the Base Line and East of the Principal Meridian. Commence at No. 1, being the southeast corner of T. IN. — R. IE., and proceed as with townships situated "north and west," except that the random or trial lines will be run and measured ivest, and the true lines east, throwing the excess over or deficiency under four hundred and eighty chains on the west end of the line, as required by law ; wherefore the surveyor will commence his measurement with the length of the deficient or excessive half-section boundary on the west of the township, and thus the remaining measurements will all be even miles and half miles. 237. In running random township exteriors, if such random lines fall short or over-run in length, or intersect the eastern or western boundary, as the case may be, of the township, at more than three chains and fifty links north or south of the true corner, the lines must be retraced, even if found necessary to re-measure the meridional boundaries of the township. 238. The exterior lines of townships having been established and duly marked, each township is divided into 36 squares, called Sections, by meridians one mile apart, and by east and west lines at the same distance from each other. The sections of a township are numbered from 1 to 36, beginning at the north- east angle and proceeding as shown in the annexed diagram : To describe a section accurately, we say, for example, section num- ber 5, in township number 4 north, in range 3 west of a known meridian. \ 6 5 4: 3 2 1-- 7 8 9 10 11 12 18 17 16 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 239. The sections are divided Fig. 117. w SEC. II.] PUBLIC LANDS. 211 into half -sections, quarter-sections, and even into eighths of sections. The following table shows the contents of a township and its subdiyisions : 1 township = 36 sections = 23040 acres. 1 section = 640 acres. ^ section = 320 acres. J section =160 acres. -J section = 80 acres. 240. As any excess or deficiency of measurement is, by law, to be thrown on the extreme tier of sections and half-sections contiguous to the north and west boundaries of townships, such sections and half-sections are sold as containing only the quan- tity expressed in the returns and plots, respectively, and all others as containing the complete legal quantity. The government surveyors are rarely required to subdivide a section into quarters, as that work properly belongs to the county surveyors. Note. — The student is referred, for more detailed information on Government Surveys, to : Instructions to tlie Surveyors-General of Public Lands of the United States, prescribed by the Commissioner of the General Land Office ; Clevenger's Government Sui-veying ; Burt's Solar Compass. BOOK VI. TRIGONOMETRICAL SURVEYING. SECTION I. MAKING THE SU RVEY. 241. Trigonometrical Surveying, or Triangulation, is the method of determining the position of points on the surface of the earth by the apphcation of the principles of Trigonometry — each successive point being determined by the intersection of two lines which make known angles with a given line. It may be used in small or extensive surveys ; it does not necessarily take into account the curvature of the earth, though always used in the great surveys, such as the U. S. Coast Survey, in which that is considered. 242.* In the construction of a true map of any large territory, three things are necessary. The first of these is to ascertain the exact relation to each other, as to distance and direction, of the leading features of the country ; selecting such high points as may be seen from the greatest distance, and con- necting the whole by a chain of triangles of the largest dimen- sions practicable, and of the forms most convenient for accuracy of computation. The series must rest upon a base of which the geographical position has been established with mathematical accuracy. The points which form the several angles of the chain will thus * Report of N. Y. State Survey, 1878. SEC. I.] MAKING THE SURVEY. 213 be fixed with equal exactness as points of the general surface of the earth. With the sides of this principal chain of triangles as bases, a net of smaller triangles is then to be constructed occupying the interior of the larger, and resting their angles upon the most conspicuous objects observable from the principal stations. And within these still a third series is to be formed, connecting as many of the less important points within those of the second order as the objects of the survey may seem to require. Every angle of every series of these triangles marks a geograph- ical point determined with the same degree of precision as that of the original base from which the triangulation began. The base on which the New York State Survey rests is a side 23 miles long of one of the large triangles of the United States Coast Survey, a magnificent work which twenty years since pushed its operations to the head of tide-water upon the Hudson Eiver. The bases of the United States Coast Survey itself are lines of some miles in length, determined in position by astro- nomical observation, and actually measured on the ground from end to end, by means of apparatus of extreme delicacy, con- structed especially for the purpose. 243. The specimen of triangulation. Fig. 118, is from a survey, made by Professor Eees of Columbia College, of Otsego Lake, N. Y., for the N. Y. State Survey, and will serve to illus- trate the method of making and plotting such a survey. 244. Before commencing a trigonometrical survey, an ex- amination of the entire territory should be made for the purpose of selecting a base line and proper points for stations ; this examination should be more or less elaborate according to the nature and extent of the survey. The proper distribution and combination of the triangles, so as to adapt them to the survey in hand, require great judgment and care, and the selection of proper trigonometrical points is a u i^ < J > o > hi DC a> CO X UJ h 1- e: O ir Q < u. < h O co 0. >- > u z > X D CO SEC. I.] MAKING THE SrKYEY. 215 very important part of the preliminary operations. The selec- tion should be so made that each point may command a view of the greatest practicable number of surrounding trigonometrical point objects, and that each angle at the point shall be as near as may be 60°. A triangle which has an obtuse or a very acute angle will experience a greater change of form, for a given error, than one which is nearly equilateral ; and since the accuracy of each triangle depends upon the preceding ones, it is evident that the introduction of a single "ill-conditioned" triangle might vitiate the whole survey. No angle less than 30° or more than 120° should be used; and even such angles should not be admitted when the locality can be so chosen as to pre- vent it, 245. If the tri angulation is to be over a limited extent of country which has already been covered by a net-work of Primary Triangles, a side of one of these triangles should be used as a base. It is never good practice to measure a base line, when a side of a triangle of a previous survey is available ; but if no such side can be obtained, then the selection of a proper site for a base-line forms one of the first objects of the preliminary reconnaissance. It should, if possible, be fixed on an open plain, free from surface encumbrance or freed from such. It must be so chosen that the surrounding signals may be distinctly seen from its extreme points ; and hence those signals which mark points of the adjacent triangulation, should be selected with reference to the base. The length of the base should be suited to the magnitude of the survey. 246. In measuring a base-line, every possible precaution should be taken to insure accuracy. The line measured should be straight, to effect which it should be ranged out with the transit. The ends of the base should be marked by a stone sunk in the ground, with a copper bolt let into it and the exact point 216 ELEMENTS OF SURVEYING. [book VI. of beginning and ending fixed by the intersection of two lines cut into the head of the bolt. The measurement may be made with steel tape or rods. If a tape is used, it should be carefully drawn out each time to its standard length, and should be compared with a standard both before and after measurement, and correction made for its variation, if any, from standard. The mean of several measure- ments should be taken for the correct measurement. If the measurement has been made on an incline, instead of on a level, the measured distance should be reduced to the horizontal distance by multiplying the inclined distance measured by the cosine of the angle of inclination. 247. For a description of the base-line apparatus used in the U. S. Coast and Geodetic Survey, see Reports of that Survey for 1854, 1857, and 1880. 248. The alignment of the measuring tape, or rods, both vertically and horizontally, or in the line of the slope if the measurement be not horizontal, is of the greatest importance, since there is no compensation of errors, a faulty alignment always resulting in a measured length greater than the true length. 249. Having carefully measured the base, it is then necessary to reduce the measurement to the sea level. Let L = measured base. Let I = reduced base. Let R = radius of earth. Let h = average height of measured base above sea level ; tia. ng. then, I : L :: R : R+h; SEC. I.] MAKING THE SURVEY. 217 whence, l=^ = L (-^|^) = L (l-^'^ • But h is so small as compared with R that we may, without sensible error, make R+h = R; whence we have, 250. The trigonometrical stations are marked by signals, which may be constructed in a great variety of ways, depending upon the locality of the stations and the lengths of the sides of the triangles. Sometimes a signal has to be raised above the level of the adjacent country, in which case it is constructed of timbers, and upon the apex is placed a vertical staff bearing a flag. The exact trigonometrical point is determined by a plumb-line suspended from the apex of the signal. A temporary signal may be constructed with three or four pieces of scantling framed and braced, as shown in the annexed figure, with a short pole projecting from the apex. The plumb determines the point B, which is the exact trigonometrical point over which the theodolite is L, to be placed. Where the sides of the triangles // are not very great, a pole, planted vertically and d surmounted by a flag, will answer as a signal. ^^^' ^^• In order to distinguish the different signals, the flags which they bear should be different from each other. They may be formed by arranging stripes of white and red, according to some prearranged plan, aud the flags of the different stations should be entered in a book. For the purpose of future reference, the trigonometrical point at each station, as B, should be indi- 218 ELEMENTS OE SURVEYING. [bOOK VI. cated by a permanent mark. If the point falls upon a rock, a hole may be drilled to show the locality ; or if not, a mark-stone may be sunk under the point, deep enough to be beyond the reach of accident. A record of the monument should be pre- served, together with its reference to some of the permanent objects in the neighborhooa, 251. A Heliotrope is necessary in long sights, and is always of great service in short sights in directing the observer to the station-mark. It consists essentially of a small mirror, so directed by an assistant as to throw a beam of sunlight into the telescope of the distant observer. Let a silvered glass, about 3 inches square, be mounted on a board in a manner similar to the telescope mount of a transit so as to have a motion about a horizontal axis and at the same time about a vertical axis ; in front of this, at a distance of two or three feet, mount a board with a hole in it, across which hole threads are to be stretched at right angles to each other, and adjust this hole over the station by a plumb-line. At the centre of the back of the mirror scrape away the silver, making a small sight-hole ; if now an assistant, sighting through the hole in the mirror, moves it so that the cross threads come in line with the distant station, it will be easy to keep the beam upon the observer by properly inclining and revolving the mirror. As the light reflected from so large a mirror would be too intense to observe with the telescope, it is necessary to make the cross-thread hole in the board quite small, not more than |- inch diameter for distances not exceeding 5 miles, about one inch diameter for distances of ten miles, and so on. A small pocket mirror will be found very useful as a means of telegraphing instructions, by combinations of flashes according to a system previously agreed upon. It can be directed to the observing station (or to the observed station) with sufficient 6EC, I.] MAKING THE SURVEY. 219 accuracy for signalling, by setting a vertical staff in line with the distant station and causing flashes to travel up and down the staff. 252. The extent of the survey, and the standard of accuracji to which the results are required to conform, must determine the size and perfection of the instrument to be employed in the measurement of angles. The angles of the primary triangles of the United States Coast and Geodetic Survey are measured with theodolites, whose horizontal circles are 24 or 30 inches in diameter; and to eliminate, as much as possible, every source of error, great numbers of observations are made at each station, the readings being made on different points of the arc by different verniers. Usually from 40 to 60 observations are made for each angle — one measurement, with the telescope direct, and one with it reverted, constituting a complete observation. With these precautions, it has been found that the error in a primary triangle (where the sum of its three angles has been compared with 180°), has fallen much within 3 seconds. The error of 3 seconds has been adopted as the highest admissible limit of error in such triangles. 253. The theodolite does not differ essentially in the principles of its construction and use from the transit, which has already been described. It is fitted with many appliances, for accuracy in the observation and the reading of angles, which it is unnecessary to describe here. Fig. 121 is a representation of the 8 to 12 inch theodolite used in the U. S. Coast and Geodetic Survey, taken from the Report of that Survey for 1880. 2W ELEMENTS OF SURVEYIXG. [BOOK VI. Fig. 181. SEC. I.] MAKING THE SURVEY. 221 Fig. 122. 254. To illustrate the principle of repetition in ttie measure- ment of angles, suppose the of the vernier to coincide with the of the limb, and the telescope to be directed, from the station A, Fig. 122, upon one of the objects, as the signal at B. Clamp the limb and, unclamping the vernier plate, direct the telescope on the second object, as the signal at E. If we now clamp the vernier plate and, unclamping the limb, direct the telescope on the signal at B, the line (0°, 180°) of the limb will make with AB 'du angle equal to BAE. Again clamp the limb and, unclamping the vernier plate, direct the telescope on the signal at E. The reading will evidently be equal to twice the angle BAE; and if we repeat the operation, the reading will be three times the angle, and so on. After ten repetitions, if we add 360° each time the of the vernier passes the of the limb, the final reading will be ten times the li 222 ELEMEl^TS OF SURVEYING. [bOOK VI. angle BAB, affected with the joint errors of the ten observations, and one- tenth of this will be the reading required to a greater degree of accuracy than could probably be attained by a single observation. 255. The method of reading angles and recording notes is as follows: (1.) First mark one vernier A and the other B by pasting these letters upon the vernier plate. Set vernier A at 0° and clamp it. Direct the cross-wires to the left of station B and then by a careful movement to the right bring the sight nearly upon station B, heing careful not to pass the station, and perfect the bisection by the lower tangent screw. Read both verniers, reading B to minutes and seconds only, and make the entry as in the table subjoined. Now loosen the vernier clamp and turn carefully to the right till nearly upon station E, completing the bisection by the vernier tangent-screw. Now make a " check " mark in column "vernier ^," as in table. Next loosen the lower clamp and turn the telescope to the right till station B is nearly reached, and repeat the previous operations, making *^ check" in "vernier A " column as before. In like manner repeat the operations for the third time, and enter the final reading upon station E, as in the table. If the motion of vernier A has h^en forward, in the direction of the graduation of the limb, the first reading must be marked negative, as in the table ; when the motion of vernier A is back- ward, the final reading must be marked negative. In the case just described the motion has been forward, therefore from the mean of the last readings (217° 15' 15"), we subtract the mean of the first readings (0' 10"), and divide the result by the number of repetitions (3) obtained for the mean angle 72° 25' 1".7, SEC. I.] . MAKING THE SURVEY. 223 (2.) Set vernier A at 180°, and having turned the telescope to the right of E, make three repetitious of the angle subtended by BB, working from right to left. In this case vernier A will pass over the 360° mark, which fact is noted in the table under the first "vernier ^" reading. As vernier A worked backward, the last reading, that taken upon B, must be negative. (3.) Kevolve the telescope upon its horizontal axis, and turn the telescope about the vertical axis towards B. Set vernier A at 90° and repeat three times from B to E. (4.) Set vernier A at 270° and repeat three times from B to B. The final mean of the four angles is 72° 24' 59". 2. Both verniers are read to eliminate error of centering of graduated limb. Kepetitions are made to eliminate errors of graduation. Eeadings from left to right, and from right to left, are taken to counteract errors due to torsion, and personal error of bisection. The telescope is reversed on its horizontal axis to eliminate error due to inequality of axis points, and error of colli mation. 224 ELEMENTS OF SUEYEYING. [book VI. Angles taken at Station A, July 17, 1883. Sta. Obs. B Vernier A. B. ± Calculation. Mean Ang. 0° 0' 0" 0' 20" — 217° 15' 15" E 217° 15' 10" 15' 20" 10 3)217 15 5 72° 25' 1".7 E 180° 0' 0" 0' 10" 180° 0' 5" 360 v/f/v/ 360° 540 5 322 45 15 • B 322° 45' 10" 45' 20" — 3)217 14 50 72° 24' 56".7 B 90° 0' 0" 0' 0" — 307° 14' 50" E 307° 14' 40" 15' 0" 90 3)217 14 50 72° 24' 56".7 E 270° 0' 0" 0' 20" 270° 0' 10" B 52° 45' 0" 45' 10" 52 45 5 3)217 15 5 72° 25' 1".7 4) 99' 56".8 Final mean. 72° 24' 59".2 SEC. I.] MAKING THE SURVEY. 225 256. If several stations can be read in succession, a simple modification of the method is practicable. Suppose the observing station to be E of Fig. 122, and we wish to read A, H, B, (7, &c. Sight A with the vernier set at 0° ; unclamp the vernier and turn to the right till ^Tis bisected and enter the reading opposite H in the table below ; unclamp the vernier and sight B, and enter its reading; continue the motion to the right, reading each station in turn. Suppose to be the station on the extreme right ; now set the vernier at 180° and work from (7, around to the left, to A, entering the readings in the table in a reverse order, from the bottom to the top of the page. Eeverse the telescope, and with 90° as the first reading work from A to C, and then with 270° as the first reading work from (7 to A. The notes are given below ; the readings on station A subtracted from the corresponding readings on H, B, and G, give the angular distances of these points from the line EA. Angles taken at Station E. Sta. Vernier A. B. Mean Read. Angles with Line EA. Mean Angles with EA. A 0° 0' 0' 0' 20" 0° 0' 10" 72 10 72 5 90 10 90 5 162 20 10 162 15 H 40° 0' 10" 0' 20" 40° 0' 15" 40° 0' 5" 112 10 112 5 40 130 10 30 130 20 40 15 202 10 10 202 10 39 59 55 40° 0' 3".75 B 66" 30' 0" 30' 20" 66° 30' 10" 66° 30' 0" 138 29 40 30 20 138 30 66 29 55 156 29 40 30 10 156 29 55 QQ 29 50 228 30 30 228 30 66 29 45 66° 29' 52".5 Z2b ELEMENTS OF SURYETiyG. [boos YI. Sta. Vernier A. B. Mean Read. Angles with Line EA. Mean Ar>sles with EA. c 108" 0' 0" 0' 10" 108° 0' 5" 107° 59' 55" 180 20 180 10 108 5 198 10 10 198 10 108 5 270 10 2:0 5 107 59 50 107°57'58".75 Having found by these methods the angles of any triangle, one of its sides being the base-line or a kno^m side of a triangle already computed, we can find the sides of the new triangle. First subtract the sum of the angles from 180° and apply ^ of the error to each angle; next treat the triangle as a plane triangle and compute the two required sides. 257. The spherical excess in triangles of a Primary System is seldom more than about 6", and triangles whose sides are not more than ten miles long may be regarded as plane triangles without sensible error. 258. It sometimes happens that a steeple, tower, or other prominent object must be used as a station, and in most cases it is impossible to set the instrument over the centre of such station. In such cases a "reduction to the centre" is necessary. Let be the position of the instrument, C the centre of a circular tower which marks the station, and DCG the desired angle, D being the right-hand object and G the left- hand one. Measure (in some one of the many ways for indirect measurement) the distance OC^ r, and take the angle GOC= y, always measuring the angle y (called the angle of direction) from the left-hand object, iig. 123. SEC. T.] MAKING THE SURVEY. 227 Gy and estimating it towards the left, from 0'^ to 300", as in the figure. The angle 1 being exterior to the triangle DOl we have I =z 0-{-a, and from the tnangle GIC we have /— C -\- (3, from which we deduce C=0-\-(a-fi) (1) We now need to determine the correction {a — /3). In the triangle JJOC we have (calling the side DC, D) sin a := r ^\n [0 -\- y) , . , __. • • -• i n -^— ^— (Art. 32), or as a is, m practice, always very small we may substitute for its sine its value in seconds, making sin a = (a)'' sin 1'', which substituted above skives (a)"=—,--i — rrr^* \ / ° ^ ^ /> sm 1 In like manner (calling the side GO, G) we have (/3)" = 7* sm 11 -jr—. — 4n' Substituting? these values in formula (1), we obtain G sin 1 ° ' ( D sin 1 6^ sin 1 ) ^ ^ This formula is general if care is taken to estimate the angle y with reference to the positions of the stations D, G, and C, as above directed, the signs of the trigonometric functions also being observed (see Davies' Leg., Trig. Art. 58). The angle is measured in the usual manner. The angle y, in the case supposed, is obtained by measuring the angle which each tangent to the tower through makes with the line OG, and taking their half sum. In formula (2) the sides D and G are unknown. In triangles whose sides are not less than 3000 feet, r being relatively very small, it will be sufficient to calculate these sides from the known side and the uncorrected observed angles, and substitute such computed values in formula (2). When the sides are less than 3000 feet, r being relatively large, the values of D and G are obtained by two approximations. First plot a triangle 0' D' G' upon as large scale as practicable, 228 ELEMElfTS OF SUEVEYIKG. [BOOK VI. using the known side and the observed angles; then plot the angle y, and lay off O'C = nr, n being either 10, 20, 30, &c., so that nr may measure between -^ and ^ of the length of the shorter of the two plotted sides O'U and 0' G\ in which case 0' D' C the plotted angle O'D'C will be equal to ^x«, or a = , 0' G' C ' and (i = These values substituted in formula (1) give a value of C which may be used in the computation of the sides D and G, and these computed values of D and G must then be substituted in formula (2). 259. An error of observation may also arise from the unequal illumination of the face of the object presented to the observer, called error due to phase ; thus, if an observer at A sight a tower at D, the sun being m the direction S, he will direct his telescope to dj the middle of the illuminated portion, instead of to D. This error, and also that due to irradiation in the case of ordinary signals, may be avoided by observing when the sun is obscured, or by throwing a shadow upon rods and other commonly used signals. It is hardly worth while to intro- duce a formula for correction of phase in this connection. (For the methods used in the work of the U. S. Coast and Geodetic Survey, see "Field Work of the Triangulation," 1877, by Richard D. Cutts, assistant.) 260. When, in connection with a trigonometrical survey on shore, a harbor is to be surveyed (see Fig. 122), for the purpose of ascertaining the channels, their depth and width, the positions of shoals, and the depth of water thereon, other means must be used, and other examinations made, in addition to those already described, If SEC. I.] MAKII^G THE SURVEY. 239 Let buoys be anchored on the principal shoals and along the edges of the channel ; and using any one of the lines already determined as a base, let the angles subtended by lines drawn from its extremities, to the buoys respectively, be measured with the theodolite. Then, there will be known, in each triangle, the base and the angles at the base, from which the distances to the buoys are easily found ; and hence, their positions become known. Having made the soundings and ascertained the exact depth of the water at each of the buoys, several points of the harbor are established, at which the precise depth of the water is known ; and by increasing the number of the buoys, the depth of the water can be found at as many points as may be deemed necessary. 261. If a person with a theodolite, or transit, be stationed at each extremity of the base-line, it will not be necessary to establish buoys. A boat, provided with an anchor, a sounding- line, and. a signal-flag, has only to throw the anchor, hoist the signal-flag, and make the sounding, while the persons at the extremities of the base-line measure the angles. From these data, the precise place of the boat can be determined. 262. There is another method of determining the places at which the soundings are made, that admits of great despatch, and which, if the observations are made with care, affords results sufficiently accurate. Having established, trigonometrically, three points which can be seen from all parts of the harbor, and having provided a sextant, let the sounding be made at any place in the harbor, and at the same time the three angles subtended by lines drawn to the three fixed points, measured with the sextant. The problem, to find, from these data, the place of the boat at the time of the sounding, is the same as Example 6, Art. 203. 230 ELEMENTS OF SURVEYIN'G. [BOOK VI. It is only necessary to measure two of the angles, but it is safest to measure the third also, as it affords a verification of the work. The great rapidity with which angles can be measured with the sextant by one skilled in its use, renders this a most expe- ditious method of sounding and surveying a harbor. For description of the sextant and its method of use, see Appendix B. 263. There is yet another method of finding the soundings, which, although not as accurate as those already explained, will, nevertheless, afford results approximating nearly to the truth. It is this: — Let a boat be rowed, with uniform speed, across the harbor, from one extremity to the other of any of the lines determined trigonometrically. Let soundings be made continually, and let the precise time of making each be carefully noted. Then, knowing the length of the entire line, the time spent in passing over it, as also the time of making each of the soundings, we can easily find the points of the line at which the several soundings were made ; and hence, the depth of water at those points becomes known. 264. If a person stationed on shore with a theodolite or transit, takes the bearing of the boat, at every second or third sounding, determined by hoisting a flag, it will fix the positions of the soundings with great accuracy. Soundings may thus be made along any number of known lines, and a comparison of the depths found, on different lines, at or near their points of intersection, will show with what degree of accuracy the work has been done. Sounding-lines should be made of strong cord, and divided into feet or fathoms, by different colored rags or other marks. The lead is shaped like the frustum of a cone, with the base B hollowed out, to hold some grease. The land or mud of the SEC. I.J MAKING THE SURVEY. 231 Fig. 125. bottom adheres to the grease, and thus shows the nature of the bottom, which should be entered in the field- book, and laid down upon the map. As the cord is liable to change its length, it should be compared, from time to time, with some standard. In tide-waters, the exact time of each sounding is to be noticed, and an assistant should note the height of the tide at regular intervals, upon a tide-gauge. The tide gauge is permanently placed at some convenient point of the harbor, and its point is referred, by means of a spirit-level, to some fixed bench- mark, on a level with mean low-water mark, to which all the soundings must be reduced. 265. Having plotted the work done with the theodolite or transit, as also the outline of the harbor traced with the compass, it remains to delineate the bottom of the harbor ; and this is done by means of horizontal curves, hereafter explained (Art. 332), which are used to represent broken or undulating ground. Let the plane of reference be taken through low-water mark, or to coincide with the surface of the water at low tide. The accuracy with which the bottom of the harbor is to be delineated, will guide us in fixing the distance between the horizontal planes of section. The first horizontal plane should be passed ac a aistance below the shallowest point that has been sounded, equal to the number of feet fixed upon for the distance between the planes of section ; and the curve, in which it intersects the bottom of the harbor, determined as in Art. 335. And similarly, for the other horizontal planes of section. Having thus delineated the bottom of the harbor, and noted on the map the distance of each intersecting plane below the 232 ELEMEN'TS OF SURVEYING. [BOOK VI. plane of reference, let such lines be drawn as will indicate the channels, shoals, sunken rocks, and direction of the current. In the example given in Fig. 122, soundings have been made in three directions, from the sand-bar in the harbor, and also from the rocky shore across to the light-house. 266. When a large extent of territory, or a long line of sea- coast is to be surveyed, it becomes necessary to consider the curvature of the earth's surface ; this branch of surveying is called Geodetic surveying. The operations necessary to the successful execution of a Geodetic Survey, require the minutest attention, and when performed, numerous corrections are to be applied to the meas- ured lines and angles, on account of the various causes of error incident to such operations. To investigate those causes of error, and to deduce rules for correcting the errors, in all cases, would exceed the limits of this treatise. We have, therefore, attempted nothing more than an outline of the operations in a trigonometrical survey, in which the Plane-Table and Compass are used in con- nection with the Theodolite or Transit, and in which the curvature of the earth is not considered. SEC. II.] FILLIis^G UP THE SUEVEY. 233 SECTION II. FILL! NG U P TH E SU RVEY. After the triangulation is completed, the interior may be filled up by the aid of the Compass, or Plane-Table, or both. By the Compass. 267. The use of the compass, in determining points and lines, by means of offsets, has been already explained (Art. 123). We will apply these principles in the example of the harbor, Fig. 122. Place the compass at A., and take the bearing of the line AE, which is S 12° W. Enter this bearing at A. Then measure along the line AH any distance, sls Aa equal to 130 yards, and make an offset to the lake, which we measure and find to be 50 yards. Enter the 130 in the middle column, and as the lake lies on the right 234 ELEMENTS OF SURVEYIl^G. [bOOK VI. (in going from A to E), we insert the 50 in the right-hand column. We then measure along the line AB to b, 350 yards from A. Here we make a second offset to the lake, and find it to be equal to 100 vards. Having entered the distances in the notes, we measure to q, the point where the line AF touches the creek, and we enter the distance from A, 415 yards. At d, we lay off an offset on the left, to the pond, 70 yards ; at e, an offset to the mouth of the creek, 150 yards ; and at E, where the course terminates, an offset to the lake, of 160 yards. The entire distance from A to E is 800 yards. At E, we take the bearing to H, which is N 50° E. Haying measured along this line to /, 315 yards, we make an offset to the pond, on the left, of 50 yards, and to the shore, on the right, of 90 yards. Having entered these distances, we recommence the notes at 315, below, which we suppose to be at the bottom of the second page. Having reached ff, the extremity of the course, we enter the entire distance from E, 680 yards. We next take the bearing to /, S 52° E. We then measure the dis- tances to m, n, p, and /, and enter them, together with the offsets, as in tlie notes. It is also well to make, in the columns on the right and left, such sketches of the ground, fields, houses, creeks, and rivers, as will afford the means of making an accurate delineation on paper. By the Plane-Table. 268. The plane-table. Fig. 127, consists of two parts: a rectangular board, and a tripod to which it is firmly secured. Directly under the rectangular board are four milled screws which pass through sockets inserted in a horizontal brass-plate ; these screws are worked against a second horizontal plate, for SEC. 1I.J FILLING UP THE SURVEY. 235 the purpose of leveling the table ; the table having a ball-and- socket motion, similar to the limb of the transit. Fig. 12T. Between the upper horizontal plate and the table, there Is a clamp-screw, similar to the clamp-screw of the transit, which being loosened, the table can be turned freely about its axis. There is, also, a small tangent screw, by which the smaller motions of the table are regulated, after the clamp-screw is made fast. A set of brass clamps for fastening the paper to the table ; a clamp with an attachment on one side for fastening a plumb- line, and, on the other, a pin immediately over the point of attachment of the plumb-line, for marking on the paper a point 236 ELEMEifTS OF SURVEYING. [BOOK VI. directly over a station of the ground ; an Alidade and a Declina- tor accompany the Plane-Table. The Alidade is a brass ruler, from one to two feet long, with a fiducial or true straight edge. On the upper face of the ruler are two spirit-levels at right angles to each other, and, near the middle of the ruler, a brass standard w^hich carries a telescope, like the telescope of the transit, with a vertical arc for measuring vertical angles, and micrometer wires for measuring distances (see Art. 206). The telescope is so placed with regard to the edge of the ruler beneath, that its line of collimation, when properly adjusted, is parallel to that edge and in the same vertical plane. The Declinator is a metal box, containing a magnetic needle with a range of about 10° on each side of the 0. It is used to orient the table, i. e., to place it at any point of the field in the same position with respect to the points of the compass that it had at other stations in the same survey ; or, in other words, to place it in such position at a new station that a line previously drawn upon the paper shall be parallel to the line of sight which it represents. The orientation may be effected as folloAvs : While the table is in position for drawing the first lines of the survey, place upon it the declinator, with one of the longer edges of its box set along a line drawn on the paper for the purpose, and note the reading of the needle ; when the table is removed to a new position, turn it till the declinator, placed along the line as before, gives the same reading. Figure 127 represents the plane-table and its accompaniments as used on U. S. Coast Survey. 269. The plane-table is used to determine the shorter lines of a survey in extent and position. Having placed a paper on the table, examine the objects and lines which are to be determined, and select for a base such a IP SEC. II.] FILLING UP THE SURVEY. 237 line of the triangulation that most of the objects can be seen from its extremities. Then place the plane-table over one ex- tremity of the base ; make it truly horizontal, and turn it until the larger part of the paper lies on the same side of the base with the objects. Then, tighten the clamp-screw, and mark with a pointed pin the point of the paper directly over the station, which point is determined most accurately by suspending a plumb from the lower side of the table. Press the pin firmly on this point, bring the fiducial edge of the ruler against it, and sight to the other extremity of the base-line, and mark, with the pin or pencil, the direction of the line on the paper. Sight, in like manner, to every other object, and draw on the paper the corresponding lines, numbering them from the base-line, 1, 2, 3, 4, &c. Then, with a pair of dividers, take from the scale a certain number of equal parts, to represent the base, and lay off the distance on the base-line from the place of the pin. Take up the table, carry it to the other extremity of the base, and place the point of the paper corresponding to that extremity, directly over it. Place the fiducial edge of the ruler on the base-line, and turn the table, by means of the tangent screw, until the sights are directed to the first station. If, however, in bringing the table to this position, the corresponding point of the paper has been moved from over the extremity of the base-hne, move the legs of the tripod until it is brought back to its place. Let the table be then leveled, after which, place the ruler again on the base-line, and bring the table to its proper position, by the tangent-screw, and continue the adjustment until the extremity of the base-line, on the paper, is directly over the station, and in the same vertical plane with the base-line, on the ground. Then direct the sights to 238 ELEMENTS OF SUKVETIXG. [book VI. Fig. 128. all the objects sighted to, from the other station, and mark the lines 1, 2, 3, 4, &c., from the base-line, as before. The intersections of the corresponding lines 1,1, 2,2, 3,3, 4,4, &c., determine, on the paper, the positions of the several objects, and a reference of these lines to the scale of equal parts, determines the true distances. 270. Let it be required, for exam- ple, to determine, by means of the plane-table, the relative positions of several houses. From station A, and on one of the lines of the triangulation, as AB, measure the base-line AJV, which we will suppose equal to 300 yards. Place the plane-table at A, and sight to the corners of the houses, and mark the lines 1, 2, 3, 4, &c. Then remove the table to iV, and sight to the same corners as before, and draw the lines as in the figure. The points at which they intersect the corresponding lines, before drawn, determine the corners of the houses. The front lines of the houses may then be drawn on the paper. Draw lines at right angles to the front lines, and on them lay off the depths of the houses, with the same scale as that used for the base line. To find the length of any line drawn on the paper, as the line 1, drawn through A, for example, place the dividers at A and extend them to the other extremity of the line, and theq apply the line to the scale. The length of the line 1 is equal to 198 yards. 271. In this example, we de- termine from the base-line CD, the positions of the points F, E, and H. p SEC. II.] FILLING UP THE SURVEY. 239 Changing the Paper. 272. When one paper is filled, and there is yet more work to be done, let the paper be removed, and a second paper put on the table ; after which, the table may be used as before. Now, in order that the .two papers may be put together and form one entire plan, it is necessary that two points, determined on the first paper, be also determined on the second ; and then, by placing the lines joining these points, one on the other, all the lines on the two papers will have the same relative position as the corresponding lines on the ground ; and the same for as many papers as it may be necessary to use. If different scales are used, the corresponding points will not join, and then the work must be reduced to the same scale, before the papers can be put together. In the first example, the position of the point F was deter- mined, in order to unite the first paper with the second. In the second example, we sighted from C and D, the ex- tremities of the base-line, to the points N and F, determined on the first paper ; we thus determined the line NF on the second paper. Placing the line NF of the one paper on NF of , the other, we have the following plan : PiQ. 130. In this plan, all the points and lines are accurately laid down. Any number of papers may be joined in a similar manner. 240 ELEMENTS OF SURVEYING. [BOOK VI. 273. I'or further description of the plane-table and its use, see ^^ A Treatise on the Plane-Table and its use in Topographical Surveying/' by E. Hergesheimer, U. S. Coast and Geodetic Sur- Yej, Report for 1880, Appendix No. 13. SECTION III. PLOTTING THE T R I A N G U L AT I N . The sides of the triangles having been completed, the work may then be plotted, as already explained, either by means of the circular protractor, or by the method of chords. The Circular Protractor. 274. This instriiment consists of a brass circular limb, Fig. 131, of about six inches in diameter, with a movable index ABf having a vernier at one extremity A, and a milled screw at the other extremity B, with a concealed cog-wheel that works with the cogs of the limb, and thus moves the index AB about the centre of the protractor. At the centre of the protractor ^ is a small circular glass plate, on which two lines are cut ; the point of their intersection is the exact centre of the instru- ment. The limb is generally divided to half- degrees ; the de- grees are numbered from to 360. At the point, and at the opposite extremities of the diameter passing through that point, are small lines on the inner edge of the Hmb ; the two extremities of the diameter, perpendicular to this latter, are designated in the same way. Two angular pieces of brass, each having a small and sharp steel pin at its extremity, are fastened to the index, and revolve freely around the lines ab and cd. The smaU screws, SEC. III.] PLOTTIKa THE TRIAJ^GULATIOI^. 24: Pig. 131. a, 5, c, and d, move them in the directions of the lines ab, cd, fcr the purpose of bringing the steel pins exactly into the line which passes through the of the index and the centre of the protractor. To adjust them to their places, place the centre of the pro- tractor oyer a marked point, and the of the index to the of the limb. Then mark the place of the index by the pins; after which, turn the index 180°, and see if the pins will mark 242 ELEMENTS OF SURYEYIl^O. [BOOK VI. the same points as before. If they do, the index is adjusted ; if they do not, correct the error with the screws «, h, c, and d. 275. To lay off an Angle with the Protractor. — Let its centre be placed over the angular point, and the diameter pass- ing through and 180'', on the given line. Turn the screw that works the index, until the of the vernier coincides with the division corresponding to the given angle; then let the angular brass pieces be turned down ; the points dotted by the steel pins will show the direction of the required line. If this line does not pass through the angular point, the pins are out of place, and must be re-adjusted. 276. First Method of Plotting. — Suppose it were required to make the plot of the harbor, Fig. 122, on a scale of 450 yards to an inch. Divide the length of the base-line AB, which is equal to 1140 yards, by 450, and the quotient 2.53 will express the length which is to represent the base-line on the paper. Draw an indefinite line AB, to represent the base; and having chosen any point, as A, for the first station, lay off 2.53 inches to B. The other extremity of the base-line wiU thus be determined. Then, place the circular protractor at A, and lay off the angle BAB, and then the angle BAG. Next, place the pro- tractor at B, and lay off the angles ABB and BBC. The intersection of the lines AB and BB will determine the station B. Let the protractor be then placed at this point, and all the angles of station B laid down. The point G. where BG intersects AG, and the point C, where BC intersects BC, will then be found. By placing the protractor at C and G, we can determine the points D and F, when th.e place, on the paper, of all the stations will be known. I SEC. III.] PLOTTING THE TRIANGULATIOl^r. 243 To unite the work done with the compass, spread the com- pass-notes before you, and draw through A a line to represent the meridian. The course AE lies to the west of this meridian, and makes an angle of 12° with it. Then, lay off from the scale the distances Aa, Ah, Aq, Ac, Ad, Ae, and at the several points erect perpendiculars to AE. Lay off, on these perpendiculars, the lengths of the offsets, and the curve traced through the points so determined, will be the margin of the lake. At E, draw a parallel to the meridian through A, and lay down the course EH, which is easterly, and makes an angle of 50° with the meridian. Then, lay down the several distances to the offsets, and draw the offsets and lay off their lengths. Do the same for the course HI, and all the compass-work will be plotted. The work done with the plane-table is united to the work done with the transit, by simply reducing it to the same scale, and then placing the line ^iV^ on the paper of the plane-table, upon the line AN, drawn on the plot of the triangnlation. 277. Second Method of Plotting.— Place the centre of the protractor near the centre of the paper, and draw a line through the points and 180°. This line will have the same position with the circular protractor that the base-line AB had with the limb of the transit. Then lay off, from the point, an arc equal to the direction from A to E, also an arc equal to the direction A G, and through the centre point, and the points so determined, draw lines. Lay off in succession, in a similar manner, the directions taken at all the stations ; and through the centre point, and the points so determined, draw lines, and designate each by the letters of the direction to which it corresponds. 344 ELEMENTS OF SURVEYING. [BOOK VI. Now, since all the lines drawn on the paper Imve the same position with the circular protractor, as the corresponding lines on the ground have with the limb of tlie transit, it follows that each direction will be parallel to its corresponding line upon the ground. Hence, an}' line may be drawn parallel to that passing through and 180°, to represent the base-line AB. Having drawn such a line, and marked a point for the station A, lay off the length of the base, and the extremity will be the station B. Through A and B, so determined, draw parallels respectively to the lines corresponding to the directions AB and BB, and the point of intersection will determine station B, Through B and B, draw parallels to the lines which correspond to the directions BO, CB, and their point of intersection will determine station C. Through and B, draw lines parallel to the lines corresponding to the directions CB and BD, and the point of intersection will determine D. In a similar manner we may determine the stations F and G. Method of Chords. 278. The chord of a given arc is equal to the sine of half the arc with double the radius. For, let DAF be any given angle. And AH a hue bisecting it. Let DKC be the chord of the arc DC, described with a given radius, and HF parallel to CD, the sine of half the given angle, to a radius AF = "2 AC. Since AF='2AC, we have, from similar triangles, HF = 2KCj but DC=2Ka hence HF = CD. 279. To lay off an Angle.— To avoid, as far as possible, the SEC. III.] PLOTTING THE TRIANGULA TlOi?". 245 use of fractions, let us suppose the radius of the table of natural sines to be 1 ten, or 10 inches. Take, from a scale, 5 equal parts, with which, as a radius, from the centre A, describe an arc CD. Take from the table the natural sine of half the arc, and remove the decimal point one place to the right; the result will express the sine of half the arc to the radius 10, or the chord of the arc to the radius 5. From the same scale, take this sine in the dividers, and from C, as a centre, describe an arc cutting (W in D ; draw AD, and CAD will be the angle required. This is the most accurate of all the methods of laying off an angle, and it may also be applied advantageously to the second method of plotting, thus : Draw a fine straight line, generally in the direction of the meridian or of the base-line of the survey; and also a line perpendicular to it. From the point of intersection, as a centre, with a radius of 5 equal parts of the scale, describe the cir- cumference of a circle cutting the straight lines in the points marked and 90°. To lay off an angle, as, for instance, the angle 14° 29'. The half of it is T 14' 30", the natural sine of which is 0.12605, or 1.26 to the radius of 10 inches. Set off from to h, as in the figure, this distance taken from the scale, and through the two points h, h, thus determined, draw a straight line. This line should pass through the centre, and will make wdth the line (0, 0) the angle 14° 29' ; and any line on the paper drawn parallel to it, will make with the line (0, 0) the same angle. The further application is obvious. BOOK VII. LEVELING. SECTION I. DEFINITIONS AND PRINCIPLES. 280. Leveling is the art of determining the relative dis- tances of points from the centre of the earth. 281. A line whose points are all equally distant from the centre of the earth, is called a line of true level ; and a surface, all whose points are equally distant from the centre of the earth, as the surface of still water, is called a level surface. 282. One point is said to be above another, when it is farther from the centre of the earth ; and this difference of distance from the centre, is called the difference of level between the two points. - 283. A straight line drawn tangent to a line of true level, at any point, is a horizontal line, and is called the hne of apparent level. Thus, Fig. 135, if C is the centre of the earth, and ^^F a line of true level, ABD is the line of apparent level. This is the line of level determined by an instrument. The difference be- tween the apparent and true level of the points A and E, is BE, the excess of the secant CB, of the arc AE, over the radius CE, Fm. 135. SEC. I.] DEFINITIONS. 247 284. To find a general formula for computing this excess, we have (Geom., B. IV, Prop. XXX), AB' = BE{BE + 2JEC); but, since the arc AU is very small in comparison with the radius of the earth, the arc AE will not differ sensibly from the tangent AB ; the diameter 2BG may, for the same reason. be taken for the secant {BF-}-2BC); hence, AB" == BBx2BC, or, dividing by 2BC, ^^=2W • • • • w- K we take the mean diameter of the earth to be 7919 miles, formula (1) gives 'AB^ ^^~7919 • • • • (^)^ ^®^^®' The departure of the apparent from the true level, starting from a given point, is equal to the square of the distance to the second point, divided hy the diameter of the earth. If, in formula (2), we give to AE, in succession, every value from 1 chain to any given number of chains (say 100), and reduce, at the same time, both terms of the fraction to inches, a table may be computed as on next page. Observe, that when the distance ^^ = 80 chains = 1 mile, that BE is = 8.001 inches, or two-thirds of a foot, very nearly ; and for any other distance, d, in miles, we have, 12 : 6?2 :: | of a foot : |^2. hence, we have the following easy rule for finding the correc- tion of curvature in feet : The correction for curvature, iii feet, is equal to tiuo-thirds of the square of the distayice in miles. 248 ELEMENTS OF SURVEYING. [book VII. Table showing the differences, in inches, between the true and apparent level, for distances between 1 and 100 chains. Chains. Inches. Chains. Inches. Chains. Inches. Chains. Inches. 1 .001 26 .845 51 3.255 76 7.221 2 .005 27 .911 52 3.380 77 7.412 3 .011 28 .981 53 3.511 78 7.605 4 .020 29 1.051 54 3.645 79 7.802 5 .031 30 1.125 55 3.781 80 8.001 6 .045 31 1.201 56 3.925 81 8.202 7 .061 32 1.280 57 4.061 82 8.406 8 .080 33 1.360 58 4.205 83 8.612 9 .101 34 1.446 59 4.351 84 8.832 10 .125 35 1.531 60 4.500 85 9.042 11 .151 36 1.620 61 4.654 86 9.246 12 .180 37 1.711 62 4.805 87 9.462 13 .211 38 1.805 63 4.968 88 9.681 14 .245 39 1.901 64 5.120 89 9.902 15 .281 40 2.003 65 5.281 90 10.126 16 .320 41 2.101 66 5.443 91 10.351 17 .361 42 2.208 67 5.612 92 10.587 18 .405 43 2.311 68 5.787 93 10.812 19 .451 44 2.240 69 5.955 94 11.046 20 .500 45 2.531 70 6.125 95 11.233 21 .552 46 2.646 71 6.302 96 11.521 22 .605 47 2.761 72 6.480 97 11.763 23 .661 48 2.880 73 6.662 98 12.017 24 .720 49 3.004 74 6.846 99 12.246 25 .781 50 3.125 75 7.032 100 12.502 SEC. II.] H^STEUMEi^-TS. 249 SECTION II. I NSTRU M ENTS. 285. The Y Level. — A Level is an instrument used to indicate a horizontal line, and also, to determine the difference of level of any two points on the surface of the earth. to £ o W 250 ELEMENTS OF SUKVEYING. [BOOK VII. The part of the instrument shown in Fig. 136, rests on a tripod, to which it is made fast, and on which it is leveled by means of two leveling plates, such as are described in the account of the Transit, Art. 177. The telescope rests on vertical supports called, from their shape, Y's or Wyes, and is confined in the wyes by loops or clips, which are fastened by tapering-pins. The telescope has at each end a ring of bell-metal, turned very truly and both rings of exactly the same diameter ; by these it revolves in the wyes, or can be at pleasure clamped in any posi- tion when the clips of the wyes are brought down upon the rings, by pushing in the tapering-pins. It has a rack and pinion movement to both object-glass and eye-piece. A spirit-level or ground bubble-tube is attached to the under side of the telescope, and furnished at the different ends with screws for movement in both horizontal and vertical directions. The level scale which extends over the whole length is gradu- ated to tenths of an inch, and figured at every fifth division, counting from zero at the centre of the aperture of the tube through which the glass vial appears ; the scale is set close to the glass. The wyes are perpendicular to a level-bar, to which they are screwed fast ; each wye has two nuts, both adjustable with the ordinary steel pin. Connected with the level-bar is the head of the tripod-socket. The tripod-socket is compound ; the interior spindle D, Fig. 137, upon which the whole instrument is supported, is made of steel, and nicely ground, so as to turn evenly and firmly in a hollow cylinder of bell-metal; this again has its exterior surface fitted and ground to the main socket EE of the tripod-head. The bronze cylinder is held upon the spindle by a washer and screw, the head of the last having a hole in its centre, through which the string of the plumb-bob is passed. SEC. II.] INSTRUMENTS. 251 The upper part of the instrument, with the socket, may thus be detached from the tripod-head; and this also can be un- &A screwed from the legs, so that both may be conveniently packed in the box. A little under the upper parallel plate of the tripod-head, and in the main socket, is a screw which can be moved into a corres 252 ELEMENTS OF SURyEYi:5sG. [BOOK Til. ponding groove, turned on the outside of the hollow cylinder, and thus made to hold the instrument in the tripod when it is carried upon the shoulders. Before using the Level, it must be adjusted. The adjust- ment consists in bringing the different parts to their proper places. The line of collimation is the axis of the telescope. With this axis, the line drawn through the centre of the eye-glass and the intersection of the cross-wires, within the barrel of the telescope, ought to coincide. FIRST ADJUSTMENT. To fix the intersection of the cross-wires in the axis of the telescope. 286. Having screwed the tripod to the instrument, extend the legs and place them firmly. Then loosen the clamp-screw and direct the telescope to a small, well-defined, and distant object. Then slide the eye-glass till the cross-wires are seen distinctly ; after which adjust the object-glass to its proper focus, when the object and the cross- wires will be distinctly seen. Xote now the precise point covered by the intersection of the cross-wires. Having done this, revolye the telescope in the Y's half round, when the attached level will come to the upper side. See if, in this position, the horizontal wire appears above or below the point, and in either case, loosen the one and tighten the other of the two screws which work the horizontal wire, until it has been carried orer half the space between its last position and the observed point; bring it the rest of the way by the leveling screws. Carry the telescope back to its place ; direct again the intersection of the cross-wires to the point, and repeat the operation till the horizontal wire neither ascends SEC. II.] Iiq-STKUMENTS. 253 nor descends while the telescope is revolved. A similar process will arrange the vertical wire, and the line of collimation is then adjusted. SECOND ADJUSTMENT. To make the axis of the attached level parallel to the line of collimation. 287. Turn the leveling-screws until the bubble of the leyel stands at the middle of the tube. Then open the loops and reverse the telescope. If the bubble still stands at the middle of the tube, the axis of the level is horizontal ; but if not, it is inclined, the bubble being at the elevated end. In such case, raise the depressed or depress the elevated end by means of the small screw provided for the purpose, half the inclination ; and then with the leveling screws bring the level to a horizontal position. Eeverse the telescope in the Y's and make similar corrections again ; and proceed thus until the bubble stands in the middle of the tube, in both positions of the telescope ; the axis of the Jevel is then horizontal. Let the telescope be now revolved in the Y's. If the bubble continues in the middle of the tube, the axis of the level is not only horizontal, but also parallel to the line of collimation. If, however, the bubble recedes from the centre, the axis of the level is inclined to the line of collimation, and must be made parallel to it by means of two small screws, which work hori- zontally. By loosening one of them and tightening the other, the level is soon brought parallel to the line of collimation ; and then, if the telescope be revolved in the Y's, the bubble will continue at the middle of the point of the tube. It is, however, difficult to make the first part of this adjustment, while the axis of the level is considerably inclined to the line of collimation ; for, even if the level be truly horizontal in one position of the telescope, after it is reversed there will be but one corresponding ii 254 ELEMEi^TS OF SUKYEYIIs'G. [BOOK VIL position in which the bubble will stand at the middle of the tube. This suggests the necessity of making the first part of the adjustment with tolerable accuracy ; then, having made the second with care, re-examine tlie first, and proceed thus till the adjustment is completed. THIRD ADJUSTMENT. To make the level and the line of coUimation perpendicular to the axis of the instrument, or parallel to the level-bar. 288. Loosen the clamp-screw and turn the bar until the level comes directly over two of the leveling screws. By means of these screws, make the level truly horizontal. Then, turn the level 180° upon its vertical axis ; if, during the revolution, it continue horizontal, it must be at right angles to the axis of the instrument about which it has been revolved. But if, after the revolution, the level be not horizontal, rectify half the error with the screws at M and R, Fig. 137, and half with the leveling screws. Then place the bar over the other two leveling screws, and make the same examinations and corrections as before ; and proceed thus, until the level can be turned entirely around without displacing the bubble at the centre. When this can be done, it is obvious that the level and the line of collimation are at right angles to the axis of the instrument about which they revolve ; and since the axis is carefully ad- justed by the maker, at right angles to the bar, it follows that the line of collimation, the level, and the bar, are parallel to each other. It is always necessary to examine the adjustments frequently, in order to secure satisfactory resr.lts. SEC. II.] INSTRUMENTS. 255 LEVELING RODS. 289. The leveling rods are used to determine the points at which a given horizontal line intersects lines that are perpendicular to the surface of the earth, and to show the distances of such points of intersection from the ground. There are three kinds of rods used by En- gineers, known as the New York, Philadelphia, and Boston or Yankee rods. The Philadelphia Rod is divided to tenths, and reads to two-hun- dredths of a foot. The New York and Boston rods are divided to hundredths of a foot, and read by verniers to thousandths. They are all sliding rods. 290. New York Rod.— This rod, which is shown in Fig. 138, is cut in two parts so that both ends may be exhibited. It is made of maple or satin-wood, in two pieces, sliding one from the other, always in the same direction, so that the same end is always held on the ground, and the graduations start from that point. The graduations are made to tenths and hun- dredths of a foot, the tenth figures being black, and the feet marked with a large red figure. A target is used to indicate where the hori- zontal line cuts the rod. The face of the target is divided into quad- rants, by a horizontal and a vertical diameter; and these diameters are the boundaries of alter- nate colors with which the diagonal quadrants are painted. The opening in the face of the target is a fig. i38. 11 256 ELEMI5NTS OF SURVEYING. [book VII. little more tliaii a tentli of ji foot long, so Uuit in any position a tenth, or a foot figure, ciin be seen on the surface of the rod. The right edge ol' the opening is chamfered, and divided into ten equal spaces, corresponding with nine-huiidredths on the rod ; the divisions start from the horizontal line whicli separates the colors of the face. The vernier reads to thousandths of a foot. For heights less than six and a hall* feet, the target is moved along the sliding part, to which it is slightly attached by springs, and to which it may be permanently attached by a clamp-screw, and the reading is made by the vernier on the target. When a greater height is required, the horizontal line of the target is fixed at that point, and (he upper half of the rod, car- rying the target, is moved out of the lower, the reading being now obtained by a vernier on the graduated side, up to an eleva- tion of twelve feet. TESTS OF ADJUSTMENT. 291. There is a method of testing the adjustments of the Y level, which ought not to be neglected, since all the results depend on the accuracy of the instrument. The method is this: F B ' "^wi;:,'.'."!!'.'.'.".]!]] is PiQ. 189. E The level being adjusted, place it at any convenient point, as G (Fig. 139). At equal distances of about 300 feet in opposite 8EC. IT.] IKSTEUMENTS. 257 directions from thie instrument, drive two pegs firmly into the ground and take readings of the rod upon them. The difference of the readings will he the difference of level of the tops of the pegs, even though the level be out of adjustment. Now set the level at about 50 feet beyond either f><^g^ nearly in line with them, and again take rod-readings upon them ; the difference of these new readings, corrected for curvature of the earth, should equal the difference of level of the pegs as found before. If the readings are not what they should U^, the adjustment may ?je perfected thus: Call the further peg «, the nearer peg h, and the position of the instrument c, (Fig. 140). Let R be the first reading at «, and W that at 6 ; let r be the second reading at a, corrected for curvature, and r' the corresponding reading at b, also corrected ; let (R _ R') -(r-r'}= ±D; then we have ab (= GOO ft.) : ac (= 650 ft.) : : ± D : ± d. Add the correction d to r, and having set the target to this reading on a, bring the horizontal wire to coincide with it by the afl justing screws. 8ee also that the bubble is in the middle of the run at the same time. Ifc 258 ELEME^'TS or SUBTEYIXG. fBOOK Vn. SECTION III. LEVE LING IN THE FIE LD. 292. The operatious of leveling may be undertaken . 1st. For the purpose of determining the difference of level between two given points ; 2d. For the purpose of obtaining a section or profile along a given line, as in the preliminary surveys for railroads and canals ; And, as will be described hereafter, 3d. For the purpose of determining the contonr lines in a topographical survey ; 4:thly. For the purpose of determining the volume of any given mass of earthwork or masonry ; as the measurement of excavations and embankments for canals and railroads ; and, 5thly. For the purpose of determining and indicating boundaries for filling and excavation ; such as setting slope stakes, &c. Difference of Level between Two Points. 293. TVhen it is proposed to find the difference of level of any two objects, or stations, all levels made in the direction of the station at which the work is begun, are called, for the sake of distinction merely, hach-sigJits ; and levels taken in the direc- tion of the other station, foresights. Before going on the field with the level, rule three columns, as below, and head them, stations, back-sights, fore-sights. SEC. III.] LEVELING IN THE FIELD. 259 FIELD NOTES. Stations. + Back-Sights. — Fore-Sight3. 1 10 3 2 11.6 3 6.8 4.9 4 3.9 8.3 Sums . . . 32.3 16.2 16.2 Dif. of leve I . . 16.1 Find the difference of level between any two points, as A and G (Fig. 141). 294. The level being adjusted, place it at any point, as B, as nearly in the line joining A and G, as may be convenient. h Place a leveling rod at A. Make the level horizontal by means of the leveling screws; turn the telescope to the rod at J, and direct the rodman to raise the target until the horizontal line 360 ELEMENTS OF SUKVEYII^G. [BOOK VII. ah pierces its centre ; then note the distance Ah (equal to 10 feet in the present example) and enter it in the column of back-sights opposite station 1. Send the rodman forward to some point, as N, in the pro- posed direction, and sight to the rod as before; enter the dis- tance Na, equal to 3 feet, in the column of fore-sights opposite station 1 {B). Then remove the level to a convenient point, as C (2). Direct the rodman to run up the vane to the proper height; then make the back-sight, and enter it, Nd = 11.6 feet, in the column of back-sights, opposite station 2. Let the rod- man then be sent forward to a convenient point, as if, and make the fore-sight to /; and enter Mf = 0, in the column of fore-sights, opposite station 2 (C). Remove the level, in succes- sion, to D and U, and make similar levels at those points, and enter the results in the column of back-sights and fore-sights, opposite station 3 (D), and 4 (U). It is evident from the figure, that the difference of level JVF, between A and iV, is equal to the back-sight hA, diminished by the fore-sight aN; also that the difference of level between iV and if, is equal to the back-sight dJV, diminished by the -fore-sight 0, and since each set of observations is entirely inde- pendent of every other set, we may infer that the difference of level hehveen two consecutive points, as determined hy the same position of the level, is equal to the hack-sight, diminished hy the fore-sight. If the fore-sight is greater than the back-sight, the difference will be affected with a minus sign, a result which shows that the second point is lower than the first ; and. Generally, the difference of level hetiveen any two points, deter- mined as ahove, is equal to the sum of the hack-sights diminished hy the sum of the fore-sights. If the result is plus, the second point is higher than the first ; if negative, it is lower. In the example given, the difference of level between A and 6^, is 16 feet and 1 tenth. SEC. III.] LEVELIXG IN" THE FIELD. 261 295. In the above example, we did not regard the difference between the true and apparent level. If it is necessary to ascertain the result with extreme accuracy, this difference must be considered ; and then, the horizontal distances between the level, at each of its positions, and the rods, must be measured, and the apparent levels diminished by the differences of level ; which differences can be found from the table. EXAM PLE. Stat. 1 Back-sts. Distances. Fore-st. Distances. Cor. back-sts. Cor. for-sts. 9.8 20 ch. 1.6 32 ch. 9.7583 1.4933 2 8.7 25 ch. 2.4 28 ch. 8.6349 2.3183 3 5.2 18 ch. 3.1 16 ch. 5.1663 3.0734 4 10.3 29 ch. 1.9 87 ch. 10.2124 1.1115 5 11.0 45 ch. 2.5 72 ch. 10.7891 1.9600 44.5610 8.9565 In this example, the first column shows the stations ; the second, the back-sights ; the third, the distances from the level in each of its positions to the back rod ; the fourth, the fore- sights ; the fifth, the distances from the level to the forward rod ; the sixth and seventh, are the columns of back and fore-sights, corrected by the difference of level. The corrections are thus made : The difference of level in the table corresponding to 20 chains, is 5 tenths of an inch, or .0417 of a foot nearly; which being subtracted from 9.8 feet, leaves 9.7583 feet for the corrected back-sights; this is entered opposite station 1 in the sixth column. The difference of level corresponding to 32 chains, is 1.280 inches, or .1067 feet, nearly ; which being subtracted from the apparent level, 1 foot 6 tenths, leaves 1.4933 feet, for the true fore-sight from station 1. The other corrections are made in the same manner. li 263 ELEMENTS OF SURVEYING. [bOOK VIL The sum of the back-sights being 44.5610 feet, and the sum of the fore-sights 8.9565 feet, it follows that the difference, 35.6045 feet, is the true difference of level. 296. In finding the true from the apparent level, we have not regarded the effect caused by refraction on the apparent elevation of objects, as well because the refraction is different in different states of the atmosphere, as because the corrections are for short distances inconsiderable. The error occasioned by refraction is opposite to, and tends to diminish, the error occa- sioned by the curvature of the earth. If desired, it may be corrected by diminishing the effect of the earth's curvature by one-seventh of itself. 297. The small errors that would arise in regarding the apparent as the true level, may be avoided iy placing the leveling rods at equal distances from the level. In such case, it is plain, 1st, that equal corrections must be made in the fore and back- sights; and, 2dly, that when the fore and back-sights are diminished equally, the result, which is always the difference of their sums, will not be affected. This method should always be followed, if practicable, as it avoids the trouble of making corrections for the difference of true and apparent level. The differences between the true and apparent level, being very inconsiderable for short distances, if only ordinary accuracy is required, it will be unnecessary to make measurements at all, Care, however, ought to be taken, in placing the levehng rods, to have them as nearly equidistant from the level, which can be effected by the rodman, by pacing the distance from back-sight rod to level, and from level to fore-sight rod ; and if the distances are unequal, let the next distances also be made unequal ; that is, if the back-sight is the longer in the first case, let it be made proportionably shorter in the second, and the reverse. SEC. IV.] SECTION LEVELING. 263 SECTION IV. SECTION LEVELING. 298. In the surveys which precede the construction of roads, raih'oads, canals, dikes, or other similar earthworks, the surveyor must make such measurements as are necessary to enable him to estimate the volume of the material to be removed. In addition, therefore, to the horizontal measurements made in connection with the location of the work, vertical dimensions, or heights, are also necessary, and are taken at every important change in the inclination of the surface along the line of the survey. These heights are taken by the level and rod, and are simply vertical distances of points along the surface above an assumed level line called the datum line. 299. In the survey of a long line of railway or canal, one of whose termini is in the vicinity of tide-water, the datum line is usually assumed at the level of mean high -water. In cases of surveys entirely inland, the datum line is taken at some convenient depth below the beginning point of the survey, and at such a distance that it shall be below the entire line on the surface. For such surveys, the system of notes described in the preceding section is insufficient. 300. As the survey progresses, fixed points of reference, called tenches, are located in the vicinity of the line. Permanent objects are usually selected for benches ; such as rocks, build- ings, or trees, and at such distances from the line of the work as to be undisturbed by the subsequent construction. II 264 ELEMENTS OF SURVETIKG. [BOOK TIL 301. Temporary benches, employed merely while changing the position of the leveling instrument, are called turning points. In either case, a well-defined point must be provided — one not easily disturbed by a blow, and, moreover, one upon which the rod can be held vertically. 302. In order to understand the field operations and the mode of keeping notes, it will be necessary to comprehend the principle involved in all leveling practice. Suppose that the depths of different points of the bottom of a shallow lake are required ; these could be readily obtained by measuring the distances from the surface of the water to the bottom by means of a sounding line, or in winter by cutting holes in the ice and measuring the distances to the bottom with a pole. In this illustration the various points on the bottom are located with reference to the plane of the surface of the water. The practice of leveling is identical in principle. The hori- zontal surface of reference, from which to deduce relative heights, is generated by the line of collimation as the telescope is revolved about the vertical axis, and the '^soundings" to points below this plane of reference are made with a leveling rod, whose lower end rests upon the point to be located, and whose target is moved so that the plane of reference shall cut its horizontal line. When the instrument is set up at a new station, the distance of its new plane of reference above or below the previous one, must be determined in order to secure continuity of the work. 303. The following example will exhibit the method of re- cording the notes of a section level. The datum line is assumed to be thirty feet below the first bench. When the field-book BEC. IV.J SECTION" LEVELING. 265 is Qf the ordinary pocket size, the seven columns of notes will generally occupy two opposite pages ; the first five being upon the left-hand page. 2\.Gyitt dboreDahnn. ^ench Dist. + Sight. Ht. of Ins. - Sight. Surface Height. Grade Height, Remarks. Bench. 1 1.637 31.637 1 / ' 2.1 1.8 30. 29.5 29.8 Bench on top of fence-post 30 ft. north of stake. 160 0.9 30.7 2 3.4 28.2 3 10.8 20.8 T. P. 1.910 22.134 11.413 20.224 ! 4 5.8 16.3 5 9.0 13.1 , 5S0 10.4 11.7 6 9.8 12.3 7 10.6 11.5 The bench haying been selected and marked, its location is described in the column of remarks. The level is adjusted in some convenient place in the vicinity, 266 ELEMENTS OF SURVEYING. [BOOK VIL and the reading of the rod is taken upon the bench. In the above example it is 1.637. As the bench is 30 feet above the assumed datum line, the height of the instrument (or line of collimation) above this datum line is 31.637 feet. The reading is recorded against Bench, in the column of + sights, and the ''height of instrument" is recorded in its proper column, in the same line. By referring to the above diagram it will be readily seen, that to obtain the height of the different points 0, 1, \^, &c., above the datum line, it is only necessary to take the readings of the rod, at these stations, and subtract them from 31.637. Such readings, therefore, are appropriately termed ininus sights, and are recorded in the 4th column. As these readings are taken only to the nearest tenth of a foot, they are taken much more rapidly than the bench readings. The subtractions by which the surface heights are found, may be worked in the field or not, as the surveyor chooses. The unit of measurement, in the column of distances, is usually the engineer's chain of 100 feet. Headings are taken at intermediate points (as at 160 feet in the above example) when there are abrupt changes in the inclination of the surface. 306. When it becomes necessary to change the position of the level, such measures must be taken as will insure the exact "height of instrument," in the new position. To effect this, a carefully-selected hard point is found (not necessarily on the exact line of the survey, but as far forward as convenience and accuracy will permit), and a reading of the rod is taken upon it, to thousandths. If likely to be used for a single occasion only, it is called a " turning -poiyit,^^ and marked T. P. in the distance column ; otherwise it is called a Bench, and its location is described in the column of remarks. SEC. IV.] SECTION^ LEVELIN^G. 267 A turning-point is taken between stations 3 and 4, in the above example. The reading of the rod, upon it, is 11.413. This is recorded in the — sight column, and the surface height of the point is at once found as before, and recorded in the column of " Heights. " The level is next carried forward to a new position, adjusted, and directed again upon the rod still held upon the turning- point. The reading is taken to thousandths. This, when added to the height of the turning-point, evidently gives the height of instrument in its new position. It is recorded, therefore, as a -|- sight. The survey is now continued by taking — sights at the various points along the line until it becomes again necessary to change the position of the level. In the above example, the reading of the rod upon the turning-point, from the second position of the level, is 1.910. The height of the point upon which the rod stands is 20.224. The sum of these, or 22.134, is the "Height of Inst." for the second set of — sights. The successive subtractions of the readings from the Height of Instrument, give the surface heights as before. The most extended section levels are but repetitions of this process. 305. The rules for taking and recording field-notes in section leveling are as follows: I. The " distances " recorded in the first column arc the horizontal measurements, in chains, frovv the beginnin£ of the survey to the points whose heights a,re to he dete?^- mined. TJie heights are taken at each ivhole chain, and at such intermediate points as the irregularities of the surface require. II. The first reading of the rod, after each setting of 268 ELEMENTS OF SURVEYING. [BOOK VII the level, is upon a bench or turning-point, and is a '^ -\- SIGHT ; '' all other readings are " — sights/' III. The + sight, added to the height of the point upon which the reading was taken, gives the " Height of Instru- ment." IV. The — sights taken, at any position of the level, subtracted from the "Height of Instrument" for that position, give the corresponding " Surface Heights." V. All the + SIGHT readings, and the last — sight of each set, being upon benches or turning-points, are taken to thousandths of a foot. The remaining '' — sights " are taken to tenths only. Note. — It will be observed that when the column of " surface heights " is complete, the second, third, and fourth columns of the field-notes are no longer needed. The first and fifth columns, which together contain the horizontal and vertical measurements for the line of work, afford all the data necessary for mapping the profile and determining the grade-line. 306. The location of the benches should be so described in the column of "remarks," that any particular bench may be found at any time, by referring to the field-notes. The impor- tance of this is apparent when it is remembered that the process of construction destroys or removes the stakes along the line of the survey, and that the question of the completion of the work can be determined only by reference to the benches. It is obvious, also, that they should be established somewhat off the line of the survey. The distance apart, of regularly established benches, should be governed by the above-mentioned uses of them. Any turning-point may be profitably made a bench (when it can be made permanent), by carefully recording, so as to admit of its identification. SEC. IV.] SECTION LEVELIITG. 269 In order to eliminate the effects of curvature and refraction, *Hurning-points," whether used as benches or not, should be taken at equal distances from the instrument when practicable, or compensation should be secured by proportional distances, as in Art. 297. 307. In conducting a section level through a rocky district, turning-points in abundance are found at hand, and cause no delay in their preparation, whereas a bench in the same section, requires marking and locating. In leveling through flat and level sections of country, although the engineer can get ^* sights" for long distances, a proper regard for accuracy will induce him to limit the distance, between successive positions of the level, to about six hundred feet. Under such circumstances, each turning-point is made a bench. 308. The methods of establishing benches are various. In a rocky section, some conspicuous point is marked either by drilling or grooving the rock. In villages or cities, stone steps, or projecting courses of masonry to dwellings, curb-stones, and fence-posts afford good benches, and admit of easy identification. In sections where trees abound, a notch is cut in the side of a trunk near the root, in such a manner as to leave a pro- jecting point upon which the rod may be held vertically. A nail driven full length into the projection, gives it the necessary firm- ness for a bench. In marshes or prairies, where there are neither rocks nor trees, the engineer is compelled to resort to long stakes, firmly driven into the ground to such a depth as to be undisturbed by the frost ; no portion of the stakes being allowed to project above the surface. The top of each is trimmed to a kind of blunt point, into which a nail is driven its full length. A re-survey of a route, to detect possible errors in leveling, is accomplished by taking the heights of the '* benches" only, and is called a " check level." 370 ELEMENTS OF SURVEYING. [BOOK VII. Drawing the Profile. 309. When the " section level " of a line of work has been completed, the "profile" is next to be drawn. The method of doing this is very simple. A horizontal line to represent the dattim line is first drawn, and the distances from the first column of notes are laid off along it, to a convenient scale; this for ordinary working draw- ings is about two hundred feet to an inch. The "surface heights" corresponding to these distances are next laid off at right angles to the datum line, and above it, but to a scale usually ten times as great as that employed for the horizontal distances; that is, an inch upon the vertical lines represents one-tenth as many feet as upon the datum line. A line joining the upper extremity of the verticals, is the profile. By thus employing two different scales, the irregularities of the surface are made more apparent to the eye, and the sub- sequent adjustment of the *^ grade-line" is rendered much easier and more accurate. 310. Every earthwork of importance requires, in addition to the working profiles, a general map, in which the plan drawn from the transit survey is represented upon the same sheet as the profile. The horizontal distances of both portions of the map being drawn to the same scale, and one being placed directly above the other, corresponding points in plan and profile are readily compared. In published maps of this kind, representing extended works, and drawn for convenience to a very small scale, the vertical scale of the profile is frequently several hundred times as great as the horizontal. Establishment of the Grade. 311. The determination of the height which the finished road or canal shall have above the datum line at different points, is called "Establishing the grade." SEC. IV. J SECTION LEYELIN-Q. 271 The position and inclination of grade-lines are influenced by a variety of circumstances : 1st. The character of the work. A street admits of an in- clination of five, or even eight feet in a hundred, and requires about one foot for its drainage, while a rise of two feet in a hundred upon a railroad is exceedingly rare. A canal is, of course, level, the change of height being effected by abrupt transitions at the locks. 2d. The economy of construction. It is desirable to make the earth excavated, form the required embankments, or, in the language of the engineer, " to make the cuttings balance the fillings." It is, however, sometimes more economical to throw away, or "make a spoil bank" of the earth of an excavation, than to transport it the required distance for the embankment. Embankments, for similar reasons, are often constructed of earth obtained outside of the road limits (''borrowing pits") ; or, when such means are not available, are often made of timber framing (trestle-work). 3d. The natural obstacles, which render the construction difficult ; such as rocky ledges, marshes, lakes, streams, and quicksands. In any case, the engineer determines, by inspection of the maps, at what points the grade-line shall intersect the natural surface. Thus the inclination of the grade, and, con- sequently, its height above the datum line, for each "distance," are easily found. Another column of notes is now made, recording these " Grade Heights ; " each being placed against the corresponding surface height. 312. The. folio wing example, with its accompanying diagram, illustrates the method of establishing a grade and recording the notes. It will be observed that the profile, with its " dis- 273 ELEMEI^TS OF SURVEY I ]SrG. [book VII. tances" and "surface heights," are the same as in the preceding problem. We will suppose it is required to establish, in the following profile, a gi*ade-line whose inclination shall not exceed 3 in 100 ; the grade to begin at station 0, at the surface. Fio. 143. DiBt. +s. H. of Ins. -S. H. of Sur. H. of Gr. Cut. Fill. Rem. 29.5 29.5 1 29.8 26.8 3.0 ^60 30.7 25.2 5.5 2 28.2 24.1 4.1 Oat 2" 3 20.8 21.4 0.6 T.P. 4 16.3 18.7 2.4 5 13.1 16.0 2.9 550 11.7 14.7 3.0 6 12.3 13.3 1.0 7 11.5 Oat 6" . It is an easy matter to represent any required inclination of grade on the profile map; nothing more being necessary than to lay off the proper distances on two different verticals. SEC. IV.] SECTION" LEVELING. 273 and draw a line through the points of measurement. For in- stance : a grade of 3 in 100, running downward from station 0, would intersect the vertical at 6, eighteen feet lower, and the vertical at 7, twenty-one feet lower. Moreover, by consulting the notes, we find that a grade- line from 0, whose height is 29.5 feet, ending at the surface at 7, whose height is 11.5 feet, descends 18 feet in 700, or 2.57 in 100. Either of these lines would fulfil the required conditions. The first would, however, require in its construction a large excess of excavation over the embankment (as may be seen by drawing a faint line in the diagram). The second would give an excess of embankment. It is best, generally, that the cutting should be slightly in excess, as nearly all kinds of earth shrink a little in the process of removal. The cuttings and fillings of the profile may be balanced with tolerable accuracy, by stretching a thread across the pro- file so as to intersect at the point, and then varying the in- clination, until the areas cut off by the profile line on opposite sides of the thread appear equal.* The column of Grade Heights must now be filled. It is easily and rapidly done. The height of Grade, at 0, is, by the conditions, 29.5. At station 1, it must be 2.7 lower, or 26.8 ; at 1.60, 4.3 lower, or 25.2; and at 2, 5.4 lower, or 24.1, &c. The remaining columns of *^cut" and ''fill" contain simply the differences between corresponding "surface" and "grade * The advantage of a thread over a ruler lies in the fact, that while using the thread, the areas on both sides of it are seen at once. In the present example, a line from 0, descending 2.7 to 100, seems to accomplish the desired purpose. The line being drawn, the "cut" and "fill" areas are measured, to determine if they are properly balanced. The complete computation of the earthwork, by which the exact position of the grade- line is determined, is explained in a following section. 274 ELEMEl^TS OF SURVEYII^G. [BOOK VII. heights." Where the surface is higher than the grade-line, the construction requires a " cutting ; " when the established grade- line is higher than the surface, an embankment, or '^ filling," is necessary. The notes in the final column, indicate the points where the grade-line intersects the natural surface. Such are called zero points. The distances are of importance in the computation of the earthwork. The above notes literally signify that either cut or fill is 0, at 2.87, also at 6.52. These distances are obtained with sufficient accuracy for ordinary purposes by a measurement of the profile map. When the cuttings and fillings are recorded in the proper columns, the notes belonging to the section-level are complete.* Note. — It will be observed that the first set of notes on page 265, did not contain the columns for cut and fill. The practice in keeping the notes differs with the work to be performed. In extensive railway surveys, it is convenient to rule the pages of the note-book as in the first example ; carrying out the field-notes to the extent of the surface heights, at least ; then transfer to another book, the "distances," "surface heights," and "grade heights," ruling columns for "cut," "fill," and "remarks." * The following calculation may be employed In the more important cases. The triangles formed by the verticals (cut or fill), the grade-line, and the surface-line are similar, and give the following proportion : The sum of the cut and fill, : the cut, : : the distance from cut to fill, : distance from the cut to point. Fill may be substituted for cut in the second and fourth terms. The application to the first zero point, in the above notes, is as follows : 4.1+0.6 : 4.1 :: 100 : required dist., or 87. In the second case in the notes, the cut necessary to the calculation is wanting, but U easily supplied, by determining the height of grade in the usual way at station 7. SEC. IT.] SECTION LEVELING. 275 These transferred notes are recorded in ink, and reserved for use in mapping and computations. 313. Most road or canal surveys are made on several trial- lines before one is finally adopted. The profile of each line is care- fully drawn, and the cost of construction approximately estimated. When the route is finally selected, and the section levjels satisfactorily completed, the exact width of the earthwork, bdth in excavations and embankments, is carefully staked out and the amount of material to be moved in the progress of construc- tion, accurately measured. The method of conducting this work is explained in a follow- ing section. 314. Before closing the subject of section leveling, we will consider the profile represented in the figure, and the set of field- 10.912 \jDahnnX/t'ne80fee% hetaw firsl- JSen^, 3*0 Fig. 144. 6 6«> notes appended, which are only partially completed, and whicli will afford some examples for practice. 276 ELEMENTS OF SUKVEYING. [book VIL Dist. + s. Ht. of Ins. - S. Surface Heights, Remarks. Bench. 1.032 3.2 1 3.8 2 5.3 3 8.9 350 10.3 4 9.0 - 5 4.8 T. P. 11.815 2.346 T. P. 10.942 2.318 6 9.7 640 6.4 7 2.1 1. What of the level 2. What 3. What level r ) 4. What 5. What level '. J 6. What 7. t( 8. a 9. a 10. iC 11. Write and 6^- is the " Height of Instrument " for the first position ? Ans. 31.032. is the height of the first T. P. ? Ans, 28.686. is the "Ht. of Inst." for th^ second position of the A71S. 40.501. is the height of the second T. P. ? Ans. 38.183. is the '^Ht. of Inst.'' for the third position of the Ans. 49.125. is the Height of Surface at ? atO? Ans. 27.8. at 3? 22.1. at 350? 20.7. at 5? 26.2. ate? 39.4. at 7? 47.0. the "Surface Heights" for the distances 1, 2, 4, SEC. V.l CROSS-SECTIOi^ LEVELING. 277 SECTION V. CROSS-SECTION LEVELING. 315. All earthworks, whether excayation or embankment, unless held in position by retaining walls, require to be con- structed with a sloping surface, the inclination of which depends upon the kind of earth. If, in a railway-cutting, for instance, the banks which bound it be left too nearly vertical, when first constructed, the weather- ing influences, to which they are subjected, soon cause the ma- terial to slide down, until the whole slope gradually assumes a much lower inclination. After a time, however, the tendency to roll or slide is checked by the friction of the particles themselves, and the slope thus formed will withstand the ordinary effects of sun, wind, and rain. The inclination thus assumed is called the " natural slope " of that kind of earth.* 316. Slopes are expressed mathematically by the ratio of E F S' ^^■^iiai^^HB^^'' Fig. U5. their horizontal to their vertical dimensions, and which is called the ratio of slope. * Thi? slope is determined, experimentally, by drying a portion of the earth, and then pouring it from a slight elevation upon a level surface. The heap thus formed is a rather flat cone, whose sides stand at the lowest inclination they would be liable to assume under the action of atmospheric influences. The angle with the horizontal plane will be somewhere between 25° and 45°. 278 ELEMEi^TS OF SURVEYING. [BOOK VII. In the diagram, which represents a road-cutting, the ratio of ES to AE, or of FS' to BE, is the ratio of slope. In practice, the slope at which earthworks are allowed to stand vary from 1 to 1, or 45° (as in very coarse material), to 2 to 1, or 26° 34', in very fine sand. A slope of li to 1 (33° 41') is found to be so far suitable for all ordinary excavations or embankments, that it is common, in the absence of an examination of the material, to adopt it as the ratio of slope throughout. Setting Slope Stakes. 317. It is evident that the width of natural surface of ground, required in the construction of a road, will vary with the depth of excavation or embankment. As often, therefore, as it is found necessary to determine the depth of the cutting or filling, in the section level, it is also necessary to mark the boundaries of the width of the work, on the natural surface. This is done by stakes called Slope Stakes, and the field-work necessary to determine their position, and to measure the section taken across the road of which the Slope Stakes indicate the boundaries, is called " Cross- Section Leveling," or " Cross-Section Work." 318. A party of five may be usefully employed in setting Slope Stakes ; viz., a leveler, rodman, axeman, and two tape- men. The rod, for cross-section work, is a ruder instrument than that employed in the section level. It should be at least fifteen feet long, with the feet and tenths plainly marked. It requires no target, the leveler himself reading the rod in the act of sighting. The field-book is ruled as shown below. SEC. v.] CROSS-SECTIOi^ LEYELIN'G. 279 Dist. Left. Centre cutfings. Right. The left-hand column contains the distances taken from the section-level notes. The third column is for the cut or fill, corresponding to the distance in the first column ; these numbers also being taken from the notes of the section level. A filling, it should be remarked here, is designated as a minus cutting in the field-notes. The second and fourth columns are for the horizontal and vertical measurements of the cross-section. 319. The examples following will illustrate the method of measuring the section and recording the notes : ^ Let Figure 147 represent a section across a road excavation. AB being the bed of the road, and 8C8' the line of the natural surface. The road-bed is supposed to be 16 feet wide, the centre cutting 12.4 feet, and the ratio of slope IJ to 1. The level being set up and adjusted in a convenient place, the rod is first hold by the centre stake at C, and a reading taken. 280 ELEMENTS OF SURVEYING. [BOOK VII. In the present example, the reading is 5.4. The line AB forms a convenient datum line, and the height of the instru- ment above this line, is evidently 12.4 + 5.4 = 17.8 ft. This is noted down, for the moment, on a reversed page of the note-book, or on a spare slip of paper ; neither the height of instrument or rod readings being matters of permanent record in cross-section work. It is evident that if the rod be held at different points along the surface, and the readings subtracted from tlie ^* height of instrument," the remainders will be the heights of these points above the datum line AB. These heights are technically called cuttings, although iu the case ot ST and S' V, no actual excava- tion is proposed. The reading at B is supposed to be 5.5. The cut is therefore 17.8—5.5 = 12.3. The horizontal distance from the centre is 8 feet. For each cutting there will be a horizontal measurement, and these two must be recorded together. The form adopted is that of a fraction in which the numera- tor is the cutting and the denominator the distance. The record of this measurement would be, therefore, J-|^, in the column marked ^Heft." The points A and ^, of the cross-section are appropriately termed the angles, and as the points B and F, directly over them, become new starting-points for horizontal measurements, it is important to distinguish them, in some way, in the notes. (Right and left in the actual survey are determined by the direction in which the survey progresses, and in which the centre stakes are numbered.) A common method is the one adopted in our notes — ^to sub- stitute for the number which represents the half width of the road, the letter A. The hind-chainman now takes his position at B, and i SEC. v.] ckoss-sectio:n' levelijs'g. 281 the remaining distances to the left are measured from this point. A change in the surface-line at // requires notice. The reading of the rod 6.2 indicates a cut of 11.6. This, with the distance from E., 10 feet, is duly recorded. There being no other material change in the surface line beyond H, there remains to be determined on this side only the intersection 8, of the surface and slope. It is found by trial. When found, it is evident that the ratio of the distance to the " cut/' must be the same as the ratio of slope. In the present example, the distance must be 1} times the cut. Suppose a trial reading taken at 25 feet out, is 3.2. The height, or cut, is (17.8—3.2) = 14.6. 1^- times this is only 21.9 feet. The distance tried, 25 feet, is too great. Suppose a second trial at 22 feet out, with a rod reading of 3.8. The cut is 14. 1^ times this is 21. Still too far out. A third trial, at 21 feet out, and a reading of 4, gives a cut of 13.8 ft. This multiplied by 1^ gives 20.7 feet. The measured distance is slightly too great, but in ordinary practice this ap- proximation would be considered near enough. The record for the slope-stake *S^ would therefore be |^:f.* In proceeding from the centre to the right, we find a pomt K between the centre and the angle, that requires attention. The distance in such a case is taken from the centre instead of the angle. CKh 3 ft. and the rod reading 5.3 gives a cut of 12.5. The reading at the angle-stake F is 5.8, giving a cut of 12 feet. If the surface-Hne from i^were level, the distance FS' would be 12 X 1|^ = 18 feet ; but as the ground descends, the distance is less. * The readings and distances in this example have been made to correspond to a rise of 282 ELEMENTS OF SURVEYING. [book VII. A trial at 11 ft. with a reading of the rod of 10.4, indicates a cut of 7.4. This multiplied by IJ gives 11.1, which is very nearly right ; y^^ is therefore the record for the location of S'. The completed notes for this cross-section are as follows : Diet. Left. Centre Cut. Right. 13.8 11.6 12.3 20.7 10 A 12.4 12.5 12 7.4 3 A 11.1 320. We will now give a similar example, illustrating the method of staking out embankments : Fig. 147. Dist. Left. Centre Cut. Right. — 15.8 —12.8 23.7 A -11.7 -8.6 —8.4 A 12.6 The filling at the centre is assumed to be 11.7 ft., which appears in the column of "centre cut," with its appropriate sign. The reading of the rod, at the centre, as shown by the dia- gram, is 7.2. SEC. v.] • CROSS-SECTIOK LEVELIN"G. 283 The sum of the reading and the centre cut (7.2 — 11.7) is —4.5, which is the '^height of instrument" referred to the line AB. The readings at all other points along the line S8' must be subtracted from this "height of instrument," as in the pre- ceding example. The several remainders are the corresponding ** cuttings." The reading at angle stake Ey 8.3, subtracted from — 4.5, gives —12.8, for the cut. For the slope stake 8, we will suppose a trial distance of 20 feet from E, and a rod reading at the trial point of 10.8 feet. The cut is therefore — 15.3 ; this multiplied by IJ gives 22.95.* The trial distance therefore, 20 feet, is not enough. A trial of 24 feet out, we will suppose to give a rod reading of 11.3 ft., which corresponds to a cut of — 15.8 ft. The ratio applied to this, gives for the proper corresponding distance out, 23.7 ft., which is nearly correct. The distance at which the trial was made is slightly too great. It is evident that if the slope stake be set at the calculated distance, 23.7 ft., the record of — — ^ may be made without involving an error of more than /vO. i a tenth of a foot, in either cut or distance. On the right, the reading at the angle stake F is 4.1. The cut, therefore, is —8.6 feet. As the surface rises but little from F to S' the trial distance for the slope stake is taken at 12 feet ; (it should be 12.9 ft., if the surface were level) ; the reading is supposed to be 3.9 feet. This gives the cut —8.4, which should correspond to a distance out of 12.6 feet. It is evident, that considering the rise in the surface and the rod readings at F and 8', the rod reading at ;S^' * The sign is disregarded in the product. It may be well to notice, however, that the ratio of slope in embankments is considered to be IJ to —1. 284 ELEMENTS OF SURVEYING. [BOOK VIL would not vary a tenth if moved from 12 feet to 12.6. The — 8 4 record, therefore, for S' is — tttt^- i/c.b Note. — It is not necessary to take rod readings at the angle stakes. If desired, these readings can be omitted and rod read- ings taken at the centre stake, and right and left from it at each change of inclination of surface. It is often convenient, how- ever, particularly for beginners, to take the angle stake readings, and hence in the rule these readings are assumed to be taken. 321. The rules for conducting and recording cross-section work, whether for excavation or embankment, are as follows : I. Prepare the fleld-hooh by riding columns for Dis- tances and Centre Cuttings, leaving wider spaces on each side of the latter column for the record of the various measurements to the left and right of the centre stake. Transfer from the section-level notes the distance and corresponding cut or fill, for each stalce of that survey. Filling in the cross-section notes is designated as minus cutting. II. Having set the level in convenient proximity to a proposed cross- section, take a reading of the rod at the centre-stake. Add this reading to the centime cutting, (regarding the sign of the latter), to obtain the ''height of instrument." III. Lay off half the ividth of the road-bed each side of the centre, and mark the distances, temporarily, with stakes. Tlxese are the angle stakes. IV. Proceed to take rod readings at the angle stakes, and beyond them outward (on a line at right angles to the direction of the line of the road), at each change of inclination of the surface. Subtract each reading from I SEC. v.] CROSS-SECTION" LEVELIl^G. 285 the height of instrument ; the remainder is the cutting, or vertical distance of the point -measured, from the pro- posed road-bed. V. Record each cutting, together ivith its horizontal distance from the nearest angle stake, in the form of a fraction expressing the ratio of the distance to the cut- ting. Each fraction being recorded in its proper column either " right " or " left " of the centre. Points between the centre and angle stahe, are located by measurements from the centre. VI. To find the position of the slope staJce : Measure off a trial distance from the angle stake, and determine the cut as before. Multiply the cut by the ratio of height to base of the proposed slope. If the trial distance be greater than this product, the assumed point is too far out, and vice versa. Repeat the trial until the ratio of the distance to the cut expresses the ratio of slope. 322. The cutting at the angle stake is, in cases of a tolerably uniform surface, a good guide to the distance to the slope stake. Thus, when the angle cutting of an excavation is 16 feet and the ratio of slope 1^ to 1, the distance out, for a level surface, would be 24 feet ; but if the ground in that distance rise 2 feet (and which in practice may be determined pretty correctly by the eye), then the horizontal distance must be in- creased by something more than IJ- times 2 feet. When the surface descends, the estimated distance out, for a level surface, should in like manner be diminished. In em- bankments the conditions are reversed ; the steeper the rise, the shorter the distance out. 323. The following examples will serve to elucidate the sub- ject still further: 286 ELEMENTS OF SURVEYING. [book VII. 9.7 10.0 110 Fig. 148. Diet. Left. Centre Cut. Right. —4. -4.5 6. A — 4.9 — 6.1 A -6.4 4 -7.4 11.1 Figure ].48 represents an embankment cross-section, in which, by reason of the small depth of filling, the height of instrument is a positive quantity. The centre cut is —4.9 ; reading of the rod at the centre, 8.5 ; the sum of these, or "height of instrument," is 3.6. The remaining rod readings are given on the line through the instrument. 324. In the example of the following diagram (Fig. 149), the cross-section is partly in excavation and partly in embank- ment. The ratio of slope is 2 to 1. The centre cut is 2.4. The centre reading is 7.9 ; height of instrument, 10.3. The reading at H, is 6.3; at E, 6.1; at S, 5.4. The point, K, is easily found in practice, it being that point on the surface line where the reading of the rod exactly equals the height of instrument. The reading at F is 13.2; and at S', 15.8. From these readings the cuttings may be found, and the notes completed as below. SEC. v.] CROSS-SECTIOK LEVELING. 287 Fig. 149. Diet. Left. Centre Cat. Right. 4.9 9.8 4.2 A 4.0 3.8 2.4 A -2.9 A — 5.5 11 325. In the following example, there is a regular rise in the surface-line of one foot in eight. The ratio of slope in the excavation is to be 1 J to 1 ; height of instrument, 14.2. In seeking for the position of the slope-stake S', a distance out of 13 feet is tried ; the reading of the rod at the trial point is 4.8. Fig. 150. How does this point compare with the true position of S' ? Ans. Not far enough out. What is the result of a trial at 16 feet out and a reading of 4.4 ? Ans. Too far out. 9.6 What is the true cut and distance at S' ? Ans. 14.4 288 ELEMENTS OF SURVEYING. [BOOK VIL 49 Find the position of S. Ans. ^4-. Note 1. — It sometimes happens in very hilly sections, that it is impracticable to sight to all the necessary points of a single cross-section from one position of the level. In such a case, it is only necessary to work from the centre as far as the surface will permit, then establish a turning-point, precisely as in section leveling ; change the position of the level so as to proceed with the work, and determine the new height of instrument, from which the readings are to be subtracted as before. Note 2. — The degree of accuracy desirable to be attained in setting the slope stake, varies with the kind of earth to be " staked out," so that no exact rule can be laid down. A principle, in quite general use, permits the stake to be set when the calculated distance varies from the trial distance by less than a foot. The limit of error should never be greater than this, but in rock and the harder kinds of earthwork, it should be made much less. SECTION VI. COMPUTATION OF EARTHWORK. 326. Before the work of construction of a railroad or canal commences, the calculation of the earthwork must be com- pleted. The cross-section levels afford the necessary data. These surveys have divided the proposed work into blocks of 100 feet, or less, in length, and which are appropriately termed prism oids. SEC. VI.] COMPUTATION^ OF EARTHWORK. 289 Different methods are employed for estimating their cubic con- tents. The most accurate, though the most laborious, is the prismoidal formula (Leg., Mensuration, page 129), vol. = g(^ + ^'4-4if). B and B' representing the areas of the end sections of the prismoid, M the area of a section midway between them, and I the entire length of the solid. The principal difficulty in applying this formula lies in finding the dimensions of the middle section. 327. We will show the application of the formula by an example of road excavation. To simplify the problem, we will suppose such a degree of regularity in the ground surface that the angle cuttings may be omitted. The length is supposed to be 100 feet. The other dimen- sions are given in the diagram. The areas of the end sections are easily found. It is only necessary, in each case, to add together the areas of the trape- zoids composing the whole end figure, as represented in the diagram, and subtract therefrom the sum of the triangles which lie outside the section. The dimensions of these triangles are always expressed in the cross-section notes, by the records for the slope stakes. The area of B is thus found to be 104.8 sq. feet, and of i?', 116 sq. feet. Now, if a section of this prismoid be taken midway between the two ends, each of its several dimensions must be an arith- metical mean of the corresponding measurements of the end sections. Thus, the centre cutting is found to be 5 ft; the dis- 290 ELEMEI^TS OF SURVEYING. [book Vll. tance from the angle to slope stake, on the left, 9.9 ft., ( — ^-^) ; (7 V I K \ -^-^ — ), &G 10.8 "^^: ""■^ ^ — _, 6.6 \. i S. XI 9.9 8. 8. 3.9 2.6 M Fig. 151. The area of if is 111.3 sq. ft. Vol. of the prismoid = - x 100 (104.8 + 116 + 445.2), = 11100 cuhic feet. 328. In appljdng the prismoidal formula to an example in cv^hich one end section has more given dimensions than the other, the calculator is frequently in doubt how he shall average these dimensions to obtain the middle section. As a rule, each cut- ting of the most irregular section should be averaged with the cutting nearest opposite to it in the other section. We will illustrate this by an example ; representing the sec- tions by the field-notes only. SEC. VI.] COMPUTATIOK OF EARTHWORK. 29] 1 j Dist. Left. Centre Cut. Right. 2 2.60 17.2 25.8 11.6 17.4 16.8 A 11.2 A 16 2 15.8 10.4 13 A 10 A 10 8.4 8 12.6 8 12 The half-width of the road, for which A is given in the notes, is to be considered as 8 feet. The length of the prismoid is expressed by the difference of the given distances, or 60 feet. The dimensions of the middle section are found as follows : The centre cut is half the sum of the given similar dimen- 15.8 + 10.4 -,ow . sions, ^ = 13.1 feet. 11.5 On the right, the average of the angle cuttings gives —^ ; for the next measurement, both cut and distance must be aver- aged ; it is, ix(10 + 8)=9 J-x(8 + 12) = 10 or 10 The last term in the upper section must be averaged with the last in the lower, thus : Jx( 8+ 8.4) = 8.2 S^ Jx (12 + 12.6) = 12.3 ^^ 12.3* On the left, the measurement — of the upper section, must be averaged with the centre cutting of the lower, being nearest opposite to that point. We have, ^x (16 + 10.4) = 13.2 13^ i.x( 2+ )= 1 ^"^ 1 ' 14 At the angle, in like manner, we have -j-; and finally at 14 4 the slope stake j^r^- The complete dimensions being 292 ELEMENTS OF SURVEYIKO. [book VII. 14.4 14 13^ 2176 A T~ 13.1 11.5 9 8.2 10 12.3 The area of the upper section, after subtracting the triangles, as before, is 543.47 sq. feet. The area of the lower end section is 325.44. The middle section contains 429.8 sq. feet. The volume of the prismoid is I X 60 (543. 47 + 325. 44 + 4 x 429. 8) = 25881.1 cubic feet, or 958.56 cubic yards. 329. The following method* of computing the contents of consecutive volumes between regular cross-sections of excavation and embankment, contained by uniform slopes, will be found of service. The accompanying diagram represents the cross-section of a railroad cut, h being half the width of road-bed, c centre-height, •/""/ I V h Fig. 152. r elevation of right slope-stake above grade, r' its horizontal distance from nearest side of road-bed, I elevation of left slope- stake, V its horizontal distance from nearest side of road-bed, and * " Formulae for Railroad Earthwork," by J. Woodbridge Davis, C. E., Ph. D. SEC. VI.] COMPUTATIOK OF EARTHWORK. 293 w the entire top-width or horizontal distance between slope- stakes. The area of this section is evidently ^rh + ilb + ^c {r' + J)) + ^c {V -{-h). r I Let 8 denote the ratio of slope ; then S = — = j,^ ^nd r = Sr', I = SI'. Substituting and reducing, Area Section = ^Sb {r' -f V) + ic (r + Z' + 2^). Adding and subtracting S/y^ does not change its value ; .-. Area Section = ^Sb {r' -\-l' + 2b) — Sb'^ + ic (/ + r + 25), or Area Section = ^lo {c-\-Sb)—Sb\ Supposing w', d to represent width and centre at next station, the area of its cross-section may be expressed by a formula similar to the above; half the sum of these, multiplied by the distance, />, between, and divided by 27, gives a near approximate of the volume in cubic yards ; Vol. = ltvc-\-'w'c'-^Sb {w-{-w')—ASb^] y^. Add two consecutive volumes of equal length by means of the general formula, w", d , representing the width and centre of third cross-section : Vol. = [wc + 22dd + w"c" + Sb(w + 2id + iv") - SSb^] ■^. By continual addition we may get a formula for the sum of any number of consecutive volumes ; but, letting n denote the number of volumes, we may at once indite a general formula for the calculation of any number of volumes consecutive. Thus we have ^^'" — \ j^Sb(iv-\-2w'-{-&c.-\-2w,^io,+,) - 4:Sb^n ] 108* Divide and multiply by 2 to convert the formula into more convenient shape, call w'dj w"d', &c., mid-products, wc and Wn+iCn+i end-products, and we have 294 Vol = ELEMENTS OF SURVEYIi^G. mid-products + ^ e7id-products -]-Sb {mid-2vidtlis-\-^ end-icidths) — 2Sb^ X no. of vols. [book vil Let us illustrate this formula by applying it to the following extract from a field-book, containing columns of stations, centre cuts, left and right heights and distances of slope-stakes, the road-bed being 18 feet, slope 1 to 1, distance apart of stations, 100 feet : station. Left. Centre. Right. 1 A^T 3.0 ■h\ 2 A^ 5.1 m 3 1 oO 6.4 ^h 4 81 ni 7.2 T^.V 5 m 9.0 ^h 6 16? 6.7 T«A OPERATION Stations. Widths. Centre. Products. 1 11.55 X 3.0 — 34.65 2 . 27.8 X 5.1 = 141.78 3 31.5 X 6.4 =: 201.60 4 34.1 X 7.2 z= 245.52 5 38.0 X 9.0 z= 342.00 6 15.85 X 6.7 = 106.195 9 ov Sbx 16S.S0 1071.745 1429.2 -810. 9)169094.5 6)18788.3 3131.38 cu. yds. SEC. VI.] COMPUTATION OF EARTHWORK. 295 330. The foregoing method of calculating earthwork is approximate. To find the true contents we use a formula of correction, which is obtained in the following manner. Let ^w {c + Sb) - 8h\ iw' (c + Sb) - SI^ be the end areas of a volume of earthwork. Then ^ (tu + w'), i{c + c') are evidently the width and centre of mid-section, and its area is iiw-^w') (^^ + Sb)-Sd^. Multiply this by 4, add thereto the end areas, multiply all by ^ D, and divide by 27, to obtain in cu. yds. True vol - P (^^ + e^V) + (?..' + ^^/.)\_^^ irue VOL - y _^^sb{w + w')-12Sb^ 7 324 This formula would prove very unwieldly to carry through the calculations for series of volumes, especially in the consideration of intermediate stations. The same results may be obtained in a simpler manner by using the difference between true and approximate contents as a correction. Subtracting the approx- imate volume, [wc-\-w'c'-}-Sb(w-}-w')—4:Sb^]^-s, from the true, we have, after reduction, {w—w'){c'-c)D 324 ' for the error or correction. 296 ELEMENTS OP SUEVEYING. [BOOK VII. ^^^- ^^^' We see by the formula that to correct a volume, 987 the difference of widths, found by a subtraction in — 481 one direction, must be multiplied by the difference 208 of centres resulting from a subtraction in the opposite direction, this product multiplied by the 1449 length of volume and divided by 324. Applying 12 ) 3827 this rule to the second volume of example, we 9 ) 319 have width at station 2 [27. S]—2uidth at station 3 3)35.4 [31.5] =—3.7; centre station 3 [6A]— centre sta- Zu^ Cor. ^^^^ ^ t^-^] = ^•^- -^•'^ X +^-^ 3131.38 Approx. con'ts. ^^^^ = -^^^' ^^^ ^^'' ^^ ^^ «.^« ^~ rr. ,, extra column. Treat each volume 3119.57 True con ts. in like manner, remembering that the first and last numbers in column of widths are half-widths. We here represent the column of corrections. The sum, — 3827, divided by 324, is —11.81 cu. yds. This added to the approx- imate result yields for the true answer 3119.57 cu. yds. It may happen, in practice, that some of the volumes involved need no correction whatever, which fact will be apparent from the formula by inspection and without actual labor. BOOK VIII. TOPOGRAPHICAL SURVEYING, 331. Besides the surveys that are made to determine the area of land and the relative positions of objects, it is frequently necessary to make minute and careful examinations for the purpose of ascertaining the form and accidents of the ground, and to make such a plan as will distinguish the swelling hill from the sunken valley, and the course of the rivulet from the unbroken plain. This branch of surveying is called Topography. In surveys made with a view to the location of extensive works, the de- termination of the slopes and irregularities of the ground is of the first importance ; indeed, the examinations would other- wise be useless. 332. The manner of ascertaining these irregularities is, to suppose the surface of the ground to be intersected by a S3^stem of horizontal planes at equal distances from each other ; the curves determined by these secant planes, being lines of the surface, will indicate its form at the places of section, and, as the planes are nearer or more distant from each other, the form of the surface will be, more or less, accurately ascer- tained. If such a system of curves be determined, and then pro- jected or let fall on a horizontal plane, it is obvious that the curves on such plane will be nearer together or farther apart, as the ascent of the hill is steep or gentle. 298 ELEMENTS OF SUKVEYING. [bOOX VIII. If, therefore, such intersections are made, and the curves so determined are accurately delineated on paper, the map will give such a representation of the ground as will show its form, its inequalities, and its striking characteristics. 333. The subject divides itself, naturally, into two parts : 1st. To make the necessary examinations and measurements on the field ; and, *^d. To make the plot or the delineations on paper. When the area is extended and the contour planes are widely separated, points along the ridges and summits, and also along the valley bottoms, are located by the Transit or Plane Table, and their heights above a datum plane are determined by means of Transit angles of inclination, or by the Y Level, or, in rapid reconnaissance, by the Barometer. We shall, perhaps, be best understood, by giving an example or two, and then adding such general remarks as will extend the particular cases to others that may occur. EXAMPLE FIRST. 334. Let A, Fig. 153, be the summit of a hill, the contour •of «vhich it is required to determine and represent. At A, let a stake be driven, and let the axis of the transit, or level, be placed directly over the nail which marks its centre. From A, measure any line down the hill, as AB, using the telescope of the transit, or level, to arrange all its points in the same vertical plane. Great care must be taken to keep the measuring chain horizontal, for it is the horizontal distances that are re- quired. At different points of this line, as a, h, c, d^ &c., let stakes be driven, and let the horizontal distances Aa, ai, he, and cd, be carefully measured. In placing the stakes, reference must be had to the abruptness of the declivity, and the accuracy BOOK VIII.] TOPOGRAPHICAL SURVEYING. 299 with which the surface is to be delineated ; their differences of level ought not to exceed once and a half, or twice, the distance between the horizontal planes of section. Having placed stakes, and measured all the distances along the line AB, run another line down the hill, as AC, placing stakes at the points e, f, g, and h, and measuring the horizontal distances Ae, ef, fg, and gli, Eun also the line AD, placing stakes at i, I, m, and n, and measuring the horizontal distances Ai, il, Im, and mn. Each line, AB, AC, AD, running down the hill, from A, may be regarded as the intersection of the hill, by a vertical plane ; and these secant planes are to be continued over all the ground which is to be surveyed. If the work is done with a transit, or with a level having a compass, the angles DAB and BAC, contained by the vertical secant planes, can be measured ; if it is done with a level, having no needle, let any 300 ELEMENTS OF SURVEYIN'G. [book VIIL of the distances ae, If, ai, hi, &c., be measured with the chain, and there will then be known the three sides of the triangles Aae, Ahf, Aai, AN, &c. Let, now, the difference of level of all the points marked in each of the lines AB, AD, AC, be determined, being careful to hold the rod upon a point near each stake which represents the general surface, and not at the bottom of a hole nor upon the top of a mound, a precaution to be observed in every leveling operation. Let now the heights of all the points marked on each of the lines AB, AD, and AC, be found with reference to the datum plane, which, near the coast, may be the plane of mean low water, or a plane assumed so that it will lie below the lowest point to be delineated. In the present example, of a single slope only, the results of the measurements and leveling are : Line AB. Distances. Aa = 4:0 feet. ab = 50 " be = 30 " cd = 4:6 '' Heights above datum plane. A = 64: feet. « = 52 " ^ = 44 " c = 35 " d = U " Distances. Ae = 28 feet. ef = 45 '' fg = 55 " gh = 49 <( Line AC. Heights above datum plane. A = 64: feet. e = 53 " / = 44 " ^r = 32 " h = 18 (t BOOK VIII.] TOPOGRAPHICAL SURVEYING. 301 Line AD. Distances. Heights above datum plane. Ai = 25 feet. ^ = 64 feet. il = 55 " i = 55 '' Im = 38 " I = 4:2 '' mn = 48 " m = 35 '' n =z 21 '' Angle CAB = 25°, Angle DAB = 30°. These data are sufficient, not only to find the intersections of horizontal planes with the surface of the hill, but also for delineating such curves of section on paper. Plot of Work. 335. Having drawn, on the paper, the line AB, lay off the angle BAG =25°, and the angle BAD = 30°. Then, from a convenient scale of equal parts, lay off the distances Aa, ab, be, cd, Ae, ef,fg, gh, Ai, il, Im, and mn. Let the horizontal planes be passed at distances of 8 feet apart, in which case the point A, in the example given, will lie in the eighth contour plane, counting from the datum plane. Since A is the highest point of the hill, and the difference of level of the points A and «, is 12 feet, the first plane, reckoned downwards, will intersect the line traced on the ground from A to B, between A and a. Regarding the descent as uniform, which we may do for small distances, without sensible error, we have this proportion : as the difference of level of the points A and a, is to the horizontal distance Aa, so is 8 feet, to the horizontal distance from A to where the first hori- zontal plane will cut the line from A to B. This distance being thus found, and laid off from A to o, gives o, a point of the curve in which the seventh plane intersects the ground. The points at which it cuts the line from A to C, and the line from 302 ELEMENTS OF SURYEYIXG. [book VIII. A to D, are determined similarly, and three points in the seventh curre are thus found. The graphic operations are greatly facilitated by the aid of a sectoral scale of equal parts, of which Fig. 154 is a repre- sentation. Fig. 154. It consists of two arms, or sides, which open by turning round a joint at their common extremity. On each arm of the sector, there is a diagonal line that passes through the point about which the arms turn : these diagonal lines are diyided into equal parts. The advantage of the sectoral scale of equal parts, is this : Let it be proposed to draw a line upon paper, on such a scale that any number of parts of the line, 40 for example, shall be represented by one inch on the paper, or by any part of an inch. Take the inch, or part of the inch, from the scale of inches, on the sector ; then, placing one foot of the dividers at 40, on one arm of the sector, open the sector until the other foot reaches to the corresponding number on the other arm ; then, lay the sector on the table without varpng the angle. Now, if we regard the Hues on the sector as the two sides of a triangle, of which the line 40, measured across, is the base, it BOOK YIII.] TOPOGRAPHICAL SURVEYIl^G. 303 is plain, that if any other line be likewise measured across the angle of the sector, the bases of the triangles, so formed, will be proportional to their sides. Therefore, if we extend the dividers from 50 to 50, this distance will represent a line of 50, to the given scale ; and similarly for other lines. In the example before ns (^ig. 153), let it be borne in mind, that the descent from A to a, is 12 feet, and that it is required, upon the supposition of the descent being uniform, to find that part of the distance corresponding to a descent of 8 feet. Take the distance from A to a, in the dividers, and open the arras of the sector until the dividers will reach from 12 on the line of equal parts, on one side, to 12 on the line of equal parts on the other. Then, without changing the angle, extend the dividers from 8 on one side to 8 on the other ; this will give the proportional distance to be laid off from A to o. Or, if the dividers be extended from 4 to 4, the proportional distance may be laid off from a to o. If the distances to be taken from the sector fall too near the joint, let multiples of them be used. 336. The sixth plane is to pass 8 feet below the first, that is, 16 feet below A, or 4 feet below a, a being 12 feet below A. Take the distance ai, in the dividers, and extend the sector, so that the dividers will reach from 8 to (the descent from a to b being 8 feet) 8, or from 80 to 80 ; then, the distance from 4 to 4, or from 40 to 40, being laid off from a to p, gives p, a point of the sixth curve. The difference of level between a and b being 8 feet, and the difference of level between a and p being 4 feet, the dif- ference of level between p and b must also be 4 feet ; hence, the fifth plane will pass 4 feet below b, and q, determined as above, is a point of the fifth curve, and so on. After having determined the points in which each contour line cuts 304 ELEMEIiTTS OF SUBVEYING. [book VIII. the lines diverging from A, let the contour lines be drawn through them, so as to indicate the surface of the hill. The numbers (64), (56), &c., show the vertical distances of the respective planes above the plane of reference. 337. Having drawn the horizontal curves, the next thing to be done is so to shade the drawing that it may represent accu- rately the surface of the ground. This is done by drawing a system of small broken lines, as in the figure, perpendicular in direction to the horizontal curves already described. In all topographical representations of undulating ground, the lines of shading are drawn perpendicular to the horizontal curves, to indicate the direction of the flow of water down the declivity. A profile along either of the diverging lines may be plotted by the rules already given. Fig. 155 shows the profile along the line AB, B d EXAMPLE SECOND. 338. The following example will illustrate the methods em- ployed in making a topographical survey, where great accuracy is required. By means of a transit or level, range a line of stakes A, B, C, D, E, &c., Fig. 156, along one side or through the middle of the ground to be surveyed, at equal and convenient distances from each other, say 100 feet apart. Mark, with a piece of red chalk, on each stake in this row, one of the letters of the alphabet, A BOOK VIII.] TOPOGRAPHICAL SURVEYING. 305 A, B, G, D, E, &c., in their order. At A, range a line of stakes, perpendicular to AE, planting the stakes at intervals of 100 feet; and mark them with the letters Aq, A^, A^, &c., which are read A zero, A one, A two, &c. D E A. 1 A. 2 A A 3 4 B 1 B z B B 3 4: C c 1 C 2 C C 3 4 D Eo D \ El D 2 a E2 D D 3 4 E3 Ea Fig. 156. At ^, range a line of stakes also perpendicular to AE, and at distances of 100 feet from each other, and designate them B^, B^, ^3, &c. Do the same at (7, D, E, &c., until all the stakes are placed, dividing the area to be surveyed, into squares of 100 feet on a side. The letters and figures should be plainly marked on a smooth face of each stake, for facility of reference. If this system of notation be followed, the stakes may be recorded without danger of confusion. 339. The following is the form of a field-book, used in topo- graphical leveling : 306 ELEMENTS OF SURVEYIKG. [book VIII. Field Notes. Benches. + Sights. Height Inst. — Sights. stations. Heights. Bench 4.136 12.142 12.1 E. 0.0 1.9 D. 10.2 C3 11.906 22.441 1.607 Ca 10.535 6.0 E. 16.4 6.9 Es 15.5 6.8 D3 15.6 1.9 Di 20.5 3.7 0. 18.7 6.8 B4 15.6 1.7 C. 20.7 B3 11.914 33.448 0.907 Bs 21.534 8.0 E„ 25.4 4.0 B. 29.4 4.1 D„ 29.3 1.9 c. 31.5 5.0 A3 28.4 9.9 A. 23.5 1.7 I>4 31.7 1.1 E4 32.3 0.1 A. 33.3 Co 11.813 45.225 0.036 Co 33.412 4.8 B. 40.4 2.6 Bo 42.6 Ai 8.925 52.669 1.481 A, 43.744 3.2 Ao 49.5 340. In the example taken for illustration the point E^ is the lowest point. Set up the level and take a reading upon a bench, which has BOOK VIII.] TOPOGRAPHICAL SURVETIKG. 307 been determined to be 8.006 feet above some plane of reference, the sea level for instance. Suppose the reading to be 4.136, which is entered in the " + Sight " column. The " Height of Instrument" will therefore be 12.142, as entered in the proper column. Next take readings upon E^y Dc^, and Cg, reading C^ to thousandths as it will be used as a "turning point" (Art. 304). Kemove the level to a new position and read C^ again, 11.906, which gives a new " Height of Instrument " 22.441. From this position read as many stations as possible, and then use B^ as a " turning point," and so on. Plotting the "Work. 341. Draw, on a piece of paper, a straight line AE. From a scale of equal parts, set off distances AB, BC, &c., each to represent 100 feet. Erect perpendiculars to AE, at each of the points A, B, G, &c., and then set off the distances from A to 1, from 1 to 2, &c., each to represent 100 feet ; and through the points 1, 2, 3, and 4, draw parallels to AE. These, by their intersections with the lines drawn through A, B, C, &c., will determine the position of the stakes Aq, Ac,, &c.; and write in red ink on the plot, the height above the plane of reference of each stake, taken from the column of total differences in the field-book. Let us suppose that the horizontal planes are to be taken at distances of 6 feet. We may find the points in which the contour lines intersect the sides of the rectangles, as in Example First. A very convenient scale for finding the points in which the contour lines cut the sides of the rectangles may be constructed thus: Upon any line as AB, Fig. 157, erect equidistant perpendiculars as at 0, 1, 2, &c. Parallel to AB draw lines, alternately heavy and light, as at 1, 2, 3, &c. Suppose we wish to find where the 12-foot plane cuts the side of a rectangle, C^G^ for example. The height of G^ is 10,5, and 308 ELEMEN^TS OF SURVEYING. [book VIII. of G^ it is 20.7, from which data we find the rise from C^ to G^ to be 10.2. The rise from G^ to the 12-foot plane is 1.5. 13 12 U 10 " -r /' 9 8 •J / ,/-'' 6 .S 4 3 2 1 /' S/-'- — A / j 12 3 4 15 6 9 f 8 9 1 F [Q. 1 57. Now fasten a fine thread at A on the scale, and stretch it to cut 10.2 on the vertical scale, as ^C in Fig. 157. Look along the horizontal line 1.5, as dotted, and its intersection with AG will be at 8, distant 16 ft. on the horizontal scale from A, or G^. The tenths on the vertical and horizontal scales may be estimated by the eye with sufficient accuracy. Engineer's Section Paper may be advantageously used for the above purpose. 342. If only a rough plot is needed, the Surveyor may take the plot thus commenced, into the field, and by the eye trace the contour lines on the map. If we note where the lines of the rectangles cut fences, roads, streams, &c., we can, by joining the points, obtain a plot of the ground. BOOK VIII.] TOPOGRAPHICAL SURVEYmG. 309 The coutour lines may be traced on the ground as follows : Set up the instrument and read a staff placed upon a bench, and determine the height of instrument above the datum plane. Sup- 2a5 15.6 20J' 31.7 32.9 Fig. 158. pose it to be 11.432. If we wish to mark out the contour line six feet above the datum plane, we set the target of the leveling rod at 11.432—6 = 5.432, and direct the rodman, by signals, up or down the hill, till the horizontal wire of the telescope coincides with the horizontal line of the vane. The foot of the staff is then 6 feet above the datum plane. Let a stake, marked 6, be driven here, and direct the rodman around the hill, until a second position shall be found, when the horizontal wire of the telescope will cut the vane, and drive there another stake, marked 6 ; and so on, until a sufficient number of stakes have been driven to determine the curve (6). Then, let the line of stakes, marked 6, be surveyed with the compass and chain, and plotted. Other contour lines may be found in a similar manner. 310 ELEMENTS OF SURVEYING. [book VIII. 343. A practical application of this method is also called for when a surveyor is required to determine, in advance, the area of land which will be submerged by the construction of a dam. lu this case the contour plane is fixed by the proposed height of the dam. Having set up at any point, a reading is taken upon the rod held upon a point at the proposed height of the dam, and the rodman is then directed to points along the shore of the proposed pond, or reservoir, which shall give the same reading. In this way the contour of the pond, when there is no overflow at the dam, can be determined. If the calculated depth of water upon the dam at the maximum overflow be added to the rod reading used above, the high-water contour may be also traced out. 344. When the plane of reference is so chosen that points of the work fall on different sides of it, all the references on one side are called positive, and those on the other, negative. The curves having a negative reference are distinguished by placing the minus sign before the number ; thus — ( ). Shading and Delineation. 345. Fig. 159 represents a piece of ground sloping towards i>, which is the lowest point ; and through this point the plane of reference is supposed to pass. The following table indicates the heights of the several points above the plane of reference. Ft. Ft. Ft. Ft. c above D, 2 H above D, 7 V above D, 9 B about D, 12 d " D, 4 h '' D, 1 q " D, 9 L '' D, 13 }i '' D, 4 s '' D, 1 c '' D, 9 '' D, 14 t A 4 f '^ D,% n '' D, 11 a ^^ D, 15 9 '' D, 5 I '' A 8 i " D, 12 F '' D, 15 I '' D, 5 I " Z), 9 m '' D, 12 E '' D, 17 A above D, 20 feet. BOOK vrii.] TOPOGRAPHICAL SURVEYING. 311 The first horizontal plane is passed 2 feet above D, and the curve of intersection with the surface passes through c. The second secant plane is passed at 3 feet above D, and intersects the i./liiiiiiii'i:iiN!ntm surface, in the curve uvy and also near d, which is one foot above the curve. All the other secant planes are passed at three feet from each other ; and, comparing the height of each point above D, with the curves lying nearest, on either side, the positions of all the points, with respect to the curves, and with respect to each other, are easily seen. 346. The manner of shading the map, so as to indicate the hills and slopes, consists in drawing the lines of shading per- pendicular to the horizontal curves, as already explained (Art. 337). These shading lines are drawn close together, when the slope is abrupt, and further apart, as it grows more gentle. Figure 159 indicates the method of shading. 347. In topographical surveys, great care should be taken to leave some permanent marks, with their levels written on them 312 ELEME^^TS OF SURVEYIXG. [BOOK VIII. in a durable manner. For example, if there are any rocks, let one or more of them be smoothed, and the vertical distance from the plane of reference marked thereon ; or let the vertical distance of a point on some prominent building be ascertained and marked permanently on the building. Such points should also be noted on the map, so that a person, although un- acquainted with the ground, could by means of the map go upon it and trace out all the points, together with their differ- ences of level. 348. Besides representing the contour of the ground, it is often necessary to make a map which shall indicate the woodland, the marsh, roads, ditches, etc. For this, certain characters, or conventional signs, have been agreed upon, as the representatives of things, and when these are once tixed in the mind, they readily suggest the objects for which they stand. Those which are given in the four following pages have been adopted by the TJ. S. Coast and Geodetic Survey, and are used in all plans and maps made under its direction. It is very desirable that a uniform method of delineation should be adopted, and it is, therefore, recommended that the conventional signs given in the accompanpng plates be carefully studied and uniformly followed. BOOK VIII.] TOPOGRAPHICAL SURVEYING. 313 PiQ. 160.— Sparsely settled Town, Salt Marsh, Pine Woods, Ditches, Fences, and Undefined Roads. ^^^^^Ml lii^JfT_- SM"^ g£L Fig. IQl .— Blockiug of Cities, Large Buildings, Suburban Vilius and Grounds, Fresh Marsh. 314 ELEMEN"TS OF SURVEYING. [book VIII. Fig. 162.— Eailroads, Canals, Iron bridges, Eocky-cliffs, Mid-river drift, Water-worn Rocks. Mixed Woods over hill curves. Fig. 163.— Eroded drift banks, with boulders set free ; and scrub deciduous woods. BOOK VIII.] TOPOGRAPHICAL SFRVEYIN"G. 315 Fig. 164.— Heavy Oak Woods, Reclaimed Marsh, and Orchards. Fio. 165.— Fresh Water Pond, Meadow Grass, Sage Brush, and Arroyos. 316 ELEMENTS OF SURVEY! XO. [book VIII. Fig. 166.— Sand and Shingle Beaches, Eroded Earth Banks, Roads, Fences, Shaded Road- sides, Hill-shading. BOOK IX. RAILWAY CURVES. M 349. After the route of a railroad or canal has been deter- mined by reconyiaissance, the centre line of the work is estab- lished by a transit (see Section III, B. IV). 350. This preliminary survey establishes a succession of straight lines, of greater or less length, according to the obsta- cles to be avoided or the advantages to be gained, arising from the nature and the contour of the ground. The angle formed, at each change in the direction of the route, is carefully measured and recorded. In the final survey or location, these angles are replaced by curves ; and in order that the change in direction shall be as gradual as practicable, the straight lines of direction are made tangents to the curves at their point of meeting. The preliminary survey is termed, by the engineer, " running out tangents." 351. We will proceed to describe the method of locating curves, first giving the mathematical principles applicable to the subject. Let AD and DB, Fig. 168, be two tangents to the arc of a circle, AB. Draw the radii AC, BC, and the secant CD. The following relations are easily deduced. The tangents AD and DB are equal (Leg., Bk. Ill, Prob. 14). The angles A and B are right angles (Leg., Bk. Ill, Prop. 9), consequently the angles C and D, of the quadrilateral ADBC, must be sup- 318 ELEMENTS OF SUKVEYING. [book IX. (1) plements of each other. The angle TDB, therefore, must be equal to the angle ACB. The right-angled triangles ADC and BDC are equal (Leg., Bk. I, Prop. 17) ; hence, the angle DC A is equal to DCB, and each equal to ^TDB. Let the radius AC hQ rep- resented by r, the distance AD by d, and the angle TDB by a. Then will (Trig., Art. 66), d =z r tang ^a. The angle TDB is the angle formed by two straight lines of the preliminary survey, and is carefully measured by the engineer, in locating tangents. From formula (1), we can determine the value of d, for any given values of a and ?•; and hence we can determine at what point on the tangent, laid off from i>, the curve of any given radius must commence. It is evident, both from the diagram and the formula, that for any given angle between the tangents, the greater the radius of the curve, the greater will be the distance cut off between the intersection of the two tangents and the point of tangency. It is sometimes necessary to give a particular value to d* In such case, we use the formula, r = d coi \a. (2) 352. The work of laying out or locating a curve in the field is somewhat simplified, if the curve have such dimensions that one chain, of 100 feet, have an arc corresponding to an exact number of degrees. BOOK IX.] KAILWAT CURVES. . 319 The radii of such curves are easily calculated. Thus, a circle in which one degree of arc measures one chain, will have a circumference of 360 chains, or of 36,000 feet, and con- sequently, a radius of -r — —-t::-^. = 5729.58 feet. In a circle in which two degrees of arc correspond to a chain, the radius will be only half as great, or 2864.79. When three degrees of arc measure one chain, the radius is 5!?M8 = 1909.85 feet. o The number of degrees, corresponding to one chain, of a TSiWwaj cur\ef is csilled the 'Ulegree of curvature.'* The radius of a one-degree curve is 5729.58 feet; of a two- degree curve, 2864.79 feet, &c. Representing the degree of curyature by c, we have the formula, 5729.58 ,^, r = -^-l (3) r being expressed in feet, and c being the number of degrees corresponding to 100 feet of arc, or in common practice to 100 feet chord. 353. Apply the preceding formulas (1), (2), (3), to the fol- lowing examples: 1. If the angle TDB, of the tangents, be 45° 10', what dis- tance must be laid off from the intersection D, to the point of tangency, to admit of a 4° curve ? From formula (3), we have, r = ^^^^^ = 1432.39 feet. 4 Substituting this value of r in formula (1), we have, d = 1432.39 tang 22° 35' = 595.76 feet. 2. If the angle a be 30°, and the distance d be 600 feet, what is the radius? Ans. 2239.2 feet. 320 ELEMENTS OF SURVEYING. [book IX. 3. What is the degree of curvature in the last example ? Formula (3) gives 5729.58 C =: = 2.°5587 = 2° 33' 31". 4. The angle a being 20° 21', what is the value of d for a one-degree curve ? Ans. Location of Curves by one Transit. 354. The location of curves, according to the most common method, consists in laying off, at the point of tangency A, such angles as shall just subtend one chain of arc. If the arcs Av, vw, tux, &c.. Fig. 169, represent arcs of one chain each, the angles AGv, vCw, &c., are each equal to the degree of curvature. The angles DAv, vAw, wAx, are each equal to one-half the degree of curvature (Leg., Bk. Ill, Prop. 7, Sch., and Prop. 21). The operations in the field are very simple. The party should consist of a transitman, two chainmen, and an axe-man. BOOK IX.] RAILWAY CURVES. 321 The transit is set and adjusted at a tangent point, as A, and directed along the tangent toward D. An angle equal to half the degree of curvature is deflected from AD toward the side on which the curve is to run. The hind-chainman holds his end of the chain at A. The fore- chainman, keeping the chain carefully extended, is directed by the transitman into line with the axis of the telescope. This locates the point v on the curve. From the line Av, another deflection is now made, of the same angle as before. The chainmen move forward ; the hind- chainman stopping at v, while the fore-chain man, keeping the chain extended, is directed by the transitman as before, and a second stake, w, is fixed on the curve. By continuing the process of deflecting angles equal to half the degree of curvature, and causing these angles to subtend measured distances of one chain each, the entire curve is located. 355. The last deflection on the curve rarely corresponds to an entire chain ; it is, therefore, less than the others. Its amount can be readily calculated, when it is remembered that the sum of all the deflections, or the angle DAB, is exactly equal to one-half the angle a. Hence, if from ^ the sum of the deflection angles laid off be subtracted, the remainder will be the final deflection angle, called the sub-deflection angle. The corresponding sub-chord is such part of 100 ft. as the sub-deflection angle is of the deflection angle. In practice the sub-chord should always be laid off, using the sub-deflection angle, to check the work ; the final peg, thus located, should fall upon the previously determined tangent-peg. By this method of laying out curves, any error in chaining any one chord is carried into all succeeding portions of the curve. 322 ELEMENTS OF SUEVEYIKG. [bOOK IX. 356. It is sometimes necessary to remove the transit from the transit- point to some other point on the curve, before the location has been completed. In such a case, the direction of the tangent to this new point should be determined. Suppose x, Fig. 169, to be a located point Dn the curve to which the transit has been transferred, and from which new points beyond x are to be located. Adjust the transit and direct the telescope to A. Lay off the angle Axt, equal to BAx (the sum of the deflections made in locating V, w, and x), — xt is the tangent. By revolving the telescope, the tangent is produced to s, from which deflections may be made as at first. Note 1. — The selection of the radius is governed by cir- cumstances. Curves of the longest possible radius are, in rail- roads, always the most desirable ; but the larger the radius for any particular pair of tangents, the greater the distance by which the curve will depart from the intersection of the tangents. It may happen, therefore, that too large a radius may lead to an obstacle, which the angle in the first survey was made to avoid. The map, therefore, of the preliminary survey, should include so much of the topography of the adjacent section, that the radii of the curves may be selected by an inspection of the map. Note 2. — It will be observed that it is the chord, and not the arc, that is measured for each deflection, when locating in the field ; the difference, in railway curves, of proper dimen- sions, does not lead to sensible error. For curves of a short radius (less than two thousand feet), the error may be diminished, by locating the stakes at half- chain distances, deflecting, of course, half the calculated deflec- tion angle. BOOK IX.] RAILWAY CUEVES. 3^3 Location of Curves by two Transits. 357. The surface, over which it is necessary to locate a curve, may be of such a character as to render it impracticable for the chainmen to make their measurements ; if, however, the various points are accessible to the axeman, as in the case of marshes, shallow lakes, or bays, the stakes may be accurately located by the simultaneous deflections of two transits. The method is based on the following geometrical principle : Fig. 170. Let A and B be the two tangent points of the curve AvB, and D the intersection of the tangents. If from any point v, on the curve, the lines vA, vB, be drawn, then the sum of the angles vAB and vBA is measured by one- half the arc AB, and is therefore equal to one-half the angle a, or to either of the entire angles A or B. To locate the curve in the field, a transit is set at each of the tangent points A and ^, and the deflection angle is deter- mined as in the first method. The transitman at A deflects, in the iffeual way, one deflec- tion angle from the tangent AD. At the same time, the transit- man at B deflects the same angle from the chord BA, or what amounts to the same, he deflects the difference between this angle and -J^a, from the tangent BD. The lines of sight of the 324 ELEMEN"TS OF SURVEYING. [book IX. two telescopes now intersect at a point v, on the curve, one chain from A. The flagman, directed at the same time by both transitmen, is readily brought to the location of the point. By a repetition of this process the entire curve is located. Location of Curves by the Chain alone. 358. It is sometimes convenient to locate a curve without using the transit. In such case, the following method is gen- erally employed. Let A represent the point of tangency, G the centre, and V, 10, X, located points of the curve, one chain apart. From V, draw vu perpendicular to the tangent, and it will be the first offset, which denote by o. Denote the length of the chain by ?, and the radius A C, by r. If, now, we suppose A G to be prolonged till it meets the circumference in some point, on the other side of the centre (7, and this point then to be jomecr with v, and vn then drawn parallel to the tangent, we shall have (Leg., Bk. IV, P. 23), 1% Av^ = 2r • An ; hence, o = ^ (4) If, DOW, we prolong Av till vt := Av, and join t and w, rv BOOK IX.] RAILWAY CURVES. 325 tw will be the second offset, and will be double vu. For, the triangles in the figure, whose vertices are C, and whose bases are the equal chords Av, viv, &c., are isosceles and equal. Now, in any one of the triangles, the sum of the two angles at the base and the vertical angle C, is equal to two right- angles. But, since Avt is a straight line, tvw-\-wvC-j-CvA, is also equal to two right angles. Therefore, tvtu is equal to any one of the equal angles at C, and is, consequently, double the angle uAv, which is half the angle C. Since the triangle tuvt is isosceles, if vp be drawn perpen- dicular to the base, it will bisect both the baseband the vertical augle, making tp =piv. But the right-angled triangles Auv and vtp are equal (Leg., Bk. I, P. 6); hence, ho = 2vu. Denoting the second offset by o', we have, c- = ^. (5) 359. The practice is as follows : Having calculated the first offset 0, fix one end of the chain at A, Fig. 171, and stretch the chain to v, measuring uv = o perpendicular to the tangent ; the measurement of o may be made with a short supplemeiitary steel tape, its end division being marked to hundredths of a foot, or with a rod marked with two divisions each equal to o. Next hold one end of the chain at v and align a pin at t, on Av pro- longed, making vt = Av ; then swing the chain towards w till the measured distance tw =: 2o :=■ o'. Prolong vw as before and find a new point x, and so continue till all the full chords have been laid down. The last chord is usually a suh-chord and its offset must be calculated separately. Suppose it is required to run a curve of 1000 feet radius, the angle of intersection, «, Fig. 170, being 48°. From the formula d = r tan — we find DA = 4-45.2, and then locate the tangent peg at A, and also that at B. 326 ELEMEISTS OF SURVEYING. [BOOK IX. We also find o = 5 and o'= 10. The angle at the centre sub- tended by the 100-ft. chain is found by the relation L^^ — r sine \ angle at centre (Leg., Trig., Art. 64). In the case supposed we find the angle subtended to be 5° 44'. The number of full 48° chords is equal to o .., -, which gives 8 full chords, and a remain- ing angle at the centre of 2° 8'. To find the sub-chord we have sine (?-^) X 1000 x 2 = 37.24 feet. The final offset is ^^^^^ = 1.39. r The peg at the end of the sub-chord should fall at the second tangent peg, if no errors have been made in chaining. Note. — In employing this method of locating curves, the aligning by which the chords are produced should be done with much care, as any error in locating a stake involves much greater and increasing errors in succeeding stakes. This is called, by engineers, ** the method by offsetting from tangent ^nd chords produced." EXAMPLES. 1. What are the tangent and chord offsets, for a curve of 2000 feet radius ; the stakes to be 100 feet apart ? Ans. From tangent, 2.5 ft.; from chord produced, 5 ft. 2. Find offsets for a one-degree curve. Ans, Tangent, .87 ft.; chord, 1.74 ft. 360. Let it be required to run out a curve of 500 ft. radius, the stations being 25 feet apart on the curve. As the chord of the angle i is to be 25 ft, we have (Fig. 172), BOOK IX.] BAILWAY CURVES. 327 i 124 ^„^ ^^"' 2 = 500 = '^^^ from which we find { = 2° 52'. Now, Tx :='bs ^^ sine i x 500 = 25 feet ; and xb—Ts — 500 — (cos i x 500) = .63 ft. Measure from T 25 feet to x^ and at x lay off the per- pendicular xh = .63 ft., thus locating the point b on the curve. To locate c we have Ty = tvc = sin 2 ^ x 500 = 49.95 ft., and yc=Tw = 500— (cos 2i x 500) = 2.50 ft. j; ■ — — •- ' W^ '"^ r^ / y/ / jfc Ji\ \ l\ 1 — i i s w Fig. 172. The next tangent distance to z equals sine 3^x500, and the offset at z equals 500— (cos 3ix500), and so on for succeeding points. When the offsets become too long to be readily and correctly set out, a new tangent is located thus: Prolong cd to li making dh = cd, and then bisect Tie {e having been already located) in/ and range a line df, which line will be tangent to the curve at d. The correctness of the new tangent should always be checked by locating from it the third or fourth station counting back towards T. The same computed values that were used in locating the points already fixed may be again used in locating new points from the new tangent. Laying off the Ordinates. 361. The methods described thus far for locating railway curves, apply to points 100 feet apart. This is sufficiently 328 ELEMENTS OF SURTETIXG. [BOOK IX. accurate for the earthwork. In laying the track, however, stakes every ten or twelve feet are necessary. These are set by drawing" the chain or tape in a straight line between the 100-ft stakes, and measuring from it, offsets, as often as desirable, to the intermediate points of the curve. The length of these offsets, or ordinates, is calculated in the following manner : Let VW represent a 100-ft. chord of a railway curve, of which C is the centre. Draw the diameter HK parallel to VW, and drop the perpendicular VL. Then, FX2 = EL X LK. (Legendre, Bk. IV., Prop. 23, cor. 2). Since EL = r — 50, and LE = r + 50, the value of VL is readily calculated for known yalues of the radius. Let NM be an ordinate, at any distance from VL, say 10 feet. Then, Xm = EM X MX; whence, NM'^ = (r — 40) (r -4- 40). Havinsr determined NM, subtract VL from it, and we have Nt, one of the ordinates required. BOOK IX.] RAILWAY CURVES. 329 In this manner, by calculating the full ordinate to the diameter, and subtracting FZ, any desired number of offsets are determined for the half chain VF. For FWy the ordinates have the same length, but are located in the inverse order. The middle ordinate, FE, is found by subtracting VL from the radius. E XA M PLE. Determine the ordinates 10 feet apart on a 100-foot chord, for a two-degree curve. Eadius, 2864.79 feet. Ans. At 10 feet = .15 feet. At 20 " = .28 « At 30 " = .36 " At 40 " = .42 " Middle ordinate = .43 " Reversed and Compound Curves.* 362. Two curves, of the same or different radii, may join each other and have a common tangent at the point of junction If the curves lie on opposite sides of the common tangent, they form a reversed curve, and their radii may be equal or unequal ; if they lie on the same side of the common tangent, they have unequal radii and form a compound curve. Thus ABC (Fig. 174) is a reversed curve, and ABD a compound curve. The point of contact of the common tangent is called, in a reversed curve, the reversing point, and, in a compound curve, the common tan- gent point or the point of compound curvature. Fiq. m. * Taken, by permission and with slight alterations, from Henck's " Field Book for Railroad Engineers,'" ^30 ELEMENTS OF SURVEYING. [book IX. Fig 175. The reversing point of a reversed curve contained be- tween parallel tan- gents is in the line ^ ^ joining the tangent points. Thus, in the curve ACB, Fig. 175, contained between the parallel tangents HA and BKj the reversing point, G, is in the line AB joining the points of contact A and B, A reversed curve is of use in certain track-work near stations, constructing turnouts, &c., though it is not, or ought not to be, used on the ordinary running track. 363. Given the perjjendicular distance letween two parallel tangents BD = b (Fig. 175), the distance hetiveen the two tangent points AB = a, and the first radius, EG ^ R, of a reversed curve uniting the tangents HA and BK, to find the chords AG ^= a' and GB = a", and the second radius GF = R', Draw the perpendiculars EG and FL (Fig. 1 75). Then the right-angled triangles ABD and EAG have the angle BAD = AEG, since each is one-half AEG, and are, therefore, similar ; hence. AB or Since a BD : h a EA : AG, R : ^a\ 2Rb a a" -\- a' = a, a = a a . (1) (2) To find R' , we have, from the similar triangles ABD and FBL, aih : : R' : \a". BOOK IX.] RAILWAY CURVES. 331 „, aa (3) Any three of the quantities a, a', a", b, R, R, being given, the others may be found from equations (1), (2), (3). EXAMPLES. 1. 5 = 8, « = 160, i? = 900 ; find a', a", R'. a' = 90 ; a" = 70 ; R' — 700. 2. ^ = 8, a' = 90, a" =z 70 ; find a, R, R', d. R=: 900, R' = 700, ^^ z= 8 ; find «, a', a", 364. Given the line AB := a (Fig. 176), ivJiich joins the fixed tangent points A and B, the angle DAB = A, and the angle ABG =z B, to find the common radius EG=GF=:R of a reversed curve to unite the tangents HA and BL. From the triangle ABK, Fig. 176, we have (Art. 32), AE . EK : : sin AKE : sin EAK', but since EAK= 90° -^ sin EAK = cos ^; hence, denoting angle AKE by K, we have, EK -sin K= R- cos A, (1) In like manner, from the triangle BFK, we have, FK'Sm K= R ' cos B. (2) From (1) and (2), by addition, we have, (EK+FK) sin K = 2R sin K ==: R (cos A +cos B). (3) H Fig. 176. 332 ELEMEITTS OF SURVEYING. . [BOOK IX. . • . sin ^ = 1^ (cos A + COS B). (4) But cos ^4-cos ^ = 2 cos J (A-{-B) cos J (A—B), (see Legendre, Trig. Art 67) ; hence, sin X= cosi{A-\- B) cos ^ (A—B). (5) Having found K, we have the angle AEK= E — 180°-(^+^^^) = 180°-(^+90°-^) :=90°-f^-^. (6) In like manner, the angle BFK = F- 90° + B—K. (7) From the triangle AEK, we have AE : AX : : sin ^ : sin E, .-. i?sin ^=1 ^^sin JT. (8) In like manner, from triangle BFK, we have, i? sin i^ = BK sin K (9) From (8) and (9), by addition, we have R (sin ^+sin F) = (AK+BK) s'm K =z a sin K. (10) But, sin ^+sin i^ = 2 sin J (E+F) cos ^ (E—F), (see Legendre, Trig. Art, 67) ; hence, 2R sin i (E+F) cos J (E—F) = a sin K. . D_ i^ sin X '• sin^(^+i^)cosi('^-i^)* ^ ^ Substituting in (11) the values of ^ and F from (6) and (7), we have p ^a^n_K , . ~ cos [^-i (^ + ^)] COS i {A-B) ^^^ Example.— Given a = 1500, A = 18°, ^ = 6° ; find i?. From equation (4), K is found to be 76° 36' 10", and then from (12), R is found to be 1710.48. BOOK IX. J RAILWAY CUEVES. 333 365. If one branch of a compound curve he produced until the tangent at its extremity is parallel to the tangent at the extremity of the second branch, the common tangent point of the two arcs is in the straight line produced which passes through the tangent points of these parallel tangents. Let ACB, Fig. 177, ^ be a compound curve uniting the tangents HA and BK, The radii UG and FC, being perpen- dicular to the common tangent at C, the point of compound curvature, are in the same straight line. Continue the curve p AC to D, where its tan- fig. itt. gent OD becomes parallel to BK, and consequently the radius ED parallel to FB, Then if the chords CD and CB be drawn, we have the angle CFD = CFB, whence BCD, the half- supplement of CFD, is equal to FCB, the half-supplement of CFB. But FCD can not be equal to FCB, unless CD coincides with CB; therefore, the line BD produced passes through the common tangent point (7. 366. To find a limit in one direction of each radius of a compound curve. Let AI and BI, Fig. 177, be the tangents of the curve ; draw IM bisecting the angle AIB ; draw ^X and ^il/ perpendicular respectively to AI and BI, meeting IM in L and M. Then the radius of the branch commencing on the shorter tangent, A I, must be less than AL, and the radius of the branch commencing on the longer tangent, BI, must be greater than BM. 334 ELEMENTS OF SURVEYING. [bOOK IX. For, suppose the shorter radius equal to AL, and hence, /iV' equal to Al; join LJV, then the equal triaugles AIL and iVTZ give AL = LN', so that the curve, if continued, will pass through N, Avhere its tangent will coincide with IN. Then (Art. 365) the common tangent point would be the intersection of the straight line through B and N with the first curve ; but in this case there can be no intersection, and therefore no com- mon tangent point. Suppose, next, that this shorter radius is greater than AL, and continue the curve till its tangent becomes parallel to BI. In this case, the extremity of the curve will fall outside the tan- gent BI in the line AN produced, and a straight line through B and this extremity will again fail to intersect the curve already drawn. As no common tangent point can be found when the shorter radius is taken equal to AL, or greater than AL, no compound curve is possible. This radius must, therefore, be less than AL. In like manner it may be shown that the radius of the other branch of the curve must be greater than BM. If the tangents AI and BI, and the intersection angle / are known, then . AL =: AI tan J J; BM = BI tan ^L These values are, therefore, the limits of the radii in one direction. 367. If nothing were given but the position of the tangents and the tangent points, it is evident that an indefinite number of different compound curves might connect the tangent points ; for the shorter radius might be taken of any length less than the limit found above, and a corresponding value for the greater could be found. Some other condition must, therefore, be intro- duced, as, for example, in the following problem: BOOK IX.] RAILWAY CURVES. 335 368. Given the line AB = a (Fig. 177), which joins the fixed tangent poijits A and B, the angle BAI ^= A, the angle ABI=z B, and the first radius AE = E, tofiiid the second radius BF= R' of a compound curve to unite the tangents HA and BK. Let the first curve be ruQ with the given radius from A to D, where its taugent DO becomes parallel to BI ; then the common tangent point G is in the line BD produced, and the chord GB= GD^BD. The angle GAD, formed by a tangent and a chord, is measured by half the arc AGD ; hence, R sin GAD = R sin IAD =z ^AD (see Legendre, Trig. Arts. 64 and 30) ; hence, AD = 2R sin IAD. (1) In the triangle GAD, since GA and GD are equal, angle AGD = 1SO°—20AD; hence, GAD=IAD=90°—iAGD=90°—iAIBj but from the triangle AIB, ^AIB = 90°—^ {A-}-B); whence the angle IAD = ^(A-\-B), (2) From (1) and (2), we have AD =:^2R sin i (A -\-B). (3) Then in the triangle ABD, we have AB = a, AD =: 2R sin J (A-\-B), and the included angle DAB = lAB-IAD = A-i {A-{-B) = i (A-B) ; whence, we have the proportion (Legendre, Trig. Art. 45), AB-^AD : AB—AD :: teiu i(ADB-\-ABD) : tsini(ADB-ABD). (4) i {ADB + ABD) = i {180°— DAB) ; i {ADB—ABD) may be found from (4); and from the half-sum and half-difference thus obtained, the angles ADB and ABD may be found. These 336 ELEMENTS OF SURVEYING. [bOOK IX. angles being known the side BD may be found from the proportion (Art. 32), sin ABD : sin DAB :: AD : BD. Tlie angle CBI = B—ABD. Again, as above, CB = 2i^' sin OBI, and CD = 2R sin CDO = 2E sin CBI. Substituting these values of CB and CD, in the equation CB= CD + BD, we have 2R' sin CBI = 2E sin CBI-^BD. HR'-R)^^-c-B-i- («) When the angle B is greater than A, that is, when the greater radius is giyen, the solution is the same except that the angle DAB =z ^[B—A), and CBI is found by subtracting the supple- ment of ABD from B. We shall also find CB = CD—BD; hence, ^'^^-2-si^^' (^^ or 2 (R-E) = -^^. ^ ' sm C7^/ Note. — If more convenient, the point D may be determined in the field by laying off the angle IAD — ^ {A -\- B), and measuring the distance AD ■= 2R sin ^ (A-\-B). BD and CBI may then be measured, instead of being calculated as above. Example, a = 950, A = 8% B = r, R = 3000 ; find R'. AD = 2x 3000 sin ^ (8° + 7°) = 783.16 BOOK IX.] KAILWAY CURVES. 337 and DAB = ^ (8°-7°) = SO'. Then, to find ABD, we have log (AB—AD), 166.84 = 2.222300 log tan i (ADB + ABD), 89° 45' = 2.360180 (a. c.) log {AB-^AD), 1733.16 = 6.7611 61 log tan i {ADB—ABD), 87° 24' 17" = 1.343641 .-. ^^Z) = 2°20'43"; whence, CBI = B—ABD = 4° 39' 17". Next, to find BD, log AD (783.16) = 2.893849 log sin DAB (30') = 7.940842 (a. c.) log sin ABD (2° 20' 43") = 1.3880 52 log BD (167.01) = 2.2227 43 Then, from (6), we have log BD (167.01) = 2.222743 (a. c.) log CBI (4° 39' 17") = 1.0907 08 log 2 (B'-B)y 2058.03 = 3.3 13451 .-. R'-R = 102dM; R' = 3000 + 1029.01 = 4029.01. BOOK X. MINING SURVEYING. SECTION I. DEFINITIONS AND GENERAL PRINCIPLES." 369. Mining Surveying comprises all the operations neces- sary to determine the relatiye positions of the parts of a mine with respect to each other, and also with respect to the surface of the earth. 370. The general principles involved in this branch of sur- veying are the same as those used in surface surveying, but, from the nature of the case, certain modifications are required. Stations are designated by lamps instead of flags, and lamp- stands instead of flag-rods, or by plumb-lines properly suspended and lighted ; station points, if temporary, are marked by cross- lines chipped in the rock, or sometimes by simple chalk lines, and, if permanent, by iron pegs driven into holes drilled for the purpose. The compass is rarely used for underground work, and ought never to be used for any but rough work, because of the inaccuracies to which it may lead. A great deal of the mining in the U. S. is done in the far West, where the magnetic declina- tion is not known ; and, further, the declination is seriously affected by the state of the atmosphere, the presence of iron ores, magnetic pyrites, &c. The transit, which is the principal angular instrument em- SEC. I.] DEFINITIONS. 339 ployed, and the only necessary one, differs from the ordinary transit in having a diagonal eye-piece, to permit observations to be made when the telescope is directed vertically upward, and also an arrangement for illuminating the cross-wires. A method of lighting the cross-wires is by the reflector shown in Fig. 178. This is an elliptical piece of brass, silver-plated on the under side, and inclined at an angle of 45° to its ring, which is fitted to the object end of the telescope ; the hole in the reflector admits the use of the telescope, while a light held near the under surface illuminates ^^^' i'*^- the cross-wires. The transit should be furnished with a solar attachment (see Appendix A) for establishing the true meridian and running out lines with reference thereto ; and should also have an extension tripod to adapt it for use in mountain surveys, where one or more legs must be shortened, and for mines, where in many places a short tripod is indispensable. 371. Traversing is the operation of running a zigzag line, from one point to another. The elements of the traverse are straight lines, determined by their lengths and by their in- clinations to certain fixed planes. In mining surveying, three such planes are used ; the first, is either a meridian plane through the origin of the traverse, or a vertical plane through the first course; the second, is a horizontal plane through the origin ; and the third, is a vertical plane through the origin, and perpendicular to the other two. 372. "Working, or Reducing the Traverse, is the opera- tion of finding the length and direction of a single line, equivalent to the zigzag, that is, starting from, and terminating at, the same points. Such a line is called the resultant of the traverse. 340 ELEMENTS OF SURVEYING. [book X. The zigzag line is run along the subterranean openings of a mine. Such openings, when vertical, are called shafts, and when not vertical, tunnels. SECTION II. METHOD OF LOCATING CLAIMS. 373. The methods employed in locating claims on the surface are various, and in general very crude, except when a surveyor, usually a U. S. Deputy Mineral Surveyor, is called upon to make the survey. Prospectors are usually without suitable instruments to lay off their claims with any degree of accuracy. We will. -%^ S5^^" ^, 3l ^. ,,--^- is^ ^■82'"-^... 20^^ T 9 XR 5T\^ Second Standurd. Fis. 179. TaralUl 'S % SEC. II.] METHOD OF LOCATING CLAIMS. 341 therefore, only consider the method employed by the Deputy Mineral Surveyors, who are compelled to follow the instructions received from the Land-office. 374. The mining claim shown in Fig. 179 has the dimensions allowed by the U. S. Mineral Laws, viz., 1500 feet in the direc- tion, or on the ^^ strike," of the vein, by 300 feet on each side of the middle of the vein at the surface. An ideal case is assumed, the exact strike is supposed to have been determined, and the side lines of the claim run parallel to it. Such cases are the exception. Generally, the strike is not accurately determined by the prospector, and claims are, in both direction and dimen- sions, usually but a rough approximation. The courts are lenient and allow considerable latitude in the matter. The prospector may alter the boundaries of, or *' swing," his claim after the strike has been determined, provided that no trespass is committed ; but when the course cannot be altered without interference with other claims, the boundaries already established must be adhered to. When a claim is made and staked out, a record of the same must be placed on file in the office of the Eecorder having charge of the records of mining locations in the district where the claim is situate ; and in case the boundaries of such claim are altered, a record of the amended location must be filed in like manner. *^A failure to make and record the location in accordance with the law and regulation in force at the date of the location, will defeat the claim ; and if it is not made with such definite- ness as to operate as notice to all persons seeking to acquire rights to mining lands, it will be void for uncertainty " (General Land Office Instructions). 375. When mineral discoveries are made and located on sur- veyed land, the surveys must conform to the public survey and 342 ELEMENTS OF SURVEYIl^G. [BOOK X. be connected with or ^^tied" to it, so that there will be no diffi- culty in finding some established corner. In Fig. 179, the points 1 and 2 are tied to the N. W. cor. of the quarter-section in T. 9 N. R. 5 W. 376. When discoveries are made on unsurvejed land, U. S. locating monuments are established to which the claims are tied in the same manner. These monuments may be natural objects, as mountain peaks, permanent rocks, or they may be artificially constructed, in which case they should be strong and without liability to disturbance. The method of tying to such a monument is shown in Fig. 179. When a corner is properly tied in the manner mentioned, the courses are staked out, 1500 ft. along the length of the vein by 300 ft. on each side of the same. The end lines must be parallel to each other in accordance with the Mining Law, but the side lines need Dot be so, though the survey of a lode ^'must be substantially a parallelogram." 377. Before commencing the survey for a patent, the original stakes are found, if possible ; that is, if no amended location cer- tificate has been filed. In the latter case, the stakes marking the amended location are the ones to be followed. When no stakes can be found, the surveyor will be guided by the locator or some one familiar with the boundaries of the property, provided the boundaries then pointed out do not differ from the record of the claim on file in the Recorder's office. Even in the latter case the surveyor can, at the wish of the owner, and when there is no interference with other locations, alter the claim (swing it), file an amended location notice in the Recorder's office, and send forward the description and maps of the altered claim for patent. 378. In making a survey, it is frequently found that there are conflicting claims, that is, two or more claims cross or lap SEC. II.] METHOD OF LOCATING CLAIMS. 343 over each other. An example is shown in Fig. 180, which is a plot made from actual claims on Treasure Hill, Black Hills, Fig. 180. Dakota. When claims conflict, priority of location must govern as to the ownership of the surface in dispute, and the U. S. Deputy Surveyor in making his plot, etc., is required, by the land-office instructions, to show and deduct the area in conflict from the subsequent location, — as is shown in the figure. Should this not be satisfactory to the owner, his appeal is to the courts. 344 ELEMENTS OE SURVEYIN'G. [BOOK X. 379. When the survey for patent is completed, the surveyor drives strong stakes, generally about 4 inches to 5 inches square, and 4 to 5 feet long, firmly into the ground, and marks them M. S. (Mineral Survey) No. 1, M. S. No. 2, etc. No less than 4 stakes, one in each corner, can be used to mark a claim, and in some cases more are needed. In Fig. 180, for example, it is readily seen that four stakes would not be sufficient to trace out the boundaries of the Golden Seal, Ocean Wave, Elkhorn, or Fenian claims. 380. When claims are surveyed for patent, they are numbered in the order of their application. Lot No. 1, No. 2, No. 3, etc. Figure 180 shows that but 3 of the claims represented were patented at the time the survey was made, viz.. Placer Lot No. 120 and 55, and Esmeralda Lot No. 226. The Esmeralda being patented, it will be seen that the areas of the other claims in conflict with it have been deducted, so also of the others in the order of their priority. 381. With regard to errors and corrections in field and office work, in a survey for patent, the following instructions are given to Deputies (see Special Instructions to Deputies by the Survey- ors-General of Montana and New Mexico) : " The error in balancing must not exceed two feet in one thousand feet, and this will be allowed only on complicated sur- veys or when the ground is very rough." In the notes of survey returned, must be a table of area cal- culated by double meridian distances; in this table must be shown first the balance of the survey as actually run upon the ground, and immediately thereafter the 'corrected latitudes and departures. The correction must be made, either by distributing the eiTor among the courses, in proportion to the length of each course, by the proportion — The sum of the lengths of aU the courses is to the total error in latitude or departure as the length SEC. II.] METHOD OF LOCATING CLAIMS. 345 of each course is to the correction of its latitude or departure (see Art. 127) ; or by the following method, which is considered preferable : *^ You can correct the balance by computing a closing course and distance. This can be made on the last line of the survey when it is one of the longest lines, but otherwise should be made upon the longest line unless there should be a line that could not be run accurately, and in that case close on it, when you use this method. This corrected line will be the true line to be used in the notes and on plot. In notes you will state immediately fol- lowing (parenthetically) the course and distance of the line as actually run upon the ground. This is the better way, as the actual error made in the survey can be detected by reference to this line only." " The maximum error that will be allowed in a lode claim is 0.03 acres." 382. The instructions of the U. S. General Land Office to U. S. Deputy Mineral Surveyors are embraced in a series of decisions and letters to the U. S. Surveyors-General, relating to particular cases. Based upon these decisions and letters, and upon what in their judgment and experience seemed necessary, the Surveyors-General have generally issued printed instructions to their deputies. A copy of the instructions of the Surveyor-General of Colorado to Deputy Mineral Surveyors, with sample Field Notes and sample Plot furnished therewith, is given in Ap- pendix 0. 346 ELEMENTS OF SURVEYING. [BOOK X. SECTION III. UNDERGROUND TRAVERSING, ETC. 383. Let it be required to sink a shaft from the surface to connect with a tunnel, which is driven into a hillside. In mauy cases the needle cannot be used on account of disturbing influences, such as the proximity of iron and its ores, or magnetic pyrites. The vernier readings of the transit are then employed, and indeed this method is preferable in all cases. Ascertain the course of the tunnel and the position of the point where it is desired to make the connection. Establish a station -stake at any convenient distance from the mouth of the tunnel that will command a sight of its centre point. If the centre line of the tunnel is a straight line, or nearly so, a sight may be taken from the established station to the point of inter- section. In general, however, this cannot be done, the line of sight being cut off by deflections in the course, thus requiring a traverse. 384. Where curves occur, stations should be established by driving nails in the centre of the cap-timber, overhead in the tunnel. If there are no timbers, holes should be drilled in the rock and filled with wooden spuds into which nails are driven. Plumb-lines are then suspended from these nails, lighted by lamps suitably placed, and used for sighting to with the transit. If desired, the selected stations may be marked by iron pegs driven into holes drilled in the floor of the tunnel. In this case, guiding lamps placed on stands similar to tripods, but with sliding pieces carrying the lamps and fixed in position by clamp- screws^ are used for sighting to. By the sliding and clamp SEC. III.] UKDERGROUKD TRAYERSIN^G. 347 arrangement, the height of the lamp may be made equal to that of the line of sight of the transit telescope. As the same methods will apply in either case, the suspended plumb-lines will be assumed as giving, in general, better results. 385. To place the transit accurately in position at a station, take a block of lead with a steel point in the centre projecting out about half an inch ; place it on the floor of the tunnel, with the steel point upward, immediately under a plumb-bob let down from the station overhead ; the plumb-line may then be removed and the transit set in position over the steel point. 386. Figure 181 shows a somewhat extreme case, chosen to illustrate the method of traversing. Take a station at A, out- side and 100 feet (say) from the mouth of the tunnel. Fix a stake at Ay and drive a nail in the top of it for convenience and accuracy of sighting. Calling the outside station No. 1, and the one at the mouth of the tunnel No. 2, take the line between stations 1 and 2 as the meridian of the survey, find the azimuths of the several successive courses, 2 to 3, 3 to 4, &c., with respect to this meridian, precisely as directed in Art. 194. The bearing of the several courses with respect to this assumed meridian may then be found and tabulated as directed in Art. 195. If necessary, measurements are made to determine the cross- section of the tunnel, its height or breadth, at a station, or at any desired point of the course, and the results entered in the column of remarks in the field-book. 387. The distances between the stations are now measured with great care. A chain should not be used in these measure- ments ; a light steel tape is much better. The points corres- ponding to the ends of the tape are marked by chalk lines on the rock, or in some other convenient manner. The distances may 348 ELEMEin^S OF SUBVEYrN'G. [book X. be measured horizontally, or along the slope Of the tunnel, in which latter case they must be reduced to horizontal distances. OVi3ruBnj^ ^ 388. When distances between stations are measured along the slope, the angle of elevation or depression of the course must be determined — and may be determined as follows: SEC. III.] UNDERGROU"N^D TRAVERSING. 34^ Let the nail on staff at A (Fig. 181) be at the same distance above the ground at A that the axis of the vertical limb of the transit is above the floor of the tunnel at station 2 ; when the telescope is pointed to the nail at A, the reading of the vertical limb will be the angle of inclination of the first course, in eleva- tion, if the telescope points downward — in depressio7i, if it points upward. Let the point of a plumb-bob, suspended from the nail overhead at station 3, at the same distance above the floor of the tunnel at 3 that the axis of the vertical limb of the transit is above the floor at 2 ; when the telescope is directed to the point of the plumb-bob, the reading of the vertical limb will be the angle of inclination of the second course, in elevation, if the telescope points upward — in depression, if it points downward. In like manner are obtained the angles of elevation or depression of the several successive courses to the end of the traverse. 389. The method of recording the observations and measure- ments made in a traverse, is shown in the following Field Book. stations. Angles of Inclination. Azimuth with Course 1.2. Distance in feet. Remabks. Elevation. Depression. 1 2 3 4 0° 0' 0° 0' 0° 0' 0° 0' 340° 300° 102 100 150 180 Sta. 1, at staff, outside tunnel. Sta. 2, at iron peg, at mouth of tunnel overhead. 5 280° 200 At Sta. 5, breadth 6 J ft. 6 325° 120 7 275° 162 At sta. 7, height 7J ft. 8 265° 120 At iron peg. 9 350 ELEMENTS OF SURVEYING. [book X. The above is a record of the traverse shown in Fig. 181. The distances measured are supposed to be horizontal distances, and the angles of inclination are, therefore, zero. 390. The method of reducing the traverse is as follows . From the azimuths of the several courses with the given course taken as meridian, find the bearings (see Art. 195) and proceed as shown in the following Office Form. o Slope in Feet. Bearing with Course 1.2. Length of Course. Latitude. Departuke. Eleva- tion. Depres- sion. N. s. E. w. 1 North 102 102 2 North 100 100 3 N20° W 150 140.95 51.30 4 N60° W 180 90 155.88 5 N80° W 200 34.72 196.96 6 N35" W 120 98.30 68.83 7 N 85° W 162 14.12 161.38 8 9 S 85° W 120 10.46 119.54 fl 580.09 10.46 00 753.89 es OQ N52°56'W 944.68 10.46 00 569.63 753.89 Hence the resultant course A G, from 1 to 9, has a northing of 569.63 feet and a westing of 753.89; to find its length and bearing, we have in the triangle ABC, hence AB = 753.89 and BC = 569.63 ; AC = VaB^ ^'M^ = 944.89. SEC. III.] UNDERGROUND TRAVERSING. 351 tmBAC BC AB 569.6 3 753.89 BAG= 37° 4'; .-. bearing oi AG = 90° - 37° 4' = 52° 56'. 391. If the distances between stations are measured on the slope, instead of horizontally, the angle of elevation or depres- sion of each course must be observed, and recorded in the proper column of the field book. The distances on the slope must then be reduced to horizontal distances in finding the resultant course. The method of record and reduction in such a case, and when the magnetic bearing of the course, from which azimuths are measured, is given, will be apparent from the following Example. Eequired the resultant of the traverse of which the notes are contained in the following Field Book. stations. Angles op Inclination. Elevation. 1°30' 2° 00' Depression. 2° 30' 3° 15' 3° 30' 4° 15' Azimuth with Course 1.2. Distance in feet. 0° 0' 307 176° 15' 402 228° 30' 240 297° 367 246° 15' 409 249° 45' 200 Remarks. Sta. 1, at iron peg, centre of shaft. At iron peg. Let the magnetic bearing of the course, 1.2, from which azimuths are measured, be S 23° W. fla 352 ELEMEl^TS OF SURVEYII^a. [book X. Find, from the bearing of the course taken as meridian and the azimuths of the several courses, the bearings of the several courses, as in Art. 196. The traverse is then reduced as in the following Office Form. Slope in "Feet. Reduced length of course. Latitttde. Depabttjre. Course. Eleva- tion. Depres- sion. Bearing. N. s. E. \V 1 13.4 S 23° W 306.7 282.3 119.8 2 22.8 S 191° W 401.4 378.9 132.3 3 14.7 S 71i° W 239.6 76.0 227.2 4 27.2 N40° W 366. 280.4 235.3 5 10.7 S 89^° W 408.9 5.4 408.9 6 7. N87i°W 199.9 9.6 199.7 17.7 78.1 17.7 290. 742.6 290. 1323.2 0.0 Result- ant. 60.4 S71°7'W 1406.9 452.6 1323.2 The length of the course on the slope, multiplied by the sine of the angle of inclination, gives the distance the course rises or falls, in feet (if the measurements are made in feet) ; and the length multiplied by the cosine of the angle of inclination, gives the reduced length of the course, that is, the length that would have been found had the course been measured on a horizontal line. The bearing and reduced length of each course being found, the latitudes and departures of the several courses are found as before. In this example, the resultant course descends 60.4 feet, its southing is 452.6 feet, and its westing 1323.2; its bearing, S. 71° 7' W., and length on the horizontal, 1406.98, are then found as before. 392. The difference of level of two points underground may SEC. III.] UNDERGR0U:N^D TRAVERSIl^a. 353 be found as shown in the example given in Art. 391, or by running a line of levels between the points as directed in Art. 294. Leveling rods for underground use should be painted and graduated to feet, tenths, and hundredths of a foot, and about 3|- to 5 feet in length. When reading the graduations of the rod through a telescope, the flame of two lamps, one from each side, should be brought to bear on its face so as to light it and not obstruct the line of sight of the observer. Method of Plotting the Underground Traverse on the Surface. 393. To plot the traverse on the surface of the earth, the first course, from which azimuths were measured, must first be laid down, in direction and horizontal length, and its two ends marked with suitable pegs. If the first station has been taken outside the tunnel, as in the traverse shown in Fig. 181, the first course is already, or may easily be, pegged out on the surface. If, however, the first course is within the tunnel, as in example given in Art. 391, the first course must be connected with thf surface. There are two principal methods of making the con- nection : 394. FIRST METHOD. A straight-edge, AB^ is mounted on two trestles, and from it are suspended two plumb lines, E and F, as far apart as the breadth of the shaft will permit. To prevent agitation from currents of air, the . bobs are permitted to dip into buckets of water, at Pio, igg. 354 ELEMENTS OF SURVEYING. [book X. the bottom of the shaft ; the transit being at the second station, K, and the telescope turned in the direction of the first station, D, the straight-edge is moved by an assistant until both are seen in line from K ', their plane then passes through the first course; and if the line ^^ be prolonged to M and L, the line ML will be directly over the first course, and consequently its bearing will be that of the first course. By measuring the line E^ the depth of the shaft may be found. Fig. 183. 395. SECOND METHOD. Let the transit, provided with a diagonal eye-piece, be planted over the station D, at the foot of the shaft, and after being leveled, let it be directed on the sta- tion K. Then, without changing the plane of vision, let the transit be directed to the top of the shaft, and let an assistant plant two flag rods, one at A and the other at B, both in the plane of vision, and let the line AB be prolonged to L and M, as before. The line XJ/ will be in the same vertical plane with the first course, DK. Hence, as before, we may determine the bearing of the first course of the traverse. 396. Having, on the surface, the direction of the first course and the position of Station 1 of the traverse. Station 2 of the traverse may be found by laying off from Station 1, in the proper direction, a horizontal distance equal to the length of the first course. The position of Station 2 on the surface will thus be determined. Put the transit over the surface position of Station 2, make the zero of the vernier coincide with the zero of the SEC. III.] U2fDERGR0UND TRAVERSIKQ. 355 horizontal limb and clamp the vernier plate ; direct the telescope to surface Station 1 and clamp the limb ; revolve the telescope on its horizontal axis, unclamp the vernier plate and revolve it around to the right, through the azimuth of the course, from 2 to 3 ; the line of sight of the telescope will then be in the direc- tion of the traverse line from Station 2 to Station 3 ; measure from surface Station 2, in the direction found, a horizontal distance equal to the horizontal distance of the traverse line from Station 2 to Station 3 ; the extremity will be the surface position of Station 3. Clamp the vernier plate and remove the transit to surface Station 3; reverse the telescope on its hori- zontal axis, loosen the lower clamp and sight surface Station 2 ; tighten the lower clamp and revolve the telescope on its axis ; unclamp the vernier plate and turn it through the azimuth of the course from 3 to 4 ; the line of sight of the telescope will then lie in the direction of the traverse line from Station 3 to Station 4 ; determine the surface position of Station 4 by measurement from surface Station 3, as before (see Art. 196). Proceed, in like manner, to mark out on the surface the suc- cessive underground stations and courses, till the required point (9 in Fig. 181) is found. 397. The slope rod is a convenient instrument for use in making surface measurements. A device for such a rod, for which caveat was entered by A. J. Rigby of New York in 1883, consists essentially, as shown in Fig. 184, of a horizontal bar 10 to 12 feet long, made to move up and down a vertical bar 6 to 8 feet long, to which it may be clamped in any position. The horizontal bar. A, carries a scale, graduated to hun- dredths of a foot and reading by vernier to thousandths of a foot, a spirit level E, by which the bar may be made truly horizontal, and an extension bar C, which may be drawn out 4 to 6 feet, 356 ELEMENTS OF SUKYETING. [book X. making the extended bar 14 to 18 feet long ; when the bar is drawn out, the reading is made by a vernier on the extension bar, as shown in the figure, in a manner similar to that in which the ^ SorixantdZ Bar S TerticdlBar. C Esiension^ JBar. X> Clamp Plabe. E SpiritXievel. F Clcanp Screws. r ^^■. 9 22'] c v. Fig. 184, extended leveling rod is read (see Art. 290). The vertical bar, B, also carries a graduated scale, reading by vernier to thou- sandths of a foot. The two clamp-screws, FF, working against a clamp-plate, fasten the horizontal to the vertical bar and keep it firmly in position, both horizontally and vertically, against its bearings on the vertical bar. The manner of using the slope-rod is shown in Fig. 185. Pegs are driven into the ground at convenient stations along the line which is to be measured, the exact point on each being marked by the intersection of cross-lines on the top of the peg ; if the selected station is on a rock, and a peg can not be driven, the cross-lines are marked in the rock to note the station. The front edge of the vertical bar is placed at the point marked on one peg and the end of the horizontal bar, extended if necessary, to the point marked on the other ; the horizontal bar is moved SEC. ni.] UNDERGROUND TRAVERSING. 357 slowly up and down the vertical bar till the bubble of the spirit- level E is in the middle of its run, and then clamped in position. Fig. 185. The reading on the horizontal bar will give the true horizontal distance between the two stations, and the reading on the vertical bar will give their difference of level. 398. A better method than repeating the traverse on the surface will appear from a consideration of the diagram, Eig. 181. It will be seen that the algebraic sums of the latitudes and departures respectively of the several courses, with respect to the first course taken as a meridian, are equal to the latitude and departure of the last or station 9. The latitude of 9 is the perpen- dicular of a right-angled triangle and the departure is the base ; the distance A C, from 1 to 9, is equal to the square root of the sum of the squares of the latitude and departure of 9. The angle BA C may be found by the relation tan A = -^-j (Art. 35). Having found the angle BAC, the bearing of AC, with respect to the course 1.2 taken as meridian, is equal to 90° — BAC. Place the transit over station 1, level it, make the zero of the vernier coincide with the zero of the horizontal limb. 358 ELEME>sTS OF SURVEYII^G. [book X. I SEC. III.] CKDEKGEOUND TRAVEKSING. 359 clamp the vernier plate, sight to station 2 and clamp the limb ; unclamp the vernier plate, turn it through the bearing, as found, of A Cy and clamp it ; the line of sight of the telescope will be in the line A C. Measure the horizontal distance, as found, from A to Cy and the point 9 will be accurately determined. Again, from 1 measure off the departure of 9 at right angles to the assumed meridian, from the end of which, and at right angles to it, measure off the latitude ; the end of this latter line will be the required point 9. The surveyor should be satisfied with nothing less than per- fect coincidence of the points so determined. Fig. 186 shows sectional maps of work performed in this manner. 399. To produce a line, previously marked out on the sur- face between two shafts, in the same relative direction below ground, so as to form a heading or tunnel from one shaft to another, proceed in the following manner : Set up the transit at the bottom and in the centre of the shaft, as near as can be estimated by the eye, as at a, Eig. 187, and, after the instrument has been accurately leveled and /^'^ ^^X^ the zero of the vernier made / to coincide with the zero of / the horizontal limb, sight -^ ^tz up the shaft and make the ^V cross- wires of the telescope \ bisect a mark J, in the line AB at the surface ; revolve the telescope vertically a little until the vertical wire strikes the opposite side of the shaft at c. Measure the angle cde, or its equal Fig. 187. cae -^ , and the radius of the shaft ae. The exact deviation of the centre of the transit from the vertical plane of the line AB at the surface, may then be found by calculation thus: Let the 360 ELEMENTS OF SUEVEYING. [BOOK X. angular deviaJon, or angle cae, be 3° 10', and the radius of the shaft 60 inches, then nnp sin ^ X «e = .27639 x 60 = 16.5, the deviation required in inches. Remove the transit from a along the line ss' , and toward .s', a distance of 16.5 inches ; it will then occupy its true position, or be in the same vertical plane as the line AB at the surface. The transit may now be adjusted and leveled, and the tele- scope pointed up the shaft; when, if the preceding operation has been properly performed, the vertical wire will exactly pass through the marks previously fixed at e and b (H. D. Hoskold). Eevolving the telescope down, its line of sight will give the true direction below ground of the line AB at surface. Method of Plotting the Traverse on Paper. 400. To plot the traverse on paper, we first plot the plan by the usual method of plotting compass-work, using the bear- ings and the reduced lengths of the courses. This gives the general direction of the horizontal projection of the traverse run ; and from the measurements for cross-section, the breadths of the tunnel on each side may be plotted, and thence a complete plan of the mine may be constructed. We next plot the profile of the traverse, using, as in railroad plotting, two scales, one for horizontal distances, and the other and larger one for vertical distances. The relation between the two scales will depend upon the circumstances of the case. Sometimes, both may be equal. The profile represents the undulation of the traverse, without reference to its horizontal deviations. Let us conceive vertical planes to be passed through all the courses. These will intersect each other in vertical lines. Take the one through the first course, as the one on which the profile is to SEC. IV.] PRACTICAL APPLICATIONS. 361 be delineated. Then, beginning with the plane through the last course, conceive the other planes to be revolved, in order, each about its intersection with the preceding one, to coincide with it, and so on till all are brought into coincidence with the fixed one. The lines of the traverse will then be situated in one plane, and a plot of them, in this position, will be the profile required. The distances from the traverse to the floor and roof of the tunnel, at different points, enable us to com- plete the profile. SECTION IV. PRACTICAL APPLICATIONS. 401. To determine the position and depth of a mine shaft, CB, Fig. 188, which is to connect with a tunnel AB, Fig. 188. If the course of the tunnel is not a straight line it must be traversed. Assuming it to be a straight line, the bearing is obtained with the transit. The length of the tunnel is then accurately measured with a steel tape or a rod. The surveyor DOW stakes out the course on the surface, then beginning at the 362 ELEMENTS OF SURVEYING. [book X. mouth of the tunnel the first peg is driven on a level with the door, and the measurement of tlie tunnel, say 1000 feet, is laid off on the course established on the surface. If the slope rod is used, the sum of the vertical measurements or readings will be the vertical distance of (7 above A. Should the tunnel at B be higher or lower than at A, the difference subtracted from or added to the sum of the vertical readings, will be the required depth of the shaft. If the slope-rod is not used, a line of levels must be run to find the difference of level between A and C. 402. To find the distance necessary to connect the main shaft GFsmd the tunnel FF (Fig. 189). Fig. 189. Determine the depth of the shaft BF, by suspending a chain from the surface or by a steel tape with a weight attached. Measure the length of the tunnel FH^ already completed, and, with the slope rod, or otherwise, obtain the horizontal distance DI, and the difference of level between the points I) and G. This difference of level added to the depth of the shaft DF ■= the total depth of the shaft GF. Find the depth of the shaft GJ already completed. Then GF— GJ = JF, and FF—FH = HF, the required distances. In this case the tunnel is assumed to be SEC. IV.] PRACTICAL APPLICATIONS. 363 straight and the position of the shaft determined as in the preceding problem. 403. To find the depth of a mine shaft AB, Fig. 190, the distance from the outcrop at C and the dip of the vein CB being given. Determine the difference of level AD = 100 ft. The horizontal distance CD = 600 ft. Angle DCB = 60°. Then DB = CD tan 60° = 1039.2 ft. And AB = AD + DB = 100 4- 1039.2 = 1139.2. To find the distance from the outcrop to the point of intersection, or the distance CB, we have (Da- vies' Leg., Trig., Art. 37), COS 60 ^ BY CONSTRUCTION. Draw to any suitable scale the distance CD = 600 feet ; from C draw the line CD, making an angle of 60° with the line CD ; from the point D draw a line perpendicular to CD, until it inter- sects CB at B; produce it upward to A, until distance from D to ^ = difference of elevation between C and A. Measure the line ^^ on the scale, which gives the required depth. Measure also CB on the scale, which is the distance from the outcrop to the point of intersection. 404. Given the depth of the shaft DB = 588.5 feet. Fig. 191, and the horizontal distance FD from the outcrop to the top 364 ELEME]S"TS OF SURVETIJs^G. [book X. of shaft = 500 feet, and the angle GFD = 60°, to find the length of a cross-cut UG from*'' the bottom of the shaft to intersect the vein ; also the distance from the outcrop to the point of intersection. Find the horizontal distance from the outcrop F, at which a shaft of 588.5 feet will intersect the vein, or the distance FH. This is readily found; for assuming it to be the base of a right-angled tri- angle, we have (Davies' Leg., Trig., Art. 37), Fig. 191. FII = 588.5 tan 60' 339.78. If a depression or elevation from the outcrop occurs at the point, it must be subtracted from or added to the depth of the shaft ; for example, if there is a depression of 50 ft. it will only be neces- sary to sink 538.5 ft. to the point of intersection. The horizontal distance so found, subtracted from the real distance of the shaft from the outcrop = length of cross-cut = 160.22, and FG = VDE^ + {FD - GEf = 679.5. BY CONSTRUCTION. Draw the line FD = 500 feet to any convenient scale. From F draw the line FG, making the angle DFG = 60°. Draw DF = 588.5 ft., and from F draw FG parallel to FD. The dis- tances can now be taken from the scale. SEC. IV.] PRACTICAL APPLICATION'S. 365 405. Given the angle LJK = 60°. The distance JL=1!00 ft., and the difference of level between J and ff = 100 ft. Required the depth of a shaft from 5" that will intersect a cross-cut of 350 ft. from the point K, Fig. 192. Fig. 192. From the point K draw to JL a line parallel to III ; it "will intersect JL 700—350 = 350 ft. from the point /. This line is the perpendicular of a right-angled triangle, and we have it = 350 tan 60°. Add to this the difference of elevation, which is the required depth. BY CONSTRUCTION. Draw the line JX=700 ft., and from / draw JE, making the angle LJIC = 60°. From K set off the distance KI, parallel to JL, and make it 350 ft. From / raise the perpendicular IL and extend it to JI, niaking the distance LH = 100 ft. of the scale. The distance IH is now taken from the scale. Problems of this kind occur only where there are inclined shafts. 366 ELEMENTS OF SURVEYING. [book X. 406. Given the distance OL = 400 ft., and the angle LON= 60° ; to find the depth of a shaft at the point of intersection with the vein, and to find the length of a cross-cut that will again inter- sect the vein when the shaft is continued 250 ft. below the point of its intersection, Fig. 193. Draw the horizontal line OL = 400 ft. Also ON, making the angle LOW = 60°. Then 400 tan 60° = depth of shaft at the point of intersection. Construct the shaft and extend it to the point Jf, 250 ft. below the point of inter- section, and draw the cross-cut MW. Then by similar triangles, M A" Fig. 193. OL : LP :: MN : PM\ OL X PM MN = LP BY CONSTRUCTION. Draw OL = 400 ft.; also ON, making the angle LO]Sr=60° ; and LM to the point M, 250 ft. below the point of intersection P. From the scale take off LP and MNthe required distances. In the preceding problems the angle of 60° was employed for convenience of construction, but the principles would l)e true for any other angle. 407. Figures 194, 195, 196, represent plan, longitudinal and transverse sections of a developed mine. The development consists of a vertical shaft, an inclined shaft following the dip of the vein, and 6 one-hundred-foot levels. SEC. IV.] PRACTICAL APPLICATIONS. 367 368 ELEMENTS OF SURYETING. [book X. In opening mines it is considered good practice to follow the vein for a considerable distance in depth, to be fully satisfied of its continuity, before sinking a vertical shaft for deep working. Should the vein be vertical, however, the prospecting shaft may be made the working shaft. The depth to which a shaft is sunk on the dip of the vein will depend upon the Engineer in Fig 19c. charge and upon the characteristics of the vein. A careful engineer will not incur the expense of sinking a deep working shaft to intersect an inclined vein, until he has followed the vein for such a distance that the possibility of its terminating, or " pinch- ing out," is quite remote. Should the vein be irregular and not possess well-defined fissure characteristics, the greater the neces- sity for care in this respect and, in cases of great uncertainty, it would even be advisable to have the working shaft follow the dip SEC. rv.] PEACTICAL APPLICATIONS. Fig. 196. 370 ELEMENTS OF SUEVEYIJ^G. [BOOK X. of the vein. If, on the other hand, the vein is found to be strong and regular, possessing well-defined fissure characteristics to a depth of 150 to 250 feet, the working vertical shaft might be sunk with comparative safety. No rule can be laid down, how- ever, and the Engineer must always exercise his own judgment. In the example here given, we have assumed the inclined shaft to be sunk to the same depth as the vertical, for if originally sunk to the first cross-cut, 440 feet (Fig. 196), it would be car- ried down to the point of intersection and used for ventilation. The method of surveying and plotting such a mine is simply an application of the principles already explained. 408. The calculation of ore-reserves does not come strictly within the province of the surveyor, yet after completing the survey and plot, he is frequently required to make the calculation. We will therefore consider methods of making it. In practice the methods employed are various. No general rule can be given, as each expert has a system of his own, and dif- ferent engineers will not agree, within wide limits, as to the quantity of ore-reserves in the same mine. One may assume as the measure of the ore in sight a rectangular block limited by the outcrop, the depth of the shaft or shafts, and the extreme points of the levels, diminished by the amount extracted. Others, but one-half or one-third of this quantity. The former would be considered an excessive estimate in all cases. The lat- ter too low when the vein possesses great strength and regularity, though even this estimate may be too high, when the conditions are the reverse. The surveyor must exercise his own judgment, exercising caution, however, as the calculation is an important one. Assume the development shown in Figs. 194, 195, 196, and let it be required to calculate the ore-reserves when the bounding lines are assumed at the extreme ends of the level drifts, or the SEC. IV.] PRACTICAL APPLICATION'S. 371 rectangular block ABCD, Fig. 195, and when the average cross- section of the vein is 6 feet, and a cubic foot of the vein matter in place weighs 150 lbs. Ore stopes, or steps made in mine- workings by and for the extraction of ore, are generally very irregular, the repre- sentation here being an ideal one. Suppose the stope-faces to be 11 feet apart and 8 feet high, and that the inclined shaft has extracted 10 x 6 ft. of vein matter, and the levels 7 x 6 f t. We see that the inclined shaft has exposed the vein for 440 -f 115 + 115 = 670 ft.; deducting say 15 ft. for inequality of surface, we should have a rectangular block 655 x (400 + 350) X 6 in width = 2947500 cubic feet of ore: to be deducted from this, we have The inclined shaft 655 x 10 x 6 = 39300 cubic feet. 1st level (150 + 200) x 7 x 6 2d '' 260 X 7 X 6 3d '' 520 X 7 X 6 4th '' 240 X 7 X 6 5th '' 345 X 7 X 6 6th " 500 X 7 X 6 Stoped out on 1st level east, roughly estimated, 3400 iC li west a 2d " ti i( 3d " east It 6th " west <( = 14700 = 10920 = 21840 = 10080 = 14490 = 21000 nated, 3400 '• 6500 7000 « 20000 12000 Total, 182000 cu. ft. (say). Deducting this from 2947500, we have, 2765500 " '' Dividing by 13^, the number of cu. ft. required for a ton, and we have 204852 tons of ore in sight. Another method of calculation is as follows: The longest 372 ELEMEI^TS OP SURVEYING. [BOOK X. drift east is 400 ft. and the shortest 100 ft. Assume the bounding line in this direction to be at a distance east of the shaft. The longest drift west is 350 ft. and the shortest 100 ft.; take the bounding line in this direction at a distance 100 + ^^» - ^»» ^ m ft., or the rectangular block alcF, Fig. 195. Calculate the ore- reserves, when the other data are the same as before. This latter method is recommended by competent engineers as the fairest and most reliable for all parties concerned. 409. Fig. 197 shows a longitudinal section, and Fig. 198 a transverse section of a deposit mine with mill connections, the mill to be erected at the point A. From a consideration of the diagram, it is evident that the most convenient method for the transportation of the ore from the mine to the mill would be by a tunnel driven into the mountain, at the end of which is a bin, made in the solid rock and inclined to the tunnel at any convenient angle at which ore will slide into cars ; the cars to be run into the tunnel on a track and directly under iron doors which are worked by rack and pinion. The bin is to connect with the ore-chamber by a chute inclined at an angle of 45°, as shown in the diagram. The lower or mill tunnel should have a slope of 2 inches in 10 leet, so that the loaded cars would descend by the force of gravity, the last car in a train having a brake with which to regulate the speed. The chute should be 12 to 15 feet from the edge of the tunnel, to admit of constructing the inclined bin for the discharge of the ore into the cars. The point in the ore- chamber, at which it is desired to sink the chute, and the mouth of the lower tunnel being selected, drive a peg to the centre point SEC. IV.] PRACTICAL APPLICATIONS. 373 374 ELEME2>'TS OF SUEYEYIi^G. [BOOK X. of the proposed tunnel floor and drive a nail in the peg, and rei^eat the operation at the point where the chute is to be sunk. Xow make a careful traverse between these points ; the direction of a line which will run from the mouth of the tunnel directly under the point selected for the chute can now be found, as explained in the section on traversing, and the course that will carry the tunnel 12 or 15 feet from the bottom of the chute may be determined. In driving the tunnel, holes should be drilled in the roof and wooden spuds driven in on which to hang plumb-bobs, the surveyor using great care to have the plumb-bobs suspended in the proper course as a guide to the miners. In starting the chute, a large ^rooden triangle should be made, one of the angles of which is the same as the angle of the chute, to be used by the miners as a guide, until sufficient depth is attained to hang, in the proper line, plumb-bobs, the points of which are on the required angle. APPENDIX A. THE SOLAR COMPASS. ( With some omissions, from Messrs. W. and L. E. Ourley's Manual of Engi- neering and Surveying Instruments, ^I^th Edition, 18S3.) This instrument, so ingeniously contrived for readily determining a true meridian or north and south line, was invented by William A, Burt, of Mich- igan, and patented by him in 1836. It has since come into general use in the surveys of U. S. public lands, the principal lines of which are required to be run with reference to the true meridian. The arrangement of its sockets and plates is similar to that of the Survey- ors' Transit, except that the sight vanes are attached to the under plate or limb, and this revolves around the upper or vernier plate on which tlie solar apparatus is placed. The limb is divided to half degrees, is figured in two rows, and reads by the two opposite verniers to single minutes. The Solar Apparatus.— The Solar Apparatus is seen. Fig. 1, in the place of the needle, and in fact operates as its substitute in the field. It consists mainly of three arcs of circles, by which can be set off the lati- tude of a place, the declination of the sun, and the hour of the day. These arcs, designated in the cut by the letters a, b, and c, are therefore termed the latitude, the declination, and the hour arcs respectively. The Latitude Arc, a, has its centre of motion in two pivots, one of which is seen at d, the other is concealed in the cut. It is moved either up or down within a hollow arc, seen in the cut, by a tangent screw at/, and is securely fastened in any position by a clamp-screw The Latitude arc is graduated to quarter degrees, and reads by its vernier, ^5, to single minutes ; it has a range of about thirty-five degrees, so as to be adjustable to the latitude of any place in the United States. The Declination Arc, &, is also graduated to quarter degrees, and has a range of about twenty-eight degrees. Its vernier, v, reading to single minutes, is fixed to a movable arm, h, hav- ing its centre of motion at the end of the declination arc at g ; the arm is moved over the surface of the declination arc, and its vernier set to any read- ing by turning the head of the tangent screw k. It is also securely clamped in any position by a screw, concealed in the engraving. Solar Lenses and Lines. — At each end of the arm, h, is a rectangu- lar block of brass, in which is set a small convex lens, having its focus on the ELEMENTS OF SURVEYI2q"G. [app. a. surface of a little silver plate A, Fig. 2, fastened by screws to the inside of the opposite block. On the surface of the plate are marked two sets of lines intersecting each other at right angles ; of these h b are termed the hour lines, and c c the equatorial lines,, as "/> rrj having reference respectively to the hour of the day and ^ ' ■ the position of the sun in relation to the equator. Fig. 2. In Fig. 1 the equatorial lines are those on the lower block, parallel to the surface of the hour arc c ; the hour lines are of course those at right angles to the first. Equatorial Sights. — On the top of each of the rectangular blocks is seen a little sighting-piece, termed the equatorial sight, fastened to the block by a small milled head-screw, so as to be detached at pleasure. A pp. A.] THE SOLAR COMPASS. 3 They are used, as will be explained hereafter, in adjusting the different parts of the solar apparatus. The Hour Arc, c, is supported by the two pivots of the latitude arc already spoken of, and is also connected with that arc by a curved arm, as shown in the figure. The hour arc has a range of about 120°, is divided to half degrees, and figured in two series ; designating both the hours and the degrees, the middle division being marked 12 and 90 on either side of the graduated lines. The Polar Axis. — Through the centre of the hour arc passes a hollow socket, p, containing the spindle of the declination arc, by means of which this arc can be moved from side to side over the surface of the hour arc, or turned completely round, as may be required. The hour arc is read by the lower edge of the graduated side of the decli- nation arc. The axis of the declination arc, or indeed the whole socket p, is appropri- ately termed the polar axis. The Adjuster. — Besides the parts shown in the cut, there is also an arm used in the adjustment of the instrument as described hereafter, but laid aside in the box when that is effected. The parts just described constitute properly the solar apparatus. Besides these, however, are seen the needle-box, n, with its arc and tan- gent screw, t, and the spirit levels, for bringing the whole instrument to a horizontal position. The Needle-Box n, has an arc of about 36" in extent, divided to half degrees, and figured from the centre or zero mark on either side. The needle is raised or lowered by a lever shown in the cut. The needle-box is attached by a projecting arm to a tangent- screw, t, by which it is moved about its centre, and its needle set to any variation. This variation is also read off by the vernier on the end of the projecting arm, reading to three minutes a graduated arc, attached to the plate of the compass. The Levels seen with the solar apparatus have ground glass vials, and are adjustable at their ends like those of other instruments. The edge of the circular plate on which the solar work is placed, is divided and figured at intervals of ten degrees, and numbered, as shown, from to 90 on each side of the line of sight. These graduations are used in connection with a little brass pin, seen in the centre of the plate, to obtain approximate bearings of lines, which are not important enough to require a close observation. Lines of Refraction —The inside faces of the sights are also gradu- ated and figured, to indicate the amount of refraction to be allowed when the sun is near the horizon. These are not shown in the cut. Principles of the Solar Compass. — The interval between two equatorial lines c c, in Fig. 2, as well as between the hour lines 6 h, is just sufficient to include the circular image of the sun as formed by the solar lens on the opposite end of the revolving arm h, Fig. 1. When, therefore, the instrument is made perfectly horizontal, the equa- 4 ELEMENTS OF SURVEYING. [aPP. A. torial lines and the opposite lenses being accurately adjusted to each other by a previous operation, and the sun's image brought within the equatorial lines, his position in the heavens, with reference to the horizon, will be de- fined with precision. Suppose the observation to be made at the time of one of the equinoxes ; the arm h, set at zero on the declination arc h, and the polar axis p placed exactly parallel to the axis of the earth. Then the motion of the arm h, if revolved on the spindle of the declination arc around the hour circle c, will exactly correspond with the motion of the sun in the heavens, on the given day and at the place of observation ; so that if the sun's image was brought between the lines c c, in the morning, it would continue in the same position, passing neither above nor below the lines, as the arm was made to revolve in imitation of the motion of the sun about the earth. In the morning, as the sun rises from the horizon, the arm h will be in a position nearly at right angles to that shown in the cut, the lens being turned towards the sun, and the silver plate on which his image is thrown directly opposite. As the sun ascends, the arm must be moved around, until, when he has reached the meridian, the graduated side of the declination arc will indicate 12 on the hour circle, and the arm h, the declination arc h, and the latitude arc a, will be in the same plane. As the sun declines from the meridian the arm 7i must be moved in the same direction, until at sunset its position will be the exact reverse of that it occupied in the morning. Allowance for Declination. — Let us now suppose the observation made when the sun has passed the equinoctial point, and when his position is affected by declination. By referring to the Almanac, and setting off on the arc his declination for the given day and hour, we are still able to determine his position with the same certainty as if he remained on the equator. When the sun's declination is south, that is, from the 22d of September to the 20th of March in each year, the arc 6 is turned toward the plates of the compass, as shown in the engraving, and the solar lens, o, with the silver plate opposite, are made use of in the surveys. The remainder of the year, the arc is turned from the plates, and the other lens and plate employed. When the Solar Compass is accurately adjusted, and its plates made per- fectly horizontal, th- latitude of the place, and the declination of the sun for the given day and hour, being also set off on the respective arcs, the image of the sun cannot he brought betioeen the equatorial lines until the polar axis is placed in the iMne of thz meridian of the place, or in a position parallel to Ihi axis of the earth. The slightest deviation from this position will cause the image to pass above or below the lines, and thus discover the error. We thus, from the position of the sun in the solar system, obtain a certain direction absolutely unchangeable, from which to run our lines, and measure the horizontal angles required. This simple principle is not only the basis of the construction of the Solar Compass, but the sole cause of its superiority to the ordinary or magnetic in- strument. For in a needle instrument the accuracy of the horizontal angles APP. A.] THE SOLAR COMPASS. 5 indicated, and tlieref ore of all tlie observations made, depends upon •' the deli- cacy of the needle, and the constancy with which it assumes a certain direc- tion, termed the magnetic meridian." The principal causes of error in the needle, briefly stated, are the dulling of the pivot, the loss of polarity in the needle, the influence of local attraction, and the effect of the sun's rays, producing the diurnal variation. From all these imperfections the solar instrument is free. The sights and the graduated limb being adjusted to the solar apparatus, and the latitude of the place and the declination of the sun also set off" upon the respective arcs, we are able not only to run the true meridian, or a due east and west course, but also to set off the horizontal angles with minuteness and accuracy from a direction which never changes, and is unaffected by at- traction of any kind. To Adjust the Solar Compass.— The adjustments of this in- strument, with which the surveyor will have to do, are few in number, and will now be given in order. 1st. To Adjust the Levels. — Proceed precisely as directed in the account of the other instruments described, by bringing the bubbles into the centre of the tabes by the leveling screws of the tripod, and then reversing the instrument upon its spindle, and raising or lowering the ends of the tubes until the bubbles will remain in the centre during a complete revolution of the instrument. 2d. To Adjust the Equatorial Lines and Solar Lenses — First detach the arm 7i from the declination arc by withdrawing the screws shown in the cut from the ends of the posts of the tangent-screw k, and also the clamp-screw, and the conical pivot with its small screws by which the arm and declination arc are connected. The arm h being thus removed, attach the adjuster in its place by replac- ing the conical pivot and screws, and insert the clamp-screw so as to clamp the adjuster at any point on the declination arc. No\y level the instrument, place the arm 7i on the adjuster, with the same side resting against the surface of the declination arc as before it was detached. Turn the instrument on its spindle so as to bring the solar lens to be adjusted in the direction of the sun, and raise or lower the adjuster on the declination arc, until it can be clamped in such a position as to bring the sun's image as near as may be between the equatorial lines on the opposite silver plate, and bring the image precisely into position by the tangent of the latitude arc or the leveling-screws of the tripod. Then carefully turn the arm half way over, until it rests upon the adjuster by the opposite faces of the rectangular blocks, and again observe the position of the sun's image. If it remains between the lines as before, the lens and plate are in adjust- ment ; if not, loosen the three screws which confine the plate to the block, and move the plate under their heads, until one-half the error in the position of the sun's image is removed. Again bring the image between the lines, and repeat the operation until it will remain in th.e same situation, in both positions of the arm, when the ad- justment will be completed. To adjust the other lens and plate, reverse the arm end for end on the ad EBi 6 ELEMENTS OF SURVEYING. [APP. A. juster, and proceed precisely as in the former case, until the same result is attained. In tightening the screws over the silver plate, care must be taken not to move the plate. This adjustment now being complete, the adjuster should be removed, and the arm h, with its attachments, replaced as before. 3d. To Adjust the Vernier of the Declination Arc— Hav- ing leveled the instrument, and turned its lens in the direction of the sun, clamp to the spindle, and set the vernier 'o, of the declination arc, at zero, by means of the tangent-screw at k, and clamp to the arc. See that the spindle moves easily and yet truly in the socket, or polar axis, and raise or lower the latitude arc by turning the tangent-screw/, until the sun's image is brought between the equatorial lines on one of the plates. Clamp the latitude arc by the screw, and bring the image precisely into posi- tion by the level ing-screws of the tripod or socket, and without disturbing the instrument, carefully revolve the arm 7i, until the opposite lens and plate are brought in the direction of the sun, and note if the sun's image comes be- tween the lines as before. If it does, there is no index error of the declination arc ; if not, with the tangent-screw h, move the arm until the sun's image passes over half the error ; again bring the image between the lines, and repeat the operation as before, until the image will occupy the same position on both the plates. We shall now find, however, that the zero marks on the arc and the ver- nier do not correspond, and to remedy this error, the little flat-head screws above the vernier must be loosened until it can be moved so as to urake the zeros coincide, when the operation will be completed. 4th. To Adjust the Solar Apparatus to the Compass Sights. — First, level the instrument, and with the clamp and tangent- screws set the main plate at 90° by the verniers and horizontal limb. Then remove the clamp-screw, and raise the latitude arc until the polar axis is by estimation very nearly horizontal, and if necessary, tighten the screws on the pivots of the arc, so as to retain it in this position. Fix the vernier of the declination arc at zero, and direct the equatorial sights to some distant and well-marked object, and observe the same through the compass sights. If the same object is seen through both, and the ver- niers read to 90° on the limb, the adjustment is complete ; if not, the correc- tion must be made by moving the sights or changing the position of the. verniers. It should be remarked that as the solar work is attached permanently to the sockets, and this adiustment is made by the maker, it will need no fur- ther attention at the hands of the surveyor except in case of serious accidents. The other adjustments are of course also made in the. process of finishing the instrument, and are liable to very little derangement in the ordinary use of the Solar Compass. To Use the Solar Compass.— Before this instrument can be used at any given place, it is necessary to set off upon its arcs both the declination of the sun as affected by its refraction for the given day and hour, and the latitude of the place where the observation is made. APP. A.] THE SOLAR COMPASS. 7 To Set Off the Declination. — The declination of the sun, given in the ephemeris of the Nautical Ahnanac from year to year, is calculated for apparent noon at Greenwich, England. To determine it for any other hour at a place in the U. S., reference must be had, not only to the difference of time arising from the longitude, but also to the change of declination from day to day. The longitude of the place, and therefore its difference in time, if not given directly in the tables of the Almanac, can be ascertained very nearly by reference to that of other places given, which are situated on, or very nearly on, the same meridian. It is the practice of surveyors in the States east of the Mississippi to allow a difference of six hours for the difference in longitude, calling the declination given in the Almanac for 13 M., that of 6 A. M., at the place of observation. Beyond the meridian of Santa Fe, the allowance would be about seven hours, and in California, Oregon, and Washington Territory about eigJit hours. Having thus the difference of time, we very readily obtain the declination for a certain hour in the morning, which would be earlier or later as the. lon- gitude was greater or less, and the same as that of apparent noon at Green- wich on the given day. Thus, suppose the observation made at a place, say, five hours later than Greenwich, then the declination given in the Almanac for the given day at noon, affected by the refraction, would be the declination at the place of observation for 7 o'clock A. M. ; this gives us the starting- point. To obtain the declination for the other hours of the day, take from tlie Almanac the declination for apparent noon of the given day, and, as the dec- lination is increasing or decreasing, add to or subtract from the declination of the first hour the difference for one hour as given in the ephemeris, which will give, when affected by the refraction, the declination for the succeeding hour ; and proceed thus in making a table of the declination for every hour of the day. To Set Off the Latitllde. — Find the declination of the sun for the given day at noon, at the place of observation as just described, and with the tangent-screw set it off upon the declination arc, and clamp the arm firmly to the arc. Observe in the Almanac the equation of time for the given day, in order to know about the time the sun will reach the meridian. Then, about fifteen or twenty minutes before this time, set up the instru- ment, level it carefully, fix the divided surface of the declination arc at 12 on the hour circle, and turn the instrument upon its spindle until the solar lens is brought into the direction of the sun. Loosen the clamp-screw of the latitude arc, and with the tangent-screw raise or lower this arc until the image of the sun is brought precisely between the equatorial lines, and turn the instrument from time to time so as to keep the image also between the hour lines on the plate. As the sun ascends, its image will move below the lines, and the arc must be moved to follow it. Continue thus, keeping it between the two sets of lines until its image begins to pass above the equatorial lines, which is also the moment of its passing the meridian. Now read off the vernier of the arc, and we have the latitude of the place. tel 8 ELEMENTS OP SUEVEYIKG. [aPP. A. which is always to be set off on the arc when the compass i& used at the given place. It is the practice of surveyors using the Solar Compass to set off, in the manner just described, the latitude of the point where the survey begins, and to repeat the observation and correction of the latitude arc every day when the weather is favorable, there being also nearly an hour at mid-day when the sun is so near the meridian as not to give the direction of lines with the certainty required. To Run Lines with the Solar Compass.— Having set off in the muuuer just given the latitude and declination upon their respective arcs, the instrument being also in adjustment, the surveyor is ready to run lines by the sun. To do this, the instrument is set over the station and carefully leveled, the plates clamped at zero on the horizontal limb, and the sights directed north and south, the direction being given, when unknown, approximately by the needle. The solar lens is then turned to the sun, and with one hand on the in- strument, and the other on the revolving arm, both are moved from side to side, until the sun's image is made to appear on the silver plate ; when by carefully continuing the operation, it may be brought precisely between the equatorial lines. Allowance being made for refraction, the line of sights will indicate the true meridian ; the observation may now be made, and the flag-man put in position. When a due east and west line is to be run, the verniers of the horizontal limb are set at 90°, and the sun's image kept between the lines as before. The Solar Compass being so constructed that when the sun's image is in position the limb must be clamped at in order to run a true meridian line, it will be evident that the bearing of any line from the meridian may be read by the verniers of the limb, precisely as in the ordinary magnetic compass the bearing of lines are read from the ends of the needle. Use of the Needle. — in running lines, the magnetic needle is always kept with the sun ; that is, the point of the needle is made to indicate on the arc of the compass-box, by turning the tangent-screw connected with its arm on the opposite side of the plate. By this means the lines can be run by the needle alone in case of the temporary disappearance of the sun ; but, of course, in such cases the surveyor must be sure that no local attraction is exerted. The variation of the needle, which is noted at every station, is read off in degrees and minutes on the arc, by the edge of which the vernier of the needle-box moves. Allowance for the Earth's Curvature.— When long lines are run by the Solar Compass, either by the true meridian, or due east and west, allowance must be made for the curvature of the earth. Thus, in running north or south, the latitude changes about one minute for every distance of 92 chains 30 links, and the side of a township requires a change on the latitude arc of 5' 12", the township, of course, being six miles square. APP. A.] THE SOLAR COMPASS. 9 This allowance is of constant una where tho surveyor fails to got an obser- vation on the sun at noon, and is a very close approximation to the truth. In running due east and vveHt, as in tracing the standard parallels of lati- tude, the sights are set at 90 on the limb, and the line is run at right angles to the meridian. If no allowance were raxule for the earth's curvature, these lines would, if 8ulHcif;ntly produrx-d, reach tlie e'piator, to which thoy are constantly tending. Of Ofmrfid, in running short lines either east or wefrt, the variation from the parallel would be so small as to be of no jjractical imwrtance ; but when long siglits are taken, the c^^rroction should be made by taking fore and back sights at every station, noticing the error on the ba^,*k sight, and sftting off onohalf of it on the fore sight on the side towards the pole. Time of Day by the Sun.— The time of day is best ascertained by the Solar Compass when the sun is on the meridian, as at the time of making the observation for latitude. The time thus given is that of apparent nrx)n, and can be reduced to mean time by merely applying the equation of time as directed in the Almanac, and adding or subtracting as the sun is slow or fast. The time, of course, can also be taken before or after noon, by bringing the sun's image between the hour lines, and noticing the position of the divided edge of the revolving arm, with referencd by the equation of time as just decribed. Caution as to the False Image.— in using the compass upon the iun, if the revolving arm be turned a little one side of its proper position, a false or reflected image of the sun will apyjear on the silver j)late in nearly the same place as that fxxupied by the true one. It is caused by the reflec- tion of the true image frrmi the surface of the arm, and is a fruitful source of error to the inexperienf;ed surveyor. It can, however, be readily distin- guished from the real image by being much less bright, and not so clearly defined. Approximate Bearings. — When the bearin/Brs of lines, such as the course of a stream, or the Fxjundaries of a forest, are not desired with the cer. tainty given by the verniers and horizontal limb, a rough approximation of the angle they make with the true meridian is obtained by the divisions on the outside of the circular plate, In this ojxjration, a pencil, or thin straight edge of any sort, is held per- pendicularly against the circular edge of the plate, and moved around until it is in range with the eye, the brass centre-pin, and the object observed. The bearing of the line is then read off' at the point where the pencil is placed. Time for Using the Solar Compass.— The Solar Compass, like the ordinary instrument, can be usf^d ut all seasons of the year, the most favor- able time l>eing, of course, in the summer, when the declination is nortli, and the days are long, and more generally fair. It is best not to take the sun at morning and evening, when it is within lO :e:lements of surveyin'G. [app. a. half an hour of the horizon, nor at noon, for about the same interval, before and after it passes the meridian. Allowance for Refraction. — The proper allowance to be made for refraction in setting oflPthe declination of the sun upon the Solar Compass has long been a source of perplexity to the surveyor ; we have, accordingly, given the subject a good deal of attention, and here publish a table, by which the amount of refraction for any hour of any day of the year can be ascertained, and set off with a degree of accuracy which is all that can be desired. A TABLE OF MEAN REFRACTIONS IN DECLINATION. To apply on the declination arc of Solar Attachment of either Compasses or Transits. Computed by Edward W. Arms, C. E., for W. & L. E. Gurlet, Troy, N. Y. 1^ o DECLINATIONS. For Latitude 30°. + 20° + 15° + 10° + 5° 0° 5° —10° —15° —20° Oh. 10" 15'' 21" 27" 33'' 40" 48" 57" 1'08" 2 14 19 25 31 38 46 54 T05 118 3 20 26 32 39 47 55 1'06 119 136 4 32 39 46 52 1^06 1'19 135 157 2 29 5 I'OO I'lO r24 r52 2 07 244 3 46 5 43 13 06 For Latitude 33° 30' . Oh. 13" 18" 24" 30" 36" 44" 52" 1'02" 1'14" 2 17 22 28 35 42 50 I'OO 111 126 3 23 29 35 43 51 I'Ol 113 128 147 4 35 43 51 roi 1'13 127 146 2 13 2 54 5 1'03 1'15 1'31 153 2 20 3 05 4 25 7 36 For Latitude 35°. Oh. 15" 21" 27" 33" 40" 48" 57" 1'08" 1'21" 2 20 25 32 38 46 55 105 1 18 135 3 26 33 39 47 56 1'07 121 1 38 2 00 4 39 47 56 1'07 1'20 136 159 2 32 3 25 5 1'07 1'20 1'38 2 00 2 34 1 3 29 5 14 10 16 1 For Latitude 37° 30'. Oh. 18" 24" 30" 36" 44" 52" 1'02" i 1'14" 1'29" o 22 28 35 42 50 I'OO 112 126 145 3 29 36 43 52 1'02 114 129 149 2 16 4 43 51 I'Ol 1'13 127 149 2 14 2 54 4 05 5 I'll 1'26 154 2 10 2 49 3 55 6 15 j 14 58 APP. A.] THE SOLAE COMPASS. 11 DECLINATIONS , < Fob Latitude 40°. o + 20° + 15° + 10° + 5° 0° 48" -5° — 10° 1'08" -15° -20 Oh, 21" 27" 33" 40'' 57" 1'21" 1'33" 2 25 32 39 46 52 1'06 119 135 157 8 33 40 48 57 ro8 121 138 2 02 2 36 4 47 55 1'06 119 136 158 2 30 3 21 4 59 5 1'15 1'31 151 3 20 3 05 4 25 7 34 25 18 For Latitude 42° 30'. Oh. 24" 30" 36" 44" 52" 1'02" 1'14" 1'29" 1'49" 2 28 35 39 50 I'OO 1 12 126 1 45 2 11 3 36 43 52 102 113 129 149 2 17 2 59 4 50 I'OO I'll 126 144 2 10 2 49 3 55 6 16 5 ri6 136 158 2 30 3 22 5 00 9 24 For Latitude 45°. Oh. 27" 33" 40" 48" 57" 1'08" 1'21" 1'39" 2 32 39 46 52 1'06 1 19 135 157 3 40 47 56 1'07 121 138 2 00 2 34 4 54 1'04 1'16 133 154 2 24 3 11 4 38 5 1'23 141 2 05 2 41 3 40 5 40 12 02 2'02" 2 29 3 29 815 For Latitude 47° 30' Oh 30" 36" 44" 52" 1'02" 1'14" 1'29" 1'49" 2 35 42 50 I'OO 112 1 26 145 2 01 3 43 51 I'Ol 113 128 147 2 15 2 56 4 56 1'09 123 140 2 05 2 40 3 39 5 37 5 1'27 1 46 2 12 2 53 4 01 6 30 16 19 2'18" 2 51 4 08 1118 For Latitude 50°, Oh. 2 3 4 5 33" 38 47 1'03 130 40' 46 56 1'44 151 48" 55 1'06 129 2 19 57' 1'06 119 148 3 04 1'08" 118 136 2 16 4 22 1'21" 135 2 29 2 58 7 28 VSd" 157 2 31 4 18 2410 2'02" 2 28 3 23 6 59 2'36" 3 19 5 02 19 47 For Latitude 52 °30'. Oh. 36" 44" 52" 1'02" 1'14" 1'29" 1'49" 218" 305" 2 43 50 59 111 126 142 2 23 2 49 3 55 3 50 I'OO I'll 126 145 2 11 2 51 2 58 6 22 4 1'05 118 135 2 10 2 28 319 4 53 8 42 5 134 156 2 27 3 16 4 47 8 52 12 ELEMENTS OF SURVEYING. [a PP. A. Oh. 2 .3 4 5 DECLINATIONS. For Latitude 55' + 20° + 15° + 10° + 5° 0° — 5° — 10^ — 15° 40' 48" 57" I'OS" 1'21" 1'39" 2' 02" 2'36" 46 55 1'05 1 18 1 34 156 2 80 3 15 55 106 1 19 1 35 1 58 2 80 8 21 4 58 I'lO 1 23 142 2 06 2 43 344 5 49 12 41 137 2 01 2 84 3 28 5 15 10 18 -20^ 3'33' 4 47 9 19 For Latitude 57 30' Oh. 44" 52" 1'02" 1'14" 1'29" 1'49" 2'18" 305' 2 50 59 1 11 1 25 143 2 09 2 47 3 51 3 58 I'lO 1 24 142 2 07 2 43 3 45 5 50 4 I'll 1 25 1 43 2 10 2 50 8 55 6 14 14 49 5 141 2 06 2 42 342 5 46 13 26 4'37" 6 04 12 47 Explanation of the Table of Refractions.— The table is cal- c'.ilated for latitudes between 30 "" and 50' at intervals of 2^°, that being as near a!3 is required. The declination ranges from to 20° both north and south, the + declina- tions beincr north, and — south, and is given for every five degrees, that being sufficiently near for all practical purposes. The hour angle in the first column indicates the distance of the sun from the meridian in hours, the refraction given for hours being that which aflfects the observed declination of the sun when on the meridian, commonly known as meridional refraction ; the refraction for the hours just before and after noon is so nearly that of the meridian, that it may be called and allowed as the same. When the table is used, it must be borne in mind that when the declina- tion is north or -f in the table, the refraction is to be added ; when the decli- nation is south or — , the refraction must be subtracted. It will be noticed that the refraction in south or — declination increases very rapidly as the sun nears the horizon, showing that observations should not be taken with the sun when south of the equator, less than one hour fro]ar axis only being directed above instead of below, as in the solar compass. A little circular disc of an inch and a half in diameter, and having a short round pivot projecting above its uf>per surface, is first securely screwed to the telescope axis. Upon this pivot rests the enlarged base of the polar axis, which is also firmly connected with the disc by four capstan head screws passing from the under side of the disc into the base already named. These screws serve to adjust the polar axis. The hour circle surrounding the base of the polar axis is easily mov- able about it, and can be fastened at any point desired by two flat- head screws above. It is divided to five minutes of time ; is figured from I. to XII., and is read by a email index fixed to the declination circle, and moving with it. A hollow cone, or socket, fitting closely to the polar axis and made to move snugly upon it, or clamped at any point desired by a milled- head screw on top, furnishes, by its two expanded arms below, a firm support for the declination arc, which is securely fastened to it by two large screws, as shown. The declination arc is of about five inches radius, is divided to quarter degrees, and reads by its vernier to single minutes of arc, the divisions of both vernier and limb being in the same plane. The declination arm has the usual lenses and silver plates on the two opposite blocks, made pre- cisely like those of the ordinary solar compass, but its vernier ^is outside the block, and more easily read. The declination arm has also a clamp and tangent movement, as shown in the cut. The arc of the declination limb is turned on its axis and one or the other solar lens used, as the sun is north or south of the equator ; the cut above shows its position when it is north. The latitude is set off by means of a large vertical limb having a radius of two and a half inches ; the arc is divided to twenty minutes, is figured from the centre, each way, up to 80% and is read by its vernier to single minutes. PACV*** Fig. 3, Showing a Transit with Solar Attachment. 14 ELEMEin^S OP SURVEYING. [aPP. A. It has also a clamp-screw inserted near its centre, by which it can be set fast to the telescope axis in any desired position. The vernier of the vertical limb is made movable by the tangent-screw attached, so that its zero and that of the limb are readily made to coincide when, in adjusting the limb to the level of the telescope, the arc is clamped to the axis. The usual tangent movement to the telescope axis serves, of course, to bring the vertical limb to the proper elevation, as hereafter described. A level on the under side of the telescope, with ground vial and scale, is indispensable in the use of the Solar attachment. To Find the Latitude — First level the instrument very carefully using the level of the telescope until the bubble will remain in the centre during a complete revolution of the instrument, the tangent movement of the telescope being used in connection with the leveling screws of the parallel plates, and the axis of the telescope firmly clamped. Next clamp the vertical arc so that its zero and that of its vernier coincide as near as may be, and then bring them into exact line by the tangent-screw of the vernier. Then, having the declination of the sun for 12 o'clock of the given day as affected by the meridional refraction carefully set off upon the declination arc, note also the equation of time and fifteen or twenty minutes before noon, the telescope being directed to the north, and the object end lowered until, by moving the instrument upon its spindle and the declination arc from side to side, the sun's image is brought nearly into position between the equatorial lines. Now bring the declination arc directly in line with the telescope, clamp the axis firmly, and with the tangent screw bring the image precisely between the lines and keep it there with the tangent screw, raising it as long as it runs below the lower equatorial line, or, in other words, as long as the sun continues to rise in the heavens. When the sun reaches the meridian the image will remain stationary for an instant and then begin to rise on the plate. The moment the image ceases to run below is of course apparent noon, when the index of the hour arc should indicate XII., and the latitude be deter- mined by the reading of the vertical arc. It must be remembered, however, that the angle through which the polar axis has moved in the operation just described is measured from the zenith instead of the horizon as in the ordinary solar, so that the angle read on the vertical limb is the complement of the latitude. The latitude itself is readily found by subtracting this angle from 90° ; thus at Troy, the reading of the limb being found as above directed to be 47° 16', the latitude will be 90° -47° 16' =42° 44'. It will be noticed that with this apparatus the latitude of any place can be most easily ascertained without any index error, as in the usual solar compass. To run lines with the Solar Attachment — Having set off the complement of the latitude of the place on the vertical arc, and the decli- nation for the given day and hour as in the solar, the instrument being also carefully leveled by the telescope bubble, set the horizontal limb at zero and APP. A.] THE SOLAR COMPASS. 15 clamp the plates together, loosen the lower clamp so that the transit moves easily upon its lower socket, set the instrument approximately north and south, the object end of the telescope pointing to the north, turn the proper solar lens to the sun, and with one hand on the plates and the other on the revolving arm, move them from side to side until the sun's image is brought between the equatorial lines on the silver plate. The lower clamp of the instrument should now be fastened, and any fur- ther lateral movement be made by the tangent screw of the tripod. The necessary allowance being mado for refraction, the telescope will be in the true meridian, and being undamped, may be used like the sights of the ordi- nary solar compass, but with far greater accuracy and satisfaction in estab- lishing meridian lines. Of course when the upper or vernier plate is un- damped from the limb, any angle read by the verniers is an angle from the meridian, and thus parallels of latitude or any other angles from the true meridian may be established as with the solar compass. The bearing of the needle, when the telescope is on the meridian, will also give the variation of the needle at the point of observation. If the instrument has a movable compass circle, the variation of the needle can be set off to single minutes, the needle kept at zero, or " with the sun," and thus lines be run by the needle alone when the sun is obscured. APPENDIX B. THE SEXTANT.* (By Prof. J. K. Rees, Columbia College.) This instrument is especially useful to tbe scientific exiJorer on account of its portability and siniplicity of manipulation. It requires no fixed su])- port, and furnishes data with the least expenditure of the time of the ob- server. The accuracy of fixed instruments is not to be expected from it, since it is lu^ld in the hand and is of small dimensions. The Principle of the Instrument.— The optical principle upon which the sextant is made is : — If a ray of light suffers two successive letiections in the same plane by two plane mirrors, the angle between the first and last directions of the ray is twice the angle of the mirrors. Let / and H be two plane mirrors perpendicular to the plane of the paper, — which is taken as the plane of reflection. A ray of light from A is reflected, first from the mir- ror 1 in the direction GO, then by the mirror fi" along OT. The angle between the first and last direction of the ray after these two reflections is ATO. Draw GN and OM nor- mal to the mirrors / and H respectively. Then NKO equals the angle of the mirrors. From the law of the reflection of light it is known that the angle AGN oxi = tLTigleNGO; and also, GOMot i' = MOT. Fig. 4. ♦ For a fiill description of this instrument, see Chauvenet'e Spherical and Practical Aetronomy, published by Lippiucott io 1863, APP. B.] THE SEXTANT. 17 ce angle angle A10 = 2i - 2i' ; NKO = i - i' ; ATO = 2NK0. Q. E. D. II Suppose now that the glass His unsilvered on the upper half; then a ray of light coming from B will pass through this unsilvered portion to T, and the angle ATO will measure the angniar distance of ^ from B as seen at T. To apply this principle the mirror /, revolving about a pivot at C, has at- tached to it an arm or bar II), which, as the mirror is turned, moves over a graduated arc BR. The mirror H is fixed in position. There will be one position of the index arm where the two mirrors will be parallel. Then since the angle bet wean the first and last directions of the ray of light which is reflected by both mirrors is zero, or the two directions are parallel, the indi- cated point of the graduated arc is marked zero. The graduations are then continued to the left, calling each degree two degrees, in order to read off at once the required angle. The best form of the common sextant is seen in the accompanying cut, furnished by Messrs. Stackpole & Brother, N. Y. : The frame is of brass » constructed so as to com- bine strength with light- ness ; the graduated arc, inlaid in the brass, is usu- ally of silver. The divi- sions of the arc are ordi- narily 10' each, which are subdivided by the vernier to 10". The handle A by which it is held in the hand is of wood. The mirrors / and H are of plate glass silvered. The upper half of the glass H is left unsilvered in or- der that the direct rays from a distant object may not be interrupted. To give greater distinctness to the images a small telescope T is placed in the line of sight OT. The telescope is supported in a ring B, which can be moved in a direction at right angles to the plane of the sextant. Thus the axis of the telescope can be directed either towards the silvered or the unsilvered part of the mirror. This motion changes the plane of reflection, which, however, remains always parallel to the plane of the sextant ; the use of the motion being merely to regulate the relative brightness of the direct and reflected images. The vernier is read with the aid of a glass G which is attached to the index bar. The central mirror 1, or index glass, is fastened in a brass frame which is firmly attached to the index bar by three screws. This glass is generally set by the maker so as to be perpendicular to the plane of the sextant. There are no adjusting screws usually connected with it. The fixed mirror E, or horizon glass (so called because through it the horizon is observed in taking Fig. 5. 18 ELEMENTS OF SURVEYING. [APP. B. altitudes), is specially provided with screws bv which its position with respect to the sextant plane may be rectified. At P and Q are colored glasses of different shades, which may be used separately or in combination, to defend the eye from the intense light of the SOB. Common Adjustments of Ordinary Sextant. 1. The Index Glass must he perpendicular to the plane of the sextant. 2. The Horizon Glass must also be perpendicular to the plane of the sextant. 3. The central sight line of the telescope must be parallel to the plane of the sextant. 4. The true zero of the arc must befcnmd. 1. AdjUStmeilt of the Index Glass.— Bring the vernier to about the middle of the graduated arc ; then, placing the eye a little above the plane of the sextant and near the index glass, examine the direct and re- flected images of the graduated arc. If the one appears to run into the other the index glass is perpendicular to the plane of the sextant, and the adjust- ment is complete. If the reflected image appears too high or too low the glass leans forward or backward. The glass may then be adjusted to perpendicularity by placing a piece of paper under one edge of the plate by which the glass is held to the index arm, first loosening the screws ; or the glass may be taken out of the frame, and the supports against which the glass leans may be filed so as to bring the glass, when set back, perpendicularly to the plane of the sextant. 2. Adjustment of the Horizon G-lass.— This glass must also be perpendicular to the plane of the sextant. The index glass having been adjusted to perpendicularity, if it is fotmd that in any one position the hori- zon glass is parallel to the index glass, then the hori2on glass is perpendicular to the plane of the sextant. In order to test this parallelism, put in the telescope and direct it to a star or any distant, well-defined terrestrial object. Move the index bar until the direct and reflected images are in the field of view, then clamp the vernier, and by moving the tangent screw cause one image to pass the other ; if they pass exactly one over the other the adjustment is complete ; if they pass one at the side of the other, the horizon glass must be adjusted. There are ad- justing screws attached to the glass whereby it can be inclined to or from the sextant plane, and also turned around an axis perpendicular to the sextant plane. By means of the first set of screws the adjustment for perpendicu- larity can be made, and by means of the second set the position of the zero of the limb can be altered to a small extent. 3. Adjustment of the Telescope.— The sight line of the tele- scope is the line from the centre of the field of viewtlirough the centre of the object glass. This line must be parallel to the plane of the sextant. In order to test for this, choose two distant objects like the sun and moon, 90' to 120' apart ; direct the telescope to one of these objects, holding the plane of the sextant so as to pass through both ; then moving the index bar, bring the second object into the field of view ; clamp the vernier, turn the tangrent APP. B.] THE SEXTANT. 19 screw until the two objects are tangent to each other on the thread of the telescope nearest to the instrument. Then by moving the instrument, cause the objects to come on tlie thread farthest from the instrument. If the tangency is still perfect, the adjustment is complete. If the objects separate upon the thread farthest from the instrument, tiien the object end of the telescope droops towards the plane of the instrument ; if the images overlap, then the telescope inclines upward from the plane of the instrument. The adjustment is made by means of the screws that work into the collar which carries the telescope. 4. Index Correction.— When the two glasses are parallel the zero of the graduated limb should coincide with the zero of the vernier. This adjustment must be very carefully looked after before taking any observa- tions, because it is an adjustment that is liable to change. Rather than make this adjustment accurately every time an observation is made it is prefer- able to determine the place of the true zero of the graduated arc and allow for the correction. This correction is simply the distance between the gradu- ated zero of the instrument and the reading of the vernier when the two mirrors are parallel. This correction is minus when it is on the graduated arc towards the increasing numbers of the graduation, and plus when on the opposite side. The index correction may be determined by observations upon a star, or a distant, well-defined, terrestrial object, or upon the sun. First, by a star : Direct the telescope to a star of the third or fourth mag- nitude ; move the index bar until the reflected image of this star comes into the field of view ; then clamp the vernier ; turn the tangent screw until the direct and reflected images of the star are exactly in coincidence ; take the reading of the vernier; apply the proper sign, and the arc reading is the index correction. Second, by a distant, well-defined terrestrial object, or the reflection of sunlight from — for example — the bulb of a thermometer or a drop of water. This can be observed in the same way as a star, although not giving as accu- rate results. Third, by the sun : Turn on the colored glasses until the light from the sun is diminished sufficiently to suit the eye ; bring the direct and reflected images of the sun into the field of view ; clamp the vernier ; turn the tangent screw until one image of the sun is tangent to the other ; take the reading ; turn the tangent screw until the contact is broken ; bring the images back to tangency ; take the reading again ; in this way make five readings : then turn the tangent screw until the images change places and tangency is made on the other side ; take the same number of readings here. In order to read always from the same end of the vernier, call the zero of the vernier 360°, and read the vernier accordingly. Take tbe mean of the readings in the first and second cases, add them together, and divide by two. Subtract the result from 360°, and the difierence will be the index correction. In order to avoid the effect of refraction it is best to measure the horizontal diameter of the sun. To check the observations compute the diameter of the sun from data given in the Nautical Almanac, for the day of the observation, and compare it with the diameter of the sun as obtained from the obser- vations. 20 ELEMEl^TS OF SUKVEYIN^G. [aPP. B. Note.— Let E be tlie true reading of the vernier when the mirrors are parallel; let 8 be the diameter of the sun ; let r be the reading of the vernier when contact is made on the left of the zero of the instrument, and r' the reading when the contact is made on the right. Then, r = R + 8 r' = B- 8. Hence, R — and 8 = r + r' r — r' June 1st, 1883.— 10 a. m. Observed sun for index correction. ON ARC. OFF ARC. 360° 34' 10" 10" 5 10 359 30 50 45 45 55 50 360 34 07 = r means 359 30 49 = r' 359 30 49 360 34 07 2)720 4 56 2) 1 3 18 360 2 28 Observed diam. sun, 31 39. Index correction. Calculated diam. sun, 31 36.7 -2' 28" Difiference, . . . , 3.3 To Measure the Angular Distance between two objects with the Sextant. Turn the eye-piece until two of the reticule threads of the telescope are parallel to the plane of the instrument ; then direct the telescope to the fainter of the two objects ; move the plane of the instrument until it passes through the two objects ; revolve the index bar until the reflected image of the second object appears in the field of view ; clamp the vernier ; with the tangent screw make the two images coincide. The reading of the vernier with the index correction applied will be the angular distance of the two objects. The index glass must always be on the side towards the second object. The coin- cidence must be made at the middle point of the field of view. Care must be taken to have the relative brightness of the two objects about the same. Altitudes of objects may be obtained in a similar way, holding the plane of the sextant vertical. APPENDIX C. INSTRUCTIONS TO DEPUTY U. S. MINERAL SURVEYORS. Surveyor General's Office, Denver, Colorado, June 1, 1880. Sir : — The following instructions have been prepared for your government in making mineral surveys : 1. You can make no official survey without an order from this office. 2. To procure an order for survey, tlie application should be made in writing, accompanied with a copy of the certificate of location of the claim to be surveyed, duly certified by the proper recorder, and inclosing a deposit of $35 for office work. Be careful to state plainly the name of the person or persons who desire the survey. 3. An order will not be issued unless in the judgment of this office the certificate of location is in accordance with law and regulations. 4. For the accommodation of claimants residing remote from a United States Depository, this office will see that the deposit is made when the proper funds are received. Send Post Office order, draft or check, on Denver, as other drafts cannot be used without discount. 5. Should the original certificate of location be found defective, and the description does not cover the ground claimed, an additional or amended cer- tificate may be filed as provided by section 13 of the laws of Colorado, ap- proved February 13th, 1874, and a certified copy of the amended certificate must be sent to this office, and your survey dated subsequent to the filing of the same in the proper recorder's office. 6. Certificates of location are frequently sent, which any well-informed deputy should know are too indefinite ; examine them carefully before send- ing, and it will save yourselves and this office much trouble, and remember that your survey MUST agree with certificate of location. 7. "If the records of locations made prior to the passage of the mining act of May 10th, 1872, are not sufficiently definite and certain to enable you to make a correct survey therefrom, you should, after reasonable notice in writing, to be served personally, or through the United States mail, on the applicant for survey and adjoining claimants (whose residence or Post Office address you may know or can ascertain by the exercise of reasonable dili- gence), take the testimony of neighboring claimants and other persons who are familiar with the boundaries thereof as originally located and asserted by the locators of the claim, and after having ascertained by such testimony the 22 ELEME^-TS OF SUEVEYING. [aPP. C. boundaries as originally established, you should make a survey in accordance therewith, and transmit full and correct returns of survey, accompanied by the copy of the record of location, the testimony, and a copy of the notice served on the claimant and adjoining proprietors, certifying thereon when, in what manner, and on whom, service was made." 8. The act of Congress of May 10th, 1872, expressly provides that " the location must be distinctly marked on the ground, so that its boundaries can be readily traced," and " that all records of mining claims hereafter made shall contain the name or names of the locators, the date of location, and such a description of the claim or claims, located by reference to some natural object or permanent monument, as will identity the claim." 9. " These provisions of the law must be strictly complied with in each case to entitle a claimant to a survey and patent, and therefore should a claim- ant under a location made subsequent to the passage of the mining act of May 10th, 1873, who has not complied with said requirements in regard to mark- ing the location upon the ground, and recording the same, apply for a survey," I ''will decline to order it." 10. " The only relief for a party under such circumstances will be to make a new location in conformity to law and regulations, as no case will be ap- proved by this office unless these and all other provisions of law are substan- tially complied with." 11. Comers must be established in as permanent a manner as possible, and should consist of rock in place, or tree if they are found at the exact point ; otherwise a STONE, not less than two feet long, set one foot in the ground ; or POSTS, not less than four inches in diameter, planted two feet in the ground and protruding not less than two and not more than four feet above ground ; both stones and posts must be protected with a mound of stone or earth in addition to the planting. 12. Each corner to be marked No. 1, 3, etc., also the number of the sur- vey, as you proceed with the work, giving in field notes, bearings and distances from each corner to rocks or trees, if any such are at convenient distance, marking same wdth number of corner and number of survey, and describe marks. Wooden posts and trees must be marked with a scribe, and rocks with a chisel. Give, from at least two corners of the survey, two or more bearings to well known points, such as mountain peaks, confluence of streams, etc., both in field notes and on plats. If any portion of this section cannot be complied with, so state it in field notes. 13. Note all objects crossed by your line of survey, such as previous sur- veys, lodes, ditches, roads, ravines, etc. , etc. , and show them upon plat. " If in running the exterior boundaries of a claim it is found that two surveys conflict, the plats and field notes should show the extent of the conflict, giv- ing the area embraced in both surveys, and also the courses and distances from the established corners at which the exterior boundaries of the respec- tive surveys intersect each other. In notes give area as follows : Total area, 5. 94 acres. Less area in conflict with surveys Nos. 13 and 17, . . 3.00 acres. Leaving net area, 3.94 acres. On plat give net area only. APP. c] i:n'structions to u. s. mineral surveyors. 23 14. Connect corner No. 1 of your survey with some corner of the public survey if the claim is located within two miles of such public survey. From corner No. 1 beginning you will proceed with the survey of the claim, giving distances in feet and true courses, as you proceed establishing a corner at each angle of the survey. If an official survey bas been made in the same vicinity, run a connecting line to some corner of the same. Surveys must be made to close exactly. Mention particularly all adjoining claimants. 15. In referring to other surveys give in your field notes the name and number of each survey, and name of claimant or owner. When referring to a survey more than once it is unnecessary to repeat more than the number. 16. See Section 2320, Revised Statutes of the United States, in regard to width of lode claims " on each side of the middle of the vein at the surface." When the locator does not determine, by exploration, where the middle of the vein at the surface is, his discovery shaft must be assumed to mark such point." 17. You will give the quarter section, township and range in which the claim is situated, in notes and on plat, showing section lines on your plat in black, and quarter section lines in red. 18. In districts where there are no public surveys within two miles of your survey, and no locating monument previously established within one hundred chains of such survey, you will establish a LOCATING MONU- MENT at some well-known point, a rock in place being preferable ; but if such cannot be found, then erect a large substantial mound of rocks, describing the same fully in your notes, and hereafter chisel upon it a name, using this as a starting point for your surveys, and when the public surveys reach the locality, run a connecting line from the monument to a corner in said surveys, thus connecting all claims surveyed from the monument. 19. Note all improvements upon the claim, such as shafts, drifts^ adits, cuts, buildings, etc. , giving the extent of same. Show improvements on plat and locate them in field notes, by course and distance, in a direct line from some corner of the survey. 20. After describing fully the improvements (stating if the excavations are in dirt or rock) placed on the claim by the claimant or his grantors, say that " the value of the said improvements, together with the labor expended on the said claim by the claimant (or his grantors, as the case may be) is not less than five hundred dollars." If $500 in improvements has not been ex pended at time of survey the work maybe performed and certificate of Sur- veyor-General filed at the land office during the 60 days of advertising. To obtain this certificate the deputy must send his affidavit stating that the sum of $500 has been expended, describing improvements fully. In estimating the value of improvements only actual mining improvements should be con- sidered. 21. " From an examination of the returns of surveys of mining claims, I am satisfied that in many instances the deputy surveyors certify to the value of improvements without ascertaining whether such improvements are made by the claimant or his grantors, or other persons." 22. " No improvements should be included in the estimate unless they have been made by the applicant for survey, or by those from whom he de- rives his title, and so stated in your notes." 23. " The value of improvements made upon other locations, or by other 24 ELEMENTS OF SUEVEYI^S'G. [APP. C. persons, should not be taken into consideration, but excluded by deputies in their'estimate of improvements upon tlie claim." 24. Following description of improvements made by the claimant, locate the improvements made by other persons, in the manner described in Sec. 19. 25. State all facts coming to your knowledge in regard to adjoining conflicting claimants, whether their claims are surveyed or not. 26. Your field notes should be complete in themselves, leaving nothing to be explained by letter ; and as they are bound in book form after approval you shoidd leave margin for binding. Fold notes so as to leave as little blank paper as possible, and stitch them together. 27. If you know your survey does not agree with that of other deputies you should communicate with such deputy before sending your survey to this ofiice, and try to reconcile all discrepancies ; if it cannot be adjusted, re- port to this office, and a joint survey will be ordered upon knowing that this section has been complied with. 28. When a joint survey is ordered, the deputy who discovers the sup- posed error will be directed to call upon the deputy supposed to be in error, and see if the discrepancy cannot be reconciled. If they cannot agree, then they shall make a joint survey within ten days after the date of the order. When said survey is completed they shall make a joint affidavit to the field notes, and forward them to this office. The deputy found to be in error shall pay all expenses, including $10 per day to the deputy whose work is found to be correct. If both deputies are found to be in such error as to require amended field notes of their former survey, then each deputy shall pay one- half the expenses of the joint survey. Any deputy refusing or neglecting to appear for the joint survey, within the ten days named, or who shall refuse or neglect to pay the expenses as above indicated, will be suspended ; and if the refusal or neglect shall extend to twenty days, his commission as deputy will be revoked. 29. In preparing plats make the top North, and color only the ground not in conflict. Mark upon the plat all corners (thus, " Cor. No. 1," etc.), courses to mountain peaks, courses and distances to all connecting lines and upon the boundaries of the survey. Plat survey upon as large a scale as paper will admit of, if practicable not less than 200 feet to the inch. Be careful that your plats and notes agree. 30. An applicant has the right to abandon, from his application for patent, any part or portion of the premises embraced in the survey, but in case he does abandon any portion of the premises embraced by his application and survey, it will be necessary that an amended survey be made upon applica- tion of the claimant, and as such amended survey involves as much office work as the original survey, the usual deposit should accompany the claimant's application for the amended survey. 31. After this date surveys will be approved in the order in which they are received, deputies being required to act promptly in all official matters, that the approval of a survey may not be unnecessarily delayed. In case of unreasonable delay on the part of the deputy to correct errors, the survey will be stricken from the files, and the applicant notified. 32. In order that all returns may be made uniform, blank plats and field note paper wiU be furnished from this office for your use, which blanks you wiU fill up as the case may require. You will be careful that the names of APP. C] IKSTRUCTIONS TO U. S. MII^ERAL SUEVEYORS. 25 claimants and number of survey agree with the order for survey and certifi- cate of location! You will make one copy of the plat and notes and transmit the same to this oflBce, prepaying the postage or express, in full ; otherwise they will not be received. Do not fold plats for transmission, but roll them. As your plat and notes come under the head of •' written matter," they require letter postage. 33. Unless plats and notes are prepared in a neat and workmanlike man ner they will not be accepted. Plats and notes are often found to be incor- rect from negligence, carelessness and ignorance. Many deputies appear to depend upon this oflBce to detect and correct these errors, as it saves them the trouble. Hereafter such surveys will be returned with the simple statement that " they are incorrect." 34 A solar transit must be used in all official surveys being guided by the solar apparatus, and not the needle, unless the courses are deflected from a meridian astronomically established. On account of local attractions a needle instrument will not be accepted as reliable. State in field notes the kind of instrument used, and the manner in which the courses are taken. 35. " You are informed that the employment of Deputy Mineral Survey- ors as attorneys in mineral claims, directly or indirectly, is absolutely prohib- ited," and you will make no survey in which yoa are interested. 36. Be particular to write all proper names plainly. 37. Deputies changing their residence should notify this office of such change. 38. I shall expect you to make yourself thoroughly familiar with all the mining laws, National and State, as well as with these instructions, and I am sure that errors will be less frequent, and this office as well as yourself will be relieved of much annoyance. 39. All official communications must be addressed to the Surveyor Gen- eral, and not to clerks in his office. Very respectfully, Albert Johnso?^, Surveyor General of Colorado. Sample Field Notes furnished Deputies. Survey No. 500, District No. 3. FIELD NOTES Of the survey of the claim of ... .John H. Marshall, upon the Excelsior Lode, in. . . . California Mining District, Lake Coimty, Colorado. Surveyed by ...... . George D. Williams, ; U. S. Deputy Mineral Surveyor. Survey begun October 23d, 1881, and completed > " 38th, .1881, u ELEMEirrS OF SUEVEYnrO. [app. c. Fee(. DEscEIPTIO^' OF Survey. Beginning at Cor. Xo. 1, a spruce post, 5 ft. long, 4 ins. diam., set 2 ft. in the ground, with mound of stone, marked 5 Jo whence the W. 34, cor. in sec. 16, T. 8 S., R. 80 W. of the 6th Prin. Mer. bears, S. 84= 2Q' W. 1847. 4 ft. A spruce, 6 ins. diam., marked B. T. gJo, bears S. 78° 32' W. 32.9 ft. Thence N. 11° 53' E. Vap. 14= 15' E. 300. To Cor. No. 2. A spruce post 4 ft. long. 4 ins. dia., set 2 ft. in ground, with mound of earth marked -Iq, whence, Holr Cross Mt. bears X. 41-37' W. ; Mt. Leon bears X. 14= 29' W. (Xo other bearings available.) Thence S. 78^ 7' E. Var. 14= 15' E. To road running S. Easterly. 224.27 Intersect line 4-1 of Sur. No. 495, Treasury lode, John P. Jones et al. claimants, N. 15= E. 79.91 ft. from Cor. No. 4. 952.72 Intersect line 4-1 of Sur. Xo. 227, Silver Dollar Lode, Henrv J. Smith, claimant, at S. 3= W. 401.02 ft. from Cor. No. 1. 1256.36 Intersect line 2-3 of Sur. No. 227, S. 3' W. 447 9 ft. from Cor. No. 3. 1500. To Cor. No. 3. APP. C] SAMPLE FIELD KOTES. 27 Feet. 149.5 198. 300. 112. 196.75 500.39 911. 1500. A granite stone 25 x 11 x 7 ins., set 1 ft. in ground, with mound of stones, chiseled 5^0, whence a spruce 6 ins. dia., marked B. T-. ^%^, bears S. 80" 23' E. 20.4 ft.; B. R. 5^0 chiseled on rock in place, bears N. 44° E. 30 ft. Thence S. 11° 53' W. Var. 14° 55' E. Intersect line 3-4 of Sur. No 495, at N. 75° W., 218 ft. from Cor. No. 3. To G-ulch ; course N. E. To Cor. No. 4. A quartz stone 24 x 13 x 8 Ins., set 1 ft. in ground, and mound of stone, chiseled ^%^, whence B. R. g^^ chiseled on large boulder, bears N. 88° 22' W. 25.4 ft. Mt. Leon bears N. 19° 22' W. Sherman Peak bears S 84° 45' E. Cor. No 3, Sur. No. 495, bears N. 69° 23' E. 258 ft. Cor. No. 3, Sur. 227, bears S. 18° 9' W. 743.9 ft. Thence N. 78° 7' W. Var. 14° 55' E. To Gulch. Intersect line 2-3 of Sur. No. 227, N. 3° E., 748.45 ft. from Cor. No. 3. Intersect line 4-1 of Sur. No. 227, N. 3° E., 795.34 ft. from Cor. No. 4. To road. To Cor, No. 1, place of beginning. 28 ELEMElfTS OF SURVEYING. [APP. a Akea. Containing 10.32 acres. Less conflicts with .... Sur. No. 227=2.13 acres " " 495=2.46 " 4.58 " acres. Leaving net area 5.74 Location. This survey is located in the N. 3.^ and S. E. U of Sec. 16. in T. 8 S R SOW. Impkovements Upon this claim consist of : A discovery shaft, 5' x 3', 275 ft. deep, timbered, which bears from Cor No. 1, N. 64° 59' E. 250.6 ft. A log shaft-house, 56' x 40', the S. W. corner of which bears from Cor No. 1, N. 65= 26' E. 207 ft. A shaft 4' X 4', 25 ft. deep, which bears from Cor. No. 1, N. 88° 39 E. 765 ft. An open cut 3' x 10', which bears from Cor. No. 3, S. 58° 45' W. 264 ft. JL hereby certify that the value of said improvements, together with the labor expended on the said claim by the claimant and his grantors, is not less than Five Hundred Dollars ($500). Improvements made by other parties are as follows : A shaft 4' x 6', 78 feet deep, which bears from Cor. No. 3, N. 83° 37' W. 389 ft. Adjoining Claims. Surveys Nos. 495 and 227. No others known. Instbuments Used. A Gurley Mountain Transit, with solar attachment (or the instrument used), and steel tape (or the measure used). All courses determined by the use of the solar apparatus. Address of Applicant, JOHN H. MARSHALL, P. O. Box 743, Leadville, Colo. [Tlien follow a list of the assistants employed in making the survey, with their aflBdavit as to its conformity to instructions, the affidavit of the Deputy Mineral Surveyor, and the certificate of the Surveyor General.] A pp. C] SAMPLE PLAT. 29 9/ &B.!iS«XKT8iS.«)ir Fig. 6. Survey No. 500. Mineral District No. 3. PLAT of tlie claim of John H. Marshall upon the Excelsior Lode, California Mining District, Lake, County, State OF Colorado. Containing an area of. . . .5.74. . . .acres. Scale of , . . . 200... feet to the inch. Variation. .. .East. Surveyed by Geo. D. Williams, U. S. Deputy Mineral Surveyor. October 28, 1881. The Original Field Notes of the Survey of the Claim of. upon the from which this plat has been made, have been examined and approved, aud are on file in this office ; and I hereby certify that they furnish such an accurate description of said Mining Claim as will, if incorporated into a patent, serve fully to identify the premises, and that such reference is made therein to natural objects and per- manent monuments as will perpetuate and fix the locus thereof. I further certify that the value of the labor and improvements placed thereon by the applicant, or his grantors, is not less than Five Hundred Dollars, and that said improvements consist of : as appears by the report of the Deputy Surveyor. And I further certify that this is a correct Plat of said Mining Claim, made in conformity with said field notes of the survey thereof. U. S. Surveyor General's Office, Denver, Colorado. 18... XJ. S. Surveyor General for Colorado. A TABLE LOGARITHMS OF NUMBERS. Remark. ^In the following table, in the nine right hand columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's, to catch the eye, and to indicate that from thence the two figures of the Logarithm to be taken from the second column, stand in the next line below. 2 A TABLK OF LOGARITHMS FROM 1 TO 1U,0U0. N. 1 I a 3 4 5 6 7 8 i 9 D. loo oooooo 0434 0868 i3oi 1734 2166 2598 3029 3461 3891 43s loi I 43a I 4751 5i8i 5609 6o38 6466 6894 7321 7748 8174 428 loa : 8600 9026 3259 Q45i 9876 •3oo •724 1147 1570 1993 24i5 424 io3 012837 368o 4100 4521 4940 536o 5779 6197 6616 419 104 7033 745 1 7868 8284 8700 2841 9116 3252 953j 3664 9947 •36i •775 416 io5 021 189 i6o3 20X6 2428 4075 4486 4896 412 106 53o6 5715 6125 6533 6942 735o 77^7 8164 8571 8978 408 IS o384 033424 0789 •195 •600 1004 1408 1812 2216 2619 3021 404 3826 4227 4628 5029 543o 5830 623o 6629 7028 400 109 7426 7825 8223 8620 9017 9414 9811 •207 •602 •998 396 no 1041 393 1787 5182 2576 2969 3362 3755 4148 4540 4932 It III 5323 5714 6io5 6495 •38o 6885 7275 1153 7664 8o53 8442 883o iia 9218 9606 3463 ^i •766 1538 1924 2309 2694 386 ii3 053078 423o 4613 ^, 5378 9185 5760 6142 6524 382 114 ! 6905 7286 7666 8046 8426 9563 3333 9942 3709 •320 379 ii5 060698 4458 1075 1452 1829 2206 2582 2958 4o83 376 116 4832 5206 5580 5953 6326 6699 7071 7443 7815 372 \\l 8186 8557 8928 9208 9668 3352 ••38 •407 4o85 •776 1145 i5i4 369 071882 225o 2617 6276 2985 3718 445i 4816 5x82 366 119 5547 5912 6640 7004 7368 7731 8094 8457 8819 363 1 20 079181 q543 3i44 35o3 •266 •626 •987 i347 1707 2067 2426 36o lai 082785 3861 4219 4576 4934 5291 5647 6004 357 132 636o 6716 7071 7426 7781 8i36 8490 8845 9198 9552 3071 355 123 0905 093422 •258 •611 •963 i3i5 1667 2018 2370 2721 35i 124 3772 4122 4471 4820 5169 55i8 5866 62i5 6562 349 125 6010 100^71 7257 7604 7951 8298 8644 8990 2434 9335 9681 3119 ••26 346 126 0715 1059 i4o3 1747 2091 2777 3462 343 \ll 3804 4146 4487 4828 5169 8565 55io 585i 6191 6531 6871 340 7210 7549 7888 8227 8903 9241 9579 9916 •253 338 129 110590 0926 1263 1599 1934 2270 2605 2940 S275 3609 335 i3o Ii3g43 4277 4611 4944 5278 56ii 5943 6276 6608 6940 333 i3i 7271 7603 7934 8265 8595 8926 9256 9586 3198 •245 33o' l32 1 205-14 0903 123l i56o 1888 2216 2544 2871 3525 328 i33 3832 4178 45o4 483o 5i56 5481 58o6 6i3i 6456 6781 325' 1 34 7io5 7429 7753 8076 8399 8722 9045 9368 9690 ••12 323 i35 i3o334 o65d 0977 1298 1619 If^t 2260 258o 2900 3219 640J 321 1 36 3539 3858 4177 4496 4814 545 1 5769 6086 3i8 »37 i38 6721 7037 7354 ■7671 7987 83o3 8618 8934 9249 9564 3i5 0870 •194 •5o8 •822 ii36 i45o 1763 2076 2389 2702 3i4 139 14301 5 3327 3639 3951 4263 4574 4885 5196 55o7 58i8 3x1 140 146128 6438 6748 7o58 7367 7676 7985 8294 86o3 8911 309 141 9210 9527 9835 •142 •449 •756 io63 1370 1676 1982 307 142 152288 2694 2900 32o5 35io 38i5 4120 4424 4728 5o32 3o5 143 5336 5640 5943 6246 6549 6852 7i54 7457 7759' 8061 3o3 144 8362 8664 8965 9266 9567 9868 •168 •469 •769 1068 3oi 145 i6i368 1667 1967 2266 2564 2863 3i6i 3460 3758 4o55 299 146 4353 465o 4947 5244 5541 5838 6i34 6430 6726 7022 297 295 "^2 7317 7613 7008 8203 8497 1434 8792 9086 9380 9674 9968 2895 170262 o555 0848 1141 1726 2019 23 1 1 26o3 293 149 3i86 3478 3769 4060 435 1 4641 4932 5222 55i2 . 5802 291 i5o I 7609 I 638i 6670 6959 7248 7536 7825 8ii3 8401 8689 289 i5i I8I844 9264 9552 9839 •126 •41 3 !^99 •985 1272 l558 287 285 l52 2129 4975 24i5 2700 5542 2985 3270 35DD 3839 4123 4407 1 53 4691 5259 5825 6108 6391 6674 6956 7239 283 1 54 7521 7803 8084 8366 8647 8928 9209 9490 9771 ••5i 281 i55 190332 0612 0892 368i 1171 i45i 1730 45i4 2010 2289 2567 j 2846 270 i56 3 1 25 3403 3959 4237 4792 5069 5346 5623 278 \ll 5899 6176 6453 6729 7oo5 7281 7556 7832 ^IT 8382 276 8657 8932 9206 9481 9755 ••29 •3o3 •577 33o5 •85o 1124 274 159 201397 1670 1943 3216 2488 2761 3o33 3577 3848 272 N. I 2 ' 4 5 6 7 8 9 D. TABLE OF LOGARITHMS FROM 1 TO 10,000. N. I 3 3 4 5 6 7 8 9 D. i6o 204120 4391 4663 4934 5304 5475 5746 6oi6 6386 6556 271 i6i 6826 7006 9783 7365 7634 7904 Hl^ 8441 ^21° X 9247 269 162 95i5 ••5 1 •3i9 •D86 •853 1121 1888 1931 4679 267 266 i63 112188 3454 3730 5373 3986 3252 35i8 3783 4049 43i4 164 4844 5109 5638 5902 6166 643o 6694 6957 7221 264 i65 7484 7747 8010 8273 8536 8798 9060 9823 9585 9846 ' 262 166 J10108 0370 o63i 0893 ii53 1414 1675 1986 3196 2456 261 ;s 3716 2976 3336 3496 3755 401 5 4374 4588 4792 5o5i , 259 5309 5568 5836 6084 6342 6600 6858 7ii5 7873 7680 258 169 7887 8144 8400 8657 8913 9170 9426 9682 9988 •193 256 170 230449 0704 0960 35o4 I3l5 1470 1724 1979 4517 2284 3488 3743 354 171 3996 5528 325o 3757 6285 401 1 4264 4770 5o33 5376 253 172 5781 6o33 6537 6789 7041 7292 7544 7795 353 173 8046 8297 8548 8799 9049 9299 9550 9800 ••5o •3oo 35o 174 240549 0709 1048 I3Q7 i546 1795 2044 2298 3541 3790 240 175 3o38 3286 3534 37§3 4o3o 6745 4525 4773 5019 5366 348 176 55i3 5759 6006 6353 6499 8q54 1395 6991 7287 7482 7728 346 172 178 7973 8219 8464 8709 9198 i63» 9443 9687 2125 9982 2868 •176 345 25o420 0664 0008 ii5i 1881 2610 243 179 2853 3096 3338 358o 3822 4064 43o6 4548 4790 5o3i 243 180 255273 55i4 5755 5996 83o8 6237 6477 6718 6958 7198 7439 9888 341 181 7679 7018 o3io 8i58 8637 1023 8877 9116 9355 9594 389 182 260071 0548 0787 1263 i5oi 1789 1076 4J46 2214 338 1 83 245i 3688 3935 3i63 3399 3636 3873 4109 4582 337 184 4818 5o54 5390 5525 5761 5996 6232 6467 6702 6987 335 i85 7172 7406 7641 7875 811O 8344 8578 8812 9046 9279 384 186 95i3 9746 9980 •3l3 •446 •679 •912 1144 1877 1609 388 188 271842 3074 4389 33o6 3538 2770 5o8i 3ooi 3233 8464 8696 8927 383 4i58 4620 485o 53ii 5542 5772 6002 6282 33o 189 6462 6692 6921 7i5i 7880 7609 7888 8067 8296 8525 339 190 278754 281033 8982 9211 943o 1715 9667 9895 •128 •85i •578 •806 328 191 1261 1488 1942 2169 2896 4656 2623 3849 8075 327 192 33oi 3537 3753 3979 42o5 443 1 4883 5i07 5832 226 193 5557 5783 6007 6232 6456 6681 6905 7180 7354 7578 225 194 7802 8036 8249 8473 8696 8920 9143 9866 i8i3 9812 223 195 290035 0357 0480 0703 0925 1 147 1869 1591 2084 222 196 2256 3478 4687 2699 2920 3i4i 3363 3584 38o4 4025 4246 221 197 4466 4907 5i27 5347 5567 5787 6007 8.9^ 6226 6446 220 198 6665 6884 7104 7333 7542 7761 7979 8416 8635 210 199 8853 9071 9289 9507 9725 9943 •161 •378 •595 •8i3 218 200 3oiu3o 1347 1464 1681 1898 2114 233 1 3547 3764 2980 317 201 3196 535 1 3413 3628 3844 4069 4275 4491 6689 4706 4931 5i36 3l6 202 5566 5781 5996 62H 6425 6854 7068 7282 3l5 203 9630 7710 9843 7924 8137 835i 8564 8778 8991 9304 9417 3l3 204 ••56 •368 •481 •693 •906 1118 i33o 1 542 3131 2o5 311754 1966 2177 4389 3389 2600 2812 8028 3384 3445 8656 311 1 206 3867 4078 6180 4499 4710 4920 5i3o 5340 555i 5760 3I0{ a 5970 6390 6599 6809 8898 7018 7227 7436 7646 7854 3091 3o8, 8o63 8373 o354 8481 8689 9106 9314 9533 9780 9988 ao9 320146 o563 0769 0977 1 184 1891 1598 i8o5 2012 207 aio 322219 3436 3633 3839 3046 3252 3458 3665 3871 tru 206 211 4282 4488 4694 4899 6930 5io5 53io 55i6 5731 5936 2o5 312 6336 6541 6745 7155 735n 7563 7767 9805 7973 8176 204 ai3 838o 8583 8787 8991 9194 9898 9601 •••8 •211 2o3 214 33o4i4 0617 0819 1033 1225 1427 1680 1883 3084 2236 202 2l5 2438 3640 3843 3o44 3246 3447 8649 385o 4o5i 4253 i 202 216 4454 4655 4856 5o57 5257 5458 5658 5859 6059 6260 30I III 6460 6660 6860 7060 7260 7459 7659 7858 8o58 8257 •246 1 200 8456 8656 8855 9054 9253 945 1 9650 9849 ••47 2028 ti9 340444 0643 0841 1039 1237 1435 i632 i83o 2225 1 N. • 2 3 1 4 5 6 7 8 9 16 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. oil 2 3 4 5 6 7 8 9 D. 220 342423' 2620 J817 4392 4589 , 4785 3oi4 3212 3409 36o6 38o2 3999 4196 196 221 4981 5178 7i35 5374 5570 5766 5962 6i57 222 6353 6549 6744 6939 7330 7525 7720 7015 8x10 195 223 83o5 85oo 8694 8889 9083 9278 9472 9666 9860 ••54 194 224 350248 0442 o636 0829 X023 1216 1410 i6o3 1796 1989 193 225 2i83 2375 1 2568 2761 2954 3x47 3339 3532 3724 3916 193 226 4io8j 43oi 1 4493 4685 4876 5o68 ' 5260 5452 5643 5834 192 227 228 6026 6217 7935; 8125 6408 6599 6790 698X 7x72 7363 7554 7744 191 83i6 85o6 8696 8886 9076 9266 9456 J 9646 190 189 ! 229 9835; ••25 •2l5 •404 •593 •783 •972 ix6x i35o 1 539 23o 361728 1917 2io5 2294 2482 267 X 2859 3o48 3236 3424 188 23 1 36i2; 38oo 3988 4176 4363 455x 4739 4926 5xx3 53oi . 188 232 5488! 5675 5862 6049 6236 6423 66x0 6796 6983 7x69 18^ 233 73561 7542 7729 9687 7915 8101 ' 8287 9o58 '■ •i4i 8473 8659 •5x3 8845 9o3o 234 92161 9401 9772 1622 •328 •698 •883 i85 235 371068 1253 1437 j^o6 1991 383 X 2175 236o 2544 2728 4565 184 236 2912 3096 4932 3280 3464 «5647 401 5 4198 4382 184 ^^1 238 4748 5ii5 5298 : 5481 5664 5846 6029 6212 6394 i83 6577 8398 6759 858o 6942 7124 8943 7306 7488 7670 9487 7852 8o34 8216 182 239 8761 9124 9306 9668 9849 ••3o 181 240 380211 0392 0573 0754 2557 4353 0934 ixx5 1296 1476 1 656 1837 3636 181 241 2017 38i5 3ga5 2377 2737 29x7 3097 3277 3456 180 242 4174 4533 4712 4891 5070 6856 5249 5428 \]t 243 56o6 5785 7568 5964 6142 632 X 6499 6677 8456 7o34 881 X 7212 8989 244 7390 7746 9520 7923 8xox 8279 8634 178 245 9166 9343 9698 9875 ••5 1 •228 •4o5 •582 •759 2521 176 246 390935 1112 1288 1464 1641 18x7 3575 \n^ 2169 2345 247 2697 2873 3o48 3224 3400 375x 55oi 3926 410X 6025 176 248 4452 4627 4802 4977 5x52 5326 5676 585o 175 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 25o 397940 8114 8287 8461 8634 8808 898X 9154 9328 95ox 173 25l 9674 9847 ••20 •192 •365 •538 •71X •883 xo56 X228 173 352 401401 1573 1745 1917 3635 2089 226X 2433 26o5 2777 2949 4663 172 253 3l2I 3292 3464 3807 3978 5688 4149 4320 4492 171 254 4834 5oo5 5176 5346 55x7 5858 6029 6x99 6370 171 255 6540 6710 6881 705 1 8749 7221 7391 756x 7731 7001 95o5 8070 109 256 8240 8410 8579 89x8 9087 9207 9426 9764 257 258 9933 •102 •271 •440 •60Q 2293 •777 •946 1114 1283 X45i 169 41 1620 1788 1956 2124 246X 2629 2796 2964 3i32 160 259 33oo 3467 3635 38o3 3970 4x37 43o5 4472 4639 4806 167 260 414973 5i4o 5307 6973 5474 8798 5641 58o8 5974 7638 6141 63o8 6474 8x35 166 261 6641 6807 7306 7472 7804 7970 262 83oi 8467 8633 8964 9129 9295 9460 9625 9701 1439 i65 263 9956 •121 •286 •451 «*6x6 •781 •945 IIIO 1275 i65 264 431604 1788 1933 2097 3737 226X 2426 2590 2754 4392 2918 4555 3082 164 265 3246 3410 3574 3901 4o65 4228 4718 164 266 4882 5o45 5208 5371 5534 5697 586o 6023 6x86 6340 7973 9591 i63 267 268 65ii 6674 6836 6999 7x61 8783 7324 8944 7486 7648 7811 162 8x35 8297 8459 ••75 8621 9106 9268 9429 162 »69 9752 9914 •236 •398 •559 •720 •881 1042 X203 161 270 43 I 364 i525 i685 1846 2007 2167 2328 2488 2649 2809 161 271 6i63 3i3o 3290 3450 36x0 3770 3930 4090 5685 4249 4409 160 272 4729 4888 5048 5207 6798 8384 5367 5526 5844 6004 i59 273 6322 6481 6640 6957 7xx6 870X 7275 8859 7433 7592 i58 274 7751 9333 7909 8067 8226 8542 9017 9175 •752 2323 275 9491 9648 9806 9964 •122 •279 i852 •437 •594 1 58 276 440909 1066 1224 i38i i538 1695 2009 2x66 i57 V,l 2480 2637 2793 2050 45x3 3 106 3263 3419 3576 3732 3889 [U 4045 4201 4357 4669 4825 4981 6537 5x37 5293 5449 6848 7oo3 279 N. 56o4 6760 5915 6071 6226 6382 6692 1 55 D. I L 2 3 4 5 6 7 8 9 A TABLE OF LOGARITHMS FBOM 1 TO 10,000. N. I 3 3 4 5 6 7 8 9 D. a8o 447158 8706 73i3 7468 7623 7778 7933 8088 8242 83^7 8552 1 55 a8i 8861 9015 9170 9324 9478 9633 9787 9941 ••95 1 633 1 54 383 450249 . o4o3 0557 071 1 0865 1018 1172 1826 1479 i54 283 1786 1940 2093 2247 2400 2553 2706 2859 8012 3i65 1 53 384 33i8 3471 3624 3777 3930 4082 4235 4887 4540 ' 4693 1 53 385 4845 4997 5i5o 53o2 5454 56o6 5758 5910 6062 1 6314 1 53 386 6366 65i8 6670 6821 6973 7125 7276 8789 7428 7579 7781 i5a 387 ■ 7882 8o33 8184 8336 8487 8638 8940 9091 9242 i5i 38a 9392 9543 9694 9845 9995 •146 •296 •447 •597 •748 i5i 289 460898 1048 1 198 1348 1499 1649 1799 1948 2098 2248 i5o 390 462398 2548 2697 2847 2997 3146 3296 3445 3594 3744 i5o 391 3893 4042 4191 4340 4490 4639 4788 4986 5o85 5234 149 292 5383 5532 568o 5829 5977 6126 6274 6428 6571 6719 149 293 6868 7016 7164 73i2 7460 7608 7756 7904 8o52 8200 148 294 8347 8495 8643 8790 8938 9085 9233 9880 9527 9675 148 295 9822 9969 •116 •263 •410 •557 2020 •704 •85i •998 1 145 147 296 471292 1438 1 585 1732 1878 2171 2818 2464 2610 146 297 2736 2903 3o4Q 3195 3341 3487 3633 3779 8925 5881 f 4071 146 298 4216 4362 45o8 4653 4799 4944 6897 5090 5285 5526 146 299 5671 58i6 5962 6107 6202 6542 6687 6882 6976 145 3oo 477121 8566 7266 871 1 741 1 8855 7555 ^999 7700 7844 7989 8188 8278 8433 145 3oi 9143 9287 9431 9575 9719 9868 144 3o2 480007 oi5i 0294 0438 o582 0725 0869 1012 ii56 1299 2781 144 3o3 1443 1 586 1729 1872 2016 2159 2802 2445 2588 143 3o4 2874 3oi6 3i59 33o2 3445 3587 8780 8872 401 5 4157 143 3o5 43oo 4442 4585 4727 4869 5oii 5i53 5295 5437 5579 142 3o6 5721 5863 6oo5 6147 6289 643o 6572 6714 6855 6997 142 3o7 7i38 7280 7421 7563 7704 7845 7086 9896 8127 8269 8410 141 3o8 855i 8692 8833 8974 9114 9255 9587 9677 9818 141 3o9 9958 ••99 •289 •38o •520 •661 •801 •941 1081 1233 140 3io 491362 l502 1642 1782 1922 2062 2201 2841 2481 3631 140 3ii 2760 2900 3o4o 3179 3319 3458 8 597 3787 3876 401 5 189 3l2 4i55 4294 4433 4572 471 1 485o 4989 5x28 5267 5406 189 3i3 5544 5683 5822 5960 6099 6238 6876 65i5 6653 6791 189 3i4 6930 83ii 7068 7206 7344 7483 7621 77^9 7897 8o35 8173 i38 3i5 8448 8586 8724 8862 8999 •374 9187 9275 9412 9550 1 38 3i6 9687 9824 9962 ••99 •236 •5ii •648 •785 •922 187 3i7 5oio59 1196 1333 1470 1607 1744 1880 2017 2i54 2291 187 3i8 2427 2564 27CO 2837 2073 4335 3109 8246 3382 35i8 3655 i36 319 3791 3927 4o63 4199 4471 4607 4743 4878 5oi4 186 330 5o5i5o 5286 5421 5557 5693 5828 5964 7816 6099 6234 6870 1 36 331 65o5 6640 6776 691 1 7046 7181 7451 7586 8984 7721 135 332 7856 1% 8126 8260 8395 853o 8664 8799 9068 i35 323 9203 9471 9606 9740 9874 •••9 •143 •277 •4n i34 324 5io545 0679 o8i3 0947 I081 ' I2l5 1849 1482 1616 1750 i34 325 1 883 2017 2l5l 2284 2418 ! 255l 2684 2818 2951 8084 ;33 326 3218 3351 3484 3617 3750 3883 4016 4149 4382 4414 1 33 337 4548 4681 48i3 4946 5079 ! 52 1 I 64o3 6535 5844 5476 5609 5741 i33 328 5874 6006 6139 6271 6668 6800 6983 7064 182 339 7196 7328 7460 7592 7724 7855 7987 8119 835i 8382 182 33o 5i85i4 8646 8777 8909 9040 9171 9808 9434 9566 9697 i3i 33i 9828 9959 ••90 ' •221 •353 •484 •6i5 •745 •876 1007 i3i 332 521138 1269 2575 1400 1 i53o 1661 1792 1922 2o53 3i83 2814 i3i 333 2444 2705 2835 2966 3096 8226 8856 3486 36i6 i3o 334 3746 3876 4006 , 4i36 4266 4396 4526 4656 4785 4915 i3o 335 5o45 5174 6409 53o4 ■ 5434 5563 5693 5822 5951 6081 1 6210 j 129 336 6339 6598 , 6727 7888 1 8016 6856 6985 7II4 7243 7872 7501 129 337 m 7630 7759 9045 8145 8274 8403 853 1 8660 8788 128 9174 1 9302 9430 9559 9687 9815 9943 ••72 339 53o2oo o338 0456 i o584 0713 ; 0840 I i 0968 109/* "7 1 1228 i85i 138 N. 1 3 3 4 5 6 8 -1 9 a A. TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 340 1 2 3 4 5 6 7 8 9 D. 531479 2754 1607 1734 1862 i'990 2117 2345 2372 25oo 2627 128 341 2882 3009 3i36 3264 3391 35i8 3645 3772 3899 127 342 4026 41 53 4280 4407 4534 4661 4787 4914 5o4i 5167 "2 126 I 343 5204 5421 5547 5674 6937 58oo 5927 6o53 6180 63o6 6432 ! 344 6558' 6685 681 1 7063 V^2 8448 73i5 8574 7441 8699 9954 ^82? J^?I 126 ! 345 7819 7945 8071 8197 9452 8322 126 346 9076 9202 9327 9578 9703 9829 ••79 •204 125 ¥2 * io329 0455 o58o 0705 o83o 0955 1080 I205 i38o 1454 125 348 'in 1825 1704 1829 1953 2078 2203 2827 2452 2576 2701 125 349 2950 3074 3199 3323 3447 3571 8696 3820 J944 124 350 144068 4192 43i6 4440 4564 4688 4812 4986 5o6o 5i83 124 35i 5307 543 1 5555 5678 58o2 5925 6049 6172 6296 6419 124 352 6543 6666 6789 6913 7086 7159 8389 7282 7405 8685 7529 875s 7652 8881 123 353 7775 7898 8021 8144 8267 85i2 123 354 9003 9126 9249 9371 9494 9616 9789 9861 9984 •106 123 355 550228 o35i 0473 0595 0717 0840 0962 1084 1206 1828 122 356 i45o 1572 1694 1816 1938 2060 2181 2808 2425 2547 122 357 358 2668 2790 291 1 3o33 3i55 3276 3398 35i9 8640 8762 121 3883 >loo4 4126 4247 4368 4489 4610 4781 4852 iul 121 359 5094 52i5 5336 5457 5578 5699 5820 5940 6061 121 360 5563o3 6423 6544 6664 6785 6905 7026 8228 7146 8840 7267 7337 120 36i 7507 8709 8829 7748 8948 7868 7988 8108 8469 8589 120 362 9068 9188 9808 9428 9548 9667 •863 9787 120 563 9907 ••26 •146 •265 •385 •5o4 •624 •748 •982 119 364 56IIOI 1221 1 340 1459 1578 1698 1817 1986 2o55 2174 119 365 2293 2412 253i 265o 2769 2887 3oo6 3i25 3244 8362 119 366 3481 36oo 3718 3837 3955 4074 4192 48ii 4429 4548 119 367 368 4666 4784 4903 502I 5i39 5257 5376 6555 5494 56i2 5780 110 5848 5966 6084 6202 6320 6437 6678 6791 6909 118 369 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 568202 83i9 8436 8554 8671 8788 8905 9028 9140 9257 117 371 9374 9491 9608 9725 9842 9959 ••76 •198 •809 •426 117 372 570543 0660 0776 0893 lOIO 1 1 26 1243 1359 1476 i5q2 117 373 1709 1825 1942 2o58 2174 2291 2407 3568 2523 2689 2765 116 374 2872 2988 3io4 3220 3336 3452 8684 3800 3915 116 375 4o3i 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 376 5i88 53o3 5419 5534 5650 5765 588o 5996 6111 6226 ii5 m 6341 7492 6457 7607 6572 7722 6687 7836 8983 6802 7951 8?66 7082 8181 7147 8295 7262 8410 nil ii5 ii5 379 8639 8754 8868 9097 9212 9826 9441 9555 9669 114 380 579784 580925 9898 10J9 ••12 •126 •241 •355 •469 1608 •583 •697 .836 •811 114 38i ii53 1267 i38i i4q5 263 1 1722 28.58 1950 114 382 2o63 2177 2291 2404 25i8 2745 '-972 3o85 114 383 433 1 33i2 3426 3539 3652 3765 4896 3879 3992 4io5 4218 ii3 384 WiA 4557 4670 4783 5009 5l22 5285 5348 ii3 385 5461 5574 5686 5799 6925 5912 6024 6187 6250 6862 6475 ii3 386 6587 6700 68i2 7037 7U9 8272 7262 8384 8496 7486 8608 7599 0720 9888 112 387 388 7711 8832 7823 8944 7935 8047 8160 112 9o56 9167 9279 9391 95o3 9615 9726 •842 112 389 995d ••bi •173 •284 •396 •5o7 •619 •780 •953 iia 390 591065 1176 2288 1287 1399 i5io 1621 1782 2843 1848 1^55 2066 Hi 391 2177 2399 25lO 2621 2732 2954 8064 8175 III 392 3286 3397 35o8 36i8 3729 3840 3950 4o6i 4171 4282 III 393 4393 45o3 4614 4724 4834 4945 5o55 5i65 5276 5886 110 394 5496 56o6 5717 5827 5987 6047 6157 6267 6377 6487 HO 395 6597 2695 9883 6707 6817 6927 7037 7146 8243 7256 7866 7476 8572 7586 8681 110 396 7805 8900 7914 8024 8i34 8858 8462 no in 9009 91 19 9228 9337 9446 9556 9665 X 109 398 9992 •101 •210 •319 •428 •537 1625 •646 •755 1843 109 399 600973 1082 1191 2 1299 1408 i5i7 1734 I95I 109 N. 3 I 3 4 5 6 7 8 9 D. A TABLE OF LOGARITHMS FROM ] TO 10,000. N. 602060 J144 4226 53o5 400 401 402 4o3 404 , 405 1 7455 406 d526 2169 3253 4334 54i3 409 410 411 412 4i3 414 41 d 416 41 B 419 420 421 422 423 424 425 426 427 428 429 43o 43 1 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 45o 45 1 452 453 454 456 456 538i 6489 7562 459 N. 9594 610660 1723 612784 3842 4897 5960 7000 8048 9093 620136 1176 2214 623249 4282 53i2 6340 7366 8389 9410 630428 1444 2457 633468 4477 5484 6488 7490 8489 94S6 640481 1474 2465 643453 4439 5422 6404 7383 836o 9335 65o3o8 1278 2246 653213 5i38 6098 7056 801 1 8965 660865 i8i3 8633 9701 0767 1829 2890 5ooc 6o55 7io5 8i53 9198 0240 1280 23i8 3353 4385 541 5 6443 7468 8491 95i2 o53o 1545 2559 3569 4578 5584 6588 7590 8589 9586 o58i 1573 2563 355i 4537 5521 65o2 7481 8458 9432 o4o5 1375 2343 33oo 4273 5235 6194 7152 8107 9060 ••11 0960 1907 2277 336i 4442 5521 6596 7669 8740 9808 0873 1936 5io8 6160 7210 S257 9302 0344 i384 2421 3456 4488 55i8 6546 7571 8593 9613 o63i 1647 2660 3670 4679 5685 6688 7690 8689 9686 0680 1672 2662 365o 4636 5619 6600 8555 953o o5o2 1472 2440 34o5 4369 533 1 6290 7247 8202 9155 •106 io55 2002 2386 3469 455o 5628 6704 IIV^ 9914 0979 2042 3l02 4159 52i3 6265 9406 0448 1488 2525 3559 4591 5621 6648 7673 8695 97i5 0733 1748 2761 3771 5785 6789 7700 8780 9785 0779 1771 2761 3749 4734 5717 6698 7676 8653 9627 0599 1569 2536 35o2 4465 5427 6386 7343 8208 9260 •201 ii5o 2096 1494 4658 5736 681 1 7884 8954 ••21 1086 2148 3207 4264 53i9 6370 73 1 5 7420 8362 I 8466 9511 o552 1592 2628 3663 4695 5724 6751 7775 8797 9817 o835 1849 2862 3872 4880 5886 7890 8888 9885 0879 1871 2860 3847 4832 58i5 6796 7774 8750 9724 0696 1666 2633 3598 4562 5523 6482 7438 8393 9346 •296 1245 219] 4 s6o3 3686 4766 5844 6919 i 7991 9061 •128 I 2254 33i3 4370 5424 I 6476 7525 8571. 9615 o656 i6q5 2732 3766 4798 5827 6853 7878 8900 9919 0986 1951 2963 3973 4981 5986 6989 7990 8988 9984 0978 1970 2959 3946 4931 5913 6894 7872 8848 9821 0793 1762 2730 3695 4658 5619 6577 7534 8488 9441 .301 1339 2286 2711 3794 4874 5961 7026 8098 9167 •234 1298 236o 3419 447^ 9719 0760 1799 2835 3869 4901 5929 6956 7980 9002 ••21 io38 2052 3064 4074 5o8i 6087 7089 8090 9088 ••84 1077 2069 3o58 4044 5029 601 1 6992 7969 8945 9019 0890 1859 2826 3791 4754 5715 6673 7629 8584 9536 •486 1434 238o 2819 3902 4982 6o5o 7133 8205 9274 •341 i4o5 2466 3525 458i 5529 I 5634 658i I 6686 7620 i 7734 8676 8780 9824 0864 1903 2939 3973 5004 6o32 7o58 8082 9104 •123 1189 2i53 3i65 4175 5182 6187 7189 8190 9188 •i83 1177 2168 3i56 4143 5127 6110 7089 8067 9043 ••16 0987 1956 2923 3888 485o 58io 6769 7725 8679 9631 •58i 1 529 2475 2928 4010 5089 6166 7241 83i2 938i •447 i5ii 2572 363o 4686 5740 % 8884 9928 0968 2007 3o42 4076 5107 6i35 7161 8i85 9206 •224 1241 2255 3266 4276 5283 6287 7290 8200 9287 •283 1276 2267 3255 4242 5226 6208 8i65 9140 •ii3 1084 2o53 3oi9 3984 4946 5906 6864 7820 8774 9726 •676 1623 2569 8 3736 4792 5845 6895 7943 .8989 ••32 1072 2110 3146 4179 5210 6238 7263 8287 9308 •326 1 342 2356 3367 4376 5383 6388 7390 8389 9387 •382 1375 2366 3354 4340 5324 63o6 7285 8262 9237 •210 1181 2l5o I 3ii6 I 408c i 5o42 i 6002 I 6960 i 7916 8870 9821 •771 1718 2663 D. :.o8 lod 108 108 107 107 107 3o36 4118 5197 6274 7348 8419 9488 •554 i 107 1617 j 106 2678 ; 106 106 106 io5 io5 io5 io5 104 104 104 104 io3 io3 io3 io3 102 102 102 102 lOi 101 100 100 100 100 99 99 99 99 99 99 9^ 9? 95 96 97 97 97 97 97 96 95 ^? 95 D. 8 A TABLE OF LOGARITHMS FBOM 1 TO 10,000. N. ° I 3 3 4 5 6 7 8 9 D. 460 662758 2852 2047 3o4i 3i35 323o 3324 3418 35i2 3607 94 461 3701 3795 4735 3S89 3983 4078 4172 4266 436o 4454 4548 94 46 a 4642 483o 4924 5862 5oi8 5lI2 5206 IIV, 5393 : 5487 94 463 558i 5675 5769 5956 6o5o 6143 6331 ! 6424 94 464 65i8 6612 6700 6799 6892 6986 ^1? 7173 7266 ] 7360 8199 1 8293 94 465 7453 7546 7640 8572 8665 7826 792c 8852 8106 93 466 83d& 8479 8759 8945 9875 9o38 9i3i 9224 93 46t 931-' 9410 95c3 9596 9689 97S2 ••60 •'53 93 46d 670546 0339 043 1 o524 0617 0710 0802 0695 0988 i ic8o 93 469 1173 1265 i358 i45i 1 543 i636 1728 1821 1913 1 3005 93 470 ^72098 2190 2283 2375 2467 25t>0 2652 2744 3836 2929 95 4-1 302I 3ii3 32o5 3297 3390 3482 3574 3666 3758 3b5o 92 472 3o42 4861 4o34 4126 4218 43io 4402 4494 4586 4677 4769 93 4-3 4953 5o45 5i37 5228 5320 5412 55o3 5595 5687 92 474 m 5370 5962 6876 7789 6o53 6145 6236 6328 6419 65ii 6603 92 475 6785 6068 7059 7i5i 7242 7333 8245 7424 7516 8427 91 476 7607 nbgS 8609 7881 7972 8882 8o63 81 54 8336 9> 477 85i8 8700 8791 8973 9064 9155 9246 9337 9* 47^ 9428 9519 9610 9700 9791 9882 9973 ••63 •i54 •243 9» 479 68o336 0426 o5i7 0607 0698 0789 0879 0970 1060 ii5i 91 43o 681241 i332 1432 i5i3 i6o3 1693 1784 1874 1064 2o55 90 4S1 2145 2235 2326 2416 25o6 2596 26S6 2777 2867 3057 90 482 3 047 3 137 3227 3317 3407 3497 3587 3677 3767 3857 90 483 3q47 4845 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 484 4935 5o25 5ii4 5204 5294 6189 5383 5473 5563 5652 % 485 5742 5-3i 5921 68i5 6010 6100 6279 6368 6458 6547 486 6636 6726 6904 ti 7o33 7172 8064 7261 8i53 735i 7440 833i 89 48? 7529 8420 7618 8598 9486 7796 ^5 8242 89 85o9 86^7 8776 8o53 9042 9i3i 9220 89 489 9309 9398 9575 9664 9753 9841 9930 ••19 •107 89 490 690196 1081 0285 0373 0462 o55o 0639 0728 0816 0905 0093 80 491 1170 1258 i347 1435 i524 1612 1700 1789 1877 88 492 1965 2847 2o53 2142 223o 23i8 2406 2494 2583 2671 2759 88 493 2935 38i5 3o23 3iii 3199 4078 3287 3375 3463 355i 3639 88 494 3727 3903 3991 4166 4254 4342 443o 4517 88 495 46o5 4693 4781 4S63 4956 5o44 5i3i 5219 5307 53q4 88 496 5482 556q 5657 5744 5832 5919 6007 6094 6182 6369 87 497 6356 6444 653 1 6618 6706 6793 6880 W9 7o55 r. ?" 498 7229 7317 S188 7404 7491 7578 7663 7752 7926 8796 87 499 8101 8275 8362 8449 8535 8622 8709 8883 87 5oo "^11 9057 9144 923i 9317 9404 9491 9578 9664 975i 87 5oi 9924 ••11 ••93 •184 •271 •3d8 •444 •53 1 •617 87 502 700704 0790 0877 0963 1827 io5o Ii36 1232 i3o9 1395 1483 86 5o3 iks 1604 1741 1913 ;^ 2086 2172 2238 3344 86 5o4 243 1 25i7 2603 2689 2775 in^ 3o33 3ii9 33o5 86 5o5 3291 3377 3463 3549 3635 3721 3893 3979 4o65 86 5o6 4IDI 4236 4322 4408 4494 4579 4665 4731 4837 4922 86 507 5oo8 5094 5179 5265 535o 5436 5522 5607 5693 5778 86 5o8 5864 5949 68o3 6o3o 6120 6206 6291 6376 6462 6547 6633 85 5*9 6718 6888 6974 7059 7144 7229 73i5 7400 7485 85 5io 707570 8421 7655 7740 7826 791 1 8?46 8081 8166 825i 8336 85 5ii 85o6 8591 8676 8761 8931 9015 9100 9185 85- 5ii 9270 9355 9440 9524 9609 9694 9779 9863 9948 ••33 85 5i3 710117 0202 0287 0371 04d6 0540 062D 0710 0-94 0879 85 5i4 0963 1807 1048 Il32 1217 i3oi 1 385 1470 1 554 i639 1723 84 i 5i5 1S92 1976 2060 2144 2229 33i3 23q7 3238 2481 3566 84 5i6 265o 2734 23i3 2902 29S6 3o-o 3i54 3323 3407 84 5id 3491 3575 3659 3742 3826 3910 4833 4078 4162 4346 84 433o 4414 4497 458i 4665 4749 4916 5ooo 5o84 84 5i9 5167 525i 1 1 I 5335 a 5418 55o2 5586 5669 5753 5836 5930 84 N. 3 4 5 6 7 « 9 D. A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. I 2 3 4 5 6 7 8 9 D. 520 716003 6087 6170 6254 6337 6421 65o4 6588 6671 6754 7587 8410 83 521 6838 6921 7004 7088 8oo3 7254 7338 7421 8253 7504 8336 83 522 ■7671 85o2 ?585 6668 7920 8751 9580 8086 8169 83 523 8834 8917 9745 9000 9083 9165 9248 83 524 9331 9414 9497 9663 9?^? 99" 9994 ••77 83; 525 720159; 0242 032D 0407 1233 0490 0573 o655 0738 0821 j 0903 U 526 0086 1811 1068 ii5i i3i6 1398 1481 i563 1646 ' 1728 62 527 528 1893 1975 2o58 2140 2222 23o5 2387 2469 ■ 2552 82 2634 2716 2798 2881 2963 3045 3127 3948 3209 3291 3374 8a 539 3456 3538 3620 3702 3784 3866 4o3o 4II2 4194 82 53o 724276 4358 4440 4522 4604 4685 4767 4849 4931 5oi3 82 53 1 5095 5176 5258 5340 5422 55o3 5585 5667 5748 5830 82 532 5912 5993 6075 6i56 6238 6320 6401 6483 6564 6646 8a 533 6727 6809 7623 6890 6972 7053 7134 7216 7297 7379 7460 81 534 7541 7704 7780 7866 7948 8029 8110 8191 8273 81 535 8354 8435 85i6 8597 8678 8759 8841 8922 9003 9084 81 536 9165 9246 9327 •i36 9408 9489 9570 9651 9782 9813 9893 81 537 538 9974 730782 ••55 •217 •298 •378 u86 •459 •540 •621 •702 81 o863 0944 1024 iio5 1266 1 347 1428 i5o8 81 539 1589 1669 1730 i83o 1911 1991 2072 2l52 2233 23i3 81 540 732394 2474 2555 2635 2715 2796 2876 2956 3o37 3117 80 541 3197 3278 3358 3438 35i8 3598 3679 ^Ia9 4560 3839 3919 80 542 3099 4079 4160 4240 4320 4400 4480 4640 4720 80 543 4800 4880 4960 5o4o 5l20 5200 5270 5359 5439 55i9 80 544 5599 5679 5739 5838 5918 5998 6078 6i57 6237 6317 7113 80 545 6397 6476 6556 6635 6715 6795 6874 6954 7034 80 546 7193 7272 7352 8146 7431 8225 75ii 7590 8384 ■7670 8463 854^ 7829 8622 7908 79 547 li] S^7 83o5 8701 79 548 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 549 9672 965i 9731 9810 9889 9968 ••47 •126 •2o5 •284 79 55o 74o363 0442 052I 0600 0678 0757 o836 0915 0994 1073 79 55i Il52 I23o 1 309 1388 1467 1546 1624 1703 178? i860 79 552 1939 2018 2096 2882 2175 2254 2332 241 1 2489 2568 2647 75 78 553 272D 35io 2804 2961 3o39 3823 3ii8 3196 3275 3353 343 1 554 3588 3667 3745 3902 3980 4o58 4i36 42i5 78 555 4293 4371 5i53 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 523i 5309 5387 5465 5543 5621 5699 5777 78 557 558 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 6634 6712 6790 6868 6945 7023 7101 7n? 7256 7334 8110 78 559 7412 7489 7567 7645 7722 7800 7878 7955 8o33 78 56o 748188 8266 8343 8421 8498 8576 8653 8731 8808 8885 77 56i 8963 9040 9118 9195 9272 935o 9427 9004 9582 9659 77 562 9736 9814 9891 9960 ••45 •123 •200 •277 1048 •354 •43 1 77 563 75o5o8 o586 0663 0740 0817 0894 0971 II25 1202 77 564 1279 2048 i356 1433 i5io i587 1664 1741 1818 1895 197a 77 565 2125 2209 2279 2356 2433 2609 2586 2663 2740 35o6 77 566 2816 2893 297© 3o47 3i23 3200 3277 3353 343o 77 567 568 3583 366o 3736 38i3 3889 3966 4041 4119 4883 4195 4272 77 4348 4425 45oi 4578 4654 4730 4807 40CX) ' 5o36 6 569 5lI2 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 755875 6636 5951 6027 6788 7548 83o6 6io3 6180 6256 6332 6408 6484 656o 76 571 6712 6864 6940 7016 7092 785i 7168 7244 8oo3 7320 8079 76 572 l^, 7472 8230 7624 7700 8458 7775 7927 76 573 8382 8533 8609 8685 8761 8836 76 574 8912 8988 9063 9139 9214 9290 9366 9441 9517 9592 76 575 9668 9743 9819 9894 9970 ••45 •121 •196 •272 •347 75 576 760422 0498 I25l 0573 0649 0724 0799 0875 0930 1025 IIOI 75 m 1176 i326 1402 1477 i552 1627 2378 1702 1778 i853 75 1928 2679 2003 2078 2i53 2228 23o3 2453 2529 2604 75 579 2754 2829 2904 2978 3o53 3i28 32o3 3278 3353 75 N. I 2 3 4 5 6 7 8 9 D. i LO A TABLE OF LOGARITHMS FROM I TO 10,000. N. 58o I I 2 3 4 5^617 1 ^ 9 "bTI 75 763428 35o3 3578 3653 3727 38o2 ' 3877 i 8952 ! 4027 1 4101 58i 4176 425i 4826 4400 4475 455o ■ 4624 4699 ! 4774 1 4848 75 582 4923 4998 5072 5i47 5221 5296 5370 5443 : 5520 ' 5594 6264 1 6338 1 75 583 5669 6413 5743 58i8 5892 5966 6041 ; 6ii5 1 6190 6859 1 6988 '' u 584 6487 6562 6686 6710 6785 7007 , 7082 '. -4 ^85 7i56 7i3o 1 7^04 7879 7458 1 8x94 8984 7527 7601 8843 7675 7749 i 7«23 ,4 586 7898 7972 i 8046 B120 8268 8416 ! 8490 1 8564 ■ 74 587 8638 8712 8786 8860 9008 9082 9i56 9280 g3o3 74 588 9377 9451 1 9525 9599 9673 9746 9820 1 9894 0681 9968 ! ••42 74 589 770II3 0189 0263 o336 0410 0484 0557 0705 i 0778 74 590 770852 0926 0999 1078 1 146 1220 1293 1867 1440 i5i4 74 591 1587 1661 1734 1808 i88i 1955 2028 2102 2175 2248 73 592 2322 2395 2468 2542 26i5 2688 2762 2885 2908 2981 73 593 3o55 3128 8201 8274 3848 8421 3494 3567 8640 3718 ' 73 594 3786 3860 3988 4006 4079 4i52 4225 4298 4371 4444 73 595 4517 4590 4668 4786 4809 4882 4955 5o28 5ioo 5178 73 596 5246 5319 5392 5465 5588 56io 5683 5756 5829 5902 73 597 5974 6047 6120 6198 6265 6338 641 1 6488 6556 6629 73 598 6701 6774 6846 6919 6992 7064 7187 7209 7282 8006 7354 8079 73 599 7427 7499 7572 7644 7717 7789 7862 7984 72 600 778i5i 8224 8296 8868 8441 85i3 8585 8658 8780 8802 72 601 8874 ^1P 9019 9091 9168 9286 9808 9880 9452 9524 72 602 9596 9669 9741 9818 9885 9957 ••29 •lOl •173 •245 72 6o3 780817 0889 0461 o588 o6o5 0677 0749 0821 0898 0965 72 604 loS-) 1 109 1181 1258 1824 1896 1468 1540 1612 1684 72 6o5 1755 1827 1899 1971 2042 2114 2186 2258 2829 2401 72 606 2473 2544 2616 2688 2739 2881 2902 2974 8046 8117 72 607 3189 3260 3882 8408 8473 3546 36i8 8689 8761 3832 71 608 3904 3975 4046 4118 4189 4261 4832 44o3 f^r 4546 71 609 4617 4689 4760 4881 4902 4974 5o45 5n6 5187 5259 71 610 785330 5401 5472 5543 56i5 5686 5757 5828 5899 5970 11 611 6041 6112 6188. 6254 6325 6896 6467 6588 6609 6680 11 612 6751 6822 6898 6964 7o85 7106 7177 7248 7819 7890 7» 6i3 7460 8168 7531 7602 7673 7744 7815 7885 7956 8027 8098 711 614 8289 83io 838 1 8451 8522 8598 8663 8784 88c4 7^ 6i5 8875 8946 9016 9087 9157 9228 9299 9869 9440 95io 71 616 9581 9631 9722 9792 9868 9988 •••4 ••74 •144 •2l5 70 618 790285 o356 0426 0496 0367 0687 0707 0778 0848 0918 70 0988 1059 1 1 29 1 199 1269 1840 1410 1480 i55o 1620 70 619 1691 1761 1881 1901 1971 2041 2111 2l8l 2252 2822 70 620 792392 2462 2582 2602 2672 2742 2812 2882 2952 3o22 70 621 3092 8162 3281 33oi 8871 3441 35ii 358i 365i 8721 70 622 3790 386o 8980 4000 4070 4189 4209 4279 4349 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5043 5ii5 70 624 5i85 5254 5824 5393 5468 5582 56o2 5672 5741 58ii ^ 6a5 588o 5949 6010 6088 6i58 6227 6297 , 6366 6436 65o5 626 6574 6644 6713 6782 6852 6921 699c 7060 7129 7198 69 ^^1 6sd 7268 7337 7406 747^ 7545 7614 7688 6374 7752 7821 7890 69 7960, 8029 8098 816- 8236 i 88o5 8443 85i3 1 8582 69 629 865i 8720 8789 8858 8927 , 8996 9065 9184 9208 1 9272 69I 63o 799341 9409 0098 9478 9547 9616 . 9685 9754 9828 9892 1 9961 : ^i 63i 800029 0167 0286 o3o5 0878 0442 o5ii 03b0 ; 0648 69 632 0717: 0786 o854 0928 0992 , 1061 1 1 29 1 198 1266 j i885 691 633 i4o4i 1472 1 541 1609 2295 ! 1678 [ 1747 i8i5 1884 1952 2021 ! 69 634 2089 2 1 58 2226 2868 1 2482 ! i5oo 2568 2687 2705 1 ^ 635 2774 2842 2qio 2979 j 3o47 3ii6 3i84 3252 3821 8889 1 68 636 3457 3525 35q4 3662 i 3780 8708 8867 8985 4008 4071 1 68 637 638 4139 4208 4276 4344 4412 4480 ; 4548 i 4616 4685 4753 : 68 4821 4889 4q57 i 5o25 5098 5i6i 5220 5297 ; 5365 5433 i 68 639 55oi 5569 5637 i 1 5705 5773 5841 5908 5976 6044 1 61 1 2 68 D N. I 3 3 4 5 1 6 7 ! 8 __9j A TABLE OF LOGARITHMS FROM 1 TO 10,000. 11 N. I 2 3 4 5 6 7 8 9 D. ; 64o 806180 6248 63i6 6384 645 1 65i9 6587 6655 6723 6790 68 641 6858 6926 6994 7061 7129 7197 7264 7882 7400 8143 68 642 7535 7603 1670 8346 7738 8414 7806 7873 7941 8008 0076 6d) ' 643 8211 8279 8953 8481 8549 8616 8684 8751 8818 67] i 644 8886 9021 9088 9i56 9223 9290 9358 9425 9492 67! 1 645 9560 9627 9694 9762 9829 9896 9964 ••3 1 ••98 1 •i65 67. ! 646 810233 o3oo 0867 0484 o5oi 0569 0686 0708 0770 0887 i5o8 671 1 647 0904 0971 1089 U06 1178 1240 1807 1874 1441 67' i 648 1575 1642 1709 1776 1843 1910 2679 1977 2044 2111 2178 6? ! 649 2245 23l2 2379 2445 25l2 2646 2718 2780 2847 67 65o 812913 2980 3047 3ii4 3i8i 3247 83i4 338i 3448 35i4 67 65 1 3o8i 3648 3714 3781 3848 8914 8981 4048 4114 4181 67 652 4248 43i4 438i 4447 45i4 458 1 4647 4714 4780 4847 67 653 4913 4980 5o46 5ii8 5170 5246 58i2 5378 5445 55ii 66 654 5D78 5644 5711 5777 5843 5910 5976 6042 6109 6175 66 655 6141 63o8 6374 6440 65o6 6578 6689 6705 7367 8028 6771 6838 66 656 6904 6970 7o36 7102 7169 7285 7801 7483 7499 66 657 7565 8226 7631 6292 7698 8358 7764 7880 7896 8556 7962 8094 8160 66 658 8424 8490 8622 8688 8754 8820 66 659 8885 8951 9017 9083 9149 92i5 9281 9846 9412 9478 66 660 819544 9610 9676 9741 9807 9873 n^i •••4 ••70 •i36 66 ! 661 820201 0267 o333 0899 0464 o53o oSoS 0661 0727 0792 66 662 o858 0924 0989 I ODD 1 1 20 1 186 I25l 1817 1882 1448 66 663 i5i4 1579 1645 17IO 1775 1 841 1906 1972 2087 2io3 65 664 2168 2233 2299 2364 243o 2495 256o 2626 2691 2756 65 665 2822 2887 2902 3oi8 8088 8148 8218 8279 3844 3409 65 666 3474 3539 36o5 3670 8785 38oo 3865 3980 8996 406 1 65 667 668 4126 4I9I 4256 4821 4886 445 1 45i6 458 1 4646 471 1 65 4776 4841 4906 4971 5o36 5ioi 5i66 5281 5296 536i 63 669 5426 5491 5556 5621 5686 5751 58 J. 5 588o 5945 6010 65 670 826075 6140 6204 6269 6884 6899 6464 6528 6598 6658 65 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7805 65 672 7369 7434 7499 7068 7628 7692 7757 7821 7886 7951 65 673 8oi5 8080 8144 8209 8273 8888 8402 8467 858 1 8595 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9289 64 675 9804 9368 9432 9497 9561 9625 9690 9754 9818 9882 64 676 9947 ••11 ••75 •189 •204 •268 •832 •896 •460 •525 64 t]l 83o589 0653 0717 0781 0845 0909 0978 1087 1102 1 166 64 i^23o 1294 i358 1422 i486 i55o 1614 1678 1742 1806 6a, 679 1870 1934 1998 2062 2126 2189 2253 2817 2881 2445 64 680 832309 2573 2687 2700 2764 2828 2892 2956 8020 3o83 6A 681 3 147 32II 8275 3388 8402 3466 853o 3598 3657 3721 64 682 3784 3848 3912 8975 4089 4108 4166 4280 4294 4357 64 683 4421 4484 4548 4611 4675 4789 4802 4866 4929 4008 64 684 5o56 5l20 5i83 5247 58io 5378 5437 55oc 5564 1 5627 63 685 5691 5754 5817 588 1 5944 6007 607; 6184 6197 6261 63 686 6324 6387 645 1 65i4 6577 6641 6704 6767 6880 6894 63 687 6957 7588 7020 7088 7146 7210 7278 7886 7899 7462 7525 63 688 7652 77i5 77-'8 8408 7841 7904 8534 7967 8080 8098 8i56 63 I 689 8219 8282 8345 8471 8597 8660 8728 8786 63 1 690 83884Q 8912 8975 9088 9101 9164 9^?I 9289 9852 041 5 63 691 9478 9541 9604 9667 9729 9792 9855 9918 9981 j ••43 63 692 840106 0169 0282 0294 o857 0420 0482 o545 0608 ; 0671 63 693 0733 0796 0850 0921 0984 1046 1 109 1172 1284 1297 63 1 694 1359 1985 1422 1485 1 547 1610 1672 1735 1797 i860 ! 1922 63 1 695 2047 2110 2172 2235 2297 2860 2422 2484 2547 63 ' 696 2609 2672 2734 2796 2859 1921 2988 8046 8108 8170 6a S 3233 3295 3857 3420 3482 3544 36o6 8669 8781 8798 6a 3855 3918 8980 4042 4104 4166 4229 4291 4358 1 441 5 6a 699 N. 4477 4539 4601 4664 4726 4788 485o 4912 4974 ' 5o36 63 0, I 2 1 3 4 5 6 7 ■ 8 9 L2 A TABLE OF L0GAEITHM3 FROM 1 TO 10,000. N. I 2 3 4 5 6 7 8 9 1)7 ; 700 845098 5i6o 5222 5284 5346 5408 5470 5532 5594 5656 62 701 5718 5780 5842 5904 5966 6585 6028 6090 6i5i 6213 1 6275 62 702 6337 6399 6461 6523 6646 6708 6770 6832 ] 6894 62 703 6955 7017 1 7079 ' 7141 7202 7264 7326 7388 7449 ' 75ii 62 704 7073 7634 7396 ' --ToS 7819 7881 7943 8004 8066 ^128 62 ! 705 8189 825i 8312 8374 8435 8497 8559 8620 8682 1 3743 6a 706 88o5 8866 8928 8989 905 1 9112 9174 9235 9297 1 p358 61 ]o^ 9419 9481 9542 9604 9665 9726 9788 9849 991 1 o524 9972 61 85oo33 0095 oi56 0217 0279 o34o 0401 0462 0585 61 709 0646 0707 0769 o83o 0891 0962 j 1014 1075 ii36 1 197 61 710 851258 l320 i38i 1442 i5o3 1 564 1625 1686 1747 1809 61 711 1870 2400 I93I 1992 2o53 2114 2175 2236 2297 2358 2419 61 '''^ 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 3090 3i5o 32II 32T2 3333 3394 3455 35i6 3577 3637 61 714 3698 3759 3820 388i 3941 4002 4o63 4124 4i85 4245 61 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 716 4913 4974 5o34 5095 5i56 5216 5277 5337 5398 5459 61 718 5519 5580 5640 5701 5761 5822 5882 5943 6oo3 6064 61 6124 6i85 6245 63o6 6366 6427 6487 6548 6608 6668 60 719 6729 6789 685o 6910 6970 7o3i 7091 7i52 7212 7272 60 720 857332 7393 7453 75i3 8116 7^74 7634 7694 7755 78i5 8417 9018 7875 60 721 7935 7995 8o56 8176 8236 8297 8357 8477 60 722 8537 8697 9198 8657 8718 8778 8838 8898 8958 9078 60 723 9i38 9258 9318 9370 9439 9499 9559 9619 9679 60 724 9739 9799 9859 9918 9978 ••38 ••98 •i58 •218 •278 60 725 86o338 0398 0458 o5i8 0678 0637 0697 0757 0817 i4i5 0877 60 726 0937 0996 io56 1116 1176 1236 1295 i355 1475 60 727 728 1 534 1594 1654 1714 1773 1 833 1893 1952 2012 2072 60 2l3l 2191 225l 23lO 2370 243o 2489 2549 2608 2668 60 729 2728 2787 2847 2906 2966 3o25 3o85 3i44 3204 3263 60 1 730 863323 3382 3442 35oi 356i 3620 368o 3739 3799 3858 59 71 3917 3977 4o36 4096 4i55 4214 4274 4333 4392 4452 59; ll^ 43II 4570 463o 4689 4748 4808 4867 4926 4985 5o45 59 733 5io4 5i63 5222 5282 5341 5400 5459 55i9 5578 5637 59I 734 56q6 5755 58i4 5874 5933 5992 6o5i 6110 6169 6228 59 1 '^ii 6287 6878 6346 64o5 6465 6524 6583 6642 6701 6760 6819 59 j 736 6937 6qq6 7055 7114 7173 7232 7291 7350 7409 59 ]ll 7467 7026 J 7644 7703 7762 7821 7880 7939 7998 59 8o56 8ii5 8174 8233 8292 835o 8409 8468 8527 8586 59 739 8644 8703 8762 8821 8879 8938 8997 9o56 9114 9173 59 740 869232 9290 9349 9408 9466 9525 9584 9642 9701 9760 ^ 741 9818 9877 9935 9994 ••53 •hi •170 •228 •287 •345 59 742 870404 0462 052I 0579 o638 0696 0755 o8i3 0872 0930 i5i5 58 743 0989 1047 1 106 1 164 1223 1281 1 339 1398 1456 58 744 1573 i63i 1690 1748 1806 1 865 1923 1981 2040 2098 58 745 2i56 22l5 2273 233i 2389 2448 25o6 2564 2622 2681 58 746 2739 2797 2855 2913 2972 3o3o 3o88 : 3146 1 3204 3262 58 747 3321 3379 3437 4018 3495 3553 36ii 3669 : 2727 I 3785 3844 58 748 3902 3960 4076 4i34 4192 425o 43o8 4366 4424 i 58 749 4482 4540 4598 4656 4714 4772 4830 4888 4945 5oo3 58 75o 875061 5ii9 5i77 5235 5293 535i 5409 1 5466 5524 5582 58 75i 5640 5698 5756 58i3 5871 5929 5987 6045 6102 1 6i6o 58 752 6218 6276 6333 6391 6449 65o7 6564 6622 6680 1 6737 58 753 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 58 754 7371! 7429 7487 7D44 7602 7659 7717 7774 7832 7889 8464 58 755 7947 8004 ' 8062 ' 8119 8177 8234 8292 ; 8349 8407 ?7 756 8522 8579 I 8637 8694 8752 8809 8866 1 8924 8981 90 J9 ^7 ^h 9096 9153 9211 9268 93a5 9383 9440 9497 9555 9612 i^ 758 9669 9726 9784 9841 9898 9o56 o528 ••i3 ••70 •127 •i85 1 57 759 880242 0299 o356 041 3 0471 o585 0642 0699 0756 1 ^7 I 3 3 4 5 6 7 8 9 D. A TABLE OP LOGARITHMS FROM 1 TO 10,000. 18 N. I 760 I 2 3 1 ^ I 5 1 6 7 8 9 D. 57 880814 0871 0928 0985 1042 1099 ii56 12l3 1271 1828 ZJ' r385 1442 1499 1 556 1618 1670 1727 1784 1 841 1898 57 ' ^? io55 2D25 2012 2069 2126 2188 2240 2297 2354 241 1 2468 57 1 763 258i 2638 2695 2752 2809 2866 2928 2980 8087 57 764 8098 3i5o 8207 8264 8321 3877 8434 8491 8348 36o5 57 i 765 1 366 1 8718 3775 8882 8888 8945 4002 4069 4ii5 4172 57 /66 4229 4285 4342 4899 4455 45i2 4569 5i85 4625 4682 4780 571 1 767 i 76B 4795 536 1 4852 4909 4965 5531 5022 5078 5192 5248 i 53o5 57 i 5418 5474 5587 5644 570c 6821 58i3 5870 U 1 769 5926 5983 6089 6096 6i52 6209 6265 6878 1 6484 i 770 886491 7064 6547 6604 6660 6716 6773 6829 6885 6942 6998 56 ' 771 7111 7167 7223 - 7280 7^^ 73^2 7449 75o5 756i 56 772 7617 8179 7674 7780 8292 ViA^ 7842 8404 7898 llib 801 1 8067 8128 56 773 8236 8460 8573 8629 8685 56 774 8741 8797 8858 8909 8965 9021 9077 9184 9190 9246 56 •,75 9802 9358 9414 9470 9526 9582 9688 9694 9730 9806 56 776 9862 9918 9974 ••3o ••86 •141 •197 •253 •809 •865 56 777 77S 890421 0477 io85 o588 o589 0645 0700 0756 0812 0868 0924 56 0980 1091 1 147 1705 1208 1209 i3i4 1870 1426 1482 56 779 1537 1598 1649 1760 1816 1872 1928 1988 2089 56 780 ^'6t 2i5o 2206 2262 2817 2878 2429 2484 2540 2595 56 7?' 2707 2762 2818 2878 2929 2985 8040 8096 8i5i 56 7?? 3207 8262 8818 3378 8429 3484 3D40 8505 865i 8706 56 783 3762 8817 8878 3928 8984 4089 4094 4i5o 42o5 4261 55 784 43i6 4871 4427 4482 4538 4598 4648 4704 4759 4814 55 7?^ 4870 4925 4980 5o86 5091 5146 5201 5257 58i2 5867 55 ! 786 5423 5478 5d83 5588 5644 5699 5754 5809 5864 5920 55 i 7^7 5975 6526 6o3o 6o85 6140 619* 6201 6806 6861 6416 6471 55 1 788 658i 6686 6692 6747 6802 6857 6912 6967 7022 55 ; 789 7077 7182 7187 7242 7297 7352 7407 7462 7517 7572 55 1 790 897627 7682 8281 7737 8286 7792 7847 8896 7902 7o5j 85o6 8012 8067 86i5 8122 55 i 791 8176 8841 8451 856i 8670 55 792 8725 8780 8885 8800 8944 8999 9054 9109 9164 9218 55 M93 9273 9828 9888 9^^I 9492 9547 9602 9656 97II 9766 55 794 9821 9875 9980 9985 ••89 ••94 •149 •203 •258 •3l2 55 i 795 900867 0422 0476 o53i o586 0640 0695 0749 0804 0859 55 796 0913 0968 1022 1077 ii3i 1 186 1240 1293 1849 1404 55 ]'^ 1458 i5i8 1 567 1622 1676 1781 1785 1840 1894 1948 54 2003 2057 2112 2i66 2221 2275 2829 2884 2488 2492 8o36 54 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 54 Boo 908090 8144 8199 3258 8807 3861 8416 3470 3524 3578 54 3oi 3633 3687 8741 3795 8849 3904 8953 4012 4066 4120 54 3o2 4174 4229 4288 ^t^l 4891 4445 4499 4558 4607 5i48 4661 54 3o3 4716 4770 4824 4878 4982 4986 5o4o 5094 5202 54 804 5256 58io 5364 5418 5472 5526 558o 5634 5688 5742 541 8o5 5796 6335 585o 5904 5958 1 6on 6066 6119 6178 6227 6281 54 806 6889 6443 6497 655i 6604 6658 6712 6766 5820 54 807 808 6874 6927 6981 7089 7143 7196 7734 7250 7804 7858 54 741 1 7465 7319 1578 8no 7626 8168 7680 8217 7787 ! 7841 8824 8878 7805 8481 54 809 7949 8002 8o56 8270 54 810 908485 8539 8592 8646 8609 8753 9235 9289 8807 8860 8914 8961 54 811 9021 9074 9128 9181 9842 9896 9449 93o3 54 812 9556 9610 9668 9716 9770 9823 9877 9980 9984 ••37 53 8i3 910091 0144 0197 0781 025l 0804 o858 041 1 0464 o5i8 0571 53 814 0624 0678 0784 o838 0891 0944 0998 io5i 1 104 53 8i5 ii58 1211 1264 i3i7 187. 1424 1477 i53o i584 1687 53 816 1690 1743 1797 i85o 1908 1956 2009 2o63 2116 2169 53 ^'7 818 2221 2275 2828 2881 2485 2488 2541 2594 2647 8178 2700 53 2753 2806 2859 2918 2066 8019 8072 3i25 8281 53 819 3284 3337 8890 8443 , 3496 3549 8602 3655 8708 3761 53 H. I 3 3 i 4 5 6 7 8 9 D. 14 A TABLE OF LOGARITHMS FROM I TO 10,000. N. 820 831 822 823 824 825 826 827 828 829 83o 83 1 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 85o 85 1 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 861 868 869 870 871 872 873 8:4 875 876 877 878 879 N. 9i38i4 4343 i 48721 54001 59271 6454 6980; 7D06' 8o3o 8555 919073: 9601 ; 920123! 0645! 1166 1686 2206 2725 3244 3762 924279 4796 53i2 5828 6342 6857 7370 7883 8396 8908 929419! 9930 930440 1 0949 1 I458| 1966 2474 2981 3487; 3993 9344981 5oo3 5507 601 1 65i4 7016 7518 8019 8520 9020 939519 940018 o5i6 !0I4 i5ii 2008 25o4 3 000 3495 3989 3867 4396 4925 5453 5980 6507 7033 7558 8o83 8607 9i3o 9653 0176 0697 1218 1738 2258 2777 3296 38i4 433 1 4848 5364 5879 6394 6908 7422 7935 8447 8959 9470 9981 0491 1000 1 509 2017 2524 3o3i 3538 4044 4549 5o54 5558 6061 6564 7066 7568 8069 8570 9070 9569 0068 o566 1064 i56i 2o58 2554 3o49 3544 4o38 2 3 3920 3973 4449 4302 4977 5o3o 55o5 5558 6o33 6o85 6559 7085 6612 7i38 761 1 8i35 7663 8188 8659 8712 9183 9706 0228 0749 1270 1790 23lO 2829 3348 3865 4383 4899 5413 5931 6445 6959 7473 7986 8498 9010 9521 ••32 o542 io5i i56o 2068 2575 3082 3589 4094 4599 5io4 56o8 6111 6614 7117 7618 8119 8620 9120 9619 Olio 0616 i III4 I6II 2107 26o3 3099 3593 4088 4026 4555 5o83 56ii 6i38 6664 7190 7716 8240 8764 9235 I 5287 9758 '9810 0280 I o332 0801 I o853 l322 1842 2362 2881 3399 3917 4434 4951 5467 5982 6497 701 1 7524 8037 8549 9061 9572 ••83 0592 II02 161O 2I18 2626 3i33 3639 4145 4650 5i54 5658 6162 6665 7167 7668 8169 8670 9170 9660 0160 ■ 0666 :i63 660 2157 2653 3148 3643 4137 1374 1894 2414 2933 345i 3969 4486 5oo3 55i8 6o34 6548 7062 7576 8088 8601 9112 9623 •i34 0643 ii53 1661 2169 2677 3i83 3690 4195 4700 52o5 5709 6212 6715 7217 7718 8219 8720 9220 971Q 0218 0716 -2l3 I7I0 2207 2702 3198 36o2 4186 4070 4608 5i36 5664 6191 6717 7243 7^68 8293 8816 9340 9862 o384 0906 1426 1946 2466 2985 3do3 4021 4538 5o54 5570 6o85 6600 7114 7627 *8i4o 8652 9163 9674 •i85 0694 1204 1712 2220 2727 3234 3740 4246 4751 5255 5759 6262 6765 7267 7769 8269 8770 9270 9769 0267 0765 1263 1760 2256 2752 3247 3742 4236 4i32 4660 5189 5716 6243 6770 7295 7820 8345 8869 9392 9914 0436 0958 1478 1998 25i8 3o37 3555 4072 4589 5io6 5621 6137 665i 7165 7678 8703 9215 9725 •236 0745 1254 1763 2271 2778 3285 3791 429b 4801 53o6 5809 63i3 68i5 7317 7819 8320 8820 9320 9819 o3i7 o8i5 i3i3 1809 23o6 2801 3297 4285 4184 4713 5241 5769 6296 6822 7348 7873 8397 8921 9444 9967 9489 lOIO i53o 2o5o 2570 3089 3607 4124 4641 5i57 5673 6188 6702 7216 7730 8242 8754 9266 9776 •287 0796 l303 I8I4 2322 2829 3333 3841 4347 4852 5356 5860 6363 6865 7367 7869 $370 8870 9369 9869 o367 o865 i362 1859 2355 2851 3346 3841 4335 8 D. 4237 I 4290 4766 4819 5294 1 5347 5822 I 5875 6349 I 6401 6875 6927 7400 745i 7925 7978 8450 ! 85o2 8973 I 9026 9496 ••19 o54i 1062 i582 2102 2622 3i4o 3658 4176 4693 5209 5725 6240 6754 7268 7781 8293 88o5 9317 9827 •338 0847 i356 i865 2372 2879 4902 5406 5910 6413 6916 9549 ••71 0593 1114 1634 2i54 2674 3192 3710 4228 4744 5261 5776 6291 68o5 7319 7832 8345 8857 9368 9?79 •389 0898 1407 1913 2423 2930 3386 , 3437 3392 J943 4397 4448 4953 5457 5960 6463 6966 7418 i 7468 7919 I 7969 8420 I 8470 8920 I 8970 9419 9469 9918 0417 0915 1412 1909 24o5 2901 3396 3890 4384 8 9968 0467 0964 1462 1958 2455 2950 3445 3939 4433 53' 53 53 53 53 :>3 53 5a 52 52 5i 5i 5i 5i 5i 5i 5i 5i 5i ^']±} A TABLE OF LOGARITHMS FROM 1 TO 10,000. 15 N. I 3 3 4 5 6 1 7 8 9 49 88o 944483 4532 1 458 I 463 1 4680 1 4729 4779 ! 4828 4877 4927 88i 4976 5o25 1 5o74 5i24 5173 5222 5272 5321 5370 5419 49 882 5469: 55i8 1 5567 56i6 5665 5715 5764 58i3 5862 5912 49 883 5961 6010 1 6o59 6108 6157 6207 6256 63o5 6354 6403 49 884 6452 65oi 655i 6600 6649 6698 6747 6706 6845 6894 49 885 i 6y43 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 m. 7434 7483 i532 8022 75§I 8070 7630 7679 7728 1 7777 7826 7875 49 ' 887 7924 841 3 7973 81 19 8168 8217 [ 8266 ' 83 1 5 1 8364 49 338 8462 85 1 1 856o 8609 8657 8706 8755 i 8804 S853 49 1 889 8902 8951 8999 9048 9097 9146 i 9195 9244 9292 9341 49 890 949390 9439 9488 953w ; j585 9634 9683 9731 9780 9829 49 891 .9?]? 9926 9975 ••24 -73 •121 •170 •219 •267 •3i6 49 892 95o36d 0414 0462 o5ii o56o 0608 0657 0706 0754 o8o3 49 893 : o«5i oqoo 0949 0997 1046 1095 1 143 1192 1240 1289 49 894 1338 1 386 1436 1483 i532 i58o 1629 1677 1726 1775 49 895 1823 1872 1020 1969 2017 2066 2114 2i63 2211 2260 48 896 23o8 2356 24o5 2453 25o2 255o 2599 2647 2606 2744 48 897 2792 2841 2889 3373 2938 2986 3o34 3o83 3i3i 3ioo 3228 48 898 3276 3325 3421 3470 35i8 3566 36i5 3663 37U 48 899 3760 38o8 3856 3905 3953 4001 4049 4098 4146 4194 48 900 954243 4291 4339 4387 4435 4484 4532 458o 4628 4677 5i58 48 901 4725 4773 4821 4869 4918 4966 5oi4 5o62 5iio 48 902 5207 5255 53o3 535i 5399 5447 5495 5543 5592 5640 48 903 5688 5736 5784 5832 588o 5928 5976 6024 6072 6553 6120 48 904 6168 6216 6265 63i3 636i 6409 6457 65o5 6601 48 905 6649 6697 7176 6745 6793 6840 6888 6936 6984 7082 7080 48 906 7128 7224 7272 7320 7368 7416 7464 7012 7559 48 907 8086 7655 7703 775i 7799 7847 7894 7942 8421 ^468 8o38 48 908 8i34 8181 8229 8277 8325 8373 85i6 48 909 8564 8612 8659 8707 8755 88o3 885o 8898 8946 8994 48 910 959041 9089 9137 9185 9232 9280 9328 9375 9423 9471 48 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 912 ,9995 ••42 ••go •i38 *i85 •233 •280 •328 •376 •423 48 913 960471 o5i8 o566 o6i3 0661 0709 0756 0804 o85i 0899 48 914 0946 0994 1 041 1089 ii36 1184 I23l 1279 1753 i326 1374 47 915 1421 1469 i5i6 1 563 1611 1 658 1706 1801 1848 47 916 1895 1943 1990 2o38 2o85 2l32 2180 2227 2275 282-2 47 9n 2369 284i 2417 2464 25ll 2559 2606 2653 2701 2748 2795 47 918 2890 2937 2985 3o32 3079 3i26 3174 3221 3268 47 919 33i6 3363 3410 3457 35o4 3552 3599 3646 3693 3741 47 920 963788 3835 3882 3929 3977 4024 4071 4118 4i65 4212 47 921 4360 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 022 4731 4778 4825 4872 4919 4966 5oi3 5o6i 5io8 5i55 47 923 5202 5249 5296 5343 5390 5437 5484 553 1 5578 5625 47 924 5672 5719 5766 58i3 d86o 5907 5954 6001 6048 6095 47 925 6142 6l8q 6236 6283 6329 6376 6423 6470 65i7 6564 47 926 661I 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 til 7080 7127 7173 7220 7267 73 14 736i 7408 7454 7501 47 7548 7595 8oi6| 8062 7642 8109 7688 7735 8203 7782 8249 7829 8296 7875 8343 7922 795? 47 9?9 81 56 8390 8436 47 980 968483 853o 8576 8623 8670 8716 8763 8810 8856 8903 47 931 89501 8996 9043 1 909c 1 9i36 91 83 9229 9276 9323 9869 47 932 ; 94161 9463 i gSoc 9556 9602 9649 9695 9742 9789 9835 1 47 934 9882 Q70347 9928 0393 9975 0440 ••21 0486 ••68 o533 •ii4 0579 •161 0626 •207 0672 •254 •3oo 0765 ii o35 : 0812 o858 0904 1 0951 0997 1044 1090 1137 ii83 1229 46 936 1276 i322 1 i369 1 i4i5 1461 i5o8 1 554 1601 1647 1693 46 937 1740 1786 1 i832 1879 1925 1971 2018 2064 2110 2x57 46 93^ 2203 2249 22q5 2342 2388 2434 2481 2527 2573 2619 46 939 N. 2666 2712 , 2758 2804 285 1 2897 2943 2989 3o35 3082 46 I 2 1 3 4 5 6 7 8 9 D. 26 16 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 1 I 2 3 4 5 6 7 8 9 "46 940 973128 3174 3220 3266 33i3 3359 34o5 345 1 3497 3543 941 35qo 4o5i 3636 3682 3728 %ii 3820 3866 3oi3 3959 4oo5 46 94a 45?? 4143 4189 4281 4327 4374 1 4420 4466 46 943 45i2 4604 465o 4696 5i56 4742 4788 ; 4834 ' 4880 4926 5386 46 944 4972 5432 5oi8 5o64 5iio 5202 5248 5294 5340 46 945 5478 5524 5570 56i6 5662 5707 5753 5790 5845 46 946 5891 6396 6854 5983 6029 6075 6121 6167 6625 6212 6258 63o4 46 947 948 635o 6442 6488 6533 , 6579 6671 6717 6763 ! 46 6808 6900 7358 6946 6992 i 7037 7083 7129 7175 7220 46 949 7266 7312 74o3 7449 7495 7541 7586 7632 7678 46 95o 977724 8181 7769 7815 8272 7861 7906 7952 8409 7998 8043 8089 8i35 46 95 1 8226 83i7 ! 8363 8454 85oo 8546 8591 46 962 8637 8683 8728 8774 > 8819 8865 891 1 8956 9002 9047 46 953 9093 9i38 9184 9230 9275 9321 9366 9412 9457 95o3 46 954 9548 9594 9639 9685 9730 9776 9821 9867 K^ 9958 46 955 980003 0049 o5o3 0094 0140 oi85 023l 0276 0322 0412 45 956 0458 o549 0594 0640 o685 0730 0776 0821 0867 45 957 958 0912 i366 0957 ioo3 1048 1093 1 139 1 184 1229 1 683 1275 l320 45 141 1 1456 i5oi 1 547 1592 1637 1728 1773 45 959 1819 1864 1909 1954 2000 2045 2090 2i35 2181 2226 45 960 982271 23i6 2362 2407 2452 2497 2543 2588 2633 2678 45 961 2723 2769 2814 2859 2904 2949 2994 3 040 3o85 3i3o 45 962 3175 3220 3265 33io 3356 3401 3446 3491 3536 358i 45 963 3626 3671 3716 3762 3807 3852 3897 3q42 4392 3987 4o32 45 964 4077 4122 4167 4212 4257 43o2 4347 U^l 4482 45 965 4527 4572 4617 4662 4707 4752 4797 4842 mi 4932 5382 45 966 4977 5022 5067 5ll2 5i57 5202 5247 5292 5337 45 967 968 5426 5471 55i6 556i 56o6 565 1 5696 5741 5786 5830 45 5875 5920 6369 5965 6010 6o55 6100 6144 6189 6234 6279 45 969 6324 641 3 6458 65o3 6548 1 6593 6637 6682 6727 45 970 986772 6817 6861 6906 6951 6996 7040 7085 7i3o 7175 45 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 8024 7622 8068 45 972 7666 8ii3 7711 8i57 2756 8202 7800 7845 8291 7890 838 1 797? 8425 45 973 8247 8336 8470 85i4 45 974 8559 8604 8648 8693 8737 8782 8826 ^S7^ 8qi6 936i 8960 45 975 9000 9049 9094 9i38 1 9i83 9227 9272 93i6 94o5 45 976 945o 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 978 9895 99h 0383 9983 ••28 ••72 •117 •161 •206 •25o •294 44 9903^9 0428 0472 o5i6 o56i o6o5 o65o 0694 1137 0738 44 979 0783 0827 0871 0916 0960 1004 1049 1093 1182 44 980 991226 1270 i3i5 1359 i4o3 1448 1492 1 536 i58o 1625 44 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 983 2111 2i56 2200 2244 2288 2333 2377 2421 2465 25o9 44 983 2554 25q8 3oJ9 2642 2686 2730 2774 2819 2863 2?°I 2951 3392 3833 44 984 2995 3o83 3127 3172 32i6 3260 33o4 3348 44 985 3436 3480 3524 3568 36i3 3657 3701 3745 3789 44 986 3877 3921 436i 3965 4009 4o53 4097 4141 4i85 4229 4273 44 ! 981 4317 44o5 4449 4493 4537 458i 4625 4669 47i3 a 988 4757 4801 4845 4889 4q33 4977 5o2I 5o65 5io8 5i5» 44 989 5196 5240 5284 5328 5372 5416 5460 55o4 5547 5591 44 990 995635 5679 5723 5767 58ii 5854 5898 5942 5986 6o3o 44 991 6074 6117 6i6i 62o5 6249 6293 6337 638o 6424 6468 44 992 65i2 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 6949 6993 7037 7080 7124 7168 7212 7255 7209 7343 44 994 7386 7430 7474 7517 7561 7605 7648 8o85 7692 7736 8172 ll]l 44 995 7823 83o3 8347 7954 8390 7908 8434 8041 8129 44 996 8259 8477 8521 8564 8608 8653 44 997 998 86q5 8739 8782 8826 8869 8913 8o56 9392 9000 9043 9087 44 91J1 9174 9218 9261 93o5 9348 9435 9479 991^ 9522 44 999 9565 9609 9652 9696 9739 9783 9826 9870 9957 43 1). N. I 3 3 4 5 6 7 8 9 A TABLE OP LOGAKITHMIC SINES AND TANGENTS FOB EVZBl DEGREE AND MINUTE OF THE QUADEANT. Kemark. The minTites in the left-hand column of eaes page, increasing downwards, belong to the degrees at the top ; and those increasing upwards, in the right-hand column, belong to the degrees below. 18 ^0 DEGREES.; A TABLE OF LOGARITHMIC M. Sine I D. 1 Cosine 1 D. Tang. D. 1 CotAHg. lO-OOOOOO i 0- 000000 i Infinite. 60 I 6-403726 5017-17 ' 2934-85 000000 -00 i 6-468726 1 5017-17 1:8-586274 5o 285244 58 , 3 764756 000000 -00 ! 764756 2984-88 3 940847 2082-81 000000 -00 940847 2082-81 059153 5-7 12-984214 56 4 7-065786 i6i5 IT 000000 • 00 7-065786 i6i5-i7 5 162696 i3i9-68 000000 -00 162696 i3i9.69 837804 j 55 1 758122 ' 54 1 6 241877 iii5-75 9-999999 •01 ! 241878 1110-78 2 308824 : 066 -58 999999 -01 i 308825 096-53 852-54 691175 1 53 i ; 688188 ; 52 ' 3668iG 1 852-54 999999 01 1 866817 9 . 417968 762-68 , 999999 999998 .01 417970 762-68 58208c , 5i i 10 468725 689-88 •01 468727 689-88 586278 5o II 7-5o5ii8 629-81 9.999998 .01 7 '5051 20 629-81 li ■494880 ^9 48 IS 542906 579-86 999997 .01 542909 579-88 407091 i3 577668 1 586-41 999997 .01 577672 536-42 422828 1 47 14 609853 499-38 999996 .01 609857 499-89 890148 , 46 i5 689816 • 467-14 1 999996 .01 689820 467-15 860180 45 i6 667845 488-81 999995 .01 667849 488-82 832i5i U ;j 694173 418-72 999995 • 01 694179 4i3^73 8o582i 43 718997 891-35 999994 •01 719004 891-36 280997 42 19 742477 871-27 353.15 999998 -01 742484 371-28 257O16 41 20 764754 999998 -01 764761 35i^36 235289 40 21 7-785943 806146 836-72 9.999992 •01 7-785951 806 1 55 386 • 73 I ?• 214049 89 ' 22 821.75 999991 .01 821-76 198840 38 23 825451 808 -o3 999970 9999^ 999988 -01 825460 3o8-o6 174540 36 1 24 848984 2o5-47 383-88 •02 848944 295-49 i56o56 25 861662 •02 861674 288-90 188826 35 36 878695 895085 278-17 263-28 999988 -02 878708 278-18 121292 34 u 999987 999986 -02 895099 263-25 104901 33 910879 258-09 •02 910894 9261^4 254-01 089106 82 29 926119 245-38 999985 -02 245-40 078866 3i 3o 940842 287-33 999983 -02 940858 287-35 059142 3o 3i 7-955082 229-80 9-999982 -02 7^955100 229-81 12*044900 20 28 32 968870 222-78 999981 •02 968889 222-75 081111 33 982233 216-08 999980 .02 982253 216-10 017747 27 34 995198 209-81 999979 -02 995219 200-88 208-92 004781 26 35 8-007787 203-90 198.31 999977 999976 -02 8-007809 11-992191 25 36 020021 -02 020045 198-88 979955 24 ll 081Q19 043DOI 198-02 999975 •02 081945 198-05 968055 23 188-01 999978 -02 048027 188-08 956473 22 3o 054781 188.25 999972 -02 054809 188-27 945191 31 40 065776 178.72 999971 .02 o658o6 178-74 934194 20 41 8 -07650c 174-41 9-999969 -02 8-076531 174-44 11-928460 9i8ooi \l 42 086965 170-81 999968 -02 086997 170-84 43 097183 166-89 999966 -02 097217 166-42 902783 892707 \l 44 107167 I 16926 162-65 999964 •o3 107202 162-68 45 , :59-o8 999968 •o3 116968 159-10 888087 i5 46' 1 2647 1 155..66 999961 •o3 i265io 155-68 878490 14 47 i i358io 157.3S 999950 •08 i3585i i52-4i 864149 i3 48 ; 144953 149-24 999958 •o3 144996 149^27 855oo4 12 49 153907 146-22 999956 •o3 158952 146-27 846048 II 56 162681 143.33 999954 •o3 162-727 143-36 887278 10 5i 3.171280 140-54 9-999952 •o3 8-171828 140-57 11-828672 t 52 1 179713 187-86 135.29 999950 • o3 170768 j 188086 187-90 135-32 820287 53 187985 999948 •o3 811964 808844 I 54 196102 182.80 999946 • 03 196156 ' 182-84 55 204070 180-41 999944 •o3 204126 180-44 795874 5 56 2x1895 ) 219581 128-10 999942 •04 211953 128-14 \ 788047 4 57 125.87 999940 •04 219641 125-90 i -780859 772800 3 58 227184 128.72 999988 • 04 22-7195 i 128-76 3 59 234557 24i855 121-64 999986 •04 284621 1 121-68 -765379 I 66 119-68 999984 • 04 241921 [ 119-67 ■758079 Cosine D. Sine 1 Cotang. D. i Tang. M. (89 DEORSEB.) SINES AND TANGENTS. (1 DEGREE.^ u 8iDe D Cosino D. Tang. D Cotnng. 8-241855 119 -63 9.999934 -04 8-241921 119 -67 11-758079 60 I 249033 "I ii5 • 68 999932 -04 249102 117 -72 -84 750898 U a 256094 .80 999929 -04 256i65 ii5 743835 3 263042 ii3 .98 999927 -04 263ii5 114 -02 736885 57 4 269881 112 •21 999925 -04 269956 112 -25 i 730044 j 56 5 276614 110 • 50 999922 -04 276691 110 •54 1 723309 55 6 283243 108 • 83 999920 -04 283323 108 •87 716677 54 I 289773 107 • 21 999918 -04 289856 107 io5 -26 710144 1 53 296207 io5 65 999915 -04 296292 -70 703708 52 <; 302546 104 i3 999913 04 302634 104 -18 697366 5i 10 308794 102 66 999910 04 308884 102 -70 691 I 16 5o II 8.314904 lOI 22 9.999907 999905 04 8-3i5o46 101 .26 11-684954 ^9 12 321027 99 82 04 32II22 99 .87 678878 672886 48 i3 327016 98 47 999002 04 327II4 98 .5i 47 14 332924 95 14 999899 o5 333o25 97 .19 666975 46 i5 338753 86 999897 o5 338856 95 90 661 144 45 i6 344504 94 60 999894 o5 344610 94 65 655390 44 \l 35oi8i 93 38 999801 o5 350289 93 43 6497 1 1 43 355783 92 i? 999888 o5 355895 92 24 6441 o5 42 19 36i3i5 91 999885 o5 36i43o 91 08 638570 41 ao 366777 89 90 999882 o5 366895 89 95 633 I o5 40 ai 8.372171 88 80 9.999879 o5 8-372292 88 85 11-627708 39 aa 377499 ll 72 999876 o5 377622 87 77 622378 38 23 382762 67 999873 o5 382889 86 72 617111 ll 24 387962 85 64 999870 o5 3880Q2 85 70 61 1908 25 393101 84 64 999867 o5 393234 84 70 606766 35 36 398179 83 66 999864 o5 398315 83 71 601685 34 11 4o3i99 82 71 999861 o5 4o3338 82 t 596662 33 408161 81 ll 999858 o5 4o83o4 81 591696 32 29 4i3o68 80 999854 o5 4i32i3 80 91 586787 3i 3o 417919 79 96 999851 06 418068 80 02 581932 3o 3i 8-422717 72 09 9-999848 06 8-422860 79 14 ii-577i3i 29 32 427462 78 23 999844 06 427618 78. 3o 572382 28 33 432 1 56 ]l 40 999841 06 4323i5 77- 45 567685 ll 34 436800 57 999838 06 436962 76 63 563o38 35 441394 75 77 999834 06 441 56o 75 83 558440 25 36 445941 74 99 999831 06 446110 75. o5 5538Q0 24 3? 450440 74 22 999827 06 45o6i3 74- 28 549387 23 36 454893 73 46 999823 06 455070 73- 52 544930 540619 aa 39 459301 463665 72 73 999820 06 459481 463849 72- 79 21 40 72 00 999816 06 72- 06 536i5i 20 41 8-467985 71 29 9-999812 • 06 8-468172 71- 35 11-531828 ;? 42 472263 70 60 999809 06 472454 70- 66 52-7546 43 476498 69 91 999803 06 476693 69- 98 523307 \i 44 480693 69 24 999801 06 480892 69- 3i 519108 45 484848 68 59 999797 • 07 485o5o 68- 65 5i4o5o i5 46 488963 67 94 999793 • 07 489170 68- 01 5io83o 14 s 493040 67 3i 999700 - 07 493250 67- 38 506750 i3 497078 5oio8o 66 60 999786 ■ 07 497293 66- ii 502707 12 49 66 08 999782 - 07 501298 66- i5 498702 II 5o 5o5o45 65 48 999778 . 07 505267 65- 55 494733 10 5i 8 -508074 64 89 9.999774 - 07 8-50Q200 64- 39 11-490800 § 52 512067 64 3i 999769 - 07 51^098 64- 486902 53 516726 63 75 999765 - 07 516961 63- 82 483o39 I 54 52o55r 63 19 999761 - 07 520790 63- 26 479210 475414 55 524343 62 64 999757 • 999753 - 07 524586 62- 72 5 56 528102 62 II 07 528349 62. 18 47i65i 4 5^ 531828 61 58 999748 - 07 532080 61- 65 467920 3 535523 61 06 999744 07 535779 61- i3 464221 2 59 539186 60 55 999740 • 07 539447 543084 60- 62 460553 1 60 542819 60 -04 999735 07 60-12 456916 Cosine D. Sind 1 Cotang. D. Tang 1 6 (83 D SOB J:B8.) 20 (2 DEGREES.) A TABLE OF LOGARITHMIC M. Bino D. Cosino D. Tang. D. OotAPg. 1 _ 1 o 8.542819 60.04 9.999735 -07 8.548084 60-12 11 -456916 1 60 I 546422 59.55 999781 .07 546691 59 •62 453309 , 59 a 549995 553539 59-06 999726 .07 550268 59 •14 449782 58 3 58-58 999722 .08 558817 58 •66 446188 57 4 557054 58.11 999717 .08 557886 58 .19 442664 56 5 56o54o 57.65 999718 .08 560828 57 •73 439172 55 6 563999 57.19 999708 .08 564291 57 •27 4357OQ 482278 54 I 567431 56-74 999704 .08 567727 56 .82 53 570886 56.80 999699 .08 571187 56 •38 428863 52 9 574214 55.87 999604 .08 574520 55 •95 425480 5i 10 577566 55.44 999689 .08 577877 55 •52 422123 5o II 8.580892 55.02 9.999685 .08 8-581208 55 10 II -418702 49 12 584193 54.60 999680 -08 5845i4 54 68 415486 48 i3 587469 54.19 999675 -08 587795 54 27 4l2205 ii 14 590721 53.79 53.39 999670 -08 591031 53 .87 408949 i5 598948 999665 -08 594288 53 47 405717 402 5o8 45 i6 597152 53.00 999660 -08 597492 53 08 44 \l 600882 52.61 999655 -08 600677 52 70 899328 43 608489 606628 52-28 999650 -08 608880 52 32 896161 42 19 51.86 999645 -09 606978 5i 94 893022 41 20 609784 5i 49 999640 .09 610094 5i 58 389906 40 31 3.612828 51-12 9.999635 -09 8-618189 5i 21 11-386811 89 32 615891 618987 50.76 999629 -09 616262 5o 85 888788 38 33 50.41 999624 -09 619818 5o 5o 880687 3? 24 621962 5o.o6 999619 -09 622843 5o i5 377657 874648 25 624965 49-72 999614 -09 625352 49 81 35 26 627948 49.88 999608 -09 628840 49 47 871660 34 11 680911 633854 49-04 999608 -09 681808 49 i3 368692 33 1 48.71 999597 -09 634256 48 80 365744 862816 32 1 ?9 686776 689600 48-89 999592 .09 687184 48 48 3i 3o 48.06 999586 .09 640098 48 16 359907 3o 3i 8.642563 47-75 9.999581 .09 8-642982 47 84 11-357018 It 32 645428 47-43 999575 -09 645858 47 53 354147 351296 33 648274 47-12 999570 -09 648704 47 22 11 34 65iio2 46-82 999564 .09 65i587 46 91 848468 35 6539 II 46-52 999558 '10 654852 46 61 345648 25 36 656702 46-22 999553 •10 657149 46 3i 342851 24 ^7 3S 659475 662230 45-92 999547 -10 659928 46 02 340072 23 45-63 999541 -10 662689 45 73 387811 22 39 664968 45-35 999535 • 10 665483 45 44 384567 21 40 667689 45.06 999529 • 10 668160 45 26 881840 20 41 8.670398 44.79 9.999524 -10 8-670870 44 88 11-829180 19 42 678080 44-51 999518 -10 673563 44 61 826487 18 43 675751 44-24 999512 -10 676289 44 34 828761 \l 44 678405 681043 43-97 999506 .10 678900 44 17 821100 45 43.70 999500 .10 681544 43 80 3 I 8456 i5 46 688665 43.44 999498 .10 684172 686784 43 54 ; i5828 14 I 47 ; 48 686272 43.18 999487 .10 43 28 ii32i6 i3 j 688868 42.92 999481 •10 689881 43 o3 810619 13 49 691488 42.67 999475 .10 691968 42 77 808087 II 5o 698998 42.42 999469 -10 694529 42 52 805471 ro 5i 8-696548 42-17 9.999463 -11 8-697081 42 28 11 802919 9 ! 5i 699078 701589 41-92 999456 .11 699617 42- o3 3oo383 8 53 41.68 999450 .11 702189 41- It 297861 1 295354 54 704090 41.44 999443 j • II 704646 41- 55 706577 41.21 999437 • 11 707140 41- 32 292860 5 56 709049 40.97 999481 • II 709618 4i- 08 200882 4 u 71 1 507 718952 716383 40-74 40. 5i 999424 999418 • II • II 712088 714534 40- 40- 85 62 287917 285465 3 3 59 40.29 999411 • II 716072 719896 40- 40 288028 I 6o 718800 40'06 999404 • II 40-17 280604 CoBino D. Sine Cotang. D. Tang. M.] (87 p SGR ESS.) SINES AND TANGENTS (3 DEGREES., 21 M. Sine D. Cosine D. Tang. D. Cotang. 60 8.718800 4o«o6 9.999404 II 8.719896 40-17 1 1 - 280604 I 721204 39 84 999898 •II 721806 89-95 278194 ^ 3 723595 39 6a 999891 11 724204 89 •74 89.52 275796 3 730688 39 41 999884 • 11 726588 278412 U 4 89 19 999878 •II 728959 89-80 27 1 041 5 88 98 999871 •11 781817 89-09 268683 55 6 733027 88 77 999864 •12 783663 38-89 266337 54 735354 88 u 999357 ; •12 735996 38-68 264004 53 737667 88 999830 '12 70.8817 38.48 261688 i 5a 1 9 739969 38 16 999843 12 740626 38-27 259874 5i 10 742259 37 96 999886 12 742922 38-07 257078 5o II 8-744536 37 76 9.999829 12 8.745207 37-87 1 1 - 254793 it la 746802 37 56 999822 •12 747479 87-68 252521 i3 749055 37 37 999815 12 749740 37-49 25o26o 47 14 751297 753528 37 17 98 999808 •12 751989 87-29 248011 46 i5 36 999801 12 754227 87-10 245778 45 i6 755747 36 79 999294 12 756453 86-92 243547 44 \l 757955 36 61 999286 12 758668 86.73 241882 43 76oi5i 86 42 999279 12 760872 36-55 289128 42 19 762337 36 24 999272 12 768065 36.36 286935 41 20 76451 I 36 06 999265 12 765246 36.18 284754 40 21 8.766675 35 88 9-999257 12 8-767417 769578 36-00 11.232588 89 22 768828 35 70 999250 l3 35.88 280422 88 23 770970 35 53 999242 18 771727 35-65 228278 226134 ^I 24 773101 85 35 999235 l3 778866 35-48 36 25 775223 85 18 999227 18 775995 35-3i 224oo5 35 26 777333 35 01 999220 18 778114 780222 35-14 221886 34 11 779434 34 84 999212 18 34-97 219778 33 78i524 34 67 999205 18 782820 84.80 217680 32 V^ 7836o5 34 5i 999197 •18 784408 34.64 215592 3i 3o 785675 34 3i 999189 18 786486 34-47 2i85i4 3o 3i 8.787736 34 18 9-999181 18 8.788554 34.31 I I- 21 1446 29 32 789187 791828 34 02 999174 i3 790618 84-15 209887 28 33 33 86 999 I 66 i3 792662 33.99 207888 27 34 793359 33 70 999 1 58 18 794701 38.88 205299 26 35 795881 33 54 999 I 5o i3 796781 83.68 208260 25 36 797894 88 8q 999 I 42 i3 798752 800768 33.52 201248 24 u 799897 801892 88 26 999184 i3 33.87 199287 23 38 08 999 1 26 i3 ^2765 33.22 197285 22 39 808876 82 93 099118 i3 804758 33.07 195242 21 40 8o5852 32 78 9991 10 i3 806742 82.92 198258 20 41 8-807819 32 63 9.999102 i3 8-808717 82-78 11-191283 19 42 809777 32 49 999094 14 8io683 82.62 189817 18 43 811726 32 34 999006 14 812641 82.48 187359 n 44 818667 82 19 o5 999077 14 814589 32-33 1 8541 1 16 j 45 815599 82 999069 14 816529 32-10 188471 i5 ' 46 817522 3i 91 999061 14 818461 32-o5 181539 179616 14 J? 819486 3i 77 999053 14 820884 81-91 i3 821843 3i 63 999044 14 822298 81-77 3i-63 177702 175795 173891 12 49 828240 3i ii 999086 14 824205 II 5o 825i3o 3i 999027 14 826108 3i-5o 10 5. 8-827011 3i 22 9.999019 14 8-827992 3i-36 11-172008 ? 52 828884 3i 08 999010 14 829874 31-23 170126 53 880749 80 95 999002 14 831748 3i-io 168252 I 54 882607 3o 82 998998 14 888618 80-96 166887 55 884456 3o 69 998984 14 885471 30-83 164529 5 56 886297 888180 3o 56 998976 14 887821 30-70 30-5-7 3o-45 J162679 160887 4 S 3o 43 998967 i5 889168 3 889956 3o 80 998958 i5 ^840998 842825 159002 a ^ 841774 843585 3o 17 998950 i5 3o.3a 157175 I 60 3o-oo 998941 i5 844644 30-19 155356 Cosine D. Sine Cotang. 1 D. Tang. M. (86 DEGRESS.) 22 (4 DEGREES.) A TABLE OF LOaARITHMIC M. Sine D. Coeino D. Tang. B. Colang. bo 8.843585 3o.o5 9-998941 • 15 8-844644 80.19 11- 155356 I 845387 847183 848971 85o75i 8525a5 29.92 998982 •i5 846455 80.07 153545 5o 2 29.80 998923 .i5 848260 1 29.05 i5i74o 56 3 29-67 998914 .15 85oo57 29.82 149943 1 148154 U 4 29.55 998905 .15 85 1 846 29.70 1 5 29.43 998896 -15 853628 29-58 146872 55 6 854291 29.81 998887 .15 855403 29.46 144597 54 I 856049 29.19 998878 .15 857171 29-35 142829 53 857801 29.07 998869 .15 858982 29-28 141068 52 9 859546 28.96 998860 • 15 860686 29-11 189814 5i 10 861283 28.84 998851 -15 862488 29-00 187567 5o II 8-863oi4 28.78 9-998841 .i5 8-864178 28-88 11-135827 t 12 864738 28.61 098882 .i5 865906 28-77 184094 i3 866455 28.50 998828 .16 867682 28-66 182868 47 14 868 I 65 28.80 998818 .16 869351 28.54 ! 180649 46 i5 869868 28-28 998804 .16 871064 28.48 \ 128986 45 i6 871565 28-17 998705 ■16 872770 28.82 , 127280 44 \l 873255 28.06 998785 .16 874469 28.21 I2553i 43 874938 27.05 998776 .16 876162 28.11 128888 42 19 876615 27.86 998766 .16 877849 28.00 122l5l 41 20 878285 27.78 998757 .16 879529 27.89 1:0471 40 21 8.879949 881607 883258 27.68 9.998747 .16 8-881202 mt 11- I 18708 117181 110470 39 38 22 27.52 998788 .16 882869 23 27.42 998728 .16 884580 27.58 U 24 884903 27.81 998718 -16 886 I 85 27.47 ii88i5 25 886542 27.21 998708 -16 887888 27.87 112167 35 26 888174 27.11 998699 .16 889476 27.27 iio524 34 11 889801 27.00 998609 -16 891 1 12 27.17 108888 33 891421 26.90 998679 •16 892742 27.07 107258 32 29 898035 26.80 998669 .17 894866 26.97 io5684 3i 3o 894643 26.70 998659 •17 895984 26.87 104016 3o 3i 8-896246 -26.60 9-998649 •17 8-897596 26.77 11.102404 20 28 32 807842 26.51 998689 .17 899208 26-67 100797 33 899432 26.41 998629 •17 900808 26-58 099197 U 34 901017 26.31 998619 .17 902898 26.48 097602 35 902596 26.22 998609 •n 908987 26-38 096018 25 36 904169 26.12 998599 •17 905670 26-29 094480 24 ^1 905786 26-08 998589 •17 907147 26-20 092853 23 38 907297 25.98 998578 •17 908719 26-10 091 281 22 39 908853 25.84 998568 •17 910285 26-01 089715 21 40 910404 25.75 998558 .17 91 1846 25-92 088 1 54 20 41 8.911940 25.66 9.998548 •n 8-918401 25-88 T» .086599 19 18 42 913488 25-56 998537 .17 914951 25-74 o85o49 43 9l5022 25.47 998527 .17 916495 25-65 o835o5 \l 44 9i655o 25.38 ' 998516 .18 918084 25-56 081966 45 918073 25.29 998006 .18 919568 25-47 080482 i5 46 919591 25.20 998495 .18 921096 25-88 078904 14 % 921103 25.12 ! 998485 .18 922619 25.80 077381 . 1-3 1 923610 25 o3 998474 .18 924186 25.21 *5864 12 <9 9«4ii2 24-94 1 998464 • 18 925649 25-12 074851 II 5o 925609 24-86 i 998453 .18 927156 25 -o3 072844 10 5i 8.927100 24-77 9-998442 .18 8-928658 24-95 II 071842 ? 52 92858T 980068 24-69 998481 .18 980155 24-86 069845 53 24-60 1 998421 .18 981647 24-78 068853 I 54 981544 24-52 ! 998410 .18 988184 24-70 066866 55 988015 24-43 1 998899 .18 984616 24-61 065884 5 56 984481 24-35 i 998888 .18 086098 24-53 068907 062435 4 ll 935942 24-27 1 998877 -18 987565 24-45 3 987898 938850 24-19 998866 -18 989082 24-87 060968 059606 3 59 24-11 998855 .18 940494 941902 24-80 I 60 940296 24-o3 998844 .18 24-21 058048 1 Cosine D. 1 Sine Cotang. D. Tang. M. (85 DEOR EES.) SIXES AND TANGENTS. (5 DEGRBB., t 22 M Bind D. Cosine D. Tang. D. Cotang. ! 8 043 JOv*) 24 -03 9.998344 .19 8-941962 24-21 M- 068048 60 I 23 .04 998333 .19 948404 24- 13 066696 u 2 943174 23 -07 998822 .19 944852 24 -06 066148 3 944606 28 •79 9988 1 1 .19 946295 23^97 068705 u 4 946084 28 •71 99880c -19 947734 23^90 062266 5 947456 28 -68 998289 .19 949168 23.8s 060882 55 6 Q48874 23 -55 998277 .19 960397 23.74 049403 54 ! 7 960287 28 -48 998266 .19 962021 I 23-66 047979 53 ; 8 951696 23 -40 998255 .19 968441 1 28-60 046669 52 9 96J100 23 -32 998243 .19 964866 28-5i 046144 5i 10 954499 23 -25 998282 .19 966267 23-44 048733 5o II 8.955894 23 I? 9 998220 •19 8-967674 23-37 11-042826 49 n 957284 958670 960052 23 •10 998209 19 969075 ! 23-29 23-23 040926 48 13 23 02 998107 .19 960478 089527 S 14 22 95 998186 .19 961866 i 23.14 o38i34 15 961429 22 88 998174 .19 968266 23.07 086746 45 16 962801 22 80 998168 .19 964689 23-00 o3636i 44 \l 964170 22 73 998151 .19 966019 22. o3 088981 43 965534 22 66 998189 • 20 967894 22.86 082606 42 19 966893 22 59 998128 •20 968766 22.79 081284 41 20 968249 22 02 998116 .20 970188 22.71 029867 40 21 8-969600 22 44 9-998104 •20 8-971496 22.65 11-028604 i% 22 970947 22 88 998092 •20 972836 22-67 027145 38 23 972289 22 3i 998080 •20 974209 22-51 026791 il 24 978628 22 24 998068 •20 976660 22.44 024440 36 35 974962 22 n 998056 .20 976906 22.37 028094 021762 35 26 976298 22 10 998044 .20 978248 22-3o 34 11 977619 22 o3 998082 •20 979686 22-23 020414 33 978941 21 97 998020 .20 980921 22-17 019079 32 29 980259 981578 21 90 998008 .20 982261 22-10 017749 016428 3i 3o 21 83 997996 •20 988677 22-04 3o 3i 8-982888 21 77 9.997985 .20 8-984899 21.97 ii-oi5ioi ^2 32 984189 21 *?? 997972 .20 980217 21.91 018788 28 33 985491 21 63 997959 .20 987682 21.84 012468 u 34 9^^789 21 ^7 997947 .20 988842 21.78 011168 35 988088 21 5o 997935 •21 990149 21-71 009861 35 36 989374 21 44 997922 •21 991461 21-65 008649 24 1 990660 21 38 997010 .21 992760 21-58 . 007260 23 991943 21 3i 997897 .21 994046 21-62 006965 22 39 998222 21 25 997885 .21 996887 21.46 004663 21 4o 994497 21 19 997872 .21 996624 21-40 008876 20 41 8-995768 21 12 9-997860 .21 8.997908 21.34 11-002092 It 42 997086 21 06 997847 997886 •21 999188 21-27 000812 43 998299 21 00 •21 9.000466 21-21 10-999635 998262 ■2 444 999560 20- q4 997822 •21 001788 21-l6 45 9'OOo8i6 20- 87 997809 •21 008007 21-09 996998 i5 46 002069 20 82 997797 •21 004272 21 -o3 996728 U 47 008818 20- 76 997784 • 21 oo5534 20-97 994466 i3 48 004563 20- 70 997771 •21 006792 20-91 998208 12 49 oo58o5 20- 64 997758 •21 008047 009298 20-85 991968 II 5o 007044 20- 58 997745 •21 20-80 990702 10 5i 9-008278 20- 52 9-997782 •21 9-010646 20-74 10-989454 I 52 009510 20- 46 997719 •21 011790 018081 20-68 988210 53 010787 20- 4e 997706 •21 20-62 986969 I 54 01196a 20- 34 997698 .22 014268 20-56 98678-2 55 018182 j 20- 29 997680 .22 oi56o2 20-5l 984498 5 56 014400 20- 23 997667 .22 016782 20-45 988268 4 u oi56i3 1 20- 17 997654 .22 017960 019188 20-40 982041 3 016824 20- 12 997641 .22 20-33 980S17 2 59 018081 20- 06 997628 .22 020408 20-28 978380 I ^ 019335 20-00 997614 .22 021620 20-23 Cofcine D. Sine Cotang. D. Tang^ M. ^84 DEGR SES.'^ u yfi DEGREES.) A TABLE OF LOGARITHmC M Sine D. Coeine D. Tang. D. Gotang. 60 9.019235 30.00 9.997614 .22 9.021620 20.23 10.978880 I 020435 I9-q5 997601 • 22 022884 20 •n 977166 5o 976966 58 a 021682 19.89 997588 .22 024044 20 .11 3 022825 19.84 997574 .22 02525l io 06 974749 57 978645 1 56 4 024016 19-78 997661 .22 026455 20 •00 5 0252o3 19-78 997547 .22 027655 028852 19 •95 972845 55 6 026386 19-67 997534 .28 19 -90 971148 54 I 027567 19-62 997620 •23 080046 19 -85 969964 53 028744 19.57 997507 .28 081287 082420 19 •79 968768 967676 5a 9 029918 i9-5i 997493 .28 19 •74 5i IC 081089 19.47 997480 .28 088609 19 69 966891 5o II 9 -032257 19-41 9.997466 -28 9-034791 19 -64 10.966209 S 12 o3342i 19.36 997452 -23 035969 »9 -58 964081 i3 034582 19.80 997489 -23 087144 19 •58 962866 47 14 035741 086896 19.25 997425 -23 o388i6 19 48 961684 46 i5 19.20 99741 1 -28 089485 19 .43 960616 45 i6 088048 19.15 997897 .28 04065 I 19 • 88 960849 44 n 039197 19.10 997888 .28 041818 19 ■33 968187 43 i8 040842 1Q.05 997869 .28 042978 19 •28 967027 42 19 041485 18.99 997355 .28 o44i3o 19 23 966870 41 20 042625 18.94 997841 .28 045284 19 18 964716 40 21 9-043762 044895 18.89 9.997827 997318 •24 9.046484 19 18 10.968666 U 22 18.84 •24 047582 19 08 9624x8 23 046026 18.79 18.75 697209 •24 048727 19 08 961278 9501J1 ll 24 047154 997285 •24 049860 18 98 25 048279 18.70 997271 •24 o5ioo8 18 08 948992 35 26 049400 i8.65 997257 -24 o52i44 18 89 947866 34 11 o5o5i9 o5i635 18.60 997242 •24 053277 18 84 946728 33 18.55 997228 •24 054407 18 79 945598 3a 29 052749 053859 18. 5o 997214 •24 055585 18 74 944466 3i 3o 18.45 997199 -24 o56659 18 70 943341 3o 3i 9-054966 18.41 9.997185 •24 9-057781 18 65 10.942219 29 28 32 o56o7i 18.36 997170 •24 068900 18 69 941 100 33 057172 0582TI J059367 18.31 997156 •24 060016 18 55 980984 27 34 18.27 997141 •24 061180 18 5i 988870 26 35 18.22 997127 •24 062240 18 46 987760 25 36 060460 I8.I7 18. i3 997112 •24 063348 18 42 9366fj 24 ll o6i55i 997008 •24 064453 18 37 986647 28 062689 18.08 997088 .25 065556 18 3i 984444 22 39 068724 064806 18.04 997068 -25 066655 18 28 933345 21 40 17.99 997053 .25 067762 18 24 982248 20 41 9-065885 17-94 9.997089 •25 9.068846 18 19 10-981154 19 42 066962 17.00 997024 .25 069988 18 i5 980062 18 43 068086 17.86 997009 .25 071027 18 10 928978 927887 926808 »7 44 069107 17-81 996994 .25 072118 i8- 06 16 45 070176 n-ii 996979 .25 078197 18. 05 i5 46 071242 17.72 996964 •25 074278 17 P 926722 14 % 072806 17.68 996949 •25 1)75356 17 924644 i3 078866 17.68 ^96934 •25 076482 n- b 928668 12 49 074424 17.59 996919 .25 o775o5 078576 n- 84 922496 11 5o 075480 ^7.55 996904 •25 17. 80 921424 13 1 5i 9.076533 17.50 9.996889 .25 9.079644 n- 76 10 '920356 § 52 077588 078681 17-46 996874 .25 080710 n- 72 919290 53 17-42 996858 .25 081778 082888 n- 63 918227 I 54 079676 080719 17-88 996848 .25 n- 917167 55 17-38 996828 .25 088891 17- 59 916100 916063 5 56 081759 17.29 17.25 996812 .26 084947 17 55 4 U 088882 996707 .26 086000 17 5i 914000 3 17.21 996782 .26 087050 17 tl 912960 a 59 084864 17.17 17 i3 996766 .26 088098 n- 911002 910066 I 66 085894 996751 .26 089144 17- 38 Coeine __D. Sine Cotan^. ~'d. Tanfir. (83 DSORBES.) 1 SINES AND TANGENTS. (7 DEGREES.) 2 Sine D. CoBuie D. Tang. D. Cotang. 9-085894 i7-i3 9-996751 .26 Q 089144 17.38 10-910856 60 I J 086922 087947 1 088970 17.09 17-04 996735 996720 • 26 -26 090187 O9122S 17-34 17-80 909813 908772 u 3 I7-00 16-96 996704 -26 092266 17.27 907784 90669b u 4 089990 996688 • 26 098802 17.22 5 091008 16.92 996678 -26 094886 1719 17.15 9o5664 55 6 1 092024 1 16.88 996657 -26 095867 904688 54 I 098087 1 16.84 996641 -26 096895 17. 11 908605 53 094047 16.80 996625 .26 097422 17.07 17.08 902578 901554 52 Q 095o56 16.76 996610 • 26 098446 5i 10 096062 16.73 996594 .26 099468 16.99 900532 5o II 9-097065 16.68 9-996578 -27 9-100487 16-95 10-899518 S 12 098066 16-65 996562 .27 ioi5o4 16-91 898496 i3 099065 16-61 996546 .27 io25i9 16.67 897481 47 14 100062 16-57 16.53 996580 .27 io8532 16-84 896468 46 i5 ioio56 996514 .27 104542 16-80 895458 45 i6 102048 16-49 996498 .27 io555o 16-76 894450 44 \l 108087 104023 i6-45 996482 .27 io6556 16-72 898444 43 16-41 996465 .27 107559 io856o 16-69 892441 42 »9 loSoio 16-88 996449 .27 16-65 891440 4i 30 105992 16-84 996488 .27 109559 16-61 890441 40 at 9-106978 107901 i6-3o 9-9964x7 .27 9-iio556 16-58 10-889444 It 22 16-27 996400 •27 iii55i 16-54 888449 23 108927 16-23 996884 .27 1 1 2543 i6-5o 887457 u 24 I 0990 I 16-19 996868 •27 I 13588 16-46 886467 25 110873 16-16 996351 .27 I 14521 16-43 885479 884493 85 26 II 1842 16-12 996885 .27 11 5507 16-89 84 11 1 1 2809 16-08 996818 •27 116491 i6-36 888509 33 118774 i6-o5 996802 .28 I 17472 118452 16-82 882528 32 29 114787 II 5698 16-01 996285 .28 16-29 881548 3i So 15-97 996269 .28 I 19429 16-25 880571 3o 3i 9-ii6656 15.94 9-996252 • 28 9 - 1 20404 16-22 10-879596 It 32 117613 15.90 996285 .28 121877 16-18 878623 33 I 18567 15.87 996219 .28 122848 16. i5 877652 27 34 119519 i5.88 996202 .28 128817 16-11 876688 26 35 120469 i5.8o 996185 .28 124284 16-07 875716 25 36 121417 15.76 996168 .28 125249 16-04 874751 24 ll 122862 15.73 996151 .28 126211 16-01 878789 23 128806 15-69 996184 .28 127172 15.97 872828 22 39 124248 i5.66 9961 17 -28 128180 15.94 871870 21 40 125187 15.62 996100 -28 129087 15-91 8-70913 20 4i 9-126125 15.59 9-996088 -29 9.180041 15-87 10-869959 '.t 42 1 27060 15.56 996066 -29 180994 15-84 869006 43 127998 i5.52 996049 -29 181944 i5-8i 868o56 \l 44 128525 15.49 996082 -29 182893 i5-77 867107 45 129854 15.45 996015 .29 133889 15-74 866161 i5 46 180781 15.42 995908 •29 184784 15.71 15.67 865216 14 s 181706 182680 i5-39 15.35 995980 •29 185726 864274 863338 i3 993968 .29 186667 i5-64 la 49 i8355i 1532 995946 .29 187605 188542 i5.6i 862895 S61458 II 5o 134470 i5'29 995928 .29 i5.58 10 5i 9.135387 i5-25 9-995911 .29 9-189476 15.55 io.86o524 t 52 186808 15-22 995894 -29 140409 i5.5i 859591 858660 53 187216 188128 15.19 995876 •29 I 41 840 15.48 I 54 i5.i6 995359 .29 142269 15.45 857781 8568o4 55 189087 l5-12 995841 .29 148196 15.42 5 56 139044 i4o85o i5-09 995828 .29 144121 15.39 i5-35 855879 4 U i5-o6 995806 -29 145044 854956 3 141754 142655 i5-o3 995788 -29 145966 i5-32 854034 a 55 15-00 995771 -29 146885 15-29 853ii5 I 60 143555 14.96 995753 .29 147803 i5-26 852197 M. Gofiine D. 1 Sine Ootan^. D. Tiui^. (82 DBGB EBB.) 86 kS DEGREES.) A fABLE OF LOGAlilTHMlC M. Sine D. Cosine D. Tang. ! ^^ Cot&Dg. r"^ o 9-143555 14-96 9-995753 •3o 9 -147503 15^26 :o-853i97 60 1 144453 14-98 990735 -3o 148718 15-28 831282 59 85o368 58 a i4534<) 14-90 993717 -3o 149682 l5-20 3 146243 14-87 990699 -3o i3o544 if-17 849436 57 4 I47i36 14-84 995681 • 3o i5i454 i5-i4 848546 56 5 148026 i4-8i 995664 -3o 152363 i5-ii 847687 55 6 14801 5 149002 14-78 995646 -3o 153269 i5.o8 846781 54 I 14-75 995628 -3o 154174 i5.o5 845826 53 i5o686 14-72 995610 -3o 1 55077 l5-02 844928 52 9 r5i569 14-69 993591 -3o 155978 14-99 844022 5i 10 1 5245 1 14-66 995573 •3o 156877 14-96 848128 5o i> 9' I J33o 14-63 9 •995555 .30 9-157775 14-93 10-842225 49 12 1 54208 14-60 995537 .30 108671 U^oo 14-87 841329 48 i3 i55o83 14-57 995519 .80 159565 840435 47 U i55o57 I 56830 14-54 995301 .81 160457 14.84 889548 888653 46 i5 i4-5i 995482 -81 161847 14-81 45 i6 157700 I 58569 14-48 995464 •81 162286 14.79 887764 44 \l 14-45 993446 •31 168128 14.76 886877 43 i5943d 14-42 993427 .81 164008 14-73 835992 42 «9 i6o3oi 14-39 993409 •3i 164892 14.70 835 108 41 20 161 164 14-36 993890 ■3i 165774 14.67 884226 40 21 9' 162025 14-38 9-995372 • 31 9- 166634 14-64 10-833346 39 38 22 162885 i4'8o 995358 .81 167532 i4-6i 882468 S3 168743 14-27 995334 • 3i 168409 14-58 881591 J7 24 164600 14-24 993316 • 31 169284 I4-35 880716 36 25 165454 14-22 993297 -31 170157 14-53 829843 35 26 i663o7 14-19 993278 • 31 171029 i4-5o 828971 34 11 i67i59 14- 16 995260 'i' 171809 172767 14-47 828101 83 168008 i4-i8 995241 .32 14-44 827288 82 29 168856 14-10 995222 •32 178684 14-42 826866 81 3o 169702 14-07 995208 .82 174499 14-89 825501 3o 3i 9-170547 i4-o5 9-995184 .82 9-175362 14-36 10-824688 ^ 33 171889 14-02 9951 65 .82 176224 14-33 828776 33 172280 18-99 995146 .82 177084 i4-3i 822916 27 34 178070 13-96 995127 .82 177942 14-28 822058 26 35 178908 i3-94 995108 .82 178799 14-25 821201 25 36 174744 18-91 995089 •32 179655 14-28 820845 24 U 175578 13-88 993070 .32 i8o5o8 14-20 819492 818640 28 176411 13-86 99305 I .82 i8i36o 14-17 22 39 177242 12-88 Q93032 82 182211 14.15 817789 21 4o 178072 i3-8o 995oi3 .32 188009 14.12 816941 20 41 9-178900 18-77 9-994993 .82 9-188907 14.09 10-816098 ;? 43 179726 18.74 994974 .82 184752 14-07 815248 43 i8o55i 18.72 ! 994935 •32 185597 14-04 814408 n 44 1 18187.' 18.69 994935 •32 186439 14-02 8i356i 16 45! 182196 13-66 994916 .33 187280 18-99 812720 i5 46 , i83oi6 i3-64 994896 •33 188120 18-96 811880 14 8 183884 18-61 i 994877 ■88 188958 18-98 811042 i3 i3465i 18-5^ , 994857 -83 189794 18-91 18-89 810206 " ; 49 185466 i3.5b ; 994838 .83 i 190629 800871 ; 11 te 186280 i3-:>3 994818 •33 191462 18-86 8o6588 10 5i 9- 10-092 i3.5i 9-994798 •88 9-192294 18-84 10-807706 t 5> 187908 13.48 1 9947-^9 .33 198124 :3-8i 806876 53 188712 1 18-46 994759 .88 198953 18-79 806047 I 54 i8o5i9 1 13-48 994789 33 194780 j 13-76 ., 8o5220 55 190823 1 18.41 994719 33 193606 13-74 804394 5 56 191180 i3-38 994700 88 196480 18-71 808570 4 57 191988 1 13-86 994680 88 197253 18-69 802747 3 X 192784 ! 18.88 ! 994660 38 198074 i3-66 801926 2 55 198534 ' i3-3o c-4640 •33 19S894 1 18-64 801106 I 66 194382 i 18-28 :g,"20 • 33 199718 j i3-6i 800287 . 1 Coftine j D. 1 Sine , Cotan^. ' D. t Tanff. (81] DEGR] EB£.) i SINES AND TANGENTS. (9 DEGREE.') 27 I'm. Bine D. Ooeine D. Tang., D. Cotang. 9-194333 i3-28 9.994620 .33 9.199713 i3-6i 10-800287 60 I a 195129 195920 i3-26 i3-23 994600 994580 .33 • 33 200520 201845 13.59 13-56 799471 798655 u 3 196719 13-21 994560 .34 202159 13-54 797841 57 4 197511 i3-i8 994540 .34 202971 i3-52 797029 56 5 198302 13-16 994519 .34 208782 18.49 796218 55 6 I 9909 I i3.i3 994499 •?4 204592 13-47 795408 54 I 199879 13-11 994479 .34 205400 i3-45 7q46oo 53 200666 i3-o8 994456 .34 206207 18-42 798798 52 9 20i45i 13-06 994438 -34 207018 13.40 792987 5i 10 202234 i3-o4 994418 -34 207817 13-88 792183 5o II 9«2o3oi7 i3-oi 9-994397 -34 9-208619 13-85 10-791881 49 12 203797 204577 205354 12-99 994377 .34 209420 18.33 790580 48 i3 12-96 994357 .34 210220 13. 3i 780780 47 U 12-94 994336 -84 21IO18 18.28 788982 46 15 2061 3 I 12-02 9943x6 -34 2ii8i5 18^26 788185 45 16 206906 12-89 994295 •34 2I26I1 i3^24 787889 44 ;? 207679 12.87 994274 •85 2i34q5 18^21 786596 43 208452 12-85 994254 -85 214198 i3'i9 785802 42 I? 209222 12-82 994233 -85 214989 18-17 78501 1 41 20 209992 12-80 994212 .35 215780 i8-i5 784220 40 21 9-210760 211326 12.78 9.994191 .35 9-216568 l8-I2 10.788482 ^2 22 12-75 9941 7 1 -35 217856 i3.io 782644 38 23 212291 12-73 9941 5o -35 218142 18.08 78x858 37 24 2i3o55 12.71 994120 -35 218926 i3.o5 781074 86 25 2i38i8 12.68 994108 .35 219710 i3.o8 780290 85 26 11 214579 215338 12-66 12-64 994087 994066 .35 .35 220492 221272 18.01 12-99 77q5o8 778728 34 33 216097 12-61 994045 .35 222052 12-97 777948 32 29 216854 12-59 994024 .35 222880 12-94 777170 3i 3o 217609 12-57 994oo3 -35 228606 12.92 776394 3o 3i 9-218363 12.55 9.998981 -35 9-224882 12-90 10-7756x8 ll 32 219116 12.53 998960 .35 225i56 12-88 774844 33 219868 12'5o 998939 -35 225929 12-86 774071 27 34 220618 12.48 998918 -35 226700 12-84 773300 26 35 22i367 3221l5 12.46 998896 -36 227471 12-81 772529 25 36 12.44 99G875 -36 228289 12.79 771761 24 ll 222861 12-42 998854 -36 229007 12.77 770998 23 223606 12.39 998882 • 36 229773 12-75 770227 22 39 224349 12.37 998811 .36 280589 12.78 769461 768698 21 40 225092 12.35 998789 -36 281802 12.71 20 41 9-225833 12-33 9.998768 -86 9-282065 12-69 10-767935 19 42 226573 12-3l 998746 -86 282826 12-67 767174 18 43 227311 22S048 12-28 998725 -86 283586 12-65 766414 17 44 12-26 998708 -36 284845 12-62 765655 16 45 228784 12-24 998681 -36 285io3 12-60 764897 i5 46 229518 12-22 998660 -36 285SJ9 12-58 764141 14 J 48 230252 12.20 998688 .36 286614 12.56 768886 i3 230984 12-18 998616 -36 287868 12-54 762682 12 49 231714 12-16 998594 -37 288120 12-52 761880 11 5o 232/i/i/i 12-14 993572 •37 288872 12 -5o 761128 TO 5. 9.233172 12-12 9.993550 .37 9-289622 12.48 10-760878 ? 52 233899 12-09 998528 .37 240871 12-46 759629 53 234626 12-07 993506 -37 241 1 18 12-44 758882 I 54 23534Q 12-05 998484 •37 241 865 12-42 758i35 55 23607J 12. o3 998462 •37 242610 12-40 757890 5 56 236795 12-01 998440 •37 248354 12-38 756646 1 4 u 2375i5 11-99 993418 •37 244097 244889 12-86 755oo3 1 3 238235 11-97 11.95 998896 .37 12-34 755161 2 1 59 238953 993374 993351 •37 245579 12-32 754421 I 60 239670 11.93 •37 246819 12-30 753681 1 1 Coeino D. Sine Cotang. D. Tang. M. (80 DSOI IXSB.) 28 (10 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotang. 60 9-239670 240386 11.93 9.993351 .37 9' 246819 12-30 10.753681 I II'OI 993329 •37 247067 12.28 762943 5? 2 241 lOI 11.89 993307 .37 248530 12.26 762206 3 241814 11.87 993285 .37 12-24 751470 760736 u 4 242626 11-85 993262 •37 249264 12.22 5 243237 11-83 993240 :U 249998 260780 12.20 760002 55 6 243947 244656 11.81 993217 12.18 749270 54 I 11.79 993196 .38 261461 12-17 12-l5 748539 53 245363 WJ,l 993172 • 38 262191 747809 52 9 246069 246775 998149 • 38 262920 1213 747080 5i 10 11.73 993127 ' • 38 268648 12-11 746862 5o 11 9-247478 248181 11-71 9.993104 .38 9.264874 12-09 10-745626 S la 11.69 998081 .88 266100 12-07 12-05 744900 i3 248883 11.67 998069 .38 266824 744176 743463 % 14 249583 11.65 998086 .38 266647 12-03 i5 250282 11-63 998013 .38 267269 12.01 742781 45 i6 260980 11-61 992990 .38 267990 268710 12.00 742010 44 \l 25167-' 11.59 992967 .38 11.98 741290 43 262373 11.68 992944 .38 269429 11.96 740671 42 19 253067 11.66 992921 .38 260146 11-94 789864 41 20 253761 11-64 992898 .38 260868 11.92 789137 40 21 9.264453 11-62 9.992876 .38 9.261678 11.90 10-788422 ^ 22 256i44 11-50 992862 .38 262292 11-89 787708 23 255834 11.48 992829 .39 268006 11-87 11-85 786996 u 24 256623 11-46 992806 .39 268717 786283 25 267211 11.44 992788 .39 264428 11-83 786672 35 26 267898 11.42 992769 .39 265i38 11-81 784862 34 11 258583 11.41 992786 .39 266847 11.79 784153 33 269268 11.39 992713 .89 266565 11.78 788445 32 29 259961 11.37 992690 .39 267261 11-76 732739 782033 3i 3o 260633 11-35 992666 .39 267967 11-74 3o 3i 9-26i3i4 11-33 9-992648 .39 9.268671 11-72 10-781829 780626 It 32 261994 11-31 992619 .39 269876 11-70 33 262673 263351 11-30 992696 19 270077 11-69 729928 ll 34 11-28 992672 .39 270779 11-67 11-65 729221 35 264027 264703 266377 11.26 992649 .39 271479 720621 25 36 11-24 992626 •39 272178 11-64 727822 24 ll 11-22 992601 .39 272876 11-62 727124 23 266061 11-20 992478 .40 278678 11-60 726427 22 39 266723 267396 11-19 992464 .40 274269 11-68 726731 21 40 11-17 992480 .40 274964 11-67 726086 20 41 9-268065 ii-i5 9.992406 .40 9.276668 11.56 10-724842 \t 42 268734 I1-I3 992882 .40 276861 11-63 728649 43 269402 11-11 992869 .40 277043 ii-5i 722967 722266 \l 44 270060 270736 II-IO 992835 .40 277784 278424 11-60 45 11.08 99281 1 1 .40 11.48 721676 720887 i5 46 271400 11.06 992287 1 .40 279118 11-47 14 il 272064 11.06 992268 .40 279801 280488 11.46 720199 i3 272726 II.03 992289 .40 11.43 719612 12 49 273388 11.01 992214 .40 281174 11.41 718826 II 5o 274049 10.99 992190 .40 281868 11.40 718142 10 5i 9.274708 10.98 9.992166 .40 9.282542 11.38 10-717468 I 52 275367 10.96 992142 .40 288226 11.36 716775 53 276024 10.94 992117 •41 288907 11.35 716093 7 54 276681 10.92 992098 -41 284688 11-33 716412 1 6 1 55 277337 10.91 992069 •41 286268 ii-3i 71473a 5 56 277991 278644 10.89 992044 •41 286947 ii-3o 714063 4 u 10.87 992020 •41 286624 11-28 718876 3 279297 279948 280399 10.86 991996 •41 287801 11-26 712690 712023 2 59 10.84 991971 •41 287977 11-26 I 60 10.82 991947 •41 288662 11-23 711348 - Ck>Bino D. Sine Cotang. D. Tang. M. (79 DEGREES.) SINKS AND TANGENTS. (11 DEGREES.) 29 M. Sine D. Coeine D. Tang. D. Cotan^. 9 '280599 281240 10-82 9 991947 •41 9-288652 11-23 10.711848 60 I 10 81 991922 •41 289826 11-22 710674 u ) 281897 10 79 991897 991878 •41 289999 11-20 710001 3 282544 10 11 •41 290671 11-18 709829 u 4 288190 283836 10 991848 •41 291842 11-17 ii-i5 7o8658 5 10 •74 991828 •41 292018 707987 55 6 284480 10 72 991799 •41 292682 11-14 707818 54 I 285124 10 71 991774 •42 298350 11-12 7o665o 53 385766 10 69 991749 •42 294017 ii-ii 7o5q83 52 9 286408 10 tl 991724 •42 294684 11-09 7053 1 6 5i 10 287048 10 991699 •42 295849 11.07 704651 5o 11 9.287687 288826 10 64 9-991674 •42 9' 296018 11-06 10-708987 49 48 12 10 63 991649 •42 296677 11-04 703328 i3 288964 10 61 991624 •42 297889 298001 II-03 702661 47 14 289600 10 5o 991599 •42 11-01 7°^?99 701888 46 i5 290286 10 58 991574 •42 298662 11-00 45 i6 290870 10 56 991549 •42 299822 10.98 700678 44 \l 291504 10 54 991524 ♦42 299980 10-96 700020 699862 698705 43 292187 292768 298J99 10 53 991498 •42 800688 10-95 42 19 10 5i 991473 •42 801295 10-98 41 20 10 5o 991448 •42 301961 10-92 698049 40 21 9 '294029 10 48 9.991422 •42 9-802607 10-90 10-697898 696789 ^ 22 294658 10 46 991897 .42 808261 10-89 23 295286 10 45 991372 .43 308914 304367 3o52i8 10-87 10-86 696086 U 24 295918 10 43 991346 .43 695433 25 296539 10 42 991821 .43 10-84 694782 35 26 297164 10 40 991295 •43 3o5869 10-83 694181 34 11 297788 10 39 991270 '43 3o65i9 10-81 693481 33 298412 10 ll 991244 .43 807168 10-80 692882 32 29 299084 10 991218 •43 807815 10-78 692185 3i 3o 299655 10 34 991193 .43 808468 10-77 691537 3o 3i 9.800276 10 32 9'99ii67 .43 9*809109 10-75 10-690891 It 32 800895 10 81 991141 .43 809754 10-74 690246 33 3oi5i4 10 29 991115 .43 810898 10-78 689602 688958 6883 1 5 ll 34 802182 10 28 991090 .43 811042 10-71 36 802748 10 26 991064 .43 3ii685 10-70 10-68 25 36 808864 10 25 991088 .43 812827 687673 24 ll 803979 804598 10 23 991012 .43 312967 818608 10-67 10-65 687088 686892 685753 23 10 22 990986 •43 22 39 3o5207 10 20 990960 •43 314247 814885 10-64 21 40 3o58i9 10 19 990934 •44 10.62 6851 i5 20 41 9 '306430 10 17 9 '990008 •44 9-3i5528 10.61 10-684477 19 42 807041 10 16 990882 •44 816159 10-60 688841 18 43 807650 808259 10 14 990855 •44 816793 817430 818064 10-58 688205 \l 44 10 i3 990829 990808 •44 10-57 10-55 682570 45 808867 10 11 •44 681986 i5 46 809474 3 10000 10 10 990777 •44 818697 10-54 68i3o3 14 ii 10 08 990750 •44 819829 10'53 680671 i3 8io685 10 ll 990724 •44 819961 iO'5i 680089 12 49 811289 811898 10 990697 •44 820592 iO'5o 679408 11 5c 10 04 990671 •44 821222 10-48 678778 10 5i 9-812495 10 o3 9-990644 •44 9-82i85i i0'47 10 '45 10-678149 t 52 818097 818698 10 01 990618 •44 822479 677521 53 10 00 99o5qi 99o5o5 •44 828106 10.44 676894 I 54 314297 9 98 •44 328788 10-43 676267 55 314897 815495 9 97 990538 •44 324858 10-41 675642 ' 5 56 9 96 9905 11 •45 824983 10-40 675017 4 5j 816092 816689 9 94 990485 •45 825607 10-89 674893 3 58 9 93 990458 .45 826281 10-87 10-36 678769 3 59 817284 9 91 990481 •45 326853 678147 672525 I 6o 817879 9.90 990434 •45 827475 10-35 CJosine D. Sine Cotan^. D. Tang. ^ M. (78 DBGRBEB.) ■60 (12 DEGREES.^ A TABLE OF LOGARITHMIC H. Sine D. Cosine D. Tang. D. Cotanfr. o 9.317870 318473 319066 9.§8 9-990404 •45 9-327474 328096 10-35 10-672526 60 I 990878 99085 I •46 10.33 671905 n 2 9.87 •46 828716 10-32 671285 3 319658 9-86 990824 -45 829334 io-3o 670666 U 4 320249 9.84 990297 •45 829963 330670 331187 33i8o3 10-29 670047 5 320840 0-83 990270 •45 10-28 669480 55 6 321480 9-82 990248 •45 10-26 668813 54 I 322019 9-8o 990215 •45 10-26 668107 53 322607 9-79 990188 •45 882418 10-24 667682 5a 9 10 328194 328780 9-77 0-76 990161 990184 •46 •45 383o33 333646 10-23 10.21 666967 666354 5i So It 9 '824866 9.75 9-990107 .46 9-384269 10-20 10-665741 ^ 12 324950 325534 9.73 990079 -46 884871 10-19 666129 664618 i3 9.72 990062 •46 336482 10-17 % 14 3261 17 9.70 990025 .46 336098 10-16 668907 668298 662689 i5 826700 9.69 989997 .46 886702 io-i5 45 i6 327281 9.68 989970 •46 337811 io-i3 44 \l 827862 828442 9.66 989942 .46 337919 33852-1 339133 10-12 662081 43 9-65 989915 .46 10-11 661473 42 19 829021 9.64 989887 .46 10-10 660867 41 20 829699 9-62 989860 -46 339789 10-08 660261 40 21 9'88oi76 330758 331829 3^1908 9-61 9-989882 •46 9-840844 10-07 10-00 10-669666 It 22 9-60 989804 .46 340948 669062 668448 23 9-58 989777 .46 341562 10-04 u 24 It 989749 •47 842166 10 -08 667845 25 332478 338o5i 989721 •47 342767 10-02 667243 35 26 9.54 989698 •47 343358 10-00 656642 34 11 338624 9.53 989665 •47 843968 9.90 9.98 666042 33 334195 9-52 989687 •47 344568 666442 3a 29 334766 9.50 989609 •47 346167 345766 9-97 9.96 664843 3i 3o 335837 9.49 989682 •47 664246 3o 3i 9 •335906 9.48 9.989663 •47 9 •346353 9.94 10-653647 S 32 886475 9.46 989626 •47 346949 347545 9-98 668061 33 337048 9.45 989497 •47 9-92 662455 11 34 337610 888176 9.44 989469 •47 348141 9-91 661869 35 9.43 989441 •47 348735 9-92 651265 25 36 338742 389806 9-41 989418 •47 849829 9-88 660671 24 U 9.40 989884 •47 349922 riz 660078 649486 23 339871 9.89 989866 •47 85o5i4 22 39 340434 VM 089828 •47 861106 9-85 648894 21 40 340996 989800 •47 361697 9-83 648808 20 41 9-34i558 9.35 9-989271 •47 9-862287 352876 853465 9-82 10.647718 \t 42 342119 9.34 989243 •47 9-81 647124 43 342679 9-82 989214 •47 9-80 646535 »7 44 348289 9.81 989186 1 •47 864063 9-79 646947 6 45 343797 344355 9'8o 969167 ' 989128 :il 864640 9-77 9.76 645360 1 5 1 46 9.29 356227 3558i3 644773 644107 14 S 344912 9.27 989100 .48 9-75 i3 345469 9-26 989071 .48 366898 866982 357666 9-74 648602 i* 49 846024 9-25 989042 -48 9-78 648018 II 5o 346679 9-24 989014 .48 9.71 642434 10 5i 9.347134 ^.22 9.988985 .48 9.368149 9-70 10-641861 t 52 347687 9-21 988966 .48 868781 9-09 641269 53 348240 9'20 988927 988898 .48 869313 9.55 640687 I 54 348792 349843 9.19 .48 369898 |:^ 640107 689526 688047 638366 55 9.17 9.16 988869 .48 860474 5 56 349898 988840 .48 861068 9-65 4 U 350443 9*15 988811 •49 861682 9-63 3 350992 35 1 540 9-14 988782 •49 862210 9-62 637790 a 5q 9.18 988768 •49 862787 363364 9-61 687213 636636 I 66 352088 9-11 988724 .49 9-60 Cosine D. Sine Cotang. D. 1 T«ng. _ Jkl (77 DXOR8X8.) StPTES AND TANGENTS. (13 DEGREES.) SI Sino D. Cosine D. Ttog, D. Cotang. 60 '» 352088 9-11 Q. 988724 -49 9-363364 9-60 10-636636 I 352635 9 10 988695 -49 363940 3645 I 5 9-59 636o6o U 2 353i8i 9 S 988666 .49 9-58 635485 3 353726 9 988636 .49 365o90 ^12 9-55 634910 634336 U 4 354271 9 :S 988607 .49 365664 5 354815 9 988578 .49 366237 9.54 633763 55 e 355358 9 04 988548 .49 366810 9.53 633190 54 I 355901 9 o3 988519 •49 367382 9.52 632618 53 356443 9 02 988489 .49 367953 9.51 632047 631470 52 9 356984 t 01 988460 .49 368524 9.50 5i 10 357024 99 988430 -49 369094 9.49 630906 5o II o> 358064 8 98 9.988401 •49 9.369663 9.48 io.63o337 629768 it 12 3586o3 8 97 988371 .49 370232 9-46 i3 359141 8 96 988342 .49 370799 9.45 629201 47 14 359678 8 95 988312 .50 371367 9.44 628633 46 i5 36o2i5 8 93 988282 -5o 371933 9-43 628067 45 i6 360752 8 92 988252 -5o 372499 9.42 627501 44 \l 361287 8 91 988223 -5o 373c64 9-41 626936 626371 43 361822 8 90 988193 -5o 373629 9.40 42 «9 362356 8 89 988163 -50 374193 374756 9.30 625807 41 20 362889 8 88 988133 -5o 9.3^ 625244 40 21 9-363422 8 87 85 9-988103 -5o 9.375319 9.37 10-624681 39 22 363954 8 988073 -5o 375881 9.35 6241 19 38 23 364485 8 84 988043 -5o 376442 9.34 623558 37 24 365oi6 8 83 988013 .5o 377003 9.33 622997 622437 36 25 365546 8 82 987983 -5o 377563 9.32 35 26 366075 8 81 987953 -5o 378122 9.31 621878 34 11 366604 8 80 987522 -5o 378681 9-30 621319 33 367131 8 79 987892 -5o 379239 9.20 620761 32 ?9 367659 368i85 8 77 987862 -5o 379797 9.28 620203 3i 3o 8 76 987832 -5i 38o354 9-27 619646 3o 3i 9'3687ii 8 75 9-987801 .51 9.380910 9. 20 10-619090 It 32 369236 8 74 987771 -5i 38 1466 9-25 618534 33 369761 8 73 987740 -5i 382020 9-24 617980 ll 34 370285 8 72 987710 .51 382575 9-23 617425 616871 35 370808 8 71 987679 -5i 383129 9-22 25 36 371330 8 70 987640 987618 -5i 383682 9-21 6i63i8 24 ll 371852 8 69 -51 384234 9.20 615766 23 372373 8 tl 987588 -5i 384786 385337 385888 9-lQ 6i52i4 22 39 372894 8 987557 -5i 9.18 614663 21 40 373414 8. 65 987526 -5i 9.17 614112 20 41 9-373933 8. 64 9-987496 • 51 9.386438 9-15 io-6i3562 \t 42 374452 8 63 987465 .5i 386987 387536 388084 9-14 6i3oi3 43 374970 375487 376003 8 62 987434 .5i 9-13 612464 \l 44 8 61 987403 .52 9-12 611916 611369 45 8 60 987372 .52 388631 9-11 i5 46 376519 377035 8 U 987341 -52 389178 9-10 610822 14 S 8 987310 -52 389724 9.0Q 610276 609730 i3 377549 378063 8 57 987270 -52 390270 9.08 12 49 8 56 987248 -52 390815 9.07 609185 608640 11 5o 378577 8 54 987217 -52 391360 9.06 10 Si 9-379089 8 53 9.987186 -52 9.391903 9-05 10-608097 607553 ? 52 379601 8 52 987155 -52 392447 9.04 53 38oii3 8 5i 987124 .52 392989 9.03 60701 1 I 54 380624 8 5o 987092 .52 393531 9.02 606469 55 i8ii34 8 49 987061 •52 394073 9.01 605927 6o5386 5 56 381643 8 48 987030 .52 394614 9.00 h 4 t2 382152 8 % 986998 -52 396154 604846 3 382661 8 986967 •52 395694 396233 6o43o6 2 59 383 168 8 45 986936 •52 tu 603767 1 60 383675 8-44 986904 .52 396771 603229 1 Cosine D. Sine 1 CotADg. D. Tansr. JL_ ^ (76 DEOKEES.) 32 (14 DEGREES.; A lAJiLE OF LOaARITTrVfTU M. Sire D. Coaiii« D Tang. D. Cotang. ! 9-333675 8-44 9 • 9S6004 9.S-6.373 52 9-396771 8.96 10-603339 60 384182 8-43 53 397309 8 .96 602691 5o 3846.87 8-42 9S6.341 53 397846 8 .90 602104 58 1 4 380192 385697 8-41 8-40 986809 9.S6773 .53 ■ 53 398383 398919 8 8 •94 .93 601617 57 1 601081 56 ■ 1 J 386201 8-39 986746 53 399450 8 .92 600045 ; 55 , f 6 386704 8-33 956714 53 399990 8 .91 600010 54 I 387207 387709 8.37 8-36 9S6683 986601 53 ■53 400024 401 o58 8 8 599476 i 53 ^ 598942 5a 1 - 9. 388210 8.35 986619 53 401591 B ■ 80 598409 ! 5i 597876 ! 5o lO 38871 I 8.34 986087 1 53 402124 8 .87 II 9.38921 1 8-33 9-986555 53 9-402656 8 .86 10.597344 49 596813 3 13 38971 1 8-32 986523 ■ 53 403187 8 .85 i3 390210 8.3i 936491 53 403718 8 .84 096282 47 14 390708 8.3o 936409 .53 404349 8 .83 590^51 46 i5 391206 8.28 986427 53 404778 8 .82 595222 45 16 391703 8.27 936395 53 4o53o-8 8 •81 594692 44 17 392199 8.26 986363 54 405836 8 ■80 594164 43 i8 392690 8.25 986331 54 406364 8 ■78 593636 42 19 393191 393685 8.24 986299 1 54 406892 8 593108 41 , 30 8.23 986266 54 407419 8 ■77 592581 40 1 21 9-3941-9 8.22 9.986234 54 9.407945 8 ■76 10- 592000 3o 091520 38 591003 3] 33 394673 8.21 986202 54 408471 8 •75 23 390166 8.20 986169 04 408997 8 74 24 390608 8.19 986137 54 409021 8 74 25 396100 8.18 986104 04 410045 8 73 589950 35 36 396641 8.17 936072 54 410569 8 72 58q43i 34 u 397132 8.17 986039 54 41 1092 8 71 588908 33 397621 8.16 986007 54 4ii6i5 8 70 588385 I 32 1 29 3981 I I 8-15 985974 54 412137 8 ^ 587863 1 3i 3o 398600 8.14 98594a 54 412608 8 587342 i 3o 3i 32 9.399088 8.i3 8-12 ^■&? 55 55 9.413179 413699 8 8 H 10.586821 29 586301 28 33 400062 8. II 985843 55 414210 8 65 585781 27 34 400049 8-10 98581 I 1 55 414738 8 64 585262 26 35 4oio3o 8.09 8.08 985778 ' 55 415257 8 H 584743 20 36 401020 985745 55 415775 8 63 584225 24 U 4o2oo5 8.07 980712 1 55 416293 8 62 583707 23 402489 8.06 980679 1 55 416810 8 61 583i90 22 39 402972 8.00 985646 55 417326 8 60 582674 21 4o 4o34o5 8.04 980613 55 417842 8 59 582158 ao , 41 9-403938 8-o3 9 ■980080 ', 55 9-418358 8- 58 10.58x642 19 18 42 404420 8-02 935547 i 55 418873 8 U 581127 58o6i3 43 404901 400382 8-01 985514 55 419387 8 17 ' 44 8.00 985480 55 419901 8 55 580099 16 45 400862 7-p 985447 55 42041 5 8 55 5795S0 i5 46 406341 985414 56 420927 8 54 579073 1 4 578,^60 i i3 i S 406820 7-97 985380 56 431440 8 53 407299 7.96 985347 j 56 431953 8 5a 578048 1 n I 49 407777 7-95 985314 ' 56 432463 8 5i 577537 ' 11 i 5o 408204 7-94 980280 56 422974 8 5o 577026 1 10 1 5i 9.4^731 7-94 9-985247 06 9.423484 8 S I0'5765i6 5 1 52 409207 7.93 9302 1 3 06 423993 S 576007 8 53 409682 7.92 935180 56 434003 8 48 075407 574989 I 54 410107 7.91 985146 56 435oii 8 % 5t 4io632 985ii3 56 435519 8 574481 5 1 56 41 1 106 7.89 980079 56 436037 8 45 573973 57 411579 7.88 935040 56 426534 8 44 573466 5d 41 3002 7.87 98501 I 56 437041 8 43 572959 572453 571948 S 412024 412996 7.8^ 7-85 984978 984944 56 56 437547 428003 8 8. 43 A2 1 Coeme D. :?ine Ccl&ng. D. Tar«. ilL i t'To Dl EGR EES.) I SINES AND TANGENTS. (15 DEGREES.; as M. Sina D. Ootiine D. Tanff. D. Cotang. 9 412996 7-85 9-984944 'i^ 9.428052 8.42 10-571948 60 I 4i34J'' 7-84 984910 .57 428557 8.41 571443 5? 3 41393d 7-83 984876 •57 429062 8.40 570938 3 414408 7-83 984842 •57 429566 8.39 8.38 570434 57 4 414878 7.82 984808 17 430070 569930 56 5 415347 4i58i5 7.81 984774 .57 430573 8.38 56q427 55 6 7.80 984740 •57 431075 B.37 568925 54 I 416283 7.70 984706 17 43 1577 3.36 568423 53 416751 7.78 984672 17 432079 8.35 567921 52 9 417217 l]l 984637 984603 .57 432580 8.34 567420 5i 10 417684 •57 433080 8.33 566920 5o II 9'4i8i5o 7.75 9.984569 •57 9.433580 8.32 10-566420 49 la 4i86i5 7-74 984535 17 434080 8.32 565920 48 i3 419079 7-73 984500 17 434570 8.3i 565421 % 14 419544 7.73 984466 •57 .58 435078 8.3o 564922 i5 420007 7-72 984432 435576 8.29 564424 45 i6 420470 7-71 984397 .58 436073 8.28 563927 44 \l 420933 421395 ]:Z 984363 .58 436570 8.28 563430 43 984328 .58 437067 8-27 8-26 562933 42 19 421857 4223i8 7-68 984294 • 58 437563 562437 41 20 7-67 984259 .58 438059 8-25 561941 40 21 9-422778 ]t 9.984224 .58 9-438554 8.24 10.561446 39 22 423238 984190 .58 439048 8-23 560952 38 23 423697 7-65 984155 • 58 439543 8.23 560457 37 24 424106 7-64 984120 .58 44oo36 8-22 559964 36 25 424615 7-63 984085 .58 44o529 8-21 559471 35 26 425073 7-62 984050 .58 441022 8.20 558978 558486 34 U 425530 7.61 984015 .58 44i5i4 8.19 33 425987 7 -60 983981 • 58 442006 8.19 557994 32 \9 426443 7-6o 983946 .58 442497 8.18 557303 3i 3o 426899 7-59 98391 I .58 442988 8.17 557012 3o 3i 9-427354 7-58 9.983875 .58 9-443479 8.16 10-556521 !§ 32 427809 428263 ]t 983840 .59 443968 8.16 556o32 33 983805 .59 44/1458 8.i5 555542 27 34 428717 7-55 983770 .59 444947 8-14 555o53 26 35 429170 7-54 983735 .59 445435 8-13 554565 25 36 429623 7-53 983700 .59 445923 8-12 554077 24 \L 430075 7-52 983664 .59 44641 1 8.12 553589 23 43o527 430978 7-52 983629 .59 446898 8.11 553102 22 39 7-5i 983594 .59 447384 8-10 552616 21 40 431429 7.50 983558 .59 447870 8.09 552i3o 30 41 9.431879 7-49 9.983523 .59 9.448356 8.09 8.08 io.55i644 \% 42 43232Q 7-49 983487 .59 448841 55ir59 43 432778 7.48 983452 .59 449326 8.07 550674 ^1 44 433226 7-47 983416 .59 449810 8.06 550190 16 45 433675 7-46 983381 .59 450294 8.06 549706 i5 46 434122 7.45 983345 .59 450777 8.o5 540223 14 % 434569 7-44 983309 983273 .59 451260 8.04 548740 i3 435oi6 1-U .60 451743 8.o3 548257 547773 13 i^ 435462 7-43 983238 .60 452225 8.02 II 5o 435908 7.42 983202 .60 452706 8-02 547294 10 5i 0-436353 7-41 9.983166 .60 9.453187 453668 8-01 10.546813 t 5s 436798 7-40 983 i3o .60 8.00 546332 53 437242 7-40 983094 .60 454 14S 7.99 545852 I 54 437686 438129 7.39 983o58 .60 454628 7.99 545372 55 7-38 983022 .60 455107 7.98 544893 5 56 438572 7.37 982986 .60 455586 7-97 544414 4 % 439014 7-36 982950 .60 456064 7-96 543936 3 439456 7-36 982914 .60 456542 7.96 543458 3 ^ 439807 440338 7-35 982878 .60 457019 7-95 542981 542504 1 6o 7-34 982842 .60 457496 7-94 1 Cosine D. Sine Cotaug. i D. Tang. M. (74 DSGRXBS.) S4 C16 DEGREES.) A TABLE OF LOGARITHMIC M. 8ino D. Cosine D. Tang. D. Cotang. 60 9-440338 7-34 9-982842 -60 j 9-457496 7-94 10 • 542504 I 440778 7-33 982805 .60 457973 7-93 542027 5I a 441218 7-32 982769 982733 -61 458449 7-93 54i55i 3 441658 7-3i -61 458923 7-92 541075 U 4 442096 442535 7-3i 982696 .61 ! 459400 7-91 C4000U 5 7-3o 982660 •61 ' 459875 7.90 540125 5t> 6 442973 7-29 982624 -61 j 460349 7-00 539651 54 i 443410 7-28 982587 •61 ! 460823 7-09 539177 53 443847 7-27 982551 •61 : 461297 7^88 538703 5a 9 444284 ]:ll 982514 -61 ! 461770 7-88 538230 5i 10 444720 982477 .61 462242 7.87 537758 5o II 9-445i55 7-25 9-982441 .61 9-462714 7.86 10-537286 49 12 445590 7-24 982404 .61 ! 463 186 7-85 536814 48 i3 446025 7-23 982367 -61 463658 7-85 536342 47 14 446459 7-23 982331 .61 1 464129 7-84 535871 46 i5 446893 7-22 982254 .61 464599 7-83 535401 45 i6 447326 7-21 982237 •61 i 465069 7-83 534931 44 17 447759 448191 7-20 982220 •62 1 465530 7-82 534461 43 i8 7-20 982183 .62 466008 7-81 533992 42 19 448623 7-19 982146 •62 466476 7-80 533524 41 20 449054 7-18 982109 -62 466945 7-80 533o55 40 21 9-449485 IM 9-982072 -62 9-467413 7-79 10.532587 39 22 449915 45o345 982035 •62 467880 7-78 532120 38 23 7-16 981998 -62 468347 7-78 53 1 653 37 24 450775 7-15 981961 -62 468814 7-77 531186 36 25 45 1 204 7-14 981924 •62 469280 7-76 530720 35 26 45i632 7-i3 981886 -62 469746 7-75 530254 34 27 452060 7-i3 981849 -62 470211 7-75 529789 529324 33 28 452488 7-12 981812 -62 470676 7-74 32 29 452915 7-11 981774 -62 471 141 7-73 528859 528395 3i 3o 453342 7-10 981737 -62 471605 7-73 3o 3i 9-453768 7-10 9-981699 -63 9-472068 7-72 10.527932 ll 32 454194 ?:S 981662 -63 472532 7-71 527468 33 454619 981625 •63 472995 7-71 527005 27 34 455o44 7-07 981387 •63 473437 7-70 526543 26 35 455469 455893 7-07 981549 •63 473919 7.69 526081 25 36 7.06 981512 -63 474381 7-69 5256iq 24 ll 4563 1 6 7-05 981474 •63 474842 7-68 525i58 23 456739 7-04 981436 -63 475303 7-67 524697 22 39 457162 7-04 981399 -63 475763 ]:n 524237 21 40 457584 7-o3 981361 -63 476223 523777 ao 41 9 -458006 7-02 9-981323 •63 9-476683 7-65 10-523317 IQ 42 458427 458848 7-01 981285 •63 477142 7-65 522858 18 1 43 7-01 981247 •63 477601 7-64 522399 \l 44 459268 7-00 981209 •63 478o5q 7-63 521941 45 459688 6. 90 981171 •63 478517 7-63 521483 i5 46 460108 6-98 981133 -64 478975 7-62 521025 14 S 460527 6-98 981095 -64 479432 7-6i 520568 i3 460946 46 I 364 6-97 981037 -64 479889 7.61 520111 la 49 6.96 981019 -64 480345 7.60 519655 ir 5o 461782 6-95 980981 -64 480801 7-59 519199 10 5i 9 462199 6-95 9-980942 • 64 9-481257 7-59 ■■ 10-518743 I 52 462616 6-94 980904 -64 481712 7-58 1 518288 53 463o32 6-93 980866 -64 482167 7-57 517833 I 54 463448 6.93 980827 -64 482621 7-57 517379 516925 55 463864 6-92 980789 -64 483073 7-56 5 56 464279 6-91 980750 -64 483529 7^55 ! 516471 4 U 464694 6-00 980712 -64 483982 7-55 ' 5i6oi8 3 465 I 08 6-90 980673 -64 484435 7-54 ! 5i5565 2 1 59 465522 6.89 980635 -64 484887 7^53 1 5i5ii3 I 60 465935 6-88 980596 -64 485339 7^53 1 5i466i ; .1 •^., Cosiiie D. Sine Cotang. D. 1 Tang. 1 M. j (73 DEGR EK8.) SINS8 AND TANGENTS. (17 DEGREES.) 36 '^M. Sine D. Cosine D. Tang. D. Cotang. 60 ! O I 9 -465035 466348 6-88 6-88 9-980596 980558 -64 .64 9-485339 485791 7-55 7-52 10.514661 514209 518758 a 466761 6-87 980519 -65 486242 7.51 3 467173 467535 6-86 980480 .65 486698 7-5i 5i33o7 57 4 6-85 980442 • 65 487148 7-5o 512857 ' 56 5 467996 468407 6-85 980403 -65 487598 7-49 512407 55 6 6-84 980364 -65 488043 7-49 5iio57 54 I 468817 6-83 980325 -65 488492 7-48 5ii5o8 53 469227 6-83 980286 -65 488941 7-47 5iio;)9 52 9 469634 470040 6-82 980247 -65 489890 7-47 5io6io 5i lO 6-81 980208 • 65 489888 7-46 510163 5o II 9.470455 6-80 9-980169 -65 9-490286 7-46 10-509714 49 48 12 470863 6-80 980130 • 65 490733 7-45 509267 i3 47 1 271 6-70 6.78 980091 980002 -65 491 180 7-44 508820 S 14 471679 -65 491627 7-44 508878 i5 472086 6.78 98001 2 -65 492078 7-43 507927 45 i6 472492 6-76 979973 -65 492519 7-43 507481 44 \l 472898 979034 -66 492965 7.42 507085 43 473304 6.76 979895 -66 498410 7-41 506590 42 »9 473710 6-75 979855 -66 498854 7.40 506146 41 20 4741 I 5 6-74 979816 -66 494299 7-40 5o570i 40 21 9'4745iQ 474Q23 475327 6-74 9.979776 -66 9-494748 7.40 io-5o5257 39 22 6-73 979737 979608 -66 495186 7-89 504814 38 23 6-72 -66 495680 7-88 504870 37 i »4 475730 6-72 .66 496073 7-37 508927 36 j 25 476133 6.71 979618 .66 49651 5 7-37 5o3485 35 26 476536 6-70 979579 .66 496957 7.36 5o8o43 34 11 476938 477^40 6-69 979539 -66 497^99 7-36 502601 38 6.69 979499 979459 -66 497841 498282 7-35 5o2i59 32 ^9 477741 478142 6-68 -66 7-34 501718 3i 3o 6-67 979420 -66 498722 7-34 501278 3o 3i 9-478542 6.67 9-979380 -66 9-499168 7-33 10 .500887 It 32 478042 479^42 6.66 979340 -66 499608 7.38 500897 49995S 499519 33 6-65 979800 -67 5ooo42 7-82 U 34 479741 480140 6-65 979260 -67 5oo48i 7-81 35 6-64 979220 -67 5oOQ20 5oij59 7-81 499080 23 36 480539 6-63 979180 -67 7-30 498641 24 ll 480937 481334 6.63 6-62 979140 979100 1^ .67 501797 5o22J5 7-30 7.20 498208 497765 497828 33 23 39 481731 6.61 979059 -67 502672 7.28 31 4o 482128 6-61 979019 -67 5o8io9 7-28 496891 20 4i 9-482525 6-60 9.978979 -67 9-5o3546 7.27 10-496454 \t 42 482921 4833 1 6 6-59 978989 -67 508982 7-27 496018 43 6.59 978898 .67 5o44i8 7-26 495582 \l 44 483712 6-58 978858 .67 5o4854 7-25 495146 45 484107 6-57 978817 -67 505289 7-25 4947 1 1 i5 46 484501 6-57 6-56 978777 9787^6 .67 505724 7-24 494276 14 s 484895 t 5o6i59 506598 7-24 498841 i3 485289 6-55 978696 978655 7.28 493407 12 1 49 485682 6-55 • 68 507027 7-22 49297^ II 5o 48607^^ 6-54 978615 -68 507460 7-22 492040 ro 5i 9.486467 6-53 9-978574 -68 9-507898 7-21 10-492107 I 52 486860 6-53 978533 -68 5o8326 7-21 491674 53 487251 6-52 978408 .68 508759 7-20 491241 I 54 487643 488-)34 6.5i 978452 .68 509191 7.19 490809 55 6-5i 978411 .68 509622 ?::§ 400878 5 56 488424 6-5o 978870 • 68 5ioo54 489946 4 U 488814 6-5o 978829 978288 68 5 I 0485 7-18 489O15 3 489204 6-49 .68 5iooi6 7-17 489084 3 59 489593 4B9982 6-48 978247 .68 5ii346 7-i6 488654 I 66 6-48 978206 -68 511776 7.16 488224 Cosine D. Sine D. Cotang. _ D, Tang._ M. 17 (72 DEGRluKS.) 36 (18 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine ' i D. Tang. D. Cotacg. 1 60 9-489082 4903] I 6-48 9-978206 1 .68 9-511776 7-16 10-488224 I 6.48 978165 1 •68 512206 7.16 487794 5q a 490709 6.47 6-46 978124 ' • 68 512635 7.15 487365 5tl 3 491147 978083 ' •69 5i3o64 7-14 486936 57 4 491535 6-46 978042 1 •^ 513493 7-14 4865o7 56 5 491922 492308 6-45 978001 •69 oi3q2i 514349 7-13 486079 55 6 6-44 977900 .69 7-i3 485651 54 I 492605 493081 6-44 977018 ; .69 514777 7.12 485223 53 6-43 977835 .69 510204 7-12 484796 5a 9 493466 6-42 .69 5i563i 7-11 484369 483943 5i 10 49385i 6-42 977794 .69 516057 7-10 5o II 9-494236 6.41 9.977752 .69 9.016484 7.10 10.483576 49 12 494621 6.41 9777" .69 016910 7-09 483090 48 i3 495oo5 6-40 977660 977628 .69 517335 ?:S 482665 47 14 493388 6.39 .69 517761 482239 481815 46 i5 495772 49604 6.39 977586 .69 5i8i85 7.08 45 i6 6.38 977544 .70 518610 ?:S 481390 44 17 496537 6-37 9775o3 .70 519034 480066 480542 43 i8 496019 497301 6.37 977461 .70 519458 7.06 42 19 6-36 977419 .70 519882 7-05 480118 41 ao 497683 6-36 977377 .70 52o3o5 7.05 479695 40 31 9.498064 6.35 9-977335 .70 9.520728 7.04 10.479272 It 2a 498444 6.34 977293 .70 52ii5i 7 -03 478849 23 498825 6-34 977231 • 70 521573 7.03 478427 478000 37 U 499204 6.33 977209 • 70 521995 7.03 36 25 499584 6-32 977167 • 70 522417 7.02 477583 35 26 499963 6-32 977120 .70 522838 7.02 477162 34 11 5oo342 500721 6.3i 6-31 977083 977041 .70 .70 523259 523680 7-01 7.01 476741 476320 33 33 1? 501090 001476 6.3o 6.29 976999 976907 .70 .70 024100 524520 7.00 6.99 475900 475480 3i 3o 3i 9-5oi854 6.29 9.976014 97683o .70 9.524939 t:^ 10.475061 '9 i 32 5o223l 6.28 •71 52535q 474641 28 33 502607 6-28 .71 520778 6.98 474222 11 34 502984 5o336o 6-27 976787 976745 .71 526197 6.97 4738o3 35 6.26 •71 5266i5 ?;^ 473385 25 36 5o3735 6.26 976702 •71 527033 472967 24 U 5o4iio 6.25 976660 •71 527451 6.96 472049 a3 5o4485 6.25 976617 •71 527868 528285 6.95 472132 22 39 504860 6-24 976574 .71 6.95 471715 21 40 5o5234 6.23 976532 .71 528702 6.94 471298 20 41 9 -505608 6-23 9.976489 .71 9-529119 6.93 10.470881 ^2 42 505981 6-22 976446 •71 029530 6.93 470465 18 43 5o6354 6.22 976404 .71 529900 53o366 6.93 47oo5o ]l 44 506727 6.21 976361 1 •71 6.92 469634 45 507099 6.20 676318 ' •71 530781 6.91 460219 i5 , 46 5oi47i 6-20 976275 1 .71 531196 6.91 468804 14 S 507843 5o82i4 6.19 6.19 976232 i 976189 : .72 .72 53i6ii 532025 6.90 6. go 6.^ 468389 467075 467061 i3 13 49 5o8585 6.18 976146 1 .72 532439 532853 u i 5o 508956 6.18 976103 .72 6.89 467147 LOJ 5i 9.509326 6.17 6-i6 9.976060 .72 9-533266 6.88 10.466734 466321 l\ 52 509696 976017 .72 533679 6.88 53 5ioo65 6.16 975974 • 72 534092 6.87 465908 h 54 5x0434 6-i5 975o3o , 975887 .72 534504 6.87 465496 465084 55 5io8o3 6.i5 .72 534916 535328 6-86 5 56 511172 6-14 975844 •72 6-86 464672 464261 4 n 5ii54o 6-i3 975800 .72 535739 6-85 3 1\IV^ 6-i3 975757 .72 536 i5o 6-85 463850 3 59 6-12 975714 970670 •72 536561 6-84 463439 463028 I 66 512642 6-12 .72 536972 6-84 Cosine D. Sine ! D. Co tang. D. lang. (71 D£(^R££6.) SINES AND TANGENTS. (19 DEGREES.) 37 M." Sine D. Cosine D. Tang. D. CotAn^. 9-512642 6-12 9-975670 .73 9-536972 537882 6.84 10.4680:8 60 I 5 I 3009 513375 t -II 975627 .73 6 83 462618 U a 6 ■11 975583 .73 537792 588202 6 -83 462208 3 5i374i 6 • 10 975539 "^i 6 .82 461708 461889 57 6 514107 6 09 975496 975452 "^l 58861 1 6 82 56 5 514472 6 09 -73 589020 6 81 460980 460371 55 6 514837 6 • 08 975408 •73 589429 6 81 54 I 5l5202 6 .08 975365 •7? 589887 540245 6 .80 460163 53 5i5566 6 .07 975321 -73 6 .80 459755 52 9 5 I 5980 6 :S 975277 975233 •73 540658 6 ■79 459347 458989 5i to 516294 6 .73 541061 6 •79 5o II 9.516657 6 o5 9'975i89 975145 •73 9.541468 6 78 10-458532 ^ la 517020 6 o5 •73 541875 542281 6 .78 458125 i3 517382 6 04 975101 •7? 6 77 457719 ii 14 517745 6 04 975057 975oi3 .73 542688 6 U 457812 i5 518107 6 o3 .73 548094 6 406906 45 i6 518468 6 o3 974969 •74 543499 6 76 456301 44 \l 518829 6 02 974925 •74 548905 6 75 456095 43 519190 5i95di 6 01 974880 •74 544810 6 75 4556Q0 42 19 6 01 974836 •74 544715 6 74 455285 41 20 519911 6 00 974792 •74 545119 6 74 454881 40 21 9.520271 6 00 9.974748 •74 9 545524 6 73 10.454476 39 22 52o63i 5 99 974703 •74 545928 6 73 454072 38 23 520990 5 99 974659 •74 546881 6 72 458669 U 34 521349 5 98 974614 •74 546785 6 72 458265 a5 521707 5 98 974570 •74 547188 6 71 452862 35 a6 522066 5 97 974525 •74 547540 6 71 452460 34 11 522424 5 96 974481 •74 547948 548845 6 70 452057 33 522781 5 96 974436 •74 6 ll 45 1 655 32 29 523 i38 5 95 974391 •74 548747 6 451253 3i 3o 523495 5 95 974347 •75 549149 6 69 45o85i 3o 3i 9-523852 5 94 9-974302 "'^ 9.549550 6 68 io.45o45o 29 32 524208 5 94 974257 •7f 549951 55o352 6 68 450049 28 33 524564 5 93 974212 •75 6 67 449648 27 34 524920 5 93 974167 •75 550752 6 U 449248 a6 35 525275 52563o 5 92 974122 .75 55ii52 6. 448848 25 36 5 91 974077 •75 55i552 6- 66 448448 24 ll 525q84 526339 526693 5. 91 974082 •75 551952 6- 65 448048 23 5 90 973987 "^i 552851 6 65 447649 32 39 5 90 973942 .75 552750 6- 65 447250 ai 40 527046 5. 89 978897 .75 553 149 6. 64 446851 20 41 9-527400 5 89 9.978852 "^i 9.553548 6. 64 10-446452 J? 4a 527753 528105 5. 88 978807 .75 558946 6- 63 446054 43 5 88 978761 -75 554344 6- 63 445656 17 44 528458 5 87 978716 .76 554741 6- 62 445259 16 1 45 528810 5 87 978671 .76 555189 6- 62 444861 i5 46 529161 5 86 978625 -76 555586 6- 6i 444464 14 4*2 5295 I 3 5 86 973580 -76 555988 556320 6. 61 4/14067 i3 529864 5 85 978585 -76 6 60 448671 12 49 53021 5 5 85 978489 -76 556725 6 60 448275 II 5o 53o565 5 84 973444 -76 557121 6 59 442879 10 5i 9 53091 5 5 84 9-973898 -76 9-557517 6 59 10.442488 I 53 531265 5 83 978852 .76 557913 5583o8 6. 59 442087 53 53i6i4 5 82 978807 .76 6 58 441692 I 54 53io63 532JI2 5 82 978261 .76 558702 6 58 441298 55 5 81 978215 .76 559097 6 57 440903 5 56 532661 5 81 978169 .76 559491 6. U 440300 4401 i5 4 U 533009 5 80 978124 '76 559885 6- 3 533357 5 80 978078 978033 -76 560270 560678 56io66 6. 56 489721 2 59 533704 5 79 •77 6. 55 480827 438934 1 60 534052 5.78 972986 •77 6.55 M. L-, Cosine E . Sine i D. Cotan^. D. Tanir (70 DEORSSS.^ 88 (20 DEGREE8J A TABLE OF LOGARITHMIC M. Sine D. Cosine | 1 D. Tang. D. Cotang. ( 9 -534052 5.78 9.972986 1 •77 9-56io66 6.55 10-438034 ^ I 53439^ 5-77 972040 1 •77 561439 6.54 438541 i 59 a 534743 5-77 972894 •77 56i85i 6-54 ^38149 58 3 535oo2 536438 5-77 972848 •77 562244 6.53 437756 u 4 5-76 972802 •77 562636 6.53 437364 5 535783 5.76 972755 '77 563023 6.53 436972 55 6 536i2o 5-75 972709 '77 563419 6-52 43608 I 54 2 536474 5-74 972663 ' •77 5638 1 1 6-52 436180 435798 53 5368 1 8 5-74 972617 •77 564202 6-51 52 9 537163 5.73 972070 •77 564002 564983 6-51 435408 5i lO 537507 5.73 972524 •77 6.5o 435017 5o II 9-537851 '5-72 9-972478 , •71 9-565373 6.5o 10-434627 S 12 538194 5-72 972431 .78 565763 6.49 434237 i3 538538 5.71 972385 .78 566 1 53 6-49 433847 47 14 538880 5-71 972338 , •'?? 566542 5.40 433408 46 i5 53^223 5-70 972291 , "'? 566932 6-48 433068 45 i6 539565 5-70 972245 ■ •''? 567320 6-48 432680 44 17 539907 5-69 972198 [ •''o 567709 6-47 432291 43 i8 540249 5-6q 9721D1 .78 568oq8 6-47 431902 42 19 540590 5-68 972105 1 .78 5684^6 6.46 43ioi4 41 20 540931 5-68 972058 j .78 568873 6-46 431127 40 31 9-541272 5-67 9-9720x1 .78 9.569261 6.45 10-430739 39 97 54i6i3 5.67 971964 •7? 569648 6-45 43o3o2 38 33 541953 5-66 971917 •■'q 570035 6.45 429965 37 24 542 2o3 542632 5-66 971870 "^^0 570422 6-44 429078 36 i5 5-65 971823 ' •7? 570809 6-44 429191 35 36 542971 ' 5-65 971776 ' .78 571190 6-43 428805 34 11 543310 5-64 971729 i •79 571081 6-43 428419 33 543649 5-64 971682 • •79 571967 6-42 428033 32 29 543987 , 5-63 971635 •79 572352 6-42 427648 3i So 544325 I 5-63 971588 •79 572738 6.42 427262 3o 3i 9-544663 1 5-62 9-971540 •79 9.573123 6.41 10-426877 29 34 540000 , 5-62 971493 ; •79 573507 6.41 426493 28 33 545338 5-6i 971446 •79 573892 6-40 426108 U 34 545674 5.61 971398 •79 574276 6-40 425724 35 54601 1 5.60 97i3di •79 574660 6-39 420340 25 36 546347 5.60 97i3o3 •79 070044 6-39 424906 24 U 546683 5-39 971256 •79 575427 6.39 424073 23 547019 5-50 971208 < •79 570610 6-38 424190 23 39 547354 5-58 971161 ' •79 576193 6.38 423807 21 40 547689 5-58 . 971113 ; •79 576576 6.37 1 423424 20 41 9.548024 5-57 9-971066 1 .80 9.576958 6.37 10.423041 10 47 548359 5«57 971018 i .80 577341 6.36 i 422659 id 43 548693 5-56 970970 ', .80 577723 6.36 422277 11 44 549027 5-56 970922 -80 578104 ■ 6-36 421896 16 45 549360 5-55 970874 .80 078486 ■ 6-35 42i5i4 A 46 549693 5-55 970827 .80 578S67 ; 6.35 421 i33 14 % 550026 5-54 970779 -80 579248 j 6.34 420752 i3 55o359 5-54 970731 -80 579629 ! 6.34 420371 13 49 550692 5-53 970683 : .80 , 580009 ! 6.34 419991 II 66 55io24 5-53 970635 ; .80 58o389 6.33 ! 419611 10 5i g-55i356 5-52 9-970586 i .80 ' 9-080769 : 6.33 10-419231 t 53 55i68t 552018 5-52 970538 ' -80 , 581149 1 6.32 4i885i 53 5-52 970490 .80 58i528 ! 6-32 418472 1 7 54 552349 5-51 970442 .80 581907 j 6-32 418093 6 55 552680 5-5i 970394 .80 082286 6-31 417714 5 56 553oio 5-5o 970345 1 .81 582665 6-31 417335 4 58 553341 5-5o 970297 -81 583043 6-30 416957 416578 3 553670 5-49 970249 • 81 583422 i 6-30 3 59 554000 1 5.49 970200 ' .81 583800 1 6.29 416200 I 66 ■ 554329 \ 5-48 970.52 : .81 : 584177 ; 6-29 415823 • ^^^» Cosine ', D. Sine D. . Cotang. t i>- : Tan«. _ M. (69 DEQRSSS.) SINKS AUD TANGENTS. (21 DEGREES.) 39 M. Sine D. Cosine D. Tang. D. Cotang. 60 o 0' 554329 554658 5.48 9-970152 -81 9 •584177 584555 6-29 10-4x5823 1 5.48 970103 .81 6 11 4x5445 ^ 2 554987 5-47 970055 -81 584Q32 585^09 6- 4i5o68 58 3 5553x5 5-47 5-46 970006 969957 -81 6 28 414691 57 4 555643 -81 585686 6 27 414314 56 5 6 555971 556299 556626 5.46 5.45 969S60 .81 -81 586062 586439 6 6. 27 27 413038 4i356i 55 54 i 5.45 96981 1 •8x 5868x5 6- 26 4i3i85 53 556953 5-44 969762 -8x 587190 6 26 412810 52 9 557280 5.44 969^65 -8x 587566 6 25 412434 5i lO 557606 5.43 -8x 587941 6 25 412059 5o II Q. 557932 558258 5-43 9-969616 .82 9-5883x6 6- 25 10 41 1684 g !2 5-43 969567 969518 -82 588691 6- 24 41 1 309 i3 558583 5-42 -82 589066 6- 24 4xoo34 4io56o 47 14 558909 5.42 969469 -82 589440 6- 23 46 i5 559234 5.41 969420 -82 589814 6- 23 410186 45 i6 550558 5-41 969370 -82 590188 6- 23 409812 44 \l 559883 5.40 969321 -82 590562 6- 22 409438 43 560207 5.40 969272 -82 590035 591^08 6- 22 409065 42 19 56o53i 5.39 969223 -82 6- 22 408692 41 ao 56o855 5.39 969173 -82 59X68X 6 21 408319 40 21 9-561178 5-38 9-969x24 .82 9 592054 6- 21 XO -407946 407574 39 22 56i5oi 5.38 969075 .82 592426 6. 20 38 23 561824 5.37 960025 968976 .82 592798 6 20 407202 U S4 56s 146 5.37 5.36 .82 593170 6 19 406829 23 562468 968926 968877 • 83 593542 6 XO x8 406458 35 26 562790 5.36 .83 5939x4 6 406086 34 11 563II2 5.36 968827 .83 594285 6 x8 405715 405344 33 563433 5.35 968777 968728 .83 594656 6 18 32 19 563755 5-35 -83 595027 595398 6 »7 404973 3i 3o 564075 5.34 968678 -83 6 17 404602 So 3i 9-564396 5.34 9.968628 -83 9-595768 6 \l 10.404232 29 32 564716 5-33 968578 -83 596138 6 4o3862 28 33 565o36 5.33 968528 -83 596508 6 16 403492 ll 34 565356 5.32 968479 -83 596878 6 x6 4o3i22 35 565676 5-3a 968429 -83 597247 6 x5 402753 25 36 565995 5663 14 5-3i 968379 .83 5976x6 6 i5 402384 24 ll 5-3i 968329 968278 .83 597085 598 J 54 6 i5 40201 5 23 566632 5-3i .83 6 14 401646 22 39 566951 5-3o 968228 -84 598722 6 14 401278 21 40 567269 5.30 968x78 .84 599091 6 x3 400909 20 41 9.567587 5-29 9-968128 .84 9-599459 6 i3 10 -400541 \t 42 567904 568222 5-29 5-28 968078 .84 599827 6 x3 400173 43 968027 .84 600194 6 xs 399806 17 44 568539 5.28 967977 .84 6oo562 6 X2 399438 . 16 1 45 568856 5.28 967027 967876 -84 600929 6 IX 39007 X .15 46 569172 5-27 .84 601296 6 XI 398704 14 fi 569488 5-27 5-26 967826 .84 60x662 6 XI 398338 iJ 569804 967775 .84 602029 602395 6 XO 397971 12 ^ 570120 5-26 967725 .84 6 10 397605 II 5o 570435 5-25 967674 .84 60270X 6 XO 397239 10 5i 9.570751 5-25 9-967624 .84 9-6o3i27 603490 6o3858 6 09 10-396873 I 5a 571066 5-24 967573 -84 6 09 396507 53 571380 5.24 967522 .85 6 .s 396142 I 54 571695 5.23 967471 .85 604223 6 395777 55 572009 572323 572636 5-23 967421 .85 604588 6 ■08 395412 5 56 5-23 5.22 967370 967319 967268 -85 -85 6o4q53 6o5Jn 6 6 •07 .07 395047 394683 4 3 572950 5«22 -85 6o5682 1 6 •07 394318 2 ^ 573263 5-21 967217 -85 606046 6 •06 393954 393590 I 60 573575 5-21 967166 -85 6064x0 6-06 L , Gosino D. Sine D. Cotang. D. Tang. (08 DEORSSS.^ iO ^22 DB3^REES.) A TABLE OF LOGARITHMIC m:. Sine D. Cosine D. Tang. D. Cotang. 1 60 9"573575 573868 5-31 9-967166 .85 9-606410 6.06 10-398590 I 5.30 9671 i5 .85 606778 6.06 39822- 892860 5? 3 574200 5.30 967064 .85 607187 6.o5 3 574512 5-19 967013 .85 607500 6.o5 892500 u 4 574834 5-19 966961 -85 607863 6 04 392187 891775 5 575 I 36 5-19 5-i8 966910 • 85 608225 6.04 55 1 6 5?5^J 966859 • 85 6o8588 6.04 891412 54 I 5.18 966808 .85 6o8o5o 609^12 6-o3 391050 i 53 1 576069 5-17 966756 .86 6-03 890688 53 9 576379 576669 5-17 5-16 966705 .86 609674 6 o3 800826 5i 10 966653 -86 610086 6-02 389964 5o II 9.576099 577300 577618 5.16 9-966603 -86 9.610807 6-02 10 889608 ^ 13 5.16 966550 -86 610739 6.02 889241 i3 5.i5 966499 •86 611120 6-01 388880 ii 14 577937 578336 5.i5 966447 966895 .86 61 1480 6-01 388520 i5 5.14 .86 611841 6.01 888159 45 i6 578545 5.14 966844 .86 612201 6.00 887709 887489 44 \l 578853 5-i3 966292 -86 6i256i 6.00 43 579162 5-13 966240 • 86 612921 6-00 887079 886719 386359 43 «9 579470 5.i3 966188 .86 618281 5.99 41 20 579777 5.13 966186 • 86 618641 5-99 40 31 9 -580085 5-13 9-966085 •87 9.614000 5-98 10-886000 ^ 23 580892 5. II 966088 .87 6i485q 614718 5.98 385641 23 580699 58roo5 5. II 965981 •87 5.98 385282 U 24 5. II 965928 965876 -87 61 5077 6i5485 5-97 884928 884565 25 58i3i2 5-10 •87 5-97 35 26 58i6i8 5-10 965824 -87 615798 6i6i5i i:^ 884207 34 U 58iQ24 5.09 965772 •87 888849 33 582229 582535 5-09 965720 •87 6i65o9 5-96 888491 388 1 33 33 ?9 5.09 5.08 965668 •87 616867 5-96 3i 3o 582840 9656 1 5 •87 617224 5.95 882776 3o 3i 9-583145 5.08 9.965568 .87 9.617582 5.95 10.382418 ^ 32 583449 5.07 96551 1 •87 611989 618295 5-95 882061 33 583754 5.07 5.06 965458 .87 5.94 881705 U 34 584058 965406 ii 6i8652 5.94 881848 35 584361 5.06 965853 619008 5-94 380992 380686 35 36 584665 5.06 965801 .88 619864 5.98 34 ll 584968 5.o5 965248 .88 619721 5.93 880279 33 585272 5.o5 965195 .88 620076 620482 5.98 879924 879568 33 J9 5g5574 5-04 965148 .88 5.92 31 40 585877 5-o4 965090 .88 620787 5.92 879218 30 41 9- 586179 586462 5-03 9.965087 .88 9.621142 5.92 10-378858 :? 42 5.o3 964984 .88 621497 621852 5.91 3785o3 43 586783 5-o3 96498 1 .88 5.91 378148 17 44 587085 5-03 964879 .88 622207 5.90 377793 877489 377085 376781 376877 16 45 587386 5-03 964826 .88 622561 5.90 i5 46 587688 5-01 964773 .88 622915 5.90 5.89 5.89 i4 S 587989 588289 5-01 5-01 964719 964066 .88 .89 628269 628628 •3 f2 49 588590 5-00 9646x8 .89 628076 624880 5.89 5.88 876024 II 5o 588890 5-00 964560 .89 875670 P 5i 9 589100 589489 4-99 9.964507 .89 9.624688 5-88 10-875317 t 52 4-99 964454 .89 625086 5.88 874964 374012 53 589780 590088 4-99 4.98 964400 .89 625888 5.87 I 54 964847 .89 625741 5.87 374259 55 56 590887 590686 4.98 4-97 964294 964240 .89 .89 626093 626445 5.87 5.86 878907 373555 5 4 U 590984 4-97 964187 964183 .89 626797 5.86 378208 3 591282 4-97 4-96 .89 627149 5.86 372851 9 59 591580 964080 .89 627501 5.85 372499 372140 I 6o 591878 4-0 964026 -89 627852 5-85 Cofline D. Sine D. Cotniig. D. Tang. J M. (67 DEORSXS.) SINES AND TANGENTS . (23 EEGRBKS.) 41 M. Sine D. Cosine D. ! Tang. D. Cotang. 9.591878 ' 4.96 9.964026 .89 : 9-627852 1 5.85 10.372148 60 I 592176 4.95 968972 .89 628208 i 5.85 371797 u a 592473 4.95 968919 .89 628554 5-85 871446 3 592770 4-95 1 963865 ■90 628905 5.84 871095 u 4 ! 598067 593363 4-94 ' 96881 1 90 1 629255 5.84 870745 5 4.94 968757 .90 629606 5.83 870394 55 6 593659 598955 4.98 968704 •90 629956 680806 5.83 870044 54 I 4.98 968650 .90 j 5-83 369694 53 594251 4.93 968596 .90 63o656 5.83 869844 5a 9 594547 4-92 963542 .90 63ioo5 5.82 860995 5i IC 594842 4.92 968488 .90 63i355 5.82 368645 5o II 9 595137 4-91 9.968484 .90 9.681704 5-82 10.368296 § 12 595432 4-91 968879 .90 682o58 5-8i 367947 i3 595727 4-91 968825 .90 682401 5-8i 867399 367260 % F4 596021 4.90 968271 .90 682750 5.81 i5 5963 1 5 4-00 968217 .90 688098 5-80 366902 366558 45 i6 596609 596903 4.89 968168 .90 683447 688795 5-8o 44 \l 4-89 968108 .91 5-8o 3662o5 43 597196 4.89 4-88 968054 .91 684148 5-79 365857 42 19 597400 962999 .91 684490 5-79 365510 41 20 597783 4-88 962945 .91 684888 5.79 365i62 40 21 9.598075 4-87 9.962890 •91 9.685i85 5-78 10 .3648 1 5 It 22 598868 4-87 962886 .91 635582 5-78 364468 23 598660 4-87 962781 .91 685879 5-78 3641 21 ll 24 598952 4-86 962727 .91 686226 5.77 868774 25 599244 4-86 962672 .91 686572 5-77 868428 35 26 599536 4-85 962617 .91 686919 5-77 368081 34 ^ 599827 6001 18 4-85 962662 .91 687265 5-77 362785 33 4-85 962508 .91 687611 5-76 362889 32 29 600409 4.84 962453 .91 687955 638802 5-76 362044 3i 3o 600700 4-84 962898 .92 5-76 361698 3o 3i 9.600900 601280 4-84 9.962848 .92 9 '688647 5.75 io.36i353 20 32 4-83 962288 .92 688992 5.75 361008 28 33 601570 4-83 962288 .92 689337 5-75 860668 ll 34 601860 4-82 962178 .92 689682 5-74 360818 35 602 1 5o 4-82 962128 .92 640027 5-74 359978 25 36 602489 4-82 962067 .92 640871 5-74 359629 24 U 602728 4-8i 962012 .92 640716 5.73 359284 23 608017 6o83o5 4-8i 961957 .92 641060 5.73 358940 358596 22 39 4-8i 961902 .92 641404 5.73 21 40 603594 4- 80 961846 .92 641747 5.72 358253 20 41 9.608882 4-8o 9.961701 961735 .92 9.642091 5-72 10.357909 357566 \l 42 604170 4-79 .92 642484 5.72 43 604457 604745 4-79 961680 •92 642777 5-72 357228 \l 44 4-79 961624 .93 648120 5.71 356880 45 46 6o5o32 605819 4-78 4-78 961569 96i5iJ .93 .93 648468 648806 5.71 5.71 35653? 356194 855852 it} 14 S 6o56o6 4-78 961458 .93 644148 5-70 i3 605892 4-77 961402 .93 644490 5.70 855510 13 19 606179 4-77 961846 .93 644882 5.70 355i68 II 5o 606465 4-76 961290 •93 645174 5.69 354826 10 5: 9.606751 4-76 0-961235 .93 9.645516 5.69 K>. 354484 I 53 607086 1 4-76 961 170 961123 .93 645857 5.69 ' 854143 53 607822 1 4-75 •9? 646199 5.69 35880I I 54 607607 1 4-75 961067 93 646540 5-68 353460 55 607892 4-74 961011 .93 646881 5-68 353119 , 352778 352488 5 56 608177 4-74 960955 .93 647222 5-68 4 57 608461 4-74 960899 960843 .93 647562 5-67 3 58 608745 1 4-73 •94 647908 648243 5-67 352097 351757 2 ^ 609029 609813 4-73 960786 •94 5.67 I 60 4-73 960780 •94 648583 5.66 1 351417 Cosine ! D. Sine 1 D. Cotan^. 1 D. • Tang. M. (66 DSOl IBES.) 12 (84 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotang. 60 o 9'6o93i3 4-73 9 • 960780 •94 9-648588 5-66 10.351417 I 609597 609880 4 .72 • 960674 -94 648928 5 66 35107-7 350737 35c39S 85oo58 So i 4 ■ 72 960618 .94 649268 5 66 5§ 3 610164 4 •72 960561 •94 649602 5 66 ^ ' 4 610447 4 •71 96o5o5 .94 649942 5 65 5 610729 4 •71 960448 •94 65o28i 5 65 349719 55 1 6 611012 4 .70 960892 .94 65o62o 5 65 349880 54 1 I 61 1 294 4 .70 960835 .94 650959 5 64 349041 53 ; 61 1 576 4 .70 960279 .94 65i297 65 I 636 5 64 848703 ! 5a 9 6ii858 4 .6q 960222 .94 5 64 348364 1 5i , 10 612140 4 69 960165 •94 651974 5 63 348026 5o II 9-612421 4 69 0-960109 -95 9-652812 5 63 10-847688 it 12 612702 4 68 960052 .95 65265o 5 63 847350 i3 612983 4 68 959995 .95 652988 658826 5 68 347012 11 14 618264 4 .67 959988 -95 5 62 346674 i5 613545 4 67 959882 .95 653668 5 62 846887 45 i6 618825 4 67 959825 -95 654000 5 62 846000 44 \l 6i4io5 4 66 Q59768 .95 654337 5 61 845668 43 6i4385 4 66 9597 II -95 654674 5 61 845826 42 19 614665 4 66 959654 .95 655oii 5 61 844989 41 20 614944 4 65 959596 .95 655348 5 61 844652 40 31 9'6i5228 4 65 9-959589 .95 9-655684 5 60 10-844316 1? 22 6i55o2 4 65 959482 .95 656o2o 5 60 348980 23 615781 4 64 959425 .95 656356 5 60 848644 u 24 616060 4 64 959868 -95 656692 5 59 843808 25 6i6338 4 64 959810 .96 657028 5 59 342972 85 26 616616 4 63 959253 .96 657864 5 .59 342686 34 11 616894 4 68 959195 -96 657699 5 .59 842801 83 617172 4 62 959188 -96 658o34 5 -58 341966 32 19 617450 4 62 959081 .96 658869 5 •58 841681 3i 3o 617727 4 62 959028 .96 658704 5 •58 341296 3o 3i 9'6i8oo4 4 61 9-958965 .96 9-659089 5 •58 10-340961 ll 32 618281 4 61 958908 -96 659878 5 57 840627 33 6i8558 4 61 958850 -96 659708 5 57 340292 889958 U 34 618834 4 60 958792 .96 660042 5 57 35 619110 4 60 958784 -96 660876 5 57 889624 25 36 619886 4 60 958677 -96 660710 5 56 33q2oo 24 ll 619662 4 59 958619 -96 661048 5 56 388957 23 619988 4 59 958561 .96 661877 5 56 888628 22 39 620218 4 59 9585o8 •97 661710 5 55 888290 21 40 620488 4 58 958445 •97 662043 5 55 337957 20 41 9-620763 4 58 9-958887 •97 9-662876 5 55 10-387624 \t 42 621088 4 57 958829 •97 662709 5 54 887291 336958 43 621818 4 57 958271 •97 668042 5 54 17 44 621587 4 57 958218 •97 668875 5 54 336625 I^ 4i' 621861 4 56 9581 54 •97 668707 5 54 886293 i5 4f. 622135 4 56 958096 958088 •97 664089 5 58 335961 i4 % 622409 4 56 •97 664371 5 58 335629 i3 622682 4 55 957979 •97 664708 5 53 335297 12 49 622956 4 55 957921 •97 665o85 5 53 884965 II 5o 628229 4 55 957868 •97 665366 5 52 884684 10 5i 9-6285o2 4 54 9-957804 •97 9-665697 5 52 10 •884803 « 52 628774 4 54 957746 .98 666029 5 52 338971 d 53 624047 4 54 957687 957628 .98 666360 5 5i 338640 I 54 624819 4 58 .98 666691 5 5i 333809 55 6245,91 4 53 957570 .98 667021 5 5i 332979 5 56 624868 4 53 957511 .98 667852 5 5i 882648 4 U 625i35 4 52 957452 .98 667682 5 5o 332818 3 625406 4 52 957898 957835 .98 668018 5 5o 331987 2 59 625677 625948 4 52 -98 668848 5 5o 331657 33i326 I 6o 4»5i 957276 -98 668672 5 5o M^ Goeino D. Sine D. Cotang. D. Tang. (60 DEGRESS.^ SINES AND TANGENTS. ^25 DEGREES.) 4fi M. Sine 1 D. Cosine D. 1 Tang. D. i Cotang. o| 9-625948 4-5i 9.957276 .98 9-668673 5'5o ' I0'33i327 60 I 626219 4-5i 957217 .98 669002 5.49 ! 330998 ^ 7 i 626490 4-5i 957158 .98 669332 5-49 330668 3 626760 4-5o 957099 .98 669661 5'4q 33o339 57 4 627030 4-5o 957040 .98 669991 5.48 330009 56 5 627300 4-5o 956981 .98 670320 5.48 i 329680 55 6 627570 4-49 ■■ 956921 •99 670649 5.48 ' 329351 54 2 627840 628109 628378 4-49 ! 956862 •99 670977 671306 5'48 32Q023 53 4-49 1 956803 •99 5.47 328694 52 9 4-48 1 956744 •99 671634 5.47 328366 5i 10 628647 4.48 956684 •99 671963 5'47 328037 5o 11 9'6289i6 4-47 9.956625 •99 9.672291 5.47 10-327709 '^ n 629185 4-47 i 956566 •99 672619 5.46 327381 a 629453 4-47 , 4-46 . 9565o6 •99 672947 5.46 327053 47 14 629721 956447 •99 673274 5'46 326726 46 i5 629989 4-46 956387 •99 673602 5'46 326398 45 i5 630257 4.46 956327 •99 673929 5'45 326071 44 17 63o524 4-46 g56268 •99 674257 5'45 325743 43 18 630792 4-45 956208 I'OO 674584 5.45 325416 43 »9 631039 4-45 956148 1. 00 674910 5'44 325090 41 30 63i326 4-45 956089 I -00 675237 5-44 324763 40 21 9-63i593 4-44 9.956029 I'OO 9-675564 5-44 10-324436 It 22 63i85g 4.44 955969 I -00 675890 5-44 324110 23 632I2D 4-44 955909 I -00 676216 5'43 323784 U 24 632302 4-43 955849 1. 00 676543 5'43 323457 25 632658 4-43 955789 I'OO ! 676869 5'43 323i3i 35 26 632923 4-43 955729 I -00 677194 5.43 322806 34 s 633189 4-42 955669 I'OO 67-/520 5.42 322480 33 633454 4-42 955609 I'OO 67 1846 670171 5.42 322154 32 29 633719 4-42 955548 I'OO 5.42 321829 3i 3o 633984 4-41 955488 I'OO 678496 5-43 32i5o4 3o 3i 9-634249 4-41 9.955428 I'OI 9-678821 5'4i I0'32n79 29 32 634514 4-40 955368 I'OI 679146 5'4i 320854 28 33 634778 4.40 955307 I'OI 679471 5'4i 320529 320205 11 34 635042 4.40 955247 I'd 679795 680120 5.41 35 635306 4.39 955186 I'OI 5.40 319880 25 36 635570 4-39 955126 I'OI 680444 5'4o 319556 24 37 635834 4-3o 4-38 955o65 I'OI 680768 5-40 319232 23 38 636097 955oo5 I'OI 681092 5'4o 318908 22 39 636360 4-38 954944 I'OI 681416 5'39 3 1 8584 21 40 636623 4-38 954883 I'OI 681740 5.39 318260 20 41 9-636886 4-37 9 954823 I'OI g. 682063 5.39 10.317937 19 42 637148 4-37 954762 I'OI 682387 5.39 317613 19 43 6374U 4-37 954701 I-OI 682710 5-38 317290 17 44 637673 4-37 95464c I'OI 683o33 5.38 316967 16 45 637935 4-36 954579 I'OI 683356 5.38 ? 16644 i5 46 638197 4-36 954518 I '02 683679 5.38 3i632i 14 47 638458 4-36 954457 I '02 684001 5.37 3 1 5999 i3 48 638720 4-35 954396 954335 I '02 684324 5'37 315676 13 49 638981 4-35 I '02 684646 5.37 3 1 5354 II 5c 639242 4-35 954274 I '02 684968 5.37 3i5o32 JO 5i 9 -639503 4-34 9.954213 I '02 9-685290 5.36 10-3147/0 3 I 4388 I 5j 639764 4-34 954152 1.02 685612 5.36 53 640024 4-34 954090 1.02 685934 5.36 3 1 4066 7 54 640284 4-33 954029 I '02 686255 5.36 3 1 3745 6 55 640544 4-33 953968 1. 02 686577 686898 5-3% 3 1 3423 5 56 640804 4-33 953006 953845 1.02 5.35 3i3io2 4 57 641064 4-32 1-02 687219 5.35 312781 3 58 641324 4-32 953783 1-02 687540 5.35 312460 3 59 64 1 584 4-32 953722 I -03 687861 5.34 3i2i39 3ii8i8 I 60 64184a 4-3i 953660 I -03 688182 5 34 Cosine D. Sine D. Cotang. D. l&n?. _M^ (64 [ DBGR ESS.) 14 (28 DEGREES.) ^ TABLE OF LOGARITHMIC HL I Sine D. Cosine I ). Tang. D. Cotar.g. 9-641842 4-3i 9-953660 I o3 9-688182 5.34 io-3ii8i8 1 60 ; I 642 1 01 4 3i 953599 I o3 688502 5 34 811498 5o 2 642860 4 3i 953537 I o3 688828 5 34 811177 58 3 642618 4 3o 953475 I o3 689148 5 33 810857 57 i 4 642877 4 3o 953413 I o3 689468 5 83 3io537 •X) ' 5 643 1 35 4 3o 953352 I o3 689788 5 33 iI02I7 55 i 6 643393 4 3o 953290 I o3 690108 5 83 809897 54 i 2 643650 4 29 953228 I 08 690428 5 88 809577 53 643908 4 29 953166 I o3 690742 5 82 809258 . 52 9 644165 4 29 953104 I 08 691062 5 32 .08988 5i 1 IC 644423 4 28 953042 I 08 691881 5 82 808619 5o II 9-644680 4 28 9-952980 1 04 9-691700 5 3i io.3o83oo S 12 644936 4 28 952018 I 04 693019 5 81 807981 i3 645 I o3 4 27 952855 I 04 692888 5 3i 807662 47 14 645450 4 37 952798 I 04 692656 5 81 807844 46 i5 645706 4 27 952781 I 04 692975 5 3i 807025 45 i6 645962 4 26 952669 I 04 698298 5 80 806707 44 \l 646218 4 26 952606 1 04 698612 5 80 806888 43 646474 4 26 952544 I 04 698980 5 3o 806070 42 19 646729 4 25 952481 I 04 694248 5 80 805752 41 20 646984 4 25 952419 04 694566 5 29 805484 40 21 9-647240 4 25 9-952856 I 04 9-694888 5 29 io-3o5ii7 3q ' i ^? 647494 4 24 952294 I 04 695201 5 29 804799 38 ! a3 647749 648004 4 24 952281 I 04 695518 5 29 804482 37 ' i 24 4 24 952168 I o5 695886 5 20 804164 36 1 1 25 648258 4 24 952106 I o5 696153 5 28 808847 35 26 648512 4 23 952048 I o5 696470 5 28 3o858o 34 . ?J 648766 4 23 951980 1 o5 696787 5 28 808218 33 , 649020 4 23 951917 I o5 697103 5 28 802807 32 ?9 649274 4 22 95 1 854 I o5 697420 5 27 3o258o 3i 3o 649527 4 22 951791 I o5 697786 5 27 802264 3o i 3i 9-649781 4 22 9-951728 I o5 9-698053 5 27 10.801947 ll 32 65oo34 4 22 95 1 665 I o5 698869 5 27 801681 33 050287 4 21 951602 I o5 698685 5 26 3oi3i5 u\ 34 65o539 4 21 951389 I o5 699001 5 26 800999 35 650792 4 21 951476 I o5 6q98i6 5 26 800684 25 1 36 65 1044 4 20 951412 1 o5 699682 5 26 800868 24 ll 65x297 4 20 951849 I 06 699947 5 26 8ooo53 23 65 1 549 4 20 951286 I 06 700268 5 25 299787 22 39 65i8oo 4 19 951222 I 06 700578 5 25 299422 21 40 652o52 4 19 951159 I 06 700898 5 25 299107 20 4i 9-6523o4 4 19 9951096 I 06 9-701208 5 24 10-298792 ^2 18 42 652555 4 18 951082 I 06 701528 5 24 298477 43 652806 4 18 900968 I 06 701887 5 24 298168 \l 44 653o57 4 18 950905 I 06 702152 5 24 297848 45 6533o8 4 18 950841 I 06 702466 5 24 297534 i5 1 46 653558 4 n 950778 I 06 702780 5 23 2q7220 14 ! 47 6538o8 4 17 950714 I 06 708095 5 23 29690J !3 1 48 654059 4 \i 95o65o I 06 708409 5 23 296091 12 1 49 654309 4 95o586 I 06 •p3723 5 23 296277 II 5o 654558 4 16 95o522 I 07 704086 5 22 295964 10 5i 9-654808 4 16 9-950458 I 07 9-704350 5 32 15-295650 g 52 655o58 4 16 95o8q4 I 95o3jo I 07 704668 5 22 295337 295023 8 53 655307 4 i5 07 704977 5 22 I 54 655556 4 i5 950266 I 07 705290 5 22 294710 294807 55 6558o5 f i5 950202 I 07 7o56o3 5 21 5 56 656o54 4 14 95oi38 I 07 705916 5 21 294084 4 u 656302 4 14 950074 I 07 706228 5 21 298772 3 656551 4 14 gSooio I 07 706541 5 21 298459 2 59 656799 4 i3 949045 I 07 706854 5 21 298146 I 60 657047 4i3 949881 I 07 707166 5-20 292834 CJosine D. Sme I ). Cotang. J). Tana^. M. i (63 DEORBE8.) SINES AND TANGENTS. (27 DEGREES.) 46 Sine D. Cosine D. Tfing. D. Cotang. 1 9.657047 657293 4-i3 9-949881 1-07 9.707166 5-20 10-292884 60 I 4-i3 949816 1-07 707478 5 20 292522 ^ 3 657542 4-12 949752 1-07 707790 5 20 292210 3 657700 4-12 949688 I -08 708102 5 •20 291898 11 4 6500^7 4-12 949628 i-o8 708414 5 19 291586 5 658284 4-12 949558 i-o8 708726 5 19 291274 5D 6 658531 4-11 949494 1.08 709087 5 19 290968 54 I 658778 4-11 949429 I -08 709849 5 19 290651 53 659025 4-11 949864 1-08 709660 5 10 290840 52 9 659271 4*10 949800 1.08 709971 5 18 290029 5i 10 659517 4*10 949235 I -08 710282 5 18 289718 5o II 9 '659763 4*10 9-949170 i-o8 9.710593 5 18 10-289407 289096 49 12 660009 4*09 949105 i-o8 710904 5 18 48 i3 660255 4-09 949040 1-08 711215 5 18 288785 47 14 66o5oi 4*09 948975 1.08 7ii525 5 17 288475 46 i5 660746 4-09 948910 i-o8 711886 5 17 288164 45 i6 660991 4-o8 948845 1.08 712146 5 17 287854 44 \l 661286 4-o8 948780 1-09 712456 5 17 287544 43 661481 4-o8 948715 1-09 712766 5 16 287284 42 19 661726 4-07 948650 1-09 718076 5 16 286924 41 20 661970 4-07 948584 1-09 718886 5 i6 286614 40 21 9-662214 4*07 9-948519 1-09 9.718696 5 16 10.286804 89 22 662459 4-07 948454 1-09 714005 5 16 283995 88 23 662703 4-o6 948888 1-09 714814 5 i5 285686 37 24 662946 4-o6 948828 1-09 714624 5 i5 285876 36 25 663190 4-o6 948257 1-09 714933 5 i5 285067 35 26 663483 4-o5 948192 1-09 715242 5 i5 284758 34 11 663677 4-o5 948126 1-09 7i555i 5 14 284449 33 663920 4-o5 948060 1-09 7i586o 5 14 284140 32 ?9 664163 4-o5 947995 1-10 716168 5 14 288882 3i So 664406 4-04 947929 I-IO 716477 5. 14 283523 3o 3i 9.664648 4-04 9-947863 I'lO 9.716785 5. 14 10.288215 29 32 664891 4-04 947797 9477^1 1-10 717098 5. i3 282907 28 33 665 I 33 4-o3 I-IO 717401 5. i3 282399 27 34 665375 4-o3 047665 I-IO 717709 5. i3 282291 26 35 665617 4-o3 947600 I-IO 718017 5. i3 281988 25 36 665859 4-02 947538 1-10 718825 5. i3 281670 24 ll 666100 4-02 947467 1-10 718688 5. 12 281867 23 666342 4-02 947401 l-IO 718940 5. 12 281060 22 39 666583 4-02 947835 1-10 719248 5. 12 280752 21 40 666824 4-01 947269 1-10 719555 5. 12 280445 20 41 9-667065 4'oi 9-947208 I-IO 9.719862 5. 12 10-280188 \t 42 667805 4*01 947186 1-11 720169 5. 11 279881 43 667546 4*01 947070 l-ll 720476 720783 5. 11 279524 '7 44 667786 4*00 947004 1-11 5- II 270217 16 45 668027 4*00 946937 l-II 721089 ; 11 278911 ! 1 5 46 668267 4-00 946871 1-11 721896 5. II 278604 14 % 6685o6 ^■99 946804 1-11 721702 5. 10 278298 i3 668746 ^•99 946788 I'll 722009 5- 10 277685 la 49 668986 ^*99 946671 l-Il 722813 5. 10 II 5o 669225 3-99 946604 I-ll 722621 5. 10 277879 10 5i 9-669464 3.98 9.946538 1-11 9.722927 5. 10 10.277073 ? 52 669708 3-98 946471 I-II 728282 5. 09 276768 53 669942 3-98 946404 l-ll 723538 5. 09 276462 7 i 54 670181 3-97 946887 l-I] 728844 5. 09 276156 6 55 670419 ^97 946270 1-1] 724149 5. 09 275851 5 56 670658 ?'97 946208 1-12 724454 5. S 275546 4 u 670896 671184 X*97 946186 1-12 724759 725o65 5 275241 3 3-96 946069 1-12 5 08 274935 274631 3 59 671872 3-96 946002 1-12 725369 5 08 I 6o U-.J 671609 3-96 945935 1-12 725674 5.08 274826 Cofline D. Sine D. Cotang. D. Tang. M. (62 DEGRSJIS.) i6 (28 DEGREES.) A TABLE OF LOGARITHMIC M. Biiio D. Cosine D. Tang. D. Cotang. 60 9.671609 3.96 9 •945085 943868 1*13 9.725674 5 08 10.274336 1 , 671847 3 .95 I-I3 725979 5 08 374031 So 9 672084 3 .95 943800 1-13 ! 726284 5 • 07 3787x6 58 3 672321 3 .95 945788 I.I3 ,26588 ' 5 .07 2784x2 57 273108 56 4 672558 3 .93 945666 I'13 736893 5 •07 5 672795 673082 3 .94 945598 945531 I.X2 737197 5 •07 272803 55 6 3 .94 1.13 737501 5 •07 272499 54 272195 53 I 673268 8 .94 943464 113 737805 5 .06 673505 3 .94 945896 1-13 728109 5 •06 27X89X 53 9 673741 3 .93 945328 I.l3 738413 5 .06 271588 5i 10 6^39- 3 •93 945261 113 7387x6 5 1 .06 271284 5. II 9 674313 8 •93 9.945198 113 9.729020 5 .06 10.270980 S 13 6744/18 8 92 945125 i-i3 729823 1 ^ .o5 270677 i3 674684 3 93 945o58 ii3 729626 ' 5 •o5 270874 47 14 674919 3 92 944990 i.i3 729929 780283 i 5 .05 270071 46 i5 675x55 3 92 944922 i.i3 1 5 .05 269767 45 i6 675390 3 91 944854 i.i3 780535 5 .o5 269465 44 \l 675634 3 91 944786 i.i3 780888 5 •04 269162 268859 43 575359 3 91 944718 i.x3 781x41 5 04 43 ^9 676094 3 91 944650 i.i3 78x444 5 .04 268556 41 20 676328 3 90 944582 1.X4 78x746 5 •04 268254 40 31 9.676563 3 90 9.944514 1-14 9-782048 5 04 10.267952 ^ 33 676796 3 90 944446 114 73235X 5 o3 267649 33 677080 3 90 944877 1.14 732653 5 o3 267847 u 24 677264 8 °9 944809 1-14 782935 5 o3 267045 25 677498 6777^1 8 89 944341 1.14 788257 5 o3 266743 35 36 3 89 944172 1.14 788558 5 o3 266442 34 11 677964 3 88 944104 1. 14 788860 5 03 266140 33 678197 3 88 9/r'io86 X.14 784162 5 03 265888 32 29 678430 3 88 943067 943899 1.14 784468 5 02 265537 3i 3o 678663 3 88 1. 14 734764 5 02 265236 3o 3i 9.678895 3 87 9.948880 X.14 9 -735066 5 03 10.264984 ^ 33 679128 8 87 948761 1.X4 735867 5 02 264688 33 679860 8 87 943698 x.i5 785668 5 OX 264882 U 34 679393 3 87 943624 i.i5 735969 5 01 264081 35 679824 3 86 948555 x.i5 786269 5 01 26878 X 25 36 68oo56 3 86 943486 i.i5 786570 5 01 268480 24 37 680288 3 86 948417 i.i5 736871 5 OX 268129 33 38 680319 3. 85 948848 x.i5 787171 5 00 262829 23 39 680700 3. 85 948279 i.i5 787471 5 00 262529 21 40 680982 3 85 943210 x.x5 737771 5 00 262239 30 41 9.681213 8. 85 9.948141 x.i5 9 -78807 1 5 00 10.261939 :? 42 681443 8. 84 948072 i.i5 788871 5 00 261629 43 681674 3- 84 948008 1x5 788671 4- 99 26x829 17 j 44 681905 3. 84 942984 x.i5 738971 4- 99 26x029 j 16 i 45 682135 3. 84 942864 x.i5 789271 4- 99 260729 ; i5 i 46 682365 3- 88 942795 X.16 789570 4 99 260480 ! 14 ! il 682595 3. 88 942726 i.x6 789870 4 99 260180 j 3 i 682825 8. 88 942656 I -16 740160 4- 99 259881 \ la 1 49 683o55 3. 88 942587 1-16 74046S 4 9^ , 259532 II 5o 688284 3- 82 9435x7 1.16 740767 4 98 259288 le 5i 9 6835i4 3. 82 9.942448 1-16 9.741066 4 98 10.258934 I 52 683743 8. 82 942878 I -16 741865 4- 98 258635 53 683972 8. 82 942808 1. 16 741664 4- 98 258386 I 54 684201 8. 81 942289 i-i6 741962 4- 97 258o38 55 684480 3. 81 i 942169 1.16 742261 4- 97 357789 5 56 684658 3. 81 i 942099 1. 16 742559 4- 97 257441 4 5^ 684887 3- 80 : 942029 i-i6 ' 743858 4- 97 257x43 ■ 3 J 685ii5 3. 80 : 941959 x.i6 ' 7481 56 4- 97 356844 2 59 685343 3. So i 941889 I.X7 743454 4- 97 256546 I 60 685571 3.80 941819 1. 17 748753 4-96 356348 1 Coeiue D. ! Sine D. Cotang. D. 1 Tang. 1 M. (61 DEaRESS.) I SINES AND TANGENTS. (29 DEGREES.; 47 M. Sine 1 D. Cosine D. Tang. D. Cotang. r 1 o 9-685571 3.80 9-941819 I-I7 9-743752 4.96 10-256248 60 I 685799 3- 79 941749 1. 17 744o5o 4 .96 255950 5o 255652 i 58 2 686027 3. 79 941679 117 744348 4 .96 3 686254 3- 79 941609 1. 17 744645 4 .96 255355 ' 57 255o57 ; 56 4 686482 3- ]l 941539 1.17 744943 4 96 5 686709 3- 941469 1. 17 745240 4 9^ 254760 i 55 6 686936 3- 78 941398 117 745538 4 9^ 254462 54 I 687163 3- 78 941328 I-I7 745835 4 9? 254165 53 687^89 687616 3. 78 941258 1.17 746132 4 9^ 253868 52 9 3- 77 941187 1. 17 746429 4 9^ 253571 5i 10 687843 3. 77 941 117 1.17 746726 4 95 253274 5o II 9-688069 688295 3 77 9-941046 1.18 9-747023 4 94 10-252977 % 12 3. 77 940975 1.18 747319 4 94 252681 i3 688521 3 76 940905 1. 18 747616 4 94 252384 47 14 688747 3 76 940834 1.18 747913 4 94 252087 46 i5 688972 3 76 940763 i-i8 748209 4 94 251791 45 \6 689198 3 76 940693 1. 18 7485o5 4 93 251495 44 \l 689423 3 75 940622 1. 18 748801 4 93 251199 250900 43 689648 3 75 94055 I 1x8 749097 4 93 42 ^ 689873 3 7? 940480 i-i8 749393 4 93 2 50607 41 20 690098 3 75 940409 1. 18 749689 4 93 25o3ii 40 21 9690323 3 74 9 -940338 1. 18 9-749985 4 93 io.25ooi5 ^? 22 690548 3 74 940267 1-18 750281 4 92 249719 23 690772 3 74 940196 i-i8 750576 4 92 249424 3l 24 690996 3 74 940125 1-19 750872 4 92 249128 36 25 691220 3 73 940054 1.19 751167 4 02 248833 35 26 691444 3. 73 939982 119 751462 4 92 248538 34 27 28 691668 3 73 939911 119 751757 4 92 248243 33 691892 3 73 939840 I 19 752o52 4 91 247948 32 ?9 6921 1 5 3 72 939768 1. 19 752347 4 9» 247653 3i 3o 692339 3 72 939697 119 752642 4 91 247358 3o 3i 9 692562 3 72 9-939625 1-19 9.752937 4 9» 10.247063 ?? 32 692785 3 71 939554 I -19 75323i 4 91 246769 33 693008 3 71 939482 1. 19 753526 4 91 246474 \L 34 693231 3 7» 939410 I 19 753820 4 90 246180 35 693453 3 71 939339 1-19 7541 1 5 4 90 245885 25 36 693676 3 70 939267 1-20 754400 4 90 245591 24 U 693898 3 70 939195 1-20 754703 4 90 245297 23 694120 3 70 939123 1-20 754997 4 90 245oo3 22 39 694342 3 70 939052 1-20 755291 4 90 244709 244415 21 40 694564 3 69 938980 1-20 755585 4 89 20 41 9 -694786 3 69 9.938908 1-20 9.755878 4 ^9 10-244122 \% 42 695007 3 69 938836 1-20 756172 4 ^9 243828 43 695229 3 69 938763 1-20 756465 4 ?9 243535 \l 44 695450 3 68 938691 1-20 756759 4 ^9 243241 45 695671 3 68 938619 1-20 757052 4 ^ 242948 i5 46 695892 3 68 938547 1-20 757345 4 88 242655 14 f 47 48 6961 i3 3 68 938475 1-20 757638 4 88 242362 i3 696334 3 67 938402 1-21 757931 4 88 242069 13 ^9 690554 3 67 938330 1-21 758224 4 88 241776 11 5o 690775 3 67 938258 1-21 758517 4.88 241483 ID 5i 9-696995 3 t>7 9-938185 i'21 9.758810 4.88 10-241190 I 53 697215 3 -66 9381 i3 1-21 759102 4.87 240898 53 697435 3 -66 938040 1-21 759395 4-87 240605 I 54 607654 3 -66 937967 I-2I 759687 4.87 24o3i3 55 697874 698094 3 -66 937895 1-21 759979 4.87 240021 i> 56 3 .65 937822 I-2I 760272 4.87 239728 4 U 698313 3 -65 937749 1-21 760564 4.87 239436 3 698532 3 -65 937676 I-2I 760856 4.86 239144 238852 a 59 698751 3 -65 937604 ' -21 761148 4.86 I 60 698970 3 _ J .64 937531 I-2I 761439 4-86 238561 CoBine 0. ' Sine D. Cotang. 1 D. Tang. _M._^ 27 (60 DBORBBS.) 48 (SO DEGREES.) A TABLE OF L0GARlTHMi:5 M. Bino D. Cosine 1 D. Tan^. D. Ootan^f. ~ o 9 •698970 699189 3.64 9-937531 1 I •21 9-761439 4-86 10-238561 60 I 3 .64 937458 I •22 761781 4 -86 288269 5o 3 699407 3 •64 937385 I •22 762028 4 -86 287977 56 2876?^ 57 3 699626 3 64 937312 I •22 762814 4 -86 4 699844 3 63 987288 I -22 762606 4 •85 387894 56 5 700062 3 63 937165 I ■22 762897 4 -85 287103 55 6 700280 3 63 987092 I -22 768188 4 •85 286812 54 7 7P0498 3 63 987019 I •22 7634-9 4 -85 236521 53 700716 3 63 986046 I -22 768770 4 -85 236280 52 9 700933 3 62 986872 I -22 764061 4 •85 285939 235648 5i 10 7oii5i 3 62 986799 I -22 764352 4 -84 5o II 9-701368 3 62 9.986725 1 22 9-764643 4 -84 10.235357 it 12 701585 3 62 986652 I -23 764988 4 84 285067 i3 701802 3 61 986578 ' I -23 765224 4 84 284776 284486 s 14 702019 3 61 9865o5 I 23 765514 4 84 'i 702236 3 61 98648 1 I 23 7658o5 4 84 284195 45 i6 702452 3 61 986857 1 23 766095 4 84 288905 44 \l 702669 702885 3 60 986284 I -23 766385 4 83 2836i5 43 3 60 986210 I 23 766675 4 83 283825 42 19 7o3ioi 3 60 986186 1 23 766965 4 88 233o35 41 20 703317 3 60 986062 I 23 767255 4 83 282745 40 21 9 -703 533 3 59 9.985988 I 23 9-767545 4 83 10-282455 39 38 22 703749 3 59 985914 I 23 767884 4 88 282166 23 703964 3 59 985840 I 23 768124 4 82 281876 11 24 70417? 704395 3 59 935766 I 24 768418 4 82 23 1 587 25 3 ll 985692 I 24 768708 4 82 281297 281008 35 26 704610 3 935618 I 24 768992 4 82 34 U 704825 3 58 935543 I 24 769281 4 82 280719 33 7o5o4o 3 58 985469 1 935893 I 24 769570 4 82 280480 32 29 7o5254 3 58 24 769860 4 81 280140 3i 3o 705469 3 57 935320 I 24 770148 4 81 22q852 3o 3i 9 •705683 3 ^7 9.985246 I 24 9-770487 4 81 10-229568 2Q 28 32 705898 3 ?7 985171 I 24 770726 4 81 229274 33 7061 1 2 3 57 985097 I 24 771015 4 81 228985 27 34 706326 3 56 935022 I 24 771808 4 81 228697 228408 26 35 706539 706753 3 56 984948 I 24 771592 4 81 25 36 3. 56 984878 I 24 771880 4 80 228120 24 u 706967 3 56 984798 I 25 772168 4 80 227882 23 707180 3. 55 984728 I 25 772457 4- 80 227543 22 39 707393 3 55 984649 J 25 772745 4 80 227255 21 40 707606 3. 55 984574 I 25 778083 4 80 226967 20 41 9.707819 3. 55 9-984499 I 25 9-778821 4- 80 10.226679 [I 42 708032 3 54 984424 I 25 778608 4- 79 226892 43 708245 3- 54 984849 I 25 778896 4- 79 226104 1 17 2258i6 ! 16 44 708458 3 54 984274 » 25 774184 4- 79 ^f 708670 3 54 984195 I 25 774.;7i 4 79 225529 i5 46 708882 3 53 984123 i I- 25 774759 4 79 225241 I4 % 709094 3- 53 984048 I 25 775046 4- 79 224954 I J 709306 3 53 988975 i I 25 775838 4- 224667 '2 1 i^ 709518 3. 53 933898 1 I 26 775621 4- 78 224879 -I ! 5o 709730 3. 53 988822 I 26 775908 4- 78 224092 10 5i 9.709941 3 52 9-933747 I- 26 9-776195 4- 78 io.2238o5 i 52 ;ioi53 3. 52 988671 I- 26 776482 d- 78 2235i8 53 710364 3 52 983596 I 26 776769 4- 78 228281 l\ 54 710575 71.0786 3. 52 938520 I 26 777055 4- 78 222945 55 3 5i 988445 ! I- 26 777342 4- 78 222658 5 1 56 7x0997 711 208 3- 5i 988860 1 - 933298 I • 26 777628 4^ 77 222872 4 U 3 5i 26 777915 A- 77 222085 3 711419 3. 5i 988217 I- 26 778201 4- 77 221799 22l5l2 3 59 711629 3 5o 988141 I* 26 778487 4- 77 I 60 711839 3.50 988066 I « 26 778774 4-77 221226 Cosine D. Sine r ►, Cotang. Tang. M. (59 DE GRI BE8.) SINES AND TANGENTS. C31 DEGREES.) 49 M. Sine D. Cosine D. • Tang. D. CotftTlff. 9-711839 3-5o 9-933066 1-26 9-778774 4-77 10-221226 60 I 7i2o5o 3-50 982990 1-27 779060 4-77 4-76 220940 5o 220654 58 2 712260 3-50 932914 1-27 779346 3 712469 3.49 982838 1.27 779682 4-76 220868 57 220082 5o 4 712679 7128B9 713098 3-49 982762 1.27 779918 4-76 5 3-49 982685 1-27 780208 4-76 2x9797 ; 55 6 3-49 982609 1-27 780489 4-76 2195XX 54 i 7i33o8 3.49 93253w 1-27 780775 4-76 2X0225 2x8940 53 7i35i7 3.48 982457 1.27 781060 4-76 53 9 713726 3-48 932880 1-27 781846 4-75 2x8654 5i 10 713935 3.48 982804 1-27 781681 4-75 2x8369 5o II 9.714144 3.48 9-982228 1.27 9-781916 4-7^ 10-218084 49 48 la 714352 3-47 982151 1-27 1.28 782201 4-75 217799 i3 714561 3-47 982075 782486 4-75 2x7514 % 14 71476Q 714978 3-47 981998 1-28 782771 4-75 2x7229 i5 3-47 981921 1-28 788056 4-75 216944 45 i6 7i5i86 3-47 98x845 1-28 788841 4-75 216659 44 \l 715394 3.46 981768 1-28 788626 4-74 216874 43 7i56o2 3.46 981691 1-28 788910 4-74 2x6090 42 »9 715809 3.46 981614 1-28 784195 4-74 2i58o5 41 20 716017 3.46 981587 1-28 784479 4-74 2X5521 40 ai 9-716224 3.45 9-981460 1.28 9-784764 4-74 io-2i5286 39 22 716432 3.45 981888 1.28 785048 4-74 214952 88 23 716639 3.45 981806 1.28 785832 4-73 2x4668 ll 24 716846 3-45 981229 1-29 7856x6 4-73 2x4884 25 717053 3.45 981152 1-29 785900 4-73 214x00 35 26 717259 3.44 98x075 1-29 786x84 4-73 2x38x6 34 s 717466 3-44 980998 1-29 786468 4-73 2x3532 33 717673 3-44 980921 1-29 786752 4-73 218248 32 ?9 717870 7180S5 3-44 980848 1-29 787086 4-73 212964 3i 3o 3.43 980766 1-29 787819 4-72 212681 3o 3i 9-718291 3.43 9-980688 1-29 9-787608 4-72 10-2x2897 It 32 718497 3-43 980611 1-29 787886 4.72 2X2XX4 33 718703 3-43 98o538 1-29 788170 4-72 2I1880 27 34 718909 3.43 980456 1-29 788453 4-72 2xx547 26 35 719114 3.42 930878 1-29 788786 4-72 2x1264 25 36 719320 3.42 980800 1-30 7890x9 4-72 2x0981 24 ll 719525 3-42 980228 i-3o 789802 4-71 2x0698 23 719730 3-42 980145 i-3o 789585 4-71 2X04x5 22 39 719935 3-41 980067 i-3o 789868 4-71 2x0x82 21 40 720140 3.41 929989 I -30 7901 5i 4-71 209849 20 41 9-720345 3-41 9'9299Xi 1.30 9-790488 4-71 10-209567 19 42 720549 3-41 929888 1-30 7907x6 4-71 209284 43 720754 3-40 929755 1.80 790999 4-71 209001 \l 44 720958 3-40 929677 1-30 79x281 4-71 2007x9 45 721162 3.40 929599 1-80 79x563 4-7® 208487 i5 46 72i366 3-40 929521 1-80 791846 4-70 2081 54 14 47 721570 3-40 929442 X.30 792x28 4-70 207872 i3 4S 721774 3.39 929864 1-81 7924x0 4-70 207590 12 49 721978 722181 3-39 929286 1-81 792692 4-70 207808 11 5o 3-39 929207 1-31 792974 4-70 207026 10 5i 9-722385 3-39 9-929129 1-31 9-798256 4-70 10-206744 t 52 722588 3.39 929060 1-31 798538 4.69 206462 53 722791 3-38 928972 i-3i 798819 4.69 206181 I 54 722994 3-38 928898 i-3i 794x01 4.69 205899 55 723197 3-38 9288x5 i.3i 794888 4-69 2o56i7 5 56 723400 3-38 928786 i-8i 794664 4-69 205336 4 ^ 7236o3 3-37 928657 i-3i 794945 4-69 2o5o55 3 7238o5 3-37 928578 i-3i 796227 4.65 204773 a 59 724007 3-37 928499 1-31 795508 4.68 204492 I 6o 724210 3-37 928420 1-81 795789 4-68 2042 II CoBine D. Sine D. Cotang. 1 D. Tang. _r , (58 t DEGR SIS.) 50 (32 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotang. 1 1 9-724210 3.37 9-928420 I 32 9-795789 4-68 10-204211 60 1 I 724412 3 37 928842 I 32 796070 4 68 208980 u 2 724614 3 36 928263 I 82 796851 4 68 208649 3 724816 3 86 928188 I 82 796682 4 68 208868 57 4 725017 3 86 928104 I 32 796913 4 68 208087 56 5 725219 3 36 928025 I 32 797194 4 -68 202806 55 6 725420 8 85 927946 I 32 797475 4 -68 202525 54 I 725622 3 35 927867 I 82 797755 4 68 202245 53 725823 3 35 927787 I 927708 1 32 798086 4 67 201964 52 9 726024 3 35 32 798816 4 67 201684 5i n 726225 3 3a 927629 I 82 798596 4 67 201404 5o II 9-726426 3 34 9-927549 I 82 9-798877 4 67 IO-20II23 49 12 726626 3 34 927470 I 33 799157 4 67 200848 48 i3 726827 3 34 927890 I 83 799437 4 67 2oo568 47 14 727027 3 34 927810 I 83 799717 4 67 200288 46 i5 727228 3 84 927281 I 88 799997 800277 4 66 200008 45 i6 727428 3 33 927151 I 83 4 66 199728 44 \l 727628 3 33 927071 1 88 800557 4 66 199443 43 727828 8 33 926991 I 83 800886 4 66 I99r64 42 19 728027 3 33 92691 1 I 83 801 1 16 4 66 198884 41 20 728227 3 33 926881 I 38 801896 4 66 198604 40 21 9-728427 3 32 9-926751 I 88 9-801675 4 66 10-198825 39 22 728626 3 32 926071 I 38 801955 4 66 198045 88 23 728825 3 32 926591 I 38 802284 4 65 197766 u 24 729024 3 32 9265ii I 34 8o25i3 4 65 197487 197208 25 729323 3 3i 926481 I 34 802792 4 65 35 26 729422 3 3i 92685i I 34 808072 4 65 196928 34 11 729621 3 3i 926270 I 34 8o885i 4 65 196649 33 729820 3 3i 926190 I 34 808680 4 65 196870 32 29 780018 3 3o 926110 I 34 808908 4 65 196092 3i 3o 730216 3 3o 926029 1 34 804187 4 65 195813 3o 3i 9 -73041 5 3 3o 9-925949 I 3/ 3^ "^4 9-804466 4 64 10-195584 ll 32 73o6i3 3 3o 925868 I 804745 4 64 195255 33 780811 3 3o 925788 1 34 8o5o23 4 64 I94Q77 194698 27 34 781009 3 29 925707 I 34 8o53o2 4 64 26 35 781206 3 29 925626 I 34 8o558o 4 64 194420 25 36 781404 3 29 925545 I 35 8o5859 4 64 194141 24 ll 781602 3 29 925465 I 35 806187 80641 5 4 64 198868 23 781799 8 29 925384 I 35 4 63 198585 22 39 781996 3 28 925808 I 35 806698 4 63 198807 21 40 782198 3 28 923222 I 35 806971 4 63 198029 20 41 9-782890 3 28 9-925i4i i 35 9-807249 4 68 10-192751 10 42 782587 3 28 925o6o I 35 807527 4 63 192478 43 782784 3 28 924079 I 35 807805 4 68 192195 17 44 782980 3 27 924897 I 35 808083 4 63 191917 16 45 733177 3 27 924816 I 35 808861 4 63 191689 i5 46 788878 3 27 924735 I 86 808688 4 62 191862 14 47 788560 788765 3 27 924654 I 36 808916 4 62 191084 i3 48 3 27 924572 1 36 809198 4 62 190807 11 49 788961 3 26 924491 I 36 809471 4 62 190529 11 5o 784157 3 26 924409 I 36 809748 4 62 190252 10 5i 0- 784853 3 26 9-924828 I 36 9-8ioo25 4 62 10-189975 ? 52 734549 3 26 924246 I 36 810802 4 62 189698 53 734744 3 25 924164 I 36 8io58o 4 62 189420 7 54 734939 3 25 924088 I 36 810857 4 62 189143 6 55 785i35 3 25 924001 I 36 811184 4 61 188866 5 56 785830 3 25 928919 I 36 811410 4 61 188590 4 U 785525 3 25 928887 I 86 81 1687 4 61 188818 3 735719 3 24 928755 . I 37 81 1964 4 61 188086 2 59 735914 3.24 928678 I 37 812241 4 61 187750 187483 I 60 1 786109 3-24 923591 I 37 812517 4-6i M. Cofliue D. Sine I ). Cotang. D. Tanff. (57 DEGREES.) snrsa and tangents. (33 degrees.) 51 ^M. Sine D. Cosine D. Tang. D. Cotan^- 60 9-736109 7363o3 3.24 9.923591 1.37 9.812517 4-6i 10.187482 I 3-24 923509 1.37 812794 4-6i 187206 U a 736498 3-24 923427 923345 i.3i 818070 4.61 1 86980 3 736692 736886 3-23 1.87 818847 4-60 186653 U 4 3-23 923263 1.87 818628 4.6c 186877 5 737080 3.23 928181 1.87 818899 814175 814452 4-60 186101 55 6 1 737274 3-23 928098 1.87 4-6o 185825 54 , 2 737467 3-23 923016 1.87 4.60 185548 58 1 737661 3-22 922933 1.37 814728 4.60 185272 52 9 737855 738048 3-22 92205l III 8i5oo4 4.60 184996 5i iO 3-22 922768 815279 4-60 184721 5o II 9-738241 3-22 9.922686 1.38 9.815555 4.59 io.i8/j/|/i5 it i« 738434 3-22 922603 1.88 81 588 1 4.59 184169 188898 i3 738627 8.21 922520 1.88 816107 4-59 47 14 738820 3.21 922438 1-88 816882 4.59 188618 46 i5 739013 3.21 922355 1.88 8i6658 4.59 188842 45 i6 739206 3.21 922272 1.88 816988 4.59 188067 44 \l 739398 3.21 922189 1.88 817209 4.59 182791 43 739500 3*20 922106 1.88 817484 4.59 i825i6 42 19 739783 3.20 922028 1.88 817759 818085 4-5o 182241 41 20 739975 3-20 921940 1.88 4-58 181965 40 21 9.740167 3.20 9.921857 1.39 9.818810 4-58 10.181690 39 22 -40359 3.20 921774 1.89 8i8585 4.58 i8i4i5 88 23 74o55o 3.19 921691 1.89 818860 4-58 181140 37 24 740742 3.19 921607 1.39 819185 4.58 i8o865 36 25 740934 3.19 921524 1.89 819410 4.58 iSoSgo 35 26 741 125 3-19 921441 1.89 819684 4.58 180816 34 11 74i3i6 3-19 3.18 921357 1.89 819959 4.58 180041 33 741 5o8 921274 1.39 820284 4.58 179766 8£ 29 741699 3.18 921190 1.89 82o5o8 4.57 179492 3i 3o 741889 3.18 921107 1.89 820788 4.57 179217 3o 3i 9 -742080 3-18 9.921023 1.89 9-821057 4.57 10.178943 29 32 742271 3.18 920989 1.40 821882 4.57 178668 28 33 742462 3.17 920856 1.40 821606 4.57 178894 27 34 742652 3-17 920772 1.40 821880 4-57 178120 26 35 742842 3.17 920688 1.40 822154 4.57 177846 25 36 743o33 3-17 920604 1.40 822429 4.57 177571 24 11 743223 3.17 920520 1.40 822700 4.57 177297 23 743413 3-16 920486 1.40 822977 4-56 177028 22 39 743602 3-16 920352 1.40 828250 4.56 176750 21 40 743792 3-i6 920268 1.40 828524 4-56 176476 20 41 9-743982 3.16 9-920184 1. 40 9.828798 4.56 10.176202 \t 42 744171 3.16 920099 1.40 824072 4-56 175928 43 744361 3-i5 920016 1.40 824845 4-56 175655 \l 44 744550 3-i5 919981 1.41 824619 824898 4-56 175881 45 744730 744928 3.i5 919846 1.41 4-56 175107 i5 46 3-i5 919762 1.41 825166 4.56 174884 14 % 745117 745306 3-i5 919677 1.41 825489 825713 4.55 174561 i3 3.14 919598 1.41 4.55 174287 12 49 745494 3.14 919508 1.41 825986 4.55 174014 II 5o 745683 3-14 919424 1. 41 826259 4-55 173741 10 5i 9.745871 746o5o 3-14 9.919889 1.41 9.826532 4-55 10.173468 Q 52 3.14 919254 1. 41 826805 4-55 178195 53 746240 3-13 919169 1. 41 827078 4-55 172922 I 54 746436 4-13 919085 1.41 827851 4.55 172649 55 746624 3.i3 919000 1. 41 827624 4.55 172376 5 56 746812 3-13 910915 1.42 827897 828170 4.54 172108 4 U 746909 747 « 87 3.i3 918880 1.42 4.54 171880 3 3.ii 918745 1.42 828442 4-54 171558 a 59 747374 3-12 918659 1.42 828715 4.54 171285 1 60 747562 3.12 918574 1.42 828987 4-54 171013 Coeine D. Sine D. Cotang. D. Tang. 18 (56 DEGREES.) 52 C34 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. , Tang. D. 1 Cotang. 9-747562 3-12 9-918574 1.42 9-828987 4.54 10-171013 60 I 747749 3-12 ' 918489 1-42 829260 4-54 I 170740 5o 170468 58 a 747936 3-12 918404 ! 1.42 829532 4-54 3 748123 3-11 i 9i83i8 i 1-42 829805 4-54 170195 57 4 748310 3-11 j 918233 1-42 83oo77 4-54 169923 56 5 748497 3-II 918147 1-42 83o349 4-53 169651 55 6 748683 3ii 1 918062 1-42 83o62i 4-53 169379 54 I 748870 3-11 917976 1-43 830893 4-53 169107 168835 53 749036 3-10 i 917891 1-43 83ii65 4-53 52 9 749243 3-10 I 917805 1-43 831437 4-53 168563 5i 10 749429 3-10 917719 1-43 831709 4-53 168291 5o II 9-749615 3-10 9-917634 1-43 9-831981 4-53 10-168019 S 12 749801 3-10 917548 1-43 832253 4-53 167747 167475 i3 749987 3-09 917462 1-43 832325 4-53 2 14 750172 3-09 917376 1-43 832796 4-53 167204 i5 75o358 3-09 917290 1-43 833o68 4-52 166932 45 i6 700543 3.09 917204 1-43 833339 4-52 166661 44 \l 750729 3-oc 917118 1-44 833611 4-52 166389 166118 43 750914 3.08 917032 1.44 833882 4-52 42 19 751099 3-o8 916946 1-44 8341 54 4-52 165846 41 20 751284 3-08 916059 1-44 834425 4-52 165075 40 21 9-751469 3 -08 9-916773 1-44 9-834696 4-52 io-i653o4 1? 22 75 I 654 3-o8 916687 1-44 834967 835238 4-52 i6oo33 23 75i83q 3-o8 916600 1-44 4-52 164762 u 24 75202J 3-07 9i63i4 1-44 835309 4-52 164491 25 732208 3-07 916427 1-44 830780 4-5i 164220 35 26 752392 3-07 916341 1-44 836o5i 4-5i 163949 34 11 732376 3.07 916234 1-44 836322 4-5i 163678 33 752760 3.07 916167 1-45 836593 4-5i 163407 i63i36 33 29 752944 3.06 916081 1-45 836864 4-5i 3i 3o 753128 3.06 915994 1-45 837134 4-5i 162866 3o 3i 9.753312 3-o6 9-913907 1-45 9-837405 4-5i 10-162595 29 28 32 •753495 3 -06 913820 1-45 837675 4-5i 162325 33 753679 3-06 913733 1-45 837946 4-5i 162054 27 34 753862 3-o5 915646 1-45 838216 4-5i 161784 26 35 754046 3-o5 915559 1-45 838487 4-5o i6ioi3 25 36 734229 3-o5 915472 1-45 838757 4-5o 161243 24 u 754412 3-o5 915385 1-45 83go27 4-5o 160973 23 i •754595 3 -05 915297 1-45 839297 839568 4-5o 160703 22 39 754778 3-04 915210 1-45 4-5o 160432 21 40 754960 3-04 9i5i23 1.46; 839838 4-5o 160162 20 41 9-755143 3-04 9-9i5o35 1-46 1 9-840108 4-5o 10-159892 19 109622 18 42 755326 3-04 914948 1-46 1 840378 4-5o 43 755508 3-04 914860 1-46 840647 4-5o 159353 17 159083 16 ' 44 755690 3-04 914773 1 1-46 840917 4-49 45 755872 3-o3 914683 1 1.46 841187 4.49 i588i3 i5 ■ 46 756o54 3 -03 914598 1 1-46 841457 4.49 158543 14 S 756236 3-03 ' 914010 ; 1-46 1 841726 4-49 158274 i3 756418 3-03 914422 j 1-46 841996 1 4-49 1 i58oo4 12 49 756600 3-o3 914334 1 1-46 842266 i 4-49 ; 157734 II 5o 706782 3-02 914246 1-47 842535 1 4.49 157465 10 5i 9-756963 3-02 9-914158 j 1-47 9 842805 i 4.49 10-157195 156926 8 55 757144 3-02 914070 i 1-47 843074 i 4-49 1 53 757326 3-02 ■ 913982 i 1-47 843343 1 4.49 1 I 56657 7 ' 54 757507 3-05 913894 i 1-47 843612 ' 4-4Q . 4-48 156388 6 55 757688 3-01 9i38o6 1-47 843882 1 i56ii8 i 5 56 757869 3-01 913718 1-47 1 844i5i 1 4-48 155849 4 ll 758o5o 3-01 9i363o 1 1-47 8/i/i/i20 4-48 i5558o ' 3 758230 3*01 913541 1-47 844689 4-48 I i553ii 3 59 75841 I 3*01 913453 1-47 844958 : 4-48 i55o42 I 6o 753591 3*01 913365 1-47 845227 ; 4.48 154773 Coeine D. Sine D. ! Cotang. 1 D. ! Tang. ' M. ; (55 DSGRS ,S8.) SINttS AND f AISTQENTS. (35 DEGREES.; 53 M. Sino D. Coftine D. Tang. D. Cotang. 60 o-7585oi 3'0i 9.913365 1-47 9.845227 845496 4-48 10.154773 1 54504 I fslll 3 00 918276 9181S7 1-47 4.48 58 1 3 00 1-48 845764 4.48 154236 3 759132 3 00 918099 1-48 846033 4.48 158967 1 53698 i5848o U 4 759312 3 00 918010 1.48 846802 4-48 5 759492 3 00 912922 1.48 846570 4-47 55 6 759672 759852 2 99 912883 1-48 846889 4-47 i53i6i 54 I 2 99 91 51744 912655 1.48 847107 847876 4-47 152898 53 760031 2 99 1.48 4-47 152624 52 9 7602 1 1 2 99 912566 1.48 847644 4-47 152356 5i 10 760890 2 99 912477 1.48 847918 4-47 152087 5o II Q- 760560 2 98 9.912888 1.48 9-848181 4-47 io-i5i8i9 it 13 760748 2- 98 912299 1-49 848449 4-47 i5i55i i3 760927 2 98 912210 1-49 848717 848986 4-47 i5i283 47 14 761 106 2- 98 912121 1-49 4-47 i5ioi4 46 i5 761285 2- 98 912081 1-49 849254 4-47 150746 45 i6 761464 2- 98 91 1942 1-49 849522 4-47 4.46 150478 44 \l 761642 2" 97 9u858 1-49 849790 l5o210 43 761821 2- 97 911763 1-49 85oo58 4.46 149943 43 «9 761999 2< 97 91 1674 911 584 1-49 85o825 4.46 149675 41 20 762177 2- 97 1-49 85o593 4.46 149407 40 21 9-762356 2- ll 9-9II495 1-49 9 -850861 4.46 10-149139 It 22 762534 2- 911405 1-49 851129 4.46 14887 1 33 762712 762889 2. 96 9ii3i5 i.5o 851896 4.46 148604 ti 24 2. 96 911226 i.5o 85 1664 4.46 148386 25 768067 2- 96 911186 1-50 851981 4.46 148069 35 36 768245 2- 96 91 1046 i-5o 852199 4-46 I 4780 I 34 11 768422 2- 96 910956 i.5o 852466 4.46 147534 33 768600 2- 95 910866 I -50 852788 4.45 147267 32 29 768777 2- 95 910776 i.5o 853001 4.45 146999 146733 3i 3o 768954 2- 95 910686 i.5o 853268 4-45 3o 3i 9.764181 2 95 9.910596 i.5o 9-853585 4.45 10-146465 20 32 764808 2. 95 9io5o6 i.5o 858802 4-45 146108 1459 J I 2d 33 764485 2- 94 9io4i5 i.5o 854069 4.45 u 34 764662 2- 94 910825 i.5i 854386 4.45 145664 35 764888 2- 94 910235 iSi 854603 4.45 145397 i45i3o 25 36 76501 5 2 94 910144 iSi 854870 4.45 34 ll 765191 2 94 910054 1-51 855187 4.45 144868 23 760867 2 94 909968 i.5i 855404 4.45 144596 22 h 765544 2 93 909878 909782 1-51 855671 4-44 144829 21 4o 765720 2 93 1-51 855938 4.44 144062 20 4i 9.765896 2 93 9-909691 1-51 9-856204 4.44 10-143796 \t 42 766072 2 93 909601 i.5i 856471 4-44 143520 148268 142996 142780 43 44 766247 760428 2 2 93 93 909510 909419 i-5i i.5i 856787 857004 4-44 4-44 ]l 45 766598 2 92 909828 1-52 857270 4-44 i5 46 766774 2 92 909287 1-52 857537 4-44 142463 14 48 766949 2 92 909 1 46 1-52 857808 4.44 142197 141901 i3 767124 3 92 909055 1-52 858069 4-44 12 i 49 767800 2 92 908064 1-52 858836 4.44 141664 II 5c 767475 2 91 908878 1-52 858602 4.43 141398 10 5i 9 • 767649 2 9« 9-908781 1.52 9-858868 4-43 io-i4ii8a t 52 767824 3 91 908690 1-52 859184 4-43 140866 53 tivn 2 91 908599 1-52 859400 4-43 140600 I 1 54 2 91 9o85oT 908410 1-52 859666 4-43 140884 55 768348 3 90 1-53 859982 4-43 140068 5 56 768522 2 90 908824 1.53 860198 4-43 139802 4 u 768697 2 90 908233 1.53 860464 4.43 189536 3 768871 3 ■ 90 908141 1.53 860780 4-43 189270 3 59 769045 2 .90 908049 1.53 860995 4-43 189005 188739 I 60 769219 3.90 907958 1.53 861261 4-43 Cosine D. Sine D. Cotang. D. Tang. M —^-j (64 DEGREES.) 54 (36 DEGREES.) A TABLE OF LOGARITHMIC "m. Sine D 1 Cosine D. Tang. D. Cotang. — -J 9.769310 769393 a -90 I 9-907958 I i 907866 I .53 ' 9-861261 4.43 10-138739 60 I 2 .89 .53 j 86i527 4 •43 138473 1 59 3 769566 2 89 907774 I -53 861792 862058 4 •4a ' i38ao8 1 58 3 769740 a .89 907682 I -53 4 -42 13794a 1 57 137677 1 56 4 769913 a .89 907590 I -53 862828 4 •42 5 770087 2 .89 ' 907498 I 53 862389 4 42 187411 55 6 770260 a 88 907406 I 53 862854 4 •42 1 87 146 1 54 2 770433 2 88 ■ 907314 I 54 868119 4 -42 i3688i 53 770606 2 88 907222 I 54 868385 4 42 i366i5 5a 9 770779 2 88 907129 . I 54 868650 4 •42 I 36350 5i 10 77095a 2 88 ' 907037 ■ I 54 8639x5 4 •42 i36o85 5o II 9.771125 2 88 9-906945 I 54 ' 9-864180 4 42 10-135820 S la 771298 2 87 9o6o52 1 54 864445 4 42 135555 i3 771470 2 87 906760 I 54 864710 4 42 135290 ii 14 771643 2 87 906667 I 54 864975 '4 41 i35o25 i5 771815 2 87 906575 I 54 865240 4 41 184760 45 i6 771987 2 87 906482 I 54 8653o5 4 41 184495 44 \l 772159 2 87 906889 I 55 865770 4 4i 134280 43 1 772331 2 86 906296 I 55 866o35 4 41 188965 42 19 7725o3 2 86 906204 I 55 866800 4 41 133700 I 41 i ao 772675 a 86 9061 1 1 I 55 866564 4 41 133436 40 31 9-772847 773018 2 86 9-906018 I 55 9-866829 4 41 10.133171 39 aa 2 86 905925 I 55 867094 867358 4 41 182906 i 38 i a3 773190 2 86 9o583a . i 55 4 41 182642 u 24 773361 2 85 905739 I 55 867628 4 41 182877 a5 773533 2 85 903643 I 55 867887 4 4i 182113 35 26 773704 a 85 9o555a i 55 868i52 4 40 181848 34 11 773875 2- 85 905459 I 55 868416 4 40 i3i584 33 774046 2 85 9o5366 I 56 868680 4 40 i3i320 33 29 774217 2 85 905272 I 56 868945 4 40 i3io55 3i 3o 774388 2 84 905179 I 56 869209 4 40 180794 3o 3i 9-774558 2 84 9 -905085 I 56 9-869473 4 40 io.i3o527 *S 32 774729 774899 2 84 904992 I 904898 I 56 869787 4 40 180268 ad 33 2 84 56 870001 4 40 129999 129735 27 34 775070 2- 84 904804 i I 56 870265 4 40 26 35 775240 2- 84 9047" ! I- 56 870529 4 40 129471 25 36 775410 2- 83 904617 ! I- 56 870793 4 40 139207 24 ll 775580 2- 83 904523 j I- 56 871057 4 40 128943 23 775750 2- 83 904429 I- 904335 1 I- 57 871821 4 40 128679 128415 23 39 775920 2- 83 57 871585 1 4 40 21 40 776090 2- 83 904241 i I- 57 871849 4- 39 ia8i5i 20 41 9-776259 2- 83 9-904147 i I- 57 9-872112 4- 39 10-127888 :? 4a 776420 2- 82 904053 1 I' 57 872876 4 39 127634 43 77659§ 2- 82 908959 1 I' ^7 872640 4 39 127860 I 17 1 44 776768 2- 82 908864 i I- 57 872908 4- 39 127097 1 16 1 45 776937 2- 82 908770 ' I. 57 878167 4 39 126833 i5 46 777106 2- 82 908676 I- 57 878480 4 39 126570 14 tl 777275 2- 81 9o858i I- 57 878694 4 39 126806 i3 777444 2- 81 908487 I- 57 878957 4- 39 126043 j 12 1 49 777613 2- 81 908892 I- 58 874220 : 4- 39 125780 11 1 50 777781 3' 81 908298 I- 58 874484 4- 39 i255i6 10 5i 9.777950 2- 81 9-908208 I- 58 9-874747 4- 39 ioi25a53 t 5? 7781 19 2' 81 908108 I- 58 875010 ! 4- It 124990 53 778287 2- 80 908014 I- 58 875273 ' 4- 134727 I 54 778455 2- 80 902919 ! I- 58 875536 4- 38 134464 55 778624 2- 80 902824 1 I- 58 875800 ; 4- 38 134300 5 1 56 778792 2- 80 902729 ! I- 58 876063 j 4- 38 133987 4 u 778960 2- 80 902634 ! I- 58 876826 1 4 38 138674 3 7-79128 a- 80 902539 I- 59 876589 , 4- 38 133411 3 59 779295 a- ■'9 902444 I- 59 876851 4- 38 ia3i49 I 60 779463 2-79 902349 1 I- 59 8771U 4-38 133886 CkMsme D. Sine i D. Cotang. 1 D. Tang. '^L^ (53 DBGRESS.) SINES AND TANGENTS. (37 DEGREES.; 5d M. o Sine D. Cosino I >. Tang. D. Cotang. Q- 779463 2-79 9-902840 1 90225j I 59 9-877"4 4-38 10-122886 60 I 779631 2- 79 59 877377 4- 88 122623 i? a 779798 2- 79 902158 1 59 877640 4 88 122860 3 780133 2- 79 902068 I ^9 877908 4 88 122097 57 4 2- "79 901967 I 59 878165 4 88 121835 56 1 5 780300 2- 78 901872 I ?9 878428 4 88 121572 55 6 780467 2- 78 901776 I 59 878691 4 88 121809 54 I 780634 2- 7^ 901681 1- 59 878953 4 37 121047 53 780801 2- 78 901 585 I- 59 879216 4 37 120784 120322 5a 9 780968 2- 7? 901490 I 59 879478 4- 37 5i 10 781134 2< 78 901394 I 60 879741 4- 37 120259 5o II 9.781301 2< 77 9-901298 1 60 9-880008 4- 37 10-119997 49 12 781468 2 77 901202 1 60 880265 4 37 119785 48 i3 781634 2< 77 901 106 I 60 880528 4 37 119472 47 14 781800 2< 77 OOIOIO I 60 880790 4 37 119210 46 i5 781966 2- 77 900914 I 60 88io52 4 37 118948 45 i6 782182 2- 77 900018 I 60 88i3i4 4- 37 1 1 8686 44 \l 782298 2- 76 900722 I 60 881576 4 37 118424 43 782464 2- 7^ 900626 1 1 60 881889 4 37 118161 42 »9 782630 2- 76 900529 I 900433 1 ■ 60 882101 4 37 117899 41 20 782796 2- 76 61 882868 4 36 117687 40 21 9.782961 2- 76 9-900387 I- 61 9-882625 4 36 10-117875 39 22 788127 2- 76 900240 1 • 61 882887 4 86 117118 38 23 788292 788458 2- 75 900144 1 61 888x48 4 86 116852 37 24- 2- 75 900047 1 • 61 883410 4 36 116590 36 25 788628 2- 75 899951 1 61 888672 4 86 116828 35 26 788788 2- 75 899854 I • 61 888984 4 36 1 16066 34 \l 788953 2- 75 899757 I 61 884196 4 36 ii58o4 33 784118 2- 75 899660 I 61 884457 4 86 115543 32 29 784282 2- 74 899564 1 ■ 61 884719 4 36 ii528i 3i 3o 784447 2- 74 899467 1 62 884980 4 36 Il5020 3o 3i 9-784612 2- 74 9-899870 1 62 9-885242 4 86 10-114758 ^ 32 784776 2 74 899278 1 62 8855o8 4 86 1 14497 114235 33 784941 2 74 899176 1 62 885765 4 36 27 34 785io5 2 74 899078 1 62 886026 4 86 118974 26 35 785269 785488 2 73 898981 1 62 886288 4 86 118712 25 36 2< 73 898884 I 62 886549 4 85 ii345i 24 11 785597 2 73 898787 1 62 886810 4 85 118190 23 785761 2 73 898689 I 62 887072 4 85 112928 22 39 785925 2 73 898592 I 62 887888 4 35 112667 21 40 786089 2 73 898494 I 63 887594 4 85 1 1 2406 20 41 9-786252 2 72 9-898897 I 68 9-887855 4 35 10-112145 19 18 42 786416 2 72 898299 I 68 8881 16 4 35 111884 45 786579 2 72 898202 1 68 888877 4 35 111623 17 U 786742 2 72 898 I Od 1 68 888689 4 35 111861 16 45 786906 2 72 898006 1 68 888900 4 35 11 1 100 i5 46 787069 2 72 897908 1 68 889160 4 35 110840 14 47 787282 2 71 897810 1 68 889421 4 35 110579 i3 48 787895 2 71 897712 1 68 889682 4 35 110810 la 49 787557 2 71 897614 1 68 889943 4 35 110057 II 5o 787720 2 V 897516 I 63 890204 4 34 •09796 10 5i 9-787888 788045 2 71 9-897418 1 64 9 • 890465 4 34 ro- 109535 t 52 2 71 897820 I 64 890725 4 34 109275 53 788208 2 71 897222 I 64 890986 4 34 109014 7 54 788870 788532 2 70 897128 1 64 891247 4 34 108753 6 55 2 70 897025 1 64 891507 4 34 108498 108282 5 56 788694 788856 2 70 896926 I 64 891768 4 34 4 5i 2 .70 896828 1 -64 892028 4 34 107972 3 5o 789018 2 .70 896729 1 64 892289 4 34 1077 I I a 59 789180 a .70 896681 I 64 892549 4 .34 107451 I 60 789342 a-69 896582 1 64 892810 4.34 107190 Goaine D. Sine D. CoUng. D. Tang. M. (52 DSORKI8.) 56 (38 DEGRKES.) A TABLE OF LOGARltmttO ^ Sine D. Cosine ; D. Tang. D. Cotwig. 9-789343 3.69 9.896532 1 -64 9-892810 4-34 10.107190 60 I 789504 3 69 896433 ! 1 • 65 893070 4.34 106930 5o 3 789665 2 ^ 896335 1 1 .65 893331 4.34 106669 59 3 789827 3 .69 896236 1 I -65 893591 4.34 106409 67 i 789988 2 69 896137 1 1 89603d 1 • 65 893801 4 34 1 06 149 56 5 790149 2 ^ •65 8941 1 1 4-34 106889 i 55 6 7903 10 2 68 895939 , 1 • 65 894371 4.34 106629 I 54 io536? I 53 1 i I 790471 2 • 68 895840 1 • 65 894632 4-33 790632 2 • 68 893741 I -65 894892 893152 4-33 10610& 53 i 9 790793 790954 2 .68 895641 1 • 65 4-33 104848 5i ! ic 2 .68 895542 1 •65 895412 4-33 104688 5o II 9.791115 3 68 9.895443 I -66 9-895672 4.33 10.104328 49 12 791275 3 67 895343 I -66 895932 4-33 104068 48 i3 791436 2 67 895244 1 • 66 896192 4.33 io38o8 47 14 791596 2 .67 895145 I • 66 896432 4.33 103648 46 i5 791757 2 .67 895045 1 • 66 896712 4.33 103288 45 i6 791917 3 .67 894045 1 I -66 89697 1 4-33 io3o29 44 17 792077 3 67 894846 i 1 • 66 897231 4.33 102769 43 i8 792237 2 66 894746 I • 66 897491 4-33 102609 43 19 792397 2 66 894646 1 • 66 897751 4-33 102249 41 20 792557 2 66 894546 1 .66 898010 4.33 101990 40 21 9.792716 792876 2 66 9-894446 1 • 67 9-898270 4-33 10.101730 ^ 22 2 66 894346 1 • 67 898530 4.33 101470 23 793o35 2 66 894246 I .67 898789 4.33 101211 37 24 793iq5 2 65 894146 1 67 899049 4.32 100961 36 25 793354 2 65 894046 1 67 899308 4-32 100692 100432 35 26 793514 2 65 893946 1 -67 899568 4-32 34 U 793673 3 65 893846 1 67 899827 4.32 100173 33 793832 2 65 893745 1 67 900086 4.32 099914 32 29 793991 2 65 893645 I 67 900346 4.32 099664 3i 3o 7941 5o 2 64 893544 1 67 900605 4-32 099396 3o ''' 3i 9.794308 2 64 9-893444 I 68 9 • 900864 4.32 io^o99i36 090876 39 ' 32 794467 2 64 893343 I 68 901 1 24 4.32 38 33 794626 2 64 893243 I 68 90i383 4.32 098617 098368 11 ; 34 794784 2 64 893142 I 68 901642 4-32 35 794942 2 64 893041 I 68 901901 4.32 098099 25 36 795^01 3 64 892940 I 68 902160 4.32 097840 24 u 795259 3 63 892839 I 68 902419 4.32 097681 33 ; 795417 3 63 892739 I 68 902679 4.32 097321 33 i 39 795575 3 63 892638 I 68 902938 4.33 097062 31 1 40 795733 3 63 892536 I 68 903197 4-3i 096803 30 1 41 9.795891 3 63 9-892435 I 69 9-903455 4.31 10 •096646 \i 42 796049 2 63 892334 I 69 903714 4.31 096286 43 796206 2 63 892233 1 69 903973 4.31 096027 096768 \i\ 44 796364 2 62 892132 ^ I 69 904232 4.31 45 796521 2 62 892030 1 I 69 904491 4.31 096609 16 ! 46 796679 2 62 891029 1 I 69 904730 4-31 096260 14 ' s 796836 2 62 801827 I 69 903008 4-31 094993 094733 i3 1 796993 797 I 5o 2 62 89*1726 I 69 906267 4-31 12 1 49 3 61 891624 1 69 906626 4.31 094474 11 1 5o 797307 3< 61 891523 I 70 906784 4.31 094216 10 j i' 9-797464 2- 61 9.891421 1 70 9 - 906043 4-31 10.093967 093698 ?i 53 797621 3 6i 891319 I 70 9o63o2 4.31 53 797777 3 61 891217 1 70 906660 4-31 093440 7 54 797934 798091 3 61 891115 I 70 90681Q 4-31 093181 6 ' 55 2 61 891013 I- 70 907077 4-31 092923 5 i 5e 798247 3 61 89091 1 I . 70 907336 4-31 092664 4 U 798403 2 60 890809 1 • 70 907694 4.31 092406 3 798660 2- 60 890707 1 I 70 907862 908111 4.31 092148 3 59 798716 3" 60 8oo6o5 1 1- 70 4-30 091889 1 60 798871 3.60 890603 1 I 70 908369 4.30 O9163I CoBine D. Sine i r ). Cotang. 1 D. Taxifr* 1 ^* (51 DEGREES.) SINKS AND TANGEl ^S (39 DEGREES *J &? M." ] Sine D. Cosine D. Tang. D. CotAng. 60 9.798872 2'6o 9.890503 I .70 9.908369 4-3o 10-0191631 I 799028 2-60 890400 j 1 •71 908628 4.30 091372 U 7 799184 2 -60 890198 1 •71 908886 j 4-3o 091114 3 799339 2-59 890195 1 •71 909144 4 -30 090856 ll 4 799493 2-59 , 890093 ; I •71 '' 909402 4-3o 1 090398 56 5 799651 2-59 889990 i 1 •71 909660 4-3o 1 090340 55 6 799806 2-59 889888 ! I •71 909918 4-3o 39008s : 54 ' I 799962 2-59 889785 ' I •71 910177 : 4-3o 089823 ' 53 8001 17 2-59 889682 . I •71 910435 4-3o 089565 52 9 800272 2-58 889579 i 1 •71 910693 4-3o 089307 5i 10 800427 2-58 889477 i I •71 910951 4-3o 089049 5o \i 9800582 2-58 9.889374 ' I .72 9-911209 4-3o 10-088791 49 12 800737 800092 2.58 889271 1 I • 72 91 1467 4-3o 088533 48 i3 2-58 889168 ' I -72 911724 4-3o 088276 47 14 801047 2.58 889064 i 1 .72 91 1982 4-3o 088018 46 i5 801201 2.58 888961 ; I -72 912240 4-3o 087760 45 i6 8oi356 2.57 888858 , I .72 912498 4-3o 087502 44 \l 8oi5ii 2.57 888755 I I 72 9127D6 4-3o 087244 43 80 I 665 2.57 888651 i I 72 9i3oi4 4-29 086986 42 »9 80181Q 801973 2.57 888548 ! I 72 913271 4-29 086729 41 20 2.57 888444 1 I 73 913529 4-29 086471 40 21 9-802128 2.57 9-888341 i I 73 9-913787 4-29 io-o862i3 39 22 802282 2.56 888237 I 73 914044 4-29 085956 38 23 802436 2.56 888134 1 73 914302 4-29 085698 37 24 802 58q 802743 2-56 888o3o I 73 914560 4-29 085440 36 25 2-56 887926 I 73 914817 4-29 o85i83 35 26 802897 2.56 887822 I 73 915075 4-29 084925 34 27 8o3o5o 2-56 887718 I 73 915332 4-29 084668 33 28 803204 2.56 887614 I 73 915590 4-29 084410 32 0^ 803357 2.55 887510 I 73 915847 4-29 084153 3i So 8o35ii 2.55 887406 I 74 916104 4-29 083896 So 3i 9 -803664 2.55 9.887302 I 74 9-916362 4-29 10-083638 ll 32 803817 2.55 887198 I. 74 916619 4-29 o8338i 33 803970 2.55 887093 I 74 916877 4-29 o83i23 27 34 804123 2.55 886989 ; I- 74 917134 4-29 082866 26 35 804276 2.54 886885 ! I. 74 917891 4.29 082609 25 36 804428 2-54 886780 1 1. 74 917648 4.29 082352 24 ll 804581 2.54 886676 1 1- 74 917905 4-29 082095 23 804734 2.54 886571 I. 74 918163 4.28 081837 22 h 804886 2.54 886466 ! I- 74 918420 4-28 o8i58o 21 4o 8o5o39 2.54 886362 I . 75 918677 4-28 o8i323 20 4i g-805191 2-54 9.886257 I. 75 9-918934 4-28 10-081066 19 18 42 805343 2.53 886i52 i 1- 75 919191 4.28 080809 43 805495 2-5S 886047 I • 75 919448 4.28 o8o552 n 44 805647 2.53 885942 I . 75 919705 4-28 080295 16 45 803799 2.53 885837 I . 75 919962 4-28 o8oo38 i5 46 805951 2.53 885732 I. 75 920219 4-28 079781 U tl 806 io3 2.53 885627 I. 75 920476 4-28 079524 i3 806254 2.53 885522 I- 75 920733 4-28 079267 12 49 806406 2-52 885416 1. 75 920990 4-28 079010 II 5c 806557 2-52 885311 1. 76 921247 4-28 078753 10 5: 9 806709 2-52 9-885205 I. 76 , 9.921503 4.28 10-078497 t 52 806860 2.52 885100 1. 76 921760 4-28 078240 53 80701 I 2-52 884994 1 • 76 922017 4-28 077983 1 54 807163 2-52 884889 I . 884783 I . 76 922274 4-28 ! 077726 6 55 807314 2.52 76 922530 4-28 077470 5 56 807465 2.5l 884677 1 . 76 922787 4-28 077213 4 u 807615 2-51 884572 1 - 76 923044 4-28 076956 3 807766 2.5l 884466 I . 76 923300 4-28 076700 2 ^ te? 2.5l 884360 1 . 76 923557 4-27 076443 I 60 2.5l 884254 I . 77 9238i3 4-27 076187 Coeino D. Sine r >. ' Cotaiig. D. i Tang. IL (60 DB OKI 1KB.) 58 (40 DEGREES.) A TABLE ] OF LOGARITHMIC ~m' i Sine D. j Cosuio 1 D. j Tting. D. 1 Cotang. 60 9-808067 2-5l i 9.884254 1-77 9-928813 i 4-27 10.076187 I j 808318 i 3-5l ' 884148 1-77 1 924070 , 4-27 073980 5q 3 8o8368 ' 3-5l ' 884042 1-77 924827 I 4-27 075673 ! 5$ 3 8o85i9 2 -Do 883936 1-77 1 924583 1 4-27 i 075417 57 ; ! 075160 56 ' 4 808669 2 00 883829 1-77 1 924840 4-27 5 808819 2 -50 883723 1-77 925056 4-27 j 074904 55 , 6 ! 808969 2 -50 883617 ' 1-77 923832 4-27 074648 54 i I 809119 2 -So 883510 1-77 025609 4-27 074891 1 53 074135 52 809269 2 • 5o 883404 1.77 925863 1 4-27 9 809419 2-49 i 883297 1.78 926122 ' 4-27 078878 5i IC 809569 3-49 j 883191 1.78 926878 ; 4.27 078622 5o II 9'8o97i8 i 2-49 9 -883084 ' '-78 9-926684 4-27 10-073866 49 13 809868 : 2.49 882077 ; 1-78 926890 4-27 078110 48 i3 810017 : 2.49 882871 1.78 927U7 4-27 072853 47 U 810167 2.4Q 882764 1.78 927408 4-27 072597 46 i i5 8io3i6 2.48 882657 1-78 927659 ! 4-27 072841 45 : i6 810465 2-48 88255o 1.78 927915 4-27 072085 44 1 \l 810614 3-48 882443 ' 1.78 928171 4-27 071829 071573 43 ! 810763 2-48 882336 1.79 928427 4-27 42 »9 810913 2-48 882229 1-79 928688 4-27 071817 41 30 811061 2-48 882121 1-79 928940 4-27 071060 40 31 9-811210 2-48 9-882014 1-79 9.929196 4.27 10-070804 39 38 33 8ii358 2-47 881907 1-79 929432 4-27 070548 33 8ii5o7 2-47 88 1 799 1-79 929708 4-27 070292 0700J6 37 24 8ii655 2-47 881692 1.79 929964 4-26 36 35 811 804 2-47 88 1 584 1-79 980220 4-26 069780 35 ! 36 811952 2-47 881477 1.79 980473 4-26 069525 34 3^ 812100 2-47 881369 1.79 980781 4-26 069260 33 812248 2-47 881261 1.80 980987 4-26 0600 li 32 29 812896 2.46 881 i53 1.80 981248 4-26 068757 3i 3o 812044 2-46 881046 1-80 981499 4-26 o685oi 3o 3i 9-8i2692 2-46 9-880938 1-80 9.981755 4-26 10-068245 20 i 28 33 812840 3-46 88o83o 1-80 982010 4-26 067990 067784 33 812988 2-46 880722 1.80 982266 4-26 27 : 34 8i3i35 2-46 880613 I -80 982522 4-26 067478 26 : 35 813283 2-46 88o5o5 1-80 i 982778 4-26 067222 25 36 8i343o 2-45 880397 I -80 ' 988033 4-26 066967 24 ll 813578 3-45 880289 i-8i 988289 4.26 066711 23 813725 2-45 880180 i-8i 1 988543 4-26 066455 23 39 813872 3.45 880072 1.81 988800 4-26 066200 31 i 40 814019 2-45 879963 1-81 ' 984056 4-26 065944 30 1 4^ 1 9-814166 2-45 9.879855 i.8i 9.984811 4-26 10-065689 965433 18 i 42 8i43i3 2-45 879746 1. 81 i 984567 4-26 43 1 81 1460 2-44 879637 i.8i 984823 4-26 i o65i77 17 ' 44 814607 2-44 ' 879529 1.81 983078 4-26 ' 064922 16 45 814753 2-44 i 879420 . 1. 81 985838 4-26 064667 i5 i 46 814900 2-44 ! 879811 1. 81 j 935589 4-26 1 06441 1 14 i 47 81 5046 2-44 8^9202 1.82 ' 985844 4-26 : 0641 56 i3 48 815193 2-44 870093 878084 i 1.82 986100 4-26 068900 12 49 815339 2-44 1-82 986855 4-26 068645 1 5o 81 5485 2.43 878875 1.82 986610 4-26 068890 5i o-8i563i 2-43 ; 9-878766 i 1.82 9.986866 4-23 10 068184 j ? 53 815778 i 2-43 1 878656 1.82 987121 4-23 062879 53 815924 3-43 \ 87S547 1-82 ! 987876 4-23 062624 7 ! 54 816069 : 3-43 : 878488 1.82 937682 4-25 062868 6 ' 55 ' 816215 2-43 i 878828 1 1.82 087887 4-25 002Il3 f ! " 1 5e ; 8i636i 1 3-43 1 878219 i 1.83 988142 4-25 1 06 1 858 4 \ 57 ' 8i65o7 2-42 ! 878109 ' 1-83 988898 4-23 06160a 3 58 gi6652 2-42 877999 ' 1.83 i 988638 4-23 , 061847 2 i 59 i 816798 ■ 2-42 877890 1.83 988908 4-25 061092 : I j 060837 1 : Tang. ! M. ! 60 i 816943 1 2-42 877780 [ 1.83 989168 1 4-23 Cosino D. ■ Sine D. 1 Cotang. 1 D. (49 DEQRE ES) SINES AND TANGENTS. (41 DEGREES.; 59 M. Sine D. Cosine D. Taug. D. Cotaiig. g. 816943 2-42 9.877780 1-83 9.939163 4-25 10-060837 60 1 817088 2 •42 877670 1-83 939418 4 •25 o6o582 59 2 817333 2 •42 877560 1.83 939673 4 •25 060827 58 3 817379 2 •42 8-, 745o 1-83 939928 4 .25 060072 57 4 817024 2 •41 8', 7340 1-83 940183 4 •25 059817 56 5 817668 2 •41 8-, 723c 1.84 940438 4 •25 059362 55 i 6 817813 2 •41 877120 1.84 940694 i •25 059806 54 ' I 817958 ^ ■41 877010 1.84 940949 4 •25 oSooSi 53 8i8ic2 2 •41 876899 876789 1.84 941204 4 •25 058796 52 9 8itf247 2 41 1.84 941458 4 •25 358542 5i lO 818392 2 41 876678 1.84 941714 4 •25 058286 5o II 9.8i8536 2 40 9.876568 1-84 9.941968 4 •25 10 .058032 49 la 818681 3 40 876457 1.84 942223 4 25 057777 48 i3 818825 2 40 876347 1-84 942478 4 25 057522 47 i4 818960 81911J 2 40 876236 1.85 942733 4 25 057267 46 i5 2 40 876125 1.85 942988 4 25 057012 45 i6 819257 2 40 876014 1.85 943243 4 25 056757 44 17 819401 2 40 875904 1-85 943498 4 25 o565o2 43 i8 819545 2 39 875703 1-85 943752 4 25 056248 42 19 819689 2 39 875682 1-85 944007 4 25 055993 41 20 819832 3 39 875571 1-85 944262 4 25 055788 40 at 9.819976 2 39 9.875459 1-85 9-944517 4 25 10.055488 39 22 820120 2 39 875348 1.85 944771 4 24 055229 38 23 820263 2 39 875237 1.85 945026 4 24 054974 37 24 820406 2 39 875126 1.86 945281 4 24 054719 36 25 82o55o 2 38 875014 1.86 945535 4 24 054465 35 26 820693 820836 2 38 874903 1.86 945790 4 24 054210 34 11 2 38 874791 1.86 946045 4 24 058955 33 820979 2 38 874680 1.86 946299 4 24 053701 32 29 821122 2 38 874568 1.86 946554 4 24 053446 3i 3o 821265 2- 38 874456 1.86 946808 4 24 058192 3o 3i 9.821407 3 38 9.874344 1.86 9.947063 4 24 10.052987 29 32 82i55o 2 38 874232 1.87 947318 4 24 052682 28 33 821693 2 37 874121 1.87 947372 4 24 052428 27 34 821835 2 37 874009 i.b7 947826 4 24 052174 26 35 821977 2 37 873896 1.87 948081 4- 24 051919 25 36 822120 2 37 873784 1.87 948336 4 24 o5i664 24 37 822262 2« 37 873672 1.87 948390 4- 24 o5i4io 23 38 822404 2 37 873560 1.87 948844 4- 24 o5ii56 22 39 822546 2 37 873448 1.87 949099 4- 24 oSogoi 21 40 822688 2 36 873335 1.87 949353 4- 24 050647 20 41 9.822830 2 36 9.873223 1.87 1.88 9.949607 4- 24 10 .050398 \l 42 822972 2 36 873110 949862- 4- 24 o5oi88 43 823114 2 36 872998 1.88 950116 4- 24 049884 17 44 823255 2 36 872885 1.88 950370 4- 24 049680 16 45 823397 2 36 872772 1-88 950625 i- 24 049875 i5 46 823539 2 36 872639 1.88 950879 4- 24 049121 14 % 823680 2 35 872547 1.88 951 i33 4- 24 048867 i3 823821 2 35 872434 1.88 95 1 388 4- 24 048612 12 49 823963 2 35 872321 1.88 951642 4- 24 048358 II 5o 824104 2 35 872208 1.88 951896 4- 24 048104 10 5i 9.824245 2 35 9-872095 1.89 9.952150 4- 24 10-047850 55 824386 2 35 871981 1.89 952405 4- 24 047595 53 824527 2 35 871868 1.89 932650 4- 24 047841 7 ^•4 824668 2 34 871755 1.89 95291J 4- 24 047087 6 5f 824808 2 34 871641 1.89 953167 4- 23 046833 5 5t 824949 2 34 871528 1.89 953421 4- 23 046579 4 '!2 8250Q0 2 34 871414 1.89 953675 4- 23 046325 3 82523c 2 34 871301 1.85 953929 9541 8J 4- 23 046071 a ^ 825371 2 34 871187 1.89 4- 23 045817 I 60 825511 2.34 871073 1.90 954437 4-23 045563 Cosiuo D. Sine D. Cota]\g. _ J ^•_J Tang. M. (48 DEaRXXo.) 80 (42 DEGREES.; A TABLE OF LOQaEITHMIC M. o Sine D. Cosine i D. Tang. D. CotAng. 1 9-8255ii 2-34 9-871073 I .90 9-954487 ! 4'28 10 045300 60 I 2 82565i 825751 2 2 33 33 870960 ! I 870846 : I .90 .90 954691 954945 i 4 4 •23 -28 045309 59 o45o53 • 58 3 825931 2 33 870782 . I .90 955200 4 •23 044800 5? 1 4 S26071 2 33 870618 I .90 955454 4 -23 044546 S6 5 82621 1 2 33 870504 I .90 955707 4 •23 044298 55 6 826351 2 33 870890 I -90 955961 4 -23 044089 54 . I 826491 2 33 870276 I 90 956215 4 -23 048783 53 826631 2 .33 870161 I -90 956469 956728 4 -23 048381 52 9 826770 2 32 870047 I 91 4 23 048277 048028 5i IC 826910 2 32 869933 j I .91 956977 4 -23 5o II 9-827049 2 32 9-869818 I I 91 9-957281 4 23 10.042769 0425 1 5 49 48 12 827189 827828 2 32 869704 I 91 957485 4 23 i3 2 32 869589 I -91 937789 4 23 042261 47 14 827467 2 32 869474 I 91 937998 4 23 042007 46 i5 827606 2 32 869860 I 91 958246 4 23 041754 45 i6 827745 2 32 869245 I 91 9585oo 4 23 04 1 5oo 44 \l 827884 2 3i 869130 I .91 958754 4 -23 041246 43 828023 2 3i 869015 1 I 868900 I .92 959008 4 -23 040992 42 »9 828162 2 3i 92 959262 4 23 040788 41 20 828301 a 3i 868785 I 92 959516 4 -23 040484 40 21 9-828439 2 3i 9-868670 1 I 92 9.959769 4 28 10-040281 39 22 828578 2 3i 868555 1 92 960028 4 23 089977 38 23 828716 2 3i 868440 j I 92 960277 4 23 089728 37 24 828855 2 3o 868324 1 I 92 960581 4 28 089469 36 25 828993 829131 2 3o 868209 1 I 92 960784 4 28 089216 35 26 2 3o 868098 I 92 961088 4 23 088962 34 11 829269 2 3o 867078 1 93 961291 4 23 088709 33 829407 2 3o 867862 I 93 961545 4 28 088453 32 29 829545 2 3o 867747 ; I 93 961799 4 23 088201 3i 3o 829683 2 3o 867681 I 93 962052 4 28 087948 3o 3i 9.829821 2 29 9-867515 1 93 9-962806 4 23 10-087694 29 32 829959 2 29 867899 i I 93 962560 4 23 087440 28 33 83oo97 2 29 867288 , I 93 962818 4 23 087187 27 34 830234 2 29 867167 i I 93 968067 4 23 086933 26 35 83o372 2 29 867051 ' 1 93 968820 4 23 086680 25 36 83o5o9 2 29 866935 i I 866819 ! J 866708 i I 94 963574 4 23 086426 24 37 830646 2 29 94 968827 4 28 086178 23 38 830784 2 29 94 964081 4 28 085919 22 39 830921 2 28 866586 1 I 94 964885 4 23 o35663 21 40 83io58 2 28 866470 1 I 94 964588 4 22 o354ia 20 41 9-83;i95 2 28 9-866353 i 94 9 - 964842 4 22 io-o35i58 10 18 42 83i332 2 28 866287 I 94 965095 4 22 o349o5 43 831469 2 28 866120 I 94 965349 4 22 084651 \l 44 83 1 606 2 28 866004 I 95 965602 4 22 084898 45 831742 2 28 865887 I 95 965855 4 22 084145 i5 46 831879 2 28 865770 1 I 95 966105 4 22 088891 14 S 83201D 2 27 865653 : I 95 966862 4 22 033688 i3 832152 2 27 865586 j I 95 966616 4 22 088384 13 , 49 832288 2 27 865419 ! I 95 966869 4 22 o83i3i II 5o 832425 2 27 865302 I 95 967123 4- 22 082877 10 5i 0-832561 2 27 9-865185 I 95 9-967876 4- 22 10-082624 I 52 882697 2 27 865o68 1 I- 95 967629 4- 22 082871 53 832833 2 27 864950 i I- 95 967888 968186 4- 22 082117 I 54 832969 833103 2 26 864833 1 I- 96 4- 22 o3i864 55 2 26 864716 I- 96 968889 968648 4- 22 081611 5 56 833241 2 26 864598 I- 96 4- 22 o3i357 4 ll 833377 2 26 864481 I- 96 068896 4- 22 081104 3 833512 2 26 864363 I - 96 969149 969408 4- 22 o8o85i 3 59 833648 2 26 864245 I - 96 4- 22 o3o597 I 60 833783 2-26 864127 I- 96 969656 4-22 080844 Coeine D. Sine E Cotang. D. Tang. ^, (47 DEGREES.) SINES -IND TANGENTS. (43 DEGREES.) 61 M. Sine D. Cosine D. Tang. D. Cotar g. 9 833783 2-26 9-864127 1 -96 9.969656 4-22 10-080844 60 I 833919 2-25 864010 I 96 969909 4 -22 080091 029888 u 3 8i^o54 2-25 868892 1 97 970162 4 22 3 834 I 8q 2-25 868774 1 97 ^70416 4 22 029584 u 4 834325 2-25 868656 1 I 97 970669 4 22 029881 6 834460 2-25 868588 '■ 1 97 970922 4 22 029078 55 6 834595 834''lo 2-25 868419 1 97 97XX75 4 22 028825 54 3 2-25 863301 1 97 971429 4 22 02857X 53 834805 2-25 868x88 I 97 971682 4 22 028818 52 9 834999 835i34 2 24 868064 I 97 971985 4 22 028065 5i 10 2-24 862946 1 98 972188 4 22 027812 5o II 9.835269 8354o3 2-24 9.862827 X 98 9-972441 4 22 10-027559 49 12 2-24 862709 I 98 972694 4 22 027806 48 i3 835538 2-24 862590 1 9b 97294^. 4 22 027052 47 14 835672 2-24 862471 I 98 978201 4 22 026799 46 i5 835807 2-24 862853 I 98 973454 4 22 026546 45 i6 835941 2-24 862284 i I 98 978707 4 22 020298 44 17 886075 2-28 862xx5 1 98 978960 4 22 026040 43 i8 836209 2-23 861996 1 1 98 974218 4 22 025787 42 19 836343 2-28 861877 ' I 98 974466 4 22 025584 41 20 836477 2-28 861758 X 99 974719 4 22 025281 40 21 9-83661 1 2-23 9-861688 i X 99 9-974973 4 22 I0-025027 39 22 836745 2-23 86x5:9 i x 99 975226 4 22 024-;74 39 23 836878 2-23 861400 I 99 975479 4 22 024521 37 24 837012 2-22 861280 I 99 975782 4 22 024268 86 25 837146 2-22 86x161 I 99 975985 4 22 0240x5 35 26 837279 2-22 861041 1 99 976288 4 22 028762 34 11 837412 2-22 860922 I 99 976491 4 22 >285o9 33 S37546 2-22 860802 1 99 976744 4 22 023256 32 ?9 S37679 2-22 860682 2 00 976997 4 22 028008 3i So 837812 2-22 86o562 2 00 97725o 4 22 022750 3o 3i 9-837945 2.22 9.860442 2 00 9.977508 4 22 10-022497 29 32 838078 2.21 860822 2 00 977756 4 22 022244 28 33 838211 2-21 860202 2 00 978009 4 22 02199X 27 34 838344 2-21 860082 2 00 978262 4 22 021788 26 35 838477 2-21 859062 2 00 9785x5 4 22 02x485 25 36 8386x0 2-21 859842 2 00 978768 4 22 021282 24 37 38 838742 2-21 859721 2 01 979021 4 22 020979 23 888875 2-21 859601 2 01 979274 4 22 020726 23 39 889007 2-21 859480 2 01 979527 4 22 020478 21 40 889140 2-20 859860 2 01 979780 4 22 020220 20 41 9-889272 2-20 9.859289 2 ox 9.980083 4 22 10-0x9967 \t 42 889404 2-20 859119 2 ox 980286 4 22 019714 43 889586 2-20 858998 2 01 980538 4 22 019462 n 44 889668 2-20 858877 2 01 980791 4 2X 019209 16 45 889800 2-20 858756 2 02 981044 4 21 0x8956 x5 46 889982 2-20 858635 2 02 981297 4 21 0x8708 14 % 840064 2-19 8585x4 2 02 98x550 4 2X 018450 i3 840196 2-19 858898 2 02 981808 4 21 0x8197 12 49 840828 2-19 858272 2 02 982056 4 21 0x7944 11 5o 840459 2-19 858i5i 2 02 982809 4 21 017691 10 5i 9-840591 2-19 9-858029 2 02 9-982562 4 21 10017488 9 52 840722 2-19 857908 2 02 982814 4 21 0x7x86 § 53 840854 2.19 857786 2 02 988067 4 2X 016988 I 54 840985 2.10 2.16 857665 2 o3 988820 4 21 0x6680 55 841116 857543 2 o3 988578 4 23 0x6427 5 5t 841247 841878 2-18 857422 2 o3 988826 4 21 0x6174 4 u 2-18 857800 2 o3 984079 4< 21 015921 3 84: 509 2.18 857178 2 857056 2 o3 98488 X 4 21 015669 a 59 841640 2.18 o3 984584 4 21 0i54i6 I 60 841771 2.18 856984 2 o3 984887 4-21 oi5i63 Cosine D. Sine r r Cotanor. D. Tang. M. (46 DBORESS.) S2 (44 DEGREES.) A TABLE OF LOGARITHSJIC M. Sine D. Cosine D. Tang. D. Cotang. io-oi5i63 1 o 9-841771 2.18 9.856934 2-o3 9-984837 4-21 60 I 841902 3.18 856812 2 o3 985090 4 31 014910 U 3 842033 2 18 856690 2 04 985343 4 21 014657 3 842163 2 17 856568 2 04 9855^6 4 21 014404 u 4 842294 2 17 856446 2 04 985848 4 21 oi4i52 5 842424 2 17 856323 2 04 986101 4 21 013899 55 6 842555 2 17 856201 2 04 986354 4 21 013646 54 I 842685 2 17 856078 2 04 986607 4 21 013393 53 842815 2 17 855956 2 04 986860 4 21 oi3i4o 52 9 842946 2 17 855833 2 04 987112 4 21 012888 5i 10 843076 2 n 8557 1 1 2 o5 987365 4 21 012635 5o II 9.843206 2 16 9-855588 2 o5 9-987618 4 21 10-012383 45 13 843336 2 16 855465 2 o5 987871 4 21 012129 48 i3 843466 2 16 855342 2 o5 988123 4 21 311877 47 14 843595 2 16 855219 2 o5 988376 4 21 011624 46 i5 843725 2 16 855096 2 o5 9S8629 4 21 011371 45 i6 843855 2 .16 854Q73 2 o5 988882 4 21 011118 44 \l 843984 2 16 854850 2 o5 989134 4 21 010866 43 844114 2 i5 854727 2 c6 989387 4 21 oio6i3 42 19 844243 2 i5 854''-'-3 2 06 989640 4 21 oio36o 41 30 844372 2 i5 854480 2 06 989893 4 21 010107 40 31 9 -844502 2 i5 9-854356 2 06 9.990145 4 21 10-009855 39 33 844631 2 i5 854233 2 06 990398 4 21 009602 38 23 844760 2 i5 854109 2 06 990601 4 21 009349 37 24 844889 2 i5 853986 2 06 990903 4 21 009097 36 25 845018 3 i5 853862 2 06 99ii56 4 21 008844 35 36 845147 3 i5 853738 2 06 991409 4 21 008591 34 11 845276 3 14 853614 2 07 991662 4 21 008338 33 845405 3 14 853490 2 07 991914 4 21 008086 ?2 29 845533 2 14 853366 2 07 992167 4 21 007833 3i 3o 845662 2 14 853242 2 07 992420 4 21 007580 3o 3i 9-845790 2 14 9-853ii8 2 07 9-992672 4 21 10-007328 29 32 845919 2 14 802994 2 07 992925 4 21 007075 38 33 846047 2 14 852869 2 07 993178 4 21 006822 27 i 34 846175 2 14 852743 2 07 993430 4 21 006570 36 35 846304 2 14 852620 2 07 993683 4 21 oo63i7 25 36 846432 2 i3 852496 2 08 993936 4 21 006064 2»^ ll 846560 2 i3 852371 2 08 994189 4 21 Do58 1 1 23 846688 2 i3 852247 2 08 994441 4 21 X)5559 22 39 846816 2 i3 852122 2 08 994694 4 21 oo53o6 21 i 4o 846944 2 i3 851997 2 08 994947 4 21 oo5o53 20 1 41 9-847071 2 i3 9.851872 2 08 9.995199 4 21 10-004801 1 19 10 . 43 847199 2 i3 851747 2 08 995402 4 21 004548 43 847327 2 i3 85x622 2 08 995705 4 21 004295 n i 44 847454 2 12 851497 2 09 995957 4 21 004043 16 , 45 847582 2 12 85i372 2 09 996210 4 21 003790 i5 : 46 847709 2 12 851246 2 09 996463 4 21 003537 14 47 847836 2 13 85ii2i 2 09 996715 4 21 003285 i3 j 48 847964 2 12 800996 2 09 996968 4 31 oo3o3: 13 ; 49 848091 2 12 850870 2 09 997221 4- 21 002779 '1 : 5o 848218 3 12 850745 2 09 997473 4- 21 002527 ro ; 5i 9-848345 3 12 9-850619 85o493 2 09 9.997726 4- 21 IO-002;274 ? 53 848472 2 II 2 10 997979 4- 21 002021 53 848599 2 II 85o368 2 10 998231 4 21 001760 I 54 848726 2 " 850242 2 10 998484 4- 21 ooi5i6 55 84S852 3 II 85oii6 2 IC 998737 4 21 001263 5 56 ll 848979 849106 3 2 II II 849990 849064 2 2 10 10 998989 999242 4- 4 21 21 OOIOIl ooopS ooooo5 4 3 849233 3 II 849738 3 10 999490 4 21 2 59 849359 a II 8496 1 1 2 10 999748 4 21 00O253 1 60 849485 3-II 849485 2-10 10- 000000 4-31 10- 000000 u 1 Coeine D. Sine D. Cotang. D. T^ftMr. JLj (45 DEGREB8.) A TABLE OF NATURAL SINES. 98 M Deg. 1 Deg. 2 Deg. 8 Deg. 4 Deg. M S. 00000 c. s. S. as. S. as. S. as. s. as. Uuit. 01745 99985 03490 99980 o5234 99868 06976 99756 60 I 00029 I -0000 01774 99984 o35i9 99988 o5263 99861 07005 99754 59 3 ooo58 I -0000 oi8o3 99984 03548 99987 00292 99860 07084 99753; 58; 3 00087 1 -0000 oi832 99983 08077 99986 o532i 99858 07068 99750 57 i 4 001 16 I -0000 j 01862 99983 08606 99935 o535o 99857 07092 99748 56' 5 00145 1. 0000 j 01891 99982 o8635 99984 o5879 99855 07121 99746 551 6 00175 I • 0000 ! 01920 99982 08664 99988 o54o8 99854 07i5o 99744 54 j 7 00204 i.ooool 01940 99981 08698 99982 05487 99802 07179 99742 53 8 00233 I .0000 01978 99980 08728 99981 05466 99801 07208 99740 52 i 9 00262 1.0000 02007 99980 08752 99980 05495 99849 07287 99788 5i 10 00291 I. 0000 02o36 99979 08781 99929 o5524 99847 07266 99786 5o II 00320 99999 o2o65 99979 08810 99927 05553 99846 07295 99734 49 13 oo34o 99999 02094 99978 08889 99926 05582 99844 07824 99781 48 i3 00378 99999 02123 99977 08868 99925 00611 99842 07353 99729 47 14 00407 99999 02l52 99977 08897 99924 o564o 99841 07882 99727 46; i5 00436 99999 02181 99976 08926 99923 o566q 99889 07411 99725 45 i6 00465 99999 02211 99976 08955 99922 05698 99888 07440 99723 44i \l 00493 99999 02240 99970 08984 99921 05727 99886 07469 99721 43 i oo524 99999 02269 99974 04018 99919 05756 99884 07498 99719 42 1 «9 oo553 99998 02298 99974 04042 99918 05785 99833 07527 99716 41' 30 00582 99998 02327 02356 99973 04071 99917 o58i4 99881 07556 99714 40. 31 006 1 1 99998 99972 04100 99916 o5844 99829 07585 99712 39! 33 00640 99998 02385 99972 04129 99915 05878 99827 07614 99710 38 33 00660 99998 02414 99971 04159 99913 C5902 99826 07643 99708 37 34 00698 99998 02443 99970 04188 99912 05981 99824 07672 99705 36! 35 00727 99997 02472 99969 04217 99911 05960 99822 07701 99703 351 36 00756 99997 025oi 99960 04346 99910 05980 99821 07780 99701 341 11 00785 99997 o253o 99968 04275 99909 06018 99819 07759 99699 331 00814 99997 02560 99967 o43o4 99907 06047 99817 99810 07188 07817 99696 32' 39 00844 99996 02589 99966 04333 99906 06076 99694 3ij 3o 00873 99996 02618 99966 04862 99905 o6io5 99818 07846 99692 30 j 3i 00902 99996 02647 99965 04391 99904 06184 99812 07875 99689 29! ?8' 33 00931 99996 02676 99964 04420 99902 06168 99810 07904 l& 33 00960 99995 02705 99963 o444q 99901 06192 99808 07988 271 34 00989 99995 02734 99963 04478 99900 06221 99806 07962 99683 26 1 35 01018 99990 02763 99962 04507 99898 o635o 99804 07991 99680 25, 36 01047 99990 02792 99961 04536 99897 06279 99808 08020 99678 24 37 01076 99994 02821 99960 04565 99896 06800 9,9801 08049 99676 23 38 oiio5 99994 02850 99959 04594 99894 06887 99799 08078 99678 23 39 01134 99994 02879 99959 04628 99898 06866 99797 08107 99671 21 40 01164 99993 02908 99958 04653 99892 06395 95795 08186 99668 20 41 01193 99993 02938 99957 04682 99890 06424 99793 081 65 99666 19 43 01222 99993 02967 99956 047 1 1 Q9880 06453 99792 08194 99664 18 43 0125l 99992 02996 99955 04740 99888 06482 99790 08228 99661 17 44 01280 99992 o3o25 99904 04760 99886 o65ii 99788 08252 99659 16 45 01 309 99991 o3o54 99953 04798 99885 06540 99786 08281 99657 i5 46 oi338 99991 o3o83 99952 04827 99883 o65oo 99784 o83io 99654 14 47 01367 9999' 03ll2 99952 04856 99882 06598 99782 08889 99652 i3 48 01396 99990 o3i4i 99951 04885 99881 06627 99780 08868 99649 12 49 01425 999QO 08170 9q95o 04914 99879 06656 99778 08897 99647 11 5o 01404 99989 0319Q 99940 04948 99878 06685 99776 08426 99644 10 fi 01483 99989 03228 99948 04972 99876 06714 99774 08455 99642 0. 53 oi5i3 99989 03257 99947 o5ooi 99875 06743 99772 08484 99689 81 53 01542! 99988 03286 99946 o5o3o 99878 06778 06802 99770 o85i3 99687 7 54 01571 99988 o33i6 99945 oSoSg 99872 99768 08542 99685 6 55 01600 99987 03345 99944 o5o88 99870 06881 99766 08571 99683 5 56 01620 99987 03374 99943 o5ii7 99869 06860 99764 08600 99680 4 u oi658 99986 o34o3 99942 o5i46 99867 06889 99762 08629 99627 99625 3 01687 j 99986 03433 99941 o5i75 99866 06918 99760 08658 a 59 M 017161 99985 03461 C. S. 99940 o52o5 99864 06947 as. 99758 08687 as. 99623 s I c. s. i s. S. as. s. s. 89 Deff. 88 J Deg. 87 Deg. 86jDeg^__ 85 Deg. 54 A TABLE OF NATURAL SINES- M 6 Deg. 6 Deg. 1 V Deg. 8 Deg. 9 Deg. r] S. 1 c. s. S. C. S. ! S. ; C. S. S. C S s. 1 c s. M| 08716 99619 j 10453 99452 12187 99255JJ 13917 99027 1 5643 i 98769! 60 1 I 08745 99617 1 10482 99449 12216 99231 i| 13946 99023 1 5672 i 98764 i? 3 08774 99614 io5ii 99446 1 12245 99248; 13975 99019 15701 1 98760 3 08803 99612 io54o 99443 12274 99244' 14004 99015 i573o! 98755 57 4 o883i 99609 10569 99440 i23o2, 99240J i4o33 99011 i5758| 98751 56 5 08860 99607 10597 99437 . i233i: 99237; 14061 99006 15787 98746 55 6 08880 99604 10626 99434 ' 12360 992331 14090 99002 i58i6! 98741 54 I 08918 99602 10655 9943 1 , 12389 9923o'; 14119 98998 15845; 98737 53 08947 99399 ; 10684 99428 1 12418 99226 j 14148 98994 15873 98732 52 9 08976 99396 1 07 1 3 99424 j 12447 99222'! 14177 98990 15902! 98728 5i| 10 09005, 99394 i 10742 99421 j 12476 99219J i42o5 98986 15931 98723 5oi II 09034 99591 10771 99418 j i25o4 99213;. 14234 .)8982 15959 98718 49 12 09063 j 99388 10800 99415 1 12533 99211; 14263 98978 15988J 98714 48 i3 09092 99386 10829 99412 1 12362 99208! 14292 98973 16017 98709 47 14 091 21 99583 10858 99409 12591 99204I 14320 98969 16046 98704 46 i5 09150 99580 10887 99406 12620 99200! 14349 98963 16074 98700 45 i6 09179I 99578 10916 99402 1 12640 99197I! 14378 98961 i6io3 98695 44 \l 092081 99375 10945 99399 1 12678 99io3;i 14407 98937 16132} 98690 43 09237 99572 10973 99396 1 12706 99189! 14436 98953 16160 98686 42 19 092661 99570 11002 99393 12735 99186I T4464 98948 16189I 9S681 41 20 092951 99367 iio3i 99300 12764 991821 14493 98944 16218 98676 40 21 09324, 99364: 11060 993S6 12793 99178I 14522 98940 16246 98671 39 22 09353 1 99362 j 11089 99383 12822 99173! i455i 98936 i62t5| 98667 38 23 09382 99339 11118 99380 i285i 99171 14580 98931 i63o4 98662 37 24 0941 1 1 99536 11 147 99377 12880 99167 14608 98927 16333 98657 36 25 09440 99553 11176 99374 1 2908 99163 14637 98923 i636i 98652 35 26 09469 99551 11205 99370 12937 99160 14666 98919 16390 98648 34 11 094981 99548 11234 99367 12966 991 56 14695 98914 16419 98643 33 09527 99345 09556 99542 11263 99364 12995 99152 14723 98910 16447 98638 32 29 11291 99360 i3o24 99148 14752 98906 16476 98633 3i 3o 09383 99540 ll320 99357 i3o53 99144 14781 98902 i65o5 98629 3o 3i 09614 995371 11349 99354 i3o8i 99141 14810 98897 16533 98624 29 32 09642 99534 99531 99528 11378 99351 i3iio 99137 14838 98893 16562 98619 28 33 09671 11407 99347 i3i39 99133 14867 98889 16591 98614 27 34 09700 11436 99344 i3i68 99129: 14896 98884 16620 98609 26 35 09720 99526 11465 99341 i3i97 99125 14923 98880 16648 98604 25 36 09758 99323 11494 99337I 13226 99122 14954 98876 16677 98600 24 ll 09787 99520 ii523 99334' 13234 99118 14982 98871 16706 98595 23 09816 99517 ii552 99331 i 13283 991 14 i5oii 98867 16734 98590 32 39 09845 99514 ii58o 993271 i33i2 99110 i5o4o 98863 16763 98585 21 40 09874 9951 1 j 11609 99324 13341 99106 15069 98858 16792 98580 20 41 09903 99508 ii638 99320 13370 99102 i5o97 98834 168201 98575 19 42 0993 2 1 993061 11667 99317 13399 99098 i5i26 98849 168491 98570 10 43 0906 1 1 995o3l 1 1696 09314 13427 99094 I3i55 98845 16878] 98565 \l 44 09990' 903001 1x723 99310 13456 990QI i5i84 98841 16906! 98561 45 10019 994971 11754 99307 13485 99087 1 13212 98836 16935 98556 i5 46 10048 99494 11783 993o3 i35i4 99083 1 5241 98832 16964 985511 14 4"' 1 10077I 99491 11812 99300 13543 99079 15270 988271 16992 98546' i3 i8 1 10106^ 99488! 1 1 840 99297 13572 99075 15292 98823- 17021' 98541, 12 49 ioi33! 994851 11869 99293 i36oo 99071 15327 98818^ i7o5o: 985361 ir 1 5o 10164' 99482! 11898, 99290] 13629 99067 1 15356 98814, 17078 98531 10' 5i 10192; 99479' 11927' 99286 13658 99063 j 15385 98809' 17107' 98326 § 52 1022 1 99476 11956 99283 1 130S7 99059: 13414 98805 1 I7i36i 98521 53 io25o| 99473 11985 99279, 13716 99033 134421 98800J 17164' 98516 I 54 102791 99470 12014 99276 13744 9905 1 134711 9879^1 17193I 98311 55 io3o8i 99467' 12043 99272 13773, 99047 i55oo 987911 17222J 98506 5 56 io337J 99464^ 12071 99269 i38o2 99043 i5529| 98787: 17250' 98501 4 ll io366i 99461 12100 99265 i333i! 99039! 15557I 98782 173 Q 98496 3 10395, 994581 12129 99262 i386o 99033 j 1 5586 9877SI 17308 98491 17336 98486 2 59 104241 99455; c. s. i S. 1 i2i58 C. S. ! 99258 S. 13889 9903 1 1 i56i5 98773; I "M C. S. 1 s. 1 C. S. s- ! c. s. s. 841 )eK. 1 88 Deg. 1 82 Deg. 1 81 Deg 1 80 Deg A TABLE OF NATURAL SINES. 66 M 10 Deg. 11 Deg. 12 Deg. 18 Deg. 14 Deg. u S. c. s. S. c. s. S. 20791 C. S. 97815 S. 22495 C. S. S. C. S. 17365 98481 1 908 1 98163 97437 24192 97080 601 I 17393 98476 19109 98157 20820 97809 22523 9743o 24220 97023 u a 17422 98471 19138 98152 20848 97803 22552 97424 24249 97015 3 1 745 1 98466 19167 98146 20877 97797 2258o 97417 24277 97008 57 4 17479 9B46I 19195 98140 20905 97791 22608 97411 243o5 97001 56 5 17508 98455 19224 98135 20933 97784 22637 97404 24333 96994 55 6 17537 98450 19252 98129 20962 97778 22665 97898 24862 96987 54 I 17565 98445 19281 98124 20990 97772 22693 97891 24890 96980 53, 17594 98440 19309 98II8 21019 97766 22722 97384 24418 96973 52! 9 17623 98435 19338 98112 21047 97760 1 22750 97378 24446 96966 5i 10 17651 98430 19366 98107 21076 97754 22778 97871 24474 96959 5o II 17680 98425 19395 98101 2 II 04 97748 22807 97865 245o3 96952 g: 12 17708 98420 19423 98096 2ll32 97742 22835 97358 24531 96945 i3 17737 98414 19452 98090 21161 97735 22863 9735i 24559 96987 47 1 i4 17766 98409 1 948 1 98084 21189 97729 22892 97345 24587 96980 46' i5 17794 98404 19509 98079 21218 97723 22920 97338 24615 96928 45 i6 17823 gS3gg 19538 98073 21246 97717 22948 973.^1 24644 96916 44 \l 17852 983q4 19566 98067 21275 97711 22977 978^5 24672 96909 43 17880 983^^0 19595 98061 2i3o3 97705 23oo5 97318 24700 96902 42; 19 17909 98383 19623 98056 2i33i 97698 23o33 973 1 1 24728 96894 41 j 20 17937 98378 19652 98o5o 2i36o 97692 23062 97304 24756 96887 40^ 21 17966 98373 19680 98044 2i388 97686 23090 97298 24784 96880 89 1 22 17995 98368 19709 98089 98033 21417 97680 23ii8 97291 24818 96873 38; 23 18023 98362 19737 21445 97673 23146 97284 24841 96866 37 i 24 i8o52 98357 19766 98027 21474 97667 23175 97278 24869 96858 36 25 18081 98352 19794 98021 2l5o2 97661 23203 97271 24897 96851 35 26 18109 98347 19823 98016 2i53o 97655 2323l 97264 24925 96844 34 11 i8i38 98341 19851 98010 21559 97648 28260 97257 24953 96887 33 18166 98336 19880 98004 21587 97642 28288 9725i 24982 96829 32 1 29 18195 98331 19908 97998 21616 97636 28816 97244 25oio 96822 3ii 30 18224 98325 19937 97992 21644 97630 28345 97287 25o38 96815 3o! 3i 18252 98320 19965 97987 21672 97623 28878 97280 'x5ob6 96807 It 32 18281 983 1 5 19994 97981 2I70I 97617 28401 97228 25094 9680L 33 i83o9 983x0 20022 97975 21729 97611 23429 97217 l5l22 96708 27 34 18338 983 04 2oo5i 97969 2175s 97604 23458 97210 25i5i 96786 26 35 18367 98299 20079 97963 21786 97598 28486 97208 25i79 96778 25 36 18395 98294 20108 97958 2I8I4 97592 285i4 97196 20207 96771 24 11 18424 98288 2oi36 97952 21843 97585 28542 97189 25235, 96764 23 18452 98283 2oi65 97946 21871 97579 28571 97182 25268 96756 22' 39 18481 98277 20193 97940 21899 97573 23599 97176 25291 96749 ^I i 30 i85o9 98272 20222 97934 21928 97566 28627 97169 25820 96742 20' 41 i8538 98267 20250 97928 21956 97560 23656 97162 25848 96734 19! 42 18567 98261 20279 97922 21985 97553 28684 97155 20876 96727 18' 43 18595 98256 2o3o7 97916 220l3 97547 28712 97148 25404 96719 \l 44 18624 98250 20336 97910 22041 97541 28740 97141 25482 96712 45 18652 98245 2o364 9790D 22070 97534 28769 97134 25460 96705 i5 46 18681 98240 20393 97899 22098 97528 28797 97127 25488 96697 14 47 18710 98234 20421 97893 22126 97521 28825 97120 255i6 96690 i3 48 1*738 98229 98223 20400 97887 22l55 975i5 28858 97118 25545 96682 it 49 18767 20478 97881 22l83 97508 28882 97106 25578 96675 II 5o 18795 98218 2o5o7 97875 22212 97502 28910 97100 256oi 96667 10 5i 18824 98212 20535 97869 22240 97496 289881 97098 25629 96660 9 52 18852 98207 2o563 97863 22268 97489 28966 97086 25657 96653 53 18881 98201 20592 97857 22297 97483 28995 97079 25685 96645 I H 18910 98196 20620 97851 22325 97476 24028 97072 25718 96688 55 18938 98190 20649 97845 22353 07470 24o5i 97065 25741 96680 5 56 18967 98185 20677 97830 97833 22382 97463 24079 97o58 25769 96623 4 u 18995 98179 20706 22410 97457 24108 97o5i 25798 96615 3 19024 98174 20734' 97827 22438 97450 24i36| 97044 25826, 9660B 2 59 M 19052 as. 98168 207631 97821 22467 97444 24164 C. S. 97087 25854 C. S. 96600 I S. C. S. 1 s. c. s. S. 1 S. 1 s. "M r» Deg. 78 Deg. J 77 ] Deg. 76 Deg. 76 Deg. 56 A TABLE OF NATURAL SINES. 16 Deg. M S. c. s. 9 10 II la i3 14 i5 i6 1:2 ! 19 30 21 22 23 24 25 26 II ?o i 3i I 33 I 34 ; 35 36 37 38 39 40 41 42 43 44 45 S n 5i 53 53 54 55 56 u M 25882 25910 23988 25966 25994 26022 26o5o 26079 26107 26i35 26163 26191 26219 26247 26275 263o3 2633i 26359 26387 26415 26443 26471 265oo 26528 26556 26584 26612 26640 26668 26696 26724 26752 26780 26808 26836 26864 26892 26920 26948 26976 27004 27032 27060 27088 27116 27144 • 37172 27200 272281 27256 37284! 27312] 27340 27368I 273961 27424; 27452 27480 27508 27536 C. S. 965o3 96585 96378 96570 96562 96555 96547 96540 96532 96524 96517 96509 96502 96494 96486 96479 96471 96463 96456 96448 96440 96433 96425 96417 96410 96402 96394 96886 96879 96871 96363 96855 96847 96840 96882 96324 96816 96808 96801 96298 96285 96277 96269 96261 96253 96246 96288 96280 96222 96214 96206 96198 96190 96182 96174 96166 96x58 96150 96142 96184 16 Deg. 74 Dog. 27564 27592 27620 27648 27676 27704 27781 27759 27787 27815 27848 27871 27899 27927 27955 27988 28011 28089 28067 28095 28128 28i5o 28178 28206 28284 28262 28290 28818 28846 28874 28402 28429 28457 28485 285i8 28541 28569 28597 28625 28652 28680 0. S. 17 Deg. C. S. 18 Deg. 2 28786 28764 28792 28820 28847 28875 28908 28981 28959 28987 29015 29042 29070 29098 29126 29154 29182, 29209 C. S. I 78 Deg. 96126 1 96118; 96110! 96102 ' 96094 ' 96086. 96078:1 96070 1 96062 t 96054 9604611 9608711 96029 96021- 96018 j 96005 95997 95972 95964 95956 95948 95940! 95981! 95928 95915 95907 95898 958 958 95874 95865 95857 95849 95841 95882 95824 95816 90807 95799 95791 95782 95774 95766 95757 95749 95740 95782 95724 95715 95707 95698 95600 95681 95678 95664 95656 95647 95689I 29287 ^9265 29298 29821 29848 29876 29404 29482 29460 29487 295i5 29548 29571; 29599 29626 29654; 29682I 29710 29787 29765 29798 29821 29849 29876 29904 29982 29960 29987 8ooi5 30048 80071 30098 30126 30154 30182 30209 80287 80265 30292 80820 30848 80876 8o4o3 30481 30459 80486 8o5i4 3o542 80570 80597 80625 3o653 30680 80708 80786 80768 80791 80819 80846 80874 ' C. S. 1" 95680 93622 93618 956o5 95596 95588 95579 95571 95562 95554 95545 95586 95528 95519; 955111 95502! 95408 954S5 95476 95467 95459 95450 95441 95488 95424 95415 95407 95808 95809 95380 95872 95868 95354 95845 95887 95828 95819 95810 95801 95298 95284 95275 95266 95257 95248 95240 95281 95222 95213 95204 95195 95186 95177 95168 95159 95i5o 95142 95i38 95124 95ii5 S. 80902 80929 80957 80985 81012 81040 81068 8 1 095 81128 3ii5i 81178 3 1 206 3i238 81261 81289 3i8i6 81844 8187 81899 81427 81454 81482 8i5io 3i587 3i565 81598 31620 81648 81675 81708 81780 81758 81786 81818 81841 81868 81896 81928 81951 81979 82006 82084 82061 82089I 82116; 82144! 82171I 82199 82227 82234 82282 82809 82887 82864 32892; 82419 82447 32474 32502 82529 "cTsT' c. s. 19 Dog. S. I s.c. M 93097; 95079 95070 95061 95o52 95043 95o83 95024 95oi5; 95oq6; 949971: 94988, 9497911 94970 94961 il 94952 94948 94988 94924 94915 94906 948, 94888 94878 94869 94860 9485 1 94842 94882 94828 94814 94805 94795 94786 94777 94768 94758 94749 94740 94780 94721 94712 94702 94698 94684 94674 94665 94656 94646 94687 94627 94618 94609 94399 94590 94580 94571 94561 S. 32557 32584' 82612, 82689 82667 82694 82722 82749 32777! 82804; 82832i 32859 82887 82914 82942 82969 32997 880241 33o5i 88079! 83io6! 38i34 83i6i 88189 83216 88244 88271 88298 33826 88858 88881 88408 33436 33468 88490 835i8 33545 33573 88600 88627 38655 88682 88710 38787 88764 88792 88819 88846 88874 88901' 33929' 88956! 88983, 3401 1, 34088 84065 84098 34120 84147 34175 ■cTs: ■ 94552 94542 94533 94523; 57! 94514: 56 • 94504I 55' 94495; 54! 94485 53 94476 94466 94457 94447 94488 94428 94418 94409 52 5i| 5o tv, 47 I 46 45 441 43; 421 41 1 40 j III 36 94899 94800 94380 94870 94861 94351 94842 94882 94822 94818] 35 94808 94298 94284 94274 94264 94254 94245 94235 94225 94215 94206 94196 94186 94176 94167 94157 94147 94187 94127 94II8 94108 14 94098 1 3 $4088; 13 94078 II 94068] ID 94o58 94049 94089 94029 94019 94009 98999 98989 93979 S. 34 33 32 3i 3o 25 24 23 22 21 20 I( il i5 li 72 Deg. 71 Deg. 70 Deg. A TABLE OF NATURAL SINES. dl M 20 Deg. 21 Deg. 22 Deg. 23 Deg. r 24 Deg. M 60 S. C. S. S. c. s. 98358 S. I c. s. S. 89078 C. S. r s. TTsT 91355 34202 93969 i 35837 37461 92718 92o5o [ 40674 I 34229 93959'; 35864 98848 87488 92707 89100 92089 40700 91843 U 2 34257 93949 35891 98887 375i5 92607 89127 92028 40727 9i33i 3 34284 93939 ! 30918 98827 87542 92686 89153 92016 40753 91819 571 4 343 1 1 98929 35945 98816 87569 91675 89180 92005 40780 91807 56 5 34339 03919 35978 98806 87595 92 64 39207 91904 40806 91295 55 6 ! 34366 98909 86000 98295 37022 92053 89284 91982 40833 91288 54 I 34393 93899 36027 98285 87649 92642 1 89260 91971 40860 91272' 53 34421 93889 36o54 98274 87676 92681 89287 91900 40886 91260; 52 9 34448 93879 36081 98264 87703 92620 89814 91948 40918 91248 5i ic 34475 93869 86108 98253 87780 9260Q 39841 91986 40989 91286 5o ,11 345o3 98859 86185 98248 87757 92508 89867 91925 40966 01924 49 12 3453o 98849 86162 98282 37784 92587 89894 91914 40992 91212 48 i3 34557 98889 36190 98222 87811 92576 89421 91902 41019 91200 47 14 34584 98829 36217 982II 87888 92565 89448 91891 41045 91188 46 i5 34612 93819 36244 98201 87865 92554 89474 91879 41072 9II76 45 i6 34639 98809 36271 98190 87892 92543 89501 91868 41098 9I164 44 \l 34666 98709 36298 93180 37919 92582 89528 9i856 41125 9ii52 43 34694 98789 36825 98169 87946 92521 89555 91845 4ii5i 91140 42 19 34721 9^779 36852 98159 87978 92510 89581 91888 41178 91128 41 20 34748 98769 86879 98148 37999 92499 89608 91822 41204 91116 40 21 34775 98750 36406 98187 88026 92488 89635 91810 41281 91104 39 22 34803 98748 36434 98127 38o53 92477 89661 91799 41257 91092 38 23 34830 98788 86461 98116 88080 92466 89688 91787 41284 91080 37 24 34857 98728 36488 98106 88107 92455 89715 91775 41810 91068 36 25 34884 98718 365i5 98095 88184 92444 89741 91764 41887 9io56 35 26 34912 98708 36542 98084 38i6i 92482 39768 91752 41868 91044 34 27 34939 98698 86569 98074 88188 92421 89795 91741 41890 91082 33 28 1 34966 93688 86596 98063 382x5 92410 89822 91729 41416 91020 32 29 34993 98677 86628 98052 88241 92890 89848 91718 41448 91008 81 So 35c2i 98667 86650 98042 88268 92888 89875 91706 41469 90996 3o 3i 35048 98657 86677 98081 88295 92877 89902 91694 41496 90984 ^2 32 35075 98647 86704 98020 88822 92866 89928 91688 4l522 90972 28 33 35io2 98687 86781 98010 88849 92355 89955 91671 41549 90960 27 34 35i3o 98626 86758 92999 88876 92843 89982 91660 41575 90948 26 35 35i57 98616 86785 92988I 88408 92882 40008 91648 41602 90986 25 36 35i83 98606 36812 92978 88480 92821 4008 5 91686 41628 90924 24 37 352II 98596 86889 92967 88456 92810 40062 91625 41655 90911 33 38 35239 98585 36867 92956 38488 92299 40088 91618 4I68I 90899 22 39 35266 93575 86894 92945 385io 92287 4oii5 91601 41707 90881 ai 40 35293 98565 86921 92985 88587 92276 4oi4i 91590 41734 90875 20 41 35320 98555 86948 92924 38564 92265 40168 91578 41760 90868 IQ 42 35347 98544 86975 92918 88591 92254 40195 91566 41787 9085 1 18 43 35375 98534 87002 92902 88617 92243 40221 91555 41818 90889 \l 44 35402 93524 87029 92892 38644 92281 40248 91548 41840 90826 45 35429 93514 37056 92881 88671 92220 40275 9i58i 41866 90814 i5 46 35456 93508 87083 92870 88698 9220Q 40801 91519 41892 90802 14 47 35484 98498 87110 92859 38725 921Q8 40828 9i5o8 4i9'9 90790 i3 48 355ii 98488 87187 92849 38752 92186 4o855 91496 41945 90778 12 ^9 35538 93472 87164 92888 38778 92175 40881 91484 41972 90766 II 5o 35565 98462 87191 92827 388o5 92164 40408 91472 41998 90753 10 5i 3559^ 98451 87218 92816 88882 92152 40484 91461 42024 90741 q 55 35619 93441 37245 92805 88859 92141 40461 91449 42o5i 90729 53 35647 98431 87272 92794 88886 92180 40488 91487 42077 90717 7 54 35674 98420 87299 92784! 38912 92119 4o5i4 91425 42104 90704 6 55 35701 98410 37826} 927781 38989 92107 40541 91414 42180 90692 5 56 357:8 98400 873581 92762 88966 92096 4o567 91402 42 1 56 90680 4 57 35755 98889 87880 92751 88998 92085 40594 91890 42183 90668 3 58 i 35782 98870 87407 92740 89020 92078 40621 91878 42209 42235 90655 2 59 35oio 98868 37434 92729 39046 92062 40647 C. S. 91866 S. 90643 S. I M C. S. S. c. s. i s. C. S. S. "cTsT I 69 Beg. 68 Deg. 67 Deg 1 66 Deg. 66 Deur. 19 68 A TABLE OF NATURAL SINES. M 25 J Dfcg. 26 Deg. 27 Deg. 28 Deg. 29 Deg. 1 S C. S. 9063 1 S. c. s. S. 1 c. s. S. C. S. 88295 s. 1 c. s. 42262 43837 898791! 45399 89IOI ! 46947 48481' 87462 60 I 42288 906x8 43863 89867!! 45425 89087 i 46973 8828X 485o6 87448 59 3 423i5 90606 43889 89804 4545 X 89074 46999 88267 48532 87434 58 3 42341 Q0594 43916 8984X 45477 89061 47024 88254 48557 87420 5- 4 42367 90082 43942 89828^1 455o3 89048 ' 47o5o 88240 48583! 87406 >^ 5 42394 90069 43968 898161! 45529 89035 ! 47076 88226 48608 8739 X 55 6 42420 90557 43994 89803'! 45554 89021 471OX 882x3 48634 87377 54 7 42446 90045 44020 89790,1 45580 89008 47127 88199 48659 87363 53 8 42473' 90532 44046 89777J 45606 88995 47153 88x85 48684 87349 5j 9 42499 90520 44072 89764 45632 88981;; 47178 88x72 48710 87335 5i iO 42525 90507 44098 89752II 45658 88968 ' 47204 88x58 48735 87321 5o II 42552 90495 44124 89739 45684 88955:! 47229 88144 4876X 87306 49 12 42578 90483 44i5x 89726 45710 88942'! 47200 88i3oi| 48786 87292 48 i3 42604 90470 44177 897x3 45736 88928'; 47281 881x7 48811 87278 47 14 4263 1 90458 442o3 89700 45762 88915, 47306 88xo3 48837 87264 46 i5 42657 90446 44229 89687 45787 88902 47332 88089 48862 87250 45 i6 1 4 2683 90433 44255 89674 458x3 88888 47358 88075 48888 87235 44 \l 42709 90421 44281 89662 45839 88875! 47383 88062 48qi3 8722X 43 42736 90408 44307 89649' 45865 88862'! 47409 88048 48938 87207 42 39 42762 90396 44333 89636' 45891 88848,1 47434 88034 48964 87193 41 20 42788 9o3»3 44359 89623 45917 88835 47460 88020 48989 87178 40 21 428x5 90371 44380 89610! 45942 88822 47486 88006 490x4 87164 39 38 22 42841 90358 44411 89O97! 45968 88808 47511. 87993 49040 87i5o 23 42867 90346 44437 89084! 45994 88795 47537; 87979 49065 87136 37 24 42894 90334 44464 89571 46020 88782!! 47562 87965 49090 87x21 36 25 42920 903 2 X 44490 89558 46046 88768 47588 87951 49116 87107 35 26 42946 90309 445x6 895451 46072 88755 47(514 87937 49141 87093 34 27 42972 90296 44542 890321 46097 88741 47639 87923 49166 87079 33 28 42999 90284 44568 89519 46123 88728 47665 87909 49192 87064 32 29 43025 90271 44594 89506 46149 887x5 47690 87896 49217 87o5o 3i 3o 43o5i 90259 44620 89493 46x75 8870X 47716 87882 49242 87036 3o 3i 43077 90246 44646 89480 4620X 88688 47741 87868 49268 87021 29 32 43x04 90233 44672 89467 46226 88674 47767 87854 49293 87007 28 33 43i3o 90221 44698 89454! 46202 88661 47793 87840 49318 86993 27 34 43 1 56 90208 44724 89441!! 46278 88647 478x8 87826 49344 86978 26 35 43x82 90x96 447^0 89428 46304 88634! 47844 878x2 49369 86964 25 36 43209 90183 44776 89415 46330 88620!! 47869 87798 49394 86949 24 37 4323 d 9017X 44802 89402 46355 88607 47895 87784 494x9 86935 23 38 43261 90x58 44828 89389I 46381 88593 47920 87770! 49445 86921 22 39 43287 90x46 44854 89376I 46407 8858oi 47946 87756 49470 86906 2X 40 433x3' 90x33 44880 89363 46433 88566! 47971 87743 49495 86892 20 41 43340 90120 44906 89350 46458 88553: 47997 87729 4952 X 86878 XO 42 43366 90x08 44932 89337 46484 88539 48022 877x5 49546 86863 x8 43 43392 90095 449 08 89324 465x0 88526 48048 87701 49571 86849 17 44 43418 90082 44984 893 II 1 46036 885x2, 48073 87687 49596 86834 16 45 43445 90070 45oio 89298 4656 X 88499! 48099 87673 49622 86820 i5 46 43471 90057 45o36 89285 46587 88485 48x24 87659 49647 868o5 14 ^J 43497 90045 45062 89272 466x3 884721! 48x5o 876451 49672 86191 i3 4a 43523 90032 45o88 89259 4663g 88458 j 48x75 8763X 49697 86777. [2| 49 43549 900x9 45x14 89245 46664 88445 48201 87617 49723 86762! II 56 43575 00007 45x40 89232 46690 8843 X 48226 87603 49748 1 86748 5i 43602 45x66 892x9 467x6 88417 48252 87589 49773 86733- ol 52 43628 8998 X 45x92 89206 46742 88404' 48277 87570 4Q798 867x9 S 53 43654 89968 452x8 89193 46767 88390 483o3 3756x!! 4q824, 86704 I 04 43680 89956 45243 89x80 46793 88377 48328 87546 ' 40849* 86690 55 43706 89943 40269 89167 468x9 88363 48354 87532 49874 86675 5 56 43733 89930 45295 891531 46844 88349 48379 875x8; 40899' 86661 4 57 ■ 5S 43759 899x81 453 2 X 89140; 46870 88336 48400 87504^ 40924 86646 3 43i85 89905 45347 89127' 46896 88322! 48430 87490, 4q95o, 86632 X 59 438x1 89892 45373 89114' 4692 x' 883o8j 48456 87476 49975 86617 C. S. 1 s. I M 1 M e.g. S. C. S. s. ! c. s. . s. C. S. S. 1 04] >eg. i 08 I ^i._l 62 I ^cg. J 611 )e^. 6:> Dog. A TABLE OF NATURAL SINES. 69 M 80 ] Deg. ~C. S. 31 : Deg. 32 Deg. 83 Deg. 84 Deg. ' M 60 S. S. C. S. 857.7 S. 52992 C. S. S. 1 c. s 54464 83867 S. c. s. 82904 o 5oooo 866o3 5i5o4 84806 569x9 I 5oo25 86588 5x529 83702 53017 84789 54488 8385x 66943, 82887 5o 2 5oo5o 86573 5i554 83687 53o4i 84774 545i3 83836 66968! 82871' 58 3 50076 86559 5 1 579 85672 53o66 84769 54537' 83819 ■J6992! 82866 57 4 5oioi 86544 5 1 604 85657 5309 X 84743 54661 838o4 66016! 82839 56 5 50126 86530 51628 85642 53xx5, 84728 54686! 83788 5604c 82822 55 6 5oi5i 865 1 5 5x653 85627 53 1 4c 84712 54610 83772 66064 82806 54 6 50176 865ox 51678 85612 53x64 84697 54636 83766 56o88 82790 53 5020I 86486 51703 85597 53189 8468 X 5463g 83740 56xx2 82773 52 9 50227 86471 5x728 85582 532X4 84666 54683 83724 56x36 82767 5i lO 50252 86457 5x753 85567 53238 84660 54708 83708 66160 82741 5o II 50277 86442 5x778 85531 53263 84635 54732 83692 66x84 82724 49 12 5o3o2 86427 5i8o3 85536 53288 84619 64766 83676 66208 82708 48 i3 5o327 86413 5x828 85521 533x2 84604 5478 X 83660 56232 82692 47 i4 5o352 86398 5x852 855o6 53337 84688 64806 83645 66266 82676 46 i5 5o377 86384 51877 85491 5336X 84673 54829 83629 56280 82669 45 i6 5o4o3 86369 51902 85476 53386 84667 64864 836 1 3 563o5 82643 44 \l 50428 86354 5x927 85461 5341 1 84642 54878 83697 56329 82626 43 50453 86340 5x952 85446 53435 84526 54902 8368i 56353 82610 42 19 50478 86325 5x977 8543 1 53460 845 XX 54927 83665 56377 82693 41 20 5o5o3 863x0 52002 83416 53484 84495 6496 X 83649 6640 X 82677 40 21 5o528 86295 52026 85401 53509 84480 64976 83633 66426 8266X 39 22 5o553 8628X 52o5i 85385 53534 84464 54999 836i7 66449 82644 38 23 50578 86266 52076 85370 53558 84448 66024 836oi 66473 82628 37 24 5o6o3 8625x 52X01 85355 53583 84433 55048 83485 56497 826x1 36 25 50628 86237 52126 85340 53607 84417 66072 83469 56621 82496 35 26 5o654 86222 52x5x 85323 53632 84402 66097 83453 66646 82478 34 ^1 50679 86207 52x75 85310 53656 84386 55i2x 83437 66669 82462 33 28 50704 86192 52200 85294 5368x 84370 55x45 83421 66693 82446 32 29 50729 86178 52225 85279 53706 84356 66169 834o5 666x7 82429 3x 3o 50754 86x63 5225o 85264 53730 84339 66x94 83389 5664X 82413 3o; 3i 50779 86x48 52275 85249 53764 84324 562x8 83373 66665 82396 29 32 5o8o4 86x33 52299 85234 53779 84308 66242 83366 66689 82380 28 33 50829 86119 52324 852x8 53804 84292 66266 83340 66713 82363 27 34 5o854 86104 52349 85203 53828 84277 66291 83324 66736 82347 26 35 50879 86089 52374 85x88 53853 8426X 553i6 833o8 56760 82330 25 36 50904 86074 52399 85x73 53877 84246 65339 83292 66784 823x4 24 H 50929 86069 52423 85x57 53902 84230 55363 83276 66808 82297 23 38 50934 86045 52448 85x42 53926 84214 56388 83260 56832 82281 22 39 50979 86o3o 52473 85x27 5393X 84198 66412 83244 66856 82264 2X 40 5 1 004 8601 5 52498 85ix2 53976 84182 66436 83228 66880 82248 20 41 51029 86000 52522 86096 64000 84167 66460 832X2 66904 8223l XO 42 5io54 83985 52547 86081 54024 841 5 1 55484 83x96 66928 822x4 18 43 5io79 85970 52572 86066 64049 84i35 55509 83x79 66962 82x98 nt 44 5x104 83966 52697 85o5i 54073 84120 56533 83x63 56976 82x81 i6j 45 5x1 29 85941 52621 85o35 64097 84104 55557 83x47 67000 82165 i5\ 46 5ix54 85926 52646 86020 54x22 84088 6658x 83x3i 57024 8ii48 ui 47 5xx79 8391 X 52671 86oo5 54146 84072 556o6 83xi5 57047 82l32 i3 48 5 1 204 85896 52696 84989 54171 84067 55630 83098 83o82 57071 82x16 12 P 5X229 8588i 52720 84974 54195 84041 55664 57096 82098 II 5o 5x254 85866 52745 84969 54220 84026 66678 83o66 57119 57143 82082 XO 5i 5x279 8585x 52770 84943 54244 84009 66702 83o6o 82065 52 5i3o4 85836 52794 84928 54269 83994 66726 83o34 57167 82048 53 5x329 8582X 528x9 84913 54293 83978 56760 83ox7 67x91 82032 7 54 5x354 858o6 52844 84897 54317 83962 66776 83oox 672x6 820x5 6 55 5x379 85792 52869 84882 54342 83946 66790 55823 82986 67238 8x999 81982 5 56 5 1 404 85777 52893 84866 54366 83930 82969 67262 4 U 51429 85762 52918 8486 X 5439X 839x5 66847 82953 67286 81965 3 51454 85747 52943 84836 64416 83899 83883 S. 5587 X 82936 67310 81949 2 §9 1 5i479 85732 52967 84820 5/i/|/io 55896 C. S. 82920 67334 C. S. 81932 I M 1 M C. S. S. C. S. 58 I S. C. S. S. S. 69 I )e^ 57 De^. 1 56 Deg. 6 55 Deg. ! 70 A TABLE OF NATURAL SINES. 85 Deg. 86 Deg. 37 Deg. 38 Deg. 89 Deg. M J^' " : C. S. S. 58779 ! C. S. S. C. S. S. C. S. S. C. S. M 57358 81915 80902 60182 79864 61666 78801 6':932 77715 bo I 57381 1 81899 588o2 80886 60206 79846 61689 78788 62966 77696 ^ 2 57405 81882 58826 80867 60228 79829 61612 78766 i 62977 77678 3 57429! 8 I 865 57453 1 81848 58849 80860 60261 79811 61686 78747 68000 77660 67 4 58873 80833 60274 79793 61668 78729 1 68022 77641, 56 5 57477 8i832 58896 80816 60298 79776 61681 7871 1 68046 77623 55 t 57501 1 8i8i5 58920 80799 60821 79768 61704 78694 68068 77606 54 % 57524 ' 81708 58943 80782 60844 79741 61726 78676 68090 77686 53 8 57548 i 81782 58967 80765 60867 79728 61749 78668 63ii3 77668 52 9 57572 1 81765 58990 80748 60890 79706 61772 78640 68i36' 77660 5i IC 57596 i 81748 59014 80730 60414 79688 61795 78622 68168 77531 5o II 57619 1 81731 59037 80713 60487 79671 61818 78604 68180 77613 49 13 57643 i 81714 59061 80696 60460 79668 61841 78686 632o3 77494 48 i3 57667 81698 59084 80679 60488 79636 61864 78668 63226 i 77476 47 14 57691 81681 59108 8^632 60606 79618 61887 78660 68248 77468 46 i5 57715 81664 59131 80644 60629 79600 61909 78682 68271 77439 46 i6 57738 81647 69154 80627 6o553 79688 61982 78614 63293 77421 44 \l 57762 8i63i 59178 80610 60676 79666 61966 78496 633 1 6 77402 43 57786 81614 59201 80693 60699 79547 61978 78478 65338 77384 42 19 57810 81 597 59225 80676 60622 79680 62001 78460 63361 77866 41 20 57833 8i58o 59248 80668 60646 79612 62024 78442 68383 77347 40 21 57857 8 1 563 59272 80641 60668 79494 62046 78424 68406 77829 39 22 57881 81546 59295 80624 60691 79477 62069 78406 68428 77810 38 23 57904 8i53o 59318 80607 60714 79469 62092 78887 68461 77292 37 24 57928 8i5i3 59842 80489 60788 79441 62116 78869 68473 77273 36 25 57952 81496 59365 80472 60761 79424 62i38 78861 68496 77266 85 26 57976 81479 59389 80466 60784 79406 62160 78888 68618 77286 34 13 57999 81462 59412 80438 60807 79888 62183 78816 68640 77218 33 58023 81445 59436 80420 6o83o 79871 62206 78297 68663 77199 32 29 58047 81428 59459 80403 60868 79363 62229 78279 68685 77181 3i 3o 58070 81412 59482 8o386 60876 79886 62261 78261 68608 77162 3o 3i 58094 81395 59606 8o368 60899 79818 62274 78243 68o3o 77144 29 32 58ii8 81378 69629 8o36i 60922 79800 62297 78226 63663 77126 28 33 58i4i 8i36i 59662 80334 60946 79282 62820 78206 68676 77107 27 34 58i65 81 344 69676 8o3i6 60968 79264 62342 78188 68698 77088 26 35 58189 81327 69699 80299 60991 79247 62866 78170 68720 77070 25 36 58212 8i3io 69622 80282 61016 79229 62888 78162 68742 77061 24 37 58236 81293 69646 80264 6io38 7921 1 62411 78184 68766 77088 23 38 58260 81276 69669 80247 61061 79198 62433 78116 68787 77014 »a 39 58283 81259 59693 80280 61084 79176 62466I 78098 68810 76996 21 40 583o7 81242 69716 80212 61107 79168 62479 78079 68832 76977 20 i 41 58330 81225 59739 80196 61180 79140 62602 78061 68854 76969 19 42 58354 81208 69763 80178 6u53 79122 62624 78043 68877 76940 18 43 58378 81191 69786 80160 61176 79106 62647 1 78026 68899 769211 17 44 58401 81174 59609 80143 61 199 79087 62670 78007 68922 76908 16 45 58425 8ii5'7 69832 80126 61222 79069 62692: 77988 68944 76884 i5 4& 58449 8.140 69866 80108 61245 79061 62616! 77970 68966 76866 14 ^J ' 58472 81123 59879 80091 61268 7go33 62688! 77962 63989 76847 i3 i 48 ' 58496' 81106 69902, 80073 61291 79016 62660 77934 640 II 76828 12 1 ir , 585i9 81089 69926 80066 61814 78998 62688^ 77916 64033 76810 11 1 5c ; 58543 1 81072 3io55 69949 8oo38 618871 78980 62706 77897 64066 76791 10 5i i 58567 O9972 8oG2I 61860' 78962 62728 77879 64078 76772, 9 5a 1 53590 8io38 69995 8ooo3 61888 78944 62751I 77861 64 1 00 1 76764 8 53 58614 81021 60019 79986 61406 78926 627741 77843 641 281 76735 I 54 53637 81004 60042 79968 61429 78908 62796 77824 64145 76717 76698 55 5866 1 80987 60066 79961 61461 78891 62819 77806 64167 5 56 58684 80970 60089 799^4 61474 78873 62842 77788 64 I 901 76679 4 ll 58708 80953 601 1 2 79916 61497 78855 62864 77769 642121 76661 3 58731 80936 6oi36| 79809 61620 78887 62887 77761 64284 76642 2 59 M 58755 80919 60168 C. S. 79881 61643 78819 62909 77733 s. 64266! 76623 I M 1 C. S. 1 S. S. C. S. S. C. S. C. S. ! S. 54 Deg 53 Deg. i 52 Deg. 1 51 Deg. 1 50 Deg. A TABLE OF J!^ATURAL SINES. n 9 !0 II la i3 M i5 i6 i8 20 31 33 23 24 35 26 37 38 39 3o 3i 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 <8 5i 52 53 54 55 56 U 59 M 40 I>eg. 8. C. S. 7615 76135 76116 76097 76078 76o5g 64379 64301 64333 64346 64368: 643 90 I 64413 64435' 64457] 64479 64001 j 64524' 64546! 64568 1 64590 6461 2 1 64635 64657 64679 64701 64723, 64746 64768^ 64790 64812! 64834' 64856 64878; 64901! 64923 64945 J 64967 64989' 65oii 65o33 65o55 65o77 65o99 65l22 65 1 44 65i66, 65 1 88 65210 65232 65254 65276 65298 65320 65342 65364 65386 654o8 6543o 65452 65474 65496 655i8 65540 65562 65584 656o6 48 Deg. 766eg. H Deg. i L>eg. o ? '5J Lat. 50.99 Dep. Dep. Lat. Dep. Lat. Dep. 0.89 509" r 1.11 .50.98 1.34 ~50M 1.56 52 51,99 0.91 51.99 1.13 51.98 1.36 51.98 1..59 52 53 52 99 0.92 52.99 1.10 52.98 1.39 52.98 1.62 53 64 53 99 0.94 53.99 1.18 53.98 1.41 53.97 1.65 54 55 o4 99 0.96 54 . 91' 1.20 54.98 1.44 54.97 1.68 55 56 55.99 0.98 55.99 1.22 55.98 1.47 55.97 1.71 56 57 56.99 0.99 56.99 1.24 56 98 1.49 56.97 1.74 57 58 57.99 1.01 57.99 1.27 .^7.98 1.52 57.97 ..77 58 59 58.99 1.03 58.99 1.29 58.98 1.54 58.97 1.80 59 60 61 59.99 1.05 1.06 59.99 1.31 1.33 59.98 60.93 1.57 1.60 59 . 97 1.83 1.86 60 61 60.99 60.99 60.97 62 61.99 1.08 61.99 1.35 61.9S 1.62 61.97 1.89 62 63 62.99 1.10 62.99 1.37 62.98 1.65 62.97 1.92 63 64 63.99 1.12 63.98 1.40 63.93 1.68 63.97 1,95 64 65 64.99 1.13 64.98 1.42 64.98 1.70 64.97 1 99 65 66 65.99 1.15 65.98 1.44 85.98 1.73 65.97 2 02 66 67 66.99 1.17 66.98 1.46 66.98 1.75 66.97 2.05 67 68 67.99 1.19 67.98 1.48 67.98 1.78 67.97 2.08 68 69 68.99 1.20 68.98 1.51 68.98 1.81 68 97 2.11 69 70 71 69.99 70.99 1.22 1.24 69.98 1.53 69.98 1.83 69. 97 2.14 2.17 70 71 70.98 1.55 70.98 1.86 70.8S 62 61.61 6.34 51.58 6.56 51.56 6.79 51.53 7.01 52 63 62.60 6.46 52.58 6.69 52.55 6.92 52.52 7.16 63 64 53.60 6.58 53.57 6.81 53.54 7.05 53.51 7.28 64 65 54.59 6 70 54.56 6.94 54.53 7.18 54.50 r.42 55 56 65.58 6.82 £5.55 7.07 55.52 7 31 55.49 7.55 56 67 66.58 6.95 56.54 7.19 56.51 7 44 56.48 7.69 57 58 57.57 7.07 57.54 7.32 57.50 7.57 67.47 7.82 58 59 58.56 7.19 58.53 7.45 58.50 7.70 58.46 7.96 59 60 59.55 7.31 7 43 59.52 7.57 59.49 7.83 59.45 8.09 60 "fil 60.55 60.51 7.70 60.48 7.96 60.44 8.23 62 61.54 7.56 61.50 7.82 61.47 8.09 61.43 8.36 62 63 62.53 7.68 63.50 7.95 62.46 8.22 62.42 8.50 63 64 63.52 7.80 63.49 8.08 63.45 8.35 63.42 8.63 64 65 64.52 7.92 64.48 8.20 64.44 8.48 64.41 8.77 65 66 65.51 8.04 65.47 8.33 65.44 8.61 65.40 8.90 66 67 66 50 8.17 66.46 8.46 66.43 8.75 66.39 9.04 67 68 67.49 8.29 67.46 8.58 67.42 8.88 67.38 9.17 68 69 68.49 8.41 68.45 8.71 68.41 9.01 68.37 9.30 69 70 71 69.48 70.47 8.63 69.44 8.83 69.40 70.39* 9.14 69.36 70.35 9.44 70 71 8.65 70.43 8.96' 9.27 9.57 72 71.46 8.77 71.42 9.09 71.38 9.40 71.34 9.71 72 73 72.46 8.90 72.42 9.21 72.38 9.53 72.33 9.84 73 74 73.45 9.82 73.41 9.34 73.37 9.66 73.32 9.9S 74 75 74.44 9.14 74.40 9.46 74.36 9.79 74.31 10.11 75 76 75.43 9.26 75.39 9.59 75.35 9.92 75.31 10.25 76 77 76.43 9.38 76.38 9.72 76.34 10.05 76.30 10.38 77 78 77.42 9.51 77.38 9.84 77.33 10.18 77.29 10.52 78 79 78.41 9.63 78.37 9.97 78.32 10.31 78.28 10.65 79 80 81 79.40 9.75 79.36 10.10 79.32 10.44 79.27 80.26 10.79 80 81 80.40 9.87 80.35 10.22 80.31 10.67 10.92 82 81.39 9.99 81.34 10.35 81.30 10.70 81.25 11.06 82 83 82.38 10.12 82.34 10.47 82.29 10.83 82.24 11.19 83 84 83.37 10.24 83.33 10.60 83.28 10.96 83.23 11.33 84 85 84.37 10.36 84.32 10.73 84.27 11.09 84.22 11.46 85 86 85.36 10.48 85.31 10.85 85.26 11.23 85.21 11.60 86 S7 86.35 10.60 86.30 10.98 86.26 11.36 86.21 11.73 87 38 87.34 10.72 87.30 11.11 87.25 11.49 87.20 11.87 88 89 88.34 10.85 88.29 11.23 88.24 11.62 88.19 12.00 89 90 91 89.33 10.97 89.28 11.36 89.23 11.75 89.18 12.14 90 91 90.32 11.09 90.27 11.48 90.22 11.88 90.17 12.27 92 91.31 11.21 91.26 11.61 91.21 12.01 91.16 12.41 92 93 92.31 11.33 92.26 11.74 92.20 12.14 92.15 12.54 93 94 93 30 11.46 93.25 11.86 93.20 12.27 93.14 12 68 94 95 94.29 11.58 94.24 11.99 94.19 12.40 94.13 12.81 as 96 95.28 J 11.70 95.23 12.12 95.18 12.53 95.12 12.95 96 S7 96.28 11.82 96.22 12.24 96.17 12.66 96.11 13.08 97 98 97.27 11.94 97.22 12.37 97.16 12.79 97.10 13.22 98 99 98.26 12.07 98.21 12.49 98.15 12.92 98.10 13.35 99 lao 8 99.25 12.19 99.20 12.62 99.14 13.05 99.09 13.49 :oo Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. i i3 .9 83 1 )eg. 821 Deg. 82| Deg. m Deg. 5 88 TEAVBllSE TABLE. 5 8 Deg. 8i Deg. 6^ Deg. 8J Deg. 5 ? 1 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 3 « 0.99 0.14 0.99 0.14 0.99 0.15 0.99 O.lo' 1 2 1.98 0.28 1.98 0.29 1.98 0.30 1.98 0..30 a 3 2.97 0.42 2.97 0.43 2.97 0.44 2.97 0.46 3 4 3.96 0.56 3.96 0.57i 3.96 0.59 3.95 0.61 i 5 4.95 0.70 4.95 0.72 4.95 0.74 4.94 0.76, 5 6 5.94 0.84 5.94 0.86 5.93 0.89 5.93 0.91 6 7 6.93 0.97 6.93 1.00 6.92 1.03 6.92 1.06 7 81 7.92 1. 11 7.92 1.15 7.91 1.18 7.91 1 22 S S 8.91 1.25 8.91 1.29! 8.90 1.33 8.90 1.37 9 10 11 9.90 10.89 1.39 1 1.53 i 9.90 1.43! 9.89 10.88 1.481 1.63i 9. 83 1.52 1.67 10 11 10.89 1.58 10.87 12 11.88 1.67; 11.88 1.72 11.87 1.77 11.86 1.83 12 13 12.87 1.81 ! 12.87 1.87 12.86 1.92 12.85 1.98 13 14 13.86 1.95 13.86 2.01 13.85 2.07 13.84 2.13 14 15 14.85 2.09 1 14.85 2.15 14.84 2.22! 14.83 2.28 15 16 15.84 2.23 15.84 2.30 15.82 2.36 1 15.81 2.43 16 17 16.83 2.37 16.83 2.44i 16.81 2.51 i 16.80 2.59 17 IS 17.82 2.51 17.81 2.58 1 17.80 2.66 17.79 2.74 18 19 18.82 2.64 18.80 2.73 18.79 2.81 18.78 2.89 19 20 21 19.81 2.78 19.79 2.87 19.78 2.96 19.77 3.04 20 20.80 2.92 20.78 3.01 20.77 3.10 20.76 3.19 21 22 21.79 3.06 21.77 3.16 21.76 3.25 21.74 3.35 i 22 23 22.78 3.20 22.76 3.30 22.75 3.40 22.73 3.50 1 23 24 23.77 3.34 23.75 3.44 23.74 3.55 23.72 3.65 1 24 25 24.76 3.48 24.74 3.59 24.73 3.70 24.71 3.80 1 25 26 25.75 3.62 25.73 3.73 25.71 3.84 25.70 3.96 1 26 27 26.74 3.76 26.72 3.87 26.70 3.99 26.69 4.11 27 28 27.73 3.90 27.71 4.02 1 27.69 4.14| 27.67 4.26 i 28 29 23.72 4.04 28.70 4.16 28.63 4.29 1 28.66 4.41 29 30 29.71 30.70 4.18 4.31 29.69 30.68 4.30 4.45 29.67 4.43 '29.65 4.56 i 30 30.66 4.53 30.64 4.72 31 32 31.69 4.45 31.67 4.59 ; 31.65 4.73 : 31.63 4.87 32 33 132.68 4.59 32.66 4.74 i 32.64 4.88 j 32.62 5.02 33 34 33.67 4.73 33.65 4.88 33.63 5.03 1 33.60 5.17 1 34 3o 34.66 4.87 34.64 5.02 34.62 5.17 34.59 5.32 35 36 35.65 5.01 35.63 5.17 35.60 5.32 35.58 5.^8 36 37 36.64 5.15 36.62 5.31 36.59 5.47 36.57 5.63 1 37 38 37.63 5.29 37.61 5.45 37.58 5.62 37.56 5.78 38 39 38.62 5.43 38.60 5.60 38.57 5.76 38.55 5.93 39 40 41 39.61 40.60 5.57 5.71 i 39-59 5.74 39.56 5.91 6.06 39.53 6.08 40 i 40.58 5.88 40.55 40.52 6.24 41 42 41.59 5.85 ,41.57 6.03 141.54 6.21 |I41.51 6.39 42 43 42.58 5.99 42.56 6.17 142.53 6.36 42.50 6.54 43 44 43.57 6.12 43.54 6.31 43.52 6.50 43.49 6.69! 44 if, 44- 56 6.26 44.53 6.46 44.51 6.65 1 44.48 6.85 45 46 45.55 6.40 45.52 6.60 45.49 6.80 t 45.46 7.00 46 17,46.54 6.54 46.5 6.74 46.48 6.95 1 46.45 7.15 47 iS 47.53 6.68 47.50 6.89 47.47 7.09 47.44 7.30 1 48 \ 49 48.52 6.82 48.49 7.03 48.46 7.24 48.43 7.45 49 60 s s 49.51 6.96 49.48 7.17 49.45 7.39 49.42 7.61 50 Dep. 83 1 Lat. deg. Dep. 811 Lat. Deg. Dep. Lat. 1 Dep. Lat, 9 U S 8U Deg. m Deg. S .2 TRAVERSE TABLE 89 c S' a o 51 3 Deg. 8i LaL Deg. 8iDeg. 81 Deg. 1 1 a 9 51 Lat. 50.50 Dep. 7.10 Dep. Lat. Dep. Lat. Dep. 50.47 7.32 50.44 7.54 50.41 7.76 52 51.49 7.24 51.46 7.46 51.43 7.69 51. .39 7.91 52 63 62.48 7.38 52.45 7.61 52.42 7.83 52.38 8.06 53 54 53,47 7.52 53.44 7.75 53 41 7.98 53.37 8.21 54 55 54.46 7.65 54.43 7.89 54,40 8.13 54.36 8.37 55 56 55.46 7.79 55.42 8.04 55.38 8.28 55.35 8.52 56 57 56.45 7.93 56.41 8.18 56.37 8.43 56.34 8.67 57 58 57.44 8.07 57.40 8.32 57.36 8.57 57.32 8.82 58 59 58.43 8.21 58.39 8.47 58.35 8.72 58.31 8.98 59 (50 61 59.42 60.41 8.35 59.38 8.61 59.34 8.87 59.30 9.13 "9.28 60 61 8.49 60.37 8.75 60.33 9.02 60.29 62 61.40 8.63 61.36 8.90 61.32 9.16 61.28 9.43 62 63 62.39 8.77 62.35 9.04 62.31 9.31 62.27 9.58 63 64 63.38 8.91 63.34 9.18 63.30 9.46 63.26 9.74 64 65 64.. 37 9.05 64.33 9.33 64.29 9.61 64.24 9.89 65 66 65.36 9.19 65.32 9.47 65.28 9.76 65.23 10.04 66 67 66.35 9.32 66.31 9.61 66.26 9.90 66.22 10.19 67 68 67.34 9.46 67.30 9.76 67.25 10.05 67.21 10.34 68 69 68.33 9.60 68.29 9.90 68.24 10.20 68.20 10.50 69 70 71 69.32 9.74 69.28 10.04 69.23 10.35 69.19 10.65 70 71 70.31 9.88 70.27 10.19 70.22 10.49 70.17 10.80 72 71.30 10.02 71.25 10.33 71.21 10.64 71.16 10.95 72 73 72.29 10.16 72.24 10.47 72.20 10.79 72.15 11.10 73 74 73.28 10.30 73.23 10.62 73.19 10.94 73.14 11.26 74 75 74.27 10.44 74.22 10.76 74.18 11.09 74.13 11.41 75 78 75.26 10.58 75.21 10.91 75.17 11.23 75.12 11.56 76 77 76.25 10.72 76.20 11.05 76.15 11.38 76.10 11.71 77 78 77.24 10.86 77.19 11.19 77.14 11.53 77.09 11.87 78 79 78.23 10.99 78.18 11.34 78.13 11.68 78.08 12.02 79 80 81 79.22 11.13 79.17 11.48 79.12 11.82 79.07 12.17 80 "81 80.21 11.27 80.16 11.62 80.11 11.97 80.06 12.32 82 81.20 11.41 81.15 11.77 81.10 12.12 81.05 12.47 82 83 82.19 11.55 82.14 11.91 82.09 12.27 82.03 12.63 83 84 83.18 11.69 83.13 12.05 83.08 12.42 83.02 12.78 84 85 84.17 11.83 84.12 12.20 84. 0-^ 12.56 84.01 12.93 85 86 85.16 11.97 85.11 12.34 85. 0(? 12.71 85.00 13.08 86 87 86.15 12.11 86.10 12.48 85.04 12.86 85.99 13.23 87 88 87.14 12.25 87.09 12.63 87.03 13.01 86.98 13.39 88 89 88.13 12.39 88.08 12.77 88.02 13.16 87.96 13.54 89 90 01 89.12 12.53 89.07 12.91 89.01 13.30 88.95 89.94 13.69 90 91 90.11 12.66 90.06 13.06 90.00 13.45 13.84 92 91.10 12.80 91.05 13.20 90.99 13.60 90.93 14.00 92 ' 93 92.09 12.94 92.04 13.34 91.98 13.75 91.92 14.15 93 94 93.09 13.08 93.03 , 13.49 92.97 13.89 92.91 14.30 \H 95 94.08 13.22 94.02 13.63 93.96 14.04 93 89 14.45 i^5 96 95.07 13.36 95.01 i 13.78 94.95 14.19 94.88 14.60 96 97 96.06 13.50 96.00 ! 13.92 95.93 14.34 95.87 14.76 97 98 97.05 13.64 96.99 : 14.06 96.92 14.49 96.86 14.91 98 99 98.04 13.78 97.98 14.21 97.91 14.63 97.85 15.06 99 100 6 O a a *^ m a 99.03 13.92 98.97 14.35 98.90 14.78 98.84 15 21 100 s eO 5 Dep. Lat. Dtp. Lat. Dep. Lat. Dep. Lat. 82 1 Dog. 811 De^. SHDeg. 81i Dog. 90 TRAVtBSE TABLE s 3 a 9 Deg. H Deg. H Deg 9f Deg s Lat. Dep. Lat. Dep. Lat. Dep. Lat. 0,99 Dep. o 2 1 0.99 0.16 0.99 0.16 0.99 0.17 0.17 I 2 1.98 0.31 1.97 0.32 1.97 0.33 1.97 0.34 2 3 2.96 0.47 2.96 0.48 2.96 0.60 2.96 0.51 3 4 3.95 0.63 3.95 0.64 3.95 0.66 3.94 0.68 4 5 4.94 0.78 4.93 0.80 4.93 0.83 4.93 0.86 5 6 5.93 0.94 5.92 0.96 5.92 0.99 5.91 1.02 6 7 6.91' 1.10 0.91 1.13 6.90 1.16 6.90 1.19 7 8 7.90 1.25 7.90 1.29 7.89 1.32 7.88 1.35 8 9 8.89 1.41 8.88 1.45 8.88 '.49 8.87 '52 9 10 11 9.88 1.56 9.87 1.61 9.86 X.65 9.88 1.69 10 11 10.86 1.72 10.86 1.77 10.86 1.82 10.84 1.86 12 11.85 1.88 11.84 1.93 11.84 1.98 11.83 2.03 12 13 12.84 2.03 12.83 2.09 12.82 2.16 12.81 2.20 13 U 13.83 2.19 13.82 2.25 13.81 2.31 13.80 2.37 14 1.5 14.82 2.35 14.80 2.41 14.79 2.48 14.78 2.54 16 16 16.80 2.5C 15.79 2.57 15.78 2.64 15.77 2.71 16 17 16.79 2.66 16.78 2.73 16.77 2.81 16.75 2.88 17 18 17.78 2.82 17.77 2.89 17.75 2.97 17.74 3.05 18 19 18.77 2.97 18.75 3.05 18.74 3.14 18.73 3.22 19 20 21 19.75 3.13 19.74 3.21 19.73 3.30 19.71 3.39 20 21 20.74 3.29 20.73 3.38 20.71 3.47 20.70 3.56 22 21.73 3.44 21.71 3.64 21.70 3.63 21.68 3.73 22 28 22.72 3.60 22.70 3.70 22.68 3.80 22.67 3.90 23 24 23.70 3.75 23.69 3.86 23.67 3.96 23.65 4.06 24 25 24.69 3.91 24.67 4.02 24.66 4.13 24.64 4.23 25 2fi 25.68 4.07 25.66 4.18 25.64 4.29 25.62 4.40 26 27 26.67 4.22 26.65 4.34 26.63 4.46 26.61 4.. 57 27 28 27.66 4.38 27.64 4.50 27.62 4.62 27.60 4.74 28 29 28.64 4.54 28.62 4.66 28.60 4.79 28.58 4.91 29 30 29.63 4.69 29.61 4.82 29.59 4.95 29.57 6.08 30 31 30.62 4.85 30.80 4.98 30.57 5.12 30.55 6.25 31 32 31.61 5.01 31.58 5.14 31.66 5.28 31.64 6.42 32 33 32.59 5.16 32.57 5.30 32.55 5.46 32.52 6.59 33 34 33.58 5.32 33.56 5.47 33.53 5.61 33.51 5.76 34 35 34.57 5.48 34.54 5.63 34.52 5.78 34.49 6.93 35 36 35.56 5.63 35.53 5.79 35.61 5.94 35.48 6.10 36 37 36.54 5.79 36.62 6.96 36.49 6.11 36.47 6.27 37 38 37.53 5.94 37.51 6.11 37.48 6.27 37.46 6.44 38 39 38.52 6.10 38.49 6.27 38.47 6.44 38.44 6.60 39 40 41 39.51 6.26 39.48 6.43 39.45 6.60 39.42 6.77 40 41 40.50 6.41 40.47 6.69 40.44 6.77 40.41 6.94 42 41,48 6.57 41.45 6.75 41.42 6.92 41.39 7 11 42 43 42 47 6.73 42.44 6.91 42.41 7.10 42.38 7 28 43 44 43,46 6.88 43.43 7.07 43.40 7.26 43.36 7.45 44 V5 44.45 7.04 1 44.41 7.23 44.38 7.43 44.35 7.62 45 46 45.43 7.20 ' 45.40 7.39 45.37 7.59 45.34 7.79 16 47 46.42 7.35 46.39 7.55 46.36 7.76 46.32 7.96 i7 4t; 47 41 7.51 47.38 7.72 47.34 7.92 47.31 8.13 18 49 48.40 7.67 48.36 7.88 48.33 8.09 48.29 8.30 49 50' 49.38 7.82 49.35 8.04 49.32 Dep. 8.25 49.28 8.47 50 s s Dep. Lat. Dep. Lat Lat. Dep. Lat. § 11 .a 81 Deg. 801 Dog. 801 Dog. 8(H Deg. & TRAVERSE TABLE. 91 o 6l 52 53 64 66 56 67 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 86 86 87 88 89 90 91 92 93 94 95 96 97 98 99 too « 9 Deg. 1 9^ Deg. H Deg. 9| Deg g- p a ? 51 52 53 54 55 56 57 58 59 60 6] 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 32 93 94 95 96 97 98 99 iOO 6 ed Q Lat 50,37 61.36 52.36 53.34 54.32 55.31 56.30 57.29 58.27 59.26 60.25 61.24 62.22 63.21 64.20 65.19 66.18 67.16 68.15 69.14 Dep. Lat. 50.34 51.32 52.31 53.30 54<28 65.27 56.26 67.25 58.23 59.22 Dep. Lat. Dep. Lat. Dep. 7.98 8.13 8.29 8.45 8.60 8.76 8.92 9.07 9.23 9.39 8.20 8.36 8.62 8.68 8.84 9.00 9 16 9.32 9 48 9 ^4 9.81 9.97 10.13 10.29 10.45 10.61 10.77 10.93 11.09 11.25 50.30 51.29 52.27 53.26 54.25 55.23 56.22 57.20 68.19 69.18 8.42 8.58 8.75 8.91 9.08 9.24 9.41 9.57 9.74 9.90 60.26 51.25 52.23 63.22 54.21 56.19 56.18 67.16 68.15 59.13 8.64 8.81 8.98 9.14 9.31 9.48 9.66 9.82 9.99 10.16 9.. 54 9.70 9.86 10.01 10.17 10.32 10.48 10.64 10.79 10.95 60.21 61.19 62.18 63.17 64.15 65.14 66.13 67.12 68.10 69.09 ■60.16 61.15 62.14 63.12 64.11 65.09 66.08 67.07 68.05 69.04 10.07 10.23 10.40 10.56 10.73 10.89 11.06 11.22 11.39 11.55 60.12 61.10 62.09 63.08 64.06 5.05 66.03 67.02 68.00 68.99 10.33 10.60 10.67 10.84 11.01 11.18 11.35 11.52 11.69 11.85 70.13 71.11 72.10 73.09 74.08 75.06 76.05 77.04 78.03 79.02 80.00 80.99 81.98 82.97 83.95 84.94 86.93 86.92 87.90 88.89 11. li 11.26 11.42 11.58 11.73 11.89 12.05 12.20 12.36 12.61 70.08 71.06 72.05 73.04 74.02 75.01 76.00 76.99 77.97 78.96 11 41 11.57 11.73 11.89 12.06 12.22 12.38 12.. 54 12.70 12.86 70.03 71.01 72.00 72.99 73.97 74.96 75.94 76.93 77.92 78.90 11.72 11.88 12.05 12.21 12.38 12.54 12.71 12.87 13.04 13.20 69.97 70.96 71.95 72.93 73.92 74.90 76.89 76.87 77.86 78.84 12.02 12.19 12.36 12.53 12.70 12.87 13.04 13.21 13.38 13.55 12.67 12.83 12.98 13.14 13.30 13.45 13.61 13.77 13.92 14.08 14.24 It. 39 14.65 14.70 14.86 15.02 16.17 16.33 16.49 16.64 79.95 80.93 81.92 82.91 83.89 84.88 85.87 86.86 87.84 88.83 89.82 90.80 91.79 92.78 d3.76 94.75 95.74 96.73 97.71 98.70 13.02 13.18 13.34 13.50 13.66 13.82 13.98 14.16 14.31 14.47 79.89 80.88 81.86 82.85 83.83 84.82 86.81 86.79 87.78 88.77 13.37 13.53 13.70 13.86 14.03 14.19 14.36 14.62 14.69 14.86 79.83 80.82 81.80 82.79 83.77 84.76 86.74 86.73 87.71 88.70 13.72 13.89 14.06 14.23 14.39 14.56 14.73 14.90 15.07 16.24 89.88 90.87 91. 8t 92.84 93.83 94.82 95.81 96.79 97.78 98.77 14.63 14.79 14.95 16.11 15.27 15.43 16.59 16.76 16.91 16.07 89.76 90.74 91.72 92.71 93.70 94.68 95.67 96.66 97.64 98.63 15.02 16.18 16.36 16.61 15.68 15.84 16.01 16.17 16.34 16.50 89.69 90 67 91.66 92.64 93.63 94.61 95.60 96.68 97.67 98.66 16.41 16.58 15.75 15.92 16.09 16.26 16 43 16.60 16.77 16.93 Dep. Lat. Dep. Lat. Dep. Lat Dep. lat. 1 81 Deg. i i 801 Deg. 80^ Deg. m Deg. 92 FRAVERSE TABLE P 10 Deg. m Deg. 1 10| Deg. |0| Deg. C 5' 5 P 1 Lat. Dep. T.JT Lat. Dep. Lat j 0.98 Dep. 1 0.18 Lat. "orgs' D€p. 1 0.98 0.98 0.18 0.19 2 1.97 0.35 1.97 0.30 1.97' 0.36 1.96 0.37 % 3 , 2.95 0.52 2.95 0.53 2.95 0.55 2.95 56 3 4! 3.94 0.69 3,94 0.71 , 3.93, 0.73 3.93 0.75 4 5 4 . 92 0.87 1 4.92 0.89 4.92' 0.91 4.91 0,93 5 6 5.91 1.04 1 5.90 1.07 5.90 1.09 5.89 i.l2 6 7 6.89 1.22 1 6.89 1.25 6.88, 1 28 S.88 I 31 7 8 7. 88 1.39; 7.87 1.42 7.87' 1 46 ?.86 1.49 8 9 8.86 1.56! 8.86 1.60 8.85 1.84. 8.84 1.68 10 9.85 11 10 83 1.74 1 9.84 1.91 10.82 1.78 1.96 9.83; 10.82 1 1.82' 2.00 9.82 1.87 2.05 lO 11 10.81 12 11 82 2.08 : 11.81 2.14 11.80' 2.19 11.79 2.24 12 13 12.80 2.26 12.79 2.31 12.78 2.37 12.77 2.42 13 14 13.79 2; 43 i 13.78 2.49 i 13.77 2 . 55 13.75 2.61 14 15 14.77 2.60 i| 14.76 2.67' 14.75 2 . 73 14.74 2.80 Vj 16 15.76 2.78 '\ 15.74 2.85 1 15.73 2.92 i 15.72 2.98 16 17 16.74 2.95 : 16.73 3.03 1 16.72 3.10 1 16.70 3.17 17 18 17.73 3.13 ; 17.71 3.20 17.70 3.28' 17. e8 3.36 18 19 18.71 3.30 ; 18.70 3.38 18.68 3.46 18.67 3.54 19 20 19.70 3.47 19.68 3.56 19.67 3.64, 19.65 3.73 20 21 20.68 3.65 20.66 3.74 20.65 3.83 20.63 3.92 21 22 21.67 3. 82 21.65 3.91 21.63 4.01 21.61 4.10 22 23 22.65 3.99 22.63 4.09 22.61 4.19 22.60 4.29 23 24 23.64 4.17 " 23.62 4.27 23 . 60 4.37 23 . 58 4.48 24 25 24.62 4.34 24.60 4.45 24.58 4.56 24.56 4.66 25 26 25.61 4.51 25.59 4.63 25.56 ; 4.74 25.54 4.85 26 27 26.59 4.69 26.57 4.80 26.55 4.92 26.53 5.04 27 28 27.57 4.86 27.55 4.98 27.53 i 5.10 27.51 5.22 28 29 28.56 5.04 2S.54 5.16 28.51 i 5.28 28.49 5.41 29 30 31 29 . 54 30.53 5.21 5.38 29.52 5.34 29.50 ! 80.48, 5.47 5.65 ; 29.47 5.60 30 30.51 5.52 30.46 5.78 31 32 31.51 5.56 31.49 5.69, 31.46 5.83 31.44 5.97 32 33 32.50 5.73 32.47 5.87! 32.45 6.01 32.42 6.16 33 34 33.48 5.90 33.46 6.051 33.43 6.20 33.40 6.34 34 35 J4 4" 6.03 ; 34.44 6 . 23 34.41 6.38 34.39 6.53 35 36 35.45 6.25 35.43 6.41 35.40 ; 6.56 35.37 6.71 38 ; 37 36.44 6.42 36.41 6.58 36.38: 6.74 36.35 6.90 37 38 37.42 6.60 37.39 6.76 37.36 6.92 37.33 7.09 38 39 38.41 6.77 38.38 6.94 38.35 7,11 38.32 7.27 39 1 40 41 39.39 40.38 6.95 39.36 7.12 40.35 7.12 7.30 39.33 40.31 7.29 7.47 39.30 7.46 40 41 40.28 7.65 42 41.36 7.29 41.33 7.47 41.30 7.65 41.26 7.83 42 43 42.35 7.47 42.31 7.65 42.28 7.84 142.25 S.02 43 44 43.33 7.64 43.30 7.83 43.26 8.02 1 43.23 8.21 44 45 ,44.32 7.81 44.28 8.01 44. 2f 8.20 44.21 8.39 45 46 45.30 7.99 45.27 8.19, 45.23 8.38 45.19 8.58 46 47 46.29 8.16 46.25 . 8.36 46.21 8.57 i46.l8 8.77 47 48 47.27 8.34 47.23 8.54 47.20 8.75 47.16 8.95 48 43 48.26 8.51 48.22 ; 8.72 48.18 8.93 48.14 9.14 49 ' 6<» 49.24 8.68 Lat. 49.20 1 Dep. 8.90 49.16 9.11 Lat. 49.12 1 9.33 Lat. 50 o 1 i S l>ep. Lut. Dep. Dep. ■mi '5 so 1 [)eg. m Deg. 791 Deg. 791 Deg Q , 1 ^O^X^B^ 1 TKAVERSE TABLE. 93 Dista 10 Deg, lOi Deg. lOi Deg. 101 Deg. O 5- 3 ? 5J 3 P Lat. Dep. Lat. Dep. 9.08 Lat. Dep. Lat. Dep, 51 50.23 8.86 50.19 50.15 9.29 50 J 9.51 52 51.21 9.03 51.17 9.25 51.13 9.48 61.09 9 70 52 53 52.19 9.20 52.15 9.43 .52.11 9.66 .52.07 9.89 53 54 53.18 9.38 53.14 9.61 53.10 9.84 63 . 05 10.07 64 55 54.16 9.55 54.12 9.79 54.08 10.02 04.03 10.26 5.'i 56 55.15 9.72 55.111 i' 96 55.06 10.21 55.02 10.45 56 57 56.13 9.90 56.09! 10.14 56 . 05 10.39 56.00 10.63 57 58 57.12 10.07 57.07 1 10.32 57 . 03 10.57 .56 . 98 10.82 58 69 58.10 10.25 58.06 ! 10.50 58.01 10.75 67.96 11.00 69 60 61 59.09 60.07 10.42 10.59 59.04! 10.68 59.00 10.93 68.95 11.19 60 '61 60.03 10.85 59.98 11.121 69.93 11.38 62 61.06 10.77 61.01 ; 11.03 60.96 11.30 60.91 11.56 62 63 62.04 10.94 61.99 i 11.21 61.95 11.48 61.89 11.75 63 84 63.03 11.11 62.98 ! 11.39 62.93 11.66 6S.88 11.94 64 65 64.01 11.29 63.96 i 11.57 63.91 11.85 63.86 12.12 65 66 65.00 11.46 64.95 11.74 64.89 12.03 64.84 12.31 66 67 65.98 11.63 65.93 11.92 65.88 12.21 65.82 12.60 67 68 66.97 11.81 66.91 12.10 66.86 12.39 66.81 12.68 68 69 67.95 11.98 67.90 12.28 67.84 12.57 67.79 12.87 69 70 71 68.94 "69.92 12.16 12.33 68.88 12.46 68.83 69.81 12.76 68.77 69.76 13.06 70 7] 69.87 12.63 12.94 13.24 72 70.91 12.50 70.85 : 12.81 70.79 13.12 70.74 13.43 72 73 71.89 12.68 71.83 12.99 71.78 13.30 71.72 13.62 73 74 72.88 12.85 72.82 13.17 72.76 13.49 '72.70 13.80 74 75 73.86 13.02 73.80 : 13.35 73.74 13.67 73.68 13.99 76 76 74.85 13.20 74.79 13.52 74.73 13.85 74.67 14.18 76 77 75.83 ! 13.37! 75.77! 13.70 75.71 14.03 76.65 14.36 77 78 76.82 1 13.54 76.76 i 13.88 76.69 14.21 76.63 14.65 78 79 77.80 1 13.72 77.74 14.06 77.68 14.40 77.61 14.74 79 80 81 78.78 13.89 14.07 78.72 14.24 14.41 78.66 14.58 78.60 14.92 80 81 79.77 79.71 79.64 14.76 79.58 15.11 82 80.75 14.24 80.69 , 14.59 80.63 14.94 80.50 15.29 82 83 81.74 14.41 81.68 14.77 81.61 15.13 81.54 15.48 83 84 82.72 14.59 82.66 14.95 82.59 15.31 82.63 15.67 84 85 83.71 i 14.76 83.64 15.13 83.58 15.49 83.51 16.85 85 86 84.09 14.93 84.63 15.30 84.56 15.67 84.49 16.04 86 87 85.68 15.11 85.61 , 15.48 85.54 15.85 85.47 16.23 87 88 86.66 i 15.28 86.60 15.66 86.53 16.04 83.46 16.41 88 89 87.65' 15.45 87.58 1 15.84 87.51 16.22 87.44 16.60 89 90 91 88.63 1 15.63 88.56 j 16.01 16.19 88.49 16.40 88.42 16.79 16.97 90 i'^91 89.62] 15.80 89.55 89.48 [16.58 89.40 92 90.60 1 15.98 90.53 16.37 90.46 16.77 90.39 17.16 92 93 91.59 1 16.15 91.52 16.55 91.44 16.95 91.37 17.35 53 94 92.57 16.32 92.50, 16.73 92.43 17.13 92.35 17.53 94 95 93.56 16.50 93.48 16.90 93.41 17.31 93.33 17.72 95 96 '94.54 16.67 94.47 17.08 94.39 17.49 94.32 17.91 96 97 95- 63 16.84 95.45 17.26 95.38 17.68 95.30 18.09 1 97 98 96.51 17.02 96.44, 17.44 96.36 17.86 96.28 18.28 ! 98 99 97.50 17.19 97.42' 17.62 97.34 18.04 97.26 18.47 ! 99 100 o § ca 98.48 Dep. I 17.36 98.40 i 17.79 98.33 18.22 98.25 18.65 ! .00 6 go 1. Q Lat. Dep. i Lat. Oep. Lat. Dep. Lat. 1 80 Deg. 791 Deg. i 79i Deg. 79i Deg. 94 TRAVERSE TABLE. 1 s • 11 Deg. lU Deg. IH Deg. Ill Deg. ■ B s 1 Lat. Dep. Lat. Dep. Lat. Dep. 0.20 Lat. Dep. 1 0.98 0.19 0.98 0.20 0.98 0.98 0.20 2 1.96 0.38 1.96 0.39 1.96 0.40 1.96 0.41 3 3| 2.94 0.57 2.94 0.59 2.94 0.60 2.94 0.61 a 1 3.93 0.76 3.92 0.78 3.92 0.80 3.92 0.82 4 5 4.91 0.95 4.90 0.98 490 1.00 4.90 1.02 5 € 5.89 1.14 6.88 1.17 6 88 \ .20 5.87 1.22 ft 7 6.87 1.34 6.87 1.37 6.86 1.40 6.85 1.43 7 8 7.85 1.53 7.85 1.56 7.84 1.59 7.83 1.63 8 9 8.83 1.72 8.83 1.76 8.82 1.79 8.81 1.83 9 10 11 9.82 1.91 9.81 10.79 1.95 9.80 1.99 9.79 2.04 10 11 10.80 2.10 2.15 10.78 2.19 10.77 2.24 12 11.78 2.29 11.77 2.34 11.76 2.39 11.75 2.44 12 13 12.76 2.48 12.75 2.54 12.74 2.59 12.73 2.66 13 14 13.74 2 67 13.73 2.73 13.72 2.79 13.71 2.86 14 15 14.72 2,86 14.71 2.93 14.70 2.99 14.69 3.06 15 16 15.71 3.05 15.69 3.12 15.68 3.19 15.66 3.26 16 17 16.69 3.21 16.67 3.32 16.66 3.39 16.64 3.46 17 18 17.67 3.43 17.65 3.51 17.64 3.59 17.62 3.66 18 19 18.65 3.63 18.63 3.71 18.62 3.79 18.60 3.87 19 20 21 19.63 3.82 19.62 3.90 19.60 3.99 19.58 4.07 20 21 20.61 4.01 20.60 4.10 20.58 4.19 20.56 4.28 22 21.60 4.20 21.58 4.29 21.56 4.39 21.54 4.48 22 23 22.58 4.39 22.56 4.49 22.54 4.. 59 22.52 4.68 II 24 23.56 4.58 23.54 4.68 23.52 4.78 23.50 4.89 25 24.54 4.77 24.62 4.88 24.50 4.98 24.48 5.09 25 26 25.52 4.96 25.50 5.07 25.48 5.18 25.46 5.30 26 27 26.50 5.15 26.48 5.27 26.46 5.38 26.43 5.50 27 28 27.49 6.34 27.46 5.46 27-44 5.58 27.41 5.70 28 29 28.47 5.53 28.44 5.66 28.42 5.78 28.39 5.91 29 30 31 29.45 5.72 29.42 5.85 29.40 5.98 29.37 6.11 30 30.43 5.92 30.40 6.05 30.38 6.18 30.35 6.31 31 32 31.41 6.11 31.39 6.24 31.36 6.38 31.33 6.52 32 33 32.39 6 30 32.37 6.44 32.34 6.58 32.31 6.72 33 34 33.38 6.49 33.35 6.63 33.32 6.78 33.29 6.92 34 35 34.36 6.68 34.33 6.83 34.30 6.98 34.27 7.13 35 36 35.34 6.87 35.31 7.02 35.28 7.18 35.25 7 33 36 37 36.32 7.06 36.29 7.22 36.26 7.38 36.22 r.53 37 38 37.30 7.25 37.27 7.41 37.24 7.58 37,20 7.74 38 39 38.28 7.44 38.25 7.61 38.22 7.78 38.18 7.94 39 40 39.27 7.63 39.23 7.80 39. 20^ 7.97 39.16 40.14 8.15 40 41 41 40.25 7.82 40.21 8.00 40.18 8.17 8.35 42 41.23 8.01 41.19 8.19 41.16 8.37 41.12 8.56 45 43,42.21 8.20 42.17 8.39 42.14 8.57 42.10 8.76 43 41 ; 43. 19 8.40 43.15 8.58 43.12 8.77 43.08 8.9G 4) 45 1 44 17 1 8.59 44.14 8.78 44.10 8.97 44.06 9.16 45 46*45.15 8.78 45.12 8.97 45.08 9.17 45.04 9.37 46 47 146.14 8.97 46.10 9.17 46.06 9.37 46.02 9.57 47 48 j 47.12 9.16 47.08 9.36 47.04 9.57 46.99 9.78 18 49 148.10 9.35 48.06 9.50 48.02 9.77 47.97 9.98 49 50^ 149.08 9.54 49.04 Dep. 9,75 49.00 9.97 48.95 10.18 50 ' 8 a ei Dep. Lat. 1 Lat. Dep. Lat. Dep. Lat. 79] Deg 78J De^. 'H Deg. 78i Deg. T1«A VERSE TAWLE. 95 o U Deg. lU Deer. IH Deg. Ill Deg. O a. Pi 3 ~~ n 51 LaU Dep. Lat. C/ep. ' 9.95 Lat. Dep. Lat. Dep. 51 50.06 9.73 60.02 49.98 10.17 49.93 1 10.39 52 51.04 9.92 51.00 10.14 50.96 10.37 50.91 i 10.59 62 53 52.03 10.11 51 98 10.34 51.94 10.57 51.89 10.79 53 5'i 53.01 10.30 52.96 10.. 53 52.92 10.77 62.87 11.00 54 55 53.99 10.49 53.94 10.73 53.90 10.97 63.85 11.20 65 56 64.97 1 10.69 54.92 10.93 64.88 11.16 54.83 11.40 56 57 55.95 1 10,88 55.90 11.12 55.86 11.36 55.81 11.61 5? 58 56.93 1 11.07 56.89 11.32 56.84 11.56 56.78 11.81 68 59 57.92 11.26 57.87 11.51 57.82 11.76 67.76 12.01 69 60 61 58.90 59.88 11.45 .58.85 11.71 58.80 11.96 58.74 12.22 12.42 60 61 11.64 59.83 11.90 59.78 12.16 69.72 62 60.86 11.83 60.81 12.10 60.76 12.36 60.70 12.63 62 63 61.84 12.02 61.79 12.29 61.74 12.56 61.68 12.83 63 64 62.82 12.21 62.77 12.49 62.72 12.76 62.66 13.03 64 65 63.81 12.40 63.75 12.68 63.70 12.96 63.64 13.24 66 66 64.79 1 12.59 64.73 12.88 64.68 13.16 64.62 13.44 66 67 65.77 12.78 65.71 13.07 65.66 13. J6 65.60 13.64 67 68 66.75 12.98 66.69 13.27 66.63 13.56 66.68 13.85 68 69 67.73 13.17 67.67 13.46 67.61 13.76 67.65 14.05 69 70 71 68.71 13.36 68.66 13.66 68.69 13.96 14.16 68.63 14.25 70 "71 69.70 13.55 69.64 13.85 69.57 69.61 14.46 72 70.68 13.74 70.62 14.05 70.55 14.35 70.49 14.66 72 73 71.66 13.93 71.60 14.24 71.53 '4.55 71.47 14.87 73 74 72.64 14.12 72.58 14.44 72.51 14.75 72.45 16.07 74 75 73.62 14.31 73.56 14.63 73.49 14.95 73.43 15.27 76 76 74.60 14.50 74.54 14.83 74.47 15.15 74.41 16.48 76 77 75.59 14.69 75.52 15.02 75.45 15.35 76 39 15.68 77 78 76.57 14.88 76.50 15.22 76.43 15.55 76.37 15.88 78 79 77.55 15.07 77.48 15.41 77.41 15.76 77.34 16.09 79 80 81 78.53 15.26 78.46 15.61 78.39 79.37 16.95 78.32 16.29 80 "81" 79.51 15.46 79.44 15.80 16.15 79.30 16.49 82 80.49 15.65 80.42 16.00 80.35 16.35 80.28 ' 16.70 82 83 81.48 15.84 81.41 16.19 81.33 16.55 81.26 16.90 83 84 82.46 16.03 82.39 16.39 82.31 16.75 82.24 17.11 84 85 83.44 16.22 83.37 16.58 83.29 16.95 83.22 17.31 86 86 84.42 16.41 84.35 16.78 84.27 17.15 84.20 17.51 86 87 85.40 16.60 85.33 16.97 85.25 17.35 86.18 17.72 87 88 86.38 16.79 86.31 17.17 86.23 17.64 86.16 17.92 88 89 87.36 16.98 87.29 17.36 87.21 17.74 87.14 18.12 891 90 91 88.35 17.17 88.27 89.25 17.56 88.19 17.94 88.11 18.33 901 89.33 17.36 17.75 89.17 18.14 89.09 18.53 91 92 90.31 17.55 90.23 17.95 90.15 18.34 90.07 18.74 92 93 91.29 17.75 91.21 18.14 91.13 18.64 91.05 18.94 93 94 92.27 17.94 92.19 18.34 92.11 18.74 92.03 19.14 : 94 95 93.25 1 18.13 93.17 18.53 93.09 18.94 93.01 19.35 95 96 94.24 18.32 94.16 18.73 94.07 .9.14 93.99 19.55 96 37 95.22 18.51 95.14 18.92 95.05 19.34 94.97 19.75 97 98 96.20 18.70 96.12 19.12 96.03 19.64 95.95 19.96 98 99 97.18 18.89 97.10 19.31 97.01 19.74 96.93 20.16 99 100 .3 Q 98.16 19.08 98.08 19.61 97.99 Dep. 19.94 97.90 20.36 100 Dop. Lat. Dep. Lat. Lat. Dep. Lat. .a ', 79 D eg. II 781 Deg. 78i Deg. rSi Deg. 96 TRAVERSE TABLE 5' 12 Deg 12i Deg. 12^ Deg. 12^ Deg. q 8 i • Lat. Dep. ~oT2r Lat. Dep. Lat, Dep. Lat. DepL * 1 0.98 0.98 0.21 0.98 0.22 0.98 0.22 1 2 1.96 0.42 1.95 0.42 1.95 0.43 1.95 0.44 « 3 2.93 0.62 2.93 0.64 2.93 0.65 2.93 0.66 3 4 3.91 0.83 3.91 0.85 3.91 0.87 3.90 0.88 4 5 4.89 1.04 4.89 1.06 4.88 1.08 4.88 1.10 6 6 5.87 1.25 5.86 1.27 5.86 1.30 5.85 1.32 3 7 6.85 1.46 6.84 1.49 6.83 1.52 6.83 1 54 7 8 7.83 1.66 7.82 1.70 7.81 1.73 7.80 1 77 8 9 8.80 1.87 8.80 1.91 8.79 1.95 8.78 1.99 9 10 11 9.78 2.08 9.77 2.12 9.76 2.16 2.38 9.75 10.73 2.21 2.43 10 11 10.76 2.29 10.75 2.33 10.74 12 11.74 2.49 11.73 2.55 11.72 2.60 11.70 2.65 12 13 12.72 2.70 12.70 2.76 12.69 2.81 12.68 2.87 13 14 13.69 2.91 13.68 2.97 13.67 3.03 13.65 3.09 14 15 14.67 3.12 14.66 3.18 14.64 3.25 14.63 3.31 15 16 15.65 3.33 15.64 3.39 15.62 3.46 15.61 3.53 16 17 16.63 3.53 16.61 3.61 16.60 3.68 16.58 3.75 17 18 17.61 3.74 17.59 3.82 17.. 57 3.90 17.56 3.97 18 19 18.58 3.95 18.57 4.03 18.55 4.11 18.53 4.19 19 20 21 19.56 4.16 19.54 4.24 19.53 4.33 19.51 4.41 20 20.54 4.37 20.52 4.46 20.50 4.55 20.48 4.63 21 22 21.52 4.57 21.50 4.67 21.48 4.76 21.46 4.86 22 23 22.50 4.78 22.48 4.88 22.45 4.98 22.43 5.08 23 24 23.48 4.99 23.45 5.09 23.43 5.19 23.41 5.30 24 25 24.45 5.20 24.43 5.30 24.41 5.41 24.38 5.52 25 26 25.43 5.41 25.41 5.52 25.38 5.63 25.36 5.74 26 27 26.41 5.61 26.39 5.73 26.36 5.84 26.33 5.96 27 28 27.39 5.82 27.36 5.94 27.34 6.06 27.31 6.18 28 29 28.37 6.03 28.34 6.15 28.31 6.28 28.28 6.40 29 30 29.34 6.24 29.32 6.37 29.29 6.49 29.26 6.62 30 31 31 30.32 6.45 30.29 6.58 30.27 6.71 30.24 6.84 32 31.30 6.65 31.27 6.79 31.24 6.93 31.21 7.06 32 33 32.28 6.86 32.25 7.00 32.22 7.14 32.19 7.28 33 34 33.26 7.07 33.23 7.21 33.19 7.36 33.16 7.50 34 35 34.24 7.28 34.20 7.43 34.17 7.58 34.14 7.72 35 36 35.21 7.48 35.18 7.64 35.15 7.79 35.11 7.95 36 37 36.19 7.69 36.16 7.85 36.12 8.01 36.09 8.17 37 38 37.17 7.90 37.13 8.06 37.10 8.22 37.06 8.39 38 39 38.15 8.11 38.11 8.27 38.08 8.44 38.04 8.61 39 1 40 41 39.13 40.10 8.32 39.09 8.49 8.70 39.05 8.66 8.87 39.01 1 8.83 40 1 ' 9 05 'i\ 8.52 40.07 40.03 139.99 42 41.08 8.73 41.04 8.91 41.00 9.09 140.96 9.27 1 iU 43 ■ 42.06 8.94 4?. 02 9.12 41.98 9.3i 141.94 a. 49 ; 43 44 43.04 9.15 43.00 9.34 42.96 9.52 42.92 9.71 44 1 45 44.02 9.36 43.98 9.55 43.93 9.74 45 89 9.93 4i 46 44.99 9.56 44.95 9.76 44.91 9.96 44.87) 10.15 1 4fi 47 45.97 9.77 45.93 9.97 45.89 10.17 45.84 1 iO.37 1 47 48 46.95 9.98 46.91 10.18 48.86 10.39 46.82 10-59, 48 48 47.93 10.19 47.88 10.40 47.84 10.61 47.79 10 81 i 49 . 60 48.91 10.40 48.86 10.61 48.81 Dep. 10.82 48.77 Dep. 11.08 1 50 1 I Dep. 1 Lat. Dep. Lat. Lat. Lat. 1 » • c: 78 Deg, 771 Deg. 771 Deg. 77i Deg. SO TRAVERSE TABLE. 97 ? 5: 12 Dsg. m Deg. 12i Dog. 12J Deg. B o a ~6l Lat. 49.89 Dep. 10.60 Lat. 49.84 Dep. Lat. Dep. Lat. Dep. 10.82 49.79 11.04 49.74 11.26 62 50.86 10.81 50.82 11.03 50.77 11.25 .50.72 11.48 62 63 61.84 11.02 61.79 11.25 51.74 11.47 51.69 11.70 63 64 52.82 11.23 52.77 11.46 52.72 11.69 52.67 11.92 54 66 63.80 11.44 53.75 11.67 53 70 i 11.90 53.64 12.14 55 66 54.78 11.64 54.72 11.88 54.67 12.12 54.62 12.36 56 57 55.75 11.85 55.70 12.09 55.65 12.34 55.59 12.58 57 58 56.73 12.06 56.68 12.31 56.63 12.55 56.57 12.80 58 59 57.71 12.27 57.66 12.52 57.60 12.77 57.55 13.02 59 60 61 58.69 12.47 58.63 12.73 58.58 12.99 58.52 13.24 60 '61 59.67 12.68 59.61 12.94 59.55 13.20 59.50 13.46 62 60.65 12.89 60.59 13.16 60.53 13.42 60.47 13.68 62 63 61.62 13.10 61.57 13.37 61.51 13.64 61.45 13.90 63 64 62.60 13.31 62.54 13.58 62.48 13.85 62.42 14.12 64 66 63.58 13.51 63.52 13.79 63.46 14.07 63.40 14.35 65 66 64.56 13.72 64.50 14.00 64.44 14.29 64.37 14.. 57 66 67 65,54 13.93 65.47 14.22 65.41 14.50 65.35 14.79 67 68 66.51 14.14 66.45 14.43 66.39 14.72 66.32 15.01 68 69 67.49 14.35 67.43 14.64 67.36 14.93 67.30 15.23 69 70 71 68.47 14.55 68.41 14.85 68.34 15.15 68.27 15.45 70 71 09.45 14.76 69.38 15.06 69.32 15.37 69.25 16.67 72 70.43 14.97 70.36 15.28 70.29 15.58 70.22 15.89 72 73 71.40 15.18 71.34 15.49 71.27 15.80 71.20 16.11 73 74 72.38 15.39 72.32 15.70 72.25 16.02 72.18 16.33 74 75 73.36 15.59 73.29 15.91 73.22 16.23 73.15 16.55 75 76 74.34 15.80 74.27 16.13 74.20 16.45 74.13 16.77 76 77 75.32 16.01 75.25 16.34 75.17 16.67 75.10 16.99 77 78 76.30 16.22 76.22 16.55 76.15 16.88 76.08 17.21 78 79 77.27 16.43 77.20 16.76 77.13 17.10 77.05 17.44 79 80 '81 78.25 16.63 78.18 79.16 16.97 78.10 17.32 78.03 17.66 80 81 79.23 16.84 17.19 79.08 17.. 53 79.00 17.88 82 80.21 17.05 80.13 17.40 80.06 17.75 79.98 18.10 82 83 81.19 17.26 8i.ll 17.61 81.03 17.96 80.95 18.32 83 84 82.16 17.46 82.09 17.82 82.01 18,18 81.93 18.54 84 85 83.14 17.67 83.06 IS 04 82.99 18.40 82.90 18.76 85 86 84.12 17.88 84.04 18.25 83.96 18.61 83.88 18.98 86 87 85.10 18.09 85.02 18.46 84.94 18.83 84.85 19.20 87 88 86.08 18.30 86.00 18.67 85.91 19.05 85.83 19.42 88 89 87.06 18.50 86.97 18.88 86.89 IS. 26 86.81 19.64 89 90 91 88.03 18.71 87.95 19.10 87.87 19.48 87.78 19.86 90 89.01 18.92 88.93 19.31 88.84 19.70 86.76 20.08 91 92 89.99 19.13 89.91 19.52 89.82 19.91 89.72 20.30 92 93 90.97 19.34 90.88 19.73 90.80 20.13 90.71 20.52 93 94 91. 9e 19.54 91.86 19.94 91.77 20.35 91.68 20.75 94 95 92.92 19.75 92.84 20.16! 92.75 20.56 92.66 20.97 95 96 93.90 19.96 93.81 20.37 93.72 20.78 1 93.63 21.19 96 97 94.88 20.17 94.79 20.58 94.70 20.99 94,61 21.41 97 98 95.86 20.38 95.77 20.79 95.68 21.21 95.58 21.63 98 99 96.84 20.58 96.75 21.01 96.65 21.43 96.56 21.85 99 100 • § c M 97.81 20.79 Lat 97.72 21.22 97.63 21.64 97.53 22.07 100 9 a a S CO Dep. Dep. Lat. Dep. Lat. Dep. Lat. 78 Deg. 771 Deg 77 1 Deg. 77i Deg. 98 trav>:rs£ table. 1 c S' o ? T! 13 Deg. 1 134 Deg. ISA Deg. 13} Deg. C Lat. 1 Dep. 1 0.23 Lat. ! Dep. ! Lat. Dep. 1 Lat. 1 Dep. 1 t 0.97 0.97 0.23 0.97 0.23 1 0.97 0.24! 2 1 1.95 0.45 1 1.95 ! 0.46 1.95 0.47 1.94 0.48- 2 J i 2 92 0.67 2.92 1 0.69 2.92 1 0.70 2.91 0.711 3 4 3.90 0.90 3.89 0.92 3.89 0.93 3.89 0.95 4 fi 4.87 1.12 4.87 1 1.15 4.86 1.17 4.86 1.19 5 fi 5.85 1.35 5.84; 1.38 5.83 1.40 5.83 1.43: 6 7 6.82 ].57 6.81 1.60 6.81 1.63 6.80 1.66 7 8 7.80 1.80 7.79 ! 1.83 7.78 1.87 i 7.77 1.90 j 8 9 8.77 2.02 8.76 2.06 ; 8.75 2.10 ; 8.74 2.14! 9 10 1 11 9.74 10.72 2.25 2.47 9.73 2.29 j 9.72 2.33 9.71 2.38: 10 11 10.71 2.52 j 10.70 2.57 1 10.68 2.61 12 11.69 2.70 11.68 2.75 11.67 2.80 i 11.66 2.85 12 13 12.67 2.92 12.65 2.98 1 12.64 3.03; 12.63 3.091 13 141 13.64 3.15 13.63 , 3.21 ' 13.61 3.27 13.60, 3.33 14 1.5 14. &i 3.37 14.60 3.44 14.59 3.50; 14.57 3.57 15 16| 15.59 3.60 15.57 3.67 15.56 3.74 15.54, 3.80; 16 17 16.57 3.82 16.55 3.90 16.53 3.97 16.51' 4.04' 17 18 17.54 4.06 17.52 4.13 , 17.50 4.20 17.48 4.28 18 19 18.51 4.27 18.49 4.35 18.48 4.44 18.46 4.52 19 20' 19.49 21 20.46 4.50 4.72 19.47 4. 58 19.45, 4.67 19.43 4.75 1 4.99 ; 20 21 20.44 4.81 20.42 1 4.90 20.40 22 21.44 4.95 21.41 5.04 21.39! 5.14 21.37 5.23 22 23 22.41 5.17 22.39 5.27 22.36: 5.37 22.34 5.47 23 24 23.38 5.40 23.36 5.50 23.34 5.50 23.31 5.70 24 25 24.36 5.62 24.33 5.73 24.31 5.84 24.28 5.94 25 26 25.33 5.85 25.31 5.96 125.28 6.07 25.25 6.18 26 27 ; 26.31 6.07 26.28 6.19 26.25 6.30 26.23; 6.42 27 28' 27.28 6.30 27.25 6.42 .27.23 6.54 27.20 6.66 28 29 28.26 6.52 28.23 6.65 '28.20 6.77 129.17 7.00 28.17 1 6.89 29 30 29.23 6.75 1 29.20 ; 30.17 6.88 7.11 29.14' 7.13 30.11, 7.37 30 31 31 1 30.21 6.97 '30.14 7.24 32 31.18 7.20 31.15 7.33 131.12 7.47 31.08! 7.61 32 33 32.15 7.42 32.12 7.56 32.09 7.70 32.05, 7.84 33 34 33.13 7.65 ! 33.09 7.79 , 33.06 7.94 33.03 8.08 34 35 34.10 7.87 , 34.07 8.02 1 34.03 8.17 34.00 8.32 35 36 35.08 : 8.10 1 35.04 8.25 35.01 35.98 8.40 34.97 8.56 36 37 36.05 8.32 i 36.02 8.48 8.64 35.94 1 8.79 37 38 37.03 8.55 36.99 8.71 36.95 8.87 36.91 9.03 38 39 38.00 8.77 37.96 8.94 37.92 9.10 37.88 9.27 39 40 38.97 9.00 3S.94 9.17 1 38.89 9.34 38.85 9.51 40 41 39.95 9.22 39.91 , 9.40 39.87 9.57 39.83 9.75 41 42 40.92 D.45 40.88 ; 9.63 40.84 9.80 40.80 9.98 42 43 41.90 9 67 41.86 9. 86 41.81 10.04 41.77 10.22 43 44 42.87 9.90 42.83 10.03 42.7^ 10.27 42.74 10.46 , 44 45 ' 43.85 .0.12 43.80 10.31 4«.76 10.51 43.71 ; 10.70 i 45 46 4-1.82 10.36 44.78 ,10.54 44.73 10.74 44.68 10.93 46 47 45.80 10.57 45.75 , 10.77 45.70 10.97 45.65 11.17 47 48 46 77 10.80 '46.72 ; 11.00 46.67 11.21 46.62 11.41 48 49 47.74 11.02 47.70 11.23 47.65 11.44 47.60 11.65 49 50 48.72 11.25 4S.67 11.46 148.62 11.67 48.57 11.88 j ^ 8 ' S 1 1 2 ' Dep. ! Lat. Dep. 1 Lat Dep. Lat. Dep. Lat. 1 "^^ i Deg. 76} Deg. 1 76-5 Deg. 76} Deg. rRAVERSE TABLU- 99 c 5* P 61 13 Dcg m Deg. 13A Deg. 131 Deg. o a Lat. Dep. Lat. Dep. 11.69 Lat. 49.59 Dep. Lat. 49.64 Dep. 49.69 11.47 49.64 11.91 12.12 51 62 50.67 11.70 50.62 11.92 60.56 12.14 60.61 12.36 ' 52 53 51.64 11.92 51.59 12.16 51.54 12.37 61.48 12.60 , 53 54 62.62 12.15 52.66 12.38 62.51 12.61 62.45 12.84 64 13. 07 , 55 55 53.59 12.37 53.54 12.61 63.48 12.84 53.42 56 54.56 12.60 54.51 12.84 54.45 13.07 64.40 1.3.31 1 56 67 55.54 12.82 55.48 13.06 56.43 13.31 56.37 13.55^ 67 58 56.51 13.05 56.46 13.29 66.40 13.54 66.34 13.79! 58 59 57.49 13.27 57.43 13.52 67.37 13.77 67.31 14 ,02 59 60 "6T 60 61 58.46 13.. 50 58.40 13.76 58.34 59.31 14.01 14.24 68.28 59.25 14,26 "14.50 59.44 13.72 69.38 13.98 62 60.41 13.95 60.35 14.21 60.29 14.47 00.22 14.74 62 63 61.39 14.17 61.32 14.44 61.26 14.71 61.19 14.97 63 64 62.36 14.40 62.30 14.67 62.23 14.94 62.17 15.2 64 65 63.33 14.62 63.27 14.90 63.20 15.17 63.14 16.45 65 66 64.31 14.85 64.24 16.13 64.18 15.41 64.11 16.69 66 67 65.28 15.07 65.22 16.36 66.15 16.64 65.08 16.93 67 68 66.26 15.30 66.19 16.69 66.12 15.87 1 66.05 16.16 68 69 67.23 15.52 67.10 16.81 67.09 16.11 67.02 16.40 69 70 71 68.21 69.18 15.75 15.97 68.14 16.04 68.07 16.34 67.99 68.97 16.64* 16.88 70 71 69.11 16.27 69.04 16.67 72 70.15 16.20 70.08 16.50 70.01 16.81 69.94 17.11 72 73 71.13 16.42 71.06 16.73 70.98 17.04 70.91 17.36 ; 73 74 72.10 16.65 72.03 16.96 71.96 17.28 71.88 17.. 59' 74 76 73.08 16.87 73.00 17.19 72.93 17.60 72.85 17.8gl 75 76 74.05 17.10 73.98 17.42 73.90 17.74 73.82 18.06 76 77 75.03 17.32 74.96 17.66 74.87 17.98 74.79 18.30 77 78 76.00 17.55 76.92 17.88 75.84 18.21 75.76 18.54 78 79 76.98 17.77 76.90 18.11 76.82 18.4-1 76.74 18.78 79 80 81 77.95 78.92 18.00 77.87 18.34 77.79 18.68 77.71 19.01 8(, 81 18.22 78.84 18.67 78.76 18.91 78.68 19.25 83 79.90 18.45 79.82 18.79 79.73 19.14 79.65 19.49 82 83 80.87 18.67 80.79 19.02 80.71 19.38 80.62 19.73 83 84 81.85 18.90 81.76 19.26 81.68 19.61 81.59 19.97 84 85 82.82 19.12 82.74 19.48 82.65 19.84 82.66 20.20 85 86 83.80 19.. S5 83.71 19.71 83.62 20.08 83.64 20.44 86 87 84.77 19.67 84.68 19.94 84.60 20.31 84.51 20.68 87 88 85.74 19.80 85.66 20.17 86.57 20.54 85.48 20.92 88 89 86.72 20.02 86.63 20.40 86.64 20.78 86.45 21.15 89 90 91 87.69 20.25 87.60 88.68 20.63 20.86 87.51 21.01 87.42 88.39 21.39 90 88.67 20.47 88.49 21.24 21.63 91 92 89.64 20.70 89.66 21.09 89.46 21.48 89.36 21.87 92 93 90.62 20.92 90.62 2^ 32 1 90.43 21.71 90.33 22.10 93 94 91.59 21.15 91.60 21.54 1 91.40 21.94 91.31 22.34 94 1 95 92.57 21.37 92.47 21.77! 92.38 22.18 92.28 22.58 95 96 93.54 21.60 93.44 22.00 ' 93.35 22.41 93.25 22.82 96 97 94 51 21.82 94.42 22.23 i 94.32 22.64 94.22 23.06 97 98 95 49 22.05 96.39 22.46 ! 96.29 22.88 95.19 23.29, 98 1 99 96 46 22.27 96.36 22.69 96.26 23.11 96.16 23.63 99 00 e u c a ■ 97.44 32.50 97.34 22.92 97.24 23.34 97.13 23 77 100 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. c d o i 7?] Deg. 761 Dog. 76:1 Dog. 76i Deg. o? 100 TRAVERSE TABLE. 5 5* o o a ~1 14Deg. 14i Deg. 14i] Deg. 141 Deg. 5' a ? 1 Lat. Dep. ~072T Lat. 0.97 Dep. Lat. Dep. Lat. Dep. 0.97 0.25 0.97 0.25 0.97 0.25 2 1.94 0.48 1.94 0.49 1.94 0.50 1.93 0.51 2 d 2.91 0.73 2.91 0.74 2.90 0.75 2.90 0.76 3 4 3.88, 0.97 3.88 0.98 3.87 1.00 3.87 1.02 4 6 4.851 1.21 4.85 1.23 4.84 1.25 4.84 1.27 5 6 5.82 1.45 5.82 1.48 5.81 1.50 5.80 1.53 6 7 6.79 1.69 6.78 1 72 6.78 1.75 6.77 1 78 7 8 7.76 1.94 7.75 1.97 7.75 2.00 7.74 2.04 8 9 8.73 2.18 8.72 2.22 8.71 2.25 8.70 2.29 9 10 11 9.70 10.67 2.42 9.69 2.46 9.68 2.50 9.67 2.55 2.80 10 11 2.66 10.66 2.71 10.65 2.75 10.64 12 11.64 2.90 11.63 2.95 11.62 3.00 11.60 3.06 12 13 12.61 3.15 12.60 3.20 12.59 3.25 12.57 3.31 13 14 13.58 3.39 13.57 3.45 13.55 3.51 13.54 3.56 14 15 14.55 3.63 14.54 3.69 14.52 3.76 14.51 3.82 15 16 15.52 3.87 15.51 3.94 15.49 4.01 15.47 4.07 16 17 16.50 4.11 16.48 4.18 16.46 4.26 16.44 4.33 17 18 17.47 4.35 17.45 4.43 17.43 4.51 17.41 4.58 18 19 18.44 4.60 18.42 4.68 18.39 4.76 18.37 4.84 19 20 21 19.41 4.84 19.38 4.92 19.36 5.01 19.34 5.09 20 21 20.38 5.08 20.35 5.17 20.33 5.26 20.31 5.35 22 •21.35 5.32 21.32 5.42 21.30 5.51 21.28 5.60 22 23 22.32 5.56 22.29 5.68 22.27 5.76 22.24 5.86 23 24 23. S9 5.81 5.91 23.24 6.01 23.21 6.11 24 25 24.26 6.05 24.23 6.15 24.20 6.26 24.18 6.37 25 26 25.23 6.29 25.20 6.40 25.17 6.51 25.14 6.62 26 27 26.20 6.53 26.17 6.65 26.14 6.76 26.11 6.87 27 28 27.17 6.77 27.14 6.89 37.11 7.01 27.08 7.13 28 29 28.14 7.02 28.11 7.14 28.08 7.26 28.04 7.38 29 30 31 29.11 30o08 7.26 29.08 7.38 29.04 7.51 29.01 7.64 7.89 30 31 7.50 30.05 7.63 30.01 7.76 29.98 32 31.05 7.74 31.02 7.88 30,98 8.01 30.95 8.15 32 33 32.02 7.98 31.98 8.12 31.95 8.26 31.91 8.40 33 34 32.99 8.23 32.95 8.37 32.92 8.51 32.88 8.60 34 35 33.96 8.47 33.92 8.62 33.89 8.76 33.85 8.91 35 36 34.93 8.71 34.89 8.86 34.85 9.01 34.81 9.17 36 37 36.90 8.95 35.86 9.11 35.82 9.26 35.78 9.42 37 38 36.87 9.19 36.83 9.35 36.79 9.51 36.75 9.67 38 39 37.84 , 9.44 37.80 9.60 37.76 9.76 37.71 9.93 39 40 41 38.81 1 9.68 38.77 9.85 38.73 10.02 10.27 38.68 10.18 40 il 39.78 9.92 39.74 10.09 39.69 39.65 10.44 42 40.75 10.16 40.71 10.34 40.66 10.52 40.62 10.69 42 43 41.72 ! 10.40 41.68 10.58 41.63 10.77 41.58 10.95 43 44 42.69 ! 10.64 42.65 10.83 42.60 11.02 42.55 11.20 44 45 43.66 10.89 43.62 11.08 43.57 11.27 43.52 11.46 45 46 44.63 11.13 44.58 11.32 44.53 11.52 44.48 11.71 46 47 45.60 11.37 45.55 11.57 45.50 11.77 45.45 11 97 47 48 1 46.57 11.61 46.52 11.82 46.47 12.02 46.42 12.22 48 49 147.54 11.85 47.49 12.06 47.44 12.27 47.39 12 48 49 60 ,48.51 12.10 48.46 12.31 48.41 12.52 48.35 12 73 50 id i 1 Dep Lat. Dep. 751 Lat. Deg. Dep. Lat. Dep. Lat. 1 76 Deg. 76i Deg. 75i Deg. V . V, thaverse table. 101 5 s p 51 14 Dog. U\ Deg. U\ Deg. L 14| Deg. C a' g .1 51 Lat. Dep. Lat. Dep. Lat. Dep. at. Dep. 49.49 12 .34 49 .43 12.55 49 .38 12.77 49 .32 12.98 62 50.46 12 58 50 .40 12.80 50 .34 13.02 50 .29 13.24 63 53 51.43 12 82 51 .37 13.05 51 .31 13.27 51 .25 13.49 53 54 52.40 13 06 52 .34 13.29 52 .28 13.52 52 .22 13.75 54 55 53.37 13 31 53 .31 13.54 53 .25 13.77 53 .19 14.00 55 56 54.34 13 55 54 28 13.78 54 .22 14.02 54 .15 14.26 56 6" 55.31 13 79 55 25 14.03 55 18 14.27 55 .12 14.51 57 5S 56.28 14 03 56 .22 14.28 56 .15 14.52 56 .09 14.77 58 59 57.25 14 27 57 18 14.52 57 .12 14.77 57 .06 15.02 59 60 61 58.22 14 52 58 .15 14.77 58 .09 15.02 58 .02 15.28 60 6] 59.19 14 76 59 .12 15.02 59 06 15.27 58 .99 15.. 53 62 60.16 15 00 60 09 15.26 60 03 15.52 59 .96 15.79 62 63 61.13 15 24 61 06 15.51 60 99 15.77 60 .92 16.04 63 64 62.10 15 48 62 03 15.75 61 96 16.02 61 .89 16.29 64 65 63.07 15 72 63 00 16.00 62 93 16.27 62 .86 16.55 65 66 64.04 15 97 63 97 16.25 63 90 16.53 63 .83 16.80 66 67 65.01 16 21 64 94 16.49 64 87 16.78 64 79 17.06 67 68 65.98 16 45 65 91 16.74 65 83 17.03 65 76 17.31 68 69 66.95 16 69 66 88 16.98 66 80 17.28 66 73 17.57 69 70 71 67.92 16 93 67 85 17.23 67 77 17.53 67 69 17.82 70 71 68.89 17 18 68 82 17.48 68 74 17.78 68 66 18.08 72 69.86 17 42 69 78 17.72 69 71 18.03 69 63 18.33 72 73 70.83 17 66 70 75 17.97 70 67 18.28 70 59 18.59 73 74 71.80 17 90 71 72 18.22 71 64 18.53 71 56 18.84 74 75 72 77 18. 14 72 69 18.46 72 61 18.78 72 53 19.10 75 76 73.74 18 39 73 66 18.71 73 58 19.03 73 50 19.35 76 77 74.71 18 63 74 63 18.95 74 55 19.28 74 46 19.60 77 78 75.68 18 87 75 60 19.20 75 52 19.53 75 43 19.80 78 79 76.65 19 11 76 57 19.45 76 48 19.78 76 40 20.11 79 80 81 77.62 78.59 19 35 77 54 19.69 77 45 20.03 77 36 20.37 80 81 19 60 78 51 19.94 78 42 20.28 78 33 20.62 82 79.56 19 84 79 48 20.18 79 39 20.53 79 30 20.88 82 83 80.53 20 08 80 .45 20.43 80 36 20.78 80 26 21.13 83 84 81.50 20 32 81 42 20.68 81 32 21.03 81 23 21.39 84 85 82.48 20 56 82 .38 20.92 82 29 21,28 82 20 21.64 85 86 83.45 20 81 83 .35 21.17 83 26 21.53 83 17 21.90 86 87 84.42 21 05 84 .32 21.42 84 23 21.78 84 13 22.15 87 88 85.39 21 29 85 29 21.66 95 20 22.03 85 10 22.41 88 89 86.36 21 5ft 86 .26 21.91 86 17 22.28 86 07 22.66 89 90 91 87.33 21 77 87 .23 22.15 87 .13 22.53 87 03 22.91 90 91 88.30 22 01 88 .20 22.40 88 10 22.78 88 00 23.17 92 89.27 22 26 89 .17 22.65 89 07 23.04 88 97 23.42 92 93 90.24 22 50 90 .14 22.89 90 .04 23.29 89 94 23.68 93 94 91 21 22 .74 91 .11 23.14 91 01 23.54 90 90 23.93 94 95 92 18 22 .98 92 .08 23.38 91 97 23.79 91 87 24.19 95 9L 93 15 23 22 93 .05 23.63 92 94 24.04 92 84 24.44 96 97 94. 12 23 .47 94 .02 23.88 93 91 24.29 93 80 24.70 97 98 95.09 23 .71 94 .98 24.12 94 88 24.54 94 77 24.95 98 99 96.06 23 95 95 .95 24.37 95 85 24.79 95 74 25.21 99 IOC 87.03 24 .19 96 .92 24.62 96 .81 25.04 96 .70 25.46 iOO » c a TO Dep. L at. Dep. Lat. Dep. Lat. Dep. Lat. 76 1 >eg. 75| Dog 1 75i Deg. 76iDeg 102 TRAVERSE TABLE. s g p 1 15 Deg. 15i Deg. 15^ Deg. 151 Deg. 3 s 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.97 0.26 0.90 0.26 0.96 0.27 0.96 0.27 2 1.93 0.52 1.93 0.53 1.93 0.53 1.92 0.54 2 3| 2.90 0.78 2.89 0.79 2.89 0.80 2.89 0.81 a 4 3.86 1.04 3.86 1.05 3.85 1.07 3.85 1.09 4 5 4.83 1.29 4.82 1.32 4.82 1.34 4.81 1.36 5 6 5.80 1.55 5.79 1.58 5.78 1.60 5.77 1.63 6 7 0.76 1.81 6.75 1.84 6.75 1.87 6.74 1.90 7 8 7 73 2.07 7.72 2.10 7.71 2.14 7.70 2.17 8 9 8.69 2.33 8.68 2.37 8.67 2.41 8.66 2.44 9 10 11 9.66 2.59 9.65 2.63 9.64 2.67 9.62 2.71 10 10.63 2.85 10.61 2.89 10.60 2.94 10.59 2.99 11 12 11.59 3.11 11.58 3.16 11.56 3.21 11.55 3.26 12 13 12.56 3.36 12.54 3.42 12.53 3.47 12.51 3.53 13 14 13.52 3.62 13.51 3.68 13.49 3.74 13.47 3.80 14 15 14.49 3.88 14.47 3.95 14.45 4.01 14.44 4.07 15 16 15.45 4.14 15.44 4.21 15.42 4.28 15.40 4.34 16 17 16.42 4.40 16.40 4.47 16.38 4.54 16.36 4.61 17 18 17.39 4.66 17.37 4.73 17.35 4.81 17.32 4.89 18 19 18.35 4.92 18.33 5.00 18.31 5.08 18.29 5.16 19 20 21 19.32 5.18 19. 3C 5.26 19.27 5.34 19.25 5.43 20 20.28 5.44 20.26 5.52 20.24 5.61 20.21 5.70 21 22 21.25 5.69 21.23 5.79 21.20 5.88 21.17 6.97 22 23 22.22 5.95 22.19 6.05 22.16 6.15 22.14 6.24 23 24 23.18 6.21 23.15 6.31 23.13 6.41 23.10 6.51 24 25 24.15 6.47 24.12 6.58 24.09 6.68 24.06 6.79 25 26 25.11 6.73 25.08 6.84 25.05 6.95 25.02 7.06 26 27 26.08 6.99 26.05 7.10 26.02 7.22 25.99 7.33 27 28 27.05 7.25 27.01 7.36 26.98 7.48 26.95 7.60 28 29 28.01 7.51 27.98 7.63 27.95 7.75 27.91 7.87 29 30 31 28.98 7.76 28.94 7.89 28.91 8.02 28.87 8.14 30 31 29.94 8.02 29.91 8.15 29.87 8.28 29.84 8.41 32 30.91 8.28 30.87 8.42 30.84 8.55 30.80 8.69 32 33 31.88 8.54 31.84 8.68 31.80 8.82 31.76 8.96 33 34 32.84 8.80 32.80 8.94 32.76 9.09 32.72 9.23 34 35 33.81 9.06 33.77 9.21 33.73 9.35 33.69 9.50 35 36 34.77 9.32 34.73 9.47 34.69 9.62 34.65 9.77 36 37 35.74 9.58 35.70 9.73 35.65 9.89 35.61 10.04 37 38 36.71 9.84 36.66 10.00 36.62 10.16 36.57 10.31 38 39 37.67 10.09 37.63 10.26 37.58 10.42 37.54 10.59 39 40 41 38.64 10.35 38.59 10.52 38.55 10.69 38.50 10.86 to 41 39.60 10.61 39.56 10.78 39.51 10.96 39.40 11.13 42 40.57 10.87 40.52 11.05 40.47 11.22 40.42 11.40 42 43 41.53 11.13 41.49 11.31 41.44 11.49 41.39 11.67 43 44 42.50 11.39 42.45 11.57 42.40 11.76 42.35 11.94 U 45 43.47 11.65 43.42 11.84 43.36 12.03 43.81 12.21 45 16 44.43 11.91 44.38 12.10 44.33 12.29 44.27 12.49 4G 47 45.40 12.16 45 35 12.36 45.29 12.56 45.24 12 76 47 48 46.36 12.42 46.31 12.63 46.25 12.83 46.20 13.03 48 49 47.33 12.68 47.27 12.89 47.22 13.09 47.16 13.30 49 50 ,48.30 12.94 48.24 13.15 48.18 13.36 48.12 13.. 57 50 § ^ s 1 CD Dep. Lat. Dep. Lat Dep. Lat. Dep. Lat. 75 Dog. 741 Deg. U^ Deg. • 744 Deg. TRAVERSE TABLE. 103 D a a a 61 15 Dcg. 15i Deg 15A Deg. 16J Deg. 1 o p '61 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 13.84 49.26 13.20 49.20 13.41 49.15 13.63 49.09 62 50.23 13.46 50.17 13.68 50.11 13.90 .50.05 U.U 52 63 61.19 13.72 51.13 13.94 51.07 14.16 51.01 14,39 53 64 62.16 13.98 52.10 14.20 52.04 14.43 51.97 14.60 54 55 53.13 14.24 53.06 14.47 53.00 14.70 .52.94 14.93 5£ 66 54.09 14.49 54.03 14.73 53.96 14.97 53.90 15.20 56 57 55.06 14.75 54.99 14.99 54.93 15.23 54.86 15.47 67 58 56.02 15.01 55.96 15.26 55.89 15.50 55.82 15.74 58 59 56.99 15.27 56.92 15.52 56.85 15.77 56.78 16.01 59 60 61 57.96 15.53 57.89 15.78 57.82 58.78 16.03 57.75 16.29 16.56 60 61 58.92 15.79 58.85 16.04 16.30 58.7] 62 59.89 16.05 59.82 16.31 59.75 16.57 59.67 16.83 62 63 60.85 16.31 60.78 16.57 60.71 16.84 60.63 17.10 63 64 61.82 16.56 61.75 16.83 61.67 17.10 61.60 17.37 64 65 62.79 16.82 62.71 17.10 62.64 17.37 62.56 17.64 65 66 63.75 17.08 63.68 17.36 63.60 17.64 63.52 17.92 66 67 64.72 17.34 64.64 17.62 64.56 17.90 64.48 18.19 67 68 65.68 17.60 65.61 17.89 65.53 18.17 65.45 18.46 68 69 66.65 17.86 66.57 18.15 66.49 18.44 66.41 18.73 69 70 71 67.61 18.12 87.54 18.41 67.45 18.71 67.37 19.00 70 71 68.58 18.38 68.50 18.68 68.42 18.97 68.33 19.27 72 69.55 18.63 69.46 18.94 69.38 19.24 69.30 19.54 72 73 70.51 18.89 70.43 19.20 70.35 19.51 70.26 19.82 73 74 71.48 19.15 71.39 19.46 71.31 19.78 71.22 20.09 74 75 72.44 19.41 72.36 19.73 72.27 20.04 72.18 20.86 75 76 73.41 19.67 73.32 )9.99 73.24 20.31 73.15 20.63 76 77 74.38 19.93 74.29 20.25 74.20 20.58 74.11 20.90 77 78 75.34 20.19 75.25 20.52 75.16 20.84 75.07 21.17 78 79 76.31 20.45 76.22 20.78 76.13 21.11 76.03 21.44 79 80 81 77.27 20.71 77.18 21.04 77.09 78.05 21.38 77.00 21.72 80 81 78.24 20.96 78.15 21.31 21.65 77.96 21.99 82 79.21 21.22 79.11 21.57 79.02 21.91 78.92 22.26 82 83 80.17 21.48 80.08 21.83 79.98 22.18 79.88 22.53 83 84 81.14 21.74 81.04 22.09 80.94 22.45 80.85 22.80 84 85 82.10 22.00 82.01 22.36 8]. 91 22.72 81.81 23.07 85 86 83.07 22.26 82.97 22.62 82.87 22.98 82.77 23.34 86 87 84.04 22.52 83.94 22.88 83.84 23.25 83.73 23 62 87 88 85.00 22.78 84.90 23.15 84.80 23.52 84.70 23 89 88 89 85.97 23.03 85.87 23.41 85.76 23.78 85.66 24 16 89 90 91 86.93 23.29 86.83 23.67 86.73 24.05 86.62 24 43 24.70 90 "91 87.90 23.55 87.80 23.94 87.69 24.32 87.58 92 88.87 23.81 88.76 24.20 88.65 24.59 88.55 24.97 92 93 89.83 24.07 89.73 24.46 89.62 24.85 89.51 25.24 93 94 90.80 24.33 90.69 24.72 90.. 58 25.12 90.47 25.52 94 95 91.76 24.59 91.65 24.99 91 34 25.39 91.43 25.79 95 96 92 73 24.85 92.62 25.25 92.51 i 25.65 92.40 26.06 96 97 93.69 25.11 93.58 25.51 93.47 25.92 93.36 26.33 97 98 94.66 25.36 94., 55 25.78 94.44 26.19 91:. 32 26.60 98 99 95.63 25.62 95.51 26.04 95.40 26.46 95.28 26.87 99 100 § e d .a 96.59 25.88 Lat. 1 36.48 26.30 96.36 26.72 Lat. 96.25 27.14 100 1 4-) Dep. Dep. Lat. Dep. Dep. Lat. 75 Deg. 1 741 Dcg. 74^ Deg. 74i Deg. 104 TRAV£U8£ TABLE. 1 Ol 5 1 r* P 16 Deg. 1 I6i Deg. \^\ Deg. i 16| Deg, 1 1 s \ 1 1 1 Lat. 0.96 Dep. Lat. Dep. Lat. D©p. 28 Lat. Dep. i 0.28 0.96 0.28 0.96 0.96 0.29 2 1 9'^ 55 1.92 0.56 1.92 0.57 1.92 0.58 2 3 2.88 C 83 2.88 0.84 2.88 0.85 2.87 0.86 3 4 3.85] 1.10 3.84 1.12 3.84 1.14 3.83 1.15 4| 6 4.81 1.38 4.80 1.40 4.79 1.42 4.79 1.44 5 6 5.77 1.65 5.76 1.68 5.75 1.70 5.75 1.73 6 7 6.73 1.93 0.72 1.96 6.71 1.99 6.70 2.02 7 8 7.69 2.21 7.68 2.24 7.67 2.27 7.66 2.31 8 9 8.65 2.48 8.64 2.52 8.63 2.56 8.62 2.59 9 10 11 9.01 10.. 57 2.76 9.60 2.80 9.59 2.84 9.58 2.88 10 11 3.03 10.56 3.08 10.55 3.12 10.53 3.171 12 11.64 3.31 11.52 3.36 11.51 3.41 11.49 3.46 12 13 12.50 3.58 12.48 3.64 12.46 3.69 12.45 3.75 13 14 13.46 3.86 13.44 3.92 13.42 3.98 13.41 4.03 14 15 14.42 4.13 14.40 4.20 14.38 4.26 14.36 4.32 15 16 15.38 4.41 15.36 4.48 15.34 4.54 15.32 4.61 16 17 16.34 4.69 16.32 4.76 16.30 4.83 16.28 4.90 17 18 17.30 4.961 17.28 5.04 17.26 5.11 17.24 5.19 18 19 18.26 5.24 18.24 5.32 18.22 5.40 18.19 5.48 19 20 21 19.23 5.51 19.20 5.60 19.18 5.68 19.15 5.76 20 '21 20.19 5.79 20.16 5.88 20.14 5.96 20.11 6.05 22 21.15 6.06 21.12 6.16 21.09 6.25 21.07 6.34 22 23 22.11 6.34 22.08 6.44 22.05 6.53 ! 22.02 6.63 23 24 23.07 6.62 23.04 6.72 23.01 6.82 1 22.98 6.92 24 25 24.03 6.89 24.00 7.00 23.97 7.10 23.94 7.20 25 20 24.99 7.17 24.96 7.28 24.93 7.38 24.90 7.49 26 27 25.95 7.44 25.92 7.56 25.89 7.67 25.85 7.78 27 28 26.92 7.72 20.88 7.84 26.85 7.95 26.81 8.07 28 29 27.88 7.99 27.84 8.11 27.81 8.24 27.77 8.36 29 30 31 28.84 29.80 8.27 28.80 8.39 28.76 8.52 1 28 . 73 29.68 8.65 8.93 30 31 8.54 29.76 8.67 29.72 8.80 32 30.76 8.82 .30.72 8.95 30.68 9.09 30.64 9.22 32 33 31.72 9.10 31.68 9.23 31.64 9.37 31.60 9.51 33 34 32.68 9.37 32.64 9.51 32.60 9.66 32.56 9.80 34 35 33.64 9.65 33.60 9.79 33.56 9.94 33.51 10.09 35 36 34.61 9.92 34.56 10.07 34.52 10.22 34.47 10.38 36 37 35.57 10.20 35 52 10.35 35.48 10.51 35.43 10.66 37 38 36.53 10.47 36.48 10.63 36.44 10.79 36.39 10.95 38 39 37.49 10.75 37.44 10.91 37.39 11.08 37.35 11.24 39 40 41 38.45 11.03 38.40 11.19 11.47 38.35 11.36 38.30 11.53 ' 40 41 39.41 11.30 39.36 39.31 11.64 39.26 11.82 42 40.37 11.58 40.32 11.75 40.27 11.93 40.22 12.10 42 43 41.33 11.85 41.28 12.03 41 23 12.21 41.18 12.39 43 44 43.30 12.13 42 . 24 12.31 42. 19 12.50 42.13 12.68 1 44 45 43.26 12.40 43.20 12.59 43. 15 12.78 43.09 12.97 45 46 44.22 12.68 44.16 12.87 44.11 13.06 44.05 13.26 46 47 45.18 12.95 45.12 13.15 45.00 13.35 45.01 18.55 47 48 46.14 13.23 46.08 13.43 46.02 13.63 45.96 13.83 48 49 47.10 13.51 47.04 "».71 46.98 13.92 46.92 14.12 49 60 .3 48.06 13.78 Lat. 48.00 13.99 47.94 14.20 Lat. 47.88 Dep. 14.41 50 o { i i s Dep. Dep. Lat. Dep. LaU 74 Deg. 73| Deg. 73^ Deg. 734 Dog. TKA.VKnSl! TABLE. 105 f 51 16 Deg. 16i Deg. 16A Deg m Deg. r* Lat. 49,02 Dep. Lat. Dep. Lat. Dep. Lat. 48.84 Dep. 14.70 O o 51 14.06 48.96 14.27 48.90 14.48 52 49.99 14.33 49.92 14.55 49 86 1 14.77 49.79 14.99 52 53 50.95 14.61 .50.88 14 83 .50.82 1 15.05 1 50.75 15.27 53 54 51.91 14.88 51.84 15 11 51.78 15. .34 1 51.71 15.66 64 55 52.87 15.16 .52.80 15.39 .52.74 15.62 .52.67 16.85 55 66 ,'S3.93 15.44 .53.76 15.67 .53.69 15.90 .53.62 16.14 56 57 64.79 15.71 54 . 72 15.95 .54.65 16.19 .54.58 16.43 57 58 55.75 15.99 55.68 16.23 .55.61 16.47 55.. 54 16.72 58 69 56.71 16.26 .56.64 16.51 .56 . 57 16.76 56.. 50 17.00 59 60 "61 57.68 58.64 16.54 .57.60 16.70 57.. 53 .58.49 17.04 17.32 57.45 58.41 17.29 17.58 60 '61 16.81 .58.56 17.07 62 59.60 17.09 59 . 52 17.35 .59.45 17.61 .59.37 17.87 62 63 60.56 17.37 50 48 17.63 60.41 17.89 60.33 18.16 63 64 61.52 17.64 61.44 17.91 61.36 18.18 61.28 18.44 04 65 62.48 17.92 62.40 18.19 62.32 18.46 62 . 24 18.73 65 66 63.44 18.19 63.30 18.47 63.28 18.74 63.20 19.02 60 67 64.40 18.47 64.32 18.75 64.24 19.03 64.16 19.31 67 68 65.37 18.74 65.28 19.03 65.20 19.31 65 . 1 1 19.60 68 69 66.33 19.02 66.24 19.31 66.16 19.60 66.07 19.89 69 70 71 67.29 1 19.29 19.57 67.20 19.59 67.12 68.08 19.88 67.03 20.17 70 71 68.25 68.16 19.87 20 . 1 7 67.99 20.46 72 69.21 19.85 69.12 20.15 69.03 20.45 68.95 20.75 72 73 70.17 20.12 70.08 20.43 69.99 20 . 73 69.90 21.04 73 74 71.13 20.40 71.04 20.71 70.95 21 02 70.86 21.33 74 75 72.09 20.67 72.00 20.99 71.91 21.30 71.82 21.61 75 76 73.06 20.95 72.96 21.27 72.87 21.. 59 72.78 21.90 76 77 74.02 21.22 73.92 21.. 55 73.83 21.87 73.73 22.19 77 78 74.98 21. .50 74.88 21.83 74.79 22.15 74 . 69 22.48 78 79 75.94 21.78 75.84 22.11 75 . 75 22.44 75.65 22.77 79 80 81 76.90 22.05 76.80 22.39 76.71 22.72 23.01 1 76.61 S77..56 23.06 80 81 77.86 22.33 77.76 22.67 77.66 23.34 82 78.82 22.60 78.72 22.95 78 . 62 23.29 78 . 52 23.63 82 83 79.78 22.88 79.68 23.23 79.. 58 23 . 57 '79.48 23.92 83 84 80.75 23.15 80.64 23..^ I 80 . .54 23.86 80.44 24.21 84 85 81.71 23.43 81.60 23 . 79 81.50 24.14 81. .39 24.. 50 85 86 82.67 23.70 82.56 24.07 82.46 24.43 82.35 24.78 86 87 83.63 23.98 83.. 52 24.35 83.42 24.71 j 83.31 25.07 87 88 84.59 24.26 84.48 24.62 84.38 24.99 84.27 25.36 88 89 85.55 24.53 85.44 24.90 85.33 25 , 28 85.22 25.65 89 90 91 86.51 24.81 86.40 25.18 86.29 25.. 56 86.18 87.14 25.94 90 87.47 25.08 87.. 36 25.46 87.25 25.85 26.23 ^ 92 88.44 25.36 88.32 25.74 88.21 26.13 88.10 26.51 92 93 89.40 25.63 89.28 26.02 89.17 26.41 89.05 26.80 y3 94 90.36 25.91 90.24 26.30 90.13 26.70 90.01 27.09 94 95 91.32 26.19 91.20 26.. 58 91.09 26.98 90.97 27.38 95 96 92.28 26.46 92.16 26.86 92.05 27.27 91.93 27.67 96 97 93.24 26.74 93.12 27.14 93.01 27.. 55 92.88 27.95 97 98 94.20 27.01 94.08 27.42 ^93.96 27.83 93.84 28.24 98 99 95.16 27.29 95.04 27.70 94.92 28.12 94.80 28.. 53 99 100 96.13 27.56 96.00 27.98 95.88 28.40 95.76 28.82 100 i c Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. (d .9 I 74 Deg. 73} Deg. ^^ Deg. 7'H Deg. o 106 THAVERSE TABLB. 5 CB f 3 17 Deg. 17i Deg. n^Deg. i 1 I7i Deg. y s S 8, Lat. D«p. Lat. Dep. Lat. Dep. Lat. Dep. ' 0.30 1 0.96 0.29 0.95 0.30 0.95 0.30 0.95 *i 1.91 0.58 1.91 0.59 1.91 0.60 1.90 0.61 a 3 2.87 0.88 2.87 0.89 2.86 0.90; 2.86 0.91 9 4 3.83 1.17 3.82 1.19 3.81 1.20 3.81 1.22 4 5 4.78 1.46 4.78 1.48 4.77 1.50 4.76 1.52 6 6 5.74 1.75 5.73 1.78 5.72 1.80 5.71 1.83 6 7 6.69 2.05 6.69 2.08 6.68 2.10! 6.67 2 13 7 8 7.65 2.34 7.64 2.37 7.63 2.41 7.62 2.44 8 9' 8.61 2.63 8.60 2.67 8.58 2.71 i 8 57 2.74 9 10 9.56 2.92 9.55 2.97) 9.54 3.01 9.. 52 3.05 10 11 11 10.52 3.22 10.51 3.26 10.49 3.31 10.48 3.35 la 11.48 3.51 11.46 3.56 1 11.44 3.61 11.43 3.66 12 13 12.43 3.80 12.42 3.85 i 12.40 3.91 12.38 3.96 13 14 13.39 4.09 13.37 4.15 i 13.35 4.21 1 13.33 4.27 14 15 14,34 4.39 14.33 4.45! 14.31 4.51 ! 14.29 4.57 15 16 15.30 4.68 15.28 4.74 i 15.26 4.81 15.24 4.88 16 17 16.26 4.971 16.24 5.04 16.21 5.11 16.19 5.18 17 18 17.21 5.26 17.19 5.34 17.17 5.41 17.14 5.49 18 19 18.17 5.56 18.15 5 63 18.12 5.71 18.10 5.79 19 20 21 19.13 5.85 19.10 5.93 19.07 6.01 19.05 6.10 20 21 20.08 6.14 20.06 6. S3 20.03 6.31 20.00 6.40 22 21.04 6.43 21.01 6 52 20.98 6.62 20.95 6.71 22 23 21.99 6.72 21.97 6.82 21.94 6.92; 21.91 7.01 23 24 22.95 7.02 2:i.92 7.12; 22.89 7.22 22 . 86 7.32 24 2.^ 23.91 7.31 23.88 7 41 23.84 7.. 52 23.81 7.62 25 26 24.86 7.60 24.83 7.71 24.80 7.82 24.76 7.93 26 27 25.82 7.89 25.79 8.01, 25.75 8.12 25.71 8.23 27 28 26.78 8.19 26.74 8.30 26-70 8.42 26.67 8.54 28 29 27.73 8.48 27.70 8.60 27.66 8.72 27.62 8.84 29 30 31 28 . 69 8.77 ; 28.65 9.06 129.61 8.90 28.61 29.57 9.02 28.57 9.15 JO 31 29.35 9.19 9.32 29.52 9.45 32 30.60 9.36 i! 30.56 9.49 30.52 9.62' 30.48 9.76 32 33 31.56 9.65 1 31.. 52 9.79 1 31.47 9.92 31.43 10.06 33 34 32.51 9.94 32.47 ia.08 1 32.43 10.22 32.08 10.37 34 35 33.47 10.23 33.43 10.38 1 33.. 38 10.52 33.33 10.67 35 36 34.43 10.53 1 .34.. 38 10.68! 34.33 10.83 34.29 10.98 36 37 35.38 10.82 35.34 10.97] 35.29 11.13 35 24 11.28 37 38 36.34 11.11 36.29 1 11.271 36.24 11.43 36.19 11.58 38 39 37.30 11.40 37.25 11.57 1 37.19 1 1 . 73 37.14 11.89 39 40 41 as. 25 11.09 ; 33.20 11.86 38.15 12.03 38.10 12.19 40 41 39.21 11.99 39.16 12.16 39.10 12.33 39.05 12.50 42 40.16 12. 2S 40.11 12.45 40.06 12.63 40.00 12.80 42 43 41.12 12.57 41.07 12.75 41.01 12.93 40.95 13 11 43 44 42. OS 12.86 42.02 13.05 41.96 13.23 41.91 13 11 44 45 143. 3 13.16 42.98 13.34 42.92 13.53 42.86 13 72 45 46 ! 43.99 13.45 43.93 13.64 43.87 13.83 43.81 14.02 46 47 44.95 18.74 44.89 13.94 44.82 14.13 44.76 14.. 33 47 48 ,45.90 14.03 45.84 14.23 45.78 14.43 45.71 14 63 48 49 ) 46.86 14.33 46.80 14.53 46.73 14.73 46.67 14.94 , 49 50 147.82 i Dep. a 73 14.62 Lat. Deg. 47.75 14.83 47.69 15.04 47.62 15.24 50 a 1 Dep. Lat. Dep. Lat Dep. 72i Lat. Der 721 Deg. 791 Deg. TRAVEBSB TABLS. 107 g s a ? 5l 17 Deg. m Deg, 17A Deg. 1?| Deg. O S p "51 Lat. 48.77 Dep. 14.91 Lat. Dep. 15.12 Lat. Dep. Lat. Dop. 48.71 48.64 15.34 48.67 16.55 62 49.73 15.20 49.66 15.42 49.59 15.64 49.52 15.85 52 53 50.68 15.50 50.62 15.72 50.55 15.94 50.48 16.16 53 54 51.64 15.79 51.57 16.01 51.50 16.24 51.43 16.46 54 55 52.60 16.08 52.53 16.31 52.45 16.54 52.38 16.77 55 56 53.55 16.37 53.48 16.61 53.41 16.84 53.33 17.07 56 67 54.51 16.67 54.44 16.90 .54.36 17.14 54.29 17.38 57 58 55.47 16.96 55.39 17.20 55.32 17.44 55.24 17.68 58 59 56.42 17.25 56.35 17.50 56.27 17.74 56.10 17.99 59 60 61 57.38 17.54 57.30 17.79 57.22 18.04 57.14 18.29 60 61 58.33 17.83 58.26 18.09 58.18 18.34 58.10 18.60 62 59.29 18.13 59.21 18.39 59.13 18.64 .59.05 18.90 62 63 60.25 18.42 60.17 18.68 60.08 18.94 60.00 19.21 63 64 61.20 18.71 61.12 18.98 61.04 19.25 60.95 19.51 64 65 62.16 19.00 62.08 19.28 61.99 19.55 61.91 19.82 65 66 63.12 19.30 63.03 19.57 62.95 19.35 62.86 20.12 66 67 64.07 19.59 63.99 19.87 63.90 20.15 63.81 20.43 67 68 65.03 19.88 64.94 20.16 64.85 20.45 64.76 20.73 68 69 65.99 20.17 65.90 20.46 65.81 20.75 65.72 21.04 69 70 71 66.94 20.47 66.85 20.76 66.76 21.05 66.67 21.34 70 "71 67.90 20.76, 67.81 21.05 67.71 21.35 67.62 21.65 72 68.85 21.05! 68.76 21.35 68.67 21.65 68.57 21.95 72 73 69.81 21.34 1 69.72 21.65 69.62 21.95 69.52 22.26 73 74 70.77 21.64 70.67 21.94 70.58 22.25 70.48 22.56 74 75 71.72 21.93 71.63 22.24 71.53 22.55 71.43 22.86 75 76 72.68 22.22 72.. 58 22.54 72.48 22.85 72.38 23.17 76 77 73.64 22.51 73.54 22.83 73.44 23.15 73.33 23.47 77 78 74.59 22.80 74.49 23.13 74.39 23.46 74.29 23.78 78 79 75.55 23.10 75.45 23.43 75.34 23.76 75.24 24.08 79 80 81 76.50 77.46 23.39 76.40 23.72 76.30 24.06 76.19 24.39 80 23.68 77.36 24.02 { 77.25 24.36 77.14 24.69 81 82 78.42 23.97 78.31 24.32 78.20 24.66 78.10 25.00 82 83 79.37 24.27 79.27 24.61 79.16 25.96 79.05 25.30 83 84 80.33 24.56 80.22 24.91 \ 80.11 25.26 80.00 25.61 84 85 81.29 24.85 81.18 25.21 81.07 25.56 80.95 25.91 85 86 82.24 25.14 82.13 25.50 82.02 25.86 81.91 26.22 86 87 83.20 25.44 83.09 25.80 82.97 26.16 82.86 26.52 87 88 84.15 25.73 84.04 26.10 83.93 26.46 183.81 26.83 88 89 85.11 26.02 85.00 26.39 84.88 26.76 84.76 27.13 89 90 91 86.07 87.02 26.31 85.95 26.69 185.83 27.06 27.36 85.72 27.44 90 91 26.61 86.91 26.99 86.79 86.67 27.74 92 87 98 26.90 87.86 27.28 87. ?4 27.66 87.62 28.05, 92 1 93 88.94 27.19 88.82 27.58 88.70 127.97 88.57 2?. 35 93' 94 89.89 27.48 89 . 77 27.87 89.65 '28.27 89.53 28.66 94 95 90.85 27.78 90.73 28.17 90.60 ■28., 57 90.48 28.96 95 96 91 82 28.07 91.68 28.47 91.56 28.87 91.43 29.27 96 97 92. 7< 28.36 92.64 28.76 92.51 29.17 92.38 ! 29.5? 97 98 93.72 28.65 93.59 29.06 93.46 129.47 93.33 29.88 98 99:94.67 28.94 94.55 29.36 94.42 29.77 94.29 30.18 99 100 S e • 1 95.63 29.24 ^5.50 29.65 Lat. 95.37 30 07 95.24 30.49 100 s a a CO Dep. Lat. Dep. Dep. Lat. Dep. Lat. 73 Deg. 72i Deg. 72^ Dog. 7Si Deg. 108 THAVERSE TAULE. 5 e P 1 ' 18 Deg. 1 m Deg. 1 l^ Deg. I8| Deg. m s ' Lai. "0.95 Dep. Lat- i Dep. 1 ' 0.95 ' 0.31 Lai. . 0,95 Dep. Lau 1 ^^P- 0.31 0.32 0.95 0.32. 1 2 1.90 0.62 1.90 0.63 1 1.90 0.63 ' 1.89 9.64 3 3 1 2.85 0.93 2.85 3.94 2 84 0.95 ' 2.84 0.96 3 4 3.80 1.24 , 3.80 1.25 3 79 1.27, 3.79 1 29 4 5 4.76 1 1.55 4.75 1.57 4.74 1.59 4.73 1.61 6 6 5.711 1.85 5.70 1.88 5.69 1.90 ,5.68 1.93 6 7 6.66 2.16 6.65 2.19 6.64 2.22 1 6.63 2.25 7 8 7.61 2.47 7.60 2.51 7.59 2.54 7.58 2.57 B 9 8.56} 2.78 S.55 2.82 1 8.. 53 2.86 1 1 8.52 2.89 9 10 11 9.51 3.09 10.46. 3.40 9.50 3.13 ' 9.48 ■ 10.43 3.17 3.49! 9.47 10.42 3.21 ! 10 3.54 11 10.45 3.44 12 11.41 3.71 11.40 3.76 i 11.38 J 81 I ' 11.36 3.86 12 13 12.36 4 02 12.35 4.07 12.33 4 12 12.31 4.18 13 14 13.31 4 33 13.30 4.38 ; 13.28 4.4i 13.26 4.50 14 15 14.27 1 4.64 14.25 4.70 14.22 4.76 14.20 4.82 15 16 15.22 4.94 15.20 5.01 : 15.17 5.08 15.15 5.14 16 17 16.17, 5.25 16.14 5.32 1 16.12 5.39 16.10 5.46 17 18 17.12 ' 5.56: 17.09 5.64 17.07 5.71 17.04 5.79 18 19 18.07 5.87 18.04 5.95 18.02 6.03 17.99 6.11 19 20 21 19.02^ 6.18 19.97^ 6.49 13 99 19.94 6.26 6.58 18.97 6.35 18.94 19.89 6.43 20 6.75 21 19.91 6.66 22 20.92' 6.80, 20.89 6.89 20.86 6.98 20.83 7.07 22 23 21.87' 7.11! 21.84 7.20 21.81 7.30 21.78 7.39 23 24 22.83' 7.42 22.79 7.52 22 . 76 7.62 22.73 r 71 24 25 23.78 ; 7.73 23.74 7.83' 23.71 7.93 23.67 8.04 25 26 24.73 8.03 24.69 8.14 24,66 8.25 24.62 8.36 26 27 25.68 , 8.34 25.64 8.46 25.60 8.57 ' 25.57 8.68 27 28 26.63 8.65 26.59 8.77 26.55 8.88, 26.51 9.00 28 29 27.58 ! 8.96 27.54 9.08 27.50 9.20 27.46 9.32 29 30 31 28.53 9.27 29.48 1 9.58 28.49 9.39 28.45 29.40 9.52 9.84 28.41 29.35 9.64 3C 9.96 31 29.44 9.71 32 30.43; 9.89. 30.39 10.02 .30.35 10.15 30.30 10.29 32 33 31.38 1 10.20 31.34 10.33 31.29 10.47 31.25 10.61 33 34 .32.34 ]0.51 32.29 10.65 32 . 24 10.79 32.20 10.93 34 3.=i 33.29 10.82 33.24 10.96 33.19 11.11 33.14 11.25 35 36 34.24 11.12 34.19 11.27 .34.14 11.42 34.09 11.57 36 37 35 19 11.43 35.14 11.59 35.09 1 1 . 74 35.04 11.89 37 3S 36.14 11.74 36.09 11.90 36.04 12.06 35.98 12.21 38 39 37.09 12.05 37.04 12.21 36 . 9S 12.37 36.93 12.54 .39 40 38.04 12.36 37.99 12.53 37.93 12.69 37.88 38.82 12.86 ' 40 13.18 ; 41 41 38.99 12.67 38 . 94 12.84 3S.SS 13.01 1 42 39.94 12.98 39.89 13.15 39.83 13.33 39.77 13.50 42 43 40.90 13.29 40.84 13.47 40.78 13 64 40.72 13.82 43 44 41.85 13.60 41.79 13.78 41.73 13.96 41.66 14.14 U it 42.80 13.91 42.74 14.09 42.67 14. 2S 42.61 14.46 45 46 43.75 14.21 43.69 14.41 43.62 14.60 43.56 14.79 46 47 44.70 14.. =52 44.64 14.72 1 44.57 14.91 44.51 15.11 47 i8 45.65 14.83 45.59 15.03! 45.52 15.23 45.45 15.43 48 49 46.60 15.14 46.54 15.35 46.47 15.55 46.40 15.75 49 50 47.55 15.45. 47.48 15,66 47.42 15.87 47.35 16.07 50 8 ; Dep. Lat. Dep. ' Lai. bep. Lat. : Dep. Lat. g 1 ' 1 72 Deg. 711 Deg. Tii 1 Deg. 1 7U Deg. 1 1 TRAVITRSK TABLK. 109 1' a a a 61 18 Dog. 18i Deg. 18^ Deg. 181 Deg. 51 Lat Dep. Lat. Dep. Lat. Dep. 16.18 Lat. 48.29 Dep. 76.^9 48.50 15.76 48.43 15.97 48.36 52 49.45 16.07 49.38 16.28 49.31 16.50 49.24 16 71 52 53 50.41 16.38 50.33 16.60 50.26 16.82 50.19 17.04 63 54 51.36: 16.69 51.28 16.91 51.21 17.13 51.13 17.36 54 55 52.31 17.00 52.23 17.22 52.16 17.45 52.08 17.68 55 56 53.26 17.30 53.18 17.54 53.11 17.77 53.03 18.00 1 56 57 54.21 17.61 54.13 17.85 54.05 18.09 53.98 18.32 i 57 58 55.16 17.92 55.08 18.16 55.00 18.40 54.92 18.64 ' 58 69 56.11 18.23 5().03 18.48 55.95 18.72 55.87 18.96 59 60 61 57.06 58.01 18.54 56.98 18.79 56.90 19.04 56.82 19.29 60 61 18 85 57.93 19.10 57.85 19.36 57.76 19.61 62 58.97 19.16 58.88 19.42 58.80 19.67 58.71 19.93 62 63 59.92 19.47 59.83 19.73 59.74 19.99 59.66 20.25 63 64 60.87 19.78 60.78 20.04 60.69 20.31 60.60 20.57 64 65 61.82 20.09 61.73 2G.36 61.64 20.62 61.55 20.89 65 66 62.77 20.40 62.68 20.67 62.59 20.94 62.50 21.22 66 67 63.72 20.70 63.63 20.98 63.. 54 21.26 63.44 21.54 67 68 64.67 21.01 64.58 21.30 64.49 21.58 64.39 21.86 68 69 65.62 21.32 65.53 21.61 65.43 21.89 65.34 22.18 69 70 71 66.57 21.63 66.48 21.92 22.23 66.38 67.33 22.21 66.29 22.50 70 71 67.53 21.94 67.43 22.53 67.23 22.82 72 68.48 22.25 68.38 22.55 68.28 22.85 68.18 23.14 72 73 69.43 22.56 69.33 22.86 69.23 23.16 69.13 23.47 73 74 70.38 22.8? 70.28 23.17 70.18 23.48 70.07 23.79 74 75 71.38 23.18 71.23 23.49 71.12 23.80 71.02 24.11 75 76 72.28 23.49 72.18 23.80 72.07 24.12 71.97 24.43 76 77 73 23 23.79 73.13 24.11 73.02 24.43 72.91 24.75 77 78 74.18 24.10 74.08 24.43 73.97 24.75 73.86 25.07 78 79 75.13 24.41 75.03 24.74 74.92 25.07 74.81 25.39 79 80 81 76.08 24.72 75.98 25.05 75.87 25.38 75.75 25.72 80 81 77.04 25.03 76.93 25.37 76.81 25.70 76.70 26.04 82 77.99 25.34 77.88 25.68 77.76 26.02 77.65 26.36 82 83 78.94 25.65 78.83 25.99 78.71 26.34 78.60 26.68 83 84 79.89 25.96 79.77 26.31 79.66 26.65 79.54 27.00 84 85 80.84 26.27 80.72 26.62 80.61 26.97 80.49 27.32 85 86 81.79 26.58 81.67 26.93 81.56 27.29 81.44 27.64 86 87 82.74 26.88 82.62 27.25 82.50 27.61 82.38 27.97 87 88 83.69 27.19 83.57 27.. 56 83.45 27.92 83.33 28.29 88 89 84.64 27.50 84.52 27.87 84.40 28.24 84.28 28.61 89 90 91 85.60 27.81 85.47 28.18 85.35 28.56 85.22 28.93 90 91 86.55 28.12 86.42 28.50 86.30 28.37 86.17 29.25 92 87.50 28.43 87.37 28.81 87.25 29.19 87.12 29,57 92 93 88.45 28.74 8y.32 29.12 88.19 29.51 88.06 29.89 93 94 89 40 29.05 89.27 29.44 89.14 29.83 89.01 30.22 94 95 90 35 29.36 90.22 29.75 90.09 30.14 89.96 30.54 95 96 91.30 29.67 91.17 30.06 91.04 30.46 5,0.91 30.86 96 97 92.25 29.97 92.12 30.38 91.99 30.78 91.85 31.18 97 98 93.20 30.28 93.07 30.69 92.94 31.10 92.80 31.50 98 9C 94.15 30.59 94.02 31.00 93.88 31.41 93.75 31.82 99 IOC g at 95.11 30.90 94.97 31.32 Lat. 94.83 31.73 94.69 32 14 .00 § a cd 2 Dep. Lat. Dep. Dep. Lat. Dep. Lat, 72 Deg. 711 Deg. ! n^ Deg. 7U Deg. no TRAVERSE TABLE. 5 19 Deg. 19i Deg. 19i Deg. 19J Dog. § Lat. Dop. 0.33 Lat. Dep. Lat. 0.94 Dep. 0.33 Lat. 0.94 Dep. 0.34 e.96 0.94 0.33 Z ..H9 0.65 1.89 0.66 1.89 0.67 1.88 0.68 2 ? 2.84 0.98 2.83 0.99 2.83 l.UO 2 82 1.01 8 4 3 . 78 . 1 30 3.78 1.32 3.77 1.34 3.76 1 35 4 5 4.73 1.63 4.72 1.66 4.71 1.67 4.71 1.69 6 6 6.67 1.95 6.66 1.98 6.66 2.00 6.66 2.03 fi 7 6.62 2.28 6.61 2.81 6.60 2.34 6.69, 2.37 7 8 7.56 2.60 1 7.65 2.64 7.54 2.67 7.63 2.70 S 9 8.51 2.93 8.60 2.97 8.48 3.00 8.47 3.04 9 10 11 9.46 10.40 3.26 3.58 9.44 10.38 3.30 3.63 9.43 10.37 3.34 9.41 3.38 10 11 3.67 10.35 3.72 12 11.35 3.91 11.33 3.96 11.31 4.01 11.29 4.06 12 13 12.29 4.23 12.27 4.29 12.25 4.34 12.24 4.39 13 14 13.24 4.56 13.22 4.62 13.20 4.67 13 18 4.73 14 15 14.18 4.88 14.16 4.95 14.14 5.01 14.12 5.07 15 16 15.13 5.21 15.11 6.28 15.08 6.34 16.06 5.41 16 17 16.07 5.53 16.06 5.60 16.02 5.67 16 00 5.74 17 18 17.02 6.86 16.99 6.93 16.97 6.01 16.94 6.08 18 19 17.96 6.19 17.94 6.26 17.91 6.34 17.88 6.42 19 20 21 18.91 6.51 18.88 6.69 18.86 19.80 6.68 18.82 6.76 20 21 19.86 6.84 19.83 6.92 7.01 19.76 7.10 22 20.80 7.16 20.77 7.25 20.74 7.34' a0.71 1 7.43 22 23 21.75 7.49 21.71 7.58 21.68 7.68! 21.66 1 7.77 23 24 22.69 7.81 22.66 7.91 22.62 8.01 22.69, 8.11 24 25 23.64 8.14 23.60 8.24 23.67 8.36 23.63 8.45 25 26 24.58 8.46 24.66 8.57 24.51 8.68 24.47 1 8.79 26 27 25 . 53 8.79 25.49 8.90 25.45 9.01 25.41 9.12 27 28 26.47 9.12 26.43 9.23 26 . 39 9.35 26.35 9.46 28 29 27.42 9r44 27.38 9.56 i 27 . 34 9.68 27.29 9.80 29 30 28.37 9.77 28.32 9.89 28 . 28 10.01 28.24 10.14 30 21 29.31 10.09 29.27 10.22 29.22 10.35 29. 18 10 48 31 32 30.26 10.42 30.21 10.55 30.16 10.68 30.12 10 81 32 33 31.20 10.74 31.16 10.88 31.11 11.02 31.06 11 16 33 34 32.15 11.07 32.10 11.21 32.05 1 1 . 36 32.00 11 49 34 35 33.09 11.39 33.04 1 1 . 54 32.99 11.68 32.94 11.83 35 36 34.04 1 1 . 72 33.99 11.87 33.94 12.02 33.88 12.17 36 37 34.98 12.05 34.93 12.20 34.88 12.36 34.82 12.50 37 38 35.93 12.37 35.88 12.53 35.82 12.68 35.76 12.84 38 39 36.88 12.70 36.82 12.86 36.76 13.02 36.71 13.18 39 40 37 82 13.02 37.76 13.19 37.71 13.35 37.65 13.62 40 41 38.77 13.35 38.71 13.52 38.05 13.69 38.59 13.85 41 42 39 71 13.67 39.65 13.86 39.69 14.02 39.63 14.19 42 43 40.66 14.00 40.60 14. :8 40 . 53 14.36 40.47 14. 53 43 44 41.60 14.32 41.54 14.51 41.48 14.69 41.41 14 87 44 45 42.55 14.66 42.48 14.84 42.42 16.02 42.36 15.21 45 46 ,43.49 14.98 43.43 15.17 43.36 16.36 43.29 16.64 46 47 44.44 16.30 44.37 15.50 44.30 16.69 44.24 16.88 47 48 45.38 16.63 45.32 15.83 45.25 16.02 46. L8 16.22 48 49 46.33 x6.96 46.26 16.16 46.19 16.36 46.12 16.66 49 50 8 a 47.28 16.28 47-20 16.48 Lat. 47.13 16.69 Lut. Deg. 47.06 16.90 60 Dcp. Lat. Dep. Dcp. 70^] Dep. Lai. s| 71 Deg. 701 Deg. 7(H Deg. G • THA.VEK6B TABLE. Ill s 61 19 Deg. 19i Deg. 19| Deg. 19| Deg. O ? Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o ? 51 48.22! 16.60 1 48.15 16.81 48.07 17.02 48.00 17.23 63 49.17; 16.93 | 49.09 17.14 49.02 17.30 48.94 17.57 52 63 50 11 17.26 1 50.04 17.47 49.96 17.69 49.88 17 91 53 64 51.06 17.58 50.98 17.80 50.90 18.03 .50.82 18 25 54 55 52.00 17.91 51.92 18.13 1 51.85 18.36 51.76 18.59 55 56 52.95 18 23 52.87 18.46 52.79 18.69 52.71 18.92 56 57 53.89 18.56 53.81 18.79 53.73 19.03 53.65 19.26 57 58 54.84 18.88 54.76 19.12 54.67 19.36 54.59 19.60 58 59 55.79 19.21 55.70 19.45 55.62 19.69 55.53 19.94 59 60 61 56.73 19.53 56.65 57.59 19.78 56.56 57.50 20.03 56.47 20.27 60 61 57.68 19.86 20.11 20.36 57.41 20.61 62 58.62 20.19 58.53 20.44 58.44 20.70 58.35 20.95 62 63 59.57 20.51 59.48 20.77 59.39 21.03 59.29 21.29 63 64 60.51 20.84 60.42 21.10 60.33 21.36 60.24 21.63 64 65 61.46 21.16 61.37 21.43 61.27 21.70 61.18 21.96 65 66 62.40 21.49 62.31 21.76 62.21 22.03 02.12 22.30 66 67 63.35 21.81 63.25 22.09 63.16 22.37 63.06 22.64 67 68 64.30 22.14 64.20 22.42 04.10 22.70 64.00 22.98 68 69 65.24 22.40 65.14 22.75 65.04 23.03 64.94 23.32 69 70 71 66.19 22.79 66.09 23.08 65.98 23.37 65.88 23.65 70 71 67.13 23.12 67.03 23.41 66.93 23.70 66.82 23.99 72 68.08 23.44 67.97 23.74 67.87 24.03 67.76 24.33 72 73 69.02 23.77 68.92 24.07 68.81 24.37 68.71 24.67 73 74 69.97 24.09 69.86 24.40 69.76 24.70 69.65 25.01 74 75 70.91 24.42 70.81 24.73 70.70 25.04 70.59 25.34 75 76 71.86 24.74 71.75 25.06 71.64 25.37 71.53 25.68 76 77 72.80 25.07 72.69 25.39 72.58 25.70 72.47 26.02 77 78 73.75 25.39 73.64 25.72 73.53 26.04 73.41 26.36 78 79 74.70 25.72 74.58 26.05 74.47 26.37 74.35 26.70 79 80 81 75.64 26.05 75.53 26.38 75.41 26.70 75.29 27.03 80 81 76.59 26.37 76.47 26.70 76.35 27.04 76.24 27.37 82 77.53 26.70 77.42 27.03 77.30 27.37 77.18 27.71 82 83 78.48 27.02 78.36 27.36 78.24 27.71 78.12 28.05 83 84 79.42 27.35 79.30 27.69 79.18 28.04 79.06 28.39 84 85 80.37 27.67 80.25 28.02 80.12 28.37 80.00 28.72 85 86 81.31 28.00 81.19 28.35 81.07 28.71 80.94 29.06 86 87 82.26 28.32 82.14 28.68 82.01 29.04 81.88 29.40 87 88 83.21 28.65 83.08 29.01 92.95 29.37 82.82 29.74 88 89 84.15 28.98 84.02 29.34 83.90 29.71 83.76 30.07 89 90 91 85.10 29.30 84.97 29.67 84.84 30.04 84.71 30.41 90 31 86.04 29.63 85.91 30.00 85.78 30.38 85.65 30.75 92 86.99 29.95 86.86 30.33 86.72 30.71 86.59 31.09 92 93 87.93 30.28 87.80 30.66 87.67 31.04 87.. 53 31.43 9.9 94 88.88 30.60 88.74 30.99 88.61 31.38 88.47 31.76 94 95 89.82 30.93 89.69 31.32 89.55 31.71 89.41 32.10 95 96 90.77 31.25 90.63 31.65 90.49 32.05 90.. 35 32.44 96 97 91.72 31.58 91.58 31.98 91.44 32.38 91.29 32.78 97 98 92.66 31.91 92.52 32.31 92.38 32.71 92.24 33.12 98 99 93.61 32.23 93.46 32.64 93.32 33.05 93.18 33.45 99 .DC 94.65 32.56 94.41 32.97 94.26 33.38 94.12 33.79 100 i Q s K o 1 11 ■■■ Dep. L&t. Dep. Lat. Dep. Lat. Dep. Lat. «^ Oeg. 701 Deg. 70^ Deg. 70i Deg. 112 TRAVERSE TABLE. 5' s a ? ~1 20] Deg. m Deg. 20^ Deg. 201 Deg. S' p ' 1 Lat. Dep. Lat Dep. Lat. Dep. Lat. Dep. 0,ZI 0.94 0.34 0.94 0.35 0.94 0.35 0.94 2 1.88 0.68 1.88 0.69 1.87 0.70 1.87 0.71 i 3 2.82 1.03 2.81 1.04 2.81 1.05 2.81 1.06 3 4 3.76 1.37 3.75 1.38 3.75 1.40 3.74 1.42 i 5 4.70 1.71 4.69 1.73 4.68 1.75 4.68 1.77 5 G 5.64 2.06 6.63 2.08 5.62 2.10 5.6^ 2.13 6 7 6.68 2.39 6.67 2.42 6.66 2.45 6.55 2 48 7 8 7.52 2.74 7.61 2.77 7.49 2.80 7.48 2.83 8 9 8.46 3.08 8.44 3.12 8.43 3.15 8.42 3.19 9 10 11 9.40 3.42 9.38 3.46 9.37 10.30 3.50 9.35 3.54 lO 11 10.34 3.76 10.32 3.81 3.85 10.29 3.90 12 1 11.28 4.10 11.26 4.15 11.24 4.20 11.22 4.25 12 13 12.22 4.45 12.20 4.50 12.18 4.56 12.16 4.61 13 14 13.16 4.79 13.13 4.86 13.11 4.90 13.09 4.96 14 15 14.10 5.13 14.07 6.19 14.06 6.25 14.03 6.31 15 16 16.04 5.47 16.01 5.64 14.99 6.60 14.96 6.67 16 17 16.97 6.81 15.96 5.88 15.92 5.95 15.90 6.02 17 18 16.91 6.16 16.89 6.23 16.86 6.30 16.83 6.38 18 19 17.85 6.60 17.83 6.58 17.80 6.66 17.77 6.73 19 20 21 18.79 6.84 18.76 6.92 18.73 7.00 ! 18.70 7.09 20 21 19.73 7.18 19.70 7.27 19.67 7.35 19.64 7.44 22 20.67 7.62 20.64 7.61 20.61 7.70 20.57 7.79 22 23 21.61 7.87 21.58 7.96 21.54 8.05 21.51 8.15 23 24 22.55 8.21 22.52 8.31 22.48 8.40 22.44 8.50 24 25 23.49 8.56 23.45 8.65 23.42 8.76 23.38 8.86 26 26 24.43 9.89 24.39 9.00 24.35 9.11 24.31 9.21 26 27 25.37 9.23 25.33 9.35 25.29 9.46 25.25 9.57 27 28 26.31 9.58 26.27 9.69 26.23 9.81 26.18 9.92 28 29 27.26 9.92 27.21 10.04 27.16 10.16 27.12 10.27 29 30 31 28.19 10.26 28.15 10.38 28.10 10.51 28.06 10.63 30 31 29.13 10.60 29.08 10.73 29.04 10.86 28.99 10.98 32 30.07 10.94 30.02 11.08 29.97 11.21 29.92 11.34 32 33 31.01 11.29 30.96 11.42 30.91 11.66 30.86 11.69 38 34 31.95 11.63 31.90 11.77 31.85 11.91 31.79 12.06 ,34 35 32.89 11.97 32.84 12 11 32.78 12.26 32.73 12.40 35 36 33.83 12.31 33.77 12.46 33.72 12.61 33.66 12.75 36 37 34.77 12.65 34.71 12.81 34.66 12.96 34.60 13.11 37 38 35.71 13.00 35.66 13.16 36.59 13.31 35.54 13.46 38 39 36.65 13.34 36.59 13.60 36.53 13.66 36.47 13.82 39 40 41 37.59 13.68 14.02 37.53 13.84 37.47 14.01 37.41 14.17 40 41 38.53 38.47 14.19 38.40 14.36 38.34 14.53 42 39.47 14,36 39.40 14.64 39.34 14.71 39.28 14.88 42 43 40.41 14.71 40.34 14.88 40.28 15.06 40.21 15.23 43, 44 41.35 15.06 41.28 16.23 41.21 15.41 41.15 15.59 44 45 42.29 16.39 42.22 15.68 42.15 16.76 42.08 :5.94 45 46 143.23 16.73 43.16 15.92 43.09 16.11 43.02 16 30 46 4- f 44.17 16.07 44.09 16.27 44.02 16.46 43.95 16 65 47 48 45. ix 16.42 46.03 16.61 44.96 16 81 44.89 17.01 48 49 46.04 16.76 45.97 16.96 46.90 17.16 45.82 17.36 49 60 i a .2 Q 46.98 17.10 46.91 17.31 46.83 17.51 46.76 17.71 50 « a ed Dep. Lat. Dep. 691 Lat. Deg. Dep. Lat. Dep. Lat. 701 Oeg. 69i Deg. 69i Deg. TRAVERSE TABLE. 113 3 ? 51 20Deg. 20t De?. 20ADeg 201 Deg. 5 m § a 51 Lat. Dep. Lat. Dep. ■T7.'65 Lat. 47777 Dep. Lat. Dep. 47.92 17.44 47.85 17.86 47.69 18.07 52 48.86 17.79 48.79 18.00 48.71 18.21 48.63 18.42 52 53 49.80 18.13 49.72 18.34 49.64 18.56 49.56 18.78 53 54 50.74 18.47 50.66 18.69 1 50.58 18.91 50.50 19.13 54 55 51.68 18.81 51.60 19.04 51.52 19.26 51.43 19.49 55 56 52.62 19.15 52.54 19.38 52.45 19.61 52.37 19.84 56 57 53.56 19.50 53.48 19.73 53.39 19.96 53.30 20.19 57 58 54.50 19.84 54.42 20.07 54.33 20.31 54.24 20.55 58 59 55.44 20.18 55.35 20.42 55.26 20.66 55.17 20.90 59 60 61 56.38 20.. 52 56.29 20.77 56.20 21.01 56.11 21.26 60 '61 57.32 20.86 57.23 21.11 57.14 21.36 57.04 21.61 62 58.26 21.21 58.17 21.46 58.07 21.71 57.98 21.97 62 63 59.20 21.55 59.11 21.81 59.01 22.06 58.91 22.32 63 64 60.14 21.89 60.04 22.15 59.95 22.41 59.85 22.67 64 65 61.08 22.23 60.98 22.50 60.88 22.76 60.78 23.03 65 66 62.02 22.57 61.92 22.84 61.82 23.11 61.72 23.38 66 67 62.96 22.92 62.86 23.19 62.76 23.46 62.65 23.74 67 68 63.90 23.26 63.80 23.54 63.69 23.81 63.59 24.09 68 69 64.84 23.60 64.74 23.88 61". 63 24.16 64.52 24.45 69 70 71 65.78 23.94 24.28 65.67 24.23 65.57 66.50 24.51 65.46 24.80 70 '71 72 66.72 66.61 24.57 24.86 66.39 25.15 72 67.66 24.63 67.55 24.92 67.44 25.21 67.33 25.51 73 68.60 24.97 68.49 25.27 68.38 25.57 68.26 25.86 73 74 69.54 25.31 1 69.43 25.61 69.31 25.92 69.20 26.22 74 75 70.48 25.65 70.36 25.96 70.25 26.27 70.14 26.57 75 76 71.42 25.99 71.30 26.30 71.19 26.62 71.07 26 . 93 76 77 72.36 26.34 72.24 26.65 72.12 26.97 72.01 27.28 77 78 73.30 26.68 73.18 27.00 73.06 27.32 72.94 27.63 78 79 74.24 27.02 74.12 27.34 74.00 27.67 73.88 27.99 79 80 81 75.18 1 27.36 1 75.06 27.69 74.93 75.87 28.02 74.81 28.34 80 81 76.12 27.70 75.99 28.04 28.37 75.75 28.70 82 77.05 28.05 76.93 28.38 76.81 28.72 76.68 29.05 82 .^3 77.99 28.39 77.87 28.73 77.74 29.07 77.62 29.41 83 84 78.93 28.73 78.81 29.07 78.68 29.42 78.55 29.76 84 85 79.87 29.07 79.75 29.42 79.62 29.77 79.49 30.11 85 86 80.81 29.41 80.68 29.77 80.55 30.12 80.42 30.47 86 87 81.75 29.76 81.62 30.11 81.49 30.47 81.36 30.82 87 88 82.69 30.10 82.56 30.46 82.43 30.82 82.29 31.18 88 89 83.63 30.44 83.50 30.80 83.36 31.17 83.23 31.58 89 90 91 84.57 30.78 84.44 31.15 84.30 31.52 84.16 31.89 90 91 85.51 j 31.12 85.38 31.50 85.24 31.87 85.10 32 24 92 86.45 1 31.47 86.31 31.84 86.17 32.22 86.03 32.59 92 93 87.39 1 31.81 87.25 32.19 87. Ix 32.57 86.97 32.95 93 94 88.33 32.15 88.19 32.54 88.05 32.92 87.90 33.30 94 95 89.27: 32.49 89.13 32.88 88.98 33.27 88.84 33.66 95 96 90.21 32.83 90.07 33.23 89.92 33.62 89.77 34.01 96 97 91.15 33.18 91.00 33.57 90.86 33.97 90.71 34.37 97 98 92.09 33.52 91.94 33.92 91.79 34.32 91.64 34.72 98 99 93.0 J 33.86 92.88 34.27 92.73 34.67 92.58 35.07 99 £00 1 93.97 34.20 93.82 34.61 93.67 35.02 93.51 35.43 100 U s 3 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 70 Deg. 69| Deg. 69^ De^ 69i Ueg 114 TRAVERSE TABLE. o 21 Deg. 2U Deg. 211 Deg. 211 Deg. 5 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 9 a ? 0.93 0.33 0.93 0.36 0.^3 0.37 0.93 0.37 2 1.87 0.72 1.86 0.72 1.86 0.73 1.86 0.74 2 8 2.80 1.08 2.80 1.09 2.79 1.10 2.79 1.11 3 4 3.73 1.43 3.73 1.45 3.72 1.47 3.72 1.48 4 6 4.67 1.79 4.66 1.81 4.65 1.83 4.64 1.85 5 6 f 5.60 2.15 5.59 2.17 6.58 2.20 5.57 2.22 6 7 6.54 2.51 6.52 2.54 6.61 2.57 6.50 2.59 7 8 7.47 2.87 7.46 2.90 7.44 2.93 7.43 2.96 8 9 8.40 3.23 8.39 3.26 8.37 3.30 8.36 3.34 9 10 9.34 3.58 9.32 3.62 9.30 3.67 9.29 3.71 10 11 11 JO. 27 3.94 10.25 3.99 10.23 4.03 10.22 4.08 12 11.20 4.30 11.18 4.35 11.17 4.40 11.15 4.45 12* 13 12.14 4.66 12.12 4.71 12.10 4.76 12.07 4.82 13 14 13.07 5.02 13.05 5.07 13.03 5.13 13.00 5.19 14 15 14.00 5.38 13.98 5.44 13.96 5.. 50 13.93 5.56 15 16 14.94 5.73 14.91 5.80 14.89 5.86 14.86 5.93 16 17 15.87 6.09 15.84 6.16 15.82 6.23 15.79 6.30 17 18 16.80 6.45 16.78 6.52 16.75 6.60 16.72 6.67 18 19 17.74 6.81 17.71 6.89 17.68 6.96 17.65 7.04 19 20 21 18.67 7.17 18.64 7.25 18.61 7.33 18.58 7.41 20 21 19.61 7.53 i 19.57 7.61 19.54 7.70 19.50 7.78 22 20.54 7.88' 20.50 7.97 20.47 8.06 20.43 8.15 22 23 21.47 8.24i 21.44 8.34 21.40 8.43 21.36 8.52 23 24 22.41 8.60 22.37 8.70 22.33 8.80 22.29 8.89 24 25 23.34 8.96 23.30 9.06 23.26 9.16 23.22 9.26 25 26 24.27 9.32 24.23 9.42 24.19 9.53 24.15 9.63 26 27 25.21 9.68 25.16 9.79 25.12 9.90 25.08 10.01 27 28 26.14 10.03 26.10 10.15 26.05 10.26 26.01 10 38 28 29 27.07 10.39 27.03 10.51 26.98 10.63 26.94 10.75 29 30 28.01 10.75 27.96 10.87 27.91 11.00 27.86 11.12 30 31 31 28.94 11.11 28.89 11.24 28,84 11.36 28.79 11.49 32 29.87 11.47 29.82 11.60 29.77 11.73 29.72 11.86 32 33 30.81 11.83 30.76 11.96 30.70 12.09 30.65 12.23 33 34 31.74 12.18 31.69 12.32 31.63 12.46 31.58 12.60 34 35 32.68 12.54 32.62 12.69 32.56 12.83 32.51 12.97 35 36 33.61 12.90 33.55 13.05 33.50 13.19 33.44 13.34 36 37 34.54 13.26 34.48 13.41 34.43 13.56 34.37 13.71 37 38 85.48 13.62 35.42 13.77 35.36 13.93 35.29 14.08 38 39 36.41 13.98 36.35 14.14 36.29 14.29 36.22 14.45 39 40 41 37.34 14.33 37.28 14.50 37.22 14.66 37.15 14.82 40 41 38.28 14.69 38.21 14.86 38.15 15.03 38.08 15.19 42 39.21 15.05 39.14 16.22 39.08 15.39 39.01 15.56 42 43 40.14 15.41 40.08 15.58 40.01 15.76 39.94 16.93 43 44 41.08 15.77 41.01 15.95 40.94 16.13 40.87 16.30 44 45 42.01 16.13 41.94 16.31 41.87 16.49 41.80 16.68 45 46 42.94 16.48 42.87 16.67 42.80 16.86 42.73 17.05 46 47 43.88 16.84 43.80 17.03 43.73 17.23 43.65 17.42 47 48 44.81 17.20 44.74 17.40 44.66 17.59 44.58 17.79 48 49 45.75 17.56 45.67 17.76 45.59 17.96 45.51 18.16 49 50 8 46.68 17.92 46.60 18.12 Lat. 46.62 18.33 46.44 18.63 Lat. 50 8 Dep. Lat. Dep. Dep. Lat. Dep. 1 Q 69] Deg. 68| Deg 681 Deg. 6Q\ Dog. T&AVERSE TABLE. 115 a 21 Deg 2U Deg. 21| Deg. 211 Deg. O 1 ^ Hi I • Lat. Dep. Lat. 47.53 Dep. Lat. Dep. Lat. Dep. 3 18 90 51 51 47.61 18.28 18.48 47.45 18.69 47.37 52 48.55 18.64 48.46 18 85 48.38 19.06 48.30 19.27 i 52 53 49.48 18.99 49.40 19.21 49.31 19.42 49.23 19.64 53 54 50.41 19.35 50.33 19.57 50.24 19.79 50.16 20.01 54 55 51 35 19.71 51.26 19.93 51.17 20.16 51.08 20.38 55 56 52 28 20.07 52.19 20.30 52.10 20.52 52.01 20.75 56 57 53 2] 20.43 53.12 20.66 53.03 20.89 52.94 21.12 5? 58 54.15 20.79 54.06 21.02 53.96 21.26 53.87 21.49 58 59 55.08 21.14 54.99 21.38 54.89 21.62 54.80 21.86 59 60 56.01 21.50 55.92 21.75 55.83 21.99 55.73 22.22 60 61 61 56.95 21.86 56.85 22.11 56.76 22.36 56.66 22.60 62 57, 88 22.22 57.78 22.47 57.69 22.72 57.59 22.97 62 63 58.82 22.58 58.72 22.83 58.62 23.09 58.52 23.35 63 64 59.75 22.94 59.65 23.20 59.55 23.46 59.44 23.72 64 65 60.68 23.29 60.58 23.56 60.48 23.82 60.37 24.09 65 66 61.62 23.65 61.51 23.92 61.41 24.19 61.30 24.46 66 67 62.55 24.01 62.44 24.28 62.34 24.56 62.23 24.83 i 67 1 68 63.48 24.37 63.38 24.65 63.27 24.92 63.16 25.20 68 69 64.42 24.73 64.31 25.01 64.20 25.29 64.09 25.57 69 70 65.35 25.09 65.24 25.37 65.13 25.66 65.02 25.94 70 71 7) 66.28 25.44 66.17 25.73 66.06 26.02 65.95 26.31 72 67.22 25.80 67.10 26.10 66.99 26.39 66.87 26.68 72 73 68.16 26.16 68.04 26.46 67.92 26.75 67.80 27.05 73 74 69.08 26.52 68.97 26.82 68.85 27.12 68.73 27.42 74 75 70.02 26.88 69.9^ 27.18 69.78 27.49 69.66 27.79 75 76 70.95 27.24 70.83 27.55 70.71 27.85 70.59 28.16 76 77 71.89 27.59 71.76 27.91 71.64 28.22 71.52 28.53 77 78 72.82 27.95 72.70 28.27 72.57 28.59 72.45 28.90 78 79 73.75 28.31 73.63 28.63 73.50 28.95 73.38 29.27 79 80 74.69 75.62 28.67 74.56 29.00 74.43 29.32 74.30 29.64 80 81 81 29.03 75.49 29.36 75.36 29.69 75.23 30.02 82 76.55 29.39 76.42 29.72 76.29 30.05 76.16 30.39 82 83 77.49 29.74 77.36 30.08 77.22 30.42 77.09 30.76 83 84 78.42 30.10 78.29 30.44 78.16 30.79 78.02 31.13 84 85 79.35 30.46 79.22 30.81 79.09 31.15 78.95 31.50 85 86 80.29 30.82 80.15 31.17 80.02 31.52 79.88 31.87 86 87 81.22 31.18 81.08 31.53 80.95 31.89 80.81 32,24 87 88 82.16 31 54 82.02 31.89 81.88 32.25 81.74 32.61 88 89 83.09 31.89 82.95 32.26 82.81 32.62 82.66 32.98 89 90 84.02 32.25 83.88 32.62 83.74 84.67 32.99 83.59 33.35 33.72 90 91 91 84.96 32.61 84.81 32.98 33.35 84.52 92 85.89 32.97 85.74 33.34 85.60 33.72 85.45 34.09 92 93 86.82 33.33 86.68 33.71 86.53 34.08 186.38 34.46 , 93 94 87.76 33.69 87.61 34.07 87.46 34.45 ! 87.31 34.83 1 94 95 88.69 34.04 88.54 34.43 88.39 34.82 i 88.24 35.20 . 95 96 89.62 34.40 89.47 34.79 89.32 35.18 89.17 35.57 i 96 97 90.56 34.76 90.40 35.16 90.25 35.55 90.09 35.94 97 98 91.49 35.12 91.34 35.52 91.18 35.92 91.02 36.31 98 99 92.42 35.48 92.27 35.88 ;1 92.11 36.28 91.95 36,69 99 1 lOO i § ♦J .2 Q 93.36 Dop. 35.84 93.20 Dep. 36.24 93.04 36.65 92.88 37.06 100 Lat. Lat. 1 Dep. Lat. Dep. Lat. i d c c (5 69 Deg. 681 Deg. 1 68^ Deg. 1 68i Deg 22 TRATERSE TABLE. C 22 Deg. 22i Dcg. 22^ Deg. 221 Dog. "1 9 s S O 9 Lai. Dep. Lat. Dep. Lai. 0.92 Dep. '' 0.33 t Lai, 0.92 Dep. 0.39 1 1 0.93 0.37 0.93 0.38 1 2 1.35 0.75 1.85 0.76 1.85 0.77 1.84 0.77! 2 3 2.78 1.12 2.73 1.14 2.77 1.15 2.77 1.16 3 4 3.71 1.50 3.70 1.51 3.70 1.53 3.69 '.55 4 4.64 1.37 4.63 1.89 4.62 1.91 4.61 i.93 5 6 5.56 2 . 25 0. 00 2.27 5.. 54 2.30 5.53 2 . 32 6 7 6.49 2 . 62 6.48 2.65 6.47 2.63 6.46 2.71 1 7 fl 7.42 3.00 7.40 3.03 7.39 3.06 7 38 3.09 8 9 8.34 3.37 3 . 33 3.41 3.31 3.44 3.30 3.48 9 10 11 9.27 3.75 9.26 3.79 9.24 3.83 9 . 22 3.87 iO " ll 10.20 4.12 10.18 4.17 10.16 4.21 10.14 4.25 12 11.13 4.50 11.11 4.54 11.09 4.59 11.07 4.64 12 13 12.05 4.37 12.03 4.92 12.01 4.97 11.99 5.03 13 14 12.93 5.24 12.96 5.30 12.93 5.36 12.91 5.41 14 15 13.91 5.62 13.33 5.68 13.86 5.74 13.33 5.80 15 16 14.33 5.99 14.31 6.06 14.78 6.12 14.76 6.19 16 17 15.76 6.37 15.73 6.44 15.71 6.51 15.63 6.57 17 18 16.69 6.74 16.66 6.32 16.63 6.39 16.60 6.96 18 19 17.62 7.12 17.59 7.19 17.55 7.27 17.52 7.35 19 20 21 18.54 7.49 13.51 7.57 13.43 7.65 13.44 7.73 20 13.47 7.87 19.44 7.95 19.40 8.04 19.37 8.12 21 22 20.40 8.24 20 . 36 8 . 33 20 . 33 8.42 20.29 8.51 22 23 21.33 3 . 62 21.29 3.71 21.25 8.80 21.21 3.39 23 24 22 . 25 8.99 22 .21 9.09 22.17 9.18 22.13 9.23 24 25 23.18 9.37 23 . 14 9.47 23.10 9.57 23.05 9.67 25 26 24.11 9.74 24 06 9.34 24.02 9.95 23.93 10.05 26 27 25.03 10.11 24.99 10.22 24.94 10.33 24.90 10.44 27 25 25.96 10.49 25.92 10.60 25.37 10.72 25.32 10.33 23 20 26.39 10.36 26.34 10.93 26.79 11.10 26.74 11.21 29 30 27.82 31 23.74 11.24 27.77 11.36 27.72 11.43 27.67 11.60 30 11.61 23.69 11.74 23 . 64 11.36 28.59 11.99 31 32 29.67 11.99 29.62 12.12 29 . .56 12.25 29.51 12.37 32 33 30.60 12.36 30 . 54 12.50 30 . 49 12.63 .30.43 12.76 33 34 31.52 12.74 31.47 12.37 31.41 13.01 31.35 13.15 34 35 32.45 13.11 32.39 13.25 32.34 13.39 32.28 13.53 3C 36 33.33 13.49 33.32 13.63 33.26 13.73 33.20 13.92 36 37 34.31 13.36 34.24 14.01 34.18 14.16 34.12 14.31 37 33 35 . 23 14.24 35.17 14.39 35.11 14.. 54 35.04 14.70 33 39 36.16 14.61 36.10 14.77 36.03 14.92 35.97 15.03 39 40 37.09 14. 9S 37.02 15.15 36.96 15.31 36.39 37.31 15.47 15. 36 40 ' 41 41 33.01 15.36 37.95 15.52 37.33 15- 69 42 33.94 15.73 33.37 15.90 33.30 16.07 33 . 73 16.24 42 43 39.37 16.11 39.30 16.23 39.73 16.46 39.65 16.63 43 U 40.80 16.43 40.72 16.66 40.65 16.84 40 . 58 17.02 44 45 41.72 16.86 41.65 17.04 41.57 17.22 41.50 17.40 45 46 42.65 17.23 42.57 17.42 12 . 50 17.60 42.42 17.79 46 47 43.53 17.61 43.50 17.30 43.42 17.99 43 . 3-1 13.13 47 48 44.50 17.93 44.43 13.18 44.35 18.37 44.27 18.56 4^ 49 45.43 13.36 45.35 18.55 45.27 18.75 45.19 18.95 49 50 46.36 i j Dep. 18.73 46.23 13.93 46.19 19.13 46.11 Dep. 19.34 Lat. 50 d c S Lat, , Dep. Lai. Dep. 671 Lat. Deg. 3 5 68] 67} Deg. 67} Deg. TRAVERSE TABLE. 117 a S' f o ? 22Deg. 22{ Dog. 22| Deg. 22| Deg. a 1 a tl Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 61 47.29 19.10 47.20 19.31 47.12 19.52 47.03 19.72 62 48.21 19.48 48.13 19.69 48.04 19.90 47.95 20.11 62 63 49.141 19.86 49.05 20.07 48.97 20.28 48.88 20.60 63 64 60.071 20.23 49.98 20.45 49.89 20.66 49.80 20.88 54 65 61.00 20.60 50.90 20.83 50.81 21.05 .50.72 21.27 55 66 61.92 20.98 51.83 21.20 51.74 21.43 51.64 21.66 66 57 52.86 21.35 52.76 21.58 52.66 21.81 52.57 22.04 57 68 63.78 21.73 63.68 21.96 53.59 22.20 53.49 22.43 58 59 54.70 22.10 54.61 22.34 54.51 22.58 54.41 22.82 59 60 61 55.63 22.48 56.53 22.72 65.43 56.36 22.96 55.33 23.20 60 61 56.56 22.85 56.47 23.10 23.34 56.25 23.69 62 57.49 23.23 57.38 23.48 57.28 23.73 57.18 23.98 62 63 58.41 23.60 58.31 23.85 58.20 24.11 58.10 24.38 63 64 59.34 23.97 .59.23 24.23 59.13 24.49 59.02 24.75 64 65 60.27 24.35 60.16 24.61 60.05 24.87 59.94 25.14 65 66 61.19 24.72 61.09 24.99 60.98 25.26 60.87 25.52 66 67 62.12 25.10 62.01 25.37 61.90 25.64 61.79 25.91 67 68 63.05 25.47 62.94 25.75 62.82 26.02 62.71 26.30 68 69 63.98 25.85 63.86 26.13 63.75 26.41 63.63 26.68 69 70 71 64.90 26.22 64.79 26.51 64.67 26.79 64.55 27.07 70 71 65.83 26.60 65.71 26.88 65.60 27.17 65.48 27.46 72 66.76 26.97 66.64 27.26 66 52 27.55 66.40 27.84 72 73 67.68 27.35 67.56 27.64 67.44 27.94 67.32 28.23 73 74 68.61 27.72 68.49 28.02 68.37 28.32 68.24 28.62 74 75 69.54 28.10 69.42 28.40 69.29 28.70 69.17 29.00 75 76 70.47 28.47 70.34 28.78 70.21 29.08 70.09 29.39 76 77 71.39 28.84 71.27 29.16 71.14 29.47 71.01 29.78 77 78 72.32 29.22 72.19 29.53 72.06 29.85 71.93 30.16 78 79 73.25 29.59 73.12 29.91 72.99 30.23 72.85 30.55 79 80 81 74.17 29.97 74.04 30.29 73.91 30.61 73.78 30.94 80 81 75.10 30.34 74.97 30.67 74.83 31.00 74.70 31.32 82 76.03 30.72 75.89 31.05 75.76 31.38 75.62 31.71 82 83 76.96 31.09 76.82 31.43 76.68 31.76 76.54 ,32.10 83 84 77.88 31.47 77.75 31.81 77.61 32.15 77.46 32.48 84 85 78.81 31.84 78.67 32.19 78.53 32.53 78.39 32.87 85 86 79.74 32.22 79.60 32.56 79.45 32.91 79.31 33.26 86 87 80.60 32.59 80.52 32.94 80.38 33.29 80.23 33.64 87 88 81.69 32.97 81.45 33.32 81.30 33.68 81.15 34.03 88 89 82.52 33.34 82.37 33.70 82.23 34.06 82.08 34.42 89 90 83.45 33.71 83.30 34.08 83.16 34.44 83.00 34.80 90 91 91 84.37 34.09 84.22 34.46 84.07 34.82 83.92 35.19 9i 85.30 34.46 85.16 34.84 85.00 35.21 84.84 36.68 92 93 86.23 34.84 86.08 35.21 85.92 35.59 85.76 35.96 93 94 87.16 35.21 87.00 36.59 86.84 35.97 86.69 36.35 94 95 88. 08 36.59 87.93 35.97 87.77 36.35 87.61 36.74 1 9& 96 89.01 36.96 88.86 36 35 88.69 36.74 88.53 37.12 1 96 M- 89.94 36.34 89.78 36.73 89.62 1 37.12 89.45 37 51 97 9S 90 86 36.71 90.70 37.11 90.54 37.50 90.38 37.90 98 9r, 91.79 37.09 91.63 37.49 91.46 37.89 91.30 38.28 99 )00 92 72 37.46 92.65 37.86 92.39 Dep. 38.27 92.22 38.67 Lat. 100 i S Dep. Lat. Dep. 1 Lat. Lat. Dep. 68 Deg. 671 Deg. er^Deg. (JTi Deg. 118 TRAVEESE TABUE. ? ij 23 Deg. 1 23i Deg. 23^ Deg. m Deg. w g ii ? ' ,-v i' ? LaU ! Dep. Lat. Dep. ; Lat. j Dep. Lat. Dep. • ft a 1 0.92 0.39 0.92 0.39 0.92 0.40 0.92 0.40, 2 1.84 0.?8 1.84 0.79 1.83 0.80 1.83 0.81 2 3 2.76 1.17 2.76 1.13 2,75 1.20 2.75 1.21 3 4 S.63 1.56 3.6S 1.5-3 3.67 1.59 3.66 1.61; i 6 4.60 1.95 4.59 1.97 4.59 1.99 4.53 2.011 T 6 5.52 2.34 5.51 2.37' 5.50 2.39 5.49 2.42 1 6 7 6.-i4 2 74 6.43 2.76 1 6.42 2.79' 6.41 2.82 7 8 7.36 3.13 7.35 3.16 7.34 3.19 7.32 3.22 8 9 8.28 3.52 8.27 3.55 8.25 3.59 8.24 3.62 9 10 9.20 3.91 9.19 3.95 9.17 3.99 9.15 4.03 11 10.13 4.30 10.11 4. .34 10.09 4.39 10.07 4.43 11 12 11.05 4.69 11.03 4.74 11.00 4.78 10.93 4.33' 12 13 11.97 5.03 11.94 5.13 li.92 5.18 11.90 6.24 13 14 12.89 5.47 12.86 5.53 12.84 5.58 12.81 5.64 14 15 13.81 5.86 13.78 5.92 13.76 5.98 13.73 6.04 15 16 14.73 6.25 14.70 6.32 14.67 6.38 14.64 6.44 16 17 15.65 6.64 15.62 6.71 15.59 6.78 15.56 6.S5 17 IS 16.57 7.03 16.54 7.11 16.51 7.13 16.48 7.25 18 19 )7.49 7.42 17.46 7.50 17.42 7.55 17.39 7.65 19 20 IS. 41 • 7.81 18.38 19.33 8.21 ' 19.29 7.S9 18.34 8.29 19.26 7.97 18.31 8.05 8.37 19.22 8.46 20 21 21 22 20.25 8.60 20.21 S.6S 20.13 8.77 20.14 S.S6 22 23 21.17 8.99 21.13 9. OS 21.09 9.17 21.05 9.26 23 24 22.09 9.38 ' 22.05 9.47 22.01 9.57 21.97 9.67 24 25 23.01 9.77 22.97 9.87 22.93 9.97 22.83 10.07 25 26 23.93 ^0.16 23.89 10.26 23.84 10.37 23.80 10.47 26 27 24.85 .0.55 24.81 10.66 24.76 10.77 24.71 10.87 27 23 25.77 .0.94 25.73 11.05 25.68 11.16 25.63 11.23 .T£5 29 26.69 U.33 26.64 11.45 26.59 11.56 26.54 11.63 29 30 27.62 11.72 27.56 11.84 27.51 2S.43 11.96 27.46 12.08 12.36 28.37 12.49 30 31 23.54 12.11 2^.4S 12.24 31 32 29.46 12.50 29. 4U 12.63 29.35 12.76 29.29 12.89 a2 33 30.38 12.89 30.32 13.03 30.26 13.16 30.21 13.29 33 34 31.30 13.28 31.24 '13.42 31.13 13.56 31.12 13.69 34 35 32.22 13.68 32.16 13.82 32.10 13.96 32.04 14.10 35 36 33.14 14.07 33.03 14.21 33.01 14.35 32.95 14.50 36 37 34.06 14.46 34.00 14.61 33.93 14.75 33.37 14.90 37 3S 34.93 14.85 34.91 15.00 34.85 15.15 34. 7S 15.30 .38 39 35.90 15.24 35.83 15.39 35.77 15.55 35.70 15.7] 39 40 36. S2 15.63 36.75 15.79 36.68 15.95 36.61 16.11 40 41 37.74 16.02 37.67 16.13 37.60 16.35 37.53 16.51 11 42 3S.66 16.41 3S.59 16. 5S 3S.52 16.75 3S.44 16.92 42 43 39. 5S 16. SO 39.51 16.97 39.43 17.15 39.36 17.32 i i3 41 40 50 17.19 40.43 17.37 40.35 17.54 40.27 17.72 ! 44 4c 41 i2 17.58 41.35 17.76 41.27 17.94 4].:- l'.:2 4.5 46 42.34 17.97 42.26 18.16 42.13 18.34 42.:: l-..:3 46 47 4:3.26 18.36 43.18 18.55 43.10 15.74 43.02,15.93 I 47 48 41.18 18.76 44.10 18.95 44.02 19.14 43.93.19.3,3 i 48 t9 45.^0 19.15 4.5.02 19. .34 44.94 19.54 44.-5 19.73 ' 49 50 4'..<;3 19.54 45.94 19.74 45.85 19.94 45.77,20.14 , 50 8 Dep. Lat. Dep. 1 '" H Lat. Dep. Lat. Dop. Lat, 5 j; - 67 Deg ' 66} Deg. 66^ Deg. 66i Deg. it^ TRAVERSE TA.BLfc. 119 a a a 51 23 Deg. 23i Deg. 23^ Deg. 23| Deg. a 3 o ? 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat Dep. 46.95 19.93 46.86 20.13 46.77 20.34 46.68 20.54 52 47 87 20.32 47.78 20 . 53 47.69 20.73 47.60 20.94 53 53 48,79 20.71 48.70 20 . 92 48.60 21.13 48.51 21.35 53 54 49.71 21.10 49.61 21.. '52 49.52 21.53 49 43 21.75 b4 55 50.63 21.49 50.53 21.71 50.44 21.93 50.34 22.15 55 56 51.55 21.88 51.45 22.11 51.36 22.33 51.26 22.55 56 57 52.47 22.27 52.37 22.50 52.27 22.73 52.17 22.96 57 58 53.39 22 . 66 53.29 22.90 53.19 23.13 53.09 23.36 58 59 54.31 23.05 .54.21 23 . 29 54.11 23.53 54 00 23.76 59 60 61 55.23 23.44 55.13 23.68 55.02 23.92 54.92 24.16 60 56.15 23.83 56.05 24.08 55.94 24.32 55.83 24.57 61 62 57.07 24.23 56.97 24 47 56.86 24.72 56.75 24.97 62 63 57.99 24.62 57.88 24.87 57.77 25. J 2 57.66 25.37 63 64 58.91 25.01 58.80 25.26 58.69 25 . 52 58.58 25.78 64 65 59.83 25.40 59.72 25.66 59.61 25.92 59.50 26.18 65 66 60.75 25 . 79 60.64 26.05 60.53 26.32 60.41 26.58 66 67 61.67 26.18 61.56 26.45 61.44 26.72 61.33 26.98 67 68 62.59 26.57 62.48 26.84 62.36 27.11 62.24 27.39 68 69 63.51 26.96 63.40 27.24 63.28 27.51 63.16 27.79 69 70 71 64.44 27.35 64.32 65.23 27.63 64.19 27.91 64.07 28.19 70 71 65.36 27.74 28.03 65.11 28.34 64.99 28.59 72 66.28 28.13 66.15 28.42 66.03 28.71 65.90 29.00 72 73 67.20 28.52 67.07 28.82 66 . 95 29.11 66.82 29.40 73 74 68.12 28.91 67.99 29.21 67.86 29.51 67.73 29.80 74 75 69.04 29.30 68.91 29.61 68.78 29.91 68.65 30.21 75 76 69.96 29.ro 69.83 30.00 69.70 30.30 69.56 30.61 76 77 70.88 30.09 70 . 75 30.40 70.61 30.70 70.48 31.01 77 78 71.80 30.48 n.67 30.79 71.53 31.10 71.39 31.41 78 79 72.72 30.87 72 . 58 31.18 72.45 31.50 72.31 31.82 79 80 81 73.64 31.26 73.50 31.58 73.36 31.90 73.22 32.22 80 74.56 31.65 74.42 31.97 74.28 31i.30 74.14 32.62 81 82 75.48 32.04 75.34 32.37 75.20 32 . 70 75.06 33.03 82 83 76.40 32.43 76.26 32.76 76.12 33.10 75.97 33.43 83 84 77.32 32.82 77.18 33.16 77.03 33.49 76.89 33.83 84 85 78.24 33.21 78.10 33.55 77.95 .33.89 77.80 34.23 85 86 79.16 33 . 60 79.02 33.95 78.87 34.29 78.72 34.64 86 87 80.08 33.99 79.93 34.34 79.78 34.69 79.63 35.04 87 88 81.00 34.38 80.85 34.74 80.70 35.09 80.55 35.44 88 89 81.92 34.78 81.77 35.13 81.62 35.49 81.46 35.84 89 90 91 82.85 83.77 35-17 82.69 35.53 82.54 35.89 82.38 36.25 90 '91 35.56 83.61 35.92 83.45 36.29 83.29 36.65 1 92 84.69 35.95 84.53 36.32 84.37! 36.68 84.21 37.05 92 93 85.61 36.34 85.45 36.71 85.29 37.08 85.12 37.46 , 93 94 86.53 36.73 86.37 37.11 86.20 37.48 86.04 37.86 ' 94 95 87.45 37.12 87.29 37.50 87.12 37.88 86.95 38.26 95 96 88.37 37.51 88.20 37.90 88.04 38.28 87.87 38.66 ; ^6 97 89.29 37.90 89.12 38.29 88.95 38.68 88.79 39.07 i )1 98 90.21 38.29 90.04 33.68 89.87 39.08 89.70 39.47 98 99 91.13 38.68 90.96 39.08 90.79 39.48 90.62 39.87 99 100 92.05 39.07 91.88 39.47 91.71 39.87 91.53 40.27 100 i\ Dep. Lat. Dep. Lat. Dep Lat. Dep Lat. i s 67 Deg. 66| Deg. 661 Deg. ... .| m Deg. : ,1 , . „> 120 TRAVERSE TABLE. CD 24 Dog. 24i Deg. 24A Dog. 1 S4| Deg. 1 g o o 1 1 Lat. 1 Dep. Lat. Dep. Lat, Dop. Lat. Dep. I 0.91 1 0.41 0.91 0.41 0.91 0.41 0,91 0.42 2 1.83 0.81 1.82 0.82 1.82 0.83 1.82 0.84 2 3 2.74 1.22 2.74 1.23 2.73 1.24 2 72 1.26 3 4 3.65 1.63 3.65 1.64 3.64 1.66 3.63 1.67 4 6 4.67 2.03 4.56 2.05 4.55 2.07 4.54 2.09 5 6 5.48 2.44 5.47 2.46 5.46 2.49 6.45 2.51 6 7 6.39 2.85 6.38 2.87 6.37 2.90 6.36 2.93 7 8 7.31 3.25 7.29 3.29 7.28 3.32 7.27 3.35 8 9 8.22 3.66 8.21 3.70 8.19 3.73 8.17 3.77 9 10 11 9.14 4.07 4.47 9.12 4.11 9.10 4.15 9.08 4.19 /O 11 10.05 10.03 4.52 10.01 4.56 9.99 4.61 12 10.96 4.88 10.94 4.93 10.92 4.98 10.90 5.02 12 13 11.88 6.29 11.85 5.34 11.83 5.39 11.81 5.44 13 14 12.79 5.69 12.76 5.75 12.74 5.81 12.71 5.86 14 15 13.70 6.10 13.68 6.16 13.65 6.22 13.62 6.28 15 16 14.62 6.51 14.59 6.57 14.56 6.64 14.53 6.70 16 17 15.53 6.92 15.50 6.98 15.47 7.05 15.44 7.12 17 18 16.44 7.32 16.41 7.39 16.38 7.46 16.35 7.54 18 19 17.36 7.73 17.32 7.80 17.29 7.88 17.25 7.95 19 20 21 18.27 8.13 18.24 8.21 18.20 8.29 18.16 8.37 20 21 19.18 8.54 19.15 8.63 19.11 8.71 19.07 8.79 22 20.10 8.95 20.06 9.04 20.02 9.12 19.98 9.21 22 23 21.01 9.35 20.97 9.45 20 . 93 9.54 20.89 9.63 23 24 21.93 9.76 21.88 9.86 21.84 9.95 21.80 10.05 24 25 22.84 10.17 22.79 10.27 22.75 10.37 22.70 10.47 25 26 23.75 10.58 23.71 10.68 23.66 10.78 23.61 10.89 26 27 24.67 10.98 24.62 11.09 24.57 11.20 24.52 11.30 27 28 25.58 11.39 25.53 11.50 25.48 11.61 25.43 11.72 28 29 26.49 11.80 26.44 11.91 26.39 12.03 26.34 12.14 29 30 31 27.41 12.20 27.35 12.32 27.30 12.44 12.86 27.24 12.. 56 30 31 28.32 12.61 28.26 12.73 28.21 28.15 12.98 32 29.23 13.02 29.18 13.14 29.12 13.27 29.06 13.40 32 33 30.15 13.42 30.09 13.55 30.03 13.68 29.97 13.82 33 34 31.06 13.83 31.00 13.96 30.94 14.10 30.88 14.23 34 35 31.97 14.24 31.91 14.38 31.85 14.51 31.78 14.65 35 36 32.89 14.64 32.82 14.79 32.76 14.93 32.69 15.07 36 37 33.80 15.05 33.74 15.20 33.67 15.34 33.60 15.49 37 38 34.71 15.46 34.65 15.61 34.58 15.76 34.51 15.91 38 39 35.63 15.86 36.56 16.02 35.49 16.17 35.42 16.33 39 40 41 36.54 16.27 86.47 16.43 36.40 37.31 16.59 17.00 36.33 16.75 37.23 ; 17.16 40 41 37.46 16.68 37.38 16.84 42 38.37 17.08 38.29 17.25 38.22 17.42 38.14 17.58 12 43 39.28 17.49 39.21 17.60 39.13 17.83 39.05 18.00 id 44 40.20 17.90 40.12 18.07 40.04 18.25 39.96 18.42 44 45 41.11 18 30 41.03 18.48 40.95 18.66 40.87 18.84 45 46 42.02 18.71 41.94 18.89 41.86 19.08 41.77 19.26 46 47 42.94 19.12 42.85 19.30 42.77 19.49 42.68 19,68 47 i8 43.85 19.52 43.76 19.71 43.68 19.91 43.59 20.10 48 49 44.76 19.93 44.68 20.13 44.59 20.32 44.50 20.51 19 60 45.68 120.34 46.59 20.54 45.50 Dep, 20.73 45.41 Dep. 20^93 Lat. 50 V a d ♦J CD 5 s Dop. Lat. Dep. Lat. Lat. 66 Deg. 65! Deg. 1 65^ D'^Sr. 654 Deg TRAVEliSE TABLE. 121 P 9 o p 61 24Deg. 24i Deg. 24i Deg. 241 De^,. 9. n ? 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 46.59 20.74 46.50 20.95 46.41 21.15 46.32 21.35 52 47.50 21.15 47.41 21.36 47.32 21.56 47.22 21.77 52 53 48.42 21.56 48.32 21.77 48.23 21.98 48.13 22.19 53 64 49.33 21.96 49.24 22.18 49.14 22.39 49.04 22.61 54 56 50.24 22.37 50.15 22.59 50.05 22.81 49.95 23.03 56 56 51.16 22.78 51.06 23.00 50.96 23.22 50.86 23.44 66 57 52.07 23.18 51.97 23.41 51.87 23.64 51.76 23.86 57 58 52.99 23.59 52.88 23.82 52.78 24.05 52.67 24.28 58 59 53.90 24.00 53.79 24.23 53.69 24.47 53.58 24.70 59 60 61 54.81 24.40 54.71 24.64 54.60 24.88 54.49 26.12 60 61 55.73 24.81 55.62 25.05 55.51 25.30 55.40 25.54 62 56.64 25.22 56.53 25.46 56.42 25.71 56.30 25.96 62 63 57.55 25.62 57.44 25.88 57.33 26.13 57.21 26.38 63 64 .58.47 26.03 58.35 26.29 58.24 26.54 58.12 26.79 64 65 59.38 26.44 59.26 26.70 59.15 26.96 59.03 27.21 65 66 60.29 26.84 60.18 27.11 60.06 27.37 59.94 27.63 66 67 61.21 27.25 61.09 27.52 60.97 27.78 60.85 28.05 67 68 62.12 27.66 62.00 27.93 61.88 28.20 61.75 28.47 68 69 63.03 28.06 62.91 28.34 62.79 28.61 62.66 28.89 69 70 71 63.95 28.47 63.8-2 28.75 63.70 29.03 63.57 29.31 70 71 64.86 28.88 64.74 29.16 64.61 29.44 64.48 29.72 72 65.78 29.28 65.65 29.57 65.52 29.86 65.39 30.14 72 73 66.69 29.69 66.56 29.98 66.43 30.27 66.29 30.. 56 73 74 67.60 30.10 67.47 30.39 67.34 30.69 67.20 30.98 74 75 68.52 30.51 68.38 30.80 68.25 31.10 68.11 31.40 75 76 69.43 30.91 69.29 31.21 69.16 31.52 69.02 31.82 76 77 70.34 31.32 70.21 31.63 70.07 31.93 69.93 32.24 77 78 71.26 31.73 71.12 32.04 70.98 32.35 70.84 32.66 78 79 72.17 32.13 72.03 32.45 71.89 32.76 71.74 33.07 79 80 81 73.08 74.00 .32.54 72.94 32.86 72.80 33.18 72.65 33.49 80 81 32.95 73.85 33.27 73.71 33.59 73.56 33.91 83 74.91 33.35 74.76 33.68 74.62 34.00 74.47 34.33 82 83 75.82 33.76 75.68 34.09 75.. 53 34.42 75.38 34.75 83 84 76.74 34.17 76.59 34.50 76.44 34.83 76.28 35.17 84 85 77.65 34.57 77.50 34.91 77.35 35.25 77.19 35.59 85 86 78.56 34.98 78.41 35.32 78 26 35.66 78.10 36.00 86 87 79.48 35.39 79.32 35.73 79.17 36.08 79.01 36.42 87 88 80.39 35.79 80.24 36.14 80.08 36.49 79.92 36.84 88 89 81.31 36.;sO 81.15 36.55 80,99 36.91 80.82 37 26 99 90 9: 82.22 36.61 82.06 36.96 81.90 82.81 37.32 81.73 37.68 90 91 83.13 37.01 82.97 37.38 37.74 82.64 38.10 92 84.05 37.42 83.88 37.79 83.72 38.15 83.55 38.. 52 92 93 84.96 37.83 84.79 38.20 84.63 38.57 84.46 38.94 93 94 85 87 38.23 85.71 38.61 85.54 38.98 85.37 39.35 94 95 86 79 38.64 86.62 39.02 86.45 39.40 86.27 39 77 95 96 87.70 39.05 1 87.53 39.43 87.36 39.81 87.18 40.19 96 97 88.61 39. 45 1 88.44 39.84 88.27 40.23 88.09 40.61 97 98 Q9.53 139.06 1 89.35 40.26 89.18 40.64 89.00 41.03 98 99 90.44 40.27 90.26 40.66 90.09 41.05 89.91 41.45 99 tOO 91.35 Dep. 40.67 Lftt. 91.18 Dep. 41.07 91.00 41.47 90.81 41.87 100 Lat. Dep. Lat. Dep. Lat. 66 Dt/T. 65| Deg. 65i Deg. 65i Dog. \ :! ( 122 TRAVERSE TAJRLE. ■ ' p 3 a a T 25 Deg. 25\ Deg. 25i Deg 251 Deg. § Lat. Dop. Lat. Dep. Lat. Dep. Lat. Dep. 0.91 0.42 9.90 0.43 0.90 0.43 0.90 0.43 2 1.81 0.85 1.81 0.85 1.81 0.86 1.80 0.87 2 3 2.72 1.27 2.71 1.28 2.71 1.29 2.70 1.30 3 4 3.63 1.69 3.62 1.71 3.61 1.72 3.60 1.74 4 6 4.53 2.11 4.52 3.13 4.51 2.15 4.50 2.17 (> 6 5.44 2.54 5.43 2.56 5.42 2.58 5.40 2.61 6 7 6.34 1 2.96 6., S3 2.99 6.32 3.01 6.30 3.04 7 8 7.25 3.38 7.24 3.41 7.22 3.44 7.21 3.48 8 9 8.16 3.80 8.14 3.84 8.12 3.87 8.11 3.91 9 10 11 9.06 9.97 4.23 9.04 4.27 9.03 4.31 9.01 " 9.91 4.34 4.78 10 11 4.65 9.95 4.69 9.93 4.74 12 iO.88 5.07 10.85 5.12 10.83 5.17 10.81 5.21 12 13 . 11.78 5.49 11.76 5.55 11.73 5.60 11.71 5.65 13 14 12.69 5.92 12.66 5.97 12.64 6.03 12.61 6.08 14 16 13.59 6.34 13.57 6.40 13.54 6.46 13.51 6.52 In 16 14.50 0.76 14.47 6.83 14.44 6.89 14.41 6.95 16 17 15.41 7.18f 15.38 7.25 15.34 7.32 15.31 7.39 17 18 16.31 7.61 16.28 7.68 16.25 7.75 16.21 7.82 18 19 17.22 8.03! 17.18 8.10 17.15 8.18 17.11 8.25 19 20 21 18.13 19.03 8.45 Ij 18.09 8.53 18.05 8.61 18.01 8.69 9.12 20 21 8.871; 18.99 8.96 18.95 9.04 18.91 22 19.94 9.30 19.90 9.38 19.86 9.47 19.82 9.56 ! 22 23 20.85 9.72 20.80 9.81 20.76 9.90 20.72 9.99 23 24 21.75 10.14 21.71 10.24 21.66 10.33 21.62 10.43 24 25 22.66 10.57 22.61 10.66 22.56 10.76 22.52 10.86 25 20 23.56 10.99 23.52 11.09 23.47 11.19 23.42 11.30 26 27 24.47 11.41 24.42 11.52 24.37 11.62 24.32 11.73 27 28 25.38 11.83 25.32 11.94 25.27 12.05 25.22 12.16 28 29 26.28 12.26 26.23 12.37 26.17 12.49 26.12 12.60 29 30 31 27.19 12.68 27.13 12.80 27.08 12.92 27.02 13.03 30 31 28.10 13.10 28.04 13.22 27.98 13.35 27.92 13.47 32 29.00 13.52 28.94 13.65 28.88 13.78 28.82 13.90 32 33 29.91 13.95 29.85 14.08 29.79 14.21 29.72 14.34 33 34 30.81 14.37 30.75 14.50 30.69 14.64 30.62 14.77 34 35 31.72 14.79 31.66 14.93 31.59 15.07 31.52 15.21 35 30 32.63 15.21 32.56 15.36 32.49 15.50 32.43 15.64 36 37 33.53 15.64 33.46 15.78 33.40 15.93 33.33 16.07 37 38 34.44 16.06 34.37 16.21 34.30 16.36 34.23 16.51 3ft 39 1 35.35 16.48 35.27 16.64 35.20 16.79 35.13 16 94 39 40 4- 36.25 16.90 36.18 17.06 36.10 17.22 17.65 36.03 17.38 40 41 37.16 17.33 37.08 17.49 37.01 36.93 17.81 42 38.06 17.75 37.99 17.92 37.91 18.08 37.83 18.25 42 43 38.97 18.17 38.89 18.34 38.81 18.51 38.73 18.68 43 44 39.88 18.60 39.80 18.77 39.71 18.94 39.63 19.12 44 45 40.78 19.02 40.70 19.20 40.62 19.37 40.53 19. £5 45 46 41.69 19.44 41.60 19.62 41.52 19.80 41.43 19.98 46 47 42,60 19.86 42.51 20.05 42.42 20.23 42.33 20.42 47 4ft 43.50 20.29 43.41 20.48 43.32 20.66 43.23 20.86 4fi 49 44.41 20.71 44.32 20.90 44.23 21.10 44.13 21.29 ' 49 1 50 45.32 21.13 45.22 21.33 45.13 21.53 45.03 21.72 60 § c d Dep. Lat. Dep. Lat Dep. Lat. Dep. Let. d 1 .a 65 Deg. 64| Dea. 64i Dog. 64i Deg. TRAVERSE TARLR. 123 S' «— • p a a a 61 25 Dog. 25i Dog 25i Deg. 251 Dog. 1 o 1' ? 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 46.22 21.65 46.13 21.75 46.03 21.96 46.94 22.16 52 47.13 21.98 47.03 22.18 46.93 22.39 46.84 22.59 52 53 48.03 22.40 47.94 22.61 47.84 22.82 47.74 23.03 53 54 48.94 22.82 48.84 23.03 48.74 23.25 48.64 23.46 54 55 49.85 23.24 49.74 23.46 49.64 23.68 49.64 23.89 56 66 50.75 23.67 60.65 23.89 50.54 24.11 50.44 24.33 561 57 51 66 24.09 61.65 24.31 51.45 24.54 51.34 24.76 57 58 52.67 24.51 52.46 24.74 52.35 24.97 52.24 25.20 58 69 53.47 24.93 53.36 25.17 53.25 25.40 53.14 25.63 69 60 61 54.38 25.. 36 54.27 25.59 26.02 54.16 25.83 54.04 26.07 60 ~61 55.28 25.78 55.17 55.06 26.26 .54.94 26.60 62 56.19 26.20 56.08 26.45 55.96 36.69 56.84 2F.94 62 63 57.10 26.62 56.98 26.87 56.86 27.12 .06 . 74 5,7 37, 63 1 64 58.00 27.05 57.89 27.30 57.77 27.56 57.64 27.80 64 65 58.91 27.47 58.79 27.73 58.67 27.98 68.. 55 28.24 66 66 59.82 27.89 59.69 28.15 59.57 28.41 59.45 28.67 66 67 60.72 28.32 60.60 28.58 60.47 28.84 60.35 29.11 67 68 61.63 28.74 61.50 29.01 61.38 29.27 61.25 29.64 68 69 62.54 29.16 62.41 29.43 62.28 29.71 62.15 29.98 69 70 71 63.44 29.68 63.31 29.86 63.18 30.14 63.06 30.41 70 71 64.35 30,01 64.22 30.29 64.08 30.67 63.95 30.85 72 65.25 .30.43 65.12 30.71 64.99 31.00 64.85 31.28 72 73 66.16 30.85 66.03 31.14 65.89 31.43 65.75 31.71 73 74 67.07 31.27 66.93 31.57 66.79 31.86 66.65 32.15 74 75 67.97 31.70 67.83 31.99 67.69 32.29 67.55 32.58 75 76 68.88 32.12 68.74 32.42 68.60 32.72 68.45 33.02 76 77 69.79 32.54 69.64 32.85 69.50 33.15 69.35 33.45 77 78 70.69 32.96 70.55 33.27 70.40 33.58 70.25 33.89 78 79 71.60 33.39 71.45 33.70 71.30 34.01 71.16 34.32 79 80 81 72.50 33.81 72.36 73.26 34.13 34.55 72.21 34.44 72.06 34.76 35.19 80 81 73.41 34.23 73.11 34.8/ 72.96 82 74.32 34.65 74.17 34.98 74.0] 35.30 73.86 35.62 82 83 75.22 35.08 75.07 35.41 74. 9i 35.73 74.76 36.06 83 84 76.13 35.50 75.97 35.83 75.82 36.16 75.66 36.49 84 85 77.04 35.92 76.88 36.26 76.72 36.59 76.56 36.93 85 86 77.94 36.35 77.78 36.68 77.62 ,37.02 77.46 37.36 86 87 78.85 36.77 78.69 37.11 78.52 37.45 78.36 37.80 87 88 79.76 37.19 79.59 37.54 79.43 37.88 79.26 38.23 88 89 80.66 37.61 80.50 37.96 80.33 38.32 80.16 38.67 89 90 91 81.67 38.04 81.40 38.39 81.23 38.75 81.06 39.10 90 91 82.47 38.46 82.31 38.82 82.14 39.18 81.96 39.63 92 83.38 38.88 83.21 39.24 83.04 39.61 82.86 39.97 92 93 84.29 39.30 84.11 39.67 83.94 40.04 83.76 40.40 93 94 85.19 39.73 85.02 40.10 84. S4 40.47 84.67 40.84 94 96 86.10 40.15 85.92 40.52 85.75 40.90 85 67 41.27 95 96 87.01 40.57 86.83 40.95 86.65 41.33 86,47 41.71 96 97 87.91 40.99 87.73 41.38 87.55 41.76 87.37 42.14 97 98 88.82 41.42 88.64 41.80 88.45 42.19 88.27 42.. 58 98 99 89.72 41.84 89.64 42.23 89.36 42.62 89.17 43.01 99 100 o c 5 dO.63 Dep. 42.26 90.46 42.66 90.26 43.05 90.07 43 44 100 o e a .2 O Lat. Dep. Lat. Dep. Lat. Dep. Lat. 65Deg. 64f Deg. 64i Deg. 64i Deg. 124 TRAVElUsf lABIilC 1 S 1 g 26 Deg. 26i Deg. t 26i Deg. 1 261 Deg. 1 2 s* 9 3 ? ~l Lat. Dep. 0.44 Lat. Dep. Lat. Dep. Lat. Dep. 0.90 0.90 0.44 0.89 0.45 0.80 0„45l 2 1.80 0.88 1.79 0.88 1.79 0.89 1.79 0.90 2 3 3 70 1.32 2 69 1.33 2.68 1.34 2.68 1.35 3 4 3.60 1 75 3.59 1.77 3.58 1.78 3.57 1.80 \ I E 4.49 2 19 4.48 2.21 4.47 2.23 4.48 2.25 5 1 6 5.39 2.63 5.38 2.85 5.37 2.68 5.36 2.70 6 1 7 6.29 3.07 6.28 3.10 6.26 3.12 6.25 3.15 7 1 8 7.19 3.51 7.17 3.54 7.16 3.57 7.14 3.80 8 1 9 8.09 3.95 8.07 3.98 8.05 4.02 8.04 4.05, 9 1 10 " 11 8.99 4.38 8.97 4.42 8.95 4.46 8.93 4.50 10 1 9.89 4.82 9.87 4.87 9.84 4.91 9.82 4.95 11 12 10.79 5.26 10.76 5.31 10.74 5.35 10.72 5.40 12 13 11.68 5.70 11.66 5.75 11.63 5.80 11.61 5.85 13 14 12.58 6.14 12.58 8.19 12.53 8.25 12.50 6.30 14 15 13.48 6.58 13.45 C 63 13.42 6.69 13.39 6.75 15 16 14.38 7.01 14.35 7.08 14.32 7.14 14.29 7.20 16 17 15.28 7.45 15.25 7.52 15.21 7.59 15.18 7.65 17 18 16.18 7.89 16.14 7.96 18.11 8.03 18.07 8.10 18 19 17.08 8.33 17.04 8.40 17.00 8.48 16.97 8.55 19 20 21 17.98 8.77 17.94 8.85 9.29 17.90 8.92 17.86 9.00 20 21 18.87 9.21 18.83 18.79 9.37 18.75 9.45 22 19.77 9.64 19.73 9.73 19.09 9.82 19.65 9.90 22 23 20.67 10.08 20.63 10.17 20.58 10.28 20.. 54 10.35 23 24 21.57 10.52 21.52 10.61 21.48 10.71 21.43 10.80 24 25 22.47 10.96 22.42 11.06 22.37 11.15 22.32 11.25 25 26 23.37 11.40 23.32 11.50 2S.27 11.80 23.22 11.70 26 27 24.27 11.84 24.22 11.94 24.18 12.05 24.11 12.15 27 28 25.17 12.27 25.11 12.38 25.06 12.49 25.00 12.60 28 29 26.06 12.71 26.01 12.83 25.95 12.94 25.90 13.05 29 30 ol 26.98 13.15 26.91 13.27 26.85 13.39 13.83 26.79 13.50 30 31 27.86 13.59 27.80 13.71 27.74 27.68 13.95 32 28.76 14.03 28.70 14.15 28.64 14.28 28.58 14.40 32 33 29.66 14.47 29.60 14.60 29.53 14.72 29.47 14.85 33 34 30.56 14.90 30.49 15.04 30.43 15.17 30.36 15.30 34 35 31.46 15.34 31.39 15.48 31.32 15.62 31.25 15.75 35 36 32.36 15.78 .32.29 15.92 32.22 18.06 32.15 16.20 36 37 33.26 16.22 33.18 16.36 33.11 16.51 33.04 16.65 37 38 34.15 16.66 34.08 16.81 34.01 16.96 33.93 17.10 38 89 35.05 17.10 34.98 17.25 34.90 17.40 34.83 17.55 39 40 41 35.95 17.53 35.87 17.69 35.80 17.85 35.72 18.00 40 4f 36.85 17.97 36.77 18.13 36.69 18.29 36.61 18.45 42 '37.75 18.41 37.67 18.58 37.59 18.74 37.51 18.90 42 43 38.65 18.85 38.57 19.02 38.48 19.19 38.40 19.35 43 44 39.55 19.29 39.46 J9.46 39.38 19.63 39.29 19.80 44 45 40.45 19.73 40.36 19.90 40.27 20.08 40.18 20.25 45 46 41.34 20.17 41.26 20.35 41.17 20.53 41.08 20.70 46 47 42.24 20.60 42.15 20.79 42.06 20.97 41.97 21.15 47 48 43.14 21 04 43.05 21.23 42.96 21.42 42.86 21.60 48 19 44.04 2. .48 43.95 21.67 43.85 21.86 43.76 22.05 49 60 i c a 44.94 Dop. 21.92 44.84 22.11 44.75 22.31 44.65 22.50 60 .1 Q Lat. Dep. Lat. Dep. Lat. Dep. Lat. 64 Deg. 631 1 Deg. 63^ Deg. 63t Deg. TRAVERSE TABLE. 126 3 s a ? "in" 96Deg. m Deg. 26^ Deg. 26» Deg. 1 5J Lat Dep. Lat. Dep. Lat. 45.64 Dep. 22.76 Lat, Dep. 32.96 45.84 22.36 45.74 22.56 46 84 62 46.74 22.80 46.64 23.00 46.54 23.20 46.43 23.41 52 53 47.64 23.23 47.53 23.44 47.43 23.65 47-33 23.86 63 54 48.53 23.67 48.43 23.88 48.33 24. 09 48 22 24.31 54 55 49.43 24.11 49.33 24.33 49.22 24.54 49.11 24.76 55 56 50.33 24.55 50.22 24.77 50. J2 24.09 50.01 25.21 56 67 51.23 24.99 51.12 25.21 51.01 25 43 50.90 25.66 57 58 52.13 25.43 52.02 25.65 51.91 25 88 51.79, 26.11 58 69 53.03 25.86 52.92,26.09 52.80 26-33 52.69 26.-56 59 60 61 53.93 26.30 53.81 26.54 53.70 26-77 27.22 53.58 27.01 60 61 54.83 26.74 54.71 26.98 54.59 54.47 27.46 62 55.73 27.18 55.61 27.42 55.49 27.66 55.36 27.91 62 63 56.62 27.62 56.50 27.86 56.38 28.11 56.26 28.36 63 64 57 52 28.06 57.40 28.31 57.28 28.56 57.15 28.81 64 65 58.42 28.49 58.30 28.75 58.17 29.00 58.04 29.26 65 66 59.32 28.93 59.19 29.19 59.07 29.45 58.94 29.71 66 67 60.22 29.37 60.09 29.63 59.96 29.90 59.83 30.16 67 68 61.12 29.81 ••>0.99| 30.08 1 60.86 30.34 60.72 30.61 68 69 62.02 30.25 61.88 30.52 61.75 30.79 61.62 31.06 69 70 71 62.93 30.69 62.78 63.68 30.96 62.65 31.23 62.51 31.51 70 71 63.81 31.12 31.40 63.54 31.68 63.40 31.96 72 64.71 31.56 64.57 31.84 64.44 32.13 64.29 32.41 72 73 65.61 32.00 65.47 32.29 65.33 32.57 65.19 32.86 73 74 66.51 32.44 66.37 32.73 66.23 33.02 66.08 33.31 74 75 67.41 32.88 67.27 33.17 67.12 33.46 66.97 33.76 75 76 68.31 33.32 68.16 33.61 68.01 33.91 67.87 34.21 76 77 69.21 33.75 69.06 34.06 68.91 34.36 68.76 34.66 77 ;8 70.11 34.19 69.96 34.50 69.80 34.80 69.65 35.11 78 r9 71.00 34.63 70.85 34.94 70.70 35.25 70.55 35.56 79 SO 81 71.90 .35.07 35.51 71.75 35.38 71.59 35.70 71.44 36.01 80 81 72.80 72.65 35.83 72.49 36.14 72.33 36.46 82 73.70 35.95 73.54 36.27 73.38 36.59 73.22 36.91 82 83 74.60 36.38 74.44 36.71 74.28 37.03 74.12 37.36 83 84 75.50 36.82 75.34 37.15 75.17 37.48 75.01 37.81 84 85 76.40 37.26 76.23 37.59 76.07 37.93 75.90 38.26 85 86 77.30 37.70 77.13 38.04 76.96 38.37 76.80 38.71 86 87 78.20 38.14 78.03 38 48 77.86 38.82 77.69 39.16 87 88 79.09 38.58 78.92 38.92 78.75 39.27 78.58 39.61 88 89 79.99 39.01 79.82 39.36 79.65 39.71 79.48 40.06 89 90 91 80.89 39.45 80.72 81.62 39.81 40.25 80.54 40.16 80.37 40.51 40.96 90 91 81.79 39.89 81.44 40.60 81.26 92 82.69 40.33 82.51 40.69 82.33 41.05 82.15 41.41 92 93 83.59 40.77 83.41 41.13 83.23 41.50 83.05 41.86 93 94 84 49 41.21 84.31 41.58 84.12 41.94 83.94 42.31 94 95 85.39 41.65 85.20 42.02 85.02 42.39 84.83 42.76 95 96 86.28 42.08 86.10' 42.46 85.91 42.83 85.73 43.21 96 97 87.18 42.52 87.00 42.90 86.81 43.28 86.62 43.66 97 98 88.08 42.96 87.89 43.34 87.70 43.73 87.51 44.11 96 99 88.98 43.40 88.79 43.79 88.60 44.17 88.40 44.56 99 100 1 89.88 43.84 89.69 44.23 89.49 44.62 89.30 45.01 Lat. 100 • Dep. Lat Dep. Lat. Dep. Lat. Dep. 64Dcg. 63| Deg, e^Deg. 63i Deg. 126 rKAVEJt:E TABLE. 2 s I 1 27 Deg. 27i Deg. 27^ Deg. 271 Deg. ft S I Lat. Dep. 0.45 Lat Dep. Lat, 0.89 Dep. J. 46 Lat Dep. 0.47 0.8S 0.89 0.46 0.88 2 1.78 0.91 1.78 ).i2 1.77! 0.92 1.77 0.93 il 3 2.67 1.36 2.67 1.37 2.66 1.39 5« 65 1.40 3 4 3.56 1 82 3.56 1.83 3.56 1.85 'J. 64 1.86 4 5 4.40 2 27 4.45 2.29 4.44 2.31 4.42 2.33 S 6 5.35 2.72 5.33 2.75 5.32 2.77 5.31 2.79 fl 7 6.24 3.18 6.22 3.21 6.2] 3.23 6.19 3.26 7 8 7.13 3.63 7.11 3.66 7.10 3.69 7.08 3.72 8 9 8.02 4.09 8.00 4.12 7.98 4.16 7.96 4.19 9 10 11 8.91 4.54 8.89 9.78 4.58 5.04 8.87 4.62 8.85 4.66 10 11 9.80 4.99 9.76 5.08 9.73 6.12 12 10.69 5.45 10.67 5.49 10.64 5.54 10.62 5.69 12 13 11.58 5.90 11.56 5.95 11.53 6.00 11.50 6.05 13 14 12.47 6.36 12.45 5.41 12.42 6.46 1 12.39 6.52 14 15 13.37 6.81 13.34 6.87 13.31 6.93 13.27 6.98 15 IH 14.26 7.26 14.22 7.33 14.19 7.39 14.16 7.45 16 17 15.15 7.72 15.11 7.78 15.08 7.85 16.04 7.92 17 18 16.04 8.17 16.00 8.24 15.97 8.31 15.93 8.38 18 19 16.93 8.63 16.89 8.70 16.85 8.77 16.81 8.86 19 20 21 17.82 9.08 17.78 9.16 17.74 9.23 9.70 17.70 9.31 20 21 18.71 9.53 18.67 9.62 18.63 18.58 9.78 22 19.60 9.99 19.56 10.07 19.51 10.16 19.47 10.24 22 23 20.49 10.44 20.45 10.53 20.40 10.62 20.35 10.71 23 24 21.38 10.90 21.34 10.99 21.29 11.08 21.24 11.17 24 25 22.28 11.35 22.23 11.45 22.18 11.54 22.12 11.64 26 26 23.17 11.80 23.11 11.90 23.06 12.01 23.01 12.11 26 27 24.06 12.26 24.00 12.36 23.95 12.47 23.89 12.67 27 28 24.95 12.71 24.89 12.82 24.84 12.93 24.78 13.04 28 29 25.84 13.17 25.78 13.28 25.72 13.39 26.66 13.50 29 30 31 26.73 13.62 26.67 13.74 26.61 13.85 26.65 13.97 30 31 27.62 14.07 27.56 14.19 27.50 14.31 27.43 14.43 32 28.51 14.53 28.45 14.65 28.38 14.78 28.32 14.90 32 33 29.40 14.98 29.34 15.11 29.27 16.24 29.20 15.37 33 34 30.29 15.44 30.23 15.57 30.16 15.70 30.09 15.83 84 35 31.19 15.89 31.12 16.03 31.05 16.16 30.97 16.30 36 36 32.08 16.34 32.00 16.48 31.93 16.62 3 '.86 16.76 36 37 32.97 16.80 32.89 16.94 32.82 17.08 •3i.74 17.23 37 38 33.86 17.25 33.78 17.40 33.71 17.55 33.63 17.69 3S 39 34.75 17.71 34.67 17.86 34.59 18.01 34.61 18.16 39 40 41 35.64 18.16 35.56 18.31 35.48 18.47 35.40 18.62 40 41 36.63 18.61 36.45 18.77 36.37 18.93 36.28 19.09 42 37.42 19.07 37.34 19.23 37.25 19.39 37.17 19.66 42 43 38.31 19.52 38.23 19.69 38.14 19.86 38.05 20.02 43 44 39.20 I1& 98 39.12 20.15 39.03 20.32 38.94 20 49 44 45 40.10 20.43 40.01 20.60 39.92 20.78 39.82 20 95 45 46 40.99 20.88 40.89 21.06 40.80 21.24 40.71 21.42 46 47 41.88 21.34 41.78 21.52 41.69 21.70 41.69 21.88 47 48 42.77 21.79 42.67 21.98 42.58 22.16 4^.48 22.35 48 49 43.66 22.25 43.56 22.44 43.46 22.63 43.36 22.82 49 60 44.55 Dep. 22.70 1 I^at 44.45 22.89 44.35 23.03 4*t.25 23 28 Lat. 50 Dep. Lat. Dep. Lat Dep. ^ (§ 63] Deg. 62| Deg. 6^ Deg 6ai Deg. Q TRAVERSE TABLE. 127 a 51 27 Dcg. m Dog. 271 Dog. 27| Dog. Distance, lo Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 15.44 23.15 45.34 23.35 45.24 23.55 45.13 23.75 55 46.33 1 23.61 46.23 23.81 46.12 24.01 46.02 24.21 52 53 47.22 24.06 47.12 24.27 47.01 24.47 46.90 24.68 53 54 48.11 24.52 48.01 24.73 47.90 24.93 47.79 25.14 51 56 49.01 1 24.97 48.90 25.18 48.79 25.40 48.67 25.61 55 56 49.90 1 25.42 49.78 25.64 49.67 25.86 49.56 26.07 56 5? 50.79 25.88 50.67 26.10 50.56 26.32 .50.44 26.54 57 1 58 51.68 26.33 51.56 26.56 51.45 26.78 51.33 27.01 58 59 52.57 26 . 79 52.45 27.01 52.33 27.24 52.21 27.47 59 60 61 53.46 27.24 53.34 27.47 53.22 27.70 53.10 27.94 60 61 54.35 1 27.69 54.23 27.93 54.11 28.17 53.98 28.40 62 55.24 28.15 55.12 28.39 54.99 28.63 54.87 28.87 62 63 56.13 28.60 56.01 28.85 55.88 29.09 55.75 29.33 63 64 57.02 29.06 56.90 29.30 56.77 29.55 56.64 29.80 64 65 57 92 29.51 57.79 29.76 57.66 30.01 57.52 30.26 65 66 58.81 29.96 58.68 30.22 58.54 30.48 58.41 30.73 66 67 59.70 30.42 .59.56 30.68 59.43 30.94 59.29 31.20 67 68 60.59 30.87 60.45 31.14 60.32 31.40 60.18 31.66 68 69 61.48 31.33 61.34 31.59 61.20 31.86 61.06 32.13 69 70 71 62.37 31.78 62.23 32.05 62.09 32.32 61.95 32.59 33.06 70 71 63.26 32.23 63.12 32.51 62.98 32.78 62.83 72 64.15 32.69 64.01 32.97 63.86 33.25 63.72 33.52 72 73 65.04 33.14 64.90 33.42 64.75 33.71 64.60 33.99 73 74 65 . 93 33.60 65.79 33.88 65.64 34.17 65.49 34.46 74 75 66.83 34.05 66.68 34.34 66.53 34.63 66.37 34.92 75 76 67.72 34.50 67.57 34.80 67.41 35.09 67.26 35.39 76 77 68.61 34.96 68.45 35.26 68.30 35.55 68.14 35.85 77 78 69.50 35.41 69.34 35.71 69.19 36.02 69.03 36.32 78 79 70.39 35.87 70.23 36.17 70.07 36.48 69.91 36.78 79 80 81 71.28 36.32 71-12 72.01 36.63 70.96 71.85 36.94 70.80 71.68 37.25 80 81 72.17 36.77 37.09 37.40 37.71 82 73.06 37.23 72.90 37 . 55 72.73 37.86 72.57 38.18 82 83 73.95 37.68 73.79 38.00 73.62 38.33 73.45 38.65 83 84 74.84 38.14 74.68 38.46 74.51 38.79 74.34 39.11 84 85 75.74 38.59 75 . 57 38.92 75.40 39.25 75.22 39.58 85 86 76.63 39.04 76.46 39.38 76.28 39.71 76.11 40.04 86 S7 77.52 39.50 77.34 39.83 77.17 40.17 76.99 40.51 87 88 78.41 39.95 78.23 40,29 78.06 40.63 77,88 40.97 88 89 79.30 40.41 79.12 40.75 78.94 41.10 78.76 41.44 89 90 91 80.19 10.86 80.01 41.21 79.83 41.56 42.02 79.65 41.91 42.37 90 "91 81.08 41.31 80.90 41.67 80.72 80.53 92 81 97 41.77 81.79 42.12 81.60 42.48 81.42 42.84 92 93 82.86 42.22 82.68 42.58 82.49 42.94 82.30 43.30 1 93 1 94 83,75 42.68 83.57 43.04 83.38 43.40 83.19 43.77 94 1 95 1 84.65 43.13 84.46 43.50 84.27 43.87 84.07 44.23 95 1 96 ' 85 54 43.58 85.35 43.96 85.15 44.33 84.96 44.70 96 1 97' 80.43 44.04 86.23 44.41 86.04 44.79' 85.84 45.16 97 98 87.32 44.49 87.12 44.87 86.93 45.25 86.73 45.63 98 99 i 88.21 44.95 88.01 45.33 87.81 45.71 87.61 46.10 99 100 89.10 45^40 Lat. 88.90 45.79 88.70 46.17 Lat. 88.50 46.56 100 1 s 1 Dep. Dep. Lat. Dep. Dep. Lat. 63 I )eg. 621 Deg. 62^ Deg. 62i Deg. 128 TRAVERSE TABLE. 5' P a p 28Deg. 28i Deg. 36- Deg. 281 Deg. B- » S s 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. 1 Dep. 1 0.88 0.47 0.88 0.47 0.88 0.48 0.88 ' 0.48 2 1.77 0.94 1.76 0.95 1.76 0.95 1.75 0.96 2 3| 2.65 1.41 2.64 1.42 2.64 1.43 2.63 1.44 3 4 3.53 1.88 3.52 1.89 3.52 1.91 3.61 1.92 4 6 4.41 2.35 4.40 2.37 4.39 2.39 4.38 2.40 5 e 5.30 2.82 5.29 2.84 5.27 2.86 5.26 2.89 6 7 6.18 3.29 6.17 3.31 6.15 3.34 6.14 3.37 7 8 7.06 3.76 7.05 3.79 7.03 3.82 7.01 3.86 8 9 7.95 4.23 7.93 4.26 7.91 4.29 7.89 4.33 9 10 11 8.83 4.69 8.81 4.73 8.79 4.77 8.77 4.81 10 " 11 9.71 5.16 9.69 5.21 9.67 5.25 9.64 5.29 12 10.60 5.63 10.57 5.68 10.55 5.73 10.52 5.77 12 13 11.48 6.10 11.45 6.15 11.42 6.20 1 11.40 6.25 13 14 12.36 6.67 12.33 6.63 12.30 6.68 \ 12.27 6.73 14 15 13.24 7.04 13.21 7.10 13.18 7.16! 13.15 7.21 15 16 14.13 7.51 14.09 7.57 14.06 7.63 14.03 7.70 16 17 15.01 7.98 14.98 8.05 14.94 8.11 ! 14.90 8.18 17 18 15.89 8.45 15.86 8.52 15.82 8.59 15.78 8.66 18 19 16.78 8.92 16.74 8.99 16.70 9.07 16.66 9.14 19 20 21 17.66 18.54 9.39 17.62 18.50 9.47 17.58 9.54 10.02 17.53 9.62 20 21 9.86 9.94 18.46 18.41 10.10 22 19.42 10.33 19.38 10.41 19.33 10.60 19.29 10.68 22 23 20.31 10.80 20.26 10.89 20.21 10.97 20.16 11.06 23 24 21.19 11.27 121.14 11.36 21.09 11.45 21.04 11.64 24 25 22.07 11.74 22.02 11.83 21.97 11.93 21.92 12.02 26 26 22.96 12.21 22.90 12.31 22.85 12.41 22.79 12.61 26 27 23.84 12.68 23.78 12.78 23.73 12.88 23.67 12.99 27 28 24.72 13.15 24.66 13.25 2-1:. 61 13.36, 24.55 13.47 28 29 25.61 13.61 25.55 13.73 25.49 13.84 25.43 13.95 29 30 26.49 14.08 26.43 14.20 26.36 27.24 14.31 14 79 26.30 14.43 30 31 31 27.37 14.55 27.31 14.67 27.18 14.91 32 28.25 15.02 28.19 15.15 28.12 15.i7i 28.06 15.39 32 33 29.14 15.49 29.07 15.62 29.00 \b.ll 1 28.93 16.87 33 34 30.02 15.96 29.95 16.09 29.88 16.22' 29.81 16.35 34 35 30.90 16.43 130.83 16.57 30.76 16.70 30.69 16.83 36 36 31.79 16.90 131.71 17.04. 31.64 17.18 31.56 17.32 36 37 32.67 17.37 : 32.59 17.51 32.52 17.65 32.44 17.80 37 38 33.55 17.84 33.47 17.99 33.39 18.13 33.32 18.28 38 39 34.43 18.31 ! 34.35 18.46 34.27 18.61 34.19 18 76 39 40 41 35.32 18.78 35.24 36.12 18.93 35.15 19.09 19.56 35.07 35.95 19.24 40 41 36.20 19,25 19.41 36.06 19.72 42 37.08 19.72 37.00 19.88 36.91 20.04 36.82 20.20 42 43 '37.97 20.19 137.88 20.35 37.79 20.52 37.70 20.68 43 44 38 S5 20.66 138.76 20.83 38 67 20.99 38.58 21.16 44 45 39 73 21.13 39.64 21.30 39.55 21.47 39.45 21.64 it 4? 40.62 21.60 40.52 21.77 40.43 21.95 40.33 22.13 16 47 41.50 22.07 41.40 22.25 41.30 22.43 41.21 22.61 47 4^5 42.38 22.53 42.28 22.72 42.18 22.90 42.08 23.09 48 49 '43.26 23.00 43.16 23.19 43.06 23.38 42.96 23.57 49 50 i 44.15 23.47 44.04 23.67 43.94 23.86 43.84 24.05 50 d I>ep. L&t. Dep. Lat. Dep. Lat. Dep. Lat. 1 c .s 62 1 )eg. 6lf Deg. 64] Oeg. 1 m Deg. s TRAVERSE TABLK. 129 5 a S • 'ft' 28Deg. 88iDog. 28i Deg. m Deg. s* ? "51 Lat. Dop. Lat. Dep. Lat. Dop. Lat. Dep. 45.03 23.94 44.93 24.14 44.82 24.34 44.71 24.53 52 1 45.91 24.41 45.81 24.61 45.70 24.81 45.59 25.01 ^3 63 46.80 24.88 1 46.69 25.09 46.58 25.29 46.47 25.49 53 64 47.68 25.35 1 47.57 25.56 47.46 25.77 47.34 25.97 54 55 48.56 25.82 48.45 26.03 48.33 26.24 48.22 26.45 55 66 49.45 26.29 49.33 26.51 49.21 26.72 49.10 26.94 56 57 50.33 26.76 50.21 26.98 50.09 27.20 49.97 27.42 57 58 51.21 27.23 51.09 27.45 50.97 27.68 50.85 27.90 58 59 52.09 27.70 1 51.97 27.93 51.85 28.15 51.73 28.38 59 60 61 52.98 28.17 52.85 28.40 52.73 28.63 52.60 28.86 60 61 53.86 28.64 53.73 28.87 53.61 29.11 53.48 29.34 62 54.74 29.11 54.62 29.35 54.49 29.58 54.36 29.82 62 63 55.63 29.58 55.50 29.82 55.37 30.06 55.23 30.30 63 64 56.51 30.05 56.38 30.29 56.24 30.54 56.11 30.78 64 65 57.39 30.52 .57.26 30.77 57.12 31.02 56.99 31.26 65 66 58.27 30.99 58.14 31.24 58.00 31.49 57.86 31.75 66 67 59.16 31.45 59.02 31.71 58.88 31.97 58.74 32.23 67 68 60.04 31.92 59.90 32.19 59.76 32.45 59.62 32.71 68 69 60.92 32.39 60.78 32.60 60.64 32.92 60.49 33.19 69 70 71 61.81 32.86 33.33 61.66 62.54 33.13 61.52 62.40 33.40 61.37 33.67 70 71 62.69 33.61 33.88 62.25 34.15 72 63.57 33.80 63.42 34.08 63.27 34.36 63.12 34.63 72 73 64.46 34.27 64.30 34.55 64.15 34.83 64.00 35.11 73 74 65.34 34.74 65.19 35.03 65.03 35.31 64.88 35.59 74 75 66.22 35.21 66.07 35.50 65.91 35.79 65.75 36.07 75 76 67.10 35.68 66.95 35.97 66.79 36.26 66.63 36.56 76 77 67.99 36.15 67.83 36.45 67.67 36.74 67.51 37.04 77 78 68.87 36.62 68.71 36.92 68.. 55 37.22 68.38 37.52 78 79 69.75 37.09 69.59 37.39 69.43 37.70 69.26 38.00 79 80 81 70.64 37.56 70.47 ?7.87 70.31 71.18 38.17 38.66 70.14 38.48 38.96 80 71.52 38.03 71.35 38.34 71.01 81 82 72.40 38.50 72.23 38.81 72.06 39.13 71.89 39.44 82 83 73.28 38.97 73.11 39.29 72.94 39.60 72.77 39.92 83 84 74.17 39.44 73.99 39.76 73.82 40.08 73.64 40.40 84 85 75.05 39.91 74.88 40.23 74.70 40.56 74.52 40.88 85 86 75.93 40.37 75.76 40.71 75 58 41.04 75.40 41.36 86^ 87 76.82 40.84 76.64 41.18 76.46 41.51 76.28 41.85 87 88 77.70 41.31 77.52 41.65 77.34 41.99 77.15 42.33 88 89 78.58 41.78 78.40 42.13 78.21 42.47 78.03 42.81 89 90 91 79.47 42.25 79.28 42.60 79.09 42.94 78.91 43.29 90 91 80.35 42.72 80.16 43.07 79.97 43.42 79. /S 43.77 92 81.23 43.19 81.04 43.55 80.85 143.90 80.66 44 25 92 93 82.11 43.66 81.92 44.02 81.73 44.38 81.54 44 73 93 94 83.00 44.13 82.80 44.49 82.61 44.85 82.41 ^.fl 91 95 83.88 44.60 83.68 44.97 83.49 45.33 83.29 45. u9 95 96 84.78 45.07 84.57 45.44 84.37 45.81 84.17 146.17 96 97 85.65 45.54 85.45 45.91 85.25 46.28 85.04 1 46.66 97 98 86.53 46.01 86 33 46.39 86.12 46.76 85.92 47.14 98 99 87.41 46.48 87.21 46.86 87.00 47.24 86.80 47.62 99 100 — r 88.29 46.95 88.09 47.33 87.88 47.72 87.67 Dep. 48.10 too § 1 Dep. Lat. Dep. Lat. Dep. 1 Lat. 1 Lat. 62 Deg. 611 Deg. 61i Deg. 6U Deg. 130 TRAVERSE TABLE. a S' § S 29 Deg. 29i Deg. 29i Deg. 29i Deg. C 5* s Lat. Dep. Lat. Dep. Lat. 1 Dep. Lat. j Dep. ■ " 0.87 1 48 0.87 : 0.49 i 0.87 0.49 0.87 0.50 1 2 1.75 1 0.97 1.74 1 0.98 1.74 0.98 1.74 0.99 \ 3 2.62 1 1.45 2.62 1 1.47 ; 2.61 1.48 2.60 1.49 3 4 I 3.50 1 1.94 I 3.49 1 1.95 3.48 1.97 : 3.47 1.98 4 6 4 37| 2.42 4. 36 2.44 4.35 2.46 ' 4.34 2.48 5 6 5.25 2.91 5.23 2.93 5.22 2.95 1 5.21 2.98 6 7 6.12 3.39 6.11 3.42 6.09 3.45 6.08 3.47! 7 8 7.00 1 3.88 6.98 3.91 6.96 3.94 6.95 3.97 8 9 7.87 1 4.36; 7.85 4.40 7.83 4.43 7.81 1 4.47 ^ 9 10 11 8.75 4.85; 8.72 9.60 4.89 8.70 4,92 8.68 1 9.55 i 4.96 10 5,46 11 9.62 5.33 5.37 9.57 5.42 12 10.50 5.82 10.47 5.86 10.44 5.91 10.42 5.95 12 13 11.37 6.30 11.34 6.35 11.31 6.40 11.29, 6.45 13 14 12.24 6.79 12.21 6.84 12.-18 6.89 12.15! 6.95 14 15 13.12 7.27 13.09 7.33 13.06 7.39 13.02 7.44 15 16 13.99 7.76 13.96 7.82 13.93 7.88 13.89 7.94 16 17 14.87 8.24 14.83 8.31 14.80 8.37 14.76 8.44 17 18 15.74 8.73 15.70 8.80 15.67 8.86 15.63 8.93 18 19 16.62 9.21 16.58 9.28 16.. 54 9.36 16.50 9.43 19 20 21 17.49 18.37 9.70 10.18 17.45 18.32 9.77 10.26 17.41 18.23 9.85 10.34 17.30 9.92 ! 20 21 18.23 10.42 1 22 1 19.24 10.67 19.19 10.75 19.15 10.83 19.10 10.92 22 23 20.12 11.15 20.07 11.24 1 20 . 02 11.33 19.97 11.41 2« 24 ,20.99 11.64 20.94 11.73 20.89 11.82 20.84 11.91 24 25 ,21.87 12.12 21.81 12.22 21.76 12.31 21.70 12.41 ! 25 26 22.74 12.60 22.68 12.70 22.63 12.80 22.57 12.90 26 27 23.61 13.09 23.56 13.19 23.50 13.30 23.44 13.40 27 28 24.49 13.57 24.43 13.68 24.-37 13.79 i 24.31 13.89 1 28 29 25.36 14.06 25.30 14.17 25.24 14.28 ■ 25.18 14.39 I 29 30 20.24 14.54 26.17 1 14.66 i 15.15 26.11 14.77 26.05 26.91 14.89 15.38 30 31 31 27.11 15.03 27.05 26.98 15.27 32 27.99 15.51 27.92 1 15.64 27.85 15.76 27.78 15.88 32 33 28.86 16,00 28.79 16.12 28.72 16.25 28.65 16.38 33 34 1 29 . 74 16.48!, 29.66 1 16.61 29.59 16.74 (29.52 16.87 ■ 34 35 130.61 16.97 30.54 17.10 30.46 17.23 30.39 17.37 35 36 31.49 17.45 31.41 17.59 31.33 17.73 ' 31.26 17.86 36 37 32.36 ! 17.94 32.28 18.08 32.20 18.22 1 32.12 18.36 37 38 33.24 i 18.42 33.15 18.57 33.07 18.71 i 32.99 18.86 38 39 34.11 ! 18.91 34.03 19.06 33.94 19.20 33.86 19.35 39 40 34.98 ! 19.39 34.90 35 . 77 19.54 34.81 35.68 19.70 20.19 34.73 35.60 19.85 20.34 40 41 41 135.86 19.88 20.03 42 ' 36 . 73 20.36 36.64 20.52 36.55 20.68 1 36.46 1 20.84 42 43 37.61 20.85" 37.52 21.01 137.43 21.17 37.33 21.34 43 44 38.48 21.33 38.39 21.50 1 38.30 21.67 ! 38.20 ' 21.83 44 45 '39.36 21.82 i 39.26 21.99 139.17 22.16 39.07 22 . 33 45 46 40.23 22.30 1; 40.13 122.48 ; 40 . 04 22.65 39.94 22.83 46 47 41.11 22.79 141.01 122.97 ! 40.91 23.14 40.81 23.32 47 4« '41.98 23.27 I 41.88 23.45 41.78 23.68 i 41.67 ' 23.82 1 48 49 1 42 86 23.76 ; 42.75 23.94 42.65 24.13 1 42.54 i 24.31 ' 49 50 Ii3.73 24.24 43.62 24.43 43.52 24.62 Lat. 43.41 ! 24.81 50 i Dep. 1 Lat. Dep. Lat Dop. Dep. 1 Lat. o a 61 Dcg. 1 601 Deg. 60J Deg. 6(H Deg. TRAVERSE TAKLE. 131 p . P 51 29 Deg. 29i Deg. 29i Deg. 29| Deg. 5 5' C3 a p Lat. 44.61 Dep. 24.73 Lat. Dep. Lat. Dep. 25.11 Lat. Dep. 25731 44.50 24.92 44.39 44.28 52 45.48 25.21 45.37 25.41 45.26 25.61 45.15 25.80 52 63 46.35 25.69 46.24 25.90 46.13 26.10 46.01 26.30 63 r>4 47.23 26.18 47.11 1 26.39 47.00 26.59 46.88 26.80 54 65 48.10 26.66 47.99 : 26.87 | 47.87 27.08 47.75 27.29 55 56 48.98 27.15 1 48.86 27.36 48.74 27.58 48.62 27.79 66 ' 57 49.85 27.63! 49.73 27.85 49.61 28.07 49.49 28.28 57 bS 50.73 28.12 50.60 28.34 50.48 28.56 50.36 28.78 58 59 51.60 28.60 51.48 28.83 51.35 29.05 51.22 29.28 69 60 61 52.48 53.35' 29.09 29.57 52.35 29.32 52.22 53.09 29.55 30.04 52.09 52.96 29.77 30.27 60 61 53.22 29.81 62 54.23 30.06 54.09 30.29 53.96 30.53 53.83 30.77 62 63 55.10 30 . 54 54.97 30.78 54.83 1 31.02 1 54.70 31.26 63 64 55.98 31.03 55.84 31.27 55.70 31.52 55.56 31.76 64 65 i,t>.85 31 51 56.71 31.76 56.57 32.01 56.43 32.25 65 66 ^7.72 32 . 00 57.58 32.25 57.44 32.50 57.30 32.75 66 67 .58.00 32.48 .58.40 32.74 .58.31 32.99 58.17 33.25 67 68 59.47 32.97 59.33 33.23 59.18 33.48 59.04 33.74 68 69 60.35 33.45 60.20 33.71 60.05 33.98 1 59.91 34.24 69 70 71 61.22 62.10 33.94 34.42 61.07 34.20 34.69 60.92 61.80 .34.47 34.96 60.77 34.74 70 71 61.95 61.64 35.23 72 62.97 34.91 62.82 35.18 62.67 35.45 ! 62.51 35.73 72 73 63.85 35.39 63.69 35.67 63.54 35.95 63.38 36.22 73 74 64.72 35.88 64.56 36.16 64.41 36.44 64.25 36.72 74 75 65 . 60 36.36 65.44 36.65 65.28 36.93 65.11 37.22 75 76 66.47 36.85 66.31 37.14 66.15 37.42 65.98 37.71 76 77 67.36 37.33 67.18 37.62 67.02 37.92 66.85 38.21 77 78 68.22 37.82 68.05 38.11 67.89 38.41 67.72 38.70 78 79 69.09 38.30 68.93 38.60 68.76 38.90 68.59 39.20 79 80 81 69.97 38.78 69.80 39.09 39.58 69.63 39.39 39.89 69.46 39.70 80 81 70.84 39.27 70.67 70.50 70.32 40.19 82 71.72 39.75 71.54 40.07 71.37 40.38 71.19 40.69 82 83 72.59 40.24 72.42 40.56 72.24 40.87 72.06 41.19 83 84 73.47 40.72 73.29 41.04 73.11 41.36 72.93 41.68 84 85 74.34 41.21 74.16 41.53 73.98 41.86 73.80 42.18 85 86 75 . 22 41.69 j 75.03 42.02 74.85 42.35 74.67 42.67 86 87 76.09 42.18 75.91 42.51 75.72 42.84 75.53 43.17 1 87 88 76.97 42.66 76.78 43.00 76.59 43.33 76.40 43.67 88 89 77.84 43.15 77 . 65 43.49 77.46 43.83 77.27 44.16 89 90 91 78.72 43.63 78 . 52 79.40 43.98 44.46 78.33 44.32 44.81 78.14 44.66 90 1 79.59 44.12 79.20 79.01 45.16 91 92 80.46 44.60 80.27 44.95 80.07 45.30 79.87 45.65 95 93 81.34 45.09 81.14 45.44 SO. 94 45.80 80.74 146.15 09 94 82.21 45.57 82.01 45.93 81. SI 46.29 81.61 46.64 91, 47.14 951 95 83.09 46.06 82.89 46.42 82.68 46.78 82.48 96 89.96 46.54 83.76 46.91 83.55 47.27 83.35 1:7,64 1 96 97 84.84 47.03 84.63 47.40 84.42 47.77 84.22 48 13 97 98 85.71 t 47.51 85.50 47.88 85.29 48.26 85.08 48.63 98 99 86.59 48.00 86.38 48.37 86.17 48.75 85.95 49.13 99 iOO s 87.46 Dep. 48.48 87.25 48.86 87.04 Dep. 49.24 86.82 Dep. 49.62 Lat. 100 o o a ed 1 Lat. Dep. Lat. Lat. il 61 Dt^. 601 Deg. 601 Deg. 60^ Deg. 132 TRAVERSE TADIB. s* s • I SODeg. SOiDeg. SOf Deg. 301 Deg. § s Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.61 0.87 0.50 0.85 0.50 3.86 0.51 0.86 1 % 1.73 1.00 1.73 1.01 1.72 '.02 1.72 1.02 3 3 2.60 1.50 2 59 1.51 2.58 1.52! 2 58' 1.53 3 4 3.46 2.00 3.46 2.02 3.45 2.03 3 44 2.05 i 5 4.33 2.50 4.32 2.52 4.31 2.54 4 30 2.56 fi 6 5.20 3.00 5.18 3.02 5.17 3.05 5.16 3.07 5 7 6.06 3.50 6.05 3.53 6.03 3.55 6.02 3.58 7 8 6.93 4.00 6.91 4.03 i 6.89 4.06 11 6.88 4.09 8 9 7.79 4.50 7.77 4.53 1 7.75 4.571 7.73 4.60 9 lU 11 8.66 9.53 5.00 8.64 5.04 1 8.62 5.08 1 8.59 5.11 10 5.50 9.50 5.54 1 9.48 5.58 i 9.45 5.62 12 10.39 6.00 10.37 6.05 10.34 6.09' 10.31 6.14 12 13 11.26 6.50 11.23 6.55 11.20 6.60 11.17 6.65 13 14 12.12 7.00 12.09 7.05 12.06 7.11 12.03 7.16 14 15 12.99 7.50 12.96 7.56 12.92 7.61 12.89 7.67 15 16 13.86 8.00 13.82 8.06 13.79 8.12 13.75 8.18 16 17 14.72 8.50 14.69 8.56 14.65 8.63 14.61 8.69 17 18 15.59 9.00 15.55 9.07 15.51 9.14 15.47 9.20 18 19 16.45 9.50 16.41 9.57 16.37 9.64 16.33 9.71 19 20 21 17.32 10.00 17.28 10.08 ! 17.23 10.15 10.66 17.19 10.23 10.74 20 21 18.19 10.50 18.14 10.58 18.09 18.05 22 19.05 11.00 19.00 11.08 18.96 11.17 18.91 11.25 22 23 19.92 11.50 19.87 11.59 19.82 11.67 19.77 11.76 23 24 20.78 12.00 20.73 12.09 20.68 12.18 20.63 12.27 24 25 21.65 12.50 21.60 12.59 21.54 12.69 21.49 12.78 25 26 22.52 13.00 22.46 13.10 22.40 13.20 22.34 13.29 26 27 23.38 13.50 23.32 13.60 23.26 13.70 23.20 13.80 27 28 24.25 14.00 24.19 14.11 24.13 14.21 24.06 14.32 28 29 25.11 14.50 25.05 14.61 24.99 14.72 24.92 14.83 29 30 31 25.98 15.00 25.92 15.11 25.85 15.23 25.78 15.34 30 31 26.85 15.50 26.78 15.62 26.71 15.73 26.64 15.85 32 27.71 16.00 27.64 16.12 27.57 16.24 27.50 16.36 32 33 28.58 16.50 28.51 16.62 28.43 16.75 28.36 16.87 33 34 29.44 17.00 29.37 17.13 29.30 17.26 29.22 17.38 34 35 30.31 17.50 30.23 17.63 30.16 17.76 30.08 17.90 35 36 31.18 18.00 31.10 18.14 31.02 18.27 30.94 18.41 36 37 32.04 18.50 31.96 18.64 31.88 18.78 31.80 18.92 37 38 32.91 19.00 32.83 19.14 32.74 19.29 32.66 19.43 38 39 33.77 19.50 33.69 19.65 33.60 19.79 33.52 19.94 39 40 41 34.64 20.00 34.55 20.15 34.47 20.30 34.38 20.45 40 '41 35.51 20.50 35.42 20.65 35., 33 20.81 35.24 20.96 42 36.37 21.00 136.28 21.16 36.19 21.32 36,10 21.47 42 43 37.24 21.50 !37 14 21.66 37.05 21.82 36.95 21.99 43 44 38.11 22.00 38.01 22.17 37.91 22.33 37.81 22.50 44 45 38.97 22.50 38.87 22.67 38.77 22.84 38.67 123.01 15 46 39.84 23.00 39.74 23. ]7 39.63 23.35 39.53 23.52 46 4r 40.70 23.50 40.60 23.68 40.50 23.85 40.39 24.03 4? 48,41.57 24.00 41.46 24.18 41 36 24.36 41.25 24.54 48 49 42.44 24.50 42.33 24.68 42.22 24.87 42.11 25.05 4S 60 s .a 43.30 25.00 43.19 25.19 43.08 125.38 42.97 25.66 60 B 00 s Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 60 1 )eg. 691 Deg. 59i Deg. 59i Deg. TRAVERSE TABLE. 133 9 r*- P 51 30Deg. 30i Deg. m Deij. 30| Deg. Q B a o a 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 44.17 25.50 44.06 25.69 43.94 25.88 43.83 26.08 52 45.03 26.00 44.92 26.20 44.80 26.39 44.69 26.69' 52 53 ' 45.90 26 50 45.78 26.70 45.67 26.90 45.56 27.10 5,3 54 46.77 27 00 46.65 27.20 46.53 27.41 46.41 27.61 64 65 47.63 27.50 47.51 27.71 47.39 27.91 47.27 28.12 65 56 48.60 28.00 48.37 28.21 48.26 28.42 48.13 28.63 56 57 49.36 28.50 49.24 28.7:> 49.11 28 98 48.99 '29.14 67 68 50 23 29.00 50.10 29. 2i 149.97 29.44 49.85 29.65 58 69 51.10 29.50 50.97 29.72 150.84 29.94 50.70 30.17 59 60 61 51.96 52.83 3U.00 51.83 30.23 61.70 30.45 51.56 30.68 60 61" 30.50 52.69 30.73 52.. 56 30.96 52.42 31.19 62 53.69 31.00 53.56 31.23 53.42 31.47 63.28 31.70 62 63 54.56 31.50 54.42 31.74 54.28 31.97 64.14 32.21 63 64 55.43 32.00 55.29 .32.24 55.14 32.48 55.00 32.72 04 65 56.29 32.50 56.15 32.75 56.01 32.99 55.86 33.23 65 66 57.16 33.00 57.01 33.26 .56.87 33.50 .56.72 33.75 66 67 58.02 33.50 57.88 33.75 t>7.73 34.01 57.58 34.26 67 68 58.89 34.00 68.74 34.26 58.59 34.61 58.44 34.77 68 69 59.76 34.60 59.60 34.76 59.45 35.02 69.30 35.28 69 70 71 60.62 35.00 60.47 35.26 35.77 60.31 35.53 60.16 61.02 35.79 36.30 70 71 61.49 35.. 50 61.33 61.18 36.04 72 62.35 36.00 62.20 36.27 62.04 36.64 61.88 36.81 72 73 63.22 36.50 63.06 36.78 62.90 37.05 62.74 37.32 73 74 64.09 37.00 63.92 37.28 63.76 37.56 63.60 37.84 74 75 64.95 37.50 64.79 37.78 64.62 38.07 64.46 38.35 75 76 65.82 38.00 65.66 38.29 65.48 38.57 65.31 38.86 76 77 66.68 38.50 66.52 38.79 66.35 39.08 66.17 39.37 77 78 67.55 39.00 67.38 39.29 67.21 39.69 67.03 39.88 78 79 68.42 39.. 50 68.24 39.80 68.07 40.10 67.89 40.39 79 80 81 69.28 40.00 69.11 69.97 40.30 68.93 40.60 68.76 40.90 80 70.15 40.50 40.81 69.79 41.11 69.61 41.41 81 82 71.01 41.00 70.83 41.31 70.65 41.62 70.47 41.93 82 83 71.88 41.. 50 71.70 41.81 71.52 42.13 71.33 42.44 83 84 72.75 42.00 72.66 42.. 32 72.38 42.63 72.19 42.96 84 85 73.61 42.50 73.43 42.82 73.24 43.14 73.06 43.46 86 86 74.48 43.00 74.29 43.32 74.10 43.66 73.91 43.97 86 87 75.34 43.50 76.15 43.83 74.96 44.16 74.77 44.48 87 88 76.21 44.00 76.02 44.33 76.82 44.66 76.63 44.99 88 89 77.08 44.50 76.88 44.84 76.68 46.17 76.49 45.61 89 90 91 77.94 45.00 77.76 45.34 77.66 46.68 77.35 46.02 90 91 78.81 45.50 78.61 45.84 78.41 46.19 78.21 46.63 92 79.67 46.00 79.47 46.35 79.27 46.69 79.07 47.04 92 93 80.64 46.50 80.34 46.85 80.13 47.20 79.93 47.55 93 94 81.41 47.00 81.20 47.35 '80.99 47.71 80.78 48.06 94 95 82.27 47.50 82.06 47.86 81.86 48.22 81.64 48.57 95 96 83.14 48.00 82.93 48.36 82.72 48.72 82.50 49.08 96 9? 84.00 48.50 83.79 48.87 83.58 49.23 83.36 49.60 97 98 84.87 49.00 84.66 49.37 84.44 49.74 84.22 60.11 98 95 85.74 49.5fl 85.62 49.87 86.30 50.25 86.08 60.62 99 too § c a «-» 86.60 Dep. 50.00 86.38 60.38 86.16 60.76 86.94 51.13 100 O c s ■n c Lat. Dep. Lat. Dep. Lat. Dep. Lat. 60 Dog. 591 Deg. 59i Dog. 59i Deg. 134 TEAVER5E! TABLE >^ 31 ] D«g- a ■^ Lv., -'^1- i 5D u..5i 8 in 1.03 3 2.57 1 . 55 4 3.43 2 . L' 5 5 i.-2j '.; . ? 6 5 . 14 3.:^ 7 6 , 'JL' 3 . -:■ . :> ~ -^ - 4 _ , U 4,0^ lu * .0 , . .0 11 - 43 5 . oT 12 ■ ^. V 13 ... 14 '"* 7 ' 3i\ D«^. 3U I>«g. 3U Dej. 14 15 16 12.00 12. S6 13.71 7.7c 8.24 26 27 28 29 30 31 32 33 34 35 36 37 38 39 41' 42 43 44 45 46 IS 49 50 ■^ 1 57 12 36 ^ ]_ 43 12 «5 22 29 13 C ^ 23 14 13 ,_- '_ 24 00 14 -r'J 24 86 14 y-t 25 71 15 45 26 57 15 97 ■^ i 43 ' P -i *^ • »* :I 29.14 30 . 00 30.86 31.72 32.57 33.43 ^4.29 35.14 36.00 35.86 37.72 33 . 57 39.43 40.29 41.14 42.00 42.86 i I 18.03 18.. 54 19.06 19.57 20.09 20 . 60 21.12 2l!63 22.15 22 . P '^ 23! 15 23.-:^ 24 . ■; : 24.72 25 . 24 25.75 La: Dep. Lat, Dep. Lat. Dep. U,8c 1.71 2.56 3.42 13 0.52 1.04 i.55 2 .OS 2.59 3.11 ■^ . - 3 85 0.52 1 71 1.04 r> - ~, 1.57 -'_ 2.09 _ ■^ J 2.61 5 ■ .-> 3.13 5 97 3.66 •3 82 4.18 "" 67 4.70 53 13, 53 30 17.10 10. 3S 17.95 1 -^ . SI - r . 6 6 20.52 21.37 ■:: 23 25 . 65 10.89 11.41 1 11.93!, 12.45 12.97 ! 13.49 ' 14.01 14.53 ' 15.04 15.56 29 92 30 1 5 31 63 32 49 33 34 34 20 26 . 50 1 6 . 08 18.16 18.68 19.19 19.71 20 . 23 20.75 21.27 21.79 22.31 22.83 23 . 34 23.86 24.38 24.90 25.42 25.94 35 05 35 91 36 / b 3- 62 :;S ^~ Sr 33 41.89 i2.75 ^.35 : .23 .1.08 11.94 12.79 13.64 14.49 1 S 35 : : . 20 13.76 19.61 20.46 21.32 22.17 23.02 23.87 24.73 25.58 26.43 27. 2-8 ,-.14 2S.99 29.84 30 . 70 31.55 32.40 33 . 25 34.11 34.96 35. SI 36.66 37 . 52 3S.37 39.22 40.07 4J.93 5 . 22 5.75 6.27 6.791 7.31 7-84 8.36 8.88 1 9.401 9.93 I 10.45 10,97 11.49 12.02 12.54 13.06 13.5.3 14.11 14.63 1 15.15 15.67 16.20 16.72 17.24 17.76, 13.29 1 13.81 1 19.33 19.85 20 . 38 20 . 90 21.42 21.94 22.47 22 . 99 23.51 24.03 24.56 25.03 25.60 26.12 o Dep. I Lat. Dep. Lat. Dep. Lat. 0.S5 1.70 2 . 55 3.40 4.25 5. 10 5.95 6.80 7.65 3.50 9.. 35 10.20 11.05 11.90 12.76 13.61 14.46 15.31 16.16 17.01 17.86 13.71 19.56 20.41 21.26 22 . 1 1 22.96 23.81 24.66 25.51 26.36 27.21 28.06 23.91 29.76 30.61 31.46 32.31 33.16 34.01 34.86 35.71 .?6 . 57 37.42 33 . 27 39.12 39.97 40.82 41.67 42.52 Deo. 0.53 1 .05 1 . 53 2.10 2.63 3.16 3.68 4.21 4.74 5.26 5. 79 6.31 6.84 7.37 7.89 •i.42 ^.95 9.47 10.00 10.52 11.05 11.58 12.10 12 . 63 13.16 13.68 14.21 14.73 15.26 15.79 16.31 16.84 17.37 17.89 13.42 13.94 19.47 20.00 20.52 21 05 21.57 22.10 22 . 63 23.15 23.63 24,21 24.73 25.26 25.78 26.31 Lat. 3 p "T 2 3 4 5 6 7 S 9 10 11 12 13 14 15 16 17 18 19 20 23 24 25 29 _30 31 32 33 34 35 36 37 33 39 40 '41 42 43 44 4.5 46 47 iS 49 50 e o c 59 Dey 58| Dee. 58i Deg. 38t nf<7. TRAVERSE TABL£ 135 9. OB a a 1 31 Deg. 3U Deg. 31^ Deg. 31} Deg. O 5' a ? "51 Lat. Dop. Lat. Dep. Lat. Dep. Lat. Dep. 43.72 26.27 43.60 26.46 43.48 26.65 43.37 26.84 62 44.57 26.78 44.46 26.98 44.34 27.17 44.32 27.36 62 63 45.43 27.30 45.31 27.49 45.19 27.69 45.07 27.89 5fl 64 46.29 27.81 46.17 28.01 46.04 28.21 45.92 28.42 64 55 47.14 28.33 47.02 28.53 46.90 28.74 46,77 28.94 56 56 48.00 28.84 47.88 29.05 47.75 29.26 47.62 29.47 66 57 48.86 29.36 48.73 29.57 48.60 29.78 48.47 29.99 67 58 49.72 29.87 49 58 30.09 49.45 30.30 49.32 30.52 68 59 50.57 30.39 50.44 30.61 50.31 30.83 50.17 31.05 69 60 61 51.43 30.90 51.29 31.13 51.16 31.35 51.02 31.57 60 61 52.29 31.42 52.15 31.65 52.01 31.87 51.87 32.10 62 53.14 31.93 53.00 32.16 52.86 32.39 52.72 32.63 62 63 54.00 32.45 53.86 32.68 53.72 32.92 53.57 33.16 63 64 54.86 32.96 54.71 33.20 54.57 33.44 54.42 33.68 64 65 55.72 33.48 55.57 33.72 55.42 33.96 55.27 34.20 66 66 56.57 33.99 56.42 34.24 56.27 34.48 56.12 34.73 66 67 57.43 34.51 57.28 34.76 57.13 35.01 56.98 35.26 67 68 58.29 35.02 58.13 35.28 57.98 35.53 57.82 35.78 68 69 59.14 35.54 58.99 35.80 58.83 36.05 58.67 36.31 69 70 71 60.00 36.05 59.84 36.31 59.68 36.57 59.52 36.83 70 71 60.86 36.57 60.70 36.83 60.54 37.10 60.37 37.36 72 61.72 37.08 61.55 37.35 61.39 37.62 61.23 37.89 72 73 62.57 37.60 62.41 37.87 62.24 38.14 62.08 38.41 73 74 63.43 38.11 63.26 38.39 63.10 38.66 62.93 38.94 74 75 64.29 38.63 64.12 38.91 63.95 39.19 63.78 39.47 75 76 65.14 39.14 64.97 39.43 64.80 39.71 64.63 39.99 76 77 66.00 39.66 65.83 39.95 65.65 40.23 65.48 40.52 77 78 66.86 40.17 66.68 40.46 66.51 40.75 66.33 41.04 78 79 67.72 40.69 67.54 40.98 67.36 41.28 67.18 41.57 79 80 81 68.57 41.20 68.39 41.50 68.21 41.80 68.03 42.10 80 81 69.43 41.72 69.25 42.02 69.06 42.32 68.88 42.62 82 70.29 42.23 70.10 42.54 69.92 42.84 69.73 43.15 82 83 71.14 42.75 70.96 43.06 70.77 43.37 70.58 43.68 83 84 72.00 43.26 71.81 43.58 71.62 43.89 71.43 44.20 84 85 72.86 43.78 72.67 44.10 72.47 44.41 72.28 44.73 85 86 73.72 44.29 73.52 44.61 73.33 44.93 73.13 45.25 86 87 74.57 44.81 74.38 45.13 74.18 45.46 73.98 46.78 87 88 75.43 45.32 75.23 45.65 75.03 45.98 74.83 46.31 88 89 76.29 45.84 76.09 46.17 75.88 46.50 75.68 46.83 89 90 91 77.15 78.00 46.35 46.87 76.94 46.69 76.74 47.02 76.53 47.36 90 91 77.80 47.21 77.59 47.55 77.36 47.89 92 78.86 47.38 78.65 47.73 78.44 48.07 78.23 48.41 92 93 79.72 47.90 79.51 48.25 79.30 48.59 79.08 48.94 93 94 80.57 48.41 80.36 48.76 80.15 49.11 79 93 49.47 94 95 81.43 48.93 81.22 49.28 81.00 49.64 80 78 49. 9S 95 90 82.29 49.44 82.07 49.80 81.85 50.16 81.63 50.52 96 97 83.15 49.96 82.93 30.32 82.71 50 .68 82.48 5:. 04 97 98 84.00 50.47 83.78 50.94 83.56 51.20 83.33 51.57 98 99 84.86 50.99 84.64 51.36 84.41 51.73 84.18 52.10 99 100 Q 85.72 51.50 85.49 51.88 85.26 52.25 85.04 62.62 100 g 01 °1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 59 I )eg. 581 Deg. 58i Deg. 58i Deg. 136 TRAVERSE TABLE, B S 32 Deg. 32i Deg. 3^ Deg. 321 Deg. S' p 3 r> a 1 Lat. Dep, Lat. Dep. Lat. Dep. Lat. Dep. 1 " 0.85 0.63 0.85 0.53 0.84 0.54 0.84 0.64 2 1.70 1.06 1.69 1,07 1.69 1,07 1.68 1.08 2 3 2.54 1.69 2.54 1.60 2.53 1.61 2.52 1.62 3 4 3.39 2.12 3.38 2.13 3.37 2.15 3.36 2.16 4 5 4.24 2.65 4.23 2.67 4.22 2.69 4.21 2.70 6 6 5.09 3.18 6.07 3.20 5.06 3.22 5,05 3.25 6 7 6.94 3.71 6.92 3.74 6.90 3.76 5.89 3.79 7 8 6,78 4.24 6,77 •4.27 6.75 4.30 6.73 4.33 8 9 7.63 4.77 7,61 4.80 7.59 4.84 7.67 4.87 9 10 8.48 6.30 8.46 6.34 8.43 5.37 8.41 5.41 10 11 9.33 6.83 9.30 5.87 9.28 5.91 9.25 6.95 11 12 10.18 6.36 10.15 6,40 10.12 6.45 10,09 6.49 12 13 11.02 6.89 10.99 6.94 10.96 6.98 10.93 7.03 13 14 11.87 7.42 11.84 7.47 11.81 7.52 11.77 7.57 14 15 12.72 7.96 12.69 8.00 12.65 8.06 12.62 8.11 15 16 13.57 8.48 13.63 8.54 13.49 8.60 13.46 8.66 16 17 14.42 9.01 14.38 9,07 14.34 9.13 14.30 9.20 17 18 16.26 9.64 15,22 9,61 16,18 9.67 15.14 9.74 18 19 16.11 10.07 16.07 10.14 16.02 10.21 15,93 10,28 19 20 16.96 10.60 16.91 10.67 1.6.87 10.75 16,82 10.82 20 21 21 17.81 11.13 17.76 11.21 17,71 11.28 17.66 11.36 22 18. 6e 11.66 18.61 11.74 19.55 11.82 18.50 11.90 22 23 19.61 12.19 19.45 12.27 19.40 12,36 19.34 12.44 23 24 20.35 12.72J 20.30 12.81 20.24 12.90 20.18 12.98 24 25 21.20 13.25 21.14 13.34 21.08 13.43 21.03 13.52 25 26 22.05 13.78 21.99 13.87 21.93 13.97 1 21.87 14,07 26 27 22.90 14.31 22.83 14.41 22.77 14.51 22.71 14.61 27 28 23.75 14.84 23.68 14.94 23.61 15.04 23.55 15.15 28 29 24.59 16.37 24.. 53 16.47 24.46 15.58 24.. 39 15.69 29 30 25.44 15.90 26.37 16.01 25.30 16.12 25.23 16.23 30 31 26.29 16.43 26.22 16.54 26.15 16.66 26.07 16,77 31 32 27.14 16.96 27.06 17.08 26.99 17.19 26.91 17.31 32 33 27.99 17.49 27.91 17.61 27.83 17.73 27.75 17.85 33 34 28.83 18.02 28.75 18.14 28.63 18.27 2a. 60 18.39 34 35 29.68 18.55 29.60 18.68 29.52 18.81 29.44 18.93 35 36 30.53 19.08 30.45 19.21 30.36 19.34 30.28 19.48 36 37 31.38 19.61 31.29 19.74 31.21 19.88 31.12 20,02 37 38 32.23 20.14 32.14 20.28 32.05 20.42 31.96 20.56 38 39 33.07 20.67 32.98 20.81 32.89 20.95 32.80 21.10 39 40 33.92 21.20 33.83 21.34 33.74 21.49 33.64 21.64 40 41 34.77 21.73 34.67 21.88 34.58 22.03 34.48 22.18 41 42 35.62 22.26 35.52 22.41 35.42 22.57 35.32 22.72 42 43 36.47 22.79 36.37 22.95 36.27 23.10 36.16 23.26 43 44 37.31 23.32 37.21 23.48 37.11 23.64 37.01 23.80 44 45 38.16 23.86 38.06 24.01 37.96 24.18 37,85 2^.34 45 46 39.01 24.38 3S.90 24.56 38.80 24.72 38.69 24.88 46 47 39.86 24.91 39.75 25.08 39 ..64 25.25 39.53 25.43 47 AS 40.71 25.44 40.59 25.61 40.48 26.79 40.37 25.97 48 49 41.55 26.97 41.44 26.15 41.33 26.33 41.21 26.51 49 50 42.40 26.50 42.29 26.68 42.17 26.86 42.05 27.05 60 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat "i Q 58 Deg. 571 Deg-. 57^ Deg. 57J Deg. Q TRAVESSE TABLE. 137 a 5' «-► p ? 51 32Deg. 32i Deg. 32i Deg. 321 Deg. s 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 43.25 27«03 43.13 27,21 43,01 27.40 42.89 27.59 52 44.10 27.56 43.98 27.75 43.86 27.94 43.73 28.13 62 53 44.95 28.09 44.82 28,28 44.70 28.48 44.58 28,67 63 54 45.79 28.62 45.67 28.82 45,54 29.01 45.42 29.21 64 55 46.64 29.15 46.51 29.35 46.39 29.55 46.26 29.76 56 56 47.49 29.68 47.36 29.88 47.23 30.09 47,10 30.29 66 57 48.34 30.21 48.21 30.42 48.07 30.63 47.94: 30.84 67 68 49.19 30.74 49.05 30.95 48.92 31.16 48.78 31.38 58 59 50.03 31.27 49.90 31.48 49.76 31.70 49.62 31.92 99 60 61 50,88 31.80 50.74 32.02 50.60 32.24 50.40 32.46 60 61 51.73 32.33 51.59 32.55 51.45 32.78 51.30 33.00 62 52.58 32.85 52.44 33.08 52.29 33.31 52.14 33.54 62 63 53.43 33.38 53,28 33.62 53.13 33.85t 62.99 34,08 63 64 54.28 33.91 54.13 34.15 53.98 34.39 53.83 34.62 64 65 55.12 34.44 54.97 34.68 54.82 34.92 54.67 35.16 65 66 55.97 34.97 55.82 35.22 55.66 35.40 55.51 35.70 66 67 56:82 35.50 56.66 35.75 56.51 36.00 56.35 36,25 67 68 57.67 36.03 57.51 36.29 57.35 36.64 67.19 36.79 68 69 58.52 36.56 58.36 36.82 58,19 37.07 ,58.03 37.33 69 70 71 59.36 37.09 59.20 37,35 59.04 37.61 58,87 57.87 70 71 60.21 37,62 60,05 37.89 59.88 38.15 59,71 38.41 72 61.06 38.15 60.89 38.42 60.72 38.69 60,65 38.95 72 73 61.91 38.68 61.74 38.95 61.57 59.22 61.40 39. 4S 73 n 62.76 39.21 62.58 39.49 62.41 39.76 62.24 40.03 74 75 63.60 39.74 63.43 40.02 63.25 40.30 63.08 40.57 76 76 64.45 40.27 64.28 40.55 64.10 40,63 63.92 41.U 76 77 65.30 40.80 65.12 41.09 64.94 41.37 64.76 41.65 77 78 66.15 41.33 65.97 41.62 65.78 41.91 €5.00 42.20 78 79 67.00 41.86 66.81 42.16- 66.63 42.45 66.44 42.74 79 80 81 67.84 42.39 67.66 42.69 67.47 42.98 67.28 43.28 80 81 68.69 42.92 68.50 43.22 68.31 43.52 68.12 43.82 82 69.54 43.45 69.35 43.76 69.16 44.06 68.97 44.36 82 83 7D.39 43.98 70.20 44.29 70.00 44.60 69.81 44.90 83 84 71i24 44.51 71.04 44.82 70.84 45.13 70.65 45.44 84 85 72.08 45.04 71.89 45.36 71.69 45.67 71.49 45.98 85 86 72.93 45.57 72.73 45.89 72.53 46.21 72.33 46.52 86 87 73.78 46. LO 73.58 46.42 73.38 46.75 73.17 47.06 87 88 74.63 45.63 74.42 46.96 74.22 47.28 74.01 47.61 88 89 75.48 47.16 75.27 47.49 75.06 47.82 74.85 48.15 89 :90 91 76.32 47.69 76.12 48.03 75.91 48.36 48.89 75.69 48.69 90 91 77.17 48.22 76.96 48.56 76.75 76.53 49.23 92 78.02 48.75 77.81 49.09 77.59 49.43 77.38 49 77 92 93 78.81' 49.28 78.65 49.63 78.44 49.97 78.22 50.31 93 94 79,72 49.81 79.50 50.16 79.28 50.51 79.06 50.85 94 95 80.56 50.34 80. B4 .50.69 80.12 51.04 79.90 51.39 95 96 81.41 50.87 81.19 51.23 80.97 51.58 80.74 51.93 96 97 82.26 51.40 8^.04 51.76 81.81 52.12 81.58 .52.47 97 98 83.11 51.93 82.88 52.29 82.65 52.66 82.42 53.02 98 99 83.96 .52.46 83.73 52.^3 83.50 63.19 83.26 53.56 99 100 o .a Q 84.80 52.99 84*57 53.3& 84.34 53.73 84.10 54.10 100 © o c ei Dep. Lai. Dep. Lat. Dep. Lat. Dep. Lat. 68 Deg. .671 Deg. 57i Deg. 57i Deg. 138 TRAVERSE TABLE. pi 3 a 9 ~1 33 Deg. 33i Deg. 33i Deg 331 Deg. o X ? ■3 1 Lat. _ 1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. O.SC 0.84 0.54; 0.84 0.55 0.83 0.55 b.83 2 1.68 1.09 1.67 1.10 1.67 1.10 1.66 1.11 2 3 2.52 1.63 2.51 1.64 2.. 50 1.66 2.49 1 67 3 4 3.35 2.18' 3.35 2.19 3.34 2 21 3.33 2.22 4 5 4 19 2 72 4.18 2.74 4.17 2.76 4.16 2.78 5 fl 5 03 3 27 5.02 3.29 5.00 3.31 4.99 3.33 6 7 . 5.87 3 81 ' 5.85 3.84 5.84 3.86 5.82 3.89 7 8: 6.71 4.36 6.69 4.39 , 6.67 4.42 6.65 4.44 8 9 7.55 4.90 7.53 4.93 7.50 4.97 7.48 5.00 f) 10 11 8.39 9.23 5.45 8.36 5.48 6.03 8.34 9.17 5.52 6.07 8.31 5.56 10 11 5.99 9.20 9.15 6.11 12 10.06 6.54 10.04 6.58 10.01 6.62 9.98; 6.67 12 13 10.90 7.08 10.87 7.13 10.84 7.18 :! 10.81 7.22 13 14 11.74 7.62 11.71 7.68 ' 11.67 7.73!: 11.64 7.78 14 15 12.58 8.17; 12.54 8.22 ■ 12.51 8.28 :j 12.47 8.33 15 16 13.42 8.71 ' 13.38 8.77 1 13.34 8.83, 13.30 8.89 16 17 14.26 9.26 14.22 9.32 14.18 9.38 ,: 14.13 9.44 17 18 15.10 9.80 15.05 9.87 , 15.01 9.93': 14.97 10.00 18 19 15.93 10.35 15.89 10.42 15.84 10.49 j 15.80 10.56 19 20 16.77 10.89) 11.44 16.73 10.97 11.51 1 16.68 11.04 16.63 11.11 20 21 21 17.61 17.56 17.51 11.59 17.46 11.67 22 18.45 11.98 18.40 12.06 1 18.35 12.14 11 18.29 12.22 22 23 19.29 12.53 19.23 12.61 ! 19.18 12.69 ' 19.12 12.78 23 24 20.13 13.07; 20.07 13.16 20.01 13.25 : 19.96 13.33 24 25 20. P7 13.62 20.91 13.71 20.85 13.80 20.79 13.89 25 26 21.81 14.16 21.74 14.26 21.68 14.35 1 21.62 ; 14.44 26 27 22.64 14.71 22.58 14.80 22.51 14.90 22.45 15.00 27 28 23.48 15.25 23.42 15.35 23.35 15.45 23.28 15.56 28 29 24.32 15.79 24.25 15.90 24.18 16.01 ! 24.11 16.11 29 30 25.16 16.34 25 . 09 16.45 25.02 25 . 85 16.56 24.94 16.67 30 "31 31 26.00 16.88 25.92 17.00 17.11 25.78 17.22 32 26.84 17.43 26.76 17.55 26.68 17.66 26.61 17.78 32 33 27.68 17.97 27.60 18.09 27.52 18.21 27.44 18.33 33 34 28.51 18.52 28.43 18.64 28.35 18.77 28.27 18.89 34 3.'i 29.35 19.06 29.27 19.19 29.19 19.32 29.10 ' 19.44 35 36 30.19 19.61 30.11 19.74 I 30.02 19.87 29.93 i 20.00 36 37 31.03 20.15 30.94 20.29 ! 30.85 20.42 1 30.76 20.56 37 38 31.87 20.70 31.78 20.84 31.69 20.97 : 31.60 21.11 38 39 32 71 21.24 32.62 21.38 32.52 21.53 32.43 21.67 39 40 41 33.55 21.79 33.45 34.29 21.93 22.48 33.36 34.19 22.08 33.26 22.22 40 41 34.39 22.33 22.63 34.09 22.78 42 35.22 22.87 35.12 23.03 35.02 23.18 : 34.92 i 23.33 *2 43 1 36.06 23.42 35.96 23.58 35. S6 23.73 ; 35.75 23.89 43 44 136.90 23.96 36.80 24.12 36.69 24.29 , 36.58 24.45 44 45 37.74 24.51 37.63 24.67 37.52 24.84 37.42 25 00 45 46 38.58 25.05 38.47 25.22 38.36 25.39 38.25 25.56 46 47 39.42 25.60 39.31 25.77 39.19 25 . 94 39.08 126.11 47 iS 40.26 26.14 40.14 26.32 40.03 26.49 39.91 1 26.67 48 49 41.09 26.69 40.98 26.87 40.86 27.04 40.74 27.22 1 49 50 41.93 27.23 41.81 27.41 41.69 Dep. 27.60 41.57 127.78 ! 50 ! 6 i g OD 5 i s .a Dep. Lat. Dep. Lat Lat. Dep. Lat. 1 57 Deg. 56| Deg. 56i Deg. 56J Deg. TRAVERSE TABLE. 139 o a a § 61 33 Dog. 33i Deg. 33^ Deg. 33i Deg. a 5' <-► g CD 61 Lat. Dep. Lat. Dep. Lat. Dep. 28.15 Lat. Dep. 42.77 27.78 42.65 27.96 42.53 42.40 28.33 52 43.61 28.32 43.49 28.51 43.36 28.70 43.24 28.89 52 53 44.45 28.87 44.32 29.06 44.20 29.25 44.07 29.45 53 64 45.29 29.41 46.16 29.61 45.03 29.80 44.90 30.00 64 55 46.13 29.96 46.00 30.16 45.86 30.36 46.73 30.56 56 56 46.97 30.50 46.83 30.70 46.70 30.91 46.56 31.11 66 57 47.80 31.04 47.07 31.26 47.63 31.46 47.39 31.67 67 1 68 48.64 31.59 48.50 31.80 48.37 32.01 48.23 32.22 6ft 1 59 49.48 32.13 49.34 32.35 49.20 32.56 49.06 32 78 ( 59 60 61 50.32 32.68 50.18 32.90 50.03 50.87 33.12 1 33.67 49.89 33.33 60 61 51.16 33.22 51.01 33.45 50.72 33.89 62 52.00 33.77 61.85 33.99 61.70 34.22 51.55 34.46 62 63 52.84 34.31 52.69 34.54 52.53 34.77 62.38 35.00 63 64 53.67 .34.86 53.52 35.09 53.37 36.32 63.21 35.56 64 65 54.51 135.40 54.36 35.64 54.20 35.88 54.05 36.11 65 66 55.351 35.95 55.19 36.19 56.04 36.43 54.88 36.67 es 67 66.19 36.49 56.03 36.74 56.87 36.98 55.71 37.22 67 68 57.03 37.04 66.87 37.28 56.70 37.53 56.64 37.78 68 69 57.87 37.68 57.70 37.83 57.54 38.08 57.37 38.33' 69 70 71 68.71 38.12 68.64 38.38 68.37 38.64 58.20 38.89 39.45 70 71 59.55 38.67 69.38 38.93 59.21 39.19 59.03 72 60.38 39.21 60.21 39.48 60.04 39.74 59.87 40.00 72 73 61.22 39.76 61.05 40.03 60.87 40.29 60.70 40.66 73 74 62.06 40.30 61.89 40.67 61.71 40.84 61.53 41.11 74 75 62.90 40.85 62.72 41.12 62 . .54 41.40 62.. 36 41.67 76 76 63.74 41.39 63.56 41.67 63.38 41.95 63.19 42.22 76 77 64.58 41.94 64.39 42.22 64.21 42.50 64.02 42.78 77 78 65.42 42.48 65.23 42.77 66.04 43.05 64.85 43.33 78 79 66.25 43.03 66.07 43.32 65.88 43.60 65.69 43.89 79 80 81 67.09 43.57 66.90 67.74 43.86 44.41 66.71 44.16 66.52 44.45 80 81 67.93 44.12 67.54 44.71 67.35 45.00 82 68.77 44.66 68.58 44.96 68.38 45.26 68.18 45.56 82 83 69.61 46.20 69.41 45.51 69.21 45.81 69.01 46.11 83 84 70.46 45 . 76 70.25 46.06 70.05 46.36 69.84 46.67 84 85 71.29 46.29 71.08 46.60 70.88 46.91 70.67 47.22 86 86 72.13 46.84 71.92 47.15 71.71 47.47 71.51 47.78 86 87 72.96 47.38 72.76 47.70 72.55 48.02 72.34 48.33 87 88 73.80 47.93 73.69 48.25 73.38 ' 48.57 73.17 48.89 88 89 74.64 48.47 74.43 48.80 74.22 49.12 74.00 49.45 89 90 91 75.48 49.02 49.56 76.27 76.10 49.35 75.05 75.88 49.67 50.23 74.83 50.00 90 91 76.32 49.89 75.66 60.56 93 77.16 .50.11 76.94 .50.44 76.72 50.78 76.60 51.11 92 J 93 78.00 50.66 77,77 ! 50.99 77.55 51.33 77.33 51.67 931 94 78 83 i 51.20 78.61 61.. 54 78.39 51.88 78.16 52.22 94 95 79.67 151.74 79.46 52.09 79.22 52 43 78.99 62.78 95 96 80.61 52.29 80.28 52.64 80.05 52.99 79.82 63.33 96 97 81.36 52.83 81.12 .53.18 80.89 53.54 180.65 63.89 97 98 82.19 i 53.37 81.96 53.73 81.72 54.09 81.48 64.45 981 09 83.03 1 63.92 82.79 54.28 82.65 54.64 82.32 65.00 99 1 100 i 83.87 154.46 83.63 54.83 83.39 66.19 83.16 Dep. 65.66 _00 a Dep. Lat. Dep. Lat. Dep. Lat. Lat. 57 Deg. 56^ Deg. 56^ Deg. 56i Dog. 140 TBAVERSE TABLS* 5 34Deg. Mi Deg. 34^ Deg. 34| Deg. O P 3 ? 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.83 0.56 0.83 0.56 0.82 0.57 0.82 0.57 2 1.66 1.12 1.65 1.13 1.65 1.13 1.64 1.14 2 8 2.49 1.68 2.48 1.69 2.47 1.70 2.46 1.71 3 4 3.32 2.24 3.31 2.25 3.30 2.27 3.29 2.28 4 5 4.15 2.80 4.13 2.81 4.12 2.83 4.11 2.85 5 6 4.97 3.36 4.96 3.38 4.94 3.40 4.93 3.42 6 7 5.80 3.91 5.79 3.94 5.77 3.96 5.75 3.99 7 8 6.63 4-47 6.61 4.50 6.59 4.-53 6.57 4.56 8 9 7.46 5.03 7.44 5.07 7.42 5.10 7.39 5.13 9 10 11 8.29 5.59 8.27 5.63 8.24 5.66 8.22 5.70 10 9.12 6.15 9.09 6.19 9.07 6.23 9.04 6.27 11 12 9.95 6.71 9.92 6.75 9.89 6.80 9.86 6.84 12 13 10.78 7.-27 10.75 7.32 10.71 7.36 10.68 7.41 13 14 11.61 7.83 11.57 7.88 11.64 7.93 11.50 7.93 14 15 12.44 8.39 12.40 8.44 12.36 8.50 12.32 8.55 15 16 13.26 8.95 13.23 9.00 13.19 9,06 13,15 9.12 16 17 14.09 9.51 14.05 9.57 14.01 9.63 13.97 9.69 17 18 14.92 10.07 14.88 10.13 14.83 10.20 14.79 10.26 18 19 15.75 10.62 15.71 10.69 15.66 10.76 15.61 10.83 19 20 21 16,58 11.18 16.53 11.26 16.48 11.33 16.43 11.40 20 17.41 11.74 17.36 11.82 17.31 11.89 17.25 11.97 21 22 J8.24 12.30 18.18 12.38 18.13 12.46 18.08 12.54 22 23 19.07 12.86 19.01 12.94 18.95 13.03 18.90 13. IL 23 24 19.90 13.42 19.84 13.51 19.78 13.59 19.72 13.68 24 25 20.73 13.98 20.66 14.07 20.60 14.16 20.54 14.25 25 26 21.55 14.54 21.49 14.63 21.43 14.73 21.36 14.82 26 27 22.38 15.10 22.32 15.20 22.25 15.29 22.18 15.39 27 28 23.2] 15 66 23.14 15.76 23.08 15.86 23.01 15.96 28 29 24.04 16.22 23.97 16.32 23.90 16.43 23.83 16.53 29 30 24.87 16.78 24.80 16.88 24.72 16.99 24.63 17.10 30 31 31 25.70 17.33 25.62 17.45 25.55 17.56 25.47 17.67 32 26.53 17.89 26.45 18.01 26.37 18.12 26.29 18.24 32 33 27.36 18.45 27.28 18.57 27.20 18.69 27.11 18.81 33 34 28.19 19.01 28.10 19.14 28.02 19.26 27.94 19.38 34 35 29.02 19.. 57 28.93 19.70 28.84 19.82 28.76 19.95 35 36 29.85 20.13 29.76 20.26 29.67 20.39 29.58 20.52 36 37 30.67 20.69 30.58 20.82 30.49 20.96 30.40 21.09 37 38 31.50 21.25 31.41 21.39 31.32 21.52 31.22 21.66 38 39 32.33 21.81 32.24 21.95 32.14 22.09 32.04 22.23 39 40 41 33.16 22.. 37 33.06 33.89 22.51 .32.97 22.06 32.87 ^2.80 40 33.99 22.93 23.07 33.79 23.22 33.69 23.37 41 42 34.82 -23.49 34.72 23.64 34.61 23.79 34.51 23.94 42 43 35.65 24.05 35.54 24.20 35.44 24.36 35.33 24.51 43 44 36.48 24.60 36.37 24.76 36.26 24.92 36.15 25.08 44 45 37.31 25.16 37.20 25.33 37.09 25.49 36.97 25.65 45 46 38.14 25.72 38.02 25.89 37.91 26.05 37.80 20.22 46 47 38.96 26.28 38.85 26.45 38.73 26.62 38.62 26.79 47 48 39.79 26.84 39.68 27.01 39.56 27.19 39.44 27.36 48 49 40.62 27.40 40.50 ^7.58 40.38 27.75 40.26 27.93 49 50 g 41.45 27.96 41.33 28.14 41.21 28.32 41.08 28.50 60 Dep. Lat, Dep. Lat. Dep. Lat. Dep. Lat. d o c 5 56 Deg^. 55| Deg. 55|1 Deg. 55i Deg. TRAVEKSE TABUS. 141 61 34 Deg. 34i Deg. 34A Deg. 341 Dig. D O a 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 42.28 28.52 42.16 28.70 42.03 28.89 41.90 29.07 52 43.11 29.08 42.98 29.27 42.85 29.45 42.73 29.64 52 53 43.94 29.64 43.81 29.83 43.68 30.02 43.55 30.21 63 54 44.77 30.20 44.64 30.39 44.50 30.59 44.37 30.78 64 55 45.60 30.76 45.46 30.95 45.33 31.15 45.19 31.35 55 56 46.43 31.31 46.29 31.52 46.15 31.72 46.01 31.92 56 57 47.26 31.87 47.12 32.08 46.98 32.29 46.83 32.49 67 58 48.08 32.43 47.94 32.64 47.80 32.85 47.66 33.06 68 59 48.91 32.99 48.77 33.21 48.62 33.42 48.48 33.63 59 60 61 49.74 35.66 49.60 33.77 49.45 33.98 49.30 34.20 60 61 50.57 34.11 60.42 34.33 50.27 34.55 .50.12 34.77 62 51.40 34.67 51.26 34.89 51.10 35.12 50.94 35.34 62 63 62.23 35.23 52.08 35.46 51.92 35.68 51.76 35.91 63 64 53.00 36.79 52.90 36.02 52.74 36.25 62.59 36.48 64 65 53.89 36.3.5 53.73 36,58 63.67 36.82 53.41 37.05 65 66 54.72 36.91 54.55 37.15 64.39 37,38 54.23 37.62 66 67 55.55 37.46 .65.38 a7.71 65.22 37,95 55.05 38.19 67 68 50,37 38.03 56.21 38.27 56.04 38.52 55.87 38.76 68 69 57.20 38.58 67.03 38.83 56.86 39.08 56.69 39.33 69 70 71 58.03 39.14 67.86 39.40 67.69 39.66 57.52 39.90 70 71 58.86 39.70 58.69 39.96 58.61 40.21 58.34 40.47 72 59.69 40.26 69.51 40.. 52 59.34 40.78 59.16 41.04 72 73 60.52 40.82 60.34 41.08 60.16 41.35 59.98 41.61 73 74 61.35 41.38 61.17 41.65 60.99 41.91 60.80 42.18 74 76 62.18 41.94 61.99 42.21 61.81 42.48 61.62 42.75 75 76 63.01 42.50 62.82 42.77 62.63 43.05 62.45 43.32 76 77 63.84 43.06 63.66 43.34 63.46 43.61 63.27 43.89 77 78 64.66 43.62 64.47 43.90 64.28 44.18 64.09 44.46 78 79 65.49 44.18 05.30 44.46 65.11 44.75 64.91 45.03 79 80 81 66.32 44.74 66.13 45.02 65.93 45.31 65.73 45.60 80 81 67.15 45.29 66.95 46.59 66.75 45.88 66.>55 46.17 82 67,98 '45.86 67.78 46.15 67.58 46.46 67.37 46.74 82 83 68.81 46.41 68.61 46.71 68.40 47.01 68.20 47.31 83 84 69-. 64 46.97 69.43 47.28 69.23 47.58 69.02 47.88 84 85 70.47 47.63 70.26 47.84 70.05 48.14 69.84 48.45 85 86 71.30 48.09 71.09 48.40 70.87 48.71 70.66 49.02 86 87 72.13 48.65 71.91 48.96 71.70 49.28 71.48 49.59 87 88 72.96 49.21 72.74 49.53 72.52 49.84 72.30 50.16 88 89 73.78 49.77 73.57 50,09 73.36 50.41 73.13 50.73 89 90 91 74.61 50.33 74.39 60.65 74.17 60.98 73.95 51.30 90 91 75.44 50.89 75.22 51.22 75.00 61.54 74.77 51.87 92 76.27 51.45 76.05 51.78 75.82 62.11 75.69 52.44 92 93 77.10 52.00 76.87 52.34 76.64 52.68 76.41 53.01 93 94 77.93 52.56 77.70 52.90 77.47 53.24 77.23 53.58 94 95 78.76 63.12 78.53 53.47 78.29 53.81 78.06 54.15 95 96 79.69 53.68 79.36 54.03 79.12 54.37 78.88 54.72 96 97 80.42 54.24 80.18 54.59 79.94 54.94 79.70 55.29 97 98 81.25 54.80 81.01 55.15 80.76 55.51 80.52 .55.86 98 99 82.07 55.36 81.83 55.72 81.59 56.07 81. .34 56.43 99 100 i s CO 82.90 66.92 82.66 56.28 82.41 56.64 82.16 57.00 100 Lat. Dep. Lat. Dep. Lat. Dep.. Lat, Dep. 41.78 29.25 41.65 29.43 41.52 29.62 41.39 29.80 52 42.60 29.83 42.47 30.01 42.33 30.20 42.20 30.38 52 63 43.42 .30.40 43.28 30.59 43.15 30.78 43.01 30.97 53 54 44.23 30.97 44.10 31.17 43.96 31.86 43.82 31.55 54 55 45.05 31.55 44.92 31.74 44.78 31.94 44.64 32.13 55 56 45.87 32.12 45.73 32.32 45.59 32.52 45.45 32 . 72 56 57 46.69 32.69 46.55 32.90 46.40 33.10 46.26 33.. 30 57 58 47.51 .33.27 47.37 33.47 47.22 33.68 47.07 33.89 58 59 48.33 33.84 48.18 34.05 48.03 34.26 47.88 34.47 59 60 61 49.15 34.41 49.00 34.63 48.85 34.84 48.69 35 . 05 60 61 49.97 34.99 49.82 35.21 49.66 35.42 49.51 35 . 64 62 50.79 35 . 56 60.63 35.78 50.48 36.00 50.32 36.22 62 63 51.61 36.14 51.45 36.36 51.29 36.58 61.13 36.81 63 64 52.43 36.71 52.27 36.94 .52.10 37.16 51.94 37.39 64 65 53.24 37.28 53.08 37.51 .52.92 37.75 52.75 37.98 65 66 54.06 37.86 53.90 38.09 53.73 38.33 53.50 38 . 56 60 67 54.88 38.43 54.71 38.67 54.55 38.91 54.38 39.14 67 68 55.70 39.00 55.53 39.55 55.36 39.49 "55.19 39.73 68. 69 56.. 52 39.58 56.35 39.82 56.17 40.07 56.00 40.31 69 70 71 57.. 34 40.15 57. IG 40 40 56.99 40.65 .56.81 40.90 70 71 58.16 40.72 57.98 40.98 57.su 41.23 57.62 41.48 72 58 . 98 41.30 58.80 41.55 58 . 65 41.81 58.43 42.07 72 73 59.80 41.87 59.61 42.13 59.43 42.39 .59.24 42.65 73 74 60.62 42.44 60.43 42.71 60.24 42.97 60.06 43.23 •74 75 61.44 43.02 61.25 43.29 61.06 43.55 60.87 43.82 75 76 62.26 43.59 62.06 43.86 61.87 44.13 61.68 44.40 76 77 63.07 44.17 62.88 44.44 62.69 44.71 62.49 44.99 77 78 63.89 44.74 63.70 45.02 63.50 45.29 63.30 45.57 78 79 64.71 45.31 64.51 45 . 59 04.32 45 . 88 64.11 46.16 79 80 81 05.. 53 66.35 45.89 65.33 46.17 65.13 46.46 64 . 93 65.74 46 . 74 80 81 46.46 66.15 46 . 75 65 . 94 47.04 47.32 82 67.17 47.03 66.90 47.33 66.76 47.62 66.55 47.91 82 83 67.99 47.61 67.78 47.90 67.57 48.20 67.36 48.49 83 84 08.81 48 . 1 8 68.60 48.48 68.39 48.78 68.17 48.08 84 85 C9.63 48.75 69.41 49.06 69.20 49.36 08.98 49.66 85 86 70.45 49.33 70.23 49.63 70.01 49.94 69.80 50.25 86 87 71.27 49.90 71.05 50.21 70.83 50.. 52 70.61 60.83 87 88 72.09 50 47 71 .86 50.79 71.64 51.10 71.42 51.41 88 89 72.90 51.05 72.68 51.37 72.46 51.68 72 . 23 52.00 89 90 91 73 . 72 51.62 73 . 50 51.94 73.27 74.08 .52.26 52.84 73.04 52.58 90 91 74.54 .52.20 74.31 52,. 52 73.85 53.17 92 75.36 52.77 75.13 53.10 74.90 53.42 74.66 53.75 92 93 70.18 .53.34 75.95 53.67 75.71 54.01 75.48 64.34 93 94 77.00 .53.92 76.76 54.25 76.. 53 -54 . 59 76.29 54 . 92 94 95 77.82 .54.49 77.58 64.83 77.34 65.17 77.10 65.50 95 96 78.64 55.06 78.40 .55.41 78.16 55.75 77.91 56.09 96 97 79.46 55.84 79.21 55.98 78 97 56.33 78.72 56.67 97 98 80.28 .50.21 80.03 56.. 56 70.78 50.91 79.53 67.28 98 99 81.10 56.78 80.85 57.14 80.60 57.49 80 .35 57.84 99 J 00 d u c a a 81.92 ,57.36 81.66 57.71 SI. 41 58.07 81.16 58.42 100 t) c a *-> m s Dep. Lat. Dep. Lat. Dep. Lat. Dep. L.it. 55 Deg. 54| Deg. 54^ Deg. 54i Deg.^ 144 TRAVERSE TABLE. 9 S' i 1 tiS Deg. m Deg. 36^ Dog. 36i Deg. 5 St § Lat. Dep. 6.69 Lat. Dep. Lat. Dop. Lat. Dep. 0.81 0.81 0.59 0.80 0.59 0.80 0.60 2 1.62 1.18 1.61 1.18 1.61 1.19 1.60 1 20 8 3 2.43 1.76 2.42 1.77 2.41 1.78 2.40 1.79 3 4 3.24 2.35 3.23 2.37 3.22 2.38 3.20 2.39 4 5 4.05 2.94 4.03 2.96 4.02 2.97 4.01 2.99 r» 6 4.85 3.53 4.84 3.55 4.82 3.57 4.81 3.59 6 7 5.66 4.11 5.65 4.14 5.63 4.16 5.611 4.19 7 8 6 47 4.70 6.45 4.73 6.43 4.76 6.41 4.79 8 9 7.28 5.29 7.26 5.32 7.23 5.35 7.21 5.38 9 10 11 8.09 5.88 8.90 6.47 8.06 5 91 8.04 5.95 8.01 8.81 5.98 10 11 8.87 6.50 8.84 6.54 6.58 13 9.71 7.05 9.68 7.10 9.65 7.14 9.61 7.18 12 13 10.52 7.64 10.48 7.69 10.45 7.73 10.42 7.78 13 14 11.33 8.23 11.29 8.28 11.25 8.33 11.22 8.38 14 15 12.14 8.82 12.10 8.87 12.06 8.92 12.02 8.97 15 16 12.94 9.40 12.90 9.46 12.86 9.52 12.82 9.57 16 17 13.75 9.99 13.71 10.05 13.67 10.11 13.62 10.17 17 18 14.56 10.58 14.52 10.64 14.47 10.71 14.42 10.77 18 19 15.37 11.17 15.32 11.23 15.27 11.30 15.22 11.37 19 20 21 16.18 1 11.76 16.13 11.83 16.08 11.90 12.49 16.03 16.83 11.97 20 21 16.99 12.34 16.94 13.43 16.88 12.56 22 17. SO 12.93 17.74 13.01 17.68 13.09 17.63 13.16 22 23 18.61 13.52 18.55 13.60 18.49 13.68 18.43 13.76 23 24 19.42 14.11 19.35 14.19 19.29 14.28 19.23 14.36 24 25 20.23 14.69 20.16 14.78 20.10 14.87 20.03 14.96 25 26 21.03 15.28 20.97 15.37 20.90 15.47 20.83 15.56 26 27 21.84 15.87 21.77 15.97 21.70 16.06 21.63 16.15 27 28 22.65 16.46 22.58 16.56 23.51 16.65 33.44 16.75 28 29 23.46 17.05 23.39 17.15 23.31 17.25 33.34 17.35 29 30 31 24.27 17.63 24.19 17.74 24.12 17.84 34.04 17.95 30 31 25.08 18.22 25.00 18.33 24.92 18.44 34.84 18.55 32 25.89 18.81 25.81 18.92 25.72 19.03 35.64 19.15 32 33 26.70 19.40 26.61 19.51 26.53 19.63 26.44 19.74 33 34 27.51 19.98 27.42 20.10 27.33 20.22 27.24 20.34 34 35 28.32 20.57 28.23 30.70 28.13 20 . 82 38.04 20.94 35 36 29.12 21.16 29.03 31.29 28.94 21.41 38.85 21.54 36 37 29.93 21.75 29.84 21.88 39.74 23.01 29.65 22.14 37 38 30.74 22.34 30.64 22.47 30.55 32.60 30.45 22.74 38 39 31.55 22. P2 1 31.45 23.06 31.35 33.20 31.35 23.33 39 40 41 32.36 23.51 32.26 33.06 23.85 33.15 33.96 33.79 34.39 33.05 23.93 24.53 40 41 33.17 24.10 24.24 32.85 42 33.98 24.69 33.87 24.83 33.76 34.98 33.65 25.13 42 43 34.79 25.27 34.68 25.43 34.57 35.58 34.45 25.73 43 44 35.60 25.86 35.48 26.02 35.37 36.17 35.36 26.33 44 45 1 36.41 36.45! 36.29 26.61 36.17 26.77 36.06 26.92 45 46 ! 37.21 27.04 37.10 27.20 36.98 27.36 36.86 27.52 46 47 38.02 27.63 37.90 27.79 37.78 27.96 37.66 28.12 47 48 38.83 28.21 38.71 28.38 38.59 38.55 38.46 28.72 48 49 39.64 28.80 39.52 28.97 39.39 29.15 39.26 29.32 49 50 40.45 29.39 Lat. 40.32 29.57 40.19 29.74 40.06 29.92 50 Dep. Dep. Lat. Dep. Lat. Dep. Lat. 1 Q , 54Deg. 53} Deg. 53i Deg. 63i Deg. .S 1 TRAVERSE TABLE. 146 o P 3 P 51 1 36Deg. 36i Dog. 36^ Deg. 361 Deg. P 51 Lat. j Dep, Lat. Dop. Lat Dep Lat. Dep. 30.61 141.26 29.98 41.13 30.16 41.00 30.34 40 86 52 42 07 30.56 41.94 30.75 41.80 30.93 41 67 31.11 52 53 42.88 31.15 42.74 31.34 42.60 31.53 42.47 31.71 53 54 43.69 31.74 43.55 31.93 43.41 32.12 43.27 32.31 54 55 44.50 '32.33 44.35 32.52 44.21 32.72 44.07 32.91 56 56 45.30 32.92 45.16 33.11 45.02 33.31 44.87 33.51 56 57 46.11 33.50 ' 45.97 33.70 45.82 33.90 46.67 34.10 57 58 46.92 34.09 46.77 34.30 46.62 34.60 46.47 34.70 58 69 47.73 34.68 47.58 34.89 47.43 35 09 47.27 35.30 59 GO 61 48.54 35.27 48.39 35.48 48.23 35.69 48.08 35.90 60 61 49.35 35.85 49.19 36.07 49.04 36.28 48.88 36.60 62 50.16 36.44 50.00 36.66 49.84 36.88 49.68 37.10 62 63 50.97 37.03 50.81 37.25 60.64 37.47 50.48 37.69 63 64 51.78 37.62 51.61 37.84 51.45 38.07 51.28 38.29 64 65 52.59 38,21 52.42 38.44 52.25 38.66 62.08 38.89 65 66 53.40 38.79 53.23 39.03 53.05 39.26 62.88 39.49 66 67 54.20 39.38 54.03 39.62 53.86 39.85 63.68 40.09 67 68 65.01 39.97 .54.84 40.21 54.66 40.45 54.49 40.69 68 69 55.82 40.56 55.64 40.80 56.47 41.04 65.20 41.28 69 70 71 56.63 41.14 66.45 41.39 56.27 41.64 56.09 41.88 70 71 57.44 41.73 57.26 41.98 67.07 42.23 66.89 42.48 72 58.25 42.32 58.06 42.57 57.88 42.83 67.69 43.08 72 73 59.06 42.91 58.87 43.17 68.68 43.42 58.49 43.68 73 74 59.87 43.50 59.68 43.76 69.49 44.02 59.29 44.28 74 75 60.68 44.08 60.48 44.35 60.29 44.61 60.09 44.87 76 76 61.49 44.67 61.29 44.94 61.09 46.21 60.90 45.47 76 77 62.29 45.26 62.10 45.53 61.90 45.80 61.70 46.07 77 78 63.10 45.85 62.90 46.12 62.70 46.40 62.60 46.67 78 79 63.91 46.43 63.71 46.71 63., 50 46.99 63.30 47.27 79 80 81 64.72 47.02 64.52 47.30 64.31 47.59 64.10 47.87 80 81 65.53 47.61 65.32 47.90 65.11 48.18 64.90 48.46 82 66.34 48.20 66.13 48.49 65.92 48.78 66.70 49.06 82 83 67.15 48.79 66.93 49.08 66.72 49.37 66.60 49.66 83 84 67.96 49.37 67.74 49.67 67.. 52 49.97 67.31 50.26 84 85 68.77 49.96 68.55 50.26 68.33 60.66 68.11 60.86 85 86 69.58 50.55 60.35 50.85 69.13 61.15 68.91 61.46 86 87 70.38 51.14 70.16 51.44 69 94 51.75 69.71 62.05 87 88 71.19 51.73 70.97 52.04 70.74 62.34 70.51 52.65 88 89 72.00 52.31 71.77 52.63 71.64 52.94 71.31 63.26 89 90 91 72.81 73.62 52.90 72.58 53.22 72.35 53.53 72.11 63.86 90 91 53.49 73.39 53.81 73.16 54.13 72.91 64.45 92 74.43 54.08 74.19 64.40 73.95 54.72 73.72 55.05 92 93 1 75.24 .54.66 75.00 54.99 74.76 55.32 74.52 55.64 93 J 94 76.05 55.25 76.81 65.. 58 75.56 55.91 75.32 56.24 94 96 , 76- 86 55.84 76.61 66.17 76.37 56.51 76.12 56.84 95 96 ' 77.67 56.43 77.42 56.77 77.17 57.10 76.92 57.44 96 97 78.47 57.02 78.23 57.36 77.97 67.70 77.72 58.04 97 98 79.28 67.60 79.03 57.95 78.78 58.29 78.62 68.64 98 99 80.09 58.19 79.84 58.54 79.58 58.89 79.32 59.23 99 100 6 80.90 58.78 80.64 59.13 80.39 59.48 80.13 59.83 100 Dep. L&t. Dep. Lat. Dep. Lat Dep Lat. 1 54 Deff. 531 Dog, 53i Deg. 53i Deg. _f. 146 TRAVERSE TADLK. O I 37 Deg. 37i Deg. 37j Deg. STi Deg, Lat. Dep. 0.30 1.60 2 . 40 3.:a 3.99 4.79 5.59 6.39 7.19 7.99 8.73 9. 53 10. a3 11.13 11.93 12.73 13.53 U.a3 15.17 15.97 2r 16.77 22 17.57 23 13.37 24 19.17 25 19.97 26 20.75 27 21.56 23 22.33 29 23.16 30 23.93 1 2 3 4 5 6 7 8 9 11 12 13 U 15 16 17 18 19 20 0.60 1.20 1.31 2.41 3.01 3.61 4.21 4.31 5.42 6.02 6.62 7.22 7.32 3.43 9.03 9.63 10.23 10.33 11-43 12.04 12.64 13.24 13.34 14.44 15.05 15.65 16.25 16.35 17.45 13.05 Lat. Dep. 0.30 0.61 1.59 1.21 2.39 1.32 3.13 2.42 3.93 3 . •j3 4.73 3 63 O.Ol 4.24 6.37 4.34 7.16 5 45 Lat. Dep. Lat. Dep 9.55 10.35 11.14 11.94 12.74 13.53 14.33 15.12 15.92 16.72 17.51 13.31 19.10 19.90 7.26 7.37 3.47 9.03 9.63 10.29 10.90 1 1 . 50 12.11 12.71 13.32 13.92 14.53 15.13 2') TO 15 74 "^ _ _^ j -0 34 oo 29 :o 95 23 03 17 00 23 33 15 16 0.79 1.59 2 . 33 3.17 3.97 4.76 5.55 6.-35 7.14 7.93 3.73 9.52 10.31 11.11 11.90 12.69 13.49 14.23 15.07 15.37 16.66 17.4.5 13.25 19.04 19.33 20.63 21.42 22.21 23! 01 23.30 0.61 1.22 ].33 2 43 3.04 3.65 4.26 4.87 5.43 6.09 6.70 7.31 7.91 8.52 9.13 9.74 10.35 10.96 11.57 12.13 12.73 13.39 14.00 14.61 15.22 15.33 16.44 17.05 17.65 13.26 0.79 1. 58 2.37 3.16 3.95 4.74 5.53 6.33 7.12 7.91 0.61 1.22 1.84 2.45 3.06 3.57 4.29 4.90 5.51 6.12 3.70 9.49 10.2-3 11.07 11.36 12.65 13.44 14.23 15.02 15.31 6.73 7.35 7.96 3.57 9.18 9.30 10.41 1 1 . 02 11.63 12.24 16.60 17.40 13.19 13.93 19.77 20.56 21.35 22.14 22.93 23 . 72 12.36 13.47 14.03 14.69 15.31 15.92 16.53 17.14 17.75 13.37 2 3 4 6 7 3 9 10 11 12 13 14 15 16 17 13 19 11 21 22 23 24 25 26 27 28 29 30 TRAVERSE TABI^. 147 o a- Deg. 37i Deg. 37i Deg. m i)^g^ p. on i 3 O a JvT Lat. Dep. Lat. Dep. Lat. Dep. 31.06 Lat. Dep. 40.73 30.69 40.60 30.87 40.46 40.33 31.22 51 52 41.53 31.29 41.39 31.48 41.25 31.66 41.12 31.84 52 53 42.33 31.90 42.19 32.08 42.05 .32.26 41.91 32.46 53 54 43.13 32.50 42.98 32.69 42.84 32.87 42.70 33.06 54 55 43.92 33.10 43.78 33.29 43.63 33.48 43.49 33.67 56 56 44.72 33.70 44.58 33.90 44.43 34.09 44.28 34.28 66 67 45.52 34.30 45.37 34.60 45.22 34.70 45.07 34.90 57 58 46.32 34.91 46.17 35.11 46.01 36.31 45.86 36.61 58 59 47.12 35.51 46.96 35.71 46.81 35.92 46.66 36.12 59 60 '61 47.92 36.11 36.71 47.76 36.32 47.60 36.53 47.44 36.73 60 61 48.72 48.66 36.92 48.39 37.13 48.23 37.35 62 49.52 37.31 49.35 37.53 49.19 37.74 49.02 37.96 62 63 50.31 37.91 50.16 38.13 49.98 38.36 49.81 38.57 63 64 51.11 38.52 50.94 38.74 50.77 38.96 .50.60 39.18 64 65 51.91 39.12 61.74 39.34 61.57 39.57 61.39 39.79 66 66 52.71 39.72 52.54 39.96 52.36 40.18 62.19 40.41 66 67 53.51 40.32 53.33 40.56 53.15 40.79 52.98 41.02 67 68 54.31 40.92 64.13 41.16 63.95 41.40 53.77 41.63 68 69 55.11 41.53 54.92 41.77 54.74 42.00 64.56 42.24 69 70 '71 65.90 56.70 42.13 42.73 55.72 42.37 65.53 66.33 42.61 43.22 65.35 42.86 70 "71 56.62 42.98 66.14 43.47 72 57.50 43.33 67.31 43.68 57.12 43.83 66.93 44.08 72 73 58.30 43.93 58 . 1 1 44.19 57.91 44.44 57.72 44.69 73 74 59.10 44.53 58.90 44.79 58.71 46.06 68.61 45.30 74 75 59.90 45.14 59.70 46.40 69.60 45.66 59.30 45.92 75 76 00.70 45 . 74 60.60 46.00 60.29 46.27 60.09 46.63 76 77 61.49 46.34 61.29 46.61 61.09 46.87 60.88 47.14 77 78 62.29 46.94 62.09 47.21 61.88 47.48 61.67 47.76 78 79 63.09 47.54 62.88 47.82 62.67 48.09 62.46 48.37 79 80 81 63.89 64.69 48.15 63.68 48.42 49.03 63.47 48.70 63.26 64.05 48.98 80 81 48.75 64.48 64.26 49.31 49.69 82 65.49 49.35 65.27 49.63 66.05 49.92 64.84 60.20 82 83 66.29 49.95 66.07 50.24 65.86 50.53 66.63 50.81 83 84 67.09 50.55 66.86 .50.84 66.64 61.14 66.42 51.43 84 85 67.88 51.15 67.66 51.45 67.43 51.74 67.21 52.04 86 1 86 68.68 51.76 68.46 62.06 68.23 62.36 68.00 52.65 861 87 69.48 52.36 09.26 .62.66 69.02 52.96 68.79 53.26 87 88 70.28 52.96 70.05 63.27 69.82 63.67 69.58 63.88 88 89 71.08 53.56 70.84 63.87 70.61 64.18 70.37 64.49 89 90 91 71.88 54.16 71.64 64.48 71.40 54.79 71.16 55.10 90 91 72.68 54.77 72.44 56.08 72.20 65.40 71.95 56.71 92 73.47 55.37 73.23 65.69 72.99 66.01 72.74 56.32 92 93 74.27 55.97 74.03 56.29 73.78 56.61 73.63 56.94 93 94 75.07 56.67 74.82 66.90 74.68 57.22 74.32 67.55 94 95 75.87 57.17 75.62 57.50 75.37 57.83 75.12 68.16 95 96 76.67 57.77 76.42 68.11 76.16 68.44 76 91 68.77 96 97 r7.47 58.38 77.21 58.7] 76.96 69.06 76 70 69.39 97 98 78.27 58.98 78.01 59.32 77.75 69.66 77.49 60.00 98 99 79.06 .59.68 78.80 59.92 78.54 60.27 78.28 160.61 99 100 i S 79.86 60.18 79.60 Dep. 60.53 79.34 60.88 79.07 61.22 100 1 Dep. Lat. Lat. Dep. Lat. Dop. Lat. 53 I Deg. 52J Deg. 52i Deg. 52i Deg. 24 148 TRAVERSE TABLB. I 1 3fl Deg. m i^eg. 38i Deg. 381 Deg. 1 Distance."" La4. "0:79" Dep. 0.62 Lat. Dep. Lat. Dep. Lat. Dep. 0.79 0.62 0.78 0.62 0.78 0.63 2 1.58 1.23 1.57 1.24 1.57 1.24 1.56 1.25 2 3 2.36 1.85 2.36 1.86 2.35 1.87 2.34 1.88 3 4 3.151 2.46 3.14 2.48 3.13 2.49 3.12 2.50 4 5 3.94' 3.08 1 3.93; 3.10 3.91 3 11 3.90 3.13 6 6 4.73 3.69 4.71 1 3.71 4.70 3.74 4.68 3.76 3 7 5.52 4.31 5.50 4.33 5.48 4.36 5.46 4.38 7 8 6.30 4.93 6.28 4.95 6.26 4.98 6.24 5.01 8 9 7.09 5 54 7.07 5.57 7.04 5.60 7.02 rj.63 g 10 11 7.88 6.16 6.77 7.85 6.19 7.8£ 6.23 7.80 b.iixt ID 8.67 8.64 6.81 8.61 6.85 8.58 6.89 11 12 1 9.46 7.39 9.42 7.43 9.39 7.47 9.36 7.51 12 13 10.24 8.00 10.21 8.05 10.17 8.09 10.14 8.14 13 14 11.03 8.62 10.99 8.67 10.96 8.72 10.92 8.76 14 .'5 11.82 9.23 11.78 9.29 11.74 9.34 11.70 9.39 15 16 12.61 9.85 12.57 9.91 12.52 9.96 12.48 10.01 16 17 13.40 10.47 13.35 10.52 13.30 10.58 13.26 10.64 17 18 14.18 11.08 14.14 11.14 14.09 11.21 14.04 11.27 18 19 14.97 11.70 14.92 ; 11.76 14.87 11.83 14.82 11.89 19 20 21 15.76 12.31 15.71 12.38 15.65 1 12.45 15.60 12.62 20 16.55 12.93 16.49 13.00 16.43 i 13.07 16 38 13.14 21 22 17.34 13.54 17.28 13.62 17.22 13.70 17.16 13.77 22 23 18.12 14.16 18.06 14.24 18.00 14.32 17.94 14.40 23 24 18.91 14.78 18.85 14.86 18.78 14.94 18.72 15.02 24 25 19.70 15.39 19.63 15.48 19.57 15.56 19.50 15.65 25 26 20.49 16.01 20.42 16.10 20.35 16.19 20.28 16.27 26 27 21.28 16.62 21.20 16.72 21.13 16.8] 21.06 16.90 27 28 22.06 17.24 21.99 17.33 21.91 17.43 121.84 17.53 28 29 22.85 17.85 22.77 17.95 22.70 18.05 22.62 18.15 29 30 31 23.64 18.47 23.56 18.57 23.48 18.68 23.40 18.78 30 31 24.43 19.09 24.34 19.19 24.26 19.30 24.18 19.40 32 25.22 19.70 25.13 1 19.81 25.04 19.92 24.96 20.03 32 33 26.00 20.32 25.92: 20.43 25.83 20.54 25.74 20.66 33 34 26.79 20.93 26.70 21.05 26.61 21.17 26.52 21.28 34 35 27.58 21.55 27.49 i 21.67 27.39 21.79 27.30 21.91 35 36 28.37 22.16 28.27 i 22.29 28.17 22.41 28.08 22.53 36 37 29.16 22.78 29.06 22.91 28.96 23.03 28.86 23.16 37 38 29.94 23.40 29.84 23.53 29.74 23.66 29.64 23.79 38 39 30.73 24.01 30.63 24.14 30.52 24.28 30.42 24.41 39 40 31.52 24.63 31.41 24.76 31.30 24.90 31.20 25.04 40 41 32.31 25.24 32.20 25.38 ,32.09 25.52 31.98 25.66 41 42 33.10 25.86 32.98 26.00 32.87 26.15 32.76 26.29 42 43 33.88 26.47 33.77 1 26.62 33.65' 26.77 33.53 26.91 43 44 34.67 27.09 34.55 27.24 34.43 27.39 34.31 27.54 44 45 35 46 27.70 35.34 27.86 35.22 38.01 35.09 28.17 46 4€ 36.25 28.32 36.12 1 28.48 36.00 28.64 35.87 28.79 46 47 37.04 28.94 36.91 29.10 36.78 29 26 36.65 29.42 47 48 ' 37 82 29.55 37.70 29.72 37.57 29,88 37 43 30.04 , 48 49 38.61 .30.17 38.48 30.34 38.35 30.60 38.21 30.67 1 49 60 • 39.40 30.78 39.27 30.95 39.13 31 13 Lat. 38.99 Dep. 31.30 i a Dep. Lat. Dep. Lat. Dep. Lat. ' 52 Deg. 511 Deg. Slh Deg. Bl\ Dog. .9 TRAVERSE TABLE. 149 a 51 38 Deg. 381 Deg. 38^ Deg. 38| Deg. d P o Lat. Dep. I^at. D ep. Lat. Dep. Lat. D ep. 40.19 31 .40 40 .05 31 .57 39.91 31 .75 39 .77 3] .92 ,51 52 40.98 32 .01 40 .84 32 .19 40.70 32 .37 40 .55 32 .55 62 53 41.76 32 .63 41 .62 32 .81 41.48 32 .99 41 .33 33 .17 53 54 42.55 33 .25 42 .41 33 .43 42.26 33 .62 42 .11 33 .80 64 65 43.34 33 .86 43 .19 34 .05 43.04 34 .24 42 ,89 34 .43 55 56 44.13 34,48 43 .98 34 .67 43.83 34 .86 43 .67 35 .05 56 57 44.92 35 .09 44 .76 35 .29 44.61 35 .48 44 .45 35 .68 57 58 45.70 35 .71 45 .55 35 .91 45.39 36 .11 45 .23 36 .30 58 59 46.49 36 .32 46 .33 36 .53 46.17 36 .73 46 .01 36 .93 59 60 61 47.28 36 .94 47 .12 37 .15 46.96 37 .35 46 .79 37 .56 60 48.07 37 .56 47 .90 37 .76 47.74 37 .97 47 ..57 38 .18 61 62 48.86 38 .17 48 .69 38 .38 48.52 38 .60 48 .35 38 .81 62 63 49.64 38 .79 49 .47 39 .00 49.30 39 .22 49 .13 39 .43 63 64 50.43 39 .40 50 .26 39 .62 50.09 39 .84 49 .91 40 .06 64 65 51.22 40 .02 51 .05 40 .24 50.87 40 .48 50 .69 40 .68 65 66 52.01 40 .63 51 .83 40 86 51.65 41 .09 51 .47 41 .31 66 67 52.80 41 .25 52 .62 41 48 52.43 41 .71 52 .25 41 .94 67 68 53.68 41 .86 53 .40 42 10 53.22 42 .33 53 .03 42 .56 68 69 54.37 42 .48 54 .19 42 72 54.00 42 95 53 .81 43 .19 69 70 71 55.16 43 .10 54 .97 43 43 34 96 54.78 43 58 54 .59 43 .81 70 71 55.95 43 .71 55 .76 55.57 44 .20 55 .37 44 4 72 56.74 44 .33 56 .54 44 57 56.35 44 82 58 .15 45 .07 72 73 57.52 44 94 57 33 45 19 57.13 45 44 56 .93 45 .69 73 74 58.31 45 56 58 11 45 81 57.91 46 07 57 71 46 .32 74 75 59.10 46 17 53 90 46 43 58.70 46 69 58 49 46 .94 75 76 59.89 46, 79 59 68 47 05 59.48 47 31 59 27 47 ..57 76 77 60.68 47 41 60 47 47 67 60.26 47 93 60 05 48 20 77 78 61.46 48 02 61 25 48 29 61.04 48 56 60 .83 48 82 78 79 62.25 48 64 62 04 48 91 61.83 49 18 61 61 49 .45 79 80 81 63.04 49 25 62 83 49 53 62.61 49 80 62 39 50 07 80 81 63.83 49 87 63 61 50 15 63.39 50 42 63 17 50 70 82 64.62 50 48 64 40 50 77 64.17 51. 05 63 95 51 33 82 83 65.40 51 10 65 18 51 38 64.96 51. 67 64 73 51 95 83 84 66.19 51 72 65 97 52 00 65.74 52. 29 65 51 52 58 84 85 66.98 52 33 66 75 52 62 66.52 52. 91 66 29 53 20 85 86 67.77 52. 95 67. 54 53. .24 67.30 53. 54 67. 07 53. 83 86 87 68.56 53. 56 68. 32 53. 86 68.09 54. 16 67. 85 54. 46 87 88 69.34 54. 18 63. 11 54. 48 68.87 54. 78 68. 63 55. 08 88 89 70.13 54. 79 G9. 89 55. 10 69.65 55. 40 69. 41 55. 71 89 90 91 70.92 55. 41 70. 68 .55. 72 70.43 56. .56. 03 65 70, 19 56. 33 90 91 71.71 56. 03 71. 46 56. 34 71.22 70. 97 5G. 96 92 72.50 56. 64 72. 25 56. 96 72.00 57. 27 71. 75 57. 58 92 93 73.28 .57. 26 73. 03 57. 58 72.78 57. 89 72. 53 .58. 21 93 94 74.07 57. 87 73. 82 58. 19 73.57 58. 52 73. 31 .58. 84 94 95 74.86 58. 49 74. 61 58. 81 74.35 59. 14 74. 09 59. 46 95 96 75.65 59. 10 75. 39 59. 43 75.13 59. 76 74. 87 60. 09 96 97 76.44 59. 72 76. 18 60. 05 75.91 60. 33 75. 65 60, 71 97 98 77.22 60. 33 76. 96 60. 67 76.70 61. 01 76. 43 61. 34 98 99 78.01 60. 95 77. 75 61. 29 77.48 61. 63 77. 21 61. 97 99 100 •-> « 5 78.80 61. 57 78. 53 61. 91 78.26 62. 25 77. 99 62. 59 100 oJ u c «j «-» s Dep. Lat. Dep. Lat. Dep. Lat. 1 Dep. Lat. 52 Deg. 511 Deg • 5li Deg. 5 liDeg 150 TRAVEESE TAUL&. ? 39 Deg. 1' 39i Deg. 3^ Deg. 39j Deg. ■ q 1 ] Lftt ' Dep. Lat. : Dep. La-t. Dep. j La.t. ! Dep. s s "^1 1 0.78 0.63 0.77 : 0.63 0.77 0.64 1 0.77] 64 2' 1.55 1.26 1.55 1.27 1.54 1.27 1.54 1.28 2 3 2.33 1.89 2.32 1.90 2.31 1,91 2.311 1.92 3 4 3.11 2.52 3.10 , 2.53 3.09 2.54 3.08 2.56 4 5 3.89 3.15 3.87 3.16 3.86 3.18 3.84 3.20 5 6 4.66 3.78 4.65 3. SO 4.63 3.82 4.61 3.84 6 7 5.44 4.41 5.42 4.43 5.40 4.45 \ 5.38 ! 4.48 7 8 6.22 5.03, 6.20 5.06 6.17 5.09 j 6.15 1 5.12 8 9 6.99 5.68 6.97, 5.6« 6.94 5.72 1 6.92 ^ 5.75 9 10 7.77 6.29 7.74 6.33 7.72 6.36 7.69 6.39 8.46 7.03 10 11 11 8 . 55 6.92 8 . 52 6.96 8.49 7.00 12 9.33 7.55 9.29 7.59 9.26 7.63 9.23! 7.67 12 13 10.10 8.18 10.07 1 8.23 10.03 8.27 9.99 8.31 13 14 10.88 8.81 10.84 1 8.86 10.80 i 8.91 10.76 8.95 14 15 11.66 9.44 11.62 9.49 11.57 9.54 11.53 9.59 15 16 12.43 10.07 12.39 10.12 12.35 10.18 12.30 10.23 16 17 13.21 10.70 13.16 10.76 13.12 i 10.81 1 13.07 10. a? 17 18 13.99 11.33 13.94 11.39 13.89 i 11.45 13.84 11.51 18 19 14.77 11.96 14.71 12.02 14 66 12.09 14.61 12.15 19 20 15.54 12.59 15.49 12.65 15.43 12.72:15.33 12.79 20 21 21 16.32 13.22 16.26 13.29 16.20 13.36 16.15 13.43 22 17.10 13.84 17.04 13.92 16.93 13.99 16.91 14.07 22 23 17.87 14.47 17.81 14.55 17.75 14.63 17.63 14.71 23 24 18.65 15.10,18.59 15.18 18.52 15.27 18.45 15.35 24 25 19.43 15.73 19.36 15.82 19.29 15.90 19.22 15.99 25 26 20.21 16.36 20.13 16.45 20.06 16.54 19.99 16.63 26 27 20.98 16.99 20.91 17.08 20.83 ; 17.17 ,20.76 17.26 17.72 21.61 17.81 i 21.53 17.90 27 S3 21.76 17.62 21.68 28 29 22.54 18.25 22.46 18.35 22.33 18.45 22.30 18.54 29 30 23.31 18.88 23.23 18.93 23.15 19.03 '23.07 19.18 30 31 31 24.09 19.51 24.01 19.61 23.92 19.72 23.83 19.82 32 24.87 20.14 24.78 20.25 24.69 20.35 24.60 20.46 32 33 25.65 20.77 25.55 20.88 25.46 120.99 25.37 21.10 33 34 26.42 21.40 26.33 21.51 26.24:21.63 26.14 21.74 M 35 27.20 22.03 27.10 22.14 27.01 22.26 26.91 22.33 35 36 27.98 22.66 27.83 22.78 27.73 22.90 27.63 23.02 36 37 28.75 23.23 28.65 23.41 23.55 23.53 23.45 23.66 37 as 29.53 23.91 29.43 24.04 29.32 24.17 29.22 24.30 33 39 30.31 24.54 30.20 24.68 30.09 24.31 29.93 24.94 39 iO 31.09 41 31.88 25.17 30.93 25.31 ■^■1 -« 25.44 ; 30.75 25.53 ' 31.. 52 26.22, to 41 25.50 31.75 25.9-i 26.03 42 32.64 26.43 32.52 26.57 .34. -x: 26.72 32.29 26.86 1 y 43 33.42 27.06 33.. 30 27.21 33.18 '27. 35 33.06 27.50 43 44 34.19 27.69 .34.07 27.84 .33.95 27.99 33.83 23.14! 44 45 .^i.97 23.. 32 34. S5 28.47 34.72 28.62 34.60 28.77 1 45 16 35.75 28.95 35.62 29.10 .35.49 29.26 35.37 29.41 46 47 36.53 29.53 36.40 29.74 36.27 29.90 36.14 30.05; 47 te 37.30 30.21 37.17 30.37 37.04 30.53 '36.90 30.69; 43 49 38.08 30.!^ 37.95 31.00 37.81 31.17 37.67 31.33 49 50 33.86 31.47 .38.72 31.64, 33.53 31.80 ) 38.44 : 31 .97 i 50 d u ■ 3 I' Dep. Lat Dep. | Lat. !' Dep. Lat. j! Dep. Lat. 51 r ( )eg. 50^ Deg. 5O5 De?. ! i 50i Deg. 1 TRAVEJISE TABLE. 151 f 39 Dog. 39i Dcg. 39i Dog. 391 Dog. s 61 Lat Dep. Lat. Dep. Lat. Dep, Lat. Dep. i 51 39.63 32.10 39.49 32.27 39.35 32.44 39.21 32.61 ^2 40.41 32.72 40.27 32.90 40.12 33.08 39.98 33 25 62 03 41.19 33.35 41.04 33.53 40.90 33.71 40.75 33 89 53 64 41.97 33.98 41.82 34.17 41.67 134.35 41.52 34.53 54 66 42.74 34 61 42.59 34.80 42.44 34.98 42.29 35.1? 55 66 43.62 36.24 43.37 35.43 43.21 35.62 43.06 35.81 56 67 44.30 35.87 44.14 36.06 43.98 36.26 43.82 36.45 57 68 45.07 36.60 44.91 36.70 44.75 36.89 44.69 37.09 58 69 46.85 37. IS 45.69 37.33 45.53 37.53 45.36 37.73 59 60 61 46.63 37.76 38.39 46.46 37.96 46.30 38.16 46.13 38.37 60 61 47 41 47.24 38.60 47.07 38 86 46.90 39.01 62 48.18 39.02 48.01 39.23 47.84 39.44 47.67 39. 65' 62 63 48.96 39.65 48.79 39.86 48.61 40.07 48.44 40.28 63 64 49.74 40.28 49.56 40.49 49.38 40.71 49.21 40.92 64 66 50.61 40.91 50.34 41.13 50.16 41.35 49.97 41.56 65 66 51.29 41.54 51.11 41.76 50.93 41.98 50.74 42.20 66 67 52.07 42.16 51.88 42.39 51.70 42.62 51.51 42.84 67 68 62.85 42.79 52.66 43.02 52.47 43.25 52.28 43.48 68 69 53.52 43.42 53.43 43.66 53.24 43.89 53.05 44.12 69 70 71 54.40 44.05 54.21 44.29 54.01 4.53 53.82 44.76 70 71 55.18 44.68 54.98 44.92 54.79 45.16 54.59 45.40 72 55.95 45.31 55.76 45.55 55.. 56 45.80 55.36 46.04 72 73 56.73 45.94 56.53 46.19 56.33 46.43 56.13 46.68 73 74 57.51 46.57 57.31 46.82 57.10 47.07 56.89 47.32 74 75 58.29 47.20 58.08 47.45 57.87 47.71 57.66 47.96 75 76 .59.06 47.83 58.85 48.09 58.64 48.34 58.43 48.60 76 77 59.84 48.46 59.63 48.72 59.42 48.98 59.20 49.24 77 78 60.62 49.09 60.40 49.35 60.19 49.61 59.97 49.88 78 79 61.39 49.72 61.18 49.98 60.96 50.25 60.74 50.52 79 80 81 62.17 50.35 61.95 62.73 50.62 61.73 62.50 60.89 61.51 51.16 80 81 62.95 50.97 51.25 51.52 62.28 51.79 82 63.73 51.60 63.50 51.88 63.27 52.16 63.04 52.43 82 83 64.50 5'x{.23 64.27 52.51 64.04 52.79 63.81 53.07 83 84 65.28 52.86 65.05 53.15 64.82 53.43 64.58 53.71 84 85 66.06 53.49 65.82 53.78 65.59 54.07 65.35 54.35 85 86 66.83 54.12 66.60 54.41 66.36 54.70 66.12 54.99 86 87 67.61 54.75 67.37 55.05 67.13 55.34 66.89 55.63 87 88 68.-^9 55.38 68.15 55.68 67.90 55.97 ff7.66 56.27 88 89 69.17 56.01 68.92 56.32 68.67 56.61 68.43 56.91 89 90 91 69.94 56.64 69.70 56.94 69.45 70.22 57.25 69.20 57.55 90 91 70.72 57.27 70.47 57.58 57.88 69.96 58.19 92 71.50 57.90 71.24 58.21 70.99 58.52 70.73 58.83 99 93 72.27 58.53 72.02 58.84 71.76 59.16 71.50 59. 4~ 93 94 73.05 59.16 72.79 59.47 72.53 59.79 72.27 60.11 94 95 73.83 59.79 73.57 60.11 73.30 60.43 73.04 60.75 95 96 74.61 60.41 74.34 60.74 74.08 61.06 73.81 61.39 96 97 75.38 61.04 75.12 61.37 74.85 61.70 74.58 62.03 97 98 76.16 61.67 75.89 62.01 75.62 62.34 75.35 62.66 98 99 76.94 62.30 76.66 62.64 76.39 62.97 76.12 63.30 99 100 « 77.71 62.93 77.44 63.27 77.16 63.61 76.88 63.94 100 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 51 Deg. 50| Deg. 50i Deg. SOiDeg. 153 TRAVERSE TABIF. 5 s 3 o ? 1 40 Deg. 40^ Deg. 40i Deg. 401 Deg. 5* r» P 3 p Lat. Dep, 0.64 Lat. Dep. ' Lat. Dep. Lat, Dep. 0.77 0.76 0.65 0.76 0.65 0.76 0.65 1 2 1.53 1.29' 1.53 1.29 1.52 1.30 1.52 1.31 2 3 2.30 1.93: 2.29 1.94 2.28 1.95 2.27 1.96 3 4 3.06 2.57 :■ 3.05 2.58 3.04 2.60 3.03 2.61 4 6 3. S3 3.21 3.82 3.23 3.80 3.25 3.79 3.26 6 6 4.60 3.80 i 4.58 3.88 4,56 3.90 4.55 3.92 6 7 5.36 4.50 5.34 4.52 5.32 4.55 5.30 4.57 7 8 6.13 5.14 6.11 5.17 6.08 5.20 6.06 5.22 8 9 6.89 5.79 i 6. 87 5.82 0.84 5.84 6.82 5.87 9 10 11 7.66 6.43! 7.63 8.40 6.46 j 7.60 6.49 7.58 6.53 10 11 8.43 7.07, 7.11 8.36 7.14 1 8.33 7.18 12 9.19 7.71 9.16 7.75 9.12 7.79 1 9.09 7.83 12 13 9.96 8.36 9.92 8.40 9.89 8.44! 9.85 8.49 13 14 10.72 9.00 1 10.69 9.05 10.65 9.09 j 10.61 9.14 14 In 11.49 9.64: 11.45 9.69 11.41 9.74 1 11.36 9.79 15 16 12.26 10.28 12.21 10,34 12.17 10.39 12.12 10.44 16 17 13.02 10.93 12.97 10.98 12.93 11.04 12.88 n.io 17 18 13.79 11.57 13.74 11.63 13.69 11.69 13.64 11.75 18 19 14.55 12.21 14.50 12.28 14.45 12.34 14.39 12.40 19 20 15.32 12.86 15.26 12.92 15.21 12.99 15.15 15.91 13.06 20 21 21 16.09 13.50 16.03] 13.57 15.97 13.64 i 13.71 22 16.85 14.14 16.79 14.21 16.73 14.29; 16.67 14.36! 22 23 17.62 14.78 17.55 14.86 17.49 14.94 1 17.42 15.01 23 24 18.39 15.43 18.32 15.51 18.25 15.59 1 18.18 15.67 24 25 19.15 16.07 19.08 16.15 19.01 16.24 18.94 16.32 25 26 19.92 16.71 19.84 16.80 19.77 16.89 19.70 16.97 26 27 20.68 17.36 20.61 17.45 20.53 17.54 20.45 17.62 27 28 21.45 18.00 21.37 18.09 21,29 18.18 21.21 18.28 28 29 22.22 18.64 22.13 18.74 22.05 18.83 21.97 18.93 29 30 31 22.98 19.28 22 . 90 19.38 22.81 19.48 22.73 19. .58 30 31 23.75 19.93 23.66 20.03 23.57 20.13 23.48 20.24 32 24.51 20.57 24.42 20.68 24.33 20.78 24.24 20.89 32 33 25.28 21.21 25.19 21.32 25.09 21.43 25.00 21.54 33 34 26.05 21.85 25.95 21.97 25.85 22.08 25.76 22.19 34 35 26.81 22.50 26.71 22.61 26.61 22.73 26.51 22.85 •35 30 27.58 23.14 27.48 23.26 27.37 23.38 27.27 23.50 36 37 28.34 23.78 28.24 23.91 28.13 24.03 28.03 24.15 37 38 29.11 24.43 29.00 24.55 28.90 24.68 28.79 24.80 38 39 29.88 25.07 29.77 25.20 29.66 25.33 29.54 25.46 39 40 41 30.64 25.71 26.35 30.53 25.84 30.42 25.98 30.30 26.11 40 41 31.41 31.29 26.49 31.18 26. C3 31.06 26.76 42 32.17 27.00 32.06 27.14 31.94 27.28 31.82 27.42 42 43 32.94 27.64 32.82 27.78 32.70 27.93 32.58 28.07 43 44 33.71 28.28 33.58 28.43 33.46 28.58 33.33 28.72 44 45 34.47 28.93 34.35 29.08 34.22 29.23 34.09 29.37 45 46 35.24 29.57 35.11 29.72 34.98 29.87 • 34.85 30.03 46 47 36.00 30.21 1 35.87 30.37 35.74 30.52 35.61 30.68 47 48 36.77 30.85 ; 36.64 31.01 36.50 31.17 36.36 131.33 48 49 37.54 31.60 1 37.40 31.66 37.26 31.82 37.12 31.99 49 60 g 38.30 32.14 38.16 32.31 38.02 32.47 37.88 32.64 50 e3 c Q Dep. ( Lat. Dep. Lat. Dep. Lat. Dep, Lat. 50 Deg. 491 Deg. 49i Deg. 49i Deg. TKAV£BSE TABLC. 153 b 03 P 3 n o ~6i 40 Deg, 40i Deg. 40i Deg. 40| Deg. Q to' r* P 3 n 9 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 39.07 32.78 38.92 32.95 38.78 33.12 38.64 33.29 52 39.83 33.42 39.69 33.60 39.. 54 33.77 39.39 33.94 52 63 40.60 34.07 40.45 34.24 40.30 34.42 40.15 34.60 53 54 4K37 34.71 41.21 34.89 41.06 35.07 40.91 35.25 54 55 42.13 35.35 41.98 35.. 54 41.82 35.72 41.67 35.90 55 56 42.90 36.00 42.74 36.18 42.. 58 36.37 42.42 36.55 56 57 43.66 36.64 43.50 36.83 43.34 37.02 43.18 37.21 57 58 44.43 37.28 44.27 37.48 44.10 37.67 43.94 37.86 68 59 45.20 37.92 45.03 38.12 44.86 3S.32 44.70 38.51 59 60 61 45.96 38.57 45.79 38.77 45.62 38.97 45.45 39.17 60 46.73 39.21 46.56 39.41 46.38 39.62 46.21 39.82 61 62 47.49 39.85 47.32 40.06 47.15 40.27 46.97 40.47 62 63 48.26 40.50 48.08 40.71 47.91 40.92 47.73 41.12 63 64 49.03 41.14 48.85 41.35 48.67 41.56 48.48 41.78 64 65 49.79 41.78 49.61 42.00 49.43 42.21 49.24 42.43 65 66 50.56 42.42 50.37 42.64 60.19 42.86 50. oa 43.08 66 67 51.32 43.07 51.14 43.29 50.95 43.51 50.76 43.73 67 68 52.09 43.71 51.90 43.94 51. 7r 44.16 51.51 44.39 68 69 52.86 44.35 52.66 44.58 52.47 44.81 52.27 45.04 69 70 71 53.62 45.00 53.43 45.23 63.23 45.46 53.03 45.69 70 71 54.39 45.64 54.19 45.87 63.99 46.11 63.79 46.35 72 55.16 46.28 54.95 46.52 54.75 46.76 54.. 54 47.00 72 73 55.92 46.92 55.72 47.17 55.51 47.41 55.30 47.65 73 74 56.69 47.. 57 56.48 47.81 56.27 48.06 56.06 48.30 74 75 57.45 48.21 57.24 48.46 57.03 48.71 56.82 48.96 75 76 58.22 48.85 58.01 49.11 57.79 49.36 .57.57 49.61 76 77 58.99 49.49 58.77 49.75 58.55 50.01 .58.33 50.26 77 78 59.75 50.14 59.53 50.40 59.31 50.66 59.09 50.92 78 79 60.52 50.78 60.30 51.04 60.07 .51.31 59 .85 51.57 79 80 81 61.28 51.42 61.06 51.69 60.83 51.96 60.61 52.22 80 81 62.05 52.07 61.82 52.34 61.59 52.61 61.36 52.87 82 62.82 52.71 62.59 52.98 62.35 53.25 62.12 53.53 82 83 63.58 53.35 63.35 53.63 63.11 53.90 62.88 .54.18 83 84 64.35 53.99 64.11 54.27 63.87 54.55 63.64 54.83 84 85 65.11 54.64 64.87 54.92 64 63 55.20 64.39 55.48 85 86 65.88 55.28 65.64 55.57 65 39 55.85 65.15 56.14 86 87 66.65 55.92 66.40 56.21 66 16 56.50 65.91 56.79 87 88 67.41 56.57 67.16 56.86 66 92 57.15 66.67 .57-44 88 89 68.18 .57.21 67.93 57.50 67 68 57.80 67.42 58.10 89 90 91 68.94 57.85 68.69 58.15 68.44 58.45 68.18 58.75 90 69.71 58.49 69.45 58.80 69.20 59.10 68.94 59.40 91 92 70.48 59.14 70.22 59.44 69.96 59.75 69.70 60.05 92 93 71.24 59.78 70.98 60.09 70.72 60.40 70.45 60,71 93 94 72.01 60.42 71.74 60.74 71.48 61.05 71.21 61.36 94 95 72.77 61.06 72.51 6L.38 72.24 61.70 71 .97 62.01 95 96 73.54 61.71 73.27 62.03 73.00 62.35 72.73 62.66 96 97 74.31 6^.35 74.03 62.67 73.76 63.00 73.48 63.32 97 98 75.07 62.99 74.80 63.32 74.52 63.65 74.24 63.97 98 99 75.84 63.64 75.56 63.97 75:28 64.30 75.00 64.62 99 100 i O 76.fi0 64.28 76.32 64.61 76.04 64,94 75.76 65.28 100 D^p. Lat. Dcp. tat. Dep. Lat. Dep. Lat. o c •-» m - 50 Deg. 491 Deg. 49i Deg. 494 Deg. 154 TBAVEESi: TABLE. 2" 1 41 Deg. 4U Deg. '. i ' 41 i Deg. 41| Deg. a P 3 r. c 1 Lat, Dep. ; Lat. ! Dep. ' Lat. ! Dep, | Lat Dep. 0.75 0.66 0.75 0.66 0.75 0.66 0.75 0.67 1.51 1.31 1.50 1.32 1.50 1.33 1.49 1.33 2 3 2.26 1.97 2.26 1.95 2.25 1,99: 2.24 2.00 3 4 3.02 1 2.62 3.01 2.64 3.00 2,65 2.98 2.66 4 5 3.77! 3.2s 3.76 3.30 3.74; 3.31 3,73 3.33 5 6 4.53, 3.94 4.51 3.96 4.49 3.9S 4.48 4.00 6 7 5.2s! 4.59 5.26 4.62 5.24; 4.64 5.22 4.66 7 8 6.04 5.25 6.01 5.27 5 99; 5.30: 5.97 5.33 8 9 6.79 5.90 6.77 5.93 6.74 5,?6; 6.71 5.99 9 10 11 7.55 1 6.53 7.. 52 6.59 7.49 8.24 6.63 7.46 6.66 10 11 S.30! 7.22 8.27 7,25 7.29 8.21 7.-32 12 9.06 i 7.57 9.02 7.91 8.99 7.95: 8.95 7.99 13 13 9. SI 1 8.53 9.77 8.57 9.74 8.61 1 9.70 8.66 13 14 10.57; 9.15 10.53 9.23 10.49 9.28 1 10.44 9.32 14 15 li.32: 9.S4 11.28 9.89 11.23 9,94 11.19 9.99 15 16 12.0s 10.50 12.03 10.55 11.95 i 10.601 11.94 10.65 16 IT 12.53 11.15 12.78 11.21 12.73 ; 11.26 1 12. B8 11.82 17 IS 13. 5S ; 11. SI 13.. 53 11.87 13.45 : 11.93: 13.43 n.99 18 19 14.34 12.47 14.2S 12.53 14.23 12.59! 14.18 12.65 19 20 ' 15.09 13.12, 15.04 13.19- 14.95 13.25; 13.91 ; 14.92 13.:^ 20 21 21 15. S5 13.75 15.79 13.55 15.73 15.67 13.98 22 ' 16.60 14.43 16.54 14.51 16. 4S 14.53! 16.41 14.65 22 23 17.35 15.C>9 17.29 15.16 17.23 ' 15.24- 17.16 15.32 23 24 IS. 11 15.75 IS. 04 15. S2 17.97 1 15.90 17.91 15.93 24 25 1S.S7 16.40 IS. 80 16. 4S 18.72 1 16.57 f 18.65 16.65 25 26 19.62 17.06 19.35 17.14 19.47; 17.23 1 19.40 17.31 26 27- 20.35 17.71 20.30 17.50 20.22 i 17.89!'; 20.14 17.98 27 2S| 21.13 IS. 37 CI .05 l"^ .4-: 20.97 ' 13. .55; 20,89 18.04 28 29 i 21. S9 19.03 .--.-. _ ."^ ■ - -- 21 .72 19.22 ' 21.6-4 19.31 29 30 31 22. C4 19.65 23.40 20.34 - i . .; ' 1 0.7? 22.47 ; 19.83 , 23 . 22 ' 20 . 54 ; 22.38 19.98 30 31 23.31 ; 20.44 23.13 20.6^1 32 , 24.15 20.99 24.06 '21.10 23.97 ' 21.20 1 23.87 21.31 32 33. 24.91 21.65 24.81 ; 21.76 24.72 ; 21.87 24.62 21.97 33 34 25.66 22.31 25,56 ; 22.42 25.46 22. .53 25.37 22.64 34 35 26.41 22.96 26.31 23.05 26.21 23.19 26.11 23.31 35 36 27.17 23.62 27.07 23.74 25.96 23.S5 1 126.86 23.97 36 37 27.92 24.27 27.82 2-1.40 27.71 124.52! i27.ao 24.64 37 38 2-S.68 24.93 28.57 25.05 28.46 : 25.15 ; 28.35 25.. 30 38 39 29.43 25.59 29.32 25.71 29.21 , 25.54: 29.10 25.97 39 40 41 30.19 26.24 30.94 26.90 30.07 i 26.37 30.83 j 27.03 29.96 I 25.r>6 , 30 .'71 l27.17j i 29.84 26.64 40 41 30.. 59 27.3i3 42 31.70 27.55 31..5-8 127.69 31.45 27. S3 31.33 27.97 42 43 32.45 23.21 32.33' 28.35 32.21 2S.49 32.08 2-8.63 43 41 33.21 2S.87 33.08 29.01 32.95 29.16 132.83 29.30 44 45 33.96 29.52 33.83 20. -:7 33.70 29.82 1 33.57 29.97 45 46 ;U.72 30. IS 34.58 ::.;; 34.45 30.45 •34,32 30.63 4^> 47 35.47 30. S3 35.34 C;'.c<9 35.20 31.14 35.06 31.30 47 48 35. 2i^ 31.49 36.09 ' 31.65 35.95 31.81' 35.81 31.96 48 49 36.93 32.15 35.84 32.31 36.70 ,32.47 36.56 32.63 49 50 u c 37.74. 32.80 .37.59 i 32.97 37.45 ,33.13 Dep. 1 Lat, 37.30 33.29 50 c /^ Dep. 1 Lat. Dep. ; Lat. Dep. Lat. 49 Deg, m Deg, , m Deg. 48^ Deg. 1 TRAVERSE TABLE. 165 o p s o o 51 41 Deg. 41i Deg. 41^ Deg. 41 1 Deg. m' 3 o p 51 Lat. Dep. Lat. 38.34 Dep. Lat. Dep. Lat. Dep. 38.49 33.46 33.63 38.20 33.79 38.05 33.96 52 39.24 34.12 39.10 34.29 38.95 34.46 .38.79 34.63 52 53 40.00 34.77 39.85 34.95 39.69 35.12 39.54 35.29 53 54 40.75 35.43 40.60 35.60 40.44 35.78 40.29 35.96 54 55 41.51 36.08 41.35 36.26 41.19 36.44 41.03 36.62 55 56 42.26 36.74 42.10 36.92 41.94 37.11 41.78 .37.29 56 67 43.02 37.40 42.85 37.58 42.69 37.77 42.53 37.96 57 5S 43 . 77 33.05 43.61 38.24 43.44 38.45 43.27 33.62 68 59 44.53 38.71 44.36 38.90 44.19 39.09 44.02 39.29 59 (;o 61 45.28 39.36 45.11 45.86 39.56 44.94 39.76 44.76 39.95 60 46.04 40.02' 40.22 45.69 40.42 45.51 40.62 61 62 4().79 40.68 46.61 40.88 46.44 41.08 46.26 41.28 62 63 47.55 41.33 47.37 41.54 47.18 41.75 47.00 41.95 63 64 48.30 41.99 48.12 42.20 47.93 42.41 47.75 42.62 64 65 49.06 42.64 48.87 42.86 48.68 43.07 48.49 43.28 65 66 49.81 43.30 49.62 43.52 49.43 43.73 49.24 43.95 66 67 50.57 43.96 50.37 44.18 50.18 44.40 49.99 44.61 67 68 51.32 44.61 51.13 44.84 50.93 45.06 50,73 45.28 68 69 52.07 45.27 51.88 45.49 51.68 45.72 51.48 45.95 69 70 71 52.83 45.92 52,63 46.15 52.43 46.38 52.2,2 .5^.97 46.61 70 71 53.58 46.58 53.38 46.81 53.18 47.05 47.28 72 54.34 47.24 54.13 47.47 53.92 47.71 53.72 47.94 72 73 55.09 47.89 54.88 48.13 54.67 48.37 54.46 48.61 73 74 55.85 43.55 55.64 48.79 55.42 49.03 55.21 49.28 74 75 56.60 49.20 56.39 49.45 56.17 49.70 .55.95 49.94 75 76 57.36 49.86 57.14 50.11 56.92 50.. 36 56.70 50.61 76 77 53.11 50.52 57.89 50.77 57.67 51.02 57.45 51.27 77 78 58.87 51.17 58.64 51.43 .58.42 51.68 58.19 51.94 78 79 59.62 51.83 59.40 52.09 59.17 52.35 58.94 52.60 79 80 81 60.33 52.48 60.15 52.75 59.92 53.01 59.68 53.27 80 81 6 1 . r3 53.14 60.90 53.41 60.67 53.67 60.43 53.94 82 61.89 53.80 61.65 54.07 61.41 54.33 61.18 54.60 82 83 62.64 54.45 62.40 54.73 62.16 55.00 61.92 55.27 83 84 63.40 55.11 63.15 55.33 62.91 55.66 62.67 55.93 84 85 64.15 55.76 63.91 56.04 63.66 56.32 63.41 56.60 85 86 64.90 56.42 64.66 56.70 64.41 56.99 64.16 57.27 86 87 65.66 57.08 65.41 57.36 65.16 67.65 64.91 57.93 R7 88 66.41 57.73 66.16 58.02 65.91 58.31 65.65 58.60 88 89 67.17 58.39 66.91 58.68 66.66 58.97 66.40 59.26 89 90 91 67.92 59.05 67.67 59.34 67.41 59.64 67.15 59.93 90 68.68 59.70 68.42 60.00 68.15 60 .30 67.89 60.60 91 92 69.43 60.36 69.17 60.66 68.90 60.96 68.64 61.26 92 93 70.19 61.01 69.92 61.32 69.65 61.62 69.38 61.93 93 94 70.94 61.67 70.67 61.98 70.40 62.29 70.13 62.59 94 95 71.70 62.33 71.43 62.64 71.15 62.95 70.88 63.26 95 96 72.45 62.98 72.18 63.30 71.90 63.61 71.62 63.92 96 97 73.21 63.64 72.93 63.96 72.65 64.27 72.37 64.59 97 98 73.96 64.29 73.68 64.62 73.40 64.94 73.11 65,26 98 99 74.72 64.95 74.43 65.28 74.15 65.60 73.86 65.92 99 100 o 1 .2 75.47 65. 6L 75.18 65.93 74.90 66.26 74.61 66.. 59 100 u ,53 O Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 49 Deg. 48| Deg. ^8^ DQg. 48i Deg. 156 TBA VERSE TABLE. P 3 O 42 Deg. 424 Deg. 42^ Deg, 421 Deg. 5 re' P p 1 Lat. Dep. Lat. Dep. Lat. Dep. 1 1 Lat. Dep. 1 0.74 0.67 0.74 0.67 0.74 0.68 0.73 0,63 2 1.49 1.34 1.48 1.34 1,47 1.35 1.47 1.36 2 3 2.23 2.01 2.22 2.03 2.21 2.03 2.20 2.04 3 4 2.97 2.68 2.96 2.69 2.95 2.70 2.94 2.72 4 6 3.72 3.35 3.70 3.36 3.69 3.38; 3.07 3.39 5 6 4.46 4.01 4.44 4.03 4.42 4.05 1 4.41 4.07 6 7 5.20 4.68 5.18 4.71 5.16 4.73: 5.14 4.75 7 8 5.95 5.35 5.92 5.. 33 5.90 5.40 5.87 5.43 8 9 6.69 6.02 6.66 6.05 6.64 6.08 6.61 6.11 9 10 11 7.43 6.69 7.40 6.72 7,37 6.76! 7.34 6.79 10 8.17 7.36 j 8.14 7.40 8.11 7.43 1 8.08 7.47 11 12 8.92 8.03! 8.88 8.07 8,85 8.11 1 8.81 8.15 12 13 9.66 8.70 9.62 8.74 9.58 8.78, 9.55 8.82 13 14 10.40 9.37 10.36 9.41 10.32 9.46 10.28 9.. 50 14 15 11.15 10.04 11.10 10.09 11.06 10.13 11.01 10.18 15 16 11.89 10.71 11.84 10.76 11.80 10.81 11.75 10.86 16 17 12.63 11.38 12.58 11.43 12.. 53 11.48 12.48 11.-54 17 18 13.33 12.04 13.32 12.10 13.27 12.16 13.22 12.22 IS 19 14.12 12.71 14.06 12.77 1 14.01 12.84 13.95 12.90 19 20 21 14.86 13.38 14.80 13.45 14.75 13.51 14.69 13.53 20 15.61 14.05 15.. 54 14.12 15.48 14.19 15.42 14.25 21 22 16.35 14.72 16.23 14.79 16.22 14.86 16.16 14.93 22 23 17.09 15.39 17.02 15.46 ; 16,96 15.54 k 16.89 15.61 23 24 17.84 16.06 17.77 16.14 17.69 16.21 , 17.62 16.29 24 25 18.58 16.73 18.51 16.81 18.43 16.89 18.36 16.97 25 26 19.32 17.40 19.25 17.48 19.17 17.57 19.09 17.65 26 27 20.06 18.07 19.99 18.15 19.91 18.24 19.83 18.33 27 2S 20.81 18.74 20.73 18.83 20.64 18.92 20.. 56 19.01 28 29 21.55 19.40 21.47 19.50; 21.38 19.59 21.30 19.69 29 30 22.29 20.07 122.21 20.17 j 22-12 20.27 22.03 20.36 30 31 31 23.04 20.74 122.95 20.84' 22.86 20.94 22.76 21.04 32 23.78 21.41 23.69 21.52 1 23.59 21.62 23.50 21.72 32 33 24.. 52 22.08 24.43 22.19 i 24.-33 22.29 24.23 22.40 33 34 25.27 22.75 25.17 22.86! 25.07 22.97 24.97 23.03 34 35 26.01 23.42 25.91 23.53 1 25.80 23.65 25.70 23.76 .35 36 26.75 24.09 26.65 24.21 26.. 54 24.. 32 26.44 24.44 36 37 27 50 24.76 27.39 24.88 27-23 25.00 27.17 25.12 37 38 28.24 25.43 28.13 25.55 23.02 25.67 27.90 25.79 38 39 28.98 26.10 28.87 26.22 23.75 26.35 28.64 26.47 39 40 41 29.73 26.77 29.61 26.89 29.49 27.02 29.. 37 27,15 40 41 30.47 27.43 i 30.35 27.. 57 30.23 27.70 30.11 27.83 42 31.21 28.10 31.09 28.24 30.97 28.37 30.84- 28.51 42 43 31.96 28.77 31.83 28. 9L 31.70 29.05 31.58 29.19 43 44 32.70 29.44 32.57 29, 5-3 32.44 29.73 32-31 29.87 44 45 33.44 30.11 33.31 30.26 .33.18 30.40 33.04 30.. 55 45 46 34.18 30.78 34.05 30.93 33.91 31.08 33.78 31.22 46 47 34.93 31.45 34.79 31.60 34.65 31.75 34.51 31.90 47 48 35.67 32.12 35.63 32.27 35.39 .32.43 35.25 32.53 48 49 36.41 32.79 136.27 32.95 36.13 33.10 35^98 .33.26 49 SO 37.16 33.46 ,37.01 33.62 36.85 33.78 .36.72 .33.94 50 d 8 c Dep. Lat Dep. Lat. Dep. Lat, Dep. Lat 48 Deg. 47i Deg. 47h Deg, 41\ Deg. TBAVER8E TABUS 157 b w o ? "61 42 Deg. 42| Deg, 42i Deg, 421 Deg. 1* a a n 61 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 37.90 34.13 37.75 34.29 37.60 34.46 37.45 34.62 52 38.64 34.79 38.49 34.96 38.34 35.13 38.18 35.30 52 53 39.39 35.46 39.23 35.64 39.08 35.81 38.92 35.98 53 64 40.13 36.13 39.97 36.31 39.81 36.48 39.65 36.66 64 55 40.87 36.80 40.71 36.98 40.56 37.16 40.39 37.33 55 56 41.62 37.47 41.45 37.65 41.29 37.83 41.12 38.01 56 57 42.36 38.14 42.19 38.32 42.02 38.51 41.86 38.69 57 68 43.10 33.81 42.93 39.60 42.76 39.18 42.59 39.37 58 69 43.85 39.48 43.67 39.67 43.50 39.86 43.32 40.05 59 60 61 44.59 40.15 44.41 40.34 44.24 40.54 44.06 40.73 60 61 45.33 40.82 45.15 41.01 44.97 41 .21 44.79 41.41 62 46.07 41.49 45.89 41.69 45.71 41.89 45,53 42.09 62 63 46.82 42.16 46.63 42,36 46.45 42.56 46.26 42.76 63 64 47.58 42.82 47.37 43,03 47.19 43.24 47.00 43.44 64 65 48.30 43.49 48.11 43.70 47.92 43.91 47.73 44.12 65 66 49.05 44.16 48.85 44.38 48.66 44.59 48.47 44.80 66 67 49.79 44.83 49.59 45.05 49.40 45.26 49.20 45.48 67 68 50.53 45.50 50.33 45.72 50.13 45.94 49.93 46.16 68 69 51.28 46.17 51.07 46.39 .50.87 46.62 50.67 46.84 69 70 71 52.02 46.84 51.82 47.07 51.61 47.29 51.40 47.52 70 71 52.76 47.51 52.56 47.74 52.35 47.97 52.14 48.19 72- 53.51 48.18 53.30 48.41 53.08 48.64 6U.87 48.87 72 73 54.25 48.85 54.04 49,08 53.82 49.32 53.61 49.. 55 73 74 54.99 49.52 54.78 49.76 54.66 49.99 54.34 50.23 74 75 55.74 50.18 55.52 50.43 55.30 50.67 65.07 50.91 75 76 56.48 .50.85 56.26 51.10 56.03 61.34 55.81 51.59 76 77 67.22 51.52 57.00 6.1.77 56.77 62.02 56.. 54 52.27 77 78 67.97 52.19 57.74 52.44 57.61 52.70 57.28 52.95 78 79 58.7.1 52.86 58.48 53.12 58.24 53.37 58.01 53.63 79 80 81 59.45 53.53 59.22 59.96 53.79 58.98 54.05 58.75 54.30 80. 81 GO. 19 54.20 54.46 59.72 54.72 59*48 54.98 82 60.94 54.87 60.70 55.13 60.46 55.40 60.21 55.66 82 83 61.68 55.54 61.44 55.81 61.19 56.07 60.95 56.34 83 84 62.42 56.21 62.18 56.48 61.93 66.75 61.68 57.02 84 85 63.17 56.88 62.92 .57.15 62.67 57.43 62.42 57.70 85 86 63.91 57.55 63.66 57.82 63.41 68.10 63.15 58.38 86 87 64.65 58.21 64.40 58.50 64.14 58.78 63.89 59.06 87 88 65.40 58.88 65.14 59.17 64.88 59.45 64.62 59.73 88 89 66.14 59.55 65.88 59.84 65.62 60.13 65.35 60.41 89 90 91 66.88 60.22 66.62 60.51 66.35 60.80 66.09 61.09 90 91 67.63 60.89 67.36 61.19 67.09 61.48 66.82 61.77 92 68.37 61.56 68.10 61.86 67.83 62.15 67.56 62.45 92 93 69.11 62.23 68.84 62.53 68.57 62.83 68.29 63.13 93 94 69.86 62.90 69.58 63.20 69.30 63.51 69.03 63.81 94 95 70.60 63.57 70.32 63.87 70.04 64.18 69.76 64.49 95 96 71.34 64.24 71.06 64.55 70.78 64.86 70.49 65.16 96 97 72.08 64.91 71.80 65.22 71.52 65.53 71.23 65.84 97 98 72.83 65.57 72.54 65.89 72.25 66.21 71.96 66.52 98 99 73.57 66.24 73.28 66.56 72.99 66.88 72.70 67.20 99 100 6 o a 74.31 66.91 74.02 67.24 73.73 67.56 73»43 67.88 100 8 1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. >48 Degi 47} Deg. 47i Deg. 47i Deg, 158 TRAVESSE TAEIT:. 5' P 3 o 1 ! 43Deg. 43i Deg. 43i Deg. 431 Deg. D « ? 1 Lat. Dep. Lat. 1 Dep. Lat. Dep. Lat. Dep. 0.73 0.63 0.73 0.69 i 0.73 0.69 0.72 0.69 2 1.46 1.36i 1.46 1.37 ! 1.45 1.33 1.44 1.38 2 3 2.19 2.05^ 2.19 2.06 1 2.13 2.07 i 2.17 2.07 3 4 2.93 2.73 2.91 2.74 2.90 2.75 j 2.39 2.77 4 5 3.66 3.41 3.64 1 4.37 3.43 3.63 3.44 3.61 3.46 5 6 4.39 4.09 4.11 4.35 4.13 4.33 4.15 6 7 5.12 4.77 5.10 4.80 i 5.08 4.82 5.06 4.34 7 8 5. 85 5.46 5.83 5.43 ; 5.80 5.51 5.73 5.53 8 9 6. 58 6.14 , 6.56 6.17 ' 6.53 6.20 6.50 6.22 9 10 11 7.31 6. 32 I 7.23 6.85 ' 7.25 6.83 7.22 i 6 92 10 8.04 7.50 1 8.01 7.54 1 7.93 7.57 7.95 7.61 11 12 8.73 8.18 8.74 8.22 8.70 8.26 8.67 8.30 12 13 9.51 8.87 1 9.47 8.91 9.43 8.95 9.39 8.99 13 14 10.24 9.55 10.20 9.59 ; 10.16 9.64 10.11 9.63 14 15 10.97 10.23 10.93 10.23 ■ 10.88 10.33 ! 10.84 10.37 15 16 11.70 10.91 11.65 10.96 11.61 11.01 ! 11.56 11.06 16 17 12.43 11.59 ' 12.33 11.65 12.33 11.70. 12.28 11.76 17 IS 13.16 12.23 ; 13.11 12 . 33 L3.06 12.39 13.00 12.45 18 19 13.90 12.98 13.84 13.02 ; 13.78 13.03 13.72 13.14 19 20 14.63 13.64; 14.57 13.70 i 14.51 13.77; 14.45 13.83 20 21 21 15.36 14.32^ 15.30 14.39 15.23 14.46 15.17 14.52 22 16.09 15.00. 13.02 15.07 15.98 15.14! ,15.89 15.21 22 23 16.32 15.69 16.75 15.76 ; 16.63 15.83 16.61 15.90 23 24 17.55 16.37 17.48 10.44 17.41 16.52 17.34 16.60 24 2o IS. 23 17.05 13.21 17.13 ■ 18.13 17.21 : 18.06 17.29 25 26 19.02 17.73 13.94 17.81 '13.86 17.90, 18.73 17.93 26 27 19.75 18.41 • 19.67 IS 50 '19.59 18.59 19.50 18.67 27 23 20.43 18f.l0 20.39 19.19 20.31 19.27, 20.23 19.36 28 29 21.21 19.78, 21.12 19.87 21.04 19.96; 20.95 20.05 29 30 31 21.94 20.46; 21.85 20.56 21.76 20.65 ; 21.67 20.75 30 31 22.67 21.14, 22.58 21.24' 22.49 21.34; 22.39 21.44 32 23.40 21. 82- 23.31 21.93 : 23.21 22.03 23.12 22.13 32 33 24.13 22.51 ; 24.04 22.61 ; 23.94 22.72 , 23.84 22.82 33 34 24.87 23.19; 24.76 23.30 24.66 23.40 24.56 23.51 34 3o 25.60 23.87: 25.49 23.93 25.39 24.09 . 25.23 24.20 35 36 26.33 24.55! 26.22 24.67: 26.11 24.73 26.01 24.89 36 37 27.06 25.23 : 26.95 25.35] 26.84 26.47 26.73 25.59 37 38 27.79 25.92; 27.63 26.04 27.56 26.16: 27.45 26.28 33 39 2S.52 26.60' 23.41 26 . 72 i 23.29 26.85 23.17 26.97 39 40 41 29.25, 27.28; 29.13 27.41 ; 29.01 27.53 ; 28.89 29 . 62 27.66 40 41 29.99 27.96, 29.86 23.09' 29.74 23.22 23.35 42 30.72 28.64: .30.59 23.73 '[ 30.47 23.91 ; 30.34 29.04 42 43 31.45 29.33 31.32 29.48 ! 31.19 29.60 31.06 129.74 43 44 32.13 30.01 ' 32.05 30.15: 31.92 30.29 ; 31.73 30.43 44 45 32.91 30.69! 32.73 30.83 32.64 30.98 , 32.51 131.12 45 46 33.64 31.37; 33.51 31.52 33.37 31.66 ; 33.23 31.81 46 47 34.37 32.05' 34.23 32.20 34.09 32.35 : 33.95 32.50 47 43 35.10 132.74! 34.96 32.89 34.82 33 . 04 . 34.67 33.19 43 49 35.84 33.42' 35.69 33.57, 35.54 33.73 ; 35.40 33,83 49 50 o w C «i ■ 5 36.57 j 34.10 36.42 34.26 ; 30.27 34.42 ! 36.12 34.53 50 S ■-> ■5 Dep. 1 Lat. ' Dep. Lat Dep. Lat. I Dep. Lat. 47 Deg. 461 Deg. II 46^ Deg. 46i Deg. TRAVERSE TABLE. 159 o CD 51 43Deg. 43i Peg. 43i Deg. 431 Deg. 5 5' o o 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 37.30 34.78 37.15 34.94 36.99 35.11 36.84 35.27 52 38.03 35.46 37.88 35.63 37.72 35.79 37.56 35.96 52 63 38.76 36.15 38.60 36.31 38.44 36.48 38.29 36.65 53 54 39.49 36.83 39.33 37.00 39.17 37.17 .39.01 37.34 64 55 40.22 37.51 40.06 37.69 39.90 37.86 39.73 38.03 65 56 40.96 38.19 40.79 38.37 40.62 38.55 40.45 38.72 66 57 41.69 38.87 41.52 39.06 41.35 39.24 41.17 39.42 67 58 42.42 39.56 42.25 39.74 42.07 39.92 41.90 40.11 68 59 43.15 40.24 42.97 40.43 42.80 40.61 42.62 40.80 69 60 61 43.88 40.92 43.70 41.11 43.52 41.30 43 34 41.49 60 61 44.61 41.60 44.43 41.80 44.25 41.99 44.06 42.18 62 45.34 42.28 45.16 42.48 44.97 42.68 44.79 42.87 62 63 46.08 42.97 45.89 43.17 45.70 43.37 45.51 43.57 63 64 46.81 43.65 46.62 43.85 46.42 44.05 46.23 44.26 64 65 47.54 44.33 47.34 44.54 47.15 44.74 46.95 44.95 65 C6 48.27 45.01 48.07 45.22 4*7.87 45.43 47.68 45.64 66 67 49,00 45.69 48.80 45.91 48.60 46.12 48.40 46.33 67 68 49.73 46.38 49.53 46.59 49.33 46.81 49.12 47.02 6S 69 50.46 47.06 50.26 47.28 50.05 47.50 49.84 47.71 69 70 71 51.19 47.74 .50.99 47.96 50.78 48.18 50.57 48.41 70 71 51.93 48.42 51.71 48-65 51.50 48.87 61.29 49.10 72 52.66 49.10 52.44 49.33 52.23 49.56 52.01 49.79 72 73 53.39 49.79 53.17 50.02 62.95 50.25 .52.73 50.48 73 74 54.12 50.47 53.90 50.70 53.68 50.94 53.46 51.17 74 75 54.85 51.15 54.63 51.39 64.40 51.63 54.18 51.86 75 76 55.58 51.83 55.36 52.07 55.13 52.31 .54.90 52.55 76 77 56.31 52.51 56.08 52.76 55.85 53.00 55.62 53.25 77 78 57.05 53.20 56.81 53.44 56.58 53.69 56.34 53.94 78 79 57.78 53.88 57.54 54.13 57.30 54.38 57.07 54.63 79 80 81 58.51 54.56 58.27 54.81 58.03 55.07 57.79 58.51 55.32 56.01 80 81 59.24 55.24 59.00 55.50 58.76 65.76 82 59.97 55.92 59.73 56.18 59.48 56.45 59.23 56.70 82 83 60.70 50.61 60.45 56.87 60.21 57.13 .59.96 57.40 83 84 61.43 57.29 61.18 67.56 60.93 57.82 60.68 58.09 84 85 62.17 57.97 G1.91 58.24 61.60 58.51 61.40 58.78 85 88 62.90 58.65 62.64 58.93 62.38 59.20 62.12 59.47 86 87 63.63 59.33 63.37 59.61 63.11 59.89 62.85 60.10 87 88 64.36 60.02 64.10 60.30 63.83 60.58 63.57 60.85 88 89 65.09 60.70 64.82 60.98 64.56 61.26 64.29 61.54 89 90 91 65.82 01.38 65.55 '66'. 28 61.67 65.28 61.95 65.01 62.24 90 91 66.55 62.06 62.35 66.01 62.64 65.74 62.93 92 67.28 02.74 67.01 03.04 66.73 63.33 66.46 63.62 92 93 68.02 63.43 67.74 63.72 07.46 64.02 67.18 64.31 93 94 68.75 64.11 68.47 64.41 68.19 164.71 67.90 65.00 94 95 09.48 64.79 09.20 65.09 68.91 65.39 68.62 65.69 95 96 70.21 65.47 69.92 65.78 69 . 64 66.08 69.35 66.39 96 97 70.94 66.15 70.65 66.46 70.30 66.77 70.07 67.08 97 98 71.67 66.84 71.37 67.15 71.09 67.46 70.79 67.77 98 99 72.40 67.52 72.11 67.83 71.81 68.15 71.51 68.46 99 100 o a a «-> to 73.14 88.20 72.84 68 . 52 72.54 68.84 72 . 24 69.15 100 o o *-> 03 Q Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 47 Deg. .461 Deg. 1 46-^- Deg. 464 Deg. 30 160 TRAVEKSE TABLE. 5 so p 3 O p 44 Deg. 44i Deg. 44i Deg. 44| Deg. 45 Deg. O 55' <-► p 3 o 9 Lai. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.72 0.69 0.72! 0.70 0.71 0.70 0.71 0.71 0.71 0.71 1 2 1.44 1.39 1.43 1.40 1.43 1.40 1.42 1.41 1.41 1.41 2 3 2.16 2.08 2.15 2.09 2.14 2.10- 2.13 2.11 2.12 2.12 3 4 2.88 2.78 2.87 2.79 2.85 2.80 2.84 2.82 2.83 2.83 4 5 3.60 3.47 3.58 3.49 3.57 3.50 3.55 3.52 3.54 3.54 5 6 4.32 4.17 4.30 4.19 4.28 4.21 4.26 4.22 4.24 4.24 6 7 5.04 4.86 5.01 4.88 4.99 4.91 4.97 4.93 4.95 4.95 7 8 5.75 5.56 5.73 5.58 5.71 5.61 5.68 5.63 5.66 5.66 8 9 6.47 6.25 6.45 6. 28 G.42 6.31 6.39 6.34 6.36 6.36 9 lU 11 7.19 7.91 6.95 7.16 6.98 7.13 7.01 7.71 7.10 7.04 7.07 7.07 10 11 7.64 7.88 7,68 7.85 7.81 7.74 7.78 7.78 12 8.63 8.04 8.60 8.37 8.56 8.41 8.52 8.45 8.49 8.49 12 13 9.35 9.03 9.31 9.07 9.27 9.11 9.23 9.15 9.19 9.19 13 14 10.07 9.73 10,03 9.77 9.99 9.81 9.94 9.86 9.90 9.90 14 15 10.79 10.42 10.74 10.47 10.70 10.51 10.65 10.56 10.61 10.61 15 16 11.51 11.11 11.46 11.16 11.41 11.21 11.36 11.26 11.31 11.31 16 17 12.23 11.81 12.18 11.86 12.13 11.92 12.07 11.97 12.02 12.02 17 18 12.95 12.50 12.89 12.56 12.84 12.62 12.78 12.67 12.73 12.73 18 19 13.67 13.20 13.61 13.26 13.. 55 13.32 13.49 13.38 13.43 13.43 19 20 21 14.39 13.89 14.33 13.96 14.26 14.02 14.20 14.08 14.14 14.14 20 21 15.11 14.59 15.04 14.65 14.98 14.72 14.91 14.78 14.85 14.85 22 15.83 15.28 15.76 15.35 15.69 15.42 15.62 15.49 15.56 15.56 22 23 16.54 15.98 16.47 16.05 16.40 16.12 16.33 16.19 16.26 16.26 23 24 17.26 16.67 17.19 16.75 17.12 16.82 17.04 16.90 16.97 16.97 24 25 17.98 17.37 17.91 17.44 17.83 17.52 17.75 17.60 17.68 17.68 25 26 18.70 18.06 18.62 18.14 18.54 18.22 18.46 18.30 18.38 18.38 26 27 19.42 18.76 19.34 18.84 19.26 18.92 119.17 19.01 19.09 19.09 27 28 20.14 19.45 20.06 19.541 19.97 19.63,19.89 19.71 19.80 19.80 28 29 20.86 20.15 20.77 20.24 20.68 20.33 20.60 20.42 20.51 20.51 29 30 21.58 20.84 21.49 20.93 21.40 22.11 21.03 21.73 21.31 ,22.02 21.12 21.21 21.21 30 21.92 31 31 22.30 21.53 22.21 21.63 21.82 21.92 32 23.02 22.23 22.92 22.33 22.82 22.43 22.73 22.53 22.63 22.63 32 33 23.74 22.92 23.64 23.03 23.54 23.13 23.44 23.23 23.33 23.33 33 34 24.46 23.62 24.35 23.72 24.25 23.83 '24.15 23.94 24.04 24.04 34 35 25.18 24.31 25.07 24.42 24.96 24.53 24.86 24.64 24.75 24.75 35 36 25.90 25.01 25.79 25.12 25.68 26.23 125.. 57 25.34 25.46 25.46!36l 37 26.62 25.70 26.50 25.82 26.39 25.93 26.28 26.05 26.16 26.16 37 38 27.33 26.40 27.22 26.52 27.10 26.63 [26.99 26.75 26.87 26.87 38 39 28.05 27.09 27.94 27.21 27.82 27.34 27.70 27.46 27.58 27.58 39 40 41 28.77 29.49 27.79 28.65 27.91 28 . 53 28.04 28.41 (29.12 28.16 28.28 28.28 40 41 28.48 29.37 28.61 29.24 28.74 28.86 28.99 28.99 42 30.21 29.18 30.08 29.31 29.96 29.44 ,29.83 29.57 29.70 29.70 42 43 30.93 29.87 30.80 30.00 30.67 30.14 30.54 30.27 30.41 30.41 43 44 31.65 30.56 31.52 30.70 31.38 30.84 31.25 30.98 31.11 31.11 44 45 32.37 31.26 32.23 31.40 .32.10 31.54 31.96 31. 6S 31.82 31.82 45 46 33.09 31.95 32.95 32.10 32.81 32.24 32.67 32.38 32.53J32.53 46 47 33.81 32.05 33.67 32.80 33.. 52 32.94 33.38 33.09 33.23 33.23 47 48 34.53 33.34 34.38 33.49 34.24 33.64 34.09 33.79 .33.94 33.94 48 49 35.25 34.04 3.5.10 34.19 34.95 34.34 34.80 34.50 34.65 34.65 49 50 § 5 35.97 Dep. 34.73 35.82 34.89 35.66 35.05 35.51 35.20 36.36 35.36 50 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 6 c •a o 46 Deg. 45| Deg. 45^ Deg. 45k Deg. 45 De^., TSAVBBSE TABLE 161 Ul' p> r> 9 51 52 53 54 55 66 57 58 69 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 .75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 8 B 2 Q 44Deg. 44i Deg. 44i Deg, 44| Deg. 45 Deg. CO r* P» 3 9 61 62 63 54 65 56 57 58 59 60 61 62 63 04 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 S d 5 Lat. Dep. Lat. Dep. Lai. Dep. Lat. Dep. 35.90 36.61 37.31 38.02 38,721 39,42; 40.13 40.83 41.54 42.24 Lat. Dep. 36.69 37.41 38.12 38.84 39.56 40.28 41.00 41.72 42.44 43.16 35.43 36.12 36.82 37.51 38.21 38.90 39.60 40.29 40.98 41.68 36.53 37.25 37.96 38.68 39.40 40.11 40.83 41.55 42.26 42.98 35,59 36.29 36.98 37.68 38.38 39.08 39.77 40.47 41.17 41.87 36.38 37.09 37.80 38.52 39.23 39.94 40.66 41.37 42.08 42.79 43.51 44.22 44.93 45.65 46.36 47.07 47.79 48.60 49.21 49.93 60.64 51.35 52.07 52.78 53.49 54.21 54.92 55.63 56.35 57.06 35.75 36.45 37.15 37.85 38.55 39,25 39.95 40.65 41.35 42.05 36.22 36.93 37.64 38.35 39.06 39.77 40.48 41.19 41.90 42.61 36.06 36,77 37,48 38,18 38,89 39.60 40.31 41,01 41.72 42,43 36.06 36.77 37,48 38.18 38.89 39.60 40,31 41,01 41.72 42.43 43. 8S 44.60 45.32 46.04 46.76 47.48 48,20 48.92 49.63 50.35 42.37 43.07 43.76 44.46 45.15 45.85 46.54 47.24 47.93 48.63 43.69 44.41 45.13 45.84 46.56 47.28 47.99 48.71 49.42 50.14 42.57 43.26 43.96 44.66 45,36 46.05 46.75 47.45 48.15 48.85 42.76 43.46 44.16 44.86 45.56 46.26 46.96 47.66 48.36 49.06 43.32 44.03 44,74 45.45 46.16 46.87 47.58 48,29 49.00 49.71 42.94 43.65 44.35 45.06 45.76 46.46 47.17 47.87 48.58 49.28 43,13 43.84 44.55 45.25 45.96 46.67 47.38 48.08 48.79 49.50 43.13 43.84 44.55 45.25 45.96 46.67 47.38 48.08 48.79 49.50 50.20 50.91 51.62 52.33 53.03 53.74 54.45 55.15 55,86 56.57 51.07 51.79 52.51 53.23 53.95 54.67 55.39 56.11 56.83 57.55 49.32 50.02 50.71 51.40 52.10 ,52.79 53.49 54. 18 54.88 55.57 56.27 56.96 57.66 58.35 59.05 59.74 60.44 61.13 61.82 62.52 50.86 51.67 52.29 53.01 53.72 54.44 55.16 55.87 56.59 .57.30 49.54 50.24 50.94 51.64 52.33 53.03 .53.73 54.43 55.13 55.82 49.76 50.47 51.17 51.87 ,52.57 53.27 53.97 54-67 55.37 56.07 50.42 51.13 51-84 52.55 53.26 53,97 54.68 ;55.39 156.10 56.81 49.98 50.69 51.39 52.10 52.80 53.51 54.21 54.91 55.62 56.32 50.20 50.91 51.62 52.33 53.03 53.74 54.45 55.15 55.86 56.67 57.28 57.98 58.69 59.40 60.10 60.81 61.52 62.23 62.93 63.64 58.27 58.99 59.71 60.42 61.14 61.86 62.58 63.30 64.02 64.74 58.02 58.74 59.45 60.17 60.89 61.60 62.32 63.03 63.75 64.47 56.52 57.22 57.92 58.61 59.31 60.01 60.71 61.41 62,10 62.80 57.77 58.49 59.20 59.91 60.63 61.34 62.05 62,77 63.48 64.19 64.91 65.62 66.33 67.05 67.76 68.47 69.19 69.90 70.61 71.33 56.77 57.47 58.18 58.88 59.58 60.28 60.98 61.68 62.. 38 63.08 57.52 58.24 •68.95 159.66 160.37 i6l.08 161.79 162.50 163.21 63.92 64.63 65.34 66.05 66.76 67.47 68.18 68.89 69.60 70.31 71.02 57.03 57.73 58.43 59.14 59.84 60.65 61.25 61.95 62,66 63.36 57.28 57.98 58.69 59:40 60.10 60.81 61.52 62.23 62.93 63,64 64.35 65.05 65.76 66.47 67.18 67.88 68.59 69.30 70.00 70.71 65.46 66.18 66.90 67.62 68.34 69.06 69.78 70.50 71.21 71.93 63.21 63.91 64.60 65.30 65.99 66.69 67.38 68.08 68.77 69.47 65.18 65.90 66.62 67.33 68.05 68.76 69.48 70.20 70.91 71.63 63.50 64.20 64.89 65.59 66.29 66.99 67.69 68.38 69.08 69.78 63.78 64.48 65.18 65.89 66.59 67.29 67.99 68.69 69.39 70.09 64.07 64.77 65-47 66.18 66.88 67,59 68,29 68.99 69.70 70.40 64.35 66,05 65,76 66,47 67.18 67.88 68.59 69.30 70.00 70.71 Dep. 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Cloth, i2mo, 302 pages . . . $1.15 A new edition of this popular work for beginners in the study and for the general reader. The book has been entirely rewritten, and improved by the addition of many new illustrations and interesting descriptions of the latest phases and discoveries of the science. In contents and dress it is an attractive volume either for the reader or student. Dana's New Text-Book of Geology By James D. Dana. Cloth, i2mo, 422 pages . . . $2.00 A text-book for classes in secondary schools and colleges. This standard work has been thoroughly revised and considerably enlarged and freshly illustrated to represent the latest demands of the science. Dana's Manual of Geology By James D. Dana. Cloth, 8vo, 1087 pages. 1575 Illustrations .... $5.00 Fourth revised edition. This great work was thoroughly revised and entirely rewritten under the direct supervision of its author, just before his death. It is recog- nized ^s a standard authority in the science both in Europe and America, and is used as a manual of instruction in all the higher institutions of learning. Le Conte's Compend of Geology By Joseph Le Conte, LL.D. Cloth, i2mo, 399 pages . $1.20 Designed for high schools, academies and all secondary schools. Steele's Fourteen Weeks in Geology By J. DoRMAN Steele, Ph.D. Cloth, i2mo, 280 pages . $1.00 A popular book for elementary classes and the general reader. Andrews's Elennentary Geology By E. B. Andrews, LL.D. Cloth, i2mo, 283 pages . $1.00 Adapted for elementary classes. Contains a special treatment of the geology of the Mississippi Valley. Nicholson's Text-Book of Geology By H. A. Nicholson, M.D. Cloth, i2mo, 520 pages . $1.05 A brief course for higher classes and adapted for general reading, Willianas's Applied Geology By S. G. Williams, Ph.D. Cloth, i2mo, 386 pages . . $1.20 A treatise on the industrial relations of geological structure; and on the nature, occurrence, and uses of substances derived from geological sources. Copies of any of the above books will be sent prepaid to any address ^ on receipt of the price ^ by the Publishers : American Book Company New York ♦ Cincinnati ♦ Chicago (93) Text-Books in Astronomy Bowen's Astronomy by Observation By Eliza A. Bowen. Boards, quarto, 94 pages. Colored Maps and Illustrations $1 .00 An elementary text-book for schools, and especially adapted for use as an atlas to accompany any other text-book in astronomy. Careful directions are given when, how and where to find the heavenly bodies, and the quarto pages admit star maps and views on a large scale. Gillet and Rolfe's Astronomies By T. A. Gillet and W. J. Rolfe. First Book in Astronomy. Short Course. 220 pages . . $1.00 Astronomy. 415 pages 1.40 These books have been prepared by practical teachers and contain nothing beyond the comprehension of pupils in secondary schools, Lockyer's Astronomies By J. N. LocKYER, F.R.S. Astronomy. (Science Primer Series.) 136 pages . 35 cents Elementary Lessons in Astronomy, 312 pages . . . $1.22 The aim throughout these books is to give a connected view of the whole subject rather than to discuss any particular parts of it, and to supply facts and ideas founded thereon, to serve as a basis for subsequent study. Ray's New Elements of Astronomy By Selim H. Pe-\body, Ph.D., LL.D. Cloth, i2mo, 352 pages $1 .20 The elements of astronomy, with numerous engravings and star maps. In the revised edition, the scope and method of the original is retained, with the addition of all the results of established discovery. The book treats of the facts, principles, and processes of the science, presuming only that the pupil is acquainted with the simplest principles of mechanics and physics. Steele's New Descriptive Astronomy By J. D0R3IAN Steele, Ph.D. Cloth, i2mo, 338 pages , $1.00 This book is written in the same interesting and popular manner as other books of the Steele Series, and is intended for the inspiration of youth rather than for the information of scientific scholars. The book conforms to the latest discoveries and approved theories of the science. It supplies an adequate course in astronomy for all secondary schools and college preparatory classes. Copies of any of tJu above books will be sent prepaid to any address, on receipt of the price^ by the Publishers •' American Book Company New York ♦ Cincinnati ♦ Chicago (94) Physical Geography Appletons' Physical Geography By John D. Quackenbos, John S. Newberry, Charles H. Hitchcock, W. Le Conte Stevens, Wm. H. Dall, Henry Gannett, C. Hart Merriam, Nathaniel L. Britton, George F. Kunz and Lieut, Geo. M. Stoney. Cloth, quarto, 140 pages ....... $1 .60 Prepared on a new and original plan. Richly illustrated with engrav- ings, diagrams and maps in color, and including a separate chapter on the geological history and the physical features of the United States. The aim has been to popularize the study of Physical Geography by furnishing a complete, attractive, carefully condensed text-book. Cornell's Physical Geography Boards, quarto, 104 pages . . . , . . $1,12 Revised edition, with such alterations and additions as were found necessary to bring the work in all respects up to date. Hinman's Eclectic Physical Geography Cloth, i2mo, 382 pages $1.00 By Russell Hinman. A model text-book of the subject in a new and convenient form. It embodies a strictly scientific and accurate treatment of Physiography and other branches of Physical Geography. Adapted for classes in high schools, academies and colleges, and for private students. The text is fully illustrated by numerous maps, charts, cuts and diagrams. Guyot's Physical Geography Cloth, quarto, 124 pages $1.60 By Arnold Guyot. Thoroughly revised and supplied with newly engraved maps, illustrations, etc. A standard work by one of the ablest of modern geographers. All parts of the subject are presented in their true relations and in their proper subordination. Monteith's New Physical Geography Cloth, quarto, 144 pages ....... $1.00 An elementary work adapted for use in common and grammar schools, as well as in high schools. Copies of any of the above books will be sent prepaid to any address ^ on receipt of the price ^ by the Publishers: American Book Company New York ♦ Cincinnati ♦ Chicago (q8) STORER AND LINDSAY'S Elementary Manual of Chemistry By F. H. STORER, S.B., A.M., and W. B. LINDSAY, A.B., B.S. Cloth, 12mo, 453 pages. Illustrated. Price, $1.20 This work is the lineal descendant of the " Manual of Inorganic Chemistry" of Eliot and Storer, and the ''Ele- mentary Manual of Chemistry " of Eliot, Storer and Nichols. It is in fact the last named book thoroughly revised, rewritten and enlarged to represent the present condition of chemical knowledge and to meet the demands of American teachers for a class book on Chemistry, at once scientific in statement and clear in method. The purpose of the book is to facilitate the study and teaching of Chemistry by the experimental and inductive method. It presents the leading facts and theories of the science in such simple and concise manner that they can be readily understood and applied by the student. The book is equally valuable in the classroom and the laboratory. The instructor will find in it the essentials of chemical science developed in easy and appropriate sequence, its facts and generalizations expressed accurately and scientifi- cally as well as clearly, forcibly and elegantly. " It is safe to say that no text-book has exerted so wide an influence on the study of chemistn' in this countn^ as this work, originally written by Eliot and Storer. Its distinguished authors were leaders in teaching Chemistr}' as a means of mental training in general edu- cation, and in organizing and per- fecting a system of instructing students in large classes by the experimental method. As revised and improved by Professor Nichols, it continued to give the highest satisfaction in our best schools and colleges. After the death of Pro- fessor Nichols, when it became necessary to revise the work again, Professor Lindsay, of Dickinson College, was selected to assist Dr. Storer in the work. The present edition has been entirely rewritten by them, following throughout the same plan and arrangement of the previous editions, which have been so highly approved by a generation of scholars and teachers. " If a book, like an individual, has a history, certainly the record of this one, covering a period of nearly thirty years, is of the highest and most honorable character," — From The American Jotirnal of Science, Copies o/ this book will be sent prepaid to any address^ on receipt qf the prict^ by the Publishers : New York (99) American Book Company ♦ Cincinnati ♦ Chicago Burnet's Zoology FOR HIGH SCHOOLS AND ACADEMIES BY MARGARETTA BURNET Teacher of Zoology, Woodward High School, Cincinnati, O. Cloth, 12mo, 216 pages. Illustrated. Price, 75 cents This new text-book on Zoology is intended for classes in High Schools, Academies, and other Secondary Schools. While sufficiently elementary for beginners in the study it is full and comprehensive enough for students pursuing a regular course in the Natural Sciences. It has been prepared by a practical teacher, and is the direct result of school-room experience, field observation and laboratory practice. The design of the book is to give a good general knowl- edge of the subject of Zoology, to cultivate an interest in nature study, and to encourage the pupil to observe and to compare for himself and then to arrange and classify his knowledge. Only typical or principal forms are described, and in their description only such technical terms are used as are necessary, and these are carefully defined. Each subject is fully illustrated, the illustrations being selected and arranged to aid the pupil in understanding the structure of each form. Copies of Burnefs School Zoology will be sent prepaid to any address ^ on receipt of the price ^ by the Publishers: American Book Company New York ♦ Cincinnati ♦ Chicago C102) Laboratory Physics Hammers Observation Blanks In Physics By William C. A. Hammel, Professor of Physics in Maryland State School. Boards, Quarto, 42 pages. Illustrated. 30 cents These Observation Blanks are designed for use as a Pupil's Laboratory Manual and Note Book for the first term's work in the study of Physics. They combine in convenient form descriptions and illustrations of the appa- ratus required for making experiments in Physics, with special reference to the elements of Air, Liquids, and Heat; directions for making the required apparatus from simple inexpensive materials, and for performing the experiments, etc. The book is supplied with blanks for making drawings of the apparatus and for the pupil to record what he has observed and inferred concerning the experiment and the principle illustrated. The experiments are carefully selected in the light of experience and arranged in logical order. The treatment throughout is in accordance with the best laboratory practice of the day. Hon. W. T. Harris, U. S. Commissioner of Education, says of these Blanks: " I have seen several attempts to assist the work of pupils engaged in the study of Physics, but I have never seen anything which promises to be of such practical assist- ance as Hammel's Observation Blanks." Specimen copies of the above book will be sent prepaid to any address, on receipt of the price ^ by the Publishers : American Book Company New York ♦ Cincinnati ♦ Chicago (103) Standard Text-Books in Botany Gray's How Plants Grow. (Introductory Book) . Gray's How Plants Behave For Beginners in Primary and Intermediate Schools. Gray's Lessons in Botany. (Revised) .... Gray's Field, Forest and Garden Botany. (Flora) . Gray's School and Field Botany. (The Standard Text-Book) For Students in High Schools, Academies and Seminaries. Gray's Manual of Botany. (Flora) Gray's Lessons and Manual. (In one volume) . For Advanced Students, Teachers, and Practical Botanists. Coulter's Botany of the Rocky Mountains A flora adapted to the mountain section of the United States. Gray and Coulter's Text-Book of Western Botany . Being Gray's Lessons and Coulter's Manual bound in one volume. Gray's Structural Botany Goodale's Physiological Botany Dana's Plants and their Children Herrick's Chapters on Plant Life Hooker's Botany. (Science Primer Series) .... Hooker's Child's Book of Nature. Part I. Plants Steele's Fourteen Weeks in Botany .... Wood's How to Study Plants Same as above work, with added chapters on Physiological and Sys- tematic Botany. Wood's Lessons in Botany. (Revised) .... Wood's New Annerican Botanist and Florist. (Revised) . Wood's Descriptive Botany ...... Beinp- the flora of the American Botanist and Florist. Wood's Class Book of Botany A standard work for Advanced Classes and for the Student's Library. Younnans's First Book in Botany Younaans's Descriptive Botany . . . Bentley's Physiological Botany A sequel to Youmans's Descriptive Botany. Willis's Practical Flora A valuable supplementary aid to any text-book in the study of Botany. 80 cents 54 cents 94 cents $1.44 $1.80 $1.62 $2.16 $1.62 $2.16 $2.00 $2.00 65 cents 60 cents 35 cents 44 cents $1.00 $1.00 90 cents $1.75 $1.25 $2.50 64 cents $1.20 $1.20 $1.50 Copies of the above books will be sent, prepaid^ to any address on receipt of the price by the Publishers : American Book Company New York ♦ Cincinnati ♦ Chicago (loo) Physiology and Hygiene Kellogg's First Book in Physiology and Hygiene Cloth, i2mo, 174 pages 40 cents Kellogg's Second Book in Physiology and Hygiene Cloth, i2mo, 2gi pages 80 cents These two books constitute an entirely new and well graded series for the study of Physiolog}' and Hygiene in schools. The subjects are treated in a natural and logical order and arranged in a form suitable for class instruction. The important subjects of sanitation and temperance are thoroughly treated from a scientific and physiological standpoint. Snnith's Prinner of Physiology and Hygiene Cloth, i2mo, 174 pages 30 cents Snnith's Elementary Physiology and Hygiene Cloth, i2mo, 225 pages ....... 50 cents A complete and symmetrical series in which the important facts of Physiology and Hygiene are presented in an interesting manner. The Primer is designed for beginners in the study and the second book for classes in the intermediate grades. Steele's Hygienic Physiology Cloth, i2mo, 400 pages $1 .00 This standard text-book has been thoroughly revised and consider- ably enlarged. It contains all the excellent and popular features that have given Dr. Steele's Science Series such wide use in schools throughout the country. The Same, abridged. Cloth, i2mo, 192 pages . . 50 cents Tracy's Essentials of Anatomy, Physiology and Hygiene Cloth, i2mo, 345 pages $1.00 A practical, thorough and scientific text-book of an advanced grade for the use of classes in High Schools, Academies, Normal Schools, and for private students. Johonnot and Bouton's How We Live Cloth, i2mo, 178 pages 40 cents An elementary text-book for beginners in which special attention is given to the laws of Hygiene. Walker's Health Lessons Cloth, i2mo, 194 pages 48 cents A book for beginners, presenting the subjects in an interesting and readable form suitable for supplementary readings. Copies of any of the above books will be sent prepaid to any address^ on receipt of the price, by the Publishers: American Book Company New York * Cincinnati « Chicago (95) lb D 78