ELEMENTS OF SURVEYING AND LEVELLING; DESCRIPTIONS OF THE IISrSTRUiyiENTS AND THE NECESSARY TABLES. By CHAKLES D.AYIES, LL.D., AUTHOR OP A rULL COUBSB Oj" MATHEMATICS. < A. S. BAEITES & COMPANY, NEW YORK AND CHICAGO. 1870. DAVIES' COUESE OF MATHEMATICS. Davies' Primary Arithmetic and Table-Book — Designed for Beginners; containing tlie elementary tables of Addition, Subtraction, Multiplication, Division, and Denominate Numbers; with a large number of easy and practical questions, both mental and written. Davies' First Lessons in Arithmetic — Combining the Oral Method with the Method of Teaching the Combinations of Figures by Sight. Davies' Intellectual Arithmetic — An Analysis of the Science of Numbers, with especial reference to Mental Training and Development. Davies' New School Arithmetic — Analytical and Practical. Key to Davies' New School Arithmetic. Davies' Grammar of Arithmetic — An Analysis of the Language of Numbers and the Science of Figures. Davies' University Arithmetic — Embracing the Science of Numbers, and their Applications according to the most Improved Methods of Analysis and Cancellation. Key to Davies' University Arithmetic. Davies' Elementary Algebra — Embracing the First Principles of the Science. Key to Davies' Elementary Algebra. Davies' Elementary Geometry and Trigonometry — With Applications in Mensuration. Davies' Practical Mathematics — With Drawing and Mensuration applied to the Mechanic Arts. Davies' University Algebra — Embracing a Logical Development of the Science, with graded examples^ Davies' Bourdon's Algebra — Including Sturm's and Homer's Theorems, and practical examples. Davies' Legendre's Geometry and Trigonometry — Revised and adapted to the course of Mathematical Instruction in the United States. Davies' Elements of Surveying and Levelling — Containing descriptions of the Instruments and necessaiy Tables. Davies' Analytical Geometry — Embracing the Equations of the Point, the Straight Line, the Conic Sections, and Smfaces of the first and second order. Davies' Differential and Integral Calculus. Davies' Descriptive Geometry— With its application to Spherical Trigo- nometry, Spherical Projections, and Warped Surfaces. Davies' Shades, Shadows, and Perspective, Davies' Logic and Utility of Mathematics— With the best methods of In- struction Explained and Illustrated. Davies' and Peck's Mathematical Dictionary and Cyclopedia of Mathe- matical Science— Comprising Definitions of all the terms employed in Mathematics— an Analysis of each Branch, and of the whole, as forming a single Science. Entered according to Act of Congress, in the year one thousand eight hundred and seventy, by Charles Davies, in the Clerk's Office of the District Court of the United States for the houthera Charles District of New York V LlTTLK, Kennie & Co., Stereotvpers, 615 and 617 Broadway, New York. 3 PREFACE 7 The Elements of Surveying, first published in 1830, was designed as a text-book for the pupils of the Military Academy, and in its preparation little regard was had to the supposed wants of other institutions. The work, however, was received by the public with more favor than was anticipated, and soon became a leading text- book in the Colleges, the Academies, and the higher grade of Schools. For the purpose of adapting it more fully to the supposed wants of these institutions, many changes have been made since its first publication ; and the present edition will be found to differ, in many respects, from those which have pre- ceded. It has been the intention to begin with the very elements of the subject, and to combine those elements in the sim- plest manner, so as to render the higher branches of Plane Surveying comparatively easy. All the instruments needed for plotting have been carefully described; and the uses of those required for the measurement of angles are fully ex- plained. The conventional signs adopted by the Topographical Bureau, which are now used by the United States Engi- neers in all their Charts and Maps, are given in plates 5 and 6. Should these signs be generally adopted in the country, it would give entire uniformity to all maps and dehneations 4 PREFACE. of ground, and would establisli a common language by which all the peculiarities of soil and surface could be accurately represented. A full account is also 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. This method was originated by CoL Jared Mansfield, whose great acquirements in science introduced him to the notice of President Jefferson, by whom he was appointed Surveyor-general of the Northwestern Territory, in the early part of the present century. Among the changes which have been made in the present edition, and which must be very acceptable to Practical Surveyors, are the methods of laying down Railroad Curves, Section Levelling for Excavation and Embankment, and the article on Mining Engineering. For the first of these, I am mainly indebted to Professor Plympton, of the Brooklyn Polytechnic Institute, who has happily combined science and art, both in his methods of teaching and in his practical operations in the field ; and for the latter, to Pro- fessor Peck, of Columbia College, to whom education and science are indebted for much valuable labor. FiSHKiLL- ON- Hudson, June, 1870. CONTENTS. BOOK I. LOGARITmiS AND TRIGONOMETRY. SECTION I. LOGAEITHMS. rAOK Of Logarithms 9 €reneral Piinciples 11 Table of Logarithms 12 Use of Table 13 Multiplication by Logaritlims 17 Division by Logarithms 18 Arithmetical Complement 19 Raising of Powers and Extraction of Roots 23 SECTION II. Geometki€Al Cokstkuctioks. Diyiders 23 Ruler and Triangle 23 Scale of Equal Parts 25 Diagonal Scale of Equal Parts 26 Scale of Chords 1 27 Semicii-cular Protractor 28 Sectoral Scale of Equal Parts 29 Guntefs Scale 81 Solution of Problems 31 b CONTENTS. SECTION III. PlAKE TrIGOI^OMETRY. paqb Plane Trigonometiy 37 Division of the Circumference 37 Definitions of tlie Trigonometrical Lines 38 Table of Natural Sines 39 Table of Logarithmic Sines 40 Theorems 44-47 Solution of Triangles 47-56 Solution of Right- Angled Triangles 56-57 BOOK 11. PLANE SURVEYING. SECTION L Measurement of Lines and Angles. Definitions 58-60 Measurement of Lines and Angles 60 Measurement of Distances 60 Measurement of a Horizontal Line 61-64 Standard of Measure 65 Measurement of Angles 65 Of the Theodolite 66-73 Verniers 73-75 ]\Ieasurement of a Horizontal Angle with the Theodolite 75 Measurement of a Vertical Angle 76 Measurements with the Tape or Chain 78-84 Applications to Heights and Distances 84-94 SECTION IL Area or Contents of Ground. Area or Contents of Ground 94-97 Problems relating to Area or Contents of Ground 97-108 CONTENTS. . 7 SECTION III. Compass Sueyeyi2hG. page Definitions lOS-110 Surveyor's Compass 110 Work on the Field 112-117 General Example 117 Traverse Table 121 Balancing Work 125-129 Double Meridian Distances and Area 129-140 Problems 140-145 Laying Out and Dividing Land * 145-153 Public Lands 153-157 Variation of the Xeedle 157-162 To find the U'ue Meridian 163 SECTION lY. Teiak"gulatio:s". Definitions and General Remarks 167-170 Uses of the Theodolite 170-173 Filling up the Survey 173 Use of Compass 173-175 Plane Table and Uses *^ 175-181 Plottmg Work 181 Circular Protractor 181 Various Methods of Plotting 182-186 Maritime Surveying 18&-189 BOOK III. LEVELLING AND ITS APPLICATIONS. SECTION I. Levelldsg. Definitions and Principles 190-193 Description of the Y Level 192-196 Levelling Rods 196 '■^vellmg in the Field. 199-206 S CONTENTS. SECTION IL Topographical SuRVEYiifG. pao» Definitions and Principles 206 Examples and Plotting 207-219 SECTION III. Railway Curyes. Definitions and Principles 219-222 Location of the Curve 222-224 Location by tlie Chain alone 224-227 Location by two Transits 227 Laying off the Ordinates 229 SECTION IV. Section Levellii^^g. Definitions and Principles 230-236 Drawing the Profile 236 Establishment of the Grade 237-243 Cross-Section Levelling 244 Setting Slope Stakes 245-254 Computation of Earthwork 255-258 SECTION V. MlN"IN^G Eiq"QIJ?'EERIi^G. Definitions and General Notions 259 Traversing— Compass— Miner's Scmich-cle 260 Field Book 262 Method of Traversing with the Theodolite 263 Field Book 264 Modes of Connecting with Surface Survey 265 Reducing the Traverse— Office Book 267 Method of Plotting the Traverse oh the Surface 268 Method of Plotting the Traverse on Paper 269 ELEMENTS OF SURYEYING. BOOK I. LOGARITHMS AND TRIGONOMETEY. SECTION I. LOaARITHMS. 1. The Logakithm of a number is the exponent of the power to which, it is necessary to raise a fixed number, to produce the given number. The fixed number is called the hase of tM system. Any positive number, except 1, may be taken as the base of a system. In the common system, the base is 10. 2. If we denote any positive number by n, and the corresponding exponent of 10 by p, we shall have the exponential equation, lOP = w (1.) In this equation, p is, by definition, the logarithm of n, which may be expressed . thus, p = log n (2.) . 3. From the definition of a logarithm, it follows that, the logarithm of any power of 10 is equal to the exponent of that power : hence, the formula, log {lOy =z log n = p. • . • . . (3.) 10 ELEMENTS OF SURVEYING. [BOOK I. If a number is an exact power of 10, its logaritlim is a whole mtmler. If a number is not an exact power of 10, its logaritlim will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally. The entire part of a logarithm is called the cJiaracf eristic j the decimal part is called the tnantissa. 4. If, in Equation (3), we make p successively equal to 0, 1, 2, 3, &c. ; and then equal to — 1, — 2, — 3, &c., we may form the following Table. log 1 = log 10 = 1 log .1 = -1 log 100 = 2 log .01 z=z -2 log 1000 = 3 log .001 = -3 &c., &c. &c., &C. When a number lies between 1 and 10, its logarithm lies 'between and 1; that is, it is equal to 0, plies a decimal; if a number lies between 10 and 100, its logarithm is equal to 1, plus a decimal ; if between 100 and 1000, its logarithm is equal to 2, plus a decimal ; and so on : hence, we have the following EuLE. — The characteristic of the logarithm of any whole num- ber is positive, and numerically 1 less than the number of places ■of figures in the given number. When a decimal fraction lies betweern .1 and 1, its logarithm lies between — 1 and 0, that is, it is equal to — 1, plus a deci- mal; if a number lies between .01 and .1, its logarithm is equal to — 2, phis -.a decimal ; if between .001 and .01, its 'logarithm is equal to — 3, plus a decimal; and so on: hence, the following SEC. I.] LOGAEirmiS. 11 KuLE. — The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the numher of (fs that immediately follow the decimal point. The characteristic alone is negative, the mantissa heing always positive. This fact is indicated by writing the nega- tive sign over the characteristic: thus, 2.3T1465, is equivalent to — 2 4- .371465. !N"OTE. — It is to be observed, that the characteristic of a mixed number is the same as that of its entu-e part. Thus, the mixed number T4.103 lies between 10 and 100 ; hence, its loga- rithm lies between 1 and 2, as does the logarithm of 74. GeXERAL PeDsCIPLES. 5. Let m and 7i denote any two numbers, and p and q their logarithms. We shall have, from the definition of a loga- rithm, the following equations, 10^ = m (4.) lO*? = n (5.) Multiplying (4) and (5), member by member, we have, lOP+2 = mn; whence, by the definition, j5 + ^ = log {/U/l) (6.) That is, the logarithm of the product of two nximlers is equal to the sum of the logarithms of the numbers. 6. Dividing (4) by (5), member by member, we have, whence, by the definition. n p - q = ^°^(f) (^•) 12 ELEMENTS OF SURVEYING. [BOOK 1. That is, the logarithm of a quotient is equal to the logarithm of the dividend diminished hy that of the divisor, 7. Raising both members of (4), to a power denoted by t, we huye, 10^' = m'; 'whence, by the definition, pt = log m* (8.) That is, the logarithm of any poiuer of a numler, is equal to the logarithm of the number, multiplied iy the exponent of the potver. 8. Extracting the root, indicated by r, of both members of (4), we have, IC = y m; whence, by the definition, f = log ^. ..... . (9.) That is, the logarithm of any root of a number, is equal to the logarithm of the number divided by the index of the root. The preceding principles enable us to abbreviate the opera- tions of multiplication and division of numbers, by the addition and subtraction of their logarithms. Table of Logaeithms. 9. A Table of Logarithms, is a table by means of which we can find the logarithm corresponding to any number, or the number corresponding to any logarithm. In the table appended, the complete logarithm is given for all numbers from 1 up to 100. For other numbers, between 100 and 10,000, the mantissas alone are given ; the characteristic may be found by one of the rules of (Art, 4). SEC. l] logaeithms. 1$ Before explaining the uses of tlie table, it is to be shown that the mantissa of the logarithm of any number is not changed by multiplying or dividing the number by any exact power of 10. Let n denote any number whatever, and 10^ any power of 10, j9 being any whole number, either positive or negative. Then, in accordance with the principles of (Arts. 5 and 3), we shall have, log ( ^i X 10^) = log n + log 10^ = i? + log n\ but y is, by hypothesis, a whole number; hence, the decimal part of the log ( ?j X 10^ ), is the same as that of log n \ which tvas to be proved. Hence, in finding the mantissa of the logarithm of a num- ber, we may regard the number as a decimal, and move the decimal point to the right or left, 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 2.00357, is the same as that of 2003.57. EXAMPLES. log 327 is 2.514548 log 32.7 te 1.514548 log 3.27 ie 0.514548 log .327 te T.514548 log .0327 (( 2.514548 Using the Table. 1*. To find, from the table, the logarithm of a number less than 100. 10. Look on the first page, in the column headed "N," for the given number; the number opposite is the logarithm required. Thus, log 67 = 1.826075. 14 ELEMENTS OF SURVEYING. [BOOK I. 2°. To find the logarithm of a number between 100 and 10,000. 11. Find the characteristic by the first rule of (Art. 4). To find the mantissa, look in the column headed "N," for the first three figures of the number; then pass along a hori- zontal line until you come to the column headed with the fourth figure of the number; at this place will be found four figures of the mantissa, to which, two other figures, taken from the column headed " 0," are to be prefixed. If the figures found stand opposite a row of six figures, in the column headed "0," the first two of this row are the ones to be prefixed; if not, ascend the column till a row of six figures is found; the first two, of this row, are the ones to be prefixed. If, however, in passing back from the four figures, first found, any dots are passed, the two figures to be prefixed must be taken from the line immediately below. When the figures first found, faU at a place where dots occur, the dots must be replaced by O's, and the figures to be prefixed must be taken from the line heloiu. Thus, Log 8979 = 3.953228 > Log 3098 = 3.491081 Log 2188 = 3.340047 3°. To find the logarithm of a number greater than 10,000. 12. Find the characteristic by the first rule of (Art. 4). To find the mantissa, place a decimal point after the fourth figure (Art. 9), thus converting the number into a mixed num- ber. Find the mantissa of the entire part, by the method last given. Then take from the column headed "D," the corre- sponding talular difference, and multiply this by the decimal part and add the product to the mantissa just found. The result will be the required mantissa. SEC. I.] LOGAEITHMS. 15 It is to be obseryed that when the decimal part of the product just spoken of is equal to or exceeds .5, Tre add 1 to the entire part; otherwise the decimal part is rejected. EXAMPLE. To find the logarithm of 672887. The characteristic is 5. Placing a decimal point after the fourth figure, the number becomes 6728.87. The mantissa of the logarithm of 6728 is 827886, and the corresponding num- ber in the column "D," is 65. Multiplying 65 by .87, we have 56.55; or, since the decimal part exceeds .5, 57. "VTe add 57 to the mantissa already found, giving 827943, and we finally haye, log 672887 = 5.827943. The numbers in the column " D" are the differences between the logarithms of two consecutiye whole numbers, and are found by subtracting the number under the heading "4" from that under the heading "5." In the example last giyen, the mantissa of the logarithm of 6728 is 827886, and that of 6729 is 827951, and their differ- ence is 65 ; 87 hundredths of this difference is 57 : hence, the mantissa of the logarithm of 6728.87^ is found by adding 57 to 827886. The principle employed is, that the differences of numbers are proportional to the differences of their logarithms, when these differences are small. 4°. To find the logarithm of a decimaL 13. Find the characteristic by the second rule of (Art. 4). To find the mantissa, drop the decimal point, and thus consider the decimal a whole number. Find the mantissa of 16 ELEMENTS OF SURVEYING. [BOOK I. the logarithm of this number, and it will be the mantissa required. Thus, log .0327 = 2.51454:8 log 378.024 = 2.577520 5°. To find the number corresponding to a given logarithm. 14. The rule is the reverse of those just given. Look in the table for the mantissa of the given logarithm. If it cannot be found, take out the next less mantissa, and also the corre- sponding number, which set aside. Find the difference between the mantissa taken out and that of the given logarithm; an- nex as many O's as may be necessary, and divide this result by the corresponding number in the column "D." Annex the quotient to the number set aside, and then point off, from the left hand, a number of places of figures equal to the character- istic plus 1: the result will be the number required. If the characteristic is negative, the result will be a pure decimal, and the number of O's which immediately follow the decimal point, will be one less than the number of units in the characteristic. EXAMPLES. 1. Let it be required to find the number corresponding to the logarithm 5.233568. The next less mantissa in the table is 233504; the corre- sponding number is 1712, and the tabular difference is 253. OPEEATION. Given mantissa, 233568 Next less mantissa, . . . 233504 . . 1712 253 ) 6400000 ( 25296. .-. The required number is 171225.296. SEC. i.] LOGARITHMS. 17 The number corresponding to the logarithm 2.2335G8 is .0171225. 2. What is the number corresponding to the logarithm 2.785407? Ans. .06101084. 3. "What is the number corresponding to the logarithm L846741 ? Ans. .702653. Multiplication by Logarithms 15. From the principle proved in (Art. 5), we deduce the following KuLE. — Find the logarithms of the factors, and tahe their sum; then find the numler corresponding to the resulting logarithm,, and, it loill he the product required. EXAMPLES. 1. Multiply 23.14 by 5.062. OPERATIOlf. log 23.14 . . . 1.364363 log 5.062 . . . 0.704322 2.068685 .-. 117.1347, product 2. Find the continued product of 3.902, 597.16, and' 0.0314728. OPEEATION". log 3.902 . . . 0.591287 log 597.16 . . . 2.776091 log 0.0314728 . . . 2.497936 1.865314 .-. 73.3354, product. Here, the 2 cancels the + 2, and the 1 carried from the decimal part is set down. 18 ELEMENTS OF SURVEYING. [liOOK I. 3. Find llic contimicd product of 3.580, 2.104(5, 0.8372, and 0.0294. Ans. 0.1857615. Dn^isioN BY LoGARirnMs. 16. From the principle proved in (Art. 6), we have the following Rule. — Find the logari/hms of the dividend and divisor, and subtract the latter from the former; then find the nnmher corresponding to the resulting logarithm, and it will he the quotient required, EXAMPLES. 1. Divide 24163 .by 4567. OTERATIO Jf. loff 24163 . . . 4.383151 loff 4567 . . . 3.659631 0.723520 5.29078, quotient Divide 0.7438 by 12.9476. OPERATION". log 0.7438 . . . T.871456 loiT 12.9476 . . . 1.112180 2.759207 .-. 0.057447, quotient. Here, 1 taken from 1, gives 2 for a result. The sub- traction, as in this case, is always to be performed in the algebraic sense, 3. To divide 0.06314 by .007241. log 0.06314 . . . 2.800305 loir 0.007241 . . . 3.859799 0.940506 .-. 8.7198, quotient. SEC. I.] LOGABITHMS. 19 Here, 1 carried from the decimal part to the '6, changes it to 2, which being taken from 2, leaves for the char- acteristic. 4. To divide 37.149 by 523.76. log 37.149 . . . 1.569947 log 523.76 . . . 2.710133 2.8.50814 .-. 0.0709274^ quotient 5. Divide 37.149 bv 523.76. Ans. 0.0709274. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of AEITHiCETICAL COMPLEMENT. 17. The Arithmetical Complement 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 of a logarithm may be written out ly commencing at the left hand and suUracting each figure from, 9, until the last significant figure is reached, which must he talcen from 10. The arithmetical complement is denoted by the symbol (a. c). Let a and h represent any two logarithms whatever, and a — b their difference. Since we may add 10 to, and sub- tract it from, a — h, without altering its value, we have, a -h = a ^ {10 -h) -10. . . . ( 10.) But, 10 — h is, by definition, the arithmetical complement of h: hence, Equation (10) shows that the difference between two logarithms is equal to the first, plus the arithmetical com- plernent of the second, minus 10. 20 ELEMENTS OF SUEYEYING. [book I. Hence, to divide one number by another by means of the arithmetical complement, vre have the following EuLE. — Fi/id the logarithm of the dividend, and the arith- metical complement of the logarithm of the divisor, add them together, and diminish the sum lyK); the numler corresponding to the resulting logarithm will be the quotient required. EXAMPLES. 1. DiTide 32T.5 by 22.07. OPEEATION-. log 327.5 . . . 2.515211 (a. c.) log 22.07 . . . S.65619S 1.1TU09 .-. 14.839, quotient 2. Divide 0.7438 by 12.9476. log 0.7438 . . . 1.871456 (a. c.) log 12.9476 . . . S.887811 2.759267 .-. 0.057447, quotient. In this example, the sum of the characteristics is 8, from which, taking 10, the remainder is 2. 3. Divide 37.149 bv 523.76. log 37.149 . . (a. c.) log 523.76 . . 1.569947 7.280867 2.850814 .-. 0.0709273, quotient 4. Multiply 35S8S4 by 5672, and divide the product by 89721. SEC. I.] LOGARITHMS. 21 log 358884 log 5672 (a. c.) log 89721 OPERATIOK. . . 5.554954 . . 3.753736 . . 5.047106 4.355796 22688, result 5. Find x in the proportion, 3976 : 7952 : 5903 (a. c.) log 3976 log 7952 log 5903 log X OPERATION. . . 6.400554 . . 3.900476 . . 3.771073 4.072103 X = 11806. The operation of subtracting 10 is always performed men- tally. XoTE 1. — In finding any term of a proportion by logarithms — obserye that, 1. The sum of the logarithms of the extremes, is equal to the sum of the logarithms of the means : 2. Tlie logarithm of the fourth term, is equal to the arith- metical cmnplement of the logarithm of the first term added to the logarithms of the mean terms: 3. Tlie logarithm of either mean term, is equal to the arith- metical complement of the logarithm of the other mean added to the logarithms of the extremes, I^OTE 2. — If any proportion, as a \ I) '.'. c '. X, be changed into the form of an equation, thus, ax = Ic, tliQn (a. c.) log a + log I + log c — 10 = log x ; that is, 22 FT.E^fF.NTS OF SURYEYING. [BOOK I. The arithmetical compleme7it of the logarith?n of the multiplier of the unlcnown factor, ^;?«s the logarithms of the two other factors, minus 10, is equal to the logarithm of the unknown factor. KoTE 3. — If the logarithm, whose arithmetical complement is taken, exceeds 10, subtract it from 20, and reject 20 in the final operation. Eaising to Powers by Logarithms. 18. To raise a number to any power. From the principle proved in (Art. 7), we have the following Rule. — Find the logaritlim of the numher, and multiply it hy the exponent of the power ; then find the numler correspond- ing to the resulting logaritlim^ and it will le the power required: EXAMPLES. 1. rind the 5th power of 9. log 9 . . . 0.954243 4.771215 2. rind the 7th power of 59049, power. Ans. 2097152. Extracting Eoots by Logarithms. 19. To find any root of a number, from the principle proved in (Art. 8), we haye the following Rule. — Find the logarithm of tlie numler, and divide it ty the index of the root ; then find the number corresponding to the resulting logarithm, and it tcill be the root required. example. 1. 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. sec: n.] GEOMETRICAL CONSTEUCTIOKS. 23 SECTION n. GEOI^IETRICAL CONSTRUCTIONS. 20. Before explaining the method of constructing geomet- rical problems, we shall describe some of the simpler instru- ments and their uses. DIVIDERS. 21. The dividers is the most simple and useful of the instruments used for drawing. It consists of tw*o legs ha, Ic, which may be easily turned around a joint at t. 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 ofif on CD, a distance equal to AB. For this purpose, place the forefinger on the joint of the dividers, and set one ^ ' ' -^ foot at A: then extend, with the thumb ^| I ^ E and other fingers, the other leg of the dividers, until its foot reaches the point 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, RULER AKD TRIANGLE. 22. A Ruler of convenient size is about twenty inches in length, two inches wide, and a fifth of an inch in thickness. 24 ELEMENTS OF SURVEYING. [BOOK I. 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. "We can solve, with the ruler and triangle, the two follow- ing problems. I. To draw through a given point a line which shall be parallel to a given line. 23. Let C be tlie 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, J^ B so that one of the sides of the triangle shall coincide exactly with A B : 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. Leaving the thumb of the left hand on the ruler, extend the fingers upon the triangle and hold it firmly, and with the right hand, mark with a pen or pencil, a line through C: this line will be parallel to AB. SEC. n.] GEOMETEICAL CONSTEUCTIONS. 25 n. To draw through a given point a line which shall be perpen- dicular to a given line. 21. Let AB he the given line, and D the given point. Place the hypothennse of the triangle against the edge of the ruler, as before. Then place the ruler and triangle so that j 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 A B, SCALE OF EQUAL PAETS. I I 1 ! I I ' I I I L 25. A scale of equal parts is formed by dividing a line of a given length, into equal portions. If, for example, the line ad, 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 ^cnit 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, at, 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 extend the other to .6, which marks the sixth of the small divisions : the dividers will then embrace the required distance. 26 ELEMENTS OF SURVEYING. [BOOK I. DIAGONAL SCALE OF EQUAL PARTS. HJ' WW w 03 -^ UL / •08 i9'_j 1 ■oi J_4_U .06 1 / Tl 05- M'-M / .04 /MM// .03 M N N / 1 •02 1 U 1 i f .0, il / / / / / a. I .2.3.^.5.6.7.8.9 k 26. This scale is thus constructed. Take db for the unit of the scale, which may be one inch, J, i, or f of an inch, 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 division 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 glu 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 hun- dredth (.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 extend the other to that figure between a and b which designates the tenths. If two or more units are required, SEC. n.] GEOMETRICAL CONSTEUCTIONS. 27 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, extend the dividers to that line between ad and he which designates the tenths: the dis- tance so embraced will be the one required. For example, to take off the distance 2.34, we place one foot of the dividers at Z, and extend the other to e: and to take off the distance 2.58, we place one foot of the dividers at ]) and extend the other to q, XoTE 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. XoTE 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. SCALE OF CHORDS. J)30 jp 27. If, with auy radius, as AC^ we describe the quadrant CD, and then divide it into 90 equal parts, each part is called a degree. If through C, and each point of division, a chord be drawn, and the lengths of these chords be accurately laid off on a 28 ELEMENTS OF SURVEYING. [book I. scale; such a scale is called a scale of chords. In the figure, the chords are drawn for every ten degrees. The scale of chords being once constructed, the radius of the circle from which the chords were obtained, is known; for, the chord marked 60, is always equal to the radius of the circle. A scale of chords is generally laid down on the scales which belong to cases of mathematical instruments, and is marked CHO. m. To lay off, at a given point of a line, with the scale of chords, an angle equal to a given angle. 28. Let AB be the line, and A the given point. JTake from the scale the chord of 60 degrees, and with this radius and the point ^ as a centre, describe the arc BC. Then ^ take, from the scale, the chord of the given ^ angle, say 30 degrees, and with this distance as a radius, and B as a centre, describe an arc cutting BC in C. Through A and C, draw the line AC, and BAG will be the required angle. SEMICIECULAR PROTRACTOR. C 0^- A B 29. This instrument is used to lay down, or protract angles. It may also be used to measure angles included be- tween lines, already drawn upon paper. SEC. II.] GEOMETRICAL CONSTRUCTIONS. 29 It consists of a brass semicircle, ABC, divided to half de- grees. 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. IV. To lay ofif an angle with a Protractor. 30. Place the diameter AB on the line, 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 Ay and mark the extremity of the aiO 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. SECTORAL SCALE OF EQUAL PARTS. 31. The sector is an instrument generally made of ivory or Irass. It consists of two arms, or sides, which open, by turning round a joint, at their common extremity. There are several scales laid down on the sector : those, how- ever, which are chiefly used in drawing lines and angles, are, the scale of chords already described, and the scale of equal parts now to be exj^lained. 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 divided into equal parts. 30 ELEMENTS OF SURVEYING. [BOOK I. On the sectors which belong to the cases of English instru- ments, the diagonal lines are designated by the letter L, and numbered from the centre of the sector, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, to the two extremities. On the sectors which belong to cases of French instruments, they are designated, "Les parties egales," and numbered 10, 20, 30, 40, &c., to 200. On the Eng- lish sectors there are 20 equal divisions between any two of the lines numbered 1, 2, 3, &c., so that there are 200 equal parts on the scale. 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 varying the angle. Now, if we regard the lines on the sector as the two sides of a triangle, of which the line 40, measured across, is the base, it 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. V. Required to lay down a line of sixty-seven feet, to a scale of twenty feet to the inch. Take one inch from the scale of inches : then place one foot of the dividers at the twentieth division, and open the sector until the dividers will just reach the twentieth division on the other arm: the sector is then set to the proper angle; after which the required distance to be laid down on the paper is SEC. n.] GEOMETRICAL CONSTRUCTIONS. 31 found by extending tlie dividers from the sixty-seventh division on one arm, to the sixty-seventh' division on the other. gutter's scale. 32. This is a scale of two feet in length, on the faces of which several scales are marked. The face on which the divi- sions of inches are made, contains, however, all the scales neces- sary for laying down lines and angles. These are, the scale of equal parts, the diagonal scale of equal parts, and the scale of chords, all of which have been described. SOLTJTIOi^ OF PROBLEMS. I. At a given point, in a given straight line, to erect a perpendicular to the line. 33. Let A be the given point, and BC the given line. From A, lay off any two distances, AB and A C, equal to each other. . Then, from the points JB and C, as centres, with a radius greater than BA, describe two arcs — intersecting each other at D : draw AD, and it will be the perpendicular required. D n. From a given point, without a straight line, to let fall a perpen- dicular on the line. 34. Let A be the given point, and BD the given line. From the point A, as a centre, with a radius sufficiently great, describe an arc cutting the line BJ) in the two points B and J) : then, mark a point ^, equally dis- tant from the points JEJ and B, and draw AU: AE will be the perpendicular required. 32 ELEMENTS OF SURTEYING. [BOOK I. m. At a point, in a given line, to make an angle equal to a given angle. 35. Let A be tlie given point, AU the given line, and IKL the given angle. From the vertex K, as a centre, with ^ ^ any radius, describe the arc IL, termi- y^ nating in the two sides of the angle. ^ I J" From the point A, as a centre, with a distance AE, eqnal to KI, describe the arc ED ; then take the chord LI, with which, from the point ^ as a centre, describe an arc cutting the indefinite arc DE, in D ; draw AD, and the angle EAD will be equal to the given angle K. TV. To divide a given angle, or a given arc, into two equal parts. 36. Let A CB be the given angle, and AEB the arc which measures it. ^ From the points A and B as centres, ^ describe, with the same radius, two arcs cutting each other in J>; through D and v/ the centre C draw CD : the angle A CE ^ will be equal to the angle ECB, and the arc AE to the arc EB. V. Through a given point, to draw a parallel to a given line. 37. Let A be the given point, and B C the given line. From ol as a centre, with a radius -A* p greater than the shortest distance from B-\ ^^\^ A to BC, describe the indefinite arc \ ^,.-'''' \ ED. Then, from the point E as a ^^ ' ^ centre, with the same radius, describe the arc AF; make EB = AF, and draw AB : thon will AB be the parallel required. SEC. II.] GEOMETRICAL CONSTEUCTIOXS. 33 VI. Two angles of a triangle being given, to find the third. 38. Draw tlie indefinite line DEF. q ^ At the point E, make the angle DEC equal to one of the given angles, and T) J? TP the angle CEH equal to the other: the remaining angle REE will be the third angle required. Vn. To represent, on paper, a line of a given length, so that any num- ber of its parts shall correspond to the unit of the scale. 39. Suppose that the given line were 75 feet in length, and it were required to draw it on paper, on a scale of 25 feet to the inch. The length of the line, 75 feet, being divided by 25, will give 3, the number of inches which will represent the line on paper. Therefore, di'aw the indefinite line AB, on which lay off from C, a distance A C equal to 3 inches : A C will represent the given line of 75 feet, drawn to the required scale. XoTE. — This problem explains the manner of representing a line upon paper, so that a given number of its parts shall, correspond to the unit of the scale, whether that unit be an. inch or any part of an inch. When the length of the line to be laid down is given, and it has been determined how many parts of it are to be repre- sented on the paper by a distance equal to the unit of the scale, we find the length which is to be taken from the scale by the following EuLE. — Divide the length of the line ly the nnmler of jimrts which is to ie represented ly the unit of the scale: the quotient will shoio the niimler of units which is to be taken from the scale. 34: ELEMENTS OF SURVEYING. [BOOK I. E X A 31 P L E S. 1. If a line of 640 feet is to be laid do^vn on paper, on a scale of 40 feet to the incli; what length must be taken from the scale? 40) 640 (16 inches. 2. If a line of 357 feet is to be laid down on a scale of 68 feet to the unit of the scale (which we will suppose half an inch), how many parts are to be taken? . j 5.25 parts, or ( 2.625 inches. 3. A line of 384 feet is di'awn on paper, on a scale of 45 feet to the inch ; what is its length on the paper ? A71.S. 8.53 inches. !N"OTE. — When the length of a line on the paper is given, and it is required to find the true length of the line which it represents, take the line in the dividers and apply it to the scale, and note the number of tmits, and parts of a unit, to which it is equal. Then multiply this number by the number of parts which the unit of the scale represents, and the product will be the length of the line. For example, suppose the length of a line drawn on the paper was found to be 3.55 inches, the scale being 40 feet to the inch : then, 3.55 X 40 = 142 feet, the length of the line. VUL Having given two sides and the included angle of a triangle, to describe the triangle. 40. Let the line J3 = 150 feet, and C = 120 feet, be the given sides ; and A = 30 degrees, the given angle : to describe the triangle on a scale of 200 feet to the inch. Draw the indefinite line DG, and at the point D, make the SEC. n.] GEOMETKICAL CONSTEUCTIONS. 35 :i:igle GDII equal to 30 degrees: then luy of£ DG equal to 150 feet, equal to three-quarters of an inch, and DH equal to 120 feet, equal to six-tenths of an inch, and draw GH : then, DHG will he the required triangle. IX. The three sides of a triangle being given, to descrihe the triangle. 41. Let A, B and C be the sides. ^Make DE equal to the side A. From the point Z> as a centre, with a radius equal to the second side B, describe an arc : from E as a centre, with a radius equal to the third side C, describe another arc, intersecting the former in F ; draw DF and EF, and DFE will be the triangle required. X. Having given two sides of a triangle and an angle opposite one of them, to describe the triangle. 42. Let A and B be the giyen sides, and C the given angle, which we will suppose to be opposite the side B. Draw the indefinite line DF: then, at any point of it, as D, make the angle FDE equal to the angle C : take DE = A, and from the point E, as a centre, with a radius equal to the other given side, B, describe an arc, cutting DF in F; draw EF: will DEE be the required triangle. If the angle C is acute, and the side B less than A, then the arc described from the centre E with the radius EF = B will cut the side DF in two poiuts, F and G, then 36 ELEMENTS OF SURVEYING. [BOOK I. lying on the same side of D : hence, there will be two triangles, DEF and DEG, either of which will satisfy all the conditions of the problem. XI. The adjacent sides of a parallelogram, with the angle which they contain, being given, to describe the parallelogram. 43. Let A and B be the given sides, and G the given angle. Draw the line DH, and lay off DE equal to A ; at the point D, make the / /; ■' angle EDF =. G; take DF^ B: de- ^Z ^ scribe two arcs, the one from F, as A 1 / J§) < ^ — a centre, with a radius FG = A, the other from E, as a centre, with a radius EG = BF ; through the point G, where these arcs intersect each other, draw FG, EG : then, DEGF will be the parallelogram required. XII. To find the centre of a given circle, or arc. 44. Take three points, A, By C, anywhere in the circum- ference, or in the arc : draw AB, BG ; bisect these two lines by the perpen- diculars, BE, FG : the point 0, where these perpendiculars meet, will be the centre sought. i^ A similar construction serves for making a circumference pass through three given points. A, B, 0, and also for describing a circumference, about a given triangle. For, if we join the points by the straight lines AB, BG, and AC, and bisect either two of them, by perpendiculars, their point of intersection, 0, will be the centre of the required circle. SEC. in.] PLANE TRIGONOMETRY. 37 SECTION III. PLANE TRIGONOMETRY. 45. 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 thi^ee angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. 46. An 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. 47. If two lines be drawn through the centre of a circle, at right angles to each other, they will divide the circum- ference into four equal parts, each of which is called a quadrant. 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 T 22' 33", is read, 7 degrees, 22 minutes, and 33 seconds. A quadrant contains 324,000 seconds, and an arc of 7° 22' 33" contains 26,553 seconds; and any arc of a quadrant may be expressed in seconds. 48. The com^ilement of an arc is what remains after sub- tracting the arc from 90°. Thus, the arc EB is the comple- ment of AB. 38 ELEMENTS OF SUEYETING. [book I. 49. The siq)j>Iej7ient of au arc is what remains after subtracting the arc from 180^ Thus, GJ^ is the supplement of the arc ABF. 50. The si7ie of an arc is the per- pendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, JBD is the sine of the arc A^. 51. The cosine of an arc is the part of the diameter inter- cepted between the foot of the sine and centre. Thus, OD is the cosine of the arc AJB. 52. The tangent of an arc is the line which touches it at one extremity, and limited by a line drawn through the other extremity and the centre of the circle. Thus, A C is the tan- gent of the arc AJy. 5Z. The seccmt of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, OC is the secant of the arc AJj. 54. The four lines, JBD, OB, AC, OC, depend for their Yalues on the arc AB and the radius OA ; they are thus written : For BB, we write sin AB For OD " " cos AB For .4 C " '• tan AB For 0(7 '-' « sec AB 55. If ABE be a quadrant, or 90°, then EB will be the complement of AB. Let the lines ET and IB be drawn per- pendicular to OE, Then, SEC. m.] PLANE TRIGONOMETRY. 89 ET, the tangent of EB, is called tlie cotangent oi AB ; IB, the sine of EB, is the cosine of AB ; OT, the secant of EB, is the cosecant of AB. In general, if A is any arc or angle, T\'e have, cos A = sin (90° - A) cot A = tan (90=^ - A) cosec A = sec (90° — A) 56. If we take an arc, ABEF, greater than 90°, its sine will be Fff; OR will be its cosine ; ^ § its tangent, and OQ its secant. But FR is the sine of the arc GF, which is the sup- plement of ^i^, and OH is its cosine; hence, tJie sine of an arc is equal to the sine of its siqji^lement ; and tlie cosine of an arc is equal to the cosine of its siqyple^nent.^ Furthermore, AQ is the tangent of the arc AF, and OQ is its secant : GL is the tangent, and OL the secant of the sup- plemental arc GF. But since AQ is equal to GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tangent of its supplement; and the secant of an arc is equal to the secant of its suppletnent.^ Table of Natural Sines. 57. A ]SrATURAL SiXE, COSIXE, TaXCtE2>"T, or COTA^'GEXT. is the sine, cosine, tangent, or cotangent of an arc whose radius is 1. A Table of Xatural Sixes is, therefore, a table showing the values of the sines, cosines, tangents, and cotangents of all * These relations are between the numerical rallies of the trigonometrical lines; the algebraic signs, -which they have in the diflferent quadrants, are not considered. 40 ELEMENTS OF SUR\^YING. [BOOK I. the arcs of a quadrant, diyided either to minutes or seconds. The Table of Natural Sines, beginning at page 63, of the tables, shoAvs the values of the sines and cosines only. Table of Logaeithmic Sines. 58. In this table are arranged the logarithms of the nu- merical values of the sines, cosines, tangents, and cotangents of all the arcs of a quadrant, calculated to a radius of 10,000,000,000. The logarithm of this radius is 10. In the first and last horizontal lines of each page, are written the degrees whose sines, cosines, &c., are expressed on the page. The vertical columns on the left and right are columns of minutes. To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given angle. 59. If the angle is less than 45°, look for the degrees in the first horizontal line of the difierent pages : when the degrees are found, descend along the column of minutes, on the left of the page, till you reach the number showing the minutes: then pass along a horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be: the number so indicated is the logarithm sought. Thus, on page 37, for 19° 55', we find, sine 19° 55' 9.532312 cos 19° 55' • . . 9.973215 tan 19° 55' 9.559097 cot 19° 55' 10.440903 If the angle is greater than 45°, search for the degrees along the bottom line of the different pages: when the number is found, ascend along the column of minutes on the right-hand side of the page, till you reach the number expressing the minutes: then pass along a horizontal line into the column SEC. III.] TUCsE TPJG0X0:5IETRY. 41 designated tang, cot, sine, or cosine, as the case may be: the nnmber so pointed out is the logarithm required. 60. The cohimn designated sine, at the top of the page, is designated by cosine at the bottom: the one designated tang, by cotang, and the one designated cotang, by tang. The angle found by taking the degrees at the top of the page, and the minutes from the left-hand vertical column, is the complement of the angle found by taking the degi'ees at the bottom of the page, and the minutes from the right-hand column on the same horizontal line with the first. Therefore, sine, at the top of the page, should correspond with cosine, at the bottom; cosine "with sine, tang with cotang, and cotang with tang, as in the tables. If the augle is greater than 90^, we have only to subtract it from 180°, and take the sine, cosine, tangent, or cotangent of the remainder, or supplement. Column of Diflferences. 61. The column of the table next to the column of sines, and on the right of it, is designated by the letter D. This column is calculated in the following manner : Opening the table at any page, as 4,2, the sine of 24:° is found to be 9.609313; that of 24° 1', 9.609597: their difference is 284; this being divided by 60, the number of seconds in a minute, gives 4.T3, which is entered in the column D. Xow, supposing the increase of the logarithmic sine to be proportional to the increase of the arc, and it is nearly so for 60", it follows, that 4.73 is the increase of the sine for 1". Similarly, if the arc were 24° 20', the increase of the sine for 1" would be 4.65. The same remarks are applicable in respect of the column D, after the column cosine, and of the column D, between the 42 ELEMENTS OF SURVEYING. [BOOK I. tangents and cotangents. The column D, between the columns tangeuts and cotangents, answers to both of these columns. Xow, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before ; then, multiply the cor- responding tabular difference by the seconds, and add the product to the number first found, for the sine of the given arc. Thus, if we T^ish the sine of 40^ 26' 28". The sine 40° 2G' 9.811952 Tabular difference 2.47 .... ^s'umber of seconds 28 .... Product . . 69.16 to be added, 69.16 Gives for the sine of 40° 26' 28", 9.812021. The decimal figures at the right are generally omitted in the last result; but when they exceed five-tenths, the figure on the left of the decimal point is increased by 1 ; the loga- rithm obtained is then exact, to within less than one unit of its right-hand place. The tansrent of an arc. in which there are seconds, is found in a manner entirely similar. In regard to the cosine and cotangent, it must be remembered, that they increase while the arcs decrease, and decrease as the arcs are increased; conse- quently, the proportional numbers found for the seconds, must be subtracted, not added. EXA:^rPLES. 1. To find the cosine of 3° 40' 40". The cosine of 3° 40' 9.999110 Tabular difference .13 I^umber of seconds 40 Product . . 5.20 to be subtracted, 5.20 Gives for the cosine of 3° 40' 40" . . 9.999105. SEC. in.] PLANE TRIGONOMETEY. 43 2. Find the tangent of 37^ 28' 31". A^is. 9.884592. 3. Find the cotangent of ST 57' 59". Ans. 8.550356. To find the degrees, minutes, and seconds answering to any given logarithmic sign, cosine, tangent, or cotangent. 62. Search in the table, in the proper colunin, and if the number is found exactly, the degrees will be shown either at the top or bottom of the page, and the minutes in the side column, either at the left, or right. But, if the number cannot be found in the table, take from the table the degrees and minutes answering to the nearest less logarithm, the logarithm itself, and also the corresponding tabular difference. Subtract the logarithm taken from the table from' the given logarithm, annex two ciphers to the remainder, and then diyide the remainder by the tabular difference : the quotient will be seconds, and is to be connected with the degrees and minutes before found : to be added for the sine and tangent, and suUracted for the cosine and cotangent. EXAMPLES. 1. Find the arc answering to the sine 9.880054 Sine 49° 20', next less in the table 9.879963 Tabular difference ..... 1.81) 91.0jO (50". # Hence, the arc 49° 20' 50" corresponds to the giyen si ire 9.880054. 2. Find the arc whose cotangent is . 10.008688 cot 44° 26', next less in the table 10.008591 Tabular difference 4.21) 97.00 (23". ■ Hence, 44° 26' - 23" = 44° 25' 37" is the arc answering to the giyen cotangent, 10.008688. 4^4 ELEMENTS OF SURVEYING. [BOOK I. 3. Find the arc answering to tangent, 9.979110. A?is. 43° 37' 21". 4. Pind the arc answering to cosine, 9.944599. A71S. 28° 19' 45". THEOREM I. The sides of a 2^^ctne triangle are proportional to the sines of their opposite angles. 63. Let ABC be a triangle; then will CB : (7^ : : sin ^ : sin B, For, with ^ as a centre, and A J) equal to the less side BC, as a radins, describe the arc JDI: and o with ^ as a centre and the equal radius B G, describe the arc CL, and draw JDJE ^ and CF perpendicular to AB : now BE is the sine of the ^ angle A, and CF is the sine of B, to the same radius AB or B C. But by similar triangles, AD : DE '.: AO : CF. But AB being equal to BC, we have BC : sin yl : : AC : sin B, or BC : AC :: sin A : sin B. By comparing the sides AB, AC, in a similar manner, we should find, AB : AC :: sin C : sin B. THEOREM II. Li any triangle, tlie sum of the tico sides containing either angle, is to their differeiice, as the tangent of half the sum of the tivo other a?igles, to the tangent of half their difference. SEC. III.] PLANE TEIGONOMETEY. 45 64. Let A GB be a triangle : then will AB ^ AC \ AB - AC : : tan \{C + B) : tan ^{C - B), "With ^ as a centre, and a radius A C, the less of the two given sides, let the semicircumference IFCJE be described, meeting AB in I, and BA produced, in M Then, BE will be the sum of the sides, and BI their difference. Draw CI and AF. Since CAE is an outward angle of the triangle A CB, it is equal to the sum of the inward angles C and B (Bk. I., Prop. XXV., Oor. 6). But the angle CIE being at the circumference, is half the angle CAE at the centre (Bk. III., Prop. XVIII.) ; that is, half the sum of the angles C and B, or equal to ^{C + B). The angle ^i^C = ACB, is also equal to ABC + BAF; therefore, BAF = ACB - ABC But, ICF=z i {BAF) =i{ACB-ABC), ov i{C- B). With I and C as centres, and the common radius IC, let the arcs CJD and Z6r be described, and draw the lines CE Mid. IH perpendicular to IC, The perpendicular CE will pass through E, the extremity of the diameter IE, since the right angle ICE must be inscribed in a semicircle. But CE is the tangent of CIE =. i{C + B); and III is the tangent of ICB = ^ {C — B), to the common radius CI But since the lines CE and IIT are parallel, the triangles BCE and BUI are similar, and give the proportion, BE : BI '.'. CE : IH, or by placing for BE and BI, CE and IH, their values, we have, AB-\-AC: AB--AC:: tan i{C + B) : tan HC-B). 46 ELEMENTS OF SUBVEYING. [book I. THEGEEM III. In any plane triangle, if a line is drawn from the vertical angle perjoendicular to the base, dividing it into tzvo segments : then, the luhole lase, or sum of the segments, is to the sum of the other two sides, as the difference of those sides, to the differ- ence of the segments. 65. Let BAC be a triangle, and AD perpendicnlar to tlie base BC ; then will BC : CA + AB :: CA — AB For, AB^ = in? + ZO^ (Bk. IV., Prop. XI.) ; and JC^ = X>6'^ + Alf- by subtraction, -JC^ _ AB^ ^ 6^ _ BD^, n 7) But, since tlie difference of the squares of two lines is equal to the rectangle contained by their sum and difference (Leg., Bk. lY., Prop. X.), we haye, ~AC^ - AB" = {AC + AB) .{AC- AB) and -(JW - 'DB^ = {CD + BB) . {CD - DB) therefore, {CD + DB) . {CD - DB) = {AC+ AB) . {AC- AB) hence, CD + DB : AC + AB :: AC - AB : CD - DB. THEOREil IT. Tn any right-angled ^lane triangle, radius is to the tangent of eitlier of the acute angles, as the side adjacent to the side opposite. 66. Let CAB be the proposed triangle, and denote the radius by R : then will R : ioxiC '.: AC : AB. For, with any radius as CD, describe the arc DH, and draw the tangent DC. SEC. III.] PLANE TEIGONOMETEY. 47 Prom the similar triangles CDG and CAB, wc have, CD : DG :: CA : AB; lience, R : tan C :: CA : AB. By describing an arc, with ^ as a centre, we could show in the same manner that, E : tsin B :: AB : AC. THEOREM Y. 1)1 every rigid-angled plane triangle, radius is to tlie cosine of either of the acute angles, as the hy2)othemcse to the side adjacent. 67. Let ABC be a triangle, right-angled at B: then will B : QO^ A :: AC : AB. For, from the point A as a centre, with any radius as AB, describe the arc DF, which will measure the angle A ; and draw BJS perpendicular to AB : then will ^^ JS'F AE be the cosine of ^. The triangles ABE and A CB, being similar, we have, AB : AE '.: AC : AB ; that is, R '. co^ A :. AC : AB. General Principles. 68. The relations between the sides and angles of plane triangles, demonstrated 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 a side, before the others can be determined. If the three angles only are given, it is plain, that an in- definite number of similar triangles may be constructed, the 48 ELEMENTS OF SUETEYrN'G. [BOOK I. 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, as given, one of the four following cases, will always be pre- sented. I. "\Vhen two angles and a side are given. II. TVhen two sides and an opposite angle are given. III. TTlien two sides and the included angle are given. IV. When the three sides are given. CASE I. 69. When two angles and a side are given. 1. In a plane triangle, ABC, there are given the angle A = 58° 07', the angle B = 22^ 3T', and the side AB = c = 408 yards; to final C, a and ^.* GEO AIETEIC ALLY. Draw an indefinite straight line, AB, and from the scale of equal parts lay off AB equal to 408. Then at A, lay off an -^^ » ^^^^^ angle equal to 58° 07', and at B an angle equal to 22° 37', and draw the lines AC and BC : then wiU ABC be the triangle. The angle C may be measured either with the protractor or the scale of chords, and will be equal to 99° 16'. The sides AC and BC may be measured by referring them to the scale of equal parts. We shaU find .46' = 158.9 and ^C'=351 yards. TKIGOXOMETEICALLT. Add the given angles together, and subtract their sum from 180 degrees. The remaining parts of the triangle can then be found by Theorem I. * The sides lying opposite the angles A, B and C. arc denoted by a, b and c. SEC. in.] PLANE TRIGONOMETRY. 49 To the ano^le A = oS° O A*' add the angle ^ = 22° 37' their sum , .... -. = 80° 44' taken from 180° 00' leaves C 99° 16', of which, as it exceeds 90°, we use the supplement 80° 44'. To find BC = a. From Theorem I., we have, sin (7 : sin J. : : c : a. Applying logarithms, we have, (a. c.) log sin C (99° 16') 0.005705 log sin A (58° 07') 9.928972 log c (408) 2.610660 log a 351.024 2.545337 In like manner, sin C : sin j5 : : c : d. Applying logarithms, we have, (a. c.) log sin C (99" 16') 0.005705 log sin B (22° 37') 9.584968 log c (408) 2.610660 log b 158.976 2.201333 A71S. C = 98° 16', a = 351.024, and d = 158.976. KoTE. — The logarithm of the fourth term of a proportion is obtained by adding the logarithm of the second term to that of the third, and subtracting from their sum the logarithm of the first term. But to subtract the first term is the same as to add its arithmetical complement and reject 10 from the 50 ELEMENTS OF SURVEYING. [BOOK I. Slim (Sec. I., Art. 17) : hence, the arithmetical complement of the first term added to the logarithms of the second and third terms, minus ten, will give the logarithm of the fourth term. 2. In a triangle ABC, there are given A = 38° 25', B = 57° 42', and c = 400 : required the remaining parts. Ans, C = 83° 53', a = 249.974, h = 340.04. CASE II. 70. When two sides and an opposite angle are given. In a plane triangle, ABC, there are given AC = b =^ 216, CB = a = 117, the angle A = 22° 37', to find the other parts. GEOMETRICALLY. Draw an indefinite right line ABB' : from any point, as A, draw AC, making BAC = 22° 37', and make AC = 216. ,, , . . With G 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. TEIGOXOMETEICALLY. From Theorem I., we have, a : b : : sin ^ : sin ^ By applying logarithms, we have, (a. c.) log a (lir) ..."... 7.931814 log b (216) 2.334454 log sin A (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 consequence of the first proportion being true for either of the ano-les ABC, or AB'C, which are supplements of each other, SEC. m.] PLANE TEIGONOMETBY. 51 aiid therefore, have the same sine (Art. 56). So long as the two triangles ACB and ACB' exist, the ambiguity will con- tiune. But if the side CB, opposite the given angle, is greater than AC, the arc BB' 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. TThen 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 he impossible to fulfil the conditions of the problem. In the example under consideration, there are two solutions, the first corresponding to B' = 45° 13' 55", and the second to ABC = 134° 46' 05". First case. A 22° 37' B' 45° 13' 55" O 180° - 67° 50' 55" = 112° 09' 05". Thus, in the triangle ACB', sin B' : sin C : : l : c, and applying logarithms, (a. c.) log sin B' (45° 13' 55") .... 0.148764 log sin C (112° 09' 05") .... 9.966700 log b (216) 2.334454 log c 281.785 2.449918 Second case, A 22° 37' B 134° 46' 05" (7 ...... . 180° - 157° 23' 05" = 22" 36' 55". 52 ELEMENTS OP SURVEYING. [BOOK I. Thus, in the triangle ACB, sin B \ m\ C '. h : c. and applying logarithms, (a. c.) log sin B (134° 46' 05") . . . 0.148764 log sin C (22° 36' 55") . . . 9.584943 log 1) (216) 2.334454 log c 116.993 2.068161 2. Given two sides of a triangle, 50 and 40 respective]y, and the angle opposite the latter, equal to 32° : required the remaining parts of the triangle. A71S. 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 third angle to 9° 28' 59", and the remaining side to 12.436. CASE III. 71. When two sides and their included angle are given. Let ABC be a triangle; AB and BC, ^ the given sides, and B the given augle. Since B is known, we can find the sum of the two other angles: for, j^- — ' h A -[- C = 180° — B, and ^{A-\-0)=l (180^ - B). We next find half the difference of the angles A and C, by Theorem IL, viz., B0+ BA : BC- BA : : tan i {A + C) : tan i{A- C), in wliich we consider BC greater than BA, and therefore A is greater than C ; since the greater angle must be opposite the greater side. SEC. III.] PLANE TRIGONOMETRY. 63 Having found half the difference of A and C, by adding it to the half sum, -J (yl + (7), we obtain the greater angle, and by subtracting it from half the sum, we obtain the less. That is, i{A -\-C) + i{A-C) =A, and i{A +C) -i{A^C) = a Having found the angles A and (7, the third side A C may be found by the proportion, sin A : sin B : : a : b. EXAMPLES. 1. In the triangle ABC, let BC = 540, AB = 450, and the included angle B = 80° : required the remaining parts. GEOMETEICALLT. Draw an indefinite right line BO, and from any i>oint, as B, lay off a distance BC = 540. At B make the angle CBA = 80° : draw BA, and make the distance BA = 450 ; draw AC; then will ABC be the required triangle. TRIGONOMETRIC ALLY. BC +BA = 540 + 450 = 990; and BC- BA = 540 - 450 = 90. A + C= 180° - B = 180° - '80° = 100°, and therefore, ^{A+C)=i (100°) = 50°. To find i (a — O). By Theorem 11. , we have, BC + BA : BC-BA :: tani{A + C) : tan l(^-C). Applying logarithms, we have, (a. c.) log (BC + BA) (990) . . . 7.004365 log (BC - BA) (90) . . . 1.954243 log tan i (^ + (7) 50° . . . 10.076187 log tan i (A - C) 6° 11' . . 9.034795 54 ELEMENTS OF SURVEYING. [BOOK I. Hence, 50^ + 6° 11' = 5G" 11' = A; and 50" - 6° 11' = 43° 49' = a To find the third side AO. sin C : sin ^ : : c : b. Applying logarithms, we have, (a. c.) log sin C (43° 49') 0.159672 log sin B (80°) 9.993351 log c (450) 2.653213 log i 640.082 2.806236 2. Given two sides of a plane triangle, 1686 and 960, and their included angle 128° 04': required the other parts. Ans. Angles, 33° 34' 39"; 18° 21' 21"; side 2400. CASE IV. 72. Having given the three sides of a plane triangle to find the angles. Let fall a perpendicular from the angle opposite the greater side, dividing the given triangle into two right-angled triangles : then find the difference of the segments of the base by Theo- rem 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. Then, since the greater segment belongs to the right-angled triangle having the greater hypothenuse, we have two sides and the right angle of each of two right-angled triangles, to find the acute angles. EXAMPLES. 1. The sides of a plane triangle being given; viz., BC = 4:0, AC = 34, and AB = 25: required the angles. SEC. in.] PLANE TBIGONOMETKT. 55 GEOMETEICALLY. With the three given lines as sides, construct a triangle as in Prob. IX. Then measure the angles of the triangk- eiiher with the protractor or scale of chords. TEIGOXOMETEICALLT. BC : AC -{- AB :: AC - AB : CD - BD. That is, 40 : 59 : : 9 : ^' ^ =13.275. 40 -f 13.2:5 Then, ^ = 26.63:5 = CD, 40-13.2:5 And, -^ = 13.3625 = BD. In the triangle DAC, to find the angle DAC. AC \ DC w sin D : sin DAC. Applying logarithms, ^e have, (a. c.) log .IC (34) 8.468521 log i)C (26.63:5) 1.425493 log sin D (90') 10.000000 lo^ sin DAC 51=^ 34' 40'' . . 9.894014 In the triangle BAD, to find the angle BAD. AB : BD :: sin D : sin BAD. Applying logarithms, we have, (a, c.) log AB (25) 8.602060 log BD (13.3625) 1.125887 log sin D (90") 10.000000 log sin BAD 32' IS' 35" . . . 9.:2:947 56 ELEMENTS OF SURVEYING. [BOOK I. Hence, 90° - i)v4C = 90° - 51° 34' 40" = 38° 25' 20" = C, and, 90° - BAD = 90° - 32° 18' 35" = 57° 41' 25" = B, and, BAD + DAC = 51° 34' 40" + 32° 18' 35" = = 83° 53' 15" = A. 2. In a triangle, of whicli the sides are 4, 5, and 6, what are the angles ? Ans. 41° 24' 35"; 55° 46' 16"; and 82° 49' 09". SOLUTION OF EIGHT-AKGLED TRIANGLES. 73. The unknown parts of a right-angled triangle may be found by either of the four last cases; or, if two of the sides are giyen, by means of the property that the square of the hypothenuse is equal to the sum of the squares of the two other sides. Or, the parts may be found by Theorems IV. and V. EXAMPLES. 1. In a right-angled triangle JBA G, there are given the hypothenuse BC— 250, and the base AC = 240 : required the other parts. C To find the angle B. By Theorem I., we haye, a : h : : sin ^ : sin ^. Applying logarithms, we haye, (a. c.) log a (250) 7.602060 log I (240) 2.380211 log sin A (90°) 10.000000 log sin B 73° 44' 23" .... 9.982271 But, (7 = 90° - J? == 90° - 73*= 44' 23" = 16° 15' 37". SEC. in.] PLANE TPJGONOMETRY. 67 To find side AB. We have from Theorem IV., H : tun C :: d : c. Applying logarithms, we have, (a. c.) log i2 (90°) 0.000000 log tan C (16° 15' 37'') . . . 9.464889 log b (240) 2.380211 lo^ c 70.0003 1.845100 2. In a right-angled triangle JBAC, there are given, .4C7=384, and ^ = 53° 08'; required the remaining parts. Ans, AB = 287.96 ; BO = 479.979; C = 36° 52'. BOOK II. PLANE SURYEYIKG. SECTION I. MEASUREMENT OF LINES AND ANGLES. 1. Surveying, in its most extensive signification, com- prises all the operations necessary for finding: 1st. The area or contents of any portion of the surface of the earth; 2d. The lengths and directions of the bounding lines; and, 3d. For making, on paper, an accurate delineation, both of the surface and boundaries; which delineation is called a Map. 2. Plane Suevet is that branch in which the curvature of the earth is neglected; as it may be when the survey is limited to small portions of the surface. 3. Geodesic Surveying is when the curvature of the earth is taken into account, as it must be in all extensive surveys. 4. A Horizontal Plane, is a plane parallel to the water- level. If the plane passes through a point on the surface of the ^arth, it is tangent to the surface, and also perpendicular to the radius passing through the point of contact. SEC. I.]' LI^'ES AND ANGLES. 59 5. A Horizontal Li^'e, is a plane parallel to the water- level, or parallel to a horizontal plane. 6. A Yeetical Plaxe, is a plane pei-pendicular to a hori- zontal plane. 7. A Vertical Lixe, is a line pei'pendicnlar to a hori- zontal plane. 8. Oblique Lixes, are those which are inclined to a hori- zontal plane. Thns, AB and DC are hori- zontal lines; BC and AB are yertical lines; and AC and BD are oblique lines. 9. The Horizontal Distance between two points, is the horizontal line intercepted between the two yertical lines passing through those points. Thus, BC or AB, is the hori- zontal distance between the two points A and C, or between the points B and D. 10. A Horizoxtal Angle, is one, whose sides are hori- zontal: its plane is also horizontal. A horizontal angle is always equal to the angle included hetween hco vertical ^planes 2)assing through the angular point and the tico objects which ■ subtend the angle. li. A A'ertical Axgle, is one, the plane of whose sides is vertical. 12. Ax Axgle of Eleyatiox, is a vertical angle having one of its sides horizontal, and the inclined side above the horizontal side. Thus, in the last figure, BAC is the angle of elevation from A to C. 13. An Angle of Depression, is a vertical angle having one of its sides horizontal, and the inclined side under the 60 ELEMENTS OF SURVEYING. " [BOOK H. horizontal side. Thus, DC A is the angle of depression from C io A, 14. Ax Oblique An-gle, is one, the plane of whose sides is oblique to a horizontal plane. 15. All lines, which can be the object of measurement, must belong to one of the classes aboye named, viz.: 1st. Horizontal lines; 2d. Vertical lines; 3d. Oblique lines. 16. All angles may also be divided into three classeSj viz. : 1st. Horizontal angles; whose sides are horizontal. 2d. Vertical angles; which include angles of elevation and angles of depression; and, 3d. Oblique angles, or those included by oblique lines. Measueemext of Lin"es and Akgles. 17. It has been shown (Bk. L, Art. 45), that at least one side and two of the other parts of a plane triangle must be giyen, or known, before the remaining parts can be found, by calculation. When, therefore, distances are to be found, by trigonometrical calculations, two preliminary steps are necessary : 1st. To measure certain lines on the ground; and, 2d. To measure, the necessary angles. Measurementt oe Distan"ces. 18. Any tape, rod, or chain, divided into equal parts, may be used as a measure; and this is called the unit of measure. The unit of measure may be a foot, a yard, a rod, or any other ascertained distance. The measure in general use, is a chain of four rods or sixty-six feet in length; it is called Gunter's chain, from the name of the inventor. This chain is composed of 100 links. SEC. I.] MEASUKEMENT OF DISTANCES. 61 Every tentli link, from either end, is marked by a small attached brass plate, which is notched, to designate its number from the end. The division of the chain, into 100 equal parts, is very convenient, since the divisions, or links, are decimals of the whole chain, and in the calculations are treated as such. TABLE. 1 chain = 4 rods = 66 feet = 792 inches = 100 links. 1 link is equal to 7.92 inches. 80 chains = 320 rods = 1 mile. 40 chains = ^ mile. 20 chains = i mile. 19. Besides the chain, there are needed for measuring, ten marking-pins, which should be of iron, each about ten inches in length and an eighth of an inch in thickness. These pins should be strung upon an iron ring, and this ring should be attached to a belt, to be passed over the right shoulder, sus- pending the pins at the left side. To measure a horizontal line. 20. The point where the measurement is to begin is usually located by a staff or stake temporarily placed for the purpose ; or by some one of the many permanent marks by which the angles in a boundary are fixed. The other extremity of the line must be provided with a staff or flag which can be easily seen. The fore-chain man, with the pins, and one handle of the chain in his right, starts off on the line, drawing out the chain to its full length. Both chainmen now examine it care- fully, to see if there are any "kinks" at the junction of the links. Having adjusted the chain for use, the fore-chainman resumes his place, to be directed by the hind-chainman, so that the measurement shall be made exactly along the estab- 62 ELEMENTS OF SURVEYING. [book n. lished line. To facnitu-te this, and to insure the correct align- ment of the pin, at its proper distance, the chain and one pin should be held firmly in the right hand, as represented in the figure. While the pin is being aligned, it should be held by the fore-chainman as far from the body as possible, so that the yiew of the flag be left unobstructed. To ac- complish 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 hind-chainman 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, the order "down" is given, when the fore-chain- man 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. After one or two chains have been measured, on any line, the fore-chainman 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 hind-chainman. When the distance is more than ten chains, the pins, when exhausted, should be returned to the fore-chainman — the dis- tance noted — and the chaining recommenced at the place of the tenth pin. SEC. I.] MEASUREMENT OF DISTANCES. 63 All distances should be measured liorizontuily : Hence, when the ground slopes, one end of the chain must be ele- vated. 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. Suppose the chaining to be w^j Mil. The fore-chainman draws the chain out to its full length, as in any other case, and then returns to within such a distance of the hind- chainman, that when the chain is drawn out to that length horizontally, it shall not be too high to be held conveniently. The hind-chainman holds his end of the chain carefully over the point or station, by means of the plumb-line, while he directs the fore-chainman in the usual manner. 64 ELEMENTS OF SURVEYING. [book n. The point fixed in this manner, by the fore-chainman, must not be marked by a marking-pin, but by a small peg or nail. At the order, "Down," the fore-chainman does not go forward in the usual manner, but waits until the hind-chainman 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 hind-chain- man, who uses the plumb, as before, and aligns the fore-chain- man, 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. In chaining down-hill, the method is essentially the same. The fore-chainman 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 only to the flag^ where it is held by the fore- chainman, until the hind-chainman comes forward to the last pin, and counts the links. In measuring up the hill from A to 0, or down the hill from C to A, we measure the horizontal distances a t, c d, and / C, and their sum is the liorizontal dis- tance between A and C. Two staves are often used w^th the chain, in the measure- ment of lines. Each of these should be about six feet in length, and have a spike in the lower end to aid in holding it firmly, and a horizontal strip of iron to prevent the chain from slipping ofi": each stalf is to be passed through the ring at the end of the chain. SEC. I.] MEASUEEMEXT OF DISTANCES. 65 Sta>'"dard. 21. As the length of the chain may vary, from heat or cold, or become changed from other causes, it should be com- pared from time to time, with a standard, kept for the purpose. To facilitate this comparison, let two stakes be driven in the ground, distant from each other one chain, and let nails be driven in the heads of the stakes to mark the exact length of the standard. Marks made upon the coping of a wall will answer the same purpose. If it is found that any line has been measured with a chain, either too short or too long, the measured dis- tance may be corrected by the following proportion: As the length of the standard the length of the chain the measured distance the true distance. For the correction of areas we have this proportion:. As the square of the length of Aie standard the square of the length of the chain, the area found the true area. Measurement op A^tgles. 22. "We come next to the measurement of ^ngles,^. and" for this purpose, several instruments are used. T^e one, however, which affords the most accurate results, and /which indeed can alone be relied on for nice or extended operations, is called a Theodolite. This instrument, only, will be described at present; others will be subsequently explained. 5 66 ELEMENTS OF SURVEYING. [BOOK II. The Theodolite. 23. PI. 1. — The theodolite is an instrument used to measure horizontal and vertical angles. It is usually placed on a tripod ABC, which enters, by means of a screw, the lower horizontal plate DEy and becomes firmly attached to the body of the instrument. Through the horizontal plate DE, four small hollow cylinders are inserted, which receive four screws with milled heads, that w^ork against a second horizontal plate, FG. The upper side of the plate DE terminates in a curved sur- face, which encloses a ball, that is nearly a semi-sphere, with the plane of its base horizontal. This ball, which is hollow, is firmly connected with the smaller base of a hollow conic frustum, that passes through the curved part of the plate DE, and screws firmly into the curved part of the second horizontal plate FG. A hollow conic spindle passes through the middle of the ball, and the hollow frustum with which it is connected. To this spindle, a third horizontal and circular plate HI, called tlie limb of the instrument, is permanently attached. Within this spindle, and concentric with it, there is a second spindle, called the inner, or solid spindle. To this latter, is united a thin circular plate, called the vernier plate, which rests on the limb of the instrument, and supports the upper frame-work. The two spindles terminate at the base of the spherical ball, where a small screw enters the inner one, and presses a w^asher against the other, and the base of the ball. On the upper surface of the plate FG, rests a clamp which goes round the outer spindle, and which, being compressed by the clamp-screw K, is made fast to it. This clamp is thus connected with the plate FG. A small cylinder a, is fastened to the j^late FG : through this cylinder a thumb-screw L passes, and works into a small cylinder h, connected with the clamp. The cylinders SEC. I.] OF THE THEODOUTE. 67 h jind «, admit of a motion round their axes, so that the screw L may work through them freely. Directly above the clamp, is the lower telescope MN. This telescope is connected with a hollow cylinder, which is worked freely round the outer spindle, by the thumb-screw P having a pinion working into a concealed cog-wheel, that is perma- nently fastened to the limb of the instrument. By means of a clamp-screw Q, the telescope is made fast to the limb, when it will have a common motion with the limb and outer spindle. The circular edge of the limb is chamfered, and is generally made of silver, and on this circle the graduation for horizontal angles is made. In the instrument described, the circle is divided into degrees and half-degrees; the degrees are numbered from to 360. On the circular edge of the vernier plate, is a small plate of silver, called a vernier ; this plate is divided into 30 equal parts, and numbered from the line marked to the left. Two levels, at right angles to each other, are attached to the vernier plate by small adjusting screws; one of the levels is seen in the figure. The vernier plate turns freely around with the inner spindle. It is made fast to the limb of the instrument by the clamp- screw S; after which the smaller motions are made by the tangent-screw T. There is a compass on the vernier plate, that is concentric with it, the uses of which will be explained under the head. Compass. The frame-work which supports the horizontal axis of the vertical semicircle UV and the upper telescope, with its attached level, rests on the vernier plate, to which it is made fast by three adjusting screws, placed at the angular points of an equilateral triangle. The vertical semicircle UV, is called the vertical limh ; its motions are governed by the thumb-screw Z, which has a pinion that works with the teeth of the vertical 68 ELEMENTS OF SURVEYING. [BOOK n. limb. On the face of the vertical limb, opposite the thumb- screw Z, the limb is divided into degrees and half-degrees: the degrees are numbered both ways from the line marked 0. There is a small plate resting against the graduated face of the vertical limb, called the vernier; it is divided into 30 equal parts, and the middle line is designated by 0. On the other face of the vertical limb, are two ranges of divisions, commencing at the point, and extending each way 45°. The one shows the vertical distance of any object to which the upper telescope may be directed, above or below the place of the instrument, in 100th parts of the horizontal dis- tance: the other, the difference between the hypothenusal and base lines — the hypothenuse being supposed to be divided into one hundred equal parts: therefore, by mere inspection, wo can ascertain the number of links, which must be subtracted from every chain of an oblique line, to reduce it to a true horizontal distance. The supports of the upper telescope are called the wyes, and designated ys. Two loops, turning on hinges, pass over the telescope, and are made fast by the pins c and d; these loops confine the telescope in the Y^s. By withdrawing the pins, and turning the loops on their hinges, the telescope may be removed for the purpose of being reversed in position; and in both situations, the telescope can be revolved in the Y^s about its axis. In the telescopes attached to the theodolite, are two prin- cipal lenses, one at each end. The one at the end where the eye is placed, is called the eye-glass, the other the object-glass. In order that the axis of the telescope may be directed to an object with precision, two spider's lines, or small hairs, are iixed at right angles to each other, and placed within the barrel of the telescope, and at the focus of the eye-glass. The vertical hair is moved by two small horizontal screws, one of SEC. I.] OF THE THEODOLITE. 69 which, /, is seen in the figure ; and the horizontal hair, by two yertical screws, g and li. Adjustments of the Theodolite. 24. 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. FiKST ADJUSTMENT. — To fix tlw intersection of tlie sjiidefs lines in tlie axis of tlie telescope, luliicli is called tlie line of collimation. Having screwed the tripod to the instrument, extend the legs, and place them firmly. Then loosen the clamp-screw .S*, of the vernier plate, and direct the telescope to a small, well- defined, and distant object. By means of a small pin i, on the under side of the telescope, slide the eye-glass till the spiders lines are distinctly seen; then with the thumb-screw X, which forces out and draws in the object-glass, adjust this glass to its proper focus, when the object, as well as the spider's lines, will be distinctly seen : after which, by the tangent-screw T and the thumb-screw Z, bring the intersection of the sj^iders lines exactly upon a well-defined point of the object. Having done this, revolve the telescope in the Y^s half round, when the attached level mn will come to the u^^per side. See if, in this position, the horizontal hair appears above or below the point; and in either case, loosen one, and tighten the other, of the two screws that work the horizontal hair, till the horizontal hair has been carried over half the space between its last position and the observed point. Carry the telescope back to its place ; direct again the intersection of the spiders lines to the point, and repeat the operation till the horizontal hair neither ascends nor descends, while the telescope 70 ELEMENTS OF SUKYEYING. [BOOK H. is revolved. A similar process ^ill arrange the vertical hair, and the line of collimation is then adjusted. Secoxd ADJUSTMENT. — To mctlce the axis of the attached level of the ui^per telescope, ^parallel to the line of collimation. Turn the vernier plate, till the telescope comes directly over two of the levelling screws, between the plates DE and FG. Turn these screws contrary ways, keeping them firm against the plate FG, till the bubble of the level mn stands at the middle of the tube. Then, open the loops, and reverse the telescope. If the bubble still stands in the middle of the tube, the axis of the tube is horizontal: but if not, it is inclined, the bubble being at the elevated end. In that case, by means of the small vertical screws m and n, at the ends of the level, raise the depressed end, or depress the elevated one, half the inclination : and then, with the levellins: screws, brinc( the level into a horizontal position. Ee verse the telescope in the Y's, and make the same correction again ; and so on, until the bubble stands In the middle of the tube, in both positions of the telescope : the axis of the level is then horizontal. Let the telescope be now revolved in the Y's. If the bubble continue in -the middle of the tube, the axis of the level is not only horizontal, but also parallel to the line of collimation. If, hov\-- ever, the bubble recede from its 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 antagonistic screws (one of which is seen at js), which work horizontally. 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 in the middle of the tube. It is difficult to make the first part of this adjustment, while the axis of the level is considerablv inclined to the line SEC. I.] OF THE THEODOLITE. 71 of collimation ; for, if the level were truly horizontal in one position of the telescope, when the telescope is reversed, the bubble would not stand in the middle of the tube, except in one position of the level. This suggests the necessity of making the first part of the adjustment with tolerable accuracy; then, liaving made the second with care, let the first be re-examined, and proceed thus till the adjustment is completed. Third adjustment. — To mcike the axes of tlie levels, on the liml), ijeryendicular to the axis of the instrument. This adjustment is effected, partly by the levelling screws, and partly by the thumb-screw Z. Turn the vernier plate, until the upper telescope comes directly over two of the levelling screws, then turn the screws contrary ways, till the upper telescope is horizontal; after which, turn the vernier plate 180°, and if the bubble of the level remains in the middle of the tube, one line of the limb is horizontal. But if the bubble recedes from the centre of the level, raise the lower, or depress the upper end, one-half, by the levelling screws, the other by the thumb-screw Z, till it is brought into a horizontal position. Turn the vernier plate again 180°, and if the level be not then horizontal, make it so, by dividing the error as before, and repeat the operation until the line of the limb is truly horizontal. Then turn the vernier plate 90°, and level as before. The limb ought now to be truly horizontal; but, lest the first horizontal line may have been changed, in ob- taining the second, it is well to bring the telescope and level, two or three times over the levelling screws, until an entire revolution can be made without displacing the bubble from the middle of the tube. As this can only take place when the level revolves around a vertical line, it follows that the limb will then be horizontal, and the axis of the instrument vertical. Then, by means of the small screws at the ends of the levels, 72 ELEMENTS OF SURVEYING. [BOOK II. bring the bubbles to the centres, and the axes of the levels will be perpendicular to the axis of the instrument. Fourth adjustment. — To make the axis of tlie vertical limb perpendicular to the axis of the instrument. Bring the intersection of the spider's lines of the upper telescope upon a plumb-line, or any well-defined vertical object, and move the telescope with the thumb-screw Z : if the inter- section of the spider's lines continues on the vertical line, the axis is horizontal. Or, the adjustment may be efiected thus: Direct the inter- section of the spider's lines to a well-defined point that is considerably elevated: then turn the vertical limb, until the axis of the telescope rests on some other well-defined point, upon or near the ground: reverse the telescope, and turn the vernier plate 180° ; then, if in elevating and depressing the tele- scope, the line of collimation passes through the two points before noted, the axis is horizontal. If it be found, by either of the above methods, that the axis is not horizontal, it must be made so by the screws which fasten the frame-work to the vernier plate. There are two important lines of the theodolite, the positions of which are determined with great care by the maker, and fixed permanently. First, the axis of the instrument is placed exactly at right angles Avith the limb and vernier plate ; and unless it have this position, the vernier plate will not revolve at right angles to the axis, as explained in the third adjust- ment. Secondly, the line of collimation of the upper telescope is fixed at right angles to the horizontal axis of the vertical limb. We can ascertain whether these last lines are truly at right angles, by directing the intersection of the spider's lines to a well-defined point ; then removing the caps which confine the SEC. I.] VERNIERS. 73 horizontal axis in its supports, and reversing the axis : if the intersection of the spider's lines can be made to cover exactly the same point, without moving the vernier plate, the line of collimation is at right angles to the axis. If the theodolite be so constructed that either of the Y's admits of being moved laterally, so as to vary the angle between the horizontal axis and the line of collimation, these lines may be adjusted at right angles to each other, if they have not been so placed by the maker. The lower telescope, being used merely as a guard, requires no adjustment, although it is better to make the axis, about which its vertical motions are performed, horizontal, or per- pendicular to the axis of the instrument ; and this is easily effected by means of the two small screws h and /, which work into the slide A', that is connected with the horizontal axis. Having explained the methods of properly adjusting the theodolite, we Avill now explain the particular uses of its several parts in the measurement of angles. Verniers. 25. Before explaining the vernier, as applied to the the- odolite, we shall discuss the general theory of verniers. A Vernier is a contrivance for measuring smaller arcs than those into which the limb of an instrument is divided. ,' It is a graduated scale, so arranged, as to cover an exact number of equal spaces on the primary scale or Umi, to v/hich it is applied. 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 ; 74 ELEMENTS OF SURVEYING. [BOOK II. the principle, however, is the same in all, and may be illus- trated by a simple diagram. S £> iO 1J J2 13 I'h iS 16 17 IS 19 a\ \ j \ I \ \ \b i j n ! 1 i 1 1 1 1 1 1 JJ Let yl^ be any Ihnb or scale of equal parts, one of which let us suppose equal to d. 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 l. The difference between a space on the limb and a space on the vernier, is therefore equal to one-tenth of 1), or -^^1). This is the least space that can be measured b}' means of the vernier, and is called the least count ; hence. The least count of a vernier is equal to one of the divisions of the limh divided ly the numher of spaces on the vernier. Eeadi:n'GS. 26. The true reading of an instrument, for any position ol 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 ad- vanced along the limb a distance equal to -^-qI), and the reading will become 9 + -^^h. If w^e 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 yV ^j and the reading becomes ^ + -^^1 ; when 3 coincides with division 12, the reading will become 9 -f y%&, and so on, till SEC. I.] ' MEASUREMENT OF ANGLES. 75 finally, when the point 10 coincides with 19 of the limb, the distance 9 will have been increased by-ff &, 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 Imd in ihe direction of the graduation iip to the division tine next preceding the j^oi/it of the vernier ; this is called the reading on the limb. Look along the vernier till a line is found to coincide tvith a line of the Umh : multiply the number of this first line by the least count of the vernier: this is the reading on the vernier: the sum of these two readings is the reading of the instrument. In the theodolite described, the limb is divided into half- degrees, and 30 spaces on the vernier cover 29 spaces on the limb. Hence, the least count of this instrument is 3V of a half-degi*ee, or 1'. Fig. 2, Plate 1, exhibits the vernier of the horizontal limb, and Fig. 3, the vernier of the vertical limb. To Measure a Hoeizoxtal Axgle with the Theodolite. 27. 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 uj^per end of the tripod. Having made the limb truly level, place the of the vernier at 0, or 360° of the limb, and fasten the clamp-screw S of the vernier plate. Then, facing in the direction between the lines which subtend the angle to be measured, turn the limb with the outer spindle, until the telescope points to the object on the left, very nearly. Clamp the hmb T\ith the clamp-screw K, and by means of the tangent-screws L and Z, bring the intersection of the spiders lines to coincide exactly with the object. 76 ELEMENTS OF SURVEYING. [BOOK II. Having loosened the clamp-screw ft of the lower telescope MN, direct it with the thumb-screw P to the same object at which the npper telescope is directed; then tighten the clamp- screw Q. This being done, loosen the clamp-screw 8 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 lower telescope having been made fast to the limb, will indicate any change of the position of the limb, should any have taken place; and, as the accuracy of the measurements depends on the fixedness of the limb, the lower telescope ought to be often examined, and, if its position has been altered, the limb must be brought back to its place by the tangent-screw L. Note. — It is not necessary to place the point of the vernier at the point of the limb, previously to commencing the measurement of the angle, but convenient merely; for, what- ever be the position of this point on the limb, it is evident that the arc which it passes over is the true measure of the hori- zontal angle. If, therefore, its place be carefully noted for the first direction, and also for the second, the difierence of these two readings will be the true angle, unless the point of the vernier shall have passed the point of the limb, in which case the greater reading must be subtracted from 360°, and the remainder added to the less. Measurement oe Vertical Angles. 28. We shall first explain the method of determining the index error. Having levelled the horizontal limb, direct the telescope to some distinctly marked object, as the top of a chimney, and read the instrument. Reverse the telescope in the Y's, and turn the vernier plate 180°, and having directed SEC. I.] MEASUREMENT OF VERTICAL ANGLES. 77 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. TVhen the reading, found with the eye end of the telescope nearest the yernier, 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 tlie reading of tlie limb iclien tlie telescope is directed to a fixed object, first with the eye end of the telescope nearest the vernier, and then with the telescope and vernier plate both reversed. Take half the difference of these readings, and affect it vAtli 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. 29. Having determined the index error, let the axis of the telescope be directed to any point 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 coiTectly 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. 78 ELEMENTS OF SURYEYING. [BOOK II. Measurements with the Tape or Chaix o:n^ly. 30. It often happens that instruments for the measurement of angles cannot be easily obtained ; we must then rely entirely on the tape or chain. AVe now propose to explain tlie best methods of determining distances, without the aid of instruments for the measurement of horizontal or vertical angles. I. To trace, on the ground, the direction of a right line, that shall be perpendicular at a given point, to a given right line. FIRST METHOD. 31. Let BC he the given right line, and A the given point. Measure from A, on the line BC, two equal distances AB, AC, one on each side of the point A. Take a portion of the chain or tape, greater than '^^ AB, and place one extremity at B, and with the other, trace the arc of a circle on the ground. Then remove the end which was at B to C, and trace a second arc intersecting the former at B. The straight line, drawn through D and A, will be per- pendicular to BC at A, SECOND METHOD. 32. Having made AB = AC, take any portion of the tape or chain, considerably greater than the distance between B and C. Mark the middle point of it, and fasten its two extremities, the one at B and the other at C. Then, taking the chain by the ^ middle point, stretch it tightly on either side of BC, and place a staff at I) or E : DAE will be the perpendicular required. SEC. I.] MEASUREMENTS WITH THE CHAIN. 79 THIRD METHOD. C 33. Let AB be tlie given line, and C the point at which the perpendicular is to be drawn. From ^ the point C, measure a distance CA equal ^ to 8. With G as a centre, and a radius equal to 6, describe an arc on either side Qi AB : then, with ^ as a centre, and a radius equal to 10, describe a second arc K intersecting the one before described, at E: then draw the line EC, and it will be perpendicular to AB at C. Note. — Any three lines, having the ratio of 6, 8, and 10, form a right-angled triangle, of which the side corresponding to 10 is the hypothenuse. FOURTH METHOD. 34. Let AD be the given right line, and D the point at which the perpendicular is to be drawn. Take any distance, on the tape or chain, and place one extremity at i), and fasten the other at some point, as E, "* 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 tlie 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 AE, prolonged, at F. Then place a staff at F ; ADF will be a right angle, being an angle in a semicircle. Note. — There is a very simple instrument, used exclusively in laying off right angles on the ground, which is called the 80 ELEMENTS OF SURVEYING. [BOOK IL Sukyeyor's Cross. 35. This iustriiment consists of two bars, AB and CD, PI. 2, Fig. 1, permanently fixed at right angles to each other, and firmly attached at E, to a pointed staff, which serves as a support. Four sights are screwed firmly to the bars, by means of the screAvs a, 1), c, and d. As the only use of this instrument is to lay ofi* right angles, it is of the first importance that the lines of sight be truly ajt right angles. To ascertain if they are so, let the bar ^^ 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 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 giyen line, the other will mark the direc- tion of a line perpendicular to it, at the point where the in- strument is placed. n. Prom a given point -without a straight line, to let fall a per- pendicular on the line. 86. Let C be the given point, and AB the given line. From Cj measure a line, as CA, ^ to any point of the line AB. From ^ A, measure on AB any distance as AF, and at F erect FE perpendicu- ^ •*' lar to AB. Haying stationed a person at A, measure along the perpen- diculaj: FE until the forward staff is aligned on the line Ad C SEC. I.] MEASUREMENTS TVITH THE CHAIN. 81 then measure the distance AE. JSTow, bj similar triangles, we have, AE '. AF :: AC : AD, in whicli all the terms are known except AD, which may, therefore, be found. The distance AD being laid off from A, the point D, at which the perpendicular CD meets AB, be- comes known. If we wish the length of the perpendicular, we use the proportion, AE : EF :: AC : CD, in which all the terms are known, excepting CD: therefore, CD is determined. III. To determine the horizontal distance from a given point to an inaccessible object. FIEST METHOD. 37. Let A be an inaccessible object, and E the point from> which the distance is to be measured. At E, lay off the right angle AED, | and measure in the direction ED, jp any conyenient distance to D, and ^-- ,'-'' I place a staff at D. Then measure ^r^-''" f from E, directly towards the object n ^^--^ • A, a distance EB of any conyenient D F ^ length, and at B, lay off a line BC perpendicular to EA, Measure along the line BC, until a person at D shall range the forward staff on the line DA. Xow, DF is known, being equal to the difference between the two measured lines DE and CB. Hence, by similar triangles, DF : FC :: DE : EA, in which proportion all the terms are known, except the fourth, which is therefore found. 82 ELEMENTS OF SURVEYING. [book II. -^A:- r^.. SECOND 3IETnOD. 38. At the point E, lay off BB perpendicular to the Ime BA, and measure along it any convenient distance, as BB. At B lay off the right angle f EBB, and measure any distance %/ in the direction BB. Let a person I) it D align a staff on BA, ^'hile a second person at B aligns it on BE : the staff will thus be fixed at C. Then measure the distance BC. The two triangles BCB and CAB being similar, we have, BC : BB :: CE : EA, in which all the terms are known, except the fourth, which is, therefore, found. THIED METHOD. 39. Let B be the given point, and A tlie inaccessible object; it is required to find i>^. Measure any horizontal base-line, as BC. Theu, having placed staves at B and C, measure any conve- nient distances BB and CE, such that the points B, B, and A, shall be in the same right line, as also, the points E, C, and A ; then meas- d / ^.-^^-^ \ s ure the diagonal lines BC and BB. o*^ ^^ K'ow, in the triangle BBC, the three sides are known, there- fore, the angle BCB can be found. In the triangle CBB, the tliree sides are also known, therefore the angle CBB can be determined. These angles being respectively subtracted from 180"^, the two angles ACB and ^li? (7 become known; and hence, SEC. L] MEASUREMENTS. 83 ill the triangle ^i^ 6', we have two angles and the included side, to fxud the side JBA. POURTH METHOD. 40. Let AC be the distance required. Lay off the right angle CAB, and measure AB, any con- venient distance. At B lay off the right angle CBI), and fix the point D, carefully, in line with AC. Measure AD. Then, AD:AB::AB:AC.'.AC = AD. (Legendre, Bk. IV., Prop. 23.) XoTE. — When such problems occur in practice, the distance AC is usually a portion of a longer line, so that the line CAD is well marked by stakes or pins, before AB is measured. IV, To prolong a line beyond an obstacle. 41. Let OA be the line to be 'prolonged. Lay off GAB = 120°, or CAB = G0°. Meas- ure AB, of such length as to permit BC to be 'measured without meeting the obstruc- tion. Make ABC =60°, and measure BC, equal to AB. If A be not in sight from C, make the angle BCF equal to 120'', and resume the survey of the line. ^C is equal to AB or BC. XoTE. — This method may be employed in the absence of any angular instruments, by constructing an equilateral triangle with the chain. Holding together the end of the chain with 84 ELEMENTS OF SURVEYING. [BOOK II. tlie 90-link point, let two assistants draw out at the 30, and at the 60 point, until the three lines are straight. V. To find the altitude of an object, when the distance to the vertical line passing through the top of it is known. 42. Let CD be the altitude required, and AC the known distance. From A, measure on the line A C, any convenient dis- tance AB, and place a staff vertically at B. Then placing ^ ,---'''[ ^^^^^^^^-^^--^^^ the eye at A, sight to the '^^ • B C object B, and let the point, at which the line AB cuts the staff BE, be marked. Measure the distance BE on tlie staff; then, AB : BE :: AC. : CD, whence CB becomes known. If the line AC cannot be measured, on account of inter- vening objects, it may be determined by calculation, as in the last problem, and then, having found fhe horizontal distance, the vertical line is readily determined, as before. Applications to Heights axd Distaxces. I. To determine the horizontal distance to a point which is inaccess- ible by reason of an intervening river.* 43. Let C be the point. Measure along the bank of the riyer 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 G from both of them. Then measure the horizontal angles CAB and CBA, with an instrument adapted to that purpose. * Read, definitions, from 3 to 14, pages t9 and S9. SEC. I.] HEIGHTS AND DISTANCES. 85 Let us suppose that we have measured AB = 600 yards; CAB = A = 5r 35", and CBA = B = 64:° 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 G (57° 34') 0.073649 log sin A (57° 35') 9.926431 log AB (600) 2.778151 log BC 600.11 2.778231 To find the distance AC sin C : sm B : : AB : AC, and applying logarithms, we have, (a. c.) log sin C (57° 34') 0.073649 log sin B (64° 51') 9.956744 log AB (600) 2.778151 log AC 64:3 Ad 2.808544 n. To determine the altitude of an inaccessible object above a given horizontal plane. FIEST METHOD. 44. Suppose D to be an inaccess- x> ible object, and BC the horizontal --^"'''I^^ plane from which the altitude is to Q^j..i .=^i.Z..^ be measured: then, if we suppose D (7 n^ -^ y,^^ to be a vertical line, it will represent Nr"*/^''' the required distance. ^ Measure any horizontal base-line, as BA; and at the ex- tremities B and A, measure the horizontal angles CBA and CAB. Measure, also, the angle of elevation BBC. Then, in the triangle CBA, there will be known, two angles 86 ELEMENTS OF SURVEYIKG. [BOOK II. and the side AB ; the side BG can therefore be found by calculation. Having found BC, we shall haye, in the right- angled triangle DBC, the base BC and the angle at the base, to find the perpendicular DC, which measures the altitude of the point D, aboye the horizontal plane BC. Let us suppose that we haye found, by measurement, BA — 780 yards. The horizontal angle CBA = i? = 41° 24', the horizontal angle CAB = A = 96° 28', and the angle of eleyation DBC = 10° 43'. To find, in the triangle BCA, the horizontal distance BC. The angle BCA = C = 180° - (^1 + B) = 42° 08'. Then, sin C : sin ^ : : AB : BC; and applying logarithms, we haye, (a. c.) log sin C (42° 08') 0.173369 log sin A (96° 28') 9.99?228 log AB (780) 2.892095 log BC 1155.29 yards . . . 3.062692 In the right-angled triangle DCB, to find DO. We haye, from Theorem IV., E : tan DBC : : BC : DC. Applying logarithms, we haye, (a. c.) log R (90°) 0.000000 log tan DBC (10° 43') . . . 9.277043 log BC (1155.29) . . . 3.062692 loo- DC 218.64 .... 2.339735 Note 1.— It might, at first, appear, that the solution which we haye giyen, requires that the points B and A should be in SEC. I.] HEIGHTS AND DISTANCES. 87 the same liorizontal plane; bat 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 throngh A should cut the ver- tical linei^C, at any point, as E, above or below C, AB would still be the horizontal distance between B and A, and AE, would be the horizontal distance between A and C. If at A, we measure the angle of elevation at the point D, we shall know in the right-angled triangle DAE, the base AE, and the angle at the base; from which the perpendicular DE can be determined. Let us sux)pose that we had measured the angle of elevation DAE, and found it equal to 20° 15'. First : In the triangle BAG, to find AC, or its equal AE. sin C \ sin B :: AB : AC or AE. Applying logarithms, we have, (a. c.) log sin C (^2° OS') 0.1T33C9 log sin B (41° 21') 9.820406 log AB (TSO) 2.892095 log AE 768.9 2.885870 In the right-angled triangle DAE, to find DE. TTe have, from Theorem IT., E : tan A : : AE : DE : hence, (a. c.) log E (90°) ..'.... 0.000000 ; log tan A (20° 15") .... 9.566932 log AE (768.9) 2.885870 lo^ DE 283.66 2.452802 88 ELEMENTS OF SUKVEYIXa. [book II. Now, since DC is less than DB, it follows that the station B is above the station A, That is, DE - DC = 283.66 - 218.64= 65.02 = EC, which expresses the vertical distance that the station B is above the station A Note 2. — It should be remembered, that the vertical dis- tance which IS obtained by the calculation, is estimated from a horizontal line passing through the eye, at the time of ob- servation. Hence, the height of the instrument is to be added, in order to obtain the true result. SECO:NrD METHOD. 45, 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 outward angle DBC is equal to the sum of the angles A and ADB, it follows that, the angle ADB is equal to the difference of the angles of elevation at A and B. Hence, we can find all the parts of the triangle ADB. Having found DB, and knowing the angle DBC, we can find the altitude DC. 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 us suppose that we have measured the base-line and the two angles of elevation, and found, SEC. I.] HEIGHTS AND DISTANCES. 89 AB = 975 yards, A = 15° oG', and DBC = 27° 29'; required the altitude DC. First: ADB = BBC - A = 27° 29' - 15° 36' = 11° 53'. In the triangle ADB, to find BD = c. sin D : sin ^ : : AB : DB ; hence, (a. c.) log sin D (11° 53') .... 0.686302 log sin A (15° 36') .... 9.429623 log AB (975) . 2.989005 ' log BD 1273.3 3.104930 In the triangle DEC, to find DO = b. We have, Theorem I.. E : sin B :: BD : DC; hence, (a. c.) log B (90°) 0.000000 log sin B (27° 29') .... 9.664163 log C (1273.3) 3.104930 log DC 587.61 2.769093 m. To determine the perpendicular distance of an object below a given horizontal plane. 46. Suppose C to be directly over the giyen object, and A the point f 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 AC, BC, to be drawn. The oblique lines from A and B, to the object, are the hypothenuses of two right-angled triangles, of which AC, BC, are the bases. The perpendiculars of these 90 ELEMENTS OF SUEVEYING. [BOOK H. triangles are the distances from the horizontal lines AC, BC, to the object. If we turn the triangles about their bases AC, BC, until they become horizontal, the object, in the first case, will fall at C, and in the second at C". Measure the horizontal angles CAB, CBA, and also the angles of depression C'AC, C'BC. Suppose that we have measured, and found AB = 6T2 yards; BAC=n° 29'; ABC =39° 20'; angle of depression C'AC = 27° 49', and C'BC =19° 10'. First: In the triangle ABC, the horizontal angle ACB = 180° - {A + B) = 180° - 111° 49' = 68° 11'. To find the horizontal distance AO sin C : sin jS : : AB : AC ; hence, (a. c.) log sin C (68° 11') 0.0322T5 log sin^ (39° 20') 9.801973 ,. log AB (672) 2.827369 log AC 458.79 ...... 2.661617 To find the horizontal distance EC. sin C : sin A :: AB : BC; whence, (a. c.) log sin C (68° 11') 0.032275 log sin A (72° 29') 9.979380 log AB (672) 2.827369 lo^ BC 690.28 2.839024 In the right-angled triangle CAC, to find CC. We have, Theorem IV., E : tan A :: AC : CC; whence, (a. c.) log E (90°) ^ 0.000000 log tan A (27° 49') 9.722315 log AC 458.79 2.661617 los CC 242.06 ..... 2.383932 SEC. I.] HEIGHTS AND DISTAIJCES. 91 In the triangle CBC", to find CC" = b. We have, Theorem IV., E : tun B :: BC : CC"; whence, (a. c.) log E (90°) ....... 0.000000 log tan B (19° 10') .... 9.541061 log BC (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 aboye station B. PEOBLEMS. 1. Wanting to know the distance between two inaccessible objects, which lie in a direct leyel line from the bottom of a tower 120 feet in height, the angles of depression are nieasnred from tli^MI 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 be- tween two trees, A and B, w^hich could not be directly measured because of a pool which occupied the intermediate space, the distances of a third point C from each of them were measured, and also the included angle ACB : 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 92 ELEMENTS OF SURVEYING. [book n. hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then measur- ing in a direct line 180 feet farther from the hill, the angle of eleyation of the top of the tower was 33° 45': required the height of the tower. Ans. 83.998. 4. TTanting to know the hori- zontal distance between two inac- cessible objects U and W, the fol- lowino: measurements were made. ■':iw 'AB = 536 yards BAW= 40° 16' yiz., - . WAB = 57° 40' ABB = 42° 22' BBW= 71° 07'; required i -he distance BW. 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 B, 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 B a staff was set up. From C a distance CF was measured, not in the direction BC, equal to 200 yards, and from B a distance BB equal to 200 yards, and the following angles taken, fAFC = 83° 00', BBB = 54° 30', Tiz., iACB= 53° 30', BBC = 156° 25', ACF= 54° 31', BBB = 88° 30'. Ans. AB = 345.467 yards. SEC. I.] HEIGHTS AND DISTANCES. 93 6. From a station P there can be seen three objects, A, B, and C, whose distances from each other are known : yiz., AB = 800, AC = 600, and BC = 4:00 yards. Now, there are measured the horizontal angles, APC = 33° 45', and BPC = 22" 30' : it is required to find the three distances, PA, PC, and PB, ^-. GEOMETRICALLY. With the three given sides construct the triangle AB C. Then, at A lay off the angle PAD = 22° 30', and at P the angle APP = 33° 45', and note P, the point at which the two lines intersect. Through the points A, P, and B de- scribe the circumference of a circle, and through C and P draw the line CPPj the point P in which it intersects the circumference, will be the position of the station. By obserying the equal angles in the figure, the trigono- metrical solution is not difficult. "We find, [ PA = 710.193 yards. Ans. jsr The offset-courses, for the lines LM, MN, and 2s 0, are re- spectively Im, m'n, and )iJ^ JST 0< Such offsets, in field surveys, may generally be measured by the flag-staff, and the right angle may be determined with sufficient accuracy by the eye. It is, of course, immaterial, so far as accuracy is concerned, upon which side of the line the offset is taken, whether it be outside or inside the field. 116 ELEMENTS OF SURVEYING. [BOOK II. 80. 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, inyolves 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 yariable, eyen in the same instrument, tliat a change of practice is yery desirable. Many surveyors, to insure a higher degree of accuracy, measure the angles of a field with the theodolite or transit, and then, haying determined the bearing of one side w^ith sufficient accuracy, calculate the others by a method to be shown in a subsequent article. 81. 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. 82. The best method of recording the notes of the survey of any line, from which numerous ofi*sets are measured, is to record the distances along the line in a central column of the left-hand page of the note-book, beginning the record at the bottom of the page, and reserving each right-hand page for a diagram of the survey, and such remarks as are necessary. The advantage of beginning at the bottom of the page is this: that when standing on the line to be surveyed, and look- ing 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 SEC. m.] COMPASS SUEYEnNG. 117 taken, a diagram is an important aid to a ready interpretation of the other notes. Gexeral Example. 83. To explain the method, in full, of making a compass survey and recording the notes, ^e will take an example of a farm, in which, in addition to the usual survey of the boundary, snch other measurements are made as to enable us to make a correct map of the whole. Page 119, represents a farm to be sui*veyed, and page 11 S, the notes which are made, in the operations on the field. Beginning with the comer marked A, the bearing of the line AB, is taken. In most cases, oflTsets from both A and B would be taken, in order that the survey may be clear of the fence, but such ofi'sets are not recorded. The record of the bearing of the first course is entered at the right of the column (page 118), 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 1.30 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. to Station A FenceN6GhtW S~¥'l- ^^^^ ,''' OAK T REIT ,.0v' IN FAST SIDE Slfdion A 2i- (^S^S^IAT MAP OF FAIUI. 120 ELEMENTS OF SUETEYES-G. [BOOK H. At B, the bearing of the northernmost chimney of the farm- honse is taken (X. T3° E.) Such bearings serre two purposes. They aid in the location of the objects observed, npon 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, Tvithin the botmdary, should be selected, and their bearings, from different angles, carefully noted. The chimney of the farm-house and the oak-tree in the comer 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 offeets cannot be correctly measured. At E, the bearing of the oak-tree is taken (INT. 1^° E.) On the course between E and F, a marsh is encountered, which the chainmen pass,. by an oflfeet course. At F, another beai-ing is taken of the oak-tree (S. 44^' 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 cotirse from H to J, the turnpike is again crossed : the intersection of both sides, together with the beating, are carefully noted. From / to K, the intersection and bearing of the fence be- tween the potato and the wheat field, are recorded. The course from jST to J. closes the survey. To locate the buildings about the farm-house, a few measure- ments would be necessary; but they may begin with the point already located by the bearings taken — the chimney nearest the north end of the house. SEC. III.] COMPASS SURVEYING. 121 The dimensions of the buildings, their distances apart, and the direction of one side of each being taken, sufficient data is afforded for locating them, correctly, upon the map. K"0TE. — 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. When the angle between adjacent sides is taken with the Transit or Theodolite, the work can only be done at the corners of the field; and when, as frequently 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 aflibrds the means of taking the proper bearing without going to the angle. Furthermore, the bearings may be meas- ured with the compass, by placing it at the alternate stations only. COXTEXTS OF GkOUXD. 84. Haying explained the necessary operations on the field, we shall now proceed to show the manner of computing the contents of ground. THE TEAVERSE TABLE AXD ITS USES. 85. This table shows the latitude and departure corre- sponding 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 followins: is the method of deducing the formulas for ■to computing a traverse table; by means of these formulas and a 6 122 ELEMENTS OF SURYEYEs'G. [BOOK H. table of natural sines, the latitude and departm-e of a course may be computed to any desirable degree of accuracy. Let AD represent any course, and XAD — ACB, expressed in degrees and minutes, be its bearing. Let AC be the unit of measure of the course, and also the radius of the table of natural sines (Bk. L, Sec. III., Art. 14). Draw DE and ^ CB parallel to XS, and AE perpendicular to AS, Then will DE be the latitude, and AE the departure of tlie course, and CB the cosine, and AB the sine of the hearing, to the radius AC=1. From similar triangles we have these proportions, AC \ CB w AD : DE, or 1 : cos of the bearing : : course : latitude; and AC \ AB w AD \ AE, or 1 : sin of the bearing : : course : departure. Whence, lat. = course X cos of the bearing, dep. = course X sin of the bearing. We have then the following practical rule for computing the latitude and departure of any course. Look in a tahle of natural sines for tJie cosine and sine of the bearing. MuUipJif each ty the length of the course, and the first product icill le the latitude, and the second will he the departure of the given course. EXAMPLES. 1. The bearing is 65^ 39', the course 69.41 chains: what is the latitude, and what the departure? SEC. III.] COMPASS SURVEYING. 123 Natural cosine of 65° 39' 41231 Length, of the course 69.41 Product, which is the Dif. of Lat. . . 28.6184371 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 departui-e? 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 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 traverse 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. 86. The latitudes or departures of courses IN of different lengths, but which have the same bearing, are proportional to the ...^ lengths of the courses. Thus, in the ^ figure, the latitudes AG, AC, or the de- partures GF, CB, are to each other as the g courses AF, AB, ..... -vF E 124 ELEMENTS OF SURVEYING. [BOOK H. 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 depart- ures of the courses taken separately. Hence, if we have any number greater than 100, as 614, we haye only to recollect that, 610 + 4= 614; and also, that the latitude and departure of 610, are ten times .as great, respectively, as the latitude and departure of 61 : that is, equal to the latitude and departure of 61 multiplied by 10; or, to such latitude and departure with the decimal point removed one place to the right. EXAMPLES. 1. To find the latitude and departure for the bearing 29|-°, and tlie course 614. Latitude for 610 . . . 530.90 Latitude for _f • • • 3.48 Latitude for 614 .. . 534.38 Departure for 610 . . 300.40 Departure for 4 . . 1.97 Departure for 614 . . 302.37 In this example, the latitude and departure answering to the bearing 29-1-°, and to the distance 61, are first taken from the table, and the decimal point removed one place to the right : this gives the latitude and departure for the distance 610; the latitude and departure answering to the same bearing and the distance 4, are then taken from the table and added. 2. To find the latitude and departure for the bearing 62|°, and the course 7855 chains. Latitude for 7800 . 3602.00 | Departure for 7800 . 6919.00 Latitude for 55 . 25.40 Departure for 55 . 48.79 Latitude for 7855 . 3627.40 | Departure for 7855 . 6907.70 SEC. III.] COMPASS SUEVEYING. 125 ]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 ofi" 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 . . 24.88 Latitude for _^ . . .38 Latitude for 37.57 . . 25.26 Departure for 37.00 • . 27.39 Departure for .57 . .42 Departure for 37.57 . 27.81 bala:s'ciis"g the woek. 87. Having explained the use of the traverse table, we can proceed to compute the area of the ground. The field-notes having been completed, rule a new table, as on next page, with four additional columns, two for latitude, and two for departure. Then find, from the traverse table, the latitude and de- parture of each course, and enter them in the proper columns opposite the station. Then add up 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 remain- der is called the erro?' i?i latitude. This error takes the name of that column which is the least. For example, if the sum of the northings is less than the sum of the southings, the error is called, eiTO?' in northing : but if the sum of the southings is less than the sum of the northings, the error is called, e7T0?' in southing. And similarly for the departures. 126 ELEMENTS OF SURVEYING. [book II. This error for liiLitude or departure must be distributed among tlie latitudes or departures of all the courses, in pro- portion to the length of each course, observing to add the correction, when applied to the deficient column, and to sui- tract it, when applied to the other. We will illustrate this, by the example of (Art. 75). stations. Bearings. Dis. Dif. Lat. Dep. Balance. + S. E. + W. 1 Lat. Dep. 1 n; 31F w. 10.40 8.87 5.43 + 8.86 -5.44 2 1^.62° E. 9.20 4.32 .... 8.13 + 4.31 + 8.12 3 S. 36° E. 7.60 6.15 4.47 -6.15 + 4.46 4 S. 451-° W. 10. 7.01 7.13 -7.02 -7.14 Sum of angles, 37.20 13.19 13.16 12.60 12.56 13.16 12.56 Error in southina* . . . .03 .04 Error in westing. The error in southings, 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 westings is similarly distributed among the eastings and west- ings. Eor this, two new columns are formed, called, the iaJanced latitudes and departures ; and to these columns the latitudes and departures are transferred, after the corrections have been made: the north latitudes being marked +, and the south lati- tudes — , 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 the northings of courses 1 and 2, and adding 1 link to the southing of course 4. This produces a balance. SEC. m.] COMPASS SURVEYING. 127 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. XoTE. — When a knowledge of the conditions under which tlie survey was made, enables us to determine that errors were more likely to occur at certain points, it is doubtless best to apply the corrections to those courses where it seems probable the errors were made. 88. 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 close the '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 T.io chains, or ^-g- of the perimeter. 89. 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 tlie ordinary methods of balancing, among all the courses, the error in area will be larger than the error in perimeter. 90. When the error is so large that a re-survey becomes necessary, the balancing 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 128 ELEMENTS OF SUEVEYING. [BOOK II. 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 Departure. When the error in the survey has been a single one, of dis- tance only, 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 Xorth and West, or South and East, should be examined. 01. The surveyor should take every possible precaution against errors 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 will 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 lines to meet at a common point. 92. One source of error, in 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. 93. If each of the angles of the survey, included between two consecutive courses, be calculated by the method explained in Article 000, the bearings may then be verified by comparing SEC. m.] AEEA OF GEOUXD. 129 the sum of these angles -with, the sum of the interior angles of any polygon (Leg., Bk. I., Prop. 25). The same verification may also be made when the angles are measured with the theodolite or transit. 94. 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 Xorth 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 Meeidia^^ Distances. 95. 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 surveys 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 princijoal station, and the course which begins at this point, the first course. Care, lioiuever, must be talcen,. not to confound this with the course which hegins at station 1,. and luhicli is the first course that is entered in the field-notes. It has already been remarked (Art. 71), that all departures in the direction east are considered as plus, and all departui'es in the direction west as minus. 96. To dedtice a rule for finding the dotible meridian dis^ tance of any course. Let SX be the assumed meridian. Let jB (7 represent any course, and AB the preceding course; also, let D and E be their middle points. Draw EH, Z>6^, and CM, 130 ELEMENTS OF SURVEYING. [book n. perpendicular to the assumed meridian ]^S. Draw also J/, BX, and BL, par- allel to XS. Then 2DG is the double meridian distance of the course BC, and 'ZBB=2KG, is the double meridian dis- tance of the course AB. Xow, WG=2 GK + '2EL -f 2LD ; but 2KL = IL is the departure of the course AB, and %LB — MC is the departure of the course BC : ' N H M a consequently. 2GD = 2GK -{- IL + 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 iirst two terms become zero. We therefore have the following EuLE. — I. The double meridian distance of the first course is equal to its departure: II. The double meridian distance of the second course is equal to the double meridian distance of the first course, 2jIus its departure, plus the departure of the second course : III. The double meridian distance of any course is equal to the double meridian distance of the preceding course, plus its departure, jjIus the departure of the course itself. XoTE. — It should be recollected that ^^/^/-s is here used in its algebraic sense, and that, when the double meridian distance 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 haA'e contrary algebraic signs ; hence, their algebraic sum will be expressed by their numerical difference, with the si^n of the greater prefixed. SEC. m.] AEEA OF GROUND. 131 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 yerification of the work is therefore obtained, by comparing this double meridian distance with the departure. Area. 97. Let us re&ume the example of Art. 75. We will first write the differences of latitude and the double meridian distances of the courses, in the following table. stations. Dif. of Latitude. D. M. D. Area. + 1 Area. 1 + cB + Ua 'ZcAB 2* + Bs + ^P 2BsC 3 -yD + 2/i7i 2visCD 4 -Df + 'Zed 2cmDA It is evident, that cB multiplied by 2I)a = cA, will giye double the area of the triangle cAB. But cB and ba are both plus; hence, the product will be pjlus, and must be put in the column of plus areas. Double the area of the triangle BsC, is equal to Bs multiplied by 2qp, which product is also plus. The area of the trapezoid 7ns CD is equal to yD = ms multi- plied by nil (Geom., Bk. IV., Prop. VII., S.) ; hence, double 132 ELEMENTS OF SURVEYING. [book II. the area is equal to yD into '^nli. But since yD (being a south- ing) is minus, and 2nli 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=mc multiplied by 2de : but, since Df is negative and 2cle 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 ABCD: and since the same may be shown for any other figure, we have, for finding the areas, the following general Rule. — I. Multiply the double meridian distance of each course ly its oiorthing or soutliing, observing that like signs in the multiplicand and multiplier give plus in the product, and that unlihe signs give minus in the product. II. Place all the products ivhich have a plus sign, in one col- umn, and all the prodMcts which have a minus sign, in another. III. Add up the columns separately and tahe the difference of their sums: this difference will be double the area of the land. 98. We will now make the calculations of this example, in numbers, from the field-notes, which are the following: stations. Bearings. Distances. Dif. Lat. Dep. V). M. D. 1 N 31i° W 10.40 + 8.86 -5.44 + 1S.02 — 7.14 — 5.44 + 5.44 2* N 62° E 9.20 + 4.31 + 8.12 + 8.12 3 S 36° E 7.60 -6.15 + 4.46 + 8.12 +4.46 + 20.70 4 ,S 45rW 10. -7.02 -7.14 4.46 —7.14 + 18.02 SEC. m.] AREA OF GROUND. 133 We see, from inspecting the notes, that 2 is the most westerly, and 4 the most easterly station. Either of them may, therefore, be taken for the principal station. Let us assume 2 for the principal station, through which, the assumed meridian passes, and distinguish it by a star, thus *. Having done so, we enter the departure 8.18 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. sta. Bearings. Dist. Dif. Lat. D. M. D. Area. j Area. i 1 N 31i° W 10.40 + 8.86 + 5.44 48.1984 2 :N"62° e 9.20 + 4.31 + 8.12 34.9972 3 S 36° E 7.60 -6.15 + 20.70 127.3050 4 S 45rW 10. -7.02 + 18.02 126.50041 83.1956 253.8054 Area in sq. ch. Dividino: bv 10 83.1956 2)170.6098 . 85.3049 8.53049 4 2.12196 40 Ans. SA. 2E. 4.88 P. 4.87840 134 ELEMENTS OF SURVEYING. [book n. Plotting. 99. It now only remains to make a plot of the ground. For this purpose, draw any line, as NS, to represent the meridian passing through the principal station; and on this line take any point, as J?, to repre- sent that station. FIRST METHOD. ' Having fixed upon the scale on which the plot is to be made, lay ofi" from B, on the meridian, a distance JBs equal to the difference of latitude of the second course, and at s erect a perpendicular to the meridian, and make it equal to the departure of that course: then draw BC, which will be the second course. Through C draw a meridian, and make Cf equal to the difference of latitude of the third course, and through / draw a perpendicular f^, and make it equal to the departure of that course: draw CD, and it will be the third course. Lay down, in the same manner, the courses DA and AB, and the entire plot will be completed. SECOND METHOD. The work may be plotted in another manner, thus. At the principal station B, lay off an angle equal to the bearing from B to C, which will give the direction oi BC. Then, from the scale of equal parts, make BC equal to the second course: this will give the station C. Through C draw a meridian, and lay off an angle equal to the bearing from C to D, and then lay off the course CD. Do SEC. III. AEEA OF GROUND. 135 the same for the bearing at D and the course DA ; also, for the bearing at A and' the course AB, and a complete plot of the ground will thus be obtained. If the work is all right, the first line AB, that was run, and the last line plotted, will exactly close the figure. This plot is made on a scale of 10 chains to an inch. By Table oe Natural Sines. 100. If the land surveyed be very valuable, and very great accuracy is necessary in the computation of the area, it may be w^ell to calculate it by means of the Table of Natural Sines (Bk. I., Art. 57). In this Table the degrees are marked at the top and bottom of each page, and the minutes at the left and right. The table being calculated to the radius 1, the sines and cosines are expressed in decimals of 1. When the Table is used for the com- putation of areas, the cosines are the differences of latitude, and the sines the differences of departure, to the distance 1. Hence, if the cosine and sine be taken from the table, for any course, and then multiplied respectively by the length of the course, the products will be the latitudes and departures for that course and distance. Let us resume the last example. stations. Bearings, Distances. LATITUDE. DEPARTURE. N + s E + w 1 2 3 4 N 3ir W N62° E S 36° E S 45i° W 10.40 9.20 7.60 10. 8.8675 4.3191 6.1486 7.0091 8.1231 4.4672 5.4340 7.1325 "We first take from the Table of Natural Sines, page 69 (of Tables), the cosine and sine of 31 T? which are .85264 and .5225. 136 ELEMENTS OF SURVEYING. [book II. We then multiply them by the distance 10.40, and the products 8.867456 and 5.43400 are the latitudes and departures of the course ; which we enter in the table, omitting the decimals after the fourth place. We find, in a similar manner, the latitudes and departures of the other courses ; after which the work is balanced and wrought, as with the Traverse Table. EXAMPLEI 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. Bearings. Dist. .Dif. ■ Lat. Dep. BALANCKD. D.M.D. + AREA. + AREA. N + S E + W Lat. Dep. 1 N 46^0 W 20.76 14.29 15.06 + 14.30 -15.04 15.04 215.0720 2 N 51|o E 13.80 8.54 10.84 + 8.55 + 10.86 10.86 92.8530 3 E 21.35 21.35 + 21.37 43.09 4 S56o E 27.60 15.44 22.88 -15.43 + 22.90 87.36 1347.9648 5 s 3310 W 18.80 15.72 10.31 —15.71 —10.29 99.97 1570.5287 6 N 74^0 W 30.95 8.27 29.83 + 8.29 —29.80 59.88 496.406-2 E 31.10 rror.... 31.16 31.10 .06 55.07 55.20 55.07 .13 Error. An s. 105 A . 2i?. S04.3312 2918.4935 804,3312 2)2114.1623 33 P. 1057.08115 Plot of the ground. Note. — When the bearing is due East or due West, the error in latitude is . nothins:, and the corrections for latitude must SEC. III.] AREA OF GROUND. 137 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. 2. Eequired the contents and plot of a piece of land, of which the following are the field-notes. stations. Bearings. Distances. i S 34° W 3.95 ch. 2 S 4.60 3 S 36i° E 8.14 4 N 59^ E 3.72 5 ]^^ 25° E 6.24 6 N 16° W 3.50 < N 65° W 8.20 Ans. 10 A. OE. 6 P. 3. Eequired the contents and plot of a piece of land, from the following field-notes. stations. Bearings. Distances. 1 S 40° W 70 rods. 2 N 45° W 89 3 N 36° E 125 4 .N" 54 5 S 81° E 186 6 S 8° W 137 7. W 130 Ans. 207 A. 3 E. 33 P. 138 ELB2JXXTS OF SUEYEYING. [book n. 4. Required the contents and plot of a piece of land, from the following notes. Stations. Bearings. Distances. 1 S 40^° E 31.80 ch. 2 :N^ 54° E 2.08 3 N 29^ E 2.21 4 ]S^ 28r E 35.35 5 X or w 21.10 6 S 47^ W 31.30 Ans. 92 A. 3 i?. 32 P. 5. Eeqnired the area of a suryey, of which the following are the field-notes. Stations. Bearings. Distances. 1 X 42^ E 5.00 ch. East. 400 3 X 9= E 4.00 4 S 69^ E 5.56 5 S 36^ E T.OO 6 S 42° W 4.00 .■v S 75° W 10.00 8 S 39° W 7.50 If, in this example, we assume 1 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 P. It should, however, be remarked, that in all the examples the answers may be slightly varied by distributing the corrections. SEC. III.] AREA OF GEOUND. 139 6. What is the area of a survey of which the following are the field-notes ? Make the plot. stations. Bearings, Distances, 1 N 75° 00' E 54.8 rods. 2 N 20° 30' E 41.2 3 East. 64.8 4 S 33° 30' W 141.2 5 S 76° 00' W 64.0 6 Xorth. 36.0 7 S 84° 00' W 46.4 8 N 53° 15' W 46.4 9 N 36° 45' E 76.8 10 K 22° 30' E 56.0 11 S 76° 45' E 48.0 12 S 15° 00' W 43.4 13 S 16° 45' W 40.5 In this survey 4 is the most easterly and 9 tire most westerly station. The area is equal to 110^. 2i?. 23 P. It may vary 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. 1 S 46i°. E 80 rods. 2 S 51f° W 55.20 3 West. • 85 4 N 56° W 110.40 5 N 33i° E 75.20 6 S 74i° E 123.80 A71S. 104: A. IB. 16 P. 140 ELEMENTS OF SURVEYING. [book n. 8. Eequired the area of the farm, of which the survey- notes are given on page 117. stations. Bearings. Distances. 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 43° 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. Problems. I. To determine the bearing and distance from one point to another, when they are so situated that one cannot be seen from the other. 101. Let A and (7 be 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 con- venient length in the direction toward C, and take the bearing with the com- pass. 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 C, and measure the distance from 4 to C. Then, the dir-^ence between the sum of the northings and SEC. III.] AREA OF GROUND. 141 the sum of the southings will be denoted by AB ; and the difference between the sum of the eastings and the sum of the yrestings, by BC. The base AB, and the perpendicular BC oi the right-angled triangle ABC, are then known. The angle at the base, BAC, is the bearing from A to 0; or the equal alternate angle at C is the bearing from C to A, and the hypothenuse AC,- is the distance. Haying 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. 1 N 61° W 40 ch. 19.39 34.98 2 N 42° W 41. 30.47 27.43 3 N 12° E 16.10 15.75 3.35 4 N 47° E 32.50 22.16 23.77 AB-- = 87.77 27.12 62.41 27.12 ch. CB = = 35.29 To find the angle BAC, or the bearing from A to C. Eadius : tan A : : AB : BC, or, AB : BC :: B : tan A; that is, applying logarithms, (a. c.) log AB (87.77) 8.056654 log BC (35.29) 1.547652 log i? 10. log tan A 21° 54' 12" 9.604306 142 ELEMENTS OF SURVEYING. [book n. To find the distance AC sm A : R :: BO : AC; Applying logarithms, (a. c.) log sin A 21° 54' 12 0.428242 log 72 10. log BO (35.29) 1.547652 log AC 94.6 1.975894 Hence, the bearing and distance are both found. Note 1. — Had any of the courses run south, AB would have been equal to the sum of the northings, minus the sum of the southings. XoTE 2. — 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, tivo omissions may in general be supplied by calculation. It is far better, however, if possible, to take all the notes on the field. For, 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. EXAMPLES. 1. In a survey w^e have the following notes: stations. Bearings. Distances. 1 2 3 4 N 314° W :N" 62f° E Lost. S 45i°W 10 ch. 9.25 Lost. 10.40 SEC. in.] AEEA OF GROUND. What is the bearing distance from station 3 to 4 ? 143 (Bearing, S 38° 52' E. Ans. } (Distance, 7.03 ch. 2. In a survey we have the following notes : stations. Bearings. Distances. 1 s m° E * 3L80 ch. 2 ISr 54° E 2.08 3 Lost. Lost. 4 N 281° E 35.35 5 N 57° W 2L10 6 S 47° W 3L30 What is the bearing and distance from 3 to 4 ? A71S. Bearing, N 34° 47' E. Distance, 2.19 ch. n. To determine the angle included between any two courses, when their bearings are known. 102. 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 AG isN 26° W AH is N 65° W (Lli?=39° Principles. When the meridional letters are alike, and those of departure also alike, the differ- ence of the leavings is the angle between the courses. lU ELEMENTS OF SURVEYrN'G. [book II. AB is N 46° E AC is ]sr 26° ^y CAB = 72° AC is N 26° W AD is S 66° W CAD = 180° - 92° = 88^ AC is N" 26° W AF is S 66^ E CAF= 180°-40° = U0° ■^ When tlie meridional letters I are alike, and those of departure unlike, the sum of the hearings iS the angle between the courses. When the meridional letters are unlike, and those of departure J^ alike, the angle letween the courses is equal to 180°, minus the sum of the hearings. When the meridional letters are unlike, and those of departure > also unlike, the angle hetween the courses is ec[ual to the difference of the hearings taken from 180°. !N"oTE. — 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 bearing of one of them must be reversed before the calcu- lation for the angle is made. EXA3IPLES. 1. The bearings of two courses, from the same point, are K 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 X 39° W, and S 48° W: what is the angle included between them ? Ans. 87 3. The bearings of two adjacent courses, in going round a piece of land, are S 85° W, and X 69° W : what is the angle included between them? Ans. 154°. SEC. III.] OF DIVIDING LAND. 145 4. The bearings of two adjacent courses, in going ronnd a piece of land, are N" 55° 30' E, and S 69° 20' E : what is the angle included between them ? Aiis. 124° 50'. Latin^g out and dividing Land. 103. 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 giyen 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. PROBLEM I. 104. To lay out a given quantity of land in a square form. Rule. — Reduce tlie^ given area to square chains, or square rods: tlien extract ilie square root, and the result will he the side of tlie required square. This square leing described on the ground, luill te the figure required. 1. To trace a square which shall contain 15^. OR. 12 P.. First, 15 ^1 = 60 i^ = 2400 P ; hence, Add, 12 P 15^ OP 12P=: 2412 P; the square root of which is 49.11, nearly. Therefore, if a square be traced on the ground, of which the side is 49.11 rods, it will be the required figure. 10 146 ELEMENTS OF SUEYEYIXG. [BOOK II. 2. To trace a square which shall contain 176.4. 1 H. 24: F. First, 176^1 = 17G0 square chains, IE = 2.5 " " hence, 24P == 1.5 " " 17G^ 1^= 24:. P = 176-1 square chains: the square root of which is 42. Hence, if a square he traced on the ground, of which the side is 42 chains, 'it will be the required figure. PEOBLE^I II. 105. 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 scjuare rods, ly the given side of the reciuired rectangle, and the quotient will le the other side. Then, trace the rectangle on the ground. 1. To lay off 240 acres in a rectangular form, one of the sides being given, and equal to 80 rods. . First, 240 A = 2400 square chains — 38400 square rods. Then, 80)38400(480 rods; which is the required side of the rectangle. 1N"0TE. — A great number of similar problems might be pro- posed. The solution of them does not, however, properly belong to surveying. The laying out of the ground, and tracing of lines, after the figure and area have been determined, are the only parts which really appertain to a practical treatise. The manner of tracing lines having been already explained, it seems unnecessary to add the numerous examples often given under this head of the subject. SEC. III.] OF DIVIDING LAND. 147 PROBLEM IIL 106. 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 11. Let ABC be any triangular field. Diyide the side BC into two parts, sucli that (Geom., Bk. IV., Prob. I.) BD : DC :: m : n; and draAV 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, 0.) : hence, the triangle is divided into parts having the ratio of m to n. PROBLEM IV. 107. To run a line parallel to one side of a triangular field, that shall form with the parts of the two other sides a triangle equal to the — part of the field. Let CBA represent a triangular field, and CA the side parallel to which the dividing line is to be drawn. On the side BC describe a ^' -'-.^ semicircle: then divide BC at D, so that BD : BC w m : n. At D, erect the perpendicular DC to the diameter BC, and draw BC Then, with B as a centre, and BG as Sb radius, describe the arc of a circle cutting BC at E. Through E draw EF parallel to CA, and it will divide the triangle in the required ratio. For. (Geom., Bk. IV., P. 23), a.i' 148 ELEMENTS OF SURVEYING. [BOOK 11. 'BG' = ^^ = BCxBD; or, BE'' ^BC^X-^; whence, BE^ : BG^ :: BD : BC : m : n. But, since the triangles BEF, BOA are similar, BE^ : BC^ :: BEF : BOA. Wherefore, from equality of ratios, BEF : BCA :: m : n; whence, BEF =- X BCA. n I^OTE 1. — The points E and F may easily be found by computation. For, since BE^ = BG^ = BCx BE, and BD = - XBG, we have, 'BE' = mrx~; or, BE^BC V —- I lYl In like manner, BF = BAy — . EXA3IPLE. Let it be required to divide the triangular field CAB, in which ^(7=9 ch., AB = 11 ch., and CB = 7 ch., into two such parts that ABE shall be one-fourth of the whole field. In this case, we have m = 1, n = 4, and |/Z^ = /I 1 n '4 2 hence, AE = 4: ch. 50 1., and AD = 6 ch. 50 1. SEC. m.] OF DIVIDING LAND. 149 peoble:m y. 108. To run a Hne 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 at page 136. That field contains 105^. 2 E. S3 P., and the following is a plot of it. N W C G 71 D ^\3 \ \ \ / ' ' / 1 1 "^X,^ ^^ 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, 26.4 2R 31 P. It is seen, by examining the field, that the division line will probably terminate on the course CD. Therefore, draw a line from A to (7, 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 on page 136 : hence, the bearing and distance from C to .1 can be calculated by Art. 101 : they are, in this example, Bearing, S 19° 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 14.4 0/i 26 P. 150 ELEMENTS OF SURVEYIKG. [BOOK H. It is now evident that the diyision line must fall on the right of the closing line AC, and must cut off an area ACH, equal to the difference between that already cut off, and the given area : that is, an area equal 26^ 2i? 31P given area, 14.4 OR 26 P area already cut off, to ... . 12^1 2E 6F. \ Since the bearing of the next course CD, and the bearing of the closing line AC 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 AC, and the angle ACC at the base, the length AG, the perpendicular let fall on the course CD can be found, and is 22.82 chains. Since the area of a triangle is equal to its base multiplied by half its altitude, it follows, that the base is equal to the area divided by half the altitude. Therefore, if the area 12 A 2R dP be reduced to square chains, and divided by 11.41 chains, which 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 CR, equal to 10.95 chains, and then run the line AH, it will cut off, from the land, the required area, viz., 26.4 2R 31 P. 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 ^a 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 than 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 DK SEC. III.] OF DIVIDING LAND. 151 Note 3. — When the point from which the division line is to be draAvn 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 course, having the same bearing. The manner of making the calculation will then be the same as before. PKOBLEM VI. - 109. 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, w^hen 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, page 138, of which the plot is on the next page. Let it be required to cut off from this area, 50 acres, by a line whose bearing shall be S 60° E, or K 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 Bo. 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 K 81° 13' E. 152 ELEMENTS OF SURVEYING. [book II. Now, in the triangle AJ3 5 we have one side and the angles, to find the remaining parts. AB is found to be 28.88 6 chains and B 6 to be 22.81 chains. We haye now the com- plete field-notes of the area cnt off. 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. A S 60° E 28.88 ch. B N 28rE 22.81 5 N 57V W 21.10 6 S 47° W 25.00 SEC. m.] PUBLIC LANDS. 153 Suppose the lines A 6 and ^ 5 be prolonged till tliey meet at some point, as V. Calculate A V and B V, also the area ABV. AV is found to be 92.19 chains and BV 88.18 chains. The area of the triangle ABV, is 127.29 acres. Let M2i represent the line sought. Then, we have two similar triangles, with all the sides of the one, and the areas of each, known ; for, FJ/LY must contain 8.5029 acres less than AVB. Then, AM and BX are easily determined. '^ The complete notes of the area to be cut off, are :m S 60° E 27.89 j N X 28r E 19.82 5 X b:' w 21.10 6 S 47^ ^ 21.87. XoTE. — Fields are so Tariously shaped that it is difficnlt to give rules that will apply to all cases. It is by practice alone that facility is obtained in that branch of snryeying relating to the division of estates. We have given only a few examples that may serve as general guides, in the appli- cation of the principles of Plane Geometry, to such cases as may arise. Public Lands. 110. Soon after the organization of the present govern- ment, 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 what is called the public lands, or public domain. Previous to the year 1802, these lands were parcelled out without reference to any 154 ELEMENTS OF SURVEYING. [BOOK II. general plan, in consequence of which the titles often con- flicted Avith each other, and in many cases, seyeral grants covered the same area. In tlie year 1802, the following method of surveying the public lands, was adopted by Colonel Jared Mansfield, then Surveyor-General of the Northwestern Territory. 111. The country to be surveyed is first divided, by meridians, six miles distant from each other ; and. then again, by a system of east and west lines, also six miles from each other. The country is thus divided into equal squares, which are called toivnsliips. Hence, each township is a square, six miles on a side, and contains thirty-six square miles. 112. For the purpose of illustration, we have obtained from the general land-office the accompanying map, which represents a considerable portion of the State of Arkansas. The principal meridian in this survey is called the 5th 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 j^orth, 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 tow]i- ships lying in the first row, east of the 5th meridian: and Eange 1 West, will apply to all lying in the first row to the Avest of it. The small figures designate the rows of townships, reckoned north and south from the base-line, and the ranges reckoned east and west from the 5th meri- S location of Zand Office T Bovndarg of LandTiistricts o + Zd72 ds ofjere d for Sd Je iM.^^\WV^tl%aH .S^.HM 156 ELEMENTS OF SUEYEYING. [book n. dian. Thus, township 1 Xorth, range 4 West, has its exact place designated, and may be immediately located. 113. The principal meridians, and the principal base-lines are established by astronomical observation, and the lines of subdivision run with the compass. For convenience in making surveys, and for the purpose of designating particular localities, a state or large tract, is often divided into parts called "Districts." There are three such districts in the map before us, the Lawrence County District, the Arkansas District, and the Mississippi District, the boundaries of which are designated by the full dark lines. 114. Each township is divided into equal squares, by me- ridians one mile apart, and by east and west lines at the same distance from each other. Hence, each township is divided into 36 square miles, each one of which is called a section. The sections of a township are numbered from 1 to 36, beginning at the northeast angle, and each con- tains 640 acres. The diagram exhibits the 36 sections of a township. \ 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 To describe a section accurately, we say, section number 5, in tovrnship number 4 north, in range 3d west of a SEC. III.] VARIATION OF THE NEEDLE. 157 knoT\'ii meridian; the one, for example, drawn through the moutli of White river. The description fixes precisely the place of the section. Go to the 3d range of townships, west of the known meridian, find township number 4 north, in this range, and lastly, section number 5 of that township. The corners of the sections should be marked by permanent corner-posts, or by lines blazed on trees. 115. The sections are divided into half sections, quarter sections, and even into eighths of sections. The following table shows the contents of a township, and its subdivisions: 1 township = 36 sections = 23040 acres. 1 section = 640 acres. ■J- section = 320 acres. ^ section =160 acres. ■J section = 80 acres. VaEIATION of THE NeEDLE. 116. The angle which the magnetic meridiaa makes with the true meridian, at any place on the surface of the earth, is called the variation of the needle at that place. The variation is east, when the northern end of the nee- dle, after settling to a state of rest, lies on the east of the true meridian; and west, when it lies on the west side of that meridian. 117. The variation is different at different places, and even at the same place it does not remain constant for any length of time. The variation is ascertained by comparing the magnetic with the true meridian. 118. If we suppose a line to be traced through those jwints, where the needle points directly north and south, such a line is called the line of no variation. 158 ELEMENTS OF SURVEYING. [BOOK II. By referring- to a map of the United States, sucli line is easily traced; for, in the year 1870, it passed, very nearly, through Ealeigh, in the State of North Carolina, Cleveland, in the State of Ohio, and crossed the Saut of St. Mary's at the lower end of Lake Superior. If a compass, at that time, had been placed anywhere on this line, the needle would have pointed due north and south : hence, this line was then the line of no variation. At all points on the surface of the earth, the north end of the needle inclines toward the U?ie of no valuation: hence, for all points east of this line, the variation is West; and for all points west of it, the variation is Fast. Places where the Variation is West. West. + Latitude. Longitude Nearest Place. Latitude. Longitude Nearest Place. 0° 85- 00' 78° 10' Raleigh ± 41° 30' 81° 45' Cleveland ± V 36° 00' 77" 20' Plymouth + 40° 80' 80° 00' Pittsburgh - 2° 37° 00' 76° 25' Richmond - 42° 00' 80° 10' Erie - 3° 39° 00' 76° 40' Anapolis — 40° 40' 78° 00' Harrisburgh — 4° 40° 00' 76° 25' Harrisburgh — 48° 20' 79° 00' Buffalo ± 5° 40° 00' 75° 25' Wilmington — 43° 00' 78° 00' Buffalo - 6° 40° 00' 74° 10' Trenton — 48° 00' 77° 10' Oswego + 7° 41° 00' 74° 00' New York — 48° 00' 76° 00' Oswego ± 8° 41° 20' 73° 00' New Haven + 44° 00' 76° 00' Oswego — 9° 42° 10' 73° 00' Hartford - 48° 00' 74° 00' Albany — 10° 42° 00' 71° 35' Providence ± 44° 00' 74° 00' 11° 42° 00' 70° 25' Boston — 44° 00' 78° 00' Montpelier ± 12° 44° 00' 72° 00' Montpelier ± 45° 40' 74° 00' Montreal + 13° 44° 00' 71° 00' Portland + 46° 00' 73° 30' Montreal + 14° 44° 10' 70° 00' Portland — 46° 20' 73° 00' 15° 44° 20' 69° 00' Augusta — 46° 20' 72° 00' 16° 45° 00' 69° 00' 47° 00' 72° 00' Quebec + 17° 45° 00' 68° 00' 47° 00' 71° 00' Quebec + 18° 45° 00' 67° 00' St. Andrews 47° 00' 70° 00' SEC. III.] VARIATION OF THE NEEDLE. 159 119. The table on the last page, and the ODe which follows, on tlie next, have been constructed from the magnetic chart accompanying the Annual Report of the Coast Survey of 1865, and all the magnectic meridians are calculated for the year ISTO. On that chart, we find the meridian of no variation. It passes through Ealeigh, in the state of Xorth Carolina (very nearly), and through Cleveland, in the state of Ohio. East of it, is marked the magnetic meridian of one degree : that is, the magnetic meridian, at any point of which, the variation is 1 degree west. Two points of this meridian are noted in the table : viz., the point whose latitude is 36° north, and longitude 77° 20' west ; and also the point whose latitude is 40° 30', and Igjigitude 80°. Marking these two points, on a map of the United States, the magnetic meridian of one degree variation, west, may be drawn. And in a similar manner any magnetic meridian (of which two points are indicated by the table), may be drawn. To aid in the ready selection of any point, indicated in the table, the prominent place, nearest to it, is written in the adjoining column ; and when the place lies east of the point (and consequently the point icest of the place), the sign + is annexed; and the sign — , when the place lies west of the point, and the sign ±, when the meridian j)asses through, or very near tlie place. Thus, in the meridian of 1° variation, Plymouth, in Xorth Carolina, is east of the first point, and Pittsburgh, west of the second. 160 ELEMENTS OF SURVEYING. [book II. Places where the Variation is East. East. Latitude. Longitude. Name of Place. Latitude. Longitude. Name of Place. 0° 35° 00' 7S° W Ealei;?h 41° 30' 81° 45' Cleveland 1° 34° 00' 79° 10' Wilmington + 41° 00' 88° 00' Detroit + 2° 33° OO' 80° 10' Charleston + 43° 00' 84° 30' Lansing — 4° 33° 00' 86° 30' Milledgeville + 40° 00' 85° 30' Indianapolis — 6° 31° 00' 87° OC Mobile — 42° 00' 88° lO' Chicago + 7° 30° 00' 89° 20' New Orleans — 43° 00' 89° 25' Madison, In. + 9° 29° 00' 95° 00' Galveston + 38° 00' 91° 20' St. Louis + 13° 32° 00' 115° 00' San Diego — 36° 00' 104° 00' Santa Fe — 16° . 37° 15' 122° 00' San Francisco — 40° 20' 112° 00' Salt Lake City ■ - 21° 46° 00' 124° OO' Astoria + This table is interpreted like the preceding.- At San Diego, on the Pacific, the variation is 13° 25' East; at San Francisco, it is 16° 30'; and at the month of the Columbia riyer, it is 21° 10' East. Methods of ascertai^^in^g the Vaeiatiok. 120. The best practical method of determining the true meridian of a place, is by observing the north star. 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 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. To the eye of an observer, this star is continually in motion, and is due north but twice in 24 hours; and is then said to be on the meridian. Now, when it departs from the meridian, it apparently moves east or west, for 12 hours, and then returns to the meridian again. When at its greatest SEC. m.] VARIATION OF THE NEEDLE. 161 distance from the meridian, east or west, it is said to be at its greatest eastern or western elongation. The following tables show the times of its greatest eastern and western elongations : Easteei^ Elongations. Days. April. May. June. July. August. Sept. 1 H. M, 18 32 H. M. 16 35 H. M. 14 33 H. M. 12 35 H. M. 10 34 H. M. 8 32 7 18 09 16 11 14 09 12 12 10 10 8 09 13 17 45 15 47 13 46 11 48 9 47 7 45 19 17 22 15 24 13 22 11 25 9 23 7 22 25 16 58 15 00 12 59 11 01 9 00 6 58 "Westekn" Elongations. Days. October. Nov. Dec. Jan. Feb. March. H. M. H. M. H. M. H. M. H. ai. H. M. 1 18 24 16 22 14 24 12 21 10 18 8 28 7 18 01 15 59 14 00 11 57 9 55 8 04 13 17 37 15 35 13 37 11 33 9 31 7 41 19 17 13 15 11 13 13 11 10 9 07 7 17 25 16 50 14 48 12 49 10 46 8 44 6 53 The eastern elongations are put down from the first of April to the first of October; and the western, from the first. of October to the first of April: the time is computed from 12 at noon. The western elongations in the first case, and the eastern in the second, occurring in the daytime, cannot be used. Some of those put down are also invisible, occurring in the evening, before it is dark, or after daylight in the morning. In such case, if it be necessary to determine the 11 162 ELEMENTS OF SURVEYING. [book II. meridian at that particular season of the year, let G hours 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. Azimuth Table. Years. Lat. 32° Azimuth. Lat. 34° Azimuth. Lat. 30° Azimuth. Lat. 3S° Azimuth. Lat. 40° Azimuth. Lat. 42° Azimuth. Lat. 44° Azimuth. 1870 1° 37f' 1°40' 1° 421' 1° 451' 1° 481' 1° 511' 1° 551' 1871 1° 37F 1° 3^' 1° 42' 1° 44|' 1° 47}' 1° 51' 1° 54|' 1872 1° 37' 1° 391' 1° 41f ' 1° 441' 1° 471' 1° oOf 1° 541' 1873 V 36V 1° 38|' 1° 411' 1°44' 1°47' 1° 501' 1° 53}' 1874 r 361' 1° 381' 1° 40J' 1° 431' 1° 461' 1° 49|-' 1° 531' 1875 1 oof 1° 38' 1° 401' 1° 431' r 46' 1° 491' 1° 53' 187G 1° 35|' 1° 37f' 1° 40' 1° 42|-' 1° 45|' 1° 49' 1° 521' 1877 1°35' 1° 371' 1° 39i' 1° 421' 1° 451' 1° 481' 1° 52' 1878 1° 34J' 1° 37' 1° 391' 1°42' 1° 44|' 1°48' 1° 51}' 1879 1° 341' 1° 361' 1° 39' 1° 411' 1° 441' 1° 47|' r 511' 1880 1° 34' 1° 36' 1° 381' 1° 41' r 44' 1° 471' r 50}' The above table exhibits the angle which the meridian plane makes with the vertical plane passing through the pole- star, when at its greatest eastern or western elongation ; such angle is called the azimuth. The mean angle only is put down, being calculated for tlie first of July of each year. The use of the above tables in finding the true meridian will soon appear. To FIND THE TRUE MERIDIAN WITH THE THEODOLITE. 121. Take a board, of about one foot square, paste white paper upon it, and perforate it through the centre; the diam- eter of the hole being somewhat larger than the diameter of SEC. m.] VARIATION OF THE NEEDLE. 163 the telescope of the theodolite. Let this board be so fixed to a vertical staff, as to slide np 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 tTrenty-fiye minutes before the time of the greatest eastern or western elongation of the pole-star, as shown by the tables of elongations, let the theodolite be placed at a conyen- ient point and lerelled. Let the board be placed about one foot in front of the theodolite, 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-hairs in the telescope of the theodolite. Then, let the vertical spider's line be brought exactly upon the pole-star, and, if it is an eastern elongation that is to be observed, and if 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 spider's line 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 attained its greatest elongation, great care should be taken that the instrument be not afterward moved. Xow, if it be not convenient to leave the instrument in its place until daylight, let a staff, with a candle or small lamp upon its upper extremity, be arranged at thirty or forty yards from the theodolite, and in the same vertical plane with the axis of the telescope. This is easily effected, by revolving the vertical limb about its horizontal axis, without moving 164 ELEMENTS OF SURVEYING. [BOOK H. the yernier plate, and aligning the staff to coincide with the vertical hair. Then mark the point directly under the theodo- lite; 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, take the azimuth correspond- ing to the year and nearest latitude. If the observed elonga- tion 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 theodolite, gives the true meridiau. To FIND THE TEUE MeEIDUN WITH THE COMPASS. 122. 1. Drive two posts firmly into the ground, in a line nearly east and west ; the uppermost ends, after the posts are driven, being about three feet above the surface, and the posts about four feet apart: then lay a plank, or piece of timber three or four inches in width, and smooth on the upper side, upon the posts, and let it be pinned or nailed, to hold it firmly. 2. Prepare a piece of board four or five inches square, and smooth on the under side. Let one of the compass-sights be placed at right angles to the upper surface of the board, and let a nail be driven through the board, so that it can be tacked to the timber resting on the posts. 3. At about twelve feet from the stakes, and in the direc- tion of the pole-star, let a plumb be suspended from the top of an inclined stake or pole. The top of the pole should be of such a height that the pole-star will appear about six inches below it ; and the plumb should be swung in a vessel of water to prevent it from vibrating. SEC. m.] VAEIATION OF THE NEEDLE. 165 This being done, about twenty minutes before the time of elongation, place the board, to which the compass-sight is fastened, on the horizontal plank, and slide it east or west, until the aperture of the compass-sight, the plumb-line, and tlie star, are brought into the same range. Then if the star depart from the plumb-line, move the compass-sight, east or west, along the timber, as the case may be, until the star shall attain its greatest elongation, when it will continue behind the plumb-line for several minutes; and will then recede from it in the direction contrary to its motion before it became stationary. Let the compass-sight be now fastened to the hor- izontal plank. During this observation it will be necessary to have the plumb-line lighted: this may be done by an assistant holding a candle near it. Let now a staff, with a candle or lamp upon it, be placed at a distance of thirty or forty yards from the plumb-line, and in the same direction with it and the compass-sight. The line so determined, makes, with^ the true meridian, an angle equal to the azimuth of the pole-star ; and, from this line, the variation of the needle is readily determined, even without tra- cing the true meridian on the ground. - Place the compass upon this line, turn the sights in the direction of it, and note the angle shown by the needle. !N"ow, if the elongation, at the time of observation, was west, and the north end of the needle is on the west side of the line, the azimuth, plus the angle shown by the needle, is the true varia- tion. But should the north end of the needle be found on the east side of the line, the elongation being west, the difference between the azimuth and the angle would show the variation: and the reverse when the elongation is east. Note. — The variation of the needle should always be noted on every survey made with the compass, and then if the land 166 ele:iiexts of sueyeying. [book ii. be surYejed at a future time, the old lines can always be re-run. 123. It lias been found by obserYation, that heat and 'cold sensibly affect the magnetic needle, and that the same needle will, at the same place, indicate different lines at different hours of the day. If the magnetic meridian be obserYed early in the morning, and again at different hours of the day, it will be found that the needle will continue to recede from the meridian as the day adYances, until about the time of tlie highest temperature, when it will begin to return, and at CYening will make the same line as in the morning. This change is called the diurnal variation, and Yaries, sometimes, during the summer season, from one-fifth to one-fourth of a degree. 124. A Yery near approximation to a true meridian, and con- sequently to the Yariation, may be had, by remembering that the pole-star Yery nearly reaches the true meridian, when it is in the same Yertical plane with the star Alioth in the tail of the Great Bear, which lies nearest the four stars forming the quadrilateral. The Yertical position can be ascertained ^ by means of a plumb-line. To see the i spider's lines in the field of the telescope, I at the same time with the star, a faint' 1 light should be placed near the object-glass. "When the plumb-line, the star Alioth, and the north star, fall on the Yertical spider's line, the horizontal limb is firmly clamped, and the telescope brought down to the '^ horizon; a light, seen through a small aper- i ture in a board, and held at some distance by an assistant, is then moYcd according to signals, until it is > it i / \ > / > i -■>f--. / ~^'" V SEC. IV.] TEIANGULATION. 167 covered by the intersection of the spider's lines. A picket driven into the ground, nnder the light, serves to mark the meridian line for reference by day, when the angle formed by it and the magnetic meridian may be measured. SECTION IV. TRIANGUL ATION. 125. When a large extent of territor}^, 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 Geodesic surveying. 126. The operations necessary to the successful execution of a Geodesic 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 far exceed the limits of an elementary treatise. We shall, therefore, attempt nothing more than a brief outline of the operations in a trigonometrical survey, in which the Plane-Table and Compass are used in con- nection with the Theodolite, and in which, the curvature of the earth is not considered. We shall then explain the methods of mapping, or plotting, such a survey. The example will be limited to the survey of the harbor, delineated in plate 6. 127. After having made a preliminary examination, or re- connaissance of the territory to be surveyed, suitable stations are selected at the most prominent points, and these points are marked by staves or signals. 168 ELEMENTS OF SURVEYING. [BOOK II. A hase-line is then measured. The length of the base will, in general, depend upon the magnitude of the survey, and each extremity is marked by a signal. The next step is the triangulation. At each extremity of the base, the angles between the base, and the lines drawn to each of the visible signals, are carefully measured by means of a theodolite. The sides of the triangles, thus obtained, serve as new bases upon w^hicli other triangles may be formed, and so on, until the entire area is covered by a net-work of triangles. This system of triangles is called the primary system, and the operation of forming them is called the primary triangula- tion. Within the primary triangles, and depending upon them, a system of smaller triangles is formed in the same manner, called the secondary system; and if the extent or importance of the work should demand it, the secondary may be sub- divided into tertiary triangles. Having completed the triangulation, the characteristics of the surface, such as roads, streams, villages, boundaries, &c., are filled in by means of the compass, plane-table, or some of the methods already explained. 128. Before commencing a trigonometrical survey, an ex- amination of the entire territory should be made, for the pur- pose of selecting a location for the 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 gTeat judg- ment and care, and but few rules can be given for the selection of trigonometrical points. Those points should, in general, be chosen in such a manner, that they may be distinctly visible from each other, and so that the triangles formed, by uniting them, may be, as nearly as possible, equilateral. SEC. IV.] TRIANGULATION. 169 It is easily seen, that a triangle which has an obtuse or a very acute angle, will experience a greater change of form, for a given error, than one wiiich is nearly equilateral ; and since the accuracy of each triangle depends upon the preceding ones, it is further evident, that the introduction of a single ill-conditioned triangle, might vitiate the whole survey. Except in extreme cases, no angle, less than 30°, should be used; and even angles of 30° should not be admitted when the locality can be so chosen as to prevent it. The base is usually much shorter than the sides of the primary triangles; these sides, however, should be increased as rapidly as is consistent with the above remarks. 129. The stations are marked by signals, which may be con- structed 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 traced, 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 to be placed. Where the sides of the triangles are not very great, a pole, planted vertically, and 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, and the flags of the different stations 170 ELEMENTS OF SURVEYING. [BOOK II. should be entered in a book. For the purpose of future reference, the trigonometrical point, at each station, as B, should be indi- 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 enougli 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 neighborhood. 130. The measurement of a base-line on which the accuracy of the entire survey depends, is one of the most difficult opera- tions of surveying, and one, for the successful accomplishment of which, art and science have been strongly taxed. The selec- tion 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. It must be so chosen, that the sur- rounding 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 liase, should, in a measure, depend upon the magnitude of the survey. Theodolite. 131. The theodolite is generally used for measuring the angles of a trigonometric survey. The extent of the survey, and the standard of accuracy to which the results are required to conform, must determine the size and perfection of the in- strument to be employed. The angles of the primary triangles of the United States Coast Survey are measured with theodo- lites, whose horizontal circles are 24 or 30 inches in diameter; and to eliminate, as much as j^ossible, every source of error, great numbers of observations are made at each station, the readings being made on different points of the arc by difierent SEC. IV.] TRIANGULATION. 171 yerniers. Usually from 4:0 to 60 observations are made for each angle — one measurement, ^vitli the telescope cHrect, and one with it reverted, constituting a complete observation. AVitli 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. 132. To illustrate the manner of carrying on a minor triaugulation, let us refer to the plan of the harbor [plate 6], in which AB is the measured base, 11-40 yards ; C, D, E, &C;, triaugulation points, at which signals have been erected. Com- mence the triaugulation at A, the west end of the base ; and for convenience in plotting, it would be well to make the line, passing * through the point and 180° parallel, in each position of the instrument, to the base AB. Having brought the of the vernier to the of the limb, clamp the vernier plate, and direct the upper telescope to the signal at B, and clamp the limb. Enter the observations as in the following tahle : Obseryatioxs at Station A. Name of Station. Vernier I. Vernier II. Mean. Station B 00° 00' 00" 00' 00" 00° 00' 00" Station E 72° 21' 55" 25' 5" 72° 25' 00" Station Ct 138° 31' 56" 35' 4" 138° 35' 00" &c. &c. &c. &c. Having recorded the reading of the first vernier, and the minutes and seconds of the second vernier, unclamp the vernier plate, and direct the telescope to the station at E, and record both verniers, as before. Again unclamp the vernier plate, 172 ELEMENTS OF SURVEYING. [BOOK H. and direct the telescope on the signal at G ; and then read and record, as before. Having determined the angles subtended by all the signals visible from A, let the theodolite be removed to B. Bring the of the vernier one, to 180° on the limb, and direct the telescope on the signal at A — the line (0°, 180°) will then be parallel to its first position, and the limb may be clamped. Read, now, the angles to the signals at A, E, C, &c., and record as before. If the theodolite is how removed to the station E, the line (0°, 180), may be made parallel to its first position, by adding 180° to the reading of the first vernier, from A to E, and then directing the telescope on the signal at A. The line (0°, 180°), will thus be made parallel to AB, and the readings may be made and recorded as before; and so on* until all the stations have been visited, and the angles measured. From the field records, the angles BAE, EAG, ABE, EBG, &c., may be easily deduced, the whole may be plotted on paper, or the several sides may be computed trigonometrically. It may be observed that the line (0°, 180°), has been made parallel to the lase-line at each station. 133. To illustrate this principle of repetition, suppose the of the vernier to coincide with the of the limb, and the tele- scope to be directed, from the station A, 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, an 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. SEC. IV.] TRIANGULATION. 173 the reading will be three times the angle, and so on. After fen repetitions, if we add 360° each time the of the vernier passes the of the limb, the final reading will be ten times the angle BAE, 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. 134. The method of reading angles, by this mode, is as follows : Angles at station A, between signals B (left), and E (right). June StJi, 1870. No. of Repe- titions. Vernier I. Vernier n. Mean of Verniers. / 1 72° 24' 55" 25' 5" 72° 25' 00" 2 144° 49' 55" 50' 0" 144° 49' 58" 3 217° 14' 50" 15' 10" 217° 15' 00" 4 289° 39' 50" 40' 00" 289° 39' 55" Mean reading 4)289° 39' 55" 72° 24' 59" Having measured the necessary angles, the parts of all the triangles, formed by joining the stations, three and three, may be readily calculated by methods heretofore explained. riLLIN"G UP THE SUEVET. 135. After the triangulation is completed, the interior may be filled up by the aid of the Compass, on the plane-table, or both. Use of the Compass. 136. The use of the Compass, in determining points and lines, by means of offsets, has been already explained (Bk. II., 174 ELEMENTS OF SURVEYING. [book II. Art. 83). Wc will apply these principles, in the example before us. Place the compass at A (plate 6), and take the bearing of the line AF, which is S 12° W. Enter this bearing at A. Then measure along the line AE any distance, as Aa equal to 130 yards, and make an offset to the lake, which wo measure and find to be 50 yards. Enter the 130 in the middle column, and as the lake lies on the right (in going from A to E), we insert the 50 in the right-hand column. "We then measure along the line AB to h, 350 yards from A. Here we make a second offset to the lake, and find it to be equal to 100 yards. Haying entered the distances in the notes, we measure to q, the point where the line AE crosses 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 tlie course terminates, an offset to the lake, of 160 yards. The entire distance from ^ to ^ is 800 yards. SEC. lY.] THE PLANE-TABLE. 175 At E, we take the beariug to //, wliicli 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 H, 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 the 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. The Plajte-Table. 137. The plane-table (PL 3, Fig. 1) consists of two parts: a rectangular board CDBA, and a tripod EHG, to Avhich 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 the purpose of levelling the table ; the table having a ball-and- socket motion, similar to the limb of the theodolite. For the purpose of levelling the table, a small detached spirit-level is used. This level being placed over the centre, and also over two of the levelling screws, the screws are turned contrary ways until the level is horizontal; after which, it is placed over the other two screAvs, and made horizontal i]i the same manner. Between the upper horizontal plate and the table, there is a clamp-screw, similar to the clamp-screw of the theodolite, which being loosened, the table can be turned freely about 176 ELEMENTS OF SURVEYING. [BOOK II. its axis. There is, also, a small tangent-screw, by wliicli the smaller motions of the table are regulated, after the clamp- screw is made fast. Neither of these screws can be seen in the figure. The upper side of the table is bordered by four brass plates, about one inch in width, and the centre of the table is marked by a small pin, F. About this centre, and tangent to the sides of the table, conceiye a circle to be described. Suppose the circumference of the circle to be divided into degrees and parts of a degree, and radii to be drawn through the centre and the points of division. The points in which these radii intersect the outer edge of the brass border, are marked by lines on the brass plates, and the degrees are numbered in the direction from left to right, from the point L to the point /, 180°, and from the point / to the point X, 180°. In some plane-tables, however, they are numbered from to 360°. There are, generally, diagonal scales of equal parts cut on the plates BLG and AIB, the use of which will be explained hereafter. Near the two other edges of the table, two small grooves are made, into which the plates of brass BB and CA are fitted, and these plates are drawn to their places by means of milled screws, which pass through the table from the under side, and screw firmly into the plates. The heads of two of the screws, Q and 8, are seen in the figure, as also one of the plates and its two screws in Fig. 3. The object of these plates is to confine a sheet of paper on the table. By loosening the screws, and pressing them upward, the plates are raised above the surface of the table; the edges of the paper can then be placed under them: then, by turning the screws back again, the plates are drawn down and the paper held tightly. Tig. 1 represents the table with the paper partly put upon it: one SEC. IV.] THE PLANE-TABLE. 177 edge of the paper has been placed under the plate DB, and the screws S and Q tightened. The paper, before being put on, should be moistened, in order to expand it; and then, after it has been dried, it will fit closely to the table. A ruler, AB (Fig. 2), with open vertical sights, is used with the plane-table. This ruler has a fiducial edge, which is in the same vertical plane with the hairs of the sights. A ruler with a telescope, and a vertical limb, similar to the vertical limb of the theodolite, is sometimes used with the plane-table. A com- pass, also, is often attached to the table, to show the bearings of the lines. Uses oe the Plake-Table. 138. The plane-table is used for two distinct objects. 1st. For the measurement of horizontal angles: and, t 2dly. For the determination of the shorter lines of a survey,. both in extent and position. To MEASUKE A HOKIZOKTAL AkGLE. 139. Place, by means of a plumb, the centre of the table- directly over the angular point : then level the table ; after • which, place the fiducial edge of the ruler against the small; pin at the centre: direct the sights to one of the objects, and. note the degrees on the brass plate; then turn the ruler and' sights to the other object, and note the degrees as before. If ' the ruler has not passed over the point, the difference of the readings is the angle sought; but, if it has, take the larger, sum from 180°, and to the remainder add the smaller^ their sum is the required angle. To DETEKMINE Lliq"ES 11^ EXTEi^T AKD POSITION". 140. Having placed a paper on the table, examine the objects and lines which are to be determined, and; select, for a. 178 ele:u:ents of suryeying. [book ir. base, such a line of the triangulation that most of the objects can be seen from its extremities. Then place the plane-table, with its centre, nearly, though not accurately, 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 ]3aper. 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 jiniv of dividers, take from the scale a certain number of equal ^arts, 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 tlie 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-line, move the legs of the tripod until it is brought back to its place. Let the table be then levelled, 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 witli the base-line, on the ground. Then direct the sights to SEC. IV.] THE PLANE-TABLE. 179 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. 141. Let it be required, for example, 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 JV, and sight to the same corners as before, and draw the lines as in the figure. The points at which they intersect the cor- responding 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 then apply the line to the scale. The length of the line 1 is equal to 198 yards. 142. In this example, we de- termine from the base-line CD, the positions of the points I^, E, and // i 180 ELEMENTS OF SURVEYING. [book n. Changing the Paper. 143. "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. ISTow, 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 IsF on the second paper. Placing the line NF of the one paper on NF of the other, we have the folloY/ing plan. In this plan, all the points and lines are accurately laid down. Any number of papers may be joined in a similar manner. SEC. IT.] CIECULAR PEOTEACTOE. 181 144. The principal use of the plane-table is for the interior filling up of trigonometrical surveys: it is also used with advan- tage, Tvhen only a plot of a field is wanted. It ought not be used for the determination of long lines, nor can it be relied on for determining extended areas. Plotting the Teiaxgulatiox. 145. 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 Cieculae Peoteactoe. 146. This instrument consists of a brass circular limb (PL 2, Fig. 4), of about six inches in diameter, with a movable index AB, having a vernier at one extremity A, and a milled screw at the other extremity B, with a concealed cog-wheel that works Avith the cogs of the limb, and thus moves the index AB about the centre of the protractor. • At the centi'e 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 limb ; 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 shai-p steel pin at its extremity, are fastened to the index, and revolve freely around the lines ah and cd. The small screws, «, J, c, and d, move them in the directions of the lines ab, cd, for the purpose of bringing the steel pins exactly into the line which passes through the of the index and the centre of the protractor. 182 ELEMENTS OF SUE^TYING. [BOOK II. 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 the same points as before. If they do, the index is adjusted ; if they do not, correct the error with the screws a, h, c, and d. To LAY OFP AS" AXGLE WITH THE PeOTKACTOE. 147. Let its centre be placed over the angular point, and the diameter passing through and 180^, on the given line. Turn the screw that works the index, until the of the A'ernier 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. FiEST Method of Plottixg. 148. Suppose it were required to make the plot of the harbor 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 (Bk. I., Art. 39). 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 will thus be determined. Then, place the circular protractor at A, and lay off the angle BAU, and then the angle BAG. Xext, place the pro- tractor at B, and lay off the angles ABB and BBC. The SEC. IV.] CIRCULAR PROTRACTOR. 183 intersection of the lines AE and BE will determine the station E. Let the protractor be then placed at this point, and all the angles of station E laid down. The point G, w^here EG intersects AG, and the point C, where EC intersects BC, will then be found. By placing the protractor at C and G, we can determine the points D and F, when the place, on the paper, of all the stations will be known. 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, Ab, 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 Eff, 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 EI, and all the compass-work will be plotted. The work done with the plane-table is united to the work done with the theodolite, by simply reducing it to the same scale, and then placing the line ^iV on the paper of the plane-table, upon the line AJS^, drawn on the plot of the triangulation. Secoin-d Method of Plotti^s-g. 149. Place the centre of the protractor near the centre of the paper, and draw a line through the points and 180°. This 184 ELEMENTS OF SURVEYING. [BOOK II. line will have the same position with the circular protractor that the base-line AB had with the limb of the theodolite. Then lay off, from the point, an arc equal to the direction from A to E, also an arc equal to the direction AG, and tlirough 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. Now, since all the lines drawn on the paper have the same position with the circular protractor, as the corresponding lines on the ground have with the limb of the theodolite, it follows that each direction will be parallel to its corresponding line upon the ground. Hence, any 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 AE and BE, and the point of intersection will determine station E, Through B and E, draw parallels to the lines which correspond to the directions BC, CE, and their point of intersection will deter- mine station C. Through (7 and E, draw lines parallel to the lines corresponding to the directions CE and ED, and the point of intersection will determine D. In a similar manner we may determine the stations F and G, Method op Chords. 150. The chord of a given arc is equal to the sine of half the arc with double the radius. SEC. IV.] METHOD BY CHORDS. 185 For, let DAF be any given angle, and AH a line bisectiug it. Let DC be the chord of the arc CD, 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= 2 AC, we have, from similar triangles, HF-. 2KC; but DC=2KC, hence HF = CD. To LAY OFF AH Als-QLE. 151. To avoid, as far as possible, the 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 CD 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 18 G ELEMENTS OF SUEYEYING. [BOOK II. circumference 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 0.126005, or 1.26 to the radius of 10 inches. Set off from to l, as in the figure, this distance taken from the scale, and through the two points l, h, thus determined, draw a straight line. This line should pass through the centre, and will make with 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 obyious. Maeitime Sueveyik"g. 152. When, in connection with a trigonometrical survey on shore, a harbor is to be surveyed, as in the example, for the purpose of ascertaining the channels, their depth and w^dth, 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. 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 meas- ured with the theodolite. Then, there will be known, in each triangle, the base and angles at the base, from w^hich the dis- tances 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. SEC. lY.] MARITIME SUEYETING. 187 153. If a person with a theodolite, or with any other instru- ment adapted to the measurement of horizontal angles, be sta- tioned at each extremity of the base-line, it will not be neces- sary 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. 154. There is another method of determining the places at which the soundings are made, that admits of great despatch, and which, if the obserrations are made with care, affords results sufficiently accurate. Haying 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, p. 93. 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. The sextant is not described, nor are its uses explained in these Elements, because its construction combines many philo- sophical principles, with which the Surveyor cannot be sup- posed conversant. 155. 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 188 ELEMENTS OF SURVEYING. [BOOK II. 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 care- fully 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. 156. If a person stationed on shore with a theodolite, takes the bearing of the boat, at every second or third sounding, de- termined by hoisting a flag, it will fix the positions of the sound- ings 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 diff'erent 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 bot- tom 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 sound- ing 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 con- venient point of the harbor, and its point is referred, by SEC. IT.] MARITIME SURYEYTNG. 189 means of a spirit-leyel, to some fixed bench-mark, on a level with mean low-water mark, to which all the soundings must be reduced. 157. Haying plotted the work done with the theodolite, as also the outline of the harbor traced with the compass, it re- mains to delineate the bottom of the harbor; and this is done by means of horizontal curves, hereafter explained, (Bk. III., Art. 00), w^hich 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 at a distance 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 Book III., Art. 00. 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 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 Plate 6, 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. BOOK III. LEVELLma AND ITS APPLICATIONS. SECTION I. OF LEVELLING. 1. Levellin"G is the art of determining the relative dis- tances of points from the centre of the earth. 2. 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. 3. 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. 4. A straight line drawn tangent to a line of true level, at any point, is a horizontal line, and is called the line of apparent level. Thus, (PI. 4, Fig. 1), if C is the centre of the earth, and AEFsl line of true level, ABD is the line of apparent level. This is the line of level determined by an instrument. The difference between the apparent and true of the points A and F, is BE, the excess of the secant of the arc AF, over the radius. f SEC. I.] OF LEVELLING. 191 5. To find a general formula for computing tliis excess, we have (Geom., B. IV., Prop. XXX.), ZB^ = BE {BE + ^EC) ; but, since the arc ^^ is very small in comparison with the radius of the earth, the arc AE will not differ sensibly from the tangent AB ; the diameter 2^(7 may, for the same reason, be taken for the secant {BE + 2EC) : hence, ~AE' = BEX 2EC, or, dividing by 2EC, ^^=¥c «• If we take the mean diameter of the earth to be 7919 miles, AE^ formula (1) gives BE = — — - (2), hence, Tlie departure of tlie apparent from tlie true level, darting from a given point, is equal to the square of the distance to the second point, divided iy 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 below. Observe, that w^hen the distance AE = 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, 1^ : ^^ :: f of a foot : f^^; hence, we have the following easy rule for finding the correc- tion of curvature in feet : Tlie correction for curvature, in feet, is equal to two-thirds of the square of the distaiice in miles. 192 ELEMENTS OF SURVEYING. [book in. Table showing the differences, in inches, between the true and apparent level, for distances betw^een 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 o o .011 28 .981 53 3.511 78 7.605 4 .020 29 1.051 54 3.045 79 7.802 5 .031 30 1.125 55 3.781 80 8.001 G .045 31 1.201 56 3.925 81 8.202 7 .061 32 1.280 57 4.061 82 8.406 8 .080 33 1.300 58 4.205 83 8.612 9 .101 34 1.446 59 4.351 81 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 l.7il 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 Iksteuments. 6. Before proceeding further in the discussion of the prin- ciples of leyelling, we will describe some of the instruments used, and first, The Y Level. 7. 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. SEC. I.] THE Y LEYEIi. 193 The part of the instrument shown in (PL 4, Fig. 2), rests on a tripod, to which it is permanently attached at Z. HE is a horizontal brass plate, through which four levelling-screws with milled heads are passed, and worked against a second horizontal plate, GG. Two of these screAvs, K and /, are seen in the figure. iS' is a clamp-screw, which, being loosened, allows the upper part of the instrument to turn freely around its axis. Q is a tangent-screw, by means of which the upper part of the instrument is moved gently, after the clamp-screw S has been made fast. EE is a horizontal bar, perpendicular to which are the wyes, designated Y's, that support the telescope LB. This telescope is confined in the Y's by the loops r, r, which, are fastened by the pins p and p. The object-glass B, is adjusted to its focus by the screw X, the eye-glass L slides out and in, freely. The screws /, /, work the slide which carries the horizontal hair; and two horizontal screws, only one of which, a, is seen, work the slide that carries the yertical hair. CD is an attached spirit-leyel. The screw N elevates and depresses the Y nearest the eye-glass. In some instruments this Y is elevated and depressed, by means of two screws at M and M, Before using this level, it must be adjusted. The adjust- ment consists in bringing the different parts to their proper places. The line of CGlUmation is the axis of the telescope. With this axis, the line drawn through the centre of the eye-glass and the intersection of the spider's lines, within the barrel of the telescope, ought to coincide. First ADJusT:^iEis-T. To fix the intersection of the spider's lines in the axis of the telescope. 8. Having screwed the tripod to the instrument, extend the legs, and place them firmly. Then loosen the clamp-screw S, 13 194 ELEMENTS OF SURVEYDTG. [BOOK III. and direct the telescope to a small, well-defined, and distant object. Then slide the eye-glass till the spider's lines are seen distinctly; after which, with the screw X, adjust the object-glass to its proper focus, when the object and the spider's lines will be distinctly seen. Note now the precise point covered by the intersection of the spider's lines. Having done this, revolve the telescope in the Y's, half round, when the attached level CD will come to the upper side. See if, in this position, the horizontal hair 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 hair, until it has been carried over half the space between its last position and the observed point. Carry the telescope back to its place; direct again, by the screws at 3f and R, the inter- section of the spider's lines to the point, and repeat the opera- tion, till the horizontal hair neither ascends nor descends while the telescope is revolved. A similar process will arrange the -vertical hair, and the line of collimation is then adjusted. SECOiy'D Adjustment. To make the axis of the attached level CD parallel to the line of collimation. Turn the levelling-screws M and R, until the bubble of the level DC 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 h, half the inclination ; and then with the screws, at M and R, bring the level to a horizontal position. Reverse the telescope in the Y's, and make similar corrections again ; and proceed thus, until the bubble stands in the middle SEC. I.] THE Y LEVEL. 195 of the tube, in both positions of the telescope ; the axis of the leyel is then horizontal. Let the telescope be now reyolved 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; one of these screws is seen at q. 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 con- siderably 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 position in which the bubble will stand at the middle of the tube. This suggests the necessity of making the first part of the adjust- ment with tolerable accuracy; then, having made the second with care, re-examine the first, and proceed thus till the adjust- ment is completed. Third Adjustmeio'. To make the level CD and the line of collimation perpendicular to the axis of the instrument, or parallel to the horizontal bar EE. Loosen the clamp-screw 8, and turn the bar EE, until the level DO comes directly over two of the levelling screws. By means of these screws, make the level CD truly horizontal. Then, turn the level quite round; 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 196 ELEMENTS OF SUKYEYDsG. [BOOK m. the revolution, the level CD be not horizontal, rectify half the error witli the screws at J/ and B, and half with the levelling screws. Then place the bar EE over the other two levelling 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 DC 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 bai* EE, it follows, that the line of collimation, the level DC, and the bar EE, are parallel to each other. The level is now adjusted. TThen used, however, it is best to re-examine it every day or two, as the work will be erroneous unless the instrument is accurately adjusted. LeyellixCt Eods. 9. The levelling rods are used to determine the points at which a given horizontal line intersects lines that are perpen- dicular to the surface of the earth, and to show the distances of such points of intersection from the ground. There are two kinds of rods used by engineei*s, known as the Boston and Xew York rods. They are both sliding rods, divided into feet, tenths, and hundredths of feet ; and the readings, by means of a vernier, are made to thousandths of a foot. 10. The Boston Eod is fonned by two pieces of hard wood, about six feet and a half in length, the one sliding through grooves, along the other, in both directions. A vane or target, six-tenths of a foot in width, divided into four equal rectangles by a horizontal and vertical line, passing SEC. I.] NEW YOEK ROD. 197 through the centre, is permanently connected with one ex- tremity of the sliding piece. The two diagonal rectangles of the target, are usually painted black or red, and the other two white, so that the centre and the horizontal dividing line, may be distinctly seen. There are two yerniers, one at either end of the second piece, by means of which the read- ings, indicating the height of the target, are read. When the height is less than six feet, the reading is made by the yernier at the top of the rod; and when it is greater than six feet, the rod is reversed, which brings the other vernier to the top ; the target is then run up to the re- quired height, and the reading made as before. New York Rod. ^ 11. This rod, which is shown in the en- graving, is cut in two parts, so that, both ends may be exhibited. It is made of satin-wood, in two pieces, like the former, but 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 M point. In this rod, as in the other, a target ' is used to indicate where the horizontal 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 the alternate colors with which the diagonal quad- rants are painted. The opening, in the face of the target, is a little more than the tenth of a foot. The right edge of the opening is J-^^ ELEMENTS OF SURVEYING. [BOOK in. chamfered, and divided into ten equal spaces, corresponding to nine hundredths on the rod: hence, the vernier reads to thou- sandths of a foot. For heights less than six feet, the target is moved along the sliding part, to Avhich 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 the height exceeds six feet, the slide and target are run np to the requisite height, and the reading is made by the vernier at the top of the staff. Tests of Adjustment. 12. 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: The level being adjusted, place it at any convenient point, as G (PI. 4, Fig. 4). At eqnal distances of about 100 yards, on either side, and in the same line with the level, place the levelling rods, Cb, BF. Make the level horizontal with the levelling screws. Then, turn it toward either rod, as BF, and run the vane np or down, as required, until the intersection of the hairs strikes the centre : then make the slide fast, and note carefully the height of the vane. Turn the level half round, and do the same in respect of the staff CI. Let the telescope be now reversed in the Y's. Sight again to the rod BF, and note the exact height of the vane. Let the telescope be now turned half round, and the same be done for the rod Ch. \i the two heights last observed are equal to those first noted, each to each, the line of collimation is perpendicular to the axis of the instrument; and if the bubble has, at the same time, preserved its place at the middle point of the tube, the instrument is truly adjusted. SEC. I.] LEVELLING. 199 For, had the line of collimation been inclined to the axis of the level, it would, in the first instance, have taken the direction AF or Ad; and when turned half round, it would have taken the direction AE or Ad. The telescope being re- versed in the Y's, and again directed to the staff BF, the line of collimation would take the direction Ad or AF, and when turned to the staff Ch, it would take the direction Ah oy AF: and the two distances BF, Bd, or Cb, CE, can only be equal to each other when the line of collimation falls on the hori- zontal line gf, LeVELLIKG IK THE FlELD. 13. The operations of levelling 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 : 3d. For the purpose of determining the contour lines in a topographical survey, as described in Section II.; and, stilly. 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 bound- aries for filling and excavation; such as setting stope stakes, &c. DlEFEEEKCE OF LeVEL BETWEEN" TWO PoiNTS. 14. When 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 200 ELEMENTS OF SURVEYING. [book m. distinction merely, lach-sigUts ; and levels taken in the direc- tion of the other station, fore-sights. Before going on the field with the level, rule three columns, as below, and head them, stations, back-sights, fore-sights. Field Notes. stations. + Back-sights. — Fore-Sights. 1 2 3 10 11.6 6.8 3.9 3 4.9 8.3 Sums -32.3 16.2 Dif. of level . . 16.1 16.2 EXAMPLES. Find the difference of level between any two points, as A and G (PL 4, Fig. 5). The level being adjusted, place it at any point, as B, as nearly in the line joining A and G, as may be convenient. Place a levelling rod at A. Make the level horizontal by means of the levelling screws ; turn the telescope to the rod at A, and direct the rodman to raise the target until the horizontal line ab pierces its centre; then note the distance Ab (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 iV", in the pro- posed direction, and sight to the rod as before; enter the dis- tance iV«, 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, JS^d = 11.6 feet, SEC. I.] LEYELLIXG. 201 in the column of back-sights, opposite station 2. Let the rod- man then be sent forward to a convenient point, as J/, and make the fore-sight to /; and enter Mf = 0, in the column of fore-sights, opposite station 2, (C). Eemoye the level, in succes- sion, to D and B, and make similar levels at those points, and enter the results in the columns of back-sights and fore-sights, opposite stations 3, (D), and 4, (E). It is evident from the figure, that the difference of level ^F, between A and X, is equal to the back-sight dA, diminished by the fore-sight aXj also that the difference of level between Ji and J/, is equal to the back-sight ciy, diminished by the foresight 0, and since each set of observations is entirely inde- pendent of every other set. we may infer that the difference of level leticeen two consecutive points, as determined l)y tlie same position of the level, is equal to tlie lacTc- sight, diminished lij tTie fore-sight. If the fore-sight be greater than the back-sight, the difference will be affected with a minus sign, a restilt which shows that the second point is lower than the first : and. Generally, the difference of level letv;een any two points, deter- mined as above, is equal to tlie sum of the ladc-sights diminished hj 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 G^, is 16 feet and 1 tenth. 15. In the above example, we did not regard the difference between the true and apparent level. If it be necessary to ascertain the result with extreme acctiracy, 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. 202 ELEMENTS OF SUKVEYING. [book m. EXAMPLE. Stat. 1 2 3 4 5 Back-sts. Distances. Fore-st. Distances. Cor. back-sts. Cor.for-sts. 9.8 8.7 5.2 10.3 11.0 20 ch. 25 ch. 18 ch. 29 ch. 45 ch. 1.6 2.4 3.1 1.9 2.5 32 ch. 28 ch. 16 ch. 87 ch. 72 ch. 9.7583 8.6349 5.1663 10.2124 10.7891 1.4933 2.3183 3.0734 0.1115 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 leyel to the forward rod; the sixth and seventh, are the columns of back and fore sights, corrected by the difference of leyel. 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 th6' true fore-sight from station 1. The other corrections are made in the same manner. 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. 16. 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 SEC. I.] LEVELLING. 203 in different states of the atmosphere, as because tlie correc- tions are inconsiderable in themselves. 17. The small errors that would arise in regarding the apparent as the true level, may be avoided dy i^lacing tlie level- ling 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 be required, it will be unnecessary to make measurements at all. Care, however, ought to be taken, in placing the levelling rods, to have them as nearly equidistant from tlie level as can be determined by the eye; 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 pro- portionably shorter in the second, and the reverse. LEYELLIis-G FOK SeCTIO:S". 18. Having decided upon the line along which a section is to be taken, let a permanent mark be made at the beginning of the line : this is called, a hench-mark. A bench-mark is made by drilling a hole in a rock, or by painting upon a rock or fence, or sometimes by driving a stake in the ground, with its upper end marked by a nail-head. Bench-marks should be made from time to time along the line, to serve as checks, in case a re-survey should become necessary. 20-1 ELEMENTS OF SURVEYING. [book in. The operations in the field are similar to those in the last example, and the field-notes are kept in the same manner, except that a new column is added for bearings, when it is necessary to make a plot of the line of survey. The total distance of each point, above or below the starting-point, may be computed, and written in a separate column, paying par- ticular attention to the signs. We annex an example, in which the heights are estimated in feet, and decimals of a foot. EXAMPLE. sta- tion. Distances in feet. B. Sight. F. Sight. Dif. between B. S. and F. S. Total Dif. of Level. Kemakks. 1 650 2.35 14.55 - 12.20 - 12.20 Commenced at hench-mark A. 2 700 3.56 9.58 - 6.02 - 18.22 3 750 10.34 6.21 + 4.13 - 14.09 4 650 14.55 0.25 + 14.30 + 0.21 5 600 9.98 1.67 + 8.31 + 8.52 6 650 3.62 14.54 - 10.92 - 2.40 B.M. 1.23 13.45 - 12.22 - 14.62 Bench-mark on rock. 7 500 2.23 - 12.05 - 9.82 - 24.44 Terminating at B. HT. on oak- 8 750 6.20 19.55 - 13.35 - 37.79 tree. The fifth column shows the diflTerence of level between any two consecutive positions of the levelling rod, and is found by subtracting the fore-sight from the corresponding back-sight, and giving to the remainder the proper sign. The sixth column shows the distance of each point, above or below the bench- mark A, and is obtained by continual additions of the numbers in column 5. Thus, (-12.20) + (-6.02) and so on. - 18.22; (- 18.22) + 4.13 = - 14.09; It will be seen that the point of termination is 37.79 feet "below the starting-point. SEC. I.] LEVELLING. 205 Profile of Sectioi^. 19. The vertical distances being generally very small as compared with the horizontal distances, two different scales be- come necessary in plotting a profile. In order that the vertical distances may be fully exhibited in the plan, the scale used for them is much larger than is used for the horizontal lines. This becomes absolutely necessary where long lines of profile, with a gentle slope, are to be plotted, as is always the case in the trial section of a railroad suryey. We shall illustrate the manner of plotting a profile section, by drawing the section determined by the field-notes just given. 20. Draw a horizontal line AK, called a datum line, and assume some point as A, to represent the point of beginning: lay ofi", on the datum line, distances equal to the measured 650 ^ ^/'''^^m ■B__ Y00_ _C _750 D 650 _^^ioO J.^e^^y G 500 H___7_5q__K distances 650, 700, 750, &c., feet to K, using in this case a scale of 1500 feet to 1 inch. At the points B, C, D, E, &c., thus determined, erect perpendiculars, making them equal, on a scale of 25 feet to the inch, to the corresponding difi'erences of level taken from the field-book ; through the points thus found, draw the irregular line APLM, and it will represent the surface of the ground along the line of level. 206 ELEMENTS OF SURVEYING. [BOOK m. The bench-mark, between stiitions 6 and 7, is not plotted, as it is supposed to be out of the line of the section, and no distances are measured to it. SECTION n. TOPOGRAPHICAL SURVEYING. 21. 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 w.ould other- wise be useless. 22. The manner of ascertaining these irregularities is, to suppose the surface of the ground to be intersected by a system 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 w^ill be nearer together or farther apart, as the ascent of the hill is steep, or gentle. SEQ. n.] TOPOGRAPHICAL SUEYEYING. 207 If, therefore, such intersections be made, and the curves so determined be 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. 23. The subject divides itself, naturally, into two parts: 1st. To make the necessaiy examinations and measurements on the field; and, 2d. To make the plot, or the delineations on paper. For the former of these objects, the theodolite is the best instrument; the common level, however, will answer all the purposes, though it is less convenient. 24. Before going on the field, it is necessary to provide a number of wooden stakes, about two feet in length, with heads. These stakes are used to designate particular points, and are to be driven to the surface of the ground. A nail should then be driven into the head of each of them, to mark its centre. 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 all others that can occur. ExAiTPLE First. 25. Let A, (PL 4, Fig. 6), be the summit of a hill, the contour of which it is required to determine and represent. At A, let a stake be driven, and let the axis of the theodolite, or level, be placed directly over the nail which marks its centre. From Ay measure any line down the hill, as AB, using the telescope of the theodolite, 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 «, d, c, d, &c., let 208 ELEMENTS OF SUKYEYING. [book m. stakes be driven, and let the horizontal distances A a, ab, he, and cdy be carefully measured. In placing the stakes, reference must be had to the abruptness of the declivity, and the accuracy 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 li, 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 theodolite, 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 of the distances ae, hf, ai, hi, &c., be measured with the chain, and there will then be known the three sides of the triangles Aae, Ahf, Aai, Ahl, &c. Let, now, the difference of level of all the points marked in each of the lines AB, AD, AC, be determined. In the present example the results of the measurements and levelling, are — Line AB. Distances. Difference of Level. Aa = 40 feet A above a 12 feet ad =50 " a above ^ 8 " he =30 " h above c 9 " cd =46 « c above d 11 " SEC. n.] TOPOGRAPHICAL SrEYEYINQ. 209 Line AC. Distances. Difference of Leyel. Ae = 28 feet ef =45 - fg =o5 " gJi =49 " A above e, 11 feet e above /, 9 " / above g, 12 " g above 7i, 14 ^^ Line ^Z). Distances. Difference of Level. Ai =25 feet il =55 " Im = 38 *•' W7i=48 " A above /, • 9 feet I above /, 13 " I above ??2, 7 " 7n above n, 14 " 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 op Woek. Having drawn, on the paper, the line AB, lay off the angle BAC = 25°, and the angle BAD = 30°. Then, from a conven- ient scale of equal parts, lay off the distances Aa, ai, ic, cd, Ae, ef, fg, gli, Ai, il, hn, and mii. Let the horizontal planes be passed at a distance of eight feet from each other. Since A is the highest point of the hill, and the difference of level of the points A and a, is 12 feet, the first plane, reckoning 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 ;210 ELEMENTS OF SURVEYING. [BOOK HI. curve in wliicli the first plane intersects the ground. The points at which it cuts the line from A to C, and the line from A to D, are determined similarly, and three points in the first curve are thus found. The graphic operations are greatly facilitated by the aid of the sector. 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 arms 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 ofiT from a to o. If the distances to be taken from the sector fall too near the joint, let multiples of them be used; as for instance, on the French sectors, let the arms be extended until the dividers reach from 120 on the one, to 120 on the other, then 80 or 40 "will be the proportional numbers. Other multiples may be used, though it is generally more convenient to multiply by 10. 20. The second 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 ab, in the dividers, and extend 'the sector, so that the dividers will reach from 8 to (the 'descent from a to i 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 second curve. The difference of level between a and 5, being 8 feet, and the difference of level between a and j9, being 4 feet, the dif- ference of level between p and d, must also be 4 feet; hence, SEC. n.] TOPOGRAPHICAL SURVEYING. 211 the third plane will pass 4 feet below I, and q^ determined as above, is a point of the third curve, and so on. After having determined the points in which each contour line cuts the lines diverging from A, let the contour lines be drawn through them, so as to indicate the surface of the hill. The numbers (8), (16), &c., show the vertical distances of the re- spective planes below the point A, 27. 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. A profile along either of the diverging lines may be plotted by the rules already given (Art. 20). The diagram shows the profile along the line AB. Bd Example Secokd. 28. The following example will illustrate the methods em- ployed in making a topographical survey, where great accuracy is required. By means of a theodolite or level, range a line of stakes A, B, C, D, E, &c., along one side, or through the middle of the ground to be surveyed, at equal and convenient distances from each other, say 50 feet apart. Mark, with a piece of red chalk, on each stake in this row, one of the letters of the alphabet, 212 ELEMENTS OF SUEVEYING. [book m. A, B, C, D, Ef &c., in their order. At A, range a line of stakes, perpendicular to AE, planting the stakes at intervals of 50 feet ; and mark them with the letters ^> ^> ^> &c., which are 1, 2, 3, ' read A first, A second, A third, &c. ^ J5 D A 1 i A 3 i i B 1 3 Z B 3 h J} B 4, i c 1 C 2 C 3 c c 4 £ 2 El J) Z m D 3 a .E3 D J) 4 S 14 Hj At i?, range a line of stakes also perpendicnlar to AE, and at distances of 50 feet from each other, and designate them ■?' ■?' ?' ^^' -^^ ^^® ^^^® ^^ ^' -^f ^' ^^'f ^^^^^ ^^^ *^^^ stakes are placed, dividing the area, to he surveyed, into squares of 50 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. The next operation is to determine the difference of level between each stake, and some fixed horizontal plane, which is called 2b plane of reference. If the sea is near, the plane of mean low water, may be taken as the plane of reference. If not, assume the horizontal plane, passing through the lowest point SEC. n.] TOPOGRAPHICAL SURVEYING. 213 of the ground to be surveyed, and make a permanent bench- mark at the point of beginning. If the lowest point cannot be easily determined, assume such a plane of reference as shall pass quite below the lowest point of the ground. The following is the form of a field-book, used in topo- graphical levelling. Field Is'otes. Back-sights. Fore-sights. Difference. Total dif. of level above E 3 Bexabks. Olqect. Beading. Object. Reading. Object E3 Beading. 0.000 E3 11.432 D3 1.211 + 10.221 D3 10.221 04 0.S97 -1- 0.314 C4 10.535 Check 10.535 C4 11.112 E2 5.281 + 5.831 E2 16.366 E4 6.154 - 0.873 E4 15.493 D4 &.001 + 0.153 D4 15.646 D2 1.1S2 + 4.819 D2 20.465 C3 2.917 - 1.735 C3 18.730 B5 6.080 — 3.163 B5 15.567 C5 0.921 -f 5.159 C5 20.726 B4 11.8S2 B4 El 0.113 8.019 + 0.S08 + 3.863 B4 El 21.5a4 25.397 Check 10.999 21.534 B3 3.990 + 4.029 B3 29.426 Dl 4.118 - 0.128 Dl 29.298 02 1.8S0 + 2.238 02 31.536 A4 5.000 - 3.120 A4 28.416 A5 9.928 — 4.928 A5 23.488 D5 1.675 + 8.253 D5 31.741 ' E5 1.111 + 0.564 E5 32.305 A3 0.108 + 1.003 A3 a3.308 CI 0.004 + 0.104 01 33.412 Check 11.878 01 11.149 B2 4.181 + 6.968 B2 40.380 33.412 Bl 2.008 + 2.173 Bl 42.553 A2 0.817 -f- 1.191 A2 43.744 Check 10.a32 43.744 A2 10.102 Al 4.332 -t- 5.570 Al 49.514 Check 5.770 49.514 214: ELEMENTS OF SURVEYING. [BOOK III. In the example, which we have taken for ilhistration, the stake ^y is the lowest point, and let us assume the plane of reference to pass through that point. Set up the level at some convenient point, as a, take the reading of a leyelling rod, set up at ^> (11.432), and enter this reading as a back-sight. Then take the readings of the rod, placed at as many stakes as can be reached from the position a of the level, entering them as fore-sights, and endeavor to make the last reading as small as possible. At this last stake, Y> drive a small peg for a bench-mark, or check. If we subtract the fore-sight, (^)> (1.211), from the back- sight (^)j (11.432), the difference, (10.221), is entered in the column headed difference. But this is not only the difference of level between (^) and (^)> but is also the height of \P), (10.221), above the plane of reference through (^) : hence, we enter this difference, under the column, headed total diff. of level, as well as in the column of differences for any two consecutive stations. If, now, we subtract the fore-sight (^)j (0.897), from the fore-sight {D), (1.211), the difference, (0.314), is evidently the height of (^) above (^) ; and if we now add this difference to the previous total, (10.221), we shall have the height of (^ above (f )^ (10.535). Move the level to a second point h, and take a back-sight to the bench-mark, (C4), and fore-sights, to as many stakes as possible. Subtracting the fore-sight (^)> (5.281), from the back- sight (^)? (11.112), we get the difference of level between {^) and (^)> (5.831) ; this being added to the previous total, gives the height of \^) above the plane of reference, through (^)? — namely, 16.366 feet. In subtracting the fore-sight (f )^ (6.154), from the fore-sight (■^)> (5.281), we find a negative result, (— 0.873 ), which shows that SEC. II.] TOPOGRAPHICAL SURYEYING. 215 (^) is below i-^j- "We then enter this difference, with its 4 2 negative sign, in the colnmn of differences : and to get the total, ice add it, luith its algebraic sign, to the preyious total, giving (15.493), and so on, for the remaining parts of the examjDle. As a check on the accuracy of our computation, subtract the fore-sight (C'4) from the back-sight (^3), and the differ- ence will give the height of (C4), (10.535), above the plane of reference. Again, subtract the fore-sight (^4) from the back-sight (C4), and add the remainder to the height of (C4), and we shall find the height of (^4), (21.534), which should agree with the height found under the heading, total diff. of level; and so on for each time the level is moved. Each back-sight indicates a new position of the level. Plotting the TToek. 29. 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 50 feet. Erect perpendiculars to AE, at each of the points A, B, C, &c., and then set off the distances fi*om A to 2, from 2 to 3, &c., each to represent 50 feet; and through the points 2, 3, 4, and 5, draw parallels to AE. These, by their iutersections with the lines drawn through A, B, C,&;c., will determine the position of the stakes ^j ^? &c. ; and write in red ink on the xolot, 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. TVe may find the points in which the contour lines intersect the lines at right angles, as in Example Eirst. 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 216 ELEMENTS OF SURYEYING. [book ni. where the lines at right angles, cut fences, roads, streams, &c. we can, by joining the points, obtain a plot of the ground. 30. The contour lines may be traced on the ground, aa follows : Set up the level at a, and observe that the back-sight, to the stake, placed at (^3), gave a reading of 11.432. Depress the yane equal to the distance between the horizontal secant planes, that is, 6 feet, which is done by placing it at the reading 5.432. Then direct the rodman, by signals, up or down the hill, till the horizontal hair of the telescope coincides with the horizontal line of the vane. The foot of the staff is then 6 feet above the first point. 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 hair of the tele- scope 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. SEC. II.] TOPOGRAPHICAL SUEVEYCsG. 217 31. 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 haviug a negative reference are distin- guished by placing the minus sign before the number ; thus - ( ). Shadixo Ais-D Delixeatiox 32. Figure 7 (PI. 4), represents a piece of ground sloping towards D, 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 2? above D, 9 B above A 12 d " A 4 k " A 7 q " D, 9, i: " A 13 h " D, 4 s " D, 7 C - A 9 a " A 14 t A 4 / " A 8 n " A 11 a « A 15 9 " D, 5 / « A 8 i " A 12 i^ " A 15 I D,b 1) ^'' D, 9 A above j m " A 12 [>, 20 feet. ^ " A 17 The first horizontal plane is passed 2 feet above A 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 surface, in the curve uv, 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. 33. The manner of shading the map, so as to indicate the hills and slopes, consists in drawing the lines of shading per- 218 ELEAEENTS OF SUEYEYING. [BOOK IH. pendicular to the horizontal curyes, as ah-eady explained. These shading lines are drawn close together, when the slope is abrupt, and further apart, as it grows more gentle. Fig. 7 indicates the method of shading. 34. In topographical surveys, great care should be taken to leave some ^:'e?'???fi??e?2^ marlcs, with their levels written on them 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 ^^lane 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. • 35. Besides representing the contour of the ground, it is often necessary to make a map which shall indicate the culti- vated field, the woodland, the marsh, and the winding river. For this, certain characters, or conventional signs, have been agreed upon, as the representatives of things, and when these are ouce fixed in the mind, they readily suggest the objects for which they stand. Those which are given in Plates 5 and 6, have been adopted by the Engineer Department, and are used ir all plans and maps made by the United States Engineers. It is very desirable that a uniform method of delineation should be adopted, and none would seem to be of higher authority than that established by the Topographical Bureau. It is, therefore, recommended, that the conventional signs given in Plates 5 and 6, be carefully studied and uniformly followed. SEC. m.] RAILWAY CUBYES. 219 SECTION in. RAILWAY CURVES. 36. Tlie preliminary suiTey of a railroad establishes a suc- cession of straight lines, of greater or less length, according to the obstacles 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." 37. We will proceed to describe the method of locating curves, first giving the mathematical principles applicable to the subject. Let AD and DB (next fig.) 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. III., Prob. 14). The angles A and B are right angles (Leg., Bk. III., Prop. 9), consequently the angles C and D, of the quadrilateral ADBC, must be sup- 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. L, Prop. 17); hence, the angle DCA is equal to DCB, and each equal to \TDB. 220 ELEMENTS OF SURVEYING. [book III. Let the radius AO he represented by r; the distance AD by d, and the angle TDB by a. Then will (Trig., Art. 66), d = r tang ^a (1) 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 r; and hence we can determine, at what point on the tangent, laid off from D, 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 cot ^a ...... (2) SEC. III.] EAILWAY CURVES. 221 38. The work of laying out or locating a curTe in the field is somewhat simplified, if the curye have such dimensions that one chain, of 100 feet, haye an arc corresponding to an exact number of degrees. The radii of such curyes are easily calculated. Thus, a circle in which one degree of arc measures one chain, will haye a circumference of 360 chains, or of 36,000 feet, and con- sequently, a radius of o i ^ i ^ = 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 5729.58 ,Q^QQ. . . — - — = 1909.85 feet. o The number of degrees corresponding to one chain, of a railway curve, is called the ^^ degree of curvature" The radius of a one-degree curye is 5729.58 feet ; of a two- degree curye, 2864.79 feet, &c. Eepresenting the degree of curvature by c, we have the . 1 5729.58 .^. formula, r — (3) r being expressed in feet, and c in degrees. Apply the preceding formulas (1), (2), (3), to the following 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 222 ELEMENTS OF SURYEYING. [book m. 2. If the angle a be 30°, and the distance d be 600 feet, what is the radius ? Ans, 2239.2 feet. 3. What is the degree of curvature in the last example ? Formula (3) gives 5729.58 c = = 2.558° = 2° 33' 29". 4. The angle a being 20° 21', what is the value of d for a one-degree curve ? Ans. The Location of the Cukye. 39. 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, wx, &c., represent arcs of one chain each, the angles ACv, vCio, &c., are each equal to the degree of curvature. The angles DAv, vAw, ivAx, are each equal to one-half the degree of curvature. (Leg., Bk. III., Prop. 1^8.) The operations in the field are very simple. The party should consist of a transitman, two chainmen, and an axe- man. SEC. in.] RAILWAY CUBVES. 223 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 ho]ds 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-chainman, 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. 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. It is sometimes necessary to remove the transit from the transit-point to some other point on the curve, before the loca- tion has been completed. In such a case, the direction of the tangent to this new point should be determined. Suppose a; to be a located point on 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 Axf, equal to DAx, (the sum of the deflections made in locating Vj w, and x), — xt is the tangent. By revolving the telescope. 224 ELEMENTS OF SURTEYIXG. [BOOK in. j the tangent is produced to 5, from which deflections may be made as at first. j ]N"OTE 1. — The selection of the radius is goyerned by cir- j cumstances. Curves of the longest possible radius are, in rail- i roads, always the most desirable; but the larger the radius for j 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. i 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. i Tor 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- t tion angle. | LocATio^T OF Curves by the Chai^' aloxe. I 40. It is sometimes convenient to locate a curve without using the transit. In such case, the following method is gen- \ erally employed. i Let A represent the point of tangency, C the centre, and ; V, w, X, located points of the curve, one chain apart. From V, draw vti perpendicular to the tangent, and it wiU be the first offset, which denote by o. Denote the length of the ' SEC. m.] EAILWAY CUEYES. 225 chain by c, and the radius A C, by r. If, now, we suppose AC to be prolonged till it meets the circumference in some point, on the other side of the centre C, and this point then to be joined with r, and vn then drawn parallel to the tangent, we shall have, (Leg., Bk. lY., P. 23), Av^ = 2r . An ; hence, o = 2r- (4) K, now, we prolong Av, till vt = Av, and join t and w, iiu will be the second offset, and will be double vii. For, the triangles in the figure, whose vertices are C, and whose bases are the equal chords Av^ vw, &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, tv20 + wvC + CvA, is also equal to two right-angles. Therefore, tvw \^ 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 wot is isosceles, if vjp be drawn perpen- dicular to the base, it will bisect both the base and the vertical angle, making tjp — piu. But the triangles Auv and vtp are equal : hence, tw = 2vu. Denoting the second offset by o', we have, 0' = (5) 15 226 ELEMENTS OF SURVEYING. [BOOK IH. | i Hence, the practical operation consists in calculating the i first offset, which is perpendicular to the tangent, by. formula (4), then locating v, on this offset, and at a chain's distance from A. Having fixed v, prolong Av, and lay off ^^ equal to one chain. Then the second, and all subsequent offsets, being double the first, we locate w by knowing its distances from v and tj and similarly for all other offsets. 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 and 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? A71S, From tangent, 2.5 ft.; from chord produced, 5 ft. 2. Eind offsets for a one-degree curve. Ans. Tangent, .87 ft.; chord, 1.74 ft. Another chain method, applicable to short curves. 41. Measure off, on the tangent, •any convenient distance, as Aa, and offset, at right angles to this tangent, the distance a v. If we denote the known radius of curva- ture by r, the distance measured on the tangent from A by d, and the offset av, by o, we have the formula, = r - Vr' - d' .... (6) GEC. m.] RAILWAY CURVES. 227 Then, by substituting for d in the formula, different dis- tances from A, the values of the corresponding offsets are found. The formula is easily deduced. Foj', draw the radii Cv and CA, and vn parallel ,to the tangent Aa, Then, nAav is a rect- angle, and in the right-angled triangle Cvn, we have, Cv^ = Cn^ + nv-; or, 'T = {r — oy + d'j from which o = r- Vr'-d' (7) AxoTHEB Method. 42. Still another method may be employed in curves where the centre is in sight from different points along the tangent. It is of use chiefly in staking out circular walks, drives, or lake borders in parks. The measurement is made along the tangent as in the last case, but the offset is measured directly toward the centre by the formula, = Vr' + d' -r (8) This expression is easily verified. examples. 1. Find the offsets to be made at right angles to the tan- gent, at 50, 100, and 150 feet from the tangent point, in a curve of 1000 feet radius. Am. 1.25, 5.02, and 11.32. 2. Find the offsets from tangent toward the centime, at 20, 40, and 60 feet on the tangent ; radius being 200 feet. Ans. .99, 3.96, 8.81 (increasing as dr). Location of Curves by two Trak'sits. 43. The surface over which it is necessary to locate a curve, may be of such a character as to render it impracticable for the 228 ELEMENTS OF SURVEYING. [BOOK HI. chainmen to make their measurements; if, howeyer, 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 prin- ciple : 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 B, and the deflection angle is deter- mined as in the first method. The transitman at A, deflects in the usual way, one deflec- tion angle from the tangent AD. At the same time, the transit- man at B defleci;3 the same angle from the chord BA, or what amounts to the same, he deflects the difierence between this angle and ia, from the tangent BI). The lines of sight of the 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 repetitioii of this process the entire curve is located. SEC. m.] RAILWAY CURVES. 229 LAYIlifG OFF THE OrDINATES. II. The methods described thus far for locating railway curves, applj^ to points 100 feet apart. This is sufficiently 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: ZM 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, VD = HL X LK, (Legen- dre, Bk. IV., Prop. 23, cor. 2). Since IIL = r— 50, and LK = r + 50, the value of VL is readily calculated for known values of the radius. Let N2f be an ordinate, at any distance from VL, say 10 feet. Then, NM' = m£xMK; whence, Nj\P ={r- 40) (r + 40). Having determined mf, subtract VL from it, and we have A7, one of the ordinates required. In this manner, by calculating the full ordinate to the 230 ELEMENTS OF SURVEYING. [book in. diameter, and subtracting VL, any desired number of offsets are determined for the half chain VF, For FW, the ordinates haye the same length, but are located in the inverse order. The middle ordinate, FF, is found by subtracting VL from the radius. EXAMPLE. Determined the ordinates 10 feet apart on a 100-foot chord, for a two-degree curve. Kadius, 2864.79 feet. Ans. At 10 feet . . At 20 " . . At 30 " . . At 40 " . . Middle ordinate . = .15 feet. = •28 (( = .36 (( = .42 a — .43 (( SECTION lY. SECTION LEVELLING. 45. In the surveys which precede the construction of roads, railroads, 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. SEC. IV.] SECTION LEVELLING. 231 46. 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. 47. As the survey progresses, fixed points of reference, called henclies, 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. 48. Temporary benches, employed merely while changing the position of the levelling instrument, are called turning 2Jomts. 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. l^OTE. — The order of the surveys, on a line of road or canal, after the route has been determined by reconnoissance, is 1st. The Transit survey, establishing the centre line of the work. 2d. The section Level, or the measurement of the profile of the centre line. 3d. The cross-section worh, or the measurements, with the level, across the centre line, to the full width of the road, for the purpose of determining the intersection of the slo2oes with the natural surface; and also, to afibrd data for the estimate of the amount of earth to be removed, or filled. 49. The following example will exhibit the method of re- cording the notes of a section level. The datum line is assumed 232 ELEMENTS OF SURYETING. [BOOK m. to be thirty feet below the first bench. When the field-book is of the ordinary pocket size, the seven columns of notes will generally occupy two opposite pages; the first five being upon the left-hand page. Beml 3/. 657 ft above Datum Dist. + Sight. Ht. of Ins. — Sight. Surface Height. Grade Height. Remarks. Bench. 1 1.637 31.637 2.1 1.8 30. . 29.5 29.8 Bench on top offence -post 30 ft. north of stake. -1 60 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 5^° 10.4 11.7 6 9.8 12.3 7 10.6 • 11.5 The bench having been selected and marked, its location is described in the column of remarks. The level is adjusted in some convenient place in the vicinity, 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 SEC. IT.] SECTION LEVELLING. 233 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 dififerent points 0, 1, 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 mimis 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. Readings are taken at intermediate points (as at 160 feet in the above example) when there are abrupt changes in the inclination of the surface. 50. 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-point,^^ and marked T. P. in the distance column; otherwise it is called a Bench, and its location is described in the column of remarks. A turning-point is taken between stations 3 and 4, in the above example. The reading of the rod, upon it, is 11.413. 234 ELEMENTS OF SURVEYING. [BOOK IH. This is recorded in the — sight column, and the height of the point is at once found (by subtracting this reading from Height of Instrument), and recorded in the column of "Heights." The level is next carried forward to a new position, ad- justed, and directed again upon the rod at the turning-point. The reading is taken to thousandths. This, when added to the height of the turning-point, evidently gives the height of instru- ment 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. The rules for taking and recording field-notes in section levelling are as follows: I. The " distances'^ recorded in the first column are the hori- zontal measurements, in chains, from the leginning of the survey to the points luhose heights are to he determined. The 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 tlie level, is upon a lench or turning-point, and is . « " + sight :" all other readings are " — sights." III. Tlie -^ sight, added to the height of the point upon which the reading was taken, gives the ''Height of Instrument." SEC. IV.] SECTION LEVELLING. 235 IV. The — sights taken, at any position of the level, sub- tracted from the ^^ Height of InstrumenV^ for that position, give the corresponding ^^ Surf ace Heights" V. All the + SIGHT readings, and the last — sight of each set, being upon benches or turning-points, are talcen to thousandths of a foot. The remaining " — sights" are talcen to tenths only, ISToTE. — 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. 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 importance 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. In conducting a section level through a rocky district, turn- ing-points in abundance are found at hand, and cause no delay in their preparation, whereas a bench in the same section, re- quires marking and locating. In levelling through flat and level sections of country, although the engineer can get "sights" for long distances, a 236 ELEMENTS OF SUKYEYING. [BOOK HI. proioer 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. 51. 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 identi- fication. 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 levelling, is accomplished by taking the heights of the "benches" only, and is called a "cheek level." Deatving the Peofile. 52. 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 datum line, is first drawn, and the distances from the first column of notes are laid off SEC. IT.] SECTION LEVELLING. 237 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. 53. 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 op the Gkade. 54. 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." The position and inclination of grade-lines are influenced by a variety of circumstances: 238 ELEMENTS OF SURYEYING. [BOOK HI. 1st. The chanictcr of the work. A street admits of an in- clination of fiye, or eyen 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 either 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 lieiglit. The following 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- tances" and "' surface heights," are the same as in the preceding problem. SEC. lY.] SECTION LEVELLING. 239 "We will suppose it is required to establish, in tlie following profile., a grade-line wliose inclination shall not exceed 3 in 100 ; the grade to begin at station 0, at the surface. 260 . Dist. .s. H. of Ins. — S. H.ofSur. H.ofGr. Cut. Fill. RE3I. 29.5 29.5 1 29.8 26.8 3.0 160 30.7 25.2 5.5 2 28.2 24.1 4.1 at 2" 3 20.8 21.4 0.5 T.P. 4 16.3 18.7 2.4 5 13.1 16.0 2.9 5" 11.7 14.7 3.0 6 12.3 13.3 1.0 7 11.5 at 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 yerticals, and draw a line through the points of measurement. For in- stance : a grade of 3 in 100, running downward from station 0, 240 ELEMENTS OF SURVEYING. [BOOK in. 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 w^ould, 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 pro- cess 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 oppo- site 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 heights." Where the surface is higher than the grade-line, the construction requires a "cutting;" when the established grade- * The advantage of a thread over a ruler lie? in thfe 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, seem? to accomplish the desired purpose. The line being drawn, the '"cut" and "fiU" areas are measured, to deter- mine 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 the next section. SEC. IV.] SECTION LEYELLING. 241 line is higlier 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 232, 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." These transferred notes tve recorded in ink, and reserved for use in mapping and computations. ♦ 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 Burface-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 is easily suprjlied, by determining the height of grade in the usual way at station 7. 16 242 ELEMENTS OF SURVEYING. [book m. 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 esti- mated. When the route is finally selected, and the section levels satisfactorily completed, the exact width of the earthwork, both 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 the next section. Before closing the subject of section levelling, we will con- sider the profile represented in the figure, and the set of field- 70.942 notes appended, which are only partially completed, and which will afibrd some examples for practice. SEC. IT.] SECTION LEVELLING. 243 Dist. + s. Ht. of Ins. -S. Surface Heights. Eemarks. 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 6" 6.4 7 2.1 1. What is the of the level ? 2. What is the 3. What is the level ? 4. What is the 5. What is the level ? 6. What is the 9. (( n 10. (( a 11. Write the (( and 6". " Height of Instrument" for the first position A71S. 31.032. height of the first T. P.? A7is. 28.686. " Ht. of Inst." for the second position of the Ans, 40.501. height of the second T. P. ? Ans. 38.183. " Ht. of Inst." for the third position of the Ans. 49.125. Height of Surface at ? at 3 ? at 3^° ? at 5 ? " " at 6 ? at 7 ? Surface Heights" for the distances 1, 2, 4, 71S. 27.8. 22.1. 20.7. 26.2. 39.4. 47.0. 2M ELEMENTS OP SURYEYING. [book III. Ceoss-Section" Levellin^g. 55. All earthworks, whether excavatiou or embankment, un- less held in position by retaining walls, require to be constructed 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.* Slopes are expressed mathematically by the ratio of their horizontal to their vertical dimensions, and which is called the ratio of slope. In the diagram, which represents a road-cutting, the ratio of ES to AE, or of FS' to BF, 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. * This slope is determined, experimentally, "by dryino; 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 he liable to assume under the action of atmospheric influences. The angle with the horizontal plane will be somewhere between 25° and 45°. SEC. IV.] CROSS-SECTION LEVELLING. 245 A elope of H 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. SETTii^^G Slope Stakes. 56. 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 StaJces, 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 Levelling," or " Cross-Section Work." 57. A party of five may be usefully employed in setting Slope Stakes; viz., a leveller, 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 leveller himself reading the rod in the act of sighting. The field-book is ruled as shown below. Diet. Left. Centre cuttings. Eight. 246 ELEMENTS OF SURVEYING. [book m. 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. The examples following will illustrate the method of meas- uring the section and recording the notes. Let figure represent a section across a road excavation. AB being the bed of the road, and 8CS' the line of the natural surface. The road-bed is supposed to be 16 feet w^de, the centre cutting 12.4 feet, and the ratio of slope 1^ to 1. The level being set up and adjusted in a convenient place, the rod is first held by the centre stake at C, and a reading taken. 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 reserved page of the note-book, or on a spare slip of paper; neither the height SEC. IT.] CROSS-SECTION LEYELLING. 247 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 the "height of instrument," the remainders will be the heights of these points above the datum line AB. These heights are technically called cuttings, although in the case of ST and S' V, no actual excava- tion is proposed. The reading at B is supposed to be 5.5. The cut is therefore 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, -^^, in the column marked "left." The points A and B, of the cross section are appropriately termed the angles, and as the points F 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 F, and the remaining distances to the left are measured from this point. A change in the surface-line at H requires notice. The reading of the rod 6.2 indicates a cut of 11.6. This, with the distance from F, 10 feet, is duly recorded. 248 ELEMENTS OF SURYEYING. [BOOK IH. There being no other material change in the surface line beyond //, there remains to be determined on this side only the intersection S, 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) r^ US. 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 mnltiplied 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 if;f .* In proceeding from the centre to the right, we find a point 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. CK is 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-line from F were level, the distance FS' would be 12 X li = 18 feet ; but as the ground descends, the distance is less. A trial at 11 ft. with a reading of the rod of 10.4, indicates a cut of 7.4. This multiplied by 1^ gives 11.1, which is very nearly right: -j^^y is therefore the record for the location of S'. * The readings and distances in this example have heen made to correspond to a rise of one foot in five from H to S. The exact record for S is 2 u:i:f • SEC. B\] CROSS-SECTION LEVELLING. 249 The completed notes for this cross section are as follows: Dist. Left. Centre Cut. Eight. 13.8 20.7 11.6 10 12.3 A" 12.4 12.5 3 12 A 7.4 11.1 58. We wiU now give a similar example, illustrating the method of staking out embankments. us Dist. Left. Centre Cut. Pdght. - 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. 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 SS', must be subtracted from this "height of instrument," as in the pre- 250 ELEMENTS OF SURVEYING. [BOOK IH. ceding example. The several remainders are the corresponding " cuttings." The reading • at angle stake E, 8.3, subtracted from — 4.5, gives — 12.8, for the cut. For the slope stake S, 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 X5 8 ^ may be made without involving an error of more than /vd.7 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 8', 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 ;S", the rod reading at S' would not vary a tenth if moved from 12 feet to 12.6. The ^.4 record, therefore, for S' is 12.6 50. The rules for conducting and recording cross-section work, whether for excavation or embankment, are as follows: I. Prepare the field-looh lij ruling columns for Distances * 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 li to — 1. SEC. IT. J CROSS-SECTION LEVELLING. 251 and Centre Cuttings, leaving u'ider spaces on either side of the latter column for the record of the various ineasurements to the left and right of the centre stake. Transfer from the section- level 7iotes the distances and corresponding cut or Jill, for each stake of that survey. Filling in the cross-section notes is desig- nated as minus cutting. II. Having set the level in convenient proximity to a pro- posed cross section, take a reading of the rod at the centre stake. Add this reading to the centre cutting, (regarding the sign of the latter), to oltain the *' height of instrument.'^ III. Lay off half the width of the road-led each side of the centre, and mark the distances, temporarily, luith stakes. TJiese are the angle stakes. IT. 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 the height of iiistrument ; the remainder is the cutting, or vertical distance of the point measured, from the proposed road-bed. Y. Record each cutting, together ivith its horizontal distance from the nearest angle stake, in the form of a fraction express- ing the ratio of the distance to the cutting. Each fraction being recorded in its proper column either ^'righf' or "left;' of the centre. Points between the centre and angle stake, are located by measuremetits from the centre. VI. To find the position of the slope stake : 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 252 ELEMENTS OF SURYEYING. [book m. trial tintil the ratio of the distance to the cut expresses the ' ratio of slope. \ 1 60. The cutting at the angle stake is, in cases of a tolerably ; uniform surface, a good guide to the distance to the slope i 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 , 1 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 1|- times 2 feet. ] When the surface descends, the estimated distance out, for | a level surface, should in like manner be diminished. In em- I bankments the conditions are reversed; the steeper the rise, the 'i shorter the distance out. j I 61. The following examples will serve to elucidate the sub- ! ject still further. Dist. Left. Centre Cut. Eight. - 4. - 4.5 6. A -4.9 -6.1 A -6.4 4 - 7.4 11.1 The diagram represents an embankment cross section, in which, by reason of the small depth of filling, the height of instrument is a positive quantity. SEC. IV.] CROSS-SECTION LEYELIilNG. 253 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 in- strument. . 62. In the example of the following diagram, the cross section is partly in excavation and partly in embankment. 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 8, 5.4. The point, K, is efjsily 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 ;iS^', 15.8. From these readings the cuttings may be found, and the notes completed as below. Dist. Left. Centre Cut. Right. 4.9 4.2 4.0 9.8 A 3.8 2.4 - 2.9 - 5.5 .4 A 11 . 63. 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 IJ to 1 ; height of instrument, 14.2. In seeking for the position of the slope-stake S', a distance 23 254 ELEMENTS OF SURVEYING. [book in. out of 13 feet is tried; the reading of the rod at the trial point is 4.8. 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 14.4 4.9 What is the true cut and distance at S' ? Ans, Find the position of ^S'. Ans. 7.3 Note 1. — It sometimes happens, in yery hilly sections, that it is impracticable to sight to all the necessary points of a single cross section from one position of the leyel. 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 levelling; 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. sec. iv.] cross-section levelling. 255 Computation of Eaethwork. 64. 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 prismoids. 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. = ^ (B + B' + 4J/) B and B' representing the areas of the end sections of the prismoid, 31 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. ^Ye 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 diagi'am, 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. 256 ELEMENTS OF SUKYEYING. [book III. The area of B is thus found to be 104.8 sq. feet, and of B', 116 sq. feet. 7.2 \2.6 M 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- tance from the angle to slope stake, on the left, 9.9 ft, f — '-- J ; the cutting on the extreme left is 6.6 ft, f -^^—^ — j &c. The area of 31 is 111.3 sq. ft Vol. of the prismoid = ^ X 100 (104.8 + 116 + 445.2), = 11100 cubic feet 65. Two other methods of computation are of frequent use among engineers. They are less laborious than the prismoidal rule, but they are also less accurate. SEC. lY.] CROSS-SECTION LEVELLING. 257 The first is known as the "Arithmetical Average" method. The rule is : Multiply half tlie sum of the end areas hj the length of the jprismoid. Applied to the example just solved, we should have, Tol. = 100 X ^°^-^ + "'^- = 11040 cu. ft. 66. The other method is called the " Mean Average" method. The rule is expressed thus : Add together the two end areas and their geometrical mean proportional. Multiply this sum hy one-third of the lengtli of tlie prismoid. This method, applied to the example, gives for the volume: vol. = -i X 100 (104.8 + 116 +\/l04.8x 116) o = 11035 cubic feet. Note. — "When the difference between the end areas is con- siderable, the mean aVerage method gives better results than the method of arithmetical average. The last mentioned rule has been largely employed by the engineers of the public works of the State of ISTew York. Tables based upon the prismoidal formula, or modifications of it, are much used by engineers in earthwork computations. 67. In applying the prismoidal formula to an example in which 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. 17 258 ELEMENTS OF SURVEYING. [book III. We will illustrate this by a final example of earthwork : rep- resenting tlie sections by the field-notes only. 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 1 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 ■sions. 2 13.1 feet. 11.5 On the right, the average of the angle cuttings gives — r^ ; A. for the next measurement, both cut and distance must be aver- :aged 'y it is, i.X(10 + 8) = 9 _9_ iX (8 + 12) = 10 ^^' 10' The last term in the upper section must be averaged with the last in the lower, thus: iX ( 8+ 8.4)= 8.2 8.2 or • • iX (12 + 12.6) = 12.3 ' 12.3 -I n 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 m iX( 2+ )= 1 ^"^^ 1 SEC. v.] MINING SURYEYING. 259 14 At the angle, in like manner, we have -^ ; and finally at 14 4 the slope stake ^. The complete dimensions being 21.6 d. 1 ^9. I 13.1 14.4 14 13.2 21.6 A 1 11.5 9 8.2 A 10 12.3* The area of the upper section, after subtracting the tri- angles, 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 Yolume of the prismoid is 1 X 60(543.47 + 325.44 + 4 X 429.8) =25881.1 cubic feet, or 958.56 cubic yards. SECTION V. MINING SURVEYING. Definitions and general Notions. 68. Mining Surveying comprises all the operations neces- sary to determine the relative positions of the parts of a mine with respect to each other, and also, with respect to the surface of the earth. 69. 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, both in the instruments employed and in the manner of using them. Stations are designated by lamps instead of flags, and lamp- 260 ELEMENTS OF SUKYEYING. [BOOK IH. stands instead of flag-rods ; 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. Lines are measured along the slope, instead of on the horizontal, the chainmen being guided by lamps instead of rods. Angles are measured by instruments specially devised for underground use ; the comjmss, when used, is generally of a widely different form from the ordinary sur- vey or's compass; the theodolite, which is the principal angular instrument employed, differs from the ordinary theodolite 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 hairs. These modifica- tions will be more fully described in a subsequent article. Traversing. 70. Teayersixg is the operation of running a zig-zag 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 tliird, is a vertical plane through the origin, and perpendicular to the other two. 71. 'WoRKiis'G, or KEDUCii^G THE TRAYEESE, is the operation of finding the length and direction of a sing-le line, equivalent to the zig-zag, that is, starting from, and terminating at, ihe same points. Such a line is called the resultant of the traverse. The zig-zag line is run along the subterranean openings of a mine. For the sake of uniformity, such openings, when ver- tical, will be called sliafts, and when not vertical, galleries. SEC. v.] MINING SUEVEYING. 261 Method of traversing with the Compass and Semicircle. 72. In ruuning a subterranean traverse with, the magnetic needle, a form of compass is used, called the Jliner's Compass. It consists of a compass-box like that of the Surveyor's Com- pass, except that the graduation extends from 0° around, by the left, to 36-0°. This box is weighted at the bottom, and mounted on a universal joint, like the Mariner's Compass. The ring that supports the universal joint is provided Tvith. hooks for suspending it from a wire stretched along the gallery. When so suspended, the box assumes a horizontal position, the diameter through the point of the graduated circle falling immediately under the wire ; if the compass be so suspended that the end of this diameter points backward, and the 180° end in the direction the traverse is being run, the reading at the north end of the needle is the angular distance from the north point, around by the east to the direction of the course : this angle is called the Azimuth of the Course. Miner's Semicircle. 73. The slope of the Course is measured by the Minei^s Se7nicircle. This is a graduated semicircle, with hooks, for sus- pending it from a stretched wire, and a plumb-line attached at the centre of the circle. When suspended, the plane of the semicircle is vertical, and its diameter is parallel to the wire; the point of the graduation is at the extremity of the radius which is perpendicular to the wire, and, as the divisions are numbered both ways, the reading of the plumb-line gives the slope. As it is impossible to stretch a wire so that it shall be perfectly straight, the slope measured at different points will not be the same; a fair result will be obtained by measuring the- slope at each end of the wire, and taking the average as the true slope. 262 ELEMENTS OF SUEYEYING. [book ni. 74. To run a traverse, with the compass and semicircle, a copper wire is stretched from the place of beginning, say at the bottom of the shaft, to some convenient point of the gallery, and both ends made fast to iron hooks, driven into the walls of the gallery; the compass is suspended from this, at some point near its middle, and after it comes to rest, the azimuth of the course is read off and recorded; the semicircle is then suspended, first, near one end of the wire and then near the other, and the average reading is recorded, care being taken to note whether the slope is in elevation or dejoression ; the length of the ware, from hook to hook, is then measured and entered in the field-book. If necessary, measurements are made to determine the cross section of the gallery, at any desired point of the course, and the results entered in the column of remarks. The wire is then detached from the hooks and car- ried forward along the gallery, one end being made fast at the extremity of the first course, and the other at some con- venient point, generally on the opposite side of the gallery; the same measurements are made as before and recorded under the proper headings, and so on, to the end of the traverse. The method of entering the field-notes, is shown in the following Field Book. Slope. Azimuth. Length in feet. Eemakks. Course. Elevation. Depression. 1 2 3 4 5 6 2° 30' 3° 15' 3° 30' 4° 15' 203° 199° 15' 251° 30' 300° 00' 269° 15' 272° 45' 307 402 240 367 409 200 Stat. 1, at iron peg, centre of shaft. At iron peg. 1° 30' 2° 00' SEC. v.] MINING SURVEYING. 263 Method of Traversing with the Theodolite. 75. In traversing with the theodolite, it becomes necessary to illuminate the cross hairs of the diaphragm. This is accom- plished by a diagonal mirror, placed in front of the object-lens of the telescope. The mirror is supported by a stem projecting from a ring that surrounds the tube of the telescope, and has an opening in the prolongation of the tube, so as not to in- tercept the rays of light from the object. A lamp, placed on one side of the theodolite, illuminates the remainder of the mirror, and, if properly situated, a portion of its light is reflected into the tube of the telescope and thrown on the cross hairs. Sometimes the theodolite is famished with three tripods exactly alike, upon each of which the instrument is mounted in turn, the other two serving to support the guiding lamps, which, are so constructed that the flame of the lamp shall be as far above the top of the tripod as the axis of the vertical limb. Sometimes a single tripod is used, in which case the guiding lamps are placed on stands similar to the tripods, but with sliding pieces carrying the lamps, and fixed in position by clamp-screws. By this arrangement, the height of the lamp may be made equal to that of the theodolite. The lamps are protected by glass shades, of any color that may be desired. To run a Traverse with a Theodolite having three Tripods. 76. Select a place for the first station, say at the foot of the shaft, and mark it permanently; set up a tripod over it, and place a station lamp on it : select a second station at some suitable point, generally at a turn in the gallery, set up a tripod over it, and on it put the theodolite : at fhe third station, still further along the gallery, set up the third tripod, and on it put a station lamp. The theodolite being levelled and the horizontal limb clamped, direct the telescope to the lamp at the first station, and read both limbs; the reading of 264: ELEMENTS OF SURVEYING. [BOOK III. the Tertical limb will be the slope of the first course, in eleva- tio7i, if tlie telescope point downward — in de^jression, if it point upward; then nnclamp the vernier plate, and direct the telescope to the lamp at the third station, reading both limbs as before ; the reading of the yertical limb will be the slope of the second course, in elevation, if the telescope point upward — in depressioUy if it point downward ; subtract the first reading on the horizontal limb from the second (the latter increased by 360° if the of the vernier has passed the of the limb), and the difference will be the horizontal angle between the first and second courses. The distance from the first to the second station is measured by chainiDg along the slope, the allignment being made by means of lamps, and the points corresponding to the ends of chains marked, by chalk-lines on the rock, or in some other convenient manner. The theodolite is then transferred to the third tripod, its place being taken by a station lamp, and the first tripod and lamp are carried along the gallery to mark the place of the fourth station; the same observations and measurements are made as before, and so on to the end of the traverse. The observations are recorded as shown in the fol- lowing Field Book. I Sta- tion. Slope. Reading on hor. linib. Horizontal Angle. Dist. in feet. Kemarks. Elevation. Depression. Back. Fore. 1 2 2° 30' 3° 15' 107° 20' 283° 35' 176° 15' 307 Station 1, at iron peg, centre of shaft. 3 3° 30' 271° 15' 143° 30' 232° 15' 402 4 4° 15' 109° 8' 337° 38' 228° 30' 240 5 1^ 80' 225° 15' 14° 30' 149° 15' 367 6 2^00' 15° 20' 198° 50' 183° 30' 409 7 . 200 At iron peg. When the bearing of the first course is known, the azimuth of that, and of the following courses, may bo found, and the SEC. v.] MINING SURVEYING. 265 field-book, thus reduced to the form given in Art. 72. Some- times, the first course is assumed as a line of departure, and the azimuths of the following courses are calculated with refer- ence to it. Modes of connecting with Surface Survey. 77. In order to determine the bearing of the first course, and also to be able to trace out the plan of the mine on the ground, it is often desirable to connect the underground traverse with the surface survey. There are two principal methods of making the connection. ..^^^^^^ FIRST METHOD. A straight-edge, AB, is mounted on two trestles, and from it are suspended two plumb lines, B 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 the bottom of the shaft; the theodolite being at the 2GG ELEMENTS OF SURVEYING. [book in. 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 oyer 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. SECON-D METHOD. Let the theodolite, provided with a diagonal eye-piece, be planted over the station JD, at the foot of the shaft, and after being levelled, let it be directed on the station K. Then, with- out changing the plane of vision, let the theodolite be directed to the top of the shaft, and let an assistant plant two flag rods, one at ^ and the other at B, both in the plane of vision, and let the line AB be prolonged to L and M, as before. Tlie line LM 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. ^ SEC. Y.] MINING SURVEYING. 267 Reducing the Traverse 78. We may now find the azimuths of the yarious courses. Suppose the bearing of the first course to be S. 23° W. ; then will its azimuth be equal to 180^+23°, or, 203°; the azimuths of each of the following courses, in order, may be found by the following EuLE. — Suhtract 180° from the liorizontal angle, ohserviiig that the remainder may le either + or — ; add tlie result to the preceding azimuth; if the sum is negative, add 360° to it; if positive and greater than 360°, subtract 360° from it; the result thus found is the azimuth required. Thus, in the given example, the horizontal angle between the first and second courses- is 176° 15'; this diminished by 180°, gives — 3° 45', which added to the preceding azimuth, 203°, gives 199° 15' for the azimuth of the second course. In like manner, we find the azimuth of the third course, 251° 30', of the fourth, 300°, of the fifth, 269° 15', and of the sixth, 272° 45'. These are the same as are given in the example, in Article 72. Having found the data, as given in Article 72, we proceed to reduce the traverse as shown in the following Office Foe:m:. Course. Slope in ft. Bearing. Reduced lenjith of course. Latitude. Departure. Eleva- tion. Depres- sion. X. s. E. W. 1 2 3 4 5 6 10.7 7. 13.4 22.8 14.7 27.2 s. 23° w. s. \W w. s. 71i° w. N. 60° w. s. 89r w. X. 87i° w. 306.7 401.4 239.6 366. 408.9 199.9 183.0 9.6 282.3 379.0 76.0 5.3 .... .... 119.8 132.0 227.2 317.0 408.8 199.7 17.7 78.1 17.7 192.6 742.6 192.6 .... 1404.5 0.0 Result- ! ant. . . • . 60.4 s. 68i' w 1508.3 .... 550.0 .... 14045 268 ELEMENTS OF SUETETING. [BOOK m. The form requires but little explanation. From the azi- muth, that is, the angle of the course from the north point around by the east, the bearing is easily deduced. The length of the course on the slope, multiplied by the sine of the slope, gives the distance the course rises or falls, in feet, and the length multiplied by the cosine of the slope, 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 course being found, the latitudes and departures of the coui'ses are found in the usual manner. In the example given, the resultant course descends 60.4 feet, its southing is 550 feet, and its westing 140-1.5 ; hence, its bearing and length on the horizontal, may easily be found by known methods. If the depth of the shaft is known, this, added to the de- pression of the resultant, in feet, gives the distance of the last station below the horizontal plane through the mouth of the shaft. Method of Plotting the Traverse on the Surface. 79. To plot the traverse on the surface of the earth, we lay down the direction of the first course, as already shown: on this, measure off, in the usual manner, the reduced length of the first course, and mark the end of this distance, by a peg ; plant the compass over the peg, set it so that the reading of the needle is equal to the bearing of the second course, and in this direction measure off the reduced value of the second course, and so on to the end. Then will the several pegs be exactly over the corresponding stations in the mine. _ 80. Suppose it were required to sink a shaft, so as to strike the gallery at station 6, in the example given in Article 76. It has been shown how to locate the point exactly over the station, (Art. 75). Let a line of levels be run from the mouth SEC. Y.] MIKING SURVEYING. 269 of the first shaft, to this point, and find the difi'erence of level corresponding to the two points. This with the depth of the sta- tion, in the mine, below the mouth of the shaft, will make known the depth of the shaft. This is a problem that frequently oc- curs in mining. A similar problem often arises in railroad tunnelling. For example, if a tunnel is to be driven through a hill, it is often desirable to sink shafts, intermediate to the end headings, so as to strike the tunnel. Oftentimes, these shafts are used as starting-points for portions of the tunnel which are intended to meet the parts that are being opened from the headings. Method of Plotting the Traverse on Paper. 81. 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 gallery, on each side, may be plotted, and thence a complete plan of the mine may be constructed. We next j^lot 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 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 270 ELEMENTS OP SURVEYING. [BOOK HI. 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 trayerse to the floor and roof of the gallery, at difierent points, enable us to com- plete the profile. A TABLE LOGARITHMS OE NUMBERS FROM 1 TO 10,000. N. Log. N. Log. N. Log. N. Log. I 0-000000 26 1-414973 5i 1.707570 76 I -880814 2 o-3oio3o 27 i-43i364 52 I -716003 77 1-886491 3 0-477I2I 28 1-447158 53 1-724276 78 1-892095 4 0- 602060 29 1-462398 54 1-732394 79 1-897627 5 0-698970 30 1-477121 55 1-740363 80 1-903090 6 o-778i5i 3i I -491362 56 1.748188 81 1-908485 I 0-845098 32 i-5o5i5o u 1-755875 82 i-9i38i4 0-903090 33 i-5i85i4 1-763428 83 1-919078 9 0.954243 34 1-53x479 1-544068 59 1-770852 84 1-924279 10 I • 000000 35 60 1-778151 85 1-929419 II 1-041393 36 i-5563o3 61 1.785330 86 1-934498 12 1-079181 11 1-568202 62 1-792392 87 1-939519 1-944483 i3 I- 1 13943 1-579784 63 imi 88 14 1-146128 39 1-591065 64 89 1-949390 i5 1-176091 40 I -602060 65 1-812913 90 1-954243 i6 i-ao4i2o 41 1-612784 66 1.819544 91 1-959041 17 1-230440 1-255273 42 1-623249 1-633468 67 1-826075 92 1.963788 i8 43 68 1-832509 93 1.968483 ^9 1-278754 44 1-643453 69 1-838849 1-845098 94 1.973128 20 I -301030 45 1-653213 70 95 1.977724 21 I-3222I9 46 1-662758 71 I-85I258 96 1.982271 22 1-342423 47 1-672098 1-681241 72 1-857333 97 1-986772 23 I. 361728 48 73 1-863323 98 1.991226 24 I-3802II 49 I -690196 74 1-869232 1-875061 99 1.995635 25 1-397940 5o 1-698970 75 100 2-000000 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 in stead 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 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. I 2 3 '^ ' 6 7 8 9 1 D. 100 000000 0434 0868 i3oi 1734 2166 2598 3029 3461 3891 ; 43a 101 4321 4751 5i8i 5609 6o38 6466 6894 732T 7748 8174 428 102 8600 9026 3259 9431 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 0116 3252 til 9947 •36 1 •775 416 io5 021189 i6o3 2016 2428 2841 4075 4486 4896 412 io6 53o6 5715 6125 6533 6942 7350 11^1 8164 8571 8978 408 107 9384 033424 0789 3826 •195 •600 1004 1408 1812 2216 2619 302I 404 108 4227 4628 5029 543o 5830 623o 6629 7028 400 109 7426 7825 8223 8620 9017 9414 981 1 •207 •602 •998 396 no 041393 1787 2182 2576 lU 3362 3755 4148 4540 4932 393 389 III 5323 57.4 6io5 6495 7275 7664 8o53 8442 883o 112 9218 053078 9606 3463 9993 •38o •qbb ii53 1 538 1924 23o9 2694 386 ii3 3846 423o 46i3 4996 5378 5760 6142 6524 382 114 6905 7286 7666 8046 8426 88o5 9i85 9563 3333 9942 •320 379 ii5 060698 4458 1075 1452 1829 2206 2582 2958 3709 4o83 376 116 4832 5206 5580 5953 6326 6699 7071 7443 78.5 372 \\l 8186 8557 8928 9298 f^l ••38 •407 •776 1.45 i5i4 369 071882 225o 2617 2985 3718 4o85 445 1 4816 5i82 366 L19 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 I/O 079181 9543 9904 •266 •626 •987 4576 1347 1707 2067 2426 36o 121 082785 3i44 35o3 386i 4219 4934 5291 5647 9198 6004 357 122 636o 6716 7071 7426 7781 8i36 8490 8845 f^\ 355 123 9905 093422 •258 •611 •963 i3i5 1667 2018 2370 2721 35i 124 3772 4t22 4471 4820 5169 55i8 5866 62i5 bbb-} 349 125 6910 7257 7604 7951 8298 8644 8990 9335 9681 ••26 346 126 100371 0715 1059 i4o3 1747 2091 2434 2777 3.19 3462 343 \ll 38o4 4146 4487 4828 5169 55io 585 1 6191 653 1 6871. 340 7210 7549 7888 8227 8565 8903 9241 9579 9916 •253^ 338 129 1 10590 0926 1263 1599 1934 2270 2605 2940 3275 3609* 335 i3o II 3943 4277 461 1 4944 5278 56ii 5943 6276 6608 6940 333 i3i 7271 7603 7934 8265 8595 8926 9256 9586 99'5 •245 33o i3i 1205-^4 0903 I23l i56o 1888 2216 2544 2871 3198 3525 328 i33 3852 4178 45o4 483o 5i56 5481 58o6 6i3i 6456 6781 325 I34 7105 7429 7753 8076 8399 8722 9045 9368 9690 ••12 323 i35 i3o334 0655 0977 1298 1619 1939 5i33 2260 258o 2900 3219 321 i36 3539 3858 4177 4496 4814 545 1 5769 6086 6403 3i8 III 6721 7037 7354 7671 7987 83o3 8618 8934 9249 9564 3i5 9879 i43oiD •194 •5o8 •822 ii36 i45o 1763 2076 2389 2702 3i4 139 3327 3639 3951 4263 4574 4885 5196 5507 58i8 3ii 140 146128 6438 6748 7o58 7367 7676 7985 8294 86o3 891 1 309 141 1 5228? 9527 9835 •142 •449 •756 io63 1370 1676 19S2 307 142 2594 2900 32o5 35,0 38 1 5 4120 4424 4728 5o32 3o5 143 5336 5640 5943 6246 6549 6852 7154 7457 7739 8061 3o3 144 8362 8664 8q65 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 297 147 7317 7613 lit 82o3 8497 8792 9086 9380 9674 9968 295 148 170262 o555 1141 1434 1726 2019 23ll 26o3 2A95 293 149 3i86 3478 3769 4060 435i 4641 4932 5222 55x2 5802 291 i5o 176091 638i 6670 6959 9S39 7248 7536 7825 8ii3 8401 8689 289 i5i 8977 181844 9264 9552 •126 •4i3 •699 •985 1272 i558 287 l52 2129 24i5 2700 2985 5825 3270 3555 3839 4123 4407 285 1 53 4691 4970 5259 5542 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 2010 2289 2567 2846 'il 1 56 3125 3403 3959 4237 45i4 4792 5069 5346 5623 i57 5899 6176 6453 6729 7005 7281 7536 7832 ^'?'' 83.S2 276 i58 8607 8932 1670 9206 9481 9755 ••29 •3o3 •577 •85o 1124 274 159 N. 201397 1943 2216 2488 2761 3o33 33o5 3577 3848 9 272 I 2 3 ^ 5 6 7 8 TABLE OF LOGARITHMS FRO:m: 1 TO 10,000. N. j I 2 1 3 4 5 1 6 j 7 1 8 9 D. 271 l6o |2o4i2o 4391 4663 4934 5204 5475 ' 5746 6016 6286 6556 161 6826 7096 7365 7634 7904 8173 8441 8710 8970 9247 269 162 95 1 5 9783 ••5 1 •3.9 •586 •853 1 121 i388 i654 1921 267 163 212188 2454 2720 2986 3252 35 1 8 3783 j 4049 43i4 4579 166 164 4844 5 1 09 5373 5638 5902 6166 643o 6694 6957 i 7221 264! l65 7484 7747 8010 8273 8536 8798 9060 9823 9585 ! 9846 262 166 i >oio8 0370 o63i 0892 ii53 1414 1675 1986 2196 2456 i 261 167 2716 2976 3236 3496 3755 401 5 1 4274 4533 4792 5o5h 25a 168 5309 5568 5826 6084 6342 6600 6858 7115 7872 7680 i 25a 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 •193 256 170 330449 0704 0960 12l5 1470 1724 1979 2234 2488 2742 254 171 2996 325o 35o4 3757 40 1 1 4264 4517 4770 5o23 5276 253 172 5528 5781 6o33 6285 6537 67S9 7041 7292 7544 7795 252 173 8046 8297 8548 8799 9049 9299 9350 9800 ••30 •3oo ' 25o 174 240549 0799 1048 1297 1 546 1793 2044 2293 2541 2790 249 175 3o3S 3286 3534 3782 4o3o 4277 4525 4772 5019 5266 248 176- 55i3 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 \]l 7973 8219 8464 8709 8934 1395 9198 9443 9687 ttl •176 245 25o420 0664 0Q08 ii5i 1633 1881 2125 2610 243 179 2853 3096 3338 3580 3822 4064 43o6 4548 4790 5o3i 242 180 255273 55i4 5755 5996 6237 6477 6718 6958 7198 7439 241 181 7679 7918 o3io 8i58 83qS 0787 8637 8877 9116 9355 9594 9833 23o 23S 182 260071 0548 1025 1263 i5oi 1789 4346 2214 i83 245 1 2688 2925 3i62 3399 3636 3873 4109 4582 287 184 4818 5o54 5290 5525 5761 5996 6232 6467 6702 6987 235 1 85 7172 7406 7641 7875 8IIO 8344 8578 8812 9046 9279 284. i 186 95i3 9746 9980 •2l3 •446 •679 •912 II44 1877 1609 233 j 187 188 271842 2074 4389 23o6 2533 2770 3ooi 3233 3464 3696 3927 282! 4i58 4620 4S5o 5o8i 53 II 5542 5772 6002 6232 280 189 6462 6692 6921 7i5i 7380 7609 7838 8067 8296 8525 229. 190 27^^754 8982 9211 ',f.t 9667 9895 •123 •35i •578 •806 228 191 281033 1261 1488 1942 2169 2396 2622 2849 8075 227 192 33oi 3527 3753 3979 42o5 443 1 4656 4882 5107 5332 226. 193 5557 5782 6007 6232 6456 6681 6905 7180 7354 7578 225 194 7S02 8026 8249 8473 8696 8920 9143 9866 9589 i8i3 9S12 223 195 290035 0257 04B0 0702 0925 1 147 1869 1591 2084 222 196 2256 2478 2699 2920 3i4i 3363 3584 38o4 4025 4246 221 197 4466 4687 4907 5i27 5347 5567 5787 6007 8198 6226 6446 220 198 6665 6S84 7104 7323 7542 7761 7979 8416 8635 2ia 218 199 8853 9071 9289 9507 9725 9943 •161 •378 •595 •8i3 2CO 3oio3o 1247 1464 1681 1898 2114 233 1 2547 2764 2980 217 201 3 1 96 3412 3628 3844 4039 4275 4491 4706 4921 5i36 21^ 202 53 5 I 5566 5781 5996 6211 6425 6639 6854 7068 7282 2l5 203 7496 204 9630 7710 7924 8^37 835i 8564 8778 8991 9204 9417 2l3 9843 ••56 •268 •481 •693 •906 1118 i33o i542 212! 2o5 3 1 1754 1966 2177 2389 2600 2812 3o23 3234 3445 3656 2ll| 206 3867 4078 4289 4499 4710 4920 5i3o 5340 5551 5760 2I0J III 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 12 8o63 8272 8481 8689 8898 9106 9814 9522 9730 9988 209 :320I46 o354 o562 0769 0977 1 184 1891 1598 i8o5 2012 207 210 322219 2426 2633 2839 3046 3252 3458 3665 3871 4077 sd6 2H 4282 4488 4694 4899 5io5 53io 55i6 5721 5926 6i3i :o5 ai2 6336 6541 6745 6950 7x55 7359 7563 7767 7972 8176 2041 3l3 838o 8583 8787 8991 9194 9398 9601 9805 •••8 •211 2o3J 214 1330414 b6i7 0819 1022 1223 1427 i63o i832 2o34 2236 202] 2l5 2438 2640 2842 3o44 3246 3447 3649 385o 4o5i 4253 202 216 4454 4655 4856 5o57 5257 5458 5658 5859 6059 6260 201 l\l 6460 6660 6860 7060 7260 7459 7659 7858 8o58 ! 8237 1 200 j 8456 8656 8855 9054 9253 945 1 9650 9849 ••47 ■: •246 I90J 219 340444 0642 0841 1039 1237 1435 i632 i83o 2028 ; 2225 1 I93-I N. 1 I. 2 ' 4 5 6 7 8 1 9 ^ 15 A TABLE OF LOGARITHMS FROM 1 TO 10,000. |n. 1 1 2 3 4 5 6 7 8 9 dTI 220 342423 2t)20 2817 3oi4 3212 3409 36o6 38o2 3999 4196 \U 221 4392 4389 4785 4981 5178 5374 5570 5766 5962 6i?7 ! 222 63d3 6549 6744 ^ 7135 7330 7525 7720 79i5 8110 195 ! 223 83o5 85oo 8694 9083 9278 9472 9666 9860 ••54 1^4 1 224 350248 0442 0636 0829 1023 1216 1410 i6o3 1796 1989 193 t 225 2i83 2375 2568 2761 2954 3i47 3339 3532 3724 3916 193 1 226 4108 43oi 4493 4685 4876 5o68 5260 5452 5643 5834 19a 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 228 7935 8i25 83i6 85o6 8696 8886 9076 9266 9456 9646 190 229 9835 ••25 •210 •404 •593 •7S3 •972 1161 l330 1539 189 23o ''^ lU 2io5 2294 2482 2671 2859 3048 3236 3424 188 23l 3988 4176 4363 455i 4739 4926 5ii3 53oi 188 232 5488; 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 233 7356! 7542 7729 79ID 8101 8287 8473 8639 •5i3 8845 9o3o 186 234 9216; 9401 9087 9772 995^ •328 •698 •883 i85 235 371068 1253 1437 1622 J 806 I'Sl 2175 236o 2544 2728 184 236 ) 2912 3096 4748 4932 3280 3464 3647 401 5 4198 4382 4565 184 237 5ii5 5298 5481 5664 5846 6029 6212 6394 i83 238 6577 6759 6942 7124 7306 7488 7670 7852 8o34 8216 182 239 8398 858o 8761 8943 9124 9306 9487 9668 9849 ••3o 181 240 3802II 0392 0573 0754 0934 iii5 1296 1476 i656 1837 181 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 242 38i5| 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 243 56o6 5780 5964 6142 6321 6499 6677 6856 7034 7212 8989 178 244 7390 7568 7746 7923 8101 8279 8456 8634 88n 178 245 9166 9343 9^^° 9698 9875 ••5 1 •228 •4o5 •582 •759 177 246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2321 176 247 2697 4432 2873 3048 3224 3400 3575 37DI 3926 4101 4277 17^ 248 4627 4802 4977 5i52 5326 5doi 5676 5850 6025 175 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 25o 397940 8114 8287 8461 8634 8808 8981 9154 9328 95oi 173 25l 9674 9847 ••20 •192 •365 •538 •711 •883 io56 1228 173 252 401401 i573 1745 1917 2089 2261 2433 26o5 2777 2949 172 253 3l2I 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 254 4834 5oo5 5176 5346 5517 5688 5858 6029 6199 6370 171 255 6540 6710 6881 7o5i 8749 7221 7391 756i 7731 7901 8070 170 256 8240 8410 8579 8918 9087 92D7 9426 Ul 9764 169 257 9933 •102 •271 •440 •609 2293 •777 •946 1114 I45i \tt 258 411620 1788 1956 2124 2461 2629 2796 2964 3i32 259 33oo 3467 3635 38o3 3970 4i37 43oD 4472 4639 4806 167 260 414973 5i4o 5307 5474 5641 58o8 ^n^ 6141 63o8 6474 167 261 6641 6807 6973 lf,t 7306 7472 7638 7804 7970 8i35 166 262 83oi 8467 8b33 8964 9129 9295 9460 9625 9791 i65 263 9956 •121 •286 •45i •616 •781 •943 2390 IIIO 1275 1439 i65 264 421604 1788 1933 2007 3737 2261 2426 2754 2918 3082 164 265 3246 3410 3574 3901 4o65 4228 4392 4555 4718 164 266 4882 5045 5208 5371 5d34 5697 586o 6023 6186 6349 i63 267 65ii 6674 6836 6999 7161 8783 7324 8944 7486 7648 781 1 7973 162 268 8i35 8297 8459 8621 9106 ni^ 9429 9591 162 259 9752 9914 ••7! •236 •398 •559 •720 •881 1042 I203 161 270 43 1 364 1 525 i685 1846 2007 2167 2328 2488 2649 2809 161 271 6i63 3i3o 3290 3450 36io 3770 3930 56^5 4249 4409 160 272 4729 4888 5048 5207 5367 5526 5844 6004 1 59 273 6322 6481 6640 6708 6957 7116 7275 7433 7592 is? 274 7751 7909 8067 9648 8226 8384 8542 8701 8859 9017 9173 275 9333 9491 9806 tsi •122 •279 •437 •594 •752 1 58 276 440909 1066 1224 i38i 1695 i852 2009 2166 2323 1 57 27T 27$ 2480 2637 2793 4357 2950 3io6 3263 3419 3576 3732 3889 1 57 4045 4201 4313 4669 4825 4981 5i37 5293 5449 7003 1 56 279 5604 5760 5915 6071 6226 6382 6537 6692 6848 i55 N. I 2 3 4 5 6 7 8 9 D. A TABLE OF LOGARITHMS FROM 1 TO ICjOOO. N. I 2 3 4 5 6 7 8 9 D. 3S0 447158 73i3 7468 7623 7778 7933 8088 8242 8397 8552 1 55 281 8706 8861 9oi5 9170 9324 9478 9633 9787 9941 ••95 1 633 1 54 282 450249 o4o3 0557 071 1 0865 1018 1172 i326 1479 i54 283 1786 1940 2093 2247 2400 2553 2706 2809 30I2 3i65 1 53 284 33i8 3471 3624 3777 3930 4082 4235 4387 4540 • 4692 1 53 285 4845 4997 5i5o 53o2 5454 56o6 5758 5910 6062 1 6ji4 l52 286 6366 65i8 6670 6821 6973 7125 8638 7276 8789 7428 7579 7731 X52 287 7882 8o33 8184 8330 8487 8940 9091 9-42 x5i 288 9392 9543 9694 9845 9995 •146 •296 •447 •597 •748 x5x 289 460898 I048 1198 i348 1499 1649 1799 1948 2098 2248 x5o 290 462398 2548 2697 2847 2997 3146 3296 4788 3445 3594 3744 i5o 291 3893 4042 4191 568o 4340 4490 4639 4936 5o85 5234 149 292 5383 5532 5829 5977 6126 6274 6423 6571 6719 149 293 6868 7016 7164 7312 7460 7608 7756 7904 8o52 8200 148 294 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 X48 295 9822 9969 •116 •263 •410 •557 •704 •85i •998 1 145 J 47 296 471295 2756 1438 i585 1732 1878 2025 2171 23i8 2464 2610 X46 297 2903 4362 3o49 3195 4653 3341 3487 3633 3779 5235 3925 4071 146 298 4216 45o8 tin 4944 5090 538i 5526 146 299 5671 58i6 5962 6107 6397 6542 6687 6832 6976 145 3oo 477121 7266 8711 741 1 7555 7700 7844 7989 8i33 8278 8422 145 3oi 8566 8855 8999 9143 9287 9431 9575 9719 9863 144 302 480007 oi5i 0294 0438 0582 0725 0869 1012 ii56 1299 144 3o3 1443 1 586 1729 1872 2016 2159 23o2 2445 2588 273i 143 3o4 2874 3oi6 3i59 4585 33o2 3445 3587 3730 3872 4oi5 4157 X43 3o5 43oo 4442 4727 4869 5oii 5i53 5295 5437 5579 142 3o6 5721 5863 6oo5 6147 6289 643 6572 6714 6855 6997 142 3o7 7i38 855i 7280 7421 7563 7704 7845 7986 9396 8127 8269 8410 141 3oa 8692 8833 8974 9114 9255 9537 9677 9818 141 3o9 9958 ••99 •239 •38o •520 •661 •801 •941 1081 1222 140 3io 491362 l502 1642 1782 1922 2062 2201 2341 2481 2621 140 3ii 2760 2900 3 040 3179 3319 3458 3597 4989 3737 3876 401 5 x39 3l2 4i55 4294 4433 4572 47II 485o 5128 5267 5406 139 3i3 5544 5683 5822 5960 6099 7483 6238 6376 65i5 6653 6791 i3q i3g 3i4 6930 83ii 7068 7206 7344 7621 7759 7897 8o35 8173 3i5 8448 8586 8724 8862 8999 9137 9275 9412 9550 i38 3i6 9687 9824 9962 ••99 •236 •374 •5ii •648 •785 •922 137 3i7 3i8 5oio59 1 196 1333 1470 1607 1744 1880 2017 2X54 2291 3655 137 2427 2564 27CO 2837 2973 4335 3109 3246 3382 35i8 i36 3i9 3791 3927 4o63 4199 4471 4607 4743 4878 5oi4 i36 3 20 5o5i5o 5286 5421 5557 5693 5828 5o64 73i6 6099 6234 6370 i36 321 65o5 6640 6776 691 1 7046 7181 745f 7586 7721 i35 322 7856 1% 8126 8260 8395 8530 8664 8799 8934 9068 i35 323 9203 9471 9606 9740 9874 •••9 •143 •277 •4x1 x34 324 5io545 0679 o8i3 0947 1081 12l5 1 349 1482 1616 1750 x34 325 1 883 2017 2l5l 2284 2418 i 255i 2684 2818 2951 3o84 :33 326 3218 335 1 3484 36i7 3750 ; 3883 1 4016 4i4q 4282 4414 x33 327 4548 4681 48i3 4946 5079 i 52II 5344 5476 5609 5741 x33 328 5874 6006 6139 6271 64o3 6535 6668 6800 6932 7064 8382 I32 329 7196 7328 7460 7592 7724 7855 7987 81 19 825i l32 33o Ii85i4 864i 8777 8909 9040 9171 93o3 9434 9566 9697 i3i 33 1 9828 9959 ••90 •22 X •353 •484 •6i5 •745 •876 X007 i3x 332 52u38 257? 1400 i53o i66i 1792 1922 2053 2i83 23x4 x3i 333 2444 2705 i 2835 2966 3096 3226 3356 3486 36x6 x3o 334 3746 3876 4006 41 36 4266 4396 4526 4656 4785 491 5 i3o 335 5o45 5174 6469 53o4 5434 5563 6985 5822 595 1 6081 1 6210 1 X29 336 6339 6598 6727 6856 7II4 7243 7372 7501 X29 337 7630 7759 7888 8016 8145 8274 8402 853 r 8660 8788 lit 338 8917 9045 9174 9302 943o 9559 9687 9815 9943 ••72 339 N. 530200 o328 0456 o584 0712 0840 0968 1096 1223 i35x 128 » 2 3 4 ' 1 6 7 ~ ' D. A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. j I 2 3 4 5 6 7 8 9 2627 D. laS "3^ 531479 1607 1734 1862 1990 2117 2245 2372 25oo 341 2754 28a2 3009 3i36 3264 3391 35i8 3645 3772 3899 127 342 4026 4i53 4280 4407 4534 4661 4787 4914 5o4i 5.67 127 343 5294 5421 5547 5b74 58oo 5927 6o53 6180 63o6 6432 126 344 6558 66>:)5 68u 6937 7063 7189 73i5 7441 7567 7693 126 34:3 7819 7Q45 8071 8.97 8322 8448 8574 8699 8825 8951 126 346 9076 9202 9327 9402 9578 97o3 9829 9954 ••79 •204 123 347 f^oJ29 0455 o58o 0705 o83o 0955 1080 I205 i33o 1454 123 34rf 1 579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 349 "825 2950 ^074 3199 3323 3447 3571 3696 3820 1944 124 350 i 44068 4192 43i6 4440 4564 4688 4812 4936 5o6o 5i83 124 331 5307 543 1 5555 5678 5d02 5925 6049 6.72 6296 6419 124 352 6543 6666 6789 6913 7o36 7159 7202 74o5 7529 8758 76J2 123 353 7775 7«9^ 8021 8144 8267 83 89 85i2 8635 8881 123 354 90')3 9126 9249 9371 9494 9616 9739 9861 9984 •106 123 350 550228 o35i 0473 0595 0717 0840 0962 1084 1206 i328 J22 350 1450 1572 1694 1816 1938 2060 2181 23o3 2425 2547 122 357 2668 2790 2911 3o33 3.55 3276 3398 3519 3640 3762 121 35rf 3883 4004 4126 4247 4368 44H9 4610 4731 4852 4973 121 359 5094 521 5 5336 5457 5578 5699 5820 5940 6061 6182 121 360 5563o3' 6423 6544 6664 6785 6905 7026 7146 7267 7387 120 361 7507 7627 7748 7^68 79^8 8108 8228 8349 8469 8589 "120 362 8709 8S?9 8948 9068 9.88 9308 9428 9548 9667 9787 120 363 9907 •*26 •146 •205 •385 •5o4 •624 •743 •863 •982 119 364 56 1 1 1 1221 1 340 1459 1578 1698 1817 1936 2o55 2174 119 36J 2293 2412 253i 26JO 2769 2887 3oo6 3i25 3244 3362 119 366 3481 3600 3718 3.^37 3930 4074 4192 43ii 4429 45^8 119 367 4666 1 4784 4903 502I 5i39 5257 D376 5494 56i2 5730 (18 36tf 5848; 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 369 70261 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 5682021 83i9 8436 8554 8671 8788 8905 9023 91 40 9207 •17 371 9374' 9491 9608 9725 9842 9959 -76 •193 •309 •426 »i7 372 570543 1 0060 0776 0893 1010 1126 1243 1359 1476 1592 117 373 1709: I«25 1942 2008 2174 2291 2407 3568 2523 2639 2755 116 374 20721 2088 3io4 3220 3336 3452 3684 3800 3915 116 370 4o3i 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 376 5i88 53o3 5419 5534 5o5o 5765 588o 5996 6111 6226 ii5 377 6341 6457 6572 6687 6802 6917 7o32 7147 7262 7377 ii5 37rf 7492 7607 7722 7836 7951 8ob6 8181 8295 8410 8525 ii5 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 579784 9S98 ••12 •126 •241 •355 Zl •583 •697 •811 114 381 580925! 1039 II 53 1267 1 38 1 2631 1722 1 836 1950 114 382 20631 2177 2291 2404 25i8 2745 2858 2972 3o85 114 383 3iQ9; 33i2 3^26 3j39 3652 3765 3879 3992 4105 42 i 8 ii3 384 433 1 4444 4557 4670 4783 4896 D009 5,22 5235 53^j ii3 385 5461 5574 5686 5799 5912 6024 6,37 6250 6362 6475 ii3 386 6587 6700 6812 6925 7037 7'49 7262 8384 6496 7486 7599 112 387 7711 7823 7935 8047 8160 8272 8608 8720 112 388 8832 3o44 9o56 9167 9279 9391 95o3 96i5 9726 983S ri2 389 9950 ••61 •n3 •284 •396 •5o7 •619 •730 •842 •953 112 39c 591065 1176 1287 1399 i5io 1621 2843 1843 t955 2066 Hi 391 2177 2288 2399 25 10 2621 2732 2954 3064 3175 III 39s 3286 3397 35o8 36i8 3729 3840 3950 406 1 4171 4282 111 393 4393 45o3 4614 4724 4834 4945 5o55 5i65 5276 5386 110 394 54961 56o6 5717 5827 5937 6047 6i57 6267 6377 ^^uZ no 395 65971 6707 6817 ^9:7 7037 7146 7256 7366 7476 7D86 IlOi 396 76951 7805 79'4 So24 8i34 8243 8353 8462 8572 8681 no 397 8791 8900 9009 9119 9228 9337 9446 9556 9665 m^ 109 398 9883 9992 •lOI •2n •3i9 •428 •537 •646 •z^5 •864 109 399 600973 1082 1191 1^99 1408 15.17 1625 1734 1843 I95i 109 K. I 2 3 4 5 6 7 8 9 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. I 2 3 j 4 j 5 1 6 1 7 ! 8 1 9 JD. 108 400 602060 2169 2277 2386 j 1494 2603 271 1 j 2819 1 2928 3o36 401 3i44 3253 336i 3469 3577 ; 3686 3794 3902 ! 4010 4118 108 402 4226 4334 4442 455o 4658 ' 4766 4874 4982 ; 5089 1 5197 1 io8 4o3 53o5 54i3 5521 5623 5736 ; 5844 •5931 bo59 i 6166 6274 108 404 63Si 6489 6596 6704 681 1 i 6919 7026 7'33 , 7241 7348 107 4o5 -7455 7062 7669 7777 7884 i 7991 8954 9061 8098 ! 8205 j 83i2 8419 107 406 8526 8633 8740 8847 9167 9274 1 9381 9488 107 407 9594 9701 9808 9914 ••21 j •128 •234 •341 1 '447 •554 1 ro7 408 61066c 0161 0873 0979 1086 1192 1 1298 i4o5 i5ii 1617 2678 1 106 409 1723 1S29 1936 2042 2148 2254 236o 2466 2572 106 410 6r^784 2890 2996 3l02 3207 33i3 3419 3525 363u 3736 106 411 3842 3947 4o53 41 59 4264 4370 4475 458 1 4686 4792 106 412 4897 5oo3 5io8 52i3 i 5319 5424 5529 5634 ' 5740 5845 io5 4i3 5900 6o55 6160 6265 ' 6370 6476 658i 6686 6790 6895 io5 414 7000 7io5 7210 8257 73i5 7420 7525 8571 7629 7734 7839 8884 7943 io5 41^) 8048 8i53 8362 8466 8676 8780 8989 io5 416 9093 9198 9302 9406 9511 9615 9719 9824 9928 ••32 104 417 620136 0240 o344 0448 o552 o656 0760 0864 0968 1072 104 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 104 419 2214 23i8 2421 2520 2628 2732 2835 2939 3042 3146 104 420 623249 3353 3456 3559 3663 3766 3869 3973 4076 4179 io3 421 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 io3 422 53 1 2 54i5 55i8 5621 5724 5827 5929 6o32 6i35 6238 io3 423 6J40 6443 6546 6648 6751 6853 6956 7o58 7161 7263 io3 424 7366 7468 7571 8593 7673 7775 7878 7980 8082 8i85 8287 102 423 83^9 8491 8695 8797 8900 9002 9104 9206 93o3 102 426 9410 9512 9613 97i5 9817 9919 ••21 •123 •224 •326 102 427 630428 o53o o63i 0733 o835 0936 io38 1 139 1 241 i342 103 428 1444 1 545 1647 1748 1849 1951 2052 2i53 2255 2356 lOi 429 2457 2559 2660 2761 2862 2963 3o64 3i65 3266 3367 101 43o 633468 3569 3670 3771 3872 3973 4074 4175 4276 4376 100 43 1 4477 4578 5584 4679 4779 48tfo 4981 5o8i 5182 5283 5383 100 432 5484 568d 57b5 58d6 5986 60S7 6187 6287 6388 100 433 6488 6588 6688 6789 6889 6989 7089 7189 8190 7290 It 100 434 7490 7590 7690 7790 7890 7990 8090 8290 99 435 8489 8589 86^9 8789 8ab8 89^8 90t,8 9188 9287 9387 99 436 94b6 9586 9686 9783 9885 9984 ••84 •i83 •283 •382 99 437 640481 o58i 0680 0779 0879 0978 1077 1177 1276 1375 99 438 1474 2465 1573 1672 1771 1871 1970 2069 2168 3o58 3i56 2267 2366 99 439 2563 2662 2761 2860 2959 3255 3354 99 440 643453 3551 365o 3749 3847 3946 4044 4143 4242 4340 98 441 4439 4537 4636 4734 4832 4931 5029 1 5127 5226 5324 98 442 5422 5521 56i9 5717 58iD 5913 6011 1 6110 6208 63o6 98 443 6404 65o2 6600 6698 6796 6594 6992 1 7089 7187 7285 98 444 7383 7481 7579 7676 7774 7872 8848 7969 8067 8i65 8262 98 445 8360 8458 8555 8653 8730 8945 9043 9140 9237 97 1 446 9335 9432 953o 9627 9724 9821 9919 ••16 •ii3 1 •210 07 447 65o3o8 o4o5 0302 0399 0696 0793 0890 0987 1084 1 181 97 448 1278 1375 1472 1 569 i6t)6 1762 1809 1956 2o53 2i5o 97 449 2246 2343 2440 2536 2633 2730 2826 2923 3019 3ii6 91 45o 653213 3309 34o5 35o2 3598 3695 3791 3888 3984 4o8c 96 45i 4177 4273 4369 4465 4562 4658 4754 485o 4946 5o42 06 452 5i38 5235 533 1 5427 5523 5619 5715 58io 5906 6002 96 453 6098 6194 6290 6386 6482 ^?77 6673 6769 6864 6960 96 454 7056 7i52 7247 7343 7438 7334 8488 7629 7725 7820 8774 It 96 455 Son 8107 8202 8298 8393 8584 8679 95 456 8965 9060 9155 9230 9346 9441 9536 963 1 9726 9821 9^ & 9916 ••11 •106 •201 •296 •391 •486 . •58i •676 1 •771 95 660865 0960 io55 ii5o 1245 1339 1434 ' 1529 1623 . 1718 95 459 N. i8i3 1907 2002 2096 2191 22^6 5 228o 1 6 1 2473 256q ■■ 2663 ^1 _^_ I 2 .^. '11 ' i 'j>. A TABLE OF LOGARITHMS FROM 1 TO 10,000. 460 662758 2852 2 3 4 5 6 7 8 9 ± 2947 3o4i 3i35 323o 3324 3418 35i2 454J 94 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 94 462 4642 4736 483o 4924 5oi8 5lI2 5206 5299 5393 5487 94 463 558i 5675 5769 5362 5906 6o5o 6143 6237 633 1 6424 94 464 65i8 6612 6705 6799 6892 69S6 7079 7173 8106 7266 7360 94 465 7453 8386 7546 7640 8572 8665 7826 792c 8oi3 8199 8293 93 466 8479 8759 9689 8852 8945 9038 9i3i 9224 93 467 468 9317 9410 95o3 9596 97S2 9875 9967 ••60 •i53 93 670246 0339 043 1 OJ24 0617 0710 0802 0895 09S8 1080 93 469 1173 1265 i353 I45i 1 543 i636 1728 1821 1913 2005 93 470 672098 2190 2283 2375 2467 256o 2652 2744 2836 2929 92! 4^1 3021 3ii3 32o5 3297 3390 3482 3574 3666 3758 385o 92 /i7i itr, 4034 4126 421S 43io 4402 4494 4586 4677 4769 92 4^3 4953 5o45 5i37 5228 5320 5412 55o3 5595 5687 92 474 ut. 5870 5962 6o53 6145 6236 6328 6419 65ii 6602 92 475 6785 6876 6q68 7059 7.5i 7242 7333 7424 7516 91 476 7607 til 77^9 7881 7972 8o63 8i54 8245 8336 8427 9» 477 85i8 8700 8791 8882 8973 9064 9155 9246 9337 91 47^ 9428 9519 9610 9700 9791 9882 9973 ••63 •I 54 •245 91 479 68o336 0426 o5i7 0607 0698 0789 0879 0970 1060 ii5i 9» 480 681 241 i332 1422 i5i3 i6o3 1693 1784 1874 1964 2o55 90 4S1 2145 2235 2326 2416 25o6 2596 26S6 2777 2867 2957 90 482 3047 3i37 3227 3317 3407 3497 3587 3677 3-67 3857 90 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4736 90 484 4845 4935 5o25 5iu 5204 5294 6189 53S3 5473 5563 5652 90 485 5742 5o3i 5921 6010 6100 6279 6368 6458 6547 89 486 6636 6726 68i5 6904 6994 7o83 8064 7261 735i 7440 89 ill 7529 7618 7707 7796 7886 797? 8i53 8242 833 1 89 8420 85o9 8398 8607 8776 8863 8q53 9841 9042 9i3i Q220 89 489- 9309 9398 9486 9575 9664 9753 9930 ••19 •107 89 490 690196 0285 0373 0462 o55o 0639 0728 0816 0905 0993 tl 491 1081 1170 1258 1347 1435 i524 1612 1700 1789 1877 492 ■It^ 2O03 2142 223o 23i8 2406 2494 2583 2671 2759 88 493 Itl 3o23 3iii 3199 3287 3375 3463 355i 3639 88 494 3727 3903 3991 4078 4166 4254 4342 443o 4517 8S 495 46o5 4693 4781 4b68 4936 5o44 5.3i 5219 5307 5394 881 496 5482 5569 5607 5744 5832 5919 6007 6094 6182 6269 87 497 6356 6444 653 1 6618 6706 6793 68S0 6968 7o55 7142 87 498 7229 7^'7 7404 7491 7578 7660 7752 7839 8709 7926 8014 87 499 8188 8275 8362 8449 8535 8622 8796 8883 87 5oo 698970 9057 9144 9231 9317 9404 9491 9578 9664 975i ?' 5oi 9^38 9924 ••11 ••98 •184 •271 •333 •444 •53 1 •617 87 502 700704 0790 0877 0963 io5o ii36 1222 i3o9 1395 1482 86 5o3 1 568 16^4 1741 1827 1913 1999 2086 2172 2258 2344 86 5o4 243 1 25i7 2603 2689 2775 2861 2947 3o33 3119 32o5 86 5o5 3291 3377 3463 3549 3635 3721 38o7 3893 3q^9 4o65 86 5o6 4i5i 4236 4322 4408 4494 4579 4665 475i 4837 4922 86 507 5oo8 5094 5179 5265 5350 5436 5522 5607 5693 5778 86 5o8 5864 5949 6o35 6120 6206 6291 6376 6462 6547 6632 85 509 6718 68o3 6888 6974 7059 7144 7229 73i5 7400 7485 85 5io 707570 8421 7655 7740 7826 791 1 7996 8081 8166 825i 8336 85 5ii 85o6 8591 8676 8761 8846 8931 9015 9100 9185 85 D12 9270 9355 9440 9024 9609 9694 9779 9863 9948 ••33 85 5i3 710117 0202 0287 0371 0456 0540 0625 0710 0794 .0879 1723 85 5i4 0963 1807 1048 Il32 1217 i3oi 1385 1470 1 554 1639 84 1 5i5 1892 1976 2060 2144 2229 23i3 2397 2481 2566 84} 5i6 265o 2734 2818 2902 2986 3826 3070 3i54 3238 3323 3407 84! 5i7 5i8 3491 3575 3659 3742 3910 3994 4078 4162 4^46 84 433o 4414 4497 458 1 4665 4749 4833 4916 5ooo 5o84 84 519 5167 525i 5335 5418 55o2 5586 5669 5753 5836 5930 84 D. 1 2 3 4 5 6 7 8 9 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N, I 2 3 4 5 6 7 1 8 9 83 520 716003 6087 6170 6254 6337 6421 65o4 6588 6671 6754 521 6838 6921 7004 7088 8oo3 7254 7338 7421 8253 7504 8336 7587 8419 9248 83 522 7671 7754 7837 8668 7920 8086 8169 83 523 b5o2 8585 8751 8834 8917 9000 9083 9165 83 524 9331 9414 9497 9580 9663 9745 9828 991 1 9994 ••77 83 520 720J59, 0242 o325 0407 0490 0573 o655 0738 0821 0903 83! 526 0986 io63 ii5i 1233 i3i6 1398 1481 i563 1646 1728 82 527 1811 1S93 1975 2o58 2140 2222 23o5 2387 2469 2552 82 528 2634 27.6 2798 2881 2963 3045 3127 3209 3291 3374 8a 529 3456 3538 3620 3702 3784 3866 3948 4o3o 4112 4194 82 53o 724276 4358 4440 4522 4604 4685 4767 4849 4931 5oi3 82 53 1 5095 5176 5258 5340 5422 55o3 5585 5667 6483 5748 5830 82 532 5912 5993 6075 6i56 6238 6320 6401 6564 6646 82 533 6727 6809 6890 6972 7053 7134 7216 7297 8110 7379 7460 81 • 534 7341 7623 8435 7704 7735 7866 8678 7948 8029 8191 8273 81 535 8354 85i6 8597 8759 8841 8922 90o3 9084 81 536 9i65 9246 9327 9408 9489 9570 965i 9732 9813 9893 81 537 9974 ••55 •i36 •217 •298 •378 •459 •540 •621 •702 81 538 730782 o863 0944 1024 iio5 u86 1266 i347 1428 i5o8 81 539 1589 1669 1750 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 3759 3839 3919 80 542 3999 4079 4160 4240 4320 4400 4480 456o 4640 4720 80 543 4800 4880 4960 5o4o 5l20 5200 5279 5359 5439 55i9 80 544 5599 5679 5709 5838 5918 5998 6078 6157 6237 63i7 80 545 6397 6476 6556 6635 6715 6795 6874 6954 7034 7ii3 80 546 7193 7272 7352 7431 75ii 8384 7670 U$ 7829 7908 79 547 1% 8067 8146 8225 83o5 8463 8622 8701 79 548 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 549 9572 965i 9731 98x0^ 9889 9968 ••47 •126 •205 ^284 79 55o 740363 0442 052I 0600 0678 0757 o836 0915 0994 1073 79 55i 1132 I23o 1 309 i388 1467 1546 1624 1703 178-2 i860 79 552 1939 2018 2096 2175 2254 2332 241 1 2489 2368 2647 ?^ 553 2725 2804 2882 2961 3o39 3ii8 3196 3275 3353 3431 554 3510 3588 3667 3745 3823 3902 3980 4o58 4i36 42i5 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5i53 523i 5309 5387 5465 5543 5621 5699 5777 78 557 5855 5933 601 1 6089 6167 6245 6323 6401 6479 6556 78 558 6634 6712 6790 6868 6945 7023 7101 7'79 7256 8a33 7334 78 559 7412 7489 7567 7645 7722 7800 7878 7903 8110 78 56o 748188 8266 8343 8421 8498 8576 8653 8731 8808 8885 77 56 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 562 9736 9814 9891 9968 ••45 •123 •200 •277 •354 •43 1 77 563 75o5o8 o586 0663 0740 0817 0894 0971 1048 II25 1202 77 564 1279 2048 1 356 1433 i5io i587 1664 1741 1818 1895 2663 1972 77 565 2125 2202 2279 2356 2433 2 509 2586 2740 77 566 2816 2893 3?]6 3o47 3.23 3200 3277 3353 343o 35o6 77 567 3583 366o 38i3 3889 3966 4042 4119 4195 4272 77 568 4348 4425 45oi 4578 4654 4730 4807 4883 4960 5o36 76 569 5lI2 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 755875 5951 z°^J 6io3 6180 6256 6332 6408 6484 656o 76 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 7396 nil l^^i 7624 8382 7700 7775 785i 7927 8oo3 8079 76 573 8i55 83o6 8458 8533 8609 8685 8761 9517 8836 76 54 89,2 8988 9063 9139 9214 9290 9366 9441 9592 76 575 9668 9743 98.9 9894 9970 ••45 •121 •196 •272 •347 75 576 760422 0498 0373 0649 0724 0799 0875 0930 1025 XIOI '^\ 577 1 176 I25l i326 1402 1477 i552 1627 2378 1702 1778 1 853 75 578 1928 2oo3 2078 2i53 2228 23o3 2453 2529 3278 2604 75 579 2679I 2754 2829 2904 2978 3o53 3i28 32o3 3353 75 N. I 3 3 4 5 6 7 8 9 D. 10 A TABLE OF LOOARITHMS FROM L TO 10,000. 1 N. 1 ° I ' 2 3 3653 1 A 5 6 i 7 8 9 1 1>. 5So 763428 35o3 3578 ' 3727 38o2 3877 , 3902 4027 4101 1 75 i^' 4176 4201 4326 4400 ; 4475. 4300 4624 1 4099 4774 , 4848 ' 75 ! 582 4Q23' 499S 5072 5i47 1 5221 5296 5370 5445 1 5)20 5594 1 75 1 583 ) 5609 ! ^743 58 1 8 5892 1 5966 6041 6ii5 6190 6264 6338 ' 74 ! 58/. 1 6413 6487 6562 6636 ^ 6710 I 6785 6859 6933 7007 ' 7082 74 1 585 1 7106 7230 7304 1 7379 ' 7453 ; 7527 ' 7601 7675 7749 7«23 . (4 586 1 -jS^Sl 7972 8046 812c , 8194 j 86381 8712 8786 1 8^00 . 8934 8268 8342 8416 8490 1 8564 74 587 9008 9082 9i56 9230 c2u3 74 588 9377 94^1 9525 9099 ! 9^73 9746 ' 9820 9S94 ■9968 , ••42 : 74 589 770115 0189 ; 0263 o336 0410 0484 0557 ' o63i 0705 0778 74 5^0 770S52 0926 0999 1073 1 146 1220 1293 1367 1440 i5i4 74 59. 1587 1 66 1 1734 1808 I88I 1955 2028 2102 2175 2248 1 73 5g2 2322 2395 2468 2542 26i5 2688 2762 2S35 2908 1 29^' 73 593 3o55 3128 3201 3274 3348 3421 3494 3567 3640 3713 ' 73 594 3786 3860 3933 4006 4079 4i52 4225 4298 4371 4444 73 595 4517 4590 4663 4736 4809 4882 1 4q55 5o28 5ioo 5i73 73 596 5246; 5319 5392 5465 5538 56 10 5683 5756 5829 5902 73 597 59741 6047 6120 6193 6265 6338 6411 6433 6556 6629 73 598 6701 6774 6846 6919 6992 7064 7i3t 7209 7282 8079 73 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 72 600 778i5i 8224 8296 8363 8441 85i3 8585 8658 8730 8802 72 601 8874' 8947 9019 9091 9163 9236 9308 9380 94 J 2 9324 ! 72 602 9596, 9"^9 9741 9813 9:^85 9957 ••29 1 'SOI •'73 •243 72 6o3 780317} o389 0461 0533 o6oD 0677 0749 0821 0X93 0965 1 72 604 io37 1109 1181 1253 1324 1396 1468 1540 1612 1 1684 72 6o5 1755 1827 1899 '971 2042 2114 2186 2.158 2329 ' 2401 72 606 2473i 2544 2616 2688 2709 2531 2902 2974 3o46 3.17 72 607 3189 3260 3332 3403 3470 3546 36i8 36S9 44o3 376. 3832 7> 608 3904 3975 4046 4118 4189 4261 4332 4473 4546 71 609 4bi7 4609 4760 483 1 4902 4974 5o45 5ii6 5i87 52J9 71 610 785330 5401 5472 5543 56i5 5686 5757 5828 5899 5970 71 611 6041 6112 6i83 6254 6325 6396 6467 6538 6609 6680 7» 6.2 6751 6822 6893 6964 7o35 7.06 7177 7248 7319 7390 7» 1 6i3 7460 753i 7602 7673 7744 7815 7885 7956 8027 8098 7» 614 8168 8239 83 10 838 1 8451 8522 8593 8663 8734 8804 7« 6i5 8875 8946 9016 9087 9'57 9228 9299 9369 9440 95 10 7» 616 gSSi 9601 9722 9792 9863 9933 •••4 .••74 •144 •2l5 70 617 7902S5 o356 0426 0496 o567 0637 0707 0778 0848 0918 70 618 0988 1009 1129 I '99 1269 i34o 1410 1480 i55o 1620 70 619 169 1 1761 i83i 1901 1971 2041 21U 2l8l 2252 2322 70 620 792392 2462 2532 2602 2672 2742 2812 2882 2932 3022 70 621 3092 3i62 ' 323i 1 33oi 3371 3441 35ii 358i 365i 3721 70 622 3790 3 860 3930 4000 4070 4.39 4209 4279 4349 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5. ,5 70 624 5i85 5234 5324 5393 5463 5532 56o2 5672 5741 58 n 70 ! 62D 588o 5949 6019 6088 6.58 6227 6297 6366 6436 65c5 69 626 6574 6644 67.3 6782 6352 6921 6990 7060 7129 7198 69 627 7263 7337 7406 1 7475 7545 7614 -7683 7752 7821 7890 ^9 628 7c6o 8029 8098 8167 8858 8236 83o5 8374 8443 85i3 8582 69 629 865 1 8720 8789 8927 8996 9065 9134 9203 9272 69 63o 799341 9409 9478 9547 9616 9685 9754 9323 9892 9901 69 63 1 800029 0098 0167 0235 o3o5 0373 0442 o5ii o58c 1 0648 69 632 0717 0786 o854 0923 0992 1061 ^'29 1 198 1266 1335 69 633 1404 1472 i54i 1609 1678 1747 i8i5 1884 1932 1 2021 69 634 2089 21 58 2226 2295 ■ 2363 ! 2432 25oo 2563 2637 1 270? ^ 635 2774 2842 2910 1 2979 3o47 1 3ii6 3i84 3232 1 3321 1 3389 636 3457 3525 3594 1 3662 : 3730 ! 3798 3867 3935 4oo3 4071 68 637 4139 4208 4276 ! 4344 4412 i 4480 4548 4616 4685 4753 68 482, 4889 4957 5o25 5093 ! 5i6i r^t 5297 1 5365 5433 68! 639 55oi 5569 5637 5705 5773 1 5841 5976 ■ 6044 6112 _68j D. N. I , 1 3 4 ; 5 6 7 ! 8 9 1 A. TABLE OF ] LOGARITH.MS FROX 1 TO 10,000. IJ N. 1 1 I 2 3 4 1 5 1 ^ 6 1 7 ' 8 6700 68 640 806180 6248 63 16 1 6384 6451 1 6319 1 6587 6655 6723 641 6858 6926 6994 1 7061 7129 ' 7197 7264 7332 7400 ' lAb-] 68 642 7535 7603 1 76-»o 7738 8279 1 8346 1 8414 7806 j 7873 7941 8008 8076 1 8143 68 643 8211 8481 ' 8549 8616 8684 8751 8818 67 644 8886 8953 j 9021 1 9088 9i56 ' 9223 9290 9358 9423 ; 9492 67 645 9060 9627 ! 9694 ! 9762 1 9829 1 9896 9964 1 ••3 1 1 "V 1 •i63 67 646 810233 o3oo 1 o3o7 1 0434 I o5oi 0569 c63t) 1 0703 j 0770 0837 67 647 0904 0971 1 1039 ' 1106 1 1173 i 1240 i3o7 1 1374 \ 1441 i5o8 67 648 ID75 1642 1 1709 1776 1843 1910 1977 2044 j 2111 2178 67 649 2245 23i2 , 2379 2443 1 25 1 2 2379 2:^46 2713. 278c 2847 67 65o 812913 2980 3o4/ 3114 3i8i 3247 33i4 338 1 3448 35i4 67 r,Di 3581 3648 37.4 3781 3848 3914 3981 4C48 41 14 4181 67 65:! 4248 43i4 438i 4447 45i4 458 1 4047 4714 4780 4847 67 653 49"3 4980 5046 5ii3 5179 5246 53i2 5378 5445 55ii 66 654 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 66 655 6241 63o8 6374 6440 65o6 6573 6639 6705 677 r 6838 66 656 6904 6970 7o36 7102 7169 7235 7301 7367 7433 8094 7499 66 657 7565 8:26 763i 7698 77(>4 783o 7896 7962 8028 8100 66 658 8292 8358 8424 8490 8536 8022 8688 8754 8820 66 659 8885 8951 9017 9063 9149 92i5 9281 9346 9412 9478 66 660 819544 9610 9676 9741 9807 9873 9939 •••4 ••70 •i36 66 661 820201 0267 o333 0399 0454 o53o 0593 d66i 0727 0792 66 662 0858 0924 0939 1035 1120 1186 I25l i3i7 i382 1448 66 663 i5i4 ID79 1645 1710 1775 1841 1906 1972 2037 2103 65 664 2168 2233 2299 2304 243o 2493 256o 2626 2601 2736 65 665 2822 2887 2952 3oi8 3o83 3148 32i3 3279 3344 3409 65 666 3474 3539 36o5 3070 3735 38oo 3865 3930 3996 4061 65 667 4126 4I9I 4256 4321 4386 445 1 43i6 458i 4646 471 1 65 668 4776 4841 4906 4971 5o36 5ioi 5i66 523i 5296 536i 65 669 5426 5491 5556 5621 5685 5751 58.' 3 588o 5945 6010 65 670 826075 6140 • 6204 6269 6334 6399 6464 6528 6593 6658 65 671 6723 6787 6852 69.7 6981 7046 71U 7175 7240 73o3 65 672 7369 7434 7499 7063 7628 7692 7737 7821 8467 7886 7951 65 673 8oi5 8080 8144 8209 8273 8338 8402 853 1 8395 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 675 9J04 9368 9432 9497 956i 9625 9690 9754 9818 9882 64 676 9947 ••11 ••75 •139. •204 •268 «332 •396 •460 '•525 64 677 83o5b9 0653 0717 0781 0845 0909 0973 1037 1102 1 166 64 678 I23o 1294 i358 1422 i486 i55o i6i4 1678 1742 1806 64 679 1870 1934 1998 2062 2126 2189 2253 2317 238i 2445 64 680 832509 2573 2637 2700 2764 2828 2892' 2956 3020 3oS3 64 681 3147 32II 3275 3338 3402 3466 3530 3593 3657 3721 64 682 3784 3848 39.2 3975 4o39 41 o3 4166 423o 4294 4357 64 683 4421 4484 4548 461 1 4675 4739 4802 4866 4929 4993 64 684 5o56 5l20 5i83 5247 53 10 5373 5437 55oo 5564 5627 63 685 5691 5754 58i7 5881 5944 6577 6007 6071 6i34 6197 6261 63 686 6J24 6387 645 1 65i4 6641 6704 6767 6830 6894 63 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 75?5 63 688 7588 7652 77i5 7778 7841 7904 7967 8o3o 8093 8i56 63 689 8219 8282 8345 8408 8471 8534 8397 8660 8723 8786 63 690 838849* 8912 8975 9o38 9101 9164 92?] 9289 9918 9352 9413 63 691 9478 9341 9604 1 9667 9729 9792 9853 9981 ••43 63 692 840106 0169 0232 1 0294 0357 0420 0482 o543 0608 0671 63 693 0733 0796 0859 ; 0921 0984 1046 1 109 1172 1234 1297 63 694 1359 1422 1485 : 1 547 1610 1672 1735 1797 i860 i 1922 63 693 1985 2047 2II0 1 2172 2235 3297 2360 2422 2484 2547 62 696 2609 2672 3233 3295 2734 : 2796 2859 1921 2983 3046 3io8 3170 6a 697 3357 1 3420 3482 3544 36o6 3669 3731 3793 62 698 3855 3918 3980 1 4042 4104 4166 4229 4291 4353 44i5 62 699 N. 4477 4339 4601 4664 4726 i 4788 485o 4912 4974 5o36 62 C I 2 1 3 4 5 1 6 I ^ J 8 1 9 1>. L2 A TABLE CF L0GAKITHM3 FROM 1 TO 10,000. N. I 2 3 4 i ' 6 , 1 a ! 9 1 5656 i>- : 700 845098I 5 1 60 5222 5284 5346 5408 5470 5532 1 5594 62 701 5718 5780 5842 5904 5966 6028 6090 6i5i 62i3 1 6275 52 702 6337! 6399 6461 6523 6585 6646 6708 6770 6832 ! 68g4 62 703 6955 7017 7079 .1 7141 7202 7264 7326 7388 7449 75ii 62 704 m 7634 1 7396 ■ ! "-'58 7819 7881 -943 8004 8066 812S 62 705 8201 ?3l2 8374 8433 8497 8539 8620 8682 8743 62 706 88o5 8866 8928 8989 905 1 9II2 9174 9235 9297 9358 61 708 9419 9481 9D42 9604 9665 9726 I 9788 9849 99" 997= 61 85oo33 0095 01 d6 0217 0279 1 o34o 1 0401 0462 0324 0585 6i 709 0646 0707 0769 o836 0891 0932 1014 1075 ii36 1197 61 710 851258 l320 i38i 1442 i5o3 1 564 1625 1686 1747 1809 61 711 2480 iqBl 1992 2o53 2114 2175 2236 2297 2358 2419 61, 712 2341 2602 2663 2724 2785 2846 2907 2968 3029 61, 713 3090 3i5o 32II 3272 3333 3394 3455 35i6 3377 3637 61 1 714 3698 3759 3820 388i 3941 4002 4o63 4124 4i85 4245 61; 7'5 4306 4367 442S 4488 4549 4610 4670 473 1 4792 4852 61 7'6 4913 4974 5o34 5095 5i56 5216 9277 5337 5398 5459 61 r.i 5519 558o 5640 5-01 5761 5822 5382 5943 6oo3 6064 61 1 6124 6i85 6245 63o6 6366 6427 6487 6348 66o3 1 666S 60 719 6729 6789 6850 6910 6970 703 1 7091 ^ 7i52 7212 7272 60I 720 857332 7393 7453 75i3 7574 7634 7694 ! 7755 78i5 7875 60; 721 7935 7995 8o56 8116 8176 8236 8297 i 8357 84.7 9018 8477 60 722 8537 8597 8657 8718 8778 8838 8898 8953 9078 60 723 9i38 9198 9258 9318 9379 9978 9439 9499 9559 9619 9679 60 724 86^338 9799 9859 99.8 ••38 ••98 •i58 •218 ! ^278 60 72D 0398 0458 o5i8 0578 0637 0697 0757 ; 0817 0877 60 726 oq37 z io56 1116 1176 1236 1293 1355 I4l3 1475 60 ]^ 1 534 1 654 1714 1773 i833 1893 2489 1952 2012 2072 60 2l3l 2I9I 225l 23lO 2370 243o 2549 2608 2668 60 729 2728 2787 2847 2906 2966 3o25 3o83 3i44 3204 3263 60, 730 863323 3382 3442 35oi 356i 3620. 3680 3730 3799 3853 59' 731 3917 3977 4o36 4096 4i55 4214 42-;4 4333 4392 4452 59, 732 401 1 4070 463o 4689 4748 4808 4867 4926 4985 5045 59 733 5io4 5i63 5222 5282 5341 5400 5459 5319 5578 5637 6228 5o' 734 1% 5755 58i4 5874 5933 5992 6o5i 6110 6169 5^' 735 6346 64o5 6465 6524 6583 6642 6701 6760 6819 59 736 6878 6937 6996 7055 7114 7173 7232 -1/-W1 7350 7409 59 737 7467 7526 7083 8174 7644 7703 7762 7821 7S80 7939 59 738 8o56 8ii5 8233 8292 835o 8409 846S 8327 59 739 8644 8703 8762 8821 8879 8938 8997 9o56 9114 9173 59, 740 869232 9290 9349 9408 9466 9525 9584 9642 9701 9760 59 741 9818 9S77 9935 9994 ••53 •hi •170 •22S •287 •345 1^ 742 870404 0462 0021 0579 0638 1^', 0755 ) oSi3 0872 0930 743 0989 1047 1106 Ii64 1223 i339 j 1398 1456 13l5 58 1 744 ID73 i63i 1690 1748 1806 1 865 1923 19S1 2040 2098 2681 58! 745 2i56 22l5 2273 233 1 23.S9 2448 25o6 i 2554 2622 58: 746 2739 2797 2855 2913 2972 3o3o 3o88 : 3.46 3204 3262 58; ?s 3321 33?9 3437 3495 3553 36ii 3669 i 2127 3785 3844 58, 3902 3960 4018 4076 4i34 4192 425o 43o8 4366 4424 581 749 4482 4540 4598 I 4656 4714 4772 4830 488S 4y45 5oo3 581 750 875061 5ii9 5698 5.77 5235 5293 535i 5409 5466 5524 5582 58 1 75i 5640 5756 53 1 3 5871 5929 5987 1 6045 6102 6160 58 752 621 8 6276 5333 6391 6449 65o7 6564 6622 6680 6737 581 753 6795 6853 6910 6968 i 7026 70S3 7141 7199 7256 1 73U 58; 734 7371 T429 7487 7544 1 7602 ! 7639 7717 8349 7832 z^^^ 58 j 755 7947 8004 { 8062 8119 ! 8177 i 8234 ' 8292 1 8407 8464 57! 756 8522 8579 ! 8b37 8694 ! 8752 I 8809 i 8«66 ' 8924 8981 1 90 J9 P' 757 9096 9153 9211 925s ' 9325 1 93S3 ! 9440 ' 9497 ; 93" 9^'? 57 1 758 9669 9726 9784 , 9^41 i 9S98 ! 9956 i ••i3 ••70 ^127 ^183 ^'^i N. 880242 0299 o356 1 0413 j 0471 o528 1 o585 j 0642 j 0699 j 0706 1 37 i ' I : 2 : 3 ' 4 ; 5 ' 6 i 7 i 8 1 9 D. A TABLE OF LOGARITHMS FEOit 1 TO 10,000. 13 760 I 2 3 \ 4 ! 5 i 6 1 1 7 8 9 D. 880814 0871 0928 0985 1042 1099 Ii56 I2l3 1271 i328 "57 761 i385 1442 1499 1 556 i6i3 1670 1727 1784 1841 1898 57 762 1955 2012 2069 2126 2i83 2240 2297 2354 241 1 2468 57 763 2525 258 1 2638 2695 2702 2809 2366 2923 2980 3o37 57 764 3093 3i5o 3207 3264 3321 3377 3434 3491 3548 36o5 57 765 366 1 3718 3775 3832 3888 3945 4002 4C39 4ii5 4172 37 766 4229 4285 4342 4399 4455 45i2 4569 4625 4682 4739 57 767 4795 4852 4909 4960 5o22 5078 5i35 0192 5248 5303 57 768 536 1 5418 5474 5531 5587 5644 5700 5757 58i3 5870 U 769 5926 5983 6039 6096 6i52 6209 6265 6321 6378 ) 6434 770 S86491 6547 6604 6660 6716 ^1^1 6829 6885 6942 6998 56 771 7054 7111 7167 7223 7280 7336 7392 7449 7303 736. 56 772 7617 7674 773o 77S6 7842 7898 8460 7935 8011 8067 8123 56 773 8179 8236 8292 8348 8404 85i6 8573 8629 8685 56 774 874. 8797 8853 8909 8965 9021 9077 9'34 9190 9246 56 775 9302 9358 0414 9470 9526 9582 9638 9730 9806 56 776 9862 9918 i 9974 ••3o ••86 •141 •197 •233 •309 •365 56 777 890421 0477 o533 o589 0645 0700 0756 0812 0868 0924 56 778 0980 io35 1091 1 147 I203 1259 i3i4 1370 1426 1482 56 779 1537 1593 1649 1703 1760 1816 1872 1928 1983 2039 56 780 892095 2i5o 2206 2262 23i7 2373 2429 2484 2540 2595 56 781 265i 2707 2762 23i8 2873 2929 2985 3o4o 3096 3:31 56 782 3207 3262 33i8 3373 3429 3484 3 340 3595 365i 3706 56 783 3762 3817 3873 3928 3984 4039 4094 4130 42o5 4261 55 784 43 16 4371 4427 4482 4533 4593 4648 4704 4759 4814 55 785 4870 4925 4980 5o36 0091 5146 5201 5257 53i2 5367 55 786 5423 5478 5D33 5588 5644 0699 5754 5809 5864 5920 55 7?7 5975 6o3o 6oS5 6140 6195 6251 63o6 636i 6416 6471 55 788 6026 658i 6636 6692 6747 6802 6857 6912 6967 7022 55 789 7077 7i32 7187 7242 7297 7352 7407 7462 7317 7372 55 790 897627 7682 823i & 7792 7847 7902 7957 8012 8067 8122 55 791 8176 8341 8396 8451 83o6 8561 86i5 8670 55 792 8725 8780 8835 8890 8944 8999 9054 9:09 9164 9218 55! 793 9273 9328 9383 9437 9492 9547 9602 9636 971 1 9766 55 794 9821 9875 9930 ••39 ••94 •149 •203 •258 •3l2 55 795 900367 0422 0476 ?d3i o586 0640 0695 0749 0804 0859 55 796 0913 0968 1022 1077 ii3i 1186 1240 1295 1349 1404 55 797 1458 IDl3 1 567 1622 1676 1731 1785 1840 1894 1948 54 798 2oo3 2037 2112 2166 2221 2275 2329 2384 2438 2492 54 1 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3o36 54 1 Boo 903090 3i44 3199 3253 3307 3361 3416 3470 3524 3578 541 3oi 3633 3687 3741 3795 3849 3904 3953 4012 4066 4120 54 i 302 4174 4229 4283 4337 4391 4443 4499 4553 4607 4661 541 3o3 4716 4770 4824 4878 4932 4986 5o4o 5094 5i48 5202 541 804 5256 53io 5364 5418 5472 5526 558o 5634 5688 5742 54 8o5 5796 585o 5904 5958 6012 6066 6119 6658 6173 6227 6281 54; 806 6335 6389 6443 6497 655i 66o4 6712 6766 6820 54! ^ 6874 6927 6981 7035 7089 7143 7196 7250 73o4 7358 54 741 1 7465 7519 7573 7626 7680 7734 7787 7841 7895 843 1 54 809 7949 8002 8o56 8110 8i63 8217 8270 8324 8378 54 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 54 811 9021 9074 9128 9181 9235 9589 9342 9396 9449 95o3 54 812 9556 9610 9663 9716 9770 9823 9877 9930 9984 ••37 53 8i3 910091 0144 0197 025l o3o4 o358 041 1 0464 o5i8 0571 53 814 0624 0678 0731 0784 o838 0891 0944 0998 io5i 1 104 53 8i5 ii58 1211 1264 i3i7 137. 1424 1477 1 530 i584 1 637 53 816 1690 1743 1797 i85o 1903 1956 2009 2o63 2116 2169 53 817 818 2222 2275 2328 238i 2435 2488 2541 2594 2647 3178 2700 53 2753 2806 2859 2913 2966 3019 3072 3i25 3231 53 819 J284 3337 3390 3443 I 3496 3549 36o2 3655 3708 3761 53 N. __°_ I 2 3 ! 4 5 6 7 8 9 D. 14 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 1 I 1 2 3 4 i 5 6 7 8 1 9 820 9i38i4 3867 ; 392c 1 3975 4026 4079 4i32 4184 4237 4290 53 821 4343 4396 4449 1 45o2 4555 4608 4660 4713 4766 48.9 53 822 4872 4925 4977 1 5o3o 5o83 5i36 5189 5241 5294 5347 53 823 5400 5453 ' 55o5 5558 56ii 5664 5716 5769 5822 5»75 53 824 5927 59S0 1 6o33 6ob5 6i38 6191 6243 6296 6349 6401 53 825 6454 65o7 6559 66i2 6664 6717 6770 6822 6875 6927 :-3 826 6960 7033 ' 7ob5 7.38 7190 7243 7295 7348 7400 7453 531 827 7506 7558 761 1 7663 7716 7768 7820 7873 7925 7978 5^ 828 8o3o 8o83 ' 81 35 8 188 8240 8293 8345 8397 8450 85o2 52 829 8355 8607 1 8659 8712 1 8764 8816 8869 8921 8973 9026 52 83o 919078 9i3o 9183 9235 1 9287 9340 9392 9444 9496 9549 52 83 1 9601 9653 9706 9758 ^ qbio 9862 9914 9967 ••19 ••71 •52 832 920123, 0176 0228 0280 1 o332 o384 1 0436 0489 054. 0593 52 833 06^5 0697 0749 0801 1 o653 0906 0908 1010 .062 :ii4 52 834 1 1 66 1 2 1 8 1270 l322 i374 1426 1478 i53o .582 i634 52 835 1 686 1738 1790 1842 ■1894 :946 1998 2o5o 2.02 2.54 52 836 2206: 2258 2310 2362 2414 2466 25i8 2570 2622 2674 52 837 2725' 2777 2529 2d8i 2933 2985 3o37 3089 3 140 3.92 52 838 3244 3296 3348 3399 3451 35o3 3555 3607 3658 3710 52 839 3762, 3Hi4 3865 3917 3969 4021 4072 4124 4176 4228 52 840 924279 433i 4383 4434 4486 4538 4589 4641 4693 4744 52 841 479^ 4848 4«99 4951 5oo3 5o54 5io6 5i57 5209 5261 02 842 53i2| 5364 5410 5467 55i8 5570 5621 5673 5725 5776 52 843 3028; 5b79 5931 0902 j 6o34 6ob5 6137 6i88 6240 6291 5i 844 6342' 6394 6445 6497 6548 6600 665 1 6702 6754 6bo5 5i 845 6bJ7 6908 6959 7011 7062 7114 7i65 7216 7268 73.9 5i 846 7370 7422 7473 7524 ! 7576 7627 7678 7730 7781 7832 8345 5i 847 7bt)3 7935 7986 8037 8o»8 8140 9191 8242 8293 5i 848 8396 8447 8498 8549 8601 8652 8703 8754 88o5 8857 5i! 849 89081 8959 9010 906*1 9112 9163 92i5 9266 93.7 9368 5i| 85o 929419" 9470 9521 9572 9623 9674 97.25 9776 9827 9879 5ii 85 1 9y3o 9981 ••32 ••63 •i34 •i85 •236 •287 •33S •389 5i 852 930440 1 0491 o542 0592 0643 0694 0745 0796 0847 0898 i'\ 853 0949 1000 io5i 1102 1153 1204 1254 l30D .356 1407 5i 854 1458 1 509 i56o 1610 1661 1712 1763 1814 1865 1915 5i 855 1906 2017 2068 2118 2169 2220 2271 2322 2372 2423 5i 856 2474 2524 2575 2026 2677 2727 2778 2829 ^^^> 2930 5i 857 2981 3o3i 3o«2 3i33 3i83 3234 3285 3335 3386 3437 5i 858 3487 3538 3589 3039 3690 3740 3791 3841 3»92 J943 5i 859 3993 1 4044 ■4094 4145 4.95 4246 4296 4347 4397 4448 5i 860 934498 1 4549 4599 4650 4700 4731 4801 4852 4002 4953 5o 861 5oo3, 5o54 5io4 5i54 52o5 5255 53 06 5356 5406 5457 5o 862 5oo7; 5558 56o8 5658 5709 5759 5809 5860 5910 5960 ^ 863 6011 6061 6!I1 6162 6212 6262 63i3 6363 64.3 6463 5o 864 65 14 6564 6614 6665 6715 6765 68i5 6865 6916 6966 5o 865 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 5o 866 7518 7568 8019 8069 7618 7668 7718 7769 & 7869 79'9 7969 5o 867 8,19 6169 ; 8219 8269 8370 8420 S470 5o 863 8520 8570 8620 8670 8720 8770 8820 G870 8920 8970 5o! 869 9020| 9070 9120 9,70 9220 9270 9320 9369 9419 9469 5o 870 939519 9569 9619 9669 9719 9769 98.9 9869 9918 9968 r\ 871 940018 006S ou8 0168 1 02l8 0267 o3i7 o367 0417 0-,67 ^° 872 o5i6 o566 0616 0666 i 0716 0765 o8i5 o865 09.5 0964 5oj 873 1014 1064 1114 ii63 j I2i3 1263 i3i3 i362 1412 1462 5o' 84 i5ii i56i 1611 1660 I 1710 1760 1809 1 859 1909 1958 5oi 875 2008 2o58 2107 2i57 ! 2207 2256 23o6 2355 24o5 2455 5o: 876 25o4 2554 26o3 2653 1 2702 2752 2801 285 1 2901 3396 2950 5o 877 3ooo 3049 3099 3148 ! 3198 3247 3297 3346 3445 59^ 878 3495 3544 3593 3643 ; 3592 3742 1 3791 3841 3.S90 3q3Q do; 879 3989 4o38 4088 4137 1 4186 .4236 i 4285 4335 43«4 4433 59 N. j I a 3 i 4 5 i 6 7 8 ^ D.\ A TABLE OF LOGARITHMS FROM 1 TO 10,000. 15 1 N. I 1 2 3 4 I 3 6 7 8 9 D. 49 8So 944483 , 4532 1 45Si 463 1 ! 4680 1 4729 4779 4828 |4877 4927 88 1 4976 ' 3025 3074 3124 5173 5222 1 5272 5321 5370 5419 49 882 546g '5518 5567 i 56i6 5665 5715 1 3764 5^13 3662 5912 49 883 596, 60 1 c 6009 6io3 I 6157 6207 6256 63o5 6354 6.io3 49 884 • 6432 ■ 6301 ' 655i 6600 j 6649 6698 6747 6796 6845 6894 49 885 i 6943 : 6992 1 704 t 7090 7140 7189 7233 7287 1 7336 7335 1 49 886 1 7434 ' 74&3 i 7532 75ai 763o 7679 7723 7777 1 7^26 7873 1 49 887 ! 7924 ! 8413 i 7973 8022 ] 8070 81 19 8i63 8217 8266 83 1 5 1 8364 1 49 888 8462 85 1 1 856o [ 8609 8657 1 8706 8755 : S804 8853 I 49 889 i 8902 8931 ; 8999 1 9^48 j 9097 J146 ' 9195 , 9244 9292 [ 9341 49 890 949390 9439 ' 9488 gojo 9585 9634 9683 9731 9780 9829 49 891 9376 9926 9975 ••24 ••73 •121 •170 •219 •267 •3i6 49 892 930.J63 0414 0462 o5ii o56o 0608 o657 0706 0754 o-.o3 49 893 0001 0900 0949 i 0997 1046 1093 1143 1192 1240 1289 49 894 i338 i386 1435 14^3 i532 i5:io 1629 1677 1726 1775 it 895 lb23 1872 IQ20 1 1969 20IT 2066 2II4 2i63 2211 2260 896 23oti 2356 2403 ! 2433 25o2 255o 2599 2647 2096 2744 48 897 2792 2841 2839 2933 2986 3o34 3os3 3i3i 3i8o 3228 48 898 3276 3325 3373 342. 3470 35i8 3566 36i5 3663 37.1 48 899 3700 3do8 3856 3905 3953 4001 4049 ; 4098 4146 4194 48 900 904243 4291 4339 4387 4435 4484 4532 ' 458o 4628 4677 48 901 4723 4773 4821 4809 4918 4966 5oi4 5o62 5iio 5i58 48 902 D207 5255 53o3 535, 5399 5447 5495 5543 5592 5b4o 48 903 5o»8 5735 5784 1 5o32 SSdo 5928 5976 6024 6072 6120 48 904 6i6d 6216 6265 I 63 1 3 6361 6409 6457 65o5 6553 6601 48 905. 60^9 6697 6745 1 6793 1 6846 68^8 6936 6934 7082 7080 48 ^06 7120 7176 7224 7272 7320 7363 7416 7464 7512. 7339 48 907 7007 7655 7703 7731 7799 7847 7894 7942 7990 8o33 48 908 8006 8i34 8181 8229 8277 8325 8373 8421 8406 85i6 48 909 8564 8612 8659 8707 8755 88o3 885o &898 8946 8994 48 910 939041 90S9 9137 9i85 9232 9280 9328 9375 9423 9471 48 911 9018 9566 9614 9561 9709 9737 9804 9S52 9900 9947 48 912 9^93 ••42 ••90 •i33 •183 •233 •2iO •323 •J7& •423 48 913 9^0^71 03 1 8 o566 06.3 0661 0709 0706 0S04 o35i o«99 48 914 09-^6 0994 1 04 1 1089 Ii36 1184 123l 1279 i326 1374 47 li5 1421 1469 i5i6 1 i563 1 1611 i658 1706 1753 1801 1848 47 916 1693 1943 1990 203S 2o85 2l32 2i3o 2227 2275 232-2 47 9n 2369 2843 2417 2464 25n 2559 2606 2653 2701 2748 2795 47 918 2890 ; 2937 2985 3o32 3079 3i26 3i74 3221 3268 47 919 33i6 3363 . 3410 3457 35o4 3552 3599 3646 3693 3741 47 920 963788 3835 ' 3882 3929 3977 4024 4071 41 18 41 65 4212 47 921 4260 4307 ! 4354 4401 1 4448 4493 4542 4590 4637 4684 47 922 4731 4778 ■ 4825 4872 1 4919 4966. 5oi3 5o6i 5io3 5.55 47 923 3202 5249 ■ 3296 5343 5390 5437 54^4 553 1 5578 5625 47 924 5672 3719 ; 5766 58i3 5%o 3907 3954 6001 6048 6095 47 925 6142 6189 ; 6236 62S3 6329 6376. 6423 6470 65 17 6564 47 926 66II 6658 1 6705 6752 6-' 99 6843 6.S92 6939 74o3 6986 7033 47 927 928 7080 7127 7173 7220 7267 73i4 736. 7454 7501 47 7348 7595 7642 7688 7735 77S2. 7829 7875 7922 7969 47 r-9 8016 8062 8109 8i56 8203 8249 8296 8343 8890 8436 47 930 08483 853o • 8576 8623 8670 8716 8763 8810 8856 8903 47 931 8900 8996 9043 9090 9i36 9x83: , 9229 9276 9823 ' 9369 47 ^}l 9416 9463 , 9309 9556 9602 9649- 9693 9742 9789 9^35 47 933 9^82 9928 9973 ••21 ••63 •ii4 •161 •207 1 •254 j ^300 4? 934 Q70347 0893 0440 0486 o533 0379 0626 0672 0719 1 0765 40 q3d 0812 o853 0904 0951 0997 1044 1090 1.37 ii33 j 1229 46 936 1276 i322 i369 I4i5 1461 i5o8 1 554 1601 1647 1693 46 9^:^ 1740 1786 i i832 1879 1925 2342 1 2383 1 97 1 20l3 2064 2110 2i57 46 938 22o3 2249 ' 2295 2434 2481 2027 2573 1 2619 46 939 2666 2712 i 2758 2804 ! 285i 1 2897 2943 2989 3o35 j 3082 46 N. I '1 2 j 3 1 4 ! 5 1 6 7 8 9 1 I>. 26 16 A TABLE OF LOGARITHMS FKOil 1 TO 10,000. K. 940 94a 943 944 945 946 947 948 949 930 901 932 954 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 976 97J 979 982 9? 983 9? 989 990 99« 99» 99^ 994 996 997 998 999 973i2d' 0174 3590' 3636 403I 4097 4312 4338 4972; 5oi8 5432 5478 5891 6330P 6808} 7266! 7312 97T724 .. 8181 82; 8637 8683 9093 9138 9548 9394 980003 0049 0438 o5o3 0912 0957 i366 141 1 1819 1864 982271; 23i6 2723 2769 3175, 3220 3626 3671 40771 4122 4327 4572 4977 1 0022 5426 5471 58751 5920 6324 6369 986772! 6817 7219' 7264 7666 771 1 8ii3 8137 8339 8604 9003 9049 9430 94c, 9: oi3i 956: >:■ Uzo 3220 3682 4143 4604 5o64 5324 5983 6442 7358 7815 8272 8728 9184 9639 0094 o549 ioo3 1456 1909 2362 2814 3265 3716 4167 4617 5067 55i6 5965 64i3 6861 7309 ?I^ c:c-. 3266 3728 4189 465o 5iio 5570 6029 6488 6946 74o3 7861 83i7 9774 9230 9685 0140 0594 1048 i5oi 1954 2407 2859 33io I 9 33i3 I 33>G 3774 382 4235 1 42 4696 ■- 3136 56i6 60-5 :; 6533 c; ■ "- 3^0- 3^.3 • _ - - ; -3 Iir2 D. 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 45 45 45 -3 _5 43 45 45 45 45 45 44 44 44 44 i4 43 A TABLE OF LOGAEITHMIC SINES AND TANGENTS FOR EVERY DEGREE AND MINUTE OF THE QUADEANT. Eemaek. The minutes in the left-hand column of eacli 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 D. ' Cosine D. Tang. D Cor&ng. 1 o 10.000000 0- 000000 1 Infinite. : 60 I 6-463726 5017.17 000000 •00 6-463726 5017.17 13-536274 2 764736 2934-85 000000 .00 764756 2934 • 83 235244 3 940H47 2082.31 000000 .00 940S47 2082 -3i 059153 57 4 7'065786 1615.17 000000 •00 7.065786 i6i5 n 12-934214 637304 56 s 162696 1319-68 000000 -00 162696 i3i9 ni5 ^i : 55 6 241877 III5-75 9.999999 •01 241878 758122 ; 54 I 308824 966-53 999999 •01 30.S825 §?? 53 691 175 i 53 3668 1 6 852-54 999999 •01 366817 54 633 1 83 1 52 9 417968 762-63 999999 999998 •01 417970 762 63 582o3o 5i 10 463725 689-88 •01 463727 689 88 536273 5o II 7.5o5n8 629.81 9.999998 • 01 7.505 120 629 81 12-494880 40 43 12 542906 579-36 999997 •01 542909 579 33 457091 i3 577668 536-41 999997 -01 577672 536 42 422328 47 14 609853 499-38 999996 -01 609857 499 39 i5 390143 I 46 i5 689816 467.14 999996 •01 639820 467 36oi8o 45 i6 667845 438-81 999995 •01 667H49 438 82 332 i5i 44 'I 694173 413-72 999993 •01 694179 4i3 73 3o582i 43 718997 391-35 999994 •01 719004 39, 36 280997 i 42 •9 742477 371-27 999993 -01 742484 371 28 257316 ! 41 20 764754 353-15 999993 •01 764761 35i 26 235239 40 21 7-785943 806146 336-72 9.999992 •01 7.785951 336 73 13.214049 193845 u 22 321.75 999991 •01 806 1 55 321 76 23 825451 3o8-oj 999990 999980 999988 -0] 825460 3o8 06 174540 ll 24 843934 itu •02 843944 295 283 49 i56o56 25 861662 •02 861674 90 138326 35 26 878695 895085 273-17 999988 •02 878708 273 18 121292 34 27 263-23 999987 • a2 895099 263 25 104901 33 28 910879 lloll 999986 •02 & 254 01 089106 073866 32 29 9261 19 999985 •02 245 40 3i 3o . 940842 237-33 999983 •02 940858 237 35 059142 3o 3i 7-955o82 229-80 9.999982 •02 7.955100 229 81 12.044900 li 32 968870 222-73 999981 •02 968889 222. 75 o3iiii 33 982233 216-08 999980 •02 982253 216. 10 017747 27 34 995108 8.007787 209-81 203.90 198.31 999979 •02 995219 209 83 004781 26 35 999977 999976 •02 8-007809 2o3 92 11-992191 23 36 020021 •02 020043 198 33 9-99P5 968055 24 U 031919 193-02 999975 -02 031945 o5 23 043 5o I I88-OI 999973 -02 043327 188 o3 956473 22 2q 054781 183-25 999972 •02 054809 1 83 27 945191 21 46 065776 178-72 999971 •02 o658o6 178 74 934194 2a 41 8 -076500 174-41 9.999969 •02 8-076531 174 44 11-928469 \l 42 086965 170-31 999968 -02 0S6997 170. 34 9i3oo3 43 097183 166-39 162-65 999966 •02 097217 166. 42 ^02^83 n 44 107167 999964 •o3 107202 162- 68 8&037 16 45 116926 159-08 999963 •o3 116963 159. 10 i5 46 1 2647 1 153-66 999961 •o3 i265io l53. 68 873490 14 47 i358io 152-38 999959 999958 •o3 i3585i l52. 41 864149 i3 48 144953 149-24 •o3 144Q96 153952 149- 27 855004 12 49 153907 146-22 999950 •o3 146- 27 846048 11 5o 162681 143.33 999954 •o3 162727 143- 36 837273 10 5i 3-171280 140-54 9.999952 •o3 8^i7i328 140- 57 11-82S672 I 52 179713 137-86 999950 •03 179763 i88o36 i37- T. 820237 53 187985 135.39 999948 •03 i35. 81 1964 I 54 196102 i32.8o 999946 .03 iq6i56 l32. 84 808844 55 2040-'0 130-41 999944 .o3 204126 i3o. 44 795«74 788047 5 56 liiSoi 128.10 999942 .04 211953 219641 128. 14 4 57 219581 125.87 999940 .04 125- 90 780359 3 58 227134 123.72 999938 • 04 227195 123. 76 772803 2 59 234557 121.64 999936 •04 234621 121. 68 765379 I. 66 241855 119. 63 999934 • 04 241921 119-67 758079 Cosine D. Sine 1 1 1 Cotang. D. ' Tang. M. (89 DEGREES.) SINES AND TANGENTS. (1 DEGKEE.) 19 M. Siue D Cosine D. Tang. D Cotimj?. 60 o 8.24185:5 119 63 9.999934 .04 8.241921 119 .67 11-758079 I 249033 117 68 999932 .04 249102 117 •72 ll 2 256094 ii5 80 999929 • 04 256i65 ii5 •84 3 263o42 ii3 98 999927 • 04 263ii5 114 .02 736885 57 4 269881 112 21 999925 • 04 269956 112 •25 73oo.i4 , 56 ! 5 276614 no 5o 999922 .04 276691 no •54 723309 55 6 283243 108 83 999920 •04 283323 108 .87 716677 54 7 289773 107 21 ■ 9999 '8 • 04 289806 107 .26 710144 53 8 296207 io5 65 999915 • 04 296292 io5 .70 703708 52 9 302546 104 i3 999913 .04 3o2634 104 .18 697366 5i 10 308794 102 66 999910 .04 308884 102 .70 691116 5o II 8.314904 lOI 22 9.999907 .04 8.3i5o46 101 .26 11-684954 % 12 321027 l^ 82 999905 .04 32II22 ^ .87 678878 i3 327016 47 999902 • 04 327114 .5i 672886 47 14 332924 97 14 999899 c5 333025 97 .19 666975 46 i5 338753 9D 86 999897 o5 338856 95 90 661144 45 i6 344304 94 60 999894 o5 344610 94 65 655390 44 \l 35oi8i 93 38 999891 o5 3502S9 93 43 6497 « I 43 355783 92 •i? 999888 o5 35589D 92 24 644 1 o5 42 19 36i3i5 % 999885 o5 36i43o 9' 08 638570 41 20 366777 90 999882 o5 366895 89 95 633 I o5 40 21 8.372171 88 80 9.999879 o5 8.372292 88 85 11-627708 It 22 377499 87 72 999876 o5 377622 382889 87 77 622378 23 382762 86 67 999873 o5 86 72 617111 37 24 387962 85 64 999870 o5 388092 393234 85 70 611908 36 25 393101 84 64 999%7 o5 84 70 606766 35 26 398179 83 66 999864 o5 398315 83 71 601685 34 27 4o3i99 82 71 999861 o5 ' 403338 82 It 596662 33 28 408161 81 77 999858 o5 4o83o4 81 591696 32 29 4i3o68 80 86 999854 o5 4i32i3 80 91 586787 3i So 4I79J9 79 96 999851 06 418068 80 02 581932 3o 3i 8.422717 ?? 09 23 9.999848 06 8.422869 ?^ 14 11.577131 It 32 427462 999844 06 427618 3o 572382 33 432156 77 40 999841 06 43231 5 77 45 567685 27 34 436800 76 57 999838 06 436962 76 63 563o38 26 35 441394 75 77 999834 06 . 441360 75 83 558440 25 36 445941 74 99 999831 06 446110 75 o5 553890 24 37 450440 74 22 999827 06 45o6i3 74 28 549387 23 38 454893 73 46 999823 06 455070 73 52 544930 22 39 439301 72 73 999820 06 459481 463849 72 79 540019 21 40 463665 72 00 999816 06 72 06 536i5i 20 41 8-467985 71 29 9.999812 06 8.468172 71 35 11.531828 \l 42 472263 70 60 999809 06 472454 70 66 527546 43 47^498 69 91 999805 06 476693 480892 69 t 523307 17 44 480693 tl 24 999801 06 tl 519108 16 45 484848 59 999797 07 485o5o 65 5 I 4950 5io83o i5 46 488963 67 94 999793 07 489170 493250 68 01 i4 47 493040 67 3x 999790 999786 07 67 38 506750 i3 48 497078 66 69 07 497293 66 76 502707 12 49 5oio8o 66 08 999782 07 501298 66 i5 498702 n 5o 5o5o45 65 48 999778 07 505267 65 55 494733 10 5i 'VX 64 89 9-999774 07 8.509200 64 t 11-490800 486902 t 52 64 3i 999769 07 513098 64 53 516726 63 75 999765 07 516961 63 82 483o39 7 54 52o55i 63 19 999761 07 520790 63 26 479210 6 55 524343 62 64 999757 07 524586 62 72 475414 5 56 528102 62 II 999753 " 07 528349 62 18 47i65i 4 u 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 60 62 460553 1 60 542819 60-04 99973d 07 543084 60.12 456916 Cosine D. Sine 1 Cotang. D. Tang 16 (88 DEGREES.) 20 (2 DEGREES.) A TABLE OF LOGARITHMIC M. Sino D. Cosine D. Tang. D. Cotaug. 1 o 8.542HI9 60-04 9.999735 -07 8-548084 60.12 It .456916 i 60 I 546422 59.55 999781 -07 546691 59 •62 453809 I 59 2 Itml 59.06 58.58 999726 .07 550268 •14 449732 58 3 999722 .08 558817 58 •66 446188 57 4 557054 58.11 999717 -08 557886 58 I'i 442664 56 5 56o54o 57-65 999718 .08 560828 57 439172 433709 482278 55 6 563999 57.19 999708 .08 564291 57 27 54 I 567431 56.74 999704 -08 567727 56 82 53 570836 56-80 999699 -08 571187 56 38 428863 ! 52 i 9 574214 55.87 999689 .08 574520 55 t 425480 5i IC 577566 55.44 .08 577877 55 422128 5o II 8.580892 55.02 9.999685 .08 8-58i2o8 55 10 11-418792 it 12 584193 54-60 999680 -08 584314 54 68 413486 i3 587469 54-19 999675 -08 587795 54 27 4l2205 47 14 590721 53.79 999670 .08 59.061 53 87 408949 46 i5 593948 53.39 999665 .08 594288 53 47 405717 4025(58 45 i6 597T52 58.00 999660 -08 597492 58 08 44 \l 6oo332 52.61 999655 -08 600677 52 70 899828 43 603489 606623 52-23 999650 .08 608889 606978 52 32 396161 42 »9 51-86 999645 .09 5i n 898022 3^9906 41 20 609734 5i 49 999640 .09 610094 5i 40 21 8-612823 51.12 9.999635 .09 8-618189 5i 21 1I-3868II 39 22 610891 6189^7 50.76 999629 •Of; 616262 5o 85 388788 3i 23 50-41 999624 -09 619818 5o 5o 880687 37 24 621962 5o-o6 999619 -09 622843 5o i5 377657 36 25 624965 49-72 999614 -09 625832 49 81 374648 35 26 627948 49-38 999608 .09 628840 ^ 47 371660 34 27 630911 633854 49-04 48-71 999608 -09 681808 a 18 368692 33 28 999397 -09 634256 80 365744 362816 32 29 686776 48.39 & •09 687184 48 48 3i 3o 689680 48.06 .09 640093 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 643853 47 58 354147 33 648274 47-12 999570 -09 648704 47 22 351296 27 34 65iio2 46.82 999564 -09 65i587 46 91 348468 26 35 6589II 46.52 999558 •10 654352 46 61 345648 25 36 656702 46-22 999553 -10 657149 46 3i 342851 24 37 659475 45.92 999547 . -10 639928 46 02 340072 23 38 662280 45.63 999541 •10 662689 45 73 887311 22 39 664968 45.35 999535 -10 665438 45 44 334567 21 40 667689 45-06 999529 -10 668 160 45 26 881840 20 41 8.670893 678080 44-79 9-999524 -10 8-670870 44 88 11.829180 826487 ]i 42 44-51 999518 -10 673563 44 61 43 675751 44-24 999312 -10 676289 44 34 828761 17 44 678405 43-97 999506 •10 678900 44 17 32II00 16 45 681043 43-70 999500 -10 681344 43 80 3 1 8456 i5 46 683665 43.44 999408 999487 -10 684172 48 54 3i5828 14 tl 686272 43.18 -10 686784 48 28 3i32i6 i3 688863 42-92 999481 -10 689881 43 08 810619 12 49 691488 42-67 999475 •10 691968 694029 42 77 808087 II 5d 698998 42.42 999469 •10 42 52 3o547i :o 5i 8 696548 42.17 9.999463 • II 8-697081 42 28 II. 802919 3oo383 t 52 699078 41.92 999456 -11 699617 42 o3 53 701589 41.68 999400 • n 702189 41 79 297861 1 54 .704090 41-44 999443 -11 704646 41 53 295354 6 55 706577 41-21 999487 -II 707140 41 82 292860 5 56 709049 40-97 999431 -11 709618 41 08 290882 28546? 4 u 7a 1 507 718952 7^6383 40.74 40 -51 999424 999418 -11 -11 712083 714534 40 40. 85 62 3 2 59 40.29 99941 1 • II 716072 719396 40. 40 288028 I 60 718800 40.06 999404 -11 40-17 280604 M. Cosine J), Sme Cotang. D- 1 Tang. (87 DEGREES.) SINES AND TANGENTS, (3 DEGKEES., 21 "m. fcjiiie D. Co.--iiie 1>. Tang, D. Cotang. 60 8-7i88oo 40.06 9.999404 .11 8.719896 40-17 11.280604 I 721204 39.84 999398 .11 721806 89-95 278194 It 2 723595 39.62 999391 •11 724204 39-74 275796 3 725972 728337 780688 39.41 999384 • II 726588 89.53 278412 57 4 39-10 38-98 999378 • II 728959 781817 89-80 271041 56 5 909871 •11 89.09 268683 55 6 733027 38-77 999864 .12 788668 88-89 266887 54 I 735354 38-57 999357 •12 785996 38-68 264004 53 737667 38-36 999350 •12 788817 38-48 261683 52 9 739969 38.16 999343 •12 740626 38-27 259874 5i 10 742259 37-96 999886 •12 742922 38-07 257078 5o II 8.744536 mt 9.999829 • 12 8.745207 37-87 11-254793 ^2 12 746802 999822 •12 747479 37-68 252521 48 i3 749055 37-37 999813 .12 ■ 749740 37-49 25o26o 47 14 751297 37.17 999808 .12 751989 87-29 24801 1 46 i5 753528 36.98 999801 .12 754227 87.10 245773 .45 i6 755747 36.79 999294 .12 756458 86.92 248547 44 \l 757955 36.61 999286 .12 758668 36.78 241882 43 76oi5i 36-42 999279 .12 760872 36-55 289128 42 19 762337 36-24 999272 .12 768065 36.86 286985 41 20 7645 1 1 36 -06 999265 .12 765246 36.18 284754 40 21 8.766675 35.88 9.999257 •12 8.767417 36-00 11-282583 39 22 768828 35.70 999250 .13 769578 35-88 280422 88 23 770970 35.53 999242 •l3 771727 35-65 228278 ll 24 773101 35.35 999283 .13 778866 35-48 226184 25 773223 35.18 999227 .13 775995 35-3i 224005 85 26 777333 35-01 999220 •i3 778114 35-14 221886 84 11 7^9434 34-84 999212 •i3 780222 34-97 219778 38 •78i524 34-67 999205 • i3 782820 34-80 217680 32 29 7836o5 34-51 999197 .18 784408 34-64 215592 3i 3o 785675 34-31 999189 •i3 786486 34-47 2i85i4 3o 3i 8.787736 34-18 9.999181 • i3 8.788554 34-3i 11-211446 29 32 789787 34-02 999 '74 • 18 790613 34- 15 209887 28 33 791828 33-86 999166 .i3 792662 38-99 207888 27 34 793859 33-70 999158 .13 794701 33-83 205299 26 35 795881 33-54 9991 5o .i3 796781 33-68 208269 201248 25 36 797894 33-39 999142 .18 798752 33-52 24 ll 799897 33-23 999134 .18 800768 33-37 199287 197285 23 001892 33-08 999126 .i3 802765 33-22 22 39 803876 32.93 099118 • 13 804758 33-07 195242 21 40 8o5852 32.78 9991 10 • 13 806742 32-92 198258 20 41 8.807819 32-63 9.999102 • i3 8.808717 32-78 11.191283 189817 \l 42 809777 32-49 999094 •14 810688 82-62 43 811726 32.34 999086 -14 81 2641 32-48 187859 17 H 8i366t 32.19 32.05 999077 •14 814589 32-33 185411 16 45 815599 999069 •14 816529 82-19 32-o5 183471 i5 46 •817522 31.91 999061 •14 8 1 8461 181589 14 tl 819436 31-77 999053 -14 820884 81-91 179616 i3 821343 3i.63 999044 •14 822298 31-77 177702 12 49 823240 3 1. '49 999086 •14 824205 31-63 175795 11 5o 825i3o 31.35 999027 •14 826108 3i-5o 178897 10 5. 8.827011 3l.22 9.999Q19 •14 8-827992 829874 31-36 11.172008 I 5j 828884 3i.o8 999010 •14 31-23 170126 53 830749 3o.o5 30.82 •14 881748 3i.io 168252 7 54 832607 •14 8386i3 30-96 3o-83 166887 6 55 834456 3o.6g •14 885471 164529 5 56 836297 30.56 998976 •14 887821 30-70 162679 4 u 838i3o 30.43 998967 998958 .i5 889168 80-57 160887 3 839956 3o.3o .i5 840998 30-45 159002 2 59 liuii 3ovi7 998950 • 15 842825 30-32 157175 I 6o 3o.oo 998941 .15 844644 80-19 155356 Cosine D. : Sine Cotang. i D. Tang. IVL (86 DEGREES.) (4 DEGREES.:* A TABLE OF LOGARITHMIC M. Sine D. CJoeine D. Tang. D. Cotang. P 8-843585 3o.o5 9.998941 • 15 8-844644 30-19 11-155356 I 845 3S7 29-92 99H932 •ID 846455 30-07 153545 u 3 847183 2980 99^923 • 15 848260 29.05 29.82 i5i74o 3 84S971 29-67 9989,4 •i5 85oo57 ',imi ! 5 4 85o75i 29-55 998905 .i5 85 I 846 29.70 5 852525 29-43 998806 998887 • 15 853628 29.58 146372 ! 55 6 854291 29-3l .15 8554^3 29-46 144597 1 54 7 856049 29-19 99SS78 .15 857171 29-35 142829 ! 53 141068 52 8 857801 TJ, 998S69 •15 858932 29-23 9 859546 998S60 •15 8606S6 29.11 139314 5i IC 861283 28.84 998851 •i5 862433 29-00 137567 5o II (j-863oi4 28-73 9-99MS41 .15 8.864173 28-88 11.135827 t 12 864733 28-61 998832 .i5 865906 28 -t7 134094 i3 866455 28.50 998823 .16 867632 28-66 132368 47 14 868 165 28.39 99S813 .16 869351 28.54 130649 46 i5 869*^68 28-28 99S804 .16 871064 28.43 128936 45 i6 871565 28 17 99S795 16 872770 28-32 12-230 44 17 873255 28-06 99^785 16 874469 1 28-21 12^531 43 18 8749^8 27-95 908776 .16 876162 28-11 123838 42 19 876615 27.86 998766 .16 877849 28-00 12?l5l 41 20 878285 27-73 998757 • 16 879529 27.89 1:0471 40 21 8-819949 27^63 9-998747 .16 8-881202 27- 79 11.118^98 39 22 as 1607 27.52 998738 .16 8«2869 27-68 117l3l 38 23 883258 27.42 99«728 .16 884530 27-58 1 1 5470 37 24 884903 27-31 99^718 .16 886i85 27-47 ii3Si5 36 25 886542 27-21 99S-708 .16 887833 07.37 112167 35 26 8S8174 27-11 99S699 .16 -889476 27-27 II0524 34 27 8S9801 27-00 9986H9 .16 891112 27-17 108888 33 28 891 42 1 26-90 99S679 .16 892742 27.07 10^258 32 1 29 893035 26 -So 99^^669 -n 894366 26-97 105634 3i So 894643 26-70 998659 •n 895984 26-87 104016 3o 3i 8-896246 26-60 9 -99^649 •17 8.897596 26.77 11-102404 S 32 807843 26-51 998639 •17 899203 26-67 100797 33 899432 26.41 998629 •n 900803 26-58 099197 27 34 901017 26.31 99% 1 9 •n 902398 26-48 09-602 26 35 902596 26-22 998609 •17 903987 26-38 096013 25 36 904169 26-12 998599 •17 905570 26-29 094430 24 ll 905736 26-03 998589 •»7 907147 26-20 092853 23 907297 25-93 998578 •n 908719 26-10 091 281 22 39 908853 25-84 998568 •n 910285 26-01 oS?i54 21 40 910404 25-75 998558 •17 911846 25-92 20 41 'V^U 25-66 9-998548 •17 8.913401 25-83 i,.d86599 'R 42 25-56 998537 •'7 914951 25--4 o'Jno4Q 18 43 9l5o2-2 25-47 91^8527 •17 916495 918034 25-65 o^^'f'^o^ '7 44 9i655o 25-38 998516 •i8 25-56 o8f96'> 16 45 91S073 25-29 998506 •18 919568 25-47 080432 ! i5 46 9 1 9591 25-20 99^495 •18 921096 25-38 0-8904 ; :4 47 921.03 25-12 99«485 •18 922619 25 -3o °'ll^' i3 48 922610 25 -o3 99«474 •18 924136 :5-2i 073864 la 49 934112 24-94 99«464 .18 925649 25-12 074351 ! II 5o 935609 24-86 99«453 •18 927156 25-o3 072844 j 10 5i 8 92-rioo 24-77 9 •99-3442 .»8 8-928658 24-95 24-86 II 07134: j t 1 52 928587 24-69 998431 .18 9301 55 069845 068353 53 930068 24-60 99*^431 .18 931647 24-78 1 54 93 1 544 24-52 998410 .18 933,34 24-70 066866 5 55 93301 5 24-43 99S399 .18 934616 24-6> 0653.84 5 56 934481 24.35 998388 .]8 o36oq3 24-53 o63qo7 4 u 935942 24-27 998377 .18 937565 24-45 062435 3 937398 24-19 998366 .18 939032 24-37 060968 a 59 9^8850 24-11 99<^355 .18 940404 24 -30 o5o5o6 1 60 940296 24 -o3 99^^344 ■18 9419^2 24-21 008048 [ I I Cosine D. Sine i 1 Cotang D. Tang. 1 M. (85 DEGREES.) SINES AND TANGENTS. (5 DEGREE.) 23 M. Sine D. Cosine D. Tang. D. Cotiuig. 8-040296 24 -03 9.998344 .19 8.941952 24-21 1 1 -058048 60 I 941738 23 :l^ 998333 .19 948404 24.13 056596 u 2 943174 28 998822 •19 944852 24.05 o55i48 3 944606 23 •79 998811 .19 946295 947734 23-97 o537o5 u 4 94.6084 23 •71 998800 .19 23.90 052266 5 947456 23 .63 998289 .19 949168 23-82 o5o832 55 6 948874 23 .55 998277 .19 950597 28.74 349408 54 7 930287 28 ■48 998266 .19 952021 23.66 047979 ! 53 8 951696 28 •40 998255 .19 953441 28-60 046559 52 9 953100 23 .32 998243 .19 954856 28.51 045144 5i 10 954499 23 •25 998282 .19 956267 23.44 048788 5o 12 8. 955^94 957284 23 23 •17 .lO 9.998220 998209 •19 .19 8.957674 959075 23-37 28.29 11-042826 040925 8 i3 938670 23 .02 998197 .19 960473 28-23 089527 47 U 960052 22 .tl 998186 .19 961866 28-14 088184 46 1 5 961429 22 998174 .19 968255 23.07 036745 45 i6 962801 22 .80 998163 .19 964689 23.00 o858bi 44 \l 964170 22 .73 998151 .19 966019 22.93 22.86 088981 43 965534 22 • 66 998.89 998128 •20 967394 968766 082606 42 ^9 966893 22 .59 •20 22-79 081284 41 20 968249 22 .52 998116 •20 970133 22.71 029867 40 2l 8.969600 22 •44 9.908104 •20 8-971496 22-65 11-028504 ll 22 970947 22 38 998092 998080 •20 972835 22.57 027145 23 972289 22 3i •20 974209 22.51 025791 37 24 978628 22 24 998068 •20 975560 22.44 024440 36 25 974q62 22 •7 998056 •20 976906 22.87 028094 35 26 976293 22 10 998044 •20 978248 22.30 021752 34 11 977619 22 o3 998082 •20 979586 22-23 020414 33 978941 21 97 998020 •20 980921 22-17 019079 32 ?9 980259 981573 21 r. 998008 •20 982251 22-10 01^749 3i So 21 997996 •20 988577 22-04 016428 3o 3i 8.982883 21 11 9.997985 •20 8-984899 21-97 ii.oi5ioi 29 28 32 984189 21 70 997972 •20 986217 21-91 018788 33 985491 21 63 997959 •20 987582 21-84 012468 27 34 ^a 21 57 997947 •20 988842 21.78 oiii58 26 35 21 5o 997935 •21 990149 21.71 ooq85i 008549 25 36 989374 21 44 997922 •21 99:451 21.65 24 57 990660 21 88 997910 •21 992750 2t-58 007250 23 38 991943 2i 81 997897 •21 994045 21-52 005955 22 39 998222 21 25 997885 •21 995887 21-46 004663 21 4o 994497 21 ^9 997872 •21 996624 21.40 008876 20 4i 8.995768 21 12 9.997860 •21 8.997908 21-34 11-002092 \t 42 997086 998299 21 06 997847 •21 999188 21-27 000812 43 21 00 997885 •21 9-000465 21-21 10-999535 \l 44 999D60 20 94 997822 •21 001788 21-15 99H262 45 9.000816 20 87 997809 •21 008007 21-09 996998 i5 :i6 002069 008818 20 82 997797 •21 004272 21-03 995728 14 47 20 76 997784 •21 005534 20-97 994466 i3 48 004568 20 70 997771 •21 OU6792 20-91 998208 12 49 oo58o5 20 64 997758 .21 008047 009298 20-85 991953 II 53 007044 20 58 997745 .21 20-80 990702 13 5i 9.008278 20 52 9.997782 .21 J9. 01 0546 20-74 10-989454 I 52 oog5io 20 46 997719 •21 011790 018081 20-68 988210 53 010787 20 40 997706 •21 20-62 986969 I 54 011962 20 34 997698 •22 014268 20-56 985782 55 018182 20 29 997680 •22 oi55o2 20- 5l 984498 5 56 014400 20 28 997667 •22 016782 20-45 988268 4 ll 0i56i3 20 17 997654 •22 017919 20-40 982041 3 016824 20 12 997641 .22 019188 20-88 980817 2 ^ 018081 20 06 997628 .22 020408 20-28 979597 97§38o : 60 019235 20-00 997614 .22 021620 20-28 Cotiue D. Sine Cotanc^. P. _J'l'^'^ _ ~£\ (84 DEGREES.) 24 v6 DEGREES.) A TABLE OF LOGARITHMIC M Sine D. Cosine D. Tang. D. Cotang. 9-019235 20-00 9-997614 -22 9-021620 20-23 10-978380 60 I 020435 19 % 997601 •22 022834 20 •17 977166 ' 59 1 3 02I632 19 997588 •22 024044 20 11 975956 58 3 4 022825 024016 19 19 84 78 997574 997561 -22 •22 02525l 026455 20 20 06 00 974749 973545 U 5 025203 19 73 997547 •22 027655 19 95 972345 55 6 026386 19 67 997534 •23 028852 19 90 971148 54 I 027367 19 62 997520 •23 o3oo46 19 85 969954 53 968763 52 028744 19 57 997507 •23 o3i237 19 79 9 029918 19 5i 9974q3 997480 •23 o32425 19 74 967575 5i ic 031089 19 47 •23 o336o9 19 69 966391 5o II 9-o32257 19 41 9-997466 •23 9-034791 19 64 10-965209 % 12 o3342i 19 36 997452 •23 035969 19 58 964031 i3 034582 19 3o 997439 997425 •23 037144 19 53 962856 47 14 035741 19 25 •23 o383i6 19 48 961684 46 I') C36896 19 20 99741 1 •23 039485 19 43 96051 5 45 lb o38o48 '9 i5 997397 •23 04065 1 19 38 95g349 44 I? 039197 19 10 997383 •23 041813 19 33 938187 43 i8 040342 It o5 997369 •23 042973 19 28 937027 42 19 04 1 485 99 997355 •23 044 i3o 19 23 933870 41 20 042625 18 94 997341 • 23 045284 19 18 954716 40 21 9.043762 18 89 9-997327 •24 9-046434 10 i3 10-953566 ll 22 044895 18 84 99731 3 •24 047582 '9 08 932418 23 046026 18 79 697209 •24 048727 19 o3 95,273 37 24 047154 18 73 997283 •24 049869 o5ioo8 18 98 9501 3i 36 25 048279 18 70 997271 •24 18 t 948992 35 26 049400 18 65 997257 •24 o52i44 i8 947836 34 ^J o5o5i9 18 60 997242 •24 053277 18 84 946723 33 o5i63d 18 55 997228 •24 054407 18 79 9455c}3 33 29 052749 18 5o 997214 •24 055535 18 74 944465 3i So . 053859 18 45 997199 •24 056659 18 70 943341 3o 3i 9-054966 18 41 9-997185 •24 9-057781 18 65 10-942219 \l 32 056071 18 36 997170 •24 058900 18 S 94U00 33 lU^', 18 3i 997156 • 24 060016 18 9399«4 27 34 18 27 997141 •24 061 i3o 18 5i 938870 26 35 059367 18 22 997127 •24 062240 18 46 937760 25 36 060460 18 17 997112 •24 063348 18 42 936652 24 37 o6i55i 18 i3 997098 •24 064453 18 37 935547 23 38 062639 18 08 997083 •25 065556 18 3? 934444 22 39 063724 18 04 997068 •25 066655 18 28 933345 21 40 064806 n 99 997033 •25 067752 18 24 932248 20 41 9-065885 17 94 9-997039 •25 9-068846 18 19 10-931 i54 \l 42 066962 17 90 997024 •25 069938 18 i5 930062 43 o68o36 17 86 997009 •25 071027 18 10 928973 17 44 069107 17 81 996994 -25 0721 13 18 06 927H87 16 45 070176 n 77 996979 •25 073197 18 02 926803 i5 46 071242 17 72 996964 •25 °7427S 17 97 923722 14 47 072306 17 68 996949 •25 073336 17 93 924644 i3 48 073366 n- 63 996934 •25 076432 17 ^9 923568 12 49 074424 17- 59 996919 •25 0775o5 n 84 922495 II 5o 075480 17 55 996904 •25 078576 n 80 921424 13 5i 9.076533 17 5o 9.996889 • 25 9-079644 17 76 io-92o356 I 5a 077583 17 46 996874 •25 0807 1 '-7 72 919290 53 078631 17 42 996858 •25 081773 17 67 918227 I 54 079676 17 38 996843 •25 0S2833 17 63 917167 55 080719 17 33 996828 •25 083891 17 59 916109 5 56 081759 17 29 996812 • 26 084947 n 53 9i5o33 4 57 082797 083832 17 25 996707 .26 086000 17 5i 914000 3 58 17 21 996782 .26 087050 088098 17 % 912950 a 59 084864 17 »7 996766 .26 17 91 1902 910856 I 66 085894 17 i3 996751 .26 089144 17 38 Cosine 1). Sine. Cotansr. D. Tang. xM. (83 DEGREES.) SINES AND TANGENTS. (7 DEGREES.) 2c M. Sine D. Cosine D. 1 Tang. 1 ^' Cotaiig. 1 o 9.085894 17-13 9-996751 .26 9-089.44 17-38 io-9io856 60 1 086922 17.09 996735 .26 09018/ 17-34 909813 So 908772 [ 5§ a 087947 088970 17-04 996720 .26 09122S i7-3o i 17.00 996704 •26 092266 17-27 TXs u 4 089990 16.96 996688 .26 093302 17-22 5 091008 16.92 16-88 996673 .26 094336 17.19 17-15 905664 ; 55 6 092024 996657 .26 095367 904633 i 54 7 093037 16.84 996641 .26 OCJ6395 17-11 9o36o5 ': 53 8 094047 16.80 996625 .26 1 097422 17-07 905578 52 Q 090036 16.76 996610 .26 098446 17-03 901354 i 5i 10 096062 16.73^ 996094 .26 099468 16.99 900032 Do II 9-097065 16. 6S" 9-996578 .27 9-100487 16.95 10-899313 49 89849^) 1 48 897481 ! 47 12 098066 16.65 996062 •27 ioi5o4 16.91 i3 099065 16.61 996346 •27 I023I9 16.87 14 100062 16.57 996530 •27 103532 16.84 896468 i 46 i5 ioio56 16.53 996514 •27 104542 16-80 895458 i 45 i6 102048 16-49 996498 •27 io555o 16-76 894450 1 44 17 io3o37 16-40 996482 •27 106556 16.72 893444 ; 43 i8 io4o25 16.41 996465 •27 107559 16.69 892441 42 19 io5oio 16.38 996449 •27 io856o 16.63 • 891440 1 41 20 105992 16-34 996433 •27 109559 16.61 890441 1 40 21 9.106973 i6-3o 9-996417 •27 9-110556 16-58 10-889444 ' 39 888449 38 22 107931 16-27 996400 • 27 11135l 16-54 23 108927 16-23 996384 •27 112343 16. 5o 887457 37 24 I 0990 I 1 10873 16.19 996368 •27 113533 16-46 8S6467 36 25 16-16 996331 •27 114521 16-43 883479 35 26 111842 16-12 996335 •27 115307 16-39 884493 , 34 27 1 1 2809 16-08 996318 •27 116491 i6-36 883309 882528 33 28 113774 16. o5 9g63o2 .28 117472 16-32 32 29 1 14737 i6-oi 996285 .28 118432 16-29 16-25 881548 3i 3o 115698 i5-97 996269 .28 119429 880571 3o 3i 9.116656 15-94 9-996232 .28 9-120404 16-22 '"■1%^ \ It 32 m6i3 15-90 15-87 996235 .28 121377 16.18 33 1 1 8567 996219 .28 122348 16. i5 l%ll i U 34 119319 15-83 996202 .28 125317 16-11 35 1 20469 i5.8o 996185 .28 124284 16-07 875716 25 36 121417 15.76 996168 .28 123249 16-04 874751 . 24 37 122362 15.73 9961 5i .28 1 262 II 16-01 873789 23 872828 23 38 i233o6 15.69 996134 .28 I27172 15-97 39 124248 i5.66 996117 .28 i28i3o 15-94 871870 21 40 120187 15-62 996100 .28 129087 15-91 870913 20 41 g.126125 \l-M 9-996083 • 29 9'i3oo4i 15-87 10-869939 19 42 127060 996066 •29 1 30994 15-84 43 127993 i5.52 996049 .29 i3i944 132893 i5.8i 86^056 I7 867107 16 ,44 128925 129834 15.49 15.45 996032 .29 15.77. 45 9960 1 5 .29 133839 15-74 866161 46 130781 15.42 993998 •29 1347H4 15.71 860216 14 47 i3i7o6 15.39 995980 •29 135726 15-67 864274 1 3 48 1 32630 15.33 993963 • 29 136667 i5-64 863333 12 49 133551 i5.32 995946 • 29 i376o5 i5-6i 862395 II 50 134470 15.29 995928 .29 138542 15-58 861438 10 5i 9.135387 i363o3 15.25 9.995911 993894 .29 9-139476 15-55 io-86o524 I 52 l5-22 •29 140409 i5-5i 839591 858660 53 137216 15-19 995876 •29 141340 15-48 I 54 138128 i5.i6 995839 •29 142269 i5-45 857731 55 139037 I5.I2 995841 •29 143196 i5-42 856S04 5 56 139044 140830 15.09 995823 .29 144121 15-39 830879 i 4 II i5-o6 993806 •29 145044 15-35 854956 1 3 141754 i5-o3 993788 •29 1 43966 15-32 854034 ' 2 59 142655 1 15-00 995771 •29 146885 15-29 853.15 ; » 60 143555 1 14-96 i 995753 1 •29 147803 15-26 852197 ! Cosine < D. Sine ' Cotiinor. D. 1 Tims. (82 DEGKEE8.) 26 (8 DEGREES.) A ' FABLE OF LOGAKJIH MIC ii.'mg. ~M. j Sine : D- Cosine D. Tang. D. i o j 9-143555 ■ 14-96 9-995753 •30 9-i47?o3 15-26 :o 852197 60 I I444J3 14-93 995735 -30 1 148718 13.23 85.2^2 U 2 ! 145340 14-90 993717 -30 ! 149632 15-20 850 368 3 , 146^43 14-87 995099 •30 i i5o544 15..7 8494^6 57 4 1471 36 14-84 993681 •3o i5i454 15-14 84N346 56 5 ' 14^026 14-81 993664 •3o 152363 i5-ii 84-637 55 6 i48oi5 : 14-78 9Q5646 •3o 153269 i5-o8 846-^31 54 ' 7 .49302 ■ 14-75 1 995628 .30 154174 i5.o5 843«26 53 i ' 1D0686 14-72 j 993610 •3o i55o77 l5-02 844923 5a 9 i5i569 1469 1 995591 •3o 155978 14-99 84402? 31 10 i5245i 1 14-66 ! 995573 •3o 156877 14-96 843 1 i3 30 15 9-1 J33o j 14-63 1 9-995555 • 30 9- 157775 14-93 10-842225 8: 12 1 54208 1 14-60 1 993337 •30 158671 \iX 841329 i3 i55o«3 14-57 1 993319 •3o I 5g565 840435 47 U 155957 i56B3o 14-54 993301 • 3i 160457 14-84 839343 83%33 46 ID 14-51 9954«2 -31 16 1 347 14-81 45 16 137700 14-48 995464 -31 162236 14-79 83-7-64 44 »7 158569 14-45 993446 .3i i63i23 14-76 8368-'7 43 i8 159435 14-42 993427 •31 164008 14-73 83^992 42 »9 i6o3oi 14-39 093409 •3i 164^92 14-70 83 3 108 41 20 161164 14-36 ' 995390 -3i 165774 14-67 834226 40 21 9-162025 14-33 9-9933-2 .3i 9-166654 14-64 10-833346 It 22 162885 i4-3o 995353 .31 167532 14-61 832468 23 163743 14-27 993334 .3i 168409 14-58 83(391 37 24 164600 14-24 993316 -31 1692S4 14-55 83o^)6 36 25 165454 14-22 993297 -3i 170157 14-53 829843 35 26 i663o7 14-19 99^278 -31 171029 14- 5o 82 V I 34 27 167,59 168008 14-16 995260 .31 I7i«99 14-47 82«I0I 33 28 U-i3 99^241 .32 172-67 14-44 82-233 32 29 168856 14-10 993222 .32 173634 14-42 826366 3i 3o 169702 14-07 993203 .32 174499 14-39 825501 3o 3i 9- 170547 i4-o5 9-995i«4 •32 9-175362 14-36 10-824638 It 3c 17.3R9 14-02 993165 •32 176224 14-33 8237-6 33 172230 13-99 995146 •32 1770^4 14-31 822916 27 34 1730-0 13-96 993127 .32 177942 14-28 82305s 26 35 173908 13-94 993108 .32 178799 14-23 821201 25 36 174744 13-91 13-88 9950^^9 .32 I7y655 .4-23 820345 24 37 175578 990070 .32 i8o5o8 14-20 819492 23 38 ■ 176411 13-86 995o5i .32 i8i36o 14-17 81^^640 22 39 . 177242 i?-83 Q95o32 •32 182211 14-15 8.7-^9 21 4o 178072 i3-8o 995oi3 .32 i83o59 14-12 8i6q4l 20 41 9-178900 13-77 9 994993 .32 9-183907 14-09 10-816093 ',1 42 179726 13-74 994974 .32 184752 14-07 813248 43 i8o55i 13-72 994935 .32 185597 14-04 814403 }-l 44 181374 13-69 994935 .32 186439 14-02 813^61 16 45 182196 13-66 994916 .33 1872S0 .3-99 812-20 i5 46 i83oi6 13-64 994V .33 188120 .3-96 8n«8o 14 47 183834 i3-6i 994''^77 .33 188908 .3-93 811042 .3 48 18465 I i3-5> 994857 • 33 189794 .3-91 810206 13 49 185466 i3-5b 994H38 .^3 190629 13-89 8093-1 II 5o 186280 13-53 994^18 .33 191462 .3-86 808533 10 5i 9-187092 i3-5i 9-994-98 .33 9-192294 .3-84 io-8o--o6 I 52 187903 13-48 9947-9 .33 193124 .3-8. 806^76 53 188712 13-46 994^39 .33 193953 13-79 806047 I 54 189519 13-43 994-39 1 .33 1947^0 .3.76 8o5220 55 190325 13-41 994-19 .33 195606 i3-74 804394 5 56 191 i3c i3-33 994-00 .33 196430 13-71 8o357o 57 191933 13-36 994680 .33 io-'253 .3-69 802747 58 192734 13-33 994660 •33 198074 i3-66 801926 59 193534 i3.3o 994640 •33 198894 13-64 801106 6o 194332 13-28 994620 • 33 199713 i3-6i 800287 .¥f. Cosine I D. Sii.e i Cotanir. D. i Tui.'?. (81 I 3EGR] EES.) SINES AND TAXGEXTS. (9 DEGREE.) 27 M.' Sine D. Co3ine D. Tang, D. Cotang. 9-194332 13.28 9-994620 .33 9.199713 i3-6i 10-800287 60 1 I 193129 i3 26 994600 .33 200329 i3 59 799471 ' 59 2 193923 i3 23 994580 .33 201343 i3 56 798655 58 3 196719 i3 21 994360 34 202159 :3 34 797841 57 797029 56 4 197311 i3 18 ^94540 •34 202971 i3 52 5 19S302 i3 16 994319 •34 203782 i3 49 796218 , 55 6 199091 i3 i3 994499 •34 204392 i3 47 795408 1 54 7 199S79 i3 1 1 994479 •34 2o54oo i3 43 794600 53 8 200666 i3 08 994439 •34 206207 i3 42 793-93 ; 5.2 9 20 1 43 1 1 3 06 994438 •34 207013 i3 40 792987 i 5i- 10 202234 i3 04 994418 •34 207817 i3 38 792x83 5o II 9 2g3oi7 i3 01 9-994397 •34 9^2o86i9 i3 35 io-79i38i 49 12 203797 12 99 994377 •34 209420 i3 33 790380 ! 48 i3 204377 12 96 994337 •34 210220 i3 3i 789780 788982 47 14 203354 12 94 994336 •34 211018 i3 28 46 i5 2o6i3i 12 92 994316 •34 2ii8i5 i3 26 788185 45 16 206906 12 89 994295 •34 212611 i3 24 787389 44 I? 207679 12 87 994274 •35 2i34o5 i3 21 786393 ; 43 1 i8 208432 12 83 994234 .35 214198 2149^9 i3 19 785802 42 19 209222 12 82 994233 .35 i3 17 785011 41 20 209992 12 80 994212 .35 215780 i3 i5 784220 40 21 9-210760 12 78 9-994191 .35 9-216568 i3 12 10^783432 ll 22 2ll526 12 75 994171 .35 217356 i3 10 782644 23 212291 12 73 994 1 5o .35 218142 i3 08 781858 37 24 2i3o55 12 71 994129 .35 218926 i3 o5 781074 i 36 25 2i38i8 12 68 994108 .35 219710 i3 o3 780290 35 26 214379 215338 12 66 994087 .35 220492 i3 01 779508 34 778728 33 27 12 64 994066 .35 221272 12 99 28 216097 12 61 994043 .35 222032 12 97 777948 1 32 29 216854 12 59 994024 • 35 222830 12 94 777170 i 3i 3o 217609 12 57 9Q4oo3 .35 2236o6 12 92 776394 j 3o 3i 9-218363 12 55 9.993981 .35 9-224382 12 90 io^7756i8 29 32 219116 12 53 993960 .35 223156 12 88 774844 28 33 219868 12 5o 993939 .35 225929 12 86 774071 27 34 2206.8 12 48 993918 .35 226700 12 84 773300 26 35 22i367 12 46 993M96 • 36 227471 12 81 772329 1 23 36 2221l5 12 44 993875 • 36 228239 12 79 771761 ; 24 37 222S61 12 42 993854 • 36 229007 12 77 770993 23 38 223606 12 39 993832 • 36 229773 12 75 770227 22 39 224349 12 37 99381 1 .36 23o539 12 73 769461 21 40 223092 12.35 993789 •36 23l302 12 71 20 41 9-225833 12-33 9-993768 • 36 9-232065 12 69 10.767935 \l 42 226573 12-3l 993746 •36 232826 12 67 767174 43 22731 1 12-28 993725 •36 233586 12 65 766414 ! 17 44 22^^048 12-26 993703 •36 234343 12 62 765655 ! 16 45 228784 12-24 993681 • 36 235 io3 12 60 764897 j 1 5 46 229318 12-22 993660 •36 235SJ9 12 58 764141 1 i4 47 230232 12-20 993638 • 36 236614 12 56 763386 ! i3 48 230984 12-18 993616 -36 23^368 12 34 762632 ! 12 49 231714 12-16 993394 •37 238120 12 52 761880 II 5o 2 32444 12-14 993372 •37 238872 12 5o 761 128 1 ID 5. 9-233172 12-12 9-993550 • 37 9-239622 12 48 10.760378 t 52 233899 12 09 9ry3328 •37 240371 12 46 750629 758882 53 234623 12-07 993306 •37 241118 12 44 7 54 235349 I2-o5 993484 •37 241865 12 42 738135 6 55 236073 12-03 993462 •37 242610 12 .40 757390 ' 5 56 236795 12-01 993440 •37 243354 12 -38 736646 ' 4 57 2373l5 n-99 993418 .37 244097 12 -36 755903 3 58 23^^235 11-97 993396 •37 244839 12 •34 755161 2 59 53M953 1 "-95 993374 .37 243379 12 -32 734421 1 60 239670 1 11-93 993351 •37 246319 12-30 753681 Coaine [_J 0. Sine Cot524 42 299980 10-96 700020 43 i8 292137 292768 10 53 991493 42 3oo638 10-95 699362 42 19 10 5i 991473 42 301295 10-93 698705 41 20 593399 10 5o 991448 42 301901 10-92 698049 40 21 9-294029 294608 10 48 9-991422 42 9.302607 10-90 10-697393 ll 22 10 46 991397 42 3o326i 10-89 696739 23 290286 10 45 991372 43 3o39i4 10-87 10-86 696086 37 24 290913 10 43 991346 43 304567 695433 36 25 296539 10 42 991321 43 3oo2i8 10-84 694782 35 26 297164 10 40 991295 43 300869 10-83 6941 3 1 34 27 297788 10 39 991270 43 3o65i9 307168 10.81 693481 33 28 298412 10 ll 991244 43 10-80 692832 32 29 299034 10 991218 43 307815 10.78 692185 3i 3o 299655 10 34 991 193 43 3o8463 10.77 691537 3o 3i 9-300276 10 32 9-991167 43 g. 309109 10.75 10-690891 ll 32 300895 10 3i 991 i4i 43 309754 10.74 '& 33 3oi5i4 10 29 28 991115 43 310398 10-73 27 34 302l32 10 991090 43 311042 10-71 688958 26 35 302748 10 26 991064 43 3ii685 10-70 6883 1 5 25 36 3o3364 10 25 991033 43 312327 10-68 687673 24 ll ^S5? 10 23 991012 43 312967 10-67 687033 23 10 22 990986 43 3i36o8 10-65 686392 1 22 39 300207 10 20 990960 43 314247 10-64 685753 i 21 40 300819 10 19 990934 44 3x4885 10-62 68511 5 20 41 9-3o643o 10 17 9-990908 990882 44 9-315023 io-6i 10-684477 19 42 307041 10 16 44 3i6i59 10- 60 683841 18 43 307600 10 14 990355 44 316795 10-58 683205 \l 44 308259 10 i3 990829 44 3 17430 10.57 682570 45 3o8367 10 II . 990803 44 318064 10-55 681936 68i3o3 i5 46 309474 10 10 990777 44 318697 10-54 14 ^l 3ioo3o 10 08 990750 44 319329 10-53 680671 i3 48 3 1 0685 10 07 990724 44 319961 10-51 680039 12 49 311289 311893 10 o5 990697 44 320092 io-5o 679408 ' II 5c 10 04 990671 44 321222 10-48 678778 i 10 5i 9 312493 • 10 o3 9-990644 44 9-32i85i 10-47 10-678149 1 9 52 3 1 3097 3.3698 10 01 990618 44 322479 10-45 677521 1 8 53 10 00 990591 44 323io6 10-44 676894 1 7 67626"' 6 54 3(4297 9 98 990565 44 323733 10-43 55 314897 9 97 99o538 44 324353 10-41 670642 5 56 310495 9 96 9905 n 45 324953 10-40 675017 / 4 57 3!6Dq2 9 94 9904^0 45 320607 10-39 674393 1 3 53 3 16639 9 93 990453 45 326231 10-37 673769 i 2 5'9 317284 9 91 9904 3 1 45 326853 10-36 673147 1 1 6o 317879 9.90 9904 4 45 327475 10-35 672520 ; r:. Cosin3 D. Sine Cotuiie. D. Tan?. M. 1 (78 DEGREES,) 30 (12 DEGREES., A TABLE OF LOGARITHMIC M.l SiLO D. Cosine D. Tanor- D. Cotang. 1 o 'tl^t 9-90 9-990404 1 45 9-327474 10-35 10-672526 i 60 I 9 88 990378 45 328095 10 33 671905 ! DQ 3 3igo66 9 87 99035 1 45 32S7,5 10 32 671285 1 58 3 3io658 9 86 990324 45 329334 10 3o 670666 57 4 320249 9 84 990297 ■ 45 329953 10 29 670047 ! 56 5 320840 9 83 990270 45 33o57c 10 28 669430 55 66881 3 54 6 321430 9 82 990243 : 45 351187 10 26 I 322019 9 80 990215 45 33iHo3 10 25 668197 53 322607 9 79 990188 , 45 3 3241 8 10 24 667582 52 9 325r94 323780 9 77 990161 • 45 3 ?3o33 10 23 666967 5i 10 = 76 990>34 ! 45 333646 10 21 666354 5o n 9-324366 9 75 9-990107 46 5 334259 10 20 io.665i4i -^2 48 12 324950 9 73 990079 46 3i4S7l 10 '9 665 1 29 664518 i3 325534 9 72 99)o52 ! 46 3354H2 10 • 7 47 14 3261 17 9 70 990025 46 3 56093 10 16 663907 46 i5 326700 9 69 9^Q997 46 336702 10 i5 603298 45 i6 327281 9 68 9Syo-o 46 33i3ii 10 i3 662689 44 17 327862 9 66 9^9942 46 33^9,9 10 12 662081 43 i8 328442 9 65 9^9913 46 33S527 10 II 661473 42 19 329021 9 64 989S87 46 339133 10 10 660867 41 20 329599 9 62 989S60 46 339739 10 08 660261 40 21 9-330176 9 61 9.989832 46 9-346344 10 07 10.639636 3§ 22 330753 9 60 9S9S04 46 340948 10 06 639032 23 33 1 329 9 58 9^9111 46 34i5d2 10 04 658448 37 24 33 1903 9 57 989749 47 342155 10 o3 637845 36 25 332478 9 56 9>^972l 47 342737 10 02 637243 35 26 333o5i 9 54 989693 47 343358 10 00 636642 34 11 333624 9 53 9S9665 47 343958 9 ^ 636042 33 334195 9 52 989637 47 344558 9 655442 32 29 334766 9 5o 989609 ^7 343157 9 97 654843 3i So 335337 9 49 989582 47 343755 9 96 634245 3o 3i 9-335906 9 48 9.9S9553 47 9-346353 9 94 10-653647 11 32 336475 9 46 9^'9525 47 34^949 9 93 653o3i 33 337043 9 45 9S9497 47 347543 9 92 652455 U 34 337610 9 44 9^9469 47 348141 9 91 65 1 839 35 338176 9 43 9'^944i 47 34S735 9 90 88 631265 25 36 338742 9 41 9^9413 47 349329 9 65o67i 24 11 339306 9 40 9^'9384 47 319922 9 87 630078 23 '339871 9 39 989356 47 33o5i4 9 86 649486 648M94 23 39 340434 9 37 9S9328 47 331106 9 85 21 40 340996 9 36 989300 47 351697 9 83 6483o3 00 41 9-341558 9 35 9.989271 47 9-3322S7 9 82 10-647713 '2 42 3421 19 9 34 9H9243 47 332876 9 81 647124 18 43 342679 9 32 9^9214 47 353465 9 80 646535 n 44 343239 9 3i 9S9186 47 334033 9 79 643947 16 45 343797 9 3o 989137 989128 47 334640 9 77 645360 i5 46 344355 9 29 48 33^227 9 ^6 644773 i4 47 344912 9 27 9S9100 48 3 3381 3 9 75 644187 i3 48 345469 9 26 98qo7 I 48 3Vi398 9 74 643602 12 49 346024 9 25 9^9042 48 3369S2 9 73 643018 11 i 5o 346579 9 24 989014 48 357366 9 71 642434 10 5i 9-347134 9 22 9.988985 48 9-358149 9 70 10 -641851 i 52 347687 9 21 9SS956 48 338731 9 69 68 641269 53 348240 9 20 9889^7 48 339313 9 640687 I 54 348792 9 19 988898 48 339S93 9 67 640107 55 349343 9 17 988869 48 360474 9 66 639526 5 56 349^93 9 16 988840 48 36I033 9 65 63S947 4 u 350443 9 i5 Q88811 49 36i632 9 63 638368 3 350992 9 14 9N8782 49 3622 10 9 62 6J-'790 i a 59 35i54o 9 i3 988753 49 362787 9 61 63-213 ' I 66 352088 9-11 9S8724 1 49 363364 9-6o 636636 1 Cosine J). Sine 1 Cotanj;. r ). Taixg. ;_^j (77 DEGREES.) SINES AND TANGENTS. (13 DEGKEES.) 31 f^" Sine D. CosiuPi D. Tang. D. Cotang. 9.352088 911 Q 9^'^724 49 9-363364 9-60 10.636636 60 I 352635 9 10 988695 49 368940 9 59 686060 1 5q 635485 1 58 2 353i8i 9 oq 988666 49 3645 1 5 9 58 3 353726 9 08 988636 49 365o90 9 57 684910 ; 57 4 354;?! 9 07 988607 49 365664 9 55 634336 1 56 354815 9 o5 988578 49 366237 9 54 633768 1 55 6 35535S 9 04 988548 49 366810 9 53 688190 1 54 I 355goi 9 o3 988019 49 367382 9 52 682618 1 53 356443 9 02 988489 49 367953 9 5i 682047 : 52 9 356q84 t 01 988460 49 368524 9 5o 681476 5i 10 357524 99 988430 49 369094 9 49 680906 5o II 9.358064 8 9S g. 98840 I 49 9-369653 9 48 io-63o337 49 12 3586o3 8 97 98S371 49 3-'o232 9 46 629768 48 i3 359141 8 96 988342 49 370799 9 45 629201 628633 46 14 359678 8 95 988312 5o 371867 9 44 i5 36021 5 8 93 98^282 5o 371933 9 43 628067 45 l6 36oi52 8 92 98«212 5o 372499 9 42 627501 44 n 3612^7 8 91 988223 5o 373r64 9 41 626986 626871 43 i8 361M22 8 % 9S8I93 5o 878629 374193 9 40 42 19 362356 8 988163 5o 9 89 625807 41 20 362S89 8 88 988133 5o 374756 9 38 625244 40 21 9-363422 8 87 9.988103 5o 9-375319 9 37 10.624681 U 22 363954 8 85 98S073 5o 375881 9 35 624119 623558 23 364485 8 84 988043 5o 376442 9 34 37 24 365oi6 8 83 988018 5o 877008 9 33 622997 36 25 3b5546 8 82 9S7983 5o 377563 9 32 622487 35 26 366075 8 81 987953 5o 378122 9 3i 621878 34 27 366604 8 80 9 -(792 2 5o 378681 9 3o 621819 33 28 367i3i 8 79 987S92 5o 879289 9 l^ 620761 32 ?9 361619 8 77 9S7S62 5o 879197 9 620208 3i 3o 368i85 8 76 987832 5i 38o354 9 27 619646 3o 3i 9-368711 8 75 9.987801 5i 9-380910 9 10 10-619090 29 32 36q236 8 74 987771 5i 381466 9 25 618534 28 33 369-61 8 73 9H7740 5i 382020 9 24 617980 ; 27 34 370285 8 72 9877 '0 5i 382575 9 23 617425 26 35 370S08 8 71 9H7679 5i 383129 9 J2 616871 25 36 37i33o 8 70 95-649 5i 383682 9 21 616818 , 24 il 371852 8 69 9S7618 5i 384234 9 20 615766 j 23 38 372373 8 67 987588 5i 384786 9 ;? 6i52i4 22 39 372894 8 66 987557 5i 385337 9 614668 21 40 373414 8 65 987526 5i 385888 9 17 614112 20 41 9.373933 8 64 9-987496 5i 9-386438 9 i5 io-6i3562 19 6i3oi3 1 18 42 3744D2 8 63 987465 5i 386987 9 14 43 374970 8 62 987434 5i 387D36 9 i3 612464 i 17 44 375487 8 61 987403 52 388084 9 12 611916 16 45 37Coo3 8 60 987372 52 388631 9 II 611J69 ' i5 46 376519 8 59 9B734I 52 889118 9 ID 610822 ; 14 s 377035 8 58 987310 52 389724 9 S 610276 i3 377549 8 57 987279 9^7248 52 390270 9 609780 1 12 49 378063 8 56 52 •390815 9 07 609185 608640 II 5o 378577 8 54 987217 52 391860 9 06 10 5i 9.3^0089 8 53 9.987186 52 9.391908 9 o5 10-608097 t 52 379601 8 52 987155 52 892447 9 04 607553 53 38oii3 8 5i 987124 1 02 892989 9 o3 607011 7 54 380624 8 5o 987092 52 393081 9 02 606469 6 5- 38ii34 8 49 987061 52 894078 9 01 605927 5 5t 38(643 8 48 987030 52 894614 00 6o5386 4 U 382152 8 47 986998 52 890154 8 ^ 604846 3 38^661 8 46 986967 52 890694 8 604806 2 59 383 168 8 45 986986 52 396233 8 07 608767 I 60 383675 8-44 986904 -52 396771 8-96 608229 ' Cosine D. Si-e Cotanor. D. Tang. ^L 'ii<0 (76 DEGREES.) 82 (14 DEGEEES.) A TABLE OF LOGARITHMIC Sire T). Cosine J). Taiig. D. Cotong. 1 9-383675 8-44 '■^4 -52 9.396771 8.96 10-603229 ' 60 I 384182 8 •43 .53 397309 8 96 602691 I 59 3 384687 8 •42 986S41 • 53 3^7846 8 95 602 1 54 58 3 385i92 8 41 986S09 986778 .53 398383 8 94 601617 57 4 385697 8 40 -53 3989.^ 8 93 ^01081 56 5 386201 8 39 986746 .53 399455 8 92 600545 55 6 386704 8 38 986714 .53 399990 4oo524 8 9' 600010 54 7 387207 8 37 9S6683 -53 8 r. m' 53 8 387709 8 36 986651 -53 401058 8 52 9 388210 8 35 986619 .53 401591 8 88 598409 5i 10 3887 1 1 8 34 986087 -53 402124 8 87 597876 5o 11 9-389211 8 33 9.986555 -53 9.402656 8 86 10.597344 ^ 12 3897 1 1 8 32 986523 .53 403187 8 85 596813 i3 390210 8 3i 986491 .53 403718 8 84 596282 47 14 390708 8 3o 986439 • 53 404240 8 83 595751 46 i5 391206 8 28 986427 .53 404778 8 82 593222 45 i6 391703 8 27 986395 -53 4o53o8 8 81 594692 44 \l 392195 39269D 8 26 986363 •54 405836 8 80 594164 43 8 25 986331 •54 406364 8 ]l 593636 42 19 lllsi 8 24 986299 •54 406892 8 593108 41 20 8 23 986266 .54 407419 8 77 592581 40 21 9.394179 8 22 9.986234 •54 9-407945 8 76 10.592055 39 22 394673 8 21 9S6202 •54 408471 8 75 591529 38 23 395166 8 20 986169 -54 408997 409521 8 74 591003 ll 24 395658 8 li 986137 -54 8 H 590479 25 396150 8 986104 •54 410045 8 73 589953 35 26 396641 8 17 986072 •54 410569 8 72 589431 588908 34 11 397132 8 17 986039 •54 41 1092 8 71 33 397621 8 16 986007 •54 4ii6i5 8 70 588385 32 29 398111 8 r5 983974 .54 412137 8 69 587863 3i 3o 398600 8 14 983942 •54 412658 8 68 587342 3o 3i 9.399088 8 i3 9-985909 .55 9-4i3i79 8 u 10.586821 29 28 32 399575 8 12 983876 -55 413699 8 586301 33 400062 8 II 985843 .55 414219 414738 8 65 585781 ll 34 400349 8 10 983811 • 55 8 64 583262 35 401 o3d 8 ^i 983778 • 55 4i5257 8 64 584743 25 36 401320 8 983745 • 55 415775 8 63 584225 24 37 4o2oo5 8 07 985712 • 55 416293 8 62 583707 23 38 402489 8 06 985679 .55 416810 8 61 583190 22 39 402972 8 o5 985646 • 55 417326 8 60 582674 21 40 403455 8 04 9856i3 .55 417842 8 59 582158 20 1 41 9.403938 8 o3 9.985580 -55 9-418358 8 58 10.581642 \t 42 404420 8 02 983347 .55 418873 8 ll 581127 43 4o4qoi 8 01 985514 -55 419387 8 58o6i3 17 44 405382 8 00 985480 -55 419901 8 55 580009 , 16 1 .^5 4o5862 9^ 985447 -55 420413 8 55 579585 i5 46 406341 985414 -56 420927 8 54 579073 57a^6o 14 47 48 406820 97 985380 • 56 421440 8 53 i3 407299 96 985347 • 56 421952 8 52 578048 n 49 407777 95 985314 -56 422463 8 5i 577537 ir 5o 408254 n 94 985280 • 56 422974 8 5o 577026 10 5i 9.408731 94 9.985247 • 56 9.423484 8 it 10.576516 I 5s 409207 7 93 983213 • 56 423993 8 576007 53 409682 92 985180 -56 424303 8 48 575497 574989 I 54 410157 91 983146 -56 423011 8 47 55 4io632 i 983113 -56 425519 8 46 574481 b 56 411106 983079 • 56 426027 8 45 573973 4 u 41 1379 983043 • 56 426534 8 44 573466 3 412052 !i 98301 1 -56 427041 8 43 572930 572453 a 59 412524 984978 • 56 427547 8 43 I 66 412996 7^85 984944 •56 428052 842 571948 Cosine D. Sine 1 Cotanor. D. Tang. Lmu (75 DEGREES.) SINES AND TANGENTS. (15 DEGREES.) 33 n. Sine D. Cosine j D. Tang. D. Cotuilg. o 9 412996 7-85 9 984944 W 9.428052 8.42 10-571948 1 60 I ^ 413467 7-84 984910 984876 •57 428557 8 41 571443 ■ 5q a 413938 7-83 •57 429062 8 40 570933 1 58 3 414408 7-83 984S42 •57 429566 8 ll 570434 1 57 4 414878 7-82 984808 •57 430070 8 569930 56 5 415347 7-81 984774 •57 430573 8 38 569427 55 6 4i58i5 7-8o 984740 'V 431075 8 37 568925 54 I 416283 ?:?? 984706 •57 43 1 577 3 36 568423 53 416731 984672 •57 432079 8 35 567921 52 9 417317 7-77 984637 'P 432580 8 34 567420 1 5i 10 • 417684 7-76 984603 •57 433080 8 33 566920 1 5o II 9'4i8i5o 7-75 9.984369 •57 9.433580 8 32 10-566420 8 IS 4i86i5 7-74 984535 •57 434080 8 32 560920 i3 419079 7-73 984300 •57 434579 8 3i 565421 47 14 419544 7-73 984466 •57 435078 8 3o 564922 46 i5 420007 7-72 984432 .58 435576 8 29 564424 45 i6 420470 7-71 984397 .58 436073 8 28 563927 44 \l 420933 421395 7-70 984363 • 58 436570 8 28 563430 43 984328 .58 437067 8 ^1 562933 42 19 421857 4223i8 7-68 984294 .58 437563 ■ 8 26 562437 41 20 7-67 984209 .58 438059 8 25 561941 40 21 9-422778 7.67 9.984224 .58 9.433554 8 24 10-561446 It 22 423238 7-66 984190 .58 439048 8 23 560902 23 423697 7-65 984155 .58 439543 8 23 560437 37 24 424i56 7-64 984120 : .58 440036 8 22 309964 36 25 424615 7-63 984085 I .58 440529 8 21 509471 35 26 420073 7-62 984000 .58 441022 8 20 553978 558486 34 27 425530 7-61 984015 .58 44i5i4 8 19 33 28 423987 7-6o 983981 .58 442006 8 \l 557994 32 29 426443 7-60 983946 .58 442497 442988 8 5575o3 3i 3o 426899 7.59 98391 1 .58 8 17 557012 3o 3i 9-427354 7-58 9.983875 .58 9-443479 443968 8 16 10-556521 ll 32 428263 7-57 983840 .59 8 16 556o32 33 7-56 9838o5 .59 444458 8 i5 555542 27 34 428717 7-55 983770 .59 444947 8 *i4 5ooo53 26 35 429170 7-54 983735 .59 445435 8 i3 504565 25 36 429623 7-53 983700 1 .39 445923 8 12 554077 24 II 430075 7-52 983664 .59 44641 I 8 12 553589 23 43o527 7-52 983629 .59 446898 8 II 553102 22 39 430978 7-5i 983594 .59 447384 8 10 552616 21 40 431429 7.50 983558 .59 447870 8 09 5o2i3o 20 41 9.431879 7-49 9.983523 .59 9-448356 8 0^ 10 -55 1 644 \l 42 432329 7-49 983487 .59 448841 8 55 1 1 09 43 432778 7-48 983452 .59 449326 8 07 550674 n 44 433226 7-47 983416 .59 449810 8 06 500190 16 45 433675 7-46 983331 .59 450294 8 06 549706 i5 46 434122 7.45 983345 .59 450777 8 o5 540223 14 "^l 434569 7-44 983309 .59 451260 8 04 543740 i3 48 435016 7-44 983273 .60 451743 8 o3 548207 12 i^ 435462 7-43 983238 .60 452225 8 02 547775 11 5o 435908 7-42 983202 .60 432706 8 02 547294 10 5i 9-436353 7-41 9.983166 .60 9-453187 8 01 10 .54681 3 I 52 436798 7-40 983 i3o .60 453668 8 00 546332 53 437242 7-40 983094 .60 454148 99 540832 7 I 54 43:636 7.39 7-38 983008 .60 454628 ^ 545372 6 L^ 433129 983022 .60 455107 544S93 5 1 56 438072 7-37 982986 .60 455586 97 544414 4 1 Jz 43?oi4 7-36 982950 .60 456064 96 543936 3 58 439406 7-36 982914 .60 456542 96 543458 2 ^ 439^97 7.35 982878 .60 457019 457496 95 542981 I 60 440338 7-34 982842 .60 7-94 542504 1 1 Cosine 1 D. Sine i Cotang. D. Tansr. "ST (74 DEGREES.) 34 (IG DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotaiig. 60 9-440338 7-34 9-982842 60 9-457496 7-94 io-5425o4 I 440778 33 982805 60 457973 •93 542027 59 58 2 441 218 32 . 982769 61 458449 •93 54i55i 3 441608 3i 982733 61 458925 92 541075 57 4 442096 3i 082696 61 439400 9' 540600 56 5 442535 3o 982660 61 459875 90 540125 55 6 442973 29 982624 61 460349 90 539631 54 7 443410 28 982587 61 460823 539177 53 & 443847 27 982551 61 461297 88 538703 52 9 444284 27 982014 6i 461770 88 538^30 5i 10 444720 26 982477 61 462242 87 537758 5o 11 9-445i55 25 9-982441 61 9-462714 86 10-537286 49 12 445590 24 982404 61 463 180 85 536814 48 i3 446025 23 982367 61 463658 85 536342 47 14 446459 23 982331 61 464129 84 535871 46 ID 446S93 22 982294 61 464399 83 535401 45 i6 447326 21 982237 61 463069 83 534931 44 n 447739 20 982220 62 463539 .466008 82 534461 43 i8 448191 20 982183 62 81 533992 42 19 448623 19 982145 62 466476 80 533524 41 20 449054 18 982109 62 466945 80 533o55 40 21 9-449485 17 9-982072 62 9-467413 79 io-53258y 39 22 4499 '5 16 9S2035 62 467880 78 532120 38 23 45o345 16 981998 62 468347 78 531653 37 24 450775 i5 98 1 96 1 62 468814 77 531186 36 25 431204 14 981924 62 469280 76 530720 35 26 45i632 i3 981886 62 469746 73 530234 34 11 452060 i3 981849 62 47021 1 75 5297H9 33 452488 12 981812 62 470676 74 529324 32 29 452915 II 981774 62 47M4J 73 528839 3i 3o 453342 7 10 981737 62 471605 73 528895 3o 3i 9-453768 . 10 9-981699 63 9.472068 72 10-527932 29 32 454194 09 981662 63 472532 7> 527468 2S 33 454619 08 981625 63 472993 7J 527005 27 34 455o44 07 981587 63 473437 70 526543 26 35 455469 07 981549 63 473919 69 526081 25 36 455893 06 981 5i 2- 63 474381 69 523610 24 37 38 4563 1 6 o5 981474 63 474842 68 525i58 23 456739 04 981436 63 4753o3 67 524697 22 39 457162 04 98i3q9 63 473768 67 524237 21 40 457584 o3 981861 63 476223 66 523777 20 41 9 -458000 02 9-981323 63 9-476683 65 10 •5233 1 7 19 42 458427 458848 01 . 981285 63 477 '42 65 522S58 18 43 01 9B1247 63 477601 64 522899 17 44 459268 00 981209 63 478059 63 521941 16 45 45968S 6 99 981171 63 478517 63 52i4>s3 i5 46 460108 6 98 981133 64 47«973 62 521025 14 47 460527 6 98 981095 64 479432 61 520368 i3 48 460946 6 97 981037 64 479f^«9 61 520111 n 49 461 364 6 96 981019 64 480345 60 519635 II 5o 461782 6 95 980981 . 64 480801 39 519199 10 5) 9.462199 6 95 9.980942 04 9-481257 39 io-5i8743 t 5a 462616 6 94 980904 • 64 481712 58 5i82e8 53 463o32 6 93 9S0866 • 64 482167 57 517833 1 54 463448 6 93 080827 • 64 482621 57 517379 6 55 463864 6 92 980789 • 64 488075 56 516925 5 56 464279 6 91 980750 • 64 483529 55 5i647i 4 u 464694 6 90 980-' 1 2 64 483982 55 5i6oi8 3 465 1 08 6 % 980673 . 64 484435 54 5 15565 2 59 465522 6 980635 • 64 484887 53 5i5ii3 I 1 60 465935 6-88 980596 . 64 485339 7^53 5 I 4661 Cosine D. Sine 1 1 Cotang. D. Tau^. IT* (73 DEGREES.) SINES AND TANGENTS. (17 DEGREES.) 35 "m. Sine D. Cosine D. Taug. D. Coiang. o 9-465935 466348 6-88 9-980596 64 9-485339 7-55 io-5i466i 60 I 6-88 980558 64 485791 52 514209 ; 59 5i3758 ; 58 2 466761 6-87 980519 65 486242 5i 3 467173 6-86 980480 65 486693 5i 5 1 3307 1 57 4 467535 6-85 980442 65 487143 5o 512857 56 5 467996 6-85 980403 65 487593 49 512407 55 6 468407 6-84 980364 65 488043 S 511957 54 5u5o8 1 53 I 468817 6-83 980325 65 488492 46g227 6-83 980286 65 488941 47 5no59 52 , 9 469637 6-82 980247 65 489390 47 5io6io 5i 10 470046 6.81 980208 65 489838 46 510162 5o II g. 470455 6.80 9-980169 65 9-490286 46 io-5o97i4 40 509267 ! 48 12 470863 6-8o 9S0130 65 490733 45 i3 471271 6-70 6.78 980091 65 491 180 44 508820 47 14 47(6^9 980052 65 491627 44 5o8373 46 i5 472086 6-78 980012 65 492073 43 507927 45 i6 472492 6-77 979973 65 492519 43 507481 44 n 472898 6.76 979934 979895 66 492965 42 5o7o35 43 i8 473304 6.76 66 493410 41 5o65go 42 19 473710 6-75 979835 66 493354 40 506146 41 30 4741 1 5 6-74 979816 66 494299 40 5o57oi 40 21 9-474519 6-74 9'97'5776 66 9-494743 40 io-5o5257 li 22 474923 6-73 979737 66 495186 It 504814 23 475327 6.72 & 56 495630 504370 37 24 473730 6-72 56 496073 37 503927 36 25 476133 6-71 979618 66 496515 37 5o3485 35 26 476536 6-70 979579 66 496957 497399 36 5o3o43 34 11 476933 6-69 979539 66 36 5o26oi 33 477340 6-69 6-68 979499 66 497841 498282 35 5o2i59 32 29 477741 979439 66 34 501718 3i 3o 478142 6-67 979420 66 498722 34 501278 3o 3i 9-478542 6.67 g- 979380 66 9-499'63 33 io.5oo837 ll 32 478942 6.66 979340 66 499603 33 5oo397 499958 33 479342 6-65 979300 67 5ooo42 32 ll 34 479741 6-65 979260 67 5oo48i 31 499319 35 480140 6-64 979220 67 500920 5oi359 31 499080 25 36 480539 6-63 979180 67 30 498641 24 ll 480937 6-63 979 '40 67 501797 30 498203 23 481334 6-62 979100 67 5o2235 It 497765 22 39 48 1 73 1 6.61 979059 67 502672 497328 21 40 482128 6-61 979019 67 5o3io9 28 496891 20 41 g. 482525 6-60 9-978979 67 g-5o3546 27 10-496454 It 42 482921 6-59 978939 978898 67 503982 27 496018 43 4833 16 67 5o44i8 26 495582 17 44 483712 978858 67 504854 25 495146 16 45 484107 6^57 978817 67 505289 25 4947 1 1 i5 46 484501 6.57 978777 67 5o5724 24 494276 14 % 484S95 6-56 978736 67 5o6i59 506593 24 493841 i3 435289 6-55 978696 68 23 493407 12 49 485682 6-55 978655 68 507027 22 492973 11 5o 486075 6-54 978615 68 507460 22 492540 10 5i 9-486467 6-53 9-978574 68 9-507893 21 10-492107 t 52 486S60 6-53 978533 68 5o8326 21 491674 53 487251 652 9784q3 68 508759 20 491 241 7 f4 487643 6.5i 978452 68 509191 19 49080Q 6 55 488034 6-5i 97841 1 68 509622 \l 490378 5 56 488424 6-5o 978370 68 5ioo54 489946 4 U 488814 6-5o 978329 68 510485 18 4895.5 3 489204 6-49 978288 68 510916 5ii346 !^ 489084 488654 a 59 4895^3 6-48 978247 68 I L, „. 6-48 978206 68 511776 7^16 488224 Coskie D. Siae D. Cotano^. D. Tanc-. M. 17 (72 DEGREKS.) 86 (18 DEGllEES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. (!otang. 9-489082 6.48 9-978206 • 68 9.5n776 7.16 10-488324 60 I 490J71 6 -48 978165 -68 512206 16 487794 5? 2 490739 6 47 978124 -68 512685 i5 487865 3 491147 6 -46 978088 .69 518064 14 486936 486507 57 4 491535 6 -46 978042 .69 518493 14 56 5 491922 6 .45 978001 .69 518921 514349 i3 486079 55 6 492808 6 44 977959 977918 .69 i3 485651 54 7 6 492695 498081 6 44 69 514777 12 485223 53 6 43 977877 69 5i52o4 12 484796 5a ; 493466 6 42 977835 69 5i563i II 484869 5i IC 498851 6 42 977794 69 516057 10 483943 5o !I 9-494236 6 4i 9-977752 69 9.516484 10 io-4835i6 49 12 494621 6 41 977711 69 516910 517335 09 488090 48 l3 495005 6 40 977669 977628 69 It 482665 47 14 49^388 6 39 ^ 517761 482239 46 i5 493772 6 39 38 977586 69 5i8i85 08 48i8i5 45 i6 496154 6 977544 70 518610 07 481890 44 n 496537 6 37 977508 .70 519084 06 480966 480O42 43 i8 496919 497801 6 37 977461 .70 519458 06 42 ^9 6 86 977419 -70 519882 o5 480118 41 20 497682 6 36 977377 70 52o3o5 o5 479695 40 21 9-498064 6 35 9-977335 70 9-520728 04 10-479272 478849 ll 22 498444 6 34 977293 70 52ii5i 08 23 498825 6 34 977231 70 521573 o3 478427 37 24 499204 6 33 977209 70 521995 03 478005 36 25 499584 6 32 977167 70 522417 02 477583 35 26 499963 6 32 977125 70 522838 02 477162 34 11 5oo342 6 3i 977083 70 528259 01 476741 33 500721 6 3i 977041 70 528680 01 476820 32 29 501099 6 3o 976999 70 524100 00 475900 3i 3o 501476 6 29 976907 70 524520 6 99 475480 3o 3i 9-5oi854 6 29 9.976914 70 9.524989 6 99 10-475061 It 32 50223l 6 28 976S72 71 525359 6 98 474641 33 502607 6 28 976880 71 52577§ 6 98 474222 27 34 502984 5o386o 6 27 976787 71 526197 6 97 478808 26 35 6 26 976745 71 52661 5 6 97 473385 25 36 508785 6 26 976702 71 527038 6 96 472967 24 37 5o4iio 6 25 976660 71 527451 6 96 472549 23 38 5o4485 6 25 976617 71 527868 6 95 472182 22 39 5o486o 6 24 976574 71 528285 6 90 471715 21 40 5o5284 6 23 976532 71 528702 6 94 471298 20 41 9-5o56o8 6 23 9-976489 71 9-529119 6 93 10-470881 19 42 505981 6 22 976446 71 529085 6 93 470465 18 43 5o6854 6 22 976404 71 529950 6 93 47oo5o 17 44 506727 6 21 , 976861 71 53o366 6 92 469684 16 45 507099 6 20 676318 71 580781 6 91 469219 i5 46 507471 6 20 976275 71 531196 6 91 468804 14 47 507848 508214 6 19 976282 72 531611 6 90 468889 467975 467561 i3 48 6 ;? 976189 72 532025 6 % 12 49 5o8585 6 976146 72 582489 6 II 5c 008956 6 18 976108 72 532853 6 89 467147 .0 5i 9.509826 6 17 9-976060 72 9.538266 6 88 10-466734 § 52 509696 6 16 976017 72 583679 6 88 466821 53 5ioo65 6 16 975974 72 534092 6 87 465908 7 54 510434 6 i5 975980 72 534504 6 87 465496 465o84 6 55 5io8o3 6 i5 975887 72 534916 6 86 5 ^6 511172 6 14 975844 72 535828 6 86 464672 4 11 5ii54o 6 18 975800 72 535789 6 85 464261 3 511907 5l22l5 6 i3 975757 72 536i5o 6 85 463850 5 59 6 12 975714 72 536561 6 84 468489 468028 I 66 512642 6-12 975670 72 536972 6 84 Cosine D. Sine D. Cotaug. D. lang. Jk {11 DEaREES.) SINES AND TANGENTS. (19 DEGREES.) 37 M. SJJie D. Coiiiie D. Taag. D. Cotang. 9.512642 6.12 9.975670 73 0-536972 6.84 10.4630'. 8 60 I 5x3000 6-11 973627 73 537382 6 83 462618 u a 513375 6. II 975583 73 537792 6 83 462208 3 5i374i 6-10 975539 73 538202 6 82 46x798 57 4 514107 6.09 975496 73 53861 X 6 82 461389 56 5 514472 6. 09 6.08 975452 73 539020 6 81 460980 55 6 5 I 483 7 973408 7? 539429 6 8x 460371 54 I 5l5202 6.08 975365 73 539837 6 80 460163 53 5 1 5566 6.07 975321 73 540245 6 80 439733 52 9 5i593o 6.07 975277 '7? 540653 6 79 439347 ; 5i 1 10 516294 6.06 975233 73 541061 6 79 458939 5o II 9.516657 6.o5 9.975189 7? c. 541 468 6 78 xo. 458532 S 13 517020 6.05 975143 75 541875 6 78 458x25 i3 5173S2 6.04 975101 7^ 542281 6 77 437719 47 14 517745 6.04 975057 H ■ 542688 6 77 437312 46 i5 518107 6.03 975oi3 73 543094 6 76 436906 45 i6 518468 6 -03 974969 74 543499 6 76 43650X 44 \l 518829 6.02 974923 74 543905 544310 6 75 436095 43 519190 6.01 974S80 74 6 75 4556Q0 455285 42 19 5i955i 6.01 974836 74 544715 6 74 41 20 519911 6.00 974792 74 545x19 6 74 45488 X 40 21 9.520271 6.00 9.974748 74 9 545524 6 73 10.454476 It 22 52063 1 5.99 974703 74 545928 6 73 454072 23 520990 til 974639 74 546331 6 72 453669 37 24 521349 974614 74 546735 6 72 453263 36 25 521707 5.98 974570 74 547138 6 71 432862 35 26 522066 5.97 974323 74 547540 6 71 432460 34 27 522424 5.96 974481 74 547943 548345 6 70 432037 33 28 522781 5.96 974436 74 6 70 431655 32 29 523 138 5.95 974391 74 548747 6 69 451253 3i 36 523495 5.95 974347 75 549x49 6 69 45o85x 3o 3i 9.523852 5.94 9.974302 75 9.549350 6 68 xo.45o45o It 32 524208 5-94 974257 75 549931 6 68 450049 33 524564 5.93 974212 75 55o332 6 67 449648 27 34 524920 5.93 974167 75 550752 6 67 449248 26 35 525275 5.92 974122 75 55ii52 6 66 448848 25 36 525630 5.91 974077 75 55x552 6 66 448448 24 u 525984 5.91 974032 75 55x932 6 65 448048 23 526339 5.90 973987 n 55235X 6 65 447649 22 39 526693 5.90 973942 973897 7^ 552750 6 65 447250 2X 40 527046 5.^9 75 553x49 6 64 446851 20 41 9.527400 5.89 9-973852 75 9-553548 6 64 10.446452 \t 42 527753 528lOD 5.88 973807 V 553946 554344 6 63 446054 43 5.83 973761 75 6 63 445656 X7 | 44 528458 5.87 973716 76 55474X 6 62 445259 x6 45 528810 5.87 973671 76 553139 6 62 44486 X i5 46 529161 5.86 973625 76 555536 6 61 444464 14 47 529513 5.86 973580 76 555933 556329 6 61 444067 x3 48 529864 5.85 973535 76 6 60 44367 X 12 49 53021 5 5.85 973489 76 556725 6 60 443275 XI 5o 53o565 5.84 973444 76 557121 6 59 442879 10 5i Q-53o9i5 5.84 9.973398 76 9.5575x7 6 59 10.442483 i 5i 531265 5.83 973332 76 557913 6 U 442087 53 53i6i4 5.82 973307 76 5583o8 6 441692 I 54 531963 532312 5.82 973261 76 558702 6 58 441298 55 ii' 9732x5 76 539097 6 57 440903 5 56 532661 5.81 973169 76 55949 X 6 57 440309 4 u 533009 5.80 973x24 76 559885 6 56 440X13 3 533357 5.80 973078 76 560279 560673 6 56 43972 X 2 59 533704 tl^ 973o32 77 6 55 439327 I 60 . 534002 972986 77 56 I 066 6.55 43893J I. - Ccr'-a« D. Sine D. Cotanor. D. Taiior 1 M. 1 (70 DEGREES.) '68 (20 DEGRE] Eb.; A TABLE OF LOGARITHMIo' ~sr Suie D. Cosine D. Tan?. D. Cotan?. c , 9- 53403 > 5.78 9-972986 •77 ' Q-56io66 6.55 10-43^34 l6o* 1 534399 5-77 972940 •77 561459 6-54 433541 i 59 438.49 1 58 1 534745 5-77 972 x>4 •77 56.85. 6-54 3 1 535092 5-77 972^48 •77 562244 6-53 437-'56 1 57 437364 i 56 4 535t3:i 5.76 972S02 •77 562636 6-53 5 535733 5-76 972755 77 563028 6-53 .436972 55 6 536129 5.75 972709 77 563419 6-52 436531 54 7 536474 5-74 972663 •77 563S11 6-52 436.^9 53 S 536Si8 5-74 972617 •77 564202 6-5i 435793 52 9 537163 5.73 972070 •77 564592 6-5i 435408 5i lo 537507 5.73 972524 •77 564953 6-50 4350.7 5o II 9.537851 '5-72 9-972478 'n 9-565373 6-5o 10.434627 it 12 53S194 5-72 972431 .78 565763 6-49 434237 i3 53S533 5-71 972 385 .78 566153 6-49 433S47 47 14 538880 5-71 972338 •^? 566542 tit 433458 46 i5 539223 5-70 972291 .78 566932 433o63 45 l6 539565 5-70 972245 •''? 567320 6-43 432680 44 n 539907 5-69 972.93 '''i 567709 6-47 432291 43 i8 540^49 5-69 972i5i •■'? 563oo3 6-47 43.902 42 19 540590 5-68 972105 .78 5634^6 6-46 43nj4 41 20 54og3i 5-63 972058 .78 563873 6-46 431127 40 21 9041272 5-67 9-97201 1 .78 Q. 569261 6-45 10-430739 39 22 54.6-3 5-67 971964 .78 569648 6-45 43o352 38 23 541953 5-66 97'9'7 'l^o 570035 6-45 429965 37 24 542293 5-66 971370 .78 570422 6-44 429378 36 25 542632 5-65 971823 "^i 570809 6-44 429191 35 26 542971 5-65 971776 .78 571.93 6-43 4233o5 34 27 5433.0 5-64 97.729 •79 57i53i 6-43 42S419 33 28 543649 5-64 9716S2 •79 571967 572352 6-42 428033 32 29 543987 5-63 971635 •79 6-42 427648 3i 36 544325 5-63 971588 •79 5-72738 6-42 427262 3o 3i 9-544663 5-62 9-971540 •79 9-573123 6-41 10-426S77 It 32 545ooo 5-62 971493 •79 573507 6.41 426493 33 545338 5-6i 971446 •79 573892 640 426.08 27 34 545674 5-6i 971398 •79 574276 6-40 425724 26 35 546011 5.60 971351 •79 574t)6o 6-39 425340 25 36 546347 5-60 97i3o3 •79 575044 6-39 424956 24 37 546683 5-59 971256 •79 575427 6-39 424573 23 33 5470«9 5-59 971208 •79 57d3io 6-38 424190 23 39 547^54 5-58 971161 •79 576.93 6-38 423307 21 40 5476^9 5-58 971113 •79 576576 6-37 423424 20 41 9.548024 5-57 9-971066 .80 9-576953 6-37 10-42304! •9 42 548359 548693 5.57 971018 .80 577341 6-36 422659 18 } 43 5-56 97097c .80 577723 6-36 4222^7 17 44 549027 5-56 970922 .«o 573104 6-36 42lb«)) 16 45 549060 5.55 970574 8c 573456 6-35 42i5i4 i3 46 549-'393 5.55 970827 .80 5-3^67 6-35 421.33 14 47 35oo2o 5.54 970779 .80 579248 6-34 42075s i3 48 560359 5-54 970731 1 .80 579029 6-34 420371 12 1 49 500692 5.53 970683 ■ • 80 D50009 6-34 419991 II 5o 55.024 5-53 970635 .80 58o3S9 6-33 419011 10 5i 9-551356 3-52 9-970586 .80 9-580769 6-33 io-4io23i 4i8«5i t 53 55i657 5-52 970533 .80 581149 6-32 53 55k).8 5-52 970490 • 80 58i528 6-32 418472 7 54 552349 5-5i 970442 1 .80 581907 6-32 418093 6 55 5526ik) 5-51 970894 1 .80 5b2286 6-3i 4I77U 5 56 553010 5-50 970345 1 .81 582665 6-3i 4.7335 4 57 55iUi 5-5o 970297 .81 583043 6-3o 416957 3 58 553670 5-49 970249 1 .81 583422 6-3o 416578 2 59 554000 5-49 770200 1 .81 583doo 6-29 416200 I 66 554329 5.48 97c 32 1 .81 584177 6-29 413823 Cosine D. SLae D. COUUi:i?. D. Tang. (69 I )EGR EE3.) SINES AND TANGENTS. (21 DEGREES.) 39 M. Sine D, Cosine | D. Tang. D. Cotang. 60 o 9.554329 554658 5-48 9970152 .81 9-534177 6.29 10-415823 I 5 48 970103 .81 584555 6 It 40445 It 3 5549B7 5 47 970055 .81 584932 6 4i5o68 3 55531 5 5 47 970006 .81 585309 6 -28 414691 57 4 555643 5 46 969907 .81 585686 6 27 414314 56 5 555971 5 46 969909 .81 586062 6 27 413938 55 6 556299 5 45 969860 .81 586439 6 27 4i3o6i 54 I 556626 5 45 96981 1 • 81 586810 6 26 4i3i85 53 556953 5 44 969762 .81 587190 6 26 412810 52 9 557280 5 44 9697*4 .81 587066 6 25 412434 5i lo 557606 5 43 969663 .81 587941 6 25 412059 5o II 9.557932 5 43 9 • 9696 1 6 .82 9-5883i6 6 25 10-411684 8 12 558258 5 43 969567 1 .82 588691 6 24 411309 i3 558583 5 42 969518 .82 589066 6 24 410934 47 14 558go9 5 42 969469 1 .82 589440 6 23 410060 46 i5 559234 5 41 969420 1 .82 589814 6 23 410186 45 i6 559558 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 590935 6 22 409065 408692 42 19 56o53i 5 39 969223 .82 59i3o3 6 22 41 20 56o855 5 39 969173 .82 591681 6 21 408319 40 21 9.561178 5 38 9-969124 .82 9-592054 6 21 10-407946 It 22 56i5oi 5 38 969075 .82 592426 6 20 407074 23 561824 5 37 969025 968976 .82 592798 6 20 407202 37 24 562146 5 ll .82 593170 6 19 406S29 36 25 562468 5 968926 .83 593542 6 19 406453 35 26 562790 5 36 968877 .83 593914 6 18 406086 34 »^ 563II2 5 36 968827 .83 5942:35 6 18 4057 1 5 33 563433 5 35 968777 .83 594656 6 18 405344 32 ^9 563755 5 35 968723 .83 595027 6 17 404973 3i 3o 564075 5 34 968678 .83 595398 6 n 404602 3o 3i 9.564396 5 34 9.96S628 .83 9-595768 6 17 10.404232 ll 32 564716 5 33 968578 .83 596133 6 16 4o3862 33 565o36 5 33 968528 .83 596508 6 16 403492 27 34 565356 5 32 968479 .83 596878 6 16 4o3i22 26 35 565676 5 32 96S429 .83 597247 6 i5 402753 25 36 565995 5 3i 968379 .83 597616 6 i5 402384 24 ll 566314 5 3i 968329 • S3 597985 6 i5 4020 1 5 23 566632 5 3i 96S278 .83 59^354 6 14 401646 22 39 566951 5 3o 96S228 • 84 598722 6 14 401278 21 40 567269 5 3o 968178 .84 599091 6 i3 400909 20 41 9-567587 5 29 9-968128 .84 9-599459 6 i3 10 -40054 1 \l 42 567904 5 29 968078 .84 099827 6 i3 400173 43 56S222 5 28 968027 .84 600194 & 12 399806 \l 44 56S539 5 28 967977 .84 6oo562 6 12 399438 45 568856 5 28 967927 .84 600929 6 11 399071 398704 ID 46 569172 5 27 967^76 .84 601290 6 11 :i ii 569488 5 ll 967826 .84 601662 6 II 398338 i3 569 S04 5 967773 .84 602029 6 10 397971 12 P 570120 5 26 967725 .84 602395 6 10 397605 II 5o 570435 5 25 967674 .84 602761 6 10 397239 10 5i 9.570751 5 25 9-967624 .84 9-6o3i27 6 09 10-396873 I 52 571066 5 24 967573 .84 603493 6 09 896507 53 57j38o 5 24 967522 .85 6o3858 6 ll 396142 7 ?^ 571695 5 23 967471 .85 604223 6 390777 6 63 572009 572323 5 23 967421 .85 604588 6 08 _". ' ' ' 390412 5 56 5 23 967370 • 85 604953 6o53i7 6 07 390047 4 i^ 572636 5 22 967319 .85 6 07 394683 3 572950 5 22 967268 .85 6o5682 6 07 394318 2 ^ D73263 5 21 967217 .85 606046 6 06 393904 I (to 573575 5.21 967166 .85 , 606410 6.06 393090 Cosine D. Sine ' D. Cotiincr. ._i^J Tu/ig. (68 DEGREES.li 40 (22 DB3^REES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotang. 1 9.573575 5.21 9-967166 85 9-606410 6.06 10-393590 60 1 573888 5 20 9671 i5 85 606773 6 06 392863 ! 5? 2 574200 5 20 967064 85 607137 6 o5 3 574512 5 19 967013 85 607500 6 o5 3925^0 57 4 574824 5 »9 966961 85 607863 6 04 392137 56 5 575i36 5 966910 85 608225 6 04 391775 55 6 575447 5 18 966859 966808 85 6o8588 6 04 391412 54 I 575758 5 18 85 608950 6 o3 391050 i 53 576069 5 17 966756 86 609312 6 o3 390688 52 9 576379 5 17 966705 86 609674 6 o3 3oo326 5i 10 576689 5 16 966653 86 6 10036 6 02 3899O4 5o II 9-576999 577309 5 16 9-966602 86 9-610397 6 02 10 389603 ii 12 5 16 966550 86 610759 6 02 389241 i3 577618 5 i5 966499 86 611120 6 •01 388880 47 H 577927 5 i5 966447 86 61 1480 6 •01 388520 46 i5 578236 5 14 966395 86 611841 6 01 388,59 45 i6 578545 5 14 966344 86 612201 6 • 00 387799 44 17 578853 5 i3 966292 86 6i256i 6 -00 387439 43 i8 579162 5 i3 966240 86 612921 6 •00 387079 42 19 579470 5 i3 966188 86 613281 5 •99 386719 41 20 579777 5 12 966136 86 6i364i 5 99 386359 40 21 9-58oo85 5 12 9.966085 87 9-614000 5 98 10^ 386000 39 22 58o392 5 II 966033 87 6i435o 614718 5 98 385641 38 23 580699 5 II 960981 87 5 98 385282 37 24 58ioo5 5 II 965928 87 61 5077 5 97 3^4923 36 25 58i3i2 5 10 965876 87 6 1 5435 5 97 384565 35 26 58. 618 5 10 965824 87 615793 5 97 3S4207 34 27 581924 5 09 965772 87 6i6ni 5 96 383849 33 28 582229 5 09 965720 87 6i65o9 5 96 3S349I 32 29 582535 5 09 963668 87 616867 5 96 3S3i33 3i 3o 582840 5 08 9656 1 5 87 617224 5 95 3S2776 ' 3o 3i 9-583145 5 08 9-965563 87 9-617582 5 95 10.382418 It 32 583449 5 07 9655 11 87 617939 5 95 382061 33 583754 5 07 965458 87 618295 5 94 38i7o5 27 34 584058 5 06 965406 87 61 8652 5 94 381348 26 35 584361 5 06 965353 88 619008 5 94 380992 25 36 584665 5 06 965301 88 619364 5 93 38o636 24 ll 584968 5 o5 965248 88 619721 5 93 3S0279 23 585272 5 o5 960195 88 620076 5 93 3-9924 22 39 585574 5 04 965143 88 620432 5 92 379068 21 40 585877 5 04 965090 88 620787 5 92 379213 20 41 9-586179 5 o3 9-965037 88 9.621142 5 92 10.378858 \l 42 586482 5 f)3 964984 88 621497 5 91 3785o3 43 586783 5 o3 964931 88 621802 5 91 378148 17 44 587085 5 02 964879 88 622207 5 90 377793 16 45 587386 5 02 964826 88 622061 5 90 377439 i5 46 587688 5 01 964773 88 622915 5 % 377085 14 47 587989 5 01 964719 88 623269 5 376731 i3 48 588289 5 01 964666 89 623623 5 89 376377 12 49 588590 5 00 964613 89 623976 5 89 376024 II 5o 588890 5 00 964560 89 624330 5 88 375670 1 10 1 5i 'ia 4 99 9.964507 89 9.624683 5 88 10-375317 t 52 4 99 964454 89 620036 5 88 374964 53 589789 590088 4 99 964400 89 625388 5 87 374612 7 54 4 98 964347 89 625741 5 87 374209 6 55 590387 4 9S 064294 89 626093 5 87 373907 5 56 590.^86 4 97 964240 89 626445 5 86 3-3555 4 U 5Q0984 4 97 964187 89 ' 626797 5 86 373203 3 591282 4 97 964133 89 627149 5 86 372851 2 59 591580 4 96 964080 89 627501 5 85 372499 372148 I 66 591878 4-96 964026 89 627852 5 85 Cosii e D. Sine D. Cotang. D. Tang. 1 M. | (67 DEGREES.) SINES AND TANGENTS. (23 EEGREES.) 41 M. Sine 1 D. ! Cosine D. 1 Tang. J). Cotang. o Q. 591878 4.96 9.964026 .89 9-627852 5-85 10-872148 60 1 592176 4.95 968972 .89 628203 5.85 371797 5? 2 592473 4-95 963919 .89 628554 5-85 371446 3 592770 4-95 968863 .90 628905 5-84 371095 57 4 593067 4.94 968811 <90 629235 5.84 370745 56 5 5q3363 4-94 968757 .90 629606 5-83 370894 55 6 593559 4-93 968704 .90 629956 5-83 370044 54 7 593933 4.93 968650 • 90 680806 5-83 869694 53 8 594231 4.93 968596 .90 680656 5-83 360844 368995 52 9 594547 4-92 968542 .90 63ioo5 5-82 5i 10 594842 4-92 968488 .90 68i855 5-82 368645 5o II 9 595137 4-91 9.968434 • 90 9-681704 5-82 10-368296 It 12 573432 4-91 968879 .90 682053 5-81 367947 13 593727 4-91 968823 .90 682401 5-81 367599 47 M 596021 4-90 968271 .90 682730 5-8i 367230 46 i5 5963 1 5 4.90 968217 .90 688098 5-80 366902 45 16 llZt 4-89 968168 .90 688447 5-8o 366553 44 \l 4-89 963 1 08 .91 688795 5-8o 366205 43 597196 968034 .91 684143 5.79 365857 42 19- 597490 4-88 962999 .91 684490 5-79 365510 41 20 5977^3 4-88 962945 .91 684888 5-79 365i62 40 21 9.598075 4-87 9.962890 962886 •91 9-635i85 5.78 io-3648i5 It 22 59^368 4-87 .91 635582 5-78 364468 23 598660 4-87 962781 .91 683879 5-78 364I2I 37 24 598932 4.86 962727 .91 686226 5-77 868774 36 25 599244 4-86 962672 .91 636572 5-77 368428 35 26 599336 4-85 962617 .91 686919 5-77 363o8i 34 '^I 599:^27 4-85 962362 .91 687263 5-77 362785 33 28 600118 4-85 962308 .91 68761 1 5.76 362889 32 29 600409 4-84 962438 .91 687936 5.76 362044 3i 3o 600700 4-84 962898 .92 638302 5-76 361698 3o 3i 9.600990 4-84 9.962843 .92 9-638647 5.75 10-361353 11 32 601 2do 4-83 962288 .92 688992 5-75 361008 33 601370 4-83 962288 .92 689887 5-75 36o663 27 34 601860 4-82 962178 .92 68968^ 5-74 36o3i8 26 35 602 I 30 4-82 962128 .92 640027 5-74 359973 25 36 602439 4-82 962067 .92 640871 5-74 359629 24 il 602728 4-8i 962012 .92 640716 5.73 359284 358940 23 38 6o3oi7 4.81 961957 .92 641060 5-73 22 39 6o33o3 4-8i 961902 961846 .92 641404 5.73 358596 21 4o 603594 4-8o .92 641747 5.72 358253 20 41 9.603882 4-8o 9.961791 .92 9-642091 5-72 10.357909 19 42 604170 4-79 961785 .92 642484 5-72 337566 18 43 604437 4-79 961680 .92 642777 5-72 337228 n 44 604743 4-79 4-78 961624 .93 648120 5-71 356880 16 45 6o3o32 961569 .98 648468 5-71 356^'» , iJ 1 46 6o53i9 4-78 9615.3 .93 648806 5-71 356104 14 47 603606 4-78 96.438 .93 644148 5.70 355832 i3 48 603892 4-77 961402 .93 644490 5.70 3555 10 12 49 606179 4-77 961846 •93 644882 5.70 355i68 li 5o 606465 4-76 961290 •93 645174 5-69 354826 10 5i 9.606731 4-76 C-96I233 .93 9-643316 5-69 10-354484 I 52 607036 4-76 961179 .98 643857 5-69 854143 53 607822 4-75 961.28 •93 646199 5.69 353801 7 54 607607 4-75 961067 .93 64-'>340 5-68 353460 6 55 607S92 4-74 96101 I •93 646881 5-68 3331.9 352778 5 56 608.77 4-74 960933 .98 647222 5.68 4 57 608461 4-74 960899 •93 647362 5.67 352438 3 58 60S745 4-73 960843 .94 647908 5.67 352097 2 h 6ogo2g 4-73 960786 .94 648243 5.67 331737 I 6o 609313 4-73 900780 ij94_ 648583 5.66 331417 1 Co^.ne i D. Sine 1 r. Cotaiisr. 1 D. Tang. IT (66 DEGREES.) 42 (24 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Taug. D. Coiang. j 9-6o93i3 4-73 9-960730 94 9-648583 5-66 >o-35i4i7 1 60 I 609397 4 72 960674 94 648923 5 66 351077 1 5o 330737 1 sa 3 609880 4 72 960618 94 649263 5 66 3 6 1 1 64 4 72 96056 1 94 649602 5 66 330398 57 4 610447 4 71 96030: 94 649942 5 65 330038 1 56 5 6S0729 4 71 960448 94 6302 Si 5 65 3497 '9 1 55 6 6 ! 1 ; 2 4 70 960392 94 65o6ic 5 65 349380 1 54 7 61 1294 4 70 960335 94 630909 5 64 349041 i 53 8 61 1076 4 in 960279 94 65 1297 5 64 348703 ' 52 9 6ii858 4 6q 960222 94 63i6J6 5 64 34S364 ' 5 1 10 612140 4 69 960165 94 651974 5 63 348026 50 1 II 9-612421 4 69 Q • 960 I 09 95 9-652312 5 63 10-347688 ii 12 612702 4 68 96003 J 93 652630 5 63 347330 i3 612983 4 68 939995 95 652988 5 63 347012 47- 14 613264 4 67 909933 95 653326 5 62 346674 46 i5 613545 4 67 939S82 95 653663 5 62 346337 45 )6 6i3S25 4 67 939S25 95 654000 5 62 346000 44 »7 6i4io5 4 66 039768 95 654337 5 61 343663 43 i8 6i4385 4 66 939711 95 654674 5 6i 345326 42 J9 614665 4 66 939654 95 655011 5 61 3449^9 41 20 614944 4 65 939396 95 655348 5 61 344652 40 21 9-6i5223 4 65 9 •939339 95 9-655684 5 60 io-3443i6 ^ 22 6i55o2 4 65 939482 95 656020 5 60 3439^0 23 615781 4 64 939425 95 656356 5 60 343644 u 24 616060 4 64 939368 95 636692 5 59 343308 25 6 1 6338 4 64 939310 96 657028 5 59 3429^2 35 26 616616 4 63 939233 96 657364 5 59 342636 34 27 616894 4 63 909195 96 657699 5 59 342301 33 28 617172 4 62 939138 96 658o34 5 58 341966 32 ?9 617450 4 62 9390R1 96 658369 5 58 34r63i 3i So 617727 4 62 939023 96 658704 5 58 341296 3o 3. 9-618004 4 61 9-958965 96 9-659039 659373 5 58 10-340961 ll 32 018281 4 61 93S908 96 5 57 340627 33 6i8558 4 61 938850 96 659708 5 57 340292 ll 34 618834 4 60 938792 96 660042 5 57 339938 35 619110 4 60 938734 96 660376 5 57 339624 25 36 619386 4 60 938677 96 6607 1 5 56 339290 24 ll 619662 4 59 958619 96 661043 5 56 33S937 23 619938 4 59 938561 96 661377 5 56 33S623 22 39 620213 4 ll 9385o3 97 661710 5 55 338290 21 40 620488 4 958445 97 662043 5 55 337957 20 41 g-620763 4 58 9-958387 97 9-662376 5 55 10-337624 \l 42 621038 4 57 93S329 97 662709 5 54 337291 43 62i3i3 4 57 938271 97 663042 5 54 336938 n 44 621587 4 57 9^8213 97 663375 5 34 336625 16 4>> 621861 4 56 958134 97 663707 5 54 336293 i5 46 ()22l35 4 56 938006 97 664039 5 53 333961 14 47 622409 4 56 938038 97 664371 5 53 335629 i3 48 622682 4 55 937979 97 664703 5 53 333297 12 49 622956 4 55 937921 957863 9^ 665o35 5 53 334965 II 5o 623229 4 55 97 665366 5 52 334634 10 5i 9-623502 4 54 9-937804 97 9-665697 5 52 io-3343()3 t 52 623774 4 54 957746 98 666029 5 52 333971 53 624047 4 54 957687 98 666360 5 5i 33364C 7 54 624319 4 53 937628 98 666691 5 5i 333309 6 55 624591 4 a3 957570 98 667021 5 5i 332979 ' 5 56 624863 4 53 937511 98 667352 5 5i 332648 4 57 623135 4 32 957432 93 667682 5 5o 3323 1 8 - ■ 3 58 625406 4 52 957393 98 66801 3 5 5o 331987 1 59 625677 4 52 957335 9^ 668343 5 5o 33 1 657 I 6o 623948 4.31 957276 98 668672 5-5o 33i328 1 Ckwino i D. Sine 1 D. Cotanor. D- .. Tang. M. {Qo DEGREES.) SINES AND TANGENTS. (25 DEGREES.) la Sine } 9-625948 1 D. , CosillQ i D. 1 Tansr. D. i Cotang. j 4-5. 9-937276 .98 9-668673 5-50 1 io-33i327 60 I 626219 1 4-5i 9)7217 .98 669002 5-49 330Q98 59 330668 58 3 626490 4-5i 957158 .98 669332 5-49 3 6367^0 4-50 1 937099 .98 669661 5.49 5-48 33o339 57 4 6y7o3o 4-30 1 937040 .98 66999, 330009 55 5 627800 4-00 i 936981 .98 670320 5-48 329680 , 55 6 627570 4-49 • 9 '^ 69 2 1 • 99 670649 5-48 329351 54 7 627840 4-49 ! 956S62 •99 670977 5-48 329023 53 8 6.2S109 628378 4-49 1 936803 •99 671306 5-47 32%94 52 9 4-48 906744 •99 671634 5-47 328366 5i lo 628647 4-48 936684 •99 67:963 5.47 328037 5o (I 9-628916 4-47 9.936625 •99 )-67229i 5.47 10-327709 ii 12 629185 4-47 [ Q 36 366 •99 672619 5-46 32738, i3 629453 4-47 , 936306 •99 67 29 17 5.46 327053 47 14 629721 4.46 1 956447 •99 673274 5-46 326726 46 i5 629989 4-46 936387 •99 673602 5.46 326398 45 l6 630237 4.46 956327 •99 673929 5-45 326071 44 17 63o524 4-46 936268 •99 674237 5.45 325743 43 i8 630792 4-45 936208 I -00 674584 5.45 325416 42 »9 63 10^9 4-45 936148 I -00 674910 5-44 325090 41 20 63i326 4-45 936089 I -00 675237 5.44 324763 40 21 9-63i593 4-44 9-956029 I- 00 9-675564 5-44 10-324436 U 22 631839 632125 4.44 935969 I -00 675890 5.44 324110 23 4-44 935909 I -00 676216 5-43 323784 37 24 C32392 4-43 933849 I -00 676543 5.43 323457 36 25 632658 4-43 955789 I -00 676S69 5-43 323i3i 35 26 632923 4-43 955729 I -00 677194 5-43 322806 34 27 633189 4-42 955669 I-OO 677520 5.42 322480 33 28 633454 4-42 935609 955548 I -00 677846 5.42 322154 32 29 633719 4-42 I -00 678171 5.42 321829 3i 3o 633984 4-41 955488 I -00 678496 5.42 32i5o4 3o 3i 9-634249 4-41 9.955428 1 -01 9.678821 5-41 10-321,79 11 32 634514 4.40 955368 I -01 679146 5.41 320854 33 634778 4-40 955307 I-OI 679471 5-41 320529 32o2o5 27 34 635042 4-40 933247 l-OI 679795 5.41 26 35 6353o6 4-39 955186 I-OI 6S0120 5-40 319880 25 36 635570 4-39 955126 I-OI 680444 5-40 319556 24 37 635834 4-39 955o65 I-OI 680768 5.40 319232 23 38 636097 4-38 955oo5 I-OI 681092 5.40 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 3i?26o 20 4i 9-636886 4-37 9 954823 I -01 9.682063 5.39 10-3,7937 \l 42 637148 4-37 934762 I -01 682387 5-3o 5-38 317613 43 6374.1 4-37 954701 I-OI 682710 317290 \l 44 637673 4-37 954640 I-OI 683o33 5-38 316967 45 637935 4-36 9^4570 934518 I-Ol 683356 5-38 316644 i5 46 638197 638458 4-36 1-02 683679 5-38 3 16321 U 47 4-36 954457 1-02 684001 5.37 3,5999 i3 48 1 638720 4-35 934396 1-02 684324 5-37 3 1 5676 12 49 1 63Mg8i 4-35 934335 1-02 684646 5.37 3 15354 II 5o 1 639242 4.35 954274 1-02 684968 5-37 3,5o32 10 5i 9-6395o3 4-34 9-954213 1-02 9-685290 5-36 10-3,4710 t 52 639764 4-34 954152 1-02 685612 5-36 3,4388 53 640024 4-34 954090 1-02 685934 5-36 3,4066 I 54 640284 4-33 954029 053968 1-02 686255 5-36 3,3745 55 640544 4-33 1-02 686577 5-35 3,3423 5 56 640804 4-33 5]?f5 1-02 686898 5.35 3,3,02 4 5-? 641064 4-32 1-02 687219 5-35 3,2781 3 68 641324 4-32 933783 1-02 687540 5.35 312460 2 5g 641584 4-32 933722 i-o3 ^87861 5-34 3,2,39 3,1818 I 6o 641843 4-3i 953660 I -03 688182 5.34 M. Cosine D. 1 Sine 1 D. Cotan.of. D. Tans-. (61 DEGR EES.^i 44 (26 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine E ►. Tang. D. Cotang. 1 — 1 o 9-641842 4-3i 9-953660 I o3 9-688182 5.34 io-3n8i8 60 i I 642101 4 3i 953599 I o3 688502 5 34 311498 U a 642360 4 3i 953537 I o3 688823 5 34 311177 3 642618 4 3o 953475 I o3 689143 5 33 3 1 0857 57 4 642877 4 3o 953413 : o3 689463 5 33 3io537 56 5 643 i35 4 3o 953352 I o3 689783 5 33 310217 55 6 643393 4 3o 953290 I o3 690103 5 33 309897 54 7 643650 4 29 953228 1 o3 690423 5 33 309577 53 8 643908 4 29 953166 I o3 690742 5 32 309258 .08938 52 9 644165 4 29 953104 I o3 691062 5 32 5i ic 644423 4 28 953042 I o3 691381 5 32 308619 5o II 9.6446S0 4 28 9-952980 I 04 9-691700 5 3i i3-3o83oo ii 12 6449-36 4 28 952918 I 04 692019 692338 5 3i 307981 i3 64Mc3 4 27 952855 I 04 5 3i 307662 47 14 6i:)/,;)o 27 932793 I 04 692656 5 3i 307344 46 i5 645706 4 27 932731 I 04 692975 5 3i 307025 45 i6 64S962 4 26 932669 I 04 693293 5 3o 306707 44 >7 646218 4 26 932606 I 04 693612 5 3o 3o6388 43 i8 646474 '4 26 952544 I 04 693930 5 3o 306070 42 »9 646729 4 25 952481 I 04 694248 5 3o 3o5752 41 20 646984 4 25 952419 04 694566 5 29 3o5434 40 21 9-647240 4 25 9-932356 I 04 9-694883 5 29 io-3o5ii7 ll 23 647494 4 24 952394 I 04 693201 5 29 304799 23 647749 4 24 952231 I 04 6q55i8 5 29 304482 37 24 648004 4 24 952168 I o5 695836 5 29 28 304164 36 25 648258 4 24 952106 I o5 696153 5 3o3847 35 26 648512 4 23 952043 I o5 696470 5 28 3o353o 34 27 648766 4 23 951980 I o5 696787 5 28 3o32i3 33 28 649020 4 23 951917 I 95 1 854 I o5 697103 5 28 3o2'<97 3o258o 32 ?9 649274 4 22 o5 697420 5 27 3i 3o 649327 4 22 951791 I o5 697736 5 27 302264 3o 3i 9-649781 4 22 9-951728 I o5 9.69S053 5 27 10.301947 ll 32 65oo34 4 22 95 1 665 I o5 698369 5 27 3oi63i 33 600287 4 21 951602 I o5 69H685 5 26 3oi3i5 27 34 65o539 4 21 951539 I o5 699001 5 26 300999 26 35 650792 4 21 951476 I o5 699316 5 26 300684 25 36 65 1 044 4 20 951412 I o5 699632 5 26 3oo368 24 ll 651297 4 20 951349 I 06 699947 5 26 3ooo53 23 65 1 549 4 20 931286 I 06 700263 5 25 299737 22 39 65 1 800 4 •9 951222 I 06 700578 5 25 299422 21 1 40 652032 4 19 951159 I 06 700893 5 25 299107 20 4i 9-6523o4 4 ;? 9-951096 I 06 9-701208 5 24 10.298792 18 42 652555 4 95io32 I 06 701323 5 24 29S477 43 652806 4 18 930968 I 06 701837 5 24 298163 '7 44 653o57 4 18 o5o9o5 I 950841 I 06 702132 5 24 297848 16 45 6533o8 4 18 06 702466 5 24 297534 i5 46 653558 4 17 950778 I 06 7027S0 5 23 297220 14 % 6538o8 4 17 950714 I 06 703095 5 23 29690.'. i3 654059 4 n 930650 . I 06 703409 5 23 296391 12 49 6543o9 4 16 93o586 I 06 703723 5 23 296277 II 5o 654558 4 16 95o522 I 07 704036 5 22 295964 10 5i 9 654808 4 16 9-950458 I 07 9-704330 5 22 10-293650 i 52 655o58 4 16 950394 I 07 704663 5 22 293337 53 655307 4 i5 93o33o I 07 704977 5 22 293023 7 54 655556 4 i5 950266 I 07 705290 5 22 2947 IP 6 55 6558o5 4 i5 930202 I 07 703603 5 21 29439" 294084 5 56 656o54 4 t4 9501 38 I 07 703916 5 21 4 57 656302 4 14 930074 I 07 706228 5 21 293772 3 58 656551 4 14 950010 I 07 706541 5 21 2Q3459 2 59 656799 4 i3 949945 I 07 706R54 5 21 293146 I 66 657047 4-i3 949881 I 07 707166 5-20 292834 M. ' Cosine 1 D. Sine 1 I ). Cotang. D. Tang. (63 DEGREES.) SINES AND TANGENTS. (27 DEGREES.) 45 M. Sine D. Cosine D. Tang. D. Cotang. o 9.657047 4'i3 9.949881 1.07 9.707166 5.20 10-292834 60 I 657295 4-i3 949816 1.07 707478 5.20 292522 59 2 607542 4-12 949732 1-07 707790 708102 5-20 292210 58 3 65779G 4-12 949688 1-08 5.20 291898 57 4 658o37 4-12 949623 I -08 708414 5.19 291386 56 5 658284 4-12 949338 1-08 708726 5.19 291274 53 6 658531 4-II 949494 1.08 709037 5.19 290963 54 ? 658778 4-II 949429 I -08 709349 5.19 290651 53 659025 4-11 949364 1-08 709660 5.19 290340 32 9 659271 4-10 949300 1-08 709971 5.18 290029 5i lo 659517 4-10 949235 1-08 710282 5.18 289718 5o n 9.659763 4-10 9-949170 1-08 9.710593 5.18 10.289407 it 12 660009 4-09 949103 i-o8 7 1 0904 5.18 ^W i3 660233 4-09 949040 948975 1.08 711215 5.18 288783 47 14 66o5oi 4'OQ I -08 711325 5.17 288475 46 i5 660746 4-09 948910 948845 1.08 711836 5.17 288164 45 i6 660991 4-o8 1-08 712146 5.17 287854 44 n 661236 4-o8 948780 1.09 712436 5.17 287544 43 i8 661481 4-o8 948713 1.09 712766 5.16 287234 42 19 661726 4-07 948630 1.09 713076 5.16 286924 41 20 661970 4*07 948584 1.09 713386 5.16 286614 40 21 9.662214 4-07 9.948319 1.09 9.713696 5.16 io.2863o4 It 22 662439 662703 4-07 948454 1-09 7 1 4oo5 5.16 283995 23 4-o6 948388 1-09 714314 5.i5 285686 37 24 662946 4-o6 948323 1-09 714624 5.i5 285376 36 25 663190 4-o6 948257 1.09 714933 5.i5 283067 35 26 663433 4-o5 948192 1-09 715242 5.i5 284738 34 !J 663677 4*o5 948126 1.09 7i555i 5.14 284449 33 663920 4-o5 948060 1-09 7 1 5860 5.14 284140 32 29 664163 4-o5 947995 I-IO 716168 5.14 283832 3i 3o 664406 4-04 947929 l-IO 716477 5.14 283523 3o 3i 9.664648 4-04 9.947863 1. 10 9.716785 5-14 IO-2832I5 29 32 664891 4-04 947797 I-IO 717093 5.i3 282907 1 28 1 33 665 1 33 4-o3 947731 I-IO 717401 5.i3 282399 27 34. 665375 4-o3 047665 l-IO 717709 718017 5.i3 282291 26 35 665617 4-o3 947600 l-lO 5.i3 281983 25 36 665859 4-02 947333 I-IO 718325 5.i3 281670 24 ll 666100 4-02 947467 I-IO 718633 5.12 281367 23 666342 4-02 947401- 1. 10 718940 5.12 281060 22 39 ■ 666583 4-02 947335 I-IO 719248 5.12 280732 21 40 666824 4-01 947269 I. 10 719355 5.12 280445 20 41 9.667065 4*01 9.947203 I. 10 9-719862 5.12 io.28oi38 10 18 42 667305 4-01 947136 l-II 720169 5.11 279831 43 667346 4-01 947070 I. II 720476 5. 11 27C-.24 •7 44 667786 4*00 947004 I. II 720783 5. II 279217 16 45 668027 4-00 946937 I- 11 721089 5.11 278911 i5 46 668267 4-00 946871 I . II 721396 5.11 278604 14 47 6685o6 3.99 946804 I. II 721702 5.10 278298 i3 48 668746 3-99 946738 I. II 722009 7223i5 5.10 277991 12 49 668986 3-99 946671 I-II 5.10 277683 II 50 669225 3.99 946604 I. II 722621 5.10 277379 JO 5i 9.669464 3.98 9-946538 I. II 9.722927 5.10 10-277073 t 52 669703 3.98 946471 I. II 723232 5.09 276768 53 669942 3.98 946404 I'll 723538 5.09 276462 7 54 670181 3.97 946337 I- n 723844 5.09 276i56 6 55 670419 670658 3.97 946270 1-12 724149 5.09 273831 5 56 3-97 946203 1.12 724434 5.09 5.08 273346 4 ll 670896 671134 3-97 946 1 36 I. 12 724739 275241 3 3.96 946069 I-I2 723o65 5.08 274935 3 59 671372 3.96 946002 1-12 725369 5.08 274631 I 66 671609 3.^6 945935 1-12 725674 5-08 274326 Cosine D. Sine D. Cotang. D. Tang. M. (62 DEGREES.) 46 (28 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tang. D. Cotang. 60 o 9 67 1609 3-96 9.945935 1. 12 9.725674 5 08 ic. 274326 I 67i'S47 3 95 943868 1.12 725979 5 08 274021 U a 6720S4 3 95 945800 I-I2 726284 5 07 273;i6 3 672321 3 95 945733 I-I2 ,26588 5 07 273412 u 4 672558 3 9: 940666 I-I2 726892 5 07 273io5 5 672795 673o32 3 94 945598 1-12 727197 5 07 272S03 55 6 3 94 945531 I • 12 727501 5 07 272499 54 I 673268 3 94 945464 I • l3 727805 5 06 272195 '.i3 673505 3 94 945396 1-13 728109 5 06 27.891 32 9 673741 3 93 945328 I.i3 728412 5 06 27 . 588 5i 10 6739- 3 93 945261 i.i3 728716 5 06 27.284 50 II 9 674213 3 9? 9-945193 i-i3 9-729020 5 06 10-270980 % 12 674448 3 92 945i25 i.i3 729323 5 o5 270677 i3 674684 3 92 945o58 1-13 729626 5 o5 270374 47 14 674919 3 92 944990 i.i3 729929 5 o5 270071 46 i5 675i55 3 92 944^22 i.|3 730233 5 o5 269167 45 i6 675390 3 91 944«54 1-13 73o535 5 o5 269 ',65 44 17 675624 3 91 944786 1-13 73o838 5 04 269162 43 18 670859 3 91 944718 1-13 731141 5 04 268si59 42 19 67601;/, 3 9' 944650 1-13 731444 5 04 268556 41 20 676328 3 90 944582 1-14 731746 5 04 268254 40 2. 9-676562 3 90 9-944514 I-I4 9-732048 5 04 10.267952 39 22 676796 3 90 044446 1-14 732351 5 o3 267649 38 23 677o3o 3 90 944377 1-14 732653 5 o3 267347 u 24 677264 3 89 944309 I -14 732955 5 o3 267043 25 677498 3 89 944241 1.14 733257 5 o3 266743 35 26 677731 3 89 944172 1-14 733558 5 o3 266442 34 11 677964 3 88 944104 1.14 733860 5 02 266 1 40 33 678197 3 88 944036 I-I4 734162 5 02 265838 32 29 678430 3 88 943967 1-14 734463 5 02 265537 3i 3o 678663 3 88 943899 i-:4 734764 5 02 265236 30 3i 9.678895 3 87 9-043830 1-14 9-735o66 5 03 10.264934 It 32 679128 3 87 94376. 1-14 735367 5 02 264633 33 679360 3 87 943693 i-i5 735668 5 01 264332 27 34 679392 3 87 943624 1-15 735969 5 01 s64o3i 26 35 679^24 3 86 943555 I. .5 736269 5 01 263731 25 36 68oo56 3 86 9434B6 i.i5 736570 5 01 263 i3o 24 37 680288 3 86 943417 i-i5 736871 5 01 263 1 29 23 38 68o5i9 3 85 943348 i-iS 737171 5 00 262^29 22 39 680750 3 85 943279 I • i5 737471 5 00 262129 21 40 680982 3 85 943210 i-i5 737771 5 00 262229 20 41 o.68i2i3 3 85 9-943i4t i-i5 9-738071 5 00 10-261929 19 42 681443 3 84 943072 i-i5 738371 5 00 261629 18 43 681674 3 84 943oo3 I- 15 738671 4 99 261329 17 44 681905 3 84 942934 i-i5 738971 4 99 261029 16 45 682135 3 84 942^64 i-n 739271 4 99 260729 i5 46 682365 3 83 942793 I..6 739570 4 99 260430 14 47 682095 3 83 942726 1-16 739870 4 99 260 1 3o i3 48 682825 3 83 942656 116 740169 4 99 25qS3i 12 49 683o55 3 83 942087 1-16 74046& 4 98 209^32 II 5o 683284 3 82 942517 1.16 740767 4 98 259233 10 5i 9-683514 3 82 9-942448 i.;6 9.741066 4 98 10-258934 t 52 683743 3 82 942378 i-i6 741365 4 98 25S635 53 683972 3 82 9423o8 1-16 741664 4 98 208336 7 5-4 684201 3 8i 942239 i-i6 741962 4 97 258o38 6 5c 684430 3 8i 942169 i-i6 742261 4 97 257739 5 56 684658 3 81 942099 i-i6 742509 4 97 20-441 4 57 684887 3 80 942029 i-i6 74285:i 4 97 207143 3 58 685n5 3 80 941909' i-i6 743.06 4 97 256844 a 59 685343 3 So 941889 I-I7 743454 4 97 206546 i 60 685571 3.80 941819 1-17 743752 4.96 206248 CoBiue D. >-!ne D. Cotang. D. Tang. (61 DEGREES.) SINES AND TANGENTS. (29 DEGREES.) 47 M. Sine D. Cosine D. Tang. 1 ^• Cotang. 1 o 9 685571 3-8o 9-941819 1-17 9-743752 4-96 io^a56a48 60 I 685799 3 79 941749 117 744o5o 4.96 255950 u 2 686027 3 79 941679 I -17 744348 4.96 255652 3 686254 3 79 941609 1-17 744645 4-96 255355 57 4 686iS2 3 ]i 941539 1-17 744943 4.96 255o57 56 5 686709 3 941469 1 -17 745240 4.96 254760 55 6 686936 3 78 941398 1-17 745538 4-9? 214462 54 2 687163 3 78 941328 1-17 745835 4-9= 214165 53 1 687389 3 78 941258 1-17 746132 4.91 253868 52 9 687616 3 77 941187 1-17 746429 4-95 213571 5i 10 687843 3 77 941117 1.17 746726 4-95 253274 5o II 9-688069 3 77 9-941046 1..8 9-747023 4-94 10 252977 % 12 68H2Q5 3 77 940975 1.18 747319 4-94 252681 i3 688.121 3 ^6 940905 1-18 747616 4-94 252384 47 14 6S8747 3 76 940834 1-18 747913 4.94 252087 46 i5 68S972 3 76 940763 1-18 748209 4-94 251791 45 i6 6S9198 3 76 940693 1-18 7485o5 4.93 211493 44 \l 689423 3 75 940622 1-18 748801 4.93 211199 43 6S9648 3 75 94055 1 i-i8 749097 4.93 210Q03 42 19 6^9^73 3 75 9404^^0 I -18 749393 749689 4-93 210607 41 20 690098 3 75 940409 i-iS 4.93 25o3ii 40 21 9-690323 3 74 9-940338 1-18 9-749985 4-93 io-25ooi5 39 22 690548 3 74 940267 i-i8 750281 4-92 249719 38 1 23 690772 3 74 940196 1-18 750176 492 249424 37 24 690996 3 74 940125 1-19 750872 4-92 249128 36 25 691220 3 73 940054 1-19 751167 4-92 24i^33 35 26 691444 3 73 939982 1-19 751462 4-92 248538 34 H 691668 3 73 939911 1-19 751757 4-92 248243 33 1 28 691S92 3 73 939840 1-19 752012 4-91 2479 i« 32 ^9 692115 3 72 93(p68 1-19 752347 4-91 247653 3i 3o 692339 3 72 939697 1-19 752642 4-91 247358 3o 3i 9-692562 3 72 9-939625 1-19 9-712937 4-91 10- 247063 29 28 32 692785 3 71 939^54 1-19 753231 4.91 246769 33 693008 3 71 939482 1-19 753126 4-91 246474 11 34 693231 3 71 939410 1-19 753820 4.90 246180 35 693453 3 71 939339 1-19 754115 4.90 245885 25 36 693676 3 70 93926^ 1-20 754409 4.90 245591 24 11 693 S98 3 70 939195 I -20 754703 4.90 24''297 23 694120 3 70 939123 1-20 754907 4.90 245oo3 22 39 694342 3 70 939012 1-20 755291 4.90 244709 21 40 694564 3 69 936980 1-20 755585 4.89 244413 20 41 9-694786 3 69 9.93S90S 1-20 9-755878 4.89 10-244122 \t 42 693007 3 69 93S836 1-20 756172 4.89 243828 43 695229 3 69 938763 1-20 756465 4.89 243535 17 44 695450 3 68 938691 1-20 716719 4-89 243241 16 43 695671 3 68 938619 I -20 717012 4.89 2429 IS i5 46 695H92 3 68 938547 1-20 757345 4-88 242655 14 % 6961 i3 3 63 938475 1-20 757638 4-88 242362 i3 690334 3 67 938402 1-21 757931 "^■fo 242069 12 P 690554 3 67 93>^33o I-2I 758224 4-88 241776 II 5o 6967 7* 7 67 938258 I-2I 758517 4.88 241483 10 5i 9-696995 3 <57 9 -93s 1 85 I-2I 9-758810 4.88 10-241190 I 53 69-215 3 66 9381.3 1-21 759102 4-87 240893 53 697435 3 66 938040 1-21 759395 4.87 240605 7 54 697654 3 66 937967 1-21 719687 4-87 24o3i3 6 5f 697874 3 66 937^95 1-21 759979 4-87 24002 1 5 56 69^094 3 •65 937822 1-21 760272 4-87 239728 4 5^ 6gS3i3 3 • 65 9^7749 1-21 760164 4-87 239436 3 5fc 6q8532 3 • 65 937676 1-21 760856 4-86 239144 2 59 698751 3 -65 93-J604 1-21 761148 4-86 238852 I 6o 698970 3 .64 937531 1-21 761439 4-86 238561 Cosine Ij X Sine D. Cotansr. D. Tamr. _^-.. 27 (60 DEGREES.) 48 (30 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine j 1>. T.ai-. D. Cotang. 9.698970 3.64 9.937531 I 21 9-761489 4.86 10.238561 60 I 699189 3-64 937458 I 2i 76I78I 4-86 288269 It 2 699407 3.64 937385 I 22 762023 4-86 237977 3 699626 3.64 937312 I 22 762814 4.86 287686 57 4 699844 3-63 987238 I 22 762606 4.85 137894 56 5 700062 3-63 987165 I 22 762897 4-85 237103 55 6 700280 3-63 987092 I 22 768188 4-85 286812 1 54 1 7 700498 3.63 987019 I 22 7684-9 4.85 236321 53 8 700716 3-63 986946 I 986872 I 22 768770 4.85 286280 52 9 700933 3-62 22 764061 4.85 283989 235648 5i 10 70ii5i 3.62 986799 I 22 764352 4-84 5o II 9.701368 3-62 9.986725 I 22 9.764643 4-84 10.235357 49 12 701585 3-62 986632 1 23 764933 4.84 285067 48 i3 701802 3.61 986578 1 23 765224 4.84 284776 47 U 7020T9 3-6i 9865o5 I 23 765514 4-84 284486 46 i5 702236 3.61 986431 I 23 7658o5 4-84 284193 45 i6 702452 3.61 986837 I 23 766095 4-84 288905 44 \l 702669 3-6o 986284 I 23 766385 4.88 2836i5 43 702883 3-60 986210 I 23 766675 4-83 233823 42 '9 7o3ioi 3.60 986186 1 23 766965 4-83 283o35 41 20 703317 3-6o 986062 I 23 767255 4-83 282745 40 21 9.703533 3.59 9.983988 1 23 9.767545 4-83 10.282455 ll 22 703749 3.59 98 5^40 I 23 767884 768124 4-83 282166 23 703964 3.59 23 4.82 281876 37 24 704179 •704395 3.59 983766 I 24 768413 4.82 23 1 587 36 25 tu 983692 I 24 768703 4.82 281297 35 26 704610 985618 I 24 »J 4.82 281008 34 27 704825 3.58 935543 I 24 4-82 280719 33 28 700040 3.58 935469 I 24 769570 4.82 280480 32 29 705254 3-58 935393 1 24 769860 4.81 280140 3i 3o 705469 3.57 935320 I 24 770148 4-8i 22QS52 3o 3i 9.705683 3.57 9-935246 1 24 9.770437 4.81 10.229563 It 32 705898 3.57 985,71 I 24 770726 4-81 220274 228985 33 706112 3.57 983097 I 24 771015 4-81 'I 34 706326 3.56 935022 I 24 77i3o3 4-81 228697 26 35 706539 3.56 934948 I 24 771592 771880 4-81 228408 25 36 706753 3-56 984878 I 24 4.80 228120 24 11 706967 3.56 93479^ ^ 25 772168 4.80 227882 23 707180 3.55 984723 I 25 772457 4-80 227543 22 39 707393 3.55 984649 I 25 772745 4-80 227255 21 40 707606 3.55 934374 I 25 773o33 4-8o 226967 20 41 9.707819 3.55 9-934499 I 25 9.773321 4-80 10.226679 19 42 708032 3.54 934424 I 25 778608 4-79 226892 18 43 708245 3.54 934849 I 25 773806 774184 4.79 226104 'I 44 708458 3.54 934274 1 25 4-79 2258i6 16 45 708670 3-54 934199 I 25 774.;] I 4-79 225329 i5 46 708882 3-53 984128 1 25 774759 4.79 225241 14 47 709094 3.53 984048 I 25 775046 4-79 224954 i3 48 709806 3.53 988973 I 25 775333 i]i 224667 i 12 j 49 709518 3.53 988S98 1 26 775621 224379 1 II 1 5o 709730 3.53 988822 I 26 775908 4-78 224092 " 1 5i 9.709941 3.52 9.933747 I 26 9.776195 4.78 io.2238o5 I 52 710153 3.52 988671 I 26 776482 4.78 2235i8 53 710364 3.52 933596 I 26 776769 4.78 228281 I 54 710575 3.52 988520 1 26 777053 4-78 222943 j 55 710786 3.5i 983445 I 26 777342 4-78 222658 5 ' 56 710997 711208 3.5i 988869 I 988293 I 26 777628 4-77 222872 4 i ll 3.5i 26 777915 4-77 222083 3 711419 3.5i 9882.7 I 26 778201 4-77 221799 3 59 711629 3.50 988141 I 26 778487 4-77 22l5l2 * 60 ]iml 3.50 088066 1 .26 778774 4-77 221226 , j Cosine 1 D. Sine 1 I ). Cotansr. _2.-.__ ___Tan2^ M. 1 (59 DEGREES.) SINES AND TANGENTS. (31 DEGREES.) 49 M. Sine D. 1 Cosine D. 1 Tang. 1 D. Cotang. 60 9-711839 3.5o 9.933066 1-26 9-778774 4-77 10.221226 I 7i2o5o 3-50 932990 1-27 779060 4-77 220940 ^5^ 2 712260 3.5o 932914 1.27 779346 4-76 220654 3 712469 3-49 932838 1-27 779632 4-76 2 20368 57 4 712679 3-49 932762 1-27 779918 4-76 220082 56 5 712889 71J098 3.49 932685 1-27 780203 4-76 219797 55 6 3-49 932609 1-27 780489 4-76 219311 54 I 7i33o8 3-49 3-48 932533 1-27 780773 4-76 219225 53 7i35i7 932457 1-27 781060 4-76 218940 52 9 713726 3.48 932380 1-27 781346 4-75 218634 5i ic 713935 3-48 932304 1-27 781631 4-75 218369 5o II 9-714144 3.48 9.93'2228 1.27 9.781916 4.75 10-218084 8 13 714352 3.47 93qi5i 1-27 782201 4-75 217799 i3 714361 3.47 932075 1-28 782486 4-75 217514 47 14 714769 3.47 931998 1-28 782771 4-75 217229 46 i5 7i497« 3.47 93 1 92 1 1.28 783o56 4-75 216944 45 i6 7i5i86 3.47 931845 1.28 783341 4-75 216659 44 !^ 715394 3.46 931768 1-28 783626 4-74 216374 43 7i56o2 3-46 931691 1-28 783gio 4-74 216090 42 »9 713809 3-46 93.614 1.28 784195 4-74 2i58o5 41 20 716017 3.46 931D37 1-28 784479 4-74 2l552I 40 21 9-716224 3-45 9.931460 1-28 9-784764 4-74 io.2i5236 U 22 716432 3-45 93 1 383 1-28 785048 4-74 214952 23 716639 3-45 93i3o6 1.28 785332 4-73' 214668 11 24 716846 3.45 931229 1.29 7856i6 4-73 214384 25 717053 3-45 93 I 132 1.29 • 783900 4-73 214100 35 26 717259 3-44 931075 1.29 786184 4-73 2i38i6 34 11 717466 3.44 930998 1-29 786468 4-73 213532 33 717673 3-44 930921 1-29 786752 4-73 213248 32 ?9 717879 3-44 930843 1-29 787036 4.73 212964 3i So 718085 3.43 930766 1.29 787319 4-72 212681 3o 3i 9-718291 3.43 9.930688 1-29 9.787603 4-72 10.212397 11 32 718497 3.43 930611 1.29 787886 4-72 212114 33 718703 3.43 93o533 r-29 788170 4-72 2ii83o 27 34 718909 3.43 93 0456 1.29 788453 4-72 211547 26 35 719114 3.42 930378 1-29 788736 4-72 211264 23 36 719320 3-42 93o3oo i-3o 789019 4-72 210981 24 37 719^25 3.42 930223 i-3o 789302 4-71 210698 23 38 719730 3-42 930145 1-30 789585 4-71 2 1041 5 22 39 719935 3.41 930067 i-3o 789868 4-71 210132 21 40 720140 3-41 929989 1-30 79oi5i 4-71 209849 20 4i 9-720345 3-41 9-92Q91I 1.30 9.790433 4-71 10.209567 19 42 720549 3.41 929833 1.30 790716 4-71 209284 10 43 720754 3-40 929755 i.3o 790999 4-71 209001 ^I 44 720958 3-40 929677 1.30 791281 4-71 208719 16 45 721162 3-40 929399 1-30 791563 4-70 208437 i5 46 721366 3-40 929521 I -30 791846 4-70 2081 54 14 47 721570 3-40 929442 1-30 792128 4-70 207872 i3 48 721774 3.39 929364 1-31 792410 4-70 207390 13 4o 721978 3.39 1 929286 1.31 792692 4-70 207308 II 5o j 722181 3.39 929207 1. 31 792974 4-70 207026 10 5i i 9-722385 3.39 9-929129 1-31 9-793256 4-70 10-206744 9 52 722588 3-39 929050 1-31 793538 4-69 206462 8 53 722791 3-38 I -'3 1 793819 4-69 206181 I 54 722994 3-38 i-3i 794101 4-69 205899 55 723197 3-38 928815 i.3i 794383 4-69 2o56i7 ' 5 56 723400 3-38 928736 i-3i 794664 4-69 205336 4 § 723603 3-37 928657 1-31 794945 4-69 2o5o55 3 7238o5 3-37 928578 1-31 795227 4-69 4-68 204773 2 59 724007 3.37 928499 i.3i 795508 204492 I 6o 724210 3.37 928420 i.3i 795789 4-68 204211 Coeine D. Sine D. Cotan?. I D. >.Tan^- lir (58 DEGREES.) 50 (32 DEGREES.) A TABLE OF LOGARITHMIC M. Siuo D. Conine B. Tung. D. Cotani?. 60 9-724210 3.37 9-928420 I 32 9.795789 • 4-68 IO-20i2ll I 724412 37 928342 I 32 796070 4 68 2o393o 5o 2 724614 36 928263 I 32 796351 4 68 2o36',9 5S 1 3 724816 36 928183 I 32 796632 4 68 203368 57 4 723017 36 928104 1 32 796913 4 68 2o3o^7 56 5 720219 36 928025 I 32 797 « 94 4 68 202S06 55 6 725420 35 927946 I 32 79-7475 4 68 2O2020 54 I 720622 35 927867 I 32 797755 4 63 202240 53 725823 35 927787 I 32 79S036 4 67 2019^)4 52 9 726024 35 927708 I 32 7yH3i6 4 67 20 1 684 5i lo 726225 3 3o 927629 1 32 798596 4 67 201404 5o II 9.726426 3 34 9-927549 I 32 9.798877 4 67 10-201 123 t 12 726626 . 3 34 927470 1 33 799157 4 67 200H43 i3 726827 3 34 927390 I 33 799437 4 67 200563 47 14 727027 3 34 927310 I 33 799717 4 67 2002S3 46 i5 727228 3 34 927231 I 33 799997 4 66 200003 45 i6 727428 3 33 927i5r I 33 800277 4 66 199723 44 n 727628 3 33 927071 I 33 8oo507 4 66 199443 43 i8 727828 3 33 926991 I 33 8oo836 4 66 I9QI64 42 19 728027 3 33 92691 1 I 33 80M16 4 66 I9S8S4 41 20 728227 3 33 926831 1 33 801396 4 66 198604 40 21 9-728427 3 32 9-926751 I 33 9-801675 4 66 10-108325 39 22 728626 3 32 926671 1 33 801905 4 66 198045 38 23 728825 3 32 926591 I 33 802234 4 65 197766 37 24 729024 3 32 926011 I 34 8o25i3 4 65 197487 36 25 729223 3 3i 926431 I 34 802792 4 65 197208 35 26 729422 3 3i 926351 I 34 803072 4 65 196928 34 27 729621 3 3t 926270 I 34 8o335i 4 65 196649 33 28 729820 3 3i 926190 I 34 8o363o 4 65 1963-0 32 2g 730018 3 3o 926110 I 34 803908 4 65 196092 3 1 3o 730216 3 3o 926029 I 34 804187 4 65 195813 3o 3i 9•^3o4I5 3 3o 9-925949 1 k 9-804466 4 64 10.195534 It 32 7306 1 3 3 3o 804745 4 64 195255 33 7308 1 1 3 3o 925788 I 34 8o5o23 4 64 194977 27 34 731009 3 29 925707 I 34 8o53o2 4 64 194698 26 35 731206 3 29 925626 I 34 8o5o8o 4 64 194420 25 36 73i4o4 3 29 925545 I 35 8o5809 4 64 194141 24 37 731602 3 29 923465 I 35 806137 4 64 I93S63 23 38 731799 3 29 28 925384 1 35 8064 1 5 4 63 193585 22 39 731996 3 9253o3 I 35 806693 4 63 193307 21 40 732193 3 28 920222 1 35 806971 4 63 193029 20 41 9.732390 732567 3 28 9-925141 I 35 9.807249 4 63 10-192751 \l 42 3 28 920060 I 35 807527 4 63 192473 43 732784 3 28 924979 I 35 807805 4 63 192195 17 44 732980 3 27 924897 1 35 808083 4 63 191917 16 45 733177 3 27 924816 I 35 8o836i 4 63 191639 i5 46 733373 3 27 924735 I 36 8o8638 4 62 191362 14 47 733569 3 27 924604 I 36 808916 4 62 1910S4 i3 48 733760 3 27 924572 1 36 809193 4 62 190^07 12 49 733961 3 26 924491 I 36 809471 4 62 190029 II 5o 734107 3 26 924409 I 36 809748 4 62 190202 10 5i 9.734353 3 26 9-924328 I 36 9-810025 4 62 10-189970 t 52 734049 3 26 924246 I 36 8io3o2 4 62 189698 53 734744 3 25 924164 I 36 8io58o 4 62 189420 1 54 734q39 3 25 9240S3 I 36 810857 4 62 189143 6 55 735i3o 3 25 924001 I 36 8iii34 4 61 188866 5 56 735330 3 25 923919 I 36 811410 4 61 i885qo 4 U 735525 3 25 923837 I 36 811687 4 61 1 883 1 3 3 735719 3 •24 923755 I 37 811964 4 61 i88o36 2 59 735914 3.24 923673 1 37 812241 4 61 187759 6o 736109 3-24 923591 I 37 812517 4-6i 187483 Cosine D. Sine 1 J). Cotang. D. Tang. M. (5*7 DEGREES.) SINES AND TANGENTS. (33 DEGREES.) 51 M. Sine 1 D. 1 Cosine D. Tang. D. Cotang. 60 "•?& 3-24 9-923591 I 37 9-8i25i7 4-61 10-187482 I 3- 24 923509 I 37 812794 4 6i 187206 It 2 736498 3 24 923427 I 37 8i3o7o 4 61 186930 3 736602 736886 3 23 923345 I 37 813347 4 60 186653 57 4 3 23 923263 I 37 8i3623 4 6c 186377 56 5 737080 3 23 923181 I 37 813899 814175 814452 4 60 186101 1 55 ! 6 737274 3 23 923098 I 37 4 60 185825 54 I 737467 3 23 923016 I 37 4 60 185548 53 737661 3 22 922933 I 922851 I 37 814728 4 60 185272 52 9 737855 3 22 37 8i5oo4 4 60 184996 5i 10 738048 3 22 922768 1 38 815279 4 60 184721 5o 11 9-738241 3 22 9-922686 I 38 9-815555 4 59 10-184445 it 12 738434 3 22 922603 I 38 8i583i 4 59 184169 i3 738627 8 21 922520 I 38 816107 4 59 183893 47 14 738820 3 21 922438 I 38 8i6382 4 59 i836i8 46 i5 739013 3 21 922355 I 38 8i6658 4 59 183342 45 i6 739206 3 21 922272 I 38 816933 4 59 183067 44 n 739398 3 21 922189 1 38 817209 4 59 182791 43 i8 739390 3 20 922106 I 38 817484 4 59 i825i6 42 19 739783 3 20 922023 I 38 817759 818035 4 ll 182241 41 20 739975 3 20 921940 I 38 4 181965 40 21 9-740167 3 20 9-921857 I 39 9-8i83io 4 58 10-181690 39 22 740359 3 20 921774 I 39 8i8585 4 58 i8i4i5 38 23 74o55o 3 19 921691 1 39 818860 4 53 181140 37 24 740742 3 19 921607 I 39 819135 4 58 i8o865 36 25 740934 3 19 921524 I 39 819410 4 58 180590 35 26 74II25 3 19 921441 I 39 819684 4 58 i8o3i6 34 27 74i3i6 3 \l 921357 I 39 819959 4 58 180041 3J 28 741 5o8 3 921274 I 39 820234 4 58 179766 32 29 741699 3 18 921190 I 39 82o5o8 4 57 179492 3i 3o 741889 3 18 921107 I 39 820783 4 57 179217 3o 3i 9-742080 3 18 9-921023 I 39 9-821057 4 57 10-178043 It 32 742271 3 18 920039 I 40 821332 4 57 178668 33 742462 3 n 920856 I 40 821606 4 57 178394 27 34 742652 3 17 920772 I 40 821880 4 57 178120 26 35 742842 3 17 920688 I 40 822154 4 57 177846 25 36 743o33 3 17 920604 I 40 822429 4 57 177571 24 37 743223 3 17 920520 1 40 822703 4 57 177297 23 38 743413 3 16 920436 I 40 822977 4 56 177023 22 39 743602 3 16 920352 1 40 823250 4 56 176750 21 40 743792 3 16 920268 I 40 823524 4 56 176476 20 41 9.743982 3 16 9-920184 1 40 9-823798 4 56 10-176202 \t 42 744171 3 16 920099 I 920015 1 40 824072 4 56 175928 43 744361 3 i5 40 824345 4 56 175655 \l 44 744550 3 i5 919931 I 919846 I 41 824619 824893 4 56 175381 45 744730 744928 3 i5 41 4 56 175107 i5 46 3 i5 919762 1 41 825166 4 56 174834 14 ^^ ]%ll 3 i5 919677 I 41 825439 825713 4 55 174561 i3 3 14 919593 I 41 4 55 174287 12 49 745494 3 14 919508 I 41 825986 4 55 . 174014 II 5o 745683 3 14 919424 I 41 826259 4 55 173741 iO 5i 9-745871 3 14 9-919339 I 41 9-826532 4 55 10-173468 t 52 746059 746248 3 14 919254 1 41 826805 4 55 173195 53 3 i3 919169 I 41 827078 4 55 172922 I 54 746436 4 i3 919085 I 41 827351 4 55 172649 55 746624 3 i3 919000 I 918015 I 918830 I 41 827624 4 55 172376 5 56 746812 3 i3 42 827897 4 54 172103 4 57 746999 747187 3 .i3 42 828170 4 54 171830 3 58 3 -12 918745 I 42 828442 4 54 171558 2 59 747374 3 -12 918659 I •42 828715 4 54 171285 I 60 747562 3-12 918574 I -42 828987 4-54 171013 1 Cosine D. Sine I ). Cotang. D. Tang. 1l 18 (56 DEGREES.) 52 (3-t DEGREES.) A TABLE OF LOGARITHMIC ["mT 1 Sine 1 ! ^' 1 Cosine D. j Tang. I>. 1 Cotang. i o 9-74- '62 , 3.12 9-918574 1-42 9.828987 4-54 10-171013 60 1 I 747749 3.12 918489 1-42 829260 4-54 170740 5q 170468 58 2 747936 3-12 918404 1-42 829532 4-54 3 748123 3. II 9i83i8 1-42 829805 4-54 170195 57 4 7483 10 3-II 918233 1-42 830077 4-54 169923 56 5 74^3683 3.11 918147 1.42 83o349 4-53 169651 55 6 3-11 918062 1-42 83o62i 4-53 169379 54 I 748870 3-11 917976 1-43 830893 4-53 169107 53 749056 3.10 917891 1-43 83ii63 4-53 168835 52 9 749243 3-10 917S05 1-43 831437 4-53 168563 5i xo 749429 3-ic 917719 1-43 83i709 4-53 16S291 5o II 9-749615 3-10 9-917634 1-43 9S^'98i 4-53 10-168019 4q 12 749801 3-10 917548 1-43 832253 4-53 167747 167475 48 |3 749987 3.09 917462 1.43 832525 4-53 47 U 750172 3.09 917376 1.43 832796 4-53 167204 46 ID 730358 3.0^ 917290 1-43 833o68 4-52 166932 , 45 i6 75o543 3.09 917204 1.43 833339 4-52 166661 44 \l 750729 730914 i-S 917118 917032 1-44 1-44 833611 833882 4-52 4-52 166389 166118 43 42 19 701099 3-08 916946 1-44 834154 4-52 165846 , 41 20 751284 3-08 916839 1-44 834425 4-52 165575 40 21 9-751469 3-o8 '■& 1-44 9-834696 4-52 io-i653o4 u 22 75i654 3-08 1-44 834967 4-52 i65o33 23 24 751839 752023 3-o8 3.07 916600 9i65i4 1-44 1-44 835238 835509 4-52 4-52 164762 16449X 37 36 23 752208 3.07 916427 1-44 835780 4-5i 164220 35 26 752392 3.07 916341 1-44 836o5i 4-5i 163949 163678 34 27 752576 3-07 916254 1-44 836322 4-5i 33 28 752760 3.07 916167 1-45 836593 4-5i 163407 32 29 752944 3-o6 916081 1-43 836864 4-5i i63i36 3i 3o 753128 3-06 915994 1.45 837134 4-5i 162866 3o 3i 9-753312 3-06 9.913907 915820 1-45 9 -837405 4-5i 10-162595 2I 32 753495 3.06 1-45 837675 4-5i 162325 33 753679 3-06 915733 1-45 837946 4-5i 162054 27 34 753862 3-o5 915646 1-45 838216 4-5i 161784 26 35 754046 3-05 915339 1-43 838487 4-5o i6i5i3 25 36 754229 3.o5 915472 1-45 838737 4-5o 161243 24 37 754412 3-05 91 5385 1-45 839027 4-50 160973 23 38 754595 3-e5 915297 1-45 ?^??I 4-5o 160703 22 39 754778 3-04 9i52io 1-43 §^9363 4.50 160432 21 40 754960 3.04 915123 1.46 339833 4-5o 160162 20 41 9-755143 3.04 9-9i5o35 1.46 9-840108 4-50 10.159892 It 42 755326 3-04 914948 914860 1-46 840378 4-50 159622 43 755508 3 -04 1-46 840647 4-5o 159353 17 44 755690 3-04 914773 1-46 840917 4-49 J5^8?3 16 45 755872 3-03 914685 1.46 841187 4.49 i5 46 756o54 3-03 914598 1-46 841457 4.49 158543 14 % 756236 3 -03 914510 1.46 841726 4.49 158274 i3 756418 3-o3 914422 1-46 841996 4.49 i58oo4 la 49 756600 3-o3 914334 1-46 842266 4.49 157734 11 5o 756782 3-02 914246 1-47 842535 4.49 157465 10 5i 9-756963 3-02 9-9i4i58 1-47 9 842805 4-49 10-157195 t 52 757144 3-02 914070 1-47 843074 4-49 156926 53 757326 3-02 913982 913894 1-47 843343 4-49 156657 7 54 757507 3-02 1-47 843612 4.49 4-43 156388 1 6 55 757688 3-01 913806 1-47 843882 i56ii8 ■ 5 56 757869 3.01 913718 1-47 844i5i 4.48 155849 i 4 y 758o5o 3-01 9i363o 1-47 844420 4-48 155580 ; 3 758230 3-01 913541 1-47 844680 844958 4.48 i553ii 2 59 7584II 3-01 913453 1-47 4-43 i55o42 i 66 753591 3oi 913365 1-47 845227 4-43 154773 Cosine D. Sine 1 D. 1 Cotang. } D. 1 Tang. < M. (55 DSGRI :e8.^ SINES AND TANGENTS. (35 DEGREES.) 53 M, Siue D. Cosine I ). Tang. D. Cotang. 60 9.758591 3.01 9.913365 1 47 9.845227 4.48 10.154773 I 758772 3 00 918276 I 47 845496 4.48 1 54504 It 2 758952 3 00 918187 I 48 845764 4.48 154286 3 759132 3 00 918099 I 48 846033 4-48 153967 153698 57 4 739312 3 00 918010 I 48 846802 4-48 56 5 759492 3 00 912922 1 48 846370 4.47 I 53480 55 6 759672 2 99 9I2S33 I 48 846889 4-47 i58i6i 54 I 759852 2 99 915=744 I 48 847107 4-47 152893 53 76003 1 2 99 912655 I 48 847876 4-47 152624 52 9 76021 1 2 99 912566 I 48 847644 4-47 152856 5i 10 760890 2 99 912477 I 48 847913 4-47 152087 5o II 9-760569 2 98 9-912888 I 48 9-848181 4-47 io.i5i8i9 tl 12 760748 2 98 912299 I 49 848449 4-47 i5i53i i3 760927 2 98 912210 I 49 848717 4-47 i5i283 47 14 761 106 2 98 912121 I 49 848986 4-47 i5ioi4 46 i5 761285 2 k 912081 I 49 849254 4-47 150746 45 i6 761464 2 98 91 1942 I 49 849322 4-47 150478 44 \l 761642 2 97 911858 I 49 849790 4-46 l5o2IO 43 761821 2 97 911768 1 49 850038 4-46 149942 42 19 761999 2 97 911674 I 49 85o325 4-46 149673 41 20 762177 2 97 9II584 I 49 850598 4-46 149407 40 21 9.762356 2 97 9.911495 I 49 9 -850861 4-46 IO.I49I39 I4887I u 22 762534 2 96 911405 I 49 851129 4-46 23 762712 2 96 911815 I 5o 851896 4.46 148604 3t 24 762889 2 96 911226 I 5o 85i664 4-46 148386 36 25 768067 2 96 9iii36 I 5o 85iq3i 4-46 148069 35 26 768245 2 96 91 1046 I 5o 852i99 4-46 147801 34 27 763422 2 96 910906 I 5o 852466 4.46 147534 33 28 763600 2 95 910866 I 5o 852733 4-45 147267 32 29 763777 2 95 910776 I 5o 858001 4-45 146999 3i 3o 768954 2 95 910686 I 5o 858268 4-45 146782 3o 3i 9.764181 2 95 9-910596 I 5o 9-853585 4-45 10.146465 11 32 764808 2 95 9io5o6 I 5o 853802 4-45 146198 145981 33 764485 2 94 910415 I 5o 854069 854386 4-45 27 34 764662 2 94 910825 I 5i 4-45 145664 26 35 764888 2 94 910285 I 5i 854608 4-45 145397 25 36 765oi5 2 94 910144 I 5i 854870 4-45 i45i3o 24 37 765191 2 94 910034 I 5i 855187 4-45 144863 23 38 765367 2 94 909968 I 5i 855404 4-45 144596 22 39 765544 2 93 909878 I 5i 855671 4-44 144329 21 4o 765720 2 93 909782 I 5i 855988 4-44 144062 20 4i 9.765896 2 93 9.909691 I 5i 9-856204 4.44 10- 148796 \l 42 766072 2 93 909601 I 5i 856471 4-44 143529 143263 43 766247 2 93 909510 I 5i 856737 4-44 17 44 766423 2 93 '^u : 5i 857004 4-44 142996 142730 16 45 766598 2 92 52 837270 4-44 i5 46 766774 2 92 909237 I 52 857587 4-44 142463 14 47 766949 2 92 909146 I 52 857803 4-44 142197 141931 i3 48 767124 2 92 909055 I 908964 I 908878 I 52 838069 4-44 12 49 767800 2 92 52 858336 4.44 I4i664 11 5o 767475 2 91 52 858602 4-43 141898 10 5i 9.767649 2 91 9-908781 I 908690 1 52 9-858868 4-43 10.141182 I 52 767824 2 91 52 859184 4-43 140866 53 767999 2 91 908399 I 52 859400 4-43 140600 7 54 768173 2 91 908507 I 908416 I 52 839666 4-43 140834 6 55 768848 2 90 53 859982 4-43 140068 5 56 768522 2 90 908824 I 53 860198 4-43 189802 4 1 768697 2 90 908288 I 53 860464 4-43 189536 3 768S71 2 90 908141 I 53 860780 4-43 189270 2 59 769043 2 90 908049 I 907958 I 53 860995 4-43 I 39005 18^739 I 60 769219 2-90 53 861261 4-43 Cosine D. Sine I ). Cotan£f. D. Tang. ^. (54 DEGREES.) 51 (36 DEGREES.) A TABLE OF LOGARITHMIC M. Sino D, Cosine 1 ^• Tang. 1 D. Cotang. 60 o 9-769219 2-90 9-907958 907866 1.53 9-861261 4-43 io-i3873q I 769393 2 .89 1-53 861 527 4-43 138473 % 2 769566 2 89 907774 1-53 861792 4-42 138208 3 769740 2 .89 907682 1-53 8620D8 4-42 137942 57 A 769913 2 .89 907590 1-53 862323 4-42 137677 56 5 770087 2 89 907498 1-53 862589 4-42 137411 55 6 770260 2 88 907406 1-53 862854 4-42 137146 54 7 770433 2 88 907314 1-54 863! 19 4-42 13688 1 53 8 770606 2 88 907222 1-54 863383 4-42 i366i5 52 9 770779 2 88 907129 1-54 863650 4-42 136350 5i 10 770952 2 88 907037 1-54 863915 4-42 i36o85 5o II 9-771125 2 88 9-906945 1-54 9.864180 4-42 10-135820 % 12 771298 2 87 906852 1-54 864445 4-42 135555 l3 771470 2 87 906760 1-54 864710 4-42 135290 47 14 771643 2 87 906667 1-54 864975 4-41 135025 46 i5 771815 2 87 906575 1-54 865240 4-41 134760 45 i6 771987 2 87 906482 1-54 8655o5 4-41 134495 44 ;? 772159 2 87 906389 1-55 865770 4-41 134230 43 772331 2 86 906296 1-55 866o35 4-41 133965 42 19 7725o3 2 86 906204 1-55 866300 4-41 133700 41 20 772675 2 86 9061 1 1 1-55 866564 4-41 133436 40 21 9-772847 773018 2 86 9-906018 1-55 9-866829 4-41 10-133171 3^ 22 2 86 905925 1-55 867094 4-41 132906 23 773190 2 86 905832 1-55 . 867358 4-41 132642 V. 24 773361 2 85 905739 1-55 867623 4-41 132377 25 773533 2 85 905643 1-55 867887 4-4i I32ii3 35 26 773704 2 85 905552 1-55 868i52 4-40 131848 34 27 773875 2 85 905439 1-55 868416 4-4o I3i584 33 28 774046 2 85 905366 1-56 868680 4-4o i3i32o 32 29 774217 2 85 905272 1-56 868945 4.40 i3io55 3i 3o 774388 2 84 9o5i79 1-56 869209 4-40 130794 3o 3i 9-774558 2 84 9 -905085 1-56 9-869473 4.40 io.i3o527 29 32 774729 2 84 904992 1-56 869737 4.40 i3o263 28 33 774899 2 84 904898 1-56 870001 4.40 12973? 27 34 775070 2 84 904804 1-56 870265 4-40 26 33 775240 2 84 90471 1 1-56 870529 4.40 1 2947 1 25 36 775410 2 83 904617 1-56 870793 4.40 129207 24 V. 775580 2 83 904523. 1-56 871057 4-40 128943 23 775750 2- 83 904429 1-57 871321 4-40 128679 128415 22 39 775920 2- 83 904335 1-57 871585 4-40 21 40 776090 2- 83 904341 1-57 871849 4-39 I28i5i 20 4i 9-776259 2 83 9-904147 1-57 9-872112 4-39 10-127888 :i 42 776429 77659S 2 82 904053 1-57 872376 4-39 127624 43 2- 82 903959 1-57 872640 4-39 1273^10 '7 44 776768 2- 82 903864 1-57 872903 4-39 1270^17 16 45 776937 2- 82 903770 1-57 873167 4-39 126833 i5 46 777106 2- 82 903676 1-57 873430 4-39 126570 14 47 777275 2- 81 9o358i 1-57 873694 4-39 i263o6 i3 48 777444 2 81 903487 1-57 1-58 873957 4-39 126043 12 49 777613 2 81 903392 874220 4-39 125780 II 5o 777781 3 81 903298 1-58 874484 4.39 I255i6 10 5i 9.777950 2 81 9-9o32o3 1-58 9-874747 4-39 I0'i25253 2 52 778119 2- 81 903 1 08 1-58 875010 4-39 124990 8 53 778287 2 80 9o3oi4 1-58 875273 4-38 124727 I 54 778455 2 80 902919 1-58 875536 4-38 124464 55 778624 2 80 902824 1-58 875800 4-38 124200 5 56 778792 2 80 902729 1-58 876063 4-38 133937 4 % 778960 2 80 902634 1-58 876326 4-38 123674 3 779128 2 80 902539 1-59 876589 4-38 1 2341 1 2 59 779295 2 ^9 902444 ■ 1.59 876851 4-38 I23i49 1 66 779463 2-79 902349 1-59 877114 4-38 122886 Cosine D. Sine D. Cotancr. D. Tang (53 DEGREES.) SINES ANT) TANGENTS. (37 DEGREES.) 55 M. Sine D. Cosiuo I ). Tang. D. Cotang. o 9-779463 2-79 9.902349 I 902253 I 59 9-877114 4-38 10-122886 60 I 779631 2 79 ^9 877377 4 38 122623 u 2 779798 2 79' 902158 1 ^9 877640 4 38 122860 3 779966 2 79 902063 I ^ 877903 4 38 \l]^l 57 4 780133 2 ?? 901967 1 59 878165 4 38 56 5 780300 2 901872 I 59 878428 4 38 121572 55 6 780467 2 "^^ 901776 I ^ 878691 4 38 121809 54 7 780684 2 78 901681 1 59 878953 4 37 121047 53 8 780801 2 78 901585 I 59 879216 4 37 120784 52 9 780968 2 78 901490 I 59 879478 4 37 120522 5i 10 781134 2 78 901394 1 60 879741 4 37 120259 5o II 9.781301 2 77 9.901298 , I 60 9.880003 4 37 10- 1 19997 49 12 781468 2 77 ■ 901202 I 60 880265 4 37 119735 48 i3 781634 2 77 901 106 I 60 880528 4 37 119472 47 14 781800 2 77 901010 I 60 880790 4 37 119210 II 8948 46 i5 781966 2 77 900914 I 60 88io52 4 37 45 i6 782132 2 77 C00818 I 60 881814 4 37 1 1 8686 44 n 782298 2 '^i 900722 I 60 881576 4 37 118424 43 i8 782464 2 76 900626 I 60 881889 4 37 11B161 42 19 782630 2 76 900029 I 60 882101 4 37 1 1 7899 41 20 782796 2 76 900433 I 61 882868 4 36 117687 40 21 9-782961 2 76 9.900337 I 61 9.882625 4 36 10-117375 39 22 783,27 2 76 900240 I 61 882887 4 36 117118 38 23 783292 783458 2 75 900144 I 61 888148 4 86 116852 37 24 2 75 61 888410 4 36 116590 36 25 783623 2 '^i 61 888672 4 36 116828 35 26 783788 2 75 61 888984 4 36 116066 34 27 783953 2 7^ 899757 I 61 884196 4 36 ii58o4 33 28 784118 75 899660 I 61 884457 4 36 1x5543 32 H9 784282 74 899564 I 61 884719 4 86 ii528i 3i v>o 784447 74 899467 I 62 884980 4 36 ll5020 3o 3i 9.784612 2 74 9.899370 I 62 9.885242 4 36 10.114758 20 28 32 784776 ^ 74 899273 I 62 8855o3 4 36 1 14285 33 784941 2 74 899176 I 62 885765 4 86 11 34 785io5 74 899078 I 62 886026 4 36 113974 35 785069 73 898981 I 62 886288 4 36 118712 25 36 785433 73 898884 I 62 886549 4 35 1x8451 24 3? 785597 73 898787 I 62 886810 4 35 1x8x90 23 38 785761 73 898689 I 62 887072 4 35 1x2928 22 39 785925 "7^ 898592 I 62 887888 4 35 1x2667 21 40 786089 73 898494 I 63 887594 4 35 1x2406 20 41 9.786252 72 9.898397 I 63 9.887855 4 35 10.112x45 11 42 786416 72 898299 I 63 888116 4 85 11x884 43 786579 72 898202 I 63 888377 4 85 IXX623 17 44 786742 2 72 898104 I 63 888689 4 35 ixx36i x6 45 786906 2 72 898006 I 63 888900 4 85 IXXIOO i5 46 ''87069 2 72 897908 I 63 889160 4 35 1x0840 14 47 787232 2 71 897810 I 63 889421 4 35 1x0579 i3 48 787395 2 71 ■ 897712 I 63 889682 4 85 110818 12 49 787557 2 71 897614 I 68 889943 4 35 '1x0057 XI 5o 787720 2 71 897516 I 63 890204 4 34 109796 10 5i 9.787883 2 71 9.897418 I 64 9.890465 4 34 10- 109535 I 52 788045 2 71 897820 I 64 890725 4 34 X09275 53 788208 71 897222 I 64 890986 4 34 X 090 14 108753 7 54 788370 70 897123 1 64 891247 4 34 6 55 788532 70 897025 1 64 891507 4 34 108493 5 56 788694 70 896926 I 64 891768 4 34 108282 4 u 788856 70 896828 I 64 892028 4 34 107972 3 789018 70 896729 I 64 892289 4 34 I077IX 2 59 789180 70 896681 I 64 892549 4 34 I0745X 6o 789342 69 896582 I 64 89281C 4-34 X07190 Cosine "1 ). Sine i D. Cotaiig. D. . Tung. _ Jkj (52 DEGREES.) 56 (38 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine I ). Tung. D. Ootang. 9-789342 2-69 Q- 896532 I -64 9-8928.0 4-34 10-107190 60 I 789504 2 6q ■ 896433 1 -65 898070 4.34 106980 U 2 789665 2 69 896335 1 -65 898881 4.84 106669 3 789827 789988 2 69 896286 1 -65 898091 8988D1 4.34 106409 57 4 2 -69 896187 1 -65 4.34 106149 56 5 790 '49 2 ^? 896088 I -65 8941 11 4.34 105889 55 6 7903 10 2 890089 I 895840 1 -65 894871 4.34 105629 54 I 790471 2 -68 -65 894632 4-38 10536? 1 53 1 790632 2 -68 893741 I -65 894892 895152 4.33 io5ioS 52 9 790793 2 68 895641 1 -65 4-38 104848 5i 10 790954 2 68 895542 1 .65 895412 4-33 104588 5o II 9-791II5 2 68 9-895443 1 -66 9-895672 4-33 10-104828 1? 12 791275 2 67 895343 I -66 890982 4-33 104068 i3 791436 2 67 895244 1 -66 896192 4-38 108808 47 14 79i5o6 79'757 2 67 895145 1 -66 896452 4-33 103548 46 i5 2 67 895045 1 -66 896712 4-33 108288 45 i6 791917 2 67 894945 I 66 896971 4-33 108029 44 n 792077 2 67 894846 1 66 897281 4.33 102769 43 i8 •792237 2 66 894746 1 66 897491 4-83 102009 42 19 792397 2 66 894646 1 66 897701 4-33 102249 41 20 792557 2 66 894546 1 66 898010 4-38 101990 40 21 9-792716 2 66 9-894446 I 67 9-898270 4-38 10-101730 38 22 792876 2 66 894846 1 67 898080 4-33 101470 23 793o35 2 66 894246 1 67 898789 4-38 10.2.1 37 24 793195 793354 2 65 894146 1 67 899049 899808 4-82 10095. 36 25 2 65 894046 1 67 4-32 . 100692 ■ 100432 35 26 79^514 2 65 898046- 1 898846 1 67 899568 4.82 34 11 798673 2 65 67 899827 4-32 100178 33 793832 2 65 898745 1 67 900086 4-82 099914 32 29 793991 7941 5o 2 65 898645 1 67 900846 4-82 099654 3i 3o 2 64 893544 I 67 900605 4-82 099895 3o 3i 9 -794308 2 64 9-898444 I 68 9.900864 4-32 10-099186 2Q 32 794467 2 64 898848 1 68 90.124 4-82 098876 28 33 794626 2 64 898243 1 68 901888 4-82 098617 27 34 794784 2 64 898142 1 68 901642 4-82 098358 26 35 794942 2 64 898041 1 68 901901 4-32 098099 25 36 795-01 2 64 892940 I 68 902160 4-82 097840 24 37 795259 2 63 892889 1 68 902419 4-32 097581 23 38 795417 2 63 892789 1 68 902679 4-82 097821 22 39 795575 2 63 892688 1 68 902988 4.82 097062 21 40 795733 2 63 892536 I 68 908197 4-3i 096808 20 41 9-795891 2 63 9-892435 I 69 9.908455 4-3i 10-096545 \l 42 796049 2 63 892884 1 69 9087.4 4-3i 096286 43 796206 2 63 892288 1 69 908978 4.8. 096027 17 44 796864 2 62 892182 1 69 904282 4-3i 090768 16 45 790D2I 2 62 892080 I 69 904491 4-3i 095009 i5 46 796679 2 62 891029 I 80.827 I 89S726 I 69 904750 4-31 095250 14 % 796886 62 69 900008 4- 3 1 094992 094733 i3 796993 62 69 900267 4-3i 12 49 797.5c 61 891624 1 69 900526 4-3i 094474 11 5o 797807 61 89.528 1- 70 905784 4-31 094216 10 5i 9.797464 61 9.891421 I- 70 9.906043 4-31 10.098957 I 52 797621 61 891819 1 70 906302 4-31 098698 53 797777 61 891217 1- 70 9o656o 4-3i 098440 7 54 797984 2 61 8911.5 1- 70 906819 4.3. 098181 6 55 798091 2 61 89.0.3 1- 70 907077 4-3i 092928 5 56 798247 2 61 8909.1 1- 70 907886 4-81 092664 4 57 798403 2 60. 890809 I • 70 907594 4-8i 092406 3 58 798060 2 60 890707 1 - 70 907852 4-3i 092.48 2 59 798716 2 60 8qo6o5 1 - 70 9081.1 4-3o 091889 1 66 798872 2 60 89o5o3 1 - 70 908869 4-3o 091681 Ooaine fj ). Sine ]] Cotang. D. Tcxig. 1 :&L 1 (51 DEGREES.) SINES AND TANGENTS. (39 DEGREES./ 67 M. Sine D. Cosine D. Tang. D. Cotang. 9.79S872 2 -60 9.890503 1.70 9.908369 908628 4-30 10-091631 60 I 799028 2-6o 890400 1. 71 4-30 091372 U 2 799184 2 -60 890'o8 I-7I 908886 4-3o 09 1 1 1 4 3 799339 799495 2.59 8901^5 1.71 909144 4-3o . 090856 57 4 2.59 890093 1. 71 909402 4-3o 090598 56 5 799631 2.59 889990 1. 71 909660 4-3o 090340 55 6 799806 2.59 889888 1-71 909918 4-3o 090082 089823 54 I 799962 2-59 889785 1-71 910177 4-3o 53 8001 17 2.59 889682 1. 71 910435 4-3o 089565 52 9 800272 2.58 889579 1.71 910693 4-3o 089307 5i 10 830427 2-58 889477 1. 71 910951 4-3o 089049 5p II 9.800582 2-58 9.889374 1.72 9.911209 4-3o 10.088791 a 12 800737 2.58 889271 1.72 91 1467 4-3o 088533 i3 800892 2-58 889168 1.72 911724 4-3o 088276 47 14 801047 2-58 88qo64 1-72 911982 4-3o 088018 46 i5 801201 2.58 888961 1.72 912240 4-3o 087760 45 i6 8oi356 2.57 888858 1.72 912498 4-3o 087502 44 n 8oi5ii 2.57 888755 1-72 912756 4-3o 087244 43 i8 80 1 665 2.57 888651 1-72 9i3oi4 4-29 086986 42 19 801819 801973 2.57 888548 1.72 913271 4-29 086729 41 20 2.57 888444 1.73 913529 4-29 086471 40 21 g. 802128 2.57 9-888341 1.73 9.913787 4-29 10. 086213 It 22 802282 2.56 888237 1.73 914044 4-29 085956 23 802436 2.56 888134 1.73 914302 4-29 085698 37 24 802589 802743 2-56 888o3o 1.73 914560 4-29 085440 36 25 2.56 887926 887822 1.73 914817 4-29 o85i83 35 26 802897 2.56 1.73 915075 4-29 084925 34 ^J 8o3o5o 2.56 887718 1.73 915332 4-29 084668 33 803204 2.56 887614 1.73 915590 4-29 084410 32 ?9 803357 2.55 887510 1.73 915847 4-29 084153 3i 3o 8o35u 2-55 887406 1-74 916104 4-29 083896 3o 3i 9.803664 2.55 9.887302 1-74 9-916362 4-29 10.083638 29 28 32 803817 2.55 887198 1-74 916619 4-29 o8338i 33 803970 2.55 8870^3 1-74 916877 4-29 o83i23 27 34 804123 2.55 1-74 917134 4-29 082866 26 35 804276 2.54 886885 1-74 917391 4-29 082609 25 36 804428 2.54 886780 1-74 917648 4-29 082352 24 ll 804381 2.54 886676 1-74 917903 4-29 4.28 082095 23 804734 2.54 886571 1-74 918163 081837 22 h 804B86 2.54 886466 1-74 918420 4-28 o8i58o 21 4o 8o5o39 2.54 886362 1.75 918677 4.28 o8i323 20 4i Q-8o5i9i 2.54 9.886257 1.75 9-918934 4.28 10-081066 \l 42 805343 2.53 886 1 52 1.75 919191 4.28 080809 43 805495 2.5S 886047 1.75 919448 4.28 o8o552 17 44 8o5647 2.53 885942 1.75 919705 4.28 080295 o8oo38 16 45 805799 2-53 885837 1.75 919962 4-28 i5 46 805951 2.53 885732 1.75 920219 4.28 079781 14 47 806 I o3 2.53 885627 1.75 920476 4-28 079524 i3 48 806254 2.53 885522 1.75 920733 4-28 079267 12 49 806406 2-52 885416 1.75 920990 4-28 079010 078753 u 5c 806557 2-52 88531 1 1.76 921247 4-28 to 5i 9 806709 2.52 9-885205 1.76 9-921503 4.28 10-078497 t 52 806860 2.52 885100 1.76 921760 4.28 078240 53 807011 2.52 884994 884889 884783 1.76 922017 4-28 077983 ■< 54 807163 2-52 1-76 922274 4.28 077726 6 55 807314 2.52 1-76 922530 4-28 077470 5 l^ 807465 2.5l 884677 1.76 922187 4-28 077213 4 ll 807615 2.5r 884572 1-76 923044 4-28 076936 . 3 807766 2.5l 884466 1.76 923300 4-28 076700 a h 807917 2.5l 884360 1.76 923557 4-27 076443 I 60 808067 2.5i 884254 1.77 9238i3 4-27 076187 Cosine D. Sine D. Ootanfi:. D. ! Tang. (50 DEOiUCKS.) 68 (40 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. 1 Cosine 1 D. Tang, 1 1^- CoUuig. 60 9-808067 2-5l 9.884254 1-77 9 -92381 3 4-27 10-076187 I 808218 2-5l 884148 1-77 924070 4-27 075980 5o 2 8o8368 2-5l 884042 1-77 924327 4-27 075673 5§ 3 8oS5i9 2-5o 883936 1-77 92. ! _... (49 DEGRE ES.) SINES AND TANGENTS. (41 DEGREES.) 59 Mr Sine D. 1 Cosiiie D. Taug. D. Cotaug. 60 0.816943 2.42 9.877780 1 1-83 9-989163 4-25 10-060887 I 817088 2-42 877670 1 1-83 989418 4-25 o6o582 59 2 817233 2-42 877360 1-83 989673 4-25 060827 58 3 817379 2-42 877430 1-83 989928 4-25 060072 5? 4 817324 2-41 877340 1-83 940183 4-25 059S17 56 5 817668 2-41 877230 1-84 940438 4-25 059062 55 6 817813 2-41 877120 1-84 940694 4-25 059806 54 I 817938 :-4i 877010 1-84 940949 4-25 009051 53 SiSinZ 2-41 876809 1-84 941204 4-25 058796 52 9 1 Sj«247 2-41 876789 1 1-84 941458 4-23 058342 5i 10 818392 2-41 876678 1-84 941714 4-25 058286 5o II 9.8i8536 2.40 9-876568 1-84 9-941968 4-25 io.o58o32 49 12 818681 2-40 876457 1.84 942228 4 25 037777 48 i3 818825 2-40 876847 1-84 942478 4-25 057522 47 i4 818969 819113 2 -40 876286 . 1-85 942733 4-25 007267 46 i5 2.40 876125 1-85 942988 4-25 057012 43 i6 819237 2-40 876014 1-85 948243 4-25 006757 44 \l 819401 2-40 873904 1-85 948498 4-25 006002 43 819545 2-39 875793 1-85 943732 4-25 056248 42 19 819689 2-39 873682 1-85 944007 4-25 055998 055788 41 20 819832 2-39 873371 1.85 944262 4-25 40 21 9-819976 2-39 9-875439 875348 1.85 9-944317 4-25 10-055483 38 22 820120 2-39 1-85 944771 4-24 000229 23 820263 2.39 875287 1-85 945026 4-24 004974 37 24 820406 2-39 875126 1.86 943281 4-24 004719 36 23 82o55o 2-38 875014 1-86 945535 4-24 054465 35 , 26 820693 2-33 874903 1.86 945790 4-24 054210 34 11 820836 2-38 874791 1-86 946045 4-24 053955 33 820979 2-33 874680 1-86 946299 4-24 053701 32 ?9 821122 2-38 874368 1.86 946554 4-24 053446 3i 3o 821265 2-33 . 874436 1.86 946808 4-24 053192 30 3i 9-821407 2-33 9-874344 1-86 9-947063 4-24 10.002987 11 32 82i55o 2-38 874282 1.87 947818 4-24 052632 33 821693 821835 2.37 874121 1.87 947372 4-24 052428 11 34 2.37 874009 1-87 947826 4-24 052174 35 821977 2.37 873896 1-87 948081 4-24 051919 25 36 822120 2-37 873784 1-87 948336 4-24 o5i664 24 u 822262 2.37 873672 1.87 948590 4-24 o5i4io 23 822404 2.37 873560 1.87 948844 4-24 o5iio6 22 39 822346 2.37 873448 1.87 949099 4-24 000901 21 40 822688 2-36 873335 1-87 949833 4-24 050647 20 41 9.822830 2-36 9-873223 1-87 9-949607 4-24 10-000893 \l 42 822972 2-36 878110 1-88 949862 4-24 o5oi38 43 823114 2-36 872998 1.88 9501 16 4-24 049884 17 44 823255 2-36 8728S5 1.88 950870 4-24 049630 16 45 823397 2-36 872772 1.83 950625 4-24 049875 i5 46 823539 2-36 872659 1.83 950879 95ii33 4-24 0491 21 14 47 48 823680 2-35 872547 1-88 4-24 048867 i3 823821 2-35 872434 1.83 95 I 388 4-24 048612 12 49 823963 2-35 872821 1-83 931642 4-24 043358 II 5(? 824104 2-35 872208 1.88 951896 4-24 048104 10 5i 0-824245 2-35 9.872095 1.89 9-932x50 4-24 10-047850 Q 52 824386 2-35 ?7'98i 1-89 952405 4-24 047395 8 53 824327 824668 2-35 871 803 1.89 952659 952918 4-24 047341 7 54 2-34 871755 1.89 4-24 047087 6 5c 824808 2-34 871641 1-89 953167 4-23 046833 5 56 824949 2-34 87.528 1-89 953421 4-23 046079 4 h 825090 ! 2-34 871414 1-89 953675 4-23 046320 3 58 82523c 2-34 871801 1-89 953929 934183 1 4-23 046071 2 1 ^ 825371 2-34 871187 i-sl 4-23 045817 I 60 8255II 2-34 871073 1-90 934437 ! 4-23 045563 Cofiine D. Sine D. Cotang. D. Tanff. ^ (48 DEQUE KS.) (42 DEGREES.; A TABLE OF LOGARITHMIC M. o Sine D. Cosine I X Tung. D. Cotang. 1 9.825511 2-34 9.871073 1 .90 9-954437 4.28 10 -045300 1 60 I 825651 2 .33 870960 1 .90 954691 4-23 ^ i U 2 823791 2 33 870846 I -90 954945 4-23 3 825981 2 33 870782 I -90 955200 4-23 044800 u 4 826071 2 33 870618 I .90 955454 4-28 044546 5 82621 1 2 33 870504 1 -90 933707 4-28 044298 55 6 826351 2 33 870890 1 -90 953961 4-23 044089 54 7 826491 826631 2 83 870276 1 .90 9562 1 5 4-23 048783 53 8 2 83 870161 I .90 956469 956728 4-28 04853 1 52 9 826770 2 82 870047 1 .91 4-23 048277 5i ic 826910 2 82 869988 I -91 956977 4-23 048028 5o 11 9-827049 2 32 9-869818 1 91 9.937231 4-23 10.042769 t 12 827189 2 32 8b97o4 1 91 957483 4-23 o425i5 i3 827328 2 32 869389 I 91 937789 937998 4-23 042261 47 14 827467 2 82 869474 I 91 4-23 042007 46 i5 827606 2 32 869860 1 91 958246 4-23 041754 45 i6 827745 2 32 869245 1 91 958500 4-28 o4i5oo 44 17 827884 2 81 869180 I 91 958754 4-28 041246 43 i8 828023 2 81 869015 1 92 959008 4-28 040992 42, »9 828162 2 3i 868900 I 92 939262 4-23 040788 41 20 828301 2 3i 868785 I 92 959516 4-23 040484 40 21 9-828439 2 3i 9-868670 I 92 9-959769 4-23 10-040281 U 22 828578 2 81 868555 I 92 960023 4-28 089977 23 828716 2 3i 868440 I 92 960277 4-^3 089728 37 24 828855 2 3o 868824 I 92 96053 1 4-23 089469 36 25 828993 2 3o 868209 1 868093 1 92 960784 4-28 089216 35 26 829131 2 3o 92 961088 4-23 088962 34 27 829269 2 3o 867978 I 93 961291 4-28 088709 33 28 829407 2 3o 867862 I 98 961343 4-23 088453 32 29 829545 2 3o 867747 I 93 961799 4-23 088201 3i So • 8296S3 2 3o 867681 I 93 962032 4-23 087948 3o 3i 9.829821 2 29 9-867515 I 93 9-962806 4-28 10-087694 It 32 829939 2 29 867899 1 93 962360 4-23 087440 33 880097 2 29 867283 I 93 962818 4-23 087187 27 34 830234 2 29 867,67 I 93 968067 4-23 086933 26 35 83o372 2 29 867031 I 93 968820 4-23 086680 25 36 830309 2 29 866935 I 866819 I 94 968374 4-23 086426 24 37 880646 2 29 94 968827 4-23 086178 23 38 880784 2 29 866708 I 94 964081 4-23 083919 22 39 880921 2 28 866586 I 94 964885 4-23 085665 21 40 83io58 2 28 866470 I 94 964588 4-22 o354i2 20 41 9-88f io5 2 28 9-866353 i 94 9-964842 4-22 io-o35i58 \l 42 881882 2- 28 866287 I 94 963095 4-22 o349o5 43 881469 2 28 866120 I 94 963849 4-22 084651 n 44 881606 2 28 866004 1 95 963602 4-22 084898 16 45 831742 2 28 865887 I 95 965855 4-22 034145 i5 46 881879 2 28 865770 I 95 966103 4-22 088891 088688 14' 47 882013 2 27 865653 I 95 966862 4-22 i3 48 832132 2 27 865586 1 95 966616 4-22 038884 12 49 882288 2 27 863419 I 95 966869 4-22 o33i3i 11 5o 832425 2 27 865302 I 95 967123 4-22 082877 10 5i Q-88256I 2 27 9-865185 I 95 9-967376 4-22 10-082624 ? 52 882697 2 27 865o68 I 95 967629 4-22 082871 53 882888 2 27 864950 I 95 967888 4-22 082117 7 54 882969 2 26 864883 I 96 968186 4-22 081864 6 55 833103 2 26 864716 I 96 968889 4-22 081611 5 56 833241 2 26 864598 I 864481 I 96 968643 4-22 o3i357 4 11 888877 2 26 96 068896 4-22 081104 3 833512 2 26 864363 I 96 969149 4-22 o3o85i 2 09 833648 2 26 864245 I 96 969408 4-22 o8o5q7 ' I 60 833783 2-26 864127 I 96 969636 4-22 080844 Cosine D. Sine r ). Cotansr. D. Tauar. 1 JsL^ (47 DEGREES.) SIXES -iND TANGEXT3. (43 DEaKEES.) 61 M. Siue D. Cosine D. Tang-. D. Cotang. 60 o 833783 2.26 9-864127 1.96 9.969656 4-22 io-o3o344 I 833919 2-25 864010 1.96 969909 4-22 030091 ^ 2 &34054 2-25 S63892 1-97 970162 4-22 029S38 3 834189 2-25 863774 1-97 970416 4-22 029584 57 4 834323 2-25 863656 1-97 970669 4.22 029331 56 5 834460 2-25 863538 1-97 970922 4-22 029078 55 6 834595 2-25 863419 1-97 971175 4-22 028825 54 I 834730 2-25 863301 1-97 971429 4-22 028571 53 834865 2-23 863 I 83 1-97 971082 4.22 o283i8 52 9 834999 2 24 863o64 1-97 971935 4-22 028065 5i 10 835134 2-24 862946 1-98 972188 4-22 0278x2 5o II 9-835269 835403 2-24 9-862827 1.98 9-972441 4-22 10-027559 it '^ 2-24 862709 1.98 972694 4-22 027306 i3 835538 2-24 . 862590 1.98 972948 4-22 02705.2 47 14 835672 2-24 862471 1.93 973201 4-22 026799 46 ID 835807 2-24 862353 1.98 973454 4-22 026546 45 i6 835941 2-24 862234 1.98 973707 4-22 025293 44 I? 836075 2-23 8621 1 5 1.98 973960 4-22 026040 43 i8 836209 2-23 861996 1-98 9742x3 4-22 025787 42 19 836343 2-23 861877 1.98 974466 4-22 025534 41 20 836477 2-23 861758 1.99 974719 4-22 025281 40 21 9 -83661 1 2-23 9-86i638 1.99 9.974973 4.22 10-025027 3^ 22 836-45 2-23 86i5i9 1-99 975226 4.22 024-74 23 836S78 2-23 861400 1.99 975479 4.22 024521 37 24 837012 2-22 861280 1-99 975732 4-22 024268 36 25 837146 2-22 861161 1-99 975985 4-22 0240x5 35 26 837279 837ii^ 2-22 861041 1-99 976238 4.22 023762 34 H 2-22 860922 860802 1-99 976491 4.22 >235o9 33' 28 837546 2-22 1-99 976744 4-22 023256 32 ^9 837679 2-22 860682 2-00 976997 4-22 023oo3 3i So 837812 2-22 86o562 2-00 977250 4-22 022750 3o V 9-837945 2-22 Q. 860442 2-00 9 -977503 4-22 10-022497 29 32 838078 2-21 ^ 86o322 2-00 977756 4.22 022244 28 33 8382II 2-21 860202 2-00 978009 4.22 021991 27 34 838344 2-21 860082 2-00 978262 4.22 021738 26 35 838477 2-21 809062 2-00 9785x5 4.22 021485 25 36 8386x0 2-21 859842 2-00 978768 4-22 021232 H ll 838742 2-21 859721 2-01 979021 4-22 020979 23 838875 2-21 859601 2-01 979274 4-22 020-26 22 39 839007 2-21 859480 2-01 979527 4-22 020473 21 40 839140 2-20 859360 2-01 979780 4.22 020220 20 41 9-839272 2-20 9-859239 2-01 9 -930033 ^ 980286 4.22 10-019967 ]l 42 839404 2-20 8591 19 2-01 4.22 OI9714 43 839536 2-20 85S998 2-01 980538 4.22 019462 17 44 839668 2-20 858877 2-01 980791 4.21 0X9209 16 45 839800 2 -20 858756 2-02 981044 4.21 0x8956 i5 46 839932 2-20 858635 2-02 98 1 297 4-21 018703 14 47 840064 2-19 8585i4 2-02 981550 4.21 018450 i3 48 840196 2.19 858393 2-02 981803 4.21 OI8197 12 49 840328 2-19 858272 2-02 9S2056 4.21 017944 11 5o 840459 2-19 858i5i 2-02 982309 4.21 017691 10 5i 9-840591 2.19 9-858029 857908 2-02 9-982562 4-21 10-0X7438 i 52 840722 2-19 2-02 982814 4-21 017186 53 840854 2-19 857786 2-02 983067 4-21 016933 7 54 840985 2.19 857665 2-o3 9S3320 4-21 016680 6 55 841116 2.18 857543 2-o3 983573 4-21 0x6427 5 54 841247 2.18 857422 2-03 983826 4-21 016174 4 ll 841378 2.18 857300 2.o3 9840-9 4.21 015921 3 84 509 2.18 857178 2-o3 984331 4-21 0x5669 2 ? 84; 540 2.18 857056 2-03 984584 4-21 oi54i6 I 6o 841771 2.18 856934 2.o3 984837 4-21 0i5i63 Cosine D. Sine IX Cotan?. D. Taiigr. M. (46 DEGREES.) 62 (U DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. T;in,'. D. Cotang. o 9-841771 2.18 9.856934 2-03 9-984837 421 io-oi5i63 60 I 841902 2.18 856812 2 o3 985090 4 21 014910 i? 2 842033 2 18 856690 2 04 985343 4 21 014657 3 842163 2 n 856568 2 04 985596 4 21 014404 57 4 842294 2 17 856446 2 04 985848 4 21 oi4i52 56 5 842424 2 17 856323 2 04 9861.1 4 21 013899 55 6 842555 2 17 850201 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 1 3 1 40 52 9 842946 2 17 855833 2 04 987112 4 21 0128S8 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 IO-012382 ii 12 843336 2 16 855405 2 o5 9S7871 4 21 012129 i3 843466 2 16 855342 2 o5 988123 4 21 011S77 47 14 843595 2 16 855219 2 o5 9S8376 4 21 01 1624 46 i5 843725 2 16 855096 2 o5 988629 4 21 0!l3-I 45 i6 843855 2 16 854973 2 o5 9888^2 4 21 OIII18 44 17 843984 2 16 854850 2 o5 9^9134 4 21 010866 43 18 844114 2 i5 854727 2 06 9893S7 4 21 oio6i3 42 19 844243 2 i5 834603 2 06 989640 4 21 oio36o 41 20 844372 2 i5 854480 2 06 989893 4 21 010107 40 21 9-844502 2 i5 9-854356 2 06 9-990145 4 21 10-009855 It 22 844631 2 i5 854233 2 06 •990398 4 21 009602 23 844760 2 i5 854109 2 06 99065 I 4 21 009349 37 24 844^89 2 i5 853986 853862 2 06 990903 4 21 009097 008844 36 25 845018 2 i5 2 06 9911 56 4 21 35 26 845147 2 i5 853738 2 06 991409 4 21 oo85qi 34 27 845276 2 14 853614 2 07 991662 4 21 OOS338 33 23 845405 2 14 853490 2 07 99'9«4 4 21 0080S6 32 29 845533 2 14 853366 2 07 992167 4 21 007S33 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 ll 32 845919 2 14 852994 2 07 992923 4 21 00-075 33 846047 2 14 852<'569 2 07 993178 4 21 006822 27 34 846175 2 14 852745 2 07 993430 4 006070 26 35 846304 2 14 852620 2 07 993683 4 oo63i7 25 36 846432 2 i3 852496 2 08 993936 4 006064 24 ll 846860 2 i3 852371 2 08 994189 4 oo58ii 23 846688 2 i3 852247 2 08 994441 4 005559 22 39 846816 2 i3 852122 2 08 994694 4 oo53o6 21 4o 846944 2 i3 851997 2 08 994947 4 oo5o53 30 41 9.847071 2 i3 9-85i872 2 08 9.995199 4 10-004801 \l 42 847199 2 i3 851747 2 08 995432 4 004548 43 847327 2 i3 85i622 2 08 995705 4 004295 \l 44 847454 2 12 851497 2 09 995957 4 004043 45 847582 2 12 85i372 2 09 996210 4 003-90 i5 46 847709 2 12 851246 2 09 996463 4 003537 14 47 847836 2 12 851121 2 09 996715 4 oo328d i3 48 847964 2 12 850996 2 09 99.696S 4 oo3o32 12 49 848091 2 12 850870 2 09 99-221 4 0027-9 M 5o 848218 2 12 850745 2 09 997473 4 002027 10 5i 9-848345 2 12 9-85o6i9 2 09 9.997726 4 IC- 002.274 t 52 848472 2 II 850493 2 10 99-9-9 4 002021 53 848599 2 II 85o36S 2 10 99«23i 4 001769 I 54 848726 2 II 85o242 2 10 99^^484 4 2 1 001016 55 848852 2 II 85oii6 2 IC ,998737 4 001263 5 56 848979 2 II 849990 2 10 99^^989 4 OOIOI I 4 ,57 849106 2 II 849«64 2 10 999242 4 000-08 3 58 849232 2 II 84973s 2 10 999493 4 ooo5o5 3 59 849359 2 II 84961 1 2 10 999-48 4 21 000253 I 66 849485 2-II 849485 2- 10 10- 000000 4-21 1 • 000000 I Cosine D. Sine D. Cotancr. D. TalK^ JVL (45 DEGREES.) A TABLE OF NATURAL SINES. 63 Beg. 1 Deg. 2 Beg. 3Deg. 4 Deg. M M S. c.s. S. c. s. S. O.S. S. C.S. S. C.S. o 00000 Unit. 01745 99985 08490 99989 o5234 99868 06976 997 56 j 6a I 00029 I -0000 01774 99984 o35i9 9998H o5263 99S61 07005 99734! 39 2 ooo58 I '0000 oi8o3 99984 08048 99987 05292 99860 07084 99752 53 3 00087 I. 0000 01832 99988 03577 99986 o532i 99858 07068 99750 57 4 001 16 I -0000 01862 99988 08606, 99935 o535o 99837 07092 99748 56 5 00145 I -0000 01891 999^2 03635 99934 05370 99S55 07121 99746 55 6 j 00175 I -0000 01920 99982 08664 99933 o54oc 99834 07i5o 99744 54 7 00204 1. 0000. 01949 99981 08698 99982 05437 99852 07179 99742 53 8 00233 1. 0000 01978 99980 037281 99981 05466 99851 07208 99740 52 9 00262 1. 0000 02007 99980 08752 99980 05495 99849 07237 99738! 5i ID 00291 I .0000 02036 99979 08781 99929 o5524 99847 j 07266 99786} 5a II 00320 99999 020651 99979 o38io 99927 o5553 99846 07295 99734 it 12 00849 99999 02094 99978 08889 99926 o5582 99844 07824 99731 i3 .00378 99999 02123 99977 08868 99925 o56ii 99842 07353 99729 47 14 00407 99999 02l52 99977 08897 99924 o564o 99841 07882 9Q727 46 I? 00436 99999 O2181 99976 08926 99928 05669 99889 0741 1 99725 45 16 00465 99999 022II 99976 08955 99922 05698 99888 07440 99723 44 \l 00495 99999 02240 99973 08984 99921 05727 99886 07469 07498 99721 43 00324 99999 02269 99974 0401 3 99919 99918 o5756 99834 99719 42 19 00553 99998 02298 99974 04042 05785 99833 07527 99716 41 20 oo5S2 99998 02827 99978 04071 99917 o58i4 99881 07556 99714 40 21 0061 1 99998 02856 99972 04100 99916 05844 99829 07585 99712 ii 22 00640 99998 02885 99972 04129 99915 05873 99827 07614 99710 23 00669 99998 02414 99971 04159 04188 99913 C5902 99826 07643 99708 37 24 00698 99998 02443 99970 99912 05981 99824 07672 99705 36 25 00727 99997 02472 99969 04217 99911 05960 99822 07701 99708 35 26 00756 99997 025oi 99969 04246 99910 05089 06018 99821 07780 99701 34 U 00785 99997 02530 99968 04275 99909 99819 07759 99699 33 00814 99997 02560 99967 04804 99907 06047 99817 07788 99696 32. 29 00844 99996 02589 99966 04888 99906 06076 99815 07817 99694 3i 3o 00873 99996 02618 99966 1 04862! 99905 o6io5 99S13 07846 99692 3a 3i 00902 99996 02647 99963 ! 048911 99904 06 1 34 99812 07875 99689 It 32 00981 99996 02676 99964 j 04420' 99902 06168 99810 07904 99687 33 00960 99993 02705 99963 04440 99901 06192 99808 07988 99685 27 34 00989 99995 02784 99968 04478, 9990a 04507 j 99898 06221 998061 07962 99683 26. 35 01018 99995 02768 99962 o6?5o 99804 07991 99680 25 36 01047 99995 02792 99961 04536! 99897 06279 o63o8 99808 08020 99678 24 37 01076 99994 02821 99960 j 04365! 99896. 99801 08049 99676 23 38 oiio5 99994 o285o 999391 04594 99^94 06887 99799 08078 99673 2i 39 01134 99994 02879 99939! 04628; 99898 06366 99797 0S107 99611 21 40 01164 99993 1 02008 99958 04658! 99892 06895 99793 081 36 99668 20 41 01193 99993 02988 99937 04682 99890 06424 99793 o8i65 99666 J 9 42 01222 99993 02967 999,-)6 047 1 1 99880 04740 99888} 06453 99792 08194 99664 18 43 0125l 99992 02996 999551 06482 9979a 99788 08228 99661 17 44 O12S0 99992 08025 99934 04769' 99886I o65ii 08252 99639 16 45 oi3o9 99991 o3o54 99953 0479B 99885 06540 99786 08281 99657 ID 46 01338 99991 o3o83 99932 1 04827 99883 06569 06598 99784 08810 99654 14 ^l 01 367 99991 03lI2 99932 i 04856 99882! 99782 08339 99632 i3 48 01396 99990 08141 999311 04885 99881 1 06627 997801 08868 99649 12 49 01 423 99990 08170 99930 04914' 998-79! 06656! 99778 0S897 99647 n 5o OI404 999^9 08190 99949' 04948! 99878! o6685j 99776 08426 99644 10 5i 014S3 999S9 03228 99948: 04972 99876 06714, 99774 08455 99642 1 52 oi5i3 999H9 99988 08237 99947 j o5ooi 99873 06748, 99772 08484 99539 53 01 542 08286 99946; o5o3o 99873 06773 99770 o85i8 99687 7 54 oi5ti 99988 08816 99943 o5o59 o5o88 99872 06802 99768 08542 99635 6 55 01600 99987 03345 99944 99870 06881 09766 08571 99682 5 56 01629 999S7 08874 99943 o5ii7 99869 06860 99764 1 0S600 99680 4 ll 01658 99986 08408 99942 o5i46 99867 06889 99762 0S629 99627 1 58 01687 99986 08482 99941 o5i75 99866 06918 99760 08658 99625 2 59 M 01716 99985 08461 C.S. 99940 o52o5 99864 06947 C.S. 99758 S. 08687 C.S. 99622 S. "u C. S. 1 s. S. C.S. s. 89 Deo:. 88 bejr. 87 Dear. 1 86 Deg. 1 85 1 >?. 64 A TABLE OF NATURAL SINES. i M 5 Deg. 1 6 Deg. 7 l»eg. 1 8 Deg. 9 Deg. s. 1 c. s. 1 ^• 1 c. s. S. 1 C. S. 1 s. 13917 C. S S. 1 c s. Im o 087161 99619 10453 99452 121871 99255 99027 1 15643 98769 60 I 08745' 99617 10482 99449 12216 99251 13946 99023 15672; 98764 59 2 08774 99614 io5ii 99446 12245 99248 13975 99019 15701 1 98760 58 3 0SS03 99612 io54o 99443 12274! 99244 14004 990 ID i573oi 98755! D7I 4 o883i 99609 10569 10597 99440 12302; 99240 14033 99011 ID758 98751! 56 1 5 08860 99607 99437 12331 99237 14061 99006 15787 98746 55 6 08889 08918 99604 10626 99434 12360 99233 14090 99002 1D816 98741 54 I 99602 10655 99431 i238o| 99230 12418 99226 14119 98998 15845 98737 53 08947 99599 10684 99428 14148 98994 15873 98732 52 9 08976' 99596 10713 99424 12447 99222 14177 & 15902 987.28 5! 10 O90o5| 99394 10742 99421 12476 99219 i25o4' 99213 i42o5 15931 98723 5o 11 090341 99591 10771 99418 14234 98982 15959' 98718 % 12 09063 99588 10800 99415 12533 99211 14263 98978 1598^' 98714 i3 09092 99586 10829 10858 99412 12562 99208 14292 98973 16017 98709 47 14 09121 99583 99409 12591 99204 14320 98969 16046 98704 46 ID 09i5o 99580 10887 99406 12620 99200 14349 98965- 16074 98700 45 i6 09179 09208 99578 10916 99402 12640 99197 14378 98961 i6io3 98695 44 ;? 99575 10945 99399 12678 99193 14407 98957 i6i32 98690 43 09237 99572; 10973 99396 12706 99189 14436 98953 98948 16160 98686 42 19 09266 99570; 11002 99393 12735 99186 14464 16189 98681 41 20 09293! 99567' iio3i 99390 99386 12764 99182 14493 98944 1621S 98676 40 21 093241 99364: 11060 12793 99178 14522 98940, 16246 98671 u 22 09353' 99362 11080 11118 99383 12822! 99175 14551 98936 i62-'5 98667 23 093S2 99559 99380 i285i| 99171 14580 98931 i63o4 98662 37 24 0941 1 99556j 11147 99377 12880 99167 14608 98927! 16333 98657 36 25 09440 99553 11176 99374 1 2908 99163 14637 98923 i636i 98652 35 26 09469 9955i 11205 99370 12937 99160 14666 98919: 16390 98648 34 27 0949S; 99348^ 11234 99367 129661 99 1 56 j 14695 98914; 16419 9S643 33 28 09327 99545 11263 99364' 12995; 99152 14723 98910 16447 98638 32 29 09556 99542! 11291 99360, i3o24 99148'; 14752 98906 16476 98633 3i So 09535 99540 ll320 99357i i3o53 99144' 14781 98902 i65o5 98629 5o 3i 09614 99537! 11349 99354] i3o8i 99141' 14810 98897 16533 98624 29 32 09642 99534; 11378 99351 1 i3iio 9913- 14S38 98893 16562 98619I 28 33 09671 99531: 11407 99347 i3i39 99133^ 14S67 98889 16591 98614 27 34 09700 99528; 11436 99344; i3i68 99129I 14896 ^To^\ 16620 98609 26 35 09720 09758 99526; 11465 99341 I3i97 99123; 14925 98880 16648 98604 25 36 99523 11494 99337 13226 99122I 14954 98876 16677 98600 24 37 09787 99520 ii523 99334 13254 99118; 14982 98871 16706 98595 23 38 09816 99517; 11552 99331 13283 99114! i5oii 98867 16734 llUl 22 39 09843 99514; ii58o 99327 13312 99110I i5o4o 98863 16763 21 40 09874 9951 1 1 11609 11638 99324I 13341 99106: 15069 98858 16792 98580 20 41 09903 1 99508, 99320 13370 99102! i5o97 98854' 16820 98575 19 98570 18 42 09932 99506 1 11667 99317 13399 99098; 13126 98849 16849 16878 43 09961 99503; 11696 9^314 13427 99094' i5i55 9??45| 98565 17 44 09990 99300: 11725 99310 13456 99091! 99087; i5i84 98841 16906 98561 16 45 10019 99497 11754 99307 13485 l5212 98836 16935 98556 i5 46 10048 99494 11783 993o3 i35i4 99083; 15241 98832! 16964 985Dii 14 47 10077 99491 11812 99300 13543 99079; 15270 "^fo't' 16992 985461 i3 48 10106 99488 11840 99297 13572 99073 15292 98823 17021 9S541! 12 49 ioi35 99485 11869 11898, 99293! i36oo 99071' 15327 98818 i7o5o! 98536] 11 5o 10164' 99482! 99290; 99286! 13629 99067 15356 98814' 17078 98531; 10 5i 10192' 99479' 11927: 13653! 99063 15335 93809' 171071 98526 t 52 1022 1 99476 11956 99283! 13687- 99059 15414 98803 I7i36| 98521 53 10250 99473 11985; 99279; 13716 99053 15442 98800 17164 9S5.6 7 54 10279 99470 12014 992761 13744 9905 I; 15471 98796 17193I 98311 6 55 io3o8 99467' 12043 992721 13773 99047 i55oo 98701 98787 17222! 98506 5 56 10337 99464 12071; 99269! i38o2 99043 15529 17250I 98301 4 u 10366 99461 1 2 100 99263] i333i qqo3q 15557 98782; 17279' 98496 3 10395 99458! 12129 99262! i586o 99035 15586 98778! 1730S; 98491 17336 Q8486 2 59 M 10424; 99^55; c. s. i S. 1 121 58 99258 C. S. 1 s. 13889 9903 1 [ i56i5 98773 S. ! I C. S. 1 s. C. S. c. s. s. 84 Deer. 1 83 I )ecr. 1 82 Deer. 81 Deg ! 80 1 )e« 1 A TABLE OF NATURAL SINES. 65 M 10 Deg. 11 Deg. 1 12 Deg. 1 13 De-. 1 . li Deg. j S. c. s. S. c. s. S. as. 1 978i5; s. 1 c. s. 1 s. c. s. M 17365 98481 1 19081 98168: 20791 22495! 9743711 24192 97080 '6oi 17393 98476 \l\?s 98157^ 20820 97809 22023 97480 24220 97028: DO 97015 58 2 17422 98471 98152 20848 97808, 22552 97424 24249 3 17451 98466 98461 19167 98146 20877 97797! 2258o 974171 24277 97008 57 4 17479 19193 98140 20905 97791 1 22608 97411 24805 97001 56 5 17508 984551 19224 98i85i 209331 977»4 22687 97404 24383 96994 55 6 1-537 98450! 19252 98129! 20962 97778 22665 97398 24862! 96987I 54 7 17565 98445 19281 98124! 20990! 97772 22698 97391: 97384' 24390 96980; 53 8 17594 98440 19309 98118 21019' 97766 22722 24418 969781 02 24446 96966! 5i 9 17623 98435 19338 98112 21047 97760^ 22750 97878 10 17651 98430 19366 981071 21076 97754' 22778 97871!! 24474! 96959! 5ol n 17680 98425; 19895 98101 2 II 04! 97748' 22807 97865 - 240081 96902] 40 1 12 17708 98420 19423 98096; 21182 97742; 22885 97358 24581 96945; 48 1 i3 17737 9^414 19452 980901 98084 21 161 97735; 22868 97351; 24509 96937 47 i4 17766 98409! 19481 21189 97729; 22892 97345 245S7 96980 46 i5 17794 98404 19509 98079 21218 97723 22920 97338 24615 96923 45 i6 17823 98399 19538 98073 21246 97717 22948 97331 24644 96916 44 \l 17852 98394! 98389 98383 19066 98067 21275 97711 22977 97325 24672 96909 43 17880 19595 98061 2i3o3 97705 23oo5 97318 24700 42 19 17909 19623 98o56 2i38i 97698 28088 97311 24728 96894 41 20 17937 98378 19652 98o5o| 21860 97602! 97686! 28062 97304 24756 96887 40 21 17966 98373 19680 98044; 21388 28090 97298 24784 96880 89 22 17995 98368 19709' 98089 21417 97680! 28118 97201 972b4 24818 96873 38! 23 18023 98362 19737 98033] 21445 97673; 28146 24841 9-6866 37 1 24 i8o52 98357 19766 980271 21474 97667 28175 97278 24869 96808 361 25 18081 98352 19794 9802 1 j 2l502 97661 23203 97271 24897 96851 35 26 18109 98347 19823 98016 2i53o 97655 23281 97264 24925 96844 34 27 i8i38 98341 19851 98010 21559 97648 28260 97257 24953 96887 33 28 18166 98336 19880 98004! 2 1 587 97642 28288 97251 24982 96829 32 29 18195 98331 19908 97998' 2I6I6 97636! 283i6 97244 25oio 96822 31 30 18224 98325 19937 97992 21644 97680 28840 97287 25o38 96815 3o 3i 18252 98320 19965 97987 21672 97628 28873 97280 25o66J 96^07 29 32 18281 983i5 19994 97981 21701 97617 28401 97228 250941 9680L 28 33 i83o9 983io 20022 97975 21729 9761 1 28429 97217 25l22 96793 27 34 18338 983o4 2oo5i 97969' 21758 97604 28458 97210 25i5i 96786 26 35 18367 98299 20079' 97963 21786 97598 23486 97208 20179 96778 25! 36 18395 & 20108, 97958 20i36' 97952 21814 97592 285i4 97196 97189 25207 96171 24 1 37 18424 21843 97585 23542 25235| 96764 23 38 18452 98283 20165 97946 21871 97579 97073 23571 97182 25263 96756 22 39 1 848 1 98277 201931 97940 2 1890 21928 28599 97176 25291 1 96-49 21 3o i85o9 18538 98272 20222I 97934 97566 28627 97169 20820' 96742 20 41 98267 2025o' 97928 21956 97560 28656 97162 25348: 96784 \t 42 18567 98261 20279, 97922 21985 97558 97347 28684 97155 25376!. 96727 43 18595 98256 2o3o7 97916 220l3 28712 97148 25404! 96719 17 44 186241 98250 20336 97910 22041 97541 28740 97141 25482! 96712 16 45 18652 98245 20364 97905 22070 97534 23769 97184 25460| 96705 i5 46 18681 98240 20893 97899 22098 97528 23797 97127 25488' 96697 14 47 18710 98234 20421 97893 22126 97521 28825 97120 255i6! 96690 25545. 96682 i3 48 18738 98229 2o45o 97887 22155 975i5 28853 97113 12 49 18767! 98223 20478 97881 22183 97508 28882 97106 255781 96675 11 5o 18795; 98218 2o5o7 97875 22212 97502 23910J 97100 25601 96667 10 5i 18824! 98212 20535! 97869 20563 97863 22240 97496 289881 97098 23966! 97086 25629! 96660 ? 52 18852! 98207 22268 97489 256571 96653 53 18881} 98201 20592! 97857 22297 97483 28995; 97079 25685 96645 7 54 18910' 98196 20620 97851 22825 97476 24028 97072 25718; 96688 61 55 18938: 98190 20649 97845 22353 97470 24o5i 97065 25741 1 96680 5 55 18967' 98 1 85 20677I 97889 20706! 97833 22882 97463 24079 97o58 24108^ 97o5i 25769 96623 25798 96615 4 57 18995 981-9 22410 97457 ^i 58 19024 98174 1 20734' 97827 22488 97450 24186 97044 25826 q66o8 2 59 M 19052 98168 1 20763 97821 2 2467 1 97444 24164 97087 25854 96600 I C. S. 1 s. 1 C. S. 1 s. C. S. 1 s. C. S. 1 s. C. S. i s. H 79 Deg. 1 78: Deor. 77 Deg. 76 Deg. 1 75] Deg. -.- 6b A TABLE OF NATURAL SINES. 15 Deg. 16 Leg. 17 Deg. 1 IS Deg. 19 Deg. M S. 1 c. s. S. c. s. S. C. S. 1 '^• C. S. S. 1 S.C. 25882 96593 27364 96126 29287 9563o!' 80902 95106 32557 94552 "67, I 25910 96585 27592 96118 29263 93622 ! 30929 95097 32584 94542 It 2 25938 96578 27620 96110 29298 95618 80957 95088 I 32612 94533 3 25966! 96570 27648 96102 29821 956o5 i 80985 95079 1 82689 94528 57 4 j 20994 96362 27676 96094 29848 95396 ' 3lOI2 95588' 31040 95070 j 82667 94514 56 5 I 26022, 96555 27704 96086 29876 95061 1 32694 94304 55 6 •. 26c5o' 96547 27731 96078!! 29404 95579! 3io68 95o52 32722 94495 54 7 j 26079: 96540 27759 96070 29482 95571 i 81095 95048 32749 94483 53 8 261071 963^2 27787 96062 29460 95562' 31128 95088 32777 94476 52 9 26i35| 96524 27815 96054 29487 95554 1 3ii5i 95024 82804 94466 5i 10 26i63j 96517 27843 96046 29315 95545 31178 95oi5 82882 94437 5o 11 261 91 ! 96009 27871 96087 29348 95586 3.206 95006 32859 94447 49 12 26219; 96502 27899 96029 2937, 9552S 8.238 94997 32887 94438 48 i3 26247 i 96494 27927 96021 29599 93519 81261 94988 82914 94428 47 14 26275! 964H6 27933 96013 29626 95511 81289 94979 82942 94418 46 i5 263o3 96479 27983 96005 29654 93502 3i8.6 94970 32969 94409 45 16 2633 1 96471 28011 95997 29682 95408 81844 94961 32997 94899 44 17 26339 96463 28089 95989 29710 93485 8.37-. 94952 33024 94890 43 18 26887 96456 28067 95981 29787 954^6 31899 94948 33o5i 94880 42 ^9 264J5 96448 28095 95972 29765 95467 31427 94988 88079 94370 41 20 26443 96440 28128 95964 29793 95439 81454 94924 33 106 94861 40 21 26471 96433 28i5o 95956 29821 95450 81482 949.5 38184 q485i 39 22 26500 96425 28178 95948 29849 95441 3.5io 94906 33i6i 94342 38 23 26528I 96417 28206 93940 29876 95483 8.537 94897 88189 94882 37 24 26556,' 96410 28284 95981 29904 95424 3i565 94888 88216 94822 36 25 26584 96402 28262 95923 29982 93415 81598 94878 33244 948.3 35 26 26612 96394 28290 95915 29960 95407 81620 94869 3327. 94308 34 27 26640 963S6 2b8i8 95907 29987 95898 316/13 94860 88298 94298 33 28 26668 96379 28846 9589b 3ooi5 95889 81675 9485, 33826 94284 32 29 26696 96371 28874 95890 30043 958801 81708 94842 38858 •94274 3i 3o 26724 96363; 28402 95882 30071 95372 31780 94882 33381 94264 3o 3i 26752 96355 28429 95874' 30098 95868 3.758 94828 88408 94234 It 32 26780 96347 28457 95S65 30126 95854 8.786 94814 33436 94245 33 26808 96340 28485 95857 30154 93845 8.8.8 94805 33468 94285 27 34 26836 96332 285i8 95849 30182 93887 81841 94795 33490 94223 26 35 26864 96324 28541 95341 30209 95828 3.868 94786 33518 94215 25 36 26892 96816 28569 95882 30287 93819 81896 94777 38545 94206 24 37 26920 96808 28597 95824 30265 958.0 3.928 94768 33573 94.96 23 38 26948 96801 28625 95816 30292 95801 8.951 94758 88600 94186 22 39 26976 96293, 28652 95807 80820 93298 81979 94749 88627 94.76 21 40 27004 96285 28680 95799 30348 95284 82006 94740 33655 94.67 20 41 27032 96277 28708 95791 30376 95275 82084 94730; 38682 94137 \l 42 27060 96269 28786 95782 30408 95266 82061 94721 33710 94147 43 27088 96261 28764 95774 80481 93257 32089I 94712 38787 94.37 '1 44 27116 96253 28792 95766 80459 95248 82116 94702 88764 94.27 16 45 27144 96246 28820 95737 30486 95240 32144; 94693 33792 94118 i5 46 27172 96288 28847 95749 3o5i4 95281 32171 94684' 33819 94108 u. 47 27200 96230I 28875 95740 30542 95222 82.99 94674: 33846 94098 i3 48 27228, 96222 28908 93782 30570 952.8 82227 94665 88874 94ob8: 12 49 27256 96214' 2898. 93724 30597 95204! 82254 94656 8890. 94078' II 5c 272841 96206: 2S959 95715 3o6?5 93195, 82282 94646 88929 94068 10 5i 273l2j 9619S 289S7 95707 3o653 95186 82809 94637 88936 94038 I 52 27340i 961901 29015 95698 306S0 9517- 32887 94627 33988 94049 53 27368j 96182} 29042 95690 30708 95.68 82864 94618 34011 94089 1 54 27396! 96174' 29070 95681 30786 95159 82892 94609 34038 94029 6 55 27424; 96166 29098; 95678 30763 95i5o; 82419 94599 34065 940.9 5 56 27452; 96158 29126! 93664 30791 951421 82447 94390 34093 94009 4 57 27480I 961 5o1 29154! 936561 30819 95188 32474 94580 34120 98999 3 58 27508, 96142, 27536 96134 291821 Q3647 30846 95124' 82502 94571 34147 989S9 Z 59 29209 95689! C. S. 1 s. 30874 95ii5j 82529 94561 34175 989-9 S. )eg. 11 M c. s. 1 s. C. S. 72 1 S. )eg. C. S. S. C. S. 70 1 74Deg. 1 73 Deg. 71 Deg. A TABLE OF NATURAL SINES. 67 M 20 Deg. 21 Deg. 22 Deg. 23 Deg. 24 Deg. M S. C.S. S. 35837 C.S. 93358 S. |C. S. S. C.S. S. 1 c. s. o 34202 93969 37461 92718 39073 92o5o 40674 91355 60 I 34229 93959 35864 93348 37488 92707 39100 92039 40700 91343 5? 2 34237 93949 35891 93337 37515 92697 39127 Q2028 40727 40753 9i33i 3 34284 93939 35918 93327 37542 92686 39153 92016 91 3 19 57 4 343 1 1 93929 35945 93316 37569 9:675 39180 92005 40780 91307 56 5 3<339 Q3919 35973 93306 37595 92 64 39207 91994 40806 91295 55 6 3i366 93909 36000 93295 37022 92053 39234 91982 40833 91283 54 I 34393 36027 93285 37649, 92642 39260 91971 40860 91272 53 34421 93889 36o54 93274 37676 92631 39287 91959 40886 91260] 52 9 34448 93879 36o8i 93264 37703 92620 39314 91948 40913 Q1248 5i ic 34475 93869 36io8 93253 37730 92609 39341 91936 40939, 91236 5o ,u 345o3 93859 36i35 93243 ^77^7 90598 39367 91925 40966 91224 g 12 34530 93849 36162 93232 37784 92587 39394 91914 40992 91212 i3 34557 ?3839 36190 93222 37811 92076 39421 91902 41019 91200 47 i4 34584 93829 36217 93211 37838 92060 39448 91891 41045 91188 46 i5 34612 93819 36244 93201 37865 92554 39474 91879 41072 91176 45 i6 34639 93809 36271 93190 37892 92543 39501 91868 41098 91 164 44 \l 34666 93799 36298 93180 37919 92532 39528 9 1 856 41125 9ii52 43 34694 93789 36325 93169 37946 92521 39555 91845 4ii5i 91140 42 19 34721 93779 36352 931 59 37973 92510 39581 91833 41178 91128 41 20 34748 93769 36379 93148 37999 92499 92488 39608 91822 41204 91116 40 21 34775 93750 93748 36406 93i37 38026 39635 91810 4i23i 91104 l^ 22 34803 36434 93127 38o53 92477 39661 91799 41257 91002 91080 23 34830 93738 36461 93116 38o8o 92466 39688 91787 41284 37 24 34857 93728 36488 93106 38107 92455 39715 91775 4i3io 91068 36 25 34884 93718 365i5 93095 38i34 92444 39741 91764 41337 9io56 35 26 34912 93708 36542 93084 38i6i 92432 39768 91752 41363 91044 34 27 34939 93698 93688 36569 93074 38188 92421 39795 91741 41390 91032 33 28 34966 36596 93o63 382i5 92410 3q822 91720 91718 41416 91020 32 29 34993 93677 36623 93o52 38241 92399 39848 41443 91008 3i 30 35021 93667 36650 93042 38268 92388 39875 91706 41469 90996 3o 3i 35048 93657 36677 93o3i 38295 92377 39902 91694 41496 90984 It 32 35075 93647 36704 93020 38322 92366 39928 91683 4l522 90972 33 35102 93637 36731 93010 38349 92355 39955 91671 41549 90960 27 34 35i3o 93626 36758 92999 92988 38376 92343 39982 91660 41570 90948 26 35 35i57 93616 36785 3 8403 92332 40008 91648 41602 90936 25 36 35i83 93606 368i2 92978 38430 92321 4oo35 91636 41628 90924 24 37 352II 93596 36839 92967 38456 92310 40062 91625 4i655 90911 23 38 35239 93585 36867 92956 38483 •92299 40088 91613 41681 ?ol? 22 39 35266 93575 36894 92945 385io 92287 4oii5 91601 41707 21 40 35293 93565 36921 92935 ??S'' 92276 40141 91590 41734 90875 20 41 35320 93555 36948 92924 38564 92265 40168 91578 41760 90863 \l 42 35347 93544 36975 92913 38591 92254 40195 91566 41787 9085 1 43 35375 93534 37002 92902 38617 92243 40221 91555 41 81 3 90839 17 44 35402 93524 37029 92892 92881 38644 92231 40248 91543 41840 90826 16 45 35429 93514 37056 38671 92220 40275 9i53i 41866 90814 i5 46 35456 935o3 37083 92870 38698 92209 4o3oi 91519 41892 90802 14 47 35484 93493 93483 37110 92859 38725 92108 40328 9i5o8 41919 41945 90790 i3 48 355II 37137 92849 38752 92186 40355 91496 90778 12 49 35538 93472 37164 92838 38778 92175 4o38i 91484 41972 90766 II 5o 35565 93462 37191 92827 388o5 92164 40408 91472 41998 90753 10 5i 35592 93452 37218 92816 38832 92l52 40434 91461 42024 907 4 i ? 52 35619 93441 37245 92805 38859 92I4I 40461 91449 42o5i 90729 53 35647 93431 37272 92794 92784 38886 92i3o 40488 91437 42077 90717 7 54 30674 93420 37299 38912 92119 4o5i4 91425 42104 90704 6 55 35701 93410 37326 92773 38939 92107 4o54i 91414 42i3o 90692 5 56 35728 93400 37353 92762 38966 92096 40567 91402 42156 90680 4 57 35755 93389 37380 92751 38993 92ort5 40594 91390 42183 9°5^? 3 58 35782 93379 93368 37407 92740 39020 92073 40621 91378 42209 42235 C.S. 90655 2 9q 35810 37434 C.S. 92729 39046 92062 40647 91366 90643 I C.S. S. S. C.S. S. C.S. S. S. "M 69 De^. 68 Peg. 67 Deg 1 66 Deg. \ 65 Deg. 1 19 6S A TABLE OF NATURAL SIXES. M 25 Dc-g. 26 Deg. 1 27 Deg. i 28 Deg. 29 Dog. 1 S. ' C. S. S. C. S. || S. i c. s. 1} s. c. s. 1 S. C. S. M o 42262 9063 1 43837 89S79 1 45399 89IOI j 46947 882951 4B481 87462 60 I 42288i 90618 43863 898671 40420 89087! 46973 88281 485o6 87448 5o 87434 53 3 423 1 5j 90606 43SS9 89804: 40451 89074 46999 88267' 48532 3 42341 90094 43916 89S4I 1 45477 89061 47024' 88204 48557 87420 57 4 43367 9o5S2 43942 89828! 40003 89048 47o5o; 88240 48583 87406 56 5 42394 90569 43968 898161 40529' 89035 i 47076 88226; 48608! 87391 55 6 42420 90507 43994 89S03 ' 40554 89021! 47101 88213; 48634 87377 54 I 42446 90545 44020 S9-90 ! 40080 89008^ 47127 89777 40606 889951 47153 88199' 48659 87353 53 424731 90532 44046 88i85 48684 87349 52 9 42499 90320 4252DJ 90007 44072 89764: 40632 88981 1 47178 88172: 48710 87335 5i 10 4409S 89752 1 45658: 88968 47204 88153; 48735 87321 5o II 42552' 90495 44124 89739 45684 88955 47229 88144: 48761 87306 it 12 425781 904S3 42604I 904-0' 44i5i 89726 457101 88942 47250 88i3oi 48786 87292 i3 44177 89713 45736 88928 ! 47281 88117: 4881 1 87278 47 14 42631 90458' 44203 89700 1 45762 88915 47306 88io3, 48837 87264 46 i5 42607 90446 44229 89687 J 45787 88902: 47332 88089 1 48862 87250 45 i6 42683 90433 44255 89674' 458i3 88888 47358 88075 48888 87235 44 \l 42709 90421 1 44281 89662 45839 88875 45860! 88862 47383 88062! 48913 87221 43 42736 90408! 44307 89640 47409 88048, 48938 87207 42 *9 42762 90396! 903831 44333 89636 45891! 88848 47434 88034 ' 48964 87193 41 20 42788 44359 89623; 45917 88835 474601 88020 48989 87178 40 21 42815 90371 443S5 89610 1 43942 88822 47486 1 88006^ 49014 87164 39 38 22 42841 90358! 4441 1 89097 45968 88808 47511! 87993 49040 87100 23 42S67 90346: 44437 89584: 4O994 88795 47537I 87979 490&5 87136 Q-, 24 42894 90334' 44464 89571: 46020 88782 47562: 87960 49090 87121 36 25 42920 9o32i: 44490 89558 46046 88768 1 4708S' 87951: 49116 87107 35 26 42946 90309' 44016 89545 46072 88755 4-614; 87937 49141 87093 34 27 28 42972 90295^ 44542 89532 46097 88741' 47639 87923; 49166 87079 33 42999 90284 1 44568 89519; 46123 88728' 47660; 87909; 49192 87064 32 29 4302D 90271 i 44594 895o6i 46149 887i5j 47690! 87896] 49217 87o5o 3i 3o 43o5i 90259J 44620 89493, 46170 88701; 47716 87882 49242 87036 30 3i 43077 90246; 44646 89480' 46201 88688! 47741 1 87868: 49268 87021 29 32 43104 90233 44672 89467; 46226 88674: 47767; 87854 49293 87007 23 33 43i3o 90221: 44698 89404 46252 88661' 46278 88647i 47793; 87840 47818 87826; 49318 86993 27 34 43 1 56 90208: 44724 89441' 49344 86978 26 35 43182 90196' 44700 8942S: 46304 88634' 47844 87812! 49369 86964 25 36 43200 90183 44776 89415; 46330 88620: 47869; 8779S! 49394 86949 24 37 38 43235 901711 44802 89402: 46355 88607' 47890^ 87784; 49419 86930 23 43261 901 58 1 44828 893S9' 46381. 88593 47920 87770; 49440 S6921 22 39 43287 90146 'i 44854 89376^ 46407 88580 4-946 87756. 49470 86906 21 40 433i3l 90i33:i 44S80 89363 46433 88566 46408 88553 47971^ 877431 49490 86892 20 41 43340 90120 44906 89350' 47997 87729' 49521 86878 ;i 42 43366 90108 1 44932 89337: 46484I 88539; 48022 877151 49546 86863 43 43392 900^5; 90082: 44958 89324; 465io 88526 48048 87701! 49571 86849 17 44 43418 44984 89311I 46536 885i2 48073 87637; 49596 86334 16 45 43445 90070 45oio 89293 46561 88499, 48099^ 876731 49622 86320 i5 46 43471 90057 1 45o36 89285; 46587 88485; 48124' 87659! 49647 868o5 14 S 43497 90045 1 40062 89272 46613 88472 43i5o 87645; 49672 86791 i3 43523 90032: 45o88 89209' 46639 88458; 48175 87631 11 49697 K"'!" 12 49 43549I 9C019; 40114 89245, 46664 88445] 48201 87617, 49723 86762 11 00 43573, 90007; 40140 89232! 46690 8843 1 i 48226, 87603! 49748 86748 5i 43602 899^4 45i66 89219I 46716 8841 7 j 48252 87589! 49773 86733 g 52 43628 S99bij 45192 89206I 46742 88404; 48277! 875701 49798 86719 fe 53 43554 89968 , 40218 46767 46793 88390 483o3 87561 49824 86704 I 04 43680! 89906 1 40243 88377 43323 87546; 49849 49874 86690 55 43706 89943, 40269 89167 46819 88363, 48354 87532; 86670 5 56 43733 89930 45290 89i53i 45844 88349 48379 87518 49899 86661 i U g?i? 89918J 45321 89U0; 46870 88336 48400 87504! 49924 86646 3 89905; 45347 89127, 46896' 88322! 48430 87490' 49939 86632 2 59 43811 89892 45373 891 14 46921! 883o8i 48406 87476: 49970 86617 I M M c. s. S. C. S. S. 1 c. s. 1 s. 1 C. S. S. i| c. s. S. C4Dosr. 63 ] 3c?. i 62] 3eg. I 61 Deg. [| 60 ] ^•_. A TABLE OF NATUKAL SINES. 69 M 3U l>eg. b2 l>eg. 33 Deg. 3^ Deg. M S. C.S. S. C.S. 85717 S. C. S. S. I C.S S. C.S. o 50000 i 866o3 5i5o4 52992 84805 54464 8386^ 559x9! 82904 "60 ' 5oo25 86588 5x529 83702 53ox7' 84789 54488 8385x 55943; 82887 It 2 5oo5o 86573 5 1 554 83687 53o4i 84774 545 1 3 83835 55968, 8287X 3 50076 86559 5x579 85672 53o66 84759 54537 ' 83819 ■-5992' 82855 ll 4 5oioi 86544 5 1 604 85637 53091! 84743 5456 X ' 83804 56oi6. 82839 5 50126 86530 51628 85642 53xx5, 84728 54586 ' 83788 56040' 82822 55 6 5oi5i 865 1 5 5x653 85627 53x40 84712 546x0 83772 56o64' 82806! 54 I 50176 86501 5x678 856x2 53x64 84697 54635 83756 56o88 82790! 53 5020I 86486 5i7o3 85397 53189 84681 546^9 83740 56ii2 82773: 53 9 50227 86471 5x728 85582 532X4 84666 54683 83724 56x36 82757! 5x lO 50252 86407 5x753 85567 53238 84650 54708 83708 56i6o 8274X 5o II 50277 86442 5x778 8535X 53263 84635 54732 83692 56x84 827241 49 562o3 82708! 40 12 5o3o2 86427' 5x8o3 85536 53288 84619 54756 83676 i3 5o327 86413 5x828 85521, 53312 84604 5478X 83660 56232 826921 47 14 5o352 86398 86384 5x852 83306 53337 84583 54805 83645 56256 82675 46 i5 5o377 51877 83491 53361 84573^ 54829 83629 56280 82659 45 i6 5o4o3 86369' 5x902' 85476 53386 84557! 54854 836x3 563o5 82643! 44 \l 50428 86354 5X927 8546 X 5341 1 ' 84542 54878 54902 83397 83581 56329 82626! 43 50453 86340 5x952 85446 53435 84526 56353 82610! 42 19 50478 86325, 5x977 8543 1 53460 845xi; 54927 83565 56377 825931 41 20 5o5o3 863x0 52002 83416 53484 84495: 5495X 83549 5640X 82577 40 21 5o528 86295 52026 85401 53509 84480 54970 83533 56425 8256x1 39 22 5o553 86281; 52o5x' 85385 53534 84464 54999 835x7 56449 82544' 38 23 50578 86266, 52076 85370 53558 84448 55o24 835oi 56473 825281 37 24 5o6o3 8625i 52x01' 83353i 53583 84433 55048 83485 56497 825xi| 36 25 50628 86237 52126; 853401 53607 84417 55072 83469 56521 82495: 35 26 5o654 86222 52x5x 85325. 53632 84402 55097 83453 56545 824781 34 27 50679 86207 52x75! 853X0; 53656 84386 55i2x 83437 56569 82462 33 28 50704 86192 52200 85294; 5368 X 84370 55x45 83421 56593 82446; 32 29 50729 86178: 52225 83279' 53705 84355 55169 834o5 566x7 82429! 3x1 3o 50754 86x63, 52250 85264j 53730 84339; 55x94 833:39 56641, 824x3 3o 3i 50779 86x48 52275 8p249 53754 84324 552x8 83373 56665 82396 ll 32 50804 86x33; 52299' 85234 53779 84308; 55242 83356 56689 82380 33 50829 861x9 52324' 852x8 53804 84292' 55266 83340 567x3 82363 27 34 5o854 86104 5234Q 85203 53828 84277! 55291 83324' 56736 82347 26 35 50879 86089 52374 85 1 88 53853 8426X 553i5 833o8| 56760 82330 25 36 50904 86074 52399 85x73 53877 84245: 55339 83292 56784 823x4 24 37 50929 86009 52423 85x57 03902 84230 55363 83276 568o8 82297 56832 8228X 23 38 50954 86045 52448 85x42 53926 842x4 55388 83260 22 39 50979 86o3o' 52473 85x27 5395X 84198; 84182! 55412 83244 56856 82264 2X 40 5 1 004 86oi5 52498 85ix2 53975 55436 83228; 5688o 82248 20 41 51029 86000 52322 85096 85o8i 54000 841671 55460 83212! 56904 8223x XQ x8 42 5io54 85985 52547: 54024 84i5i: 55484 83x95 56928 822x4 43 5io79 85970' 52572 85o66 54049 84x35 55309 83x79 56952 82198 56976! 82i8x 17' 44 51104 85956; 52597j 85o5i 54073 84120: 55533 83x63 161 45 5X129 8594X 5262 xj 85o35 54097 84104; 55557 83x47 57000: 82x65 15; 46 5ii54 85926 52646' 85o2o 54x22 84088' 55581 83i3i 57034 82148 14 47 5ii79 859XX 85896' 526-7x1 83oo5i 54x46 84072 556o5 83xi5 57047! 82X32 x3 48 5i2o4 52696; 84989 54x71 84057 5563o 83098 57071! ?2XX5 I2| 49 51229 85881 ; 52720 84974 54195 84041' 55654 83o82 57095 82098 IX ! 57119' 82082 xo 5o 51234 85866: 52745 84959 54220 84023; 55678 55702 83o66 5i 51279 85851 ' 52770 84943 54244 84009' 83o5o 57x43 82o65j 9 52 5i3o4 85836 52794 84928! 54269 83994 55726 83o34 57x67 82048 81 53 5i329 8582 x! 52819 849x3, 84897 84882! 54293 83978 55750 83017 57I9X: 82032 1 54 5i354 858o6, 52844 543x7 83962 55775 83oox 572x5i 82oi5 6 55 5i379 85792 52869 54342 83946! 55799 82985 57238: 81999 5 56 5x404 85777 52893 84866! 54366 83930! 55823 82969 57262' 81982 4 ^2 51429 80762; 52918 84851 5439 X 83899! 83883' S._ 1 55847 82953 57286! 81965 3 58 5x454 80747; 52943 84836! 544x5 53871 82936 573x0 81949 2 59 5x479 85732; 52967 84820 54440 55895 8292,0 57334 81932 "m M C.S. s. 1 C. S. i S. C.S. C.S. S. C. S. 1 S. 59] 3e^. I 58 De?. 1 57 Dee. 1 56 Deer. 1 55 Eeg. 70 A TABLE OP NATURAL SINES. M 35 Deg. 36 Deg. 37 Dcg. 38 Deg. 39 Deg. M S. C. S. 81915 S. C. S. S. 60182 jC. S. j 79864 S. C. S. S. 0. S. 57358 58779 80902 ^566 7880. 62932 77715 "60 I 57381 81899 588o2 80885 60203 79S46 6.589 78783 62955 77696 59 2 57405 81882 58826 80867 ' 60228 , 79829 1 61612 78165 62977 1 -77678 58 i 3 57429 8i865 58849 8o85o ' 60231 1 79811 6.635 78747 63000 77660 1 57 4 57453 81848 58873 80833 60274 ! 79793 6.658 78729 63022 77641 56 5 57477 8 1 83 2 58896 80816 60298 i 79776 6.68. 787.1 63045 77623 ! 55 6 1 57501 8i8i5 58920 80799 60321 79758 61704 78694 63o68 i 77603 1 54 7 i 57524 8179^ 58943 80782 6o344 79741 6.726 78676 63090 ! 775% ! 53 8 ; 57548 81782 58067 80763 60367 79723 6.749 78658 631.3 1 77568 52 9 57572 81765 58990 80748 60390 79706 6.772 78640 63.35 77550 5i ic 57596 81748 59014 80730 60414 79688 61795 78622 63 1 58 77531 5o 11 1 57619 81731 59037 80713 60437 79671 618.8 78604 63. 80 775i3 49 12 57643 81714 59061 80696 60460 79653 61841 78586 63203 77494 48 i3 57667 81698 59084 80679 60483 ! 79635 61864 78568 63225 77476 47 14 57691 81681 59108 8.6j2 6o5o6 I 796J8 6.887 78550 63248 77438 46 ID 57715 81664 59131 80644 60529 79600 61909 78532 63271 77439 45 16 57738 81647 59154 80627 6o553 79583 6.932 785.4 63293 77421 44 17 57762 8i63i 59178 80610 60576 79365 6.955 78496 633.6 77402 43 18 57786 81 614 59201 80393 60399 79547 61978 78478 63338 77384 42 19 57810 81397 59223 80376 60622 79530 62001 78460 6336. 77366 41 20 57833 8i58o 59248 80338 60643 79312 62024 78443 63383 77347 40 21 57857 8i563 59272 8o54i 60668 79494 62046 78434 63406 77339 39 38 22 57881 81546 59293 80334 60691 79477 62069 78403 63438 77310 23 57904 8i53o' 59318 80307 60714 79459 62092I 78387 63451 77292 37 24 57938 8l5l3: 59342 80489 60738 79441 62. .5 78369 63473 77273 36 25 57952 81496; 59365 80472 60761 79434 62.38 7835. 63496 77255 35 26 57976 81479' 593S9 80455 60784 79406 62160 78333 635.8 77236 34 27 57999 81462, 59412 80438 60807 79388 62.83 783.5 63540 77218 33 28 58023 8i445i 59436 80430 6o83o 79371 622061 78297 63563 77.99 32 29 58047 81428: 59439 80403 60853 79353 62229 78379 63585 77.8. 3i 3o 58070 81412: 59482 8o386 60876 79335 6225i 78261 636o8 77.62 3o 3 1 58094 81395^ 59306 8o368 60899 793^8 622-^' 78243 63o3o 77'44 29 28 32 58ij8 81378: 59329 8o35i 60922 79300 622971 7S225 63653 77.25 33 58i4i 8i36i: 59552 80334' 60945 79282I 62320 78206 63675 77.07 27 34 58i65 8i344' 59376 8o3i6^ 60968 792641 79247I 63342 78.88 63698 77088 26 35 58189 813275 59399 80299 60991 62365 78.70 63730 . 77070 25 36 58212 8i3io' 59622. 8028 a; 6101 5 79239: 62388 78.52 63743 77031 24 37 582.36 81293; 59646 80264 6io38 79311! 62411! 78.34 63765 77033 23 38 58260 81276! 59669 59693, 80247 61061 79J93| 62433 781 16 63787 77014 22 39 58283 81239! 8o23o 61084 79.76 62456 78098 638.0 76996 21 40 583o7i 81242! 59716! 80212! 61 107 79i5S 62479 78079 63832 76977 20 41 58330 8l225| 59739' 80195! 6ii3o 79 "40 62502 78061 63854 76939 ;? 42 58354 8i2o8[ 59763: 80178! 6ii53 791231 62324 78043 63877 76940 43 58378' 81191 59786' 80160 61176 79105 62347 78025 63^99 7^03. 17 44 58401 1 81 174! 59S09 80143 61199 79087] 62570 78007 63922 7<>.,)3| 16 45 58425, 8ii57| 59832 80125 61222 79069 62392; 77988 63944 70)34 i5 46 54449' 811401 59836 80108 61245 7905 1 626.5 77970 63966 76866 14 47 584721 31123^ 59879 80091 61268 790331 62638 77932 63989 76847 i3 48 58496; 81106! 59902, 80073 61291 790i5i 62660 77934 640.1 76828 12 49 58519! 81089? 59926 8oo56 6i3i4 789931 62683 77916 64033 768.0 II 5o 58543' 81072; 59949I 8oo38 6.337 6i36o 78980I 62706 77897 64o56 76791 10 5. 58067! 8l033| 59972! 80021 78962! 62728; 77879 64078 76772 I 52 58390! 8io38 59995 8ooo3 6i383 78944I 6275.1 77861 64100 76754 53 586 1 4| 81021 60019 799S6 61406 78926 62774 77843 641 231 76735 7 54 53637! 81004 60042 79968 61439 78908 62796: 77824 64145 767.7 6 55 58661 i 80987 6oo65l 79931 6i45i 78891 628.9! 77806! 64167 76698 5 56 58684' 80970 60089 79934' 61474 78873 62842 77788 64190 76679 4 57 58708 80953 60112 799i6{ 61497 78855 63864 77769 64212 76061. 3 58 587311 80936! 6oi35| 79899I 6i52o 78837 63887 7775.1 64234 76642 2 59 58755 80919J S. 60 1 58 7988. S. 61 543 78819 62909 C. S. 77733 64256 76623 S. I M M c. s. 1 O.S.I C. S. S. S. C. S. 1 54 Deof. 1 53 Deg. 1 52 I )esr. 51 Deg. 1 50 Deg. 1 A TABLE OF NATUKAL SINES. Tl _M^ 40 Deg. s. c. s. ! 41 Deg. 1 ^^ Deg. 1 ^3 Deg. 1 44 Deg. M 60 S. 1 c. s. 656o6 75471 s. 1 c. s. S. IC. s. S. |C,S. 642^9 76604 66913; 743 14 68200 73i35 69466, 71934 1 6430II 76086 i 65628 75452 ] 66935; 742951 68221; 73116^1 69487' 7I9I4 ! 5g 3 64323; 76367 6563C 73433 i 66956I 74276 68242 73096 j 69308! 71894 5? 3 64346 76548 , 63672 75414:1 66978; 74236 68264 73076 6g529 71873 ;s 4 64368: 7653o ! 65694 75395 66999: 74237 68285 , 73036 69549 7«853 5 64390' 765 1 1 63716 75375 6-02 li 74217 683o6 73o36 1 69370 ' 71833 55 6 64412 76492 ! 65-38 75356 670431 74108 68327 730.6 ' 6959, ! 7i3i3 ' 54 7 64435 76473 : 63759 75337 67064' 74178 68349 ' 72996 ; 696.2 ! 71792 ' 53 8 644571 76455 1 65781: 75318;! 67086 74139 68370 i 72976 i 69633 ; 71772 52 9 64479 76436 658o3 75299 67107 74139 68391 I 72937 69654 i 71752 5i 10 645oi| 76417 1 65825, 73280 67129 74120 6S412 72937 69675 71732 5o II 64524' 76398 ' 65847 75261 67151 74100 68433 ; 72917 69696 71711 % 12 64346 76380 1 65869 75241 67172 74080 68455 i 72897 69717 71691 i3 64568 76361 i 63891 75222 67194 74061 68476 ; 72877 69737 71671 ii U 64590 76342 ' 63913 75203 67215 74041 68497 : 72857 69758 7i65o i5 64612 76323 1 65935, 75184 67237 74022 685 18 i 72837- 69779 7i63o 45 i6 64635 76304 1 65956 75i65 67258! 74002 68539 ' 72817 69800 71610 44 17 64657 76286 : 639-8 75146 67280 739^3 68361 72-97 69821 71590 43 i8 64679 76267 66000 75126 67301 73963 68382 1 72777 69842' 71369 42 19 64701 76248 66022: 75107 67323 73944 686o3 1 72757 69862 71549 41 20 64723 76229 66044 75088 67344 73924 68624 1 72737 69883 7.529 40 21 64746 76210 66066 75069 67366 73904 68645 72717 69904 7i5o8 ii 22 64-68 76192 66088 75o5o 67387 73885 6S666 1 72697 69925 71488 23 64790 76173 66109; 75o3o 67409 73865 68688 72677 69946 71468 37 24 64812' 76154 66i3i| 75011 67430 73846 6S709 72657 69966 71447 36 s5 64834 76135 661 53; 74992 67452 73826 6S730 72637 69987 71427 35 26 64856 76116 66175, 74973 67473 73806 68751 726n i 70008 71407 34 27 64878. 76097 661971 74953 67495 -3787 68772 72597 70029 71386 33 28 64901 ' 76078' 66218. 74934, 67516 73767 68793 72577 70049 71366 32 29 64923 76059 66240, 74915, 67538 73747 68814 72557 70070 71343 3i 3o 64945 76041, 66262, 74896, 67559 73728, 68835 72537 70091 7i325j 3oi 3i 64967' 76022 66284 74876 67580 73708 68857 72517 70112 7i3o5 29I 28 1 32 64989 76003 663o6| 74857: 6760? 73688: 68878 72497 7oi32 71284 33 65oii 75984 66327I 74838." 67623 73669! 68899 72477 701 53 71264 27 34 65o33 75965 66349 74818 67645 73649' 68920 72457 70174 71243 26 35 65o55 75946 66371, 74799: 67666 73629 68941 72437 70193 71223 25 36 65o77 75927 66393 74780, 67688 73610, 68c62 72417 70215 71203 24 ?Z 65o99 73908 66414 74-60. 67709 73590; 68983 72397 70236 71182 23 38 63122 75S89 66436 74741 67730 73370 69004 72377 70257 71162 22 39 65i44 75870 66458 74722j 67752 73551, 69025 72357 70277 71141 21 40 65 1 66 75831 66480 74703 j 67773 73531 69046 72337 70298 71121 20 41 • 65i88 75832 66301: 74683; 67795 735: 1 1 69067 72317 70319 71100 19 42 65210 75Si3 66523 74664' 67816 73491! 69088 72297I 70339 71080 18 43 65232 75794 66545 74644 67S37 734721 69109' 72277! 7o36o 71059 17 44 65234 75775 66566 74625, 67859 73452' 69130 72257 7o38i 71039, l5 45 65276 75756 665S8j 74606 67880 73432 69i5i 72236 70401 71019 i5 46 65298 75738 66610I 745861 67901 73412! 69172 72216 70422 70998 14! H 65320 75719 66632 745671 67923 73393 69193, 721961 70443 70978 i3i 48 65342 75699 66653 74548 67944 73373 692141 72176 70463 70957 12^ ^9 65364 75680 66675, 74528 67965 73353 692351 72i56j 70484 70937! 11 1 5o 65386 75661: 66697; 74509 V&l 73333: 69256 72i36i 7o5o5 70916; lei 5i 63408 75642 66718, 74489! 73314: 69277 721 i6j 7o525 70896I g; 52 6543o 73623 66740' 74470 68029' 73294 69208, 72093 70546 70875! 81 53 65452 75604 66762, 74451 68o5i, 73274 69319 72075| 70567 7o855i 7i 54 65474 755^5 66783 74431 68072 73234: 69340 720551 70587 7o834| ^I 55 65496 75566 668o5] 74412; 68093 73234 69361 72o35| 70608 708,31 5I 56 655i8 75547 668271 74392! 68ii5 732i5 693SS 72oi5i 70628 7 0649 i 707931 4! 57 65540 75528 66848 74373; 68i36: 73.95 69403 71995; 71974 70772; 31 58 65562 75509 66870 74353 1 6807, 73175 69424 70670; 70752 3i i»9 i 65584 75490 66891 1 74334! 68179' 73i55 69445 71954 70690' 70731 x| 60 • 656o6 75471: 66913: 74314 C.-S. 1 s. 43 Deg. 1 68:00! C. S. 1 73i35 69466, 7.934I 70711; 707 II 0; M 1 C. S. ! S. C. S. • ^- 1 C. S. 1 S. I "Mi - 49 De?. ! 47 Deg. 1 40 1 )eg. 11 45 Deg. 1 I 72 TRaVERSF taklf. ?■ = i Deg. i Deg. IDeg. ^ Lai. Dep. La:. Dep. Lat. Dpp r "fToo" v.vu '-' . •'' 1 1 . 0'.i "o.of 1 2 2.U0 V . l< 1 . . . ■;; •:.oo 0.03 1 2 3 3JJ0 0.01 1 1 _ t'rj J.uO 0.04 1 3 4 4.U0 0.02 -; . ':' • ' 3 0.05 ' 4 5 5.M 0.02 '-' . - -r o.uO 07 • 5 6 6.00 0.03 ; 5 6.00 : 08 - 6 7 7.00 0.03 7 ^ [ . "5 7.00 ' 0.09 7 8 8.00 0.03 . , 7 8.00 0.10 8 9 9.00 0.04 , .1;^; 9.00 ( 0.12 9 JO 10.00 0.04 ■- . 'J .- 10.00 0.13 . 10 11 11.00 0.05 O.iO 11.00 -o:t4 11 12 12.00 • 0.05 '. Z , . 0. :o 12.CK) 0.16 12 13 13.00 0.06 ".11 13.00 i 0.17 13 14 14.00 0.06 'J . 1 c 14.00 , O.IS 14 15 15.00 0.07 . . 3 15.00 0.20 15 16 16.00 0.97 '.' . 1 r 16.W 0.21 16 17 17.00 0.07 ■ 7 ^ " i;, , ; - 17.00 0.22 17 IS IS.OO OS . IS -■ .' L' , 1 G IS. 00 0.24 18 19 19.00 o.ns ! 19.00 U.17 . 19.00 0.25 19 20 20.00 0.09 1 20.00 0.17 20.00 0.26 20 21 21.00 0.09 21.00 0.18 21.00 C.27 2l 22 22.00 0.10 22.00 0.19 ' 22.00 0.29 22 23 » 23.00 ; 0.10 : 23.00 i 0.20 23.00 0.30 23 24 \ 24.00 0.10 24.00 0.21 24.00 0.31 24 25 I 25.00 0.11 25.00 0.22 25.00 0.33 25 26 i 26.00 O.Il 26.00 0.23 26.00 0.34 26 27 1 27.00 0.12 27.00 0.24 27.00 0.35 27 28 28.00 0.12 28.00 0.24 2S.0O 0.37 2> 29 29.00 0.13 29.00 0.25 29.00 0.3S 29 30 30.00 0.13 30.00 0.26 30.00 0.39 30 '31 31.00 0.14 31.00 0.27 31.00 0.41 31 32 32.00 0.14 32.00 0.2S 32.00 0.42 32 33 33.00 0.14 33.00 0-29 33.00 0.43 33 34 a4.oo 0.15 : 34.00 0.30 34.00 0.45 34 35 3.5. UO 0.15 35.00 0.31 3.^.00 0.46 35 36 3S.00 0.16 36.00 0.31 36.00 0.47 36 37 37.00 0.16 37.00 0.32 37.00 0.48 37 3S 3S.00 0.17 38.00 0.33 3S.00 0.50 38 39 39. UO 0.17 39.00 0.34 39.00 0.51 39 40 40.00 0.17 40.00 0.35 40.00 0.52 40 41 41.00 0.18 41.00 0.36 41.00 0..54 4! 42 42.00 0.18 42.00 0.37 42.0<:) . 55 4-: , 43 43.00 0.19 43.00 0.33 43.00 O.rS 43 44 ; 44.00 0.19 44.00 0.38 44.00 0.5S 44 « 45 45.00 0.20 45.00 0.39 45.00 0.59 45 1 46 1 46.00 0.20 46.00 0.40 46.00 0.60 46 47 47.00 0.21 47.00 0.41 47.00 0.62 47 i 4^ 4S.00 0.21 48.00 0.42 48.00 \ 0.6a ; 4s 49 49. IM) 0.21 49.00 0.43 49.00 0.64 : 49 M 50.00 i 0.22 50.00 O.-U 50.00 0.65 r>o 1 — Q Dep. Lat. Dep. 894 Lat. Deg. Dep. Lat. 831 Deg. 89t Deg. TKAVEliSE TAHLE. 73 ."3 g 1 ^ r)eg. i k Deg. IDeg. i Lat. Dep. Lat. Dep. Lat. 1 Dep. '51 51.00 1 0.22 51.00 0.45 61.00 i 0.67 61 52 52.00 i 0.23 62.00 0.45 52.00 0.68 5i 53 53.00 0.23 53.00 0.46 ! 53.00 0.69 1 53 54 54.00 0.24 54.00 0.47 64.00 0.71 ■ 54 55 55.00 0.24 55.00 0.48 55.00 0.72 1 65 5G 56.00 0.24 56.00 1 0.49 56.00 0.73 ' 56 57 57.00 0.25 57.00 1 0.50 57.00 0.75 1 57 58 58.00 0.25 58.00 '■ 0.51 57.99 0.76 68 59 59. OC 0.26 59.00 0.51 56.99 0.77 . 59 eK) 60. OC o.2e 60. OC 0.52 59.99 0.79 60 "61 61.00 0.27 61.00 0.63 i 60.99 0.80' "61 62 62.00 0.27 62.00 0.54 61.99 0.81 62 63 63.00 0.27 63.00 0.55 62.99 0.82 63 64 64.00 0.28 64.00 0.56 63.99 0.84 64 65 65.00 0.28 65.00 0.57 64.99 0.85 65 66 66.00 0.29 1 66.00 0.58 65.99 0.86 66 67 67.00 0.29 ! 67.00 0.58 66.99 0.88 67 68 68.00 0.30 1 68.00 0.59 1 67.99 0.89 68 69 69.00 0.30 i 69.00 0.60 i 68.99 0.90 69 70 70.00 0.31 1 70.00 0.61 69.99 0.92 70 71 71.00 0.31 1 ' 71.00 0.62 70.99 0.93 ■71 72 72.00 0.31 i 72.00 0.63 71.99 0.94 72 73 73.00 0.32 1 73.00 0.64 72.99 0.96 73 74 74.00 0.32 74.00 0.05 73.99 0.97 74 75 75.00 0.33 75.00 0.65 74.99 0.98 75 76 76.00 0.33 76.00 0.66 75.99 0.99 76 77 77.00 0.34 77.00 0.67 1 76.99 l.Gl 77 78 78.00 0.34 78.00 0.68 77.99 1.02 78 79 79.00 0.34 79.00 0.69 78.99 1.03 79 80 80.00 0.35 80.00 0.70 79.99 1.05 80 81 81.00 0.35 81.00 0.71 80.99 1.06 81 82 82.00 0.36 82.00 0.72 81.99 1.07 82 83 83.00 0.36 83.00 0.72 82.99 1.09 83 84 84.00 0.37 84.00 0.73 83.99 1.10 84 85 85.00 0.37 85.00 0.74 84.99 1.11 85 86 86.00 0.38 88.00 0.75 85.99 1.13 86 87 87.00 0.38 87.00 0.76 85.99 1.14 87 88 88.00 0.38 88.00 0.77 1 87.99 1.15 88 89 89.00 0.39 89.00 0.78 1 88.99 1.16 89 90 91 90.00 91.00 0.39 0.40 90.00 91.00 0.79 89.99 90.99 1.18 1.19 90 91 0.79 92 92.00 0.40 92.00 0.80 91.99 1.20 92 93 93.00 0.41 93.00 0.81 92.99 1.22 93 94 94.00 0.41 94.00 0.82 93.99 I 23 94 95 95.00 0.41 95.00 0.83 94.99 1.24 9o 96 96.00 0.42 96.00 0.84 95.99 1.26 95 97 97.00 0.42 97.00 0.85 96.99 1.27 97 98 98.00 0.43 1 98.00 0.86 97.99 1.28 9S 99 99.00 0.43 i 99.00 0.86 98.99 1.30 99 100 100.00 _0.j44 100.00 0.87 99.99 1.31 100 i ei "re Dep. 89}] Lat. Degr. Dep. Lat. Dep. 39U Lat. )eg. g c 5 89^ Deg. 74 TRAVI RSE TAKLE 1 p iDeg. U Deg. . 1 U Deg. 1 1| i -•^■^. 3 Lat. 1.00 Dep. 0.02 I Lat. 1.00 Dep. Lat. 1 D".p "koo ^H Lat. Dep. 1 1 0.02 1 l.L'Oi 0.03 1 2 2.00 0.03 2.00 0.04 1 2.00 1 f Jr i.ool 0.06 2 3 3.00 0.05 3.00 0.07 ! 3 '»o! O.f'.- fi.OO 09 . 3 4 i.OO 0.07 4.00 0.09 i 0.1 O.in 4.00 0.12: 4 5 5.00 0.09 5.00 0.11 1 5.00 0.13 i; 5.00 0.15' 5 6 6.00 0.10 6.00 0.13 6..0 ^rifij! 6.00, 18 6 7 7.00 0.12 7.00 0.15 7.00 ' 18 |l 7.00; 0.21 7 8 8.00 0.14 0.16 ! 8.00 0.17 8.00 0.21 j 8.00 1 0.25 S 9 9.00 9.00 0.20 ! 9.00 0.24'; 9.00! 0.23' 9 10 10.00 17 10.00 0.22 0.24 10.00 11.00 0.26^ 10.00: 0.28 10.99 0.31 10 0.34 ; 11 11 11.00 0.19 il.OO !•:> 12.00 0.21 ;: 12.00 0.26 12.00 0.31 ; 11.99 0.37 12 13 13.00 0.23 , 13.00 0.28 13.00 0.34 : 12.99 0.40 13 li 14.00 0.24 , 14.00 0.31 ' 14.00 0.37 13.99 0.43 14 lo 15.00 0.26 1 15.00 0.33 14.99 0.39 i 14.99 0.46 1.5 16 16.00 0.28 16.00 0.35 , 15.99 ; 0.42 15.99 0.49 10 17 17.00 0.30 1 17.00 0.37 16.99; 0.45 16.99 0..52 17 IS IS. 00 0.31 :• 18.00 0.39 17.99 0.47! 17.99 0..55 IS 19 19.00 0.33 19.00 0.41 18.99 ' 0..50 18.99 0.58 19 'ZO 20.00 0.35 20.00 21.00 0.4^1 0.46" 19.99 0.52 ' 19.99 20.99 ■ 0.£5 20.99 0.61 0.64 20 "21 •21 21.00 0.37 j 22.00 0.38 21.99 0.48 21.99' 0..58 •; 9,1.99 : 22.99 60 || 22.99 0.67 22 23 23.00 0.40 22.99 0..50 0.70 23 24 24.00 0.42 23.99 24.99 25.99 0.52 23.99 0.62 j! 23.99 0.73 24 25 25.00 0.44 0.55 24.99 0.G5f 24.99! 0.76 25 20 2G.00 0.45 0.57 25.99! 0.68 1 25.99; 0.79 26 27 27.00 0.47 1 26.99 59 '26.99 0.71 ! 26.99 i U.S3 27 2S 2S.00 0.49 27.99 0.61 27.99 0.73 ' 27.99 i 0.36 23 29 29.00 0.51 28.99 0.63 28.99 0.76 128.99] 0.89 29 30 30.00 31.00 0.52 0..54 29.99 30.99 0.65 0.68 29.99 0.79 29.99 1 0.92 30 30.99 0.81 30.99 0.95 , ail 32 32.00 0.56 31.99 0.70 31.99 ' 0.84 ' 31.99 0.93 32 33 32.99 0.58 32.99 0.72 32.99 0.R6 32.98 i.Ol 33 34: 83.99 0.59 33.99 0.74 33. 9d U.99 33.98 1.04 34 35 31.99 0.61 34.99 0.76 '34.99 0.92 34.98 1.07 35 36 '■ ?o . 99 0.63 35.99 0.79 35.99 , 0.94 35.98 1.10 36 37 36.99 0.65 36.99 O.Sl 36.99 ' 0.97 36.98 ].13 37 3S 37.99 0.66 37.99 0.83 37.99 1 0.99 37.98 1.16 33 39 3S.99 0.68 38.99 0.85 38.99 1 1.0-i 33.98 1.19 39 40 39.99 0.70 ; 39.99 0.87 39.99 1 1.05 39.93 ^ . 22 40 41 40.09 0.72 40.99 0.S9 40;99 1.07 40.93 1 25 41 42 41.99 0.73 41.99 , 0.92 41.99 1.10 41.93 1 20 42 43 42.99 0.75 42.99 ; 0.94 42.99 1.13 42.93 i I. 31 43 44 43.99 0.77 43.99 : 0.96 43.99 1.15 43.93 1 l.o4 14 45 44.99 0.79 44.99 0.98 44.99 1.13 44.98 1.37 45 46 45.99 0.80 45.99 ' 1.00 45 . 99 1.20 45.98 1.40 46 47 46.99 0.82 46.99 ; 1.03 46.99 1.23 46.98 1.44 47 4S 47.99 0.84 47.99 , 1.05 47.98 1.26 47.93 1.47 43 49 4S 99 0.86 48.99 1.07 48.98 : 1.28 48.98 1 . 50 49 1 50 49 99^ 0.87 Lat. 49.99 1.09 49.93 1 1.31 , 49.98 j" Dep. l.c3 i 50 1 1 Dep. Dep. Lat. , Dep. , Lat Lat. if 89] Deg. 88? Deg. 1 88| Deg. !' 88J Deg. 1 2 TRAVERSE TABLE 75 5 w p '51 1 1 Deg. 1 1 n Dog. 1 H Deg. 1| Deg. ?• '5T Lat. '50^99 Dep. 0.89 Lat. Dep. 'ITn Lat. Dep. Lat. .50.98 Dep. 1^079"8" 1.34 1.56 52 51 99 0.91 51.99 1.13 51.98 1.36 51.98 1.59 52 53 52 99 . 92 i 52.99 1.16 52.98 1.39 52.98 1.62 53 54 53 99 0.94 .53.99 1.18 53.98 1.41 53.97 1.65 51 55 54 99 0.96 .54.9- 1.20 54.98 1.44 .54.97 1.68 551 5(5 5.3.99 0.98 55.99 1.22 55.98 1.47 55.97 1.71 56 1 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 57.9.8 l.,52 57.97 1.77 58 59 58.99 1.03 58 . 99 1.29 58.98 1.54 58.97 1.80 59 60 61 59.99 60.99 1.05 LOG 59. S9 60.99 1.31 1.3''. 59.98 1.67 59.97 1.83 60 61 60.93 1.60 60.97 1.86 62 61.99 1.08 61. P9 1.35 61. 9S 1.02 61.97 1.89 62 63 62.99 1.10 62.99 1.37 62. &8 1.65 62.97 1.92 63 64 63.99 1.12 63.98 1.40 63.93 1.68 63.97 1,95 641 65 64.99 1.13 64.98 1.42 64.08 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 6S 97 2.11 C9 70 tT 69.99 70.99 1.22 1.24 69.98 1..53 69.98 1.83 69 97 2.14 2 ]7 70 71 70.98 1.55 70.98 1.86 70. 'j7 72 71.99 1.26 71.98 1.57 71.98 1.88 71.97 2.20 72 73 72.99 1.27 72.93 1.59 72.97 1.91 72.97 2.23 73 74 73.99 1.29 73.98 1.61 73.97 1.94 73.97 2.26 74 75 74.99 1.31 74.98 1.64 74.97 1.96 i 74.97 2.29 75 76 75.99 1.33 75,98 1.66 75.97 1.99 75.96 2.32 76 77 76.99 1.34 76.98 1.68 76.97 2.02 76.96 2.35 77 78 77.99 1.36 77.98 1.70 77.97 2.04 77.96 2.38 78 79 78,99 1.38 78.98 1.72 78.97 2.07 78.96 2.41 79 80 79.99 1.40 79.98. 1.75 79.97 2.09 79.96 2.44 80 81 80.99"" 1.41 80.98 1.77 80.97 2.12 80.96 2.47 '81 82 81.99 1.43 81.98 1.79 81.97 2.15 81.96 2.. 50 82 83 82.99 1.45 82.98 1.81 82.97 2.17 82.96 2.. 53 83 84 83.99 1.47 83 . 98 1.83 83.97 2.20 83. 9G 2.57 84 85 84.99 1.48 84.98 1.S5 84.97 2.23 84.96 2. CO 85 86 85.99 1.50 85.98 1.88 85.97 2! 25 85.96 2.63 86 87 86.99 1..52 86.98 1.90 86.97 2.28 86.96 2.66 87 88 87.99 1.54 87.98 1.92 87.97 2.30 87.96 2.69 88 89 88.99 1.55 88.98 1.94 88.97 2.33 88.96 2.72 89 90 91 89.99 1.57 89.98 "90:98" 1.96 i.99 89.97 2.36 h'9.96 2.75 ' 2.78 90 91 90.99 1.59 90.97 2.38 90.96 92 91.99 1.61 91 98 2.01 9i.9r 2.41 91.96 2.81 92 93 92.99 1.62 92.98 2.03 92.9" 2.13 92.96 2.84 93 94 93.99 1.64 93.98 2.05 93.97 2.46 93.96 2.87 94 95 94.99 1.66 94.98 2.07 94.97 2.49 94.96 2.90 95 96 95 99 1.68 95.98 ■ 2.09 95.97 2.51 95.96 2.94 98 97 96,99 1.69 96.98 2.12 96.97 2.. 54 96.95 2.96 97 98 97.99 1.71 97.98 2.14 97.97 2.. 57 97.95 2.99 i 9& 99 9H.98 1.73 98.98 2.16 98.97 2.. 59 98.95 3 02 1 '^'^ 100 CO Q 99.98 Dep. 1.75 i Lat. 99.98 2.18 Lat. 99.97 2.62 99.95 S.O.') ■100 i « Dep. 1 Dep Lat. Dep. 88i I"u.>; 89 Dejr. 881 Deg. 881 Deg. Dec 1 rt iu 76 TRAVERSE TARLE. 2 Deg. i 2i Deg. H Deg. 2| Deg. fC Lai. Dep. Lat. Dep.j Lat. Dep Lat. Dep. n 'T 1.00 0.03 1.00 0.04 1 1.00 0.04 1.00 0.05 "1 2 2.00 0.07 2.00 0.08' 2.00 0.09 2.00 0.10 2 3 , 3.00 0.10 3.00 0.12 3.00 0.13 3.00 0.14 2 4 1 4.00 0.14 4.00 0.16 4.00 0.17 4.00 0.19' 4 r. .' 5.00 0.17 5.00 0.20 5.00 0.22 4.99 0.24 5 6 6.00 0.21 6.00 0.24 6.99 0.26 5.9.^ 0.29' 6 7 7.00 0.24 6.99 0.27; 6.99 0.31 6.99 0.34. 7 8 7.99 0.28 I 7.99 0.31 i 7.99 0.35 7.99 0.38 8 9 8.99 0.31 ! 8.99 0.3c 1 8.99 0.39 8.99 0.43 9 10 1 9.99 0.35 i 9.99 0.39 9.99 0.44 9.99 0.48 lO 11 ; 10.99 0.38 10.99 0.43; 10.99 0.48 10.99 0.53 Ti 12 ; 11.99 42 • 11.99 0.47: 11.99 0..52 11.99 0.58 12 13 1 12.99 0.45 1 12.99 0.51 i 12.99 0..57! 0.61 ! 12.99 0.62 13 14 i 13.99 0.49 : 13.99 0..55! 13.99 13.98 0.67 14 15 14.99 0.52! 14.99 0.59' 14.99 0.65 0.70 14.98 0.72 15 16 1 15.99 0..56! 15.99 0.63' 15.99 15.98 0.77 16 17 16.99 0.59' 16,99 0.67 16.98 0.74 i 16.98 0.82 17 18 1 17.99 0.63 1 17.99 0.71! 17.98 0.79 17.98 0.86 18 19 1 18.99 0.66 18.99 0.75! 18.98 0.83 1! 18.98 0.91 19 20 19.99 0.70 i 19.98 0.79 19.98 0.87 19.98 0.96 20 21 20.99 0.73 20.98 0.82: 20.98 ' 0.92 20.98 1.01 21 22 ;21.99 0.77! 21.98 0.86 21.98 0.96 21.97 1.06 22 23 22.99 0.80 1 22.98 0.90' 22.98 1.00 22.97 1.10 23 24 23.99 0.84 1 23.98 0.941 23.98 1.05 '23.97 1.15 24 25 24.98 0.871 24.98 0.93! 24.98 1.09 24.97 1.20 25 26 25.0'^ 0.91 25.98 1.02 1 25.98 1.13 25.97 1.25 26 27 : 26.98 0.94 26.98 1.061 26.97 1.18 26.97 1.30 27 28 ! 27.98 0.98 27.98 1.10 27.97 1.22 .27.97 1.34 28 29 '28.98 1.01 28.98 1.14 28.97 1.26 23.97 1.39 29 30 29.98 1.05 ! 29.98 1.18 29.97 1.31 29.97 1.44 30 31 31 130.98 1.08 1 30.98 1.22 30.97 1.35 30.96 1.49 32 ,31.98 1.12 31.98 1.26 31.97 1.40 31.96 1..54 32 33 32.98 1.15 32.97 1.30 32.97 1.44 32.96 1.58 33 34 33.98 1.19' 33.97 1.33 33.97 1.48 33.96 1.63 34 35 34.98 1.22 34.97 1.37 34.97 1.53 34.96 1.68 35 36 35.98 1.26 35.97 1.41 35.97 1.57 35.96 1.73 36 37 30.98 1.29 36.97 1.45 36.96 1.61 36.96 1.78 37 38 37.93 1.33 37.97 1.49 37.96 1.66 .37.96 1.82 38 39 38.98 1.36 38.97 1.53 38.96 1.70 ,38.96 1.87 39 40 139.98 41 ! 40.98 1.40 1.43 39.97 1 67 139.96 1.75 ,39.95 1.92 40 40.97 1.61 40.96 1.77 40.95 1.97 41 42 41.97 1.47 41.97 1.65 41.96 1.83 i 41.95 2.02 42 43 42.97 1.60 42.97 1.69 42.96 1.88 1 42 . 95 2.C6 43 44 43.97 1.54 43.97 1.73 43 . 96 1.92 143.95 2.11 44 45 44.97 1.57 44.97 1.77 44.96 1.96 144.95 2.16 45 40 ' 45.97 1.61 45.96 1.81 45.96 2.01 145.95 2.21 46 47 : 46.97 1.64 46.96 1.85 46.96 2.05 ; 46.95 2.25 47 48 47.97 1.68 147.96 1.88 147.95 2.09 47.95 2.30 iS 49 48.97 1.71 48.96 1.92 ,48.95 2.14 : 48.94 2.35 49 .'iO 49.97 1.74 49.96 1.96 Lat. 49.95 2.18 149.94 2.40 50 i 1 o u ! Dep 88] Lat. Jeg. Dep. Dep. 8'^ Lat. Deg. Dep. Lat. c3 .2 ' m Deg. 87i Deg. TIIAVERSE TABLE. 77 p 3 ? 51 1 2Deg. 1 n Deg. li 1 n Deg. 21 Deg. 5 51 Lat. Dep. ~n7s' Lat. Dep. Lat. Dep. Lat. Dep. 2.45 50.97 50.90 2.00 .50.95 2.22 50.94 52 51.97 J. 81 51.90 2.04 51.95 2.27 51.94 2.50 ! 52 58 52.97 1.85 52.96 2. OS .52.95 2.3] 52.94 2.. 54 i 53 51 53.97 1.88 53.96 2.12 53.95 2.36 1 53.94 2.59 C4 ^5 ' 54.97 1.92 54.96 2.16 54.95 2.40 54.94 2,64 55 56 55 . 97 1.95 55.96 2.20 55.95 2.44 55.94 2 69 56 57 150.97 1.99 56.98 2.24 56.95 2.49 56.93 2.73 1 57 5S 157.96 2.02 57.96 2.28 57.94 2.. 53 .57.93 2.78 58 59 ! 58.98 2.06 58.95 2.32 58.94 2.57 58.93 2.83 59 Gl 59.98 2.09 59 . 95 2.30 59 . 94 60.94 2.62 2.66 59.93 2.88 60 60.90 2.13 60,95 2.39 60.93 2.93 62 61.90 2.16 61.95 2.43 61.94 2.70 61.93 2.97 62 63 62.98 2.20 62.95 2.47 62.94 2.75 62.93 3.02 63 64 63.98 2.23 63.95 2.51 63.94 2.79 63.93 3.07 64 65 64.96 2.27 64.9.7 2.55 64.94 2.84 64.93 3.12 65 66 65.93 2.30 65.95 2.. 59 65.94 2.83 65.92 3.17 66 67 66.96 1 2.34 00.95 2.63 60.94 2.92 66.92 3.21 67 63 67.98 1 2.37 67.95 2.67 67.94 2.97 67.92 3.20 68 69 68.98 2.41 68.95 2.71 68.93 3.01 68.92 3.31 69 70 71 69.96 2.44 69.95 2.75 69.93 3.05 89.92 70.92 3.30 3.41 70 71 70.96 2.48 70.95 2.79 70.93 3.10 72 71.96 2.51 71.94 2.83 71.93 3.14 71.92 3.45 72 73 72.96 2.. 55 72.94 2.87 72.93 3.18 72.92 3.. 50 73 74 73.95 2.58 73.94 2.91 73.93 3.23 73.91 3.55 74 75 74.95 2.63 74.94 2.94 74.93 3.27 74.91 3.60 75 76 75.95 2.65 75.94 2.98 75.93 3.31 75.91 3.65 76 77 76.95 2.69 76.94 3.02 76.93 3.36 76.91 3.70 77 78 77.05 2.72 77.94 3.00 77.93 3.40 77.9) 3.74 78 79 78.95 2.76 78.94 3.10 78.92 3.45 78.91 3.79 79 80 79.95 2.79 79.94 3.14 79.92 3.49 79.91 3.84 80 81 80.95 2.83 80.94 3.18 80.92 3.53 80.91 3.89 81 82 81.95 2.86 81.94 3.22 81.92 3.58 81.91 3.93 82 83 82.95 2.90 82.94 3.26 82.92 3.62 82.90 3.98 83 84 83.95 2.93 83.94 3.30 83.92 3.66 83.90 4.03 84 85 84.95 2.97 84.93 3.34 84.92 3.71 84.90 4.08 85 86 85.95 3.00 85.93 3.38 85.92 3.75 8,s 9^ 4.13 86 87 86.95 3.04 86.93 3.42 86.92 3.79 86.90 4.17 87 88 87.95 3.07 87.93 3.45 87.92 3.84 87.90 4.22 88 89 88.95 3.11 88.93 3.49 88.92 3.88 88.90 4.27 89 90 89.95 3.14 89.93 3.53 89.91 3.93 89.90 4.32 1 90 91 90.95 3.18 90.93 3.57 90.91 3.97 90.90 4.37 1 91 92 91.94 3.21 91.93 3.61 91.91 4.01 91.89 4.41 92 93 92.94 1 3.25 92.93 3.65 92.91 4.06 92.89 4.46 93 94 93.94 3.28 93.93 3.69 93.91 4.10 93.89 4.51 94 95 94.94 3.32 94.93 3.73 94.91 4.14 94.89 4.56 95 96 95.94 3 35 95.93 3.77 95.91 4.19 95.89 4.61 98 97 96.94 3.39 96.93 3.81 96.91 4.23 90.89 4.65 97 98 97.94 3.42 f 97.92 3.85 97.91 4.27 97.89 4.7V 93 99 98.94 3.. 46 98.92 3.89 98.91 4.32 98.89 4.75 99 100 o a 99.94 3.49 99.92 3.93 99.91 4.36 99.88 _4.8_0_ Lat. 100 8 Q Dep. Lat. Dep. Lat. Dep. Lat. Dep. 88 Deg. 87| Deg. - - 8711 3eg. 87i Deg. 78 TRAVERSK TABLE. 3 o p 3 Deg. 3i Deg. 3i Deg. 3^ Deg. a ' I Lat. Dep. Lat. Dep. Lat. 1 Dep. Lat. 1.00 Dep. 06 1.00 0.05 ! 1.00 ~"o:o6" 1.00 i .06 2 2.00 0.10 2.00 0.11 2.00' 0.12 2.00 -.3 2 3 3.00 0.16 3.00 0.17 2.99: 0.18 2.99 ^.iO 3 4 3.99 0.21 3.99 0.23 3.99 0.24 3.99 0.26 1 5 4.99 0.26 4.99 0.28 4.99 0.31 4.99 0.33 5 6 5.99 0.31 5.99 0.34 5.99 0.37 5.99 0.39 o 7 6.99 0.37 6.99 . 40 6.99 0.43 0.49 6.99 0.46 "t 8 7.99 0.42 7.99 0.45 7.99 7.98 0.52 s 9 8.99 0.47 8.99 0.51 8.93 0..55 8.98 O.r/J 9 10 11 9.99 0.52 9.93 0.57 9.98 0.61 9 98 . 65 11 10.98 0.58 10.98 0.02 10.98 0.67 10". 98 ~0?f2" 12 11.98 0.63 11.98 0.68 11.98 0.73 11.97 0.7S 12 13 12.98 0.68 12.93 0.73 12.98 0.79 12.97 0.85 13 14 13.98 0.73 13.93 0.79 13.97 0.85 13.97 0.92 14 I.=i 14.98 0.79 14.93 0.85 14.97 0.92 14.9? 0.98 15 16 15. 9S 0.84 15.97 0.91 15.97 0.98 15.97 1.05 16 17 16.98 0.89 16.97 0.96 16.97 1.04 16.96 l.ll 17 IS 17.98 0.94 17.97 1.02 17.97 1.10 17.96 1.18 18 19 18.98 0.99 18.97 1.08 ' 18.96 1.16 1 18.90 1.24 19 20 21 19.97 1.05 19.97 1.13 i 19.96 1.22 19.96 1.31 _20 21 20.97 1.10 20.97 1.19 120.96 1.28 20 . 96 1.37 22 21.97 1.15 21.96 1.25 21.96 1.34 21.95 1.44 22 23 22.97 1.20 22.96 1.30 22.96 1.40 22.95 1.50 23 24 23.97 1.26 23.96 1.36 23.96 1.47 23.95 1.57 24 25 24.97 1.31 24.96 1.42 24.95 1.53 24.95 1.64 25 26 25.96 1.36 25.96 1.47 25.95 1.59 25.94 1.70 26 27 26.96 1.41 26.96 1.53 26.95 1.65 20.94 1.77 27 28 27.96 1.47 27.95 1.59 27.95 1.71 27.94 1.83 28 29 23.96 1..52 23.95 1.04 28.95 1.77 1 28.94 1.90 29 39 31 29.96 1.57 29.95 1.70 : 1.76| 29.94 l.S3i 29.94 1.96 JO 31 30.96 1.62 30.95 30.94 1.89 30.93 2.03 32 31.96 1.67 31.95 1.81 31.94 1.95 31.93 2.09 32 33 32.95 1.73 .32.95 1.87 32.94 2.01 32.93 2.16 33 34 33.95 1.78 33.95 1.93 1 33.94 2.08 33.93 2 . 22 34 35 34.95 1.83 34.94 1.93 ' 34.93 2.U 34.92 2.29 35 36 35.95 1.88 35.94 2.04 i 35.93 2.20 35.92 2.35 36 37 36.95 1.94 36.94 2.10i 36.93 2.26 36.92 2.42 37 33 37.95 1.99 37.94 2.15 . 37.93 2.32 37 . 92 2.49 38 39 33.95 2.04 .33.94 2.21 38.93 2.38 1 33.92 2.55 39 40 39.95 2.09 39.94 2.27 39 . 93 2.44 39.91 2.62 40 41 40.94 2.15 40.93 2.32 40.92 ~~2~,~b0 40.91 2 . 68 "41 42 41.94 2.20 4U93 -2. .38 \ 41-92 2.56 41.91 2.75 42 43 42.94 2.25 42 . 93 2.44 1 42.92 2.63 42.91 2.81 43 44 43.94 2 . 30 43.93 2.49 j| 43.92 2.69! 43.91 2.88 44 45 44.94 2.36 44.93 2.55 |i 44.92 2.75 i 44.90 2.94 45 46 45.94 2.41 45 . 93 2.61 i 45.91 2.81 1 ^5.90 3.01 JG 47 46.94 2.48 46.92 2.66 ;l 46.91 2.87 i 46.90 3.07 4? 48 47.93 2.51 47.92 2.72 1 47.91 2.93 i 47.90 3.14 48 49 48.93 2.56 48.92 2.78 n 48.91 2.99 ; 48 . 90 3.20 19 50 o o c .2 Q 49.93 Dep. 2.62 Lat. 49.92 2.83 '149.91 3.05 49^ 3.2-7 Lat. Deg. 50 h 1 Dep. L.t. Dep. Lat. Dep. 80i 87 E >cg. 86^ Deg. 1 ;;6| Deg. TRAVERSE TABLE. 79 p 3 Deg. 3i Deg. 1 . Deg. Dep. ^ 21 Deg. . § 51 Lat Dep. i 2.G7 Lat. Dep. Lat. Lat. Dep. 3.34 61 "50; 93 '50.92 2.89 50.90 3-11' 50.89 fi2 51.93 i 2.72 ;oJ.92 2.95 51.90 3.17! 51.89 3.40 52 5S 52.93 2.77 li'2.91 3.00 52.90 3.24 52.89 3.47 53 54 53.93 2.83 !.53.91 3.06 53.90 3.30 53.88 3.53 64 55 5-4.92 2.88 54.91 3.12 54.90 3.36 il 54.88 3.60 65 6C 55.92 2.93 155.91 3.17 .55.90 3.42:155.88 3.66 56 57 5fi.92 2.98 6G.91 3.23 56.89 3.48 56.88 3 73 57 5S 57.92 3.04 57.91 3.29 57.89 3.54:! 57.88 3.79 58 59 58.92 3.09 58.91 3.34 58.89 3.60 58.87 3.86 59 60 59.92 3.14 ; 59.90 60.90 3.40 3.46 59.89 3.66 1 .59.87 60.87 3.92 3.99 60 6] 61 6U.92 3.19 60.89 3.72' 62 6 J. 92 3.24 ;61.90 3.51 61.88 3.79 i 61.87 4.05 62 63 62.91 3.. 30 ,62.90 3.57 ,62.88 3.85i 62.87 4.12 63 64 63.91 3.35 63.90 3.63 ! 63.88 3.91 ! 63.86 4.19 64 65 04.91 3.40 64.90 3.69 ; 64.88 3.97! 64.66 4.25 65 6G 65.91 3.45 65.89 3.74 65.88 4.03 1 65.86 4.. 32 66 07 66.91 3.51 66-. 89 3.80 ,66.88 4 .09 i 66.86 4.38 67 68 67.91 3.. 56 67.89 3.86 1 67.87 4.15, 67.85 4.45 68 69 68.91 3.61 68.89 3.91 168.87 4.211 68.85 4.51 69 70 71 69.90 70.90 3.66 69.89 3.97 ; 69.87 170.87 4.271 4.33! 69.85 4.58 70 71 3.72 70.89 4.03 70.85 4.64 72 71.90 3.77 71.88 4.08 ,71.87 4.40; 71.85 4.71 72 73 72.90 3.82 72.88 4.14 72.86 4.46 1 72.84 4.77 73 74 73.90 3.87, 73.88 4.20 73.86 4. .52 1 73.84 4.84 74 75 74.90 3.93 74.88 4.25 74.86 4.58 74.84 4.91 75 76 75.90 3.98 75.88 4.31 75 86 4.64 75.84 4,97 76 77 76.89 4.03 76.88 4.37 76.86 4.70 76.84 5.04 77 78 77.89 4.08 j 77.87 4.42 77 85 4.76 77.83 5.10 78 79 78.89 4.131 78.87 4.48 78.85 4.82 78.83 5.17 79 80 79.89 4.19! 79.87 4.54' r9.85 4.88 79.83 5.23 80 81 80.89 4.24 1 80.87 4. .59 1 80.85 4.94 80.83 5.30 81 82 81.89 4.29; 81.87 4.66 81.85 5.01 81.82 5.36 82 83 82.89 4.341 82.87 4.71 82.85 5.07 82.82 5 43 83 84 83.88 4.40 83.86 4 76 83.84 5.13 83.82 6.49 84 85 84.88 4.45 84.86 4.82 '84.84 5.19 84.82 5.56 85 86 85.88 4.. 50 85.86 4 88 85.84 5.25 85.82 6.62 86 87 86.88 4..55i 86.86 4.93 86.84 5.31 86.81 5.69 87 88 87.88 4.61: 87.86 4.99 87.84 5.37 87.81 5.76 88 89 88.88 4.66 88.86 5.05 88.83 5.43 88.81 6.82 89 90 89.88 4.71 : 89.86 5.10 89.83 5.49 89.81 5.89 90 91 90.88 4.76; 90.85 5.16 90.83 5.56 90.81 6.95 91 92 91.87 4.81 91.85 5.22 91.83 5.62 91.80 6.02 92 93 92.87 4.87 92.85 5.27 92.83 5.68 92.80 6.08 93 94 93.87 4.92 1 93.8.5 5.33 ; 93.82 5.74 93.80 6.15 94 9c 94. 8r 4.97 i 94.85 5.39:194.82 6.80 94.80 6.21 95 96 95.87 5.02; 95.85 5.4-/ 1195.82 6.86 95.70 6.28 96 97 90.87 5.08; 96.84 5 :(» 96.82 5.92 96.79 6.34 97 98 '97.87 5.13i 97.84 5..'>6 97.82 5.98 97.79 6.41 93 99 98.86 5.18| 98.84 5.61 : 98.82 6.04 98.79 6.47 99 100 Q ^9.86^ Dep. 5.23 Lat. 99.84 5.67 1 99.81 6.10 99.79 6.54 100 i Dep. Lat. Dep. Lat. Dep. Lat. 8,1 ^eg. 86| Deg. 83i 1 Deg. 86i Deg. 80 TRAvrrsr tattle. p 3 o 4 Deg. ' 1 4^ Deg. H Deg. • 1 1 41 Deg. Lat. ■ Dep. E' a a a Lat. Dep. 1 Lat. Dep. Lat. Dep. ~1 "TToo" 0.07 i 1.00 0.07 1.00 0.08 1.00 0.08 i 2 2.00 0.14 1.99 0.15 1.99 16 ! 1.99 0.17 s 3| 2.99 0.21 ! 2.99 0.22: 2.99 24 1 2.99 0.25 3 4 1 a 99 1 0.28 3.99 0.30 1 3.99 31 1 3.98 0.23 1 5 4. 99 1 0.35 4.99 0.37 4.98 0.39 1 4.98 0.41 5 6 , 5.99, 0.42 5.98 0.44 5.98 0.47 5.98 0.50 5 7 6.98 0.49 6.98 0.52 6.98 0..55 6.97 0.5S 7 8 7.98 0.56 7.98 0.59 7.98 0.63 7.97 0.06 8 9 8.98 0.63 8.98 0.67 8.97 0.71 8.97 0.75 9 10 1 9.98 0.70 9.97 0.74 9.97 0.78 9.97 0.83 10 Hi 10.97 0.77 10.97 0.82 10.97 0.86 10.96 0.91 ii 12 1 11.97 0.84 11.97 0.89 11.96 0.94 11.96 0.99 12 13 1 12.97 0.91 12.96 0.96 12.96 1.02 12.96 1.08 J3 14 13.97 0.98 13.96 1.C4 13.96 1.10 13.95 1.16 14 15 14.96 1.05 14.96 1.11 14.95 1.18 14.95 1.24 15 16 15.96 1.12 15.96 1.19 15.95 1.26 15.95 1.32 16 17 16.96 1.19 16.95 1.26 16.95 1.33 16.94 1.41 17 18 1 17.96 1.26 17.95 1.33 17.94 1.41 17.94 1.49 18 19 18.95 1.33 18.95 1.40 18.94 1.49 18.93 1..57 19 20 21 19.95 1.40 19.95 1.48 1.56 19.94 1.57 19.93 1.66 20 21 20.95 1.46 20.94 20.94 1.65 20.93 1.74 22 21.95 1.53 21.94 1.63 21.93 1.73 21.92 1.82 22 23 22.94 1.60 22.94 1.70 22.93 1.80 22.92 1.90 23 24 23.94 1.67' 23.93 1.78 23.93 1.88 23.92 1.99 24 25 24.94 1.74 24.93 1.85 24.92 1.96 24.91 2.07 25 26 25.94 1.81 25.93 1.93 25.92 2.04 25.91 2.15 26 27 20.93 1.88 26.93 2.00 26.92 2.12 26.91 2.24 27 28 27.93 1.95 27.92 2.08 ' 27.91 2.20 27.90 2.32 28 29 28.93 2.02 28.92 2.15 28.91 2.28 28.90 2.40 29 30 29.93 2.09 29.92 30.91 2.22 29.91 2.35 29.90 2.48 30 '31 31 30.92 2.16 2.30 30.90 2.43 30.89 2.57 32 31.92 2.23 31.91 2.37 31.90 2.51 31.89 2.65 ^12 33 32.92 2.30 32.91 2.45 32.90 2.69 32.89 2.73 33 34 33.92 2.37 33.91 2.52 33.90 2.67 33.88 2.82 34 35 34.91 2.44 34.90 2.59 34.89 2.75 34.88 2.90 35 36 35.91 i 2.51 35.90 2.67 35.89 2.82 35.88 2.98 36 37 36.91 1 2.58 36.90 2.74 36.89 2.90 36.87 3.06 37 38 37.91 2.65 37.90 2.82 37.88 2.98 37.87 3.15 38 39 38.90 2.72 38.89 2.89 38.88 3.06 38.87 3.23 39 40 39.90 1 2.79 39.89 2.96 39.88 3.14 39.86 3.31 40 41 40.90 2.86 40.89 3.04 40.87 3.22 40.86 3.40 "41 42 41.90 2.93 41.88 3.11 41.87 3.30 41.86 3.48 42 43 42.90 3.00 42.88 3.19 42.87 3.37 42.85 3.56 43 44 43.89 3.07 43.88 3.26 43.86 3.45 43.85 3.64 44 45 44.89 1 3.14 44.88 3.33 44.86 3.53 44.85 3.73 4r. 46 45.89 i 3.21 45.87 3.41 45.se 3.61 45.84 i 3.81 4G 47 46.89 3.28 46.87 3.48 46.86 3.69 46.84 1 3.89 4? 48 47.88 3.35 47.87 3.56 47.85 48.85 3.77 47.84 1 3.97 1 4^ 49 48.88 8.42 48.87 3.63 3.84 48.83 1 4.06 49 _60 49.88 3.49 49.86 3.71 49.85 3.92 49.83 ! 4.14 50 Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 Lat. 86 Deg. 85f Deg. 851 1 Deg. 85J Deg. ei TRAVERSE TABLE. 81 D 05 P 3 ? 61 4 Dog. 4i Deg. H Deg. 4| Deg. & 3 ? 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 50. 8S 3.5G 50.86 3.78 50.84 4.00 50.82 4.22 o2 51.87 3.63 51.86 3.85 51.84 4.08 51.82 4.31 52 53 52.87 3.70 52.85 3.93 52.84 4.16 52.82 4.39 53 51 53.87 3.77 53.85 4.00 53.83 4.24 53.81 4.47 54 55 54.87 3.84 54.85 4.08 54.83 4.32 54.81 4-55 55 56 55.86 3.91 55.85 4.15 55.83 4.39 55.81 4.64 56 57 156.86 3.98 56.84 4.22 56.82 4.47 56.80 4.72 57 58 57.86 4.05 57.84 4.30 57.82 4.55 57 80 4.80 58 59 58.86 4.12 58.84 4.37 58.82 4.63 58.80 4.89 59 60 61 59.85 4.19 59.84 4.45 59.82 4.71 59.79 4.97 60 61 60.85 4.26 60.83 4.52 60.81 4.79 60.79 5.05 62 61.85 4.32 61.83 4.59 61.81 4.86 61.79 5.13 62 63 62.85 4.39 62.83 4.67 62.81 4.94 62.78 5.22 63 64 63.84 4.46 63.82 4.74 63.80 5.02 63.78 5.30 64 65 64.84 4.53 64.82 4.82 64.80 5.10 64.78 5.38 65 66 65.84 4.60 65.82 4.89 65.80 5.18 65.77 5.47 66 67 66.84 4.67 66.82 4.97 66.79 5.26 66.77 5.55 67 f)8 67.83 4.74 67.81 5.04 67.79 5.34 67.77 5.63 68 69 68.83 4.81 68.81 5.11 68.79 5.41 68.76 5.71 69 70 71 69.83 4.88 69.81 5.19 69.78 5.49 69.76 5.80 5.88 70 71 70.83 4.95 70.80 5.26 70.78 5.57 70.76 72 71.82 5.02 71.80 5.34 71.78 5.65 71.75 5.96 72 73 72.82 5.09 72.80 5.41 72.77 5.73 72.75 6.04 73 74 73.82 5.16 73.80 5.48 73.77 5.81 73.75 6.13 74 75 74.82 5.23 74.79 5.56 74.77 5.88 74.74 6.21 75 76 75.81 5.30 75.79 5.63 75.77 5.96 75.74 6.29 76 77 76.81 5.37 76.79 5.71 76.76 6.04 76.74 6.38 77 78 77.81 5.44 77.79 5.78 77.76 6.12 77.73 6.46 78 79 78.81 5.51 78.78 5.85 78.76 6.20 78.73- 6.54 79 80 81 79.81 5.58 79.78 80.78 5.93 79.75 6.28 79.73 6.62 80 81 80.80 5.65 6.00 80.75 6.36 80.72 6.71 82 81.80 5.72 81.78 6.08 81.75 6.43 81.72 6.79 82 83 82.80 5.79 83.77 6.15 82.74 6.51 82.71 6.87 83 84 83.80 5.86 83.77 6.23 83.74 6.59 83.71 6.96 84 85 84.79 5.93 [84.77 6.30 84.74 6.67 84.71 7.04 85 86 85.79 6.00 85.76 6.37 85.73 6.75 85.70 7.12 86 87 86.79 6.07 86.76 6.45 85.73 6.83 86.70 7.20 87 88 87.79 6.14 87.76 6.52 87.73 6.90 87.70 7.29 88 89 88.78 6.21 88.76 6.60 88.73 6.98 88.70 7.37 89 90 91 89.78 6.28 89.75 90.75 6.67 89.72 7.06 89.69 7.45 90 91 90.78 6.35 6.74 90.72 7.14 90.69 7.. 54 92 91.78 6.42 91.75 6.82 91.72 7.22 91.68 7.62 92 93 92.77 6.49 92.74 6.89 92.71 7.30 92.68 7.70 93 94 93.77 6.56 93.74 6.97 93.71 7.38 93.68 7.78 94 95 94.77 6.63 94.74 7.04 94.71 7.45 94.67 7.87 95 96 95.77 6.70 95.74 7.11 95.70 7.53 95.67 7.95 96 f)7 196.76 6.77 96.73 7.19 96.70 7.61 96.67 8.03 97 98 97.76 6.84 97.73 7.28 97.70 7.69 97.66 8.12 as 99 98.78 6.91 98.73 7.34 98.69 7.77 98.66 8.20 99 100 0) 99.76 Dep. 6.98 Lat. 99.73 7.41 39.69 7.85 99.66 8.28 m o Dep. Lat. Dep. Lat. Dep. Lat. .2 86 I 3eg. 1 m Deg. 851 Deg. 85i Deg. 82 TRAVERSE TABLE. 5 3 o o 5D eg- \ 5i Deg. H Deg. 5} Deg. 1 1 C p Lat. "roo Dep. 0.09 Lat. j Dep. Lat. Dep. Lat. Dep. 1 1.00 0.09 1.00 0.10 0.99 O.ir' 1 2 1.99 0.17 1.99 0.18 1.99 0.19 1 . 99 O.i;) 2 3 2.99 0.26 2.99 0.27 2.99 0.29 2 . 98 0.30 3 4 3.98 0.35 3.98 0.37 3.98 0.38 3.98 0.40 4 5 4.98 0.44 4.98 0.46 4.98 0.48 4.97 0..50 5 6 5.98 0.52 5.97 0.55 5.07 0.58 5.97 0.60 6 ?( 6.97 0.61 0.97 0.64 6.97 0.67 6.96 0.70 7 8i 7.97 0.70 7.97 0.73 7.06 0.76 7.96 0.80 8 9 8.97 0.78; 8.96 0.82 8 96 '0.86 8.95 0.90 9 10 11 9.9o 10.96 0.87, 9.96 0.92 9.95 0.96 9.95 1.00 1^' 11 6.96 10.95 1.01 10.95" 1.05 10.94 KIO 12 11.95 1.05 1 1 1 . 9o 1.10 1 1 1 . 94 1.15 11.94 1.20 12 13 12.95 1.13 1 12.95 1.19 12.94 1.25 12.93 1.30 13 14 13.95 1.22! 13.94 1.28 13.94 1.34 13.93 1.40 14 15 14.94 1.31 14.94 1.37 14.93 1.44 14.92 1..50 15 16 15.94 1.39: 15.93 1.46 15.93 1.53 15.92 1 . 60 lb 17 16.94 1.48 16.93 1.56 16.92 1.63 16.91 1.70 17 18 17.93 1.57 17.92 1.65 17.92 1.73 17.91 1.80 18 19 18.93 1.66 18.92 1.74 18.91 1.82 18.90 1.90 19 20 '21 19.92 1.74 19.92 1.83 19.91 20 . 90 1.92 19.90 2.00 20 21 20.92 1.83 20.91 1.92 2.01 20.89 2.10 1 22 21.92 1.92 21.91 2.01 21.90 2.11 1 21.89 2.20 22 23 22.91 2.00! 22.90 2.10 22.89 2'. 20 22.88 2.30 23 24 23.91 2.09 23.90 2.20 23.89 2.30 23.88 2.40 24 25 24.90 2.18 24.90 2.29 24.88 2.40 24.87 2.50 25 26 25.90 2.27 25.89 2.38 25.88 2.49 i 25.87 2.60 26 27 26.90 2.35 26.89 2.47 26.88 2.59 26.86 2.71 27 28 27.89 2.44 27.88 2.56 27.87 2.08 27.86 2.81 28 29 28.89 2.53 28.88 2.65 28.87 2.78 28.85 2.91 29 30 31 29.89 30.88 2.61 29.87 2.75 29.86 30.86 2.88 2.97 29.85 3.01 30 2.70 30.87 2.84 30.84 3.11 31 32 31.88 2.79 31.87 2.93 31.85 3.07 31.84 3.21 2Z 33 32.87 2.88 32.86 3.02 32.85 3.16 32.83 3.31 33 34 33.87 2.96 33.86 3.11 33.84 3.26 33.83 3.41 34 35 34.87 3.05 34.85 3.20 34.84 3.35 34.82 3.51 3.> 36 35.86 3.14 35.85 3.29 35.83 3.45 35.82 3.61 36- 37 36.86 3.22 36.84 3.39 36.83 3.55 36.81 3.71 37 38 37.86 3.31 37.84 3.48 37.83 3.64 37.81 3.81 38 39 38.85 3.40 38.84 3.57 38.82 3.74 38.80 3.91 39 40 39.85 3.49 39.83 3.66 39.82 3.83 39.80 4.01 40 41 40.84 3.57 40.83 3.75 40.81 3.93 40.79 4.11 41 42 41.84 3.66 41.82 3.84 41.81 4.03 41.79 4.21 42 43 42.84 3.7B 42.82 3.93 42.80 4.12 42.78 4 31 43 44 43.83 3.83 43.82 4.03 43.80 4.22 43.78 4 il 44 45 44.83 3.92 44.81 4.12 44.79 4.31 44.77 4 51 45 40 45.82 4.01 45.81 4.21 45.79 4.41 45.77 4 61 46 47 46.82 4.10 46.80 4.30 46.78 4.50 46.76 4.71 47 48 47.82 4.18 47.80 4.39 47.78 4.60 47.76 4.81 45 49 48.81 4.27 48,79 4.48 48 . 77 4.70 48.75 i 4.91 49 fJO 49.81 4.36 49.79 Dep. 4.58 1 ?.at. 49.77 Dep. 4.79 Lat. 49 . 75 5.01 Lat. f 03 l>oi>. 1 Lat. Dep. 85 Deg. 84J Deg. 84^ Deg. 84i Deg. 1 '■ TRAvEHSE TA±lLE. 83 OS 3 o o 61 5Deg. 5i Deg. H Deg. ■n Deg. i C ■ 1 p '51 Lat. Dep. Lat. 1 Dep. Lat. Dep. Lat. Dep. 5.11 50.81 4.44 |i 50.79 4.67 50.77 4.89 50 74 52 51.80 4.53 [51.78 4.76 '51.76 4.98 51 74 6.21 52 53 52.80 4.62! 52.78 4.85 j 52.76 5.08 52 73 5.31 5o 64 53.79 4.71 53.77 4.94 ' 53.75 5.18 53 73 5.41 54 55 54.79 4.79 54.77 5.03 54.75 5.27 54.72 5.51 55 56 55.79 4.88 55.77 5.12 55.74 5.37 55.72 5.61 56 57 56.78 4.97 56.76 5.22 156.74 5.46 56.71 5.71 57 58 57.78 5.06 57.76 5.31 J 57.73 5.56 57.71 5.81 . 58 59 58.78 5.14 58.75 5.40 j 58.73 5.65 58.70 5.91 59 60 61 59.77 60.77 5.23 5.32 59.75 60.74 5.49 159.72 5.75 59.70 6.01 : 60 5.58 60.72 5.85 60.69 6.11 , 61 62 61.76 5.40 61.74 5.67 61.71 5.94 61.69 6.21 62 63 62.76 5.49 62.74 5.76 62.71 6.04 62.68 6.31 63 64 63.76 5.58 63.73 5.86 63.71 6.13 63.68 6.41 64 65 64.75 5.67 64.73 5.95 64.70 6.23 64.67 6.51 65 66 65.75 5.75 65.72 6.04 65.70 6.33 65.67 6.61 66 67 66.75 5.84 66.72 6.13 66.69 6.42 66.66 6.71 6? 68 67.74 5.93 67.71 6.22 67.69 6.52 67.66 6.81 68 69 68.74 6.01 68.71 6.31 68.68 6.61 68.65 6.91 69 70 71 69.73 6.10 69.71 6.41 69.68 70.67 6.71 6.81 69.65 7.01 70 70.73 6.19 70.70 6.50 70.64 7.11 71 72 71.73 6.28 71.70 6.59 71.67 6.90 71.64 7.21 1 72! 73 72.72 6.36 72.69 6.68 72.66 7.00 72.63 7.31 73 74 73.72 6.45 73.69 6.77i 73.66 7.09 73.63 7.41 74 75 74.71 6.54 74.69 6.86i 74.65 f 7.19 74.62 7.51 75 76 75.71 6.62 75.68 6.95 75.65 7.28 75.62 7.61 76 77 76.71 6.71 76.68 7.05 76.65 7.38 76.61 7.71 77 78 77.70 6.80 77.67 7.14 77.64 7.48 77.61 7.81 78 79 78.70 6.89 1 78.67 7.23 78.64 7.57 78.60 7.91 79 80 81 79.70 6.97' 79.66 7.32 79.6b 7.67 .79.60 8.02 80 80.69 7.06 '80.66 7.41 80.63 7.76 80.59 8.12 81 82 81.69 7.15 181.66 7.50 81.62 7.86 81.59 8.22 82 83 82.68 7.23 82.65 7.59 82.62 7.96 82.58 8.32 83 84 83.68 7.32 83.65 7.69 83.61 8.05 83.58 8.42 84 85 84.68 7.41 84.64 7.78 84.61 8.15 84.57 8.52 85 86 85.67 7.50 85.64 7.87 85.60 8.24 85.57 8.62 86 87 86.67 7.58 86.64 7.96 86.60 8.341 86.56 8.72 87 88 87.67 7.67 87.63 8.05 87.59 8.43 87.56 8.82 8S 89 88.66 7.76 88.63 8.14 88.59 8.53! 83.55 8.92 89 90 91 89.66 90.65 7.84 7.93 89.62 8.24 89.59 8.63! 8.72 i 89.55 9.02 00 90.62 8.33 90.58 90.54 9.12 91 92 91.65 8.02 91.61 8.42 91.58 8.82 1 91.54 9.22 92 93 92.65 8.11 92.61 8.51 1 92.57 8.91 1 92.53 9.32 93 94 93.64 8.19 93.61 8.601 93.57 9.01 93.53 9.42 94 95 94.64 8.28 94.60 8.69! 94.56 9.11 1 94.52 9.52 95 96 95.63 8.37 95.60 8.78' S5.56 9.20 95.52 9.62 96 97 96.63 8.45 96.59 8.88 1 96.55 9.30 96.51 9.72 97 98 97.63 8.54 97.59 8.97 j 97.55 9.39I 97.51 9.82 98 99 98.62 8 63 98.59 9.06 i 93.54 9.49! 93.50 9.92 991 100 99.62 8.72 99.58 9.15 1 99.54 9.58! 99.50 "Dep. 10.02 Lat 100 1 Dep. Lat. Dep. Lat.| ! Dep. Lat.| 851 )eg. 84} Deg. 84i Deg. 84J Deg 20 84 TRAV7RSE TA-BLE. fa ! 6 Deg. e\ Dcg. 1 1 6| Deg. 1 6a Deg. 7.' Lat. Dep. Lat. Dep. j Lat. 0.99 Dep. o.ui Lat. Dep. C3 1 I 0,99 0.10 ' 0.99 0.11 0.99 ~'07l2"i T a ■ 1 99 0.21 1 1.99 0.22 1 1.99 0.23 1.99 24' 2 3, 2 9S 0.31 1 2.98 0.33! 2.93 0.34; 2.98 35 ! 3 4 ' 3 98 3.41 ; 3.93 44 i 3.97 0.45' 3.97 0.47 1 4 5! 4.97 0.52 4.97 -).54i 4.97 0.57! 4.97 0.59 i 3 6|. 5.97 0.63 5.96 0.80, 5.96 0.63 5.96 0.71 : 8 7 6.96 0.73 6.96 0.76; 6.96 0.79 6.95 0.82! 7 8 7.96 0.S4 7.95 i 0.87 j 7.95 0.91 1 7.94 0.94, S 9 8.95 0.94 8.95 1 0.93 8.94 1.02! 8.94 LOG I 9 10 11 9.95 10.94 1.05 9.94! 1.09 i 9.94 1.13: 1.25! 9.93 10.92 1.181 10 1.29 ' 11 1.15 10.93 1.20 1 10.93 12 11.93 1.25 11.93 1.31 i 11.92 1.36] 1.47! 11.92 1.41 12 13 12.93 1.36 12.92 1.42 12.92 12.91 1.53; 13 14 13.92 1.46 13.92 1.52 13.91 1.59 13.90 1.60' 14 15 14.92 1.571 14.91 1.63 i 14.90 1.70! 14.90 1.76 15 16 15.91 1.67! 15.90 1.74, 15.90 1.81 1 15.89 1.88 ^ 16 17 16.91 1.78, 16.90 1.85 16.89 1.92 16.88 2 00 1 17 18 17.90 1.88 i 17.89 1.96, 17.88 2.04 17.88 2.12 13 19 18.90 1.99 ' 18.89 2.07 18.83 2.15 18.87 2.23 19 20 19.89 2.09: 19.88 2.18 19.87 2.26 19.86 2.35 20 21 20.88 2.20' 20.88 2.29 20.87 2.38 20.85 2.47 21 22 21.88 2.30; 21.87 2.40 21.86 2.49 21.85 2.59 22 23 22.37 2.40 22.86 2.50 22.85 2.00 22.84 2.70 23 24 23. S7 2.51 ! 23.86 2.61 23.85 2.72 '23.83 2.82 . 24 25 24.88 2.61 1 24.85 2.72 24.84 2.83 24.83 2.94 25 26 25.86 2.72 j 25.85 2.83 25.83 2.94 25.82 3.06 26 27 26.85 2.82 26.84 2.94 ,26.83 3.06 '26.81 3.17 27 2S 27.85 2.93 27.83 3.05 27.82 3.17 .27.81 3.29 23 29 28.84 3.03 28.83 3.16 28. 81 3.28 '28.80 3.41 ' 29 30 29.84 3.14 3.24 29.82 3.27 29.81 3.40 29.79 3.53 30 3.64 31 '31 30.83 30.82 3.37 130.80 3.51 30.79 32 31 82 3.34 31.81 3.48 31.79 3.62 31.78 3.76 32 33 32.82 3.45 32.80 3.59 i 32.79 3.74 32.77 3.83 1 33 34 33.81 3.55 33-80 3.70 133.78 3.85 33.76 4.00 1 34: 35 34.81 3.66 34.79 3.81 34.78 3.96 34.76 4.11 35 36 35.80 3.76 35.79 3.92 35.77 4.08 35.75 4.23 ! 36 37 36.80 3.87 ,36.78 4.03 1 36.76 4.19 36.75 4.35 1 37 38 37.79 3.97 ' 37.77 4.14 37.76 4.30 37.74 4.47 : 3S 39 38.79 4.08 138.77 4.25 33.75 4.41 38.73 4.53; 39 40 39.78 4.18 [39.76 4.35 39.74 4.53 i39.72 4.70! 40 41 40.78 4.29 140.76 "4 46 140.74 4.64 ! 40.72 4.82 j 41 42 41.77 4.39 141.75 4.57 141.73 4.76 '41.71 4.941 42 43 42.76 4.49 42.74 4.68 i 42.72 4.87 42.70 5.05 1 43 44 43.76 4.60 43.74 4.79 43.72 4.93 : 43.70 5.17 44 45 44.75 4.70 44.73 4.90 44.71 5.09 144.69 5.29 1 45 46:45.75 4.81 45.73 5.01 45.70 5.21 145.68 5.41 . 46 47 46.74 4.91 46.72 5.12 46.70 5.32 ,46.07 5 52 ; 47 48 47.74 5.02 47.71 5.23 47.69 5.43 147.67 5.64 43 49 43.73 1 5.12 48.71 6.34 148.69 5.55 j 48.06 5.76 49 50_ 49.73 ! 5.23 49.70 5.44 149.63 5.66 149.65 5.88 50 u 1 1 Dcp. |Lat. Dep. Lat. Dep. Lat. Dep. Lat. 84 Deg. 831 Deff. 831 Deg. m Deg. ._ TRAVERSE TABLE. 85 5 ST 6 Deg. 6iDeg. 6^ Deg 61 Dog. § ~51 o - Lat. Dep. Lat. Dep. ^5.55| J at. 50.67' Dep. 1 5.77, Lat. Df^p. 51 50.72 1 5.33 50.70 60.65 ~579b' 52 51.72 5.44 51.69 5.66 i 51.67 5.59 G1.64 6.n 52 53 52.71 5.. 54 52.68 5.77 .52.66 6.0JI 5^.63 6.2? 53 54 53.70 5.64 53.68 5.88 1 53.60 G.r-i 53.63 6.35 54 55 54.70 5.75 54.67 5.99! 54.65 6.r.3: 54.62 6.16 5£ 56 55.69 5.85 55.67 6.101 55.64 6 341 55.61 6..5S 56 57 56.69 5.96 56.66 6.21 56.63 6.15! 50.60 6.70 57 58 57.68 6. 06 57.66 6.31 1 57.63 6.57i 57.60 6.82 58 59 58.68 6.r/j58.65 6.42 58.62 6.68! 58.53 6.93 59 60 59.67 6.27:59.641 6.53 59.61 6.79 1 59.68 7.05 60 61 60.67 6.38 ; 60.64 6.64! 60.61 6.91 i 60.58 7.17 61 62 61.66 6. 48 ll 61.63 6.75 61.60 7.02 i 61.57 7.29 62 03 62.65 6.59 62.63 6.86 02.60 7.131 62.56 7.40 63 64 63.65 6.69 l! 63.62 6.97 63.59 7.25 63.56 Y.52 64 65 64.64 6.79 ij 64.61 7.08 64.58 7.36 64.55 7.64 65 66 65.64 6.90 65.611 7-191 65.58 7.47 65.54 7.76 66 67 66.63 7.00; 66.60 7.29 { 66.57 7 58 66.54 7.88 67 68 67.63 7.11 I 67.60 7,40 1 67.56 7.70 67.53 7.99 68 69 68.62 7.211 68.59 ?.51 68.56 7.81 68.62 8.11 69 70 '71 69.62 70.61 7.32 i 69.58 7.62 69.55 7.92 69.51 8.23 8.35 70 71 7.42 1 70.58 7.73 70.54 8.041 70.51 72 71.61 7.53 71.57 7.84 71.54 8.15 71.50 8.46 72 73 72.60 7.63 72.57 7.95 72.53 8.26 72.49 8.58 73 74 73.59 7.74 73.. 50 8.06 73.52 8.38 73.49 8.70 74 75 74.59 7.84 74.55 8.17 74.52 8.49 , 74.48 8.82 75 76 75.58 7.94 75.. 55 8.27 75.51 8.60 75.47 8.93 76 77 76.58 8.05 76.54 8.38 76.51 8,72 76.47 9.05 77 78 77.57 8.15 77.. 54 8.49 77.50 8.83 77.46 9.17 78 79 78.57 8.26 78.53 8.60 78.49 8.94 78.45 9.29 79 80 81 79.56 8.36 79.53 80.52 8.71 8.82 79.49 9.06 79.45 9.40 80 81 80.56 8.47 80.48 9.17 80.44 9.52 82 81.55 8.571 81.51 8.93 81.47 9.28 81.43 9.64 82 83 82.55 8.68; 82.51 9.04 82.47 9.40 ; 82.42 9.76 8? 84 83.54 8.78- 83.50 9.14 83-46 9.51 83.42 9.87 84 85 84.53 8.88; 84.50 9.25 84.45 9.62 84.41 9.99 86 86 85.53 8.99 85.49 9.36 85.45 9.74 85.40 10.11 86 87 86.52 9.09 86.48 9.47 86.44 9.85 86.40 10.23 87 88 87.52 9.20 87.48 9.58 87.43 9.96 87.89 10.34 88 89 88.51 9.30 88.47 9.69 88.43 10.08 88.38 10.46 89 90 91 89.51 9.41 89.47 9.80 89.42 10.19 89.38 90.37 10.58 90 91 90.50 9.51 90.46 9.91 90.42 10.30 10.70 92 91.50 9.62 91.45 10.02 91.41 10.41 ; 91.36 10.81 92 93 92.49 9.72 92.45 10.12 92.40 10.53 92.3b 10.93 93 94 93.49 9.83 93.44 10.23 93.40 10.64 1 93.35 11.05 94 95 94.48 9.93 94.44 10.34 94.39 10.75 : 94.34 11.17 95 96 95.47 10.03 95.43 10.45 95.38 10.87 95.33 11.28 96 97 96.47 10.14 96.42 10.56 96.38 10.98 196.33 11.40 97 93 97.46 10.24 97.42 10.67 97.37 11.09 ii 97.32 ? 1 . 52 98 99 98.46 10.35 98.41 10.78 98.36 11.21 1198.31 11.64 no 100 99.45 10.46 99.41 10.89 99.36 11.32 l| 99.31 il.75 100 6 o 1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 841 )eg. 83| Deg. 8^ Deg. 83i Deg. 86 TRAVERSE TABLE- 5 w fa a n ? 7Deg. n Deg. n^ Deg 7.1 Deg. o Lat. Dep. Lat. D«p. Lat. Dep. Lat. Dep. ~1 ■"0799" 0.12 0.99 0.13 0.99 0.13 0.99 0.13 ] 2 1.99 0.24 1.98 0.25 1.98 0.26 1.98 0.27 s 3 2.98 0.37 2.98 0.38 2.97 0.39 2.97 0.40 s 4 3.97 0.49 3.97 0.50 3.97 0.52 3.96 0.54 4| 5 4 96 0.61 4.96 0.63 4.96 0.65 4.95 0.67 6 6 5.96 0,73 5.95 0.76 5.95 0.78 5.95 0,81 7 6.95 0.85 0.94 0.88 6.94 0.91 6.94 0.94 7 8 7.94 0.97 7.94 1. 01 7.93 1.04 7.9a 1.08 8 9 8.93 1.10 8.93 1.11 8.92 1.17 8.92 1.21 9 10 9.93 10.92 1.22 j 1 . 34 i 9.92 1.26 9.9.1 1.31 9.91 1..35 10 10.91 1.39 10.91 1.44 10.90 1.48 -ir 12 11.91 1.46! 1 1 . 90 1.51 11.90 1.57 11.89 1.G2 1 12| 13 12. GO 1.58; 12.90 1.64 12.89 1.70 12.88 1.75 13, 14 13.90 1.71 13.89 1.77 13.88 1.83 13.87 1.89 14 15 14.89 1.83 14.88 1.89 14.87 1.96 14.86 2.02 15 16 15.88 1.95 1 15.87 2.02 15.86 2.09 15.85 2.16 16 17 16.87 2.07! 16.86 2.15 16.85 2.22 16.84 2.29 17 18 17.87 2.19| 17.86 2.27 17.85 2.35 17.84 2.43 18 19 18.86 2.32 1 18.85 2.40 18.84 2.48 18.83 2.56 19 20 21 19.85 2.44 2.56' 19.84 20.83 2.52 " 2.65 19.83 2.61 19.82 2.70 20 21 20.84 20.82 2.74 20.81 2.83 22 21.84 2.68 1 21.82 2.78 21.81 2.87 21.80 2.97 22 23 22.83 2.80' 22.82 2.90 22.80 3.00 22.79 3.10 23 24 23.82 2.921 23.81 3.03 23.79 3.13 23.78 3.24 24 25 24.81 3.05 24.80 3.15 24.79 3.20 24.77 3.37 25 26 25.81 3.17 25 . 79 3.28 25.78 3.39 25.76 3.51 26 27 26.80 3.29 26.78 3.41 26.77 3.52 26.75 3.64 27 28 27.79 3.41 27.78 3.53 27.76 3.65 27.74 3.78 28 29 28.78 ,^ . §3 28.77 3.66 28.75 3.79 28.74 3.91 29.1 80 31 29.78 3.06 29.76 3.79 29.74 3 92 29.73 30.72 4.05 4.18 30 30.77 3.78 .30.75 3.91 30.73 4.05 31 32 31.76 3.90 31.74 4.04 31.73 4.18 31.71 4.32 32 33 32.75 4.02 32.74 4.16 32.72 4.31 32.70 4.45 33 34 33.75 4.14 83.73 4.29 33.71 4.44 33.69 4.58 34 35 34.74 4.27 34.72 4.42 34.70 4.57 34.68 4.72 35 36 35.73 4.39 35 .7 1 4.54 35.69 4.70 35.67 4.85 ae 37 36.72 4.51 36.70 4.67 .36.68 4.83 36.66 4.99 37 38 37.72 4.63 37.70 4.80 37.67 4.96 37.65 5.12 38 39 38.71 4.75 38.69 4.92 38.67 5.09 38.64 5.26 39 40 39.70 4.87 39.68 5.05 39.66 5.22 39.63 5.39 40 41 40.70 5.00 40.67 5.17 40.65 5.35 40.63 6.. 53 "41" 42 41.69 5.12 41.66 5.30 41.64 6.48 41.62 5.66 42 43 42.68 5.24 42.66 5.43 42.63 5.61 42.61 5.80 4» 44 43.67 5.36 43.65 5.. 55 43.62 5.74 43.60 6.93 44 45 44 67 5.48 44.64 5.68 44.62 6.87 44.69 6.07 45 46 , 45.66 5.61 45.63 5.81 45.61 6.00 45 68 6.20 46 47 46,65 5.73 46.62 6.93 46.60 6.13 46.57 6.34 47 48 47.64 5.85 47.62 6.06 47.59 6.27 1 47.56 6.47 48 49 48 . 63 5.97 48.61 6.18 48.58 6.40 ; 48.55 6.61 49 50 49.63 6.09 49.60 6.31 49.57 6.53 49.54 6.74 50 Dep» Lat. Dep. Lat. Dep. Lat. Dep. Lat. 6 o 63] Deg. 82| Deg. 821 Deg. 824 Deg. TRAVEFcSE TABLE. 87 o 3 O a "51 7Deg. n 1 3eg. U Deg. 7| D6g. 1 a § '51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 6,83 50.62 6.22 50.59 6.44 1 ~5f 56" 0.66' 50.53 52 51.01 6.34 51.. 58 6.56 u 51.56 6.69^2.55 6.79 51.53 7,01 < 52 53 52.60 6.46 52.58 6.92 1 52.52! 7.15 53 54 53.60 6.53 53.57 6.81 53.54 7,05 1 53.51 7.28 54 55 54.59 6. 70 54.56 6.94 54.53 7.18i 54. fO 7.42 55 56 55.58 6.82 i;5.55 7.07 55.52 7 31 55.49 7.55 56 57 56.58 6.95 56.54 7.19 56.51 7 4-tl 56.43 7.69 57 53 57.57 7.C7 57.54 7.32 57.50 7.57! 57.47 7.82 53 59 5S.56 7.19 58.53 7.45 53.50 7.70 1 58.46 7.96 59 60 61 59.55 60.55 7.31 59.52 60.51 7.57 1 59.49 7.83 1 59.45 8.09 60 ' ei 7.43 7.70 60.48 7.96 60.4^4 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 62.50 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 64.4-1 8.48 64.41 8.77 65 66 65.51 8.04 05.47 8.33 65.44 8.61 65.40 8.90 66 67 06 50 8.17 66.46 8. 40 66.43 8.75 66.39 9.04 67 6S 67.49 8.29 67.46 8.58 67.42 8.88 ►67.38 9.17 68 69 63.49 8.41 63.45 8.71 63.41 9.01 63.37 9.30 69 70 71 69.43 8.53 69.44 8.83 69.40 9.14 69.36 9.44 70 71 70.47 8.65 70.43 8.96 70.39 9.27 70.35 9.57 72 71.48 8.77 71.42 9.09 71.38 9.40 71.34 9.71 72 73 72.46 8.90 72.42 9.21 V2 33 9.. 53 72.33 9.84 73 74 73.45 9.02 73.41 9.34 73.37 9.66 73.32 9.98 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 9.33 176.33 9.72 76.34 10.05 76.30 10.38 77 78 77.42 9.51 77.33 9.84 77.33 10.18 1 77.29 10.52 78 79 78.41 9.63 78.37 9.97 1 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 1 10.79 80 81 80.40 9.87 80.35 10.22 80.31 10.57 '80.26! 10.92 82 81.39 9.99 '81. .34 10.35 81.30 10.70 81.25 11.06 82 83 82.33 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 10.96 183.23 11.33 84 85 84.37 10.36 84.32 10.73 84.27 11.09 184.22 11.46 85 86 85.36 10.48 85.31 10.85 185.26 11.23 185.21 11.60 86 87 86.35 10.60 86.30 10.93 86.26 11.30 86.21 11.73 87 83 87.34 10.72 87.30 11.11 187.25 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 11.83 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 95 96 95.23 11.70 95.23 12.12 95.18 12.53 95.12 1 12.95 96 97 96.23 11.82 96.22 12.24 '96.17 12.66 1 96.11 1 13.03 97 9S 97.27 11.94 97.22 12.37 ! 97.16 12.79 97.10 13.22 9S 93 93.26 12.07 93.21 12.49 98.15 12.92 '93.10 13.35 99 100 99.25 12.19 99.20 12.62 99.14 13.05 1 99.09 13.49 100 J 1 Dep. 83 1 Lat. 1 Dep. Lat. 1 Dep. Lat. Dep. Lat. 821 Deg. jl 821 Deg. i ^ 82V Deg. 88 TRAVERSE TATTLE. 3 8 Deg. 8i Deg. H D«g. H Deg. § Lai. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 0.99 0.14 0.99 0.141 0.99 0.15 0.99 0.15 ' 1 2 1.98 0.28 1.98 0.29 i 1.98 0.30 1.98 0.30 2 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 0.57 3.90 0.59 3.95 0.61 4 5 4.95 0.70 4.95 0.72 4.95 0.74 4.94 0.78 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 1 8 I 7.92 1.11 7.92 1.15 7.91 1.18 7.91 1 22 *? 1 9 8.91 1.25 8.91 1.29 8.90 1..33 8.90 1.37 9 10 9.90 1.39 9.90 10.89 1.43 1.58 9.89 1.48 9.88 1.52 10 ' 11 Jl 10.89 1 . 53 10.88 1.63 10.87 1.67 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 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 15.81 2.43 16 17 16.83 2.37 16.83 2.44 16.81 2.51 16.80 2.59 17 IS 17.82 2.51 17.81 2.58 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 "27 19.81 2.78 19.79 2.87 19.78 20.77 2.96 19.77 3.04 20 "21 20.80 2.92 20.78 3.01 3.10 20.76 3.19 22 21.79 3.06 21.77 3.16 21.76 3.25 21.74 3.35 22 23 22.78 3.20 22.76 3.30 22 . 75 3.40 22 . 73 3.50 23 24 23.77 3.34 23.75 3.44 23.74 3.55 23.72 3.65 24 25 24.76 3.48 24.74 3.59 24.73 3.70 24.71 3.80 25 2G 25.75 3.62 25.73 3.73 25.71 3.84 ! 25.70 3.96 26 27 26.74 3.76 26.72 3.87 26.70 3.99 126.69 4.11 27 2S 27.73 3.90 27.71 4.02 27.69 4.14 27.67 4.26 28 29 28.72 4.04 28.70 4.16 28.68 4.29 28.66 4.41 29 30 29.71 4.18 29.69 4.30 29.67 4.43 129.65 4.56 30 31 30.70 4.31 30.68 4.45 30.66 4.58 i 30.64 4.72 31 32 31.69 4.45 31.67 4.59 31.65 4,73 131.63 4.87 32 33 32.68 4.59 32.66 4.74 32.64 4.88 32.62 5.02 33 34 33.67 4.73 33.65 4.88 33.63 5.03 33.60 5.17 34 35 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 35.00 5.32 135.58 5.48 36 37 36.64 5.15 36.62 5.31 36.59 5.47 ! 36.57 5.03 37 38 37.63 5.29 37.61 5.45 37.58 6.62 37.56 5.78 38 39 138.63 5.43 38.60 5.60 38.57 5.76 38.55 5.93 39 40 130.61 5.57 5.71 39-59 40. o8 5.74 39.56 5.91 39.53 6.08 40 41 41 40.60 5.88 40.55 6.06 40.52 6.24 42 41.59 5.85 41.57 6.03 41.54 6.21 41.51 6.39 42 43 42.. 58 5.98 42.56 6.17 42.53 6.36 42.50 6.54 43 1 44 43.57 6.12 43.54 6.31 43.52 6.. 50 143.49 6.69 44 4 5 44.56 6.26 44.53 6.46 44.51 6.65 144.48 6.85 45 46 45.55 6.40 45.52 G.60 45.49 6.80 145.46 7.00 46 47(46.54 6.54 46.5 6.74 46.48 6.95 146.45 7.15 47 48 47.53 6.68 47.50 6.89 47.47 7.09 47.44 7.30 43 49 48.52 6.82 48.49 7.03 48.46 7.24 48.43 7.45 49 50 3 2 49.51 6.96 49.48 7.17 49.45 7.39 49.42 7.61 50 Dep. 82 1 Lat. 3eg. Dep. Lat. D>ep. Lat. Dep. Lat. .2 Oil Deg. 81| 1 Deg. 8U Deg. TKAVERSE TABLE 89 8Dcg. 1 ni Deg. 8^ Deg. 8| Deg. 2 3 o ? Lat. Dep. T,at,. Dep. Lat. Dep. Lat. Dep. 51 50. 50 T.lo" 50.47 7.32 50.44 7.54 50.41 7.76 "51 (32 51.49 7.24 51.46 7.46 51.43 7.69 51.39 7.91 52 63 52.48 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 69 GO 59.42 8.35 59.38 8.61 59.34 8.87 59.30 9.13 60 61 60.41 8.49 60.37 8.75 60.33 9.02 60.29 9.28 61 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 60 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 73.08 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 85.0ti 12.71 85.00 13.08 86 87 86.15 12.11 86.10 12.48 86.04 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 91 89.12 90.11 12.53 89.07 12.91 89.01 13.30 88.95 89,94 13.69 90 91 12.66 90.06 13.06 90.00 13.45 13.84 92 91.10 12.80 91.05 13.20 90.99 13.50 190.93 14.00 92 93 92.09 12.94 92.04 i3.34 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 94 95 94.08 13.22 1 94.02. 13.63 93.96 14.04 93.89 14.45 9-5 96 95.07 13.36 195.01 13.78 94.95 14.19 94.88 14.00 96 97 96.06 13.50 i 96.00 1 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. S5 15.00 99 100 c i5 99.03 13.92 98.97 Dup. 14.35 93.90 Dep. 14.78 98.84 15.21 100 "i B "en Q Dep. Lat. Lat. Lat. Dep. Lat. 82 Dog. Oil Deg. 8U Deg. 8U Deg. 90 TRAVERSE TABLE 5 9 E >eg. 9i Deg. H Deg. 91 Leg 1 r. p Lat Dep. 1 Lat. Dep. Lat. Dep. i - i Lat. 1 Dep. 1. 0.99 0.16 Ik 99" 0.16 1 0.99 0.17 0.99 1 0.17 "l 2 1.9S 1 0.31 1.97 0.32 i 1.97 0.33 1.97 ; 0.34 2 3 2.96 0.47 2.96 0.4S 2.96 0.50 2.96 0.51 3 4 3.95 1 0.63 3.95 0.64 3.95 0.66 3.94 0.68 4 5 4.94 1 0.78 4.93 0.80 4.93 0.83 4.93 0.85 5 6 5.93 ; 0.94 5.92 0.96 5.92 0.99 5.91 1.02 6 7i. 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 1 7.89 1.32 7.88 1.35 8 9 8.89 1.41 8.88 1.45 8.88 ' of 8.87 ' 52 11 9.83 1.56 9.87 1.61 1 9.86 i 10.85 TsT S.8G 1.09 10 n 10.86 1.72 1 10.86 1.77 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 2.15 12.81 2.20 13 1-4 13.83 2.19 13.82 2.25 13.81 2.31 13.80 2 37 14 15 14.82 2.35 14.80 2.41 14.79 2.48 14.78 2.54 15 16 15.80 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 16.7f, 2.88 17 18 17.78 2.82 17.77 2.89 17.75 2.97 17.74 3.05 18 19 13.77 2.97 18.75 3.05 18.74 3.14 18.73 3.22 19 20 19.75 3.13 19.74 3.21 19.73 3.,30j 19.71 3.39 20 21 "21 i 20.74 3.29 20.73 3.33 20.71 3.47' 20.70 3.56 22 21.73 3.44 21.71 3.54 21.70 3.63 21.68 3.73 22 23 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 26 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 2S 27.66 4.38 27.64 4.50 1 27.62 4.62 i 27.00 4.74 28 29 i 28.64 4.54 23.62 4.66 23.60 4.79 ! 28.. 58 4.91 29 so' 29.63 31 1 30.62 4.69 29.61 4.82 29.59 4.951 29.57 .30.55 5.03 5.25 30 31 4.85! 30.80 4.98' 30.57 5.I2I 32 131.61 5.01 31.53 5.14 31.56 5.28 3 1.. 54 5.42 32 33 1 32.59 5.16 32.57 5.30 32 . 55 5.45 5.61 32.52 5.59 33 34 33.53 6.32 33.56 5.47: 33.53 33.51 5.76 34 35 34.57 5.43 1 34.54 5.63: .34.52 5.78! 34.49 5.93 35 36 35.56 5.63 i 35.. 53 5.79; 35.51 5.94 35.48 6.10 36 37 36.54 5.79 36.52 5.95 36.49 6.11 36.47 6.27 37 3S 37.53 5.94 37.51 6.11 37.48 6.27 37.45 6.44 38 39 38.52 6.10 33.49 6.27I 38.47 6.44 38.44 6.60 39 40 39.51 6.26 i 39.48 6.43; 39.45 6.60 39.42 6.77 40 "ir! 40.50 6.41 j 40.47 6.59 ] 40.44 6.77 i 40.41 6.94 41 42 41.48 6.57 41.45 6.75 1 41.42 6.92! 41.39 7.1 1 42 43 ! 42.47 6.73! 42.44 6.91 42.41 7.10, 42.38 7.28 43 44 1 43.46 6.88 1 43.43 7.07 43.40 7.26 ! 43.36 7.45 44 45 j 44.45 7.O4I 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 1 47 1 46.42 7.35 j 46.39 7.55 46.36 7.76 46.32 7.G6! 17 1 4& 147.41 7.51 47.33 7.72 47.34 7.92 47.31 8.13 18 49' 48.40 7.67 43.36 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 = 'l Dep. Lat. Dep. Lat. Lat.' Dep. Lat. 5 3 a 1 81] i Deg. 1 1 80| Deg. 1 1 801 Deg. m Deg. C3 .5 TIIAVKKSK TAF5LK 91 p 9 Beg. i 9i Deg. r 9^ Deg. i 1' 9.i Deg. V.' p 3 o eg. 1 Dep. i 791 L.t. Deg. Dep. Lat. Dep. Lat. 1 ' CO 5 1 80 J 791 Deg. 791 Deg. TKAVERSE TABLE. 93 c 51 10 Deg. lOi Deg. lOi Deg. 1 103 Deg. i .51 Lat. Dep. 8.86 Lat. Dep.j Lat. Dep. Lat. Dep , .50.10 i 9.51 50.23 50.19 9.08 ■160.15 ~9T29~| 52 51.21 9.03 51.17 9.25 j 51.13 9.48 51.09! 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 53.05 10.07 j 51 55 54.16 9.55 54.12 9.79 54.08 10.02 54.03 10.20 sri 56 55.15 9.72 1155.11 9.96 1 55.06 10.21 55.02 10.45 j 56 57 56.13 9.90 i 56.09 10.141 56.05 10.39 56.00 10.63 I 57 58 57.12 10.07 57.07: 10.32 1 57.03 10.57 56.98 10.82 58 59 58.10 10.25 58.06 10.50 58.01 10.75 57.96 11.00 59 60 59.09 10.42 59.04 10.68 1 10.59 60.03 10.85 1 59.00 10.93 58.95 59.93 11.19 11.38 00 ~61 61 60.07 59.98 11.12 1 62 61.06 10.77 161.01 11.03 60.96 11.30 60.91 11.56 62 63 62.04 10. 94161. 99 11.21 11.11162.93 11.39 61.95 11.48 61.89 11.75 63 64 63.03 62.93 11.66 62.88 11.94 64 65 64.01 11.29 j 63.96 11.57 163.91 1 11.85! 63.86 12.12 65 66 65.00 11.46 1164.9') 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 12.50 67 68 66.97 11.81 |! 66.91 12.10 1 66.86 12.39 66.81 12.68 68 69 67.95 11.98 1 67. 90 12.28 67.84 12.57! 67.79 12.87 69 70 71 68.94 69.92 12.16; 68.88 12.46! 12.33 1 69.87 12.63} 68.83 12.76; 12.94i 68.77 13.06 70 7] 69.81 69.75 13.24 72 70.91 12.50: 70.85 12.81 j 70.79 13.12 1 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 j 73.80 ; 13.35 73.74 13.67! 73.68 13.99 75 76 74.85 13.20 174.79 13.52 74.73 13.85 1 74.67 14.18 76 77 75.83 13.37 75.77 13.70 1 75.71 14.03 75.65 14.36 77 78 76.82 13.54 76.76 13.88 76.69 14.21 76.63 14.55 78 79 77.80 13.72 77.74 14.06 77.68 14.40 77.61 14.74 79 80 81 78.78 13.89 78.72 1 14.24 78.66 14.58 78.60 14.92 80 81 79.77 14.07 79.71 ' 14.41 79.64 14.76 179.58 15.11 82 80.75 ; 14.241 80.69 14.59 80.63 14.94 80.50 15.29 82 83 81.74 14.41 81.68 14.77 81.01 15.13 81.54 15.48 83 84 82.72 14.59 82.66; 14.95 82.59 15.31 82.53 15.67 84 85 83.71 14.76 83.64, 15.13 83.58 15.49 83.51 15.85 85 86 84.69 1 14.93 84.63 15.30 84.56 15.67 84.49 16.04 86 87 85.68 15.11 85.61 1 15.48 85.54 15.85 85.47 16.23 87 88 86.66 j 15.28 86.60 ! 15.66 86.53 16.04 8'3.46 16.41 88 89 87.65' 15.45 87.58' 15.84 87.51 16.22 87.44 1 16.60 89 •90 91 88.63; 15.63 88.56 ! 16.01 16.19 88.49 16.40 88.42 16.79 16.97 90 91 89.62* 15.80 89.55 89.48 16.58 89.40 92 90.60 15.98 90.53 16.37 90.46 16.77 90.39 1 17.16 92 93 91.59 16.15 91.52 16.55 91.44 16.95 91.37 1 17.35 S3 94 92.57 i 16.32 92.50 16.73 92.43 17.13 92.35 1 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 95.53 16.84 95.45 17.26 95.38 17.68 95.30 18.09 ' 97 98 96.51; 17.02 96.44 17.44 96.36 17.86 96.28 IS. 28 ' 98 99 1 97.50 17.19 97.42 17.62 97.34 18.04 97.26 1 18.47 ' 99 100 u i5 98.48 : 17.36 98.40 17.79 98.33 18.22 98.25 18.65 .00 c Dep. 1 Lat. Dep. j Lat. Dep. Lat. Dep. Lat. 80 Deg. 79} Deg. 791 Deg. 79i Deg. ! 94 TRAVERSK TABLE. 1 11 Deg. lU Deg. Hi J)eg. 11. f Deg. K SS 3 2 1 Lat. Dep. Lat. Dep. Lat. 0.98 Dep. 0.20 I Lat. 1 Dep. ' 1 " 0'.9S 0.19 0.93 0.20 ' 0.98! 0.20 2 1.96 0..38 1.96 0.39 1.96 0.40 1.96 0.41 2 3 2.94 0.57 2.94 0.59 2.94 0.60 i 2.94 i 0.61 d 1 3.93 0.76 3.92 0.73 3.92 0.80 . 3.92 : 9.82 1 5 4.91 0.95 4.90 0.98 4.90 1.00 4.90 1.02 5 6 5.89 i.u' 5.88 1.17 5.88 1.20 5.87 1.22 6 7 6.87 1..34, 0.87 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 9.82 1.91 9.8] 1.95 9.80 1.99 9.79 2.04 10 11 10.80 2.10 "10.79' 2.15 10.78 2.19 10.77 2.24 11 12 11.73 2.29 11.77 2.34 11.7-6 2.39 11.75 2.44 12 13, 12.76 2.48 12.75 2.54 12.74 2.59 12.73 2.65 13 14 j 13.74 2 67 13.73 2.73 13.72 2.79 13.71 2.85 14 15 ; 14.72 2.80 14.71 2.93 14.70 2.99 14.69 3.06 15 16 i 15.71 3.05 15.69 3.12 15.68 3.19 15.66 3.26 16 17 16.69 3.24 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 1 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 ~4.23 21 22 1 21.60 4.20 21.58 4.29 21.56 4.39 21.54 4.43 22 23 1 22.58 4.39 22.56 4.49 22.54 4.59: 22.52 4.63 23 24! 23.56 4.58 ,23.54! 4.63 1 23.52 4.78, 23.50 4.39 24 25 24.54 4.77 1 24.52 4.88 24.50 4.98 24.43 5.09 25 26 25.52 4.96 25.50 5.07 25.43 5.18 25.46 5. 3 J 26 27 20.50 5.15 26.43 5.27 26.46 5.38 26.43 5 . 50 27 28 j 27.49 5.. 34 27.46 5.46 27.44 5.53 27.41 5 . 70 28 29 28.47 5.53 28.44 5.66 23.42 5.78. 23.39 5.91 29 30,29.45 5.72 5.92 29.42 30.40 5.85 6.05 29.40 5.93' 29.37 6.11 30 31 30.43 30.38 6.18 30.35 -6:ji 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 i 34.33 6.83 34.30 6.93:134.27 7.1:5 35 36 35.34 6.87 135.31 7.02 35.28 7.18 35.25 7.33 36 37 36.32 7.06 136.29 7.22 36.26 7.33 36.22 7.53 37 38 ! 37.30 7.25 137.27 7.41 37.24 7.53 37.20 7.74 3S 39 i 33.23 7.4-1: 38.25 7.61 38.22 7.78 33.18 7.94 39 40 39.27 7.63 39.23 1 7.80 39.20 7.97 39.18 8.15 40 41 40.25 7.82 40.21 8.00 40.18 8.17 '40.14 8.35 41 42 41 23 8.01 41.19 8.19 41.16 8.37 41.12 8.55 42 43 42.21 8.20 1 42.17 8.39 42.14 8.57 42.10 8.76 43 44 143.19 8.40 43.15 8.53 43.12 8.77 j 43.08 8.96 44 4.^ [44.17 8.59 1 44.14 8.78 44.10 8.97 44.06 9.16 45 46 45.15 8.78 i 45.12 1 8.97 45.03 9.17 45.04 9.37 16 47 : 46.14 8.97 J46.10 9.17 46.06 9.37 46.02 9.57 47 48 ; 47.12 9. 16 ii 47.08 9.36 47.04 9 . 57 46.99 9.7S 43 49 148.10 9.35 143.06 9.-^0 4b . 02 9.77 47.97 9. 98 49 5^ 49.08 1 9.54 1 49J()^:_9^75jj49.00 9.97 Lat. 48.95 Dep. 10.18 Lat. 50 5 Dep. 79 1 Lat. Deg. Dep. : Lat. 1 Dep. 1 _ .. . V'^Desr. ' 8^ 1 Deg. 1 781 Deg. 1 TRA^KSE TABLE. 95 o o 11 Deg. IH De?. 1 IH Deg- 111 Deg, i C S fo* 3 ? Lat. Dep. Lat. Dep. Lat. Dep. Lat. 1 Dep. "61 50.06) 9.73 50.02 9.95 49.98 10.17 49.93 10.39 ~51 52 51.04 9.92 51.00 10.14 50.96 10.37 50.91 i 10.59 52 53 52.03 10.11 51.98 10.34 51.94 10.57 51.89 ! 10.79 53 5i .153.01 10.30 52.96 10.53 52.92 10.77 52.87 11.00 54 55 153.99 10.49 53.94 10.73 53.90 10.97 53.85 11.20 55 56 54.97 10.69 54.92 10.93 64.88 11.16 54.83 11.40 56 57 55.95 10.88 55.90 11.12 55.86 11.36 55.81 11.61 57 58 56.93 11.07 56.89 11.32 56.84 11.56 56.78 1 11.81 58 59 57.92 11.26 57.87 11.51 57.82 11.76 57.76 12.01 59 60 61 58.90 1 11.45 58.85 11.71 53.80 11.96 58.74 12.22 12.42 60 61 59.88 i 11.64 169.83 11.90 59.78 12.16 59.72 63 60.86 1 11.83 60.81 12.10 60.76 12.36 60.70 ! 12.63 62 63 61.84 1 12.02 61.79 12.29 61.74 12.56 1 61.63 12.33 63 64 62.82 ! 12.21 62.77 12.49 62.72 12.76 i 62.66 13.03 64 65 63.81 12.40 63.75 12.63 63.70 12.96 63.64 13.24 65 66 64.79 12.59 64.73 12.88 64.63 13.16 64.62 13.44 66 67 65.77 12.78 65.71 13.07 65.66 13.36 65.60 13.64 67 68 66.75 12.98 66.69 13.27 66.63 13.56 66.58 13.85 68 69 67.73 13.17 67.67 13.46 67.61 13.76 67.55 ' 14.05 69 70 71 68.71 69.70 13.36 68.66 13,66 68.59 13.96 68.53 ; 14.25 70 71 13.55 69.64 13.85 69.57 14.16 69.51 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 15.07 74 75 73.62 ! 14.31 73.56 14.63 73.49 14.95 73.43 15.27 75 76 74.60 14.50 74.54 14.83 74.47 15.15 74.41 15.48 76 77 75.59 1 14.69 75.52 15.02! 75.45 15.35 75 39 15.68 77 78 76.57 14.88 76.50 15.22: 76.43 15.55 76.37 15.83 78 79 77.55 1 15.07 77.43! 15.411 77.41 15.75 77.34 16.09 79 80 81 73.53 79.51 15.26 78.46 15.61 78.39 15.95 78.32 16.29 80 81 15.46 79.44 15.80 79.37 16.15 79.30 16.49 82 80.49 15.65 80.42 16.00 80.35 1 16.35 1 80.28 16.70 82 83 81.48 15.84 81.41 16.19 81.33 1 16.55 81.26 16.90 83 84 82.46 16.03 82.39 16.39 82.31 16.75 j 82.24 17.11 84 85 83.44 16.22 83.37 16.58 83.29 16.95 1 83.22 17.31 85 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- 85.18 1 17.72 87 83 86.. 38 16.79 86.31 17.17 86.23 17.54 86.16 17.92 83 89 87.36 16.98 87.29 17.36 87.21 17.74 87.14 18.12 1 89 90 91 88.35 17.17 88.27 17.56 88.19 17.94 88.11 18.33 1 90 89.33 17.36 89.25 17.75 1 89.17 18.14 89.09 13.53 91 92 90.31 ! 17.55 90.23 17.95 90.15 18.34 90.07 13.74 92 i 93 i 91.29 ! 17.75 91.21 18.14 91.13 18.54 91.05 18.94 931 94 92.27 1 17.94 92.19 18.34 92.11 1 18.74 92.03 19.14 94 95 93.25 18.13 93.17 13.53 II 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 97 95.22 18.51 95.14 18.92 1 95.05 19.34 94.97 19.75 97 98 96.20 ' 18.70 96.12 19.12 1 96.03 19.54 95.95 19.96 98 99 97.18 18.89 97.l0i 19.31 97.01 19.74 96.93 20.16 99 100 Q 1 98.16 19.08 93.081 19.51 97.99 19.94 97.90 20.36 lOOl Dep. I Lat. Dep. Lat. 1 Dep. 1 Lat. Dep. Lat. 5 79 Deg. 11 781 Deg. 78| Deg. 78J Deg. 96 TRAVFT?SK TAPLE. p 12Deg m Deg. m De^. 12| Deg. X 1 Lat. Dep. 0.21 Lat. Dep. Lat. Dep. Lat. 0.98 Dep. 0.22 1 '0.98 0.98 0.21 0.98 0.22 2 1.96 0.42 1.95 0.42 1.95 0.43 1.95 0.44 2 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 i 5 4.89 1.04 4.89 1.06 4.88 1.08 4.88 1.10 5 6 5.87 1.25 5.S6 1.27 5.86 1.30 5.85 1.32 S 2 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 9i 8.80 1.87 8.80 1.9i: 8.79 1.95 8.78 1.99 9 10 ! 9.78 2.08 1 9.77 2.12, 9.76 10.74 2.16 9.75 2.21 10 'll i 10.76 2.29 10.75 2.33 2.38 10.73 2.43 11 12 11.74 2.49 11.73 2.55 1 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 i 15.65 3.33 15.64 3.39 15.62 3.46 15.61 3. .53 1 16 17 16.63 3.. 53 16.61 3.61 16.60 3.68 16.58 3.75 1 17 18 17.61 3.74 17.59 3.82 17.. 57 3.90 17. C6 3.97 ; 13 19 18.. 58 3.95 18.57 4.03 18.55 4.11 18.53 4.19 ■ 19 20 19.. 56 4.16 19.54 4.24 19.53 4.33 19.51 4.41 ! 20 21 20.54 4.371 20 . 52 4.46 20.50 4.55 1120.48 4.63 1 21 22 21.52 4.57' 21.50 4.67 21.48 4.76 121.46 4.86 22 23 22.50 4.78 i 22.48 4.88 22.45 4.93 ; 22.43 5.08 23 24 23.48 4.99 1 23.45 5.09 23.43 5.19 123.41 5.30 24 25 24.45 5.20 1 24.43 5.30 24.41 5.41 t 24.33 5.. 52 25 2G 25.43 5.41 11 25.41 5.52 25.33 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 23 27.39 5.82 27.36 5.94 27.34 6.06. 27.31 6.18 23 29 28.37 6.03 28.34 6.15 23.31 6.23 1 28.23 6.40 ■ 29 30 129.34 6.24 1 29.32 6.37 29.29 6.49 29.26 30.24 6.62 ' 30 6.84 31 31 30.32 6.45' 30.29 6.58; 30.27 6.71 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 1 33 34 33.26 7.07 33.23 7.21 33.19 7.36 33.16 7.50 1 34 35 34.24 7.28 34.20 7.43 34.17 7.53 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 1 36.09 8.17 37 38 37.17 7.90 37.13 8.06 37.10 8.22! 37.06 8.39 33 39 38.15 8.11 39.11 8.27 33.08 8.44 33.04 8.61 39 40 139.13 8.32 39.09 8.49 8.70 39.05 8.66 8.87 39.01 8.83 40 41 i 40.10 8.52 40.07 40.03 39.99 9.05 41 42 ; 41.08 8.73 141.04 8.91 41.00 9.09 1 40.96 9.27 42 43 42.06 8.94 14?. 02 9.12 41.98 9.31 i 41.94 9.49 43 44 43.04 9.15 '43.00 9.34 42.96 9.52 j 42.92 9.71 1 44 45 44.02 9.36 43.93 9.55 43.93 9.74 1 43.89 9.93 , 45 46 44.99 9.56 44.95 9.76 44.91 9.96 44.87 10.15 1 46 47 45.97 9.77 45 . 93 9.97 ' 45.89 10.17 45.84 10.371 47 1 48 46.95 9.98 46.91 10.18 48.86 10.39 46.82 10.59 ig 49 47.93 10.19 47.88 10.40 47.84 10.61 47.79 10.81 49 5C i i 48.91 10.40 48.86 10.61 48.81 10.82 Lat. 43.77 i 11.03 50 Dep. Lat. Dep. 771 Lat. Deg. Dep. ; Dep. Lat. 78 Deg. 77| Deg. II m Deg. 5 TRAVERSE TABLE. 97 o P 51 12 Deg. 121 Deg. 12i Deg. 121 Deg. 1 C 1 "51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 49.89 TO^GO" 49.84 10.82 49.79 i 11.04 49.74 11.26 52 50.86 10.8] 50.82 11.03 50.77 11.25 50.72 11.48 52 53 51.84 11.02 .'^1.79 11.25 51.74 11.47 51.69 1 11.70 63 54 52.82 11.23 52.77 11.46 52.72 11.69 52.67 11.92 54 55 63.80 11.44 .53.75 ! 11.67 53.70 11.90 53.64 12.14 55 56 54.78 11.64 54.72 ' 11.88 54.67 12.12 54.62 12.36 5G 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 i 12.55 56.57 12.80 58 59 57.71 12.27 57.66 12.52 57.60 I5i.77 57.55 13.02 59 60 61 58.69 12.47 58.63 12.73 12.94 58.58 12.99 58.52 13.24 60 '61 59.67 12.68 59.61 159.55 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 1 64 65 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 G7 68 G6.51 14.14 66.45 14.43 166.39 14.72 66.32 15.01 68 69 67.49 14.35 67.43 14.64 ; 67.36 14.93 67.30 15.23 ogr 70 71 68.47 14.55 68.41 14.85 ! 68.34 15.15 68.27 15.45 70 71 69.45 14.76 69.. 38 15.06 169.32 15.37 69.25 15.67 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 15.49 171.27 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 15.91 173.22 16.23 73.15 16.55 75 76 74.34 15.80 74.27 16.13 174.20 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 79.23 16.84 17.19 79.08 17.53 79.00 17.88 81 82 80.21 17.05 80.13 17.40 80.06 17.75 79.98 18.10 82 83 81.19 17.26 81.11 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 18.04 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 84.83 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 88.93 19.10 87.87 19.48 87.78 19.86 90 89.01 18.92 19.31 88. 8i 19.70 88.76 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 91.95 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 1 95 96 93.90 i 19.96 1 93.81 20.37 93.72 20.78 93.63 21.19 j 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 1 99 99 96.84:20.58 96.75 21.01 96.65 21.43 96.66 21.85 99 100 97.91 120.79 97.72 21.22 97.63 21.64 97.53 22.07 Lat. 100 i Q Dep. Lat. Dep. Lat. Dep. Lat. Dep. 78 Deg. 1 77J Deg 77^ Deg. TTi Deg. 98 TRAVKKSE TABLE. I 13 Deg. 134 Deg. m Deg. I3.f Deg. n o Lat. 1 Dep. 1 1 Lat. Dep. Lat. Dep. Lat. Dep. 0.97 0.23 0.97 0.23 0.97 0.23 0.97 0.2i 2 1.95 0.45 1.95 0.46 1.95 0.47 1.94 0.48 2 3 2 92 0.67 2.92 0.69 2.92 .0.70 2.91 0.71 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 1.15 4.86 1.17 4.86 1.19 5 6 5,85 1.35 5.84 1.38 5.83 1.40 5.83 1.43 6 7 6.82 1.57 0.81 1.60 6.81 1.63 6. 80 1.66 7 8 7.80 1.80 7.79 1.83 7.78 1.87 7.77 1.90 8 13.30 49 50 (48.30 12.94 48.24 13.15 ■43.18 1 13.36 ||48.12 13.57 50 6 c "1 1 j Dep. 1 Lat. ij Dep. Lat Dep. Lat. ! Dep. 1 Lat. 75 Deg. 1 74i Deg. 741 Deg. 74.i Deg. s i TRAVEKSE TABLE. 103 P. P 3 o a "51 15 Dcg. 15| Deg 1 15^ Deg. 15J Deg. 1 51 Lat. Dep. 13.20 Lat. Dep. Lat. Dep. 49.15; 13.63' Lat. Dep. 49.26 49.29 13.41 49.09 13.84 52 50.23 13.46 50.17 13.68 50.11 13.90, .50.05 U.ll 52 53 51.19 1 13.72 51.13! 13.941 51.07 14.16! 51.01 U,39 53 64 52.16 1 13.98 52.10 ' 14.20 52.04 14.43 51 97 14. 6G 54 65 53. IS 14.24 53.06 ! 14.47 53.00 14.70 52.94 14.93 5£ 56 54. U9 , 14.49 ii 54.03 14.73 53.96 14.97 .53.90! 15.20 56 67 55.06 i 14.75 1 ,54.99 1 14.99, 54.93 15.23 1 54.86 \ 15.47 57 68 56.02 i 15.01 55.96 1 15.26 1 55 . 89 15. .50 1 55.82 i 15.74 58 59 56.99 115.27 56.92 15.52 \ 56.85 15.77 56.78 . 16.01 59 60 57.96 ' 15.53 57.89 i 15.78 | 57.82 16.03 1 57.75 1 16.29 60 61 58.92 ; 15.79 1 58.85 1 16.04! 58.78 1 18.30 1 58.71 i 16.56 61 62 59.89 16.05 59.82 j 16.31 i 59.75 16.57 1 59.67 16.83 62 63 60.85 16.31 60.78 i 16.57 1 60.71 16.841 60.63 17.10 63 64 61.82 16.56 61.75 16.83; 61.67 17.10! 01.60 17.37 64 65! 62.79 16.82 62.71 17.101 62.64 17.371 62.56 17.64 65 66 , 63.75 17.08 63.68 17.35 1 63.60 17.64 63.52 17.92 66 67 1 64.72 17.34 64.04 17.62 1 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 66.571 18.15; 66.49 18.4-11 66.41 18.73 69 70 1 "^1 67.61 18.12 67.54: 18.41 1 68.-50 i 18.68 i 67.45 18.711 67.37 19.00 70 71 68.58 18.38 68.42 i 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 1 70.43 , 19.20, 70.35' 19.51 1 70.26 19.82 73 74 71.48 19.151 71.39 19.46 1 71.31 19.78; 71.22 20.09 74 75 72.44 19.41 72.36 19.73 1 72.27 20.041 72.18 120.86 75 76 73.41 19.67 73.32 19.99, 73.24 20.31 1 73.15 1 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 1 75.25 20.52 ; 75.16 20.84 1 75.07 21.17 78 79 76.31 20.45 76.22 1 20.78 1 76.13 121.11 1 76.03 21.44 79 80 81 77.27 20.71 77.18 : 21.04 78.15 '21.31 i 77.09 21.38 1 77.00 21.72 80 "SI 78.24 20.96 78.05 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 8'1- 85 82.10 22.00 '82.01 22.36 81.91 22.72 181.81 23.07 85 86 83.07 22.26 82.97 22.62 82.87 22.98 1 82.77 23.34 86 87 84.04 22.52 ; 83.94 122.88 83.84 23.25 83.73 23.62 87 88 85.00 22.78. 84.90 23.15 84.89 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 23.29 186.83 23.67 1 86.73 24.05 ' 86.62 24.43 90 91 87.90 ! 23.55 87.80 123.94 '87.69 24.32 87.58 24.70 92 88.87 123.81 88.76 24.20 '88.65 24.59 : 88.55 24.97 92 93 89.83 124.07 89.73 1 24.46 189.62 24.85 i 89.51 25.24 93 94 90.80 24.33 90.69 ; 24.72 190.58 '25.12 190.47 25.52 94 95 91.76 24.59 91.65 124.99 1 91 54 25.39 : 91.43 ; 25.79 9.^ 96 92 73 124.85 92.62' 25.25 192.51 ,25.65 92.40 120.06 96 97 93.69 25.11 93.58 125.51 93.47 25.92 ' 93.36 26.33 97 98 94.66 25.36 194.55 125.78 94.44i26.l9l9t. 32 26.60 9S 99 95.03 25.62 ■95.51 126.04 95.40 i 26.46 ' 95.28 26.87 90 100 § .2 96.59 25.88 36.4.S '26.30 96.36 126.72 : 96>25 | 27.14 100 i ~ Dep. Lat. Dep. Lat. Dep. Lat. | Dep. Lat. 75 Deg. 1; 741 Ueg. 74^ Deg. U\ Deg. 104 TRAVERSE iABLE. II C : 16 Deg. \\ iQk Deg. 16^ Deg. 16| Deg. 1 a 1 Lat. ' ! \ m Dep. ! 1 Lat. Dep. Lat. Dep. ~0 23"! Lat. Dep. ! 0.96 0.28' 0.96 0.28 0.96 0.96 0.29 2 1 92 55 i 1.92 0.56 1.92 0.57 1.92 0.58 2 3 2.S8 C 83 'i 2.88 0.84 2. 83 0.85 2.87 0.86 3 4 3.85 1.10: 3.84 1.12 1 3.84 i.u; 3.83 1.15- i 5 4.81 1.33 i 4.80 1.40 4.79 1.42! 4.79 1.44 1 5 6 5.77 1.65: 5.76 1.63 5.75 1.70. 5.75 1.73 1 6 7 6.73 1.93i 0.72 1.96 1 6.71 1.99 1 0.70 2.03 7 8 7.69 2.21 : 7.68 2.24 ! 7.67 2.27 1 7.66 2.31 8 9 8.65 2.48: 8.64 2.52 i 8.63 2.56 1 8.62 2.59 9 10 11 9.01 2.76 9.60 2.80 1 9.59 2.84! 9.58 2.83 10 10. .57 3.03 ! 10.56 3.08 : 10.55 3.12 10.53 3.17 1 IT 1 o 11.54 3.31 'i 11.52 3.36 : 11.51 3.41 11.49 3.46 12 13 12.50 3.. 58:1 12.48 3.64 i 12.46 3.69 12.45 3.75 13 14 13.46 3.86 i 13.44 3.92 ll 13.42 3.98 13.41 4.03 14 15 14.42 4.13 1 14.40 4.20 1 14.33 4.26: 14.36 4.32 15 16 15.38 4.41 !| 15.36 -4.48 15.34 4.54 i 15.32 4.61 16 17 16.34 4.69 i 16.32 4.76 16.30 4.83 ;i 16.28 4.90 17 18 17.30 4.961 17.28 5.04 17.26 5.11 ! 17.24 5.19 18 19 18.23 5.24: 18.24 5.32 18.22 5.40 1 18.19 5.48 19 20 19.23 5.51! 19.20 5.60 19.18 5.08 :' 19.15 5.76 6.05 20 21 21 20.19 5.79 1 20.16 5.88 20.14 5.96 1 20.11 23 21.15 6.06 1| 21.12 6.16 21.09 6.25 21.07 6.34 22 23 22.11 6.34!, 22.08 6.44 122.05 6.53 22.02 6.63 23 21 23.07 6.62; 23.04 6.72 123.01 6.82, 23.98 6.92 24 25 24.03 6.89 24.00 7.00 23.97 7.10 23.94 7.20 25 26 24.99 7.17 24.96 7.28 ; 24.93 7.38 24.90 7.49 26 27 25 . 95 7.44 25.92 7.56 125.89 7.67. 25.85 7.78 27 28 26.92 7.72 26.88 7.84 26.85 7.95 1 26.81 8.07 23 29 27.83 7.99 27.84 8.11 27.81 8.24 27.77 8.36 29 30 31 23.84 8.27 23.80 8.39 28.76 8.53 I 28.73 8.65 30 29.80 8.54 29.76 8.67 29.72 8.80 29.63 8.93 31 32 30.76 8.83i 30.72 8.95 30.63 9.09 30.64 9.22 32 33 31.73 9.io; 31.68 9.23 31.64 9.37 31.60 9.51 33 34 33.68 9.37 33.64 9.51 33.60 9.06 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 1 34.56 10.07 34.53 10.22 31.47 10.38 36 37 35.57 10.20 1 35 53 10.35 35.43 10.51 35.43 10.66 37 38 36.53 10.47 35.48 10.63 36.44 10.79 36.. 39 10.95 33 39 37.49 10.75 37.44 10.91 37.39 11.08 37.35 11.24 39 40 41 38.45 39.41 11.03 38.40 11.19 38.35 11.36 38.30 11.53 40 41 11.30 39.36 11.47 39.31 11.64 39.30 11.82 42 40.37 11.58 40.33 11.75 40.27 11.93 40.23 12.10 42 43 41.33 11.85 41.23 12.03 41.23 13.21 41.18 12.39 1 43 41 42.30 13.13 13 . 24 12.31 42.19 12.50 43.13 13.68 ; 44 1 45 43.23 13.40 43.20 12.59 43.15 12.78 43.00 13.07 45 1 46 44.22 12.63 44.16 12.87 44.11 13.08 44.05 13.23 1 46 47 45.18 12.95 45.12 13.15 45. OG 13.35 45.01 13.55 47 48 46.14 13.23 46.03 13.43 148.02 13.63 45.96 13.83 1 48 49 47.10 13.51 47.04 13.71 146.93 13.93 46.92 14.12 1 49 50 48.03 13.78 43.00 13.99 47.91: 14.20 47.83 14.41 1 50 6 Dep. j Lat. Dep. i 731 Lat. Deg. Dep. Lat. Dep. Lat. i i '1 74 Deg. ■^ Deg. 734 Deg. TKA.VERSE TABLE. 105 ^1 p 3 o o 5l i 16 Deg. m Oeg. 16A Deg i 161 Deg. K 3 o CD 51 Lat. ! Dep. j 49.02 14.06 Lat. Dep. 14.27 Lat. Dep. Lat. Dep. 48.96 48.90 14.48 48.84 1 14.70 52 1 49.99 14.33 49.92 14.55 49 86; 14.77 49.79 14.99 52 53. .50.95 14.61 50.88 14 83! 50.82 i 15.05] 50.75 15.27 53 54 51.91 1 14.88 1 51.84 15 11 !! 51.78 1 15.34! 51.71 15.56 54 55 1 52.87 ' 15.16 1 52.80 15.39 !i 52.74 1 15.62 1 52.67 15.85 55 56 .53.83 15.44 53.76 15.67 53.69 15.90 1 53.62 16.14 56 57 .54.79 '15.71 1 54.72 15.95 54.65 16.19 54.53 16.43 57 53 55.75 15,99 1 55.68 16.23 55.61 16.471 55.54 16.72 58 59 56.71 16.26 56.64 16.51 56.57 1 16.76! 56.50 17.00 59 60 1 "61 ' 57.68 16.54| 57.60 .58.56 16.79 1 57.53} 58.49 17.04] 17.32 i 57.45 17.29 60 '61 .58.64 16.81 1 17.07 58.41 1 17.58 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 1 17.64 61.44 17.91 61.36 18.18i 61.28 18.44 64 65 ; 62.48 1 17.92 i 62.40 18.19 62.32 18.46 1 62.24 18.73 65 66 163.44 18.19 1 63.33 18.47 63.28 18.74 ; 63.20 19.02 60 67 164.40 18.47! 64.32 18.75 64.24 19.03 164.16 19.31 67 68 65.37 18.741 65.28 19.03 65.20 19.31 ; 65.11 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 ! 19.29 , 67.20 68.16 19.59 19.87 67.12 68.08 19.88 67.03 20.17 70 71 68.25 19.57, 20.17 1 67.99 20.46 72 69.21 19.85 69.12 20.15 69.03 20.45 i 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 21.221 73.92 21.55 73.83 21.87 73.73 22.19 77 78 74.98 21.50 1 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 77.86 22.05 76.80 77.76 22.39 76.71 22.72 76.61 23.06 23.34 RO 81 22.33 22.67 77.66 23.01 77.56 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 S3 84 80,75 23.15 80.64 23.51 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 66 87 83.63 23.98 83.52 24.35 83.42 24.71 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 80.51 24.81 86.40 25.18 86.29 25.56 86.18 87.14 25.94 26.23 90 91 87.47 25.08 87.36 25.46 87.25 1 25.85 92 ,88.44 25.36 88.32 25.74 88.21 26.13 ^88.10 26.51 92 93 189.40 25.63 89.28 26.02 89.17 26.41 89.05 26.80 y3 94 !90.£6 25.91 90.24 26.30 90.13 26.70 : 90.01 27.09 9i 95 91.32 26.19 91.20 i 26.58 91.09 26.98 90.97 27 .38 95 96 02.28 26.46 92.16 26.86 ; 92.05^27.27 91.93 27.67 9G 97 93.24 26.741 93.12 27.14! 93.01 27.55 02.88 27.95 i^'i 98 194.20 27.011 94.08 27.42 1 93.96 27.83 93.84 28.24 [lis 99 195.16 127.29 95.04 27.70; 94.92 28.12 ,94.80 28.53 99 100 Q 196.13 Dep. 27.56' 96.00 [27.98 95.88 128.40 195.76 28.82 ' 3 00 a ! ° Lat. Dep. Lat. Dep. 1 Lat. Dep. Lat. 74] Deg. 73f Deg. 73i Deg. ij '^M Deg. 106 TRAVERSE TABLE. 17 Deg. 17i Deg. ni Dog. 17| Deg. o o ,•1 J. at. Dep. j Lat. Dep. Lat. 0T95 Dep. 0.30 Lat. Dep. 1 0.95 0.29 1 0.95 0.30 0.95 0.30 2' 1.91 0..58 1.91 0.59 1.91 0.60 1.90 0.61 2 3 2.87 0.88 2.87 0.89 2.86 0.90 1 2.86 0.91 3 4 3.83 1.17 3.82 1.19 3.81 1.20 3.81 1 22 4 5; 1.78 1.46 4.78 1.48 4.77 1..50 4.76 1 . .52 5 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 8 57 2.74 9 10 9.56 2.92 9.55 2.97 9.54 3.01 9.52 3.05 10 11 "ii 10.52 3 . 22 10.51 3.26 10.49 3.31 10.48 3.35 12 11.48 3.51 11.46 3.56 11.44 3.61 11.43 3.66 12 13 12.43 3.80 12.42 3.85 12.40 3.91 12.. 38 3.96 13 14 13.39 4.09 13.37 4.15 13.35 4.21 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 15.26 4.81 15.24 4.88 16 17 16.26 4.97 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 5.40 18 19 18.17 5.56 18.15 6 63 1 18.12 5.71 18.10 5.79 19 20 21 19.13 5.85 19.10 20.06 .5.93 6.23! 19.07 20.03 6.01 6.31 19.05 6.10 20 21 20.08 6.14 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^.92 7.12 22.89 7.22 22.86 7.32 24 U 23.91 7.31 23.88 7 41 23.84 7.52 23.81 7.62 25 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 XC^.7S 8,19 26.74 8.30 '46.70 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 3f 28 . 69 29.65 S.77 9.06 28.65 29.61 8.90 28.61 9.02 28.57 9.15 30 31 9.19 29.57 9.32 29.52 9.45 32 30.60 9.36 30.56 9.49 30.52 9.62 30.48 9.76 32 33 31.56 9.65 31. .52 9.79 31.47 9.92 31.43 10.06 33 34 82.51 9.94 32.47 10.08 32.43 10.22 32. C8 10.37 34 35 33.47 10.23 33.43 10.38 33.38 10.52 33.33 10.67 35 36 34.43 10.53 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 11.27 36.24 11.43 36.19 11.58 38 39 37.30 11.40 37.25 11.57 37.19 11.73 37.14 11.89 39 40 41 38.25 11.09 38.20 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 12.28 40.11 12.45 40.06 12.63 40.00 12.80 42 43 41.12 1 12.57 41.07 12.75 41.01 12.93 40.95 13.11 43 44 ,42.08 12.86 42.02 13.05 41.96 13.23 41.91 13.41 44 45 43.03 ! 13.16 42.98 13.34 42.92 13.53 42.86 13.72 45 46 13.99 : 13.45 43.93 13.64 43.87 13.83 43.81 14.02 46 47 ,44.95 1 18.74 44.89 13.94 44.82 14.13 44.76 14,33 47 48 145.90 : 14.03 45.84 14.23 45.78 14.43 45.71 14 63 48 49 146.86 14.33 46.80 14.. 53 46.73 14.73 46.67 14.94 49 50 *f5 47.82 14.62 47.75 Dep. 14.83 47.69 15.04 Lat. 47.62 15.24 50 ■J Dap. Lat. Lat. Dep. Dep. Lat. 73 Deg. 721 Deg. 72| Deg. 72i Deg. TRAVERSE TABLE. 107 o P P '51 1 17 Deg. 1 ] in Deg. 1 1 17A Dfcg. 171 De.. 3 Lat. Dep. 14.91 1 Laf.. 48.71 Dep. Lat. Dep. \ Lat. Dep. 48.77 15.12 48.64 15.34 1 48.57 15.55 51 52 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 15.91 50.48 16.16 5^ 5i 51.64 15.79; 51.57 j 16.01 51.50 16.24 1 51.43 16.46 54 55 52.60 16.08 '■ .52.53 I 16.31 .52.45 1 16.54 52.38 16.7?! 55 56 53.55 16.37 53.48 ; 16.61 53.41 16.84 1.53.33 17.07, 56 1 57 54.51 16.67 .54.44 16.90 .54.36 17.14 54.29 17.33 i 57 1 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 58.33 17.54 1 57.30 i 17.79 57.22 18.04 5?'.]4 53.10 18.29 60 17.83 58.26 . 18.09 58.18 1 18,34 18.60! 61 62 59.29 18.13 1 59.21 13.39 j 59.13 18.64 59.05 18.90 j 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 1 61.04 19.25^ 60.95 19.51 64 65 62.16 19.00 62.08 19.28 61.99 : 19.55 01.91 19.82 65 66 63.12 19.30 63.03 19.57 62.95 i 19.35" 62.86 20.12 66 67 64.07 19.59 63.99 19.87 63.90 120.15 63.81 20.43 67 6S 65.03 19.88 64.94 120.16 ' 64.85! 20.45 64.76 20.73 68 69 65.99 20.17 65.90 120.46 | 65.81 20.75 65.72 21.04 69 70 71 6'S . 94 67.90 20.47 68.85 120.75 ! 67.81 1 21.05 1 66.76 21.05 1 67.71 121.35, 66.67 21. .34! 70 20.76 67.62 21.65 71 72 68.85 21.05 63.76 21.35 68.67 21.65! 63.57 21.95 72 73 69.81 21.34 69.72 121.65 69.62 121.95 1 69.52 22.26 73 74 70.77 21.64 70.67 121.94 70.58 22.25 ; 70.43 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 72.43 22.85 72.33 23.17 78 77 73.64 22.51 73.54 22.83 1 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 24.03 79 80 76.50 23.39 76.40 23.72 76.30 24.06 76.19 24.39 1 80 81 77.46 23.68 ; 77.36 24.02 77.25 24.36 77.14 124.69 | 81 &2 78.42 23.97 78.31 24.32 78.20 24.66 78.10 25.00 1 82 83 79.37 24.27 1 79.27 24.61 79.16 25.96 179.05 1 25.30 1 83 84 80.33 ! 24.56 180.22 24.91 80.11 25.26 80.00 25.61 1 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 i 86 87 83.20 25.44 183.09 25.80 82.97 26.16 83.93 26.46 84.88 26.76 82.86 26.52 87 88 84.15 25.73 184.04 26.10 83.81 26.83 88 89 35.11 26.02!! 85.00 26.39 i 84.76 27.13 89 ■ 90 91 86.97 87.02 26.31 126.61 i 85.95 26.69 26.99 '85.83 27.06 86.79:27.36 [85.72 1 86.67 27.44 1 90 186.91 27.74 1 91 92 1 87 98 1 26.90 1 87.86 27.28 87.'?'4| 27.66 87.62 28.05 1 92 '.i2 '88.94 127.19 i 88.82 27.58 88.70 27.97 188.57 2?. 35! 93 94 89.89 27.48 89.77 27.87 89.65 28.27 89.53 i 23.66 94 95 190.85 27.78 1 90.73 23.17 90.60 28. ,57 190.48 28.96 1 95 96 ■91 8! 28.07 1191.63 28.47 91.56 28.87 ; 91.43 29.27 1 96 97 i 92.7c 23.36 Ij 92.64 23.76 192.51 29.17 192.33 29.57! 97 98 '93.72 28.65 93.59 29.06 '93.46 29.47 i 93.33 29.88 ! 9S 99 94.67 28.94 ij 94.55 29.36 94.42 29.77 94.29 30.18 i 99 100 |^.63_ 29.24 !|i)5. 50 29.65 95.37 30.07 95.24 30.49 100 6 c Dep. Lat. Dop. i Lat. Dep. 1 Lat. \ Dep. Lat. c ci 73 Deg. i 1 72| Deg. 72^ Deg. 72i Deg. 108 TRAVERSE TABLE. 5 18 Deg. m Derr. 1 1 18^ Deg. 181 Deg. ^ p 1 ,' tr Lat. Dep. Lat. 1 Dep. Lat. Dep. Lat. Dep. r3 ? 1 "0:95" 0.31 0.95 0.31 0.95 0.32 0.95 0.32 1 2 ].90 0.62 1.90 0.63 1.90 0.63 1.89 0.6i 1 * 3 2.85 0.93 2.85 0.94 2 84 0.95 2.84 0.96 3 4 3.80 1.24 3.80 1.25 3 79 1.2? 3.79 1 29 i 4 5 4.76 J.. 55 4.75 1.57 4.74 1.59 4.73 1.61 ' 5 : 6 5.71 1.85 5.70 1.88 5.69 1.90 5.68 1.93 6 J 7 6.66 2.16 6.65 2.19 6.64 2.22 6.63 2.25 7 8 7.61 2.47 7.60 2.51 7.59 2.54 7.58 2.r-7 8 9 8.56 2.78 i S.55 2.82 8.53 2.86 8.52 2.89 1 9 10 9.51 3.09 9.50 3.13 9.48 3.17 9.47 3.21 10 11 10 ,46 3.40 16.45 3.44 10.43 1 3.49 10.42 " 3.54" fl 12 11.41 3.71 11.40 3.76 11.38 i J 81 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 4.4il|13.26 4.. 50 14 15 14.27 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 1 5.08 1: 15.15 5.14 16 * 17 16.17 5.25 16.14 5.32 16.12 5.39 , 16.10 5.46 17 18 17.12 5.56 17.09 5.64 17.07 5.71 1 17.04 5.79 18 i 19 18.07 5.87 18.04 5.95 18.02 6.03 ! 17.99 6.11 19 i 20 21 19.02 6.18 ]'3 99 6.26 18.97 6.35 i 18.94 19.89" 6.43 ; 20 6.75 i 21 19.97 6.49 19.94 6.58 19.91 6.66 22 20.92 6.80 20.89 6.89 20.86 6.98 120.83 i 7.07 [ 22 I 23 21.87 7.11 |21.84 7.20 21.81 7.30 i 21.78 7.39 ! 23 24 '22.83 7.42 i 22.79 7.52 22.76 7.62! 22.7.3 7 71 24 25 23.78 7.73 23.74 7.83i 23.71 7.93 1 23.67 8.04 ' 25 26 24.73 8.03 24.69 8.14 2^1.66 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 1 26.59 8.77 26.55 8.88 26.51 9.00 28 29 27.. 58 8.9^1 27.54 9.08 27.50 9.20 1 27.46 9.32 £9 30 3J 28.53 9.27| 28.49 29.44 9.39 28.45 9.52 9.84 28.41 9.64 30 29.48 9.58 9.71 29.40 29.35 9.96 Ml 32 30.43 9.89 30.39 10.02 30.35 10.15 1 30.30 10.29 32 33 31.38 10.20 31.34 10.33 31.29 10.47 31.25 10.61 S3 34 .32.34 10.61 32.29 10.65 32.24 10.79 32.20 10.93 34 35 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 11.42 134.09 11.57 36 37 35.19 11.43 35.14 11.59 35.09 11.74 35.04 11.89 37 38 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 38.98 12.37 36.93 12.54 39 40 38.04 12.36 37.99 12.53 .37.93 12.69 13.01 37.88 38.82 12.86 13.18 40 41 41 138.99 12.67 38.94 12.84 38.88 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 44 45 42.80 13.91 42.74 14.09 42.67 1 U.28 42.61 14.40 45 40 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 44.57 1 14.91 44.51 15.11 47 18 45.65 14.83 45.59 15.03 45.52 15,23 45.45 15.43 48 19 46.60 15.14 46.54 15.35 46.47 1 15.55 1 46.40 15.75 49 51) 47.55 g ; Dep. 15.45 Lat. )eg. 47.48 J^5 , 6 6_ 47.42 15.87! 47.35 Dep. 16.07 La.. _50 1 Dep. •''11 Lai. Deg. Dep. Lat. 1 .1 72 1 7H Deg. 7U Deg. TItAVETlSE TAI5LK 109 v3? p ? 18 Deg. m Deg. 1 l«i Deg. 181 Deg. 7 Lat. ! Dep. Lat. Dep. j Lat. 1 Dep. 1 Lat. Dep. i 51 i8.50 15.76' 48.43 15.97 .48.36 116.18 48729^116739 51 52 49.45 16.07 1 49.33 16.23 49.31 1 16.50 49.24 '16 71 52 53 50.41 16.38 j 50.33 16.60 50.26 i 16.82 50,19 17.04 53 54 51.36 16.69 51.23 16.91 51.21)17.13 51.13 17.36 54 65 52.31 17.00 52.23 17.22 52.16 . 17.45 52.03 117.08 oF. 56 53.28 17.30 i 53.13 17. .54 53.11 17.77, 53.03 ; 18.00 56 57 .54.21 ; 17.61 1 54.13 17.85 i 54.05 ; 18.09 I 53.93 113.32 57 53 55.10 1 17.92 55.08 18.16 55.00 : 18.40; 54.92 i 18.6-i ' 53 59 .56.11 !l8.23 53.03 18.48 55.95 18.72 ^ 55.87 1 18.96 ■ 59 GO 61 57.06 1 18.541 36.93 18.79 .56.90 19.04 1 56.82 ! 19.29 60 i 58.01 i 18 85 ! 57.93 1 19.10 57.85 19.36 57.76 19.61 '61 62 .58.97 10.16 53.68 j 19.42 58.80 ! 19.67 58.71 19.93 62 63 59.92 19.47 1 59.83 1 19.73 59.74 ; 19.99 59.66 20.25 : 63 61 60.87: 19.78 60.73 1 20.04 60.69 20.31 60.60 : 20.57 i G4 65 61.82 ' 20.09 61.73 1 20.36 61.64 20.62 1 61.55 120.89 65 66 62.77 20.40 62.68 i 20.67 62.59 120.94: 62.50 21.22 : 66 67 63.72 20.70 63.63 20.98 i! 63.54 121.26 63.44 21.54 67 68 64.67 21.01 64.58 21. .30 64.49 21.53 64.39 21.86 1 63 69 65.62 21.32 65.53 21.61 65.43 21.89' 65.34 122.13 . 69 70 71 66.57; 21.63 66.43 21.92 66.33 22.21 66.29 j 22.50 : 70 67.23 122.82 : 71 67.53 21.94 67.43 22.23 67.33 22.53 72 63.48 122.25 63.38 22.55 68.28 22.85 68.18 123.14 , 72 73 69.43 i 22.56 i 69.. 33 22.86 69.23 23.16 69.13 23.47 i 73 74 70. 3S ! 22.87 70.28 23.17 70.18 23.48 70.07 123.79 74 75 71.33 i 23.18 71.23 23.49 71.12 23.80 71.02 24.11 75 j 76 72.28 i 23.49 72.18 23.80 72.07 24.12 71.97 24.43 76 77 73 23 '23.79 73.13 24.11 1 73.02 24.43 i 72.91 24.75 77 78 74.18 24.10 174.03 24.43 i 73.97 24.75 I 73.86 25.07 78 79 75.13 24.41 175.03 24.74! 74.92 25.07 74.81 25.39 ' 79 80 81 76.08 77.04 24.72! 25.03 1 75.98 25.05 i 75.87 25.33 : 75.75 25.72: SO 76.93 25.37! 76.81 125.70 76.70 26.04 81 82 77.99 25.34 177.88 25.68' 77.76 126.02 77.65 26.36 82 83 78.94' 25.65 !| 78.83 25.99 78.71 26.34 78.60 26.68 , S3 84 79.89; 25.96 !| 79.77 26.31 79.66 26.65: 79.54 27.00 ; 84 85 80.84; 26.27 li 80.72 28.62 80.61 26.97 80.49 27.32 ' 85 86 81.79, 26.58 !|81. 67 26.93 i 81.56 27.29 , 81. M 27.64 86 87 82.74! 26.88 j 82.62 27.25 1 82.50 27.61 82.33 27.97; &7i 88 83.69 ! 27.19 1 83.57 i 27.56 i 83.45 27.92 83.33 28.29 i 88 1 89 84.64: 27.50 184.52 '27.87! 84.40 1 23.24 84.23 28.61 ! S9e 90 91 85.60; 27.81 ! 85.47,23.18! 85.35 123.56 , 85.22 28.93 90 86.55 128.12 1 86.42 1 23.50: 86.30 123.87: 86.17 29.25 91 92 87.50 128.43 I 87.37 28.81 1 87.25 : 29.19 87.12 29.57 I 92 93 88.45 128.74 1 88.32 29.12; 88.19 29.51 88.06 29.89 ! 93 94 89.40 129.05 89.27 1 29.441 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 91.171 30.06 91.04 30.46. b0.91 30.86 96 97 92.25 29.97 92.12 1 30.38 i 91.99 30.78 91.85 1 31.13 97 93 93.20 30.28 93.07| 30.69 ' 92.94 31.101 92.80 131.50 9.:^ 99 94.15, 30.59 94.02 31.00 ; 93.88 31.41 1 93.75 131.82 99 IOC c s 95.11 i 30.90 94.97 1 31.32 j 94.83 131.73 94.69 132 14 .00 Dep. 1 Lat. Dep. Lat. Dep. ! Lat. | Dep. Lat. "i 72 Deg. 711 Deg. 'H Deg. TH Deg. .a Q 110 TRAVFRSE TABLE. ST 9 o o 19 Deg. m Deg. 1 19^ Deg. 1 191 Deg. » fD Lat. Dep. 0.33 Lat. Dep. 0.33 Lat. Dep. Lat. 0.94 Dep. 0,34 1 , 0.95 0.94 0.94 0.33 2l 1.89 0.65 1.89 0.66 1.89 0.67 1.88 0.68 2 2 2.84 0.98 2.83 0.99 2.83 l.UO 2 82 1.01 3 4 ; 3.78 1.30 3.78 1.32 3.77 1.34 3.76 I 35 4 6 4.73 1.63 4.72 1.65 4.71 1.67 4.71 1.69 5 6 6.67 1.95 5.66 1.98 5.66 2.00 5.65 2.03 8 7 6.62 2.28 6.01 2.51 6.60 2.34 6.59 2.37 7 8 7.56 2.60 7.55 2.64 7.54 2.67 7.53 2.70 8 9 8.51 2.93 8.50 2.97 8.48 3.00 8.47 3.04 9 10 9.46 3.26 9.44 "10.38 3.30 9.43 3.34 9.41 3.. 38 3.72 10 ~11 11 10.40 3.58 3.63 10.37 3.67 10.35 13 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 5.28 15.08 5.34 15. oe 5.41 16 17 16.07 5.. 53 16.05 5.00 16.02 5.67 16 00 5.74 17 18 17.02 5.86 16.99 5.93 10.97 6.01 j 16.94 6.08 18 19 17.96 6.19 17.94 6.26 17.91 6.34 17.88 6.42 19 20 18.91 6.51 18.88 6.59 18.85 19.80' 6.68 1 18.82 19.76 6.70 7.10 20 21 21 19.86 6T84 19.83 6.92 7.011 22 20.80 7.16 20.77 7.25 , 20.74 7.34 1 20.71 7.43 22 23 21.75 7.49 21.71 7.58 21.68 7.68, 21.05 7.77 23 24 22.69 7.81 22.66 7.91 22.62 8.01 22.59 8.11 24 25 23.64 8.14 23.60 8.24 23.57 8.35 23.53 8.45 25 26 24.58 8.46 24.55 8.57 24.51 8.68 24.47 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 I 26.39 9.35 26.35 9.46 28 29 27.42 9.44 27.38 9.56 ! 27.34 9.08 27.29 9.80 29 30 31' 28.37 29.31 9.77 10.09 28.32 9.89 28.28 10.01 j 28.24 29.18 11/.14 10 48 30 3i 29.27 10.22! 29.22 10.35 1 32 30.26 10.42 30.21 10.55 1 30.16 10.68 30.12 10 81 32 33 31.20 10.74 31.15 10.88 31.11 11. 02, 31.06 11 15 33 34 32.15 J1.07 32.10 11.21 ! 32.05 11.35: 33.00 11 49 34 35 33.09 11.39 33.04 11.54i 32.99 11.68 1 32.94 11.83 35 36 34.04 11.72 33.99 11.87 33.94 12.02! 33.88 12.17 36 37 34.98 12.05 34.93 12.20 34.88 12.35 i 34.82 12.50 37 38 35.93 12.37 35.88 12.53 35.82 12.68 35.76 12.84 33 39 36.88 12.70 36.82 12.86 36.76 13.02 36.71 13.18 39 40 4f 37.82 13.02 37.76 38.71 13.19 37.71 13.35 37.65 13.52 13.8:^ 40 '41 38.77 13.35 13.52 38.05 13:69 38.59 42 39.71 13.67 39.65 13.85 39.59 14.02 ! 39.53 14.19 42 43 40,66 14.00 40.60 14. :d 40.53 14.35 i; 40.47 14.5:5 43 44 41.60 14.32 41.54 14.51 41.48 14.69 41.41 14.87 44 45 42.55 14.65 42.48 14.84 42.42 15. Of; 42.35 15.21 45 46 43.49 14.98 43.43 15.17 43.36 15.33 43.29 15.. 54 46 47 44.14 15.30 44.37 15.. 50 44.30 15.69 1 44.24 15.SS 47 48 45.38 15.63 45.32 15.83 1 45.25 16.02 : 45.18 16.22 48 49 46.33 15.95 46.26 16.15} 46.19 16.36 46.12 16.56 49 50 47.28 16.28 47.20 16.48 47.13 16.89 La. 47.06 Dep. 1.0.90 Lhi. 50 § l! I CO 5 Dep. 711 Lat. Dep. Lat. Dep. 701 1 De?. 70,: >8- 70.1 Deg. TRAVERSE TABLE. Ill "51 19 Deg. 191 Deg. 1 19A Deg. 19| Deg. "51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 48.22 16.60 48.15 16.81 48.07 17.02 48.00 17.23 52 49.17 16.93 49.09 17.14 49.02 17.30 48.94 17.57 52 5:j 50.11 17.26 50.04 17.47 j 49.96 17.69 1 49.88 17.91 53 54 51.06 17.58 50.98 1 17.80 1 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 1 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 19.00 58 59 55.79 19.21 55.70 19.45 55.62 19.69 55.53 19.94 59 60 61 50.73 19.53 1 56.65 19.78 56.56 57.50 20.03 20.86 56.47 20.27 00 61 57.68 19.86 57.59 20.11 .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 01.27 21.70 61.18 21.96 65 66 02.40 21.49 62.31 21.76 62.21 22.03 62.12 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 64.10 22.70 64.00 22.93 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 68.82 23.99 72 08.08 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 76.35 26.70 27.04 75.29 76.24 27.03 80 81 76.59 26.37 76.47 26.70 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 79.42 127.35 79.30 27.69 79.18 23.04 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 23.00 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 8.3.21 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 86.04 29.30 84.97 29.67 84.84 1 30.04 84.71 30.41 90 91 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 87.93,3(J.28 87.80 30.66 87.67 31.04 87.53 31 .43 93 94 88.88 1 30.60 88.74 30.99 88.61 31.38 88.47 31.76 94 95 89.82 i 30.93 89.69 31.32 89.55 31.71 89.41 32.10 95 96 90.77 1 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 IOC 04.55 32.56 94.41 32.97 94.26 33.38 94.12 Dep. 33.79 100 1 Dep. Lat. Dep. Lat. Dep. Lat. Lat. 71 Deg. 701 Deg. 70J Veg. 70i Deg. 112 TRAVERSr T.\nT,E. 1 c 3 ? " 1 20 Deg. 20i Deg. 20 1 Deg. 201 Deg. 1 Lat. 0.94 Dep. Lat. Dep. Lat. Dep. Lat. Dep. o^T 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 2 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 2.05 5.63 2.08 5.62 2.10 5.61 2.13 6 7 6.58 2.39 6.57 2.42 6.56 2.45 6.55 2.48 7 8 7.52 2.74 7.51 2.77 7.49 2.80 7.48 2.83 6 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 10.32 3.46 3.81 9.37 10.30 3.50 9.35 3.54 lO ~11 10.34 3.76 3.85 10.29 3.90 13 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 4.55 12.10 4.61 13 14 13.16 4.79 13.13 4.85 13.11 4.90 13.09 4.96 14 15 14.10 5.13 14.07 5.19 14.05 5.25 14.03 5.31 15 16 15.04 5.47 15.01 5.54 14.99 6.60 14.96 5.67 16 17 15.97 5.81 15.95 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 6.50 17.83 6.58 17.80 6.65 17.77 6.73 19 20 18.79 6.84 18.76 6.92 18.73 7.00 18.70 7.09 20 21 21 19.73 7.18 19.70 7.27 19.67 7.35 19.64 7.44 22 20.67 7.52 20.64 7.61 20.61 7.70 20.57 7.79 '^2 23 21. C] 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 8.55 23.45 8.65 23.42 8.76 23.38 8.86 25 26 24.43 8.89 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 9.68 26.27 9.69 26.23 9.81 26.18 9.92 28 29 27.25 9.92 27.21 10.04 27.16 10.16 27.12 10.27 29 30 28.19 10.26 28.15 10.38 28.10 10.51 28.05 10.63 30 '31 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 11.56 30.86 11.69 33 34 31.95 11.63 31.90 11.77 31.85 11.91 31.79 12.05 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 35.65 13.15 35.59 13.31 35.54 13.46 38 39 36.65 13.34 36.59 13.50 36.53 13.66 36.47 13.82 39 40 41 37.59 38.53 13.68 37.53 13.84 37.47 14.01 37.41 14.17 40 41 14.02 38.47 14.19 38.40 14.36 38.34 14.53 42 39.47 14.36 39.40 14.54 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 15.05 41.28 15.23 41.21 15.41 41.15 15.. 59 44 45 42.29 15.39 42.22 15.58 42.15 15.76 42.08 15.94 45 46 43.23 15.73 43.16 15.92 43.09 16.11 43.02 16. .30 46 47 44.17 16.07 44.09 16.27 44.02 16.46 43.95 16.65 47 48 45.11 16.42 45.03 16.61 44.96 16.81 44.89 17.01 48 49 146.04 16.76 45.97 16.96 45.90 17.16 45.82 17.36 49 50 u c 46.98 17.10 46.91 17.31 46.83 17.51 46.76 17.71 _50 6 o c d Dep. Lat. Dep. 69| Lat. Deg. Dep. Lat. Dep. Lat. 70 Deg. 69i Deg. 69i Deg TRAVEKSE TABLE. 113 1 20 Deg. 20$ Deg. 201 Deg. Dep. 201 Deg, Ci o 3 P Lat. Dep. Lat. Dep. Lat. Lat. Dep. 51 47.92 17.4-4 1 47,35 1 17". 65 43.79 1 18.00 1 47.771 17.86 47.69 18.07 51 52 1 43.86 17.79 48.711 18.211 43.63 18.42 52 53' 49.80 18.13 t 49.72; 1S.34I 49.64 i 13.56 49.56! 13.73 53 54 50.74 18.47: 50.06 18.69 1 50.58 1 18.91 ! 50.50 1 19.13 ! 54 55 51.63 18.81 1 51.60' 19.04^ 51.52 i 19.26. 51.43 19.49 55 56 i 52.62 19.15 1 52.54 19.33 52.45! 19.61 52.37 19.84 56 57 53.56 19.50 1 53.48 1 19.73, 53.39! 19.96 53.30 20.19 57 5S 54.50 19.84 1 54.42 20.07 1 54.33 ' 20.31 54.24 20.55 53 59 55.44 20.18' 55.35 20.42 i 55.26 20.66 !l 55.17 20.90 59 61 56.33 20.52 20.86 i 56.29 20.77 i 56.20 57.14" 21.011 56.11 21.26 21.61 60 61 57.32 57.23 21.11 21.36; 57.04 62 58.26 21.21 ' 53.17 21.46 58.07 21.71 ; 57.93 21.97 62 63 59.20 1 21.55 1 60.14121.89' 59.11 21.81 59.01 22.06 1 58.91 22.32 63 64 60.04 22.15 59.95 22.41 59.85 22.67 64 65 61.03 22.23 60.93 22.50 60.88 22.76 60.73 23.03 65 66 62.02 1 22.57 61.92 22.84; 61.82 23. Hi 61.72 23.33 66 67 62.96: 22.92 62.86 23.19' 62.76 23.46; 62.65 23.74 67 6S 63.90 23.26 63.80 23.54 1 63.69 23.81; 63.59 24.09 68 69 64.84 23.60 64.74 23.88' 6t.63 24.16; 64.52 24.45 69 70 71 65.78 23.94 65.67 24.23 65.57 66.50 24.51 : 24.86 65.46 24.80 70 71 66.72 i 24.23 65.61 9A.57 66.39 25.15 72 67.66] 24.63 67.55 24.92 67.44 25.21 67.33 25. 5L 73 73 68.60 i 24.97 68.49 25.27 68.33 25.57; 63.26 25.86 73 74 69.54 j 25.31 ; 69.43 25.61 69.31 25.92 69.20 26.22 74 75 70.48 25.65 70.36 25.96 1 70.25 26.27 70.14 26.57 75 76 71.42 1 25.99 71.30 26.30 1 71.19 26.62 71.07 26.93 76 77 72.36 1 26.34 72.24 26.65 72.12 26.97 72.01 27.23 77 78 73.30 26.63 73.18 27.00 73.06 27.32 72.94 27.63 73 79 74.24; 27.02 74.12 27.34 74.00 27.67 73.83 27.99 79 80 81 75.18 1 27.36; 76.12 i 27.70 75.06 27.69 74.93 23.02 174.81 23.34 80 81 75.99 1 28.04 75.87 1 23.37 75.75 23.70 82 77.05 23.05 76.93 28.33 76.81 28.72 76.63 29.05 82 S3 77.99 28.39 77.87 23.73 77.74 29.07 77.62 29.41 83 84 73.93 23.73 78.81 29.07 78.68 29.42 73.55 29.76 84 85 79.87 i 29.07 79.75 29.42 79.62 29.77 i 79.49 30.11 85 86 80.81 29.41 80.68 1 29.77 80.55 30.12 80.42 30.47 86 87 81.75 29.76 81.62 30.11 81.49 30.47 181.36 30.82 87 88 82.69 30.10 82.56 I 30.46 82.43 30.82 182.29 31.^18 83 89 83.63 30.44 83.50 1 30.80 83.36 31.17 83.23 31.53 89 90 91 84.57 30.73 84.44 1 31.15 84.30 31.52 184.16 131.89 90 91 185.51 1 31.12 85.33 31.50 85.24 31.87 85.10 32.24 92 ! 86.45 i 31.47 86.31 31.84 86.17 32.22 ! 86.03 32.59 92 93 1 87.39 i 31.81 1 87.25 132.19 87.11 32.57 ! 86.97 32.95 93 94 83.33' 32.15 83.19 , 32.54 83.05 132.92 ! 87.90 33.30 94 95 89.27 32.49 89.13' 32.83 83.93 '33.27 183.84 33.65 95 96 ; 90.21 32.83 90.07 1 33.23 89.92 33.62 i 89.77 34.01 96 97 i 91.15! 33.18 91.00 33.57 90.86 1 33.97 190.71 34.37 97 98 92.09 33.52 91.94 33.92 91.79 34.32 91.64 .34.72 9S 99 93.03 1 33.86 92.88 34.27 92.73 34.67 92.58 135.07 1 99 lOO 1 93.97; 34.20 93.82! 34.61 93.67 35.02 93.51 35.43 ! 100 11 i Dep. 70 [ Lat. Dep. Lat. Dep. 1 Lat. Dep. Lat. Deg. 69| Deg. 69| Deg. 69+ Deg 114 TRAVEJiSK TABLE. p 3 n o 21 Deg. 2U Deg. 21 1 Deg. 21| Deg. a 1 Lat. 0.93 Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.36 0.93 0.36 0.93 0.37 0.93 0.37 2 1.87 0.72 1.86 0.72 1.86 0.73 1.86 0.74 2 3 2.80 1.08 2.80 1.09 2.79 1.10 2.79 1.11 3 i 3.73 1.43 3.73 1.45 3.72 1.47 3.72 1 1.48 4 li 4.67 1.79 4.66 1.81 4.65 1.83 4.64 1.85 5 b 5.60 2.15 5.59 2.17 5. 58 2.20 5.57 2.22 6 V 6.54 2.51 0.52 2.54 6.51 2.57 6.50 2.59 7 8 7.47 2.8? 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 4.03 9.29 3.71 4.08 10 "ll 11 10.27 3.94 10.25 3.99 10.23 10.22 12 11.20 4.30 11.18 4.35 11.17 4.40 I 11.15 4.45 12 13 12.14 4.66 12.12 4.71 12.10 4.76 12.07 i.82 13 14 13.07 5.02 13.05 5.07 13.03 5.131 13.00 5.19 14 15 14.00 5.38 13.93 5.44 13.96' 5.50 1 13.93 5.56 15 16 14.94 5.73 14.91 5.80 1 14.89 5.86 14.86 5.93 16 U 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 i 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 18.67 7.17 18.64 7.25 18.01 7.33 18.58 1 7.41 20 21 19.61 7.53 ; 19.57 7.61 19.54 7.70 19.50 1 7.78 1 21 22 20.54 7.88 1 20.50 7.97 20.47 8.06 20.43 1 8.15 22 23 21.47 8.24 ; 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 1 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 i 24.15 9.63 26 27 25.21 9.68 25.16 9.79 25.12 1 9.90 ! 25.08 10.01 27 28 26.14 10.03 26.10 10.15 26.05, 10.26 ! 26.01 10 33 28 29 27.07 10.39 27.03 10.51 26.98 ■: 10.63 26.94 10.75 29 30 28.01 10.75 27.90 10.87 27.91 : 11.00 27.86 11.12 30 31 28.94 11.11 28.89 11.24 28.84, 11.36 28.79 11.49 31 32 29.87 11.47 29.82 11. GO 29.77 11.73 1 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 1 12.46 31.58 12.60 34 35 32.68 12.54 32.62 12.69 32.58 i 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 1 13.56 34.37 13.71 37 38 35.48 13.62 35.42 13.77 35.36 1 13.93 35.29 14.08 38 39 36.41 13.98 36.35 14.141 36.29 14.29 36.22 14.45 39 40 41 37.34 14.33 37.28 14.50 37.22' 14.66 1 37.15 14.82 40 41 38.28 14.69 33.21 14.86 38.15. 15.03 38.08 1 15.19 42 39.21 15.05 39.14 15.22 39.08 i 15.39, 39.01 15.56 42 43 40.14 15.41 40.08 15.58 40.01 1 15.76 ' 39.94 1 15.93 43 44 41.08 15.77 41.01 15.95 40.94! 16.13' 40.87 1 16.30 44 45 42.01 16.13 41.94 16.31 41.87 i 16.49 41.80 1 16.63 1 45 46 42.94 16.48 142.87 16.67 42.80! 16.86 42.73! 17.05 1 46 47 43. S8 16.84 j 43.80 17.03 43.73: 17.23 43.65 1 17.42 \ 47 48 44.81 17.20 144.74 17.40 44.66 i 17.59 44.58 1 17.79 48 49 45.75 17.56 145.67 17.76 45.59 17.96 45.51 1 18.16 49 50 .a 46.63 17.92 146.60 18.12 48.52 18.33 46.44 Dep. 18.53 50 Dep. Lat. Dep. Lat. Dep. 1 Lat. Lat. 69 Deg. 68| Degr 68i Deg. en\ Deg. TJiAVERSE TABLE. 115 1 5i 21 Deg. 21i Deg. 21A Deg. 1 211 Deg. 5' Lat. Dep. 1 Lat. Dep. Lat. Dep. Lat. Dep. i 47.61 18.28 47.53 18.48 47.45 18.69 47.37 18>90 51 52 48.55 18.64 1 48.46 18.85 48.38 19.06 48.30 19.27 52/ 53 49.48 18.99 49.40 19,21 49.31 19.42 49.23 19.64 5;i 64 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 1 52.10 20.52 52.01 20.75 .5f, 5? 53 2] 20.43 53.12 20.66 53.03 20.89 52.94 21.12 57 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 61 56.01 21.50 21.86 55.92 21.75 55.83 21.99 55.73 22.23 60 61 56.95 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 23.40 59.44 23.72 64 65 60.68 23.29 60.58 23.56 60.48 23.82 60.37 24.09 65 60 61.62 23.65 61.51 23.92 01.41 24.19 61.30 24.46 66 67 62.55 24.01 62.44 24.28 62.34 24.56 62.23 24.83 67 68 63.48 24.37 63.38 24.65 63.27 24.92 63.16 25.20 es 69 64.42 24.73 64.31 25.01 64.20 25.29 64.09 25.57 69 70 71 65.35 25.09! 65.24 25.37 65.13 25.66 65.02 25.94 70 71 66.28 25.44 66.17 25.73 66.06 26.02 65.95 26.31 72 67.22 25.80 1 67.10 26.10 66.99 26.39 66.87 26.68 72 73 68.15 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. 8S 69.90 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 81 74.69 28.67 74.56 29.00 74.43 29.32 74.30 29.64 80 75.62 29.03 75.49 29.36 75.36 29.69 75.23 30.02 81 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 91 84.02 32.25 83.88 32.62 83.74 32.99 83.59 33.35 90 91 84.96 32.61 84.81 32.98 84.07 33.35 84.52 33.72 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 85.53 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 91 95 88.69 34.04 88.54 34.43 88.39 34.82 88.24 35.20 95 96 89.02 34.40 89.47 34.79 89.32 35.18 89.17 35.57 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 35.52 191.18 35.92 91.02 36,31 98 99 92.42 35.48 92.27 35.88 : 92.11 36.28 91.95 36.69 99 100 93.36 35.84 93.20 36.24 II 93.04 36.65 92.88 37.06 100 o Dcp. Lat. Dep. Lat. I| Dep. Lat. Dep. J-at. 69 Deg. 681 Deg. 1 68^ Deg. 68i Dog 22 116 TRAVERSE TABLE. 1 I" 22 Dog. 2Si Deg. 22^ Deg. 22? Deg. X ~1 Lat. Dep. Lat Dep. j Lat. Dep. Lat, Dep. 6.93 0.37 1 0.93 0.38 0.92 0.38 0.92 0.39 o 1.85 0.75' 1.85 0.76 1.85 0.77 1 1.84 0.77 2 3 2.78 1.121 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 1 5 4.64 1.87 1 4.63 1.89 4.62 1.91 j 4.61 i.93 5 6 5.56 2.25 5.55 2.27 i 5.54 2.30 5.53 2.32 6 7 6.49 2.62 6.48 2.65 1 6.47 2.68 6. 40 2.71 7 8 7.42 3.00 7.40 3.03 7.39 3.06 7 38 3.09 8 9 8.34 3.37 8.33 3.41' 8.31 3.44 8.30 3.48 9 1 10 9.27 3.7c 9.26 3.79 9.24 3.83 9.22 3.87 10 -ll 11 10.20 4.12 10.18 4.17 10.16 4.21 10.14 4.25 12 1 11.13 4.50 '11.11 4.54! 11.09 4.59 11.07 4.64 12 13 12.05 4.87 12.03 4.92 12.01 4.97 11.99 5.03 13 14 12.98 5.24 12.96 5.30 12.93 5.36 12.91 5.41 14 15 13.91 5.62 13.88 5.08 : 13.86 5.74! 13.83 5.80 15 16 14. S3 5.99 14.81 6.06 14.78 6.12 14.76 6.19 j 16 17 15.76 6.37 15.73 6.44 15.71 6.51 15.68 6.57 t 17 18 16.69 6.74 16.66 6.82 16.63 6.89 16.60 6.96 18 19 17.62 7.12 ! 17.59 7.19 |i 17.55 7.27 17.52 7.35 ! 19 20 18.54 7.49 18.51 7.57 j 18.48 19.40 7.65 8.04 18.44 7.73' 20 21 19.47 7.87 1,19.44 7.95! 19.37 8.12 i 21 22 20.40 8.24! 20.36 8.33 |! 20.33 8.42 20.29 8.51 1 22 23 21.33 8.62!' 21.29 8.71 l; 21.25 8.80 21.21 8.89 1 23 24 22.25 8.99 22.21 9.09 22.17 9.18 :' 22.13 9.23 1 24 2.5 23.18 9.37 23.14 9.47 I 23.10 9.57 23.05 9.67 1 25 26 24.11 9.74 ,24.06 9.84 24.02 9.95 23.98 10.05 ! 26 27 25.03 10.11 124.99 10.22 24.94 10.33 24.90 10.44: 27 28 25.96 10.49 25.92 10.60 ; 25.87 10.72, 25.82 10.83 ' 23 20 26.89 10.86 26.84 10.98 ! 26.79 11.10 26.74 11.21 1 29 30 27.82 11.24: 27.77 23.69 11.36 ; 27.72 11.74 1 23.64 11.48! 27.67 11.86 28.59 11.60 30 11.99. 31 31 28.74 11.61 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 i 30.49 12.63 30.43 12.76 I 33 34 31.52 12.74 31.47 12.87 1 31.41 13.01 31.35 13.15 34 35 32.45 13.11 32.39 13.25 1 32.34 13.39 32.28 13.53 2i, 36 33.33 13.49 33.32 13.63 : 33.26 13.73 33.20 13.92 36 37 34.31 13.86 34.24 14.01 1 34.18 14.16 34.12 14.31 37 33 35.23 14.24 35.17 14.39 1 35.11 14.54 135.04 14.70 33 39 36.16 14.61' 36.10 14.77 36.03 14.92 35.97 15.08 39 40 37.09 14. 9S 15.36 37.02 37.95 15.15 15.52 36.96 15.31 15.69 ; 36.89 15.47 40 41 i 33.01 37.83 [37.81 15.86 41 42 38.94 15.73 38.87 15.90 33.80 16.07 33.73 18.24 42 43 39.87 16.11 39.80 16.23 : 39.73 16.46 39.65 16.63 43 4^1 40.80 16.48 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 -13.50 17.80 43.42 17.99 ,43.34 13.13 47 48 144.50 17.93 44.43 18.18 : 44.35 18.37 1 44.27 18.56 •iS 49 145.43 18.36 45.35 18.55 : 45.27 18.75 145.19 18.95 49 60 46.36 18.73 46.28 18.93 1 46.19 19.13 Lat. ,46.11 19.34 50 1 Dep. Lat. Dep. Lat. Dep. Dep. Lat. Q 68 1 Deg. 671 Deg. 674 Deg. 67i Deg. TRAVERSE TABLE. 117 22 Deg. m Deg. 22^ Deg. j 1! 22| Deg. § o CO Ti Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 47.29 19.10 47.20 19.3li 47.12 19.52 47.03 19.72 52 48.21 19.48 48.13 1 19.69 i 48.04 19.90 47.95 20.11 52 53 49.14 19.85 49.05 20.07 48,97 20.28 48.88 20.50 53 54 50.07 20.23 49.98 20.45 49.89 20.66 49.80 20.88 54 55 51.00 20.60 50.90 20.83 50.81 21.05 50.72 21.27 r.5 56 51.92 20.98 51.83 21.20 51.74 21.43 51.64 21.66 56 57 52.85 21.35 52.76 21.58 52.66 21.81 52.57 22.04 57 58 53.78 21.73 53.68 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 55.53 22.72 55.43 56.36 22.96 55.33 23.20 60 61 56.56 22.85 56.47 23.10 23.34 56.25 23.. 59 62 57.49 23.23 57.38 23.48 57.28 23.73 57.18 23.98 62 6.J 58.41 23.60 58.31 23.85 58.20 24.11 58.10 24.36 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 80.66 32.59 80.52 32.94 80.38 33.29 80.23 33.64 87 88 81.59 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 91 83.45 33.71 83.30 34.08 83.15 34.44 83.00 34.80 90 91 84.37 34.09 84.22 34.46 84.07 34.82 83.92 35.19 9£ 85.30 34.46 85.15 34.84 85.00 35.21 84.84 35.58 92 93 86.23 34.84 86.08 35.21 85.92 35.59 85.76 35.96 93 94 1 87.16 35.21 87.00 35.59 86.84 35.97 86.69 36.35 94 95 ;8S.08 35.59 87.93 135.97 87.77 36.35 87.61 36.74 95 96 89.01 35.96 88.85 36 35 88.69 33.74 88.53 37.12 96 97 : 89.94 36.34 89.78 36.73 89.62 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 99 91.79 37.09 91.63 37.49 191.46 37.89 91.30 38.28 99 100 92.72 37.46 92.55 37.86 92.39 38.27 92.22 38.67 100 1 o y a Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 1 s ! 68 Deg. 67| Deg. &7^Deg. 67i De^. 118 TRAVKRSE TABLE. 23 Deg. 23i Deg. 23i Deg. 231 Deg. CO p Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. i 0.92 0.39 0.92 0.39 0.92 0.40 0.92 0.40 "■"l 2 1.84 0.78 1.84 0.79 1.83 0.80 1.83 0.81 2 3 2.78 1.17 2.76 1.18 2.75 1.20 2.75 1.21 3 4 S.68 1.56 3.68 1.58 3.67 1.59 3.66 1.61 4 5 A. GO 1.95 4.59 1.97 4.59 1.99 4.58 2.01 5 6 5.52 2.34 5.51 2.37 5.50 2.39 5.49 2.42 6 7 6.44 2 74 6.43 2.76 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 4.30 9.19 3.95 9.17 3.99 9.15 10.07 4.03 4.43 '0 11 'll 10.13 10.11 4.34 10.09 4.39 12 11.05 4.69 11.03 4.74 11.00 4.78 10.98 4.83 12 13 11.97 5.08 11.94 5.13 11.92 5.18 11.90 5.24 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 6.85 17 18 J6.57 7.03 16.54 7.11 16.51 7.18 16.48 7.25 18 19 ]7.49 7.42 17.46 7.50 17.42 7.58 17.39 7.65 19 20 18.41 7.81 18.38 7.89 18.34 19.26 7.97 18.31 8.05 20 21 19.33 8.21 19,29 8.29 8.37 19.22 8.46 21 22 20.25 8.60 20.21 8.68 20.18 8.77 20.14 8.86 22 23 21.17 8.99 21.13 0.08 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 22.88 10.07 25 26 23.93 iO.iO 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 28 25.77 .0.94 25.73 11.05 25.68 11.16 25.63 1.1.28 28 29 26.69 U.33 26.64 11.45 26.69 11.56 26.54 11.68 29 30 31 27.62 11.72 27.56 11.84 27.51 11.96 27.46 12.08 30 28.54 12.11 28.48 12.24 28.43 12.36 28.37 12.49 31 32 29.46 12.50 29.40 12.63 29.35 12.76 29.29 12.89 32 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 31.18 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 33.08 14.21 33.01 14.35 32.95 14.50 36 37 34.06 14.46 34.00 14.61 ,33.93 14.75 33.87 14.90 37 38 i 34.98 14.85 34.91 15.00 34.85 15.15 34.78 15.30 38 39 35.90 15.24 35.83 15.39 35.77 15.55 35.70 15.71 39 40 36.82 15.63 36.75 15.79 36.68 15.95 36.61 16.1! 40 41 37.74 16.02 37.67 16.18 37.60' 16.35 37.53 16.51 \l 42 38.66 16.41 38.59 16.58 38.. 52 16.75 38.44 16.92 12 43 39.58 16.80 39.51 16.97 39.43 17.15 39.36 i 17.32 43 44 40.50 ! 17.19 40.43 17.37 40.35 17.54 40.27 17.72 44 45 ! 41.42 17.58 41.35 17.76 41.27 17.94 41.19 18.12 45 46 42.34 17.97 42.26 18.16 42.18 18.34 42.10 18.53 46 47 43.26 18.36 43.18 18.55 43.10 18.74 43.02 18.93 47 48 44.18 1 18.76 44.10 18.95 44.02 19.14 43.93 19.33 48 19 45.10 19.15 45.02 19.. 34 44.94 19.54 44.85 19.73 49 50 46.03 1 19.54 45.94 19.74 45.85 19.94 145.77 20.14 50 § i 1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. C 1 67 Deg 661 Deg. 66i Deg. 66i Deg. TRAVERSE TA.BLE. ' 119 p n o 23Deg. 23i Deg. 23i Deg. 23| Deg. o P 3 51 Lat. Dep. Lat. Dep. Lat. Dep. Lat 1 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 47.60 120.94 52 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 21.32 49.52 21.53 49.43 121.75 54 55 50,63 21.49 50.53 21.71 50.44 21.93 50.34 122.15 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 5S 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 00 123.76 , 591 60 61 55.23 23.44 55.13 23.63 55.02 23.92 54.92 24.16 60 56. 15 23.83 56.05 24.03 55.94 24.32 55.83 24.57 Tl 62 57.07 24.23 56.97 24 47 56.85 24.72 56.75 24.97 62 63 57.99 24.62 57. 8S 24.87 57.77 25.12 57.66 25.37 63 64 53.91 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 20.72 61.33 26.93 67 68 62.59 28.57 62.48 26.34 62.36 27.11 62.24 27.39 68 69 63.51 26.96 63.40 27.24 63.23 27.51 63.16 27.79 69 70 71 64.44 65.. 36 27.35 64.32 65.23 27.63 64.19 27.91 64.07 28.19 70 71 27.74 23.03 65.11 23.31 64.99 23.59 72 66.28 23.13 66.15 23.42 66.03 23.71 65.90 29.00 72 73 67.20 28.52 67.07 23.82 66.95 29.11 66.82 29.40 73 74 63.12 23.91 67.99 29.21 67.86 29.51 1 67.73 29.80 74 75 69.04 29.30 68.91 29.61 63.78 29.91 68.65 30.21 75 76 69.96 29.70 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 70.43 31.01 77 78 71.80 30.48 ?l-.67 30.79 71.53 31.10 71.39 31.41 i 73 79 72.72 30.87 72.53 31.18 72.45 31.50 72.31 31.82 1 79 80 81 73.64 31.26 73.50 31.53 73.36 31.90 73.22 32.22 1 80 74.56 31.65 74.42 31.97 74.28 35i.30 74.14 32.62 81 82 75.43 32.04 75.34 ,32.37 75.20 32.70 75.06 33.03 82 83 70.40 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 1 85 86 79.16 33.60 79.02 33.95 78.-87 1 34.29 78.72 34.64! 86 87 80.08 33.99 79.93 34.34 79.78 1 34.69 79.63 35.04 87 88 81.00 34.33 80.85 34.74 80.70 i 35.09 80.55 35.44 88 89 81.92 34.78 81.77 35.13! 81.62' 35.49 81.46 i 35.84 89 90 91 82.85 35.17 82.69 35.53 i 82.54; 35.89 82.33 36.25 90 91 83.77] 35.56 83.61 35.92 83.45: 36.29 1 83.29 36.65 92 84.69! 35.95 84.53 36.32 84.37 136.63 1 84.21 37.05 92 93 85.61 : 36.34 85.45 36.71 j 85.29 37.08 1 85.12 37.46 93 94 86.53 36.73 86.37 37.11 1 86.20 37.48; 86.04 37.86 ' 94 95 87.45 37.12 87.29 37.50 i 87.12 37.83! 86.95 38.26 95 96 88.37 37.51 83.20 37.90] 88.04 33.28 i 87.87 ,38.66 ^6 97 89.29,37.90 89.12 3S.29i 83.95 38.63 88.79 39.07 97 98 90.21 39.29 90.04 33.68^ 89.87 39.03 89.70 39.47 98 99 91.13 33.63 1 90.96 39.08! 90.79 39.48 90.62 39.87 99 ^00 92.05 Dep. 39.07 91.83 39.47 1 91.71 139.871 91.53 40.27 100 Lat. Dep. Lat. Dep 1 Lat. Dep Lat. a 3 67 Deg. i 661 Deg. 661 Deg. 66,^ Deg. ; 120 TRAVERSE TABLE. o ? 24 Deg. 24i Deg. 24^ Deg. 1 24| Deg. 1 C 5' §" a a Lat. Dep. Lat. Dep. Lai. Dep. Lat. Dep. ^ 0.91 : 0.41 0.91 0.41 0.91 0.41 0.91 0.42 '~l 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 5 4.57 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 i 2.85 6. 38 2.87 6.37 2.90 6.36 2.93 7 8 7.31 1 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 9.141 4.07 9.12 4.11 9.10 4.15 9.08 4.19 lO 11 10.05; 4.47 10.03 4.52 10.01 4.56 9.99 4.61 11 12 10.96 i 4.88 10.94 4.93 10.92 4.98 10.90 5.02 12 13 11.88 1 5.29 11.85 5.34 11.83 5.39 11.81 5.44 13 14 12.79 1 5.69 12.76 5.75 12.74 5.81 12.71 5.86 14 15 13.70 1 6.10 13.68 6.16 i 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 i 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 J7.36 7.73 17.32 7.80 17.29 7.88 17.25 7.95 19 20 21 18.27 8.13 19.18 , 8.54 18.24 8.21 18.20 19.11 8.29 18.16 19.07 8.37 20 21 19.15 8.63 8.71 8.79 "2 20.10 1 8.95. 20.06 9.04 20.02 9.12 ' 19.98 9.21 22 23 21.01 1 9.35 , 20.97 9.45 20.93 9.54 20.89 9.03 23 24 21.93 1 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 i 11.39 ' 25 . 53 11.50 i 25.48 11.61 25.43 11.72 28 29 26.49 1 11.80 26.4-1 11.91 ' 26.39 12.03! 26.34 12.14 29 30 31 27.41 • 12.20 1 28.32 12.61 1 27.35 28.26 12.32 27.30 28.21 12.44' 12.86 ■■ 27.24 12.56 30 31 12.73! 28.15 12.93 32 29.23 13.02 • 29.18 13.14i 29.12 13.27: 29.06 13.40 32 33 30.15 13.42' 30.09 13.55: 30.03 13.63 : 29.97 13.82 33 34 31.06 13.83 31. 00113. 96; 30.94 14.10 ! 30.88 14.23 34 35 31.97 14.24 31.91 14. 3S 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 15.4fi ' 34.65 15.61 i 34.58 15.76 i 34.51 15.91 38 39 1 35.63 15.86 ; 35.56 ! 16.02 .35.49 I6.17i .35.42; 16.33 39 40 ; 36.54 16.27 i 36.47 i 16.43 36.40 16.59 ! 36.33 ' 16.75 40 411 37.46 16.68 37.38; 16.84! 37.31 1 17.00 i 37.23 17.16 41 42 38.37 17.03 33.29 ; 17.25 I 38.22; 17.42 38.14 17.58 42 43! 39.28 17.49 39.21 17.60 i 39.13 17.83; 39.05 1 13.00 43 441 40.20 17.90 40.12 ! 13.071 40.04; 18.25 1 39.96 : 18.42 44 45 41.11 18 30 ; 41.03 ; 18.48 i 40.95 18.66 40.87! 18.84 45 46 i 42.02 '18.71 41.94 18.89 41.86 ! 19.03 41.77 1 19.26 46 47 ' 42.94 19.12 42.85 19.30 42.77 19.49 42.63 19. 6S 47 48 -43.85 19.52 43 . 70 19.71 43.63 ' 19.91 43.59 ' 20.10 48 49 '44.76 19.93 44.68 20.131 44.59 ' 20 . 32 44.50 i 20.51 49 50; 1 ! 45.68 20.34 45.59 20.54 45.50 , 20.73 45.41 1 Dep. 20.93 Lat. _50 6 1 Q Dep. i Lat. Dep. Lat. Dep, Lat. 66 r )eg. 651 Deg. 1 651 1 D'^.g. 65J Deg. "A. TKAVEUSE TABLE. 121 5 P 51 24 Deg. 24i Deg. 24i Deg. 241 Dej. 1 ? 51 Lat. 46.59 Dep. r Lat. Dep. Lat. Dep. Lat. Dep. 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 54 49.33 21.96 49.24 22.18 49. H 22.39 49.04 22.61 54 55 50.24 22.37 50.15 22.59 50.05 22.81 49.95 23.03 55 56 51.16 ! 22.78 51.06 23.00 50.96 23.22 50.86 23.44 56 57 52.07 123.18 51.97 23.41 151.87 123.64 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 i 53.69 24.47 53.58 24.70 59 60 "61 54.81 24.40 54.71 24.64 j 54.60 24.88 54.49 25.12 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 "7l 63.95 28.47 63. &2 28.75 63.70 29.03 63.57 29.31 29.72 70 71 64.86 28.88 64.74 29.16 64.61 29.44 04.48 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 31.52 169.02 31.82 76 77 70.34 31.32 70.21 31.63 70.07 31.93 169.93 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 i 71.74 33.07 79 80 81 73.03 32.54 72.94 32.86 72.80 33.18 i 72.65 33.49 80 81 74.00 32.95 73.85 33.27 73.71 33.59 i 73.56 33.91 82 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 1 78.10 36.00 86 87 79.43 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 1 79.92 36.84 88 89 81.31 36.20 81.15 36.55 80,99 36.91 i 80.82 1 37 26 39 90 "91 82.22 '36.61 82.06 36.96 81.90 82.81 37.32 37.74 : 81.73 37.68 33.10 90 91 83.13 37.01 82.97 37.. 38 [82.64 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 33.94 93 94 85 87 3S.2? 85.71 38.61 85.54 38.98 85.37 39.35 94 95 86 79 38.6/i 86.62 39.02 86.45 39.40 86.27 39 77 95 96 87.70 39.05' 87.53 39.43 87.36 39.81 87.18 40.19 96 97 68.61 3&.45 38.44 39.84 88.27 40.23 88.09 40.61 97 98 39.5:3 ! 39.0-3 89.35 40.25 89.18 40.64 89.00 41.03 98 99 •^0.44 40.^7 90.26 40.66 90.09 41.05 89.91 41.45 99 ''7 3 •Jl.SO Dep. 4ti).L7 L?t. 91.18 Dop. 41.07 91.00 41.47 90.81 41.87 lOO 6 c Q Lat. Dep. Lat. Dep. Lat. 66 E > Dcp. Lat. Dep. 63| Lat. Deg. Dep. Lat. Dep. Lat. 1 ^ 64 Deg. 63^ Doer. 63t Deg. TfiAVERSE TABLE. 125 -3 So' p -1 52 53 54 55 5 3 57 53 59 60 26 Deg. 261 Deg. 1 26^ Deg. 2ex Deg. \ a J.at, 1 Dep. Lat. Dep. ! Lat. 1 Dep. 1 i 45.64' 22.76 46.54 23.20 '47.43; 23.65 ,48.33 24.09 '49.22 24.54 50.12 24.09 .51.01,25-43 51.91 25 S3 52.80: 26.. 33 ,53.70 26-77 1.54.59 ' 27.22 155.49 1 27.66 56.33 ! 23.11 57.28 23.56 58.17 29.00 59.07 29.45 59.96 29.90 60.36 30.34 61.75 30.79 62.65 31.23 Lat, Dop. 1 i 22.90 ' 51 23.41 5'Z\ 23.83 53 24.31 5i 24.76 1 55 25.21 ! 56 25.66 i 57 26.11 : 53 26.-56 1 59 27.01 i 60 45.84 1 22.36 : 46.74 122.80 47.64 i 23.23 43.53! 23.67 49.43: 24.11 50.33 ■ 24.55 51.23, 24.99 52.13 125.43 53.03 ! 25.86 53.93:26.30 45.74 46.64 47.53 48.43 49.33 50.22 51.12 52.02 52.92 53.81 22.56 23.00 23.44 23.83 24.33 24.77 25.21 25.65 26.09 26.54 46 . 54 46.43 47.33 48 22 49.11 50.01 .50.90 51.79 52.69 53.53 61 62 63 64 65 66 67 63 C9 70 54.33 23.74 55.73 27.13 56.62 27.62 57.52 23.06 53.42 23.49 59.32 23.93 60.22 29.37 01.12 29.81 62.02 30.25 62.92 30.69 154.71 1 26.93 . 55.61 ! 27.42 56.C0; 27.88 57.40; 28.31 53.30' 23.75 59.19; 29.19 60.09 1 29.63 60.99 30.03 61.83 30.52 62.73 30.95 54,47 27.46 ^ 61 5-5.36 ' 27.91 ' 62 56.26 ; 23.36 [ 63 57.15 1 23.31 64 53.04 29.26 65 53.94 29.71 ' 66 59.83 30.16 i 67 60.72' 30.61 1 63 61.63 JGl. 06 69 62.51 131.51 1 70 71 63.81 72 64.71 73 65.61 74 ; 66.51 75 ; 67.41 76 63.31, 77 i 69.21 73 70.11 ro 71.00 SO 71.90 31.12 1 63.63, 31.40 31.56 164.57; 31.84 32.00 165.47; 32.29 32.44 1 66.37, .32.73: 32.83 67.27! 33. 17 33..32| 63.16 i 33.61 33.75 '69.06 ! 34.06 34.19 69.96 i 34.50 34.63 70.85; 34.94 35.07 j 71.75' 35.38 63.54 31.63 64.44 32.13 65.33 32.57 66.23! 33.02 67. 12 '33.46 63.01 33.91 63.91 34.36 69.80 34.30 70.70 35.25 71.59, 35.70 63.40 64.29 65.19 66.03 66.97 67.87 63.76 69.65 70.55 71.44 72.33 73.22 74.12 75.01 75.90 76.80 77.69 78.58 79.48 80.37 31.96 - 71 32.41 72 32.36 , 73 33.31 1 74 33.76 1 75 34.21 i 76 34.66 77 35.11 . 73 35.56 79 36.01 SO 81 1 72.30, 82 73.70 83 74.60' 84 75.50 85 76.40 86 77.30 87 73.20 88 79.09. 89 79.99 90 80.89 35.51 35.95 36.33 36.82 37.26 37.70 38.14 33.53 .39.01 39.45 172.65 73.54 74.44 75.34 76.23 77.13 178.03 i 78.92 179.82 80.72 35.83: 36.27' .36.71 ! 37.15 37.59 33.04; 33 43 33.92 39.36 39.81 72.49' 36.14 73.33 36.59 74.28 1 37.03 75.17: 37.43 76.07' 37.93 76.96' 38.37 77.86 .38.82 73.75' 39.27 79.65 39.71 30.54 40.16 81.44 40.60 82.33 41.05 83.23 41.50 84.12 41.94 85.02 42.39 85.91 42.83 86.81 43.23 37.70 43.73 83.60 44.17 89.49,4.1.62 36.46 : 81 36.91 i 82 37.36 83 37.81 . 84 33.26 ' 85 33.71 . 86 39.16 87 39.61 , S3 40.06 89 40.51 ; 90 91 92 93 94 95 96 97 98 99 100 § c 2 m Q 81.79 82.69 83.59 84.49 85.39 86.23 87.18 88.03 88.93 89.83 39.89 40.33 40.77 41.21 41.65 42.03 42.52 42.96 43.40 43.84 181.62 82.51 ; 83.41 84.31 85.20 ,86.10 87.00 ,87.89 ,88.79 89. 6S 40.25 40.69 41.13 41.53 42.02 42.46 42.90 43.34 43.79 44.23 81.26 40.96 91 82.15 41.41 92 83.05 41.80 93 83.94 42.31 ' 94 84.83 42.76 95 85.73 43.21 96 86.62 43.66 97 87.51 44.11 9S 83.40 44.56 99 89.30 45.01 100 Dep.; Lat. S , 63i Deg. Q Dep. Lat. Dep. Lat. Dep. ! Lat. 63| Deg. 64 Deg. 63J Dog. 126 TEAVEKSE TABLE. 5 Si 27 Deg. 27iDeg. 27^ Deg. 1 27i Deg. 1 1 Lat. Dep. Lat. Dep. Lat. Dep. 1 Lat Dep. "n 0.8& 0.45 0.89 0.46 1 0.89 j.4fi C.K% G.47 1 2 1.78 0.91 1.78 0.92 1 1.77 0.92 1.77 0.93 2 3 2.67! 1.36 2.67 1.37 1 2.66 1.3y! {! G5 1.40 3 4 3.56 i 1.82 3.. 56 1.83 1 3..^5 1.85 i L.Ci 1.86 4 5 4.4.;! 2.27 4.45 2.29 4.44 2.31 1 4.42 2.33 5 6 5.35 2.72 5.33 2.75 5.32 2.77 5.31 2.79 G 7 6.24 3.18 6.22 3.21 1 ^ 6.21 3.23 6.19 3.26 7 8 7.13 3.63 7.11 3.66 5 7.10 3.69 7.08 3.72 8 9 8.02; 4.09 8.00 4.12 1 7.98 4.16' 7.96 4.19 S 10 11 8.91 ! 4.54 9.80 4.99 8.89 4.58 8.87 4.62 1 8.85 4.66 5.12 10 11 9.78 5.04 1 9.76 5.08 1 9.73 12 10.69 5.45 10.67 5.49 10.64 5.54' 10.62 5.. 59 12 13 11.. 58 5.90 11.56 5.95 1 11.53 6.00! U..50 6.05 13 14 12.47 6.36 12.45 6.41 1 12.42 6.46 12.39 6.52 14 15 13.37 6.81 13.34 6.87 13.31 6.93 13.27 6.9S 15 it- 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 1 15.04 7.92 17 18 16.04 8.17 16.00 8.24 1 15.97 8.31 15.93 8. .38 18 19 16.93 1 8.63 16.89 8.70 1 16.85 8.77 16.81 8.85 19 2« 21 17.82 18.71 9.08 9.53 17.78 9.16 17.74 18.63 9.23 17.70 9.31 20 21 18.67 9.62! 9.70 18.58 9.78 22 19.60 9.99 19.56 10.07 1 19.51 10.16 19.47 10.24 22 23 20.49 10.44 20.45 10.53 20.10 10.62 1 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 n.64 25 26 23.17 11.80 23.11 11.90! 23.06 12.01 23.01 12.il 26 27 24.06 12.26 i 24.00 12.36 23.95 12.47! 23.89 12.57 27 28 24.95 12.71 24.89 12.82 24.84 12.93 24.78 13.04 28 29 25.84 13.17 1 25.78 13.28 25.72 13.39 25.66 13.50 29 30 -31 26.73 13.62 126.67 27.56 13.74 14.19 26.61 13.85 26.55 13.97 30 31 27.62 14.07 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 1.'..24 29.20 15.37 33 34 30.29 15.44 30.23 15.57 .30.16 •5.70 30.09 15.83 34 35 31.19 15.89 31.12 16.03 31.05 16.16 30.97 16.30 35 36 32.08 16.34 32.00 16.48 31.93 10.62 i • .86 16.76 36 37 32.97 16.80 32.89 16.94 32.82 17.08 Si. 74 17.23 37 38 33.86 17.25 33.78 17.40 33.71 17.55 33.63 17.69 38 39 34.75 17.71 34.67 17.86 34.59 18.01 34.51 18.16 39 40 35.64' 18.16 !: 35.56 18.31 35.48 18.47 35.40 18.62 40 41 36.53 18.61 ! 36.45' iS.77 36.37 18.93 36.28 19.09 41 42 37.42 19.07 37.34 19.52 ! 38.23 19.23 37.25 19.39 37.17 19.56 42 43 38.31 19.69 38.l4j 19.86 38.05 20.02 43 44 39.20 19.98 139.12 20.15 39.03 20.32 33.94 20.49 44 45 40.10 20.43 140.01 20.60 39.92 20.78 39.82 20.95 45 46 40.99 20.88 140.89 21.06 140.80' 21.24 40.71 21.42 46 47 41.88 i 21.34 41.78 21.52 41.691 21.70 41.59 21.88 47 4S 42.77 21.79 42.67 21.98 42.. 58 ! 22.16 42.48 22.35 48 49 43.66 ,22.25 43.56 22.44 43.45 22.63 43.36 22.82 49 50 44.55 ! 22.70 144.45 22.89 44.35 23. Oj 4t.25 Dep. 23 28 Lat. 50 ci o 1 Dep. 1 Lat. Dep. Lat, Dep. Lat .2 Q II 63 Deg. 621 Deg. 621 Deg 6'* Deg. TRAVERSE TABLE. 127 p 27 Deg. 27i Deg. 27i Deg. 27| Deg. a a 3 Lat. Dep. 1 Lat. 1 Dep. Lat. Dep. i Lat. Dep. 51 Il5.4i 123.15 45.34 23.35 '45.24 23.55 45.13 1 23.75 'sT 62 16.33 23.61 '46.23 ,23.81 ; 48.12 24.01 46.02 124.21 52 53 47.22 24.06 47.12 24.27 47.01 24.47 46.90! 24.68 53 54 4S.11 24.52 43.01 24.73 47.90 24.93 47.79 1 25.14 54 55 49.01 24.97 43.90; 25,18 43.79 25.40; 48.67 25.61 55 56 49.90 25.42 49.73 I 25.64 49.67 25.86' 49.56 26.07 56 57 50.79 1 25.83 ' 50.67 26.10 50.56 26.32 .50.44 126.54 57 58 51.63 1 26.33 51.56 26.56 51.45 26.73 51.33 27.01 58 59 52.57 126.79 52.45 27.01 ,52.33 27.24 52.21 127.47 59 60 61 53.46' 27.24 54.35 27.69 1 53.34 27.47 54.23 27.93 ,53.22 27.70 1 .53.10 127.94 53.98 1 23.40 60 61 54.11 23.17 62 55.24 28.15 55.12, 23.39 54.99 23.63 .54.87 j 23.37 62 63 56.13: 23.60 56.01 ! 23.85 55.83 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 , 53.63 30.22 53.54 30.43 53.41 130.73 66 67 59.70 .30.42 59.56 ; 30.68 159.43 30.94 59.29 31.20 67 63 60.59 30.87 60.45 31.14 160.33 31.40 60.18 31.66 68 69 61.48 31.33 : 61.34: 31.59 161.20 3].86' 61.06 32.13 69 70 71 62.37 31.78 i 62.23: 32.05 162.09 32.32 61.95 32.59 70 71 63.26 32.23 , 6?. 12 ,32.51 162.98 32 . 73 62.83 133.06 72 64.15 32.69 [G4.01 32.97 : 63.86 33.25 63.72 1 33.52 72 73 65.04 33.14 ! 64.90 33.42 164.75 33.71 64.60 33.99 73 74 65.93 33.60 1 65.79 133.83 '65.64 34.17 65.49 134.46 74 75 66.83 34.05 66.63 34.34 1 66.53 34.63 66.37 1 34.92 75 76 67.72 34.50 : 67.57 34.80 ! 67.41 35.09 67.26 35.39 76 77 63.61 34.96 ; 68.45 35.26 163.30 35.55 63.14 35.85 77 78 69.50 35.41 1 69.34 35.71 I 69.19 136.02' 69.03 136.32 78 79 70.39 35.87 1 70.23 136.17 11 70.07 36.43 69.91 136.73 79 80 81 71.28 36.32 36.77 71. 12 j 36.63 I: 70.96 36.94 37.40 70.80 137.25 80 81 72.17 72.01 37.09 Ij 71.85 71.63 37.71 82 73.06 37.23 72.90 37.55 1 72.73 37.86! 72.57 33.13 82 83 73.95 37.68 73.79 38.00 ij 73.62 33.33 73.45 33.65 83 84 74.84 38.14 74.68 33.46 ii 74.51 33.79 74.34 1 39.11 84 85 V5.74 38.59 75.57 33.92 Ii 75.40 39.25 75.22 39.58 85 86 76.63 39.04 76.46 39.38 i 76.23 39.71 ' 76.11 40.04 86 87 77.:.2 139.50 77.34 39.83 77.17 40.17 76.99 40.51 87 88 78.41,39.95 78.23 40.29 1,78.06 140.63 77.88 40.97 88 89 79.30:40.41 79.12 i 40.75 ji 78.94 141.10 73.76 41.14 89 90 91 80.19' 10.86 81.08 41.31 i SO. 01 i41.21 80.90 ,41.67 i 79.83 141.56 79.65 41.91 90 91 180.72,42.02 80.53 42.37 92 81 97 41.77 1 81.79 42.12!; 81.60 '42.48 81.42 i 42.84 92 93 82.86 42.22 1 82.68 42.53 i 82.49 42.94 82.30 i 43.30 93 94 83.75 42.68 t 83.57 43.04 I 83.33 43.40 83.19 43.77 94 95 84.65 43.13 84.46 ; 43.50 | 84.27 43.87 84.07 44.23 95 96 85.54 43.. 53 85.35 '43.96 85.15 144.33 84.96 44.70 • 96 97 86.43 44.04^ 86.23 i 44.41 86.04 1 44.79 85.84 45.16 97 9S 87.32 44.49- 87.12 144.87 86.93 145.25 86.73! 45.63 98 99 83.21 44.9') 1 88.01 45.33 i 87.81 45.71 i 87.61146.10 99 iOO i 3 89.10 Dep. 45.40 i 88.90 45.79 88.70 46.17; 88.50 146.56. Dep. 1 Lat, , 100 .5 I.at. Dep. Lat. Dep. Lat. 1 63Deg. 62i Deg. 621 Deg. , 62i Deg, 128 TRAVERSE TARLE. 1' p 23 Deg. 28i Beg. 28|- Deg. 281 Deg. a.' '"l Lat. Dep. Lat. 0.88 Dep. 0.47 Lat. Dep. Lat. 0.88 Dep. 0.48 "T 0.88 0.47 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 3.62 1.89 3.52 1.91 3.51 1.92 4 5 : 4.41 2.35 4.40 2.37 4.39 2.39 4.38 2.40 5 6 : 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 3.85 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 9.69 4.73 5.21 8.79 9.67 4.77 5.25 8.77 4.81 10 11 9.71 5.16 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 11.40 6.25 13 14 12.36 6.57 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 30 21 17.66 9.39 17.62 9.47 17.58 18.46 9.54 10.02 17.53 9.62 20 21 18.54 9.86 18.50 9.94 18.41 10.10 22 19.42 10.33 19.38 10.41 19.33 10.50 19.29 10.58 22 23 20.31 10.80 20.26 10.89 20.21 10.97 20.16 11.06 23 24 21.19 11.27 21.14 11.36 21.09 11.45 21.04 11.54 24 25 22.07 11.74 22.02 11.83 21.97 11.93 21.92 12.02 25 26 22.96 12.21 22.90 12.31 22.85 12.41 22.79 12.51 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 24.61 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 15.37 28.06 15.39 32 33 29.14 15.49 29.07 15.62 29.00 15. 7e 28.93 15.87 33 34 30.02 15.96 29.95 16.09 29.88 16.22 29.81 16.35 34 35 30.90 16.43 30.83 16.57 30.76 16.70 30.69 16.83 35 36 31.79 16.90 31.71 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 35.32 18.78 35.24 18.93 35.15 19.09 35.07 19.24 40 41 41 36.20 19,25 36.12 19.41 36.03 19.56 35.95 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 37.88 20.35 .37.79 20.52 37.70 20. 08 43 44 38.85 20.66 38.76 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 45 46 40.62 21.60 40.52 21.77 40.43 21.95 40.33 22.13 46 47 41.50 22.07 41.40 22.25 41.30 22.43 41.21 22.61 47 48 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 g 1 44.15 23.47 44.04 23.67 43.94 23.86 43.84 24.05 50 '1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. 1 Lat. 62 1 )eg. 61| Deg. 61^ Deg. 6U Deg. TRAVERSE TABLE. 129 OD 13 O a 28 Deg. 28i Deg. 28i Deg. 281 Deg. 'rr. i 51 Lat. Dep. Lat. 44.93 Dep. Lat. Dep. Lat. Dep. 45.03 23.94 24.14 44.82 i 24.34 44.71 24.53 52 45.91 24.41 i 45.81 24.61 i 45.70 124.81 45.59 25.01 52 53 46.80 24.88 1 46.69 25.09 46.58 1 25.29 46.47 25.49 53 54 47.68 25.35! 47.57 25.56 47.46 125.77 47.34 25.97 54 55 48.56 25.82' 48.45 26.03 48.33 26.24 48.22 26.45 55 56 49.45 26.29 26.76 49.33 26.51 49.21 '26.72! 49.10 26.94 56 57 50.33 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 52.93 28.17 i 52.85 1 28.40] .52.73 28.63 52.60 28.86 60 61 53.86 28.64 53.73 23.87 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 50.24 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 61.60 33.13 61.52 33.40 61.37 33.67 70 71 62.69 62.54 33.61 62.40 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 i 65.75 36.07 75 76 67.10 35.68 66.95 35.97 66.79 36.26 166.63 36.56 70 77 67.99 36.15 67.83 36.45 67.67 36.74 1 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 V9 80 81 70.64 37.56 70.47 37.87 70.31 38.17 70.14 38.48 38.96 80 81 71.52 38.03 71.35 38.34 71.18 38.65 71.01 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 79.78 43.77 92 81.23 43.19 81.04 43.55 80.85 43.90 80.66 44.25 92 93 82.11 43.66 81.92 44.02 81.73 44.38 81.54 44.73 93 94 183.00 44.13 82.80 44.49 82.61 44.85 82.41 45. PI 94 95 83.88 44.60 83.68 44.97 83.49 45.33 83.29 45.19 95 95 84.78 45.07 84.57 45.44 84.37 45.81 84.17 46.17 90 97 '85.65 45.54 85.45 45.91 85.25 46.28 85.04 46.66 97 98 186.53 46.01 86 33 46.39 86.12 46.76 85.92 47.14 98 99 87.41 146.48 87.21 46.86 87.00 47.24 86.80 47.62 99 100 § a S .3 Q 88.29 |46.95 88.09 47.33 87.88 47.72 87.67 48.10 100 i c a Dep. Lat. Dep. Lat. Dep, Lat. Dep. Lat. 1 1 62 Deg. 61| Deg. eU Deg. 6ii Deg. 1 130 TRAVERSE TAELE. § 1 29 Deg. 1 2n Deg. 29i Deg. 29J Deg. Lat. "0787 Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.48 0.87 0.49 0.87 0.49 0.87 0..'J0 2 1.75 0.97 1.74 0.98 1.74 0.98 1.74 0.99 2 3 2.62 f 1.45 2.62 1.47 2.61 1.48 2. GO 1.49 3 4 3.50 1.94 3.49 1.95 3.48 1.97 3.47 1.98 4 5 4 37 2.42 4.36 2.44 4.35 2.46 4.34 2.48 5 6 5.25 2.91 6.23 2.93 5.22 2.95 5.21 2.98 6 7 6.12 3.39 0.11 3.42 6.09 3.45 6.08 3.47 7 8 7.00 ( 3.88 6.98 3.91 6.96 3.94 6.95 3.97 8 9 7.87 4.36 7.85 4.40 7.83 4.43 7.81 4.47 9 10 11 8.75 9.62 4.85 5.33 8.72 4.89 8.70 4,92 8.68 4.90 10 11 9.60 5.37 9.. 57 5.42 9.55 5.46 12 10.50 5.82 10.47 5.88 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 17.49 9.70 10.18 17.45 18.32 9.77 10.26 17.41 18.28 9.85 10734 17.30 9.92 20 21 21 18.37 18.23 10.42 22 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 20.02 11.33 19.97 11.41 23 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 24.31 13.89 28 29 25.36 14.06 25.30 14.17 25.24 14.28 25.18 14.39 29 30 26.24 14.54 26.17 14.06 26.11 26.98 14.77 15.27 26.05 14.89 30 31 31 27.11 15.03 "27.05 15.15 26.91 15.38 32 27.99 15.51 27.92 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 29.74 16.48 29.66 16.61 29.59 16.74 29.52 16.87 34 35 30.61 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 32.12 18. .36 37 38 33.24 18.42 33.15 18.57 33.07 18.71 32.99 18.86 38 39 34.11 18.91 34.03 19.06 33.94 19.20 33.86 19.35 39 40 41 34.98 19.39 34.90 19.54 34.81 19.70 34.73 19.85 40 41 35.86 19.88 35.77 20.03 35.68 20.19 35.60 20.34 42 36.73 20.36 36.64 20.52 36.55 20.68 36.46 20.84 42 43 37.61 20.85 37,52 21.01 37.43 21.17 37.33 21.34 43 44 38.48 21.33 38.39 21.50 38.30 21.67 38.20 21.83 44 45 '39.36 21.82 39.26 21.99 39.17 22.16 39.07 22.33 45 46 1 40.23 22.30 40.13 22.48 40.04 22.65 39.94 22.83 46 47 41.11 22.79 41.01 22.97 40.91 23.14 40.81 23.32 47 48 41.98 23.27 41.88 23.45 41.78 23,68 41.67 23.82 48 •^9 42.86 23.70 42.75 23.94 42.65 24.13 42.54 24.31 49 50 i3.73 24.24 Lat. 43.62 24.43 43.52 24.62 43.41 _24.^1 50 8 a Dep. Dep. Lat. Dep. Lat. Dep. Lat. 61 Deg. 601 Deg. eOh Deg. 60i Deg. 1 TRAVERSE TABLE. 131 o * 1 29 Beg. 29i Deg. 29A Deg. 291 Deg. D "51 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 25731 "51 144.61 1 24.73 1 44.50 24.92 44.39 ■25. IT 44.28 62 45.48 25.21 45.37 25.41 45.26 25.61 45.16 25.80 62 63 46.35 25.69 46.24 25.90 46.13 26.10 46.01 26.30 53 d4 47.23 26.18 47.11 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 65 56 48.98 27.15 48.86 27.36 48.74 27.68 48.62 27.79 66 67 49.85 27.63 49.73 27.85 49.61 28.07 49.49 28.28 57 58 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 61.22 29.28 69 60 61 52.48 29.09 29.57 52.35 29.32 52.22 29.55 52.09 52.96 29.77 60 61 53.35 53.22 29.81 53.09 30.04 30.27 62 54.23 30.00 54.09 30.29 53.96 30.63 53.83 30.77 62 63 55.10 30.54 54.97 30.78 54.83 31.02 54.70 31.26 63 64 55.98 31.03 .55.84 31.27 65.70 31.62 65.56 31.76 64 65 56.85 31.51 56.71 31.76 56.57 32.01 66.43 32.25 65 66 57.72 32.00 57.58 32.25 57.44 32.60 67.30 32.75 66 67 .58.60 32.48 58.46 32.74 58.31 32.99 58.17 33.25 67 68 .59.47 32.97 59.33 33.23 59.18 33.48 69.04 33.74 68 69 60.35 33.45 60.20 33.71 60.05 33.98 ,59.91 34.24 69 70 71 61.22 33.94 61.07 34.20 60.92 34.47 60.77 34.74 70 62.10 34.42 61.95 34.69 61.80 34.96 61.64 35.23 71 72 62.97 34.91 62.82 35.18 62.67 35,46 62.61 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 76 76 66.47 36.85 66.31 37.14 66.15 37.42 66.98 37.71 76 77 67.35 37.33 67.18 37.62 67.02 37.92 66.86 3§.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 70.84 38.78 69.80 39.09 69.63 39.39 69.46 39.70 80 81 39.27 70.67 39.58 70.50 39.89 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 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 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 78.14 44.66 90 91 79.59 44.12 79.20 44.81 79.01 46.16 92 80.46 44.60 80.27 44.95 80.07 45.30 79.87 45.65 92 93 81.34 45.09 81.14 45.44 80.94 45.80 80.74 46.15 93 94 82.21 45.57 82.01 45.93 81.81 46.29 81.61 46.64 94 95 83.09 46.06 82.89 46.42 82.68 46.78 82.48 47.14 95 96 83.96 46.54 83.76 45.91 83.55 47.27 83.35 47.64 96 97 84.84 147.03 84.63 47.40 84.42 47.77 84.22 4S.13 97 98 85.71 47.51 85.50 47.88 85.29 48.26 86.08 48.63 98 99 86.59 48.00 86.38 48.37 86.17 48.76 85.95 49.13 99 100 l1 87.46 48.48 87.25 48.86 Lat. 87.04 1 49.24 86.82 49.62 Lat. 100 1 Dep. Lat. Dep. Dep. Lat. Dep. 61 Deg. 601 Deg. 601 Deg. 60J Deg. 23 132 TRAVERSE TABLE. 1, 1 30 Deg. 30i Deg. 30^ Deg. 301 Deg. rs T Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.87 0..50 0.83 0.50 0.86 0.51 0.86 o.sf 2 1.73 1.00 1.73 1.01 1.72 1.02 1.72 1.02 2 3 2.00 1.50 2 59 1.51 2.58 1.52, 2 58 1.53 3 4 3.40 2.00 3.46 2.02 3.45 2.03 3 44 2.05 4 5 4.33 2.50 4.32 2.52 4.31 2.54 4 30 2.56 5 6 5.20 3.00 5.18 3.02 I 5.17 3.051 5.16 3.07 6 7 6.06 3.50 6.05 3.53 6.03 3.55 6.02 3.58 1 8 6.93 4.00 6.91 4.03 6.89 4.06 6.88 4.09 8 9 7.79 4.50 7.77 4.53 7.75 4.57] 7.73 4.60 9 10 11 8.66 5.00 8.64 9.50 5.04! 5.54 8.62 5.08 1 8.59 5.11 .0 :1 9.53 5.50 9.48 5.58 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. IS 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 17.19 10.23 20 21 18.19 10.50 18.14 10.58 18.09 10.66 18.05 10.74 22 19.05 11.00 19.00 11.08 18.96 11.17i 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 35.33 20.30 34.38 20.45 40 41 35.51 20.50 35.42 20.65 20.81 35.24 20.96 42 36.37 21.00 36.28 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 23.01 45 46 39.84 23.00 39.74 23.17 39.63 23.35 39.53 23.52 46 47 40.70 23.50 40.60 23.68 40.50 23.85 40.39 24.03 i"? 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 49 60 g 43.30 25.00 43.19 25.19 43.08 25.38 42.97 25.56 50 Dep. Lat, Dep. Lat. Dep. Lat. Dep. Lat. 6 o c 2 GO] Dcg. 691 Deg. 59 1 Deg. 59i Deg. TRAVERSE TABLE. 133 c p 51 30 Deg, 30i Deg. 30i Dc^. 301 Dog. C B n a Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 44.17 25.50 44. p6 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 26.59 52 53 45.90 26.50 45.78 26.70 45.67 26.90 45.55 27.10 63 54 46.77 27.00 46.65 27.20 46.53 27.41 46.41 27.61 54 55 47.63 27.50 47.51 27.71 47.39 27.91 47.27 28.12 5.T 56 48.50 28.00 48.37 28.21 48.25 28.42 48.13 28.63 56 57 49.36 28.50 49.24 28.72 49.11 28 93 48.99 29.14 67 68 50 23 29.00 50.10 29.22 49.97 29.44 49.85 29.65 58 59 51.10 29.50 50.97 29.72 50.84 29.94 50.70 30.17 59 60 61 51.96 30.00 51.83 52.69 30.23 51.70 30.45 51.56 30.68 60 61 52.83 30.50 30.73 52.56 30.96 52.42 31.19 62 63.69 31.00 53.56 31.23 53.42 31.47 53.28 31.70 62 63 54.56 31.50 54.42 31.74 54.28 31.97 54.14 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 33.25 ,56.87 33.50 .56.72 33.75 66 67 58.02 33.50 57.88 33.75 57.73 34.01 57.58 34.26 67 68 58.89 34.00 58.74 34.26 58.59 34.51 .58.44 34.77 68 69 59.76 34.50 59.60 34.76 59.45 35.02 59.30 35.28 69 70 71 60.62 35.00 60.47 35.26 60.31 35.53 60.16 35.79 70 71 61.49 35.. 50 61.33 35.77 61.18 36.04 61.02 36.30 72 62.35 36.00 62.20 36.27 62,04 36.54 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 65.65 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 39.59 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 40.30 68.93 40.60 68.75 40.90 80 81 70.15 40.50 69.97 40.81 69.79 41.11 69.61 41.41 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 72.56 42.32 72.38 42.63 72.19 42.95 84 85 73.61 42.50 73.43 42.82 73.24 43.14 73.05 43.46 85 86 74.48 43.00 74.29 43.32 74.10 43.65 73.91 43.97 86 87 75.34 43.50 75,15 43.83 74.96 44.16 74.77 44.48 87 88 76.21 44.00 76.02 44.33 75.82 44.66 75.63 44.99 88 89 77.08 44.50 76.88 44.84 76.68 45.17 76.49 45.51 89 90 91 77.94 45.00 77.75 45.34 77.55 45.68 77.35 46.02 90 91 78.81 45.50 78.61 45.84 78.41 46.19 78.21 46.53 92 79.67 46.00 79.47 46.35 79.27 46.69 79.07 47.04 92 93 80.54 46.50 80.34 46.85 80.13 47.20 79.92 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 81.85 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 97 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 99 85.74 49. 5i) 85.52 49.87 85.30 60.25 85.08 50.62 99 100 § e s .2 86.60 Dep. 50.00 86.38 50.38 j 86.16 50.75 85.94 51.13 Lat. 100 d V c O Lat. Dep. Lat. Dep. Lat. Dep. 60 Dog. 591 Deg. 69i Deg. 594 Deg. 134: TRAVERSE TABLE P. 31 Deg. 3U Dog. 3U Deg. 311 Deg. a i I .2 Lat. "07S6' Dep. Lat. 0.85 Dep. Lat. 1 Dep. Lat. Dep. 0.51 0752" 0.85 0.52 0.85 0..53 1 fi : 1.71 1.03 1.71 1.04 1.71 1.04 1.70 1.05 2 3 1 2.57 1.55 2.56 1.56 2.56 1.57 2.55 1.58 3 4, 3.43 2.06 3.42 2.08 3.41 2.09 3.40 2.10 4 5 4.29 2.58 4.27 2.59 4.26 2.61 4.25 2.63 5 e . 6.14 3.09 5.13 3.11 5.12 3.13 5.10 3.16 6 7 6.00 3.61 5.98 3.63 5.97 3.66 5.95 3.68 7 8 6.86 4.12 6.84 4.15 6.82 4.18 6.80 4.21 8 9 7.71 4.64 7.09 4.67 7.67 4.70 1 7.65 4.74 9 10 8.57 5.15 8.55 5.19 8. .53 I 5.22 8.50 5.26 10 11 9 43 5.67 9.40 5.71 9.38 5.75] 9.35 5.79 11 12 10.29 6.18 10.26 6.23 10.23 6.27 10.20 6.31 12 13 11.14 6.70 11.11 6.74 11.08 6.79 11.05 6.84 13 14 12.00 ! 7.21 1 11.97 7.26 11.94 7.31 11.90 7.37 14 15 12.86 7.73 12.82 7.78 12.79 7.84 12.76 7.89 1 15 16 13.71 8.24 13.68 8.30 13.64 8.36 13.61 8.42' 16 17 14.57 8.76 14.53 8.82 14.49 1 8.83 14.46 8.95 17 18 15.43 9.27 15.39 9.34 15.35 1 9.40 15.31 9.47 18 19 16.29 9.79 16.24 9.86 16.20} 9.93 16.16 10.00 19 20 21 17.14 10.30 17.10 10.38 17.05 10.45 17.01 10.52 20 18.00 10.8:', 1 17.95 10.80 17.91 10.97 17.86 11.05 21 22 18.86 11.33 18.81 11.41 18.76 11.49 18.71 11.58 22 23 19.71 11.85 19.66 11.93 19.61 12.02 19.56 12.10 23 24 20.57 12.36 20.52 12.45 20.46 12.54 20.41 12.63 24 25 21.43 12.88 21.37 12.97 21.32 13.06 '21.2.6 13.16 25 26 22.29 13.39 22.23 13.49 22.17 13.53 i22.ll 13.68 26 27 23.14 13.91 23.08 14.01 23.02 ' 14.11 22.96 14.21 27 28 24.00 14.42 23.94 14.53 23.87 i 14.63 23.81 14.73 28 29 24.86 14.94 24.79 15.04 24.73 15.15 ,24.66 15.26 29 30 31 25.71 15.45 25.65 15.50 25.58 ^6.43 15.67 16.20 125.51 126.36 15.79 30 26.57 15.97 26.5a 16.08 16.31 31 32 27.43 1 16.48 27.36 16.60 27.28 16.72 •27.21 16.84 32 33 28.29 17.00 28.21 17.12 28.14 17.24 ! 28.06 17.37 33 34 29.14 17.51 29.07 17.64 28.99 17.76 128.91 17.89 34 35 30.00 18.03 29.92 18.16 29.84 18.29 29.76 18.42 35 36 30.86 18. .54 30.78 L8:.68 30.70 18.81 30.61 18.94 36 37 31.72 19.06 31.63 19.19 31.55 19.33 31.46 19.47 37 38 32.57 19.57 32.49 19.71 32.40 19.85 I33..3I 20.00 38 : 39 33.43 20.09 33.34 20.23 1 33.25 20.38 |33.16 20.52 39 40 4l' 34.29 20.60 34.20 20.75 34.11 20.90 134.01 21 On 40 35.14 21.12 35.05 21.27 34.96 21.42 34.86 21.57 U 42 36.00 21.63 35.91 21.79 35.81 ,21.94 35.71 22.10; 12 I 43 36.86 22.15 36.76 22.31 36.66 22.47 .^,6.57 22.63 43 44 37.72 22.66 37.62 122.83 37.52 22.99 37.42 23.15 44 45 38.57 23.18 38.47 23.. 34 38.37 23.51 38.27 23.68 45 46 39.43 23.69 39.33 23.86 39.22 24.03 139.12 24. SI 46 47 40.29 24.21 40.18 24.38 40.07 24.56 '39.97 24.73. 47 I 48 41.14 24.72 41 04 24.90 40.93 25.08 40.82 25.26 48 49 42.00 25.24 41.89 25.42 41.78 25.60 41.67 25.78 49 _50^|42.86 25.75 42.75 25.94 42.63- 26.12 42.52 26.31 60 5 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 5&Deg. 58| Deg. 58i Deg. 58J Dpg, TRAVERSE TABLE 135 o 3 p "51 31 Deg. 2H Deg. 3li Deg. 3U Deg. p Lat. Dcp. Lat. Dep. Lat. Dep. Lat. Dep. 43.72 126.27 '43.60 26.46 43.48 26.05' 43.37 26.84 52 44.57 126.78 44.46 26.98 44.34 27.17 44.22 27.36 52 53 45.43 I27.30 45.31 i 27.49 |i 45. 19. 27.69^ 45.07 27.89 5P; 54 46.29 :27.81 46.17! 28.01 46.04; 23.21 45.92 23.42 54 55 47.14 123.33 47.02 28.53 46.90 1 28.74 46, >7 28.94 55 56 48.00 '23.84 47.83 29.05 |i 47.75 29.26. 47.62 (29.47 56 57 43.86 29.36; 48.73:29.57 48.60 29.78' 48.47 129.99 57 5S 49.72 29.87, 49.53 30.09 49.45 30.30 : 49.32! 30.52 58 59 50.57 30.39! 50 44 30.61 50.31 30.83 .50. 17 i 31.05 59 60 61 51.43 30.90 i 52.29 31.42' 51.29 31.13 51.16 31.35 1 51.02 131.57 60 61 52.15 31.65 ; 52.01 |31.87 51.87 132.10 62 53.14 31.93 1 53.00 32.16, 52.86 32.39 52.72 1 32.63 62 63 54.00 32.45 53.86 32.68 1 53.72 132.92 53.57 133.15 63 64 54.86 32.96 54.71 33.20 154.57 33.44 54.42 133.63 64 G5 55.72 33.43 55.57 33.72 i 55.42 33.96 55.27 '34.20 65 66 56.57 33.99 56.42 34.24 56.27 34.43 56.12 34.73 66 67 57.43 34.51 57.23 34.76 57.13 35.01 56.93 .35.26 67 63 53.29 35.02 53.13 35.23 57.93 35.53 57.82 35.73 68 69 59.14 35.54 53.99 35.80 53.83 36.05 53.67 36.31 69 70 71 60.00 36.05 60.86 36.57 59.84 36.31 , 59.63 ' 36.57 59.52 36.83 60.37 37.36 70 71 60.70 36.83 60.54 i 37.10 72 61.72 37.03 61.55 37.35 61 .39 | 37.6-:4 61.23 37.89 72 73 62.57 37.60 62.41 37.87 62.24 j 33. 14 62.08 33.41 73 74 63.43 33.11 63.26 33.89 63.10 33.66 62.93 33.94 74 75 64.29 33.63 64.12 33.91 ; 63.95 1 39.19 63.73 39.47 75 76 65.14 39.14 64.97 39.43 ' 64.80 i 39.71 64.63 39.99 76 77 66.00 39.66, 65.83 39.95 , 65.65 40.23 65.43 40.52 77 78 C6.S6 40.17 i 66.68 40.46 66.51 40.75 66.33 41.04 78 79 67.72 i 40.69 67.54 40.93 ; 67.36 41.23 67.13 41.57 79 80 68.57 41.20 : 68.39 41.50 ; 63.21 41.80 , 63.03 42.10 80 81 69.43 41.72 ■ 69.25 42.02 ■.: 69.06 42.32 63.83 1 42.62 81 62 70.29 ; 42.23 i 70.10 42.54 69.92:42.84 69.73 43.15 82 i 83 71.14,42.75 70.96 43.06 70.77 : 43.37 70.53 : 43.68 83 84 72.00 i 43.26 71.81 43.53 ,71.62 43.39 71.43 144.20 84 85 72.86 ! 43.78. 72.67 44.10 72.47 44.41 72.23 44.73 85 86 73.72 44.29 73.52 4-1.61 73.33 44.93 73.13 45.25 86 87 74.57 44.81 74.33 45.13 74.18 : 45.46 73.93 45.73 87 88 75.43 45.32 75.23 45.65 75.03 45.93 74.83 46.31 88 89 76.29 45.34; 76.09 46.17 75.33 46.50 75.63 46.83 89 90 91 77.15 46.35 76.94 46.69 76.74 47.02 77.80 47.21 77.59 ,47.55 76.53 47.36 77.33 ; 47.89 90 '91 78.00 46.87 92 78.86:47.33 : 78.65 47.73 i: 78.44 43.07 73.23 43.41 92 93 79.72 ! 47.90 1 79.51 48.25 179.30 48.59 79.03 43.94 93 94 80.57 '43.41 i 80.36 43.76 80.15 49.11 79.93 49.47 94 U5 81.43 48.93 1 81.22 49.23 81.00 49.64 80.78 j 39 32.33 21.81 32.24 21.95 32.14 22.09 32.04 22.23 39 40 33.16 i 22.37 33.06 22.51 32.97 33.79 22: 06 32.87 22.80 40 41 41 33.99 1 22.93 33.89 23.07 23.22 33.69 23.37 42 34.82 23.49 34.72 23 64 34.61 23.79 34.51 23.94 4^ 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 70 36.26 24.92 36.15 25.03 44 45 37.31 25.16 37.20 25 .33 37.09 25.49 36.i,7 25.65 45 46 .38.14 25.72 38.02 25 89 37.91 26.05 37.80 126.22 46 47 38.96 26.28 38.85 26.45 38.73 26.62 38.02 126.79 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 27.58 40.38 27.75 40.26 27.93 49 50 41.45 27.96 41.33! 28.14 41.21 28.32 41.08 28.50 60_ 1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 56 Deg. 55J Deg. 55^ Deg. 5h^ Deg. TRAVEKSE TABUS 141 34Deg. 34i Deg. 34ADeg. 341 Deg. c i Lat. i Dep. Lat. Dep. ji Lat. ii Dep. ! Lat. Dep. T] ! 42.23 128.52: 42.16 28.70 42.03 28.89 i 41.90 29.07 51 62 ; 43.11 ■ 29.08 ij 42.98 29.27 ii 42.85 1 29.45 42.73 29.64 52 63' 43.94 29.64 1 43.81 29.83 143.68 '■ .30.02 1 43. .55 30.21 53 54 44.77 30.20 1 44.64 30.39 ! 44.50! 30.59 j 44.37' 30 78 | 54 55 45.60 30.76 1 45.46 30.95:145.33 i 31.15 45.19 31.35 1 55 56 40.43 31.31 '46.29 31.52 '46.15 : 31.72 46 01 31.92 i 56 57 47.26 31.87 47.12 32.08; 46. 9S! 32 29 46.83 32.49 ' 57 58 43.08 32.43 47.94 32.64 Ii 47.80 132.85 47.66 33.00 ' 53 59 48.91 ; 32.99 48.77 33.21 ! 48.62 33.4,2 48.48 33.63 i 59 60 49.74 33.55 49.60 61 50.57,34.11:50.42 33.77 1)49.45 33.98 49.30 34.20 60 '61 34.33 '50.27 34.55 50.12 34.77 1 62 51.40 34.67 !| 51.25 34.89;;51.10 35.121 50.94 35.34 62 63 52.23 35.23 jl 52.03 35.46 !j 51.92 35.68 51.76 35.91 63 64 53.06 35.79 52.90 36.02 152.74 36.25 1 52.59 36.48 64 65 53.89 36. 35 1 53.73 36.53 1' 53.57 36.82 i 53.41 37.05 i 65 66 54.72 36.91 ; 54.5c 37.15 ''54.39 37.38; 54.23! 37.62 66 67 55.55 37.46 55.38 37.71 ;l 55.22 i 37.95 i 55.05 38.19! 67 08 56.37 38.03 56.21 38.27, 56.04 38.52! 55.87 38.76 68 69 57.20 38.58 i 57.03 38.83 56.86 39.08 1 56.69 39.33 69 70 " 71 5S.03 39. 14 {157.86 58.86 39.70 '58.69 39.40 j 39.96! 57.69 39.65 1 40.21 1 57.52 39.90 70 71 58.51 58.34 40.47 72 59.69 40.26 ' 59.51 40.52! 59.34 40.78 i 59.16 41.04 72 73 60.52 40.82 160.04 41.08 1 60.16 41.35: 59.98 41.61 73 74 61.35 ,41.38 ]\ 61.17 41.65 60.99 41.91 i 60.80 42.18 74 75 62.18 41. 94i| 61.99 42.21 61.81 42.48 61.62 42.75 75 76 63.01 1 42.50 1162.82 63.84 i 43.06 II 63.65 42.77 62.63 43.05 62.45 43.32 76 77 43.34 63.46 43.61 63.27 43.89 77 78 64.66 i 43.62 ii 64.47 43.90 64.28 44.18 64.09 44.46 . 78 79 65.49 [44. 18 1165.30 44.46 65.11 44.75 64.91 45.03 79 80 81 66.32 67.15 44.74 66.13 45.02 45.59 65.93 45.31 65.73 45.60 80 81 45.29 66.95 66.75 45.88 66.55 46.17 82 67.98 45.85 1 67.78 46.15 67.. 58 46.45 67.37 46.74 82 83 68.81 46.11 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 47.53 1! 70.26 47.84 70.05 48.14 69.84 48.45 85 86 71.30 48.09 l! 71.09 48.40 70.87 48.71 70.66 49.02 86 87 72.13 48.65 1171.91 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 11 73.57 50.09 i 73.35 50.41 73.13^ 50 73 89 90 91 74.61 50.33 1 74.39 50.65 1 74.17 50.98 ; 73.95, 51.30 90 91 75.44 50.89 75.22 51.22 1 75.00 51.54 174.77 51.87 92 76.27 51.45 76.05 51.78 175.82 52.11 175.59 52.44 92 93 77.10 1 52.00 76.87 52.34 176.64 52.68 76.41 53.01 93 94 77.93 52.56 77.70 52.90 j 77.47 53.24 177.23 53.58 94 95 78.76 53.12 78.53 53.47 78.29 53.81 .78.06 54.15 l)n 96 79.59 53.68 79.35 54.03 179.12 54.37 .78.88 54.72 90 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 93 99 82.07 55.36 81.83 55.72 81.59 56.07 :si..34 56.43 99 100 1 82.90 55.92 82.66 56.28 182.41 56.64 1 82.16 57.00 100 a 'oa Dep. Lat, Dep. Lat. Dep. Lat. j Dep. I Lat. 56 Deg. 551 Deg. 554 Deg. 55i Deg, 142 TRAVERSfi TABLE. § 35 Deg. 35i Deg. 35i Deg. 351 Deg. 5 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. 0.81 0.58 1 0.82 0.57 0.82 0.58 0.81 1 0.58 2 1.64 1.15 1.63 1.15' 1.63 1.16 1.62 1.17 2 3 2.46 1.72 2.45 1 1.73. 2.44| 1.74 2.43 1.75 3 4 3.28 2.29 3.27 1 2.31 1 3.26! 2.32 3.25 2.34 i 4 5 4.10 2.87 4.08 2.89 1 4.07 1 2.9ff| 4.06 2.92 1 5 6 4.91 3.44 4.90 3.46 1 4.83 3.48 4.87 3.51, 3 7 5.73 4.01 5.72 4.04 1 5.70 4.06 5.68 4.09 7 \ 8 6.55 4.59 6.53 4.62 6.51 4.65 6.49 4.67 S i 9 7.37 5.16 7.35 5.19 1 7.33 5.23 7.30 6.26 9 j 10 8.19 5.74 8.17 5.77 1 8.14 5.81 8.12 8.93 5.84 6.43 10 11 " 11 9.01 6.31 8.93 6.35 8.96 6.39 1 12 9.83 6.88 9.80 6.93 9.77 6.97 9.74 7.01 12 13 10.65 7.46 10.62 7.50 10.. 58 7.55 10.55 7.60 13 M 11.47 8.03 11.43 8.08 11.40 8.13 11.36 8.18 14 15 12.29 8.60 12.25 8.66 12.21 8.71 12.17 8.76 15 16 13.11 9.18 13.07 9.23 13.03 9.29 12.99 9.35 16 17 13.93 9.75 13.83 9.81 13.84 9.87 13.80 9.93 17 18 14.74 10.32 14.70 10.39 14.65 10.45 14.61 10.. 52 18 19 15.56 10.90 15.52 10.97 15.47 11.03 15.42 11.10 19 20 21 16.33 11.47 16.33 11.54 16.28 11.61 IS. 23 11.63 20 17.20 12.05 17.15 12.12 17.10 12.19 17.04 12.27 21 22 18.02 12.62 17.97 12.70 17.91 12.78 17.85 12.85 22 23 18.84 13.19 18.78 13.27 18.72 13.36 18.67 13.44 23 21 19.66 13.77 19.60 13.85 1 19.54 13.94 19.48 14.02 24 25 20.48 14.34 20.42 14.43 20.35 14.52 120.29 14.61 25 26 21.30 14.91 21.23 15.01 21.17 15.10 21.10 15.19 26 27 22.12 15.49 22.05 15.58 21.98 15.68 21.91 15.77 27 28 22.94 16.06 22.87 16.15 22.80 16.26 22.72 16.36 28 29 23.76 16.63 23.68 16.74 23.61 16.84 23.54 16.94 29 30 24.57 17.21 17.78 24.50 17.51 24.42 17.42 124.35 25.16 17.53 30 ■31 "31 25.39 25.32 17.89 25.24 18.00 13.11 32 26.21 18.. 35 26.13 18.47 26.05 18.58 25.97 18.70 32 33 27.03 18.93 26.95 19.05 26.87 19.16 26.73 19.23 33 34 27.85 19.50 27.77 19.62 27.68 19.74 27.59 19.86 34 35 28.67 20.08 28.58 20.20 28.49 20.32 128.41 20.45 35 36 29.49 20.65 29.40 20.78 29.31 20.91 ! 29.22 21.03 36 37 30.31 21.22 30.22 21.35 30.12 21.49 130.03 21.62 37 38 31.13 21.80 31.03 21.93 30.94 22.07 30.84 22.20 33 39 31.95 22.37 31.85 22.51 31.75 22.65 31.65 22.79 39 40 32.77 22.94 32.67 23.09 23.66 32.56 23.23 23.81 32.46 33.27 23.37 23:95 40 41 41 33.59 23.52 33.48 33. 3S 42 i 34.40 24.09 34.30 24.24 34.19 24.39 [34.09 24.54 42 43 35.22 24.66 35.12 24.82 35.01 21.97 34.90 25.12 43 44 36.04 25.24 35.93 25.39 35.82 25.55 35.71 ; 25.71 44 45 36.86 25.81 36.75 25.97 36.64 26.13 36.52 26.29 45 46 137.68 26.38 37.57 26.55 37.45 26.71 137.33 26.88 46 47 1 33.50 26.96 38.38 27.13 38.26 27.29 38.14 27.46 ; 47 48 39.32 27.53 39.20 j 27.70 1 39.08 27.87 38.96 28.04 i 43 49 40.14 28 . 1 1 40.02; 28.28 39.89 ; 23.45 39.77 23.63, 49 50 40.96 28.68 40.83 1 28. 86 40.71 [29.04 ; 40.58 29.21 _50 c 3 6 o a a .2 Q Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 55] Deg. 54J Deg. 54.1 Deg. 544 Oeg. TJIAVEESE TABLE. 143 5 1 a CD 5l 35 Deg. 35i Deg. 1 35i Deg. -351 Deg. a" "51 Lat. 41.78 Dep. Lat. Dep. Lat. Dep. Lat. Dep. 29.80 29.25 41.65 129.43 41.52 29.62 41.39 62 42.60 29.83 42.47 30.01 42.33 30.20 42.20 30.38 62 63 43.42 3'?. 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 31.36 43.82 3 J. 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 43.33 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 35.64 60 61 49.97 34.99 49.82 35.21 49.66 35.42 49.51 62 50.79 35.56 50.63 35.78 50.48 36.00 50.32 36.22 62 63 51.61 36.14 51.45 36.36 51.29 36.58 51.13 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 05 66 54.06 37.86 53.90 38.09 53.73 38.33 53.56 38.56 66 67 54.88 38.43 54.71 38.67 54.55 38.91 54.38 39.14 1 67 I 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 57.16 40 40 56.99 40.65 .56.81 40.90 70 71 58.16 40.72 57.98 40.98 157.80 41.23 57.62] 41.48 72 58.98 41.30 58.80 41.55 58.62 41.81 53.43 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 1 77 63.07 44.17 62.88 44.44 i62.69 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 64.32 45.88 64.11 46.16 79 80 "81 65.53 45.89 65.33 46.17 65.13 46.46 64.93 46.74 80 66.35 46.46 66.15 46.75 65.94 47.04 65.74 47.32 81 82 67.17 47.03 66.96 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 68.81 48.18 68.60 48.48 68.39 48.78 68.17 49.08 84 85 69.63 48.75 69.41 49.06 69.20 49.36 68.98 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 50.83 87 88 72.09 50 47 71.86 50.79 71.64 51.10 71.42 51.41 83 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 74.31 51.94 52.52 73.27 52.26 73.04 ,52.58 90 91 74.54 ^2.20 74.08 52.84 73.85 53.17 92 75.36 52.77 75.13 53.10 74.90 53.42 74.66 53.75 92 93 76.18 53.34 75.95 53.67 75.71 54.01 75.48 54.34 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 54.83 77.34 55.17 77 10 55.50 95 96 78.64 55.06 78.40 55.41 78.16 55.75 77.91 56.09 96 97 79.46 55.64 79.21 55.98 7S 97 56.33 78.72 .56.67 97 98 80.28 56.21 80.03 56.. 56 79.78 56.91 79.53 57.28 98 99 81.10 56.78 80.85 57.14 80.60 57.49 80.35 f7.84 99 J GO 1 m Pi 81.92 57.36 81.66 57.71 81.41 58.07 81.16 £8.42 100 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. s c d 55 Deg. 541 Deg. 1 54i Deg. 54i Deg. 144 TRAVERST! TABLE. S6 Deg. 36i Deg. 36^ Deg. 361 Deg. 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. "^ 0.81 0.69 0.81 0.59 i 0.80 0.59 0.80 0.60 "l 2 1.62 1.18 1.61 1.18 1.61 1.19 1.60 1 20 5i 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 5 6 4.85 3.53 4.84 3.55 4.82 3.57 4.81 3.59 6 7 5.66 1 4.11 5.65 4.14 5.63 4.16 5.61 4.19 7 8 6.47! 4.70 6.45 4.73 6.43 4.76 6.41 4.79 8 9 7.28 1 5.29 7.26 5.32 7.23 5.35 7.21 5.3S 9 10 11 8.09 1 5.88 8.06 5.91 8.04 5.95 8.01 5.98 10 "ll 8.90, 6.47 8.87 6.50 8.84 6.54 8.81 6.58 12 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 16.1.8 11.76 16.13 11.83 16.08 11.90 16.03 11.97 20 21 16.99 12.34 16.94 12.42 16.88 12.49 16.83 12.56 22 17.80 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 22.51 16.65 22.44 16.75 28 29 23.46 17.05 23.39 17.15 23.31 17.25 23.24 17.35 29 30 31 24.27 17.63 24.19 17.74 24.12 17.84 24.04 17.95 30 31 25.08 18.22 25.00 18.83 24.92 18.44 24.84 18.55 32 25.89 18.81 25.81 18.92 25.72 19.03 25.64 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 20.70 28.13 20.82 28.04 20.94 35 36 29.12 21.16 29.03 21.29 28.94 21.41 28.85 21.54 36 37 29.93 21.75 29.84 21.88 29.74 22.01 29.65 22.14 37 38 30.74 22.34 30.64 22.47 30.55 22.60 30.45 22.74 38 89 31.55 22.92 31.45 23.06 31.35 23.20 31.25 23.33 39 40 ~41 32.36 23.51 32.26 23.65 32.15 23.79 32.05 23.93 40 41 33.17 24.10 33.06 24.24 32.96 24.39 32.85 24.53 42 33.98 24.69 33.87 24.83 33.76 24.98 33.65 25.13 42 43 34.79 25.27 34.68 25.43 34.57 25.68 34.45 25.73 43 44 35.60 25.86 35.48 26.02 35.37 26.17 35.26 26.33 44 45 30.41 26.45 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 28.55 38.46 28.72 48 49 39.64 28.80 39*. 52 28.97 39.39 29.15 39.26 29.32 49 d 1 40.45 29.39 40.32 29.57 40.19 29.74 40.06 29.92 50 d u a Dcp. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 541 )eg. 53} I >e. 53i Deg. 53i Deg. TRAVERSE TABLE. 145 s ? 61 36 Deg. 36i Deg. 36i Deg. 361 Deg. "51 Lat. Dep. Lat. Dep. Lat Dep Lat. Dep. 141.26 '29.98 41.13 30.16 41.00 30.34 40 86 30.51 52 42 07:30.56 41.94 30.76 41.80 30.93 41 67 31.11 52 53 42.88 131.15 142.741 31.34 42.60 31.53 42.47 31.71 53 54 43.69 131.74 43.56 31.93 43.41 132.12 43.27 32.31 54 55 44.50 ! 32.33 44.35 32.52 44.21 32.72 44.07 32.91 55 56 45.30 1 32.92 45.16 33.11 46.02 33.31 44.87 33.51 5G 57 46.11 33.50 45.97 33,70 46.82 33.90 45.67 34.10 57 58 46.92 34.09 46.77 34.30 46.62 34.60 46.47 34.70 58 59 47.73 34.68 47.58 34.89 147.43 35 09 47.27 35.30 59 60 '61 48.54! 35.27 48.39 35.48 148.23 i 35.69 48.08 35.90 60 61 49.35 35.85 49.19 36.07 ! 49.04 36.28 : 48.88 36.50 62 50.16 36.44 50.00 36.66 149.84 36.88 [49.68 37.10 62 63 50.97 37.03 50.81 37.25 150.64 37.47 50.48 37.69 63 S4 51.78 37.62 61.61 37.84 ,161.45 38.07 51.28 38.29 64 65 52.59 38,21 52.42 38.44 62.25 3S.66 62.08 38.89 65 66 53.40 38.79 53.23 39.03 153.05 39.26 52.88 39.49 66 67 54.20 39.38 64.03 39.62 153.86 39.85 53.68 40.09 67 68 55.01 39.97 54.84 40.21 154.66 40.45 54.49 40.69 68 69 55.82 40.56 55.64 40.80 156.47 41.04 55.29 41.23 69 70 71 56.63 41.14 50.45 41.39 156.27 41.64 66.09 41.88 70 71 57.44 41.73 57.20 41.98 i 67.07 ; 42.23 166.89 42.48 72 58.25 42.32 58.06 42.57 i 57.88 142.83 J 57.69 43.08 72 73 69.06 42.91 58.87 43.17 j 68.68 43.42 il 58.49 43.68 73 74 59.87 43.50 1 59.68 43.76 59.49 44.02 '69.29 44.28 74 75 60.68 44.08 60.48 44.36 60.29 44.61 60.09 44.87 75 76 61.49 44.67 1 61.29 44.94 61.09 45.21 60.90 45.47 70 77 62.29 45.26 62.10 45.63 61.90 45.80 61.70 46.07 77 78 63.10 45.85 62.90 46.12 62.70 46.40 62.50 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 65.70 49.06 82 83 67.15 48.79 1 66.93 49.08 66.72 49.37 66.50 49.66 83 84 67.96 49.371 67.74 49.67 67.52 49.97 67.31 50.26 84 85 68.77 49.96 1 68.55 60.26 68.33 50.56 68.11 60.86 85 86 69.58 50.55; 60.35 50.85 69.13 51.15 68.91 61.46 86 87 70.38 51.141 70.16 61.44 69.94 51.75 69.71 52.06 87 88 71.19 51.73 1 70.97 52.04 70.74 52.34 70.51 52.65 88 89 72.00 52.31 j 71.77 62.63 71.54 52.94 71.31 53.25 89 90 91 72.81 52.90] 72.58 73.39 63.22 72.35 53.53 72.11 53.85 90 91 73.62 53.49 j 63.81 73.16 54.13 72.91 54.45 92 74.43 64.08 74.19 64.40 73.96 54.72 73.72 65.05 92 93 1 75.24 154.66 1 75.00 54.99 74.76 55.32 74.62 65.64 93 94 76.05 '55.25! 75.81 65.58 75.56 55.91 76.32 50.24 94 95, 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 57.70 77.72 58.04 97 98 1 79.28 67.60 79.03 57.95 78.78 58.29 78.52 68.64 98 99; 80.09 58.19 79.84 58.54 79.58 58.89 79.32 59.23 99 100 1 1 Q 80.90 158.78 80.64 59.13 80.39 59.48 80.13 59.83 100 5 Dep. Lat. 1 Dep. Lat. Dep. Lat. Dep Lat. 54 Deg. 531 Degr. 53i Deg. 53i Dog. 146 TRAVERSE TABLE. 1 37 Deg. 1 37^ Deg. 37i Deg. 371 Deg. C n" J Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep "oTeT 0.80 0.60 0.80 0.61 0.79 0.61 0.79 2; 1.60 1.20 1.59 1.21 1.59 1.22 1.58 1.22 2 3; 2.40 1.81 2.39 1.82 2.38 1.83 ; 2.37 1.84 3 4 1 3.19 2.41 3.18 2.42 3.17 2 43 i 3.16 2.45 4 51 3.99 3.01 3.98 3.03 3.97 3.04: 3.95 3.06 R 6 4.79 3.61 4.78 3 63 4.76 3.65! 4.74 3.67 6 7 5.59 4.21 5.57 4.24 5.. 55 4.26 6.53 4.29 7 8 6.39 4.81 6.37 4.84 6.35 4.87 1 6.33 4.90 8 9 7.19 5.42 7.16 5 45 7.14 5.48 1 7.12 5.51 9 10 11 7.99 6.02 7.98 6.05 7.93 8.73 6.09 6.70 7.91 6.12 10 11 8.78 6.62 8.76 6.66 8.70 6.73 12 9.58 7.22 9.55 7.26 9.52 7.31 9.49 7.35 12 13 10.38 7.82 10.35 7.87 10.31 7.91 10.28 7.96 13 14 11.18 8.43 11.14 8.47 11.11 8.52 11.07 8.57 14 15 11.98 9.03 11.94 9.03 11.90 9.13 11.86 9.18 15 16 12.78 9.63 12.74 9.68 12.69 9.74 12.65 9. 80 16 17 13.58 10.23 13.53 10.29 13.49 10.35 13.44 10.41 17 18 14.38 10.83 14.33 10.90 14.23 10.96 14.23 11.02 18 19 15.17 11.43 15.12 11.50 15.07 11.57 15.02 11.63 19 20 21 15.97 12.04 15.92 12.11 15.87 12.18 15.81 12.24 20 21 16.77 1 12.64 16.72 12.71 16.66 12.78 16.60 12.80 22 17.57 13.24 17.51 13.32 17.45 13.39 17.40 13.47 22 23 18.37 13.84, 18.31 13.92 118.25 14.00 18.19 14.08 23 24 19.17 14.44 19.10 14.53 119.04 14.61 18.98 14.69 24 25 19.97 15.05 19.90 15.13 19.83 15.22 19.77 15.31 25 26 20.76 15.65 20.70 15.74 20.63 15.83 20.56 15.92 26 27 21.56 16.25 21.49 16.34 21.42 16.44 21.35 16.53 27 28 22.38 16.85 22.29 16.95 22.21 17.05 22.14 17.14 28 29 23.16 17.45 23.08 17.55 23.01 17.65 22.93 17.75 29 30 31 23.96 18.05 23.88 18.16 23.80 18.26 23.72 18.37 30 31 24.76 18.06 24.68 18.76 24.59 18.87 24.51 18.98 32 25.56 19.26 25.47 19.37 25.39 19.43 25.30 19.59 33 33 20.35 19.86 26.27 19.97 26.18 20.09 26.09 20.20 33 34 27.15 20.46 27.06 20.58 26.97 20.70 26.88 i 20.82 34 35 27.95 21.06 27.86 21.19 27.77 21.31 27.67 121.43 35 36 28.75 21.67 28.66 21.79 28.56 21.92 28.46 1 22.04 36 37 29.55 22.27 29.45 22.40 29.. 35 22.52 29.26 i 22.65 37 38 30.35 22.87 30.25 23.00 30.15 23.13 30.05 1 23.26 33 39 31.15 23.47 31.04 23.61 30.94 23.74 30.84! 23.88 39 40 "41 31.95 24.07 31.84 24.21 31.73 24.35 31.03 24.49 32.42 25.10 40 41 32.74 24.67 32.64 24.82 32 53 1 24.96 42 33.54 25.28 33.43 25.42 33 32 35.57 33.21 1 25.71 ' 42 43 34.34 25.88 34.23 26.03 34.11 26.18 34.00; 26.33, 43 44 35.14 20.48 35.02 26.63 34.91 26.79 34.79 26.94 44 45 35.94 27,08 35.82 27.24 35.70 27.39 35.53 27.. 55 45 40 36.74 127.68!! 36.62 27.84 36.49 1 28.00 36.37 28.16 46 47 37.54 28.29 1 37.41 28.45 37.29 ; 28.61 37.16 28 77 47 48 3S.33 23.89 I 38.21 29.05 38.08 ! 29.22 37.95 29.39 . 48 49 39.13 29.49 .39.00 29.66 38.87 29.83 38.74 30.00 i 49 50^ 8 1 U4 39.93 30.09 39.80 30.26 39.67 ,30.44 39.53' 30.61 50 1 .a Dep. 1 Lat. Dep. Lat. Dep. Lat. Dep. 1 Lat. 53 Deg. 521 Deg. 52i Deg 52i Deg. 1 TRAVERSE TABLE. 147 3 ? 61 3? Deg. 374 Deg. 37i Deg. 371 Deg. 5 1 61 Lat. Dop. Lat. Dep. Lat. Dep. Lat. 40.33 Dep. 31.22 40.73 30.69 40.60 30.87 40.46 31.05 52 41.53 31.29 41.39 31.48 41.25 31.66 41.12 31.84 62 63 42.33 31.90 42.19 32.08 42.05 32.26 41.91 32.45 53 64 43.13 32.50 42.98 32.69 42.84 32.87 42.70 33.06 54 65 43.92 33.10 43.78 33.29 43.63 33.48 43.49 33.67 55 66 44.72 33.70 44.58 33.90 44.43 34.09 44.28 34.28 66 67 45.52 34.30 45.37 34.50 45.22 34.70 45.07 34.90 67 68 46.32 34.91 40.17 35.11 46.01 35.31 46.86 85.51 58 59 47.12 35.51 46.96 35.71 46.81 35.92 46.65 36.12 69 60 "~61 47.92 36.11 47.76 36.32 47.60 36.53 47.44 36.73 60 '61 48.72 36.71 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 50.15 38.13 49.98 38.35 49.81 38.57 63 64 51.11 38.52 60.94 38.74 50.77 38.96 .50.60 39.18 64 65 51.91 39.12 51.74 39.34 51.67 39.57 51.39 39.79 66 66 52.71 39.72 52.64 39.95 62.36 40.18 52.19 40.41 66 67 53.51 40.32 53.33 40.65 53.15 40.79 62.98 41.02 67 68 54.31 40.92 54.13 41.16 53.95 41.40 53.77 41.63 08 69 65.11 41.53 54.92 41.77 54.74 42.00 54.56 42.24 09 70 '71 65.90 42.13 65.72 42.37 65.53 66.33 42.61 65.35 42.86 70 '71 56.70 42.73 66.62 42.98 43.22 56.14 43.47 72 67.50 43.33 57.31 43.58 57.12 43.83 66.93 44.08 72 73 58.30 43.93 .58.11 44.19 57.91 44.44 57.72 44.69 73 74 59.10 44.53 68.90 44.79 .58.71 45.05 68.51 45.30 74 75 59.90 45.14 59.70 45.40 .59.60 45.66 69.30 45.92 75 76 60.70 45.74 60.60 46.00 60.29 46.27 60.09 46.53 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 47.75 78 79 63.09 47.64 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 63.47 48.70 63.26 43.98 80 81 48.75 64.48 49.03 64. 2T) 49.31 64.05 49.59 82 65.49 49.35 65.27 49.63 65.05 49.92 64.84 50.20 82 83 66.29 49.95 66.07 60.24 65.85 50.53 65.63 50.81 83 84 67.09 50.55 66.86 .50.84 66.64 61.14 66.42 61.43 84 85 67.88 51.15 67.66 51.45 67.43 51.74 67.21 52.04 1 86 86 68.68 51.76 68.46 52.06 68.23 62.35 68.00 62.65 i 86 87 69.48 52.36 69.25 52.66 69.02 52.96 68.79 63.26. 87 88 70.28 52.96 70.05 53.27 69.82 63.57 69.58 53.88 88 89 71.08 53.56 70.84 53.87 70.61 54.18 70.37 54.49 89 90 91 71.88 54.16 71.64 64.48 71.40 64.79 71.16 65.10 90 72.68 54.77 72.44 65.08 72.20 55.40 71.96 65.71 91 92 73.47 55.37 73.23 55.69 72.99 56.01 72.74 .56.32 92 93 74.27 55.97 74.03 66.29 73.78 56.61 73.53 56.94 93 94 75.07 66.57 74.82 66.90 74.58 67.22 74.32 67.65 94 95 75.87 57.17 75.62 57.60 75.37 67.83 75.12 58.16 95 90 76.67 57.77 76.42 68.11 76.16 58.44 75 91 68.77 96 97 r7.47 68.38 77.21 68.71 76.96 69.05 76 70 69.39 97 98 78.27 58.98 78.01 69.32 77.75 59.66 77.49 60.00 98 99 79.06 .69.58 78.80 69.92 78.64 60.27 78.28 60.61 99 100 d 1 79.86 60.18 79.60 60.53 79.34 60.88 79.07 61.22 100 i Q Dcp. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 53 Deg. 521 Deg. 52^ Deg. 52i Deg. 24 148 TRAVERSE TABLE. 3ft Dag. \ 1 38i Deg. 38i Deg. 1 381 Dtg. X "1 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. i* 0.79 "0.62" 0.79 1 0.62 0.78 0.62 0.78 0.63 1 2 1.58 1.23 1.57 1 1.24 1.57 1.24 1.56 1.25 2 3 1 2.36 1.85 2.36 1.86 2.35 1.87 2.34 1.88 3 4; 3.15 2.46 3.14 2.48 3.13 2.49 3.12 2.50 4 5 3.94 3.08 3.93 3.10 3.91 3 11 3.90 3.13 5 6 4.73 3.69 i 4.71 3.71 4.70 3.74 4.63 3.76 3 7 5.. 52 4.31 1 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 3 9 7.09 6 54! 7.07 5.57 7.04 5.60': 7.02 5.63 n 10 7.88 6.161 7.85 8.64 6.19 6.81 7.83 6.23 !i 7.80 6.20 "11 8.67 6.77 8.61 1 6.85 II 8. .58 6.89 12, 9.46 7.39 9.42 7.43 9.39 1 7.47I! 9.36 7.51 12 13 1 10.24 8.00 10.21 8.05 10.17i 8.09!! 10.14 8.14 13 14 11.03 8.62 10.99 8.67 10.96 i 8.72;; 10.92 8.70 14 75 11.82 9.23 11.78 9.29 11.74] 9.34!' 11.70 9.39 15 16 j 12.61 9.85 12.57 9.91 12.52 9.96,12.48 10.01 1 16 17 1 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 14.091 11.21 1 14.04 11.27 i 18 19 : 14.97 11.70 14.92 1 11.76 14.87 i 11.83 1 14.82 11.89 1 19 20 115.76 21 1 16.55 12.31 ; 12.93 15.71 1 12.38 15.65! 12.45:' 15.60 12.52 i 20 16.49 13.00 16.43 1 13.07 16.38] 13.14 i 21 22 ' 17.34 13.54 17.28 13.62 17.22 1 13.70 17.161 13.77 22 23 i 18.12 14.16 18.06 14.24 18.00 1 14.32 17.941 14.40 23 24 118.91 14.78 18.85 14.86 1 18.78 14.94 18.72 115.02 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 21.23 16.62 21.20 16.72 21.13 16.8] 21.06 16.90 27 28 22.06 17.24 21.99 17.331 21.91 17.43 21.84 1 17.53 ' 23 29 22.85 17.85 22.77 17.95 22.70 18.05 22.62 1 18.15 ; 29 30 23.64 18.47 23.56 18.57 23.48 1 18.63 24.26 1 19.30 23.40; 13.78 ! 30 24.13 ] 19.40 i 31 31 24.43 19.09 124.34 1 1^.19 32 25.22 19.70 25.13 ! 19.81 25.041 19.92 24.96 [20.03 ' 32 33 26.00 20.32 , 25.92' 20.43!! 25.83 1 20.54 25.74 1 20.66 1 33 34 ! 26.79 20.93 26.70 21.05 26.61 ! 21.17 26.52 121.23 | 34 35 27.58 21.55 27.49 121.67 27.39! 21.79 27.30 21.91 ! 35 36 23.37 1 22.16 28.27 122.29 23.17 22.41 28.08 22.53 \ 36 37 29.16 22.78 29.06 i 22.91 28.96 23.03 23.86 23.16 1 37 38 29.94 23.40 ' 29.84 1 23.53 29.74 23.66 '29.64! 23.79 i 38 39 30.73 24.01 ! 30.63 24. Uji 30.52 24.23 30.42 124.41 ' 39 40 31.52 24.63 1 31.41 ! 24.76 31.30 24.90 31.20 1 25.04 40 41 132.31 25.24 i 32.20 25.38 32.09 25.52 ■■ 31.98 25.66 : 41 42 33.10 25.86 132.98 26.00 ' 32.87 26.15 32.76 26.29 1 42 43 33.88 26.47 133.77 26.62 33.65 26.77 33.53 26.911 43 44 134.67 127.09 34.55 27.24,1 31.43 27.39 34.31 27.54 44 45 35 46 127.70 35.34 27.86 35.22 28.01 ii35.09 28.17 1 45 46 36.25 (28.32 36.12 28.48 1! 36.00 28.641 35.87 28.79 46 47 37.04 128.94 36.91 ; 29.10 36.78 2J.26: 36.65 ,2;^. 42 47 48 '37.82 129.55 37.70 1 29.72 37.57 29.83 1' 37.43 J 30.04 1 4^ 49 i 38.61 30.17 38.48 30.34 38.35 30.50 1-38.21 30.67 i 49 60 139.40^ [30.78 39.27 30.95 39.13 31 13i;38.S9 31. 3Q i! 1 1 8 1 .52 Q Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 52. Deg. -511 Deg. 5U Deg. 5U Do-. TRAVERSE TABIE. 149 Ti 38 Deg. m Deg. 38i Deg. 3S| Deg. C I Lat. Dep. Lat. 40.05 .Dep. 31.57 1 Lat. j Dep. Lat, Dep. 40.19 131.40 39.91 31.75 39.77 1 31.92 i 61 1 62 40.98 132.01 40.84 1 32.19 ! 40.70 132.3?' ! 40.55 132.55 ; 52 63 41.76 32.63 41.62 32.81 1 41.48 132.99 1:41.33 33.1?' 53 64 42.55 33.25 42.41 '33.43 ! 42.26 33.62 li 42.11 : 33.80 54 65 43.34 33.86 43.19 134.05 43.04 34.24 142.89 34.43 ,')5 66 44.13 34.48 43.98 I 34.67 43.83 34.86 ; 43.67 35.05 56 57 44.92 35.09 44.76 i 35.29 4-4.61 1 35.48 44.45 135.68 .57 68 45.70 35.71 45.55 j 35.91 145.39 36.11 l! 45.23 36.30 58 69 46.49 36.32 46.33 136.53 146.17 136.73'! 46.01 36.93 i 59 60 61 47.28 36.94 47.12 137.15 146.96 137.35 147.74 , 37.97 ; 46.79! 37.56 60 61 48.07 37.56 47.90 37.76 47.57 1 38.18 62 48.86 38.17 48.69 33.38 j 48.52 i 38.60 '48.35 1 33.81 62 63 49.64 38.79 49.47 139.00 49.30 139.22 '49.13! 39.43 63 64 50.43 39.40 50.26 39.62 150.09 39.84 149.91 40.06 64 65 51.22 40.02 51.05 140.24 150.87 40.46 '50.69 40.68 i 65 66 52.01 40.63 51.83 140.86 51.65 41.09 51.47 41.31 , 66 67 62.80 41.25 52.62 141.48 52.43 41.71 152.25 41.94 1 67 68 53.58 41.86 1 53.40 1 42.10 : 53.22 42.33 ] 53.03 i 42.56 ! 63 | 69 54.37 42.48 ! 54.19 142.72 54.00 142.95 153.81 43.19 69 1 70 71 55.16 43.10 54.97 43.34 55.76 43.96 54.78 143.58 55.57 144.20 '54.59 43.81 70 55.95 43.71 55.37 44 44 71 72 56.74 44.33 56.54! 44.57 56.35 44.82 56.15 45 07 ! 72 73 157.52 44.94 57.33 45.19 57.13 145.44 '56.93 45.69! 73 74 58.31 45.56 [58.11 45.81 157.91 148.07 157.71 46.32 j 74 75 59.10 46.17 58.90 46.43 58.70 146.69 158.49 46.94 1 75 76 59.89 46.79 59.08 47.05 59.48 47.31 159.27 47.57! 76 77 00.68 47.41 60.47 ., 47.67 '69.26 47.93 160 05 48.20' 77 78 61.46 48.02 61.25 148.29 61.04 43.56 !60 83 1 48.82 1 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 50.15 62.61 49.80 50.42 62.39 1 50.07! 80 63.83 49.87 63.61 63.39 63.17 50.70 i 81 82 64.02 50.43 64.40 50.77 64.17 51.05 63.95 51.33 1 82 83 65.40 51.10 65.18 51.38 64.96 151.67 1 64.73 51.95 1 83 84 66.19 51.72 52.33 65.97 52.00 65.74 152.29 65.51 52.58 i 84 85 60.98 66.75 152.62 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 1 87 88 69.34 54.18! 69.11 54.48 68.87 54.78 68.63 65.08 1 88 89 70.13 54.79 i 69.89 55.10 69.65 55.40 69.41 55.71 89 90 91 70.92 55.41 i 56.03; 70.68 55.72 70.43 56.03 70.19 .'V6.33I 90 71.71 71.46 56.34 71.22 56.65 70.97 £6.96 i 91 92 72.50 56.64' 72.25 56.96 72.00 57.27 71.75 57.53 ; 92 93 73.23 57.26: 73.03 57.58 72.73 157.89 72.53 53.21 93 94 74.07 57.87 1 73.82 '58.19 73.57 153.52 73.31 58.84 1 94 95 .'4.86 i 58.49 1 74.81 '58.81 74.35 159.14 74.09 59.46' 95 96 75.65 i 69.10 1 75.39 i 59.43 75.13 159.76 74.87 60. 09 96 97 76.44 159.72! 76.18 160.05 75.91 160.38 75.65 60.71 97 93, 77.22 i6u.33l 76.96 60.67 76.70 61.01 76.431 61. 341 98 99 73.01 60.95 77.75 61.29 77.48 61.63 77.21 1 61.97 1 99 1 100 M 78.80 61.57 78.53 61.91 78.26 '62.25 1 77.99 62.59; 100 Dep. Lat. Dep. Lat., Dep. Lat. Dep. 1 Lat. i c 52 1 >eg. 511 Deg. ! 51 i Deg. 5U] ^g 1 (5 150 TRAVERSE TABLE. 2 5 P 3 s ' 1 39 Deg. 394 Deg. m Deg. 39t Deg. O £• 3 O ? ^1 Lat. 0.78 Dep. Lat. Dep. II Lat. Dep. Lat. Dep. 0.63 0.77 0.63 1 0.77 0.64 0.77 64 2 1.55 1.26 1.55 1.27 j 1.54 1.27 1.54 1.28 2 3 2.33 1.89 2.32 1.90 ! 2.31 1.91 2.31 1.92 3 4 3.11 2.52 3.10 2.53 3.09 ( 2.54 3.08 2.56 4 6 3.89 3.15 3.87 3.16; 3.86 3.18 3.84 3.20 6 8 4.66 3.78 4.65 3.80] 4.83 3.82 4.61 3.84 ft 7 5.44 4.41 5.42 4.43 !l 5.40 4.45 5.38 4.48 7 8 6.22 5.03 6.20 5.06 |! 6.17 5.69 6.94 5.09 6.15 5.12 8 9- 6.99 5.68 6.97 5.72 6.92 5.75 9 \0 11 7.77 6.29 6.92 7.74 6.33 i! 7.72 6.36 7.69 6.. 39 10 11 8.55 8.52 6.96; 8.49 7.00 8.46 7.03 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 8.23 11 10.03 8.27 9.99 8.31 13 14 1 10.88 8.81 10.84 8.86 h 10,80 9.49 1 11.57 8.91 10.76 8.95 14 15 11.66 9.44 11.62 9.54 11. .53 9.59 15 16 12.43 10.07 12.39 10.12 11 12.35 10.18 12.30 10.23 16 17 13.21 10.70 13.16 10.76 ll 13.12 10.81 13.07 10.87 17 18 ! 13.99 11.33 13.94 11.39 |; 13.89 11.45 13.84 11.51 18 19 1 14.77 11.96 14.71 12.02 I 14 06 12.09 14.61 12.15 19 20 I 15.54 12.59 15.49 12.65 }i 15.43 12.72 15.38 16.15 12.79 13.43 20 21 21 ' 16.32 13.22 16.28 13.29 i' 16.20 13. .36 22 17.10 13.84 17.04 13.92 i 16.93 13.99 16.91 14.07 22 23 17.87 14.47 17.81 14.55 1 17.75 14.63 17.63 14.71 23 24 18.65 15.10 ,18.59 15.18 1 18.52 15.27 18.45 15,35 24 25 19.43 15.73 '19.36 15.82 1 19.29 15.90 19.22 15.99 25 26 20.21 ; 10.36 20.13 16.45 '20.06 16. .54 19.99 16.63 20 27 20.98 1 16.99 20.91 17.03 20.83 17.17 20.76 17.26 27 23 21.76 1 17.62 '21.63 17.72 21.61 17.81 21.53 17.90 28 29 22.54 i 18.25 22.46 18.35 22.33 18.45 22.30 18.54 29 30 31 23.31 i 18. 8S 23.23 18.93 23.15 19.03 23.07 19.18 30 31 24.09 19.51 24.01 19.61 23.92 19.72 23.83 19.82 32 24.87 i 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 20.99 25.37 21.10 33 34 26.42 21.40 26.33 '21.51 26.24 21.63 26.14 21.74 34 35 27.20 22.03 27.10 1 22.14 27.01 22.26 26.91 22.33 35 36 27.98 22.66 27.83! 22.78 27.78 22.90 27.63 23.02 36 37 28.75 23. 2S , 23.65:23.41 28.55 23.53 23.45 23.66 37 38 29.53 23.91 29.43 24.04 29.32 24.17 29.22 124.30 33 39 30.31 24.54 30.20 24.68 30.09 24.81 29.93 124.94 39 40 41 3U09 25.17 31.86 25.80 1 30.98 25.31 30.86 25.44 30.75 125. 58 40 41 31.75 25.94 31.64 26.08 31.52 26.22 42 32.64 126.43 , 32.52 26.57 32.41 26.72 32.29 26.86' 12 43 33.42 127.06 33.30 27.21 3a. 18 27.35 33.06 27.50 43 44 34.19 27.69 .34.07 27.84 33.95 27.99 33.83 28.14 44 45 34.97 28.32 34.85 23.47 34.72 23.62 34.60 28.77 45 46 35.75 ,23.95 35.62 29.10 35.49 29.26 35.37 29.41 46 47 36.53 ,29.58 36.40 29.74 36.27 29.90 1 36.14 30.05 47 48 37.30 30.21 37.17 30.37 37.04 30.53 36.90 130.69 43 49 33.03 130.^4 1 37.95 31.00 37.81 131.17 37.67 131.33 1 49 50 S 2 33.86 31.47 33.72 31.64J 33.53 1 31. 80 3a.44 31.9V 50 OQ Dep. Ut. Dep. L^t. Dep. 1 Lat. Dep. Lat. 51 Deg. 501 1 3eg. 50i Dog. 50i Deg. TRAVEJISK TABLE. 151 s ? 51 39 Deg. 39i Deg. 39i Deg. 391 Deg, O I- s n a "51 Lat. Dep. Lat. Dep. Lat. 39.35 Dep. 32.44 Lat. Dep. 39.63 32.10 39.49 32.27 39.21 32.61 52 40.41 32.72 40.27 32.90 40.12 33.08 39.98 33. 25 52 £3 41.19 33.35 41.04 33.53 40.90 ' 33.71 40.75 33 89 53 54 41.97 33.98 41.82 .34.17 41.67 134.35 41.52 34.53 54 65 42.74 34 61 42.59 34.80 42.44 34.98 42.29 .35.17 55 56 43.52 1 36.24 43.37 35.43 43.21 35.02 43.06 35.81 66 57 44.30, 35.87 44.14 36.06 43.98 36.26 43.82 36.45 57 68 45.07 36.50 44.91 36.70 44.75 36.89 44.59 37.09 58 59 45.85 37.13 45.69 87.33 45.53 37.63 45.36 37.73 59 60 61 48.63 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 38.80 46.90 139.01 62 48.18 39.02 48.01 39.23 47.84 39.44 47.67 39.65 62 63 48.96 S9.G5 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 65 50.61 40.91 50.34 41.13 50.16 41.35 49.97 41.56 65 66 51.29 41.64 51.11 41.76 50.93 41.98 60.74 42.20 06 67 52.07 42.16 61.88 42.39 51.70 42.62 61.51 42.84 67 68 52.85 42.79 62.66 43.02 62.47 43.25 52.28 43.48 08 69 53.52 43.42 53.43 43.66 63.24 43.89 53.05 44.12 69 70 71 54.40 44.05 54.21 44.29 54.01 44.53 53.82 44.76 70 71 55.18 44.68 64.98 44.92 54.79 45.16 54.59 45.40 72 55.95 45.31 65.76 45.55 55.56 45.80 55.36 46.04 72 73 66.73 45.94 56.53 46.19 66.33 46.43 56.13 46.68 73 74 67.51 46.57 57.31 46.82 67.10 47.07 56.89 47.32 74 75 68.29 47.20 58,08 47.45 57.87 47.71 .57.66 47.06 75 76 .69.06 47.83 68.85 48.09 68.64 48.34 68.43 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 50.62 61.73 62.50 60.89 61.51 62.28 51.16 80 81 62.95 50.97 62.73 51.25 51.62 61.79 82 63.73 51.60 63.50 61.88 63.27 52.16 63.04 62.43 82 83 64.60 6'J.23 64.27 52.51 64.04 5^.'>'9 63.81 53.07 83 84 65.28 52.86 66.05 63.15 64.82 53.43 64.58 53.71 84 85 66.06 53.49 65.82 63.78 65.59 54.07 65.35 64.35 85 86 66.83 64.12 66.60 54.41 66.36 64.70 66.12 .54.99 86 87 67.61 54.75 67.37 56.05 67.13 65.34 66.89 65.63 87 88 68.39 55.38 68.15 65.68 67.90 66.97 67.66 56.27 88 89 69.17 66.01 68.92 .'56.32 68.67 56.61 168.43 56.91 89 90 91 69.94 66.64! 69.70 56.94 69.45 70.22 67.25 57.88 169.20 57.65 90 '91 70.72 57.27 j 70.47 .57.58 i 69.96 68.19 92 71.50 57.90; 71.24 68.21 70.99 68.52 70.73 68.83 92 93 72.27 58.53, 72.02 68.84 71.76 59.16 71.50 59.47 93 94 73.05 59.16 72.79 59.47 72.53 59.79 72.27 60.11 94 95 73.83 69.79 1 73.57 60.11 73.30 60.43 73.04 60.75 95 90 74.61 60.411 74.34 60.74 74.08 61.06 73.81 61.39 9G 97 76.38 61.041 76.12 61.37 74.85 61.70 ■74.. 58 62.03 97 98 76.16 61.671 75.89 62.01 75.62 62.34 76.35 62.66 98 99 76.94 62.30 76.66 62.64 76.39 62.97 76.12 63.30 99 ^00 1 ' 77.71 62.93 77.44 63.27 77.16 63.61 76.88 63.94 100 6 1 1 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 51 Deg. 501 Deg. 50^ Deg. 50\ Deg. 152 TRAVEKSE TABLE o B o n 1 40 Deg. 404 Deg. 40i Deg. 40J Deg. p I Lat. 0.77 Dep. 0.64 Lat. Dep. Lat. Dep. Lat. Dep. ! 0.76 0.65 0.76 0.65 0.76 0.65 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 1.951 2.27 1.96 3 4 3.06 2.57 3.05 2.58 3.04 2.60 ; 3.03 2.61 4 5 3.83 3.21 3.82 3.23 3.80 3.25 ! 3.79 3.26 5 6 4.60 3.86 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 6.03 5.20 : 6.06 5.22 8 9 6.89 5.79 0.87 5.82 6.84 5.84! 6.82 5.87 9 10 11 7.66 6.43 7.63 6.46 7.60 6.49 i 7.14| 7.58 6.53 10 11 8.43 7.07 8.40 7.11 8.36 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 1 9.92 8.40 9.89 8.44; 9.85 8.49! .3 14 10.72 9.00! 10.69 9.05 10.65 9.09 10.61 9.141 14 15 11.49 9.64 11.45 9.69 1 11.41 9.74 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 1 12.93 11.04 12.88 11.10 17 18 13.79 11.57 13.74 11.63 i 13.69 11.69 13.64 11.75 18 19 14.55 12.21 14.50 12.28 1 14.45 12.34 1 14.39 12.40 19 20 15.32^ 12.86 15.26 12.92 15.21 12.99 i 13.64 1 15.15 13.06 20 21 21 16.09 13.50 16.03 13.57 15.97 15.91 13.71 22 16.85 14.14 16.79 14.21 16.73 14.29 1 16.67 14.35 22 23 17.62 14.78 17.55 14.86 17.49 14.94 j 17.42 15.01 23 24 18.39 15.43 18.32 15.51 18.25 15.59 ! 18.18 15.67 24 25 19.15 16.07 19.03 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 22.98 19.28 22.90 19.38 22.81 19.48 22.73 19.. 08 30 31 31 23.75 19.93 23.66 20.03 23.-57 120.13 23.48 20.24 32 24.51 20.57 24.42 20.68 24.33 120.78 24.24 20.89 32 33 25.28 21.21 2i^,i9 21.32 25.09 1 21.43 25.00 21.54 33 34 26.05 21.85 25.95 21.97 25.85 122.08 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 29.11 124.43 29.00 24.55 28.90 24.68 28.79; 24.80 38 39 29.88 125.07 29.77 25.20 29.66 25.33 29.54 25.46 39 40 30.64 j 25.71 30.53 25.84 30.42 25.98 30.30 26.11 40 41 31.41 26.35 ! 31.29 26.49 31.18 26. G3 31.06 26.76 "41 42 32.17 27.00 32.06 27.14 31.94 27.28 31.82 27.42 42 43 132.94' 27.64 32.82 27.78 32.70 27.93 32.58 28.07 , 43 44 133.71 i 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 1 29.23 34.09 29.37 45 46 35.24 29.57 35.11 29.72 34.98 29.87 34.85 130.03 ! 46 47 36.00 1 30.21 35.87 30.37 35.74 30.52 35.61 i 30.08 1 -17 48 36.77 [30.85 36.64 31.01 36.50 31.17 36.36 {31.33 48 49:37.54 31.50 37.40 31.66 37.26 31.82 37.12 J31.99 49 50^ 38.30 32.14 J8.16_ 32.31 38.02 132.47 37.88 32.64 60 6 V (S 1 c ai -a Dep. Lat. Dep. Lat. Dep. Lat. Dep IL.. 50 Deg. 49| Deg. 49h Deg. m Deg. TRAVERStf TABLE. 153 in 51 40 Deg, 40i Deg. 40^ Dog. 40| Deg. 3 ? 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 53 40.60 * 34.07 40.45 34.^4 40.30 34.42 40.15 34.60 53 54 41.37 34.71 41.21 34.89 41.06 85.07 40.91 35.25 34 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 1 561 57 43.66 .36.64 43.50 ^6.83 43.34 37.02 43.18 37.21 57 58 44.43 37.28 44.2V 37.48 44.10 37.67 43.94 37.86 68 59 45.20 37.02 45.03 38.12 44.86 .38.32 44.70 38.51 59 60 61 45.96 38.57 45.79 38.77 45.62 38.97 39.62 45.45 39.17 60 46.73 39.21 46.56 39.41 46.38 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 50.19 42.86 50.00 43.08 66 67 51.32 43.07 51.14 43.29 50.95 43.51 50.76 43.73 67 6S 52.09 43.71 51.90 43.94 51.71 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 53.23 45.46 53.03 45.09 70 71 54.39 45.64 54.19 45.87 53.99 46.11 53.79 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 8i 61.28 62.05 51.42 61.06 51.69 60.83 61.59 51.96 52.61 60.61 52.22 80 '81 52.07 61.82 52.34 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 1 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 b8 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 69.20 58.45 68.18 58.75 90 69.71 58.49 69.45 58.80 j 59.10 68.94 69.40 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 60.7] 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 61.38 72.24 61.70 1 71.97 62.01 95 96 73.54 61.71 73.27 62.03 73.00 62.35 72.73 62.66 1 96 97 74.31 62.35 74.03 62.67 73.76 63.00 73.48 63.32 97 08 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 6 u c .2 76.60 64.28 76.32 Dop. 64.61 76.04 04.94 75.76 65 28 100 Dep. Lat. Lat. 1 Dep. 1 Lat. Dep. Lat. 50 Deg. 1 491 Deg. 49 i Deg. 49i Deg. 1.54 TSAVEBSE TABLE. p r; 41 De^. 4U Deg. 4U Deg. ! 4l| Deg. 3' 1 1 La:. Dep. Lat. D ?p. Lat. Dep. 3 Lat. j Dep. ^ 0.75 0.66 0.75 "o" C6 0.75 0.66 0.75 1 67 ~ j 2 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 93 2.25 1.99 I 2.24. 2.00 3 4 3.02 2.62 3.01 2 64 3.00 2.65 2.98 i 2.66 4 c 3.77 3. 23 1 3.76 3 3> 3.74 3.31 3.73 1 3.33 5 6 4 53 3.94 1 4.5. - 4.49 3.98 4.48 4.00 6 7 5 28 4.59 ' 5.2'^ , 5.24! 4.64 5.22 4.66 7 8 6.04 5.25 ' 6.'V -~ 5 99i 5.30: 5.97 1 5.33 8 9 6.79 5.90 G.r: .: 6.74 1 5.96; 6.71; 5.99 9 10 11 7.55 8.30 6.56 7 2*^ 7.5-: 7.49! 6.63; 7.46! 6.66 10 11 8.27 ,' S.24 7.29 'i 8.21 7.32 12 9.06 7!87 9.0-: rl 3.99 7.95;; 8.95 7.99 12 13 9.81 8.53 9.77 r7 9.74 8.61 ; 9.70 8.66 )3 14 10.57 9. IS 10.53 -3 10.49 9.28 ■ 10.44 9.32 14 15 11.32 9.84 11. 2S y >? 11.23 9.94 f 11.19 9.99 15 16 12. OS 10.. 50 12.03 ': 11.93 10.60 11.94 10.05 16 17 12.83 11.15 12.7^ ,: 12.73 11.26 12.68 11.32 17 18 13. 5S 11.81 13.. = 3 -' 13.43 11.93 ; 13.43 11.99 13 19 14.34 12.47 14.-:- :•; ry 14.23 12.59 ; 14.18; 12.65 19 20 15.09 13.12 15.04: ■ Q : 14.93 13.25 [ 14.92 1 13.32 20 21 15.85 13. 7S 15.79 13. So 15.73 13.91: 15.67 13.98 21 no 16.60 14.43 16.54 14. 51 16.48 14.58 16.41 14.65 22 23 24 17.36 13.11 15.09 17.29 15. 16 17.23 15.24; 17.16 15.32 15.90 17.91 15.98 23 24 15,: ' 1 ■* . ' -i : " ^; 17.97 25 IS. 87 le.:: > . - - 4^ IS. 72 16. .57 18.65 16.65 25 26 19.62 17.. 6 1 -^ . J •' : ^ , ^ 19.47 17.23 19.40.17.31 26 27 20. 3S 17.71 20.3^ - 20.22 17.89 ,20.14 17.98 27 2S 21.13 IS. 37 21. Or '. i . ^' CO. 97 lS..55i 20.89 18.64 28 29 21. S9 19.03 '2\S : :: •:i.72 19.22 21. at 19.31 29 30 31 22.64 19.63 2i.5- ^^ •:2.47 19.33 22.33 19. 9S •::'.-4 •:3.I3 20.64 30 "31 23.40 20.34 ^.3^ 1- C:'.-C 32 24.15 20.99 24.0: : : , : b'." •::.-:: C3.S7 21.31 32 33 24.91 21.65 24. S ', : , : 14-; , .-: :4. 62 21.97 3S 34 25.66 22.31 25, c"^ 4: :^l ::..53 25.37 22.64 34 35 •2^.41 22.96 23.3: ■:: . ^ :■ .: 1 19 26.11 23.31 35 S6 27.17 23.62 27 .''T ■^ : . ^^ i'.,' i.<5 26.36 23.97 36 37- 27.92 24.27 27. SC ; ;. . . CT.Tl ':4.52 27.60 24.64 37 3S 2S.6^ 24.93 r> ^ ."7 :S.4o '25 AS 2S. 35 25.30 38 39 29.43 25.59 29 is-: Co . 71 -ZB.^l •;'.^4 ':9. 10 (25.97 39 40 30.19 26.24 30.07 26. 37 29. P5 •: .^ Cy. 34 26.64 40 41 30.94 26.90 30.33 27 . 03 30.71 - .:: 30.59 27.30 41 42 31.70 27.55 31.53 27. 69 31.4-3 •:.-.-: 81.33-27.97 42 43 32.45 23.21 32.33 23 . 35 32.2: :-.: ^C.OS 2-S.03 43 44 .33.21 2S.S7 33. 0> "3^ '•: -^CP' - 3:. 33 29.30 44 45 33.96 29.52 33.-? ; r ^ : - -1^1 53. 57. 29. 97 45 46 34.72 30.13 34..= ^ : i.:^ : ^- 34.32 30.63 46 47 35.47 30. S3 35.3^ '■.-, :: ?o.06' 31.30 47 48 36.23 31.49 36. 0> : - 35.31 !31 96 48 49 38. 9S 32.15 35. S. ' -l : :^ 35.56 32.63 49 50 1 37.74 32.80 Dep. '. Lat. 37.5. r~ 3^.45 33.13 37.30 33.29 Dep. 1 Lat. 50 ' i c ti .J Dep L -:, Dep. Lat. Dej. 491 >-■ 48.' Der ! 48r m Deg. TRAVERSE TABLE. 155 41 Deg. 4U Deg. Lat Dep. | Lat. , Dep. 52 ; 39 . 57 43 53 '43 5e 144 61 I 46 62 146 63 147 64 [48 65 I 49 66 !49 67 : 50 68 51 49 33.46 24 34.12 00; 31.77 75 35.43 51 136.0S 26 36.74 02' 37.40 77 38.05 53 3S.71 25 39.36 69 70 40 . 02 40.68 41.33 41.99 06 j 42 . 64 81 '43.30 57 43. 9G 32 44.61 07 45.27 83 45.92 71 53, 72 i 54, 73; 55, 74155, 75 ■ 56 , 76 I 57, 77 58, 78 I 53, 79 1 59 , 80 I 60, 58 46.58 .34 47.24 09 47.89 85 48.55 60 49.20 36,49.86 11 ! 50.52 87 51.17 62 51.83 38 52.48 81 ; 61. 82^61. 83 162. 84i63. 85 64. 86 64. 87 65. 83 66. 89 67. _90 6]_. 91 63. 92 I 69. 93, 70. 94 70, 95 '71. 96 172. 97 73. 93 73. 99 , 74. 13 53.14 53.80 54.45 55.11 15 55.76 90 56.42 66 57.08 41 57.73 17 58.39 92 59.05 63 59.70 43 60.36 19 61.01 94 61.67 70 62.33 45 62.98 21 63.64 96 64.29 72 64.95 100 • 75.47: 65.61 38.34 39.10 39.85 40.60 41.35 42.10 42.85 43.61 44.36 45.11 45.86 46.61 47.37 48.12 48.87 49.62 50.37 51.13 51.88 52.63 53.38 54.13 54.88 55.64 56.39 57.14 57.89 .58.64 59.40 60.15 33. 34. 34. 35. 36. 36. 37. 38. 33, 39. 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 41i Deg. Lat. 63 29 95 60 26 92 58 24 90 56 22 88 54 20 86 52 IS S4 49 33.20 38.95 39.69 40.44 41.19 41.94 42.69 43.44 44.19 44.94 45.69 46.44 47.18 47.93 48.68 49.43 50! 93 51.68 52.43 Dep. 33.79 .34.46 35.12 35 . 78 36.4-4 37.11 37.77 33.43 39.09 39.76 40.42 41.08 41.75 42.41 43.07 43.73 44.40 45.06 45.72 46.38 411 Deg. 60.90 61.65 62.40 63.15 63.91 64.66 65.41 I 57, 66.16 153, 66.91 1 53, 67.67 1 59, 68.42 69.17 69.92 70.67 71.43 72.18 72.93 ' 63, 73.68 '64, 74.43 165, 75.18 65 i 53.18 i| 53.92 154.67 55.42 I 56.17 : 56.92 ' 57.67 ! 53.42 I 59.17 ! 59.92 i 60.67 ! 61.41 ; 62.16 ' 62.91 ' 63.66 : 64.41 65.16 65.91 66.66 67.41 • 68.15 68.90 69.65 70.40 71.15 ; 71.90 72.65 73.40 74.15 74.90 47.05 47.71 48.37 49.03 49.70 50.36 51.02 51 .68 52.35 53.01 53.67 54.33 55 . 00 55.66 56.32 56.99 57.65 53.31 53.97 59.64 60.30 60.96 61.62 62.29 62.95 63.61 64.27 64.94 65.60 66.26 Lat. I Dep. 3 I » 33.05 I 33.96 ' 51 38.79 34.63 52 39.. 54 35.29 53 40.29 35.96 ' .54 41.03 36.62 55 41.73 37.29 56 42.53 37.96 57 43.27 33.62 44.02 39.29 44.76 39.95 45.51 46.26 47.00 47.75 48.49 49.24 49.99 50.73 51.43 5-:2.22 52.97 53.72 54.46 55.21 55.95 56.70 57.45 53.19 58.94 59.68 40 . 62 41.28 41.95 42 . 62 43.28 43 . 95 44.61 45.28 45.95 46.61 47.28 47.94 48.61 49.28 49.94 50.61 51.27 51.94 52.60 53.27 60.43 61.18 61.92 62.67 63.41 64.16 64.91 65.65 66.40 67.15 67.89 68 . 64 69.38 70.13 70.88 71.62 72.37 73.11 73.88 74.61 53.94 54.60 55.27 55.93 56.60 57.27 57.93 58.60 59.26 59.93 60.60 61.26 61.93 62 . 59 63.26 63.92 64.59 65.26 65.92 66.59 lOi^ Dep. I Lat. ;, Dep. | Lat. i Dep. j Lat. : D^p. I Lat. 49 Deg. 48| De^ 48^ Deg. 48i Deg. 156 TRAVERSE TABLE. 5 a n 9 42 Deg. 424 Dog. 42h Deg. — — 42| Deg ' T Lat. Dep. 1 0.67 Lat. 0.74 Dep. Lat. Dep. Lat. Dep. 1 0.74 0.67 0.74 0.68 0.73 0.68 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 2.02 2.21 2.03 2.20 2.04 3 4 2.97 2.68 2.96 2.69 2.95 2.70 2.94 2.73 4 5 3.72 3.35 3.70 3.36 3.69 3.38 3.67 3.39 5 6 4.46 4.01 4.44 4.03 4.42 4.05 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 5.93 5.38 5.90 5.40 5.87 5.43 8 9 6.69 6.03 6.66 6.05 6.64 6.08 6.61 6.11 9 10 if 7.43 8.17 6.69 7'. 3 6 7.40 6.72 7.37 6.76 7.43 7.34 6.79 10 8.14 7.40 8.11 8.08 7.47 11 12 8.93 8.03 8.88 8.07 8.85 8.11 8.81 8.15 12 13 9.66 8.70 9.63 8.74 9.58 8.78 9.. 55 8.82 13 14 10.40 9.37 10.36 9.41 10.33 9.46 1 10.28 9.50 14 15 11.15 10.04 11.10 10.09 11.06 10.13' 11.01 10.13 15 16 11.89 10.71 11.84 10.76 11.80 10.81 i 11.75 10.86 10 17 12.63 11.38 13.58 11.43 12.53 11.48 12.48 11.54 17 18 13.38 13.04 13.33 12.10 13.27 12.16 1 13.22 13.32 13 19 14.12 13.71 14.06 12.77 14.01 13.84! 13.95 13.90 19 20 21 14.86 13.38 14.80 13.45 14.75 13.51 1 14.69 13. 5S 20 15.61 14.05 15.54 14.12 15.48 14.19; 15.42 14.25 21 22 16.35 14.73 16.28 14.79 16.22 14.86 1 16.16 14.93 22 23 17.09 15.39 17.02 15.40 16.96 15.54 1 16.89 15.61 23 24 17.84 16.06 17.77 16.14 17.69 16.21 1 17.62 16.29 24 25 18.58 16.73 18.51 16.81 18.43 16.89 1 18.36 16.97 25 26 19.33 17.40 19.25 17.48 19.17 17.57 1 19.09 17.65 26 27 20.06 18.07 19.99 18.15 19.91 18.24 1 19.83 18.. 33 27 28 20.81 18.74 20.73 18.83 20.64 18.92 1 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 22.21 20.17 22.12 30.37 22.03 20.36 30 31 31 23.04 20.74 22.95 20.84 23.86 20.94 22.76 21.04 32 23.78 21.41 23.69 21.53 23.59 21.62 23.50! 21.72 32 33 24.52 23.08 24.43 22.19 24.33 22.29 24.23 22.40 33 34 25.27 22.75 25.17 22.86 25.07 22.97 24.97 23.08 34 35 26.01 23.42 25.91 23.53 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 27.28 25.00 27.17 25.12 37 38 28.24 25.43 28.13 25.. 55 28.03 25.07 27.90 25.79 38 39 28.98 26.10 28.87 26.23 28.75 26.35 28.64 26.47 39 40 29.73 26.77 29.61 26.89 29.49 27.02 29.37 1 27.15 40 41 30.47 27.43 30.35 27.57 30.23 27.70 130.11 27.83 41 42 31.21 28.10 31.09 28.24 30.97 28.37 130.84 28.51 42 43 31.96 28.77 31.83 28.91 31.70 29.05 131.58 29.19 43 44 33.70 29.44 32.57 29.58 32.44 29.73 132.31 129.87 44 45 33.44 30.11 33.31 30.26 33.18 39.40 133.04 30.. 55 45 46 134.18 30.78 34.05 30.93 33.91 31.08 1 33.78 31.22 46 47 34.93 31.45 34.79 31. GO 34.65 31.75 134.51 31.90 47 48 35.67 33.12 35.53 32.27 35.39 32.43 35.25 32.. 58 48 49 36.41 32.79 36.27 32.95 36.13 33.10 .35.98 33.26 49 50 1 37.16 Dop. 33.46 37.01 33.62 36.86 33.78^ .36.72 Dep. 33.94 Lat. .50 o c 2 .2 e Lat. Dep. Lat. Dep. Lat. 48 Deg. 47| Deg. 47i Deg. 47i Deg. TBAV3SRSE TABLE. 157 "5i 42 Deg. 1 42i De^. 42i Deg. 1 42| Deg. s ? 51 Lat Dep. 34.13 Lat. Dep. Lat. De,. Lat. Dep. 37.90 37.75 34.29 37.60 34.46 37.45 34.62 53 3S.64 34.79 38.49 34.96 38.34 35.13 38.18 35.30 32 53 39.39 35.46 39.23 35.64 39.08 35.81 38.92 35.93 63 54 40.13 36.13 39.97 36.31 39.81 36.48 39.65 36.66 54 55 40.87 36.80 40.71 36.93 40 55 37.16 40.39 37.33 55 66 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 58 43.10 38.81 42.93 39.00 42.76 39.18 42.59 39.37 53 59 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 40.541 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 47.56 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 52.56 47.07 51.61 47.29 51.40 47.52 70 71 52.76 47.51 47.74 52.35 47.97 52.14 48.19 72 .53.51 48.18 53.30 48.41 .53.08 48.64 ,52.87 48.87 72 73 54.25 48.85 54.04 49.08 53.82 49.82 53.61 49.55 73 74 54.99 49.52 .54.78 49.76 54.56 49.99 54.34 .^0.23 74 75 55.74 50.18 55.52 50.43 55.30 50.67 155.07 50.91 75 76 56.48 .50.85 56.26 51.10 56.03 51.34 155.81 51.59 76 77 57.23 51.52 57.00 51.77 56.77 .52.02 156.54 52.27 77 78 57.97 52.19 57.74 52.44 57.51 52.70 57.28 52.95 78 79 58.71 52.86 58.48 53.12 58.24 53.37 58.01 53.63 79 80 81 59.45 ,53.53 59.22 53.79 58.98 54.05 58.75 54.30 80 81 60.19 54.20 59.96 54.46 59.72 54.72 159.48 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 56.07 160.95 56.34 83 84 62.42 56.21 62.18 56.48 61.93 56.75 161.68 57.02 84 85 63.17 56.88 62.92 57.15 62.67 57.43 162.42 57.70 85 86 63.91 57.55 63.66 57.82 63.41 58.10 63.15 58.38 86 87 ?4.65 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 60.89 66.62 60.51 66.35 60.80 66.09 61.09 90 91 67.63 67.36 61.19 67.09 161.48 166.82 61.77 92 68.37 61.56 68.10 61.86 67.83 62.15 167.56 62.45 92 93 69.11 62.23 68.84 62.53 68.57 63.83 i 68.29 63.13 93 94 69.86 62.90 69.58 63.20 69.30 163.51 69.03 63.81 94 95 70.60 63.57 70.32 63.87 70.04 164.18 1 69.76 64.49 95 96 71.34 64.24 71.06 64.55 70.78 ' 64.86 70.49 65.16 9G 97 72.08 64.91 71.80 65.22 71.52 ; 65.53 171.23 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 u a S .2 O 74.31 66.91 74.02 67.24 73.73 67.56 173.43 67.88 100 6 Q Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 48 Deg. 471 Deg. -■ 47^ Deg. 1 47i Deg. 158 TRAVERSE TABLE. c DO 1 P 1 43 Deg. A2\ Deg. 43i Deg. 431 Deg. 5- g p 1 Lat. Dep. Lat. 0.73 Dep. 0.69 Lat. Dep. Lat. 0.72 Dep. 0.73 0.68 0.73 0.69 0.69 2 1.46 1.36 1.46 1.37 1.45 1.38 1.44 1.38 2 3 2.19 2.05 2.19 2.06 2.18 2.07 2.17 2,07 8 4 2.93 2.73 2.91 2.74 2.90 2.75 2.89 1 2 77 4 5 3.66 3.41 3.64 3.43 3.63 3.44 3.61 3 46 5 6 4.39 4.09 4.37 4.11 4.35 4.13 4.33 4.15 6 7 5.12 4.77 5.10 4.80 5.08 4.82 5.06 4.84 7 8 5.85 5.46 5.83 5.48 5.80 5.51 5.78 5.53 8 9 6.58 6.14 6.56 6.17 6.53 6.20 6.50 6.22 9 10 7.31 6.82 7.28 6.85 7.25 6.88 7.22 6 92| 10 11 8.04 7.. 50 8.01 7.54 7.98 7.57 7.95 7.61 ! 11 12 8,78 8.18 8.74 8.22 8.70 8.26 8.67 8.30 1 12 13 9.51 8.87 9.47 8.91 9.43 8.95 9.39 8.99' 13 14 10.24 1 9.55 10.20 9.59 10.16 9.04 10.11 9.68! 14 15 10.97 1 10.23 10.93 10.28 10.88 10.33 10.84 10.37 15 16 1 1 . 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 18 13.16 12.28 13.11 12.33 13.06 12.39 13.00! 12.45 18 19 13.90 12.98 13.84 13.02 13.78 13.08 13.72. 13.14 19 20 14.63 13.54 14.57 13.70 14.51 J3.77 14.45 1 13.83 20 21 15.36 14.32 15.30 14.39 15.23 14.46 15.17 14.. 52 21 22 16.09 15.00 1tj.02 15.07 15.96 15.14 15.89 15.21 22 23 16.82 15.69 16.75 15.76 16.68 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 25 18.28 17.05 18.21 17.13 18.13 17.21 18.06 17.29! 25 26 19.02 17.73 18.94 17.81 18.86 17.90 18.78 1 17.93 1 26 27 19.75 18.41 19.67 18 50 19-53 18.59 19.50! 18.67 27 28 20.48 19.10 20.39 19.19 20.31 19.27 20.23; 19.36 1 28 29 21.21 19.78 21.12 19.87 21.04 19.96 20.95 '20.05 29 30 21.94! 20.46 21.85 20.56 21.76 1 20.65 21.67 120.75 30 :ii 22.67 21.14 22.58 21.24 22.49 21.34 22.39; 21.44 "31 32 23.40 21.82 23 31 21.93 23.21 22.03 23.12 22.13 1 32 1 33 24.13 22.51 24.04 22.61 23.94 22.72 23.84 22.82 33 34 24.87 23.19 24. 7C. 23.30 24.66] 23.40 24.66 23.51 34 35 25.60 23.87 25.49 23.98 25.39 24.09 25.28 24.20 35 36 26.33 24.55 26.22 24.67 26.11 1 24.78 26.01 24.89 36 37 27.06 25.23 26.95 25.35 26.841 25.47 26.73 i 25.59 37 3S 27.79 25.92 27.68 26.04 27.56 26.16 27.45 i 26.23 t 38 | 39 28.52 26.60 28.41 26.72 28.29 26.85 28.171 26.97! 39) 40 41 29.25 29.99 27.28 29.13 27.41 29.01 27.53 28.89 27.66 40 27.96 29.86 28.09 29.74 23.22 29.62 i 28.35 41 42 30.72 28.64 30.59 28.78 30.47 23.91 30.34 29.04; 42 43 31.45 29.33 31.32 29.46 31.19 29.60 31.06 29 . 74 i 43 44 32.18 30.01 32.05 50.15 31.92 30.29 31.78 30.43 1 44 45 32.91 30.69 32.78 30.83 32.64 30.93 1 .32.51 31.12 j 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 48 35.10 32.74 34.96 32.89 34.82 1 33.04 3-^.67 ??.19 48 49 35.84 33.42 35.69 33.57 35.54 33.73 35.40 3.1 88 1 49 50 36.. 57 34.10 36.42 34.26 30.27 34,42 3ti.l2 l^..i9 1 60 J i4 'oo Q Dep. Lat. Dep. Lat. 1 Dep. Lat. Dep. Lat. i 47 Deg. 46f Deg. 46 i Deg. 46^ Deg TRAVERSE TABLE. 1B9 43Deg. 43| Deg. 431 Deg. 43| Deg. i Lat. Dep. Lat. 1 Dep. Lat. Dep. Lat. Dep. 51 ,37.30 34.73 37.15 .34.94 36.99 35.11 1 36.84 35.27 51 52 33.03 35.46 37.88 '35.63 37.72 35.79 1 37.56 35.96 52 53 38.76 36.15 38.60 '36.31 38.44 36.48 1 33.29 36.65 53 54 39.49 36.83 39.33 37.00 39.17 37.17! 39.01 37.34 54 55 40.22 ' 37.51 40.06 137.69 39.90 37.86 39.73 38.03 55 56 40.96:38.19 40.79 38.37 40.62 38.55 40.45 38.72 56 57 41.69 38.87, 41.. 52 39.06 41.35 39.24 41.17 39.42 57 58 42.42 39.56 42.25 39.74 42.07 39.92 41.90 40.11 68 69 43.15 40.24 42.97 40.43 42.80 40.61 42.62 40.80 59 60 61 43.88 40.92 43.70 44.43 41.11 41.80 43.52 41.30 43.34 41.49 60 61 44.61 41.60 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 44.20 64 65 47.54 44.33 47.34 44.54 47.15 44.74 46.95 44.95 65 66 48.27 45.01 43.07 45.22 47.87 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 1 49.53 46.59 49.33 46.81 49.12 47.02 68 69 50.46 47.06 50.26 47.28 50.05 47.50 49.84 47.71 69 70 71 51.19 47.74 1 50.99 47.96 50.78 48.18 50.57 48.41 49.10 70 71 51.93 48.42 51.71 148.65 ti 51.50 48.87 51.29 72 52.66 49.10 52.44^49.33 52.23 49.56 52.01 49.79 72 73 53.39 49.79' 53.17 150.02 .52.95 50.25 52.73 50.48 73 74 54.12 50.47 53.90 150.70 53.68 50.94 53.45 51.17 74 75 54.85 51.15 54.63 51.39 54.40 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 53.69 156.34 53.94 78 79 57.78 53.88 57.54 54.13 57.30 54.38 157.07 54.63 79 80 58.51 54.56 58.27 54.81 58.03 .55.07 157.79 55.32 80 81 81 59.24 55.24 59.00 155.50 58.76 55.76 158.51 56.01 82 59.97 55.92 59.73 |50.18 59.48 56.45 159.23 56.70 82 83 60.70 .50.61 60.45 50.87 60.21 57.13 1 59.96 57.40 83 84 61.43 57.29 61.18 57.56 60.93 57.82 160.68 58.09 84 85 62.17 57.97 01.91 58.24 61.60 58.51 161.40 58.78 85 88 62.90 58.65 62.64 58.93 62.38 59.20 162.12 59.47 86 87 03.63 59.33 63.37 59.61 63.11 59.89 62.85 1 60.16 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 65.82 61.38 65.55 161.67 65.28 61.95 165.01 62.24 90 91 91 66.55 162.06 66.28 162.35 66.01 62.64 65.74 62.93 92 167.28 ; 02.74 67.01 63.04 66.73! 63.33 66.46 63.62 92 93 j 68.02 63.4? 67.74 63.72 67.46 1 64.02 67.18 64.31 93 94 i 68.75 64.11 f 68.47 64.41 68.19 '64.71 67.90 65.00 94 95 169.48 64.79 69.20 65.09 ! 68.91 165.39 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 60.46 1 70.36 66.77 70.07 67.08 97 98 171.67 66.84 71.37 67.15 171.09 67.46 170.79 67.77 98 99 i 72.40 ; 67.52 72.11 67.83 71.81 68.15 171.51 68.46 99 100 73.14 68.20 72.84 Dep. 68.52 72.54 68.84 72.24 69.15 100 1 I Dep. Lat. Lat. Dep, Lat. Dep. Lat. 47 Deg. 461 Deg. 1 461 Deg. 46J Deg. 160 TRAVERSi:: TABLE. ~1 44Deg. U\ Deg. 44 i Deg. 44J Deg. .45 Deg. C 0; p 3 3 I Lat. Dep. Lat. Dep. Lat. 0.71 Dep. "oTto Lat. Dep. Lat. Dep. 0.72 0.69 0.72 0.70 0.71 0.71 0.71 0.71 "Z 1.44 1.39 1.43 1.40 1.43 1.40 1.42 1.41| 1.41 1.41 a a 2.16 2.08 2.15 2.09 2.14 2.10 2.13 2.11 2.12 2.12 a 4 2.88 2.78 2.87 2.79 2.85 2.80 2.84 2.82! 2.83 2. 83 4 i £ 3.60 3.47 3.58 3.49 3.57 3 50 3.55 3..52i 3.. 54 3.64 ti 4.32 4.17 4.30 4.19 4.28 4.21 4.26 4.22! 4.24 4.24 € 7 5.04 4.86 5.01 4.88 4.99 4.91 4.97 4.93 4.95 4.95 1 « 5.75 5.56 6.73 5.58 5.71 5.61 6.68 6.63 5.66 5.66 8 9* 6.47 6.25 6.45 6.28 0.42 6.31 6.39 6.34 6.36 6.36 9 10 11 7.19 7.91 6.95 7.16 7.88 6.98 7.13 7.01 7.10 7.04 7.07 7.071 10 7.78ill 7.64 7.68 7.85 7.71 7.81 7.74 7.78 12 8.63 8.34 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.I0I 9.19 9.19 13 14 10.07 9.73 10.03 9.77 9.99 9.81 9.94 9.861 9 90 9.90 14 15 10.79 10.42 10.74 10.47 10.70 10.51 10.65 10.561 10.61 10.61 15 16 11.51 11.11 11.46 11.16 11.41 11.21 11.36 11.261 11.31 11.31 16 17 12.23 11.81 12.18 11.86 12.13 11.92 12.07 11.971 12.02 12.02117 18 12.95 12.50 12.89 12.56 12.84 12.62 12.78 12.671 12.73 12.73118 19 13.67 13.20 13.61 13.26 13.55 13.32 13.49 13.38! 13.43 13.43119 20 14.39 13.89 14.33 13.96 14.65 14.26 14.02 14.20 14.08: 14.78^ 14.14 14.85 14.14120 14.85I2I 21 15.11 14., 59 15.04 14.98 14.72!!14.91 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 16.2623 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 26118.70 18.06 18.62 18.141 18.54 18.22 18.46 18.30 18.38 18.3826 27 19.42 18.76 19.34 18.841 19.26 18.92 [19.17 19.01 19.09 19.0927 28 20.14 19.45 20.06 19.541 19.97 19.63 19.89 19.71 19.80 19.80i28 29 20.86 20.15 20.77 20.24 20.68 20.33 20.60 20.42 20.51 20.51i29 30 31 21.68 22.30 20.84 21.49 20.93 21.40 22.11 21.03 21.73 21.31 21.12 21.21 21.92 21.21J30 21.53 22.21 21.63 22.02 21.82 21.92 31 32 23.02 22.23 22.92 22.33 22.82 22.43 22.73 22.53 22.63 22.63132 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 23.831 24. 15123.94 24.04 24.04 34 35 25.18 24.31 25.07 24.42 24.96 24.53124.86 24.64 24.75124. 75'35 36 25.90 25.01 25.79 25.12 25.68 25.23 25.. 57 25.34 26.46 25.46136 37 26.62 25.70 26.50 25.82 26.39 25.93 26,28 26.05 26.16 26.16137 38 27.33 26.40 27.22 26.52 27.10 26.63 26.99 26.75 26.87 26.87138 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 28.48 28.65 27.91 28.53 28.04!28.41 28.16 28.86 29.57 28.28 ,?8. 28140 29.37 28.61 29.24 28.74 29.12 28.99 28.9941 42 30.21 29.18 30.08 29.31 29.96 29.44[29.83 29.70 29.70142 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 31.68 131.82:31.82:45 46 33.09 31.95 32.95 32.10 32.81 32.24 32 67132.38 [32.53 32.53 40 47 33.81 32.65 33.67 32.80 33.52 32.94133 38 33.09 i33.23|33.23 4/ 48 34.53 33.34 34.38 33.49 34.24 33.64 34 09 33.79 1.33.94 33. 94'4S 49 35.25 34.04 3.5.10 34.19 34.95 34.34I34 30 .34.50 i34.65 34.65 49 60'35.97 34.73 35.82 34.89 35.66 35.05i35.51 35.20 136.36 35. 36 j50i 1 Dcp. Lat. Dep. Lat. Dep. Lat. Dep. Lat. j Dep. Lat. 6 46Deg. 45| Deg. 45i Deg. 45i Deg. 45] Deg. JQ TBAVERSE TABLB. 161 Plate '> i- '■"", "/■ '- \ A. .^ \ I %■ 'J. ■J^/y J \ --- jcl \ /5: g . 6-S : H^M=^ d ■ '^ JV 7 ^r^ u-.. \ " l> "-^^ \ a Ft M ion 'obacco P. A A A I L hioar P. Jtirc Plant f^ Ploughed L. Gardens Orchards Piatt CoUonP. -- -^ .y -^ Vmo vanl ^«; «- -ir- i DETAILS OF LEAVES Oak .'^- ^ ' jfy Fruit (lusi- '/'^'' -//>-* _'' Fresh Marsh Heath Oak AVoQfls Salt Marsh 3Ieadoi\'s // mD^'DS Iss. Scarped road Stone Fence Tiirnpiltt" i-oad (V;::s,-v:^: LogLivii^i. V v^ X WoodeiJ Fe ncf A/V\ JM/Wr// J'„H- fhl//,il/ /i>r I'll' /1,1/Uri/ /rr Mv 0/riyi /-llrfrn/ amni^r/;, J}//,'., JJo. /I,/,,,/,,,/ J-'fltm/iy '"'""":" J/r„ r Mi// ar/.i />,. .w. A., S/,,m /',: .r.s-/i ,., Hi: ■rii-r v/ Jhsl 1'///, ' Tr/,/1 „r„„l ■,;■„/ /; Ti/.i .,/,/, a I) h O Hill ■V '^j> * «,* "(■> ** f HiSiilDS Jas- _.s 'L^ ' ' , S'»,,iffiii iiiiikfBi iriibSAi/.s, v\-\^,\\\. - — . v-s-X/'/v-v £lare_G.=- S:uia t-l.Mltl isli weirs \ (^ >aiid always covered i >^i&tn.lsoTnetirm.s Lai Dire ctiuii olthe Tiirrent Jiocks sometiiues L SaVtt'^'aii^e-mai Lfflock- z-' U'^Mij^ )v^ Mil ] I 0>N ■"^^ Teaiii1>oat fecrv Vb ^0 \5^^ LIBRARY OF CONGRESS illlllillllllllllll. 020 365 744 4