Education Library ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY, WITH THEIR APPLICATIONS TO MENSURATION, SURVEYING, AND NAVIGATION. BY ELIAS LOOMIS, LL.D., if PKOPESSOS OF NATURAL PHILOSOPHY AND A8TEONOMT I1C TALE COLLEGE, AND AUTIIOE A u COCESE OF MAT1IEMATICS." TWENTY- FIFTH EDITION. NEW YORK: HARPER & BROTHERS, PUBLISHERS, FRAXKLIN SQUARE. 1873. <= * LOOMIS'S SERIES OF TEXT-BOOKS. ELEMENTARY ARITHMETIC. 166 pp., 33, cents. TREATISE ON ARITHMETIC. 352 pp., $1 03. ELEMENTS OF ALGEBRA. Revised Edition. 281 pp., $1 05. Key to Elements of Algebra, for Use of Teachers. 128 pp., $1 05. TREATISE ON ALGEBRA. Revised Edition. 384 pp., $1 17. Key to Treatise on Algebra, for Use of Teachers. 219 pp., $1 17. ELEMENTS OF GEOMETRY. Revised Edition. 388 pp., $1 17. ELEMENTS OF TRIGONOMETRY, SURVEYING, AND NAVIGATION. 194 pp., $1 17. TABLES OF LOGARITHMS. 150 pp., $1 17. The Trigonometry and Tables, bound in one volume. 360 pp., $1 75. ELEMENTS. OF ANALYTICAL GEOMETRY. Revised Edition. 261 pp., $1 17. DIFFERENTIAL AND INTEGRAL CALCULUS. Revised Edition. 309 pp.,$l 17. The Analytical Geometry and Calculus, bound in one volume. 570 pp., $2 05. ELEMENTS OF NATURAL PHILOSOPHY. 351 pp., $1 25. ELEMENTS OF ASTRONOMY. 254 pp., $1 17. PRACTICAL ASTRONOMY. 499 pp., $1 75. TREATISE ON ASTRONOMY. 351 pp., $1 75. TREATISE ON METEOROLOGY. 308 pp., $1 75. Entered, according to Act of Congress, in the year one thousand eight hundred and fifty-eight, by HARPER & BROTHERS, in tho Clerk's Office of the District Court of the Southern District of Ne\v York. PREFACE. *TiiE following treatise constitutes the third volume of ii course of Mathematics designed for colleges and high schools, and is prepared upon substantially the same model as the works on Algebra and Geometry. It does not profess to embody every thing which is known on the subject of Trigonometry, but it contains those principles which are most important on account of their applications, or their connection with other parts of a course of mathematical study. The aim has been to render every principle intelligible, not by the repetition of superfluous words, but by the use of precise and appropriate language. "Whenever it could conveniently be done, the most important principles have been reduced to the form of theorems or rules, which are distinguished by the use of italic letters, and are designed to be committed to memory. The most im- portant instruments used in Surveying are fully described, and are illustrated by drawings. The computations are all made by the aid of natural num- bers, or with logarithms to six places ; and by means of the accompanying tables, such computations can be performed with great facility and precision This volume, having been ttsed by several successive classes, has been subjected to tho ssverest scrutiny, and the present edition embodies all the al- terations which have been suggested by experience in the re- oitaHoo zcom. L7 CONTENTS, . ___ _ __ i*] U < BOOK I. THE NATURE AND PROPERTIES OF LOGARITHMS. r-< Nature of Logarithms ................................................. 1 Description of the Table of Logarithms .................................. 9 Multiplication by Logarithms .......................................... 1 . Division by Logarithms ............................................... 1C Involution by Logarithms ............................................. 17 Evolution by Logarithms ....... . ...................................... 17 Proportion by Logarithms ............................................. 18 BOOK II. PLANE TRIGONOMETRY. Kines, Tangents, Secants, &c., defined .................................. 20 Explanation of the Trigonometrical Tables ............................... 23 To find Sines and Tangents of small Arcs ............................... 29 Solutions of Right-angled Triangles ..................................... 32 Solutions of Oblique-angled Triangles ................................... 3G Instruments used in Drawing .......................................... 42 Geometrical Construction of Triangles .................................. 46 Values of the Sines, Cosines, &c., of certain Angles ....................... 48 Trigonometrical Formula) .............................................. 52 Computation of a Table of Sines, Cosines, &c ............................. 57 BOOK III. MENSURATION OF SURFACES AND SOLIDS. Areas of Figures bounded by Right Lines ................................ 5S> Area of a Regular Polygon ............................................. 64 Quadrature of the Circle and its Parts ........ : .......................... 6G Mensuration of Solids ................................................. 71 Rail-way Excavations or Embankments ................................. 77 Regular Polyedrons ................................................... 81 The three Round Bodies ............................................... 84 Area of a Spherical Triangle ........................................... 8S BOOK IV. SURVEYING. Definitions .......................................................... HG Instruments for measuring Angles ........................... . .......... 91 ri CONTENTS. +* Explanation of the Vernier ............................................ 94 Description of the Theodolite ......................................... 9o Heights and Distances ............................................... 97 The Determination of Areas .......................................... 303 Plotting a Survey .................................... . ............... 101 The Traverse Table ...... ............................................ 1 06 To find the Area of a Field ........................................... 109 Trigonometrical Surveys ..................... ..... ___ ......... ....... 114 Variation of the Needle .............................................. 117 Leveling ........................................................... 119 Topographical Maps ................................................. 123 Setting out Rail-way Curves .......................................... 127 Surveying Harbors .................................................. 130 The Plane Table .................................................... 132 To determine the Depth of Water ..................................... 133 BOOK V. NAVIGATION. Definitions, &c ...................................................... 135 Plane Sailing ....................................................... 138 Traverse Sailing ..................................................... 141 Parallel Sailing ..................................................... 144 Middle Latitude Sailing ........ . ...................................... 1 46 Mercator's Sailing ................................................... 149 Nautical Charts ..................................................... J 53 BOOK VI. SPHERICAL TRIGONOMETRY. Right-angled Spherical Triangles Napier's Rule of the Circular Parts .................................... 158 Examples of Right-angled Triangles ................................... 160 Oblique-angled Spherical Triangles .................................... 163 Examples of Oblique-angled Triangles ............. ............ ........ 165 Trigonometrical Formulae .......... .................................. 171 Bailing 'on an Arc of a Great Circle .................................... 17 TElGOIOMETliY. BOOK I. THE NATURE AND PROPERTIES OF LOGARITHMS. ARTICLE L Logarithms are numbers designed to dimmieii th 3 labor of Multiplication and Division, by substituting in their stead Addition and Subtraction. All numbers are regarded a* O powers of some one number, which is called the base of the system ; and the exponent of that power of the base which is equal to a given number, is called the logarithm of that number. The base of the common system of logarithms (called, from their inventor, Briggs' logarithms) is the number 10. Hence all numbers are to be regarded as powers of 10. Thus, since 10 1, is the logarithm of 1 in Briggs' system; 10' = 10, 1 " " 10 " 10 5 =100, 2 " " 100 " ' 10 3 =1000, 3 " " 1000 " 10 4 =10000, 4 " " 10,000 " &c., &c., &c. ; whence it appears that, in Briggs' system, the logarithm oi every number between 1 and 10 is some number between and 1, i. e., is a proper fraction. The logarithm of every num- ber between 10 and 100 is some number between 1 and 2, i. e.j is 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e., is 2 plus a fraction, and so on. (2.) The preceding principles may be extended to fractions by means of negative exponents. Thus, since 10-'= 0.1, 1 is the logarithm of 0.1 in Briggs' system ; 10- 2 =0.01, -2 " 0.01 " 10-' =0.001, -3 " " 0.001 " " U)- 4 =0.0001 -4 " 0.0001 H TRIGONOMETRY. IIciioc it appears that the logarithm of every number between 1 and 1 is some number between and 1, or may be rep resented by 1 plus a fraction ; the logarithm of every num- ber between 0.1 and .01 is some number between 1 and 2, or may be represented by 2 plus a fraction; the logarithm of every number between .01 and .001 is some number be- tween 2 and 3, or is 'nual to 3 plus a fraction, and soon. The logarithms of most numbers, therefore, consist of an in teger and a fraction. The integral part is called the charac teristic, and may be known from the following RULE. The characteristic of the logarithm of any number greater than unity, is one less than the number of integral figures in the given number. Thus the logarithm of 297 is 2 plus a fraction ; that is, the characteristic of the logarithm of 297 is 2, which is one less than the number of integral figures. The characteristic of the logarithm of 5673.29 is 3 ; that of 73254.1 is 4, &o. The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which its first significant figure is removed from the place of units. Thus the logarithm of .0046 is 3 plus a fraction; that is, the characteristic of the logarithm is 3, the first significant figure, 4, being removed three places from units. (3.) Since powers of the same quantity are multiplied by adding their exponents (Alg., Art. 50), The logarithm of the product of two or more factors z.s fqual to the sum of the logarithms of those factors. Hence we see that if it is required to multiply two or more numbers by each other, we have only to add their logarithms : the sum will be the logarithm of their product. "We then look- in the table for the number answering to that logarithm, in order to obtain the required product. Also, since powers of the same quantity are divided by sub- tracting their exponents (Alg 1 ., Art. 66), The logarithm of the quotient of one number divided lit an- LOGARITHMS. S othei, in equal to the difference of the logarithms, of those numbers. Hence we see that if we wish to divide one number by an- other, we have only to subtract the logarithm of the divisor from that of the dividend ; the difference will be the logarithm 3f their quotient. (4.) Since, in Briggs' system, the logarithm of 10 is 1, it any number be multiplied or divided by 10, its logarithm will be increased or diminished by 1 ; and as this is an integer, it will only change the characteristic of the logarithm, without affecting the decimal part. Hence The decimal part of the logarithm of any number is the same as that of the number multiplied or divided by 10, 100, 1000, &c. Thus, the logarithm of 65430 is 4.815777 , " 6543 is 3.815777; " 654.3 is 2.815777; " 65.43 is 1.815777; " 6.543 isO.S15 r ;77; " " ,6543 is 1.815777 ; " " .06543 is 2.815777; " " .006543 is 3.815777. The minus sign is here placed over the characteristic, to show that that alone is negative, while the decirial part of the logarithm is positive. TABLE OF LOGARITHMS. (5.) A table of logarithms usually contains the logarithms of the entire series of natural numbers from 1 up to 10,000, and the larger tables extend to 100,000 or more. In the smaller tables the logarithms are usually given to five or six decimal places ; the larger tables extend to seven, and sometimes eight or more places. In the accompanying table, the logarithms of the first 100 numbers are given with their characteristics ; but, for all other numbers, the decimal part only of the logarithm is given, whilo the character stic is left t> be supplied, according to the in Ait, 2. 10 TRIGONOMETRY. (6.) To fend Hie Logarithm of any Number between 1 and 100 Look on the first page of the accompanying table, along the column of numbers under N., for the given number, and against it, in the next column, will be found the logarithm with its characteristic. Thus, opposite 13 is 1.113943, which is the logarithm of 13 ; " 65 is 1.812913, " 65. To find the Logarithm of any Number consisting of three Figures. Look on one of the pages of the table from 2 to 20, alon^ the left-hand column, marked N., for the given number, and against it, in the column headed 0, will be found the decimal part of its logarithm. To this the characteristic must be pre- fixed, according to the rule in Art. 2. Thus the logarithm of 347 will be found, from page 8, 2.540329 ; " 871 " " 18, 2.940018. As the first two figures of the decimal are the same for sev eral successive numbers in the table, they are not repeated for each logarithm separately, but are left to be supplied. Thus the decimal part of the logarithm of 339 is .530200. The first two figures of the decimal remain the same up to 347 ; they are therefore omitted in the table, and are to be supplied. To find the Logarithm of any Number consisting of foiu Figures. Find the three left-hand figures in the column marKecl JN , as before, and the fourth figure at the head of one of the other columns. Opposite to the first three figures, and in the col- umn under the fourth figure, will be found four figures of the logarithm, to which two figures from the column headed are to be prefixed, as in the former case. The characteristic must be supplied according to Art. 2. Thus the logarithm of 3456 is 3.538574 ; " " 8765 is 3.942752. In several of the columns headed 1, 2, 3, &c., small dots are tound in the place of figures. This is to show that the two figures which are to be prefixed from the first column have ahangod. and th n ,y are to be taken from the horizrntal line di- LOGARITHMS. (1 reefcly Idow. The place of the dots is to "be supplied with oi phers. Thus the logarithm of 2045 is 3.310693 ; " " 9777 is 3.990206. The two leading figures from the column must also be taken f.*om the horizontal line below, if any dots have "been oassed over on the same horizontal line. Thus the logarithm of 1628 is 3.211654. To find the 'logarithm of any Number containing' more than four Figures. (7.) By inspecting the table, we shall find that, within cer- tain limits, the differences of the logarithms are nearly propor- tional to the differences of their corresponding numbers. Thus the logarithm of 7250 is 3.860338 ; " " 7251 is 3.860398 ; " " 7252 is 3.860458 ; " " 7253 is 3.860518. Here the difference between the successive logarithms, called the tabular difference, is constantly 60, corresponding to a dif- ference of unity in the natural numbers. If, then, we sup- pose the logarithms to be proportional to their corresponding numbers (as they are nearly), a difference of 0.1 in the num- bers should correspond to a difference of 6 in the logarithms , a difference of 0.2 in the numbers should correspond to a dif- ference of 12 in the logarithms, &c. Hence the logarithm of 7250.1 must be 3.860344 ; " 7250.2 " 3.860350; " " 7250.3 " 3.860356. In order to facilitate the computation, the tabular difterenco is inserted on page 16 in the column headed D., and the pro- portional part for the fifth figure of the natural number is given at the bottom of the page. Thus, when the tabular difference is 60, the corrections for .1, .2, .3, &c., are seen to be 6, 12, 18, &c. If the given number was 72501, the characteristic of its log. arithm would be 4, but the decimal part would be the same as for 7250,1. If it were required to find the correction for a sixth fissure 12 TRIGONOMETRY. m the natural number, it is readily obtained from the Propor- tional Parts in the table. The correction for a figure in tha sixth place must be one tenth of the correction for the samo figure if it stood in the fifth place. Thus, if the correction for .5 is 30, the correction for .05 is obviously 3. As the differences change rapidly in the first part of the ta- Lie, it was found inconvenient to give the proportional parts for each tabular difference ; accordingly, for the first seven pages, they are only given for the even differences, but the pro- portional parts for the odd differences will be readily found by inspection. Required the logarithm of 452789. The logarithm of 452700 is 5.655810. - The tabular difference is 96. Accordingly, the correction for the fifth figure, 8, is 77, and for the sixth figure, 9, is 8.6, or 9 nearly. Adding these cor- rections to the number before found, we obtain 5.655896. The preceding logarithms do not pretend to be perfectly exact, but only the nearest numbers limited to six decimal places. Accordingly, when the fraction which is omitted ex- ceeds half a unit in the sixth decimal place, the last figure must be increased by unity. Required the logarithm of 8765432. The logarithm of 8765000 is 6.942752 Correction for the fifth figure, 4, 20 " " sixth figure, 3, 1.5 " " seventh figure, 2. 0.1 Therefore the logarithm of 8765432 is 6.942774. Required the logarithm of 234567. The logarithm of 234500 is 5.370143 Correction for the fifth figure, 6, 111 sixth figure, 7, 13 Therefore the logarithm of 234567 is 5.370267. To find the Logarithm of a Decimal Fraction (8.) According to Art. 4, the decimal part of the logarithm of any number is the same as that of the number multiplied 9r divided by 10, 100, 1000, &c. Hence, for a decimal frac LOGARITHMS. 13 lion, we find the logarithm as if the figures were integers, and prefix the characteristic according to the rule of Art. 2 EXAMPLES. The logarithm of 345.6 is 2.538574 ; 87.65 is 1.942752 : " " 2.345 is 0.370143; " .1234 is 1.091315 ; " " .005678 is 3.754195. To find the Logarithm of a Vulgar Fraction. (9.) We may reduce the vulgar fraction to a decimal, and find its logarithm by the preceding article ; or, since the value of a fraction is equal to the quotient of the numerator divided by the denominator, we may, according to Art. 3, subtract the logarithm of the denominator from that of the numerator; *,ho difference will be the logarithm of the fraction. Ex. 1. Find the logarithm of T 3 F , or 0.1875. From the logarithm of 3, 0.477121, Take the logarithm of 16, 1.204120. Leaves the logarithm of T 3 F , or .1875, 1.273001. Ex. 2. The logarithm of ^ is 2.861697. Ex. 3 The logarithm of iff is 1.147401. To find the Natural Number corresponding to any Logarithm. (10.) Look in the table, in the column headed 0, for the first, two figures of the logarithm, neglecting the characteristic ; the other four figures are to be looked for in the same column, 01 in one of the nine following columns ; and if they are exactly found, the first three figures of the corresponding number will be found opposite to them in the column headed N., and the fourth figure will be found at the top of the page. This number must be made to correspond with the characteristic of the given logarithm by pointing off decimals or annexing ciphers. Thus the natural number belonging to the log. 4.370143 is 23450; " " 1.538574 is 34.56. If the decimal part of the logarithm can not be exactly found ill the table, look for the nearest less logarithm, and take out L 4 the lour figures of the corresponding natural number as be- fore ; the additional figures may be obtained by means of the Proportional Parts at the bottom of the page. Required the number belonging to the logarithm 4.368399. On page 6, we find the next less logarithm .368287 The four corresponding figures of the natural number are 12335. Their logarithm is less than the one proposed by 112. The tabular difference is 186 ; and, by referring to the bottom of page 6, we find that, with a difference of 186, the figure corresponding to the proportional part 112 is 6. Hence the five figures of the natural number are 23356 ; and, since the characteristic of the proposed logarithm is 4, these five figures aie all integral. Required the number belonging to logarithm 5.345678. The next less logarithm in the table is 345570, Their difference is 108 The first four figures of the natural number are 2216. With the tabular difference 196, the fifth figure, correspond- ing to 108, is seen to be 5, with a remainder of 10. To find the sixth figure corresponding to this remainder 10, we may multiply it by 10, making 100, and search for 100 in the same- line of proportional parts. "We see that a difference of 100 would give us 5 in the fifth place of the natural number. Therefore, a difference of 10 must give us 5 in the sixth place of the natural number. Hence the required number is 221655 In tho same manner we find tin number corresponding to log. 3.538672 is 3456.78 ; " 1.994605 is 98.7654; " " " 1.647817 is .444444 MULTIPLICATION BY LOGARITHMS. (11.) According to Art. 3, the logarithm of the product oi two or more factors is equal to the sum of the logarithms of those factors. Hence, for multiplication by logarithms, we have the following RULE. Add the logarithms of the factors ; the sum will be the log srithm of their product. EK, 1. Required the product of 57,98 by 18. LOGARITHMS. 13 The logarithm of 57.98 is 1.763278 " 18 is 1.255273 The bgarithm of the product 1043.64 is 3.018551 Ex. 2. Required the product of 397.65 by 43.78. Ans., 17409.117. Ex. 3. Required the continued product of 54.32, 6.543, and 12.345. The word sum, in the. preceding rule, is to be understood in its algebraic sense ; therefore, if any of the characteristics of the logarithms are negative, we must take the difference be- tween their sum and that of the positive characteristics, and prefix the sign of the greater. It should be remembered that the decimal part of the logarithm is invariably positive ; hence- that which is carried from the decimal part to the character- istic must be considered positive. Ex. 4. Multiply 0.00563 b^ 17. The logarithm of 0.00563 is 3.75050S 17 is 1.230449 Product, 0.09571, whose logarithm is 2.980957. Ex. 5. Multiply 0.3854 by 0.0576. Ans. 0.022199. Ex. 6. Multiply 0.007853 by 0.00476. Ans., 0.0000373S. Ex. 7. Find the continued product of 11.35, 0.072, and 0.017 (12.) Negative quantities may be multiplied by means of logarithms in the same manner as positive, the proper sign being prefixed to the result according to the rules of Algebra. To distinguish the negative sign of a natural number iron* th negative characteristic of a logarithm, we append the letVr n to the logarithm of a negative factor. Thus the logarithm of -56 we write 1.748188 n. Ex. 8. Multiply 53.46 by -29.47. The logarithm of 53.46 is 1.728029 -29.47 is 1.469380 n. Product, -1575.47, log. 3.197409 n. Ex. 9. Find the continued product of 372.1, -.0054, and -175.6. Ex. 10. Find the continued product of -0 137, 7.689, and - 0376 Ifi TRIGONOMETRY. DIVISION BY LOGARITHMS (13.) According to Art. 3, the logarithm of the quotient of ono number divided "by another is equal to the difference of the logarithms of those numbers. Hence, for division "by log. arithms, we have the following RULE. From the logarithm of the dividend, subtract the logarithm of the divisor ; the difference will be the logarithm of tht quotient. Ex. 1. Required the quotient v \f 888.7 divided by 42.24 The logarithm of 888.7 is 2.948755 42.24 is 1.625724 The quotient is 21.039, whose log. is 1.323031. Ex. 2. Required the quotient of 3807.6 divided by 13.7. Ans., 277.927. The word difference, in the preceding rule, is to be under, tood in its algebraic sense ; therefore, if the characteristic of one of the logarithms is negative, or the lower one is greater than the upper, we must change the sign of the subtrahend, and proceed as in addition. If unity is carried from the deci- mal part, this must be considered as positive, and must bn united with the characteristic before its sign is changed. Ex. 3. Required the quotient of 56.4 divided l-y 0.00015. The logarithm of 56.4 is 1.7oi279 " " 0.00015 is 4.176091 The quotient is 376000, whose log. is 5.575188. This result may be verified in the same way as subtraction in common arithmetic. The remainder, added to the subtra- hend, should be equal to the minuend. This precaution should always be observed when there is any doubt with regard to the sign of the result. Ex. 4. Required the quotient of .8692 divided by 42.258. Ans. Ex. 5. Required the quotient of .74274 divided by .00928 Ex. 6. Required the quotient of 24.934 divided by .078541 Negative quantities may be divided by means of logarithms Li G A R IT II M S. -17 in the same manner as positive, the pr >per sign being prefixed to the result according to the rules of Algebra. Ex. 7. Required the quotient of -79.54 divided by 0,08321 Ex. 8. Required the quotient of -0.4753 divided by -36.74. INVOLUTION BY LOGARITHMS. (14.) It is proved in Algebra, Art. 340, that the logarithm of any power of a number is equal to the logarithm of that number multiplied by the exponent of the power. Henco, to involve a number by logarithms, we have the folk wing RULE. Multiply the logarithm of the number by the exponent of the power required. Ex. 1. Required the square of 428. The logarithm of 428 is 2.631444 2 Square, 183184, log. 5.262888. Ex. 2. Required the 20th power of 1.06. The logarithm of 1.06 is 0.025306 20 20th power, 3.2071, log. 0.506120. Ex. 3. Required the 5th power of 2.846. [t should be remembered, that what is carried from the dec- imal part of the logarithm is positive, whether the characteris- tic is positive or negative. Ex. 4. Required the cube of .07654. The logarithm of .07654 is 2.883888 3 Cube, .0004484, log. 4.651664. Ex. 5. Required the fourth power of 0.09874. Ex. 6. Required the seventh power of 0.8952. EVOLUTION BY LOGARITHMS. (15.) It is proved in Algebra, Art. 341, that the logarithm ol any root of a number is equal to the logarithm of that num 5er divided by the index of the root. Hence, to extract th< cot of a number by logarithms, we havo the following B 18 T R I G IN M E T R Y. RULE. Duide the logarithm of the number by the index oj Int root required. Ex. 1. Required the cube root of 482.38. The logarithm of 482.38 is 2.683389. Dividing by 3, we have 0.894463, which corresponds to 7.842, which is therefore the root required. Ex. 2. Required the 100th root of 365. Ans., 1.0608. When the characteristic of the logarithm is negative, and i* not divisible by the given divisor, we may increase the char- acteristic by any number which will make it exactly divisible, provided we prefix an equal positive number to the decimal part of the logarithm. Ex. 3. Required the seventh root of 0.005846. The logarithm of 0.005846 is 3.766859, which may be writ- ten 7+4.766859. Dividing by 7, we have 1.680980, which is the logarithm oi 4797, which is, therefore, the root required. This result may be verified by multiplying 1.680980 by 7, the result will be found to be 3.766860. Ex. 4. Required the fifth root of 0.08452. Rx. 5. Required the tenth root of 0.007815. PROPORTION BY LOGARITHMS. (16.) The fourth term of a proportion is found by multiply ing together the second and third terms, and dividing by the first. Hence, to find the fourth term of a proportion by loga lithms, Add the logarithms of the second and third terms, and from their sum subtract the logarithm of the first term. Ex. 1. Find a fourth proportional to 72.34, 2.519, and 357.4S Ans., 12.448. (17.) When one logarithm is to be subtracted from anothej, it may be more convenient to convert the subtraction into au addition, which may be done by first subtracting the given leg. arithm from 10, adding the difference to the other 8 ml aftorwnrd rejecting the 10. LOGARITHMS. 19 The difference between a given logarithm and 10 is called its complement ; and this is easily taken from the table by be- ginning at the left hand, subtracting each figure from 9, ex- cept the last significant figure on the right, which must be subtracted from 10. To subtract one logarithm from another is the same as to fidd its complement, and then reject 10 from the result For it b is equivalent to 10 b + a 10. To work a proportion, then, by logarithms, we must Add the complement of the logarithm of the first term f<* 'he logarithms of the second and third terms. The characteristic must afterward be diminished by 10. Ex. 2. Find a fourth proportional to 6853, 489, and 38750 The complement of the logarithm of 6853 is 6.164119 The logarithm of 489 is 2.689309 " " 38750 is 4.588272 The fourth term is 2765, whose logarithm is 3.441700. One advantage of using the complement of the first term in working a proportion by logarithms is, that it enables us to exhibit the operation in a more compact form. Ex. 3. Find a fourth proportional to 73.84, 658.3, and 4872 Ans. Ex. 4 Find R fcurth proportional to 5.745, 781.2, and 512* BOOK II. PLANE TRIGONOMETRY, (18.) TRIGONOMETRY is the science which leaches how to de- termine the sevwt** parts of a triangle from having certain parts given. Plane Trigonometry treats of plane triangles ; Spherical Trigonometry treats of spherical triangles. (19.) The circumference of every circle is supposed to Le divided into 360 equal parts, called degrees ; each degree into 60 minutes, and each minute into 60 seconds. Degrees, min- utes. and seconds are designated by the characters , ', '' Thus 23 14' 35" is read 23 degrees, 14 minutes, and 35 sec- onds. Since an angle at the center of a circle is measured by the arc intercepted by its sides, a right angle is measured by 90, two right angles by 180, and four right angles are measured by 360. (20.) The complement of an arc is what remains after sub tr acting the arc from 90 Thus the *rc DF is the complement of AF. The complement of 25 15' is 64 45'. In general, if we represent any arc >y A, its complement is 90 A. Hence, if an arc is greater than 90, its complement must be negative. Thus, the complement of 100 15' is -10 15'. Since the two acute an- gles of a right-angled triangle are to- E gether equal to a right angle, each of them must be the plemont of the other. (21.) The supplement of an arc is vhai remains alter sub- tracting the arc from ISO 3 . Thus the arc BDF is the supple* nnnt of the arc AF. The supplement of 25 15' is 154 45'. In general, 'f we represent any arc by A, its supplement is PLANF TRIGONOMETRY. 21 ISO - A Hence, if an arc is greater than 180, its supple- ment must be negative. Thus the supplement of 200 i^ 20 ! . Since in every triangle the sum of the three angles is 180, either angle is the supplement of the sum of the other two. (22.) The sine of an arc is the perpendicular let fall from one extremity of the arc on the radius passing through the other extremity. Thus FGr is the sine of the arc AF, or of th( : angle ACF. Every sine is half the chord of double the arc. Thus the sine FG- is the half of FH, which is the chord of the arc FAH, double of FA. The chord which subtends the sixth part of the circumference, or the chord of 60, is equal to the radius (Geom.j Prop. IV., B. VI.) ; hence the sine of 30 is equal tn half of the radius. (23.) The versed sine of an arc is that part of the diametei intercepted between the sine and the arc. Thus GrA is the. versed sine of the arc AF. (24). The tangent of an arc is the line which touches it ai one extremity, and is terminated by a line drawn from the. cenlsr through the other extremity. Thus AI is the tangent of the arc AF, or of the angle ACF. (25.) The secant of an arc is the line drawn from the cen- ter of the circle through one extremity of the arc, and is Urn lied by the tangent drawn through the other extremity. Thus CI is the secant of the arc AF, or of the angle ACF. (26.) The cosine of an arc is the sine of the complement of that arc. Thus the arc DF, being the complement of AF, FK is the sine of the arc DF, or the cosine of tho arc AF. The cotangent of an arc is the tangent of the complement of that arc. Thus DL is the tangent of the arc DF, or the co- tangent of the arc AF. The cosecant of an arc is the secant of the complement o( that arc. Thus CL is the secant of the arc DF, or the coso- cant of the arc AF. In genera], if we repr ent any angle by A, cos. A=sine (90-A). cot. A^tang. (90 -A), cosec. A=sec. (90 A). Since, in a right-angled triangle, either of the acute angle* 23 TRIGONOMETRY. is the complement of the other, the sine, tangent, and secant <>f cne of these angles is the cosine, cotangent, and cosecan of the other. (27.) The sine, tangent, and secant of an arc are equal te fcho sine, tangent, and secant of its supplement. Thus FG ii the sine of the arc AF, or of its sup- plement, BDF. Also, AI, the tan- gent of the arc AF, is equal to BM, the tangent of the arc BDF. And CI, the secant of the arc AF, is equal to CM, the secant of the arc BDF. The versed sine of an acute angle, ACF, is equal to the radius minus the cosine CG. The versed sine of an obtuse angle, BCF, is equal to ra- dius pins the cosine CG ; that is, to BG. (28.) The relations of the sine, cosine, &c., to each othei, may be derived from the proportions of the sides of similai triangles. Thus the triangles CGF, CAI, CDL, being similar, we have, 1. CG : GF : : CA : AI ; that is, representing the arc by A, and the radius of the circle by R, cos. A : sin. A : : R : tang. A. R sin. A Whence tang. A . cos. A "2. CG : : CF : CA : CJ; that is, cos. A : R : : R : sec. A. R a Whence sec. A= - cos. A 3 GF : CG : : CD : DL; that is, sin, A : cos. A : : R : cot. A. _ R cos. A Whence cot. A= : r . sm. A 4 GF : CF : : CD : CL ; that is, sin. A : R : : R : cosec. A. T>3 Whence cosec. A= r- sin. A 5. AI : AC : : CD : DL ; that is, tang. A : R : : R : cot. A. R a Whence tang. A. = - r-. cot, A The preceding values of tangent and cotangent, secant and cosecant will be frequently referred to hereafter, and should bf carefully committed to memory. PLANE T R i c o N j M E T R v. Also, in the right-angled triangle CGF, we find CGr*-|- 3F'=- UF; that is, sin. 2 A+cos. 2 A=R 2 ; or, The square of the sine of an arc, together with the square. tf its cosine^ is equal to the square of the radius. Hence sin. A VR 2 cos. 2 A. And cos. ,A= VR a sin. 2 A. (29 ) A table of natural sines ^ tangents, &c., is a table giv- ing the lengths of those lines for different angles in a circle whose radius is unity. Thus, if we describe a circle with a radius of one inch, and divide the circumference into equal parts of ten degrees, we Khali find the sine of 10 equals 0.174 inch ; " " 20 " 0.342 " P " " 30 " 0.500 " " " 40 " 0.643 " " " 50 " 0.766 " " " 60 " 0.866 " " " 70 " 0.940 " " " 80 " 0.985 " u u 90 " 1.000 " ^ ^ .K = \ \ N w \ 31? y 1 If \ve draw the tangents of the same arcs, we shall find the tangent of 10 equals 0.176 inch ; tt 20 " 0.364 " tt 30 " 0.577 " tt 40 " 0.839 " tt 45 " 1.000 " tl 50 " 1.192 " It 60 " 1.732 " tt 70 u 2.747 " It 80 " 5.671 tt 90 " infinite. Also, if we draw the secanf s of the same wcs, we shall find that the secant of 10 equals 1.015 inch; " " 20 " 1.064 " " 30 " 1.155 " tt 40 " 1.305 ' 70" 23 TRIGONOMETRY. the secant of 50 equals 1.556 inch ; " " 60 " 2.000 " u 70 " 2.924 " < " 80 " 5.759 " " " 90 infinite. In the accompanying table, pages 116-133, the sines, co- sines, tangents, and cotangents are given for every minute of the quadrant to six places of figures. (30.) To find from the table the natural sine, cosine, c, } of an arc or angle. If a sine is required, look for the degrees at the top of the page, and for the minutes on the left ; then, directly under the given number of degrees at the top of the page, and opposite to the minutes on the left, will be found the sine required. Since the radius of the circle is supposed to be unity, the sims of every arc below 90 is less than unity. The sines are ex- pressed in decimal parts of radius ; and, although the decimal point is not written in the table, it must always be prefixed As the first two figures remain the same for a great many numbers in the table, they are only inserted for every ten min- utes, and the vacant places must be supplied from the two leading figures next preceding Thus, on page 120, the sine of 25 11' is 0.425516 ; page 126, " " 51 34' is 0.783332, &c. The tangents are found in a similar manner. Thus the tangent of 31 44' is 0.618417 ; " " 65 27' is 2.18923. The same number in the table is both the sine of an arc arid the cosine of its complement. The degrees for the cosines must be sought at the bottom of the page, and the minutes on the right. Thus, on page 130, the cosine of 16 42' is 0.957822 ; on page 118, " " 73 17' is 0.287639, &c. The cotangents are found in the same manner. TKus the cotangent of 19 16' is 2.86089 ; " 54 53' is 0.703246. It is not necessary to extend the tables beyond a qu?nl (Art. 27). Thus PLANE T RIG oyoy ETR r. . ^ the sine of 143 24' is 0.596225 ; ' cosine of 151 23' is 0.877844; " tangent of 132 36' is 1.08749 ; " cotangent of 116 7' is 0.490256, &o. (31.) If a sine is required for an arc consisting of degrees, minutes, and seconds, we must make an allowance for the sec- onds in the same manner as was directed in the case of loga- rithms, Art. 7 ; for, within certain limits, the differences ol tho sines are proportional to the differences of the corresponding arcs. Thus the sine of 34 25' is .565207 ; " ; < 34 26' is .565447. The difference of the sines corresponding to one minute ol arc, or 60 seconds, is .000240. The proportional part for 1 ' is found by dividing the tabular difference by 60, and the quo- tient, .000004, is placed at the bottom of page 122, in the col- umn headed 34. The correction for any number of seconds will be found by multiplying the proportional part for 1" by the number of seconds. Required the natural sine of 34 25' 37". The proportional part for 1", being multiplied by 37, becomes 148, which is the correction for 37". Adding this to the sinn jf 34 25', we find the sine of 34 25' 37" is .565355. Since the proportional part for 1" is given to hundredths of a unit in the sixth place of figures, after we have multiplied by the given number of seconds, we must reject the last two fig- ures of the product. Tn the same manaer we find the cosine of 56 34' 28" is .550853. It will be observed, that since the cosines decrease while the arcs increase, the correction for the 28" is to be subtracted from the cosine of 56 34'. In the same manner we find the natural sine of 27 17' 12" is 0.458443 ; " " cosine of 45 23' 23" is 0.702281 ; " tangent of 63' 32' 34" is 2.00945 : " cotangent of 81 48' 56" is 143825 2<> TRIGONOMETRY. (32.) To find the rur/iber of degrees, minutes , and second* belonging to a given sine or tangent If the given sine or tangent is found exactly in the table, the corresponding degrees will be found at the top of the page, and the minutes on the left hand. But when the given num- ber is not found exactly in the table, look for the sine or tan- gont which is next less than the proposed one, and take out the corresponding degrees and minutes. Find, also, the dif- ference between this tabular number and the number proposed, and divide it by the proportional part for 1" found at the hot f.om of the page ; the quotient will be the required number of seconds. Required the arc whose sine is .750000. The next less sine in the table is .749919, the arc correspond- ing to which is 48 35'. The difference between this sine and that proposed is 81, which, divided by 3.21, gives 25. Hencr the required arc is 48 35' 25". Tn the same manner we find the arc whose tangent is 2.00000 is 63 26' 6". If a cosine or cotangent is required, we must look for the number in the table which is next greater than the one pro posed, and then proceed as for a sine or tangent. Thus the arc whose cosine is .40000 is 66 25' 18" ; " " " cotangent is 1.99468 is 26 37' 34". (33.) On pages 134-5 will be found a table of natural se- oants for every ten minutes of the quadrant, carried to seven places of figures. The degrees are arranged in order in tho first vertical column on the left, and the minutes at the top nf the page. Thus the secant of 21 20' is 1.073561 ; " 81 50' is 7.039622. if a secant is required for a number of minutes not given in the table, the correction for the odd minutes may be found by means of tho last vertical column on the right, which shows the proportional part for one minute. Let it be required to find the secant of 30 33' The secant of 30 30' is 1.160592. The correction for 1' is 198.9, which, multiplied by 3. bo- PLANE TR IGONOMETIIT. 27 comes 597. Adding this to the number before found, we ob. tain 1.161189. For a cosecant, the degrees must be sought in the right- hand vertical column, and the minutes at the bottom of th page. Thus the cosecant of 47 40' is 1.352742 ; 38 33' is 1.604626. (34.) When the natural sines, tangents, &c., are used in pro portions, it is necessary to perform the tedious operations of multiplication and division. It is, therefore, generally prefer- able to employ the logarithms of the sines ; and, for conven- ience, these numbers are arranged in a separate table, called logarithmic sines, &c. Thus the natural sine of 14 30' is 0.250380. Its logarithm, found from page 6, is 1.398600. The characteristic of the logarithm is negative, as must be the case with all the sines, since they are less than unity. Te avoid the introduction of negative numbers in the table, we in- crease the characteristic by 10. making 9.398600, and this is the number found on page 38 for the logarithmic sine of 14 30'. The radius of the table of logarithmic sines is therefore, properly, 10,000,000,000, whose logarithm is 10. (35.) The accompanying table contains the logarithmic sinea and tangents for every ten seconds of the quadrant. The de- grees and seconds are placed at the top of the page, and the minutes in the left vertical column. After the first two de- grees, the three leading figures in the table of sines are only given in the column headed 0", and are to be prefixed to the numbers in the other columns, as in the table of logarithms of numbers. Also, where the leading figures change, this change is indicated by dots, as in the former table. The correction for any number of seconds less than 10 is given at the bottom uf the page. (36.) To find the logarithmic sine or tangent of a given arc. Look for the degrees at the top of the page, the minutss on the left hand, and the next less number of seconds at the tor>; then, under the seconds, and opposite to the minutes, will be tound four figures, to which the three leading figures are to b 2% TlUGONOMETR*. prefixed from tht column headed O x/ ; to this add the proper tional part for the odd seconds at the bottom of the page. Required the logarithmic sine of 24 27' 34". The logarithmic sine of 24 27' 30" is 9.617033 Proportional part for 4" is 18 Logarithmic sine of 24 27' 34" is 9.617051. Required the logarithmic tangent of 73 35' 43". The logarithmic tangent 73 35' 40" is 10.531031 Proportional part for 3" is 23 Logarithmic tangent of 73 35' 43" is 10.531054. When a cosine is required, the degrees and seconds must be sought at the bottom of the page, and the minutes on the right, and the correction for the odd seconds must be subtracted from the number in the table. Required the logarithmic cosine of 59 33' 47". The logarithmic cosine of 59 33' 40" is 9.704682 Proportional part for 7" is 25 Logarithmic cosine of 59 33' 47" is 9.704657. So, also, the logarithmic cotangent of 37 27' 14" is found to be 10.115744. It will be observed that for the cosines and cotangents, the seconds are numbered from 10" to 60", so that if it is re* quired to find the cosine of 25 25' 0" we must look for 25 C 24' 60" ; and so, also, for the cotangents. (37.) The proportional parts given at the bottom of each page correspond to the degrees at the top of the page, in- creased by 30', and are not strictly applicable to any other number of minutes ; nevertheless, the differences of the sines change so slowly, except near the commencement of the quad- rant, that the error resulting from using these numbers for every part of the page will seldom exceed a unit in the sixth decimal place. For the first two degrees, the differences change so rapidly that the proportional part for 1" is given for *ach minute in the right-hand column of the page. The cor- rcction for any number of seconds less than ten \\ill be foum by multiplying the proportional part for 1" by the given nun> ber of seconds. Required the logarithmic sine of 1 17' Go". PLANS TRIGONOME TRY. 2& The Lgarithmic sine of 1 17' 30" is 8.352991. The correction for o" is found by multiplying 93.4 by 3, which gives 280. Adding this to the above tabular number, we obtain for the sine of 1 17' 33", 8.353271. A similar method may be employed for several of the first degrees of the quadrant, if the proportional parts at the bottom of the page are not thought sufficiently precise. This correc tioa may, however, be obtained pretty nearly by inspection, from comparing the proportional parts for two successive de- grees. Thus, on page 26, the correction for 1", corresponding to the sine of 2 30', is 48 ; the correction for 1", correspond- ing to the sine of 3 30', is 34. Hence the correction for 1", corresponding to the sine of 3 0', must be about 41 ; and, in the same manner, we may proceed for any other part of the table. (38.) .Near the close of the quadrant, the tangents vary su rapidly that the same arrangement of the table is adopted as for the commencement of the quadrant. For the last, as well as the first two degrees of the quadrant, the proportional part to 1" is given for each minute separately. These proportional parts are computed for the minutes placed opposite to them, increased by 30", and are not strictly applicable to any other number of seconds ; nevertheless, the differences for the most part change so slowly, that the error resulting from using these numbers for every part of the same horizontal line is quite small. When great accuracy is required, the table on page 114 may be employed for arcs near the limits of the quadrant. This table furnishes the differences between the logarithmic sinca and the logarithms of the arcs expressed in seconds. Thus the logarithmic sine of 5', from page 22, is 7.162696 the logarithm of 300" (=-5') is 2.477121 the difference is 4.685575. This is the number found on page 114, under the heading log. sine A log. A", opposite to 5 min. ; and, in a similar man- ner, the other numbers in the same column are obtained. These numbers vary quite slowly for two degrees ; and henc, to find I he logarithmic sine of an arc less than two degrees we 30 T R I G N M E T R 7. but to add ths logarithm of the arc expressed in seconds to thf appropriate number found in this table. Required the logarithmic sine of 7' 22 v . Tabular number from page 114, 4.685575 The logarithm of 442" is 2.645422 Logarithmic sine of T 22" is 7.330997. The logarithmic tangent of an arc less than two degrees la found in a similar manner. Required the logarithmic tangent of 27' 36". Tabular number from page 114, 4.685584 The logarithm of 1656" is 3.219060 Logarithmic tangent of 27' 36" is 7.904644 The column headed log. cot. A+log. A", is found by adding f.hc logarithmic cotangent to the logarithm of the arc expressed in seconds. Hence, to find the logarithmic cotangent of an arc less than two degrees, we must subtract from the tabular num- ber the logarithm of the arc in seconds. Required the logarithmic cotangent of 27' 36". Tabular number from page 114, 15.314416 The logarithm of 1656" is 3.219060 Logarithmic cotangent of D 27' 36" is 12.095356. The same method will, of course . furnish cosines and cotan gents of arcs near 90. (39.) The secants and cosecants are omitted in this table, since they are easily derived from tho cosines and sines. Wo R 2 have found, Art. 28, secant = : ; or, taking the logarithms, cosine ' log. secant =2. log. R log. cosine 20 log. cosine. R 2 Also, cosecant = - , sine or log. cosecant =20 log. sine. That is, The logarithmic secant is found by subtracting the logo** rilhmic cosine from 20; and the logarithmic cosecant is found by subtracting the logarithmic sine from 20. Thus we have found tho logarithmic sine of 24 27' 34' tc be 9.617051. Hence the logarithmic c .secant of 24 27' 34" is 10.382949 PLANE TRIGONOMETRY, 3\ The logarithmic cosine of 54 12' 40" is 9.76700S Hence the logarithmic secant of 54 12' 40" is 10.232992 (40.) To find the arc corresponding- to a given logarithms, line or tangent. If the given number is found exactly in the table, the cor- responding degrees and seconds will be found at the top of the page, and the minutes on the left. But when the given num- ber is not found exactly in the table, look for the sine or tan- gent which is next less than the proposed one, and take out the corresponding degrees, minutes, and seconds. Find, also, the difference between this tabular number and the number proposed, and corresponding to this difference, at the bottom of the page, will be found a certain number of seconds which is to be added to the arc before found. Required the arc corresponding to the logarithmic sine 9.750000. The next less sine in the table is 9.749987. The arc corresponding to which is 34 13' 0". The difference between its sine and the one proposed is 13, corresponding to which, at the bottom of the page, we find 4" nearly. Hence the required arc is 34 13' 4". In the same manner, we find the arc corresponding to loga- rithmic tangent 10.250000 to be 60 38' 57". When the arc falls within the first two degrees of the quad- rant, the odd seconds may be found by dividing the difference between the tabular number and the one proposed, by the pro- portional part for 1". We thus find the arc corresponding to logarithmic sine 8.400000 to be 1 26' 22" nearly. We may employ the same method for the last two degrees of the quadrant when a tangent is given ; but near the limits of the quadrant it is better to employ the auxiliary table on page 114. The tabular number on page 114 is equal to log. sin. A log. A". Hence log. sin. A tabular number =log, A" ; that is, if we subtract the corresponding tabular number on page 114, from the given logarithmic sine, the remainder will be the logarithm of the arc expressed in seconds. Required the arc corresponding to logarithmic sine 7.000000. We see, from page 22, that the arc must be nearly 3' ; tn.0 ( >or respond ing tabular number on page 114 is 4.685575 32 T K I G O N O M. K T R V. The difference is 2.314425, which is the logarithm of 206."265. Hence tho required arc is 3' 26. "265. Required the arc corresponding to log. sine 8.000000. We see from page 22, that the arc is ahout 34'. The co/ responding tabular number from page 114 is 4.685563, which; subtracted from 8.000000, leaves 3.314432, which is the log- arithm of 2062. "68. Hence the required arc is 34' 22."68. In the same manner, we find the arc corresponding to loga nthmio tangent 8.184608 to be 52' 35". SOLUTIONS OF RIGHT-ANGLED TRIANGLES. THEOREM I. (41.) In any right-angled triangle, radius is to the hypoth* enuse as the sine of either acute angle is to the opposite. side, or the cosine of either acute angle to the adjacent side. Let the triangle CAB be right angled at A, then will R : CB : : sin. C : BA : : cos. C : CA. From the point C as a center, with a radius equal to the radius of the tables, C describe the arc DE, and on AC let fall the perpendicular EF Then EF will be the sine, and CF the cosine of the angle C Because the triangles CAB, CFE are similar, we have CE : CB : : EF : BA, or E, : CB : : sin. C : BA. Als.1, CE : CB : : CF : CA, r.r R : CB : : cos. C : CA. THEOREM II. (42.) In any right-angled triangle, radius is to either sidt as the tangent of the adjacent acute angle is to the opposite side, or the secant of the same angle to the hypothenuse. Let the triangle CAB be right angled at A, then will R : CA : : tang. C : AB : : sec. C : CB. From the point C as a center, with a radius equal to tho radius of the tables, c PLANE TRIGONOMETRY. H3 describe the arc BE, and from the point D draw DF perpen- dicular to CA. Then DF will "be the tangent, and CF the se- cant of the angle C. Because the triangles CAB, CDF aro similar, we have CD : CA : : DF : AB, or R : CA : : tang. C : AB. Also, CD : CA : : CF : CB, or R : CA : : sec. C : CB. (43.) In every plane triangle there are six parts : three sides and three angles. Of these, any three being given, provided one of them is a side, the others may be determined. In a right-angled triangle, one of the six parts, viz., the right anglo, is always given ; and if one of the acute angles is given, the other is, of course, known. Hence the number of parts to be considered in a right-angled triangle is reduced to four, any two of which being given, the others may be found. It is desirable to .have appropriate names by which to des ignate each of the parts of a triangle. One of the sides ad- jacent to the right angle being called the base, the other side adjacent to the right angle may be called the perpendicular. The three sides will then be called the hypothenuse, base, and perpendicular. The base and perpendicular are sometimes called the legs of the triangle. Of the two acute angles, that which is adjacent to the base may be called the angle at the base, and the other the angle at the perpendicular. We may, therefore, have four cases, according as there are given, 1. The hypothenuse and the angles : 2. The hypothenuse and a leg ; 3. One leg and the angles ; or, 4. The two legs. All of these cases may be solved by the tfro preceding iheo rcms. CASE I. (44.) Given the hypothenuse and the angles, to find the base and perpendicular. This case is solved by Theorem I. Radius : hypothenuse : : sine of the angle at the base : per- pendicular ; : : cosine of the angle at ihe base : base C 34 TRIGONOMETRY. Ex. 1, Given the hypothenuae 275, and the angle at the bas 57 23', to find the base and perpendicular. The natural sine of 57 23' is .842296 ; " cosine " .539016. Hence 1 : 275 : : .842296 : 231.631=AB. 1 : 275 : : .539016 : 148.229=AC. The computation is here made by natural numbers. If we work the proportion by loga- rithms, we shall have Radius, 10.000000 Is to the hypothenuse 275 2.439333 As the sine of C 57 23' 9.925465 To the perpendicular 231.63 2.364798. Also, Radius, 10.000000 Is to the hypothenuse 275 2.439333 As the cosine of C 57 23' 9.731602 To the base 148.23 2.170935, Ex. 2. Given the hypothenuse 67.43, and the angle at tin* perpendicular 38 43', to find the base and perpendicular. Ans. The base is 42.175, and perpendicular 52.612. The student should work this and the following examples both by natural numbers and by logarithms, until he has made himself perfectly familiar with both methods. He may then p.mploy either method, as may appear to him most expeditious. CASE II. (45.) Given the hypothenuse and one leg-, to find the angles and the other leg-. This case is solVed by Theorem I. Hypothenuse : radius : : base : cosine of the ang-le at the base Radius : hypothenuse : : sine of the angle at the base : perpendicular. When the perpendicular is given, perpendicular must b substituted for base in this proportion. Ex 1. Given the hypothenuse 54.32, and the ba' M E T R Y. . AD : BE : : DF : BF ; that is, OB-f-CA : CB-CA : : tang. -~ : tang. THEOREM III. (51.) If from any angle of a triangle a perpendicular ot drawn to the opposite side or base, the whole base will be 'tc the sum of the other two sides, as the difference of those twn sides is to the difference of the segments of the base. For demonstration, see Geometry, Prop. 31, Cor., B. IV. (52). In every plane triangle, three parts must be given ic enable us to determine the others ; and of the given parts, one> at least, must be a side. For if the angles only are given, i hese might belong to an infinite number of different triangles in solving oblique-angled triangles, four different ca^s may therefore be presented. There may be given, 1. Two angles and a side ; 2. Two sides and an angle opposite one of them ; 3. Two sides and the included angle ; or, 4. The three sides. We shall represent the three angles of the proposed triangle by A, B, C, and the sides opposite them, respectively, by a, b, c CASE I. (53.) Given two angles and a side, to find the tliiid angle ind the other two sides. To find the third angle, add the given angles together, and subtract their sum from 180. The required sides may be found by Theorem I. The pro- jjortion will be, The sine of the angle opposite the given side : the given side : : the sine of the angle opposite the required side : the re- quired side. Ex. 1. In the triangle ABC, there are given the angle A, 57 15', the angle B, 35 30', and the side c, 364, to find the ther parts. The sum of the given. angles, subtracted A PLANE TRIGONOMETRY 3y 180, leaves 87 15' for the angle C. Then, to find the side ff on a scale : this is called a line of chords. Since the chord of 60 is equal to radius, in order to lay down an angle, we take from the scale the chord of 60, and with this radius describe an arc of a circle. Then take from the scale the chord of the given angle, and set it off upon the former arc. Through these two points of division draw lines to the center of the circle, and they will contain the required angle. The line of sines, commonly marked SIN., exhibits the lengths of the sines to every degree of the quadrant, to the same ra- dius as the line of chords. The line of tangents and the line of secants are constructed in the same manner. . Since the sine of 90 is equal to radius, and the secant of is the same, the graduation on the line of secants begins where the line of sine* ends. On the back side of the plane scale are often found lines rep icsenting the logarithms of numbers, sines, tangents, &c. Thin is called Grunter's Scale. (62.) Y. The Sector is a very convenient instrument in drawing. It consists of two equal armc, mova- ble about a pivot as a center, having several scales drawn on the faces, some single, oth- ers double. The single scales are like those upon a common Gunter's scale. The double scales are those which proceed from the center, each being laid twice on the same face of the instrument, viz., once on each leg. The double scales are a scale of lines, marked Lin. or L. ; the scale of chords, sines, &c. On each arm of the sector there is a diagonal line, which diverges from the central point like the radius of a circle, and these diagonal lines are divided into equal parts. The advantage of the sector is to enable us to di? k \v'a'lin 46 TfllGONOMETRv upon paper to any scale ; as, for example, a scale of 6 feel to the inch. For this purpose, take an inch with the dividers from the scale of inches ; then, placing one foot of the dividers at 6 on one arm of the sector, open the sector until the othe? foot reaches to the same number on the other arm. Now, re- garding the lines on the sector as the sides of a triangle, of which the lino measured from 6 on one arm to 6 on the oth- er arm is the base, it is plain that if any other line be measured across the angle of the sector, the bases of the triangles thus formed will be pro- portional to their sides. Therefore, a line of 7 feet will be rep- resented by the distance from 7 to 7, and so on for other lines. The sector also contains a line of chords, arranged like the line of equal parts already mentioned. Two lines of chords are drawn, one on each arm of the sector, diverging from the central point. This double line of chords is more convenient than the single ono upon the plane scale, because it furnishes chords to any radius. If it be required to lay down any angle, as, for example, an angle of 25, describe a circle with any convenient radius. Open the sector uo that the distance from CO to 60, on the line of chords, shall be equal to this radius. Then, preserving the same opening of the sector, place one foot of the dividers upon the division 25 on one scale, and extend the other foot to the same number upon the other scale : this distance will be the chord of 25 degrees, which must be set off upon the circle first described. The lines of sines, tangents, &c., are arranged in the same manner. (63.) By means of the instruments now enumerated, all the cases in Plane Trigonometry may be solved mechanically. The sides and angles which are given are laid down accord- ing to the preceding directions, and the required parts are then measured from the same scale. The student will do well to exercise himself upon the following pr ?blems : I. Given the angles and one side oj a triangle, to find, by construction, the other two sides PLANE TRIGONOMETRY. 47 Draw an indefinite straight line, and from thu scale ui equa. parts lay off a portion, AB, equal to the given side. From each extremity lay off an angle equal to one of the adjacent an- gles, by means of a protractor or a scale of chords. Extend the two lines till they intersect, and measure their lengths upon the same scale of equal parts which was used in laying off the base. Ex. 1. Given the angle A, 45 30', the angle B, 35 20', and the side AB, 43 rods, to construct the triangle, and find the lengths of the sides AC and BC. The triangle ABC may be constructed of any dimensions whatever ; all which is essential is that its angles be made equal to the given angles. We may construct the triangle upon a scale of 100 rods to an inch, in which case the side AB will be represented by 4.32 inches ; or we may construct it upon a scale of 200 rods to an inch ; that is, 100 rods to a half inch, which is very conveniently done from a scale on which a half inch is divided like that described in Art. 60 ; or we may use any other scale at pleasure. It should, however, be remembered, that the required sides must be measured upcn the same scale as the given sides. Ex. 2. Given the angle A, 48, the angle C, 113, and the side AC, 795, to construct the triangle. II. Given twi> sides and an angle opposite one of them, tn find the other parts. Draw the side which is adjacent to the given angle. From one end of it lay off the given angle, and extend a line indefin- itely for the required side. From the other extremity of the first side, with the remaining given side for radius, describe an arc cutting the indefinite line. The point of intersection will determine the third angle of the triangle. Ex. 1. Given the angle A, 74 45', the side AC, 432, and the side BC, 475, to construct the triangle, and find the other parts. Ex. 2. Given the angle A, 105, the side BC, 498, and the side AC, 375, to construct the triangle. III. Given two sides and the included angle, to find the itlwr parts. *8 TRIG GNOME TR\ Draw one of the given sides. From one end of it lay off the given angle, and draw the other given side, making the re- quiied angle with the first side. Then connect the extremities of the two sides, and there will be formed the triangle required Ex. 1. G-iven the angle A, 37 25', the side AC, 675, and the side AB, 417, to construct the triangle, and find the othci parts. Ex. 2. Given the angle A, 75, the side AC, 543, and the side AB, 721, to construct the triangle. IV. Given the three sides, to find the angles. Draw one of the sides as a "base ; and from one extremity of the base, with a radius equal to the second side, describe an arc of a circle. From the other end of the base, with a radius equal to the third side, describe a second arc intersect- ing the former ; the point of intei section will be the third an- gle of the triangle. Ex. 1. Given AB, 678, AC, 598, and BC, 435, to find tho angles. Ex. 2. Given the three sides 476, 287, and 354, to find tho angles. Values of the Sines, Cosines, Sfc., of certain Angles (64.) We propose now to examine the changes which ths tines, cosines, &c., undergo in the dif- ferent quadrants of a circle. Draw two diameters, AB, DE, perpendicu- lar to each other, and suppose one of them to occupy a horizontal position, the other a vertical. The angle ACT) is called the first quadrant, the angle DCB the second quadrant, the angle BCE the third quadrant, and the an- gle EGA the fourth quadrant; that is, the first quadrant is above the horizontal diameter, and on the right of the vertical diameter ; the second quadrant is above the horizontal diame- 2cr, and on the left of the vertical, and so on. Suppose one extremity of the arc remains fixed in A, while the other extremity, marked F, runs round the entire circum- foronce in the direction ADBE. PLANE T R i G o N o M E T R Y. 49 Whm the point F is at A, or when the arc AF is zero, the tine is zero. As the point F advances toward D, the sine in- creases ; and when the arc AF becomes 45, the triangle CFQ being isosceles, we have FGr : CF : : 1 : y/2 (Geom., Prop. 11, Cor. 3, B. IV.) ; or sin. 45 : R : : 1 : ^2. [fence, sin. 45 D = -=Rv/2. \/Z The sine of 30 is equal to half radius {Art. 22). Also, since sin. A= VR 2 cos. 2 A, the sine of 60, which is equal to the co sine of 30, = Vll'-R*= VfRJR V3. The arc AF continuing to increase, the sine also increase till F arrives at D, at which point the sine is equal to the ra dius ; that is, the sine of 90 = R. As the point F advances from D toward B, the sines dimin- ish, and become zero at B ; that is, the sine of 180 = 0. In the third quadrant, the sine increases again, becomes iqual to radius at E, and is reduced to zero at A. (65.) "When the point F is at A, the cosine is equal to ra- dius. As the point F advances toward D, the cosine decreases, and the cosine of 45 = sine 45 JR v/2. The arc continuing to increase, the cosine diminishes till F arrives at D, at which point the cosine becomes equal to zero. The cosine in the sec- ond quadrant increases, and becomes equal to radius at B ; in the third quadrant it decreases, and becomes zero at E ; in the fourth quadrant it increases again, and becomes equal to ra- dius at A. (66.) The tangent begins with zero at A, increases with the arc, and at 45 becomes equal to radius. As the point F ap- proaches D, the tangent increases very rapidly ; and when the difference between the arc and 90 is less than any assignable quantity, the tangent is greater than any assignable quantity Hence the tangent of 90 is said to be infinite. In the secoud quadrant the tangent is at first infinitely great, and rapidly diminishes till at B it is reduced to zero. In the third quadrant it increases again, becomes infinite at E, and is reduced to zero at A. The cotangent is equal to zero at D and E, and is infinite at A and B. /f>7.) The secant begins with radius at A, increases through 50 T R I G N M E T R "i . the first quadrant, and becomes infinite at D : diminishes in the second quadrant, till at B it is equal to the radius ; increases again in the third quadrant, and becomes infinite at E ; decreases in the fourth quadrant, and becomes equal to the B radius at A. The cosecant is equal to radius at D and E, and is infinite at A and B. (68.) Let us now consider the al- gebraic signs by which these lines are to be distinguished. Jn the first and second quadrants, the sines fall above the diame- ter AB, while in the third and fourth quadrants they fall be- low. This opposition of directions ought to be distinguished by the algebraic signs ; and if one of these directions is re- garded as positive, the other ought to be considered as nega- tive. It is generally agreed to consider those sines which fall above the horizontal diameter as positive ; consequently, those which fall below must be regarded as negative. That is, the sines are positive in the first and second quadrants, and nega- tive in the third and fourth. In the first quadrant the cosine falls on the right of DE, but in the second quadrant it falls on the left. These two linos should obviously have opposite signs, and it is generally agreed to consider those which fall to the right of the vertical diam- eter as positive ; consequently, those which fall to the left must be considered negative. That is, the cosines are positive in the first and fourth quadrants, and negative in the second and third. (69.) The signs of the tangents are derived from those o f the sines and cosines. For tang. = ' (Art. 28). Henco when the sine and cosine have like algebraic signs, the tan- gent will be positive ; when unlike, negative. That is, the tan gent is positive in the first and third quadrants, and negative in the second and fourth. T>2 Also, cotangent =- (Art. 28) ; hence the tangent and cotangent have always the same sign ELANE TRIGONOMETRY. 51 T>2 We have seen that sec, = - ; hence the secant must hav cos. the same sign as the cosine. TJ8 A] so, cosec. = : hence the cosecant must have the same sin. ' sign as the sine. (70.) The preceding results are exhibited in the following Uoles, which should be made perfectly familiar : First quad. Second quad. Third quad. Fourth quad fiine and cosecant, -t- + Cosine and secant, -f- + Tangent and cotangent, -f -f Sine, Cosine, Tangent, Cotangent, Secant, Cosecant, (71.) In Astronomy we frequently have occasion to considei greater than 360. But if an entire circumference, or any number of circumferences, be added to any arc, it will termin- ate in the same point as before. Hence, if C represent an en- tiro circumference, or 360, and A any arc whatever, we shall have sin, A=sin. (C-fA)=sin. (2C+A)=sin. (3C+A) = , &c. The same is true of the cosine, tangent, &c. We generally consider those arcs as positive which are esti- mated from A in the direction ADBE. If, then, an arc were estimated in the direction AEBD, it should be considered as negative ; that is, if the arc AF be considered positive, AH must be considered negative. But the latter belongs to the fourth quadrant; hence its sine is negative. Therefore, sin ( A) = sin. A. The cosine CG i. the same for both the arcs AF and AH- Hence, cos. ( A)=cos. A AJso, tang. ( A) = tang. A. A nd cot, ( A) = cot. A 90 180 270 360- f'B -R -fR -R +R GO GO it, 00 00 00 +R GO -R GO +R 00 +R GO -R GO TRIGONOMETRY TRIGONOMETRICAL FORMULAE. (72.) Expressions for the sine and cosine of the sum ana difference of two arcs. Let AB and BD represent any two given arcs ; take BE equal to BD : it is required to find an D expression for the sine of AD, the sum, and of AE, the difference of these arcs. Put AB=a, and BD=6; then AD= a -f b, and AE = ab. Draw the chord DE, and the radius CB, which may be represented by R. Since DB is by construction equal to BE, DF is equal to FE, and therefore DE is perpendic- ular to CB. Let fall the perpendicular EG, BH, FI, and DK upon AC, and draw EL, FM parallel to AC. Because the triangles BCH, I 1 CI are dnular, we have CB : CF : BH : FI ; or R : cos. b : : sin. a : FI. Therefore, sin. a cos. b ~~~- Also, CB : CF : : CH : CI ; or R : cos. b : : cos. a : CI. Therefore, CI=- cos. a cos, b The triangles DFM, CBH, having their sides perpendicula! each to each, are similar, and give the proportions CB : DF : : CH : DM ; or R : sin. b : : cos. a : DM. Hence .____ cos. a sin. b DM= --- -- . Also, CB : DF : : BH : FM ; or R : sin. b : : sin. a : FM sin. a sin. b [fence But and Also, and Hence, FI+DM=DK =sin. (a+bj ; CI-FM=CK =cos. (a+b). FI-FL =Ea=sin. (a-b) ; CI+EL = Ca=cos. (a-b). T . sin. a cos. b+cos. a sin. b sm. (a+ b)= ^ cos. a cos b sin. a sin. b cos (a+b)= R - - ... (1) PLANE TRIGONOMETRY. 53 , . sin. a cos. b cos. a sin. b sm. (a-b} = - -g - (3) cos. a cos. 6+sin. a sin. & *.(*-*)= -^ - (4) (70.) Expressions for the sine and cosine of a double are. If, in the formulas of the preceding article, we make b=a fco first and second will become 2 sin. a cos. a sm. a= cos. 2a= =r it cos. # sn. Making radius equal to unity, and substituting the values oi sin. #, cos. a. &c., from Art. 28, we obtain 2 tans, a sn. a rt cos. 2^= 1 4- tang. V 1 tanff. 2 # ^ . 1+tang. a (74.) Expressions for the sine and cosine of half a give* arc. Tf we put \a for a in the preceding equations, we obtain 2 sin. \a cos. A# sm. a= g , cos. ^a~ ? : n. 2 Aa cos. a= - - ^ . K We may also find the sine and cosine of ^ in terms of a. Since the sum of the squares of the sine and cosine is equal ; the square of radius, we have cos. 2 ^#+sin. 2 J# = R 2 . And, from the preceding equation, cos. *i sin. S ^#=R cos. a. If we subtract one of these from the other, we have 2 sin. 9 J#=R 2 R cos. &. And, adding the same equations, 2 cos. 2 ^ 5b TRIGONOMETRY. known to be 3.14159. This being divided successively by ISC and 60, gives .0002908882 for the arc of one minute, which may be regarded as the sine of one minute. The cosine of 1'= /I^sin?= 0.9999999577. The sines of very small angles are nearly proportional to the angles themselves. We might then obtain several other sines by direct proportion. This method will give the sines correct to five decimal places, as far as two degrees. By the follow- ing method they may be obtained with greater accuracy for the entire quadrant. By Art. 75, we have, by transposition, sin. (a+b)=2 sin. a cos. 6 sin. (& 6), cos. (a+b)=2 cos. a cos. b cos. (a b). If we make a=b, 26, 36, &c., successively, we shall hav? sin. 26=2 sin. b cos. b ; sin. 36=2 sin. 2b cos. b sin. b sin. 46=2 sin. 36 cos. b sin. 26, &c., &c. cos. 26=2 cos. 6 cos. 61 ; cos. 36=2 cos. 26 cos. 6 cos. 6 : cos. 46=2 cos. 36 cos. 6 cos. 26, &c., &c, Wl jnce, making 6=1', we have sin. 2' =2 sin. V cos. 1' =.000582 sin. 3'=2 sin. 2' cos. I'-sin. 1'=.000873; sin. 4'=2 sin. 3' cos. I'-sin. 2'=.001164, &c., &c. cos. 2'=2 cos. V cos. 1'- 1 =0.999999 ; cos. 3' = 2 cos. 2' cos. I'-cos. l'=0.999999 ; cos. 4' =2 cos. 3' cos. I'-cos. 2' =0.999999, &c., &c. The tangents, cotangents, secants, and cosecants aio ca^i!]? iari"wJ from the sines and Dosines. Thus, sin. V cos. V tang. r= T- ; cot. l'=- r ; cos. 1' sin. 1 ' 1 1 sec. 1' -r- j cosec. 1 =- ~ ; cos. 1 ' am 1' ' )U OK III. MENSURATION OF SURFACES. (80.) THE arsa of a figure is the space contained within thf lino or lines by which it is bounded. This area is determined by finding how many times the figure contains some other sur- face, which is assumed as the unit of measure. This unit is commonly a square ; such as a square inch, a square foot, a square rod, &c. The superficial unit has generally the same name as the linear unit, which forms the side of the square. Thus, the side of a square inch is a linear inch ; " " of a square foot is a linear foot ; u " of a square yard is a linear yard, &o. There are some superficial units which have no correspond- ing linear units of the same name, as, for example, an acre. The following table contains the square measures in com- mon use : Table of Square Measures. Sq. Inches. Sq. Feel. 144 = 1 5^. yards 1296= 9 - 1 Sq . Rods . 39204- 272|= 30= 1 5 . c; ,, 627264= 4356 = 484 = 16= 1 AM 6272640= 43560 = 4840 = 160= 10= 1 M 4014489600=27878400 =3097600 =102400=6400=640=1 PROBLEM I. (81.) To find the area of a parallelogram. RULE I. Multiply the base by the altitude. For Ihe demonstration of this rule, see Geometry, Prop 5 B. IV 60 TRIGONOMETRY. Ex. 1. What is the area of a parallelogram whoso base ia 17.5 rods, and the altitude 13 rods ? Ans., 227.5 square rods. Ex. 2, What is the area of a square whose side is 315 7 foot ? Ans., 99666.49 square feet. Ex. 3. What is the area of a rectangular board whose length is 15,25 feet, and breadth 15 inches ? Ans., 19.0625 square feet. Ex. 4. How many square yards are there in the four sides of a room which is 18 feet long, 15 feet broad, and 9 feet high 1 Am., 66 square yards. (82.) If the sides and angles of a parallelogram are given, the perpendicular height may be found by D c Trigonometry. For DE is one side of a right-angled triangle, of which AD is the hypothenuse. Hence, R : AD : : sin. A : DE ; AD X sin. A from which Therefore, the R ABxADXsin. A R Hence we derive RULE II. Multiply together two adjacent sides, and the sine of tht included angle. Ex. 1. What is the area of a parallelogram having an angl of 58, and the including sides 36 and 25.5 feet ? Ans. The area = 36 X 25.5 X. 84805 (natural sine of 58)=^ 778.508 square feet. The computation will generally be most conveniently per- formed by logarithms. Ex. 2. What is the area of a rhombus, each of whose sides is 21 feet 3 inches, and each of the acute angles 53 20' ? Ans., 362.209 feet. Ex. 3. How many acres are contained in a parallelogram one of whose angles is 30, and the including sides are 25.3*1 and 10.4 chains? A.ns., 13 acres, 29.12 rods MENSURATION or SURFACES. 61 PROBLEM II. (83.) To find the area of a triangle. RULE I. Multiply the base by half the altitude. For demonstration, see Geometry, Prop. 6, B. IV Ex. 1. How many square yards are contained in a iriaLgle whose base is 49 feet, and altitude 25 feet ? Ans., 68.736. Ex. 2. "What is the area of a triangle whose base is 45 feet, and altitude 27.5 feet ? Ans., 618.75 square feet. (84.) When two sides and the included angle are given, \ro may use RULE II. Multiply half the product of two sides by the, sine of tfis, included angle. The reason of this rule is obvious, from Art. 82, since a tri- angle is half of a parallelogram, having the same base and at- titude. Ex. 1. What is the area of a triangle of which two sides are 45 and 32 feet, and the included angle 46 30' ? Ans. The area=45x 16 X. 725374 (natural sine of 46 30')=- 522.269 feet. Ex. 2. What is the area of a triangle of which two sides are 127 and 96 feet, and the included angle 67 15' ? Ans. (85.) When the three sides are known, we may use RULE III. From half the sum of the three sides subtract each side sev- erally ; multiply together the half sum and the three remain* ders, and extract the square root of the product. Demonstration. Let , &, c denote the sides of the tri- angle ABC ; then, by Geometry, Prop. 12, B, IV., we have BC 2 =AB 2 +AC 2 -2ABx U), or a'--= 2 -j-r; 2 -2f:X AD ; whence, TRIGONOMETRY But CD 2 - AC 2 -AD 2 ; honcc ( ^ =b ,_( CD =- But /At area= The quantity under the radical sign being the difference ol two squares, may be resolved into the factors 2&c+(6 3 -fc a a*) and 2bc (tf+c n ~ a?) ; and these, in the same manner, may be resolved into (b+c+a)x(b+c #), and (a+b c)x(a b+c). Hence, if we put S equal to - , we shall have & the arf>a= \/S(S-a) (~S-&) (S-c). Ex. 1. "What is lite area of a triangle whose sides are 125, 173, and 216 feet ? Here S-257, S-6=84, S a=132, S-c=:4l. Hence the area= V257 X 132 X 84 X 41 = 10809 square feet. Ex. 2. How many acres arc contained in a triangle whose sides are 49, 50.25, and 25.69 chains? Ans.j 61 acres, 1 rood, 39.68 perches. Ex. 3. "What is the area of a triangle whose sides are 234, 289, and 345 feet? Ans. (86.) In an equilateral triangle, one of whose sides is a, tho expression for the area becomes _oV3 4 ; that is, the area of an equilateral triangle is equal to on .ourth the square of one of its sides multiplied by the square root of 3. Ex. What is the area of a triangle whose sides are each 37 feet? Ans., 592.79 feet MENSURATION OF SURFACES. 6S PROBLEM III. (87.) To find the area of a trapezoid. RULE. Multiply half the sum of the parallel sides into their per ptndicular distance. For demonstration, see Geometry, Prop. 7, B. IV. Ex. 1. "What is the area of a trapezoid whose parallel sidea are 156 and 124, and the perpendicular distance between them 57 feet? Ans., 7980 feet. Ex. 2. How many square yards in a trapezoid whose par- allel sides are 678 and 987 feet, and altitude 524 feet ? Ans. PROBLEM IV. (88.) To find the area of an irregular polygon. RULE. Draw diagonals dividing the polygon into triangles, and find the sum of the areas of these triangles. Ex. 1. What is the area of a quadrilateral, one of whoso diagonals is 126 feet, and the two perpendiculars let fall upo it from the opposite angles are 74 and 28 feet ? Ans., 6426 feet. Ex. 2. In the polygon ABODE, there are given EC=205, EB=242, AF=65, DGr = 114, and DH=110, to find the area. Ans. (89.) If the diagonals of a quadrilateral are given, the area may be found by the following RULE. Multiply half the product of the diagonals by the sine of the angle at their intersection. Demonstration. The sines of the four angles at E are all equal tc each other TRIGONOMETRY. since fat adjacent angles AED, DEC are the supplements 01 each other (Art. 27). But, according to Ihe Rule, Art. 84, the area of the Mangle ABE=iAExBEXsine E ; " " AED=|-AExDEXsine E; " " BEC=iBExECXsine E; " " DEC=i-DExECXsineE. Therefore, the area of ABCD=J(AE-fEC)X(BE + ED)Xsine E =jACxBDXsine E. Ex. 1. If the diagonals of a quadrilateral are 34 and 56 rods, and if they intersect at an angle of 67, what is the area? Ans., 876.32. Ex. 2. If the diagonals of a quadrilateral are 75 and 49, what is the area ? Ans. and the angle of intersection is 42, PROBLEM Y. (90.) To find the area of a regular polygon. RULE I. Multiply half the perimeter by the perpendicular let fall from the center on one of the sides. For demonstration, see Geometry, Prop. 7, B. VI. Ex. 1. "What is the area of a regular pentagon whose side is 25, and the perpendicular from the center 17.205 feet ? Ans., 1075.31 feet. Ex. 2. What is the area of a regular octagon whose side is 53, and the perpendicular 63.977 ? Ans. (01.) "When the perpendicular is not given, it may "be con. puted from the perimeter and number of sides. If we divide 360 degrees by the number of sides of the polygon, the quotient will be the angle ACB at the center, subtended by one of the sides. / c The perpendicular CD bisects the side AB, and the angle ACB. Then, in the triangle ACD, we have (Art. 42), R : AD : cot. ACD : CD ; that is, A HE.\S A ATI ON OF SURFACES. 66 Radius is to half of one of the s\des of the polygon, as the cotangent of the opposite angle is to *he perpendicular from t\s center. Ex. 3. Find the area of a regular hexagon whose side is 32 inches. The angle ACD is T V of 360=30. Then R : 16 : : cot. 30 : 27.7128= CD, the perpendicular ; and the area=27.7128x!6x6=2660.42S8. Ex. 4. Find the area of a regular decagon whose side is 4b feet. Ans., 16280.946. (92.) In this manner was computed the following tahle of tha areas of regular polygons, in which the side of each poly- g<-Ti is supposed to be a unit. TABLE OF REGULAR POLYGONS. Names. Sides. Areas. Triangle, 3 0.4330127. Square, 4 1.0000000. Pentagon, 5 1.7204774. Hexagon, 6 2.5980762. Heptagon, 7 3.6339124 Octagon, 8 4.8284271 Nonagon, 9 6.1818242. Decagon, 10 7.6942088. Undecagon, 11 9.3656399. Dodecagon, . 12 11.1961524. By the aid of this table may be computed the area of any other regular polygon having not more than twelve sides. For, since the areas of similar polygons are as the squares of their homologous sides, we derive RULE II. Multiply the square of one of the sides of the polygon by the area of a similar polygon whose side is unity. Ex. 5. What is the area of a regular nonagon whose side is 63 ? Ans., 24535.66. Ex. 6. What is the area of a regular dodecagon whose side is 54 feet? Ans., 32647.98 feet. E 6tt TRIGONOMETRY. PROBLEM VI (93.) To find the circumference of a circle from its diamete* RULE Multiply the diameter by 3.14159. "For the demonstration of this rule, see Geometry, Prop. 13, Cor. 2, B.VI. When the diameter of the circle is small, and no great ac- curacy is required, it may be sufficient to employ the value of TT to only 4 or 5 decimal places. But if the diameter is large, and accuracy is required, it will be necessary to employ a corresponding number of decimal places of TT. The value of TT to ten decimal places is 3.14159,26536, and its logarithm is 0.497150. Ex. 1. "What is the circumference of a circle whose diame- ter is 125 feet ? Ans., 392.7 feet. Ex. 2. If the diameter of the earth is 7912 miles, what in its circumference ? Ans., 24856.28 miles. Ex. 3. If the diameter of the earth's orbit is 189,761,000 miles, what is its circumference ? Ans., 596,151,764 miles. To obtain this answer, the value of TT must be taken to at least eight decimal places. PROBLEM VII. (94.) To find the diameter of a circle from its circum- ference. RULE I. Divide the circumference by 3.14159. This rule is an obvious consequence from the preceding To divide by a number is the same as to multiply by its re- ciprocal ; and, since multiplication is more easily performed than division, it is generally most convenient to multiply by the reciprocal of TT, which is 0.3183099. Hence we have RULE II. Multiply the circumference by 0.31831. MENSURATION OF SURFACES. 67 Ex. 1. What is the diameter of a circle whose circumference b 875 foot ? Ans., 278.52 feet Ex. 2. If tho circumference of the moon is 6786 miles, what is its diameter ? Ans., 2160 miles. Ex. 3. If the circumference of the moon's orbit is 1,492,987 miles, what is its diameter ? Ans., 475,233 miles PROBLEM VIII. (95.) To find the length of an arc of a circle. RULE I. As 360 is to the number of degrees in the arc, so is the cu- cumference of the circle to the length of the arc. This rule follows from Prop. 14, B. III., in Geometry, where it is proved that angles at the center of a circle have the same ratio with the intercepted arcs. Ex. 1. "What is the length of an arc of 22, in a circle whose diameter is 125 feet? The circumference of the circle is found to be 392.7 feet. Then 360 : 22 : : 392.7 : 23.998 feet. Ex. 2. If the circumference of the earth is 24,856.28 miles what is the length of one degree ? Ans., 69.045 miles. RULE II. (96.) Multiply the diameter of the circle by the number of degrees in the arc, and this product by 0.0087266. Since the circumference of a circle whose diameter is unity is 3.14159, if we divide this number by 360, we shall obtain the length of an arc of one degree, viz., 0.0087266. If we multiply this decimal by the number of degrees in any arc, we shall obtain the length of that arc in a circle whose diameter is unity ; and this product, multiplied by the diameter of any other circle, will give the length of an arc of the given num- ber of degrees in that circle. tiS TRIGONOMETRY. Ex. 3. What is the length of an arc of 25, in a circle whose radius is 44 rods ? Ans., 19.198 rods. Ex. 4. What is the length of an arc of 11 15', in a circle whose diameter is 1234 feet ? Ans., 121.147 feet. (97.) If the number of degrees in an arc is not given, it m? ? be computed from the radius of the circle, and either the chord or height of the arc. Thus, let AB be the chord, and DE the height of the arc ADB, and C the center of the circle. Then, in the right-angled tri- angle ACE, AC R I AE : sin * ACE) : .(CE : cos. ACE, either of which proportions will give the number of degrees in half the arc. . If only the chord and height of the arc are given, the diam- eter of the circle may be found. For, by Geometry, Prop. 22, Cor., B. IV., DE : AE : : AE : EF. Ex. 5. What is the length of an arc whose chord is 6 feet, in a circle whoso radius is 9 feet ? Ans., 6.117 feet. PROBLEM IX. (98.) To find the area of a circle. RULE I. Multiply the circumference by half the radius. For demonstration, see Geometry, Prop. 12, B. VI. RULE II. Multiply t/ie square of the radius by 3.14159, See G-eometry, Prop 13, Cor. 3, B. VI. Ex, 1. What is the area of a circle whose diameter is iH feet? Ans., 254.469 feet. MENSURATION OF SURFACES fi8 Ex 2. What is the area of a circle whose circumference is 74 feet? Ans., 435.766 feet. Ex. 3. What is the area of a circle whose radius is 125 yards ? Ans., 49087.38 yards PROBLEM X. (99.) To find the area of a sector of a circle. RULE I Multiply the arc of the sector by half its radius. See Geometry, Prop. 12, Cor., B. VI. RULE II. As 360 is to the number of degrees in the arc, so is the area nf the circle to the area of the sector. This follows from Geometry, Prop. 14, Cor. 2, B. III. Ex. 1. What is the area of a sector whose arc is 22, in a circle whose diameter is 125 feet ? The length of the arc is found to be 23.998. Hence the area of the sector is 749.937. Ex. 2. What is the area of a sector whose arc is 25, in a circle whose radius is 44 rods ? Ans., 422.367 rods. Ex. 3. What is the area of a sector less than a semicircle, whose chord is 6 feet, in a circle whose radius is 9 feet ? Ans., 27.522 feet PROBLEM XI. (100.) To find the area of a segment of a circle. RULE. Find the area of the sector which has the same arc, and also the area of the triangle formed by Ike chord of the seg- ment and the radii of the sector. Then take the sum of these areas if the segment is greater than a semicircle, but take their difference if it is less fO TRIGONOMETRY It is obvious that the segment AEB is equal to the sum of the sector ACBE and the triangle ACB, and that the segment ADB is equal to the difference "between the sector ACBD and the triangle ACB. Ex. 1. What is the area of a segment whose arc contains 280, in a circle whoso diameter is 50 ? The whole circle = 1963.495 The sector = 1527.163 The triangle 307.752 The segment = 1834.915, Ans. Ex. 2. What is the area of a segment whose chord is 20 A eet, and height 2 feet ? Ans., 26.8788 feet. Ex. 3. What is the area of a segment whose arc is 25, in a circle whose radius is 44 rods ? Ans. (101.) The area of the zone ABHGr, included between two parallel chords, is equal to the difference between the segments GDI! and ADB. Ex. 4. What is the area of a zone, one side of which is 96, and the other side 60, and the distance between them 26 ? Ans., 2136.7527. The radius of the circle in this example will be found to be 50. PROBLEM XII. (102.) To find the area of a ring- included between the cir eumferences of two concentric circles. RULE. Take the difference betiveen the areas of the two circles; or, Subtract the square of the less radius from the square of the greater, and multiply their difference by 3.14159. For, according to Geometry, Prop. 13, Cor. 3, B. VI., the area of the greater circle is equal to TT R a , and the area of the smaller, TT r*. Their difference, or the area of the ring, b IT (R 3 y 2 ). MENSURATION OF SOLUS. 7. Ex. 1. The diameters of two concentric circles are 60 and 50. What is the area of the ring included "between their cir- cumferences ? Ans., 863.938. Ex. 2. The diameters of two concentric circles are 320 and 280 What is the area of the ring included between their cir- cumferences ? Ans., 18849.55 PROBLEM XIII. (103.) To find the area of an ellipse. RULE. Multiply the product of the semi-axes by 3.14159. For demonstration, see Greometry, Ellipse, Prop. 21. Ex. 1. What is the area of an ellipse whose major axis i* 70 feet, and minor axis 60 feet ? Ans., 3298.67 feet. Ex. 2. What is the area of an ellipse whose axes are 340 and 310? Ans., 82780.896 PROBLEM XIY. (104.) To find the area of a parabola. RULE. Multiply the base by two thirds of the height. For demonstration, see Greometry, Parabola, Prop. 12. Ex. 1. What is the area of a parabola whose base is 18 feet, and height 5 feet ? Ans., 60 feet. Ex. 2. What is the area of a parabola whose base is 525 feet, and height 350 feet ? Ans., 122500 feet MENSURATION UF SOLIDS. (105.) The common measuring unit of solids is a cubc t vhose faces are squares of the same name ; as, a cubic inch a cubic foot, &c. This measuring unit is not, however, of 72 TRIGONOMETR* necessity a cube whose faces are squares of the same name*. Thus a bushel may have the form of a cube, but its faces can only be expressed by means of some unit of a different denom- ination. The following is The Table of Solid Measure. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard. 4492J cubic feet = 1 cubic rod. 231 cubic inches = 1 gallon (liquid measure). 268.8 cubic inches = 1 gallon (dry measure). 2150.4 cubic inches = 1 bushel. PROBLEM I. (106.) To find the surface of a right prism. . RULE. Multiply the perimeter of the base by the altitude for the convex surface. To this add the areas of the two ends when the entire surface is required. See G-eometry, Prop. 1, B. VIII. Ex. 1. What is the entire surface of a parallelepiped whose altitude is 20 feet, breadth 4 feet, and depth 2 feet ? Ans., 256 square feet. Ex. 2. What is the entire surface of a pentagonal prism whose altitude is 25 feet 6 inches, and each side of its base 3 feet 9 inches ? Ans.j 526.513 square feet. Ex. 3. What is the entire surface of an octagonal prism whose altitude is 12 feet 9 inches, and each side of its base 1? feet 5 inches ? Ans., 302.898 square f jot. PROBLEM II. (107.) To find the solidity of a pnsm. iiULE. Multiply the area of the base by the altitude. See Geometry, Prop. 11, B. VIII. MENSURATION OF SOLIDS 7C Ex. 1. What is the solidity of a parallelepiped whose alii, tude is 30 feet, breadth 6 feet, and depth 4 feet ? Am., 720 cubic feet. Ex. 2.. What is the solidity of a square prism whose altitude is 8 feet 10 inches, and each side of its base 2 feet 3 inches ? Am., 44 1 f cubic feet. Ex. 3 What is the solidity of a pentagonal prism whoso a. titude is 20 feet 6 inches, and its side 2 feet 7 inches ? Ans.) 235.376 cubic feet. PROBLEM III. (108.) To find the surface of a regular pyramid. RULE. Multiply the perimeter of the base by half the slant height for the convex surface. To this add the area of the bast when the entire surface is required. See Geometry, Prop. 14, B. VIII. Ex. 1. What is the entire surface of a triangular pyramid whose slant height is 25 feet, and each side of its base 5 feet? Ans.^ 198.325 square feet. Ex. 2. What is the entire surface of a square pyramid whose slant height is 30 feet, and each side of the base 4 foot'.' Ans.) 256 square fe'jt. Ex. 3. What is the entire surface of a pentagonal pyra- mid \vhose slant height is 20 feet, and each side of the base 8 feet? Ans.) 165.484 square feet. PROBLEM IY. (109.) To find the solidity of a pyramid. . RULE. Multiply the area of the base by one third of the altitude. See Geometry, Prop. 17, B. VIII. Ex. 1. What is the solidity of a triangular pyramid whu* altitude is 25 f eet, and each sid 3 of its base 6 feet ? Ans., 129,904 cubic feet 7 4 TRIGONOMETRY. Ex. 2. What is the solidity of a square pyramid whose slant height is 22 feet, and each side of its "base 10 feet ? Ans., 714.143 cubic feet. Ex. 3. What is the solidity of a pentagonal pyramid whosw altitude is 20 feet, and each side of its base 3 feet ? Ans., 103.228 cubic feet. PROBLEM V. (110.) To find the surface of a frustum of a regular pyr- amid. RULE. Multiply half the slant height by the sum of the perime- ters of the two bases for the convex surface. To this add the areas of the two bases when the entire surface is re- quired. See Geometry, Prop. 14, Cor. 1, B. VIII. Ex. 1. What is the entire surface of a frustum of a square pyramid whose slant height is 15 feet, each side of the greater" base being 4 feet 6 inches, and each side of the less base 2 feet 10 inches ? Ans. 9 248.278 square feet. Ex. 2. What is the entire surface of a frustum of an oc- tagonal pyramid whose slant height is 14 feet, and the sides of the ends 3 feet 9 inches, and 2 feet 3 inches ? Ans., 428.344 square feet. PROBLEM VI. (111.) To find the solidity of a frustum of a pyramid. RULE. Add together the areas of the two bases , and a mean pro- portional between them, and multiply the sum by one thira of the altitude. See Geometry, Prop. 18, B. VIII. When the pyramid is regular, it is generally most conven- ient to find the area of its base by Rule II., Art. 92. If we put a to represent one side of the lower base, and b one side of the upper base, and *.he tabular number from Art. 92 b> MENSURATION c F SOLIDS. 72 T, the area of the lower base will be a 2 T ; that of the upper base will be & 2 T ; and the mean proportional will bo a&T. Hence, if we represent the height of the frustum -by /*, its so- lidity will be Ex. 1. "What is the solidity of a frustum of an hexagonal pyramid whose altitude is 15 feet, each side of the greater end being 3 feet, and that of the less end 2 feet ? Ans., 246.817 cubic feet. Ex. 2. What is the solidity of a frustum of an octagonal pyramid whose altitude is 9 feet, each, side of the greater end being 30 inches, and that of the less end 20 inches ? Ans., 191.125 cubic feet Definition. (112.) A wedge is a solid bounded by five planes, viz., a rec- tangular base, ABCD, two trape- zoids, ABFE, DCFE, meeting in an edge, and two triangular ends, ADE, BCF. The altitude of the wedge is the perpendicular drawn from any point in the edge to the plane of the base, as EH. PROBLEM VII. (113.) To find the solidity of a wedge. RULE. Add the length of the edge to twice the length of the base, and multiply the sum by one sixth of the product of the height of the ivedge and the breadth of the base. Demonstration. Put L=AB, the length of the baso ; " /=EF, the length of the edge ; " =BC, the breadth of the base ; " /i=EH, the altitude of the wedge. Now, if the length of the base is equal to that of the e f(5 TRIGONOMETRY. it is evident that the wedge is half of a pri&m of the same base and height. If the length of the base is greater than that of the edge, let a plane, EGrI, be drawn parallel to BCF. The wedge will le divided into two parts, viz., the pyramid E- AIGrD, and the triangular prism BCF Or. The solidity of the former is equal to M(L /), and that of the latter is \blil. Their sum is If the length of the base is less than that of the edge, the wedge will be equal to the difference between the prism pyramid, and we shall have which is equal to the same result as before. Ex. 1. What is the solidity of a wedge whose base is 3C inches long and 5 inches broad, its altitude 12 inches, and the length of the edge 2 feet ? Ans.y 840 cubic inches. Ex. 2. What is the solidity of a wedge whose base is 40 inches long and 7 inches broad, its altitude 18 inches, and ths length of the edge 30 inches ? Ans., 2310 cubic inches. Definition. (114.) A rectangular prismoid is a solid bounded by six planes, of which the two bases are rectangles having their cor- responding sides parallel, and the four upright sides of the sol- id are trapezoids. PROBLEM VIII. To find the solidity of a rectangular prismoid. RULE. Add together the areas of the two bases, and four times tnt area of a parallel section equally distant from the bases, and multiply the sum by one sixth of the altitude. Demonstration. Put L and B= length and breadth of one baso ; MENSURATION OF SOLIDS. 77 Put / and b = length and breadth of the other base; M " m= length and breadth of middle sec.; ^ - ~~ " h =the altitude of the prismoid. -- ' > It is evident that if a plane be made to pass through the opposite edges of the upper and lower bases, the prismoid will be divided into two wedges, whose bases are the bases of the prismoid, and whose edges are L and L The solidity of these wedges, and, consequently, that of the prismoid, is But, since M is equally distant from L and /, we have 2M=L+Z, and 2m=~B+b ; hence 4Mw=(L+/) (B+6)=BL+B/+6L+W. Substituting 4Mm for its value in the preceding expression, we obtain for the solidity of the prismoid Ex. 1. "What are the contents of a log of wood, in the form of a rectangular prismoid, the length and breadth of one end oeing 16 inches and 12 inches, and of the other 7 inches and 4 inches, the length of the log being 24 feet ? Ans., 16 J- cubic feet. Ex. 2. What is the solidity of a log of hewn timber, whoso nids are 18 inches by 15, and 14 inches by 11 J, its length be- ing 18 feet? Ans., 26fi cubic feet. PROBLEM IX. To compute the excavation or embankment for a rail-way (115.) By the preceding rule may be computed the amount if excavation or embankment required in constructing a rail- oad or canal. If we divide the line of the road into portions jo small that each may be regarded as a straight line,' and suppose an equal number of transverse sections to be made, the excavation or embankment between two sections may be regarded as a prismoid, and its contents found by the pre ceding rule. Let ABCD represent the lower surface of the supposed ex- oavation, which we will assume to be parallel to the horizon ; tind let EFG-H represent the upper surface of the excavation TRIGONOMETRY. projected on a horizontal plane. Also, let E'A'B'F', G'C'I) H represent the vertical sections at the extremities. If we suppose ver- tical planes to pass through the lines AC, BD, the middle part of the ex- cavation, or that contained between these vertical planes, will be a rect- angular prismoid, of which A'B'KI will be one base, and C'D'ML the other base. Its solidity will there- fore be given by Art. 114. The parts upon each side of the middle prismoid are also halves of rectangular prismoids ; or, if the two parts are equal, they may be regarded as constituting a second prismoid, one of whose bases is the sum of the triangles A'E'I, B'F'K ; and the other base is the sum of the triangles C'Gr'L, D'H'M. There- fore the volume of the entire solid is equal to the product of one sixth of its length, by the sum of the areas of the sections at the two extremities, and four times the area of a parallel and equidistant section. Ex. 1. Let ABCDEFGr represent the profile of a tiact of land selected for the line of a rail- way ; and suppose it is re- quired, by cutting and embankment, to reduce it from its pres- ent hilly surface to one uniform slope from the point A to the point GK The distance AH is 561 feet ; the distance DK is 820 feet; " " HI is 858 feet; " " KL is 825 feet; " " ID is 825 feet ; " " LG- is 330 feet. The perpendicular BH is 18 feet ; the perpendicular KE is 19 feel " CI is20feet; " " LF in 8 feet The annexed figure repre- sents a cross section, showing the form of the excavation. The base of the cutting is to MENSURATION OF Sc LIDS. 7$ be 50 feet wide, the slope 1J horizontal to 1 perpendicular; that is, where the depth ad is 10 feet, the width of the slope c.d at the surface will be 15 feet. Calculation of the portion ABH. Since BH is 18 feet, the length of cd in the cross section will be 27 feet, and c/, the breadth at the top of the section, will be 104 feet. We accordingly find, by Art. 87, the area of the trapezoid forming the cross section at BH equal to 1^X18=1386 fat For the middle section, the height is 3 feet, cd is 13.5 feet, and cf is 77 feet. The area of the cross section is therefore equal to The solid ABH will therefore be equal to fr*-\ (1386+4x571.5) -^-=343332 cubic feet, or 12716 cubic yards. Calculation of the portion BCIH. Since CI is 20 feet, the length of cd is 30 feet, and cf is 1 (0 feet. The area of the section at CI is therefore equal to For the middle section, the height is 19 feet, cd is 28.5 foet, and cf is 107 feet. The area of the cross section is therefore equal to 1^X19= 1491.5. The solid BCIH will therefore be equal to oro (1386+1600+4x1491.5) -g-= 1280136 cubic feet, or 47412.4 cubic yards. Calculation of the portion CID. The height of the middle section is 10 feet ; therefore cf is 80 feet, and the area, of the cross section is TRIGONOMETRY. The solid C1D will therefore be equal to DQ K (1600+4X650) -^-=577500 cubic fcetj 02 21388.9 cubic yards. The entire amount of excavation therefore is, ABH= 12716.0 cubic yards. BCIH=47412.4 " CDI=21388.9 " Total excavation, 81517.3 " The following is a cross section, showing the ftrm of the era bankment. The top of the embankment ^ j is to be 50 feet wide, the slope 2 to 1 ; that is, where the height ad is 10 feet, the base cd is to be 20 feet. Calculation of the portion DKE. Since EK is 19 feet, the length of cd is 38 feet, and cf is 126 feet. The area of the cross section at EK is therefore equal to 1^X19=1672. # For the middle section, the height is 9.5 feet ; cd is therefore 19 feet, and cf is 88 feet. The area of the cross section is therefore 88+50 The solid DKE is therefore equal to 820 (1672+4 X 655.5) -g-= 586846.7 cubic feet, or 21735.1 cubic yards! Calculation of the portion KEFL. Since LF is 8 feet, cd is 16 feet, and cf is 82 feet, area of the section at LF is therefore equal to MENSURATION V.F SOLIDS. 81 The height of the middle section is 13.5 feei; therefore cd b 27 feet, and cf is 104 feet. The area of the cross: section is therefore equal to 1^x13.5=1039,5. The solid KEFL will therefore be equal to (1672+528+4xl039.5)-g-= 874225 cubic feet, or 32378.7 cubic yards. Calculation of tke portion LFGr. The height of the middle section is 4 feet ; therefore cf is 66 *eet, and the area of the cross section is equal to 66+50 The solid LFGr will therefore be equal to OOQ (528 +4 X 232) -g-= 80080 cubic feet, or 2965.9 cubic yards. The entire amount of embankment therefore is DKE =21735.1 cubic yards. KEFL=32378.7 " LFG-= 2965.9 Total embankment, 57079.7 " Ex. 2. Compute the amount of excavation of the hill ABCD from, the following data : The distance AH is 325 feet ; the perpendicular BH is 12 feet ; " " HI is 672 feet; " " CI is 13 feet, " " ID is 534 feet. The base of the cutting to be 50 feet wide, and the slope 1^ horizontal to 1 perpendicular. Am., 33969 cubic yards PROBLEM X. (116.) To find the surface of a regular polyedron. RULE. Multiply the area of one of the faces by the number oj V S2j TRIGONOMETRY. faces; or, Multiply the square of one of the edges by tht surface of a similar solid whose edge is unity. Since all the faces of a regular polyedron are equal, it is evident that the area of one of them, multiplied by their num- ber, will give the entire surface. Also, regular solids of the same name are similar, and similar polygons are as the squares of their homologous sides (Geom., Prop. 26, B. IV.). The fol- lowing table shows the surface and solidity of regular poly- edrons whose edge is unity. The surface is obtained by mul- tiplying the area of one of the faces, as given in Art. 92, by the number of faces. Thus the area of an equilateral trian- gle, whose side is 1, is 0.4330127. Hence the surface of a regular tetraedron = .4330127 X 4= 1.7320508, and so on for the other solids. A Table of the regular Polyedrons whose Edges are unity Names. No. of Faces. Surface. Solidity. Tetraedron, 4 1.7320508 0.1178513. Hexaedrpn, 6 6.0000000 1.0000000. Octaedron, 8 3.4641016 0.4714045 Dodecaedron, 12 20.6457288 7.6631189. Icosaedron, 20 8.6602540 2.1816950. Ex. 1. What is the surface of a regular octaedron whose rilges are each 8 feet? Ans., 221.7025 feet. Ex. 2. What is the surface of a regular dodecaedron whose se is 12 feet ? Ans., 2972.985 feet PROBLEM XI. (117.) To find the solidity of a regular polyedron. RULE. 3Iulliply the surface by one third of the perpendicular let from the center on one of the faces: or, Multiply the i.ube of one of the edges by the solidity of a similar polyedron, whose edgt is unity. Since the * ices of a r.xnilar polyedron are similar and equal. MENSURATION OF SOLID s. 83 and the solid angles are al] equal to each other, it is evident that the faces are all equally distant from a point in the solid called the center. If planes be made to pass through the cen- cer and the several edges of the solid, they will divide it into as many equal pyramids as it has faces. The base of each pyramid will be one of the faces of the polyedron ; and since their altitude is the perpendicular from the center upon one of the faces, the solidity of the polyedron must be equal to the areas of all the faces, multiplied by one third of this perpen- dicular. Also, similar pyramids are to each other as the cubes ol their homologous edges (Geom., Prop. 17, Cor. 3, B. VIIL). And since two regular polyedrons of the same name may be divided into the same number of similar pyramids, they must be to each other as the cubes of their edges, (118.) The solidity of a tetraedron whose edge is unity, may ne computed in the following manner : Let C ABD be a tetraedron. From one angle, C, let fal 1 . a perpendicular, CE, on the opposite face; c draw EF perpendicular to AD; and join CF, AE. Then AEF is a right-angled tri- angle, in which EF, being the sine of 30, is one half of AE or BE ; and therefore FE is one third of BF or CF. Hence the cosine of the angle CFE is equal to J ; that A. is, the angle of inclination of the faces of the polyedron is 70 31' 44". Also, in the triangle CAF, CF is the" sine of 60, which is 0.866025. Hence, in the right-angled triangle CEF, knowing one side and the angles, we can compute CE, which is found to be 0.8164966. Whence, knowing the base ABD (Art. 92), we obtain the solidity of the tetraedron 0.1178513. In a somewhat similar manner may the solidities of the other regular polyedrons, given in Art. 116, be obtained, Ex. 1. What is the solidity of a regular tetraedron whoso edges are each 24 inches ? Ans., 0.9428 feet. Ex. 2. "What is the solidity of a regular icosaedron whoso ylges are each 20 feet ? Ans. 17453.56 fret #4 TRIGONOMETRY. THE THREE ROUND BODIES. PROBLEM I. (119.) To find the surface of a cylinder. RULE. Multiply the circumference of the base by the altitude fm the convex surface. To this add the areas of the two endi when the entire surface is required. See G-eometry, Prop. 1, B. X. Ex. 1. What is the convex surface of a cylinder whose alti- tude is 23 feet, and the diameter of its base 3 feet? Ans., 216.77 square feet. Ex. 2. What is the entire surface of a cylinder whoso alti tude is 18 feet, and the diameter of its base 5 feet ? Ans. PROBLEM II. (120.) To find the solidity of a cylinder. RULE. Multiply the area of the base by the altitude. See Geometry, Prop. 2, B. X. Ex. 1. What is the solidity of a cylinder whose altitude i? IS feet 4 inches, and the diameter of its base 2 feet 10 inches? Ans., 115.5917 cubic feet. Ex. 2. What is the solidity of a cylinder whose altitude is 12 feet 11 inches, and the circumference of its bass 5 feet 3 inches ? Ans., 28.3308 cubic feet. PROBLEM III. (121.) To find the surface of a cone. RULE. Multiply the circumference of the base by half the side for the convex surface ; to which add the area of the base ivhen (he entire surface is required See Geometry. Prop. 3, B. X. MENSURATION OF SOLIDS. 85 Ex. 1. What is the entire surface of a cone whose side is 10 feet and the diameter of its base 2 feet 3 inches ? Am., 39.319 sqiure feet. Ex. 2. What is the entire surface of a cone whose side ia 15 feet, and the circumference of its base 8 feet ? Ans., 65.093 square feet. PROBLEM IV. (122.) To find the solidity of a cone. RULE. Multiply the area of the base by one third of the altitude. See Geometry, Prop. 5, B. X. Ex. 1. What is the solidity of a cone whose altitude is 12 feet, and the diameter of its base 2 J feet ? Ans., 19.635 cubic feet. Ex. 2. What is the solidity of a cone whose altitude is 25 fcet, and the circumference of its base 6 feet 9 inches ? Ans. PROBLEM V. <123.) To find the surface of a frustum of a cont. RULE. Multiply half the side by the sum of the circumferences of the two bases for the convex surface ; to this add the areas of the two bases when the entire surface is required. See Geometry, Prop. 4, B. X. Ex. 1. What is the entire surface of a frustum of a cone, the diameters of whose basrs are 9 feet and 5 feet, and whose hide is 16 feet 9 inches ? Ans.) 451.6036 square feet. Ex. 2. What is the convex surface of a frustum of a cone whose side is 10 feet, and the circumferences of its bases 6 feet and 4 feet ? Ans.^ 50 square feet. PROBLEM VI. 1124.) To find the solidity of a frustum of a cone. #6 "TRIGONOMETRY. RULE Add together the areas of the two base*, and a mean prv- portional between them, and multiply the sum by one third of the altitude. See Geometry, Prop. 6, B. X. If we put R and r for the radii of the two bases, then TrR' will represent the area of one base, Trr* the area of the other, and TrRr the mean proportional between them. Hence,, if wo represent the height of the frustum by h, its solidity will be Ex. 1. What is the solidity of a frustum of a cons whose altitude is 20 feet, the diameter of the greater end 5 feet, and that of the less end 2 feet 6 inches ? Ans., 229.074 cubic feet Ex. 2. The length of a mast is 60 feet, its diameter at the greater end is 20 inches, and at the less end 12 inches : what is its solidity ? Ans., 85.521 cubic feet PROBLEM VII. (125.) To find the surface of a sphere. RULE. Multiply the diameter by the circumference of a great cir- cle ; or, Multiply the square of the diameter by 3.14159 See Geometry, Prop. 7, B. X. Ex. 1. Required the surface of the earth, its diameter be- ing 7912 miles. Ans., 196,662,896 square miles, Ex. 2. Required the surface of the moon, its circumference being 6786 miles. Ans. PROBLEM VIII. (126.) To find trie solidity of a sphere. RULE. Multiply the surface by one third of the radius ; or, Mul* tiply the cube of the diameter by ITT] that is, by 0.5238. See Geometry, Prop. 8, B. X. Where creal accuracy is required, the value cf $w must ba MENSURATION OF S o L i D s. taken to more than four decimal places. Its value, correct to ten decimal places, is .52359,87756. Ex. 1. What is the solidity of. the earth, if it be a sphere 7912 miles in diameter ? Ans., 259,332,805,350 cubic miles. Ex. 2. If the diameter of the moon be 2160 miles, what is its solidity? Ans. PROBLEM IX. (127.) To find the surface of a spherical zone. RULE. Multiply the altitude of the zone by the circumference o) a great circle of the sphere. See Geometry, Prop. 7, Cor. 1, B. X. Ex. 1. If the diameter of the earth be 7912 miles, what is the surface of the- torrid zone, extending 23 27' 36" on each ide of the equator ? Ans., 78,293,218 square miles Let PEP'Q, represent a meridian of the earth; EQ, the equator ; P, P' the poles ; AB one of the tropics, and GH one of the polar circles. Then PK will represent the height of one of the frigid zones, KD the height of one of the temperate zones, and CD half the height of the torrid zone. Each of the angles ACE, CAD, and GCK is equal to 23 27' 36". Tn the right-angled triangle ACD, R : AC : : sin. CAD : CD. Also, in the ight-angled triangle CGrK, R : CO : : cos. GCK : CK, Then PK=PC-KC. Where great accuracy is required, the sine and cosine ol 23 27' 36" must be taken to more than six decimal places The following values are correct to ten decimal places : Natural sine of 23 27' 36"=. 39810,87431. " cosine of 23 27' 36" :=. 91733,82302. 88 TRIGONOMETRY. Ex. 2. If the polar circle extends 23 27' 36" from the pole find the convex surface of either frigid zone. Ans., 8,128,252 square miles. Ex. 3. On the same suppositions, find the surface of each oi the temperate zones. Ans.. 51,056,587 square miles. PROBLEM X. (128.) To find the solidity of a spherical segment with one base, RULE. Multiply half the height of the segment by the area of the base, and the cube of the height by .5236, and add the two products. See Geometry, Prop. 9, B. X. Ex. 1. What is the solidity of either frigid zone, supposing the earth to be 7912 miles in diameter, the polar circles ex- tending 23 27' 36" from the poles ? Ans., 1,292,390,176 cubic miles. (129.) The solidity of a spherical segment of two bases is (he difference between two spherical segments, each having a single base. Ex. 2. On the same supposition as in Ex. 1, find the solid- ity of either temperate zone. Ans., 55,032,766,543 cubic miles Ex. 3. Find the solidity of the torrid zone. Ans., 146,682,491,911 cubic miles PROBLEM XL (130.) To find the area of a spherical triangle. RULE. Compute the surface of the quadrantal triangle, or one eighth of the surface of the sphere. From the sum of the three angles subtract two right angles ; divide the remainder by 90, and multiply the quotient by the quadrantal triangh Sco Geometry, Prop. 20, B. IX. MENSURATION OF SOLIDS. 89 Ex, 1. "What is the area of a triangle on a sphere whose di ameter is 10 feet, if the angles are 55, 60, and 85 ? Ans.j 8.7266 square feet. Ex. 2. If the angles of a spherical triangle measured on the surface of the earth are 78 4' 10", 59 50' 54", and 42 5' 37", what is the area of the triangle, supposing the earth a sphere, of which the diameter is 7912 miles ? Ans.j 3110.794 square miles. If the excess of the angles above two right angles is ex- pressed in seconds, we must divide it by 90 degrees also ex- pressed in seconds ; that is, by 324,000. PROBLEM XII. (131.) To find the area of a spherical polygon. RULE. Compute the surface of the quadrantal triangle. From the sum of all the angles subtract the product of two right angles by the number of sides less two ; divide the remainder by 90, and multiply the quotient by the quadrantal triangle See G-eometry, Prop. 21, B. IX. Ex. 1. What is the area of a spherical polygon of 5 sides on a sphere whose diameter is 10 feet, supposing the sum of tbe angles to be 640 degrees ? Ans.j 43.633 square feet 62 33' 13" ; Q/ ^)fiff Ex. 2. The angles of a spherical r~~ ^ p, Q,, ' polygon, measured on the surface < ^ ^ . _ ; ^ /; | of the earth, are ^. .Q, _ /; | 155 19' 12". Required the area of the polygon. Ans. t 5690.477 square miles BOOK IV. SURVEYING. (182.) THE term Surveying includes the measurement ol heights and distances, the determination of the area of portions of the earth's surface, and their delineation upon paper. Since the earth is spherical, its surface is not a piano sur* lace, and if large portions of the earth are to be measured, the curvature must be taken into account ; but in ordinary sur- veying, the portions of the earth are supposed to be so small that the curvature may be neglected. The parts surveyed are therefore regarded as plane figures. (133.) If a plummet be freely suspended by a line, and al- lowed to come to a state of rest, this line is called a vertical tine. Every plane passing through a vertical line is a vertica. plane. A line perpendicular to a vertical line is a horizontal line. A plane perpendicular to a vertical line is a horizontal plane. A vertical angle is one the plane of whose sides is vertical A horizontal angle is one the plane of whose sides is hori- zontal. An angle of elevation is a vertical angle having one side horizontal and the other an amending, line, as the angle BAD. An angle of depression is a vertical angle having one side horizontal and the other a descending line, as the an- gle CDA. (134.) When distances are to be found by trigonometrical computation, it is necessary to measure at least one line upon the ground, and also as many angles as may be necessary to render three parts of every triangle known SURVEYING. 91 la the measurement of lines, the unit communly employed ty surveyors is a chain four rods or sixty-six feet in length, called Gunter's Chain, from the name of the inventor. This chain is divided into 100 links. Sometimes a half chain i* used, containing 50 links. Hence, 1 chain 100 links =66 feet; Irod = 25 links =16J feet; 1 link =7.92 inches = f of a foot nearly. (135.) To measure a horizontal line. To mark the termination of the chain in measuring, ten iro* pins should be provided, about a foot in length. Let the person who is to go foremost in carrying the chain, and who is called the leader, take one end of the chain and the ten pins ; and let another person take the other end of the chain, and hold it at the beginning of the line to be measured. When the leader has advanced until the chain is stretched tight, he must set down one pin at the end of the chain, the other person taking care that the chain is in the direction of the line to be measured. Then measure a second chain in the same manner, and so on until all the marking pins are ex- hausted. A record should then be made that ten chains ha\ r e been measured, after which the marking pins should be re- turned to the leader, and the measurement continued as be- fore until the whole line has been passed over. It is generally agreed to refer all surfaces to a horizontal plane. Hence, when an inclined surface, like the side of a hill, is to be measured, the chain should be maintained in a horizontal position. For this purpose, in ascending a hill, the hind end of the chain should be raised from the ground until it is on a level with the fore end, and should be held vertically over the termination of the preceding chain. In descending a hill, the fore end of the chain should be raised in the same manner. INSTRUMENTS FOR MEASURING ANGLES. In measuring angles, some instrument is used which con- tains a portion of a graduated circle divided into degrees and minutes These instruments may be adapted to measuring >2 TRIGONOMETRY. either horizontal o: vertical angles. The instrument most fro quently employed for measuring horizontal angles is called THE SURVEYOR'S COMPASS. (136.) The piincipal parts of this instrument are a compas&. box, a magnetic needle, two sights, and a stand for its support, The compass-box, ABC, is circular, generally about six inches in diameter, and at its center is a small pin on which the mag- netic needle is balanced. The circumference of the box is di- vided into degrees, and sometimes to half degrees ; and the de- grees are numbered from the extremities of a diameter both ways to 90. The sights, DE, FGr, are placed at right angles to the plane of the graduated circle, and in each of these there is a large and small aperture for convenience of observation The instrument, when used, is mounted on a tripod, or a single staff pointed with iron at the bottom, so that it may be firmly placed in the ground. Sometimes two spirit levels, H and K, are attached, to indi- cate when the plane of the graduated circle is brought into a horizontal position. (137.) When the magnetic needle is supported so as to turn freely, and is allowed to come to a state of rest, the direction it assumes is called the magnetic meridian, one end of the needle indicating the north point and the other the south. A horizontal line perpendicular to a meridian is an east an;i west line SURVEYING. All the meridians passing through a survey of moderate ex- tent, are considered as straight lines parallel to each other. The bearing or course of a line is the angle which it makes with a meridian passing through one end ; and it is reckoned from the north or south point of the horizon, toward the east or west. Thus, if NS represent a meridian, and the angle NAB is 40, then the bearing of AB from the point A is XT 40 to the west of north, and is written N. 40 \V., and read north forty degrees west. ] The reverse bearing of a line is the bearing taken from the other end of the line. The forward bearing and reverse bearing of a line are equal angle?, but lie between di- rectly opposite points. Thus, if the bearing of AB from A is N. 40 W., the bearing of the same line from B is S. 40 E. (138.) For measuring vertical angles, the instrument com- monly used is A QUADRANT. It consists of a quarter of a circle, usually made of brass, and its limb, AB, is divided into degrees and minutes, numbered from A up to 90. It is furnish- ed either with a pair of plain sights or with a telescope, CD, which is to be directed toward the object observed. A plumb line, CE, is suspended from the center of the quadrant, and in- dicates when the radius CB is brought into a vertical position. To measure the angle of elevation, for example, of the top of a tower, point the telescope, CD, toward the tower, keeping the radius, CB, in a vertical position by means of the plumb line, CE. Move the telescope until the given object is seen in the middle of the field of view. The center of the field is in- ilicated by two wires placed in the focus of the object-glass of TRIGONOMETRY the telescope, one wire being vertical and the other horizonta When the horizontal wire is made to coincide with the sum mit of the tower, the angle of elevation is shown upon the arc AB by means of an index which moves with the telescope. As the arc is not commonly divided into parts smaller than half degrees, when great accuracy is required, some contriv* ance is needed for obtaining smaller fractions of a degree. This is usually effected by a vernier. (139.) A Vernier is a scale of small extent, graduated iu such a manner that, being moved by the side of a fixed scale, we are enabled to measure minute portions of this scale. The length of this movable scale is equal to a certain number of parts of that to be subdivided, but it is divided into parts one more or one less than those of the primary scale taken for the length of the vernier. Thus, if we wish to measure hundredths of an inch, as in the case of a barometer, we first divide an inch into ten equal parts. "We then construct a vernier equal in length to 11 of these divisions, but divide it into 10 equal jarts, by which means each division on the vernier is T Vth longer than a division of the primary scale. Thus, let AB be the upper end of a barometer tube, the mer sury standing at the point C ; the scale is rlivided into inches and tenths of an inch, and the middle piece, numbered from 1 to 9, is the vernier that slides up and down, having 10 of its divisions equal to 11 divisions of the scale, that is, to } jths of an inch. Therefore, each division of the vernier is T y ths of an inch ; or one division of the vernier exceeds one divi- ! IIIHIIil 6 HH 2 9 sion of the scale by T j th of an inch. Now, as the sixth division of the vernier (in the figure) coincides with a division of the scale, the fifth division of the ver- nier will stand th of an inch above the nearest division o/ the scale ; the fourth division fifths of an inch, and the to* of the vernier will be T w ths of an inch above the next lowei division of the scale ; i. e., the top of the vernier coincides with 29 G6 inches upon the scalo. in practice, therefore, we ob- SURVEYING. 93 serve what division of the vernier coincides with a division of the scale ; this will show the hundredths of an inch to be added to the tenths next below the vernier at the top. A similar contrivance is applied to graduated circles, to ob- tain the value of an arc with greater accuracy. If a circle ia graduated to half degrees, or 30', and we wish to measure sin- gle minutes by the vernier, we take an arc equal to 31 divi- sions upon the limb, and divide it into 30 equal parts. Then each division of the vernier will be equal to Jths of a degree, while each division of the scale is f ths of a degree. That is, each space on the vernier exceeds one on the limb by 1'. In order, therefore, to read an angle for any position of the vernier, we pass along the vernier until a line is found coin- ciding with a line of the limb. The number of this line from the zero point indicates the minutes which are to be added to the degrees and half degrees taken from the graduated circle. Sometimes a vernier is attached to the common surveyor's compass. (140.) An instrument in common use for measuring both horizontal and vertical angles is THE THEODOLITE. The theodolite has two circular brass plates, C and D (see fig. next page), the former of which is called the vernier plate, and the latter the graduated limb. Both have a horizontal motion about the vertical axis, E. This axis consists of two parts, ono external, and the other internal ; the former secured to the graduated limb, D, and the latter to the vernier plate, C, so that the vernier plate turns freely upon the lower. The edge of the lower plate is divided into degrees and half degrees, and this is subdivided by a vernier attached to the upper plate into single minutes. The degrees are numbered from to 360. The parallel plates, A and B, are held together by a ball which rests in a socket. Four screws, three of which, #, a, a, are shown in the figure, turn in sockets fixed to the lower plate, while their heads press against the under side of the uppc* plate, by which means the instrument is leveled for observe tnn. The whole rests upon a tripod, which is firmly aU&clit>d to the body of the instrument. TRIGONOMETRY. To the vernier plate, two spirit-levels, c, c, are attached ai right angles to each other, to determine when the graduated limb is horizontal. A compass, also, is placed at F. Two frames, one of which is seen at N, support the pivots of tho horizontal axis of the vertical semicircle KL, on which the tel- escope, GrH, is placed. One side of the vertical arc is divided into degrees and half degrees, and it is divided into single min- utes "by the aid of its vernier. The graduation commences at the middle of the arc, and reads "both ways to 90. Under and parallel to the telescope is a spirit-level, M, to show when tho telescope is brought to a horizontal position. To enable us to direct the telescope upon an object with precision, two lines called wires are fixed at right angles to each other in the focus of the telescope. To measure a Horizontal Angle with the Theodolite. (141.) Place the instrument exactly over the station from which the angle is to be measured ; then level the instrument by means of the screws, a, #, bringing the telescope over each pair alternately until the two spirit-levels on the vernier plate retain their position, while the instrument is turned entirely round upon its ay is. Direct the telescope to one cf the object- SURVEYING. 97 to be observed, moving it until the cross-wires and abject co- incide. Now read off the degrees upon the graduated limb, and the minutes indicated ""ry the vernier. Next, release the upper plate (leaving the graduated limb undisturbed), and move it round until the telescope is directed to the second ob- ject; and make the cross-wires bisect this object, as was done by the first. Again, read off the vernier ; the difference bo- tween this and the former reading will be the angle required. The magnetic bearing of an object is determined by simply reading the angle pointed out by the compass-needle when the object is bisected. To measure an Angle of Elevation with the Theodolite. (142.) Direct the telescope toward the given object so that it may be bisected by the horizontal wire, and then read off the arc upon the vertical semicircle. After observing the ob- ject with the telescope in its natural position, it is well to re* volve the telescope in its supports until the level comes upper- most, and repeat the observation. The mean of the two meas . ures may be taken as the angle of elevation. By the aid of the instruments now described, we may de- termine the distance of an inaccessible object, and its height above the surface of the earth. HEIGHTS AND DISTANCES. PROBLEM I. (143.) To determine the height of a vertical object situated i:n a horizontal plane. Measure from the object to any convenient distance in a slraight line, and then take the angle of elevation subtended hy the object. C, If we measure the distance DE , and the angle of elevation CDE, there will be given, in the right-angled triangle CDE, the base and the angles, to find the perpendicular CE (Art. 46). To this we must add the height of the in- strument, to obtain the entire height i>f the object a T ;*ove the plane AB. Gr T R I G O IV O M E T R v Ex. 1. Having measured AB equal to 100 feet from the c bottom of a tower on a horizontal plane, I found the angle of elevation, CDE, of the cop to be 47 30', the center of the quadrant being five feet above the ground. What is the / height of the tower 1 / B, : tang. GDE : : DE : CE = 109.13. To which add five feet, and we obtain f -U- the height of the tower, 114.13 feet. A Ex. 2. From the edge of a ditch 18 feet wide, surrounding a fort, the angle of elevation of the wall was found to be 62 40'. Required the height of the wall, and the length of a lad- der necessary to reach from my station to the top of it. Ans. The height is 34.82 feet. Length of ladder, 39.20 feet. PROBLEM II. (144.) To find the distance of a vertical object whor-e hetgitt is known. Measure the angle of elevation, and we shall have given the angles and perpendicular of a right-angled triangle to find the base (Art. 46). Ex. 1. The angle of elevation of the top of a tower whoso height was known to be 143 feet, was found to be 35. What was its dis- tance 1 Here we have given the angles of the triangle ABC, and the side CB, to find AB. Ans., 204.22 feet. i If the observer were stationed at the top of the tower BC, he might find the length of the base AB by measuring the angle of depression DC A, which is equal to BAG: Ex. 2. From the top of a ship's mast, which was 80 feet above the water, the angle of depression of another ship's hull *-as found to be 20. What was its distance 1 Ans., 219.80 feet SURVEYING. 99 PROBLEM III. (145.) To find the height of a vertical object standing on an inclined plane. Measure the distance from the object to any convenient sta- tion, and observe the angles which the base-line makes with lines drawn from its two ends to the top of the object. If we measure the base-line AB, and the two angles ABC, BAG, then, in the triangle ABC, we shall have given one side and the angles to find BC. Ex. 1. Wanting to know the height of a tower standing on an inclined plane, BD, I measured from the bottom of the tower a distance, AB, equal to 165 feet ; also the angle ABC, equal to 107 18', and the angle BAG, equal to 33 35'. Required the height of the object. sin. ACB : AB : : sin. BAG : BC=144.66 feet. The height, BC, may also be found by measuring the dis- tances BA, AD, and taking the angles BAG, BDC. The dif- ference between the angles BAG and BDC will be the angle ACD. There will then be given, in the triangle DAG, one side and all the angles to find AC ; after which we shall have, in the triangle ABC, two sides and the included angle to find BC. Ex. 2. A tower standing on the top of a declivity, I meas- ured 75 feet from its base, and then took the angle BAG, 47 r 60' ; going on in the same direction 40 feet further, I took the angle BDC, 38 30'. What was the height of the tower ? Am., 117.21 feet. PROBLEM IY. (146.) To find the distance of an inaccessible object. Measure a horizontal base-line, and also the angles between tills line and lines drawn from each station to the object. Lei -<3 the obieot inaccessible from A and R Then> if the din 100 T ii > c o N o M L r R y. sin. C : AB tance between the stations A and B be measured, as also the angles at A and B, there will be given, in the triangle ABC, the side AB and the an- gles, to find AC and BC, the distances of the object from the two stations. Ex. 1. Being on the side of a river, and wanting to know the distance to a house which stood on the other side, I measured A B 400 yards in a right line by the side of the river, and founr that the two angles at the ends of this line, formed by the other end and the house, were 73 15' and 68 2'. What was the distance between each station and the house ? The angle C is found to be 38 43'. Then sin. A : BC-612.38; sin. B : AC-593.09. Ex. 2. Two ships of war, wishing to ascertain their distance from a fort, sail from each other a distance of half a mile, when they find that the angles formed between a line from one tc the other, and from each to the fort, are 85 15' and 63 4S' What are the respective distances from the fort ? Ans., 4584.52 and 4596.10 yards. PROBLEM V. (147.) To find the distance between two objects separated by an impassable barrier. Measure the distance from any convenient station to each of the objects, and the angle included between those lines. If we wish to know the distance between the places C and B, both of which are accessible, but sep- arated from each other by water, we may c measure the lines AC and AB, and also the angle A. We shall then have given two sides of a triangle and the included angle to find the third side. Ex. 1. The passage between the two nbjects C and B being obstructed, I measured from A to C 735 rods, and from A to B 840 rods ; also, the angle A, equal tn 55 40'. What is the distance of the places C and B ? Ans., 741.21 rod- SURVEYING. 101 Ex. 2. In order to find the distance between two cbpcts, C -id B, which could not be directly measured, I measured i om C to A 652 yards, and from B to A 756 yards ; also, the angle A equal to 142 25'. What is the distance between tho objects C and B ? Ans. PROBLEM VI. (148.) To find the height of an inaccessible object above a horizontal plane. First Method. Take two stations in a vertical plane pass- ing through the top of the object ; measure the distance be- tween the stations and the angle of elevation at each. If we measure the base AB, and the angles DAG, DBC, then, since CBA is the sup- plement of DBC, we shall have, in the triangle ABO, one side and all the angles to find BC. Then, in the 3b right-angled triangle DBC, we shall have the hypothenuse and the angles to find DC. . Ex. 1. What is the perpendicular height of a hill whose an- gle of elevation, taken at the bottom of it, was 46 ; and 100 yards farther off, on a level with the bottom of it, the angle was 31 ? Ans., 143.14 yards. Ex. 2. The angle of elevation of a spire I found to be 58, *nd going 100 yards directly from it, found the angle to be only 32. What is the height of the spire, supposing the in- strument to have been five feet above the ground at each ob- Rervation ? Ans., 104.18 yards. (149.) Second Method. Measure any convenient base-line, also the angles between this base and lines drawn from each of its extremities to the foot of the object, and the angle of elevation at one of the stations. Let DC be the given object. If we measure the horizontal base-line AB, and the angles CAB, CBA, we can compute tha iistanco BC. Also, if we observe the angle of elevation CBD, IOL T R 1 G O N D M E T R Y. we shall ha/e given, in the right-angled triangle BCD, tka base and angles to find the perpen- dicular. Ex. 1. Being on one side of a river, and wanting to know the height of a * spire on the other side, I measured 500 yards, AB, along the side of the river, and found the angle ABC =74 14', and BAC-49 23' ; also, the an- gle of elevation CBD=11 15'. Required the height of th Ans., 271.97 feet. Ex. 2. To find the height of an inaccessible castle, I meas- ured a line of 73 yards, and at each end of it took the angle of position of the object and the other end, and found the one to be 90, and the other 61 45' ; also, the elevation of the castle from the latter station, 10 35'. Required the height of the castle Ans., 86.45 feet. PROBLEM VII. (150.) To find the distance between two inaccessible objects Measure any convenient base-line, and the angles between this base and lines drawn from each of its extremities to each of the objects. Let C and D be the two inaccessible objects. If we meas- nre a base-line, AB, and the an- gles DAB, DBA, CAB, CBA, then, in the triangle DAB, we shall have given the side AB and all the angles to find BD ; also, in the triangle ABC, we shall have one side and all the angles to find BC ; and then, in the triangle BCD, we shall have two sides, BD, BC, with thfl included angle, to find DC. Ex. 1. "Wanting to know the distance between a house and a mill, which were separated from me by a river, I measured a base-line, AB, 300 yards, and found the angle CAB=58 20', CAD-37 , ABD=53 30', DEC = 45 15'. What is tta distance of the house from the mill ? Ans., 479.80 yards. SURVEYING 103 Ex. 2. Wanting to know the distance between two inaccessi lie objects, C and D, I measured a base-line, AB, 28.76 rods, and found the angle CAB -33, CAD =66, DBA=59 45', and BBC =76. What is the distance from C to D ? Ans., 97.C96 rods THE DETERMINATION OF AREAS. (151.) The area or content of a tract of land is the horizon- tal surface included within its boundaries. When the surface of the ground is broken and uneven, it is very difficult to ascertain exactly its actual surface. Hence it has been agreed to refer every surface to a horizontal plane ; and for this reason, in measuring the boundary lines, it is nec- essary to reduce them all to horizontal lines. The measuring unit of surfaces chiefly employed by survey- ors is the acre, or ten square chains. One quarter of an acre is called a rood. Since a chain is four rods in length, a square chain contains sixteen square rods ; and an acre, or ten square chains, con- tains 160 square rods. Square rods are called perches. The area of a field is usually expressed in acres, roods, and perches, designated by the letters A., R., P. When the lengths of the bounding lines of a field are given in chains and links, the area is obtained in square chains and square links. Now, since a link is T | 7 of a chain, a square link will be T ^> tar-cos, as given in the field-book SURVEYING. H)c Statfons | Bearings. Distances. 1 N. 45 E. 9.30 chains. 2 a 60 E. 11.85 " 3 S. 20 W. 5.30 " 4 S. 70 W. 10.90 5 N. 31 W. 9.40 " Draw NS to represent a meridian line ; in NS take any con' venient point, as A, for the first station, and lay off an angle, NAB, equal to 45, the bear- ing from A to B, which will give the direction from A to B. Then, from the scale of equal parts, make AB equal to 9.30, the length of the first side : this will give the sta- tion B. Through B draw a second meridian parallel to NS ; lay off an angle of 60, and make the line BC equal to 11.85. Proceed in the same manner \vith the other sides. If the survey is correct, and the plotting accurately performed, the end of the last side, EA, will fall on A, the place of beginning. This plot is made on a scale of 10 chains to an inch. (154.) To avoid the inconvenience of drawing a meridian through each, angle of the field, the sides may be laid down from the angles which they make with each other, instead of the angles which they make with the meridian. Reverse one of the bearings, if necessary, so that both bearings may run from the same angular point ; then the angle which any two contiguous sides make w r ith each other may be determined from the following RULES. 1. If both courses are north or south and both east or west, subtract the less from the greater. 2. If both are north or south, but one east and the other west, add t\en: together. 1.06 TRIGONOMETRY. 3. If one is nortli and the other south, but both east or west subtract their sum from 180. 4. If one is north and the other south, one east and the othoi \vest, subtract their difference from 180. Thus the angle CAB is equal to \AB-NAC. The angle CAD is equal to NAG V-NAD. The angle 'DAF is equal to 180 -(NAD+SAF). The angle CAF is equal to 180 (SAF-NAC). In the pieceding example we ac- cordingly find the angle ABC=105. DEA-101 . BCD=100. EAB=104. CDE=130. With these angles the field may he plotted without drawing parallels. EXAMPLE 2. The following field notes are given to protract the survoy: Stations. Bearings. Distances. 1 N. 50 30' E. 16.50 chains. 2 S. 68 15' E. 14.20 3 S. 9 45' E. 8.45 " 4 S. 21 0' W. 6.84 5 S. 73 D 30' W. 12.31 6 N. 78 15' W. 9.76 7 N. 15 30' W. 11.55 " THE TRAVERSE TABLE. '155.) The accompanying traverse table shows the difference of latitude and the departure to four decimal places, for dis- tances from 1 to 10, and for hearings from to 90, at inter- vals of 15'. If the bearing is less than 45, the angle will be found on the left margin of one of the pages of the table, and the distance at the top or bottom of the page ; the difference SURVEYING. 107 oi latitude will be found in the column headed Lat. at the top of the page, and the departure in the column headed Dep. If the bearing is more than 45, the angle will be found on the right margin, and the difference of latitude will be found in the column marked Lai. at the bottom of the page, and the departure in the other column. The latitudes and departures for different distances with the same bearing are proportional to the distances. Therefore the distances may be reckoned as tens, hundreds, or thousands, if the place of the decimal point in each departure and difference of latitude be changed ac- cordingly. Ex. 1. To find the latitude and departure for the course 45* and the distance 93. Under distance 9 on page 141, and opposite 45, will be found latitude 6.3640 and departure 6.3640. Hence, for dis- tance 90, the latitude is 63.640, and adding the latitude for the distance 3, viz., 2.121, we find the latitude for distance 93 to be 65.761. Ex. 2. To find 1 ^- latitude and departure for the course 60 a and the distance 11.35. Departure for 10 is 8.6603. " " 1 is .8660. " " .8 is .6928. " " .05 is .0433. The latitude for 10 is 5.0000. " " " 1 is .5000. u " " .8 is .4000. " " " .05 is .0250. Latitude for 11.85 is 5.9250. Depart, for 11.85 is 10.2624 Ex. 3. To find the latitude and departure for the courso 20 and the distance 5.30. Ans. Latitude 4.98, and departure LSI. The traverse table may be used not only for obtaining de- parture and difference of latitude, but for finding by inspection the sides and angles of any right-angled triangle ; for the lati- tude and departure form the two legs of a right-angled trian- gle, of which the distance is the hypothenuse, and the courso is one of the acute angles. In this manner we find the latitude and departure for each side df the field given in Example 1, page 105, tc be as in the following table : 108 JL R I G O N M E T R Y. Oc urses. Dis- tances. Latitude. Departure. Cor. Lat. Cor. Dep. Balanced. N. s. E. W. N. S. E. W. 1 N. 45 E. 2S. 60 E. 3S. 20 W. 9.30 11.85 5.30 6.58 5.92 4.98 6.58 10.26 1.81 +.01 +.01 +.01 .01 6.58 5.93 4.98 6.59 10.27 1.80 4 S. 70 W. 10.90 3.73 10.24 .01 3.73 10.23 5N. 31 W. 940 8.06 4.84 .01 8.06 4.83 Perimeter 46.75 1464J14.63 16.84 16.89 14.64 14.64 1686 16.86] (156.) When a field has been correctly surveyed, and the latitudes and departures accurately calculated, the sum of the northings should be equal to the sum of the southings, and the sum of the eastings equal to the sum of the westings. If th/; northings do not agree with the southings, and the eastings with the westings, there must be an error either in the survey or in the calculation. In the preceding example, the north- ings exceed the southings by one link, and the westings ex- ceed the eastings by five links. Small errors of this kind are unavoidable; but when the error does not exceed one link to a distance of three or four chains, it is customary to distribute the error among the sides by the following proportion : As the perimeter of the field, Is to the length of one of the sides, So is the error in latitude or departure, To the correction corresponding to that side, This correction, when applied to a column in which the sum of the numbers is too small, is to be added ; but if the sum of the numbers is too great, it is to be subtracted. We thus obtain the corrections in columns 8 and 9 of the preceding table ; and applying these corrections, we obtain the balanced latitudes and departures, in which the sums of the northings and southings are equal, and also those of the east- ings and westings. As the computations are generally carried to but two deci- mal places, the corrections of the latitudes and departures are only required to the nearest link, and these corrections may often be found by mere inspection without stating a formal proportion. Thus, in the preceding example, since the depart- ures require a correction of five links, and the field has five sides which are not very unequal, it is obvious that we must make a correction of one liiiiv on each side. SURVEYING. It is the opinion of some surveyors that when the error in latitude or departure exceeds one link for every five chains of the perimeter, the field should he resurveyed ; hut most sur veyors do not attain to this degree of accuracy. The error, however, should never exceed one link to a distance of two 01 three chains. (157.) To find the area of the field. Let ABODE he the field to he measured. Through I . A, the most western station, draw the meridian NS, and upon it let fall the perpen- diculars BF, CG, DH, EL Then the area of the re- quired field is equal to FBCDEI-(ABF+AEI). But FBCDEI is equal to the sum of the three trape- zoids FBCG-, GCDH, HDEI. Also, if the sum of the parallel sides FB, GO he multiplied "by FG, it will give twiou the area of FBCG (Art. 87). The sum of the sides GO, DH, multiplied hy GH, gives twice the area of GCDH ; and the sum of HD, IE, multiplied hy HI, gives twice the area of HDEI. Now BF is the departure of the first side, GO is the sum of (he departures of the first and second sides, HD is the alge- braic sum of the three preceding departures, IE is the algebraic sum of the frur preceding departures. Then the sum of the parallel sides of the trapezoids is ohtained hy adding together the preceding meridian distances two hy two ; arid if these sums are multiplied hy FG, GH, &o., which are the corre- sponding latitudes, it will give the 'double areas of the trape- (158.) It is most convenient to leduce all these operations a tabular form, according to the following RULE, Having arranged the balanced latitudes and departures in uo TRIGONOMETRY. their appropriate columns, draw a meridian through the mosi eastern or western station of the survey, and, calling' this the first station, form a column of double meridian distances. . The double meridian distance of the first side is equal ta its departure ; and the double meridian distance of any side is equal to the double meridian distance of the preceding side, plus its departure, plus the departure of the side itself. Multiply each double meridian distance by its correspond* ing northing or southing, and place the product in the column of north or south areas. The difference between the sum of the north areas and the sum of the south areas will be double, the area of the field. It must be borne in mind that by the term plus in this rule is to be understood the algebraic sum. Hence, when the double meridian distance and the departure are both east or both west, they must be added together ; but if one be east and the other west, the one must be subtracted from the other. The double meridian distance of the last side should always be equal to the departure for that side. This coincidence af- fords a check against any mistake in forming the column of double meridian distances. The preceding example will then be completed as follows : N. s. i:. vv D.M.D. N. Areas. S. Areas. | 1 2 3 4 5 6.5 8.06 5.93 4.98 3.73 6.59 10.27 1.80 10.23 4.83 6.59 23.45 31.92 19.89 4.83 43.3622 38.9298 139.0585 158.9616 74.1897 1 82.2920 372.2098 1 1 Twice the figure FBCDEI is 372.2098 square chains. Twice the figure FBAEI is 82.2920 The difference is ..... 289.9178 " Therefore the area of the field is 144.9589 square chains, or 14.49589 acres > which is equal to 14 acres, 1 rood, 39 perches. Ex. 2. It is required to find the contents of a tract of land of which the following are the field notes : SURVEYING. 11 Sta- tions. Bearings. Distances. 1 N. 30' E. 16.50 chains. 2 S. 68 15' E. 14.20 3 S. 9 45' E. 8.45 < 4 S. 21 0' W. 6.84 5 S. 73 30' W. 12.31 6 N. 78 15' W. 9.76 7 N. 17 0' W. 11.64 Calculation. Courses. Dist. Dif. Lat. Departure. CO, Balanced. D.M. D. N. Areas. S. Areas. N. s. K. | VV. N. 10.47 1.97 11.11 s. K. \v. 1 N. 5(P 3(y E. 2 S. 68 15' E. 3 S. 90 45' E. 4 S. 21 0' W. 5 S. 730 30' W. 6 N. 78 15' W. 7N.170 VW. 16.50 14.20 8.45 6.84 12.31 9.76 11.64 10.50 1.99 11.13 5.26 8.33 ^.39 3.50 12.73 13.19 1.43 2.45 11.80 9.56 3.40 .03 .03 .01 .01 .02 .02 .02 5.29 8.34 6.40 3.52 12.70 13.16 1.42 2.46 11.82 9.58 3.42 12.70 38.56 53.14 52.10 37.82 16.42 *, 132.9690 32.3474 37.9962 203.9824 443.1876 333.4400 133.1264 79.70,23.6-123.48 27.35 27.21 2:5.55 23.55 27.28 27.28 205.3126 1113.7364 \ Error .14 | Error .14 203.3126 Am., 45 A., 2 R., 3 P. Ex. 3. Required the area of a tract of land of which the follow- ing are the field notes : 2)910.4238 455.2119 Sta- tions. Bearings. Distances. 1 N. 58 45' E. 19.84 chains. 2 N. 39 30' E. 10.45 3 S. 45 15' E. 37.26 4 S. 52 30 W. 21.53 5 S. 34 0' E. 9.12 " 6 S. 66 15' W. 27.69 " 7 N. 12 45' E. 24.31 8 N. 48 15' W. 24.60 ' " Ans., 130 A., 2R.,23P, Ex. 4. Required the area of a piece of land from the follow- Lag field notes : TRIGONOMETRY [ Ptations. Bearings. Distances. 1 N. 5 15' E. 15.17 chains. ! 2 3 N. 45 45' E. N. 32 0' W. 16.83 14.26 " 4 N. 88 30' E. 19.54 " 5 S. 28 15' E. 17.92 " 6 S. 40 45' W. 9.71 " 7 8 9 S. 31 30' E. S. 14 O'W. S. 82 45' W. 22.65 " 18.39 " 24.80 " 10 N. 23 15' W. 26.31 Ans., 173 A., R., 23 P. En fi . Required the area of a field from the followin Stations. Bearings. Distances. 1 N. 32 15' E. 28.74 chains. 2 N. 17 45' E. 21.59 " 3 S. 81 30' E. 13.38 " 4 S. 9 45' W. 11.92 " 5 S. 43 0' E. 19.65 " 6 N. 25 30' E. 17.26 7 S. 78 15' E. 18.87 " 8 S. 5 45' W. 31.41 9 S. 37 30' W. 26.13 10 N. 69 0' W. 23.86 11 S. 74 15' W. 20.91 12 N. 27 30' W. 23.20 " Ans., 304 A., 2 R., 9 P. Rv. 6. Required the area of a field from the following Stations. Bear in ITS. Distances. 1 N. 36 15' E. 24.73 chains. 2 N. 7 45' E. 11.58 " 3 N. 79 30' E. 15.39 4 S. 86 45' E. 20.56 5 S. 12 15' W. 18.14 " 6 S.. 25 0' E. 21.92 " 7 S. 58 30' W. 29.27 " 8 N. 34 0' W. 19.81 " 9 N. 81 15' W. 21.24 .. 1.79 A., 1 R., fi f SURVEYING. 113 (159 ) The field notes from which the area is to be com- puted may "be imperfect. There may be obstacles which pre- vent the measuring of one side, or the notes may be defaced so as to render some of the numbers illegible. If the bearing?, and lengths of all the sides of a field except one are given, the remaining side may easily be found by calculation. For the difference between the sum of the northings and the sum of the southings of the given sides will be the northing or south- ing of the remaining side ; and the difference between the sum of the eastings and the sum of the westings of the given sides will be the easting or wssting of the remaining side. Having, then, the difference of latitude and departure of the required side, its length and direction are easily found by Trigonome- try (Art. 47). Ex. Griven the bearings and lengths of the sides of a tract of land as follows : Stations. Bearings. Distances. ] 1 N. 18 15' E. 8.93 chains. 2 N. 79 45' E. 15.64 " 3 S. 25 0' E. 14.27 " 4 Unknown. Unknown. 5 N. 87 30' W. 18.52 chains. 6 N. 41 15' W. 12.18 Required the bearing and distance of the fourth side. Ans., S. 15 33' E., distance 8.62 chains. (160.) There is another method of finding the area of a fieJd which may be practiced when great accuracy is not required It consists in first drawing a plan of the field, as in Art. 153 then dividing the field into triangles by diagonal lines, and measuring the bases and perpendiculars of the triangles upon the same scale of equal parts by which the plot was drawn. Thas, if we take Ex. 1, and draw the diagonals AC, AD, the field will be divided into three triangles, whose area e easily found when we know H 114 TRIGONOMETRY. the diagonals AC, AD, and the perpendiculars BF, DO, EIL The diagonal AC is found by measurement upon the scale of equal parts to be 16.87 ; the diagonal AD is 15.67 ; the perpen- licular BF is 6.30 ; DG- is 4.92 ; and EH is 6.42. Hence the triangle ABC =--16.87x3.15= 53.14 "' " ADC-16.87X2.46= 41.50 = 15.67x3.21= 50.30 the figure ABCDE --144.94 sq. chains. This method of finding the area of a field is very expedi- tious, and when the plot is carefully drawn, may afford results sufficiently precise for many purposes. (161.) To survey an irregular boundary by means of off- sets. When the boundaries of a field are very irregular, like a river or lake shore, it is generally best to run a straight line, coming as near as is convenient to the true boundary, and measure the perpendicular distances of the prominent points of the boundary from this line. Let ABCD be a piece of land to be surveyed ; the lar^d be- ing bounded on the east by a lake, and on the west by & creek We select stations A, B, C, D, so as to form a polygon which shall embrace most of the proposed field, and find its area. We then measure perpendiculars aa', bb', cc', &c., as also the distances A#, ab, be, &c. Then, consider- ing the spaces Aaa 1 , abb'a', &c., as triangles or trapczoids, their area may be computed ; and, adding these areas to the figure ABCD, we shall obtain the area of the proposed field nearly. (162.) To determine the bearing 1 and distance from onf point to another by means of a series of triangles. When it is required to find the distance between two points remote from each oth?r, we form a series of triangles such tha* SURVEYING. D the first and second triangles may have one side in common ; the second and third, also, one side in common ; the third and fourth, &c. "vVe then measure one side of the first triangle for a base line, and all the angles in each of the triangles. These data are sufficient to determine the length of the sides of each triangle ; for in the first triangle we have one side and the angles to find the other sides. When these are found, we shall have one side and all the angles of a second triangle to find the other sides. In the same manner we may calculate the dimensions of the third triangle, the fourth, and so on. "Wo shall illustrate this method by an. example taken from the Coast Survey of the United States. The object here is to make a survey of Chesapeake Bay and its vicinity ; to determine with the utmost precision the position of the most prominent points of the country, to which subordinate points may be referred, and thus a perfect map of the country be obtained. According- ly, a level spot of ground was select- ed on the eastern side of the bay, on Kent Island, where a base line, AB, <)f more than five miles in length, was measured with every precaution. A station, C, was also selected upon the other side of the bay, near An- napolis, so situated that it was visi- ble from A and B. The three angles of the triangle ABC were then meas- ured with a large theodolite, after which the length of BC may be com- puted. A fourth station, D, is now taken on the western shore of the bay, visible from C and B, and all the angles of the tri- angle BCD are measured, when the line BD can be computed. A fifth station, E, is now taken on an island near the eastern shore, visible both from B and D, and all the angles of the tri- angle BDE are measured, when DE can be computed. Also, all the ang.es of the triangle DEF are measured, and EF i.t computed. Then all the angles of the triangle EFG- are meas 1 J b* TRIGONO M'E T R Y. ured, and FG- is computed. So, also, all the angles of the tri. angle FGH are measured, and GH is computed ; and thus a chain of triangles may he extended along the entire coast of rhe United States. To test the accuracy of the work, it is common to measure a side in one of the triangles remote from the first "base, and compare its measured length with that de- duced by computation from the entire series of triangles. This line is called a base of verification. Such a hase has been measured on Long Island ; and, indeed, several bases have been measured on different points of the coast. These are all 3onnected by a triangulation, and thus the length of a side in any triangle may be deduced from more than one base line, and the agreement of these results is a test of the accuracy of the entire work. Thus the length of one of the sides of a tri- angle which was twelve miles, as deduced from the Kent Island base, differed only twenty inches from that derived from the Long Island base, distant two hundred miles. The superiority of this method of surveying arises from the circumstance that it is necessary to measure but a small num- ber of base lines along a coast of a thousand or more miles in extent ; and for these the most favorable ground may be se- lected any where in the vicinity of the system of triangles. All the other quantities measured are angles; and the pre- cision of these measurements is not at all impaired by the in- equalities of the surface of the ground. Indeed, mountainous countries afford peculiar facilities for a trigonometrical survey, :since they present heights of ground visible to a great distance, and thus permit the formation of triangles of very large di- mensions. (163.) To divide an irregular piece of land into any two griven parts. We first run a line, by estimation, as near as may be to th*. required division line, and compute the area thus cut off. li this is found too large or too small, we add or subtract a tri- angle, or some other figure, as the case may require. Sup- pose it is required to divide the field ABCDEFGHI into two equal parts, by a line IL, running from the corner I to the opposite side CD, We first draw a line from I to D, and com- pute the area of the part DEFGHI ; and, knowing the area SURVEYING. ir of the entire 'field, we learn the area which must be contained in the triangle DIL, in order that IL may divide the field into two equal parts. Having the bearings and distances of the sides DE, EF, &c., we can compute the bearing and distance of DI. Thus the an- gle TDK is known ; and, having the hypothenuse ID, we can com- pute the length of the perpendicu- lar IK let fall on CD. Now the base of a triangle must be equal to its area divided by half the al- titude. Hence, if we divide the area of the triangle DIL by half of IK, it will give DL. In a similar manner we might proceed if it was required to divide a tract of land into any two given parts. Jj Variation of the Needle. (164.) The line indicated by a magnetic needle, when free- ly supported and allowed to come to a state of rest, is called the magnetic meridian. This does not generally coincide with the astronomical meridian, which is a true north and south line. The angle which the magnetic meridian makes with the true meridian is called the variation of the needle, and is said to be east or west, according as the north end of the needle, points east or west of the north pole of the earth. The variation of the needle is different in different parts of the earth. In some parts of the United States it is 10 west. and in others 10 east, while at other places the variation has every intermediate value. Even at the same place, the varia- tion does not remain constant for any length of time. Hence it is necessary frequently to determine the amount of the varia- tion, which is easily done when we know the position of tin? true meridian. The latter can only be determined by astro- nomical observations. The best method is by observations of the pole star. If this star were exactly at the pole, it would always be on the meridian ; but, being at a distance of about 118 T R J G O K O M E T R Y. a degree and a hall from the pole, it revokes about the polo ia a small circle in a little less than 21 hours. In about six hcmis from its passing the meridian above the pole, it attains its greatest distance west of the meridian ; in about six hours more it is on the meridian beneath the pole ; and in about six hours more it attains its greatest distance east of the meridian. If the star can be observed at the instant when it is on the meridian, either above or below the pole, a true north and south line may be obtained. (165.) The following table shows the time of the pole star's passing the meridian above the pole for every fifth day of the year : t.1 TInj. Srti Tfaj. irtfi DH.V. I'ith Day. 21 st Hay. 2fYtli 17ar. f Janoary . . February. . March . . . April . May.. June .... JB)V . li. in. 6 20 P.M. 4 J8 2 28 " 20 " 10 28 A.M. 8 26 " 6 28 " k m. 6 P.M. 358 " 2" 8 " 07 " 10 9 A.M. 87" 69" 5 41 P.M. 339 " 1 49 11 47 A.M. 9 49 * 7 47 " 5 49 " b. m. 5 21 P.M. 3 19 " 1 29 < 11 27 A.M. 929 " 7 27 " 5 29 " h. m. 5 1 P.M. 30" 19" 11 8 A.M. 99" 78" 5 10 * h. ro. f 4 42 P.M. 2 40 " 50 " 10 48 AJH. 8 50 6 4.8 " 4 50 " August . . . September . October . . November . IXeceinber . 4 27 225 " 26 10 21 P.M. 8 23 " 47" 2 5 " 7 " 10 1 P.M. g 3 " 3 47 " 1 45 " 11 43 P.M. 9 41 " 743 3 27 " 1 26 " 11 24 P.M. 9 22 " 7 24 " 38" 16" 11 4 P.M. 92" 7 4 2 48 " 46 * 10 44 P.M. 8 42 " 6 44 " If the pole star passes the meridian in the daytime, it can not be observed without a good telescope ; but ll k 58 m " after the dates in the above table, the star will be on the meridian below the pole, and during the whole year, except in summer, the pole star may be seen with the naked eye on the merid- ian either above or below the pole. These observations are best made with a theodolite, but they may be made with a common compass. At 5' 1 ' 59 m> after the dates in the above table, the star will have attained its greatest distance west of the meridian ; and 5 h- 59 ra< before these dates, it will be at its greatest distance east of the meridian. In summer, therefore, we may observe the greatest eastern elongation of the pole star, at which time the star is 1 48' east of the true meridian fox all places in the neighborhood of New York. Making this al- lowance, a true meridian is easily obtained ; after which, the variation of the needle is determined by placing a compass upon this line, turning the sights in the same direction, and noting the angle shown by the needle. The following table shows the angle which the plane cf tHc SURVEYING. Ill meridian make 3 with a vertical plane passing through the pole star, when at its greatest eastern or western elongation, for any latitude from 30 to 44. Lat. 30 1 37' Lat. 34 I Lat. 3G 1 39' 11 42' Lat. 38 Lat. 40 1 48' Lat. 42 1 51' Lat. 44 1 55' 1 35' (166.) Tha variation of the needle, in 1870, for several parts of the United States, was as follows : Burlington,Yt. . Boston, Mass. . Albany, KY. . New Haven, Ct. New York City Philadelphia . . Washington City 11 7' W. 11 0' W 8 40' W. 8 15' W. G 50'W. 5 40' W. 2 40' W. Buffalo, N.Y. . . 3 54' Cleveland, Ohio . Detroit, Mich. Charleston, S. C. Cincinnati, Ohio Mobile, Ala. . . St. Louis, Mo. . 1 1 O'E. 35' E. 3 IS'E. C 44' E. 8 O'E. Since 1870, the variation in New England has increased about four minutes annually ; in New York and Pennsylvania it has increased from threo to four minutes annually. In the Western States it decreases about two minutes annually, and in the Southern States it decreases about two minutes annually. LEVELING. (167.) Leveling- is the art of determining the difference oi level between two or more places. The surface of an expanse of tranquil water, or any surface parallel to it, is called a level surface. Points situated in a level surface are said to be on the same level, and a line traced on such a surface is called a line of true level. On account of the globular figure of the earth, a level sur- face is not a plane surface. It is nearly spherical ; and in the common operations of leveling it is regarded as perfectly so. Hence every point of a level surface is regarded as at the same d istance from the center of the earth ; and the difference of level of two places is the difference between their distances from the center. A line of apparent level is a straight line tangent to the sur- face of the earth. Thus, if AB ^represent the surface of the ocean, the two places A and B are said to be on the same level; but if All 120 TRIGONOMETRY be drawn tangent to the, arc AB at A, then AD is a line o apparent level. ^ T> This is the line which is indicated by a leveling instrument placed at A. The theod- olite may be employed for tracing horizontal lines ; but if nothing further were required, there would be no occasion for graduated cir- cles, and several parts of the theodolite might be dispensed with. A leveling instrument, therefore, usually consists of a large spirit level attached to a telescope, mounted upon a stand in a manner similar to the theodolite. (168.) The surveyor should also be provided with a pair oi leveling' staves. A leveling staff consists of a rectangular bar of wood six feet in length, di- A vided to inches and sometimes tenths of an inch, and having a groove running its entire length. A smaller staff of the same length, called a slide, also divided into inches, is in- serted in this groove, and moves freely along it. At the upper end of the slide is a rectangu- lar board called a vane, AB, about six inches wide. The vane is divided into four equal parts by two lines, one horizontal and the other vertical. Two opposite parts of the vane are painted white, and the other two black, in order that they may be distinguished at a great distance. To find the difference of level between any two points. (169.) Set up the leveling staves perpendicular to the hori- zon, and at equal distances from the leveling instrument Having adjusted the level by means of the proper screws, turn the telescope to one of the staves, and direct an assistant to slide up the vane until the line AB coincides with the center of the telescope, and note the height of this line from the ground. Turn the telescope to the other staff, and repeat the same operation. Level in the same manner from the second station to the third, from the third to the fourth, &c. Then the difference between the sum of the heights at the back sta- tions and at the forward stations will be equal to the difference nf level between the first station and the last. PURVEYING. 121 Jf we wish to level fiom A to E, we set up the stages at a sonvenient distance, AC, and midway be- tween them place the level B. Observe where the line of lev- el, FGr, cuts the rods, and note the heights AF, CGr. Theii Difference is the difference of level between the first and second stations. Take up the level and place it at D, midway be- tween the rods C and E, and observe where the line of level, [II, cuts the rods, and note the heights CH, El. Then FA CGr=the ascent from A to C, and CH-EI =the ascent from C to E. Therefore (FA+CH) (CG-+EI)=the entire ascent from A to E ; and in the same manner we may find the difference of level for any distance ; that is, the difference between the sum of the heights at the back stations and at the forward stations is equal to the difference of level between the first sta- tion and the last. (170.) The following is a copy of the field notes for running a level from A to E : Back sights. Feet. Inches. 4 5 10 Fore sights. Feet. Inches. 3 2 2 6 11 7 1 Sum 31 5 .am 20 7 The back sights being greater in amount than the forward sights, it is evident that E is higher than A by 10 feet 10 inches. The heights indicated by the leveling staves are sometimes read off by the assistant, but it is better for the observer to read off the quantities himself through the telescope of his leveling instrument. This may easily be done provided the graduation of the staff is perfectly iistinct ; and in that case ii 122 TRIGONOMETRY. is only necessary to rely upon the assistant to hold the staff perpendicularly. To enable him to do this, a small plummet is suspended in a groove cut in the side of the staff. (171.) It must be observed that the lines GrF, HI are linei of apparent level, and not of true level; nev- ertheless, we shall ob- tain the true differ- ence of level between A and E by this method if the leveling instrument is placed midway between the leveling staves, because the points Gr and F will in that case be at equal distances from the earth's cen - ter. If the level is not placed midway between the staves, then we must apply a correction for the difference between the true and apparent level. (172.) To find the difference between the true and apparent level. Let C be the center of the earth, AB a portion of its surface, and AD a tangent to the earth's surface at A ; then BD is the difference between the D true and apparent level for the distance AD. Now, by Greom., Prop. 11, B. IV., Hence and CD = VAC 2 +AD 2 , BD-v / AC 8 +AD 2 -BC. If we put R = BC, the radius of the earth, and A=BD, the difference between the true and ap- parent level, we shall have that is, to find the difference between the true and apparent level for any distance, add the square of the distance to the square of the earth's radius, extract the square root of thf. sum, and subtract the radius of the earth. If BD represent a mountain, or other elevated object, then AD will represent the distance at which it can be seen in con- sequence of tho curvature of the earth. SURVEYING. 125 Ex 1. If the diameter of the earth be TD12 miles, and if Mount ^Etna can be seen at sea 126 miles, what is its height? Ans.y 2 miles. Ex. 2. If a straight line from the summit of Chimborazo touch the surface of the ocean at the distance of 179 miles. what is the height of the mountain ? Ans., 4.05 miles. From the preceding formula we obtain that is, tf But in the common operations of leveling, h is very small in comparison with the radius of the earth, and A 2 is very small in comparison with 2R/L If we neglect the term h\ we have / whence h ^-; xilt that is, the difference between the true and apparent level is nearly equal to the square of the distance divided by the di ameter of the earth. Ex. 1. What is the difference between the true and apparent level for one mile, supposing the diameter of the earth to be 7912 miles? Ans., 8.008 inches, or 8 inches nearly. Ex. 2. What is the difference between the true and apparent level for half a mile ? Ans., 2 inches. d' In the equation h = ^o", since 2R is a constant quantity, h varies as d 2 ; that is, the difference between the true and ap- parent level varies as the square of the distance. Hence, the difference for 1 mile being 8 inches, Ft In the difference for 2 miles is 8x2'= 32 inches= 2 8. " " 3 " 8x3'= 72 " = 6 0. " " 4 " Sx4 2 =128 " =10 8. 5 " 8x5 2 =200 " =16 8. 6 8x6 2 =288 -24 0, &CL Topographical Maps. (173.) It is sometimes required to determine and represent upon a map the undulations and inequalities in the surface of 124 TRIGONOMETRY. a tract of land, Such a map should give a complete view oi the ground, so as to afford the means for an appropriate loca tion of buildings or extensive works. For this purpose, \va suppose the surface of the ground to he intersected by a num her of horizontal planes, at equal distances from each othei The lines in which these planes meet the surface of the ground, being transferred to paper, will indicate the variations in the inclination of the ground ; for it is obvious that the curves will be nearer together or further apart, according as the ascent is steep or gentle. Thus, let ABCD be a tract of broken ground, divided by a stream, EF, the ascent being rapid on each bank, the ground swelling to a hill A E at Gr, and also at H. It is required to represent these inequalities upon paper, so as to give an exact idea of the face of the ground. The low- est point of the ground is at F. Suppose the tract to be intersected C ^ D by a horizontal plane four feet above F, and let this plane in- tersect the surface of the ground in the undulating lines marked 4, one on each side of the stream. Suppose a second horizon- tal plane to be drawn eight feet above F, and let it intersect the surface of the ground in the lines marked 8. Let other horizontal planes be drawn at a distance of 12, 16, 20, 24, &o., feet above the point F. The projection of these lines of te\el upon paper shows at a glance the outline of the tract, We perceive that on the right bank of the stream the ground rises more rapidly on the upper than on the lower portion of the map, as is shown by the lines of level being nearer to on another. On the right bank of the stream the ascent is unin terrupted until we reach Gr, which is the summit of the hill. Beyond G the ground descends again toward B. On Ihe left oank of the stream the ground rises to H ; but toward A the level line of 12 feet divides into two branches, and between them the ground is nearly level. (174.) The surveys requisite for the construction of such a map may be made with a theodolite or common level. The object is to trace a series of level lines upon the surface of the ground. For this purpose we may select any point on the surface of a hill, place the level there, and run a level line around the hill, measuring the distances, and also the angles, at every change of direction. "We may then select a second point at any convenient distance above or below the former, and trace a second level line around the hill, and so on for as many curves as may be thought necessary. Such a method, however, would not always be most convenient in practice. (175.) The following method may sometimes be preferable: Set up the level on the summit of the hill at Gr, and fix the vane on the leveling staff at an elevation of four feet in addition to the height of the telescope above the ground. Then direct an assistant to carry the leveling staff, holding it in a vertical posi- tion, toward K, till he arrives at a point, as a, where the vano appears to coincide with the cross wires of the telescope. This will determine one point of the curve line four feet be- low G-. The assistant may then proceed to the line GrB, and afterward to GrL, moving backward or forward in each of those directions till he finds points, as d and g-, at which the vane coincides with the cross wires of the telescope. The horizonta] distance between Gr and a, Gr and d, Gr and g*, must then be measured . If the leveling staff is sufficiently long, the vane may be fixed on it at the height of eight feet, in addition to the height of the telescope at GT ; and the assistant, placing himself in the directions GrK, GrB, GL, must move till the vane appears to coincide with the cross wires as before. The horizontal dis- tances ab, de, gh, must then be measured, and stakes driven into the ground at b, e, and h. The level must now be removed to b ; and the vane being fixed on the staff at a height equal to four feet, together with the height of the instrument from the ground at b, the as- sistant must proceed in the direction Z>K, and stop at c T R I G N M E T R Y the vane coincides with the cross wires ; then thb horizonta distance jf c from A. E & must be meas- ured. In a like manner, the op- erations may be continued from b or c as far as nec- essary toward K ; then, commenc- ing at e, and aft- erward at /*, they may be continu- ed in the same u jr D way toward B and L respectively. The angles which the di- rections GK, GB, GL make with the magnetic meridian being found with the compass, these directions may be represented on paper. Then the measured distances G#, ab, &c. ; Gt/, de, &c. ; G*, gli, &c., being sot off on those lines of direction, ourves drawn through , (/, g- ; b, e, h ; c, /, &, &c., will show the contour of the hill. The map is shaded so as to indicate the hills and slopes by drawing fine lines, as in the figure, perpendicular to the hori- zontal curves. (176.) Another method, which may often be more conven- ient than either of the preceding, is as follows : From the sum- mit of the hill measure any line, as GK, and at convenient points of this line let stakes be driven, and their distances from G be carefully measured. Then determine the difference of level of all these points ; and if the assumed points do not fall upon the horizontal curves which are required to be delineated, we may, by supposing the slope to be uniform from one stake f> another, compute by a proportion the points where the hori- zontal curves for intervals of four feet intersect the line GK. The same may be done for the lines GB and GL, and for othe lines, if they should be thought necessary. (177.) If the surface of the ground is gently undulating, it rnay be more convenient to run across the tract a number of lines parallel to one another. Drive stakes at each extremity SURVEYING. 12* oi these lint^, and also at all the points along them wrier* there is any material change in the inclination of the ground, and find the difference of level between all these stakes, and their distances from each other. Then, if we wish to draw upon a map the level lines at intervals of 4, 6, or 10 feet, we may compute in the manner already explained the points where the horizontal curves intersect each of the parallel lines. The curve lines are then to be drawn through these points, ac cording to the judgment of the surveyor. (178.) If it is required to draw a profile of the ground, i ** example, from vl to K, draw a straight line, G'K, to represent a horizontal line to which the heights are referred, and set off GV, Gr'b' t G-'c', &o., equal to the distances of the stations K ' ^' ^ G from the beginning of the line. At the points G', a', b f , &c., erect perpendiculars, G'G, a' a, &c., and make them equal to the heights of the respective stations. Through the tops of these perpendiculars draw the curved line GK, and it will b3 the profile of the hill in the direction of the line GK. On setting- out Rail-way Curves. (179.) It is of course desirable that the line of a rail-way should be perfectly straight and horizontal. This, however, ia seldom possible for any great distance ; and when it becomes necessary to change the direction of the line, it should be done gradually by a curve. The curve almost universally employ, ed for this purpose is the arc of a circle, and such an arc may '^e traced upon the ground by either of the following methods. First Method. "When the center of the circle can be seen trorn every part of the curve. Let AB, CD be two straight portions of a road which it is desired to connect by an arc of a circle. Set up a theodolite at B and another at C, and from each point range a line at right angles to the lines AB and CD respectively ; and at the inter- section of these lines, E, which will be the center of the circle, erect a signal which can be seen from any point between B and C. The stations must be so chosen that BF equals CF; then 128 TRIGONOMETRY. 4F on th^se linos drive stakes at equal distances, ,, a 2 , 3 , mencing from the points B and C. If r represents the radius A of the circle, and d the distance between the points ,, a a , # 3 , &c., then (Art. 172), will be the distance which must be set off from the first point a,, in the direction fl^E, to ob- tain a point of the circular arc. In like manner, will be the distance to be set off from the point o 2 , in the ai rection & 2 E ; and, generally, will be the distance to be set off at the nth points from B and C. For example, let r be one mile, or 5280 feet, and d equa to 100 feet ; then, V5280 2 +100'-52SO=.94 feet, will be the distance a l b l . In a similar manner, we find at 3 , or 200 feet from B, the offset will be 3.79 feet o 3 , or 300 " " " 8.52 " a,, or 400 " " " 15.13 " 3,or500 fi , 23.62 (180.) Second Method. When A the center of the circle can not be seen from every part of the curve, the offsets may be set off perpendic- ularly to the tangent BF, in which case they must be computed from the formula For, in the annexed figure, that is, EH =/?=?. SURVEYING. 12^ If r=5280 feet, we shall find the offsets at intervals of 100 feet to be a \^\~ -95 feet. a a b 2 = 3.79 " a 3 b~ 3 = 8.53 A'= 15.17 " a A =23.73 For small distance*., the offsets will be given with sufficient accuracy by the formula see Art. 172. It is very common for surveyors, after they have found tho first point, b , , of the curve, to join the points B, b , , and produce the line BZ> l to the distance d, and from the end of this line set off an offset to determine the point b 2 ; then, producing the line #,& 2 , set off a third offset to determine the point & 3 , and so on. The objection to this method is, that any error com- mitted in setting out one of the points of the curve will occa- sion an error in every succeeding one. Whenever this method, therefore, is employed, it should be checked by determining the position of every fourth or fifth point by independent compu- tation and measurement. (181.) Third Method. Where the radius of the curve ia MPp.ll, nbce a theodolite at B, and point its telescope toward C. Place another theodolite at C, and point its telescope to- ward E, the point of intersection of the lines AB, CD produced. Then, if the former be moved through any number of degrees toward a , , and the latter the same number of degrees toward a , , the point a , will be a point of the curve, for the angle 80,0 will be equal to BCD (Geom., Prop. 16, B. III). In the same manner, a a , a 3 , &c., any number of points cf the curve, *nay be determined. It will be most convenient to move the I TRIGONOMETRY. theodolites each time through an even number of degrees, foi example, an arc of two degrees, and a stake must be driven at each of the points of intersection a l , a 2 , & 3 , &c. The ac- curacy of this method is independent of any undulations in the surface of the ground, so that in a hilly country this method may be preferable to any other. When the position of one end of the curve is not absolutely determined, the engineer may proceed more rapidly. Suppose it is required to trace an arc of a circle having a curvature of two degrees for a hundred feet. Place a theodolite at C, the point where the curve commen- ces, and lay off from the line CE, toward B, an angle of two degrees, and in the direction of the axis of the instrument set off a distance of 100 feet, which will give the first point a 1 oi the curve. Next lay off from CE an angle of four degrees, and from a l set off a distance of 100 feet, and the point where this line cuts the axis of the instrument produced will be the sec- ond point a 2 . In the same manner, lay off from CE an angle of six degrees, and from a 2 set off a distance of 100 feet, and the point where it cuts the axis of the instrument produced will be the third point a z . All the points a, , a 2 , # 3 , etc., thus determined lie in the circumference of a circle (Geom., Prop. 15, B. III.). Circles thus drawn are generally made with a curvature of one or two degrees, or some convenient fraction of a degree, for every hundred feet. This method is very ex tensively practiced in the United States. Surveying 1 Harbors. (182.) In surveying a harbor, it is necessary to determine the position of the most conspicuous objects, to trace the out- line of the shore, and discover the depth of water in the neigh- hnrlicod of the channel. A smooth, level piece of ground is SURVEYING. chosen, on which a base line of considerable length is meas- ured, and station staves are fixed at its extremities. "We also erect station staves on all the prominent points to be surveyed, forming a series of triangles covering the entire surface of the harbor. The angles of these triangles are now measured with a theodolite, and their sides computed. After the principal p ints have been determined, subordinate points may be ascer- t incd by the compass or plane table. Let the following figure be a map of a harbor to be survey- ed. "We select the most favorable position for a base line, which is found to be on the right of the harbor, from A to B. "We also erect station flags at the points C, D, E, F, and G. Having carefully measured the base line AB, we measure the three angles of the triangle ABC, which enables us to compute the remaining sides. "We then measure the three angles of the triangle ACD, and by means of the side AC, just computed, we are enabled to compute AD and CD. "We then measure the three angles of the triangle CDF, and by means of the side CD, just found, we are enabled to compute CF and DF. Pro- ceeding in the same manner with the triangles CEF, DFGr, we are enabled, after measuring the angles, to compute the sides. (183.) Having determined the main points of the harbor, we nay proceed to a more detailed survey by means of the chain 132 TRIGONOMETRY. B and compass. If it is required to trace the shore, TICK, wa commence at H, and observe the bearings with the compass-, and measure the distances with the chain. Where the shjre is undulating, it is most convenient to run a straight line fo* a considerable distance, and at frequent intervals measure off- sets to the shore. When a great many objects are to be represented upon a map, the most convenient instrument is The Plane Table. (184.) The plane table is a board about sixteen inches square, designed to receive a sheet of draw- ing paper, and has two plates of brass upon op- posite sides, confined by screws, for stretching and retaining the paper upon the board. The margin of the board is divided to 360 degrees from a cen- ter C, in the middle of the board, and these are subdivided as minutely as the size of the table will admit. On one side of the board there is usually a diagonal scale of equal parts. A compass box is sometimes attached, which renders the plane table capable of answering the purpose of a survey- or',^ compass. The ruler, A, is made of brass, as long as the diagonal of the table, and about two inches broad. A perpendicular sight* vane, B, B, is fixed to each extremity of the ruler, and the eye looking through one of them, the vertical thread in the other is made to- bisect any required distant object. To the under side of the table, a center is attached with a ball and socket, or parallel plate screws, like those of the the- udolite, by which it can be placed upon a staff-head ; and the table may be made horizontal by meins of a detached spirit level SURVEYING 133 (185.) To prepare the table for use, it mast be covered with arawing paper. Then set up the instrument at one of the stations, for example, B (see fig. on p. 131), and fix a needle in the table at the point on the paper representing that station, and place the edge of the ruler against the needle. ; Then di- rect the sights to the station A, and by the side of the ruler draw a line upon the paper to represent the direction of AB. Then, with a pair of dividers, take from the scale a certain number of equal parts to represent the base, and lay off this distance on the base line. Having drawn the base line, move the ruler around the needle, direct the sights to any object, as L, and keeping it there, draw a line along the edge of the ru- ler. Then direct the sights in the same manner to any other objects which are required to be sketched, drawing lines in their respective directions, taking care that the table remains steady during the operation. Now remove the instrument to the other extremity of the base A, and place the point of the paper corresponding to that extremity directly over it. Place the edge of the ruler on the base line, and turn the table about till the sights are directed to the station B. Then placing the edge of the ruler against the needle, direct the sights in succession to all the objects ob served from the other station, drawing lines from the point A in their several directions. The intersections of these lines with those drawn from the point B will determine the posi- tions of the several objects on the map. In this manner the plane table may be employed for filling in the details of a map ; setting it up at the most remarkable spots, and sketching by the eye what is not necessary should be more particularly determined, the paper will gradually be- come a representation of the country to be surveyed. To determine the Depth of Water. (186.) Let signals be established on the principal shoals ami along the edges of the channel, by erecting poles or anchoring buoys, and let their bearings be observed from two stations of the survey. Then in each triangle there will be known one side and the angles, from which the other sides may be com- puted, and their positions thus become known. Then 124 TRIGONOMETRY.. tain the precise depth of water at each of the buoys, ana pr<~ ceed in this manner to determine as many points as may be thought necessary. If an observer is stationed with a theodolite at each extremi- ty of the base line, we may dispense with the erection of per- manent marks upon the water. One observer in a boat may make a sounding for the depth of water, giving a signal at the same instant to two observers at the extremities of the base line. The direction of the boat being observed at that instant from two stations, the precise place of the boat can be com- puted. In this way soundings may be made with great ex- pedition. There is also another method, still more expeditious, which may afford results sufficiently precise in some cases. Let a boat be rowed uniformly across the harbor from one station tc another, for example, from D to Gr (see fig. on p. 131), and let a series of soundings be made as rapidly as possible, and the instant of each sounding be recorded. Then, knowing the en- tire length of the line DGr, and the time of rowing over it, we may find by proportion the approximate position of the boat at each sounding. If the soundings are made in tide waters, the times of high water should be observed, and the time of each sounding bo recorded, so that the depth of water at high or low tide may be computed. In the maps of the United States Coast Survey, the soundings are all reduced to low-water mark, and the num- ber of feet which the tide rises or. falls is noted upon the map (187.) The results of the soundings may be delineated upon u map in the same manner as the observations of level on page 124. We draw lines joining all those points where the depth of water is the same, for example, 20 feet. Such a line is seen to be an undulating line running in the direction from E to Gr. We draw another line connecting all those points where the depth of water is 40 feet. This line runs somewhat to the east of the former line, but nearly parallel with it. We draw other lines for depths of 60 feet, &e. The lines being thus drawn, a mere glance at the map will show nearly the de| \V of water at any point of the harbor. SOOK V. N A. V I G A T I 6 N. (136.) NAVIGATION is the art of conducting a ship at sea from cue port to another. Tliure are two methods of determining the situation of a vessel at sea. The one consists in finding by astronomical ob- servations her latitude and longitude ; the other consists in measuring the ship's course, and her progress every day from the time of her leaving port, from which her place may be computed by trigonometry. The latter method is the one tn be now considered. (189.) The figure of the earth is nearly that of a sphere, and in navigation it is considered perfectly spherical. The earth's axis is the diameter around which it revolves once a day. The extremities of this axis are the terrestrial pole* ; one is called the north pole, and the other the south pole. The equator is a great circle perpendicular to the earth's axis. Meridians are great circles passing through the poles of the earth. Every place on the earth's surface has its own meridian. (190.) The longitude of any place is the arc of -the equator intercepted between the meridian of that place and some as- sumed meridian to which all others are referred. In most countries of Europe, that has been taken as the standard me- ridian which passes through their principal observatory. The English reckon longitude from the Observatory of Greenwich ; and in the United States, we have usually adhered to the En- glish custom, though we believe the time has come when longi- tude should be reckoned from the Observatory of Washington. Longitude is usually reckoned east and west of the first me ridian, from to 180. The difference of longitude of two places is the arc of the equator included between their meridians. It is equal to the 1'JG TRIGONOMETRY difference c f their 'longitudes if they are on the same sido of the first meridian, and to the sum of their longitudes if on op. posite sides. (1.91.) The latitude of a place is the arc of the meridian pass- ing through the place, which is comprehended between that place and the equator Latitude is reckoned north and south of the equator, from IT to 90. Parallels of latitude are the circumferences of small circles parallel to the equator. The difference of latitude of two places is the arc of a me- ridian included between the parallels of latitude passing through those places. It is equal to the difference of their latitudes if they are on the same side of the equator, and to the sum of their latitudes if on opposite sides. The distance is the length of the line which a vessel de- scribes in a given time. The departure of two places is the distance of either place from the meridian of the other. If the two places are OK the same parallel, the departure is the distance between the places. Otherwise, we divide the distance AB into portion? A&, be, cd, &c., so small P that the curvature of the earth may be neglected. Through these points we draw the meridians P, PC, &c.,-and the par- allels be, cf, &c. Then the departure for A.b is eb, for be it is fc ; and the whole departure from A to B is eb+fc+gd+hR ; that is, the sum of the departures corresponding to the small portions into which the distance is divided. Distance, departure, and difference of latitude are measured in nautical miles, one of which is the 60th part of a degree at the equator. A nautical mile is nearly one sixth greater than an English statute mile. The course of a ship is the angle which the ship's pa tli makes with the meridian. A ship is said to continue on the same NAVIGAI ION course when she cut3 every meridian which she crosses at the same angle. The path thus described is not a straight line, but a curve called a rhumb-line. The course of a ship is given by the mariner's compass (192.) The mariner's compass consists of a circular piece ol paper, called a card, attached to a magnetic needle, which is balanced on a pin so as to move freely in any direction. Di rectly over the needle, a line is drawn on the card, one end ol which is marked N, and the other S. The circumference is divided into thirty-two equal parts called rhumbs or points* each point being subdivided into four equal parts called quarter points. The points of the compass are designated as follows, begin- ning at north and go- ing east: north, north by east, north-north- east, northeast by north, northeast, and so on, as shown in the annexed figure. The interval be- tween two adjacent points is 11 15', which is the eighth part of a quadrant. On the inside of the compass-box a black line is drawn perpen- dicular to the horizon, and the compass should be so placea that a line drawn from this mark through the center of the card may be parallel to the keel of the ship. The part of the card which coincides with this mark will then show the point of the compass to which the keel is directed. The compass is suspended in its box in such a manner as to maintain a hori- zontal position, notwithstanding the motion of the ship. The following taije shows the number of degrees am 1 Ro- utes corresponding to each point of the compass ; Y R I G O N O M E T K T North. Pts. Pts. South. N. by E. N.byW. 1 11- 15' 1 S.byE. S.byW. N.N.E. N.N.W. 2 22 30' 2 S.S.E. S.S.W. N.E.byN. N.W.byN. 3 33 45' 3 S.E.byS. SW.byS. N.E. N.W. 4 45 0' 4 S.E. S.W. N.E. by E. N.W.byW. 5 56 15' 5 S.E.byE. S.W.byW. E.N.E. W.N.W. 6 67 30' 6 E.S.E. W.S.W. E.byN. W.byN. 7 78 45' 7 E.byS. W. by S. East. West. 8 90 0' 8 East. West. (193.) The ship's rate of sailing is measured by a log-line. The log-line is a cord about 300 yards long, which is wound round a reel, one end being attached to a piece of thin board called a log. This board is in the form of a sector of a circle, the arc of which is loaded with lead sufficient to give the board a vertical position when thrown upon the water. This is de- signed to prevent the log from being drawn along after the vessel while the line is running off the reel. The time is measured by a sand-glass, through which the sand passes in half a minute, or the 120th part of an hour. The log-line is divided into equal parts called knots, each of which is 50 feet, or the 120th part 'if a nautical mile. Now, since a knot has the same ratio to a nautical mile that half a minute has to an hour, it follows, that if the motion of a ship is uniform, she sails as many miles in an hour as she does knots in half a minute. If, then, seven knots are observed to run off in half a minute, the ship is sailing at the rate of seven miles an hour. PLANE SAILING. (194.) Plane sailing is the method of calculating a ship's place at sea by means of the properties of a plane triangle. The particulars which are given or required are four, viz., the N L V I G A T I N. 139 Hence the difference of lati- distance, course, difference of latitude, and departure. CK these, any two being given, the others may be found. Let the figure EPQ represent a portion of the earth's sur. face, P the pole, and EQ, the equator. Let AB be a rhumb-line, or the track described by a ship in sailing from A to B on a uniform course. Let the whole distance be divided into portions A&, be, &c., so small that the curvature of the earth may be neglected. Through the points of division draw the meridians P&, PC, &c., and the parallels eb, ft, &c. Then, since the course is every where the same, each of the angles ePJ), fbc, &c., is equal to the course. The distances Ae, bf, &c., are the differences of lati- tude of A and , b and c, &c. Also, eb, fc, &c., are the de- partures for the same distances, tude from A to B is equal to Ar+bf+cg+dh, and the departure is equal to eb+fc+gd+hE. Construct the triangle A'B'C' so that A'b'e' shall be equal to A.be, b'c'f shall be equal to bcf, c'd'g' equal to cdg, and d'E'h' equal to t/B/i. Then A'B' represents the distance sail- -A- ed, B'A'C' the course, A'C' the difference of latitude, and B'C the departure ; that is, the distance, dif- ference of latitude, and departure are cor- rectly represented by the hypothenuse and sides of a right-angled triangle, of which the angie opposite to the departure is the course. Of these four quantities, any two being given, the others may be found. Plane sailing does not assume the earth's surface to be a plane, and does not involve any error even in great distances. ,' Departure 110 TR IG OJS'CMETRY. EXAMPLES. 1. A ship sails from Yera Cruz N.E. by If. 74 miles. Re> quired her departure and difference of latitude According to the principles of right-angled triangles, Art. 44, Radius : distance : : sin. course : departure. : : cos. course : diff. latituat, The course is three points, or 33 45' ; hence we obtain Departure =41.11 miles. Diff. latitude=61.53 miles. 2. A ship sails from Sandy Hook, latitude 40 28' N., upon a course E.S.E., till she makes a departure of 500 miles. "What distance has she sailed, and at what latitude has she arrived 1 By Trigonometry, Art. 44, Sin. course : departure : : radius : distance, : : cos. course : diff. latitude- Ans. Distance =541.20 miles. Diff. latitude=207.11 miles, or 3 27'. Hence the latitude at which she has arrived is 37 V N. 3. The bearing of Sandy Hook from Bermuda is N. 42 56' W., and the difference of latitude 486 miles. Required tho distance and departure. By Trigonometry, Art. 46, Radius : diff. latitude : : tang, course : departure* : : sec. course : distance. Ans. Distance =663.8 miles. Departure =452.1 miles. 4. A ship sails from Bermuda, latitude 32 22' N., a distance of 666 miles, upon a course between north and east, until she finds her departure 444 miles. What course has she sailed, and what is her latitude ? By Trigonometry, Art. 44, Distance : radius : : departure : sin. course, Radius : distance : : cos. course : diff. latitude. Ans. Latitude^ 40 38' N. Course =N. 41 49' E 5. The distance from Vera Cruz, latitude 19 12' N., to. Pen. sacola, latitude 30 19' N., is 820 miles. Required the beai- ing and departure. NAVIGATION. Ill By Trigonometry, Art. 45, Distance : radius : : diff. latitude : cos. course, Radius : distance : : sin. course : departure. Ans. Bearing =N. 35 34' E. Departure =476. 95 miles (5. A ship sails from Sandy Hook upon a course between south and east to the parallel of 35, when her departure was "00 miles. Required her course and distance. By Trigonometry, Art. 47, Diff. latitude : radius : : departure : tc^ng. course, Radius : diff. latitude : : sec. course : distance. Ans. Course S. 42 27' E. Distance 444.5 miles TRAVERSE SAILING. (195.) A traverse is the irregular path of a ship when sail- ing on different courses. The object of traverse sailing is to reduce a traverse to a single course, when the distances sailed are so small that the curvature of the earth may be neglected. "When a ship sails on different courses, the difference of latitude is equal to the difference between the sum of the northings and the sum of the southings ; and, neglecting the earth's curvature, the departure is equal to the difference between the sum of the eastings and the sum of the westings. If, then, the difference of latitude and the departure for each course be taken from the traverse table, and arranged in appropriate columns, the difference of latitude for the whole time may be obtained exactly, and the departure nearly, by addition and subtraction ; and the corresponding distance and course may be determined as in plane sailing. EXAMPLES. 1. A ship sails on the following successive tracks: 1. N.B. 23 miles. 2. E.S.E. 45 " 3. E.byK 34 " 4. North 29 5. K by W. 31 6. N.N.E. 17 Find the course and distance for the ivholo traverao. 142 T R I G O N M E T R V. We form a table as below, entering the courses from thn table of rhumbs, page 138, and then enter the latitudes and departures taken from the traverse table. Traverse Table. No. | Course. ~ Distance. N. s. E. w. N. 45 E. 23 16.26 16.26 2 S. 67 30' E. 45 17.22 41.57 3 N. 78 45' E. 34 6.63 33.35 4 North. 29 29.00 5 N. 11 15' W. 31 30.40 6.05 6 N. 22 30' E. 17 15.71 6.51 Sum of columns . . . 98.00 17.22 97.69 6.0,5 17.22 6.05 Diff. latitude . . . . =80.78 N. Dep. =91.64 E. Hence the course is found by plane sailing N. 48 36' E., Rnd the distance =122.2 miles. The proportions are Diff. latitude : radius : : departure : tang, course, Radius : diff. latitude : : sec. course : distance. 2. A ship leaving Sandy Hook makes the following courses and distances : 1. S.E. 25 miles. 2. E.S.E. 32 3. East 17 " 4. E.byS. 51 " 5. South 45 " 6. S.byE. 63 " Required her latitude, the distanco made, and the direct flour se. Ans. Latitude=3S 1' N. Distance =193.7 miles. Course =S. 40 47' E. 3. A ship from Pensacola, latitude 30 19', sails on 1he fol lowing successive coursss : 1 South 48 miles. 2 S.S.W. 23 " a. SAY. 32 NAVIGATION. 4 S.W. by S. 76 miies. 5. West 17 " 6. W.S.W. 54 Required lier latitude, direct course, and distance. .4ras. Latitude- 27 23' N, Course = S.3839'W. Distance =225.0 miles. 4. A ship from Bermuda, latitude 32 22', sails on the fol- lowing successive courses : 1. N.B. 66 miles. 2. N.N.B. 14 3. N.B. by E. 45 " 4. East 21 . 5. E.byN. 32 " Required her latitude, direct course, and distance. Ans. Latitude^ 33 53' N Course =N. 57 22' E Distance =168.4 miles. (196.) When the water through which a ship is moving ha a a progressive motion, the ship's progress is affected in the same manner as if she had sailed in still water, with an additional course and distance equal to the direction and motion of the current. \ Ex. 5. If a ship sail 125 miles N.N.E. in a current which sets W. by N. 32 miles in the same time, required her true course and distance. Form a traverse table containing the course sailed by the ship and the progress of the current, and find the difference of latitude and departure. The resulting course and distance is found as in the preceding examples. Traverse Table. Courses. Distance. N. E. w. ' N. 22 30' E N. 78 45' W. 125 32 115.49 6.24 47.84 31.39 Biff, latitude . . . =121.73 47.84 31.39 31.39 Departure -16.45 E. 144 TRI fi ON o ME TII Y. Hence the course is found by plane sailing -"N. 7 42' E., anc the distance =122.8 miles. Ex. 6. A ship sails S. by E. for two hours at the rate of 9 miles an hour ; then S. by W. for five hours at the rate of 8 miles an hour ; and during the whole time a current sets W. by N. at the rate of two and a half miles an hour, Required the direct course and distance. Ans. The course is S. 21 D 51' W. Distance 57.6 miles. PARALLEL SAILING. (197.) Parallel sailing" is when a ship sails exactly cast ui west, and therefore remains constantly on the same parallel of latitude. In this case the departure is equal to the distance sailed, and the difference of longitude may be found by the fol- lowing THEOREM. The cosine of the latitude of the parallel is to radius, as the distance run is to the difference of longitude. Let P be the pole of the earth, C the center, AB a portkvn of the equator, and DE any parallel of lati- tude ; then will CA be the radius of the equator, and FD the radius of the parallel. Let DE be the distance sailed by the ship on the parallel of latitude, then the difference of longitude will be measured by AB, the arc intercepted on the equator by the merid- ians passing through D and E. Since AB and DE correspond to the equal angles ACB, DFE, they are similar arcs, and are to each oilier as their radii. Hence FD : CA : : arc DE : arc AB. But FD is the sine of PD, or the cosine of AD, that is, the cosine of the latitude, and CA is the radius of the sphere ; hence Cosine of latitude : R : : distance : diff. longitude. Cor. Like portions of different parallels of latitude are if ftAch other as tho cosines of the latitudes. NAVIGATION. 115 The length of a degree of longitude in different parallels may ce computed by thL theorem. A degree of longitude at tha equator being 60 nautical miles, a degree in latitude 40 may be found by the proportion R : cosine 40 : : 60 : 45.96, the required length. The following table is computed in the same manner. (198.) Table showing the length of a degree of longitude (or each degree of latitude. Lat. Miles. Lat. Miles. Latj Miles. Lat. Miles. lat.! Miles. Lat. Miles. 1 59.99 16 57.68 31 51.43 46 41.68 61 29.09 76 14.52 2 59.96 17 57.38 32 50.88 47 40.92 62 28.17 77 13.50 3 59.92 18 57.06 33 50.32 48 40.15 63 27.24 78 12.47 4 59.85 19 56.73 34 49.74 49 39.36 64 26.30 79 11.45 5 59.77 20 56.38 35 49.15 50 38.57 65 25.36 80 10.42 6 59.67 21 56.01 36 48.54 51 37.76 66 24.40 81 9.39 7 59.55 22 55.63 37 47.92 52 36.94 67 23.44 82 8.35 8 59.42 23 55.23 38 47.28 53 36.11 68 22.48 83 7.31 9 59.26 24 54.81 39 46.63 54 35.27 69 21.50 84 6.27 10 59.09 25 54.38 40 45.96 55 34.41 70 20.52 85 5.23 11 58.90 26 53.93 41 45.28 56 33.55 71 19.53 86 4.19 12 58.69 27 53.46 42 44.59 57 32.68 72 18.54 87 3.14 13 58.46 28152.98 43 43.88 58 31.80 73 17.54 88 2.09 14 58.22 29 52.48 44 43.16 59 30.90 74 16.54 89 1.05 15 57.96 30 51.96 45 42.43 60 30.00 75 15.53 90 0.00 Let ABC represent a right-angled triangle ; then, by Trig, onomotry, Art. 41, cos. B : R : : AB : BC. But, by the preceding Theorem, we have cos. lat. : R : : depart. : diff. long., trom which we see that if one leg of a right-angled triangle represent the dis- ' epa tance run on any parallel, and the adjacent acute angle be made equal to the degrees of latitude of that parallel, then the hypothenuse will represent the difference of longitude. EXAMPLES. J. A ship yails from Sandy Hook, latitude 40 28' N., longi- tude 74 1' W., 618 miles due east. Required her present longitude. Cos. 40 28' : R : : 618 : 812'.3=13 32', the dLTomuoe of longitude. K 146 TRIGONOMETRY. This, subtracted irom 74 1', leaves 60 29' W., the longi- tudo required. 2. A ship in latitude 40 saLs due east through nine degrees of longitude. Required the distance run. Ans. 413.66 miles. 3. A ship having sailed on a parallel of latitude 261 miles, finds her difference of longitude 6 15'. What is her latitude ? Ans. Latitude 45 54' 4. Two ships in latitude 52 N., distant from each other 9* miles, sail directly south until their distance is 150 miles What latitude do they arrive at? Ans. Latitude 13 34' MIDDLE LATITUDE SAILING. (199.) By the method just explained may be found the lon- gitude which a ship makes while sailing on a parallel of lati- tude. When the course is oblique, the departure may be found by plane sailing, but a difficulty is found in converting ihia departure into difference of longitude. If a ship sail from A to B, the departure is equal to eb + fc , which is less than AC, but p greater than DB. Navigators have as- sumed that the departure was equal to the distance between the meridians PA, PB, measured on a parallel EF, equidis- tant from A and B, called the middle lati- tude. The middle latitude is equal to half the sum of the two extreme latitudes, if both are north or both south ; but to half their difference, if one is north and the other south. The principle assumed in middle latitude sailing is not per fectly correct. For long distances the error is considerable; but the method is rendered perfectly accurate by applying to the middle latitude a correction which is given in the accom panying tables, page 149. (200.) It has been shown that when a ship sails upon ac oblique course, the distance, departure, and difference of lati tude may be represented by the sides of a right-angled trian NAVIGATION. 147 gle. The difference of longitude is derived from the departure, in the same manner as in parallel sailing, the ship being sup- posed to sail on the middle latitude parallel. Hence, if we combine the triangle ABC for plane sailing with the triangle BCD for parallel sailing, we shall obtain a triangle ABD, by which all the cases of middle latitude sailing may be solved. In the triangle BCD, Cos. CBD :BC : : R : BD ; that is, cosine of middle latitude is to the departure, as radius is to the difference of longitude. In the triangle ABD, since the angle D is the complement of CBD, which represents the middle latitude, we have Sin. D : AB : : sin. A :*BD; that is, cosine of middle latitude is to the distance, as the sine of the course is to the difference of longitude. In the triangle ABC, we have the proportion AC : BC : : R : tang. A. But we have before had the proportion Cos. CBD : BC : : R : BD. The means being the same in these two proportions, we hav* Cos. CBD : AC : : tang. A : BD ; that is, cosine of middle latitude is to the difference of lati> tude, as the tangent of the course is to the difference of lon- gitude. The middle latitude should always be corrected according to the table on page 149. The given middle latitude is to be looked for either in the first or last vertical column, opposite to which, and under the given difference of latitude, is inserted the proper correction in minutes, which must be added to the middle latitude to obtain the latitude in which the meridian distance is exactly equal to the departure. Thus, if the mid- : 'le latitude is 41, and the difference, of latitude 14, the cor- rection will be found to be 25', which, added to the middk 'atituds, gives th<5 corrected middle latitude 41 25'. 148 TRIGONOMETRY. EXAMPLES. 1. Find the bearing and distance of Liverpool, latitude 53* 1 22' N., longitude 2 52' W., from New York, latitude 40 42' N M longitude 74 1' W. Here are given two latitudes and longitudes to find tht course and distance. The difference of latitude is , . . . 32 40' = The difference of longitude is .... 71 The middle latitude is 47 To which add the correction from p. 149 The corrected middle latitude is ... 47 C Then, according to the third of the preceding theorems, Diff. lat. : cos. mid. lat. : : diff. long'. : tang: cour. r e='N. 75 16' E To find the distance by plaiie sailing, Cos. course : diff. latitude : : R : distance=298SA miles 2. A ship sailed from Bermuda, latitude 32 22' N., longi. aide 64 38' W., a distance of 500 miles, upon a course W.N W Required her latitude and longitude at that timo. By plane sailing, R : distance : : cos. course : diff. latitude^ 191.3. Therefore the required latitude is 35 33' , the middle latitude ....... 33 58' ; and the corrected middle latitude . . . . . 33 59 Then we have Cos. mid. lat. : distance : : sin. course : diff. long-. =557'.!. Therefore the longitude required is 73 55'. 3. A ship sails southeasterly from Sandy Hook, latitude 40 28' N., longitude 74 1' W., a distance of 395 miles, when her latitude is 34 40' N. Required her course and longitude Ans. Course S. 28 14' E. Longitude 70 5' W. 4. A ship sails from Brest, latitude 48 23' N., longitude 4 29' W., upon a course W.S.W., till her departure is 556 miles. Required the distance sailed and the place of the ship. Ans. Distance 601.8 miles. Latitude 44 33' N. Longitude 17 57' 'VV. NAVIGATION. MERCATOR'S SAILING (201.) Mercator's sailing is a method of computing differ- ence of longitude on the principles of Mercator's chart. On this chart, the meridians, instead of converging toward the poles as they do on the globe, are drawn parallel to each other, by which means the distance of the meridians is every when* 150 12,0 QO* 60* 30 0" SO" 60* .90 i5o 60' 43 30' 15 15 ' 30' 45 75 made too great except at the equator. To compensate foi this, in order that the outline of countries may not be toe much distorted, the degrees of latitude are proportionally en- larged, so that the distance between the parallels of latitude increases from the equator to the poles. In latitude 60 the distance of the meridians is twice as great, compared with a degree at the equator, as it is upon a globe, and a degree of latitude is here represented twice as great as near the equator The diameter of an island in latitude 60 is represented twice as great as if it was on the equator, and its area four times too great. In latitude 70 32' the distance of the meridians is three times too great, in latitude 75 31' four times too great and so on, by which means the relative dimensions of coun- tries in high latitudes is exceedingly distorted. On this ac- count it is not common to extend the chart beyond latitude 75. (202.) The distance of any parallel upon Mercator's chart from the equator has be^n computed, and is exhibited in tha 150 TRIGONOMETRY. accompanying tables, pages 142-8, which is called a Table of Meridional Parts. This table may be computed in the fol lowing manner : According to Art. 197, cosine of latitude is to radius, as the departure is to the difference of longitude ; that is, as a part of a parallel of latitude is to a like part of the equator, or any meridian. But by Art. 28, cosine : E- : : R : secant ; hence V of a parallel : V of a meridian : : K : sec. latitude. But on Mercator's chart the distance between the meridians is the same in all latitudes ; that is, a minute on a parallel of latitude is equal to a minute at the equator, or a geographical mile. Hence the length of one minute, on any part of a me* ridian, is equal to the secant of the latitude. Thus, The first minute of the meridian = the secant of 1' ; second " " = 2', third " " " 3', &c., &c. The table of meridional parts is formed by adding togetlni the minutes thus found. Thus, Mer. parts of l'=sec. 1' ; Mer. parts of 2'= sec. l'+ec. 2' ; Mer. parts of 3' see. I'+sec. 2'-fsec. 3' ; Mer. parts of 4'=sec. I'+sec. 2'+sec. 3'+sec. 4', &c., &c., &c. Since the secants of small arcs are nearly equal to radius or unity, if the meridional parts are only given to one tenth nf a mile, we shall have The meridional parts of l'=1.0 mile ; <: 2'=2.0 " it a 3'=3.0 " " " " 4'=4.0 " &o., as shown in the table on page 142. At 2 33' the sum of the small fractions omitted becomes greater than half of one tenth, and the meridional parts ol 2 33' is 153.1 ; that is, the meridional parts exceed by ona tenth of a mile the minutes of latitude. At 3 40' the excess ia two tenths of a mile ; at 4 2 1 7 the excess is three tenths - NAVIGATION. 151 and as the latitude increases, the meridional pares increase more rapidly, as is seen from the table. An arc of Mercator's meridian contained between two par- allels of latitude is called meridional difference of latitude, It is found by subtracting the meridional parts of the less latitude from the meridional parts of the greater, if both are north or south, or by adding them together if one is north and the other south. Thus, The lat. of New York is 40 42' ; meridional parts=2677.8, < New Orleans 29 57' : " 1884.9. The true diff. of lat. is 10 45' ; mer. diff. lat. is 792.9 If one latitude and the meridional difference of latitude bj given, the true difference of latitude may be found by reversing this process. Thus, The meridional parts for New Orleans . . . =1884.9. Meridional difference of latitude between New York and New Orleans Therefore the meridional parts for New York =2677.8, /ind the corresponding latitude from the table is 40 42'. (203.) If we take the figure ABC for -i- ?on i E Diff. Long. plane sailing, as on page 139, and pro- duce AC to E, making AE equal to the meridional difference of latitude, then will DE rp present the difference of lon- gitude corresponding to the departure BC. FPT we have seen (Art. 202) that the dr parture is to the difference of lon- gitude as radius is to the secant of lati- tude, which is also the ratio of the true difference of latitude to the meridional difference of latitude. Now, from the similarity of the triangles ABC, ADE, we have AC : AE : : BC : DE ; that is, the true difference of latitude is to the meridional dif> ference of latitude, as the departure is to the difference of longitude. Also, in 'he triangle ADE, we have E : tan. A : : AE : DE ; 1 52 TRIGONOMETRY. that is, radius is to the tangent of the course, as the ?ncri. 2. Find the bearing and distance from Nantucket Shoals, in latitude 41 4' N., longitude 69 55' W., to Cape Clear, in lati- tude 51 26' N., longitude 9 29' "W. Ans. Course N. 76 E. Distance 2572.9 miles 3. A ship sails from Sandy Hook a distance of 600 miles upon a course S. by E. Required the place of the ship. The difference of latitude may be found by plane sailing, the difference of longitude by Mercator's sailing. Ans. Latitude 30 39'.5 N. Longitude 71 36'.7 W. 4. A ship sails from St. Augustine, latitude 29 52' N., lon- gitude 81 25' "W., upon a course N.E. by E., until her lati- tude is found to be 34 40' N. What is then her longitude, and what distance has she run ? Ans. Longitude =72 55' W. Distance =518.4 miles. 5. A ship sails from Bermuda upon a course N.W. by W until her longitude is found to be 69 30' W. What is then her latitude, and what distance has she run ? Ans. Latitude 35 4' N. Distance 291.6 miles. 6. A ship sailing from Madeira, latitude 32 38' N , long). tude 16 f 5' W., steers westerly until her latitude i* \ f f 2 N.. NATIGATION. L5 and her departure 2425 miles. Required her course, distance find longitude. Ans. Course N. 79 37' W Distance 2465.3 miles. Longitude 67 9'.3 W. 7. Find the bearing and distance from Sandy Hook, latitude 40 28' N., longitude 74 1' W., to the Cape of G-ood Hopp latitude 34 22' S., longitude 18 30' E. Ans. Course Distance CHARTS. (204.) The charts commonly used in navigation are plane charts, or Mercator^s chart. In the construction of the former, the portion of the earth's surface which is represented is sup- posed to be a plane. The meridians are drawn parallel to each other, and the lines of latitude at equal distances. The dis- tance between the parallels should be to the distance between the meridians, as radius to the cosine of the middle latitude of the chart. A chart of moderate extent constructed in this manner will be tolerably correct. The distance of the merid ians in the middle of the chart will be exact, but on each sido it will be either too great or too small. When large portions of the earth's surface are to be repre- sented, the error of the plane chart becomes excessive. To obviate this inconvenience Merca tor's chart has been con- structed. Upon this chart the meridians are represented by parallel lines, and the distance between the parallels of latitude is proportioned to the meridional difference cf latitude, as rep- resented on page 149. We have seen that the meridional difference of latitude is to tha difference of longitude as radius is to the tangent of tho cour.se. Hence, while the course remains unchanged, the ratio of the meridional difference of latitude to the difference of lon- gitude is constant ; and, therefore, every rhumb line will bo represented on Mercator's chart by a straight line. TMs property renders Mercator's chart peculiarly cciavwciO to navigators 1>4 TRIGONOMETRY. The preceding sketch affords a very incomplete view of the present state of the science of navigation. The most accurate method of ascertaining the situation of a vessel at sea is by means cf astronomical observations. For these, however, the t must be referred to some treatise on Astronomy. BOOK VI. SPHERICAL TRIGONOMETRY. (205.) SPHERICAL trigonometry teaches how to determino the several parts of a spherical triangle from having certain parts given. A spherical triangle is a portion of the surface of a sphere, bounded hy three arcs of great circles, each of which is less than a semicircumference. RIGHT-ANGLED SPHERICAL TRIANGLES. THEOREM I. (206.) In any right-angled spherical triangle, the sine of the hypothenuse is to radius, as the sine of either side is to the sine of the opposite angle. Let ABC be a spherical triangle, right-angled at A ; then will the sine of the hypothenuse BC be to radius, as the sine of the side AC is to the sine of the angle ABC. Let D be the center of the sphere ; join AD, BD, CD, and draw CE per- pendicular to DB, which will, there- fore, be the sine of the hypothenuse BC. From the point E draw the straight line EF, in the plane ABD, perpendicular to BD, and join CF. Then, because DB is perpendicular to the two lines CE, EF, it is perpendicular to the plane CEF; and, conse quently, the plane CEF is perpendicular to the plane ABD (Geom., Prop. 6, B. VII.). But the plane CAD is also per- pendicular to the plane ABD ; therefore their line of common section, CF, is perpendicular to the plane ABD ; hence CFD, CFE are right angles, and CF is the sine of the arc AC. Now, in the right-angled plane triangle CFE, CE : radius : CF : sine CEF 156 TRIGONOMETRY. But since CE and FE are both at right angles to DB, the angle CEF is equal to the inclination of the planes CBD, ABD; that is, to the spherical angle ABC. Therefore, sine BC : R : : sine AC : sine ABC. (207.) Cor. 1. In any right-angled spherical triangle, tk* sines of the sides are as the sines of the opposite angles. For, by the preceding theorem, sine BC : R : : sine AC : sine ABC, and sine BC : R : : sine AB : sine ACB ; therefore, sine AC : sine AB : : sine ABC : sine ACB. Cor. 2. In any right-angled spherical triangle, the cosine of zither of the sides is to radius, as the cosine of the hypothenusc is to the cosine of the other side. Let ABC be a spherical triangle, right-angled at A. Do- scribe the circle DE, of which B is the pole, and let it meet the three sides of the triangle ABC produced in D, E, and F. Then, because BD and BE are quadrants, the arc DF is per- pendicular to BD. And since BAC is a right angle, the arc AF is per- pendicular to BD. Hence the point F, where the arcs FD, FA intersect each other, is the pole of the arc BD (Geom.. Prop. 5, Cor. 2, B. IX.), and the arcs FA, FD are quadrants. Now, in the triangle CEF, right-angled at the point E, ac. cording to the preceding theorem, we have sine CF : R : : sine CE : sine CFE. But CF is the complement of AC, CE is the complement of BC, and the angle CFE is measured by the arc AD, which is the complement of AB. Therefore, in the triangle ABC, we have cos. AC : R : : cos. BC : cos. AB. Cor. 3. In any right-angled spherical triangle, the cosine of either of the sides is to radius, as the cosine of the angle opposite to that side is to the sine of the other angle. For, in the triangle CEF, we have sine CF : R : : sine EF : sine ECF. But sine CF is rqual to cos. CA. EF is the complement at SPHERICAL TRIGONOMETRY. 1^7 ED, which measures the angle ABC, that is, sine EF is equa, to cos. ABC, and sine ECF is the same as sine ACB ; there- fore, cos. AC : R : : cos. ABC : sine ACB THEOREM II. (208.) In any right-angled spherical triangle, the sine oj either of the sides about the right angle is to the cotangent of the adjacent angle, as the tangent of the remaining side is to radius. Let ABC "be a spherical triangle, right-angled at A ; then will the sine of the side AB be to the cotangent of the angle ABC, as the tangent of the side AC is to radius. Let D he the center of the sphere ; join AD, BD, CD ; draw AE perpen- dicular to BD, which will, therefore, be the sine of the arc AB. Also, from the point E in the plane BDC, draw the straight line EF perpendicular to BD, meeting DC produced in F, and join AF. Then will AF "be perpendicular to the plane ABD, because, as was shown in the preceding theorem, it is the com- mon section of the two planes ADF, AEF, each perpendiculai to the plane ADB. Therefore FAD, FAE are right angles, and AF is the tangent of the arc AC. "Now, in the triangle AEF, right-angled at A, we have AE : radius : : AF : tang. AEF. But AE is the sine of the arc AB, AF is the tangent of ti, -, arc AC, and the angle AEF is equal to the inclination of tH planes CBD, ABD, or to the spherical angle ABC ; hence sine AB : R : : tang. AC : tang. ABC. And "because, Art. 28, R : cot. ABC : : tang. ABC : R ; therefore, sine AB : cot. ABC : : tang. AC : R. (209.) Cor. 1. In any right-angled spherical triangle, thi cosine of the hypothenuse is to the cotangent of cither of ikt oblique angles, as the cotangent of the other oblique ang ' i. to radius lf>8 TRIGONOMETRY. Let ABC be a spherical triangle, right-angled at A. scribe the circle DEF, of which B is the pole, and construct the com- plemental triangle CEF, as in Cor. 2, Theorem I. Then, in the triangle CEF, ac- cording to the preceding theorem, we tiave sine CE : cot. ECF : : tan. EF : R. But CE is the complement of BC, EF is the complement of ED, the measure of the angle ABC ; and the angle ECF is equal to ACB, being its vertical angle; hence cos. BC : cot. ACB : : cot. ABC : R. Cor. 2. In any right-angled spherical triangle, the cosine of either of the oblique angles is to the tangent of the adjacent side, as the cotangent of the hypothenuse is to radius. For, in the complemental triangle CEF, according to tho preceding theorem, we have sine EF : cot. CFE : : tan. CE : R; hence, in the triangle ABC, cos. ABC : tan. AB : : cot. BC : R. Napier' s Rule of the Circular Parts. (210.) The two preceding theorems, with their corollaries, are sufficient for the solution of all cases of right-angled spheri- cal triangles, and a rule was invented by Napier by means of which these principles are easily retained in mind. If, in a right-angled spherical triangle, we set aside the right angle, and consider only the five remaining parts of the trian- gle, viz., the three sides and the two oblique angles, then the two sides which contain the right angle, and the complements of the other three, viz., of the two angles and the hypothenuse, are called the circular parts. Thus, in the triangle ABC, right-angled at A, the circular parts are AB, AC, with the complements of B, BC, and C. When, of the five circular parts, any on'e is taken for the middle part, then, of the remaining four, the two which are immediately adjacent to it on the right and left are called the SPHERICAL TRIGONOMETRY. 159 adjacent parts ; and the other two, each of which is separated from the middle by an adjacent part, are called opposite parts. In every question proposed for solu- tion, three of the circular parts are concerned, two of which are given, and one required ; and of these three, the middle part must be such that the other two may be equidistant from it; that is, may be either both adjacent or both opposite parts. The value of the part required may then be found by the following RULE OF NAPIER. (211.) The product of the radius and the sine of the middle part, is equal to the product of the tangents of the adjacent parts, or to the product of the cosines of the opposite parts. It will assist the learner in remembering this rule to remark, that the first syllable of each of the words tangent and adja- cent contains the same vowel a, and the first syllable of tho words cosine and opposite contains the same vowel o. It is obvious that the cosine of the complement of an angle is the sine of that angle,- and the tangent of a complement is a cotangent, and vice versa. In the triangle ABC, if we take the side b as the middle part, then the side c and the complement of the angle C are the adjacent parts, and the complements of the angle B and of the hypothenuse a are the opposite parts. Then, according to Napier's rule, R sin. =tan. c cot. C, which corresponds with Theorem II. Also, by Napier's rule, R sin. &=sin. a sin. B, which corresponds with Theorem I. Making each of the circular parts in succession tke muldla part, we obtain the ten following equations : R sin. #=sin. a sin. B=tan. c cot. C. R sin. c=sin. a sin. C=tan. b cot. B. R cos. B= cos. b sin. C = cot. a tan. c. R cos. a =co8. b cos. c cot. B cot. C. R cos. C = cos. c sin. B = oot. a tan. ft. tGO TRIGONOMETRY. (212.) In order to determine whether the quantity sought is less 01 greater than 90, the algebraic sign of each term should be preserved whenever one of them is negative. If the quantity sought is determined by means of its cosine, tangent, or cotangent, the algebraic sign of the result will show whether this quantity is less or greater than 90 ; . for the cosines, tan- gents, and cotangents are positive in the first quadrant, and negative in the second. But since the sines are positive in both the first and second quadrants, when a quantity is determined by means of its sine, this rule will leave it ambiguous whether ihe quantity is less or greater than 90. The ambiguity may. however, generally be removed by the following rule. In every right-angled spherical triangle, an oblique angle and its opposite side are ahvays of the same species ; that is, both are greater, or "both less than 90. This follow? from the. equation R sin. &=tan. c cot. C ; where, sirr// sin. b is always positive, tan. c must always have the same sign as cot. C ; that is, the side c and the opposite angle C both belong to the same quadrant. (213.) When the given parts are a side and its opposite an- gle, the problem admits of two solutions ; for two right-angled spherical triangles may always be found, having a side and its opposite angle the sama' in both, but of which the remaining sides and the remaining angle of the one are the supplements of the remaining sides and the remaining angle of the other. Thus, let BCD, BAD be the halves of two great circles, and let the arc CA be drawn perpendicu- lar to BD ; then ABC, ADC are two right-angled triangles, having the side B AC common, and the opposite angle B equal to the angle D; but the side DC is the supplement ol BC, AD is the supplement of AB, and the angle ACD is trie supplement of ACB. EXAMPLES. 1, In the right-angled spherical triangle ABC, there are given a = 63 56' and & 40. Required the other sile c, anr' the angle? B and C. {SPHERICAL TRIGONOMETRY. 161 Gild the side c the circular parts concerned are the two legs and the complement of the hypothenase; and it is evident that if the complement of a be made the mid- dle part, b and c will be opposite parts ; hence, by Napier's rule, R cos. acos, b cos. c ; or, reducing this equation to a proportion, cos. b : R : : cos. a : cos. c=54 59' 49". To find the angle B. Here b is the middle part, and the complements of B and u are opposite parts ; hence R sin. b=cos. (comp. a)Xcos. (comp. B)=sin. a sin. B, or sin. a : R : : sin. b : sin. B=45 41' 25". B is known to be an acute angle, because its opposite side is less than 90. To find the angle C. Here the complement of C is the middle part; also b and the complement of a are adjacent parts ; hence R cos. C=cot. a tan. b, or R : tan. b : : cot. a : cos. 0=65 45' 57". Ex. 2. In a right-angled triangle ABC, there are given tho hypothenuse a=91 42', and the angle B 95 6'. Required the remaining parts. To find the angle C. Make the complement of the hypothenuse the middle part; Jien R cos. a=cot. B cot. C. Whence = 71 36' 47". To find the side c. Make the complement of the angle B the middle part ; an 4 we have R cos. B=cot. a tan. c. Whence c=71 32' 14'. To find the side b. Make the side b the middle part ; then L 162 TRIGONOMETRY. R sin. b=sin. a sin. B. Whence =95 22' 30". b is known to be greater than a quadrant, Lecause its opposite angle is obtuse. Ex. 3. In the right-angled triangle ABC, the side b is 20 4', and its opposite angle B 36. Required the remaining parts. ( a =48 22' 52", or 131 37' 8" Am. \ c -42 19' 17", or 137 40' 43". ( C=64 14' 26", or 115 45' 34". This example, it will be seen, admits of two solutions, con- formably to Art. 213. Ex. 4. In the right-angled spherical triangle ABC, there are given the side c, 54 30', and its adjacent angle B, 44 50'. Required the remaining parts. ( C=65 49' 53" Ans. ) a =63 10' 4'. (b =38 59' 11" Why is not the result ambiguous in this case ? Ex. 5. In the right-angled spherical triangle ABC, the side b is 55 28', and the side c 63 15'. Required the remaining narts. (a =7 5 13' 2". Ans. ) B=58 25' 47 ( C = 6727' 1'. Ex. 6. In the right-angled spherical triangle ABC, there are given the angle B=69 20', and the angle C=58 16'. Re- quired the remaining parts. / a=76 30' 37". Ans. ]b = 65 28' 58". (c =55 47' 46". (214.) A triangle, in which one of the sides is equal to a quadrant, may be solved upon the same principles as right- angled triangles, for its polar triangle will contain a right rally proportional to the tangents of the angles at the bas? For, by Theorem II., sin. AD : R : : tan. CD : tan. A. Also, sin. BD : R : : tan. CD : tan. B. Hence sin. AD : sin. BD : : tan. B : tan. A. Cor. 4. The cotangents of the two sides are proportions >.o the cosines of the segments of the vertical angle. For, by Theorem II., Cor. 2, cos. ACD : cot. AC : : tan. CD : R. Also, cos. BCD : cot. BC : : tan. CD : R. Hence cos. ACD : cos. BCD : : cot. AC : cot. BC. THEOREM IV. (217.) If from an angle of a spherical triangle a perpen- (Uvular be drawn to the base, then the tangent of half the sum of the segments of the base is to the tangent of half the sum of the sides, as the tangent of half the difference of the sides is to the tangent of half the difference of the segments of th* base. Let ABC be any spherical trian- gle, and let CD be drawn from C perpendicular to the base AB ; then tan. i(BD+AD) : tan. ^(BC+AC) : : tan. J(BC-AC) : tan. i(BD-AD). Let BC=o, AC =6, BD=m, and ATh=. Then, by Theorem III., Cor. 1, cos. a : cos. b : : cos. m : cos. . SPHERICAL TRIGONOMETRY 163 Whoncs, G-eom., Prop. 7, Cor., B. IT., Gets. 6+C03. a : cos. b cos. a : : cos. w-fcos. m : cos, n cos. m But by Trig. , Art. 76, cos. &+GOS. a : cos. b cos. a : : cot. \(a-\-b) : tan. ^(a &). Also, by the same Art., cos. +cos. m : cos. n cos. m : : cot. J(m+w) : tan. J(i w). Therefore cot. |(#+&) : cot. J(w-r-ft) : : tan. ^(a b) : tan. $(mn). But, since tangents are reciprocally as their cotangents, Art. 88, we have cot. ^(a+b) : cot. \(m-\-ri) : : tan. \(m+n) : tan. ^(a+b) Hence tan. (w+w) : tan. \(a+b) : : tan. \(a b) : tan. %(m n). (218.) In the solution of oblique-angled spherical triangle*, six cases may occur, viz, : 1. (riven two sides and an angle opposite one of them. 2. Griven two angles and a side opposite one of them. 3. Griven two sides and the included angle. 4. Griven two angles and the included side. 5. Given the three sides, ^5. Griven the three angles. CASE I. (219.) Given two sides and an angle opposite one of them^ io find the remaining 1 parts. In the triangle ABC, let there be given the two sides AC and BC, and the angle A opposite one of them. The angle B may be found by Theorem III. sin. BC : sin. AC : : sin. A : sin. B. From the angle C let fall the per- pendicular CD upon the side AB. The triangle ABC is divided into two right-angled triangles, in each of which there is given the hypothenuse and the angla at the base. The remaining parts may then be found by Na- rule. Ex. 1. In the oblique-angled sphericil triangle ABC, the I b'ti TRIGONOMETRY. *ide AC=70 10 30", BC=80 5' 4", and the angle A^53 f 1.5' 7". Required the other parts. sin BC : sin. AC : : sin. A : sin. B=31 34' 38" Then, in the triangle ACD, R cos. AC = cot. A cot. ACD. Whence ACD =77 27' 47". Also, in the triangle BCD, R cos. BC=cot. B cot. BCD. Whence BCD= 83 57' 29". Therefore ACB=161 25' 16". To find the side AB. sin. A : sin. ACB : : sin. BC : sin. AB=145 5' 0". When we have given two sides and an opposite angle, there are, in general, two solutions, each of which will satisfy the conditions of the problem. If the side AC, the angle A, and the side opposite this angle are given, then, with the latter for radius, de- scribe an arc cutting the arc AB in the points B and B'. The arcs CB, CB' will be equal, and each of the tri- angles ACB, ACB' will satisfy the B ' conditions of the problem. There is the same ambiguity in the numerical computation. The angle B is found by means of its sine. But this may be the sine either of ABC, or of its supplement AB'C (Art. 27). In the preceding example, the first proportion leaves it ambiguous whether the angle B is 31 84' 38', or its supplement 148 25' 22". In order to avoid false solutions, we should remember that the greater side of a spherical triangle must lie opposite the greater angle, and conversely (Geom., Prop. 17, B. IX.). Thus, since in the pre- ceding example the side AC is less than BC, the angle B must be less than A, and, therefore, can not be obtuse. If the quantity sought is determined by means of its cosine, tangent, or cotangent, the algebraic sign of the result will show whether this quantity is less or greater than 90 ; for the cosines, tangents, and cotangents are positive in the first quad- rant, and negative in the second. Hence the algebraic sign SPHERICAL TRIGONCMETR\. 16* of each term of a proportion should be preserved whenever ona of them is negative. Ex. 2. In the spherical triangle ABC, the side 0=124 53', 6=31 19', and the angle A=16 26'. Required the remain- ing parts. ( B= 10 19' 34" Ans. } C=171 48' 22" ( c =-155 35' 22' CASE II. (220.) Given two angles and a side opposite one of them^ to find the remaining parts. In the triangle ABC let there be given two angles, as A and B, and the side AC opposite to one of them. The side BC may be found by Theorem III. sin. B : sin. A : : sin. AC : sin. BC. From the unknown angle C draw CD perpendicular to AB ; then will the triangle ABC be divided into two right-angled triangles, in each of which there is given the hypothenuse and the angle at the base. Whence we may proceed by Napier's rule, as in Case I. Ex. 1. In the oblique-angled spherical triangle A.BC, there are given the angle A=52 20', B=63 40', and the sidu /;=S3 25'. Required the remaining parts. sin. B : sin. A : : sin. AC : sin. BC=61 19' 53". Then, in the triangle ACD, cot. AC : R : : cos. A : tan. AD =79 18' 17". Also, in the triangle BCD, cot. BC : R : : cos. B : tan. BD=39 3' 8". Hence AB=118 21' 25". To find the angle ACB. sin. BC : sin. AB : : sin. A : sin. ACB=127 26' 47". When we have given twc angles and an opposite side, there are, in general, two solutions, each of which will satisfy the conditions of the problem. If the angle A, the side AC, and 168 TRIGONOMETRY. the fcngle opposite this side are given, then through the point C there may generally be drawn two arcs of great cir- cles CB, CB', making the same angle with AB, and each of the triangles ABC, AB'C will satisfy the condi- tions of the problem. There is the same ambiguity in the numerical computation, since the side BC is found by means of its sine (Art. 27). In the preceding example, however, there is no ambiguity, because the angle A is less than B, and, therefore, the side a must be less than Z>, that is, less than a quadrant. Ex. 2. In the oblique-angled spherical triangle ABC, the angle A is 128 45', the angle C=3035 / , and BC = 6S50'. Required the remaining parts. It will be observed that in this case the perpendicular BD 5 drawn from the angle B, falls without the triangle ABC, and therefore the side AC is the difference between the segment* CD and AD. ( AB=37 28' 20 ' Am. \ AC -40 9' 4". ( B =32 37' 58". CASE III. (221.) Given two sides and the included angle, to find tk remaining parts. In the triangle ABC let there be given two sides, as AB AC, and the included angle A. Let fall the perpendicular CD on the side AB ; then, by Napier's rule, R cos. A=tan. AD cot. AC. Having found the segment AD, the segment BD becomes known ; then, by Theorem III., Cor. 3, sin. BD : sin. AD : : tan. A : tan. B. The remaining parts may now be found by Theorem III. Ex. 1. In the spherical triangle ABC, the side AB = 73 20 , AC =41 45', and the angle A^30 30'. Required the remain- ing par's SPHERICAL TRIGONOMETRY. 16U cot. AC : cos. A : : R : tan. AD=37 33' 41". Hence . ' BD=35 46' 19". sin. BD : sin. AD : tan. A. : tan. B=31 33' 43". Also, by Theorem III., Cor. 1, cos. AD : cos. BD : : cos. AC : cos. BC-40' J3 X /x . Then, by Theorem III., sin. BC : sin. AB : : sin. A : sin. ACB=131 S' 47' . Ex. 2. In the spherical triangle ABC, the side AB=78 15', A.C=56 20', and the angle A=l?0. Required the othti parts. ( B -=48 57' 29". Ans. } C =62 31' 40". 7' 45' CASE IV. (222.) Given two angles and the included side, to find ttit remaining parts. In the triangle ABC let there be given two angles, as A and ACB, and the side AC included be- tween them. From C let fall the per- pendicular CD on the side AB. Then, by Napier's rule, R cos. AC=cot. A cot. ACD. Havirigr found the angle ACD, the angle BCD becomes known ; then, by Theorem III., Cor. 4, cos. ACD : cos. BCD : : cot. AC : cot. BC. The remaining parts may now be found by Theorem 111. Ex. 1. In the spherical triangle ABC, the angle A=32 10', the angle ACB =133 20', and the side AC =39 15'. R* quired the other parts. cot. A : cos. AC : : R : cot. ACD=64 1' 57" Hence BCD=69 18' 3". Then cos. ACD : cos. BCD : : cot. AC : cot. BC=45 D 20' 43' , Also, by Theorem III., Cor. 2, sin. ACD : sin. BCD : : cos. A : cos. B=28 15' 47". Then, by Theorem III., sin. B : sin ACB : : sin, AC : sin. AB-^76 23' Ti". 1 7 T R I G N O to E I' R V Ex. 2. In the spherical triangle ABC, the angle A=125 20 \ the angle C=48' J 30', and the side AC=83 13'. Recmired f he remaining parts. ( AB= 56 39' 9". Ans. }BC = 11430'24". : ( B =62 54' SS . CASE V. (223.) Given the three sides of a spherical triangle, to find the angles. In the triangle ABC let there be given the three side*. From one of the angles, as C, draw CD perpendicular to AB. Then, by The- orem IV., tan. JAB : tan. 4(AC+BC) : tan. i(AC-BC) : tan. l(AD-BD). Hence AD and BD become known ; then, by Napier's rule, R cos. A=tan. AD cot. AC. The other angles may now be easily found. It is generally most convenient to let fall the perpendicular upon the longest side of the triangle. Ex. 1. In the spherical triangle ABC, the side AB=112 3 25 , AC =60 20', and BC=81 10'. Required the angles, tin. 56 12J-' : tan. 70 45' : : tan. 10 25' : tan. 19 24' 26''. Hence AD=36 48' 4", and BD=75 36' 56". Then R : tan. AD : : cot. AC : cos. A=64 46' 36". Also, R : tan. BD : : cot. BC : cos. B=52 42' 12". Then sin. AC : sin. AB : : sin. B : sin. ACB=122 11' 6". Ex. 2. In the spherical triangle ABC, the side AB=40 35 AC -39 10', and BC=71 15 . Required the angles. ( A=130 35' 55". Ans. )-B= 30 25' 34' ( C- 31 26' 32". CASE VI. (224.) Given the three angles of a spherical triangle, to find the side*. If A, B, C are the angles of the given triangle, and a, b. c its sides, then 180 -A, 180 -B, and 180 -C are the sides SPHERICAL TRIGONOMETRY. 17i ci its polar triangle, whose angles may be found by Case V Then the supplements of those angles will be the sides a, b, ( of the proposed triangle. Ex. 1. In the spherical triangle ABC, the angle A=125 C 34', B=98 44', and 0=61' 53'. Required the sides. The sides of the polar triangle are 54 26', 81 16', and 118 7'. f'rom which, by Case V., the angles are found to be 134 6' 21", 41 28' 17", and 53 34' 47". Hence the sides of the proposed triangle are AB=45 53' 39", BC=138 31' 43", and AC-126 25' 13" Ex. 2. In the spherical triangle ABC, the angle A=109 3 55', B=116 38' ; and C=120 43'. Required the sides. ( a= 98 21' 20' . Ans. 6=109 50' 10". TRIGONOMETRICAL FORMULAS. (225.) Let ABC be any spherical triangle, and from the angle B draw the arc BD perpendicular to the base AC. Represent the sides of the trian- gle by a, b, c, and the segment AD by x ; then will CD be equal to b x. By Theorem III., Cor., 1, cos. c : cos. a : : cos. x : cos. (b x) cos. b cos. re + sin. b sin. x : : cos. x : ^5 (Trig., Art. 72), formula (4). Whence R cos. a cos. re cos. b cos. c cos. rc+sin. b cos. c sin. x t or, dividing each term by cos. x, and substituting the value rf (Art. 28\ we obtain V COS. X R 8 cos. #=R cos. b cos. c-f- sin. b cos. c tan, But by Theorem II., Cor. 2, we have I cos. A_^cos. A sh cot. c cos. c R cos. A cos. A sin. c tan. x=- -- (Art, 28). f\r\4r /* r*f\<3 / * * 172 TRIGONOMETRY. Hence R 2 cos. a=R cos. b cos. c+sin. b sin. c cos. A, (1; from which all the formulas necessary for the solution of spheri- cal triangles may be deduced. Tn a similar manner we obtain R 2 cos. &=R cos. a cos. c+sin. a sin. c cos. B, (2) R a cos. c=R cos. a cos. #+sin. a sin. b cos. C. (3) These equations express the following Theorem : The square of radius multiplied by the cosine of either side of a spherical triangle, is equal to radius into the product of thz cosines of the two other sides, plus the product of the sines of those sides into the cosine of their included angh (226.) From equation (1) we obtain, by transposition, R 2 cos. a R cos. b cos. c cos. A= : - : , sin. b sm. c a formula which furnishes an angle of a triangle when the three sides are known. If we add R to each member of this equation, we shall have _ R 2 cos. &+R sin. b sin. c R cos. b cos. c R + COS. A = : ; : sin. b sin. c O f>/^Q _1_ A But, by Art. 74, R+cos. A= ^~~' And, by Art. 72, formula (2), by transposition, R sin. b sin. c R cos. b cos. c= R 2 cos. (+c). Hence, by substitution, we obtain 2 cos. 2 JA_R 2 (cos. a cos. (b+c)) R sin. b sin. c _2R sin, ^(a+b+c) sin, ^(b+c-a) sin. b sin. c by Art. 75, formula (4). If, then, we put s=^(a+b+c), that is, half the sum of tha sides, we shall find _ . /sin. s sin. (s a) cos. AA=RV = i ' (4) sm. b sin. c By subtracting cos. A from R instead of adding, we shal obtain, in a similar manner, sn . = sin. b SPHERICAL TRIGONOMETRY. 17 Either formula (4) or (5) may be employed to compute thr angles of a spherical triangle when the three sides are knovfn, and this method may be preferred to that of Art. 223. Ex. 1. In a spherical triangle there are given a =63 50', 6=80 19', and c=120 47'. Required the three angles. Here half the sum of the sides is 132 28' =s. Also, s-a=68 38'. Using formula (4), we have log. sine s, 132 28' . . 9.867862 log. sine (s- a), 68 38' . . 9.969075 -log. sine 6, 80 19' comp. 0.006232 -log. sine c, 120 47' comp. 0.065952 Sum 19.909121 log. cos. JA, 25 45' 19" 9.954560. Elence the angle A=51 30' 38". The remaining angles may be found by Theorem TIL, or by formulas similar to formula (4). cos. %n. sin. a sm. c ., _ /si cos. 1C . sin. a sin. b We thus find the angle B= 59 16' 46", and = 131 28' 36". Ex. 2. In a spherical triangle there are given #=115 20', />=57 30', and c=S2 28'. Required the three angles. ( A=126 35' 2". Arcs. } B= 48 31' 42". ( C= 61 43' 58". (227.) By means of the polar triangle, we may convert the preceding formulae for angles into formulae for the sides of a triangle, since the angles of every triangle are the supplements of the sides of its polar triangle. Let, then, a', b', c', A', B', C' represent the sides and angles of the polar triangle, and we shall have A=180 #', B = 1SO b', C = a=180-A', &=180-B', c= Therefore sin. JA=sin. (90 \a') =cos. cos. |A =cos. (90-|a') =sin. 174 TRIGONOMETRY. sin. b=s'm. (180-B'}=sin. B', sin. c=sin. (180-C ; )=sin. C'. Also, if we put S'=half the sum of the angles tf the poia* triangle, we, shall have or s=270-S', whence sin. s= cos. S', sin. (s-a)=sin. [90-(S'-A')]=cos. (S'-A'^ sin. (s-b)=cos. (S'-B'), sin. (s-c)=cos. (S'-C'). By substituting these values in formula (5), Art. 226, and omitting all the accents, since the equations are applicable to any triangle, we obtain . _ cos. (S-B)cos.(S-C) cos. Ja=R\ - ^ ^ -- ^ -- '; (6) sin. B sm. C and formula (4) becomes sS cos. (S-A) sin. ia=R\- p . v n - - ; , (7) sin. B sin. C which formulae enable us to compute the sides of a triangle when the three angles are known ; and this method may bo preferred to that of Art. 224. In a similar manner, by means of the polar triangle, we rlerive from formula (1), Art. 225, the equation R a cos. A=cos. a sin. B sin. C R cos. 13 cos. C ; (8) that is, the square of radius multiplied by the cosine of either angle of a spherical triangle, is equal to the product of the sines of the two other angles into the cosine of their included side, minus radius into the product of their cosines. Ex. 1. In a spherical triangle ABC, there are given A 130 30', B=30 50', and 0=32 5'. Required the Hire* sides. Here half the sum of the angles is 96 42 30''=^S. Also, S-A=-3347'30", S-B= 65 52' 30", S-C= 6/37'30' Using formula (6), we have SPHERICAL TRIGONOMETRY. .75 log, cos. (S-B), 65 52' 30" . 9.611435 log. cos. (S-C), 64 37' 30" . 9.631992 -log. sin. B, 30 50' comp. 0.290270 -log. sin. C, 32 5' comp. 0.274781 Sum 19.808478 log. cos. \a, 36 40' 1" 9.904239. Hence the side a=73 20' 2". The remaining sides may be found by Theorem TIL, or by formulas similar to formula (6). 4 /cos. (S-A)cos. (S-C) COS. \b ]^ V : : ~ , sin. A sin. C . _ /cos. (S-A) cos. (S-B) cos. ^C=RV ^ T' 5 sin. A sin. i3 We thus find the side 6=40 13 12", and c=42 0' 12". Ex. 2. In the spherical triangle ABC, the angle A=129 : 30', B=54 35', and C = 63 5'. Required the three sides. ( 0=120 57' 5". Ans. ]b= 64 55' 37" (c = 82 19' 0". (228.) Formula (1), Art. 225, will also furnish a new test for removing the ambiguity of the solution in Case I. of oblique angled triangles. For we have o O . - R 2 cos. a R cos. b cos. c COS. A== : ; : . sm. b sin. c Now if cos. a is greater than cos. b, we shall have R 2 cos. >R cos. b cos. c, or the sign of the second member of the equation will be the same as that of cos. a, since the denominator is necessarily positive, and cos. c is less than radius. Hence cos. A and cos. a will have the same sign ; or A and a will be of the sama species when cos #>cos. b, or sin. a steered upon a direct course for these points successively. Ex. 2. Required the course and distance from Nantucket Shoals to G-ibraltar, in latitude 36 6' N., longitude 5 2f "V 7 ., on the shortest route. Ans. The course is N. 73 29' E Distance 2974.1 miles. Ex. 3. Required the course and distance from Sandy Hook, in latitude 40 28' "N., longitude 74 V W., to Madeira, in latitude 32 38' K, longitude 16 55' W., on the shortest route. Ans. The course is N. 80 53' E. Distance 2744.1 miles. Ex. 4. Required the course and distance from Sandy Hoolf to St. Jago, in latitude 14 54' N., longitude 23 30' "W., on the shortest route. Ans. The course is S. 74 46' E Distance 3037.6 miles. Ex. 5. Required the course and distance from Sandy Hook to the Cape of Good Hope, in latitude 34 22' S., longitude 18 30' E., on the shortest route. Ans. The course is S. 63 48' E Distance 6792 miles. EXAMPLES FOR PRACTICE. PLANE TRIGONOMETRY. Prob. 1. Given the three sides of a triangle, 627, 718.9, and 1140, to find the angles. Ans. 29 44' 2", 34 39' 26", and 115 36' 32". Prob. 2. In the triangle ABC, the angle A is given 89 45 ; 43", the side AB 654, and the side AC 460, to find the remain- ing parts. Ans. BC = 798; the angle B = 35 12' 1", and the angle C = 552 X 16". Prob. 3. In the triangle ABC, the angle A is given 56 12" 45", the side BC 2597.84, and the side AC 3084.33, to find the remaining parts. Ans. B = 80 39' 40", C = 43 T 35", c = 2136.8 ; or, B = 99 2020, C=24 2655, c = 1293.8. Prob. 4. In the triangle ABC, the angle A is given 44 13' 24", the angle B 55 59' 58". and the side AC 368, to find the remaining parts. Ans. C = 79 46' 38", AB = 436.844, and BC = 309.595. Prob. 5. In a right-angled triangle, if the surn of the hy- pothenuse and base be 3409 feet, and the angle at the base 53 12 X 14", what is the perpendicular ? Ans. 1707.2 feet. Prob. 6. In a right-angled triangle, if the difference of the hypothenuse and base be 169.9 yards, and the angle at the base 42 36' 12", what is the length of the perpendicular ? Ans. 435.732 yards. Prob. 7. In a right-angled triangle, if the sum of the base and perpendicular be 123.7 feet, and the angle at the base 58 19' 32", what is the length of the hypothenuse ? Ans. 89.889 feet. Prob. 8. In a right-angled triangle, if the difference of the base and perpendicular be 12 yards, and the angle at the base 38 V 8", what is the length of the hypothenuse ? Ans. 69.81 yards. 180 TRIGONOMETRY. Prob. 9. A May-pole, 50 feet 11 inches high, at a certain time will cast a shadow 98 feet 6 inches long; what, then, is the breadth of a river which runs within 20 feet 6 inches of the foot of a steeple 300 feet 8 inches high, if the steeple at the same time throws its shadow 30 feet 9 inches beyond the stream ? Ans. 530 feet 5 inches. Prob. 10. A ladder 40 feet long may be so placed that it shall reach a window 33 feet from the ground on one side of the street, and by turning it over, without moving the foot out of its place, it will do the same by a window 21 feet high on the other side. Required the breadth of the street. Ans. 56.649 feet. Prob. 11. A May-pole, whose top was broken off by a blast of wind, struck the ground at the distance of 15 feet from the foot of the pole ; what was the height of the whole May-pole, supposing the length of the broken piece to be 39 feet ? Ans. 75 feet. Prob. 12. How must three trees, A, B, C, be planted, so that the angle at A may be double the angle at B, the angle at B double the angle at C, and a line of 400 yards rnay just go round them ? Ans. AB = 79.225, AC = 142.758, and B = 178.017 yards. Prob. 13. The town B is half way between the towns A and C, and the towns B, C, and D are equidistant from each other. What is the ratio of the distance AB to AD ? Ans. As unity to <\/3. Prob. 14. There are two columns left standing upright in the ruins of Persepolis ; the one is 66 feet above the plain, and the other 48. In a straight line between them stands an ancient statue, the head of which is 100 feet from the summit of the higher, and 84 feet from the top of the lower column, the base of which measures just 74 feet to the centre of the figure's base. Required the distance between the tops of the two columns. Ans. 156.68 feet. Prob. 15. Provo that tan. 45-) = . Prob. 16. One angle of a triangle is 45, and the perpendic- ular from this angle upon the opposite base divides the base into two parts, which are in the ratio of 2 to 3. "What are the EXAMPLES FOR PRACTICE. 181 parts into which the vertical angle is divided by this perpen- dicular? Ans. 18 26' 6" and 26 33' 54". Prob. 17. Prove that sin. 3b 3 sin. & 4 sin. 3 b. Prob. 18. One side of a triangle is 25, another is 22, and the angle contained by these two sides is one half of the angle op- posite the side 25. What is the value of the included angle? Ans. 39 58' 51". Prob. 19. One side of a triangle is 25, another is 22, and the angle contained by these two sides is one half of the angle opposite the side 22. What is the value of the included angle ? Ans. 30 46' 28".. Prob. 20. Two sides of a triangle are in the ratio of 11 to 9, and the opposite angles have the ratio of 3 to 1. What are those angles? Ans. The sine of the smaller of the two angles is -J, and of the greater f-f ; the angles are 41 48 X 37" and 125 25' 51". Prob. 21. One side of a triangle is 15, and the difference of the two other sides is 6 ; also, the angle included between the first side and the greater of the two others is 60. What is the length of the side opposite to this angle ? Ans. 57. Prob. 22. One side of a triangle is 15, and the difference of the two other sides is 6 ; also, the angle opposite to the greater of the two latter sides is CO C . What is the length of said side ? Ans. 13. Prob. 23. One side of a triangle is 15, and the opposite an- gle is 60 ; also, the difference of the two other sides is 6 What are the lengths of those sides ? Ans. 11.0712 and 17.0712. Prob. 24. The perimeter of a triangle is 100 ; the perpendic- ular let fall from one of the angles upon the opposite base is 30, and the angle at one end of this base is 50. What is the length of the base ? 182 TRIGONOMETRY. MENSURATION OF SURFACES AND SOLIDS. Prob. 1. The base of a triangle is 20 feet, and its altitude 18 feet. It is required to draw a line parallel to the base so as to cut off a trapezoid containing 80 square feet. What is the length of the line of section, and its distance from the base of the triangle? -Ans. Length 14.907 feet ; distance from base 4.584 feet. Prob. 2. The base of a triangle is 20 feet, one angle at the base is 63 26', and the other angle at the base is 56 19 X . It is required to draw a line parallel to the base, so as to cut off a trapezoid containing 109 square feet. What is the length of the line of section, and its distance from the base of the tri- angle ? Ans. Length 12.070 feet; distance from base 6.797 feet. Prob. 3. In a perpendicular section of a ditch, the breadth at the top is 26 feet, the slopes of the sides are each 45, and the area 140 square feet. Required the breadth at bottom and the depth of the ditch. Ans. Breadth 10.77 feet ; depth 7.615 feet. Prob. 4. The altitude of a trapezoid is 23 feet ; the two par- allel sides are 76 and 36 feet ; it is required to draw a line par- allel to the parallel sides, so as to cut off from the smaller end of the trapezoid a part containing 560 square feet. What is the length of the line of section, and its distance from the shorter of the two parallel sides ? Ans. Length 56.954 feet; distance 12.048 feet. Prob. 5. From the greater end of a trapezoidal field whose parallel ends and breadth measure 12, 8, and 10| chains re- spectively, it is required to cut off an area of six acres by a fence parallel to the parallel sides of the field. What is the length of the fence, and its distance from the greater side. Ans. Length of fence 9.914 chains; distance from greater side 5.476 chains. Prob. 6. There are three circles whose radii are 20, 28, and 29 inches respectively. Required the radius of a fourth circle, whose area is equal to the sum of the areas of the other three. Ans. 45 inches, Prob. 7. In constructing a rail-road, the pathway of which EXAMPLES FOR PRACTICE. 183 is 24 feet broad, it is necessary to make a cutting 40 feet in depth; what must be the breadth of the cutting at top, sup- posing the slopes of the sides to be 65 ? Ans. 61.305 feet. Prob. 8. The sides of a quadrilateral field are 690 yards, 467 yards, 359 yards, and 428 yards; also, the angle contained between the first and second sides is 57 30', and the angle be- tween the third and fourth sides 96 42'. Required the area of the field. Ans. 212184 square yards. Prob. 9. There are two regular pentagons, one inscribed in a circle, and the other described about it ; and the difference of the areas of the pentagons is 100 square inches. Required the radius of the circle. Ans. 8.926 inches. Prob. 10. What is the length of a chord cutting off one third part of a circle, whose diameter is 289 feet. Ans. 278.67 feet. Prob. 11. The area of a triangle is 1012 ; the length of the side a is to that of b as 4 to 3, and c is to b as 3 to 2. Re- quired the length of the sides. Ans. a = 52.470, 6 = 39.353, c = 59.029. Prob. 12. The area of a triangle is 144, the base is 24, and one of the angles at the base is 30. Required the other sides of the triangle. Ans. 24 and 12.4233. V- Prob. 13. Seven men bought a grinding-stone of 60 inches diameter, each paying one seventh part of the expense. What part of the diameter must each grind down for his share ? Ans. The 1st, 4.4508 inches; 2d, 4.8400 inches; 3d, 5.3535 inches ; 4th, 6.0765 inches ; 5th, 7.2079 inches ; 6th, 9.3935 inches; 7th, 22.6778 inches. Prob. 14. The area of an equilateral triangle is 17 square feet and 83 square inches. What is the length of each side ? Ans. 76.45 inches. Prob. 15. The parallel sides of a trapezoid are 20 and 12 feet, and the other sides are 15 and 17 feet. Required the area of the trapezoid. Ans. 240 square feet. 184 TRIGONOMETRY. Prob. 16. How many square yards of canvas are required to make a conical tent which is 20 feet in diameter and 12 feet high? Am. 54.526 square yards. Prob. 17. The circumference of an hexagonal pillar is 7 feet, and the height 11 feet 2 inches. Required the solid contents of the pillar. Ans. 39.488 cubic feet. Prob. 18. The base of the great pyramid of Egypt is a square whose side measures 746 feet, and the altitude of the pyramid is 450 feet. Required the volume of the pyramid. Ans. 83,477,400 cubic feet. Prob. 1 9. A side of the base of a frustum of a square pyra- mid is 25 inches, a side of the top is 9 inches, and the height is 20 feet. Required the volume of the frustum. Ans. 43.102 cubic feet. Prob. 20. Three persons, having bought a sugar-loaf, would divide it equally among them by sections parallel to the base. It is required to find the altitude of each person's share, suppos- ing the loaf to be a cone whose height is 20 inches. Ans. 13.8672, 3.6044, and 2.5284 inches. Prob. 21. If a cubical foot of brass were to be drawn into wire of one thirtieth of an inch in diameter, it is required to de- termine the length of the said wire, allowing no loss in the metal. Ans. 55003.94 yards ; or 31 miles 443.94 yards. Prob. 22. How high above the surface of the earth must a person be raised to see one third of its surface ? Ans. The height of its diameter. Prob. 23. If a heavy sphere, whose diameter is 4 inches, be let fall into a conical glass full of water, whose diameter is 5, and altitude 6 inches, it is required to determine how much wa- ter will run over. Ans. 26.272 cubic inches. Prob. 24. The capacity of a cylinder is a cubic feet, and its convex surface is b square feet. Required the dimensions of the cylinder. Ans. Radius of base = -, and altitude = , Prob. 25. A triangular pyramid, the sides of whose base are EXAMPLES FOR PRACTICE. 185 13, 14, and 15 inches respectively, and whose altitude is 16 inches, is cut, at the distance of 2 inches from the vertex, by a plane parallel to the base. Required the volume of the frustum of the pyramid. Ans. 447.125 cubic inches. Prob. 26. The altitude of a cone is 10 inches, and the radius of its base is 5 inches. At what distance from the base must a plane pass parallel to the base, so as to cut off a frustum whose capacity is 20 cubic inches ? Ans. 0.2614 inches. SURVEYING. Prob. 1. The angle or elevation of a spire I found to be 39 27', and going directly from it 225 feet on a horizontal plane, I found the angle to be only 24 38'. What is the height of the spire, and the distance from its base to the second station ? Ans. Height 233.02 feet, distance 508.18 feet. Prob. 2. Wishing to know the distance of an inaccessible ob- ject, I measured a horizontal base-line 1328 feet, and found the angles at the ends of this line were 84 23' and 43 19'. What was the distance of the object from each end of the base-line ? Ans. 1151.44 feet, and 1670.35 feet. Prob. 3. Wishing to know the distance between two inacces- sible objects, C and D, I measured a base-line, AB, 3784 feet, and found the angle BAD = 47 32', the angle DAC = 39 53', the angle ABC =46 34 X , and the angle CBD^38 V. What is the distance from C to D ? Ans. 3257.36 feet. Prob. 4. Suppose a light-house built on the top of a rock ; the distance between the place of observation and that part of the rock which is level with the eye, and directly under the build- ing, is 1860 feet ; the distance from the top of the rock to the place of observation is 2538 feet, and from the top of the build- ing 2550 feet. Required the height of the light-house. Ans. 17 feet 7 inches. Prob. 5. At 85 feet distance from the bottom of a tower, standing on a horizontal plane, the angle of its elevation was found to be 52 30'. Required the altitude of the tower. Ans. HOjJ feet 186 TRIGONOMETRY. Prob. 6. At a certain station, the angle of elevation of an in- accessible tower was 26 30' ; but, measuring 225 feet in a di- rect line toward it, the angle was then found to be 51 30'. Required the height of the tower, and its distance from the last station. Ans. Height 186 feet, distance 147 feet. Prob. 7. To find the distance of an inaccessible castle gate, I measured a line of 73 yards, and at each end of it took the an- gle of position of the object and the other end, and found the one to be 90, and the other 61 45'. Required the distance of the castle from each station. Ans. 135.8 yards, and 154.2 yards. Prob. 8. From the top of a tower 143 feet high, by the sea- side, I observed that the angle of depression of a boat was 35. What was its distance from the bottom of the tower ? Ans. 204.22 feet. Prob. 9. I wanted to know the distance between two places, A and B, but could not meet with any station from whence I could see both objects. I measured a line CD = 200 yards ; from C the object A was visible, and from D the object B was visi- ble, at each of which places I set up a pole. I also measured FC=200 yards, and DE=:200 yards, and at F and E set up poles. I then measured the angle AFC = 83, ACF = 54 31', ACD = 53 30', BDC = 156 25', BDE = 54 30', and BED=: 88 30'. Required the distance from A to B. Ans. 345.5 yards. Prob. 10. From the top of a light-house, the angle of depres- sion of a ship at anchor was 3 38', and at the bottom of the light-house the angle of depression was 2 43'. Required the horizontal distance of the vessel, and the height of the promon- tory above the level of the sea, the light-house being 85 feet high. Ans. Distance 5296.4 feet, height 251.3 feet. Prob. 11. An observer, seeing a cloud in the west, measured its angle of elevation, and found it to be 64. A second observ- er, situated half a mile due east from the first station, and on the same horizontal plane, found its angle of elevation at the same moment of time to be only 35. Required the perpendic- ular height of the cloud, and its distance from each observer. Ans. Perpendicular height 935.75 yards, distances 1041.1 and 1631.4 yards. EXAMPLES FOR PRACTICE. 187 Prob. 12. An observer, seeing a balloon in the north,, meas- ured its angle of elevation, and found it to be 36 52'. A second observer, situated one mile due south from the first station, and on the same horizontal plane, found its angle. of elevation at the same instant to be only 30 58'. Required the perpendicular height of the balloon, and its distance from each observer. Ans. Perpendicular height 3.003 miles, distances 5.006 and 5.837 miles. Prob. 13. From a window near the bottom of a house which seemed to be on a level with the bottom of a steeple, I found the angle of elevation of the top of the steeple to be 40 ; then from another window, 21 feet directly above the former, the like angle was 37 30 X . "What was the height and distance of the stee- ple ? Ans. Height 245.51 feet, distance 292.59 feet. Prob. 14. "Wanting to know my distance from an inaccessi- ble object, P, on the other side of a river, and having no instru- ment for taking angles, but a chain for measuring distances, from eacli of two stations, A and B, which were taken at 300 yards asunder, I measured in a direct line from the object P 60 yards, viz., AC and BD each equal to 60 yards ; also, the diag- onal AD measured 330 yards, and the diagonal BC 336 yards. What was the distance of the object P from each station A and B ? Ans. AP^ 321.76 yards, BP = 300.09 yards. Prob. 15. Having at a certain (unknown) distance taken the angle of elevation of a steeple, I advanced 60 yards nearer on level ground, and then observed the angle of elevation to be the complement of the former. Advancing 20 yards still nearer, the angle of elevation now appeared to be just double of the first. Required the altitude of the steeple. Ans. 74.162 yards. Prob. 16. In a garrison there are three remarkable objects, A, B, C, whose dis- tances from each other are known to be, AB 213, AC 424, and BC 262 yards. I am desirous of knowing my position and- distance at a station, P, from which I ob- served the angle APB, 13 30', and the angle CPB, 29 50'. Ans. AP^ 605.7122, BP^ 429,6814, CP= 524.2365. 188 TRIGONOMETRY. Prob. 17. Supposing the object B to be on the opposite side of the line AC (see figure to Prob. 16), and that the distances of the objects were, AB 8 miles, AC = 12 miles, and BC = 7-J miles ; also, the angle APB = 19, and the angle CPB = 25. It is re- quired to find the distances AP, BP, and CP. Ans. AP = 9.4711 miles, BP = 16.3369 miles, CP = 16.8485 miles. Prob. 18. In a pentangular field, beginning with the south side, and measuring round toward the east, the first or south side was 27.35 chains, the second 31.15 chains, the third 23.70 chains, the fourth 29.25 chains, and the fifth 22.20 chains; also, the diagonal from the first angle to the third was 38.00 chains, and that from the third to the fifth was 40.10 chains. Required the area of the field. Ans. 117 A. 2 R. 39 P. Prob. 19. The following are the dimensions of a five-sided field, ABCDE: the side AB = 19.40 chains, and the angle B 110 30'; the side BC = 15.55 chains, and the angle C 117 45' ; the side CD = 21.25 chains, and the angle D 91 20' ; and the side DE = 27.41 chains. Required the area of the field. Ans. 66 A. 2 Jt. 24 P. Prob. 20. From a station, H, near the middle of a field, ABCDEF, from which I could see all the angles, I measured the distances to the several corners, and measured the angles formed at H by those distances, as follows : Distances. Angles. AH, 43.15 chains ; AHB, 60 30'. BH, 29.82 " BHC, 47 40 CH, 35.61 " CHD, 49 50 DH, 50.10 " DHE,57 10 EH, 46.18 EHF,64 15 FH, 36.06 FHA, 80 35 Required the area of the field. Ans. 412 A. 1 R. 17 P NAVIGATION. Prob. 1. From a ship at sea I observed a point of land to bear east by south, and, after sailing northeast 12 miles, I ol> EXAMPLES FOR PRACTICE. 189 served again, and found its bearing to be southeast by east. How far was the last observation made from the point of land ? Ans. 26.07 miles. Prob. 2. If a ship in latitude 50 N., sails 52 miles in the di- rection southwest by south, what latitude has she arrived in, and how much farther to the west ? Ans. Latitude 49 16/8 N. ; west, 28.9 miles. Prob. 3. Two ships sail from the same port ; the one sails east-northeast 85 miles, the other sails east by south till the first ship bears northwest by west. What is the distance of the sec- ond ship from the port, and also from the first ship ? Ans. From the port, 184.7 miles ; from the first ship, 123.4 miles. Prob. 4. Two ports lie east and west of each other ; a ship sails from each, namely, the ship from the west port sails north- east 89 leagues, and the other sails 80 leagues, when she meets the former. Required the latter ship's course, and the distance between the two ports. Ans. Course, N. 38 8 X W. ; distance, 112.3 leagues. Prob. 5. Two ships sail from a certain port ; the one sails south by east 45 leagues, and the other south-southwest 64 leagues. What is the bearing and distance of the first ship from the second ? Ans. Bearing, T\ T . 65 44' E. ; distance, 36.5 leagues. Prob. 6. A ship sailing northwest, two islands appear in sight, of which the one bears north, and the other west-north- west ; but, after sailing 20 leagues, the former bears northeast, and the latter west by south. What is tlio distance asunder of the two islands? Ana. 32.38 leagues. Prob. 7. To a vessel sailing on a certain course, a headland was observed to bear due west ; four hours after which it was seen at west-southwest ; and six hours after this, the vessel con- tinuing to run at the same rate, its bearing was found to be south-southwest. What was the vessel's course at the time ? Ans. N. 42 35' W. Prob. 8. Two ships of war, intending to cannonade a fort, are, by the shallowness of the water, kept so far from it that they suspect their guns can not reach it with effect. In order, there- fore, to measure the distance, they separate from each other 500 190 TRIGONOMETRY. rods; then each ship observes the angle which the other ship and the fort subtend, which angles are 38 16 X and 37 9'. What, then, is the distance between each ship and the fort ? Ans. 312 rods and 320 rods. Prob. 9. A ship from the latitude 42 18' N., sails southwest by south until her latitude is 40 18' N. "What direct distance has she sailed, and how many miles has she sailed to the west- ward? Ans. Distance run 144.3 miles, and has sailed to west- ward 80.2 miles. Prob. 10. A ship having run due east for three days, at the rate of eight knots an hour, finds she has altered her longitude 15 degrees. What parallel of latitude did she sail on ? Ans. Latitude 50 12'. Prob. 11. A ship in latitude 43 30' N., and longitude 44 W., sails southeasterly 532 miles, until her departure from the meridian is 420 miles. Required the course steered, and the latitude and longitude of the ship. Ans. Course S. 52 8' E., latitude, 38 3/5 N., longitude 34 45' W. Prob. 12. A ship from latitude 43 20' N., and longitude 52 W., sails E.S.E. until her departure is 745 miles. Required the distanca sailed, and the latitude and longitude of the ship. Ans. Distance 808.4 miles, latitude 38 11/5 N., longitude 35 35' A7. Prob. 13. If the height of the mountain called the Peak of Teneriffe be 4 miles, and the angle taken at the top of it, as formed between a plumb-line and a line conceived to touch the earth in the horizon, or farthest visible point, be 87 25' 55 X/ , it is required from hence to determine the magnitude of the whole earth, and the utmost distance that can be seen on its surface from the top of the mountain, supposing the earth to be a per- fect sphere. Ans. Distance 178.458 miles, diameter 7957.793 miles. Prob. 14. Required the course and distance from St. Jago, one of the Cape Verd islands, in latitude 14 56 / N., to the island of St. Helena, in latitude 15 45'' S., their difference of longitude being 30 12'. Ans. Course S. 44 12' E., distance 2567.8 miles, EXAMPLES FOR PRACTICE. 191 Prob. 15. A ship from the latitude of 49 57 X K, and longi- tude of 30 W., sails S. 39 W., till she arrives in the latitude of 45 31 X N. Required the distance run, and the longitude of the ship. Ana. Distance 342.3 miles, longitude of ship 35 21 X "W. Prob. 16. Find the bearing and distance from San Francisco, latitude 37 48 X N., longitude 122 28' W., to Jeddo, latitude 35 40 X N., longitude 139 40 / E., by Mercator's sailing. Ans. Course S. 88 26 X W., distance 4705 miles. Prob. 17. Find the bearing and distance from San Francisco to Batavia in Java, latitude 6 9 X S., longitude 106 53 X E., by Mercator's sailing. Ans. Course S. 70 12 X W., distance 7783 miles. Prob. 18. Find the bearing and distance from San Francisco to Port Jackson, latitude 33 51 X S., longitude 151 14 X E., by Mercator's sailing. Ans. Course S. 48 18 X W., distance 6462 miles. Prob. 19. Find the bearing and distance from San Francisco to Otaheite, latitude 17 29 X S., longitude 149 29' W., by Mer- cator's sailing. Ans. Course S. 24 44 X W., distance 3652 miles. Prob. 20. Find the bearing and distance from San Francisco to Valparaiso, latitude 33 2 X S., longitude 71 41 X W., by Mer- cator's sailing. Ans. Course S. 33 47 X E., distance 5354 miles. SPHERICAL TRIGONOMETRY. Prob. 1. In the right-angled spherical triangle ABC, there are given the angle C 23 27 X 42 /x , and the side b 10 39 X 40 /x . Required the angle B, and the sides a and c. (a = 11 35 X 49 X/ . Ans. } c = 4 35 X 26 XX . (B=6658 X l xx . Prob. 2. In the spherical triangle ABC, the side BC-90 , the side AB = 32 57 X 6 /x , and the side AC = 66 32 X . Required the angles. (A = 132 2 X 44 XX . Ans. JE= 42 56 X 12 X '. (C= 2349'26 X/ 192 TRIGONOMETRY. Prob. 3. In the right-angled spherical triangle ABC, there are given the angle B = 47 54' 20", and the angle C = 61 50' 29". Required the sides. (a = 61 4' 56". Ans. ) 6 =40 30' 20". (Crr 50 30' 30". Prob. 4. In the spherical triangle ABC, the side AC = 90, the side AB = H5 9', and the angle B = 10i 40'. Required the remaining parts. ( BC = 113 IS 7 7". Ans. ] A = 115 54' 46". (C = 117 33' 49". Prob. 5. In the spherical triangle ABC, the angle A =130 5' 22", the angle = 36 45' 28", and the side AC =44 13' 45" Required the remaining parts. (AB = 51 6' 12". Ans. JBC:=84 14' 29". ( B = 3226 / 6". Prob. 6. In the spherical triangle ABC, the angle A=33 15' 7", B = 31 34' 38", and C = 161 25' 17". Required the sides. (a= 80 5' 4". * b= 70 10 X 30". c = 145 5 X 2". >. 7. In the spherical triangle ABC, the side AB = 112 22 X 58", AC = 52 39' 4", and BC = 89 16 X 53". Required the angles. ( A= 70 39 X 0". Ans. < B = 48 36 X 0". ( C =119 15 X 0". Prob. 8. In the spherical triangle ABC, the side AB = 76 35 36", AC = 50 10' 30", and the angle A = 34 15' 3". Re- quired the remaining parts. (B = 42 15' 13". Ans. one less than the number of integral figures in the given number. Thus the logarithm of 297 is 2 plus a fraction ; that is, the character- istic of the logarithm of 297 is 2, which is one less than the number ot integral figures. The characteristic of the logarithm of 5673.29 is 3 ; ihat of 73254.1 is 4, &c. The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which its first significant figure is removed from the place of units. Thus the logarithm of .0046 is 3 plus a fraction ; that is, the char- acteristic of the logarithm is 3, the first significant figure 4 being removed three places from units. The accompanying table contains the logarithms of all numbers from 1 to 10,000 carried to 6 decimal places. To find the Logarithm of any Number between 1 and 100. Look on the first page of the table, along the column of numbers under N, for the given number, and against it, in the next column, will be found the logarithm, with its characteristic. Thus, opposite 13 is 1.113943, which is the logarithm of 13; 65 is 1.812913, " " 65. To find i.ht Logarithm of any Number consisting of three Figures. Look on one of the pages from 2 to 20, along the left-hand column marked N, for the given number, and against it, in the column headed 0, will be found the decimal part of its logarithm. To this the character- stic rnnst be prefixed, according to the rule already given. Thus the logarithm of 347 will be found, from page 8, to be 2.540329 ; " " 871 " " " 18, " 2.940018. As the fiist two figures of the decimal are the same for several suc- cessive numbers in the table, they are not repeated for each logarithm separately, but are left to be supplied. Thus the decimal part of th logarithm of 339 is .530200. The first two figures of the decimal remain the same up to 347 ; they are therefore omitted in the table, and are to be supplied. To find the Logarithm of any Number consisting of four Figures. Find the three left-hand figures in the column marked N as before, EXPLANATION OF THE TABLES. vii and the fourth figure at the head of one of the other columns. Opposite to the first three figures, and in the column under the fourth figure, will he found four figures of the logarithm, to which two figures from the column headed are to be prefixed, as in the former case. The char- acteristic must be supplied by the usual rule. Thus the logarithm of 3456 is 3.538574 ; " 8765 is 3.942752. In several of the columns headed 1, 2, 3, &c., small dots are found in the place of figures. This is to show that the two figures which are to be prefixed from the first column have changed, and they are to be taken from the horizontal line directly below. The place of the dots is to be supplied with ciphers. Thus the logarithm of 2045 is 3.310693 ; " " 9777 is 3.990206. The two leading figures from the column must also be taken from the horizontal line below, if any dots have been passed over on the same horizontal line. Thus the logarithm of 1628 is 3.211654. To find the Logarithm of any Number containing more than four Figures. By inspecting the table, we shall find that within certain limits the log- arithms are nearly proportional to their corresponding numbers. Thus the logarithm of 7250 is 3.860338 ; " 7251 is 3.860398 ; " 7252 is 3.860458 ; " 7253 is 3.860518. Here the difference between the successive logarithms, called the tabular difference, is constantly 60, corresponding to a difference of unity in the natural numbers. If, then, we suppose the logarithms to be pro- portional to their corresponding numbers (as they are nearly), a differ- ence of 0.1 in the numbers should correspond to a difference of 6 in the logarithms ; a difference of 0.2 in the numbers should correspond to a difference of 12 in the logarithms, &c. Hence the logarithm of 7250.1 must be 3.860344; " 7250.2 " 3.860350; " " 7250.3 " 3.860356; &c., &c. in order to facilitate the computation, the tabular difference is insert- ed on page 16 in the column headed D, and the proportional part for the fifth figure of the natural number is given at the bottom of the page, Thus, when the tabular difference is 60, the corrections for .1, .2, .3 ; &c., are seen to be 6, 12, 18, &c. If the given number was 72501. the characteristic of its logarithm would be 4, but the decimal part would be the same as for 7250.1. EXPLANATION OF THE TABLES. If it were required to find the correction for a sixth figure in the nat- ural number, it is readily obtained from the Proportional Parts in the table. Thus, if the correction for .5 is 30, the correction for .05 is ob- viously 3. As the differences change rapidly in the first part of the table, it was found inconvenient to give the proportional parts for each tabular difference ; accordingly, for the first seven pages they are only given for the even differences, but the proportional parts for the odd differences will be readily found by inspection. Required the logarithm of 452789. The logarithm of 452700 is 5.655810 The tabular difference is 96. Accordingly, the correction for the fifth figure, 8, is 77, and for the sixth figure, 9, is 8.6, or 9 nearly. Adding these corrections to the num- ber before found, we obtain 5.655896. The preceding logarithms do not pretend to be perfectly exact, but only the nearest numbers having but six decimal places. Accordingly, when the fraction which is omitted exceeds half a unit in the sixth deci- mal place, the last figure must be increased by unity. Required the logarithm of 8765432. The logarithm of 8765000 is 6.942752 Correction for the fifth figure 4, 20 " sixth figure 3, 1.5 " " seventh figure 2, 0.1 Therefore the logarithm of 8765432 is 6.942774. Required the logarithm of 234567. . The logarithm of 234500 is 5.370143 Correction for the fifth figure 6, 111 " " sixth figure 7, 13 Therefore the logarithm of 234567 is 5.370267. To find the Logarithm of a Decimal Fraction. The decimal part of the logarithm of any number is the same as that of the number multiplied or divided by 10, 100, 1000, &c. Hence, for a decimal fraction, we find the logarithm as if the figures were integers, and prefix the characteristic according to the usual rule. EXAMPLES. The logarithm of 345.6 is 2.538574 ; " 87.65 is 1.942752; 2.345 is 0.370143; " .1234 is L091315; .005678 is 3.754195. EXPLANATION OF THE TABLES ix The minus sign is placed over the characteristic to show that this alone is negative, while the decimal part of the logarithm is positive. To find the Logarithm of a Vulgar Fraction. Wo may reduce the vulgar fraction to a decimal, and find its loga- rithm by the preceding rule ; or, since the value of a fraction is equal to the quotient of the numerator divided by the denominator, we may sub- tract the logarithm of the denominator from that of the numerator; the difference will be the logarithm of the quotient. Required the logarithm of T 3 F , or 0.1875. From the logarithm of 3, 0.477121, Subtract the logarithm of 16, 1.204120. Leaves logarithm of T 3 F , or .1875, f. 273001. Tn the same manner we find the logarithm of / T is 2.861697; " " jf is 1.147401. To find the natural Number corresponding to any Logarithm. Look in the table in the column headed for the first two figures of the logarithm, neglecting the characteristic ; the other four figures ara to be looked for in the same column, or in one of the nine following col- umns ; and if they are exactly found, the first three figures of the corre- sponding number will be found opposite to them in the column headed N, and the fourth figure will be found at the top of the page. This num- ber must be made to correspond with the characteristic of the giver, logarithm by pointing off decimals or annexing ciphers. Thus the natural number belonging to the logarithm 4.370143 is 23450 ; " " " " 1.538574 is 34.56. If the decimal part of the logarithm can not be exactly found in the table, look for the nearest less logarithm, and take out the four figures of the corresponding natural number as before ; the additional figures may be obtained by means of the Proportional Parts at the bottom of the page. Required the number belonging to the logarithm 4.368399. On page 6, we find the next less logarithm .368287. The four corresponding figures of the natural number are 2335. Their logarithm is Jess than the one proposed by 112. The tabular difference is 186; and, by referring to the bottom of page 6, we find that, with a difference of 186, the figure corresponding to the Piopor- tional Part 112 is 6. Hence the five figures of the natural number are 23356 ; and, since the characteristic of the proposed logarithm is 4, these five figures are all integral. Required the number belonging to logarithm 5.345678. The next less logarithm in the table is .345570 Their difference is 108. x EXPLANATION OF THE TABLES. The first four figures of the natural numljer are 2216. With the tabular difference 196, the fifth figure corresponding to 108 s seen tc be 5, with a remainder of 10, which furnishes a sixth figure 5 nearly. Hence the required number is 221655. In the same manner we find the number corresponding to logarithm 3.538672 is 3456.78 ; 1.994605 is 98.7654; " " " T.647817 is .444444. TABLE OF NATURAL SINES AND TANGENTS, pp. 116188. This is a table of natural sines and tangents for every degree and minute of the quadrant, carried to six places of figures. Since the ra- dius of the circle is supposed to be unity, the sine of every arc below 90 is less than unity. These sines are expressed in decimal parts of the radius ; and, although the decimal point is not written in the table, it must always be prefixed. The degrees are arranged in order at the top of the page, and the minutes in the left hand vertical column. Directly under the given number of degrees at the top of the page, and opposite to the minutes on the left, will be found the sine required. The twn leading figures are repeated at intervals often minutes. Thus the sine of 6 27' is .112336; 28 53' is .483028. The same number in the table is both the sine of an arc and the co- sirie of its complement. The degrees for the cosines must be sought at the bottom of the page, and the minutes on the right. Thus the cosine of 62 25' is .463038 ; " " 84 23' is .097872. Ji a sine is required for an arc consisting of degrees, minutes, and sec- onds, it may be found by means of the line at the bottom of each page, which gives the proportional part corresponding to one second of arc. Required the sine of 8 9' 10". The sine of 8 9' is .141765. By leferring to the bottom of page 116, in the column under 8, we find the correction for I" is 4.80 ; hence the correction for 10" must be 48, which, added to the number above found, gives for the sine of 8 9' 10", .141813. In the same manner we find the cosine of 56 34' 28" is .550853. It will be observed, that since the cosines decrease while the arcs in- crease, the correction for the 28" is to be subtracted from the cosine of 56 34'. The arrangement of the table of natural tangents is similar to that of the table of sines. The tangents for arcs less than 45. are all less than radius, and consist wholly of decimals. For arcs above 45, the tan- gents are all gi eater than radius and contain both integral and decima. EXPLANATION OF THE T A D L K s. x i figures. The proportional parts at the bottom of each page enable us readily to find the correction for seconds. Thus the natural tangent of 32 29' IS" is .636784 ; " 74 3 35' 55" is 3.63014. To find the Number of Degrees, Minutes, and Seconds belonging to a given Sine or Tangent. If the given sine or tangent is found exactly in the table, the corre- sponding degrees will be found at the top of the page, and the minutes on the left hand. But when the given number is not found exactly ii) the table, look for the sine or tangent which is next less than the pro- posed one, and take out the corresponding degrees and minutes. Find, also, the difference between this tabular number and the number pro- posed, and divide it by the proportional part for I" found at the bottom of the page ; the quotient will be the required number of seconds. Required the arc whose sine is .750000. The next less sine in the table is .749919, the arc corresponding tu which is 48 35'. The difference between this sine and that proposed is 81, which, divided by 3.21, gives 25. Hence the required arc is 4S G 35' 25". Tn the same manner we find the arc whose tangent is 2.000000, to be 63 26' 6". TABLE OF NATURAL SECANTS, pp. 134-5. This is a table of natural secants for every ten minutes of the quad- rant carried to seven places of figures. The degrees are arranged in order in the first vertical column on the left, and the minutes at the top of the page. Thus the secant of 21 20' is 1.073561 ; " 81 50' is 7.039622. If a secant is required for a number of minutes not given in the table, the correction for the odd minutes may be found by means of the last vertical column on the right, which shows the proportional part for one minute. Let it be required to find the secant of 30 33'. The secant of 30 30' is 1.160592. The correction for 1' is 198.9, which, multiplied by 3, gives 597. Adding this to the number before found, we obtain 1.161189. For a cosecant, the degrees must be sought in the right-hand vertical column, and the minutes at the bottom of the page. Thus the cosecant of 47 40' is 1.352742. TABLE OF LOGARITHMIC SINES AND TANGENTS, pp. 21115. This is a table of the logarithms of the sines and tangents for every ten seconds of the quadrant, carried to six places of decimals The de xii EXPLANATION OF THE TABLES. grees and seconds are placed at the top of the page, and the minutes in the left vertical column. After the first two degrees, the three leading figures in the table of sines are only given in the column headed 0", and are to be prefixed to the numbers in the other columns, as in the table of logarithms of numbers. Also, where the leading figures change, this change is indicated by dots, as in the former table. The correction for any number of seconds less than 10 is given at the bottom of the page. To find the Logarithmic Sine or Tangent of a given Arc. Look for the degrees at the top of the page, the minutes on the left hand, and the next less tenth second at the top ; then, under the seconds, and opposite to the minutes, will be found four figures, to which the three leading figures are to be prefixed from the column headed 0" ; to this add the proportional part for the odd seconds from the bottom of the page. Required the logarithmic sine of 24 27' 34". The logarithmic sine of 24 27' 30" is 9.G17033. Proportional part for 4" is 18. Logarithmic sine of 24 27' 34" is 9.617051. This is the logarithm of .414049 found in the table of natural sines on page 120. The natural sine being less than unity, the characteristic of its logarithm is negative. Tc obviate this inconvenience, the char- acteristics in the table nave all been increased by 10; or the logarith- mic sines may be regarded as the logarithms of natural sines computed for a radius of 10,000,000,000. Required the logarithmic tangent of 73 35' 43". The logarithmic tangent of 73 35' 40" is 10.531031. Proportional part for 3" is 23. Logarithmic tangent of 73 3.V 43" 10.531054. When a cosine is required, the degrees and seconds must be sought at the bottom of the page, and the minutes on the right, and the correc- tion for the odd seconds must be subtracted from the number in the table, Required the logarithmic cosine of 59 33' 47". The logarithmic cosine of 59 33' 40" is 9.704682. Proportional part for 7" is 25. Logarithmic cosine of 59 33' 47" is 9.704657. So, also, the logarithmic cotangent of 37 27' 1 4" ic found to be 10.115744. The proportional parts given at the bottom of each page correspond to the degrees at the top ol the page increased by 30', and are not strictly applicable to any other number of minutes ; nevertheless, the differences of the sines change so slowly, except near the commence- ment of the quadrant, that the error resulting from using these numbers for every part of the page will seldom exceed a unit in the sixth deci- mal place. For the ftrsf two degrees, the differences change so rapidly E X r 1 A N A T f O N OF THE T A B L E S. X 1 1 1 that the proportional part for 1" is given for each minute in tho right- hand column of the page. The correction for any number of seconds less than ten will be found by multiplying the proportional part for V by the given number of seconds. Required the logarithmic sine of 1 17' 33". The logarithmic sine of 1 17' 30" is 8.352991. The correction for 3" is found by multiplying 93.4 by 3, which gives 280. Adding this to the above tabular number, we obtain the sine of 1 17' 33", 8.353271. A similar method may be employed for several of the first degrees of the quadrant, if the proportional parts at the bottom of the page are net thought sufficiently precise. This correction may, however, be ob- tained pretty nearly by inspection from comparing the proportional parts for two successive degrees. Thus, on page 26, the correction for 1", corresponding to the sine of 2 30', is 48 ; the correction for 1", cor- responding to the sine of 3 30', is 34. Hence the correction for 1", corresponding to the sine of 3 0', must be about 41 ; and in the same manner we may proceed for any other part of the table. Near the close of the quadrant, the tangents vary so rapidly, that the same arrangement of the table is adopted as for the commencement of the quadrant. For the last as well as the first two degrees of the quad- rant, the proportional part to 1" is given for each minute separately. These proportional parts are computed for the minutes placed opposite to them, increased by 30' , and are not strictly applicable to any other number of seconds ; nevertheless, the differences for the most part change so slowly, that the error resulting from using these numbers for every part of the same horizontal line is quite small. When great ac- curacy is required, the table on page 114 may be employed for arcs near the limits of the quadrant. This table furnishes the differences be* tween the logarithmic sines and the logarithms of the arcs expressed in seconds. Thus the logarithmic sine of 5' is 7.16269G ; the logarithm of 300" (=5') is 2.477121 ; the difference is 4.G85575. This is the number found on page 114, under the heading log. sine A log. A", opposite 5 min. ; and in a similar manner the other numbers in the same column are obtained. These numbers vary quite slowly for two degrees ; and hence, to find the logarithmic sine of an arc less than two degrees, we have but to add the logarithm of the arc expressed in seconds to the appropriate number found in this table. Required the logarithmic sine of 7' 22". Tabular number from page 114, 4.685575. The logarithm of 442" is 2.645422 Logarithmic sine of 7' 22" is 7.330997. xiv EXPLANATION OF THE TABLES. The logarithmic tangent of an arc less than two degrees is found in a sim : .!ar manner. Required the logarithmic tangent of 21' 36". Tabular number from page 114, 4.G85584. The logarithm of 1656" is 3.219060. Logarithmic tangent of 21' 36" is 7.904644. The column headed log. cot. A+log. A" is found by adding the log- arithmic cotangent to the logarithm of the arc expressed in seconds. Hence, to find the logarithmic cotangent of an arc less than two degrees, we must subtract from the tabular number the logarithm of the arc in seconds. Required the logarithmic cotangent of 21' 36". Tabular number from page 114, 15.314416. The logarithm of 1656" is 3.219060. Logarithmic cotangent of 21' 36" is 12.095356. The same method will, of course, furnish cosines and cotangents ot arcs near 90. The secants and cosecants are omitted in this table, since they arc easily derived from the cosines and sines. The logarithmic secant is found by subtracting the logarithmic ccsine from 20 ; and the logarithmic cosecant is found by subtracting the loga- rithmic sine from 20. Thus we have found the logarithmic sine of 24 21' 34" to be y.617051. Hence the logarithmic cosecant of 24 27' 34" is 10.382949. The logarithmic cosine of 54 12' 40" is 9.767008. Hence the logarithmic secant of 54 12' 40" is 10.232992. To find the Arc corresponding to a given Logarithmic Sine or Tangent. If the given number is found exactly in the table the corresponding degrees and seconds will be found at the top of the page, and the min- utes on the left. But when the given number is not found exactly in the table, look for the sine or tangent which is next less than the pro- posed one, and take out the corresponding degrees, minutes, and sec onds. Find, also, the difference between this tabular number and the number proposed, and, corresponding to this difference, at the bottom of the page will be found a certain number of seconds, which is to be added to the arc before found. Required the arc corresponding to the logarithmic sine 9.750000. The next less sine in the table is 9.749987. The arc corresponding to which is 34 13' 0". The difference between its sine and the one proposed is 13, corre- sponding to which, at the bottom of the page, we find 4" nearly, Hence the required arc is 34 13' 4". EXPLANATION OF THE TABLES, xv In the same manner we find the arc corresponding to logarithmic tan- gent 10.250000, to be 60 38' 57". When the arc falls within the first two degrees of the quadrant, the odd seconds may be found by dividing the difference between the tab- ular number and the one proposed, by the proportional part for 1". We thus find the arc corresponding to logarithmic sine 8.400000, to he 1 26' 22" nearly. We may employ the same method for the last two degrees of the quadrant when a tangent is given ; but near the limits of the quadrant it is better to employ the auxiliary table on page 114. If w r e subtract the corresponding tabular number on page 114 from the given logarith- mic sine, the remainder will be the logarithm of the arc expressed in seconds. Required the arc corresponding to logarithmic sine 7.000000. We see, from page 22, that the arc must be nearly 3' ; the correspond- ing tabular number on page 114 is 4.685575. The difference is 2.314425 ; which is the logarithm of 206."265. Hence the required arc is 3' 26."265. In the same manner we find the arc corresponding to logarithmic tangent 8.184008, to be 52' 35". TABLE FOR THE LENGTHS OF CIRCULAR ARCS, p. 135. This table contains the lengths of every single degree up to 60, and at intervals often degrees up to 180 ; also for every minute and second up to 20. The lengths are all expressed in decimal parts of radius. Required the length of an arc of 57 17' 44."8. Take out from their respective columns the lengths answering to each of these numbers singly, and add them all together thus : 57 . . . . . . 0.9948377 17' ... . . . .0049451 40" . . . . . .0001939 4" . . . . . . .0000194 0."8 . . . . . . .0000039 The sum is 1.0000000. That is, the length of an arc of 57 17' 44. "8 is equal to the radius of the circle. TRAVERSE TABLE, pp. 136-141. This table shows the difference of latitude and the departure to four decimal places for distances from 1 to 10, and for bearings from to 00, at intervals of 15'. If the bearing is less than 45, the angle will be found on the left margin of one of the pages of the table, and the dis- tance at the top or bottom of the page ; the difference of latitude will xvi EXPLANATION OF THE TABLES. be found in the column headed lat. at the top of the page, and the de- parture in the column headed dep. If the bearing is more than 45, the angle will be found on the right margin, and the difference of latitude will bo found in the column marked lat. at the bottom of the page, and the departure in the other column. The latitudes and departures for different distances with the same bearing, are proportional to the dis- tances. Therefore the distances may be reckoned as tens, hundreds, or thousands, if the place of the decimal point in each departure and differ- ence of latitude be changed accordingly. Required the latitude and departure for the distance 32.25, and the bearing 10 30'. On page 136, opposite to 10 30', we find the following latitudes and departures, proper attention being paid to the position of the decimal points. Distance. Diff. Lat. DC p. 30 29.498 5.467 2 1.966 .364 .2 19? .036 .05 .049 .009 32.25 SlTnO 57876. TABLE OF MERIDIONAL PARTS, pp. 142-148 This table gives the length of the enlarged meridian on Mercator's Chart to every minute of latitude expressed in geographical miles and tenths of a mile. The degrees of latitude are arranged in order at the top of the page, and the minutes on both the right and left margins. Under the degrees and opposite to the minutes are placed the merid 'onal parts corresponding to any latitude less than 80. Thus the meridional parts for latitude 12 23' are 748.9 ; " " " 57 42' are 4260.5. TABLE OF CORRECTIONS TO MIDDLE LATITUDE, p. 149. This table is used in Navigation for correcting the middle latitude. The given middle latitude is to be found either in the first or last verti- cal column, opposite to which, and under the given difference of latitude, K; inserted the proper correction in minutes, to be added to the middle latitude to obtain the latitude in which the meridian distance is accu- rately equal to the departure. Thus, if the middle latitude is 41, and the difference of latitude 14, the correction will be found to be 25', which, ndded to the middle latitude, gives the corrected middle latitude 41 23' A TABLE Of LOGARITHMS OF NUMBERS FROM 1 TO 10,000. N. Log. N. Log. N. Log. N. Log. i 2 3 4 5 o.oooooo o.3oio3o 0.477121 0.602060 0.698970 26 27 28 o 9 3o .4i4973 .43i364 .447i58 .462398 .477121 5i 52 53 54 55 i. 7 o 7 5 7 o i .716003 i .724276 1.732394 i. 7 4o363 76 77 78 79 80 .880814 .886491 .892096 .897627 .903090 6 7 8 9 10 0.778151 0.845098 0.903090 0.954243 I. 000000 3i 32 33 34 35 .491362 .5o5i5o .5i85i4 .53x479 .544o68 56 5 7 58 5 9 60 1.748188 1.755876 1.768428 i .770852 i. 778161 8! 82 83 84 85 . 908486 .913814 .919078 .924279 .929419 ii 12 i3 1.4 i5 I. 041893 I .079181 1.113943 J. I46I28 1.176091 36 3 7 38 39 4o .5563o3 .568202 .579784 .5qio65 .602060 61 62 63 64 65 1.786330 i .792392 1.799341 i. 806180 i .812913 86 87 88 89 90 .934498 .939619 . 9 44483 .949390 .964243 16 17 18 '9 20 I.204I20 i.23o449 1.255273 i .278754 i.3oio3o 4i 42 43 44 45 .612784 .623249 .633468 .643453 .653213 66 67 68 69 70 1.819644 i .826076 1.832609 i.83884 9 1.846098 9 1 92 9 3 9 i 96 .969041 . 9 63 7 88 . 9 68483 .973128 .977724 21 22 23 24 r 1.322219 1.342423 1.361728 i.38o2ii 1.397940 46 47 48 49 5o .662758 .672098 .681241 .690196 .698970 7i 72 7 / 74 75 1.861268 1.867332 1.863323 1.869232 1.876061 96 97 98 99 IOO .982271 .986772 .991226 995635 2.OOOOOO N.B. In the following table, the two leading figures in the first column of logarithms are to be prefixed to all the numbers of the same horizontal line in the next nine columns ; but when a point (.) occurs, its place is to be supplied by a cipher, and the two leading figures are to be taken from the next lower line. LOGARITHMS OF .N u M B E K s. ~N~] j 1 2 3 4 || 5 6 7 8 9 D. 100 oooooo o434 0868 i3oi i?34 2166 25 9 8 3029 346i 38 9 i 43a 101 432i 4 7 5i SiSi 5609 6o38 6466 68 9 4 7 32I 7748 8i 7 4 428 102 8600 9 026 9 45i 9876 .3oo 7 24 1147 i5 7 o I 99 3 24i5 424 io3 oi283 7 325 9 368o 4ioo 452i 4 9 4o 53Co 5 779 6197 6616 419 io4 7 o33 7 45i 7868 8284 8700 9 ii6 9 532 9 9 4 7 .36i . 77 5 4x6 io5 02Il8 9 i6o3 2016 2428 2841 3252 3664 4o 7 5 4486 48 9 6 412 1 06 53o6 5 7 i5 6i25 6533 6942 7 35o 77 5 7 8i64 85 7 i 8 97 8 4o8 X0 7 9 384 9789 .i 9 5 .600 ioo4 i4o8 1812 2216 2619 3021 4o4 1 08 o33424 3826 4227 4628 5029 543o 583o 623o 6629 7028 4oo io 9 7 426 7 8 2 5 8223 8620 9017 9 4i4 9811 .20 7 .602 . 99 8 3 9 6 no o4i3 9 3 1787 2182 2 5 7 6 2969 3362 3 7 55 4i48 454o 4 9 32 3 9 3 in 5323 5 7 i4 6io5 6495 6885 7 2 7 5 7 664 8o53 8442 883o 38 9 112 9218 9606 999 3 .38o .766 n53 i538 1924 2309 2694 386 n3 o53o 7 8 3463 3846 4230 46i3 4996 53 7 8 5 7 6o 6142 6524 382 n4 6905 7286 7666 8o46 8426 88o5 9 i85 9 563 9942 .320 379 ii5 060698 io 7 5 i452 1829 2206 c582 2958 3333 3 7 o 9 4o83 3 7 6 116 4458 4832 5206 558o 5 9 53 6326 6699 7 o 7 i 7443 7 8i5 3 7 3 ii 7 8186 8557 8928 9298 9668 ..38 4<> 7 .776 ii45 i5i4 36 9 ix8 071882 225o 2617 2 9 85 3352 3 7 i8 4o85 445i 48i6 5i82 366 1x9 5547 5 9 I2 6276 664o 7004 7 368 77 3i 8094 845 7 8819 363 I2O 9181 9 543 9904 .266 .626 . 9 8 7 i34 7 1707 206-7 2426 36o 121 082785 3i44 35o3 386i 4219 45 7 6 4934 5291 564 7 6oo4 35 7 N. | 1 2 3 4 || 5 6 7 8 | 9 D. 434 43 87 i3o 1 7 4 2I 7 260 3o4 34 7 3ox 432 43 86 i3o i 7 3 216 259 302 346 38 9 43o 43 86 129 I 7 2 2l5 258 3oi 344 38 7 428 43 86 128 I 7 I 214 25 7 3oo 342 385 426 43 85 128 I 7 213 256 298 34i 383 424 42 85 127 I 7 O 212 254 297 33 9 382 422 42 84 127 169 211 2 53 295 338 38o 420 42 84 126 1 68 210 252 294 336 3 7 8 4x8 42 84 125 167 209 25l 293 334 3 7 6 4i6 42 83 125 1 66 208 250 291 333 3 7 4 4x4 4i S3 124 166 2O 7 248 290 33i 3 7 3 4l2 4i 8-2 124 i65 206 24 7 288 33o 371 4io 4i 82 123 1 64 2O5 246 287 328 36 9 4o8 4i 82 122 i63 2O4 245 286 326 36 7 4o6 4i 81 122 162 203 244 284 325 365 4o4 a 4o 81 121 162 202 242 283 323 364 402 e3 4o 80 121 161 201 a4i 281 322 362 g 4oo ^ 4o 80 120 1 60 2OO 24O 280 320 36o a 398 g 4o 80 II 9 160 199 23 9 279 3i8 358 CD 3 9 6 .2 < 4o 79 II 9 iSQ 198 238 277 8x7 356 | 3 9 4 g. 3 9 79 1X8 1 58 I9 7 236 276 3i5 355 " 3 9 2 8 3 9 78 118 i5 7 196 235 274 3x4 353 3 9 o &H 3 9 78 117 1 56 i 9 5 234 2 7 3 3l2 35i 388 3 9 78 116 i55 i 9 4 233 272 3io 349 386 3 9 77 116 i54 i 9 3 232 270 309 34 7 384 38 77 ii5 1 54 I 9 2 230 269 3o 7 346 38 2 38 76 ii5 i53 i 9 i 229 267 3o6 344 38o 38 76 4 l52 I 9 O 228 266 3o4 342 3 7 8 38 76 Ii3 i5i i8 9 22 7 265 302 34o 3 7 6 38 75 Ii3 i5o 188 226 263 3oi 338 3 7 4 37 75 112 i5o 187 224 262 299 33 7 3 7 2 3? 74 112 i4 9 1 86 223 260 298 335 3 7 o 37 74 III i48 i85 222 25 9 296 333 368 37 74 110 i4 7 1 84 221 258 294 33i 366 1 3 7 73 110 1 46 i83 220 256 2 9 3 329 364 36 73 I0 9 146 182 218 255 2 9 I 3a8 362 36 72 I0 9 i45 181 2I 7 253 2 9 O 826 36o i 36 72 1 08 1 44 180 216 352 288 324 LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 7 8 J 9 ID.] I2S oS636o 6716 7071 7426 77 8i 8i36 8490 8845 9198 9662 355 123 99 o5 .268 .611 . 9 63 i3i5 1667 2018 2370 2721 3071 35i 124 093422 3 77 2 4l22 447i 4820 6169 55i8 5866 6215 6662 349 125 6910 7267 7604 796! 8298 8644 8990 9335 9681 ..26 346 126 100371 0716 1069 i4o3 i 7 4 7 2091 2434 2777 3119 3462 343 127 38o4 4i46 4487 4828 6169 55io 585i 6191 65.? i 6871 3/i i 128 7210 7649 7888 8227 8565 8903 9241 9679 9916 .253 338 129 110690 0926 1263 1699 1934 2270 2606 2940 32 7 5 3609 335" i3o 3 9 43 4277 46n 4944 6278 56n 5 9 43 6276 6608 6940 333 i3i 7271 7603 79 34 8266 85 9 5 8926 9266 9686 99 i5 .245 33o 132 130674 o 9 o3 I23l 1660 1888 2216 2644 2871 3198 3525 328 i33 3852 4178 45o4 483o 5x56 548 1 58o6 6i3i 6456 6781 3 2 5 1 34 7106 742 9 77 53 8076 83 99 8722 9045 9 368 9690 . .12 3 2 3 :35 i3o334 o655 o 977 1298 i6i 9 I 9 3 9 2260 2680 2900 32I 9 321 1 36 3539 3858 4i 77 4496 48i4 5i33 545 1 6769 6086 64o3 3i8 i3 7 6721 7 o3 7 7 354 7 6 7 i 79 8 7 83o3 8618 8 9 34 9249 9 564 3i5 1 38 9879 .194 .5o8 .822 n36 i45o i 7 63 2076 2389 2702 3i4 i3 9 i43oi5 332 7 363 9 3961 4263 45 7 4 4885 6196 6607 6818 3n i4o 6128 6438 6 7 48 7 o58 7 36 7 7 6 7 6 79 85 8294 86o3 8911 309 Ui 9219 9627 9 835 .142 44 9 . 7 56 io63 1370 i6 7 6 1982 3o 7 i4a 162288 2694 2 9 OO 3 2 o5 35io 38i5 4l2O 4424 4 7 28 5o3 2 3o5 i43 5336 564o 5 9 43 6246 6549 6862 7 i54 7 45 7 77 5 9 8061 3o3 i44 8362 8664 8 9 65 9266 9 56 7 9868 .168 46 9 .-769 1068 3oi i45 i6i368 1667 I 9 6 7 2266 2664 2863 3i6i 346o 3 7 58 4o55 299 i46 4353 465o 4 9 4 7 6244 554i 5838 6i34 643o 6726 7022 297 i4 7 7 3i 7 7613 79 o8 8ao3 84 97 8792 9086 938o 9674 9968 296 i48 170262 o555 o848 n4i i434 1726 2019 23ll 26o3 2896 293 N. 1 2 3 4 5 6 7 8 9 D. 358 ' 36 7 2 107 i43 i 7 9 ai5 261 286 322 356 36 7* I0 7 i4a 178 214 249 286 320 354 35 71 1 06 142 I 77 212 248 283 3ig 352 35 7 1 06 i4i i 7 6 211 246 282 3i 7 35o { 35 7 o io5 i4o i 7 5 2IO 245 280 3i5 348 35 7 o io4 i3 9 i 7 4 209 244 278 3i3 346 35 69 io4 1 38 i 7 3 208 242 2 77 3n 344 34 69 io3 1 38 I 7 2 206 241 2 7 5 3io 342 34 68 io3 i3 7 I 7 I 206 239 2 7 4 3o8 34o 34 68 102 i36 I 7 2O4 238 272 3o6 338 34 68 101 1 35 I6 9 203 23 7 270 3o4 336 34 6 7 101 1 34 168 2O2 235 269 302 334 33 67 IOO 1 34 i6 7 2OO 234 267 3oi 332 g 33 66 IOO i33 166 I 99 232 266 299 33o 33 66 99 132 i65 I 9 8 23l 264 297 X 328 5 33 66 98 i3i 1 64 197 230 262 296 f 3a6 c , o 33 65 98 i3o i63 196 228 261 293 324 32 65 97 i3o 162 194 22 7 269 292 S 322 g. 32 64 97 129 161 193 225 268 290 320 2 32 64 96 128 1 60 192 224 266 288 3i8 ^ 32 64 9 5 127 169 191 223 264 286 3i6 32 63 9 5 126 168 190 221 253 284 3i4 3i 63 94 126 167 188 22O 261 283 3l2 3i 62 94 126 166 i8 7 218 260 281 3io 3i 62 9 3 124 155 186 2I 7 248 2 7 9 3o8 3i 62 92 123 1 54 i85 216 246 2 77 3o6 3i 61 92 122 i53 1 84 214 245 276 3o4 3o 61 9 1 122 162 182 2l3 243 2 7 4 302 3o 60 9 1 121 i5i 181 211 242 2 7 2 3oo 3o 60 9 I2O 1 56 180 210 240 2 7 298 3o 60 89 119 149 i -79 209 238 268 296 3o 5 9 89 118 i48 178 2O 7 23 7 266 294 i 29 5 9 88 118 i4 7 176 206 235 265 LOGARITHMS OF NUMBER:*. N. 1 2 3 4 5 6 7 8 | 9 D. 1 4y 173186 3478 3769 4o6o 435i -, 464 1 4932 5222 55i2 58o2 291 160 6091 638i 6670 6 9 5 9 7248 7 536 7825 8ii3 84oi 8689 289 i5i 8977 9264 9 552 9839 .126 .4i3 .699 .986 I2 7 2 i558 287 1 5?. 181844 2129 2 4i5 2700 2 9 85 3370 3555 383 9 4i23 44o 7 2 85 i53 46 9 i 497 5 525 9 5542 5825 6108 63 9 i 66 7 4 6 9 56 7 23 9 2 83 1 54 7621 7 8o3 8o84 8366 8647 8 9 28 9209 9 4 9 o 977i ..5i 281 i55 ioo332 0612 0892 1171 i45i 1730 2OIO 2289 256 7 2846 279 1 56 3i25 34o3 368i 3 9 5 9 423 7 45i4 4792 5o6 9 5346 5623 278 1*7 6900 6176 6453 6729 7000 7281 7556 7 832 8io 7 8382 276 168 8667 8932 9206 948i 97 55 ..29 .3o3 .5 77 .85o 1124 2 7 4 i5 9 201897 1670 1943 2216 2488 2761 3o33 33o5 35 77 3848 272 ;6o 120 43 9 i 4663 4934 5204 5475 5746 60:6 6286 6556 271 161 6826 7096 7365 7634 794 8i 7 3 844 1 8 7 io 8979 9 24 7 269 162 9 5i5 9783 ..5i .319 .586 .853 I 121 i388 i654 1921 267 i63 212188 2454 2720 2986 3252 35i8 3 7 83 4o4 9 43i4 45 7 9 266 1 64 4844 5 1 09 53 7 3 5638 6902 6166 643o 66 9 4 6 9 5 7 7 22I 264 i65 7484 7747 8010 8273 8536 8798 9060 9 323 9 585 9846 262 166 220108 0370 o63i 0892 n53 i4i4 i6 7 5 ig36 2I 9 6 2456 261 167 2716 2976 3236 3496 3 7 55 4oi5 42 7 4 4533 4792 5o5i 269 168 53o 9 5568 5826 6o84 6342 6600 6858 7 u5 7 3 7 2 7 63o 258 169 7887 8i44 84oo 865 7 8 9 i3 9 I 7 9426 9682 99 38 .i 9 3 256 170 230449 0704 0960 I2l5 1470 1724 i 979 2234 2488 2 7 42 254 171 2996 325o 35o4 3 7 5 7 4on 4264 45i 7 4 77 o 5o23 52 7 6 253 172 5528 5 7 8i 6o33 6285 653 7 6 7 8 9 7 o4i 7 2 9 2 7 544 779 5 262 i 7 3 8o46 8297 8548 8799 9 o4 9 9 2 99 9 55o 9800 ..5o .3oo 260 174 240649 0799 io48 1297 1 546 i 79 5 2044 2293 254i 2 79 249 i 7 5 3o38 3 2 86 3534 3782 4o3o 4277 4525 4 77 2 5019 5266 248 176 55i3 5769 6006 6262 6499 6 7 45 6 99 i 7237 7 482 77 28 246 177 797 3 8219 8464 8709 8 9 54 9198 9 443 9 68 7 9932 .176 f f 178 260420 o664 0908 n5i i3 9 5 i638 1881 2125 2368 2? X> 243 179 2853 3096 3338 358o 3822 4o64 43o6 4548 4 7 9o 5o3i 242 180 52 7 3 55i4 5 7 55 5 99 6 623 7 6477 6718 6 9 58 7 i 9 8 743 9 24 1 181 7679 7918 8:58 83 9 8 863 7 8877 9116 9 355 9^94 9833 239 N. 1 2 3 4 5 6 7 | 8 D. 292 f 29 58 88 117 i46 i 7 5 204 234 263 290 29 58 87 116 i45 i 7 4 203 232 261 288 29 58 86 IID i44 i 7 3 202 230 269 286 29 57 86 n4 i43 172 2OO 22 9 2D 7 284 28 5 7 85 n4 142 170 I 9 9 22 7 2 56 282 28 56 85 n3 i4i 169 197 226 254 280 28 56 84 112 i4o 168 196 224 252 278 28 56 83 III i3 9 167 i 9 5 223 25o 276 28 55 83 no i38 1 66 i 9 3 221 248 2?4 j: 27 55 82 no i3 7 i64 I 9 2 2I 9 247 272 -g 27 54 82 io 9 i36 i63 I 9 O 218 245 i 270 ^ 27 54 81 1 08 i35 162 I8 9 216 243 268 - 27 54 80 107 1 34 161 188 2l4 241 l< 266 |. 27 53 80 106 i33 1 60 186 213 239 I 264 26 53 79 1 06 i3a i58 i85 211 238 3 262 o^ 26 52 79 io5 i3i i5 7 i83 2IO 236 260 I 26 52 78 io4 i3o i56 182 208 234 258 Pk 26 52 77 io3 129 i55 181 206 232 2 56 26 5i 77 IO2 128 1 54 i 79 2O5 230 2 54 25 51 76 IO2 127 152 i 7 8 203 229 262 25 5o 76 101 126 i5i i 7 6 2O2 22 7 25o 25 5o 75 IOO 125 i5o i 7 5 2OO 225 248 25 5o 74 99 124 149 i 7 4 198 223 246 25 49 74 9 8 123 i48 I 7 2 197 221 a44 24 49 73 98 122 i46 I 7 I i 9 5 22O 242 24 48 73 97 121 i45 169 194 218 240 24 48 72 96 120 i44 1 68 192 216 LOGARITHMS OF NUMBERS. N. i 2 3 4 5 6 7 8 9 D. 182 260071 o3io o548 0787 IO25 1263 i5oi i 7 3 9 1976 22l4 2 38 i83 2461 2688 2925 3i6 2 33 99 3636 38 7 3 4109 4346 458a 2:7 i84 48i8 5o54 5290 5525 5 7 6i 5 99 6 6232 646 7 6702 6937 235 i85 7172 7406 7641 7 8 7 5 8110 8344 85 7 8 8812 9046 9279 234 186 9 5i3 9746 9980 .213 .446 .679 . 9 I2 n44 i3 77 1609 233 187 271842 2074 23o6 2538 2770 3ooi 3233 3464 36 9 6 3927 232 188 4:58 438 9 4620 485o 5o8i 53n 5542 5 77 2 6002 6232 230 189 6462 6692 6921 7 i5: 738o 7609 7 838 8o6 7 8296 85 2 5 229 190 8 7 54 8982 9211 9439 9667 9 8 9 5 .123 .35i .5 7 8 .806 228 191 28io33 I2DI i488 i 7 i5 1942 2169 23 9 6 2622 2849 8076 227 192 33oi 352 7 3753 3979 42o5 443 1 4656 4882 5107 5332 226 193 555 7 5 7 82 6007 6456 6681 6 9 o5 7 i3o 7354 7 5 7 8 225 u;4 7802 8026 8249 8473 8696 8920 9 i43 9 366 9 58 9 9812 223 i 9 5 290035 025 ? o48o 0702 0925 u4 7 i36 9 iSgi i8i3 2o34 222 196 2256 2478 2699 2920 3i4i 3363 3584 38o4 4o25 4246 221 197 4466 4687 4907 5127 5347 556 7 5 7 8 7 6oo 7 6226 6446 22O 198 6665 6884 7104 7323 7542 7761 7979 8198 84i6 8635 2I 9 199 8853 9071 9289 9 5 7 9725 9943 .161 .3 7 8 .5 9 5 .8i3 218 200 3oio3o 1247 1681 1898 2Il4 233i 2764 2980 2I 7 201 3i 9 6 34i2 3628 3844 4o59 4275 44 9 i 4706 4921 5i36 216 202 535i 5566 5781 5 99 6 6211 6425 663 9 6854 7068 7282 2!4 203 7496 7710 7924 8i3 7 835i 8564 8778 8901 9204 9417 2!3 204 9 63o 9 843 ..56 .268 .48i .6 9 3 . 9 o6 1118 i33o 1 542 212 205 3n 7 54 1966 2177 238 9 2600 2812 3o23 3234 3445 3656 211 206 386 7 4o 7 8 4289 4499 4710 4920 5i3o 534o 555i 5760 210 207 5970 6180 63 9 o 65 99 6809 7018 7227 7436 7 646 7854 2O 9 208 8o63 8272 848 1 8689 8898 9106 9522 97 3o 99 38 208 209 32oi46 o354 o562 0769 0977 n84 i3 9 i 1598 i8o5 2OI2 2O 7 210 2219 2426 2633 2889 3o46 3252 3458 3665 38 7 i 40 77 2O6 211 4282 4488 46 9 4 4899 5io5 53io 55i6 5721 5926 6i3i 2O5 212 6336 654i 6 7 45 6950 7 i55 7 35 9 7 563 7767 7972 8176 2O4 213 838o 8583 8787 8991 9194 9 3 9 8 9 6oi 9805 ...8 .211 203 2l4 33o4i4 0617 0819 IO22 1225 l42 7 i63o i83 2 2034 2236 2O2 2l5 2438 2640 2842 3o44 3246 344 7 3649 385o 4o5i 4253 2O2 216 4454 4655 4856 5o57 525 7 5458 5658 585 9 6o5 9 6260 201 217 646o 6660 6860 7060 7260 7459 7 65 9 7 858 8o58 8 2 5 7 2OO 218 8456 8656 8855 9054 9 253 945i 9849 ..4 7 .246 I 99 219 34o444 0642 o84i roSg I23 7 i435 i632 i83o 2028 2225 i 9 8 220 2423 2620 2817 3oi4 3212 3409 36o6 38o2 3 999 4196 i 97 N. 12 3 4 5 6 7 8 9 D. 238 r 24 48 7 1 9 5 119 i43 167 190 214 236 24 47 7 1 94 118 1 42 i'65 189 212 234 23 47 70 94 117 i4o 1 64 187 211 232 23 46 70 9 3 116 i3 9 162 186 209 230 23 46 69 9 2 n5 i38 161 1 84 2O 7 228 23 46 68 9 1 n4 i3 7 1 60 182 205 226 23 45 68 9 n3 i36 i58 181 203 224 S 22 45 67 9 112 1 34 i5 7 i 79 202 o5 222 ce 22 44 67 89 III i33 i55 i 7 8 2OO B U 220 " 22 44 66 88 no 132 1 54 176 I 9 8 i c c | &* J 218 .2' 22 44 65 87 io 9 i3i i53 i?4 196 216 | 22 43 65 86 108 i3o i5i I 7 3 194 214 21 43 64 86 107 128 i5o 171 I 9 3 212 p^ 21 42 64 85 1 06 I2 7 i48 170 2IO 21 42 63 84 io5 126 i4 7 1 68 189 208 21 42 62 83 io4 125 i46 166 187 2O6 21 4i 62 82 io3 124 i44 i65 i85 204 20 4i 61 82 IO2 122 i43 i63 1 84 (j 202 20 4o 61 81 101 121 i4i 162 182 j 20O 20 4o 60 80 IOO I2O i4o 1 60 180 I I 9 8 20 4o 5 9 79 99 I ! 9 i : 3g 1 58 i 7 8 LOGARITHMS OF N D M B E 11 i N - 1 2 3 4 5 6 7 8 9 D. 221 344392 458 9 4?85 4 9 8i 5i 7 8 53 7 4 55 7 o 5 7 66 5962 6:67 106 222 6353 654 9 6 7 44 6 9 3 9 7 i35 7 33o 7 5 2 5 7720 79 i5 8110 ,95 223 83o5 85oo 8694 888 9 9 o83 9 2 7 8 94 7 2 9666 9 86o ..54 i 9 4 224 35o248 o442 o636 o8a 9 1023 1216 i4io i6o3 i 79 6 i 9 8 9 I 9 3 225 2i83 2 3 7 5 2568 2 7 6l 2 9 54 3i4 7 333 9 3532 3 7 24 3 9 i6 226 4io8 43oi 44 9 3 4685 48 7 6 5o68 5260 5452 5643 5834 I 9 2 227 6026 6217 64o8 65 99 6 79 o 6 9 8i 7172 7 363 7 554 7744 I 9 I 228 79 35 8i25 83i6 85o6 86 9 6 8886 9 076 9266 9 456 9 646 I 9 O 229 9 835 ..25 .215 .4o4 .5 9 3 . 7 83 972 1161 i35o i53 9 189 230 361728 1917 2105 22 9 4 2482 26 7 I 285 9 3o48 3236 3424 188 23l 36i2 38oo 3988 4i 7 6 4363 455i 4 7 3 9 4926 5n3 53oi 23a 5488 56 7 5 5862 6o4 9 6 2 36 6423 6610 6 79 6 6 9 83 7i6 9 187 233 7 356 7542 7729 79 i5 8101 828-7 84 7 3 865 9 8845 9 o3o 1 86 234 9216 9401 9 58 7 9772 99 58 .i43 .328 .5i3 .6 9 8 .883 i85 235 371068 1253 i43 7 1622 1806 1991 2I 7 5 236o 2 544 2728 1 84 236 2912 3o 9 6 3280 3464 364 7 383i 4oi5 4i 9 8 4382 4565 23 7 4748 4932 5n5 52 9 8 548 1 5664 5846 6o2 9 6212 63 9 4 i83 238 65 77 6 7 5 9 6 9 42 7124 7 3o6 7 488 7 6 7 o 7 85 2 8o34 82:6 182 239 8398 858o 8 7 6i 8 9 43 9 I24 9306 9487 9 668 9 84 9 ,.3o 181 240 38o2ii o3 9 2 o5 7 3 o 7 54 o 9 34 ui5 1296 i4 7 6 i656 i83 7 241 2017 2197 2 3 77 255 7 2 7 3 7 2 9 I 7 3o 97 32 77 3456 3636 180 242 38i5 3 99 5 4i 7 4 4353 4533 4 7 I2 48 9 i 5o 7 o 5 2 4 9 5428 i 79 243 56o6 5?85 5 9 64 6142 632i 64 99 66 77 6856 7 o34 7 2I2 178 44 7 3 9 7 568 7746 7923 8101 82 79 8456 8634 88n 8 9 8 9 245 9166 9 343 9 52O 9 6 9 8 9 8 7 5 ..5i .228 .4o5 .582 . 7 5 9 177 246 390935 III2 1288 i464 i64i i8i 7 i 99 3 2i6 9 2345 2521 176 247 2697 2873 3o48 3224 34oo 35 7 5 3 7 5i 3926 4ioi 42 77 248 4452 4627 4802 4977 5i5 2 5326 55oi 56 7 6 585o 6o25 i 7 5 249 6199 63 7 4 6548 6 7 22 68 9 6 7 o 7 i 7 245 7 4i 9 7 5 9 2 7766 i 7 4 25o 794o 8114 828-7 846 1 8634 8808 8 9 8i 9 i54 9 328 9 5oi i 7 3 25! 9674 9847 . .20 . I 9 2 .365 .538 . 7 n .883 io56 1228 262 4oi4oi i5 7 3 i 7 45 1917 2o8 9 2261 2433 26o5 2 777 2 9 4 9 I 7 2 2 53 3l2I 3292 3464 3635 38o 7 3 97 8 4i4 9 4320 44 9 2 4663 I7 1 254 4834 5oo5 5i 7 6 5346 55i 7 5688 5858 6o2 9 6i 99 63 7 o 255 654o 6710 6881 7o5i 7 22I 7 3 9 i 7 56i 77 3i 79 oi 8070 170 256 8240 84io 85 79 8 7 49 8 9 i8 9 o8 7 9 25 7 9 4s6 9 5 9 5 97 64 i6 9 257 99 33 .102 .2 7 I .44o .6o 9 777 . 9 46 iu4 1283 i45i 2 58 411620 I 7 88 i 9 56 2124 22 9 3 2461 262 9 2 79 6 2964 3i32 168 25 9 33oo 346 7 3635 38o3 3 97 o 4i3 7 43o5 44 7 a 463 9 48o6 167 260 4973 5i4o 53o 7 54 7 4 564i 58o8 5 97 4 6i4i 63o8 64 7 4 261 664 1 6807 6 97 3 7 i3 9 7 3o6 7 4 7 2 7 638 7804 7970 8i35 1 66 262 83oi 846 7 8633 8 79 8 8 9 64 9 I2 9 9 2 9 5 9 46o 9 625 979 i i65 263 9956 .121 .286 .45i .616 .781 . 9 45 IIIO 1275 i43 9 N, 1 2 3 4 || 5 6 7 8 9 D. 196 f '20 3 9 5 9 78 9* 118 i3 7 i5 7 J 7 6 194 *9 3 9 58 78 97 no 1 36 i55 i 7 5 ?92 J 9 38 58 77 96 n5 1 34 1 54 178 190 J 9 38 57 76 9 5 ii4 i33 i5a 171 188 I 9 28 56 75 94 ii3 132 i5o 169 186 g 9 3 7 . 56 74 9 3 112 i3o 1 49 167 <5 1 84 18 37 55 74 9 2 no ,I2 9 i4 7 1 66 g 182 -3 18 36 55 73 9 1 IO 9 I2 7 i46 1 64 2- 180 < 18 36 54 72 90 1 08 126 i44 162 i?3 S 18 36 53 71 9 io 7 125 142 160 5 176 g, 18 35 53 70 88 1 06 123 i4i i58 174 I J 7 35 52 7 87 io4 122 i3 9 i5 7 172 17 34 52 69 86 io3 120 i38 1 55 170 '7 34 5i 68 85 IO2 II 9 i36 i53 168 *7 34 5o 67 84 101 118 i34 i5i 166 17 33 5o 66 83 100 116 1 33 x4 9 164 16 33 49 66 fa 9 8 n5 i3i 1 48 LOGARITHM? OF NtM B E R S. 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N. 1 2 3 4 5 6 7 8 | 9 D. 3l2 494 i 55 4294 4433 45 7 2 4711 485o 4 9 8 9 5i 2 8 5267 54o6 i3 9 3i3 5544 5683 5822 5 9 6o 6o 99 6238 63 7 6 65i5 6653 6791 X 3i4 6 9 3o 7068 7206 7 344 7 483 7621 77 5 9 7897 8o35 8i 7 3 i38 ( 3i5 83n 8448 8586 8724 8862 8 999 9 i3 7 9275 9412 9 55o 3i6 9687 9824 9962 99 .236 3 7 4 .5n .648 : 7 85 .922 i3 7 3i 7 SoioSg 1196 i333 1470 i6o 7 i 7 44 1880 2017 2i54 2291 3i8 2427 2564 2700 2 83 7 2 97 3 3io 9 3246 238 35i8 3655 i26 3i 9 3 79 i 3927 4o63 4i 99 4335 44 7 i 4607 4743 48 7 8 5oi4 320 5i5o 5 2 86 5421 555 7 56 9 3 58 2 8 5 9 64 6o 99 6234 63 7 o 321 65o5 664o 6776 6 9 n 7 o46 7 i8i 7 3i6 7 45i 7 586 77 2I i35 322 7 856 799 * 8126 8260 83 9 5 853o 8664 8 799 8 9 34 9068 323 9 203 9 33 7 9471 9606 97 4o 9 8 7 4 ...9 .i43 2 77 .4ii 1 34 324 5io545 0679 o8i3 0947 1 08 1 I2l5 1 349 1482 1616 I 7 5o 325 i883 2017 2l5l 2284 2418 255i 2684 2818 2g5i 3o84 i33 326 32i8 335i 3484 36i 7 3 7 5o 3883 4oi6 4i4 9 4282 44i5 32 7 4548 468 1 48:3 4946 5o 79 5211 5344 5476 56o 9 5 7 4i 328 58 7 4 6006 6i3 9 6271 64o3 6535 6668 6800 6 9 32 7064 132 329 7196 7828 7460 7 5 9 2 77 24 7 855 7987 8119 8 2 5i 8382 33o 85:4 8646 8777 8909 9 o4o 9171 93o3 9 434 9 566 9 6 97 i3i 33i 9828 99 5 9 ..90 .221 .353 .484 .6i5 . 7 45 .8 7 6 IOO 7 33 2 52ii38 1269 i4oo i53o 1661 I 79 2 1922 2o53 2i83 23i4 333 s444 25 7 5 2705 2835 2966 3o 9 6 3226 3356 3486 36i6 i3o 334 3746 3876 4oo6 4i36 4266 43 9 6 45 2 6 4656 4 7 85 4gi5 335 5o45 5i 7 4 53o4 5434 5563 56 9 3 5822 5 9 5i 6081 6210 129 336 633 9 646 9 65 9 8 6727 6856 6 9 85 7114 7243 7 3 7 2 7 5oi 33 7 " 7 63o 77 5 9 7888 8016 8i45 8 27 4 8402 853i 8660 8 7 88 338 8917 9o45 9174 9302 943o 9 55 9 9687 9 8i5 9943 ..72 128 339 53o2oo o328 o456 o584 7 I2 o84o 0968 io 9 6 1223 i35i 34o 1479 1607 1734 1862 1990 2117 2245 2372 25oo 2627 34i 2 7 54 2882 3009 3i36 3264 3391 35i8 3645 3 77 2 38 99 127 342 4026 4i53 4280 44o7 4534 466i 4 7 8 7 4 9 i4 5o4i 5i6 7 343 5 29 4 542i 5547 56 7 4 58oo 5 9 2 7 6o53 6180 63o6 6432 126 344 6558 6685 6811 6 9 3 7 7 o63 7189 7 3i5 744i 7 56 7 7 6 9 3 345 7819 7945 8071 8197 832s 8448 85 7 4 86 99 88 2 5 8951 346 9076 9202 9 32 7 9452 9 5 7 8 97 3 9829 99 54 79 .204 125 347 5^0339 o455 o58o 0705 o83o o 9 55 1080 I2O5 i33o i454 348 i5 79 1704 1829 i 9 53 20 7 8 2203 2327 2452 2576 2701 349 2825 2950 3o 7 4 3i 99 3323 3447 35 7 i 36 9 6 3820 3o44 124 35o 4o68 4192 43i6 444o 4564 4688 4812 4 9 36 5o6o 5i83 35i 53o 7 543 1 5555 56 7 8 58o2 5 9 25 6049 6172 6296 6419 352 6543 6666 6 7 8 9 6 9 i3 7 o36 7i5 9 7282 7 4o5 7 5s 9 7652 123 353 7775 7898 8021 8i44 8 2 6 7 838 9 85i2 8635 8 7 58 8881 354 9003 9126 9249 9 3 7 i 9 4 9 4 9 6i6 9739 9861 9 9 84 .106 355 55o2 2 8 o35i o4 7 3 o5 9 5 o 7 : 7 o84o 0962 1084 1206 1328 122 356 i45o 1572 i6 9 4 1816 I 9 38 2060 2181 2 3o3 2425 2547 35 7 2668 2790 2 9 II 3o33 3i55 32 7 6 33 9 8 35i9 364o 3762 121 358 3883 4oo4 4l26 4247 4368 448 9 46io 4 7 3i 4852 4973 35 9 5094 52i5 5336 545 7 55 7 8 56 99 5820 5940 6061 6182 36o 63o3 6423 6544 6664 6 7 85 6905 7026 7 i46 7267 7 38 7 120 36 1 7 5 7 7627 77 48 7868 79 88 8108 8228 8349 8469 8589 N. 1 2 3 4 || 5 6 | 7 8 9 D. 1 38 i4 28 4i 55 69 83 97 no 124 i36 i4 2 7 4i 54 68 82 96 109 122 1 34 fe i3 2 7 4o 54 67 80 94 107 121 f 132 fit i3 26 4o 53 66 79 92 1 06 119 03 , i3o -3 a , i3 26 39 52 65 78 9 1 io4 117 1 128 -2 i3 26 38 5i 64 77 9 IO2 ii5 Qi 126 | i3 25 38 5o 63 76 88 101 ii3 124 12 25 37 5o 62 74 87 99 112 j 122 & 12 24 37 40 61 73 85 98 110 (iao 12 24 36 48 60 72 84 96 108 UOGARITHMS OF NUMBERS. N. i 2 3 4 5 6 7 8 9 D. 1 " 362 558 7 o 9 882 9 8948 9068 9 i88 9 3o8 9428 9548 9 667 9787 120 363 997 ..26 .i46 .265 .385 .5o4 .624 . 7 43 .863 . 9 82 364 56noi 1221 1 34o i459 i5 7 8 1698 1817 I 9 36 2o55 2174 119 365 22 9 3 24l2 253i 265o 276 9 2887 3oo6 3i25 3244 3362 366 348 1 36oo 3 7 i8 383 7 3 9 55 4074 4192 43ii 4429 4548 5<5 7 4666 4 7 84 4903 5021 5i3 9 5 2 5 7 5376 5494 56i2 5 7 3o 118 368 5848 5 9 66 6084 6202 6320 643 7 6555 66 7 3 6 79 i 6 9 o 9 36 9 7026 7 i44 7262 7379 7497 7614 77 32 7 849 79 6 7 8o84 3 7 o 8202 83i 9 8436 8554 8671 8788 8905 9023 9 i4o 9 25 7 117 3 7 i 9 3 7 4 9 4 9 i 9608 9725 9 842 99 5 9 ..76 .193 .3o 9 .426 3 7 2 5 7 o543 0660 0776 0893 IOIO 1126 1243 i35 9 1476 I0 9 2 3 7 3 I 7 o 9 i825 1942 2o58 2174 2291 2407 2523 263 9 2 7 55 116 3 7 4 28 7 2 2 9 88 3io4 322O 3336 3452 3568 3684 38oo 3 9 i5 3 7 5 4o3i 4i4 7 4263 4379 44 9 4 46 10 4726 484i 4957 5072 3 7 6 5i88 53o3 54i9 5534 565o 5 7 65 588o 5 99 6 6111 6226 n5 377 634i 645 7 6572 6687 6802 6917 7o32 7 i4 7 7262 7377 3 7 8 7 4 9 2 7 6o 7 7722 7836 7 9 5i 8066 8181 82 9 5 84io 85 2 5 3 79 863 9 8 7 54 8868 8983 997 O2I2 9 3 2 6 9 44i 9 555 9 66 9 n4 38o 9784 9 8 9 8 . . 12 .126 .241 .355 .46 9 .583 .6 9 7 .811 38i 58o 92 5 io3 9 ii53 1267 i38i i495 1608 I 7 22 i836 i 9 5o 382 2o63 2I 77 2291 2404 25i8 263i 2 7 45 2858 2 9 72 3o85 383 3i 99 33i2 3426 353 9 3652 3 7 65 3879 3 99 2 4io5 4218 ii3 384 433i 4444 455 7 4670 4783 48 9 6 5009 5l22 5235 5348 385 546i 55 7 4 5686 5 799 5 9 I2 6024 6i3 7 625o 6362 6475 386 658 7 6700 6812 6 9 25 7 o3 7 7149 7262 7 3 7 4 7 486 7 5 99 112 38 7 77 ii 7823 79 35 8047 8160 8272 8384 84 9 6 8608 8720 388 8832 8 9 44 go56 9167 9279 9 3 9 i 9 5o3 9 6i5 97 26 9 838 38 9 99 5o ..61 .i 7 3 .284 .3 9 6 .507 .619 . 7 3o .842 . 9 53 3 9 o , 5 9 io65 2:76 1287 i3 99 i5io 1621 1732 i843 i 9 55 2066 III 3 9 i ! 2i 77 2288 2 3 99 25lO 2621 2732 2843 2 9 54 3o64 3i 7 5 3 9 2 3 2 86 33 97 35o8 36i8 3729 384o 3950 4o6i 4171 4282 3 9 3 43 9 3 45o3 46i4 4724 4834 4945 5o55 5i65 52 7 6 5386 IIO 3 9 4 54 9 6 56o6 6717 582 7 5 9 3 7 6047 6i5 7 626 7 63 77 648 7 3 9 5 65 97 6707 6817 6 9 2 7 7 o3 7 7146 7 256 7 366 7476 7 586 3 9 6 7 6 9 5 7 8o5 7914 8024 8i34 8243 8353 8462 8572 8681 3 9 7 8 79 i 8 9 oo 9009 9 II 9 9228 9337 9446 9 556 9 665 9774 109 3 9 8 9 883 999 2 .101 .210 .3i 9 .428 .53 7 .646 . 7 55 .864 3 99 6oo 97 3 1082 1191 I2 99 i4c8 i5i7 1625 i 7 34 i843 i 9 5i 4oo 2060 2l6 9 2277 2386 2494 26o3 2 7 II 28l 9 2928 3o36 1 08 4oi 3i44 3 2 53 336i 346 9 35 77 3686 3 79 4 3 9 O2 4oio 4n8 402 4226 4334 4442 455o 4658 4 7 66 48 7 4 4 9 82 5o8 9 5i 97 4o3 53o5 54 1 3 552i 5628 5 7 36 5844 5 9 5i 6o5 9 6166 62 7 4 4o4 essi 648 9 65g6 6704 6811 6919 7026 7 i33 7241 7 348 I0 7 4o5 7 455 7562 7669 7777 7 884 7991 8098 82o5 83i2 84i 9 4o6 8526 8633 8740 8847 8 9 54 9061 9 l6 7 9 274 938i 9 488 4o 7 9 5 9 4 9701 9808 99 i4 . .21 .128 .234 .34i .447 .554 4o8 610660 0767 o8 7 3 979 1086 1192 1298 i4o5 i5ii 161-7 1 06 N. | 1 2 3 4 || 5 6 7 8 9 D. IIO 12 24 36 48 60 71 83 9 5 I0 7 118 12 24 35 47 5 9 7 1 83 94 1 06 117 12 23 35 47 5 9 7 82 94 io5 zi6 12 23 35 46 58 7 81 9 3 104 n5 12 23 35 46 58 69 81 9 2 104 8 n4 - II 23 34 46 5? 68 80 9 1 io3 1 n3 | II 23 34 45 57 68 79 9 102 *c 112 II 22 34 45 56 6 7 78 9 101 p III Q. II 22 33 44 56 67 78 8 9 IOO 110 2 II 22 33 44 55 66 77 88 99 109 ^ II 22 33 44 55 65 76 87 98 1 08 II 22 32 43 54 65 76 86 97 107 i ii 21 32 43 54 64 75 86 96 10 LOGARITHMS OP NUMBERS. N. 1 2 3 4 5 6 7 8 9 D. 409 611723 l82 9 i 9 36 2O42 2148 2254 236o 2466 25 7 2 2678 1 06 4io 2784 28 9 o 2 99 6 3l02 3207 33i3 34i9 35 2 5 363o 3 7 36 4n 3842 3 9 47 4o53 4i5q 4264 43 7 o 4475 458i 4686 4-792 412 48 97 5oo3 5io8 52i3 53i 9 5424 552 9 5634 5 7 4o 5845 io5 4i3 5 9 5o 6o55 6160 6265 6370 64 7 6 658i 6686 6 7 9o 68 9 5 4i4 7000 7io5 7210 7 3i5 7420 7525 7629 77 34 7 83 9 7943 4i5 8o48 8i53 8 2 5 7 8362 8466 85 7 i 8676 8780 8884 8989 4i6 9 o 9 3 9 i 9 8 9 3o2 9 4o6 95n 9 6i5 9719 0824 9928 .Tall 104 4ry 62oi36 O24O o344 o448 o552 o656 0760 o864 0968 1072 4i8 1176 1280 1 384 i488 1592 i6 9 5 1799 I 9 o3 2OO 7 2IIO 419 22l4 23i8 2421 2525 2628 273,2 2835 2 9 3 9 3o42 3i46 420 324 9 3353 3456 355 9 3663 3 7 66 3869 3 97 3 4o 7 6 4179 io3 4-21 4282 4385 4488 45 9 i 46 9 5 4 7 98 4901 5oo4 5io 7 5210 422 53i2 54i5 55i8 562i 5724 582 7 5929 6o32 6i35 6238 428 634o 6443 6546 6648 6751 6853 6956 7 o58 7161 7263 424 7366 7 468 7571 7 6 7 3 7775 7878 7980 8082 8i85 8c8 7 102 425 8389 84 9 i 85 9 3 86 9 5 8797 8900 9002 9 io4 9206 9 3o8 426 9 4io 9 5l2 9 6i3 9 7 i5 9817 9919 . .21 .123 .224 .326 427 63o428 o53o o63i o 7 33 o835 o 9 36 io38 n3 9 1241 1 342 428 i444 i545 1647 1748 1849 ig5i 2O52 2i53 2255 2356 101 429 2457 255 9 2660 2761 2862 2963 3o64 3i65 3266 336 7 43o 3468 356 9 36 7 o 3 77 i 38 7 2 3 97 3 4074 4i 7 5 4276 43 7 6 43i 4477 45 7 8 46 7 9 4779 488o 4981 5o8i 5i8 2 5 2 83 5383 IOO 432 5484 5584 5685 5 7 85 5886 5986 6087 6i8 7 628 7 6388 433 6488 6588 6688 6789 6889 6989 7089 7 i8 9 7 2 9 o 7 3 9 434 74 9 o 75 9 o 7600 779 7890 799 8090 8i 9 o 8290 838 9 435 848 9 858 9 868 9 8789 8888 8988 9088 9 i88 9 28 7 9 38 7 99 436 9 486 9 586 9 686 9785 9 885 9984 ..84 .i83 .283 .382 43 7 64o48i o58i 0680 0779 0879 0978 1077 1177 I2 7 6 i3 7 5 438 1474 1573 1672 1771 1871 i9 7 o 2069 2168 2267 2366 43 9 2465 2563 2662 2761 2860 2 9 5 9 3o58 3i56 3255 3354 44o 3453 355i 365o 3 7 4 9 384 7 3 9 46 4o44 4i43 4242 434o 98 44i 443 9 453 7 4636 4 7 34 483 2 4g3i 5029 5127 5226 5324 442 5422 552i 56i 9 5717 58i5 5 9 i3 60 1 1 6110 6208 63o6 443 64o4 65o2 6600 6698 6796 68 9 4 6992 7o8 9 7187 7 285 444 7 383 748 1 7 5 79 7676 7774 7 8 7 2 7969 8067 8i65 8262 445 836o 8458 8555 8653 8 7 5o 8848 8 9 45 9 o43 9i4o 9 23 7 97 446 9 335 9 432 9 53o 9627 9724 9821 9919 ..16 .n3 .210 447 65o3o8 o4o5 o5o2 0599 0696 o 79 3 0890 o 9 8 7 1084 1181 448 1278 1375 1472 i56g 1666 I 7 62 i85 9 i 9 56 2o53 2i5o 449 2246 2343 2440 2536 2633 2 7 3o 2826 2 9 23 3019 3n6 45o 32i3 33o 9 34o5 35o2 35 9 8 36 9 5 3 79 i 3888 3 9 84 4o8o 96 45i 4177 42 7 3 436 9 4465 4562 4658 4 7 54 485o 4946 5o42 45 2 5i38 5235 533i 5427 5523 5619 5 7 i5 58io 5906 6002 453 6o 9 8 6i 9 4 62 9 6386 6482 65 77 66 7 3 6 7 6 9 6864 6960 454 7 o56 7162 7247 7343 7438 7 534 7 62 9 7725 7820 7916 455 Sou 8107 8202 8298 83 9 3 8488 8584 86 79 8 77 4 88 7 o 9 5 456 8 9 65 9 o6o 9 i55 925o 9 346 944i 9 536 9 63i 9726 9821 45 7 99 i6 . .11 . 1 06 .201 .296 .3 9I .486 .58i .676 . 77 i N. I 2 3 4 5 6 7 8 9 1). 106 ii 21 32 42 53 64 74 85 9 5 io5 ii 21 32 42 53 63 74 84 9 5 1 104 g 10 21 3i 42 52 62 73 83 9 4 io3 10 21 3i 4i 52 62 72 82 9 3 g 102 _ 10 2O 3i 4i 5i 61 7i 82 92 ^ 101 g , *" j O 10 20 3o 4o 5i 61 7 1 81 9 1 ,0) ' fe 100 tj IO 2O 3o 4o 5o 60 7 80 9 6 99 c. IO 2O 3o 4o 5o 5 9 69 79 So S 8 I IO 20 29 3 9 49 5 9 69 78 88 97 * IO *9 29 3 9 49 58 68 78 87 96 IO '9 2Q 38 48 58 67 77 86 LOGARITHMS OF IN UMBERS. N. 1 2 3 4 1 5 6 7 8 9 -*H 458 66o865 0960 io55 ii5o 1245 i33 9 i434 l52 9 1623 1718 9H 409 i8i3 1907 2OO2 2O 9 6 2I 9 I 2286 2 38o 24 7 5 256 9 2663 46o 2 7 58 2852 2947 3o4i 3i35 323o 3324 34i8 35i2 3607 94 46i 3701 3 79 5 388 9 3 9 83 4078 4172 4266 436o 4454 4548 462 4642 4736 483o 4 9 24 5oi8 5lI2 5206 52 99 53 9 3 548 7 463 558i 56 7 5 5769 5862 5 9 56 6o5o 6i43 623 7 633i 6424 464 65i8 6612 6705 6 799 6892 6 9 86 779 7173 7266 736o 465 7453 7546 7640 77 33 7826 79 20 8oi3 8106 8i 99 82 9 3 93 466 8386 8479 85 7 2 8665 8 7 5 9 8852 8 9 45 9 o38 9 i3i 9224 46 7 9 3i 7 9410 9 5o3 9 5 9 6 9689 0782 9 8 7 5 99 6 7 ..60 .i53 468 670246 oSSg o43i OD24 0617 0710 0802 o8 9 5 o 9 88 1080 46 9 1173 1265 i358 i45i j543 i636 1-728 1821 I 9 i3 2OO5 4yo 2098 2190 2283 2375 2467 256o 2652 2 7 44 2836 2 9 2 9 92 4yi 3021 3n3 32o5 32 97 3390 3482 35 7 4 3666 3 7 58 385o 472 3 9 42 4o34 4126 4218 43io 4402 44 9 4 4586 46 77 47 6 9 4?3 486i 4 9 53 5o45 5i37 5228 5320 54i2 55o3 55 9 5 568 7 4?4 5 77 8 5870 5962 6o53 6i45 6236 6328 64i 9 65n 6602 475 66 9 4 6785 6876 6 9 68 7 o5 9 7i5i 7242 -,333 74a4 7 5i6 9i 476 7607 7698 7789 7881 797 2 8o63 8i54 8245 8336 8427 477 85i8 86o 9 8700 8 79 i 8882 8973 9 o64 9 i55 9 2/|6 933 7 478 9428 9 5i 9 9610 9 7oo 979 i 9.882 9973 ..63 .154 .245 479 68o336 0426 o5i7 0607 o6 9 8 0789 o8 79 o 97 o 1060 n5i 48o 1241 i33 2 l422 i5i3 i6o3 1693 i 7 84 i8 7 4 i 9 64 2o55 90 48 1 2i45 2235 2326 2416 25o6 2596 2686 2777 286-7 20.57 482 3o47 3i37 3227 33i 7 34o 7 34 9 7 358 7 36 77 3 7 6 7 385 7 483 3 9 47 4o3 7 4127 421 7 43o 7 43 9 6 4486 45 7 6 4666 4 7 56 484 4845 4 9 35 5o 2 5 5n4 52o4 52 9 4 5383 5473 5563 5652 485 5742 583i 5921 6010 6100 6189 62 79 6368 6458 654 7 89 486 6636 6726 68i5 6 9 o4 6 99 4 7 o83 7 I 7 2 7261 7 35i 7 44o 48 7 7 52 9 7618 777 779 6 7 886 797 5 8o64 8i53 8242 833i 488 8420 85o 9 85 9 8 8687 8 77 6 8865 8 9 53 9 042 9 i3i 9 220 48 9 9309 9 3 9 8 9 486 9 5 7 5 9 664 9753 9 84i 99 3o ..i 9 .I0 7 490 690196 0285 o3 7 3 0462 o55o o63g 7 28 0816 o 9 o5 o 99 3 491 1081 1170 1258 1 347 i435 i5 2 4 1612 1700 i 7 8 9 i8 77 88 492 i 9 65 2o53 2l42 223o 2 3i8 2406 24 9 4 2583 26 7 I 2 7 5 9 493 2847 2 9 35 3o23 3in 3i 99 328 7 33 7 5 3463 355i 363 9 494 3727 38i5 3 9 o3 3 99 i 4o 7 8 4i66 4254 4342 443o 45i 7 495 46o5 46 9 3 4781 4868 4 9 56 5o44 5i3i 5219 53o 7 53 9 4 496 6482 5569 565 7 5744 5832 5 9 i 9 6oo 7 6o 9 4 6182 6269 87 497 6356 6444 653i 6618 6706 6 79 3 6880 6 9 68 7 o55 7142 498 7229 7 3i 7 7 4o4 7491 7 5 7 8 7665 77 52 7 83 9 79 26 8oi4 499 8 1 01 8188 8275 8362 844 9 8535 8622 8 7 o 9 8796 8883 5oo 8970 9 o5 7 9 i44 9 23l 9 3i 7 9 4o4 9 4 9 i 9 5 7 8 9664 97DI 5oi o838 9924 . . ii ..08 .184 .2 7 I .358 .444 .53i .617 502 700704 0790 0877 o 9 63 io5o n36 1222 i3o 9 i3 9 5 i48a 86 5o3 i568 i654 1741 1827 I 9 i3 i 999 2086 2172 2258 2344 5o4 243i 2617 26o3 268 9 2 77 5 2861 2 9 4 7 3o33 3119 32o5 5o5 3291 33 77 3463 354 9 3635 3 7 2I 38o 7 38 9 3 3 979 4o65 5o6 4i5i 4236 4322 44o8 44 9 4 45 79 4665 4?5i 483 7 4 9 22 607 5oo8 5o 9 4 6179 5265 535o 5436 5522 56o 7 56 9 3 5 77 8 N. 1 2 \ 3 4 5 6 7 8 9 "57" r 9 5 10 J 9 29 38 48 57 67 76 86 94 * 9 I 9 28 38 47 56 66 75 85 93 1 9 I 9 28 37 47 56 65 74 84 8 92 * 9 18 28 37 46 55 64 74 83 a g 91 1 9 18 27 36 46 55 64 73 82 90 -2 ' 9 18 27 36 45 54 63 72 81 ^3 5 89 S: 9 18 27 36 45 53 62 71 80 rH 88 2; 9 18 26 35 44 53 62 7 79 87 9 17 26 35 44 52 61 7 78 86 I 9 I? 26 34 43 5s 60 69 77 12 L O C, R I T II M 3 OF N U M B E K S. N. 1 2 3 4 5 6 7 8 9 D. 5o8 yo5864 5949 6o35 6120 6206 62 9 I 6376 6462 654 7 6632 65 609 6718 68o3 6888 6 97 4 7 o5 9 7144 7229 7 3i5 74oo 7485 5io 7 5 7 7655 774o 7826 79" 799 6 8081 8166 8 2 5i 8336 5n 8421 85o6 85 9 i 8676 8761 8846 8 9 3i 9 oi5 9 ioo 9i85 5l2 9270 9355 944o 9 524 9 6o 9 9 6 9 4 9779 9 863 99 48 ..33 5i3 710117 02O2 0287 0371 o456 o54o 0625 0710 o 79 4 0879 5i4 0963 io48 Il32 1217 i3oi i385 1470 i554 i63 9 1723 84 5i5 1807 1892 1976 2060 2i44 222 9 23i3 2 3 97 2481 2566 5i6 265o 2 7 34 2818 2 9 O2 2 9 86 307O 3i54 3 2 38 3323 34o 7 5r 7 349i 35 7 5 3659 3742 38 2 6 3 9 io 3 99 4 4o 7 8 4162 4246 5iS 433o 44i4 4497 458i 4665 474 9 4833 4 9 i6 5ooo 5o34 619 5i6 7 525i 5335 54i8 55o2 5586 566 9 5 7 53 5836 5920 620 6oo3 6087 6170 6254 633 7 6421 65o4 6588 6671 6 7 54 83 621 G838 6921 7004 7088 7171 7254 7338 7421 75o4 7 58 7 622 7671 7754 7 83 7 7920 8oo3 8086 8i6 9 8253 8336 8419 5 2 3 85o2 8585 8668 8 7 5i 8834 8 9 i 7 9 ooo 9083 9 i65 9248 624 933i 94i4 9497 9 58o 9 663 9745 9 828 9911 9994 77 525 720159 0242 o325 0407 o4 9 o o5 7 3 o655 o 7 38 0821 0903 5 2 6 0986 1068 n5i 1233 i3i6 i3 9 8 1 48 1 i563 1 646 I 7 28 82 52 7 1811 i8 9 3 1976 2o58 2l4o 2222 23o5 238 7 246 9 2552 5 2 3 2634 2716 2798 2881 2 9 63 3o45 3127 3209 32 9 i 33 7 4 529 3456 3538 3620 3702 3 7 84 3866 3 9 48 4o3o 4112 4194 53o 4276 4358 444o 4522 46o4 4685 4767 4849 4 9 3i 5oi3 53i 5o 9 5 5i 7 6 5 2 58 534o 5422 55c3 5585 566? 5 7 48 583o 532 5912 5 99 3 6075 6i56 6238 6320 64oi 6483 6564 6646 533 6727 6809 6890 6972 7 o53 7i34 7216 7 2 97 7 3 79 7 46o 81 534 ?54i 7623 774 77 85 7866 7948 8o2 9 8110 8i 9 i 8 2? 3 535 8354 8435 85i6 85 9 7 8678 8 7 5 9 884i 8922 9 oo3 9084 536 9i65 9246 9 3 27 9 4o8 9489 9 5 7 o 9 65i 9732 9 8i3 9 8 9 3 53 7 9974 ..55 .i36 .217 .298 .3 7 8 .45 9 .54o .621 .702 538 730782 o863 0944 1024 no5 1186 1266 1 347 1428 i5c8 53 9 1589 1669 1750 i83o 1911 1991 2072 2l52 2233 23i3 54o 23 9 4 s4?4 2555 2635 2715 2 79 6 2876 2956 3o3 7 3n 7 80 54i 3i 97 3278 3358 3438 35i8 35 9 8 36 79 3 7 5 9 383 9 3 9 i 9 542 3 999 4079 4i6o 424o 4320 44oo 448o 456o 464o 4720 543 48oo 488o 4960 5o4o 5l2O 5200 52 79 535 9 543 9 55i 9 544 55 99 56 79 5 7 5 9 5838 5 9 i8 5 99 8 6o 7 8 6i5 7 6 2 3 7 63i 7 545 63 97 6476 6556 6635 6715 6 79 5 68 7 4 6 9 54 7o34 7.II3 546 7 i 9 3 7272 7 352 743i 7 5n 7 5 9 7 6 7 o 774 9 7 82 9 79 o8 79 54 7 7987 8067 8i46 8225 83o5 8384 8463 8543 8622 8701 548 8781 8860 8 9 3 9 9 oi8 997 9 i 77 9 256 9 335 9 4i4 9 4 9 3 549 9672 965i 97 3i 9 8io 9 88 9 99 68 ..4 7 .126 .205 .284 55o 7 4o363 0442 O52I 0600 0678 o 7 5 7 o836 o 9 i5 o 99 4 1073 55i Il52 1230 i3o 9 1 388 i46 7 1 546 1624 1703 I 7 82 1860 552 J 9 3 9 2018 20 9 6 2175 2254 2332 2411 248 9 2568 2647 553 2725 2804 2882 2 9 6i 3o3 9 3n8 3i 9 6 3275 3353 343 1 78 554 35io 3588 366 7 3745 3823 3 9 02 3 9 8o 4o58 4i36 42i5 555 4293 43 7 i 444 9 4528 46o6 4684 4 7 6 2 484o 4 9 i 9 4997 556 5o 7 5 5i53 523i 53o 9 5387 5465 5543 562i 56 99 5 777 55 7 5855 69.33 6011 6o8 9 6167 6245 6323 64oi 6479 6556 558 6634 6712 6 79 o 6868 6 9 45 7023 7101 7i7 9 7 256 7 334 N. 1 2 3 4 || 5 6 7 8 9 D. 86 9 -7 26 34 43 52 60 69 77 85 3 9 J 7 26 34 43 5i 60 68 77 VI 84 & 8 7 20 34 42 5o 5 9 67 76 83 _ 8 *7 25 33 42 5o 58 66 75 fl\ 82 J . 8 16 25 33 4i 49 57 66 74 81 1 8 16 24 32 4i 49 57 65 73 Q 80 | 8 16 24 32 4o 48 56 64 72 79 8 16 24 32 4o 47 55 63 7 1 78 ^ , 8 16 23 3i 3 9 47 55 62 70 LOGARITHMS OF .NUMBERS. 13 N. 1 2 3 4 5 6 7 8 9 D. 55 9 7 4 7 4i2 7 48 9 7 56 7 7645 7722 7800 7878 79 55 8o33 8110 7 8 56o 8188 8266 8343 8421 8498 85 7 G 8653 8 7 3i 8808 8885 77 56i 8 9 63 9 o4o 9 n8 9 i 9 5 9272 9 35o 9 42 7 9 5o4 9 58 2 9 65 9 662 9736 9 8i4 9 8 9 i 99 68 ..45 .123 .200 2 77 .354 .43i 563 7 5o5o8 o586 o663 0740 0817 o8 9 4 0971 io48 I 125 1 202 564 I2 79 i356 i433 i5io i58 7 1 664 1741 1818 i8 9 5 I 97 2 565 2048 2125 22O2 227 9 2356 2433 25o 9 2586 2663 2740 566 2816 28 9 3 2 97 3o4 7 3i23 320O 3277 3353 343o 35o6 56y 3583 366o 3 7 36 38i3 388 9 3 9 66 4042 4u 9 4i 9 5 4272 568 4348 4425 45oi 45 7 8 4654 4?3o 4807 4883 4 9 6o 5o36 & 56g 5lI2 5i8 9 5265 534i 54i7 54 9 4 55 7 o 5646 5 7 22 5 799 5yo 58 7 5 5 9 5i 6o2 7 6io3 6180 6256 633 2 64o8 6484 656o 5 7 i 6636 6712 6 7 88 6864 6940 7016 700.2 7 i68 7244 7320 572 7 3 9 6 7472 7 548 7624 7700 7775 7 85i 79 2 7 8oo3 8079 5 7 3 8i55 823o 83o6 8382 8458 8533 86o 9 8685 8761 8836 5 7 4 8 9 ia 8 9 88 9 o63 9 i3 9 9 2l4 9 2 9 o 9 366 9 44i 9 5i7 9 5 9 2 5 7 5 9 668 9743 9819 9 8 9 4 9970 ..45 .121 . i 9 6 .272 .34? ?5 5 7 6 760422 o4 9 8 o5 7 3 o64 9 0724 0799 o8 7 5 o 9 5o 1025 IIOI 5 77 n 7 6 I25l i326 I4O2 1477 i552 1627 I 7 02 1778 i853 5 7 8 I 9 28 2003 2078 si53 2228 23o3 2378 2453 2629 2604 5 79 26 79 2754 282 9 2 9 o4 2 97 8 3o53 3i28 32o3 32 7 8 3353 58o 3428 35o3 35 7 8 3653 3727 38o2 38 7 7 3 9 52 4027 4ioi 58i 4176 425i 4326 44oo 4475 455o 4624 46 99 4774 4848 58 2 4 9 23 4 99 8 5o 7 2 5i47 5221 52 9 6 53 7 o 5445 5520 55 9 4 583 566 9 5 7 43 58i8 58 9 2 5 9 66 6o4i 6ii5 6i 9 o 6264 6338 74 584 64i3 648 7 6562 6636 6710 6 7 85 685 9 6 9 33 7007 7082 585 7 i56 723o 73o4 7379 7453 7 5a 7 7601 7675 774 9 7823 586 7898 7972 8o46 8120 8i 9 4 8268 8342 84i6 8490 8564 58 7 8638 8712 8786 8860 8 9 34 9 oo8 9082 9 :56 9 23o 93o3 588 9 3 77 9 45 1 9 5?5 9 5 99 9 6 7 3 9746 9820 9 8 9 4 99 68 ..42 58 9 770115 oi8 9 0263 o336 o4io o484 o557 o63i 0705 0778 Sgo o852 0926 999 1073 n46 I22O I2 9 3 i36 7 i44o i5i4 5 9 i 1587 1661 1734 1808 1881 i 9 55 2028 2IO2 2175 2248 73 5 9 2 2322 23 9 5 2468 254* 26i5 2688 2762 2835 2 9 o8 2081 5 9 3 3o55 3i 2 8 3201 32 7 4 3348 3421 34 9 4 356 7 364o 3713 5 9 4 3786 386o 3 9 33 4oo6 4o 79 4i5 2 4225 42 9 8 43 7 i 4444 5 9 5 45i7 45 9 o 4663 4 7 36 48o 9 4882 4 9 55 5028 5ioo 5i 7 3 5 9 6 5246 53i 9 53 9 2 5465 5538 56io 5683 5 7 56 58 29 5902 5 97 5 97 4 6047 6120 6i 9 3 6 2 65 6338 64n 6483 6556 6629 698 6701 6 77 4 6846 6 9 i 9 6992 7064 7 i3 7 7209 7282 7354 5 99 7427 74 99 7572 7644 77 i 7 7789 7862 7934 8006 8079 72 600 8i5i 8224 82 9 6 8368 844 1 85i3 8585 8658 8 7 3o 8802 601 88 7 4 8 9 47 9 OI 9 9 o 9 i 9 i63 9 236 9 3o8 9 38o 9 452 9524 602 9 5 9 6 9 66 9 97 4i 9 8i3 9 885 99 5 7 ..29 . IOI .i 7 3 .245 6o3 780317 o38 9 o46i o533 o6o5 0677 0749 0821 o8 9 3 0965 6o4 1037 no 9 1181 1253 i3 2 4 i3 9 6 i468 i54o 1612 1 684 6o5 1755 1827 i8 99 1971 2042 2Il4 2186 2258 232 9 2401 606 2473 2544 2616 2688 2 7 5 9 2 83i 2902 2 97 4 3o46 3n 7 6o 7 3i8 9 3260 3332 34o3 3475 3546 36i8 368 9 3 7 6i 3832 7* 608 3 9 o4 3 97 5 4o46 4n8 4i8 9 4261 4332 44o3 4475 4546 609 46i 7 468 9 4 7 6o 483i 4 9 02 4 9 74 5o45 5n6 5i8 7 5 2 5 9 610 533o 54oi 5472 5543 56i5 5686 5 7 5 7 5828 58 99 5970 6n 6o4i 6112 6i83 6254 6325 63 9 6 646 7 6538 66o 9 6680 N. | 1 2 3 4 5 6 7 8 9 D. 77 8 i5 23 3i 3 9 46 54 62 69 S 76 8 i5 23 3o 38 46 53 61 68 | 75 8 i5 23 3o 38 45 53 60 68 8- 74 C ' 7 T * 1 v> 23 3o 3? 44 52 5 9 67 i 73 g. 7 i5 22 2 9 3 7 44 5i 58 66 3 72 2 7 i4 22 2 9 36 43 5o 58 65 71 * 7 i4 21 28 36 43 5o 5? 64 LOGARITHMS OF LUMBERS. N. .1 2 3 4 5 6 7 8 9 | D. 612 786751 6822 68 9 3 6964 7035 7106 7i 77 7248 7 3l(; 7390 7 1 6i3 7 46o 7 53i 7602 7 6 7 3 7744 7810 7 885 7956 8027 8008 6i4 8168 8 2 3 9 83io 838i 845 1 8522 85 9 3 8663 8734 88o4 6i5 88 7 5 8 9 46 9016 9087 9 l5 7 9228 9299 9369 944o 9 5io 616 958r 965i 9722 9792 9863 99 33 ...4 .-74 .i44 .2i5 70 617 790285 o356 0426 0496 o56 7 o63 7 0707 o 77 8 o848 0918 618 0988 loSg 1129 1199 1269 i34o i4io i48o i55o 1620 619 1691 1761 i83i 1901 1971 2041 2III 2181 2252 2322 620 2392 2462 2532 2602 2672 2 7 42 0812 2882 2952 3O22 621 3092 3i62 3 2 3i 33oi 3371 344i 35u 358i 365i 3721 622 3790 386o SgSo 4ooo 4070 4i3 9 4209 4279 4349 44i8 623 4488 4558 4627 4697 4 7 6 7 4836 4906 49 7 6 5o45 5n5 624 5i85 5 2 54 5324 53 9 3 5463 5532 56o2 56 7 2 574i 58n 626 588o 5949 6019 6088 6i58 622 7 629-7 6366 6436 65o5 69 626 65 7 4 6644 6713 6 7 82 6852 6921 6990 7060 7129 7198 627 7268 733 7 7406 7 4 7 5 7545 76l4 7 683 7752 7821 7890 628 7960 8029 8098 8167 8 2 36 83o5 83 7 4 8443 85i3 8582 629 865i 8720 8789 8858 8927 8996 9065 9i34 9203 9272 63o 934i 9409 9478 9 54 7 9616 9685 9 7 54 9823 9892 9961 63i 800029 0098 0167 0236 o3o5 o3 7 3 0442 o5u o58o o648 63a 0717 0786 o854 0923 0992 1061 1129 1198 1266 i335 633 i4o4 1472 i54i 1609 1678 1747 1816 1 884 1952 2021 634 2089 2i58 2226 229 5 2363 2432 25oo ?,568 263 7 2705 635 2774 2842 2910 2 979 3o47 3n6 3i84 3252 332i 3389 68 636 345 7 35 2 5 35 9 4 3662 3 7 3o 3 79 8 386 7 SgSS 4oo3 4071 63 7 4i3 9 4208 4276 4344 44i2 448o 4548 46i6 4685 4753 638 4821 4889 49 5 7 5o 2 5 SogS 5i6i 5229 5297 5365 5433 63 9 55oi 556 9 563 7 5 7 o5 5 77 3 584i 5908 5 97 6 6o44 6112 64o 6180 6248 63i6 6384 645 1 65i 9 658J 6655 6723 6790 64 1 6858 6926 6 99 4 7061 7129 7197 7264 7 332 7 4oo 7467 642 7 535 7603 7670 7738 7806 7873 794 1 8008 8076 8i43 643 8211 8279 8346 84i4 848 1 8549 8616 8684 8 7 5i 8818 C 7 644 8886 8 9 53 9021 9088 9i56 9223 9290 9358 9 425 9492 645 9560 9627 9694 9762 9829 9896 9964 ..3i ..98 .i65 646 810233 o3oo o36 7 o434 o5oi 0569 o636 0703 0770 o83 7 647 0904 0971 1039 1106 n 7 3 1240 1307 i3 7 4 i44.i i5o8 648 i5 7 5 1642 1709 1776 i843 1910 1977 2044 2III 2178 649 2245 23l2 23 79 2445 25l2 25 79 2646 2713 2780 2847 65o 2913 2980 3o4 7 3n4 3i8i 324 7 33i4 338i 3448 35i4 65i 358i 3648 3 7 i4 3781 3848 3914 3 9 8i 4o48 4u4 4i8i 65 2 4248 43i4 438i 444? 45i4 458i 464 7 4 7 i4 4780 4847 653 4gi3 4980 5o46 5n3 5i 79 5 2 46 53i2 53 7 8 5445 55n 66 654 55 7 8 5644 5 7 n 5 777 5843 Sgio 5 97 6 6042 6109 6i 7 5 655 6241 63o8 63 7 4 644o 65o6 65 7 3 663 9 6 7 o5 6771 6838 656 6904 6970 7 o36 7102 7169 7235 7 3oi 7 36 7 7433 7499 65 7 7 565 763i 7698 7764 783o 7896 79 62 8028 8094 8160 658 8226 8292 8358 8424 8490 8556 8622 8688 8 7 54 8820 65 9 8885 8 9 5i 9017 9083 9149 9215 9 28l 9 346 9412 94 7 8 660 9544 9610 9676 97 4i 9807 9 8 7 3 99 3 9 ...4 ..70 .i36 661 820201 0267 o333 o3 99 o464 o53o o5 9 5 0661 0727 0792 662 o858 0924 0989 io55 II2O 1186 I25l i3i 7 1 382 i448 663 i5i4 i5 79 1 645 1710 i 77 5 i84i 1906 I 97 2 2037 2103 65 664 2168 2233 2299 2364 2430 24g5 2 56o 2626 2691 2756 N. ] 1 2 3 4 5 6 7 8 9 D. 7i 7 i4 21 28 36 43 5o 57 64 05 7 S 7 i4 21 28 35 42 49 56 63 W a 69 p^ 7 i4 21 28 35 4i 48 55 62 S- 68 ri 7 i4 2O 27 34 4i 48 54 61 e? a 7 i3 20 27 34 4o 47 54 60 P 66 g 7 i3 20 26 33 4o 46 53 5 9 65 * 7 i3 20 26 33 3 9 46 52 5 9 LOGARITHMS OF NUMBERS. 15 N. 1 2 3 4 5 6 7 8 9 D. 665 822822 288 7 2952 3oi8 3o83 3i48 32i3 3279 3344 34o 9 65 666 3474 3539 36o5 36 7 o 3 7 35 38oo 3865 3930 3 99 6 4o6i 667 4126 4191 4256 4321 4386 445 1 45i6 458i 4646 4711 668 4776 484 1 4906 4971 5o36 5ioi 5i66 5 2 3i 52 9 6 536i 669 5426 5491 5556 562i 5686 5 7 5i 58i5 588o 5 9 45 6010 670 6075 6i4o 6204 6269 6334 6399 6464 65 2 8 65 9 3 6658 671 6723 6787 6852 6917 6981 7 o46 7 ni 7175 7240 73o5 672 7 36 9 7434 7499 7 563 7628 7692 77 5 7 7821 7886 79 5i 6 7 3 8oi5 8o'8o 8i44 8209 82 7 3 8338 8402 846 7 853i 85 9 5 64 6 7 4 8660 8724 8789 8853 8918 8982 9046 9111 9 i 7 5 9 23 9 6 7 5 93o4 936? 9432 9497 9 56i 9625 9690 97 54 9 8i8 9 882 676 9947 ..n .. 7 5 .i3 9 .204 .268 .332 .3 9 6 .46o .525 677 83o58 9 o653 7 I 7 0781 0845 0909 97 3 1037 1 1 02 1166 678 I23o 1294 i358 1422 i486 i55o i6i4 1678 1742 1806 679 1870 1934 1998 2062 2126 2189 2253 2317 238i 2445 680 2509 25 7 3 263 7 2700 2 7 64 2828 2892 2956 3020 3o83 68 1 3i47 3211 32 7 5 3338 3402 3466 353o SSgS 3657 3721 682 3784 3848 3912 3 97 5 4o3 9 4io3 4i66 423o 4294 435 7 683 4421 4484 4548 46n 46 7 5 4 7 3 9 4802 4866 4929 4 99 3 684 5o56 5l2O 5i83 5 2 4 7 53io 53 7 3 543 7 55oo 5564 562 7 63 685 56 9 i 5 7 54 58i 7 588i 5 9 44 6007 6o 7 i 6r34 6197 6261 686 6324 638 7 645 1 65i4 65 77 664i 6 7 o4 6767 683o 68 9 4 687 6 9 5 7 7020 7 o83 7 i46 7 2IO 7 2 7 3 7 336 7399 7462 7 5 2 5 688 7588 7 652 771.5 777 8 7 84i 79 o4 796-7 8o3o 8093 8i56 689 8210 8282 8345 84o8 84 7 i 8534 85 97 8660 8723 8 7 86 690 8849 8912 8 97 5 9o38 9101 9164 9227 9289 9 35 2 9 4i5 691 9478 954i 9604 966 7 9729 979 2 9855 9918 9981 ..43 692 84oio6 0169 0232 0294 0357 O42O 0482 o545 0608 o6 7 i 693 o 7 33 0796 oSSg 0921 0984 io46 1109 1172 1234 I2 97 6 9 4 i35 9 1422 i485 i54 7 1610 l6 7 2 i 7 35 1797 1860 1922 6 9 5 i 9 85 2047 2IIO 2I 7 2 2235 229-7 236o 2422 2484 254 7 62 696 2609 2672 2734 2 79 6 2859 2921 2 9 83 3o46 3io8 3i 7 o 697 3233 3295 335 7 3420 3482 3544 36o6 3669 3 7 3i 3 79 3 698 3855 3 9 i8 3980 4042 4io4 4i66 4229 4291 4353 44i5 699 44 7 7 453 9 46oi 4664 4726 4 7 88 485o 4912 4974 5o36 700 5098 5i6o 5222 5284 5346 54o8 54 7 o 5532 55 9 4 5656 701 5 7 i8 5 7 8o 5842 5go4 5 9 66 6028 6090 6i5i 6213 62 7 5 702 633 7 6399 646 1 6523 6585 6646 6 7 o8 6770 6832 68 9 4 7o3 6955 7017 7079 7 i4i 7 202 7264 7 3 2 6 7 388 7449 7 5n 704 7 5 7 3 7634 7696 77 58 7819 7 88i 79 43 8oo4 8066 8128 7<>5 8189 8 2 5i 83i2 83 7 4 8435 8497 855 9 8620 8682 8 7 43 706 88o5 8866 8928 8989 905 1 9112 9 i 7 4 9235 9297 9358 61 707 9419 948i 9542 9604 9665 9 7 26 97 88 9849 9911 9972 708 85oo33 oogS oi56 O2I 7 02 79 o34o o4oi 0462 o524 o585 709 o646 0707 0769 o83o 0891 0952 1014 1075 n36 1197 710 1258 1320 i38i 1442 i5o3 1 564 1625 1686 1747 1809 711 i8 7 o ig3i 1992 2 o53 2Il4 2I 7 5 2236 2297 2358 2419 712 2480 254i 2602 2663 2 7 24 2 7 85 2846 2907 2968 3029 7 i3 3090 3i5o 3211 32 7 2 3333 33 9 4 3455 35i6 35 77 3637 7i4 36 9 8 3 7 5 9 3820 388i 394i 4OO2 4o63 4124 4x85 4245 7 i5 43o6 436 7 4428 4488 4549 46io 46 7 o 4?3i 4792 4852 716 4gi3 4974 5o34 5095 5i56 52i6 52 77 5337 53 9 8 545 9 717 55i 9 558o 564o 5 7 oi 5 7 6i 5822 5882 5 9 43 6oo3 6o64 718 6124 6i85 6245 63o6 6366 642 7 648 7 6548 6608 6668 60 719 6 7 2 9 6789 685o 6910 69 7 o 7 o3i -7091 7i52 7212 7272 N. | 1 2 3 4 5 6 7 8 9 P. 64 & f 6. 13 J 9 26 32 38 45 5i 58 ^ 63 S 6 i3 J 9 25 32 38 44 5o 5? 62 M 6 12 I 9 25 3i 37 43 5o 56 5 61 o | 6 12 1 8 24 3i 37 43 4 9 55 60 pL| I 6 12 18 24 3o 36 4a 48 54 10 LOGARITHMS OF NUMBERS. i N. 1 2 3 4 5 6 7 8 9 ID. 720 85 7 33 2 7 3 9 3 7 453 7 5i3 7 5 7 4 7 634 7 6 9 4 7755 7 8i5 7875 60 721 79 35 799 5 8o56 8116 8i 7 6 8236 82 97 835 7 84i 7 84 77 722 853 7 85 97 865 7 8 7 i8 8778 8838 88 9 8 8 9 58 9018 9 c 7 8 7 23 9i38 9 i 9 8 9 258 9 3i8 9 3 79 943 9 9499 9 55 9 9 6i 9 9 6 79 724 97 3 9 9799 9 85 9 9918 9978 ..38 ..98 .i58 .218 .2 7 8 726 86o338 o3 9 8 o458 o5i8 o5 7 8 o63 7 o6 97 o 7 5 7 0817 o8 77 726 o 9 3 7 0996 io56 1116 ii 7 6 1236 1295 i355 i4i5 i4 7 5 727 i534 i5 9 4 i654 1714 I 77 3 i833 i8 9 3 I 9 52 2OI2 2O 7 2 728 2l3l 2I 9 I 225l 23lO c3 7 o 243o 2 48 9 254 9 26o8 2668 729 2 7 28 2 7 8 7 2 84 7 2 9 o6 2966 3o25 3o85 3i44 3204 3263 7 3o 3323 3382 3442 35oi 356i 3620 368o 3 7 3 9 3 799 3858 5 9 7 3l 3 9 i 7 3977 4o36 4o 9 6 4i55 4214 42 7 4 4333 4392 4452 782 45x1 45 7 o 463o 468 9 4?48 48o8 486 7 4926 4985 5o45 733 5io4 5i63 5222 5282 534i 54oo 545 9 55i 9 55 7 8 563 7 7 34 56 9 6 5 7 55 58i4 58 7 4 SgSS 6992 6o5i 6110 6i6 9 6228 7 35 628 7 6346 64o5 6465 6524 6583 6642 6 7 oi 6 7 6o 68i 9 736 68 7 8 6 9 3 7 6 99 6 7 o55 7114 7 i 7 3 7 232 7291 7 35o 7 4o 9 7 3 7 7 46 7 7 526 7 585 7 644 77 o3 7-762 7821 7880 79 3 9 7998 7 38 8o56 8n5 8i 7 4 8233 8292 835o 84o 9 8468 852 7 8586 7 3 9 8644 8 7 o3 8 7 62 8821 88 79 8 9 38 8997 9o56 9 n4 9 i 7 3 7 4o 9 232 9 2 9 9 34 9 9 4o8 9 466 9 525 9 584 9642 97 oi 9 7 6o 7 4i 9 8i8 9 8 77 99 35 9994 ..53 .in .1-70 .228 .287 .345 7 42 8 7 o4o4 0462 0621 o5 79 o638 o6 9 6 o 7 55 o8i3 0872 o 9 3o 58 743 o 9 8 9 io4 7 1106 ix64 1223 1281 i33 9 i3 9 8 i456 i5i5 744 i5 7 3 i63i i6 9 o i 7 48 l8o6 i865 I 9 23 1981 2O4O 2098 745 2i56 22l5 22 7 3 2 33i 2 38 9 2448 25o6 2564 2622 2681 7 46 2 7 3 9 2797 2855 2 9 i3 2 97 2 3o3o 3o88 3i46 3204 3262 747 332i 33 79 343 7 34 9 5 3553 36ii 366 9 3727 3 7 85 3844 7 48 3 9 02 3960 4oi8 4o 7 6 4i34 4l 9 2 425o 43o8 4366 4424 749 4482 454o 45 9 8 4656 4 7 i4 4 77 2 483o 4888 4 9 45 5oo3 7 5o 5o6i 5u 9 5i 77 5235 52 9 3 535i 54o 9 5466 55 2 4 5582 7 5i 564o 56 9 8 5 7 56 58:3 58 7 i 5 9 2 9 5 9 8 7 6o45 6102 6160 762 6218 62 7 6 6333 63 9 i 6449 65o 7 6564 6622 6680 6 7 3 7 7 53 6 79 5 6853 6 9 io 6 9 68 7 O26 7 o83 7 i4i 7199 7 256 7 3i4 7 54 7 3 7 i 7 42 9 7 48 7 7 544 7602 7 65 9 77 i 7 7774 7 832 7889 7 55 7947 8oo4 8062 8119 8i 77 8234 82 9 2 8349 84o 7 8464 5? 766 8522 85 79 863 7 86 9 4 8 7 52 88o 9 8866 8924 8 9 8i 9 3 9 7 5 7 9 o 9 6 9 i53 9 2II 9268 9325 9 383 9 44o 9497 9 555 9612 7 58 9669 9726 Q 7 84 9 84i q8 9 8 9956 ..i3 ..70 .I2 7 .i85 769 880242 O2 99 o356 o4i3 o4 7 i 0028 o585 0642 o6 99 o 7 56 760 o8i4 - o8 7 i 0928 o 9 85 IO42 1099 n56 I2l3 I2 7 I i328 761 i385 i442 1499 i556 i6i3 i6 7 o I 7 2 7 i 7 84 i84i 1898 762 i 9 55 2OI2 2069 2126 2i83 2240 22 97 2354 2411 2468 7 63 2525 258i 2638 2695 2 7 52 2809 2866 2923 2980 3o3 7 7 64 3o 9 3 3i5o 320 7 3264 3321 33 77 3434 3491 3548 36o5 7 65 366i 3 7 i8 3 77 5 3832 3888 3 9 45 4OO2 4059 4n5 4l 7 2 766 422 9 4285 4342 43 99 4455 45i2 456 9 4625 4682 4 7 3 9 767 4 79 5 485 2 4909 4 9 65 5022 5o 7 8 5i35 5l 9 2 5248 53o5 768 536i 54i8 54 7 4 553i 558 7 5644 5 7 oo 5 7 5 7 58i3 58 7 o 769 5 9 26 5 9 83 6o3 9 6096 6i52 62O 9 6265 632i 63 7 8 6434 56 770 6491 654 7 66o4 6660 6 7 i6 6 77 3 682 9 6885 6942 6 99 8 771 7 o54 7 in 7 i6 7 7 223 7280 7 336 7 3 9 2 7 44 9 7 5o5 7 56i 772 76i 7 7 6 7 4 77 3o 77 86 7 842 7 8 9 8 79 55 Son 8o6 7 8i23 773 8i 79 8236 82 9 2 8348 84o4 846o 85i6 85 7 3 8629 8685 774 8 7 4i 8797 8853 8909 8 9 65 9 O2I 977 9i34 9 i 9 o 9 246 N. 1 2 3 4 5 6 7 8 9 D. 60 6 12 18 24 3o 36 42 48 54 5 9 $ 6 12 18 24 3o 35 4i 47 53 58 &. 6 12 *7 23 29 35 4i 46 52 S 5 7 o 6 II '7 23 29 34 4o 46 5i 56 6 II '7 22 28 34 3 9 45 5o LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 7 8 9 D. 77 5 889802 ! 9358 94 1 4" 9470 9526 9582 9 638 9 6 9 4 97 5o 9806 5fi 776 9862 9918 9974 ..3o ..86 .141 .197 .253 .3o 9 .365 777 890421 0477 o533 0589 o645 o 7 oo 0756 0812 0868 0924 77 8 0080 io35 1091 "4? 1203 1259 i3i4 i3 7 o 1426 1482 779 i53 7 i5 9 3 1649 1705 i 7 6o 1816 1872 I 9 28 I 9 83 2039 780 2095 2i5o 2206 2262 23l 7 23 7 3 2429 2484 254o 2 5 9 5 781 265i 2707 2762 2818 28 7 3 2929 2 9 85 3o4o 3o 9 6 3i5i 782 3207 3262 33 1 8 3373 3429 3484 354o 35 9 5 365i 3 7 o6 7 83 3 7 62 3817 3873 3928 3 9 84 4039 4o 9 4 4i5o 42o5 4261 55 74 43i6 4371 4427 4482 4538 45 9 3 4648 4 7 o4 4 7 5 9 48i4 7 S5 4870 4925 4980 5o36 5091 5t46 52OI 5 2 5 7 53i2 536 7 786 54 2 3 5478 5533 5588 5644 56 99 5 7 54 58o 9 5864 5920 787 5 97 5 6o3o 6o85 6i4o 6195 625i 63o6 636i 64i6 64 7 i 7 88 65 2 6 658i 6636 6692 6 7 4 7 6802 685 7 6 9 I2 6 9 6 7 7 O22 789 777 7 1 32 7187 7242 7297 7 352 7 4o 7 7462 7 5i 7 7 5 7 2 79 7627 7682 77 3 7 7792 7847 7 902 79 5 7 8012 8o6 7 8122 79 l 8176 823i 8286 834i 83 9 6 845 1 85o6 856i 86i5 86 7 o 79 2 8 7 25 8780 8835 8890 8 9 44 8999 9 o54 9 IO 9 9 i64 9218 79 3 0273 9828 9383 943 7 9492 9 5 47 9 6o2 9 656 9711 97 66 794 9821 9 8 7 5 99 3o 99 85 ..3 9 .. 9 4 .i4 9 .203 .258 .312 79 5 900367 0422 0476 o53i o586 o64o o6 9 5 o 7 4 9 0804 oSSg 79 6 o 9 i3 0968 1022 1077 n3i 1186 1240 I2 9 5 i34 9 i4o4 797 i458 i5i3 i56 7 1622 1676 I 7 3i i 7 85 1840 i8 9 4 i 9 48 54 79 8 2OO3 2057 2112 2166 2221 22 7 5 232 9 2384 2438 24 9 2 799 2547 2601 2655 2710 2 7 64 28l8 28 7 3 2 9 2 7 2 9 8l 3o36 800 3o 9 o 3i44 3i 99 3253 33o 7 336i 34i6 34 7 3524 35 7 8 801 3633 368 7 3 7 4i 3 79 5 3849 3 9 o4 3 9 58 4012 4o66 4120 802 4i74 4229 4283 433 7 43 9 i 4445 44 99 4553 4607 466 1 8o3 4716 4 7 7 4824 48 7 8 4932 4986 5o4o 5o 9 4 5i48 5202 8o4 5s56 53io 5364 54i8 5472 5526 558o 5634 5688 5 7 42 8o5 57 9 8 5fc5o 5 9 o4 5 9 58 6012 6066 6n 9 6i 7 3 6227 6281 806 6335 6:189 6443 6497 655i 66o4 6658 6 7 12 6766 6820 8o 7 63 7 4 6927 6981 7o35 7089 7 i43 7 i 9 6 7 25o 73o4 7 358 808 7 4n 7 465 7 5l 9 7 5 7 3 7626 7 68o 7734 7787 784i 7 8 9 5 809 7949 8002 8o56 8110 8i63 82I 7 82 7 8324 8378 843i 810 8485 853 9 85 9 2 8646 8699 8 7 53 88o 7 8860 8 9 i4 8 9 6 7 811 9 02I 9074 9 I28 9 i8i 9235 9289 9 342 9 3 9 6 9 44 9 9 5o3 812 9 556 9610 9 663 97 i6 9770 9823 9877 99 3o 99 84 ..3 7 53 8i3 9 ioo 9 i oi44 oi 9 7 025l o3o4 o358 o4n o464 o5i8 o5 7 i 8i4 0624 0678 0731 o 7 84 o838 0891 o 9 44 o 99 8 io5i no4 8i5 n58 I2II 1264 i3i 7 i3 7 i 1424 1477 i53o 1 584 i63 7 816 i6 9 o 1743 1797 i85o 1903 i 9 56 200 9 2o63 2116 2i6 9 817 2222 2275 23 2 8 238i 2435 2488 254i 2 5 9 4 2647 2 7 OO 818 2 7 53 2806 285 9 2 9 l3 2966 3019 3o 7 2 3i25 3178 323i 819 3 2 84 333 7 33 9 o 3443 3496 354 9 36o2 3655 3 7 o8 3 7 6i 820 38i4 386 7 3 9 20 3 97 3 4026 40-79 4l32 4i84 423 7 42 9 821 4343 43 9 6 4449 45o2 4555 46o8 466o 47i3 4766 48i 9 822 4872 4925 4977 5o3o 5o83 5i36 5i8 9 5 2 4i 52 9 4 534 7 823 5400 5453 55o5 5558 56n 5664 5 7 i6 5 7 6 9 5822 58 7 5 824 5927 5 9 8o 6o33 6o85 6i38 6191 6243 62 9 6 634 9 64oi 825 6454 65o 7 655 9 6612 6664 6 7 i 7 6 77 o 6822 68 7 5 6 9 2 7 826 6980 7o33 7o85 7 i38 7190 7 243 7 2 9 5 7 348 74oo 7 453 827 75o6 7 558 7611 7 663 7716 7768 7820 7873 79 s5 7978 5a 828 8o3o 8o83 8i35 8188 8240 8293 8345 83 97 845o 85o2 829 8555 8607 8659 8 7 I2 8 7 64 8816 886 9 8 9 2I 8 97 3 9 O26 83o 9078 9i3o 9 i83 9235 9 28 7 9340 9 3 9 2 9 444 9 4 9 6 9 54 9 N. | 1 | 2 3 4 [| 5 6 7 8 9 P. . f 55 3 6 ii J 7 22 28 33 3 9 44 5o 1 54 1- 5 ii 16 22 2 7 32 38 43 49 | 53 PH 5 ii 16 21 2 7 3a 37 4a 48 I 52 p; i 5 10 16 21 II 26 3i 36 4a 4? 18 LOGARITHMS OF NUMDERS. N. 1 2 3 4 || 5 6 7 8 jQDr 83i 919601 9653 9706 97 58 9810 9862 99 i4 99 6 7 ..i 9 .. 7 i 52 | 832 920123 0176 0228 0280 o332 o384 o436 o48 9 o54i o5 9 3 833 0645 0697 0749 0801 o853 0906 o 9 58 IOIO 1062 iix4 834 1166 1218 1270 1322 i3 7 4 1426 i4 7 8 i53o i582 i634 835 1686 1738 1790 1842 1894 1946 i 99 8 2o5o 2IO2 2i54 836 2206 2258 23lO 2362 24i4 2466 25i8 2570 2622 2 6 7 4 837 27^6 2777 2829 2881 2 9 33 2985 3o37 3o8 9 3i4o 3i 9 2 838 3 2 44 3296 3348 33 99 345i 35o3 3555 36o 7 3658 3 7 io 83 9 3762 38i4 3865 3 9 i 7 3 9 6 9 4O2I 4072 4i24 4176 4228 84c 4279 433i 4383 4434 4486 4538 458 9 464i 46 9 3 4 7 44 84i 4796 4848 48 99 4 9 5i 5oo3 5o54 5io6 5i5 7 52O 9 5261 842 53i2 5364 54i5 546 7 55i8 55 7 o 562i 56 7 3 5 72 5 5 77 6 843 5828 58 7 9 5 9 3i 5982 6o34 6o85 6i3 7 6188 6240 62 9 I 5i 844 6342 63 9 4 6445 649 7 6548 6600 665i 6702 6 7 54 68o5 845 685 7 6908 6 9 5 9 7 on 7062 7 u4 7 i65 7216 7268 7 3i 9 346 7 3 7 o 7422 7 4 7 3 ?524 7 5 7 6 7 62 7 7 6 7 8 77 3o 778i 7 832 84 7 7883 79 35 7 9 86 8o3 7 8088 8i4o 8i 9 i 8242 82 9 3 8345 848 83 9 6 8447 84 9 8 8549 8601 8652 8 7 o3 8 7 54 88o5 885 7 84 9 8908 8 9 5 9 9 oio 9061 9112 9i63 9 2l5 9 266 9 3i 7 9 368 85o 9419 9470 9 52I 9 5 7 2 9623 96 7 4 97 25 977 6 9 82 7 9 8 79 85i 99 3o 9981 ..32 ..83 .i34 .i85 .236 .287 .338 .38 9 852 93o44o 0491 o54a 0592 o643 0694 o 7 45 0796 o84 7 o8 9 8 853 0949 IOOO io5i 1 1 02 u53 1204 1254 i3o5 i356 i4o 7 854 i458 iSog i56o 1610 1661 I 7 I2 I 7 63 1814 i865 I 9 i5 ' 855 1966 2017 2068 2118 2169 222O 22 7 I 2322 23 7 2 2423 856 24?4 2524 2 5 7 5 2626 26 77 2 7 2 7 2 77 8 2&2 9 2 8 79 2 9 3o 85 7 2981 3o3i 3o82 3i33 3j83 3234 3285 3335 3386 343 7 858 348 7 3538 358 9 363 9 36 9 o 3 7 4o 3 79 i 384i 38 9 2 3 9 43 85 9 3 99 3 4o44 4094 4i45 4i 9 5 4246 42 9 6 434 7 43 97 4448 *6o 44 9 B 4549 4599 465o 4 7 oo 4 7 5i 48oi 4852 4 9 02 4 9 53 5o 86 1 5oo3 5o54 5io4 5i54 5 2 o5 5 2 55 53o6 5356 5406 545 7 862 55o 7 5558 6608 5658 5 7 o 9 5 7 5 9 58o 9 586o 5 9 io 5 9 6o 863 6011 6061 6111 6162 6212 6262 63i3 6363 64i3 6463 864 65i4 6564 66i4 6665 6715 6 7 65 68i5 6865 6 9 i6 6 9 66 865 7016 7066 7117 7167 7217 ?2 6 7 7 3i 7 7 36 7 7 4i8 7 468 866 7 5i8 7 568 7618 7668 77 i8 77 6 9 7819 7 86 9 79 i 9 79 6 9 867 8019 8069 8119 8169 8219 8269 8320 83 7 o 8420 84 7 o 868 85ao 85 7 o 8620 8670 8 7 20 8770 8820 88 7 o 8 9 20 8 97 o 869 9020 9070 9120 9 i 7 o 9220 9270 9320 9 36 9 9 4i 9 9 46 9 870 9 5l 9 9 56 9 9619 9 66 9 97 I 9 9769 9 8i 9 9 86 9 99 i8 99 68 871 940018 0068 0118 0168 0218 026 7 o3i 7 o36 7 0417 o46 7 872 o5i6 o566 0616 0666 o 7 i6 o 7 65 o8i5 o865 o 9 i5 o 9 64 8 7 3 1014 io64 in4 n63 I2l3 1263 i3i3 i362 1412 1462 8 7 4 i5n i56i 1611 1660 1710 i 7 6o i8o 9 i85 9 I 9 o 9 i 9 58 8 7 5 2008 2o58 2IO 7 2l5 7 22O 7 2256 23o6 2355 24o5 2455 876 25o4 2554 26o3 2653 2702 2 7 52 2801 285i 2 9 OI 2 9 5o 877 3ooo 3o49 3099 3i48 SigS 324 7 32 97 3346 33 9 6 3445 49 878 34 9 5 3544 35 9 3 3643 36 9 2 3 7 42 3 79 i 384i 38 9 o 3 9 3 9 879 3 9 8 9 4o38 4o88 4i3 7 4i86 4236 4285 4335 4384 4433 ! 880 4483 4532 458i 463i 468o 4729 4779 4828 48 77 492 7 881 4976 5o25 5o 7 4 6124 5i 7 3 5222 52 7 2 532i 53 7 o 54i9 882 546 9 55i8 556 7 56i6 5665 5 7 i5 5 7 64 58:3 5862 5912 883 5 9 6i 6010 6059 6108 6i5 7 6207 6256 63o5 6354 64o3 884 6452 65oi 655i 6600 6649 6698 6 7 4 7 6 79 6 6845 68 9 4 885 6 9 43 6992 7 o4i 7 o 9 o 7 i4o 7189 7 238 72 8 7 7 336 7 385 886 7434 7483 7 532 7 58i 7 63o 7 6 79 772 8 7777 7826 7 8 7 5 N. 1 2 3 4 5 6 7 8 9 D. {62 ,' 5 10 16 21 26 3i 36 4a 4? 5i si 5 10 i5 20 26 3i 36 4i 46 50 *' 5 IO i5 2O 25 3o 35 4o 45 4 9 * K ij 10 i5 20 25 29 34 3 9 44 LOGARITHMS OF NUMBERS. 10 K 1 2 3 4 || 5 6 7 8 9 I). 887 947924 7973 8022 8070 8119 8168 8217 8266 83i5 8364 49 888 84i3 8462 85u 856o 8609 865 7 8706 8 7 55 88o4 8853 889 8902 8 9 5i 8999 9048 9097 9 i46 9 i 9 5 9 244 9292 9 34i 890 9 3 9 9 43 9 9488 9 536 9 585 9 634 9 683 97 3i 9780 0820 891 9878 99 26 997 5 ..24 .. 7 3 . 121 .170 .2I 9 .267 .3i6 892 95o365 o4i4 0462 o5n o56o 0608 o65 7 0706 0754 o8o3 8 9 3 o85i o 9 oo 0949 0997 io46 io 9 5 n43 II 9 2 I24o I28 9 8 9 4 i338 i386 i435 i483 i532 i58o i62 9 1677 1726 1775 8 9 5 1823 1872 1920 1969 2017 2066 2Il4 2i63 2211 2260 43 896 2 3o8 a356 24o5 2453 2502 2 55o 2 5 99 2647 2696 2744 897 2792 284 1 2889 2 9 38 2986 3o34 3o83 3i3i 3i8o 3228 898 3 27 6 3325 33 7 3 3421 3470 35j8 3566 36i5 3663 3711 899 3760 38o8 3856 3 9 o5 3 9 53 4ooi 4o4 9 4o 9 8 4i46 4i 9 4 900 4243 4291 433 9 438 7 4435 4484 4532 458o 4628 46 77 901 4726 4 77 3 4821 486 9 4 9 i8 4 9 66 5oi4 5o62 5uo 5i58 902 6207 5 3 55 53o3 535i 53 99 5447 54 9 5 5543 5592 564o 903 5688 5 7 36 5 7 84 5832 588o 5 9 28 5 97 6 6024 6072 6120 904 6168 6216 6265 63i3 636i 64o 9 645 7 65o5 6553 6601 906 6649 6697 6 7 45 6 79 3 684o 6888 6 9 36 6 9 84 7032 7080 906 7128 7176 7224 7272 7320 7368 74i6 7464 75i2 7 55 9 907 7607 7 655 77 o3 775i 7799 7847 78 9 4 79 42 799 8o38 908 8086 8i34 8181 8229 8277 8325 83 7 3 8421 8468 85i6 909 8564 8612 865 9 8707 8755 88o3 885o 88 9 8 8 9 46 8994 910 9041 9089 9 i3 7 9i85 9 232 9 28o 9 328 9 3 7 5 9 423 9471 911 9 5i8 9 566 9614 9661 979 97 5 7 9 8o4 9 852 9900 9947 912 9 99 5 ..42 ..90 .i38 .1.85 .233 .280 .328 .3 7 6 .423 9 i3 9 6o47i o5i8 o566 o6i3 0661 o 7 o 9 0756 08 o4 o85i 0899 914 o 9 46 o 99 4 io4i 1089 u36 1184 I23l I2 79 i326 i3 7 4 47 916 1421 1469 i5i6 i563 1611 i658 1706 i 7 53 1801 1 848 916 i8 9 5 i 9 43 i 99 o 2038 2o85 2132 2180 222 7 2275 2322 917 236 9 2417 2464 25ll 255 9 2606 2653 2 7 OI 2748 2795 918 2843 28 9 o 2 9 3 7 2985 3o3 2 3o 79 3i26 3i 7 4 3221 3268 919 33i6 3363 34io 3457 35o4 3552 35 99 3646 36 9 3 3 7 4i 920 3788 3835 3882 3929 3 977 4024 4071 4n8 4i65 4212 921 4260 43o 7 4354 44oi 4448 44 9 5 4542 4590 463 7 4684 922 473i 4778 48 2 5 4872 4 9 i 9 4 9 66 5oi3 5o6i 5io8 5i55 923 5202 524 9 52 9 6 5343 53 9 o 543 7 5484 553i 55 7 8 56 2 5 924 56 7 2 5 7 i 9 5 7 66 58i3 586o 5 9 o 7 5 9 54 6001 6o48 6095 926 6142 6i8 9 6236 6283 63 29 63 7 6 64 2 3 6470 65i 7 6564 926 6611 6658 6 7 o5 6752 6 799 6845 68 9 2 6 9 3 9 6986 7 o33 927 7080 7127 7 i 7 3 7220 7267 73i4 736i 74o8 7454 75oi 928 7548 7 5 9 5 7642 7688 77 35 7782 7 82 9 7 8 7 5 7922 7969 929 8016 8062 8io 9 8i56 82o3 8 2 4 9 82 9 6 8343 83 9 o 8436 93o 8483 853o 85 7 6 8623 8670 8716 8763 8810 8856 8903 9 3i 8 9 5o 8 99 6 9 o43' 9090 9 i36 9 i83 9 22 9 9276 9 3 2 3 9 36 9 982 9 4i6 9 463 9 5o 9 9 556 9 6o2 9649 9 6 9 5 9742 978 9 9 835 933 9 882 9928 997 5 . .21 ..68 .n4 .161 .207 .254 .3oo 934 9 7o347 o3 9 3 o44o o486 o533 o5 79 0626 0672 7 r 9 0765 46 9 35 0812 o858 0904 o 9 5i 997 io44 io 9 o ii3 7 u83 1229 936 1276 1322 i36 9 i4i5 i46i i5o8 i554 1601 1 647 i6 9 3 9 3 7 1740 1786 i83 2 1879 I 9 25 i 97 i 2018 2064 2110 2157 9 38 2203 224 9 2295 2342 2388 2434 248 1 2527 25 7 3 2619 9 3 9 2666 2712 2758 2804 285i 2 8 97 2 9 43 2989 3o35 3o82 940 3i 2 8 3|74 3220 3266 33i3 335 9 34o5 345 1 34 97 3543 94r 35 9 o 3636 3682 3728 3 77 4 3820 3866 SgiS 3 9 5 9 4oo5 9 42 465 j 4097 4i43 4i8 9 4235 4281 432 7 43 7 4 4420 4466 9 43 45i2 4558 46o4 465o 46 9 6 4742 4788 4834 488o 4 9 26 N. | 1 2 3 4 || 5 6 7 8 9 D. g f 43 f 5 10 i4 J 9 a4 2 9 34 38 43 1 4 7 - '! 5 9 i4 '9 24 28 33 38 42 P 1 46 p; 5 9 r4 1 9 23 28 32 37 4i 20 LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 7 8 9 II 944 9?4Q72 5oi8 5o64 5no 5i56 5202 5248 52 9 4 534o 5386 46 9 45 5432 5478 5524 55 7 o 56i6 5662 5707 5 7 53 5 799 5845 946 58 9 i 5 9 3 7 5 9 83 6o2 9 6075 6121 6167 6212 6258 63o4 9^7 635o 63 9 6 6442 6488 6533 65 7 9 6625 6671 6717 6 7 63 948 6808 6854 6 9 oo 6 9 46 6992 7 3 7 7083 7i2 9 7175 7220 949 7266 7312 7358 74o3 744 9 74 9 5 754i 7 586 7 63 2 7678 9 5c 7724 7769 7 8i5 7861 7 9 o6 79 5a 7998 8o43 8o8 9 8i35 961 8181 8226 8272 83i 7 8363 84o 9 8454 85oo 8546 85 9 i 9 52 8637 8683 8728 8 7 ?4 88i 9 8865 8911 8 9 56 9 OO2 9 o47 9 53 99 3 9i38 9 i84 9 23o 9 2 7 5 9 32I 9 366 9 4i2 9 45 7 9 5o3 9^4 9 548 95 9 4 9 63 9 9 685 97 3o 9776 9821 9 86 7 99 I2 99 58 9 55 9 8ooo3 oo4 9 oo 9 4 oi4o oi85 023l 0276 0322 0367 O4l2 45 966 o458 o5o3 o54 9 o5 9 4 o64o o685 0730 0776 0821 086 7 9 5 7 O 9 I2 o 9 5 7 ioo3 1048 io 9 3 1139 1184 I22 9 1275 1320 9 58 i366 1411 i456 i5oi i547 l5 9 2 i63 7 i683 1728 1773 9 5 9 1819 i864 I 99 i 9 54 20OO 2o45 2090 2i35 2181 2226 960 2271 2 3i6 2362 2407 2452 2497 2543 2588 2633 2678 961 2723 276 9 2814 285 9 2 9 o4 2949 2 99 4 3o4o 3o85 3i3o 962 3175 322O 3265 33io 3356 34oi 3446 34 9 i 3536 358i 9 63 3626 36 7 i 3 7 i6 3 7 62 38o 7 3852 38 97 3 9 42 3 9 8 7 4o32 964 4077 4l22 4i6 7 4212 425 7 4302 4347 43 9 2 443 7 4482 965 452 7 4572 46i 7 4662 4707 4?52 4797 4842 488 7 4 9 32 966 4977 5O22 5067 5lI2 5i5 7 52O2 5247 52 9 2 5337 5382 967 5426 5471 55i6 556i 56o6 565i 5696 574l 5 7 86 583o 968 58 7 5 5 9 2O 5 9 65 6010 6o55 6100 6i44 6i8 9 6234 62 79 969 6324 636 9 64i3 6458 65o3 6548 65 9 3 6637 6682 6727 97 6772 6817 6861 6 9 o6 6 9 5i 6996 7040 7 o85 7i3o 7 i 7 5 971 7219 7264 739 7 353 7 3 9 8 7443 7 488 7 53 2 7577 7622 972 7666 7711 7756 7800 7 845 7890 7934 7979 8024 8068 973 8n3 8i5 7 8202 8247 82 9 I 8336 838i 8425 8470 85i4 974 855 9 86o4 8648 86 9 3 8 7 3 7 8782 8826 8871 8916 8 9 6o 97 5 9 oo5 9 o4 9 9 o 9 4 9 i38 9 i83 9227 9 2 7 2 9 3i6 9 36i 9 4o5 976 9 45o 9 4 9 4 9 53 9 9 583 9628 9672 97 i7 97 6i 9 8o6 9 85o 44 977 9 8 9 5 99 3 9 99 83 ..28 ..72 .117 .161 .206 .25o 2 9 4 978 99 o33 9 o383 0428 0472 o5i6 o56i o6o5 o65o o6 9 4 0738 979 o 7 83 0827 0871 o 9 i6 0960 ioo4 io4 9 io 9 3 n3 7 1182 980 1226 1270 i3i5 i35 9 i4o3 i448 l4 9 2 i536 i58o 1625 981 i66 9 1713 1758 1802 1 846 1890 i 9 35 i 9 7 9 2O23 2067 982 21 I I 2i56 22OO 2244 2288 2333 2 3 77 2421 2465 25o 9 9 83 2 554 25 9 8 2642 2686 2730 2774 28l 9 2863 2 9 7 2 9 5l 984 2 99 5 3o3 9 3o83 3127 3172 32i6 3260 33o4 3348 33 9 2 9 85 3436 348o 3524 3568 36i3 3657 3701 3 7 45 3 7 8 9 3833 986 38 7 7 3 9 2I 3 9 65 4oo 9 4o53 4o 9 7 4i4i 4i85 4229 42 7 3 987 43i 7 436i 44o5 444 9 4493 453 7 458i 4625 466 9 47'3 988 4 7 5 7 48oi 4845 488 9 4933 4977 5021 5o65 5io8 5i52 9 8 9 5i 9 6 524o 5284 5328 5372 54i6 5460 55o4 5547 55 9 i 99 5635 56 7 9 5 7 23 5 7 6 7 58i i 5854 58 9 8 5 9 42 5 9 86 6o3o 99 i 6074 6117 6161 62o5 6249 62 9 3 6337 638o 6424 6468 99 2 65i2 6555 65 99 6643 6687 6731 6774 6818 6862 6 9 o6 99 3 6 9 4 9 6 99 3 7 o3 7 7080 7124 7168 7212 7255 7299 7343 994 7 386 7 43o 7474 7517 7 56i 7605 7 648 76 9 2 7736 7779 99 5 7823 7867 7 9 io 7954 7998 8o4i 8o85 8l2 9 8172 8216 99 6 8 2 5 9 83o3 8347 83 9 o 8434 8477 852i 8564 8608 8652 997 86 9 5 8 7 3 9 8782 8826 8869 8 9 i3 8 9 56 9 ooo 9043 9087 99 s 9 i3i 9 i 7 4 9 2l8 9 26l 9 3o5 9 348 9 3 9 2 9 435 9479 9522 999 9 565 9 6o 9 9 65a 9 6 9 6 97 3 9 9783 9 826 9 8 7 o 99 i3 99 5 7 43 N. 1 2 3 4 !| 5 6 7 8 9 D._ . f 46 a f 5 9 i4 18 I! 23 28 32 37 4i fel 45 is I 5 9 i4 18 23 27 32 36 4i | 44 * 4 9 i3 l8 22 26 3i 35 4o fi I 43 fc I 4 9 i3 I 7 22 26 3o 34 39 1 TABLE OF LOGARITHMIC SINES AID TANGENTS FOR EVERY TEN SECONDS OF THE QUADRANT. 22 LOGARITHMIC SINES. Win. Sine of Degree. P. Part 0" 10" 20" 30'' 40" 50" to 1". O Inf. Neg. 5.685575 5.9866o5 6. 162696 6.287635 6.384545 5 9 I 6.463726 6.53o6 7 3 6.588665 63 9 8i 7 6855 7 5 726968 58 2 764 7 56 799518 83i 7 o3 861666 889695 916024 57 3 940847 964328 986605 7.007794 7.027997 7 .o4 7 3o3 56 4 7.065786 7-o835i5 7.ioo548 116939 i32 7 33 i4 7 9 7 3 55 5 162696 176936 190725 204089 2i 7 o54 229643 54 6 241877 253 77 6 265358 276639 28 7 635 2 9 8358 53 7 3o8824 319043 329027 338 7 8 7 348332 35 7 6 7 2 52 8 3668i6 3 7 5 77 i 384544 393145 401678 409850 5i 9 417968 425 9 3 7 433762 44i449 449002 456426 5o 10 463 7 26 470904 477966 4849i5 49i 7 54 498488 49 689.4 1 1 5o5n8 5 i 1649 5i8o83 524423 53o6 7 2 536832 48 629.4 12 542906 548897 5548o6 56o635 56638 7 5 7 2o65 4? e 579.1 i3 5 77 668 5832oi 588664 594059 5 99 388 6o4652 46 536.2 i4 6o 9 853 614993 620072 625093 63oo56 634964 45 499.2 i5 63 9 8i6 6446 i 5 64936i 654o56 658 7 oi 66329-7 44 467.0 16 667845 672345 676799 681208 6855 7 3 689895 43 438. 7 17 6 9 4i73 698410 702606 706762 7 io8 79 714907 42 4x3.6 18 718997 722999 726965 730896 734791 7 3865i 4i 391.3 '9 742478 746270 75oo3i 753 7 58 7 5 7 455 761 119 4o 371.2 20 764754 7 68358 771932 775477 778994 782482 3 9 353.1 21 7 85 9 43 789376 792782 796162 7995i5 802843 38 336.7 22 806146 809423 812677 816906 819111 822292 3? 321.7 23 8 2 545i 828586 831700 834791 83 7 86o 84090-7 36 3o8.o 24 843 9 34 846 9 3 9 849924 852889 855833 858 7 5 7 35 295.4 25 861662 864548 86 7 4i5 870262 873092 8-75902 34 283.8 26 878695 881470 884228 886968 889690 892396 33 273.1 27 8 9 5o85 8 977 58 900414 9o3o54 905678 90828-7 32 263.2 28 910879 9 i345 7 916019 9i8566 921098 923616 3i 254.0 29 926119 928608 931082 933543 9 35 9 8 9 938422 3o 245.4 1 3o 940842 943248 94564i 948020 950387 952 7 4i 29 237.5 3i 955082 957411 959727 962031 964322 966602 38 229.8 32 968870 971126 97 33 7 o 9756o3 977 824 980034 27 222.7 33 982233 984421 986598 988764 990919 993064 26 216.1 34 995198 997322 999435 8.ooi538 3.oo363i 8.oo5 7 i4 25 209.8 35 8.007787 8.009850 8.011903 013947 016981 oi8oo5 24 2o3. 9 36 020021 022027 024023 026011 02-7989 029959 23 198.3 37 oSigig 033871 o358i4 037749 039676 041^92 22 IQ3.0 38 o435oi o454oi 047294 049178 o5io54 052922 2 I 188.0 3 9 054781 o56633 o58477 o6o3i4 062142 o63 9 63 20 i83.s 4o 065776 067582 069380 071171 o 7 2955 o 7 4 7 3i J 9 178.7 4i 076500 078261 080016 081764 o835o4 o85238 1 8 i 7 4.4 42 o86 9 65 088684 090398 092104 093804 096497 i? 170.3 43 097183 098863 100537 IO22O4 io3864 io55i9 16 166.4 44 107167 108809 no444 II2O74 n369 7 Il53i5 i5 162.6 45 116926 n8532 I2Ol3l 121725 I233i3 124895 i4 iSg. i 46 126471 128042 129607 i3ii66 l32 7 2O 134268 i3 i55.6 4? i358io i3 7 348 138879 i4o4o6 i4i92 7 143443 12 i52.4 49 144953 146458 i479 5 9 149453 150943 152428 II 149.2 4 9 iWqoy i5538 2 156852 i583i6 i59 77 6 i6i23i 10 146.2 5o 162681 164126 165566 167002 i68433 169859 9 i43.3 5! 171280 172697 174109 I 7 55i 7 1-76920 1-78319 8 i4o.5 52 179713 i8no3 182488 183869 i85245 186617 7 i37-9 53 i8 79 85 i8 9 348 190707 192062 I 9 34i3 194760 6 i35.3 54 196102 197440 198774 20OIO4 2oi43o 202752 5 i32.8 55 204070 2o5384 206694 2O8OOO 209302 210601 4 i3o.4 56 2ii8 9 5 2i3i85 214472 2i5 7 55 2i 7 o34 218309 3 128.1 5 7 219681 220849 2221 l3 223374 224631 225884 2 125.9 58 227134 22838o 229622 23o86i 232096 233328 I 123.7 5 9 23455 7 235782 237003 238221 23 9 .436 24o64 7 O 121. 6 80" 50" 40" 30" 20" 10" Co-sine of S9 Degrees. Min. LOGARITHMIC TANGENT s. Min. Tangent of Degree. 0' 10" 20" 30-'' 40'' | 50" o Inf. Neg. 5.6855 7 5 5.986605 6. 162696 6.28 7 635 6.384545 6.463726 6.53o6 7 3 6.588665 6398l 7 686676 726968 2 764756 799 5i8 83i 7 o3 861666 889696 9 i6o24 3 940847 964329 9 866o5 7.007794 7 027998 7.047808 A 7.065786 7 .o835i5 7.ioo548 116939 i32 7 33 147973 5 162696 i 7 6 9 3 7 190726 204089 217064 229643 6 241878 253 777 26535 9 276640 287635 298869 7 3o8825 319044 329028 338788 348333 35 7 6 7 3 8 366817 3 V 5 77 2 384546 393146 401679 409862 9 4i797 426989 433 7 64 44i45i 449004 466428 10 463 7 2 7 470906 477968 484917 491766 498490 ii 5o5i2o 5n65i 5i8o85 624426 530676 536835 .*2 542909 548900 5548o8 56o638 666890 672068 13 577672 583204 588667 694062 699891 6o4655 14 609857 614996 620076 626097 680060 634968 i5 639820 644619 649366 654o6i 668706 663302 16 667849 672350 6768041 68i2i3 686678 689900 l l 694179 698416 702612 706768 710886 714963 18 719003 723oo5 726972 730902 734797 788668 '9 742484 746277 760087 7 53 7 65 767462 761127 20 764761 768365 771940 77 5485 779002 782490 21 785951 789384 79 2 79 796170 799624 802862 22 8o6i55 8o 9 433 812686 816916 819120 822802 23 825460 8 2 85 9 6 831710 8348oi 887870 840918 24 843 9 44 846950 84 99 35 862900 855844 868769 25 861674 86456o 867426 870274 878104 876916 26 878708 88i483 884240 886981 889704 892410 27 8 9 5o 99 897772 900428 9o3o68 906692 908801 28 910894 913471 916034 918681 921118 9 2363i 29 926134 928623 9 3io 9 8 9 3355 9 9 36oo6 9 3843 9 3o 9 4o858 943265 9 45658 948037 9 5o4o4 9 52 7 58 Si 955ioo 9 5 7 428 969745 962049 9 6434i 9 6662i 32 968889 971145 97 338 9 976622 977 844 9 8oo54 33 982253 98444i 9 866i8 988786 99 o 9 4o 99 3o85 34 995219 997343 999467 8. 001660 8.003653 8.006736 35 8 007809 8.009872 8,oi 1926 013970 016004 oi8o2 9 36 020044 O22o5l 024048 O26o35 028014 029984 37 o3i 9 45 033897 o3584o 037776 089701 041618 38 043527 045428 047321 049206 061081 o52 9 4 9 3 9 054809 o56662 068606 060342 062171 o63 99 2 4o o658o6 067612 069410 071201 072986 074761 4? o 7 653i 078293 080047 081796 o83536 086270 4a 086997 088717 090431 092187 098887 o 9 553o 43 097217 098897 100671 102289 108900 io5554 44 107203 io8845 110481 II2IIO 118784 ii535 2 45 116963 118569 120169 I2I763 i2335i i24 9 33 46 i265io 128081 129646 i3i2o6 182760 i 343o8 47 i3585i i3 7 38 9 138921 i4o447 141969 i43485 48 144996 i465oi i48ooi 149497 160987 162472 49 i53 9 52 155426 166896 i5836i 169821 161276 5o 162727 164172 i656i3 167049 168480 169906 5i 171328 172745 i 7 4i58 176666 176969 i 7 8368 52 i 797 63 i8u53 182538 183919 186296 186668 53 i88o36 189400 I 9 o 7 6o 192116 198466 194818 54 196156 197494 198829 200169 2oi485 202808 55 204126 206440 2o6 7 5o 208067 209869 210668 56 211953 213243 2i453o 216814 217098 218869 5 7 219641 220909 222I 7 4 223434 224692 >25 9 45 58 227195 228442 229686 280924 282160 2333 9 2 59 234621 235846 23 7 o68 238286 289602 240718 60" 50" 40" 30'' 20" 10" Co-tangent of 89 Degrees. P. Tart tol". ~5g 58 5 7 56 55 54 53 62 5i 5o 49 68 9 .4 48 62 9 .4 47 679.1 46 536.2 45 4 99 .2 44 467.0 43 438. 7 42 4i3.6 4 1 3 9 i.3 4o 3 7 1.2 q 353.1 38 336.7 37 321.7 36 3o8.o 35 2 9 5.4 34 283. 9 33 278.2 32 268.2 3i 264.0 3o 245.4 29 287.8 28 22 9 .8 27 222.7 26 2l6.2 25 2O 9 .8 24 208 9 28 198.8 22 193.0 21 188.0 2O i83.3 I 9 178.7 18 174.4 17 170.3 J6 166.4 i5 162.7 i4 169. i 18 166.7 12 162.4 II 149.8 IO 146.2 9 143.4 8 i4o.6 7 187.9 6 i35.3 5 i32.8 4 180.4 3 128 i 2 126.9 I 128.8 O vi in. 121.7 L o G A i. i T a M i c Mia. Sine of 1 Degree. F. Part 0" 10" 20" 30" 40" 50" to 1". o 8.24i855 8. 243o6o 8.244261 8. 2 4545 9 8.246664 8.247845 69 II 9 .6 i 249033 250218 25l4oo 262678 253 7 53 254 9 25 58 117.7 2 256094 267260 258423 269682 260739 261892 57 116.8 3 263042 264190 265334 266476 267613 26874 9 56 114.0 4 269881 271010 272137 273260 2 7 438i 276499 55 112. 2 5 276614 277726 2 7 8835 27 99 4i 281046 282146 54 II0.5 6 283243 284339 285431 286621 287608 2886 9 2 53 ioS.8 7 289773 290862 291928 2 9 30O2 294073 296141 52 107. 2 8 296207 297270 298330 2 99 388 3oo443 3oi4 9 6 5i 106.7 9 302546 3o3594 3o463 9 3o568i 306721 3o 77 5 9 5o 1 04. I 10 8.308794 8.309827 8. 3io85 7 8.3n885 8.3i2 9 io 8,3i3 9 33 4 9 102.6 ii 3i4954 316972 316987 3i8ooi 3l 9 OI2 32OO2I 48 IOI .2 12 321027 322031 323o33 324032 326029 326O24 47 99 .8 i3 327016 328007 328996 32 99 8o 33o 9 64 33i 9 45 46 98.6 14 332924 333901 334876 335848 3368i 9 33 77 8 7 45 97.1 i5 338753 33 97 i 7 340679 34i638 3425 9 6 34355i 44 96.8 16 3445o4 345456 3464o5 347352 3482 9 7 34 9 24o 43 9 4.6 l l 35oi8i 361119 352056 352 99 i 353 9 24 354855 42 9 3.4 18 355783 366710 35 7 635 358558 35 9 4 7 9 36o3 9 8 4i 92.2 '9 36i3i5 36223o 363i43 364o55 364 9 64 3658 7 i 4o 91 .0 20 8.366777 8.36 7 68i 8.368582 8.369482 8.370380 8.371277 3 9 8 9-9 21 372171 3 7 3o63 3 7 3 9 54 3 7 4843 376730 3 7 66i5 38 88.8 22 377499 3 7 838o 379260 38oi38 38ioi5 38i88 9 37 87.7 23 382762 383633 384602 3853 7 o 386236 387100 36 86.7 24 387962 388823 389682 390639 391396 3 9 224 9 35 85.6 25 393101 3 9 3 9 5i 394800 396647 396493 397337 34 84.6 26 398179 399020 399869 4oo6 9 6 4oi532 402366 33 83. 7 27 403199 4o4o3o 404869 406687 4o65i4 407338 32 82,7 28 408161 4o8 9 83 409803 410621 4n438 412264 3i 81.8 29 4i3o68 4i388o 414691 4i55oo 4i63o8 417114 3o 80.8 3o 8.417919 8.418722 8.419624 8.420326 8.42II23 8.42I92I 29 80.0 3i 422717 4235n 4243o4 425o 9 6 426886 426675 2o 79 .i 32 427462 428248 429032 42 9 8i5 430697 43i3 77 27 78.2 33 432i56 432 9 34 433710 434484 436267 436029 26 77-4 34 4368oo 43 7 56 9 438337 43 9 io3 439868 44o632 26 76.6 35 44i3 9 4 442166 442916 4436 7 4 44443i 445 i 86 24 7 5. 36 445 9 4i 4466 9 4 447446 448 i 9 6 448 9 46 44 9 6 9 4 23 7 5.o 3? 45o44o 461186 461930 462673 4534i4 454i54 22 74.2 38 454893 45563i 456368 467103 45 7 83 7 4585 7 o 21 7 3,5 3 9 459301 46oo32 460761 46i48 9 462216 462941 2O 72.7 4o 8.463665 8.464388 8.465iio 8.46583o 8.466660 8.46 7 268 I 9 72.0 4i 467985 468701 469416 47oi2 9 470841 4 7 i553 18 7 i.3 42 472263 472971 473679 474386 476091 4 7 5 79 5 17 70.6 43 476498 477200 477901 478601 479299 479997 16 69.9 44 480693 48i388 482083 482776 483467 484i58 :5 69.2 45 484848 485536 486224 486 9 io 487696 488280 i4 68.6 46 488963 489645 4 9 o326 4 9 ioo6 491686 4 9 2363 i3 67.9 47 493o4o 493716 4 9 43 9 o 4 9 5o64 4 9 5 7 36 4 9 64o8 12 6 7 .3 48 497078 497748 4 9 b4i6 4 99 o84 499760 5oo4i6 II 66.7 49 601080 601743 602406 603067 603727 5o4386 IO 66.1 5o 8.5o5o45 8.606702 8.5o6358 8.607014 8.607668 8.5o832i 9 65 5 5! 508974 609626 610276 5io 9 25 SrtSyS 612221 8 64.9 52 512867 5i35i3 614167 614801 5i5444 616086 7 64.3 53 516726 5i 7 366 618006 5i8643 619280 5i 99 i6 6 63.7 54 5 2 o55i 621186 52i8i 9 622461 523o83 523 7 i3 5 63.2 55 524343 624972 5255 99 626226 626862 52 7 4 77 4 62.6 56 628102 628726 5 29 347 52 99 6 9 530690 53i2o 9 3 62.1 57 53i828 532446 533o63 53367 9 534296 534 9 o 9 2 61.6 58 5355 2 3 536i36 536 7 47 537358 537969 5385 7 8 I 61.1 5 9 539186 53 9 7 9 4 54o4oi 541007 641612 542216 6o.5 60" 50" 40" 30" 20" 10" \t :_. 1 Do-sine of 38 Degrees Aim. LOGARITHMIC TANGENTS. Man. Tangent of 1 Degree. P. Pait 0" 10" 20" 30" 40" 50" to 1". 8.241921 8.243126 8.244328 8.246626 8.246 7 2I 8.247913 5 9 II 9 . 7 I 249102 260287 25l46 9 262648 253823 264996 58 II7. 7 2 266166 2 5 7 33i 2 584 9 4 269664 260811 261966 57 ixS.8 3 263n5 264263 266408 266649 267688 268824 56 n4.o 4 269966 271086 2 7 22l3 2 7 333 7 2 7 4458 276676 55 112. 2 5 276691 277804 2 7 8 9 i3 280020 281124 282226 54 no. 5 6 2833 2 3 284419 286612 286602 287689 288774 53 108.9 7 280.866 290935 2 9 2OI2 293086 294.167 296226 62 107.2 296292 297355 2 9 84i6 2994 7 4 3oo53o 3oi583 5i 106.7 9 302634 3o3682 3o4 7 2 7 306770 3o68u 307849 5o 104.2 10 8.3o8884 8.309917 8.3io 9 48 8.311976 8.3i3oo2 8.3i4o25 49 102.7 ii 3i5o46 3i6o65 3i 7 o8i 318096 319106 320116 48 loi.S 12 32II22 322127 323l2 9 324129 326126 326121 47 99.8 i3 32 7 Il4 328106 32 9 o 9 3 33oo8o 33io64 332o45 46 98.6 i4 333025 334oo2 334 977 335 9 5o 336921 337890 45 97.2 i5 338856 339821 34o 7 83 34i 7 43 342701 343657 44 96.9 16 3446io 345562 3465i2 347469 3484o5 349348 43 94.6 17 360289 361229 352i66 353ioi 354o35 354966 42 93.4 18 3558 9 5 356823 35 77 48 3586 7 i 369693 36o5i2 4.i 92.2 '9 36i43o 362345 36325 9 364i 7 i 365o8i 365 9 88 4o 91.1 20 8. 3668 9 5 8.367799 8.368 7 oi 8.36 9 6oi 8.370600 8.3 7 i3 97 3 9 89.9 21 372292 3 7 3i85 3 7 4o 7 6 374966 3 7 5853 3 7 6 7 38 38 88.8 i.2 377622 3 7 85o4 3 79 385 380263 38n4o 382016 37 87.8 23 382889 383 7 6o 38463o 385498 386364 38 7 229 36 86.7 24 388092 388 9 53 38 9 8i3 39o6 7 o 391626 392381 35 85. 7 25 3 9 3234 3 9 4o85 394934 395 7 82 396628 39 7 4 7 2 34 84.7 26 398316 3 99 i56 3 9999 6 4oo834 401670 402606 33 83.7 27 4o3338 404170 4o5ooo 406828 4o6655 4o 7 48o 32 82.8 28 4o83o4 409.126 409946 4io 7 65 4n583 412399 3i 81.8 29 4i32i3 414026 4i483 7 4i564 7 4i6456 4i 7 263 3o 80.9 3o 8.418068 8.418872 8.4i96 7 4 8.42o4 7 5 8.421274 8.422O 7 2 29 80.0 3i 422869 423664 4s4458 426260 426041 42683o 28 79. i 32 427618 428404 429189 4299 7 3 430766 43i536 27 -js',3 33 43 2 3i5 433o 9 3 4338 7 o 434645 4354i 9 436191 26 77.6 34 436962 43 77 3 2 4385oo 43926 7 44oo33 44o 7 9 7 25 76.6 35 44i56o 442322 443o82 44384i 444699 445355 24 7 5.8 36 446iio 446864 44 7 6i6 448368 449117 449866 7 5.o 37 45o6i3 45i35 9 462104 45 2 84 7 45358 9 45433o 22 7 4.3 38 466070 466808 456545 45 7 28i 458o<6 458 7 4 9 2 I 7 3.5 3 9 45 9 48i 460212 460942 46i6 7 o 4623 9 8 463i24 20 72.8 4o 8.46384 9 8.4645 7 2 8.466296 8.466oi6 8.466736 8.46 7 455 19 72.1 4i 468172 468889 469604 4 7 o3i8 47io3i 4 7 i 7 43 18 7 i.3 42 472454 4 7 3i63 4 7 38 7 2 4 7 45 79 476286 4 7 5 99 o i 7 70.7 43 476693 477396 4 7 8o9 7 479497 480196 16 70.0 44 480892 48i588 482283 482976 48366 9 48436o 16 6 9 .3 45 485o5o 485 7 4o 486428 487116 487801 488486 i4 68.6 46 489170 489862 490634 491216 491894 4926-73 13 68.0 47 493260 493927 494602 496276 496949 496622 12 67.4 48 497293 497963 498632 499300 499967 5oo633 II 66.8 49 601298 601962 602626 608287 5o3 9 48 5o46o8 10 66.1 5o 8.606267 8.606926 8.606682 8.5o 7 238 8.607893 8.5o854 7 9 65.5 5i 609200 609862 5io5o3 5iu53 611802 612461 8 65.o 62 5i3o 9 8 5i3 7 44 5i438 9 5i5o34 616677 5i632o 7 64.4 53 616961 617602 618241 618880 619618 620164 6 63.8 54 620790 621426 622069 622692 523324 623966 5 63.3 55 624686 626216 525844 5264 7 2 627098 52 77 24 4 62.7 56 628349 6.289-73 629696 53o2i8 53o84o 53i46o 3 62.2 67 532o8o 532698 5333i6 533933 534549 535i64 2 61.6 58 535 779 5363 9 2 53 7 oo5 53 7 6i 7 538227 53883 7 I 61.1 5 9 53 9 447 54oo55 540662 641269 54i8 7 5 642480 O 60.6 60" SO" 40" 30" 20" 10" Co-tangent of 88 Degrees, LOGARITLMIC SlNES. d 1 Sine of 2 Degrees. _ Sine of 3 Degrees. s o" rl 10* j 20" 30-'' 40" 50" ^ 0" | 10' 20" 30" 40'' 50" 018.542819 3422 4oa3 4624 5224 5823 5 9 8. 7 i88oo 9202 960!: ...4 .4oz .8o4 5 9 JJ 6422 7 oi 9 7616 8212 8807 9401 58 I 8. 7 2I2O4 i6o3 20O2 2401 2799 3i 97 58 , 2 999 5 .58 7 1179 I 77 236i sgSo 5 7 2 35 9 5 3992 4^9 4 7 85 5i8i 55 77 57 2 8.553539 4i26 47i3 53oo 5885 6470 56 3 59-72 636 7 6 7 62 7 i56 7 55o 7943 56 j 4 7054 7 63 7 8219 8801 9 38i 9961 55 4 833 7 8 7 2 9 9122 95i4 9906 2 97 55 8.56o54u mo 1696 22 7 3 2849 3425 54 A 8. 7 3o688 10-79 i46 9 i85 9 2249 2638 54 6 3999 45 7 3 5i46 5 7 i 9 6290 6861 53 6 302 7 34i6 38o^ 4l 9 2 45 7 9 4967J53 7 ?43i 8000 856 9 9 i3 7 97o4 .270 52 7 5354 5 7 4o 6126 65i2 6898 7283 52 g 8.5 7 o836 i4oi 1965 2528 3091 3653 5i 8 7 66 7 8o52 8436 8820 9203 9586 5i 9 421^ 47745334 58 9 3 645 1 7009 5o 9 9969 .35 2 . 7 34 in5 1497 1878 5o 10 7566 8122 8678 9232 9786 .34o 49 10 8. 7 42259 2 63 9 3oi 9 33 99 3778 4i5 7 4 9 1 1 12 8 53o8 9 2 4193 1 444 4 7 4i i 99 5 5 2 88 2546 5834 3096 638o 3645 6 9 25 48 47 ii 12 4536 6802 49i4 7178 52 9 3 7 554 56 7 o 79 3o 6o48 83o5 6425 8680 48 47 i3 7469 8oi3 8556 9098 9640 .181 46 i3 9 o55 9 43o 9 8o/ ; .178 .55i .924 46 i4 8.590721 1260 1799 2338 28 7 5 3412 45 i4 8. 7 5i2 97 l6 7 ol2o42 2414 2786 3i5 7 45 x5 3 9 48 4484 5019 5553 6087 6619 44 i5 3528 38 9 8 4269 463 9 5oo8 53 7 8 44 16 7 l52 7 683 8214 8 7 45 9274 980^ 43 16 5 7 4 7 6116 6484 6852 7220 758 7 43 178.600332 o85 9 i38 7 I 9 i3 243 9 2964 42 *7 7955 832i 8688 9054 9420 9786 42 18 348 9 4012 4536 5o58 558o 6102 4i 18 8. 7 6oi5i o5i6 0881 1245 1609 i 97 3 4i J 9 6623 7 i43 7662 8181 8699 9217 4o J 9 233 7 2 7 OO 3o63 3425 378 7 4i4 9 4o 20 9734 .261 .766 1282 1796 2310 39 20 45n 4872 5234 55 9 4 5 9 55 63i5 3 9 21 8.612823 3336 3848 436o 48 7 i 538i 38 21 66 7 5 7 o34 7 3 9 4 7752 8m 846 9 38 22 58 9 i 64oo 6909)7417 79 2 4 843i 37 22 8828 9 i85 9 543 9900 . 2 5 7 .6i3 3? 23 8 9 3 7 9442 9947-452 . 9 56 1459 36 23 8. 77 o 97 o i326 1687 2037 2392 2 7 4 7 36 24 8.621962 2464 2 9 65 3466 3966 4466 35 24 3ioi 3456 38io 4i63 45i 7 48 7 o 35 25 4 9 65 5464 5962 6459 6956 7453 34 25 5223 55 7 5 5 9 2 7 6279 663i 6 9 8 2 34 26 7948 8444 8 9 38 9 432 9926 .419 33 26 7 333 7 684 8o35 8385 8 7 35 9 o85 33 27 8.630911 i4o3 i8 9 42385 28 7 5 3365 32 27 9434 9783 .132 .48o .829 1177 o 2S| 3854 4342 483o53i 7 58o4 6291 3i 28 8. 7 8i524 l8 7 2 2219 2566 2912 8269 3i 29 6776 7 262 7746823o 8714 9197 3o 29 36o5 3 9 5i 4296 464 1 4 9 86 533i 3o 3o 9680 .162 .643 1124 i6o4 2084 29 3o 56 7 5 6oi 9 6363 6707 7 o5o 7 3 9 3 2Q 3i 8.642563 3o42 35203998 4475 4952 28 3i 7736 8o 7 8 8421 8762 9104 9 446 28 % 5428 5904 6379 6854 7328 7801 2 7 32 9787 .128 .468 .808 1149 i488 2 7 33 8274 8 7 47 92199690 .161 .632 26 33 8.791828 2l6 7 25o6 2845 3i83 352i 26 34 8.65uo2 i5 7 i 2040 25o8 2976 3444 25 34 385 9 4i 97 4534 4872 5208 5545 25 35 3911 43 77 4843 53o8 5 77 3 6238 24 35 588i 6218 6553 6889 7 224 7 55 9 24 36 6702 7 i65 7628 8090 8552 9014 23 36 7894 82291 8563 8897 9 23l 9 564 23 3? 9 4 7 5 99 35 .3 9 5 .855 i3i4 1772 22 37 9897 .230 .563 .896 1228 i56o 22 38 8.66223o 2688 3i45 36o2 4o58 45i3 21 38 8.801892 2223 2554 2885 3 2I 6 3546 21 3 9 4968 54?,3 58 77 633i 6 7 84 7 23 7 2O 3 9 38 7 6 4206 4536 4866 5i 9 5 55 2 4 20 4o 7689 8i4i 85929043 9 4 9 4 9944 19 4o 5852 6181 65og 683 7 7 i65 7 4 9 2 '9 4i 8.870393 0842 1291 i 7 39 2187 2634 18 4i 7819 8t 46 84 7 3 8799 9126 9 45i 18 42 3o8o 35 27 3 97 2 44i8 4863 53o7 17 42 9777 .io3 .428 . 7 53 10-78 1402 17 43 5 7 5i 6194 6638 ! 7 o8o 7522 7964 16 43 8.811726 2o5o 23 7 4 2698 3021 3344 16 44 84o5 8846 9286^9726 .166 .6o5 i5 44 366 7 3 9 8 9 43i2 4634 4 9 56 52 77 i5 45 8 68io43 i48i 191912356 2793 323o i4 45 5599 5 9 2O 6241 656i 6882 7 202 i'4 46 3665 4ioi 45364971 54o5 5838 i3 46 7 522 7841 8161 848o 8 799 9Il8 i3 4? 6272 6 7 o5 71377669 8001 8432 12 47 9 436 ; 7 55 .. 7 3 .390 .-708 1025 12 48 8863 9 2 9 3 9723 |.i52 .58i IOIO II 48 8.82i343 i65 9 1976 2292 2609 2925 II 4 9 8. 6 9 i438 1866 2293 2 7 2O 3i46 35 7 2 IO 49 324o 3556 38 7 i 4i86 45 OI 48:6 IO 5o 3998 4423 4848 5272 5696 6120 9 5o 5i3o 5444 5 7 58 6072 6385 6698 9 5i 6543 6966 7 388 7 8io 8232 8653 8 5i 7011 7 324 7 63 7 7949 8261 85 7 3 8 52 9 o 7 3 9494 9913 .333 .752 1171 7 52 8884 9196 9 5 7 9818 . 129 43 9 7 53 8.701589 2OO 7 2424 2841 3 2 58 36 7 4 6 53 8.83o 7 4 9 1060 1369 1679 1988 22 9 8 6 54 4090 45o5 4920 5335 5 749 6i63 5 54 26o 7 2 9 l5 3224 3532 384o 4i48 5 55 65 77 6990 7 402 7 8i5 8226 8638 4 55 4456 4 7 63 5o 7 o 53 77 5684 5 99 i 4 56 9049 9460 98 7 o .280 .690 1099 3 56 629-7 66o3 6909 72i5 7 52O 7825 3 5? 8, 7 n5o 7 1916 2324 27 3i 3:39 3546 2 5 7 8i3o 8435 8 7 4o 9044 9348:9652 2 58 3g52 4358 4 7 64 5i6 9 55 7 4 5 979 I 58 99 56 .260 .563 .866 1169 l4 7 2 I J.9 6383 6787 7190 7 5 9 3 799 6 3398 5 9 8.84r 77 4 2O 7 6 23 7 8 2680 2982 3283 O 60" OOP 40" 30" 20" 10" c 60" 50" | 40" 30" | 20" 10" c Co-sine of 87 Degrees. a Co-sine of 86 Degrees. S C j// 2 ,. 3" 4" 5" 6" 7" 8" 9" J 48 96 145 1S3 241 289 338 386 434 .( 1" 2" 3" 4" 5" G" 7" 8" 9" irt } 34 69 103 138 172 207 2-11 275 310 LOGARITHMIC TANGENTS. .5 Tangent of 2 Degrees. _c Tangent of 3 Degrees. s 0" 10" 20" 30" 40" 50" 0" 10" 20" 30" 40" j 50" o 6.543o84 368 7 4289 4891 5492 6092 5 9 8.yi 9 3 9 6 979 8 .2OI .6o3 ioo4 i4o5 5 9 i 6691 7289 7887 8483 9079 9674 58 i 8. 7 2i8o6 22O 7 2607 3007 3406 38o5 58 2 8 550268 0862 i454 2046 263 7 3227 5 7 2 42o4 4602 5ooo 53 97 5 79 4 6191 ^7 3 38i 7 44o5 499 3 558o 6166 6752 56 3 6588 6 9 84 738o 777 5 8170 8565 56 4 y336 7920 85o3 9085 9667 .248 55 4 8 9 5 9 9353 9746 . i4o .533 . 9 25 55 5 8.56o828 1407 1985 2563 3i4o 3 7 i6 54 5 8. 7 3i3i 7 I 7 9 2101 24 9 2 2883 32 7 3 54 6 4291 4866 544o 6oi3 6585 7 l5 7 53 6 3663 4o53 4442 483i 5220 56o8 53 7 7727 8298 8867 9 435 ...3 .570 52 7 5 99 6 6384 6771 7 i58 7 545 79 3i 52 6 8.5 7 n37 1702 2267 2832 33 9 5 3 9 58 5i 8 83i 7 8 7 o3 9088 9 4 7 3 9 858 .242 5i 9 4520 5o8i 5642 6201 6760 7 3i 9 5o 9 8.740626 1009 i3 9 3 1776 2i58 254o 5o 10 7877 8434 8990 9545 . IOO .654 49 10 2922 33o4 3685 4o66 4447 4827 4 9 ii 8- 58 1 208 1760 23l2 2864 34i4 3 9 64 48 ii 5207 5586 5 9 66 6344 6723 7101 48 12 45i4 5o62 56io 6i57 6704 7249 47 12 7479 7 85 7 8234 8611 8 9 88 9 364 47 i3 77 9 5 8339 8883 9426 9968 .5io 46 i3 974o .116 4 9 i .866 1241 i6i5 40 i4 8.5gio5i i5gi 2l3l 2670 3208 3746 45 i4 8.751989 2363 2 7 36 3109 3482 3855 45 i5 4283 4820 5355 5890 6425 6 9 5 9 44 i5 4227 45 99 4 9 7o 534i 5712 6o83 44 16 7492 8024 8556 9087 9618 .i4? 43 16 6453 6823 7 i 9 3 7562 79 3i 83oo 43 17 8.600677 I2O5 1733 2260 2787 33i3 42 l l 8668 9o36 9 4o4 9771 .139 .5o5 42 18 383 9 4363 488 7 54n 5 9 34 6456 4i 18 8.760872 1238 1604 1970 2335 2700 4i ^9 6978 7499 8019 853 9 go58 9 5 7 6 4o J 9 3o65 3429 3 79 3 4i5 7 4520 4884 4o 20 8.610094 0612 1128 1 644 2160 26 7 5 3 9 20 5246 5609 5 97 i 6333 66 9 5 7 o56 39 21 3189 3702 42i5 4728 524o 5 7 5i 38 21 74i7 777 8 8139 8499 885 9 9218 38 22 6262 6772 7281 779 8298 8806 37 22 9 5 7 8 9937 .295 .654 IOI2 1370 37 23 9 3i3 9819 .325 .83o i335 i83 9 36 23 8. 77 i 7 2 7 2o85 2442 2798 3i55 35n 36 24 8.622343 2846 3348 385o 435i 485 2 35 24 3866 4222 45 77 4 9 3 2 5287 564i 35 25 5352 585i 635o 6849 7346 7844 34 25 5 99 5 6349 6702 7o56 7409 7761 34 26 834o 8836 9 332 9827 .321 .8i5 33 26 8n4 8466 8817 9169 9520 9871 33 27 8.63i3o8 1801 2293 2 7 85 32 7 6 3 7 66 32 27 8.780222 o5 7 2 0922 1272 1622 1971 32 28 4256 4746 5 2 35 5 7 23 6211 6698 3i 28 2320 2669 3017 3365 3 7 i3 4o6i 3i 29 7i84 7 6 ?I 8i56 864i 9126 9610 3o 2 9 44o8 4 7 55 5 I 02 5448 5 79 4 6i4o 3o 3o 8.640093 0576 io58 i54o 2O2I 25O2 2 9 3o 6486 683i 7177 7521 7866 8210 29 3i 2982 3462 394i 4420 4898 53 7 6 28 3i 8554 8898 9242 9 585 9928 .271 28 32 5853 632 9 68o5 7281 7756 8 2 3o 27 32 8. 7 9o6i3 0955 1297 1639 1980 2321 2 7 33 8704 9178 9 65i .123 .5 9 5 1067 20 33 2662 3oo3 3343 3683 4023 4362 26 34 8.65i53 7 2008 2478 2947 34x6 3884 25 34 4 7 oi 5o4o 53 79 5 7 i8 6o56 63 9 4 25 35 4352 4820 5 2 86 5 7 53 6219 6684 24 35 6 7 3i 7 o69 7 4o6 77 43 8079 84i6 24 36 7149 7613 8077 854i 9004 9466 23 36 8 7 5 2 9088 9 423 97 5 9 .. 9 4 .429 23 37 9928 .38 9 .85o i3n 1771 2230 22 37 8.8oo 7 63 1098 i432 i 7 65 2099 2432 22 38 8.662689 3i48 36o6 4o63 4520 4977 21 38 2 7 65 3098 343i 3 7 63 4095 4427 21 3 9 5433 588 9 6344 6799 7 253 7707 20 3 9 4 7 58 Sogo 5421 5 7 5i 6082 6412 2O 4o 8160 86i3 9065 9 5i 7 9968 .419 '9 4o 6 7 42 7 7 2 7 402 77 3i 8060 838 9 '9 4i 8.670870 1320 1769 2218 2667 3n5 18 4i 8 7 i 7 9O46 9 3 7 4 9701 ..29 .356 18 4s 3563 4oio 445 7 4903 5349 5 79 4 17 42 8.8io683 IOIO i33 7 i663 1989 23 1 5 i? 43 6239 6684 7128 7 5 7 2 8oi5 845 7 16 43 2641 2966 32 9 I 36i6 3 9 4i 4265 16 44 8900 934i 97 83 .224 .664 uo4 i5 44 458 9 4913 5237 556o 5884 6207 i5 45 8.68i544 1983 2422 2860 32 9 8 3 7 35 i4 45 6529 6852 7174 7496 7818 8Uo i4 46 4172 46o8 5o44 548o 5 9 i5 635o i3 46 846 1 8 7 82 9 io3 9423 9744 ..64 i3 47 6784 7218 7652 8o85 85i 7 8 9 5o 12 47 8.820384 o 7 o3 1023 1 342 1661 1980 I 2 48 9381 9813 .244 6 7 4 no4 i534 II 48 2298 26l 7 2 9 35 3253 3570 3888 II 49 8.691963 2392 2820 3248 36 7 5 4io3 10 49 42o5 4522 4838 5i55 547i 5 7 8 7 IO 5o 4529 4956 538i 5807 6232 6656 9 5o 6io3 64i8 6733 7049 7 363 7678 9 5i 7081 7 5o4 7928 835i 8 77 3 9 i 9 5 8 5i 799 2 83o 7 8621 8934 9248 9661 8 52 9617 ..38 .45 9 .880 i3oo 1720 7 52 9 8 7 4 .187 .5oo .812 1124 i436 7 53 8.702139 2558 2976 33 9 5 38i2 4230 6 53 8.83i 7 48 2o5g 2371 2682 2992 33o3 6 54 4646 5o63 54 79 58 9 5 63io 6725 5 54 36i3 3924 4234 4543 4853 5i6 2 5 55 7140 7554 7967 838i 8 79 4 9206 4 55 54 7 i 5 7 8o 6o8 9 63 97 6 7 o5 7013 4 56 9618 ..3o .442 .853 1263 i6 7 4 3 56 7 32I 7629 7936 8243 855o 8857 3 57 8.712083 2493 2902 33n 3 7 i 9 4127 2 57 9i63 9470 9776 ..81 .38 7 .692 2 58 4534J4942 5348 5 7 55 6161 656 7 I 58 8.840998 i3o3 1607 I 9 I2 2216 2521 I 5 9 6972 7377 7781 8186 8589 8 99 3 O 5 9 2825 3i28 3432 3 7 35 4o38 434l O 60" 50" 40" 30" 20" 10" fi 60" 50" 40" 30" 20" 10" B* Co-tangent of 87 Degrees. 9 S Co-tangent of 86 Degrees. P PnrtJ 1//2// 3// 4 " 5 " fi " 7// 8" 9" "*j 48 97 145 193 242 290 338 387 435 ( 1 2 " 3" i" 5" 6" 7" 8" 9 ;/ m \ 35 69 104 138 173 207 242 275 311 LOGARITHMIC SINES. J Sine of 4 Degrees. y Sine of 5 Degrees. * 0" 10" 20" 30" 40" ] 50" i 0" 10" 20" 30" ' 40" 50" o 8.843585 3886 4i86 448 7 4787 5o8 7 5 9 o 8. 9 4o2 9 6 o53 7 777 IOI 7 1258 i4 9 8 5 9 I 538 7 568 7 5 9 8 7 6286 6585 6884 58 I I 7 3 i 977 22I 7 2696 2935 58 2 7 i83 748 1 7780 8078 83 7 6 86 7 3 5 7 2 3i 7 4 34i3 3652 SSgi 4i2 9 4368 5 7 3 8 97 i 9268 9 565 9862 .i5 9 .455 56 3 46o6 4844 5o83 532i 5558 5 79 6 56 4 8.85o 7 5i 1047 1 343 1639 i 9 34 222 9 55 4 6o34 62 7 I 65o8 6 7 45 6 9 82 7 2I 9 55 5 2525 28l 9 3ii4 34o8 3 7 o3 3 997 54 5 7456 7 6 9 3 79 2 9 8166 8402 8638 54 6 42 9 I 4584 48 7 8 5i 7 i 5464 5 5io 53 6 88 7 4 9 no 9 345 9 58i 9817 ..52 53 7 OO^Q 6342 6634 6926 7 2l8 52 7 8. 9 5o28 7 0522 o 7 5 7 o 99 2 1227 i46i 53 8 7 8oi 8o 9 2 8383 86 7 4 8 9 65 92 55 5i 8 i6 9 6 I 9 3o 2164 2 3 9 8 2632 2866 5i 9 9 546 9 836 .126 .4i5 . 7 o5 994 5o 9 3ioo 3333 356 7 38oo 4o33 4266 5o 10 8.86i283 1572 1861 2149 2438 2-726 49 10 44 99 4 7 32 4 9 65 5i 9 7 542 9 5662 4 9 ii 3oi4 33o2 3589 38 77 4i64 445 1 48 ii 58 9 4 6126 6358 65 9 o 6821 7 o53 48 12 4 7 38 5o24 53n 55 97 5883 6i6 9 47 12 7284 7 5i6 7747 7978 82O 9 844o 47 i3 6455 6740 7025 7 3io 7 5 9 5 7 88o 46 i3 8670 8901 9 i3i 9 362 9 5 9 2 9 822 46 i4 8i65 844 9 8 7 33 901-7 9 3oi 9 585 45 i4 8. 9 6oo52 0282 o5n 0741 o 97 o I2OO 45 15 9 868 .i5i .434 . 7 i 7 1000 1282 44 i5 1429 i658 1887 2116 2344 2 5 7 3 44 16 8.8 7 i565 1847 2129 26 9 2 29^3 43 16 2801 3o3o 3258 3486 3 7 i4 3 9 42 43 i 7 3 2 55 3536 38i 7 '4o 9 7 43 7 8 4658 42 '7 4170 43 97 4625 4852 5o8o 53o 7 42 18 4 9 38 52i8 5498 5 777 6o5 7 6336 4i 18 5534 5 7 6i 5 9 8 7 6214 644 1 666 7 4i i 9 66i5 68 9 4 7172 745 1 7729 8oo 7 4o i 9 68 9 3 7 I2O 7346 7572 7797 8023 4o 20 8285 8563 884i 9118 9 3 9 5 9 6 7 2 3 9 20 824 9 8474 8700 8 9 25 9 i5o 9 3 7 5 39 21 9949 .226 .5o3 779 io55 i33i 38 21 9 6oo 9 8 2 5 ..4 9 .274 4 9 8 . 7 23 38 22 8.881607 i883 2i58 2433 2-708 2 9 83 37 22 8. 97 o 9 47 n 7 i i3 9 5 i6i 9 1842 2066 37 23 3 2 58 3533 38o 7 4o8i 4355 462 9 36 23 228 9 2 5i3 2736 2 9 5 9 3i82 34o5 36 24 4 9 o3 5i 77 5 99 6 626 9 35 24 3628 385i 4o 7 3 42 9 6 45i8 4740 35 25 6542 68i4 7087 7359 7 63i 79 3 34 25 4 9 62 5i84 54o6 5628 585o 6071 34 26 8174 8446 8717 8988 9 25 9 9 53o 33 26 62 9 3 65i4 6735 6 9 56 7177 7 3 9 8 33 27 9 8oi ..71 .34i .612 .882 n5i 32 2 7 7619 7 83 9 8060 8280 85oi 8721 3a 28 8.891421 i6 9 o 1960 2229 2 4 9 8 2 7 6 7 3i 28 8 9 4i 9 i6i 9 38i 9 6oo 9 820 ..3 9 3i 29 3o35 33o4 3572 ! 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...4 .270 .53 7 27 32 62I 7 643 7 6656 68 7 5 7094 7 3i3 27 33 8. 9 oo8o3 io6 9 i335 1601 1867 2l32 26 33 7 532 77 5 79 6 9 8i8 7 84o6 8624 26 34 2 3 9 8 2663 2 9 28 3i 9 3 3458 3722 25 34 8842 9 o6o 9 2 7 8 9 4 9 6 9714 99 3i 25 35 3 9 8 7 425i 45i5 4779 5o43 53o6 24 35 8. 99 oi4 9 o366 o583 0801 1018 1235 24 36 55 7 o 5833 6o 9 6 635 9 6622 6885 23 36 i45i 1668 i885 2101 2 3i8 2534 23 3? 7147 7410 7672 79 34 8i 9 6 845 7 22 37 2 7 5o 2 9 66 3i82 3398 36i4 383o 22 38 8 ?I9 8 9 8o 9 242 9 5o3 9764 ..25 21 38 4o45 4261 44 7 6 4692 490-7 5l22 21 3 9 8. 9 io285 o546 0806 1066 i3 2 6 i586 20 3 9 533 7 5552 5 7 66 5981 6196 64io 2O 4o i846 2106' 2365 2624 2883 3l42 '9 4o 6624 683 9 7 o53 7 26 7 7 48i ?6 9 4 I 9 4i 34oi 366o 3 9 i8 4177 4435 46 9 3 18 4i 7908 8122 8335 8549 8 7 6 2 8 97 5 18 42 4 9 5i 52O 9 5466 5724 5 9 8i 6238 '7 42 9188 9 4oi 9614 9 8 27 ..4o .252 X 7 43 64 9 5 6752 7OO 9 7265 7522 7778 16 43 9 .ooo465 0677 0889 IIO2 i3i4 i5 2 6 16 44 8o34 820.0 8546 8801 9 5 7 9 3l2 i5 44 I 7 38 i 9 4 9 2161 23 7 3 2 584 2 79 5 i5 45 9 568 9 823 ..78 .332 .58 7 .84i i4 45 3oo 7 3 2 i8 3429 364o 385i 4o6i i4 46 8. 9 2io 9 6 i35o 1604 i858 2112 2365 i3 46 42 7 2 4483 46 9 3 4 9 o4 5u4 5324 i3 4? 26l 9 2872 3i 2 5 3378 363i 3884 12 47 5534 5 7 44 5 9 54 6164 63 7 3 6583 12 48 4i36 438 9 464 1 48 9 3|5i45 53 97 II 48 6 79 2 7002 7 2II 7 42O -7629 7 838 II 49 564 9 5 9 oo 6i52 64o3 6654 6 9 o5 10 49 8o4 7 8 2 56 8465 86 7 3 8882 9 o 9 o IO 5o 7 i56 74o 7 7 65 7 79 o8 8i58 84o8 9 5o 9 2 9 8 9 5o 7 97 i5 99 23 .i3i .338 9 5i 8658 8 9 o8 9 i58 9 4o 7 9 65 7 99 o6 8 5i 9 ,oio546 o 7 54 0961 ii6 9 i3 7 6 1 583 8 62 8. 9 3oi55 o4o4 o653 O 9 O2 n5o i3 99 7 52 I 79 199-7 220^5 2411 2618 2824 7 53 1 647 i8 9 5 2i43 23 9 I 2 63 9 2887 6 53 3o3i 3 2 3 7 3444 365o 3856 4062 6 54 3i34 338i 362 9 38 7 6 4123 436 9 5 54 4268 44 7 4 468o 4886 Sogi 52 97 5 55 46i6 4862 5io 9 5355 56oi 5847 4 55 55o2 5 7 o 7 5 9 i3 6118 6323 6528 4 56 6o 9 3 633 9 6584 683o 7 o 7 5 7320 3 56 6 7 32 6 9 3 7 7 l42 7 346 7 55i 77 55 3 57 7 565 7810 8o55 82 99 8544 8788 2 5 7 79 5 9 8i64 8368 85 7 2 8 77 6 8 9 79 2 58 9 o32 9 276 9 52O 9.764 ...7 .25l I 58 9 i83 9 38 7 9 5 9 o 9794 9997 .200 I 5 9 8. 9 4o4 9 ^ o 7 38 o 9 8i 1224 i46 7 1700. o 5 9 9 .o2o4o3 0606 o8o 9 IOI2 I2l5 i4i8 O 60" 50" 40" 30" 20" 10" 60" 50" 40" 30" ' 20" 10" a Co-tangent of 85 Degrees. .3 S Co-tangent of 84 Degrees. 1 p Po .< 1" 2" 3" 4" 5" 6" 7" 8" 9" irl \ 27 54 81 108 135 162 188 215 242 .< 1" 2" 3" 4" 5" 6" 7" 8" 9" \ I 22 44 66 88 110 132 154 177 199 LOGARITHMIC SINES. 1 .9 S tSine of 6 Degrees. 1 Sine of 7 Degrees. 0" 10" 20" 30" 40" W 0" 10" 20" | 30" 40" 1 50" 019.019235 9 435 9 635 9 835 ..35 .235 5 9 O 9.085894 6066 6237 64o 9 658o 6 7 5i 5 9 I 9.020435 o635 o834 io34 1233 i433 58 I 6922 7093 7264 7435 7606 7777 58 a| i632 i83i 2o3o 222 9 2428 2627 57 2 7947 8118 8288 845 9 8629 8800 57 1 ^ 2825 3o24 3223 342i 36i 9 38i8 56 3 8970 9140 93io 9 48o 965i 9 82O 56 4 4oi6 4214 44i2 46io 48o 7 5oo5 55 4 999 .160 .33o .5oo .669 .83 9 55 5 52o3 54oo 55 9 8 5 79 5 5 99 2 6189 54 5 9.091008 1178 1 347 i5i6 i685 i855 54 fij 6386 6583 6780 6977 7174 7370 53 6 2024 2193 2362 253o 2609 2868 53 7 7367 7763 7960 8i56 835 2 8548 52 7 3o37 32o5 3374 3542 o * 3711 38 79 J52 8 9 8 7 44 99 i8 8 9 4o .u4 9 i36 .309 9 33 2 .5o4 9 5 27 .690 97 23 8 9 4 5i 5o 8 9 4047 5o56 4216 5223 4384 53 9 i 4552 555 9 4720 5726 4S88|5i 5894]5o 10 ty.o3io8 9 1284 1479 1673 i868'2o62 49 10 6062 6229 63 9 6 6564 6 7 3i 6898 4 9 li 225 7 245 1 2645 283 9 3o33 3227 48 ii 7 o65 7232 7399 7566 77 33 7900 48 li: 3421 36i5 3809 4OO2 4196 438 9 47 12 8066 8233 83 99 8566 8 7 32 8899 47 i3 4582 4776 4969 5i62 5355 5548 46 i3 9065 9 23l 9 3 9 8 9 564 97 3o 9896 46 r4 5 7 4i 5 9 33 6126 63 1 9 65n 6703 45 i4 9. 100062 0227 o3 9 3 o55 9 0725 0890 45 i5 68 9 6 7 o88 7280 7472 7664 7 856 44 i5 io56 1221 i38 7 i552 1717 i883 44 16 8o48 8239 843i 8623 88t4 9005 43 16 2048 2213 2 3 7 8 2543 2708 28 7 3 43 17 9 i 97 9388 9 5 79 9770 9961 .152 42 17 3o3 7 32O2 336 7 353i 3696 386o 4a 18 o533 0724 0914 no5 I2 9 5 4i 18 4o 2 5 4i8 9 4353 45i 7 4682 4846 4i J 9 i485 i6 7 5 i865 2 o55 2245 2435 4o 19 5oio 5 1 74 533 7 55oi 5665 5829 4o 20 2625 28i5 3oo4 3 1 94 3383 3572 3 9 20 5992 6i56 63i 9 6483 6646 6810 3 9 21 3 7 62 3 9 5! 4i4o432 9 45i8 4707 38 21 6 97 3 7i36 7299 7462 7625 77 88 38 22 48 9 5 5o84 52 7 3 546i 565o 5838 37 22 7 9 5i 8n4 8 277 843 9 8602 8760 3? 23 6026 6214 6402 6590 6778 6 9 66 36 23 8927 99 9 252 9 4i4 9 5 77 9739 36 24 7 i54 7 342 75297717 794 8o 9 i 35 24 9901 ..63 .225 .38 7 .711 35 25 82 79 8466 8653 884o 9027 9 2l4 34 25 9.110873 io34 II 9 6 i358 1519 1681 34 26J 940O 9 58 7 97749960 .147 .333 33 26 1842 2OO3 2i65 2326 2 48 7 2648 33 27';.o5o5i9 o 7 o6 0892 1078 1264 i45o 32 27 2809 2 9 70 3i3i 32 9 2 3453 36i3 32 28 i635 1821 2007 2192 23 7 8 2563 3i 28 3 77 4 3 9 35 4o 9 5 4256 44i6 45 77 3i 29 2/49 2 9 34 3119 33o4 348 9 36 7 4 3o 29 4 7 3 7 48 97 5o5 7 52i8 53 7 8 5538 3o 3o 385 9 !4o44 4228 44i3 45 97 4782 20 3o 5698 5858 6oi 7 6177 633 7 6497 2 9 3i 4 9 66 5i5o 533555i 9 5703 588 7 28 3i 6656 6816 6 97 5 7 i35 7 2 9 4 7453 28 32 6o 7 i 6254 6438 6622 68o5 6 9 8 9 27 32 7613 7772 79 3i 8090 8249 84o8 27 33 7172 7 356 7 53 9 7722 7905 8088 26 33 856 7 8726 8884 9 o43 9202 9360 26 34 82 7 I 8454 86378820 9002 9 i85 25 34 9 5l 9 9 6 77 9 836 99 9 4 ,|5a .'3 z i 25 35 9 36 7 9 55o 97329914 ..96 .278 24 35 9.120469 0627 o 7 85 IIOI 1259 24 36 9.060460 0642 0824 1006 1188 i36 9 23 36 1417 i5 7 4 I 7 32 1890 204 7 2205 23 37 i55i I 7 32 1914 2095 2276 2457 22 37 2362 2520 26 77 2835 2992 3i49 22 38 2639 2820 3ooi|3i8i 3362 3543 21 38 33o6 3463 3620 3 777 3 9 34 4091 21 3 9 3 7 24 3904 4o85'4265 4445 4626 2O 39 4248 44o4 456i 4718 48 7 4 5o3i 2O 4o 48o6 4986 5i66:5346 5526 SyoS I 9 4o 5187 5344 55oo 5656 58i2 5 9 6 9 I 9 4i 5885 6o65 6244 6424 66o3 6 7 83 18 4i 6i25 6281 643 7 65 9 3 6 7 48 6904 18 42 6962 7 i4i 7320 7499 7678 7 85 7 '7 42 7060 7216 7 3 7 i 7527 7 682 7 838 17 43 8o36 82i5 8393,85728751 8929 16 43 799 3 8i4 9 83o4 84^9 86i4 8 77 o 16 44 9 I0 7 9286 9464 9642 9820 999 8 i5 44 8 9 25 9 o8o 92 35 9 3 9 o 9 544 9699 i5 459.070176 o354 o532 0709 0887 io65 i4 45 9 854 ... 9 .i63 .3i8 -4 7 2 .62 7 i4 46 1242 1420 1597 1774 1951 2128 i3 46 9.i3o 7 8i o 9 36 io 9 o 1244 i3 9 8 i55 2 i3 47 2 3o6 2482 265 9 2836 3oi3 3i 9 o 12 4? I 7 o6 1860 20l4 2168 2322 24 7 6 12 48 3366 3543 3 7 i 9 38 9 6 4072 4248 II 48 263o 2 7 83 2 9 3 7 3091 3244 3398 II 4 9 4424 46oo 4777 4952 5i 2 8 53o4 10 49 355i 3704 3858 4on 4x64 43i 7 IO 5o 548o 5656 583i 6007 6182 6358 9 5o 44 7 o 4623 4 77 6 4929 5o82 5 2 35 9 5i 6533 6708 6883 7o58 7 233 7 4o8 8 5i 538 7 554o 56 9 3 5845I5 99 8 6i5o 8 52 7 583 77 58 7933 8107 8282 845 7 7 52 63o3 6455 66o 7 6760(6912 7064 7 53 863i 88o5 8980 9i54 9328 9 5o2 6 53 7216 7 368 7 520 7672 7 824 7976 6 54 96 7 6 9 85o ..24 .198 .372 .545 5 54 8128 82 79 843i 8582 8 7 34 8886 5 55 9 .o8o 7 i 9 0892 1066 1239 i4i3 i586 4 55 9 o3 7 9 i88 9 34o 9491 9 642 979 3 4 56j i75 9 1932 2IO5 2278 245i 2624 3 56 9944 ..96 .247 .3 9 8 .548 .699 3 57 2797 2369 3:42 33:4 348 7 365 9 2 57 9.:4o85o IOOI n5i 1,302 i453 i6o3 2 58 3832 4oo4 4176 4348 45so 46 9 2 I 58 1754 1904(2055 22O5 2355 2 5o5 I 59 4864 5o36 5208 538o 555i 5723 5 9 2655 280612956 3io6 3256]34o5 O 60" 50" 40" 30" 20" 10" c 60* 50" | 40" 30" 20" 1C-" d Co-sine of 83 Degrees. Co-sine of 82 Degrees. 8 . C 1" 2" 3" 4" 5" 6" 7" 8" 9" p p A I" 2" 3" 4" 5" 6" 7" 8" 9" T P&Trt < - irt } 16 32 48 64 80 96 112 128 144 LOGARITHMIC TANGENTS. 31 1 Tangent of 6 Degrees. .5 Tangent of 7 Degrees. 0" 10" | 20" 30" 40" 50'' 0" 10" 20" 30" 40" 50" 9.021620 1823 2025 2227 2430 2632 5 9 o 9.089144 9318 9492 9666 9 83 9 ..i3 5 9 I 2 834 3o36 3238 343 9 364i 3843 58 i 9.090187 o36i o534 0708 0881 io54 58 2 4o44 4245 4447 4648 4849 5o5o 57 2 1228 i4oi i5 7 4 I74 7 1920 2093 57 3 5 2 5i 5452 5653 5853 6o54 6254 56 3 2266 2439 2612 2 7 84 2 9 5 7 3129 56 4 6455 6655 6855 7 o55 72 55 7455 55 4 33o2 34 7 4 364 7 38i 9 3991 4i63 55 5 7 655 7855 8o55 8 2 54 8454 8653 54 5 4336 45o8 468o 485i 5o23 5i 9 5 54 6 8852 go52 925i 9 45o 9649 9 848 53 6 536 7 5538 5710 588i 6o53 6224 53 7.9.030046 0245 o444 o64t 0841 1039 52 7 63 9 5 6567 6738 6909 7 o8o 7 25l 5s 8f i23 7 i435 i633 i83i 2029 2227 5i 8 7422 7 5 9 3 7764 79 34 8io5 82 7 6 5i 91 2425 2623 2#20 3oi 7 32i534i2 5o 9 8446 8616 8787 8 9 5 7 912-7 9 2 9 8 5o 10 3609 38o6 4oo3 4200 43 97 45 9 4 49 IO 9468 9 638 9808 9978 .i48 .3i 7 49 ii 4791 4987 5i84 538o 55 7 6 5 77 3 48 ii 9. 100487 o65 7 0827 o 99 6 1166 i335 48 12 5969 6i65 636i 655 7 6 7 53 6 9 48 47 12 i5o4 i6 7 4 i843 2OI2 2181 235o 47 i3 7i44 7 33 9 7 535 77 3o 7 926 8121 46 i3 2519 2688 285 7 3O26 3i94 3363 46 i4 83i6 85n 8706 8 9 oi 9 o 9 5 9290 45 i4 3532 3 7 oo 386 9 4o3 7 42o5 43 7 4 45 i5 9485 9679 9874 ..68 .262 .456 44 i5 4542 4 7 io 48 7 8 5o46 52i4 5382 44 16 9.o4o65i o845 1039 1232 1426 1620 43 16 555o 5 7 i8 5885 6o53 6221 6388 43 I? i8i3 2007 22OO 2 3 9 4 2 58 7 2780 42 i? 6556 6 7 23 68 9 o 7 o58 7 225 7 3 9 2 42 18 2 97 3 3i66 335 9 3552 3 7 45 3 9 3 7 4i 18 7 55 9 77 26 7893 8060 8227 83 9 4 4i 19 4i3o 4322 45i5 4707 48 99 5o 9 2 4o J 9 856o 8 7 2 7 88 9 4 9060 9 22 7 9 3 9 3 4o 20 5 2 84 5476 5668 585 9 6o5i 6243 3 9 20 9 55 9 9 7 26 9 8 9 2 ..58 .224 .390 39 21 6434 6626 6817 7 oo 9 7200 7 3 9 i 38 21 q. no556 O 7 22 0888 io54 1219 i385 38 22 7682 777 3 7 964 8i55 8346 8536 37 22 i55i I 7 l6 1882 204 7 2213 2 3 7 8 37 23 8727 8917 9108 9 2 9 8 948 9 9 6 79 36 23 2543 2 7 o8 28 7 3 3o3 9 3204 3368 36 24 9869 ..5 9 .249 43 9 .629 .818 35 24 3533 3698 3863 4028 4l92 435 7 35 25 9.o5ioo8 1197 i38 7 i5 7 6 1766 i 9 55 34 25 452i 4686 485o 5oi5 5i 79 5343 34 26 21442333 2522 2 7 II 2900 3o88 33 26 55o 7 56 7 i 5835 5 999 6i63 632 7 33 27 32 77 3466 3654 3843 4o3i 4219 32 27 6491 6655 6818 6982 7 i45 7 3 9 32 28 4407 45 9 6 4 7 84 4 9 72 5i5g 534 7 3i 28 7472 7 636 7799 7962 8126 8289 3i 29 5535 5723 5 9 io 6o 9 8 6 2 85 64 7 2 3o 29 8452 86!5 8778 8 9 4i 9104 9266 3o 3o 665 9 6347 7 o34 7221 7 4o8 7 5 9 4 29 3o 9429 g5 9 2 97 54 99i 7 79 .242 29 3i 7781 7968 8i55 834i 85 2 8 8714 28 3i 9.120404 o56 7 7 2 9 0891 io53 I2l5 28 32 8900 9086 9 2 7 3 9 45 9 9 645 9 83i 27 32 i3 77 i53 9 I 7 OI i863 2025 2187 27 33 9.060016 O2O2 o388 o5 7 3 0759 0944 26 33 2346 25io 2671 2833 2994 3i56 26 34 n3o i3i5 i5oo i685 1870 2o55 25 34 33i 7 3478 364o 38oi 3962 4123 25 35 2240 2425 2610 2 79 5 2979 3i64 24 35 4284 4445 46o6 4 7 66 4927 5o88 24 36 3348 3533 3 7 i 7 3 9 oi 4o85 4269 23 36 5249 54o 9 55 7 o 5 7 3o 58 9 i 6o5i 23 37 4453 463 7 4821 5o'o5 5i8853 7 2 22 37 6211 63 7 i 6532 6692 6852 7012 22 38 5556 5 7 3 9 5922 6106 6 2 8 9 6472 21 38 7 I 7 2 7 332 7 4g2 7 65i 7811 7971 21 3 9 6655 6838 7 O2I 7 204 73877570 2O 3 9 8i3o 8290 8449 8609 8 7 68 8928 2O 4o 77 5 2 79 35 8lI 7 83oo 8482 8664 J 9 4o 9 o8 7 9246 94o5 9 564 9723 9882 19 4i 8846 9029 9 2II 9 3 9 3 9 575 9 756 18 4i 9.i3oo4i 0200 o35 9 o5i8 o6 7 6 o835 18 42 99 38 .120 .3oi .483 .664 .846 l l 42 0994 Il52 i3n i46 9 1628 1786 J 7 43 9.071027 1208 i38 9 i5 7 o 1751 I 9 32 16 43 1944 2IO2 2261 2419 25 77 2 7 35 16 44 2Il3 2294 2 4 7 5 2655 2836 3oi6 i5 44 2893 3o5o 3208 3366 3524 368i i5 45 3i 97 33 77 3558 3 7 38 3 9 i84o 9 8 i4 45 3839 3997 4i54 43l2 446 9 4626 i4 46 4278 4458 4638 4Si 7 49975i77 i3 46 4 7 84 4941 5098 5 2 55 54i2 556 9 i3 4? 5356 5536 5 7 i5 58 9 5 607416253 12 47 5 7 26 5883 6o4o 6l 97 6353 65io 12 48 6432 6611 6 79 o 6969 7 i48 7 32 7 II 48 666 7 6823 6980 7I 36 7 2 9 2 7449 II 49 7 5o5 7 684 7862 8o4i 8219 83 9 8 IO 49 7 6o5 77 6i -7918 8o 7 4 823o 8386 IO 5o 85 7 6 8 7 54 8 9 3 2 9 IIO 9 288 9 466 9 5o 8542 8698 8854 9009 9i65 9321 9 5i 9644 9822 . . . . .177 .355 .532 8 5i 94 7 6 9632 9788 9943 ..98 .254 8 62 9.080710 0887 1064 1241 1419 i5 9 6 7 52 9.140409 o564 7 20 o8 7 5 io3o i:85 7 53 I 77 3 1950 212,6 23o3 2480 265 7 6 53 i34o i495 i65o i8o5 I 9 5 9 2Il4 6 54 2833 3oio 3i86 3362 3539 3 7 i5 5 54 2269 2424 25 7 8 2 7 33 2 88 7 3o42 5 55 3891 4o6 7 4243 44i9 45 9 5 4 77 i 4 55 3196 335o 35o4 3659 38i3 3 9 6 7 4 56 4947 5l22 52 9 8 5473 5649 5824 3 56 4l2I 4275 4429 4583 4 7 3 7 4890 3 5? 6000 6i 7 5 635o 65 2 5 6700 68 7 5 2 57 5o44 5i 9 8 535i 55o5 565 9 58i2 2 58 7o5o 7225 74oo 7 5 7 4 7749 79 2 4 I 58 5 9 66 6119 62 7 2 6425 65 79 6 7 32 I 5 9 8098 8273 8447 8621 8 79 5 8 97 o 5 9 6885 7o38 7191 7 344 7497 7 65o 60" 50" 40" 30" 20" 30" a 60" 50" 40" 30" 20" 10" Co-tangent of 83 Degrees. ffl & Co-tangent of 82 Degrees. 1 P Part 5 r/ 2 " 3 " 4// 5 " 6// 7" 8" 9" !!, ,< I" 2" 3" 4" 5" 6" 7" 8" 9" iri \ 19 37 56 75 94 112 131 150 163 j irt { 16 33 49 G5 81 98 114 130 14fi LOGARITHMIC SINES. g 1 Sine of d Degrees. .3 Sine of 9 Degrees. * p o 77 f~io" W [ 30" 40" 50" ^ 0" | 10" 20" 30" 40" ' w ~o; 9i 143555 37'>5 3855 4oo5 4i54 43o4 5 9 o 9. 194332 4465 45 9 8 4 7 3i 4864 4997 5 9 I 4453 46o3 4?52 4902 5o5i 5200 58 I 5129 5262 53 9 5 552 7 566o 5792 58 2 534 9 5498 5648 5 797 5946^095 57 2 5 9 25 6o5 7 6i8 9 6322 6454 6586 5 7 3 6243|6392J654i 6690 68396987 56 3 6719 685i 6 9 83 7 n5 7 24 7 73 79 |56 4 7i36 7 284! 7 433 7 58i 773o 7878 55 4 7 5u 7643 7775 7907 8o38 8i 7 o!55 5! 8026 8i 7 4 832318471 8619 8767 54 5 8302 8434 8565 8697 8828 8960 54 6 8 9 i5 9063 921119358 95o6 9654 53 6 9091 9 223 9 354 9 486 9 6l 7 9748 53 7 j 9 8o2 99^9! 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( 1" 2" 3" 4" 5" 6" 7 ' 8" 9' 1 \ 6 11 17 93 23 34 39 45 51 1 I 5 11 16 21 27 32 37 43 48 LOGARITHMIC TANGENTS. 1 Tangent of 20 Degrees. 50" j j Tangent of 21 Degrees. 1 0" 10" 20" 30" 40" S 0" 10" | 20" 30" 40" 50" o l/. 661666 n3i "97 1262 i3 2 8 i3 9 3|5 9 O 9 . 584177 4240 43o3 4366 4429 44c^ 5c; 1 I i45 9 i524 i655 1721 1786 58 I 4555 46i8 468 1 4744 48o6 486 9 58 2 i85i 1917 1982 2048 2Il3 2178 57 2 4 9 32 4995 5o58 5l2I 5i83 5246 57 3j 2244 2309 2375 2440 25o5 2571 56 3 5309 53 7 2 5435 54 9 8 556o 5623 56 4 2636 2701 27672832 2 8 97 2963 55 4 5686 5 7 4 9 6811 58 7 4 5 9 3 7 6000 51 5 3o 2 8 3093 3i58 3224 3 2 8 9 3354 54 5 6062 6i25 6188 625i 63 1 3 63 7 6 54 6 34 1 9 3485 355o 36i5 368o 3 7 46 53 6 643 9 65oi 6564 6627 668 9 6 7 52 53 7 38n 38 7 6 3 9 4i 4oo6 4o 7 i 4i3 7 52 7 68i5 6877 6940 7003 7o65 7128 5 2 8 4202 4267 4332 43 97 4462 452 7 5i 8 7190 7253 73i6 7 3 7 8 744 1 75o3 5r 9 45 9 3 4658 4723 4 7 88 4853 4 9 i8 5o 9 7566 7629 7691 7754 7816 7879 5o IO 9 .564 9 83 5o48 5n3 5i 7 8 5243 53o8 49 iol9.58 7 94i 8oo4 8066 8i2 9 8i 9 i 8254 49 II 53 7 3 5438 55o3 5568 5633 56 9 8 48 ii 83i6 8379 844 1 85o4 8566 8629 48 12 5 7 63 5828 58 9 3 5 9 58 6o 2 3 6088 47 12 8691 8 7 54 8816 8878 8 9 4i 9003 4 7 i3 6 1 53 6218 6283 6348 64i3 6478 46 i3 9066 9128 9191 9 253 9 3i5 9378 46 i4 6542 6607 6672 6 7 3 7 6802 6867 45 i4 944o 9502 9565 9627 45 i5 6o32 6996 7061 7126 7IQI 7256 44 i5 9814 0877 oo3o ... i 63 . 1 26 AA 16 uy^^i 7 32O 7 385 i w x 7 45o / * ^^ 7 5i5 / V 7 58o 7644 43 16 9.590188 v / / 025o vv^v o3i3 o3 7 5 o43 7 0499 ^T^T 43 17 779 7774 7 83 9 79 o3 79 68 8o33 42 17 o562 0624 0686 o 7 48 0811 o8 7 3 42 18 8o 9 8 8162 8227 82 9 2 8356 8421 4i 18 0935 0997 1060 1122 1184 1246 4i i 9 8486 855o 86i5 8680 8 7 44 88o 9 4o i 9 i3o8 1370 i433 1495 i55 7 1619 4o 20 9 .5688 7 3 8 9 38 9 oo3 9 o6 7 9 l32 9197 39 20 9.591681 1743 i8o5 1868 i 9 3o I 99 2 3 9 21 9 26l 9 3 2 6 9 3 9 o 9 455 9 5l 9 9 584 38 21 2o54 21 l6 2178 2240 2302 2364 38 22 9 648 97 i3 9777 9 842 99 o6 997 1 37 22 2426 2488 2 55o 26l2 2674 2 7 3 7 >7 23 9 .5 7 oo35 OIOO oi64 O22 9 0293 o358 36 23 2799 2861 2923 2 9 85 3o4 7 3io 9 36 24 0422 0487 o55! O6l6 0680 0744 35 24 3171 3232 32 9 4 3356 34i8 348o 35 25 0809 0873 o 9 38 IOO2 1066 n3i 34 25 3542 36o4 3666 3728 3790 3852 34 26 1 1 9 5 1259 1324 i388 i452 i5i7 33 26 3914 3 97 6 4o38 4099 4i6i 4223 33 27 i58i 1 645 1710 1774 i838 I 9 o3 32 2 7 4285 4347 44o 9 4471 4532 45 9 4 3a 28 i 9 6 7 2031 2095 2160 2224 2288 3i 28 4656 4718 4 7 8o 4842 4 9 o3 4 9 65 3i *9 2352 2417 2481 2545 2609 26 7 3 3o 2 9 5027 5o8q 5i5o 5212 5274 5336 3o 3o 9 .5 7 2 7 38 2802 2866 2 9 3o 2994 3o5 9 29 3o 9 .5 9 53 9 8 6469 552i 5583 5644 5706 29 3i 3i 2 3 3187 3 2 5i 33i5 33 79 3443 28 3i 5 7 68 583o 58 9 i 5 9 53 6oi5 6076 28 32 35o 7 35 7 i 3636 3700 3 7 64 38 2 8 27 32 6i38 6200 6261 6323 6385 6446 2 7 33 38 9 2 3 9 56 4O2O 4o84 4i48 4212 26 33 65o8 6570 663i 66 9 3 6 7 54 6816 26 34 42 7 6 434o 44o4 4468 4532 45 9 6 25 34 6878 6 9 3 9 7001 7062 7124 7i85 25 35 466o 4724 4788 485 2 4 9 i6 4 9 8o 24 35 7247 7 3o 9 7 3 7 o 7432 7493 7 555 24 36 5o44 5io8 5172 5236 52 99 5363 23 36 7616 7678 77 3 9 7801 7862 7924 o 37 542 7 54 9 i 5555 56i 9 5683 5 7 4 7 22 37 7985 8o4 7 8108 8170 823i 8293 22 38 58io 58 7 4 5 9 38 6002 6066 6i3o 21 38 8354 84i5 8477 8538 8600 8661 21 39 6i 9 3 6257 632i 6385 644 9 65i2 2O 3 9 8722 8 7 84 8845 8 9 07 8968 9029 2O 4o 9.576676 664o 6704 6767 683i 68 9 5 I Q 4o 9 .5 99 o 9 i 9 l52 9 2l3 9 2 7 5 9 336 9 3 9 8 I 9 4i 6 9 5 9 7022 7086 7i5o 7 2l3 7277 18 4i 9 45 9 9 52O 9 58i 9 643 9 7o4 9765 18 42 74o4 7 468 7 53 2 7 5 9 5 7 65 9 17 42 9 8 27 9 888 9949 . . ii ..72 .i33 17 43 7723 7786 785o 7914 7977 8o4i 16 43 9 .6ooi 9 4 0256 0317 o3 7 8 o43 9 o5oo 16 44 8io4 8168 823i 82 9 5 8359 8422 16 44 o562 0623 0684 o 7 45 0806 0868 16 45 8486 854 9 86i3 8676 8 7 4o 88o3 i4 45 9 2 9 o 99 o io5i I 112 1174 1235 i4 46 8867 8 9 3o 8994 9 5 7 9E2I 9 i84 i3 46 I2 9 6 i35 7 i4i8 1479 i54o 1601 i.& 4 7 9248 9 3n 9 3 7 5 9 438 9502 9 565 12 47 i663 1724 i 7 85 1 846 1907 1968 IS 48 9 62 9 9 6 9 2 97 55 9819 9882 99 46 II 48 2O2 9 2090 2l5l 2212 2273 2334 ii 49 9 .58oob 9 0072 oi36 0199 0262 o326 IO 49 23 9 5 2456 2 5i 7 25 7 8 2 63o 2700 IO 5o 9 .58o38 9 o453 o5i6 o5 79 0642 0706 9 5o 9.602761 2822 2883 2944 3oo5 3o66 9 5i 0769 o832 0896 o 9 5 9 1022 1086 8 5i 3127 3i88 324 9 33io 33 7 i 3432 8 52 1149 1212 1275 i33 9 1402 i465 7 52 34 9 3 3554 36i5 36 7 5 3 7 36 3797 7 53 i5 2 8 i5 9 i i655 1718 1781 1 844 6 53 3858 3 9 i 9 3 9 8o 4o4i 4102 4162 6 54 .907 1971 2o34 20 97 2l6o 2223 5 54 4223 4284 4345 44o6 446 7 452 7 5 55 2286 235o 24i3 24 7 6 2539 2602 4 55 4588 4649 4710 4 77 i 483i 4892 4 56 2665 2728 2791 2854 2917 2 9 8o 3 56 4953 5oi4 5o 7 4 5i35 5196 525 7 3 57 3o44 3107 3170 3 2 33 3296 335 9 2 5 7 53i 7 5378 543 9 55oo 556o 562i 2 58 3422 3485 3548 36n 36 7 4 3 7 3 7 I 58 5682J5742 58o3 5864 5924 5 9 85 I 5 9 38oo 3863 3926 3 9 8 9 4o52 4n4 O 5 9 6o46j6io6 6167 6228 6288 634 9 C 60" 50" | 40" 30" 20" 10" q 60" | 50" 40" 30" 20" 10'' d Co-tangent of 69 Degrees. i Co-tangent of 68 Degrees. i Ps , rf $ 1" 2" 3" 4" 5" 6" 7" 8" 9" p T> . $ I" 2" 3" 4" 5" 6" 7" 8" 9" 6 13 19 26 32 39 45 51 58 \ 6 12 If) 25 31 37 43 49 5G LOGARITHMIC SINES. i .5 Sine of 22 Degrees. d Sine of 23 Degrees. 2 0' 10" 20" 30" 40" 50" s 0" 10" 20" 30" | 40" 50" 9.5735^5 3628 368o3 7 32 3 7 84 3836 5 9 9 .5 9 i8 7 8 1928 I 977 202 7 20 7 6 2120 5 9 I 3888 3940 3 99 2 444 4096 4i48 58 I 2I 7 6 2225 22 7 5 2324 23 7 4 2423 58 2 4aoo 4252 43o4435644o8 446o 5 7 2J 24 7 3 2522 25 7 2 2621 26 7 I 2720 57 2 45ia4564 46i646684 7 so 4 77 2 56 3 2 77 2819 2869 2918 2968 3oi 7 56 4 4824 4876 492849805032 5o84 55 4 3o6 7 3n6 3i65 32i5 3264 33j4 55 c 5i36 5i8 7 5239^291 5343 53 9 5 54 c 3363 34i2 3462 35n 356i 36io|54 6 5447 5499 555o ! 56o2 5654 5 7 o6 53 6 365 9 3 79 3 7 58 38o 7 385 7 3 9 o6 S3 7 SySS 58io!586i|5 9 i3 5 9 65 6oi 7 52 7 3 9 55 4oo5 4o54 4io3 4i53 4202 5a 6069 6120 6i 7 2;6224 6276 632 7 5i 8 425i 43oi 435o 43 99 4448 44 9 8 5i 9 63 79 643i 6482 6534 6586 6638 5o 9 454 7 45 9 6 4645 46 9 5 4 7 44 4 79 3 5o II 9.576689 6999 6741 7o5i 6793 6844 7102)7154 68 9 6 7206 6 9 48 7 25 7 49 48 IO ii 9 .5 9 4842 5i3 7 48 9 i 5i86 4g4i 5 2 36 4990 5 2 85 5o39 5334 5o88 5383 49 48 12 7 3 9 7 36o 74 1 2 7464 7 5i5 7 56 7 47 12 5432 548i 553o 558o 562 9 56 7 8 47 1 3 7618 7670 7721 7773 7824 7876 46 i3 5 7 2 7 5 77 6 5825 58 7 4 5 9 23 5 97 2 46 i4 7927 7979 8o3o8o82 8i33 8i85 45 i4 6021 6o 7 o 6119 6168 621-7 6266 45 i5 8236 8288 83398391 8442 84 9 4 44 i5 63i5 6364 64i3 6462 65n 656o 44 16 8545 85 9 6 8648 8699 8751 8802 43 16 66o 9 6658 6 7 6 7 6 7 56 68o5 6854 43 37 8853 8 9 o5 89569008 9 o5 9 9 TIO 4? J 7 6 9 o3 6 9 52 7 OOI 7 o5o -7099 7i48 42 18 9162 9213 92649316 9 36 7 9 4i8 4i 18 7196 7 245 7294 7 343 7 3 9 2 7 44i 4i '9 9470 9521 9 5 7 2 9623 9 6 7 5 972 6 4o J 9 7 4 9 o 7 53 9 7 58 7 7 636 7 685 7734 4o 20 9.579777 9 828 9 88o 99 3i 9982 ..33 3 9 20 9 .5 977 83 7 83i 7 88o 79 2 9 7978 8o2 7 3 9 21 9 .58oo85 oi36 01870238 0289 o34i 38 21 8o 7 5 8124 8i 7 3 8222 82-70 83i 9 38 22 0392 o443 o4g4 o545 0596 o647 3 7 22 8368 84i7 8465 85i4 8563 8612 3 7 23 0699 o 7 5o 0801 o85 2 0903 o 9 54 36 23 8660 8 7 o 9 8 7 58 8806 8855 8904 36 24 1006 io56 1107 u58 1209 1261 35 24 8 9 5 2 9 ooi 9o5o 9098 9i4 7 9 i 9 5 35 25 l3l2 i363 1414 i465 i5i6 1567 34 25 9 244 9 2 9 3 9 34i 9 3 9 o 9 438 9 48 7 34 26 1618 i66 9 1720 1771 1822 i8 7 3 33 26 9 536 9 584 9 633 9681 97 3o 9778 33 27 1924 1975 2O25 2076 212 7 2178 32 27 9 8 2y 9 8 7 6 99 2 4 9973 . .21 ..70 32 28 2229 2280 233i 2382 2433 2484 3i 28 9 . 600118 oi6 7 0215 0264 O3l2 o36i 3i i 29 2535 2585 2636 2687 2 7 38 2 7 8 9 3o 29 o4o 9 o45 7 o5o6 o554 o6o3 o65r 3o 3o 9.582840 2890 2941 2992 3o43 3o 9 4 29 3o 9 . 600700 o 7 48 797 o845 o8 9 3 o 9 42 29 3i 3i45 3195 3246 32 97 3348 33 9 8 28 3i o 99 o io38 io8 7 n35 n84 1232 28 3 2 3449 35oo 355i 36oi 3652 3 7 o3 2 7 32 1280 i32 9 i3 77 i425 1474 1522 2 7 33 3 7 54 38o4 3855 3 9 o6 3 9 56 4007 26 33 i5 7 o i6i 9 i66 7 i 7 i5 i 7 63 1812 26 34 4o58 4i68 4i5 9 4210 4260 43u 25 34 1860 I 9 o8 i 9 5 7 2005 2o53 2IOI 25 35 436i 44i24463 45i3 4564 46i5 24 35 2i5o 2I 9 8 2246 22 9 4 2342 23 9 I 24 36 4665 47164766 48i 7 486 7 4 9 i8 23 36 a43 9 248 7 2 535 2 583 2632 2680 23 37 4968 6019 5o 7 o 5l20 5i 7 i 5221 22 37 2 7 28 2776 2824 2872 2 9 2O 2 9 6 9 22 38 5272 5322 53 7 3 5423 54 7 4 55 2 4 O T 38 3oi 7 3o65 3n3 3i6i 32O 9 325 7 21 3 9 4o 55 7 4 9. 5858 77 5625 5 9 2 7 56 7 5 5 97 8 5 7 26 6028 5 7 76 6o 79 5827 6l2 9 2O 19 3 9 4o 33o5 "353 34oi 9 .6o35 9 4 3642 "6 9 o 344 9 3 7 38 34 97 3 7 86 3546 3834 20 1 9 4i 6179 6230^280 633i 638i 643i 18 4i 3882 3 9 3o 3 97 8 4026 4074 4122 18 42 6482 6532 6582 6633 6683 6 7 33 *7 42 4i 7 o 4218 4266 43i3 436i 44o 9 n 43 6 7 83 6834 6884 6 9 34 6985 7 o35 16 43 445 7 45o5 4553 46oi 464 9 46 97 16 44 7 o85 7I 35 7 i86 7 236 -7286 7 336 i5 44 4 7 45 4 79 3 484i 4888 4 9 36 4 9 84 i5 45 7 386 7 43 7 7 48 7 7 53 7 7 58 7 7 63 7 i4 45 5o32 5o8o 5i 2 8 5176 5223 52 7 I i4 46 7688 7738 77 88 7 838 7888 7938 i3 46 53i 9 536 7 54i5 5462 55io 5558 i3 47 7989 80398089 8i3 9 81898239 12 47 56o6 5654 5 7 oi 5749 5 797 5845 12 48 8289 833 9 838 9 843 9 8489 854o II 48 58 9 2 5 9 4o 5 9 88 6o35 6o83 6i3i II 49 85 9 o 864o 8690 8 7 4o 8 7 9o 884o IO 49 61-79 6226 62 7 4 6322 6369 64i 7 IO 5o 9.588890 8g4o 8990 9 o4o 9090 9140 9 5o 9 .6o6465 65i2 656o 6608 6655 6 7 o3 9 5i 9190 9240 9290 9 34o 9 38 9 9 43 9 8 5i 6 7 5i 6 79 8 6846 68 9 3 6941 6989 8 52 9489 95399589 9 63 9 9 68 9 97 3 9 7 52 7o36 7 o84 7 i3i 7179 7227 7 2 7 4 7 53 9789 98399889 99 38 9988.. 38 6 53 7322 7 36 9 7 4i 7 7464 7512 7 55 9 6 54 9.590088 oi38oi88 023 7 O28 7 o33 7 5 54 7607 7654 77 02 7749 7797 7 844 5 55 o38 7 o43 7 o48 7 o536 o586o636 4 55 7892 79 3 9 7987 8o34 8082 8i2 9 4 56 0686 o 7 35 o 7 85 o835 o885 o 9 34 3 56 8177 8224 82 7 I 83i 9 8366 84U 3 5 7 0984 io34 1084 n33 u83 1233 2 57 846 1 85o8 8556 86o3 865i 86 9 8 2 58 1282 i332 i382 i43i i48i i53i I 58 8 7 45 8 79 3 884o 8887 8935 8 9 82 I 5 9 i58o i63o 1680 1729 i 779 1828 O 5 9 9029 977 9124 9171 9219 9 266 O 60" 50" 40" | 30" 20" 10" A 60" 50" | 40" | 30" 20" 10" d Co-sine of 67 Degrees. .3 & Co-sine of 66 Degrees. P Part* l " ~" 3/ ' 4" 5" 6" 7" 8" 9" . ( 1" 2" 3" 4" 5" 6" 7" 8" 9" i irt { 5 10 1; 20 25 31 36 41 46 irt } 5 10 15 19 24 29 34 39 44 j LOGARITHMIC TANGENTS. A Tangent of 22 Degrees. a Tangent of 23 Degrees. Si 0'' 10" 20" 30" 40" | 50" JJ 0" 10" 20" 30" 40" 50" o 9 .6o64io 6470 653i 65 9 i 6652 6 7 i3 5 9 O 9.627852 79 io 79 6 9 8028 8086 8i45 5 9 j 6 77 3 6834 68 9 4 6955 7oi5 7076 58 I 82o3 8262 8320 83 79 843 7 84 9 6 58 2 7 i3 7 7197 7 258 7 3i8 7379 7 43 9 57 2 8554 8612 86 7 i 8 7 2 9 8 7 88 8846 57 3 7 5oo 7 56o 7 62I 7681 7742 7802 56 3 8905 8 9 63 9 O22 9080 9 i38 9197 56 4 7 863 7923 79 84 8o44 8io5 8i65 55 4 9 255 9 3i4 9 3 7 2 9 43i 9 48 9 9 54 7 55 5 8225 8286 8346 84o 7 S46 7 8528 54 5 9606 9 664 97 22 97 8i 9 83 9 9 8 97 54 6 8588 8648 8 7 o 9 8769 883o S8 9 o 53 6 9966 . .14 .. 7 3 .i3i .189 .247 53 7 8 9 5o 9011 90-71 9i3i 9192 9 2D2 62 7 9 .63o3o6 o364 O422 o48i o53 9 o5 97 5s 8 9 3l2 9 3 7 3 9 433 9493 9 554 9 6i4 5i 8 o656 07 1 4 77 2 o83o o88 9 0947 5i 9 9 6 7 4 9735 979 5 9 855 9915 9976 5o 9 ioo5 io63 1122 1180 1238 I2 9 6 5o 10 9 .6ioo36 0096 oi56 0217 0277 o33 7 49 ro 9. 63j-355 i4i3 l47I 1529 i58 7 1 646 49 1 ii o3 97 o458 o5i8 o5 7 8 o638 0698 48 ii - 1704 1-762 1820 i8 7 8 9 36 1995 48 12 o 7 5 9 0819 o8 79 o 9 3 9 999 io5 9 47 12 2o53 2III 2l6 9 2227 2385 2343 47 i3 1120 1180 1240 i3oo i36o l42O 46 i3 2402 246o 25i8 2 5 7 6 2634 2692 46 i4 i48o i54o 1601 1661 1721 1781 45 1 4' 2750 2808 2866 2924 2 9 82 3o4o 45 i5 i84i 1901 i 9 6i 2021 2081 2l4l 44 i5 3o 99 3 1 5 7 32i5 32 7 3 33 3 1 338 9 44 16 22OI 2261 2321 2 38i 244i 2501 43 16 3447 35o5 3563 362i 3679 3 7 3 7 43 *7 256l 2621 2681 2 7 4l 2801 286l 42 17 3 79 5 38.5.3 3 9 n 3969 4027 4o85 42 18 2 9 2I 2981 3o4i 3ioi 3i6i 3221 4i 18 4i43 4201 425 9 43i6 43 7 4 4432 4r 19 328l 334i 34oi 346i 352i 358i 4o 19 44 9 o 4548 46o6 4664 4 7 22 4 7 8o 4o 20 9 .6i364i 3 7 oi 3 7 6o 3820 388o 3 9 4o 3 9 20 9. 634838 48 9 6 4 9 54 Son 5o6 9 5l2 7 39 21 4ooo 4o6o 4l20 4i8o 423 9 4299 38 21 5i85 5243 53oi 5359 54i6 54 7 4 38 22 435 9 44i9 44 79 453 9 45 9 8 4658 37 22 5532 55 9 o 5648 5 7 o6 5 7 63 582i 37 23 4 7 i8 4 77 8 4838 48 97 4 9 5 7 5017 36 23 58 79 5 9 3 7 5 99 5 6o52 61 10 6168 36 24 5o 77 5i36 5i 9 6 5256 53i6 53 7 5 35 24 6226 6283 634i 6399 645 7 65i4 35 25 5435 54 9 5 5555 56i4 56 7 4 5 7 34 34 25 65 7 2 663o 6688 6 7 45 68o3 6861 34 26 5 79 3 5853 5 9 i3 5 97 2 6o32 6092 33 26 6919 6 97 6 7034 7 9 2 7 i4 9 7 20 7 33 27 6i5i 6211 62 7 I 633o 63 9 o 645o 32 27 7 265 7 322 7 38o 7438 74 9 5 7 553 32 n 65o 9 6569 6628 6688 6 7 48 6807 3 1 28 7 6n 7 668 77 26 7783 7841 7899*31 29 686 7 6926 6 9 86 7046 7io5 7i65 3o 29 7956 8oi4 80-72 8i2 9 8i8 7 8244 3o 3o ^ . 6f 7 224 7284 7343 74o3 7462 7 522 29 3o 9. 638302 835 9 84i 7 8475 8532 85 9 o 29 3i 7 58 2 7 64i 77 oi 7760 7820 7879 28 3i 864 7 8 7 o5 8762 8820 88 77 8 9 35 28 32 79 3 9 7998 8o5 7 8117 8176 8236 27 32 8992 goSo 9107 9 i65 9 222 9280 27 33 82 9 5 8355 84i4 8474 8533 85 9 3 26 33 9337 9 3 9 5 9 45 2 9 5ib 9 56 7 9625 26 34 8652 8711 8771 883o 88 9 o 8949 25 34 9682 9 74o 9797 9 855 9912 9969 25 35 9008 9068 9 I2 7 9186 9246 9 3o5 24 35 9.640027 oo84 0142 oi 99 O25 7 o3i4 24 36 9 364 9424 9 483 9 543 9 6o2 9661 23 36 . 0371 042 9 o486 o544 0601 o658 23 37 97 20 9780 9 83 9 9898 99 58 ..17 22 37 0716 o 77 3 o83o 0888 0945 1002 22 38 9.620076 oi36 0195 0254 o3i3 o3 7 3 21 38 1060 1174 1232 1289 1 346 21 3 9 0432 0491 o55o 0610 o66 9 O 7 28 20 3 9 i4o4 i46i i5i8 i5 7 5 i633 1690 2O 4o 9.620787 o846 0906 o 9 65 1024 io83 I 9 4o 9.641747 i8o5 1862 1919 19-76 2034 19 4i Il42 I2OI 1261 1320 i3 79 i438 18 4i 2091 2148 22O5 2263 2320 23 77 18 42 1497 i556 1616 !6 7 5 1734 1-793 17 42 2434 24 9 I 2549 2606 2663 2720 1-7 43 1862 1911 i 97 o 2029 2088 2l4 7 16 43 2777 2834 2892 2 9 4 9 36o6 3o63 16 44 2207 2266 2325 2384 2443 2502 i5 44 3l20 3i77 3235 32 9 2 334 9 34o6 i5 45 2.56i 2620 26 79 2 7 38 2797 2856 i4 45 3463 3520 35 77 3634 36 9 i 3 7 4 9 i4 46 2 9 !5 2974 3o33 3092 3i5i 3210 i3 46 38o6 3863 3920 3977 4o34 4091 i3 47 3260. 33 2 8 338 7 3446 35o5 3564 I 2 47 4i48 42o5 4262 43i 9 43 7 6 4433 12 48 3623 3682 3 7 4i 38oo 3858 3 9 i 7 ii 48 4490 454 7 46o4 466i 47'i8 4 77 5 II 49 3 97 6 4o35 4o 9 4 4i53 4212 4271 10 49 4832 488 9 4946 5oo3 5o6o 5n 7 IO 5o 9 .62433o 4388 444 7 45o6 4565 4624 9 5o 9.645174 523i 5 2 88 5345 5402 545 9 9 5i 4683 4742 48oo 485 9 4918 4977 8 5i 55i6 55 7 3 563o 568 7 5 7 44 58oi 8 52 5o36 5094 5i53 5212 5271 533o 7 52 585 7 5 9 i4 5 97 i 6028 6o856i42 7 53 5388 5447 55o6 5565 5623 5682 6 53 6199 6256 63i3 6369 6426 6483 6 54 5 7 4i 58oo 5858 5917 5 97 6 6o35 5 54 654o 65 97 6654 6710 6-7676824 5 55 6o 9 3 6i52 6211 6269 6328 638 7 4 55 6881 6 9 38 6 99 5 7 o5i -7108 7 i65 4 56 6445 65o4 6563 6621 6680 6739 3 56 7222 7 2 79 7 335 7 3 9 2 7 44 9 75c6 3 57 6797 6856 6 9 i5 6973 7 o32 7090 2 5 7 7 56 2 7 6i9 7 6 7 6 7733 7789 7 846 2 58 7149 7208 7 266 7325 7 383 7442 I 58 79 3 7 96o 8016 8o 7 3 8i3o|8i86 I 5 9 75oi 7 55 9 7618 7676 77 35 779 3 O 5 9 8243 83oo 8356 84i3 84708526 O 60" 50" 40" 30" 20" 10" . 60" 50" 40" 30" 20" 10" S" Co-tangent of 67 Degrees. i , Co-tangent of 66 Degrees. p, J 1" 2" 3" 4" 5" 6" 7" 8" 9" . ( 1" 2" 3" 4" 5" 6" 7" &' 9''' } 6 12 18 24 30 36 42 48 54 trt 6 12 17 23 29 35 40 46 52 LOGARITHMIC SINES. 1 Sine of 24 Degrees. c | Sine of 2;> Degrees. Li 0" 10-' 20" 30" 40" 50" 2, 0" 10" 20" 30" 40'' 50" o 2 9.609313 9 5 97 9880 9361 9644 9928 94o8 9691 99 7 5 9 455 97 3 9 . .22 9502 9786 ..69 9550 9 833 .116 5 9 58 5 7 o i 2 9 .625 9 48 62I 9 64 9 o 5993(6039 6o84 62^4 63o 9 6354 6535658o 6625 6129 64oo 6670 6174 6445 6 7 i5 5 9 - 58 5 7 3 9.610164 0211 0258 o3oo o352 0399 56 3 6760 68o5 685o 68 9 5 6 9 4o 6 9 85 56 4 o44y 0494 o54i o588 o635 0682 55 4 7o3o 70 7 5 7120 7 i65 7210 7255 55 V 0729 0776 0823 0871 0918 0965 54 5 73oo 7345 7 3 9 7 435 7480 7 5 2 5 54 3 1012 1059 1106 n53 1 200 1247 53 6 7 5 7 o 7 6i5 7660 77 o5 77 5 77 9 5 53 m 1294 i34i i388 i435 1482 1529 52 7 7 84o 7 885 7929 7974 8oi 9 8o64 52 1 8 i5 7 6 1623 1670 1717 iy64 1811 5i 8 8io 9 8i54 8i99 8244 828 9 8333 5i 9 i858 1905 1952 1999 2046 2093 5o 9 83 7 8 8423 8468 85i3 8558 8602 5o TO q.6;2i4o 2187 2234 2280 2327 23 7 4 49 10 9 . 628647 8692 8 7 3 7 8 7 8 2 8826 8871 49 ii 2421 2468 25i5 ! 2562 2609 2655 48 ii 8 9 i6 8961 9006 9 o5o 9 o 9 5 9 i4o 48 12 2702 2749 2796 2843 2890 2 9 36 47 12 9 i85 9229 9274 9 3i 9 9 363 9 4o8 47 i3 2 9 83 3o3o 3077 3i24 3i 7 i 3217 46 i3 9 453 9498 9542 9 58 7 9 632 9 6 7 6 46 t4 3264 33n 3358 34o4 345i 34 9 8 45 i4 9721 9766 9810 9 855 99 oo 99 44 45 i5 3545 SSgi 3638 3685 3 7 32 3 77 8 44 i5 9989 ..34 ..78 .123 .168 .212 44 16 3825 3872 3 9 i83 9 65 4OI2 4o58 43 16 9.630257 o3oi o346 o3 9 i 0435 o48o 43 *7 4io5 4i52 41984245 4292 4338 42 J 7 o524 o56 9 o6i3 o658 0703 o 7 4 7 42 18 4385 4432 44?8 4525 45 7 i 46:8 4i 18 0792 o836 0881 O 9 25 o 97 o 1014 4i *9 4665 4711 4 7 58 48o4 485i 48 9 8 4o J 9 1069 no3 n48 II 9 2 I23 7 1281 4o 20 9.614944 4991 5o37 5o84 5i3o 5177 3 9 20 9.63i326 1370 i4i5 i45 9 i5o4 1 548 3 9 21 5223 5270 53:65363 5409 5456 38 21 i5 9 3 i63 7 1681 1726 1770 i8i5 38 22 55o2 554 9 55g5 5642 5688 5 7 35 37 22 i85g 1904 1948 I 99 2 203712081 37 23 5 78 1 6828 5874 5921 5 9 6 7 6oi3 36 23 2125 2170 22l4 225 9 23o3 234 7 36 24 6060 6106 6i53 6199 6245 6292 35 24 2392 2436 2480 2525 256 9 2613 35 25 6338 6385 643i 6477 6524 6570 34 25 2658 2702 2746 2 79 2835 ' 2 8 79 34 26 6616 6663 6709,6755 6802 6848 33 26 2923 2968 3012 3o56 3ioo 3i45 33 27 6894 6941 6987 7o33 7080 7126 32 27 3189 3233 3277 3322 3366 34io 32 28 7172 7218 7265 7311 7 35- 7 7 4o3 3i 28 3454 3498 3543 358 7 363i 36 7 5 3i 29 745o 7496 7542 7 588 7 635 7681 3o 29 3 7 i 9 3 7 64 38o8 3852 38 9 6 3 9 4o 3o 309.617727 777 3 7819 786617912 79 58 2 9 3o 9.633984 4028 4o 7 3 4117 4i6i 42o5J2 9 3i 8oo4 8o5o 80968143 8189 8 2 35 28 3i 4249 42 9 3 433 7 438i 4426 4470 28 32 ' 8281 832 7 83788419 8465 85i2 2 7 32 45i4 4558 4602 4646 46 9 o 4 7 34 27 33 8558 86o4 86508696 8742 8788 26 33 4778 4822 4866 4910 4 9 54 4 99 8 26 34 8834 8880 8926 8972 9018 9 o64 25 34 5o42 5o86 5i3o 5i74 5 2 i8 5262 25 35 9110 9i56 9202 9248 9294 9 34o 24 35 53o6 535o 53 9 4 5438 5482 55 2 6 24 36 9 386 9 43 2 9478 9524 9570 9 6i6 23 36 5570 56i4 5658 5702 5 7 46 579 O 23 3? 9662 9708 9754 9800 9846 9892 22 37 5834 58 7 7 5921 5 9 65 6oo 9 6o53 22 38 9938 9984 ..3o!.. 7 6 .121 .167 21 38 6097 6i4i 6:85 622 9 6272 63:6 21 39 9.620213 0259 o3o5o35i o3 97 o443 20 3 9 636o 64o4 6448 64 9 a 6535 65 79 2O 409.620488 o534 o58o 0626 0672 0718 *9 4o 9. 636623 6667 6 7 n 6754 6 79 8 6842 9 4i o 7 63 oSoc, o855 0901 0947 0992 18 4i 6886 6 9 3o 6 97 3 7017 7061 7io5 18 42 io38 1084 n3o ! ii75 1221 1267 J 7 4s 7 i48 7192 7 236 7280 7 323 7 36 7 J 7 43 i3i3 i358 1 4o4 i45o|i496 i54i 16 43 74xi 7 455 7 4 9 8 7542 7 586 7 62 9 16 44 i58 7 i633 1078 1724 1770 1816 i5 44 7 6 7 3 7717 77 6o 7 8o4 7 848 7891 i5 45' 1861 1907 1953 1998 2044 2089 i4 45 79 35 7979 8022 8066 8110 8i53 i4 46 2i35 2181 2226,2272 23i8 2363 i3 46 8197 8240 8284 8328 83 7 i 84:5 i3 4? 2409 2454 25oo 2546 2591 263 7 12 47 8458 85o2 8546 858 9 8633 8676 12 48 682 2728 27732819 2865 2910 II 48 8720 8 7 63 88o 7 885i 88 9 4 8 9 38 n 4g 2966 3ooi 3o47 3092 3i38 3i83 10 49 8981 9 025 9 o68 9112 9 i55 9 i 99 10 5o 9.623229 32 7 4 3320 3365 34u 3456 9 5o 9.639242 9286 9 32 9 9 3 7 3 9 4i6 9 46o 9 5i 35b2 354 7 35 9 3 3638 3683 0729 8 5i 95o3 9 546 9 5 9 o 9 633 9 6 77 97 20 8 52 3 77 4 3820 3865 3911 3 9 56 4ooi 7 52 97 64 9807 9 85i 9894 99 3 7 99 8i 7 53 4o47 4092 4i38 4i83 4228 4274 6 53 9.640024 0068 OIII oi54 oi 9 8 0241 6 54 43i9 4364 44io 4455 45oo 4546 5 54 0284 o328 o3 7 i o4i4 o458 o5oi 5 55) 45 9 i 4636 4682 47274772 48i8 4 55 o544 o588 o63i 0674 o 7 i8 0761 4 56 4863 4908 4954 4999|5o44 5o8 9 3 56 0804 o848 o8 9 i o 9 34 0978 1021 3 5? 5i35 5i8o 5225 52706316 536i 2 57 1064 1107 n5i 1194 12 } 7 1280 2 58 5406 545 1 54 9 6 554 2 |558 7 5632 I 58 1324 1367 i4io i453 i4g6 i54o I 59 5677 5722 5 7 68 58i35858 5 9 o3 5 9 i583 1626 i66 9 1712 i 7 56 1799 60" .- 50" 40" 30" . 20" 10" n 60" | 50" | 40" 30" 20" 10" d . Co- sine of 65 Degrees. Co-sine of 64 Degrees. ( 1" 2" 3" 4" 5" 6" 7" 8" 9" p p ( 1" 2" 3" 4" 5" 6" 7" 8" 9" r.Jartj 5 g H ig 23 2g 32 37 42 1 I 4 9 13 18 22 26 31 35 40 LOGARITHMIC TANGENTS. 1 Tangent of 24 Degrees. Tangent of 25 Degrees. 9 0" | 10" 20" | 30" 40" 50" * 0" 10" 20" 30" | 40" 1 50' o 9.648583 864o 8696 8 7 53 88lO 8866 5 9 O 9 . 668673 8728 8782 8837i88Q2 8947 5 9 2 8923 9263 8980 9319 9o36 9 3 7 6 9 o 9 3 9 433 9i5o 9489 9206 9 546 58 2 9002 9332 9 5 7 9 38 7 9112 9442 9167(9222 9277 9497(9552 9606 58 3 9602 g65 9 97 i5 9772 9829 9 885 56 3 9661 9716 9771 9826 98819936 56 4 9942 9998 ..55 .in .168 .224 55 4 9991 ..45 .IOO .i55 .210 .265 55 5 9.650281 0337 o3 9 4 o45o 0507 o563 54 5 9.670320 0375 0429 o484 o53 9 o5 9 4 54 6 0620 0676 o 7 33 o 7 8 9 o846 O 9 O2 53 6 0649 0703 o 7 58 o8i3 0868 9 23 53 7 0959 ioi5 IO 7 2 1128 n85 1241 52 7 0977 IO32 1087 1142 1197 I25l 52 8 1297 i354 i4io i46 7 i523 i5 79 5i 8 i3o6 i36i 1416 1470 i5 2 5 i58o 5i 9 1636 1692 i 7 4 9 i8o5 1861 1918 5o 9 i635 1689 1744 1799 i853 I 9 o8 5o 10 9.651974 2031 208 7 2143 2200 2256 49 10 9-67i 9 63 2018 2072 2127 2182 2236 49 ii 23l2 2369 2425 2481 2538 2594 48 ii 22 9 I 2346 2400 2455 25lO 2564 48 12 265o 2707 2 7 63 28l 9 28 7 5 2932 47 12 26i 9 2674 2728 2783 a838 2892 47 i3 2988 3o44 3ioi 3i5 7 32i3 3269 46 i3 2 9 47 3ooi 3o56 3m 3i65 322O 46 i4 3326 3382 3438 34 9 4 355i 3607 45 i4 32 7 4 332 9 3384 3438 34 9 3 354 7 45 i5 3663 3 7 i 9 3 77 6 3832 3888 3944 44 i5 36o2 365 7 3711 3 7 66 3820 38 7 5 44 16 4ooo 4o57 4n3 4169 4225 4281 43 16 3 9 2 9 3 9 84 4o38 4o 9 3 4i48 4202 43 7 4337 43 9 4 445o 45o6 4562 46i8 42 17 43n 4366 4420 4475 45 29 42 18 46 7 4 4 7 3i 4787 4843 48 99 4 9 55 4i 18 4584 4638 46 9 3 4747 4802 4856 4i 19 Son 5067 5i23 5179 5236 5292 4o I 9 4 9 n 4 9 65 5oi 9 5o 7 4 5i28 5i83 4o 20 9 .655348 54o4 546o 55i6 55 7 2 5628 39 20 9.675237 5292 5346 54oi 5455 55o 9 3 9 21 5684 5740 5796 5852 5908 5 9 64 38 21 5564 56i8 56 7 3 5 7 2 7 5 7 8i 5836 38 22 6020 6076 6 1 32 6188 6244 63oo 37 22 58 9 o 5 9 45 5 999 6o53 6108 6162 37 23 6356 6412 6468 6524 658o 6636 36 23 6217 6271 6325 638o 6434 6488 36 24 6692 6 7 48 68o4 6860 6916 6972 35 24 6543 65 97 665i 6 7 o6 6760 68i4 35 25 7028 7084 7140 7196 7252 73o8 34 25 6869 6923 6977 7 o32 7086 7140 34 26 7364 ?4i9 7 4 7 5 753i 7 58 7 7643 33 26 7194 7249 73o3 7 35 7 7412 7 466 33 27 7699 77 55 7811 7807 7922 7978 32 27 7520 7 5 7 4 7 62 9 7 683 7737 7791 32 28 8o34 8090 8i46 8202 8 2 58 83i3 3i 28 7 846 7900 79 54 8008 8062 8117 3i 29 8369 8425 848 1 853 7 85 9 2 8648 3o 29 8i 7 i 8225 82 79 8334 8388 8442 3o 3o 9.658704 8760 8816 8871 8927 8 9 83 2O 3o 9.6 7 8496 855o 86o4 865 9 8713 8767 2 9 3i 9039 9095 9i5o 9206 9262 93i8 2o 3i 8821 88 7 5 8929 8 9 84 9 o38 9092 28 32 9 3 7 3 9429 9 485 9540 9 5 9 6 9652 27 32 9 i46 9200 9254 93o8 9363 9417 27 33 9 7 o8 9763 9819 9 8 7 5 99 3o 9986 26 33 9471 9 5 2 5 9 5 79 9 633 9687 97 4i 26 34 9.660042 0098 oi53 0209 0265 0320 25 34 979 5 9 84 9 9904 99 58 . . 12 ..66 25 35 o3 7 6 o43i o48 7 o543 0598 o654 24 35 9.680120 0174 0228 0282 o336 o3 9 o 24 36 0710 o 7 65 0821 0877 0932 0988 23 36 o444 o4 9 8 o552 0606 0660 0714 23 37 io43 1099 n55 1210 1266 1321 22 37 o 7 68 0822 0876 o 9 3o 0984 io38 22 38 i3 77 1432 i488 1 544 i5 99 i655 21 38 1092 n46 1 200 1254 i3o8 i362 21 39 1710 1766 1821 1877 1932 1988 20 39 i4i6 1470 i524 i5 7 8 i63 2 1686 2O 4o 9.662043 2099 2i54 22IO 2265 2321 '9 4o 9.68i 7 4o i 79 4 1 847 1901 1955 2009 J 9 4i 23 7 6 2432 2487 2543 2598 2654 18 4i 2o63 2117 2171 2225 2279 2333 18 42 2709 2 7 65 2820 2876 2931 2987 17 42 238 7 J244o 2494 2548 2602 2656 17 43 3o42 3o 97 3i53 3208 3264 33i 9 16 43 2710 2764 2817 2871 2925 2979 16 44 33 7 5 343o 3485 3541 35 9 6 365i i5 44 3o33 3o8 7 3i4o 3i 9 4 3248 33o2 i5 45 3707 3762 38i8 38 7 3 3928 3 9 84 i4 45 3356 34io 3463 35i 7 35 7 i 3625 i4 46 4o3 9 4094 4i5o 42o5 4260 43i6 i3 46 3679 3 7 32 3 7 86 384o 38 9 4 3 9 4 7 i3 4 7 43 7 i 4426 4482 453 7 4592 4648 12 47 4ooi 4o55 4109 4162 4216 4270 12 48 47o3 4?58 48i4 4869 4924 4 9 79 II 48 4324 43 77 443 1 4485 453 9 45 9 2 II 49 5o35 5090 5i45 5200 5256 53n IO 49 4646 4700 4753 4807 486i 4914 10 5o 9,665366 5421 5477 5532 558 7 5642 9 5o 9.684968 5022 5o 7 5 5129 5i83 5 2 36 9 5i 56 9 8 5 7 53 58o8 5863 5 9 i8 5 97 4 8 5i 5290 5344 53 97 545i 55o5 5558 S 52 6o2 9 6o84 6i3 9 6194 6249 63o5 7 52 56i 2 5666 5 7 i 9 5 77 3 582 7 588o 7 53 636o 64i5 6470 65 2 5 658o 6636 6 53 5 9 34 5 9 8 7 6o4i 6095 6i48 6202 6 54 66 9 i 6 7 46 6801 6856 691 1 6966 5 54 6 2 55 63o 9 6363 64i6 6470 6523 5 55 7021 7 o 7 6 7132 7187 7242 7297 4 55 0577 663o 6684 6 7 3 7 6791 6845 4 56 7 35 2 7 4o 7 7462 7517 7 5 7 2 7627 3 56 6898 6952 7oo5 7 o5 9 7112 7166 3 57 7682 773 7 7792 7 84 7 79 3 7958 2 5 7 7219 7273 7326 738o 7 433 7487 2 58 8oi3 8068 8123 8178 8233 8288 I 58 7647 7701 77 54 7808 I .1? 8343 83 9 8 8453 85o8 8563 8618 5 9 7861 79 i5 7968 8021 8o 7 5 8128 60" 50" 40" 30" 1 20" 10" . 60" 50" 40" 30" 20" 10" g" Co-tangent of 65 Degrees. Co-tangent of 64 Degrees. p p $ 1" 2" 3" 4" 5" 6" 7" 8" 9" II -p -p ( 1" 2" 3" 4" 5" 6" 7" 8" 9" } 6 11 17 22 28 33 39 45 50 j r - rart j 5 n 10 22 27 33 38 43 49 50 LOGARITHMIC SINES. 1 Sine of 26 Degrees. .3 Sine of 27 Degrees. S f 0" [ 10" | 20" | 30" i 40" 50" & 0" 10' 20'' | 30"~] 40" 1 50" O I 9.641842 2101 |i885ii928 1971 2oi5 214412187 223o 2273 2 o58 2317 5 9 58 o i 9 .65 7 o47 72 9 5 7088 7336 7129 7 3 77 7171 7212 74i8! 7 46o ? 5o 3 5 9 58 2 236o 24o3 2446 24892532 2 5 7 5 57 2 7542 7 584 7 625 7666 7707 7749 5? Q 26i8'266i'27o4i2 7 47;27 9 o 2833 56 3 779 78^17872 7913 7955 7996 56 4| 2877 29202963 3oo6 3o49 3o 9 2 55 4 8037 807818119 8161 8202 8243 55 5 3i35 3178 3221 3 2 64 33o 7 335o 54 5 8284 8325 836? 84o8 8449 84 9 o 54 6 33 9 3 3436 3479 3522 3565 36o 7 53 6 853i 85 7 2 86i3 8655 86 9 6 8737 53 7 365o 36 9 3|3 7 36 3 779 3822 3865 52 7 8778 8819 8860 8901 8 9 42 8 9 83 52 8 3 9 o8 4o3 7 4123 5i 8 9025 9066 9107 9148 9 i8 9 9 23o 5i 9 10 4i65 9. 644423 42o8'425i 4465;45o8 42 94 14 3 3 7 l4 3 80 455 1 14594)4637 5o 49 9 10 9271 9 .65 9 5i 7 93i2 9 353 9 374 9 558| 9 5 99 ! 9 64o 9 435 9 68i 9476 9722 5o 49 ii 468o 4722476514808 485i 48 9 4 48 ii 9763 9 8o4| 9 845 9886 99 2 7 9968 48 12 4 9 36 4979 5022 5o65 5 I0 8 5i5o 47 12 9 .660009 oo5o 0091 Ol32 0173 02l4 47 i3 5i 9 3 5236 5279 532i 5364 54o 7 46 i3 0255 0296 o33 7 o3 7 8|o4i 9 o46o 46 i4 545o 5492 5535 55 7 8 5620 5663 45 i4 o5oi o54i o58 2 0623 o664 0705 45 i5 5706 5749 5791 5834 58. 7 7 5919 44 i5 0746 0787 0828 0869 o 9 o 9 0950 44 16 5 9 62 6oo5 6047 6090 6i33 6i 7 5 43 16 0991 1032 1073 in4 n54 1195 43 17 6218 6260 63o3 6346 6388 643i 42 i 7 1236 1277 i3i8 i35 9 i3 99 i44o 42 18 6474 65i6655 9 6601 6644 6686 4i 18 i48i l522 i563 i6o3 1 644 1 685 4i 19 6 7 2 9 6772:6814 685 7 68 99 6 9 42 4o *9 1726 1766 1807 1 848 i88 9 1929 4o 20 9.646984 70277069 7112 7*54 7 i 97 3 9 20 9.661970 201 I 2052 2092 2i33 2174 39 21 7240 7282,7325 7 36 7 74o 9 7452 38 21 22l4 2255 2296 233 7 2377 2418 38 22 7494 75377579 7 622 7664 7707 37 22 245 9 24 9 9 254o 2 58i 2621 2662 37 23 7749 7877 7 9 i 9 79 6i 36 23 2703 2 7 43 2784 2825 * 65 2906 36 24 8004 8o46 8089 8i3i 8i 7 3 8216 35 24 2946 2987 3o 2 8 3o68 3 1 09 3i4 9 35 r 8258 83oi 8343 8385 8428 84 7 o 34 25 3l90 323i 3271 33i2 3352 33 9 3 34 26 85i2 85558597 863 9 8682 8 7 24 33 26 ' 3433 3474 35i5 3555 35 9 6 3636 33 27 8766 88o9 ! 885i 88 9 3 8 9 36 8978 32 27 3677 3717 3 7 58 3 79 8 3839 3879 32 28 9020 9063 9105 9147 9 i8 9 9 232 3i 28 3920 3960 4ooi 4o4i 4082 4122 3i 2 9 9274 93i6 9 358 94oi 9 443 9 485 3o 29 4i63 4203 4244 4284 4325 4365 3o 3o 9.649527 9570 9612 9 654 9 6 9 6 97 3 9 29 3o 9.6644o6 4446 4486 452 7 456 7 46o8 29 3i 9781 98239865 997 9949 9992 28 3i 4648 468 9 4729 4769 48io 485o 28 32 965oo34 0076 0118 0160 0202 0245 2 7 32 4891 4 9 3i 4971 5012 5o5 2 5o 9 3 27 33 0287 0329 0371 o4i3 0455 0497 26 33 5i33 5i 7 3 5 2 i4 5254 52 9 4 5335 26 34 oSSg o582 0624 0666 0708 o 7 5o 25 34 53 7 5 54i5 5456 54 9 6 5536 5577 25 35 0792 08340876 0918 o 9 6o 1 002 24 35 56i 7 565 7 56 97 5 7 38 5778 58i8 24 36 1044 1086 1128 1171 I2l3 1255 23 36 585 9 58 99 5 9 3 9 5 979 6020 6060 23 37 1297 13391881 1423 i465 i5o 7 22 37 6100 6i4o 6181 6221 6261 6 3 o.i 23 38 1 549 i5 9 i ! i633 i6 7 5 1716 i 7 58 21 38 6342 6382 6422 6462 65o2 6543 21 3 9 1800 1842*1884 1926 I 9 68 2OIO 20 3 9 6583 6623 6663 6703 6 7 43 6 7 84 2O 4o 9 . 652062 2094 2 1 36 2178 2220 2262 19 4o 9.666824 6864 6904 6 9 44 6 9 84 7025 1 9 4i 23o4 2345,238 7 2429 2471 25i3 18 4i 7o65 7io5 7i45 7i85 7 225 7265 18 42 2555 2597 2638 2680 2722 2764 i 7 42 73o5 7346 7 386 7 4a6 7 466 7 5o6 17 43 2806 2848 2890 2931 2 97 3 3oi5 16 43 7546 7 586 7626 7666 7706 7746 16 44 3o5 7 3099 3i4o 3i82 3224 3266 i5 44 7786 7826 7866 79 o6 7946 7986 i5 45 33o8 3349 3391 3433 34 7 5 35i6 i4 45 8027 8067 8107 8i47 8227 i4 46 3558 36oo 3642 3683 3 ?2 5 3767 i3 46 8267 83o 7 8347 8386 8426 8466 i3 47 38o8 385o38 9 2 3 9 34 3 97 5 4oi 7 12 47 85o6 8546 8586 8626 8666 8706 12 48 4059 4ioo 4*42 4i84 4225 426 7 II 48 8 7 46 8786 8826 8866 8 9 o6 8 9 46 II 49 4309 435o43 9 2 4434 4475 45i 7 IO 49 8986 9 026 9065 9 io5 9i45 9 i85 IO 5o 9.654558 46oo 4642 4683 4 7 25 4 7 66 9 5o 9.669225 9 265 9 3o5 9 345 9 384 9 424 9 5i 48o8 485o|48 9 i 4 9 33 4 9 74 ! 5oi6 8 5i 9464 9 5o4 9544 9 584 9 624 9 663 8 52 5o58 5o99'5i4i 5i82 5224 5265 7 52 973 9743 9783 9 823 9862 99 02 7 53 53o 7 5348,53 9 o 543i 5473 55i4 6 53 9942 9982 ..22 ..61 . IOI 6 54 5556 55 97 563 9 568o 5722 5763 5 54 9.670181 O22O O26o o3oo o34o o3 79 5 55 58o5 5846;5888 5 9 2 9 5 9 7i 6012 4 55 0419 o45 9 0499 o538 o5 7 8 0618 4 56 6o54 6095,6136 6178 62i 9 6261 3 56 o658 0697 o 7 3 7 0777 0816 o856 3 5 7 63o2 6344J6385 6426 64686509 2 57 0896 0935 o 97 5 ioi5 io54 io 9 4 2 58 655i 66926633 66 7 5 6716^6757 I 58 u34 il 7 3 I2l3 1253 I2 9 2 i33 2 I 5 9 6799 684o688i 6 9 23 6 9 64 7005 O 5 9 1372 i4n i45i i4 9 o i53o 1570 60" 50" | 40" | 30" 20" I 10" d 60" 50" 40" 30" 20" 10" . Co-sine of 63 Degrees. 9 2 Co-sine of 62 Degrees. a P Part 5 l " ~" 3 " 4 " 5 " 6 " 7 " 8/ 9 " p p ( 1" 2" 3" 4" 5" 6" 7" 8" 9" I 4 8 13 17 21 25 30 34 38 111 \ 4 8 12 16 20 24 28 32 36 L O G A R I T 11 M I C T A N G E N T S. 1 Tangent of 26 Degrees. c Tangent of 27 Degrees. s 0" 10" ! 20" j 30" 40" 50" 2 0" 10" | 20" 30" 40" 50-n I 2 3 9 . 688182 85o2 8823 855686o 9 88 7 6|893o 9196,9.250 8342 8663 8 9 83 9 3o3 83 9 5 8716 9o36 9 356 844 9 8769 9090 9410 5 9 58 57 56 o i 2 3 9 . 707166 779 8102 7218 753o 7 842 8i54 7270 7582 7394 8206 7 322 7634 8258 7 3 7 4 7 686 799 8 83io 7738 8o5o 8362 5 9 58 5 7 56 4 9 463 9 5i6;95 7 o 9 623 9676 973o 55 4 84i4 8466 85i8 85 7 o 8622 86 7 4 55 5 9783 9836(9890 9943 9996 ..5o 54 5 8726 8778 883o 8882 8 9 34 8985 54 6 9.690103 oi56|o2io 0263 o3i6 0369 53 6 9 o3 7 9089 9141 9 i 9 3 9 245 9297 53 7 j o423 0476(0529 o582 o636 0689 52 7 9 34 9 9401 9 453 95o4 9556 9608 52 81 0742 o 79 5 0849 0902 o 9 55 1008 5i 8 9 66o 9712 9764 9816 9868 9919 5i 9! 1062 in5 1168 1221 1274 i3 2 8 5o 9 997 i ..23 .. 7 5 .127 .179 .23l 5o 10 9.691381 i434 i48 7 i54o i5 9 4 1 647 49 10 9. 7 I0282 o334 o386 o438 0490 o542 49 n 1700 i 7 53 1806 i85 9 1913 1966 48 n 0593 o645 0697 0749 0801 o85 2 48 12 2019 20 7 2 2125 21-78 2232 2285 47 12 0904 o 9 56 1008 1059 mi ii63 47 i3 2338 2391 2444 249 7 2 55o 2603 46 i3 I2l5 1267 i3i8 1370 1422 1474 46 1 4 2656 2 7 IO 2 7 63 2816 2869 2922 45 i4 i5 2 5 i5 77 1629 1681 I 7 32 1784 45 i5 2975 3028 3o8i 3i34 3i8 7 324o 44 i5 i836 1887 I 9 3 9 1991 2o43 2094 44 iC 32 9 3 3346 34oo 3453 35o6 355 9 43 16 2l46 2198 2301 2353 24o5 43 17 36i2 3665 3 7 i8 3771 3824 38 7 7 42 17 2456 25o8 256o 2611 2663 2 7 l5 42 18 SgSo 3 9 83 4o36 4o8 9 4142 4i 18 2 7 66 2818 28 7 o 2921 2 97 3 3o25 4i 19 4248 43oi 4354 4407 446o 45i3 4o i 9 3o 7 6 3i28 3i 79 3 2 3i 3283 3334 4o 20 9.694566 4619 4672 4724 4777 483o 3 9 20 9 . 7 i3386 3438 348 9 354i 35 9 2 3644 3 9 21 4883 4936 4989 5o42 5i48 38 21 3696 3 7 4 7 3799 385o 3 9 O2 3 9 54 38 22 52OI 5 2 54 53o7 536o 5412 5465 37 22 4oo5 4o57 4io8 4i6o 42 1 1 4263 37 23 55i8 55 7 i 5624 56 77 5 7 3o 5 7 83 36 23 43i4 4366 44i8 446 9 452i 45 7 2 36 24 5836 5888 5 9 4i 5994 6047 6100 35 24 4624 4675 4 7 2 7 4778 483o 488i 35 20 6i53 6206 6258 63ii 6364 64i7 34 25 4 9 33 4984 5o36 5o8 7 5i3 9 5i 9 o 34 26 6470 6522 65 7 5 6628 6681 6 7 34 33 26 5242 52 9 3 5345 53 9 6 5448 54 99 33 27 6787 683 9 6892 6 9 45 6998 7o5o 32 27 555i 56o2 5654 5705 5 7 5 7 58o8 32 28 7io3 7 i56 7209 7262 73i4 7 36 7 3i 28 586o 5911 5 9 62 6014 6o65 6117 3i 29 7420 7 4 7 3 7 5 7 8 763i 7 684 3o 2 9 6168 6220 62 7 I 6322 63 7 4 6425 3o 3o 9.6 977 36 7789 7842 7894 7947 8000 2 9 3o 9.716477 65 2 8 65 79 663i 6682 6 7 34 2 9 3i 8o53 8io5 8i58 8211 8263 83i6 28 3i 6785 6836 6888 6 9 3 9 6991 7042 28 32 8369 8421 8474 852 7 85 7 9 8632 2 7 32 7 o 9 3 7i45 7 i 9 6 7247 7299 7 35o 27 33 8685 8 7 3 7 8790 8843 88 9 5 8 9 48 26 33 7401 7453 7 5o4 7 555 7607 7658 26 34 9001 9o53 9106 9 l5 9 9211 9264 25 34 7709 7761 7 8l2 7863 79 i5 7 9 66 25 35 93i6 9 36 9 9422 9474 9 5 27 9 5 79 24 35 8017 8069 8120 8171 8223 8274 24 36 9632 9 685 97 3 7 979 9 8 9 5 23 36 8325 83 7 6 8428 84 79 853o 858i 23 37 9947 53 .io5 .i58 .210 22 37 8633 8684 8 7 35 8 7 86 8838 888 9 22 38 9. 7 oo263 o3i5 o368 0420 o473 o5 2 5 21 38 8940 8991 9 o43 9094 9i45 9 i 9 6 21 39 o5 7 8 o63o o683 0736 0788 o84i 2O 3 9 9248 9299 9 35o 9401 9 45 2 9 5o4 2O 4o 9. 7 oo893 09^6 0998 io5i no3 n55 J 9 4o 9.719555 9606 9 65 7 97 o8 9760 9 8ii I 9 4i 1208 1260 i3i3 i365 1418 1470 18 4i 9862 99 i3 99 6 4 ..16 ..67 .118 18 42 i5 2 3 i5 7 5 1628 1680 1733 1785 17 42 9.720169 O22O 02 7 I 0322 o3 7 4 o425 ly 43 i83 7 1890 1942 i 99 5 2047 2IOO 16 43 0476 0527 o5 7 8 o62 9 0680 0732 16 44 2l52 2204 2257 2309 2362 2414 i5 44 o 7 83 o834 o885 o 9 36 0987 io38 15 45 2466 2519 25 7 I 2623 2676 2728 i4 45 1089 i i4o 1 191 1243 1294 i345 i4 46 2 7 8l 2833 2885 2 9 38 2 99 3o42 i3 46 1396 1447 1498 i54 9 1600 i65i i3 47 3o 9 5 3i47 3i 99 3252 33o4 3356 12 47 1702 1753 1804 i855 1906 i 9 5 7 12 48 3409 346i 35i3 3566 36i8 36 7 o II 48 2009 2060 2III 2162 22l3 2264 II 49 3 7 22 3 77 5 382 7 3879 3 9 3 2 3 9 84 IO 49 2 3i5 2366 24l 7 2468 2519 2570 IO 5o 9 , 7 o4o36 4o88 4i4i 4i93 4245 4207 9 5o 9.722621 2672 2 7 23 2 77 4 2825 2876 9 5i 435o 4402 4454 45o6 455 9 46n 8 5i 2927 2978 3029 3o8o 3i3o 3i8i 8 52 4663 4715 4 7 68 4820 4872 4 9 24 7 52 3232 3 2 83 3334 3385 3436 3487 7 53 4976 5029 5o8i 5i33 5i85 5237 6 53 3538 358 9 364o 36 9 i 3742 3 79 3 6 54 55 5290 56o3 5342 5655 53 9 4 5 7 o 7 5446 5 7 5 9 54 9 8 58ii 555i 5863 5 4 54 55 3844 4149 3895 4200 3 9 45 3 99 6 4302 40474098 4353l44o3 5 4 56 5 9 i6 5 9 68 6020 6072 6124 6176 3 56 4454 45o5 4556 46o 7 4658(4709 3 57 6228 6280 6333 6385 643 7 648 9 2 5 7 4760 48io 486i 4 9 I2 4963 5oi4 2 58 654i 65 9 3 6645 6697 674 9 6801 I 58 5o65 5ii5 5i66 52I 7 5268 53i9 I 5 9 6854 6906 6 9 58 7010 7062 7114 O 5 9 5370 5420 54 7 i 5522 55 7 356 2 4 O 60" 50" j 40" 30" 20" 10" c 60" | 50" 40" | 30" 20" 10" d Co-tangent of 63 Degrees. i Co-tangent of 02 Degrees. i p Plrt < I" 2" 3" 4" 5" 6" 7" 8" 9" 1 r> 11 1G 21 2G 32 37 42 47 , ( I- 1 ' 2" 3" 4" 5" 6" 7" 8" 9" trl { 5 10 15 21 20 31 36 41 46 j Li G A R I T H M I C SlNES. d Sine of 28 Degrees. a Sine of 29 Degrees. 3 o v 10" 20" 30- 7 | 40" 50" 3 0" IO 5 ' 20" 30" ' 40" | 50" 9.671609 1649 1688 I 7 28 I 7 68 1807 5 9 9 .6855 7 i 56o 9 564 7 5685 5723l576i 5 9 I 1847 1886 I 9 26 i 9 65 2005 2045 58 I 5 799 583 7 5873 5 9 i3 5 9 5i 5989 58 o 2084 212^ 2i63 22O3 2242 2282 57 2 6027 6o65 6io3'6i4i 6i 7 8 6216 57 3 2321 236l 2400 2440 2479 25l 9 56 *3 6254 62 9 2 633o 6368 64o6 6444 56 4 5 2558 2 79 5 2 5 9 8 2835 263 7 2874 2677 2 9 l4 2716 2 9 53 2 7 56 2 99 2 55 54 4 5 6482 6 7 o 9 65i 9 6 7 4 7 655 7 6 7 85 65 9 5 6822 6633 6860 6671 6898 55 54 6 3o32 3071 3m 3i5o 3i 9 o 322 9 53 6 6 9 36 6 97 4 7 OI2 7 o4 9 7087 7125 53 7 3268 33o8 334 7 338 7 3426 3465 52 7 7i63 7 2OI 7 238 7276 7 3i4 73.'2 52 8 35o5 3544 3583 3623 3662 3702 5i 8 7 38 9 7 42 7 7 465 7 5o3 7 54i 7 5 7 8 5i 9 374i 3 7 8o 38 2 o 385 9 38 9 8 3 9 38 5o 9 7616 7654 76 9 2 77 2 9 7767 7806 5o 10 9.673977 4oi6 4o56 4o 9 5 4i34 4i 7 3 49 10 9. 687843 7 88o 79 i8 7956 799 3 8o3i 49 ii 42l3 4252 42 9 i 433i 43 7 o 44o 9 48 ii 8o6 9 8106 8i44 8182 8220 825 7 48 12 4448 4488 452 7 4566 46o6 4645 47 12 82 9 5 8333 83 7 o 84o8 8446 8483 4? i3 4684 4723 4 7 62 4802 484i 488o 46 i3 852i 855 9 85 9 6 8634 86 7 i 8 7 o 9 46 i4 4919 4 9 5 9 4998 5o3 7 5o 7 6 5n5 45 i4 8 7 4 7 8 7 84 8822 8860 88 97 8 9 35 45 i5 5i55 5i 9 4 5233 5272 53u 535o 44 i5 8 97 2 9 oio 9 o48 9086 9123 9 i6o 44 16 5390 542 9 5468 55o 7 5546 5585 43 16 9 i 9 8 9 235 9273 93n 9 348 9 386 43 i? 5624 5664 5 7 o3 5742 5 7 8i 5820 42 *7 9 423 9 46i 9 4 9 8 9 536 9 5 7 3 9 6i i 42 18 585 y 58 9 8 5 9 3 7 5 97 6 6016 6o55 4i 18 9 648 9 686 9723 9761 979 8 9 836 4i J 9 6094 6i33 6172 621 1 625o 628 9 4o I 9 9 8 7 3 9911 99 48 9986 ..23 ..61 4o 20 9.676328 636; 64o6 6445 6484 6523 39 20 9 .6ooo 9 8 oi36 0173 02 I I 0248 0286 3 9 21 6562 6601 664o 667 9 6718 6 7 5 7 38 21 'o323 o36i o3 9 8 o435 o473 o5io 38 22 23 6796 7 o3o 6835 7 o6 9 68 7 46 9 i3 71087147 6962 6991 71867225 37 36 22 23 o548 0772 o585 0809 0622 0847 0660 o884 o6 97 O 9 22 0735 o 9 5 9 3 7 36 24 7264 7 3o3 7 342 7381 7420 7 45 9 35 24 o 99 6 io34 1071 IT 08 n46 n83 35 25 7498 7 536 7 5 7 5J 7 6i4 7 653 7 6 9 2 34 25 I22O 1258 I2 9 5 1 332 1370 1407 34 26 77 3i 777 78097848 7886 792 5 33 26 1 444 1482 i5i 9 i556 i5 9 4 i63i 33 27 79 6 4 8oo3 80428081 8120 8i58 32 27 1668 I 7 o6 1743 1780 1817 i855 32 28 8197 8236 82758314 8353 83 9 i 3i 28 l8 9 2 1929 i 9 66 2004 2o4l 2078 3i 29 843o 846 9 85o8'8547 8585 8624 3o 2 9 2Il5 2i53 2I 9 O 222 7 2264 23O2 3o 3o 9 .6 7 8663 8 7 O2 8 7 4o8 779 8818 885 7 2 9 3o 9 .6 9 233 9 2 3 7 6 24l3 2450 2488 2525 29 3i 88 9 5 8 9 34 8973 9 OI2 9 o5o 9 o8 9 28 3i 2562 2 5 99 2636 26 7 4 2 7 II 2748 28 32 9128 9 l6 7 9 2o5 9 244 92 83 9 32I 27 32 2 7 85 2822 285 9 2 8 97 2 9 34 2 97 I 2 7 33 9360 9 3 99 9 438 0476 9 5i5 9 554 26 33 3oo8 3o45 3o8 2 3lI9 3i5 7 3i 9 4 26 34 9 5 9 2 9 63i 9670 9 7o8 9747 97 86 25 34 323i 3268 33o5 3342 33 79 34i6 25 35 9824 9 863 99 02 99 4o 9979 ..i 7 24 35 3453 3490 35 2 8 3565 36o2 363 9 24 36 9.68oo56 oo 9 5 oi33 0172 0210 024 9 23 36 36 7 6 3 7 i3 3 7 5o 3787 3824 386i 23 37 0288 o3 2 6 o365 o4o3 0442 o48i 22 37 38 9 8 SgSS 3 97 2 4009 4o46 4o83 22 38 o5i 9 o558 o5 9 6 o635 0673 O 7 I2 21 38 4l2O 4i5 7 4i 9 4 423i 4268 43o5 21 39 0750 o 7 8 9 0828 0866 o 9 o5 o 9 43 20 3 9 4342 43 79 44i6 4453 44 9 o 452 7 2O 4o 9.680982 IO2O io5 9 io 97 n36 n 7 4 J 9 4o 9 .6 9 4564 46oi 4638 46 7 5 4 7 . 2 4 7 4 9 J 9 4i I2l3 I25l I2 9 O i328 i366 i4o5 is; 4 4 7 86 4823 486o 48 97 4 9 34 4971 i a 42 1 443 1482 l52O i55 9 i5 9 7 i636 J 7 42 5oo 7 5o44 5o8i 5n8 5i55 5l 9 2 *7 43 i6 7 4 I 7 i3 i 7 5i 1789 1828 1866 16 43 522 9 5266 53o3 5339 53 7 6 54i3 16 44 I 9 o5 I 9 43 1981 202O 2o58 20 97 i5 44 545o 548 7 5524 556i 55 9 8 5634 i5 45 2i35 2I ? 3 2212 225o 2288 232 7 i4 45 56 7 i 5708 5745 5782 58i 9 5855 i.4 46 2365 24o3 2442 2480 25l 9 255 7 i3 46 58 9 2 5 9 2 9 5 9 66 6oo3 6o3 9 6076 i3 4? s5 9 5 2633 26 7 2 2 7 IO 2748 2787 12 47 6ii3 6i5o 6187 6223 6260 62 97 12 48 2825 2863 2 9 O2 2 9 4o 2 97 8 3oi6 II 48 6334 6370 6407 6444 648 1 65i 7 II 49 3o55 3o 9 3 3i3i 3i 7 o 3208 3246 IO 49 6554 65 9 i 6628 6664 6 7 oi 6738 10 5o 9.683284 3323 336i 33 99 343 7 34 7 5 9 5o 9 .6 9 6775 6811 6848 6885 6 9 21 6 9 58 9 5i 35i4 3552 35 9 o 3628 366 7 3 7 o5 8 5i 6 99 5 7o3i 7068 7io5 7 i4i 7178 8 52 3 7 43 3 7 8i 38i 9 3858 38 9 6 3 9 34 7 52 72i5 725i 7288 7 325 7 36i 7 3 9 8 7 53 3 97 2 4oio 4o48 4o8 7 4i25 4i63 6 53 7435 7471 75o8 7 545 7 58i 7 6i8 6 54 4201 423 9 4277 43i5 4353 43 9 2 5 54 7654 7 6 9 i 7728 77 64 7801 7 838 5 55 443o 4468 45o6 4544 4582 4620 4 55 7 8 7 4 7911 7947 7 984 8020 8o5 7 4 56 4658 46 9 6 4 7 35 4 77 3 48n 484 9 3 56 8o 9 4 8i3o 8167 8203 8240 8276 3 5 7 488 7 4 9 25 4 9 63 5ooi 5o3 9 5o 77 2 5 7 83-i 3 834 9 8386 8423 845 9 84 9 6 7, 58 5n5 5i53 5i 9 i 522 9 5267 53o5 I 58 8532 856 9 86o5 8642 8678 8 7 i5 : 5 9 5343 538i 54i 9 5457 54 9 5 5533 O 5 9 8 7 5i 8 7 88 8824 8861 88 97 8 9 34 o 60" 50" ; 40" 30" 20" 10" c 60" 50" 40" 30" | 20" 19' d Co-sine of 61 Degrees. s Co-sine of 60 Degrees. 1 ! p p .( 1" 2" 3" 4" 5" 6" 7" 8" 9" , 1" 2" 3" 4" 5" 6" 7" 8" 9" 1 irt $ 4 8 12 16 19 23 27 31 35 irt \ 4 7 11 15 19 22 2C 30 33 ) LOGARITHMIC TANGENTS. 53 A Tangent of 28 Degrees. c Tangent of 29 Degrees. 2 Q 1 ' | 10" 20" | 30" 40" 50" 0" 10" 20" 30" | 40" 50" o I 9 . 725674 5725 5 9 7 9 6o3o 5776 5827 6081 6i3i 58 7 8 6182 5928 6233 D 9 58 I 9.743752 4o5o 38o2 4099 385i 4149 SgOl 4199 3 9 5i 4248 4ooo 4298 II 2 3 62846334 6588 663Q 63856436 6690 6740 648 7 6791 653 7 6842 56 2 3 4348 4645 43 97 46 9 5 4447 4744 4496 4794 4546 4844 45 9 6 48 9 3 II 4 5 68 9 2 7 X 97 6 9 43 6994 7o45 7298 7 34 9 7 o 9 5 7399 745o 55 54 4 5 4943 524o 499 2 5290 533 9 5092 538 9 5i4i 543 9 5i 9 i 55 5488 54 6 755i 7 6o2 7 653 77 3 7754 53 6 5538 5587 563 7 5686 5736 5 7 85 53 7 7805 7 855 7906 79 5 7 8007 8o58 52 7 5835 5884 5 9 34 5 9 83 6o33 6082 52 8 8109 8i5 9 8210 8261 83u 8362 5i 8 6i32 6182 6 2 3i 6281 633o 638o 5! 9 8412 8463 85i4J8564 86i5 8665 5o 9 6429 6479 6528 65 77 6627 6676 5o ] 9.728716 8767 8817 8868 8918 8969 49 IO 9.746726 6 77 5 6825 68 7 4 6924 6 97 3 4o II 12 9020 9 3 2 3 9070 9 3 74 9121 9424 9171 9 4?5 9222 9 5 2 5 9272 9 5 7 6 48 47 ii 12 7023 7 3i 9 7072 7 36 9 7122 74i8 7171 7 468 7221 7 5i 7 7 2 7 7 56 7 48 4 7 1 3 9626 9677 9727 9778 9828 9879 46 i3 76i6| 7 665 7715 7764 7814 7 863 46 1-4 9929 9980 ..3o ..81 .132 .182 45 i4 79I 3 -7962 801 1 8061 8110 8160 45 i5 9.730233 0283 o333 o384 o434 o485 44 i5 82O 9 8 2 58 83o8 835 7 84o6 8456 44 16 o535 o586 o636 0687 o 7 3 7 0788 43 16 85o5 8555 86o4 8653 8 7 o3 8 7 5 2 43 17 o838 0889 o 9 3 9 0990 io4o 1091 42 17 8801 885i 8900 8949 8999 9048 42 18 n4i 1 191 1242 1292 i343 1393 4i 18 997 9147 9196 9245 9 2 9 5 9344 4i ! 9 1 444 i4g4 1 544 i5 9 5 i645 1696 4o i 9 9 3 9 3 9443 9492 9541 9 5 9 i 9640 4o 20 9.731746 1796 1847 1897 1948 1998 39 20 9.749689 9739 9788 9 83 7 9 886 9936 3 9 21 2048 2099 2149 22OO 2250 2300 38 21 99 85 ..34 ..84 .i33 .182 .23l 33 22 235! 2401 245 1 25O2 2552 2602 37 22 9 . 750281 o33o o3 79 0428 0478 0527 37 23 2653 2 7 o3 2? 53 28o4 2854 2904 36 23 o5 7 6 0625 0675 7 24 o 77 3 0822 36 24 2 9 55 3oo5 3o55 3io6 3i56 3206 35 24 0872 0921 00-70 ioi 9 io6 9 1118 35 25 3257 33o 7 335 7 34o8 3458 35o8 34 25 1167 1216 1265 i3i5 1 364 i4i3 34 26 3558 3609 365 9 3709 3 7 6o 38io 33 26 1462 i5n i56i 1610 i65 9 1708 33 2 7 386o 3910 3 9 6i 4on 4o6i 4m 32 27 i 7 5 7 1806 i856 igoS 1954 2003 32 28 4162 4212 4262 43i2 4363 44i3 3i 28 2O52 2IOI 2l5l 22OO 2249 2298 3i 2 9 4463 45i3 4564 46i4 4664 47i4 3o 2 9 2347 2396 2446 2495 2544 2593 3o 3o 9.734764 48i5 4865 49 1 5 4965 5oi5 29 3o 9.752642 2691 2740 2 7 8 9 2839 2888 29 3i 5o66 5n6 5i66 5 2 i6 5 2 66 53i 7 28 3i 2 9 3 7 2986 3o35 3o84 3i33 3i8 2 28 32 5367 54i7 546 7 55i 7 5567 56i8 27 32 323i 3280 333o 33 79 3428 3477 27 33 5668 5 7 i8 5 7 68 58i8 5868 5918 26 33 3526 35 7 5 3624 36 7 3 3722 3 77 i 26 34 5 9 6 9 6019 6069 6119 6169 6219 25 34 3820 386 9 3 9 i8 3967 4oi6 4o66 25 35 6269 63i 9 6370 6420 6470 652O 24 35 4n5 4i64 42i3 4262 43n 436o 24 36 65 7 o 6620 6670 6720 6770 6820 23 36 44o 9 4458 45o 7 4556 46o5 4654 23 37 6870 6921 6971 7021 7071 7121 22 37 4703 4 7 52 48oi 485o 4899 4948 22 38 7171 7221 7271 7321 7 3 7 i 7421 21 38 4997 5o46 5o 9 5 5i44 5i 9 3 5242 21 3 9 7471 7521 75 7 i 7621 7671 7721 2O 39 5291 534o 538 9 5438 548 7 5536 20 4o 9.737771 7821 7871 7921 7971 8021 19 4o 9-755585 5634 5682 5 7 3i 5780 582Q 19 4i 8071 8121 8171 8221 8271 832i 18 4i 58 7 8 5 9 2 7 5976 6o25 6074 6i23 18 42 83 7 i 8421 8471 852i 85 7 i 8621 17 42 6172 6221 6270 63i 9 6368 64i6 17 43 8671 8721 8771 8821 8871 8921 16 43 6465 65i4 6563 6612 6661 6710 16 44 8971 9021 9071 9121 9171 9221 i5 44 6 7 5 9 6808 685 7 6905 6 9 54 7oo3 i5 45 9271 9321 9371 9420 9470 9520 i4 45 7052 7 IOI 7i5o 7199 7247 7296 i4 46 47 9 5 7 o 9 8 7 o 9620 9670 9920 9969 9720 ..19 9770 ..69 9820 . 119 i3 12 46 47 7 345 7 638 7 3 9 4 7 68 7 7443 77 36 7492 77 85 7834 7 58 9 7882 i3 12 48 9 .74oi6 9 0219 0269 0319 o368 0418 II 48 79 3i 79 8 8029 8078 8127 8i 7 5 II 49 o468 o5i8 o568 0618 0668 0717 IO 49 8224 82 7 3 8322 8371 8419 8468 10 5o 9-740767 0817 0867 0917 0967 1016 9 5o 9.758517 8566 86i5 8663 8712 8761 9 5i 1066 1116 1166 1216 1265 i3i5 8 5i 8810 8858 8907 8 9 56 9005 9 o53 8 52 i365 i4i5 i465 i5i4 1 564 1614 7 52 9102 9i5i 9200 9 248 9297 9 346 7 53 1 664 1714 1763 i8i3 i863 1913 6 53 9 3 9 5 9443 9492 9541 9590 9 638 6 54 10,62 2012 2062 2112 2161 221 I 5 54 9687 9736 9785 9 833 9882 99 3i 5 55 2261 23ll 236o 24lO 2460 25lO 4 55 9979 ..28 77 .126 174 .223 4 56 255 9 2609 2659 2709 2 7 58 2808 3 56 9.760272 o32O o36 9 o4i8 o466 o5i5 3 5- 2858 2907 2 9 5 7 3007 3o56 3io6 2 57 o564 0612 0661 0710 0758 0807 2 58 3i56 3206 3255 33o5 3355 34o4 I 58 o856 0904 0953 1 002 io5o 1099 I 5 9 3454 35o4 3553 36o3 3653 3702 O 5 9 n48 1196 1245 1293 1 342 1391 O 60" 50" | 40" 30" 20" 10" c 60" | 50" 40" 30" 20" 10" d Co-tangent of 61 Degrees. Co-tangent of 60 Degrees. 1 p p ( I" 2" 3" 4" 5" 6" 7" 8" 9" I 5 10 15 20 25 30 35 40 45 p p . < 1" 2" 3 " 4 " 5 " 6// 7 " 8// 9 " irt \ 5 10 15 20 25 29 34 39 44 LOGARITHMIC SINES. .3 Sine of 30 Degrees. a Sine of 31 Degrees. "i % 0" 10" 20" 30" 40" 50" 9 0" 10" 20" 30" 40" 50" 9.698970 9006 9043. go 7 9 9116 9 l52 5 9 o 9 . 7 n83 9 i8 7 4 I 9 o 9 i 9 44 IQ7 9 20l4 5 9 I 9189 9225 9262^298 9334 9 3 7 i 58 i 2o5o 2o85 2120 2i55 2190 2225 58 2 9407 9444 9480 9 5i 7 9553 9 58 9 5 7 2 2260 220)5 2330 2365 24oo 2434 ^7 3 9626 9662 9699 9735 977 I 9 8o8 56 j 246 9 25o4 253 9 2 5 7 4 26o 9 2644 56 4 9844 9880 991-799539990 ..26 55 4 26 79 2-714 2 7 4 9 2 7 84 28i 9 2854 55 5 6 9.700062 0280 0099 0317 oi35 017110206 o353o38 9 o425 0244 0462 54 53 5 6 288 9 3o 9 8 2 9 24 3i33 2 9 5 9 3i68 2 99 4 3 2 o3 3o2 9 3 2 38 3o63 32 7 3 54 53 7 0498 o534 o5 7 i 060-7 oC43 0680 52 7 33o8 3343 33 77 3412 344? 3482 52 8 0716 0752 o 7 88 0825 0861 o8 97 5i 8 35i 7 3552 358 7 362i 3656 36 9 i 5i , 9 0933 0970 1006 1042 io 7 8 in5 5o 9 3 7 26 3 7 6i 3 79 6 383o 3865 3 9 oo 5o 10 9. 7on5i 1187 1223 1259 I2 9 6 i33 2 49 IO 9 . 7 i3 9 35 3 97 o 4oo5 4o3 9 4o 7 4 4io 9 49 ii 1 368 i4o4 i44o 1477 i5i3 i54 9 48 ii 4i44 4i79 4213 4248 4283 43i8 48 12 i585 1621 i658 1694 i 7 3o i 7 66 47 12 4352 438 7 4422 445 7 44 9 i 4526 47 i3 1802 i838 i8 7 4 1911 i 9 4 7 i 9 83 46 i3 456i 45 9 6 463o 4665 4 7 oo 4 7 35 46 i4 2019 2o55 2091 2127 2164 22OO 45 i4 4 7 6 9 48o4 483 9 48 7 3 4 9 o8 4 9 43 45 i5 2236 2272 23o8 2344 238o 2416 44 i5 4 97 8 5oi2 5o4 7 5o8 2 5n6 5i5i 44 16 2452 2488 25 2 4|256i 25 97 2633 43 16 5i86 522O 5 2 55 52 9 o 5324 535 9 43 *7 2669 2705 2 7 4i 2777 2 8i3 2 84 9 42 *7 53 9 4 5428 5463 54 9 8 5532 556 7 42 18 2885 2921 2 9 5 7 2 99 3 3o2 9 3o65 4i 18 56o2 5636 56 7 i 5 7 o5 5 7 4o 5 77 5 4i *9 3ioi 3i3 7 3i 7 3 32O 9 3245 3281 4o X 9 58o 9 5844 58 7 8 5 9 i3 5 9 48 5 9 82 4o 20 9 . 7 o33i 7 3353 3389 3425 346i 34 9 7 3 9 20 9. 7 i6oi 7 6o5i 6086 6121 6i55 6i 9 o 3 9 21 3533 356 9 36o5 364i 36 77 3 7 i3 38 21 6224 6 2 5 9 62 9 3 6328 6362 63 97 38 22 3 7 4 9 3784 3820 3856 38 9 2 3928 37 22 6432 6466 65oi 6535 65 7 o 66o4 37 23 3 9 64 4ooo4o36 4072 4io8 4i44 36 23 663 9 66 7 3 6 7 o8 6 7 42 6 777 6811 36 24 4179 42i5 425i 4287 43 2 3 435 9 35 24 6846 6880 6 9 i5 6 9 4 9 6 9 84 7 oi8 35 25 43 9 5 443 1 4466 45o2 4538 45 7 4 34 25 7 o53 7 o8 7 7 I22 7 i56 7 i 9 i 7 225 34 26 46io 4646 4682 4717 4 7 53 4 7 8 9 33 26 7 25 9 7 2 9 4 7 328 7 363 7 3 97 7 432 33 27 4825 486i 48 9 6 4 9 32 4 9 68 5oo4 32 27 7 466 7 5oo 7 535 7 56 9 7 6o4 7 638 32 28 5o4o 5o 7 5j5in 5i47 5i83 52I 9 3i 28 7 6 7 3 777 77 4i 777 6 7 8io 7 844 3i 29 5254 52 9 o 5326 5362 53 97 5433 3o 29 7 8 79 79 i3 79 48 79 8 2 8016 8o5i 3o 3o 9.705469 55o5 554o 55 7 6 56i2 5648 2 9 3o 9 . 7 i8o85 8119 8i54 8188 8223 8 2 5 7 2 9 3i 5683 57x96755 5 79 o 58 2 6 5862 28 3i 8291 8326 836o 83 9 4 842 9 8463 28 32 58 9 8 5 9 33 5 9 6 9 6oo5 6o4o 6o 7 6 2 7 32 8497 853i 8566 8600 8634 8669 27 33 6112 61476183 62I 9 6254 62 9 o 26 33 8 7 o3 8 7 3 7 8 77 2 8806 884o 88 7 4 26 34 6326 636i 63 97 6433 6468 65o4 25 34 8909 8 9 43 8977 9 oi i 9 o46 9080 25 35 653 9 6575 6611 6646 6682 6 7 i8 24 35 9114 9 i48 9 i83 9217 9 25l 9 285 24 36 6 7 53 678916824 6860 68 9 5 6 9 3i 23 36 9320 9 354 9 388 9 422 9 456 9 4 9 i 23 3? 6967 7 oo2 7 o38 7 o 7 3 7 io 9 7 i45 22 37 9 5 2 5 9 55 9 9 5 9 3 9 62 7 9 662 o6 9 6 22 38 7180 7216 7 25i 7287 7 322 7 358 21 38 973o 97 64 979 8 9 833 9 86 7 $ 9 oi 21 3 9 7 3 9 3 7429 7 464 7 5oo 7 535 7 5 7 i 2O 3 9 99 35 99 6 9 ...3 ..38 ..72 . 106 20 4o 9.707606 7642 7 6 77 77 i3 77 48 7784 I 9 4o 9. 7 2oi4o oi 7 4 0208 0242 0276 o3n J 9 4i 7810 7 855 7 89o 79 26 79 6i 7997 18 4i o345 o3 79 o4i3 o44 7 o48i o5i5 18 42; 8o32 8068,8103 8i3 9 8i 7 4 8210 !? 42 o549 o583 o6i 7 o652 0686 O 7 2O J 7 43| 8245 8j8o|83i6 835i 838 7 8422 16 43 o 7 54 o 7 88 0822 o856* o8 9 o 0924 16 44 8458 84 9 38528 8564 85 99 8635 i5 44 0958 99 2 1026 1060 io 9 4 1128 i5 45 8670 8 7 o58 7 4i 8 77 6 8811 884 7 i4 45 1162 1 1 9 6 1230 1264 J2 9 8 i332 i4 46 8882 8 9 i8:8 9 53 8 9 88 9 024 9 o5 9 i3 46 1 366 i4oo i434 1 468 i5o2 i536 i3 4? 9094 9 i3o 9 i65 9 2OO 9 236 9 2 7 I 12 47 i5 7 o i6o4 i638 l6 7 2 1706 i 7 4o 12 48 9306 9 342 9 3 77 9 4l2 9 448 9 483 II 48 i 77 4 1808 1842 i8 7 6 I 9 IO i 9 44 II 49 9 5i8 9 553 9 58 9 9 624 9 65 9 9 6 9 5 IO 49 i 97 8 2OI2 2046 2080 2Il4 2148 IO 5o 9.709730 9 7 65 9800 9 836 9 8 7 i 99 o6 9 5o 9. 7 22l8l 22l5 224 9 2283 2317 235i 9 5i 9941 9977 - 12 ..47 ..82 .118 8 5i 2385 24l 9 2453 248 7 2520 2554 8 52 9.710153 01880223 O25 9 02 9 4 o32 9 7 52 2 588 2622 2656 2690 2724 2 7 5 7 7 53 o364 0399 o435 o4 7 o o5o5 o54o 6 53 2 79 I 2825 285 9 28 9 3 2 9 2 7 2 9 6o 6 54 o5 7 5 061 1 0646 0681 o 7 i6 o 7 5i 5 54 2994 3028 3o62 3o 9 6 3i3o 3i63 5 55 0786 0822 o85 7 o8 9 2 9 2 7 0962 4 55 3i 97 323: 3265 32 99 3332 3366 4 56 0997 io32 io6 7 no3 n38 n 7 3 3 56 34oo 3434 3468 35oi 3535,356 9 3 57 1208 1243 I2 7 8 i3i3 1 348 1 383 2 ^7 36o3 3636|36 7 o 3 7 o4 3 7 38 3 77 i 2 58 5 9 1419 1629 i454 i4&9 1 664 1699 ID24 i 7 34 i55 9 i5 9 4 i 7 6 9 i8o4 I O 58 5 9 38o5 383 9 38 7 3 3 9 o6 3 9 4o 3 97 4 4oo 7 |4o4i 4o 7 5 4io 9 4i42 4i 7 6 I O 60" 50" 40" 30" | 20" 10" a 60" 50" | 40" 30" | 20" 10" | Q . Co-sine of 59 Degrees. ' Co-sine of 58 Degrees. & f < 1" '2" 3" 4" 5" 6" 7" 8" 9" jl . < 1" 2" 3" 4" 5" 6" 7" 8" 9" 1 I 4 7 11 11 18 21 25 29 32 A - lmt } 3 7 10 14 17 21 24 27 31 LOGARITHMIC TANGENTS. d | Tangent of 30 Degrees. g Tangent of 3 1 Degrees. * 1 0" 10" 20" 30" 40* 50" i 0" | 10" 20 7 30" 40" \ 50" o 9. 7 6i43 9 1 488 i53 7 i585 i634 1682 5 9 9.778774 8821 8869 8917 8964)9012 5 9 i I73l 1-780 1828 1877 1925 1974 58 I 9 o6o 9108 9 i55 9203 9 25: 9298 5s 2 2023 2071 2I2O 2168 22I 7 2266 57 2 9 346 9 3 9 4 q44i Q48 9 9537 9 584 5? 3j 23i4 2363 2411 2460 25o8 255 7 56 3 9 632 9679 9727 9775 9822 9870 56 4 2606 2654 2 7 o3 2751 28002848 55 4 99 i8 99 65 . . i3 ..61 .108 .i56 55 5j 2897 2 9 45 2994 3o43 3o 9 i 3i4o 54 5 9.780203 O25l 0299 o346 o3 9 4 o44i 54 6 3i88 323 7 3285 .3334 3382 343 1 53 6 0489 o53 7 o584 o632 0679 0727 53 7 34 79 352 35 7 6 3625 36 7 3 3722 52 7 0775 0822 0870 0917 o 9 65 ioi3 52 8 3 77 o 38i 9 386 7 3 9 i6 3 9 64 4oi3 5i 8 1060 1108 n55 1203 1250 1298 5i 9 4o6i 4no 4i58 4207 4255 43o4 5o 9 1 346 i3 9 3 i44i i488 i536 i583 5o 10 9.764352 44oo 4449 44 9 7 4546 45 9 4 49 10 9.781631 1678 1726 1774 1821 1869 4 9 r r 4643 46 9 i 4 7 4o 4788 4836 4885 48 ii 1916 1964 2OII 2o5g 2106 2i54 48 [2 4 9 33 4 9 82 5o3o 0079 5l2 7 5i 7 5 47 12 2201 2249 2296 2344 2391 243 9 47 r3 5224 5272 532i 536 9 54i8 5466 46 i3 2486 2534 258i 2629 2676 2724 46 i4 55i4 5563 56n 566o 5 7 o8 5 7 56 45 i4 2771 2819 2866 2914 2961 3009 45 i5 58o5 5853 5 9 oi 5 9 5o 5 99 8 6047 44 i5 3o56 3io4 3i5i 3l 99 3246 32 9 4 44 16 6o 9 5 6i43 6l 9 2 6240 6288 633 7 43 16 334i 3388 3436 3483 353i 35 7 8 43 17 6385 6433 6482 653o 65 7 8 6627 42 17 3626 3673 3721 3768 38i6 3863 42 18 6675 6 7 23 6772 6820 6868 6917 4i 18 3910 3 9 58 4oo5 4o53 4ioo 4r48 4i 19 6 9 65 70i3 7062 7110 7i58 7207 4o 19 4i95 4242 4290 433 7 4385 4432 4o 20 9.767255 7 3o3 7 35 2 74oo 7448 7496 39 20 9.784479 452 7 45 7 4 4622 466 9 4716 39 21 7 545 7 5 9 3 7641 7 6 9 o 77 38 7786 38 21 4764 48xi 485 9 4906 4 9 53 5ooi 38 22 7834 7883 79 3i 7979 8027 8076 37 22 5o48' 5o 9 5 5i43 Sigo 5238 5285 37 23 8124 8172 8221 826 9 83i 7 . 8365 36 23 5332 538o 5427 5474 5522 556 9 36 24 84i4 8462 85io 8558 8606 8655 35 24 56i6 5664 5 7 n 5 7 58 58o6 5853 35 25 8703 8 7 5i 8799 8848 88 9 6 8 9 44 34 25 Sgoo 5 9 48 5 99 5 6042 6090 6137 34 26 8992 9 o4o 9 o8 9 9 i3 7 9 i85 9 233 33 26 6i84 6232 6279 6326 63 7 4 6421 33 27 9281 9 33o 9 3 7 8 9426 9474 9 522 32 27 6468 65i6 6563 6610 665 7 6 7 o5 32 28 95 7 i 9 6i 9 9 66 7 97 i5 9763 9811 3i 28 6 7 5 2 6799 6847 6894 6 9 4i 6988 3 1 2 9 9860 99 o8 99 56 ...4 ..52 . IOO 3o 29 7 o36 7083 7i3o 7178 7225 7272 3o 3o 9.770148 oi 9 7 O245 0293 o34i o38 9 29 3o 9-7873I9 7 36 7 74i4 746 1 7 5o8 7556 20 3i o43 7 o485 o534 o582 o63o 0678 28 3i 7 6o3 765o 7697 7745 7792 7 83 9 28 32 0726 0774 0822 o8 7 o o 9 i 9 0967 2 7 32 7 886 79 34 7981 8028 8o 7 5 8I22J27 33 ioi5 io63 mi nSg 1207 1255 26 33 8170 8217 8264 83n 835 9 84o6 20 34 i3o3 i35i l3 99 i448 1496 1 544 25 34 8453 85oo 8547 85 9 5 8642 868 9 25 35 l5 9 2 i64o 1688 i 7 36 1784 i832 24 35 8 7 36 8 7 83 883o 8878 8 9 25 8 97 2 24 36 1880 I 9 28 1976 2024 2072 2I2O 23 36 9019 9066 9114 9161 9208 9 255 23 3? 2168 2216)2264 23l2 236i 2409 22 37 9302 9 34 9 9397 9444 9491 9 538 22 38 2457 2 5o5l 2 553 26OI 2649 2697 21 38 9 585 9 532 9679 9727 9774 9 82J 21 39 2745 2793 2841 2889 2 9 3 7 2 9 85 2O 39 9868 99 1 5 9962 ...9 ..5 7 .104 2O 4o 9-773o33 3o8i 3129 3i77 3225 32 7 3 X 9 4o 9.790151 0198 O245 0292 0339 o386 I 9 4i 332i 336 9 34i7 3465 35i2 356o 18 4i o434 o48i o528 o5 7 5 0622 o66 9 18 42 36o8 3656 3704 3 7 52 38oo 3848 17 42 0716 o 7 63 0810 0857 0905 O 9 52 17 43 3896 3 9 44 3 99 2 4o4o 4o88 4i36 16 43 0999 io46 1093 n4o 1187 1234 16 44 4i84 4232 4280 43 2 8 43 7 5 4423 i5 44 1281 i328 i3 7 5 1422 1469 i5i6 i5 45 4471 45i 9 456 7 46i5 4663 4711 i4 45 i563 1611 i658 1705 1752 i 799 i4 46 4 7 5 9 4807 4855 4902 495o 4998 i3 46 -846 i8 9 3 1940 1987 2034 2081 i3 47 5o46 5og4 5i42 Sigo 5 2 38 5 2 86 12 47 u*a 2175 2222 2269 23i6 2363 12 48 5333 538i 5429 54 77 55 2 5 55 7 3 II 48 2410 245 7 25o4 255i 2598 2645 II 49 " 562i 5668 5 7 i6 5 7 64 58i2 586o 10 49 2692 2739 2 7 86 2833 2880 2927 IO 5o 9.775908 5 9 56 6oo3 6o5i 6099 6i47 9 5o 9.792974 3021 3o68 3n5 3i6 2 3209 9 5i 6i 9 5 6243 6290 6338 6386 6434 8 5i 3 2 56 33o3 335o 33 9 7 3444 3491 8 52 6482 6529 65 77 6625 6673 6721 7 52 3538 3585 3632 36 79 3726 3 77 3 7 53 6768 6816 6864 6912 6960 7007 6 53 38i 9 3866 3 9 i3 4oo7l4o54 6 54 7 o55 7io3 7161 7 r 99 7246 7294 5 54 4ioi 4i48 4i95 4242 428 9 '4336 5 55 56 57 58 5 9 7628 79 i5 8201 8488 7 3 9 o 7 6 7 6 79 63 8249 8535 743 7 7724 8010 8297 8583 7 485 777 2 8o58 8344 863i 7 533 7819 8106 83 9 2 8678 7 58i 7867 8 1 54 844o 8 72 6 4 3 2 I O 55 56 5 7 58 59 4383 4664 4946 5227 55o8 443o 4476 47II4758 4992 SoSg 52 7 4532i 5555J5602 452345704617 48o5 4852 4899 5o865i33:5i8o 536 7 54i4546i 56495696 5742 4 3 2 I O 00" 50" 40" 30" | 20" 10" q 60" 50" | 40" l 30" | 20" | 10" a Co-tangent of 59 Degrees. .* Co-tangent of 58 Degrees. i P Tart 4 '" ~" :i " 4 " 5 " " 7 " 8 " 9 " ( I" 2" 3" 4" 5" 6" 7" 8" 9" J 5 10 14 1!) 24 20 34 3f) 43 in { 5 14 19 ^t 28 33 38 43 56 LOGARITHMIC SINES. p 1 Sine of 32 Degrees. d Sine of 33 Degrees. m 10" 20" 30" 40" 50" ii 0" 10" 20" 30" 40" 50" o 9.724210 4243 4277 43ri 4344 4378 5 9 o 9> 736io 9 6i4i 6174 6206 6288 62-71 5 9 il 44i2 4445 4479 45:3 454G 458o 58 I 63o3 6336 6368 64oo 6433 6465 58 c 46i4 464? 468i 4 7 i5 4748 4782 57 2 04 9 8 653o 6562 65 9 5 6627 665 9 57 3j 48 1 6 4849 4883 4917 495o 4 9 84 56 3 6692 6724 6 7 5 7 6 7 8 9 6821 6854 56 4j 5oi 7 5o5i 5o85 5n8 5 1 52 5i85 55 4 6886 6918 6 9 5i 6 9 83 7 oi5 7 o48 55 5 6219 5 2 53 5286 5320 5353 538 7 54 5 7080 7112 7i45 7 i 77 7 20 9 7 24l 54 6 5420 5454 5488 552i 5555 5588 53 6 7274 73o6 7 338 7 3 7 i 74o3 7 435 53 7 5622 5655 568 9 572-3 5 7 56 5 7 8 9 52 7 7 46 7 75oo 7532 7 564 7 5 97 7 62 9 5a 8 5823 5856 5890 5 9 23 5 9 5 7 5 99 o 5i 8 7661 7 6 9 3 7726 77 58 779 7822 5i 9 6024 6o5 7 6091 6124 6i58 6i 9 i 5o 9 7 855 7887 7919 79 5i 79 83 8016 5o 109.726226 6258 6292 6325 635 9 63 9 2 4 9 10 9.73-8048 8080 8112 8i45 8i 77 8209 49 ii! 6426 645 9 6493 65 2 6 656o 65 9 3 48 ii 8241 8273 83o6 8338 83 7 o 8402 48 12 6626 6660 6693 6727 6760 67 9 4 47 12 8434 8466 8499 853i 8563 85 9 5 47 i3 6827 6860 6894 6927 6961 6 99 4 46 i3 8627 865 9 8692 8724 8 7 56 8 7 88 46 i4 7027 7061 7094 7128 7161 7 i 9 4 45 i4 8820 885 2 8884 8 9 i 7 8 9 4 9 8 9 8i 45 i5 7228 7261 7294 7 3 2 8 7 36i 7 3 9 4 44 i5 9013 9 o45 9077 9 io 9 9 i4i 9 i 7 3 44 16 7428 746 1 7494 7628 756i 7 5 9 4 43 16 9206 9 238 9270 9 302 9 334 9 366 43 J 7 7628 7661 7694 7728 7761 7794 42 17 9 3 9 8 9 43o 9462 9 4 9 4 9 526 9 558 42 18 7828 7861 7 89 5485 553i 6676 6622 6667 12 48 9 i 9 3 9240 9286 9 332 9 3 7 8 9424 II 48 5 7 i3 5 7 58 58o4 5849 58 9 5 5 9 4o II 49 9471 9617 9 563 9609 9666 9702 IO 49 5 9 86 6o32 6077 6i23 6168 6214 10 5o 9.809748 9794 9 84o 9887 99 33 9979 9 5o 9 .82625 9 63o5 635o 63 9 6 644 1 64.8 7 9 5i 9.810026 0071 0118 0164 0210 0266 8 5i 6532 6678 6623 666 9 6 7 i4 6760 8 5f. o3o2 o34 9 o3 9 5 o44i o48 7 o533 7 62 68o5 685i 6896 6 9 42 6 9 8 7 7 o33 7 53 0680 0626 0672 0718 0764 0810 6 53 7078 7124 7169 7216 7260 73o6 6 54 0867 0903 o 9 4 9 o 99 5 io4i 1087 5 54 735i 7397 7442 7 488 7 533 7 5 79 5 55 1 1 34 1180 1226 1272 i3i8 1 364 4 55 7624 7670 7716 7761 7 8o6 7861 4 56 i4io i45 7 i5o3 1 549 1696 i64i 3 56 7897 7942 7988 8o33 8o 79 8124 3 5 7 1687 1733 1780 1826 1872 1918 2 57 8170 8216 8261 83o6 835i 83 97 2 58 1964 2010 2066 2IO2 2149 2196 I 58 8442 8488 8533 85 79 8624 866 9 1 5 9 224l 2287 2333 2 3 79 2426 2471 O 5o 8716 8760 8806 8861 8897 8 9 42 60" 50" 40" 30" 20" 10" 71 60" 50" 40" oO" 20" 10" j Co-tangent of 57 Degrees. 2 Co-tangent of 56 Degrees. 2 P Part 5 ]// ~" 3// 4 " 5// 6 " 7// 8// 9 " il tf l " 2 " 3// 4// 5 " 6 " 7// 8> 9 " l { 5 9 14 19 23 28 33 37 42 || 1 - rart { 5 9 14 J8 23 27 32 ,17 41 58 LOGARITHMIC SINES. c Sine of 34 Degrees. .3 Sine of 35 Degrees. r~ r*. 0" 10" 20" | 30" 40" 50" m 0" 10" 20" [ 30'' ] 40" 50" O 9.747562 7 5 9 3 7 624 7 655 7686 7718 5 9 o 9 .7585 9 i 8621 865i 868i|8 7 i2 8742 5 9 I 7749 7780 7811 7842 7874 79 o5 58 i 8772 8802 8832 886218892 8 9 22 58 79 36 79 6 7 7998 8o3o 8061 8o 9 2 5 7 2 8 9 5 2 8982 9 OI2 9O42 9 7 2 9 IO2 5? 3 4 8is3 63ic 8i54 834i 8i858 2I 6 83728403 8248 8434 8279 8466 56 55 3 4 9 l32 9 3l2 9162 9342 9 I 9 2 9 3 7 2 9222 94O2 9252 9 432 9 282 9 462 56 55 5 84 9 7 8528 855 9 !85 9 o 8621 865 2 54 5 9 4 9 2 9522 9 552 9 582 9 6l2 9 642 54 6 8683 8 7 i4 87458777 8808 883 9 53 6 9 672 9702 9732 9 7 62 9792 9822 53 8 8870 9 o56 8901 9 o87 8932 8963 9118 9149 8 99 4 9 i8o 9025 9 2I2 52 5i 7 8 9 852 9 . 76oo3i 9881 0061 9911 oo 9 i 9941 0121 997i oi5i ... I 0181 5 2 5! 9 9 243 9 2 7 4 9 3o5 9 336 9 36 7 9 3 9 8 5o 9 02 1 1 0240 0270 o3oo o33o o36o 5o 10 9-74942 9 9 46o 9 4 9 i 9 522 9 553 9 584 49 IO 9.760390 0420 o45o o48o o5o 9 o539 4 9 ii 9 6i5 96469677 9708 97 3 9 977 48 ii o56 9 o5 99 0629 o65 9 o68 9 o 7 i8 48 12 9 8oi 9832 9 863 9894 992 5 99 56 47 12 o 7 48 0778 0808 o838 0868 0898 4? i3 99 8 7 ..18 ..48 79 .110 .i4i 46 i3 9 2 7 9 5 7 0987 IOI 7 io4 7 io 7 6 46 i4 9.750172 0203 0234 0265 O2 9 6 0327 45 i4 1106 n36 1166 1196 1225 1255 45 i5 o358 o38 9 0420 o45 1 0482 o5i2 44 i5 1285 i3i5 1 345 i3 7 4 i4o4 i434 44 1.6 o543 o5 7 4 o6o5 o636 o66 7 o6 9 8 43 16 1 464 i4 9 4 i523 i553 !583 i6i3 43 l l o 7 2 9 0760 070,1 0821 o852 o883 42 7 1642 1672 1702 I 7 32 1761 1791 42 18 o 9 i4 o 9 45 o 9 76 1007 io3 7 1068 4i 18 1821 i85i 1880 1910 i 9 4o 1969 4i J 9 io 99 u3o 1161 1192 1222 1253 4o J 9 i 999 2O2 9 2059 2088 2118 2148 4o 20 9 . 7 5i 2 84 i*3z5 1 346 i3 77 i4o 7 i438 3 9 20 9 . 762177 2207 2237 226 7 2296 2326 39 21 i46 9 i5oo i53i i56i l5 9 2 1623 38 21 2356 2385 24i5 2445 2474 25o4 38 22 i654 i685 I 7 i5 i 7 46 I 777 1808 37 22 2534 2563 2593 2623 2652 2682 37 23 i83 9 i86 9 I 9 00 1931 I 9 62 I 99 2 36 23 2712 2741 2771 2801 2 83o 2860 36 24 2023 2o54 2o85 2Il5 2146 2I 77 35 24 2889. 2 9 I 9 2949 2 97 8 3oo8 3o38 35 25 22O8 2238 226 9 23oo 2 33o 2 36i 34 25 3067 3o 9 7 3i26 3i56 3i86 32i5 34 26 23 9 2 2423 2453 2484 25i5 2545 33 26 3245 32 7 4 33o4 3333 3363 33 9 3 33 27 2576 2607 26372668 26 99 2 7 2 9 32 27 3422 3452 348i 35n 354o 35 7 o 3^ 28 2760 2 79 I 2822 2852 2883 2 9 l4 3i 28 36oo 362 9 365 9 3688 3 7 i8 3 7 4 7 3i 29 2 9 44 2 97 5 3oo5 3o36 3o6 7 3o 9 7 3o 2 9 3777 38o6 3836 3865 38 9 5 3 9 25 3o 3o 9.753128 3i5 9 3i8 9 3220 3 2 5i 3281 2 9 3o 9.763954 3 9 84 4oi3 4o43 4072 4102 29 3i 33i2 3342 33 7 3 34o4 3434 3465 28 3i 4i3i 4i6i 4190 4220 4249 4279 28 32 34 9 5 3526 355 7 358 7 36i8 3648 27 32 43o8 4338 436 7 43 9 6 4426 4455 27 33 36 79 3710 3 7 4o 3 77 i 38 OI 3832 26 33 4485 45i4 4544 45 7 3 46o3 4632 26 34 3862 38 9 3 3 9 23 3 9 54 3 9 85 4oi5 25 34 4662 46 9 i 4 7 20 4 7 5o 4779 4809 25 35 4o46 4076 4107 4i3 7 4i68 4i 9 8 24 35 4838 4868 48 97 4926 4956 4 9 85 24 36 422 9 4269 4290 4320 435i 438i 23 36 5oi5 5o44 5o 7 4 5io3 Dl32 5i6 2 23 3? 44i2 4442 44 7 3 45o3 4534 4564 22 37 Sigi 5221 5 2 5o 52 79 53o 9 5338 22 38 45 9 5 46 2 5 4656 4686 4 7 i 7 4 7 4 7 21 38 536 7 53 97 5426 5456 5485 55i4 21 3 9 4778 48o8 483 9 4869 4 9 oo 4 9 3o 2O 3 9 5544 5573 56o2 5632 566i 56go 20 4o 9.7549,60 4991 5O2I 5o5 2 5o82 5n3 I 9 4o 9.765720 5 7 4 9 5778 58o8 583 7 5866 1 9 4i 5i43 5i 7 3 52o4 5 2 34 5 2 65 52 9 5 18 4: 58 9 6 5 92 5 5 9 54 5 9 84 6oi3 6042 18 42 53 2 6 53565386 54i7 5447 5478 *7 42 6072 6101 6i3o 6i5 9 6189 6218 17 43 55o8 5538 556 9 55 99 562 9 566o 16 43 6247 6277 63o6 6335 6364 63 9 4 16 44 56go 5721 5 7 5i 5 7 8i 58125842 i5 44 6423 6452 648 1 65n 654o 656 9 i5 45 58 7 2 SgoS 5 9 33 5 9 63 5 99 4 6024 i4 45 65 9 8 6628 665 7 6686 6715 6 7 45 i4 46 6o54 6o85 6n5 6i45 6176 6206 i3 46 6 77 4 68o3 6832 6862 6891 6920 i3 47 6 2 36 6267 62 97 632 7 6358 6388 12 47 6949 6 97 8 7 oo8 7 o3 7 7066 7 o 9 5 12 48 64i8 6448 6479 65o 9 653 9 65 7 o II 48 7124 7i54 7 i83 7 2I2 7241 7270 I I 4 9 6600 663o 6660 66 9 i 6 7 2i 6 7 5i IO 49 73oo 7329 7 358 7 38 7 74i6 7445 10 5o 9.756782 6812 6842 6872 6 9 o3 6 9 33 9 5o 9.76747 5 75o4 7 533 7 562 7 5 9 i 7620 9 5i 6 9 63 6 99 3 7023 7 o54 7 o84 7 n4 8 5i 7649 7 6 79 77 o8 7737 7766 779 5 8 5i 7 i44 7175 72o5 7235 7265 7295 7 52 7824 7 853 7 88 2 79 I2 79 4i 7 9 7 o 7 53* 7326 7 356 7386 7 4i6 7446 7 4 77 6 53 7999 8028 8o5 7 8086 8n58i44 6 54 7 5 7 75377567 7 5 97 7 62 77 658 5 54 8173 82o3 8232 826l 82908319 5 55 7688 7718, 77 48 7778 78087839 4 55 8348 83 77 84o6 8435 8464 84g3 4 56 7869 7899 79 2 9 7 9 5 9 7989 8019 3 56 8522 855i 858o 8600 86388668 3 5 7 58 8o5o 8 2 3o 8o8o8iio|8i4o 8260829018321 8170 8200 835i'838i 2 I 57 58 8697 8871 87268755 8 9 oo 8 9 2 9 8 7 84 8 9 58 88i38842 898-7 9016 2 I 5 9 3411 844i 8471 85oi 853i 856i O 5 9 9o45j9o 7 4 9 io3 9 i3s 9161 9 i 9 o 60" | 50" | 40" | 30" j 20" 10" ff 60" 1 50' 40" | 30" | 20" | 10" C Co-sine of 55 Degrees. Jo % Co-sine of 54 Degrees. ~ C t" 2" 3" 4" 5" 6" 7" 8" 9" irl 3 6 9 12 15 18 21 25 28 P P-,rt5 l " ~" 3 " 4// 5 " fi " ? ' 8 '' 9 " .\:Z 9 12 15 18 21 24 27 LOGARITHMIC TANGENTS. 50 jj Tangent of 34 Degrees. a \ Tangent of 35 Degrees. * 0" | 10" 1 20" | 30" 40" | 50" 3 j 0" | 10" 20" 30" | 40" | 50" I 57128987 9,260 9033(9078 9 3o5 9 35i 9 I24 9 3 9 6 9 i6 9 9 2i5 9 44^ 9 48 7 5 9 58 O I 9.845237 5496 554o 53i6 5585 536i 563o 5406 56 7 5 545 1 5 7 20 5 9 58 2 9 532 9 5 7 8 9 623 9 66 9 97 i4 97 5 9 5 7 2 5 7 64 58o 9 5854 58 99 5944 5 9 88 5 7 3 9805 985o 9 8 9 5 <; 9 4i 99 86 ..32 56 3 6o33 6078 6i23 6168 6212 625 7 56 4 5 9.830077 0349 OI22 o3 9 5 oi68|o2i3 o44o o485 0258 o53i o3o4 o5 7 6 55 54 4 5 63o2 65 7 o 634 7 66i5 6660 6436 6 7 o5 648i 6 7 5o 6526 ij 6 0621 0667 0712 o 7 5 7 o8o3 o848 53 6 6839 6884 6 9 2 9 6973 7 oi8 7 o63 53 7 0893 o 9 3 9 o 9 84 I02 9 io 7 5 II2O 52 7 7108 7 l52 7 J 97 7 242 7 28 7 7 33i 5a 8 n65 I2II 1256 i3oi 1 347 i3 9 2 5i 8 7 3 7 7421 7465 7 5io 7 555 7 6oo 5i 9 i43 7 i483 i5 2 8 i5 7 3 i6i 9 1 664 5o 9 7644 7 68 9 7734 7779 7 823 7868 5o 10 9.831709 i 7 55 1800 i845 1891 i 9 36 49 10 9.847913 79 5 7 8002 8o4 7 8o 9 2 8i36 49 ii 1981 2026 2072 2II 7 2162 2208 48 ii 8181 8226 82 7 83i5 836o 84o5 48 12 2253 2298 2343 2 38 9 2434 2479 47 12 8449 84 9 4 8539 8583 8628 86 7 3 47 i3 2525 2570 26i5 2660 2 7 o6 2 7 5l 46 i3 8717 8 7 62 88o 7 885i 88 9 6 8 9 4i 46 i4 2796 2842 2887 2 9 32 2977 3o23 45 i4 8 9 86 9 o3o 9 o 7 5 9 I20 9 i64 9209 45 i5 3o68 3n3 3i58 3204 324 9 3 29 4 44 i5 9 254 9 2 9 8 9 343 9 388 9 432 9477 44 16 3339 3385 343o 3475 3520 3566 43 16 9 522 9 566 9 6n 9 656 97 oo 9745 43 17 36n 3656 3701 3 7 4 7 3 79 2 383 7 42 17 979 9 834 9 8 79 99 24 99 68 ..i3 42 18 3882 3927 3 97 3 4oi8 4o63 4io8 4i 18 9 .85oo57 OIO2 0147 oi 9 i 0236 0281 4i I 9 4i54 4199 4244 428 9 4334 438o 4o i 9 o325 o3 7 o o4i5 o45 9 o5o4 o548 4o 20 9.834425 4470 45i5 456i 46o6 465i 39 20 9 .85o5 9 3 o638 0682 7 2 7 77 2 0816 39 2 I 4696 474i 4 7 8 7 4832 48 77 4 9 22 38 21 0861 o 9 o5 o 9 5o o 99 5 io3 9 1084 38 22 4967 5oi2 5o58 5io3 5i48 5i 9 3 37 22 II2 9 u 7 3 1218 1262 i3o 7 i352 37 23 5 2 38 5284 532 9 53 7 4 5419 5464 36 23 i3 9 6 i44i i485 i53o i5 7 5 i6i 9 36 24 55o 9 5555 56oo 5645 56oo 5 7 35 35 24 1 664 I 7 o8 i 7 53 i 797 1842 1887 35 25 5780 5826 5871 5 9 i6 5 9 Gi 6006 34 25 i 9 3i i 97 6 2O20 2o65 2110 2i54 34 26 6o5i 6096 6142 6187 6232 6277 33 26 2i 99 2243 2288 2332 2 3 77 2422 33 27 6322 636 7 64 1 2 6458 65o3 6548 32 27 2466 25ll 2555 2600 2644 268 9 32 28 65 9 3 6638 6683 6 7 28 6773 68i 9 3i 28 2 7 33 2 77 8 2823 2 86 7 2 9 I2 2 9 56 3i 29 6864 6909 6 9 54 6999 7 o44 7 o8 9 3o 2 9 3ooi 3o45 3o 9 o 3i34 3i 7 9 3223 3o 3o 9.83 7 i34 7179 7225 7 2 7 7 3i5 736o 29 3o 9.853268 33i3 335 7 3402 3446 34 9 i 29 3i 74o5 745o 7 4 9 5 7 54o 7 585 763o 28 3i 3535 358o 3624 366 9 3 7 i3 3 7 58 28 32 33 7 6 7 5 7946 7721 7991 77 66 8o36 781 1 7856 8o8i|8i26 7901 8171 2 7 26 32 33 38o2 4o6 9 384 7 4n4 38 9 i 4i58 3 9 36 4203 3980 4247 4o25 27 26 34 8216 8261 83o 7 835 2 83 97 8442 25 34 4336 438i 4425 4470 45i4 455 9 25 35 848 7 8532 85 77 86228667 8712 24 35 46o3 4648 46 9 2 4 7 3 7 4781 4826 24 36 8 7 5 7 8802 884 7 88 9 2 8937 8 9 8a 23 36 48 7 o 4 9 i5 4 9 5 9 5oo4 5o48 5o 9 3 23 37 9027 9072 9117 9162 9 20 7 9 252 22 37 5i3 7 5i8 2 5226 52 7 I 53i5 536o 22 38 9297 9 343 9388 9 433 9 4 7 8 9 5 2 3 21 38 54o4 544 9 54 9 3 553 7 5582 5626 21 39 9 568 9613 9 658 97 o3 9748 979 3 2O 39 5671 5 7 i5 5 7 6o 58o4 5849 58 9 3 2O 4o 9 .83 9 838 9883 99 28 9973 ..18 ..63 I 9 4o 9 .855 9 38 5 9 82 6026 6o 7 i 6n5 6160 I 9 4i 9.840108 oi53 oi 9 8 0243 0288 o333 18 4i 6204 624 9 62 9 3 6338 6382 6426 18 42 o3 7 8 0423 o468 o5i3 o558 o6o3 I 7 42 6471 65i5 656o 66o4 6649 66 9 3 17 43 o648 o6 9 3 o 7 3 7 7 82 082-7 0872 16 43 6 7 3 7 6 7 82 6826 68 7 i 6 9 i5 6 9 5 9 16 44 0917 0962 I00 7 IO52 io 97 1142 i5 44 7004 7 o48 7093 7 i3 7 7 l82 7226 i5 45 1187 1232 I2 77 1322 i36 7 l4l2 i4 45 7270 7 3i5 7 35 9 7 4o4 7448 74 9 2 i4 46 i45 7 l5o2 I 547 l5 9 2 i63 7 1682 i3 46 7 53 7 7 58i 7626 7 6 7 o 77 i4 77 5 9 i3 47 1727 1771 1816 1861 I 9 o6 i 9 5i 12 47 7 8o3 7 848 7 8 9 2 7936 7981 8025 12 48 1996 2041 2086 2l3l 2I 7 6 2221 II 48 8o6 9 8n4 8i58 8203 824-7 82 9 i II 49 2266 23ll 2355 240O 2445 24 9 10 49 8336 838o8424 846 9 85i3 8558 10 5o 9-842535 258o 2625 26 7 2 7 l5 2760 9 5o 9 . 8586o2 8646 86 9 i 8 7 35 8779 8824 9 5i 28o5 2849 2 8 9 4 2 9 3 9 2 9 84 3o2 9 8 5i 8868 8 9 I2 8 9 5 7 9001 9045 9 o 9 o 8 52 3o 7 4 3n 9 3i64 32O 9 3 2 53 32 9 8 7 52 9i34 9 i 7 8 9 223 926-7 9811 9 356 7 53 54 3343 36i2 3388 365 7 3433 3 7 02 3478 3 7 4 7 3523 3792 3568 383 7 6 5 53 54 9 4oo 9444 9489 9666 9 7 io9 7 55 9 533 9799 9 5 77 9 843 9622 9 888 5 5 55 3882 3927 3 97 i 4oi6 4o6i 4io6 4 55 99352.9976 ..a i ..65 . 109 . i54 4 56 4i5i 4196 424r 4285 433o 43 7 5 3 56 9.860198 0242 O28 7 o33i o3 7 5 0420 3 57 4420 4465|45io 4554 45 99 4644 2 57 o464 o5o8 o552 o5 97 o64i o685 2 58 468 9 47344779 4823 4868 4 9 i3 I 58 073007740818 0862 0907^961 I 5 9 4 9 58 5oo3 5o48 5o 9 2 5i3 7 5i82 O Sg o 99 5 io4o io84 1128 |II 7 2 I2I 7 O GO" | 50" 40" 30' 20" 10" g 60" | 50" 40" 30" i 20" 10" a Co-tangent of 55 Degrees. Co-tangent of 54 Degrees. i pp < 1" 2" 3" 4" 5" 6" 7" 8" 9" , ( 1" 2" 3'' 4" 5" 6" 7" 8" 9" iri \ 5 9 14 18 23 27 32 36 41 / 4 9 13 18 C2 27 31 3(5 10 GO LOGARITHMIC SINES?. 1 jt | Sine of 36 Degrees. _g Sine of 37 Degrees. SJ 0" | 10" | 20" 30' 40" 50" s 0" | 10" 20" 30" 40" 50" i 9.7692:9 9 3 9 3 9248)9277 9 3o6 9 42i 9 45o 9 47 9 9335936459 95089537 58 I 9- 779463 9 63i 9 4 9 i 9 65 9 9619 9 686 9 54 7 97 i4 9 5 7 5 97 42 9 6o3 977 5 9 58 2 9 566 9 5 9 5 9 624 9 653 9682 9 7 n 57 2 979 s 9826 9 854 9 882 9910 99 38 57 3 97409769 97989 82 7 9 856 9884 56 3 9966 9998 . .21 .149 77 .io5 56 4 99189942 997 i . . . . ..29 ..58 55 4 9.780188 0161 oi8 9 0216 0244 0272 55 6:9.770087 0116 oi45 oi 7 3 02O2 023l 54 5 o3oo 0828 o356 o384 o4n o43 9 54 I 6 0260 028 9 0818 o34 7 o3 7 6 o4o4 53 6 0467 0495 o523 o55i o5 7 8 0606 53 7 o433 0462 o4 9 i O520 o549 o5 7 7 52 7 o634 0662 0690 0-718 o 7 45 o 77 3 5 2 8 0606 o635 o664 o6 9 3 7 22 o 7 5o 5i 8 0801 0829 o85 7 0884 0912 o 9 4o 5i 9 779 0808 08870866 o8 9 5 0928 5o 9 0968 0996 1023 io5i io 79 1107 5o 10 9 . 77 o 9 52 o 9 8i IOIO io3 9 io6 7 1096 49 10 9 . 781184 1162 II 9 O 1218 1246 1278 49 ii 1125 n54 n83 I2II 1240 1269 48 ii 1801 1829 i35 7 1 384 l4l2 i44o 48 12 1298 1826 i355 i384 i4i3 i44i 47 12 i468 i495 i523 i55i i5 7 8 1606 47 i3 1470 i4 99 1628 i556 i585 i6i4 46 18 1 634 1662 i68 9 i 7 i 7 i 7 45 1772 46 i4 1 643 1671 1700 1729 i 7 58 i 7 86 45 i4 1800 1828 i856 1888 i 9 n I9 3 9 45 i5 i8i5 1 844 1872 1901 1980 i 9 5 9 44 i5 i 9 66 i 99 4 2O22 204 9 20 77 2IO5 44 16 i 9 8 7 2016 2o45 20 7 3 2IO2 2181 43 16 2182 2160 2188 22l5 2243 2271 43 17 2188 2217 2245 22 7 4 23o3 42 17 22 9 8 2826 2354 2881 24o 9 2437 42 18 233i 2860 2 38 9 24l 7 2446 24 7 5 4i 18 2464 2492 252O 254 7 25 7 5 2602 4i Z 9 2 5o3 2532 2 56i 258 9 2618 2646 4o i 9 2680 2658 2685 2-718 2 7 4l 2768 4o 209.772675 2704 2782 2 7 6l 2 79 2818 3 9 20 9 . 7 82 79 6 2823 2 85i 2 8 79 2 9 o6 2 9 34 3 9 21 2 84 7 2875 2 9 04 2 9 33 2 9 6l 2990 38 21 2961 2989 3oi 7 3o44 3o 7 2 3o 99 38 22 3oi8 8047 3o 7 6 3io4 3i33 3i6i 37 22 8127 3i54 8182 32IO 3 2 3 7 5265 37 23 8190 32I 9 8247 32 7 6 33o4 3333 36 23 32 9 2 3320 334 7 33 7 5 3402 343o 36 24 336i 33 9 o 34i8 344 7 34 7 6 35o4 35 24 3458 3485 35:3 354o 3568 35 9 5 35 25 3533 356i 35 9 o 36:8 364 7 36 7 5 34 25 3623 365o 36 7 8 3 7 o5 3 7 33 3 7 6o 34 26 3 7 o4 3 7 32 3 7 6i 3789 38i8 3846 33 26 3 7 88 38i5 3843 38 7 o 38 9 8 3 9 25 33 27 38 7 5 3 9 o3 3 9 32 3 9 6o 3 9 8 9 4oi 7 32 27 3 9 53 8980 4oo8 4o35 4o63 4o 9 o 82 28 4o46 4074 4io34i3i 4i6o 4i88 3i 28 4n8 4i45 4i 7 3 4200 4228 4255 3i 29 421-7 4245 42744302 433i 435 9 3o 29 4282 4810 433 7 4365 43 9 2 4420 3o 3o g. 77 4388 44i6 4445 4473 45oi 453o 2 9 3o 9. 784447 44 7 5 45o2 452 9 455 7 4584 29 3i 32 4558 4 7 29 458 7 4 7 5 7 46i54644 47864814 46 7 2 4842 4 7 oo 48 7 i 28 27 3i 82 4612 4776 463 9 48o4 466 7 483i 46 9 4 4858 4 7 2I 4886 4 9 i3 28 3 7 33 4899 4 9 28 4 9 564 9 85 5oi3 5o4i 26 33 4 9 4i 4968 4 99 5 5o23 5o5o 5o 7 8 26 34 5o 7 o 5o 9 8 5i26 5i55 5i83 5212 25 34 5io5 5i32 5i8 7 52i4 5242 25 35 6240 5268 52 97 53 2 5 5353 5382 24 35 526 9 5296 5324 535i 53 7 8 54o6 24 36 54io 5438 546 7 54 9 5 55 2 3 5552 23 36 5433 546i 5488 55i5 5543 55 7 o 23 37 558o 56o8 563 7 5665 56 9 3 5722 22 37 55 97 5624 5652 56 79 5 7 o6 5 7 34 22 38 5 7 5o 5 77 8 58o 7 5835 5863 5892 21 38 5 7 6i 5 7 88 58i6 5843 58 7 o 58 9 8 21 3 9 5 9 2O 5 9 48 5 977 6oo5 6o33 6061 2O 3 9 5 9 25 5 9 5 2 5 979 6oo 7 6o34 6061 2O 4o 4i 9 . 77 6o 9 o 6269 6118 6288 6i46 6i 7 5i62o3 63i6634463 7 2 6281 6401 18 4o 4i 9 . 7 86o8 9 6252 6116 62-79 6i43 63o 7 6i 7 o 6334 6i 9 8 6861 6225 6388 18 42 642 9 645 7 6485 65i46542 6570 17 42 64i6 6443 64 7 o 64 97 6525 6552 1-7 43 65 9 8 662 7 6655 6683 6 7 n 6789 16 43 65 7 9 6606 6634 6661 6688 6 7 i5 16 44 6 7 68 6 79 6 6824 6852 6880 6 9 o 9 i5 44 6 7 42 6 77 o 6 797 6824 685i 6878 i5 45 6 9 3 7 6 9 65 6 99 3 7 O2I 7 o5o 7078 i4 45 6 9 o6 6 9 33 6 9 6o 6 9 8 7 7 oi4 7042 i4 46 7106 7 i34 7162 -7191 7 2I 9 7247 18 46 7 o6 9 7 o 9 6 7 I23 7 i5o 7 i 77 7205 i3 47 7 2 7 5 7808 7 33i 7 35 9 7 388 74i6 12 47 7 232 7 25 9 7286 7 3i3 7 34o 7 36 7 12 48 7 444 7472 7 5oo 7 528 7 556 7 585 II 48 7 3 9 5 7 422 7 44 9 7 4 7 6 7 5o3 7 53o I I 49 7 6i3 7641 7 66 9 7697 77 25 77 53 10 49 7 55 7 7 585 7 6l2 7 63 9 -7666 7 6 9 3 10 5o 9-7777 8 i 7810 7 838 7 866 7 8 9 4 79 22 9 5o 9 . 7 8 77 2o 7747 7774 -7801 7 82 9 7 856 9 5i 79 5o 7978 8006 8o34 8062 8o 9 i 8 5i 7 883 -7910 79 3 7 79 64 799 1 8018 8 62 8119 8i47 8r 7 5 8208 8281 8269 7 52 8o45 80-72 8o 99 812-7 8i54 8181 7 53 8287 8816 8343 83 7 i 83 99 8427 6 53 8208 8235 8262 828 9 8816 8343 6 54 8455 8483 85n 853 9 856 7 85 9 5 5 54 83 7 o 83 97 8424 845 1 84 7 8 85o5 5 55 50 57 8624 8 79 2 8 9 6o 8652 8820 8 9 88 8680 8848 9 oi6 8 7 o8 8 7 36 88-768904 9044 9072 8 7 64 8 9 3 2 9 ioo 4 3 2 55 56 5 7 8532 86 9 4 8856 86698686 8721 8748 8883 8 9 io 86i3864o 8 77 588o2 866 7 882 9 8 99 i 4 3 2 58 9.128 9 i56 9 i83 9211 9289 9 26 7 I Co Do 9 oi8 9 o45 9 o 7 2 9 o 99 9 i26 91 53 I 5 9 9 2 9 5 9 323 9 35i 93-79 940-7 9 435 5 9 9 i8o 9 2o 7 9 234 9261^9288 9 3i5 O 60" 50" 40" 30" 20" 10" S 60" 50" 40" | 30" 20" 10" | ^ Co-sine of 53 Degrees. Co-sine of 52 Degrees. < 1" 2" 3" 4" 5" 6" 7" 8" 9" t it/ o" }/' 4" V fi" 7" 8" Q" ' P P-irt > 1 ~ d4 ' )0 ' 8 r.PartJ 3 6 9 11 14 17 20 23 26 i i . r an < ty K o -.i -11 if. iQ ^Q or- 4 9 13 18 22 26 31 35 40 || J - lart { 4 9 13 17 22 2G 31 35 39 LOGARITHMIC SINES. n [ Sine of 38 Degrees. c Sine of 39 Degrees. & [~ 0" | 10" 20" 30" 40" 50" & 0" 10" 20" | 30" | 40" 50" V 9.789342 9369 95o4 9 * 9665 9692 939619423 9450 955-7195849611 9 7 i 997 469 77 3 9477 9 638 9800 5 9 58 57 o I 2 9.798872 8898 9028 9054 918419210 8 9 24 9080 9236 8 9 5o8 97 6 9106 9132 9262 928-7 9002 9i58 93i3 5 9 58 57 3 9827 9804 9880 9 9 o 7j99 34 99 6i 56 3 9 33 99 365 9 3 9 i 9 4i 7 9443 9469 56 4 5 6 9988 9 .790149 o3io oi 7 6 o33 7 ..42 0203 o364 |..6 9 O230 0391 ..96 O25 7 o4i8 . 122 0284 o445 55 54 53 4 5 6 9495 9521 9547 9651196779703 98o6 9 832 9 858 9 5 7 3 97 28 9 884 9 5 99 97^4 9910 9625 97 8o 99 36 55 54 53 7 0471 0498 o525 o552 o5 79 0606 52 7 99629987 . .i3 ..3 9 ..65 52 8 9 o632 o 79 3 0659 0686 0820 o84 7 o 7 i3 o8 7 4 o 7 4o o 9 oi o 7 6 7 09 2 7 5i 5o 8 9 9.8001 17|oi43 0272 0298 0169 o324 oi 9 5 o35o 0220 o3 7 6 0246 o4oi 5i 5o 10 9. -790954 0981 1008 io34 1 06 1 1088 49 IO 9.800427 o453 0479 o5o5 o53i o556 49 1 1 ni5 n4a 1168 1195 1222 1249 48 ii o582 0608 o634 0660 0686 0-711 48 12 I2 7 5 1302 l32 9 i356 i382 1409 47 12 o 7 3 7 o 7 63 0789 o8i5 o84o 0866 47 i3 i436 i463 i48 9 i5i6 i543 1570 46 i3 0892 0918 0944 o 9 6 9 o 99 5 1 02 1 46 i4 i5 9 6 1623 i65o i6 7 6 I 7 o3 1730 45 i4 io4 7 io 7 3 1098 1124 n5o n 7 6 45 i5 i 7 5 7 i 7 83 1810 i83 7 i863 1890 44 i5 I2OI I22 7 1253 I2 79 i3o5 i33o 44 16 191-7 i 9 43 i 97 o 1997 2024 2o5o 43 16 i356 1 382 i4o8 i433 i45 9 i485 43 17 20 77 2104 2i3o 2l5 7 2184 22IO 42 17 i5n i536 i562 i588 i6i3 i63 9 42 18 223 7 2264 22 9 O 23l 7 2344 2370 4i 18 i665 1691 1716 I 7 42 1-768 1-794 4i 19 2 3 97 24z3 245o 24 77 25o3 253o 4o i 9 1819 i845 1871 :8 9 6 1922 i 9 48 4o 20 9 . 79 255 7 2583 26lO 2 636 2663 2690 3 9 20 9.8019-73 1999 2025 2o5i 20-76 2102 3 9 21 2 7 l6 2 7 43 2 77 2 79 6 2823 2849 38 21 2128 2i53 2I 79 22O5 2230 2256 38 22 28 7 6 2903 2929 2g56 2 9 82 3009 37 22 2282 230 7 2333 235 9 2384 24lO 37 23 3o35 3062 3o8 9 3ii5 3l42 3i68 36 23 2436 2461 2487 25l2 2538 2564 36 24 3i 9 5 3222 3248 32 7 5 33oi 33 2 8 35 24 2 58 9 26i5 2641 2666 2692 2 7 l8 35 25 3354 338i 34o 7 3434 346o 348 7 34 25 2 7 43 2 7 6 9 2794 2820 2846 28 7 I 34 26 35i4 354o | 356 7 35 9 3 3620 3646 33 26 2S 97 2Q22 2 9 48 2 97 4 2999 3o25 33 27 36 7 3 36 99 ! 3 7 26 3 7 5a|3 779 38o5 32 2 7 3o5o 3o 7 6 3l02 3l2 7 3i53 3i 7 8 32 28 3832 3858 3885 3911 3938 3 9 64 3i 28 3204 3229 3255 3281 33o6 3332 3i 29 3 99 i 4oi 7 4o44 40-70409-7 4l23 3o 2 9 335 7 3383 34o8 3434 345 9 3485 3o 3o 9 . 79 4i5o 4i 7 642o3 42294255 4282 2 9 3o 9.8o35n 3536 3562 358 7 36i3 3638 29 3i 43o8 4335 436i 4388 44i4 444i 28 3i 3664 368 9 3 7 i5 3 7 4o 3 7 66 3 7 9i 28 32 446 7 44934520 4546 45 7 3 4599 27 32 38i 7 3842 3868 38 9 3 3 9 i 9 3 9 44 2 7 33 4626 4652 46 7 8 4 7 o5 4 7 3i 4 7 58 26 33 3970 3 99 5 4021 4o46 4o 7 2 4o 97 26 34 4 7 84 48io483 7 4863 48 9 o 4916 25 34 4i23 4i48 4i 7 4 4199 4225 25 35 4942 4 9 6 9 4 99 5 5022 5o48 5o 7 4 24 35 4276 43oi 4327 4352 43 77 44o3 24 36 5ioi 5l2 7 5i54 5i8o 52o6 5233 23 36 4428 4454 4479 45o5 453o 4556 23 3 7 5259 5285 53 1 2 5338 5364 53 9 i 22 37 458i 46o 7 4632 465 7 4683 4 7 o8 22 38 54i 7 5443, 54 7 o 54 9 6 5522 554 9 21 38 4 7 34 4 7 5 9 4 7 84 48io 4835 486i 21 39 55 7 5 56oi 5628 5654 568o 5 7 o 7 2O 39 4886 /o3 7 4962 4988 5oi3 2O 4o 9 . 79 5 7 33 5 7 5 9 5 7 86 58i2 5838 5865 I 9 4o 9.8o5o3 9 |5o64 5^89 5n5 5i4o 5i65 I 9 4i 58 9 i 5 9 i 7 5 9 43 5 97 o 5 99 6 6022 18 4i 5191 5 2 i6 5242 526 7 5292 53i8 18 42 6o4 9 6o 7 5 6101 6127 6i54 6180 17 42 5343 5368 53 9 4 5419 5444 54 7 o "7 43 6206 6233 625 9 6285 63n 6338 16 43 54 9 5 5520 5546 55 7 i 55 97 5622 16 44 6364 63 9 o ! 64i6 6443 646 9 64g5 i5 44 564 7 56 7 3 5698 5 7 23 5 7 48 5 77 4 i5 45 652i 654 7 65 7 4 6600 6626 6652 i4 45 5 799 5824 585o 58 7 5 5900 5926 i4 46 66 79 6 7 o5 6 7 3i 6 7 5 7 6 7 83 6810 i3 46 5 9 5i 5 97 6 6002 602-7 6o52 6o 77 i3 47 6836 68626888 6 9 i4 6 9 4i 6 9 6 7 12 47 6io3 6128 6i53 6i79 6204 6229 12 48 6 99 3 7 oi 9 ' 7 o45 7 7 2 7 o 9 8 7124 II 48 6254 6280 63o5 633o 6355 638i II 49 7 i5o 7 i 7 6 -7202 7 22 9 7 255 7281 IO 49 64o6 643 1 6456 6482 65o 7 6532 10 5o 9-797 3 o 7 7 333 | 7 35 9 7 386 7 4l2 7438 9 5o 9.806557 6583 6608 6633 6658 6684 9 5i 7 464 7 4 9 o; 7 5i6 7 542 7 56 9 7 5 9 5 8 5i 6 7 09 6 7 34 6759 6 7 85 6810 6835 8 52 7621 7 64 7 7 6 7 3 7699 77 25 77 5i 7 52 6860 6885 6911 6 9 36 6961 6986 7 53 7777 7 8o4 7 83o 7 856 7882 7908 6 53 7 OII 7 o3 7 7062 7 o8 7 7 II2 7 i3 7 6 54 7934 79 6o 79 86 8012 8o38 8o65 5 54 7 i63 7188 7213 7 238 7 263 72 88 5 55 8091 81178143 8i6 9 8195 8221 4 55 7 3i4 733 9 7364 7 38 9 7 4i4 743 9 4 56 8 2 4 7 8273,8299 8325 835i 83 77 3 56 7 465 74 9 o 7 5i5 7 54o 7 565 7 5 9 3 57 84o3 842 9 ! 8455848i 85o88534 2 57 7 6i5 7641 7666 -7691 77 i6 77 4i 2 58 856o 858686128638 8664j86 9 o I 58 77 66 779 1 7816 7 842 7 86 7 7 8 9 2 I 5 9 8 7 i6 874287688794 8820J8846 O _5o_ 79 i 7 79 4s 79 6 7 -7992 8oi 7 8042 O 60" 50" 40" | 30" 20" | 10" S" 60" 50" | 40" | 30" 20" 10" j Co-sine of 51 Degrees. Co-sine of 50 Degrees. i 1. T >art$ l " Z " 3/ 4" 5" 6" 7" 8" 9" ( 1" 2" 3" 4" 5" 6" 7" 8" 9" 1 rl \ 358 11 13 16 19 21 24 I trt j 3 5 8 10 13 15 18 20 23 LOGARITHMIC TANGENTS. g | Tangent of 38 Degrees. d Tangent of 39 Degrees. 1 SJ 0" , 10" 20" 30'' 40" 50" * 0" | 10" 20" 30" | 40" 50" 019.892810 2 8 97 2 9 40 2 9 83 3o2 7 5 9 9 . 9 o836 9 |84i2 8455 84 9 8 854i 8584 5 9 i 3o 7 o 3n4 3i5 7 320O 3244 3a8 7 58 i 86288671 8714 8 7 5 7 8800 8843 58 2 333i 33 7 4 34i 7 3461 35o4 354 7 57 2 8886 8929 8972 9 oi5 9 o58 O.IOI 5 7 3 35 9 i 3634 36 7 8 3 7 2I 3 7 64 38o8 56 3 9 i44 9187 9230 9273 9 3i6 9 35 9 56 4 385i 38 9 4 3 9 38 3 9 8i 4o 2 5 4o68 55 4 9 402 9445 9 488 9 53i 9 5 7 4 9 6i 7 55 5 4ni 4i55 4198 4241 4285 4328 54 5 9 66o 97 o3 97 46 9789 9 83 2 9 8 7 5 54 6 4372 44i5 4458 45o2 4545 4588 53 6 99 i8 9961 ...5 ..48 . . 9 i .i34 53 7 4632 46 7 5 4718 4 7 62 48o5 4848 52 7 9 . 9 ioi 77 O220 0263 o3o6 o34 9 o3 9 2 5 2 8 48 9 2 4 9 35 4979 5022 5o65 5io 9 5i 8 o435 o4 7 8 0521 o564 060-7 o65o 5i 9 5i5 2 5i 9 5 5 2 3 9 5282 5325 536 9 5o 9 o6 9 3 o 7 36 779 0822 o865 o 9 o8 5o IO 9 .8 9 54i2 5455 54 99 5542 5585 562 9 49 IO 9.910951 0994 1037 1080 1123 1166 t ii 5672 5 7 i5 5 7 59 58o2 5845 588 9 48 ii I2O 9 1262 I2 9 5 i338 i38i 1424 48 12 5 9 3 2 5 97 5 601916062 6io5 6i4 9 47 12 i46 7 i5io i553 i5 9 6 i63 9 1682 47 i3 6l 9 2 6235 62 7 8 63226365 64o8 46 i3 I 7 25 1768 1810 i853 i8 9 6 i 9 3 9 46 i4 6452 64 9 5 6538 6582 6625 6668 45 i4 I 9 82 2O25 2068 21 II 2 1 54 2197 45 i5 6712 6 7 55 6 79 8 6842 6885 60.28 44 i5 2240 2283 2326 2369 24l2 2455 44 16 6971 7 oi5 7 o58 7 IOI 7 i45 7 i88 43 16 2 4 9 8 254i 2584 262 7 26 7 2 7 i3 43 17 7231 7 2 7 5 7 3i8 7 36i 7 4o4 7448 42 1 7 2 7 56 2799 2842 2&85 ! 2 9 28 29-71 42 18 7491 7534 7 5 7 8 -7621 7 664 777 4i 18 3oi4 3o57 3ioo 3i43j3i85 3228 4i i 9 77 5i 7794 7 83 7 7 88i -7924 7 9 6 7 4o i 9 32 7 I 33i4 335 7 34oo 3443 3486 4o 20 9 .8 9 8oio 8o54 8o 97 8i4o 8:83 822 7 39 20 9 . 9 i352 9 3572 36i5 3658 3 7 oi 3 7 44 39 21 82 7 83i3 835 7 84oo 8443 8486 38 21 3787 383o 38 7 3 3 9 i6 3 9 5 9 4ooi 38 22 853o 85 7 3 8616 865 9 8 7 o3 8 7 46 37 22 4o44 4o8 7 4i3o 4i 7 3 4216 425 9 3 7 23 8 7 8 9 8832 88 7 6 8 9 i 9 8 9 62 9 oo5 36 23 4302 4345 4388 443 1 44 7 4 45i 7 36 24 9049 9 9 2 9 i35 9 i 7 8 9 222 9 265 35 24 456o 46o3 4645 4688 4 7 3i 4 7 74 35 25 93o8 9 35i 9 3 9 5 9 438 9 48i 9 524 34 25 48i 7 486o 4 9 o3 4 9 46 4 9 8 9 5o32 34 26 9 568 9 6n 9 654 9 6 97 97 4i 97 84 33 26 5o 7 5 5n8 5i6i 52o3 5246 528 9 33 27 0827 0870 QQl4 0,0,57 43 32 27 5332 5375 54i8 546i 55o4 554 7 32 7 / 28 v / 9.90008-7 7 / oi3o yy*** oi 7 3 77 / 02l6 025 9 o3o3 3i / 28 55 9 o ^/^i i -j 5633 56 7 5 5 7 i8 5 7 6i * *** / 58o4 3i 2 9 o346 o38 9 0432 o4 7 6 o5i 9 o562 3o 2 9 584 7 5890 5 9 33 5 97 6 6019 6062 3o 3o 9.900605 o648 o6 9 2 o 7 35 o 77 8 0821 2 9 3o 9.916104 6147 6i 9 o 6233 62-76 63i 9 2 9 3i 0864 o 9 o8 o 9 5i o 99 4 io3 7 1081 28 3i 6362 64o5 6448 64 9 i 6533 65 7 6 2$ 32 1124 1x67 1210 1253 I2 97 i34o 2 7 32 66i 9 6662 6 7 o5 6 7 48 6 79 i 6834 27 33 i383 1426 i46 9 i5i3 i556 i5 99 26 33 6877 6919 6 9 62 7 oo5 7 o48 7 o 9 i 26 34 1642 i685 I 7 2 9 I 77 2 i8i5 i858 25 34 7177 -7220 -7262 7 3o5 7 348 25 35 1901 i 9 44 i 9 88 2031 20 7 4 2II 7 24 35 7 3 9 i 7434 7477 7 52O 7 563 7 6o5 24 36 2160 2204 224 7 22 9 O 2333 2 3 7 6 23 36 7 648 7691 7734 7777 -7820 7 863 23 37 2420 2463 25o6 254 9 25 9 2 2635 22 37 7 9 o6 7948 799 1 8o34 8o 77 8120 22 38 2 6 79 2-722 2 7 65 2808 285i 28 9 4 21 38 8i63 8.206 8248 82 9 I 8334 83 77 21 3 9 2 9 38 2 9 8i 3024 3o6 7 3no 3i53 20 39 8420 8463 85o6 8548 85 9 i 8634 2O 4o 9 . 9 o3i 97 324o 3283 33 2 6 336 9 34i2 I 9 4o 9.918677 8720 8 7 63 88o5 8848 88 9 i I 9 4i 3456 34 99 3542 3585 36 2 8 36 7 i 18 4i 8 9 34 8977 9 O2O 9 o63 9io5 9 i48 18 42 3 7 i4 3 7 58 38oi 3844 388 7 3 9 3o 17 42 9191 9 234 9277 9 32O 9362 9 4o5 i 7 43 3 97 3 4oi6 4o6o 4io3 4i46 4i8 9 16 43 9448 9491 9 534 9 5 77 9619 9 662 16 44 4232 42 7 5 43i8 4362 44o5 4448 i5 44 9705 9748 979 i 9 834 9 8 7 6 99 i 9 i5 45 44 9 i 4534 45 77 4620 4663 4707 i4 45 9962 ...5 . 48 . . 9 i .i33 .176 i4 46 4 7 5o 4793 4836 48 79 4 9 22 4 9 65 i3 46 9.920219 0262 o3o5 o348 o3 9 o o433 i3 4 7 5oo8 5o52 5o 9 5 5i8i 5224 12 47 0476 oSig o562 0604 o64 7 o6 9 o 12 48 526 7 53io 5353 53 97 544o 5483 II 48 o 7 33 0776 08 1 9 0861 o 9 o4 o 9 4 7 II 49 55 2 6 556 9 56i2 5655 56 9 8 5 7 4i IO 49 0990 io33 io 7 5 1118 1 161 1204 10 5o 9 . 9 o5 7 85 58 2 8 58 7 i 5 9 i4 5 9 5 7 6000 9 5o 9.921247 1289 i332 i3 7 5 i4i8 1 46 1 9 5i 6o43 6086 6129 61-72 6216 625 9 8 5i i5o3 1 546 i58 9 i63 2 i6 7 5 1717 8 52 63o2 6345 6388 643 1 6474 65 1 7 7 52 1760 i8o3 1 846 i88 9 i 9 3i 19-74 7 53 656o 66o3 6646 66 9 o 6 7 33 6776 6 53 2017 2060 2103 2145 2188 223l 6 54 68i 9 6862 6905 6 9 48 6 99 i 7034 5 54 2274 23i6 235 9 2402 2445 2488 5 55 7 o 77 7120 7 i63 7 20 7 7 25o 72 9 3 4 55 253o 2573 2616 265 9 2 7 O2 2 7 44 4 56 7 336 7379 7 422 7465 7 5o8 7 55i 3 56 2787 283o 2 8 7 3 2 9 l5 2 9 58 3ooi 3 57 7 5 9 4 7 63 7 7 68o 7723 77 66 7 8o 9 2 57 3o44 3o8 7 3i2 9 3l 7 2 32i5 3258 2 58 7 853 7 8 9 6 79 3 9 79 82 8025 8068 I 58 33oo 3343 3386 342 9 34 7 i 35i4 I 5 9 8xi i 8i54 8i97 8240 8283 8326 o 5 9 355 7 36oo 3642 3685 3728 3 77 i 60' 50" | 40" 30" 20-- 10" a* 60" 50" 40" 30" 20" 10" t! Co-tangent of 5 1 Degrees. Co-tangent of 50 Degrees. 2 p r t ( 1" 2" 3" 4" 5" 6" 7" 8" 9" v C I" 2" 3" 4" 5" 6" 7" 8" 9" irl \ i 9 13 17 22 26 30 35 39 I.Fart^ 4 9 13 17 01 26 30 34 39 LOGARITHMIC SINES. d Sine of 40 Degrees. o Sine of 41 Degrees. 0" 10" 20" i 30" 40" 50" s 0" 10" 20" 30" 40" 50" o . 8o8o6 7 8o 9 3 8n88i438i68 8i 9 3 5 9 O 9 .8i6 9 43 6967 6 99 i 7016 7040 7 o64 5 9 I 8218 8243 826882938318 8343 58 I 7088 71 12 7 i3 7 7161 7i85 -7209 58 2 8368 83 9 3 841984448469 8494 5 7 2 7233 72 58 -7282 73o6 7 33o 7 354 5 7 3 85i 9 8544 856 9 ;85 9 4;86i 9 8644 56 3 7379 74o3 7 42 7 745 1 7 4 7 5 7499 56 4 866 9 86 9 4 8719 8744 8769 8 79 4 55 4 7524 7 548 7 5 7 2 7 5 9 6 7620 7644 55 5 88i 9 8844 886988948919 8944 54 5 7668 7693 77 I 7 77 4i 77 65 7789 54 6 8969 8 99 4 9019 9044 9069 9094 53 6 7 8i3 7 83 7 7 86 2 7886 7910 7934 53 7 9119 91449169 9194 9 2I 9 9244 52 7 7958 79 8 2 8006 8o3o 8o55 8o 79 52 8 9269 92949319 9344 9 36 9 9394 5i 8 8io3 8l2 7 8i5i 8i 7 5 8199 8223 5i 9 9419 94449469,9494 9 5i 9 9544 5o 9 8247 82 7 2 8 29 6 8320 8344 8368 5o 10 9.809569 g5g4 9619 9643 9 668 9 6 9 3 49 10 9.818392 84i6 844o 8464 8488 85i2 4 9 ii 97i8 974397689793 9 8i8 9 843 48 ii 8536 856o 8584 86o 9 8633 865 7 48 I 2 9868 9 8 9 3| 99 i8 i99 43 99 6 7 9992 4 7 I 2 8681 8 7 o5 8729 8 7 53 8777 8801 4 7 i3 9.810017 0042 006710092 OII 7 0142 46 i3 8825 884 9 88 7 3 8897 8921 8 9 45 46 i4 0167 oi 9 i 0216 0241 0266 0291 45 i4 8969 8 99 3 9017 9041 9 o65 9 o8 9 45 i5 o3i6 o34i o366 o3 9 o o4i5 o44o 44 i5 9113 9 i3 7 9161 9i85 9 20 9 9 233 44 16 o465 o4 9 o o5i5 o54o o564 o58 9 43 16 9 25 7 9 28l 9 3o5 9 32 9 9 353 9 3 77 43 17 0614 o63 9 o664 o68 9 o 7 i3 o 7 38 42 17 9401 9 425 9449 9497 9 52I 42 18 o 7 63 07880813 o838 0862 o88 7 4i 18 9 545 9 56 9 9593 9 6i7 9641 9 665 4i i 9 0912 0937 0962 0986 IOII io36 4o '9 9689 97 i3 97 3 7 97 6i 9785 9 8o 9 4o 20 9.811061 1086 IIIO n35 1160 n85 3 9 20 9.819832 9 856 9880 99 o4 99 28 99 5 2 3 9 21 1210 1234 i25g 1284 i3o 9 i334 38 21 9976 .... ..24 ..48 ..72 .. 9 6 38 22 i358 i383 i4o8 i433 i45 7 1482 3 7 22 9.820120 oi43 0167 oi 9 i 0215 023 9 3 7 23 1507 i532 i556 i58i 1606 i63i 36 23 " 0263 028-7 o3n o335 o35 9 o382 36 24 i655 1680 I 7 o5 i 7 3o i 7 54 i 779 35 24 o4o6 o43o o454 0478 o5o2 o526 35 25 1804 1828 i853 i8 7 8 I 9 o3 I 9 2 7 34 25 o55o o5 7 3 o5 97 0621 0645 o66 9 34 26 1952 i 977 2001 2026 2o5i 2076 33 26 0693 o 7 i 7 0740 0764 0788 0812 33 27 2IOO 2125 2i5o 2I 7 4 2I 99 2224 32 27 o836 0860 o883 9 O7 0931 o 9 55 32 28 2248 22 7 3 2298 2322 234 7 23 7 2 3i 28 0979 ioo3 1026 io5o 1074 io 9 8 3i 2( ; 2396 2421 2446 24 7 24 9 5 252O 3o 2 9 1122 u46 n6 9 n 9 3 1217 1241 3o 3t> g. 8i2544 256 9 2594 26l8 2643 2668 2 9 3o 9.821265 1288 l3l2 i336 i36o 1 384 29 ?i 2692 2 7 I 7 2 7 42 2 7 66 2 79 I 28i5 28 3i 1407 i43i i455 1 479 i5o2 i5 2 6 28 32 2840 2865 2889 2914 2 9 3 9 2 9 63 2 7 32 i55o i5 7 4 i5 9 8 1621 1 645 i66 9 27 33 2988 3OI2 3o3 7 3o62 3o86:3in 26 33 i6 9 3 i 7 i6 1740 1764 1788 1811 26 34 3i35 3i6o 3i8532o 9 32343258 25 34 i835 i85 9 i883 I 9 o6 1930 i 9 54 23 35 3283 33o 7 3332 335 7 338i 34o6 24 35 1977 20OI 2025 204 9 2072 2096 24 36J 343o 3455 34 79 35o4 35 29 !3553 23 36 2120 2i44 2167 2I 9 I 22l5 2238 23 37 3578 36o2 362 7 365i 36 7 6 3 7 oo 22 3? 2262 2286 23o 9 2333 2357 238i 22 38 3 7 25 3 7 4 9 3 77 4 3 7 99 382313848 21 38 2404 2428 2452 2475 2499 25 2 3 21 3 9 8872 38 97 3921 3 9 46 3 97 o 3 99 5 2O 3 9 2546 25 7 O 2 5 9 4 2617 2 64i 2665 2O 4o 9.814019 4o44 4o68 4o 9 3 41174142 I 9 4o 9.822688 2 7 I2 2 7 36 2 7 5 9 2783 280-7 19 4i 4i66 4191 42i5 4240 4264 428 9 18 4i 283o 2854 2878 2 9 OI 2925 2 9 48 18 42 43i3 43384362 438 7 44n;4436 17 42 2972 2996 3oi 9 3o43 3o6 7 3o 9 o J 7 43 446o 4484 45og 4533 45584582 16 43 3n4 3i3 7 3i6i 3i85 3208 3232 16 44 4607 463i 4656 468o 4 7 o4 4 7 2 9 i5 44 3255 32 79 33o3 3326 335o 33 7 3 i5 45 4753 47784802 48 2? 485i ; 48 7 6 i4 45 33 97 3444 3468 3491 35i5 i4 46 4 9 oo 4924 4949 4 97 3 4 99 8 5O22 i3 46 353 9 3562 3586 36o 9 3633 3656 i3 47 5o46 5o 7 i SogS 5l20 5i445i68 12 47 368o 3 7 o4 3727 3 7 5i 3 77 4 3 79 8 12 48 5i 9 3 52I 7 5242 5266 52 9 o 53i5 II 48 382i 3845 3868 38 9 2 3 9 i5 3 9 3 9 II 49 533 9 5364 5388 5412 543 7 546i 10 49 3 9 63 3 9 86 4oio 4o33 4o5 7 4o8o IO 5o 9 .Si5485 55io 5534 5558 558356o 7 9 5o 9.824104 4l2 7 4i5i 4174 4198 4221 9 5i 5632 5656568o 5 7 o5 5 7 2 9 5 7 53 8 5i 4245 4268 42 9 2 43i5 433 9 4362 8 52 5 77 8 58o2 5826 585i 58 7 5 5899 7 52 4386 4409 4433 4456 448o 45o3 7 53 5 9 24 5 9 48 5 97 2 5 99 6 6021 6o45 6 53 4527 455o 45 7 4 45 97 4621 4644 6 54 6o6 9 6o 9 4 6118 6142 6i6 7 6191 5 54 4668 46 9 i 4 7 i5 4738 4 7 6i 4 7 85 5 55 62i5 6240 6264 6288 63i2 633 7 4 55 48o8 4832 4855 48 7 9 4902 4 9 26 4 56 636i 6385 64o 9 6434 6458 6482 3 56 4949 4 97 2 4 99 6 5oi 9 5o43 5o66 3 5 7 65o 7 653i 6555 65 79 66o4 6628 2 57 5090 5n3 5i36 5i6o 5i83 52O 7 2 58 6652 6676 6701 6725 6 7 49 6 77 3 I 58 523o 5254 5277 53oo 5324 534 7 I 5 9 679816822 68466870 6894 6919 5 9 53 7 i 53 9 4 5417 544i 5464 5488 60" | 50" | 40" | 30" 20" 10" .9 60" 50" 40" 30" 20" 10" d Co-sine of 49 Degrees. Co-sine of 48 Degrees. 3 p P .< 1" 2" 3" 4" 5" 6" 7" 8" 9" . ( 1" 2" 3" 4" 5" 6" 7" 8" 9" l.lartj 257 10 12 15 17 20 22 in \ 2 5 7 10 12 14 17 19 2t L o G A R i T ii M i c TANGENTS. Jj Tangent of 40 Degrees. .5 ?, Tangent' of 4 L Degrees. 30" | 40" | 50" t 0" 10" 20" 30" 40" | 50" 0" 10" 20" 9,923814 3856 3899 3 9 42 3 9 85 402 7 5 9 o 9.939163 9206 9248 9 2 9 i| 9 333 9 3 7 6,5 9 I I 4070 4n3 4i56 419814241 4284 58 i 94i8 946 1 95o3 9 546 9 588 9 63i . r 8 2 3 432 7 4583 436 9 4626 44i2 466 9 4455 4711 4498 4 7 54 454.0 4797 57 56 2 3 9 6 7 3 9928 9716 9971 9758 ..18 9801 9843 ..56 ..98 9886 .1*1 5? 56 4 484o 4882 492514968 5on 5o53 55 4 9.940183 0226 0268 o3n o354 o3 9 6 55 i 5 5o 9 6 5i3 9 5i8i 5224 526 7 53io 54 5 0439 o48i o524 o566 o6o 9 o65i|54 6 5352 53 9 5 5438 5481 5523 5566 53 6 0694 0786 779 0821 o864 o 9 o653 7 5609 5652 56 9 4 5 7 3 7 5 7 8o 5822 52 7 0949 0991 io34 io 7 6 1119 1161 52 8 5865 5 9 o8 5 9 5i 5 99 3 6o36 6o 79 5i 8 I204 1246 1289 1881 i3 7 4 i4:65j 9 6122 6i64 62O 7 6260 6292 6335 5o 9 i45 9 i5oi 1 544 i586 1628 1671 5o ioj9 .926878 6421 6463 65o6 654 9 6^91 49 10 9.9417*3 1756 i 79 8 i84i 1888 1926 49 ii 6634 6677)6720 6762 68o5 6848 48 ii 1968 201 I 2o53 2096 2188 2181 48 12 6890 6 9 33 69 7 6 7019 7061 7 io4 47 12 2223 2266 2808 2 35i 2893 2436 47 r3 7147 7 i8 9 7 232 7 2 7 5 7817 7 36o 46 i3 2 4 7 8 2521 2563 2606 2648 2691 46 i4 74o3 7446 7 488 7 53i 7 5 7 4 7616 45 i4 2 7 33 2776 2818 2861 2903 2 9 45 45 i5 7 65 9 7702 77 44 7787 7 83o 7 8 7 2 44 i5 2988 8080 3o 7 3 3n5 3!58 8200 44 iC 7915 7958 8001 8o43 8086 8129 43 16 3243 3 2 85 8828 33 7 o 34i3 3455 43 17 8171 8214 825 7 8299 8342 8385 42 17 3498 354o 3583 3625 366 7 3 7 io 42 18 8427 8470 85i3 8555 85 9 8 864i 4i 18 3 7 52 3 79 5 383 7 8880 8922 3 9 65 4i *9 8684 8726 8 7 6 9 8812 8854 88 97 4o J 9 4oo 7 4o5o 4092 4i35 4177 4219 4o 20 9.928940 8982^025 9068 9110 9 i53 3 9 20 9.944262 43o4 434 7 438 9 4432 44 7 4 3 9 21 9196 9238 9281 9 3 2 4 9 366 9 4o 9 38 21 45i 7 455 9 4602 4644 4686 4 7 2 9 38 22 9452 9 4 9 4J 9 537 9 58o 9 622 9 665 37 22 477 1 48i4 4856 48 99 4g4 1 4984 3 7 23 9708 97 5o 979 3 9 836 9 8 7 8 9921 36 23 5o26 5069 5ni 5i53 5i 9 6 5 2 38 36 24 9964 ...6 ..49 ..92 .i34 .177 35 24 5 2 8i 5323 5366 54o8 545 1 54 9 3 35 r 9.930220 0262' o3o5 o348 o3 9 o o433 34 25 5535 55 7 8 5620 5663 5 7 o5 5 7 48 34 26 o4?5 o5i8 o56i o6o3 o646 0689 33 26 5790 5832 58 7 5 5 9 i 7 5960 6002 33 27 0731 0774 08 1 7 o85 9 0902 0945 32 "7 6o45 6087 6i3o 6l 7 2 6214 625 7 82 28 0987 io3o io 7 3 ni5 n58 I2OO 3i 28 6299 6842 6384 642 7 6469 65n 81 29 1243 1286 i328 i3 7 i i4i4 i456 3o 29 6554 65 9 6 6689 6681 6 7 24 6 7 66 3o 3o 9.931499 1 542 i584 1627 1669 I 7 I2 29 3o 9.946808 685 1 68 9 3 6 9 36 6 97 8 7 O2I 2 9 3i| i 7 55 i 797 i84o i883 1925 1968 28 3i 7 o63 7io5 7 i48 7 i 9 o 7 233 7275 28 32 2OIO 2o53 2096 2i38 2181 2224 2 7 32 7 3i8 7860 7 402 7 445 7 48 7 7 53o 27 33 2266 2309 2 35i 23 9 4 243 7 24 7 9 26 33 7 5 7 2 7614 7 65 7 7 6 99 7742 77 84 26 34 2522 2565 26o 7 265o 2692 2 7 35 25 34 7827 7869 79" 79 54 799 6 8089 25 35 2778 2820 2863 2 9 o6 2948 2991 24 35 8081 8i23 8166 8208 825i 8293 24 36 3o33 3076 3iig 3i6i 3204 3246 23 36 8335 8878 8420 8463 85o5 8548 28 37 3289 3332 33 7 4 34i? 3459 35o2 22 37 8590 8632 86 7 5 8717 8 7 6o 8802 22 38 3545 358 7 363o 3672 3 7 i5 3 7 58 21 38 8844 8887 8929 8 97 2 9014 9o56 21 3 9 38oo 3843 3885 3 9 28 3 97 i 4oi3 2O 3 9 9099 9141 9184 9226 9268 9811 2C 4o 9.934o56 40984141 4i84 4226 4269 *9 4o 9.949353 9 3 9 6 9 438 9480 9 523 9 565 '9 4i 43n 4354 43 97 443 9 4482 4524 18 4i 9608 9 65o 9692 9735 9777 9819 18 42 456 7 46 1 o 4652 46 9 5 4737 4 7 8o '7 42 9862 994 9947 9989 ..81 .. 7 4 r? 43 4822 4865 4908 495o 4993 5o35 16 43 9. 9 5on6 0159 02OI 0243 0286 0828 16 44 5078 5i2i 5i63 5206 5 2 48 5291 i5 44 0371 o4i3 o455 0498 o54o o582 i5 45 5333 53 7 6 54i9 546i 55o4 5546 i4 45 0625 0667 7 IO O 7 52 0794 o83 7 i4 46 558 9 5632 56 7 4 5 7 i 7 5 7 5 9 58o2 i3 46 0879 0921 0964 1006 1049 1091 18 47 5844 588 7 5 9 3o 5 97 2 6oi5 6o5 7 12 4 7 n33 1176 1218 1261 1808 i345 12 48 6100 6142 6i85 6227 62 7 O 63i3 II 48 1388 i43o l4 7 2 i5i5 i55 7 1600 II 49 6355 63 9 8 644o 6483 6525 6568 IO 49 1 642 1 684 I 7 2 7 i 7 69 1811 i854 IO 5o 9 . 9 366n 6653 6696 6 7 38 6 7 8i 6823 9 5o 9.951896 1988 1981 2O23 2066 2108 9 5' 6866 6908 6 9 5! 6994 7 o36 779 8 5i 2i5o 2193 2235 22 77 2820 2862 8 52 7121 7 i64 7 2O6 7249 7 2 9 i 7334 7 52 24o5 244 7 2489 2532 25 7 4 2616 7 53 7 3 77 7 4i 9 7 462 7 5o4 7 54 7 7 58 9 6 53 2659 2 7 OI 2 7 43 2 7 86 2828 28 7 o 6 54 7 632 7 6 7 4 7717 77 5 9 7 8O2 7845 5 54 2913 2 9 55 2998 3o4o 8082 3i25 5 55 7 88 7 79 3o 7972 8oi5 8o5 7 8100 4 55 3167 8209 3252 8294 3336 33 79 4 56 8!428i85 822 7 82 7 o83i2 8355 3 56 3421 3463 35o6 354835 9 i 3633 3 57 83 9 8 844o 8483 85258568 8610 2 5 7 8675 8718 3 7 6o38o23845i388 7 2 58 8653 86 9 58 7 38 8 7 8o 88238865 I 58 39298972 4oi44o564o 99 4i4i I 5o 8 9 o8|8 9 5o;8 99 3 9 o35 9 O 7 8| 9 I2I O 5 9 4i83;4226 4268.4310 43534395 h60" 50" 40" 30" | 20" | 10" a 60" 50" 40" | 30" 20' .10" 3 Co-tangent of 49 Degrees. 9 & Co-tangent of 48 Degrees. 1' p p < 1" 2" 3" 4" 5" 6 ; ' 7" 8" 9" j 4 9 13 17 21 26 30 34 38 p p f C 1" 2" 3" 4" 5" 6'' 7' 8" 9" irt } 4 8 13 17 21 25 30 34 38 E GG LOGARITHMIC SINES. jf Sine of 42 Degrees. d Sine of 43 Degrees. m 0" 10" 20" | 30" 40" 50" s 0" 10" 20" | 30" 4(X' 1 50" o 9 .$255u 5534 5558 558i 56o4 5628 5 9 9 ,833 7 83 38o6 3828 385i 38 7 4'38q6 5 9 2 565i 56 7 5 5 79 i 58i5 56 9 8 5 7 2i 5838586i 5 7 45 5885 5 7 6858 590867 i 3 9 i 9 Ao54 3 9 4i 4077 3964 398614009 4o 99 4122 4i44 4o3 3 4167 53 5 7 3 5 9 3i 5 9 55 5 97 8 6001 6025 6o48 56 3 4i8 9 42T2 4234 4257 4280 4302 56 4 6071 6o 9 5 6118 6i4i 6i65|6i88 55 4 4325 434 7 43 7 o 43 9 2 44 1 5 443 7 55 5 6211 6235 6258 6281 63o5i6328 54 5 446o 4482'45o5 452 7 455o 45 7 2 54 6 635i 63 7 5 63 9 8642i 6444 6468 53 6 45 9 5 46i 7 464o 4662 4685 470-7 53 7 64 9 i 65i4 6538656i 6584 66o 7 52 7 4 7 3o 4752 4775 4797 4820 4842 5-3 g 663i 6654 66 77 6 7 oi 6724 6 7 4 7 5i 8 4865 488 7 4 9 io 4 9 3 2 4954 4977 5i 9 6770 6 79 4 68i 7 684o 6863 688 7 5o 9 4999 5022 5o44 5o6 7 5089 P- 5lI2 5o 10 9 .826 9 io 6 9 33 6 9 56 6 9 8o 7 oo3 7026 49 io 9 .835i34 5i5 7 5i 79 5201 5224 5246 49 1 1 7o4 9 7 o 7 3 7 o 9 6 7 n 9 7142 7165 48 ii 526q 53i4 5336 5358 538i 48 12 7 i8 9 7 2I2 7 235 7 258 7282 73o5 47 12 54o3 5436 5448 54 7 i 54 9 3 55:5 47 i3 7 328 7 35i 7 3 7 4 7 3 9 8 7 42I 7 444 46 i3 5538 556o 5583 56o5 562 7 565o 46 i4 746 7 7 4 9 o 7 53 7 7 56o 7 583 45 i4 56 7 2 56 9 5 5 7 i 7 5 7 3 9 5 7 62 5 7 84 45 i5 7606 7 62 9 7 653 7 6 7 6 76907728 44 i5 58o 7 582 9 585i 58 7 4 58 9 6 5 9 i8 44 :6 7745 77 68 7792 7 8i5 7 838 7861 43 16 5 9 4i 5 9 63 5 9 86 6008 6o3o 6o53 43 '7 7 884 797 79 3i 7954 79 77 8000 42 7 6o 7 5 6o 97 6120 6142 6i64 6187 42 18 8o23 8o46 8o6 9 8o 9 3 8n68i3 9 4i 18 62O 9 623i 6254 6276 62 9 8 632i 4i 19 20 21 8162 q. 828301 843 9 8i85 8324 8462 82088231 825482-78 834 7 83 7 o83 9 384i6 8485 85o 9 8532 8555 4o 3 9 38 '9 20 21 6343 o.S364 77 6611 6365 64 99 6633 6388 6522 6656 64io 6544 6678 6432 6566 6 7 oo 6455 658 9 6 7 22 4o 3 9 38 22 8578 8601 8624 864 7 8670 86 9 3 37 22 6 7 45 6 7 6 7 6 7 8q 6812 6834 6856 37 23 8 7 l6 8 7 3 9 8 7 6 2 8 7 86 88o 9 8832 36 23 68 7 8 6 9 oi 6 9 23 6 9 45 6 9 68 6 99 o 36 24 8855 88 7 8 8 9 oi 8 9 24 8 9 4 7 8 97 o 35 24 7 OI2 7034 7 o5 7 7079 7 IOI 7 I23 35 25i 8 99 3 9 Ol6 qo3 9 9 o62 9 o85 9 io8 34 25 7 i46 7 i68 7 i 9 o 7212 7 235 72 5 7 34 26 9 l3l 9 i54 9 i 77 9 2oo Q223 9 246 33 26 7279 7 3oi 7 324 7346 7 368 7 3 9 o 33 27 9 26 9 9 2 9 2 9 3i5 9 338 9 36i 9 384 32 27 7412 7 435 7 45 7 7479 7 5oi 32 28 9 4o 7 9 43o 9 453 9 4 7 6 9499 9 522 3i 28 7546 7 568 7 5qo 7612 7 635 7 65 7 3i 29 9 545 9 568 9 5qi 9 6i4 9 63 7 9 66o 3o 2 9 7 6 79 77 oi 7723 7746 77 68 779 3o 3o 9.&2 9 683 97 o6 97 2y 97 5 2 977 5 9798 2 9 3o 9 .83 7 8i2 7834 7 85 7 7879 79 oi 79 23 29 3i 9 82I 9 844 9 86 7 9 8 9 99 i3 99 36 28 3i 7945 79 6 7 799 8012 8o34 8o56 28 32 99 5 9 99 8 2 ...5 ..28 .. 7 4 27 32 8078 8100 8123 8i45 816-7 8i8 9 27 33 9.83000.7 OI20 0142 oi65 0188 O2II 26 33 8211 8233 8256 8278 83oo 8322 26 34! 0234 O25 7 0280 o3o3 o326 o34 9 25 34 8344 8366 8388 84io 8433 8455 25 35 0372 o3 9 5 o4i 7 o44o o463 o486 24 35 8477 84 99 852i 8543 8565 858 7 24 36 o5oq o53 2 o555 o5 7 8 0601 0624 23 36 8610 8632 8654 8676 86 9 8 8720 23 3? o646 o66 9 o6 9 2 o 7 i5 o 7 38 0761 22 37 8742 8 7 64 8 7 86 8808 883i 8853 22 38 o 7 84 o8o 7 o82 9 o85 2 o8 7 5 o8 9 8 21 38 8875 8897 8 9 iq 8 9 4i 8 9 63 8 9 85 2! 3 9 4o 0921 0944 9 .83io58 1081 o 9 6 7 no4 o 9 8 9 II2 7 IOI2 Il4 9 io35 1 172 20 39 4o 9007 9 .83 9 i4o 9029 9162 9 o5i 9 i84 97 3 99 5 9 2o6 9228 9118 9 25o 2O I 9 4i u 9 5 1218 1241 1263 1286 r3o 9 18 4i 9 2 7 2 9294 9 3i6 9 338 9 36o 9 38 2 1 8 42 1 332 i355 i3 7 8 i4oo 1423 1 446 '7 42 9 4o4 9426 9 448 9470 9 4 9 2 9 5i4 17 43 i46 9 l4 9 2 i5i4 i53 7 i56o i583 16 43 9 536 9 558 9 58o 9602 9 624 9 646 16 44 1606 1628 i65i i6 7 4 i6 9 7 1720 i5 44 9 668 9690 97 I2 9734 9756 9778 i5 45 1742 i 7 65 1788 1811 i833 i856 M 45 9 8oo 9822 9 844 9866 9 888 99 io i4 46 i8 79 I 9 02 I 9 24 i 9 47 i 97 o i 99 3 i3 46 99 32 99 54 997 6 9998 . .20 ..42 i3 47 201 5 2038 2061 2084 2106 2I2 9 12 4 7 9 .84oo64 0086 0108 oi3o Ol52 0174 12 48 2l52 2I 7 5 2197 222O 2243 2266 I I 48 oi 9 6 0218 0240 0262 0284 o3o6 II 49 2288 23ll 2334 2356 2379 2402 10 49 o3 2 8 o35o 0372 o3 9 3 o4i5 o43 7 10 5o 9 . 832425 244 7 2470 2 4 9 3 25i5 2538 Q 5o q.84o45q o48i o5o3 o5 2 5 o547 o56 9 9 5i 256i 2584 260612620 2652 2674 8 5i o5 9 i o6i3 o635 o65 7 c6 7 8 o 7 oo 52 26 97 2 7 20 2 7 42 2 7 65 2788 2810 7 52 0722 0744 0766 o 7 88 0810 o832 7 53 2833 2856 2 8 7 8 2 9 OI 2 9 24 2 9 46 6 53 o854 0876 o8 97 0919 o 9 4i o 9 63 6 54 2 9 6 9 2 99 2 3oi4 3o3 7 3o6o 3o82 5 54 o 9 85 1007 IO2 9 io5i IO 7 2 io 9 4 5 55 3io5 3i 2 8 3i5o 3i 7 3 3i 9 6 32i8 4 55 1116 n38 1160 1182 1204 1226 4 56 324i 3 2 63 3286 33o 9 333i 3354 3 56 1247 1269 I2 9 I :3i3 i335 i35 7 3 5 7 33 77 33 99 3422 3444 346 7 34 9 o 2 57 i3 7 8 i4oo 1422 1 444 i466 i488 2 58 35i2 3535 355 7 358o 36o3 3625 I 58 i5oq i53i i553 i5 7 5 i5q 7 ii6iq I 5, 3648 36 7 o 36 9 3 3 7 i6 3 7 38 3 7 6i O 5 9 i64o 1662 1 684 1-706 i 7 28|i 7 4 9 60" 50" 40" 30" 20" 10" B 60" 50" | 40" j 30" 20" i 10" C3 Co-sine of 47 Degrees. 3 Co-sine of 46 Degrees. 9 as p < 1" 2" 3'' 4" 5" G" 7" 8" 9" ,, ( 1" 2" 3" 4" 5" G" 7" 8" 9" art c 2 5 7 9 11 14 1(5 18 21 A ""I 2 4 7 9 11 13 16 18 20 LOGARITHMIC TANGENTS. 67 1 Tangent of 42 Degrees. I Tangent of 43 Degrees. 0" 10" | 20" 30" 40" 50" * 0" j 10" 20" 30" 40" 50" I 9 . 9 54437 46 9 i 448o 4 7 34 4522 4776 4564 4819 460 7 486i 4649 4 9 o3 5 9 58 o i 9.96965619698 99099951 97 4o 9994 9783 ..36 9 825 ..78 9867 . 120 5 9 58 2 4946 4988 5o3o 5o 7 3 5n5 5i5 7 57 2 9 . 9 7oi62 O205 0247 0289 o33i o3 7 3 57 3 5 200 5a4 2 5284 532 7 536 9 5-4n 56 3 o4i6 o458 o5oo 0542 o584 0627 56 4 5454|54 9 6 5538 558i 5623 5665 55 4 o66 9 7 II o 7 53 o 79 6 o838 0880 55 5 5 7 o8 5 7 5o 5 79 2 5835 58 7 7 5 9 i 9 54 5 9 22 o 9 64 IO0 7 1049 io 9 i n33 54 6 5 9 6r 6oo4 6o46 6088 6i3i Oi 7 3 53 6 n 7 5 1218 1260 1302 1 344 i386 53 7 62i5 6 2 58 63oo 6342 6385 b42 7 52 7 i4 7 i i5i3 i555 i5 97 i64o 52 8 646 9 65i2 6554 65 9 6 663 9 6681 5i 8 1682 i -724 i 7 66 1808 i85i i8 9 3 5i 9 6 7 23 6 7 66 6808 685o 68 9 3 6 9 35 5o 9 i 9 35 i 977 2OI 9 2062 2IO4 2146 5o 10 9-9'6 977 7 02O 7062 7104 7i46 7189 49 10 9 . 9 72i88 2230 22 7 3 2 3i5 235 7 2 3 99 4 9 ii 7231 7273 7 3i6 7 358 7400 7443 48 ii 244 1 2484 2526 2568 2610 2652 48 12 7485 7 5 27 7 5 7 o 7612 7654 7697 47 12 26 9 5 2 7 3 7 2779 2821 2863 2 9 o5 4 7 i3 77 3 9 7781 7823 7866 79 o8 79 5o 46 i3 2 9 48 2 99 3o32 3o 7 4 3n6 3i5 9 46 i4 7993 8o35 8077 8120 8162 8204 45 i4 3201 3243 3 2 85 33 27 33 7 o 34i2 45 i5 8247 8289 833! 83 7 3 84i6 8458 44 i5 3454 34 9 6 3538 358i 3623 3665 44 16 85oo 8543 8585 8627 8670 8 7 I2 43 16 3 7 o 7 3 7 49 3 79 i 3834 38 7 6 3 9 i8 43 17 8 7 54 8 79 6 883 9 8881 8 9 23 8966 42 17 3 9 6o 4O02 4o45 4o8 7 4i2 9 4i 7 i 42 18 9 oo8 9 o5o 9 o 9 3 9 i35 9 i 77 9219 4i 18 42 1 3 4255 42 9 8 434o 4382 4424 4i 19 9 262 9 3o4 9 346 9 38 9 943i 9473 4o 19 4466 45o 9 455i 45 9 3 4635 46 77 4o 20 9 . 9 5 9 5i6 9 558 9600 9 642 9 685 9727 39 20 9.974720 4 7 6 2 48o4 4846 4888 4 9 3o 39 21 9769 9 8l2 9 854 9 8 9 6 99 38 998i 38 21 4 97 3 5oi5 5o5 7 5o 99 5i4i 5i83 38 22 9 . 9 6oo23 oo65 0108 oi5o OI 9 2 0234 37 22 5226 5 2 68 53io 5352 53 9 4 543 7 3? 23 0277 o3i 9 o36i o4o4 o446 o488 36 23 5479 552i 5563 56o5 564 7 56 9 o 36 24 o53o o5 7 3 o6i5 o65 7 o 7 oo 7 42 35 24 5 7 32 5 77 4 58i6 5858 5 9 oi 5 9 43 35 25 0784 0826 o86 9 o 9 n o 9 53 0996 34 25 5 9 85 602-7 6o6 9 6m 6i54 6i 9 6 34 26 io38 1080 1122 n65 I2O 7 1249 33 26 6238 6280 6322 6364 64o 7 6449 33 27 I2 9 2 i334 i3 7 6 i4i8 i46i i5o3 32 27 64 9 i 6533 65 7 5 661-7 6660 6702 32 28 1 545 i58 7 i63o 1672 i 7 i4 i 7 5 7 3i 28 6744 6 7 86 6828 68 7 o 6 9 i3 6 9 55 3i 29 i 799 i84i i883 I 9 26 i 9 68 2OIO 3o 29 6997 7 o3 9 -7081 7 I23 -7166 7208 3o 3o 9 . 9 62O52 20 9 5 2137 2179 2222 2264 2 9 3o 9 . 977 25o 7 2 9 2 7334 7377 74i9 746 1 29 3i 23o6 2348 23 9 I 2433 24 7 5 25l 7 28 3i 7 5o3 7 545 7 58 7 7 63o 7 6 7 2 77 i4 28 32 256o 2602 2644 2686 2 7 2 9 2771 27 32 7756 7798 7 84o 7 88 2 7925 79 6 7 27 33 28i3 2856 28 9 8 2 9 4o 2 9 82 3o25 26 33 8oo 9 8o5i 8o 9 3 8i35 8i 7 8 8220 26 34 3o6 7 3io 9 3i5i 3i 9 4 3236 32 7 8 25 34 8262 83o4 8346 8388 843i 8473 25 35 3520 3363 34o5 3447 348 9 3532 24 35 85i5 855 7 85 99 864i 8684 8726 24 36 35 7 4 36i6 365 9 3701 3 7 43 3 7 85 23 36 8 7 68 8810 8852 88 9 4 8 9 3 7 8 979 23 37 3828 38 7 o 3 9 I2 3 9 54 3 997 4o3 9 22 37 9 O2I 9 o63 9 io5 9147 9 i 9 o 9 232 22 38 4o8i 4l23 4i66 4208 4292 21 38 9274 9 3i6 9 358 9400 9 442 9 485 21 39 4335 43 7 7 44i9 446 1 45o4 4546 20 39 9 56 9 9 6n 9 653 9 6 9 5 9738 20 4o 9. 964588 463o 46 7 3 47i5 4 7 5 7 4799 '9 4o 9.9-79-780 9 822 9 864 99 o6 9948 999 ig 4i 4842 4884 4 9 26 4 9 68 Son 5o53 18 4i 9.980033 oo 7 5 OII 7 oi5 9 0201 0243 18 42 SogS 5i3 7 5i8o 5222 5264 53o6 17 42 0286 o3 2 8 o3 7 o 0412 0454 o4 9 6 17 43 5349 53 9 i 5433 54?5 55i8 556o 16 43 o538 o58i 0623 o665 7 7 0749 16 44 56o2 5644 568 7 5 7 2 9 5 77 i 58i3 i5 44 79* o834 08-76 o 9 i8 0960 IOO2 i5 45 5855 58 9 8 5 9 4o 5 9 82 6024 6067 i4 45 1044 1086 II2 9 n 7 i I2l3 1255 i4 46 6109 6i5i 6i 9 3 6236 62 7 8 6320 i3 46 I2 97 i33 9 1382 1424 i466 i5o8 i3 47 6362 64o5 6447 648 9 653i 65 7 4 12 47 i55o i5 9 2 i634 i6 77 I 7 i 9 1761 12 48 6616 6658 6700 6742 6 7 85 682-7 II 48 i8o3 i845 i88 7 I 9 2 9 1972 2Ol4 II 49 6869 6 9 n 6 9 54 6 99 6 7 o38 7080 IO 49 2o56 20 9 8 2l4o 2182 2224 2267 IO 5o 9.96-7123 7 i65 7207 7249 7 2 9 I 7334 9 5o 9.982309 235i 23 9 3 2435 2477 25 I 9 9 5i 7 3 7 6 7 4i8 7460 7 5o3 7 545 7 58 7 8 5i 2562 2604 2646 2688 2 7 3o 2772 8 52 7629 7 6 7 2 7714 7756 7798 7 84o 7 52 2814 2 85 7 28 99 2 9 4i 2983 3o 2 5 17 53 7 883 7925 79 6 7 8oo 9 8o52 8o 9 4 6 53 3o6 7 3io 9 3i52 3i 9 4 3236 32 7 8 6 54 8i36 8178 8220 8263 83o5 834 7 5 54 3320 3362 34o4 3447 348 9 353i 5 55 838 9 843 2 8474 85i6 8558 8600 4 55 35 7 3 36i5 365 7 36 99 3 7 4z 3 7 84 4 56 8643 8685 8727 8769 8812 8854 3 56 3826 3868 3 9 io 3 9 52 3994 4o3 7 3 57! 88 9 6 8 9 38 8 9 8o 9023 9 o65 9107 2 57 4o 79 4l2I 4i63 4i$o5 424 7 428 9 2 58 9149 9 I 9 2 9 234 9276 9 3i8 9360 I 58 433 2 43 7 4 44i6 4458 45oo 4542 I 5 9 9 4o3 9445 948 7 9 5 29 9 5 7 i 9614 5 9 4584|462 7 466 9 4711 4 7 53 4795 o 60" 50" 40" 30" 20" 10" a 60" 50" 40" 30" 20" 10" Co-tangent of 47 Degrees. .9 Co-tangent of 46 Degrees. C 1" 2" 3" 4" 5" 6" 7" 8" 9" t S i" 2" 3" 4" 5" 6" 7" 8" 9" P. 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( 1" 2" 3" 4" 5" 6" 7" 8" 9" irt $ 2 4 G 9 11 13 15 17 19 irt ( 2 4 G 8 10 12 14 17 19 LOGARITHMIC TANGENTS. 60 "7 Tangent of 44 Degrees. .5 Tangent of 45 Degrees. * 0" 10" 20" | 30" 40" | 50" m 0" 10" | 20" 30" 40" | 50" o 9.984837 4879 4921 4964 5oo6l5o48 5 9 o IO.OOOOOO 0042 oo84 0126 0168 02 1 1 5 9 I 5090 5i3 2 5i74 52i6 5 2 5 9 53oi 58 I 0253 02 9 5 o33 7 o3 79 0421 o463 58 2 5343 5385 5427 546 9 55n 5553 57 2 o5o5 o54 7 o5 9 o o632 o6 7 4 o 7 i6 5 7 3 55 9 6 5638 568o 5722 5 7 64 58o6 56 3 o 7 58 0800 0842 o884 0926 0969 56 4 5848 5891 5 9 33 5 97 5 6017 6059 55 4 ion io53 io 9 5 n3 7 1179 1221 55 5 6101 6i43 6:85 6228 6270 63i2 54 5 1263 i3o5 1 348 i3 9 o 1432 i4 7 4 54 6 6354 63 9 6 6438 648o 6523 6565 53 6 i5i6 i558 1600 1642 1684 I 7 2 7 53 7 6607 6649 6691 6733 6 77 5 68i 7 52 7 i 7 6 9 1811 i853 i8 9 5 : 9 3 7 I 979 52 8 6860 6902 6 9 44 6986 7 028 7 o 7 o 5i 8 2O2I 2063 2106 2148 2190 2232 5i 9 71 12 7i54 7197 72 3 9 7 28l 7 3 2 3 5o 9 2274 23i6 2358 2400 2442 2485 5o 10:9 .987366 7407 7449 749117534 7 5 7 6 49 10 IO.OO2527 2 56 9 2611 2653 26 9 5 2 7 3 7 49 1 1 7618 7660 7702 7744 7786 7829 48 ii 2 779 2821 2864 2 9 o6 2948 2990 48 12 7871 79 i3 79 55 7997 8039 8081 47 12 3o32 3o 7 4 3n6 3i58 320O 3243 47 i3 8i23 8166 8208 825o 82928334 46 i3 3285 332 7 336 9 34n 3453 34 9 5 46 i4 83 7 6 84i8 846o 85o3 8545 858 7 45 i4 353 7 35 79 3622 3664 3 7 o6 3 7 48 45 i5 8629 8671 8 7 i3 8 7 55 8797 884o 44 i5 3790 3832 38 7 4 3 9 i6 3 9 58 4ooi 44 16 8882 8924 8966 9008 9 o5o 9092 43 16 4o43 4o85 4127 4169 4211 4253 43 i? 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T d Sine of 47 Degrees. M 1 o" 10" 20" 30" 40" | 50" j 2 0" 10" 20" | 30" 40" 50" 0)9. 85693^ 6 9 54 6 97 5 6 99 5 7 oi5 7 o36 59 o 9 .864i2 7 4i4 7 4i6 7 4i86 4206 4226 5 9 i 7o56 7076 797 7117 7 i3 7 7 i58 58 i 4245 4265 4284 43o4 432< 4343 58 2 7178 7I 9 8 7 2i 9 72 3 9 72 5 9 7 2 79 5 7 2 4363 4383 4402 4422 444 1 446i 57 3 73oo 7320 7 34o 73 6 1 7 38i 7 4oi 56 2 448 1 45oo 4520 453 9 455 9 45 79 56 4 7422 7442 7 462 7^82 7 5o3 7 5 2 3 55 t 45 9 8 46i8 463 7 465 7 46 7 6 46 9 6 55 5 7 543 7563 7 584 7 po^ 7 6 2 4 7 645 54 r 4716 4 7 35 4 7 55 4 77 4 4794 48i3 54 5 7 665 7 685 77 o5 7p6 77 46 77 66 53 6 4833 4853 48 7 2 48 9 2 4 9 n 4 9 3i 53 7 7786 7807 7 82 7 7 &4 7 7 86 7 7 888 52 7 4 9 5o 4 97 o 4990 5oo 9 5o2 9 5o48 52 8 7908 7928 79 48 79 68 7989 8009 5i 8 5o68 5o8 7 5io 7 5i26 5i46 5i65 5i 9 8029 8o4 9 8o 7 o|8o 9 o 8110 8i3o 5o 9 5i85 52o4 5224 5244 5263 5283 5o 10 9.858i5i 8171 8191 8211 8 2 3i 8252 49 10 9 . 8653 02 5322 5341 536i 538o 54oo 4 9 n 8272 8292 83128332 8353 83 7 3 48 ii 54i 9 543 9 5458 54 7 8 54 9 7 55i 7 48 12 / SSgS 84i3 84^3 8454 84 7 4 8494 47 12 5536 5556 55 7 5 55 9 5 56i4 5634 47 i3 85i4 8534 8554 85 7 5 85 9 5 86i5 46 i3 5653 56 7 3 56 9 2 5 7 I2 5 7 3i 5 7 5i 46 i4 8635 8655 86 7 5 8696 8 7 i6 8 7 36 45 i4 5 77 o 5 79 o 58 o 9 58 2 8 5848 586 7 45 i5 8756 8776 8 79 688i 7 883 7 885 7 44 i5 588 7 5 9 o6 5 9 26 5 9 45 5 9 65 5 9 84 44 16 8877 8897 8 9 i 7 8 9 3 7 8 9 58 8978 43 16 6oo4 6023 6042 6062 6081 6101 43 r? 8998 9018 9 o38 9 o58 9 o 7 8 9098 42 ij 6120 6i4o 6i5 9 6i 79 6i 9 8 62I 7 42 18- 9119 9 i3 9 9 i5 9 ' 9 i 79 9 i 99 9219 4i 18 623 7 6256 62 7 6 62 9 5 63i5 6334 4i r 9 9 23 9 9 25 9 9 2 79 9 3oo 9 320 9 34o 4o I 9 6353 63 7 3 63 9 2 6412 643 1 645o 4o 209.859360 9380 9 4oo 9 42o 9 44o 9 46o 3 9 20 9.866-4 7 o 648 9 65o 9 65 2 8 654 7 656 7 3 9 21 9480 95oi 9 52i 9 54i 9 56i 9 58i 38 21 6586 6606 66 2 5 6644 6664 6683 38 22 9601 9621 9 64i 9661 9681 9 7oi 37 22 6 7 o3 6722 6 7 4i 6 7 6i 6 7 8o 6800 37 23 9721 97 4i 97 6i l 97 8i 9 802 9 822 36 23 6819 6838 6858 68 77 68 9 6 6 9 i6 36 -4 9842 9862 9882 9902 99 22 99 42 35 24 6935 6 9 54 6 97 4 6 99 3 7 oi3 7 o32 35 25 9962 9982 . . .2 . .22 ..42 ..62 34 25 7 o5i 7071 7 o 9 o 7 io 9 7 I2 9 7 i48 34 26 9.860082 OIO2 OI22'oi42 0162 0182 33 26 7167 7187 7 2o6 7 225 7 245 7 264 33 27 O2O2 O222 O242 O262 0282 0302 32 27 7 283 7 3o3 7 322 7 34i 7 36i 7 38o 32 28 O322 0342 03620382 04O2 0422 3i 28 7 3 99 7 4i 9 7 438 7 45 7 7 4 7 6 7 4 9 6 3i 29 0442 O462 0482, o5o2 O522 0542 3o 29 75i5 7 534 7 554 7 5 7 3 7 5 9 2 7 6l2 3o 3o 9.86o562 o582 0602 0622 0642 0662 2 9 3o 9 .86 7 63i 7 65o 7 66 9 7 68 9 77 o8 77 2 7 29 3i 0682 0702 O 7 22 O 7 42 7 62 0782 28 3i 7747 77 66 77 85 7 8o4 7 8 2 4 7843 28 32 0802 0822 o842 ! o862 0882 9 02 2 7 32 7862 7 88 2 79 oi 79 20 79 3 9 79 5 9 2 7 33 0922 0941 o 9 6i o 9 8i IOOI 1021 26 33 797 8 7997 8016 8o36 8o55 8o 7 4 26 34 io4i 1061 1081 noi II2I Il4l 25 34 8o 9 3 8n3 8i32 8i5i 8i 7 o 8i 9 o 25 35 1161 1181 I2OI 1221 1240 I26O 24 35 8209 8228 8247 826 7 8286 83o5 24 36 1280 i3oo i32o i34o i36o i38o 23 36 8324 8343 8363 8382 84oi 8420 23 37 i4oo l42O i43 9 i45 9 i4 79 1499 22 37 844o 845 9 84 7 8 84 9 7 85!6 8536 22 38 i5i 9 i53 9 i55 9 i5 79 i5 99 1618 21 38 8555 85 7 4 85 9 3 8612 8632 865i 21 3 9 i638 i658 i6 7 8 : i6 9 8 I 7 i8 i 7 38 2O 3 9 86 7 o 868 9 8 7 o8 8728 8 7 4 7 8 7 66 2O 4o 9. 86:758 1777 i 797 1817 i83 7 i85 7 I 9 4o 9 .868 7 85 8804 88 2 3 8843 8862 8881 J 9 4i 1877 i8 97 I 9 i6 I 9 36 i 9 56 i 97 6 18 4i 8900 8 9 i 9 8 9 3 9 8 9 58 8 977 8 99 6 18 42 i 99 6 2016 2 o35 2o55 20 7 5 20 9 5 J 7 42 9015 9 o34 9 o53 97 3 9 9 2 9 in 17 43 2Il5 2i35 2 1 54 21 7 4 2I 9 4 22l4 16 43 9 i3o 9 i4 9 9 i68 9 i88 9 20 7 9 226 16 44 2234 2254 22 7 3 J 22 9 3 2 3i3 2333 i5 44 9245 9 264 9 283 9 302 9 32I 9 34i i5 45 2353 23 7 2 23 9 2 2412 24312 2452 i4 45 g36o 9 3 79 9 3 9 8 9 4i7 9 436 9 455 i4 46 2471 24 9 I 25ll 253i 2 55i 25 7 i3 46 94 7 4 9 4 9 4 9 5i3 9 53 2 9 55i 9 5 7 o i3 47 25 9 o 26lO 2 63o 265o 2669 268 9 12 47 9 58 9 9 6o8 9 62 7 9 646 9 665 9 685 12 48 2709 2 7 2 9 2 7 48 2 7 68 2 7 88 2808 II 48 97 o4 9723 97 42 97 6i 97 8o 9799 II 49 2827 2847 2 86 7 288 7 2906 2926 IO 49 9818 9 83 7 9 856 9 8 7 5 9 8 9 4 99 i4 IO 5o 9.862946 2 9 66 2 9 85 3oo5 3o25l3o45 9 5o 9.86 99 33 9952 997 1 999 ...9 ..28 9 5i 3o64 3o84 3io4 3i24 3i43 3i63 8 5i 9 .8 7 oo4 7 0066 oo85 oio4 0123 Ol42 8 52 3i83 32o3 3222 3242 3262 3281 7 52 0161 0180 oi 99 0218 0238 O25 7 7 53 33oi 332i 334i 336o 338o 34oo 6 53 O2 7 6 0295 o3i4 o333 o352 o3 7 i 6 54 34i9 343 9 345 9 34 7 8 3498 35i8 5 54 o3 9 o 0409 0428 o44 7 o466 o485 5 55 3538 355 7 35 77 35 97 36i6 3636 4 55 o5o4 o523 o542 o56i o58o o5 99 4 . r >6 3656 36 7 5 36 9 5 3 7 i5 3 7 34 3 7 54 3 56 0618 o63 7 o656 o6 7 5 o6 9 4 o 7 i3 3 57 3 77 4 3 79 3 38i3 3833 3852 38 7 2 2 5 7 0782 o 7 5i o 77 o o 7 8 9 0808 o82 7 2 58 3892 3 9 n 3 9 3i 3 9 5i 3 97 o 3 99 o I 58 o846 o865 o884 o 9 o3 o 9 22 o 9 4i I 5 9 4oio 402 9 4o4 9 4o6 9 4o88 4io8 5 9 o 9 6o o 979 o 99 8 ioi 7 io36 io54 O 60" 50" 40" 30" I 20" 10" c 60" 50" 40" 30" 20" 10" | e - Co-sine of 43 Degrees. 2 Co-sine of 42 Degrees. .Cl" 2" 3" 4" 5" 6" 7" 8" 9" p p t f 1" 2" 3" 4" 5" 6" 7" 8" 0" lrt } 2 4 G 8 10 12 14 16 18 1 I 2 4 6 8 10 12 14 15 17 LOGARITHMIC TANGENTS. a ' Tangent of 46 Degrees. d Tangent of 47 Degrees. S J 0" ' | 10" 20"~J~30" 40" 50" 2 0" 10" ?0" 30" 40" 50" ;j 1 o io.oi5i63 52o5 5 2 4 7 5289 533i 53 7 3 5 9 io.o3o344 o386 0429 0471 o5i3 o555 5 9 I 54i6 5458 55oo 5542 5584 5626 58 I 0597 o64o 0682 0724 0766 0808 58 2 5668 5711 5 7 53 5 79 5 583 7 58 79 5 7 2 o85i o8 9 3 o 9 35 977 1020 1062 5 7 3 5921 5 9 63 6006 6o48 6090 6i32 56 3 1 104 n46 1188 I23l 1273 i3i5 56 4 6i 7 4 6216 6258 63oi 6343 6385 55 4 i35 7 i4oo i442 i484 i5 2 6 1 568 55 5 6427 646 9 65n 6553 65 9 6 6638 54 5 1611 i653 1695 i 7 3 7 1780 1822 54 6 6680 6722 6 7 64 6806 6848 68 9 i 53 6 1864 1906 i 9 48 1991 2033 2075 53 7 6933 6 97 5 701-7 7 o5 9 7101 7i43 52 7 2117 2160 2202 2244 2286 2328 5 a 8 7186 7228 7 2 7 7 3l2 7354 7 3 9 6 5i 8 2371 24i3 2455 2497 254o 2582 5i 9 7438 748 1 7 523 7 565 7607 7 649 5o 9 2624 2666 2 7 9 2 7 5l 2 79 3 2835 5o 10 10.017691 77 33 7776 7 8i8 7860 7 9Q2 4 9 10 10.032877 2920 2 9 62 3oo4 3o46 3o8 9 4 9 ii 7944 79 86 8028 8o 7 i 8ii3 8i55 48 ii 3i3i 3i 7 3 32i5 3 2 58 33oo 3342 48 i 12 8197 823 9 8281 8323 8366 84o8 47 12 3384 3426 346 9 35n 3553 35 9 5 47 i3 845o 84 9 2 8534 85 7 6 8618 8661 46 i3 3638 368o 3722 3 7 64 38o 7 384 9 46 i4 8703 8 7 45 8787 8829 8871 8914 45 i4 3891 3 9 33 3 97 6 4oi8 4o6o 4l02 45 i5 8 9 56 8998 9 o4o 9 o82 9124 9166 44 i5 4i45 4i8 7 422 9 42 7 I 43i3 4356 44 16 9209 925i 9 2 9 3 q335 9 3 77 9419 43 16 43 9 8 444o 4482 45 2 5 456 7 4609 43 7 9462 9 5o4 9 546 o588 9 63o 9 6 7 2 42 I? 465 1 46 9 4 4?36 4778 4820 4863 42 18 9714 97 5 7 9799 9 84i 9 883 99 25 4i 18 4905 4 9 4 7 4 9 8 9 5o32 5o 7 4 5n6 4i J 9 9967 . .10 ..62 .. 9 4 .136 .178 4o J 9 5i58 52OI 5243 5 2 85 532 7 53 7 o 4o 20 IO.O20220 0262 o3o5 0347 o38 9 o43i 3 9 20 io.o354i2 5454 5496 553 9 558i 5623 3 9 21 0473 o5i5 o558 0600 0642 o684 38 21 5665 5708 5750 5 79 2 5834 58 7 7 38 22 0726 0768 0810 o853 o8 9 5 o 9 3 7 3 7 22 6919 5961 6oo3 6o46 6088 6i3o 3 7 23 0979 1021 io63 1106 n48 II 9 O 36 23 6172 62i5 6257 6299 634i 6384 36 24 1232 1274 i3i6 i35 9 i4oi i443 35 24 6426 6468 65n 6553 65 9 5 663 7 35 25 i485 1527 i56 9 1612 i654 i6 9 6 34 25 6680 6^22 6 7 64 6806 684 9 6891 34 26 1738 1780 1822 i865 1907 i 9 4 9 33 26 6 9 33 6 97 5 7018 7 o6o 7 I02 7i44 33 2 7 1991 2033 2075 2118 2160 22O2 32 27 7187 7229 7271 73i4 7 356 7 3 9 8 32 28 2244 2286 2328 2370 24l3 2455 3i 28 744o 7483 7 5 2 5 7 56 7 7 6o 9 7652 3i 2 9 2497 2539 258i 2623 2666 2 7 o8 3o 29 7694 7736 7778 7 82I 7 863 79 o5 3o 3c 10.022750 2792 2834 2877 2919 2 9 6i 29 3o 10.037948 799 8o32 8o 7 4 8117 8i5 9 29 3i 3oo3 3o45 3o8 7 3i3o 3172 3214 28 3i 8201 8243 8286 8328 83 7 o 84i3 28 32 3256 32 9 8 334o 3383 3425 346 7 27 32 8455 84 9 7 853 9 8582 8624 8666 27 33 35og 355i 35 9 3 3636 36 7 8 3 7 20 26 33 8708 8 7 5i 8 79 3 8835 8878 8920 26 34 3762 38o4 3846 388 9 3931 3 97 3 25 34 8962 9004 9047 9089 9 i3i 9174 25 35 4or5 4o5 7 4o 99 4i42 4i84 4226 24 35 9216 9 258 gSoo 9 343 9 385 9427 24 36 4268 43io 4353 43 9 5 443 7 4479 23 36 9470 9512 9 554 9 5 9 6 9 63 9 9681 23 3? 452i 4563 46o6 4648 4690 4732 22 37 9723 9766 9808 9 85o 9892 99 35 22 38 4774 48i 7 485 9 4901 4943 4985 21 38 9977 ..19 ..62 . io4 .i46 .188 21 3 9 5027 6070 5lI2 5i54 5i 9 6 5238 20 3 9 io.o4o23i 0273 o3i5 o358 o4oo 0442 2O 4o 10.025280 5323 5365 54o 7 5449 549i 1 9 4o io.o4o484 0527 0569 061 1 o654 0696 19 4i 5534 55 7 6 56i8 566o 5702 5 7 45 18 4i o 7 38 0781 0823 o865 0907 0960 18 42 5 7 8 7 582 9 5871 5 9 i3 5 9 55 5 99 8 J 7 42 O 99 2 io34 1077 ni 9 1 161 1204 *7 43 6o4o 6082 6124 6166 620 9 625i 16 43 1246 1288 i33o i3 7 3 i4i5 1457 16 44 6293 6335 63 77 64i 9 6462 65o4 i5 44 i5oo i542 i584 1627 1669 I 7 II i5 45 6546 6588 663o 66 7 3 6 7 i5 6 7 5 7 i4 45 I7 53 1796 i833 1880 1923 i 9 65 i4 46 6799 684i 6884 6 9 26 6 9 68 7 OIO i3 46 200 7 2o5o 2092 2134 2177 2219 i3 4? 7052 7 o 9 5 7 i3 7 7 i 79 7221 7 263 12 4? 226l 23o3 2346 2388 243o 2473 12 48 7 3o5 7348 7 3 9 o 7432 7474 7 5i6 I I 48 2 5i5 2 55 7 2600 2642 2684 2727 II 49 7 55 9 7601 7643 7685 7727 7770 IO 49 2 7 6 9 2811 2854 280.6 2 9 38 2980 IO 5o 10.027812 7 854 7896 7938 79 8i 8o23 9 5o io.o43o23 3o65 3107 3i5o 3192 3 2 34 9 5i 8o65 8107 8i4 9 8l 9 2 8234 8276 8 5i 32 77 33 I9 336i 34o4 3446 3488 8 52 83i8 836o 84o3 8445 848 7 852 9 7 52 353i 35 7 3 36i5 3658 3700 3742 7 53 85 7 i 86i4 8656 8698 8 7 4o 8782 6 53 3785 3827 386 9 3 9 I2 3 9 54 3996 6 54 8825 886 7 8909 8 9 5i 8 99 3 9 o36 5 54 4o3 9 4o8i 4123 4i65 4208 425o 5 55 9078 9 I20 9162 9204 9 24 7 9289 4 55 4292 4335 43 77 44i 9 4462 45o4 4 55 933i 9 3 7 3 9 4i6 9458 9 5oo 9 542 3 56 4546 458o 463i 46 7 3 4716 4758 3 5 7 9 584 9 62 7 9 66 9 9711 97 53 979 5 2 5 7 48oo 4843 4885 4927 4970 5012 2 Cft 9 838 9880 99220064 ...6 ..4 9 I 58 5o54 5o 97 SiSg 5i8i 5224 5a66 I 5 9 io.o3oo 9 i oi33 oi 7 5 0217 0260 0302 O 5 9 53o 9 535i 53g3 5436 5478 5520 O 60" 50" 40" 30" 20" 10" H 60" 50" 40" 30" ao" 10" ^ Co-tangent of 43 Decrees. s Co-tangent of 42 Degrees. .? E p ( 1" 2" 3" 4" .V 6" 7" 8" 9" P Part J l " ~" 3 " 4 " 5 " 6 " 7 " 8// ' J " / 4 8 13 17 21 25 30 34 38 > 4 8 1? r ','1 25 30 34 38 j LOGARITHMIC IS i N E s. J3 Sine of 48 Degrees. A Sine of 49 Degrees. 2 0" | 10' 20" 30" 40' 50' & 0' 10" 20" 30" 40' 50" O 9 .87io73|io 9 2 ii 1 1 i i3o 1149 1168 5 9 o 9 .8 777 8o 779 8 7 8i6 7835 7 853 7871 5o I 1187 1206 1225 1244 1263 1282 58 i 7 8 9 o 79 o8 7926 7945 79 63 7981 58 2 i3oi 1320 1339 i358 i3 77 i3 9 55 7 2 7999 8018 8o36 8o54 8o 7 2 8o 9 i 5 7 3 i4i4 i433 i452 i47i;i49o i5o 9 56 5 8io 9 8127 8i46 81648182 8200 56 4| i5 2 8 1547 i566 i585 1604 1622 55 L 82I 9 8 2 3 7 8255 8273 82 9 2 83io 55 D i64i 1660 1679 1698 1717 i 7 36 54 t 83 2 8 8346 8365 8383 84oi 84i 9 54 i 7 55 1774 1793 1811 i83o i84 9 53 6 8438 8456 8474 8492 85n 85 29 53 7 1868 1887 1906 1925 1944 I 9 62 52 7 8547 8565 8583 8602 8620 8638 5 2 8 1981 2000 2019 2o38 2057 20 7 6 5i 8 8656 86 7 5 8693 8711 8 7 2 9 8 7 47 5! 9 20 9 5 2Il3 2l32 2l5l|2I70 2l8 9 5o 9 8 7 66 8 7 84 8802 8820 8838 885 7 5o 1019.872208 2226 2245 2264 2283 2302 49 10 9 .8 7 88 7 5 88 9 3 8911 8 9 2 9 8 9 48 8 9 66 49 1 1 2^21 2340 235823772396 24i5 48 ii 8 9 84 9 OO2 9020 9 o3 9 9 o5 7 9 o 7 5 48 12 2434 2452 2471 2490,2509 2528 47 12 9 o 9 3 9 III 9129 9 i48 9 i66 47 i3 2547 2565; 25342603:2622 2641 46 i3 9 2O2 9 22O 9238 9 25 7 9 2 7 5 9 2 9 3 46 i4 2 65 9 2678269727162735 2 7 53 45 i4 9 3n 9 32 9 9 34 7 9 365 9 3S4 9 402 45 i5 2772 2791! 2810 2829 2847 2866 44 i5 9 420 9 438 9 456 9 4 7 4 9492 9 5n 44 16 2885 2904 2923 2941 2 9 6o 2979 43 16 9529 9 54 7 9 565 9 583 9 6oi 9619 43 17 2 99 8 3oi6 3o35 3o54 3o 7 3 3091 42 *7 9 63 7 9656 9 6 7 4 9692 97 io 97 28 42 18 3no 3i2 9 3i483i66 3i85 3204 4i 18 9746 9764 97 82 9800 9 83 7 4i 19 3223 324l 32603279 32 9 8 33i6 4o 19 9 855 9873 9 8 9 i 9909 9927 99 45 4o 20 9. 873335 3354 3373 3391 34io 3429 3 9 20 9 .8 799 63 998! ..18 ..36 ..54 3 9 21 3448 3466 3485 35o4 3522 354i 38 21 9.88oo 7 2 0090 0108 0126 oi44 0162 38 22 356o 35 79 3597 36i6 3635 3653 37 22 0180 0198 0216 0234 0253 0271 37 23 36 7 2 36 9 i 3 7 io:3 7 28 3747 3 7 66 36 23 0289 o3o 7 o3 2 5 o343 o36i o3 79 36 24 3 7 84 38o3 382 2 384o 385 9 38 7 8 35 24 o3 97 o4i5 o433 o45i 0469 0487 35 25 38 9 6 3 9 i5 393453953 3 97 i 3 99 o 34 25 o5o5 o523 o54i o55 9 o5 77 o5 9 5 34 26 4oo 9 4027 4o46j4o65 4o83 4lO2 33 26 o6i3 o63i o64 9 0667 0686 0704 33 27 4l2I 4i3 9 4i58 4177 4i 9 5 4214 32 27 O 7 22 o 7 4o o 7 58 0776 0794 0812 32 28 4232 425i 4270 4288 43o 7 4326 3i 28 o83o o848 0866 o8S4 O 9 02 0920 3i 29 4344 43634382 44oo 44i9 4438 3o 2 9 0938 0956 o 97 4 0992 IOIO 1028 3o 3o 9 .8 7 4456 4475 4493 45i2 453i 4549 29 3o 9.881046 io63 1081 1099 1117 n35 2 9 3i 4568 4586 46o5 4624 4642 466i 28 3i :i53 ii 7 i n8 9 1207 1225 1243 28 32 468o 46984717 4735 4754 4773 27 32 1261 I2 79 I2 9 7 i3i5 i333 i35i 27 33 479 1 48104828 4847 4866 4884 26 33 i36 9 i38 7 i4o5 1423 i44i i45 9 26 34 4903 4921 4g4o 4958 4 9 77 4996 25 34 1477 i4 9 5 l5l2 i53o 1 548 i566 25 35 5oi4 5o335o5i 5070 5o88 5io 7 24 35 1 584 1602 1620 i638 i656 1674 24 36 5i 2 6 5i445i63 5i8i 5200 52i8 23 36 1692 I 7 IO 1728 1746 i 7 63 1781 23 37 523 7 5255 5274 5293 53ii 533o 22 37 i 79 9 1817 i835 i853 1871 1889 22 38 5348 53675385 54o4 5422 544 1 21 38 1907 I 9 25 I 9 42 1960 i 97 8 1996 21 3 9 545 9 5478 5496 55i5 5534 5552 2O 3 9 20l4 2032 2o5o 2068 2086 2103 2O 4o 9.875571 5589 56o8 5626 5645 5663 19 4o 9.882I2I 2l3 9 2l5 7 2I ? 5 2I 9 3 2211 19 4i 5682 5700 5719 5 7 3 7 57565774 18 4i 222 9 2246 2264 2282 23oo 23i8 18 42 5 79 3 58ii 583o 5848 58675885 '7 42 2336 2354 23 7 I 238 9 2407 2425 17 43 5904 5922 5g4i 5 9 5 9 59786996 16 43 2443 2461 2479 2496 25i4 2532 16 44 6oi4 6o33 6o5i 6o 7 o 60886107 i5 44 255o 2568 2586 26o3 2621 2639 i5 45 6i 2 5 6i44 6162 6181 6-1 99 6218 i4 45 265 7 26 7 5 26 9 2 2710 2728 2 7 46 i4 46 6 2 36 6255 6273 62 9 I 63io6328 i3 46 2 7 64 2782 2799 2817 2835 2853 i3 47 6347 6365 6384 6402 6421 643 9 12 47 28 7 I 2888 2 9 o6 2924 2 9 42 2960 12 48 645 7 6476 6494 65i3 653i 655o II 48 2977 2 99 5 3oi3 3o3i 3o4 9 3o66 II 49 6568 6586 66o5 66 2 3 6642 6660 IO 49 3o84 3 1 02 3l2O 3i37 3i55 3i 7 3 IO 5o 9.876678 6697 6715 6 7 34 6762 6770 9 5o 9.883191 32O 9 3226 3244 3262 3 2 8o 9 5i 6789 6807 6826 6844 68626881 8 5i 32 97 33i5 3*333 335i 3368 3380 8 52 6899 69186936 6 9 54 6 97 3 6 99 i 7 52 34o4 3422 343 9 345 7 3475 3493 7 53 7010 7028 7046 7065 7 o83 j7 ioi 6 53 35io 3528 3546 3564 358i 35 99 6 54 7120 7 i38 7 i5 7 7175 7193 7212 5 54 36i 7 3635 365 2 36 7 o 3688 3 7 o5 5 55 7230 7248 7267 7285 7 3o3 7 322 4 55 3 7 2? 3 7 4i 3 7 5 9 3 77 6 3 79 4 38i2 4 56 734o 73587877 7?, 9 5 7 4i3 7 432 3 56 382 9 3847 3865 3883 3 9 oo 3 9 i8 3 5 7 745o 7 468 7 48 7 7 5o5 7 523 7 54s 2 57 3 9 36 3 9 53 3 9 7i 3 9 8 9 4oo6 4024 2 58 756o 7 5 7 8: 7 5 977 6i5 7 633; 7 652 I 58 4o42 4o6o 4077 4095 4n3 4i3o I 5 9 7670 7 688 77 o 7 i 77 25 77437762 O 5 9 4i48 4i664i83 4201 4219 4236 O Uw 50" | 40" 30" 20" 10" rt 63" 50" 40" 30" 20" 10" . Co-sine of 4 1 Degrees. .8 Co-sine of 40 Degrees. & ( i// o/. 3" 4" 5" 6" 7" 8" 9" P PartJ 1 " 2 " 3 " 4 " 5 " 6 " 7 " &> " ^ < 2 4 6 7 9 11 13 15 17 1 \ 24 5 7 9 11 13 14 1> _L O r, A R I T H M I C T A N G E A T 6. i Tangent of 48 Decrees. a Tangent of 49 Degrees. V \ 10" 20" 30" j 40" 50" 0" i(y- 20" 30" 40" 50" io.o45563 56o5 564 7 5690 5 7 32 5 77 4 5 9 o 10.060837 0879 0922 0966 1007 io5o 5 9 I 6817 585 9 5901 5 9 44 5 9 86 6028 58 I 1092 n35 II 77 I22O 1262 i3o5 58 2 3 6071 6325 6n3 636 7 6i55 6198 6409 6452 6240 64 9 4 6282 6537 57 56 r . i34 7 1602 1390 i645 1432 1688 i4 7 5 1730 i5 I? 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56 9 8 5 7 4i 5784 582 7 58 7 o 5 9 i3 22 37 I2II 1254 I2 97 i34i 1 384 l42 7 22 38 5 9 56 5 999 6o4i 6o84 6127 6170 21 38 1470 i5i4 i55 7 1600 1 643 i68 7 21 3 9 62i3 6256 62 99 6342 6385 6428 2O 3 9 1730 i 77 3 1817 1860 I 9 o3 i 9 46 20 4o 10.086471 65i4 655 7 6600 6643 6686 '9 4o 10. ioi 99 o 2o33 2076 2II 9 2i63 2206 T 9 4i 6729 6772 68i5 685 7 6 9 oo 6 9 43 18 4i 224 9 22 9 3 2336 23 79 2422 2466 18 42 6986 7 02 9 7072 7 ii5 7 i58 7201 7 42 25o 9 2552 25 9 6 263 9 2682 2 7 25 17 43 7244 7287 7 33o 7 3 7 3 74i6 7 45 9 16 43 2 7 6 9 2812 2855 28 99 2 9 42 2 9 85 16 44 7502 7545 7 588 763i 7674 7717 i5 44 3o2 9 3o 7 2 3n5 3i58 32O2 3245 i5 45 7760 7 8o3 7 846 7 88 9 79 32 797 5 i4 45 3288 3332 33 7 5 34i8 3462 35o5 i4 46 8018 8061 8io4 8i47 8i 9 o 8232 i3 46 3548 35 9 2 3635 36 7 8 3 7 22 3 7 65 i3 4? 82 7 5 83i8 836i 84o4 8447 84 9 o 12 47 38o8 385i 38 9 5 3 9 38 3 9 8i 4o25 12 4 8533 85 7 6 86i 9 8662 8 7 o5 8 7 48 II 48 4o68 4m 4i55 4i 9 8 4241 4285 II 49 8791 8834 8877 8 9 2O 8 9 63 9 oo6 10 49 43 2 8 43 7 i 44r5 4458 45oi 4545 10 5o 10.089049 9092 9 i35 9 i 7 8 9 22I 9 264 9 5o i o.io4588 463i 46 7 5 4718 4761 48o5 9 5i 9 3o 7 gSSo 9 3 9 3 9 436 9479 9 522 8 5i 4848 48 9 i 4 9 35 4 9 78 5021 5o65 8 52 9 565 9608 9 65i 9 6 9 4 9737 97 8o 7 52 6108 5i5 2 5i 9 5 5238 5282 5325 7 53 9 823 9866 9909 99 5 2 999 5 ..3 9 6 53 5368 54i2 5455 54 9 8 5542 5585 6 54110.090082 0125 0168 02 1 1 0254 02 97 5 54 56 2 8 56 7 2 5 7 i5 5 7 5 9 58o2 5845 5 55 o34o o383 0426 o46 9 O5l2 o555 4 55 588 9 5 9 32 5 97 5 6oi 9 6062 6106 4 56 o? o5 9 8 o856 o64i 0899 0684 0942 0727 o 9 85 0770 1028 o8i3 1071 3 2 56 5 7 6i4 9 64o 9 6i 9 2 6453 6236 64 9 6 62 79 653 9 6322 6583 6366 6626 ) o 58 ui4 n5 7 1200 1243 1286 i32 9 I 58 666 9 6 7 i3 6 7 56 6800 6843 6886 i 5 9 1372 i4i6 i45 9 i5o2 1 545 i588 o 5 9 6 9 3o6 9 73 7017 7060 7io3 7 i4 7 o 80" 50" | 40" 30" 20" 10" 60" 50" 40" 30" 20" 10" d Co-tangent of 39 Degrees. S Co-tangent of 38 Degrees. B 2 P Part* l " ~" 3 " 4 " 5 " 6// 7// 8 " 9 " $ 4 9 13 17 21 20 30 34 39 PPartM" 2 " 3// 4 " 5 " 6 " 7 " 8 " f> " r { 4 9 13 17 22 26 30 3.~> c9 LOGARITHMIC SINES. d 1 Sine of 52 Degrees. c Sine of 53 Degrees. S | 0" I 10" 1 20' 30" 40" 50" i 0" 10" 20" 130" 40" 50" 9.8965326649 663i 6647 6565658i 66646680 6697)6713 5 9 58 I 9 . 9 oa34 9 2444 2364 2460 238o 24 7 5 23 9 6 24 9 I 2607 2428 2523 5 9 58 2 6729(6746 6762 6 779 6 79 5 6812 57 2 253 9 2555 25 7 I 2586 2602 2618 57 3 6828 6844 6861 68 77 68 9 4 6 9 io 56 3 2634 2 65o 2666 2681 26 9 7 2713 56 4 6 9 26 6 9 43 6959 6 97 6 6 99 2 79 55 4 272 9 2 7 45 2 7 6l 2 77 6 2792 28o8J55 5 7025 74i 7 o58 7074 7 o 9 o 7107 54 5 2824 2840 2856 28 7 I 2887 2 9 o3 54 6 7123 7140 7 i56 7172 7 i8 9 7205 53 6 2 9 I 9 2 9 35 2 9 5o 2 9 66 2 9 82 2 99 8 53 7 7222 7 238 7 254 7271 7287 7 3o3 52 7 3oi4 3o2 9 3o45 3o6i 3o 77 3093 52 g 7320 7336 7 353 7 36 9 7 385 7402 5i 8 3io8 3i24 3i4o 3i56 3171 3i8 7 5i y 7 4i8 7434 7 45i 7 46 7 7483 75oo 5o 9 32o3 32I 9 3235 3 2 5o 3266 3282 5o 10,9.897516 7533 7549 7 565 7 58 2 7 5 9 8 49 10 9 . 9 o32 9 8 33i3 332 9 3345 336i 3377 4 9 ii 7614 763i 7647 7663 7680 76 9 6 48 ii 33 9 2 34o8 3424 344o 3455 48 12 7712 7729 77 45 7761 77787794 47 12 348 7 35o3 35!8 3534 355o 3566 47 i3 7810 7827 7 843 7 85 9 7876 7 8 9 2 46 1 3 358i 35 97 36i3 362 9 3644 366o 46 i4 7 9 o8 7924 7941 79 5 7 797 3 799 45 i4 3676 36 9 i 3 7 o 7 3 7 23 3 7 3 9 3 7 54 45 i5 8006 8022 80398055 8071 8088 44 i5 3 77 o 3 7 86 38o2 38i 7 3833 384 9 44 16 8io4 8120 8i368i53 8i6 9 8i85 43 16 3,864 388o 38 9 6 3 9 I2 3 9 2 7 3 9 43 43 171 8202 8218 8 2 34 825o 82678283 42 17 3 9 5 9 3 97 4 3 99 o 4oo6 4O2I 4o3 7 42 18 82 99 83i5 8332 8348 8364 838i 4i 18 4o53 4o6 9 4o84 4ioo 4n6 4i3i 4i 1 9 83 97 84i3 8429 8446 8462 4o : 9 4i4 7 4i63 4i 7 8 4i 9 4 4210 422D 4o 20 9.8 9 84 9 4 85u 852 7 8543 855 9 85 7 6 3 9 20 9 . 9 o424i 42 5 7 42 7 2 4288 43o4 43i 9 3 9 21 85 9 2 8608 86248641 865 7 86 7 3 38 21 4335 435i 4366 4382 43 9 8 44i3 38 22 868 9 8706 8722 8738 8754 8 77 o 3? 22 442 9 4445 446o 44?6 44 9 2 45o 7 37 23 8787 88o3 8819 8835 885 2 8868 36 23 4523 453 9 4554 4570 4586 46oi 36 24 8884 8900 8916 8 9 33 8 9 4 9 8 9 65 35 24 46i 7 4632 4648 4664 46 79 4695 35 25 8 9 8i 8997 9014 9 o3o 9 o46 9 o62 34 25 4711 4726 4 7 42 4 7 5 7 4773 4 7 8 9 34 26 9078 9095 9111 9 I2 7 9 i43 9 i5 9 33 26 48o4 4820 4836 485i 486 7 4882 33 27 9176 9192 9208 9224 9 240 9 256 32 27 48 9 8 4 9 i4 4 9 2 9 4 9 45 4 9 6o 49 7 6 32 28 9273 9289 9 3o5 9 32I 9 33 7 9 354 3i 28 4 99 2 5007 5o23 5o38 5o54 5o 7 o 3i 291 9370 9386 9402 <;4i8 9 434 9 45o 3o 29 5o85 5ioi 5n6 5i32 5i48 5i6! 3o 809.899467 9 483 9499 9 5i5 9 53! 9 54 7 2 9 3o 9 . 9 o5i7 9 5i 9 4 5210 5225 524i 52t'f 2O 3i 9 564 9580 9 5 9 6 9612 9 628 9 644 28 3i 5272 5288 53o3 53i 9 5334 535o 2o 32 9 66o 9677 9 6 V 3 9709 97 25 97 4i 27 32 5366 538i 53 97 54i2 5428 5443 27 33 97 5 7 977 3 9789 9806 9 822 9 838 26 33 545 9 5474 54 9 o 55o6 552i 553 7 26 34 35 9 854 99 5i 9870 g886 9902 9967(9983 9999 9918 99 35 25 24 34 35 5552 5645 5568 566i 5583 56 7 6 55 99 56 9 2 56i4 5 7 o8 563o 5723 25 24 36 9 .9ooo4 7 oo63 oo7 9 oo 9 6 0112 0128 23 36 5 7 3 9 5 7 54 5 77 o 5 7 85 58oi 58i6 23 3? oi44 or6o 0176 OI 9 2 0208 0224 22 37 5832 5847 5863 58 7 8 58 9 4 5909 22 38 0240 02*16 0272 O28 9 o3o5 0321 21 38 5 92 5 5 9 4o 5 9 56 5 97 i 5 9 8 7 6002 21 3 9 o33 7 o353 o36 9 o385 o4oi o4i 7 2O 3 9 6018 6o33 6o4 9 6o64 6080 6095 2O 4o 9.900433(0449 o465o48i o4 9 7 o5i3 I 9 4o 9 . 9 o6ni 6126 6142 6i5 7 6i 7 3 6188 I 9 4i 0629 o545 o562 0578 o5 9 4 0610 18 4i 6204 62I 9 6235 6260 6265 6281 1 8 42 o66 0642 o658 0674 o6 9 o o 7 o6 17 42 62 9 6 63i2 632 7 6343 6358 63 7 4 17 43 0722 0738 0754 0770 o 7 86 0802 16 43 638 9 64o5 6420 6436 645 1 6466 1 6 44 0818 o834 o85oo866 0882 o8 9 8 i5 44 648a 64 9 7 65i3 6528 6544 655 9 i5 45 0914 0980 o 9 46|o 9 62 o 97 8 o 99 4 i4 45 65 7 5 65 9 o 66o5 6621 6636 665 2 i4 46 fOIO 1026 IO42 io58 io 7 4 IO 9 O i3 46 6667 6683 66 9 8 6 7 i3 6 7 2 9 6 7 44 i3 47 /1 06 1122 n38 n54 1170 1186 12 47 6760 6 77 5 6 79 i 6806 6821 683 7 12 48 12O2 1218 1234 I25o 1266 1282 II 48 6852 6868 6883 68 9 8 6 9 i4 6929 II 49 1298 i3i4 i33o 1 346 i362 i3 7 8 10 49 6 9 45 6 9 6o 6 97 5 6 99 i 7006 7 O22 10 56 q qoi3 9 4 i4io i4a6 1 442 i458 1 4?4 9 5o 9 . 9 o7o37 7052 7 o68 7 o83 7 o 99 7 n4 9 5i 1490 i5o5 l52I i53 7 r553 i56 9 8 5i 7 I2 9 7i45 7 i6o 7 i 7 5 7i 9 i 7 206 8 52 i585 1601 1617 i633 i64 9 i665 7 52 7222 723 7 7 252 72 68 7 283 7298 7 53 1681 1697 I 7 i3 1729 i 7 45 1760 6 53 73i4 7 32 9 7 344 7 36o 7 3 7 5 7 3 9 i 6 54 1776 1792 1808 1824 i84o i856 5 54 7406 7421 7 43 7 7 45 2 7 46 7 7483 5 55 1872 1888 i 9 o4 1920 i 9 36 i 9 5i 4 55 7 4 9 8 7 5i3 7 52 9 7 544 7 55 9 7 5 7 5 4 56 1967 i 9 83 i 999 2Ol5 2031 2047 3 56 7 5 9 76o5 7 62I 7 636 7 65i 7 66 7 3 5 7 2o63 2079 20 9 5 2110 2126 2l42 2 57 7682 7 6 97 77 i3 7728 77 43 77 5 9 2 58 2i58 2174 2I 9 2206 2222 2238 I 58 7774 7789 7 8o5 7820 7 835 7 85i I 5 9 2253 2269 2285 2301 2317 2333 O 5 9 7866 7881 7 8 9 6 79 I2 7927 79 4a O 60" 50" 40" 30" | 20" 10" a 60" 50' 40" 30" 20" 10" d Co-sine of 37 Degrees. Co-sine of 36 Degrees. | 3 ( I" " 3" 4" P. Part ^ 2356 5" f5'' 7" 8' 9" 8 10 11 13 15 I'.PartJ 2 ~3 5 6 8 9 11 12 14 ' LOGARITHMIC TANGENTS. n Tangent of 52 Degrees. d Tangent of 53 Degrees. 3 ()'' | 10" 20" 30" 40" 50" % 0" 10" 20" So" 40" 50" 10. 107190 7 23Z 7277 7 320 7 364 7 4o 7 5 9 o 10.122885 2 9 2 9 2 97 3 3017 3o6i 3io5 5 9 I 745 1 7494 7 53 7 7 58i 7 62/ 7668 58 i 3i48 3i 9 2 3236 3280 3324 3368 58 2 7711 77 54 779 8 784i 7 885 79.28 57 2 34xi 3455 34 9 9 3543 358 7 363o 5 7 ^ 7972 8oi5 8o58 8102 8i45 8189 56 3 36 7 4 3718 3 7 62 38o6 385o 38 9 3 56 4 8232 82 7 5 8319 8362 84o6 8449 55 4 3 9 3 7 3 9 8i 402D 4o6 9 4u3 4i5 7 55 8493 8536 85 79 8623 8666 8710 54 f ; 4200 4244 4288 4332 43 7 6 4420 54 6 8753 8 797 884o 8884 8 9 2 7 8970 53 G 4463 45o 7 455i 45 9 5 463 9 4683 53 7 9014 9 o5 7 9101 9i44 9 i88 9 23l 52 7 4727 4770 48i4 4858 4 9 O2 4946 52 8 9275 9 3i8 9 36i 94o5 9 448 9492 5i 8 4990 5o34 5o 77 5l2I 5i65 5209 5i 9 9 535 9 5 79 0,622 9666 979 9753 5o 9 5 2 53 52 97 534i 5385 5428 5472 5o 10 10. 109796 9 84o 9 883 9926 997 ..i3 49 10 10. 1255:6 556o 56o4 5648 56 9 2 5 7 36 / 1 1 to. i 10057 OIOO oi44 oi8 7 0231 0274 4 i ] 5 7 8o 5823 586 7 5 9 u 5 9 55 5 999 12 o3i8 o36i o4o5 o448 0492 o535 47 12 6o43 6o8 7 6i3i 6i75i62i 9 6262 47 i3 o5 79 0622 0666 o 7 o 9 O 7 52 0796 46 q 63o6 635o 63 9 4 6438 6482 6526 46 U o33 9 o883 9 26 o 97 o ioi3 io5 7 45 4 65 7 o 66i4 6658 6701 6 7 45 6 7 8 9 45 i5 i roo n44 1187 I23l I2 7 4 i3i8 44 5 6833 6877 6 9 2I 6 9 65 7 oo 9 7 o53 44 16 i36i r4o5 i448 1492 i535 i5 79 43 G 797 7141 7 i85 7229 7273 7 3i6 43 J 7 1622 1666 1709 i 7 53 i 797 i84o ^T- 1 7 736o 74o4 7 448 74 9 2 7 536 7 58o 42 18 i884 I 9 2 7 1971 20l4 2o58 2101 4i g 7624 7668 7712 77 56 7 8oo 7844 4i J 9 2i45 2188 2232 22 7 5 2 3i 9 2362 4o 9 7888 79 3-a 7976 8020 8o63 8io 7 4o 20 10. 112406 2449 24 9 3 2536 258o 2623 J 9 21: 10. 128i5i 8i 9 5 823 9 8283 832 7 83 7 i 3 9 21 2667 2711 2 7 54 2 79 8 2841 2885 iS 21 84i5 845 9 85o3 854 7 85 9 i 8635 38 22 2 9 28 2 97 2 3oi5 3o5 9 3lO2 3i46 37 22 8679 8723 8767 8811 8855 8899 37 23 3i8 9 3 2 33 3277 332O 3364 34o 7 16 23 8 9 43 8 9 8 7 9 o3i 9 o 7 5 9119 9 i63 36 24 345i 3494 3538 358i 3625 366 9 35 24 9207 9 25l 9 2 9 5 9 33 9 9 383 9 42 7 35 25 3712 3 7 56 3 799 3843 3886 3 9 3o 34 25 9471 9 5i5 9 55 9 9 6o3 9647 9 6 9 i 34 20 3 97 4 4oi 7 4o6i 4io4 4i48 4i 9 i 33 26 9735 9779 9 823 9 86 7 9911 99 55 33 2 7 4235 42 7 9 4322 4366 4409 4453 32 ^7 9999 ..43 ..87 .i3i .i 7 5 .2I 9 32 28 4496 454o 4584 462 7 46 7 i 4714 3i 28 10. i3o263 o3o 7 o35i o3 9 5 0439 o483 3i 29 4758 4802 4845 483 9 4 9 32 4976 3o 29 o52 7 o5 7 i o6i5 o65 9 o 7 o3 o 7 4? 3o 3o IO. I I5O2O 5o63 5107 5i5o 5 1 94 5 2 38 29 30 10. 130791 o835 0879. O923 o 9 6 7 1011 29 3i 6281 5325 5368 54i2 5456 5499 a 8 3i io55 1099 n43 1187 1232 1276 28 32 5543 5586 563o 56 7 4 5 7 i 7 5 7 6i 2 7 32 l320 i364 i4o8 i452 i4 9 6 i.54o 2 7 33 58o4 5848 58 9 2 5 9 35 5 979 6023 20 J3 1 584 1628 1672 I 7 i6 i 7 6o 1804 26 34 6066 6no 6i53 6i 97 6241 6284 25 34 1 848 1892 1936 i 9 8i 2O25 2o6 9 25 35 6328 63 7 2 64i5 645 9 65o2 6546 ^4 35 2Il3 2i5 7 2201 2245 228 9 2333 24 36 6590 6633 6677 6 7 2I 6 7 64 6808 23 3G 23 77 2421 2465 25o 9 2 554 2 5 9 8 23 3? 6852 6895 6 9 3 9 6982 7 O26 7 o 7 o 22 >7 2642 2686 2 7 3o 2 77 4 2818 2862 22 38 7'il3 7i5 7 7201 7 244 7 288 7332 21 38 2 9 o6 2 9 5o 2994 3o3 9 3o83 3l2 7 21 3 9 7 3 7 5 74i9 7463 7 5o6 7 55o 7594 20 3 9 3171 32i5 3 2 5 9 33o3 334 7 33 9 i 2O 4o 10. 117637 7681 7725 77 68 7 8l2 7 856 T 9 4o 10. 133436 348o 3524 3568 36i2 3656 19 4i 7899 7943 7987 8o3o 8o 7 4 8118 18 4i 8700 3 7 44 3789 3833 38 77 3 9 2I 18 42 8161 8 2 o5 8249 8292 8336 838o 17 42 3 9 65 4oo 9 4o53 4o 97 4i42 4i86 17 43 8423 846 7 85u 8555 85 9 8 8642 16 43 423o 42-74 43i8 4362 44o6 445 1 16 44 8686 8 7 2 9 8 77 3 881-7 8860 8904 i5 44 44 9 5 453 9 4583 462 7 46 7 i 47i5 i5 45 8948 8 99 2 9o35 979 9 I23 9166 i4 45 4760 48o4 4848 48 9 2 4 9 36 4 9 8o i4 46 9210 92 54 9297 934i 9 385 9429 i3 46 5o25 5o6 9 5u3 5i5 7 5201 5245 i3 4? 9472 9 5i6 9 56o 9 6o3 9 64 7 9691 12 47 6290 5334 53 7 8 5422 5466 55io 12 48 9735 9778 9822 9 866 999 99 53 I I 48 5555 55 99 5643 568 7 5 7 3i 5 77 5 II 49 9997 ..4i ..84 .128 .172 .216 IO 49 6820 5864 5 9 o8 5 9 5 2 5 99 6 6o4i IO 5o IO. I2O25 9 o3o3 o347 o3 9 i o434 0478 9 5o io.i36o85 6i2 9 6i 7 3 62I 7 6262 63o6 9 5i O522 o565 0609 o653 o6 97 0740 8 5i 635o 63 9 4 6438 6483 652 7 65 7 i 8 52 0784 0828 0872 o 9 i5 o 9 5 9 ioo3 7 52 66i5 665 9 6704 6 7 48 6 79 2 6836 7 53 1047 io 9 i n34 1178 1222 1266 6 53 6881 6 9 25 6 9 6 9 7 oi3 7o5 7 7 I02 6 54 i3o 9 i353 i3 97 i44i i484 i528 5 54 7i46 7 i 9 o 7234 7 2 79 7 323 7 36 7 5 55 1572 1616 i65 9 I 7 o3 1747 1791 4 55 74n 7455 75oo 7544 7 588 7 63 2 4 56 i835 i8 7 8 1922 1966 2OIO 2o53 3 56 7677 7721 7765 7809 7 854 7 8 9 8 3 57 20 97 2l4l 2i85 2229 2272 2 3i6 2 57 7942 79 86 8o3i 8075 8119 8i63 2 58 2 36o 2404 2448 2491 2535 2579 I 58 8208 8252 82 9 6 834i 8385 842 9 1 5 9 2623 266 7 2 7 IO 2 7 54 2 79 8 2842 O 5 9 8473 85i8 8562 8606 865o 86 9 5 O 60" | 50" 40" 30" 20" 10" K 60" 50" 40" 30" *0" 10" d Co-tangent of 37 Degrees. Co-tangent of 36 Degrees. & T "P^i- j 1" 2" 3" 4" 5" 6" 7" 8" 9" P Port j *" 2 " 3 " 4 " 5 " 6 " 7 " S " 9 " n | 4 9 13 17 22 26 31 35 39 f. rart. -j 4 9 13 18 22 o 6 31 35 40 .s Sine of 54 Degrees. a Sine of 55 Degrees. j 1 0" 10" 20" 30" 40" | 50" | 3 0" 10" 20" 30" 40' 50" I 9-979 58 797 3 8049 8o65 7 9 888oo48oi 9 !8o34 80808095 8111 8126 c O 9 . 9 i3365|337 9 3453 3468 33 9 4 3482 34o 9 34 9 7 3423 35i2 3438 35 27 58 2 8i4i 8i568i 7 2 8187820218217 57 354i 3556 35 7 i 3585 36oo 36i5 5 7 3 82338248 8263827982948309 56 fl 363o 3644 365 9 36 7 4 3688 3 7 o3 50 4 8324 834o ( 8355 83 7 o 8385 84oi 55 4 3 7 i8 3 7 33 3 7 4 7 3 7 62 3?77 3 79 i 55 5 84i6 843i 18446 8462 8477 84 9 2 54 c 38o6 382i 3836 385o 3865 388o 54 6 8507 8523 8538J85538568 8584 53 6 38 9 4 3909 3 9 24 3 9 38 3 9 53 3 9 68 53 7 8099 86i4 862 9 8644 8660 86 7 5|5 2 7 3 9 82 3o 97 4012 4026 4o4i 4o56 52 8 8690 8 7 o5 '8721 8736 8751 8 7 66l5i 8 4070 4o85 4ioo 4n4 4i2 9 4i44 5i 9 10 8781 9.908873 87971881288278842 88888 9 o38 9 i88 9 33 885 7 8 9 4 9 5o 49 9 10 4i58 4261 4i88 42 7 6 4202 42 9 42I 7 43o5 4232 4320 5o 4 9 ii 8 9 64 9o55 8 979 .8 99 4: 9 oo 9 9 O25 0,0709085)9101 9 n6 9 o4o 9 i3i 48 47 ii 12 4334 AA22 4349 443 7 4364 445 1 4378 4466 43 9 3 448 1 4407 44 9 5 48 47 i3 9 i46 9161 917691929207 9 222 46 i3 45io 45 2 4 453 9 4554 4568 4583 46 ii 9 23 7 9 252 '9267 9283 9 2 9 8 9 3i3 45 i4 45 9 8 4612 462 7 464i 4656 46 7 i 45 i5 9 328 9 343 9 358 9 3 7 4 9 38 9 9 4o4 44 i5 4685 4700 4 7 i4 472 9 4744 4758 44 16 9 4i 9 9 434 9449 9 464 9 48o 9 4 9 5 43 16 4773 4 7 8 7 4802 48i 7 483i 4846 43 17 9 5io 9 525 ! 9 54o 9 555 9 57o 9 586 42 17 486o 48 7 5 48 9 o 4 9 o4 4 9 i 9 4 9 33 42 18 9 6oi 9 6i6 9 63i 9646 9661 9 6 7 6 4i 18 4 9 48 4962 4 9 77 4 99 2 5oo6 5021 4i i 9 9 6 9 i 97 o 7 ! 97 22 97 3 797 52 97 6 7 4o i 9 5o35 5o5o 5o64 5o 79 5o 9 4 5io8 4o 20 9 . 9 o 97 8 2 9797 9 8i2 0.827 9 843 9 858 3 9 20 9 . 9 i5i23 5i3 7 5i5 2 5i66 6181 5i 9 6 3 9 21 9873 9 888 99 o3 9918 99 33 99 48 38 21 5210 5225 5 2 3 9 5254 5268 5 2 83 38 22 99 63 99789994 ...9 ..24 ..3 9 37 22 52 97 53i2 5326 534i 5356 53 7 o 37 23 9 . 9 ioo54 0069 oo84 99 oii4 0129 36 23 5385 53 99 54i4 5428 5443 545 7 36 24 0144 oi5 9 oi 7 5 oi 9 o O205 0220 35 24 5472 5486 55oi 55i5 553o 5544 35 25 0235 O25o'o265 0280 O2 9 5 o3io 34 25 555 9 55 7 3 5588 56o2 5617 563i 34 26 o325 o34o o355 o3 7 o o385 o4oo 33 26 5646 566o 56 7 5 568 9 5704 5 7 i8 33 27 o4i5 o43o o446 o46i o4 7 6 o4 9 i 32 27 5 7 33 5 7 4 7 5 7 62 5 77 6 5 79 i 58o5 32 28 o5o6 o52i o536 o55i o566 o58i 3i 28 5820 5834 584 9 5863 5878 58 9 2 3i 2 9 o5 9 6 0611 0626 o64i;o656 o6 7 i 3o 2 9 5 97 5921 5 9 36 5 9 5o 5 9 65 5 979 3o 3o 9 . 910686 O 7 OI 0716 o 7 3i o 7 46 0761 2 9 3o 9 . 9 i5 99 4 6008 6o23 6037 6o52 6066 2 9 3i o 77 6 o 79 i 0806 0821 o836 o85i 28 3i 6081 6095 6io 9 6124 6i38 6i53 28 32 0866 0881 o8 9 6 o 9 n 9 26 o 9 4i 27 32 6167 6182 6i 9 6 6211 6225 6240 27 33 o 9 56 o 97 i 0986 IOOI 1016 io3i 26 33 6254 6268 6283 62 97 63i2 6326 26 34 io46 1061 1076 IO 9 I 1 1 06 II2I 25 34 634i 6355 636 9 6384 63 9 8 64i3 25 35 n36 n5i 1166 1181 1 1 9 6 I2II 24 35 6427 6442 6456 6470 6485 64 99 24 36 1226 1241 1256 I2 7 I 1286 i3oo 23 36 65i4 6528 6543 655 7 6571 6586 23 3 7 i3i5 i33o 1 345 i36o i3 7 5 i3 9 o 22 37 6600 66i5 6629 6643 6658 6672 22 38 i4o5 1420 i435 i45o i465 i48o 21 38 6687 6 7 or 6715 6 7 3o 6744 6 7 5 9 21 3 9 i4 9 5 i5ioi5 2 5 i54o i555 i56 9 2O 3 9 67736787 602 6816 683o 6845 2O 4o 9, 9 n584 i5 99 i6i4 i62 9 1 644 i65 9 I 9 4o 9.916859687416888 6 9 O2 6 9 i7 6 9 3i I 9 4i i6 7 4 i68 9 1704 1710 i 7 34 i 7 48 18 4i 6 9 4C 6960 6 97 4 6 9 8 9 7003 7017 18 42 i 7 63 1778 I 79 3 1808 1823 i838 '7 42 7032 7 o46 7 o6o 7 o 7 5 7 o8 9 7104 17 43 i853 1868 i883 i8 97 I 9 I2 I 9 2 7 16 43 7118 7i32 7 i4 7 7161 7 i 7 5 7 I 9 16 44 45 I 9 42 2031 I 9 57|i 97 2 2o46 2061 20-76 2OO2 2OI 7 2091 2106 i5 i4 44 45 7204 7 2 9 7218 73o4 7 233 7 3i 9 7 24 7 7 333 7261 7347 7276 7362 i5 i4 46 2121 2i36 2i5o 2i65 2180 2i 9 5 i3 46 7 3 7 6 7 3 9 o 7 4o5 7 4i 9 7433 7448 i3 47 2210 2225 2240 2255 226 9 2284 12 47 7 462 7 4 7 6 7 4 9 i 7 5o5 7 5i 9 7534 12 48 22 99 23l4 2329 2344 2358 23 7 3 II 48 7 548 7 562 7 5 7 6 7 5 9 i 7605 7610 II 49 2388 24o3 2418 2433 2448 2462 IO 49 7 634 7 648 7662 7 6 77 7601 77 5 10 5o 9.912477 24 9 2 2507 2522 253 72 55i 9 5o 9-9 I 77 I 9 7734 7748 77 62 7777 779 1 9 5i 2566 2581 2596 26ll 2625 2640 8 5i 7 8o5 7819 7834 7 848 7862 7877 8 52 2655 2670 2685 2 7 OO 2 7 l4,2 7 2 9 7 52 7891 79 5 79 i 979 34 7948 7 9 62 7 53 54 2 7 44 2833 275 9 2848 2774 2862 2 7 88 28 77 28o3 2818 28 9 2 2 9 O 7 6 5 53 54 797 6|7 99 i 8062 8076 8oo5 8oi 9 8o 9 o 8io5 8o33 8n 9 8o48 8i33 6 5 55 2 9 22 2 9 36 2g5i 2 9 66 2 9 8l 2 99 5 4 55 8147 8162 8176 8i 9 o 8204 82i 9 4 56 57 3oio 3o 99 3o25 3n4 3o4o 3o55 3069 3o84 3i 2 83i433i583i 7 3 3 2 56 8233 8247 8261 8276 82908304 83i8 8333 8347 836i 83 7 5 838 9 3 2 58 3i8 7 32O2 32i 7 3232 324 7 3261 I 58 84o4 84i8 8432 8446 846i 84 7 5 I 5 9 32 7 6 32 9 I 33o6 3320 3335335o o 5 9 8489 85o385i 7 8532 8546856o o 60" 50" 40" | 30" 20" 10" . | 60" 50" 40" 30" 20" 10" fl . Co-sine of 35 Degrees. 2 ! Co-sine of 34 Degrees. 2 P P JVt 5 l " ~" 3 " 4" 5 " 6 " T 8" 9" ,. A 1" 2" 3" 4" 5" 6" 7" 6" 9" ) 2 3 5 G 8 9 11 12 14 : " l \ 1 3 4 6 7 9 10 12 13 LoGAKIlHMIfJ T A N G F N T S. d Tangent of 54 Degrees. e Tangent of 55 Degrees. 3 0" 10-' 20" 30" 40" | 50" 0" 10" 20" 30" 40" 50" O io.i38 7 39 8 7 83 8828 88 7 2 8916 8960 r o io.i54773 48i8 4863 4908 4952 4997 5 9 I 9 oo5 9 o4 9 9 o 9 3 9i38 9182 9226 58 il 5o42 5o8 7 5i32 5i 77 5221 5266 58 2 92 7 9 3i5 9 35 9 94o3 9448 9492 5 7 2 53n 5356 54oi 5446 5490 5535 57 9 536 9 58o 9625 9669 97 i3 9758 56 3 558o 5625 5670 5 7 i5 5 7 5 9 58o4 56 4 9802 9 846 9891 99 35 9979 55 4 5849 58 9 4 5 9 3 9 5 9 84 6029 6o 7 3 55 5 10. i4oo68 OII2 oi5 7 02OI O245 0290 54 5 6118 6i63 620816253 6298 6343 54 6 o334 o3 7 8 0423 o46 7 o5n o556 53 6 6388 6432 6477 6522 656 7 6612 53 7 0600 o644 0689 o 7 33 0777 0822 52 7 6657 6 7 02 6 7 4 7 6 79 i 6836,688i 52 8 0866 o 9 io o 9 55 999 io43 1088 5i 8 6926 6971 7016 7 o6i 7 io6 7 i5i 5i 9 Il32 n 7 6 1221 1265 i3o 9 i354 5o 9 7 i 9 5 7 24o 7 285 7 33o 7 3 7 5 7420 5o 10 10.141398 1442 i48 7 i53i i5 7 6 1620 4 9 10 io.i5 7 465 7 5io 7 555 7 6oo 7 645 7689 4 9 ii 1 664 I 7 o 9 i 7 53 i 797 1842 1886 48 1 1 7734 7779 7 8 2 4 7 86 9 79 J 4 79 5 9 48 12 1931 i 97 5 2019 2064 2108 2l52 47 12 8oo4 8o4 9 8o 9 4 8i3 9 8i84 822 9 47 i3 2i 97 2241 2286 233o 23 7 4 24l9 46 i3 8273 83i8 8363 84o8 8453 84 9 8 46 i4 2463 2 5o8 2552 2596 2641 2685 45 i4 8543 8588 8633 86 7 8 8 7 23 8768 45 i5 2 7 3o 2 77 4 2818 2863 2907 2952 44 i5 88i3 8858 8 9 o3 8 9 48 8 99 3 9o38 44 16 2996 3o4i 3o85 3i2 9 3i 7 4 3218 43 16 9 o83 9128 9 i 7 3 9218 9263 9 3o 7 43 17 3263 33o 7 335i 33 9 6 344o 3485 42 17 9 35 2 9 3 97 9 442 948 7 9 532 9 5 77 42 18 352 9 35 7 4 36i8 3662 3 7 o 7 3 7 5i 4i 18 9 622 9 66 7 97 I2 97 5 7 9802 9 84 7 4i 19 3 79 6 384o 3885 3 9 2 9 3 97 4 4oi8 4o J 9 9892 9937 99 82 ..2 7 .. 7 2 .117 4o 20 10.144062 4io 7 4i5i 4196 424o 4285 3 9 20 10. 160162 020 7 0252 02 97 O342 0387 o 21 432 9 43 7 4 44i8 4463 45o 7 455i 38 21 0432 o4 77 O522 o56 7 O6l2 o65 7 38 22 45 9 6 464o 4685 4 7 29 4 77 4 4818 37 22 0703 0748 o 79 3 o838 o883 0928 3 7 23 4863 4 9 o 7 4952 4996 5o4i 5o85 36 23 o 97 3 1018 io63 1108 n53 1198 36 24 5i3o 5219 5 2 63 53o8 5352 35 24 1243 1288 i333 i3 7 8 1423 i468 35 25 53 97 544i 5486 553o 55 7 5 56i 9 34 25 i5i3 i558 i6o3 1648 i6 9 3 i 7 3 9 34 26 5664 5 7 o8 5 7 53 5 797 5842 5886 33 26 1784 1829 i8 7 4 1919 1964 2009 33 27 5 9 3i 5 97 5 6020 6o64 6io 9 6i53 32 27 ao54 2099 2i44 2189 2234 2279 32 28 6i 9 8 6242 628-7 633i 63 7 6 6420 3i 28 2325 23 7 2.4 1 5 2460 25o5 2 55o 3i 29 6465 65o9 6554 65 9 8 6643 668 7 3o 29 25 9 5 2640 2685 2 7 3o 2 77 5 2821 3o 3o 10, i46 7 32 6777 6821 6866 6 9 io 6 9 55 29 3o 10.162866 2911 2 9 56 3ooi 3o46 3091 29 3i 6 999 7 o44 7 o88 7 i33 7 i 77 7 222 28 3i 3i36 3i8i 322 7 32 7 2 33i 7 3362 28 32 7 26 7 7 3n 7 356 7 4oo 7 445 7489 2 7 32 34o 7 3452 34 9 7 3542 3588 3633 27 33 7 534 7 5 7 8 7 623 7 668 77 I2 77 5 7 26 33 36 7 8 3 7 23 3 7 68 38i3 3858 3 9 o4 26 34 7 8oi 7 846 7 8 9 o 79 35 79 8o 8024 25 34 3 9 4 9 3 99 4 4o3g 4o84 4129 4174 25 35 8o6 9 8n3 8i58 8 2 o3 824 7 8292 24 35 4220 4265 43io 4355 44oo 4445 24 36 8336 838i 8425 84 7 o 85i,5 855 9 23 36 44 9 i 4536 458i 4626 46 7 i 4716 23 37 86o4 8648 86 9 3 8 7 38 8 7 82 882 7 22 37 4762 48o 7 4852 48 97 4942 4988 22 38 88 7 i 8916 8961 9 oo5 9 o5o 9095 21 38 5o33 5o 7 8 5i23 5i68 52i3 5 2 5 9 21 3 9 9 i3 9 9184 9228 9 2 7 3 9 3i8 9 362 20 3 9 53o4 534 9 53 9 4 543 9 5485 553o 2O 4o 10. i4 9 4o 7 9 45 2 9496 9 54i 9 585 963o I 9 4o IO. 165575 5620 5666 5 7 n 5 7 56 58oi I 9 4i 9 6 7 5 9-719 9 7 64 9809 9 853 9898 18 4i 5846 5892 5 9 3 7 5 9 82 6O2 7 6073 18 42 99 43 9987 ..32 ..76 .121 .166 17 42 6118 6i63 6208 6 2 53 6299 6344 i 7 43 IO. I5O2IO 0255 o3oo o344 o38 9 o434 16 43 638 9 6434 648o 65 2 5 65 7 o 66i5 16 44 o4 7 8 o523 o568 0612 o65 7 O 7 O2 i5 44 6661 6 7 o6 6 7 5i 6 79 6 6842 6887 i5 45 o 7 46 0791 o836 0880 0925 09 7 i4 45 6 9 32 6977 7 023 7068 7 n3 7i58 i4 46 1014 1069 no4 1149 1 193 I2J8 i3 46 7204 7249 7294 7 34o 7 385 743o i3 4 7 1283 1327 l3 7 2 i4i 7 i46i i5o6 12 47 7 4 7 5 7 52I 7 566 7 6n 7 65 7 7702 12 48 i55i i5 9 5 i64o i685 i 7 3o i 77 4 II 48 7747 7792 7 838 7 883 7928 7974 II 49 1819 i864 1908 i 9 53 1998 2043 10 49 8019 8o64 8109 8i55 8200 8245 IO 5o 10. i52o8 7 2l32 2I 77 2221 2266 23ll 9 5o 10:168291 8336 838i 842 7 8472 85i 7 9 5i 2356 24OO 2445 2490 2 535 25 7 9 8 5i 8563 8608 8653 86 99 8744 8789 8 52 2624 266 9 2 7 l3 2 7 58 2 8o3 2848 7 52 8835 8880 8 9 25 8 97 i 9016 9061 7 53 28 9 2 2 9 3 7 2982 302 7 3o 7 i 3n6 6 53 9 I0 7 9152 9 i 97 9 243 9 288 9 333 6 54 3i6i 3206 325o 32 9 5 334o 3385 5 54 9 3 79 9424 9 46 9 9 5i5 9560 9 6o5 5 55 343o 34 7 4 35i 9 3564 3609 3653 4 55 965i 9696 97 42 9787 9 832 9 8 7 8 4 56 36 9 8 3 7 43 3 7 88 3832 38 77 3922 3 56 9923 9968 ..5 9 .io5 .i5o 3 57 3 9 6 7 4012 4o56 4ioi 4i46 4191 2 57 10. I 7 oi95 0241 0286 o33i o3 77 O422 s, 58 4236 4280 4325 43 7 o 44i5 446o I 58 o468 o5i3 o558 0604 0649 od 9 5 i 5 9 45o4 454 9 45 9 4 463 9 4684 4 7 28 O 5 9 o 7 4o 0785 o83i o8 7 6 0922 u 9 6 7 o (30" 50" 40" 30" 20" 10" . GO" 50" 40" 30" 20" 10 " 1 d Co-tangent of 35 Degrees. 9 Co-tangent of 34 Degrees. 1 3 p p ( 1" 2" 3" 4" 5" 6" 7" 8" 9" f -[it nil 3" 4" 5" Q" 7" g" 9" I 4 9 13 18 22 27 31 36 40 P. Part J 5 g 14 ig 03 o / 32 36 41 80 LOGARITHMIC SINES. I 1 Sine of 56 Degrees. .5 Sine of 57 Degrees. ) | 2 0' 10" 20" 30'' 40" | 50" 2 0" j 10" 20" [ 30" 40" 50' 9.918574 8588 86o3 8617 863ij8645 5 9 o 9.923591 36o5 36l9 3632 3646 366o 5 9 i 8659 86 7 4 8688 8702 8716 8 7 3o 58 i 36 7 3 368 7 3 7 oi 3 7 i4 3 7 2S 3742 58 2 8745 8 7 5 9 8 77 3 8787 8801 88i5 57 '2 3755 3 7 6 9 3 7 83 3 79 6 38io 3824 57 3 883o 8844 8858 8872 8886 8900 56 3 383 7 385i 3865 38 7 8 38 9 2 3 9 o6l56 4 8 9 i5 8929 8 9 43 8 9 5 7 8971 8 9 85 55 4J 3919 3 9 33 3 9 46 3960 3 97 4 3 9 87i55 5 9000 9014 9028 9042 go56 9070 54 51 4ooi 4oi5 4028 4042 4o55 4069 54 U 9085 9099 9113 9127 9141 9 i55 53 6 4o83 4096 4no 4124 4i3 7 4i5i 5'3 7 9169 9184 9 ! 9 8 9212 9226 9240 52 7 4i64 4i 7 8 4192 42o5 4219 4232 5a 8 9 254 9268 9282 9297 93u 9 325 5i 8 4246 4260 42 7 3 428 7 43oo 43i4 5i 9 9 33 9 9353 9 36 7 9 38i 9 3 9 5 9 4io 5o 9 4328 434i 4355 4368 4382 43 9 6 5o 10 9.919424 9 438 9 45 2 9 466 9480 9 4 9 4 49 IO 9.924409 4423 4436 445o 4464 4477 49 ii 95o8 9522 9 53 7 955i 9 565 9 5 79 48 ii 4491 45o4 45i8 453i 4545 455 9 48 12 9 5 9 3 9607 9621 9635 9649 9 663 47 12 4572 4586 4599 46i3 4626 464o 47 i3 9 6 77 9692 9706 9720 97 34 9748 46 i3 4654 466 7 468i 46 9 4 4 7 o8 4 7 2I 46 i4 97 62 9776 979 9804 9818 9832 45 i4 4735 4748 4 7 62 4776 4 7 8 9 48o3 45 i5 98^6 9860 9 8 7 5 9889 9903 9917 44 !5 48-i 6 483o 4843 485 7 48 7 o 4884 44 16 99 3i 9945 99 5 9 997 3 9987 . . . i 43 16 48 97 4 9 n 4924 4g38 4 9 52 4 9 65 43 i? 9 . 9 200l5 0029 oo43 0057 0071 oo85 42 J 7 4979 4992 5oo6 5019 5o33 5o46 42 18 oo 99 on3 0127 oi4i oi56 0170 4i 18 5o6o 5o 7 3 5o8 7 5ioo 5n4 5l2 7 4i J 9 oi84 0198 O2 1 2 0226 0240 0254 4o J 9 5i4i 5i54 5i68 5i8i 5i 9 5 5208 4o 20 9 . 9 20268 0282 0296 o3io 0324 o338 3 9 20 9.925222 5 2 35 5249 5262 52 7 6 5 2 8 9 39 21 o352 o366 o38o o3 9 4 o4o8 0422 38 21 53o3 53i6 533o 5343 535 7 53 7 o 38 22 o436 o45o o464 o4 7 8 0492 o5o6 37 22 5384 53 97 54n 5424 5438 545i 37 23 o52O o534 o548 o562 0576 oSgo 36 23 5465 5478 549i 55o5 55i8 5532 36 24 0604 0618 o632 o646 0660 0674 35 24 5545 555 9 55 7 2 5586 55 99 56i3 35 25 0688 0702 0716 0730 0744 o 7 58 34 25 5626 564o 5653 566 7 568o 56 9 3 34 261 0772 0786 0800 0814 0828 0842 33 26 5 7 o 7 5 7 20 5 7 34 5 7 4 7 5 7 6i 5 77 4 33 2 7 | 0866 0869 o883 0897 0911 0925 32 27 5 7 88 58oi 58i4 58 2 8 584i 5855 32 28 o 9 3 9 o 9 53 0967 0981 o 99 5 1009 3! 28 5868 5882 58 9 5 5908 5 9 22 5 9 35 3! 29 1023 1037 io5i io65 1079 1093 3o 29 5949 5 9 62 5 97 6 5 9 8 9 60O2 6016 3o 3o 9.921 107 II2I 1 1 34 n48 1162 1176 29 3o 9.926029 6o43 6o56 6069 6o83 6096 29 3i 1190 1204 1218 1232 1246 1260 28 3i 6110 6i 2 3 6i36 6i5o 6i63 6i 77 28 32 1274 1288 1302 i3i5 1329 1 343 27 32 6190 6203 62I 7 623o 6244 625 7 2 7 ' 33 i35 7 1371 i385 i3 99 i4i3 1427 26 33 62 7 6284 629-7 63n 63 2 4 633 7 26 34 i44i i455 i468 1482 1496 i5io 25 34 635i 6364 63 77 63 9 i 64o4 64i8 25 35 i524 i538 i55 2 i566 i58o i5 9 3 24 35 643 1 6444 6458 6471 6484 6498 24 36 1607 1621 i635li649 i663 1677 23 36 65ii 6525 6538 655i 6565 65 7 8 23 37 1691 1704 1718 1732 1746 1760 22 37 65 9 i 66o5 6618 663i 6645 6658 22 38 1 77 4 1788 1802 i8i5 1829 i843 21 38 66 7 i 6685 6698 6 7 n 6 7 25 6 7 38 21 3 9 1857 1871 i885 !8 99 ! 9 I2 1926 2O 39 6 7 5i 6 7 65 6 77 8 6 79 i 68o5 6818 2O 4o 9.921940 1954 1968 I 9 82 I99 5 2009 I 9 4o 9.926831 6845 6858 68 7 i 6885 6898 J 9 4i 2023 2037 2o5i 2o65 20 79 2092 18 4.i 6911 6925 6 9 38 6 9 5i 6 9 65 6 97 8 18 42 2106 2I2O 2134 2148 2l62 2175 17 42 6991 7 oo5 -7018 7 o3i 7 o44 7 o58 17 43 2189 2203 2217 223l 2244 2258 16 43 7 o 7 i 7084 7 o 9 8 7 in 7 I24 7 i38 16 44 2272 2286 2300 23i3 2327 234l i5 44 7 i5i 7164 7 i 77 7191 7 204 7 2I 7 i5 45! 2355 236 9 2383 23 9 6 2410 2424 i4 45 7 23l 7244 72 5 7 7 2 7 -7284 7 2 97 i4 AQ 2438 2452 2465 2479 2493 2507 i3 46 7 3io 7324 7 33 7 7 35o 7 363 7377 i3 47 252O 2534 2 548 2562 2 5 7 6 2589 12 47 7 3 9 o 74o3 74i6 743o 7443 7456 12 48 2603 2617 263i 2644 2658 2672 II 48 7 4 7 o 7483 7496 7 5o 9 7 523 7 536 I I 49 2686 2700 2713 2727 2 7 4l 2 7 55 IO 49 7 54 9 7 562 7 5 7 6 7 58 9 7 6o2 7 6i5 10 5o 9.922768 2782 2796 2810 2823 2837 9 5o 9 . 9 2 7 62 9 7642 7 655 7 668 7 68i 7 6 9 5 9 5i 285i 2865 2878 2892 2906 2920 8 5i 77 o8 7721 7734 7748 7761 7774 8 52 2 9 33 2947 2961 2 97 5 2988 3002 7 52 7787 7801 78i4 7 82 7 7 84o 7 853 7 53 3oi6 3o3o 3o43 3o57 3071 3o84 6 53 7 86 7 7 88o 7 8 9 3 79 o6 79 20 7933 6 54 3o 9 8 3lI2 3i26 3i3g 3 1 53 3i6 7 5 54 7946 79 5 9 7972 79 86 7999 8012 5 55 3i8i 3i94 32o8 3222 3235 324 9 4 55 8o 2 5 8o38 8o52 8o65 8o 7 8 8o 9 i 4 56 3263 3 277 3290 33o433i8 333i 3 56 8io4 8118 8i3i 8i44 8i5 7 8170! 3 5 7 3345 335 9 33 7 2 338634oo 34i4 2 57 8i83 8197 8210 8223 8236 824 9 2 58 3427 344 1 3455 3468 3482 3496 I 58 826382-76 8289 83o2 83i5 8328 i 59 35o 9 35 2 3 353 7 355o3564 35 7 8 5 9 8342J8355 8368 838i 83 9 4 84o7j 60" 50" 40" 30" 20" 10" C 60' j 50" 40" 30" 20" 10" j Co-sine of 33 Degrees. 3 Co-sine of 32 Degrees. 1 3 p .< 1" 2" 3" 4" 5" 6" 7" 8" 9" v , ( 1" 2" 3" 4" 5" 6" 7" 8" 9" irt $ 1 3 4 6 7 8 10 11 13 irt \ 1 3 4 5 7 8 9 11 12 LOGARITHMIC TANGENTS d Tangent of 56 Degrees. _g Tangent of 57 Degrees. 53 "0"" 10" 20" 30" 40" 50" 3 0" 10" 20'' 30" 40" | 50" ojio. 171018 io58 no3 n4 9 H94 I24o 5 9 10.187483 7 52 9 7 5 7 5 7 62I 7667 77 i3 5 9 I 1285 i33i i3 7 6 1421 i46 7 l5l2 58 I 77 5 9 7 8o5 7 85i 7 8 9 8 7944 799 58 ii i558 i6o3 1649 1694 i 7 3 9 i 7 85 5 7 2 8o36 8082 8128 8i 7 4 8220 82^7 5? 3 i.83o i8 7 6 1921 J 9 6 7 2OI2 2o58 56 3 83i3 835 9 84o5 845i 84 9 7 8543 56 4 2103 2149 2I 9 4 2239 2285 2 33o 55 4 85 9 o 8636 8682 8 7 28 87748820 55 5| 2 3 7 6 2421 2467 25l2 2 558 2 6o3 54 5 8866 8 9 i3 8 9 5 9 9 oo5 9o5i 9 o 97 54 6 2649 2694 2740 2 7 85 283i 28 7 6 53 6 9i43 9190 9 236 9282 9 328 o3y4 53 7 2922 2g6 7 3oi3 3o58 3io4 3i49 52 7 9 420 9 46 7 95i3 9 55 9 | 9 6o5 9 65i 52 8 3195 3 2 4o 3286 333i 33 77 3422 5i 8 9 6 9 8 97 44 979 9 836 9 882 9929 5! 9 3468 35i3 355 9 36o4 365o 3695 5o 9 9975 . .21 ..67 .n3 . 160 .206 5o 10 10.173741 3 7 86 3832 38 77 3 9 a3 3 9 68 49 10 10. 190252 02 9 8 o344 o3 9 i o43 7 o483 49 1 1 4014 4o6o 4io5 4i5i 4i 9 6 4242 48 II o52 9 0576 0622 0668 0714 0760 48 12 4287 4333 43 7 8 4424 446 9 45i5 47 12 0807 o853 0899 o 9 45 0992 io38 47 i3 456i 46o6 4652 46 97 4 7 43 4 7 88 46 i3 1084 n3o 1177 1223 1269 i3i5 46 U 4834 488o 4925 4 97 i 5oi6 5o62 45 i4 i362 i4o8 i454 i5oi 1 547 i5 9 3 45 i5 5107 5i53 5i 99 5244 52 9 O 5335 44 i,5 i63 9 1686 I 7 32 i 77 8 1824 1871 44 16 538i 542 7 5472 55i8 5563 56o 9 43 16 1917 J 9 63 2O10 2o56 2IO2 2l4 9 43 I? 5655 5 7 oo 5746 5 79 i 583 7 5883 42 *7 2195 2241 2287 2334 2380 2426 42 18 5928 5 974 6019 6o65 6111 6i56 4i 18 24 7 3 25l 9 2565 2612 2658 2704 4i 19 6202 624 7 6293 633 9 6384 643o 4o J 9 2751 2797 2843 28 9 o 2 9 36 2982 4o 20 10. 176476 652i 656 7 66i3 6658 6 7 o4 3 9 20 10. 193029 3o 7 5 3l2I 3i68 32l4 3260 39 21 6749 6 79 5 684i 6886 6 9 32 6 97 8 38 21 3307 3353 33 99 3446 3492 3538 38 22 7023 7 o69 7116 7 i6o 7 206 7 252 3? 22 3585 363i 36 7 8 3 7 24 3770 38i 7 37 23 7297 7 343 7 38 9 7 434 7 48o 7 526 36 23 3863 3 9 o 9 3 9 56 4OO2 4049 4095 36 24 7 5 7 i 7 6i 7 7 663 77 o8 77 54 7 8oo 35 24 4i4i 4i88 4234 42&I 432 7 43 7 3 35 25 7 846 7 8 9 i 79 3 7 79 83 8028 8o 7 4 34 25 4420 4466 45i3 455 9 46o5 4652 34 26 8120 8i65 8211 825 7 83o3 8348 33 26 4698 4745 47 9 i 483 7 4884 4g3o 33 3 7 83 9 4 844o 8485 853i 85 77 8623 32 27 4 9 77 5o23 5070 5n6 5i62 5209 32 28 8668 8 7 i4 8760 88o5 885i 88 97 3i 28 5255 53o2 5348 53 9 5 544i 548 7 3i 2 9 8 9 43 8988 9 o34 9 o8o 9126 9171 3o 29 5534 558o 5627 56 7 3 5720 5 7 66 3o 3o 10. 179217 92 63 9 3o 9 9 354 9400 9 446 29 3o IO. I958l3 585 9 5 9 o6 5 9 52 5 999 6o45 2 9 3i 9492 9 53 7 9583 9629 9 6 7 5 97 20 28 3i 6091 6i38 6184 623i 6277 6324 28 32 9766 9812 9 858 99 o4 9949 999 5 27 32 63?o 64i7 6463 65io 6556 66o3 27 33 10. i8oo4i oo8 7 Ol32 oi 7 8 0224 0270 26 33 6649 66 9 6 6742 6 7 89 6835 6882 26 34 o3i6 o36i 0407 o453 o4 99 o545 25 34 6928 6 97 5 7021 7 o68 7114 7161 26 35 o5go o636 0682 7 28 o 77 4 o8i 9 24 35 7208 7254 73oi 7347 7 3 9 4 7 44o 24 36 o865 0911 o 9 5 7 ioo3 io48 io 9 4 23 36 748 7 7 533 7 58o -7626 7 6 7 3 7719 23 37 n4o 1186 1232 12-78 i323 i36 9 22 *7 7766 7 8i3 7 85 9 7906 79 52 7999 22 38 i4i5 i46i 1507 i553 i5 9 8 1 644 21 38 8o45 8o 9 2 8i38 8i85 8232 8278 21 3 9 1690 i 7 36 1782 1828 i8 7 4 i 9 i 9 2O 3 9 8325 83 7 i 84i8 8465 85n 8558 20 4o 10.181965 2011 2057 2103 2l4 9 2195 : 9 4o 10. 198604 865i 86 97 8 7 44 8791 883 7 '9 4i 224l 2286 2332 2 3 7 8 2424 2470 18 4i 8884 8 9 3o 8977 9024 9070 9117 18 42 25i6 2562 2608 2 653 2699 2 7 45 *7 42 9164 9 2IO 9 25 7 9 3o3 9 35o 9 3 97 l l 43 2791 283 7 2 883 2 9 2 9 2 97 5 3021 16 43 9 443 9 4 9 o 9 53 7 9 583 9630 9676 16 44 3o6 7 3lI2 3i58 32o4 325o 3296 i5 44 9723 977 9 8i6 9 863 9910 9956 i5 45 3342 3388 3434 3480 3526 3572 i4 45 IO.2OOOO3 oo5o oo 9 6 oi43 0190 0236 i4 46 36i8 3664 3 7 o 9 3 7 55 38oi 3847 i3 46 0283 o33o o3 7 6 o423 0470 o5i6 i3 47 38 9 3 3 9 3 9 3 9 85 4o3i 4o 77 4l23 12 47 o563 0610 o656 0703 0750 0796 12 48 4i6 9 42i5 4261 43o 7 4353 439 9 I I 48 o843 o8 9 o o 9 3 7 o 9 83 io3o 1077 II 49 4445 4491 453 7 4583 4629 46 7 4 10 49 1123 1170 I2I 7 1263 i3io i35 7 10 5o 10.184720 4 7 66 4812 4858 4904 4 9 5o 9 5o IO.2Ol4o4 i45o i4 97 1 544 iSgi i63 7 9 5i 4996 5o42 5o88 5:34 5i8o 5226 8 5i 1684 I 7 3i 1777 1824 1871 1918 8 52 5272 53i8 5364 54io 5456 55o2 7 52 i 9 64 2OII 2o58 2105 2l5l 2I 9 8 7 53 5548 55 9 4 564o 5686 5 7 32 5 77 8 6 53 2245 22 9 2 2338 2385 2432 2479 6 54 5824 58 7 i 5 9 i 7 5 9 63 6009 6o55 5 54 25 2 6 2572 2619 2666 2713 3 7 5 9 5 55 6101 6i4 7 6i 9 3 6239 6 2 85633i 4 55 2806 2853 2900 2 9 47 2 99 3 3o4o 4 56 63 77 6423 646 9 65i5 656i 66o 7 3 56 3087 3i34 3i8i 3227 32 7 4 332i 3 57 6653 6699 6 7 45 6 7 9i 683 7 6884 2 57 3368 34i5 346i 35o8 3555 36o2 3 58 6 9 3o 6976 7 022 7 o68 7 u4 7 i6o I 58 364 9 36 9 6 3 7 42 3789 3836 3883 X 1? 7 2O6 7 252 7 2 9 8 7 344 7 3 9 o 7 43 7 5 9 3 9 3o 3977 4023 4070 4117 4i64 o ~~ GO" 50" 40" 30" 20" 10" a 60" 50" | 40" 30" 20" 10" Co-tangent of 33 Degrees. 2 Co-tangent of 32 Degrees. a p p 41" 2" 3" 4" 5" 6" 7" 8" 9" ( 1" 2" 3" 4" 5" 6" 7" 8" 9" ( 5 9 14 18 23 27 32 37 41 irt } 5 9 14 19 23 28 33 37 42 LOGARITHMIC SINES. Jj Sine of 58 Degrees. d s3 Sine of 59 Degrees. 7 "~* 8 0" 10" 20" 30" | 40" 50" s 0" 10" 20" 30" 40" 50" 9. 928420 8434 844784608473 8486 5 9 o 9.933066 3078 Sogi 3 10^ 3n6 3i2 9 5 9 I 8499 85i3 8526 853 9 8552 8565 58 i 3i4i 3i54 3i6 7 3i 79 3i 9 2 3 2 o5 58 2 85 7 8 85 9 i 86o586i8 ! 863i 8644 57 2 3217 323o 3243 3255 3268 3 2 8o 57 3 865 7 8670 8683 86 9 6 8710 8723 J56 J 32 9 3 33o6 33i8 333i 3344 3356 56 4 8 7 36 8 7 4 9 8762 87758788 8801 55 L 336 9 338i 3394|34o7 34i 9 3432 55 5 88i5 88 2 8l884i 8854 8867 888o|54 i 3445 345 7 3470 3482 34 9 5 35o8 54 6 88 9 3 8 9 o68 9 i 9 8 9 338 9 46 8 9 5 9 l53 6 3520 3533 3545 3558 35 7 i 35b3 53 7 8 97 2 8 9 85 8 99 8 9 oii 9024 9 o3 7 52 7 35 9 6 36o8 362i 3633 3646 365 9 52 8 9o5o 9 o63 9 077 9 o 9 o 9 io3 9116 5i 8 3671 3684 36 9 6 3 7 o 9 3 ? 22 3 7 34 5i 9 9129 9 i42 9 i55 9 i68 9181 9 i 9 4 5o 9 3 7 4 7 3 7 5 9 3 77 2 3 7 84 3797 38io 5o 10 9.929207 9 22O ! 9 233 9 247 9260 9273 49 IO 9 . 9 33822 3835 3847 386o 38 7 2 3885 49 ii 9286 92999312 9 3 2 5 9338 9 35i 48 ii 38 9 8|3 9 io 3 9 23 3 9 35 3 9 48 3 9 6o 48 12 9 364 93771939 9 4o3 94169429 47 12 3 97 3 3 9 85 3 99 8 4on 4o23 4o36 47 i3 9442 9 455; 9 468 9 482 9495 9 5o8 46 i3 4648 4o6i 4073 4o86 4o 9 8 4m 46 i4 9521 9 534: 9 547 9 56o 9 5 7 3 9 586 45 i4 4l23 4i36 4i48 4i6i 4i 7 4 4i86 45 i5 9 5 99 9 6l2 9 625 9 638 9 65i 9 664 44 i5 4199 4211 4224 4236 424 9 4261 44 16 9677 9 6 9 o 97 3 97169729 9 742 43 16 4274 4286 4299 43n 4324 4336 43 17 97 55 97 68 97 8i 9794 9807 9 820 42 17 4349 436i 4374 4386 43 99 44" 42 18 9833 9 846 9 85 9 9 8 7 2 9 885 9898 4i 18 4424 4436 4449 446! 4474 4486 4i '9 9911 99 24 99 3 7 99 5o 99 63 9976 4o 19 4499 45n 4524 4536 454 9 456i 4o 20 9 . 9 2 99 8 9 . . .2 . . i5 ..28 . .4i ..54 39 20 9.934574 4586 45 99 46 n 4624 4636 3 9 21 oo8o'oo 9 3 0106 on 9 Ol32 38 21 4049 466i 4674 4686 46 99 4711 38 22 oi45 oi58 0171 0184 oi 9 7 O2IO 37 22 4723 4736 4748 4761 4773 4786 37 23 O223 O236 024 9 0262 0274 0287 36 23 4798 48" 48 2 3 4836 4848 486i 36 24 o3oo o3i3 o3 2 6 o33 9 o35 2 o365 35 24 48 7 3 4885 48 9 8 4 9 io 4 9 23 4 9 35 35 25 o3 7 8 o3 9 i o4o4 0417 o43o o443 34 25 4948 4 9 6o 4 97 3 4 9 85 4997 5oio 34 26 o456 o46 9 0482 0495 o5o7 O52O 33 26 5O22 5o35 5o47 5o6o 5o 7 2 5o84 33 27 o533 o546 o55 9 0572 o585 o5 9 8 32 27 5o 97 5io 9 5l22 5i34 5i47 5i5 9 32 28 0611 0624 o63 7 o65o o663 o6 7 5 3i 28 5171 5i84 5i 9 6 520 9 5221 5 2 34 3i 29 0688 o 7 oi 0714 0727 0740 o 7 53 3o 29 5246 5258 0271 5283 5 29 6 53o8 3o 3o 9.930766 779 79 2 o8o4 0817 o83o 2 9 3o 9 . 9 3532o 5333 5345 5358 53 7 o 5382 29 3i o843 o856 o86 9 0882 o8 9 5 0908 28 3i 53 9 5 54o 7 5420 5432 5444 545 7 28 32 O 9 2I o 9 33 o 9 46 o 9 5 9 97 2 o 9 85 27 32 546 9 5482 54 9 4 55o6 55i 9 553i 27 33 o 99 8 IOII 1024 io36 io4 9 1062 26 33 5543 5556 5568 558i 55 9 3 56o5 26 34 1075 1088 I 101 in4 1127 n3 9 25 34 56i8 563o 5642 5655 566 7 56 7 9 25 35 Il52 n65 1178 1191 I2O4 1217 24 35 56 9 2 5 7 o4 5 7 i 7 5 7 2 9 5 7 4i 5 7 54 24 36 I22 9 1242 1255 1268 I28l I2 9 4 23 36 5 7 66 5778 5 79 i 58o3 58i5 5828 23 37 i3o6 i3i 9 i332 1 345 i358 1371 22 37 584o 5852 5865 5877 588 9 5902 22 38 i383 i3 9 6 i4o 9 1422 i435 1 448 21 38 5 9 i4 5 9 26 5 9 3 9 5 9 5i 5 9 63 5 97 6 21 3 9 i46o i4 7 3 i486 1 499 l5l2 i5 2 5 2O 39 5 9 88 6000 6oi3 6o25 6o3 7 6o5o 2O 4o 9 . 9 3i537 i55o i563 i5 7 6 i58 9 1601 '9 4o 9 . 9 36o62 6074 6o8 7 6o 99 6m 6124 I 9 4i 1614 1627 i64o i653 1666 1678 18 4i 6i36 6i48 6161 6i 7 3 6i85 6i 9 8 73 . 42 i6 9 i 1704 1717 1730 1742 i 7 55 17 42 6210 6222 6234 625 9 62 7 I '7 43 1768 1781 i 79 4 1806 i8i 9 i83 2 16 43 6284 62 9 6 63o8 6320 6333 6345 16 44 i845 i85 7 1870 i883 i8 9 6 I 9 o 9 i5 44 635 7 6370 638 2 63 9 4 64o6 64i 9 i5 45 I 9 2I i 9 34 i 9 47 1960 i 97 2 i 9 85 i4 45 643i 6443 6456 6468 648o 64 9 s i4 46 i 99 8 2OII 2024 2o36 2o4 9 2062 i3 46 65o5 65i 7 652 9 6542 6554 6566 i3 47 2075 208 7 2IOO 2Il3 2126 2i38 12 47 65 7 8 65 9 i 66o3 66i5 662 7 664o 12 48 2l5l 2164 2177 2189 2202 22 1 5 11 43 6652 6664 66 7 6 668 9 6 7 oi 6 7 i3 II 49 2228 2240 2253 2266 227 9 22 9 I IO 4 9 6 7 25 6 7 38 6 7 5o 6 7 62 6 77 4 6 7 8 7 10 5o 9.932304 23l 7 232 9 2342 2355 2368 9 5o j. 9 36 799 6811 6823 6836 6848 6860 9 5i 238o 23 9 3 2406 24l9 243i 2444 8 5i 68 7 2 6884 68 97 6 9 o 9 6 9 2I 6 9 33 8 52 245 7 246 9 2482 24 9 5 2 5o8 2520 7 52 6 9 46 6 9 58 6 97 o 6 9 82 6 99 4 7 oo 7 7 53 2533 2546 2 558 2.71 2584!25 9 7 6 53 7 oi 9 7o3i 7o43 7 o56 7 o68 7 o8o 6 54 2609 2622 2635 2647 26602673 5 54 79 2 7104 7 n 7 7 I2 9 7 i4i 7 i53 5 55 2685 26 9 8 2711 2724 2 7 36l2 7 4 9 4 55 7 i65 7178 7 I 9 7 202 7214 7226 4 56 2762 2774:2787 28002812^825 3 56 7 238 725i 7 263 7 2 7 5 7 28 7 7 2 99 3 5 7 2838 285o 2 863 2876 2888 2 9 OI 2 57 7 3l2 7324 7 336 7 348 7 36o 7 3 7 2 2 58 2914 2 9 26 2 9 3 9 2952 2 9 64 2 9 77 I 58 7 385 7397 7 4o 9 7 42I 7433 7446 I 5 9 2990 3oo2 3oi5 3028 3o4o 3o53 O 5 9 7458 7470 7 482 74 9 4 7 5o6 7 5i8 O 60" 50" 40" 30" 20" 10" d 60" 50" 40" 30" 20" 10" Co-sine of 31 Degrees. Co-sine of 30 Degrees. P Part* l " ~" 3 " 4" 5" 6" 7" 8" 9" ,, , ( 1" 2" 3" 4" 5" 6" 7" 8" 9" i 1 i 1 3 4 5 6 8 9 10 12 "") 12 4567 9 10 11' LOGARITHMIC TANGENTS. o o o o c i Tangent of 58 Degrees. j3 Tangent of 59 Degrees. % 0" | 10'-' 20" [ 30" 40" 50" s 0" | 10" 20" 30" 40" JjO 10.20421 1 4268 43o4 435i 43 9 8 4445 r o IO.22I226 1274 l322 i36 9 1417 i465 5 9 i 4492 453 9 4586 4633 467 9 4726 58 I l5l2 i56o l6o8 i656 1703 i 7 5i 58 2 4773 48 2 o|486 7 |4 9 i4 4 9 6i 5oo8 57 2 I 799 1 846 i8 9 4 I 9 42 1990 2037 5 7 3 5o54 5roi 5i48 5i 9 5 5242 5 2 8 9 56 3 2o85 2i33 2181 2228 2276 2324 56 4 5336 5383 543o 54 77 5524 55 7 o 55 4 2372 24l 9 246 7 25i5 2563 26lO 55 5 56i 7 5664 5711 5 7 58 58o5 5852 54 5 2658 2706 2754 2801 2849 28 9 7 54 6 58 99 5 9 46 5 99 3 6o4o 6087 6i34 53 6 2 9 45 2 99 3 3o4o 3o88 3i36 3i84 53 7 6181 6227 6274 632i 6368 64i5 52 7 3232 3279 3327 33 7 5 3423 347i 52 8 6462 65o 9 6556 66o3 665o 66 9 7 5i 8 35i8 3566 36i4 3662 3710 3 7 5 7 5i 9 6 7 44 67 9 i 6838 6885 6.982 6 979 5o 9 38o5 3853 3901 3 9 4 9 3 997 4o44 5o 10 10 .207026 7 o 7 3 7120 7167 7214 7261 4 9 10 ro.224o 9 2 4i4o 4i88 4236 4284 4332 4 9 ii 7 3o8 7 355 7402 ?44 9 74 9 6 7543 48 1 1 4379 4427 4475 4523 457i 46i 9 48 12 i3 7 5 9 7872 7 63 7 79 i 9 7 684 79 66 77 3i 8oi3 7778 8060 7825 8107 47 46 12 i3 46674714 4 9 54 5oo2 4762 5o5o 48io 5o 9 8 4858 5i45 4 9 o6 5i 9 3 47 46 i4 8i54 8201 8248 82 9 5 8342 838 9 45 i4 524i 5289 5337 5385 5433 548 1 45 i5 8437 8484 853i 85 7 8 86 2 5 8672 44 i5 552 9 55 77 56 2 5 56 7 2 5720 5 7 68 44 16 8719 8766 88i3 8860 8 9 o 7 8 9 54 43 16 58i6 5864 5912 5960 6008 6o56 43 '7 9001 9 o48 9 o 9 5 9 i43 9 i 9 o 9 23 7 42 *7 6io4 6i5 2 6200 62^8 6296 6344 42 id 9 284 9 33i 9 3 7 8 9 425 9472 9 5i 9 4i 18 63 9 2 644o 6488 6535 6583 663i 4i J 9 9 566 9 6i4 9 66i 0708 97 55 9 8o2 4o r 9 667 9 6727 6 77 5 68 2 3 6871 6 9 i 9 4o 20 io.2o 9 84 9 9 8 9 6 9.943 999 l ..38 ..85 3 9 20 10.226*567 7Oi5 7 o63 7111 7 i5 9 7 20 7 3 9 21 IO.2IOI32 oi7 9 0226 0273 0321 o368 38 21 7 255 73o3 7 35i 7399 7447 7 4 9 5 38 22 o4i5 0462 o5o 9 o556 o6o3 o65i 3 7 22 7543 7 5 9 i 7 63 9 7 688 7736 77 84 37 23 0698 o 7 45 07 9 2 o83 9 0886 o 9 34 36 23 7832 7880 -7928 797 6 8024 8072 36 24 o 9 8i 1028 1075 1122 1170 1217 35 24 8120 8168 8216 8264 83i2 836o 35 25 1264 i3n i358 i4o5 i453 i5oo 34 25 84o8 8456 85o4 8552 8601 864 9 34 26 i547 i5 9 4 i64i i68 9 i 7 36 i 7 83 33 26 8697 8 7 45 8 79 3 884i 8889 8937 33 27 i83o 1878 I 9 25 I 97 2 2OI 9 2066 32 27 8 9 85 9 o33 9 o8i 9i3o 9178 9 226 32 28 2Il4 2161 2208 2255 23o3 235o 3i 28 9274 9322 9 3 7 94:8 9 466 9 5i5 3i 29 23 97 2444 24 9 2 2 53 9 2586 2633 3o 29 9563 9611 9 65 9 9707 97 55 9 8o3 3o 3o TO. 2I268l 2728 2 77 5 2822 2870 2 9 i 7 29 3o 10.229852 9900 99 48 9996 ..44 . . 9 2 2 9 3i 2 9 64 3012 3o5 9 3io6 3i53 3201 28 3i io.23oi4o 0189 023 7 0285 o333 o38i 32 3248 32 9 5 3343 33 9 o 343 7 3484 27 32 0429 0478 o526 o5 7 4 0622 0670 27 33 3532 35 79 3626 36 7 4 3721 3 7 68 26 33 0719 0767 o8i5 o863 0911 o 9 6o 26 34 38i6 3863 3 9 io 3 9 58 4oo5 4o52 25 34 1008 io56 uo4 Il52 I2OI I24 9 25 35 4ioo 4i4 7 4: 9 4 42/12 428 9 4336 24 35 1297 1 345 i3 9 4 i442 1490 i538 24 36 4384 443 1 4478 4526 45 7 3 4620 23 36 i586 i635 i683 I 7 3i 1779 1828 23 37 4668 47i5 4762 48io 485 7 4 9 o5 22 37 1876 1924 i 97 3 2021 2069 2117 22 38 4 9 52 4999 5o47 5o 9 4 5i4i 5i8 9 21 38 2166 22l4 2262 23lO 2359 2407 21 3 9 5 2 36 5284 533i 53 7 8 5426 5473 2O 3 9 2455 25o4 2552 26OO 2648 26 9 7 2O 4o ro.2i552i 5568 56i5 5663 5710 5 7 58 r 9 4o 10.232745 2793 2842 2890 2 9 38 2 9 8 7 *9 4i 58o5 5852 5 9 oo 5 9 4 7 5 99 5 6042 18 4i 3o35 3o83 3i32 3i8o 3228 3277 18 42 6o 9 o 6137 6i84 6232 6279 6327 17 42 33 2 5 33 7 3 3422 34 7 o 35i8 356 7 *7 43 63 7 4 6422 646 9 65i 7 6564 6612 16 43 36i5 3663 3 7 I2 3 7 6o 38o8 385 7 16 44 665 9 6706 6 7 54 6801 684 9 68 9 6 i5 44 3 9 o5 3953 4OO2 4o5o 4o 99 4i47 i5 45 6 9 44 6 99 i 7 o3 9 7086 7i34 7181 i4 45 4i 9 5 4244 42 9 2 434o 438 9 443 7 i4 46 7 22 9 7276 7324 7 3 7 i 7 4i 9 7 466 i3 46 4486 4534 4582 463i 4679 4728 i3 47 75i4 756i 7 6o 9 7 656 774 775i 12 47 4776 48 2 5 48 7 3 4921 4970 5oi8 12 48 7799 7 846 7 8 9 4 79 4i 79 8 9 8o36 I I 48 5o6 7 5n5 5i64 5212 5260 53o 9 II 49 8o84 8i3i 8i 79 8226 8274 8322 10 49 535 7 54o6 5454 55o3 555i 56oo 10 5o io.2i836 9 8417 8464 85i2 855 9 8607 9 5o io.235648 56 9 6 5 7 45 5 79 3 5842 58 9 o 9 5i 8654 8702 8 7 5o 8797 8845 88 9 2 8 5i 5 9 3 9 5 9 8 7 6o36 6o84 6i33 6181 62 8 9 4o 8987 9 o35 9083 9 i3o 9178 7 52 6 2 3o 6278 632 7 63 7 5 6424 6472 7 53, 9225 9 2 7 3 9 32I 9 368 9 4i6 9 463 6 53 652i 6569 6618 6666 6 7 i5 6763 6 54 9 5n 9 55 9 9 6o6 9 654 97 oi 9749 5 54 6812 6860 6 9 o 9 6 9 5 7 7006 7 o55 5 55 9797 9844 9 8 9 2 9939 9987 ..35 4 55 7io3 7i52 7 20O 7249 7297 7346 4 56 10.220082 oi3o 0178 0225 0273 O32I 3 56 7 3 9 4 7443 7 4 9 2 754o 7 58 9 7 63 7 o 5 7 o368 o4i6 o463 o5i i o55 9 0606 2 5 7 7686 7734 7783 7832 7880 79 2 9 2 58 5 9 o654 o 9 4o 0702 o 7 4 9 io36 0797 io83 o845 n3i 0892 1179 I O 58 5 9 7977 826 9 8026 83i8 8o 7 58i23 8366|84i5 8172 8463 8220 85i2 I O 0>' 50" 40" 130" 20" 10" Q 60" 1 50" 40" 30" | 20" j 10" dj Co-tangent of 3 1 Degrees. s Co-tangent of 30 Degrees. "prarJ 1 " ~" 3// 4// 5 " 6 " 7 " 8 " 9 " Ii PPart* 1 " 2 " 3 " 4// 5// G " 7 " 8 " 9 " { 5 9 14 19 24 28 33 38 43 ! 1 - Jcirt } 5 JQ 14 19 24 29 34 39 43 SINES. 4 Sine of 60 Degrees. . Sine of 6 1 Degrees. fl 0" 10" | 20" 1 30" 40" | 50" * 0" | 10" 20" 30" 40" 50" c \;. 9 3 7 53j 7543 7555 7567 7 5 79 7 5 9 i 5 9 O 9 . 9 4i8i 9 i83i 1843 i854 1866 1878 5 9 I 7604 7 6l6 7628 7 64c 7 65 2 7664 58 I i88 9 1901 I 9 i3 192^ 1936 i 9 48 58 2 7676 7689 7701 77 i3 77 25 7737 57 2 i 9 5 9 1971 i 9 83 1994 2006 2017 5 7 3 7749 7761 7773 7786 7798 7810 56 3 2O2 9 2O52 206^ 2076 2087 56 4 7822 7834 7 846 7 858 7870 7 883 55 4 20 99 2III 2122 2i34|2i46 2l5 7 55 5 7 8 9 5 797 79 i 9 79 3i 79 43 79 55 54 5 2169 2l8o 2I 9 2 22O4 22l5 2227 54 6 7967 7979 799 2 8oo4 8016 8028 53 6 223 9 2250 2262 22 7 3 2285 22 9 7 53 7 8o4o 8o52 8o64 8076 8088 8100 52 7 23o8 2320 233l 2343 2355 2366 5 a 8 8n3 8i25 8i3 7 8i4 9 8161 8i 7 3 5i 8 23 7 8 2390 2401 2 4i3 242^ 2436 5! 9 8i85 8i97 82O 9 8221 82338245 5o 9 2448 2459 2471 2482 2494 25o6 5o 109.938258 82-70 8282 820.4 83o6 83i8 49 10 9 . 9 425i 7 252 9 254o 2552 2563 2 5 7 5 4 9 ii 833o 8342 83548366 83 7 8 83 9 o 48 ii 258 7 25 9 8 2610 2621 2633 2645 48 12 8402 84i4 84268439 845i 8463 47 12 2656 2668 267 9 2691 2702 2 7 l4 47 i3 84 7 5 848 7 84998611 8523 8535 46 i3 2726 2737 2 7 4 9 2760 2772 2 7 83 46 i4 854 7 855 9 85 7 i|8583 85 9 5 86o 7 45 i4 2 79 5 2806 2818 283o 2841 2853 45 i5 8619 863i 8643 8655 8667 86 79 44 i5 2864 2876 2887 2899 2910 2 9 22 44 16 8691 8 7 o3 8715 8727 8739 8 7 5i 43 16 2 9 3^ 2 9 45 2957 2968 2980 2 99 I 43 17 8 7 63 8776 8788 8800 8812 8824 42 17 3oo3 3oi4 3026 3o3 7 3o4 9 3o6o 42 18 8836 8848 88608872 8884 88 9 6 4i 18 3072 3o83 3o 9 5 3107 3n8 3i3o 4i 19 8908 8 9 20 8 9 32 8 9 44 8956 8 9 68 4o i 9 3i4i 3i53 3i64 3i 7 6 3i8 7 3i 99 4o 20 9.938980 8 99 2 9 oo4 9 oi6 9028 9 o4o 3 9 20 9 . 9 432io 3222 3233 3245 3256 3 2 68 3 9 21 9062 9 o64 9 O76 9 o87 9099 9 ni 38 21 3279 32 9 I 33o2 33i4 33 2 5 333 7 38 22 9123 9 i35 9 i47 9 i5 9 9171 9 i83 37 22 3348 336o 33 7 i 3383 33 9 4 34o6 37 23 9195 9 20 7 9 2i 99 23i 9243 9 255 36 23 34i7 342 9 344o 3452 3463 34 7 5 36 24 9267 9 2 79 9 2 9 i 9 3o3 9 3i5 9 32 7 35 24 3486 34 9 8 35o 9 352i 3532 3543 35 25 9339 9 35i 9 363 9375 9 38 7 9 3 99 34 2 5 3555 3566 35 7 8 358 9 36oi 36i2 34 26 9410 9 422 9 434 9 446 9 458 0.470 33 26 3624 3635 364 7 3658 3670 368i 33 27 9482 9 4 9 4 9 5o6 9 5i8 9 53o 9 542 32 27 36 9 3 3 7 o4 3 7 i5 3727 3 7 39 3 7 5o 32 28 9554 9 566 9 5 7 8 9 5 9 o 9 6oi 9 6i3 3i 28 3 7 6i 3 77 3 3 7 84 3 79 6 38o 7 38i8 3i 29 9625 9 63 7 9 64 9 9661 9 6 7 3 9 685 3o 29 383o 384i 3853 3864 3876 388 7 3o 809.939697 979 97 2i 97 33 9 744 9756 2 9 3o 9 . 9 438 99 3 9 io 3 9 2I 3 9 33 3 9 44 3 9 56 29 3i 9768 97 8o 9 7 9 2 9 8o4 9 8i6 9 828 28 3i 3967 3 97 8 3 99 o 4ooi 4oi3 4024 28 32 9840 9 852 9 863 9 875 9 88 7 9899 2 7 q 4o36 4o4 7 4o58 4o 7 o 4o8i 4o 9 3L 2 7 33 9911 99 23 99 35 9947 99 5 9 997 26 33 4io4 4n5 4127 4i38 4i5o 4i6i 26 34 9982 9994 ...6 ..18 ..3o ..42 25 34 4172 4i84 4195 4207 4218 422 9 25 35 9 , 9 4oo54 oo65 0077 oo8 9 OIOI on3 24 35 4241 4252 4264 4275 4286 42 9 8 24 36 OI25 oi3 7 0148 0160 OI 7 2 oi84 23 36 43o 9 4321 4332 4343 4355 4366 23 37 0196 0208 O22O O23l 0243 0255 22 37 4377 438 9 44oo 44i2 4423 4434 22 38 0267 02 79 O2 9 I o3o3 o3i4 o326 21 38 4446 445 7 4468 448o 44 9 i 45o3 21 3 9 o338 o35o o362 0374 o385 o3 97 2O 3 9 45i4 45 2 5 4537 4548 455 9 45 7 i 2O 4o 9 . 9 4o4o 9 0421 0433 0445 o456 o468 9 4o 9 . 9 44582 45 9 3 46o5 46i6 4627 463 9 I 9 4i o48o 04 9 2 o5o4 o5i6 o52 7 o53 9 8 4i 465o 466i 4673 4684 46 9 6 4 7 o 7 18 42 o55i o563 o5 7 5o586 o5 9 8 0610 7 42 4 7 i8 4 7 3o 4 7 4i 4752 4764 4 77 5 17 43 0622 o634 o645 0667 0669 0681 6 43 4 7 86 4 79 8 4809 4820 483i 4843 16 44 o6 9 3 0704 0716 0728 o 7 4o O 7 52 5 44 4854 4865 48 77 4888 48 99 4911 i5 45 0763 o 77 5 07 8 7 799 0811 0822 4 45 4 9 22 4 9 33 4945 4 9 56 4 9 6 7 4979 i4 46 o834 o846 o858 0870 0881 0893 3 46 49 9 5ooi 5oi3 5024 5o35 5o46 i3 47 o 9 o5 o 9 i 7 O 9 28 0940 0962 0964 2 47 5o58 5o6 9 5o8o 5092 5io3 5u4 12 48 o 97 5 o 9 8 7 0999 ion 1023 io34 I 48 5i25 5i3 7 5i48 5i5g 5i 7 i 5i82 II 49 io46 io58 1070 1081 1093 no5 O 4 9 5i 9 3 52o4 52i6 5227 5238 5 2 5o IO 5o 9.94IH7 1128 n4o Il52 n64 n 7 5 9 5o 9.945261 5272 5283 5295 53o6 53 I7 9 5i 1187 1199 121 I 1222 1234 1246 8 5i 5328 534o 535i 5362 53 7 4 5385 8 52 1258 I26 9 I28l 1293 i3o4 i3i6 7 52 5396 5407 54i9 543o 544i 5452 7 53 i3 2 8 i34o i35i i363 i3 7 5 i38 7 6 53 5464 54 7 5 5486 5497 55o 9 552O 6 54 i3 9 8 i4io 1422 i433 i445 i45 7 5 54 553i 5542 5554 5565 55 7 6 558 7 5 55 i46 9 i48o l4 9 2 i5o4 i5i5 l52 7 4 55 55 9 8 56io 562i 5632 5643 5655 4 56 i53 9 i55o i562 i5 7 4 i586 159-7 3 56 5666 56 7 7 5688 5700 5 7 n 5 7 22 3 57 i6o 9 1621 632 1 644 i656 i66 7 2 5 7 5 7 33 5744 5 7 56 5 7 6 7 5 77 8 5 7 8 9 2 58 l6 79 i6 9 i 702 i 7 i4 I 7 26 i 7 38 I 58 58oo58i2 5823 5834 5845 585? I 5 9 i74 9 1761 773 i 7 84 i 79 6 1808 O 5 9 5868 58 7 9 5890 5901 5gi3 5924 O 60" 50" 40" 30" 20" 10" 3 60" 50" 40" 30" 20" 1(X' . Co-sine of 29 Degrees. Co-sine of 28 Degrees. a p p C 1" 2" 3" 4" 5" 6" 7" 8" 0" in \l 2 4 5 6 7 8 10 11 p \ I" 2" 3" 4" 5" 6" 7" 8" 9" LOGARITHMIC TANGENTS. S5 .s 5? Tangent of 60 Degrees. .s' s Tangent of 61 Degrees. 0" 10" 20" 30" 40" 50" 0" 10" 20" 30" 40" 50" o 10. 23856i 8609 86583707 8755 88o4 5 9 10.256248 62 9 8 6347 63 97 6447 64 9 6 5 9 I 8852 8901 8 9 5o 8998 9047 9096 58 I 6546 65 9 6 6645 66 9 5 6745 6 79 4 58 2; oi44 9 : 9 3 9242 9290 9 33 9 9 388 57 2 6844 68 9 4 6 9 44 6 99 3 7043 7 o 9 3 5 7 3 9 436 9 485 9 534 9 582 9 63i 9680 56 3 7142 719.2 7242 72 9 i 7341 7 3 9 i 56 4 9728 9777 9826 9 8 7 4 99 23 9972 55 4 7 44i 7490 754o 75 9 o 7640 768 9 55 5 10.240021 0069 0118 0167 O2l5 0264 54 5 7739 7789 7 83 9 7888 7938 79 88 54 6 7 o3i3 o6o5 o362 o654 o4io 0703 0459 0752 o5o8 0800 o55 7 0849 53 52 6 7 8o38 8336 8087 8386 8i3 7 8436 8187 8486 8 2 3 7 8535 8286 8585 53 52 8 0898 0947 0995 io44 io 9 3 n4a 5i 8 8635 8685 8 7 35 8 7 84 8834 8884 5i 9 1190 1239 1288 i33 7 i385 i434 5o 9 8 9 34 8984 9 o33 9 o83 9 i33 9 i83 5o 10 io.24i483 i532 i58i 1629 1678 1.727 49 10 10.259233 9283 9 332 9 382 9 432 9482 49 1 1 1776 i825 i8 7 3 1922 1971 2O2O 48 1 1 9 532 9582 9 632 9 68i 97 3i 9781 48 12 2069 2118 2166 22l5 2264 23l3 4 7 12 9 83i 9881 99 3i 998i ..3i ..80 47 i3 2362 2411 245 9 2 5o8 2 55 7 2606 46 i3 io.26oi3o 0180 O23O 0280 o33o o38o 46 1.4 2655 2704 2 7 53 2801 285o 2899 45 i4 o43o o48o o53o o58o o62 9 0679 45 i5 2 9 48 2997 3o46 3095 3i43 3i 9 2 44 i5 0729 0779 o82 9 08.79 O 9 2 9 0979 44 16 324l 3290 3339 3388 343 7 3486 43 16 1029 1079 II2 9 n 79 I22 9 1279 43 17 3535 3584 3632 368i 373o 3779 42 '7 1329 i3 79 l42 9 1479 l52 9 i5 79 42 18 8828 38 7 7 3926 3 97 5 4024 4o 7 3 4i 18 1629 1679 I 7 2 9 1779 l82 9 1879 4i 19 4l22 4171 4220 4269 43i8 4366 4o X 9 1929 1979 202 9 2079 2I2 9 2179 4o 20 10. 2444i5 4464 45i3 4562 46zi 466o 3 9 20 10.262229 2279 232 9 2 3 79 242 9 2479 39 21 4709 47^8 48o 7 4856 4 9 o5 4954 38 21 2529 25 79 262 9 26 79 2 7 2 9 2779 38 22 5oo3 5o52 5ioi 5i5o 5199 5248 37 22 2829 2879 2 9 2 9 2979 3o2 9 3o 79 3 7 23 5297 5346 53 9 5 5444 5493 5542 36 23 3i3o 3i8o 3230 3280 333o 338o 36 24 55 9 i 564o 5689 5 7 38 5 7 8 7 5836 35 24 343o 348o 353o 358o 363o 368i 35 25 5885 5 9 34 5 9 84 6o33 6082 6i3i 34 25 3 7 3i 3 7 8i 383i 388 1 3 9 3i 3981 34 26 6180 6229 6278 6327 63 7 6 6425 33 26 4o3i 4082 4l32 4182 4232 4282 33 27 6474 65 2 3 6572 6621 6670 6720 32 27 4332 4382 4433 4483 4533 4583 32 28 6769 6818 6867 6916 6965 7014 3i 28 4633 4683 4734 4784 4834 4884 3i 29 7063 7112 7161 7211 7260 7 3o 9 3o 2 9 4934 4985 5o35 5o85 5i35 5 1 85 3o 3o ic.247358 7407 7456 7 5o5 7554 7604 2 9 3o io.265236 5286 5336 5386 5436 5487 2 9 3i 7653 7702 77 5i 7800 7849 7899 28 3i 553 7 5587 563 7 5688 5 7 38 5 7 88 28 32 7948 7997 8o46 8o 9 5 8i44 8194 27 32 5838 588 9 5 9 3 9 5 9 8 9 6o3 9 6090 27 33 8243 8292 834i 8390 8439 848 9 26 33 6i4o 6190 6240 62 9 I 634i 63 9 i 26 34 8538 858 7 8636 8685 8 7 35 8784 25 34 6442 6492 6542 65 9 2 6643 66 9 3 25 35 8833 8882 8931 8981 9o3o 9079 24 35 6 7 43 6 79 4 6844 68 9 4 6 9 45 6 99 5 24 36 9128 9178 9227 9276 9 325 9 3 7 5 23 36 7045 7096 7 i46 7i 9 6 7247 7 2 97 23 3? 9424 9 4 7 3 9522 9 5 7 2 9621 9670 22 37 7 34 7 7 3 9 8 7448 74 9 8 7549 7 5 99 22 38 9719 9769 9818 9867 9916 9966 21 38 7649 77 oo 77 5o 7800 7 85i 79 oi 21 3 9 io.25ooi5 0064 on4 oi63 O2I2 0261 2O 3 9 7952 8002 8o52 8io3 8i53 8204 20 4o io.25o3n o36o 0409 0459 o5o8 o55 7 X 9 4o 10.268254 83o4 8355 84o5 8456 85o6 r 9 4i 0607 o656 0705 o 7 55 o8o4 o853 18 4i 8556 86o 7 865 7 8708 8 7 58 88o 9 18 42 0903 0952 1001 io5i IIOO n4 9 17 42 885 9 8 9 o 9 8 9 6o 9 oio 9 o6i 9 m 7 43 1199 1248 1297 1 347 i3 9 6 i445 16 43 9 l62 9 2I2 9 263 9 3i3 9 364 9 4i4 16 44 i495 1 544 i5 9 4 i643 1692 1742 i5 44 9 465 9 5i5 9 566 9616 9 66 7 9717 i5 45 1791 i84o 1890 i 9 3 9 1989 2038 i4 45 9767 9 8i8 9 868 99 i 9 997 . .20 i4 46 2087 2137 2186 2236 2285 2335 r3 46 10.270071 OI2I 0172 0222 0273 o323 i3 47 2384 2433 2483 2532 2582 263i 12 47 o3 7 4 O424 0475 0525 0576 0626 12 48 2681 2730 2779 2829 2878 2 9 28 II 48 0677 0728 0778 o82 9 o8 79 0930 II 49 2977 3027 3076 3i26 3i 7 5 3225 10 49 o 9 8o io3i 1082 Il32 n83 1233 10 5o 10.253274 3324 3373 3423 3472 352i 9 5o 10.271284 i335 i385 i436 i486 i53 7 9 5i 3571 3620 3670 3719 3 7 6 9 38i8 8 5i i588 i638 i68 9 i 7 3 9 I 79 '0 i84i 8 52 3868 3 9 i8 3 9 6 7 4017 4o66 4n6 7 52 i8 9 i I 9 42 I 99 3 2o43 2O 9 4 2l45 7 53 4i65 42i5 4264 43i4 4363 44i3 6 53 2I 9 5 2246 22 9 7 2 34 7 23 9 8 244 9 6 54 4462 45i2 456i 46n 466i 4710 5 54 24 99 255o 26OI 2 65i 2702 275 1 5 55 4 7 6o 4809 485 9 4908 4 9 58 5oo8 4 55 2803 <,:854 2 9 o5 2 9 55 3oo6;3o5 7 4 56 5o5 7 5107 5i56 5 2 o6 5 2 56 53o5 3 56 3io8 3i58 32O 9 3260 33io!336i 3 $7 5355 54o4 5454 55o4 5553 56o3 2 57 34i2 3463 35i3 3564 36i5|3666 2 58 5 9 5652 5 9 5o 5702 6000 5 7 52 6049 58oi 6099 585i 6149 5 9 oi 6i 9 8 I 58 5 9 3 7 i6 4O2I 3 7 6 7 4072 38i8 4l22 386 9 [3 9 i 9 '3 97 o 4173^224)4276 O 60" 50" 40" no" 20" 10" d 00" 50" 40" 3<>' | 20" 10" c Co-tangent of 29 Degrees. a Co-tangent of 28 Degrees. 1 P Part$ l " ~" 3 " 4// 5 " 6 " 7 " 8 " 9// } 5 10 15 20 25 29 34 39 44 .( 1" 2" 3" 4" 5" 6" 7" 8" 9" irt \ 5 10 15 20 25 30 35 40 45 86 LOGARITHMIC SINES. I Sine of 62 Degrees. d Sine of 63 Degrees. m 0" 10" 20"' 30" 40" 50" s 0" 10" 20" 30" | 40" 50" 9.945935 5946595-7,59695980 5991 5 9 9 . 9 4 9 88i 9 8 9 2 99 02 99 i3 99 2/ 99 35 5 9 6002 601360246036,6047 6o58 58 I 99 45 99 56 99 6 7 9977 99 88 9999 58 2 6069 6080 6092 6io3 61 1^ 6i 2 5 5 7 t 9.950010 OO2O oo3i 0042 OOD2 oo63 57 C 6i3( 6147615961706181 6192 56 f ~ 007^: 0084 oo 9 5 0106 011-7 0127 56 i 620^ 6214622662376248 625 9 55 L oi38 oi4 9 oi5 9 01700181 oi 9 i 55 t 6270 6281 62 9 3 63o4 63i5 6326 54 t 0202 02 1C 022^ O234 0245 0256 54 6 633 7 6348 635 9 637i 6382 63 9 3 53 6 0266 02 77 0288 02 9 8 o3o 9 O32O 53 8 6471 64i5 64266437 6482 6493 65b4 6^1 646o 6526 52 5i 8 o33o 0394 o34i o4o5 o35 2 o4i6 o36 2 0426 o3 7 3 o43 7 o384 5a 5i 9 6538 6549656o 6571 6582 65 9 3 5o 9 o458 o46 9 o48o o4 9 o o5oi O5l2 5o 10 9.946604 66i5 6627 6638 664 9 6660 49 10 9 . 9 5o522 o533 o544 o554 o565 o5 7 6 49 ii 6671 6682 66 9 367o467i5 6-726 48 ii o586 o5 97 0607 0618 0629 o63 9 48 12 6 7 38 6749 6760 6771 6782 6 79 3 47 12 o65o 0661 0671 0682 o6 9 c 0703 47 i3 68o4 68i5 6826 683 7 684 9 6860 46 i3 0-714 0-724 o 7 35 o 7 46 o 7 56 0767 46 i4 6871 6882 68 9 36 9 o4 6 9 i5 6926 45 i4 o 77 8 0-788 799 o8o 9 0820 o83i 45 i5 6 9 3 7 6948 6 9 5 9 6 9 7o 6 9 82 6 99 3 44 1 5 o84i o852 0862 o8 7 3 0884 o8 9 4 44 16 7004 7oi5 702617037 7 o48 7 5 9 43 16 o 9 o5 o 9 i5 O 9 26 o 9 37 o 9 4 7 o 9 58 43 17 7 o 7 o 7081 70 9 2 7 io3 7114 7 I25 42 i 7 o 9 68 979 o 99 o IOOO IOII 1021 42 18 7 i36 7147 7 i58 7 i 7 o 7181 7192 4i 18 1032 io43 io53 1064 io 7 4 io85 4i 19 7 203 7214 7 225 7 236 7247 7258 4o i 9 io 9 6 1106 1117 1127 u38 n48 4o 20 9.947269 7280 7 2 9 I 7302 7 3i3 7 324 3 9 20 9 . 9 5n5 9 II 7 O 1180 1 191 I2OI 1212 3 9 21 7 335 7 346 7 35 7 7368 7 3 79 7 3 9 o 38 21 1222 1233 1244 1254 1265 1275 38 22 7 4oi 7 4l2 7 423 7434 7445 7 456 37 22 1286 I2 9 6 i3o 7 .3i 7 i3 2 8 i33 9 3? 23 7 46 7 7 4 7 8; 7 489 7 5oo 75n 7 522 36 23 i34 9 i36o i3 7 o i38i i3 9 i 1402 36 24 7 533 7545 7 556 7 56 7 7 5 7 8 7 58 9 35 e4 1412 i423 i434 1 444 i455 i465 35 25 7 6oo 7 6n -7622 7 633 7 644 7 655 34 25 i4 7 6 i486 1497 1507 i5i8 i528 34 26 7 665 7676 7687 7698 77 o 9 77 20 33 26 i53 9 i54 9 :56o i5 7 o i58i i5 9 i 33 27 77 3i 77 42 7753 77 64 777 5 77 86 32 2 7 1602 i6i3 1623 1 634 1 644 i655 32 28 7797 -7808 78i 9 783o 7 852 3i 28 i665 1676 1686 i6 9 7 I 7 o 7 1718 3i 29 7 863 7874 7 885 7896 7907 79 l8 3o 2 9 1728 i 7 3 9 i 7 4 9 1760 I 77 o 1781 3o 3o 9 . 9 47 9 2 9 7940 7 9 5i 7962 7973 7 984 29 3o 9 . 9 5i 79 i 1802 1812 1823 i833 1 844 29 3i 799 5 8006 8017 8028 8o38 8049 28 3i 1 854 i865 1875 1886 i8 9 6 J 97 28 32 8060 8071 8082 8o 9 3 8io4 8n5 2 7 32 1917 I 9 28 i 9 38 i 9 4 9 i 9 5 9 i 9 6 9 2 7 33 8126 81378148 8i5 9 8170 8181 26 33 i 9 8o i 99 o 20OI 2OII 2022 2032 26 34 8l 9 2 82o382i3 8224 8235 8246 25 34 2o43 2o53 2064 20 7 4 2085 2O 9 5 25 35 8257 8268 8279 82 9 o 83oi 83i2 24 35 2106 2116 2126 2l3 7 2l4 7 2i58 24 36 8323 8334 8344 8355 8366 83 77 23 36 2168 2179 2i8 9 22OO 2210 2221 23 3 7 8388 83998410 8421 8432 8443 22 37 223l 2252 2262 22 7 3 2283 22 38 8454 8464 8475 8486 84 9 7 85o8 21 38 22 9 4 23o4 2 3i4 2325 2335 2346 21 3 9 85i 9 853o854i 8552 8562 85 7 3 2O 3 9 2356 2 36 7 2377 2 38 7 23 9 8 2408 20 4o 9. 948584 85 9 5 8606 8617 8628 863o I 9 4o 9 . 9 524i 9 2440 2 45o 2460 2471 X 9 4i 865o 86608671 8682 86 9 3 8704 18 4i 248 1 24 9 2 25O2 25l2 2523 2533 18 42 8 7 i5 87268736 8 7 4 7 8 7 58 8 7 6 9 '7 42 2 544 2554 2565 25 7 5 2585 2 5 9 6 17 43 8780 8791 8802 8812 8823 8834 16 43 2606 2617 2627 2637 2648 2658 16 44 8845 8856886 7 8878 8888 8899 i5 44 2669 2 6 79 2 68 9 2700 2 7 IO 2720 i5 45 8910 8921 8 9 3 2 8 9 43 8 9 54 8 9 64 i4 45 2731 2 7 4l 2752 2762 2772 2 7 83 i4 46 8 97 5 8986 Sogri 9 oo8 9 oi 9 9 02 9 i3 46 2793 2803 2814 2824 2835 2845 i3 47 9040 9o5i 9 o62 9 o 7 3 9 o83 9 o 9 4 12 4 7 2855 2866 2876 2886 2 8 97 2 9 7 12 48 9io5 9116 9 I2 7 9 i38 9 i48 9 i5 9 II 48 2918 2 9 28 2 9 38 2 9 4 9 2 9 5 9 2 9 6 9 II 49 9 I 7 9181 9 I 9 2 9 2O2 9 2l3 9 224 IO 49 2 9 8o 299 3ooo 3on 3O2I 3o3i IO 5o 9.949235 9246 9 256 9 267 9 2 7 8 9 28 9 9 5o 9 >9 53o42 3o52 3o62 3o 7 3 3o83 3o 9 3 9 5i 93oo 9 3io 9 32I 9 332 9 343 D354 8 5i 3io4 3n4 3i24 3i35 3i45 3i55 8 52 9 364 9 3 7 5 9 386 9 3 97 9 4o8 9 4i8 7 52 3i66 3i 7 6 3i86 3i 97 3207 3217 7 53 9429 944o 9 45i 9 472 9 483 6 53 3228 3238 3248 325 9 326 9 3279 6 54 9494 95o5 9 5i5 9 526 9 53 7 9 548 5 54 32 9 o 33oo 33io 332i 333i 334i 5 55 9 558 9 56 9 9 58o 9 5 9 i 9 6o2 9612 4 55 3352 3362 33 7 2 3382 33 9 3 34o3 4 56 9623 9 634 9 645 9 655 9 666 96-7-7 3 56 34i3 3424 3434 3444 3455 3465 3 5/ 58 9688 9 7 52 9 6 9 8 9763 97099720 9774 9784 97 3i' 97 4i 9-795 9806 2 I 5 7 58 34?5 353 7 3485 3496 35o6J35i6 352 7 354 7 355 7 3568 35 7 8 3588 2 I 5 9 9816982-7 9 8">8 9 84 9 985998-70 O 5 9 35 99 36o 9 36i 9 362 9 '364o 365o o 60" 50" 40" 30" 20" 10" cf 60" 50" 40" 30" 20" 10" a . Co-sine of 27 Degrees. Co-sine of 26 Degrees. PP^t* 1 " 2" 3// 4 " 5" 6 " ?" 8" 9" . ( I" 2" 3" 4" 5" 6" 7" 8" 9" fft { 1 2 3 4 5 7 8 9 10 irl 12 3 456 789 LOGARITHMIC T v \ G E N r s. 87 jj Tangent of 62 Degrees. .5 Tangent of 03 Degrees. s 0" | 10" 20" 30" 40" 50" s 0" 10" 20" 30" 40" 50" o 10.274326 43 7 6 442 7 4478 452 9 458o 5 9 o 10.292834 2886 2 9 38 2 99 3o42 3o 9 4 5 9 I 463o 468i 4732 4783 4834 4885 58 i 3i46 3i 99 3 2 5i 33o3 3355 3407 58 2 4935 4986 5o3 7 5o88 5i3 9 Sigo 5 7 2 345 9 35ii 3563 36i5 366 7 3720 57 3 6240 5291 5342 SSgS 5444 5495 56 3 3 77 2 3824 38 7 6 3 9 2& 3 9 8o 4o32 56 4 5546 55 97 5647 5698 5 7 4 9 58oo 55 4 4o84 4i3 7 4189 424i 42 9 3 4345 55 5 585i 5902 5 9 53 6oo4 6o55 6io5 54 5 43 9 7 444 9 45o2 4554 46o6 4658 54 6 6i56 62O 7 6258 6309 636o 64n 53 6 4710 4 7 63 48i5 486 7 4 9 i 9 4971 53 7 6462 65i3 6564 66i5 6666 6717 52 7 5o24 5o 7 6 5i28 5i8o 5232 5 2 85 5a 8 6768 6819 6870 6920 6 97 i 7 O22 5i 8 533 7 538 9 544 1 54 9 4 5546 55 9 8 5.i 9 7 o 7 3 7 I24 7 i 7 5 7226 7 2 77 7 328 5o 9 565o 5 7 o3 5 7 55 58o 7 585 9 5912 5o 10 io.2 77 3 79 7 43o 748 1 7 532 7 583 7 634 4c; 10 10.295964 6016 6068 6121 6i 7 3 6225 49 ii 7 685 7736 7787 7 838 7 88 9 79 4o 48 1 1 62 7 8 633o 6382 6434 648 7 653 9 48 12 799 r 8o43 8094 8i45 8i 9 6 824 7 47 12 65 9 i 6644 6696 6 7 48 6801 6853 47 13 8298 8349 84oo 845 1 85o2 8553 46 i3 6905 6 9 58 7 OIO 7 o62 7 ii5 7167 46 i4 86o4 8655 8706 8 7 5 7 88o 9 8860 45 i4 7219 7 2 7 2 7 324 7377 7 42 9 7 48i 45 i5 8911 8962 9013 9064 9 u5 9166 44 i5 7534 7 586 7 638 7 6 9 i 7743 779 6 44 16 9217 9268 9320 9 3 7 i 9 422 9 4 7 3 43 16 7 848 79 oo 79 53 8oo5 8o58 8no 43 i? 9 5 2 4 9 5 7 5 9626 9 6 7 8 9729 9780 42 J 7 8i63 82i5 826 7 8320 83 7 2 8425 42 18 9831 9882 99 33 9984 ..36 ..87 4i 18 84 77 853o 8582 8635 868 7 8 7 4o 4i *9 io.28oi38 0189 0240 0292 o343 o3 9 4 4o 1 9 8 79 2 8845 88 97 8 9 4 9 9 OO2 9 o54 4o 2t_/ io.28o445 0496 o548 o5 99 o65o 0701 3 9 20 IO.2 99 I0 7 9 i5 9 9 2I2 9 264 9 3i 7 9 3 7 o 3 9 21 0752 o8o4 o855 0906 o 9 5 7 1009 38 21 9 422 9 4 7 5 9 52 7 9 58o 9 63 2 9 685 38 22 1060 mi 1162 I2l4 1265 i3i6 3 7 22 9737 979 9 842 9 8 9 5 9947 .... 37 23 i367 1419 i4 7 o l52I l5 7 2 1624 36 23 io.3ooo53 oio5 oi58 O2IO 0263 o3i5 36 24 i6 7 5 1726 i 777 1829 1880 1931 35 24 o368 0421 o4 7 3 o526 o5 7 8 o63i 35 25 1983 2034 2o85 2I3 7 2188 2239 34 25 o684 o 7 36 o 7 8 9 o84i o8 9 4 o 9 4 7 34 26 2291 2342 2 3 9 3 2445 24 9 6 2 54 7 33 26 999 1052 iio5 n5 7 1210 1263 33 27 2599 265o 2701 2 7 53 2804 2855 32 27 i3i5 i368 1421 i473 i526 i5 79 32 28 2907 2 9 58 3oo 9 3o6i 3lI2 3i64 3i 28 i63i i684 i 7 3 7 1789 1842 i8 9 5 3i 29 32i5 3 2 66 33i8 3369 3421 34 7 2 3o 29 i 9 4 7 2OOO 2o53 2106 2i58 2211 3o 3o ic.83523 35 7 5 3626 36 7 8 3 7 2 9 3 7 8o 29 3o IO. 3O2264 23i6 236 9 2422 24 7 5 252 7 2 9 3i 3832 3883 3 9 35 3 9 86 4o38 4089 28 3 1 258o 2633 2686 2 7 38 2 79 I 2844 28 32 4i4o 4192 4243 42 9 5 4346 43 9 8 27 32 2 8 97 2 9 5o 3OO2 3o55 3io8 3i6i 2 7 33 4449 45oi 4552 46o4 4655 4 7 o 7 26 33 32i3 3266 33i 9 33 7 2 3425 34 7 8 26 34 4 7 5S 48io 486i 49>3 4 9 64 5oi6 25 34 353o 3583 3636 368 9 3 7 42 3 79 4 25 35 5067 5119 5170 5222 52 7 3 5325 24 35 384 7 3 9 oo 3 9 53 4oo6 4o5 9 4lI2 24 36 53 7 6 5428 5479 553i 5582 5634 23 36 4i64 42 I 7 42 7 4323 43 7 6 4429 23 37 5686 5 7 3 7 5789 584o 58 9 2 5 9 43 22 37 4482 4535 4588 464o 46 9 3 4 7 46 22 38 5 99 5 6o46 6098 6i5o 6201 6253 21 38 4799 485 2 4 9 o5 4 9 58 5on 5o64 21 3 9 63o4 6356 64o8 645 9 65ii 6562 20 3 9 5i i 7 5170 5223 52 7 6 5328 538i 20 4o 10.286614 6666 6717 6769 6821 68 7 2 1 9 4o io.3o5434 548 7 554o 55 9 3 5646 56 99 1 9 4i 6924 6 97 5 7027 779 7 i3o 7 l82 18 4i 5 7 5 2 58o5 5858 5 9 ii 5 9 64 6oi 7 18 42 7234 7 285 7 33 7 7 38 9 7 44o 74 9 2 *7 42 6o 7 o 6128 6i 7 6 622 9 6282 6335 1 7 43 7544 7 5 9 5 7 64 7 7699 77 5i 7 802 16 43 6388 644 1 6 49 4 654 7 6600 6654 16 44 7 854 7 9o6 79 5 7 8009 8061 8n3 i5 44 6 7 o 7 6760 68i3 6866 6919 6 97 2 i5 4? 8i64 8216 8268 8319 83 7 i 8423 i4 45 7 O25 7078 7 i3i 7i84 7 23 7 7 2 9 o i4 46 84 7 5 85 2 6 85 7 8 863o 8682 8 7 33 i3 46 7344 7397 7 45o 7 5o3 7 556 7 6o9 i3 47 8 7 85 883 7 8889 8941 8992 9044 12 4 7 7 662 77 i5 77 68 7 822 7 8 7 5 79 28 12 48 9096 9i48 9199 925i 9 3o3 9 355 II 48 798i 8o34 8o8 7 8i4i 8194 8 2 4 7 II 49 9 4o 7 9 458 9 5io 9 562 9614 9666 10 4 9 83oo 8353 84o6 846o 85i3 8566 IO 5o io.289 7 i8 9769 9821 9873 9925 9977 9 5o io.3o86i 9 86 7 2 8 7 26 8779 8832 8885 9 5i 10.290029 0081 Ol32 oi84 0236 0288 8 5i 8 9 38 8 99 2 9 o45 9 o 9 8 9161 9205 8 52 o34o o3 9 2 o444 0496 o54 7 0599 7 52 9 258 9 3ii 9 364 9 4i8 9471 9524 7 53 o65r o 7 o3 0755 o8o 7 oSS^ 0911 6 53 9 5 77 963i 9 684 9737 979 9 844 6 54 0963 ioi5 1066 1118 II 7 1222 5 54 9 8 97 99 5o ...4 ..5 7 . no .i64 5 55 I2 7 4 i326 i3 7 8 i43o 1482 i534 4 55 IO.3lO2I 7 02 7 O 0324 o3 77 o43o o484 4 56 i586 i638 1690 I 7 42 1-794 i846 3 56 o53 7 oSgO o644 o6 97 o 7 5o 0804 3 5? 1898 i 9 5o 2002 2o54 2106 2i58 2 57 o85 7 0910 o 9 64 IOI 7 io 7 o 1124 2 58 22IO 2262 23l4 2366 2418 24 7 I 58 ii 77 I23l 1284 i33 7 i3 9 i i444 I 5 9 2522 25 7 4 2626 26 7 8 2 7 3o 2 7 82 O 5 9 i4 9 8 i55i i6o5 i658 1711 I7 65 60" 50" 40" 30" 20" 10" a 0" 50" 40" 30" 20" 10" a Co-tangent of 27 Degrees. .9 s Co-tangent of 26 Degrees. P P ,rt 5 l " ~" 3 " 4// 5// 6 " 7// 8 " 9 " 1 * * 1" 2 " 3 " 4// r> " (J " 7 " 8 " 9 " tri ) 5 10 15 21 26 31 36 41 46 | l ' l urt { 5 11 16 21 LOGARITHMIC TANGEJVTS. 89 J Tangent of 64 Degrees. a Tangent of 03 Degrees. s 0" 10" 20" 30" 40" 50" & 0" 10" 20" 30" 40" 50" ~~o|io.3ii8i8 1872 1925 I 979 2O32 2085 5 9 O io.33i32 7 i382 i43 7 1492 1 547 1602 5 9 I 2i3 9 2192 2246 2299 2353 2406 58 I i65 7 1712 1767 1822 i8 77 1932 58 21 246O 25i3 2567 2620 26 7 4 2 7 2 7 5 7 2 i 9 8 7 2042 2097 2I 53 2208 2263 57 3 2781 2334 2888 2941 2995 3o48 56 3 2 3i8 2 3 7 3 2428 2483 a538 2 5 9 3 56 L 3io2 3i55 3209 3263 33i6 33 7 o 55 4 2648 2703 2 7 58 28i3 2868 2 9 24 55 5, 3423 34 7 7 353o 3584 363 7 36 9 i 54 5 2979 3o34 3089 3i44 3i 99 3254 54 e 374^ 3 79 8 3852 3 9 o5 3 9 5 9 4oi3 53 6 3309 3364 3420 3475 353o 3585 53 7 4o66 4l2O 4173 4227 4281 4334 52 7 364o 36 9 5 3 7 5i 38o6 386i 3916 52 8 4388 4442 4495 4549 46o3 4656 5i 8 3971 4026 4082 4i3 7 4192 4247 5i 9 4710 4764 48i 7 48 7 i 4 9 25 4 97 8 5o 9 4302 4358 44i3 4468 4523 45-79 5o 10 io.3i5o32 5o86 5 1 39 SigS 5247 53oo 49 10 io.334634 468 9 4744 48oo 4855 4910 49 ii 5354 54o8 546i 55i5 556 9 5623 48 1 1 4g65 5021 5076 5i3i 5i86 5242 48 12 5676 5730 5784 5838 58 9 i 5 9 45 47 12 5297 5352 5408 5463 55i8 55 7 4 47 13 5999 6o53 6106 6160 6214 6268 46 i3 5629 5684 5 7 4o 5 79 5 585o 5906 46 i4 632i 63 7 5 6429 6483 653 7 65 9 o 45 i4 5961 6016 6072 6127 6182 6238 45 i5 6644 6698 6752 6806 6860 6913 44 i5 6293 634 9 64o4 645 9 65i5 65 7 o 44 16 6967 7021 7075 7129 7 i83 7 236 43 16 6625 6681 6736 6792 684 7 6903 43 J 7 7290 7344 7 3 9 8 7452 7 5o6 7 56o 42 17 6 9 58 7013 7069 7124 7 i8o 7 235 42 18 76i3 7667 7721 777 5 7 82 9 7 883 4i 18 7291 7346 7402 7 45 7 7 5i3 7 568 4i J 9 79 3 7 7991 8o45 8099 8!53 8206 4o J 9 7624 7679 77 35 779 7 846 7901 4o 20 10.318260 83i4 8368 8422 84 7 6 853o 3 9 20 10.337957 8012 8068 8i23 8i 79 8234 3 9 21 8584 8638 8692 8 7 46 8800 8854 38 21 8290 8345 84oi 8456 85i2 8568 38 22 8908 8962 9016 90-70 9124 9178 37 22 8623 8679 8 7 34 8790 8845 8901 37 23 9232 9286 9 34o 9394 9448 9502 36 23 8 9 5 7 9012 9068 9123 9 i 79 9 235 36 24 9 556 9610 9 664 97i8 977 2 9826 35 24 9 2 9 9 346 9402 9 45 7 9 5i3 9 56 9 35 25 9880 9934 9988 ..42 ..96 .i5i 34 25 9 624 9 68o 97 35 979 1 9 84 7 9902 34 26 I0.32O2O5 0259 o3i3 o36 7 0421 o475 33 26 99 58 ..i4 ..70 .125 .181 . 2 3 7 33 27 0529 o583 o63 7 0692 o 7 46 0800 32 27 10.340292 o348 o4o4 o46o o5i5 0571 32 28 o854 0908 0962 1016 I0 7 I 1125 3i 28 0627 0682 o 7 38 o 79 4 o85o 0905 3i 29 1179 1233 1287 i34i i3 9 6 i45o 3o 29 0961 1017 1073 1129 n84 1240 3o 3o io.3ai5o4 i558 1612 1666 I 7 2I 1775 29 3o 10.341296 i35 2 i4o8 i463 1519 i5 7 5 2 9 3i 1829 i883 i 9 38 1992 2046 2IOO 28 3i i63i 1687 i 7 42 i79 8 i854 1910 28 82 2i54 2209 2263 23l 7 23 7 I 2426 2 7 32 1966 2O22 2078 2i33 2189 2245 2 7 33 2480 2534 2588 2643 269 7 2 7 5! 26 33 2301 2 35 7 24i3 2469 2525 258i 26 34 2806 2860 2914 2968 3o23 3o 77 25 34 2636 2692 2 7 48 2804 2860 2916 25 35 3i3i 3i86 324o 3294 3349 34o3 24 35 2972 3028 3o84 3i4o 3196 3252 2 A 36 3457 35i2 3566 3620 36 7 5 3 7 2 9 23 36 33o8 3364 3420 34 7 6 3532 3588 2'J 3? 3 7 83 3838 3892 3 9 4 7 4ooi 4o55 22 37 3644 3700 3 7 56 38i2 3868 3924 2'< 38 4no 4i64 4219 4s 7 3 432 7 4382 21 38 3980 4o36 4092 4i48 4204 4260 21 3 9 4436 449i 4545 4599 4654 4708 2O 3 9 43i6 43 7 2 4428 4484 454o 45 9 6 2O 4o 10.324763 4817 4872 4926 4981 5o35 J 9 4o 10.344652 4708 4?64 4821 48 77 4 9 33 19 4i 5089 5i44 5198 5253 53o 7 5362 18 4i 4989 5o45 5ioi 5i5 7 52i3 5269 1 8 42 54i6 547i 55 2 5 558o 5634 568 9 l l 42 5326 5382 5438 5494 555o 56o6 17 43 5 7 43 5 79 8 5852 590-7 5 9 62 6016 16 43 5663 5719 5 77 5 583i 588 7 5 9 43 16 44 6071 6i25 6180 6234 6289 6343 i5 44 6000 6o56 6112 6168 6224 6281 i5 45 63 9 8 6453 65o 7 6562 6616 6671 i4 45 633 7 63 9 3 644 9 65o6 6562 6618 i4 46 6726 6780 6835 6889 6 9 44 6999 i3 46 66 7 4 6 7 3i 6787 6843 6899 6 9 56 i3 4? 7 o53 7108 7162 7 2I 7 7 2 7 2 7326 12 47 7 OI2 7068 7125 7 i8i 723 7 7 2 9 3 12 48 7 38i 7436 7490 7 545 7 6oo 7654 II 43 7 35o 74o6 7462 7 5i9 7 5 7 5 7 63i I I 49 7709 7764 7818 7 8 7 3 -7928 7982 10 49 7688 7744 7800 7 85 7 79 i3 7969 10 5o 10.328037 8092 8i47 8201 8 2 56 83n n 5o 10.348026 8082 8i3 9 8i 9 5 825i 83o8 9 5i 8365 8420 S475 853o 8584 8639 8 5i 8364 8421 8477 8533 8690 8646 8 52 8694 8 7 4 9 88o3 8858 8 9 i3 8968 7 52 8703 8 7 5 9 88i5 88 7 2 8928 8 9 85 7 53 9023 9077 9132 9 l8 7 9242 9297 6 53 9 o4i 9098 9i54 9211 9267 9324 6 54 935 1 9406 9 46i 9 5i6 9 5 7 i 9625 5 54 9380 9437 9493 9 55o 9606 9 1363 5 55 9680 97 35 979 9 845 99 oo 99 55 4 55 9719 9776 9832 9889 9945 . . .2 4 56 10.330009 006/4 on 9 oi 7 4 022 9 0284 3 56 io.35oo58 on5 0171 O2a8 0285 o34i 3 57 o33 9 oSg/j o448 o5o3 o558 o6i3 2 ^7 o3 9 8 o454 o5n o56 7 0624 0681 2 58 0668 0723 0778 o833 0888 o 9 43 I 58 o 7 3 7 0794 o85o 090-7 0964 I02O I 69 0998 io53 1 108 n63 1218 1272 o 5 9 1077 1 1 34 1190 1247 i3o4 i36o O 60" 50" 40" 30" 20" 10" S* 60" 50" 40" 30" 20" 10" R Co-tangent of 25 Degrees. s Co-tangent of 24 Degrees. g p p 41" 2" 3" 4" 5" 6" 7" 8' 9" 1 \ 5 11 13 122345667 LOGARITHMIC TANGENTS. 03 Tangent of 68 Degrees. d a Tangent of 69 Degrees. s 0" 10" | 20" | 30" 40" 50" 7] 2 0" | 10" | 20" 30" 40" 50" 1 o 10.393590 365II37I2 3772 3833 38 9 4 5 9 io.4i5823 5886 5 9 48 6011 6074 6i3 7 5 9 1 I 3 9 54 4oi5 4o 7 6 4i36 4i97 4258 58 i 6200 6263 6326 6389 645 2 65i5 58 2 43 1 8 43 79 444o 45oo 456i 4622 5 7 2 65 7 8 664i 6 7 o4 6 7 6 7 683o 68 9 3 5 7 9 17 26 35 43 52 61 69 78 100 LOGARITHMIC IS i N E s. pi B Sine of 76 Decrees. .= Sine of 77 Degrees. 8 0" 10" 20" 30" 40" 50" * 0" 10" 20" 30" 40" 50" o 9.986904 6909 6 9 i5 6920 6925 6 9 3o 5 9 O 9 . 9 88 7 24 8729 8 7 34 8 7 3 9 8743 8 7 48 5 9 i 6936 6941 6 9 46 6 9 5i 69 5 7 6962 58 I 8 7 53 8 7 58 8 7 63 8 7 68 8772 8777 58 2 6967 6972 6978 6 9 83 6988 6 99 3 57 2 8 7 82 8787 8 79 2 8797 8802 8806 57 3 6998 7004 7 oo 9 7014 -7019 7025 56 3 8811 8816 8821 8826 883i 8835 56 4 7o3o 7035 7 o4o 7045 7 o5i 7 o56 55 4 884o 8845 885o 8855 8880 8864 55 5 7061 7066 7 7 2 7077 7 o82 7 o8 7 54 5 8869 88 7 4 88 79 8884 888 9 88 9 3 54 6 7092 7098 7 io3 7108 7 n3 7118 53 6 8898 8 9 o3 8 9 o8 8 9 i3 8 9 i8 8 9 22 53 7 7124 7129 71 34 7 i3 9 7i44 7 i5o 52 7 8 9 2 7 8 9 3 2 8 9 3 7 8 9 42 8 9 46 8 9 5i 52 8 7 i55 7160 7 i65 7 i 7 o 7 i 7 6 7 i8i 5i 8 8 9 56 8 9 6i 8 9 66 8 97 o 8 97 5 8 9 8o 5i 9 7186 7191 7196 7 2O2 7 20 7 7 2I2 5o 9 8 9 85 8 99 o 8 99 4 8 999 9 oo4 9009 5o 10 9.987217 7222 7228 7 233 7 238 7 243 49 IO 9 . 9 8 9 oi4 9 oi8 9 O23 9 028 9 o33 9 o?S 49 ii 7248 72 53 7259 7 264 -7269 7 2 7 4 48 ii 9 o42 9 o4 7 9 o52 9 o5 7 9 o62 9 o66 48 12 7279 7284 7290 7 2 9 5 7 3oo 7 3o5 47 12 9 o 7 i 9 o 7 6 9 o8i 9 o85 9 o 9 o 9 o 9 5 47 i3 73io 73i5 7 32I 7 3 2 6 7 33i 7 336 46 i3 9100 9 io5 9 I0 9 9 n4 9 n 9 9 I24 46 i4 734i 7346 7 352 7 35 7 7 362 7 36 7 45 14 9128 9 i33 9 i38 9 i43 9148 9 l52 45 i5 7372 7377 7383 7 388 7 3 9 3 7 3 9 8 44 i5 9 i5 7 9 l62 9 i6 7 9 i 7 i 9176 9181 44 16 18 74o3 7434 7465 7 4o8 743 9 7470 74i3 7419 7444 7449 74 7 5 7 48o 7 454 7485 7 42 9 7 46o 7 4 9 o 43 42 4i 16 17 18 9186 9214 9243 9 i 9 o 9 i 9 5 921919224 9 24 79 252 9 2OO 9 228 9 25 7 9205 9233 9262 9 20 9 9 238 9 266 43 42 4i 19 7496 7 5oi 7 5o6 7 5n 7 5i6 7 52I 4o J 9 9 2 7 I 9 2 7 6 9 28l 9 285 9290 9 2 9 5 4o 20 9.987526 7 53i 7 53 7 7 542 7 54 7 7 55 2 3 9 20 9. 9 8 9 3oo 9 3o4 9 3o 9 9 3i4 9 3i8 9 323 3 9 21 7 55 7 7562 7 56 77 5 7 2 7^77 7 583 38 21 9 328 9 333 9 33 7 9 342 9 34 7 9 35 2 38 22 7 588 7 5 9 3 7 598 7 6o3 7 6o8 7 6i3 37 22 9 356 9 36i 9 366 9 3 7 o 9 3 7 5 9 38o 37 23 7618 7623 7 628 7 634 7 63 9 7 644 36 23 9 385 9 38 9 9 3 9 4 9 3 99 9 4o3 9 4o8 36 24 7649 7 654 7659 7664 -7669 7 6 7 4 35 24 9 4i 3 94i7 9 422 9 42 7 9 432 9 436 35 25 7679 7 684 7690 7695 77 oo 7706 34 25 944^ 9446 9 45o 9 455 9 46o 9 464 34 26 7710 77 i5 77 20 77 25 77 3o 7735 33 26 9469 9474 9479 9 483 9 488 9 4 9 3 33 27 77 4o 77 45 77 5o 77 56 7766 32 27 9497 9 5o2 9 5 7 9 5n 9 5i6 9 52I 32 28 7771 7776 77 8i 77 86 779 1 779 6 3 1 28 9 525 9 53o 9 535 9 53 9 9 544 9 54 9 3i 29 7801 7806 7811 7816 7821 7826 3o 2 9 9553 9 558 9 563 9 568 9 5 7 2 9 5 77 3o 3o 9.987832 7 83 7 7842 7847 7852 7867 29 3o 9.989582 9 586 9 5 9 i 9 5 9 6 9 6oo 9 6o5 29 3i 7862 7867 7872 7877 7882 7887 28 3i 9610 9 6i4 9 6l 9 9628 9 628 9 633 28 32 7892 7897 7902 7907 7912 79 i 7 27 32 9 63 7 9 642 9 64 7 9 65i 9666 9661 27 33 7922 7927 79 32 7937 7942 7947 26 33 9 665 9 6 7 o 9 6 7 5 9 6 79 9684 9689 26 34 79 53 79 58 7963 7968 797 3 7978 25 34 9 6 9 3 9 6 9 8 97 o3 9707 9712 9710 25 35 79 83 7988 79937998 8oo3 8008 24 35 97 26 97 3o 9735 97 4o 9744 -4 36 8oi3 8018 80238028 8o33 8o38 23 36 9749 97 53 97 58 97 63 97 6 7 9772 23 3 7 8o43 8o48 8o538o58 8o63 8068 22 3 7 9777 978i 97 86 979 979 5 9800 23 38 8o 7 3 8078 8o83 8088 8093 8o 9 8 21 38 9804 9 8o 9 9 8i4 9 8i8 9823 9827 21 3 9 8io3 8108 8n38n8 8i23 8128 2O 3 9 9832 9 83 7 9 84i 9840 985o 9 855 2O 4o 9.988133 8i38 8i438i48 8i53 8i58 '9 4o 9 . 9 8 9 86o 9 864 9 86 9 9873 9 8 7 8 9883 '9 4 1 8i63 8168 81738178 8i83 8188 18 4i 9 88 7 9 8 9 2 9 8 9 6 99 oi 99 o6 9910 18 42 8193 8198 8.2638208 82i3 8218 17 42 99 i5 99 i 9 99 24 99 2 9 99 33 99 38 17 43 8223 8227 8232 823 7 8242 8247 16 43 99 42 9947 99 52 99 56 99 6i 99 65 16 44 8252 8 2 5 7 8262 8 2 6 7 8272 8277 i5 44 997 9974 9979 99 84 99 88 999 3 i5 45 8282 8287 8292 82978802 8307 i4 45 9997 . . .2 . . .6 ..ii ..16 . .20 i4 46 83i2 83i 7 8322 832 7 8332 8337 i3 46 9 . 99 0025 OO2 9 oo34 oo38 oo43 oo48 i3 47 8342 8346 835i 8356836i 8366 12 47 oo52 oo5 7 006 1 0066 0070 0075 12 48 83 7 i 83 7 6 838i 83868391 83 9 6 II 48 79 0084 0088 oo 9 3 0098 OIO2 II 49 84oi 84o6 84n 84168420 8425 IO 49 OIO 7 OIII 0116 OI2O 0125 OI2 9 10 5o 9.988430 8435 844o 8445 845o 8455 9 5o 9 . 99 oi34 oi38 oi43 oi48 Ol52 0157 9 5i 846o 8465 8470 84 7 5 848o 8484 8 5i 0161 0166 OI 7 O oi 7 5 oi 79 oi84 8 52 848 9 8494 8499 85o4 8509 85i4 7 52 0188 oi 9 3 oi 97 0202 0206 O2II 7 53 85i 9 85 2 4 8529 8534 8538 8543 6 53 O2l5 O22O 0225 O22 9 0234 0238 6 54 8548 8553 8558 8563 8568 85 7 3 5 54 0243 024 7 O252 O256 0261 0265 5 55 85 7 8 8583 8587 8592 85 97 8602 4 55 O2 7 O 02 7 4 02 79 0283 0288 O2 9 2 4 56 8607 8612 8617 8622 8626 863i 3 56 O2 97 o3oi o3o6 o3io o3i5 o3i 9 3 57 8636 864i 8646 865i 8656 8661 2 57 0324 o328 o333 0337 0342 o346 2 58 8666 8670 867518680 8685 86 9 o I 58 o35i o355 o36o o364 o36 9 0373 I ' 69 86 9 5 8700 870487098714 87i 9 59 o3 7 8 o382 o386 o3 9 i o3 9 5 o4oo O 60" 50" 40" 1 30" | 20" 10" d 60" | 50" 40" 30" 20" 10" d Co-sine of 13 Degrees. Co-sine of 1 2 Degrees. 55 P T, t p " 2" 3" 4" 5" f," 7" 8" 0" . ( 1" 2" 3" 4" 5" 6" 7" 8" 9" in \ 1 122 334 4 5 in { 011223344 LOGARITHMIC T A N a R N T s. 101 d Tangent of 76 Degrees. c Tangent of 77 Degrees. s 0" 10" 20" 30" 40" 50" ij 0" 10" 20" 30" 40" (50^ o 10. 603229^319 34o8 34 9 8 3588 3678 5 9 10.636636 6 7 3 2 6828 6924 7020 7116 5 9 I 3 7 6 7 3857 3 9 4 7 4o3 7 4127 4217 58 I 7213 73oq 7 4o5 75oi 7 5 97 7694 58 2 43o6 43 9 6 4486 45 7 6 4666 4756 57 2 779 7886(7983 8o 79 8i 7 5 8272 5 7 3 4846 4 9 36 5o 2 6 5n6 52o6 52 9 6 56 3 8368 8465 856i 865 7 8 7 54 885o 56 4 5386 54 7 7 556 7 565 7 5 7 4 7 583 7 55 4 8 9 4 7 9043 9 i4o 9237 9 333 9 43o 55 5 5927 6017 6108 6i 9 8 6288 63 7 8 54 5 9 526 9623 9 720 9 8i6 99 i3 . .10 54 6 646 9 655 9 664 9 6 7 4o 683o 6 9 20 53 6 10.640107 O203 o3oo o3 9 7 o4 9 4 o5 9 i 53 7 7011 7101 7 I 9 2 7282 7 3 7 2 7463 52 7 0687 0784 0881 978 1075 1172 52 8 7553 7 644 77 34 7 8 2 5 79 i5 8006 5i 8 I26 9 i366 i463 i56o 1657 1754 5i 9 8097 8187 8278 8368 845 9 855o 5o 9 i85i 1948 2046 2143 2240 2337 5o 10 10. 6o864o 8 7 3i 8822 8 9 i3 9 oo3 994 4 9 10 10.642434 2 53i 262 9 2726 28a3 2 9 2I 4 9 IT 9i85 9 2 ? 6 9 36 7 9 45 7 9 548 9 63 9 48 ii 3oi8 3ii5 32i3 33io 34o 7 35o5 48 12 97 3o 9 82I 99 I2 ...3 .. 9 4 .186 47 12 36o2 3700 3797 38 9 5 3 99 2 4o 9 o 47 i3 10. 610276 0367 o458 o54 9 o64o 0731 46 i3 4i8 7 4285 4383 448o 45 7 8 4676 46 i4 0822 o 9 i3 ioo4 io 9 5 1186 1278 45 i4 4773 48 7 i 4 9 6 9 5o66 5i64 5262 45 1 5 1 369 i46c 1 55 1 1642 i?34 1825 44 i5 536o 5458 5555 5653 5 7 5i 584 9 44 16 1916 2008 20 99 2I 9 O 2282 23 7 3 43 16 5 9 4 7 6o45 6i43 6241 633 9 643 7 43 i? 2464 2 556 2647 2 7 3 9 2 83o 2 9 22 42 '7 6535 6633 6731 682 9 6 9 27 7026 42 18 3oi3 3io5 3i 9 6 3288 33 7 9 3471 4i 18 7124 7222 7320 74i8 75i 7 76i5 4i X 9 3562 3654 3 7 46 3837 3 9 2 9 4021 4o J 9 77 i3 7811 7 9 io 8008 8106 82o5 4o 20 I0.6l4lI2 4204 42 9 6 4388 4479 45 7 i 3 9 20 io.6483o3 8402 85oo 85 99 8697 8796 3 9 21 4663 4755 4847 4 9 38 5o3o 5l22 38 21 88 9 4 8993 9 o 9 i 9 1 9 9288 9 38 7 38 22 52i4 53o6 53 9 8 54 9 o 5582 56 7 4 37 22 9 486 9 584 9 683 9 782 9880 9979 37 23 5 7 66 5858 5 9 5o 6042 6i34 6226 36 23 io.65oo 7 8 0177 0276 o3 7 4 o473 0572 36 24 63i8 64n 65o3 65 9 5 6687 6 779 35 24 o6 7 i 0770 o86 9 o 9 68 1067 1166 35 25 6871 6 9 64 7o56 7i48 7241 7333 34 25 1265 1 364 i463 i56 2 1661 1760 34 26 7425 7 5i8 7610 7702 77 9 5 7887 33 26 i85 9 1958 2o58 2157 2256 2355 33 27 7980 8072 8i64 8 2 5 7 834 9 8442 32 27 4455 2554 2653 2752 2852 2 9 5l 32 28 8534 8627 8720 8812 8 9 o5 8997 3i 28 3o5i 3i5o 324 9 334 9 3448 3548 3i 29 9090 9 i83 9 2 7 5 9 368 9 46i 9 554 3o 2 9 364 7 3 7 4 7 3846 3 9 46 4o46 4i45 3o 3o 10.619646 97 3 9 9 832 992 5 ..17 . no 2 9 3o 10.654245 4344 4444 4544 4643 4743 2 9 3i 10. 620203 O2 9 6 o38 9 0482 o5 7 5 0668 28 3i 4843 4943 5o43 5i42 5242 5342 28 32 0761 o854 o 9 47 io4o n33 1226 27 32 5442 5542 5642 5742 5842 5 9 42 27 33 1319 1412 i5o5 i5 9 8 i6 9 i 1784 26 33 6042 6142 6242 6342 6442 6542 26 34 1878 I 97 I 2064 2if>7 225o 2344 25 34 6642 6742 6842 6 9 43 7o43 7i43 25 35 243 7 253o 2624 2717 2810 2904 24 35 7243 7344 7444 7544 7 645 77 45 24 36 2997 3o 9 o 3i84 3277 3371 3464 23 36 7845 7946 8o46 8i47 8247 8348 23 3? 3558 365i 3745 3838 3 9 32 4o25 22 37 8448 8549 864 9 8 7 5o 885o 8 9 5i 22 38 4119 4213 43o6 44oo 44 9 4 458 7 21 38 9 o52 9152 9 253 9 354 9 454 9 555 21 3 9 468i 4 77 5 486 9 4962 5o56 5i5o 20 3 9 9 656 97 5 7 9 85 7 99 58 ..5 9 .160 2O 4o 10.625244 5338 543 1 55 2 5 56i 9 5 7 i3 I 9 4o 10.660261 o362 o463 o564 o665 0766 I 9 4i 5807 5 9 oi 5 99 5 6o8 9 6i83 6277 18 4i 0867 0968 io6 9 1170 1271 1372 10 42 6371 6465 655 9 6653 6 7 4 7 684i 17 42 i473 i5 7 4 1676 1777 1878 i 9 7 9 7 43 6 9 36 7 o3o 7124 7218 7 3l2 7407 16 43 2081 2182 223 2385 2486 2 58 7 16 44 75oi 7 5 9 5 7 68 9 77 84 7878 797 2 i5 44 2 68 9 2790 28 9 2 2 99 3 3o 9 5 3i 9 6 i5 45 8067 8161 8256 835o 8444 853 9 i4 45 32 9 8 33 99 35oi 36o2 3 7 o4 38o6 i4 46 8633 8 7 28 8822 8 9 i 7 9 on 9106 i3 46 3 97 4oo 9 4in 4212 43i4 44i6 i3 47 9201 9 2 9 5 9 3 9 o 9 484 9 5 79 9 6 7 4 12 47 45i8 4620 4721 4823 4 9 25 5027 12 43 9768 9 863 99 58 ..53 i47 .242 I I 48 5i2 9 523i 5333 5435 553 7 563 9 II 49 io.63o337 o432 0527 0622 0716 08 1 1 IO 49 574i 5843 5 9 45 6047 6i4 9 6252 10 5o 10.630906 1001 1096 n 9 i 1286 i38i 9 5o 10.666354 6456 6558 6660 6 7 63 6865 9 5i 1476 i5 7 i 1666 i 7 6i i85 7 I 9 52 8 5i 6967 7070 7172 7274 7 3 77 7479 8 62 2047 2l42 2237 2332 2428 2523 7 52 7582 7 684 7787 7 88 9 7992 8094 7 53 2618 2713 28o 9 2 9 04 2 999 3o 9 5 6 53 8197 82 99 8402 85o5 8607 8710 6 54 Sigo 3 2 85 338i 3476 3572 366 7 5 54 88i3 8 9 i6 9 oi8 9 I2I 9224 9327 5 55 3 7 63 3858 3 9 54 4o4 9 4i45 4240 4 55 943o 9 532 9 635 9738 9 84i 9944 4 56 4336 4432 452 7 4623 4718 48i4 3 56 10.670047 oi5o 0253 o356 o45 9 o562 3 57 4910 5oo6 5ioi 5i 97 5 29 3 538 9 2 5 7 0666 O76 9 0872 o 97 5 1078 1181 2 58 5485 558o 5676 5772 5868 5 9 64 I 58 1285 1 388 i4 9 i i5 9 5 i6 9 8 1801 I 5 9 6060 6i56 6252 6348 6444 654o 59 1905 2008 2112 22l5 2 3i8 2422 O 60" 50" 40" 30" 20" 10" c 60" 50" 40" 30" 20" | 10" a Co-tangent of 13 Degrees. 3 Co- tangent of 12 Degrees. a 3 P ParfJ *" 2 " 33333 333 7 334o 3344 3348 O 60" 50" 1 40" 30" 20" 10" g 60" j 50" 40" 30" 'JO" ' 10" ^ Co-sine of 1 1 Decrees. 2, Co-sine of 1 Degrees. 2 p p . $ 1" 2" 3" 4" 5" 6" 7" 8" 9" irl $ 1 1 2 2 3 3 3 4 < 1" 2" 3" 4" 5" C" 7" 8" 9" P. Tart J o i ! o o o 3 ;j 4 LOGARITHMIC TANGENTS. 103 A Tangent of 78 Degrees. a Tangent of 79 Degrees. S 0" 10" 20" 30" 40" 50" 0" 10" 20" 30" 40" 50" o 10.672525 2620, 2733 2836 2 9 4o 3o43 r o 10.711348 i46o i5 7 3 i685 i 79 8 I 9 IO 5 9 I 3i47 3 2 5i 3354 3458 3562 3666 i 2023 2i35 2248 236i 2473 2586 58 2 3 7 6 9 38 7 3 3 9 77 4o8i 4i85 428 9 57 2 2 6 99 2811 2 9 24 3o3 7 3i5o 3263 5 7 3 43 9 3 44 97 46oi 4 7 o5 48o 9 4 9 *3 56 3 3376 3488 36oi 3 7 i4 382 7 3 9 4o 56 4 5017 5l2I 5225 532 9 5433 553 7 55 4 4o53 4i6 7 4280 43 9 3 45o6 46i 9 55 5 5642 5 7 46 585o 5 9 54 6o5 9 6i63 54 5 4732 4846 4 9 5 9 5072 5i85 52 99 54 6 6267 63 7 2 6476 658o 6685 6789 53 6 54i2 55 2 6 563 9 5 7 5 2 5866 5 979 53 7 68 9 4 6 99 8 7io3 7 20 7 7312 7417 52 7 6o 9 3 62O 7 6320 6434 654 7 6661 52 8 7521 7626 77 3i 7 835 7940 8o45 5i 8 6 77 5 6889 7 OO2 7 n6 723o 7344 5i 9 8i4 9 8254 835 9 8464 856 9 86 7 4 5o 9 7458 7 5 7 2 7 686 7799 79i3 8027 5o 10 8778 8883 8 9 88 9 o 9 3 9 i 9 8 9 3o3 49 10 8142 8 2 56 83 7 o 8484 85 9 8 8712 49 ii 9 4o8 9 5i3 9 6i8 97 23 9 82 9 99 34 48 ii 8826 8 9 4i 9 o55 9 i6 9 9 283 9 3 9 8 48 12 io.68oo3 9 oi44 O24 9 o355 o46o o565 47 12 9 5i 2 9 62 7 97 4i 9 856 9970 ..85 47 i3 0670 o 77 6 0881 o 9 8 7 I0 9 2 n 9 7 46 i3 IO. 7 2OI 99 o3i4 0428 o543 o658 0772 46 14 i3o3 i4o8 i5i4 i6i 9 1725 i83o 45 i4 088 7 1002 1116 I23l 1 346 i46i 45 r5 I 9 36 2042 2147 2253 235 9 2464 44 i5 i5 7 6 i6 9 i 1806 1921 2o36 2l5l 44 16 2570 26 7 6 2782 288 7 2 99 3 3o 99 43 16 2266 238i 24 9 6 2611 2726 2841 43 17 32o5 33ii 3417 35 2 3 362 9 3 7 35 42 17 2 9 5 7 3o 7 2 3i8 7 3302 34i8 3533 42 18 384i 3 9 4 7 4o53 4i5 9 4265 43 7 i 4i 1.8 364 9 3 7 64 3 99 5 4no 4226 4i '9 4477 4584 46 9 o 4 79 6 4 9 O2 5oo 9 4o i 9 4342 445 7 45 7 3 4688 48o4 4 9 20 4o 20 5n5 5221 5328 5434 5540 5647 3 9 20 5o36 5i5i 5267 5383 5499 56i5 3 9 21 5 7 53 586o 5 9 66 6o 7 3 6179 6286 38 21 5 7 3i 584 7 5o63 6o 79 6i 9 5 63n 38 22 63 9 2 64 99 6606 6 7 I2 68i 9 60,26 37 22 642 7 6543 6l55o 6 77 5 68 9 i 7008 37 23 7032 7 i3 9 7246 7 353 7460 7 56 7 36 23 7 I24 7 24o 7 356 7 4 7 3 7 58 9 7706 36 24 7 6 7 3 77 8o 7887 7994 8101 8208 35 24 7822 7938 8o55 8i 7 i 8288 84o5 35 25 83i5 8422 852 9 8636 8 7 44 885i 34 25 852i 8638 8 7 54 88 7 i 8 9 88 9 io5 34 26 895.8 9 o65 9 I 7 2 9 28o 9 38 7 9 4 9 4 33 26 9 22I 9 338 9 455 9 5 7 2 9 68 9 9 8o6 33 27 9 6oi 9709 9 8i6 99 24 .138 32 27 99 23 ..4o ;l5 7 .274 .3 9 i .5o8 32 28 10.690246 o353 o46i o568 0676 0784 3i 28 io. 7 3o625 o 7 42 0860 0977 io 9 4 I2II 3i 29 o8 9 i 999 1107 I2l4 1322 i43o 3o 2 9 i32 9 1 446 i563 1681 i 79 8 i 9 i6 3o 3o i53 7 1 645 i 7 53 2861 i 9 6 9 2077 2 9 3o 2o33 2l5l 2268 2386 25o3 2621 2 9 3ii 2i84 22 9 2 2400 2 5o8 2616 2724 28 3i 2 7 3 9 2856 2 9 74 3o 9 2 32IO 3327 28 3aj 2832 2 9 4i 3o4 9 3i5 7 3265 33 7 3 27 32 3445 3563 368i 3 799 3 9 i 7 4o35 27 33 348 1 35 9 o 36 9 8 38o6 3 9 i4 4023 26 33 4i53 42 7 I 438 9 45o 7 4625 4744 26 34 4i3i 42 3 9 4348 4456 4565 46 7 3 25 34 4862 4 9 8o 5o 9 8 5217 5335 5453 25 35 4782 48 9 o 4999 5io 7 52i6 5325 24 35 55 7 2 56 9 o 58o 9 5 9 2 7 6o46 6i64 24 36 5433 5542 565i 5 7 5 9 5868 5 977 23 36 6283 64oi 6520 663 9 6757 6876 23 37 6086 6i 9 5 63o3 6412 652i 663o 22 37 6 99 5 7 n3 7232 7 35i 7470 7 58 9 22 38 6 7 3 9 6848 6 9 5 7 7 o66 7175 7 284 21 38 77 o8 7 82 7 7946 8o65 8184 83o3 21 39 7 3 9 3 7 5o3 7612 772i 783o 79 3 9 20 3 9 8422 854i 8660 8780 88 99 9 oi8 2O 4o 8o4 9 8i58 8267 8376 8486 85 9 5 I 9 4o 9 i3 7 9 257 9376 9 4 9 6 9 6i5 9734 I 9 4i 8 7 o5 88i4 8 9 24 9 o33 9 i43 9 252 18 4i 9 854 997 3 .. 9 3 .213 .332 .452 18 42 9 362 9 4 7 i 9 58i 9 6 9 i 9 8oo 99 io 17 42 io. 7 4o5 7 i o6 9 i 0811 o 9 3i io5o 1170 17 43 IO. 7 OOO2O OI2 9 023 9 o34 9 o45 9 o56 9 16 43 I2 9 i4io i53o i65o 1770 i8 9 o 16 44 o6 7 8 o 7 88 o8 9 8 1008 1118 1228 1.5 44 2OIO 2i3o 225o 2370 24 9 O 2611 i5 45 i338 1 448 i558 1668 1779 i88 9 i4 45 2 7 3l 285i 2 97 I 3o 9 2 3212 3332 i4 46 I 999 2IO 9 22I 9 233o 2440 2 55o i3 46 3453 35 7 3 36 9 4 38i4 3 9 35 4o55 i3 47 2661 2 77 I 288l 2 99 2 3l02 32i3 12 47 4176 42 9 7 4417 4538 465 9 4779 12 48 3323 3434 3544 3655 3 7 65 38 7 6 II 48 4 9 oo 5O2I 5i42 5263 5384 55o5 II 49 3 9 8 7 4o 97 4208 43i 9 4429 454o 10 49 5626 5747 5868 5 9 8 9 6110 623i IO 5o 465 1 4762 48 7 3 4 9 84 5o 9 5 5 2 o5 9 5o 6352 6473 65 9 5 6716 683 7 6 9 5 9 9 5i 53i6 542 7 5538 564 9 5 7 6i 58 7 2 8 5i 7080 7201 7323 7444 7566 7687 8 5a 5 9 83 6o 9 4 62o5 63i6 6428 653 9 7 52 7 8o 9 79 3o 8o52 8i 7 4 82 9 5 84i7 7 53 665o 6 7 6i 6S 7 3 6 9 84 79 5 7 20 7 6 53 853 9 8661 8782 8 9 o4 9 O26 9 i48 6 54 7 3i8 7 43o 7 54i 7 653 7876 5 54 9 2 7 9 3 9 2 9 5i4 9 636 9758 9 88o 5 55 79 8 7 8o 99 8211 8322 8434 8546 4 55 io. 7 5ooo2 0124 0247 o36 9 0491 o6i3 4 56 8658 8 7 6 9 8881 8 99 3 9 io5 9 2I 7 3 56 o 7 36 o858 o 9 8o no3 1225 1 348 3 57 9 32 9 9 44i 9 553 9 665 9777 9 88 9 2 5 7 i4 7 o i5 9 3 1715 i838 i 9 6i 2o83 2 58 IO. 7 IOOOI on3 0225 o337 o449 o562 I 58 2206 232 9 2452 25 7 4 2607 2820 i 5 9 o6 7 4 0786 o8 9 8 IOII 1123 1235 O 5 9 2 9 43 3o66 3i8 9 33i2 3435 3558 o 80" 50" 40" 30" 20" 10" g bO" 50" 40" 30" 20" 10" a Co-tangent of 1 1 Degrees. Co-tangent of 10 Degrees. i P Part$ l " ~" 3 " 4 " 5 " 6// 7 " 8 " 9// , ( 1" 2" 3" 4" 5" 6" 7" S" 9" I 11 22 32 43 54 65 75 86 97 irt \ 12 23 35 47 59 70 82 94 10G 104: LOGARITHMIC SINKS. 1 Sine of 80 Degrees. c Sine of 81 Degrees SI 0" 10" 20'' 30"' 40" 50" 0" 10" 20" 30" 40" 50" i o.99335i 3355 335 9 3363 3366 3370 5 9 9.90/4620 4623 462 7 463o 4633 4637 5 9 i 33 7 4 33 77 338i 3385 3389 33 9 2 58 I 464o 4643 4647 465o 4653 465 7 58 o 33 9 6 34oo 34o3 3407 34n 34i4 57 2 466o 4663 466 7 46 7 o 46 7 3 46 7 6 57 3 34i8 3422 3426 3429 3433 343 7 56 3 468o 4683 4686 46 9 o 46 9 3 46 9 6 56 4 344o 3444 3448 345i 3455 345 9 55 4 4700 4 7 o3 4 7 o6 4 7 io 4 7 i3 4716 55 5 3462 3466 34 7 o 34 7 3 34 7 7 348i 54 5 4720 4 7 23 4 7 26 4 7 2 9 4 7 33 4 7 36 54 6 3484 3488 34 9 2 34 9 5 3499 35o3 53 6 4 7 3 9 4 7 43 4 7 46 4 7 4 9 4 7 52 4 7 56 53 / 35o6 35io 35i4 35i 7 352i 35 2 5 52 7 47 5 9 4 7 62 4 7 66 4 7 6 9 4 77 2 4 77 6 52 b 35 2 8 3532 3536 353 9 3543 354 7 5i 8 4779 4 7 8 2 4 7 85 4 7 8 9 4 7 92 47 9 5 5i 9 355o 3554 3558 356i 3565 356 9 5o 9 4798 4802 48o5 48o8 48i2 48i5 5o 10 9.993572 35 7 6 358o 3583 3587 35 9 i 49 10 9.994818 4821 48 2 5 4828 483i 4834 49 ! I 35 9 4 35 9 8 36oi 36o5 36o 9 36i2 48 ii 4838 484i 4844 4848 485i 4854 48 12 36i6 3620 3623 362 7 363i 3634 47 12 485 7 486i 4864 486 7 48 7 o 48 7 4 47 i3 3638 364i 3645 364 9 3652 3656 46 i3 48 77 488o 4883 488 7 4890 48 9 3 46 i4 366o 3663 366 7 |36 7 o 36 7 4 36 7 8 45 i4 48 9 6 4 9 oo 4 9 o3 4 9 o6 4909 4 9 i3 45 i5 368i 3685 368 9 |36 9 2l36 9 6 36 99 44 i5 49-6 49 T 9 4 9 22 4 9 26 4929 4 9 32 44 16 3 7 o3 3 7 o 7 3710 3714 3 7 i 7 3 7 2I 43 16 4 9 35 4 9 38 4 9 42 4 9 45 4948 49.5i 43 !? 3 7 25 3728 3 7 323 7 35 3 7 3 9 3 7 43 42 l l 4955 4 9 58 4 9 6i 4 9 64 4968 4 9 7i 42 18 3746 3 7 5o 3 7 53|3 7 5 7 3 7 6i 3 7 64 4i 18 4974 4 9 77 4 9 8o 4 9 84 498 7 4990 4i T 9 3768 3 77 i 3 77 53 779 3 7 82 3 7 86 4o J 9 499 3 4997 5ooo 5oo3 5oo6 5oo 9 4o 20 9.993789 3 79 3 3 7 9 7 38oo 38o4 38o 7 3 9 20 9.99.501 3 5oi6 5oi 9 5022 5o 2 5 5029 3o 21 38n 38 1 4 38L83822 3825 382 9 38 21 5o3 2 5o35 5o38 5o4i 5o45 5o48j38 22 3832 3836 384o;3843 384 7 385o 37 22 5o5i 5o54 5o5 7 5o6i 5o64 5o6 7 l3 7 23 3854 385 7 386i 3864 3868 38 7 2 36 23 5070 5o 7 3 5o 77 5o8o 5o83 5o86 36 34 38 7 5 38 7 9 3882 ] 3886 3889 38 9 3 35 24 5089 5o 9 2 5o 9 6 5o 99 5 1 02 5io5 35 25 38 97 3 9 oo 3904 39O 7 3911 3 9 i4 34 25 5io8 5lI2 5n5 5n8 5l2I 5i24 34 26 3918 3 9 2i 3925 3928 3932 3 9 36 33 26 5127 5i3i 5i34 5i3 7 5i4o 5i43 33 27 3 9 3 9 3 9 43 3g46 SgSo 3 9 53 3 9 5 7 32 27 5i46 5i5o 5i53 5i56 5i5 9 5i62 32 28 3960 3 9 64 3967*3971 3 97 4 3 97 8 3i 28 5i65 5i6 9 5l 7 2 5i 7 5 5i 7 8 5i8i 3i 29 3982 3 9 85 3989 3992 3 99 6 3 999 3o 29 5i84 5i88 5i 9 i 5i94 5i 97 5200 3o 3o 9.994003 4oo6 4oio 4oi3 4017 4020 2 9 3o 9 >99 52o3 5206 5210 5 2 i3 52i6 5219 29 3i 4024 4027 4o3i;4o34 4o38 4o4i 28 3i 5222 5225 5228 5 2 32 5235 5238 28 32 4o45 4o48 4o52 4o55 4o5 9 4062 2 7 32 524i 5244 524 7 5 2 5o 5 2 53 5257 27 33 4o66 4o6 9 4o 7 3 4o 7 6 4o8o 4o83 26 33 5260 5 2 63 5 2 66 5269 52 7 2 52 7 5 26 34 4087 4o 9 o 40944097 4ioi 4io4 25 34 5278 5282 5 2 85 5 2 88 52 9 I 52 9 4 25 35 4io8 4i ii 4n54n8 4l22 4i25 24 35 52 9 7 53oo 53o3 53o 7 53io 53i3 24 36 4129 4i32 4i364i39 4i43 4i46 23 36 53i6 53i 9 5322 5325 5328 533i 23 3? 4i5o 4i53 4i5 7 ! 4i6o 4i64 4i6 7 22 37 5334 5338 534i 5344 534 7 535o 22 38 4171 4i 7 4 4i 7 84i8i 4i84 4i88 21 38 5353 5356 535 9 5362 5365 536 9 21 3 9 4191 4i 9 5 4198 4202 42o5 4209 2O 3 9 53 7 2 53 7 5 53 7 8 538i 5384l538 7 2O 4o 9.994212 4216 4219 4223 4226 423o r 9 4o 9 . 99 53 9 o 53 9 3 53 9 6 53 99 54o3 54o6 X 9 4i 4233 4236 42404243 4247 425o 18 4i 54o 9 54i2 54i5 54i8 5421 5424 18 42 4254 425 7 4261 4264 4267 42 7 I 1 7 42 542 7 543o 5433 5436 543 9 5442 i? 43 4274 4278 4281 4285 4288 4292 16 43 5446 544 9 545 2 5455 5458 546 1 !6 44 45 42 9 5 43i6 42 9 8 43i 9 43o2 43o5 4322)4326 43o 9 432 9 4312 4333 i5 i4 44 45 5464 5482 546 7 5485 54 7 o 5488 5473 549i 5476 54 9 4 5479 54 9 7 i5 i4 46 4336 434o 4343 4346 435o 4353 i3 46 55oi 55o4 55o 7 55io 55i3 55!6 i3 4? 435 7 436o 4363 436 7 4370 43 7 4 12 47 55i 9 5522 55 2 5 55 2 8 553i 5534 12 48 43 77 438i 4384 438 7 43 9 i 4394 II 48 553 7 554o 5543 5546 554 9 555 2 II 49 43 9 8 44oi 44o4 44o8 44" 44i5 IO 49 5555 5558 556i 5564 556 7 55 7 o 10 5o 9.994418 4421 4425 4428 4432 4435 9 5o 9 <99 55 7 3 55 7 6 55 79 5582 5585 5588 9 5i 4438 4442 4445 4448 4452 4455 8 5i 55 9 i 55 9 4 55 97 56oi 56o4 5607 8 52 445 9 4462 4465 446 9 4472 44 7 6 7 52 56io 56i3 56i6 5619 5622 56 2 5 7 53 4479 4482 4486 4489 44 9 2 4496 6 53 5628 563i 5634 563 7 564o 5643 6 54 4499 45o3 45o6 4509 45i3 45i6 5 54 5646 564 9 5652 5655 5658 566i S 55 45i 9 4523 4526 453o 4533 4536 4 55 5664 566 7 56 7 o 56 7 2 56 7 5 5678 4 56 454o 4543 4546 455o 4553 4556 3 56 568i 5684 568 7 5690 56o3 56 9 6 3 5? 456o 4563 4566 45 7 o 4573 45 7 6 2 57 56 99 5 7 02 5 7 o5 5 7 o8 5 7 n 57i4 2 58 458o 4583 458 7 4590 45 9 3 45 9 7 I 58 5 7 i 7 5 7 20 5 7 23 5 7 26 5 7 29 5 7 3 2 1 5 9 46oo 46o3 46o 7 46 10 46i3 46i 7 o 5 9 5 7 355 7 385 7 4i5 7 44:5 7 4 7 5 7 5o 60" 50" 40" 30" 20" 10" ri 60-'' 50" 40" | 30" : 20" i 10" j Co-sine of 9 Degrees. g Co-sine of 8 Degrees. ^ f > 1" 2" 3" 4" 5" 6" 7" 8" 9" .I.lartJ o x ! ! 2 2 2 3 3 P p, rf $ 1" 2" 3" 4" 5" G' 1 7" 8" i,' in } o 1 i i 2 2 2 n 3 L O G A R I T II M f C T A N G E X T S. 105 |j Tangent of 80 Degrees. c 1 Tangent of 8 1 Degrees. ^ 0" 10" | 20" 30" 40" | 50" 2 0" 10" 20" 30" 40" 50" ] i I0 . 7 5368i 38o4 4544 39284051 46674791 4i 7 4 4 9 i4 4297 5o38 5 9 58 i 10.8002871042^ 1 1 06 1242 o56o l3 79 0696 i5i6 0833,0969 i652 1789 159 58- 2 5i6i 5285 54o8 5532 5655 5 779 57 2 1926 2062 2199 2336 2473 2610 5? 3 5 9 o3 6026 6i5o 62 7 4 68986522 56 O 2 7 4 7 2884 3021 3i58 32 9 5 3433 56 4 6646 6770 68 9 4 7 oi& 714217266 55 4 35 7 o 370 7 3844 3982 4119 425 7 55 5 7 3 9 o 7$i4 7 638 77 6 2 7 886 8011 54 5 43 9 4 4532 4669 48o 7 4944 5o8 2 54 6 8i35 8259 8384 85o88633 8 7 5 7 53 6 522O 5358 54 9 5 5633 5 77 i 5 9 o 9 53 7 8882 9006 9i3i 9 255 9 38o 95o5 52 7 6o4 7 6t85 6323 646 1 65 99 6788 5a 8j 9629 9110.760378 97 54 o5o3 9879 0628 .. .4 .128 0763:0878 ,a53 ioo3 5i 5o 8 9 68 7 6 77 o6 7 oi4 7 845 7983 7291 8122 8261 7568 83 99 5i 5o loj 1128 1253 i3 79 i5c4 1629 17-54 49 10 8538 86 77 8816 89549093 9282 49 n| 1880 2O05 2130 2256 238i 2507 48 1 1 9 3 7 i 95io 9649 9788 99 28j . .67 48 12 2632 2 7 58 2883 3oo 9 3i35 3 2 6o 47 12 10.810206 o345 o485 0624 0764 o 9 o3 47 i3 3386 35i2 3638 3 7 63 388 9 4oi5 46 i3 1042 1182 1322 i46i 1601 1741 46 i4 4i4i 4267 43 9 3 45i 9 4645 4771 45 i4 l88o 202O 2160 2300 2440 258o 45 i5 48 97 5o24 5i5o 5? 7 6 5402 5529^4 i5 2 7 2O 2860 3ooo 3i4o 3 2 8o 342i 44 16 5655 5 7 8i 5 9 oS 6o34 6161 6287 43 16 356i 3701 3842 3982 4122 42t>3 43 17 64i4 654o 6667 6 79 4 6920 7047 42 i 7 44o3 4544 4685 4825 4 9 66 5107 42 18 7174 7801 7427 7 554 7 68i 7808 4i 18 5248 5388 552 9 56 7 o 58n 5 9 5 2 4i 19 7 9 35 8062 8i8 9 83i6 8443 85 7 o 4o ! 9 6o 9 3 6234 63 7 5 65i 7 6658 67 99 4o 20 8698 88 2 5 8 9 52 979 920-7 9 334 3o 20 6 9 4i 7082 7228 7 365 7506(7648 3 9 21 946 1 9 58 9 97 i6 9 844 997 1 99 38 21 7789 79 3i 8073 8214 8356 84 9 8 38 22 10.770227 o354 0482 0610 o 7 3 7 o865 37 22 864o 8782 8924 9066 9 208 9 35o 37 23 o 99 3 I 121 I24 9 i3 77 i5o4li632 36 9492 9 634 9776 9918 ..61 .203 36 24 1761 i88 9 2017 2i45 22 7 3 2401 35 24 io.82o345 o488 o63o o 77 3 o 9 i5 io58 35 25 252 9 2658 2786 2914)3043 3171 34 25 1201 1 343 i486 1629 1772 I 9 i5 34 26 33oo 3428 3557 3685 38x4 33 26 2o58 22OO 2344 248 7 2 63o 2 77 3 33 2 7 4071 4200 4320 445 7 4586 47i5 32 27 2916 3o5 9 3 2 o3 3346 348 9 3633 32 28 4844 4 97 3 5l02 523i 536o 5489 3i 28 3 77 6 3920 4o63 420-7 435o 44 9 4 3i 29 56i8 5 7 4 7 58 7 6 6006 6i35 6264 3o 29 4638 4782 4925 5069 52i3 535 7 3o 3o 63 9 3 65 2 3 66526782 691 1 7041 29 3o 55oi 5645 5 7 8 9 5 9 33 6078 6222 2 9 3i 7170 7800 742 9 755 9 7 68 9 7818 28 3i 6366 65n 6655 6 799 6 9 44 7088 28 3s 7948 8078 8208 8338 8468 85 9 8 27 32 7233 7377 7522 765.7 7812 79 56 2 7 33 8728 8858 8 9 88 9 n8 9248 9378 26 33 8101 8246 83 9 i 8536 8681 8826 26 34 95o8 9 63 9 9769 9 8 99 ..29 . 160 25 34 8971 91 16 9261 9 4o 7 9 55 2 9 6 97 25 35 10.780290 0421 o55i 0682 0812 0943 24 35 9843 9988 .i34 2 79 .425 .5 7 o 24 36 1074 1204 i335 i466 i5 97 1727 23 36 10.800716 0862 1008 n53 i2 99 i445 23 37 i858 i 9 8 9 2120 225l 2382 22 37 1691 i 7 3 7 i883 2O2 9 2175 2322 22 38 2644 2 77 5 2 9 07 3o38 3169 33oo 21 38 2468 2614 2760 2 9 7 3o53 32OO 21 39 3432 3563 36 9 5 3826 3 9 5 7 4089 20 39 3346 3493 363 9 3 7 86 3 9 33 4o8o 2O 4o 4220 4352 4484 46i5 4 7 4 7 48 79 I 9 4o 4226 43 7 3 452O 466 7 48i4 4961 19 4i 5oi i 5i42 5274 54o6 5538 5670 18 4i 5i68 5 2 55 54o2 555o 56 9 7 5844 18 42 58o2 5 9 34 6066 6i 9 8 633o6463 17 42 5 99 2 6i3 9 6287 6434 6582 6720, i 7 43 65 9 5 6727 685 9 6992 7124 7 25 7 16 43 6877 7025 7172 7 32O 7468 7616 16 44 738 9 7522 7654 7787 7919 8o52 i5 44 7764 7912 8060 8208 8356 85o4 i5 45 8i85 83i 7 845o 8583 8716,8849 i4 45 8653 8801 8949 9 o 9 8 9 246 9 3 9 5 i4 46 8982 9 n5 9 248 9 38i 95149647 i3 46 9 543 9692 9 84o 9989 .138 .28-7 i3 47 9780 99 i3 ..46 .180 .3i3 .446 12 4 7 io.84o435 o584 0733 0882 io3i 1180 12 48 10.790580 o 7 i3 0847 o 9 8o ii i4 1247 II 48 1829 1479 1628 i 777 I 9 26 20 7 6 I I 49 i38i i5i5 1 648 I 7 82 1916 2o5o IO 49 2225 23 7 5 2524 2 6 7 4 2823 2 97 3 IO 5o 2i83 23l 7 245i 2 585 2719 2853 9 5o 3i23 8272 3422 35 7 2 3722 38 7 2 9 5i 2987 3l22 3 2 56 33 9 o 3524 3658 8 5i 4O22 4172 4322 44 7 2 4623 4773 3 52 3 79 3 3 9 2 7 4062 4i 9 6 433i 4465 7 52 4923 5074 5224 53 7 4 5525 ( 56 7 5 7 53 46oo 4734 486 9 5oo4 5138,5273 6 53 5826 5 977 6127 6278 642 9 658o 6 54 54o8 5543 5678 58i2 5947 6082 5 54 6 7 3i 6882 7 o33 7184 7 335l 7 486 5 55 6218 6353 6488 6623 6758 68 9 3 4 55 7 63 7 7789 7940 8091 8243 83 9 4 4 56 7029 7164 7299 7 435 75707706 3 56 8546 8697 884 9 9001 9 l52 9 3o4 3 57 7841 7977 8112 8248 8384 85i 9 2 57 9 456 9608 9760 9912 ..64 .216 2 58 8655 8791 8 9 2 7 9 o63 9 i 99 9 335 I 53 io.85o368 O52O 0672 0825 0977 II2 9 I 5 9 9471 9607 97 43 9 8 79 ..i5 .i5i O 5 9 1282 i434 i58 7 i 7 3 9 i8 9 2 2045 O .60" 50" | 40" 30" 20" 10" . 60" 50" 40" 30" ?0" I 10" Co-tangent of 9 Degrees. 1 Co-tangent of 8 Degrees. .3 P PartJ l " 2 " 3 " 4 " 5 " 6 " 7 " 8 " 9 " III J 13 26 39 52 65 78 91 103 116 p p ( 1" 2" 3" 4" 5" 6" 7" 8" 9" 1 $ ,14 29 43 58 72 8G 101 115 130 10G L o G A K i r ii M i c SINES. 1 Sine of 82 Degrees. a Sine of 83 Degrees. 51 0" 10" 20" 30" 40" 50" i 0" 10" 20" 30" 40" 50" 9.996753 5756 5 7 5 9 5 7 62 5 7 65 5 7 68J5 9 o 9 . 9 96 7 5i!6 7 53 6 7 56 6 7 58 6 7 6i 6 7 64 5 9 i 5771 5 77 3 5 77 6 5 779 5 7 8 2 5 7 85 58 I 6 7 66 6 7 6 9 6 77 i 6 77 4 6 777 6 779 58 2 5 7 88 5 79 i 5 79 4 5 797 58oo 58o3 5 7 2 6 7 82 6 7 84 6787 6 7 8 9 6 79 2 6 79 5 57 3 58o6 5809 58i2 58i5 58i8582i 56 3 6 797 6800 6802 68o5 68o 7 6810 56 4 5823 5826 5829 5832 5835|5838 55 4 6812 68i5 6818 6820 68 2 3 6825 55 5 584i 5844 584 7 585o 5853 5856 54 5 6828 683o 6833 6835 6838 684i 54 6 585 9 5862 5864 586 7 58 7 o 58 7 353 6 6843 6846 6848 685i 6853 6856 53 7 58 7 6 58 79 5882 5885 5888 58 9 il52 7 6858 6861 6863 6866 6869 68 7 i D2 8 58 9 4 58 97 58 99 5 9 02 5 9 o5 5 9 o8 5i 8 68 7 4 68 7 6 68 79 6881 6884 6886 5i 9 5 9 n 5qi4 5 9 i 7 5 9 2O 5 9 23 5 9 26 5o 9 6889 68 9 i 68 9 4 68 9 6 6899 6 9 oi 5o 109.995928 5931 5 9 34 5 9 3 7 5 9 4o 5 9 43 49 io 9 . 99 6 9 o4 6 9 o6 6 9 o 9 6 9 I2 6914 6 9 i 7 49 ii 5 9 46 5 9 49 5 9 5 2 5 9 54 5 9 5 7 5 9 6o 48 ii 6919 6 9 22 6 9 24 6 9 2 7 6929 6 9 32 48 12 5963 5 9 66 5 9 6 9 5 97 2 5 97 5 5 97 8 47 12 6 9 34 6 9 3 7 6 9 3 9 6942 6 9 44 6 9 47 47 i3 5980 5 9 83 5 9 86 5 9 8 9 5 99 2 5 99 5 46 i3 6 9 4 9 6 9 52 6 9 54 6 9 5 7 6 9 5 9 6 9 62 46 i4 5 99 8 6001 6oo3 6006 6oo 9 6012 45 i4 6 9 64 6 9 6 7 6 9 6 9 69 7 2 6 97 4 6 977 45 i5 6oi5 6018 6021 6023 6026 6o2 9 44 i5 6 979 6 9 82 6 9 84 6 9 8 7 6 9 8 9 6 99 2 44 16 6o32 6o35 6o38 6o4i 6o43 6o46 43 16 6 99 4 6 997 6 999 -7002 7 oo4 7 oo 7 43 I? 6049 6o5s 6o55 6o58 6061 6o63 42 *7 79 7 OII 7 oi4 7 oi6 7 OI 9 -7021 42 18 J 9 6066 6o83 6069 6086 6o 7 2 6o 7 5 6o8 9 6o 9 2 6078 6o 9 5 6081 6o 9 8 4i 4o 18 J 9 7024 7 o3 9 -7026 7 o4i 7 02 9 7 o44 7 o3i 7 o46 7 o34 7 o4 9 7 o36 7 o5i 4i 4o 20 9.996100 6io3 6106 6 1 o 9 6112 6n5 3 9 20 9 . 99 7o53 7 o56 7 o58 7 o6i 7 o63 -7066 3 9 21 6117 6120 6i23|6i26 6l2 9 6i3i 38 21 7068 7 o 7 i 7073 7 o 7 6 7 o 7 8 -7080 38 22 6i34 6i3 7 6l4o6i43 6i46 6i48 37 22 7083 7 o85 7 o88 7 09o 79 3 7 o 9 5 37 23 6i5i 6i54 6i5 7 6160 6162 6i65 36 23 7O 9 8 7 ioo 7 IO2 7 io5 7 io 7 7 IIO 36 24 6168 6171 6i 7 4 : 6i 77 6i 79 6182 35 24 71 12 7 n5 7X17 -7120 7 I22 7124 35 25 6i85 6188 6i 9 i 6i 9 3 6i 9 6 6i 99 34 25 7127 7 I2 9 7 l32 7 i34 7 i3 7 7 i3 9 34 26 6202 62o5 62O 7 62IO 6 2 i3 6216 33 26 7141 7 i44 7'46 7 i49 7 i5i 7 i54 33 27 6219 6221 6224 622-7 623o 6232 32 27 7i56 7 i58 7 i6i 7 i63 7 i66 7 i68 3a 28 6235 6 2 38 6241 6244 6246 D24 9 3i 28 7170 7 i 7 3 7 i 7 5 7 i 7 8 7 i8o 7182 3i 29 6262 6 2 55 625 7 6260 6263 6266 3o 2 9 7 i85 7 i8 7 7 i 9 o 7 I 9 2 7 i 9 4 7197 3o 3o 9.996269 62 7 I 62 7 4 62-77 6280 6282 2 9 3o 9-997 J 99 7 2O2 7 204 -7206 7 20 9 7211 29 3i 6285 6288 62 9 I 6293 62 9 6 6299 28 3i 7214 7 2l6 -7218 7 22I 7 223 7226 28 32 6302 63o5 63o 7 63io 63i3 63i6 27 32 7228 7 23o 7233 7 235 7 238 7240 2 7 33 63i8 632i 6324 632 7 632 9 6332 26 33 7242 7 245 7247 7 249 7 252 7254 26 34 6335 6338 634o6343 6346 634 9 25 34 72 5 7 7 25 9 7261 -7264 7 266 7268 25 35 36 635i 6368 6354 63 7 o 635 7 635 9 6362 63 7 3 i 63 7 663 79 6365 638i 24 23 35 36 7271 7 285 7273 7 28 7 7276 7 2 9 7 2 7 8 7292 7 280 7 2 9 4 7283 7297 24 23 37 6384 638 7 6390 6392 63 9 5 63 9 8 22 37 7299 7 3oi 7 3o4 7 3o6 7 3o 9 7 3n 22 38 64oo 64o3 64o6 6409 64n 64i4 21 38 7 3i3 7 3i6 7 3i8 7 320 7 323 7 325 21 3 9 6417 64i9 6422 6425 64s8 643o 20 39 7327 7 33o 7 332 7 334 7337 7339 2O to 9.996433 6436 6438644i 6444 644 7 J 9 4o 9 . 99 734i 7 344 7 346 7 348 7 35! 7 353 19 4i 644 9 6452 645564576466 6463 18 4i 7 355 7 358 7 36o 7 362 7 365 7 36 7 18 42 6465 6468 6471 64 7 4 64 7 6 6479 '7 42 7 36 9 7372 7 3 7 4 7 3 7 6 7 3 79 7 38i 17 43 6482 6484 6487 6490 6492 64g5 16 43 7 383 7 386 7 388 7 3 9 7 3 9 3 7 3 9 5 16 44 6498 65oo 65o365o665o8 65n i5 44 7397 7 3 99 7 402 7 4o4 7 4o6 7 4o 9 i5 45 65i4 65i 7 65i 9 ! 65226525 652 7 i4 45 7411 74i3 7 4i6 7 4i8 7 420 7 423 i4 46 653o 6533 65356538654i 6543 i3 46 7425 742 7 7 42 9 7432 7 434 7 436 i3 /*7 6546 6549 655i 655/655 7 655 9 12 47 743 9 744i 7 443 7445 7448 7 45o 12 48 6562 6565 656 7 65 7 o 65 7 3 65 7 5 II 48 7452 7 455 7 45 7 745 9 7 46i 7 464 II 4 9 65 7 8 658o 658365866588 65 9 i IO 49 7 466 7 468 7 4 7 i 7 4 7 3 7 4 7 5 7477 10 5o 9.996594 6696 65 99 6602 66o4 66o 7 9 5o 9.097480 7 48 2 7 484 7 48 7 7 48 9 7 4 9 i 9 5i 6610 6612 66i5|66i8j662o 6623 8 5i 7 4 9 3 -7496 74 9 8 7 5oo 7 5o2 7 5o5 8 62 66 2 5 6628 663i 6633 6636 663 9 7 52 7 5 7 7 5o 9 7 5n 7 5i4 7 5i6 7 5i8 7 53 664i 6644 6646 6649 6652 6654 6 53 7620 7 5 2 3 7 525 7 5 27 7 53o 7 53 2 6 54 665 7 6660 6662 6665 666 7 66 7 o 5 54 ?534 7 536 7 53 9 7 54i 7 543 7 545 5 55 66 7 3 66 7 5 66 7 8 6681 6683 6686 4 55 7 54 7 7 55o 7 552| 7 554 7 556 7 55 9 4 56 6688 6691 66 9 4 6696 6699 6 7 oi 3 56 7 56i 7 563 7 565 7 568 7 5 7 o 7 5 7 2 3 57 6704 6707 6 7 o 9 6 7 i2j6 7 i4i6 7 i 7 2 57 7 5 7 4 7^77 7 5 79 7 58i 7 583 7 585 2 58 6720 6 7 22 6 7 25 6 7 2 7 6730 6733 I 58 7 588 7 5 9 o 7 5 9 2 7 5 9 4' 7 5 97 7 5 99 I 5 9 6 7 35 6 7 38 6 7 4o 6 7 43;6 7 46!6 7 48 O 5 9 7 6oi 7 6o3 7 6o5 7 6o8 -7610 -7612 O &y \ 50" 40" 30" 20" | 10" n 60" 50" 40" 30" i 20" 10" i rf Co-sine of 7 Degrees. i Co-sine of 6 Degrees. C 1" 2" 3" 4" IM'artJ ! ! j 5" 6" 7" 8" 9" 12222 p p . $ 1" 2" 3" 4" 5" 6" 7" 8" 9" irt \ 1 1 1 1 2 2 2 LOGARITHMIC TANGENTS. 107 J Tangent of 82 Degrees. .s Tangent of 83 Degrees. s 0" 10" 20" 30" 40"' 50" s 0" | 10" 20" 30" 40" 50" o 30.80219,7 235o 2 5o3 2 656 2800, 2 9 62 5 9 o io. 9 io856 io3o 1205 !3 79 i553 1727 59 I 3ii5 3268 3421 35 7 5 3 7 28 388i 58 I I 9 O2 2076 225l 2426 2600 2 77 5 58 2 4o34 4i88 434i 4495 464 9 4802 57 2 2 9 5o 3i25 33oo 34 7 5 365o 3825 57 3 4956 5no 5 2 63 54i7 55 7I 5725 56 3 4ooo 4176 435! 452 7 4?02 48 7 8 56 4 58 79 6o33 6187 634i 64 9 6 665o 55 4 5o53 522 9 54o5 558i 5 7 5 7 5 9 33 55 5 68o4 6 9 58 7ii3 7267 7 422 7 5 7 6 54 5 6io 9 6285 646i 6638 6814 6990 54 6 77 3i 7886 8o4i 8i 9 5 835o 85o5 53 6 7167 7 343 7520 7 6 97 7 8 7 4 8060 53 7 8660 88i5 8970 9125 9 28o 9 436 52 7 8227 84o4 858i 8 7 5 9 8 9 36 9113 5 2 8 9 5 9 i 97 46 9902 ..5 7 .212 .368 5i 8 9 2 9 9 468 9 645 9 823 .... .178 5i 9 io.86o524 o67 9 o835 0991 n46 1302 5o 9 io. 9 2o356 o534 0712 o8 9 o 1068 1246 5o 10 i458 1614 1770 1926 2082 223 9 4o 10 1424 1602 1781 i 9 5 9 2i38 23i6 4 9 ii 2 3 9 5 2 55i 2708 2864 3020 3i77 ii 2 4 9 5 2 6 7 3 2852 3o3i 3210 338 9 48 12 3333 34 9 o 3647 38o3 3 9 6o 4n 7 4 7 12 3568 3 7 4 7 3926 4io5 4285 4464 47 i3 4274 443 1 4588 4745 4 9 02 5o5 9 46 i3 4644 48 2 3 5oo3 5i83 5362 5542 46 i4 52i6 53 7 4 553i 5688 5846 6oo3 45 i4 5722 5 9 O2 6082 6262 644a 6623 45 i5 6161 63i 9 6476 6634 6 79 2 6 9 5o 44 i5 68o3 6 9 84 7 i64 7345 7 525 7706 44 16 7107 7 265 7423 7 58i 773 9 7 8 9 8 43 16 7887 8068 8249 843o 8611 8792 43 17 8o56 8214 83 7 2 853i 868 9 8848 42 17 8973 9 i54 9 336 9 5i 7 9 6 99 9880 42 18 9 oo6 9 i65 9324 9482 9 64i 9 8oo 4i 18 io. 9 3oo62 0244 0425 060-7 o 7 8 9 0971 4i J 9 99 5 9 .118 .277 .436 .5 9 5 . 7 54 4o 19 n54 i336 i5i8 I 7 OO i883 2o65 4o 20 io.8 7 o 9 i3 1072 1232 1391 i55i I 7 IO 3 9 20 2248 2 43o 2613 2 79 6 2979 3162 3 9 21 1870 202 9 2l8 9 2 34 9 25o8 2668 38 21 3345 3528 3 7 n 38 9 4 4o 7 8 4261 38 22 2828 2 9 88 3i48 33o8 3468 362 9 37 22 4444 4628 4812 4 99 5 5l 79 5363 37 23 3 7 8 9 3 9 4 9 4109 4270 443o 45 9 i 36 23 5547 5 7 3i 5 9 i5 6o 99 6283 6467 36 24 47^1 4 9 I2 5o 7 3 5234 53 9 4 5555 35 24 665 2 6836 7021 7 205 7 3 9 o 7 5 7 5 35 25 5 7 i6 58 77 6o38 6199 636o 6522 34 25 7760 79 45 8i3o 83i5 85oo 8685 34 26 6683 6844 7006 7167 7 32 9 7 4 9 o 33 26 8870 9 o56 9241 9 427 9 6l2 979 s 33 2 7 7652 7 8i3 797 5 8i3 7 82 99 846i 32 27 99 84 .169 .355 .54i .727 .914 32 28 8623 8785 8 9 4 7 9109 9 2 7 i 9 433 3i 28 io. 9 4i ioo 1286 1472 i65 9 i845 2032 3i 2 9 9 5 9 6 9758 9921 ..83 .246 .4o8 3o 2 9 22I 9 2406 2592 2779 2 9 66 3i53 3o 3o 10.880571 0734 0896 1059 1222 i385 29 3o 334i 35 2 8 3 7 i5 4o 9 o 42 77 29 3il i548 1711 i8 7 4 2o38 22OI 2364 28 3i 4465 4653 484 1 5o28 52i6 54o4 28 32 2528 260,1 2 855 3oi8 3l82 3345 27 32 55 9 3 5 7 8i 5 9 6 9 6157 6346 6534 27 33 35o 9 36 7 3 383 7 4ooi 4i65 432 9 26 33 6723 6912 7100 ?2 8 9 7470 7667 26 34 44 9 3 465 7 4821 4985 5i5o 53i4 25 34 7856 8o45 8234 8424 86i3 88o3 25 35 5479 5643 58o8 5972 6i3 7 63o2 24 35 8 99 2 9182 9 3 7 i 9 56i 97 5i 99 4i 24 36 646 7 6632 6 79 6 6961 7127 7 2 92 23 36 io. 9 5o:3i 0321 o5n 0702 0892 io83 23 37 7 45 7 7622 7787 7953 8118 8284 22 37 1273 i464 i654 1 845 2036 2227 22 38 8449 86i5 8 7 8i 8 9 46 9 II2 9 2 7 8 21 38 2418 2609 2800 2 99 I 3i83 33 7 4 21 3 9 9444 9 6io 977 6 9942 .108 .2 7 4 2O 3 9 3566 3 7 5 7 3949 4i4i 4332 4524 2O 4o io.8 9 o44i 0607 o 77 3 0940 1106 I2 7 3 '9 4o 4716 4908 5ioi 52 9 3 5485 56 7 8 I 9 4i i44o 1606 i 77 3 1940 2107 22 7 4 18 4i 58 7 o 6o63 6 2 55 6448 664i 6834 18 42 244 1 2608 2 77 5 2942 3no 32 77 17 42 7 02 7 7220 74i3 7606 7 8oo 799 3 i 7 43 3444 36i2 3 779 3 9 4 7 4n5 4282 16 43 8i8 7 838o 85 7 4 8768 8961 9 i55 16 44 445o 46i8 4 7 86 4 9 54 5l22 52 9 o i5 44 9 34 9 9544 9738 99 3 2 . 126 .321 i5 45 5458 5626 5 79 5 5 9 63 6i3i 63oo i4 45 io. 9 6o5i5 O 7 IO ogoS io 99 1294 i48 9 i4 46 6468 6637 6806 6 9 74 7i43 7 3l2 i3 46 1 684 1879 2074 2270 2465 2661 i3 47 748 1 7 65o 7 8i 9 79 88 8i5 7 8326 12 47 2856 3o52 3247 3443 363 9 3835 12 48 84 9 6 8665 8834 9 oo4 9 J 7 3 9 343 II 48 4o3i 4227 4424 4620 48i6 5oi3 II 49 9 5i3 9 683 9 85 2 . .22 .I 9 2 .362 10 49 52O 9 54o6 56o3 58oo 5 997 6i 9 4 IO 5olio. 9 oo532 0702 o8 7 3 1043 I2l3 i384 9 5o 63 9 i 6588 6 7 85 6 9 83 7 i8o 7377 9 5i i554 1724 i8 9 5 2066 2236 240 7 8 5i 7 5 7 5 7773 7971 8i6 9 836 7 8565 8 52 2 5 7 8 2 7 4 9 2 9 20 3o 9 i 3262 3433 7 52 8 7 63 8961 9 i5 9 9 358 9 556 97 55 7 53 36o5 3776 3 9 4 7 4n 9 42 9 O 4462 6 53 99 54 .i5 2 .35i .55o 7 49 . 9 48 6 54 4633 48o5 4977 5i4 9 5320 5492 5 54 1 347 1 546 1746 1945 2i45 5 55 5664 583 7 6oo 9 6181 6353 6526 4 55 2345 2545 2 7 45 2 9 45 3i45 3345 4 56 66 9 8 6871 7o43 7216 7 388 7 56i 3 56 3545 3746 3 9 46 4i4 7 4347 4548 3 67 7734 797 8080 8253 8426 85 99 2 5 7 4?4 9 4 9 5o 5i5i 5352 5553 5 7 55 2 58 5 9 8 77 2 9 8i3 8 9 46 9987 9"9 .161 929 2 .334 9 466 .5o8 9 68a I O 58 5 9 5 9 566i57 71667368 635 9 75 7 o 656i 777 3 6762 797 5 6 9 64 8i77 I 60" 50" 40" 30" 20" 10" , 60" | 50" 40" | 30" 20" 10" s i Co-tangent of 7 Degrees. S Co-tangent of 6 Degrees. 9 p p C 1" 2" 3" 4" 5" 6" 7" 8" 9" 1 I 16 33 49 65 81 98 114 130 146 p p .( 1" 2" 3" 4" 5" 6" 7" 8" 9'' I 19 37 50 75 94 112 131 IfC 1U8 108 LOGARITHMIC SINES. .s Sine of 34 Degrees. d Sine of 85 Degrees. 3 0" i 10" | 20" 30" 40" 50" & 0" | 10" 20" 30" 40-' 50" 9 . 997 6i4 76i 7 i 7 6i9 -7621 7 623 7625 r O 9 . 99 8344 8346 8348 835o 8352 8353 5 9 I 7628 763076327634 7 636 7 63 9 58 I 8355 835 7 835 9 836i 8363 8364 58 2 7641 7 643 7 645' 7 64 7 7 65o 7 65 2 57 2 8366 8368 83 7 o 83 7 2 83 7 4 83 7 5 5 7 3 7654 7 656 7 658' 7 66i 7 663 7 665 56 ^ 83 77 83 79 838i 8383 8385 8386 56 4 7667 7669 76727674 7 6 7 6 7 6 7 8 55 L 8388 83 9 o 83 9 2 83 9 4 83 9 5 83 97 55 5 7680 7682 7686 7687 7 68 9 7 6 9 i 54 t 83 99 84oi 84o3 84o4 84o6 84o8 5{ 6 7 6 9 3 -7696 7 698 77 oo 77 02 77 o4 53 6 84io 8412 84 1 3 84i5 84i 7 84i 9 53 7 7706 77 o 977 n 77 i3 77 i5 77 i 7 52 7 8421 8422 8424 8426 8428 843o 52 8 7722 77 24 |77 26 77 28 77 3o 5i 8 843: 843318435 843 7 843 9 844o 5i 9J 7732 77 35 77 3 7|77 3 9 77 4i 77 43 5o 9 8442J8444 8446 8448 844 9 845i 5o 109.997-745 77 4 7 ; 77 5o 77 52 77 54 77 56 4 9 10 9 . 99 8453 8455 8456 8458 846o 8462 4 9 ii 77 58 77 6o 77 62j 77 65 77 6 7 7769 48 ii 8464 8465 846 7 8469 84 7 i 84 7 2 48 12 7771 777 3| 777 5 7777 77 8o 77 82 47 12 84 7 4 84 7 6 84 7 8 84 7 9 848 1 8483 47 i3 7784 77 86 77 88 779 o 779 2 7794 46 i3 8485 848 7 8488 8490 84 9 2 84 9 4 46 i4 7797 7799i7 8oi 7 8o3 7 8o5 7 8o 7 45 i4 84 9 5 84 97 84 99 85oi 85o 2 85o4 45 i5 7 8o 9 7811 78147816 -7818 -7820 44 i5 85o6 85o8 85o 9 85n 85i3 85i5 44 16 7822 7 824l 7 826 7 828 7 83o 7 833 43 16 85iO 85x8)8520 8022 85*3 85 2 5 43 i 7 7 835 783778397841 7 843 7 845 42 * 7 85 27 852 9 853o 8532 8534 8535 42 18 7 84 7 7 84 9 78627854 7 856 7 858 4i 18 853 7 853 9 854i 8542 8544 8546 4i 19 7860 7 862 7 864 7 866 7 868 7 8 7 o 4o 19 8548 854 9 855i 8553 8554 8556 4o 20 9'997872 7875:78777879 7 88i 7 883 3 9 20 9 . 99 8558 856o 856i 8563 8565 8566 3 9 21 7 885 7887 7889 7891 7 8 9 3 7 8 9 5 38 21 8568 85 7 o 85 7 2 85 7 3 85 7 5 85 77 38 22 23 7897 7910 7900 7902 7904 7912 7914:7916 7906 7918 79 o8 79 20 37 36 22 23 85 7 8 858 9 858o 85 9 o 8582 85 9 2 8584 85 9 4 8585 85 9 5 858 7 85 97 37 36 24 7922 7924 7926 7 9 2 9 79 3i 79 33 35 24 85 99 8601 8602 86o4 8606 86o 7 35 25 79 35 7 9 37 7 9 3 9 7 9 4i 79 43 79 45 34 25 86o 9 8611 8612 86i4 8616 86i 7 34 26 7947 7 9 4 9 7 9 5iJ7 9 53 79 55 79 5 7 33 26 86i 9 8621 8622 8624 8626 862 7 33 2 7 79 5 9 7 9 6i 7 9 63i7 9 65 7 9 6 7 797 32 27 862 9 863i 8633 8634 8636 8638 32 28 7972 7974797 6 7978 79 8o 79 82 3i 28 863 9 864i 8643 8644 8646 8648 3i 2 9 7984 79 8 6 79 88 799 799 2 7994 3o 29 864 9 865i 8653 8654 8656 865 7 3o 3o 9.997996 7 99 8 8000^002 8oo4j8oo6 2 9 3o 9 . 99 865 9 8661 8662 8664 8666 866 7 2 9 3i 8008 8010 8012 8oi4 8016 8018 28 3i 866 9 86 7 i 86 7 2 86 7 4 86 7 6 86 77 28 32 8020 8022 8024 8026 8028 8o3o 2 7 32 86 79 8681 8682 8684 8686 8687 2 7 33 8o32 8o34 8o36 8o38 8o4o 8042 26 33 868 9 86 9 o 86 9 2 86 9 4 86 9 5 8697 26 34 8o44 8o46 8o48 8o5o 8o52 8o54 25 34 86 99 8 7 oo 8 7 02 8 7 o4 8 7 o5 8707 25 35 8o56 8o58 8060 8062 8o64 8066 24 35 8 7 o8 8 7 io 8 7 I2 8 7 i3 8 7 i5 8717 24 36 8068 8070 8072 8074 8076^078 23 36 8 7 i8 8 7 20 8 7 2I 8 ?2 3 8 72 5 8726 23 3 7 8080 8082 8o84 8086 80888090 22 37 8 7 28 8 7 2 9 8 7 3! 8733 8 7 34 8 7 36 22 38 8092 8o 9 4 8o 9 6 8o 9 8 8100:8102 21 38 8 7 38 8 7 3 9 8 7 4i 8742 8 7 44 8 7 46 21 3 9 8io4 810681088110 8112 8114 2O 3 9 8 7 4 7 8 7 4 9 8 7 5o 8 7 52 8 7 54 8 7 55 2O 4o 9.998116 8n88i2o ! 8i22 81248126 I 9 4o 9 . 99 8 7 5 7 8 7 58 8 7 6o 8762 8 7 63 8 7 65 I 9 4i 42 8128 8i3 9 8i3o8i3i 8i338i358i3 7 8i4i '8i43 8i45 81478149 17 4i 42 8 7 66 8776 8 7 68 8777 8 7 6 9 8 779 8771 8781 8 77 3 8 7 82 8 77 4 8 7 84 18 43 8i5i 8i538i558i5 7 81598161 16 43 8 7 85 8 7 8 7 8 7 88 8 79 o 8 79 2 8 79 3 16 44 8i63 816581678168 8170 8172 i5 44 8 79 5 8 79 6 8798 8 799 8801 88o3 i5 45 8i 7 4 817681788180 81828184 i4 45 88o4 8806 8807 88o 9 8810 8812 i4 46 8186 818881908192 8i 9 48i 9 5 i3 46 88i3 88i5 8817 8818 8820 8821 i3 4 7 8197 8199 8201 8203 8205,8207 12 47 8823 8824 8826 8827 882 9 883i 12 48 8209 8211 82i382i5 82178218 II 48 8832 8834 8835 883 7 8838 884o II 49 8220 8222 8224 8226 82288230 IO 49 884i 8843 8844 8846 8848 884 9 IO 5o 5i 9.998232 8243 8 2 3482368238 8245:82478249 823 9 824i 825i 8253 | 5o 5i 9 . 99 885i 8860 8852 8861 8854 8863 8855 8864 885 7 8866 8858 886 7 9 8 52 8255 825782588260 82628264 7 52 886 9 88 7 o 8872 8873 88 7 5 88 77 7 53 8266 8268 8270 8272 82 7 3 82 7 5 6 53 88 7 8 8880 8881 8883 8884 8886 6 54 8277 8279^281 8283 82858287 5 54 888 7 888 9 8890 88 9 2 88 9 3 8895 5 55 8289 8290 8292 8294 82968298 4 55 88 9 6 88 9 8 8899 8 9 oi 8 9 O2 8904 4 56 83oo 83o2 1 83o3.83o5 8307 8309 3 56 8 9 o5 8 9 o 7 i8 9 o8|8 9 io 8911 8 9 i3 3 5 7 83n 83i3:83i583i683i8832o 2 5 7 8 9 i4 8 9 i6 8 9 i 7 8 9 i 9 8 9 20 8922 2 58 5 9 8322 8333 8324 83 2 6 8328 832 9 833i 8335 833 7 8339834i 8342 I O 58 5 9 8 9 23 8 9 25 8926 892-7 8929 8 9 3o 8 9 32:8 9 33 8 9 35 8 9 36 8 9 38 8 9 3 9 60" 50" j 40" 30" 20" 10" | ^ 60" ; 50" 40" 30" 20" 10" ^ Co-sine of 5 Degrees. j Co-sine of 4 Degrees. j < i" 2" 3" 4" 5" 6" 7" 8" 9" al l 001111122 . < 1" 2" 3" 4" 5" 6" 7" 8" 9" irt \ 1 1 1 1 1 1 o G A it i T ii M i c TANGENTS. 109 g Tangent of 84 Degrees. _ s - Tangent of 85 Degrees. ! s 0" 10" 20" 30" 40" 50" S 0" | 10" j 20" 30" 40" | 50" 10.978380 8582 8 7 85 8 9 88 9191 9394 5 9 O n.o58o48 8291 8533 8 77 6 9019 9262 5 9 I 9^97 9 8oo .206 .4io .6i3 58 I 9 5o6 9749 999 3 .236 .480 .724 58 2 10.980817 1021 1224 1428 i632 i836 57 2 ii .o6o 9 68 1212 i456 I 7 OI I 9 45 2190 "5 7 3 2o4i 2245 244 9 2 654 2858 3o63 56 3 2435 2680 2 9 25 3i 7 o 34i6 366i 56 4 3268 3472 36 77 3882 4o8 7 42 9 3 55 4 3 97 4i53 43 99 4645 48 9 i 5i38 55 r 4498 4703 4 9 o 9 5n4 53 20 5526 54 5 5384 563i 58 77 6124 63 7 i 6619 54 6 5 7 3 2 5 9 38 6i44 635o 6556 6763 53 6 6866 7 n3 7 36i 7 6o 9 7 85 7 8:o5 53 7 6969 7176 7 38 2 7 58 9 7796 8oo3 02 7 8353 8601 885o 9098 9 34 7 9596 5 2 8 8210 84i7 8624 883i 9 3 9 9246 5i 8 9 845 .'.'94 .343 .842 1092 5i 9 9454 9 662 9 86 9 77 .285 .493 5o 9 II .O 7 l342 1592 1842 2092 2343 2593 5o 10 10.990702 o 9 io 1118 l32 7 i535 1744 4 9 10 2844 3095 3346 35 97 3848 4ioo 49 1 1 1953 2162 23 7 I 2 58o 2789 2998 48 ii 435i 46o3 4855 5io 7 535 9 56n 48 12 3208 34i7 362 7 3836 4o46 4256 47 12 5864 61 16 636 9 6622 68 7 5 7128 47 i3 4466 46 7 6 4886 5o 9 6 53o 7 55i 7 46 i3 7 38i 7 635 7 888 8142 83 9 6 865o 46 i4 5 7 28 5 9 3 9 6i4 9 636o 6571 6782 45 i4 .8904 9 l5 9 9 4i3 9668 9922 .177 45 i5 6 99 3 7205 74i6 7 62 7 7 83 9 8o5i 44 i5 i i. 080432 0688 o 9 43 1199 1710 44 16 8262 8474 8686 88 9 8 9111 9 323 43 16 I 9 66 2222 24 7 8 2 7 35 2991 3248 43 17 9 535 97 48 99 6o .i 7 3 .386 .5 99 42 17 35o5 3 7 62 4oi 9 4276 4534 4791 42 18 i i .000812 1025 1238 i45i 1 665 1878 4i 18 5o49 53o 7 5565 5823 6082 634o 4i '9 2092 23o6 25i 9 2 7 33 2947 3i6i 4o Z 9 65 99 6858 7117 7 3 7 6 7 635 7894 4o 20 33 7 6 35 9 o 38o4 4oi 9 4234 4448 3g 20 8i54 84i4 86 7 4 8 9 34 9194 9454 3 9 21 4663 48 7 8 5o 9 3 53o8 5524 5 7 3 9 38 21 97 i5 997 5 .236 497 . 7 58 1 020 38 22 5 9 55 6170 6386 6602 6818 7 o34 37 22 II .O 9 I28l 1 543 i8o4 2066 2328 2590 3 7 23 725o 7 466 7 68 2 7899 8n5 8332 36 23 2853 3l ID 33 7 8 364i 3 9 o4 4i6 7 36 24 854 9 8 7 65 8 9 82 9199 9417 9 634 35 24 443o 4694 4 9 5 7 5221 5485 5 749 35 25 9 85i ..69 .286 .5o4 .722 . 9 4o 'M 25 ooi3 62 7 8 6542 6807 7 7 2 34 26 ii.oin58 1376 i5 9 4 i8i3 2031 2250 33 26 7602 7 868 8i33 83 99 8665 8 9 3i 33 27 2468 2687 2 9 o6 3i25 3344 3563 32 27 9 i 97 9 463 97 3o 9996 .263 .53o 32 28 3 7 83 4OO2 4222 444 1 466i 488i 3i 28 ii .100797 io65ii332 1600 1868 2 1 36 3r 29 5ioi 532i 554i 5 7 62 5982 6202 3o 2 9 2404 2672 2 9 4O 3209 34 7 8|3 7 4 7 Jo 3o 6423 6644 6865 7 o86 7 3o 7 7628 29 3o 4oi6 4285 4555 4824 5094 5364 2 9 3i 774 9 7971 8 524 84i4 8636 885 7 28 3i 5634 5 9 o4 6i 7 5 6445 6 7 i6 6 9 8 7 23 32 9079 9801 9746 9968 .191 27 32 7258 7 52 9 7 8oi 8072 8344 8616 27 33 II.02o4l4 o636 o85 9 1082 i3o5 i528 26 33 8888 9 i6o 9 433 97 o5 9978 .25l 26 34 1752 i 97 5 2i 99 2422 2646 2870 25 34 ii.no524 o 79 8 IO 7 I 1 345 1618 1892.26 35 3094 33i8 3542 3767 3991 4216 24 35 2167 2 7 l5 2990 3265 354o l a'4 36 444o 4665 48 9 o 5n5 534o 5565 23 36 38i5 4o 9 o 4366 4642 491-7 5i 9 3 23 37 5 79 i 6016 6242 6468 66 9 3 6919 22 37 5470 5 7 46 6o23 6299 65 7 6 6853 22 38 7 3 7 2 7 5 9 8 7 8 2 4 8o5i 8277 21 38 7 i3i 7 4o8 7 686 7963 8241 8520 21 3 9 85o4 8 7 3i 8 9 58 9i85 9412 9 64o 20 3 9 8798 9 o 7 6 9 355 9 634 99 i3 . I 9 2 2O 4o 9867 .. 9 5 .322 .55o .778 1006 T 9 4o ii . 120471 o 7 5i io3i i3n i5 9 i 1871 T 9 4i n.o3i234 1462 i6 9 i 1919 2148 23 77 18 4i 2l5l 2432 2 7 l3 2994 32 7 5 3556 18 42 2606 2 835 3o64 32 9 3 3522 3 7 5 2 17 42 3838 4n 9 44oi 4683 4966 5248 17 43 3 9 8i 4211 444 1 46 7 i 4 9 oi 5i3i 16 43 553i 58i3 6o 9 6 638o 6663 6946 16 44 536i 55 9 2 5822 6o53 6284 65i4 i5 44 7 23o 7 5i4 779 8 8082 8367 865i i5 45 6 7 45 6 977 7 208 743 9 7671 7 9 02 i4 45 8 9 36 9 22I 9792 77 .363 i-4 46 8i34 8366 85 9 8 883o 9062 9 2 9 5 i3 46 n.i3o649 o 9 35 1221 i5o8 1794 2081 13 47 9 527 9 7 6o 999 2 .225 .458 .6 9 i 13 4 7 2368 2656 2 9 43 323i 35i8 38o6 12 48 ii . o4o 9 25 n58 i3 9 i 1625 i85 9 2O 9 2 II 48 4094 4383 46 7 i 4960 5249 5538 II 49 2326 256i 2 79 5 3O29 3 2 64 34 9 8 10 49 582 7 6.117 64o 7 6697 6987 7 2 77 IO 5o 3733 3 9 68 4203 4438 46 7 3 4 9 o8 9 5o 7 56 7 7 858 8i4 9 844o 8 7 3i 9 5i 5i44 53 79 56i5 585i 6087 6323 8 5i 9 6o6 9 8 9 8 . 190 .483 9 775 8 52 655 9 6 79 5 7 o32 7268 75o5 7742 7 52 n.i4io68 i36i i654 1947 2241 2535 7 53 7979 8216 8453 8691 8928 9 i66 6 53 2829 3i23 3417 3712 4007 43oi 6 54 9 4o3 9 64i 9879 .117 .356 5 9 4 5 54 45 97 48 9 2 5i8 7 5483 5779 6o 7 5 5 55 ii.o5c832 1071 i3io 1 549 1788 2027 4 55 63 7 2 6668 6 9 65 7262 7 55 9 7 856 4 56 2266 25o6 2 7 45 2 9 85 3225 3465 3 56 8i54 8452 8 7 5o 9 o48 9 346 9 645 3 5 7 3 7 o5 3 9 45 4i85 4426 4666 4907 2 57 99 43 .242 .542 .84 1 i i4i i44o 2 58 5i48 538 9 563o 5871 6112 6354 I 58 2o4l 234l 2642 2942 3243 I 5 9 65 9 6 683 7 ^o_79 7 32I 7563 7806 5 9 3545 3846 4i48 4449 4 7 52 5o54 O 60" 50" | 40" 30" | 20" 10" d | 60" 50" 40" 30" 20" 10' 7 S* Co-tangent of 5 Degrees. .3 Co-tangent of 4 Degrees. . P PartM" 2 " 3// 4// 5 " G " 7 " 8// 9 " p . 5 .1" 2" 3" 4", 5" 6" 7" &' 9" \ 22 44 66 88 110 132 154 177 199 Lrt ) 27 54 81 108 135 162 188 215 242 110 LOGARITHMIC SINES. I Sine of 86 Degrees. a Sine of 87 Degrees. m 0" 10" 20" 30" 40" 50" i 0" 10" 20" 30" 4,. 50" o 9. 99 8 9 4i 8 9 4 2 8 9 44 8 9 45 8947 8 9 48 5 9 o 9.999404 9 4o6 9 4o7 9 4o8 9 4o 9 9 4io 5 9 1 8 9 5o 8 9 5i 895389548955 8 9 5 7 58 I 94n 9 4l2 9 4i3 9 4i4 94:5 9416 58 2 8 9 58 8960 8961 8963 8964 8966 5 7 f 94i8 9 4i 9 9 420 9 4si 9422 9423 57 3 8967 8969 8970 8971 8 97 3 8 97 4 56 : 9424 9 425 0426 9 42 7 9428 943o 56 4 8976 8977 8979 8980 8982 8 9 83 55 L 943i 9 43 2 9 433 9 434 9 435 9 436 55 5 8984 8986 8987 8989 8990 8992 54 t 9437 9 438 9 43 9 9 44o 944i 9442 54 6 8 99 3 8 99 5 899618997 8999 9000 53 6 9443 9 445 9 446 9 44 7 9448 9 44 9 53 7 9002 9003 9005 9000 9007 9009 52 7 9 45o 9 45i 9 45 2 9 453 9454 9 455 52 8 9010 9012 9013 9Ol5 9016 9017 5i 8 9 456 9 45 7 9 458 9 45 9 9460 9 46i 5i 9 9019 9020 9022 9023 9024 9026 5o 9 9 463 9 464 9 465 9 466 9467 9 468 5o 10 9.999027 9029 9o3o 9032 9o33 9034 49 10 9 999469 9 4 7 o 9 4?i 9 472 9 473 9 474 4 9 ii 9036 9 o3 7 9 o3 9 9040 904 1 9 o43 48 ii 9475 9 4 7 6 9477 9 4 7 8 9479 9 48o 48 12 904^ 9046 9047 9048 9o5o 9o5i 47 12 948i 9 48 2 9 483 9 484 9 485 9 486 47 i3 9o52 9 o54 9o55 9057 9o58 9 o5 9 46 l2 9 48 7 9 488 9489 9 4 9 o 9491 9 4 9 2 46 1.4 9061 9062 9064 9065 9066 9 o68 45 i4 9493 9 4 9 5 9 4 9 6 9497 9498 9499 45 i5 9069 9071 9072 9073 9 o 7 5 9 o 7 6 44 i5 95oo 9 5oi 9502 9 5o3 9 5o4 95o5 44 16 9077 9079 9080 9082 9083 9 o84 43 16 9 5o6 9 5 7 95o8 9 5o 9 95io gSn 43 i? 9086 90879088 9090 9091 9 o 9 2 42 *7 9 5l2 9 5i3 95i4 9 5i5 9 5i6 9 5i 7 42 18 9094 9 o 9 5 9097 9098 9099 9 IOI 4i 18 95i8 9 5i 9 9520 9 52I 9522 9523 4i J 9 9102 9103 9105 9106 9107 9 io 9 4o J 9 9 5 2 4 9 5 2 5 9526 9 5 2? 9 5 27 9 5 2 8 4o 20 9.999110 9111 9113 911^3 9 n5 9117 3 9 20 9.999529 9 53o 9 53i 9 53 2 9 533 9534 3 9 21 9118 9120 9121 9122 9124 9125 38 21 9 535 9 536 9 53 7 9538 9 53 9 9 54o 38 22 9126 9128 9129 9i3o 9132 9 i33 37 22 954i 9 542 9 543 9544 9 545 9 546 37 23 9134 9 i36; 9 i3 7 9i38 9140 9141 36 23 9547 9 548 9 54 9 9 55o 9 55i 9 55 2 36 24 9142 9 i43 9 i45 9i46 9147 9149 35 24 9 553 9 554 9 555 9 556 9 55 7 9 55 7 35 25 9i5o 9i5i 9i53 9i54 9 i55 9 i5 7 34 25 9 558 9 55 9 9 56o g56i 9 562 9 563 34 26 9i58 9159 9161 9162 9i63 9i65 33 26 9 564 9 565 9 566 9 56 7 9 568 9 56 9 33 27 9166 9167 9168 9170 9171 9172 32 27 9 5 7 9 5 7 i 9 5 7 2 9 5 7 3 9 5 7 3 9 5 7 4 3a 28 9174 9175:91769178 9179 9180 3i 28 9 5 7 5 9 5 7 6 9 5 77 9 5 7 8 9 5 79 9 58o 3i 29! 9181 9183:9184 9i85 9187 9188 3o 29 9 58i 9 582 9 583 9 584 9 585 9 586 3o 309.999189 91909192 9 i 9 3 9 i 9 4 9196 29 3o 9.999586 9 58 7 9 588 9 58 9 9 5 9 o 9 5 9 i 2 9 3i 9197 91989199 9201 9202 9203 28 3i 9 5 9 2 9 5 9 3 9 5 9 4 9 5 9 5 9 5 9 6 9 5 97 28 32 9205 9206 9207 9208 9210 9211 2 7 32 9 5 97 9 5 9 8 9 5 99 9600 9 6oi 9602 2 7 33 9212 9213 9215 9216 9217 9219 26 33 9603 9604 9 6o5 9606 9 6o6 9607 26 34 9220 9221 9222 9224 9225 9226 25 34 9608 9609 9 6io 9611 9 6l2 9 6i3 26 35 9227 9229 9230 9231 9232 9 234 24 35 9614 9614 9 6i5 9616 9 6i 7 9618 24 36 9235 92369237 9239 9240 9241 23 36 9619 9620 9 62I 9622 9 622 9623 23 37 9242 92449245 9246 9247 9249 22 37 9624 9625 9 626 962-7 9 628 9629 22 38 9250 9251 9252 9254 9255 9256 21 38 9629 9 63o 9 63i 9 63 2 9 633 9 634 21 3 9 9 25 7 9258 9260 9261 9262 9263 2O 3 9 9 635 9 635 9 636 9 63 7 9 638 9 63 9 2O 4o 9.999265 9266 9267 9268 9270 9271 19 4o 9.999640 9 64i 9 64i 9642 9 643 9644 Z 9 4i 9272 92739274 9276 9277 9278 18 4i 9 645 9 646 9 t>47 9 64 7 9 648 9649 1 8 42 9279 9280 9282 9283 9284 9 285 J 7 42 965o 9 65i 9 65 2 9 653 9 653 9 654 J 7 43 9287 9288^289 9290 9291 9 2 9 3 16 43 9655 9 656 9 65 7 9 658 9 658 9 65 9 16 44 9294 9295 9296 9297 9299 93oo i5 44 9660 9 66i 9 662 9 663 9 663 9 664 i5 45 93oi 9302 93o3 93o5 93o6 9 3 7 i4 45 9665 9 666 9 66 7 9 668 9 668 9 66 9 i4 46 93o8 93099310 9312 9 3i3 9 3i4 i3 46 9670 9671 9 6 7 2 9 6 7 2 9 6 7 3 9 6 7 4 i3 47 93i5 93169318 93i 9 9320 9321 12 47 9 6 7 5 0676 9 6 77 9 6 77 9 6 7 8 9 6 79 12 48 9322 93239325 9326 9327 9 3 2 8 II 48 9680 9 68i 9 68i 9 68 2 9 683 9 684 II 49 9329 933i 9332 9333 9 334 9 335 IO 49 9685 9 685 9 686 9687 9 688 9689 10 5o 9. 999 336 93389339 9 34o 934i 9 342 9 5o 9.999689 9 6 9 o 9 6 9 i 9692 9 6 9 3 9 6 9 3 9 5z 9343 93449346 9347 9 348 9349 8 5i 9694 9 6 9 5 9 6 9 6 9697 9 6 97 9698 8 62 935o 9 35i 9353 9 354 9 355 9 356 7 52 9699 9 7oo 97 oo 9701 97 02 97 3 7 53 9 35 7 9358 9 35 9 9 36i 9362 9363 6 53 9 7 o4 9 74 97 o5 9706 977 9707 6 54 9 364 9 365 9 366 9 36 7 9 36 9 9 3 7 5 54 9708 979 97 10 9711 9711 9712 5 55 9 3 7 i 9372 9 3 7 3 9 3 7 4 9 3 7 5 9 3 77 4 55 97 i3 9 7i4 97i4|97i5 97 i6 9717 4 56 9 3 7 8 9 3 79 9 38o 9 38i 9 382 9383 3 56 9717 97 i8 97199720 97 20 972i 3 57 9 384 9385 9 38 7 9388 9 38 9 9 3 9 o 2 5 7 9722 97 23 9723 9724 9725 97 26 2 58 9391 9392 9 3 9 3 9394 9 3 9 6 9 3 97 I 58 9726 97 2 7 97 28 9729 9729 97 3o I 59 9 3 9 8 9 3 99 94oo 94oi 9 4o2 9 4o3 O 5 9 97 3i 97 3 2 97 32 97 33 9734 9735 O 60" 50" 40" 30" 20" | 10" d CC" 50" 40" 30" 20" 10" a Co-sine of 3 Degrees. 9 & Co-sine of 2 Degrees. 1 p p . < 1" 2" 3" 4" 5" 6" 7" 8" 9" irl l 000111111 p p ( 1" 2" 3" 4" 5" 6" 7" 8" 9" '^000001111 o G A R i T ii TANGENT s. Ill : Tangent of 86 Degrees. . Tangent of 87 Degrees. 0" 10" 20" 30" 40" 50" 2 0" 10" 20" 30" | 40" 50" 1 1. 155356 565 9 5962 6265 6568 6872 5 9 ii .280604 1007 i4n 1814 2219 2623 5 9 I 7175 7479 7784 8088 83 9 3 86 97 58 i 3o 2 8 3433 383 9 4245 4652 5o58 58 = 9 OO2 II.I60837 9 3o8 ii43 9 6i3 i45o 99 i 9 1757 .224 2064 .53o 23 7 I 56 2 3 5466 7917 58 7 3 8326 6281 6689 87379^7 7098 9 558 7 5 7 997 56 4 2679. 2 9 87 32 9 5 36o3 3 9 i i 4220 55 4 11.290382 0794 1206 1619 2o33 2446 55 5 452 9 4838 5i47 545 7 5 7 66 6076 54 5 2860 3 27 5 3690 4io5 452i 4 9 3 7 54 6 6387 66 97 7008 73i8 7 62 9 7 9 4i 53 6 5354 5 77 o 6188 66o5 7 O24 53 7 8252 8564 8876 9 i88 9 5oo 9 8i3 5 2 7 7861 8280 8700 9120 9 54i 99 6 2 52 S ii . 170126 o43 9 7 52 1066 i3 79 i6 9 3 5i 8 ii.3oo383 o8o5 1227 1649 20 7 2 2496 5i 9 2008 2322 2637 2 9 5l 326 7 3582 5o 9 2919 3344 3 7 68 4i93 46i 9 5o44 5o 10 ii 38 97 5 79 5 42l3 6112 4529 643o 4845 6 7 4 7 5 1 62 5478 7065^383 49 48 10 ii 8o3 7 58 97 8466 6325 8896 6 7 5 2 9326 7 i8o 97 56 7 6o8 .187 4 9 48 12 7702 8020 8339 8658 8 977 9297 4 7 12 11.310619 io5o i483 1915 2348 2782 4 7 i3 9 6i6 99 36 .256 .5 77 897 1218 46 i3 32i6 365o 4o85 4-52O 4 9 56 53 9 2 46 i4 n.i8i53 9 1860 2182 2826 3i48 45 i4 5828 6265 6702 7140 7 5 7 8 8017 45 i5 3471 3 79 3 4n6 444o 4763 5087 44 i5 8456 88 9 6 9336 9776 .217 .65 9 44 16 54n 5 7 35 6o5 9 6384 6 79 7034 43 16 II . 32IIOO 1 543 1985 2428 2872 33i6 43 17 7 35 9 7685 Son 833 7 8663 8990 42 J 7 3 7 6i 4206 465i 5097 5543 5 99 o 42 18 9 3i 7 9 644 9971 2 99 .626 .954 4t 18 643 7 6885 7 333 7782 8 2 3i 8680 4i i 9 ii I 9 i283 1611 1940 226 9 2 5 9 8 2928 4o 19 9i3o 9 58i o .483 . 9 35 1387 4o 20 3258 3588 3 9 i8 424 9 45 79 4910 3 9 20 n.33i84o 22 9 3 2747 3201 3656 4'n i 3 9 21 5242 55 7 3 5 9 o5 6237 656 9 6902 38 21 4567 5o23 548o 5 9 3 7 63 9 4 6852 38 22 7 235 7568 79 oi 8235 8568 8902 3y 22 7 3ii 777 8229 868 9 9 i5o 9 6n 3 7 23 9 23 7 9 5 7 i 99 o6 .241 .5 77 .912 36 23 ii.34oo 7 2 o534 0996 i45 9 I 9 23 2387 36 24 II. 201248 i584 I 9 2I 2257 2 5 9 4 2931 35 24 2 85i 33!6 3 7 8i 4247 4 7 i4 5i8o 35 25 326 9 36o6 3 9 44 4282 4621 4960 34 25 5648 6116 6584 7o53 7 522 799 s 34 26 52 99 5638 5 977 63i7 665 7 6997 33 26 8463|8 9 33 9 4o5 9877 .34 9 .822 33 27 7 338 7679 8020 836i 8703 9045 32 27 11.351296 1770 2244 2719 3i 9 5 36 7 i 3a 28 9 38 7 9729 .. 7 2 .4i5 . 7 58 IIO2 3i 28 4i4 7 4624 5lO2 558o 6o5 9 6538 3i 29 ii.2ii446 1790 2134 2479 2823 3l69 3o 29 7018 74 9 8 7979 846o 8 9 42 9424 3o 3o 35i4 386o 4206 4552 48 9 8 5 2 45 29 3o 997 .3 9 o i35 9 1 844 232 9 2 9 3i 55 9 2 5 9 3 9 6 2 8 7 6635 6 9 83 7 33i 28 3i ii .362816 33o2 3 7 8 9 4277 4 7 65 5 3 54 28 32 7 68o 8029 83 7 8 8728 9 o 7 8 9428 2 7 32 5 7 44 6 2 34 6724 777 8i 99 27 33 9778 .129 .48o .83i n83 i534 26 33 86 9 2 9 i85 9679 !i 7 3 .668 n64 26 34 11.221886 2239 25 9 i 2 9 44 3 29 8 365i 25 34 1 1 .371660 2i56 2654 3i5i 365o 4149 25 35 4oo5 435 9 4 7 i3 5o68 5423 5 77 8 24 35 4648 5i48 5649 6i5o 6652 7154 24 36 6i34 648 9 6845 7202 7558 79 i5 23 36 7 65 7 8161 8665 9 J 7 9 6 7 5 .181 23 37 82 7 3 863o 8 9 88 9 346 97 5 ..63 22 37 rt. $80687 1194 1702 2210 2 ? I 9 3228 22 38 II.23o422 0782 n4i i5oi 1861 2222 21 38 3 7 38 4249 4760 5272 5 7 85 62 9 8 21 39 2583 2944 33o5 366 7 4o2 9 4391 2O 39 6811 7325 7 84o 8356 8872 9 388 20 4o 4 7 54 5n6 548o 5843 6207 65 7 I 19 4o 99 o6 424 .942 i46i 1981 25oi 19 4i 6 9 35 73oo 7 665 8o3o 83 9 6 8762 18 4i II .3 9 3022 3544 4o66 458 9 5n3 5637 18 42 9 I28 9 4 9 5 9 86i .22 9 .5 9 6 .964 17 42 6161 6687 7213 77 4o 8267 8 79 5 17 43 II.24l332 1700 2o6 9 2438 2807 3i 77 16 43 9 323 9853 .382 . 9 i3 1 444 i 97 6 16 44 354 7 3917 4288 465 9 5o3o 54oi i5 44 11.402608 3o4i 3575 4no 4645 5i8o i5 45 5 77 3 6i45 65i8 68 9 i 7264 7 63 7 i4 45 5717 6254 6792 7 33o 7 86 9 84o 9 i4 46 Son 8385 8 7 5 9 9 i34 9 5oo 9 884 i3 46 8 9 4 9 9490 .5 7 4 1117 1661 i3 47 ii .250260 o636 IOI2 i38 9 1766 2i43 12 47 II.4l22o5 2751 32963843 43 9 o 4 9 38 12 48 2521 2899 32 77 3656 4o34 44i4 I I 48 5486 6o36 6586 7 i36 7688 8240 II 49 4 79 3 5 I7 3 5553 5 9 34 63i5 6696 IO 49 8792 9346 9900' .455 1010 1 566 IO 5o 7078 7460 7 842 8224 8607 8 99 i 9 5o 11.422123 2681 324013799 4358 4 9 i 9 9 5i 9 3 7 4 9758 .142 .527 . 9 I2 129-7 8 5i 548o 6o42 66o5l 7 i68 77 33 82 9 8 s 52 11.261683 2069 2455 2842 322 9 36i6 7 52 8863 943o 9997! .565 1 1 33 I 7 02 7 53 54 55 56 4oo4 6337 8683 ii .271041 43 9 2 6727 9075 i435 4780 7117 9 467 i83o 5i6 9 7 5o8 9 86o 2225 5558 7899 .254 2620 5 9 4 7 82 9 I .64 7 3oi6 6 5 4 3 53 54 55 56 11.432273 5709 9172 11.442664 2843 6284 9752 3248 34i5 6860 .333 3834 39874560 743 7 |8oi5 .915.1497 4420 5oo 7 5i34 85 9 3 2080 55 9 5 5 4 3 $7 34i2 3809 4206 46o3 5ooo 53 9 8 2 57 6i83 6 77 3i 7 363 7954|8546 9 i38 2 58 5 9 5 79 6 8i 9 4 6195 85 9 5 65 9 4 8 99 6 6 99 3 9 3 97 7 3 9 3 9799 779 3 .202 I O 58 5 9 9732 ii.4533o9 .326 .Q2I 3908 45o8 i5i 7 5 1 09 2113 5711 2 7 I 63i3 I 60" 50" | 40" 30" 20" 10" S" 60" 50" 4'W | 30" | 20" 10" . Co-tangent of 3 Degrees Co-tangent of 2 Degrees. i f -i II nil }// AII r // p// // // Q// P. Part J 3 _ G() 1Q4 13g ]73 2Q7 ^ 276 gn p p. ( 1"2" 3" 4" 5" 6" 7" 8" 9" irt } 48 97 145 193 242 290 338 387 435 112 LOGARITHMIC SINES. 1 Sine of 88 Degrees. jj Sine of 89 Degrees. fl 0" 10" 20" 30" 40" 50" s 0" 10" | 20" 30" 40" 50" o 9.999735 9736 97 3 7 973*8 9738 9739 59 9 . 9999 34 99 34 99 35 99 35 9935 99 36 5 9 I 974o 9 74o 97 4i 97 42 9743 9743 58 i 99 36 99 36 99 3 7 99 3 7 99 3 7 99 38 58 2 9744 9745 9 746 97 46 9747 9748 5 7 2 99 38 99 3 9 99 3 9 99 3 9 99 4o 99 4o 5? 3 9748 9749 97 5o 97 5 i 97 5i 97 5 2 56 3 99 4o 99 4i 99 4i 99 4i 9942 99 42 56 4 9753 97 53 97 54 9755 9756 9756 55 4 99 42 99 43 99 43 9943 99 44 99 44 55 5 97 5 7 9758 9758 97 5 9 9760 9 76o 54 5 99 44 99 45 99 45 9945 99 46 99 46 54 6 9761 9762 97 63 9763 9764 9765 53 6 99 46 9947 9947 9947 99 48 99 48 53 7 9765 9766 9767 9767 9768 97 6 9 52 7 99 48 9949 99 4 9 9 9 4 9 99 5o 99 5o 52 8 9769 9770 9771 9772 9772 9773 5i 8 995o 99 51 99 5i r 99 51 99 52 99 52 5i 9 9774 9774 977 5 9776 9776 9777 5o 9 99 52 99 53 99 53 99 53 99 53 99 54 5o 10 9.999778 9778 9779 97 8o 9780 978i 49 10 9 . 9999 54 99 54 99 55 99 55 99 55 99 56 4 9 ii 9782 9782 97 83 97 84 9784 9785 48 ii 99 56 99 56 99 56 99 5 7 99 5 7 99 5 7 48 12 9786 9786 9787 9788 9788 9789 47 12 99 58 99 58 99 58 99 5 9 99 5 9 99 5 9 4 7 i3 979 979 9791 979 2 9792 979 3 46 i3 99 5 9 9960 99 6o 99 6o 99 6i 99 6i 46 i4 9794 9 7 9 4 979 5 979 5 9796 9797 45 i4 9961 9961 99 62 99 62 99 62 99 63 45 i5 9797 9798 9799 9799 9800 9 8oi 44 i5 99 63 99 63 99 63 99 64 99 64 9 o64 44 16 9801 9802 9 8o3 9 8o3 9 8o4 9804. 43 16 99 64 99 65 99 65 99 65 99 65 99 66 43 l l 9805 9806 9806 9 8o 7 9808 9 rfo8 42 T 7 9966 99 66 9967 99 6 7 99 6 7 99 6 7 42 18 9809 9809 9810 9 8n 9811 9812 4i 18 9968 9968 9968 99 68 99 6 9 99 6 9 4i '9 9813 9 8i3 9 8i4 9 8i4 9 8i5 9816 4o J 9 9969 9969 9970 997 997 997 4o 20 9.999816 9817 9817 9 8i8 9819 9819 3 9 20 9-99997 1 9971 997 1 997 1 997 2 9972 3 9 21 9820 9820 9 82I 9 822 9822 9823 38 21 9972 9972 9973 997 3 997 3 997 3 38 22 9824 9824 9 8 2 5 9 825 9826 9827 37 22 9973 9974 9974 9974 9974 997 5 3 7 23 9827 9828 9828 9 82 9 9829 9 83o 36 23 997 5 9975 9975 997 6 997 6 997 6 36 24 9 83i 9 83i 9 832 9 83 2 9 833 9 834 35 24 997 6 9976 9977 9977 9977 9977 35 25 9 834 9 835 9835 9 836 9836 9 83 7 34 25 9977 9978 9978 9978 9978 9979 34 26 9 838 9 838 9 83 9 9 83 9 9840 9 84o 33 26 9979 9979 9979 9979 99 8 9980 33 27 9841 9 842 9842 9843 9 843 9 844 32 27 998o 9980 99 8o 99 8i 99 8i 99 8i 32 28 9844 9 845 9846 9 846 9 84 7 9 84 7 3i 28 998i 9981 9982 99 82 99 8 2 99 82 T : Co-sine of Degree. 1 , ( 1" 2" 3" 4" 5" 6" 7" 8" 9" irt $0000 00 001 C !// 2" 3" 4" 5" 6" 7" 8" 9" in l oo ooooooo LOGARITHMIC TANGENTS. 113 F Tangent of 88 Degrees. P. Part 3 0" 10" 20" 30" 40" 50" to 1". o 11.456916 ii .457520 ii. 458i25 II.45873I 11.459338 II .459945 09 60.6 I 46o553 46n63 461773 462383 462995 4636o8 58 61.1 a 464221 464836 46545i 466067 466684 46 7 3o2 5 7 61.6 3 467920 46854o 469160 469782 470404 4 7 I02 7 56 62.2 4 47i65i 472276 472902 473528 474i56 4 7 4 7 85 55 62.7 5 4754i4, 476o44 476676 477308 47794i 4 7 85 7 5 54 63.3 6 479210 479846 480482 481120 481759 4823 9 8 53 03. 8 7 483o39 48368o 484323 484966 4856n 486256 52 64.4 8 486902 48 7 549 488198 488847 489497 490148 5i 65.o 9 490800 4 9 i453 492107 4.92762 493418 494o 7 5 5o 65.5 10 ii.494?33 11.495392 it .496052 11.496713 11.497375 11.498038 4 9 66.1 ii 498702 499 36 7 5ooo33 500700 5oi368 5o2o3 7 48 66.8 12 502707 5o33 7 8 5o4o5i 504724 5o53 9 8 506073 47 6 7 4 i3 506750 507427 5o8io6 5o8 7 85 509466 5ioi48 46 68.0 i4 5io83o 5n5i4 512199 5i2885 5i3572 5i426o 45 68.6 i5 5i495o 5i564o 5i633i 517024 5i 77 i 7 5i84i2 44 6 9 .3 16 519108 SigSoS 52o5o3 52I2O2 621903 522604 43 70.0 l l 523307 524OIO 5247i5 525421 526128 52683 7 42 70.7 18 52 7 546 528257 528969 629682 53o3 9 6 53ini 4i 7 i.3 '9 53i828 532545 533264 533 9 84 534705 535428 4o 72.1 20 n.536i5i 1 1. 5368 7 6 .11 .537602 ii. 53833o ii.539o58 ii.53 97 88 3 9 72.8 21 54o5i9 54i25i 541984 542719 543455 544192 38 7 3.5 22 54493o 5456 7 o 5464n 547i53 5478 9 6 54864: 37 7 4.3 23 549387 55oi34 55o883 55i63 2 552384 553i36 36 7 5.o 24 553890 554645 5554oi 556i5 9 556 9 i8 55 7 6 7 8 35 7 5.8 25 55844o 55 9 2o3 55 99 6 7 560733 56i5oo 562268 34 76.6 26 563o38 563809 56458: 565355 566i3o 56690-7 33 777-5 27 56 7 685 568464 56 9 245 570027 570811 5 7 i5 9 6 32 78.3 28 572382 573170 5 7 3 9 5 9 574750 5 7 5542 5 7 6336 3i 79-- 29 5 77 i3i 5 779 a8 578726 5 79 5 2 5 58o3 2 6 581128 3o 80.0 3o 11.581932 11.582737 u.583544 n.584353 n.585i63 n.585974 2 9 80.9 3i 586 7 8 7 587601 5884i7 589235 5 9 oo54 590874 28 81.8 33 591696 592520 5 9 3345 594172 5 9 5ooo 5 9 583o 2 7 82.8 33 596662 597495 598330 599166 600004 6oo844 26 83. 7 34 6oi685 602528 603372 604218 6o5o66 6o5 9 i5 25 84-7 35 606766 607619 608474 609330 610187 611047 24 85. 7 36 611908 612771 6i3636 6i45o2 615370 616240 23 86.7 3? 617111 617985 618860 6i 97 3 7 620615 621496 22 87.8 38 622378 623262 624147 625o35 625 9 24 626815 21 88.8 3 9 627708 628603 629500 630399 63i2 99 632201 20 8 9 . 9 4o n.633io5 ii .634oi2 ii .634919 n.635829 11.636741 n.637655 I 9 91.1 4i 6385 7 o 639488 640407 641329 642252 643i 77 18 9 2.2 42 644io5 645o34 645965 646899 647834 648771 17 9 3.4 43 649711 65o65 2 65i5 9 5 65 2 54i 653488 654438 16 94.6 44 655390 656343 657299 658257 65 9 2i7 660179 i5 95. 9 45 66 i i 44 662110 663079 664o5o 665023 665 99 8 i4 9 7 .2 46 666975 66 79 55 668 9 36 669920 670907 671895 i3 98.5 4? 672886 6 7 38 79 6 7 48 7 4 6 7 58 7 i 676871 677873 12 99 .8 48 678878 679885 680894 681905 682 9 i 9 683935 II 101.3 49 684954 685 97 5 686998 688024 68 9 o52 690083 IO 102.7 5o 11.691116 n .692151 11.693189 11.694230 n.6 9 5273 11.696318 9 IO4.2 5i 697366 698417 699470 700526 7oi584 702645 8 105.7 52 703708 704774 705843 706914 707988 709065 7 107.3 53 710144 711226 712311 713398 7i4488 7 i558i 6 108.9 54 716677 717775 718876 719980 721087 722196 5 no. 5 55 7 233o 9 724424 725542 726663! 727787 728914 4 112 2 56 73oo44 731176 7323i2 73345i! 734592 7 35 7 3 7 3 n4.o 5? 736885 738o35 7 3 9 i8 9 7 4o346 74i5o6 742669 2 n5.8 58 743835 745oo4 746177 7 47352 7 4853i 7 4 97 i3 I "7-7 . 5 9 750898 752087 7 532 79 754474 7 556 7 2 7 568 7 4 O 119.7 60'' 50" 40" 30" 20" 10" . Co- tangent of 1 Degree.. * H 114 AUXILIARY TABLE FOR SINES, czc. 1 J Degree. 1 1 Degree. i 3 a s log. sin. A fog. A 1 '. log. tan. A log. A". log. cot A+ log. A". 1 i 9 log. sin. A log. tan. A log. A". 1 log. A". log. cot A + log. A". o 4.6855 7 5 4.6855 7 5 i5.3i4425 60 36oo 4.685553 4.6856i 9 i5.3i438i 60 00 I 5 7 5 5 7 5 425 5 9 366o I 55 2 620 38o 5 9 120 2 5 7 5 5 7 5 425 58 8720 2 55i 622 3 7 8 58 180 3 5 7 5 5 7 5 425 5 7 3780 3 55i 623 3 77 57 24o 4 5 7 5 5 7 5 425 56 384o 4 55o 625 3 7 5 56 3oo 5 5 7 5 5 7 5 425 55 3 9 oo 5 54 9 627 3 7 3 55 36o 6 5 7 5 5 7 5 425 54 3 9 6o 6 548 628 3 7 2 54 420 7 5 7 5 5 7 5 4a5 53 4O2O 7 54 7 63o 370 53 48o 8 5 7 4 5 7 6 424 52 4o8o 8 547 63 2 368 52 54o 9 574 5 7 6 424 5i 4i4o 9 546 6331 36 7 5i 600 10 4.6855 7 4 4.6855 7 6 i5.3i4424 DO 4200 10 4.685545 4.685635 i5.3i4365 5o 660 ii 5?4 5 7 6 424 4 9 4260 1 1 544 63 7 363 49 720 12 5 7 4 5 77 423 48 4320 12 543 638 362 48 780 i3 5 7 4 5 77 423 47 438o i3 542 64o 36o 47 84o i4 674 5 77 423 46 444o i4 54i 642 358 46 900 i5 5 7 3 5 7 8 422 45 45oo i5 54o 644 356 45 960 16 5 7 3 5 7 8 422 44 456o 16 539 646 354 44 IO2O J 7 5 7 3 5 7 8 422 43 4620 17 53 9 647 353 43 1080 18 5 7 3 5 79 421 42 468o 18 538 64 9 35i 42 u4o J 9 5 7 3 5 79 421 4i 474o J 9 53 7 65i 349 4i I2OO 20 4.6855 7 2 4.68558o i5.3i442o 4o 48oo 20 4.685536 4.685653 i5.3i434 7 4o 1260 21 672 58o 420 3 9 486o 21 535 655 345 3 9 1320 22 672 58i 4i 9 38 4 9 20 22 534 65 7 343 38 i38o 23 672 58i 419 37 4 9 8o 23 533 65 9 34i 3 7 i44o 24 5 7 i 582 4i8 36 5o4o 24 532 661 33 9 36 1600 26 671 583 4i 7 35 5ioo 25 53i 663 33 7 35 1660 26 5 7 i 583 4i7 34 5i6o 26 53o 665 335 34 1620 27 5 7 o 584 4i6 33 522O 2 7 52 9 668 332 33 1680 28 5 7 o 584 4i6 32 5280 28 527 670 33oj32 1740 -9 5 7 o 585 4i5 3i 534o 2 9 526 672 3 2 8 3i 1800 3o 4.68556 9 4.685586 i5.3i44i4 3o 54oo 3o 4.685525 4.6856 7 4 i5.3:4326 3o 1860 3i 56 9 58 7 4i3 2 9 546o 3i 524 676 324 29 1920 32 669 58 7 4i3 28 5520 32 5 2 3 6 79 321 28 1980 33 568 588 412 27 558o 33 522 681 3i 9 27 2040 34 568 58 9 4n 26 564o 34 521 683 3i 7 26 2IOO 35 56 7 5 9 o 4io 25 5700 35 520 685 3i5 25 2l6o S6 56 7 5 9 i 409 24 5760 36 5i8 688 3l2 24 222O ^7 566 5 9 2 4o8 23 5820 37 5i 7 6 9 o 3io 23 228O 38 566 5 9 3 407 22 588o 38 5i6 6 9 3 3o 7 22 2340 3 9 566 5 9 3 407 21 5 9 4o 3 9 5i5 6 9 5 3o5 21 24OO 4o 4.685565 4.6855 9 4 i5.3i44o6 2O 6000 4o 4.6855i4 4. 6856 97 i5.3i43o3 20 2460 4i 565 5 9 5 4o5 I 9 6060 4i 5l2 700 3oo *9 2626 42 564 5 9 6 4o4 18 6120 42 5n 702 298 18 2580 43 564 5 9 8 4O2 *7 6180 43 5io 7 o5 295 17 2640 44 563 5 99 4oi 16 6240 44 509 707 293 16 2700 45 562 600 4oo i5 63oo 45 5oy 710 290 i5 2760 46 562 601 3 99 i4 636o 46 5o6 7 i3 28 7 i4 2820 4? 56i 602 3 9 8 i3 6420 47 5o5 7 i5 285 i3 2880 48 56i 6o3 3 97 12 648o 48 5o3 718 282 12 2940 4 9 56o 6o4 3 9 6 II 654o 49 5o2 720 280 II 3ooo 5o 4.68556o 4.6856o5 i5.3i43 9 5 IO 6600 5o 4.6855oi 4.685723 i5.3i42 77 10 3o6o 5i 55 9 607 3 9 3 9 6660 5i 499 726; 274 9 3l20 62 558 608 3 9 2 6 6720 52l 4 9 8 729 271 8 3i8o 53 558 609 3 9 i 7 6780 53 497! 73i 269 7 324o 54 55 7 611 38 9 6 684o 54 495 7 34 266 6 33oo 55 556 612 388 5 6900 55 494 7 3 7 263 I 336o 56 556 6i3 38 7 4 6960 56 492 7 4o 260 4 3420 5? 555 6i5 385 3 7020 57 491 7 43 25 7 3 348o 58 554 616 384 2 7080 58 4 9 o 7 45 255 2 354o 5 9 554 618 382 I 7140 5 9 488 748 252 I log. cos. A log. c. A". log. cot A log. c. A". log. tan. A+ log. c. A." 1 log. cos. A log. c. A". log. cot A log. c. A". log. tail. A-J- log.c.A" | 89 Degrees. 2 88 Degrees. 2 LOGARITHMIC TANGENTS. 115 1 Tangent of 89 Degrees. P. Part tol". 0" 10" 20" 30" 40" 50" o 11.758079 11.759287 II .760498 11.761714 II .762932 H.764l54 5 9 121.7 I 765379 766608 767840 769076 77o3i5 77 i558 58 123.8 2 772805 774o55 775308 77 6566 777826 779091 5 7 125.9 3 7 8o35 9 7 8i63i 782907 784186 785470 7 86 7 5 7 56 128.1 4 788047 789342 790641 791943 79325o 79456o 55 i3o.4 5 79^874 797192 7985i5 799841 801171 8o25o6 54 i32.8 6 8o3844 805187 8o6534 8o 7 885 809240 810600 53 i35.3 7 811964 8i3332 814704 816081 817462 818847 52 137.9 8 820237 821632 823o3i 824434 825842 827255 5i i4o.6 9 828672 830094 83i52o 832951 834387 835828 5o 143.4 10 11.837273 11.838724 ii .840179 n. 841639 n.843io4 H.844574 4 9 146.2 ii 846o48 847528 849013 85o5o3 851999 853499 48 149.3 12 855oo4 8565i5 858o3i 85 9 553 861079 862611 47 i52.4 i3 864149 865692 867240 868794 870354 871919 46 i55. 7 i4 873490 875067 876649 878237 879831 88i43i 45 159.1 i5 883o37 884648 886266 887890 889519 8 9 n55 44 162.7 16 892797 8 9 4446 896100 897761 899429 901103 43 166.4 17 902783 904470 906163 .907863 909569 911283 42 170.3 18 9i3oo3 914730 916464 918205 919953 921707 4i 174.4 '9 923469 925239 927015 928799 930590 9 32388 4o 178.7 20 11.934194 11.936008 ii .937829 11.939658 11.941494 11.943338 3 9 i83.3 21 945191 947o5i 948919 950795 952679 954572 38 188.0 22 9 56473 9 58382 960299 962225 964160 966103 37 193.0 23 9 68o55 970016 971986 973965 975952 977949 36 198.3 24 9 79 9 56 981971 983996 986o3o 988074 990128 35 2o3. 9 25 992191 994264 996347 998440 I2.ooo543 12.002657 34 209.8 26 12.004781 12.006915 12.009060 12.01 I2l5 oi3382 oi555 9 33 216.2 27 017747 019946 O22l56 024378 026611 028855 32 222 7 28 oSini 033379 o3565 9 O3795l o4o255 042572 3i 229.8 29 044900 047242 049596 051963 o54342 o56735 3o 2 3 7 .3 3o 12.059142 12 ,o6i56i 12.063994 I 2. 06644 I 12.068902 12.071377 29 245 4 3i o 7 3866 076369 078887 O8l4l9 083966 086529 28 254.0 32 089106 091699 094308 096932 099572 102228 27 263.2 33 104901 107590 110296 iiSoig 115760 n85i7 26 273.2 34 121292 I24o85 126896 129726 i325 7 4 i 3544o 25 283.9 35 i383 2 6 i4i23i i44i56 147100 i5oo65 i53o5o 24 295.4 36 i56o56 159082 i62i3o 165199 168290 171404 23 3o8.o 3? i7454o 177698 180880 i84o85 187314 190567 22 321.7 38 i 9 3845 197148 200476 2o383o 207210 210616 21 336.7 3 9 214049 217510 220998 2245i5 228060 2 3i635 20 353.1 4o 12.235239 12.238873 12.242538 12.246235 12. 249963 12.253723 19 371.2 4i 257516 261342 265203 269098 273028 276995 18 391.3 4a 280997 2 85o3 7 289115 293232 297388 3oi584 17 4i3.6 43 3o582i Sioioo 3l4422 3i8 7 8 7 3 2 3i 9 6 327650 16 438. 7 44 332i5i 336698 341294 345939 35o634 35538i i5 467.0 45 36oi8o 365o32 369940 374903 379924 385o64 i4 499.2 46 390143 3 9 5345 400609 4o5938 4n333 416796 i3 536.2 47 422328 427932 4336io 439362 445192 45noo 12 579.1 48 457091 463i65 46 9 325 4755 7 4 481915 488349 II 629.4 49 494880 5oi5io 5o8244 5i5o83 522O32 529094 IO 689.4 5o 12.536273 12.543572 12.550996 12.558549 12.566236 12.574061 9 5i 582o3o 590148 598421 6o6854 6i5454 624228 8 52 633i83 64232 7 651667 661212 670972 680956 7 53 691175 701641 7 i2365 723360 7 3464i 746223 6 54 758122 77 o35 7 782946 795911 809275 823o63 5 55 83 7 3o4 802027 867267 883o6i 899452 9 i6485 4 56 9 342i4 952697 972002 992206 i3.oi33 9 5 i3. 035671 3 57 i3.o5 9 i53 1.3.083976 i3.no3o5 i3.i38334 168297 200482 2 58 235244 273032 3i4425 36oi83 4n335 469327 I 5 9 5362 7 4 6i5455 712365 83 7 3o4 i4-oi33 9 5 i4.3i4425 O 00" 50" 40" 30" 20" 10" c Co-tangent of Degree. S 116 NATURAL SINES. QE! 1 2 3 4 5 6 7 8 9 r- o 000000 017452 034899 o5 2 336 o6 9 756 087156 104528 I2i86 9 i3 9 i 7 3 i56434 60 I 0291 77 43 5i 9 o 2626 070047 7446 48i8 2i58 9 46i 6722 5 9 2 o582 8o34 548! 2917 o33 7 77 35 5io 7 2447 9749 7 oo 9 58 3 0878 83 2 5 5 77 2 3207 0627 8o25 53 9 6 2735 i4oo3 7 7 2 9 6 5? 4 n64 8616 6062 3498 0917 83i5 5686 3024 o325 7 584 56 5 i454 8907 6353 3788 1207 86o5 5 97 5 33i3 o6i3 7 8 7 i 55 6 1745 9 T 97 6644 4079 1497 88 9 4 6264 36oi o 9 oi 8i58 54 7 2o36 9488 6 9 34 436 9 1788 9 i84 6553 38 9 o n8 9 8445 53 8 2327 9779 7225 466o 2078 9 474 6843 4179 i4 77 8 7 32 52 9 2618 020070 75i6 4 9 5o 2368 9763 7 l32 446 7 i 7 65 9 O2O 5i 10 002909 O2o36i 037806 o5524i 072658 o 9 oo53 I0 7 42I 124756 i42o53 i5 9 3o 7 5o ii 3200 o652 8097 553i 2 9 48 o343 77 io 5o45 a34i 9 5 9 4 49 12 3491 0942 8388 5822 3 2 38 o633 7999 5333 262 9 9 88i 48 i3 3782 1233 8678 6112 3528 O 9 22 8 2 8 9 5622 2 9 I 7 160168 47 i4 4072 i524 8969 6402 38i8 1212 85 7 8 5910 3205 o455 40 i5 4363 i8i5 9260 66 9 3 4io8 i5o2 8867 6199 34 9 3 o 7 43 45 16 4654 2106 9 55o 6 9 83 43 99 I 79 I 9 i56 6488 3 7 8o io3o 44 J 7 4945 23 97 9841 7274 468 9 2081 9 445 6776 4o68 i3i 7 43 18 5 2 36 2687 o4oi32 7564 4979 2371 9734 7 o65 4356 1604 42 *9 552 7 2978 0422 7854 526 9 2660 IIOO23 7 353 4644 i8 9 i 4i 20 oo58i8 023269 040713 o58i45 075559, o 9 2 9 5o Iio3i3 127642 i44 9 32 i62i 7 8 4o 21 6109 356o 1004 8435 584 9 3 2 3 9 0602 79 3o 5220 . 2465 3 9 22 6399 385i 1294 8726 6i3 9 352 9 o8 9 i 8219 55o 7 2 7 52 38 23 6690 4i4i i585 9&i6 642 9 38i 9 1180 85o 7 5 79 5 3o3 9 3 7 24 6981 4432 1876 93o6 6 7 : 9 4io8 i46 9 8 79 6 6o83 3326 36 25 7272 4723 2166 9 5 97 7oo 9 43 9 8 i 7 58 9 o84 63 7 i 36i3 35 20 7563 5oi4 2457 9887 7 2 99 468 7 2047 9 3 7 3 665 9 3 9 oo 34 a 7 754 53o5 2748 060177 7 58 9 4977 2336 9 66i 6 9 46 4i8 7 33 28 8i45 55 9 5 3o38 o468 7879 5267 2625 9949 7 234 44 7 4 32 29 8436 5886 3329 0758 8i6 9 5556 2 9 l4 i3o238 7 522 4 7 6i|3i 3o 008727 026177 043619 o6io4 9 07845 9 o 9 5846 Ii32o3 i3o526 i4 7 8o 9 i65o48 3o 3i 9017 6468 3910 i33 9 8 7 4 9 6i35 34 9 2 o8i5 8o 97 5334 20, 3a 9 3o8 6 7 5 9 4201 1620, 9 o3 9 6425 3 7 8i uo3 8385 56ai 28 33 9 5 99 7049 449i I 9 2O 9 32 9 6 7 i4 4070 i3 9 i 86 7 2 5 9 o8|2 7 34 9890 7340 4782 22IO 9 6l 9 7004 435 9 1680 8 9 6o 6i 9 5 26 35 010181 7 63i 5072 25oo 999 7 2 9 3 4648 i 9 68 9 248 6482 25 36 0472 7922 5363 2 79 I o8oi 99 7 583 4 9 3 7 2256 9 535 6 7 6 9 24 3 7 0763 8212 5654 3o8i o48 9 7 8 7 2 5226 2545 9 8 2 3 7o56 23 38 io54 85o3 5 9 44 33 ?I 779 8162 55i5 2833 iSoni 7342 22 3 9 1 344 8 79 4 6235 366i io6 9 845i 58o4 3l2I o3 9 8 762 9 21 4o on635 029085 o465 2 5 063952 o8i35 9 o 9 8 7 4i n6o 9 3 i334io i5o686 1679,16 2O 4i 1926 9 3 7 5 6816 4242 i64 9 9 o3o - 6382 36 9 8 o 97 3 82o3 19 42 2217 9666 7106 4532 i 9 3 9 9 320 6671 3 9 86 1261 848 9 18 43 2 5o8 99 5 7 7 3 97 4823 2228 9 6o 9 6 9 6o 42 7 4 1 548 8776 J 7 44 2799 o3o248 7688 5n3 25i8 9 8 99 7249 4563 i836 9 o63 16 45 3090 o539 7978 54c3 2808 100188 7 53 7 485i 2123 9 35o i5 46 338o 0829 8269 56g3 3o 9 8 o4 77 7826 5i3 9 2411 9 636 i4 47 36 7 i II2O 855 9 5 9 84 3388 07 6 7 8n5 5427 26 9 8 9923 i3 48 3962 i4i i 885o 6274 3678 io56 84o4 5 7 i6 2 9 86 170209 12 49 4253 1702 9140 6564 3 9 68 1 346 86 9 3 6oo4 32 7 3 0496 II 5o oi4544 031992 o4943i o66854 o84258 ioi635 n8 9 82 l362 9 2 i5356i 170783 10 5i 4835 2283 9721 - ?i45 4547 I 9 24 9 270 658o 3848 1069 9 52 5i26 25 7 4 o5ooi2 7 435 483 7 2214 9 55 9 6868 4i36 i356 8 53 54i6 2864 0302 77 25 5l2 ? 2 5o3 9 848 7 i56 4423 i643 7 54 5707 3i55 o5 9 3 8oi5 54i 7 2 79 3 120137 7 445 4 7 io 1929 6 55 5 99 8 3446 o883 83o6 5 77 3o82 0426 7733 4 99 8 2216 5 56 6289 3 7 3 7 H74 85 9 6 5 997 33 7 i 071^ 8021 5e85 25O2 4 57 658o 4027 1 464 8886 6286 366i ioo3 83o 9 55 7 2 2789 3 58 6871 43i8 i 7 55 9176 65 7 6 395o I2 9 2 85 97 586o 3o 7 5 2 5 9 7162 4609 2o45 9 466 ' 6866 423 9 i58i 8885 6i4 7 3362 I 89 88 87 86 85 84 83 82 81 80 d Natural Co-sines. 3 4.85 4.85 4.84 4.84 4.83 4.83 4.82 4.8i 4.8o 4.78 IN A T u R A L TANGENT H 11 V 1

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A i 20 21 22 23 24 25 20 27 28 29 o 342020 358368 3 7 46o 7 39o 7 3i 4o6 7 3 7 422618 4383 7 i 453990 469472 4848 10 60 i 2293 864o 48 7 6 0999 7002 2882 8633 425o 9728 5o64 5 9 2 256 7 8911 5i46 I26 7 7268 3i45 8894 45o 9 99 85 53i8 58 3 2840 9i83 54i6 i534 7 534 34o 9 9i55 4 7 68 470242 55 7 3 5 7 4 3n3 9454 5C85 1802 7799 36 7 3 9417 5o2 7 o4 99 582 7 56 5 3387 97 25 5 9 55 2O 7 8o65 3936 9678 5 2 86 o 7 55 6081 55 6 366o 9997 6224 233 7 833o 4199 99 3 9 5545 IOI2 6335 54 7 3 9 33 360268 6494 26o5 85 9 6 4463 44O20O 58o4 1268 65 9 o 53 8 4206 o54o 6 7 63 2872 8861 4726 0462 6o63 i5 2 5 6844 52 9 4479 0811 7 o33 3i4o 9127 4990 0723 6322 I 7 82 7 o 9 8 5i 10 344 7 52 361082 3 77 3o2 393407 409392 425253 440984 45658o 472038 48 7 35 2 5o I! 5o 2 5 i353 7 5 7 i 36 7 5 9 658 55i6j 1245 6839 22 9 4 7 6o6 49 12 5298 1625 7 84i 3942 99 23 5779] i5o6 7098 2 55i 7 86o 48. i3 55 7 i 1896 8no 4209 4ioi88 6042 1767; 7 35 7 2807 8n4 47 i4 5844 2167 83 79 44 77 o454 63o6 2028 7 6i5 3o63 836 7 46 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9 8 46i 7 49J477 l5 9 4 9 2424 3o 3i o48o 6 77 2 2952 9016 4 9 58 0774 6458 2007, 7414 26 77 29 32 0752 7042 3221 9283 5223 io36 6 7 i8 2265) 7670 2 9 3o 28 33 1025 7 3i3 3490 9549 548 7 1299 6979 2523; 7925; 3i83 2 7 34 1297 7 584 3 7 58 9816 5 7 5 2 i56i ?2 3 9 2780! 8181 3436 26 35 i56 9 7 854 4027 400082 6016 1823 7499 3o38| 8436! 368 9 25 36 1842 8i25 42 9 5 o349 6281 2086 77 5 9 ' 32 9 6 86 9 2 3 9 42 24 37 2Il4 83 9 5 4564 0616 6545 2348 8oi 9 j 3554 894-7 4i 9 5 23 38 2386 8665 4832 0882 6810 2610 82 79 38i2; 9 2o3 4448 23 3 9 2658 8 9 36 5ioi 1149 7074 28 7 3 853 9 4069 9458 4700 21 4o 352 9 3i 369206 38536 9 4oi4i5 4i7338 433:35 448 799 464327 4797 l3 4 9 4 9 53 2O 4i 3 2 o3 94 7 6 5638 1681 7603 33 97 9 o5 9 4584 99 68 5206 J 9 42 3475 9 7 4 7 5906 1948 7867 365 9 9 3i 9 4842 480223 545 9 18 43 3 7 4 7 3 7 ooi 7 6i 7 4 22l4 8i3i 3921 9 5 79 5ioo o4 79 5 7 n !7 44 4019 O28 7 6443 2480 83 9 6 4:83 9 83 9 535 7 o 7 34 5 9 64 16 45 4291 o55 7 6 7 n 2 7 4 7 8660 4445 450098 56i5 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4.26 4.22 I NATURAL TANGENTS, 121 1 A ] as 20 21 22 23 24 25 26 27 28 29 o 363970 383864 404026 4s44 7 5 44522 9 4663o8 487733 5o 9 5 2 5 53i7o 9 5543o 9 60 i 43oo 4198 4365 48:8 55 77 6662 8o 9 3 9 8 9 2 2o83 468 9 5 9 2 4629 4532 4 7 o3 5i6s 5926 7016 8453 5io258 2456 6070 58 3 4 9 $9 4866 5o42 55o5 62 7 5 7 3 7 i 88i3 0625 282 9 545o 5 7 4 6288 52OO 538o 5849 6624 77 25 9174 O 99 2 32o3 583i 56 5 56 1 8 5534 5 7 i 9 6192 6 97 3 8080 9 534 i35 9 35 77 6212 55 6 5 9 48 5868 6o58 6536 7 322 8434 9 8 9 5 I 7 26 3 9 5o 65 9 3 54 7 6278 6202 63 97 6880 7 6 7 i 8789 4 9 o256 20 9 3 43 2 4 6 97 4 53 8 6608 6536 6 7 36 -7224 8020 9 i44 0617 2460 46 9 8 7 355 52 9 6 9 38 6871 7 o 7 5 7 568 836 9 9499 o 97 8 2828 5o 7 2 7736 5i 10 367268 38 7 2o5 407414 42 7 9I2 448 7 i 9 46 9 854 4 9 i33 9 5i3i 9 5 535446 558ii8 5o li 7 5 9 8 7 54o 77 53 8 2 56 9068 47O2O 9 I 7 OO 3563 582i 84 99 49 12 7928 7874 8092 8601 9 4i8 o564 2061 3 9 3o 6i 9 5 8881 48 i3 8259 8209 8432 8 9 45 97 68 O 9 2O 2422 42 9 8 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7862 0776 l l 44 8534 8622 8 99 3 9 663 o654 1984 36 7 7 5 7 54 8240 1161 16 45 8866 8960 9 335 44oou 1006 2343 4o4i 6i25 86i 9 i547 i5 46 9199 9 2 97 9 6 77 o358 i35 9 2 7 OI 44o6 64 97 8 997 i 9 33 i4 47 9 532 9 634 420019 o 7 o5 1712 3o6o 477i 6868 9 3 7 6 23l 9 i3 48 9864 997i o36i io53 2o65 34i 9 5i36 7240 97 55 2 7 o5 12 49 380197 4oo3o9 070^ i4oo 2418 3778 55o2 7612 55oi34! 3o 9 2 II 5o 3?.o53o 4oo64G 421046 441748 462771 484i3 7 5o586 7 527 9 84 55o5i3 673478 IO 5i o863 0984 i38 9 20 9 5 3124 44 9 6 6232 8356 o8 9 2 3865 g 52 1196 1322 i 7 3i 2443 3478 4855 65 9 8 8728 I2 7 I 4252 3 53 i53o 1660 20 7 4 27 9 I 383i 5 2 i4 6 9 63 9 ioo i65o 4638 7 54 i863 I 997 24l7 3i3 9 4i85 5574 7 32 9 9473 2030 6026 6 55 2196 2335 2759 348 7 4538 5 9 33 7 6 9 5 9 845 24o 9 54i3 5 56 253o 26 7 3 3102 3835 48 9 2 62 9 3 8061 53o2i8 2 7 8 9 58oo 4 67 2863 3on 3445 4i83 5246 6653 842 7 o5 9 i 3i6 9 6187 3 58 3l 97 335o 3788 4532 56ooj 7 oK 8793 o 9 6^ 354 9 65 7 5 2 69 353o 3688 4i32 488o 5 9 54| 7373 9 i5 9 i336 3 9 2 9 6962 I 69 68 67 66 65 64 63 62 61 60 d Natural Co-tangents. M s i . 1 P. P. 5 53 to I". 5 '" 5.6o 5.68 5. 7 6 5.85 5. 9 5 6.o5 [ 6.16 6.28 6.4o 1 122 NATURAL SINES. n 30' | 31 32 33 34 35 36 37 38 39 o 5ooooo 5i5o38 52 99 i 9 544639 55 9 i 9 3 573576 58 77 85 6oi8i5 6i566i 62 9 320 60 I O252 5287 53oi66 4883 9434 38i5 8021 2o4 7 58 9 i 9 546 5 9 2 o5o4 5537 o4i2 5l2 7 9 6 7 5 4o53 8256 2280 6120 977 2 58 3 0706 5 7 86 o65 9 53 7 i 99 i6 4291 84 9 i 25l2 634 9 999 8 57 4 1007 6o35 0906 56i5 56oi5 7 452 9 8 72 6 2 7 44 65 7 8 630224 56 5| 1269 6284 rx5a 5858 o3 9 8 4767 8 9 6i 2 97 6 6807 o45o 55 6 i5n 6533 i3 99 6102 o63 9 5oo5 9 i 9 6 3208 7036 o6 7 6 54 7 1762 6782 i645 6346 0880 5243 9 43i 344o 7265 O 9 O2 53 8 20l4 7o3i 1891 658 9 II2I 548r 9 666 36 7 2 74 9 4 II2 7 52 9 2266 7280 2i38 6833 i36i 5 7 i 9 99 oi 3 9 o4 7722 i353 5i 10 5o25i7 517529 532384 547076 56i6o2 5 7 5 9 5 7 5 9 oi36 6o4i36 6i7 9 5i 63i5 7 8 5o ii 2769 7778 263o 7320 i843 6i 9 5 o3 7 i 436 7 8180 1804 49 12 3020 8027 2876 7563 2o83 6432 0606 4599 84o8 2O2 9 48 13 3271 8276 3l22 7807 2324 6670 0840 483i 863 7 2255 47 i4 3523 8525 3368 8o5o 2564 6 9 o8 io 7 5 5o62 8865 2480 46 i5 3 77 4 8 77 3 36i5 82 9 3 28o5 7i45 i3io 5294 9 o 9 4 2 7 o5 45 16 4o25 9022 386i 8536 3o45 7383 1 544 55 2 6 9322 2 9 3i 44 i? 4276 9271 4io6 8780 3286 7620 i 779 5 7 5 7 9 55i 3i56 43 18 4528 9 5i 9 4352 9023 35 2 6 7 858 20l3 5 9 88 9779 338i 42 '9 4779 9768 45 9 8 9266 3766 8o 9 5 2248 6220 620007 36o6 4i 20 V>5o3o 520016 534844 54 9 5o 9 564007 578332 5 9 2482 6o645i 620235 63383i 4o 21 528i 0265 Sogo 97 5a 4247 85 7 o 2716 6682 o464 4o56 3 9 22 5532 o5i3 5335 999 5 4487 8807 2 9 5i 6914 o6 9 2 4281 38 23 5 7 83 0761 558i 55o238 4727 9 o44 3i85 7i45 O 9 2O 45o6 37 24 6o34 IOIO 582 7 o48i 4 9 6 7 9 28l 34i 9 7 3 7 6 u48 473i 36 25 6285 1258 6072 0724 5207 9 5i8 3653 7 6o 7 i3 7 6 4 9 55 35 26 6535 i5o6 63 1 8 o 9 66 5447 97 55 388 7 7 838 1604 5i8o 34 2 7 6786 1754 6563 I20 9 568 7 999 2 4l2I 8069 i83i 54o5 33 28 7o3 7 2OO2 6809 i45 2 5927 58o22 9 4355 83oo 2o5 9 5629 32 29 7288 225l 7 o54 i6 9 4 6166 o466 458 9 853i 228 7 5854 3r 3o 5o 7 538 52249 9 537300 55i 9 3 7 5664o6 580703 5 9 4823 6o8 7 6i 622^15 636078 3o 3i 7789 2747 7545 2180 6646 o 9 4o 5o5 7 8 99 2 2 7 42 63o3 29 32 8o4o 2 99 5 779 2422 6886 1176 52 9 o 9 223 2 97 652 7 28 33 8290 3242 8o35 2664 7125 i4i3 5524 9 454 3i 97 6 7 5i 27 34 854: 3490 8281 2907 7365 i65o 5 7 58 9 684 3425 6976 26 35 8791 3 7 38 8526 3i49 7604 1886 5 99 i 99 i5 365 2 7200 25 36 9041 3 9 86 8771 33 9 2 7844 2123 6225 6ioi45 388o 7424 24 3 7 9292 4234 9016 3634 8o83 235 9 6458 o3 7 6 4io 7 7 648 23 38 9542 448 1 9261 38 7 6 8323 25 9 6 66 9 2 0606 4334 7872 22 3 9 9792 4729 95o6 4n8 8562 2832 6 9 25 o836 456i 8096 21 4o 5xoo43 524 9 77 53 97 5i 55436o 5688oi 583o6 9 5 97 i5 9 6no6 7 624 7 8 9 638320 20 4i 0293 5224 999 6 4602 9 o4o 33o5 7 3 9 2 i2 97 5oi6 8544 1 9 42 o543 5472 540240 4844 9 28o 354i 7625 l52 7 5243 8768 18 43 o 79 3 5 7 i 9 o485 5o86 9 5l 9 3777 7858 i 7 5 7 54 7 o 8 99 2 '7 U io43 5 9 6 7 0730 5328 97 58 4oi4 8o 9 2 1987 56 97 9 2l5 16 45 1293 6214 o 9 74 5570 9997 425o 8325 2217 5 9 23 9 43 9 i5 46 i543 646 1 I2I 9 58i2 570236 4486 8558 2447 6i5o 9 663 i4 47 I 79 3 6709 i464 6o54 0475 4 7 22 8 79 i 2677 63 77 9 886 i3 48 2o43 6 9 56 1708 6296 0714 4 9 58 0.024 2907 66o4 64ono 12 49 2293 7203 I 9 53 653 7 O 9 52 5i 9 4 9 256 3i37 683o o333 II 5o 5i 2 543 527450 542197 556 779 57ii 9 i 58542 9 5 99 48 9 6i3367 62 7 o5 7 64o557 10 5i 2792 7 6 97 2442 7021 i43o 5665 97 22 35 9 6 7284 0780 9 52 3o42 7944 2686 7262 1669 5 9 oi 99 55 3826 7 5io ioc3 8 53 3292 8191 2g3o 75o4 1907 6i3 7 600188 4o56 7737 1226 7 54 354i 8438 3i 7 4 77 45 2l46 63 7 2 O42O 4285 7963 i45o 6 55 3 79 i 8685 34i 9 79 8 7 2384 6608 o653 45:5 8189 i6 7 3 5 5C 4o4o 8932 3663 8228 2623 6844 o885 4744 84i6 i8 9 6 4 5? 4290 9179 3 9 o 7 8469 2861 779 1118 4 9 74 8642 2119 3 58 453 9 9426 4i5i 8710 3ioo 7 3i4 i35o 5 2 o3 8868 2342 2 5 9 4789 9 6 7 3 43 9 5 8 9 5 2 3338 7 55o 1 583 5432 9 o 9 4 2 565 I 59 58 U 57 56 55 54 53 52 61 50 a Natural Co-sines. S p P !ol>' 8 4.i3 4.o 9 4-o4 4.00 3. 9 5 3. 9 o 3.85 3.8o 3. 7 4 1 i\ A T u u A L TANGENT 123 !J 30 31 32 33 34 35 36 37 38 39 1 5^ 7 35o 600861 624869 649408 674509 700208 726543 753554 781286 8o 9 784 Co I 7738 1257 5 27 4 9821 4932 o64i 6987 4oio 1754 810266 5 9 2 8126 i653 56 79 650235 5355 1075 7432 446 7 2223 0748 58 3 85i4 2049 6o83 0649 5 779 i5o 9 7877 4 9 23 2692 1230 5? 4 8903 2445 6488 io63 6203 i 9 43 8322 538o 3i6i 1712 56 5 9291 2842 6894 i477 6627 2377 8767 583 7 363i 2I 9 5 55 6 9680 3239 7299 1892 7o5i 2812 9213 6294 4ioo 2678 54 7 58oo68 3635 774 23o6 7 4 7 5 3246 9 658 6751 4570 3i6i 53 8 o45 7 4o3 2 8110 2721 7900 368i 73oio4 720 9 5o4o 3644 52 9 o846 4429 85i6 3i36 83 2 4 4n6 o55o 7667 55io 4128 5i JO 58i235 604827 628921 65355i 678749 7o455i 73o 99 6 758i25 785981 814612 5o rr i6 2 5 5224 9 32 7 3 9 66 9174 4987 J443 8583 645 1 5o 9 6 49 12 2Ol4 5622 97 34 4382 9 5 99 5422 88 9 9 o4i 6922 558o 48 i3 24o3 6019 63oi4o 4797 680025 5858 2336 9 5oo 7 3 9 4 6o65 47 i4 2793 6417 o546 52i3 o45o 62 9 4 2 7 83 99 5 9 7 865 654 9 46 i5 3i83 68i5 0953 562 9 0876 6730 3 2 3o 760418 8336 ?o34 45 16 35y3 7213 i36o 6o45 1302 7166 3678 0877 8808 7 5l 9 44 17 3 9 63 7611 1767 646 1 1728 7603 4i25 i336 9 28o 8oo5 43 18 4353 8010 2174 6877 2i54 8o3 9 45 7 3 i7 9 6 97 5a 84 9 i 42 1 9 4743 84o8 258i 72 9 4 258o 8476 5O2I 2256 790225 8 97 6 4i 2P 585i34 608807 632988 657710 683007 7o8 9 i3 73546 9 762716 790697 8i 9 463 4o 21 5524 9205 33 9 6 8127 3433 9 35o 5 9 i7 3i 7 6 1170 9 9 4 9 3 9 22 SgiS 9604 38o4 8544 386o 9788 6366 3636 i643 820435 38 23 63o6 6iooo3 4ai i 8 9 6i 4287 710225 68i5 4o 9 7 2117 O 9 22 37 24 6697 o4o3 4619 9 3 79 47i4 o663 7264 4558 2590 i4o 9 36 25 7088 0802 5027 9796 5i4a I 101 77 i3 5oi 9 3o64 i8 97 35 26 7479 I2OI 5436 660214 5569 i53 9 8162 548o 3538 2384 34 27 7870 1601 5844 o63i 5 997 1977 86n 5 9 4i 4012 2872 33 28 8262 2001 6253 1049 6425 2416 9061 64o3 4486 336o 32 29 8653 24OI 6661 i46 7 6853 2854 95n 6865 4 9 6i 3848 Si 3o 589045 612801 637070 661886 687281 713293 7 3 99 6i 767327 7 9 5436 824336 3o 3.1 9 43 7 3201 7479 23o4 7709 3732 74o4n 7789 5 9 u 48 2 5 29 32 9829 36oi 7888 2723 8i38 4171 0862 8252 6386 53i4 28 33 5 9 022I 4OO2 8298 3i4i 856 7 46n l3l2 8 7 i4 6862 58o3 2 7 34 o6i3 4402 8707 356o 8 99 5 5o5o I 7 63 9 i77 7337 62 9 2 26 35 1006 48o3 9117 3 979 9425 5490 22l4 9640 7 8i3 6782 25 36 i3 9 8 52o4 9527 43 9 8 9 854 SgSo 2666 770104 8290 7272 24 37 I7 9 i 56o5 9937 48i8 690283 63 7 o 3n 7 0567 8766 7762 23 38 2184 6006 64o347 5237 0713 6810 356 9 io3i 9 242 8252 22 3 9 2577 64o8 o 7 5 7 565 7 n43 725o 4O2O i4 9 5 97 I 9 8 7 43 21 4o 5 9 2 9 7o 616809 641167 666077 691572 717691 744472 77i 9 5 9 800196 82 9 234 2O 4i 3363 7211 i5 7 8 6497 2003 8:32 4 9 25 2423 0674 9 725 1 9 42 3 7 5 7 7 6i3 1989 6917 2433 85 7 3 53 77 2888 u5i 83o2i6 18 43 4i5o 8oi5 2399 7 33 7 2863 9014 583o 3353 i62 9 0707 I 7 44 4544 8417 2810 7758 3294 9 455 6282 38i8 2107 1199 16 45 4 9 3 7 8819 3222 8179 3725 9897 6735 4283 2585 1691 i5 6 533i 9221 3633 85 99 4i56 720339 7 i8 9 4748 3o63 2i83 i4 4 7 5 72 5 9624 4o44 9020 458 7 0781 7642 52i4 3542 2676 i3 48 6 120 620026 4456 9442 5oi8 1223 8o 9 6 568o 4O2I 3169 12 49 65i4 0429 4868 9 863 545o i665 854 9 6 1 46 45oo 3662 II 5o 596908 620832 645280 670284 695881 722108 749003 776612 8o4 9 7 9 834x55 IO 5i 73o3 12351 5692 0706 63i3 2 55o 9 458| 7078 5458 4648 o 52 7698 i638J 6io4 1128 6 7 45 2 99 3 9912 7545 5 9 38 5x42 g 53 8093 2042] 65i6 i55o 7177 3436 75o366| 8012 64i8 5636 7 54 8488 2445 6929 1972 7610 38 79 082 i 1 8479 68 9 8 6i3o 6 55 8883 2849 7342 2 3 9 4 8042 43 2 3 1276 8 9 46 7379 6624 5 56 9278 3253 77 55 2817 8475 4 7 66 I 7 3i 9 4i4 7 85 9 7119 4 5 7 ^ 9674 365 7 8168 324o 8908 52IO 2187 9881 834o 7614 3 58 600069 4o6i 858i 3662 9 34i 5654 2642 780349 8821 8109 2 5 9 o465 4465 8994 4o85 9774 6o 9 8 3098 0817 9 3o3 86o4 I 59 58 57 56 55 54 53 52 51 50 J Natural Co-tangents. 1 #6.53 | 6.67 6.82 6.97 7-i4 7 .3i 7.5o 7.70 7.92 s.i4 124 NAIURAI. SINES. d i 40 41 42 43 44 45 46 47 48 49 ) 642788 656o5 9 66 9 i3i 68i 99 8 C 9 4658 7 o 7 io 7 7i 9 34o 73i354 7 43i45 7 54 7 io 60 I 3oio 6279 9 34 7 2211 4868 7 3l2 9 542 i55 2 333 9 4 9 oo 5 9 2 3233 64 9 8 9 563 2424 5o 77 7 5i8 9 744 1700 3534 5o 9 i 58 3 3456 6717 9779 2636 5 2 86 77 24 99 46 i 9 4 9 3728 5282 57 6 36 79 6 9 3 7 999 5 284 9 54 9 5 7929 720148 2147 3 9 23 54 7 2 56 5 3901 7i56 6 7 O2II 3o6i 5 7 o4 8i34 o34 9 2345 4117 5663 55 6 4124 7 3 7 5 0427 32 7 4 5 9 i3 834o o55i 2543 43i2 5853 54 7 4346 7 5 9 4 0642 3486 6122 8545 o 7 53 2741 45o6 6o44!53 8 456 9 7814 o858 36 9 8 633o 8 7 5o o 9 54 2 9 3 9 4700 6234 52 9 4791 8o33 1074 3 9 n 653 9 8 9 56 n56 3i3 7 48 9 4 6425 5i 10 645oi3 658 2 52 6 7 i289 684i23 6 9 6 7 48 7 o 9 i6i 721357 733334 745o88 7 566i5 5o ii 5236 847i i5o5 .4335 6 9 5 7 9 366 i55 9 3532 5282 68o5 49 12 5458 868 9 1721 454 7 7i65 9 5 7 i 1760 373o 5476 6 99 5 48 i3 568o 8908 i 9 36 47 5 9 7 3 7 4 9776 I 9 62 3927 56 7 o 7 i85 47 i4 59O2 9 I2 7 2l5l 4 97 i 7 58 2 99 8i 2i63 4i25 5864 7 3 7 5 46 i5 6124 9 346 2367 5i83 779 710185 2364 4323 6o5 7 7 565 45 16 6346 9 565 2582 53 9 5 7999 o3 9 o 2565 4520 625i 77 55 44 17 6568 9783 2797 56o 7 820 7 o5 9 5 2766 4717 6445 7 9 45 43 18 6790 660002 3oi3 58i8 84i5 <>799 2 9 6 7 4 9 i5 6638 8i34 42 *9 7012 O22O 3228 6o3o 8623 ioo4 3i68 5lI2 6832 83 2 4!4i 20 647233 66o43 9 6 7 3443 686242 C 9 883 2 711209 7 2336 9 7 353o 9 747025 7 585i4 4o 21 7455 o65 7 3658 6453 9 o4o i4i3 35 7 o 55o6 7218 8 7 o3 3 9 22 7677 0875 3873 6665 9 248 1617 3 77 i 5 7 o3 7412 88 9 3 38 23 7898 109.4 4o88 6876 9 455 1822 3 9?I 5 9 oo 7605 9082 37 24 8120 l3l2 4302 7088 9 663 2026 4172 6o 9 7 7798 9 2 7 I 36 25 834i i53o 45i 7 7299 9 8 7 i 2230 43 7 2 62 9 4 799 1 9 46i 35 26 8563 i?48 4732 75io 7ooo7 9 2434 4573 64 9 i 8184 9 65o 34 27 8 7 84 1966 4 9 47 7721 0287 2639 4773 6687 83 7 7 9 83 9 33 28 9006 2184 5i6i 79 32 o4 9 4 2843 4 9 74 6884 85 7 o 760028 32 29 9227 2402 53 7 6 8i44 0702 3o47 5i 7 4 7081 8 7 63 0217 3i 3o! 649448 662620 6755 9 o 688355 7oo 9 o 9 7i325o 72 53 7 4 737277 7 48 9 56 760406 3o 3i 9669 2838 58o5 8566 1117 3454 55 7 5 7 4 7 4 9 i48 o5 9 5 2O 32 9890 3o56 6oi 9 8776 1 324 3658 5 77 5 7670 9 34i 0784 2o 33 65om 3273 6233 8087 i53i 3862 5 97 5 7867 9 534 ?972 27 34 o332 349i 6448 9 i 9 8 I 7 3 9 4o66 6175 8o63 97 26 1161 26 35 o553 3 7 o 9 6662 9 4o 9 i 9 46 4269 63 7 5 8 2 5 9 99 i 9 i35o 25 36 0774 3926 6876 9 620 2i53 4473 65 7 5 8455 7 5oi i i i538 24 3? o 99 5 4i44 7O 9 o 9 83o 236o 46 7 6 6 77 5 865i o3o3 1727 23 38 1216 436i 73o4 6 9 oo4i 2567 488o 6 97 4 8848 o4 9 6 I 9 i5 22 3 9 i437 45 79 7 5i8 O25l 2774 5o83 7174 9 o43 0688 2104 21 4o 65i65 7 664796 6 777 32 6 9 o462 702981 7 i5286 7 2 7 3 7 4 73 9 23 9 750880 7622 9 2 20 4i 1878 5oi3 7946 0672 3i88 5490 7 5 7 3 9 435 1072 2480 *9 42 2098 523o 8160 0882 SSgS 5693 777 3 9 63i 1264 2668 18 43 2319 5448 83 7 3 io 9 3 36oi 58 9 6 797 2 9 82 7 i456 2856 i? 44 2 53 9 5665 858 7 i3o3 38o8 6099 8172 740023 1 648 3o44 16 45 2760 5882 8801 i5i3 4oi5 6302 8371 0218 1840 3232 i5 46 2980 6099 9 oi4 1723 4221 65o5 85 7 o o4i4 2032 3420 i4 47 32OO 63i6 9 228 i 9 33 4428 6 7 o8 8 7 6 9 o6o 9 2223 36o8 i3 48 342i 6532 9 44i 2i43 4634 691 1 8 9 6 9 o8o5 24i5 3 79 6 12 49 364i 6749 9 655 2353 484 1 7 n3 9 i68 IOOO 2606 3 9 84 II 5o 65386i 666066 67 9 868 6 9 2563 7o5o47 7 i 7 3i6 7 2 9 367 74n 9 5 7 52 79 8 764171 10 5i 4o8i 7 i83 680081 2 77 3 5253 7 5l 9 9 566 i3 9 i 2 9 8 9 435 9 9 52 43oi 7399 02 9 5 2 9 83 545 9 7721 9765 1 586 3i8i 4547 8 53 452i 7616 o5oS 3l 9 2 5665 7924 9 9 63 1781 33 7 2 4734 7 54 4?4i 7 833 0721 3402 58 7 2 8126 7 3oi62 I 9 76 3563 4921 6 55 4961 8o4 9 o 9 34 36n 6078 832 9 > o36i 2171 3 7 55 5 1 09 5 56 5i8o 8 2 65 ii47 382i 6284 853i o56o 2366 3 9 46 5296 4 5 7 54oo 8482 i36o 4o3o 648 9 8733 0758 256i 4i3 7 5483 3 58 5620 86 9 8 i5 7 3 424o 66 9 5 8 9 36 c 9 5 7 2 7 55 4328 5670 2 5 9 583 9 8 9 i4 i 7 86 444 9 6901 9 i38 n55 2 9 5o 45i 9 585 7 I 49 48 47 46 45 44 43 42 41 40 5 Natural Co-sines. 3* 3.69) 3.63 | 3.5 7 3.5a 3.46 3.4o 3.34 3.2 7 3.21 3.i5 i J NATURAL TANGENTS 125 4 i 40 41 ' 42 43 44 45 46 47 48 49 1 83 9 ioo 869287 9 oo4o4, 9 325i5 9 6568 9 I.OOOOO i.o3553 1.07237 i. 11061 I.i5o37 60 I 9 5 9 5 9798 o 9 3i) 3o5 9 625i oo58 36i3 7299 1126 5io4 5 9 2 840092 870309 i458 36o3 68i4 0116 36 7 4 7362 1191 5172 58 3 o588 0820 i 9 85 4i48 7377 oi 7 5 3734' 7 425 1256 524o 57 4 1084 i332 2 5i3 46 9 3 79 4o 0233 3 79 4 748 7 l32I 53o8 56 5 i58i i843 3o4i 5238 85o4 O2 9 I 3855 755o i38 7 53 7 5 55 6 20 7 8 2356 356 9 5 7 83 9 o67 o35o 3 9 i5 7613 i452 5443 54 7 25 7 5 2868 4o 9 8 6329 9 632 o4o8 3976 7676 i5i 7 55n 53 8 3o 7 3 338i 4627 68 7 5 9 7oi 9 6 o46 7 4o36 7738 l5&2 55 7 9 52 9 35 7 i 38 9 4 5i56 7422 0761 o5 2 5 4097 7801 1 648 5647 5i 10 844069 874407 9 o5685 937968 9 7i326 i.oo583 I.o4i58 1.07864 1.11713 i .:5 7 i5 5o ii 456 7 4920 62i5 85i5 1892 0642 4218 7927 i 77 8 5 7 83 49 12 5o66 5434 6745 9063 2458 0701 4279 799 1 844 585i 48 i3 5564 5 9 48 7275 9610 3o24 0759 434o 8o53 J 99 5 9 i 9 47 i4 6o63 6462 7 8o5 94oi58 3590 0818 44oi 8116 i 97 5 5987 46 i5 6562 6976 8336 0706 4i5 7 0876 446 1 8179 2041 6o56 45 16 7062 7491 8867 1255 4724 o 9 35 4522 8243 2106 6124 44 J 7 7562 8006 9 3 9 8 i8o3 5291 o 99 4 4583 83o6 2I 7 2 6192 43 18 8062 852i 99 3o 2352 585 9 io53 4644 8369 2238 6261 42 i 9 8562 9 3 7 2902 6427 III2 47o5 8432 2 3o3 6329 4x 20 849062 879553 9 io 99 4 9 4345i 97 6 99 6 I.OII70 1.04766 1.08496 i.i236 9 i.i63 9 8 4o 21 9563 880069 i526 4ooi 7564 I22 9 4827 855 9 2435 6466 3 9 22 85oo64 o585 2o5 9 4552 8i33 1288 4888 8622 25oi 6535 38 23 o565 1102 25 9 2 5 1 02 8 7 o3 1 347 4949 8686 256 7 66o3 37 24 1067 1619 3i25 5653 9 2 7 2 i4o6 5oio 8 7 4 9 2633 6672 36 25 i568 2i36 365 9 8204 i465 5072 88i3 26 99 6 7 4i 35 26 2070 2653 4193 6 7 56 9 8o4i3 i524 5i33 8876 2 7 65 6809 34 27 25 7 3 3i 7 i 4727 7 3o 7 o 9 83 i583 5194 8 9 4o 283i 68 7 8 33 28 3o 7 5 368 9 5261 7 85 9 i554 1642 5255 9 oo3 28 97 6947 32 29 3578 4207 5796 8412 2126 1702 53i7 9 o67 2 9 63 7016 3i 3o 854o8i 884725 9i633i 948965 9 826 9 7 1.01761 1.05378 I.o 9 i3i 1.13029 1.17085 3o 3i 4584 5244 6866 9 5i8 326 9 1820 543 9 9 i 9 5 3096 7i54 2 9 32 5o8 7 5 7 63 7402 95oo 7 i 3842 1879 55oi 9 258 3i62 7223 28 33 55 9 i 6282 7938 0624 44i4 I 9 3 9 5562 9 322 3228 7292 27 34 6o 9 5 6802 8474 n 7 8 4 9 8 7 1998 5624 9 386 32 9 5 7 36i 26 35 65 99 7321 9 oio i 7 33 556o 2057 5685 9 45o 336i 743o 25 36 7 io4 7842 9547 2287 6i34 2117 5?47 9 5i4 3428 75oo 24 3 7 j -7608 8362 920084 2842 6708 2176 58o 9 9 5 7 8 3494 756 9 23 38 8n3 8882 0621 33 97 7282 2236 58 7 o 9 642 356i 7638 22 3 9 86i 9 9 4o3 1159 7 85 7 2295 5 9 3 2 97 6 362 7 7708 21 4o 859124 889924 921697 9 545o8 9 8843 2 1.02355 i.o5 99 4 i.i36 9 4 i.i7 777 2O 4i 9 63o 890446 2235 5o64 9 oo 7 24i4 6o56 ' 9 834 3 7 6i 7 846 '9 42 86oi36 0967, 2773 562i 9 582 2474 6117 9 8 99 3828 7916 18 43 0642 1489! 33i2 6177 99 oi58 2533 6179 o 9 63 38 9 4 79 86 17 44 n48 2OI2 385i 6 7 34 o 7 35 2 5 9 3 6241 1.10027 3 9 6i 8o55 16 45 i655 2534 43 9 o 7292 i3n 2653 63o3 oo 9 i 4028 8i25 i5 46 2162 3o57 493o 7 84 9 1888 2713 6365 oi56 4o 9 5 8194 i4 47 2 66 9 358o 5470 8407 2465 2772 6427 O22O 4162 8264 i3 48 3i 77 4io3J 6oioj 8966 3o43 2832 6489 0285 422 9 8334 12 49 3685 4627 655i 9 524 362i 2892 655i o34 9 42 9 6 84o4 II 5o 864i 9 3 8 9 5i5i 927091 9 6oo83 994i99 I O2 9 52 i.o66i3 I.io4i4 1. 14363 i.i8474 10 5i 4 7 oi 5675j 7632 0642 4778 3012 6676 0478 443o 8544 9 52 5209 6199 8174 I2O2 535 7 3072 6 7 38 o543 44 9 8 86i4 8 53 5 7 i8 6724! 8715 1761 5936 3i3 2 6800 0607 4565 8684 7 54 6227 7249 9257 2322 65:5 3l 9 2 6862 0672 4632 8 7 54 6 55 6736 7774 9800 2882 79 5 3252 6 9 25 o 7 3 7 46 99 8824 5 56 7246 8299 93o342 3443 7676 33i2 6087 0802 4 7 6 7 88 9 4 4 57 7756 88 2 5 o885 4oo^ 8256 33 7 2 7o4 9 o86 7 4834 8964 3 58 8266 9 35i 1428 4565 883 7 3433 7112 o 9 3i 4 9 02 9 o35 2 ! 5 9 8776 9877 1971 5l2 7 9 4i8 34 9 3 7 i 7 4 0996 4 9 6 9 9 io5 i 1 49 48 47 46 45 44 43 42 41 -5 1 Natural Co-tangents. PT> ft o o to 1" 8.64 8.92 0.21 9 .53 -99 i .02 i .06 I . IO i.i5 1 L :. 126 A T U R A L, .4 a 50 C 51 52 53 54 55 56 57 5S ' 59 o 766044 777146 788011 79 8636 8o 9 oi 7 8i 9 i52 82 9 o38 838671 848o48 85 7 i6 7 60 i 6 2 3i 7 32 9 8i 9 o 8811 9 i88 9 3i 9 9 200 8 29 8202 7 3i 7 5 9 2 64i8 75i2 836 9 8 9 85 9 35 9 9 486 9 363 8987 8356 7 46 7 58 3 66o5 7695 8548 9 i6o 9 53o 9 652 9 525 9146 85io 7616 5? 4 6792 7878 8 7 2 7 9 335 97 oo 9 8i 9 9 688 9 3o4 8664 77 66 56 5 6979; 8060 8 9 o5 9 5io 9 8 7 i 99 85 9 85o 9462 8818 79 i5 55 6 7i65 8243 9 o84 9 685 810042 820152 83ooi2 0.620 8 97 2 8o65 54 7 7 35 2 8426 9263 9 85 9 O2I2 o3i8 oi 7 4 9778 9 I25 8214 53 8 7538 86o8| 9 44i 8ooo34 o383 o485 o33 7 99 36 9279 8364|5a 9 7725 8791 9 620 0208 o553 o65i' o4 99 84oo 9 4 9 433 85i3l5i 10 767911 77 8 97 3 7 8 979 8 8oo383 8io 7 23 820817*830661 84oa5i 84 9 586 858662 5o ii 8097 9 i56 9977 o55 7 o8 9 4 o 9 83 0823 o4o 9 9739 88zi 49 12 8284 9 338 79 oi55 o 7 3i 1064 n4 9 o 9 84 o56 7 9 8 9 3 8 9 6o 48 i3 8470 9 52O o333 o 9 o6 1234 i3i5 n46 0-724 85oo46 9 io 9 47 i4 8656 9 702 o5n 1080 i4o4 i48i i3o8 0882 oi 99 9 258 46 16 8842 9 884 o6 9 o 1254 i5 7 4 i64 7 i4 7 o io3 9 o352 9 4o6 45 16 9028 780067 0868 1428 i 7 44 ! i8i3 i63i n 9 6 o5o5 9 555 44 I 7 9214 024 9 1046 1602 i 9 i4 I 97 8 i 79 3 i354 o658 97 o4 43 18 9400 o43o 1224 1776 2084 2i44 i 9 54 i5n 0811 9 S52 42 J 9 9 585 0612 i4oi i 9 4 9 2253 23lO 2Il5 i668j o 9 64 860001 4i 20 769771 78o7 9 4 79 l5 79 802123 812423 8224 7 5 8322 77 84i825 85ni7 86oi4 9 4o 21 99 5 7 o 9 76 i 7 5 7 22 97 25 9 2 264i; 2438 I 9 82 I26 9 02 97 39 22 770142 1167 i 9 35 24 7 2762 2806 2 5 99 2i3 9 1422 o446 38 23 o328 i33 9 21 12 2644 2 9 3l 2 97 I 2 7 6o 22 9 6 i5 7 5 o5 9 4 3 7 24 o5i3 l520 22 9 2817 3ioi 3i36 2 9 2I 2452 1727 0742 36 25 0699 1702 246 7 2 99 I 32 7 33o2 3o82 26o 9 1879 o8 9 o 35 26 o884 i883 2644 3i64 343 9 346 7 3243 2-766 2032 io38 34 27 1069 2o65 2822 333 7 36o8 3632 34o4 2 9 22 2184 1186 33 28 I254 ! 2246 2999 35n 3 77 8 3797 3565 3o 79 2336 i334 32 29 i44o 2427 3i 7 6 3684 3 9 4 7 3 9 6i 3 7 25 3235 2488 1481, 3i 3o 771625 782608 7 9 3353 8o385 7 8i4n6 824126 833886 8433 9 i 85 2 64o 861620' 3o 3i 1810 2 7 8 9 353o 4o3o 4284 42 9 I 4o46 3548 2 79 2 1777 2 9 32 i 99 5 2 9 70 3 7 o 7 4203 4453 4456 420 7 3 7 o4 2 9 44 I 9 24 28 33 2179 3i5i 3884 43 7 6 4622 4620 436 7 386o 3o 9 6 2072 27 34 2364 33321 4o6i 4548 4 79 i 4785 452 7 4oi6 3248 22I 9 26 35 2549 35i3 4238 4721 4 9 5 9 4 9 4 9 4688 4172 33 99 2366 25 36 2734 36 9 3 44i5 48 9 4 5128 5n3 4848 4328 355i 25i4 24 3? 2918 38 7 4 45 9 i 5o66 52 9 6 5 27 8 5oo8 4484 3702 2661 23 38 3 I0 3 4o55 4768 5 2 3 9 5465 5442 5i68 464o 3854 2808 22 3 9 3287 4235 4 9 44 54n 5633 56o6 5328 4 79 5 4oo5 2 9 55 21 4o 773472 784416 7 9 5i2i 8o5584 8i58oi 825 77 o 835488 844 9 5i 854i56 863io2 2O 4i 3656 45 9 6 52 9 7 5 7 56 5 9 6 9 5 9 34 5648 5io6 43o8 3 2 4 9 J 9 42 384o 4776 5473 5 9 28 6i38 6o 9 8 58o 7 5262 445 9 33 9 6 18 43 4024 4 9 5 7 565o 6100 63o6 6262 5 9 6 7 54i 7 4610 3542 i? 44 4209 5i37 5826 6273 64 7 4 6426 6127 55 7 3 4-761 368 9 16 45 43 9 3 53i 7 6002 6445 6642 65 9 o 6286 5728 4 9 I2 3836 i5 46 45 77 54 97 6178 6617 68o 9 6 7 53 6446 5883 5o63 3 9 82 i4 4? 4761 56 77 6354 6788 6 977 6 9 i 7 66o5 6o38 52i4 4128 i3 48 4944 585 7 653o 6 9 6o 7 i45 -7081 6 7 64 6i 9 3 5364 4275 12 49 5i 2 8 6o3 7 6706 7132 7 3i3' 7 244 6 9 24 6348 55i5 442i II 5o 77 53i2 7 862i 7 7 9 6882 807304 8i 7 48o 82 7 4o 7 83 7 o83 8465o3 855665 86456 7 :o 5i 5496 63 9 6 7 5 7 7 4 7 5 7 648 7 5 7 i 7242 6658 58i6 4 7 i3 9 52 56 79j 65 7 6 7233 7 64 7 7 8i5 7734 74oi 68:3 5 9 66 486o 8 53 5863| 6756 7408 7818 79 82 7 8 97 7 56o 6 9 6 7 6117 5oo6 7 54 6o46 6 9 35 7584 799 8i5o 8060 7719 7122 6267 5i5i 6 55 623o 7ii4 77 5 9 8161 83i 7 8223 7878 7 2 77 64i7 52 97 5 56 64i3 72 9 4 79 35 8333 84841 8386 8o36 743 1 656 7 5443 4 57 65g6 7473 8110 85o4 865i 854 9 8i 9 5 7 585 6718 558 9 3 58 6780 7652 8285 86 7 5 8818 8 7 I2 8354 77 4o 6868 5 7 34 a 5 9 6 9 63 7 83 2 846o 8846 8 9 85 88 7 5 85i2 78 9 4 7017 588o i 39 38 37 36 35 34 33 32 31 30 d Natural Co-sines. i #3.08 3.02 2. 9 5 2.88 2.8l 2. 7 5 2.68 2.60 2.53 a. 46 NATURAL TANGENTS. 127 d 50 51 52 53 54 55 56 57 58 59 r. 19175 1.23490 1.27994 1.32704 1.37638 1.42815 1.48256 1.53986 i.6oo33 1.66428 60 I 9246 3563 8071 2 7 85 7722 2903 8349 4o85 oi3 7 6538 5 9 2 93i6 3637 8i48 2865 7807 2992 8442 4i83 0241 664 7 58 3 9 38 7 3710 8225 2946 7891 3o8o 8536 4281 o345 6 7 5 7 57 4 9457 3 7 84 83o2 3026 7976 3169 8629 43 7 9 o44g 686 7 56 5 9528 3858 83 7 9 3107 8060 3258 8722 4478 o553 6 97 8 55 G 9-99 3 9 3i 8456, 3i8 7 8i45 3347 8816 45 7 6 o65 7 7088 54 7 9669 4oo5 8533 3268 8229 3436 8909 46 7 5 o 7 6i 7198 53 8 9?4o 4079 8610 3349 83i4 3525 9003 4774 o865 7 3o 9 52 9 9811 4i53 8687 343o 83 99 36i4 9097 48 7 3 o 97 o 7 4i 9 5i 10 1.19882 1.24227 1.28764 i.335n 1.38484 1.43703 1.49190 1.54972 i.6io 7 4 i.6 7 53o 5o ii 9 9 53 43oi 8842 3592 8568 3 79 2 9284 5071 "79 764i 4 9 12 1.20024 43 7 5 8919 36 7 3 8653 388i 9 3 7 8 5i 7 o 1283 77 52 48 i3| oog5 4449 8997 3754 8738 3 97 o 94 7 2 5269 1 388 7 863 47 i4 0166 45 2 3 9074 3835 8824 4o6o 9566 5368 1493 7974 46 i5 0237 45 97 9152 3916 8 9 o 9 4i4g 9661 546 7 i5 9 8 8o85 45 16 o3o8 4672 9229 3 99 8 8 99 4 423 9 97 55 556 7 I 7 o3 8196 44 l? 3 79 4746 9 3 7 4079 9079 4329 9849 5666 1808 83o8 43 18 o45 1 4820 9 385 4i6o 9i65 44i8 9944 5 7 66 1914 8419 42 *9 0622 48 9 5 9 463 4242 9250 45o8 i.5oo38 5866 2019 853i 4i 20 1.20593 1.24969 1.29541 1.34323 i.3 9 336 1.44598 i.5oi33 i.55 9 66 1.62125 1.68643 4o 21 o665 5o44 9618 44o5 9421 4688 0228 6o65 223O 8 7 54 3 9 22 0736 5n8 9696 448 7 95o 7 4778 O322 6i65 2336 8866 38 23 0808 5i 9 3 977 5 4568 9 5 9 3 4868 0417 6265 2442 8979 37 24 0879 5268 9 853 465o 9679 4958 o5i2 6366 2548 9091 36 25 ogS i 5343 99 3i 4 7 32 9764 5o49 0607 6466 2654 9203 35 26 1023 54i? i.Sooog 48i4 985o 5i3 9 0702 6566 2 7 6o 93i6 34 27 1094 5492 0087 4896 99 36 5229 0797 6667 2866 9428 33 28 1166 556 7 0166 4978 1.40022 5320 0893 6767 29-72 954i 32 29 1238 5642 0244 5o6o oio 9 54io 0988 6868 3o 79 9653 3i 3o i aiSio 1.25717 i.3o323 i.35i4a 1.40195 i.455oi i.5io84 1.56969 i.63i85 1.69766 3o 3i i382 5 79 2 64oi 5224 028: 5592 1179 7069 3292 9879 29 32 i454 586 7 o48o 53o7 0367 5682 1275 7170 33 9 8 9992 28 33 i526 5 9 43 o558 5389 o454 5 77 3 1370 7271 35o5 i. 7 oio6 27 34 i5 9 8 6018 0637 5472 o54o 5864 i466 7 3 7 2 36i2 0219 26 35 1670 6o 9 3 0716 5554 0627 5 9 55 i562 7474 3 7 i 9 o332 25 i 36 1742 6169 0795 5637 0714 6o46 i658 7 5 7 5 3826 o446 24 3? 1814 6244 0873 5 7 i 9 0800 6137 1754 7 6 7 6 3 9 34 o56o 23 38 1886 63i 9 0952 58o2 0887 6229 i85o 7778 4o4i o6 7 3 22 3 9 1959 63 9 5 io3i 5885 o 9 74 6320 1946 7879 4i48 o 7 8 7 21 4o I.22O3I 1.26471 i.Snio 1.36968 1.41061 i.464ii 1.52043 i.5 79 8i 1.64256 1.70901 20 4i 2IO4 6546 1190 6o5i n48 65o3 2i3g 8o83 4363 ioi5 J 9 42 2176 6622 1269 6i34 1235 65 9 5 2235 8i84 447i 1129 18 43 2249 6698 1 348 6217 1322 6686 2332 8286 45 79 1244 17 44 2321 6 77 4 1427 63oo i4o 9 6778 2429 8388 468 7 i358 16 45 23g4 6849 1507 6383 1497 6870 2525 8490 4795 i473 i5 46 2467 6925 i586 6466 i584 6962 2622 85 9 3 4903 i588 i4 4? 253 9 7001 1666 6549 1672 7o53 2719 86 9 5 Son 1702 i3 48 2612 7077 1745 6633 i 7 5 9 7146 2816 8797 5l2O 1817 12 49 2685 7i53 1825 6716 1 847 7238 2913 8900 5 2 a8 I 9 32 II 5o 1.22758 1.27230 1.31904 i.36Soo 1.41934 i.4733o i.53oio 1.59002 i.6533 7 1.72047 10 5i 283i 73o6 1984 6883 2O22 7422 3107 9io5 5445 2i63 9 52 2904 7 382 2064 6967 2IIO 75i4 32o5 9208 5554 2278 8 53 2977 7458 2144 7o5o 2198 7607 33o2 9 3ii 5663 2 3 9 3 7 54 3o5o 7 535 2224 7i34 2286 7699 34oo 94i4 5 77 2 25o 9 6 55 3:23 7611 23o4 7218 a3 7 4 7792 34 9 7 9 5i 7 588i 2625 5 56 3196 7688 2384 7302 2462 7 885 35 9 5 9620 5 99 o 2741 4 5? 3270 7764 2464 7386 255o 7977 36 9 3 9723 6o 99 2857 3 58 3343 784i 2 544 7470 2638 8070 3 79 i 98261 6209 2973 2 5 9 34i6 7917 2624 7554 2726 8i63 3888 9930 63i8 3089 I 39 38 37 36 35 34 33 32 31 300 L- Natural Co-tangents. P. P. to I". 1 ' 20 1.26 i.3i i.3 7 i.44 i.5i i.5 9 1.68 1.78 1.88 128 NATURAL SINES. j 60 | 61 62 63 64 65 66 67 68 | 69 866025 874620 882948 891007 898794 9 o63o8 9 i3545 9 2o5o5 0.27184! 9 3358o!6o 6171 4761 3o84 n3 9 8922 643i 3664 0618 7 2 9 3| 3685 5 9 3 63i6 4902 3221 1270 9049 6554 3 7 82 7 32 7 402 3 7 8 9 58 j 646i 5o42 3357 1402 9176 66 7 6 3 9 oo o846 7 5io 38 9 3 57 ; 6607 5i83 3493 i534 93o4 6799 4oi8 o 9 5 9 7619 3 997 56 1 6 7 5 2 5324 362 9 1666 943i 6 9 22 4i36 IO 7 2 7728 4ioi 55 6 6897 5465 3 7 66 1798 9 558 7044 4254 n85 7836 4204 54 7 7042 56o5 3902 1929 c;685 7166 43 7 2 i2 99 79 45 43o8 53 8 7187 5 7 46 4o38 2061 9812 7280. 44 9 o 1412 8o53 44i2 5 2 9 7 33i 5886 4174 2192 99 3 9 74*1 46o 7 i525 8161 45i5 5i 10 867476 876026 884309 892323 900065 9 o 7 533 9 i4 7 25 9 2i638 928270 9 346i 9 5o ii 7621 6167 4445 2455 0192 7 655 4842 i 7 5o 83 7 8 4 7 22 4 9 12 7765 63o 7 458i 2586 0319 7777 4 9 6o 1 863 8486 4826 48 i3 7910 6447 4717 2717 o445 7899 5o 77 i 97 6 8594 4 9 2 9 47 i4 8o54 658 7 4852 2848 0572 8021 5i 9 4 2088 8 7 02 5o32 46 i5 8199 6727 4988 2979 0698 8i43 53ii 22OI 8810 5i35 45 16 8343 6867 5i23 3no o8 2 5 8265 542 9 23i3 8917 5238 44 17 848 7 7006 5258 3 2 4i o 9 5i 838 7 5546 2426 9025 534i 43 18 8632 7i46 5394 33 7 i 1077 85o8 5663 2538 9 i33 5444 42 19 8776 7286 552 9 35o2 1203 863o 5 779 2 65o 9240 554 7 4i 20 868920 8 77 425 885664 893633 901329 9 o875i 9 i58 9 6 0,22762 929348 9 3565o 4o 21 9064 7 565 5 799 3763 i455 8872 6oi3 28 7 5| 9 455 5 7 52 3 9 22 9207 77o4 5 9 34 38 9 4 i58i 8 99 4 6i3o 2986 9 562 5855 38 23 935i 7844 6069 402^ 1707 9 n5 6246 3o 9 8 9 66 9 5 9 5 7 37 24 9495 79 83 6204 4i54 i833 9 236 6363 3210 9776 6060 36 9 63 9 8122 6338 4284 1958 9 35 7 6479 3322 9 884 6162 35 26 9782 8261 6473 44i5 2084 9 4 7 8 65 9 5 3434 999 6264 34 27 9926 84oo 6608 4545 2209 9 5 99 6 7 I2 3545 9 3oo 97 6366 33 28 870069 853 9 6742 46 7 5 2335 97 20 6828 365 7 0204 6468 32 29 O2 I 2 8678 6876 48o5 2460 9 84i 6 9 44 3 7 68 o3n 65 7 o 3i 3o 870356 878817 887011 894934 902585 9 o 99 6i 9 i 7 o6o 923880 9 3o4i8 9 366 7 2 3o 3i 0499 8 9 56 7i45 5o64 2710 9 ioo82 7 i 7 6 3 99 i o524 6 77 4 2 9 32 O642 9095 7279 5i94 2836 O2O2 7 2 9 2 4lO2 o63i 6876)28 33 0785 9 233 74i3 5323 2961 o323 7408 4213 o 7 3 7 6 977 27 34 0928 9 3 7 2 7548 5453 3o86 o443 7 5 2 3 4324 oS43 7079 26 35 1071 95io 7681 5582 3210 o563 7 63 9 4435 o 9 5o 7181 25 36 I2l4 9649 7 8i5 5712 3335 0684 77 55 4546 io56 7282 24 3y i35 7 9787 7949 584i 346o o8o4 7870 465 7 1162 7383 23 38 1499 9925 8o83 5970 3585 9 24 79 86 4 7 68 1268 7485 22 3 9 1642 88oo63 8217 6099 3709 io44 8101 48 7 8 i3 7 4 7 586 21 4o 871784 880201 88835o 896229 903834 9 iii64 9 i82i6 924989 9 3i48o 9 3 7 68 7 2O 4i 1927 oSSg 8484 6358 3 9 58 1284 833i 5099 i586 77 88 '9 42 2069 o477 8617 6486 4o83 i4o3 8446 52IO i6 9 i 7 88 9 18 43 2212 o6i5 8 7 5i 66i5 4207 i523 856i 5320 1797 799 l l 44 2354 0753 8884 6 7 44 433i i643 86 7 6 543o I 9 O2 8o 9 i 16 45 2496 0891 9017 6873 4455 1762 8 79 i 554i 2008 8i 9 i i5 46 2638 1028 9i5o 7001 45 79 1881 8 9 o6 565i 2Il3 82 9 2 i4 4? 2780 1166 9283 7i3o 4 7 o3 2OOI 9 O2I 6761 22I 9 83 9 3 i3 48 2922 i3o3 94i6 7258 48 2? 2120 9 i35 58 7 i 2324 84 9 3 12 49 3o64 i44i 9549 7 38 7 495i 223 9 9 25o 5980 242 9 85 9 3 II 5o 873206 88i5 7 8 889682 8 97 5i5 9o5o 7 5 9 I2358 9 i 9 364 926090 9 32534 9 386 9 4 IO 5i 334 7 1716 9 8i5 7643 5i 9 8 2477 9479 6200 263 9 8 79 4 9 52 348 9 i853 9948 7771 5322 25 9 6 9 5 9 3 63io 2 7 44 88 9 4 8 53 363i 1990 890080 7900 5445 2715 970 7 6419 284 9 8 99 4 7 54 3 77 2 2127 02l3 8028 556 9 2834 9 82I 652 9 2 9 54 9 o 9 4 6 55 3914 2264 o345 8:56 5692 2 9 53 99 36 6638 3o58 9 i 9 4 5 56 4o55 2401 0478 8283 58i5 3072 9 2oo5o 6 7 4 7 3i63 9 2 9 4 4 5 7 4196 2538 0610 84n 5 9 3 9 3i 9 o oi64l 685 7 326 7 c;3 9 4 3 58 4338 2674 0742 853 9 6062 33o 9 02 77 6966 33 7 2 9 4 9 3 2 5 9 4479 2811 o8 7 4 8666 6i85 342 7 o3 9 i 7 o 7 5 34 7 6 9 5 9 3 I 29 | 28 27 26 25 24 23 22 21 20 d Natural Co-sines. 9 >39| 2.3! | 2.24 2.16 2.O 9 2.01 i. 9 3 1.86 1.78 1.70 NATURAL TANGENTS. 129 J s 60 61 62 63 64 65 66 67 G8 69 o 1.73205 i.8o4o5 i.88o 7 3 1.96261 a.o5o3o a.i445i 2.2460^ 2.35585 2.47509 2.6o5o 9 60 I 332i o52g 8205 6402 5i8 2 46i4 4 7 8o 5 77 6 7716 0736 5 9 2 3438 o653 833 7 6544 5333 4777 4 9 56 5 9 6 7 7924 o 9 63 58 r 3555 777 846 9 6685 5485 4 9 4o 5i3 2 6i58 8x32 II 9 O 57 j! 36 7 i 0901 8602 682 7 563 7 5io4 53o 9 634 9 834o i4i8 56 t 3 7 88 1025 8 7 34 6 9 6 9 5 79 o 5 2 68 5486 654i 8549 1 646 55 6 SgoS u5o 886 7 7111 5 9 42 5432 5663 6 7 33 8 7 58 1874 54 7 4O22 127^ 9 ooo 72 53 609^ 55 9 6 584o 6 9 20 8967 2:o3 53 8 4i4o i3 99 9 i33 7 3 9 5 624 7 5760 6018 7 n8 9177 2332 52 9 4s5 7 i524 9 266 7 538 64oo 5 9 25 6196 7 3u 9 386 256i 5i 10 !. 7 43 7 5 1.81649 i.8 9 4oo i. 97 68i 2.o6553 2.j6o 9 o 2.26374 2.3 7 5o4 2.49697 2.627 9 I 5o ii 4492 i 77 4 9 533 7 823 6 7 o6 6255 6552 7697 9807 3021 4 9 12 46io 1899 9 66 7 79 66 6860 6420 6 7 3o 7891 2.5ooi8 3252 48 i3 4 7 28 2O25 9 8oi 8110 701^ 6585 6 9 o 9 8o84 0229 3483 47 i4 4846 2i5o 99 35 8 2 53 7167 6 7 5i 7088 82 79 o44o 37i4 46 i5 4964 22 7 6 i. 9 oo6 9 83 9 6 7 32I 6 9 i 7 7267 84 7 3 o652 3 9 45 45 16 5o82 24O2 O2O3 854o 7 4 7 6 7083 7 44 7 8668 0864 4i 77 44 17 6200 2528 o33 7 8684 7 63o 7 24 9 7626 8863 1076 44io 43 18 5319 2654 o4 7 2 8828 7785 7 4i6 7806 9 o58 I28 9 4642 42 J 9 543 7 2 7 8o o6o 7 8972 79 3 9 7 58 2 79 8 7 92 53 i5o2 48 7 5 4i 20 i. 7 5556 1.82906 i. 9 o 7 4i i. 99 n6 2.08092! 2-I 77 4 9 2.28167 2.3 9 44 9 2.51715 2.65io 9 4o 21 56 7 5 3o33 o8 7 6 9261 825o 79 i6 8348 9 645 I 9 2 9 5342 3 9 22 5 794 SiSg IOI2 9406 84o5 8o84 8528 9841 2142 55 7 6 38 23 5 9 i3 3286 n4 7 9 55o 856o 8 2 5i 8710 2.4oo38 235 7 58ii 37 24 6o32 34i3 1282 9 6 9 5 8716 84i 9 88 9 i 0235 2571 6o46 36 25 6i5i 354o i4i8 9 84i 8872 858 7 9 o 7 3 0432 2786 6281 35 26 62 7 I 366 7 i554 9986 9028 8 7 55 9 254 0629 3ooi 65i6 34 27 6890 3 79 4 i6 9 o 2.00l3l 9184 8 9 23 9 43 7 o82 7 3217 6 7 5 2 33 28 65io 3 9 22 1826 02 77 934i 9 o 9 2 9 6i 9 1025 3432 6 9 8 9 3i 29 663o 4o4 9 I 9 62 o423 9498 9 26l 9 8oi 1223 3648 7225 3x 3o 1.76749 i.84i 77 I. 9 2O 9 8 2.00669 2.09654 2.i 9 43o 2.2 99 84 2.41421 2.53865 2.67462 3o 3i 6869 43o5 2235 o 7 i5 9811 9 5 99 2.3oi6 7 l620 4082 7700 29 32 6990 4433 23 7 I 0862 9969 97 6 9 o35i 1819 42 99 79 3 7 28 33 7 no 456i 25o8 1008 2.10126 99 38 o534 2019! 45i6 8i 7 5 27 34 7 23o 468 9 2645 n55 0284 2.20108 0718 2218 4734 84i4 26 35 7 35i 48i8 2 7 82 1302 0442 0278 9 O2 2418 4 9 5 2 8653 25 36 7 4 7 i 4 9 46 2 9 20 1 449 0600 o44 9 1086 2618 5170 88 9 2 24 3? 7 5 9 2 5o 7 5 3o5 7 i5 9 6 o 7 58 o6i 9 I2 7 I 2819 538 9 9 i3i 23 38 77i3 6204 3i 9 5 I 7 43 0916 0790 i456 3019 56o8 9 3 7 i 22 3 9 7834 5333 3332 1891 io 7 5 0961 i64i 322O 582 7 9 6l2 21 4o 1.77966 1.85462 i. 9 34 7 o 2.O2O3 9 2.II233 2.2II32 2.31826 2.43422 2.56o46 2 .6 9 853 20 4i 8o 77 55 9 i 36o8 2l8 7 1392 :3o4 2OI2 3623 6266 2.7O0 9 /! J 9 42 8198 5 7 20 3 7 46 2335 i55 2 1475 2I 97 3825 648 7 o335 18 43 8319 585o 3885 2483 i 7 n 1 647 2383 4O27 6707 o5 7 7 !? 44 844i 5 979 4oa3 263i 1871 1819 25 7 O 4230 6 9 28 o8i 9 16 45 8563 6io 9 4162 2 7 8o 2o3o 1992 2 7 56 4433 7i5o 1062 i5 46 8685 623 9 43oi 2 9 2 9 2190 2164 2 9 43 4636 7 3 7 i i3o5 i4 4? 88o 7 636 9 444o 3o 7 8 235o 233 7 3i3o 483 9 7 5 9 3 i548 i3 48 8929 64 99 45 79 322 7 25l I 25lO 33i 7 5o43 7 8i5 I 79 2 12 49 goSi 663o 4 7 i8 33 7 6 2671 2683 35o5 5246 8o38 2o36 II 5o 1.79174 i.86 7 6o i. 9 4858 2.o3526 2.12832 2.22857 2 .336 9 3 2.4545i 2.58261 2.72281 IO 5i 9296 68 9 i 4997 36 7 5 2993 3o3o 388i 5655 8484 2526 g 52 9419 7021 5i3 7 38 2 5 3i54 32o4 4o6 9 586o 8708 2771 8 53 9542 7 l52 52 77 3 97 5 33i6 33 7 8 4258 6o65 8 9 32 3017 7 54 9 665 7 283 54i 7 4i25 3477 3553 444 7 62 7 O 9 i56 3263 6 55 97 88 7 4i5 555 7 42 7 6 363 9 3727 4636 6476 9 38i 35o 9 5 56 9911 7 546 56 9 8 4426 38oi 3902 48 2 5 6682 9 6o6 3 7 56 4 5? i.8oo34 7 6 77 5838 45 77 3 9 63 4077 5oi5 6888 9 83i 4oo4 3 58 oi58 7 8o 9 5 979 4 7 28 4i25 4252 52o5 7095 2.60057 4261 2 69 0281 79 4i 6120 48 79 4288 4428 53 9 5 7302 0283 44 99 I 29 28 27 26 25 24 23 22 21 20 a Natural Co-tangents. i P. P fol''. 2 ' 00 2.13 2.27 2.44 2.62 2.82 3.o5 3.3i 3.6i 3. 9 5 j 130 NATURAL s s 70 71 72 73 74 75 7G> 77 78 79 o 939693 9 455i 9 9 5io57 9563o5 9 6l2&2 965926 970296 9 7437o 97 8i48 98162-7 5o I 9792 56i3 n46 6390 I 342 6001 o366 4435 8208 1 683 5o 2 9891 5 7 o8 1236 6475 l422 6o 7 6 o436 45oi 8268 i 7 38 58 3 9991 58o2 i3 2 6 65 60 1502 6i5i o5o6 4566 8^29 i 79 3 5 7 4 940090 58 97 i4i5 6644 i582 6226 o5 77 463i 838 9 i84 9 56 5 0189 5 99 i i5o5 672 9 1662 63oi 0647 46 9 6 844 9 i 9 o4 55 6 0288 6o85 i5 9 4 68i4 i 7 4i 63 7 6 0716 4 7 6i 85o 9 i 9 5 9 54 7 o38 7 6180 1684: 68 9 8 1821 645 1 0786 4826 856 9 20l4 53 8 o486 6274 i 77 3| 6 9 83 I 9 OI 6526 o856 48 9 i 8629 2o6 9 52 9 o585 6368 1862 7 o6 7 I 9 8o 6600 0926 4 9 56 8689 2123 5i 10 940684 946462 9 5i 9 5i 95 7 i5i 9 62o5 9 966675 9799 5 97 502O 97 8 7 48 9 82i 7 8 5o ii 0782 6555 2o4o| 7 235 2l3 9 6749 io65 5o85 8808 2233 4 9 12 0881 6649 2I2 9 7319 2218 6823 n34 5i4 9 886 7 228 7 48 i3 0979 6743 2218 74o4 22 97 6898 1204 5214 892-7 2342 47 i4 1078 683 7 2 3o 7 7 48 7 23 7 6 6972 I2 7 3 52 7 8 8986 23 9 6 46 i5 1176 6930 23 9 6; 7571 2455 7046 1 342 5342 9045 245o 45 16 1274 7024 2484^ 7655 2534 7120 i4n 54o6 9io5 25o5 <4 *7 1372 7ii'7 25 7 3 77 3 9 26i3 7194 i48o 547i 9164 2 55 9 43 18 i4yi 7210 2661 7 822 26 9 2 7268 1 549 5535 9223 26i3 4s J 9 1669 73o4 2750 7 906 2 77 7342 1618 5598! 9282 266 7 4i 20 o4i666 947397 9 52838 957990 9 6284 9 967415 97 i68 7 9 75662 9 7 934i 9 82 7 21 4o 21 1764 7490 2926 8o 7 3 2 9 28 748 9 i 7 55 5 7 26 9 3 99 2 77 4 39 22 1862 7583 3oi5 8i56 3oo6 7562 1824 57 9 o 9 458 2828 38 23 1960 7676 3io3 823 9 3o84 7636 i8 9 3 5853 9 5i 7 2882 37 24 2057 7768 3191 8323 3i63 7709 1961 5 9 i 7 9 5 7 5 2 9 35 36 25 2i55 7861 3279 84o6 324l 7782 2029 5 9 8o 9 634 2 9 8 9 35 26 2252 79 54 3366 8489 33i 9 7 856 2098 6o44 9 6 9 2 3o42 34 27 235o 8o46 3454 85 7 2 33 9 7 7929 2166 6107 97 5o 3o 9 6 33 8 2447 8139 3542 8654 3475 8002 2234 6170 9 8o 9 3i4 9 32 29 2544 823i 362 9 8 7 3 7 3553 8o 7 5 23O2 6233 9 86 7 32O2 3i 3o 942641 948324 9 53 7 i 7 958820 96363o 968148 9 7 23 7 o 9 762 9 6 979925 9 83255 3o 3i 2 7 3 9 84i6 38o4 8902 3 7 o8 8220 2438 635 9 99 83 33o8 29 32 2 836 85o8 3892 8 9 85 3 7 86 8293 25o6 6422 9 8oo4i 336i 28 33 2932 8600 3 979 906? 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5234 0786 6o4 9 1021 5 7 oo oo84 4173 79 66 i46o 4656 3 58 532 9 0877 6i34 IIOI 5 77 5 oi55 423 9 8026 i5i6 4 7 o 7 2 _59 542/ 0967 6220 1181 585o O225 43o5 8o8 7 l5 7 2 4 7 5 7 I 19> 18 17 ]6 15 14 13 12 11 10 d Natural Co-sines. 1 '' i.54 i.46 i.38 :.3o I. 21 i.i3 i.o5 0.97 ' 0.88 NATURAL TANGENTS. 131 1 70 71 72 73 74 75 76 77 78 79 o 2.74748 2.90421 3.07768 3.2 7 o85 3.48 7 4i 3. 7 32o5 4-oio 7 8 4.33i48 4.70463 5.14455 60 I 4997 0696 8o 7 3 7 426 9126 364o 16-76 3 7 23 1137 6266 69 a ' 5246 0971 8379 77 6 7 9609 4o 7 5 20 7 4 43oo i8i3, 6068 58 3 5496 1246 8685 8109 9894 4612 25 7 4 4379 2490. 6863 67 4 5 7 46 i523 8991 8452 3OO2 t 7Ql L\.Q3Q OO70 545 9 i 3i 7 o| 7 6 7 i 56 5 5 99 6 1799 9298 8795 0666 5388, 35 7 8 6o4oi 385i 848o 55 6 624 7 2076 9606 9i3 9 io53 6828 4o8i 6623 4534 9 2 9 3 54 7 64 9 8i 2354 9914' 9483; i44i 6268 4586 7 20 7 6219 6.20107 53 8 6760 2632 3.IO223 9829 1829 6 7 o 9 5o 9 2 7793 6906 0926 62 9 7002 2910 o532 3.30174 2219 7162 55 99 838i 65 9 5 1744 61 10 2.77254 2. 9 3i8 9 '3.io842 3.3o52i 3.62609 3. 77 5 9 5 4.0610-7 4.38 9 6 9 4.77286 5.22666 5o ii 7 5 7 3468 n53 0868 3ooi 8o4o 6616 9 56o 7978 33 9 i 49 12 7761 3 7 48 i464 1216 33 9 3 8485 7127 4.40162 86 7 3 4218 48 i3 8oi4 4028 i 77 5 i565 3 7 85 8 9 3i 7 63 9 o 7 45 9 3 7 o 5o48 47 i4 826 9 43o 9 2o8 7 1914 4179 9 3 7 8| 8162 i34o 4.80068 6880 46 i5 8523 45 9 i 2400 2264 45 7 3 9 82 7 8666 I 9 36 o 7 6 9 6716 45 16 8778 4872 2 7 l3| 26l4 4968 3.8o2 7 6 9 l82 2534 i4 7 i 7553 44 17 9 o33 5i55 302 7 2965 5364 0-726 9 6 99 3i34 2I 7 5 83 9 3 43 18 9 28 9 543 7 334i 33i 7 5 7 6i n 77 4.10216 3 7 35 2882 9235 42 19 9 545 6721 3656 36 7 o 6169 i63o 0736 4338 3690 5.3oo8o 4i 20 27 9 8O2 2.96004 3.i39 7 2 3.34023 3.5655 7 3.82083 4.ii256'444 9 42 4-84300 6.30928 4o 21 2.80060, 6288 4288 43 77 696-7 2 53 7 1778 5548 5oi3 1778 3 9 22 o3i6 65 7 3 46o5 4732 7357 2 99 2 2301 6i55 5 7 ? 7 263i 38 23 o5 7 4 6858 4922 5o8 7 7758 344 9 2826 6 7 64 6444 348 7 37 24 o833j 7144 524o 5443 8160 3 9 o6 335o 7 3 7 4 7162 4345 36 25 io 9 i 743o 5558 58oo 8662 4364 38 7 7 79 86 7882 6206 35 26 i35o 77 i 7 58 77 6i58 8966 4824 44o5 8600 8606 6070 34 27 1610 8oo4 6i9 7l 65i6l 93 7 o 6284 4 9 34 9 2l5 933o 6 9 36 33 28 1870 8292 65i 7 68 7 5 977 5 5 7 45 5465 0832 4.90066 7806 32 29 2i3o 858o 6838 7 234'3. 60181 6208 5 997 4.5o45i 0786 8677 3i 3o 2.823 9 I 2.98868 3.i 7 i59 3.3 7 5 9 43.6o588 3.866 7 i 4.i653o 4.61071 4.91616 6.39662 3o 3t 2653 9 1 58) 7 48i 7955 0996 7 i36 7064 i6 9 3 2249 5.40429 2 9 32 2 9 l4 9447 1 7804 83i7 i4o5 7 6oi 7600 23i6 2984 1 309 28 33 3176 9738 8127: 8679, 1814 8068 8i3 7 2 9 4i 3721 2192 27 34 343 9 3.00028 845il 9042 2224 8536 8676 3568 446o 3077 26 35 3702 oSig 8775! 9406 2636 9 oo4 9 2l5 4i 9 6 6201 3966 25 36 3 9 65 06 1 1 9100 9771 3o48 9 4 7 4 9766 4826 5 9 45 485 7 24 37 422 9 0903 9426 3.4oi36 346i 99 45 4.2020,8 5458 6690 6761 23 38 44 9 4 1196 97 52 o5o2 38 7 4 3. 9 o4i 7 0842 6o 9 i 7438 6648 22 3 9 4768 1489 3.200-79 0869 4289 0890 i38 7 6726 8188 7548 21 4o 2.85o 2 3 3.oi 7 83 3.2o4o6 3.4i236 3.64 7 o5 3. 9 i364 4.2i 9 33 4.57363 4- 9 8 9 4o 5.4845i 2O 4i 528 9 20 77 o 7 34 i6o4 6121 i83 9 2481 8001 9696 9 356 '9 42 5555 23 7 2 io63 I97 3 5538 23i6 3o3o 864i 5. 0046 1 6.60264 18 43 5822 266 7 1392 2343 6967 2793 358o 9 283 - 1210 1176 17 44 6089 2 9 63 I 7 22 2713 63 7 6 32 7 I 4l32 9927 1971 2090 16 45 6356 3260 2o53 3o84 6 7 96 3 7 5i 4685 4.60672 2 7 34 3007 16 46 6624 3556 2384 3456 7217 4232 523 9 1219 3499 3927 i4 4 7 6892 3854 2 7 l5 3829 7 638 4 7 i3 5 79 5 1868 4267 485 1 i3 48 7161 4i5 2 3o48 4202 8061 6196 6352 2618 6037 6777 12 49 743o 445o 338i 45 7 6 8485 568o 6 9 n 3i 7 i 6809 6706 II 5o 2.87700 3.o4 7 4 9 3.23 7 i4 3.44961 3.68909 3.96166 4.27471 4.63825 5.06584 5.5 7 638 IO 5i 7970 6049 4049 532 7 9 335 6661 8o3 2 448o 7 36o 85 7 3 9 52 8240 534 9 4383 5 7 o3 97 6i 7 i3 9 85 9 5 5i38 8139 9611 8 53 85n 564 9 4-7I9 1 6080 3. -70188 7 62 7 9169 5797 8921 6.60462 7 54 8783 5 9 5o 5o55i 6458 0616 8n 7 97 24 6458 9704 i3 97 6 55 9 o55 6262 53 9 2 683 7 io46 860-7 43o2 9 i 7121 6.10490 a344 5 56 9327 6554 5-729 7216 1476 999 0860 7786 1279 32 9 5 4 5 7 9600 685 7 6o6 7 7696 i97 9 5 9 2 i43o 8462 2069 1 4s48 3 58 9 8 7 3 7160 64o6 7977 2338 4.00086 200 1 9121 2862 52o5 2 5 9 2.90147 7464 6 7 45 835 9 2771 0682 2 5 7 3 9791 3658 6166 I 19 18 17 16 15 14 13 12 11 i 10 a Natural Co-tangents. i P. P. , or to I". 4 ' 35 4.8a 5.36 6.01 6. 79 7 . 7 3 8. 9 o 1 10.35 12.20 14.60 NATURAL SINEH. 0T 80 ] 81 82 83 84 85 86 87 88 89 oig848o8 987688 99 o268 99 2546 99 4522 996195 9-97564 99 863ol 999 3 9 i 999848 60 i 4858 77 34 o3o 9 2582 4552 6220 7584 8645 9 4oi 9 853 5 9 2 4go 9 7779 o34 9 2617 4583 6245 7604 8660 9 4n 9 358 58 3 4 9 5 9 7824 o38 9 2652 46i3 6270 7625] 8675 9 42I 9 863 57 4 Soog 7870 042 9 2687 4643 6295 76451 86 9 o 9 43i 9867 56 5 5o5 9 7 9 i5 o46 9 2722 46 7 3 6320 7664) 8705 9 44i 9872 55 6 5109 7 9 6o o5o 9 2 7 5 7 4703 6345 7684, 8710. 9 45o 9877 54 7 5i5 9 8oo5 o54 9 2 79 2 4733 63 7 o 7704 8 7 34 9 46o 9 88i 53 8 5209 8o5o o58 9 2827 4762 6395 7724 8 7 4 9 9 46 9 9 886 52 9 5 2 5 9 8o 9 4 o62 9 2862 4792 6419 7743 8 7 63 9479 9 8 9 o 5i 10 985309 988i3 9 99 o66 9 99 28 9 6 994822 996444 997763 998778 999488 999 8 94 5o ii 5358 8184 0708 2 9 3i 485i 6468 7782 8 79 2 9497 9 8 9 8 $9 12 54o8 8228 0748 2 9 66 488 1 6493 7801 8806 99 o3 48 13 545 7 8273 0787 3ooo 4910 65i 7 7821 8820 9 5i6 9907 47 i4 5507 83i 7 0827 3o34 4939 654i 7 84o 8834 9 5 2 5 9910 46 i5 5556 8362 0866 3o68 4969 6566 7 85 9 8848 9 534 9914 45 16 56o5 84o6 ogoS 3io3 4998 6590 7878 8862 9 542 9918 44 17 5654 845o 0944 3i37 5027 66i4 78 9 7 88 7 6 9 55i 9922 43 18 5 7 o3 84 9 4 0983 3171 5o56 663 7 79 i6 88 9 o 9 56o 99 25 42 19 5 7 5s 8538 IO22 32o5 5o84 6661 7934 8 9 o4 9 568 9929 4i 20 985801 9 8858 2 991061 99 3238 995n3 996685 997953 99 8 9 i 7 999 5 77 999932 4o 21 585o 8626, noo 3272 5i42 6709 797 2 8 9 3i 9 585 99 36 3 9 22 58 99 866 9| n38; 33o6 5170 6 7 32 799 8 9 44 9594 99 3 9 38 23 5 9 4 7 8713. 1177 333 9 5i 99 6 7 56 8008 8 9 5 7 9602 99 42 3? 24 5 99 6 87561 1216 3373 5227 6779 8027 8971 9610 99 45 36 25 6o45 8800 1254; 3406 5 2 56 6802 8o45 8 9 84 9618 99 48 35 26 6o 9 3 8843 1292 343 9 5284 6825 8o63 8997 9626 99 5i 34 27 6i4i 8886 i33i 3473 53i2 6848 8081 9010 9 634 99 54 33 28 6i8 9 8 9 3o 1 369 35o6 534o 6872 8o 99 9023 9642 99 5 7 32 29 6238 8 97 3 1407 353 9 5368 68 9 4 8117 9o35 9 65o 99 5 9 3i 3o 986286 9 8 9 oi6 99i445 99 35 7 2 995396 996917 99 8i35 999048" 99 9 65 7 999962 3o 3i 6334 9 o5 9 i483 36o5 5424 6 9 4o 8i53 9061 9 665 9964 29 32 638i 9 I02 l52I 3638 5452 6 9 63 8170 9 o 7 3 9 6 7 2 99 6 7 28 33 642 9 9 i45 i558 36 7 o 5479 6 9 85 8188 9086 9 68o 99 6 9 27 34 6477 9 i8 7 1596 3703 7008 82o5 9098 9687 997 1 26 35 6525 9 23o i634 3735 5535 7o3o 8223 9111 9694 9974 25 36 65 7 2 9 272 1671 3 7 68 5562 7o53 8240 9 i23 9701 997 6 24 3 7 6620 9 3i5| i7Q 9 38oo 558 9 7 o 7 5 8 2 5 7 9 i35 9709 997 s 23 38 666 7 9.357 1746 3833 5617 797 8274 9147 9716 99 8o 22 3 9 6714 9 3 99 1783 3865 5644 7110. 82 9 I 9159 9722 99 8i 21 4o 9 86 7 6 2 989442 991820 993897 99 56 7 i 99 83o8 99 9 i 7 i 999729 9999 83 2O 4i 68o 9 9484 i85 7 3 9 2 9 5698 "^63 83 2 5 918: 9736 99 85 J 9 42 6856 9526 1894 3 9 6i 5 7 25 7 i85 8342 9 i 9 4 9743 9986 1 8 43 6 9 o3 9568 i 9 3i 3 99 3 5 7 5 2 7207 835 9 9 2o6 9 7 4 9 9988 17 44 6 9 5o 9610 i 9 68 4o25 5 77 8 722 9 8375 9 2l8 9756 9989 16 45 6996 965i 2OO5 4o56 58o5 725o 83 9 2 9 22 9 97 6 2 999 i5 46 9 6 9 3 2O42 4o88 5832 7272 84o8 9 24o 9768 9992 i4 47 7090 9735 2078 4l2O 5858 7 2 9 3 8425 9 252 9775 999 3 i3 48 7 i36 9776 2Il5 4i5i 5884 7312 844 1 9 263 9781 9994 12 49 7i83 9 8i8 2l5l 4182 5911 7336 845 7 9 2 7^ 9787 999 5 II 5o 9 8722 9 9 8 9 85 9 99 2i8 7 994214 99 5 9 3 7 997357 99 8473 999 285 999793 999996 o 5i 7275 99 oo 222^ 4245 5 9 63 7378 848 9 9 2 9 6 9799 9997 9 52 7 322 99 42 2260 4276 5 9 8 9 7 3 99 85o5 980^ 9997 8 53 7368 99 83 22 9 6 4307 6oi5 7420 852i 9 3i8 9810 9998 7 54 7 4i4 99 OO24 2332 4338 6o4i 744i 8537| 9328 9816 9998 6 55 7 46o oo65 2368 436 9 6067 7462 8552 9 33 9 982: 9999 5 56 7 5o6 oio5 2404 44oo 6093 7 48 2 8568 9 35o 9827 9999 4 57 7 55i oi46 2439 443o 6118 75o3 8583 9 36o 9 832 I.OOOOO 3 I 58 7 5 97 0187 2475 446 1 6:44 7 5 2 3 85 99 9 3 7 o 9 83 7 oooo 2 5 9 7643 0228 25ll 44 9 i 6169 7544 86i4 9 38i 9 843 oooo I 9 8 7 6 5 4 3 2 1 1 Natural Co-sines. P> ?, 0.80 0.72 o.63 o.55 o.46 o.38 ' o.3o 0.21 o.i3 o.o4 ' 1 NATURAL TANGENTS. 133 1 80 81 82 83 84 85 86 87 88 89 o 5.67i286.3i3 7 5 7 .n53 7 i8.i4435 9 .5i436 n.43oi i4.3oo 7 19.0811 28.6363 5-7.2900 60 I 8094 2566j 3o42 63 9 8 4io6 4685 3607 1879 8 77 i 58.2612 69 2 9064! 3761 4553 83 7 o 6 7 9i 5072 4212 2 9 5 9 2 9 .1220 59.2659 58 3 6.70037! 4961 6071 8.20352 9490 546i 4823 4o5i 3 7 n 6o.3o58 6 7 4 ioi3: 6i65 7 5 9 4 2344 9.62205 5853 5438 5i56 6245 61.3829 56 5 1992' 7 3 7 4 9125 43451" 4935 6248 6059 6273 8823 62.4992 55 6 2974 8587 7.20661 6355 7 68ol 6645 6685 7 4o3 3o.i446 63.656 7 54 7 3960 9804 22O4 83 7 6 9.704411 7o45 7 3i 7 8546 4n6 64.85&0 53 8 4949^.41026 3 7 54 8.3o4o6 32 1 7 ; 7 448 7954 9702 6833 66.io55 52 9 5g4i 2253 53io 2446 6009 7 853 85 9 6 20.08 7 2 9 5 99 67.4019 5i 10 ~. 7 C 9 3 7 6.43484 7 .268 7 3 8.34496 9.78817 11.8262 14.9244 2O.2O56 3i.24i6 68. 7 5oi 5o 1 1 79 36 4720 844s 6555 9.81641 86 7 3 9898 3 2 53 5 2 84 7 o.i533 49 12 8 9 38 5 9 6i 7 .3ooi8 86 2 5 4482 9087 i5.o557 4465 82o5 7i.6i5i 48 i3 9944 7206 1600 8.40705 7338 95o4 1222 56 9 i 32.n8i 73.1390 47 i4 5.8o 9 53 8456 3i 9 o 2795 9.90211 9923 iSgS 6 9 3 2 42 1 3 74.7292 46 i5 1966 9710 4 7 86 4896 3ioi I2.o346 2571 8188 7 3o3 76.3900 45 16 2982 6.50970 6389 7007 6007 0772 3254 9 46o 33.o452 78.1263 44 I? 4ooi 2234 7999 9128 8 9 3i I2OI 3 9 43 21.0747 3662 79.9434 43 i&- 6024 35o3 9616 8.5i 2 5 9 10.0187 i63 2 4638 2o4 9 6 9 35 81.8470 42 19 6o5i 4777 7.41240 3402 o483 2067 534o 336 9 34.02 7 3 83.8435 4i 20 5.87080 6.56o55 7.42871 8.55555 10.0780 I2.25o5 i5.6o48 21.4704 34.36 7 8 85. 9 3 9 8 4o 21 8n4 7339 4509 7718 1080 2946 6762 6o56 7161 88.i436 3 9 22 9 i5i 8627 6:54 9 8 9 3 i38i 3390 7483 7426 35.0695 90.4633 38 23 5: 9 oi 9 i 9921 7806 8.62078 i683 3838 8211 88i3 43i3 92.9085 37 24 1236 6.61219 9 465 42 7 5 1988 4288 8 9 45 22.0217 8006 9 5.48 9 5 36 25 2283 2523 7 .5n32 6482 2294 4742 9 68 7 i64o 36.i 77 6 98.2179 35 26 3335 383i 2806 8701 2602 5i 99 i6.o435 3o8i 562 7 101.107 34 27 43 9 o 5r44 448 7 8.70931 2913 566o II 9 454i 956o 104.171 33 28 5448 6463 6176 3l 7 2 3224 6124 1962 6020 3 7 .35 79 107.426 32 29 65io 7787 7872 5425 3538i 65 9 i 2722 7 5l 9 7686 110.892 3i 3o 5. 97 5 7 6 6.691 16 7 .5 9 5 7 5 8.77689 10.3854112.7062 16.3499 22. 9 o38 38.1885 114.689 3o 3i 8646 6.70450 7.61287 9964 4172 7536 4283 23.0577 6i 77 n8.54o 29 32 9720 1789 3oo5 8.82252 4491 8oi4 5o 7 5 2137 3 9 .o568 122.774 28 33 6.00797 3i33 4 7 32 455i 48i3 84 9 6 58 7 4 3 7 i8 5o5 9 127.321 27 34 1878 4483 6466 6862 5i36 8 9 8i 6681 532i 9 655 132.219 26 35 2962 5838 8208 9186 5462 9469 7496 6 9 45 4o.4358 i3 7 .5o 7 26 36 4o5i 7199 99 5 7 8.91520 5 7 8 9 99 6 2 83i 9 85 9 3 9 i 7 4 i43.23 7 24 3? 5i43 8564 7.71715 3867 6118 1 3.o458 9 i5o 24.0263 4i.4io6 149-465 23 38 6240 9936 348o 622 7 645o o 9 58 999 i 9 5 7 9 i58 156.269 22 3 9 7 34o 6.8i3i2 5254 85 9 8 6 7 83 i46i I 7 .o83 7 36 7 5 42.4335 163.700 21 4o 6.o8444 6.82694 7 . 77 o35 9.00983 10.7119 i3.i 9 6 9 I7.i6 9 3 24.54i8 42.9641 171.886 2O 4i 9 55 2 4082 8825 33 79 7457 2480 2558J 7185 43.5o8i 180.932 J 9 42 6.10664 54?5 7 . 80622 5 7 8 9 7797 2 99 6 3432! 8 9 78 44.o66i 190.984 1 8 43 1779 68 7 4 2428 8211 8139 35i5 43i4'25.o7 9 8 6386 202.219 I 7 44 2 8 99 8278 424219.10646 8483 4o3 9 6206. 2644 45.2261 2i4.858 16 45 4o<>3 9688 6o64 3o 9 3 8829 4566 6106 45i 7 8294 229.182 i5 46 5i5i 6.91104 7 8 9 5 5554 9178 5o 9 8 7oi5 64i8 46.4489 245.552 i4 4? 628*3 2525 9734 8028 9529 5634 7934 8348 47.o853 264.44i i3 48 7419 3 9 52 7 . 9 i582 9.20616 9882 6i 7 4 8863|26.o3o 7 7 3 9 5 286.478 12 49 855 9 5385 3438 3oi6 1 1.0237 6719 9802 22 9 6 48.4i2i 312.521 I I 5o 6.19-703 6.96823 7 . 9 53o2 9 .2553o 1 1.0594 13.7267 i8.o 7 5o 26.43l6 49.1039 343.774 10 5i 6.zo85i 8268 7 i 7 6 8o58 0954 7821 1-708 6367 8i5 7 381.971 9 52 2003 9718 9 o58 9.30599 i3i6 83 7 8 26 77 845o 5o.5485 429.718 8 53 3i6o 7 .on 7 4 8.oo 9 48 3i55 1681 8 9 4o 3655 27.0566 5i.3o3 2 491.106 7 54 43si 263 7 2848 6724 2048 9 5o 7 4645 2715 52.o8o 7 672.967 6 . 55 548G 4io5 4 7 56I 83o 7 2417 14.00-79 5645 48 99 8821 687.549 5 56 6655 55 79 66 7 4 9.40904 2789 o655 6656 7117 53. 7 o86 85 9 .436 4 5? 7829 7 o5 9 8600 35i5| 3i63 1235 7 6 7 8 9 3 7 2 54.56i3 1145.92 3 58 9007 8546 8.io536 6i4i: 354o' 1821 8 7 n 28.1664 55.44i5 1718.87 2 5 9 6 Hor9 7 .ioo38l 2481 8781 3919 2411 9?55| 3 99 4 56.35o6 343 7 . 7 5 I 9 8 7 6 5 4 3 30 1 x o d Natural Co-tangents. Jl & 17.80 22. 10 28.46 3 7 .83 5.28 7.88 i3 .01 to i . * ! 7 / 134 JV A 'T u R A L SECANTS. Deg. 0' 10' 20' 30' 40' 50' P. Fait tol'. o i .000000 i . 000004 i .000017 i .oooo38 i . 000068 i .000106 89 2.5 i 000152 000207 000271 ooo343 000423 ooo5i2 88 7 .6 a 000609 000715 0008 3o 000953 001084 001224 87 I2. 7 3 001372 001529 001695 001869 OO2o5l 002242 86 I 7 .8 4 002442 oo265o 002867 003092 oo3326 003569 85 22.0 5 003820 oo4o8o oo4348 oo4625 004911 oo52o5 84 6.1 6 oo55o8 oo582o oo6i4i 006470 006808 007154 83 33.3 7 007510 007874 008247 008629 009020 009419 82 38.6 8 009828 010245 010671 011106 on55o OI2OO3 81 43. 9 9 oi2465 012936 oi34i6 013905 oi44o3 014910 80 4 9 .3 10 1.015427 1.015952 1.016487 i .017030 i. 017583 I. 018145 79 54.8 ii 018717 019297 019887 020487 021095 021713 78 60.4 12 022341 022977 023624 024280 024945 O2562O 77 66.0 i3 O263o4 026998 027702 O284i5 029138 029871 76 71,8 i4 o3o6i4 o3i366 032128 032900 033682 034474 75 77-7 i5 035276 o36o88 036910 037742 o38584 039437 74 83. 7 16 040299 041172 o42o55 042949 o43853 044767 73 89.8 *7 045692 046627 047573 048529 049496 o5o4?4 72 96.1 18 o5i462 o5246i o5347i 054492 o555 2 4 o56567 71 102.6 19 ' 057621 o58686 059762 060849 061947 o63o57 70 109.2 20 1.064178 i.o653io i.o68454 i .067609 i .068776 1.069955 69 116.1 21 071145 072347 o 7 356i 074786 076024 077273 68 123.2 22 078535 079808 081094 082392 083703 o85o25 67 i3o.4 23 o8636o 087708 089068 090441 091827 o 9 3225 66 J37-9 24 o 9 4636 096060 097498 098948 ioo4n 101888 65 i45.6 25 103378 104881 106398 107929 109473 nioSo 64 I.S3.7 26 II26O2 114187 115787 i i 7400 119028 120670 63 162.0 2 7 122326 123997 125682 127382 129096 130826 62 170.7 28 132570 i3432 9 i36io4 i3 7 8 9 3 139698 i4i5i8 61 179.7 2 9 143354 i452o5 147073 148956 i5o854 152769 60 189.1 3o 1.154701 i. i 56648 I.i586i2 i .160592 1.162589 i.i646o3 5 9 198.9 3 1 166633 168681 170746 172828 174927 177044 58 209. i 32 179178 :8i33: i835oi i8568 9 187895 190120 57 219.7 33 192363 194625 196906 199205 2Ol523 2o386i 56 230.9 34 206218 208594 210991 2i34o6 2i5842 218298 55 242.6 35 220775 223271 225789 228327 23o886 233466 54 254.8 36 j 2 36o68 238691 24i336 244oo3 246691 24 9 402 53 267.7 37 2 52i36 254892 257671 260472 263298 266l46 52 281.3 38 269018 271914 2 7 4834 277779 280748 283741 5i 295.6 39 286760 289803 292872 295967 2Q 9 o88 302234 5o 310.7 4o i i.3o54o7 i. 308607 i.3n833 i. 3 1 5087 i.3i8368 I .321677 49 326.7 4i 325oi3 328378 331771 335i 9 2 338643 342123 48 343.6 42 345633 349172 352742 356342 359972 363634 4 7 36i.5 43 36 7 32 7 371052 374809 378598 382420 3862 7 5 46 38o.5 44 390164 394086 398042 402032 4o6o57 4ion8 45 400.7 45 4i4ai4 4i8345 4225i3 426718 430960 435239 44 422.3 46 43 9 55 7 443912 4483o6 45274O 45 7 2i3 461726 43 445.3 47 466279 470874 475509 480187 484907 489670 42 46 9 .8 48 494477 499327 5o422I 509160 5i4i45 519176 4i 496.2 49 524253 529377 534549 53 97 6 9 545o38 55o356 4o 524.4 5o 1.555724 i .56n42 i .5666i2 I.572I34 i. 5 777 o8 1.583335 3 9 554.7 5i 58 9 oi6 594751 6oo542 6o6388 612291 6i825i 38 58 7 .4 52 624269 63o346 636483 642680 648938 655 2 58 37 622.7 53 66i64b 668086 674597 68n 7 3 687815 6 9 45 2 4 36 660.9 54 ! 701 3o2 708148 7i5o64 722O5I 729110 736241 35 702.2 55 743447 750727 758o84 7 655i 7 773029 780620 34 747-2 56 788292 796045 8o388i 811801 819806 827899 33 796.2 5? 836078 844348 852707 861159 869704 8 7 8344 32 849.8 58 887080 895914 904847 9i388i 923017 982258 3i 908.5 5 9 941604 95io58 960621 970294 980081 989982 3o 97?. o i 60' 50' 40' 30' 20' 1 0' Deg. Natural Co-secants. NATURAL SECANT 135 f Deg. 0' 10' 20' 30' 40' 50' P. Port tol'. i 60 2.OOOOOO 2.o:oi36 2,020393 2. 03O772 2.041276 2.061906 29 io44 6l 062665 o 7 3556 084679 096739 107036 118474 28 1123 62 i3oo54 141781 i53655 I6568I 177869 190196 27 1210 63 202680 2i5346 228168 24ll58 254320 267667 26 i3o8 64 ' 281172 294869 308760 322820 337083 35i542 25 1417 65 366202 38io65 3 9 6i3 7 4i 1421 426922 442645 24 i53 9 66 4585 9 3 474773 491187 607843 624744 541896 23 1678 67 55 9 3o5 676976 694914 6i3i26 63i6i8 660396 22 i835 68 669467 68838 7 708614 728604 7488i4 769453 21 2016 69 790428 811747 8334r 9 85545i 8 77 853 90063.6 20 2222 70 2.923804 2.947372 2.971349 2.996744 3.020669 3.o45835 '9 2461 7* 3.o 7 i553 3.097736 3.124396 3. i5i545 179198 207367 18 2740 72 236o68 2 653i5 296123 326610 356490 388082 17 3o68 ?3 42o3o4 453i 7 3 486711 620937 5558 7 i 5 9 i536 16 3458 74 627955 666162 7o3i5r 741978 781660 822226 16 3 9 25 75 863 7 o3 906126 949622 99 3 9 2 9 4.039380 4.086913 i4 44 9 2 76 4.133565 4.182378 4.232394 4.283658 3362i5 390116 i3 6190 77 4454H 602167 56o4o8 620226 681676 744821 12 6062 78 809734 876491 946169 5.oi5852 6.088628 5.i635 9 2 II 7171 79 5.24o843 5.320486 5.402633 487404 674926 665333 10 8612 80 5.758770 5.855392 5. 9 55362 6.068868 6.166067 6.277193 9 81 6.392453 6.612081 6.636329 6.766469 6.899794 7.039622 8 82 7.185297 7.337191 7.496711 7.661298 7.834433 8.016646 7 83 8.206609 8.404669 8.613790 8.8336 7 i 9.o65i5i 9.309170 6 84 9.566772 9.839123 10. 12762 10.43343 10.76849 11.10455 5 85 11.47371 11.86837 12.29126 12.74549 i3. 23472 i3. 7 63n 4 86 i4.3355 9 14.96788 16.63679 i6.38o4r I 7 .i 9 843 18. 10262 3 6; 19. 10732 2O.23O28 21.49368 22.92669 24.66212 26.46061 2 86 28.65371 3l.25 7 58 34.38232 38. 20166 42.97671 49.11406 I 89 67.29869 68.76736 85. 9 456i 114.6930 i 7 i.8883 343.7762 O 60' 50' 40' 30' 20' 10' Deg. Natural Co-secants. LENGTHS OF CIRCULAR ARCS. Degrees. Minutes. Seconds. o I .oi 7 4533 26 .453 7 856 o 61 .8901179 / I .0002909 r I . 0000048 2 .0349066 27 .4712389 62 .9076712 2 .0006818 2 .0000097 3 .0623599 28 .4886922 53 .9260246 3 .0008727 3 .ooooi45 4 .0698132 29 .5o6i455 54 .9424778 4 .oon636 4 .0000194 5 .0872666 3o .6236988 55 .9699311 5 .ooi4544 5 .0000242 6 .1047198 ft I .6410621 56 . 977 3844 6 .0017453 6 .0000291 7 .1221730 32 .5585o54 5 7 9948377 7 .0020362 7 .0000339 8 .i3 9 6 2 63 33 .6769687 58 .0122910 8 .0023271 8 .oooo388 9 .1070796 34 .6934119 5 9 .0297443 9 .0026180 9 ,oooo436 10 .1745329 35 .6108662 60 .0471976 10 .0029089 10 .0000486 ii . 1919862 36 .6283i85 65 . i 34464o ii .0031998 ii .oooo533 12 .2094395 3 7 .6467718 70 .2217306 12 .0034907 12 .0000682 :3 .2268928 38 .6632261 75 .3089969 i3 .003/816 i3 .oooo63o i4 .2443461 3 9 .6806784 So .3962634 i'4 .0040724 i4 .0000679 i5 .2617994 4o .6981317 85 .4836299 16 .oo43633 i5 .0000727 16 .2792627 4i . 7166860 90 .6707963 16 .0046642 16 .0000776 17 .2967060 42 .733o383 IOO .7453293 *7 .oo4g45i i? .0000824 18 .3i4r5 9 3 43 .7604916 IIO .9198622 18 .0062360 18 .0000873 '9 .33i6i26 44 .7679449 120 2*> 9 43 9 5i J 9 .0066269 ! 9 .0000921 20 .3490669 45 .7863982 i3o 2.2689280 20 .0068178 20 .0000970 21 .3666191 46 .8028616 i4o 2.44346io 25 .0072722 2.5 .OOOI2I2 22 .383 97 24 47 .8203047 i5o 2.6179939 3o .0087266 36 .0001454 23 .4014267 48 .83 77 58o 1 60 2.7926268 4o .0116355 4o .0001939 a/i .4188790 49 .8662113 170 2.9670.597 5o .0145444 5o .0002424 25 .4363323 5o ,8726646 1 80 3. 1416927 60 .01 745V- 60 . 0002909 136 TRAVERSE TABLE. Course Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 6. Lat. Dcp. Lat. Dep. Lat. Dep. 1 Lat Dep. Lat Dep. o / o / o i5 1. 0000 o.oo4 2.OOOC 0.008 3. OOOO o.oiS 4-OOO 0.017 5.OOO O.02I 89 45 3o oooo 008 I. 9 999 017 2 -9999 0262 3.999 o34 4-999 o43 3o 4o 0.9999 oi3 999 8 026 9997 0393 999 o52 999 o65 :5 I 9998 017 9997 o34 9 999 5 o52^ 999^1 o6 9 8 999 o8 7 89 o i5 9998 0218 9995 o436 999 3 o654 999 c 087 99 88 IO 9 45 3o 9997 0262 9993 o52; 999 o 7 85 99 86 1047 998 i3o 3o 45 9995 o3o5 9991 061 99 86 0916 998 1222 9977 i5 2 i5 2 O 9994 0349 9988 o6 9 8 99 82 1047 9976 i3 9 6 997 i 7 4 88 o . i 9992 039: 99 85 o 7 85 9977 1178 9969 i5 7 o 996 i 9 6 45 3o 999 o43 6 9981 0872 997i iSog 99 6 2 i 7 45 99 5 2 218 3o 45 0.9988 o.o48o 1.9977 0.0960 2. 99 65 o.i43 9 3. 99 54 o.i 9 i 9 4. 99 4a o.23 99 i5 3 o 9986 o523 9973 1047 99 5 9 i5 7 o 9945 20 9 ^ 99 3i 26l 7 87 o i5 99 8 4 0567 9968 n34 99 5 2 1701 99 36 2268 99 20 2835 45 3o 9981 0610 99 63 1221 99 44 i83i 9925 2442 997 3o5 2 3o 45 9979 o654 99 5 7 i3o8 99 36 1962 99 i4 26l6 9 8 9 3 32 7 o i5 4 o 9976 0698 99 5i i3 9 5 99 2 7 2093 99 o3 2 79 9878 3488 86 o i5 997 3 0741 99 45 1482 99 i8 2223 9 8 9 o 296^ 9 863 3705 45 3o 9969 o 7 85 99 38 i56 9 99 o8 2354 9877 3i38 9 846 3 9 23 3o 45 9966 0828 99 3i i656 9897 2484 9 863 33i2 9828 4i4o i5 5 o 9962 0872 99 2 4 i 7 43 9 886 26i5 9 848 3486 9810 4358 85 o i5 o. 99 58 0.0915 1.9916 o.i83o 2 . 9 8 7 4 0.2745 3. 9 83 2 o.366o 4- 9 79 o.45 7 5 45 3o 9954 0958 9908 1917 9 862 2875 9816 3834 977 47 9 2 3o 45 99 5o 1 002 9899 200^ 9 84 9 3oo6 9799 4oo8 97 48 5oo 9 i5 6 o 9945 io45 9890 20 9 I 9 836 3i36 9781 4i8i 9.726 5226 84 o T *i 9941 1089 9881 2177 9 822 3266 9762 4355 97 3 5443 45 3o 99 36 ii3a 9871 226^ 9 8o 7 33 9 6 9743 45 2 8 9679 566o 3o 45 99 3i n 7 5 9861 2 35i 979 2 35 2 6 9723 4701 9 653 5877 i5 7 o 9925 1219 9 85i 2437 977 6 3656 9702 48 7 5 9,627 6o 9 3 83 o i5 9920 1262 9 84o 252^ 97 6o 3 7 86 9680 5o48 9 6oo 63io 45 3o 9914 i3o5 9829 26ll 9743 3 9 i6 9 658 5221 9 5 7 2 6526 3o 45 0.9909 o.i349 1.9817 o.26 9 7 . 9 726 o.4o46 . 9 635 o.53o4 . 9 543 0.6743 i5 8 o 99 o3 1392 9805 2783 97 o8 4i 7 5 9611 556 7 9 5i3 6o5 9 82 o i5 9897 1435 979 3 2870 9 6 9 o 43o5 9 586 5 7 4o 9 483 7175 45 3o 9890 1478 9780 2 9 56 9 6 7 o 4434 956i 5 9 I2 9 45i 7 3 9 o 3o 45 9884 l52I 9767 3o42 9 65i 4564 9 534 6o85 9 4i8 7606 i5 9 9877 1 564 9754 3l2 9 9 63i 46 9 3 9 5o8 6 2 5 7 9 384 7822 I O i5 9870 1607 9740 3ai5 9 6io 4822 9 48o 643o q35o 8o3 7 45 3o 9863 i65o 9726 33oi 9 58 9 4g5i 945i 6602 9 3i4 8262 3o 45 9856 i6 9 3 9711 338 7 9 56 7 5o8o 9422 6 77 4 9 2 78 846 7 i5 10 9848 I 7 36 9696 3473 9 544 5209 9 3 9 2 6 9 46 9 240 8682 o o i5 .9840 0.1779 .9681 o.355 9 . 9 52I 0.5338 . 9 362 o. 7 n8 . 9 2O2 .8897 45 3o 9833 1822 9665 3645 9 4 9 8 5467 9 33o 72 8 9 9 i63 9112 3o 45 9825 i865 9649 3 7 3o 9 4?4 55 9 6 9 2 9 8 7461 9 I23 9 3 2 6 i5 II 9816 1908 9 633 38i6 9 44 9 5 7 24 9 265 7 63 2 9 o8i 9 54o 9 i5 9808 1951 9616 3 9 02 9 4a4 5853 9 23l 7 8o4 9 o3 9 9755 45 3o 9799 i 99 4 9 5 9 8 3 9 8 7 9 3 9 8 5 9 8i 9 i 97 797 5 8 99 6 9968 3o 45 979 2o36 9 58i 4o 7 3 9 3 7 r 6109 9 l62 8i46 8 9 52 .0182 1 5 12 9781 2079 9 563 4i58 9 344 623 7 9 I26 83x6 8 9 o 7 o3 9 6 8 o i5 9772 2122 9 545 4244 9 3i- 6365 9 o8 9 848 7 8862 0609 45 3o 9763 2l64 9 526 432 9 92 8 9 64 9 3 9 o52 8658 88i5 0822 3o 45 . 97 53 .22O7 . 9 5o7 .44 1 4 . 9 26o .6621 . 9 oi4 .8828 .8767 .io35 i5 i3 o 9744 225oj 9 48 7 44 99 9 23l 6749 8 9 75 8 99 8 8719 1248 7 o i5 97 34 2292 9 468 4584 9 20I 6876 8 9 35 9 i68 866 9 i46o 45 3o 9724 2334 9 447 466 9 9 i7i 7003 88 9 5 9 338 8618 1672 3o 45 97 i3 23 77 9 427 4754 9 i4o 7 i3i 8854 9 5 7 856 7 1884 i5 i4 o 97 o3 2419 9 4o6 4838 9 I0 9 7258 8812 9 6 77 85i5 2096 6 o i5 9692 2462 9 385 4 9 23 977 7 385 8 7 6 9 9 846 8462 23o8 45 3o 9681 25o4 9 363 5oo8 9 o44 7 5n 8726 i.ooiS 84o 7 2519 3o 45 9670 2546 9 34i 5o 9 2 9 OI I 7 638 8682 0184 835 2 2730 i5 r5 o ! 9 65 9 2588 9 3i 9 5i 7 6 8978 7765 863 7 o353 82 9 6 2941 5 o l! Dep. Lat. Dep. | Lat. Dep. | Lat. Dep. Lat. Dep. Lat Dist. 1. i Dist. 2. ii Dist. 3. Dist. 4. Dist. 5. Course. TRAVERSE TABLE. 137 Course, Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. 10. , i Lat. Dcp. Lat. Dep. Lat. Dep. Lat. Dep. Lat Dep. o / o / o i5 6.9999 0.0262 6.9999 o.o3o5 7.9999 o.o34 9 8.9999 o.o3 9 3 9.9999 o.o436 8 9 45 3o 9998 o524 9997 0611 9997 o6 9 8 9997 o 7 85 9996 o8 7 3 3o 45 999 5 0785 9994 o 9 i6 999 3 1047 9992 1178 999 I i3o 9 i5 I 9991 1047 9989 1222 9988 i3 9 6 9986 1571 99 85 i 7 45 89 o i5 9986 1 309 99 83 1527 9981 i 7 45 9979 I 9 63 9976 2181 45 3o 9979 1571 9976 i832 997 3 2094 9969 2356 99 66 2618 3o 45 9972 i832 9967 2i38 99 63 2443 99 58 2748 99 53 3o54 i5 2 9968 2094 99 5 7 2443 99 5i 2792 99 45 3i4i 99 3 9 34 9 o 83 o i5 99 5 4 2356 9946 2 7 48 99 38 3i4i 99 3i 3533 99 23 3 9 26 45 3o 9943 2617 99 33 3o53 , 99 24 34 9 o 99 i4 3 9 26 99 5 4362 3o 45 5.9931 0.2879 6.9919 0.3358 7.9908 0.3838 8. 9 8 9 6 o.43i8 9 . 9 885 o.4 79 8 i5 3 o 9918 3i4o 9904 3664 9 8 9 o 4i8 7 9 8 77 4710 9 863 5234 87 o i5 9904 3402 9887 3 9 68 9 8 7 i 4535 9 855 5lO2 9 83 9 566 9 45 3o 9888 3663 9869 42 7 3 9 85i 4884 9 832 54 9 4 9 8i3 6io5 3o 45 9872 3 9 24 985o 45 7 8 9 82 9 5232 9 8o 7 5886 97 86 654o i5 4 o 9 854 4i85 9829 4883 9 8o5 558i 978i 6278 9756 6 97 6 86 o i5 9835 4447 9808 5i88 97 8o 5 9 2 9 9753 6670 9725 7 4n 45 3o 9815 4708 9784 54 9 2 9753 6277 9723 7061 9 6 9 2 7 846 3o 45 9794 4968 97 6o 5 797 9725 6625 9 6 9 i 7453 9 65 7 8281 i5 5 o 9772 5229 9734 6101 9 6 9 6 6 97 2 9 658 7 844 9619 8716 85 o i5 5. 97 48 0.5490 5. 9 7o6 o.64o5 7 . 9 664 0.7320 8. 9 622 0.8235 9 . 9 58o o. 9 i5o 45 3o 9724 5 7 5i 9 6 7 8 670 9 9 632 7668 9 586 8626 9 54o 9 585 3o 45 9698 6011 9 648 7013 9 5 97 8oi5 954 7 9 oi 7 9 4 97 I.OOI 9 i5 6 o 9671 6272 9 6l 7 7 3i 7 9 562 8362 9 5 7 9 4o8 9 452 o453 84 o i5 9 643 6532 9 584 7621 9 525 8 7 o 9 9 465 979 8 9 4o6 0887 45 3o 9614 6792 9 55o 7924 9 486 9 o56 9 42I 1.0188 9 35 7 1320 3o 45 9 584 7 052 9 5i5 8228 9 445 9 4o3 9 3 7 6 o5 7 8 9 3o 7 1754 i5 7 o 9 553 7312 9 478 853i 9 4o4 97 5o 9 32 9 o 9 68 9 255 2187 83 o i5 9520 7672 9 44o 8834 9 36o i.oo 9 6 9 28o i358 9 20O 2620 45 3o 9 48 7 7832 9 4oi 9 i3 7 9 3i6 o442 9 23o i 7 4 7 9 i44 3o53 3o 45 5.9452 0.8091 6. 9 36i o. 9 44o 7 . 9 26 9 1.0788 8. 9 i 7 8 I.2l3 7 9 . 9 o8 7 1.3485 i5 8 o 9416 835o 9 3r 9 97 42 9 22I n34 9 I24 2526 9 02 7 3 9 i 7 82 o i5 9 3 79 8610 9 2 7 6 i.oo44 9 I 7 2 1479 9 o6 9 2 9 l4 8 9 65 434 9 45 3o 934i 8869 9 23l o34? 9 I2I 1825 9 on 33o3 8 9 02 4781 3o 45 9 3o2 9127 9 i85 o64 9 9 o6 9 2170 8 9 53 36 9 i 8836 5212 i5 9 9 26l 9 386 9 i38 oa5o 9 oi5 25i5 88 9 2 4o 79 8 7 6 9 5643 81 o i5 9 22O 9 645 9 o 9 o 1252 8 9 6o 285 9 883o 446 7 8 7 oo 6074 45 3o 9 i77 99 o3 9 o4o i553 8 9 o3 3204 8766 4854 862 9 65o5 3o 45 9 i33 1.0161 8989 i854 8844 3548 8700 524i 8556 6 9 35 i5 IO O 9 o88 0419 8 9 3 7 2i55 8785 38 9 2 8633 5628 .848 1 7365 80 o i5 5. 9 o42 1.0677 6.8883 1.2456 7.8723 1.4235 8.8564 i.6oi5 9 .84o4 i.77 9 4 45 3o 8 99 5 0934 8828 2 7 56 8660 4579 84 9 3 64oi 83 2 5 8224 3o 45 8 9 47 1191 8772 3o5 7 85 9 6 4 9 22 8421 6787 8245 8652 i5 II 88 9 8 1 449 8714 335 7 853o 5 2 65 8346 7 i 7 3 8 1 63 9 o8i 79 o i5 8847 1705 8655 3656 8463 5607 8271 7 558 8o 79 9 5o 9 45 3o 8 79 5 1962 85 9 5 3 9 56 83 9 4 5 9 4 9 8i 9 3 7943 799 2 99 3 7 3o 45 8 7 43 2219 8533 4255 8324 6291 8ii4 8328 795 2.o364 i5 12 O 868 9 2475 8470 4554 8252 6633 8o33 8712 7 8i5 O7 9 i 78 o i5 8634 2731 84o6 485 2 8178 6974 7 9 5i 9 o 9 6 7723 1218 45 3o 85 7 8 2986 834i 5i5i 8104 7 3i5 7867 9 48o 7 63o 1 644 3o 45 5.8521 1.3242 6.8274 r.5449 7.8027 1.7656 8.7781 i. 9 863 9 . 7 534 2.2070 i5 i3 o 8462 34 9 7 8206 5 7 4 7 79 5o 7996 7 6 9 3 2.0246 7 43 7 2 4 9 5 77 i5 84o3 3 7 5 2 8i3 7 6o44 7870 8336 7 6o4 0628 7 338 2 9 20 45 3o 8342 4007 8066 634i 779 8676 7 5i3 1010 7 23 7 3345 3o 45 8281 4261 7994 6638 7707 9015 7 42I i3 9 2 7 i34 3 7 6 9 i5 i4 o 8218 45i5 7 9 2I 6935 7624 9 354 7 32 7 1773 7 D3o 4l 9 2 7 6 o i5 8i54 4769 7 846 7 23l 7 538 9692 7 23l 2 I 54 6 9 23 46i5 45 3o 8089 5o23 7770 7 52 7 7452 2.oo3o 7 i33 2534 68i5 5o38 5o 45 8023 5276 76 9 3 -7822 7364 o368 7 o34 2 9 I^ 6 7 o5 546o i5 i5 o 7956 552 9 7616 8117 7274 0706 6 9 33 32 9 4 65 9 3 5882 7 5 o Dep. Lat. Dep. Lat. Dep. | Lat. Dep. Lat." 1 Dep. Lat Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist 10. Coursa 138 TRAVERSE TABLE. Course. Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist 5. Lat. Dep. Lat Dep. Lat. Dep. Lat Dcp. f Lat Dep. f o / i5 i5 0-9648 0.2630 1.9296 0.5261 2 .8 9 44 o. 7 8 9 i 3.85 9 i I.052I 4.8 2 3 9 I.3i52 74 45 3o 9636 2672 9273 5345 8 9 o 9 8oi 7 8545 o6 9 o 8182 3362 3o 45 9625 2714 9249 542 9 887; 8i43 84 9 8 o858 8i 2 3 35 7 2 i5 16 o q6i3 2756 9225 55i3 8838 8 2 6 9 845o IO25 8o63 3 7 82 74 o i5 9600 2798 9201 5597 8801 83 9 5 8402 n 9 3 8002 3 99 i 45 3o 9588 2840 9176 568o 8 7 65 8520 8353 i36i 7 9 4i 4201 So 45 9 5 7 6 2882 9i5i 5 7 64 8727 8646 83o3 i528 7879 44io i5 17 o 9 563 292^ 9126 5847 868 9 8 77 i 8262 i6 9 5 7 8i5 46i 9 7 3 o i5 955o 2966 9100 5931 865i 88 9 6 8201 1862 77 5i 4827 45 3o 9 53 7 3007 9074 6oi/ 8612 9 O2I 8i4 9 2028 7686 5o35 3o 45 0.9524 o.3o49 1.9048 0.6097 2.85 7 2 o. 9 i46 3.8o 9 6 I.2I 9 5 4.7620 1.5243 i5 18 o 95n 3090 9021 6180 8532 9 2 7 I 8042 236i 7 553 545 1 72 o i5 9497 3i32 8994 6263 84 9 i 9 3 9 5 79 88 2527 7485 5658 45 3o 9 483 3i 7 3 8966 6346 845o 9 5i 9 7933 26 9 2 74i6 5865 3o 45 9469 32i4 8 9 3 9 6429 84o8 9 643 7877 2858 7347 6072 i5 19 o 9 455 3^56 8910 65n 8366 9767 7821 3o23 7276 6278 71 o i5 944 1 32 97 8882 6594 8323 9 8 9 i 7?64 3i88 7204 6485 45 3o 9426 3338 8853 6676 82 79 i.ooi4 7706 3352 7 l32 66 9 o 3o 45 9412 33 79 8824 6 7 58 8235 oi38 7 64 7 35i 7 7 o5 9 68 9 6 i5 20 o 9 3 97 3420 8 79 4 684o 8i 9 i 0261 7 588 368i 6 9 85 7101 70 o i5 0.9382 0.346: 1.8764 0.6922 2.8146 i.o384 3. 7 528 1.3845 4.6 9 io 1.7306 45 3o 9 36 7 35o2 8 7 33 7004 8100 o5o6 7 46 7 4oo8 6834 7 5io 3o 45 935i 3543 8703 7086 8o54 o62 9 74o5 4172 6 7 5 7 77 i5 i5 21 9336 3584 8672 7167 8007 0751 7 343 4335 66 79 79 i8 69 o i5 9320 3624 864o 7249 7 9 6o 0873 7280 44 9 8 6600 8122 45 3o 93o4 3665 8608 733o 7 9 i3 o 99 5 7217 466o 652i 8325 3o 45 9288 3 7 o6 8576 7411 7864 1117 7 l52 4822 644o 8528 i5 22 O 9272 3y46 8544 7492 7816 1238 7087 4 9 84 635 9 8 7 3o 68 o i5 9255 3 7 86 85n 7 5 7 3 7766 i35 9 7022 5i46 6277 8 9 32 45 3o 9 23 9 382 7 8478 7654 7716 i48i 6 9 55 53o 7 6i 9 4 9 i34 3o 45 0.9222 0.3867 1.8444 o. 7 734 2.7666 1.1601 3.6888 1.5468 4.6110 1.9336 i5 23 9205 3 9 o 7 84io 7 8i5 76i5 1722 6820 562 9 6025 9 537 67 o l5 9188 3 9 4 7 83 7 6 7 8 9 5 7564 1842 6 7 5 2 57 9 o 5 9 4o 9737 45 3o 9171 3 9 8 7 834i 797 5 7512 i 9 6 2 6682 5 9 5o 5853 99 3 7 3o 45 9 i53 4027 83o6 8o55 7 45 9 2082 6612 6110 5 7 66 2.0l3 7 i5 24 9 i35 4o6 7 8271 8i35 7 4o6 2202 6542 626 9 56 77 o33 7 66 o i5 9118 4107 8235 8214 7353 2322 6470 64aq 5588 o536 45 3o 9100 4i47 8i 99 8294 7 2 99 244i 63 9 8 6588 54 9 8 o 7 35 3o 45 9081 4187 8i63 83 7 3 7244 2660 6326 6 7 46 54o 7 o 9 33 i5 26 o 9063 4226 8126 8452 7 i8 9 267 9 6252 6 9 o5 53i5 u3i 65 o i5 0.9045 0.4266 i.8o8 9 o.853i 2. 7 i34 i.27 9 7 3.6178 1.7063 4.5223 2.1328 45 3o 9026 43o5 8o52 8610 7078 2 9 l5 6io3 7220 5i2 9 i526 3o 45 9007 4344 8oi4 8689 7021 3o33 6028 7 3 7 8 5o35 I 7 22 i5 26 o 8988 4384 797 6 8767 6 9 64 3i5i 5 9 52 7 535 4 9 4o 1919 64 o i5 8969 4423 79 3 7 8846 6 9 o6 3269 58 7 5 7692 4844 2Il4 45 3o 8949 4462 7 8 99 8924 6848 3386 5 797 7 848 4 7 4 7 23lO 3o 45 8930 45oi 7860 9002 6 7 8 9 35o3 5 7 i 9 8oo4 464 9 2 5o5 i5 27 8910 454o 7820 9080 6 7 3o 3620 564o 8160 455o 2 7 00 63 o i5 8890 45 79 7780 9 l5 7 66 7 i 3736 556i 83:5 445 1 2894 45 3o 8870 46i 7 7?4o 9 235 6610 3852 548o 84 7 o 43 5 1 3o8 7 3o 45 o.885o 0.4656 1.7700 0.9312 2.655o 1.3968 3.54oo 1.8625 44249 2.3281 i5 28 o 8829 46 9 5 7 65 9 9389 6488 4o84 53i8 8779 4i4 7 34 7 4 62 o 16 8809 4733 7618 9466 642 7 4200 5236 8 9 33 4o45 3666 45 3o 8788 477 2 7 5 7 6 9 543 6365 43i5 5i53 9 o86 3 9 4i 3858 3o 45 8767 48io 7535 9 620 63o2 443o 5o6 9 9 24o 3836 4049 i5 29 o 8 7 46 4848 74 9 2 9 6 9 6 623 9 4544 4 9 85 9 3 9 2 3 7 3i 4240 61 o i5 8 72 5 4886 745o 977 2 6i 7 5 465 9 4 9 oo 9 545 3625 443 1 45 3o 8704 4924 7407 9 848 61 1 1 4773 48i4 9 6 97 i 35i8 4621 3o 45 8682 4962 7364 9.024 6o46 4886 4728 9 84 9 | 34io 48n i5 3o o 8660 5ooo 7321 I.OOOO 5 9 8i 5ooo 464 1 2.000O 33oi 5ooo 60 o Dop. Lat Dcp. Lat Dep. Lat. Dep. Lat. Dep. Lat Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 5. Course. TRAVERSE TABLE. 139 Course. Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. 10. . Lat. Dep. Lat Dep. Lat. Dep. Lat. Dep. Lat Dep. c r o / i5 i5 5. 7 88 7 1.5 7 82 6. 7 535 i.84i2 7 . 7 i83 2.IO42 8.683i 2.36 7 3 9 -64 79 2.63o3 74 45 3o 7 8l8 6o34 7 454 8 7 o 7 7 o 9 o i3 79 6727 4o5i 6363 6 7 24 3o 45 7747 6286 7372 9001 6 99 6 I 7 i5 6621 443o 6246 7 l44 1 5 16 o 7676 6538 7288 9 2 9 5 6 9 oi 2o5i 65i4 48o 7 6126 7564 74 o i5 7 6o3 6790 7 2O3 9 588 68o4 2386 64o4 5i85 6oo5 79 83 45 3o 7529 704 1 7117 9881 6 7 o6 2 7 2I 6294 556i 5882 8402 3o 45 7454 7292 7o3o 2.0I 7 4 6606 3o56 6181 5 9 38 5 7 5 7 8820 i5 17 o 7 3 7 8 7542 6 9 4i o466 65o4 33 9 o 6067 63i3 563o 9 23 7 7 3 o i5 73oi 7792 685i o 7 58 6402 3 7 23 5 9 5 2 668 9 55o2 9 654 45 3o 7223 8042 6760 1049 62 97 4o56 5835 7 o64 53 7 2 3.oo 7 i 3o 45 5.7i44 1.8292 6.6668 2.i34i 7 .6i 9 2 2.4389 8.5 7 i6 2. 7 438 9.5240 3.o486 i5 18 o 7063 854i 65 7 4 i63i 6o85 4 7 2I 55 9 5 7 8l2 5io6 O 9 02 72 o i5 6982 8790 64 79 1921 5 97 6 5o53 5473 8i85 4g 7 o i3i6 45 3o 6899 9o38 6383 2211 5866 5384 5349 855 7 483 2 I 7 3o 3o 45 6816 9286 6285 25oi 5 7 54 5 7 i5 5224 8 9 3o 46 9 3 2i44 i5 19 o 6 7 3i 9 534 6186 2 79 564i 6o45 5o 97 9 3oi 4552 2 55 7 7 I i5 6645 9781 6086 3o 7 8 552 7 63 7 5 4968 9 6 7 2 4409 2 9 6 9 45 3o 6558 2.0028 5 9 85 3366 54n 6 7 o5 4838 3.0043 4264 338i 3o 45 6471 0275 5882 3654 5 29 4 7 o33 4706 o4i3 4n8 3 79 2 i5 20 o 6382 o52I 5 77 8 394i 5i 7 5 7 362 45 7 2 0782 3 9 6 9 4202 70 o i5 5.6291 2.0767 6.56 7 3 2.4228 7 .5o55 2 . 7 68 9 8.4437 3.ii5i 9 .38i 9 3.46i2 45 3o 6200 1012 556 7 45i5 4 9 34 8oi 7 43oo i5i 9 366 7 5021 3o 45 6108 1257 545 9 48oo 48n 8343 4162 1886 35i4 542 9 i5 21 O 6016 i5o2 535i 5o86 4686 866 9 4022 2253 3358 583 7 6g o i5 5920 1746 524i 53 7 i 456i 8 99 5 388i 26i 9 3201 6244 45 3o 58 2 5 1990 5129 5655 4433 9 32O 3738 2986 3o4a 665o 3o 45 5729 2233 5oi 7 5 9 3 9 43o5 9 645 3593 335o 2881 7 o56 i5 22 O 563i 2476 4go3 6222 4i 7 5 99 6 9 3447 3 7 i5 2-718 7 46i 68 o i5 5532 2719 4 7 88 65o5 4o43 3.O2 9 2 3299 4o 7 8 2554 7 865 45 3o 5433 2961 46 7 2 6 7 88 3 9 io o6i5 3i49 4442 2388 8268 3o 45 5.533 2 2.3203 6.4554 2. 7 7 7-3776 3.o 9 3 7 8.2998 3.48o4 9.2220 3.86 7 i i5 23 5 2 3o 3444 4435 7 35i 364o 1258 2845 5i66 2o5o 9 o 7 3 67 o i5 6127 3585 43i5 7 63 2 35o3 i58o 2691 55 27 i8 79 9 4 7 4 45 3o 5o24 3 92 5 4194 79 I2 3365 1900 2535 588 7 I 7 o6 9 8 7 5 3o 45 4919 4i65 4o 7 2 8i 9 2 3225 2220 2 3 7 8 6 2 4 7 i53i 4.02 7 5 i5 24 48i3 44o4 3g48 84 7 2 3o84 2539 22I 9 6606 i355 o6 7 4 66 o i5 4706 4643 38 2 3 8 7 5o 2 9 4i 2858 2o5 9 6 9 65 n 7 6 IO 7 2 45 3o 45 9 8 4882 36 97 9 O2 9 2 797 3i 7 5 i8 97 7 322 o 99 6 i46 9 3o 45 448 9 5l2O 35 7 o 9 3o6 265i 3493 I 7 33 7679 0814 1866 i5 25 43 7 8 535 7 3442 9 583 2 5o5 38o 9 i568 8o36 o63i 2262 65 o i5 5.4267 2 .55 9 4 6.33i2 2. 9 86o 7 . 2 356 3.4i25 8.i4oi 3.8391 9 .o446 4.265 7 45 3o 4i55 583i 3i8i 3.oi36 220 7 444 1 1233 8 7 46 os5 9 3o5i 3o 45 4042 6067 3o4 9 o4n 2o56 4 7 56 io63 9100 oo 7 o 3445 i5 26 o 3928 63o2 2916 0686 1904 5o 7 o o8 9 i 9 453 8. 9 8 79 383 7 64 o i5 38i2 653 7 2 7 8l o 9 6o I 7 5o 5383 o 7 i 9 9806 9 68 7 422 9 45 3o 36 9 6 6772 2645 1234 i5 9 5 56 9 6 o544 4-oi58 9 4 9 3 4620 3o 45 35 79 7006 2509 i5o 7 i438 6008 o368 oSog 9 2 9 8 5oio i5 27 o 346o 72 3 9 23 7 O I 779 1281 63i 9 0191 0859 9 IOI 53 99 63 o i5 334i 7 4 7 2 223l 2o5i II2I 663o 0012 1209 8 9 O2 5 7 8 7 45 3o 3221 77 5 2091 2322 0961 6g4o 7 . 9 83i i55 7 8 7 oi 6i 7 5 3o 45 D.Sogg 2-7937 6.1949 3.25 9 3 7-799 3. 7 249 7 . 9 64 9 4.1905 8.8499 4.656i i5 28 o 2977 8168 1806 2863 o636 7 558 9 465 2252 8295 6 9 4 7 62 o i5 2853 83 99 l662 3i32 o4 7 i 7 866 9280 2599 8089 7 332 45 3o 2729 863o i5i 7 34oi o3o5 8i 7 3 9094 2944 7882 77 i6 3o 45 2604 885 9 i3 7 i 366 9 oi38 84 7 9 SgoS 3289 7 6 7 3 8o 99 i5 20 2477 9089 1223 3 9 3 7 6. 997 o 8 7 85 8 7 i6 3633 7 462 848i 61 o i5 235o 9 3i 7 io 7 5 4203 9800 9090 85 2 5 3 97 6 7 25o 8862 45 3o 2221 9 545 0925 44 7 o 9628 9 3 9 4 8332 43i8 7 o36 9242 3o 45 2092 9773 o 77 4 4 7 35 9 456 969-7 8i38 465 9 6820 9 622 i5 3o o 1962 3.OCOO 0622 5ooo 9282 4.0000 79 42 5ooo 66o3 S.oooo 60 o Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. 10. Oourse. 1 40 TRAVERSE TABLE. Bourse. Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 5. 1 Lat. Dep. Lat. Dep. Lat. Dep. Lat Dep. Lat Dep. o / o / So i5 0.8638 o.5o38 1.7277 1.0076 2.5 9 l5 I.5ll3 3.4553 2.Ol5l 4.3192 2.5l8 9 5 9 45 3o 8616 5o 7 5 7 233 Ol5l 584 9 5226 4465 0302 3o8i 53 77 3o 45 85 9 4 5n3 7188 O226 5782 533 9 43 7 6 o452 2970 5565 i5 h o 85 7 2 5i5o 7 i43 o3oi 5 7I 5 545i 428 7 0602 2858 5 7 5 2 5 9 o i5 854 9 5:88 7098 o3 7 5 564 7 5563 4i 9 6 o 7 5i 2746 5 9 3 9 45 3o 8520 5225 7o53 o45o 55 79 5675 4/o6 o 9 oo 2632 6125 3o 45 85o4 5262 7007 o524 55n 5 7 86 4oi4 io4 9 25i8 63n i5 32 848o 5299 6961 o5 9 8 544i 58 9 8 3922 II 97 2402 6496 58 o i5 845 7 5336 6915 0672 53 7 2 6008 3829 i345 2286 6681 45 3o 8434 53 7 3 6868 0746 53o2 6119 3 7 36 l4 9 2 2170 6865 3o 45 o.84io o.54io 1.6821 1.0819 2.5231 1.6229 3.3642 2.1639 42052 2. 7 049 i5 ^3 o 838 7 5446 6 77 3 0893 5i6o 633 9 354 7 i 7 86 1934 7 232 5 7 o i5 8363 5483 6726 0966 5o8 9 644 9 345i 1982 1814 7 4i5 45 3o 833 9 55i 9 6678 1039 5oi 7 6558 3355 20 77 1694 7 5 97 3o 45 83i5 5556 6629 ii 1 1 4g44 6667 3 2 5 9 2223 1573 7779 ID 34 o 8290 5592 658i u84 48 7 i 6776 3i62 2368 i452 7960 56 o i5 8266 56 2 8 6532 1256 4 79 8 6884 3o64 25l2 1329 8i4o 45 3o 8241 5664 6483 i328 4-724 6 99 2 2966 2656 1206 8320 3o 45 8216 5 7 oo 6433 i4oo 4649 7100 2866 2800 1082 85oo i5 35 o 8192 5 7 36 6383 1472 45 7 5 7207 2 7 66 2 9 43 0958 8679 55 o i5 0.8166 o.5 77 i 1.6333 i.i543 2.4499 i. 7 3i4 3.2666 2.3o86 4.o83 2 2.885 7 45 3o 8i4i 58o 7 6282 1614 4423 7421 2565 3228 0706 9 o35 3o 45 8116 5842 623i i685 434 7 7527 2463 33 7 o 5 79 9 2I2 i5 36 o 8090 58 7 8 6180 i 7 56 42 7 I 7634 236i 35n o45i 9 38 9 54 o i5 8064 5 9 i3 6129 1826 4193 77 3 9 2258 3652 j o322 9 565 45 3o 8o3 9 5 9 48 6077 1896 4n6 7845 2i54 3 79 3 oi 9 3 9741 3o 45 8oi3 5 9 83 6026 1966 4o38 79 5o 2o5o 3 9 33 oo63 9916 ID 87 o 7986 6018 5 97 3 2o36 3 9 5 9 8o54 1945 4o 7 3 3.9932 3.oo 9 i 53 o i5 7960 6o53 6920 2106 388o 8i5 9 i84o 4212 9800 0265 45 3o 7934 6088 586 7 2175 38oi 8263 I 7 34 435o 9668 o438 3o 45 0.7907 0.6122 i.58r4 1.2244 2.3 7 2I 1.8867 3.1628 2.448 9 3. 9 534 3.o6i i i5 33 o 7880 6i5 7 5 7 6o 23i3 364o 8470 l52O 4626 9401 0783 52 : i5 7 853 6191 5 7 o6 2382 356o 85 7 3 i4i3 4 7 64 9266 o 9 55 45 3o 7826 6225 5652 245o 34 7 8 86 7 5 i3o4 4 9 oi 9i3o 1126 3o 45 7799 6259 55 9 8 2618 33 97 8778 1 195 5o3 7 8 99 4 1296 i5 3o o 7771 6293 5543 2586 33i4 8880 1086 5i 7 3 885 7 i466 5i o i5 7744 632 7 5488 2654 3232 8981 0976 53o8 8 7 20 i635 45 3o 7716 636i 5432 2722 3i4 9 9082 o865 5443 858i 1804 3o 45 7688 63 9 4 53 77 2 7 8 9 3o65 9 i83 o 7 54 55 7 8 8442 1972 i5 4o o 7660 6428 532i 2856 2981 9284 0642 5 7 I2 8302 2139 5o o i5 0.7632 o.646i i.5 2 65 1.2922 2.28 97 1.9884 3.o529 2.5845 3.8162 3.23o6 45 3o 7604 64g4 5208 2 9 8 9 2812 9 483 o4i6 5 97 8 8020 2472 3o 45 7 5 7 6 65 2 8 5i5i 3o55 2 7 2 7 9 583 o3o3 61 10 7878 2638 i5 4i o 7 54 7 656r 5094 3l2I 264l 9682 0188 6242 77 35 28o3 4 9 o i5 75i8 65 9 3 5o3 7 3i8 7 2 555 9780 oo 7 4 63 7 4 7 5 9 2 2967 45 3o 7490 6626 4979 3252 2469 9879 2.9958 65o5 7 448 3i3i 3o 45 746 1 665 9 4921 33i8 2382 9976 9842 6635 7 3o3 8294 i5 42 743 1 6691 4863 3383 2294 2.00 7 4 9 7 26 6 7 65 7 i5 7 345 7 48 o i5 7402 6 7 24 48o4 344 7 22O 7 01 7 I 9609 68 9 5 7011 36i8 45 3o 7 3 7 3 6 7 56 4 7 46 35i2 2118 0268 9491 7024 6864 3 7 8o 3o 45 o. 7 343 o.6 7 88 1.4686 i.35 7 6 2.2O30 2.o364 2 . 9 3 7 3 2.7152 3.6 7 i6 3.3 9 4o i5 43 o 73i4 6820 462 7 364o I94l o46o 9 254 7280 6568 4ioo 4 7 o i5 7284 6852 456 7 3 7 o4 i85i o555 9 i35 7407 6419 425 9 45 3o 7254 6884 45o 7 3 7 6 7 1761 o65i 9 oi5 7534 6269 44i8 3o 45 7224 6 9 i5 444 7 383o 1671 o 7 45 88 9 5 7661 6118 45 7 6 i5 44 o 7 i 9 3 6 9 4 7 438 7 38 9 3 i58o o84o 8 77 4 7786 5 9 6 7 4 7 33 46 o i5 7i63 6 97 8 4326 3 9 56 1489 o 9 34 8652 7912 58i5 48 9 o 45 3o 7 i33 79 4265 4oi8 i3g8 I02 7 853o 8o36 5663 5o45 3o 45 7102 7 o4o 4204 4o8o i3o6 1120 84o 7 8161 55o 9 6201 i5 45 o 7 o 7 i 7071 4i42 4i42 I2l3 I2l3 8284 8284 5355 5355 45 o Dep. Lat Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat Dist. 1. Dist. 2. Dist. 3. Dist 4. Dist. 5. Course. TRAVERSE TABLE. 141 Course. Dist. 6. Dist. 7. || Dist. 8. Dist. 9. Dist. 10. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o / o r 30 ID 5.i83o 3.0226 6.o468 3.5264 6. 9 IO7 4.0302 7 . 7 745 4.5340 8.6384 5.o3 77 5 9 45 3o 1698 0452 o3i^ 55 2 8 8 9 3o o6o3 7547 56 7 8 6i63 o 7 54 3o 45 1 564 , 0678 oi58 5 79 i 8 7 53 o 9 o3 7347 6016 5 9 4i 1129 i5 3i o i43o 0902 OOO2 6o53 85 7 3 1203 7145 6353 5 7 i 7 i5o^ 5 9 o| i5 I2 9 5 1126 5. 9 844 63i4 83 9 3 1502 6 9 42 66 9 o 549i 1877 45 3o n58 i35o 9 685 65 7 5 8211 1800 6738 7025 5264 2250 3o 45 IO2I i5 7 3 9 5 2 5 6835 8028 ao 9 7 6532 7 35 9 5o35 2621 i5 32 o883 i 79 5 9 363 7094 7844 23 9 4 6324 7 6 9 3 48o5 2992 58 o 1 5 0744 2017 9 2OI 7 353 7658 268 9 6116 8o 2 5 45 7 3 336i 45 3o o6o3 2238 9 3 7 7611 7471 2 9 84 5 9 o5 835 7 433 9 3 7 3o 3o 45 5.o462 3.2458 5.88 7 3 3. 7 868 6.7283 4.3278 7 .56 9 4 4.8688 8.4io4 5.4097 i5 33 o O32O 2678 8707 8i25 70 9 4 35 7 i 548o 9 oi8 386 7 4464 57 o i5 0177 2898 854o 838i 6 9 o3 3863 5266 9 346 3629 4829 45 3o oo33 3n6 8372 8636 6711 4i55 5o5o 9 6 7 4 338 9 5i 9 4 3o 45 4.9888 3334 8203 88 9 o 65i8 4446 4832 S.oooi 3i4 7 555 7 i5 34 o 9?4a 3552 8o33 9 i44 6323 4735 46i3 o32 7 2904 5 9 i 9 56 o i5 gSgS 3 7 68 7861 9 3 9 6 6127 5o24 43 9 3 o652 265 9 6280 ,45 3o 9448 3 9 84 7 68 9 9 648 5 9 3o 53i2 4171 977 2 4i3 664i 3o 45 9299 4200 7 5i5 99 oo 5 7 3 2 56oo 3 9 48 i3oo ai65 7000 i5 35 o 9149 44i5 734i 4-oi5o 5532 5886 3724 1622 1915 7358 55 o i5 4.8 99 8 3.4629 5. 7 i65 4-o4oo 6.533i 4.6172 7 .34 9 8 5.i 9 43 8.i664 5.7715 45 3o 884 7 4842 6 9 88 o64 9 5l2 9 6456 3270 2263 l4l2 8070 3o 45 8694 5o55 6810 o8 97 4 9 26 6740 3o42 2582 n5 7 8425 i5 36 o 854: 5267 663i n45 4721 7023 2812 2 9 OI 0902 8779 54 o i5 838 7 5479 645 1 l3 9 2 45i6 7 3o5 2 58o 32i8 o644 9i3i 45 3o 823i 568 9 6270 i638 43o 9 7586 2347 3534 o386 9482 3o 45 8o 7 5 58 99 6088 i883 4ioo 7866 2Il3 384 9 OI25 9 832 i5 3y o 7918 6109 5 9 o4 2I2 7 38 9 i 8i45 1877 4i63 7.9864 6.0182 53 o i5 7760 63i8 5720 23 7 I 368o 8424 i64o 44 7 6 9600 0529 45 3o 7601 6526 5535 26i3 3468 8701 1402 4 7 8 9 9335 0876 3o 45 4.744i 3.6 7 33 5.5348 4-2855 6.3255 4.8977 7.1162 5.5ioo 7.9069 6.1222 :5 38 o 7281 6940 5i6i 3o 9 6 3o4i 9 253 9 2I 54io 8801 i566 52 O i5 7119 7146 4 9 72 333 7 2825 9 528 o6 79 5 7 i8 8532 1909 45 3o 6 9 56 735i 4 7 83 35 7 6 26o 9 9 8oi o435 6026 8261 225l 3o 45 6 79 3 V555 45 9 2 38i5 23 9 I 5.0074 oi 9 o 6333 7988 2592 i5 3o o 6629 77 5 9 44oo 4o52 2172 o346 6. 99 43 663 9 7 7 i5 2932 5i o i5 6464 7962 4207 428 9 I 9 5i 0616 9 6 9 5 6 9 43 743 9 3271 45 3o 6297 8i65 4oi4 4525 i 7 3o 0886 9 446 7247 7 l62 36o8 3o 45 6i3i 8366 38i 9 4 7 6i 1507 u55 9 i 9 6 7 55o 6884 3944 i5 4o o 5 9 63 856 7 3623 4 99 5 1284 1423 8 9 44 7 85i 66o4 4279 5o o i5 4.5 79 4 3.8767 5.3426 4.522 9 6.io5 9 5.i6 9 o 6.86 9 i 5.8i5i 7.6323 6.4612 45 3o 5624 8967 3228 546i o832 i 9 56 8437 845o 6o4i 4945 3o 45 5454 9166 3o3o 56 9 3 o6o5 2221 8181 8 7 48 5 7 56 52 7 6 i5 4i o 5283 9 364 2 83o 5 9 24 o3 77 2485 7 9 24 9045 54 7 i 56o6 49 o i5 5no 956i 262 9 6i54 0147 2748 7666 934i 5i84 5 9 35 45 3o 4 9 3 7 97 5 7 2427 6383 5. 99 i6 3oio 7406 9 636 48 9 6 6262 3o 45 4763 99 53 2224 6612 9 685 3271 7i45 9929 46o6 6588 i5 42 458 9 4.oi48 2O20 683 9 9 452 353o 6883 6.O222 43i4 6 9 i3 48 o i5 44i3 0342 1816 7066 9 2I 7 3739 6620 o5i3 4022 7 23 7 45 3o 4a3 7 o535 i6o 9 7 2 9 i 8 9 82 4047 6355 o8o3 3 7 28 7 55 9 3o 45 4-4o5 9 4.0728 5.i4o3 4-75i6 5.8 7 46 5.43o4 6.6o8 9 6.1092 7 .3432 6.7880 i5 43 o 388i 9 2O n 9 5 77 4o 85o8 456o 5822 i38o 3i35 8200 4? i5 3702 IIII o 9 86 79 63 8270 48i5 5553 1666 2 83 7 85i8 45 3o 3532 i3oi 0776 8i85 8o3o 5o68 5 2 84 I 9 52 253 7 8835 3o 45 3342 i4 9 i o565 84o6 7789 532i 5oi3 2236 2236 9 i5i i5 44 o 3i6o 1680 o354 8626 7547 5573 474i 2519 1934 9 466 46 o i5 2978 1867 oi4i 8845 73o4 58 2 3 446 7 2801 i63o 9779 45 3o 2 79 5 2o55 4. 99 28 9 o64 7060 6073 4i 9 3 3o8 2 i325 7 .oo 9 i 3o 45 2611 224l 97 i3 9 28l 68i5 632i 3 9 i 7 336i 1019 o4oi i5 45 o 2426 2426 9497 9497 656 9 656 9 364o 364o O 7 II O 7 II 45 o Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. 10 V Course. 142 MERIDIONAL PARTS. LATITUDE. Min. 1 2 3 4 5 6 7 8 9 10 11 12 Min. o.o 60.0 I20.O 180.1 240.2 3oo.2 360.7 421. 48i.6 542.2 6o3.i 664.1 7 25.3 o I I.O 61.0 21.0 81. 4i.a 01.2 61.7 22. 82.6 43.3 04.1 65.3 26.3 I 2 2.O 62.0 22.0 82. 42.2 O2.^ 62.7 23. 83.6 44.3 o5.i 66.1 2 7 .4 a 3 3.o 63.o 23.0 83. 43.2 o3.2 63. 7 24. 84-6 45.3 06. i 67.1 28.4 3 4 4.0 64-0 24.0 84. 44-2 04.^ 64.7 25. 85.6 46.3 07.1 68.2 29.4 4 5 5.o 65.o 25.0 85. 45.2 o5./ 65.7 26. 86.6 4 7 -3 08.2 69.2 3o.5 5 6 6.0 66.0 26.0 86. 46.2 o6./ 66.7 27. 87.6 48.3 09.2 70.2 3i.5 6 7 7.0 67.0 27.0 87. 47.2 07.4 67.7 28. 88.6 4 9 .3 10.2 71.2 32.5 7 8 8.0 68.0 28.0 88. 48.2 08.4 68.7 29. 89.6 5o.3 II. 2 72.2 33.5 8 9 9.0 69.0 29.0 89. 49.2 09.2 69.7 3o. 90.7 5i.4 12.2 7 3.3 34.5 9 10 10. 70.0 iSo.o 190. 250.2 3ioJ 370.7 43i. 491.7 552.4 6l3.2 6 7 4.3 7 35.6 10 ii II. 71.0 3i.o 91. 5l.2 1 1. L 71.7 32. 92.7 53.4 14.2 7 5.3 36.6 ii 12 12.0 72.0 32.0 92. 52.2 12.1 72.7 33. 9 3. 7 544 i5.3 76.3 3 7 .6 12 i3 i3.o 7 3.o 23.o 9 3. 53.2 I 3.4 7 3. 7 34.2 94.7 55.4 i6.3 77.3 38.6 i3 i4 i4o 74.0 34.o 94- 54.2 l4-4 74-7 35.2 95.7 56.4 i 7 .3 78.4 3 9 .6 i4 i5 i5.o 75.0 35.o 9 5. 55.2 i5.4 7 5. 7 36.2 96.7 57-4 i8.3 79-4 4o. 7 i5 16 16.0 76.0 36.o 96. 56.2 16.4 76.8 3 7 .2 97-7 58.4 19.3 80.4 4i. 7 16 *7 17.0 77.0 37.0 97- 57.2 i 7 .5 77.8 38.2 98.7 5 9 .4 20.3 81.4 42. 7 '7 18 18.0 78.0 38.o 98. 58.2 iS.5 78.8 39.2 99.8 6o.5 21.3 82.4 43. 7 18 '9 19.0 79.0 39.0 99. 59.2 i 9 .5 79.8 40.2 5oo.8 6i.5 22.4 83.5 44.8 '9 20 2O.O 80.0 i4o.o 200. i 260.2 320.5 38o.8 441.2 5oi.8 562.5 623.4 684.5 7 45.8 2C 21 21. 81.0 4i.o OI.I 6i.3 21.5 81.8 42.2 02.8 63.5 244 85.5 46.8 21 22 22. 82.0 42.0 02.1 62.3 22.5 82.8 43.2 o3.8 64-5 25.4 86.5 4 7 .8 22 23 23.0 83.o 43.o o3.i 63.3 23.5 83.8 44.2 o4-8 65.5 26.4 8 7 .5 48. 9 23 24 24.O 84.0 44.o 04.1 64-3 24.5 84-8 45.2 o5.8 66.6 27.4 88.6 49-9 24 25 25. 85.o 45.o o5. 65.3 25.5 85.8 46.3 06.8 67.6 28.5 89.6 5o. 9 25 26 26.0 86.0 46.o 06. 66.3 26.5 86.8 47-3 07.8 68.6 20.5 90.6 5i. 9 26 2? 27.0 87.0 47.0 07. 6 7 .3 27.5 87.8 48.3 08.9 69.6 3o.5 91.6 53.o 27 28 28.0 88.0 48.o 08. 68.3 28.5 88.8 49-3 09.9 70.6 3i.5 92.6 54.0 28 2 9 29.0 89.0 49.0 09.1 69.3 29.5 89.8 5o.3 10.9 71.6 32.5 93.6 55.o 2 9 3o 3o.o 90.0 iSo.o 2IO.I 270.3 33o.5 3 9 o.8 45i.3 5 1 1. 9 572.6 633.5 694.7 7 56.o 3o 3i 3i.o 91.0 5i.o II. I 7 i.3 3i.5 91.8 52.3 12.9 7 3. 7 34.6 9 5. 7 5 7 .i 3i 32 32.0 92.0 52. 12. 1 7 2.3 32.5 92.9 53.3 i3. 9 74-7 35.6 96.7 58.i 32 33 33.o 93.0 53.1 i3.i 7 3.3 33.5 9 3. 9 54.3 i4-9 7 5. 7 36.6 97 : 7 5 9 .i 33 34 34.o 94.0 54-1 14.1 7 4.3 34.5 94.9 55.3 i5. 9 76.7 3 7 .6 98.7 60.1 34 35 35.o 95.0 55.i i5.i 7 5.3 35.5 9 5. 9 56.3 16.9 77-7 38.6 99.8 61.1 35 36 36.o 96.0 56.i 16.1 7 6.3 36.5 96.9 5 7 .3 18.0 78.7 39.6 700.8 62.2 36 3? 3 7 .o 97.0 5 7 .i 17.1 77-3 3 7 .5 97-9 58.4 19.0 79-7 40.7 01.8 63.2 37 38 38.o 98.0 58.i 18.1 7 8.3 38.5 98.9 59.4 20. o 80.8 4i.7 02.8 64.2 38 3 9 39.0 99.0 69.1 19.1 79 .3 3 9 .6 99-9 60.4 2 I.O 81.8 42.7 o3.8 65.2 39 4o 4o.o IOO.O 160.1 220. 2 280.3 34o.6 ^00.9 46i.4 522.0 582.8 643. 7 704.9 7 66.3 4o 4i 4i.o 01. 61.1 21.2 8i.3 4i.6 01.9 62.4 23.0 83.8 44-7 o5. 9 6 7 .3 4i 42 42.0 02. o 62.1 22.2 82.3 42.6 02.9 63.4 24.0 84.8 45.8 06.9 68.3 42 43 43.o o3.o 63.i 23.2 83.3 43.6 03.9 64-4 25. 85.8 46.8 07.9 69.3 43 44 44.0 o4.o 64.i 24.2 84.3 44-6 o4>9 65.4 26.0 86.8 4 7 .8 09.0 70.4 44 45 45.o o5.o 65.i 25.2 85.3 45.6 5.9 66.4 27.1 87.9 48.8 IO.O 7 i.4 45 46- 46.o 06.0 66.1 26.2 86.3 46.6 07.0 67.4 28.1 88.9 4 9 .8 I I.O 72.4 46 47 47-0 07.0 67.1 27.2 8 7 .3 47-6 08.0 68.4 29.1 89.9 5o.8 12.0 73.4 47 48 48.o 08.0 68.1 28.2 88.3 48.6 09.0 69.5 3o.i 90.9 5i. 9 i3.i 74.5 48 49 49-0 09.0 69.1 29.2 8 9 .3 49.6 10.0 70.5 3i.i 91.9 52. 9 14.1 7 5.5 49 5o 5o.o IIO.O 170.1 230.2 290.3 35o.6 4n.o 471.5 532.1 5 9 2. 9 653. 9 7 i5.i 776.5 5o 5i 5i.o II. 71.1 3l.2 9 i.3 5i.6 12. 72.5 33.1 9 3. 9 54- 9 16.1 77 .5 5i 5 2 52. 12.0 72.1 32.2 92.4 52.6 i3.o 7 3.5 34-1 95.0 55. 9 17.1 78.6 52 53 53.o i3.o 7 3.i 33.2 93.4 53.6 i4-o 74-5 35.i 96.0 5 7 .o 18.2 7 9 .6 53 54 54.o 14.0 74.1 34.2 944 54.6 i5.o 7 5.5 36.2 97.0 58.o 19.2 80.6 54 55 55.o i5.o 7 5.i 35.2 95.4 55.6 16.0 76 5 3 7 .2 98.0 59.0 20.2 81.7 55 56 56.o 16.0 76.1 36.2 96.4 56.6 17.0 77 .5 38.2 99.0 60.0 21.2 82.7 56 57 57.0 17.0 77.1 3 7 .2 97-4 5 7 .6 18.0 7 8.5 39.2 600.0 61.0 22.3 83. 7 57 58 58.o 18.0 78.1 38.2 9B-4 58.6 19.0 79.5 4O.2 01. 62.1 23.3 84-7 58 5 9 Sg.o 19.0 79.1 39.2 99.4 5 9 . 7 20. o 8o.5 41.2. 02. 1 63.i 24-3 85.8 5 9 MERIDIONAL PARTS. 143 LATITUDE. 1 13 14 15 16 17 18 19 20 21 22 23 24 .a 53 o 786.8 848.5 910.5 972.7 io35.3 1098.2 u6i.5 1225. 1 I28 9 .2 i353. 7 i4i8.6 i484-i o i 87.8 49.5 i.i-.5 73.8 36.3 99 .3 62.5 26.2 9 o.3 54.8 i 9 .7 85. 2 I 2 88.8 5o.5 12.6 74-8 3 7 4 noo.3 63.6 2 7 .3 9 i.3 55.8 20.8 86.3 2 3 89.9 5i.6 i3.6 7 5. 9 38.4 01.4 64-7 28.3 9 2.4 56-9 21.9 8 7 .3 3 4 90.9 52.6 i4-6 76.9 3 9 .5 02.4 65.7 29.4 9 3.5 58.o 23.0 88.4 4 5 91.9 53.6 i5. 7 78.0 4o.5 o3.5 66.8 3o.4 9 4.5 Sg.o 24.1 8 9 .5 5 6 92.9 54.7 16.7 79.0 4i.6 o4.5 67.8 3i.5 9 5.6 60. i 25-1 9 o.6 ti 7 94.0 55. 7 17.7 80.0 42.6 o5.6 68. 9 32.6 9 6 -7 61.2 26-2 91.7 7 8 95.0 56. 7 18.8 81.1 43.7 06.6 70.0 33.6 97-8 62.3 27.3 9 2.8 8 9 96.0 5 7 .8 19.8 82.1 44-7 07.7 71.0 34.7 9 8.8 63.4 28.4 9 3. 9 9 10 797.0 858.8 920.8 983.2 io45.8 1108.7 1172.1 1235.8 I2 99 . 9 i364.5 i42 9 .5 i4 9 5.o 10 ii 98.1 59.8 21.9 84.2 46.8 09.8 7 3.i 36.8 iSoi.o 65.6 3o.6 9 6.i ii 12 99.1 60.9 22.9 85.2 47-9 10.8 74.2 3 7 . 9 O2.O 66.6 3i. 7 97- 2 12 i3 800.1 61.9 23.9 86.3 48. 9 11.9 75.2 39.0 o3.i 67.7 32.8 9 8.3 i3 i4 OI.2 62.9 25. 87.3 49.9 12.9 76.3 4o.o 04.2 68.8 33. 9 99-4 i4 i5 O2. 2 64-o 26.0 88.4 5i.o 14.0 77-4 4i.i o5.3 69.9 34.9 i5oo.5 i5 16 03.2 65.o 27.0 89.4 52.0 i5.o 78.4 42.2 o6.3 70.9 36.o 01.6 16 !? 042 66.0 28.1 90.4 53,1 16.1 79 .5 43.2 07.4 72.0 87.1 02.7 7 18 o5.3 67.1 29.1 9 i.5 54-1 17.1 8o.5 44-3 o8.5 7 3.i 38.2 o3.8 18 J 9 06.3 68.1 3o.i 92.5 55.2 18.2 81.6 45.4 o 9 .6 74-2 3 9 .3 o4-9 J 9 20 807.3 869.1 9 3l.2 99 3.6 io56.2 1119.2 1182.7 1246.4 i3io.6 i3 7 5.3 i44o.4 i5o6.o 20 21 08.4 70.1 32.2 9 4-6 5 7 .3 20.3 83. 7 47-5 11.7 76.4 4i.5 07.1 21 22 09.4 71.2 33.3 9 5.6 58.3 21.3 84.8 48.6 12.8 77-4 42.6 08.2 22 23 10.4 72.2 34-3 9 6 -7 69.4 22.4 85.8 49.6 i3.8 7 8.5 43.7 09.3 23 24 11.4 ^3.2 35.3 97-7 60.4 23.4 86. 9 5o.7 i4. 9 79-6 44-8 10.4 24 25 12.5 "4.3 36.3 9 8.8 6i.4 24.5 88.0 5i.8 16.0 80.7 45.8 n. 5 26 26 i3.5 7 5.3 3 7 4 99 .8 62.5 25.5 8 9 .o 62.8 17.1 81.8 46. 9 12.6 26 2 7 i4.5 7 6.3 38.4 1000.8 63.5 26.6 9 o.i 53. 9 18.1 82.8 48.o i3. 7 27 28 i5.5 77-4 39.5 oi. 9 64.6 27.6 91.1 55.o 19.2 83. 9 4 9 .i i4.8 28 2 9 16.6 78.4 4o.5 02. 9 65.6 28.7 9 2.2 56.o 20.3 ,85.o 50.2 i5. 9 29 3o 817.6 8794 9 4i.6 ioo4o 1066.7 1129.7 II 9 3.2 1257.1 i32i.4 i386.i i45i.3 1517.0 3o 3i 18.6 8o.5 42.6 o5.o 67.7 3o.8 9 4-3 58.2 22.5 87.2 52.4 18.1 3i 32 19.6 8i.5 43.6 06. i 68.8 3i.8 9 5.4 59.2 23.5 88.3 53.5 19.2 32 33 20.7 82.5 44-7 07.1 69.8 32.9 9 6.4 6o.3 24.6 89.^ 54.6 20.^ 33 34 21.7 83.6 45-7 08. i 70.9 34.o 97 .5 61.4 2 5. 7 90.4 55.6 21.^ 34 35 22.7 84.6 46. 7 9 .2 72.0 35.1 9 8.5 62.4 26.7 9 i.5 66.7 22.5 35 36 23.8 85.6 4 7 -8 IO.2 73.0 36.i 99.6 63.5 27.8 92.6 5 7 .8 23.6 36 3? 24.8 86.7 48.8 II. 3 74.1 3 7 .2 1200.7 64-6 28.9 9 3. 7 58. 9 24.7 37 38 25.8 87.7 49.9 12.3 7 5.i 38.2 01.7 65.6 3o.o 94.8 60.0 25.8 38 3 9 26.9 88.7 5o. 9 1 3.4 76.2 3 9 .3 02. 8 66.7 3i.i 9 5.8 61.1 26.9 3 9 4o 827.9 889.8 961.9 ioi4-4 1077.2 n4o.3 I2o3. 9 1267.8 i332.i i3 9 6. 9 :462.2 1 528.0 4o 4i 28.9 90.8 53.o 1 5.4 7 8.3 4i4 04.9 68.8 33.2 9 8.o 63.3 29.1 4i 42 29.9 91.8 54.o i6.5 79 .3 42.4 06.0 69.9 34.3 99 .i 644 30.2 42 43 3i.o 92.9 55.i i 7 .5 80.4 43.5 07.1 71.0 35.3 I4OO.2 65.5 3i.3 43 44 32.0 9 3. 9 56.i 18.6 81.4 44.6 08. i 72.1 36.4 OI. 66.6 32.4 44 45 33.o 94.9 5 7 .i 19.6 82.5 45.6 09.2 7 3.i 3 7 .5 O2./! 67.7 33.5 45 46 34.i 96.0 58.2 2O.6 83.5 46. 7 IO.2 74.2 38.6 o3.4 68.8 34.6 46 4? 35.i 97.0 59.2 21.7 84-6 47-7 ii. 3 7 5.3 3 9 . 7 o4-5 6 9 .8 3/>. 7 4? 48 36.i 98.0 60.2 22.7 85.6 48.8 12.4 7 6.3 4o. 7 o5.6 7-9 36.8 48 49 3 7 .2 99.1 6i.3 23.8 86.7 4 9 . 9 1 3.4 77-4 4i.8 06.7 72.0 3 7 . 9 49 5o 838.2 900.1 962.2 1024.8 1087.7 n5o. 9 i2i4.5 !2 7 8.5 1342.9 1407.8 i473.i iSSg.o 5o 5i 3g.2 01. 1 63.4 2 5. 9 88.8 52.0 i5.5 79 .5 44-0 08.8 74.2 4o.i 5i 62 4O.2 O2. 2 64.4 26. 9 89.8 53.o 16.6 80.6 45.i 9 ! 9 7 5.3 4i.a 5s 53 4i.3 03.2 65.5 28.0 9 9 54.i 17.7 81.7 46.i II. O 76.4 42.3 53 54 42.3 o4.3 66.5 2 9 .0 91.9 55.i 18.7 82.8 47.2 12. 1 77 .5 43.4 54 55 43.3 o5.3 6 7 .5 3o.i 93.0 56.2 19.8 83.8 48.3 13.2 78.6 44-5 55 56 444 o6.3 68.6 Sz.i 94.0 5 7 .2 20.9 84-9 49.4 i4-3 79-7 45.6 56 5 7 45.4 07-4 69.6 32.2 9 5.i 58.3 21.9 86.0 5o.4 1 5.4 80.8 46. 7 57 58 46-4 08.^ 70.7 33.2 96.1 5 9 .4 23.0 87.0 5i.5 i6.5 8i. 9 4 7 -8 58 5 9 47-5 09. i 71.7 34.3 97.2 60.4 24.1 88.1 52.6 i 7 .5 83.o 46-9 5 9 M E ii i D i o N A L PARTS. LATITUDE. MlE. 25 26 27 28 29 30 31 32 33 54 35 Min. i55o.o i6i6.5 !683.5 I 7 5l.2 i8i 9 4 1888.4 i 9 58.o 2028.4 200, 0,. 5 2171.5 2244.3 I 5i.i 17.6 84.6 52.3 2O.6 8 9 .5 5 9 . 2 2 9 .6 2100.7 72.7 45.5 I 2 52.2 18.7 85.8 534 21.7 90.7 60.4 3o. 7 01.9 73.9 46.8 2 3 53.3 i 9 .8 86. 9 54.6 22. 9 91.9 61.6 3r. 9 03.! 7 5.i 48.o 3 4 544 2O. 9 88.0 55. 7 24.O 9 3.o 62.7 33.i o4.3 7 6.3 49.2 4 5 55.5 22. 89.1 56.8 25.2 9 4-i 63. 9 34.3 o5.5 77 .5 5o4 5 6 56.6 23.2 90.3 58.o 26.3 9 5.3 65.o 35.5 06.7 78.7 5i.6 6 7 5 7 . 7 24.3 91.4 5 9 .i 27.5 9 6.5 66.2 36. 7 07.9 80.0 52. 9 7 8 58.8 254 92.5 60.2 28.6 97-6 6 7 4 3 7 .8 09.1 81.2 54.1 8 9 5 9 . 9 26.5 93.6 61.4 2 9'7 9 8.8 68.5 3 9 .o io.3 82.4 55.3 9 10 i56i.o 1627.6 1694.8 1762.5 i83o. 9 i8 99 . 9 i 9 6 9 . 7 2O4O.2 2III.5 2i83.6 2256.5 10 ii 62.1 28.7 9 5. 9 63.6 32.0 70-9 4x4 12.7 84.8 5 7 .8 i 12 63.2 20.8 9-7.0 64.8 33.2 02.3 72.0 42.6 13-9 86.0 5 9 .o 12 r3 644 3i.o 98.1 65. 9 34-3 o34 7 3.2 43.8 87.2 60.2 i3 i4 65.5 32.1 99 .3 67.0 35.5 o4-6 7 44 44. 9 i6.3 884 61.4 i4 i5 66.6 33.2 I 7 00.4 68.2 36.6 o5. 7 7 5.6 46.i i 7 .5 8 9 .6 62. 7 i5 16 67.7 34.3 oi.5 6 9 .3 3 7 .8 06. 9 7 6.8 4 7 .3 18.7 9 o.8 63. 9 16 '7 68.8 35.4 02.6 70.5 38. 9 08.1 77-9 48.5 19.8 02. o 65.i i 7 18 6 9 . 9 36.5 o3.8 71.6 4o.i O 9 .2 79 .i 49-7 21.0 9 3.3 66.3 18 J 9 71.0 3 7 .6 o4- 9 72.7 41.2 IO.4 80.2 5o.8 22.2 944 6 7 .5 J 9 20 1572.1 1638.8 I 7 o6.o 1773.9 1842.4 i 9 n.5 i 9 8i.4 2O52.O 21234 2195.7 2268.8 20 21 7 3.2 3 9 . 9 O 7 .l 75.0 43.5 12.7 82.6 53.2 24.6 96.9 7 o.o 21 22 74.3 4i.o o8.3 76.1 44.6 i3.8 83. 7 544 25.8 98.1 7 1.2 22 23 754 42.1 o 9 4 77.2 45.8 i5.o 84.0. 55.6 27.0 99 .3 7 2.5 23 24 76.5 43.2 io.5 784 46. 9 16.2 86.1 56.8 28.2 220O.5 73.7 24 25 77.6 44-3 ii. 6 7 9 .5 48.i i 7 .3 8 7 .3 58.o 20.4 01.7 25 26 78.7 45.5 12.8 80.6 4 9 .2 i8.5 884 5 9 .i 3o.6 o3.o 7 6.i 26 27 79-8 46.6 i3. 9 81.8 5o4 i 9 .6 8 9 .6 6o.3 3i.8 04.2 774 27 28 8o. 9 4 7 -7 i5.o 83.o 5i.5 20.8 9 o.8 6i.5 33.o o54 7 8.6 28 2 9 82.1 48.8 16.1 84.1 5 2 . 7 21. 9 9 2.O 62. 7 34.2 06.6 79-8 29 3o i583.2 i64 9 . 9 I7I7-3 1785.2 i853.8 I 9 23.I ! 99 3.I 2o63. 9 2i354 2207.8 2281.0 3o 3r 84.3 5i.o 18.4 864 55.o 24.3 9 4.3 65.i 36.6 09.0 82.3 3i 32 854 52.2 I 9 .5 8 7 .5 56.i 254 9 5.5 66.2 3 7 .8 10.2 83.5 32 33 86.5 53.3 20. 7 88.6 5 7 .2 26.6 9 6.6 6 7 4 3 9 .o II.4 84- 7 33 34 87.6 544 21.8 8 9 .8 584 27.8 97-8 68.6 4o.2 12-7 86.0 34 35 88.7 55.5 22. 9 9-9 5 9 .6 28.0 99- 6 9 .8 4i4 1 3.9 87.2 35 36 8 9 .8 56.6 24.0 9 2.1 60. 7 3o.i 2OOO.2 71.0 42.6 i5.i 884 36 37 9-9 5 7 .8 25.2 9 3.2 61.9 3i.3 01.3 7 2.2 43.8 i6.3 89-7 37 38 9 2.O 58. 9 26.3 9 4.3 63.o 32.4 02. 5 7 34 45.o i 7 .5 9-9 38 3 9 9 3.: 60.0 2 7 4 9 5.5 64-2 33.6 o3. 7 7 4-5 46.2 18.7 9 2.1 3 9 4o i5 9 4.3 1661.1 1-728.6 i7 9 6.6 i865.3 i 9 34.7 2OO4. 9 20 7 5. 7 2147.4 2219.9 22 9 3.3 4o 4i 96.4 62.2 2 9 . 7 97-8 66.5 35. 9 O6.O 7 6. 9 48.6 21.2 94.6 4i 42 9 6.5 634 3o.8 98.9 6 7 .6 3 7 .i 07.2 78.1 4 9 .8 22.4 9 5.8 42 43 976 64.5 3i. 9 1800.0 68.8 38.2 084 79 .3 5i.o 23.6 97.0 43 44 98.7 65.6 33.i 01.2 69.9 3 9 4 o 9 .6 8o.5 52.2 24.8 98.3 44 45 99-8 66.7 34.2 02.3 7 i.i 4o.5 10.7 81.7 53.4 26.0 99 .5 45 46 i6oo. 9 67.8 35.3 o3.5 7 2.2 4i. 7 11.9 82.0 54.6 27.2 23oo. 7 46 47 02. o 6 9 .o 36.5 o4.6 7 34 42. 84.o 55.8 28.5 02. 47 48 o3.i 70.1 3 7 .6 05.7 7 4.5 44.o i4^3 85.2 57.0 29.7 03.2 48 49 04.2 71.2 38. 7 o6. 9 7 5. 7 45.2 1 5.4 864 58.2 30.9 o44 49 5o i6o54 i6 7 2.3 I 7 3 9 . 9 1808.0 i8 7 6.8 i 9 464 2016.6 2087.6 2i5 9 4 2232.1 2 3o5. 7 5o 5i o6.5 7 34 4i.o 9 .2 7 8.o 4 7 .5 17.8 88.8 60.7 33.3 06.9 5i 52 07.6 7 4-5 42.1 io.3 79.2 48. 7 I 9 .0 90.0 6i. 9 34-6 08. 1 53 53 08.7 75.7 43.2 11.4 8o.3 4 9 . 9 20.2 91.2 63.i 35.8 09.4 53 54 o 9 .8 7 6.8 444 12.6 8i.5 5i.o 21.3 92.4 64.3 37.0 10.6 54 55 10. 9 77-9 45.5 i3. 7 82.6 52.2 22.5 93.6 65.5 38.2 11.8 55 56 12. 79- 46.6 I4.9 83.8 534 23. 7 94.8 66.7 3 9 4 i3.i 56 57 i3.i 80.2 4 7 -8 16.0 84. 9 54.5 96.0 67-9 4o. 7 i4.3 57 58 14.2 8i.3 48. 9 17.2 86.1 55. 7 26.0 97.1 6 9 -i 41.9 i5.5 58 5 9 i54 82.4 5o.o i8.3 87.2 56. 9 27.2 98.3 70.3 43.i 1 6.8 5 9 MERIDIONAL PARTS. 145 LATITUDE. 1 MiD |T0 37 38 39 40 41 42 43 44 45 46 m, o I23i8.o 23 9 2.6 2468.3 2545.o 2622.7 2701.6 2781.7 2863.1 2 9 45. 8 3o3o.o 3n5.6 O I 19.2 93.9 6 9 .5 46.2 24.0 02.9 83.i 64.5 47.2 3i.4 17.0 I 2 20.5 9 5.i 70.8 47-5 2 5.3 o4-3 84-4 65.8 48.6 32.8 i8.5 2 3 21. 7 96.4 72.1 48.8 26.6 o5.6 85.8 67.2 5o.o 34.2 i 9 . 9 3 4 23.0 97-7 73.4 5o.i 27.9 06.9 87.1 68.5 5i.4 35.6 21.4 4 5 24.2 98.9 74.6 5i.4 29.2 o8.3 88.5 70.0 52.8 3 7 .o 22.8 5 6 25.4 2400.2 7 5. 9 52.7 3o.5 09.6 89.8 7 i.3 54-2 38.4 24.2 6 7 26.7 01.4 77-i 54.o 31.9 10.9 91.2 72.7 55.6 3 9 .8 25.7 7 8 2-7.9 02.7 78.5 55.3 33.2 12.2 92.5 74.1 57.0 4i.3 27.1 8 9 29.1 03.9 79-7 56.6 34.5 i3.5 9 3.8 75.4 58.3 42.7 28.5 9 10 233o.4 24o5.2 2481.0 2 55 7 .8 2635.8 2714.9 2 79 5.i 2876.8 2959.8 3o44.i 3i3o.o 10 ii 3i.6 06.4 82.2 5 9 .i 3 7 .i I 6.2 9 6.5 78.2 61.1 45.5 3i.5 n 12 32.9 07.7 83.5 60.4 38.4 17.5 97-9 79 .5 62.5 47.0 32. 9 12 i3 34.i 09.0 84-8 61.7 3 9 . 7 18.9 99.3 80.9 63.9 48.4 34.3 i3 i4 35.3 10.2 86.1 63.o 4i.o 20.2 2800.6 82.3 65.3 49-8 35.8 i4 i5 36.6 ii. 5 87.4 64-3 42.3 21.5 02. o 83. 7 66.7 5i.? 3 7 .2 i5 16 3 7 .8 12.7 88.6 65.6 43.6 22.9 o3.3 85.o 68.1 52.6 38.7 16 J 7 3 9 .o i4o 89.9 66. 9 44.9 24-2 04.7 86.4 6c;.5 54-1 4o.i 7 18 4o.3 15.2 91.2 68.2 46.3 2 5.5 06.0 87.8 70.9 55.5 4i.6 18 9 4i.5 i6.5 92.4 69.5 47-6 26.8 o 7 .3 89.1 72.3 56. 9 43.o '9 20 2342.8 2417-8 24 9 3. 7 25 7 o. 7 2648.9 2728.212808.8 2890.5 2973.7 3o58.3 3i44-5 20 21 44.o 19.0 95.0 72.0 5o.a 29.5 10. 1 91.9 7 5.i 59.7 45-9 21 22 45.3 20.3 96.3 7 3.3 5i.5 3o.8 11.4 9 3.3 76.5 61.2 47-4 22 23 46.5 21.5 97.6 7 4-6 52.8 02.2 12.8 94-7 77-9 62.6 48.8 23 24 47-7 22.8 98.8 7 5. 9 54-1 33.5 14.1 96.0 .79.3 64-0 5o.3 24 25 49.0 24.0 25oo.i 77.2 55.5 34.8 i5.5 974 80.7 65.4 5i. 7 25 26 5o.2 25.3 01.4 78.5 56.8 36.2 16.8 98.8 82.1 66.9 53.2 26 27 5j.5 26.5 02.7 79.8 58.i 3 7 .5 18.2 2900.2 83.5 68.3 54-6 2 7 28 52. 7 27.8 03.9 81.1 5 9 .4 38.8 i 9 .5 oi.5 84.9 69.7 56.i 28 29 54.0 29.1 05.2 82.4 66.7 40.2 20.9 02.9 86.3 7-1.1 5 7 .5 2 9 3c 2355.2 243o.3 2 5o6.5 a583. 7 2662.0 2741.5 2822.3 2904.3 2987.7 3072.6 SiSg.o 3o 3i 56.5 3i.6 07.8 85.o 63.3 42.9 23.6 05.7 89.1 74.o 60.4 3i 32 5 7 . 7 32.9 09.0 86.3 64-6 44-2 25. 07.1 90.5 7 5.4 61.9 32 33 58. 9 34-1 io.3 8 7 .6 66.0 45.5 26.3 08.4 91.9 76.9 63.3 33 34 60.2 35.4 n.6 88.9 6 7 .3 46. 9 27.7 09.7 9 3.3 7 8.3 64-8 34 35 6i.4 36-7 12.9 90.2 68.6 48.2 29.0 II. 2 9 4-7 79-7 66.2 35 36 62 7 3 7 . 9 14.2 9 i.5 69.9 49.5 3o.4 12.6 9 6.i 81.1 6 7 . 7 36 37 63. 9 39.2 16.4 9 2.8 71.2 So.g 3i. 7 i4-o 9 7.5 82.6 69.1 37 38 65.2 4o.4 16.7 9 4.i 7 2.5 52.2 33.i i5.3 98.9 84.0 7 o.6 38 3 9 66.4 4i. 7 18.0 9 5.4 7 3. 9 53.5 34-5 16.7 3ooo.3 85.4 7 2.O 3 9 4o 236 7 .6 2443.0 2519.3 25 9 6. 7 2675.2 2754.9 2835.8 2918.1 3ooi.8 3086.9 3i 7 3.5 4o 4i 68.9 44-2 2O.5 9 8.o 7 6.5 56.2 3 7 .2 19.5 03.2 88.3 7 5.o 4i 42 7 O.2 45.5 21.8 99- 3 77.8 5 7 .6 38.6 20.9 o4,6 89.7 7 6.4 42 43 71.4 46.8, 23.1 2600.5 79.1 58. 9 3y. 9 22.3 06.0 91.2 77-9 43 44 72.6 48.o 24.4 oi. 9 8o.5 60.2 4i.3 23.6 07.4 92.6 79 .3 44 45 73-9 49.3 25.7 03.2 81.8 6i.5 42.6 25 08.8 9 4.o 80.8 45 46 7 5.i 5o.6 27.0 o4.5 83.i 62.9 44.o 26.4 IO.2 9 5.5 82.3 46 47 76.4 5i.8 28.3 o5.8 84-4 64.3 45.4 27.8 n.6 9 6. 9 83. 7 47 48 77.6 53.i 29.5 07.1 85. 7 65.6 46. 7 29.2 i3.o 9 8.3 85.2 48 49 78.9 54-3 3o.8 08.4 87.1 66.9 48.i 3o.6 i44 99-7 86.6 4 9 5o 238o.i 2455.6 2532.1 26o 9 . 7 2688.4 2768.3 2849.5 2932.0 3oi5.8 3lOI.2 3i88.i 5o 5i 81.4 56.9 33.4 II. 89.7 69.6 5o.8 33.3 17.2 02. 6 89.6 5i 52 82.6 58.i 34. 7 12.3 91.0 71.0 52.2 34-7 18.7 04.1 91.0 52 53 83.9 5 9 .4 36.o i3.6 02.3 7 2.3 53.5 36.i 20. i o5.6 92.5 53 54 85.i 60.7 3 7 .2 i4- 9 93.7 7 3. 7 54-9 3 7 .5 21.5 07.0 94.0 54 55 86.4 61.9 38.5 16.2 95.0 75.0 56.3 38. 9 22. 9 o8.4 9 5.4 55 56 87.6 63.2 3 9 .8 i 7 .5 96.3 7 6.3 5 7 . 7 4o.3 24-3 09.8 96.9 56 5? 88.9 64.5 4i.i 18.8 9.7.6 77-7 5g.o 4i-7 2 5. 7 II. 2 98.4 5 7 58 90.2 65.8 42-4 20.1 99.0 79.0 6o.5 43.i 27.1 12.7 99 8 58 69 91.4 67.0 43.6 21.4 2700.3 80.4 61.7 44.4 2^.5, 14.1 8201.3 5) K MERIDIONAL PARTS. LATITUDE. Min. 47 48' 49 50 51 52 53 54 55 56 57 3202.7 3291.5 3382.1 34 7 4.5 3568.8 3665.2 3 7 63.8 3864.6 3968.0 4o73.q 4182 6 I o42 93.0 83.6 76.0 70.4 66.8 65.4 66.3 6 9 . 7 vS. 1 - 84.5 2 5.7 9 4.5 85.i 77.6 72.0 b8.4 67.1 68.0 7 i.5 77- 86.3 3 07.1 96.0 86.7 79-i 7 3.6 70.1 68.8 6 9 . 7 7 3.2 79.3 88.1 A 08.6 97 .5 88.2 80.7 7 5.2 71.7 70.4 71.5 7 5.o 8l.T 90.0 5 I O.I 99.0 89.7 82.3 76.8 7 3.3 72.1 7 3.2 76.7 82.9 91.8 6 ii. 5 33oo.5 9 i.3 83.8 78.4 75.o 7 3. 7 7 4. 9 784 84.7 ;3-7 7 i3.o 02.0 92.8 85.4 79-9 76.6 75.4 76.6 80.2 86.4 95.5 8 14.5 o3.5 9 4.3 87.0 8i.5 78.2 77.1 7 8.3 82.0 88.2 97-3 9 i5. 9 OD.O 9 5.8 88.5 83.i 79.8 78.7 80.0 83. 7 90.0 99- 2 10 3217.4 33o6.5 33 97 .4 34 9 o.i 3584-7 368i.5 3780.4 388i. 7 3 9 85.4 4091.8 4201.0 ii 18.9 08.0 9 8. 9 91.6 86.3 83.i 82.1 83.4 87.2 93.6 02. 9 12 2O.3 o 9 .5 34oo.4 9 3.2 87.9 84-7 83. 7 85.i 88. 9 9 5.4 04.7 i3 21.8 II. 02.0 94.7 8 9 .5 86.4 85.4 86.8 9-7 97- 2 06.6 i4 23.3 12.5 o3.5 96.3 91.1 88.0 87.1 88.5 ' L 9 2.5 99- 08.4 i5 24.8 i4-o o5.o 97-9 92.7 89.6 88.8 9 0.2 9 4.2 4ioo.8 io.3 16 26.2 i5.5 06.6 99.4 9 4-3 9 i.3 9 o.4 9 2.0 9 6.o 02.6 12. 1 I 7 27.7 17.0 08. i 35oi.o 9 5. 9 92.9 9 2.1 93-7 97-7 o44 14.0 18 2 9 .2 i8.5 o 9 .6 02.6 97-5 9 4.5 9 3.8 9 5.4 99 .5 06.2 1.5.8 1 9 3o. 7 20.O ii. i 04.1 99 .i 96.2 9 5.5 4001.2 08.0 17.7 20 3232.1 3321.5 3412.7 35o5. 7 3600.7 36 97 .8 3 797 .i 38 9 8.8 4oo3.o 4io 9 .8 4219.5 21 33.6 23.0 14.2 o 7 .3 02. 3 99-4 9 8.8 3 9 oo.5 04.7 11.6 21.4 22 35.i 24.5 i5. 7 08.8 o3. 9 3 7 oi.i 38oo.5 O2. 2 06.5 i3.4 23.2 23 36.6 26.0 i 7 .3 10.4 o5.5 02. 7 02.2 o4.o 08.3 15.2 25 I 24 38.o 27.5 18.8 12.0 07.1 o44 o3.8 o5. 7 10. 17.1 27.0' 25 3 9 .5 29.0 20.4 i3.5 08.7 06.0 -o5.5 07.4 ii. 8 18.9 28.8 26 4i.o 3o.6 21. 9 i5.i io.3 07.6 07.2 o 9 .i i3.5 20.7 3o.6 27 42.5 32.1 23.5 16.7 n. 9 o 9 .3 08. 9 10.8 i5.3 22.5 32.5 28 44.o 33.6 25.0 i8.3 i3.6 1 0.0 io.5 12.5 17.1 24-3 34.4 29 45.4 35.i 26.5 19.8 i.5.i 12.6 12.2 i4-3 18.8 26.1 36.2 3o 3246.9 3336.6 3428.0 3521.4 3616.7 3714.2 38i3. 9 3 9 i6.p 4020.6 4127.9 4238.1 5i 48-4 38.i 2 9 .6 23.0 18.4 i5.8 10.6 17.7 22.4 29.7 4o.o 32 4 9 . 9 39.6 3i.i 24.6 20. o i 7 .5 i 7 .3 19.4 24-1 3i.5 4i.8 33 5i.4 4i.i 32. 7 26.1 21.6 18.9 21.2 25. 9 33.3 43. 7 34 5 2 .8 42.6 34-2 27.7 23.2 20. 8 20.6 22. 9 27.7 35.2 45.5 35 54.3 44.i 35.8 29.3 24.8 22.4 22.3 24-6 29.4 37.0 47-4; 36 55.8 45-7 3 7 .3 3o.8 26.4 24.1 24.0 26.3 31.2 38.8 4 9 -3| 37 5 7 .3 47.2 38.8 32.4 28.0 2 5. 7 20.7 28.1 33.o 4o.6 38 58.8 48. 7 4o.4 34.o 20.6 27.4 2 7 .4 2 9 .8 34.8 42.4 53*o 3 9 6o.3 60.2 41.9 35.6 3i.3 2 9 .0 2 9 .I 3i.5 36.5 44.2 54- 9 4o 3261. 7 3351.7 3443.5 353 7 .i 3632.8 3 7 3o. 7 383o.8 3 9 33.2 4o38.3 4 1 46. i 4256. 7 4i 63.2 53.2 45.o 38.7 34.4 32.3 32.4 35.o 4o.i 47-9 58.6 ! 42 64-7 54.7 46.6 4o.3 36.i 34.o 34.i 36. 7 4i.8 6o.5 43 66.2 56.2 48.i 41.9 3 7 . 7 35.6 35.8 38.4 43.6 5i.5 6a.3 44 67.7 5 7 .8 49-7 43.5 3 9 .3 3 7 .3 3 7 .5 40.2 45.4 53.4 64-2 45 69.2 5 9 .3 5.1.9 45.o 40.9 38. 9 3 9 .2 4i- 9 47.2 55.2 66.1 46 70.7 60.8 52.8 46.6 42.5 4o.6 4o. 9 43.6 4 9 .o 57.0 47 72.1 62.3 54.3 48.2 44-1 42.2 42.6 45-4 5o. 7 58.8; t 48 7 3.6 63.8 55.8 49.8 45.8 43. 9 44-3 47.1 52.5 60.7 7'-7 49 7 5.i 65.4 5 7 -4 5i.4 474 45.5 46.o 48.8 54.3 62.5 7 3.6 5o 3276.6 3366. 9 3458. 9 3553.o 3649-0 3747.2 3847-7 3 9 5o.6 4o56.i 4i64.3J4a75.5j 5i 78.1 68.4 6o.5 54-6 5o.6 48.8 4 9 .4 52.3 5 7 .8 66.1 77-4 5a 79.6 69.9 62.0 56.i 52.2 5o.5 5i.i 54.o 5 9 6 68.0 7 9 .2 53 81.1 63.6 5 7 . 7 53.8 52.1 52.8 55.8 61.4 69.8 81.1 54 82.6 7^0 65.2 5 9 .3 55.5 53.8 54-4 5 7 .5 63.2 71.6 83.o 55 84.i 7 4.5 66.7 60.9 5 7 .i 55.5 56.i 5 9 .3 65.o 7 3.5 84- 9 56 85.6 76.0 68.3 62.5 58. 7 5. 7 .i 5 7 .8 61.0 66.8 7 5.3 86.8 1 5 7 87.1 77 .5 6 9 .8 64.0 6o.3 56.8 5 9 .5 62. 7 68.5 77.1 88.6 58 88.5 79.0 7 r.4 65.6 61.9 6o.4 61.2 64-5 7 o.3 79.0 9 o.5 90.0 80.6 7 3.o 67.2 63.6 62.1 62. 9 66.2 72.1 80.8 9 2.4 5o 5 1 52 53 54 55 56 57 58 5 9 MERILIONAL TARTS. i LATITUDE. Win. 58 59 60 61 62 63 64 65 66 67 68 Min. 4294.3 4409.1 452 7 .4 4649.2 4 77 5.o 4904.9 5o39.4 5178.8 5323.5 5474.0 563o.8 I 96.2 i i.i 29.4 5i.3 77.1 07.1 4i. 7 81.2 26.0 76.6 33.5 I 2 98.1 i3.o 3i.4 53.4 79 .3 09.4 44-0 83.5 28.4 79.1 36.2 2 3 43oo.o i5.o 33.4 55.4 81.4 11.6 46.3 85.9 30.9 81.7 38.8 3 4 oi.g 16.9 35-4 5 7 .5 83.5 i3.8 48.6 88.3 33.4 84.3 4i.5 4 5 03.7 18.9 87.41 5 9 .6 85.6 16.0 5o-9 90.7 35.8 86.9 44.2 5 6 o5,6 20.8 3 9 4l 61.6 87.8 18.2 53.i 98.1 38.3 89.4 46. 9 e 7 o 7 .5 22.8 4i-4 63. 7 89.9 20.4 55.4 95.4 4o.8 92.0 49.6 7 8 09.4 24.7 43.4 65.8 92.l| 22.6 57.7 97.8 43.2 94.5 52.3 8 9 n.3 26.7 45.4 67.8 94.2 24.8 60.0 52OO.2 45-7 97.1 54.9 9 10 43i3.2 4428.6 4547-4 4669.9 4796.3 4927.0 5o62.3 52O2.6 5348.2 5499.7 565 7 .6 10 ii i5.i 3o.6 49.4 72.0 9 8.5 29.2 64-6 04.9 50.7 55o2.3 6o.3 ii 12 17.0 32.5 5i.4 7 4-i 4800.6 3i.5 66.9 07.3 53.i o4-9 63.o 12 i3 18.9 34.5 53.5 76.1 02.8 33. 7 69.2 09.7 55.6 07.4 65. 7 i3 i4 20.8 36.4 55.5 78.2 o4-9 SS.g 7 i.5 I 2. 1 58.i 10. 68.4 i4 i5 22.7 38.4 5 7 .5 8o.3 07.1 38.i 7 3.8 i4.5 60.6 12.6 71.1 i5 16 24.6 4o.3 5 9 .5 82.4 09.2 4o.3 76.1 16.9 63.i 15.2 7 3.8 16 l l 26.5 42.3 6i.5 84.5 n.4 42.6 78.4 [9.3 65.6 17.8 7 6.5 '**? iS 28.4 44-2 63.5 86.5 i3.5 44-8 80.7 21.6 68.0 20.4 79.2 18 '9 3o.3 46.2 65.6 88.6 16.7 4 7 .o 83.o 24.0 70.5 23.0 81.9 '9 20 4332.2 4448.2 456 7 .6 4690.7 4817.8 4949.2 5o85.3 5226.4 53 7 3.o 5525.6 5684-6 20 21 34.1 5o.i 69.6 92.8 20. o 5i.5 87.6 28.8 7 5.5 28.1 8 7 .3 21 22 36.o 52.1 71.6 94.9 22.1 53. 7 90.0 3l.2 78.0 30.7 90.0 22 23 3 7 . 9 54.1 7 3.6 97.0 24.3 55. 9 92.3 33.6 8o.5 33.3 92.7 23 24 3 9 .8 56.o 7 5. 7 99.1 26.4 58.2 94.6 36.o 83.o 35. 9 9 5.5 24 25 4i.8 58.o 77-7 4701.1 28.6 60.4 96.9 38.4 85.5 38.6 98.2 25 26 43.7 60.0 79-7 03.2 3o.8 62.6 99.2 4o.8 88.0 41.2 5700.9 26 27 45.6 61.9 81.7 o5.3 32.9 64.9 5ioi.5 43.2 90.5 43.8 o3.6 2-7 28 4 7 .5 63. 9 83.8 07.4 35.i 67.1 o3.8 45. 7 93.0 46.4 o6.3 28 29 49.4 65. 9 85.8 o 9 .5 3 7 .3 6 9 .4 06.2 48.i 9 5.5 49.0 09.1 29 3o 435i.3 446 7 .8 458 7 .8 4711.6 483 9 .4 4971.6 5io8.5 D250.5 5398.0 555i.6 6711.8 3o 3i 53.2 69.8 89.9 i3. 7 4i-6 73.8 10.8 52.9 5400.5 54.2 i4.5 3i 32 55.i 71.8 91.9 i5.8 43.8 76.1 i3.i 55.3 o3.o 56.8 i 7 .3 32 33 5 7 .i 7 3. 7 9 3. 9 i 7 .9 45.9 7 8.3 i5.5 5 7 . 7 o5.5 59.4 20.O 33 34 Sg.o 7 5. 7 96.0 20.0 48.i 80.6 17.8 60. i 08.1 62.1 22-7 34 35 60.9 77-7 98.0 22.1 5o.3 82.8 2O.I 62.6 10.6 64-7 2 5.5 35 36 62.8 79-7 46oo. o 24.2 52.4 85.i 22.4 65.o i3.i 67.3 28.2 36 3? 64-7 81.6 02. i 26.3 54-6 8 7 .3 24.8 67.4 i5.6 69.9 3o-9 37 38 66.7 83.6 o4-i cS.4 56.8 89.6 27.1 69.8 18.1 72.6 33. 7 38 3 9 68.6 85.6 06.1 3o.5 59.0 91.8 29.4 72.2 20.6 7 5.2 36.4 3 9 4o 4370.5 4487.6 4608.2 4 7 3 2 .6 486i.i 4994.1 5i3i.8 52 7 4. 7 5423.2 55 77 .8 5 7 3 9 .2 4o 4i 72.4 89.6 10.2 34. 7 63.3 96.3 34.i 77.1 2 5. 7 80.4 41.9 4i 42 74-3 9 i.5 12.3 36.8 65.5 98.6 36.5 79.5 28.2 83.i 44.7 42 43 7 6.3 9 3.5 i4.3 39.0 67.7 5ooo.8 38.8 82.0 3o.8 85. 7 4 7 .5 43 44 78.2 9 5.5 16.4 4i.i 69.9 o3.i 4i.i 84-4 33.3 88.4 50.2 44 45 80. i 97.5 18.4 43.2 72.0 o5.4 43.5 86.8 35.8 91.0 52.9 45 46 82.1 99.5 20. 5 45.3 74.2 07.6 45.8 89.3 38.4 93.6 55.7 46 4? 84.0 45oi.5 22.5 4 7 4 76.4 09.9 48.2 91.7 40.9 96.3 58.5 47 48 85.9 o3.4 24.6 49-5 78.6 12.2 5o.5 94.1 43.4 98.9 61.2 48 49 87.8 o5.4 26.6 5i.6 80.8 l44 52.9 96.6 46.o 56oi.6 64.o 49 5o 438 9 .8 4507.4 4628.7 4 7 53. 7 4883.o'5oi6. 7 5i55.2 5299.0 5448.5 56o4.2 5 7 66.8 5o 5i 91.7 09.4 3o. 7 55. 9 85.2 18.9 5 7 .6 53oi.5 5i.o 06.9 69.5 5i 52 93.6 11.4 32.8 58.o 8 7 .4 21.2 59.9 o3. 9 53.6 09.5 72.3 52 53 9 5.6 i3.4 34.8 60. i 89.5 23.5 62.3 o6.3 56.i 12.2 $5.i 53 54 97 .5 1 5.4 36.9 62.2 91. 7 25.8 64.6 08.8 58. 7 i4.8 77-9 54 55 99.4 17-4 39.0 64-4 9 3. 9 28.0 67.0 II. 2 61.2 i 7 .5 80.6 55 56 44oi-4 i 9 4 4i.o 66.5 96.1 3o.3 69.4 i3. 7 63.8 2O.2 83.4 56 5 7 o3.3 21.4 43.o 68.6 98.3 32.6 71.7 16.1 66.3 22.9 86.2 5 7 ' 58 o5.3 23.4 45.i 7-7 4900.5 34-9 7 4.i 18.6 68.9 2 5.5 89.0 58 59 07.2 25.4 47-2 7 2 -9 02.7 3 7 .i 76.4 21. 1 7i.4 28.2 91.8 5 9 148 MERIDIONAL PARTS. LATITUDE. Min. 69 70 71 72 73 74 75 76 77 78 79 MiB. O 6794.6 5 9 65. 9 6145.7 6334-8 6534.4 6745.7 6 97 o.3 7210.1 7 46 7 .2 77 44-6 8045.7 O I 97-4 68.8 48.8 38.i 3 7 .8 4 9 .4 7 4.2 14.2 71.7 4 9 .4 5i.o I 2 58oo.i 71.8 5!. 9 4i.3 4i.3 53.o 78.1 E'8.3 76.1 54.2 56.2 2 3 02.9 74-7 54-9 44-6 44. 7 56.6 81.9 22.5 80.6 5 9 .o 6i.5 3 4 o5. 7 77.6 58.o 4/-8 48.i 6o.3 85.8 26.6 85.o 63. 9 66.7 4 5 o8.5 80.6 61.1 5i.i 5i.6 63. 9 89.7 3o.8 8 9 .5 68.7 7 2.0 5 6 ii. 3 83.5 64.2 54.3 55.o 67.6 93.6 35.o 94.0 7 3.5 77.3 6 7 14.2 86.4 6 7 .3 5 7 .6 58.5 71.2 97.5 3o.i 9 8.5 78.4 82.6 7 8 17.0 89.4 70.4 60.8 61.9 7 4. 9 7001.4 43.3 75o3.o 83.3 87-9 8 9 19.8 92.3 7 3.5 64.1 65.3 7 8.5 o5.3 4 7 .5 07.4 88.1 9 3.2 9 10 5822.6 5 99 5.3 6176.6 636 7 .4 6568.8 6782.2 7009.2 7251.7 7511.9 77 9 3.o 8o 9 8.5 10 ii 25.4 98.2 79-7 70.6 7 2.3 85.9 1 3. i 55.8 i6.5 97-9 8io3.8 ii 12 28.2 6001.2 82.8 7 3. 9 7 5. 7 89.6 17.0 60.0 21.0 7802.8 09.2 12 i3 3i.o o4-i 85. 9 77.2 79 .2 93.2 20.9 64.2 25.5 07.7 i4.5 i3 i4 33.8 07.1 89.0 80.4 82.6 96.9 24.8 68.4 3o.o 12.6 i 9 . 9 i4 i5 36. 7 IO.O 92.1 83. 7 86.1 6800.6 28.8 72.6 34-5 i 7 .5 25.2 i5 16 Sg.S i3.o 55.2 87.0 89.6 o4-3 32. 7 76.8 39.1 22.4 3o.6 16 *7 42.3 16.0 98.3 90.3 93.0 08.0 36.6 81.1 43.6 27.3 36.0 *7 18 45.i 18.9 6201.4 93.6 96.5 u.6 4o.6 85.3 48.i 32.2 4i.3 18 '9 48.o 21.9 o4-5 96.9 6600.0 i5.4 44.5 8 9 .5 62.7 3 7 .2 46. 7 X 9 20 585o.8 6024.9 6207.7 64oo.2 66o3.5 6819.1 7 o48.5 7 2 9 3. 7 7 55 7 .3 7 842.i 8162.1 20 21 53.6 27.8 10.8 o3.4 o 7 .o 22.8 52.4 9 8.o 61.8 4 7 .i 5 7 .5 21 22 56.5 3o.8 i3. 9 06.7 io.5 26.5 56.4 7302.2 66.4 52. 62.9 22 23 5 9 .3 33.8 17.0 10. 1 i4o 30.2 6o.3 06.4 71.0 5 7 .o 68.4 23 24 62.2 36.8 20.2 i3.4 i 7 .5 33. 9 64.3 10.7 7 5.5 6i. 9 7 3.8 24 25 65.o 3 9 .8 23.3 16.7 21. 3 7 .6 68.3 1 5.0 80.1 66. 9 79 .2 25 16 67.8 42.7 26.5 20. o 24-5 4i.3 7 2.2 19.2 84-7 71.9 84-7 26 27 70.7 45.7 29.6 23.3 28.0 45.i 7 6.2 23.5 8 9 .3 7 6. 9 9 o.i 27 28 7 3.5 48.7 32. 7 26.6 3i.5 48.8 80.2 27.7 9 3. 9 81.9 9 5.6 28 29 76.4 5i. 7 35. 9 29.9 35.o 52.5 84.2 32.0 9 8.5 86. 9 8201.1 29 3o 53 79 .2 6o54-7 6239.0 6433.3 6638.5 6856.3 7 o88.2 7 336.3 7603.2 7 8 9 T -9 8206.6 3o 3i 82.1 5 7 . 7 42.2 36.6 42.1 60.0 92.2 4o.6 07.8 96.9 12. 1 3! 32 85.o 60.7 45.3 3 9 . 9 45.6 63.8 96.2 44.9 12.4 79 02.O i 7 .6 32 33 87.8 63. 7 48.5 43.2 49.1 6 7 .5 7 IOO.2 4 9 .2 17.0 o 7 .o 23.1 33 34 90.7 66.7 5i. 7 46.6 52.6 7 i.3 04.2 53.5 21.7 12. 28.6 34 35 93.6 69.7 54.8 49.9 56.2 7 5.o 08.2 5 7 .8 26.3 I 7 .I 34.1 35 36 96.4 72.7 58.o 53.3 59.7 7 8.8 12.2 Sa.l 3i.o 22.1 3 9 . 7 36 3? 99.3 7 5. 7 61.2 56.6 63.3 82.6 i6.3 66.4 35.6 2 7 .2 45.2 3? 38 5902.2 78.8 64-3 60.0 66.8 86.3 20.3 70.7 4o.3 32.3 5o.8 38 3 9 o5.o 81.8 6 7 .5 63.3 70.4 90.1 24.3 7 5.i 45.o 3 7 .3 56.3 39 4o 5907.9 6o84.8 6270.7 6466.7 6673.9 6893.9 7 i28.4 7379-4 7649.7 7942.4 8261.9 4o 4i 10.8 87.8 7 3. 9 70.0 77*4 97-7 32.4 83. 7 54.3 4 7 .5 6 7 .5 4i 42 13.7 90.8 77.1 73.4 81.0 6ooi.5 36.4 88.1 5 9 .o 5 2 .6 7 3.i 4'2 43 16.6 93.9 80.2 76.7 84-6 o5.3 4o.5 92.4 63. 7 5 7 . 7 7 8.6 43 44 19.4 96.9 83.4 80.1 88.2 09.1 44-5 9 6.8 68.4 62.8 84-3 44 45 22.3 99.9 86.6 83.5 91.7 12. 9 48.6 7 4oi.i 7 3.2 68.0 89.9 45 46 25.2 6io3.o 89.8 86.9 9 5.3 16.7 5 2 . 7 o5.5 77-9 7 3.i 9 5.5 46 47 28.1 06.0 9 3.o 90.2 9 8. 9 20. 5 56. 7 9'9 82.6 78.2 83oi.i 47 48 3i.o 09.0 96.2 9 3.6 6702.5 24-3 60.8 i4.3 8 7 .3 83.4 06. 7 48 4 9 33.9 12. 1 99.4 97.0 06. i 28.1 64-9 18.6 92.1 38.5 12.4 49 5o 5986.8 6n5.r 6302.6 65oo.4 670 9 .7 6931.9 7169. o 7 423.o 76 9 6.8 7993.7 83i8.i 5o 5i 3 9 . 7 18.2 o5.8 o3.8 13.2 35. 7 7 3.i 2 7 .4 7701.5 98-9 23.8 5i 52 42.6 21.2 09.1 07.2 16.8 93.6 77 .2 3i.8 o6.3 8oo4-o 2 9 4 52 53 45.5 24.3 12.3 10.6 20.4 43.4 81.2 36.2 IX. I 9 .2 35.i 53 54 48.4 2 7 .3 i5.5 i4.o 24.0 4 7 -2 85.3 4o.6 i5.8 i44 4o.8 54 55 5i.3 3o.4 18.7 17-4 27.7 5i.i 8 9 .5 45.1 2 o.6 I 9 .6 46.5 55, 56 54-2 33.4 21.9 20.8 3i.3 54.9 9 3.6 4 9 .5 2 5.4 24.8 52.2 56 57 5 7 .2 36.5 25.1 24.2 34-9 58.8 97-7 53. 9 30.2 3o.o 58.0 5 7 58 60. i 3 9 .6 28.4 27.6 38.5 62.6 7201.8 58.3 35.o 35.2 63. 7 58 5 9 63.o L_ 42.6 3i.6 3 1.0 4s. j 66.5 o5.6 62.8 3 9 .8 4o.5 69.4 5 9 CORRECTIONS TO MIDDLE LATITUDE. 149 Mid. Lat. 1 DIFFERENCE OF LATITUDE. Mid Lot. 2^ 3 4 5 6 7 8 9 10 11 12 13 14 15J16 IT 18 19 20 o i5 / / 2 / 3 / 5 / 7 / 9 12 i5 18 22 26 3i 36 4i 47 52 5 9 65 / 72 r5 16 2 3 4 6 9 II i4 18 21 25 3o 34 39 44 5o 56 62 69 iC 17 o 2 3 4 6 8 II i4 l l 2O 24 28 33 38 43 48 54 60 66 17 18 o 3 4 6 8 IO i3 16 2O 23 27 32 36 4i 46 52 58 64 18 J 9 o 3 4 6 8 IO i3 16 *9 22 26 3o 35 4o 45 5o 56 61 T 9 20 , o 2 4 5 7 IO 12 i5 18 22 25 29 34 38 43 48 54 60 20 21 o 2 4 5 7 9 12 i5 18 21 25 29 33 37 42 47 62 58 21 22 i O 2 4 5 7 9 12 i4 l l 21 24 28 32 36 4i 46 5i 56 22 23 o 2 3 5 7 9 II i4 1 7 20 23 27 3i 35 4o 45 5o 55 23 24 o 2 3 5 7 9 II i4 16 20 23 27 3i 35 3 9 44 49 54 24 25 o 2 3 5 7 9 11 i3 16 J 9 23 2U 3o 34 3 9 43 48 53 25 26 2 3 5 6 8 II i3 16 J 9 22 26 3o 34 38 42 47 52 26 27 o 2 3 5 6 8 II i3 16 J 9 22 25 2 9 33 37 42 47 52 27 28 o 2 3 5 6 8 10 i3 16 18 22 25 2 9 33 37 4i 46 5i 28 29 o 2 3 5 6 8 10 i3 i5 18 21 25 28 32 37 4i 46 5i 2 9 3o o 2 3 5 6 8 IO i3 i5 18 21 25 28 32 36 4i 45 5o 3o 3i o 2 3 5 6 8 IO 12 i5 18 21 24 28 32 36 4o 45 5o 3i 32 o 2 3 4 6 8 IO 12 i5 18 21 24 28 32 36 4o 45 5o 32 33 o o 2 3 4 6 8 10 12 i5 18 21 24 28 32 36 4o 45 49 33 34 o o 2 3 4 6 8 IO 12 i5 18 21 24 28 32 36 4o 45 49 34 35 o o 2 3 4 6 8 IO 12 i5 18 21 24 28 32 36 4o 45 49 35 36 2 3 4 6 8 IO 12 i5 18 21 24 28 32 36 4o 45 49 36 3? o 2 3 4 6 8 10 12 i5 18 21 24 28 32 36 4o 45 49 37 38 o 2 3 4 6 8 10 12 i5 18 21 24 28 32 36 4o 45 5o 38 3 9 o 2 3 4 6 8 IO 12 i5 18 21 24 28 32 36 4o 45 5o 3 9 4o o 2 3 5 6 8 IO 13 i5 18 21 25 28 32 36 4i 45 5o 4o 4i o 2 3 5 6 8 IO 13 i5 18 21 25 28 32 37 4i 46 5ij 4i 42 2 3 5 6 8 10 i3 i5 18 22 25 29 33 37 4i 46 5i i 42 43 o 2 3 5 6 8 10 i3 16 18 22 25 29 33 37 42 46 52' 43 44 : o 2 3 5 6 8 IO i3 16 J 9 22 25 29 33 38 42 47 52 44 i 45 2 3 5 6 8 II i3 16 X 9 22 26 3o 34 38 43 48 53 45 46 o 2 3 5 6 8 II i3 16 *9 22 26 3o 34 38 43 48 53 46 47 o 2 3 5 7 9 II i3 16 J 9 23 26 3o 35 39 44 49 54 47 48 o 2 3 5 7 9 II i4 ^7 20 23 27 3i 35 4o 44 5o 55 48 49 o 2 3 5 7 9 II i4 17 20 23 27 3i 36 4o 45 5o 56 49 5o 2 4 5 7 9 II i4 !7 20 24 28 32 36 4i 46 5i 57 5o 5i o 2 4 5 7 9 12 i4 !7 21 24 28 32 37 42 47 52 58 5i 52 o 2 4 5 7 9 12 i5 18 21 25 29 33 38 43 48 53 5 9 52 53 o 2 4 5 7 IO 12 i5 18 21 25 29 34 38 43 49 54 60 53 54 o 2 4 5 7 IO 12 i5 18 22 26 3o 34 39 44 5o 56 62 54 55 o 2 4 6 8 10 13 16 J 9 22 26 3i 35 4o 45 5i 57 63 55 56 o 3 4 6 8 IO i3 16 r 9 23 27 3i 36 4i 46 52 58 65 56 5 7 o 3 4 6 8 IO i3 16 20 24 28 32 37 42 48 54 60 66 57 .58 o 2 3 4 6 8 II i4 17 2O 24 28 33 38 43 49 55 61 68 58 5 9 o 2 3 4 6 8 II i4 l l 21 25 29 34 3 9 45 5o 57 63 70 5 9 60 o 2 3 4 6 9 II i4 18 22 26 3o 35 4o 46 52 58 65 72 60 6 1 o 2 3 5 7 9 12 i5 18 22 26 3i 36 42 47 53 60 67 75 61 62 2 3 5 7 9 12 i5 '9 23 27 32 37 43 49 55 62 70 77 62 63 o 2 3 5 7 10 12 16 20 24 28 33 3 9 44 5i 57 64 72 80 63 64 o 2 3 5 7 10 13 16 20 24 29 34 4o 46 52 5 9 67 75 83 64 05 2 3 5 7 IO i3 J 7 21 25 3o 36 4i 48 54 62 6 9 78 86 t>5 66 o 2 3 5 8 II i4 18 32 26 32 37 43 5o 57 64 72 81 9 66 67 o 2 4 6 8 II i4 18 23 28 33 3 9 45 52 5 9 67 76 85 9 4 67 68 o 2 4 6 8 12 i5 T 9 24 29 34 4o 47 54 62 70 79 39 99 68 69 o 2 4 6 9 12 16 20 25 3o 36 42 4 9 57 65 74 83 9 3 io4 ; 6 9 70 o 2 4 6 9 i3 16 21 26 32 38 44 52 60 68 7 8 88 98 no 7 7i 2 4 7 10 i3 '7 22 27 33 4o 47 55 63 72 82 9 3 io4 116 7' 72 o 3 5 7 10 i4 18 23 29 35 42 4 9 58 67 76,87 9 8 - 1 1 1 124 7 P. 150 LOGARITHMS FOR COMPUTING COMPOUND INTEREST. In computing compound interest for long periods of time, the following logarithms to more than six places. it is necessary to have Number. Logarithm. Number. Logarithm. .0026 .oo5o .0076 .OIOO .0125 .0160 .0176 .0200 .0226 .0260 .0276 .o3oo ,o325 .o35o .0876 .o4oo .00108 438i3 .00216 60618 .oo324 5o548 .oo432 i3738 .00539 5o3i9 .00646 60422 .00753 44i79 .00860 01718 .00966 33167 .01072 38654 .01178 i83o5 .01283 72247 .01389 oo6o3 .01494 03498 .01598 8io54 .01703 33393 .o425 .o45o .0476 .o5oo .o5 2 5 .o55o .o5 7 5 .0600 .0626 .o65o .0675 .0700 .0726 .0750 .o 77 5 .0800 .01807 6o636 .01911 62904 .O20i5 4o3i6 .02118 92991 .02222 2io45 .O2325 24596 .0242803760 .O253o 58653 .02632 89387 .0273496078 .02836 78837 .02938 37777 .oSoSg 73009 .o3i4o 84643 .o324i 72788 .o3342 37555 NUMBERS OFTEN USED IN CALCULATIONS. Circumference c Surface of a spb Area of a circle Area of a circle Capacity of ti sp Capacity of a sp i 3.1415926 . f a circle to diameter i ) ere to diameter i . . . . > to radius i S = 3.i4i5 9 26 .7853982 .5235988 4.1887902 0.3183099 . 2957795 206264". 8 .0174533 .0002909 .ooooo485 .00000970 .00001454 .00001939 .00002424 .00002909 .00003394 .00003879 .oooo4363 2.7182818 .4342945 1296000 864oo 528o Logarithms. o.-497i5o 9.895090 9.718999 0.622089 9-5o285o i.758i23 5.3i4425 8.241877 6.463 7 26 4.6855 7 5 4.986605 5.162696 5.28 7 635 5.384545 5.463 72 6 5.53o6 7 3 5.588665 5.639817 0.434294 9.63 77 84 6.ii26o5 4-9365i4 3.722634 to diameter i here to diameter i here to radius i Arc equal to rad Arc equal to rad Length of i deg Length of i min Sine of i seconc Sine of 2 seconc Sine of 3 seconc Sine of 4 seconc Sine of 5 seconc Sine of 6 seconc Sine of 7 seconc Sine of 8 seconc Sine of 9 seconc Base of Napier' Modulus of the 36o degrees exp 24 hours expres Number of feet ius expressed in degrees . 5 7 ius expressed in seconds ree in parts of radius ute in parts of radius s s Is 3 S S s s _ s system of logarithms common logarithms : ressed in seconds sed in seconds in one mile I THE END. 1C 49S87 M289993 Lr THE UNIVERSITY OF CALIFORNIA LIBRARY