ELEMENTS OF PLANE AND SPHERICAL. BY LEFEBURE DE FOURCY,. FWrAA.?F THI LEGION OF HONOR; PROFESSOR IN THE FACULTY OF IN STEIR -JQAEMHY OF PARIS; AND EMERITUS EXAMINER OF CANDIDATES FOR THE POLYTECHNIC SCHOOL. 79, TfiAuglttri from tt TOt tr-crh (gaiffin, BY FRANCIS H. SMITII, A.M. UIPEItINTENDENT AND PROFESSOR OF MATHEMATICS OF THE VIRGINIA MILITARY INSTITUTE. WITH TABLES OF LOGARITHMS OF NUMBERS AN.D OF.SINES. ANl COSINES FOR EVERY TEN SECONDS OF THE QUADRANT, AND OTHER USEFUL TABLES. BALTIMORE: r x IJ m ILM O R E: KLELLY, PIET S CO.. 174 ]Baltimore Street. 1871. natered accordiug to Act of. Congress, in the year 1868,.by FRANCIS IH. SMITH, In the Clerk's Office of the District Court of the United States for Virginia. p Orfi 0;LtY L& PIlS! TO GENERAL ROBERT E. LEE. IN submitting to the public a translation of Lefibure de.Fourcy's admirable treatise, on Trigonometry, I have thought it not inappropriate to associate your name with. this effort to promote the interests of education. You have given your influence, by precept and example, to this important cause, and in dedicating this work to you, I only give expression to the general sentiment of those who appreciate the weight of this great influence. The personal and official relations which it has been my privilege to sustain towards you, during. a period of more than thirty years, add a peculiar gratification to me in rendering this tribute-of affectionate regard and esteem. FRANCIS H. SMITH. VIRGINIA MILITARY INSTITUTE, Lexington, Va., September 19, 1867. (iii) PIREFACE. THE elementary treatise on Trigonometry by LefebuT7 de Fourcy has been justly regarded as one of the best ever published; and by its translation, the American student will have access to a work which will prove a valuable auxiliary to the prosecution of the course of the Higher Mathematics. This work constitutes one of the Mathematical Series of the Virginia Military Institute. The following works now comprise the series as published: Smith's Elementary A Bithnmetic, for Beginners. Smith'is Arithmetic for High Schools and Academies. Smintths Algebrw. Smith'zs Biot's Analytical Geometry. Smith's Legendre's Geometry. Smith's Ley'eb ~re de Flourcy's Trigonometry with Tables. A Descriptive Geometry will immediately follow. VIRGINIA MILITARY INSTITUTE, Lexington, Va., September 19, 1867. TABLE OF CONTENTS. INTRODUCTION. PAGN ILOGARITIHMS, TABLES. -PROPERTIES OF LOGARITHMS, MODE OF USING TABI ES....................................................................................................... 1-15 TRIGONOMIETRY. CHAPTER I. OBJECT OF TRIGONOMETRY. —METHOD OF REPRESENTING THE SIDES AND ANGLES OF TRIANGLES BY NUMIBERS...................................................... 16 DEFINITIONS OF TRIGONOMETRICAL LINES.- MANNER OF USING + AND — TO INDICATE LINES IN OPPOSITE DIRECTIONS...................................... 17 PROGRESSIVE CIIANGES IN TRIGONOMETRICAL LINES. — METHOD OF REDUCING THEM TO FIRST QUADRANT.................................. 21 OF AlRCS WHIIICtI CORRESPOND TO A GIVEN SINE, COSINE, &c....................... 2 RELATIONS OF TRIGONOMETRICAL LINES TO EACII OTHER.......................... 31 FORMULE FOR SIN (a -tb) AND COS (a — i b).................................................. 35 FOR MULTIPLICATION AND DIVISION OF ARCS........................... 38 RELATIVE TO TANGENTS................................................................. 43 " FREQUENTLY USED........................................................................ 45 GEOMETRICAL DEMONSTRATIONS OF FORMUL2E ALREADY FOUND............... 47 CHAPTER II. TRIGONOMIETRICAL TABLES. — SOLUTIONS OF TRIANGLES. CONSTRUCTION OF TRIGONOMETRICAL TABLES............................... 52 CALCULATION OF SINES AND COSINES FOR EVERY 90 TO VERIFY THE TABLES........................................................................................................... 7 ARRANGEMENT AND MANNER OF USING CALLET'S TABLES.......................... 59 RESOLUTION OF TRIANGLES.-RELATIONS BETWEEN SIDES AND ANGLES OF A PLANE TRIANGLE.................................................. 63 " RIGHT-ANGLED PLANE TRIANNGLES a........................... 67 " OBLIQUE PLANE TRIANGLES................................................ 68 APPLICATION TO EXAMPLES.......................................................... 75 (v) vi TABLE OF CONTENTS. RELATIONS BETWEEN ANGLES AND SIDES OF SPHERICAL TRIANGLES. PaoG FUNDAMENTAL FORMULA.......................................................... 82 RELATIONS BETWEEN THIIREE SIDES AND AN ANGLE................................... 84.... TWO SIDES AND THE TWO OPPOSITE ANGLES.......... 8 " TWO SIDES, THE INCLUDED ANGLE, AND TIIEANGLE OPPOSITE ONE.. 87 c" " A SIDE AND THE THREE ANGLES.............................. 87.... SIDES AND ANGLES OF POLAR TRIANGLE TO THE GIVEN TRIANGLE................................8..................... 88 PROPORTION OF THE FOUR TANGENTS................................................. 89 " " " FOUR COSINES............................................. 89 FORMULAE OF DELAMBRE.........9...................................................0............ 9 iVAPIER',S ANALOGIES.................................................................. 91 RELATIONS BETWEEN THE PARTS OF A RIGHT-ANGLED SPHERICAL TRIANGLE.......................................................................................................... 92 RAPIER'S RULES.9............................................................................ 95 APPLICATION OF NAPIER'S RULES TO SOLUTION OF RIGI-IT-ANGLED SPILERICAL TRIANGLES..................................................................... 95 RESOLUTION OF OBLIQUE SPHERICAL TRIANGLES.................................... 97 AMBIGUOUS CASES IN SPHERICAL TRIANGLES............................................... 101 SURFACE OF SPIIERICAL TRIANGLE................................,,,. 105 APPLICATIONS IN SPHERICAL TRIGONOMETRY........ l ":.................................107 CHAPTER III. FORMULAE USED IN HIGHER MIATHEMATICS.-SINES AND COSINES IN SERIES.-RESOLUTION OF BINOMIAL EQUATIONS AND EQUATIONS OF THIRD DEGREE. FORMULA OF DE MOAIVRE..........................................................111 FORMULJE FOR SIN n 0 AND COS n 0 IN TERMS OF SINE AND COSINE OF SIMPLE ARC.......................................................................................1.......... 114 FORMULME FOR (SIN O)n, (COS p)", IN TERMS OF SINE AND COSINE OF THE MULTIPLE ARCS........................................................................................... 115 DEVELOPMENT OF SINE AND COSINE INTO SERIES........................................ 117 RESOLUTION OF BINOMIAL EQUATIONS............ 118 IRREDUCIBLE CASE OF CUBIC EQUATIONS..............................123 CHAPTER IV. PROMISCUOUS EXAMPLES.'PLANE TRIGONOMETRY..................................................................................... 128 SPHERICAL TRIGONOMETRY............................................................................... 131 TABLES OF TRIGONOMETRICAL FORMUL.........1,................................ 132 T RI:GO N OME0TRY..INNTRODUCTION. LOGARITHMS. TABLE OF LOGARITIHMS OF NUMBERS.- USE. OF TABLE. 1. In the indeterminate equation a"- y, every value, positive and negative, given to x, will lead to a corresponding value of y; and, conversely, for every numerical value assigned to y there will be a corresponding value of x. If, therefore, the quantity a be arbitrarily assumed, and be supposed' to have a constant value, a fixed relation will exist between the numbers represented by x and y, and this relation is expressed by calling x the logarithm of y. The constant number a is called the base of the system of logarithms. 2. All positive numbers. may be produced by the powers of any proposed positive number greater than unity. For, if, in the equation ax -y, a being positive and >, x assume successively and continuously all possible values from 0 to 0o, it is clear that y will receive all values from I to co. Also, if x become negative and equal-x, then we have a- =y, ora,=y; and now, if x assume all possible values from 0 to co, y will receive all values from 1 to 0. H}ence, as x changes continuously from —oo to +- oo, az changes continuously from 0 to + or produces all positive numbers. If a <1, writing the equation c = y under the form. (a)-= y, it appears from what has been proved, since a >1, that by varying x, - and therefore y will assume all positive values. 3. We cannot take unity nor a negative quantity for the base: for all the powers of unity are unity; and the powers (1) 2 TRIGONOMETRY. of a negative quantity may be positive, negative, or imagi. nary, and hence are not subject to the law of continuity and do not reproduce all numbers. The logarithm of a number to a given base is the exponent of the power to which the base must be raised to become equal to the given number. 4. Since a- =a, a~ =1, and a — =-0, or a+- =0, according as a > 1 or < 1, we deduce from the definition of logparithms, 1st, that the logarithm of the base itself is unity; 2d, that the logarithm of unity is 0; and 3d, that the logarithm of 0 is negative infinity or positive infinity according as the base is > 1 or < 1. 5. When the base is >1, the equation ax = y shows, that as the number y increases, its logarithm x increases, and that the logarithms of all numbers greater than 1 are positive, and the logarithms of all numbers less than 1 are negative:' when the base is < 1, the contrary takes place, that is, the logarithms of numbers greater than 1 are negative, and those of numbers less than 1 are positive. We shall at present assume that, a being any positive quantity, if in the equation ax=y, the value of y be given, the corresponding arithmetical value of x can be found to any degree of approximation we please. (See Smith's Algebra, Chap. XII.) If y be assumed to be the exact power of the base a, the corresponding logarithms will be whole numbers,, 2, 3, 4, &c. Thus, a y, a2= y, a3 = y, a4= y3, &c. But if y be not equal to an exact power of the base, its logarithm must inuvolve a decimal part. 6. Ifnoaw yassume successive integral values beginning with I, a remraining the same, and the corresponding, values of x be computed from the equation a =-y, and registered in a Table, -n order, we shall obtain a system of logarithms to the base a. Since a may be any positive quantity, except 1, there may be an infinite number of systems of logarithms. If, however, the logarithms of all numbers in any one system be known, those corresponding to any other given base may be deduced by multiplying the former by a constant factor. 7. Although there is no limit to the number of systems of logarithms which -may be formed, there are only two systems which are used, viz, the Nrapierian system, and the Common system. LOGARITHMIS. 8 The -NAapicrian logarithms, which are always employed in analytical investigations, have for a base the incoummensurable number e- 2.7182818 The Common logarithms, which are no less constantly employed in numerical calculations, have for the base the number 10; and since the number is the base of the common scale of notation in numbers, this system has the advantagebthat its tables are far more comprehensive than tables of the same size in any other system. 8. If x represents the common logarithm of any n umber y, a being the base, and x. the Napierian logarithm of thle same number y, e being the base, and if we designate Napierian logarithms always by the dash (1); we shall have the relations ex' =, and ao — y, hence e' - ax and ex = a. Therefore _ — nap. log a = log'a. x And we have x x' X log'a and x' = x — log'a. That is, knowing the Napierian logarithm x' of any number y, we may obtain the common logarithm x of the same number y, by multiplying the Napierian logarithm x' by the multiplier log'a; or, knowing the common logarithm x of any number y, we may obtain the Napierian logarithm x' by dividing the common logarithm x by the same quantity log'a.'The common fiactor log'a is called the modulus of the system whose base is a, relative to the system whose base'is e. Therefore, the modulus of' the common system of logarithms is equal to unity divided by the Napierian logarithm of the common base. 1 1 When a = e, the factor log'a = log'e = 1. Hence, the modulus of the Nazpierian system of logarithms is unity. 9 Let y, y', y", &c. represent any numbers whatever, and ~4 TRIGONOMETRY. x, X', x', &c. their corresponding logarithms, tak-en i a system of which a is the base. We shall have a = y, am' = y", ax" =y", &c. (1 If we multiply these equations together, member lby jaember, we have a+ + $X + yXC._ t y+, &c. But, x + x + xi', &c. = log (y +y' + y", &c.) and x= log y, x' = log y', x"= log y", &c. hence log y + log y' + log y", &c. = log (y + y' + y",. &c.) (2) Therefore, the logarithm of the product of two or more numbers is equal to the sum of the logarithms of the numbers forining ethe product. 10. If we divide two of the equations (1), member by member, we have -Y:hence x-x' = log (Y) or, log y- log y' lo ( That is, the logarithm of a quotient is obtained by subtracting the logarithm of the divisor from that of the dividend. 11. If all the numbers y, y', y", &c. be equal in equation (2), and there be m of them, this equation becomes m log y =log (yi) which shows, that the logarithm of the mth power of a number is equal to m times the logarithm of this number. 12. If we extract the mth root of both members of the equation we shall have am = y. But, I = log V y, and x= log y. Hence lo = log /y. That is, the logarithm of the mth root of a number is equal to the logarithm of the number divided by the index of the root. 13. Hence, if we have to multiply two numbers together, or to divide one by the other, we may, by taking their logarithms out of the tables and adding or subtracting.them, LOGARITHMS. " and then finding in the tables the number whose logarithm is equal to the sum or difference, determine the product or quotient of the numbers, as the case may be. 14. Or, if we have to find any power or root of a number, we may, by taking its logarithm out of the tables and multiplying or dividing it by the index of the power or root, and then finding in the tables the number whose logarithm is equal to the product or quotient, determine the power or root of the proposed number. 15. We thus see, that by the aid of a Table of Logarithms, the arithmetical operations of multiplication and division of all numbers within the limits of the tables, may be replaced by the addition and subtraction of their logarithms; and those involving the powers and roots of numbers, by the single operation of multiplying or dividing their logarithms by the index of the root or power, as the case may be. 16. These advantages of logarithms in effecting numerical calculations, are especially seen in the case of large numbers; and they bring within the range of computation many arithmetical operations which, on account of their intricacy and the labor they involve, would be otherwise impracticable. Properties of Logarithms to the Base 10, or Common System. 17. If we make successively x = 1, 2, 3, 4, &c., in the equatio.n as= y, when a 10, we shall have for the corresponding values of y, 10l=10, (10)2=100, (10)3-1000, (10)4=10000, (10)5=100000, &c. And for negative values of x, -1, -2, -3, -4, -5, &c., we have (10)l-s'=.1 (lO)- =.01, (10)-3 T =.=001, (0)iJo- -. 0001, &c. Hence, in the common system of logarithms, log 10 = 1 and logt 0.1 = — 1 log 100 = 2 " log 0.01 - -2 log 1000 =3 " log 0.001 = -3 log 10000 4 " log 0.0001 = —4 &c. &c. From which we see, that those numbers only which correspond to the exact powers of 10 have entire numbers for 6 TRIGONOMETRY. their logarithms; that the logarithm of any number between 1 and 10 is some number between 0 and. 1, that is a fraction; the logarithm of any number between 10 and 100 is some number between 1 and 2, or 1 plus a fiaction; the logarithm of any number between 100 and 1000 is sgfne number between 2 and 3, or 2 plus a fraction: and so on. Again, for numbers less than 1, we see that the logarithm of any number between 1 and.1 is some number between 0 and -1; the logarithm of any number between.1 and.01 is siome number between -1 and -2; the logarithm of any number between.01 and.001 is some number between -2 and -3; and so on: which show that if the number be less than 1, its logarithm, both decimal part and integral part, will be negative. 18. The entire part. of a logarithm is called the characteristic of the logarithm. 19. Hence, from the above table we see, that the characteristic of the logarithm of a number > 1, which is expressed by a single figure, is 0; if the number be expressed by 2 figures, the characteristic is 1; if it be expressed by 3 figures, the characteristic is 2; and in general, if it be expressed by m figures, the characteristic is m -1; that is, the characteristic of the logarithmic of a number > 1 is one less than the number of significant figures in the number. 20. The characteristic of the logarithm of a number < 1 is always negative. If the fraction be expressed decimally, the characteristic of its logarithm will be always 1 added to the number of ciphers following the decimal point. Thus,.00462 lies between.001 and.01; its logarithm must be some number between -3 and -2, and is therefore -3 plus a fraction. The characteristic is -3, the number 3 being composed of 1+2, or 1 plus the number of ciphers after the decimal point. 21. A Logarithm will therefore usually consist of a whole number, followed by a decimal part less than 1; and if the number to which it corresponds be less than 1, its entire value, both decimal and integral part, will be negative. In the Tables, however, it is usual to print positive decimals only. We proceed now to show how, nevertheless, the logarithms of all numbers whatever, both the logarithms of numbers less than 1, and whose values are consequently negative, and the LOGARITHMS. 7 logarithms of numbers greater than 10 and whose values consequently have an integral as well as a decimal part, can be included in a table where only positive decimals are printed. 22. A negative logarithm may be expressed so that its characteristic only shall be negative. Let -(n+ d) be such a logarithm, n being its characteristic, and d its decimal part: thus. -(n d)= -(n+1)+ 1 -d=-(n+- 1)- d' where the new decimal part d' is positive. Hence we see, that the transformation of' a logarithm entirely negative, into one whose characteristic only is negative, is effected by increasing the characteristic by 1, and substituting for the decimal part, unity minus the decimal part. This difference may be readily formed by subtracting the last digit on the right of the decimal part from 10, and the rest from 9. And conversely, a logarithm whose characteristic only is negative may be made entirely negative by performing on the decimal part the operation just described, and diminishing the characteristic by 1. Thus, -(2.2346899) --- 3 + (1 0.2346899) = — 3 +.7653101..3.7653101. where we place the sign - above and not before the charac-. teristic, to show that the sign affects only the characteristic:and the whole is used as an. abbreviated mode of writing -3 + 0.7653101. If, again, we wish to convert this into an expression entirely negative, we have, 3.7653101 = -3 +.7653101 = -2 -(1 -.7653101) = -(2.2346899). 23. In practice, a negative logarithm is always expressed; so that its characteristic only is negative: when negative, logarithms are expressed in this manner, there are certain peculiarities in multiplying or dividing them by any quantity, which must be attended to, and will be understood by the following illustration. 24. If a logarithm whose characteristic only is negative, is to be divided by any number, we must make the charac. teristic divisible by the number, and correct the expression} by a corresponding addition. Thus, if 8. TRIGONOMETRY'. 7.3295642 is to be divided by 3, we have 7.3295642 -_9 + 2.3295642, and the quotient by 3, is 3.7765214. 25. In the common system, the same decimal part serves for the logarithms of all numbers which differ from one another only in the position of the place of units relative to significant digits. Let N. be any whole number, having n for the characteristic and d for the decimal part of the logarithm, so that log N = n + d. First, let N-x (10)". be a whole number having the same significant digits as N, but having its place of units removed miplaces to the right: then, log (N x (10)m) = log (10)'" + log N, - (m +- n) + d, which has m + n for its characteristic, and the same decimal part.as log N. Next, let N (10)m be a decimal having the same significant digits as N, but having its units' place removed m places to the left; then log (N -- (10)m) -= log (10)-' + log N = -- + n + d; now, if the proposed decimal be greater than 1, or N > (10)", and therefore. log N > m, or n not less than mn, log (N -- 0)) =- (- -m) + d; which has a positive characteristic n — in, and the same deci. mal part as log N: but if the proposed decimal be less than 1,.or N < (10)", and therefore log N < m, or n less than nm, then log (N (0)m) -(m — n) + d, so that in this case also, provided the logarithm be expressed so that the characteristic only is negative, the decimal part is the same as that of log N. 26. From the preceding observations we collect the principal advantages of Tables of Logarithms calculated to a base the same as the base of the system of notation in numbers, 10, to be; that since the characteristics of all numbers whole or fractional are known by inspection, they LOGARITHMS. 9 heed -Uot be recorded in the Tables; and the characteristic being omitted, the same record in the Tables will serve for all numbers which have the same succession of significant digits, and differ only in the position of the place of units relative to these digits: Thus, the record.5386617, which in reality expresses the logarithm of 3.4567, can be made to express the logarithms of 345670, 34567, 3456.7, 345.67, 34.567, 3.4567,.34567,.034567, or any number formed by adding ciphers to the end of the former, or to beginning of the latter immediately after the decimal point; so that every logarithm taken out of the Tables for a particular number, becomes, by simply altering its characteristic, the logarithm of an infinite variety of other numbers, that is, of all that are expressed by the same Ruccession of significant digits. Mode of using Tables of Logarithmnis for Nuwmbers. 27. Tables of Logarithms contain all whole numbers, from 1 up to a certain limit, with their logarithms, in which the characteristic is usually suppressed. The tables in common use contain the logarithms of all numbers from i to 10,000. In making use of tables of logarithms to effect numerical calculations, the two main problems which arise are Ist. Any number being given, to find, by means of the Tables, its logarithm. 2d. A logarithm being given, to filnd, by means of the Tables, the correspondinig number. 28. 1st. Any number Nbeing given, to find by the Tables its logarithm. The characteristic, whether N be a whole number or a decimal, is known by inspection. When N, leaving the decimal point out of consideration, exceeds the limits of the Tables, we must transpose the decimal point, so that the integral part may be contained in the Tables, and may be the greatest possible. Let n be this integral part, and d the remaining decimal part of the proposed number; I the decimal part of the logarithm of n, and 8 the difference between the logarithms of the numbers n 10 TRIGONOMETRY. and n + 1,'(this difference is always.set down, in the Tables,) we have log(n + 1)-log n=log (F-+')= log(l q-+ )=0, when n= o tience, the difference of the logarithms of two consecutive numbers is less in proportion as the numbers themselves are greater. We may therefore assume that the difference of the logarithms will' always be proportioned to the differences of the numbers; or that d=, hence x=d x 6, the value of x, =d X *, is what must be added to I to get the decimal part of the logarithm of n+d, which is also: the decimal part of the logarithm of the proposed number. 29. This property, that the increment of the logarithm ma:y be assumed to be proportional to the increment of -the number, is extremely important; and the application of it deserves the closest attention, as it furnishes a ready means of greatly extending the range of the Tables. We see' also.that the number n should be as great as possible, for the error in the above proportion is diminished as n increases, ~Lnd disappears when n=-oo. 30. We give specimens of the usual logarithmic Tables for numbers. Table A has the natural numbers arranged in order in the column marked N, and is limited to numbers embracing not more than three digits. If the logarithm of a number is required which contains not more than three digits, it will be found in the first column. To save the needless repetition of figures, the two first decimals of the logarithm are only written at the head of each break, and must be regarded as belonging equally to all logarithms which succeed up to the next break. If the number contain four digits, seek the number in column N corresponding to the first three digits, and then look along the top of the pages among the numbers 0, 1, 2, 3, 4, &e., for the number that corresponds to the fourth digit, the logarithm of the given number will be found at the intersection of the horizontal lines passing through the numbers in the column N, and the vertical line in which the fourth digit is found. TABLE OF LOGARITHMS OF NUMBERS. 11 The differences of the logarithms are placed in column D. If the number contain more than 4 digits, place a decimal point after the fourth figure from the left hb.nd, thus convert. ing the given number into a whole number and decimal. Find the logarithm of the entire part as just explained, and then tale the corresponding difference in the column ]D, and multiply it by the decimal part; add the product thus found to the logarithm, and the sunm will be the whole logarithm sought. The characteristic is determined by known rules, in each case. A. A TABLE OF L0GARITHIS FROMI 1 TO 10,000, N. 0 0 1 2 3 4 5 6 7 8 9 D. 280 447158 7313 7468 7623 7778 7933 8088 8242 8397 8552 155 281 8706 8861 9015 9170 9324 9478 9633 9787 9941 00 90 154 282 450249 0403 0557 0711 0865 1018 1172 1326 1479 1633 154 283 1786 1940 2093 2247 2400 2558 2706 2859 3012 3165 153 284.3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 285 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 152 286 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151.288 9392 9543 9694 9845 9995 0146 0296 0447 0597 0748 151 289 460898 1048 1198 1348 1499 1G49 1799 1948 2098 2248 150 290 462398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 291 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 292, 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 294 8347 8495-1 8643 8790 8938 9085 *9233 9380 9527 9675 148 295 9822 9969 0116 0263 0410 0557 0704 0851 0998 1145 147 296 471292 1438 1585 1732 1878 2025 2171 2318 2464 2610 146 297 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 298 4216 4362 4508 4653 4799 4944 5090 5235 5381 5526 146 299 5671 5816 5962 6107 6252 6397 6542 6687 6832 6, G 145 31. In Table B, the column N has four digits in it, and hence this table may be used to find directly and without the use of the column of differences, the logarithms of all numbers as high as those which contain five digits, the fifth digit of the given number being found as before in the top line above the logarithms. When the first three figures of the decimal part of the logarithm are common, they are omitted 2* B 12 TRIGONOMETRY. in the tables except in the first logarithm of each break, but are always to be regarded as constituting the first figures of the logarithms immediately following in the same break. B. OF NUMBERS, N. 0'1 2 3 4 5 6 7 8 9 D. Pro. 2550 406540'2 5572 5742 5913 6083 6253 6424 6599 6764 6934.51 7105 7275 7445 7615 7786 7956 8126 8296 84668637 170 52 8807 8977 9147 9317 9487 1'9658 9828 9998 0168 0338 1 17 53 4070508 0678 0848 1018 1189 1359 1529 1694 18691 2039 2 31 54 2209 2379'549 2719 2889 059 39 3229 3399 3569:3739 1,70 51 4 68 5 85 55 3909 4079 4249 4419 4589 4759 4929 5099 5269 5439 6102 56 5608 5778 5948 6118 6288 6458 6628 6798 6968 7137 7 119 57 7307 7477 7647 78171 987 8156 8326 8496 86668836 8 136 58 9005 9175 93'45 9515684 9854 0024 0194 0363 0633 59 4080703 08731042 112121382 155111721 1.891 2060 2230 2560 2400 2569 2739 2909 30o78 3248 3417 3587 3757 3926 61 4096 4265 4435 4604 4774 4944 5113 5283 5452 5622 62 5791 5961 6130 63001 6469 6639 6808 6978 7147 7317 169 63 7486 7656 57825 7094 8164 8333 8503 8672 8841 9011 1 17 64 9180 9350 9519 9688 9858 0027 0196 0366 0535 0704 2 31 3 51 65 4090874 1043 1212 1382 1551 1720 1889 2059 2228123097 4 68 66 2567 2736 2905 3074/ 3243 3413 3582 3751 3920 4089 l 101 67 4259 4428 459 7 4766 4935 6105 52741 5443 5612 15781 7 118 68 5950 6119 6288 6458 6627 6796 6965 7134 7303 7472 169 8 1135 69 7641 7810 7979 8148 8317 8486 865518824 899319162 9 152 EXAMPLE 1. —Seek the logarithm of 2967. In table A, find in column N, 296, then run along the top line of the table until you come to the number 7, the logarithml 472318 is the decimal part of the logarithm of the given number, and prefixing the characteristic, the logarithm is 2.472318. EXAMPLE 2. —Find the logarithm of 296789. Regard all the figures on the right of the fourth figure as decimal part. Find the logarithm of the entire part 2967, as before. We have log 2967, (omitting characteristic) = 472318. USE OF TABLE. 13 The column D gives 146 for the difference. Multiplying this by.89, the decimal part of the given number, we have 146 x.89 — = 146.85, or neglecting the decimal, and since it exceeds.5, let 1 be added to the entire part, and we have 147, to be added to the logarithm as before found. Thus, log.2967 = 472318 ".89 147 "296789 472465 Now, ii.troducing the characteristic, the whole logarithm becomes 5.47246(5. 32. The trouble of multiplying the difference by the decimal part may be avoided in table B, by the column of proportional parts, which contains the product of the tabular difference by ai',,-1, 2-3, &C., or of one-tenth of this tabular difference by 1, 2, 3, 4.... 9. Thus, find the logarithm of 2568946. log 25689 = 4097472 log,.4 (16.9 x 4) = 68 log.06 (1.69 x> 6) - 10 log 2568946 = 6.4097550 33. 2d. A logarithm being given, to find, by means of the Table, the corresponding number. When the decimal part I of the logarithm IL is positive, and found exactly in the tables, we have only to take out the corresponding number; and the given characteristic will determine the position of the units' place. When the decimal part I of the given logarithm is not found exactly in the tables, let n be the integer whose logarithm has a decimall part 1' immediately inferior to 1, d, the difference between the decimal part 1' and that which immediately follows it in the tables corresponding to n + 1, and d' the difference --'; making the proportion. d 1 we find the fourth term x which we must reduce to a decimal. and add it to the number n: we thus find a number whose logarithm has for a decimal part 1, and from which 14 TRIGONOMETRY. w.e,can deduce the required number, having regard to the Characteristic, as in the preceding case. When, the given logarithm is entirely negative, we must transform it so as to have the decimal part positive, as has been explained, and then proceed as above. Ex. 1. —Let the given logarithm be 5.5386617. In the column marked 0, we seek the logarithm which is next less than the decimal part L, which we find to be 5385737, opposite to the number 3456 in the column marked N: then advancing in the horizontal line to the column marked 7, we find the four last decimals 6617 of L. Hence, affixing the digit 7 to 3456, we have 34567, and since the given characteristic is 5, the required number is 345670. ~Ex. 2.-Let the given logarithm be 2.5386729. In seeking for the decimal part in the tables as above, we find that it lies between.5386617 and.5386743: if it were exactly equal to the former, the corresponding number would be 34567, but as it surpasses.5386617 by 112, there will be an increase in 84567; and as the difference of the logarithms corresponding to an increase of unity in 34567 is 126, the required increment x, admitting the common,principle of proportional parts, will be found by the proportion 1 _1 =$..X = 1 e2 0.8 99. Therefo're, the required number, neglecting the decimal point; is 34'56789, and since the given characteristic is 2, the required number is 345.6789. By means of the table of proportional parts of the tabular differences, we may avoid the trouble of dividing 112 by 126. in the table of proportional parts of the difference 126, the part next less than 112 is 101, which corresponds to.8 and there remains 11. If we put zero to the right of 11, we have 110, which differs very little from the part 113, which has 9 opposite to it: we conclude therefore, that 110 corresponds to.9, and 11 to.09; consequently the required number is composed of the digits 3456789; and taking account of the characteristic, the number is 345.6789, as before. 34. In actual calculations it is often necessary to add several logarithms together, and from this sum to subtract LOGARITHMS. 15 other logarithms. In such cases it is convenient to reduce all the operations to a single addition by the use of the arithmetical complement of the logarithms to be subtracted. 35. The arithmetical complement of a logarithm is the number which remains after subtracting the logarithm from 10. If c be the arithmetical, complement of b, we shall have c= 10 — b. Now, if we propose to subtract the logarithm b from the logarithm a, we have, evidently, since -b = c — 10, a -b a + c -10, which shows that the difference between two logarithms may be found, by adding to the first logarithm the arithmetical complement of the logarithm to' be subtracted, and then diminishing this sum by 10. The arithmetical complements of logarithms may be all formed by subtracting the last digit of each from 10, and all the other digits from 9. TRIGONTOMETRY CHAPTER I. ~THEORY OF TRIGONOMETRICAL LINES. Object of Trigonometry. The Methzod of Representing the Sides and Angles of Triangles by Nunmbers. 1. In every triangle, plane and spherical, there are six parts: three angles and three sides. To determine all the parts of a triangle, it is generally sufficient that three of these be known; but it is further necessary, when the triangle is rectilinear or plane, that at least one of these parts shall be a side. For, with three given angles an infinite number of triangles can be formed, which will be similar to each other, but not equal. It is always understood that the sum of the three angles is equal to two right angles. 2. Geometry furnishes very simple constructions for determining a triangle, by means of some of its parts, as known; but these constructions, like all graphic processes, are attended by the inconvenience of giving an inexact and therefore insufficient approximation, on account of the imperfections of the most accurate instruments. These constructions have been replaced by numerical calculations, which admit of any degree of precision that may be required. 3. The special object of Trigonometry is to give the methods of calculating all the parts of a triangle, when a suglicient number of parts is given. This is called the resolution of the triangle. Plane Trigonometry gives the various methods of solving plane triangles, and Spherical Trigonometry treats of the resolution of spherical triangles. 4. To express the dimensions of right lines by numbers, (16) THEORY OF TRIGONOMETICAL LINES. 17 they are referred to a common unit of measure, as a foot, a yard, &c.: and thus, the sides of a plane triangle are represented by a certain number of feet or yards, &c. 5. Angles are designated by the arcs which measure them. For this purpose, the circumference, whatever be the radius, is divided into a certain number of equal parts, called degrees, and then an angle or an arc is expressed by a number of degrees. 6. Formerly, geometricians agreed to divide the circnumference into 360 degrees, the qegree into 60 minutes, the minute into 60 seconds, &c. By this division, the fourth part of the circumference, or quadrant, which is the measure of a right angle, contains 90 degrees. To avoid the embarrassment from complex numbers, it has been proposed to subject angles to the decimal division, thus dividing the quadrant into 100 degrees, the degree into 100 minutes, the minute into 100 seconds, &c. 7. Degrees, minutes, and seconds, are denoted by the symbols 0', ". Thus, to represent 44 degrees, 9 minutes, 37 seconds, we write 440 9' 37". In the decimal division, if we wish to refer this arc to the quadrant, taken as the unit of Lmeasure, it would be expressed thus, 0.440937: for, in this division, degrees are hundredths of the quadrant, minutes are ten thousandths, and seconds are millionths. 8. The old division is adopted in this treatise, the circumference being divided into 360 degrees. In the formulke which are given, the letter 7S is often used to represent the semi-circumference 1800~. Definitions of Trigonometrical Lines. The Manner of using the Signs + and - to indicate the Positions of Lines when their Direction is opposed. 9. Geometricians were for a long time embarrassed by the difficulty of establishing the relations which exist between the angles and the sides of triangles; and yet this is essential to the resolution of the triangles. For, the sides being given in terms of a linear unit, and the angles in terms of the arc of a circumference, all the parts of a triangle would not be expressed in homogenous terms, and could not therefore be combined in the calculations. The use of Trigonomet 18 TRIGONOMETRY. rical Lines, which are linear fulnctions of' the angles or arcs considered, enables us to substitute right lines for the angles or arcs, and thus to establish the homogeneity between all the parts of a triangle, which is indispensable to its resolution. We will now proceed to define these trigonometrical lines.* 10. The sine of the arc'A lI, is the perpendicular I P, let fall from one extremity of the arc,'! - - /1 ~' upon the diameter which passes through the other extremity. X \ Pl A The Tangent of the arc A M, is A ~ the distance A T, intercepted on the tangent drawn at one exQ, tremity of the arc, between that extremity and the prolongation of.B' the radius 0 MS, which passes through the other extremity. The Secant of an arc A M1, is the part 0 T, of the radius produced, comprised between the centre and the tangent: IRepresenting the arc A I by x, the sine, tangent, and secant, are designated as follows,Mn P = sin x, A T = tang x, O T =- sec x. 11. If M P be produced until it meets the circumference at N, the chord M N will be double of M P, and the arc MAN double of A M, hence, the sine of an arc is half the chord which subtends double the arc. 12. If we designate by r the radius of the circle, the side of the inscribed square will be r / 2: the subtended arc to the side of the square is 900, hence, sin 45~= - r /2. * The principle of homogeneity between the parts of a triangle, to which reference is made in the text, is an important one, and will be frequently referred to in this treatise. It is a general principle, not restricted to trigonometry, but necessarily applied in every branch of analytical mathematics. It is founded upon the essential relations which must exist between the two members of every Algebraic equation, to make it the consistent expression of equality between equal magnitudes; and serves also to point out the modifications which any equation not homogeneous in j'rm, must have, that the relations expressed by the equations miay be consistent with established facts or truths. THEORY OF TRIGONOMETRICAL LINES. 19 13. In like manner, the side of the inscribed hexagon being equal to the radius, and the subtended arc being 60~, we shall have sin 300=~ r. 14. The complement of an arc or angle is the difference between that arc or angle and 90~. W hen the are is greater than 900, its complement is negative; thus, the complement of 1270 is -370. The two acute angles of a right-angled triangle are complements of each other. 15. We designate by the termns cosine, cotangent, and cosecant of an arc, the sine, the tangent, and the secant of the complement of that arc: and to express these new lines, we employ the abbreviations cos, cot, cosec; thus, by the definitions, we have cos x = sin (900 -x), cot x = tang (900 -x), cosee x = sec (900-x). 17. If we draw the radius 0 B 3t3 perpendicular to 0 A, and also - M Q, and B S, perpendicular to OB, the arc BM will have MQ for its sine, B S for its tangent, 0 S p / p for its secant: but the arc B I is A P A the complement of A M; hence, designating always A M by x, we have MQ=cos x, BS=cot x, OS= B' cosec x. We notice that M Q = O P; which shows, that the cosine of an arc is equal to the part of the radius comprised between the centre and thefoot of the sine. 18. The distance A P, comprised between the origin of the arc and the foot of the sine, is called the verse-sine; and the distance B Q, the co-verse-sine. 19. In giving to the point iI every possible position on the circumference, the trigonometrical lines take situations altogether different from those which they hare, when the are A Al is less than 900. For example, if we have the arc A N', whose complement is B M', and negative, its cosine Q M' or 0 P', is found situated on the left of the point 0, whilst before, it was on the right; such changes in the positions of the lines introdulce into the calculations difficulties, which the following proposition will explain. 3 20 TRIGONOMETRY. Let AB X be any line whatever, on which there are two points A and B, separated a distance AB = a. Assume, as known, the distance x of the point B firom any point, as M, of the line ABX, and let it be required to find the distance from the point A to the point M. IRepresenting this RA distance by z, it is evident we shall have z=a+x, or, z=a-x, according as the point M is between B and X, or between B and A: so that two different formula are required for these two positions of the point ni. But this inconvenience is avoided, and a single formula made to suffice, by giving different Algebraic signs to distances which have contrary positions with respect to the point B. Thus, if, in the first formula z a + x, we make successively x_= — B 3, and x =- B AI, we have, under the first supposition, z = a + B Al, and under the second, z= a-BMi, as should be. We thus see, that the -first formula z = a + x, will correspond to all positions of the point NM, and the second is unnecessary. We might also take x positive between B and A, and negative between B and X: in this case, the second formula z= a-x, would be used. It would be easy to multiply illustrations, but what precedes is sufficient to make apparent the importance of the following rule of Descartes. 20. If we consider on any line whatever, straight or curved, diferent distances, measured from a common origin, fixed on that line, we may introduce into the calculation distances which have opposite positions with respect to this origin, by giving those lying in one direc-tion the sign ~-, and those lying in the opposite direction, the sign -. 21. The rule by which positive distances are estimated is altogether arbitrary, but once fixed, negative distances must be reckoned in an opposite direction. With crigonometrical lines, the usage is, to consider them as positive in the posi. tions they occupy when the are is less than 90~. 22. We shall soon have occasion to make numerous applica THEORY OF TRIGONOMETRICAL LINES. 21 tions of this rule; but before proceeding further, the student must be guarded against a very common error, which consists, in assimilating the principle in question to a theorem susceptible of an d priori demonstration. It is far otherwise, and whatever considerations may have been adduced in its support, it is to be regarded as a simple agreement which must not be violated, and the utility of which is made apparent by its applications. Progressive Changes in Trigonomletrical Lines. The Aliethod of reducing them to the First Quadrant. 23. When the radius OM coincides with 0 A, the are A 3i is o, B S the sine is o, the tangent is o, and X 1 the secant is equal to 0 A: at the same time, the cotangent and _ cosecant are infinite, since the lines. P P_ _ A BS and OS increase more and more as O Al approaches O A, and. become infinite, when they coin- T cide. Thus, denoting the radius B' by r, we have sin o - o, tang o = o, see o=- r cos o = r, cot o= c, cosec o = co. 24. If the radius O 3f approaches the position O B, it is easy to see that the sine, tangent, and secant increase, while the cosine, cotangent, and cosecant diminish. When the point X, is at the middle point of AB, the arc AM is 450, the triangle O P M is isosceles, and the sine is equal to the cosine. Now this triangle gives 2 MP2= r2, hence, M P= r /2, and therefore sin 450~= cos 450= 4 r V/2 25. The triangles OAT, OBS, being also isosceles and equal to each other, the tangent and cotangent are equal to the radius, hence tang 450= cot 450= r. 26. Finally, the secant and cosecant are also equal, and the triangle OAT giving OT'=- 2 r', we have O T= r / 2, and hence 22 TRIGONOMETRY. sec 450= cosee 450= r v2. 27. When the point M reaches B, the sine is equal to' B the tangent and secant are infinite, the cosine X Q is o, the cotangent B S is also o, and the cosecant becomes equal to O B. We have then, sin 900= r, tang 900= s, see 900= 00 cos 90~= o, cot 90~= o, cosec 990= r. These values are indeed consequences from those which were found when the are was o; for, the arcs o and 90~ being complements of each other, we ought to have sin 900 = cos o, tang 900 = cot o, see 900 = cosee o, cos 90~ = sin o, cot 990 = tang o, cosee 90~ = sec o. 28. Continuing the rotation of the radius O M, let us sup. pose that it has reached the point IF': the are is then A M', and its sine M' P'. Draw WA' M parallel to A'A, and construct all the trigonometrical lines of the are A M1X as indicated in the figure, we see at once that the sines Al P and M'P' will be equal, and therefore sin A I'= sin A AM. 29. To get the tangent, produce the radius O M' below the diameter AA', we see that the tangent, which is here AT', is found in a position opposite to that which it had at first: consequently, it is negative. But the triangles O AT, OAT', being equal, give AT'= AT, hence tang AM'- - tang AM. 30. From the definition, the se_S___B S cant of the are A M' is O T'. This X.T line is no longer laid off on the ra.' dius O Ai' on the same side of the * A~ centre with the point M', but it;A P < A lies in the opposite direction, and is, consequently, negative; and since W~ ~ ~,O -- O- T, it follows that sec A N'. — see A At. 31. The cosine, cotangent, and cosecant present remarkable anal. ogies. Since the are AMi' exceeds 900, its complement is negative; and, further, as the cosine Q M' or O P' is situated on the left of the point O, we also consider this cosine as negative. The same reasoning applies to the cotangent THEORY OF TRIGONOMETRICAL LINES. 23 B S'. As to the cosecant O S', its sign is still positive, since it is on O M', on the same side of the centre as the point M'. The triangle O Q 3M and O Q Il' being equal, as well as the triangles OBS, OBS, we have Q M'= Q 1, BS'= BS; hence, cos AMI'=-cos AMI; cot A'i'= —cot AMI; cosec A M'= cosec AM l. 32. The supplement of an tarc orh angle is the difference between that arc or angle and 1800: thus, A' 3i', or its equal AM, is the supplement of A M', and we may enunciate the properties found above by saying, that two supplementary arcs have their trigonometrical lines equal and with contrary signs, with the exception of the sine and cosecant, which are also equal, but have the same algebraic signs. 33. To express these properties by equations, designate A M' by x, we shall have A = A''- - 1800 -x, and we may write sin x = sin (180~ -x), cos x= -cos (1800 —-x); tang x= — tang (1800 -x), cot x= -cot (1800- x), (1) sec x= sec (1800 -x), cosecx= cosec (180~ —x), J 34. It is also evident, that from 900 to 1800, the sine, tangent, and secant decrease; whilst, on the contrary, the cosine, cotangent, and cosecant increase. If O M coincides with OA', we have, for an arc of 180~, sin 1800 o, tang 180~ = o, sec 180~ — - r, cos 180~- r, cot 180~=-c-, cosec 180~= oo. All these values may be deduced from the relations (1), by making x = 1800. Thus, cos x = -cos (1800 -x), becomes cos 1800 = — cos o; but cos o = r; therefore, cos 180~ = - r. 35. The applications of analysis to geometry frequently involve the consideration of arcs which contain many semicircumferences. It is necessary then to give formulae for reducing all such arcs to the first quadrant. To abridge, we will consider specially the sine and cosine, which are the trigonometrical lines most used; and since every are greater than a semi-circumference is composed of an arc less than 1800, increased by an arc containing 180~ one or more times, we will, in the first place, determine the sine and cosine for the arc 180~ + x, x being < 180~. Let A M be the are designated by x, which may be taken, 3* 24 TRIGONOMETRY. Flig.. at pleasure, between o and 1800; St _ _B S adding to A 1 the semi-circlmfer-, ence hA' N' the newv arcAMA'N' will be equal to 180~ + x. The two arcs have equal sines, viz., A P A M P Cand N' P', or, what is the same. thing, 0 Q and OQ'; but since these lines. have opposite positions, - we should, in accordance with the agreement established (Art. 20), B' give to them contrary algebraic signs. The cosines O P and O P' are also equal, and must, in like manner, have different algebraic signs: hence sin (1800 + x)=- sin x, cos (180 + x) = - cos x. (2) 3d. In the next place, add 360~ to A I, it is evident we will return to the same point M of the circumference, and thlat consequently all the trigonometrical lines become the same. Hence we have, sin (360~ + x) _ sin x, cos (360 + x) = cos x. (3) 37. In general, whatever value we attribute to the are X, if' we add to x, 180~, or an odd number of semi-circumferences, its extremity will be moved from the vertex of one diameter to the opposite vertex; and hence, it is evident, that the algebraic signs of the sine and cosine are changed. But, if we add to x, 360~, or an even number of semi-circumferences, since we return to the same point of the circumference, no trigonometrical line should change its sign. 38. It remains now that we consider negative arcs, that is, arcs which are described when the radius, which at first coincided with 0 A, moves in the direction A B'A', contrary to that which it before followed. Let A M and A N be two equal and opposite ares, designated by x, and -x. It is evident that their sines X P and N P are also equal and inversely disposed. To obtain the cosines, we observe, that the complements of these arcs, 900 - x, and 900 + x, are represented by the arcs B M and B M N, of which the sines AM Q and N Q' are equal and similarly situated: hence we have sin (- x) = - sin x, cos (- x) = cos x. (4) THEORY OF TRIGONOMETRICAL LINES. 26 39. Although in the figure the arcs A fi and A N are less than 90~, these formule are nevertheless general. For, it is evident that by augmenting the two arcs at will, provided they continue equal, the sines MA P and N P do not cease to be equal and opposite, we shall therefore always have sin (- x) _ - rsin -x. With regard to the cosine, substitute in the second formula (4) arcs greater than 90~, A B A[' and A B' N', for example; and make x = A. B M', and- x = - AB'N'. The complement 90 - x of the first arce is negative, and is repre- _:1 sented in the figure by the arc B M', situated on the left of the point B, whilst the complement 900 + x of the other arce is equal P A A to B A N', and always on the right ofthe point B. ButthesinesM3'Q,Q and N'Q' of these complementary arcs are equal, and are similarly B disposed with respect to the diameter B B'; hence, we have always cos (-x) - cos x. FormulE (4) are therefore general. 40. It is well to note that the cosine of any arc whatever, positive or negative, is always represented in magnitude and position,. by the distance from the centre to the.foot of the sine. 41. Formulm (1), (2), (3), (4), may be applied to all possible arcs, positive and negative. Forbrevity, let us consider only the sine and cosine. 1st. Resuming the two formule sin x = sin (180~-x), cos x = - cos (1800 - x) (Art. 33), which were demonstrated for positive arcs comprised between o and 1800 they become, when x is changed into 1800 + x, sin (180~ -+ x) = sin (- x), cos (180~ + x) = - cos (- x), which results are evident by virtue of the relations (2) and (4). We may increase the are by additions of 180~, to infinity, and the formula will remain unchanged. If we substitute -x for + x, we see, in like manner, that the two formulae are still true. They therefore suit all possible arcs. 2d. Formulie (2) which have been demonstrated for all positive arcs, are equally applicable to negative arcs. Fort changing x into - x, they become 26 TRIGONOMETRY. sin (180~ — x) - sin (- x,) sin x cos (1800 -x) - cos (- x,) = - cos x. thus reproducing formula (1). 3d. Since the addition of 1800 to any are whatever + x or -x, only changes the algebraic sign of the sine or cosine, it follows, that the addition of 360~ should produce no change: hence, formule (3) apply also to negative arcs. 4th. Formulue (4) require no special demonstration, since it is evident that in them x may be replaced by -x. 42. Nothing is now easier than to reduce to the first quadrant the trigonometrical lines of any are whatever. Suppose x- 10290, to be an arc whose sine is required. Take 3600 from this are as many times as possible, and there remains 3090. Then by formula (3), sin x = sin 3090. Now take 180~ from 309~: by formula (2), we have sin (180~ + x) - - sin x, hence, sin x = 1sin 129~. Finally, sin x - sin 510~, since 510~, is the supplement of 1290. We might carry the reduction farther, by observing sin 570, = cos (90~ - 51~), and hence sin x= -sin 510 =-cos 390~. 43. If the given are had been x - 10290~, the algebraic sign of the sine would have been contrary to the preceding, and ve should have sin x cos 390~. Of Arcs which correspond to a given Sine, Cosine, &c. 44. The preceding discussion has shown, that there exists an infinite number of arcs which have the same trigonometrical lines. Let us now suppose one of these lines given, and let it be required to determine the arcs which correspond to it. Assume sin x =a.'On the raS' B _S dius O B, perpendicular to O A, lay X M' T off 0 Q - a, and through the point Q draw M1 X' parallel to 0 A. It is evident that we must take for; P Pe A values of x all arcs which terminate at the points M and Mlt. Designate:d/ d \1/the arc AM5~ by a, and 1800 by 2; T AM' will be equal to b in the expressions for sin (a-b) and cos (a-b). We may readily modify the constructions to correspond with these various cases, but as these cases are numerous, the following process will be found preferable. 1st. The restriction a > b may be removed from the formulm (20) and (21). For when a < b, we know (Art. 38), that we have sin (a - b) -sin (b - a), and cos (a-b) = cos (b-a). THEORY OF TRIGONOMETRICAL LINES 37 But, b being greater than a, we may obtain sin (b -a) and cos (b —a) from the formulm (20) and (21), by changing a into b, and b into a: but this substitution will only cause the sign of the first to be changed, while the second will remain the same: we shall, therefore, have, for sin (a-b) and cos (a - b), the same formula as in the case of a> b. Hence, these formulae apply to all cases in which a and b are positive, and a + b < 90~, and we may therefore give to a and b any values we please between 0 and 45~. 2d. Since the formula which embrace (a - b) may be deduced from those which express sin (a + b)j and cos (a + b), by changing b into - b, it follows, that formulae (18) and (19) are true for all values of a between 0 and 450, and for all values of b between -45~ and +450. Further, these formule also suit negative values of a, taken from 0 to -450~. For, representing by a an arc <450, and making a -- --, we shall have sin (a r+ b) = sin (-a+- b) =-sin (c- b), cos (a + b) = cos (- c + b) = cos (a - ). Thee. arcs ac and -b are within the limits for which the formule (18) and (19) have been demonstrated: therefore, -*sin cos b- - cos a sin b: sin (a +- b) - sin (a -b) = _ —., cosa Cos b + sin -a sin b cos(a +:b) - cos (a — b =Since a —--, we have (Art. 38) sin a=- sin a, CGos. = cos a, and, hence, these formulae reduce to (18) and (19). 3d. We will now prove that in formulae (18) and (19), we may extend indefinitely the positive and negative limits of a and b. Make a = 900~ + a, being any are whatever between - 450 and + 450~: we.shall have, on taking the complements, sin (a + b) = sin (90~ -+ a + b) = cos (-L- b) = cos c Cos b sin a sin b cos(C+ ) = - r cos (a Jr b) = cos (900~ + C + b) sin ( —L- ab) = sin C cos b — cosa sin b -sin (C b)= — 38 TRIGONOMETRY. But since a- 900 + a, we have, by known reductions, sin a = sin (90~ + a) = cos (- a) -- cos a, cos a cos (90~ + a) = sin (-a) = —-sin a. Therefore, we may substitute for cos a, the sin a, and for sin a, cos a, which again reduces to the formulm (18) and (19). Now, taking a between — 450 and + 450, the are 900 + a, or a, passes through all magnitudes firom 45~ to 1350, and the positive limit of a is extended to 1350. Repeating the same reasoning, it is evident that this limit may be again extended from 90~ to infinity. 74. The demonstration given (2d), to show that if the formulae (18) and (19) are true for positive values of a <45~, they are also true for the same values taken negatively, may be applied to the case in which the positive limit of a is greater than 45~. Therefore, since we have seen that they are true for all positive values of a, they are also for all negative values. 75. As to the arc b, it is evident it is subject to the same reasoning that has been applied to a, and we may also extend to infinity each of its limits. Therefore, formulae (18) and (19), as well as formula (20) and (21), are proved for all values of the arcs a and b. Formuce for the Iultiplication and Division of Arcs. 76. Hereafter, the radius will always be considered as unity, so that we shall have r —1, and under this hypothesis formule (8), (9), (10), (11), (12), (18), (19), (20), (21) become sin a cos a sin2 a - cos2 a=l, tang a — a' cot a-. a a ---- cos a sin a cos a, cosec a=- a' sin (a I b) = sin a cos b:h cos a sin b, COgeO gt Cos a cos (a f-b) = cos a cos b q: sin a sin b. 77. If now we make b = a in the expressions for sin (a + b) and cos (a + b), we have sin 2a=2 sin a cos a, (1) cos 2a = cos2 a - sin2 a, (2) which enable us to determine the sine and cosine of double of an arc, when we know the sine and cosine of that are. TIHEORY OF TRIGONOMETRICAL LINES. 39 78. Making b - 2 ct, the same expressions give sin 3 a = sin a cos 2 a - cos a sin 2a, cos 3 a =cos a cos 2 a-sin a sin 2 a; and substituting for sin 2a. and cos 2a, their values, and remembering that sin2 a + cos2 a = 1, we have sin 3 a = 3 sin a-4 sin3 a, (3) cos 3 a = 4 cos3a — 3 cos a. (4) By the same process we may deduce the expressions for sin 4 a, sin 5 a, &c., and for cos 4 a, cos 5 a, &c., or of any other multiple of a. 79. The formula for the division of arcs are readily deduced. Thus, to find the sine and cosine of half an are, make a= ~'a in formulae (1) and (2); we have 2sin i a cos -3 a =sina, (5) Cos a -sing a=cos a,-sin (6) and we have also cos'2 a-+ sin2 a = 1. (7) If we suppose the value of cos a to be known, we have only to resolve the equations (6) and (7) to get the values of sin a and cos a. We get /sin,a=G, —cos a, / 1 + cos a sin 4 a= \~/O5 cos a= 2 2. (8) 80. It is proper to note that these values of sin I a and cos 4 a should be affected with the double sign: 1. We may readily see why these two equal values with contrary signs should exist, for we observe that it is not the arc a itself which enters into these values, but its cosine, so that the formulse should make known, at the same time, the sine and cosine of the half arc for all arcs which have the same cosine. These arcs are made known by the formula, Art. 44, x = 2k 1: -'a,) in which a denotes the smallest positive are corresponding to the given cosine, 7t the semi-circumference, and k any whole number, positive or negative. We ought then to find ror sin - a and cos - a, the values included in sin (k r:i ~- a) and cos (k t =:= ~ a). 40 TRIGONOMETRY. If ik is even, k A will be a multiple of 3600, and we may suppress it without altering the value of the sine or cosine, and there will result sin (~ 4 A) = ~ sin 4 a, and cos (~ e-) = cos 4-a. If k is odd, we may still suppress k a, but the signs of the sine and cosine must be changed (Art. 35), and we have - sin (d e- a) = F sin - a, and - cos (~= -- a) = - cos - a. We see, therefore,.that we have in fact two equal values for sin - a and cos A a, affected with contrary signs. 81. If we have the sine given instead of the cosine, it will be only necessary to substitute in formula (8) for cos a, its value, /1 -sin'2 a. Since this new radical also.has the double sign —, we should have four values for each of the unknown quantities sin 4 a, and cos i a. But we may obtain'these values under another form. Resuming equations (5) and (7), 2 sin I a cos 4 a = sin a, Cos2 a + sin2 a= 1, we deduce from them the values of sin 4 a and cos I a. Adding the first equation to the second, and afterwards subtracting the first from the second, we have, when we extract the square root of both numbers of the resulting equations, cos a +asin a=I +sin a, cos a - sin a = /1 — sin a; from which we readily deduce the values sought: sinl a = / 1 + sin a - / 1-sin a, (9) cos a- 1 + sin a +- 1- sin a.'(10) Since there are two radicals, each of these expressions bhai four values, as might have been anticipated. For, these expressions should make known the sine and cosine of the half are for all arcs which have the same sine. But these arcs are made known by the formula~, (Art. 44,) x = 2 k e + a,) X = (2 k + 1) cta; hence, the expressions for sin 4 a and cos i a, must give.theQ sine and cosine for all arcs represented by THEORY OF TRIGONOMETRICAL LINES. 41 k7t - +a, and (k + 4) — ae. But we may suppress kt, taking care to retain or change the signs of the sine and cosine, according as k is even or odd. Hence, we should have for sin 4 a and also for cos 4 a, four values, viz., sin a =i sin,and sin a- =a- sin ( - ), cos a = Cos a and cosa = 4- cos( - a). We see, further, that they are equal, two and two, with contrary signs. If a = 900, we have -- a-=45 - a = 450~; and the four values are reduced to two. 82. Another observation suggests itself. Sine T -- 180~, it follows that the two arcs 4 a, and 4 - _- a, are complements of each other. Hence, the preceding values may also be presented thus: sin. a -= - sin a, sin - a = -- cos a, COS - a = COS, cos a=4- sin a. That is, the values of sin 4 a and cos 4 a are the same, and this is indicated by formule (9) and (10). 83. One difficulty remains to be explained. When the arc a and its sine are known, to ascertain which of the four solutions must be selected for sin 4 a or for cos 4 a. For brevity, let us consider sin 4 a only. Taking the radicals with their different signs, the four values of sin 4 a may be written, sin.- a = ~ 4 (/ 1 + sin a- v 1- sin a), sin 4 a= = 4- (V 1 -+- sin a + 1- sin a). In the first place, it is evident, that the two first are equal with contrary signs, and the same for the two last. If we square the first, the result is < 4, while the square of the other gives a result > 4. But we know that sin 450 = cos 450 - -; hence, neglecting the signs, the two first values are less than the sin 450, while the two last are greater than sin 45~. But, when the are is given, we may readily determine d priori whether the sine of the half arc is positive or negative, and whether it should be greater or less than 45~. Thus, all indetermination vanishes The same reasoning applies to the cosine. If, for example, we take a <900 42 TRIGONOMETRT-. sin i- a must be positive and less than sin 450: cos i- a must also be positive, but greater than 450: and those values of (9) and (10) must be taken Which fulfil these conditioi s. 84. Let us now consider the trisection of arcs. Substituting for a, 4 a, formulne (3) and (4) become sin a =- 3 sin a. —4 sin3. a, cos a 4 cos3 ~ a - 3 cos. a. If now the cos a be given, and we wish to find the value of cos 4 a, make cos a = b, cos 4 a = z, and the second of these equations becomes --- 4 z I= 0 (11) The resolution of this equation determines the value of cos I a. WVithout entering into the algebraic details, we will show d priori that the three values of z in this equation are real. Since the cosine is given, the formula of the arcs corresponding to this cosine is 2 kI = ak (Art. 44); hence, equation (11) has for its values all those which are embraced in the expression 2 k t 4: - The whole number k, can only have one of the three forms 3 n, 3 n + 1, 3 n - 1, (n being also a whole number). Make k = 3 n, k = 3 n 1, k 3 n- 1, successively; we shall have, suppressing useless circumferences, 3 n. 2 72 cos a, aZ cos =) cos (2 n ) os ) cos 3 (3 n-1).2 +zi.a = 2 ca, 22- )C; =COs -- =cosQ- t+ z cos os 2 1 -= 3) Cos (4-3 3 (2n-1) - 2 7 C 2A aL 2 7t a Z cos2 Cos ( 3 i _ ) = Cos ( ); 4)COS(- 3 3 3 3 the two last values are the same as the preceding; then there are in all three different values, as follows: ffi 2C c2 2A_ a z = COS C( - 2 = coS ( - -=o~s,0(T +)' TH-IEORY OF TRIGONOMETRICAL LINES 43 It sometimes happens that two of these values are equal to each other: thus, the first is equal to the third, when a -7t _Formuzlc relative to Tangents. 85. Let us'now find the tangent of the sum and difference of two arcs, when the tangents of these arcs are known. According to the relations established (Art. 59), we have in the first place, sin (a + b) cos (a + b)' Substituting.for sin (a + b) and cos (a + b) their values (Art. 72), we have sin a cos b + cos a sin 6 tang (a + b) = -- cos a cos b + sin a sin b' To express this value of tang (a + b) in terms of tangents only, divide the numerator and denominator by cos a cos b, there results sin a sin b Cos a cos b' tang (a + b)- sin a sin b sinasib~sin a sin b cos a cos O' But S = tang a, and - tang b; substituting these cos a cos b values, we have ta a +tang b tang (a + 9)- - i + t 6 (1) nb)= 1 —tang a tangb (1) By a like process, we find for tang (a-b) tang (a-b) - tang a- tang b 1- tang a tang b (2) 86. Let b = a, we shall have for the tangent of double arcs, tang 2a 2 tang a (3) 2 tang a tang 2- ng 2 a 87. To determine tangent of V a, in terms of tang a, substitute in (3) i a for a, we get 2 tang ~ a 1-tang a, -- -tang' a 5 D 44 TRIGONOMETRY. which reduces to this equation of the second degree tang2 ~ a + tang I a-1 —=-O (4) tanga a from which we deduce tang - a= -- 1 tang 2 a). Equation (4) having - 1 for its last f -/ 7g T term, we know, without solving it, ~M/ \< that the product of the two values of tang ~ a is equal to -1. }Hence, if A T and A T' are those values, in the A situation corresponding to their signs, we must have AT x AT'-V A, which shows that 0 A is a mean proportional between A T and A T'. and hence the angle TOT' is a right angle; or the arc M3MI[' is equal to 90~. 88. We frequently meet with the following formulse, tang ~ a (5) 1 - cos a tang ga! = i + cos a sin a tang a- cos a (6) 1-n cos a tang 3a- sin a (7) These formulhe are easily deduced from those already known. It is evident that we have, (Art. 76,) (Art. 79,) sin ~ a /1 —cos a tang a cos I a 1+ cos a, sin j a cos ~ a sin a Cos2 o a 1 cos a, sina a I —cos a tang I= a i - - a sin ~ a cos j a sin a. THEORY OF TRIGONOMETRICAL LINES. 45 Formulce frequently used. 89. The formulse for sin (a ~ b) and cos (a ~: b), (Art. 72,) lead to a great many formul1a which are often used by astronomers. We will now refer to the most important of these. Combining formulme for sin (a + b) and sin (a — b) (Art. 72), and those for cos (a + b) and cos (a-b), by addition and subtraction, we have 2 sin a cos'-= sin (a + b) -- sin (a- b), 2 cos a sin b = sin (a + b)- sin (a —b), 2 cos a cos 7, = cos (a - b)+ cos (a + b), 2 sin a sin b = cos (a - b) - cos (a + b). These formnule may be used to transform the product of a sine and cosine, or of two cosines, or of two sines, into a sum or difference of two trigonometrical lines. 90. Let p and q denote any two arcs whatever; make a + b = p, and a-. b = q; we shall have a= - (p+ q), b = (p - q). Substi tuting these values in the preceding formulI, we get sin p + sin q = 2 sin ~ (p + q) cos t (p- q), sin p- sin q = 2 cos - (p q- q) sin ~ (p- q), cosp + cos q =2 cos t (p + q) cos ~ (p-q), cos q - cos p = 2 sin I (p + q) sin 4 (p - q). These formule are of frequent use in logarithmic calcula. tions, to change a sum or difference into a product. 91. Finally, by division, and remembering that, in general, sin A 1 fn- -tang A - the last formulm lead to the fol. tos A cos A lowing, sin p + sin q sin I (p + q) cos t (p —q) tang 4 (p + q) sinp — sin q- cos 2 (p+ q- ) sin (p —q) tang ~ (p-qq sin p + sinq sinq tp-q) s~in p +-sn s _in - (P + q) _ tang - (p +- q) cos p + cos q co (p + q) sin p + sin q cos t- (p — q) cot I (p - q) coSp-cos q sinl (p —q) sin p —sin q sin t (p-q)tang(pq) cos, p + cos q cos -~ (p-q) 46 TRIGONOMETRY. sin p-sin q cos 4 (p - q) cos q —cosp sin (p + q) cQs p + cosq cos 4 (p +q) cos 1 (p —q)_ cot (p + q) cos q- os:p sin (p + q) sin 4 (p-q)- tang 4- (p-q) Among these formulhe, we observe that the first may be enunciated as follows: The sum of the sines of two arcs is to the difference of the same sines, as the tangent of half the sum of the arcs is to the tangent of'half their difference. 92. We sometimes meet with trigonometrical transformations, the origin of which we may not be able to trace. We may assure ourselves of their accuracy, by verifying them. Thus, to verify the relation sin (a + b) sin (a-b) = sin' a —sin' b, we substitute for sin (a ~+b) and sin (a-b) their values -(Art. 72), and we have'siln (a + 6) sin (a-b) = sin2 a cos2 6-cos2 a sin2 6; then putting for cos2a and cos2b, their values l —sin2a, i-sin2 b, we may readily deduce the proposed equality. i —tangs I a To verify the relaltion cos a= 1-+tang', substitute for tang.4 a its value -— s in and the second member becomes cos2 ~ a-sin" ~- a cos2 a+sin'2 a. But since cos2 a+ sin2 a —1, and cos' - a-sin2 4 acos a, (Art. 59 and 79,) these substitutions reduce the second member to cos a, as was required. As a further exercise, the student may verify the following transformaitions, cos (a + b) Cos (a —b)= co82 a -sin2 b. 1~ tang a tang (450 + a)_ 1 tang a -tang a cos a= +tang a tang 4 a THEORY OF TRIGONOMETRICAL LINES. 47 tang a tang b (a tang a + tang b + tang c = tanga tang b tang c. The last formula supposes that a+b+-c=1800, and it proves that there may be an infinite number of combinations of three quantities, such, that their sum shall be equal to their continued product. Geometrical Demonstrations of Formulwe Already Found. 93. It has been observed, that, after establishing geometrically the formulce which express sin (a + b) and sin (a —b), we have deduced the other formulwe by the ordinary processes of algebra. From this the conclusion is drawn, that the first being true for all possible arcs, the last are not less general, and in this consists the essence of all analysis. On the contrary, in geometrical constructions, the conclusions drawn from each figure are necessarily limited to the particular construction which is made. Still, as geometry serves to make the truth more apparent, and constitutes a most valuable means of mental discipline for the student, we proceed now to demonstrate in this way the principal results already obtained. 94. The sine and cosine of an arc being given, to find the sine and cosine of the double arc. Let arc AB-B C —a, and make the constructions indicated in the figure, we have sin a —AP, cos a= OP, sin 2 a= CQ= cos 2 a OQ=OH O Q- - = O AH. 0 Q H A The right-angled triangle OP A gives A PxOP o 2 -. OA OA A consequently, substituting for the lines their trigonometrical designations, and making radius O A =1, we find again the formulae (1), (2), art. 78, as follows, sin 2 a = 2 P HI = 2 sin a cos a, cos 2a-OH-AH —cos2 a-sin2 a. 5* 49 TRIGONOMETRY. 95. Raving given the cosine of an are, to find the sine and cosine of the half are. Make the arc A C = a, draw C P perpendicular to the diameter A B, F,...... and the chords A C and BC, and the /. / \\(Ds radii OD and 0E, intersecting the chords at their middle points. Assuming the radius O A 1=, we shall o P3 A have OP=cos a, AP= 1-cos a, BP =1+cos a, AC=2sin i a, B 0 cos ~ a. Since each chord is a mean proportional between the diameter and the adjacent segment, we have AC = AB x AP, or, 4sin' j a=2 (1- cos a) BC' AB x BP, or, 4 cos2 a 2 (1 + cos a). From these results we deduce the formulae previously established for sin i a, cos ~ a, /1-coa Co(1 + cos a sin ~ a - - cos a 96. The sine and cosine of an arc being given, to find the sine and cosine of three times the given arc. Let the radius O B - I, the arc AB - BC=CD=a. The isosceles triangle BO D is similar to B D F, for the angle OBD is common, and the angle B D F, which is measured by the half of B A E or B C D,./' PUBis equal to B O D. We have, therefore, BF BD B F - -) hence, B F 4 sin2 a. G / Draw P G parallel to B F, we have P G = B F = 4 sin2 a, and the similar triangles QGP, OBP, give:E QG PG EB P _ O- - hence, Q G - 4 sin' a, PQ PG -- hence, P Q = 4 sim a cos a. OP IB' THEORY OF TRIGONOMIETRICAL LINES. 49 But we have sin 3 a = D Q I)F + F G- G -— B.D + BP- QG - 3sin a-QG, cos 3 a = O Q= O P-P Q= cos a- P Q. Substituting the values just found for Q G and P Q, we have sin 3a = 3 sin a-4 sin3 a cos 3 a - cos a -4 sin.2 cos a. The first of these results corresponds with formula (3), Art. 78, and substituting for sin2 a its equivalent 1- cos2 a, the second reduces to formula (4). 97. Having given the tangents of two arcs, to find the tangent of their sum, and the tangent of their difference. ILet the radius O A-= 1, A B = a, B C = b. At the extremities of the radii O A and 0 B draw the tangents A T and B S, and let fall the perpendicular S H on OA. By hypothesis we have given BR = tang a, B S= tang b; and it is required to find B AT= tang (a + b). Since the triangles OAT, OHS are 0; AR similar, we have AT SH SH OA-O - hence, tang (a + b) S Since the triangles S H R and O B R are similar, we deduce S H from the proportion, S H OB tang a tang b SR OR' or, S a =- OR t. To find O R, we know, from an established theorem, that: we have SR'= O=R'+ OS -2OR x OH. But, SR2= (B R +BS)2= B +BS2+2BR X BS. Hence, 20RX OH=OR2-BR2 +-OS — BS2 -2BR xBS = 20 B -2BR x BS=2 —2tangatangb. Consequently, OH- 1 —tang a tang b OR. 50 TRIGONOMETRY. Substituting in tang (a + b), the values of S H and O I, we have the formula already known, Art. 85, tang a 4 tang b tang (aSrb)1 -tang a tang b. We may readily deterP mine, by a like process,the'> S Tvalue of tang (a —b). In this case, A C = a- b, and no difficulty exists in immediately deducing the exO RId[ ia 11 pression. Since R S is equal to tang a-tang b, the second term of the numerators of S H and O HI have a change of sign in this case, and we get, as before, tang a- tang b tang (a —b)1 -tang a tang b. 98. To demonstrate geometrically the formulsa sin p m+ sin q = 2 sin ~ (p + q) cos i ip - sin pq = 2 cos 4 (p + q) sin K - (p —q). O P F RQA H Take AB =p, A C = q, draw the chord B C, and the radius O D cutting the chord perpendicularly at its middle point: let fall on O A, the perpendiculars B P, C Q, D R, E F, and draw E G parallel to 0 A. From the construction we have sinp + sin q - sin p -sin q BP sinp, C Q=sin q, EF- 2 -B G= P2 AD = (p - q), D R = sin- (p + q),O R-=cos (p - +q). BID = - (p-q), B E = sin (p-q), O E = cos ~ (p-q). But the similar triangles O D R, O EF, B GE, give EF OE B1G'BE DR -D and -— D D) Rx O OR xDBE hence, EF= _ - and OBG I OD OD THEORY OF TRIGONOMETRICAL LINES. 51 Substituting for the different lines their values, doubling these expressions, and making the radius O D = 1, we obtain the required formulm. 99. To demonstrate geometrically that the sum of the sines of two arcs is to the difference of these sines, as the tangent of half the sum of the two arcs is to the tangent of half the difference of the two arcs. Make the same construction as T in the preceding; further, draw the tangent ST at the point D, and B D terminated in S and T, on the radii OA and 0 B produced: produce also B C to H. On account of the parallels, we have EF ERH DS P IRQA [s B G ~-E B = ~Y' But we have 2EF= -sin p +sin q, 2 B G = sin p-sin q, D S = tang I (p + q), D T = tang D B = tang ~ (p-q); hence sin p + sin q tang (p + q) sinlp- sin q tang 4 (p q) which was to have been demonstrated. CHAPTER II. TRIGONOMIETRICAL TABLES. SOLUTION OF TRIANGLES. Construction of Trigonomnetrical Tables. 100. To realize the advantage of substituting trigonometrical lines for the corresponding angles and arcs, we must have some ready means of knowing the numerical values of these trigonometrical lines, when the angle or are is given, and reciprocally. For this purpose tables are formed, in which the numerical values of the trigonometrical lines are written down opposite to the arcs to which they correspond. We propose now to explain the manner of calculating the numerical values of the sine, cosine, &c., for all arcs, differing from each other by 10". This is the law of variation in the arcs as calculated in the Tables of Callet. We begin by calculating the value of sin 10". For this purpose, we remark that the ratio of the circumference to the diameter is X = 3.14159 26535 89793.... When the radius is taken for unity, the semi-circumference is equal to A; and since there are 648000 seconds in 1800, we shall have, in terms of the radius, arc 10"= -- 0,00004 84813 68110. - -- (1) 648000 But, for a very small are, the sine and are are so nearly equal, we may consider the value just found for an are of 10" as a near approximate value for the sin 10". To explain this and to show the degree of approximation made by substituting the value of the arc of 10" for that of the sin 10", we shall demonstrate some preliminary principles. 101. In the first place, we shall prove, that in thefirst quadrant, an are is greater than its sine and less than its tangent. (52) TRIGONOMETRICAL TABLES.-SOLUTIONS. 53 Let A P be the sine of the arc AB, and AT its tangent. Revolve the figure - around O T, and suppose the point A falls on C. We shall have arc A C > chord A C, T and consequently are A B> A P; hence, the are is greater than the sine. We have also arc AC sine and < the tangent, the ratio a sin a tanoga tang a can neither be < I nor -.. Buttheratio may sin a sin a approach unity as near as we please; hence, the ratio has unity also for its limit. IIence, in taking the value of the arc of 10" for that of the sin 10", we have only done what this demonstration shows we had the right to do, by way of a near approximation. 103. To determine the degree of approximation, that we may reject useless decimals. We have sin a = 2 sin ~ a cos 4 sin ~ a a. But the inequality tang i a > 1 a reduces to cos I a > 4 a, and gives 2 sin 4 a > a cos ~ a; hence, sin a > a cos2 ~ a. But cos2 a = 1-sina 4 a; therefore, cos2a a > 1 -( a)2; and hence, finally, 54 TRIGONOMETRY. sin a > a — To apply the result to arc of 10", we shall have, from the value (1) when the 5th decimal figure is increased by 1, are 10" < 0.00005; hence, 1 (arc 10")3 <0.00000 00000 00032. From which we have sin 10" > arc 10" > 0,00004 84813 68110... — 0.00000 00000 00032, and sin 10" > 0,00004 84843 68078. We see that the sin 10" does not commence to differ from the arc 10" until the 13th decimal; and even then, the 13th decimal in the value of the arc is greater only by 1. Hence, we may take sin 10" = 0,00004 84813 681, and may be assured that the error is less than a unit of the 13th order. Indeed, it is evident that the preceding value becomes too small if we take 1 from its last figure, and too large, if we add I to it, for then it would exceed the arc. Placing the value of sin 10" under the radical V/ 1-sin2 10", we find the value of cos 10" to be cos 10" -0,99999 99988 248. We may now readily determine the sines and cosines of 20", 30", 40", &c., to 450~, by the aid of the known formule, sin (a + b) = sin a cos b 4- cos a sin b cos (a + b) = cos a cos b- sin a sin b. 104. The following process which is borrowed from an English Geometrician, Thomas Simpson, gives the most rapid mode.of making the required calculations. The formulke, Art. 89, give sin (a + b) = 2 cos b sin a-sin (a-b) cos (a + b)= 2 cos b cos a-cos (a —b). We may regard the arcs a —b, a, a + b, as three consecutive terms of an arithnmetiCal progression whose common ratio is b: hence, if we represent by t, t', t", these three terms, we have TRIGONOMETRICAL TABLES.-SOLUTIONS. 55 sin t" —2 cos b sin t'- sin t cos t"'-=2 cos b cos t' —cos t. The first formula shows that, having calculated the values of two consecutive sines, we will find the next sine following, by multiplying the last by 2 cos b, and the one before the last by-1, and adding the two products together. The same applies to the cosines. Thus, to find the values of the sines and cosines for all arcs, differing from each other by 10" successively, we make b= 10"; and representing by a and a the known value of sin 10" and cos 10", we shall have sin 0 = 0 cos.0=1 sin 10" = c cos 10" = P sin 20"= 2 p sin 10" cos 20"= 2 p cos 10" - sin 30" -= 2 sin 20" —sin 10" cos 30" = 2 3 cos 20" — cos 10" sin 40" = 2 3 sin 30" —sin 20" cos 40" = 2 P cos 30" - cos 20" &c., &c. &c., &c., &c. But since 24 differs very little fi'om two units, the calculation may be further abridged. Designating by k the difference 2-2i3, we shall have k = 0,00000 00023 504, and 2 3 = 2- k. Hence, the value of sin t" becomes sin t"= 2 sin t'- k sin t'-sin t and therefore, sin t" - sin t'= (sin t'- sin t) - k sin t. When the difference sin f'"-sin t' has been calculated, we may determine the sin t", by adding sin t' to this difference. But the last formula shows, that the difference sin t"-sin e is equal to the difference sin t'-sin t (a difference already calculated before arriving at the arc t",) minus the product k sin t'. Hence, the only tedious operation, which is repeated for each sine, consists in multiplying the last sine by the constant number k= 0,00000 00023 504. But the operation may itself be abridged by forming in advance the products of 23504 by the natural numbers 1, 2, 3 — to 9. We shall then have immediately the partial products which compose each product, such as k sin t, and it only remains to add them. By calculations, nearly similar, the values of the cosines are determined. 6 66 TRIGONOMETRY. In so long a series of operations, errors are multiplied, since it is impossible to preserve thirteen decimals exact throughout. To ascertain the degree of precision on which we must rely, we shall find, presently, by methods which give a certain approximation, the values of many sines and cosines; and then the number of' decimals common to those results and those which are given by the calculation just explained, will indicate, with sufficient certainty, the decimals which may be regarded as exact in the intermediate results. If' we find that the approximation is insufficient, we select as the basis of calculation an are less than 10", 1" for example, and recommence the operations. 105. In practice it is more convenient to have the logarithms of the numbers corresponding to the trigonometrical lines, rather than the trigonometrical lines themselves. But in supposing the radius to be unity, the sines and. cosines will be fractions, and their logarithms will be nega-. tive. To make these logarithms positive, we maize r 1010, which is equivalent to dividing the radius into 10,000000000; equal parts; and then the logarithms of the sines and cosines; will only be negative for angles which differ so little from, 0 or 90~, that this difference may be neglected. It is easy to transform all the results which have been, determined under the hypothesis of r = 1, to the condition r = 1010, by multiplying them by 101', or by' adding 10 to their logarithms. Indeed, when r = 1, we find the ratio of the sines and cosines to the radius, and it is evident that if we divide the radius into m equal parts, we must multiply all of these ratios by m, to know the number of parts con. tained in the sines and cosines. 106. The logarithms of the tangents are made Iknown by r sin a the formula tang a = n which gives log tang a = log sin a + (10 - log cos a); that is, we must add to the logarithm of the sine the arithmetical complement of the logarithm of the cosine. The logarithms of the cotangents are found by means of' the relation tang a cot a = r', which gives SOLUTION OF TRIANGLES. 57 log cot a = 10 + (10 - log tang a). Some tables do not contain the cotangent, but this result shows that we may easily supply them by adding 10 to the arithmetical complement of the logarithm of the tangent. Nor do the tables usually contain secants and cosecants, but their logarithms are readily calculated from those of the sines and cosines. The Tables are not usually constructed beyond 45~. For arcs greater than 450, the logarithms of the sines and tangents, are obtained from the cosines and cotangents of the complement, and reciprocally. Thus, if a > 450, we shall have, sin a= cos (900 —a). The tables are so arranged as to avoid the necessity of calculating this complement. Calculation of Sines and Cosines for every 90, to verify the Tables. 107. To effect the verifications referred to, Art. 104, we propose now to calculate the sines and cosines for arcs varying by 9~. Let sin 180= x; 2x will be the chord of 360, or the side of the regular inscribed decagon. But this side is equal to the greater segment of the radius, when it is divided into extreme and mean ratio: hence, the radius being 1, we have 4x2=- 1x (1-2x). From this we get x2+ ix= -i, and solving the equation, and neglecting the negative value of x as useless, we have x = sin 180= cos 72~= (- 1 + V5). With this value we readily find 1 -x2= — cos 180= sin 720=- \/10 + 2 /-. Substituting these values of sin 18~ and cos 180 for sin a and cos a in the formuhe for sin 2 a and cos 2 a, Art. 77, we have sin 360= cos 540= ~\10-2V5cos 36~ = sin 540 = i (1 + /5) 58 TRIGONOMETRY. Substituting now th'e same value of the sin 180 in the formulso which give the values of sin ~ a and cos ~ a in terms of the sine, Art. 81, we obtain sin 9~ =cos 81 9/3 + v5 — 5 /5 cos 90 = sin 810~ = \/3 + 5 // /5. Finally, if we substitute in the same formulhe for sin a the value sin 540 =- (1 + / 5), we deduce sin 270 = cos 630 3\/5 +/ 5- /3 cos27 = sin 630== i/5+ 5+ 3-V5 Further, remembering that sin 45 —cos 450=~- 2, we deduce the following table: sin O=cos 900 ~ — O sin 90~ cos 810= / + V5-<-5- /5sin 18~ = cos 720~ = (- 1 - 5) sin270 = cos 630= iV\5+ V5 - \/3- 5 sin 360 cos 54~= f\/ 10-2/5 sin450 = cos 450 = h 42 sin 540 =- cos 36 a- (1 5) sin 630 == cos 27~0 = /5 + 5 3 5 sin 720 = cos 18~= /10 + 2 V/5 sin 810 = cos 9~- = / 3 +- A/5 5- /5 sin 90 = cos 00 = 1. These expressions being very simple and involving only the square root, may be calculated decimally to any degree SOLUTION OF TRIANGLES. 59 of approximation desired. These are the values which are used to verify the calculations explained in Art. 114. By bisecting the arcs by; the aid of the established formula, we might deduce the values of sine and cosine of arcs of 40 30', and 20 15', and the successive multiples of 20 15' would supply new verifications. Arrangement and ~ca'nner of using the Tables of Cacllet. 107. The Tables of Callet are the best for the old division of the circumference, and those of Borda for the new or decimal division. In the Tables of Callet three tables are especially to be observed. The first contains the logarithms of numbers to 108000, and the application of this table has been explained in the introduction. The second contains the logarithms of the sines, tangent, and cosine for all arcs, differing by one minute, according to the new division; and the third contains the logarithms of the sine, cosine, tangent, and cotangent, for every 10", by the old division. Inasmuch as the instruments used, and astronomical calculations made are referred to the sexigesimal division of the circumference, the third table only will be referred to in this place. 108. This Table gives the logarithm of the sine and tangent for every second up to 50, and consequently the logarithms of the cosine and cotangent of all angles above 85~, and when the arcs are within these limits we find here the corresponding logarithms. Afterwards, we have the logarithms of the sine, cosine, tangent, and cotangent for every 10". These are written in the columns headed sine, cosine, &c. When the tang or cotang is greater than the radius, its logarithm exceeds 10. In the table the tens are suppressed, but we must not forget to restore it in practice. 109. If we only consider the degrees which are at the head of each page, w'e might suppose that the tables only extended to 450; but if we observe that the columns marked at the top sin, cos, &c., are marked at the bottom cos, sin, &c., we see that by taking the degrees and titles at the bottom of each page, at the same time ascending the columns placed on the right for minutes and seconds, we shall find the logarithms of sines and cosines from 450 to 900~. 6* E 60 TRIGONOMETRY. From what has been said, we find immediately L. sin 6~ 32' 30"-= 9,0566218 L. cot 810 46' 20"= 9,1601596. 1.10. When the given angle contains seconds and fractions of a second, it is necessary to make use of the tabular dif. ferences, and to make calculations similar -to those which have been explained in finding the logarithms of numbers. In doing this, the differences of log sin, log cos, &c., are regarded as proportional to those of the arcs; and this proportion, although not altogether exact, gives nevertheless a sufficient approximation. We remark that the same differences are common to log tang and log cotang. The following examples are given for practice in the use of the Tables. 1st. Find the log. sin 6~ 32' 37", 8. log sin 60 32' 30" (dif. 1836) 9,0566218 log for 7" 1285.2 log for 0.8 146.88 log sin 6~ 32' 37", 8 _ 9,056765 2d. Find the log cos 83~ 27' 22", 2 log cos 83~ 27' 30" (dif. 1836) 9,0566218 log for - 7" 1285.2 log for - 0.8 146.88 log cos 83~ 27' 22", 2 - = 9,0567650: 3d. Find the log tang 80 13' 52", 76 log tang 80 13' 50" (dif. 1486) 9,1603083 log for 2" 297.2 log for 0,7 104.02 log for 0,06 8.916 log tang 80 13' 52", 76 9,1603493 4th. Find the log cot 810 46' 17", 24 log cot 810 46' 10" (dif. 1486) 9,1603083 log for -2" 297.2 log for - 0,7 104.02 log for - 0,06 8.916 log cot 810 46' 7", 24 = 9,1603493. SOLUTION OF TRIANGLES. 61 111. Let it now be required to determine the angle when we know the logarithm of a sine, a cosine, &c. Thus, let log sin x = 9,0567650, find x. The next less logarithm given in the tables for logarithms of sines is 9,0566218, and this logarithm corresponds to 6~ 32' 30". The difference between this logarithm and the given logarithm is 1432, and the tabular difference, corresponding to 10", is 1836. Hence, dividing 1432 by 1836, and adding the tenths of the quotient as seconds, we find 7", 8. Therefore, the required are is x = 6032' 37", 8. The following calculations will explain this example, and others of an, analogous character. 1st. Find the angle corresponding to log sin 9,0567650. log sin x= 9,0567650 The next less log. 9,0566218 (tab. dif. 1836) 60 32' 30" 1st remainder 14320 7 2d remainder 14680 0.8 x = 6~ 32' 37", 8 2d. Find the angle corresponding to log cos 9,0567650. log cos x - 9,0567650 The next log. 9,0568054 (tab. dif. 1836) 830 26' 20" 1st remainder 4040 2 2d remainder 3680 0.2 x-830 27' 22", 2 8d. Find the angle corresponding to log tang 9,1603493. log tang x = 9,1603493 For 9,1603083 (tab. dif. 1486) 80 13' 50" 1st remainder 4100 2 2d remainder 11280 0.7 3d remainder 8780 0.06 x = 80 13' 52", 76 62 TRIGONOMETRY. 4th. Find the angle corresponding to log cot 9,1603493. log cot x = 9,1.603493 For 9,1.604569 (tab. dif. 1486) 810 46' 0" lst remainder 10760 7" 2d remainder 2580 0.2 3d remainder 6080 0.04 x 81~ 46' 7",24 112. The formul' which contain sines, cosines, &c., assume almost always that the radius has been regarded as unity. To apply the tables to them, two methods may be used. The first consists in introducing the radius r in the formulsa as in Art. 57, and then using the logarithms of the tables, always remembering that log r = 10. In the second process, we do not change the formnulf, but we retain the hypothesis of r = 1, and subtract 10 from every logarithm taken from the trigonometrical table. It answers to effect this subtraction on the characteristic only, which may, by this operation, become negative, but it is better to employ such logarithms as the table gives, and to take no account of the 10 until the end of the operation. This method of correction will be always easy, for, in the calculations, the logarithms are always added or subtracted, and it is evident that each additive logarithm will give one ten too much in the result, while each subtractive logarithm will have one ten too little. 113. The use of the arithmetical conplement enables us to substitute addition for subtraction of logarithms, as has been heretofore explained. In this case, the ten which must be subtracted from this logarithm to reduce it to the hypothesis of r 1, is compensated by that which we have to add in using the arithmetical complement. Besides, the error of a ten in a logarithm is so considerable, that it could not fail to be noticed. For illustration, the following examples are given. 1st. Let x = 419 x sin2 400, hence, log x = log 419 + 2 log sin 40~. Taking log sin 40~ from the tables, log x contains two tens too much, which must be taken from the result. SOLUTION OF. TRIANGLES. 63 log 419 = 2,6222140 2 log sin 400 19,6161350 log x - 2,2383490 If we wish to obtain x within'1, add 2 to the character. istic, and we find x= 173,12. 314 x sin 30~ 2d. Let sin x 4 411 x cos' 450' we get log sin x = log 314- log 411 + log sin 300~-2 log cos 15. Operating by the complements, the two tens which must be taken from 2 log cos 15~, are compensated by the two tens for which we take the complements. The log sin 30~ and the complement of the log 411 introduce two tens too much; but as the angle must be sought in the tables, we only take away one ten from the result, as is shown below., The arithmetical complements are designated by Lo'g. log 314 - 2,49692965 lo'g 411 7,38615818 log sin 30~ 9,6989700 2 lo'g cos 150 0,0301127 log sin x= 9,6121702 This logarithm is prepared to suit the tables. We find x — 240 10' 7". RESOLUTION OF TRIANGLES. RELATIONS BETWEEN THE SIDES AND ANGLES OF A PLANE TRIANGLE. ~114. Hereafter we will designate the angles of a triangle by the large letters A, B, C, placed at their vertices, while the sides opposite to these angles will be respectively designated by the small letters a, b, c. If a triangle be right-angled, the letter A will always correspond to the right angle, and the letter a to the hypothenuse. 115. The solution of all plane triangles depends upon the following theorems which are here enunciated.and demonstrated. 64i TRIGONOMETRY. 116. Theorem I. In every right-angled plane triangle, each side about the right angle is equal to the hypothenuse multiplied by the sine of the angle opposite to this side. [Let AB C be a triangle C \ right-angled at A. From; the point B as a centre, with,,q any radius, describe the are D E, and let fall the perpen. dicular E F. The sine of the A c F c angle B is the ratio of E F to the radius B E (Art. 54). The similar triangles B C A, B E F, give AC EF b BC B-E' and we have -— sin B, hence, b a sin B, (1) which was to be proved. 117. The angle B is the complement of the angle C;' hence, sin B = cos C, and we conclude, that in every right-angled plane triangle, each side about the right angle is equal to the hypothenuse multiplied by the cosine of the adjacent angle. 118. Theorem II. In every right-angled plane triangle, each side about the right angle is equal to the other side multiplied by the tangent of the angle opposite to the first side. For, in the triangle AB C, describe the arc D E, from B as a centre, and with any nit radius; erect D G perpendicular toAB. The ratio of D G to B D is the tangent of A. i'~- C 19the angle B, (Art. 54.) But the similar triangles give AC AG b A[B=B-D' or c-tang B, therefore, b = c tang lB, (2) which was to boe proved. 119. We may deduce this result directly from Theorem 1. Indeed, applying this theorem to each side b and c, and ob. serving that sin C = cos B, we have b a sin B and c = a cos B; SOLUTION OF TRIANGLES. 65 b sin B hence, - =cos B tang B or b = c tang B. c cos B 120. Theorem III. in every plane triangle the sides are proportional to the sines of the opposite angles. Let A and B be any two angles of the triangle AB C. Let fall the C perpendicular, CD from the vertex C on the side A B. If the perpen- / j a dicular falls within the triangle A B C, the two right-angled triangles, ACD) and B C D, will give A D B C D -b sin A, and C D ==a sin B; hence, b sin A = a sin B; a sin A 3 and we have a sin (3) b si B If the perpendicular falls on o the prolongation of B A, the triangle AC D gives CD= b sin CAD. But the angle CAD be- \ ing the supplement of C A B, we have sin CA D=sin CA B= sinA, 1 A e B and we have still the same proportion before found. 121. Theorem IV. In every plane triangle the square of either side is equal to the sum of the squares of the other two sides, minus twice twice the rectangle of these two sides, multiplied by the cosine of their included angle. That is, we have a2 = b + c — 2 b c cos A; (4) Let ABC be the given triangle; let fall the perpendicular C D on C A B. When the angle A is acute, we have, from a known theorem, c B2 = _AC-2 A2B+ — 2 ABxAD, or, a2-=b -c2- 2cxAD, But the triangle ACD gives AD A D e - b cos A, and substituting this value of AD, we have formula (4). 66 TRIGONOMETRY. When the angle A. is obtuse, we have a-2 =b + e-2 c X A D And the triangle A CD gives tC ~AD=b cos CAD. But CAD being the supplement of CAB or A, we have cos C AD=13 \b cos CAB -cos A; hence, __\__ AD- -b cos A; and substitutAL e B ing this value in that of a2, we have the equation before ftound. 122. Theorem IV is sufficient of itself to solve every plane triangle. For, it is evident we have, for each side, the three equations a2 = b2+ c2- 2 b c cos A, (5) b2= a 2+ c 2 a c cos B, (6) c2= a + 2b-2ab cos C, (7) by which we may always determine three of the six parts of a triangle, when the other three are known, except in the cases when the triangle is impossible, or when the three angles are given. 123. Theorem III., since it expresses a relation between two sides and their opposite angles, must be a consequence from the three last equations. It may be deduced as follows: The first equation gives cos A b2 + c2-a hence 4 b2 c2 (b2 + C2 — a2) "sin2 A =- I cos2 A = (b - 4 b: Vc 2a2b2 + 2a2 e+ 2he2 a —a —b4 c4 4 bl C2 and consequently, sin A V/222 a b 2 + 2 2 b2 cT-4-b4 - a 2abc The other two equations give, in like manner, the ratios sin B sin C and; but we may deduce them at once by b~~ SOLUTION OF TRIANGLES. 67 changing, in the second member of the preceding equation, first ac into b and b into a, and afterwards a into c and c into a. Since the second member is a symmetrical fhunction of a, b, c, it will remain the same when the changes are made upon these letters, and we have therefore, in accordance with Theorem III., sin A sin B sin C a b c RESOLUTION OF RIGHT-ANGLED PLANE TRIANGLES. 124. First case. Having given the hypothenuse a and an acute angle B, to find the angle C, and the two sides b and c. We have C = 900 —B. We may then determine b and c by means of Theorem I., which gives b= a sin B, c a cos B. The values of b and c, will of course be calculated by means of logarithms. 125. Second case. The side b of the right angle, and the acute angle B being given, to find C, a and c. We have again C = 900 - B. Theorem I. gives a, by the relation b = a sin B, hence, a =' and Theorem II. gives c, since we have c=b tang C, or c=b cot B., 126. Third case. The hypothenuse a, and a side b, being given, to find the other side c, and the angles B and C. From the known property of the right-angled triangle, we have C2 — a2- b2, or c = (a + b) (a —b), which makes known the value of c very readily by means of logarithms. We determine B by the relation b = a sin B (Art. 116), which b gives sin B - - and finally we have C = 900 - B. If we determine the angles first, we may still find c by the relation c = a sin C. 127. Fourth case. ilaving given the two sides b and c of the right angle, to find the hypothenuse a and the angles B and C. 7 68 TRIGONOMETRY. We first determine B, by the relation b = c tang B (Theo. II.), and this gives afterwards C = 90 — B. The hypothenuse a is made known by the relation (Theo. I.) b = a sin B. We might find a directly from the formula a v 6b2 + c2. But since the binomial b2 + C2 cannot be decomposed into factors, it is not suitable for logarithmic calculation, and it is better to find the angle B in the first place, and then make use of it to find a. RESOLUTION OF OBLIQUE PLANE TRIANGLES. 128 Case I. Having given a side, a, and two angles of a triangle, to find the three other parts. If the sum of the two given angles be taken from 180~, we'shall have the third angle. The sides b and c are then determined by Theorem III, by means of the proportions b sin B c sin C a sin A' a-sin A' 129. Case II. Having given two sides, a and b, and the angle A opposite one of them, to find the side c and the two other angles B and C. The simplest solution is, to find first the angle B, opposite sin B b the side b, by the proportion sin =- a The angles A and B being known, we have C = 1800~ e sin C (A + B); and the side c is found by the relation —in 130. This solution requires some development. The first proportion makes known sin B, tlhus, b sin A sin B —: a and fiom the tables we find for the value of B an acute angle. But the same sine corresponds also to a supplementary anole, which is obtuse: so that, calling M the angle of the tables, we shall have for B the two values B- =M, B= 180~ —l, which seem to indicate two triangles, and suggest the following important observations. SOLUTION OF TRIANGLES. 69 1st. When the known angle A is obtuse or a right angle, the two other C angles must be acute; and in this case, we take B = AM only; and in order that the triangle may be possible, it is further necessarythatwe have a> b. Thiscon- % a dition is sufficient. eC 2d. When A is acute A\. and a> b, we must have A A> B, and it is still necessary to reject the value of B= 1800~-MI.' In this case the tri-'-. angle is always possible. 3d. But when we have A acute 0 and a 1; hence, sin B > 1. Thus, the value of sin B is greater than unity; but no sine is greater than unity, hence the triangle is impossible. 70 TRIGONOMETRY. 131. We have undertaken to find the side c after determining the angle B. We may, however, find c at once, when a, b, and A are given; for, by Theo. IV., we have a2= b2+ c - 2b c cos A; hence, c2- 2 b cos A.c = a b2 This equation is of the second degree, and gives c = b cos A -./a — b2 + b' cos2 A. = b cos A ~ V/a'2 b' sin2 A. Since the side of a triangle must always be a real and positive quantity, we have only to inquire what relations between a, b, A, will give for c one or two values which fulfil these conditions: and this inquiry may be made by the student without any difficulty. 132. The preceding values of c being inconvenient for logarithmic. calculations, are of no use in trigonometry. Still, as we sometimes meet with like expreseions, we will here explain the transformations adopted by astronomers to facilitate the use of such results. Placing these values under the form c= b cos A o/ a\ l sin A As these values are supposed to be real, the quantity sin A is less than 1, and we may consider it as the sine of a an angle., which we may determine in placing b sin A sin We have then b = sin A' 1 - a -/ — = cos; and consequently, a (sin q cos A -4 sin A cos p) a sin ( i- A.) sin A sin A results which may be readily calculated by logarithms. This solution exactly corresponds with the first, for the auxiliary angle p is no other than the angle B. 133. Case lII. Having given in a triangle the tywo sides a and b and the included angle C. to find the side c, and the angles A and B. SOLUTION OF TRIANGLES. 71 a sin A By Theorem III., we have B. This proportion contains two unknown quantities A and B: but we may deduce from it, in the first place, a+b sinA+sinB a- - sin A-sin B' And we know, further, that sin A + sin B tang A (A + B) sin A sin tang (A -- B) Therefore, a +b tang - (A -+ B) a - b - tang (A - B) which shows that, in any triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite to those sides is to the tangent of half their dtfference. We have A B — 180~ - C; hence (A 3 B) = 90~ - C: therefore, the three first terms of the above proportion are known, and we may find from them the value of ~ (A-B). IKnowing now the values of the half sum and half difference of the angles A and B3, each may be determined; for, by adding the half sum to the half difference, we obtain the greater angle, and by subtracting the half difference from the half sum we get the less angle. Thus, A +B: A-B I -A+B AB13 A - + 2and + A,2 + 2 2.. 2 A and B being found, we obtain the side c, by the proportion c sin C (2) a sin A i34. This proportion requires that awe find three new lo.garlthms, viz.: those of a, sin A, sin C. We may abridge the calculation by the following process: sin A sin B sin C S;Xua we have - a - -; we mnust also have a b c sin A - sin a - b (a + b) sin C; henc c= sin C cc sinA + sinB' 1* 72 TRIGONOMIETRY. By known formule we h1ave sin A + sin B = 2 sin 4 (A + B) cos. (A-B) sin C = 2 sin 4- C cos - C. And we have also, sin A- (A - B) =- sin (90~- 4 C) - cos i C. Substituting these values in c, and making the necessary reductions, we have (a + b) sin 4C cos 4 (A- B) Since the logarithm of (a + b) is already known, this value of c may be used in preference to that given by the proportion (2), and we thus save the calculation of one logarithm. 135. We have determined the value of c after finding those of A and B. To find c directly, we make use of Theorem IV., which gives c = -/a' + b2-2 a b cos C.. But inasmuch as logarithms cannot readily be applied to this formula, we make use of an auxiliary angle. We have cos2' C + sin'2 C 1, and cos2 2 C sin2 ~ C cos C. Hence c -- (a2+ b2) (0os2 ~ C + sin2 - C)-2 a b (co82 - C-sin2 C) = / (a + b)2 sin2 4 C + (a - b)2 cos2' C. =(a + b) sin i C V/1 + (a —b)2 cot2' C v. ~ (a + b)2 Since the magnitude of a tangent is unlimited, we will make (a - b) cot 7 C tang= a + a + b Then the radical becomes.1 sin'2 q 1 + tangy2' = 1 + s' p (a + b) sin IC. Hence) c - cos q Thus, we will find successively the auxiliary angle t and the side c by means of two formula which may be readily calculated by the tables. SOLUTION OF TRIANGLES. 73 This solution differs from the preceding only in appearance: for, tang S(A + B) being equal to cot - C, the value of tang, is identical with that of tang ~ (A-B) deduced from. equation (1), and hence the last value of c is found also to be identical with the formula (3). 136. In practice, it often happens that the sides are known by their logarithms. Let us suppose that the sides a and b are thus known, and that, further, C is given, and that A and B are the only quantities to be determined. To find + (A- B), by means of the proportion (1), it would first be necessary to find a and b in the tables; but this may be avoided by making use of an auxiliary angle. Let, be this auxiliary angle, given by the assumed relab tion tang = —. We have, from known formulse, a tang (450 1) = tang 450 — tang 1- tang 4 -- tang 45~ tang + 1 + tang + Substituting for tang + its value, we have tang (450 — +) - __ But we have, Art. 133, a- b tang ~ (A-B) = + b tang i (A - B) Hence, tang I (A -B) = tang (45O~-) tang ~ (A + B). And since,~ is known, we readily find ~ (A - B). By this process we have two logarithms less to calculate than if we had determined the sides a and b. 137. Case IV. Having given the three sides a, b, c, to find the three angles. By Theorem IV. we have a2 = b2 - c- 2 b c cos A; hence b2 + c — a2 cosA 2bc In like manner, we may determine B and C. But it is necessary to find a f-,rmula more suitable fobr logarithmio calculations. TRIGONOMIETRY. Art. 78 gives the formula 2 sin2 A = 1- cos A, and substituting in it the value of cos A, we have successively i2 + C2- a2;a2 - b2-c2 + 2 bc ~ 2bc 2bc a2o- (b- c) (a- b - c) (a - b + c) 2bc 2bc!eonce sin SA= _ /(a+b-c) (a-b+c) HEbenc. sin'I, To simplify this formula, make the perimeter a + b + c — 2s, we deduce a + b-c- 2s- 2 (s-c), and an -b -c= 2s — 2 b = 2 (s-b); helnce, sin A =/(s —b) (s-c) 05 From which we form the following- important rule, that the sine of half an angle of a plane triangle is equal to the square root of half the sum of the three sides minus one of the adjacent sides, multiplied by the half sum of the three sides minus the other adjacent side, divided by the rectangle of the adjacent sides. Although the angle I A is determined by its sine, there is no ambiguity in the result, because A being an angle of a triangle, we must always have A < 1800, and I A < 900~. 138. We may determine with eqnal facility the formula which determine cog ]A and tang i A. PResuming the known relation 2 cos2-A = 1 + cos A, we deduce, by similar transformations to those just employed, cos A = /3 (s - a) If we divide the value of sin ~ A, by that of cos A,) we find for tang - A, tang 2- A-t/(- b) (s-c) 139. We know that it is not always possible to form a triangle with three sides taken at will. This impossibility is indicated by the calculation. For, if we make use of the formula sin A = /(s —b) (s-c) b1_ SOLUTION OF TRIANGLES. 75 when the triangle is possible, this expression must give a real value for sin -1 A, which must be less than 1. But if the triangle is impossible, this formula will lead to an imaginary value for sin ~ A, or to a value > 1. In order that the impossibility shall exist, it is necessary that one side be greater than the sum of the other two: let us now examine the results which the formula gives under this condition. 1st. Let b>a+ c: we shall have 2b>a+b + c hence 2b>2s and s-bc, hence a b+ c or2swill be >2c, ands-c>0. Thus, the value of sin i A is imaginary. 2d. Let c> a + b: then we shall have s —c 0, and sin ~ A will be imaginary. 3d. Leta>b+c; weshallhavea:+b+c or2s >2b+2c: hence, s > b + c, and s-b > c, s —c > b, and (s-b) (s —c) >>be: consequently the value of sin P A would be greater than 1, which would be impossible for any angle. APPLICATION TO EXAMPLES. 140. Extended trigonometrical operations require the use of various instruments, which it is not proposed to describe in this treatise. A few remarks will suffice to make intelligible the several examples which are proposed for solution. To trace a right line upon the ground-stakes are used, and they are placed from point to point, at convenient distances, and so located that when the eye rests on the first stake, all the others will be covered by it. An angle is traced on paper by means of an instrument called a protractor. This is a semi-circle divided into degrees and parts of degrees. There are many different kinds of instruments used for measuring angles either on the ground, or in space. The chief of these are the compass, the theodolite, the graphometer, &c. In general, they consist of a circle, or sector of a circle, on which is marked a fixed radius, which serves as an origin for the subdivisions, whilst a second radius movable around the centre, may take any position. The plane of the circlo may itself turn around i'ts centre. To measure the angle F 76 TRIGONOMETRY. included between two right lines which connect a given point with two other points, place the centre of the instrument at the first point, and direct the two radii upon the other two points; then read on the graduated circumiference the number of degrees included between the radii, and this will be the required angle. 141. EXAMPLE I. In a triangle A B C, right-angled at A, we have given a — 1785; 395 yds. B = 590 37' 42", it is required to find C, b, c, (124). Subtracting B from 90~, we have the angle C = 90~- 590 37' 42" - 30~ 22' 18". We have now to A c l find the sides b and c. Calculation of side b. Calculation of side c. b a sin B c = a cos B log sin 59~ 37' 42" 9,9358919 log cos 590 37' 42" 9,7038132 log 1785,395 3,2517343 log 1785,395 3,2517343 log b 3,1876262 log c 2,9555475 b = 1540,374 yards. c = 902,708 yards. 142.: EXAMPLE II. In a triangle A B C, we have given the side a - 2597,843 yards, the side b - 3084,327 yards, and the angle A = 560 12' 47", it is required to find B, C, and c. Calculation for angle B. sin B b sinA a log sin 560 12' 47" 9,9196592 There are two solutions. log 3084,327 3,4891604 lotg 2597,845 (rm,. ) 6,5853868 In 1st solution B 800 39' 43". In 2d solution B- 990 20' 17" log sin B 9,9942064 1st solution: B = 800 39' 43" 2d solution: B - 99~0 20' 17" SOLUTION OF TRIANGLES. 77 Calculation for angle C. Calculation for angle C. C - 1800 -A -B. C - 1800-A- B 1800 1800 A = 56~ 12' 47" A= 56~ 12' 47" B- 80~ 39' 43" B3= 990 20' 17" C = 430 7' 30" C =- 40 26' 56" To find the side C. To find the side C. c sin C c sin C a sin A a sin A log 2597,845 3,4146132 log 2597,845 3,4146139 log sin 430 7' 30" 9,8347972 logsin 240 26' 56" 9,6168759 log sin 560 12' 47"A.C.0,0803408 logsin 56~ 12' 47"A.C.0,0803408 iog C 3,3297512 log c 3,11182899 C= 2136,737 yards. c = 1293,689 yards. 143. EXAMPLE III. Having measured the distances a and b of the point C from two known points A and B, to find the points C. If the triangle ABC is small, we may describe two arcs of circles f:-onm the points A and B as centres, with the given distances as'6 radii. But the operation becomes impracticable when the distances are consider: able. In this case, we mea- A B sure the distance A.B, which will enable us; knowing the three sides of the triangle A B C, to calculate the angle A, (Art. 137.) The direction of the side A C is thus determined, and it only remains to lay off, on this direction, -thegiven distance. A C = b, and the point C is made known. Let a = 9459,31 yards, b = 8032,29 yards, c = 8242,58, we shall have 2 s 25734,18, s= 12867,09, s- b -4834,80, s -. = 4624;5.5I sin0A= \/~l(s-b) (s-c ) sin.1 Ab 78 TRIGONOMETRY. log (s - b) 3,6843785 log (s - c) 3,6650657 log b (Ar. Comp.) 6,0951606 log c (Ar. Comp.) 6,0839368 2 log sin 4 A 19,5285416 log sin ~ A 9,7642708 A-= 350 31' 47"1 A-=710 3' 34" 144. EXAMPLE IV. To fid the height A B of a tower, the base of whick is accessible. ~Lay off on the ground, supposed level, a base line B C, from the foot of the tower. Place the instrument at C, and measure the angle ED A formed by D A with the horizontal line D E parallel to -LJn~~ o1 _aCB. Then, in the rightnLL3ii7-*~- C.angled triangle A ED, we know the side D E, and the angle D, and we may calculate A E. Adding C D to A BI, we have the height required. Let C D = 1,10 yard, D E = 61,28 yards, D 410 31' 25", we shall have A E = 61,28 x tang 410 31' 25" log tang 410 31' 25" 9,9471690 log 61,28 1,7873188 log A E 1,7344878 A E = 54,261 yards, A B = 55,361 yards. A When the base of the tower is inaccessible, as when A B is the elevation of a hill above the surrounding ground, the foot of the perpendicular is unknown, and we a're therefore unable to measure the distance B C C. We may, neverthe SOLUTION OF TRIANGLES. 79 less, measure the angle AD)E; for, without' seeing the line A B, it. is still possible to make the plane of the circle with which the angles are measured, to pass through the vertical A B. Moreover, we will determine A D, as will be explained in the following example; and then knowing the hypothenuse A D and the angle D, we may find A ]E. 145. EXAMPLE V. Find the distance of an observer, situated at A, from a- point P, which is inaccessible, but which can be /, seen by the observer. Measure a base A B, and the angles PAB, PBA, we may then readily determine' A P. Let A B= 247,49 yards, a 1 A 62~ 41', B = 590 42'. We deduce at once the angle P = 57~0 37'; and then calculate A P, as follows. A P sin B AB -- in' log A B 2,3935577 log sin B 9,9362098 log sin P (Ar. Comp.) 0,0734087 log A P- 2,4031762 A P = 253,032 yards. 146. EXAMPLE TI. To find the distance P Q, between two inaccessible points, which may be seen by the observer. Measure the base line A B, and the angles B A P, B A QI, AB P, A B Q; we may then determine, as above, the side A P of the triangle A B Q. Besides, the angle P A Q is known; for, if the four points A,B,P. Q, are in. / the same plane, we have... PAQ=BAP- BAQ: and in every ease we - ____ may determine that angle by direct measurement../- _ Thus, in the triangle PAQ, A B 8 60 TRIGONOMIETRY. we shall know two sides and the included.angle, and we may readily determine the side P Q. Let A B = 345,29 yards, B A P = 69~ 26', B A Q = 440 31', P'AQ = -250 41', AB P = 480 15', A B Q - 102~0 14'. We deduce directly A P B = 62~ 19', A Q B - 330 15', and we may calculate A P and A Q as follows: 1st. To find AP. 2d. To find AQ. AP sinABP AQ sin ABQ AB- sin PB AB sin A Q B log; A B 2,5381840 log A B 2,5381840 log sinA B3 P 9,8727722 log sin AB Q 9,9900247 log sin A PB (A. 0.) 0,0527973 log sin A Q B (A. C.) 0,2609871 Log A P 2,4637535 log AQ 2,7891958 A P - 290,907 yards. A Q 615,454 yards. 3d. To find the angles P and Q. 4th. To find P Q. Let A Q =p, AP=q, APQ PQ sinPAQ -P,AQP=Q. We shall have q sinQ p+q=906,361, p-q=324,547. log q 2,4637535 (P -Q)= 770 9' 30": then, logsinPAQ 9,6368859 we may place log sin Q (Ar. Comp.) 0,4735196 p + q tang ~ (P + Q) log P Q 2,5741590 p. -q tang (P - Q) P Q= 375,110 yards. log tang I (P-J Q) 10,6421427 og (p — q) 2,5112776 log (p+q) (Ar.Com.) 7,0426988 log tang - (P- Q) 10,1961191 ( (P Q) = 57 31' 6" P = 134~ 40' 36" Q - 19~ 38' 24". 147. Another solution. We have obtained the distances AP and AQ, by passing through their logarithms. We may avoid this, by using the auxiliary angle e, as indicated in Art. 136. Then, having found log AP, and log AQ, we obtain the angles P and Q. as follows: SOLUTION OF TRIANGLES. 81 To find o. To find P and Q. tang P tang (P-Q)=tang ~ (P+Q) tang' -- A Q + tang (450 —4-) log A P' 2,4637535 log tang 4 (P + Q) 10,6421427 log A Q, (Ar. Comp.) 7,2108042 log tang (45~-,4) 9,5539790 log tang. 9,6745577 log tang (P - Q) 10,1961217 += 25 17' 50, 4 (P -Q) = 570 31' 6". 450 o 190 42' 5" The remainder of the solution is determined as above. 148. EXAMPLE VII. Three points, A, B, C, are situated on level ground. It is required to find the point AN, from which the distances A B and A C are seen under known angles. From the enunciation, AMB and AMC A are known. On A B, a b describe a segment of. \ a circle capable of con- AMB, and on AC, a.u segment that will contain the angle A M C: these arcs will intersect each other in A and 3M, and the point M will be determined. But this construction being impracticable on the ground, we may obtain, by calculation, the angle B A M, and the distance A M. The following method by which we find first the angles A B M and A C M, seems to be the simplest. Mlake the given parts AB=a, AC=b, BAC- A A M B = a, A M C -=; and let the unknown parts be represented as follows, A BM3 -- x, AC M - y. In the quadrilateral A B M C we have x + y = 3600 - (A + + ) thus the sum of the angles x and y is known. Let us now find their difference. The triangles A B Ml and A C M give AM sinx AM siny a sinc' b sin p. 82 TRIGONOMETRY. Placing tne values of AM I drawn friom these two propor. tions equal, we have a sin x b sin y b sin ca sin x 5ilza s inl a sin P sin y Placing a' a., the quantity a' may be readily calcusin a lated by logarithms. Hence, we have b sin x ba' sin x si sin y — sill y' from which we get a sinll y b- a' sin x- sin y and therefore, b -+ a' tang (x + y) b — a' tang 2 (x-y) Since x + y is known, we shall have the difference x-y, by this last equation, and we may determine x and y. We may then find the angle B AM = 180~ - ( + x), and the distance A i will be made known by one of the proportions (1). RELATIONS BETWEEN THE ANGLES AND SIDES OF SPHERICAL TRIANGLES. 149. Fundamental Flormula. The parts of a spherical triangle, traced on the surface of a given sphere, are known, when we know the number of degrees which each of them contains. The solution of questions relating to spherical triangles depends then upon the relations which exist between the number of degrees, or, in other words, between the corresponding trigonometrical numbers, sine, cosine, &c. Consequently, we propose, in the first place, to establish the formula which connects any angle whatever with the three sides of the given triangle, and we will afterwards show how we may deduce from this formula the solution of all cases which may be presented in spherical triangles. Angles will always be designated by the large letters O A, B, C, and the opposite sides -.. —------- - E by the small letters a, b, c. - A o Let O be the centre of the'., is//,./ sphere on which is situated the triangle A B C; draw the radii O A, OB, O C: erect on OA itA.....':1 9 the perpendiculars AD and AE, SOLUTION OF TRIANGLES. 83;one in the plane OA B, the other in the plane OAC, and let them meet in D and E the prolongation of the radii OB and OC. The angle DAE is equal to the angle A of.the spherical triangle; and taking O A for unity, we have AD =- tang c, O D = see c, A E = tang b, O E -- see b. The triangles D A E, D O E give 2 2 2 AD +AE 2 2AD x AE.cosA== D E O D2 + OE-2 2OD X O E.cosa = D E2 Taking the first equation from the second, and observing that O D2 -- A-2 O E — AE' - 1; and substituting for the-lines their trigonometrical designations, and dividing by 2, we have 1 - sec b sec c cos a + tang b tang c cos A =- 0. I sinb 1 But see b ob tang b = cos b' Cc= tang c -- cb os b''Cos C sin c n c; we have therefore, cos c cos ac sin b sin c cs A0 cos b cos C cos b cos c Getting rid of the denominator, and transposing cos a to the first member, we have cos a = cos b cos c + sin b sin c cos A, (1) which is the fundamental formula of spherical Trigonometry. 150. In the figure the sides b and c are less than 900, but it is easy to see this formula (1) is general. Suppose one of the sides, as b or A C, exceeds 90~. Complete the semi-circumferences CAC', C B C', forming thus the triangle AB C', b a_ in which the sides a' and b', or BC' and A AC', are supplements of a and b, and A'the angle B AC' is the supplement of A. -' 7 Since the sides b' and c are less than.... 900, the formula (1) is applicable to the triangle AB C', and gives cos a' - cos b' c sin b' sin c cos B A C'. But a' - 1800 -, b =1800 -b, BAC' =1800-A. Substituting these values, and changing the signs of both 8* 84 TRIGONOMETRY. members, we reproduce formula (1), hence it applies to cases in which b > 90~. If the two sides b and c exceed 900, produce them until they meet in A' and form the triangle B C A', in which the, angle A' is equal to A, and the sides b' and c' are equal to the sup-./C plements of b and c. Formula (1) C/.. / applied to this triangle must hold "B good when 1800 - b and 180~- c A / are substituted for b and c. But -c~~ these substitutions produce no change in this formula, and it is again verified. Finally, we may verify the formula in the case in which b= 900 and c= 90', either separately or together. But since the formnula is true for values of b and c very near 90', the formula must hold'good in these. articular cases. 151. If the angle A = 900: we have cos A 0, and the general formula becomes cos a = cos b cos c, that is, in a right-angled spherical triangle, the cosine of the hypothenuse is equal to the product of the cosines of the two sides containing the right angle. 152. We may apply formula (1) to each of the sides of the triangle, and we shall thus have three equations by means of which we may always determine any three parts of a spherical triangle when the other three parts are known. But in practice, it is necessary to have, separately, the various relations which exist between four parts of the triangle, taken in every possible combination. There are only altogether four distinct combinations-and we propose to examine these successively. 153. 1st. Relations between the three sides and an angle. Formula (1) gives, by permutation of the letters, the following relations: cos a -= cos'b cos c sin b sin c cos A (1) cos b = cos a cos c + sin a sin c cos B (2) cos c = cos a cos b + sin a sin b cos C (3) To calculate A, for example, by means of the three sides, we make use of the first equation, and we draw from it os os a - cos b co' c sin b Bill C SOLUTION OF TRIANGLES. 85 Then, with this value, we find friom it others better adapted to logarithms, by obtaining the values of sin 4 A, cos I A, tang - A, as we have done in plane triangles. Let us take the formula 2 Sin2i A = 1 - cos A, and substitute for cos A, the value deduced above. We have cos b cos c+sin b sin c- - eos a cos (b -c) - os a 2 sin2 ~ A == sinl b sin e sin b sill C. In the known formula cos q - cosp = 2 sin 4 (p + q) sin ~ (p - q), makep= a, q= b - c; we have cos (b -c) -cos a = 2 sin ~ (a + b - c) sin (a - b +c); hence, sin 4 A s/ sinl 4 (a + b —c) siln 4 (a- b + c) sin b sin c. If we take the formula 2 cos f'A = 1 + cos A, we shall find, by a like operation, the'value of cos 4 A, cos-~ A 4= / siln (a + b + c) sill n (b +- c-a) v- sin b sin c. Then, knowing the values of sin 4 A and cos - A., we may deduce tang ~ A. Place a + b + c = 2 s, we shall have a + b- c = 2 (s —c), a - b c = 2 (s- b), b + c - a = 2 (s - a), and the three formuli may be written, sin 4A= / sin (s -b) sin (s - c) \/.b~ si sin b sin c cos-A= /sins sin (s-a) o sAsin b sin c tang ~ A= / sill (s-b) sin (s-c) - sin s sin (s-a) By a simple change in the letters, we may deduce analo. gous formnlm for - B and 4 C. 154. It is shown in geometry that the sum of the three sides of a spherical triangle is less than a circumference of a great circle. Calling 8 the difference, we shall have s = 180~ -- ~, s -a-= 180~-(a +- ~)., &c.; and since the sine of an are is equal to the sine of its supplement, we may give the preceding formule in the following form: .86 TRIGONOMETRY. sin ~ A =1 ~sin (b + — ~ ) sin (c + ~ ) sin b sin c cos 4 A_- // sisl - i1 sin (a + ~ s) sin b sin c tang A= /sin (b + -6) sin (c + a) 2 sin 2 sin (a + ~ ) 155. 2d. Relation between two sides and the two opposite angles. To find that which corresponds to the combination a, b, A,B, it is necessary to eliminate c between (1) and (2). The method which at once presents itself, is to deduce the values of sin c and cos c, and substitute them in equation sin 2C +cos 2-=1. But the following process, analogous to that adopted Art. 123, is the most simple. Equation (1) gives cos a -- cos b cos c Cos. -- sin b sin c Hence, we have sin2A - 1_ cos 2A -- 1 (cos b cos C)2 sin" b sin2 c (1- cos2b) (1 -cos2 C) —(cos a-cos b cos c) sin b sin2 c 1- cos2 a — cos2 b-cos2 c + 2 cos a cos b cos c sinj'b Sin 2 c Therefore, sin A / 1 —Cos2 a —os2b- osc q- 2+cosa cosb cosc sin a sin asib sin c There is no ambiguity here'in the sign of the radical, inasmuch as the angles and sides being less than 900, their sines will be positive. Since the second member remains constant when we change A and a into B and b, and reciprocally, or into C and c, and the reverse, we conclude; that sin A sin B sin C (4) sin a -sin b sin c Hence, in any spherical triangle, the sines of the angles are proportionate to the sines of the oppositc sides. SOLUTION OF TRIANGLES. 87 156. 3d. Relation between two sides, the angle included by them, and the angle opposite one of them. Considering the combination a, b, A, C, eliminate cos c, between (1) and (3), and we have cos a = cosa cos2b + cos b sin a sin b cos C +- sin b sin c cosA. Transposing cos a cosI b, observing that cos a — cos a cos2 b cos a sin2 b, and dividing by sin b sin a, we have cos a sin b sin c cos A = cos b cos C +sin a sin a sin c sin C sinBut sin' henco, wo have for the required relation sin a sinA' cot a sin 6b -= cos b cos C - sin C cot A. By making the different permutations in the letters, we may deduce the following six equations: cot a sin b cos b cos C +- sin C cot A (5) cot b sin a cosa cos Co + sin C cot B (6) cot a sin G c cos c cos B + sin B cot A (7) cot c sin a = cos a cos B + sin B cot C (8) cot b sin c = cos c cos A + sin A cot B (9) cot c sin b = cos b cos A + sin A cot C. (10) 157. 4th. Relation between a side and the three angles. This is the last relation to be determined. Eliminate b and c, between the equations (1), (2), and (3). For this purpose, substitute in equation (1) the value of cos c deduced from (3), and their results: cos a sin b sin c cos A cos b cos C.sin a sin a and this relation, by means of the equations sin b sin B sin c sin C - _=. and sin a sin A sin a sin A becomes cos a sin B = cos b sin A cos B + cos A sin C. Performing the same operations on equation (2), or rather changing, in this last equation, a and A into b and B, and vice versa, we have cos b sin A = cos a sin B cos C + cos B sin C 88 TRIGONOMETRY. We have now only to eliminate cos b between the two last equations. This gives us, after making the necessary reductions, the required relation between A, B, C:and a; and if we apply this relation to each of the three angles in succession, we deduce the three following equations: ccs A cos B cos C + sin B sin C cos a (11) cos B = - cos A cos C - sin A sin C cos b (12) cos C = -cos A cos B + sin A sin B cos c (13) 158. It is easy to express sin I a, cos2 a, tang 1 a, in func tions of the angles A, B, C. We deduce from formula (11), cos A -. cos B cos C COS a(.sin B sin C. By operations analogous to those in Art. 153, we readily deduce the formula required. Placing A + B 3+ C- 1800 + s, these formule become sin ~a- ~/sin E sin (A C-s1) sin B sin C. c o iD ~a - - ~) sin (C~~ s) co a =(sin B sin C. / sin sin (A-) tang a X,2/2 The quantity E is called in geometry the spherical excess. 159. The analogy between equations (11), (12), (13) and. the fundamental formula (1) is striking, and leads to a remarkable consequence. Let us conceive a spherical triangle A' B' C', the sides of which, a', b', c', are the supplementsi of the angles A,13, C. By virtue of formula (1) we shall: have cos a' = cos b' cos c' + sin b' sin c' cos A'. But sin a'=sinA, cos a' =- cos A, sin b' = sin B, &c. Ience. - cos A = cos B cos C - sin B sin C cos A' From this equation we get for cos A' a value equal and with a contrary sign to that which equation (11) gives for cos a': hence, a = 1800 -A'. In like manner, b - 1800-B', and c = 180~- C'. Therefore, a spherical triangle being given, SOLUTION OF TRIANGLES. 89 if we form a second triangle from it, so that the sides of this new triangle shall be supplements of the angles of the first triangle, the sides of the first triangle will be also the supplements of the angles of the second triangle. 160: These triangles are said to be supplementary. Since each of these triangles may be described by taking as poles the three vertices of the other, as is shown in geometry, this property leads to a designation of these triangles as polar one to the other. 161. By the polar triangle, the formulm found above for la may be deduced from those which are known for IA (Art. 153). Indeed, we can apply these formulm to the polar triangle itself, and for this purpose it is only necessary to replace the angles by the supplements of the sides and the sides by the supplements of the angles. We thus reproduce the values of sin a., cos a, tang Ia. 162. Proportion of the four tangents. We proceed now to demonstrate some formula which are of frequent use by si n A sin a astronomers. Taking the proportion of the sines sin B sii b we deduce, in the first place, sin A + sin B sin a + sin b sin A sin B sin a- sin b. Replacing now each ratio, by a ratio of tangents, from known relations, we have tng'l- (A + B) tang 2 (a + b) tang I(A+-B) tang (a- b) This formula is called the proportion of the four tangents. 163. Proportion of the four cosines. The student may readily construct the figure for himself. From the vertex C of any triangle draw the arc of a great circle perpendicular to the side A B13. Designate the perpendicular by p, and by a and a3 the segments adjacent to the sides a and b. The two right-angled triangles will give cos a = cosp cos a, cos b = cosp cos P (Art. 150): and from these results, we deduce the proportion of the four cosines. cos a cos a cos b cos 3' 90 TRIGONOMETRY. 164. ionnrmulce of Delambre. These formul.le which bear the name of the distinguished astronomer who first published them are: sin - (A + B) = cos I C ~s 2 (a-b) ('OS 1C sin ~ (A - B) = cos C Sin (-b coms (A + -B)=sin C cos I (a+b) cos (a b) cos ~ (A - B) sin C sin 4 (a + b) sin i c. They are readily deduced by finding the expressions for the sines and cosines of the angles I (A + B), A (A - B) in functions of the sides a, b, c. For example, let us find sin X (A +- B). In the first place, we have sin ~ (A + B) = sin i-A cos i B + sin- B cos i A. Substituting for the sines and cosines, in the second member, their values expressed in terms of the sides, we have s./sin (sb) i (- C) ssin(s-b) sin b sin c / si a sin c sin g (n 3- R) = \,/if (s-b) sin (s- ______________ sin (A + B) (q\/sin(s - a) sin(s- C) sin (s-a) sin a sin sin b sin c - Psin (s —b) + sin (s —a) A/sin ssin (s - c) sin C sin ac sin b. The last radical is equal to cos 4 C. The numerator sin (s —b)+ (s-c) is a sum which mav be replaced by a product, and the denominator sin c is equal to 2 sin - c coss Ic: we have, therefore, s a + s - b s —a —sq-b sin (s-b) + sin (s-a) 2 sin cos 2 2 -- -sin c 2s in 8 c cos - c. sin - (2 s- a - b) cos - (a - b) sin 4 c cos 4 c. Recollecting that 2 s a + b + c, we have 2 s - t -b c, cos~(a —b) We and the preceding fraction is reduced to cos - We have therefore SOLUTION OF TRIANGLES. 91 sin ~ (A + B) = cos - C cos - (a- b) COS -IC, which is the first of Delambre's formulae. The three other formula may be obtained by analogous operations. 165. It might be supposed that in applying these formulw to the polar triangle we should obtain new formulm. But this is not the case.; For example: replacing A, B, C, in the 1st by the supplements of the sides, and reciprocally, it becomes sinsi(anb)= si1c o-(A B), and this is no sin -- C other than the 4th formula, under a little different form. Reciprocally, the 4th formula will reproduce the 1st. As to the 2d, it will be made to reduce to itself, with a slight change in form: and so also with the 3d. 166. Napier's Analogies. These remarkable analogies or proportions are embraced in the following formulae: cos ~ (a -b) tang ~ (A + B),= cot -- C ( cos ~ (a + b) tang -. (A B) — Cot''C sin ~ (a-b) sin (a + b) cos (A-B) Cos -(A + B) tang:i (a + b) _=tang eos - (A -3) To demonstrate these formulae directly, we should have to go over the same calculations which have just been made for the formulae of Delambre. It will therefore be more simple to deduce them at once from these formulke. If we divide sin a (A + B) by cos - (A + B), and sin ~(A —B) by cos ] (A -B) in Delambre's formule, we at once deduce the two first analogies. In like manner, by dividing the 4th formula of Delambre by the 3d, and the 2d by the 1st, we obtain the two last analogies. We may also deduce the two last analogies from the two first by means of the polar triangle. 167. Napier's Analogies serve to simplify some cases in the resolution of spherical triangles. We make use of the two first when we have two sides and the included angle given; 9 G 92 TRIGONOMETRY. and the two last, when we know two angles and the included side. 168. Relations between the parts of a right-angled spherical triangle. To deduce the formule which correspond to the particular case of the right-angled triangle, it is only necessary to make A = 90~ in the relations heretofore found, which contain this angle. In this manner, we find the following formulae: Art. 153, gives cos a = cos b cos c (a) Art. 155,' sin b = sina sinB, sin c = sin a sinC (b) Art. 156, " tangb = tang, a cos C, tang c = tang a cos B, (c) Art. 156, " tangb - sin c tang B, tang c = sin b tang C, (d) Art. 157, " cos B sin C cosb, cos C sin B cosc, (e) Art. 157, " cos a = cot B cot C. (f) We have thus six distinct formulm equally adapted to logarithmic calculation. The first gives a relation between the hypothenuse and the two sides of the right angle; the second, between the hypothenuse, a side and the angle opposite; the third, between the hypothenuse, a side and the adjacent angle; the fourth, between the two sides and the angle opposite one of them; the fifth, between a side and and the two oblique angles;.and the sixth, between the hypothenuse and the oblique angles. Thus, any two of the five parts being known, we have a formula to determine all the other parts. 169. Certain properties of right-angled triangles must be noticed here. 1st. Formula (a) requires that cos a have the sign of the product cos b cos C: but, for this purpose, it is necessary that the three cosines be positive, or only one positive. Htence, in a right-angled spherical triangle, the three sides are less than 900; or ease two of the sides are greater than 900, and the third less. 2d. The formulae (d) show that tang b has the same sign as tang B, and tang c the same sign as tang C. Hence, each side of the right angle is of the same species as the opposite angle: that is, the angle and the side are both less than 900, or both greater. SOLUTION OF TRIANGLES. 93 170. 2Vapier's Rules. The formulae given in Art. 168 are embodied with great elegance and convenience in Napier's.Rules, the enunciation and proof of which are as follows. The right angle being left out of consideration, the two sides including the right angle, and the complements of the hypothenuse and of the two other angles, are called the five circular parts. Any one of these being taken as the middle part, the two circular parts which are immediately contiguous to it, in position, are called the adjacent parts; and the other two parts, considered with reference to the same part as the mnzddle part, are called the opposite parts. Then, whatever the middle part be, whether a side, or the complement of the hypothenuse, or the complement of an angle, we have always Ist. The sine of the middle part equal to the product of the tangents of the adjacent parts. 2d. The sine of the middle part equal to the product of the cosines of the opposite parts. 1st. Let the complement of the hypothenuse, 90~-a, be the middle part, then the complements of the two oblique angles, viz. 900 - B, 90~- C, are the adjacent parts, and the two sides b and c, the opposite parts; and formula (11), Art. 157, and formula (1), Art. 149, cos A - cos B cos C + sin B sin C cos a cos a - cos b cos c + sin b sin c cos A, become, since A -= 900~, and therefore cos A - 0 and sin A =, cos a = cotB cot C, or, sin (900 - a) = tang (90 - B) X tang (90_ C), cos a = cos b cos c, or, sin (900~ -a) = cos b cos c, which prove the rules. 2d. Let the complement of the angle C, 90 —C, be the middle part; then the complement of the hypothenuse, 90~ —a, and the side b, are the adjacent parts; and the complement of the angle B, 900 —B, and the side c, are the opposite parts: and the formule, cot a sin b = cos b cos C + sin C cot A (5), Art. 156. cos C = - cos A cos B + sin A sin B cos c (13), Art. 157. become, since A = 900~, 94, TRIGONOMETRY. cos.- = cot a tang b, or, sin (90~- C) = tang (900 -a) tangb, cos C=sin Bcos-c, or, sin (900-C) = cos(90~-B) cos c. which prove the rules again, with reference to the part assumed as middle part. If 900-B, be taken for the middle part, the rules may be proved exactly in the same way, and other two similar relations found. 3d. Let the side b be the middle part, then 900-C and the side c will be the adjacent parts, and 900 - a, and 90~ B, the. opposite parts; "and the formule cot c sin b ='cos b cos A + sin A cot C, (10), Art. 156, sin A-sin b- sin a sin B. (4), Art. 155, become, when A= 90~ sin b =cot C tang c, or, sin b = tang (900 ~- C) tang c,. sib = sin a sinB, or, sinb = cos (900 —a) cos (900-B), which show that the rules hold good in this case also. If c Be; the middle.part., the rules may be proved in the same w..ay.anS d other- t;wo similar relations found. 17t1. We aiare thus furnished with ten relations amongst the fiVee.'parts,of a- right-angled triangle, each being a different combination of three of the parts; but five parts, taken three at a time, can be combined only in 10 different ways; consequently, the above Rules, when any two parts whatever are given, will supply us with formule in which each of the remaining parts is separately combined with these two given parts, and in a form adapted to the immediate application of logarithms. 172. There can evidently be only six distinct cases, viz.: I and II, when the hypothenuse is given together with an angle, or with one of the sides containing the right angle; III~ and IV,;when one of the sides of the right angle' is given together with the angle adjacent to it, or with the angle opposite to it; V and VI, when the two sides containing the right angle are given, or when the two angles are given. In applying Napier's Rules, to obtain the three unknown parts from two given ones, it is sometimes requisite to take for a middle part one of the given parts, and sometimes one of those that are sought, the sole object being SOLUTION OF TRIANGLES. 95 to -separately combine each of the unknown parts iwith the given parts. 173. A spherical triangle may be bi-rectangular or tri-rectangular, that is, two of its angles may be right angles, or all three may be right. In the last-case, the three sides are quadrants; in the first, the sides opposite to the two right an gles are also quadrants; and the third angle havingfor a measure the third side, must be expressed by the same nuniber of degrees as this side. Thus, these two cases not giving rise to any particular investigation, it will be only necessary to refer to the spherical triangle which contains one rlight angle. 174. Case 1. Having given the hypothenuse a, and an:ange B, to find b, c, and C. Taking successively b, 900 - B, and 900 - a for the middle part, we get, sin b - sin a sin B, cos B = cot a tang c, cos a - cot B cot C: C and c are determined without ambiguity; b and B- must be both greater or both less than 90~. 175. Case II. Having given the hypothenuse a, and a side b, to find c, B, and C. Taking successively 900 -a, b, and 900 —C for the middle part, we get cos a = cos b cos c, sin b -sin a sin B, cos C = cot a tang b. By these formula, c and C are determined without ambiwguity, for there is only one angle less than 1800 oorresponding to a given cosine; and B, determined by its sine, is not ambiguous since it must be of the same species as b. 176. Case III. Having given one of the sides b containing the right angle, and the angle C adjacent to it, tofind a, c, and B. Taking successively 90~- C, b, and 90~ —B for the middle part, we get cos C = tang b cot a, sin b = tang c cot C, cos B = cos b sin 0, which determine a, c, and B, without ambiguity. 177. Case IV. Having given one of the sides, b, of the right angle, and the angle B, opposite to it, to find a, c, and C. Taking b, c, 900 - B, successively for the middle part, we get sinb in ]B sin a, sin c = cot B tang b, cos B sin C cos b, 9* 96 TRIGONOMETRY. Here, since all the unknown parts are determined by their sines, and since there are always two angles less than 180~ corresponding to a given sine, an amniguity presents itself, and it is easily seen that such ought to be the case. For if. in the triangle BA C, right-angled / at A, we produce BA and BC, / until they meet in D, then taking C/ / D AD'= B A, and D C'=BC, the triangles B A C, D A' C', will be equal in all their parts; hence,:A the angle A' is right, and C' A'= CA=b. The triangle BA'C' is thus rectangular, and contains also the two given parts B and b. We may then take, at will, a < 900 or a> 900, but when the choice is made, the species of c will be given by the relation co's a = cos b cos c, and the species will be the same as that of C. 178. Case V. Having given the two sides b and c containing the right angle, to find a, B, C. Taking successively 900 — a, c, and b for the middle part, we get cos a = cos b cos c, sin e = cot B tang b, sin b = cot C tang c, which determine a, B, C, without ambiguity. 179. Case VI. Having given the two oblique angles B and C, to find a, b, e. Taking successively 900 —a, 900-B, 900-C, for the middle part, we get cos a = cot B cot C, cos B = cos b sin C, cos C = cos c sin B. These formula give no ambiguity, and if the triangle be impossible they will show it. 180. Observations. The solution of many cases of obliqueangled spherical triangles may be reduced to that of rightangled triangles. 1st. If in an oblique-angled spherical triangle, three parts be given, one of which is a side equal to 900~ the correspond. ing angle in the polar triangle will be a right angle. Besides, we shall know two of the five parts of thlis triangle, hence it may be solved by rules just given; and it is evident, SOLUTION OF TRIANGLES. 97 when the parts of the polar triangle are known, we may readily determine those of the given triangle. 2d. When a spherical triangle is isosceles, the two equal sides are considered as only one element, and the two equal angles opposite as only one element, and in this case any two parts of the triangle being given, we may determine the triangle. For this purpose we have only to draw from the vertex to the middle point of the base the arc of a grieQat circle, and the given triangle will be divided into two rightangled triangles equal in all respects, and in each of which two parts will be known, besides the right angle, and thus all the parts of the given triangle are determined by the Rules for the solution of right-angled triangles. 3d. Let ABC be a spherical triangle, in which we have a + b - 180~. C.- 1-0..D Producing a and c until they meet a in D, we shall have a + C D-= 180; hence, C D — b. But each known element of the triangle ABC makes / known one element in the isosceles B triangle AC D, and reciprocally; hence the resolution of a triangle in which the sum of two sides is equal to 1800 reduces to that of an isosceles triangle, and consequently to that of a right-angled triangle. 4th. The same remark applies to a spherical triangle in which two angles are supplements of each other; for we cannot have a + b — 1800, without having at the same time A +-B = 1800~, and vice versa. Indeed, in the isosceles triangle A CD,the angle C A D= D = B: but C AD + CAB 180~; hence, in the triangle AB C, we have also A + B — 180~. RESOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 181. Case I. Having given the three sides a, b, c, to find the three angles A, B, C. Equation (1), Art. 153, gives cos a - cos b cos c cos A -- sin b sin c; but as this formula is not adapted to logarithmic calculation, 98 TRIGO NOMETRY. we mlake use of the formulle for sin - A, cos iA, tang i A given in Art. 153. sin -,- A - / sin (s-b ) sin ( s c) sin b si - c cos - 1A/ si/ 11 sin (s- a) sin b sin c tang,,,A = / sin (s -b) sin (s -c) sinlls slln (s-a) 182. Case II. Having given twvo sides a and b, and the angle. A opposite one of them, to find c, B, and C. We obtain the angle B, opposite the side b, immediately, by the proportion sin B si sin A sin A b *. -.. hence, sin B3 sin b sin a sin a. To find c and C, we make use of Nalpier's analogies, Art. 166, which give tang c = — - tang (a-b) sin (A - B) sin - (A —B) cot ~ C = tang I (A-1B) si - (a~ + b) sin _ (a - b) The angle B being. determined by its sine, may be either acute or obtuse. Nevertheless, for certain values of the given parts a, b, A, only one triangle exists. We will resume the consideration of this case hereafter, presentino as it does an analogous discussion to that w-hich has been'made in Case II of plane triangles. We may also determine C directly by making use of the formula (5), Art. 156, cot A sin C - cos b cot C = cot a sin b..(5) For this purpose, we will first determine an auxiliary angle p, by making cot A = cos b cotp, which gives cot A cot = -cob' Then, substituting the value of cotA= cosb cot-=: o os in equation (5), this equation becomes sin (p SOLUTION OF TRIANGLES. 99 cos b (sin C cos p + cos C sin p) = cot a sin b sin p, from which we deduce tang b sinp tang a which makes k]nown c,+-. Put e +, = m, we have&C = -- n-o Having found C, we obtain the side c, by the rule that the sines of the angles are proportional to the sines of the opposite sides. But if we wish to calculate c directly, we make use of formula (1) Art. 153, cos b cos c + cos A sin b sin c = cos a. We reduce, as above, the first member to a single term, by means of the auxiliary angle p, by placing cos A sin b — cos b cot p, which gives cot t- = cos A tang b. Consequently, the equation becomes cos b (sin p cos c + cos p sin c) = cos a sin p, cos a sinl hence, sin (c + )- c) s Knowing now the value of e, we readily determine c. 183. Case III. Having given two sides, a and b, with the included angle C, to find A, B, and c. Formule (5) and (6), Art. 156, give for A and B, cot a sin b- cos b cos C cot A = sin C cot b sin a -- cos a cos C CQtB sin C By making use of auxiliary angles, we may readily reduce each numerator to a single term. But Napier's Analogies furnish a more simple solution. We have cos-} (a, —b) tang I (A + B)= cotC os - cosI(a+b6) tang,- (A-B) = cot 1 C sin - (a -b) sin 4 (a + 6) These analogies make known - (A + B) and (A —B), from which we readily deduce the values of A and B3.' 100 TRIGONOMETRY. When A and B are known, c is found by the proportion of' the sines. If we want to determine c directly, we take the well-known formula, Art. 153, cos c = cos a cos b + sin a sin b cos C, in which we make cos b cos si n b cos C cos b cot I. sin? We have, without Any ambiguity, cos b sin (a + q) cot P = tang b cos C, Cos sin sin q 184. Case IV. Having given two angles, A and B3, with the included side c, to find a, b, C. We may determine a and b by formulae (7) and (9), Art. 156, cot A sin B + cos B cos c cot a = sin c cot B sin A + cos A cos c cot b= sin c; or, better still, by Napier's Analogies: tang ~ (a - b) = tang J cos 0 (A - B) tncos (A + B) sin (A B) tang ~ (a - b) tan (rg c tang(a-b tan sin (A + B) The angle C is found by the proportion of the sines. To find C directly, we take formula (13), Art. 157, cos C = sin A sin B cos c - cos A. cos B, and placing sin B cos c = cos B cot cp, we have cos B sin (A -— ) cot _ tang B cos c, cos sin (A - sing This case is analogous to Case III., and presents no ambi guity. 185. Case V. Raving given two angles A and B, and the side a opposite one qf them, to find b, c, and C. This case is precisely analogous to Case II. It is discusse4:in the same way, and presents like ambiguities. SOLUTION OF TRIANGLES. 101 sin b sin]3 The side b is deduced from the proportion = and sill a sin A' c and C are determined by the formula employed in Case II. sin ~ (A q- B) tang - tang ~ (a- b) i (A-~) sin (A - B) cot ~ C = tang ~ (A - B) sin ( b) sin ~ (le - b) The side c is also given by the formula (7), cot a sin c - os B cos c = cot A sin B. In which, if we make cot a = cos B cot 1, we have cot a tang B sin q cots —B X osin(c — ~) tang A. Finally, the angle C is found either by the formula that the sines of the angles are proportional to the sines of the opposite sides, or by means of the equation cos a sin B sin C- cos B cos C = cos A. We may reduce the first member to a single term byplacing cos a sin B = cos B cot i, cos A sin hence, cot~ -- cos a tang B, sin (C -- ) cos B. These values make known q, C -, and 6onsequently the angle C. 186. Case YI. Raving given the three angles A, B, C, to find the three sides a, b, c. This case is solved by analogous calculations to those in Case I. Thus,.o find a, we make use of the formula, Art. 158, sin ~ a — sin ]s sin (A- si vasin B sin C — cos a -- /sin (B- z- sin (C - E) sin B sin C /sin ) sin (A-4]) tang a V sin (B - ) sin (C- We remark that the three last cases bear a strong analogy to the three first. Indeed, by the properties of the polar triangle, they may be deduced, one from the other. 102 TRIGONOMETRY. A;:MBIGUOUS CASES IN SPHERICAL TRIANGLES. 187. The only cases in which ambiguity arises, as to'the values of the computed elements, are the II. and V., and it becomes necessary to investigate the conditions to which the given elements are respectively subject, when they correspond to two triangles, or to only one, or to none at all. Let D H D', DCD', be two great circles at right angles to each other, C) I}D take C 1) < 900, and from the point C, draw the arcs of great circles to the different points of the circumference ) H D'. Produce C D until C'D is B D B equal to CD, and join C'B. The triangles C D B, C'DB are equal, since they have sides including a right equal, ~, x.~t~each to each: hence, C B C'B. But CCDC' D B; draw C' H, and produce C B until it meets C'H in I. Since the are C C' is less than a semicircumference, it must be intersected by the prolongation of C B beyond the point C', which makes it necessary that the point I should be between HI and C'. We have then C'B900: hence, if A is acute, B CD is the shortest distance from the point C to the semi-circumference,.and it is the greatest if A is oblique (187.) In the first hypothesis, the ti'iangle will be impossible" if we have a CD, which gives still sin a < sin C D. But in the right-angled triangle ACD), we have sin C D =sinb sinA; hence, in the two hypotheses, we should have sin a 1800, the point B' falls on E, or passes beyond it, and there is no longer any tri angle. We may discuss in the same manner the other hypotheses. The results are comprised in the following table. ab, or a=b, provided a+b is not 1800~ or greater than 180~, -onesolutis-o a-b=1SO00, or aq-b>180~....................................... no solution!a<900 a { aTb90~0 -a+b>1800, or a+b=1800,provided wehave not ab, or a>b, one solution a=b, or a>b................................................................. no solution { ab................................... two solutiorns b>90o a=b, or ab.......................................................................... t o solutionsx a=b, or ab, if we hab is not a+b=180, or >1800~....................... one solution b<900 -a=lb, or a1800........................................... no solution a900 a=b, or a>b........................ no solution it a+b=180~, or 18 180~........................ no solution a —.=90n........................o.................................. ininite nubr osouti a<90~, or a>90s..........................I......................... no solution SOLUTION OF TRIANGLES. 105 190. The property of the polar triangle enables us to apply these results to a triangle in which A,B, and a are given, as in Case V. It is only necessary to change a, b, and A, into A, B, and a, the sign > into <, and the sign < into>. When the conditions which are given lead to one of the cases in which there is only one solution, the calculation always indicates two. To ascertain which must be preserved, it will be only necessary to note that the greater anigles must lie opposite to the greater sides, and reciprocally. Suppose that we have given A. = 1120, a = 102~, b = 106~. In the preceding table, among the cases which correspond to A>900, we consider those in which b>900, and note in theseI that one in whiclh e have a= b, or a1800. We conclude, therefore, from the table, that we have only one solution, and since b> a, the angle B > A, hence'B -is obtuse. SURFACE OF THE SPHERICAL TRIANG'LE. 191. The surface of a spherical triangle is equal to the excess of the sum of its three angles above two right angles. This excess,. which is also called the spherical excess, may be found from the sides, by the aid of very simple formule, which will now be explained. Designating, as before, the spherical excess by s, we shall have sins = sin (~ (A- - C) - 90) = -cos ~ (A + B + C). It is required to find cos - (A + B + C). In the first place we have cos ~'(A + B) = cos ~ A cos~ B -sin { A sin ~ B. Now putting B + C for B, we readily get cos - (A + 3 + C) = cos -- A cos - B cos - C -cos ~ A sin ~ B sin ~ C -cos ~ B sin ~ A sin C - cos C sin ~ Asin i B. Substituting for the sines and cosines, their values in terms of the sides (Art. 153); and considering, in the first place, the 1st term which contains three cosines, we shall have cC=_' /sins sin (s-a) sin ssin (s-b) sins sin (s-c) V sin b sinec sin a sine sin blaink 106 TRIGONOiMETRY. s s/sins sin(s-a) sin (s-b) sin (o-c) sin a sin b sin Designating the radical which occurs in the second mem.ber by IR, the expression assumes the form R sin s cost A cos I B cos I C = i sin a sin b sin c. If we now consider the 2d term, we find os, A sin'B sinl -. / sin 8 sin (s-a) sin (s-ca) sin (s-e) sin (.s-a-) sin (s-b) - sinb sin c sin a.sin c sin a sin'.b or, Isimplifying, g-sin (sa> ) cos I A sin. B sin sin a) - sin a sin b sin c. By a single change in the letters, we readily find analogous values of the two other terms, viz.: cos B B sin A sin C = sin(s-b) sin a sin b sin c R sin (s —c) cos - C sin ~ A sin n (B -c) sin a sin-b sin c By means of these transformations we get R (sin s-sin (s —a)-sin (s-b)-sin (s-e)) cos ~(A+B+C)= sin a sin b sin c In this value the multiplier of Ri may be reduced to a single term, as follows. The two first terms being a difference of sines, we. may make use of the formula sinlp - sin q -= 2sin - (n ) c s; (p (+ q) anvd we have sin s- -sin (s-a) = 2 sin g- a cos - (b + c) For the two others, we make use of the formula sinp - sin qsin 2sin (p x. q)- cos (p - q) and we find sin (s - b) +- sin (s -c) = 2 sinl a cos - (b -c) 1Hence, the multiplier of R will be equal to 2 sin - a (cos g (b + c)- cos I (b - c) ) 50LUTION OF TRIANGLES. 107.and this last expression may itself be reduced to -4 sin 4 a:sin b sin c. As to t:he denominator, we remark that sin a sin b sinc- = 8 sin ~a cos ~a sin ~ b cos ~ b sin ~c cos Ic. Consequently, after making all the reductions, and restoring the radical designated by R, we shall have cos (A - B +B C). Then changing the sign of the result, we have the required formula, V/sinsl sin (s-a) sin (s-b) sin (s-c) sin ~ E — 2 cos - a cos ~ b cos - c. If we find the value of cos ~s, in the place of sin 4 s, we obtain the followving formula; 1 + cos a + cosb + cosc 4 cos l a cos f b cos I c, which is not convenient to be used in logarithmic calculations. The transformations required are complicated, and it would be useless to stop here to make them. Finally, we introduce, in this place, without developing the calculations, the very elegant formule of Simon Lhuillier. By means of cos 4 s, we obtain sin I., cos I s, tang I s, and we get the formulae referred to. fsin ~,iE / sin I s sin I (s- a) sin ~ (s-b) sin 4 (s-c) cos 4 a cos I b cos I c cos i,/ cos s cos 4 (s-a) cos I (s-b) cos t (s-c) cos l a cos i b cos c tang I =Vtang I s tang I (s -a) tang J (s - b) tang J (s —c) 10* 1t 108 TRI GONOMETRY. APPLICATION IN SPHERICAL TRIGONOMETRY. 192. EXAMPLE 1. To reduce an angle to the horizon. Let BAC be an angle situated in a A plane inclined to the horizon, and AD ~s'".~D~ the vertical passing through the vertex A. Draw at will the horizontal plane M N, meeting the lines AB, A C, AD, in E, F, G: the angle E G F is the horizontal F >a projection of the angle B A C, or, in other words, it is the angle B AC, reduced to the horizon. It is required to calculate the angle E G F, supposing the angles B A C, B A D, C A D, as known, being determined bv instrumental measurements. The graphic solution would be easy, for, the line AG being arbitrary, we should have sufficient elements given to construct, i l the first place, the right-angled triangles EA G, and F A G, then the triangle E A F, and finally the triangle EGF. The calculation of the angle E G F is easily -made. If we describe a sphere from A, as a centre, with any radius, the right lines A B, A C, A D, determine, by their intersection with the surface of the sphere, a spherical triangle B CD, the sides of which are known in degrees by means of the given angles, and the angle BCD equal to EGF is the required angle. The solution is readily made by using the well-known formula for sin ~ A in terms of the three sides. sin ~ A = A/ sin (s - b) sin (s —b) sin b sin c, in which we make a =BA C, b = B AD, c AD, s= i (a + b + c). Let a = 470 45t 39w, b - 690 49' 19", c = 800 17' 36", wE shall have 2 s,=197~ 52' 34", s=98~ 56' 17", s- b=290 6' 58", s —C- 180 38' 41", and we may make the following caleulalion. SOLUTION OF TRIANGLES. 109 Log sin (s — b)..... 9,6871552 Log sin (s-c)... 9,5047412 Log Sin b (Arith. Compl.). 0,0275078 Log sin c (Arith. Compl.). 0,0062623 2 Log sin A...... 19,2256665 Log sin A...... 9,6128332 A= 240 12' 27",9 J A 48 24' 56" 193. EXAMPLE II. Having given the latitudes and longitudes p of two points on the surface of the q earth, to find the distance between these points. LetAandBbethetwopoints. \/ Let Q R be the circle of the equator, C, the north pole, and C ED, C F D, the meridians passing through the points A and 3. Finally, suppose that the longitudes are reckoned from the point P in the direction P Ef. The difference of longitudes, P F —P E, is equal to the are E F, or to the angle C included between the two meridians; and the arcs A C, B C, are the complements of the given latitudes A E, B F. Thus, in the spherical triangle A B, we know the angle C, as well as the sides which contain this angle, and it is required to calculate the side A B. Case Il!i, in which two sides and the included angle are given,-givg the formule, cos b sill (a +) cot -tang b coscos — c Suppose, for example, that we wish to calculate the dietance from Brest to Cayenne. The longitude of Brest = 6~ 49' Latitude = 480~ 23' 14". The longitude of Cayenne = 540 35' Latitude = 40 36' 15". Both longitudes are west and are reckoned from the meridian of Paris; both latitudes are north. With these data, we find, in the first place, 110 TRIGONOMETRY. C= 540 35'- 6~ 49' = 47~ 46' a = 900 - 480 23' 14"= 410 36' 46" b = 90 - 40 51'1I]5" — 85~ 3' 45" We may now calculate c as follows. Calculation of the auxiliary Calculation of the side c. angle A. log cos C 9,8274671 log cos b 8,9348468 log tang b 11,0635386 log sin (a +- ) 9,8773621 i 0Arith., 08945642 log cot, 10,8910057 log sin (Comp.) 8945642 = 7? 19' 26" log cosc 9,7067731 a+ f 480 56' 12. c-=590~ 23' 54t 38. Thus, the are which mneasures the distance between Brest and Cayenne is 590 23' 54", 38. To estimate this arc in myriametres," we must remember that the fourth of the terrestrial meridian is 10,000,000 metres, or 1000 myriametres. Hence, we have the proportion x 590:23' 54" 38 1000 9:0~9 and reducing the arcs to se'ondS, we find the distance x to be 213834 x 1000' 324,000 - 65,983 myriametres. -This; last operation could have been easier, if the are c had been expressed'in centesimal degrees. For example, take in this'division an ftrc 37~ 45' 69": referring it to the quadrant, it will be expressed by 0,374569, and multiplying the number by the value of the quadrant, in terms of myriametres, we find at once, by the simple removal of the decimal point, 374,569 myriametres. CHAPTER III. FORMULLE USED IN THE HIGHER MATHEMATICS. SINES AND COSINES IN SERIES. RESOLUTION OF BINOMIAL EQUATIONS AND OF TLIE EQUATION OF THE 3D DEGREE. FORMULA OF DE MOIVRE. 195. This formula, which bears the name of the French Geometer who discovered it, is as follows: (cos t q- - 1 sin.) = cos n t q - V/- sin n t (A) It expresses that to find the value of the binomial cos t + V/-1 sin t, raised to any power, we have only, to multiply the are b by the exponent of this power. Either sign + or —may be placed before V/-1, for this would involve only the change of + t into -(. 196. Case 1st. n positive and entire. We have, by multiplication, (cosp + v' /1 sin p) (cos, + 1 sin A) = cos t cos -sin, sinl + V- 1 (sin q cos 4 +- cos T sin 4). But, from known formulae, the real part of this product is equal to cos (a _+u A), and the imaginary part is equal to I —1 sin (a + - ); hence, (cost + V/ITsin ~) (cost + /-1 sin +) =cos (t+)+ )/-1 (sin l + A). That is to say, when we multiply two expressions of the form cos ~ + - I1 sin A, we obtain a like expression, in which the two arcs are added together. If now we wish to multiply this product by a new factor of the same form, we have only to add the new are to the two others, and so on, for any number of factors. Hence, if there should be n factors; all equal to cos t -F- V-i sin A, we have (cos ) + S in )C=coSfl fsin n, (1) (111) 112 TRIGONOMETRY. 197. Case 2d. n positive and fractional. Substituting ~ for s in formula (1), we have, (cost +/- 1 sin )= cos + /-1 sin,. Now, extracting the nth root, and putting a fractional exponent in place of the radical, formula (1) is found proved for an exponent o, for, we shall have (cos + Vs/-1 sin t)- = cos + V/Z- sin -. (2) n which proves formula (1) for an exponent n In general, the expression A- signifies that the mtI power of A is taken, and then, the ntf root of this result. Hence, if We -raise cosq + V-/ sin q to the mt'" power by formula C1),:and afterwards extract the n"t root by formula (2), we ha havo (cosq +- V-l sin,) n = cos + V -1 sin- (3) h[ich proves formula (1) for any positive fractional expo. nent -. 198. Case 3d. n negative. In this case, we observe that cos:n + V- sinnq ) (cos n - V/- 1 sin n ) = cos' n + sin2-n. = 1 from which we get cos n, v -- 1l sin n, cos n, + v- 1 sin n, or;. the equivalent expression (c.s"'+/V-I sin )-"=sco (-n)+ s/-i sin ( - n), (4) ihence, formula (A) is verified when the exponent n is positive or- negative, entire or fractional; and, in general, cos - sin - is one of the n different values which t;he expression (cos ~+ V /-i sin,)j, according to the principles:of Algebra, admits of. FORMULAE USED IN HIGHER IMATHEMATICS. 113 199. There is no difficulty in assigning the remaining values; for, from what has been proved, since 7 is any are whatever, and may therefore be replaced by the general value of all arcs which have the same sine and cosine as p, viz., 2 r A+, r being any whole number, positive or negative,, it follows that cos - (2 r, +7)+ /I sin m (2 r ~+7) (B) U n is a value of (cos 2 r A+,+ -/ -i sin (2 r t+p)n, or of (cos + V/ — sin q). We will now show that the expression (B) admits of n different values, and no more. If we make r=O, 1, 2, &c. to n- 1, we get n different values;: for, if two of them were alike, as r — p, and r = q, it would m m be necessary that the arcs -(2p a+ ), (2 7q + p), should n n differ by a multiple of 2 t, or that m (p —) should be a multiple of A, which is impossible, since p and q are both less than n, and mr not divisible by n. Also, if we take for r some number beyond the limits 0 and n —1, we shall get no new value; for, if r= -n+r', in which X is any positive or negative number, and r' positive and less than n, so that r may represent any positive or negative numbers whatever;, then the above expression becomes cos (2 X m 7.+ I_ (2 7? a))A +,s/I sin (2 m7A+- (2 r', + ), n 3n or, suppressing the multiple of 2 X cos - (2 r' 7 + ) + V-1 sin (2 F Y+w), n n which, since r' is positive and less than n, is comprised among the values obtained in making r= O, 1, 2, - n —1. Consequently, the complete value of (cos 7+ / -1 sin ) is given by the formula (cos + / —1 sil B) =cos -- (2 r.+t)+ v/ — sin m (2 r 7+t), 114 TRIGONOMETRY. r being any whole number, positive or negative, not excluding zero, and the n values of the first member are found frome the second, by taking n from 0 to n- 1. We see, also, by assuming n p =p 2 t r + a, in which r is any whole number, aud: a is any angle between 0 and r, we obtain n different values of ~, every one of which furnishes a different value of cos t + /-i sill., but the same value of cos n -V/ — sin n,. FORMULl Fron sin nr AND COS nf IN TERMS OF SINE AND COSINE OF TInE SIMPLE ARc. 200. Resuming _De 3]loivre's formula cos nq+f/ — 1 si nn-=(cos + /l sin +)' (1) in which n is considered entire and positive, we have, when p is changed into -- cos n -/- V 1 sin n = (cos -v/'-1 sin T))". (2) Adding and subtracting these equations, we find these values,. (cos - V/ —1 sin p)n+ (Cos P —— 1 sin), (3) ) cos n q 2 (cos + /1 sin )L-(cos — p V-i sin p) (4) sin n (4= 2 /1' ) 201. Expanding the powers in the second member by the binomial forniula, and suppressing terms which destroy each other, we obtain s =(c n (n-) (-1)(-2)(n-3) cos n p=(cos ) 1 2 (eosl')I'-(sin l)2~ 1, 2 3 4 I; l, 2, 3, 4 (cos l),z-4 (sin rq)4, &c. (5) n )_ sn (n —) (n-2) (cos )- (sin ) sin n — (cos p)1-1 sin ~ - (COS 1)n-3 (sin e 3+ n ( —1) (n —2) (n-3) ( —4) (cosep)-s (sinq)- &c (6), 1,. 2, 3, 4, 5 These formnulm express the values of the sine and cosine of the multiple arec bn q in terms of the sine and cosine of the' simple arc 1. Their law is evident, and, like the binomial theorem fi'om which they are deduced, each is to be continued until we arrive at a term zero. FORMULAE USED IN I-IGHER MATHEMATICS. 115 FORMULI FOR (sin q()f, (cos ()n IN TEtRMS OF SINE AND COSINE OF THTE MULTIPLE ARCS. 202. In the higher mathematics, it is often necessary to express (sin,)n, and (cos q)" in terms of the sines and cosines of the multiple arcs (n ~).'When n is a positive whole number, which is the ordinary case, it may be effected as follows. Assume cos +,-/-i sin -u, cos -CS 1 sin q=v, we shall have 2 cos + = zv, 2 — 1 sin p= u-v; hence, 2n (cos q)n = (I + v)'0, (2 /-l)n (sin q)n= (u —v)"; and developing (u + v)n, (u-v)',% we get 9n n (17 -1) 2' (cos q)= +t"- + (u1v 1)-1 u- U2v2 + &c. (7) (2 v- 1) (sin e)" = u — u'-v n (n-1) u"v-2u&c. (8) If n be odd, there will be an even number of terms n + 1 in the expansions, and grouping together the terms which are equally distant from the extremes, we have for formula (7), n n (n-) 2o' (cos q)=((U +v) — n) v (u-'+v-2- V -2, 2i) 2 V2 (Un-'+ 2,n + 1: 2 v ) n (n —1) (n-2) —-+ 2) (U (u+ v). 1, 2, 3, 4 —-— n But De Mioivre's formula gives U + -n = (cos + -l sin 4))" + (cos - J/ l sin " - +cos n q+ / — sin p+ cos n q —1 sin n ~=2 cos n qr Besides, the powers of uv are equal to 1, for we have t v= (cos + /1-i sin 1) (cos,- V — sin p)=cos2 + + sin2 -=1. We see, consequently, that the expression for 2" (cos 1)" will be simplified by these reductions; and dividing all the terms by 2, it becomes 2n-1 (C()S ~)-COS q,+ncoS(n1-2),q (2) cos (n -4) -, (n — i) ( 22 — 2) —-( O 2) (9) 1, 2, 3 ------- 11 116 TRIGONOMETRY. If n be even, tne number of terms in the development will be odd, and there will be only one middle term; and if we divide the middle term into two others, each equal to half of the middle term, we shall find, by similar reductions to the pre. ceding, n n (no-i)Cs 2"na (cos,)"=cos n,+cos (n-2) (,4- 1, 2 cos (n- 4) 9 (n (n-1) (n-2) —-(n+ 1)) (10) 1, 2, 3 —------- If now we consider formula (8), the terms being alternately + and -, if n be odd, the coefficients of the terms equally distant from the' extremes will be equal, but will have contrary signs. We shall have, therefore, by the same process, used in formula (7), A U UnV2 an (n- 1), (2 / — 1)(sin )n = u -v" - u vu (u-t2 — 1~,) +, u'2 7v (?u-' —v'~4-)... n.(n-1) (n-) ) —- (n + 2) (-v) 1, 2, 3-... n To make the necessary reductions, we remark that u v- 1, and in general u- -v=- 2 V —1 sin?. The two members' will then be divisible by 2 V/1, and we shall have n In (n —l) (2 V/-l)i-) (sin p)" = sin (n ) — sin(u -v) q~+ 1, 2 sin (n —-4) q~ —- a (n -1) (n2) (n --- (n+2) si (11) -4) 1, 2, 3 s- a If n is an even number, the terms equally distant from the extremes in formula (8) have equal coefficients with the same sign, and by analogous reasoning to that used for formula (7) we find nn nSa ()n —i) i (2 V —) (sin,)' —=cos n p —l cos (n-2) + 1, 2 cos (,_-4).- 4 (f (- 1) (n-2) --- - (n-O1, 2 3 — 1)). (12) Formule (9), (10), (11), (12) are the required formula,; and they enable us to convert the powers of a cosine or of a sine into a series of terms, each of which contains the sine or cosine of a multiple arc, of the first degree only. SINES AND COSINES IN SERIES. 117 It will be observed that -in the last two formula, the imaginary factor V/-I being raised to an even power, must produce a real result equal to + 1, or - 1, according as the index of the power is even or odd. Also, for the conlvenience of the calculation, we must stop. at and exclude the first negative arc; and we must take only half of the last term when it involves the are zero. DEVELOPMENT OF THE SINE AND COSINE INTO SERIES. 203. We shall now show how Euler deduces from formula (5) and (6), Art. 201, the series which give the values of the sine and cosine in functions of the arc. We may, without; departing from the hypothesis of n being a whole number, assumne (, so that n ~ shall be equal to any given arc a. Let, therefore, t p =a, we shall have i=i-,- and with this value the formule become (,Cos -&., &. () (COS T)( )(cos p3 (Si eitl a~l~tO~t)4-1( t)_ ( 1(sin a). (a —) ( —2 ) (a —3 ) (a-4) (cs ) ). (2) (Cos q),Y5 5((2in) 1~ 2) 3, 4 Now suppose that p diminishes to zero, the number n must increase to infinity; then, these formulae will contain no traces of q and of n, and will contain a only. For, when ~=0i sin q, cos 5 = 1, and -_ =1; and for this value of A, we may also admit that the powers of cos p and of are equal to 1i however great the indices of the powers. Consequently, the above formulm become 2 4 6 CoSa=1-j- - ++ 4+ C. (3) 1cos 2- = M-z + l z,:1, 4 1, 2, 3, 4, 5, 6 3 5 7 s=, a.a Si 1 2,3+1 2, 3, 4, 5 l, 2, 3,A,45,~ 6 7 118 TRIGONOMETRY. As the number n has become infinite, these series do not terInimiate; but they are no less convenient to give approximate values of the sine and cosine when a is a sn;all friaction,which is the case in practice, although the formule, being convergeOnt, may be applied for any value of a. 202. By dividing series (4) by series (3), we obtainthe expansion of tang ca, the first four terms of which are a 2 a5 17c7 tang X = + -+- 3357 + &c. (5) In like manner, by dividing 1 by each of the series (3), (4), and (5), we may obtain the expansions for sec, cosec, cotang a. RESOLUTION OF BINOMIAL EQUATIONS, &c. 203. -De lfoivre's formula affords a ready means of resolving binomial equations of the form y"- —:A. Let a represent one of the n roots of A, and make y-= ax. The binomial equation becomes x —= 4- 1. Consider, in the first place, the case x"= _ 1. (1) Placing x cos X - C / — I sin p, the formula of De Moivre gives x"= cos n cp + -i sin n. Consequently, all the values of p are made known by the equation cos n - V/ —1 sin n p + 1 and will give values of x corresponding to equation (1). That the last equation nzay be satisfied, it is necessary that the imaginary factor V/-1 sin np disappear. Thi'i requires that in should be a multiple of the semi-circumference. It is also necessary that we have cos ntz 1, whicit requires that nq should be an even multiple of the semi-eirl cuimforence. Hence, designating the semi-circumference by i, and by 2k any even number whatever, we shall have 2kn n - = 2 kA, from which we get' k- Consequently ___ 21k7e x= cos ~-lsin n 21 RESOLUTION OF BINOMIAL EQUATIONS. 119 The reasoning will be the same for - -'1. Thus, we shall find all the values of x in the equation x_-= 1, by taking all the values of x comprised in the equation 2kt 2 k ye x = cos - V_ IT sin -- (M) 1s n Equation (1) has n values for x, and we know that all these values are unequal. But in formula (a) we may give to k every possible entire value, positive and negative; and we will now prove that in this way we may obtain n different values of x, from equation (a), and no more; and these values will therefore be the values of x in equation (1). In the first place, it is useless to give to k negative values: for, if we put - k for + k, the two values of formula (a) reduce to the same form exactly.'In the second place, it is unnecessary to take k = n or > n, for we may take from k the greatest multiple of n, which it 2k~ contains; this is equivalent to taking from the arc one or more circumferences, which changes neither the cosine nor the sine. Finally, if we take numbers k' and n- k' between 0 and n, equally distant from the extremes, the corresponding values of x will be the same. For, let k- n -', we shall have 2 (n - k') _ e (n- A), = COS +- v - 111 2 Wit - 2 i2t = cos -~ v —-1 sin n 2k' 2 1, = cos - = I/- sin - n 7 and these values are the same as those which correspond to Ic. It is therefore useless to give to k values greater than ~ n; and hence, whether we suppose n equal to an even number 2p, or to an odd number 2p + 1, we shall be limited in giving values to It, to the values k = 0, 1,2, 3 -— p. It remains now that we show that formula (a) gives all the values of x belonging to equation (1). If the equation is of the form xr:= 1, formula (a) becomes 11* 120 TRIGONOMETRY. x = cos - - i1 sinP P For the extreme numbers k = 0, and k =p, we find x=- +X,; and x= — 1. For the intermediate numbers 1,2,3, —p - 1, the are is comprised between 0 and 180~; then the sine which is a multiplier of V/- 1 does not become zero: hence, these values ofx are imaginary. Further, among these imagi nary values, none are repeated; for, in the two roots which form part of the same couple, that is, which result from the same value of k, the imaginary factor V-/1 has contrary signs; and in the couples which result from different values of k, the real parts are different, whereas they are the cosines of arcs which increase continuously from O to 180~. There are therefore 2p-2 different imaginary valutes, and two real values; in all, 2p values for the equation x2P=I,l -as there should be. If the equation (1) has the form x+1 1=l, formula (a); becomes 2k 2k__ x= cos 2 1/ - sin2 2pd -b s 2p-X 1 In this form, there will be only one real value for x;. x = + 1, which corresponds to k = 0 the other values are imaginary, and further, the total number of values is equal to the exponent 2p + 1. ELet us now consider the equation xn= — 1. (2) If we make x = cos ~ ~ V/ —1 sin,, we shall have x'= cos n ~ -1 sin n. Thus, the values of x will be made known, by determining the values of p from the equation cos n c pi /v-1 sin n P = - 1. But this condition requires that we shall have separatqly sin n-= 0O and cos n -=- 1. Hence, the are n q must be an odd multiple of A. Therefore, making n = (2 k + ), we decluce- ( and we shall have RESOLUTION OF BINOMIAL EQUATIONS. 121 x = cos (2 k J 1)rte 4 -sin (2 kA + 1)r Z. CSsin(+) n We need not take negative multiple values of a, for there would result the same values of x as when these multiple values are positive. Nor will be necessary to take k> n or even equal to n: for, in taking from k the multiple of n which it contains, we diminish the arc ( kJ 1) e one or more entire circumferences, which does not change the values of x in equation (i). The number kt = n -1 gives (2 n-1)rt -.2 (2 n- 1) n n cos - v /- 1 sin — = cos-V -: sin. n n 1 n These values are the same as those obtained by the hypothesiS k = 0; and, in general, the values of k equally distant from 0 and n —1 give the same values of x. Thus, if we make k = n — — 1', we have cos (2 n- 2 k'- 1): /- sin (2n -2 k'- 1)7 x Cos sin n n (2 Jr 1])r. (21'-+ 1)~ x:= cos -(2 F') - sin 7w n a result which corresponds with that which we have when we make kt- k'. Hence, it follows, that we shall obtain all the values of x by giving to k values which do not exceed - (n - 1). If n is an even number, 2p, we make k = 0, 1, 2, 3, - - - p -1; and if n is an odd number, 2p + 1, we make k = 0, 1, 2, 3 —-p. In the case in which n = 2p, the equation to be resolved is x2x=: 1, and the numbers k= 0, 1, 2, 3, -— p-1I give in the formula (P) the increasing arcs 3 are 5;r (2p - l)c 2p' 2p'' 2pp 21p which are all included between 0 and 1800. Consequently the series of no one of them is equal:to O, and their cosines are all unequal.;Each are thus gives two imaginary values 122 TRIGONOMETRY. for x which differ from each other by he sig;n of thn factor /- 1, and which therefore cannot be repeated. Henme x will have 2p different v-aues as it should have. When n = 2p + 1, the equation is x2P+-1' —I, and the values of k = 0, 1, 2 —-p, give from formula (3) the following arcs _% 3~ (2p +1)>t or A. 2p + 1' 2p +''2p+l The last are being equal to 7Y, gives x = —1. For each of the other arcs, x has two imaginary values, and it is easy to see, that, among all these values, none are repeated. Hence x has 2p +- 1 different values. 204. When we know the values of x from the equations.X= + 1, and xn -1, it is easy to form the real divisors of the second degree of the binomials xn —1, and x"+ 1. In the first place, formula (a) gives for the factors of the first degree of the binomial x-l1, the two expressions 2kA 2k, - cos v —- 1 sinn n and, multiplying them together, we have x2-2 x cos -- - 1 (a') This formula contains all the real divisors of the 2d degreE of the binomial x —l. To deduce these divisors, it is only necessary to substitute for k the positive numbers from 0 to I~n. In like manner, we find for the divisors of the 2d degree of the binomial x"+ 1, the formula (2 k + 1)72 x —2x cos + (I') in which, by giving to k all entire positive values from 0 to (n-l1), we shall find the divisors of the 2d degree. Since the formula (a) and (0) comprise the real values of x from the equations xx'= 1, and x — 1, it fbllows that the two last formulae, (a) and (3'), must also include the real CUBIC EQUATIONS. 123 factors of the 1st degree of the binomials x"-1, and x"+ 1. But it is to be observed that in these formulke, they are presented under the form of being raised to the 2d power. Thus, if k=O, in formula (a'), it becomes x2 —2 k+1, or (x_1)2. The factor required is therefore x-1. IRREDUCIBLE CASE OF CUBIC EQUATIONS. 205. De Moivre's formula furnishes a very direct and simple process of converting the irreducible case of cubic equa, tions to a finite real form. The solution of a cubic equation by the method of Cardan is as follows. Assuming a cubic equation, which has been deprived of its second term by known rule, we have x3+ 3px + 2q =0. (1) Place x equal to the sum of two other unknown quantities; that is, put x-a + b. We shall then have x3 = a3 +- b3 + 3 ab (a + b); that is, replacing a + b by x, and transposing, - 3 a b x -a3 - b3- 0; and, in order that this may be identical with the proposed equation, we must determine a and b so as to satisfy these conditions, viz.: ab= —p, a3 +b3 — 2 q. The problem is thus reduced to the determination of a and b from these two equations. From the first we have a3 b3 = —p3; hence, combining this with the second, we have the sum of two quantities, a3 + b3, and their product a3 b3 given, to determine these quantities; a problem which may be readily solved by the quadratic equation v2 + q v- - =0 (2) by making v = b3, in which the two values of v will be the I 124 TRIGONOMETRY. expressions for a3 and b3. Hence, solving the equation, and separating these two values, we have a3 = q - q2 + ~p3 (3) b3= _ -- qP + (4) and, since xz a + b, there results the following general expression for the values of x in the proposed equation, viz., x= (-q V2+p3)3 + (qVq2 +p3I ) which is Cardan's formula. Since the cube root of a3 may be represented indifferently by either of the three expressions -- + — 3 — 13/ 3 a, 2 ct a, 2 a, and the cube root of b3 by either of the expressions b, -- b, - -b 2 2 it would appear that the proposed equation admits of nine values. It must be remembered, however, that in all cases where we assign the root of any expression, no reference is made to the mode in which the given power was formed, the root being in fact the expression from which the given power has been actually produced. When we speak of a proposed power having a multiplicity of roots, we merely refer to the various expressions, from either of which that power mightbe produced; and as many of these as prove inconsistent with the conditions involved in the production of the power, are of course to be rejected. Now, one of the conditions in virtue of which a3 and b63 have been produced, is a6=-p; but of the nine products which the preceding expressions for a and b flrnish, six are imaginary; such a combination of values must therefore be rejected, as inconsistent with the conditions to be fulfilled; the other three products are possible. Hence, the only admissible solutions are the three following, where the product of a and b fulfils the above condition. CUBIC EQUATIONS. 125 a + bb, -1 /3 -1 + /2 2 2 -a + 2 b If the values of v in equation (2), as exhibited in (3) and (4), are real, the formula for x will consist of the cube roots of two real quantities; but, if imaginary, then the expression for x will be the cube roots of two imaginary quantities; and, consequently, such value must itself be imaginary; that iS, if the relation between p and q be such that 2 +p 3v, the expression for x will consist of two parts, each of which is imaginary. But the sum of these parts, that is, the complete express sions for x, must be real; because the preceding relation between p and q is that which necessitates the reality of all the values of the equation. The calculation, therefore, must furnish the means of effecting a reduction of the imaginary symbols. This difficulty, which has taxed the ingenuity of analysts, has been called the irreducible case of cubic equations, and is usually solved by developing each part separately, and then by combining the two, representing the aggregate by an infinite series, from which the imaginary symbol has disappeared. De Mobire's formula furnishes a ready means of effecting the required reduction. For this purpose the two cubic radicals must be first reduced to the form' cos -v +V /- 1 sin A. Since we have the condition q2 + p3 < v, p must be negative. Placing -p for p in the given equation, it becomes X3-3px + 2q= v. (1) Then we shall have q2 p3 D.( 64 6 3 19 26 32 38 45 5 58 i 63 6 3 19 25 32 38 44 50 57 t 62 6 12 19 25 3I 37 43 50 56 | 6r 6 12 I8 24 3i 37 43 49 55 160 6o. 6 12 18 24 3o 36 42 48 54 '8' LOGARITHMS OF NUMBERS. _N.| 0 | 1| 2 | 3 41 5 | 6 7 8 1 9 D. 720 857332 7393 7453 7513 7574 7634 7694 7755 78I5 7875 60 721 7935 7995 8o56 8ii6 8176 8236 8297 8357 8417 8477 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 723 9I38 9198 9258 9318 9379 9439 9499 9559 9619 9679 724 9739 9799 9859 9918 9978..38..98.x58.2I8.27t' 725 860338 0398 o458 o5i8 o578 o637 o697 0757 08I7 o877 726 0937 0996 I056 iii6 II76 1236 1295 I355 i4i5 I475 727 I534 I594 I654 I714 1773 I833 i893 I952 2012 2072 728 2I3I 2191 2251 23I0 2370 2430 2489 2549 2608 2668 729 2728 2787 2847 2906 2966 3025 3o85 3i44 3204 3263 730 3323 3382 3442 35oi 356i 3620 3680 3739 3799 3858 59 73I 3917 3977 40o36 4096 455 4214 4274 4333 4392 4452 732 45II 457o 4630 4689 4748 4808 4867 4926 4985 5o45 733 5 io4 5i63 5222 5282 534I 5400 5459 55I9 5578 5637 734 5696 5755 58i4 5874 5933 5992 6o05 6iio 6169 6228 735 6287 6346 64o5 6465 6524 6583 6642 6701 6760 68I9 736 6878 6937 6996 7055 7II4 7173 7232. 729I 7350 7409 737 7467 7526 7585 7644 77o3 7762 7821 7880 7939 7998 738 8056 8ii5 8174 8233 8292 8350 8409 8468 8527 8586 739 8644 8703 8762 8821 8879 8938 8997 9056 9I14 9173 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 74I 98I8 9877 9935 9994..53 1III.170.228.287.345 742 870404 o462 o521 0579 o638 o696 0755 o83 o872 og0930 58 743 0989 1047 io6 I I64 1223 128I 1339 1398 x456 i5I5 744 1573 i63 I69o0 I748 I8o6 I865 1923 1981 2040 2098 745 2I56 22I5 2273 233I 2389 2448 2506 2564 2622 268I 746 2739 2797 2855 29I3 2972 3o3o 3o88 3I46 3204 3262 747 3321 3379 3437 3495 3553 36ii 3669 3727 3785 3844 748 3902 3960 4oi8 4076 4i34 4I92 4250 4308 4366 4424 749; 4482. 4540' 4598 4656 4714 4772 4830 4888 4945 5003 750 5o6I 5II9 5I77 5235 5293 535i 5409 5466 5524 5582 751 5640 5698 5756 58i3 5871 5929 5987 6o45 6102 6i6o 752 6218 6276 6333 639I 6449 6507 6564 6622 6680 6737 753 6795 6853 69io 6968 7026 7083 7I4I 7I99 7256 73I4 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 755 7947 8oo4 8o62 8I91 8I77 8234 8292 8349 8407 8464 57 756 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 757 9096 9I53 92II 9268 9325 9383 9440 9497 9555 9612 758 9669 9726 9784 984I 9898 9956..I3..70.127.i85 759 880242 0299 0356 o4i3 o47I o528 o585 o642 0699 0756 760 o8I4 0871 0928 0985 1042 Iogg ii56 I2I3 I271 1328 76I1 i385 I442 I499 i556 i6i3 1670 1727 1784 I84i 1898 762 1955 20I2 2069 2126 2183 2240 2297 2354 2411 2468 763 2525 258I 2638 2695 2752 2809 2866 2923 |2980 3037 764 3093 3i5o 3207 3264 332I 3377 3434 349I 3548 3605 765 366i 3718 3775 3832 3888 3945 4002 4059 45 4I72 766 4229 4285 4342 4399 4455 45I2 4569 4625 4682 4739 767 4795 4852 4909 4965 5022 5078 5i35 5192 5248 53o5 768 536i 54I8 5474 553I 5587 5644 5700 5757 583 5870 769 5926 5983 6039 6096 6I52 6209 6265 632I 6378 6434 56 770 649I 6547 6604 6660 67i6 6773 6829 6885 6942 6998 77I 7054 711 7167 7223 7280 7336 7392 7449 7505 756I 772 7617 7674 7730 7786 7842 7898 7955 8ozI 8067 8123 773 8I79 8236 8292 8348 8404 846o 85i6 8573 8629 8685 774 874I 8797 8853 8909 8965 9021 9077 9134 9I90o 9246 N.t 0 2 1 3 d 4 ~ 5 6 1 7 8 I 9i D.) 6o, 6 I2 I 8 24 30o 36 42 48 54 |59 ~ |6 I2: I8 |24 30 35 4I 47 53- 1. 58 6 I2 I7 23. 29 35 4I 46 52 57 6 II 17 23 29 34 4 46 5I 1 56 p 1 6 2 II I 22 | 28 34 39 45 50 II.~~~ 17 45.~~~~ LOGARITHMS Op NUMBERS. 1' r O 1i [1 2 3 4 5 [ 51 6 7 D8.jT j 775 889302 93 58 9414 9470 9526 9582 9638 9694 975 9806 56 776 9862 99I8 9974..30..86.I4I.197.253.309.365 777 890421 I 0477 o533 o589 o645 0700 0756 0o812 o868 09o24 778 o0980 o35.11091 II47 I20o3 I259i i3i4 I370 426 1482 779 I537 I593 1649 1705 I760 I816 1872 1928 I983 2039 780 2095 2I50 2206 2262 2317 2373 2429 2484 2540 2595 78I 265I 2707 2762 2818 2873 2929 2985 3o4o 3096 3i5i 782 3207 3262 33i8 3373 3429 3484 35401 3595 365I 3706 783 3762 3817 3873 3928 3984 4039 4094 4150o 4205 426i 55 784 4316 437i 4427 4482 4538 4593 4648 4704 4759 48I4 785 4870 4925 4980 5036 509i 5i46 520i 5257 5312 5367 786 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 787 5975 6030 6o85 6I4o 6195 625I 63o6 636i 64i6 647I 788 6526 658I 6636 6692 6747 6802 6857 6912 6967 7022 789 7077 7I32 7I87 7242 7297 7352 7407 7462 7517 7572 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 81I22 791 8176 8231 8286 834I 8396 845i 8506 856i 86i5 8670 792 8725 8780 8835 8890 8944 8999 9054 9Io9 9164 92I8 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 794 982I 9875 9930 9985..39..94.I49.203.258.312 795 900367 0422 0476 o53I o586 o640 0695 0749 o804 o859 796 ogI3 og68 1022 1077 1131 ii86 1240 1295 I349 i4o4 797 i458 I5I3 1567 I622 1676 I73I 1785 i840 I894 I948 54 798 2003 2057 2112 2166 222I 2275 2329 2384 2438 2492 799 2547 260I 2655 27IO 2764 281i8 2873 2927 298i 3036 8oo 30901 344 3199 3253 3307 3361 3461 3470 3524 3578 80o 3633 3687 374i 3795 3849 | 39o4 3958 4012 4o66 4120 802 4174 4229 4283 4337 439I 4445 4499 4553 4607 466i 803 4716 4770 4824 4878 4932 4986 5o4o 5094 5148 5202 804 5256 53io 5364 54I8 5472 5526 558o 5634 5688 5742 805 5796 5E5o 5904 5958 6012 6o066 6119 66173 6227 6281 806 6335 6389 6443 6497 655i 66o4 6658 6712 6766 6820 807 6874 6927 698I 7035 7089 7143 7 7I96 7250 7304 7358 808 74II 7465 75I9 7573 7626 7680 7734 7787 7841 7895 809 7949 8002 8056 8.I io 8i63 8217 8270 8324 8378 8431 8Io 8485 8539 8592 8646 8699 8753 8807 886o 89I4 8967 8ii 902I 9074 9128 9181I 9235 9289 9342 9396 9449 9503 812 9556 961o 9663 9716 9770 9823 9877 9930 9984.~.37 53 8i3 910091 oi44 0197 025I o3o4 o358 o4xi o464 o5S8 o571 8I4 o624 o678 0731 0784 o838 o8gi o0944 o998 I o5i iio4 815 1158 1211 1264 1317 I371 1424 I477 JI53o I584 1637 816 69o0 1743 I797 I85o I903 I956 2009 2063 2116 2169 817 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 8i8 2753 2806 2859 2913 2966 3019[ 3072 3125 3178 3231 819 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 820 3814 3867 3920 3973 4026 4079 4I32 4i84 4237 4290 82I 4343 4396 4449 4502 4555 46o8 466o 4713 4766 4819 822 4872 4925 4977 5030 5o83 5i36 5I89 j5241 5294 5347 823 54oo 5453 55o5 5558 56i1 5664 57I6 5769 5822 5875 824 5927. 5980 6o33 6085 6i38 6191 6243 6296 6349 64o0 825 6454 6507 6559 6612 6664 67I7 6770 6822 6875 6927 826 6980 7033 7o85 7138 7I90 7243 7295 7348 7400 7453 827 7506 7558 7611 7663 77I6 7768 7820 7873 7925 7978 52 828 8030 8o83 8135 8I88 8240 8293 8345 8397 8450 8502 829 8555 8607 8659 8712 8764 88i6 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 9340 9392 9444 9496 9549 N. O 0 1 2 1 3 4 11 5 6 7_1 8 9 D. 55 6 II 17 |22 |28 |33 39 4 50 4 5 II i6 |22 |27 |32 38 43 49 II i6 21 27 32 37 42 48 B. 54~~~~~ 18 L)LOoGARITHMS OP NUMnERSo Hi s0 N1 1 2 13 1 4 11 5 1 6 7 - 8 9 83I 9I96oi 9653 9706 9758 9810 9862 99I4 9967..19.7 7 52 832 920123 0176 0228 0280 o332 o384 o436 o489 o54I o0593 833 o645 0o697 0749 0o80I o853 o090o6 o0958 IOIO I062 1114 834 ii66 1218 I270 I322 I374 1426 1478 i 530I 1582 I634 835 i686 I738 I790 I842 I894 x946 x998 2050 2102 21I54 836 2206 2258 23IO 2362 2414 2466 2518 2570 2622 2674 837 2725 2777 2829 288I 2933 2985 3037 3089 3I40 3I92 838 3244 3296 3348 3399 345i 3503 3555 3607 3658 3710 839 3762 38i4 3865 3917 3969 4021 4072 4I 14 4I76 4228 84o 4279 433i 4383 4434 4486 4538 4589 464I 4693 4744 84i 4796 48,48 4899 4951 5003 5o54 5o6 5157 5209 5261 842 5312 5364 54I5 5467 55i8 5570 5621 5673 5725 5776 843 5828 5879 5931 5982 6034 6085 6137 6i88 6240 629I 5i 844 6342 6394 6445 6497 6548 6600 665i 6702 6754 68o5 845 6857 6908 6959 7011 7062 7114 7165 72I6 7268 7319 846 7370 7422 7473 7524 7576 7627 7678 7730 778I 7832 847 7883 7935 7986 8037 8088 8I4o 8I9I 8242 8293 8345 848 8396 8447 8498 8549 86oi 8652 8703 8754 88o5 8857 849 8908 8959 go90 906I 9112 9I63 9215 9266 9317 9368 850 9419 9470 952I 9572 9623 9674 9725 9776 9827 9879 85I 9930 9981..32..83.34.85.236.287.338.389 852 930440 o49g 0542 o592 o643 o694 0745 0796 o847 o898 853 o949 000oo Io5I;o2 ii53 120o4 r254 i3o5 i356 1407 854 14581 1509 i56o i6io i66 I 7I2 I763 i814 i865 1915 855 1966 2017 2068 2118 2I69 2220 227I 2322 2372 2423 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 293o 857 298i 3o3i 3082 3i33 3I83 3234 3285 3335 3386 3437 858 3487 3538 3589 3639 3690 3740 379I 384i 3892 3943 859 3993 4o44 4094 4I45 4I95 4246 4296 4347 4397 4448 860 4498 4549 4599 465o 4700 4751 48o 4852 4902 4953 50o 86r 5003 5054 5Io4 5I54 5205 5255 5306 5356 5406 5457 862 5507 5558 5608 5658 5709 5759 5809 5860 59Io 5960 863 6o01 6o6I 6iii 6162 6212 6262 63i3 6363 64I3 6463 864 65i4 6564 66i4 6665 6715 6765 68i5 6865 69I6 6966 865 7016 7066 7II7 7I67 72I7 7267 73I7 7367 74I8 7468 866 7518 7568 7618 7668 7718 7769 7819 7869 79I9 7969 867 8019 9 8069 8119 8169 8219 8269 8320 8370 8420 8470 j868 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 869 9020 9070 9I20 9170 9220 9270 9320 9369 949 9469 870 9519 9569 9619 9669 9719 9769 9819 9869 9918 9968 871 9400o8 oo68 oII8 oi68 o0218 0267 o037 0367 04I7 o467 872 o056 o566 o6i6 o666 0716 0765 o8i5 o865 09o5 0964 873 loI4 io64 1114 ii63 I213 1263 I3I3 1362 I4I2 I462 874 i5II I56I i6ii i66o 17IO 1760 I80g9 859 1909 1958 875 2008 2058 2I07 257 2207 2256 2306 2355 2405 2455 876 2504 2554 2603 2653 2702 2752 2801 2851 2901 295 877 3ooo000 3049 3099 3i48 3198 3247 3297 3346 3396 3445 49 878 3495 3544 3593 3643 3692 3742 379I 384i 3890 3939 879 3989 4o38 4088 i4137 4i86 4236 4285 4335 4384 4433! 880 4483 4532 458i 463I 468o 4729 4779 4828 4877 4927 88i 4976 5025 5074 5I24 5I73 5222 5272 5321 537o 1549 882 5469 55I8 5567 56I6 5665 57i5 5764 58I3 5862 59I2 883 596i 60oio 6059 6io8 6i57 6207 6256 6305 6354 6403 884 6452 650o 655i 66oo 6649 6698 6747 6796 6845 6894 885 6943 6992 7041 7090 7140 7I89 7238 7287 7336 7385 886 7434 7483 7532 758I 7630 7679 7728 7777 7826 I 78751 N. 0 1 2 3 4 5 6 7 1 8 1 9 D. -. 52 5 |IO I6 2I 26 3I 36 42 47 5I 5 5 Io IOI 20 26 3I 36 4I 46 o 51 5 |Io i5 20 25 30 35 4o 45 ~9 A 1 5io 5 20 25 29 34 39 44 LOGARITHMS OF N'UMBERS.. - 2. 3 4 5 6 7 8 9 D. 887 947924 7973 0228070 81I9 8I68 8217 8266 83I5 8364 49 888 8413 8462 85ii 856o 8609 8657 8706 8755 88o4 8853 889 8902 8951 8999 9048 9097 9146 9I95 9244 9292 934I 890 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 891 9878 9926 9975..24.73.121.I70.219.267.3i6 892 950365 o4i4 o462 o5ii o56o o6o8 o657 0706 0754 o80 3 893 o85i ogoo 0949 0997 Io46 I og5 i43 1192 I240 1289 894,i338 I386 I435 I483 I532 I 580 629 1677 1726 1775 895 I823 1872 I920 1969 201 7 2066 2114 2I63 22II 2260 48 896 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 897 2792 2841 2889 2938 2986 3o34 3083 3I3I.3I8o 3228 898 3276 3325 3373 342I 3470 35i8 3566 36i5 3663 3711 899 3760o 3808 3856 3905 3953 400i 4049 4098 4i46 4194 goo900 4243 429I 4339 4387 4435 4484 4532 458o 4628 4677 90o 4725 4773j 482r 4869 49I8 4966 50o4 5062 5iio 5i58 902 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 903 5688 5736 5784 5832 588o 5928 5976 6024 6072 6120 o904 6i68 6216 6265 6313 636i 6409 6457 65o5 6553 66oi 905 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 906 728 7176 7224 7272 7320 7368 746 7464 7512 7559 907 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 908 8086 8i:34 8i8i 8229 8277 8325 8373 8421 8468 85i6 909 8564 86I2 8659 8707 8755 8803 885o 8898'8946 8994 910 904I 9089 9I37 i985 9232 9280 9328 9375 9423 9471 91i 95I8 9566 96I4 966I 9709 9757 98o4 9852 99~0 9947 912 9995..42..go90.38.I85.233.280.328.376.423 913 960471 o5i8 o566 o6i3 o66 0709 0I 756 o8o4 o85r 0899 914 o946 o994 Io4I I o89 1.36 ii84 I23I 1279 I326 1374 47 915 I421r 469 i5i6 i563 I6rII 658 1706 1753 I8o0r i48 916 I895 1943 Ig99 2038 2085 2132 2180 2227 2275 2322 9.17 2369 2417 2464 2511 2559 2606 2653 270I 2748 2795 918 2843 2890 2937 2985 3032 3079 3126 33174 3221 3268 9g9 33i6 3363 34io 3457 3504 3552 3599 3646 3693. 374I 920 3788 3835 3882 3929 3977 4024 407, 4118 4I65 4212 921 4260 4307 4354 44oi 4448 4495 4542 4590 4637 4684 922 4731 4778 4825 4872 49I9 4966 5013 5o6i 5io8 5I55 923 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 924 5672 57I9 5766 58i3 5860o 5907 5954 600ooi 648 6095 925 6142 6I89 6236 6283 6329 6376 6423 6470- 6517 6564 926 66ii 6658 6705 6752 6799 6845 6892 6939 6986 7033 927 7080 727 7I73 7220 7267 7314 7361 7408 7454 7501 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 80o6 8062 8109og 856 8203 8249 8296 8343 8390 8436 930 8483 853o 8576 8623 8670 87I6 8763 88o0 8856 8903 93I 8950 8996 9043 9090 9136 9I 83 9229 9276 9323 9369 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 933 9882 9928 9975..21.I 68.4.i6I.207.254.3oo 934 970347 o393 o44o o486 o533 0579 0o626 o672 0719 0765 46 935 08I2 o858 0904 095I 0997 1o44 1090 1137 I83 I229 936 1276 I322 1369 I4i5 I46Ix 508 i554 i6oi I647 I693 937 1740 1786 I832 I879 1925 197I 20I8 2064 2IIO 2I57 938 2203 2249 2295 2342 2388 2434 248r 2527 2573 26I9 939 a666 2712 2758 2804 2851 2897 2943 2989 3o35 3082 94o 3I28 3174 3220 3266 33I3 3359 34o5 345i 3497 3543 941 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 942 4o05 4097 4I43 4I89 4235 428I'4327 4374 4420 4466 943 4512 4558 46o4 4650 4696 4742 4788 J 4834 488o 4926 -N. 0 1 2 3!4 5 17 1 8 1 9 1JD S f48 f 5 Io | I4 119 1 24 29 34 38 43 A 47 m 5 9 I4 || 24 28 33 38 42 J: 46 p [ 5 |9 r4 i8 23 28 32 37 4I -. LOGARITHMS OF NUMBER3. NjO] 1 2 __ 4 45 6 9 T N. | 0 2 | 3 8 9 D 944 974972 5oi8 5664 511o 5I56 5202 5248 5294 534o 5386 46 945 5432 5478 5524 5570 56I6 5662 5707 5753 5799 5845 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 6350 6396 6442 6488 6533 6579 6625 667I 67I7 6763 948 68o8[ 685.4 6900 6946 6992 7037 7083 7129 7175 7220 949 7266] 73'2 7358 7403 7449 7495 7541 7586 7632 7678 950 7724 7769 7815 786I 7906 7952 7998 8o43 8089 8I35 951 818I 8226 8272 83I7 8363 8409 8454 85o0 8546 8591 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 953 og9093 9 138 91I84 9230 9275 932I 9366 9412 9457 9503 954 9548 9594 9639 9685 9730 9776 9821 9867 99I2 9958.955 980003/ 0o09 0094 oi4o o 085 023I 0276 0322 0367 0412 45 956 o458 o5o3 o549 o594 o64o o685 0730 0776 0821 0867 957 09I2 o957 ioo3 io48 I093 1139 ii84 I229 1275 1320 958 I366 14I I 456 I5oi 1547 1592 1637 I683 1728 1773 959 I81I9 i 864 190o9 954 2000 2045 2090 2135 218I 22261 960 227I 2316 2362 2407 2452 2497 2543 2588 2633 2678 96I 2723 2769 2814 2859 2904 2949 2994 3040 3085 3i3o 962 3I75 3220 3265 33io 3356 34oi 3446 349I 3536 358I 963 3626 367I 37I6 3762 3807 3852 3897 3942 3987 4032 964 4077 41I22 4167 4212 4257 4302 4347 4392 4437 4482 965 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 966 4977 50'22 5067 5I2 5157 5202 5247 5292 5337 5382 967 5426 547I 55i6 556i 5606 565i 5696 574i 5786 5830 968 5875 5920 5965 6oio 60 55 6o00 6i44 6189 6234 6279 969 6324:6369 64I3 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 686I 6906 695i 6996 7040 7085 7130 7I75 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 972 7666 77III 7756 7800 7845 7890 7934 7979 8024 8068 973 8ii3 8I57 8202 8247 829I 8336 838i 8425 8470 85I4 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 975 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 976 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 9895 9939 9983..28..72 7 i6i.2.250.294 978 990339 o383 0428 0472 oI56 o56i o605 o650 0694 0738 979 0783 0o827 087I 0916 0960 I004 I049 I093 1137 1182 980 1226 1270 I1315 I359 i403 I448 I492 i536 i58o I625 981 I669 1713 1758 1802 i846 1890 I935 1979 2023 2067 982 21II 2I56 2.200 2244 2288 2333 2377 242I 2465 2509 983 2554 2598 264a 2686 2730 2774 28I9 2863 2907 295I 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 985 3436 3480 3524 3568 36i3 3657 3701 3745 3789 3833 986 3877 392 I' 3965 4009 4053 4097 414I 4I85 4229 4273 987 43I7 436i 4405 4449 4493 4537 458I 4625 4669 4713 988 4757 48oi 4845 4889 4933 4977 5021 5065 5io8 5152 989 5196 5240 5284 5328 5372 54i6 5460 5504 5547 5591 990 5635 5679 5723 5767 58 II 5854 5898 5942 5986 603o 991 6074 6117 I 66 6205 6249 6293 6337 6380 6424.6468 992 6512 6555 6599 6643 6687 6731 6774 68i8 6862 6906 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 994 7386 7430 7474 7517 756I 7605 7648 7692 7736 7779 995 7823 7867 79IO 7954 7998 804i 8085 8I29 8172 8216 996 8259 8303 8347.8390 8434 8477 8521 8564 86o8 8652 997 8695 8739 8782 8826 8869 89I3 8956 9000 9043 9087 998 9131 9174 92I8 926I 9305 9348 9392 9435 9479 9522 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 N. 0 | 1 2 3 1 14 I 5 6 7 1 9 D.. ( 46 t 5 i9 |4 I8 23 28 32 37 41 45 5 9 I4 I8 23 27 32 36 41 44 la~ 4 9 13.. 44 4 4 9 | I i8 22 26 3I 35 4o 43 pl 4 9 1I3 |17 | 22 26 3o 34 39 E;, ~l~ri TAB LE OF LOGARITHMIC SINES AND TANGENTS FOn EVERY TEN SECONDS OF THE QUADRANT. 22 LOGARITIHMIC SIN E5. Min. ( Sine of 0 Degree. r. Puri 0l" 10 - 20' 30" 40" i to 1. o Inf. Neg. 5.685575 5.986605 6.162696 6.287635 6.384545 59 I 6.463726 6.530673 6.588665 639817 685575 726968 58 2 764756 799518 831703 86i666 889695 916024 57 3 940847 964328 986605 7.007794 7.027997 7 047303 56 4 7.0o65786 7.083515 7.1oo548 116939 132733 I47973 55 5 162696 176936 I90725 204089 217054 229643 54 6 241877 253776 265358 276639 287635 298358 53 7 308824 319043 329027 338787 348332 357672 52 8 3668i6 375771 384544 393145 4oI578 409850 5I 9 4I7968 425937 433762 441449 449002 456426 50o io 463726 470904 477966 484915 491754 498488 49 689.4 II 5o5ii8 511649 5i8o83 524423 530672 536832 48 629.4 I2 542906 548897 554806 560635 566387 572065 47 579.1 13 577668 583201 588664 594059 599388 604652 46 536.2 I4 609853 6I4993 620072 625093 630056 634964 45 499.2 15 639816 6446I5 649361 654056 658701 663297 44 467.0 I6 667845 672345 676799 68I208 685573 689895 43 438.7 17 694173 6984Io 702606 706762 7Io879 7I4957 42 413.6 I8 718'997 722999 726965 730896 734791 738651 4I 391.3 19 742478 746270 750031 753758 757455 761119 40 371.2 20 764754 768358 771932 775477 778994 782482 39 353.1 21 785943 789376 792782 796162 799515 802843 38 336.7 22 8o6i46 809423 812677 8I5906 819111 822292 37 321.7 23 825451 828586 831700 834791 837860 840907 36 308.o 24 843934 846939 849924 852889 855833 858757 35 295.4 25 861662 864548 867415 870262 873092 875902 34 283.8 26 878695 881470 884228 886968 889690 892396 33 273.1 27 895085 897758 900414 903054 905678 908287 32 263. 2 28 91o879 913457 9160o9 918566 921098 9236I6 31 254.o 29 926119 928608 93Io82 933543 935989 938422 30 245.4 30 940842 943248 945641 948020 950387 95274I 29 237.3 31 955082 957411 959727 962031 964322 966602 28 229.8 32 968870 971126 973370 975603 977824 980034 27 222.7 33 982233 984421 986598 988764 990919 993064 26 2:6.I 34 995I98 997322 999435 8.oo00538 8.oo363i 8.005714 25 209.8 35 8.007787 8.oo 985o 8.oIi9o o013947 01598I oI8oo5 24 203.9 36 020021 022027 024023 026011 027989 029959 23 198.3 37 03I919 033871 o358i4 037749 039675 04I592 22 I93.o 38 04350o o454oi 047294 049I78 o051054 052922 21 I88.o 39 054781 o56633 058477 o6o3i4 o62142 063963 20 I83.2 40 o65776 o67582 o6938o 071171 072955 07473I 9 1I78 7 4I 076500 07826I 0o8oo6 081764 o83504 085238 i8 I74.4 42 086965 o88684 og0398 092Io4 093804 095497 17 I70.3 43 097183 098863 100537 102204 io3864 105519 i6 i66.4 44 107I67 I08809 11o444 112074 113697 1153i5 i5 I62.6 45 II6926 118532 120o3I 12I725 1233I3 124895 I4 I59.1 46 126471 128042 129607 I3Ii66 i32720 134268 13 i55.6 47 i358I0 I37348 138879 I40406 141927 I43443 12 I52.4 48 144953 i46458 147959 149453 150943 152428 I I49.2 49 153907 155382 156852 I583I6 159776 161231 I0 146.2 50 I6268I I64I26 I65566 167002 i68433 I69859 9 I43.3 5I I71280 172697 174109 1755I7 176920 178319 8 140.5 52 1797I3 I8IIo3 182488 I83869 I85245 I86617 7 137.9 53 I87985 I89348 190707 192062 I93413 I94760 6 I35.3 54 196I02 197440 I98774 200I04 201430 202752 5 I32.8 55 204070 205384 206694 208000 209302 2o060I 4 i3o.4 56 211895 2I3I85 214472 2I5755 217034 218309 3 128.1 57 219581 220849 222113 223374 224631 225884 2 125.9 58 227134 228380 229622 230861 232096 233328 I I23.7 59 234557 235782 237003 238221 239436 240647 o0 121.6 60" 5'" 4(Y 30" 20" 10' Co-sine of 89 Degrees. r' LOGARITHMIC TANGENTS. 2 Minj Tangent of 0 Degree. P. Part I, 10" - 20" 30" 40" 50 to 1. o Inf. Neg. 5.685575 5.986605 6. 162696 6.287635 6.384545 59 I 6.463726 6.530673 6.588665 639817 685575 726968 58 2 764756 799518 831703 86i666 889695 916024 57 3 940847 964329 986605 7.007794 70o27998 7.047303 56 4 7,065786 70o83515 7.100548 1I6939 132733 147973 55 5 162696 176937 I90725 204089 2I7054 229643 54 6 241878 253777 265359 276640 287635 298359 53 7 308825 390o44 329028 338788 348333 357673 52 8 366817 37'5772 384546 393146 401579 409852 51 9 4I7970 425939 433764 44u15z 449004 456428 50o ro 463727 470906 477968 4P49I7 491756 498490 49 689.4 1I 505120 511651 5I8o85 J 24426 530675 536835 48 629.4 12 542909 548900 5548o8 560638 566390 572068 47 579.1I 13 577672 583204 588667 594062 599391 6o4655 46 536.2 14 609857 614996 620076 625097 63006o 634968 45 499.2 15 639820 6446I9 649366 654o6t 658706 663302 44 467.o i6 667849 672350 676804 681213 685578 689900oo 43 438.7 17 694I79 6984I6 702612 706768 710885 714963 42 413.6 I8 719003 723005 726972 730902 734797 738658 4i 391.3 19 742484 746277 750037 753765 757462 761127 4o 371.2.20 764761 768365 771940 775485 779002 782490 39 353.I 21 78595I 789384 792790 796I70 799524 802852 38 336.7 22 806I55 809433 812686 8I5915 89IgI20 822302 37 32I.7 23 825460 828596 83171 8348oi 837870 840918 36 3o8.o 24 843944 846950 849935 852900 855844 858769 35 295.4 25 861674 864560 867426 870274 873104 875915 34 283.9 26 878708 88I483 884240 88698I 889704 892410 33 273.2 27 895099 897772 900428 903068 905692 908301 32 263.2 28 910894 913471 916034 918581 921113 92363I 3I 254.0 29 926134 928623 931098 933559 936006 938439 30 245.4 30 940858 943265 945658 948037 950404 952758 29 237.3 31 955100 957428 959745 962049 964341 966621 28 229. 8 32 968889 971I45 973389 975622 977844 980054 27 222.7 33 982253 98444i 986618 988785 990940 993085 26 2I6.2 34 995219 997343 999457 8.ooI56o 8.oo3653 8.oo5736 25 209.8 35 8 007809 8.oo9872 8.0II926 013970 600oo4 018029 24 203 9 36 020044 022051 024048 026035 028014 o29984 23 198.3 37 031945 033897 o35840 037775 039701 o41618 22 193.0 38 043527 045428 047321 049205 051081 052949 21 i88.o 39 054809 o56662 o585o06 o60342 o62171 o63992 20 i83.3 4o o65806 067612 06941o 07120I 072985 074761 I9 I78.7 41 076531 078293 080047 081795 o83536 085270 i8 74.4 42 086997 088717 090431 092137 093837 o9553o 1 7 170.3 43 0972I7 098897 oo00571 102239 03900 o105554 J6 I66.4 44 I07203 108845 110481 112110I 113734 115352 15 I62.7 45 i 6963 118569 I20I69 I21763 I23351 124933 14 I 59. 1 46 126510 128081 129646 I31206 132760 i34308 I3 I55.7 47 13585I 137389 I3892I i40447 I41969 I43485 12 1 52.4 48 I44996 I4650o I48o00I 49497 I50987 i52472 II I49.3 49 I53952 I55426 I56896 i5836i 159821 161276 IO 146.2 50 I62727 164172 i656I3 I67049 I68480 169906 9 i43.4 5I 171328 172745 I74I58 I75566 176969 I78368 8 i4o0.6 52 179763 i8Ii53 182538 I839I9 I85296 i86668 7 I37, 53 i188036 I89400 190760 192I15 193466 194813 6 I35.3 54 I96I56 197494 I98829 200O59 201485 202808 5 132.8 55 204I26 205440 206750 208057 209359 210o658 4 x3o.4 56 2I1953 213243 2I4530 2158I4 2I7093 2I8369 3 128.1 57 219641 220909 222I74 223434 224692 225945 2 125.9 58 227195 228442 229685 230924 232I60 233392 1I23.8 5_99 234621 235846 237068 238286 239502 240713 0 121.7 60(Y' 50"- 40" 30" 1 e20" 10' _I~ ~ Co-tangent of 89 Degrees.. 2 4 LOGALITHMIC SI i: S Miu. ________ Sine of 1 Degree. P. Part Y', 10" 20" 1 30'" 40" [ 5(Y' o 8.24I855 8.243060 8.24426I 8.245459 8.246654 8.247845 59 II9.6 I 249033 250218 25I400 252578 253753 254925 58 I17 7 2 256094 257260 258423 259582 260739 26I892 57 ii5.8 3 263042 264190 265334 266475 267613 268749 56 II4.o 4 269881 2710IO 272137 273260 27438i 275499 55 II2.2 5 2766I4 277726 278835 279941 28I045 282145 54 ITo.5 6 283243 284339 28543I 286521 287608 288692 53 io8.8 7 289773 290852 291928 293002 294073 295141 52 I07.2 8 296207 297270 298330 299388 300443 301496 5I Io5.7 9 302546 303594 304639 3o568I 30672I 307759. 50 Io4.I Io 8.308794 8.309827 8.3I0857 8.3II885 8.3129Io 8.3I3933 49 I02.6 II 3i4954 315972 3I6987 3I8ooI 319012 320021 48 101.2 12 32I027 32203I 323033 324032 325029 326024 47 99'.8 I3 327016 328007 328995 329980 330964 331945 46 98.5 I4 332924 333901 334876 335848 3368I9 337787 45 97.I I5 338753 3397I7 340679 341638 342596 34355I 44 95.8 i6 3445o4 345456 346405 347352 348297 349240 43 94.6 17 350o8i 351119 352056 35299I 353924 354855 42'93.4 i8 355783 356710 357635 3585.58 359479 360398 4I 92.2 I9 36I3I5 362230 363I43 364055 364964 36587I 4o 9I.o 20 8.366777 8.36768I 8.368582 8.369482 8.370380 8.37I277 39 89.9 2I 372171 373063 373954 374843 375730 3766I5 38 88.8 22 377499.378380 379260 38oi38 38Io15 38I889 37 87.7 23 382762 383633 384502 385370 386236 387oo00 36 86.7 24 387962 388823 389682 390539 39I395 392249 35 85.6 25 393IO1 39395I 394800 395647 396493 397337 34 84.6 26 398179 399020 399859 400696 4oI532 402366 33 83.7 27 4o3199 404030 404859 405687 4065I4 407338 32 82.7 28 4o8i6i 408983 409803 4I0621 4ii438 4I2254 3I 8I.8 29 4i3068 4Z388o 4i469I 4i5500 4I6308 4I7II4 30 80.8 30 8.4I79I9 8.4I8722 8.419524 8.420325 8.42123 8.42I92I 29 80.0 3i 4227I7 4235II 424304 425096 425886 426675 28 79.I 32 427462 428248 429032 4298I5 430597 431377 27 78.2 33 432I56 432934 433710 434484 435257 436029 26 77.4 34 436800 437569 438337 439I03 439868 440632 25 76.6 35 441394 442I56 4429I5 443674 44443I 445.i86 24 75.8 36 44594I 446694 447446 448I96 448946 449694 23 75.0 37 450440 45ii86 45I930 452673 4534I4 454i54 22 74.2 38 454893 45563i 456368 457Io03 457837 458570 21 73.5 39 459301 460032 46076I 461489 4622I5 46294I 20 72.7 4o 8.463665 8.464388 8.465IIO 8.465830 8.466550 8.467268 I9 72.0 4I 467985 468701 4694I6 470129 47084I 47I553 I8 7I.3 42 472263 47297I 473679 474386 47509I 475795 I7 70.6 43 476498 477200 47790I 478601 479299 479997 I6 69.9 44 480693 48i388 482083 482776 483467 484i58 15 69.2 45 484848 485536 486224 4869Io 487596 488280 i4 68.6 46 488963 489645 490326 49100oo6 49685 492363 i3 67.9 47 493040 4937I5 494390 495064 495736 496408 12 *67.3 48 497078 497748 4984I6 499084 499750 50oo46 II 66.7 49 5oIo8o 501743 502405 503067 503727 504386 io 66. 50 8.505045 8.505702 8.506358 8. 50704 8.507668 8.50o8321 9 65 5 5i 508974 509625 5I0275 510925 5ri573 5I2221 8 64.9 52 5I2867 5i35i3 514I57 51480I 5I5444 5i6o86 7 64.3 53 5I6726 517366 5i8oo5 5i8643 519280 5I99I6 6 63.7 54' 52o55I 52186 521819 52245I 523083 5237I3 5 63.2 55 524343 5-4972 525599 526226 526852 527477 4 62.6 56 528I02 528725 529347 529969 530590 531209 3 62.I 57 53I828 532446 533063 533679 534295 534909 2 6i.6 58 535523 536i36 536747 537358 537969 538578 I 6i.i 59 539i86 539794 54o4o0 54I007 541612 542216 0 6o.5 60Y' 50' 40 1 30 1 20" 10 n' Co-sine of 88 Degrees I LOGARITH M I C TANGENTS. Min. _ Tangent of 1 Degree. P. Part 0" 10"' 20" 30"1 40/" 501 to 1. o 8.24192I 8.243I26 8.244328 8.245526 8.24672I 8.2479I3 59 II9.7 I 249I02 250287 251469 252648 253823 254996 58 II7.7 2 256165 25733I 258494 259654 2608II 261965 57 II5.8 3 263II5 264263 265408 266549 267688 268824 56 II4.0 4 269956 27I086 272213 273337 274458 275576 55 112.2 5 27669I 277804 278913 280020 281124 282225 54 izo.5 6 283323 284419 285512 286602 287689 288774 53 108.9 7 289856 290935 29201I2 293086 294157 295226 52 107.2 8 296292 297355 298416 299474 300530 3oi583 5i 105.7 9 302634 303682 304727 305770 3o68ii 307849 5o 104.2 1o 8.3o8884 8.3099I7 8.310948 8.3II976 8.3I3002 8.314025 49 102.7 1I 3i5o46 3i6o65 317081 3I8095 3I9I06 320115 48 ioi.3 I2 321122 322I27 323129 324129 325126 326I21 47 99.8 I3 327114 328105 329093 330080 33io64 332045 46 98.5 I4 333025 334002 334977 335950 336921 337890 45 97.2 I5 338856 339821 340783 34I743 342701 343657 44 95.9 i6 3446io 345562 346512 347459 3484o5 349348 43 94.6 17 350289 351229 352I66 3531io 354035 354966 42 93.4 i8 355895 356823 357748 35867I 359593 360512 41 92.2 I9 36i430 362345 363259 364I7I 365o8I 365988 4o 9r.1 20 8.366895 8.367799 8.368701 8.369601 8.370500 8.37I397 39 89.9 21 372292 373I85 374076 374965 375853 376738 38 88.8 22 377622 378504 379385 380263 381i4o 382015 37 87.8 23 382889 383760 384630 385498 386364 387229 36 86.7 24 388092 388953 389813 390670 39I526 392381 35 85.7 25 393234 394085 394934 395782 396628 397472 34 84. 7 26 3983I5 399I56 399996 4oo834 40I670 402505 33 83.7 7 40o3338 404170 4o5ooo 405828 406655 407480 32 82.8 2S 4o83o4 40926 40o9946 410765 4ii583 412399 3i 8x.8 29 4132I3 414026 414837 415647 4i6456 417263 30 80.9 30 8.4I8o68 8.4I8872 8.4I9674 8.420475 8.421274 8.422072 29 80.o 31 422869 423664 424458 425250 42604i 426830 28 79. I 32 4276I8 428404 429I89 429973 430755 43I536 27 78.3 33 432315 433093 433870 434645 435419 436191 26 77.5 3A 436962 437732 4385oo 439267 44oo33 440797 25 76.6?5 44i56o 442322 443082 44384i 444599 445355 24 75.8 36 446iio 446864 4476I6 448368 449117 449866 23 75.0 37 45o6i3 451359 452104 452847 453589 454330 22 74.3 38 455070 455808 456545 457281 4580o6 458749 21 73.5 39 459481 460212 460942 461670 462398 463124 20 72.8 4o 8.463849 8.464572 8.465295 8.466o06 8.466736 8.467455 19 72.I 4' 468172 468889 469604 4703I8 47I03I 471743 i8 7I.3 42 472454 473I63 473872 474579 475285 47599~ 17 70.7 43 476693 477396 478097 478798 479497 480I95 i6 70.0 44 48o892 48i588 482283 482976 483669 48436o i5 69.3 45 485o5o 48574o 486428 487115 487801 488486 i4 68.6 46 489170 489852 490534 4912I5 49,894 492573 I3 68.o 47 493250 493927 494602 495276 495949 496622 12 67.4 48 497293 497963 498632 499300 499967 500633 II 66.8 49 50I298 50o962 502625 503287 503948 504608 io 66.I 50 8.505267 8.5o5925 8.506582 8.507238 8.507893 8.5o8547 9 65.5 5i 509200 509852 5o050o3 5iii53 5I8o02 5I245I 8 65.o 52 5I3098 5i3744.54389 5r5034 5I5677 5I6320 7 64.4 53 5i696i 517602 518241 5I888o 5I95I8 520154 6 63.8 54 520790 521425 522059 522692 523324 523956 5 63.3 55 524586 525215 525844 526472 527098 527724 4 62.7 56 528349 528973 529596 5302I8 530840 53x46o 3 62.2 57 532080 532698 5333i6 533933 534549 53564 2 6I.6 58 535779 536392 537005 5376I7 538227 538837 I 6. 59 539447 540055 540662 541i69 54i875 542480 o 60.6 601Y 50W 40" 30" 20" 10", in.Co-tangent o 88 Degre Co-tangent of 88 Degrees. 26 LOGARITHM.C SI-NES. Sine of 2 Degrees. Sine of 3 Degrees. _ 10 1, 20" [-30" 40t 50" o, 0 " 10", 20o' 30-" 40"1 50'" o 8.5428I9 3422 40 23 4624 5224 5823 59 0 8.7I8800o9202 9603..4.404.804 59 1 6422 70o9 761618212 8807 940I 58 I 8.721204 I603 2002 2401 2799 3I97 58 2 9995.587 II79 I770 236I 2950 57 2 3595 3992 4389 4785 518 I 5577 57 3 8 553539 4I26 471353o00 5885 6470 56 3 5972 6367 6 762 7I56 7550 7943 56 4 7054 7637 82I9 880 I 938I 996I 55 4 8337 8729 9I22 9514 9906.297 55 5 8.560540o II9 I696 2273 2849 3425 54 5 8.730688 I079 I469 I859 2249 2638 54 6 3999 4573 5I46 57I9 6290 686I 53 6 3027 34I6 3804 4192 4579 4967 53 7 743I 8000 8569 937 9704.270 52 7 5354 5740o6126 65I2 6898 7283 52 8 8.570836 I4r)I I965 2528 30 9I 3653 5i 8 7667 8052 8436 8820 9203 9586 5i 9 42,4 4774 5334 5893 645I 7009 50 9 9969.352.734 I1I5 1497 1878 50 Io 7566 8122 8678 9232 9786.340o49 I0 8.742259 2639 30I9 3399 3778 4157 49 II8.58o892 I444 i995 2546 3096 3645 48 II 4536 49,4 5293 5670 6048 6425 48 12 4I93 474I 5288 5834 6380o 6925 47 12 6802 7I78 7554 7930 8305 8680 47 13 7469 80I3 8556 9098 9640.181 46 I3 90o55 9430o980o4.78.55I.924 46 I4 8.590721 I260 I799 2338 2875 3412 45 14 8.75I297 I670 2042 2414 2786 357 45 x5 3948 4484 50I9 5553 6087 66I9 44 i5 3528 3898 4269 4639 5008 5378 44 I6 7I52 7683 82I4 8745 9274 9803 43 6 5747 616 6484 6852 7220 7587 43 17 8.600332 o859 1387 I9I3 2439 2964 42 I7 7955 832I 8688 9054 9420 9786 42 I8 3489 40I2 453655858 5580 60241 I88.760I5I0516088I I2451609 I973 41 19 6623 7I43 7662 81 8i 8699 9217 40 1I9 2337 2700 3063 3425 3787 4149 4o 20 9734.25I.766 1282 I796 23IO 39 20 4511 4872 5234 55945955 63I5 39 2I 8.6I2823 3336 3848 436o 487I 538i 38 2I 6675 7034 7394 7752 8III 8469 38 22 5891 6400 6909 7417 7924 843I 37 22 8828 9 85 9543 9900. 257 6 I 3 37 23 8937 9442 9947.452.956 I459 36 23 8.770970 I 326 i68I 2037 2392 2747 36 24 8.621962 2464 2965 3466 3966 4466 35 24 3o0i 3456 3810 4I63 45I7 4870 35 25 4965 5464 5962 6459 6956 7453 34 25 5223 5575 5927 6279 663i 6982 34 26 7948 8444 8938 9432 9926.4I9 33 26 7333 7684 8035 8385 8735 9085 33 27 8.6309II I403 I894 2385 2875 3365 32 27 9434 9783. 32.480.829 1II77 32 28 3854 4342 4830 53I7 5804 629I 3I 28 8.78I5241 872 2219 2566 29I2 325931I 29 6776 7262 7746 8230 87I4 9I97 30 29 3605 395I 4296 464i 4986 533i 30 30 9680.I62.643 II24 I604 2084 29 3o 5675 60I9 6363 6707 7050 7393129 3I 8.642563 3042 3520o3998 4475 4952 28 31 7736 8078 8421 8762 9I0 4 9446|28 32 5428 5904 6379 6854 7328 780I 27 32 9787.128.468.808 II49 I488 27 33 8274 8747 92I9 9690.I6I.632 26 33 8.79828 2I67 25o6 2845 3I83 352I 26 34 8.65IIo 2 1571 2040|2508 2976 3444 25 34 3859 4I97 4534 4872 55208 5545 25 35 3911 4377 4843 53o8 5773 6238 24 35 588i 6218 6553 6889 7224 7559.24 36 6702 7I65 7628 8090 8552 9o04 23 36 7894 8229 8563 8897 9231 9564 23 37 9475 9935.395.855 I3I4 I772 22 37 9897.230.563..896 I228 i560 22 38|8.662230o268831453602 4058 451321 38 8.80I892 2223 2554 2885 32I6 3546|2I 39 4968 5423 5877 633i 6784 7237 20 39 3876 4206 4536 4866 5195 5524 20 40 7689 8I41 8592 9943 9494 9944'9 40o 58526i8i 69 6837 765 7492 19 4i 8.670393 0842 I291 I739 2I87 2634 I8 4, 78I9 8I46 8473 8799 9I26 9451 i8 42 3080 3527 3972 44I8 4863 53071I7 42 9777.Io3.428.753 I078 I40217 43 575i 16I94 6638 7080 7522 7964 i6 43 8.811726 2050 2374 2698 302I 3344 i6 44 84o58846 9286 9726. i66.6o5 I5 44 3667 3989 4312 4634 4956 5277 i5 45 8.68io43 I48i I919 2356 2793 3230 I4 45 5599 5920 624I 656i 6882 7202 I4 46 3665 4Ioi 4536 497 54o0 5 5838 13 46 7522 784I 81618480 8799 91813| 47 6272 6705 7I37 7569 8ooI 8432 I2 47 9436 9755.73.390.708 1025 I2 48. 8863 9293 9723.I52.58I 10IO II 48|8.82I343 I659 1976 2292 2609 2925 II 49 8.69I438 i866 2293 2720 3I46 3572 Io 49 3240 3556 3871 4I86 45o, 48i6 I O 50 3998 4423 4848 5272 5696 6120 9 50 5I30 5444 5758 6072 6385 6698 9 5S 6543 6966 7388 78io 8232 8653 8 5I 70II 7324 7637 7949 826I 8573 8 62 907319494 99I3.333.752 II71 7 52 8884 9I96|9507|98I8.129.439 7 53 8.70I589 2007 2424 284I 3258 3674 6 1538.830749 io6o 0369 I679 1988 2298 6 54 4090 4505 492o 5335 5749 6I63 5 54 2607 29I5 3224 3532 3840 4i48 5 55 657716990 7402 78i5 8226 8638 4 55 4456 4763 5070o5377 5684 599i 4 56 904919460 9870.280.690 Io99 3 56 6297 66o036909 7215 7520 7825 3 5798.71I5071I9i6 2324 273I 3I39 3546 2 57 8I30 8435 8740o90 44 9348 9652 2 58 3952 4358 4764 5I69 5574 5979 I 58 9956.260o.563.866 II69 i472 I 59 638316787 7190 7593 7996 8398 0 598.841774 2076 23782680 29823283 0 60'1 50" 40"' 301' 20" 10" d0 50"1 40" 30" 20"' 10 1 Co-sine of 87 Degrees. - Co-sine of 86 Degrees. Pt 1" 2"' 3 4 5" 6 7" 8" 911 PP tS 1" 2 3" 4" 5" 6 7" 8" 9"/ P.Part a48 96 145 193 241 289 338 386 434.P art 34 69 103 138 172 207'241 275 310 LO G A I T HMC TA N G E N T 2 7 Tangent of 2 Degrees. Tangent of 3 Degrees. o" 10" 0"' 301 40 50o"' 0"0 10" 20' 30" 1 4 o 8.543o84 3687 4289 489I 5492 6092 59 0 8. /9396 9798.20I.603 I0o4 io5 59 I 669I 7289 7887 8483 9079 9674 58 I 8.721806 2207 2607 3007 34o6 3805 58 2 8 550268 o862 z454 2046 2637 3227 57 2 4204 4602 5ooo 5397 5794 619'57 3 38 74405 4993 558o 6I66 6752 56 3 6588 6984 7380 7775 8I7o 8565 56 41 7336 7920 8503 9085 9667.248 55 4 8959 9353 9746.I40.533.925 55 5 8.560828 14o07 985 2563 340o 37I6 54 5 8.731317 I707 2IOI 2492 2883 3273 54 6 429z 4866 544o 6oi3 6585 7157 53 6 3663 4o53 4442 483i 5220 56oS 53 7 7727 8298 8867 9435...3.570 52 7 5996 6384 6771 7158 7545 793 I 52 8 8.57 I 37 1702 2267 2832 3395 3958 5 I 8 8317 8703 9088 9473 9858.242 5I 9 4520 508I 5642 6201 6760 73I950 98.7406260I oo9 I 393 1776 2 I 581254050 I 0 7877 8434 8990 9545. I00.654 49 Io 2922 33o4 3685 4o66 4447 4827 49 I I. 581208 I760 2312 2864 34I4 3964 48 II 5207 5586 5966 6344 b723 17IoI 48 I2 4514 5062 5610 6I57 67o4 7249 47 12 7479 7857 8234 861.l 8988 9364 47 I3 7795 8339 8883 9426 9968.5io 46 I3 9740. I I6.49I].866 I24, I 6I 5 46 14 8. 59 I05 1i5I 592 31 2670 3208 3746 45 14 8. 75I989 2363 2736 3 og 3482 3855 45 i5 4283 4820 5355 5890 6425 6959 44 i5 4227 4599 4970 534I 57I2 6o83 44 i6 7492 8024 8556 9087 96I8. I47 43 I6 6453 6823 7I93 7562 7931 83oo 43 I7 8.600677 I205 1733 2260 2787 33I3 42 I7 8668 9036 94041977I. I 39.505 42 i8 3839 4363 4887 54i 5934 6456 4 I 8 8.760872 I1238 I 6041 970 2335 27oo04I 9 6978 7499 80o9 8539 9o58 9576 40 I9 3o65 3429 3793 1457 4520 4884 40 20 8.6oo094 06I2 I I28 I644 2I60 2675 39 20 5246 5609 5971 6333 6695 7056j39 21 3I89 3702 42I5 4728 5240 575T 38 21 7417 7778 8I39 8499 8859 9281 38 22 6262 6772 7281 779o 8298 880 6 37 22 9578 99371.295.654 I012 i37o037 23 93I3 98I9.325.83o 1335 I839 36 23 8.771727 2085 2442 2798 3I55 351 |36 24 8.622343 2846 3348 3850 435I 4852 35 24 3866 4222 45774932 5287 564i 135 25 5352 585i 635o 6849 7346 7844 34 25 5995 6349 6702 7056 74o9 776I 34 26 8340 8836 9332 9827.321.815 33 26 8II4 8466|88I7|9169 9520 9871 33 27 8.63I308 I80 I 2293 2785 3276 3766 32 27 8.780222 o572 0922 1272 I622 I97I132 28 4256 4746 5235 5723 62I1 6698 31 28 2320 2669 30I7 3365 37I3 4o6i 3I 29 7184 767I 8I56 864, 9126 96io130 29 4408 4755|5I02|5448 5794 6i40o30 30|8.640093 o 576 oo58 8 540 2021 2502|29 30 6486 683I 7I77 7521 7866 82I0o29 31 2982 3462 394I 4420 4898 5376 28 3I 8554 8898 9242 9585 9928.271 28 32 5853 6329 6805 7281 7756 8230 27 32 8.7906i3 0955 1297 1639 I980 2321 27 33 8704 9I78 965I.123.595 Io67 26 33 2662 3oo3 3343 3683 4023 4362 26 34|8.65I537 2008 2478 2947 34I6 3884 25 34 4701 5040o5379 57I8 6056 6394 25 35 4352 4820 5286 5753 62ig 6684 24 35 673I 7069 7406 7743 8079 84i6 24 36 7149 76I3 8077 854I 9004 9466 23 36 8752 9088 9423 9759.. 94.429 23 37 9928.389.85o I 3iI I 77I 2230 22 37 8.800763 1098 1432|I 765 2099 2432 22 38 8.662689 3I48 36o6 4o63 4520 4977 2i 38 2765 3098 3431 3763 4o95 4427 21 39 5433 5889 6344 6799 7253 7707 2o 39 4758 50o90542I 575I 6082 6412 20 4o 8I60 86I3 9065 95I7 9968.4I 919 40 6742170721740217731 [80o6083891I9 4i 8.670870 I320 I 769 22I8 2667 3iI5 18 4i 8717 9046 9374 970I.. 29.356 I 8 42 3563 4010 4457 4903 5349 5794 I7 42 8.88o683 IOIo|I337|I663 I989 2315|I7 43 6239 6684 7128 7572 8o05 8457 i6 43 2641 2966 3291 36I6 394I 4265 i6 44 8900oo 934I 9783.224.664 IIo4` I5 44 4589 4913 5237 556o 5884 620o/ i5 4518.68i 544 I983 2422 2860 3298 37351I4 45 6529 6852a7I74J7496 78I8 81401I4 46 4172 4608 5044 5480o59I 5635o 3 46 846I 8782 9 o3 94239744..641 3 47 6784 72I8 7652 8o85 85I78950o 12 47 8.820384 0703 I023 I342 i66I 1980oI2 48 938I 98I3.244.674 iio4 i534 I| 48 2298 2617 2935 3253 357o 3888 II 49|8.69I963 2392 282o 3248 3675|4IO3|IO 49 4205 4522 4838 5I55 547I157871IO 5o 4529 4956 538 5880 7 6'2326656 9 50 6I0o3 64I8 6733 70497363 7678 9 51 708I 704 7928 835i 8773 9195 8 75 7992 8307 862I18934 9248 956I 8 52 96I7.. 38.459.880o 3oo 1720 7 52 9874.I87.500oo.82 I1124 436 7 53 8.70o2 39 2558 2976 3395 38I 24230 6 53 8.83I748 2059 2371 2682 2992 3303| 6 54 4646 5o63 5479 5895 63io 6725 5 54 36i3 3924`4234`4543 4853 5i62 5 55 7I40 7554 7967 838i 8794 9206 4 55 547i j578o 6089 6397 67o5 70I3 4 56 96I8..30o.442.853 I263 674 3 56 732I17629 7936 8243 855o 8857j 3 57 8.712083 2493 2902e 331I 371914I27 2 6 57 91i63 9470 9776l..8I.387.692 2 58 4,534 4942 5348 5755 6S61 65667 I 58|8.8409981I303I 607oI9721222I6252 I 59 6972 7377 7781 8i86 8589|8993 o 59 2825 3I28 3432 3735 4o38 434I o 60"' 50" 40" 30" 20 1' 6" 1( 50"1 40"1 30"1 2 10 Co-tangent of 87 Degrees. Co-tangent of 86 Degrees. P. Part 1// 2" 3"// 4" 5" 6"// 7" 8"'" P Part 1" 2" 3" 4" 5" 6" 7 S'/ 9" 48 97 145 193 242 290 338 387 435 35 69 104 Ij 173 207 242 276 311 LOGARITHMIC SINE S Sine of 4 Degrees.. I Sine of 5 Degrees. 0' 10" 20" 30"' 40"' 50"' O" 10' 20' 30" 40' 5(Y0 o18.843585 3886 4I8614487 4787|5087 59 o 8.940296 0537 0777 IOI7 I258 /498 59 I 5387 5687 5987 6286 6585 6884 58 I I 731 1977 22, 7 2457 2696 2935 58 2 7I83 748I 7780 8078 8376 8673 571 2 3 I74 34i3 3652 3891 4I29 4368 57 3 897I 9268 9565 9862. 59.455 56 3 46o6 4844 5c3 532I 5558 5796 56 4 8.85075I I047 i343 I639 I934 2229 55 4 6034 627I 6508 6745 6982 72I9 55 5 2525 28I9 3 II4 34o8 3703 3997 54 5 4 7456 7693 7929 8I66 8402 8638 54 6 4291 4584 4878 5I7I 5464 5757 53 6 8874 91IO 99345 958I 98I7..52 53?17 604g 6342 6634 6926 7218 7510 52 7 8.950287 0522 0757 0992 I227 i46I 52 8 7801 8092 8383 8674 8965 9255 5i 8 I696 1930 2I64 2398 2632 2866 5I 91 9546 9836.I26.4I5.705 994 50 9 3I00 3333 3567 3800 4033 4266 50 IO 8.861283 1572 i86I 2I149 2438 2726 49 IO 4499 4732 4965 5197 5429 5662 49 II 30143302 3589 3877 4I64 445I 48 II 5894 6I26 6358 6590 682I 7053 48 I2 4738 5024 53II 5597 5883 6I69 47 I2 7284 75I6 7747 7978 8209 8440 47 13 6455 6740 7025 7310 7595 788o046 i3 8670o890I 913I 9362 959219822 46 I4 8I65 8449 8733 9o071930I 9585 45 I14 8.960052 02.82 o5 I 074I10970oI200 45 15 9868 1.I5.434.717 IOOO I282 44 i5 I429 i658 1887 21I6 2344 2573144 I6j8.87I565 I847 2I29 2410 2692 2973143 i6 280I 3030 3258 3486 3714 3942 43 17 3255 3536138I7 409714378J4658[ 42 I 7 4I70 43974625 4852 5080o5307J42 i8 4938 5218 5498 5777 6057 6336 4I i8 5534 576I 5987 6214 644I 6667 4I 19 66I5 6894 7I72 745I 7729 8007 4o I9 6893 7120 7346 7572 7797 8023 4o 20 8285 8563 884i 9118 9395 9672 39 20 8249 8474 8700 8925 9I50 9375 39 21 9949.226.5o3.779 I0o55 33I 38 21 9600o9825..49. 274.498. 723 38 22 8.881607 i883 2I58 2433 2708 2983 37 22 8.970947I 7 I I 395 I 6 19 842 2066 37 23 3258 3533 3807 408I 4355 4629 36 23 2289 25I3 2736 2959 3I82 34o5 36 24 4903 5I77 545o 5723 5996 6269 35 241 3628 385i 4073 4296 45I8 4740o35 25, 6542 68I4 7087 7359 763I 7903 34 25 4962 5i84 54o6 5628 585o 607I 34 26 8I74 8446|8717[8988 9259 9530 33 26 629316514 6735 6956 7177 7398 33 27 980I o..7I.34I.612.882 II5I 32 27 76I9 7839 8060 8280 85o0I872I 32 28 8.89I42I I690o 96o02229 2498 2767 3I 28 894I 9161 938I 9600 9820...39 3 29 3035 33o043572 384014I08 4376 30 2918.980259 o478 o697 o 096 II35 I354 30 30, 4643149I1 157815445 5712 5979 29 3o0 I573 79I 20IO 2228 2447 2665 29 3I| 6246 6512 6778 7044 7310 7576 28 3I 2883 3ioi 33I9 3536 3754 3972 28 32 7842[81071837318638 890319168 27 32 4I89 4406 4623 4840o 50575274 27 33 9432 9697 996I.225.489.753 26 33 549I 5708 5924 6I4I 6357 6573 26 3418.90IOI7 I280 i544 1807 2070 2333 25 34 6789 7005 7221 7437 7652 7868 25 351 2596 2858 312I13383 3645 3907 24 35 8o83 8299 85I4 8729 8944 9159 24 361 4I6914430o4692|4953 5214 5475123 36 9374 9588 9803 1.. 7.232.446 23 37 5736 5997 6257165I7 6778 7038 22 37 8.990660 o0874 I0o88 13o2 i5i6 729 22 38 7297 7557178I7 8076 8335 8595 2I 38 1943 2156 2370 2583 2796 3009 2I1 39 8853 91II2|937I|9629 9888 *I46 20 39 3222 3435 3647 3860 4072 4285 20 40 8.9Io04040662 09191II77 I4341I692 I9 4o 4497 4709 492I 5I33 5345 55561 9 4i I949 2206 2462 27I9 2976 3232 i8 4I 5768 5980 6I9I 6402 66I4 6825 i8 42, 3488 3744|4000o4256 45I I 47671x7 42 70367247 7457 7668 7879 80891I7 43 5022 527715532 578716041 6296 i6 43 8299185IO 8720 89309 49350 i6 44 655o 68o057059 73I3 7566 782o0 5 44 9560 9769 9979.i88.398.607 i5 451 807318327|858o0883319086 9338 I4 459.ooo008I6 I025 I234 I443 652 I86o 14 46 959I 9843..96.348.6oo.852 I3 46 2069 2277 2486 2694 2902 3I Io 1 47|8.92IIo3 I3551I6o6 I858 2I09 2360 I2 47 33I8 3526 3733 394I 4I49 4356 12 48 26I 1286I 13 I 1213362 36 I 2 38621I I 48 4563 4771 4978 5I85 5392 5599 I 49 4II2|4362|46I21486I 5III 536o IO 49 5805 60I2162I8 6425 663I16837 IO 50o 56091 58586I71 6355 66o4 6852 9 50o 7044 7250 7456 766I 7867 8073 9 5 700oo 7348 7596 7844 8092 83398 8 Si 8278 8484J8689 8894 10oo 093o5 8 52 858718834|9081 9328 9575 9821 7 52 951097I5 99I1. I24.329.533 7 53 8.930068 o3I4o056o/o8o61Io 52 1298 6 53|9.oio737 o942 1 i46 I35o I554 1758 6 54 x544 I789 2035 2280 2525 277o 5 54 I962 2I65 2369 2572 2776 2979 5 55 3o0I5 32603504 3749 3993 4237 4 55 3I82 3385 3588 379I 3994 4197 4 56 448I14725|4969152I215456 5699 3 56 440014602 4805 5007 52091541 I 3 57 5942 6I85 6428 667I 69I4 7156 2 57 56I3 58I5 60I7 6291 642I 6622 2 58 7398 764I 7883 8I25 8366 86o8 I 58 6824 7025 7227 7428 7629 783o I 159 885o 9091 93321957398T41..551 oi 59 803 8232 8433 8633 8834 93434 o 60" 50" 1 40 1 30"' 1 20" 10" 1 60" 50"1 40" 30" 2(Y 10" Co-sine of 85 Degrees. t Co-sine of 84 Degrees. IP. Part27' 53 80 107 134 16187 "14 2841 PPat22" 3/ 45" 6" 87109 7 51 81" 97 27 53 80 107 134 16M 187 214 241 22 44 66 87 109 131 153 175 197 LOGARITHMIC TANGENTS 2 - Tangent of 4 Degrees. l Tangent of 5 Degrees. 0"_ 10"N 20"- 30" 40" 50" O 0' 10" 20" 30", 40tT 50" o 8.844644 4946 5248 555I 5852|6I54 59 o 8.94I952 2194 2437 2679 292I 3I63 59 I 6455 6757 7o58 7358 76591 7959 58 I 34o4 3646 3888 4I29 4370 46 I 58 2 8260 8560o8859 9I59 9458 9758 57 2 4852 5093 5334 5574 58I5 6o55 57 3 8.850057 o355 o654 o09 52 I 25 i548 56 3 6295 6535 6775 7oI5 7255 7494 56 4 I846 2144 244I 2738 3o35 3332 55 4 7734 7973 8212 845I 8690 8929 55 5 3628 3925 422I 45I7 483 5Io8 54 5 9168 9406 9644 9883. 21.359 54 6 5403 5699 5993 6288 6583 6877 53 6 8.950597 o834 I072 309 I 1547 I784 53 7 7I7I 7465 7759 8o53 8346 8639 52 7 202I 2258 2495 2732 2968 3205 52 8 8932 9225 9517 98Io.102.394 5I 8 344I 3677 39I3 4149 4385 462I S i 9 8.86o686 0977 I269 I56o i85I 2I42 50 9 4856 5092 532, 5562 5797 6032150 10 2433 2723 3o0I3 3303 3593 3883 49 Io 6267 6502 6736 697I 7205 743914i II 4173 4462 475 5040 o 5329 56I7 48 I 7674 7908 8I4I 8375 8609 8842i48 12 5906 6I94 6482.67697057 7344 47 I2 9075 9309 9542 9775... 8.240o47 I3 7632 791 98206 84928779 90 6546 I 3 8.960473 0705 0938 I I 70 402 I 63446'4 935i 9637 9923.208.494.779 45 14 i866 2098 2329 2561 2792 3023 45 i5 8.87Io64 1349 i633 1918 2202 2486 44 I5 3255 3486 3716 3947 4178 4408 44 i6 2770 36543337 3620 3904 4187 43 i6 4639 4869 5099 5329 5559 5789 43 17 4469 4752 5034 5317 5599 588I 42 I7 60I9 6248 6478 6707 6936 7165 42 I8 6I62 6444 6725 7006 7287 7568 4I I8 7394 7623 7852 8o8I 8309 8538 4: I9 7849 8129 8409 8689 8969 9249 4o 1I9 8766 8994 9222 9450 9678 9905 4o 20 9529 9808.. 87. 366.645.924 39 20 8.970o33 o36o o588 08 51 042 I1269 39 2I 8.88I202 1480 I759 2037 23 4 2592 38 21 I496 I723 1949 2I76 2402 2628 38 22 2869 3I47 3424 370I 3977 4254 37 22 2855 3o8i 3307 3532 3758 3984 37 23 4530 4807 5083 5358 5634 59o 036 23 4209 4435 4660 4885 5 I I o 5335 36 24 6I85 646o 6735 70 oI7285 7559 35 24 5560 5784 6009 6233 6458 6682 35 25 7833 8xo8 8382 8655 8929 9202 34 25 6906 7I30 7354 7578 780o 8025 34 26 9476 9749..22.295.567.84o 33 26 8248 8472 8695 8918 9I4I 9364 33 27 8 89 I I I 2 I 384 I 656 928 2 I 99 47 I 32 27 9586 9809..32.254.476.699 32 28 2742 3o0I3 3284 3555 3825 4096 3i 28 8.98092I xII43 364 i586 80o8 2029 31 29 4366 4636 4906 5I76 5445 57I5 30 29 225I 2472 2693 2914 3I35 3356 30 3a 5984 6253 6522 679I 7060 7328 29 3o 3577 3798 4o0 8 4238 4459 4679 29 3i 7596 7864 8 3Ia 8400 8668 8935 28 3i 4899 5iI9 5339 5559 5778 5998 28 32 920o3 9470 9737..4.270.537 27 32 62I7 6437 6656 6875 7094 73I3 27 33 8.9oo803 1o69g 335 i6oi I867 2I32 26 33 7532 7750 7969 8I87 84o6 86o4 26 34 2398 2663 2928 3I93 3458 3722 25 34 8842 9060 9278 9496 97I4 993 I 25 35 3987 425 45 5 4779 5o43 53o6 24 35 8.990o49 o366 o583 o8oI Io8 81 235 24 36 5570 5833 6096 6359 6622 6885 23 36 I45i i668 i885 21o I 23I8 2534 23 37 7147 74IO 7672 7934 8 96 8457 22 37 2750 2966 3I82 3398 36I4 3830 22 38 87I9 8980 9242 9503 9764..25 2I 38 4o45 426I 4476 4692 4907 5122 21 39 8.9Io285 o546 o806 Io66 1326 I586 20 39 5337 5552 5766 598I 6I96 64oO 20 40o 846 2Io6 2365 2624 2883 3142 I9 40 6624 6839 7053 7267 7481 7694 I9 4I 34o0I 3660 39I8 477 4435 4693 i8 4I 7908 8I22 8335 8549 8762 8975 i8 42 495I 5209 5466 5724 598I 6238 I7 42 9188 940I 9614 9827 ~.4o.252 17 43 6495 6752 7009 7265 7522 7778 i6 439.000465 0o677 0889I 02 I 34 I 526 I 6 44 8034 8290o 8546 880I 9057|93I2 I5 44 I738 949 2I6I12373 2584 27951I5 45 9568 9823.. 78.332.587.84I 4 45 3007 3218 3429 3640 385 4o6I 1i4 46 8.92Io96 I35o I6o4 I858 2112'2365 I3 46 4272 4483 4693 4904 5 I 14 5324 13 47 26I9 2872 3I25 3378 363i 3884 I2 47 5534 5744 595A 6i64 6373 6583 12 48 4I36 4389 464i 489315145 5397 11 48 6792 7002 721 7420 7629 78381 i 49 5649 5900 6I52 64o3 6654 6905 IO 1 49 8047 8256 8465 8673 8882 90901 I 50 7i5617477657 79~08 8 58884o8 9 50 9298 950797I 5 9923. I31.338 9 5I 8658 8908 9I58 9407 9657 9906 8 5I9.oio546 o754 o96I II69 I376 I583 8 52 8.93~I55 o4o04o653 0902 rI50 i399 7 52 1790 I997 2204 2411 26I8 2824 7 53J I647 I895 2I43 239I 2639 2887 6 53 3o3I 3237 3444 365o 3856 4o62 6 54 3134 338i 3629 3876:4I23 4369 5 54 4268 44744680 4886 509i 5297 5 55 46I6 4862|5Io9 5355560Io5847 4 55 55o0257o7 5913 6II8 6323 6528 4 56 6093 6339 6584 6830 7075 7320 3 56 6732 6937 7142 7346 755I 7755 3 571 7565S78I0o80558299|85448'788 2 57 7959 8164 8368 8572 8776 8979 2 58 9o32 9276 9520 9764...7| 25I I 58 9I83 9387 959o 9794 9997.200 I 59 8.940494 0738 098 1 1224 1467 1709 0 599.020403 o6o6 o809g i012 II5 i4i8 o 60YI 50" 40" 30" 201' 10"' 6( Y' 50" 40 30' 20" 10' Co-tangent of 85 Degeees. Co-tangent of 84 Degrees. I// 2// 3/" 4// 5/" 6"/ 7"/ 8/" 9/ 1"' 2/- 3of 4 / 6// 9 PPart 27 54 81 108 135 162 1.88 215 242 P. Part 22 44 66 88 110 132 154 177 199 ~3 9L o G A R LOGARITHMIC bINES. Sine of 6 Degrees. e: Sine of 7 Degrees. 0" 10" 20/" 30" 40/'" 50" 0' 10" 20" 30" 40" 50"' o..oI9235 9435 9635 9835.. 35.235 59 o 9.085894 6066 6237 6409 658o 675I 59 I 9. C)2435 o635 o834 Io 34 I233 I433 58 I 6922 7093 7264 7435 7606 7777 58 2:632 I83I 2030 2229 2428 2627 57 2 7947 81i8 8288 8459 8629 88oo 57 3 2825 3024 3223 342I 36I 38I8 56 3 89709r40o93Io 948o965I 982o056 4 40I6 4214 4412 46Io 4807 5005 55 4 9990. I6o.330.5oo.669.839 55 5 52035400 5598 5795 5992 6I89 54 59.091I008 II78 I347 I5I6 I685 I855 545 6 6386 65836780 6977 7174 7370o 53 6 2024 2I93 2362 2530 2699 2868 53 7 7567.7763 7960 8I56 8352 8548 52 7 3037 3205 3374 3542 37I I 3879152 8 8744 89409136 9332 9527 9723 5I 8 4047 42I6 4384 4552 4720 4888 5I 9 9918.114.3o.5o4.699.894 50 9 5056 5223 539i 5559 5726 5894 50 109.03Io891I284 I479 i673 I86: 2o62149 io0 6062 6229 6396 6564 673I 6898 49 II 2257 245I 2645 2839 3o33 3227 48 I I 7065 7232 7399 7566 7733 7900 48 12 342 36i5 3809 4002 496 438947 12 I2 8066 8233 8399 8566 8732 8899 47 I3 4582 4776 4969 5162 5355 5548146 13 9065 923.I 93989564 9730 989646 14 5741 5933 6126 631965 I I 6703 45 I49.1 00062 0227 o39310559 0725 o89go 45 15 6896 7088 7280 7472 7664 7856 44 I5 Io56 I22I I387I552 17I7 I883 44 i6 8048 8239 843I 8623 88i4 9005 43 I6 2048 22I3 2378 2543 27o8 2873 43 17 9197 9388 9579 9770 996I. 52 42 17 3037 3202 3367353I 3696 3860o 42 18 9.o40342 o533 0724 09I41IIo5 I29541 I8 4025 4189 4353 45I7 4682 4846 4I 19 I485 I675 I865 2055 22452435 40 19 5oio 5I74 5337 55ox 5665 5829 40 20 2625 28I5 3004 3194 3383 3572139 20 5992 6156 63I96483 6646 6810 39 21 3762 3951|4I40o4329 4518 4707138 2 6973171367299 7462 7625 7788 38 22 4895 5o84 5273 546i 565o 5838 37 22 79iI 8II4 8277 8439 8602 8765 37 23 6026 6214 6402 6590 6778 6966 36 23 8927 9090 9252 9414 9577 9739 36' 24 7154 7342 7529 77I7790 4 809I135 |24 990I..63.225.387.549. 71 135 25 8279 8466 8653 8840o 9027 9214 34 2519.1 0873I034 II961 358 I5Ig9 I68 134 261 9400 9587 9774 9960 ~ I47.333133 26 I842 2003 2165 2326 2487 2648133 27 9.0505g19070o 6 892 Io781I264 I45o032 27 2809 2970 313I13292 3453 36I3 32 28 I635 182I 2007 2192 2378 2563 3I 28 3774 3935 4095 4256 44I6 4577 3i 29 2749 2934 3II9 33o4 3489 3674 30 29 473714897 5057 52I8 5378 5538 30 30 3859 4044 42284413 4597 47882 20 30 5698.5858 60I761I77 6337 6497129 31 4966 5i50o5335 55I9|57035887.28 31 6656 68I6 6975 7I35 7294 7453 28 32 6071 6254 6438 6622 68o5 6989 27 32 7613 7772 7931 8090 8249 84o8 27 33 71721735617539 7722 7905.8088 26 33 8567:8726 8884|9043 9202 9360 26 34 827I18454 8637 8820o9002 9185,25 34 9519 96779836 9994.1I52.311i25 351 9367 9550o9732 9914..96.278 24 35 9.120469o0627 0785e1943 Ioio1 259|24 36 9.060460o0642 0824 ioo6 II88 I369 23 36 14171I57417321I890 2047|2205|23 37 I55I I732 19I4 2095 2276 2457 22 37 2362 2520 267712835 2992 3149 22 38 2639 2820 3ooi 3I8I 3362 3543 21 38 3306 3463 3620 3777 3934 44o 9I21 39 3724 3904 4085 4265 4445 4626 20 39 4248 44o4 456I147I8 487415o3i 20 4o 4806,4986 5I66 5346 5526i5705 I9 40 5I87 5344 5500oo 5656 51I2 596919 41 5885 6065 6244 6424 6603 6783I8 4I 6125 6281 6437 6593 6748 69041i8 42 6962 |741 7320 7499 7678 787 1 7 42 7060 7216 737I 7527 7682 7838 17 43 8036 82I5 8393 8572 875i 8929' I6 43 7993 8I49|830418459 8614|8770 I6 44 9107 9286 9464 9642 9820 9998I5 44 8925 9080 923519390 9544 9699 I5 45 9.o07I761o354 o532 0709 0887 io65 14 45 9854 1..9. I63.3I8.472.627 14 46 1242 142015971I7741I951 2I281I3 46|9.I3078I|0936 IO90I1244 I398|I552I3 47 2306|2482|2659|2836|30I33g9012 47 I706I860012 I42168|2322|24761I2 48 3366 3543 37I9 3896 4072 4248 II 48 2630 2783 2937 309I 3244 3398 II 49 4424 4600 4777 4952 5128 5304 10 49 355i 3704 385840 II4164|431710IO 50 548o 5656 583i 6007 6I82 635S8 9 50 4470 4623 4776 4929 5082 5235 9 5i 6533|67o0868837o058|7233|7408 8 51 53875540o5693584159986i50 8 52 758317758 793318107 828218457 7 | 52 63o036455 6607o676)691287o64 7 53 863i 8805 8980 9i54 9328 9502 6 53 72I6 7368 7520 7672 7824 7976 6 54 96769850o..24.I98.372.545 5 54 8I2888279 843I1858218734 8886 5 559.o807191o892 Io 66 1239 i4i3 i586 4 55 90379188 934o0949i 9642 9793 4 56 I1759 1932 205 5 2278 2451 2624 3 56 9944..96.247.398.548.699 3 57 2797 2969 3I42 33i4 3487 3659 2 5719. I4085 00 IIooI 1i5i302 1453 i603 2 58 383214004 41764348 452014692 I 58 I7541904 2055 2205123552505 I 59 4864 5o36 5208 538o 555I 5723 0 59 2655 2806 2956 3 Io66325634o5 o 60" 0" 40" 30" 201 10" 1 60 50" 140" 30" 120,, 10"o, Co-sine of 83 Degrees. Co-sine of 82 Degrees. 1" 2'" 3 4/' 5// 6" 71/ 8 9/" 1" 2/ 3 4" 5t 6 7 8 9" Part 32 48 64 80 18 37 55 74 92 111 129 148 166 16 32 48 64 80 96 112 128 144 LOGAR ITHMIC TANGENTS. 3 Tangent of 6 Degrees. Tangent of 7 Degrees. 0" 1 10" 20" 30" 40" 50'1 0" 10o 20// 301" 401 50 c 9. o02 I6201 823 2025 2227 2430 2632 59 o9. o89g44193 8 9492 9666 9839.~ I13 59 I 2834 3036 3238 3439 364I 3843 58 I 9.090I87 o36I o534 0708 o 88I Io54 58 2 4o44 4245 4447 4648 4849 5050 57 2 X228 i4o0I 574 I747 I920 2093 57 3 525I 5452 56535853 5853 654 6254 56 3 226612439 2612 2784 2957 3129156 4 6445 6655 6855 7055 7255 7455 55 4 3302 3474 3647 38I9 399I 4I63 55 5 7655 7855 8o55 8254 8454 8653 54 5 4336 45o8 468o 485 I 5023 595 54 6 8852 9052 9251 945o 9649 9848 53 6 5367 5538 5710 588I 60 53 6224 53 7 9.030046 0245 o444 o642 o84iI I039 52 7 6395 6567 6738 6909 7080 725I 52 81 I237 I435 i633 I83I 2029 2227 5I 8 7422 7593 7764 7934 8io5 8276 5I 9 2425 2623282o 30I71732I5 3412 50 9 8446 86I6 8787 8957 9I27 9298 50 Io 3609 3806400o3 4200 4397 4594 49 Io 9468 9638 9808 9978. I48.317 49 I 4791 4987 5I84 5380o 5576 5773 48 I9. Ioo4870 o657 o 82q o996 II66 i335148 I2 5969 6I65 636I 6557 6753 6948 47 I2 I5o4 1674 I843 2I2 2I8 12350147 I3 7144 7339 7535 7730o792681I2I 46 i3 25I9 2688 2857 3026 3I94 3363 46 i4 83I6 85ii 8706 890909599 45 929045 4 3532 3700 3869 4037 4205 4374145 i5 9485 9679 9874..68.262.456 44 iS 4542 4710 4878 5o46 52I4 5382 44 I6 9. 04065i o845 1o391g232 426 I620 43 I6 5550 5718 5885 6053 622I 6388143 I7 I8I3 2007 2200 2394 2587 278o 42 I7 6556 6723 6890 705.8 7225 7392142 i8 2973 3I66 3359 3552 3745 3937 4I1 8 7559 7726 7893 8060 8227 8394 41 19 4I30o4322 45I5 4707 4899 5092 4o0 9 8560 8727 8894 9060 9227 9393 40 20 5284 5476 5668 5859 6o05 6243 39 20 9559 9726 9892..58.224.390o39 21 6434 6626 68I7 7009:7200 739I 38 2I 9. II0556 0722 o0888 I05412I9 I385 38 22 7582 7773 7964 8I55 8346 8536 37 22 I55I I7I6 1882 20 47 22I3 2378137 23 8727 8917 9108 9298 9489 9679 36 23 2543 2708 2873 3039 3204 3368 36 24 9869..59.249.439.629.8i8 35 24 3533 3698 3863 4028 4192 4357 35 259.0o5I008oo897 1387 r5761I7661I955 34 25 4521 4686 485050o 5I517915343!34 26 2I44 2333 2522 27I 1 29oo 30 88 33 26 5507 567I 5835 5999 6i63 6327133 27 3277 3466 3654 3843 4o3i 42I9 32 27 649i 6655 68i8 6982 7I45 73o9 32 28 4407 4596|4784 4972 5I59 5347 3I 28 7472 7636 7799 7962 8I26 8289 31 29q 5535 5723 59I 6098 6285 6472 30 29 8452 86i5 8778 894I 9IO4 9266 3o 30 6659 6847 7034 722I 7408 7594 29 3o 9429 9592 9754 9917 1. 79.242 29 3r 778 I 7968 8i 55 834I 8528 87 1428 3 9. I 20404 o567 o729 0o89 io53 1 x528 32 890019086 9273 9459 9645 9831 27 32 I377 i539 I70I I863 2025 2I87 27 339,o600oo6 0202 0388 o573 0759 o094426 33 2348 250o 2671 2833 2994 3i56 26 34 ii3o i3i5 5oo00 685 1870 2055 25 34 3317 3478 3640 38oi 3962 4123 25 35 2240 2425 26O1 2795 2979 3I64124 35 4284 4445 4606 4766 4927,5o88 24 36 3348 3533 37I7 390o 4085 4269 23 36 5249 5409 5570 5730 5891 6o5I 23 37 445314637 4821 5oo051588 5372|22 37 62IiX637I 6532 6692.6852 70oI222 38 5556 5739 5922 6Io6 6289 6472 21 38 7I72 7332 7492 765I 78II 7971 21 39 6655 6838 702I17204 7387 7570 20 39 8I3o 8290 8449 8609 8768 8928 20 40 7752 7935 8II7 83oo~84828664x19 4o 9087 9246 9405 9564 9723 9882 I9 4I 8846 9029 92II9393 9575 9756 6 8 4I9. I3004Io0200 o359 o5i8 o676 o83581I 42 9938 I.20.3oI.483.664.846 17 42 0994 I52 I3II I469 1628 I7861I7 43 9.07IO27 I208 I389 i570o I75i 932 i6 43 I9442I102 226I 2419 2577 2735 i6 44 2 I I3 2294 2475 26655 2836 3oi 6 I5 44 2893 3o050o 3208 3366 3524 368i i5 45 3197 3377 3558 3738 39i8 4098I4 1 45 3839 3997 4i54 4312 4469 4626 14 46 4278 4458 4638 45817 4997 5I77 13 46 4784 494I 5098 5255 54I2 5569 i3 47 5356 5536 57I5 5895 6074 6253 I2 47 5726 5883 6o4o 6I97 6353 65Io 12 48 6432 66ii 6790 6969 7148 7327 II 48 6667 6823 6980 o736 7292 7449 II 49 7505 7684 7862 8o4i 82I9 8398 1o 49 76o05 7761 79i8 8074 823o 8386 Io 5o.8576 8754 8932:9IoI9288 9466 9 50o 8542 8698 8854 9009o9165932I 9 5i 9644|9822..... 177.355.532 8 5i 9476|963219788 99431,..98.254 8 52|9.080o7O10887 IO64|1241 i4i9|I596 7 52 9.x4o4o09o564o720 0875 io3o1;:85 7 53 I773 I95o02I16 2303 2480 2657 6 53 I34o0 495 I65o 18o5 I959 21I4 6 54 2833 30Io 3I86 3362 3539 37I5 5 54 2269'2424 2578 2733 288713042 5 55 389i14067:4243 44I9!45951477I 4 55 3I96!3350o350o43659 38i3 3967 4 56 4947 5I22 5298 5473 5649 5824 3 56 412I 4275 4429 4583 4737 4890! 3 57 6000o 675 6350o 6525 6700 6875 2 57 5044 5I98 535I 5505 5659 5812 2 58 7o5o 722517400 757417749 7924 I 58 5966 6I I96272 6425 6579 6'32 I 59 8098 8273 8447 8621 8795 8970 o 59 6885 70387I9I173447497765o1 0 60" 1 50" 40",30t 1i0''. 60 - 50" 40" 30"' 20" 10". Co-tangent of 83 Degrees. Co-tangent of 82 Degrees. p. p I 2" 3" 4"1 5/" 6" 7" 8" 9" pr s1 2 3"' 4 5"6" 7" 8" 9". Part 19 37 56 75 94 112 131 150 168 I.'art 16 33 49 65 81 98 114 130 146 32 LOG ARITHMI C S INES.. 1 _ RSine of' 8 Degrees. A Sine of 9 Degrees. S-' |10" 90Y 30" 40" 5 (i0 10" 20" 30 " 40" W0" o91435'-5 37'35 3855 400514I 5414304 59. 0o9. 94332 4465 4598 473I 48641499'159 i 4453 46031 4752 4902 5o5I 5200 58 I I29 5262 5395 5527 566o 5792 58 2 5349 549815648 579715946 6095 57 2 59256057 689 6322 6454 6586 57 3 6243 6392 654i 6690o6839 6987 56 3 67I9 6851 6983 7II517247 7379 56 4 7I367284 7433 758I 7730 7878155 4 75II 764317775 7907 8038 8I70 55 5 802618I7418323 847I 869g 8767 54 5 8302 8434 85651869718828896o54 6 89I 5 9063 92 19358 9506 9654 53 6 909I 99223 9354 9486 96 I7 9748 53 7 9802 9949.97.244.392.539 52 7 9879...II.142.273.4o4.535152 819.I5o0686 o834 098 II28 1275 I422 5I 8 9.200666 0797 0928 I059 II89 I320 5I 91 I569 7I6 i863 20IO 2I57 2304 5o 9 I45i I582 1712 I843 I973 2I04 5o Io 2,451 2597 2744 289I 3037 3I84 49 Io 2234 2365 2495 2626 2756 2886 49 II1 333013476 3623 3769 39I5 4o06 48 II 30I7 3I4713277 3407 3537 3667 48 12 4208 4354 4500o464614792 4938 47 12 3797 3927 405741I87 43I7 4447 47 13 5o83 5229 5375 552I 5666 58I2 46 I3 4577 4706 4836 4966 5095 5225 46 14 5957 6I03 6248 6394 6539 6684 45 i4 5354 5484 56I3 5743 587216002 45 I5 6830 6975 7I20 7265174I0 7555144 I5 6I3I 6260 6389165I91664816777 44 i6 7700 7845 799o 8I35 8279|842414311 6 69061703517I6417293174221755I 43 17 8569 8713 8858|9002 91471929I 42 17 7679 78081793788066 818948323 42 i8 9435 9580 9724|9868..12.I56 4I i8 845218580o87091883718966 9994 41 19 9.60301 o445 o589 0732 0876 I020 4o 19 9222 935I 9479 9607 9735 9864 4o 20 ii64 i3o8 I451I|595 1738 1882 39 20 9992 1I20.248.376.5o4.632 39 21 2025 2I69 23I2 2456 2599 2742 38 2I 9.210760 o888 IOI5 II43 I271I 399 38 22 2885 3028 3172 33i5 3458 36oo 37 22 I526 i654 I78I I 991 2037 2I64 37 23 3743 3886 4029 4I72 43i414457 36 23 2291 24I9 2546 2674 280I 2928 36 24/ 46oo 4742 4885 5027 5170153I2 35 24 3o55 3I82 33Io 3437 3564 3691 35 25 5454 5597 5739 588i 6023 6i65 34 25 38x8 3945 407I 4198 4325 4452 34 261 6307 6449 6591 6733 687570oI7 33 26 4579 4705 4832 4959 5o85 53I12 33 27 7I59 7300 7442 7584 7725 7867 32 27 5338 5465 5591 57I8 5844 5970o32 281, oo8 8I50o829I 843281574871553I3 28 60976223 6349 6475 66o0 6728.31 291 d6856 8997 9I38 9279 9420 9561I 30 29 6854 6980 7I06 [72327358 74831 3 301 9702 9843 9984 I25.265.406 29 30 760917735 7861 798781iI2 8238|29 3I,9 170547 o0687108281o09 68 IIo9249128 3 8363 8489186I51874o08866 8991 28 32 I389 I5301o670oI8IO|I950o2090o27 | 32 I 9I6 9242 936719492196I8 9743127 33. 2230,2370 2510o650o279012930o26 I| 33 986819993.'II8.243[.368.493126 34 3o7o132o 13349 3489 3629 3768 25 34/9.2206I8 0743 o868 0993 III8 1242 25 35 390840o47 4187!4326 4465 46o5 24 351 367 I492 I6I6 1741 I866 I990 24 36 4744j4883 5022 5I6i53oo005439123 36 2II5 2239 2364|2483826I2 2737|23 37 5578 5717 5856 5995 6I34 627322 22 37 2861 2985 3Io913234 3358 3482 22 38 64ri1655o 6688 6827/696617Io421I 38 3606 3730 38541397814I02 422621I 39 7242 758I175I197657 7796 7934 20 39 434914473 4597 472I 4845 4968 20 40 8072182Io08348 848618624 87621]9 || 4o 5092 52I5 5339 5462 5586 5709 I9 4I| 89oo0090389I76 93I3 945I]9589 18 114i 5833 5956 60806203 6326 64498 i8,.2 9726 9864...2. II39.276.4I4 I7 42 6573 6696 68I9 6942 7065 7I8817,.J 39.I8055I o689 0826 0963 II00 I237 I6 43 7311 |7434 75571768017803 7925]6 i | 42j I374 i5ii i648 1785 1922 2o595 i 44 8o48 8I7I 8294 84i6 8539 866i I5 4 2196 2333 2469 260o62743 2879I4 1 45 8784 8906 9029 9I5I 9274 9396 I4 46 30o6 3152 3289 3425 3562 3698 I3 46 95I8 964I 9763 9885...7.3 I3o3 47 3834|39714,10714243 4379[45I51I2 11479.230252 0o374j0496o06I8[0740 o86I2 12 48 465i 478714923|505915I951533I|II || 48 09841II0612281I3491I47I I5931II 49, 5466 5602 S738 5874 6009 6i45 Io 149 I7I5 i836 i9582079 220I 2323 Io o.50 628o 64I61655i 6686 6822 6957/ 9 So 2444 2565 2687 2808 293o 3o5i 9 |r ~o1092 7228 736374987633776833 87768 58 3723293 345 35636 3657 3778 8 |52,qo3 8o38 8173 83o8 8442 85771 7 52 3899[4020o144i[4262 4383 4504J 7 531.4),I2.8847[898I 9II69 25019385 6 53 4625s4746148671498715Io815229 6 54`Sf;9j9654 9788 9923..57|.I9I 5 54 53491547o 559i 57rI 5832 5952 5 55 9.Iqo9035 o46o o594o0728 0862 o996| 4 | 551 6073 6I93 63I3 6434 6554 6674 4 561 II30oI264]I3981I5321I6656I799 3 56 6795[69I5703517I5517275 7395i 3 57 9332o0662200oo2334 24671260I 2 57 75I517635 77551787517995 8II5| 2 58 2734 2868 3oo00 334 3268 34o0 I 58 8235 8355 8474 8594 87I4 8834| I 59 3534 3667 3800|393314066 4199 0 591 8953 907319i92 93I2 9431i 955i 0 60t 501" 40"' 3Yi 2011 10' 50" 401 30 20"' 1t I Co-sine of 81 Degrees. Co-sine of 80 Degrees. "''f 3" 9.,,.1 8i 1 2" 3" 4" " 6i 7 8"' 14 28 42 5fi 70 85 99 113 127 25 38 50 63 75 88 101t 113 P.,art LOG AR THMIC TANGENTS. 33 Tangont of 8 Degrees. { Tangent of 9 Degrees. 0// 10" 20"1 30" 40" 50" 0" 1C01 20" 30" 40" 50"/ o0 9. 147803 7955 8I08 826I 84I3 8566 59 0 9. I997I3 9849 9985.12I.257.393 59 87I8 887I19023 9I75 9328 9480 58 I 9. 200529 o665 080I 0937 I073 1209 58 2 9632 9784 9936.88.240.392 57 2 I345 I48i ii6 1752 I 888 2023 57 3 9.150544 o696 o848 0999 I I 5 I I 3o3 56 3 2I59 2294 2430 2565 270I 2836 56 4 I454 I606 1757 I909 2060 2213 55 4 297I 3107 3242 3377 3512.3647 55 5 2363 25I4 2665 28I6 2967 311 8 54 5 3782 39I8 405314I88 4322 4457154 6 3269 3420 357I 3722 3873 4023 53 6 4592 4727 4862 4996 5i3i 5266 53 7 4I 741432514475 4626 4776 4926 52 7 54oo00 5535 5669 58o4 593816073 52 8 5077 5227 5377 5528 5678 5828 5I 8 6207 6342 6476 66Io 6744 6878 51 9 5978 6I328 6278 6428 6578 6728 50 9 70I3 7I47 728I 74I5 7549 7683 5o 10O 6877 7027 7I77 7326 7476 7625 49 1IO 78I7 7950 8084 82I8 8352 8485 49 II 7775 7924 8074 8223 8372 852I 48 I I 86I9 8753 8886 9020 9I53 3 9287 48 12 867I 8820 8969 9I I8 9267 94I6 47 I2 9420 9554 9687 9820 9954..87 47 13 9565 97339862.. II. I 60o.308 46 13 9. 2 I0220 03530 o486 o6I 9o752 o885 46:4 9. I6o457 o6o5 0754 0902 io5I II99 45 i4 I oi8 I i 5 I 284 14I7 I 550o I 683 45 1 5 T347 I496 i644 1792 I940 2088144 i 5 i8i5 I948 208I 2233 2346 2478 44 I6 v36 2384 2532 2680 2828 2975 43 i 6 26I I 2743 2876 3oo8 3i4i 3273 43 17 3123 3271 34I8 3566 3713 386I 42 I 7 3405 3537 3670 3802 3934 4o66 42 I8 4oo8 4I56 4303 4450 4598 4745 4I i8 4I98 4330 4462 4594 4726 4858 4I 39 4892 5039 5I86 5333 5480 5627 4o I 9 4989 5I23 5253 5385 55I6 5648 4o 20 5774 5920 6067 62I4 636i1 6507 39 20 5780 59 I 6o43 6174 6305 6437 39 21 6654 68oo 006947 7093 7240 7386 38 2I 6568 6700 683i 6962 7093 7225 38 22 7532 7678 7825 797I 8II7 8263 37 ^ 22 7356 7487 76I8 7749 7880 8oi I 37 23 8409 8555 870I 8847 8992 9I38 36 23 8I42 8273 8403 8534 8665 8796 36 24 9284 9430 9575 972 I 9866. I2 35 24 8926 9057 9I88 93I8 9449 19579 35 25 9. 1701 57 o303 o448 0593 0739 o884 34 25 9710 9840 997I. IOI.23I.36i 34 26 I029 II74 13I9 I4641I6091 754 33 26 9.220492 0622 0752 o882 I0121142|33 27 I899 2044 2I88 2333 2478 2623 32 27 1272 1402 I532 I662 3792 I922 32 28 2767 29I2 3056 320I 3345 3489 31I 28 2052 2I82 23 II 244I 257I 2700 3I 29 3634|3778 3922 4067 42I 4355 30 29 2830 2959 3089 32I8 3348 3477 3o 30 449914643 4787 4931I 5075 52I38 29 30 3607 3736 3865 3994 4324 4253 29 3I 5362 5506 5650o 5793 5937 6080 28 3i 4382 45i I 4640 4769 4898 5027 28 32 6224 6367 65 II 6654 6797 694I 27 32 5i56 5285 54I4 5543 567 5800o27 33 708417227 7370 75I3 7656 7800 26 33 5929 6058 6I86 63i5 6443 6572 26 34 7942 8085 8228 837I 85I4 8657 25 34 6700 6829 6957 7086 72I4 7342 25 35 8799 8942 9085 9227 9370 9512 24 35 747I 7599 7727 7855 7983 8III 124 36 965519797 9939..82.224.366 23 36 8239 8368 8496 8623 875i 8879 23 37 9. 805080o650 0792|09343I076 I2I8 22 37 90goo07 9I35 9263 93o 95i089646 22 38 I3603I502 I6443I7863I927 2069 2I 38 9773 990I..29. I56.284.43I 2I 39 2211 2352 2494 2635 2777 291I8 20 39 9.230539 o666 0793 092I Io48| I75120 40 3059 320I 3342 3483 3625 3766 19 4o i302 Il(ol I557 I684 I8I I I938 I9 41 390714048 4I89 4330|447i 46I2 18 4 2065S12I92 23i9 2446j2573 26991i8 |42 4752 4893 50345I 75|53I5 5456 I7 42 282612953 3080o 320613333 3460o I7 43 5597 5737 5878 6oi8 6I58 6299 16 43 %586137I3 3839 3966 4092 42I9 i6 44 6439 6579 6720 6860 7000 73401 5 441 4345 44744598 4724 4850|49763I5 45 7280 7420 7560 7700 7840 7980 14 45 Sio0315229 5355 548i 5607 5733:4 46 8 I 20o825983998539|8678 88 I 8 I 3 46 5859 59856 I I 6237 63621 6488 I 3 47 8958|9097|9236|9376|95I5 9655| 12 47 66I4|6740|6865 69917II717242 12 48 979419933. 72.212.35I.490 II 48 7368 7493 76I9 7744 7870 7995 II 49 9. Igo90629 0768 0907 io46 II84 I323 10 49 8I20|8246|837I18496|862I187471IO 50 I462 I60IoI739 I878 20I7 2I55 9 50o 8872 8997 9I22 9247 9372 9497 9 5i 2294|2432|257I32709|2848 2986 8 5i 9622|9747|9872|9996.12I.246 8 52 3124 3262 34oi 3539 3677 38i5 7 52 9.24037I o 495 0620 0745 0869 0994 7 53 3953 409I34229 4367 4505 4642 6 53 1II81I2431i3671I492I6I61I741 6 54 4780o49i8 5o56 5I93 533I 5468 5- 54 I865 I989 2II42238 2362 2486 5 55 56065743588I|60I8|6I56 6293 4 55 2610 2734 2858 2982 33o63230 4 56 6430o6567|6705S6842|6979 7I 36 3 56 3354|3478|3602|3726|385o3974 3 57 7253 7390 7527 7664 780I 7938 2 57 4097 422I 4345 4468 4592 47IS a 58 8074|82I I8348|8484.862I 8758 I 58 4839 4962S5086132095333S5456 I i591 8894 go903I 967 9304 9440 95761 0 59 5579 5703582615949 6072]6396 0 __t__ | 50'" 40" |0" 20" 10 G 60" 50 " 40" 30" 20 " 10" Co-tangent of 81 Degrees. Co-tangent of 80 Degrees. P.Part 1" 2' 3" 4" 5" 6" 7" 8" 9" P. Pt 1" 2" 3" 4 5" 6" 7" 8" 9" 1 29 43 58 72 86 101 115 130 ar 13 26 39 52 65 78 91 103 116 U,.1 * (;: ~ ~ ~ ~, ,3 4 L.' - oGARITHMIC S:IN ES. Sine of 10 Degrees. Sine of 11 Degrees. 0" 10" 20' 1 30"1 40" 50 0" 10" 20 30 40" 50" 0 9.239670 97909909..28. I48.267 59 o 9.280599 0707 o8I 50924 IO32 I I40 39 I 9.240386 o5o5 o624 0744 o863 o0982 58 I I248 i356 i465 5 73 I68i I789 58 2 IIoI0 I1220 1339 i458 1576 I695 57 2 I 897 2005 2 I I 3 2220 2328 2436 57 3 I8I4 I 33 2052 2170 2289 2408156 3 2544 2652 2760 2867 2975 3o83 56 4 2526 2645 2763 2882 3ooi 3II9 55 4 319013298 34o6 3513 362I 3728 55 5 3237 3356 13474 3592 37I I 382 54 5 3836 3943 4o05I 458 4266 4373 54 6 3947 4065 4I84 4302 4420 4538 53 6 448o04588 469514802149~9150I7153 7 4656 4774 4892 50Io0 51285245 52 7 5I24 523 I 53381544515552 5659152 8 5363 548I 5599 57I7 5834 5952 5I 8 5766 5873 5980 6087 6I94 63oi/ 5I 9 6069 6I87 63o5 6422 6546 665750 9o 64o8 6514 662I 6728 6835 694I 50 Io 6775 6892 7009 7 I 27 7244'736 I 49 1I 7048 7I 55 726I 7368 7474 758 I 49 11 7478 7596 7713 7830 7947 8064 48 II 7688 7794 79 008007 8 I 3 8220 48 12 8I8Ii 8298 84I5 853218649 8766 47 1 21 8326 8432 8539 86451875I 8857 47 13 8883 8999 9 II.6 9233 9350 9466 46 I3 8964 9070 9I7619282 9388 9494 46 I4 9583 9700 98I6 9933.. 49.i6645 4 9600 9706 9812 99I8..24.I3o3045 i5 9.250282 0399 15 5o63 I 0748 o864|44 I5 9.290236 0342 0447 o553 o65910765 44 I6 og80 1097 I12 I3 329 I445 I56143 I6 0870 0976 o82 I 187 I293 I398 43 17 I677 I793 I9 912025 2141 2257142 1 I 7 I50416101 7151 820 1926 203I 42 I8 2373 2489 2605 2720 2836 29524I I8 2I37 2242 2347 2453 2558 2663 4I 19 3o67 3183 3299 34I4 35303364540 1I9 276812874 297913084 3189 3294 4 20 376I 3876 3992 4I07 4223 4338 39 20 3399 35o4 3609 37I4 3819 3924 39.21 4453|456814684 4799 491415029 38 2I 4029 4I 34 4239 43441444814553 38 22 5i44 5259 5374 549o 560o457I9 37 22 4658 4763 4867 4972 5077 5i8i 37 23 5834 5949 6o64 6 I 79 6294 64o09361 23 5286539I 5495 56oo 5704 58o9136 24 6523 6638 6753 6867 6982 7096 35 24 59I3160o7 6I22 6226 633o 6435 35 25 72II7326 7440 7554 7669 7783 34 25 6539 6643 6747 6852 6956 7060 34 26 - 7898 80212 8126 8241 8355 8469 33 26 7I64 7268 7372 7476 7580 7684 33 27 8583 8697 88II 8926 o9040 954 32 27 778817892 7996 8Ioo 8204 83o0832 28 9268 9382 949519609 9723|9837|3I 28 84I2185I5 86I918723 8827 893o031 29 995 I..65. I78.292.4o06.51930 29 903419I38924I 9345 9448 955230~ 3019. 2606330o747 o8600o9741 087|1201|29 30 965597591986219966..69.172 29 31 I3I4 1428 i54i i654 I768 i88I 28 3I 9.300276 3791 0482 o586 o6891 0792 18 32 I994 2I 7 2220 2334 2447 2560 27 32 0895 o0999 II02 I205 I3o8 I4iI7 33 2673 2786 2899 312 3I25 3238 26 || 33 I5I4 I6I7 1720 I823 I926 2029 26 34 335I 3464 3576 3689 3802 3915 25 34 2I32 2235 2337 2440 2543 2646 25 35 4027 4I4o 4253 4365 4478 4590 24 35 2748 2851 2954 3057 3I59 3262 24 36] 4703 48I5 4928 5040o 553 5265 23 36 3364 3467 3569 3672 3774 3877193:37 5377 5490 5602 57I4 5827 5939 22 || 37 3979 4082 4I84 4287 4389 449x 22 | 381 605i [663 6275 6387 64991661|2 | 38 459314696J4798 490~~5002 5104[21'39 6723 6835 6047 7059 717I17283 20 39 5207 5309 54II 55i3 56i5 57I7 20 40 7395 7506 76I8 7730 7841 79539 11 4 58I9 592I16023 6I25 6227 6328I9 4I| 8o65 1876 82881 8399 85I I8622 18 4I 6430o 6532 6634 6736 6837 6939 8 42 8734 8845 8957 9068 1979 929I I7 42 704I 7I42 7244 7346 7447 75491 I7 |43 9402 95I3 962419736 9847 9958 i6 43 7650 7752 7853 7955 8o56 8i58 6 |449.270069 0I800o291 o4o02 o5 I 3 0624 i5 44 8259|8360o8462 8563 8664 8766 I5 45 0735 o846o0957 Io67 II78 I289 14 1 45 8867 8968 9069 9I70 9272 9373 14 |46 i4oo00I 5 I62I I7321I842 19531I3 |46 947419575 967619777 987819979 i3 |47 20642I174 2285S2395125o05266'1I2 147 9.31oo800ooI8 I0282 o382 0483 0584 I2:48 2726 2837 2947 3o5713I68 327881II 48 0o68507868860987 io88 1II891 491 3388 3498 36o8 37I8 3829 3939~IO 1 49 I289I39olI49o 1591i I692 I792 IO 50 404914I59J4269 4379 448914598 9 5o0 1893993 2094 2Ig942294 23951 9 5I 4708148I8J4928 5o38 5I48 5257 8 5I 2495 2595 2696 2796 2896_ 2997'8 521 536715477155861569615805159I5 7 || 52 30973I97[329713397 3497 35971 7 53 60256I134,6243i635316462 6572 6 53 3698J3798 389813998J4098 4I98 6 54 668I 679o 690oo 7009 7I81 7227 5 54 4297 4397 449714597 4697 4797 5 55 7337 7446 755517664 7773 7882t 4 55 4897 4996 5096 5i96 5295 5395 4 561 799I18Ioo 82o9183I8]8427 8536 3 56 54955594 5694 579315893 59931 3 |57 8645 8753 8862 8971 9080|9I881 2 57 6092 6I92 629I 6390 6490 65891 2 |-58 9297 9406 9514 9623 973I1984o0 I 58 66896788 6887 6986 7086 7185 59 9948. 57 Ii65.274.382.49I 0o 59 7284 7383 7482 75821768I 77801 o 60~1 501 401 301 20 101 60 50" 40"f 30" 1 20rlO 10/ Co-sine of 79 Degrees.' Co-sine of 78 Degrees. " 5" 40"1" 230 320 40 50 6" 7" 8" 30'" 1... 1rt1 23 3" 45 5" 6" 7/ 8" 9"1 1S" 2/ 30 41" 5/" 6/ 7/ 8" 9'.1 11 23 34 45 57 68 80 91 102 P.iPart 10 21 31 41 52 62 72 83 93.~~~~~1.1..'4.2.a 72.3...... LOGARITHMIC TANGENTS. 3A - Tangent of 10 Degrees. Tangent of 11 Degrees.. -- 1(c'1 20" 30" I 40" 50" 0," 10", 20", 3'0" 40", 50"," 0o 9.2463I9 {6442 6565 6688 68II 6934 59 0 9.288652 8765 8877 8989 9I02 9214159 I 7057 7180 7303 742617548 7671 58 I 9326 9438 956I 9663 9775 9887 58 - 7794 79I7 8o39 8162 8285 84077 57 9999. III.223.335.447.559157 3 853o 8652 8775 8897 9020 9142 56 3 9.29067Iio783 o895 10o7 I 11I9 123I 56 4 9264 9387 9509 963I 9753 9876 55 4 I342 I454 I566:678 I789 Igo9055 5 9998. I20.242.364.486.6o8 54 5 2132I24 2236 23472459 2570 54 6 9.250730 o852 0974 1O96 I218 I339 53 61 2682 2793 2905 3ox6 3127 3239 53 7 I46I I583 I7o5 1826 1948 2070 52 7 3350 346i 3572 3684 3795 3906 52 8 219I 2313 243432556 26771279915I 81 40Io7 4128 4239 435I 4462 4573 5 I 9 2920 3o04 3I63 3284 34o5 3527 50 9 4684 4795 4905 50o6 5127 5238 50 Io 3648 3769 3890 40141:32 4253 49 IO 5349 546o 557 I 568i 5792 5903 49 I I 437414495146i6 4737 4858 4979 48 1I 60I3 6124 6235 6345 6456 6566 48 I2 5Ioo 5221 534 15462 5583 5703 47 12 6677 6787 6898 7008 7II9 7229 47 13 582415945 6o65 6I86 63o6 6427 46 13 7339 7450 7560 7670 778I 7891 46 I4 6547]6668 6788i6908 7029 7149 45 I4 800oo 8ii 8221 8332 8442 8552 45 I5 7269/7389 75Io 7630,7750 7870 44 I5 8662 8772 8882 8992 9IO2 92Ir244 16 799o818 IO08230o 8350 8470 8590 43 i6 9322 9431 954I 965I 976I 9871 43 17 8710 S830o895090go69|91899309 42 17 9980..90.200.309.4I9.529 42 i8 9429 9548 9668 97879907..27 4I I89. 300638 0748 o857 0967 I076 I 86|4I 19 9.260I46 0266 o38 o0504 0624 0743 40 19 I295 I4o5 I5 I4 1624 1733 i842 40 20 o8630982 I IO11 1220 34o 459 39 20 1951 206 62170 2279 2388 2497 39 21 I 5781 6971 I 8 I 6I 93520542 I 73138 2 I 2607276 2825 2934 30433 I 5238 22 2292 24II12530 2649 2768 2887 37 22 326I 337o 3479 3588 3697 38o5 37 23 3oo005 3I24 3243336 3480 3599 36 231 3914 4023 4I32 424I 4349 4458 36 24 3717 3836 3.954 4073 419I 43I0o35 24 4567 4675 4784 4893 500oo 5Io135 25 4428 4547 4665 4783 4902 502o 34 25 52I8 5327 5435 5544 5652 5761i34 26 5I38 5256 5375 5493 56i 15729 33 26 5869 5977 6o86 6194 6302 64io 33 L17 5847 5965 6o83 62o0 6319 6437 32 27 6519 6627 6735 6843 695I 7059 32 28 6555 6673 6790 6908 7026 7144 31| 28 7168 7276 7384 7492 7600 7708 31 29 7261 7379 7497 7614 7732 7849 30 29 7816 7923 803I 839 8247 8355 130 3o0 7967 8o84 8202 83I9 8437 8554 29 30 8463 8570 8678 8786 8893 9001 29 31 867I 8789 89o6 9023 914o4 0 9258 28 3I 9Io9 92I6 9324 9432 9539 9647 28 32 9375 9492!9609 9726 984319960127 320 9754 9862 9969~ ~ 76 1I84.291 27 33 9.270077o01940o3II0o428o05450o662 26 33 9.31o399 o0506o 6I3 o 720 o8 28 o935 26 34 o779o0895 I012 II29 1246 1362 25 34 1042 1I49 1256 x364 I47I 1578 25 35 14791I595 71271829 1945I2062 24 35 I685 I7921I899 2006 2II3 2220|24 36 2I78 2294 24II 2527 2644.2760123 36 2327 2433 2540 2647 2754 2861 23 37 2876 2992 3Io9 322'5 334i 3457 22 37 2968 3074 3I8I13288 3394 35oi 22 38 3573 3689 3805 392I 4037 4I53 2I 38 3608 37I4 382i13927 4034 4i4o 2I 39 4269 4385 450Io46I7 4733 4849 20 39 4247 4353 4460o 4566 4672 4779 20 40 4964i5o8o 5I96 53I2 5427 5543 19 40 4885 4991 5098 5204 53io 54I6 19 4I 5658 5774 5890 6oo05612I16236 i8 4i 5523 5629 5735 584i 5947 6053 i8 42 635i 6467 6582 6698 68i3 6928 17 42 6I59 6265 6371 6477 6583 6689 I7 43 7043 7I59 7274 7389 7504 7619 I6 43 6795 690I 7007 71I3 7218 7324 i6 44 7734 7849 7964 8079 8194 8309I5 i 44 7430 7536 7641 7747 7853 7958 I5 45 8424 8539 8654 8769 8884 89981I4 45 80648170o8275 838I 8486 8592|14 46 9iI3 9228 9342 945719572.96861I3 46 869718803 89089o1I3 91199224 i3 47 980I199I 5..3o.1 44.259.3731 I2 47 93309435|9540|9645 975I 9856 12 489.,28o488 o602o0717 o831 o945 Io5911 48 9961..66 1I71.277.382.4871II 49 I1741 I28814021 I5I616301I7441 IO 49 9.320592 06970o8020o907 I10oI 11171I0 50 I8581I973 2087 2201 2341 2428 9 50 I222 1326 143i i536 i64I 1746 9 5i 2542 2656 2770o2884 299831III 8 51 I85I 1955 206021i65 22692374 8 52 3225 3339 3453 3566 368o 3793 7 52 2479 2583 2688 2793 2897 3002 7 53 3907 4021 4134 4248 436i4474 6 53 3Io6 3213315.342o03524 3628 6 54 4588 470I148i5 4928 5o4i 5i54 5 54 3733 3837 3941' 4046 4I5o 4254 5 55 5268 538i 5494 560o7572o 5833 4 55 4358 4463 4567 467I 4775 4879 4 56 5947 6o6o06I73 6286 6399 6512 3 56 4983 5087 5I9I 5295 5399 55o3 3 57 6624,6737 6850o6963 70767I89 1 57 5607S57IS158i559196o02336127 2 58 7301 7414 7527 7639 7752 7865 1I58 623I 6334 6438 6542 6646 67491 59 797718o90o 82o2831584271854o o1 59 685316957 706o7I647267 7371 0 601 50" 1 40" 130" 20 10" 60" 50' 40"' 30" 20" 10" Co-tangent of 79 Degrees.' Co-tangent of 78 Degrees. p. par 1" 2" 3"1 41 5/" 6" 7/1 8'11 9"/ 11 2" 311 4/1 5/1 6" 7" 8" 91.*art...12 23 35 47 59 70 82 94 106 P. Part 11 22 32 43 54 65 75 86 97. ~ 6 LOGARITHMIC SINES. Sine of 12 Degrees.. Sine of 13 Degrees. O- 10"- 20/" 301 401" 50"" o 0f 101' 20"/ 30" 1 40Y 50" o 0931 7879 7978 8077 81I76 8275 8374 59 0 9.352088 21 79 2270 2362 2453 2544 59 I 8473 8572 867I 8769 8868 8967 58 I 2635 2726 28I7 2908 2999 3090 58 2 9066 9165 9263 9362 946I 9559 57 2 3i8I 3272 3363 3454 3545 3636 57 3 9658 9757 9855 9954..52. I 5 I 56 3 3726 38I7 3908 3999 409o 4i8o 56 4 9.320249 o348 o446 o545 o643 0742 55 4 427I 4362 4452 4543 4634 4724 55 5 o84009og3810o37 I I 351233 332 54 5 48 5 4906 4996 5087 5177 5268 54 6 I43o I528 1626 I724 1822 i921 53 6 5358 5449 5539 5630 5720 58io 53 7 20I92II7 2215 2313 24II11250952 7 590I599i1608I16I72 6262635252 8 2607 2705 2802 2900 2998 3096 5I 8 6443 6533 6623 67I3 6803 6894 5I 9 3I94 3292 3389 3487 3585 3683 5o 9 6984 7074 7164 7254 7344 7434 5o IO 9.323780o3878 3975 4073 417I 4268 49 IO 9.3575247614 77~4 7794 7884 7974 49 II 4366 4463 456I 4658 4756 4853 48 I I 8064 8 I 54 8243 8333 8423 85 I 3 48 I2 495o 5048 5I45 5243 534o 5437 47 12 8603 8692 8782 8872 8962 905I 47 I3 5534 5632 5729 5826 5923 602046 13 9141 923I 9320 94I0 9499 9589 46 I4 6II7 6215 6312 6409 6506 660345 i4 9678 9768 9858 9947..36. 126 45 I5 6700 6797 6894 699I 7087718444 544 9.3602I5 o3050 394 o484 573 o662 44 i6 728I 7378 7475 7572 7668 7765 43 6 0752 0o84I o0930 o I9 I1og9 I98 43 17 7862 79558 85 81 528248 8345 42 I7 I 287I 376 465 I 554 I 644 I 73342 I 8 8442 8538 8635 8731 8828 8924 4I I8 1822 191I 2000 20892 I78 2267 4I 19 902I 9II7 9213 93Io 9 S0240 19 2356 2445 2534 2623 27I I 2800 4 20 9.329599 96955 979I 9888 9984..8o 39 209.362889 2978 3o67 3I56 3244 3333 39 21 9. 330176 0272 o368 o465 o56i o657 38 21 3422 35I I 3599 3688 3777 3865 38 22 0753 o849 0945 Io4I I I37 1233 37 22 3954 40424I3 I 4220 4308 43.9737 23 I329 I424 1520 I616 17I2 1I808 3- 23 4485 4574 4662 475I14839 4927 36 24 90o3 I999 2095 2I9I 2286 2382 35 24 50or6 5I04 5193 528I 5369 5457 35 25 2478 2573 2669 2764 2860 2956 34 25 5546 563415722 58I o5899 5987 34 26 3o5I 3I47 3242 3337 3433 3528 33 26 6075 6I63 6251 6339 6427 65I6 33 27 3624 3719 384 39Io 4005 4oo00 32 27 6604 6692 6780 6868 6956 7044 32 28 4I95 4291 4386 448i 4576 4671 3I 28 7131 72I9 7307 7395 7483 757i 31 29 4767 4862 4957 5052 5147 5242 30 29 7659 7747 7834 7922 8oio 8 098 3o 30 9. 335337 5432 5527 5622 57I6 58 I 29 30 9.368I85 8273 836I 8448 8536 8624 29 3I 5906600oo 6096 6I9I 6285 6380 28 31 87II 8799 8886 8974 9061 9149 28 32 6475 6570 66646759 6854694827 32i. 9236 9324|94I I 9499 9586 9673 27 33 7043 7I37 7232 7326 7421 7515 26 33 976I 98489936.. 23. IO.197 26 34 7610 770~47799 7893 7988 80 82125 34 9.3702850 3720459 o546 o634 072 25 35 8I76 8271 8365 8459 8553 8648 24 35 o8o8895og982 Io69g i56 I243 24 36 8742 8836 8930o9024 9118'92 2 23 36 3301I4I71I504 59I 1678 I76523 37 9307 940I 9495 9589 9683 9777 22 37 I852 I939 2026 2II3 2200 2287 22 38 987I 9964..58.I52.246.340 21 38 2373 2460 2547 2634 2721 2807 2I 39 9.340434 0528 0621 07I5 o809o 903 20 39 2894 298I13067 3I54 324I 3327 20 40o 9.34o996 io90o I84 1277 1371 I464 I9 4o 9.373414 35oo 3587 3674 376063847 19 41 I558 1652 I745 I839 I932 2026 i8 41 3933 4020 4I0o64192 4279 4365 i8 42 2119 22I202306 2399 2493 2586 I7 42 4452 45384624 47II 4797 4883 I7 43: 2679 2772 2866 2959 3o52 3I45 i6 43 497o 5o56 5I42 5228 53I4 54oi i6 44 3239 3332 3425 35I8 36ii 3704 15 44 5487 5573 5659 5745 583I 59I7 5 45 3797 3890 3983 4076 4I69 4262 I4 45 6003 6089 6I75 6261 6347 6433 I4 46 4355 4448 454i 4634 4727 4820 3 46 65I9 66o05669I 6777 6863 6949 3 47 4912 50oo55098 5I9I 5283 5376 I2 47 7035 7I2o0 720 6 7292 7378 7464 12 48 5469 556i 5654 5747 5839 5932 II 48 7549 7635 772I 78o6 7892 7978 1i 49 6024 617 76210 6302 6395 6487 Io 49 8063 8149 8235 8320 84o6 8491 Io 50o 9.346579 6672 6764 6857 6949 704 9 50 9.378577 86628748 8833 8919 9004 9 5i 7134 7226 73I8 7410 75037595 8 5i 9089 9175 92609346 9431 95I6 8 52 7687 7779 787I 7963 8o56 8I48 7 52 960I 9687 9772 9857 9942.,.28 7 53 82408332 8424 85I6 8608 8700 6 53 9.380II3 oI98o0283 o368 o454 o539 6 54 8792 8884 8976 9067 959 925I 5 54 o624o0709o0794 o879 o9641 o49 5 55 9343|9435|9526 9618 97I0o9802 4 55 II34 I2I19 304 I389 1474 i559 4 56 9893 99851..77.I68.260.352 3 56 I6431I728 i8i3 1898 I983 2o6q1 3 57 9.350443 o535 o626 0718 08o8 09 og090I 2 57 2152 2237 2322 24o6 249I 2576 2 58 0992 io84 II75 I266 i358 1449 I 58 2661 2745 2830 2914 2999 30 84 x 59 i54 1632 1723 i8I4 1906 1997 o 59 3I68 325313337 3422 35o06359I 0'60' 50,' 40"' 30- 20" 10", 60- 50" 40"- 30" 20, 10"1' Co-sine cf 77 Degrees. Co-sinof 76 Degrees.! P t ~P 1" 2" 3/ 4" 5" 6" 7" 8" 9. P- 1" 2" 3" 4" 5" 6"1 71 811 911 P. Part 9 19 28 38 47 57 66 76 85 P.art 9 18 26 35 44 53 61 70 79 L~ ~~ R9 q 45q~17i7 LO GA R IT HM IC TANGENTS. 3 Tangent of 12 Degrees..d Tangent of 13 Degrees. { 0" 10" 20", 30", 40"'501,, 10"1 20", 301 40 1 501, 0 9.327475 7578 768277857888 7992 59 oQ. 363364 346o 3556 3652 3748 384-/ 59 I 8o95 8199 8302 84o5 8509 86I2 58' I 3940 4036 4132 4228 4324 442058 8715 88I9 8922 9025 9I2819231 57 2 45 5 46 I 4707 4803 4899 4994 57 3 9334 9438 954I 9644 9747 9850 56 3 5090 5I86 5282 5377 5473 5568 56 4 9953..56. 59.262.365.468 55 4 5664 5760 5855 595I 60 46 6I 42 55 5 9.330570 o673 0776 o879 0982 io84 54 5 6237 6333 6428 6524 6619 67154 6 II87 I290 I393 I495 I598 I70I 53 6 68io 6905 700I 70 96 7I91 7287 53 7 I8o3 I9o6 2008 2III 2213 2316 52 7 7382 7477 7572 7668 7763 7858 52 8 2418 2521 2623!2726 2828 2930 5I 8 7953 8oA8 8I43 8239 8334 8429 5I 9 3033 3I35 3237 3340 3442 3544 50 9 8524 86I9 87I4 8809890489 999 5 Io 9. 333646 3748'3851 39534o055 4 547 49 Io 9.3690949I 89 9284 9378 9473 9568 49 II 4259 436I 4463 4565 4667 4769 48 I I 9663 9758 9853 9947..42..37 48 I12 487 14973 507 5I7715279538047 129.370232 o326 042I o0 56 o6io 0705 47 13 5482 5584 5686 5788 5889 599 I 46 I3 0799 o894 o989 083 I I 78 I27246 I4 6093 6194 6296 6398 6499 66oi 45 I4 I367 I46i I556 I650 I744 1839 45 i5 6702 6804 6905 700 7 71o8[721o44 I5 I933 20282122 2216 23I I 2405 44 I6 731 74I3 75I4 76I5 771 7 7818 43 i6 2499 2593 2688 2782 2876 2970 43 17 79I9 802I 8I22 8223 8324 842642 17 3064 3I59 3253 3347 344i 3353 4a I8 8527 8628 8729 8830 893I 9062 4I i8 3629 3723 38I7 39I 4005 4099 4I 19 9I33 9234 9335 9436 9537 9638 4o 1 9 4I93 4287 4381 4475 4569 4662 4o 20 9.339739 9840 994I..42.I43.243 39 20 9.374756 4850 4944 5038 5I3I 5225 39 2I 9. 340344 o445 o546 o646 0747 o848 38 2I 53I9 54I3 5506 5600 5694 5787 38 22 0948 I049 iI5oI 250o 35i I45I 37 22 588I 5975 6068 662 62 6255 6349 37 23 I552 I652 1753 i853 I954 2054 36 23 6442 6536 6629 6723 68 6 69I0 36 24 2I55 2255 2355 2456 2556 2656 35 24 7003 7096 7I90 7283 7376 7470 35 25 2757 2857 2957 3o57 3I58 3258 34 25 7563 7656 7750 7843 7936 8029 34 26 3358 3458 3558 3658 3758 3858 33 26 8I22 82I6 8309 84o2 8495 8588 33 27 3958 4o58 4i58 4258 4358 4458 32 27 868i 8774 8867 8960 9053 9I46 32 28 4558 4658 4758 4858 4957 5057 3I 28 9239 9332 9425 95I8 96I I 9704 3 29 5I57 5257 5357 5456 5556 5656 30 29 9797 9890 9983..75. i68.26I 3o 3o 9.345755 5855 5954 6o54 6I54 6253 29 30o 9.380354 o446 o539 o6320o725 0817 29 31 6353 6452 6552 665I 67516850 28 3I 09I0 I003 I095 I 88 I2801I373 28 32 6949 7049 7o 48 7248 7347 7446 27 32 I466 I558 i65II 743 i836 I928 27 33 7545 7645 7744 7843 7942 8042 26 33 20202 I I 3 2205 2298 2390 2482 26 34 8i4I 8240 8339 8438 8537 8636 25 34 2575 2667 2759 2852 2944 3036 25 35 8735' 8834 8933 9032 9I3I 9230 24 35 3129 322I 33I3 3405 3497 3589 24 36 9329 9428 9527 9626 9724 9823 23 36 3682 3774 3866 3958 4050 4142 23 37 9922..2I.I20.2I8.3I7.416 22 37 4234432644I8451046o2469422 389,3505I4 o6I3 07I2 o8Io 0909 Ioo07 2 38 4786 4878 49705062 5 I 53 5245 2 39 Izo6 ]204 i3o3 I4o0 i500 I598 20 39 5337 5429 552I 5612 5704 5796 20 40 9.35I697 1795 18941I992 20902I189 I9 4o9 385888 5979 6071 616362546346I19 4I 2287 2385 2483 2582 2680 2778 i8 4I 6438 6529 662i 6712 6804 6895 I 8 42 2876 2974 3073 317I 3269 3367 17 42 6987 7078 7170 726I 7353 7444 17 43 3465 3563 366I 3759 3857 3955 I6 43 7536 7627 7718 78I0 790I 7992 i6 44 4053 4I5I 4249 4347 4445 4542 15 44 8084 8I75 8266 8358 8449 85401 i5 45 4640 4738 4836 4934 5o3i 5I29 I4 45 863i 8722 88I4 8905 8996 9087 14 46 5227 5324 5422 5520 56I7 5715 i3 46 9x78 9269 9360 945I 9542 9633 13 47 5813 59I0 6008 6o 5 6203 63oo001 I2 47 9724 98I5 9906 9997.4.88 I79 12 48 639816495 6593 6690o6787 68851 I 48 9.390270 o36I 0452 o543 o633 0724 1 49 6982 707971 7717274 737I 7469 Io 49 085 o090o6 0997 I087 II 78 2691 0 50o 9.357566 7663 7760 7857 7954 8052 9 50 9.39360 I45o i54 I632 1722 i8I3' 9 51 8149 8246 8343 8440 8537 8634 8 5 I90o3 1994 2085 2175 2266 2356 8 5 873 18828 8925 9022 919 92I6 7 52 2447 2537 2628 2718 2 208 2899 7 53 9313 9409 9506 9603 9700 9797 6 53 2989 3080 3I7o 3260 3351 344i 6 54 9893 9990..87. 84.280.377 5 54 353I 3622 37I2 3802 3892 3983 5 55 9.360474 ~57~ o667 o763 o86o o957 4 55 4073 4I63 425314343 4433 4523 4 56 - I053 II50o 246 I343 I439 i535 3 56 46I414704 4794 4884 4974 5064 3 57 I632 1728 825 192 20oI7 2I14 2 57 5I54 5244 5334 5424 55I4 56041 58 2210 2306 2403 2499 2595 269I 1 58 5694 5783 5873 5963 6053 6I43 1 59 2787 2884 2980 3076 3I72 3268 0 59 6233 6322 6412,6502 6592 668i 0 60" -- 50"" 5 40"" 30t 20'' - -- 60" 50/ 40" 30" 20" 10"' Co-tangent of 77 Degrees. 4 Co-tangent of 76 Degrees. - P -art 1" 2"/ 3"1 4/ 5"/ 6" 7" 8" 9i (p 1t" 2" 3/I 4"1 5" 6" 7/" 8" 9" [' t 10 20 30 40 50 60 70 80 90' ~ 9 19 28 37 46 56 65 74 83 LOGARITHMIC SINES..Sine of 14 Degrees.. Sine of 15 Degrees.' I 10"' 20'30" 140"150" of 0" 10" 20' 30"' 40' 50 o 9.383675 3760 3844 3928 40o3 4097 59 0o9.412996 3075 3I53 3232 33I 3389 59 I 4182 4266 435o 4435 45 I9 4603 58 I 3467 3546 3624 3703 3781 386o 58 2 4687 4772 4856 4940 5024 51I8 57 2 3938140oI64095 4I73 4252 433o 57 3 5I92 5277 536I 5445 5529 56I3 56 3 44o8 4486 4565 4643 4721 48oo 56 4 5697 578 I 5865 59496033 6033117 55 4 487814956 5034 5 I 2 5 I 90529 55 5 620I 6285 6369 6452 6536 6620 54 5 5347 5425 55o3 558i 5659 5737 54 6 6704 6788 6872 6955 7039 7I23 53 6 58I5 5893 5971 6049 6I27 62o0553 7 7207 7290 7374 7458 7541 7625 52 7 6283 636i 6439 65I7 6595 6673 52 8 7709 7792 7876 7959 8o43 8I27 5I 8 6751 6828 6906 6984 7062 7I40 5I 9 82Io 8294 8377 846i 8544 8627 50 9 7217 7295 7373 745I 7528 7606 50 io 9 3887II 8794 8878 896I 9044 9I28 49 I0O 9.4I7684 776I 7839 79I7 7994 8072 49 II 92I 11 9294 9378 946I 9544 9627 48 II 8I50 8227 8358382 84608537 48 12 97II 9794 9877 9960 *.43.126 47 12 86I5 8692 8770 884718925 9002 47 i3 9.39o2IO o293 o376 o459 o542 o625 46 I3 90799 g57 9234 93I2 938919466 46 4 o0708 079gI o874 957 I040 I I23 45 I4 9544 962I 969819776 9853 9930o45 5 1206 I289 I37I 1454 I537 I620 44 I5 9.420007 oo0085 o62o239o36 610393 44 i6 I703 1786 I868 I95I 2034 2II7 43 I6 0470 o548 o62510702 107791085643 I7 2199 2282 2365 2447 2530 26I3 42 17 o933 1IO1O I087 I64 124I i3i8 42 I8 2695 2778 2860 2943 3025 3Io081 8 I 395 I472 I 549 I6261703 1I 78o 41 19 3191 3273 3356 3438 3520 360 3 4o I9 I857 1933 20I o12087 2164 224I 4o 20o9.393685 3768 385o 3932 4oi 5 4097 39 20 9.422318 2394 2471 2548 2625 270I 39 21 41 79 4262 4344 4426 45o8 4591 38 21 2778 2855 293I 3oo8 3o85 3i6i 38 22 4673 4755 4837 49I9 5002 5o84 37 22 3238 33I5 339I 3468 3544 3621 37 23 5I66 5248 5330o5412 549415576 36 23 3697 3774 385o 3927400oo3 4080 36 24 5658 5740 5822 5904 5 986 6o68 35 24 4I56 4233 43094386 4462 4538 35 25 6i50|6232 63I4 6395 6477|6559 34 25 46I51469I14767 484414920 4996 34 26 664i 6723 6805 6886 6968 7050 33 26 5073 5I49 5225 530o 5378 5454 33 27 7I32 72I3 7295 7377 745817540 32 27 553o 56o6 5682 5758 5835 5911 32 28 7621 7703 7785 7866 7948 8029 3I 28 5987 6o63 6391 62I5 6291 6367 31 29 8III 8I92t8274 8355 843785I8 30 29 6443165I9 6595 667I 6747 6823 3o 30 9,398600 868i 8762 8844 8925 9007 29 30 9.426899 6975 705I 7I27 7202 7278 29 31 9088 9169 9250 9332 943139494 28 3I 7354 743017506 7582 765717733 28 32 957519657 9738 98I9g 99 981 27 32 780917885 796o08036 8 II218I87 27 3319,400062 0I44 0225 0306 0387 0468 26 33 826318339 81484985661864I 26 34 0549|0630 07I1 0792 0873 0954 25 34 87I7 8792 886818944 gOIg 9095 25 35 Io35 IIi6 II97 I277 1358 I439 24 35 9I70 9246 932I 9397 947219547 24 36 1520 i6oi I682 I762 I843 I924 23 36 9623 9698 9774 9849 9924.... 23 37 2005 2085 2I66 2247 2318 2408 22 37 9.430075 oi5o 0226 o3o0 o376 o45i 22 38 2489 2570 2650 273I 28II 2892 2i 38 o527 o602 o677 0o752 o828 o09032 39 297,e 305313I33 32I4 3294 3375 2 39 o978 io53 1128 I203 o3I278 35420 40 9.403455 3536{36i6 3697 377713857 I9 409.43429 I504 I 579 I654 I 729 I8041 I 41 3938 40o184098 4I79 4259 4339 8 4I I879 I954]2029 2IO4 2179 2254 i8 42 4420 4500o4580o 4660 474I 482I 17 42 2329 2403 2478 25531262812703 I7 43 4901 4981 50o6 5i4i 522I 5302 i6 43 2778 2853 2927 3002 3077 1352 i6 441 5382 5462 5542 5622 5702 5782 i5 44 3226 33o0I 3376 345I 3525 3600o 45 5862 5942 6022 6102 6I8I 6.26I i4 45 3675 3749 3824 3898 3973 4048 I4 46 634i 642I 65oi 658i 666i 6740 i3 46 4I22 4I97 4271 4346 4420 4495 I3 47' 6820o6900 6980 7060o 739 72I9 12 47 4569 4644 47I8 4793 4867 4942 12 48 7299 737817458 7538 76I7 7697 II 48 50I61509 5I65 523953534388 49 7777 7856 7936 80i5 8095 8174 IO 49 5462 5537 56II 5685 5760 5834 Io 5019.408254 8333 84I3 8492 8572 865I 9 50 9.435908 5982 6056 6 3i 6205 6279 9 51 8731 88Io 8889 8969 904891I27 8 5i 6353 6427 6502 6576 665o 6724 8 52 9207 928619365 94459524 7 52 67986872 6946 7020 7917094 68 7 53 9682 9762 9841 9920 9999 " 78 6 53 7242 73I6 7390 7464 7538 7612 6 54|9.4IoI57o0237o03I6 o3950o4740o553 5 54 7686 7760 7834 7908 798I 8055 5 55 0632 07II 0790 0869 0948 I027 4 55 8IS29 8203 8277 835I 8424 8498 4 56 Iio06II851I264 I3431I422 I5oo 3 56 857268646 871918793 8867894Il 3 57 I579 16581I737 18I6|I8951I973 2 57 90oI49088 196219235 9309 9382 2 58 2052 2I31 22I0 2288 2367 2446 I 58 945619530 9603 9677 9750 9824 I 591 2524 2603|2682 2760o2839129i8 o 59 9897/997I..44.ii8.I9I.265 0 60" 50" 40" 3020" 10" 60"' 50" 40"~ 30" 20"' 10". Co-sine of 75 Degrees. o-sine of 74 Degrees. I P. part 1" 22" 3" 4" 5" 6" 7 P Parti 1 2 3" 4" 5" 6" 7" 8" 9". P. Part 8 16 24 33 41 49 57 65 73 Part 8 15 23 30 38 46 53 61 68 h' I 8 16.24 57 6 4 -.- - ~ - LOG T R 1 T HAM-IC TA N G E NT S. 3 9 -_ of 1011 20" 3011I 40t.504'1 0" 10"Jo 20" 30" 1 40 1 50" 9.39677I 686I 6o50 7040 7I30 721959 O9.4280528I37822I 83058389 84'3059 I 7309 7399 7488 7578 7667 7757 58 I 8558 8642 8726 88Iol8894 8978 58 2 7846 7936 8025 8 I5 82o048294 57 2 o6219I469230o93I4 9398 9482157 3 8383 8472 85621865i 8740 883o 56 3 9566 9650 9734 98I8 9902 9986 56 4 899qoo90089098 19879276 9365 55 4 9.430070 0O54 0237 o321 o4o5 o489 55 5 9455 9544 9633 9722 98I1 9goo 54 5 0573 o657 07400 o824 o090g8 0992 54 6 9990..79.I68.257.346.435 53 6 1075 II159 243 1326 I4Io I49453 7 9.400524 o6I3 0702 079I o88o o969 52 7 1 577 i66I I745 1828 1912 I995 52 8 io58 II47 1236 I325 I4I3 1502 5I 8 2079 2I62 2246 2329 24I3 2496 5i 9 I59I i68o 1769 1857 I9462035 50 9 2580 2663 2747 2830 291424997 5 10 9.402124122I2 230I 2390 2478 2567149 IO 9.433080o 364 32471333I 3414 3497 49 11 26561274412833 2922 30O1 3099 48 II 358c 3664 3747 383o 3914 3997 48 I2 3I87 3276 3364 3453 354i 363o 47 I2 4o8o 4i63 4246 433o 44x 3 4496 47 i3 3718 3807 3895 398314072 4i60o46 I3 45 79146624745 4828 4912{4995 46 I4 4249 4337 4425 45I4 4602 4690 45 I4 5078 56 I 5244 5 327 54io 5493 45 i5 4778 4867 4955 5o43 5I3I 52I9144. 15 5576 5659 5742 5825 5907 599o 44 i6 53o8 5396 5484 5572 566o 5748 43 I6 6073 6i56 6239 6322 64o5 6488 43 I7j 583615924160I261Io00618816276142 I 7 6570665316736]68I9 690T 6984142 I8 6364 645216540 6628167I668o4 41I i8 706717I507232 73I 5 7398 748014I 19 6892 6980 7068 7I5517243 733I 4o I9 7563 7646 7728 78II 7894 797614o 20 9e4074,9 7507 7594 7682 7770 7858 39 20 o9.438o59 8i4i 8224 83o6 8389 847i139 21 7945S803318I2I 82o08829618384 38 21 855418636 87I9188oI0888418966138 22 847I 855918646 873418821 89o9 37 22 904891I3I 92139296|9378 946o 37 23 8996 9084 9171 9259 9346 9434 36 23 9543 9625 9707 9790 9872 9954 36 24 952I 960919696 97831987I 9958 35 24 9.440036 011 9 0201 0283 o365 ~447135 25 9.410045 oi33 0220 o3071o3951o482 34 25 05291 062 0694 07760858 094~13 4 26 o569So656o0743 o83109I81 o005 33 26 10221II041Ii86 I268I 3501 432|33 27 I092 11791I 266 i3531 44I I 528 32 27 I 5i4 596 1678| 17601 I842 I924132 28.I6I517021I789 I8761I963|2o5o 31 28 200620882I70|2252|23342416 3I 291 2I37|2224 2310 2397124841257i 30 29 249712579 2661 2743{2825 2907 30 309.4 12658|2745 2832 29i913005 3092 29 30 9.442988 3070 3152 3234133I5 3397129 31 3179 3266 3352 3439 3526 36I3 28 3i 3479 356o 3642 3724 38o5 3887[28 32 3699 3786 3873 3959 4o46 4132 27 32 3968|4050o4I32|42i3 4295 437627 33 4219143o064392 4479 456514652 26 33 4458 4539 462I 4702 4784 4865 26 34 4738 4825149"I 4998 5o84 5I7I 25 34 4947 1528 5IIO 5I9I 5272 5354 25 35 5257 53431543o 5516156o3 5689 24 35 5435 55I7 5598 5679 576i 5 842 24 | 361 577515862 5948 6o34 6120 6207 23 36 5923 6005 6o86 6I67 6248 6330 23 37 6293 6379 6465 5 655 6638 6724122 37 64II76492 657316654167351681722| 381 68Io 6896 6982 7068 1754 7240 2I 38 6898 6979 7~6~ 7I4I17222 7303 2I 39 7326 74 317499 75851767 17757 20 39 7384 7465 7546 7627 7708 7789 20 40o9.417842 792880oI4 8I00|8I86|8272 19 4o 19.447870~7951 803218II3 8I94{82751I9 41 8358 8444 853o 86I618701 8787 i8 4 8356 8437 85I818599 868o 8761 i8 42 8873 8959 9044 9I30 92I6 9302 I7 42 884i 8922 9003 9084 9i6419245 17 43 9387 9473 9559 964419730~9816 I6 43 9326 9407 9487 19'S 9649 9730o 6 44 990I19987 1.72. i58.244.329 I5 44 98Io1989I 99721.. I33. 2i31I5 45 9.420451 o500oo o586 67 0757 1842 I4 451 9.450294 o375 o455 o536 o6zclo697'4 |461 C927[ ioi3 IO981II84I 269 i354 i3 46 o777[o8581o938 1oi9g 0 699 I8o0 I3 47 I44o 0525 i6io I696 I78~ i|866 12 47 I260 i34i 142I I502 ~582 I662 I2 48 I952 203712122 220712292 2378 II 48 I74341823 I03 9o38420o64 2,44 |I 49 2463 2548 2633 27I8 2803 2888 IO 49 2225 2305 2385 2465 2546 2626 Io 509.422974 3o0591344 3229 33I413399 91 50 9, 4527o6 278612867 2947 30 27 3107 9 51 3484 3569 3654 3739 3824 3909 8 5i 3I87 326713347 3428 350835881 8 52 3993 4078|4I63|424814333|44I8 7 52 366813748 3828 3908 398814068l 7 53 45o034587]467214757[4842949276 6 153 4I48 4228143o438 884468 4648 6 54 5oii 5o96 5I8I 5265 535o 5435 5 54 4628 4708 478714867 494750o27 5 55 55 915604568957731 85585942 4 55 5Io071587 526715346S5426S550o6 4 56 60276I I2 6I96 628I |6365 645o 3 56 5586 5655 5745 5825 59o5 5984 3 57 6534 66I9 6703 6788 6872 6956 2 57 6o64 6I44 6223 63o3 6383 6462[ 2 58 704I 7125 7210 7294 7378 7463 I 58 6542166221670I1678I 686o 6940 I 59 7547 7631 77I5 7800 7884 7968 o0 5 70190 7I78 7258 7337 7417| 0 60" 0 40"1 30" 20" 10 6, 50" 40" 30" 20" Co-tangent of 75 Degrees. Co-tangent of 74 Degrees. ] p 1" 2" 3" 4" 5" 6" 7" 8" 9 1P 1" 2" 3/" 4" 5" 6" 711 8" 9'/ 9 17 26 35 43 52 61 69 78 art8 16 25 33 41 49 57 65 74 _.Portj -"." 3" 4// 5/1 6" 7/1 8/1 -_t 40 LOG R.ITH MI C S N rE S. Sine of 16 Degrees. Sine of 17 Degrees. 0"' 10"' 20" 30" 40"' 50" 0 10' 20"/ 30"/ 40" 30" o 9.440338 o4 I o485 o0558 o632 0705 59 o 9.465935 6004 6o-31 6I42621 I 6280o59 I o778 o852 o25 ogg998 I072 II 45 58 I 6348 6417 6486 6555 6623 6692 58 2 I2188 1292 I365 I438 I5iI I584 57 2 6761 6830 6898 6967 7036 7104 57 3 I658 I73I i8o4 1877 I95o 2023 56 3 7173 7242 73IO 7379 7448 7516 56 4 2096 2170 2243 2316 2389 2462 55 4 7585 7653 7722 779~ 7859 7928 55 5 2535 2608 2681 2754 2827 29oo00 54 5 7996 8065 8133 8202 827o 8338 54 6 2973 3046 3119 319 3 3265 3337 53 6 8407 8475 8544 86I2 8681 8749 53 7 341o 3483 3556 3629 3702 3774 52 7 88I7 8886 8954 022 909gI 9I59 52 8 3847 3920 3993 4066 4I38 4211 5I 8 9227 9296 9364 9432 9500 9569 5I 9 4284 4356 4429 4502 4574 4647 o 9 9 9637 9705 9773 9842 991o 9928 50 10o 9.444720 4792 4865 4938 50IO 5083 49 IO 9 470046 oI 14 oI82 025I 03g o0387 49 I I 5I55 522815300 5373 5445 5518 48 I 0455 o523 05o5I 659 0727 0795148 12 5590 5663 5735 58 08 588o 5953 47 12 o863 0o93I 0999 I067 I 35 I2013 47 13 6025 6097 617o 6242 63I4 6387 46 3 I27I 339 I407 1475 i543 i6ii 46 I4 6459 6531 66o046676 6748 6820 45 IA I679 1746 I814 18882 I950o2018 45 I 5 6893 6965 7037709 I7I827254 44 15 2086 2I53 222I 2289 2357 2424 44 I6 7326 7398 747~ 7542 7614 7687 43 I6 2492 2560 2628 2695 2763 283I 43 I7 7759 7831 7903 7975 8047 8I 19 42 I17 2898 2966 3034 3IoI 3169 3237 42 I8 8191 8263 8335 8407 8479 855I 4i I8 3304 3372 3439 3507 3575 3642 4I I 9 8623 8695 8767 8838 89Io 8982 4o 19 37I0 3777 3845 3912 3980 4047 4c 20 9.449054 91 26 9198 9269 934I 9413 39 20 9.47411I 5 41 82 42504317 4384 4452'39 21I 9485 9557 9628 9700 9772 9844138 21I 4519 4587 4654 472I 4789 4856 38 22 99I5 9987..59. I3o.202.274 37 22 4923 499I 5058 5125 5193 5260 37 23 9.45o345 0417 o488 o560 o632 0703 36 23 5327 5394 5462 5529 5596 5663 36 24 0775 o846 918 o989 Io6I I 132 35 24 5730 5798 5865 5932 5999 6066 35 25 I204 1275 1347 I4I8 1489 i56i 34 25 6I33 6200 6268 6335 6402 6469 34 26 1632 I704 1775 I846 I g1918 1989 33 26 6536 6603 6670 6737 680 4 687i 33 27 2o60 2I32 2203 2274 2345 2417 32 27 6938 7005 7072 7I39 7206 7273 32 28 2488 2559 2630 2702 2773 2844 3 I 28 7340 7407 7473 7540 7607 7674i31 29 2915 298613057 3I29 3200 327I 30 29 774I 7808 7875 794I 8008 8075 3o 30 9.453342 3413 3484 3555 3626 3697 29 30 9.478142 8209o8275 8342 8409 8476 29 3I 3768 3839|39Io 398I 4052 4123 28 3I 8542 8609 8676|8742 8809 8876 28 32 4194 4265j4336 4407 4477 4548 27 32 8942 9009 90769i142 9209 9275 27 33 - 4619 46901 4761 4832 4903 4973 26 33 9342 94o9 947519542 9608 9675 26 34 5o44 15I55ri86 5256 5327 5398 25 34 974I19808 9874 994I...7..7425 35 5469 5539 56Io 568I 575I15822 24 35 9.480o40o0207 027310339 o4o6 o47224| 36 5893 5963 6034 6Io4 6I75 6246 23 36 0539 o605 067I 0738 o8o4 o870 23 37 63I6 638716457 6528 6598 6669 22 37 0937 I00oo3 o69II35 I202 I268922 38 6739 68io 688o 6951 7021 7091 21 38 I334 i4oo00467 i533 1599 i665 21 39 7162 723297303 7373 7443 75I4 20 39 1731 1798 I864 I930 I996 2062 20 40 9.457584 7654|7725 7795 7865 793619 40 9.482128 2I94226I 12327 2393 2459 I9 4I 8006 8076 8I46 82I7 8287 8357 I8 4I 2525 2591 265712723 2789 2855 i8 42 8427 8497 8567 8638 87O8 8778 17 42 292I 2987 3053 3119 3I85 325I 17 43 8848 89I8 8988 9o589128 99898 i6 43 3316 3382 344835 I43580o 3646 16 44 9268 9338 9408 9478 9548 9618 iS 44 37I2 3778.3843 3909 3975 4o4I i5 45 9688 9758 9828 9898 9968..38 I4 45 4107 4I72 4238 4304 4370 4435 14 46 9.460Io8 OI.78 0248lo3I7 0387 o457 I3 46 450I 4567 4632 4698 4764 482913 47 o527 o597 o667 0736 o8o6 0876 I2 47 4895 496I 5o 26 5092 5I58 5223 I2 48 o0946 ioi5 io85 I55 1224 1294 II 48 5289 5354 5420o 5485 555I 56I7 II 49 I364 I433 503o3 573 16421 1712 I 49 5682 574858I 315879 5944 60091 I 5019 46I782 I85 1921 1990 20609 2129 9 50 9.486075 6I40 6206 6271 6337 6402 95i 2I99 2268 2338 2407 2477 2546 8 5I 6467 6533 6598 6664 6729 6794 8 52 2616 2685 2755 2824 2894 2963 7 52 6860o 6925 699017055 7121 7186 7 53 3032 3Io02317I13240o33io 3379 6 53 725173167382 7447 75I2 7577 6 54 3448 35I8 3587 3656 3725 3795 5 54 7643 7708 7773 7838 7903 7968 5 55 3864 3933 4002 4072 4141 4210 4 55 8034 80998I64 8229 8294 8359 4 56 4279 4348 44I7:4486 4556 4625 3 56 8424 8489 855486I98684 8749 3 57 4694 4763 4832 490I 4970 o5039 2 57 88I4 8879 894419000 9074 9139 2 58 5108 5I77 5246|53i5 5384 5453 I 58 9204 9269 933419399 9464|9528 I 59 5522 559i 566o 5729 5798 5866 o 59 9593 9658 9723 9788983 5399 I8 o 60"' 5-0, 40' 30'1 2-' 10" 60-'1 50," 40" 30' 20" 10 Co-sine of 73 Degrees. Co-sine of 72 Degrees. p pPart(" 1" 21 3" 4" 5" 63" 7" 8" 9 P. t 1" 2" 3/" 4/1 511" 6' 7" 8g'1 9". Part 14 21 28 36 43 50 57 64 P Part 7 13 20 27 33 40 4' 53 660,I -~ $ 4~ 3 35 76 L G A i_.v'P M t C'1 ANGENTS. 4'.i~ ~Tangent of 16 Degrees. {[ Tangent of 17 Degrees.. 0" _ 10" 1 20" 30" 40" 50" I 0" 10" 20" 30" 40' 50 0 9.457496 7576 7655 7735 7814 7894619 o 9.485339 544 5545490 5565 564o57I 59 I 7973 8052 8 I 32 821 I 8290 8370 58 I 5791 5866 5941 6oi6 6092 6I67 58 2 8449 8528 86o8 868718766 8846 57 2 6242 63I7 6392 6467 6543 66i8 57 3 8925 9004 90o83 963 9242 9321 56 3 6693 6768 6843 69I8 6993 7068 56 4 9400 9479 9558 9638 97I7 9796 55 4 7I43 7218 7293 7368 7443 75I8 55 5 9875 9954 1..33 1. 12..91.27054 5 7593 766817743 788187893 7968154 6 9.460349 o428 o507 o586 o665 0744 53 6 8o4318II8 8193 8268 8343 84I8 53 7 082310902 og98 Io601I I39 1218 52 7 8492 8567[8642 87I718792 8866 52 8 1297376 454 533I6I2 I6915I. 8 894l oI6 90IgOgI 9I6619240 93I51 5 9 1770 1849 1927 2oo006 2085 26450 9 9390 9465 9539 96I419689 9763 50o 10 9.462242 232I 2400 2478.25572636 49 I1o 9.489838 99I3 9987..62.137.2I1149 II 27I5 2793 2872 2950 302,9 30o8 48 II 9 490286 o360 0435 05I o0584 o659 48 12 3I86 3265 3343 3422 3Soi 3579 47 12 0733 o8.o8 0882 0957 io3i iio6 47 13 3658 3736 38I5 3893 3972 40o5o46 I3 1 80Io255 I329 1404 1478 I552146 14 4 28 4207 4285 4364 4442 452 I 45 I4 I627 I70I 1776 I85o I924 1I999 45 IS 4599 4677 475614834 4912 499I 44 i5 2073 2I47 2222 2296 2370 2445 44 i6 5069 5 47 5226 53o4 5382 546o143 I6 2519 2593 2668 2742 2816 289o043 I 7 5539 56I7 5695 5773 585I 593o142 17 2965 30391 3II3 387 3261 3336 42 I 8 6oo8 6o86 6i64 6242 6320 639814' I8 34io 3484 3558 3632 3706 3780 4I 19 6477 6555 6633 67 I I 6789 6867140 I9 3854 3929 4oo3 40774I 5 I 42254o 20 9.466945 7023 7IOI 7I79 7257 7335 39 20 9.494299 4373 4447 4521 4595 4669 39 21 74I37491 7569 7647 7724 7802 38 21 4743 4817489I 4965 5039 511338 22 7880 7958 8o36 8I I4 8 I 92 8269 37 22 5i 86 5260 5334 54o8 5482 5556 37 23 8347 8425 8503 858 8658 87366 36 23 5.63o 57o4 5777 585i 5925 5999 36 24 88I4 8892 8969 9047 91g25 920235 24 6073 6i46 6220 6294 6368 644I 35 25 9280 9358 9435 95I3 9591 9668 34 25 655 6589 6663 6736 68 Io 6884 34 26 9746 9823 990oI 9979..56. i34 3 26. 6957 7031 7I05 7178 7252 7326 33 2719.4702 II 0289 o366 o444 o52 I 0599132 27 7399 7473 7546 7620 7693 7767132 28 0676 0754 o83I o0909o o986 o63 3I 28 7841 7914 7988 8o6I 8i35 8208 3i 29 I141 128 I2S95 I373 I45o l528 30 29 8282 8355 8429 8502 8575 8649 3o 30 9.47I6051 6821 759 I837 I914 I99I 29 30 9.498722 879618869 8943901 619089129 31 2069 2I46 2223 2300 2377 2455 28 31 9I63 9236 9309 9383 9456 9529 28 32 2532 2609 2686 2763 2840 29I[ 27 32 9603 96769749982298969969 27 33 2995 3072 3I49 3226 33o03338o 26 3-13 9.500042 OI5 1o891 0262 o335 o4o8 26 34 3457 3534 36Ii 3688 376513842l25 34 o48I o555 o628 0701I 7740 847 25 35 39gI 3996 4073 4I50 422714304124 35 0920o0993 I0o66 I40 I2I3 1286 24 3;5 438 14458 4535 46I2 4688 4765 23 36 I3591432 I505o 578 i65I I724 23 3 3 4842 49Ig 4996 5073 154915226.22 371 I7971I870 I943120I6 2o89 2162 22,38 53o3538o 5456 5533 56io 5687 2I 38 2235 2308 238I 2453 2526 2599 21 39 5763 1584o 59I7 5993 607o 614712o 39 267212745 28I8 289I 2964 336 20 40 9.476223 6300 6377 6453 653o066o6 19 40o9.503109 3182 3255 3328 340oo34731 9 4i 6683 6759 6836 69I3 69897066 88' 1 1 41 3546 3691 369I 3764 3837 39Io I8 42 7I42172I9 7295 7372 744817524 I7 | 42 398214055 4I28 4200 4273143461 7 43 760I17677 7754 7830 7906 7983 i6 43 44I8 4491 4563 4636 4709 478I i6 44 8059 8136 82I2 8288 836518441 i5 44 485414926 4999 5072 5I44 52r7 i5 451 85I7 8593 8670 8746 8822 8899 i4 | 451 5289 5362 5434 5507 5579 5652 I4 46 8975 9051 9127 9203 9280o9356 I3 46 5724 5796 5869 594I 6o0I46o86 13 47 9432 9508 9584 9660o9737|98I31I2 | 47 6i59 623i 6303 6376 6448 6520 I2 48 988999965..4I. 117. I93.269II 48 6593J6665 6737 68I 6882 6954 I I 49 9.480345 0421 o 0497 o73 o 649 o 7251 I 49 7027 7099 77I1 7243 7316 7388 Io 50o19 48080I o877o o93 1029 I Io5 I i8i 9 5019.5074607532 76o05 7677 77491782 9 5I1 I257 I3331I4o8 I4841I56o 16361 8/ 5i 7893 796518o38 81IO 8182 8254 8 52 I7I217881I863 I939 20oI5209II 7 |52 8326 8398 8470o8542 86I4 8687 7 53 2I67 2242 23I8 23942470o2545j 6 53 8759 883I 8903 8975 9047 91I9| 6 54 262I 2697 2772 2848 2924 2999 5 54 9I91 9263 9335 940/ 9479 9551 5 55 307513I5I 3226 3302 3377 3453l 4 I155 9622 9694 9766 9838 99I1 9982| 4 56 3529 3604 368o 3755 383I 3906A 3 | 56 (.5:oo54 0I26 o I98 0269 o34I o4313 3 57 3982 4o57 4I33 42o8 428443591 2 57 o4851o557 o629 o7o00o772 o8441 2 58 4435 45io 4585 466I 4736148I21 x l58 o9gI60987 io59 II3 I 3I203 I274 I 591 4887 4962 5038 5ii3 5I88 S264o 0 59 I346 I4I8 1489 I56I i633 I704 0 - 60" 50" 401 30"1 20" 10" ( 0"o 50" 40" 30" 20" 10" Co-tangent o- 73 Degrees. Co-tangent of 72 Degrees. P. Part 1" 3" 4" 5"' 6' 7"' 8"; 9" - Part 1" 2" 3" 4"' 5"' 6' 7" 8"' 9'1. 8 15 23 31 39 46 54 62 70 7 15 22 29 37 44 51 59 66 L 3 G A R IT' H —M I C 8 I N ES. Sine of 18 Degrees. Sine of 19 Degrees.. 0 I 10" 20"/ 301 40 0" 10 " 20" 30" 40"1 50"' 09.489982..47 I112 177.241.306 59 0-9. 5,2642 2703 2764 28251 28e 2948 59 1 9 114937I o436 0500 o565 o630 o695 58 I 3009 3070 3I3I 3I92 3253 33i4 58 - 759 824 o889o953 ioi8 1082 57 2 3375 3436 3497 3558 36I9 368o 57 3 II47 I2I12 I276 I34I I4o5 I470 56 3 374I 3802 3863 3924 3985 4o46 56 4 I535 I5991I664'I7281I7931I&57155 4 4o07 4I68 4229 4289 435o 44i155 5 19221I98620o5I12Ii52I79 2244154 5 44724533 459414655 47I5 4776 54 6 2308 2373 243712502 2566 2630 53 6 4837 4898 4959 50I9 5o8o 5I4I 53 7 2695 2759 282312888 2952 30o6 52 7 5202 5262 5323 5384 5445 55o5 52 8. 308 I 3i45 320913273 3338 34025 I,. 8 5566 5627 5687 5748 5809 5869 51 9 3466 353o 3595 3659 3723 3787 5o 9 5930 599 6o5 6II112 6I72 6233 5o o109.49385I 39I5 3980o044 418 4I7249 14 9 516294 6354 64I5 6475 6536 6596149 II 4236 4300oo4364 4428 449214556 48 II 6657 67I7 6778 6838 6899 6959 48 12 462I 4685,4749 48i3 4877 494I 47 I2 7020 7080 7I4I 7201 7262 7322147 13 5oo05 50o6953315r96 5260 5324 46 I3 7382 7443 7503 7564 7624 7684 46 14 5388 5452'55I6 5580 564415708 45 I4 7745 7805 7865 7926 7986 8o46145 i5 5772 5835 5899 596316027 609I144 I51 81078'67 82271828718348 84o0844 I6 6I54162I8 6282 6346 64IO16473 43 i6 8468 8528 8589 8649 8709 8769143 17 6537 66oi 6664 6728 6792 68564 2 I7 8829 889o 8950 oo907o90 79I3o 42 i8 69I9 698370477IIO 17I74 7237 4I I8 9I90 9250o931II 9371 943I 949I 4I I9 7 7301 7365 7428 7492 7555 76,9 4o I9 9551i 961 967I 973I 979I 985, 4o 20 9.49768217746178o10787317937 8ooo039 20 9.5 99II 997I.. 3i.9I. I 5I.2II 139 21 8064 8I2718I90o8254183I71838i138 2I 9.520271 o33I 039I o45 10o5ii 057i 38 22 8444 8508185718634 8698 876 I1 37 22 o63i 69 0750 o8o 0870 o09303o 37 23 8825 8888i895I 90I5 9078 9I4I 36 23 o990o o5o 1IIIo I69I229 1I289 36 24 9204 9268 933I19394 9458 952I 35 24 1349 I409 I468 1528 i588 I648 35 25 95841 9647 9710o9774198371990oo34 25 I7071I7671I827 I8871946 2006 34 26 9963..261..9go.53.2I16.279133 26 20662I25 2I85 2245|2304|2364|33 27 9.500342 o4o5io468 o53i 0594 o658 32 27 2424 2483 2543 26021266212722132 28 072I o784 o0847 09IO 0973 io36 3I 28 2781 284I 2900 2960 30I9 3079 3I 29l. IO99 II62 I225 1I288 i35I i44130 29 3I38 3I9813257133I71337613436130 30~9.50I476 153911602 i665 17281I79I 29 30o9.52349513555 36I4 3674 373313792329~ 31 I854 I9I7 I980 2042 2Io5 2168 28 31 3852 3911 3971 4o0304089 4I49 28 32 223I 229412356 2419 2482 2545 27 32 4208 4267 4327 4386 4445 45o5 27 331 2607 267012733 2796 2858 292I 26 33 4564 4623 4683 4742 48o1 48 6o 26 34 2984 30463I10913I72 323413297125 34 49201497950o38 5097 5i56152I625 35 336o 3422 3485 3548 36io 3673 24 35 5275 5334 5393 5452 55I2 557I 24 36 3735 379813860o3923 3985 4048 23 36 563o05689 5748 5807 5866 5925 23 37 410o 4I73 4235 42988436o04423 22 37 5984 6o44 6io3 162 622I16280|22 38 4485 4548146Io04673 4735[479712I 38 6339 6398 6457 65i616575 6634[2I 39 4860o4922|4985150475IO951 5I72 20 39 6693 6752 68II 6870o6928|6987|20 4o19.505234 52961535'9 5421i5483 5545 I19 4o 9.527046 17i5 7I64 7223 7282 7341 I9 4I 56o8 567015732 5794 5857 59I9 i8 4 7400 7459 75I7 7576 7635 7694 I8 42 598I1 604316io6 6I68862306292 17 42 7753 78II 7870 7929 7.988 8047 I7 431 6354 64I6;6478 654i 66o3 6665 i6 43 8io05 864 8223 8282 8340|8399 i6 44 6727 6789 685i 69I3 697517o371 5 44 8458 85I618575 86341 892 875I i5 45 7099 7I 61 7223 7285 73471740I4 4 45 88io 8868 8927 8986 9044 903 I4 4 46 7471 7533 759576577719 9 1778i 13 461 96I 9220 9279 9337 9396 9454 13 471 7843 7905 7967 8028 88090 8I52 2 47 95I3 957I 9630 9688 9747 98051I2 48 82,418276 8338184oo0846,185:2311 48 9864199221998,..391..98.i56iI 49 858518647187091877o/883218894 Io 49 9.5302I510273 o33I o390oo418 o5o07 Io 5o~9.50895619oI7 9079 914i19202 9264 9 5o 9.530565 0o623 o682 0o740o798 0857 9 5i 932619387 944995 I I 95 729634 8 5I og9150973 Io321ogo090 481 I207 52 9696 9757 9819 988o 9942...4 7 52 1265 1323 I38III44o 1498 i556 7 5319.5Ioo65 O1271oI880o25o003 Io0373 6 53 I6I41I673 I731 I7891i8Z7 I9go55 6 54 o4341o4961o5571o6I9lo680o 742 5 54 I963 2022/2080|2I38 2i96 2254 5 55 o0831 865o09261o987 1049110 4 55 2312 2370 2428 24871 2545 2603 4 561 II72 I233 1294 i356 14I7 1478 3 56 2661 27I9 2777 2835 2893 2951 3 57 I540o i6oi 662 1724 1785 1i846 2 57 300913067 3I2513I83l324I 32991 2 581 I907 I969 2030 209I12I5222I4 I 1 58 3357 34I5 3473 353i 3589 3647 1 591 2275 2336 2397 2458 2520S225I8 o8 1 59 3704 3762 3820o3878 3936 39941 o 60"1 50" 40" 30" 20" 10" 60"" 50" 401 30" 20 - 10"t/ Co-sine of 71 Degrees. - Co-sine of 70 Degrees. P Part (1" 2" 3" 4" 5,' 6" 7"/ 81 9" - 1 2" / 3/1 4" 5" 6" 71 8" 9 1 6 13 19 25 31 38 44 50 57 Part 6 12 8 24 30 36 42 48 54 LO(GARITIIM TAN GENTS. 43 rT'angent 3f 18 Degrees.. Tangent of 19) Degrees. 0' - 10" r20' 1 30" 40" 1 50" 0" 10'" 20" 30" 4110" 50" o 9.5II776 i848 I919 I99I 2063 2I34 59 0 9. 536972 7040 7109 71 77 7245 7314 59 I 220622772349 2420 2492 2564 58 I 7382 7450 7519 7587 7655 7724 58 2 2635 2707 2778 2850 2921 2993 57 2 7792 7860 7929 7997 8o65 8i33 57 s 3o64 3I36 3207 3278 3350 3421 156 3 8202 8270 8338 84o6 8475 8543 56 4 3493 3564]3636 3707 3778 3850 55 4 86 i 8679 87478816 6 8884 8952 55 5 392 I3992 4o64 4i 35 4206 4278 54 5 9020 9088 9I56 9224 9293 9361 54 4349 4420 4492 4563 4634 4705 53 6 9429 9497 9565 9633 9701 9769 53 7 4777 4848149 I9 499o 5o6215 I 33 52 7 9837 990519973 *.4I. Io9. 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Co-tangent of 70 Degrees. P. Part 1i" 2' 311 41. 511' 6"1 7"" 88" 9 4/ 5/1 6" 711 8" 7 14 21 28 35 42 49 56 63.art 13 20 7 33 40 47 54 60 14 LoG RI rHMTC SINES Si1ne of 20 Degrees. i Sine of 21 Degrees. O" 10"' 2 30 40 5010" 20 30" 40 50 0 10 30 4 o 9. 34052 4I IO 4i67 4225 4283 434I 59 o 9.554329 4384 4439 4494 4549 4603 59 x 4399 4456 45I4 4572 4630 4687 58 i 4658 47I3 4768 4822 4877 4932 58 2 4745 4803 486i 491 4 976 50 34 57 2 4987 5042 5096 5I5 5206 5260 57 50o92 549 5207 5265 5322 538o 56 3 53i5 537o 5425 5479 5534 5589 56 4 5438 5495 5553 56io 5668 5726 55 4 5643 5698 5753 5807 5862 59I6 55 5 5783 584i 5898 5956 6oi4 607I 54 5 5971 6026 6080 6I35 6g19 6244 54 61 6129 6I86 6244 63oi 6359 64i6 53 6 6299 6353 6408 6462 65I7 657I 53 7 6474 653i 6589 6646 67o4 676I 52 7 6626 6680 6735 6789 6844 6898 52 8 68i8 6876 6933 699I 7048 7I05 5I 8 6953 7007 7062 71I6 7I7I 7225 5 9 7I63 7220 7278 7335 7392 7450 50 9 7280 7334 7388 7443 7497 7552 50o o109.537507 7564 7622 7679 7736 7794 49 I1 9.557606 7660 7715 7769 7823 7878 49 11 7851 7908 7965 8023 8o8o 8137 48 I 7932 7986 804I 8o95 8149 8204 48 12 8I94825283o98366 8423 8480 47 12 8258 83I2 8366 842I 8475 852947 i3 8538 8595 86528709 8766 8823 46 I 3 8583 8638 8692 8746 8800 8855 46 I4 8880 8938 8995 9052 91o99 166 45 I4 8909 8963 9017 907I 9I25 9180o45 15 922319280 9337 9394 945I 9508 44 I5 9234 9288 9342 9396 9450 9504 44 16 9565 9622 9679 9736 9793 9850 43 i6 9558 96I3 9667 972I 9775 9829 43 17 9907 9964..2I..78.I35.78.35.I92 42 I7' 98839937 9991.45 99.53 42 I8 9.540249 o306o 363 4200477 o5334i i8 9.560207026I o3I5 o3690o423 0o477 4I Ig o5o9 o647 o704 076I 08I8 o875 4o I9 o53I o585 o639 o693 0747 080I 4o 20 9.54093I o988 io45 I1102 1159 I 21539 209.560855 o908 o962 1oi6 I070 1124 39 21 1272 1329 1386 I442 1499 i556 38 21'I 78 I232 1286 I339 I393 I447 38 22 I6I3 1669 1726 I783 I839 1896 37 22 50oi I555 I609 I662 1716 I 770 37 23 I953 2009 2066 2I23 1279 2236 36 23 I824 I878 I93I I985 2039 2093 36 24 2293 2349 2406 2462 2519 2576 35 24 2146 2200 2254 2307 236I 2415 35 25 2632 2689 2745 2802 2858 2915 34 25 2468 2522 2576 2629 2683 2737 34 26 2971 3028 3o84 3I4I 3I97 3254 33 26 2790 2844 2898 295 30oo05 30 58 33 271 33I0 336713423 3480 3536353 93 32 27 3112 3I66 32Ig 3273 3326 3380 32 28 3649 3705 3762 38I8 3875 3931 3I 28 3433 3487 354i 3594 3648 370I 3I 29 3987 4044 41oo 4156 4234 234269 30 29 3755 3808 3862395 3969 4022 3 0 30o9 544325 4382443844 94 4550 4607 29 30o 9.564075 4129 4182 4236 4289 4343 29 3i 4663 4719 4776 48324888'494428 31 4396 4449 45o3 4556 461o 4663 28 32 5000oo 5057 5II35I69 52255281 27 32 4716477O 4823 4876 49304983 27 33 5338 5394545o0155o6 556256I8 26 33 5036 5090 5I43 5196 5250o530326 34 5674 573 5787 843 543 58995955 25 34 5356 5409 5463 55I6 5569 5622 25 35 60I I 6067 612361I79 6235 629I 24 35 5676 5729 5782 5835 5888 5942 24 36 6347 6403 6459 65I5 657i 6627 23 36 5995 6o48 6Ioi 6I54 6207 6261 23 37 6683 6739 6795 685i 6907 696322 37 63,4 6367 6420 6473 6526 6579122 38 70I9 7075 7I31 7187 7242 7298 2I 38 6632 6685 6739 6792 6845 6898|2I 39 7354 74I0 7466 7522 7578 7633 20 39 695I 7004 7057 711 7163 7216 20 40 9.547689 7745 780I 7857 79I2 7968 19 4o 9. 567269 7322 7375 7428 748I 75341 9 41 8024 8080 8I36 819I 8247 8303 i8 41 7587 7640 7693 7746 7799 785i I8 42 8359 84I48470 8526 858i 8637 I7 42 7904 7957 80I108063 811 6 8691I7 43 869.3 8748 8804 8860 8915 8971 i6 43 8222 8275 8327 838o 8433 8486 16 44 927 9082 9138 9I93 92491930 5 i5 44 8539 8592 8644 8697 8750 8803 I5 45 9360 9416 947I19527 95829638 I4 45 885689o8 8908 6I90I49067 9II9 I4 46 9693 9749 98o 9860 99i6 997I i3 46 91I72 9225 9277 9330 9383 9436 13 4719.550026 0082 oI37 0I93 o248 o3o4 12 47 9488 9541 9594 9646 9699 975212 48 o359|o4151o47010525 o58I o636 11 48 9804 9857 99Io 9,962..I5..671II 49 o692 o747o0802 o858 09I 3 o968 o 49 9.570I20OI730225 o0278 o33oo383 Io 50|9.551024 I079 II34II90 1245 i3oo 9 50 9. 57o435 o488 054I o593 o646 o698 9 51 1356, i41 1466 I52I I577 1632 8 5i 07:51 o803 o856 ogo8 096I ioi3I 8 52 I687 1742 I798 I853 I9 08 I9 63 7 52 I066 III8 II70 I223 1275 1328 7 53 2018 2074 2I292184 2239 2294 6 53 138014331 I485 I5371I590 1 642 6 54 2349 24052460 25I5 2570 2625 5 54 I6951747 I799 I852 1904 I956 5 55 2680 273512790o2845 29002955 4 55 2009 206I 2113 266 22I8 2270 4 56 30oI03065 3I21 3I76 323I13286 3 56 2323|2375 2427 2479 2532 2584 3 57 334i 3396 345i 3506 3560o 365 2 57 2636 2688 274I12793 2845 2897 2 58 367o03725 378o03835 389o3945 I 58 29503002 3o543IO6 3i58 3210 I F9 4oo0004o5514I Io4I65 42I914274 o 59' 3263 33I5 3367:34I9 347I13523 0.60"l 50"S 40"| 30" 20" 1 1 - 60"' 50" 40" 30" 20" 10" - Co.sine of 69 Degrees. " Co-sine of 68 Degrees. p 1" 2"/ 3" 411" 5" 6" 711 8" 91, 1" 2/"' 3 4/ 5" 6" 71 8" 9" 1. ar 6 11 17 23 28 34 39 45 51 a 5 11 16 21 27 32 37 43 481 LOG AR ITMIC TA N a E N T H. 44 Tangent of 20 Degrees. | Tangent of 21 Degrees. 0" 10" 20" 30" 40' 50/" O" 10"/ 20" 30" 40"' 50" __ O 9.56Io66 ii3i I 971 I2621 328 I393 59 0 9.584I77 4240 43034366 4429 4492 59 1 I459 1524 I59o I655 I72I I786 58 I 4555 46I8 468I 4744 4806 4869 58 2 I85I I197 I982 20482II3 2178 57 2 4932 4995 5058 512i 5I83 5246 57 3 2244 2309 2375 2440 2505 2571 56 3 5309 5372 543554 498 556o 5623 56 4 2636 270I 2767 2832 2897 2963 55 4 5686 5749 58II 5874 5937 6ooo 55 5 3028 3093 3I58 3224 3289 3354 54 5 6062 6125 6I88 625I 63I3 6376 54 6 34I9 3485 3550 36I5 3680 3746 53 6 6439 65oI 6564 6627 6689 6752 53 7 38II 3876 394I 4006 407I 4I37 52 7 6815 6877 6940 7003 7065 712852 8 4202 4267 433224397 4462 4527 5I 8 7190 7253 73I6 7378 744I 7503 5r 9 4593 4658 4723 4788 4853 4918 50 9 7566 7629 769I 7754 78I6 7879 5o 10 9.564983 504851135178 5243 5308 49 IO 9.587941 8004 8066 8I29 819I 8254 49 II 5373 5438 5503 5568 5633 5698 48 II 83I6 8379 844I 85o4 8566 8629 48 12 5763 5828 5893 5958 6023 6o88 47 12 8691 8754 88i6 8878 894I 9003 47 I3 6I53 62I8 6283 6348 64I3 6478 46 I3 9066 9118919I19253 93I5 9378 46.4 6542 6607 6672 6737 6802 6867 45 I4 944o 9502 9565 9627 9690 9752 45 I5 6932 6996 706 7I126719I 725 6 44 I5 9814 9877 9939..:..63.I26 44 i6 7320 7385 745o 7515 7580 7644 43 I6 9.59o1088 o25oo3I3 1375 0437 0499 43 17 7709 7774 7839 7903 7968 80 33 42 17 o0562 o624 o686 0748 o8II o873 42 18 8098 8I62 8227 8292 8356 842141 I8 0935 0997 o60o1II22 I184 I24641I 19 8486 855o086I5 868o08744880o94o 19 I3o8 13701433 I495 i557 I6I9 4o 20 9.568873 8938 900oo3 90o67 9132 997 39 209.59I68I 1743 I8o05 868 1930 I992 39 21 9261 93269390o 9455 9519 9584 38 21 2054 21 i6 2I78 2240 2302 2364 38 22 9648 97I3 9777 9842 9906 997I 37 22 2426 2488 2550 2612 2674 2737 37 23 9.57oo35 1ooo oI 64 0229 0293 o358 36 23 2799 2861 2923 2985 3047 31o9 36 24 0422 o0487 o55 o66 o68680 0744 35 24 3I71 3232 3294 3356 348 3480o 35 25 o809 o873 o0938 I002 o66 II3i 34 25 3542 3604 3666 3728 3790 3852 34 26 II95 I259 i324 1388 452 I517 33 26 3914 3976 4038 4099 416I4223 33 27 I58I I645 17IO I774 I838 19o3 32 27.4285 4347 4409 4471 4532 4594 32 28 1967 2o3 2095 2160 2224 2288 31 28 4656 47I8 4780 48424903 4965 3i 29 2352 24I7 248I 2545 2609 2673 30 29 5027 5089 5I5o 5212 5274 5336 30 30 9.572738 280o2 2866 2930 2994 305 9 29 30 9.595398 5459 5521 5583 5644 5706"29 3I 3123 3I87 3251 33i5 3379 3443 28 31 5768 583o 589 159.53 601 56076 12 32 3507 3571 3636 3700 3764 3828 27 32 6I38 6200o626I 6323 6385 6446 27 33 3892 3956 4020 4084 4I48 42I2 26 33 6508 6570 663i 6693 6754 68i6 26 34 4276 4340 44o4 44684532 4596 25 34 6878 6939 700o1 7062 7124 718525 35 4660 4724 4788 4852 49 6 4980 24 35 7247 7309 7370 743217493 7555 24 36 5o44 5I8 5I172 5236 5299 5363 23 36 76I6 7678 7739 78.oI07862 7924 23 37 5427 5491 5555 56I9 5683 5747 22 37 7985 8047 8I.o8 8170 823I 8293 22 38 58io 5874 5938 6002 6066 6i30 2I 38 8354 84I5 8477 85388600oo 866I21 39 6193 6257 6321 6385 6449 6512 2o 39 8722 8784 884589.07 8968 9029 20 40o 9.576576 6640 6704 6767 683I 6895 19 4o 9.59909I 9152 921 392759336 93981I9 4I 6959 7022 7086 7I50o 723 72 77 I8 4I 9459 9520 958i 9643 9704 9765 I8 421 734i174~417468 7532S7595 765917 1 42 9827 9888 99491'.7II'.72.133.I7 43 7723 7786 7850 7914 7977 780 4I 1 43 9.600I940o256 03170378 439 o5 I6 44 8Io418I68 823I 8295 835.9 8422 i5 44 o562 o623 o684 0o745 o80o6 868I i 45 8486 8549 86I3 8676 8740 88o3 14 45 0929 0990 io5II 1112II74 I2351I 4.46 8867 8930 8994 9057 912 91841I 3 46 12961 357 I4i8 1479I 54o I6o0 iI' 47 9248 93 1 5375 9438 95o02 9565 I2 47 i663 I724 1785 1846 1907 I968 I2 48 9629 9692 9755 98I9 9882 9946 II 48 2029 2090 21512212 2273 2334 ii 49 9.58oo009g 0072o i360199 0262 o0326 10 49 2395 2456 25I7 2578 2639 2700 10 5o 9.580o389 o453 o5I6 o579 o642 oo6 9 50 9.602761 2822 2883 2944 300 oo 366 9 51 o7669 o832 o896 o959 I022 io86 8 5i 3127 3i881324933Io 337I 3432 8 52 I149 1212 1275 I339 1402 I465 7 52 3493 3554 36i'53675 3736 3797 7 53 I528 I59I i655 1718 1781 I844 6 53 3858 3919 3980|4o4i 4102 4I62 6 54 I907 97I 20342097 2160 2223 5 54 4223 4284 4345 4406 4467 45271 5 55 2286 2350 2413 2476 2539 2602 4 55 4588 4649 4710 477I 483I 48921 4 56 2665 2728 279I 2854 2917 2980 3 56 4953 50o4 5074 5i35 5 196 5257 3 57 3044 3107 3I73233 3296 3359 2 57 53I7|5378.5439 5500|5560o5621 2 58 3422 3485 3548 36i 3674 3737 I 58 5682 5742 5803 5864 5924 59851 59 3800 38633926 3989 405214114 o 59 6046|6Io661676228628863491 o 6(Y' --' 40' 1 30" 20 Y 10 i 60"1 50" 40" 30"1 20" 10" Co-tangent of 69 Degrees. Co-tangent of 68 Degrees. P.PartS 1" 2" 31" 4" 5" 6" 7 P 8P" 9"P t 1" 2" 3" 4 5" 6 7/ 8" 9". 6 13 19 26 32 39 45 51 58. art 6 12 19 25 31 37 43 49 56 L OGARITHMIC SINES. Sine of 22 Degrees. Sine of 23 Degrees. -0" 10" 20"/ 1 30' 40" 50" 0" 10" 20" 301" 40" 50". o 9.5735715 3628 3680 3732 3784 3836 59 o 9.59I878 I928 I977 2027 2076 2126 59 I 3888 3940 3992 04o44 49614I48 58 I 2176 2225 2275 2324 2374 2423 58 2 4200 4252 43o044356 44o8 446o 57 2 2473 2522 2572 2621 2671 2720 57 3 45I 24564 46I 64668 4720 4772 56 3 2770 28I9 2869 2918 2968 3017 56 4 4824 4876 4928 4980 5032 5084 55 4 3067 3II16 365 32I5 3264 33I4 55 5 5I36 5I87 5239 5291 5343 5395 54 5 3363 34I2 3462 35II 356i 36io 54 6 5447 5499 5550 5602 5654 57o6 53 6 3659 3709 3758 3807 3857 3906 53 7 5758 58105861 5913 5965 6017 52 7 3955 4005 4054 4I03 4I53 4202 52 8 6069 6120 6I72 6224 6276 6327 51 8 425I 43oI 4350 4399 4448 4498 5I 9 6379 6431 6482 6534 6586 6638 50 9 4547 4596 4645 4695 4744 4793 5o to 9.576689 674I 6793 6844 6896 6948 49 1o 9.594842 489I 494I 499~ 5039 5088 49 II 6999 705I 7I02 7I54 7206 7257 48 lII 5I37 5I86 5236 5285 5334 5383 4812 7309 7360 7412 7464 75I5 7567 47 12 5432 548I 553o 558o 5629 5678 47 13 7618 7670 7721 7773 7824 7876 46 I3 5727 5776 5825 5874 5923 5972 46 14 7927 7979 80o3o 8o82 833 885 45 i4 602I 6070 6II919668 62I7 6266 45 I5 8236 8288 8339 839I 8442 8494 44 i5 63I5 6364 64I3 6462 65Ii 656o 44 16 8545 8596 8648 8699 8751 8802 43 I6 6609 6658 6707 6756 68o5 6854 43 17 8853 890o58956 90oo 8 90599TI 42 I7 6903 6952 7001 7050 7099 7i48 42 81 9I62 92I3 9264]9316 9367194I8 4I i8 7I96 7245 7294 7343 7392 744141 19.9470 9521 9572 9623 9675 9726 4o I9 7490 7539 7587 7636 7685 7734140 20 9.579777 982898809931I9982...33 39 20 9.597783 7831 7880 7929 7978 8027 39 2I19.5800oo 85 oi36 o087 0238 0289 o34i 38 2I 8075 8124 8173 8222 8270 8319 38 22 o392 o443 04940o545 o596 o647 37 22 8368 8417 8465 85I4 863 8612 37 23 0699 0750 o8oi o852 o09 o3 0o95436 23 8660o 8709 8758 88o6 8855 8904 36 24 Ioo5 I056 I I07 ii58 1209 126I 35 24 8952 900 90o50 9098 9I47 9I95 35 25 13I2 13631I414I 465 i5i6 I567 34 25 9244 92939341 93 93 9438 9487134 26 I6I8 I669 I720 I77I I822 I873 33 26 9536 9584 9633 968I 9730 9778 33 27 I924 1975 2025 2076 2127 2I78 32 27 9827 9876 9924 9973..21. 70132 28 2229 2280 2331 2382 2433 2484 3I 28 9.600II8 oI67 0o25 0264 0312 0361 3i 29[ 2535 2585 2636 2687 2738 2789 30 29 0409 o457 oo56 o554 o6o3 o657 30 30o 9.582840 2890!294I 2992 3o43 3094129 3019.600700 0748 0797 o845 o893 0942129 SI 3I45 3I951324613297 3348 3398128 31 09900Io38 1087 II35 I184 1232 28 2 3 4 34493500oo355 36oi 3652 3703 27 32 I280oI329 1377 1425 I474 1522 27 ~31 3754 3804|3855|390o63956 4007 26 33 I570oi6I9 I667 I7I5 I763I8 I 226 34 4o58 4io8 4i59 42IO4260 43I 125 34 I86o 0908 i957 2005 2053 2IOI25 35 436i 44I214463 453 4564 46i5 24 35 2I50 2I98 2246 2294 2342 239I 24 36 4665 47I614766 48I7 4867 49I8 23 36 2439 2487 2535 2583 2632 2680 23 37 4968 5o0I9 507o 5I2o 5I7 522I 22- 37 2728 2776 2824 2872 2920 2969 22 38 5272 5322 5373 5423 5474 552421 38 3o07 3o65 3II3 3I6I 3209 325721I 39 5574 5625 5675 5726 5776 5827 20 39 330513353 340I 3449 3497 3546 20 40g9.585877 5927 5978'6028 6079/6I 29 19 409.603594|3642 3690 3738 3786|3834 I9 41 6179 6230o6280 633i 638i 643i iS 4, 3882 3930 3978 4026 407441I22 i8 42 6482 6532 6582 6633|6683|6733 I7 42 4I70o4218 4266 43I3 436I144091I7 43 6783 6834 6884 6934 698517o35 i6 43 4457 4505 4553 46oI 4649 4697 i6 44 7085 7I3517I867236 72861733615 44 4745 4793 4844888493649369845 45 7386 7437 7487 7537 7587 7637 I4 45 5032 5080 5I28 5I76 223 52723 1 4 46 7688 773817788 7838 7888 7938 31 46 53I9 5367 54I5 5462 55Io 5558 I3 47 7989 8o3918089 8I39 8I89 8239 12 47 56o6 5654 570I 5749 5797 5845 I2 48 8289 8339 8389 8439 8489 854o II 48 5892 5940 5988 6035 6o83 6i3i iI 49 859o0 864081 8690 8740 8790 8840o 49 6 79 6226 6274 6322 6369641 7 10 So 9.5888go 8940 89o990490 409090I40 9 50o9. 606465 652 6560 6608 6655 6703 9 5i 9190 9240 9290 9340 9389 9439 8 5i 675i 6798 6846 6893 694I16989 8 52 9489 95,39'9589 9639 9689 9739 7 52 7036 7084 7131 779 7227 7274 7 53 9789 9839!9889 9938 9988..38 6 53 7322 7369 74I7 7464 7512 7559 6 541959oo88 o38 o8 8 388802370287 0337 5 541 7607 7654 7702 7749 7797 7844 5 55 0387 o437 o487 0536 o586o636 4 551 7892|7939 7987 8038080828129 4 56 o686 0735 0785 o835 808 85 0934 3 56 8I77 8224 827I 8319 8366 84I4 3 57 0984 IO341IO84|II331II83I233| 2 57 846It8508 8556 8603 865I 8698 2 58 I282 1332 I382 143I I48 I 53 i| I 58 8745 8793 8840o 8887 8935 8982 Z 59 I58o I63o!I68o 1729 1779 I828 o 59 9029 9077 9I24 9I7I 9219 9266 0 60" 50" 40" 30" 20" 10" —, 60" -' 50" 1 40" 30" 201 10"' Co-sine of 67 Degrees. Co-sine of 66 Degrees. I 1 2// 3' 41/ 5 6" 7" 8" 9 11 1tt 2" 31/ 4/ 5/1 6'1 7/I 8" 91" ar 10 15 20 25 31 36 41 46 I. 5 10 15 19 24 29 34 39 44 LOGARIT H M I C TANGE NTS. 4 Tangent of 22 Degrees.. Tangent of 23 Degrees. Of/ 10" 1 20" 301" 40" 50" 0o" 1011 20" 30 40"1 50 o 9.60f4Io 6470 653I 659I 6652 67i3 59 o09.627852 7910 796918028 8086 8I45 59 _ 6773 6834 6894 6955 70I5 7076 58 I 82o3 8262 8320o8379 8437 8496 58 2 7I37 7197 7258 7318 7379 7439 57 2 8554 8612 867I 8729 8788 8846 57 3 7500 7560 762I 768I 7742 7802 56 3 8905 8963 9022 9080 9I38 9197 56 4 7863 7923 7984 8044 81805 865 55 4 9255 93I4 9372 943I 9489 9547 55 5 8225 8286 8346 8407 8467 8528 54 5 9606 9664 9722 978I 9839 9897 54 61 8588 8648 8709 87698830o889o053 6 9956..I4..73.3I. I89. 247 53 7 89509 90II 9071 9I3I 192 9252 52 7 9.630306 03640 o422 048I 05390597 52 8 9312 9373 9433 9493 9554 96I4 5I 8 0656 07I4 0772 0830 0889 0947 5i 9 9674 9735 9795 98559959 7595 985 9956311229976 580 1238 I296 5o IO 9.6IOO36 oo96 OI56 02I7 0277 o337 49 TO 9.631355 I4I3 147I I529 I 587 646 49 II o397 o458 o58o578o638o6898 8 II 1704 1762 182o0 878 I936 I99548 12 0759 08I9 0879 0939 ~999 I059147.I2 2053 2II.1 2I69 2227 2285 2343 47 I3 II20 o I80 I24o I300 I360oI420o46 13 2402 246025 I8 25762634 2692 46 14 I48o I540o 6oI x66i 1721 178i 45 4 2750 2808 2866 2924 2982 3o4o 45 I 5 84i I9OI I96I 202I 208I 214I144 I5 3o99 3I57 32I5 3273 3333 3389 44 I6 220I 226I 232I 2381 244I 25o0I 43 6. 3447 3505 3563 362I 3679 3737 43 I7 2561 262I 2681 2741 2801 286i 42 17 3795 3853 39II 3969 4.027 4o85 42 | s8 292I 298I 3o4I 3IOI 3I6i 3221 41 i8 4I43 420I 4259 43I6 4374 4432 4I 19 328I 334 34o 3 346I 352I 358i 40 I9 449~ 4548 46o6 4664 4722 4780 40 20 9.6I364I 370I 376o 3820 388o 394o 39 20 9,.63483814896 4954 50I I 5o069 527139 21 ooo4000 40604I2o 4I8o 4239 429938 1 2I 5i85 5243 53o0 5359 54I6|5474 38 22. 4359 44I91447'9 4539 4598 4658 37 22 5532 5590 5648 5706 5763 582I 37 23 47I8 4778 4838 4897 4957 50oI736 23 587915937 5995 6052|6I11o6I68|36 24 5077 5I36 5I96 5256 53I6 5375 35 24 6226 6283 634i 6399 6457 65I4 35 25 5435 5495 5555 56I4 5674 5734 34 25 6572 6630o 6688 6745 68o3 686I 34 26 5793 5853 59I3 5972 6o32 609233 26 69I9 6976 7034 7092 7I49 7207 33 27 6I5I162II1627I 6330 6390 6450|32 27 7265 7322 7380 74381749517553I32 28 6509 6569 6628 6688 6748 68073I 28 761I 7668 7726 7783 7841 7899 3I 29 6867 6926 6986 7046 71o5 7I65 30 29 7956 80oI48072 8I29 8187 8244330 309 61r7224|7284|7343|74o037462 7522|29 3019.638302 8359 84I7|8475 8532 8590|29 31 7582 7641 7701 7760 7820 7879 28 3I 8647 8705 8762 882o08877 8935 28 32 7939 7998,8057 81i7 8I76 8236 27 32 8992 9050 9107 9I65 9222 9280 27 33 8295|8355184I4|8474 8533 8593 26 33 933719395 9452|95I0|956719625|26 34 8652 87I1 87718830 88890 89 49 25 34 9682 9740~979719855 99I219969 25 35 90oo89068 9I27 9I86 9246 9305124 35 9.640027 0084 oI42 0I99 0257|0314 24 36 9364 9424 9483 9543 9602 966I123 36 o37I 0429 0486 o544 o6oIx 658 23 37 9720 9780 9839 9898 9958..i7 22 37 07I6 0773 o83o o888 0945 1002 22 38 9.620076 o136 oz95 o254 o3I3 O373| 2 38 060 III7 II74| 232 I289 346 21 39 0432 o0491 o550o o6 o669 0728 20 39 I4o4 I46i i5I8 I575 I633 169o020 40|9.6207870o846o9po6o965 I o24 1o8319 4019.641747 I8oSI 86219I91997620o34 I9 41 II42 1201 126I I32o0 379 i438 i8 4I 209I 2548 2205 2263 2320 2377.I 8 42 I497 I556 I6I6 1675 1734 1793 17 42 2434 2491 2549 2606 2663 2720 I7 43 i852 I9II 1970 2029 2088 2I47 I6 43. 2777 2834 289212949 3oo6 3o63 I 6 44 2207 2266 2325 2384 2443 2502 15 44 3I20o 377 3235 3292 3349 3406 i5 45 2561 2620 2679 2738 2797 28564 4 45 3463 3520 3577 3634 369I 3749 I4 46 29I5 2974 3o33 3092 3I5I 3210 I3 46 38o6 3863 392o 3977 4o34 4o091 3 47 3269 3328 3387 3446 350535641 47 4I48 4205 426243 I 94376 4433 I 2 48 3623 3682 374I138oo 3858 3971|II 48 449o 4547 4604 466I 47I8 4775 II 49 3976,4035 40944I53 42I2 427I IOo 49 4832 4889 4946 5oo3 5o6o5I I7|I O 50o9.624330 4388 4447,4506 4565 4624 9 5o 9.645174 523I 5288 5345 5402 5459 9 5I 4683 4742.4800o4859 49I8 4977 5Ix 55I6 5573 563o 5687 5744 58oi 8 52 5036 5094 5I53 52I2 527I 5330 7 52 5857 59I4 5971 6028 6085 6542 7 53 5388 5447 550665565 5623 5682 6 53 6I99 625663I36369 6426 6483 6 54 574I158oo05858 59I7 5976 6o35 5 54 654o 6597 6654 675o 6767 6824 5 55 6o93 6552 6211 6269 6328 6387 4 55 688I 6938 6995 7051 7108 7i65 4 56 6445 6504 6563 662I 668o06739 3 56 7222 7279 7335 739217449 7506 3 57 6797 6856 69I5 6973 7032 7090 2 57 7562 76I9 7676 7733 7789 7846 2 58 7I49 72o8 7266 7325 7383 7442 I 58 79o3 7960o80o6 8073 8I3o08i86 I 59i 750 7559 76I8 7676 7735 7793 o 59 8243 83oo 8356 84z3 8470o 8526 o 60" 50' 1 40" 30" 20"/ 10" d 60 50" 40" 30/ 20" I10 Co-tangent of 67 Degrees. Co-tangent of 66 Degrees. P. Prt$ 1" 2" 3" 4" 5" 6" 7" 8"' 911 P. Pat" 1" 2" 3" 4" 5" 6" 7" 8g" 9" 6 12 1-8 24 30 36 42 48 54 6 12 17 23 29 35 40 46 52' L _ It - ~.~o~ _L Ib: b ~LocALOGARITHMIC SINES.. Sine of 24 Degrees. Sine of 25 Degrees. S~_h _ ~e 0 10of 20"' 30" 40" 50"" 1 201 30" 40" 507 o 9 6093I3 936I 9408 945519502 9550 59 o. 625948 5993 6039 6084 6I29 6I74 59. 9597 9644 969I 973919786 9833158 I 62I9g 6264 6309 6354 64oo 6445 58 2 9880o 9928 9975..22..691.II6157 2 6490 6535 6580 6625 667 6 7I5 5 39,6IoI6o 02II o 258 o3o5 o352 o399 56 3 6760 6805 6850 6895 694L 6985 56 4 o447 o494 o54I o588 o635 o682 55 4 7030 7075 77I20 65 72IO 7255 55 5 0729 0776 o823 o871 oI18 o965 54 5 7300 7345 7390 7435 7480 7525 54 6 IoI2 Io5 1I o61rI531I200 I247153 6 7570 76I5 7660 7705 7750 7795 53 7 I294 I34i i388 I435 1482 1529 52 7 7840 7885 7929 7974 80I 9g 8064 52 8 i576 I623 I670 I77 I 764 818II 5 8 8I098548I99g 8244 8289 8333 5I 9 I858 1905 1952 I999 20 46 20 93 50 9 8378 8423 8468 85i3 8558 8602 50 Io 96I2I4o0 287 2234 2280 2327 2374 49 IO 9.628647 8692 8737 8782 8826 887I 49 II 2421I 2468 25522562 26092655 48 II 896 896I 90oo69050o 9095 914o 48 12 2702 2749 2796 2843289029367 12 918592299274 939 9363 940847 13 2983 3030 3077 3124 31I7 3217 46 I3 9453 9498 9542 95879632 9676 46 14 3264 33 I 3358 34o4 345i 3498 45 I4 9721 9766 9810 98 985 9900 9944 45 L5 3545 359i 3638 368513732 3778 44 i5 9989..34.. 78.1 23.68.212 44 I6 3825 3872 39I8 3965 4o0I24o58143 I69.630o257 o3o0I 346 039g o435 o480o43 17 4I05 4I5214I 98 4245142921433842 1 7 o524 o569 06I3 o658 0703 0747 42 i8 43851 4432 4478 4525 457I[ 46I98 4i I 8 0792 o836 o88 o0925 097o0 oIO4 4 19 4665 47II 14758 48o4 485 I 4898 4o 19 io59 IIo3 Iz48 II92 237 I281 4o 20 9.614944 499I 5037 5o84 5I3o 5I77 39 20 9.63I326 I370.4I5 1459 I504 I548 39 21 5223 5270 53i6 5363 5409 5456 38 211 593 1637 i68I I726 I770 8i51 38 22 5502 5549 5595 5642 5688 5735 37 22 I859 1904 1948 1g992 2037208 2 37 23 578 I 5828 5874 5921 5967 6oi3 36 23 2 I 25 2 I 70 22I4 2259 2303 2347 36 24 6o6o 6Io6 1653 6I99 6245 6292 35 24 2392 2436 2480 2525 2569 26I3 35 25 6338 6385 643i 6477 6524 6570 34 25 2658 2702 2746 2790 2835|2879 34 26 66I6 666316709.6755 6802 6848 33 26 2923 2968 30I2 3o56 3ioo 3i45 33 27 6894 694I 6987 7033 7080 7I26 32 27 3I89 3233 3277 3322 3 366 34io 32 28 7172 7281 7265 73iI 7357 7403 3I 28 3454 3498 3543 3587 363i 3675 3i 29| 7450 7496 7542 758817635 7681 30 || 29 37I9 3764 38o8 3852 3896 3940 3o 30 9. 6I 67727 777317819 7866 7912 7958 29 | 30 9.633984 4028 40734I I7 4I6I 4203129 3I 8oo4 8o5o 8o968I4318I889 18235128 3I 4249 4293 4337|4381 4426 4470 28 32 828i 8327 8373 84I9 8465 85I2 27 32 45I4 4558 460224646 4690o4734 27 33) 8558 8604 865018696 8742 8788 26 33 4778 4822 4866|491o 4954I4998I26 34 88341888018926 897219018 9064 25 /| 34 5042 5086 5130|5174 5218 5262 25 35 9II10915619202 924899294 9340 24 || 35 5306 5350 539415438 5482 5526 24 36 938619432 9478 952419570 96I6 23 136 5570 56i4 5658 5702 5746 5790 23 37 9662 9708 9754 9800 9846 9892 22 | 37 5834 5877 592i 5965 6009 6o53 22 38 993819984..3o..761.I2I. I67 2I 38 6097 6I4I 6I85 6229 6272 63I612 I 39 96.602I3 0259 o3o5 o35i 0397 o443 20 39 6360o 6404 6448 6492 6535 6579 20 40o9.62o488 o534 o5801o6260o672 07I81I9 1 409.636623 6667 67II16754 6798|684219 -41 0763 o809 o855 o9oo 0947 0992z i8 4i 6886 6930 6973 707 7o061 715xo5 l421 Io38 Io84 li30 1I75 1221 I267 I7 42 7I48 7192 7236 7280 7323 7367 17 431 I33 i3581 4o41 i450o 496 i54i I6 43 74II 7455 7498 7542 7586 7629 I 6 44 I 587 i633 678 I 7241 I770 i8i6 I5 44 7673 7717 7760 7804 7848 789i i5 -455 I86I |I907 953| II9982044 2089 I4 45 7935 7979 8022 8066 8I Io 853 14 46) 2I35 218i 2226 227223818 2363 i3 1 46 8197 8240o8284 8328 837I 84I5 13 47 2409 2454 2500 2546 259I 2637 I2 1 47 8458 8502 8546 8589 8633 8676 I2 48'2682 272812773 2819 2865 29IOII 48 8720 8763 8807 885i 8894 8938 ii 491 2956 3oo003047 3092 3I38 3i83 Io 149 898I 9025 9068 9II2 9I55 9I99 1o 50 9. 623229 3274 332o 3365 34 I 3456 9 50 9.639242 9286 9329 9373 94I 6946.0 9 5i 3502 3547 3593 3638 3683 3729 8 5i 9503 9546 95901 9633 9677 97201 8 52 3774 3820o3865 39II 3956 4oo00 7 52 9764 9807 985I 9894 9937 998I 7 53 4047 409214I384I18314228 4274 6 53 9.6400240068o o I I 54 oI98 0241 6 54! 43g194364A44Io04455 45oo 4546 5 54 0284 o0328 o37I o44 o458 o50o 5 55 459I 4636 4682j4727 4772 48i8 4 55 o544 o588 o63I 0674 o 78o 0761 4 56 4863 4908 4954 4999 5o44 5089 3 | 56 o8o4 o848 o89I 0934 0978 Io 2I 3 57 5I35 5I8o 5225 527015316 536I 2 57 io64 II07 iii 1i94 I237 I280 2 |58 54o06545i 5496 5542 5587 5632 I 58 I324 I367 I4Io I453 I496 I540 I 59 567765722I572685813s58585903 o 0| 59 I583|I626I669I712I7561 799 0~ I 60" 50" 40. -230- 10"' 60"/ 50"' 40," 30"1 20tO 1 Co-sine of 65 Degrees. 1 Co-sine of 64 Degrees. 2_ P. Part 1"t 2" 3" 4" 5" 6" 7" 8" 91 p Pat 1" 2" 3" 4" 5" 6" 7// 8t' 9"I 5 9 14 18 23 28 32 37 42. art 4 9 13 18 22 26 31 35 40 LOGARITHMIC TANGENTS. 49 Taentof 24 Degrees. Tngent of 25 Degrees. [ I 0c" 10" 20"11 30" 40" 50" 0" 10"1 20"1 301 40"11 50_ O 9.648583 8640 8696 8753 88Io 8866 59 o 9.668673 87281878218837 8892 8947'59 I 892318980 90361909391g50 9206 58 I 900290579II21gI679222 9277.58 2 9263 939g 9376 9433 948919546 57 2 9332-9387 9442 9497 95529606o657 3 9602 9659 97I5 97.72 9829 9885 56 3 9661 97I6 9771 9826 988I 9936 56 4 99429998..55.....i68.224 55 4 999I..45. 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Co-tangent of 64 Degrees.! |P. Part 1" 2"t 33" 4" 5" 6" 7" 8" 911". p 1 2 3 4 5 6 7 8 P r 6 11 17 22 28 33 39 45 50 PPt11 *a 5 11 16 22 27 33 38 43 491 L~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~... 50 L o G AR I H MI C S IUN E S. I Sine of 26 Degrees. 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Co-tangent of 62 Degrees. t 1" 21t 3"1 4" 5"t 6" 71" 8i" 9 p Part 1 2/" 31 41 5" 6" 7" 8" 91 P.Partt 5 11 16 21 26 32 37 42 47 5 10 15 21 26 31 36 41 46 S,. LoOGAR IT'HMIC SINES. i tj Sine of 28 Degrees. d Sine of 29 Degrees. 0' 10"- 20" 30"1 40" 50. 0" 10" 20 1 30 40" 50" o 9.67I609oI649 I688 I728 I768 I80759 0 9.685571 560o9 56475685 5723 5761 59 I I8471 886 I926 I965 2005 2045 58 1I 5799 5837 5875 59I3 595I 5989 58 2 2084 2I24 2I63 2203 2242 2282 57 2 6027 6065 6Io3 6I4I 6I78 6216 57 3 2321 236I 2400 244o0 479 2519 56 3 6254 6292 633o 6 6368 64o6 6444 56 4 2558 2598 2637 2677 27I6 2756 55 4 6482 65I9 6557 6595 6633 667I 55 5 2795 2835 2874 29I4 2953 2992 54 5 6709 6747 6785 6822 6860 6898 54 6 3032 307 3III111 35o 3190 3229 53 6 6936 6974 7012 7049 7087 7125 53 7 3268 3308 3347 3387 3426 3465 52 7 7163 720I 7238 7276 73 14 73 5 52 8' 35o5 3544 3583 3623 3662 3702 5I 8 7389 7427 7465 750 3 754 7578 1I 9 374I 3780 3820 3859 3898 3938 50 9 76I6 7654 7692 7729 7767 78o5 50 Io 9. 673977 4o06 4o56 4095 4I34 4173 49 IO 9.687843 7880 79I8 7956 7993 803i 49 II 42I3 4252 4291 433I 4370 4409 48 II 8o69 8I06 8144 8I82 8220 8257 48 12 4448 4488 4527 4566 4606 4645 47 12 8295 8333 8370 8408 8446 8483 47 13 4684 4723 4762 4802 484i 4880 46 13 8521 8559 8596 8634 867I 8709 46 I4 49194959 4998 5037 5076 15I5 45 I4 8747 8784 8822 8868897 8935145 15 5I55 5194 5233 5272 53I 535o044 15 8972 90oIo 9048 908591I23 9160o44 i6 5390 5429 5468 5507 5546 5585 43 i6 9198 9235 9273 93I 9348 9386 43 17 5624 5664 5703 5742 578I 5820 42 17 9423 9461 9498 9536 9573 961II 42 i8 5859 5898 5937 5976 60oi6 6055 4I i8 9648 9686 9723 9761 9798 9836 4I 19 6094 6i33 6172 6211 6250 6289 4o 19 9873 9911 9948 9986..23..6I 4o 20 9.676328 6367 6466446406 6445 64846523 39 209.690098013601 OI73 02110248028639 21 6562 6601oi 664016679 6718 6757 38 2I 0323 o036I 0398 o435 0473 o5io 38 22 6796 6835 6874 6913 6952 6991 37 22 o548 o585 0622 10o660 0697 0735 37 23 7030 7069 7o0871I47 718617225 36 23 0772 o809 o847 0884 0922 o959 36 24 72647303 7342 17381 7420 7459 35 24 0996 io34 Io 71:to8 II46 ii83'35 25 7498 7536 7575 7614 7653:7692 34 25 1220 1258 I295 1332 I370 1407 34 26 7731 7770 7809 7848 7886 7925 33 26 1444 I482 1519 i556 1594 i63i 33 27 7964 8003 8042 80o8 8Io208158 32 27 I668 1706 I743 I780o18I7 i855 32 28 8197 8236 8275 83I4 8353 8391 31 28 I892 I929 1966 2004 204I 2078 3i 29 843o 8469 85o8 8547 8585 8624 30 29 2II5 2153 2190 2227 2264 2302 3o 30 9.678663 8702 874018779 8818 8857 29 30 9.692339.2376 24I132450 2488 2525 29 31 8895 8934 8973 90I2 9050 9089 928 31 2562 2599 2636 2674 2711 2748 28 32 9I28 9I67 920519244 9283 9321 27 32 2785 2822 2859 2897 2934 297I 27 33 9360|9399 9438 9476 951I59554 26 33 300813045 30823119 357 3194 26 341 9592 9631 9670 97o8 9747 9786 25 34 3231 3268 3305 3342 3379 34i6 25 35 9824 9863 9902 994o 9979..17 24 35 3453 349o 3528 3565 3602 3639 24 36 9.6800560oo95 oi33 oI72o 02I00249 23 36 3676 371375 3787 3824 3861 23 37 0288 0326 o365 o403 0442 o48I 22 37 3898 3935 3972 4009 4o46 4083 22 38 o519 o558 o596 o635 0 673 o7I1 21 |38 412o04I157 4194 4231 4268 4305 2I 39 o0750o 0789 o828 o866 090510943 20 39 4342 4379 44I6 4453 449o 4527120 40o19.8o0982 I020 io59 1097 ii36 II74 19 4o09.694564'460I 4638 4675 47. 2 4749 1i 41 I2I31I251 1290 1328 I366 14o5 i8 14 4786 4823 4860 4897 4934 497I i8 42 1443 1482 I520 1559 I597 1636 I7 42 5007 5044 5o8I 5II8 5I55 5192 17 43 167417I3 I75i 1789 1828 i866 i6 43 5229 5266 5303 5339 5376 543 16 44 I9o5 1943 198I 2020 2058 2097 I5 44 545015487 5524 556I 5598 5634 i5 45 2135 2I73 22I2 2250 2288 2327I4 14 45 567 5708 5745 5782 589 5855 14 46 2365 2403 2442 2480 25 19 25577 3 46 5892 5929 5966 6003 6039 6076 13 47 2595 2633 2672|27IO 2748 2787 I2 47 6II361I50o6187 6223 6 6297 12 48 2825 2863 2902 2940 2978 3o06 ili 48 6334 6370o6407|64446481 6517 II 49 3o5513093 3I 31 70 3208 3246 IO 49 6554 659i 6628 6664 670I 6738 Io.5o9.683284 3323 336I13399 343713475 9| 50 9.696775 68I I 6848 6885 6921 6958 9 5I 35i4 3552 359Q 3628 36673'7o5 8 51 6995 703I 7068 7Io5 7I41 7I78i 8 52 3743 378I 3819 3858 3896 3934 7 52 72I5 72517288 7325 7361 7398 7 53 3972,40Io 4048 4o87 4I254I63 6 53 7435o747Io750o87545 758 I76i8 6 54 4201 4239 4277 43I5 4353 4392 5 54 7654 7691 7728 7764 7801 7838 5 55 443014468 45o6 4544 4582 4620 4 55 7874 79II 7947 7984 802o08057 4 56 4658|4696 473514773 48114849 3 56 80948I30o81 67820o38482408276 3 57 4887 4925 4963500 l5 3950 77 2 57 83i3 8349 8386 8423 8459 8496 2 58 5II5 5I53 5191 5229 5267 53o5 I 58 8532 8569 860o58642 8678 87I5:59 5343 538I 54I9 5457 5495 5533 o 59 875i 8788 8824 886i 8897 8934 o 60' 50' 40" 30" 20" 7o"- 60 1 50" 4" 20" 10" Co-sine of 61 Degrees. @ Co-sine of 60 Degrees. P.Pa 1"Ii 2" 3"1 411 5/I 6"' 7"1 8// 9/1 P.Part 1" 3" 4 511 6" 7" 8/" 9 4 8- 1 16 19 23 27 31 35 7111519222630 33 LO ARIT H MC TANGE NT S. b 3 Tangent of 28 Degrees. Tangent of 29 Degrees. 0" 10"' 20" 30 40" 50 0" 0" 201' 30" 40" 5' 0 9.725674 df725 5776 5827 5878 5928 59 0 9.74375213802 385I 390I 395I 4000ooo 59 I 5979 6o3o 608I 6i3i 6I8216233 58 I 4050 40994I49 4I99 4248 4298 58 2 6284 6334 6385 6436 6487 6537 57 2 4348 4397 4447 4496 4546 4596 57 3 6588 6639 6690 6740 6791 6842 56 3 4645 4695 4744 4794 4844 4893 56 4 6892|6943 6994 7045 7095 7I46155 4 4943 4992 5042 5092 5I4I 5I9I 55 5 7I97 7247 7298 7349 7399 7450 54 5 5240 5290 5339 5389 5439 5488 54 6 750I 7551 7602 7653 7703 7754 53 6 5538 5587 5637 5686 5736 5785 53 7 76o5 7855 79o6 7957 8007 8o58 52 7 5835 5884 5934 5983 6o33 6082 52 8 8I09 8159 8 8IlS26I 831I 8362 5i 8 6I32 6182 623I 6281 633o 638o 5I 9 84128463 85r48564186i5 8665 5o 9 642916479 6528 6577 6627 6676 5o o 9. 7287I6 8767 88I718868 89I8 8969 49 I09. 746726 6775 6825 6874 69246973 49 I 9020 9070 912I 917I 9222 9272 48 II 7023 7072 722 71 7I 7221 7270 48 12 9323 9374 9424 9475 9525 9576 47 1 2 73I9 7369 74I8 7468 75I7 7567 47 13 9626 967.7 9727 9778 9828 9879 46 I3 7i61 7665177I5 7764 78I417863 46 14 992919980..3o..8I. 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I1912 47 7638 7687[773617785 783417882 12 48 9.740I691o 2I9 0269o03i9 o368 o4i8,I 48 793I17980 8029 8078 8I271 875I I 49 o468|o5I8|o568|o6I8|o66807|17 io 49 8224182738322 837I 84i9|8468Io 50|9.7407670o8I70o8670o9171o967 IoI6| 9 5o09.758517 8566186I518663187I2'8761 9 5i Io661III6 II66 I2I612651I3I5 8 5i 88iot885818907[8956900oo590o53 8 52 i365 I4I5 i465 i5i4 i564 i6I4 7 52 9102 9I5I 9200 9248 9297 93461 7 531 I664 I7I4 1763 i8i3 i863 i9I3 6 53 9395 9443 9492 954i 9590o96381 6 54 I962 20I22062 2112II2I6I122II 5 S54 9687 9736 9785 983398 882 993I 5 55 226I123II1 2360o24I 12460o251o 4 55 99791..281. 771.I26.174.2231 4 56 2559 2609 2659 2709 2758 28o08 3 5619. 760272|0320 0369 o4I81o466o5I 513 3 57 2858 2907 2957 3007 3o56 3io6 2 57 o564 06I2 o66i 07IO 0758 08071 2 58 3I5632o6 32r5133o053355 34o41 I 58 o856 o9o4 1953 1002 io50Io 99 I 59 3454 3504|3553 36o033653 3702| o 59 II48 II96 I245 1293I13421391 o 60" - )40" 30" 20 1 60" 50"1 40" 301 20"1 10" Co-tangmnt of 61 Degrees. Co-tangent of 60 Degrees. P. Part 1" 2/" 3/" 4"! 5" 6" 7" 8"/ 9//" 1 P / 2// 3/ 4/" 5"/ 6 7r" 8" 9": 5 1015202530354045 a 5 10 15 20 25 29 34 39 44 65 4 LOGARITHMIC SINES. F - Sine of 30 Degrees. d Sine of 31 Degrees. t-0" 10" 20" 30" 40" 50" f 10" T20" 30" 40" 5(Y o 9.698970 9006 9043'9079919I I691 52159 0o9.71 I839 1874 I909 I944 179 2014 59 1I 98919225 9262 9298 9334 1937 58 I 2050 208512I20 2I5512101 2225 58 2 9407 9444 9480o 95I79553 9589 57 2 2260 2295 2330 23651240012434157 3 9626 9662 9699 9735 977I 9808 56 3 2469 2504 2539 2574[2609 2644 56 4- 9844 9880 99 I 7 9953999o..26 55 4 2679 27I42749 278428I 9 2854155 5 9 706ooo oo99 o35 o171 o0208 o024454 5 28892924 2959 29994 3029 3o63 54 6 0 280 031 I7 o353 o389 o425 o462153 6 3098 13331368 3233238 3273 53 7 0498 o534 o057I o607 o643 o68o 52 7 3308 3343 3377 34123447 3 482 52 8 07I6 0752 0788 o825 o86I o897 15 8 35I7 3552 3587 362i 3656 369I 5I 9 o933 0970 ioo6 Io 42 Io78 iI15 5o 9 3726 376I 3796 3830 3865 3900 50 10 9 70II5II I87 I223 I29 1296 1332 49 10 9.713935 397 4oo005 4039 4074 409g 49 I I 36814o4144o0I4771 I5 3 549148 II 414414179 42 314248 4283143 8148 I21 I585 I62'I I658 I 694 1730 I 76 6 47 I21 435214387 4422 441 4 49I 4526 47 I3 I802 I838 1874 19III9471 983 46 I13 456I 45961463o04665 47oo 4735 46 I4 20I9 2o55 20gI SI27 2I641220045 i4 4769 48o414839 4873 49o8 4943145 I5 22362272 2308 2344 2380 24I6 44 i5 4978 50I2 5047 5082 5I I 6 515 44 I 6 2452 2488 2524 2561 2597 2633 43 I 6 5i86 5220 5255 5290 5324 5359 43 I7 2669 2705 274I 2777 283 2849 4142 I 7 5394 5428 5463 5498 5532 5567142 I8 2885 2921 2957 2993 3029 3o65 4I I8 5602 5636 567I 57o5 574o 5775 4I Ig 3xoI 3I37 373 3209.3245 328I 4o I9 580915844 5878 59I3 594815982 4o 20 9.70337 3353 338913425 346I13497 39 20 19.760oI7 605l 6086 6I2I 6155 6190 39 21I 3533 3569 36o5 364I 3677 37I3 38 21 6224 6259 6293 6328 6362 6397 38 22 3749 3784 3820 3856 3892 3928 37 22 6432 6466 65oI 6535 6570 66o4 37 23 3964 40o004036 4072 4io8 4144 36 23 6639 6673 6708 6742 6777 68ii 36 24 4179 42I 5 4 42542871432343359 35 24 684616880o6951 6949 698417018 35 25 4395 443ix4466 4502 4538 4574 34 25 7053 7087 7I22 7I56 7I91 7225 34 26 46 I1 4646 4682 47I714753 4789133 26 725917294 7328 736317397 7432133 27 4825 486i 4896 4932 4968 500oo4 32 27 7466 7500 7535 7569 76o4 7638 32 28 5o4o 5075 5Iii 5I47 5i83 52I1 31 28 7673 7707 774I 7776 78Io 7844 3i 29 5254 5290 5326 536 62 5397 5433 30 29 7879 79I3 79 82 80I6 805I 30 30 9.7 555155469 5555545576656I2564829 30 9.7I8085 8II91 854 8I88 8223 8257129 31 5683 57I9 5755 5790o5826 5862 28 31 829118326 836o0 8394 8429 8463 28 32 5898 5933 5969 6oo5 6o4o 6076 27 32 8497 853 8566 86oo00 8634 8669 27 33 6I I24 6I4783 62I91625416290126 33 8703 873718772 8806 8840o8874 26 34 6326 636i 6397 6433 6468 6504 25 34 8909og 8943 8977 9011 9046 9080 25 35 6539 6575|66:II 6646 6682 67I8 24 35 9II4 9I48 9I8319217 925I 9285124 36 6753 6789 6824 686o 6895 693I 23 36 9320 9354 9388 9422 9456 9491 23 37 6967 7002 7038 7073 17I09 745 22 37 9525 955919593 9627 9662 9696 22 38. 7180 726 725I1 72877322 7358 2 I 38 9730 9764 9798 9833 9867 9901 2I| 39 7393 7429 746417500 75535 757I 2 39 9935 9969...31..38 1.72. io6 20 40o 9.707606 7642 7677 77 I31774817784 I9 4o 19.720I40 OI7410281 0242 0276 03II I,9 4I 7819 7855 7890 7926 796I 7997 i8 4I o345 0379 o4I3 0447 o48i o5.51 i8 42& 8032 8068|8IO3 8I398174 82I0 I7 42 o5491o5831o6I7 o652 o686o72o0 I7 431 8245 828o083I6 835i 8387 8422 i6 43 0754 0788 0822 o856 o8go 0924 i6 44i 8458 8493 8528 8564 8599 863.r]5 I5 44 o958 0992 1026 io6o io94 II28 i5 45 8670187o51874I 8776|88II 88471I4 45 11621 i961I230 I2641 298 1332 14 46 888218918 8953 8988 9024 90591I3 46 i366 I4oo 0 434 1468 1502 i536 13 471 9094 9I30|9i65 9200 9236 927I 12 47 I570 i6o4 i638 I672 I706 I740 12 48 9306 9342 9377 94I2[9448194831 I 48 I7741 808 18421I 8761I9IO I9441I 49 958189553'9589 9624 9659 9695 Io 49 1978 20I21220462080 2114 2148I)O 5019. 70973o09765198~0098361987I 9906 9 50o9.7.22I8I12I 5|2249 2283 23171235Il 9 5I 994I 9977..I2 2.47..82. 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Part 1" 22" 3"1 4" 5" 6" 7" 8" 9"' /P Part 1 2"/ 3" 4/1 51 6"1 7' 8". 9"l t 4 7 11 14 18 21 25 29 32 3 7 10 14 17 21 24 27 31 LJOOAP IT HMIC TANGENTS. 65, Tangent of 30 Degrees..fi Tangent of 31 Degrees. 0" 10"lo 20'" 30 40 50" 0" O 10" 20' 30" 40"1 50" 09.76I4391 4881 537 I585 I634 I682 59 0 9 778774 882I 8869 89I78964 90I2159 I 173I I78o 1828 I877 I925 I974 58 I 9060 9Io8 9I55 9203 925: 9298 5t 2 o2023 2071 2I20 2I 62217 2266 57 2 9346 9394 144I q489 9537 9584 57 3 2314 2363 24II 2460 2508 255756 3. 963219679 9727 9775 982219870156 4 2606 2654 2703 275I 20Soo2848 55 4 99I819965. I3..6I. 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Sine of 32 Degrees. d Sine of 33 Degrees, f 1w0 20" 301" 40" / 5 _ 0 10"l| 20" 30". 40!,50 ol9. 7242Io 4243 4277 43 1 14344/'378 59 0o 9.736o 09614 6I 74 62066238 627I 159 II I2 2 4445 4479 45 3 4545 458o 58 I 6303 6336 6368 64oo 6433 6465158 = 464 7 4647468 4715 474814782 57 2 498 6530 6562 6595 6627 6659 57 31 4816 4849 4883 4917 495o 4984 56 3 6692 6724 6757 6789 682 6854 56' 4 5017 505I 5085 5Ii8 5I52 5I85155 4 6886169I8 6951 6983 70157048 55 5 52I9 5253 5286 532o 5353 5387154 5 7080 71 I 745 7177 7209 7241 54 6 542 5454 5488 552I 5555 5588 53 6 7274 7306 7338 737I 7403 7435 53 7 5622 5655 5689 5722 5756 5789 52 7 7467 7500 7532 7564 7597 7629 52 8 5823 5856 589 59'2335957 59 9051 8 766I 7693 7726 7758 779~ 7822 5i 9 60Io24 6057 6o9 61 24 6i58 6I91 5o 9 7855 7887 79I9 795I 7983 8oI6 5o 0 9 72622516258 6292 6325 63591639249 1I0 9.73804888 08o 8II2 8I45 8I77 8209[49 II 6426 6459 6493 6526 6560o6593 48 I i 8241 8273 8306 8338 8370 8402148 12 6,626 666o 6693 6727 6760 6794 47 12 8434 8466 8499 853i 8563 8595 47 31 6827 686o 6894 6927 696I 6994 46 I3 8627 8659 8692 8724 8756 8788 46 I4 7027 706I 7o94 7128 7I61 7194 45 I4 8820 8852 8884 89I7 8949 898I 45 i5 7228 726I 7294 7328 7361 7394 44 iS 13 9045 9077 9109 9 I41 973 144 i6 7428 7461 7494 7528 756I 7594 43 I 6 9206 9238 9270 9302 9334 9366 43' t17 7628 766I 7694 7728 776I 7794 42 I7 9398 9430 9462 9494 9526 9558 42 i8 7828 7861 7894 7928 796I 7994 4I I8 9590 9622 9654 9687 97I9 9751 4I 9I 802718o6I 8o9418127 8i6i 8I94 4o I19 9783 9815 9847 9879 99II 9999 434o 209. 72822718260 8294 832718360o8393139 2019. 739975... 7..39.71 I0o3 I3539 21 8427 846o 849318526 8560 8593 38 2I |9. 740I67o0I99 o023I 026302950 o327 38 221 8626 86591 86928726 8759 8792 37 22 0359 0391 o423 o455 o0487 o5 9 37 23 8825 8858 8892 8925 8958 8991 36 23 o0550o0582 o6I4 o646 o678 07io 36 24 9024 90581909I 9I24 9I57 19I1 35 24 0742 0774 8060 o8380870 0902 35 25 9223 9257 9290 9323 9356 9389 34 25 o934og966 0997 I029 io6i Io93 34 26 9422 9455 948919522 9555 9588 33 26 II25| I57 889g I22I I253 I284 33 27 9621 9654 968719720 9753 9787 32 27 I3i6 I348 I38o I4I2 I444 476 32 281 9820 9853 988699ggg 9952 9985 3I 28 I5o8 1539 157I1 6o3 I635 1667 3i 299. 73008oo5I 00oo 84|0II7 o 50o o83 30 29 I699 I7301 762 I794 I8261 858130 3019 7302I7 o25o 0283 jo316 o349 o382 29 3019. 74I8891I921 I953 I985 2017 2049129 31 0415 o448 o48I1o5I41o5471o580 28 31 20802 I I 2 244 2 I 76 2207 223928 32 06I33 o646 o679o07121745 0778 27 32_ 227I 2303 2335 2366 2398 2430 27 331 o8i o844o8771o9io o943 0976 26 33 2462 2493 2525 2557 2589 2620o26 34 Ioo09 10o42 IO75 io8 II4I II73 25 34 2652 2684 27I5 2747 2779 28I I 225 351 1206 1239 1272 i3o05I338 1371 24 35 2842 2874 2906 2937 2969 3oo00 124 361 i404o1437 I470oI5o3 i536 1569 23 36 303313064 3096 3I28 3I59 319I123 37 I1602 i634 i667 I70o I733 I766 22 37 3223 3254 3286 33i8 3349 338I 22 381.I799 I832 1865 i8971930o1963[2I |38 34I3 3444 3476 3508 3539 3571 2I| 39 1996 2029 2062 2o95 2I27 2I60 20 39 3602 3634 3666 3697 3729 376120 40o9.732I93.222612259]229212325 23571I9 4019.743792 3824 3855 3887|39I9 39501I9 4x1 239012423 2456 2489 2521 25541I8 41 3982|40I3 4045 4077 4Io8 4/4o i8 421 2587 2620 2653 268512718 275II7 1-42 4171 4203 4234 426642.97143291I7 43 2784 2816 2849 2882 29I5 2948 I6 1143 436I 439214424 445514487 45I81i6 44 2980o30I313046 30793 III1 3 I441I 5 44 4550o458 46 3 4644 4676 47071I5 45 3177 32io03242 3275133o8 334o04 14 45 473914770 4802 4833 4865 4896-14 46 3373 340613439 3471 35o4 3537 i3 46 4928 459499I 5022 5054 5o8513 471 3569 3602 363513667 3700 3733]I2 12 47 5II7 5I48 5I80 521 5243 5274 12 48 3765 3798 383i 3863 3896 3929| I 48 530615337 5369 54oo00543 5463 II 49 396I13994 40274o059 4092 41251 io 149 5494 5526 5557 5589 562o 565I Io 50o9.7341i57 41g9014222 42554288 4320o 9 S1o 9.745683 57I4 5746 5 777580 8 5840 1 9 5I 4353 4386 44I8 445i 4483 45I6[ 8 5I1 587i 59o3 5934 596155997 6028 8 52 45491458IA46I4 4646 4679 47II 7 152 6o0606o9I 6I22 6154 6I85 62I61 7 53 4744[4777148o094842 4874 49071 6 1153 62486279 63io 6342 6373 64o4 6 54 4939 4972500oo450 o3750 o69 5IO2 5 54 6436 6467 6498 6530o 656I 659 5 55 513515i675200oo52325265152971 4 55 6624 6655 668667I8 6749 6780[ 4 56 533015362 5395 54275460 54921 3 11 56 68I26843 687469go56937 6968 3 57 55255557 559056 56555622 56555687 2 57 69997031 706270937I24 7I56j 2 58 57I9 5752 5784 58 I7 5849 58821 I 58 7i87 72I8 724917281 73I2 7343i: 59 59I459475979 601 6044 6076 o_ 59 737417406 7437 7468 7499 75301 o 601" 5W" 40' 30"' 20" 101 1 60" 0" so,, - 40' 20" 10 Co-sine of 57 Degrees. - Co-sine of 56 Degrees. P -par p ~1" 2" ".4" 5" 6" 7" 8" 9". P t{ 1" 2" 3" 4" 5" 6" 7"' 8," 9" P. P 3 7 10 13 17 20 23 26 30 P. Part 3 6 10 13'6 19 22 25 29 _________ __ _ _ _ __ _ _ _ __ _ _ _ _ __ _ _ _ __ _ _ _ _ __ _ _ _ __ _ _ _ LOGARIT IIMIC r ANGENTS. Eb S!i Tangent of 32 Degrees. e Tangent of 33 Degrees. (Y' 1'0" 20" 30" 400" 50 0" 1OlJ' 20' 30" 40"' a 0o. 79578958365883 5930 5977 6023 59 o 9.8 12517 2563!2610 2656:2702 2748 59 I 6070 6I Ix7 6I64 62 II 6258 63o4 58 I- 2794 2840 2886 2932 2978 3024 58 t 635 I6398 6445 6492 6539 6585 57 2 307o 3II6 3I63 3209 3255 33o0 57 3 6632 6679 6726 6773 681I 6866 56 3 3347 3393 3439 3485 353I 3577 56 4 69i3 6960 7007 70 53 7o00 7147 55 4 3623 3669 3715 376I 3807 3853 55 5 7I94 7241 7287 7334 738I 7428 54 5 3899 3945 3991 4037 4083 4I29 5A 6 7474 7525I 7568 76I5 7662 7708 53 6 4576 4222 4268 43I4 436o 44o6 53 7 7755 7802 7849 7895 7942 7989 52 7 4452 4498 4544 4590 4636 4682 52 8 8036 8082 8129 8I76 8223 8269 5I 8 4728 4774 4820 486649I 2 4958 51 9 831 6 8363 8409 8456 8503 8550 5o 9 5oo4 5o5o 5096 5142 5i88 5234 5o IO09.798596 8643 8690 8737 8783 8830 49 I0 9.815280 5326 537I 54I7 5463 5509 49 I I 8877 8923 8970 90I7 9063 95i 048 II 5555 560o 5647 5693 5739 5785 48 2. 9I57 9204'9250o929719344 9390 47 1 2 583i 5877 5923 523 5969605 606 47 I3 9437 9484 9530 9577 9624 967o 46 I3 610o76I53 6I99 6245 629I 6336 46 I4 97I7 9764 985I 9857 9904 9950~45 I4 6382 6428 6474 6520 6566 66I 245 55 99997.44.. go. I37.I84 4. 230o44 I 5 6658 6704 6750o6796 68426888 44 I6 9.800277 o324 0370 04I7 o463 o5io 43 I 6 6933 6979 7025 707 7 I I7 7I63 43 17 0557 o603 o650 o697 0743 0790 42 I7 7209 7255 730I 7347 7392 7438 42 I8 o836 o883 og30 0976 5o23 Io 70 45 I8 7484 7530 7576 7622 7668 77I4 4I Ig9 I I6 I63 I209 I256 3o03 I349 40 I9| 7759 7805 785I 7897 7943 7989 40 20 9.80o396 1442 1489 I 535 5582 5629 39 2019.8I8035 8o08 85I26 8I7218218 8264 39 25I I5675 1722 5768 ]8i5 I862 1908 38 25 83Io 8356 8402 8447 8493 8539 38' 22 I955200oo 2048 2094 21441 2I87 7 22 8585 863i 8677 8722 8768 88sI4 37 23 2234 2281 2327 237412420 2467 36 23 8860 8906 8952 8997 9o43 9089 36 24 25I 32560 2606 2653 2699 2746 35 24 9I35 9585 922692729318 9364 35 2.5 2792 2839 2886 2932 2979 3025 34 25 94I0 9455 950o 9547 9593 9639 34 26 3072|3II8 3I65 32II32258 330433 26 9684 973019776 98229868 99I3 33 27 335 I3397 3444 3490 3537 3583 32` 27 9959 5..5.5I..96.142.I88 32 28 3630o 3676 3723 3769 38I6 3862 3I 28| 9.820234 o280 0325 0375 04I7 o463 3I 29 3909 3955 400oo 4o48 4o94 4I4I 30 29 o508 o55410600oo o646 o6 0737 30 30 9.804I87|4234 4280 4327|4373|4420|29 3o 9.820783 o829 08740920 0966 10I2 29 3i 4466,451345594605i4652469828 3I 1057 IIo03 149 II95 I2401I286 28 32 4745 4791 4838 4884 4930 4977 27 321 332 5377 5423 1469 i5i5 i560 27 33 5023 5070 5II6 5I63 5209 5255 26 33 I606 16525I697 I743 I7895i835 26 34 5302 5348 5395 544I 548715534 25 34 I880 I926 I972 20I7 2o63 2Io0925 35 5580o 5627 5673 57I9 576658I5224 35 2I542200 224622 292!4337 2383'.4 36 5859 5905 595I 5998 6044 609I 23 36! 2429 24742520 2566 26 I1 2657 23 37 6I37 6I83 6230 6276 6322 6369 22 371 2703 2748 2794 2840 2885 293I 22 38 64I5 6462 65o8 6554 655466o66472 38 2977 3022 3o68 343II4 3593205 2I 39 6693 6740 6786 6832 6879 6925 20 39 3251 3296 3342 3387 3433 3479 20 40o9 80697I 70I8 7064 7110 o757 7203 I9 4019.823524 3570o 366 366I 3707137531I9 41 7249 7296 7342 7388 7435 748I I8 14I' 3798 3844 3889 3935 398I 4026 i8 421 752717574'7620 7666 7753 7759 I7 42 4072 411745i63 4209 4254|4300I7 7 43 7805 7851 7898 7944 7990 8037 i6 43 4345 439i 4437 4482 4528 4573 I6 44 808318I29 8I76 8222 8268|8314 I5 44 46I9 4665`4710 4756 480I 4847 55 45 836r 8407 8453 8499 8546 8592I4 45 4893 4938 4984 5029 5075|5I20 I4 46 8638 8685 873I 8777|8823|88701I3 46 5I66 5212]5257 5303 5348 5394 13 47 89i6 8962 9008 9055 91II 9I47 52 47 5439 5485 553I 5576 5622 5667 12 48 9I93 9240:9286 93329378 942455I, 48 57i3 5758 5804 5849 58955940oII 49 9471 95I7 9563 9609 9656 97002. 10 49 5986 6032i6077 6I23 6I68 6214 0o Sog 9809748 9794 9840 9887 9933 9979 9 | So 9.826259 63o5 6350o 6396 644i 6487 9 5I 9.8oo025o0071 oI18 oI64o02Ioo0256 8 5i 6532 6578:6623 6669 6714 6760 8 52 0302 0349 o395 o44 044 87 0533 7 52 6805 685i 6896 6942 6987 7033 7 53 o58o 0626 0672 07I8 0764 o8Io 6 53 7078 7124 7I69 72I5 7260 7306 6 54 0 o857 ogo3 o949 o995 io4I Io 87 5 54 735I 7397 7442 7488 7533 7579 5 55 II34 I80o 1226 1272 I3i8 sI364 4 55 7624 7670 77i5 776I 7806 785Il 4 56 I4Io I457 i503 1549 I595|I64i| 3 56 7897 7942 7988 8033 8079 8124 3 57 I687 I733 I780:826 I872 i9I8 2 57 8I70o82i5 826I 83o6 835I 8397 2 58 I964 20Io02056 2102 2,49 2i59 I 58 844218488 8533 8579 8624 8669 1 59 224i 2287 2333 2379 2425 247I o 59 8715 8760 8806 885i 8897 8942 o 60" - 50" 40", 30, 20" 10"' 60"' M5' 4"' 0,," 2(Y' 10." Co-tangent of 57 Degrees.'' Co-tangent of 66 Degrees. P Partl 5 1"19 3/ 8 3 3"1 4/ 7 8". 9" tl 2// 3 4" 5" 6" 7" 8" 91 5 9 14 19 23 28 33 37 42 5 9 14 18 23 27 32 37 41 668 t ARI LOGARITHMIC SINES Sine of 34 Degrees. Sine of 36 Degrees. of0 10" 20" [30"4" 50 _ 0" 10" 20" 30" 40" 50" 0 9.747562 7593 76247655 7686 77I8 59 o g.7585gI 862I 865I 8681 87I2 8742 59 I 7749 7780 78 1 7842 7874 7905 58 I 8772 8802 8832888862 8892 8922 58 2 7936 7967 7998 8o3o 6180 8092 57 2 8952 8982 9OI2 9042 9072 9-02 57 3 8 813[81 54885 821618248 82799 56 3 913219I629 I92192229252 9282 56 83Ic 834I 8372 84o3 8434 8466 55 4 9321 9342 9372 9402 9432 9462 55 5 8497 8528 8559 8590o 86218652 54 5 9492 95221955219582 9612 9642 54 6 8683 8714 8745 8777 88o8 8839 53 6 9672 9702 9732 9762 9792 9822 53 7 8870 8go1 8932 8963 8994 9025 52 7 9852 988 99gI I 994I 997I... I 52 8 9056 9087 9II8 949 91I80 92I2 5 I 8 9.76003I oo6 0091 gII21 oi5i oi8i 51 9 9243 9274 9305 9336 936719398 50 9 0211 o240 0270 o30oo o330 o36 50.0 9 J 749429 9460 949I 9522 9553 9584 49 IO 9.9760390g 420 o45o 0480 0509 0539149 I I 96I5 9646 9677 97o8 9739 9770 48 II 0569 o599 o629 o659 o689o0718 48 12 g801 9832 9863 9894 9925 9956 47 I12 0748 0778 o8o8 o838 o868 o898 47 I3 9987.. i87..48..79.1IO.I4146 13 0927 o957 o987 IOI7 1047 I076 46 14 9.75o072 o203 o234 o265 o296 o327 45 I4 I06 ii36 ii66I I96 I225 I 255 45 I5 o358 o389 o420 o45I o482 o512 I44 i5 1285 1315 345 I 374 I404 I434 44 i6 0543o 574 o6o5 o636 o667 o698 43 I6 I464 1494 1523 i553 I583 16I3143 I7 0729 076o 079I 082I o852 o883 42 17 I642 1672 I 702 17321I76l 1791 42 I8 ogI4 o945 o976 I00oo 7 37 io68 4i I8 I821 i85i I88o 19IO 1940 I969 41 9Ig 099I ii3o i6i II92 1222 1253 4o 19 1999 2029 20591 2088 2II8 248 4o 209. 75I284 i3i 51 346I 377 1407 I438 39 2019.762177 220712237 22672267 96 232639 21 1469 50oo I53I i656 1592 1623 38 21 2356 2385 24I5 2445 2474 2504 38 22 I654 i685| I7I5 1746 1777 i8o8 37 22 2534 2563 2593 2623 2652 2682 37 23 I839 1869,g900oog3I I962 I992 36 23 27I2 274I 277I 280I1 2830 286 36 24 2023 2054 2085 2I 5 2146 2177 35 24 2889 29I9 2949 2978 3008 3038 35 25 2208 2238o2269 2300 2330 236I 34 25 3067 3097 3I26 3i56 3I86 32I5 34 26 2392 242312453 24841 255 2545533 26 3245 3274 33o4 333313363 3393133 27 2576 2607 2637 2668 2699 2729 32 27 3422 3452 348I 35I I 3540 3570132 28 2760 279I 2822 28 52 2883 2914 3 I 28 36oo 3629 3659 3688 37 8 374713 2) 2944 297513005 3036 3067 3097 30 29 3777 38o6 3836 3865 3895 392513o 30.753 I128 3I 59 3189 3220 325I 328I 29 30o 9. 763954 3984 4013 4o43 4072 4102 29 31 33I2 3342 3373 3404 3434 3465 28!13 4I3I 4i6I 4i90o4220 4249 4279128 32 3495 35263557 3587 36i8 36481 7 32 43o8 4338 4367 4396 4426 44 55 27 33 3679 37I 73740 3771 380I 3832 26 33 4485 45I4 4544 4573 46o3 4632 26 34 3862 3893:'3923 3954 3985 40oi 25 34 4662 469I 4720o4750 4779 4809125 35 404614076 4I07 4I37 4I168 4I98 24 I 35 4838 486814897 4926 4956 4985 24 36 4229 4259 4290~4320 435'I 4381 23 1 36 5o0I 5o4415074 Sio3 5132 5I62 23 37 4412 4442 4473 4503 4534 4564 22 37 5IgI 522S15250 5279 5309 5338 22 38 459514625 4656|4686 47I7 4747 2I 38 5367 5397 5426 5456 5485 5514 21 39 4778 4808 4839 4869 4900 4930 20 39 5544 5573 15602 5632 566i 5690 20 4o 9.754960 499I,502i 5052 5082 5II3 I9 4o 9.765720 5749 5778 5808 5837 5866 I9 41 5I4315I-73.52o4 5234 5265 5295s i8 4 5896 592515954 5984 6o0I3 6042 8 42 5326|5356;5386 54I 7 544754781 I7 1 42 6072 6IO16130o 6I59 6I8g 62I8 I7 43 55o085538&5569 5599 5629 566o0 6 | 43 6247 6277 63o6 6335 6364 6394 i6 44 5690 5728I575I 578I 5812 5842 I5 44 6423 6452648I 65I I 6540 6569915 45 5872 5903'5933 5963 5994 6024 I4 4 45 6598 6628 6657 6686 67I5 6745lI4 46 6054|6085 6I5 I 6I45 6I76 6206 13 46 6774 680316832 6862 68gI16920l 3 471 6236626762937 6327 6358 68 388 12 47 6949 697817008 7037 7066 7095 I2 48 64i8 6448 6479 6509 6539 6570 II 48 7 I 24 7541783 72 I 2 724I 7270 I I 49 66oo00663o0666o 669I 672I 675I IO | 49 7300 7329 7358 7387 7416 7445o10 5o 9.756782 68I26842 6872 6903 6933 9 50 9.767475 7541 7533 7562 7591 7620] 9 5i 6963|6993|7023|7054 7084 7II4 8 i 5I 7649 76797708 7737 7766|7795 8 52 7I44 7I75 7205 7235 7265 7295 71 52 7824 7853 7882 79I'2 7941 7970 7 53 732667356 7386174I6 7446 7477 6 53 7999 8028180571808681 I 58144| 6 541 750717537 756717597 7627 7658 5| 54 8173 8203 8232 826I 8290o83I9 5 55 7688 7718 7748 7778 7808 7839 4 55 8348 8377 84o6 844435 8 64 84931 4 56 7869 7899 7929 7959 7989 1809 3 56 8522 855 858o 86098638 8638 8668 3 571 8o5o08o8o8 I I o8140o817o08200 2 57 86971872618755 878 88I 318842 2 58Q 8230 8260 8290 8325 83531 838 I 58 887I 8900oo 8929 8958 8987199061 I 59 84II|844I|847I|850I 853I 856I 0 |59 9045 90741903 9I32 9I6I9I90o 0 I60 i50" 40" 30" 20" 110" 60"- 50/ 40/1 30 20" 10"1 Co-sine of 55 Degrees. Co-sine of 54 Degrees. 1"i 2" 3" 4/1 5" 6" 7"/ 8/1 9 1 p (p 1" 2" 3/' 4'/ 5/I 61 7'1 8" 9" ar 3 6 9 12 15 18 21 25 28 3 art 36 9 12 15 18 21 24 27 LOG AR IT H MI C A N G N1 S. 5 _ Tangent of 34 Degrees.. Tangent of 36 Degrees. 0" _ 1-0"-1 20' 30" 40" J0"1 ) " - 10 20 30/ 40 1 50 0 9.828987 9033'19078 9124 9169 g925 59 o 9.8452"27 5272 53I6 536I 54o06545I 59 I 9260 9305 9351 9396 9442 9487 58 I - 5496 5540o 5585 5630o 5675 5720 58 9532 9578 9623 9669 97I4 9759 57 2 5764 5809 5854 5899 5944 5988 57 9805 9850 9895 (94i 9986..32 56 3 6o33 6078 6123 6I68 6212 6257 56 49.830077 0122 1o I 602I 3 258 o304 55 4 6302 6347 639I 6436 648I 6526}55 5 o3491o395 o44o o485 o53I o0576 54 5 6570 66I 5 666o 6705 6750o6794154 6 o621 0o667 07I2 0757 o8o3 o848 53 6 6839 6884 6929 6973 7o08 7063 53 7 o893 0939 984 1029 o I075 112052 7 7108 7I52 7197 7242 7287 733I 52 8 II65 1211 1256 i3oi I347 I392 SI 8 737( 742I 7465 75IO 7555 7600 5r 9 1437 i483 1528I 573 1619 I664 50 9 7644 7689 7734 7779 7823 7868 5o o19.83I709 I755 i800oo845 189I 1936149 O 9.8479I 3795718002 8047 80928I 3649 II I98I 2026 2072 2 I2 7 2162 2208 48 II - 8i8i 8226 8270 83i5 836o 84o5 48 12 2253 2298 2343 2389 2434 2479 47 12 8449 8494 8539 8583 8628 8673 47 13 2525 2570 2615 2660 2706 275 1 46 i 3 8717 8762 88071885I 88961894I146 I4 2796 2842 2887 2932 2977 3023145 4 8986 9030 975 9 I 20 9 I 64 9209145 I5 3068 3113 3 I 58 3 204 324 94144 I 5 9254 9298 9343 9388 9432 9477 44 I6 3339 3385 343o 3475 352o 3566143 16 9522 9566 96 I 96569700 9745 43 17 36ii 3656 370I 3747 3792 3837 42 I7 9790 9834 9879 9924 9968..3 42 i8 3882 3927 3973 40I8 4063 41O8 41 I8 9.8500570 OI2 01470191 0236 028I 141 19 4 54 4I99 4244 428914334 438o 4o I9 o325 o37~1o4I 5 o459 o504 o548 4o 20 9. 8344254470 45 I 5 456 I 46o6465 I 39 20 9.85o593 o638 o682 0o727 0772 o8I6 39 21 469614741 4787 4832 487714922 38 2I o86I ogo5 og50 o9951 39g io84 38 22 4967 5012 5058]5I03 5148 5193 37 22 1129 I173 I218 I2621 307 135237 23 5238 5284 5329 5374 5419 59464 36 23 396 1441I485 i 530o 575 1619 36 24 5509 5555 56oo 5645 56o0 5735 35 24 I664 I708 I753 I797 1842 I887 35 25 5780 5826 587I 59I6 5961 6oo6 34 25 I931 I976 2020 2065 2110 2I54 34 26 605I S 60966,42 687 6232 6277 33 26 2199 2243 2288 2332 2377 2422 33 27 6322 6367 6412 6458 65o31 654832 27 2466 25 I 2555 2600 644 2689 32 28 6593 6638 6683 6728 6773 68I9 31 28 2733 2778 2823 2867 29I2 2956 3i 29 6864 69o09 6954 6999 7044 7089 30 29 300o 3o45 3090 3x34 3179 3223 30 30 9.837I3417I79 7225 7270 73I 57360 29 30 9. 853268 33I3 3357 3402 3446 3491 29 31 7405 7450 7495 7540 7585 7630 28 3i 3535 358o 3624 3669137I3 3758 28 32J 7675 7721 7766 7811 7856 7901 27 32 3802 3847138I 939369804025 7 33 7946 799I 8036 8o8i 8I2618I7I126 33 4069 4II414I5814203 4247 4292[26 34 8 8261826I 83078352;8397 8442 25 34 4336 438I 44251447o045I4 4559125 35 8487 8532 8577 8622 8667 8712 24 35 46o3 464814692 473714781 4826124 36 875718802 8847188921893718982 23 36 4870 49I5 4959 5oo4 5o48 5093 23 37 9027 907219I17 9I6219207 9252 22 37 53751825226 527I 535 536022 38 9297 9343 9388 9433 9478 9523 2 I 38 54o4 5449 5493 5537 5582 5626 21 39 9568 96I3 9658 970319748 9793 20 39 5671 57I5 5760 58o4 5849 5893 20 4o 9. 839838 9883 9928 9973.. I8.. 63 19 14o 9. 855938 5982 6026 607I 6I51 6I6o 09 4i 9.840108 oi53 o0198 o243o288 o333 i8 4 6204 6249 6293 6338 6382 6426 i8 4!j o378 o423 o468 o5 o3 o558 o6o3 17 42 6471 65I5 656o 66o4 6649 6693 17 43 o648 o693 0737 0782j0827 o872 i6 43 6737 6782 6826 687I 69I5 6959 16 44 09I 7 0962 1007I 052 1097 11425 I5 44 700470487093 7137 7I82 7226 5 45 I871 I232 I2771 3221:367 1412 14 45 7270 73I 573591740417448 7492 I4 46 I457 I502 I547 i592 i637 I682 13 46 7537 758I 7626 7670 77I4 7759 13 47 I727 177I i8i6 i86i 190o6 95i1 Ij2 47 7803 7848 7892 7936 798I 8025 12 48 I996 2041 2086 2I31 2I76 222 III 48 80691 8ii4 8x58 8203 8247 8291 II 49 2266 23 I 2355 2400oo2445 2490o I1 49 8336 8380o842418469185s3 85581 o 50 9 842535 2580 2625 2670o2715 2760 9 50 9.858602 8646 869I 8735 8779 8824| 9 51 2805 2849 2894 2939 2984 3029 8 5i 8868 89I2 8957 900 9oo45 9gogo 8 52 3074 3I I9 3I16493209 3253 3298 7 52 9I34 9I781922319267 93 I 19356 7 53 3343 3388 343313478 3523 3568 6 53 94oo 9444 948919533 9577 9622 6 54 3612 3657 3702 3747 3792 3837 5 54 9666 9710 9755 9799 9843 9888 5 |55 388213927 3971 4o6 4o6i406 4o6 4 55 9932 9976...2I..65.,i09.I54 4 56 4I5I1 496 424I 428514330o4375 3 56 9.860I98 o242 o287 0331 0375|0420 3 57 4420 4465 4510o455414599 4644 2 57 0464 o508 o552 o597o 64I o685! 2 58 4689 4734 4779|4823 4868 493 I 58 0730o774 08i8o0862 0907j0951 I 59 49585003oo3504850925 3715I82 0o 59 o995 140 o841128 1172 1217j 0 I60" 150" 40" 30"11 20" 10"- 60 50" 40 30" 20 1(" Co-tangent of 55 Degrees. Co-tangent of 54 Degrees. P. PartS 1" 2" 31' 4"/ 5"1 6"/ 7'/ 8" 9" P. Part 1 2" 31 4/1 5 6" 7/I 8 911 5 9 14 18 23 27 32 36 41. art 4 9 13 18I 2 27 31 36 40 j ~~~(' ~'; ~~,; j~ ~j/p: ~ 1" 2/'3// 4' 5/' /'.7/r_ _ 9 f 01 L o LO A R 1 rIMIC S I N ES. Sine of 36 Degrees. l Sine of 37 Degrees. [ O 0d" -10'" 20"1 30" 40/1 50" 10" 201 30" - 40 5" O19.769219 924819277 9306 933519364 59 9.779463-94' I 99519547 95759603 59 I 9393 9421 94r 5 9479 9508 9537 58 I 963I 9659 9686 97I4 9742 977o 58 2 9566 5995 9624 96539 682197I11 7 9798 9826 9854 9882 9910 9938 57 3 9740 9769 9798 9827 9856 9884 56 3 996619993.. 21.49..77. Io5 56 4 99 I 3 9942 997I!...... 29..5855 49.780 33016I OI89 0216o244 01o272155 59.770087 0 I I6 OI451oI73 0202 023I 54 5 o300 o328 o356 o 3841o4i I o439 54 6 o26o 028 03I810347 o376 o4o4 53 6 o467 o495 o523 o55i o578 o6o6 53 7 0433 o462 o49Io520 o549 o 577 52 7 o634 o662o69go 7I8 ~745 ~773 52 8 o606 o635 o6641o693 0722 07505 I 8 o8oi 0829 o857 o884 092 ~94o0 5 I 9 0779 o8o8 o837.~o866 o895 0923 50 9 968 og96 1023 5 I I079 I I 07 50 10 9 770952 0981 IoII03 91067 Io96 49 o109.78I I34 I162 II90o I28 I246 273 49 II I1125 II54 1I83I12II I240 I269 48 II I30I I329 1357I 384 I412 I44o 48 1 2 I298 I326 i355 i384 I4I3 1441 47 12 I468 1495 1523 I55i 1578 i6o6 47 13i I470I499 5281I556 I585 I6I4146 I3. 1634 16621 6891 7I 71745 I 772 46 14 i643 I 671.I 700i'729 I 758 I 786 15 I4 800oo 828 i856 I883 I91 I 939 45 i5 I8I5 I844 I872,I9 o I930 1959144 I5 I966 I994 2022 2049 2077 2Io5 44 I6 I987 206~ 20 4512073 2I022I3T1 43 I6 2I32 2I601 288 22I5 2243 2271 43 17 21,59 2I88 22I 712245 2274 2303 42 17 2298 2326 2354 2381 2409 2437 42 I8 2331 236012389124I712446 2475 4I i8 2464 2492 2520 2547 2575 26o02 4 I9 2503 2532 256I 2589 2618 2646 4o I39 2630 2658 2685 27I3 274I 2768 4o 20 9.772675 2704|2732l276I 2790 28I8 39 209.782796 2823 285I 2879 2906 2934 39 2I 2847 2875 2904 2933 296 12990 38 21 296I 2989 30I7 3o44 3072 3l99 38 22 3oi8 3047 30763I104 3i33 3I6I 37 22 3I27 3I54 3I82 32Io 3237 3265 37 23 3I90132I9 3247 3276 33o413333 36 23 3292 3320 3347 3375 3402 343o 36 24 336i 3390 34I8 3447 3476 35o4 35 24 3458 3485 35I3 354o 3568 3595 35 25 35331356i 3590 36i8 3647 3675 34 25 3623 365o 3678 3705 3733 3760 34 26 3704 3732 376I 3789 38I8 3846 33 26 3788 38x5 3843 3870 3898 3925 33 27 3875 3903 3932 3960 398914017 32 27 3953 3980 4008 4035 463 4090 32 28 4046 4074 4Io34I3I 4I 6o|4i88 31 28 4IIS 4I45 4173 4200 42281425513I 29 42I714245 4274 4302 433i 4359 3o 29 4282 430o 4337 43654392442030 3019 774388 44I614445 4473 45o0I 4530 29 3019.78444714475 4502 4529 4557 4584 29 31 3 4558 4587 46I5T46441467214700 28 3I 46I2 4639 4667[4694 4723 4749128 32 4729|4757|4786!48I414842|487I 27 32 4776 4804 483I14858 488614913J27 33 4899 4928 495614985 50o3 504I 26 33 494' 4968 14995 5023 5050o 507826 34 507o50o985I265I55 5I8352I12 25 34 5Io515I321 56.ol 5I87152I4 5242 25 |35 5240 5268 5297 5325 5353 5382 24 351 5269 6 5296 5324 535i 5378 54o6 24 361 54io 5438 5467 5495 552315552 23 36 5433 546i 5488 55I5 5543 5570 23 37 568o 56o8 5637 5665 5693 5722 22 37'5597 5624 5652 5679 5706 5734 22 38 5750 5778 5807 5835 5863 5892 2I 38 576I 5788 58i6 5843 5870 5898 2I 39 592o05948 5977 60oo 6033 6o6I 20 39 5925 5952 5979 6007 6034 6o6I 20 40 9.77609016I I861466I175 6203|623 I19 4019. 786o896I16 6I43 617o 6I98|6225 Ig 4I 6259 6288 63I6 6344 6372 64 i811 41 6252 6279 6307 6334 636i 6388 i8 42 6429 6457 6485 65I4 6542 6570 I7 42 64I6 6443 647016497 6525 6552 I7 43 6598 6627 6655 6683 67II 6739 I6 43 6579 66o6 6634 666I 6688 67 51 i6 44 6768 6796 6824 6852 688o 6909 I5 44 6742 6770 6797 6824 685i 6878 i5 45 6937 6965 6993 702I 7o0507078 14 45 6906 6933 6960o6987 7o04 7042'4 46 7IO617I34 176217I9I 72I917247 I3 46 7o6917096 712317150 7I77172051 3 47 7275 7303 7331 7359 7388 74i6 I2 47 7232 7259 72861733 73401 73671 I 2 48 7444 7472 7500 7528 7556 7585 II 48 739517422 744917476 7503 7530 ii 49 76I31764I 7669 7697 7725 7753 I0 49 755717585 76I2 7639 7666 76931 I 50 9. 77778I 78IO 7838 7866 7894 7922 9 5019. 78772017747 7774|780I 7829 7856 9 5i 7950|797880oo068034 806280o9I 8 5i 7883779IO 793717964 799I|80I8 8 52 8II98I478758820 3823i8259 7 521 804518072 809918I27 8I54|8S8IS 7 53 8287 83I5 8343 837I 8399j8427 6 53 8208 8235 8262 8289 83i6 8343j 6 54 8455 8483 85 I|8539 8567|8595 5 54 837018397 8424 845i 847885o05 5 55 8624 8652.8680|8708 8736|8764 4 55 853218559 8586|863 1 86 667 4 56 8792 8820 8848 8876 89o048932 3 56 8694 872I 8748 8775 8802 8829J 3 57 8960|8988 9o06 9044 9072 9100 2 57 8856[8883 89Io 8937189641899I 2 58 9128915683 I 9218 9 2399267 58 o8o459072909919I26 912653| I 59 9295 9323 9351 9379 940794 0 59 9I80o9207 9 9342 9288193I5 0 6('v 50"1 40'' 30" 20" 10 60" 1 50t 1 40 1 30" 20" 1 Co-sins of 53 Degrees.. Co-sine of 52 Degrees. I paprt 1" 2" 3/1 4" 5n 6" 7" 8/' 9" 1' t 2"r 3/ 4/" 5/" 6" 7/ 8/ 9/t ar 3 6 9 11 14 17 20 23 26 P. Part 3 5 8 11 14 16 19 22 25 I _ LOG A R I T H M IC TANGENTS. 6. Tangent of 36 Degrees. Tangent of 37 Degrees.' ].01, 20"1 30" 4 50" 10, 20"' 30, 40" 50" o 9.86126I I3o05 i3501394I1438 I482 59 09.877114715877202 7246 7290 7333 59 i I527 I57I1 i65 I659 704 I 748 58 7377 7421 7465 7509 7552 7596 58 2 1792 I837 i88i I925 I969 2014 57 2 7640 7684 7728 777I 78I5 7859 57 3 2058 2I02 2I46 2I9I 2235 2279 56 3 7903 7947 799o 8034 8078 8I22 56 4 2323 2368 2432 2456 2500 254555 4 8I65 8209 8253 8297 834I 8384 55 5 2589 2633 2677 2721 2766 28I0 54 5 8428 8472 85I6 8559 8603 8647 54 6 2854 2898 2943 2987 3o3i 3075 53 6 8691 8734 8778 8822 8866 8909 53 7 3II9 3I64 3208 3252 3296 3341 52 7 8953 8997904I 90o85 91289I72 52 8 3385 3429 3473 3537 37562 36o6 5 8 92I6 9260 9303 9347 939I 9435 5I 9 365o 3694 3738 3783 3827 387I 5o 9 9478 9522 9566 9609 9653 9697 50 I O 9. 8639 I 5 3959 40o044048 4092 4I 36 49 IO 9.879741 9784 9828 9872 99I 6 9959 49 I 4I80 4225 4269 43I3 4357 440I48 I 9 9.880003 oo4 00479I I 34 01780222148 12 4445 4490 4534 4578 4622 4666 47 12 0265 o30g o353 o397 o44o o484 47 I3 47I O4755 4799 4843 4887 493I 46 I3 o528 o57I o6I5 o659 o 703 o 746 46 I4 4975 502050o64 5IO8 5I52 5I96 45 I4 0790 o834 o877 0921 o965 ioo8 45 i5 5240 5285 5329 5373 54I 7546I144 I5 I052 Io96 II4o0 183 122727 2744 I6 5505 5549 5594 5638 5682 5726 43 I6 I3I4 i358 I402 I445 I489 I533 43 I7 5770 58 4 5858 5903 5947 5993 42 7 I577 1620 I664I 708 I75 I I795 42 i8 6035 6079612361 67 62II 6256 4I I8 I839 I882 1926 I97o0Io3 20 57 41 9Ig 63006344 6388 6432 6476 6520 40 I9 210 2I44 2i88 2232 2275 23g o40 20 9.866564 6609 6653 6697 674I 6785 39 20 9.882363 2406 2450 2494 2537 2581 39 21 6829 6873 69I7 696I 7oo6 70 50 38 23 2625 2668 27I2 2756 2799 2843 38 22 7094 7I38 7I82 7226 7270 7334 37 22 2887 2930 2974 3o0I8 3o6i 3o5 37 23 7358 7402 7446 749I 7535 7579 36 23 3I48 3I92 3236 3279 3323 3367 36 24 7623 7667 771I 7755 7799 7843 35 24 34Io 3454 3498 354i 3585 3628 35 25 7887 793I 7975 80o9 8064 8io8 34 25 3672 37I6 3759 3803 3847 3890 34 26 8I52 8I96 8240 8284 8328 8372 33 26 3934 3977402 I 4065 41o8 4I52 33 27 84I6 846o 85o4 8548 859218636 32 27 4196 423914283432646 43744432 28 8680 8724 8768 88I3 8857 89oI 33 28 4457 450I4544 4588 4632 4675 3I 29 8945 89899033 9077 9I29I 65 30 29 47I9 476248064850o 4893 4937 30 30 9.869209 9253 9297 9341 9385 9429 29 30o9.884980o 5024 5o68 5 I I I 555 5I98 29 3I 9473 95I7 956I 9605'9649 9693 28 3I 5242 5286 5329 5373 54I6 5460 28 32 9737 9781 9825 9869 9913 9957 27 32 5504 5547 559 5634 5678 5721 27 33 9187000I oo45 oo8g O33 0177022I 26 33 5765 58095852 5896 5939 5983 26 34 o265 0309 o353 0397 0441 o485 25 34 6026 6070 6 I4 6i57 620I 6244 25 35 o529 o573 06I7 o66i 7oI 50749 24 35 6288 633i 6375 643 66462 650 6 24 36 o0793 o837 o88i 0925 0969 0IO3 23 36 6549 6593 6636 668o 6723 6767 23 37 IO57 II01 II45 II89I 233 1277 22 37 68 I 6854 6898 694I 6985 7028 22 38 I32I I365 I409 I453 I497 I54I 2I 38 7072 73I5 7I59 7202 7246 7289 2I 39 I585 I629 i673 I717 I76I]I8o5 20 39 733317377 7420 7464 7507 755I 20 40 9.87I849 1893 I937 I980 2024 2068 19 4o 9.887594 7638 768I 7725 7768 7812 I9 41 2II2 2I56 2200 2244 2288 2332 I8 43 7855 7899 7942 7986 8029 8073 i8 42 2376 2420 2464 2508 2552 2596 I7 42 8 I6 8I6o 8203 8247 829I18334/I7 43 2640 2684 2727 2771 |281 2 859 I6 43 8378 842I 8465 8508 8552 8595 i6 44 2903 2947 299I 3035 3079 3I23 i5 44 8639 8682 8726 8769 88i3 8856 I5 45 3I67 32I1 3255 3299 3342 3386 I4 45 8900 8943 8987 9030 9074 9I 7 14 46 343o 3474 35i8 38562 3606 3650o 3 46 916 92o 4 9248 9291 9334 9378 i3 47 3694 3738 3781 3825 3869 39I3 I2 47 942I]9465 95o8 9552 9595 9639 12 48 3957 400o 4o45 4089 4I33 4I77I I 48 9682:97269769 98I3 98569ggoo I 49 4220 4264 430843524396 444 I 49 99439987..30o 74. I 7. I 6 I O 5019 874484 4528 4572 46I 514659 4703 9 50 9. 8902o4o247 029I o334 o378 o421 9 51 I 4747 479I 4835 4879 4923 4966 8 5i o4651o50o8 o552 o595 o639 o682 8 52 5oio 5o54 5o98 5]42 5]86523 7 7 62 0725o769 o8I2o856 o899 0943 7 53 5273 53I7 536I 54o5 5449 5493 6 53 o986 Io3o 3073 III6 II60o I203 6 54 5537 5580 5624 5668 57I2 5756 5 54 I2471 1390o I334 377 I42I I464 5 55 58oo005843 5887 593i 5975 6o09 4 55 i5o7 I55 I594 i638 i68i 1725 4 56 6o63 6Io7 650o 6I94 6238 6282 3 56 I7681 i8i I855 I898 I942 i985 3 57 6326 6370 643 6457 65oI 6545 2 57 2028 207 22I 5 2I59122202 2246 a 58 6589|6632 6676 67206764 680 58 22.8 2332 2376 24I9 24632506 1 59 6852 6895 6939 6983 7027 707 O 59 2549 2593 2636 2680 2723.3 2766 o - 601A 5-0" 40' 30" 20" 10" 60" 0 40 30" 1 20" 10" _, I Co-tanlgent of 53 Degrees.! Co-tangent of 52 Degrees. { P. Part 4 1 2 3" 418 5 6" 7"3 8" 9 P. Part / 24 3 41 " 5 6" 7'/ 8" 9" 4913 1822631 3540'4 9 13 17 22 26 31 35 39 0:2 LOGARITH MIC SINE S. Sine of 38 Degrees. [ Sine of 39 Degrees. "'- - 10"- 20" 30 40 50 0 10" 201" 30/" 40"1 1 510" 0 9.789342 9369 9396 9423 9450 9477 59 9. 7988728898 8924 8950 89769002 59 I 9504 9531 9557 9584 96 I 9638 58 I 9028 go54 go80 9Io691 329I58 58 2 966519692 9719 974697739800 57 2 9184 92IO 92369262 9287 93I3 57 3 9827 9854 9880 9907 9934 996 I56 3 9339 9365 939I 94I7 9443 9469 56 4 9988..I5..421..69..96. I22 55 4 9495 952I 9547 95739599962555 519 790I49 OI 760o203 02300o257 0284 54 5 965I 9677 9703 9728 9754 9780 54 6 o3o 0337 o364 o391 o4i8 o445 53 6 9806198329858 9884 991io9936 53 7 o47 I o4980525 o552 o579 o6o6 52 7 9962 9987.. 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652 6547 6574 66oo6626 66166 5214 45 5799 5824 5850o 5875 5900 59926 4 461 6679 67o05 673I16757 6783 68Io I3 46 595i 5976 6002 6027 6o52 6077 I3 47 6836168626888 694146941.69671I2 47 6Io316I281865316179 620416229 I2 48 699370I9 7045 7072770987124II1 48 6254 6280 63o05633o 6355 638i II 49 7i50o 776 7202 7229 72755 728IIO 49 64o6 643I 6456 6482 6507 6532 IO 50o 9797307 73337359 73867427438 9 50 9.80o6557 6583 66o866o833 6633 668 6684 9 5I 74641749o;75i6 7542 7569 7595 8 5 67o096734 6759 6785 68Io 6835 8 52 7621 764717673 7699 7725 775I 7 52 6860o 6885 69I1 6936 696I 6986 7 53 7777 78'047830 7856 7882 7908 6 53 70I17037 70 62 7087 7II2 7137 6 54 7934 7960 7986 80oI280388065 5 54 7I63178872I317238 72637288 5 55 809I8I1I78I43'8169 8i95S822i 4 55 73417339739 73673 89741474 39 4 | 56 8247 82738299 8325 835i 8377 3 56 7465 7490 755 755 75 7565 759o 3 57 8403 8429 8455 84 8508 85o8 8534 2 57 76I5 764I 7666 769 776 774I 2 58 8560 8586186I2 8638 8664' 86g90 I 58 7766 779g 78i6 7842 7867 7892 I 59 87i.684 2 8768 8794 8820 6 59 791779427967799280718042 " 50" 40" 20 10" 60" 1 50t "140" 1 30" 20" 1'l0 Co-sine-of 51 Degrees. 1, Coosine of 50 Degrees. p art 2"' 3/" 4"/ 5"1 6" 7/" 8" 9' 11 2/ 3/ 4/ 5V 6' 71 8" 9" ar4: 3 5 8 11 13 16 19 21 24 Par 3 5 8 10 13 15 18 20 23 LO G A RI T MIC TA N G E N T S. 63 -'~ k Tangent of 38 Degrees. - Tangent of 39 Degrees. - 0'".101" 20" 30" 40"'50" 0" I W' 20"t 1 30" 40 1 50 0g9.8928Io 2853 2897 2940 2983 3027 59 o09.908369 84I2 8455 8498 854I 858 59I 307 3 I I4 3 I 57 3200 3244 3287 58 I 8628 867I 8714 875718800 8843 58 2 333i 3374 34I7 346 3504 3547 57 57 2, 8886 8929 8972 90o5 9058 9IoI 57 3 359I 3634 3678 372I 3764 380 8 56 3 9144 9187 9230 9273 93I6 9359 56 4 385I 3894 3938 398I 4025 4068 55 4 9402 9445 9488 9531 9574 9617 55 5 41 4I55 4198 424I 4285 4328 54 5 9660 9703 9746 9789 98329875 54 6 4372 44I5 4458 4502 4545 4588 53 6 99I8 996I...5..48..9. 34 53 7 4632 4675 47184762 4805 4848 52 7 9. 910177 0220 o263 o3o6 o349 0392 52 8 4892 4935 4979 5022 5065 5Iog 5I 8 0435 0478 o52I o564 0607 o650o 5I 9 5i52 5195 5239 5282 5325 536950 9 0o693 0736 0779 0822 o865 o9o8 50 IO 9.895412 5455 5499 5542 5585 5629 49 Io 9.9Iog951 994I o37 I0801123 I I66 49 I 5672 57I5 575915802 5845 5889 48 I I I209 I252 1295 I338 I38I 424 48 12 5932 5975 60I96062 6 Io5 6149 47 1 2 I467I 5 IO553 I 596 I639 1682 47 I3 61926235 62786322 6365 6408 46 I3 1725 17681 8Xo Ix853 I896 I939 46 I4 6452 6495 6538 6582 6625 6668 45 I4 I982 2025|2068 2I I 2I54 2197 45 15 67I2 6755 6798 6842 6885 6928 44 I5 2240 2283 2326 2369 2412 2455 44 6 6971 7o15 7058 OI7 71I45 7I88 43 I6 2498 254I 2584 2627 2670 2713 43 17: 7231 7275 7318 7361 7404 7448 42 17 2756 2799 2842 2885 2928 297I 42 18 749' 7534 7578 7621 7664 7707 4 I 8 3oi4 3057300oo 3I43 3I85 3228 4I 19 7751 7794 7837 7881 7924 7967 4o I9 327I 33I4 3357 340013443 3486 4o 20|9.8980Io08054 8097|8140o8I83|8227|39 20 9.9I3529 3572 36i5 3658|3701 3744 39 21 8270 8313 8357 8400 8443 8486038 21 3787 3830o 3873 39163959 400oo38 22 85308573861686659 8703874637 22 4044 4087 4I30o4I73 42i6 4259137 23 8789 88321887689I9 89629005136 23 4302 43454388 443i 4474 45z1736 24 9049 90929I135 9178 9222 9265 35 24 4560o4603 4645 4688|473I 4774135 25 9308 935I 9395|9438948I9524|34 25 48I7 486o04903 4946|4989 5032134 26 9568 9611I 9654 9697 974I 9784 33 26 5075 5II8 5I6I 5203 5246 5289 33 27 9827 9870199I4 9957...... 43 32 27 5332 5375 54I8 546i 550 4 5547132 289.9oo00087 I300 173 102o6 0259 0303 3I 288 5590 5633 5675 57I8 5761 58o043I 29 o346 0389 04320476 o5i9 o562 30 29 5847 5890 5933 5976 60o9 6o62o30 30 9.900605 0o648 06920735 0778o082I 29 30o9.9I61o4|6147 6I90o6233 6276 63I9 29 3i o864 og9o8o95I 0994 1O37 Io8I 28 3I 6362 64o5 6448 649I 6533 65762S8 32 I I 241 I I 671 I 210I 2531 297I 340127 321 66I96662 6705 674856791 6834127 331 383 1426 r469 5I 31 556 1599 26 331 6877 69 I96962 7005 70487o091 26 34 I642 I685 I729 I772 i8I5 I858 25 34 71347I77 777220 7262 7305 7348 25 35 Ig90I 944 I988 203I 2074 2II7 24 35 7391 7434 7477 7520 7563 7605 24 36 2I60 22o042247 2290 2333 2376 23 36 764817691 7734 7777 7820 7863 23 37 2420 2463 2506 2549 2592 2635 22 37 7906 7948 7991 8o34 8077 8I20 22 38 2679 2722 2765 28o08285I 2894 2I 38 8I63 8206 8248 829I 8334 8372I 21 39 2938 298I13024 3067 313I 3i53 20 39 8420o8463 8506 8548 859I 8634 20 40o9.903197 3240o3283|3326|33699341219 4099I867 8720876388058848 889II9 41 3456 3499 3542 3585 3628 367I I8 4I 8934 8977 9020 9063 9io5 9148 I8 42 3714 3758 38o0 3844|3887|393017 42 919I1923499277 932093629405I7 43 3973 40oI64060o41o34146 4I89|I6 43 9448|9491 9534 9577 919662 i6 44 4232 4275143 184362 4405 4448 I 5 44 97o059748 979i 9834|98769991 I 5 45 4491 4534 4577 4620o4663 4707 i4 45 9962...5. 48..9I.I33.I76 I4 46 4750o 4793 4836 4879 4922 4965 i3 46 9.920219 0262I0305 o348 o3go o433 i3 47147oo8 o52 So9 5 3008 505 509224 I2 47 0476 05I o562 o604 o 647 0690 12 48 5267 53io 535353397 544o 5483II 48 0733 0776 081 oo8 o86 9o o 947 II 49 5526 5569 5612 5655 5698 574I Io 49 0990 io33Io75 II I8 Ii6I 1204 IO 50 9.905785 5828 587i 59I4 5957 6ooo 9 50o9.92i247 I289 I332 i375 I4I8 I46I 9 5 I 6o436o86 6129 6I72 66662 6259 8 5I.1 I5031I5461I589 I632 I675 I7I7 8 52 6302 6345 6388 643i 6474 65I7 7 52 1760o 8o3 I846 I889 193I 1974 7 53 6560o6603 6646 6690o6733 6776 6 531 2oI720o6o02o3 2I45 288 223I 6 54 68I9 6862 69o5 6948 699I 7o34 5 54 2274 23i6 2359 2402 2445 2488 5 55 7077 7120 7I63 7207 7250 7293 4 55 2530 2573 2616 2659 2702 2744 4 56 7336 7379 7422 7465 7508 755I 3 56 2787 2830o2873 29i5 2958 3ooi 3 57 7594 7637 7680 7723 77776678o9 2 57 3o44 3087 3I29 3172 32I5 3 258 2 58 7853 7896 7939 7982 8025 8o68 I 58 33o003343 3386 342913471 35I4 I 591 8ii 81548I978240o82838326 o 59 355736003642 36 853728 377I -o 60(" 50" 40" 300" 20" 10" I 60' 50"I 40" 30" 20 10 I Co-tangent of 51 Degrees. Co-tangent of 50 Degrees. P. Part 1" 2" 3" 4" 51 6" 7" 8" 91 P. Part 1" 29 3" 41 5" 6" 7' 81 911.ar 4 9 13 17 22 26 30 35 39 4 9 13 17 21 26 30 34 39 64 LOGARITHMl IC SI NES. Sine of 40 Degrees. Sine of 41 Degrees. 0. " 10" 1 20"! 30" 40/ 1 50" 0 10" 20" 3010 40" 50" o 9. 8067 893 8 I I 88 8I43 8I68 8I93159 o 9.8I6943 6967 6991 7016 7040 764~ 59 I 82I 88243 8268i8293 83I818343158 I 708871II2 7137 71I6 7I85 7209 58 8 8368 8393 849 18444 8469 8494 57 2 7233 7258 7282 73o6 7330 7354 57 3 85ig 8544 85698 594.86I9 8644 56 3 7379 74o3 7427 745I 7475 7499 56 |4' 8669 8694 87I9 8744 8769 8794 55 4 7524 7548 7572 7596 7620 7644 55 5 88I9 8844 88698894 89I9 8944 54 5 7668 7693 7717 774I 7765 7789 54 6 8969 8994 90I9 9044 9069 9094 53 6 7813 7837 7862178861791o 7934 53 7 9I I9 9I44'9I699I94 g2I9 9244 52 7 7958 7982 8006 8030 8055 8079 52 8 9269 9294 93 919344 9369 9394] 5 8 8103 8I27 8 5I 8175 8199 8223 51 9 94 199444 946919494 95I9 9544 50 9 824718272 8296 8320o8344 8368 50o I0 9.8o9569 9594 96I9 9643 9668 9693 49 Io 9.818392 84I6 844o08464 8488 8512 49 II I 97I8 9743 976819793 98I8 9843 48 nII 8536 8560 858418609 8633 8657 48 12 9868 9893i998 i9943 9967 9992 47 I2 8681 8705 8729 8753 8777 880I 47 I3 9.8IooI17oo42 oo67 oog2 OI I 7 OI42 46 13 8825s8849 8873 88978g92I 8945 46 I4 oi67 OI9I 02I6 024I o 266029I 145 i4 896918993 90 7 904I 9065 9089 45 5 0316 034 o10366 o039004oI5o 44o 44 I5 9I I3 937 9 I6I9I69 8519209923344 i6 o465 0490goo55 0540 o564 o58 9 143 I 6 9257 928I 93o5 9329 9353 9377 43 17 06I4 o639:o664 o689 07I3 073842 I 7 940I 9425j9449 9473 9497 952 142 I8 0763 0788 08I3 o838 o862 o887 4I I8 9545 9569 9593 96I7 964I 9665 4I I9 0912 ~937,0962 o986 io I IIo36 4o I 9 9689197I3 9737 9761 9785 98og 4o 20 9 8Iio6I o0861IIIO II35 II 60o I85 39 20 9.8I9832 9856 98801990499289 952 39 21 1210 I234 I 259 I284 1309o i334 38 21 9976...24..48..J72..96 38 22 I 358 I383 14o8 I433 I457 I482 37 22 9.820120 oI43 OI67 0191 0215 0239 37 23 I507 i5321I556 I58I I 6o6 I63i 36 23 0263 0287 o03I I o335 10359 o382 36 24 i 655 I 680 I705 1730 175A I779 35' 24 o4o6 t 43o o454 0478 o502 o526 35 25 I804 I828'i853 I878 I9O3 I927 34 25 oF51 o573 1597 0621 I645 o669 34 6 I 952 1977'200I 2026 205I 2076 33 26 o693 0I 7 0740 0764 0788 o8 I2 33 27 2I00 2I2512 I 50 2174 2199 2224 32 27 o836 o860o883 o09071093I 0955 32 ~s/ 2248 227312298 2322 2347 2372 3I 28 0979 I003 I026 0o5o I074 1o98 3I 29 2J96 242I2446 2470 2495 2520 30 29 II22|I I46 1169 Ii93 1217 I24i 3o 3o.8sI2544 25692594 26I8 26432668 29 30o9. 821 265| 288 312 x336 i36o I 38429 31 2692 27I712742 2766 279I 28I5 28 3I I407 43i I455 I479 I 502 526 28 3J 2840 2865 2889 29I4 2939 2963 27 32 I550 574 I 598 I 62 I 645 1669 27 31 2988.130123037 3062 3086 3I II26 33 I693I7I61 7401 7641I788 18I I26 34 3I35 3i6o03i85 3209 3234 3258 25 34 i835 18591 8831I 906 930 I 954 25 35 3283 3307:3332 3357 338I 34o6 24 35 1977 200I 2025 2049 2072 2o96 24 361 343o 3455.3479 35o4 3529 3553123 36 2I20 2I44 2I67 2191 22I5 2238 23 37' 3578 3602 3627 365i 3676 3700 22 37 2252 2286 2309 2333 23571238I 22 38 3725 3749379 74 3799 38233838482 I 38 2404 2428 2452 2475 2499 2523 21 39 3872 3897 392I 3946 397o 3995 20 39 2546 2570 2594 26I7 264I 2665 20 40o9.8I4019 4044!4068 4093 4I 7 41421I9 4o 9.822688 27I122736 2759 2783[2807 I9 4 4I66 4I9I42I5 42404264 4289I8 i8 4 2830|2854|2878 2901 2925 2948 I8 |42 43I3 43384362 4387 44 14436| 7 42 297272996 30I9 3043|3067309017 43 446o 448445o09 4533 4558 4582 i6 43 3I I4 3137 3i6i 3 I85 3208 3232 i6 |441 4607 463I 4656 4680 470414729 IS | 44 3255 3279 3303 3326 335o 3373 i5 45 4753 4778'4802 4827 485I 4876 I4 45 33971342 1 3444 3468 3491 35i5 I4 46 4900 4924!4949 4973 4998 5022 13 46 3539 3562 3586 3609 3633 3656 I3 47 5046 507I15095 5I20 5I44 5i68 1z2 47 3680o 370413727 375I 3774 3798 2 48 5193 5217 5242 5266 529o 53I51'I 48 382I 3845 3868 3892 39I513939 II 49 5339 536415388 5412 54371546 IIo 49 3963 398640o1 o403340o5740o80 10 50,9.8I5485 55Io 5534 5558 5583i56071 9 5019.824Io414I4274I51 4I7414198422II 9 ] 511 5632 565656568o057o5 5729 5753 8 5I 424514268 4292 43I5 433914362 8 ] 52j 5778 5802 5826 585i 587515899 7 52 4386 4409 4433 4456 448o450o3 7 t 53i 5924 5948 5972 5996 602I 6o45 6 53 452714550|4574 4597|462I14644 6 54 6069 6094 6ii8 6142 6 16i9il 5 54 46681469I147I5 47381476I|4785 5 55 6251 624o 6264 6288 63I62 63371 4 55 480814832 4855 48794902 4926 4 56 636I 6385 640966434 6458 6482 3 56 4949 4972 4996 50i9 5o43 5o66 3 57! 6507653I 6555657916604o6628 2 57 5o090o5I35I36 5I6o05I835207J 2 58 6652 6676 6701 67725 67496773J I 58 5230o5254 5277 53oo 5324 5347 I 59 6798 6822 6846 68 70 68946919I9 o |59 537 53941 54,7 544i15464 5488 Co-sine of 49 Degrees. Ci Co-sine of 48 Degrees. P. Part t 1" 2" 3" 4/" 5/" 61" 7"' 8 98" p. Partl 1" 2" 3/' 4/" 5" 6" 7" 8"/ 9"| 2 5 7 10 12 15 17 20 22 r1 t 2 5 7 10 12 14 17 19 21 LO G A R I T H M I C TANGENTS. 6 Tangent of 40 Degrees. 9 Tangent of 41 Degrees. l 10"lot 20" 30/' 40/" -501" f0" -10" 201" 30" 40" _o" ol9.923814 3856 3899 3942 3985 4027 59 0o9.939I63 9206 9248 929I 9333I9376 59 4070 4I3 14i56 4I98 424I 4284 58 I 94I8 946I 9503 9546 9588 963I 58 2 4327 4369 44I2 4455 4498 454o 57 2 9673 9761 9758 980I 9843 9886 57 3 4583 4626 4669 471II4754 4797 56 3 9928 9971.I3..56..98.81I4 56 4 4840148821492514968 50II 5o53 55 4 9.940I83 0226 0268 o3II o354 o396 55 5 596 39 5I8I1 5224 5267153Io54 5 0439 048I1 o524 o566 609 o65I 54 6 5352 5395 5438 548I 5523 5566 53 6 o694 0o736 0779 82I o864 ogo6 53 7 5609 5652 5694 5737 578o 5822 52 7 og949 099I io34 I076 III9 i6i 52 8 5865 5908 595I 5993 6o36 6079 5I 8 I204 I246 I289 iI I374 I4I6 5i C 6I22 6I64 6207 6250 6292 6335 5o 9 I459 i5oi i544 i586 I628 1671 50 I0o9.926378!'"' 6463 6506 6549 659I 49 10o9.9471I3 I756 I798 i84I i883 1926 49 II 6634 6677 6720 6762 68o5 6848 48 II I968 20II 2053 2096 2I38 2I81 48 I'2 6890 6933 6976 70I9 706I1 704 47 12 2223 2266 2308235I 2393 2436 47 13 7147 7189 7232 7275 73I7 7360 46 I3 2478 2521 2563 2606 2648 2691 46 I4 7403 7446 7488 7531 7574 761645 4 14 2733 2776 2818 286I 2903 2945 45 i5 7659 7702 7744 7787 7830 7872 44 i5 2988 3o3o 30730 3i 3 I58 3200 44 I6 79I5 7958 8001 8043 8086 8129 43 I6 3243 3285 3328 3370 343 3455 43 I7 87I 82I4 8257 8299 8342 8385 42 17 3498 354o 3583 3625 3667 3710 42 i8 842718470 853 855585558598 864 4 18 3752 3795 3837 3880 3922 3965 4I 19 868418726 8769 882 88548 89714o I9 400714050 4092 4I35 4I77 4219 4 20o9.928940 8982 902519068 9II09 g53 39 20 9.944262 4304 4347 4389 4432 4474 39 2I 9r96 9238 928I 9324 9366 9409 38 21 4517 4559 4602 4644 4686 4729 38 22 9452 9494 9537 9580o9622 9665 37 22 477I 48I4 4856 4899 494I 4984 37 23 97081 975o979319836 9878 992I 36 23 5026 5069 5i1i 5I53 5I96 5238 36 24 9964...6..49.. 92. 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R Co-tangent of 48 Degrees. Co-tangent of 48 egees.__ t l 2" 3'1 4" 5"' 6" 7" 8" 9 P. P 1 2 31 41 511 611 71, 8" 9.ar 4 9 13 17 21 26 30 34 38. art4 8 13 17 21 25 30 34 38 36 6LOGARITHMIC SINES. Sine of 42 Degrees. Sine of 43 Degrees. 0" 1T 20" " 30" 40" 50" ( 0" 10" 20" 30" 40 50" O 9.8255II 5534 5558 558I15604 5628 59 o 9.833783 3806 3828 385I 3874 3896 59 I 565i 5675 5698 572I 5745 5768 58 I 39I9 394 3964 3986 4oo009 432 58 2 579I 58I5 5838 586i 5885 5908157 2 4054 4077 4099 4122 4I44 4I67 57 3 593I 5955 5978 6oo00I 6025 6o48 56 3 4I89 42I2 4234 4257 4280 4302 56 4 6071 609516II8 6I4I 6i65 66 88 55 4 4325143474370 47392 4415 4437155 5 62 I I 6235 6258 628 I 6305 6328 54 5 4460 4482 4505 4527 4550 4572 54 6 635 I6375 63981642I 6444 6468 53 6 4595 4617 464o 4662 4685 4707 53 7 649 I 65I4 6538 66 6584 6607 52 7 4730 4752 4775 4797 4820 4842 52 8 663i 6654 6677 670I 6724 6747 5I 8 4865 4887 4914I932 4954 4977 5 r 9 6770 67946817 68 o 6863 68 87 0 9 49995022685o44 5067 5089 5 I I 2 5 10 9.8269IO 6933 6956 6980 7003 7026 49 10 9.835I34 5I5751I 79 5201 5224 5246 49 II 7049 7073 7096 7119 7I42 7165 48 I I 5269 529I 53I4 5336 5358 538I 48 12 7I89 72I2 7235 7258 7282 7305 47 I2 54o3 542,6 5448 547I 5493 55I5 47 13 7328 735I 7374 7398 742I 7444 46 13 5538 55601 5583 5605 5627 56501 46 i4 7467 7490 75I4 7537 7560 7583 45 14 5672 5695 57I7 5739 5762 5784 45 x5 7606 7629 7653 7676 7699 7722 44 i 5 5807 5829 585I 5874 5896 59I8 44 I6 7745 7768 7792 78I5 7838 786I 43 I 6 594I 5963 5986 6oo008 6030o 6053 43 17 7884 7907 793I 7954 7977 8000 42 I 7 6075 6097 6I20 6142 6164618 7 42 i8 802380468069 8093 811 I 6 8 39 41 I 8 62091623I 6254 6276 6298 632 I14I 19 8162 8I85 8208 8231 8254 8278 40 19 6343 6365 6388 64IO 6432 6455 40 20 9.82830O 8324 8347 8370 8393 84i6 39 20 9.836477 649916522 6544 6566 6589 39 21 8439 8462 8485 8509 8532 8555 38 21 66Ii 6633 6656 6678 6700 6722 38 22 8578 86oi 8624 8647 8670 8693 37 22 6745 6767 6789 68I2 6834 6856 37 23 87I6 8739 8762 8786 8809 8832 36 23 6878 690OI 6923 6945 6968 6990 36 24 8855 8878 890I 8924 8947 8970 35 24 70I2 7034 7057 7079 7I01 7I23 35 25 8993 90I6 9039 9062 9085 g108 34 25 746 768 790 721I2 7235 7257 34 26 9I3I g954 9I77 9200 9223 9246 33 26 7279 730o 7324 7346 7368 7390 33 27 926c9 9292 93I5 9338 936I 9384 32 27 74I2 7435 7457 7479 750I 7524 32 28 9407 9430 9453 9476 9499 9522 3I 28 7546 7568 7590 76I2 7635 7657 31 29 9545 9s568 959I 9614 9637 9660 30 29 7679 770I 7723 7746 7768 7790 30 309. 829683 9706 9729 9752 9775 9798 29 30 9.8378I 2 7834 7857 7879 7901I 7923|29 31 982I 9844 9867 9890 99I 39936 28 31 7945 7967 799080I2 8034 856 28 32 9959 9982....5..28..5i.74 27 32 8078 8Ioo 81I23 8145 8167 8189127 33 9.8300,97 OI20 oI42oI 65 O1I 88 02 1 26 33 821 18233 8256 8278 8300 8322 26 34 o234 o257 o280o0303 o3.26 0349 25 34 8344 8366 8388 84Io 8433 8455 25 35 o372 0395 0417 o44o o463 o486 24 35 8477 8499 852I 8543 8565 8587 24 36 0509 o532 o05551 0578 o6oi 0624 23 36 86io 8632 8654 8676 8698 8720 23 [ 37 0o646 066910692 o 7 o7380o76I22 37 8742 8764 8786 88o8 883i 8853 22 38 o0784o807 o829 o852 0875 0898,1 I 38 887518897 899g 8941 8963 8985|2I 39 o092 o944o09671o989 I0121 o 3520o 39 9007902990oo5I 907390o95 1II8120 40o9.83IO58 io8II Io04 I271 I149 I729.40o 9.839I40o g62 9I84 9206 9228 925o 19 41 II951I218 I24I 1263 1 286I 3o091~8 4I 9272 929493I61 9338 936o09382 I8 42 I332 I355 I378I 400oo 423 446 17. 42 940~49426 9448 9470 9492 9514I I 43 I469 1492 I 5I4 I537 i560o 583 i6 43 9536 9558 958019602 962419646 IE 44 I606 1628 I65I| I674 1697 I720oI5 44 9668 9690 97I12 9734 9756 9778 I 5 451 742 1765 I788 I8I I:I833 I856 i4 45 9800[9822 9844 986696888 99IOI4 46 I879 I90o2 924 i947 I970 I993 I3 46 9932 9954 9976 9998..20..421I 3 471 2o015 2038206 2084 2I06 2I291I2 47 9.840064 oo86 oIo8 oI30 1OI52 OI74I12 48 2I52 2I75 2I97 2220 2243 2266I 1 48 OI96 02102 240 0o262 0284 0306I 49 2288 23II 2334 2356 2379 2402|1O 49 0328 0350o0372 0393 o4I5 0437 10 50 9.832425 2447 2470 2493 255 2538 9 50o 9.840459 o48I 0503 o525 0547 o569g 9 51 256I 2584 2606 2629 2652 2674 8 5i 05gI o6I3 o635 o657 0678 0700 8 52 2697 2720 2742 2765 2788 28IO 7 52 0722 0744 0766 0788 08IO 0832 7 53 2833 2856 2878 290I 2924 29461 6 53 o854 1o876 o897 09I9 094I o963 6 54 2969 2992 30I4 3037 3o6o 3082 5 54 O985 I007 1029 io5i I072 I o094 5 55 3IO5 3I283150|3I733I196 32I8 4 55 III6 II381 I6 0 II82 I204 I226 4 56 324I 3263 3286 3309 333i 3354 3 56 I247 I269 I29I I3I3 I335 I357 3 57 3377 3399 3422 3444 3467 3490 2 57 I378 i4oo 1422 I444 i466 I488 | 58 35I2 3535 3557 3580o 3603 3625 I 58 I509 I53I 1553 1575 I597 1I69 I 59 3643670 369 37 6 3738 37 59 64 1662 I684 706 I 728 749 O i 6 I 6 50" 401" 30 20" 1 6 50 40" 30" 20" 10" Co-sine of 47 Degrees. 1 Co-sine of 46 Degrees. P.*art2 1" 2"1 3" 4' 5" 61 71 8/" 9" P. Pa 1" 21" 3" 4/1 5" 6/ 7" 8"1 9"1 2 r 57 9 11 14 16 IS 21 2 4 7 9 7 1 13 16 18 20 LOGARITH M I C TA N G E N T S. 67 ~ —~ Tangent of 42 Degrees. Tangent of 43 Degrees. 0"f 10" 20"t 30" 40" 50" 0" 10" 20" 30"' 40" 50" o 9.954437 4480 4522 4564 4607 4649 59 O 9.969656 9698 974o 9783 9825 9867 59 I 469i 4734 4776 4819 486i 4903 58 I 9909 995 I 9994..36..78.120 58 3 4946 4988 5030 5073 5II5 5157 57 2 9 970oI62 205 0247 0289 o33 1373 57 3 52oo 5242 5284 5327 5369 54i I 56 3 o4I6 o458 o5oo 0542 o584 o627 56 4. 5454 5496 5538 558I 5623 5665 55 4 o669 07I I 0753 0796 o838 880 55 5 5708 5750 5792 5835 5877 59I9 54 5 0922 og964 Ioo 007 I 49 09I I33 54 6 596 600oo46046 6088 6i3i 6I73 53 6 I I75 I2I81 I260 I302 I344 I386153 7 62I15 6258 63 342 6385 42 6385427 64 2 7 1429 147 I53 I 555 I597 I640o 52 8 6469 6512 6554 6596 6639 66839 66 8 682 1724 1766 I 8o8 I85I 1893 51 9 6723 6766 68o8 685o 6893 6935 So i93 1977 2019 2062 2I04 2I46 50 IO 9.956977 7020 7062 7Io047146 7189 49 IO 9.972188 2230 2273 2315 2357 2399 49 11 7231 7273 73I6 7358 7400 7443 48 II 244I 2484 2526 2568 26IO 2652 48 12 7485 7527 757o 76I2 7654 7697 47 12 2695 2737 2779 2821 2863 29o5 47 13 7739 7781 7823 7866 7908 795o046 I3 2948 2990 3032 3074 31I6 3159 46 14 7993 8o35 8077 8120 8162 8204 45 14 3201 3243 3285 3327 3370 3412 45 I5 8247 8289 833I 8373 84I6 8458 44 i5 3454 3496 3538 358i 3623 3665 44 i6 8500 8543 8585 8627 8670 8712 43 I6 3707 3749 379I 3834 3876 39I8 43 17 8754 8796 8839 888I 8923 8966 42 I7 3960 4002 4o45 4087 4129 4I7I 42 i8 9008 905090939135 9I77 9219 4I I8 4'213 42554298 434434382 44244 19 9262 9304 9346 9389 943 1 9473 40 I9 4466 4509 455I 4593 4635 4677 40 20o 9.959516 9558 9600 9642 9685 9727 39 20 9.974720 4762 480 4 4846 4888 493o 39 21 9769198I29854 9896 9938 998I 38 21 4973 50o5 50575099 5I4I 5i83 38 22 9.960023 oo65 oio8 0 1500192 o0234 37 22 522 5268 53 I o 5352 5394 5437 37 23 0277 o319 o36 o4o0 4 o446 o488 36 23 5479 5521 5563 56o5 5647 5690 36 24 053 o0573 o6i 5o657 070o 0742 35 24 5732 5774 58I6 5858 590I 5943 35 25 0784 o0826 o869 o91I 1095310996 34 25 5985 6027 6069 6III 6I54 6196 34 26 Io38 Io80 I122 ii65 I2071 249 33 26 6238 6280 6322 6364 6407 6449 33 27 I12921334 1376 I4I8 146i.I 503 32 27 649I 653316575 6617 6660 6702 32 28 I 545 5871I630oI672 I714 I757 31 28 6744 6786 6828 6870o693 6955 31 29 I1799 I84I 883 I926 1968 201o 30 29 6997 7039 708 7123 7I 66 7208 30 3o 9.962052 2095 2 37 2I79 2222 2264 29 3o 9.977250 7292 7334 7377 7419 7461 29 31 2306 2348 239I 2433 2475 251 7 28 31 7503 7545 7587 7630 7672 77I4 28 32 2560 2602 2644 2686 2729 2771 27 32 7756 7-798 7840 7882 7925 7967 27 33 2813 2856 2898 2940 2982 3025 26 33 8009 8o05I 893 8I35 8I78 8220 26 34 3067 3o09 3i5i 3194 3236 3278 25 34 8262 8304 8346 8388 843i 8473 25 35 3320 3363 3405 3447 3489 3532 24 35 85I5 8557 8599 864i 8684 8726 24 36 3574 36i6 3659 370I 3743 3785 23 36 8768 88io 8852 8894 8937 8979 23 37 3828 3870 3912 3954 3997 4o39 22 37 902I 9063 9I05 9I47 9I90 9232 22 38 4o84I412341 66 4208 4250 4292 2i 38 9274 9316 9358 9400 9442 9485 21 39 4335 4377 4419 446I 4504 4546 20 39 9527 9569 96I 9653 9695 9738 20 40 9.964588 4630 4673.47P5 4757 4799 19 40 9.979780 9822 9864 9906 9948 99919 I 41 4842 4884 4926 4968 5 Io I 5053 I 8 4 9 9.800330oo 7501 I7 oI 59 020I 0243 18 42 5095 5I37 5I8o 5222 5264 5306 I7 42 0286 0328 0370 0412 o454 04961I7 43 5349 539I 5433 5475 55I8 556o i6 43 o538 o58 0o623 o665 0707 0749 16 44 5602 5644 5687 5729 577I 58i3 5 44 0791 o834 o876 o091 8 0960 I00215 45 5855 5898 5940 5982 6024 6067 I4 45 1044 I086 1129 I I7I 12131 255 4 46 6o09 6I5I 6193 6236 6278 6320 13 46 129713391 382 I424 4661 508 13 47 6362 64o5 6447 6489 653,574 I2 47 I550o 1592 i634 I677 I719 176I 12 48 66I6 6658 670016742 678516827 Ii 48 I8o3 I845 1887 I9291I972 2014 11 49 6869 69 I 16954 6996 7038 080 I 49 2056 20o982 140 2182 2224 2267 I 50 9.967123 7I65 7207 7249 7291 73341 9 50 9.982309 2351 2393 2435 2477 2519 9 5I 7376 7418 7460 7603 7545 7587 8 74 2562 2604 2646 2688 2730 2772 8 52 7629 7672 7714 7756 7798 7840 7 52 2814 2857 2899 294I 2983 3025 7 53 7883 7925 7967 80og 80o528094 6 53 3067|3IO193I52 3194 3236 3278 6 54 8I36 8178 822o 8263 83o5 83471 5 54 3320 3362 34o4 3447 3489 353I 5 55 8389 8432 8474 85I6 85588600 4 55 3573 36I5 367 3 699 3742 3784 4 56 8643 8685 8727 8769 8812.8854 3 56 3826 3868 39Io 3939 94 4o37 3 57 8896 8938 8980o902390o659Io 7 2 57 4079 4I2I14I63 4205 4247 4289 2 58 9149 91929234 92769318 93601 I 58 433214374 44I6 4458 4500 4542 I 59 9403 9445 9487 952995719 9614 oi 59 45844627 4669 47II 4753,4795 o 6(0" 50" 40" 3If' I0 10" 11 60" 50" 40" 30" 20" 10' Co-tangent of 47 Degrees. Co-tangent of 46 Degrees. p t1" 2"1 3" 41' 5"1 6" 7/' 8" 911 tI/" 2" 3/ 41 5" 6" r71 8" 9" 48 13 17 21 25 303438 P t4 8 13 17 21 2530 3438 68 LOGARITHMIC SIN ES. Sine of 44 Degrees. Sine of 45 Degrees. r: 0"o;l 10" 20"1 30" 40/" 50" 0o" 10"- 2011 30' 1 4-01 5011 09.84I77 1793 1815 I8371 858 880 59 o 9. 849485 95oQ69527 9548 9569 9590o 59 I 1921I9241946I 9671I989 20I1158 I 9611 9632 9653 96749695 97I6 58 2 2033 2055 2076 2098 2120 2142 57 2 9738 9759 9780 9801 9822 9843 57 3 2163 2185 2207 2229 2250 2272 56 3 9864 9885 9906 9927 9948 9969 56 4 2294 23I6 2337 2359 2381I2403 55 4 9990..11..32..53..74..95 55 5 2424 24462468 2490 25I1 2533 54 5 9.850 116 o137 oI 58 OI 79 200 0221 i54 6 2555 2772598 2620 2642 26,63 53 6 0242 0263 0284 o30 5 0326 o347 53 7 2685 2707 2729 2750 2772 2794 52 7 o368 o388 o409 o43o o45I o472 52 8 2815 2837 2859 2880 2902 2924 5I 8 0493 o514 0535 o556 o0577 0598 5 I 9 2946 2967 2989 3o I 3032 30 54 50 9 0619 o640 o66i o682'73 0724 50 o 19.843076 309731 I9 3I4I131623184 49 10o9. 850745o0766 0787 0807 o828 o849 49 I I 3206 322732493 27 3292 3292 3314 48 I I o8700891 0 91209330 954 0975 48 12 3336 3357 3379 340I 3422 3444 47 1 2 0996 IoI07 o38 io58 Io79 IIo0047 13 3466 3487 35o91353o 3552 3574 46 13 II2I 1142 116311841205 I226 46 I4 3595 3617 3639 3660 3682 370345 4 1246 I267 1288 I 3309 I 33 3o 35145 15 3725 3747 3768 3790 381I 3833 44 i5 I372 I393 I4i3 I434 455 1476 44 I6 3855 3876 3898 3919 3941 3963 43 i6 I497 I518 I539 1559 I 58o0160I43 17 3984 4006 4027 4049 407I 409242 1 7 I622 I 643 i664 I 685 1 7o5 I726 42 I8 4114414354157 4178 4200 422241 i8 1747 I768 I 789 I8I OI8301I851 4I 19 4243 4265 4286 4308 43294351 4o I9 I872 1893 I94 I935 1955 I 976 40 20 9.844372 4394 44I6 4437 4459 448o039 20 9.85I997 20I8 20392059 20802IO39 21 4502 4523 4545 456654588 4609 38 21 2122 2143 2163 2184.205 o52226 38 22 463i 4652 4674 4696 47I7 4739 37 22 2247 2267 2288 2309 2330 2350 37 23 4760 4782 4803 4825 4846 4868 36 23 237I 2392 24I 3434 2454 2475 36 24 4889 49II 4932 4954 4975 4997 35 24 2496 2517 2537 2558 1.579 2600 35 25 5oi8 5040 506I 5083 5io4 5I26 34 25 2620 2641 2662 2683 2703 2724 34 26 5147 5I68 5190 52II 5233 5254133 26 274512766 2786 2807 2828 2849 33 27 5276 5297 5319 534o 5362 5383 32 27 2869 2890 29 II 293I 2952 2973 32 28 5405 5426 5447 5469 549o 55I2 31 28 2994 3oi4 3035 3056 3076 3097 3i 29 5533 5555 5576 5598 56I9 564o 30 29 3ii8 3139 3559 3I8o 320I 3221 30 30 9.845662 5683 5705 5726 5747'5769 29 30 9.853242 3263 3283 3304 3325 3345 29 31 5790 5812 5833 5855 5876 5897 28 3i 3366 3387 3408 3428 3449 3470 28 32 59I9 5940 5962 5983 6004 6026 27 32 349 35I I 3532 3552 35733594 27 33 6047 6069 6090 611 61336154 26 33 36I4 3635 3655 3676 3697 37I7 26 34 6175 6I97 62I8 6240 6261 6282 25 34 3738 3759 3779 38oo 382I 3841 25 35 63o4 6325 6346 6368 6389 64I0 24 35 3862 3883 3903 3924 3944 3965 24 36 6432 64536474 6496 657657 6539 23 36 3986 4006 4027 4047 4068 4089 23!37 656o 658i166o3 6624 6645 6667 22 37 41o9 4I3o 415i 4171 4I92 42I1 22 38 6688 6709 6731 6752 6773 6794 2I 38 4233 4254 4274 4295 43i5 433612I 39 68i6 6837 6858 6880 690I 6922 20 39 4356 4377 4398 44I8 4439 4459120 4o 9.846944 6965 6986 7008 7029 7o50 I9 4o 9.854480 45o00 4521 4542 4562 4583 19 4I 7071 7o9317II4 7I35 7I577178 8I8 4 46034624 4644 4665 4686 4706 I8 42 7199 7220o7242 7263 7284 73o5 17 42 4727 4747 4768 4788 4809 4829 17 43 7327 7348 7369 7391 7412 7433 i6 43 4850o4870 4891 4911 4932 4953 i6 44 7454 747617497 75I8 7539 7561 I5 44 4973 4994 5oi4 5035 5055 5076 i5 45 7582 760317624 7645 7667 7688 4 45 5096 5iI7 5137 5I58 5178 5199 I4 46 7709 7730772 77752 773 7794 785 3 46 5219 5240 5260 5281 53oi 5322 I3 47 7836 7858 7879 7900 792I 7943 12 47 5342 5363 5383 5404 5424 5445 12 48 7964 79851800oo6 802780498070 11II 48 5465 5485 5506 5526'5547 55671 I 49 8091 8I I2{8I33 8I5481768197 10 49 5588 5608 5629 5649 5670 56901 50 9.848218 8239 8260 8282 83o3 8324 9 50o9.855711 5731 5751 5772 579 2 58I3 9 5i 8345 8366 8387 8409 84301845I 8 5I 5833 5854 5874 5895 59I5 5935 8 51 8472 8493 85I4 8535 855718578 7 52 5956 5976 5997 60I 76038 6058 7 53 8599 8620|864I 8662 8683 8705 6 53 6078 6o99gg 6II9i4o06I6o 618o 6 54 8726 8747 8768 8789 88io 883I 5 54 620I 6221 6242 6262 6282 63o3 5 55 8852 8874 8895889I6 89378958 4 55 6323 6344 6364 6384 6405 6425 4 56 8979 9000 902I 9042 9063 9085 3 56 6446 6466 6486 6507 6527 6547 3 57 9io6 91279I148 9I699I9o092I I 2 57 6568 6588866o9 6629 6649 6670 2 158 9232 9253 92749295 9316;9338 Ir 58 6690 671o 673I 675 t 6771 6792 I 59 9359938o094o19422944319464 o 59 68 216832,6853 6873 6893 6914 o 50" 40" 30" 20"1 10" 60" 5 0" 40"! 30" 1 20" 1 10" Co-sine of 45 Degrees. Co-sine of 44 Degrees.' P. Part 1 2 3" 41 5" 6// 7/ 8/ 9 P. Part 1" 2" 3" 4" 5" 6" 7 18" 9 2 4 6 9 11 13 15 17 19 2 4 6 8 10 12 14 17 19 LOGARITHMIC TANGENTS. 69 Tangent of 44 Degrees. Tangent of 46 Degrees. o 0" 10" 20"1 30"1 40" 50"l t O 10" 20ft 30 40" -50" o 9.984837 4879 492I 4964 5006 5048 59 o Io.o oo42 oooo 0o84 o126 I68 02I I 59 I 5090 5132 5I74 52I6 5259 5301 58 I 0253 o295 o337 o379 0421 o463 58 2 5343 5385 5427 5469 5511 5553 57 2 o505 0547 o0590 0o632 674 076 57 3 5596 5638 5680o 5722 5764 580 6 56 3 0758 o800 0842 o884 0926 o969 56 4 5848 5891 5933 5975 60O7 6o59 55 4 ioI io53 ro95 11 37 II79 1221 55 5 6ioi 6I43 6I85 6228 627o 6312 54 5 1263 i3o05 348 1390 1432 i474 54 6 6354 6396 643816480 6523 6565 53 6 I5I61I558 1600oo 64216841 727 53 7 6607 6649 6691 6733 6775 6817 52 7 I769 i8i i853 1895 1937 1I979 52 8 686o 6902 6944 6986 7028 70701 5 8 202I 2063 2Io6 2148 2190 2232 5I 9 7II22 7154 7I97 7239 728I 7323150 9 2274 231 2358 2400 442485 50 Io 9.987365 7407 7449 749I' 7534 757649.10 10.002527 2569 2611 2653 2695 2737 49 I 7618 7660 7702 774417786 7829 48 I 2779 2821 2864 29o6 2948 2990 48 I2 7871 7913 7955 7997 8039 8o8i 47 I2 3032 3074 3II 6 3 58 3200 3243 47 I3 8123 8166 82o8 825o 8292 833446 I3 3285 3327 3369 34I I 3453 3495 46 I4 8376 8418 8460 8503 8545 8587145 4 3537 3579 3622 3664 3706 3748 45 15 8629 867I 87I3 8755 8797 884o044 15 3790 3832 3874 3916 3958 400I144 i6 8882 8924 8966 9008 9050 9092 43 16 4043 4085 4I2741I69 421 I 4253 43 I7 9I34 9177 92I9 9261 9303 9345 42 I7 4295 4337 438o 4422 4464 450 6 42 18 9387 9429 9471 95I3 9556 9598 41 i8 4548 459o 4632 4674 47I164759 41 19 9640 9682 9724 9766 9808 9850 4o I 48oi 4843 4885 4927 4969 5I I 4o 209.989893 9935 9977....6i. o0339 20 oIo.oo5o53 5o95 5I38518o05222 5264 39 219.990o45 OI87 0230 0272 o3I4 o356 38 21I 5306 5348 5390 5432 5474 55I7 38 22 0398 o440 o482 o524 0567 0609 37 22 5559 56oi 5643 5685 5727 5769 37 23 o65I o693 0735 0777 08I9 o86i 36'23 5811 5854 5896 5938 5980 6022 36 24 o0903 0946 988 o1030 I072 II4 35 24 6064 6Io6 6148 6Ig196233 6275 35 25 I156 I98 I2401I 283 I325 1367 34 25 6317 6359 640I 6443 6485 6527 34 26 1409 I45i I493 i535 1577 I620 33 26 6569 66I2 6654 6696 6738 6780 33 27:662 1704 1746 1788 i83o I872 32 27 6822 6864 6906 6949 6991 7o33 32 28 I9I4 1956 I999 2o04I 2083 225 31 28 7075 7II7 7159 7201 7243 7285 3i 29 2I67 2209 2251 2293 2336 2378 30 29 7328 7370 7412 7454 7496 7538 30 3o 9,992420 2462 2504 2546 2588 2630 29 3o Io.007580 7622 7664 7707 7749 779I 29 31 2672 275 2757 2799 284I 2883 28 3I 7833 7875 79I7 7959 8ooi 8o44 28 32 2925 2967 3009 3o5i 3094 336 27 32 8o86 8I28 8I70 882I2 8254 8296 27 33 3178 3220 3262 3304 3346 3388 26 33 8338 8380 8423 8465 8507 8549 26 34 343i 3473 35i5 3557 3599 364I 25 34 859I 8633 8675 87I7 8760o8802 i 5 35 3683 3725 3767 38io 3852 3894 24 35 8844 8886 8928 8970 9012 9054 24 36 3936 3978 4020 4062 41o4 4146 23 36 9097 9139 918I 9223 9265 9307 23 37 4I89 423I14273 43I 54357 4399122 37 9349 9391 9433 9476 9518 9560 22 38 444I 4483 4526 4568 46I 0 4652 21 38 9602 9644 9686 9728 977019813 21 39 4694 4736 4778 4820 4862 4905 20 39 9855 9897 9939 998I..231..65 20 4o 9. 994947 4989 5o3I 15073 5 15 5I557 I9 4 1I.o OIO07 oI 50o0192 o0234 0276 o318 I9 4i 5199 524I 5284 5326 5368 54io i8 41 o360 0402 o444 0487 0529 o571 i8 42 5452 5494 5536 5578 5620 5663 I7 42 o6i:3 o655 o697 0739 078I 0823 I7 43 5705 5747 5789 583i 5873 5915 I6 43 o866 og090o8 o95 0992 I034 1076 I6 44 5957 5999 6042 6084 6I26 6I68 15 44 III8 1160 I203 I245 1287 I1329 I5 45 620 6252 6294 6336 6378 6424 45 378 6421 443 4I455 I497 i54o 1582 I4 46 6463 65o5 6547 6589 663i 6673 i3 46 1624 I666 1708 17501 792 i834 3 47 67I516757680oo6842 6884 6926 I2 47 18771 I9191961 2003 204520o87 12 48 6968701o 7o52 7~94 7I36 7I79 1I 48 2129 2I71 2214 2256 2298 23401 I 49 7221 7263 73o05 7347 7389 743 io 49 2382 2424 2466 2509 255I 2593 IG 5 9 *9997473 75I5 75558 76007642 7684 9 50O.OI2635 2677 2719 276I 28o3 2846 9 51 7726 7768 781o 7852 7894 7937 8 5I 2888 2930 2972 3o0I4 3o563098 8 52 7979 80s2 8o63 8io5 8147 8189 7 52 314o 3I83 3225 3267 33o913351 7 53 8231 8273 83i6 8358 84oo 8442 6 53 3393 3435 3477 3520 356|2 3604 6 54 8484 8526 8568 86io 8652 8695 5 54 3646 3688 3730 3772 38I5 3857 5 55 8737 8779 882I 8863 8905 8947 4 55 3899 394I 3983 4o25 4o67 4109 4 56 89899o3I[9o0749116 9I58 9200 3 556 4I52 4I94 4236 4278 432014362 3 57 9242 9284 9326 9368 941o 9453 2 57 4404 4447 4489 453I 4573|46i5 2 58 9495 9537 9579 9621 9663 9705 I 58 4657 4699 474' 4784 4826 4868 x 59 9747 9789,983219874 99196 9958 o 59 49o14952 4994 503650795I21 60" i 50'' 1' 40"' 30". - 20" -— ",' 50"30 60 " 40" 30" 20" 1" I. Co-tangent of 45 Degrees. - Co-tangent of 44 Degrees. p 1" 2" 3" 4" 5"' 6" 7" 8" 9"' P. P at 1" 2" 3" 4" 5" 6" 7" 8" 91 4 8 13 17 21 25 29 34 38 ar4 8 13 17 21 25 29 34 38 10 LOGARITHMIC S I[NE. Sine of 46 Degrees. Sine of 47 Degrees. 0'' 10" 20' -30" 401 50" - O" 10" 20" 30" 40" 9. 856934 6954 6975 6995 7015 7036 59 o 9.864I27 4147 4I67 4I86 4206 4226 59 I 7056 7076'7097 7I71737 7158 58 I 4245 4265 4284 4304 4324= 4343 58 2 7178 71981729729397259 7279 57 2 4363 4383 4402 4422 444I 446I 57 3 7300 7320 734/0736I 738I 740I15611 3 448I 4500o4520 4539 455914579 56 4 7422 7442 7462 7482 7503 7523 55 4 4598 46I8 4637 4657 4676 4696 55 5 7543 7563 7584 7604 7624 7645 54 5 47I6 4735 4755 4774 4794 48I3 54 6 7665 7685 7705 7726 7746 7766 53 6 48331485314872 4892 49I I493I 53 7 77861780717827 7847 786717888152 7 495o 4970 499o 500oo9 5029 5048 52 8 7908 792817948 7968 7989 8009 51 8 5068 5087 50715 5 I26 5I46 5165 5I 9 8029 8o491807o08o9o 8 I I o 8 3o 5o 9 5i85 5204 5224 5244 526315283 50 IO 9.858I5I 81I71 8gI1 82II18823 8252 49 I 9.865302 5322 534I 536i 538o 54oo 49 II 8272 8292 8312 8332 8353 8373148 II 54I91543915458 5478 5497155I7 48 I2 8393 84i3 8433 8454 8474 8494 47 I2 5536 5556 5575 5595 56I4 5634147 i3 85i4 8534 8554 8575 8595 86i5 46 13 5653 5673 5692 5712 5731 575i 46 i4 8635 8655 8675 8696 87I618736145 I4 577o0579o15809 58281584815867145 i5 8756 8776 87961881I7 8837 8857 44 i5 5887 5906 5926 5945 5965 5984 44 i6. 88778897189I 7 8937 8958 8978143 6 6oo0046023 6042 6062 60o8 6ioI 43 I 7 8998 901 8o9038o9058 o9078 9098 42 IIo 61I420o6 4016I59616I 7916198162I7142 i8 9I919139199 1599I 7919199 92I9 41 i8 62371625616276 6295635 516334 41 19 9239 9259 9279 9300 9320 930 4o 4o I 6353 6373 6392 64I2 643i 645o 4o 20 9.859360 9380o 90094209440 9460 39 20 9.866470 6489 65091652816547 6567139 21I 9480 1950I 952I 954I 9561 9581 38 21I 6586 6606 6625 6644 6664 6683 38 22 960 19621 964I 966I 968I 970 37 22 6703 6722 674 676I 6780 68oo00 37 23 972I 974I 976I 9781 9802 9822 36 23 68I9 6838 6858 6877 6896 6961 36 24 98421986219882 9902 9922 9942135 24 6935 6954 697416993 70I3 7032 35 25 9962 9982...''.22 1. 42..62 34 25 7051 7I07I 7090 7IO9 7229 7148 34 26 9.860082 OI02 OI22 1O42 0 I62 OI82 33 26 7I6717I87 7206 7225 7245 7264 33 27 0202 0222 0242 0262 0282 o302 32 27 7283 7303 7322 734I 736I 7380 32 28 0322 o342 o362 o382 0402 0422 3I 28 7399 7419 7438 7457 7476 7496 3I 29 o442 o462 o482 o502 0522 o542 30 29 75I5 7534 7554 7573 7592 7612 30~ 30 9.860562 o582 o602 o622 o642 o662 29 3o 9.86763I 7650 7669 768977088772729 3i 0682 0702 0722 0742 0762 0782 28 31 7747 7766 7785 7804 7824 7843 28 32 o802 0822 0842 o862 o882 0902 27 32 7862 7882 790I 7920 7939 7959 27 33 og22 0941 Og6I Og8I TooI I02I 26 33 7978 79971 8o6 8o36 8o55 8074 26 34 Io4i io6i I08I II OI II2I II4I25 34 8093 8II3 8I32 8I5I 8170 8190o25 35 ii6i ii8 I I201 22I I240 I260 24 35 820918228 824718267 8286 83o0524' 36 1280 I300oo 320 I34 I360o 380 23 36 8324 8343 8363 8382 84oi 8420 23 37 I400o I420 14391 4591 I4791499 22 37 844o08459 847818497 85I6 8536122 38 I5I9 I 539 I559 I 579 I 599 I6I8 2 I 38 855518574 8593 8612 8632 865 I 121 39 I638 I658 1678 I698 1718 1738 20 39 8670 8689 8708 8728 8747 8766 20 4o 9.86758 17771 I 797 I8I71 837 1857 I 940o 9.86878518804 8823 8843 8862 888 I 9 4I' I877 i897 IgI6 1936 i956 I976 I 8 4 8900oo89I918939 8958 897718996 i8 42 I996 20oI62035 2055[2075120951 I7 42 90I51903490o53 9073 9092 9 III I7 43 2115 2I35 2I54 2I74 2I94 22I41 6 43 9g309I49 g916819I8819207 92261i6 44 2234 2254 2273 3 12293 233 2333 i5 44 9245 9264 9283 9302 932I 9341 i5 45 2353 2372 2392 24I2 2432 2452 i4 45 9360 9379 9398 94I7 9436 9455 i4 46 247I 249I1 25II 253I 255I 2570 31 46 9474 9494 95I3 9532 955i 9570oI3 47] 2590 26IO 2630 2650 2669 2689 I 2 | 47 9589 9608 9627 9646 96651 96851 I2 48 27092729 2748 2768 2788 2808 11 48 9704972319742976197897991I 49 2827 2847 2867 2887 2906 292610 |o 49 9818 983719856 9875 9894 99ggI4o 50 9.862946 2966 2985 305o 3025 3o45 9 50 9.869933 9952 997I 9990...9..28 9 5i 3064 3084 3I04 3I2413I43 3I63 811 5I 9.870047 oo 66 oo851oo 04 OI23 Io42I 8 52 3I83 3203 3222 3242 3262 328I 7 52 oi6i OI80 O1gg99 0218 238 02571 7 53 330I 3321 334 3360 3380o 3400 6 53 o276 0o295 o34 o333 o35o 037I| 6 54 34I9 3439 3459 3478 3498 35I8 5 54 o3go o409 o428 1o447 o466 o4851 5 55 3538 355713577 3597 36I6 3636 4 55 oo41o523 o542 o 56 o58o o5gg99 4 56 3656 3675 3695 37I5 3734 3754 3 | 56 o6i8 o637 o656 o675 o 694 07I31 3 57 3774 37931 383 3833 3852 3872 2 57 o732 075 0o77o 0789 o8o8 0827 2 581 3892 391 [3931 3951 3970 3990 I| 58 o846 o865 o884 ogo3o 0922 094 I 59 40 o10o2904 9 404o69408841 08 0o 59 og60o 979o gg998 1OI7 o36,Io54 o 60"t 5M0 40" 30 20" 10"lot 60" 50" 40" i 30" 1 3 20 1 0" WI Co-sine of 43 Degrees. Co-sine of 42 Degrees. P. Part 1" 2" 3" 4" 51" 6"' 7" 8" 9 1 p 21 3" 4" 5/" 6"' 7e 8"1 9"1 P Part 2 4 6 8 10 12 14 16 18 P Part 2 4 6 8 10 12 14 15 17 L:3ARITHMIC rANGENTS. 71 Tangent of 46 Degrees. Tangent of 47 Degrees. 0" 1.0" 20" 30", 40' 50 0". 1C' 20', 30' 40" 50"/ o io.oi5I63 520 5 5247 5289 533i 5373 59 o I o. o3o344 o386'429 047I 05I3 0555 59 I 54I6 5458 55oo 5542 5584 5626 58 I o597 o64o0 682 0724 o766 Q8o8 58 2 5668 57115753,5795 5837587957 21 o85 Io893 o935 0977 o 020 oio62 57 3 592i 5963 6006 6o48 6090o 632 56 3 I0o4 ii46 i88 I123I I273 I3I5 56 4 6I74 62I6 6258 630I 6343 6385 55 41 1357 4oo001I442 I484 1526 I568 55 5 6427 6469 65II 6553 6596 6638 54 5 I6ii i653 1695 11737 I780 I822 54 6 668o 6722 6764 68o6 6848 689I 53 6 i864 1906 1948 I99I 2033 2075 53 7 6933 6975 I 7017 7059 7II 7143 52 7 2 I 7 2 I 60 2202 2244 2286 2328 52 8'71I86 7228 7270 7312 7354 7396 5I 8 2371 2413 2455 2497 2540 2582 55I 9 7438 7481 7523 7565 7607 7649 5o 9 2624 2666 27o9 275I 2793 2835 50 I o. I 769I 7733 7776 78 8 7860 7902 49 I I. o32877 2920 2962 3oo4 3o46 3089 49 II 7944 7986 8028 8071 8ii3 8I55 48 ii 3131 3173 3215 3258 3300o3342 48 I2 8197 8239 828I 8323 8366 8408 47 12 3384 3426 3469 35ii 3553 3595 47 I3 8450 8492 8534 8576 86I8 866i 46 I3 3638 3680 3722 3764 3807 3849 46 14 8703 8745 8787 8829 887I 8914 45 14 389g1 3933 3976 4oi8 4o6o 4102 45 15 8956 8998 9040 9082 9124 9I66 44 15 4I45 4I87 4229 427I 4313 4356 44 i6 9209 9251 9293 9335 9377 94I9 43 I6 4398 44404482 4525 4567 4609 43 17 9462 9504 9546 9588 9630 9672 42 17 465i 4694 4736 4778 4820 4863 42 I8 97I4 9757 9799 9841 9883 9925 4I i8 4905 4947 4989 5032 5074 5II6 4I 19 9967. 10..52..94. 36.178 40 19 5158 5201 5243 5285 5327 5370 40 20 10. 020220 0262 o3o5 o347 o389 o43I 39 20 o.0354I2 5454 5496 5539 558I 5623 39 21 0473 o551 o558 o60oo o0642 o684 38 2I 5665 5708 5750 5792 5834 5877 38 22 0726 0768 o8o o853 o895 0937 37 22 59I9 596I 6oo3 6046 6088 6I30 37 23 0979 I021 io63 I 0o 6 1148 1190 36 23 6172 6215 6257 6299 634i 6384 36 24 1232 I274 I3i6 1359 I40I i443 35 24 6426 6468 65ii 6553 6595 6637 35 25 I485 I527 I569 I6I2 I654 1696 34 25 6680o 6722 6764 6806 6849 689134 26 1738 17801 822 i865 I9o7 I949 33 26 6933 6975 70I8 7060 7o 02 7144 33 27 I991 2033 2075 2118 2160 2202 32 27 7187 7229 729 7271 7314 7356 7398 32 28 2244 2286 2328 2370 2413 2455 31 28 7440 7483 7525 7567 7609 765213 3 29 2497 2539 258I 2623 2666 2708 30 29 7694 7736 7778 7821 7863 79o5 30 30o 0.022750 2792 2834 2877 2919 296I 29 3o IO.037948 799o 8o0328074 8117 8159 29 31 3003 3045 3087 313 3172 3214 28 31 8201 8243 82868328 8370 84i3 28 32 3256 3298 3340 3383 3425 3467 27 32 8455 8497 8539 8582 8624 8666 27 33 350o9 355 3593 3636 3678 3720 26 33 8708 8751 8793 8835 8878 8920 26 34 3762 38o4 3846 3889 393I 3973 25 34 896290o0490o47 90899I 31 9174 25 35 4o0I 54057 4o99 4142 41844226 24 35 9216 9258 9300 9343 9385 9427 24 36 4268 43Io 4353 4395 4437 4479 23 36 9470 9512 9554 9596 9639 9681 23 37 4521 4563 4606 4648 4690 4732 22 37 9723 9766 9808 98509892 9935 22 38 4774 4817 4859 4901 4943 4985 21 38 9977..19..62. o4. I46. i88 21 39 5027 507o 51 12 I 54 5196 5238 20 39 Io.o04023 0273 o3I5 o358 o400 o442 20 4o IO.025280 5323 5365 5407 5449 549I 19 4o o.0o4o484 o527 o569 o6II o654 o696 I9 41 5534 5576 568 566o0 5702 5745 I8 4I 0738 0781 o823 o865 0907 og50 i8 42 5787 5829 587i 59I3 5955 5998 17 42 0992 I034 I077 II19 i6i 1204 I7 43 6o4o 6082 6I24 6i66 6209 6251 i6 43 I246 I288 i33o 1373 14151 457 i6 44 6293 6335 6377 64I9 6462 6504 I5 44 i5oo0 1542 i584 I627 I669 I 7I I 5 45 6546 6588 663o 6673 67 5 6757 14 45 I753 1796 i833I 88o 1923 1965 14 46 6799 6841 6884 6926 69687010 I 3 46 2007 205020922I 342I 77 2291 3 47 7052 7095 7I37 7I79 7221 7263 12 47 2261 2303 2346 2388 2430 2473 12 48 73o057348 7390 7432 7474 7516 II 48 25I5 2557 2600 2642 2684 2727 II 49 7559 7601 7643 7685 7727 7770 Io 49 2769 2811 I2854 2896 2938 2980 IO 5o 10.027812 7854 7896 7938 798 I 8023 9 50o.o43023 3o65 3 I 07 350o 392 3234 9 5I 8o65 8I07 8149 8I92 8234 8276 8 5I 3277 33I 9336i 34o4 3446 3488 8 52 83I8 8360o 8403 8445 8487 8529 7 52 353I 3573 36I5 3658 37o003742 7 5.3 8571 8614 8656 8698 8740 8782 6 53 3785 3827 3869 39I123 3543996 6 54 8825 8867 8909 895I 89939036 5 54 40394081 4I23 4165 4208 4250 5 55 9078 9I20 9I62 9204 9247 9289 4 55 4292 4335 4377 4419 4462 4504 4 56 9331 9373 9416 9458 9500 9542 3 56 4546 4589 4631 4673 47I6 4758 3 57 9584 9627 9669 971 I 9753 9795 2 57 4800 4843 4885 4927 4970 5012 2 58 9838 9880 9922 9964..6..49 i 58 5o54 5097 539 5i8I 5224 5266 I 59 1o.o3oo091 o 33 01o75 02I70260 o 302 o 59 5309 535i 5393 5436 5478 5520 0 60" 50"- 40" 31Y' 20"11Y' 1 T 60Y' 50" 40" 30'" 20"' Co-tangent of 43 Degrees. Co-tangent of 42 Degrees. Part 1" 2" 31 411 4 5 6" 7" 8"/ 9/ 1 1' 2" 31 41/ 5// 6"// 7" 8"' 9' ar 4 8 13 17 21 25 30 34 38 Part 4 8 13 17 21 25 30 34 38 72 LOGARITHMIC SINES. J - Sine of 48 Degrees. 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Co-sine of 40 Degrees. j 1" 2" 3" 4' 5" 611 7"1 81" 9" 1"t 2" 3/ 4" 511 67' 7"1 8" 91 Part 2 4 6 7 9 11 13 15 17 Part 2 4 5 7 9 11 13 14 16 L o A R ITHMIC TA N G EN r s. 73 Tangelnt of 48 Degrees. | Tangent of 49 Degrees. 0il 10"1 20" 30" 40" -50", " 10' 20' 30"' 40" 50" olo.o45563 5605 5647 5690 5732 5774 59 o io.o60837 o8790922 o965 1007 io5o 59 I 15817 58 5 59 44 59445986 602858 I Io92a I35 11771220 I12621I305 58 2 607I 6113 6I55 6198 6240 6282 57 2 I347 I390 I4321 475 15I7 i56o 57 3 6325 6367 6409 64526494 653 56 3 I602 i645 I688 17301I773 i8i5 56 4 65 79 6621 6664 6706 6748 679 I 55 4 I858 1goo 1943 I985 2025 2070 55 5 6833 6875 6918 6960 7002 7045 54 5 2 I113 2 55 298 2241 2283 2326 54 6 7o87 7130o7172 7214 7257 7299 53 6 236824II11453 24962538258I 53 7 734I 7384 7426 7468 75II 7553 52 7 2623 26662709 2751 27942836 52 8 7595 7638 7680 7723 7765 7807 5I 8 2879 2921 2964 30o6 3049 3092 51 9 7850 7892 7934 7977 80 19 80 62 50 9 313431I77 3213I262 33043347 50 o Io0.048Io48 I46 8189 823I 8273 83I 649 Io Io0.0o63389 3432 3475 35 I 7 3560o 3602 49 II 8358 84Go 8443 8485 8528 8570 48 I I 3645 3687 3730 3773 38I5 3858 48 I 2 861 2 8655 8697 8739 8782 8824 47 1 2 3900o3943 3985 4028 407o04II 3147 I3 8867 8909 8951 89949036 9079 46 i3 4156 4984241 4283 4326 4368 46 I4 9I21 9 63 9206 924892909333 45 I4 4411 A454 4496 4539458 I 4624 45 i5 9375 94I8 9460 9502i9545 9587 44 I5 4667 4709 4752 4794 4837 4879 44 I6 9629 9672 9714 975719799 984I 43 i6 4922A 965 5007 5050 5092 535 43 17 98849926 9969. 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P 1" 2"1 3"1 4'1 5" 6"' 7" 8" 9" 1" 2" 3" 4" 5 6" 7" 8" 9" P. Part4 8 13 17 21 25 30 34 38 F.a tt 4 9 13 17 21;zG 30 34 38 A4,* L OG A RIT.1 MIC S I NES.' 2 gSine of 50 Degrees. Sine of 51 Degrees. 9 -' 7 10", 220" 30" - 40" -50" 0o" 10J" 20"/ 30' 40"' 50'" 09. 864254l4272 4289 4307 4325 4342 59 0 9. 890503 0520 53 71o554 0571 058859 I 436o 4378 4395 44I3 443I 4448 58 I o605 0622 o6391 o656 0673 obgo 58 2 4466 44833j4oI 45I9 4536 4554 57 2 3707 o724 074I 0758 1775 0792 57 3 4572 4589 4607 64625 4642 466o 56 3 o809g 826 o843 o860 o877 o894 56 4 46771469514713 4730 4748 4766155 4 09I1 0928 I 945 o945 62 0979 o9965, 5j 4783 48oI 48iS 4836 4854 4871 54 5 ioI3 o3o0 I047 io64 io8I I09o 54 6" 48891490614924 4942 4959 4977153 6 II I5 II32 II49 II66 183 I200oo53 7 4994 5012 5o3o 504715065 5082 52 7 12I7 1234 125 II268 1285 1302 52 8 5Ioo15II8 5I5 5i53 5170o5188 5I 8 I3I91 3361 3531 370 I3871 404 51 9 05 521 52 5258 5276 58 5293 5o 9 I42I i438 I455 1472 489 i 5o6 5 0o 9.8853 I 5328 5346 53641538I 539q 49 I0 9.891523 I 5401o 556 5731I 5901o607 49 xI 54I615434 545 15469 54865504 48 55o I I I624 64I 658 675 692 I 70948 12 5522 5539 5557 5574 5592 56og 947 12 I7261I743 I760 I777 1794 810 47 I3 562715644 5662 5679 5697 57I4146 13 827 I844 I86I I878 I895 I9I2 46 I4 5732 5749 5767 5784 5802 58I9145 i 4 1929I 946 I963 i980 1996 2013 4[ i5 5837 5855 5872 5890 5907 5925144 IS 2030 2047 2064 208I 2098 2II5144 i6 5942 59605977 599560o12 6030143 t6 2I32 249 2I65 2182 2I99 221 43 17 6047 6065 6082 60996117 6I34 42 17 2233 2250 2267 228412300 23I7142 I8 6I52I 66966I87 6204 6222 6239 4i 18 2334 235I 2368 2385 2402 24i9 4i 19 6257 6274,6292 63091632763 634414 91 2435 2452 2469 2486 2503 2520 4 20 9.886362 637916396 64i4[643I 6449 39 20 9.892536 2553~2570 2587 2604 262I 39 21 6466 6484!650I 6519 6536 6554 38 2I 2638 2654 267I 2688 2705 2722 38 22 657I 6588 66o6 6623 664i 6658 37 22 2739 2755 2772 2789 2806 282337 23 6676 6693167IO 6728 6745 6763136 123 2839 2856 2873 2890 2907 2924 36 24 678016798168I5 6832 685o 6867 35 24 2940.29572974 299 3oo008 3024 35 25 6885 6902169I9 6937 6954 6972 34 | 25 3o04I 3o58 3075 3092 3Io8 3I25 34 26 6989 7006 7024 704I17059 70 76 33 1|26 3I42 3I5913I76 3I92 3209 3226 33 27 7093 7III 7128 7I4617I63 7i8o032 27 3243 3259 3276 3293 33io 3327 32 28 7I98172I517232 7250 7267 7285 31 | 28 3343 336o 3377 3394 34Io 3427 3I 29 7302 7317397337 7354 737I 7389 30 29 3444 346i 3477 3494 35ii 3528 30 30 9.887406 742317441 745817475 7493129 30o 9.893544356i 3578 3595 36II 3628|29 31 75o10752817545 7562 7580 7597 28 31 3645 3662 3678 3695 3712 3728 28 32 76I4 763217649 7666 7684 770I 27 132 3745 3762 3779 3795 38I2 3829 27 33 7787736177537770 7787 7870526 [ 33 3846 3862 3879 3896 3912 3929 26 34 7822 7839!7857 7874 7891 7909125 34 3946 3963 3979 3996 4oi3 4029 25 35 7926 79431796I 797817995 8012 24 35 4046 4063 4079 4096 4113 4130 24 36 8030J8047 806418082 809981i6[23 | 36 4I46|4I634I80oj4I9614213 4230 23 37 813418i51 8I6818I85/820318220'22 37 4246|42634280 4296 431 34330|22 38 8237 8254 8272 8289 83o683068324 2 38 4346|4363|4380 439644I3 4430|21 30 834i 8358 8375 8393 84io 8427 20 39 4446 4463 448o 4496 45I3 4530120 40 9.888444 846218479 8496 85i 853II 19 4o09.894546|4563|4580 4596 46I 4629 19 4I 85 8548 8582 86oo0086I7186341 i8 4I 4646 4663 4679 4696 47I3 4729;8 42 865i 8669,8686 8703187201 8737 17 42 4746 4763 4779 4796 48I2 4829! 7 43 8755 8772 8789 88o6 8824 884I,6 43 4846 4862 4879 4896 49I2 4929 i6 44 8858 8875 8892 89IO0892718944 5 144 4945 4962 4979 4995 o50I2 5028 5 45 896I18978 899690oI3 9030 90471i4 45 5045 5062 5078 5095 5III 5128 14 46 9064908219099 9II619I3391i50o13 46 5I45|516I|5I78 5194 52II 52271I3 47 9168 9I8519202 9219 9236 9253 I2 47 5244 526I15277 5294 53Io 5327|12 48 9271 9288 9305 9322 933919356III 48 534315360o53771539354I0o 5426|II 49 9374 9391 9408 9425 944219459 io | 49 5443 545915476 5493 55o09 5526 o 5019.889477 9494195II 952819545956219 5o 9.895542 5559 5575 5592 56o8 5625 9 5i 9579 9597 96I4 963i 964819665 8 5I 564I 5658 5675 569i 57o8 5724j 8 52 9682 9699197I6 9734 975i 9768 7 52| 574I 57575774 579o 0[5807 5823 7 53 9785 9802198I9 9836 98531987I, 6 53 584o 5856 5873 5889 59o6 5922 6 54 9888 99o0519922 9939 99569973l 5 54 5939 5955 5972 5988 6oo5 602I 5 |55 99901...7..25..421..59 * 76 4 55 6038 6054 6071 6087 6I04 6201o 4 5619.89oo93 oIIo oI27 1o44160I6I.I78 3 56 6I3716I5316I7o 16861620o362I9[ 3 57 OI095 02I2 0230o247 0264 028I 2 57 6236 625216269 6285 6302 63i8J 2 58 0298 o3i5 0332 o349 o366,o383 I 58 6335 635i 6368 6384 64oo 64I7 I 59 o4000o4I7 o434 o45I o468,o486 o 59 6433 6450o6466 6483 6499 65I6 o - 60"- 50' 40" 30" 20" 10" 60" -- 50" 1 40" 30" 201" 10" Co-sine of 39 Degrees. Co-sine of 38 Degrees. IP. Pa1r " 2"1 3' 4/ 5"/ 6" 7/ 8" 911 PPt 1' 2"' 3/1 4" 5/ 6" 7/1 8" 9" ar 2 3 5 7 9 10 12 14 16 P Part 2 3 5 7 8 iO 12 13 15 L G ARIT HMIC TANGENTS. R L Tangent of 50 Degrees. Tangent of 51 Degrees. I_ o" 10" 20" 30" 40" 50" I 0" 0o" 20"J 30". 40" 50I 1 —IO.76I86 6229 6272 63i5 6358 64oo 59 oIo.o09163 I674 1717 I76oI8803 3 I84659 [ z' 6443 6486 6529 6571 66I4 6657 58 I I889 I932 I975 20o8 2o06 210o4 58 2 6700 6742 6787825 78 8 687I 693 57 2 2I47 2I9I 22342277 2320 2363 57 3 6956 699917042 7085 7127 717o 56 3 2406 2449 2492 2535 2578 262I 56 4 72I3 7256 7298 7341 7384 7427 55 4 2664 2707 2750 2793 2837 2880 55 5 7470o 7512 7555 7598 7641 7684 54 5 2923 2966 3009 3052 3095 3I38 54 6 77 i5 7769 7812 7855 7897 7940 53 6 3181 3224 3267 33Io 3354 3397 53 7 7983 8026 8069 8 I I I 8 I548 19 7 52 7 3440 3483 3526 3569 362 3655 52 8 824G0 8283 8325 82568 84i I 8454 5 8 3698 374I 3784 3828 3871 3914 5I 9 8497 8539 858 8625 8668 871 I 50 9 3957 4000 4043 4086 4129 4I 72 50 10 o 0o78753 8796 8839 8882 8925 8967 49 To 10o.0942I 5 4259 43024345 4345 88 443I 49 II[ 9o0o 9053 9096 9I399 i8 9224 48 I I 4474 45I7 4560 4603 4647 4690 48 12 9267 931o 9353 9396 9438 948I147 12 4733 4776 4819 4862 4905 4948147 I3 9524 9567 96Io 9652 9695 9738 46 i3 4992 5035 5078 512i 5i64 5207 46 i4 978I 9824 9867 9909 9952 9995 45 I4 5250 5293 5337 538o 5423 5466 45 i5 Io.o80038 008I o1 24, o66o02o9g 0252 44 I5 5509 5552 5595 5638 5682 5725 44 I6 0295 o338 o38I 042,3 0466 o050g 43 6 5768 5811 I5854 5897 5940 5984 43 17 o552 o595 o638 o68o 0723 0766 42 17 6027 6070 6 I 3 6561 699 6242 42 i8 0809 o852 o89515 0937 o980 1023 4I i8 6286 6329 6372 64i5 6458 65oI 4I 19 I066 IIog 91I52 195 I237 I28o04o 19 6544 6588 663I 6674 6717 6760o 40 20 I 8 I 323 366 1409 I452 494 I537 39 20 I. o096803 6847 6890o 6933 6976 7019 39 21 I5801 623 I666 170917522179438 21 7062 71o67 49719277235 7278 38 22 I837 i88o I923 1966 2009 2o52 37 22 7321 7365 7408 7451 7494 7537 37 23 2094 237 28022232266 2309 36 23 7580 7624 7667|77IO 7753 7796 36 24 2352 2395 2437 2480 2523 2566 35 24 7840 7883 7926 7969 80I2 8o56 35 25 2609 2652 2695 2738 2780 2823 34 25 8099 8I42 8I85 8228 827I 83i5 34 26 2866 2909 2952 2995 3038 3o8I 33 26 8358 84o0I 8444 8487 853i 8574 33 27 3123 3I66 3209 3252 3295 3338 32 127 8617 8660 8703 874718790 8833 32 28 338i 3424 3467 3509 3552 3595 3I 28 8876 89I9 8963 9006 9049 9092 3I 29 3638 368 3724 3767 38 Io 3853 30 29 9136 9 79 9222 9265 9308 9352 3o 30oIo.o83896 3938 398I14024~4067 410 29 3o0 o.o99395 9438 948i19524 9568 96I 129 31 41534196 4239 4282 4325 4367 28 31 9654 9697 9743 9784'9827 9870 28 32 44Io 4453 4496 45394582 4625 27 32 99I39957...... 43..86. I30 27 33 4668 47I 147541479714839 4882 26 331I IO10OI73 02I6 0259 o3o3 o346 0389 26 34 4925 4968 5oti 5o54 5097 5 I4 0 25 34 o432 o476 o5I9 0562 o605o0649 25 35 5i83 5226 5269 5312 5355 5397 24 35 o692 0735 0778 822 8650908 24 36 544o 5483 5526 5569 56I2 5655 23 36 095 o995 I0o38 io8I I24I ii68 23 37 5698 5741 5784 5827 5870 59I3 22 37 I211 1254 I297 i34i I384 I427 22 38 5956 5999 6041 60846127 6170 2I 138 I4701 514 i5571i600o16431i687 21 39 62I3 6256 6299 6342 6385 6428 20 39 1730 I773 I8 I71 8601I9031I946 20 4o 1o.o8647I 6514 6557660066 43 6686 9 40o I.101990 2033 20762I rI9|2I63|22o06 9 4I 6729 6772 68i5 6857 6900 6943 I8 4I 2249 2293 2336 2379 2422 2466 8 42 6986 7029 7072 77 I5 7158 720I I7 42 2509 2552 2596 2639 2682 2725 17 43 7244 7287 7330 7373 74I6 7459 I6 43 2769 28I2 2855 2899 2942 2985 i6 44 7502 7545 7588 763I 7674 77I7 i5 44 3029 3072 3II5 3I58 3202 3245 I5 45 7760 7803 7846 7889 7932 7975 I4 45 3288 3332 3375 34I8 3462 3505 i4 46 80I8 806I 8Io048I4781I90o8232 13 46 3548 3592 3635 3678 3722 3765 3 47 8275 83i8 836 8404 8447 8490 12 147 3808 385I 3895 3938 398I 4025 I2 48 8533 8576 86I9'8662 8705 8748 1 48 4o684I I I 45554I98 4241 4285 I I 49 ~ 8791 8834 8877 89208963 900oo 6 Io 49 4328 4371 44I5 4458 45oI 4545 Io 50o.o890o49gog9092I 35 9789221 9264 964 5 I o. 04588 4631 4675 4718476I14805 9 5i 9307 9350 9393 9436 9479 9522 8 5I 4848 4891 4935 4978 502 15065 8 52 9565 9608 965I 9694 9737 9780 7 52 5o 085I52 5I95 5238 5282 5325 7 53 9823 9866 990999599952 9995..39 6 53 5368154a2 5455 5498 5542 5585 6 541io.o9oo82 oI25 oI68o02IIo0254o0297 5 54 5628 5672 575 5 757595802 5845 5 55 o34o o383 0o426 o469o512 o555 4 55 5889 5932 5975 603o96062 6io6 4 56 o598 o64I o6840o727o0770 08I3 3 56 6I49|6I92 6236 6279 6322 6366 3 57 o856 o899 0o942 o985 1028 I07I 2 57 640916453 6496 6539 658536626 2 58 III4 II57 I200 I243 12861 329 I 58 6669673 6756 68o006843 6886 I 59 I372 I4I6 i49 502 i545I 588 o L9 6930o6973 70o7 7060 71o03 747 0 60" 50o' 40" - 30"8 6 1"' Tr.0 60" 50" 40" 30" 20" 10" 601 4 |, 60,'~0, 20, Co-tangent of 39 Degrees. Co-tangent of 38 Degrees.' 1" 2 311 4"1 51" 6" 7'/ 8// 9// I p 1" 2" 311 4/1 511 6/i 7"1 811 911 P. Part 9 13 17 21 26, 303439 P. art 4 9 13 17 22 26 30 35 39 76 L OGARITrHMIC SIN ES. -._"' o Sine of 52 Degrees Sine of 53 Derees 01 - o10 20"o 30"' 40" 50, o 0" 10', 20" 30" 40"/ 50" / o 9.896532 6549 6565658 I 6598 66i4 59 0o.902349 2364 2380 2396 24I2 2428 59 I 663i 6647 6664 66,80 6697 6713 58 I 2444 2460 2475 249I 2507 2523 58 2'6729 6746 6762 6779 6795 68I12 57 2 2539 2555 257I 2586 2602 2618 57 3! 6828 6844 686I 6877 6894 691o 56 3 2634 2650 2666 268I 2697 2713 56 4 6926 6943 6959 6976 6992 7009 55 4 2729 2745 2761 2776 2792 2808 55 5 7025 7041 705817074 7090 71o7 54 5 2824 2840 2856 2871 2887 2903 54 6 7123 7140 7I56 7I72 7189 720o553 6 2919 2935 2950 2966 2982 2998 53 7 7222 7238 7254 7271 7287 7303 52 7 3oi4 3029 3o45 3o6I 3077 3093 52 8 7320 7336 7353 7369 7385 7402 5I 8 3Io8 3I24 3I4o 3156 3I71 3I87 51 9 748 7434 745 I 7467 7483 75oo00 9 3203 32I9 3235 325o 3266 3282 5o IO 9.8975i6 7533 754917565 7582 7598 49 IO 9.903298 33I3 3329 3345 336i 3377 49 IIX 76I4 67631 647J7663 7680 7696 48 II 3392 34o8 3424 344o 3455 347I 48 12 7712 7729 779 7745776 7778 7794 47 I2 3487 3503 35I8 3534 355o 3566 47 I3 78IO 7827 7843|7859 7876 7892 46 I3 358i 3597 36i3 3629 3644 3660 46 I4 7908 7924 794I 7957 7973 7990 45 I4 3676 369I 3707 3723 3739 3754 45 I5 8006 8022 8039180 55 807I 8o88 44 I5 3770 3786 3802 38I7 3833 3849 44 i6 8Io4 812o 8I3681I53816g69 8 185 43 I6 3864 3880 3896 3912 39273943 43 17 8202 8218 8234 8250 8267 8283 42 17 3959 3974 3990 4006 4021 4037 42 i8 8299 83I5 8332 8348 8364 838 I 4 i8 4053 4069 4084 4oo 00 4 116 43 I 4 19 8397 84I3 8429j8446 8462 8478 4o 19 4447 4I6314I78 4194 42I O 4225 4o 20 9.898494 85Ii 852718543 855918576 39 20 9.90424I 4257 4272 4288 430o4 439 39 21 8592 8608 8624 864I 8657 8673 38 21 4335 435i 4366 4382 4398 44I3 38 22 8689 8706 8722 8738 8754 8770 37 22 4429 4445 4460 4476 4492 4507 37 23 8787 8803 8835 8852 8868 36 23 4523 4539 4554 457o 4586 460o 36 24 8884 8900 89I618933 8949 8965 35 24 46I7 4632 4648 4664 4679 4695 35 25 898 18997 9o4 90o30 9o46 9062 34 25 47I I 14726 4742 4757 4773 4789 34 26 9078 9095 9I I 9I27 9I43 g5g9 33 26 4804 4820 4836 485I 4867 4882 33 27 9176 9192 9208 9224 9240 9256 32 27 4898 494 414 929 494514960 4976 32 28 9273 9289 93o5 932I 9337 9354 3I1 28 4992 5007 5023 5038 5o54 5070 3i 29 9370 9386 9402 94i8 9434 9450 30 29 5085 5IOI 51165132 5148 5I63 30 30 9 899467 9483 9499195 I 5 953I 9547 29 3o 9.905I79 5I94 52IO 5225 524I 5257 29 3I 9564 9580 9596|9612 9628 9644 28 3I 5272 5288 5303 53I9 5334 5350 28 32 966o 9677 9693 9709 9725 974I 27 32 5366 538i 5397 5412 5428 5443 27 33 9757 9773 9789 9806 9822 9838 26 33 5459 5474 54990 550 6 552I 5537 26 34 9854 9870 9886 9902 99ggI8 9935 25 34 5552 5568 5583 5599 56I4 5630 25 35 995I 9967 9983 9999..I5..3I 24 35 5645 566I 5676 5692 5708 5723 24 369. 900047oo6300oo79oo0096 OI I 2 OI 2823 36 5739 5754 577o 5785 58oi 58I6 23 37 oI44 o6o01OI760192 o208 0224 22 37 5832 5847 5863 5878 5894 5909 22 38 o240 0o256 o272o 0289 o3o5 o321 21 38 5925 594o 5956 597I 5987 6002 2I 39 0337 o353 o369 o385 o4o0I o417 20 39 6o0I8 6033 6049 6064 6080 60959 20 4o9.9 oo4330o449 o465 o48I o497 05I3 I9 40 9.906III 6I26 6I42 6I57 6I73 61881 9 41 o529 o545 o562 o578 o594 o6io i8 41 6204 6219 6235 6250 6265 628I i8 42 o6260o642 o658 o674 o6go 07o6 I7 42 6296 6312 632716343 6358 637417 43 0722 0738 0754 0770 0786 o80802 i6 43 6389 6405 6420 6436 645i 6466 i6 44 o8I8 o834 o85o o866 o882 o898 i5 44 6482 6497 65I3 6528 6544 6559 i5 45 o9I4o930o o946 0962 0978 0994 I4 45 6575 6590 6605 662I 6636 6652 14 46 IOIOIo26 Io42 ioo58 I74 I0901 3 46 6667 6683 6698 67I36729 6744 I3 47 I0o6II22 II38II54 II17o110 861 2 47 6760 6775 679 168o6 682I16837 12 48 1202o12181 2341I250 I266 I282 I, 48 6852 6868 688316898169I4 6929 II 49 I298 I3I4 I33oI 346 I362 1378 Io 49 6945 6960 6975 699I 7006 7022 1o So 9.90I394 I4Io I426 1442 I458 i474 9 509.907037 7052 7068 7083 7099 7 II4 9 5i 1490 I505 I52I I537 r553 I569 8 5I 7129 745 7I60 77517I91 72606 52 i585160 6oi 717 633 I649 I665 7 552 7222 7237 7252 7268 7283 7298 7 53 I68i 1697 I7131I729 I745 I760 6 53 7314 7329 73 44 7360 7375 7391 6 54 I776 1792 i8o8 1824 i84o i856 5 54 7406 742I 7437 7452 7467 7483 5 55 I872 i888 1904 I920 1936 I 95I 4 55 7498 7513 7529 7544 755917575 4 56 1967jI983 I999 20I5 203I 2047 3 56 7590 7605 7621 7636 765i 7667 3 57 20631207920952IIO 2I26 2142 2 1 57 7682 76977713 7728 774317759 2 58 2I58|2I74 2190 2206 2222 2238 I 58 7774 7789 7805 7820 7835785i I 59 22531 269285 2301 2317 2333 o 59 7866 788I 7896 79I2 7927 7942 0 e0 l* 50" 401 30" 20"- 1 0" dl 60' 50 40 " 3 0 " * Co-sine of 37 Degrees.. Co-sine of 36 Degrees. p S 1" 2" 3"1 4' 5" 6"1 7'1 8" 9" IPPt 1" 2" 3" 4" 5" 6" 71 8/" 9" art 2 3 5 0 8 10 11 13 15 P.ar 2 3 5 -6 8 9 11 12 14 LOGARITHMIC'1ANGENTS. [~ Tangent of 52 Degrees. e Tangent of 53 Degrees. 0,, 10" 20" 30" 40,, 1 50,,' ( 10" 20"1 30'- 40" -__ O IO- I07190 7234 7277 7320 7364 7407 59 0 10. 122886 2929 2973 30 173061 3oI0559 I 745I 7494 7537 7581 7624 7668 58 I 3i48 3192 3236 3280 3324 3368 58 77I I 7754 7798 7841 7885 7928 57 2 34Ii 3455 3499 3543 3587 3630o57 3 7972 8o0I58o58 8I1o2 8145 8I89 56 3 3674 3718 3762 3806 3850 3893 56 4 8232 8275 8319i8362 84o6 8449 55 4 3937 398i1402514069 411 3 415755 5 8493 8536 8579 8623 8666 871o 54 5 4200 4244 4288 4332 4376 442o 54 6 8753 8797 8840o 8884 8927 897o 53 6 4463 4507 455x 4595 4639 4683 53 7 90I4 90579IOI 19I44 9188 9231 52 7 4727 4770 48i4 4858 4902 4946 52 8 9275 93I81936I 9405 9448 9492 5I 8 499~ 5034 5077 5121 516515209 5I 9 9535 9579 9622 9666 9709 9753 50 9 5253 5297 534i 5385 5428 5472 5o 10 10. I09796 9840 9883 9926 9970.1. 3 49 10 IO1. 255I6 5560 5604 5648 5692 5736 49 II 10. 1xoo57 oIoo oI44 O1 87 023 10274 48 II 5780 5823 5867 5911I 595515999 48 12 o3i8 o36i o405 o448 o492 o535 47 I 2 6043 6087 6I 316 175 6219 6262 47 13 o579 0o622 o666 0o7o9 0752 0796 46 13 63o6 635o 6394 6438 648216526 46 I4 o839 o883 0926 0970 ioi3 I057 45 14 6570 6614 6658 6701 6745 6789 45 i5 I OO I I44 I 187 I23 11274 1318 44 i5 6833 6877 6921 6965 700917053 44 i6 I36i I405o I448 1492 I535 1579 43 I16 7097 7141 7185 7229 7273 731 6 43 I7 I622 1666 1709 1753 I1797 184o 42 I7 7360 7404 7448 7492 7536 7580 42 i8 i884 I927 I971 2014 2058 2IOI 41 I8 7624 7668 7712 7756 7800 7844 41I 19 2I45 2I88 2232 2275 23I9 2362 40 19 7888 7932 7976 8020 8063 8IO7 4o 20 10.IO I2406 2449 2493 2536 2580 2623 39 20 IO. I28I5I 8r95 8239 8283 8327 8371 39 21I 2667 2711 2754 2798 2841 2885 38 21 8415 8459 85o3 8547 859I 8635 38 22 2928 2972 3o0I 5 3059 3102 3I46 37 22 8679 8723 8767 88r1 8855 8899 37 23 3189 3233 3277 332o 336l, 3407 36 23 8943 898790o3i 9075 9:19 19163136 24 345i 3494 3538 358i 3625 3669 35 24 9207 925I 9295 9339 9383 942 7135 25 3712 3756 3799 3843 3886 3930 34 25 947I 9515 9559963 9673 961969 34 26 3974 4017 406i 4Io44148 4I9I 33 26 9735 9779 9823 9867 991I 9955 33 27 423514279 4322 4366 4409 4453 32 27 9999 -.43..87 131.175 2 91 32 28 4496 454o 4584 4627 467I 47I4 3I 28 Io..30263 0307 o35i 0395 0439 o483 31 29 4758 4802 4845 4889 4932 4976 30 29 0527 0o57I o65 0659 0703 0747 3o 30o IO.I5020o 5063 5107 5I50 5194 5238 29 30 1i*z*'3079 I o835 0o879 0923 o96710o11 29?I 5281 5325 5368 5412 5456 5499 28 31 io55 1o99 ggi43 1187 1232 I1276 28 32 5543 5586 563o 5674 57I7 576I 27 32 32o0 i364 14o08 452 1496 540o 27 33 5804 5848 5892 5935 5979 6023 26 33 r584 I628 1672 7I16 1760o 8o4 26 34 6o66 6I I0 6I53 6I97 624I 6284 25 34 1848 I892 1936 1981 2025 2069 25 35 6328 6372 64I5 6459 6502 6546 24 351'It3 2157 2201 2245[2289 2333 24 36 6590 6633 6677 672I 6764 6808 23 36i 2377 2421 2465 2509 2554 2598 23 37 6852 6895 6939 6982 7026 7070 22 37i 2642 2686 2730 277412818 2862 22 38 71 I37157 7201 7244 7288 733a 21 38 29062950 29943039 3083 3127 21 39 7375 7419 7463 756755o07594 20 3q 3171 32i5 3259 3303 3347 3391 20 40o 10.I17637 7681 7725 7768 781217856 19 3013b s83436 3480 3524'3568 3612 3656 19 41 7899 7943 7987 8030 8041881i8 18 41 3700 374413789138331387713921 18 42 8i6 8 8205 8249 8292 83361838o0 I7i 42i 3965 400914053 40974I142 4186 17 431 8423 8467 85ii 8555 8598 8642.;6 43 423o 4274 4318 4362 44o6 445i I6 44 8686 8729 8773 8817 886o089o4 i 5 441 4495 4539 4583 4627 46 7 1471 5 15 45 8948 8992 9035 9o 79 9I23 9I66 14 45, 4760o 48o4 488 4892 4936 4980o 4 46 92IO 9254 9297 934I 9385 9429 i3 46j 5025 50695 5 13 5157 5201 5245 I3 47 9472 9516 95609603 9647 9691 12 47 529o05334 5378 5422 5466 550o 12 48 973519778 9822 9866 990919953 11 48 5555 5599 5643 5687 5731 5775 11 49 9997..4I..84.I28.I72,216 110 49 5820 5864 5908 5952 5996 6041 io 501I0.120259 o3o3 o347 o3g o434478 9 50o io.36o8516129 6173 62I7 269263o6 q51 o522 o565 o609 o653 o697 0740 8115 6350o6394 643816483 6527 6571: 8 52 07840o828o0872 o915 0959 I0oo3 71 52 6615 6659 6704 6748 6792 6836 7 53 Io47 1091 II34 1178 1222 1266 6 53 6881 6925 69696 7O3 37057 7102 6 54 I309o 1353 I397 I441 I484 1528 5 5154 7IP46 7190 7234 7279 7323 7367? 5 55 1572 I6I6 1659 1703 1747 I791 4 55 7411 7455 7500 7544 7588 7632] 4 56 1835 1878 1922 I966 2010 2053 3 56 7,677 7721 7765 780o978.54 7898 3 57 2097 2141 2185 2229 2272 2316 2 57 7942'798618031 8.075 819 8163 2 58 23601 2404 2448 249I 2535 25791 I 58 820:o88252 8.296 8,34i 8385 8429 1 59 262312667127Io027542798 2842 o0 59 847318518i8562 86068.650 8695 o 60" 50"1 40" 1 30' 1' 50"1 40" 1 30' 1 20" 1 10" 1 g { Co-tangent of 37 Degrees. C - C-o-tangent of 36 I)egrees. - 1" 2". 3"1 4' 5"' 6" 7,, 8", 9/, 1"/ 2", 3" 4/ 5" 6" 7,/ 8" 9 1 P. Par t 4 9 13 17 22 26 31 35 39 P. *Part.4 9 13 18- 2 26 3a1 35 40 7 8 LOGARITHM I S'I N E S.. I Sine of 54 Degrees. Sine of 55 Degrees.' t10" [ 20"1 30" 40" 1 50" 1 0" 10" 20" 30" 40" 50" 09 -907958 7973 988 8004 80o98034599 09.93365 3379 3394 3409 3423 3438 59 I 8o4918o658o8oo95 8I I I 8I26 58 I 3453 3468 3482 3497 35I23.527 58 2.8I4I 8I5618I72 8187'8202 8217 57 2 354I 3556 357I 3585 3600 36I5 57 3 8233 8248 8263'8279 8294 8309 56 3 3630 3644 3659 3674 3688 3703 56 4 8324 8340o 8355 8378385 84o0I 55 4 37I8 3733 3747 3762 3777 379I55 5 84i6 843I 8446 846218477 8492 54 5 38o6 382I 3836 385o 3865 3880 54 6 8507 8523 8538855318568 8584 53 6 3894 3909 3924 3938 3953 3968 53 7 8599 86I418629 8644 866o 8675 52 7 3982 3997 4o2 4026 4o04I 4056 52 8 8690o8705 872I58736 875I 876615I 8 4070 4085 4o00 4II4 4I29 4I44 5I 9 878I 8797 88I2 8827 8842 8857 50 9 4I584173 4I88 4202 4217 4232 50 1o 9.9o8873 8888890o3 898933 8949049 o 9.9 4246426 I 4276 4290 4305 4320 49 I I 896418979899490oo9 90o25 90o4o48 I I 4334 4349 4364 4378 4393 4407 48 12 90551907019085 9IOI9 II61913 I 47 12 4422 4437 445I 4466 448, 4495 47 I3 9I46 1I6I 1976 9192 9207 9222 46 I3 45Io 4524 4539 4554 4568 4583 46 I4 9237 925219267 928319298 93I3 45 4 4598 4612|4627 464I 4656 467I 45 I5 9328 9343 9358 9374 9389 940444 I 5 4685 4700 47I4 4729 4744 4758 44 I6 94I9 9434 9449 9464 948o0949543 I6 4773 4787 4802 48,7483 I 4846 43 I7 95o109525!9540o9555 9570 9586142 I7 4860 4875 4890 49~4 4959 4933 42 I8 96o0 96I6 9631 9646 966I 967614I I8 4948 4962149774992 oo006 6 500 2 4 59 969 I97079722 973719752 197674o I9 50o35 5050150o64 5079 5094 5Io8 4o 20 9.909782 9797 981 2 9827 9843 9858 39 20 9.9I5 I 235 I 375'525 6651815 96 39 2I 9873 9888 99o03998 9933 3 9948 38 2I 52IO 5225 5239 5254 5268 5283 38 22 9963 9978 99941..9...241..39 37 22 5297 53I2 5326 534I 5356 5370 37 23 9.9I00540 o0690484 oo84 oo II41o I29 36 23 5385 5399 54I4 5428 5443 5457 36 24 OI44 1O59 0175 0190 120510220 35 24 5472 5486 550 55I5 5530o 5544 35 25 0235 0250o0265 01o2800295o03Io34 25 5559 5573 5588 5602 56I7 563I 34 26 o325 o3401o3551 370 o385 o4oo 33 26 5646 5660o 5675 5689 5704 57I8 33 27 04I5 o430'o446 o46I o476 1049I 32 27 5733 5747 5762 5776 579I 580 5 32 28 o50o6 05210536 o55i o566 o58I 3I 28 5820 5834 5849 5863 5878 5892 3I 29 o596 o6 I 0626 o64i o656 167I 30 29 5907 5925 5936 5950 5965 5979 3o 30 9.9o 686 o 07OI!7I61o 73I 0746 0765 29 30o 9.95994 6008 6023 60376052606629 3 o776 o791 o806 0821 o8361o85I 28 3I 6o8i 60956Iog 96I24 6I38|6I53 28 32 o866 0881 o89609i I 0926094~I 27 32 6167 6I82|6I96 62I I6225 6240 27 33 0956 097I 0986 5I0 IIoi6 Io3I 26 33 6254 6268 6283 6297 63 I 26326 26 34/ Io46 Io6I I076| IO9I Iio6 I2I 25 34 634i 6355 6369 6384 6398 6413 25 35 I I36 II5I |II66| II8I 196 I25I 24 35 6427 6442 6456 6470 6485 6499 24 36 I226 1241 I256 127I 1286I 3oo100 23 36 65I4 6528 6543 6557 657I 6586 23 371 I3I5 I330 345 I 36o I3751 390 22 37 6600 66I5 6629 6643 66.5816672 22 381 405 I420 I435 I450 1465 I480 2I 38 6687 670: 6755 6730 6744 0759 2I 39 I495 i5Io I525 I 540 I555| 569 20 39 6773 6787 6802 68I 6830o 6845 20 40o 9.95I584 I599 I6I4 I629 I644 I659 I9 4o 9.9I6859 6874 6 888 6902oi697 693I I9 4 I 6741I68917041 I 7I9 7341 I748 I8 4I 69466960 697469746989 7 700 3 7 8 42 I763 1778 I793) 8o8 I823 i838 17 42 70327046 7060|7075 7089 7I14 17 43 I853 1868!I8831I897 I9I2 I927 I6 43 7II8 7I32 747 7I 65 7575 7I90 16 44 I942 i957 1972 I987 2002o20I7 I5 44 7204 72I8 7233 7247 726I 7276 I5 45 203I 2046 206I12076 2091 2I06 14 45 7290 7304 73I9 7333 7347 7362 I4 46 2I2I 2I3612I5012I65 2I80o 295I3 46 7376 73901740~574I917433:7448 I 3 47 2210 2225 2240 2255 2269 2284 I2 47 7462 7476 749I 750517519 7534 I2 48 2299 23I4 2329 2344 235812373 II 48 7548 756217576 7595 7605 76I9 II 49 2388 2403 24I8 2433 244812462 IO 49 7634 7648 7662 7677 769I 77o5 IO 50 9.9I2477124921250712522 25371255I 9 50 9.9I7719 7734:7748 7762 7777 7795 9 51 2566 258I 2596 26II12625 2640 8 5 7805 7859 7834 7848 7862 7877 8 52. 2655 2670 268512714 2729 7 52 789I 7905 79I9 7934 7948 7962 7 53 2744 2759 2774 2788 2803 28i8 6 53 7976 799I18005 8019 8033 8o48 6 54 2833 2848 2862 2877 289212907 5 54 8062 8076 80g90 8o5 8IIg98i33 5 55 2922 2936/295I12966 298I 2995 4 55 8I47 8I62 8I761890o 820418259 4 56 30io 53025 3040|3055 306930o84 3 56 8233182471826i58276 82908304 3 57 309913II43I128 3I4313i58 3173 2 57 8358j8333 8347!836T 8375 8389 2 58 3I87 3202 32I713232 3247 326I I 58 8404184I8!8432 8446;846I 8475 I 59 3276 3295 33o613320 3335 3350 o 59 8489':8503185I7 853218546 8560o 0 60" 50s 40" 30"T 20"1 0. 60 150":40-" 30"1 20, 10"_ IO" Co-sine of 35 Degrees. lfi } Co-sine of 34 Degrees. i P. Partl 1' 2 311 41" 5"1 6" 7" 8" 91 I pa 1 2" 3"1 4"/ 5' 6" 7" 8" 911 9.~2 3.5 6 8 9 11 12 14 PPart 3 4 6 7 9 10 1213 L o G AR THMIC TANGENTS,.t1 Tangent of 54 Degrees. Tangent of 65 Degrees. i0" 10" 20" 30"1 40" 50" 0" 10" 20' 30" 40" 5c0 oIxo.138739 8783 8828 8872 89I6 8960 59 o Io. 154773 48I8 4863 49o8 4952 4997 59 I 9oo005 9o4919093 9389182 9226 58 I 5042 508751I32 5I771522I 5266 58 2 9270 93I5 9'359 9403 9448 9492 57 2 53II 5356 54oi 544615490 5535 57 3 9536 9580 96259669 97139758 56 3 5580 5625 5670 57I5 5759 5804 56 4 9802 9846 9891 9935 9979..24 55 4 5849 5894 5939 5984 6029 6073 55 5 o. I400o68o112r 01571020 10245 0290 54 5 6II 61186163 6208 6253 6298 6343 54 6 o334 o378 o423 o467 o5 II o556 53 6 6388 6432 6477 6522 6567 66r 153 7 o600 o644 o689 0o733 0777 0822 52 7 6657 6702 6747 6791 6836 688I 52 8 o8660910og0955 0999 io43 xo885 I 8 6926 6971 70I617o6I 7106 715I 5I 9 II32 II76 1221 1265 I309 3I54 50 9 7I95 7240 7285 7330 7375 7420 50 10 O. 14I398 1442 4871 531 I5761 620 9 IO Io.1 57465 75I017555 7600 7645 7689 49 11 I 664 I709 I1753 I1797 I8421 886 48 II 7734 7779 782417869 7914 7959 48 I2 I93I I975 2019 2064 2Io082152 47 I2 8004 80491809481I 39 8I84 8229147 I3 2I97 224I 2286 2330 2374 24I9 46 I3 8273 8318 8363 840818453 8498 46 I4 2463 2508 2552 2596 264I 2685 45 i4 8543 8588 8633 8678 8723 8768 45 i5 2730 2774 288 2863 2907 2952 44 5 88I3 8858 8903 8948 8993 903844 i6 2996 304I13085 3I29 3I74 32I8 43 I6 9083 9I289173 9218 9263 9307 43 17 3263 3307 335i 3396 3440 3485 42 I7 9352 9397 9442 9487 9532 9577 42 18 3529 3574 36I8 3662 3707 3754I 1 i8 9622 9667 97I2 9757 9802 9847 4I I9 3796 3840 3885 3929 3974 40I8 4o I9 9892 9937 9982..27 -'72.II17 4o 20 IO. I44o02 4Io074I5I 4I96 4240 4285 39 20 0o.160162 0207 0252 0297 o0342 o387 39 21 4329 4374 44I8 4463 4507 455I 38 21 0432 0477 0522 0567 06I2 o657 38 22 4596 4642 468514729 4774 4818 37 22 0703 0748 0793 o838 o883 0928 37 23 4863 4907 4952 4996 5041 5o85 36 23 0973 ioi8 Io63 iio8 1I5311 9836 24 5I3o 5174 52 9 5263 53o8 5352 35 24 I243 1288 1333 378 I423 I468 35 25 5397 544I 5486 5530 5575 56I9 34 25 I5I3 1558 I6o3 I648 1693 1739 34 26 5664 5708 5753 5797 5842 5886 33 26 I784 1829 1874 I9I9 I964 2009 33 27 5931 5975 6020 6064 61091 653 32 27 2054 2099 2I44 2I89 2234 2279 32 28 6i98 6242 6287 633I 6376 6420 3I 28 2325 2370 24I5 2460 2505 2550 31 29 6465 6509 6554 6598 6643 6687 30 29 2595 2640 2685 2730 2775 2821 30 30 Io..046732 6777 682I16866 69I0 6955 29 3o 0o. 162866 291I 2956 3ooi 3o46 3091 29 3i 6999 7044 708817133 1777 722228 3 3I36 3i8i 3227 3272 33I7 3362 28 32 7267 73II1 7356 7400 7445 7489 27 32 3407 3452 3497 3542 3588 3633 27 33'7534 7578 7623 7668 77I2 7757 26 33 3678 3723 3768 383 3858 390426 34 78i 7846 7890 7935 7980 8024 25 34 3949 3994 4039 40 84 4129 4174 25 35 8069 8I 3 8 I 58 8203 8247 8292 24 35 4220 4265 43Io 4355 44oo00 444524 36 8336 838I 8425 8470 85I5 8559 23 36 449I 4536 458I 4626 4671 47I6 23 37 8604 8648 869318738 8782 8827 22 37 4762 8071 2 852 48974942 4988 22 38 887i 896 896I9005 9050 9095 21 38 5033 5078 5I23 5I68 521315259 2I 39 9r39 9I84 9228,9273 93I8 9362 20 39 5304 5349 5394 5439 5485 5530 20 4o Io. 149407 9452 9496 954I 9585 96301 9 4o Io. I65575 5620o 5666 57I I5756 580I I9 4I 9675 9719 9764 9809 9853 9898 I8 4I 5846 5892 5937 5982 6027 6073 i8 42 9943 9987..32..76.121.I66 I7 42 6118 6i636208 6253 6299 6344 I7 43o I0. 50200255 o30010344 o389 o434 i6 43 6389 6434|6480o6525 657066 5 I 6 44 o478 o523 o568 o 6I2 0657 0702 i5 44 666z 6706 675I 6796 6842 6887 i5 45 0746 079I o836 o880o0925 o 9701i4 45 6932 6977 7023 7068 7I137 I581I4 46 10I4 1o59 114o41149 II93 122I 8 13 46 7204 7249 7294 7340 7385 7430oI3 47 1283 1327 13 72 4 7 I46i I5065oI2 147 7475 7521 7566 761 7657 7702 I2 48 5511i 595 i64o i685 I730 I774II 148 7747 7792 7838 7883 7928 7974 1I 49.I8I9I8641I9o 8 I9531 998 2o43I1O 49 8oIg 8o648Io91 8I55 8200 8245 Io 50o Io152o87 2I32 2I77 12 12266 23II 9 50o 10.I6829i 8336 8381 8427 8472185I7 9 51 2356 2400 2445 2490 2535 2579 8 5i 8563 86o0886538699 8744 8789 8 52 2624 2669 27i3 2758 2803 2848 7 52 8835 8880o 89258975 1 90 I6 906I 7 53 2892 2937 2982o3027 307I 3116 6 153 907529197 9243 9288 9333 6 54 3I61 320o6325o03295 334o03385 5 54 9379 9424 9469 95I5 9560 96o5 5 55 343o 3474 3519 3564 3609 3653 4 55 965I 9696 9742 9787 9832 9878 4 56 3698 3743 3788 383213877 3922 3 56 9923 9968.. 4..59.Io5S.I5o 3 57 3967 40oI24o56|4I1oI 446 4191 2 157 1o.170195 0241|0286 033I 1377o0422 2 58 4236 428o0432514370 4415 446o I 58 o468 o513o0558 o604 0649 o695 I 59 4504:4549,45941463946844728 o 59 o740 0785 o83 o8760922 o0967 0 ""0" 50Y' 40" 30" 20" 10 1' 50" 40" 30' 1 20" 10 Co-tangent of 35 Degrees. 1 Co-tangent of 34 Degrees. i p a 1" 2 3t 4" 5" 6" 7" 8" 9/" s 1" 2" 3" 4" 5" 6" 7" 8" 91' P. Part} 4 9 13 18 22 27 31 36 40 P. Part 5 9 14 18 23 27 32 36 41 co LLOGARITHMIC SINES. l....Sine of 56 Degrees. - Sine of 57 Degrees. I Q' 10" 20"- 30"y 41Y 50," 0, 10" 20" 30" 40" 50 o. 918574 8588 8603 86I7 863I 8645 59 o9.923591 3605 36I93632 3646 366o 59 I 8659 8674 8688 8702 87I6 8730 58 I 3673 3687 370I 3714 3728 3742 58 2 8745 8759 8773 8787 880I 88I5 57 2 3755 3769 3783 3796 38IO 3824 57 3 8830o 8844 8858 8872 8886 8900 56 3 3837 385i 3865 3878 3892 3906156 4 8915 8929 8943 8957 897I 8985 55 4 39I9 3933 3946 3960 3974 37 981 5 5 gooo 9000149028 9042 90569070 54 5 4o00 4015 4028 4042 4055 4069 54 6i 9085 9~09991 g3 927 9I4I 9155 53 6 4083 4096 4IIO 4I24 4I37415I 53 7 9I69gI184 9I98 9212 9226 9240 52 7 4I64 4178 4I92 4205 42g 4232 52 8 9254 9268 9282 9297 93 I I9325 5 I 8 4246 426o 4273 4287 4304oo4314 5 9 9339 9353 9367 938I 9395 941o 5o 9 4328 434I 4355 4368 4382 4396 5o 10 9.9I 9424 9438 9452 9466 9480 9494 49 IO 9.9244o094423 4436 445o04464 4477 49 I I 9508 9522 9537 955I 9565 9579 48 II 449I 454 458 453I 4545 4559 48 I2 9593 9607 962I 9635 9649 9663 47 I2 4572 458645699 46 4626 464o 47 I3 9677 9692 9706 9720 9734 9748 46 I 3 4654 4667 468I 4694 4708 4721 46.4 9762 9776 9790~9804988 9832 45 14 4735 4748 4762 4776 4789 480 3 45 I5 9846 9860 9875 9889 993 99I7 44 i 5 48I6 483o 4843 4857 4870 4884 44 I6 993I 9945 9959 9973 9987... I 43 I6 4897 491 I 14924 4938 4952 4965 43 I7 9.920051 0029 oo43 oo57 007I oo85 42 I7 4979 4992 500oo6 5o9g 5o33 5o46 42 i8 oogg OI 3 OI27 14I I0:56 OI70 4I I8 5060o5073 50 87 5IOo 5II4 5I27 4I I9 o84 oIg98 02I2 0226 0240 o254 4o I9 5I41 5I54 5I68 5I8i 5I15 5208 4o 20 9.920268 0282 0296 03I o324 o338 39 20o9.925222 5235 5249 5262 5276 5289 39 2I 03520 o3660 o380 03940 o4080422 38 21 5303 53I6 5330o 53435357 537o 38 22 o436 o4500 o4640 o478 492 o506 37 22 5384 5397 54I I 5424 5438 545I 37 23 0520 o534 o548 o0562 o576 0590 36 23 5465 5478 5491 5505 55I8 5532 36 24 o604 o6i8 632 o646 o660 o0674 35 24 5545 5559 5572 5586 5599 56I3 35 25 o688 0702 07I 6 0730 0744 0758 34 25 5626 5640 5653 5667 5680o 5693 34 26 0772 0786 o8oo o84 0o828 o842 33 26 5707 5720 5734 5747 576I 5774 33 27 o856 o0869 o883 o897 09I I 0925 32 27 5788 580o 58i4 5828 584i 5855 32 28 0939 0953 0967 o981 0995 1009 31 28 5868 5882 5895 5908 5922 5225935 3 2) Io23 1Io37 io05i I65 I079 I093 30 29 5949 5962 5976 5989 6002 6oi6 3o 3,) 9.92IIo7 [121 II34 II48 II62 II76 29 30o 9.926029 6o43 6056 6069 6083 6096 29 3r II90 I204 I2I8 I232 I246 I260 28 31 6II0 6I23 6I36 6I50o6I63 6I77 28 3i I274 I2881I302 13I5 I3291 343 27 32 6196203 62I7 6236 6446257 27 33 I357 137I I385 I 399 I4I3 1427 26 33 6270 6284 6297 63II 6324 6337 26 34 I441 I455 i468 1482 I496 I5I o 25 34 635i 6364 6377 639I 64o4 64i8 25 35 I5241 I 538 1552 566 I 580 o 593 24 35 643i 6444 6458 6471 6484 6498 24 36 1 607 I621 I635 I 649 i:663 I677 23 36 65Ii 6525 6538 655i 6565 6578 23 37 I69I 1704 17x8I 732 1746 I 760 22 37 659I 66o5 66I8 663i 6645 6658 22 38 I774:1788 I802 I8I5 I829 I843 21 38 667I 6685 6698 67II 6725 6738 21 139 g I857 I87I I885 899 9 Ig2 Ig926 20 39 675I 6765 6778 679I 680 5 68I8 20 4o 9.9219401 I954 1968 I[982 I995 2009 19 40o.92683I 6845 6858 6871 6885 6898 I9 4 I 2023 2037 205I 2065 2079 2092 i8 4I 69II 6925 6938 695I 6965 6978 i8 42 21o6 2I20 2I34 2148 2I62 2175 17 42 69I 7005 7005 8 703 I 7044 7058 17 43 2189 2203 2217 223I 2244 2258 i6 43 7071 7o84 7098 7I I 1 724 7I38 i6 44 2272 2286 2300 23i3 2327 2341 I5 44 7151 7I64 7177 719 17204 72I7 I5 45 2355 2369 2383 2396 24IO 2424 I4 45 7231 7244 7257 7270 7284 7297 14 46 2438 2452 2465 2479 2493 2507 3 46 73I0o73247337 7350 7363 73771 I3 47 2520 2534 2548 2562 2576 2589 12 47 7390 74o037416 7430 7443 7456 12 48 2603 2617 2631 2644 2658 2672 1I 48 7470~7483|7496 7509 7523 7536 I I 49 2686 2700 27I3 2727 274I 2755 Io 49 7549 7562 7576 7589 7602 76I5 IO 509.922768 2782 2796 28IO 2823 2837 9 5o019.927629 7642 7655 7668 768I 7695 9 5 285I1 2865 2878 2892 2906 292o 8 5i 7708 772I 7734 7748 7761 7774 8 52 2933 2947 2961 2975 2988 3002 7 52 7787 7801 7814 7827 7840 7853 7 53 3oi6 3030 3o4313057 307I13084 6 53 7867 7880 7893 7906 7920 7933 6 54 309831I2 3I263139 3153316 7 5 54 7946 795917972 7986 7999 80I2 5 55 3I8I 3I94 3208 3222 3235 3249 4 11 55 8025 8038 8052 8065 8078 8o09 4 56' 3263 3277 3290 33o4 33i8 333i 3 56 8i0o4 8ii8 8I3 8I44 8157 8I70 3 57 334513359 3372 3386 340034I4 2 1157 8I83 8I97 8210o8223 8236 8249 2 58 3427 344i 3455 3468 3482 3496 I 58 8263 8276 8289 8302 83I5 83281 I 59 3509 3523 3537 355o 356413S78 o 59 8342 835518368 8381 8394 8407 0 60f1 50" 40't 30( 20"~ 1( 60/ 50/" 40/' 30" 20" 10" Co-sine of 33 Degrees. Co-sine of 32 Degrees. " I Part I/ 2"1 3/1 4/1 5f" 6" 7// 8" 11 P9 1.' 2" 3/" 41/ 5// 6/" 7/" 8/' 9' P.Parti 1 3 4 6 7 8 10 11 13 1 3 4.5 7 8 9 11 12 LOGAR IT HMIC TANGENTS. 81 a Tangent of 56 Degrees. A Tangent of 67 Degrees o —r.... 0 10" 20"1 30"' 40" J 50" 0"i Of 10"'20"' 30" 40"' 50" OIO.17IOI3 1058 103 Ii49 IT9g I240 59 o IO. I87483 7529 7575 7621 7667 771'- 59 1 I285 i33i I376 I42I I467 15I2 58 I 7759 7805 785 178 898 7944 799~58 2 5581 I6031I649 I6941 739 I785 57 2 8036 8082 8I28 8I74 8220 8267 57 3 ir830 1876 192I I967 20I2 2o058 56 3 83I3 8359 8405 845I 8497 8543 56 4 2Io3 2149 21942239 2285 2330 55 4 8590 8636 8682 87288774 882o 55 5 2376 2421 2467 2512 2558 2603 54 5 8866 8913 8959 9005 9051 9097 54 6 2649 2694 2740 2785 2831 2876 53 6 9143 9190 9236 9282 9328 9374 53 7 2922 2967 3o3 3o58 3I0o4 3149 52 7 9420 9467 9513 9559 96059651 52 8 3195 3240 3286 333I 3377 3422 5I 8 9698 9744 9790 9836 9882 9929 5I 9 3468 3513 3559 3604 3650369550 9 9975..21..67. II3.I60.206 50 10 0o.17374I 3786 3832 3877 3923 3968 49 10 IO.190252 0298 o344 0391 0437 o483 49 II 40I4 4060o4Io5 4 5I 4I96 4242 48 II o529 o576 0622 o6680 714 076O 48 I2 4287 4333 4378 4424 4469 45i5 47 I2 0807 o853 o899 0945 0o992 o38 47 I.3 456I 4606 4652 4697 4743 4788 46 3 I o4 I I 30 I I 77 1223 1269 I 3 I 5 46 i4 4834 488o 4925 497 5o0I6 5062 45 i4 I362 I40o8 454 I50I I547 I593 45 i5 5I07 5I53 5g199 5244 5290 5335 44 I5 I639 1686 1732I 778 1824 I87 1 4 i6 538i 5427 5472 55i8 5563 5609 43 i6 19I'7 1963 2010 2056 2102 2i49 43 17 5655 5700 5746 579i 5837 5883 42 I7 2I95 2241 2287 2334 2380 2426 42 I8 5928 5974 60o9 6065 6III 6I56 4i i8 2473 2519 2565 26I2 2658 27o 041 19 6202 6247 6293 6339 6384 643o 4o I9 2751 2797 2843 2890 2936 2982 4o 200 o. I76476 652 I 6567 66I3 6658 6704 39 20 IO.I93029 3075 31I2 3168 32I4 3260 39 21 6749 6795 684I 6886 6932 6978 38 21 3307 3353 3399 3446 3492 3538 38 22 7023 7069 7115 7160 7206 7252 37 22 3585 363i 3678 3724 377o 38I7 37 23 7297 7343 7389 7434 7480 7526 36 23 3863 3909 3956 4002 4049 4095 36 24, 7571 7617663 7708 775417800 35 24 4141 4i88 4234 4281 4327 4373 35 25 7846 7891 7937 7983 8028 8074 34 25 4420 4466 45i3 4559 460 5 4652 34 26 8I20o 865 82II 8257 8303 8348 33 26 4698 47451 79I 4837 4884 493o033 27 8394 844o 8485 853I 857718623 32 27 4977 5023 50~7~ 5ii6 562 5209 32 28 8668 87I4 8760 88o5 885 8897 3I 28 5255 5302 5348 5395 544i 5487 3i 29 8943 8988 9034 9080o9I26 917I 30 29 5534 5580o 5627 5673 5720 5766 3o 3o0 IO I7921 7|9263 93o0993549400oo9446 29 3o I o.958 I3 5859 59o6 5952 599916o45 29 3I 9492 9537 9583 9629 9675 9720 28 3I 609I 6I38 6I84 623I 6277 6324 28 32 9766 9812 9858 9904o994919995 27 32 6370 64I7 6463 65 io 6556 6603 27 33 Io. 80oo4i 0087 oI32 OI78|0224|0270 26 33 6649 6696 6742 6789 6835 6882 26 34 o3I6 o36I o407 o453 0499 o545 25 34 6928 6975 702I 7068 7114 7161 25 35 o50o o636 o682 0728 0774 o8I9 241 35 7208 7254 73 7347 7394 7440 24 36 o865o09II o957 ioo3 Io48 Io94 23 36 748717533 7580 7626 7673j7719 23 37 II4o II86 I232 I2781I3231I369 22 7"1 7766 78I37859 7906 795297999 22 38 i4i5 i46i I507 i553 I598 i644 2I 38 8o45 8092 8i38 8I85 8232 8278 21 39 I6901I736 I7821 8281I874 I919 20o 39 8325|8371 84i8848 846585II 85820 4o I0. 18I965 20I1 257 2I03 2149 2195 i9 40o 0. 9860486516 697 8744 879i 8837 19 4I 2241 2286 2332 2378 24242470 i8 41 8884 8930 8977 9024 90709I I71 I8 42 25I612562 2608 2653|2699 2745 17 42 9I64 9210 9257 93039350~9397 i7 43 2791 2837 2883 2929 2975 3021 i6 43 9443 9490 9537 9583 9630 9676 i6 44 3067|3123158 3204o3250o3296 i5 44 972319770 98169863 99Io09956 i5 45 3342 3388 3434 348o 3526 3572 I4 45 0.200oo0031o0050 oog6 O43 01900236 14 46 36i8 3664 37o9 3755 38o 3847I3 1146 0283 o330 o376 o423 o470 o5i6 i3 47 3893 3939 3985 4o3I 4077!4123 I2 47 o563 o6Io o656 0703 o750o079612 48 4I69|4215 4261 4307 435314399 II 48 o843 o890 0937 0983 I0o3oI077 I-I 49 4445 449I 4537 458314629 4674 110 49 123 I 70 I217 I263 13ioI 357 I 501 o. I8472014766 4812 4858 4904 4950 9 50 IO.20o404 4501 I497 Ix544 1591 I637 9 5i 4996 5042 5o88 5I3415 8o5226 8 5 I 684 1731 1777 I824 I87Z 19I8 8 52 5272 53I8 5364 54I o5456 5502 7 52 1964 20I 2058 21o5 2 I5 2i98 7 53 5548 5594 564o 5686 5732 5778 6 53 2245 2292 2338 2385 2432 2479 6 54 5824 587I 5917 5963 6009 6055 5 54 2526 2572 2619 2666 27I3 2759 5 55 6ioi 6147 6193 6239 6285 6331 4 55 2806 2853 2900 2947 2993 3o4o 4 56 637716423 6469 65i5 656i 6607 3 56 3087 3I34 3I8I 3227 3274 3321 3 57 6653 6699 6745 679I 6837 637884 2 57 3368 34i5 346i 35o8 355513602o 2 58 693o06976 70o227o687I1I4 7I6o I 58 3649 3696 3742 3789 3836|3883 I 59 720617252 7298 7344 739017437 0 59 39303977402344070 4II74 4 0 6y' 5" 40"' 30" 20' _10" 60Y' 50" 40" 30"' 2 10" 10" Co-tangent of 33 Degrees. _ Co-tangent of 32 Degrees. 5 /I" 2"I 3/" 41/ 5" 6&" 7// 8"// 9" 11 1" 2" 3"/ 4" 5" 6" 7" 8" 9" Part 5 9 14 18 23 27 32 37 41 P. Part5 9 14 19 23 28 33 37 42 LOGARIT E M I;SINES. Sine of 58 Degrees. Sine of 69 Degrees. 0" 10" 20" 30" 40" 50" 0"- 10" 20" 30/ 40" 50Nl 09.928420 8434844784608473848659 o 9.9330663078309 13104116 3I 2959 I 8499 85I3 8526 8539 8552 8565 58 3i4i 3i54 3I67 3I79 3192 3205 8 2 8578 859I 86o5 86I81863i 8644 57 2 32I7 3230 3243 3255 3268 32857 3 8657 8670 8683 8696 87IO 8723 56 3 3293 3306 33I8 333I 3344 3356 56 4 8736 8749 8762 8775 8788 88oi 55 4 3369 338I 3394 3407 34I9 3432 55 5 88i5 8828 884I 885418867 888o 54 5 3445 3457 3470 3482 3495 35o8 54 6 8893 8906 89I9 8933 8946 8959 53 6 3520 3533 3545 3558 357I 3583 53 7 8972 8985 8998 g90I 9024 9037 52 7 3596 36o8 362I 3633 3646 3659 52 8 9050 9063 9077 9090 9I03 91I6 5I 8 3671 3684 3696 3709137223734 51 9 9I29 9I429I55968 I8I 9I94 50 9 37473759 3772 3784 3'797 381I 50 1o0 9.9292079220.9233 92479260 927349 10 9.933822 3835384738603872388549 II 9286 929919312 9325 9338935I 48 II 38981 390 3923 3935 3948 3960 48 I2 9364 937719390 9403 9416 9429 47 12 3973 3985 3998 40II 4023 4036 47 I3 9442 9455194689482 9495 9508 46 I3 4o48 406I 4073 4o86 4098 4III 46 I4 952I 953419547 956019573 9586 45 I4 4I2314I36 4I48 4I6I 4I74 4I86 45 i5 959996I2 9625 96381965I 9664 44 15 4199 421I 4224 4236 4249 426 144 i6 9677 9699703 97i6 97299742 43 i 6 4274 4286 4299 43II 4324 4336 43 I7 9755 97681978I 979419807 9820 42 17 4349 4361 4374|4386 4399 44,1142 i8 9833 984619859 9872 9885 9898 4'I 8 4424 4436 4449 446I 4474 4486 4I I9 99II 9924:993719950o9963 9976 4o I9 4499145II 4524|4536 4549 456i 40 20 9.929989...2.I-5.. 28..4i. 54 39 209.934574 4586 4599|46II 4624 4636|39 2I 9.930067 oo80o0o93 oI0 6 oII9oI32 38 21 46491466I 4674 4686 4699 47I1138 22 OI45 OI5810I7I oi8410197 02io 37 22 4723 4736 47481476I 4773 4786|37 23 0223 0236 0249 0262 02740287 36 23 4798 48 I I 482348364848 4861 36 24 o300 03i3l03261o3391032 o365 35 24 4873 4885 4898 49r 014923 4935 35 25 0378 039I 04o0404I7 o430443 34 25 4948 4960 4973 4985 4997 50oo 34 26 o456 o46910482 0495 o5071 520 33 26 5022 5035 504715060 5072 5o84 33 27 0533 o5461o5591o572[o585 0598 32 27 509715I09 5I22 534 5I47 5I59|32 281 06 II o6241o637 o65o o663 o675 3I 28 571 71584 5I96 52091522I 52343 I 29 o688 070I107I4 0727 0740 0753 30 29 5246 5258 5271 5283 5296 53o8 30 30|9.930766 0779o0792 0804o0817 0830729 3o09.93532015333 534515358 5370|5382 29 3I o843 o856 o869 o882 o895 0908 28 3I 539515407 5420o 5432 544415457 28 32 0921 o933o946 o959 0o972 o985 27 32 5469 5482 5494155o6 5519 5 531 27 33 o998 1or I|024 Io36|I o4 I062 26 33 554315556 5568 558I 5593 56o5 26 34 o1751 o88 IIo IIII4 I1 27 1139 25 34 5681 563o 5642 5655 5667 5679125 35 Ii52 II651II78 I9II 12041217/24 35 5692 5704 57I7 5729 574x 5;,-4 24 36 I229 I2421I255 12681I281 I29/ 23 36 5766 5778 579i 58o3 58I5 5828 23 37 130o6 I3I9,332 I 345 I358 137I122 37 584015852 5865 5877 5889 5902 22 38 I383 I3961I409|I422 I435 I4481 2 38 59I4 5926 5939 595I 5963 5976 2I 39 i46o I473,148614991 512)I525120 39 5988 6000 60I3 6025 6037 6050|20 40o 9.93537 I550 I563 I576 I589 i6oi I9 4o 019.936062 6074 6087 60996, III6124 I9 4~I I6I4 I6271I64o[ 6531I6661I678 i8 41 6I36]648 6i6i 6I73 6185 6I98 i8 42 I69I 1704 I 717 1730 1742 I 755 17 4 62Io 6222 6234 6247 6259 6271I 7 43 1768 1781 I7941I806 18I9 1832 i6 43 628416296 630816320o63336345 i6 44 I845 I857JI870o1883 I896 1I09 i5 44 635716370 63821639416406641915 45 192I I9341I9471 I960 I972 I985 I4 45 643i 6443 6456 6468 648o 6492| i4 46 I99820I I 202412036/2049120621I3 46 65o5 65I7 6529 6542 6554 6566 i3 471 2075 2087 2Ioo12II312I262I38 12 47 6578 6591 6603166 516627 6640o12 48 215I1216412I7712I89 2202221251 Ii 4 6652[6664 6676|6689167oI167I31II 49 2228 2240 2253 2266 2279 229IO 1 49 6725 6738 6750o6762 6774 6787 IO 5o 9.932304 23I712329 2342 2355123681 9'l 50 9.936799 68iI 6823 6836 6848 686o/ 9 5i 2380 2393 2406 2419 243I 2444 8 5i 6872 6884 68971 6909692I 6933 8 52 24571246912482 2495 2508 2520 7I 52 694616958 6970 6982 6994/ 7007 7 53 2533 254622558 257 12584 2597 6 53 70191703I 704317056 7068 7080 6 541 26o09'2622"26352647 266026733 1 5 54! 7092 7Io4 7II7 7I29 7I4I7I53 5 55 2685 2698127I I;2724 2736 2749[ 4 55 7I65 7I78 7I907202 72I4 7226 4 56 2762 2774 2787 2800 2812 2825 3 56 7238 7251 726317275 7287 7299 3 57 2838 2850 2863 2876 28881290 2| 57 7312 7324 7336 7348 7360 7372 2 58 2914 2926 2939 2952 296412977 I 58 738517397 74o9 742174337446 59 299o03002 3oi53028 3o4o13o53 o0 59 7458 7470 74821749417506 7518 o 60"| 50"' 40" 30 20" 1 10" 1 60"t 50 40 3" 20" 1" l Co-sine of 31 Degrees. Co-sine of 30 Degrees. 8" I P. Part 1" 2" 3" 4" 5" 6" 7" 8" 9"P. Part 1 2 4 5 6 7 3456 910 1 2 4 5 6 7 9 10 1 G A R [T H MI C TAN G E N TS. 3 dI Tangent of58 Degrees. T'angent of 59 Degrees. (Y 10 20" 30" 40 1 50" _ 0" 10 20/1 30" 40"/ 50" 1 0.2042 I 4258430414351 4398 4445 59 I 0.2212261 274 I3221 3691I4I71I465'59 I ~l 4492 4539 4586 4633 4679 4726 58I I 5I 2 I56o i6o81 656 703 175I 58 2 477314820~4867 4914 961 5008 57 2 I799 i846 18941I942 1990 2037157 3 5o54 5ioi1 548 5I95 5242 5289 56 3 2085 213312I8 I2228 2276 2324 56 4 5336 5383 543o 5477 5524 557o055 4 2372 24I912467125I5 2563 26I155 5 56 I 715664 57 I I 5758 5805 5852 54 5 26581270612754 2801I 28492897 54 6 5899 5946 5993 6040 6087 6I34 53 6 2945 2993 3o4o 3o88 3r36 3I84 53 7 6 i8i 6227 6274 632I 6368 64i5 52 7 3232 3279 3327 3375 3423 3471 52 8 646265095 556 66031665o0669715I 8 35r813566 3614366213710 3757 51 9 6744 679I 6838 6885 6.932 6979 5 9 380o5 3853 390I13949 3997 4o44 5o I0o 10,207026 7073 7120 767 72I4 7261 49 I Io0. 224092 4140 488 4236 4284 4332 49 11 7387355 7402 7449 7496 7543 48 rI 4379 4427 4475 4523 457I 46I9 48 12 759017637 7684 7731 777817825147 12 4667 4714 4762 48 I 0485814906 47 I3 78721791 97966 80o38060o18I 0746 I3 4954 500215050o5098 5I45 5I93 46 14 8I 54 820I 8248 8295 8342 8389 45 14 524I 5289 5337 5385 5433 548I 45 I5 8437 8484 853i 8578 8625 8672 44 I5 5529 5577 5625 5672.5720 5768 44 I6 87 9 876688 I 3 88608907 8954 43 I 6 58I615864 5912 5 960o6008 6056 43 I7 900ooi904890o95 9I439190 9237 42 17 61o041652 6200o6248 6296 634442 i8 9284 933l 9378 9425 9472 9519141 i8 6392 644o 6488 6535 6583 663I 4I 19 9566 96I4 966I 970819755 9802 4o Ig 6679 6727 6775 6823 687I 69I9 4o 20 I0. 209849 9896 9943 999..38.. 85 39 201Io. 226967 7015 7o63 7 II 7159 7207 39 2I 10.2IOI320 I79 02n26'0273 0321 o368 38 |2I 7255 7303 7351 7399 744717495 38 22 o451 04620509 o556 0o6o3 o65I 37 |22 7543 7591 7639 7688 7736 7784 37 23 o698 0745 0792 o839 o886 o934 36 23 7832 7880 7928 7976 8024 8072 36 24 098i 1028 I075 1122 1x70 I2I7 35 [24l 8120 8I68 82I6 8264 83I2 836o 35 25 1264 i3ii 358 I4 05 453 I5oo 34 125 84o8 8456 85o4 8552 86oi 8649 34 261 547 15941 64I 1689 1736 1I783 33 126 8697 8745 8793 884I 8889 8937 33 27 I8301o878 19251I972 2019 2066132 27 898590o3390o8 19I30 917819226 32 28 2I14|2I61 2208 2255 2303 2350 3T1 28 9274[9322 9370 94I8 9466 95I5 3i 29 2397 24442492 2539 2586 2633 3o 29| 9563 96 I 9659 970719755 9803130 3o 110.212681 2728 2775 2822 2870 2917 29 30 IO.229852 9900 9948 9996..44..92 29 31 2964 30I2 305913 io6 3I53 320128 3I 0I.230o40 Io 89 0237 0285 o333 o381 28 32 3248 3295 3343 3390o3437 3484 27 32 o429 0478 o526 o574 o622 o670 27 [33[ 353213579 3626 36 74 372I13768 26 33 0719 0767 o8I5 o863 1oII o960o26 34 38I6 3863 391013958 4oo5 4552 25s 34 Ioo08I0o56 IIo4 II52 1201 1249 25 35 4Ioo 4I4',7 494 4242 4289 4336 24 135 I297 1345 1394 1442 I490 I538 24 36 4384 443 [4478 4526 4573 4620 23 l36 i586 I635 i683 173I 1779 1828 23 37 4668147I5|476214810|485714905 22 137 I87611924 I973 2021 2069 2117 22 38 4952 4999 5047 5094 5I4I 5I89 2I 38 2I66 22I4 2262 23IO 2359 2407 21 39 5236 5284 533i |5378 5426 5473 20 39 2455 2504 2552 2600 2648 2697 20 4o 10.2,5521 5568 5615 5663 57Io15758 19 4o Io. 232745 2793 2842 2890 2938 2987 19 4I 58o5 5852 5900 5947 5995 6042 i8 4jI 3o35 3c83 3I32 3i8o 3228 3277 I8 42 6090o61376184|6232|6279|6327 I7 142 332513373 3422 3470 35I8 3567 I7 43 6374 6422 646965765 64|66z2 i6 143 361 53663 37 12376o13808 3857 i6 44 6659 6706 6754 68oI 6849 6896 I51 44 390513953 4002 405014~99 4147 I5 45 6944 699I 7039 70861 7 I83 718 14145 4195 4244 4292 434o 4389 4437 I4 46 7229 7276 7324 737I 749 917466 13 46 4486 4534 4582 463i 4679 47-28 I3 471 75I41756I 7609 7656 770o4775I I2 471 4776 4825 4873 4921 4970150i8 12 48 779917846178941794 17989 8036 I I 48 506715I I5 5I64152I2 5260o53091II 49 8o8418x 3 18791822618274.83221 o1|49 5357154o6 545415503 555I 156oo1Io 50oo10.28369 84I178464 851218559 8607 9150o 10.235648 569615745157935842 5 890 9 1 SI 8654 8702 875o 8797 8845 8892 8 115I 593915987 6036 6o84 6I33j6I8I 8 52 8940|8987 903519083 9I30o9I78 7 552 623016278 6327 6375 6424A64721 7 53, 9225 9273 9321 9368 9416 9463 6 153 652I}6569 66i8 6666 67i5 67631 6 54 951 9I955S996o6 9654 97o1 9749 5 54 6812 6860o69o096957 7006 7o551 5 55 979719844198921993919987..35 41155 7Io317I52 7200 724917297 7346/ 4 5(;-10o.220082o0i 300o o 78 0225 0273 0321 3 156 739417443 749217540 7589 7637/ 3 57 o368o4I6 o463 051II 0559 o060o6 2 57 7686 7734 7783 7832 788o0 7929 2 58 o6540o7o.2o07490o7970o8450o 892j I158 7977|8026180758I23 8I7218220 I1 59 o0940o0988j1i36 io83 i3 1II791 o 59 8269 833t88366,84i5 8463 8512 o - 60-;' 50"' 40"1 30-'1 20" 10"t 60" 1 50 1 40," 1 30"- 1 20t": 10" Co-tanlgent of 31 Degreis. 1 Co-tangent of 30 Degrees. P. Part 1" 2"' 3"i 411 5" 6" 7"/ 8t 9/I P. Part / 2" 3} 41 - 6 7/1 81 9/ 5 -9 14 19 24 28 33 3 43 P. Part 5 10 14 19 24 29 34 39 43 [,': 5 9 b 4 LO GARITIMIC SINES. Sine of 60 Degrees. Sine of 61. Degrees. - 0" [ 10" 20" 1 30" 40" 50" 0" 1 10"220"30" 40" "50" o 9.93753I 7543 7555 7567 7579 7591 59 o 9.94t8I9 183I I8 43 I854 I866 I878 59 I 76 3-76I6 7628 7640 7652 7664 58 I I889 190II I93 1924 1936 1948 58 2 7676 7689 770 773 7725 7737 37 2 I959 I97I I983 I994 2006 20I7 57 3 7749 776 7773 7786 7798 78io 56 3 2029 204 205o5 2 2o 64 2076 2087 56 4 782278347834784678587870 788355 4 209912III 2I2212I3412:46 2I57 55 5 7895 7907 79I9 7931 7943 7955 54 5 2I69 2I80 2192 2204 2215 2227 54 6 7967 7979 7992 8004 80oI68028 53 6 2239 2250 2262 2273 228512297 53 7 8040 8052 8064 8076 80o88 8ioo 52 7 2308 2320 233I1 2343 2355 2366 5 8 8iI3 8I25 8I37 849 8I6I 8173 51 8 2378 2390 240I 24I3 24242436 5I 9 885 8I97 8209 822I 8233 8245 50 9 2448 2459 247I 2482 2494 2506 15 IO 9.938258 8270 8282 8294 8306 83I849 19.9425I7 2529 2540 2552252 63 2575 49 II 8330 8342 8354 8366 8378 8390 48 II 2587 2598 26IO 262I 2633 2645 48 12 8402 84i4 8426 8439 845I 8463 47 I21 2656 2668 2679 2691 2702 2714 47 I3 8475 8487 8499 85II 8523 8535 46 13 2726 2737 2749 2760 2772 2783 46 I4 8547 8559 857I 8583 8595 8607 45 i4 2795 2806 28I8 2830 284I 2853 45 i5 86I9 863I 8643 8655 8667 8679 44 i5 2864 2876 28871289912910o2922 414 I6 869I 8703 87I5 8727 8739 875I 43 I6 2934 2945 2957 2968 2980 299I 43 17 8763 8776 8788188oo00 88I2 8824 42 7 3003 3o41 3026 3.037 3049 306042 i8 8836 8848 886o 08872 8884 8896 41 i8 3072 3o83 3o95 3I 07 3II8 3I3o0 4 I9 8908 892o 8932 8944 8956 8968 4o 19 3I4I 3I53 3I6437613I87 3I99 4o 20 9.938980o8992 900490o6 9028 9040 39 20 9.9432Io13222 32331324513256 3268 39 21 9052 90o64 90769o87 9099 9 II 38 21 3279 329I 3302 33I4 3325 3337 38 22 9123 9I35 9I4719I59 9I71 9I83 37 22 3348 336o 3371 3383 3394 34o6 37 23 9I95 9207 92I91923i 9243 9255 36 23 34I7 3429 344o 3452 3463 3475 36 24 9267 9279 929I 9303 93I5 9327 35 24 3486 3498 3509 352I 3532 3543 35 25 9339 935I 936319375 9387 9399 34 25i 3555 3566 3578 3589 36o0I 362 34 26 940o 9422 9434[9446 9458 9470 33 26 3624 3635 3647 3658 3670 368i 33 27 9482 949456 9518 9530 9542 32 27 3693 3043715 37273738 375o32 28 9554 9566 9578 9590 960o 9613 3I 28 376I 3773 3784 3796 3807 38I8 3: 29 9625 9637 9649 966i 9673 9685 30 29 3830 384i 3853 3864 3876 3887 3o 3o 9.939697 9709 9721 9733 9744975629 3o 019.943899 39Io 3921 3933 3944 3956 29 31 9768 9780 979219804 98I6 9828 28 31 3967 3978 3990[ oc- o3I31 402A 28 32 9840 9852 986319875 9887 9899 27 32 4036 4047 4o58 40{iG4081j409c 27 33 99I oI0923 993519947 9959 9970 26 33 4I044115 4I27 4138 4I50o4I61126 34 99820,)94...6...8..3o.0.4225 34 4I72 4184 4I95|4207 427814229|25 35 9.940054>0o65 0077 op00890 o II3| o 24 35 424I14252 4264 4275!4286 4298124 361 OI25 fo37 oI481o060o I72 oI84|23 36 4309|432I 4332[434314355 4366,23 37 oi960208 02201 023 o0243 0255 22 37 4377 438914400o4412.442314434122 38 0267 0279 029I 0303 o3I4 0326|2I 38 444614457 4468 448o04491 45o3 21 39 o338 0350 o362 0374 o0385 0397 20o 39 45I4 4525 4537 4548 4559 457I.2o 40 9.940409 o421 o433 o445 o456 o468 19 40o 9.944582 4593 460o546I614627|4639 19 4i o480o o492 o5o4o o50 6527 0539 i8 41 4650o466i 4673 4684 4696 4707 i8 42 055i o0563 0575 o586 o598 o6Io 17 42 47I847301474514752 476414775I7 43 o622 o634 o645 1657 o669 o68i i6 43 4786 4798 4809 4820 483I14843} 6 44 o693 07o4 0716 0728 0740 07521I5 44 485414865148774888 48899|491 i5 45' 07630o775 0787 o799o08II 0822 I4 45 492249339451495614967 49791I4 46 o834 o846 o0858 0870 o88I 0893 i3 46 4995oo 500 o3 15024 5035 5o46 1I3 47 ogo5og917 o928o9g40oo952 o964 2 471 5058o5069 5080o5692 5Io35 II4 12 48 0975o0987 0999 OII 10o23 io34 ii 48 5125 5I37 5I48 1559 5171 5I82 II 49 I0o46I o58 1070 o8i io93 iio5 1 o 49 593 5204 52I605227 5238 5250 io 50 9.9411171II28 II40Ii52 II64 II75 9 50|9.94526152721 528315295530653617| 9 5i II87 II99 12 IIx222 1234 I246 8 5i 5328 534o 535i 5362 537415385 8 52 I258 I269 1281 1293x 3o4 i3i6 7 52 5396 5407 54I91543o 544i 5452 7 53 I3a8 i340o 35 iI363 1375 1387 6 53 5464 5475 5486 5497 55o0g552o 6 54 1398 I4Io 1422 i433 i445 1457 5 54 553i 5542 5554 5565 5576 5587 5 55 1469 i48o 492 i5o4| 5Ii 527 4 55 5598 56Io 5621 5632 5643 5655 4 561 539 i55o 156211574 i586 1597 3 56 566615677!5688 57oo0057II5722 3 57 I609|I62I I632 I644I 656 i667| 2 57 57331574415756 5767 5778 5789 2 58 I679 I69I I702 17141I726 1738 I 58 58oo05812 5823 583415845 5857 I 591 I749 I76I I773 1784 796 i8o8 o 59 5868 587glS8go5goI59gI3 5g24 0 60" 50" 41 40" I " 10" 60 50/ 40." 30/ 1 20". 13" H - d L.8 Co-sine of 29 Degrees.' Co-sine of 28 Degrees. P... at p 1" 2" 3" 4" 5" 6" 7" 8" 9" P 1pt''2" o'' 4" 5" 6" 7" g1 9"11 * Part 1 2 4 5 6 7 8 10 11 P. t 1 2 5 6 7 8 9 U 11.. LO G A R IT HMI TANGENTS. ii L Tangent of 60 Degrees. 9 s Tangent of.61 Degrees.' 0" 10" 20" 30" 40" 1 50" 0" 10" 20" 30" 40" 1 50"1 0 0o.23856I 8609 865818707 8755 8804 59 O IO. 256248 6298 6347 6397 6447 6496 59 ~I 8852 890o 8950 8998 9047 9096 58 I 6546 6596 6645 6695 6745 6794 58 2; 9144 9I939242 9290 9339 9388157 2 6844 6894 6944 6993 7043709357 3 9436 9485 9534 9582 963I 9680 56 3 7I42 7192 7242 729I 734I 739I 56 14 9728 9777 982698874 9923 9972 55 4 744I 7490 7540 7590 7640 7689 55 5I o. 24002 I oo6g91oI I8 016702I 50264154 5 7739 7789 7839 7888 7938 7988 54 6, 03I3 o362o04I o4591o5o8 o557 53 6 8038 80878I37 8I87 8237 8286 53 97 o605 o654 0703 0752 o8oo o849 52 7 8336 8386 8436 8486 8535 8585 52 8 0898 0947 o995 I o44 I 093 I I 42 5 I 8 8635 8685 8735 8784 883418884 5I 9 I190 I0239 I288 13371 385 1434 5o 9 8934 8984 9033 9083 9I33 9I83 50 IO 10.24I483 I532 I58I I629 I678 I 72749 IO 10.259233 9283 9332 93829432 9482 49 II I776 I825 1873I 922 1970 2020o48 II 9532 9582 9632 968I 973I 978 I48 I2 2069 2II8 2I66122 512264[23x3147 I2 983.988I 993I 998I.. 3I..8o047 I3 2362 24II 2459 2508 2557 2606 46 I3 IO.260130 OI80 0230 0o280 o330o380o 46 I4 2655 2704[2753 2801 2850 2899145 4 0430o o48o o530 o58 o62910679 45 15 2948 2997 3046 3095 3 4313 92 44 I 5 07291779 0829 o879 0929 0979 44 I6 324I 3290 3339 3388 3437 3486 43 I6 1029 I079 0I29| I 79 1229 027943 I7 3535 3584 3632 368I 373o13779 42 17 I329 I379 I429 479 I 529 1579142 IS 3828 3877 392613975 402414073141 I8 I629 I679 I729 I779 I829 I879 40 09 4I22 4171 422014269143I8 4366140 I9 19291I979 2029|2079 2129 217914( 20 10. 244415 4464145 3145624614 660|39 20 I0. 262229 2279 2329 2379242912479139 20 4709 4758 4807 4856 49o514954 38 | 2I 2529|25791262926792729277938 22 5oo3 5052 5IOI 5I50o 59915248 37 22 282912879 2929 2979 3029 3079 37 23 5297 5346 5395 5444 5493 5542 36 123 33o 3o8o13230 328o0333o 338o 36 24 559I 564o 5689 5738 5787 5836 35 1124 343o 348o 353o 358o 3630 368I 35 25 5885 5934 5984 6o33 6082 6I3I 34 25 373I 378I 383I 388I 393I 398I 34 26 6I80 62291627886327 6376|6425133 26 403i 4082 4I324824232 4282|33 27 6474 6523 6572 662I 6670 6720 32 27 4332 4382 4433 4483 4533 4583 32 28 6769 68I8 6867 69066965 7004 3i 28 4633 4683 4734J4784 4834 4884 3I 29 7063 7II21 76I72II 1 726017309130 129 4934 4985 5035 5o85 5 8513 30o IC.247357 407 7456175o0575541760429 3o 10.265236 5286 5336 5386 5436 5487 29.30 7653 7702 775I |7800 7849 7899 28 3 5537 5587 5637 5688 5738 5788 28 32 7948 7997 8o46 8095 8i44 1894 27 32 5838 5889 5939 5989 6039 6090 27 33 8243 8292 834i 8390 8439 8489 26 133 6I40o6190 6240o629I 634I 6390 26 34 8538 8587 8636 8685 873518784 25 134 644216492 6542 6592 6643 6693 25 35 8833 8882 893 1898I 90o3o9079124 35 6743 6794 6844 6894 6945 6995 24 36 9I28 9 7819227 9276 9325|93751231 36 7045 7096}7146 7196 7247 729723 37 9424 9473 9522 9572 962I 19670 22 137 7347 7398 7448 7498 7549 7599 22 38 9719 9769 98I8 9867 9906 99661 2 38 7649 7700 7750 7800 7850 79012 I 39 10.25005 o0064 01 41oI63 02oI2026I 20 139 7952 8002 8052 8Io3 8I53 82o0420 4o I0.25030I o3601o0409o 459 05o080557iJ9 4o o. 26825418304 8355 8405 8456850o6 19 4I o607 o656 o 7051 0755 o8o4 o853 8 141 8556 8607 8657 8708 8758 88o09 8 42 ogo3 og952 1oo o5I Iool I I497117 42 8859 8909g8960 90 6I 9 I I I I1 7 43 I199 0248 1I2971 347 13961 I445 6 116 43 90621922926319309364 94I4 6 44 I495 I544 I5941 643 1692 1 742 15 44 94651950 59566|96I6|9667197 7 i5 45 I791 184o 890o I 939 1989 2038 14 145 9767 98i8 9868 9919 9970..20 I4 46 2087 2037 2I86 2236 2285 2335 I3 I46 IO.27007I o I2Il 072 0222 0273 0323 I3 47 2384 2433 2483 2532 2582 263I I2 147 0374 0424 0475 0525 o 576 o626 I2 48 268I 2730 277912829 2878 29281 1I i48 0677 0728 0778 0829 08791930 I I 49 2977 3027 3076 3126 3I75 3225 IO 49 0980|Io3IIo 82I I32| I83 I233 IO 5o 10.253274 3324 337313423 3472 352I1 91150 IO.27284 i335 3851I436 I486 I537 9 5I 357I 3620 3670o37I9 3769 38I8| 8 5I I 588 I638 I689 1739 I790 I84I 8 52 3868 39I8 3967 14o7 4o661 4I6| 71152 I89 942 i993 2043 2094 2I45 7 53 4I65 4215 4264|43I414363 44i3j 611 53 2I95 22/46 22972347239 82449 6 54 4462 45I2 4561I46I 1466I|47IO| 5 54 2499 2550 260I0265I12702}2753) 5 55 4760 480914859149o81495815o081 4 155 28031285412905 2955 3006 3057 4 56 5o57 5Io71 556|5206 5256153o5 31156 3IOS83I5832o32 6o09 330336i 3 57 5355 54o4 54545155o4 5553 56o31 2 57 34I2 3463 3513 3564 36I5 3666 2 58 5652 5702 5752 58o0 585I 590I. 1|58 37I6!3767 38I8 3869 39I9:3970 I 59 595 06000o60491609916I4916I981 01159 40201407214022417342244275- 0 60" 0" 40" 30 10" 50" 40" 30/ 20"/ 10" t" Co-tangent of 29 Degrees. 1 Co-tangent of 28 Degrees. - 1" 2"' 31 4t 5 6"1 7"/ 8" 97 " 91~ 1/" 2" 3/' 4// 5," 6" 7/* 8" 9" P. art 5 10 1 250 25 29 34 39 44 P. art 5 10 15 20 25 30 35 40 45 n 6L O GARLOGARITHMIC SINE S. rSine of 62 Degrees Sine of 63 Degrees. r 0" l 1 30" t 4 50" 0" 10"1 20" 30" 1 40 50"(p 0 9. 945935 5946 5957,5969 5980 599 59 o 9.94988I 9892 9902 99I3 9924 9935 59 1 6002 6o3 6024 6036 6047 6o58 58 I 9945 9956 9967 9977 9988 9999 58 6069 6080 6092 603 6II4 6I25 57 29.9500oooo 0020003oo3oo 421oo52oo63 57 3 6I36 6147 6I59 6I70 6I8I 6I92 5-6 3 0074 oo84 oog95 o o 016 I7 [oI27156 4 6203 62I41622616237 6248 6259 55 4 oI38 O149 1~59 0170 OI 8I og1 55 5 6270 6281 6293 6304 63I5 6326 54 5 0202 o[o 3 02241o2o 34o0245 o256 54 6 6337 634816359!637I 6382 6393 53 6 0266 0277 0288 0298 0309 o320 53 7 64o4 64I51642616437 6449 646o 52 7 o330 o34i o352 o362 o373 o384 52 8 6471 6482 16493 6504 65I5 6526 5 8 0394 o405 o4i6 0426 0437 o448 5i 9 6538]6549;6560 6571 658'2 6593 So 9 o4581o469 o48o ]o49olo5oi o512 50o io 9.946604 66 51662716638 6649 666o 49 1O 9.95o522 o533 o544 o554 o565 o576 49'1I 667I 6682 6693 6704 67I5 6726 48 II o586 0597 o607 o6I8 o62g o639g48 12 6738 67491676oI677I 6782 6793 47 I2 oG50o o66I 0671 682 o693 0703 47 I3 6804 68I5 682616837 6849 6860o46 3 0o741o724 07351o746 o756 o767 46 i4 687 16882 689369~4 6915816926 45 14 07781 0788 0799 8091 o8o o83 45 I5 6937 6948 6959,6970 6982 6993 44 5 o84i o852 o862 0o873 o884 o894 44 r6 7004 70I5 70267037 7048 7059 143 I6 0905 0915 o0926 0937 0947 o958 43 17 7070~7081I7092 7P03 7II4 7I25 42 I7 o96810979 0990 1000 IOII 1102 42 i8 7I36 47 58 7I477I 70 7I81 7192 41 I 8 I032 Io43 I0o53 o64 I 074 io8S5 4I I9 7203172I47 72257236 724717258 4o I I o96 II06 III7II 271i38 II48 4 20 9.947269 7280 7291 7302 73017324 39 20 9.95II591 II70 I8o 19I 120I1 I,2 39 21 7335 734617357 7368 7379 7390 38 21I I222 I233 1244 254 I26 1 275 38 22 740I174I2 7423 7434 7445 7456 37 22 I286 1296 I307 13I7 1328 339 37 23 7467 7478 7489 7500 75II 752z236 23 349 I36o 1370 i38i I39I 1402 36 24 7533 7545 7556 7567 7578 7589 35 24 1 /2 1423 I434 I444 i455 I465 35 25 760076II 7622 7633 7644 7655 34 25 I476 I486 I497 I1507 i5I8 1528 34 26 7665 7676 7687 7698 7709 7720 33 26 I539 I549 56o i570 i58i I59I 33 27 7731 7742 7753 7764 7775 7786 32 27 I602 i6.I31i623 I634 I644 i655 32 28 7797 7808 78I9 7830 78 41 7852 31 28 x665 I676 i686 1697 1707 I718 3I 29 7863 7874 7885 7896 7907 7918 30 29 1728 1739 I749 I760 I 770 I781 3o 30 9.947929 79401795I 7962 7973 7984129 3o 9.95179II802I 81I2 I823 I833 i844 29 31 7995 8006 8017 8028 8038 8049 28 3I i854 i865 I875 I886 1896 I907 28 32 8060o807I 8082 8093 8I o4 8I 5127 32 I9171|9281I938 I949 I959 I969127 33 8I26 8I37j8148 18I59 8I7o 8I8I 26 33 I19801o9902001 20ooII 2022 2032|26 34 8192 82038213 8224 8235 8246 25 34 2043 205312064 2074 2085 2095125 35 8257 826818279 829o 83o0I.83I224 35 2io62II62126 2137 2I47 2158|24 36 832328334 8344 835568366837723 36 2I68|2I792189 2200 2210 222I23 37 8388 839984iO 84 2I84 8432 844322 37 223I 2241 2252 2262 2273 2283 22 38 8454 8464i8475 8486 8497 8508 21 38 2294|2304|2314 2325 23351234612I 39 85I9 853o0854i 8552 8562 8573 20 39 2356 2367 2377 2387 2398 2408 20 40 9.948584 85958606 861 7 862886399 Ig 40|9.952419 2429 2440 2450 2460|247I I9 41 865o 8660o867I 8682 869 93 8704 i8 41 248I 2492 2502 25 12 2523 2533 I8 42 871518726 8736|874718758187691r7 421 2544 2554[2565 2575 2585125961I7 43 8780 879I S8802 188I 2318834 i6 43 2606 2617 2627 2637 2648 2658 i6 44 8845 8856-88678-878 8888 8899 i5 44 2669 267912689 2700 2710 2720I 5 45 891018921 8932 8943 8954 8964 I4 45 2731 274I 2752 2762 2772 2783 I4 46 8975 8986)8997 9008 9019 9029 o3 46 2793 2803128I4 2824 2835 2845 13 47 9040 9051(9062 9073 9083|9094|12 47 2855 2866 2876 2886 2897129071I2 48 I05 9II60 9I27 9I38 914899. 951:9I 48 298292829382949 2959 29691II 49 9I70 9I8I 9I992 02 92I319224 0o 49 2980 2990 3000 30II 302I1303I Io 5o 9.949235 9246 9256 9267 9278 9289 9 50o 9.953042 3052 3062 3073 3o8313o93 9 5I 9300o93I0|93219332 9343193541 8 5I 31i043i4143I24 3I35 3I45|3155 8 52 93641937519386 9397940894I8 7 52, 366|3I76386 3I97 3207 3271 7 53 942919440o945I19462 947299483 6 53 3228 3238 3248 3259 3269 3279 6 54 9494 9505 9515 9526 9537 9548 5 54 3290o33oo 33io 332I 3333I334i 5 55 9558 9569 958o 959I 9602 96I2/ 4 55 3352 3362 3372 3382 3393 3403 4 56 9623/963419645|96559666196771 3 56 341x33424 343413444 3455 3~65 3 57 968819698 9709 972019731974I 2 57 347513485:3496 3506 35I613527 2 58 9752 976319774 9784 979519806] 1 58 3537]3547 3557 3568 3578!3588 59 98I679827983819849 9859!9870 0 591 359913609o36I3629 3640 365o 0 60 Y' 5" | 40/ 1 301 20" 10 60" I 50 I 40/" 30" 20" 10" Co-sine of 27 Degrees. Co-sine of 26 Degrees. P. Part 1 2 31" 41 5/ 6"1 7" 8, 9 P. 1" 2" 3" 4" 5" 6"' 7" 8" 91 1 1 2 3 4 5 7 8 9 10 1 2 3 4 5 6 7 8 9 L U G A 1 T II M IC TA. N G E'. 8 { Tangent of 62 Degrees. Tangent of 63 Degrees. (Y 1C0 20'o 302" 40'" 501 oT 10"t 20" 30" 40" 50s IO. 274326 4376 442714478 4529 4580 59 0 IO.292834 2886 2938 2990 3042 3094 59 I 4630 468r 4732 4783 4834 4885 58 I 3I46 3gg9 325 33o3 355 3407 58 2i 4935 4986 5037 5o88 5I39 5I90 57 2 3459 3511 3563 36i 5 3667 3720 57 3 5240 52I1 5342 5393 5444 5495 56 3 3772 3824 3876 3928 398o 4o32 56 4 5546 5597 5647 5698 5749 5800 55 4 4084 4I37 4I89 424, 4293 4345 55 5 585i 5902 5953 6oo4 6o55 6o55 54 5 4397 4449 4502 4554 46o6 4658 54 6 6I 56 6207 6258 6303 360o64 I 153 6 47I04763 4854867 49,I94971 53 7 6462 65I 36564 6656666677 52 7 5024 5076 5128 5I80 5232 5285 52 8 6768 689g 6870 6920 6971 7022 SI 8 5337 5389 544i 5494 5546 5598'5I 9 7073 7124 7I75 7226 7277 7328 5o 9 565o 5703 5755 5807 5859 59I2 5o 10 I. 277379 7430 7481 7532 7583 7634 49 10 Io 295964 6oi6 6o68 6I2I 6173 6225 49 11 7685 7736 7787 7838 7889 7940 48 I I 6278 633o 6382 6434 6487 6539 48 123 799I 8043 8094 8 45 8I96 8247 47 12 659i 6644 6696 6748 680i 6853 47 13 8298 8349 8400 845I 8502 8553 46 13 69o05 6958 70I0 7062 7I5 7I 67 46 |4 86o4 8655 8706 8757 88088 64886 j 4 72I9 72727324 7377 7429 748,I45 15 89I I 8962 o903 9064 91I519166 44j I5 7534 7586 7638 769I 7743 7796 44 i6 92 I 7 9268 9320 937 I 9422 9473 43 1i6 7848 7900 7953 8oo5 8o58 8I I043 I7 9524 9575 9626 9678 9729 9780 42 17 8I63 82I5 8267 8320 8372 8425 42 i8 9831 9882 9933 9984.. 36..87 4I i8 8477 853o 8582 8635 8687 8740 4i 19 IO.280I38 1O890o24o 0292 o3430394 40 Ig 8792 8845 8897 8949 9002 9054 4o 20 10. 280445 o496 o548 59g9 o65o 070139 20 10. 299107 9 I5919212 9264 93 I 7 937o039 21 o0752 o8o4o855 0906 0957 o100938 1 2 9422 9475195271958o09632 9685138 22 Io6o iiii 1162 1214 1265 i3i6 37 22 9737 9790 9842 9895 9947.... 37 23 I367 I419 1470 I52I I572 I624 36 23 Io. 300053 oo5 o58 02IO1 0263 o3iS 36 |2f4 I675 1726 I 777 I829 i88o 1931 35 24 o368 o42I o473 o526 o578 o63I 35 25 i983 2034 2085 2I37 2I88 2239 34125 o684 o736 0789 o84I o89410947 34 26 229I12342 2393 2445 2496 2547 33 26 o999 1052 II05 II57 I210 I26 3 33 27 2599 2650 2701 2753 2804 2855 32 27 i3i5 I368 I42I 1473 I526 1579 32 28 2907 2958 3oo300 o6i 3I 2 3I64 31 28 i63i i684 1737 I789 I842 1895 3 29 3215 3266 33i8 3369 342I13472 30 29 1947 2000 2053 2I6 2158 22 II 3o 30 I.283523 3575 13626 3678 3729 3780 29 30 I0.302264 2316 2369 2422 2475 2527 29 3i 3832 3883 3935 3986 4o38 4089 28 3I 2580 2633j2686 2738 279I 2844 28 32 4140o4I92J4243 4295 4346 4398 27 32 2897 2950o3002 3055 3Io8 3i6I 27 33 44491450Io4552 46o4465514707126 33 32I33266|33I9 3372 342513478126 34 4758 48io 486i 493 34964 50I6 25 34 353o 3583 3636 3689 3742 3794 25 35 5067|5 I915I70o 5222 527315325 24 35 3847 39003953 4006 4o059141I 2 2436 5376542815479S5531 55825634123 36 4I6442174270 432314376 4429 23 37 5686 5737 57891584o15 58925943 22 37 448214535 4588 464o 4693 4746 22 38 5995 6o46 6098 I50o 620I 6253 2I 38 4799 485249o5 4958 50oiI 5o64 2 I 39 6304 6356 64o8 6459 65II 6562 20139 5117 5I7o 5223 5276 5328 538I 20 40 10. 2866 146666 67I7 6769 682 I 687219 40o io.305434 5487|554o 5593 564615699 I9 4I 6924 697517027 70797I 30 7I82 I8 84I 5752 5805 5858 59I I 5964 60I7 i8 42 7234 7285 7337 7389 7440 7492 71142 6070 6123 6I76 6229 6282 6335 17 L43 7544 7595 7647 7699 775I 7802 i6 43 6388 644I 6494 6547 66oo00 6654 I6 44 7854|79o067957 8oo8o6i 8i3 IS 35, 44 6707 6760368I3 6866 699Ig6972 15 45 8i64 82I6 8268 83Ig 837I 8423 14 45 702517078 17i3I 784 7237 7290 I4 | 46 8475 8526 8578 863o 8682 8733 3 146 7344 7.397 7450 7503 7556 7609 i3 47 8785 8837 8889 894I 8992 9044 I 2 47 7662 7715 7768 7822 7875 7928 I 2 48 9096 9148 1999 925I 9303 9355 II 48 798I 8034 8087 814I 8I94 82471II 49 9407l9458195Io 9562196I419666 Ij 49 83oo08353 8406 8460o85I3 8566Io5010. 2897I8|9769|982I 9873 992519977 9 So o50 3086I918672 8726 8779 8832 8885 9 5i 10. 29002900oo8 I 32 OI84 0236 o 288 81 8938 8992 qo45 9098 1I5I 9205 8 52 o34o 0o392o444 o496 o547 o599 7 52 9258 93 I 9364 94I81947' 9524 7 53 o65I 0703 o7551 0807 o859og09 6 53 9577 963I 9684 9737 9790 98441 6 54 o963[ ioi5 o66 iii8 1I70 1222 5 1154 9897 9950o -.4..1571.I I O,64 5 55 I274 1326 I378 I43o 0482 i534 4 55 Io.3Io2I7 0270 0324 o0377 o0430 0484 4 56 I5861 638 I690 I742 1794 i846 31 56 o537 o5go o644 o697 075 0 o8o4 3 57 I 898|I 950 2002 20542 I o062 I 58 2 57 o857o9 I oo964 I017 1070~I 241 2 58 2210 2262 2314 2366 24I8 2470 I 58 II77 123I I2841 337 139I I444| I 59 2522125742626 2678|2730|2782 0 S9 I498i55i i6o5 i658 I711 1765 0 60" 50"1 40" 30" I 20" 10 1 60" 501 40/" 30"1 20" 10" Co-tangent of 27 Degrees. - Co-tangent of 26 Degrees.!I P P 1" 2" 3" 4" 5 61 7' 8 9 Part1 2 3 7 8 9. P 2rt 5 10 15 21 26 31 36 41 46 1 Pat 5 11 16 21 26 32 37 42 47 -______________________________ 1L _ ____ ____ ____ ____ ____ 88 L OGARITHMIC IN E, ~-~.d.. Sine of 64 Degrees. Sine of 65 Degrees. 0"/ 10 20/1 301" 40" 501" 1 0 10" 20/' 30"/ 40" 0 9.953660o 3670 3681369 37373712 59 o 9.957276 7286 7295 73o5 735 7325 59 I 3722 3732 3742 3753 3763 3773 58 i 7335 7344 7354 7364 7374 7384 58 2 3783 3794 3804 3814 3824 3835 57 2 7393 7403 74I3 7423 7433 7442 57 3 3845 3855 3865 3876 3886 3896 56 3 7452 7462 7472 7482 749I 750I 56 4 3906 39I7 3927 3937 3947 3957 55 4 75II 752I 753I 7540 7550 7560 55 5 3968 3978 3988 3998 4009 40I9 54 5 7570 7579 7589 7599 7609 76I9 54 6 4029 4039 4o5o 4o6o 407o0 4o8o 53 6 7628 7638 7648 7658 7667 7677 53 7 4o90 4IoI 4III 4121 43I 441 52 7 7687 7697 7707 77I6 7726 7736 52 81 4I52 4I62 4172 4I82 4192 4203 5I 8 7746 7755 7765 7775 7786 7794 51 9 4213 4223 4233 4243 4254 4264 50 9 7804 78I4 7824 7833 7843 7853 50 10 9.954274 4284 4294 4305 4315 4325 49 10o 9.957863 7872 7882 7892 7902 791]49 II 4335 4345 4356 4366 4376 4386 48 II 792I 793I 7940 7950 7960 7970 48 12 4396 4406 4417 4427 4437 4447 47 I2 7979 7989 7999 8009 8oi8 8028 47 13 4457 4468 4478 4488 4498 45o8 46 13 8o38 8047 8057 8067 8077 8o86 46 I4 451 845294539 45494559 456945 I4 8096 81o6 815 8125 8I35 8145 45 I5 4579 4589 4600 46io 4620 4630 44 i5 8I54 8164 8174 8I83 8193 820344 i6 464o 465o 466I 467I 468I 469I 43 i6 82I3 8222 8232 8242 825I 826I 43 17 470I 47II 472I 4732 4742 475242 I7 827I 8280o829013083 830983I42 i8 4762 4772 4782 4792 4802 4813 4 i8 8329 8339 8348 8358 83688-377 4I 19 4823 4833 484314853 4863 4873 4o I9 8387 8397 840618416 8426 8435 40 20 9.954883 4894 4904 49I4 4924 4934 39 20 9.958445 8455 584648474 8484 8493 39 21 4944 4954 4964 4974 4985 4995 38 2I 8503 85I3 8522 8532 8542 855I 38 22 5005 5oi5 5025 5035 5045 5055 37 22 856I 8571 8580o8590o860oo8609 37 231 5065 5075 5o86 5o96 5Io6 5II6 36 23 86986288638 86488657 8667 36 24 5I26 5I36 5I4615I56 5I66 5I76 35 24 8677 8686 8696 8706 875 8725 35 25 5I86 5I96 520715217 522715237134 25 8734 8744 8754 8763 8773 8783 34 26 5247 5257 5267 5277 5287 5297133 26 8792 8802 8812 882I 883i 884o 33 27 5307 53I7 532755337 5348 5358 32 27 8850o 8860o 8869 8879 8888 8898 32 28 5368 5378 5388|5398 5408 54I8 3I 28 89o0889178927 8937 8946 8956 3 29 5428 5438,5448 5458 5468 5478 30 29 8965:8975 8985 8994 90oo 4 go3 30 3o 9.955488 5498 55o8 55i8 55.28 553829 30o 9. 959023o33 931 042o9052o 961 29 3I 5548 5559 556915579 5589 5599'28 3I 9o80g9090 9~ 9100ogI 9II I28 28 32 56095619 5629 5639:5649 565927 32 9138 9148 9157 9167 9176 9I8627 33 5669 5679 5689 5699 5709 571926 33 9I95 9205 9259224 9234 9243 26 34 5729 5739 5749 5759 5769 5779 25 34 9253 9262 9272 9282 929I 930I 25 35 5789 5799 5809 58I95829 5839 24 35 93Io 9320 9329|9339 9348 935824 36 58 5869 5879 5889 589 9 23 36 9368 9377 9387 9396 9406 94I5123 37 5909 59I9 5929 5939 5949 5959 22 37 942519434 944419453 9463 9473122 38 5969 979 579 5989 5999 6009 6o 2I 38 9482:9492 9501 95II 952095302I 39 6029 6039 649 6o059 6069 6079 20 39 9539 9549 9558 9568 9577 9 587120 40o9.956089 6099 6I 0816II86I28 6I38 19 4o 9.959596~9606 96i5 9625 9634 9644 19 4.I 6I48 6I58 6i68 6178 6I88 6I18 i8 4I 9654 9663 9673 9682|9692 970I i8 42 628 6o8 68 6228 62386248 6258 I7 42 97II9720 97309739 9749 9758 17 43 6268 6278 6288 6298 6308 63I7 i6 43 9768 9777 9787 9796 9806 98I5 i6 44 6327 6337 6347 6357 6367 6377 I5 44 9825 9834 9844 9853 9863 9872 i5 45 6387 6397 6407 64I7 6427 6437 I4 45 9882 9899900oo 9910 9919 9929 14 46 6447 6457 6466 6476 6486 6496I 3 46 9938 9948 9957 9967 9976 9986 13 147 65o6 65i6 65266536 6546 6556 12 47 9995... 5.. I414..24..33..43 I2 48 6566 6575 6585 6595 660 5 66I5 II 48 9.960052|oo6I OC7I1oo8o oog9ooo09 II 491 6625 6635 6645S66556665 66741Io 49 oi90 o II8o01280137 o01470 o56 IO 5o 9.956684 6694 67o4 6714 6724 6734 9 5o 9.96o165 oI75 oi84 oI94 02o3o23 9 5i 6744|6754 67631677316783 6793 8 5I o222 0232 024I 025o 026o 0269 8 52 68o3 6813 6823 6833:6843 6852 7 52 o02790288 0298o3o3070317 326 7 53 68621687216882 689216902 69I2 6 53 o335 o345 o3541o364 0373 o382 6 54 692I1693I1694I 695I 696I 697I 5 54 0392 040I o4II o42oo 430 o439 5 55 698I16990 7000 7010 7020 70303 4 55 o448 o458 0467 4771o o486 o495 4 56 7o4o7o05o7o59|7o69|70 79 7089 3 56 o5051o5I4 05240o533 05420552 3 57 7099 7IO9 7118712871i38 7148 2 571 056I|O57I|0580OO589|O599 o608 2 58 7158716871777I877197 7207 I 581 o618 0627 0636o6460655 o665 I 59 7217 7227 7236 7246 727 7266 o 591 0674 o683 0693107021O7II 072 60" 50" 40" 30(Y 20" 10"11 60'1 50"S - 40" 30S"! 20" 10"' Co-sine of 25 Degrees. Co-sine of 24 Degrees. P. Part I 1 2 3 4 5 6 7 8 9 P. Part 1 2 3' 6" 7 8" 9 13 5 6 789.r9ir 1234567' 89 LOG ARITIIMIC 1'ANGENTS..8.El Tangent of 64 Derees. Tangent of 6_ Degrees. 3 0"!1 10" 20" -30" 40"1 1 50" j 0" 10"o 20/1 30" 40"- 50'" O IO.3I18IS I872 I925 1979 2032 2085159 o Io.3I327 I382 I437 1492 I547 I602 59 I 2139 2I92 2246 2299 2353 2406 58 1 1657 I712 1767 I822 I877 I932 58 2 2460 25I31256712620 2674 2727157 2 19871204212097 2153 22082263 57 3 2781I 2834 2888 294I 2995 3o48 56 3 23I8 2373 2428 2483 2538 2593 56 4 3 3Io1 355 3209 3263 33I613370o55 4 2648 2703 2758 28I3 2868 2924 55 3423 3477 3530 3584 3637 369I 54 5 2979 3034 3089 3I44 3I99 3254 54 6 3745 379813852 3939539 59 401353 6 3309 3364 3420o3475 35303585 53 7 4066 4I20 4I73 4227 428 I 4334 52 7 364o 3695 375I 38o6 386I 39I6 52 8 4388 4442 449514549 46o034656 5i 8 397,1402614082 4I37 4192 4247 5I 9 4710 4764148I7 4871 492514978150 9 4302 4358 44144 68 453234579 50 io io.315032 5o861513915193 5247 5300o49 Io Io. 334634 4689 47441480o 4855 49O 49 II 5354o5408 546i 55-I5 5569 5623 48 1i1 4965o21 15076 5I3I 5i86 5242 48 12 5676 573o 5784 5838 589I 5945 47 I2 5297 5352 54o8 5463 55I8 5574 47 I3 5999 6o53 6Io6 6I6o 62I4 6268146 13 5629 5684 574o 5795 585 59o6 46 I4 6321 6375 6429 6483 6537 6590 451 4 596i 6o6 6o072 627 6I82 6238 45 I5 6644 6698 6752 68o6 6860o69I3144 5 6293 6349 64o4 6459 65I5 657o144 I 6 6967 702I17 757 729 7183 7236 43 i6 6625 668I 6736 6792 6847 6903 43 I 7 7290 7344 7398 7452 7506 7560142 I 7 6958 7013 7069 7I24'7180 7235 42 I8 7613 7667 7721 7775 7829 7883 41 I8 729I 7346 7402 7457 7513 7568 41 19 7937 799I 8o45 80991 853 8206 4o 19 7624 7679 7735 7790 7846 790, 4o 20 i0.3I8260 8314 8368 8422 8476 853o 39 20 o. 337957180I2 8068 8I23 81798234 39 21 8584 8638 8692 8746 88oo 8854 38 21 8290 8345 84oi 8456 8512 8568 38 22 8908 8962 90I6 9070 9I24 9178 37 22 8623 8679 8734 8790 8845 890I 37 23 9232 928619340 9394 9448 9502 36 23 8957 90oI29068 9I23 9179 9235 36 24 9556 96IO19664 9718 9772 9826 35 24 9290 934619402 9457 9519956935 25 9880 9934 9988..42..96.I51 34 25 9624 9680 9735 979I 9847 9902 34 26 1o.320205 0259 o3I3 o367 042I 0475133 26 9958..I4. 70.125.i8 11,237 33 27 o529 o583 o637 o692 0746 o8oo 32 27 Io.340292 o348 0404 46o o05I5 o57I 32 28 o854 o9o8 o962 ioI6 107I 1125131 28 06270 o682 0738 o794 o85 oo9,o 31 29 1179 I2331I287 I34i 1396 I45o 30 29 096I 10171I073 II29 II84 I24 S3o 30 -o 32I 5o4 I558 I612 i666 I721 I775 29 3o0o.3412o6 13521 4o08 I463 i591 I575129 31 1829 i883 I938 I992 2046 21OO 28 31I I63I 687 1742 I798 I854 19I1O28 32 2I54 2209 2263 2317 2371 2426 27 132 966 2022 2078 2 33 2189 2245 27 33 2480o 2534 2588 2643 2697 275I 26 133 230I 2357 2413 2469 2525 2581 26 34 2806 2860 29I4 2968|3023|3077 25 34 2636 2692 2748 2804 2860 29I6 25 35 3I3i 3i86 3240 3294 3349 349134 24 35 2972 30283084 3i40o 3I96 3252 24 36 3457 3512 3566 3620o367513729 23 136 3308 3364 3420 3476 3532 3588 23 37 3783 3838 3892 3947400I 14o55 22 137 3644 370013756 3812 3868 3924 22 38 41o 41i6414219142731432714382 2I 138 3980o403614092 4148 4204 4260 21 39 4436 449I14545 4599 4654 4708 20 39 43i6 4372 4428 4484 454o 4596 20 40o 10.324763 481714872 492614981 5o35 19 4o0lo. 344652 4708147641482I 4877 4933 I9 41 5o89 5I44 5I98 52535353075362 8 414 I 4989 5045 5Ioi 5I57 5213 5269 I8 42 54I6 5471 5525 558o 563415689 17 ]42 5326 5382 5438 5494 5550o 5606 7 43 5743 5798 5852 5907 5962 6016i6 143 5663 57I9 5775 583I 5887 5943 1I6 44 6 07I61251 6i8o 6234 6289 6343 1511 44 6000 6o56 6I112 6i68 6224 6281 i5 45 6398 6453 6507 6562 66I6 6671 I4 145 6337 6393 6449 65o6 2 6668 I4 46 6726 6780o6835 68891694469991 I3 46 6674 6731 6787 6843 6899 6956 13 47 7053 7o1087I62 172 72172 7326 1I2 47 70I2 706817125 7181 7237 7293 12 48 7381 7436 7490 754517600l7654 II 1148 7350 74o067462 75I9 7575 763|1 I 49o 7709 7764 7818 7873 7928 7982 o 1149 7688 7744 7800 7857 7913 7969 10 5oI 0.328o37 8o.9281 47 820I 8256|831 I 9 5o 1o0.348026 8082 8139 8195 8251 8308 9 51 8365 8420 8475 853o 8584 8639 8l 5I 8364 8421 8477 8533 8590o8646 8 52 8694 87491880318858189I3/8968 711 52 8703 8759188I5 8872 8928 89851 7 53 9023 9077|1932 91871924299297 6| 53 904I 9098|9154 9211 I9267~9324 6 54 9351 9406I9461 95161957I19625 5 154 93801943719493 9550 96o069663 5 55 968o973519790 9845990oo9955 41155 97I9 977698328 889 9945...2 4 56 IO.330009 00oo 64o09 OI74|022910284 3 1561 o.35oo58 OII5I017I1 o28 0285 o34I 3 57 o339 o3941o448o5o31o558 1o6331 2| 57 o398 o4541oSIi o567 o624o68I 2 58 0668 072310778 08330o888|0943 I 158 0737 07941085o0907 o964O 1020 I 59 ogg998 o 531O18 I631I2I181272 11591 1077113411901 24713o04360 0 60I 50" 40" 30" 20" 10" 60" 50" 140" 30" 20 10" Co-tangent of 25 Degrees. G Co-tangent of 24 Degrees. p 1" 2" 3" 4" 5" 6" 7I" 8' 9" 1 i 1 2" 3'1 4" 5" 6" 7"1 8"' 9" ar 5 11 1,2 27 33 38 43 49 P.Part 6 11 17 22 28 33 39 45 59 0 L LOGARI T H M INES.;. | Sine of 66 Degrees.. Sine of 67 Degrees. __- _ _ 10"1- 20" 30" 1 40" 50/ 0 1 10" 10" 30"1 40" 50" o0 9 960730 0o740 o0749 0758 9768 07770 O 9.964026 4035 4o44 4053 4062 407I 59 I 0786 0796 o805 08I4 o824 o833158 I 4080 408914098 4Io6 4I I5 4124 58 2 o843 o852 o86I 087I o880 o889 57 2 43 4334421 5141 6o41694I 78 57 3 o899 ogo8 0917 0927 o936 1945 56 3 4I87 4I96142o0 4224 4222 423I 56 4 0955 o964 0973 o983 0992 102 55 4 4240 4249 4258 4267 4276 4285 55 5 IOII o020o I0o3o1I039g Io48 o58 54 5 4294 4303 43II 4320 4329 4338 54 6 Io67 1o76 io861 o95 1Io4 1III353 6 4347143561436514374 4383 4392 53 7 II23 1132 II4 I I5 I I601 I I69g 52 7 4400 44o9 4AI 4427 4436 4445 52 8 II791 88 II97 12071 I261 225 5I 8 4454144631447I 448o 448914498 51 9 I235 I 244 I253 1263 I272 I281 50o 9 4507 45I6 4525 4534 4542 455I 50 9og.g6I1290 1300 1 I3I9318 1328 I337149 I o9.96456o 4569 4578 4587 4596 46o4 49 I I I 346I356 I 365 I374 I 383 I 39348 I I 46 I 3 4622 463 1464o0464914658148 2 1402 I4II 1421 i43o I439 I448147 12 4666 46751468414693 4702147I 147 13' I458 I467 I476 i485 i495 I5o446 I 31 4720o4728 473714746 4755 4764 46 I4 I5I3 1523 I532 I54i i550o 560 45 I4 4773 478 I 4790 4799 48o8 48 I 7 45 i5 I569 I5781I5878 I597 I6o616i5 44 i5 4826 4834 4843 4852 486 14870 44 i6 1624 i6341 643 I652 I66I 1671 43 16 4879 4887 4896 4905 49I4 4923 43 17 I68o I689 I6989 I 708 I7I7 1726142 I7 493I494494494995849671497542 i8 I735 I745 I7541I763 I1772 1782 4' I8 4984 4993 5002 50II 502o 5028 4I 19 I79I I8oo 880 o9 I8I9 828 I 837 4o I 5037 5o4650o55 5o64 50721 50o8 4o 20 9.9g6846 I856,I8651I874 i883 I892 39 20o9.965090509915 o75I I16 5I255I34 39 2I I902 I9I I 920 I929 I939 1948 38 21 5I43 5i5i 5I6o 5I6g 5I78 5187 38 22 I957 i966 I975Ig985 I994 2003 37 22 5I9515204152I3 5222 523523523937 233 2012 202I 203I 12040 204912058 36 23 5248 5257 5266 5274 5283 5292 36 24 2067 2077 2o86i2o95 21o04 213 35 24 53o0I 530953I8 5327 533615344 35 51 2I23 2I32 2I41 2501 2I5912I69 34 25 5353 5362 537I 5379 5388 5397 34 z.6l 21i78 2I87 2I96 2205 2214 2224 33 26 54o6 544 15423 5432 544 15449 33 277 2233122421225I 2260 226912279 32 271 5458 546715476 5484 5493 5502 32 28 2288 2297 2306 23I512325 2334 31 28 55I I 55i9 5528 5537 554615554 3 29 2343 2352 236I 2370 2379 2389 30 | 291 5563 5572 558o 5585589 55985607 30 3c9.962398 2407 24I6 242512434 2444 29 309.965655 656245633 5642 565o 5659 29 3I 2453 2462 247I12480 248249249828 31I 5668 5676 5685 5694 5702 57 I 28 32 2508 251712526125351254412553 27 32 5720o57291573715746 575515763 27 33 2562 2572 2581s2590 2599 2608 26 331 5772 5781 5790 5798 5807 58I6 26 34 26I7 2626 263512645 2654 2663 25 34 5824 5833 5842 585o 5859 5868 25 35 2672 2681 2690 2699 27o0827I7 24 35 5876 5885 5894 5902 59II 5920 24 36 27271 2736 2745 2754 276312772123 36 5929 5937 5946 5955 5963 5972 23 37 278I12790 279912809 28I8 2827 22 37 5981 5989 5998 6007 60oi56024 22 |381 2836 2845 285412863 2872 288I/2I 38 6o33 6o4 6o050o 60596067 6076 21 39 2890o289912909 29I812927]2936]2o 39 6085 60931 6IO2|61III|6I96I28 20 40 9.962945 2954 2963,2972 298I 2990 19 ] 40o 9.96636 6I45 6I54 6I62 617I 6I8o 19 4I 2999 3008 3018'3027 3o36 3o45 i8 4I 6I88 6197 6206 6214 6223 6232 i8 42 305413o63 307230oI 13090o30o99 17 142 624 6249 6257 6266 6275 516283 I7 43 3Io8|3II71 3263I35 344 31353 3 i6 43 6292 630I 6309 63I8 6326 6335 i6 44 3I63 3I721 3i8 3I90o3I99 3208 i5 144 6344 6352 636i 6370 6378 6387 i5 45 3217 3226 323513244 3253 3262 I4 |145 6395 6404 64I3 642 1643016438 i4 46 327I1328013289 3298 3307 33i6 i3 46 6447 6456 6464 6473 64826490o I3 47 3325 3334 3343 3352 336i 3370 12 47 6499 6507 65I66525 653365 542 I2 48 3379 3388 3398 3407 34i6 3425 Ii 48 6550o 6559 65676576 65851 6593 II 49 3434 3443 3452 346i 347013479I 11o 491 6602 66io 66I9 6628 6636 6645 Io 50 9. 963488 3497 35o6 3 56 5 3524 3533 9 5o 9.966653 6662 667066079 6688 66961 9 5i 3542 355I 356o 3569 3578 3587[ 8 5i] 6705 671316722 673o06739 6748 8 52 3596 3605 364 33623 3632 364I 7 521 6756 6765 6773 678216790 6799 7 53 3650o 3659 3668 3677 3686 3695 5 53 6808 68i616825 6833 6842 685o 6 54 3704 37337223730373937393748 5 54 6859168671687616884A689316902 5 55 3757.3766 3775[3784 37933802[ 4 { 55 690o 69I916927 693616944 6953A 4 56 38i I 3820o 38293838 3847 3856[ 3 56 696I16970o6978169876995 74oo 4 1 3 57 3865 3874 3883 38892 390 39o10 2 571 70I3 7021 7o3017o38 7o47 7o55 2 58 39I9 3928 3937 3946 3955]3963 I| 581 7064 70721708I 7089 7098 17o6 I 59! 39721398I1 3990399900oo840 I71 o 59 7115 712347I327I14017497I57 0 60"1 50 40/ 3(Y' 20" 10 60"- 501 40,30/ 20l 10" Co-sine of 23 Degrees. Co-sine of 22 Degrees. I..pa rtI 1" 2/ 3// 4/ 5/ 6" 7"1 8"' 911 p 1 2'/ 3/' 4'/ 5/i 6" 7/i 8" 9Y1 I P art 1 2 3 4 5 5 6 7 8 P. art 1 2 3 3 4 5 i 78 LO G AR I T H 1C TANGE N TS. 91 S Tangent of 66 Degrees. I Tangent of 67 Degrees,; 0" n10 20" 30" 40" 00 10 20' 30"l 40' 50"1 -IO.3514I 7 I474 I5301 I587 1644I 700 59 oIO.372148 2207 2265 32412382 244 I 59 I 1757 I8I~ I8770 r927 1984 2040 58 I 2499 2558 261712675 2734 2792 58 2 2097 2I 541221 2267123241238I 57 II 2 2851I 29I o1296813027 3o85 3i44057 3 2438 24942551 2608 2665272 I 56 3 3203 326I 3320 3379 3437 3496 56 4 2778 2835 2892 2949 3oo5 3062 55 4 3555 36I3 3672 3731 3789 3848 55 5 3I1I9 376 32331329013346 34o0354 5 3907139654024 40834I424200 54 6 3460 3517 3574 363I 3687 3744 53 6 4259 43I8 4377 4435 4494 4553 53 7 38oi 385813951 3972 40294086 52 7 46124670o47.29 4788 4847 4906152 8 4I43 4I99 4256 43I3 4370~4427 5I 8 4964 502315082 5 1520oo5258 5i 9 4484 4541 4598 46551471214769 50 9 5317 5376 543515494 5553 5612{50 I O10 354826 48834944997 5o54 5 I i 19 10 10.37567 572915788 5847 59o6 596549 II 51681522515282 5339 5396 5453148 I 6024 6083 6I426200 625916318|48 12 Sio 55o 67 565624 568 5738 5795 47 12 6377 6436 6495 6554 66i3 6672|47 I3 5852 5909 5966 6023 6o8o0I 37 46 1I 3 6731 6790 6849 6908 6967 7026 46 i4 6194 6251 63091 666 6423 648o045 I4 708517144 72031726217321 7380145 i5 6537 6594 6651 6708 6765 6823 44 I5 7439 7498 7557 7616 7675 7734144 I6 6880o 6937 6994 705I 7Io81 766 43 i6 7793 7853 79I2 79797i8o3o80 89 43 17 7223 7280 7337 7394 7451 7509 42 17 8i48 820718266 8325 8384 8444 42 i8 7566 7623 7680 7737 7795 7852 41 I8 8503 8562 8621 868o 8739 8799|4I I9 7909 79i6680o24808 1I 8 I 38|8 I 95 40 1 9 8858|89I 7897619o3519o949I 5414o 20 0. 358253183Io18367 8425 8482 8539 39 20 10o.3792I3 92721933I|93909o450o9509139 21 859618654]87 I I8768 8826|8883138 21 9568 9627|9687|9746 98o059864|38 22 8940|8998|9055|9II2 9I70 9227 37 22 992499983t..42 Io02.i6i.220137 23 9284 9342]939919456195I41957I 36 23 o. 380280033903939810457051710576[36 24 9629 9686 9743 9801 9858|9916 35 24 o636 o695 0754 o84o 087310932135 25 9973..3 II..881. I45.203.26034 25 0992t io5i I10 II70l I229 1289 34 26 I. 360o38 037510433 o4900o5481o6o5133 261 I3481 407 1467 1526 15861 6451 33 27 0663 0720 0778 o835 0893 0950o32 27 1705 1764 18241 883 1943 2002 32 28 i00oo31 Io65 1123II80o 1238 129531 28 206 2121 2180 2240 2299 2359 3I 29 I353 4io I468I 5251 5831 64i 130 29 24I8 2478 2538 2597 2657 27I6 3o 3o o.36I698 I756 1813 I871 I928 I98629 13o 0Io.38277612835 289512954 30I413074129 3 20442 II 2 I 5922 17122742332o281 3I 3 31333 I 933252133 I 233721343 I28 32 2389 244712505 2562 2620 2678 27 132 3491 355o 36io 3670o3729 3789 27 33 2735 2793 285 12908 2966 3024126 j 33 3849 3908 3968 4028 4087 4147 26 34 3o08I 3393197 3255 3312|3370|25 341 4207|4266|4326|4386|444514505|25 35 3428 3486 3543 36oI 3659 37I724 1135 4565 4625 4684 4744 48o4 4864 24 36 3774 3832 383238903948 4oo54o6323 36 4923 4983 5o43 15o3 5I62 5222 23 37 4121 4179 4237 4294 4352 44Io 22 1 37 5282 5342 5402 546i 1552I1558I 22 38 4468 4526 45841464I 4699 4757 2I 138 564I1570i 5761 5820o 5880o 5940 21 39 4815|4873|4931 4989 5046 5io04 20 39 60ooo6o60 61206180o 624o 6299 20 40o 0.365I62 52201527815336 5394 5452 19 4o~Io. 386359164I9i64791653916599'66591i9 41 55io 5568 5626 5684 574I 5799 i8 4i 67I9 6779 6839 6899 695970I9 9 I8 42 5857 59I5 5973 6o3i 689 61I47 I7 42 707917 I 3917 7199 7259 73 I 9 7379 I 7 43 6205 6263 632I 6379 6437 6495 16 |43 7439 7499 7559 76I9 7679 7739 i6 44 6553 66ii 6669 6727 6785 6843 15 44 7799 7859 79I9 7979 8039 8099 i5 45 6901 696o0 7OI 87076 7134 7192 i4 45 8159 82I9 8279 8339 8399 846o 14 46 7250 73081736617424 748217540o13 |46 8520 8580 864087oo00876o08820 13 47 7598 7657 775 7773 7831 7889 1I2 47 888o 894I 9oo I0906I 9I2I 9I81 12 48 79478oo005806418I 22 8I80o8238 I I 48 924 1930219362|94221948 219542| 1 49 8296 8354 84Ir3847I 8529 85871 Io 49 9603 9663 9723 9783 9844 9904 o 5o Ib.368645|8704|876218820o8878|8937 9 11 Io.389964..24..85.I45.205.265 9 5I 8995|905391 I I19I701922819286 811 5 Io. 390o326o386446o050 7 o567o0627[ 8 52 9344 9403 946I 9591 9578 9636 7 152| o688 0748 o8o8 o86910929 0989 7 53 9694{975398 I 19869 9927|9986 6 53 io50oIIIO 1170 I23129I11352 6 54 [0o.370044 1I3 01I6I 102I9 0278|0336 5 154 1412 I472 i533 1593 16541 I714 5 55 0394 o453 o5IIIo569 0628 o6861 41155 I7751 835 189 956 201612077 4 56 07450o8030o8620o920o09781IO37[ 3 156 213712I98[2258[231912379 2440 3 57 IO95 ii54 122 1|271 1329 i388 2 57 2500 2561 262I12626822742 2803 2 58 I446I50o41 563 1 62I I68o 1738| I 158 2863 2924 2985 3o45 3Io6 3i66 I 59 I797 i8551I914 I972 2o3I 209o0 0 o591 32271328713348 3409 3469 353oj 0 60" 50" 40' i306 10" 10" = 40 f 3060" 100 40 d" Co-tangent of 23 Degrees. 1:t G Co-tangent of 22 Degrees. a PPartl 1" 2" 31" 4/" 5/" 6" 7" 8" 9" p. Part It' 2" 3/ 41" 5" 6" 7" 8g" 9' 6 12 17 23 29 35 40 46 52 12 6 18 24 30 36 42 48 54 L. AR 11 IIM SINES. Sine of 68 Degrees. 1 Sine of 69 Degrees. " Oi 1011 20" 30" 4011 50" 01 10" 20" 30" 4011 50"/ o09.967i66 7174 783,7I9I 7200 7208 59 09 970I52 016001168 OI76 OI84 OI92 59 I 7217 7225 7234 7242 725I 7259 58 I 0200 0208 0216 0224 0233 0241 58 21 7268 7276 72857293 7302 730o 57 2 0249 0257 0265 0273 028 0o289 57 3 73i9 732173 7336447353736I 56 3 0297 o3o5 o3I3 o32I o329 0337 56 4 7370 7378 7387,7395 7404174I2 55 4 o345 o353 o36Io 0370 o378 o386 55 5 7421 7429 7437 7446 7454 7463 54 5 o394 0402 o4I o o4i8 0426 o434 54 6 747I 7480 7488i7407 75o5 7514 53 6 o442 o45o o458 o466 o474 o482 53 7 7522 7531 7539;7547 7556 7564152 7 0490 o498 o5o6 0o54 0522 o53o 52 8 7573 7581 7590:7598 7607 76I 551 8 o538 o546 o554 o562 o570 o578 5I 9 7624 7632 7640 76491765717666150 9 o586 o5941o63 o 6 I I o6 9 o627 5 Io 9.967674 7683 7691 7699 7708 776 149 0 9 970635 o643 o65 I o659 o667 o675 49 II 7725 7733 774217750 7758 7767148 II o683 06gI o699 0707 07I5 0723 48 I2 7775 7784 77921780I 78099 1 78147 I2 073 0739 0747 0755 o763 077I 47 I3 7826 7834 7843 785I 7859 7868 46 I3 0779 0787 795 o8o3 o8 Ig o819 46 I4 7876 7885 17893 7901 79I17918 45 I4 o827 o835 o842 o85o o858 o866 45 i5 792717935 7943 795217960 7969144 5 o874 o882 o890 o898 og9o6 oq91444 i6 7977 798 579948002 80 I I 80 I 9 43 I6 0922 0930o o938o946 o954 o962143 I7 8027 803680o448053806I 806942 17 0 978 o986 0994 I002 IooI042 I8 807818o86:8094 8Io3 8I II 8I204I I8 Io1811026 1o341042 Io5o Io5814I I91 8I2818I36i84518I5318I6I18I7014o Ig9 o6611073 io8i s089 I097 iio5 4o 20o9.968 r78 8I8718I95S8203 82I218220o39 20 997II I3112I I29 1137 1I451 I53139 21 8228 8237 82458253 8262 81827o 381 2I I I6I I691177 ii85 I13 I200 38 22 8278 8287 8295 833 8832 8320 37 22 I20812 I6 I224 I232 I 240 I248 37 23 8329 8337 8345 8354 8362 837o 36 23 I256 1264 1272 I 280 I288 I 296 36 24/ 8379 8387 8395 8Ao4 84I2 8420 35 1 24 I303 I3ii I39 I327 1 3335 i343 35 25 8429 8437j8445 8454 8462 8470 34 25 I35i I359I 367 I375 I383 1390 34 265 8479 18487849581 2 1852o 33 26 I398 i406 I4I4 I422 I43 I438 33 27 8528 8537 8545 8553 8562 857o 32 27 I446 i454 1462 1469 I477 i485 32 28 8578 8587:8595 8O1863 2 8620 3I 28 I493 i5o0I I509 15I7 1525 I532 31 29 8628 86368645 86531866i 8670 30 29 I540 I 548 i556 I564 1572 i58o 30 30o9 96867818686 86948703 87187I 87929 30 9.97I588 [5951I 60o3 6 I II6I9g 627 29,3I 8728 873618744875762 876118769 28 3I I635 I 6431 65i i658I 6661 674128 32 8777 878618794t8802 88IO188I9 27 32 I6821 6901I698 I706 I7131 72I 27 331 8827 883518844 8852 886o 886 26 33 I729 17371 745 1753 176I I768 26 34 8877 888518893 890I 89Io 89 1 25 3 4 1776 I 7841 I792 I8oo00 I 8o8 185 25 351 8926 8934/8943 8951 8959 8967 24 35 I823 I83 18391I847 i855 I,862 24 361 8976 898428999290q00 9009 907123 36 I870 1878 i886 1894 1902 1909 23 37 9025 9033,90421 90509058 9066l22 37 I917 19251I933 1941 1949 I956 22 38 9075 90go83909I 9099 gIo8I 616 21 38 1964 I972 I980 1I 988 I995 2003 12I 39 9124|9I3219I4I19149 1I57 9165 20 39 20I I 120I 920271203412042 2050120 4019.969173|9I829 g90g9I9819206 925 i9 40 9.97205812066 2073 208I 208920997 I9 41 | 9223 923I 92391924719256 9264 I8 4I 2io05 2II22I o20s28 2I362I441I8 42 9272|9280 9288 9297 9305193I3|17 1 42 2I51 2I59 21671z175 2I8312I90l17 43 932I 9329 9338 9346 9354 9362 i6 43 2I98 2206 22I4 222I 2229 2237 i6 44 9370 9379 9387 9395 9403 94II I5 44 2245 2253 2260o2268 2276 2284 i5 45 9420 9428 9436 9444 9452 9461I 14 45 229I 2299 2307 23I5 2322 2330 I4 46( 9469 9477T9485 9493 950I 95Io 13 46 2338 2346 235423611 2369 2377 i3 47 9518 9526 9534 9542 9550o9559:12 47 2385 2392 2400 2408 24i6 2423 12 48 9567 957519583 9591 9599 9608 Ii 48 2431 2439 2447 2454 2462 2470 II 49 96I619624 96329640o 4964896571IO 49 2478 2485 2493.250I 2508 251610I 50o9.96966519673 968I 9689 9697 9705 9 50 9972524 2532 2539'25471 2555 2563 9 5I 97I4 972219797309738 9746 9754 8 5i 2570 2578 2586 2593 260I 2609 8 52 9762 977I 977919787 9795 9803 7 52 26I7 2624 2632 2640 2648 2655[ 7 53 98II998.I9 9828 9836 98449852 6 53 2663 267 1267812686 2694270I1 6 54 986019868 9876198841989319901 5 54 2709127I71272512732 274012748 5 55 9909 9917 9925 9933 994' 9949 4 55 2755 2763 277I12778 2786 2794[ 4 56 9957/99661997419982 999o19998 3 56 2802 2809128I712825 2832 2841 3 57|9.970006ooI4 002oo3ooooo38o0047 2 57 2848 2855o2863 287I 2878 2886 2 58 oo55oo0063007I 0079 00871095 I 58 2894 2901 2909 29I7 292412932 59 o__010301II 0I910I27 o0136o1044 0 59 2940129471295512963 2970!29781 0, - 60"' 1 50" 40" 1 305" 0"' 10 0 50 40 20- 1 10 1 Co-sine of 21 Degrees.. I' Co-sine of 20 Degrees. I p. P art 1, 2" 32" 4/' 5/ 6" 71" 811 9 P.11 P ]1 2l' 3" 4" 5/ 6" 71 8' S.1 2 2 3 4 5 6 7 7 ar 1 2 2 3 4 r5; fi t LOGA nTHMIC TANG E NT S. r7 Tangent of 68 Degrees. Tangent of 69 Degrees.. _ - 0" 10" 20" 30' 40"1 50", 0" 10" 201" 30' 40"- 50" 0Jo.3935o90 365I 37I2 3772 3833 3894 59 o o. 415823 5886 5948 60oi 6074 6I37 59 I 3954 40I514076 436141I97 4258 58 I 6200 6263 6326 6389 6452 65i5 58 2 43I8 4379 4440 4500 4561 4622 57 2 6578 664i 6704 6767 6830 6893 57.3 4683 4743 48o4 4865 4926 4986 56 3 6956702070 83 7I46720 7272 56 4 5047 5 5047 51I69 5229 5290 535i 55 4 7335 7398 746I 7524 7587 7650 55 5 54I2 5473 5533 5594 5655 5716 54 5 77I4 7777 7840 7903 7966 8029 54 6 5777 58385898 5959 602o0608I 53 6 8093 8I56 8219 8282 8345 8409 53 7 6I42 6203 6264 6325 6385 6446 52 7 8472 8535 8598 866i 8725 8788 52 8 6507 6568 6629 6690 675I 6812 5I 8 885I 8914 8978 9041 I4 9I 68 5 I 9 6873 6934 6995 7056 7II7 7I78 5o 9 9231929419358 9421 9484 9547 50 IoIo.393972397300736I 7422 7483 7544 49 IO IO.4196II19674 9738 980I 9864 9928 49 11 7605 7666 7727 7788 7849 7910 48 II 9991..54. I 8. I8I.245.308 48 r2 7971 8032 8093 8I54 82IS5 8276 47 I2 Io.420371 o435 o0498 o5621o60625 68947 13 8337 8399 8460 852I 8582 864346 I3 0752 o816 0o879 o943 oo6 I070o 46 I4 8704 8765 8826 8888 8949 9o1o 45 I4 ii33 Ii97 I26o0I3241I387 I45I 45 I5 9071 9I32 9I94 9255 9316 9377144 I5 5 T4 i578 I64I I7o5 I769 I832 44 I6 9438 95qo 9561 9622 9683 9744143 I6 I896 I959 2023 2086 25o0 22I4 43 17 98o069867 9928 9989..5I. II242 17 2277 234I 24o5 2468 2532 2596 42 i8 Io.4o0073o 0235 o296o357 0o4I9 04804I 8 26591272312787 2850 29I4 29784i 19 o54i 0602 o664 o0725 0787 o848 40 19 3o4i 3Io5 31699 233 3296 3360o 40 20 0Io.400909 097II o32 Io93 II55 I2I6 39 20 10.423424 3488 355J 36I5 3679 3743139 o1 I278 I339 i4oo I462 I523 I585 38 2 1 3807|3870|3934|3998|4062 4I26.38 22 I646 1707 1769 i83o 1892 I953 37 22| 4190 4253 4317 438I 4445 4509137 |23 20I5 2076 2I38 2I99 226I 2322 36 23 4573 4637 470I14764 4828 4892 36 24 2384 2445 2507 2568 2630 269i 35 24 4956 5020 5o84 5I48 5212 5276 35 25 2753 2815 2876 2938 2999 306I 34 25 5340o 5404 5468 5532 5596 5660134 26 3122 318433246 3307 3369 3430|33 26 5724;5788 5852 5916 5980 6044133 27 3492 3554 36I5 3677 37393800oo 32 27 6Io8 61726236 6300o6364 6429 32 1 28 3862 3924 3985 4047 4109 4I 70o31I 28 6493 6557 6621 6685 6749 68I3 3I 29 4232 4294 4356 4417 4479 454i 30 29 6877 6941 7006 707017I34 719813o 3o. r.4o4602 4664 /726 4788 4850o49II 29 3o 10.42726297327 739I 7455 7519 7583 29 31i 4973 5o35 5097 5I58 5220 5282 28 31 7648 7712 7776 7840 79o057969$ 28 | 32 5344 5406 5468 5529 5591i5653327 32 8033 8097 8I62 8226 8290 835527 331 57i 5777 5839 59oi 5962 6024 26 33 84I918483 8548 861208676 874I126 34 6o86 6i48 620o 6272 6334|639625 34 88o5 8869 8934 899890o62 9127025 35 6458 6520o 6582 664416706 6768 24 35 919I 92569320|9384|9449|9513324 36 6829 689I 6953 7oi57077 71039 23 36 9578 9642 9707 977I 9836 9900 23 37 720I1 72637326 7388 7450 75I2 22 3 97 965..29'..94.i58.223.287|22 38 7574 7636 7698 7760 7822 7884 2I 38 Io.4303520o4I6 o48i o545 o6io 067421 39 7946 8oo8 80 70o8I13281I95825720o 39 07391o8o3 o868 0933o09971I062o20 4o0io.4083I9 838i 8443 85o5 8567 8630 19 4o 10.43II27 II9I I256 1320 I385I45O 19 4I 86928754i88i6|8878|8940 9003 i8 4 I 514 579 I644 I708 17731 838 i8 42 90659I2 719I89 9252|93 I49376 I7 42 1902 I967,203292097261 02226 17 43 9438 950I 9563 9625 9687 9750o I643 2291 235692420 24852550 26i5 i6 44 9812 9874 9937 9999..6I. I23 15 44 2680 2744 2809 287492939 3oo00 4 5 45 Io.4Ioi86 o248|o3Io o373|o435|o498 14 45 306831333198 3263 3328 33931i4 46 o56o 0622 o685 0747 180 9 o 872 13 46 3458 3522 3587 3652937I7 3782 i3 47 0934 0997Io59 1II221I8412461I2 47 3847 39I2|3977 40424107 4I72412:48 I309 I37I i434 1496 I559 I62I ii 48 4237 43024367 443244 97 4562 II 149 1684 17461 8091 1871 I9341 9961 0 49 46-7 469247574822,488749521 0 5o Io.4I20o592I21'2I84224623~09237I 9 5ozIO.435oI75o0825I47 52I2,5277 5342 9 5II 2434 2497 2559 2622 268402747' 8 5i 5407 5473 5538 56o3 5668 5733 8 | 52 28Io02872 299352997 3o6o 3I23 7 52 5798 5863 5929 5994 6059 6I24 7 54 356I 3624 3687 3749 3812 3875 5 54 658I16646|6711 6776|6842|6907 5 55 393884000|4o63 4I26 4I893 4253 4 1553 697217037 7Io37I 687233 7299 4 56 43I4 4377 4440 450204565 4628 3|56 7364 74291749517560|7625 769i 3 57 469I 47544817 4879 4942 5005 2 57 775678227887 795280oi880o83 2 58 50685131I5I9452565319 5382| I 58 8i49 82I488279 8345S84Io 8476 I 59 5445 55o8 5571 5634 5697 576 oo 59 854i 860718672.8738 8803 8869 0 60" 50' 1 40" 30" 201 10" 1 6Y' 50"' 40". 30" 20" IY Co-tangent of 21 Degrees. Co-tangent of 20 Degrees.. r 1 /" 2" 3" 4' 5"' 6" 7 8// 9". PartI l 2" 3/i 4/1 51/ 611 71/ 84 91/ P. Part 6 12 19 25 31 37 43 49 56 ar-6 13 19 26 32 39 45 51 58 LOGAR I T HnI SINES. Sine of 70 Degrees.. Sine of 71 Degrees. _2_0" -10" 20"1 30'1 4011 50" 0"_ 10"1 20'E 30" 401' 50"_ _ 09.972986 2993 300I 3009 30I 6 3024 59 O 9.975670 67756 857756 92 5699 576 59 3032 3039 3047 3o55 3062 3070 58 I 57I4 572I 5728 5735 5743 57355743 50 58 3078 3085 3093 3IoI 3io8 3II657 2 5757 5764 577I 577915786 5793 57 3 3I24 3I3r 3039 3I46 3I54 3I62 56 3 58oo 58o8 58I5 5822 5829 5837 56 4 3I69 3177 3i85 3I92 3200 3208 55 4 5844 585I 5858 5865 5873 588o 55 5 32I5 3223 3230 3238 3246323253 54 5 5887 5894 590o 5909 59I6 5923 54 6 326 3269 376 384329I3284 329 99 53 6 5930 5938 5945 5952 5959 5966 53 7 3307 33I4 3322 333o 3337 33 3345 52 7 5974 598I 5988 5995 6002 6oio 52 8 3352 336o 3368 3375 3383 3390 5I 8 60I7 6024 6o3I 6o38 6o46 6o53 5I 9 3398 34o6 34I3 342I 3428 3436 5o 9 6o6 6067 6074 6o808 I 6089 6096 50 I09.973444 345I 3459 3466 347413482149 I0 0.976IO3 6IIO16II766I25 6I32 603949 II 348913497 3504135I2 3519g3527148 I 6I4616I53 6I60o 668 6I7516I82 48 02 3535 3542 3550 3557 3565 3572 47 I2 6I891 696 6203 62II 62I8 6225 47 O3 358o 3588 3595 36o3 36io 36i8 46 I3 6232 6239 6246 6254 626I 6268 46 04 3625 3633 364i 3648 3656 3663 45 I4 6275 6282 6289 6296 63o4 63II 45 I5 367I 3678 3686 3694 370I 3709 44 I5 63I8 6325 6332 6339 6347 6354 44 I6 37I6 3724 373I 3739 3746 3754 43 I6 636i 6368 6375 6382 6389 6396 43 17 376I 3769 3777 3784 3792 3799 42 I7 64o4 64II 64I8 6425 6432 6439 42 I8 3807 38I4 3822 3829 3837 3844 4I I8 6446 6454 646I 6468 6475 6482 4I I9 3852 3859 3867 3875 3882 389 40140 I9 6489 6496 65036o 6565Io165 525 4 209. 973897 3905 39I2 3920o392713935139 20 9.976532 6539 6546 6553 656o 6567 39 21 3942 3950 3957 3965 3972 3980 38 2I 6574 6582 6589 6596 6603 66Io 38 22 3987 3995 4002 4o0I 40I7 4025 37 22 66I7 6624 663 6638 63866 6653 37 23 4032 4o4o 4o47 4o55 4062 4~70 36 23 666o 6667 6674 668I 6688 6695 36 24 4077 4085 4092 4Io004I07 4I15 35 24 6702 6709167I6 6723 673I 6738 35 25 4I22 4I30 4I374I145 4I52 4I60 34 25 6745167521675916766 6773 6780 34 26 4I6714I7514I824I90o4I09714205133 26 6787 6794 680I 6808685 6823 33 27 42I2 4220 4227 4235 4242 4250 32 27 683o 6837 6844 685I 6858 6865 32 28 4257 4264 42714279 4287 4294 3 28 68726879 6886 68793 690 1 6907 38 29 4302 4309 43I714324 4332 4339 30 29 69I4 692I 6928 6935 6942 6950 3o 30 974347 4354 436I 4369 4376 4384|29 3o0 9. 976957 6964 697I 6978 698516992 29 3:1 439I 439914406144I41442I 4428128 3I 6999 7006 70I37020 702717034 28 32 4436 4443 445I 4458 4466 4473127 32 704I 7048 7055 7o62 7o69 7076 27 33 448o 4488 44954503 45I0o 45I826 33 7083 7090 7097 7I04 7III7I0I8 26 34 4525 4533 4540 454714555 4562 25 34 7I25 7032 7I39174617I53 7I60o25 351 457014577 458514592 4599 4607[24 35 7I671774718I 7I88 7I95 7202 24 36 46i4 4622 4629|4636 4644 465I 23 36 7209 72I6 7223 7230 7237 7244 23 37 4659 4666 46741468I 4688 4696 22 37 7250 7258 7265 7272 7279 7286 22 38 4703 47 i1 47I8 4725 4733 4740 2I 38 7293 7300 7307 73I4 732I 7328 20 39 474814755 47621477o 4777 4784120 39 7335 7342 7349 7356 736317370 20 409.97479214799 4807148 4482I 4829 I.9 4o 9.977377 7384 739I 7398 74o5174I2 I9 14I 4836 4844 485, 448584866 4873 I8 4I 7419 7426 7433 7440 7447 7454 I8 42 488o 4884888 4959349o3 49Io49 97I7 I7 42 746I 7468 7475 7482 7489 7496 I7 43 492514932 49391494714954 496I I6 43 70375IO 75I7 7524753017537 I6 441 4969 4976 49841499I 4998 5oo6 i5 44 7544 755I 7558 7565 757217579 i5 45! 5o0I35020 5028 5035 5042 505o I4 45 7586 7593 7600 7607 7614 762I 14 46 5057 5064 507215079 5086 5094 IO 46 7628 7635 7642 7648 7655 7662 O1 47 5I5Io8i 5IO II15615I23 513o 5I38 I2 47 7669 7676 7683 7690 7697 7704 I2 48! 5I45 5I52 5I60o 567 5I74 5i82 oo 48 77II 77IS 7725 7732 7739 7745 II 49 5I89 5I96 5204152I I52i8 5226|IO 49 7752 7759 7766 7773 7780 7787 I0 509.975233 5240 5248 5255 5262 52701 91 5o19.977794 780 788 788 5 782I 7828 9 5i 5277 5284 5292 299 53o6 5313 8 5I 7835 7842 7849 7856 7863 7870o 8 52 532i 5328 5335 534315350 5357 7 521 7877 7884 7890 7897 7904 79 I7 7 53 5365 5372 5379 5386 5394 540oI 6 53 79I8 7925 7932 7939 7946 7952 6 541 54o8 54I6 5423 543o 5437 54451 5 54 7959 7966 7973 7980 7987 7994 5 551 545215459 546715474 548I 5488 4 551 8oo00 8007 8o04 802I 8028 8035 4 561 5496 5503 551o 55I8 5525 5532. 3 56 8042 8049 8056 8062 806918076 3 57 5539 5547 5554J556i 5568 55761 2 57 8o83180go98o97 8IO4|8IIO8II7 2 58 55831559o0559856o0556I2 56i9 I 58 8I2418I3I 8138 8I458185218i581 I 59 56275634 56456485656 5663 0 59 8I6518I72 8I79 8-8618938I99' 0 60"1 50 401 30" 1 20/" 10 60" 50" 1 40/ 30/ 20" 1 Co-sine of 19 Degrees. Co-sine of 18 Degrees. /1 2" 3" 4" 5" 6" 7" 8/ 9" pp1 S 1" 2" 3" 4" 5" 6/ 7/ 8"i 9" | P'art 1 1] 2 3 4 4 5 6 7 * art 1 11 2 3 4 4 5 6 6 P__________________ __________________ LOGARITtHMLC TANG'ENTS, 9a Tangent of 70 Degrees. i Tangent of 71 Degrees. f0" 10" 20" 30C' 40" 50" U0" 10" 20"' 30" 40" 50' O Io.438934 gooo o90659I 3I 919619262 59 o I0.463028 30973I 65 3233 3302 3370 59 |It 9327 939319458 9524 9590 9655 58 I 3439 3507 3576 3644 37I3 378I 58 | 21 972 I 9786 9852 99898983..49 57 2 3850o 3918 3987 4055 4124 492 57 3 Io.440II5 OI80 o246 0312 0377 0443 56 3 426I 4329 4398 4466 4535 4604 56 4 o5091o574 o640o 70 6 0771 o837155 4 4672 474I 48094878 4947 o50 I 55 5 ogo3 o969 I034 0I Ioo II66 I232 54 5 5084 5I53 522X 5290o 5359 5427 54 6 I297 I363 1I429 495 i56i 1627 53 6 5496 5565 5633 5702 577I 584o 53 7 6921 I7581I8241 890 I956 2022 52 7 5908 5977160466 I 56i8416252 52 18 2087 2I53 22I9 2285 235I 2417 5i 8 632I 6q90 6459 6528 6596 6665 5I 9 2483 2549 26I 5 2681 27471283 50o 9 6734 68o036872 694I 7010 7079 5o Io 10.442879 2945 3oiI 3077 3I43 3209 49 Io 10.467147 72i6 7285 7354 7423 7492 49 II 3275 334i 3407 3473 3539 36o5 48 I 7561 7630 7699 7768 7837 7906 48 12 367I 373713803 3869 3935 4oo0047 I2 7975 8044 II3|8I882 825I 832047 i3 4067 40 3314200 4266 34332439846 3 8389 8458 8527 8597 8666|8735|46 I 4 4464 453014596 4663 472994795145 I4 88048873 8942|9011 9080o19I 5o045 I5 486I 49274994 5060 5126 5192|44 |I5 92I9 9288 9357 9426 949619565 44 i6 5259 53251539I 5457 5524 559o143 I 6 9634 9703 9772 9842 99 I 9980 43'7 565615722 57895855 592 I 598842 I 710.4700491 I 9 8802570 3270o396 42 I 8 605416I 20o68716253 6320o638614 I 8 o465 o535 o6o4|o673 o7431o08I241'9 6452 65I9 6585 6652 67I8 6784 40 19 o88I 0o95I 10201090 II59 1228 40 20 Io.44685I 69I7 6984 7050 7II7 7183 39 20 I.471298 I 367 I437 I 506 15761 645139 21 7250 73 I6 7383 7449 75I6 7582 38 21 I715 1784 i854 I923 I993 2062 38 22 7649 7751 7782 7848 79I5 798I 37 22 2I32 2201 2271 2340 24Io 248o 37 | 23 8o48 8II518i8I 8248 83I418381|36 23 2549 2619|2688|2758 2828 2897 36 24 8448185i4 58I 88647 8714[878I|35 24 2967 30373I06 3I76 3246 33I5 35 25 8847189I41898I 904819,I489I8I134 25 3385 3455|3524|3594 3664 3734134 26 9248 93I4|938I 9448 95I5|958I133 26 38033873|3943|40I3 408294I52 33 27 9648 97I519782 9848 99I5 9982 32 27 4222 4292 4362 443 2 450I 457I 32 28 I0.450049 oii6 oi83 0249 o3i6 o3833 I 28 464 4711 478I 1485, 492I 499I 3I 29 o450oo5 I 7 o584 065I 07 I 7 0784130 29 50605 I 305200o52 70o5340154 1o 3 30 o.45o85 09qI8 0985 I52 I I I9I i86 29 30 I.475480o 4& o 55562o 56 90 576o 5830 29 31 I253 1320 I387 I454 1521 1588 28 3I 5900oo 597 6040 6I o 6180o6250 28 321 6551 1722 17891 856 19231990127 32 6320o639i 646i 653I 660o] 667127 33 2057 2I24 2I91 2258 2325 2392 26'33 674I 68Ii 688II6952 7022 7092 26 34 2460 2527 2594 266I 2728 2795 25 |34 7I62 7232 7302 7373 7443 75I3 25 35 2862 2929 2997 3o64 3I3I 3198 24 35 7583 7654 772417794 7864 7935 24 36 3265 3333 340oo3467 3534 3602 23 36 8005 8075 8I46 82I6 8286 8357 23 37 3669 3736 3803 387I 3938 4005 22 37 8427 8497 8568 8638 8708 8779 22 38 4072 4 1404207 4274 4342|440921I 38 8849 8920|8990 06 I I31 920I21 39 447614544|46II14678 4746 48I3 20o 39 9272 9342 9413 94839554962420 4o01 o45488I 4948 5015 5o83 5i'o 5528 I9 40o 0.479695 9765 9836 9906 9977]..47 I9 4I 5285 5353 5420o5488 5555 5623 18 41 1o.480I 8 oI890o2599 3300 4001047 18 42 5690 5758 5825 5893 5960 6028 I7 42 0542 06I2 o683 0754 o824 o895 I7 43 60956I 63|6230o6298 6365 6433 6 1143 o966o36 I I07 I I 78 I 248 I3 16 44 650o 6568 6636 6703 6771 6839I511 44 I390oI46I I53i 1 602 673 I7/441 i 45 690616974 7042 71091777245I41445 I8I4 I885 1956 2027 2798 2I69j14 46 731217380 7448 75I5 75831765 [ I3 46 2239 23I0o238I 2452 2523 2594 1: 3 47 7719 7786[7854 7922 799o08057 12 47 2665 27362807 2877 29483I 319 2 48 8I25 8I93 8261 8329 8397 8464 I, 48 3090 3i6i 3232 33o3 3374 34451 i 49 8532 860oo8668 8736 88o4 8872 10o 49 356 3587 3658 3729 38oo 3871 |u 5o0o0.458939 900719075 9I43 11 9279 9 50o o.483943 40oI44085 4I56 4227 4298| 9 5I 9347 94I5 9483 955i 96I919687 8 5I 4369 4440o45II 14583 4654 4725 8 52 9755 9823 989I 9959..27. 95 7 552 4796 4867 4938 50 Io 508 o8 55 2 7 53 io.46o0I63 023 0299 0367 o4351050o3 6 53 5223 5295 5366 5437 55o8 558o 6 54 0571 I 06 39070707761o8440912 5 54 565i 57225794586559366008 5 55 09og80 io48 i 16 184 I253 I32I 4 55 6079 6I50o62226293 6364 6436 4 56, 389 1457 15251I 5941662 I73o0 3 156 6507 6579 665o 6722 6793 6864 3 57 I798 I867 I935 2003 2071 2140| 2 57 6936 7007 7079 750 7222 72[ 931 58' 2208 2276 2345 24I3 248 12551 I 58 7365 7436 7508 7580 7651 77238 59 26I8 26862755 2823 2891 2960o oil 59 7794178661 7937800980 8I 18i52 0 " 5 j540" 3" 2o" 1" 60" 50 4011 30 10"1 Co-tangent of 19 Degrees. lo Co-tangent of 18 Degrees.: [ I - 1"- 2" 3-i 4-/ 5-/ 6"- 7/ 8" 9/ p 1p " 2" 3" 4" 5/ 6" 7" 8" 9"-1 L art 7 13202733 40 47 54 60 r 7 14 21 28 35 42495663 L IL _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ LOGARITHMIC SINEB. Sine of 72 Degrees. Sine of 73 Degrees. _ 0"'10" 20" 30" 40" 50" 0"il 10" 20't 30"1 40" 501 9. 978206 82 3 822o 8227 8234 824I 59 o 9.9805960 o603 60 g 6I6 0622 o628 59 1 8247 8254 826I 8268 8275 8282 58 o635 o64I o648 o654 o66i o667 58 2 8288 8295 8302 8309 83I6 8322 57 2 o673 o680 o686 0693 o699 0o76 57 3 8329 8336 8343 8350 8357 8363 56 3 07I2 07I18 725 073I 0738 0744 56 4 8370 837718384 839I 8397 84o4 55 4 0750 0757 0763 0770 0776 0783 55 5 84I I 84I8 8425 843I (8438 8445 54 5 0789 0795 0802 o8o8 o8I5 082I 54 6 8452 8459 8465 8472 8479 8486 53 6 0827 o834 o840 0847 o853 0859 53 7 8493 8499 8506 85I3 8520 8527 52 7 o866 o872 o878 o885 08gI o898 52 8 8533 8540 8547 8554 8561 8567 5i 8 ~9~4 ogIo 0917 o 0923 0930 og36 9 8574 858I 8588 8594 86o0I 86o8 50 9 o942 o949 o955 o g6I o968 0974 5 Io 9.97865 8622862886358642 864949 109.98098I 0987 0993 I ooo Ioo6 I012 49 II 8655 8662 8669 8676 8682 868948 II IoIg I025 Io3i io38 Io44 105I 48 I21 8696 8703 8709 8706 8723 8730 47 I2 I057 Io63 I070 I076 I082 1089 47 3 8737 8743 8750o 875718764 8 770 46 13 I095 IIOI 0 18 III II 20 IZ 27 46 I41 877718784 879879 8797880488 I 45 14 I33 1I3g I I461I 521I58 I I65 45 I5 8817 8824[8831 8838 8844 885 144 i 5 II7I II77 11841 90 I 96 I203 44 I6 8858 8865 887I 8878 8885 8892 43 I6 1209 1215 I222 I228 I234 I24I 43 17 8898 8905 89I2|89I8|8925 8932 42 57 I247 I253 I2601 266 I272 I279 42 I8 8939 8945 8952 8959 8965 8972 4I Ii8 I285I 291 I 298 I 3oI 3 IO I 317 4I 19 89791898618992 8999 o690206 I 40 I3 323 g329 I336 I342 I348 I354 4o 20 9.9790o9 9026 9033 9039 904619053 39 20 9.98136i 1367 1373 i38o I386 2 3939 21 9059 9066 9073 90799086 9093 38 2I I399 I405 I4II 14I7 I424 1i43o 38 22 9I00 9106 9II3|9I20|9I26 9133 37 22 I436 I443 I449 i455 I46I I468 37 23 9I40o I9469I539I60o9I66 9I73 36 23 1474 1480 1487 1493'499 I5o5 36 24 9I80o9 I86 993 9200 9206 92I3 35 924 I512 i5zi8 I524 I53I i537 i543 35 25 9220 9227 9233 9240 9247 9253 34 25J 1549 I556 I562 i568 1574 i58i 34 26 9260 9267!9273 9280 9287 9293 33 26 1587 I503 I5gg99 I6o6 i612 i6i8 33 27 9300 9306 93I3 9320|9326 9333 32 927 I625 i63i I637 I643 I65o i656 32 28 9340 93461935319360 9366 9373 3I 28 1662 I668 I675 I681 i687 I693 31 29 9380 938619393 9400 9406 94I3 30 29 I700 1706 I7I2 17I8 1724 173i 3o 30 9.979420 9426 94331943994 46 9453 29 30 9 981737 1743 1749 I756 I 762 768 29 31 9459 9466 9473 9479 9486 9492 28 3I I'774 178I 1787 1793 I799 i8o6 28 32 9499 9506:95I2 95ig 9526 9532 27 32 i58:2 i8i8 1824 1830o18371i843 27 33 9539 954519552 9559 9565 9572 26 33 18491 855 i86i I868 I874 i880 26 34 9579 95851,9592 9598 9605 96I2 25 34 I886 1893 I899 I905 I9II I9I7 25 35 9618 9625 963i 9638 9645 965I 24 35 I924 1930 I936 I942 I948 I955 24 36 9658 9664 967I 9678 9684 9691 23 36 I96I i967 1973 I979!-86 I992 23 37 9697 970497I I 97I 7972497430o 37 I998 2004 20Io O 20 64o023 2029 22 38 9737 9743 9750 9757 9763 9770 2I 38 2035 204I 2047 2054 2060 2o66 21 39 9776 9783 9790 9796 9803 9809 20 39 2072 2078 2084 209I 2097 2I03 20 4019.9798I6 982219829 9836 9842 9849 I9 4019.982I092 I5 2I 22 2I2892I3412I40 19 41 9855 9862|9868 9875 988i 9888 I8 4I 2I46 2I52 2I 59 2I65 2I7I 2I77 I8 42 9895 990I 19908 991499921 9927157 I 42 2183 2I891 2 95 2202|2208 22I41I 7 43 9934 994o 9947 9954 9960 9967I6 i6 43 2220 2226 2232 2239 2245 225I i6 44 9973 9980 9986 9993 9999...6 I5 44 2257 2263 2269 2275 2282 2288 I5 45 9.98ooI.2 0oo I90026 oo320039o 0045 i4 45 2294 2300 2306 23I12 232 2324 I4 46 0052 oo58oo65 oo007 oo 0078 oo0084 3 46 233I 2337 2343 2349 2355 236 3 47 0091 0097 oI04 oIIOo0I7|0I23 I2 47 2367 2373 238012386 2392 2398 I2 48 o 30 oi361oI43 0I49|0I560I563 II 48 2404|2410 24612422 2429 2435 I I 49 oI69 OI76 OI82 0I89 0I95 0202 IO 49 244I 2447 2453 2459 2465 247I IO 5019.9802o8 02I5 10221 0228 0234 0245 9 50 9.982477 2484 2490 2496 2502 2508 9 5 I 0247 o254o026o 0267 027310280 8 51 25i4 2520o 25262532 2538 2544 8 52 o0286 0o293 0299 o3o6o3I2 o38 7 52 255i 2557 2563 2569 2575 2581 7 53 o325 1o33 o338 o3441o35o 0357l 6 53 2587 2593 2599 2605 26I1 6I7 6 54 o364 0370 0377 o383[0390 0396 5 54 262412630 2636 2642 2648 2654 5 55 o4o3 o409 o416 o422 o429 0435 4 55 2660o266d 2672!2678 2684 2690 4 56 o442 o448 o454 o46i o467 0474 3 56 2696 2702 27cqt27I5 272I 2727 3 571 o480 04870493o050010506 o5I3 2o 571 273312739 7.4-'275I12757 2763 2 58 o5I9 o525 o532 o538 o545o55i I 58 2769 2775 27SI!2787,279312799 I 59 0558 0564057I o577o583o59o o 59 2805,28I1 2SI7 282/12830 2836 o 60" 50" 1 40" 30" 0" 10' 60" 1 50": 40" 30"i 20"1 1i 2"11 4" 7 Po-si1 13" 5"6" Dg" 60 P t Co-sine of 17 Degrees. Co-sine of 16 Degrees. - r 11 21 3/1 4" 5/ 6" 7/ 9 8/ 911 2 3/ 4/ 5/ 6".'P 8 1 12 3 3 4 5 5 6 1 1 2 3 4 4 5 6 LOGARITHM C TANG E N T S. 9 1 Tangent of 72 Degrees. Tangent of 73 Degrees. 0"' 10" 2~0" 30" 40" 50" 0" 10" 20"/ 30" 4!' 50" O Io.488224 8296 8367 8439 85ii 8582 59 o I0.5I466 14736 48 I24887 4962 5038 59 I 865418726 8797 886918949I oI3158 I 5I13 5I88 5264 5339 54i5 5490 58 2 9084 9I5692289 3oo00 9371 9443 57 2 5565 564i 57I6 5792 5867 5943 57 3 95I5 9587 9659 973I 9802 9874 56 3 686094669 6245 6320 6396 56 4 9946., I8..go.I62.234.3o6 55 4 6471 6547 6623 6698 6774 6849 55 5 o.49o378 0o449 o52o 0593 o665 0737 5 4 5 6925 700I 7076 7I52 7228 7303 54 6 o80g 91o88 0o9531025 I0971 69 53 6 7379 7455 7530 7606 7682 7758 53 7 124 1I3I3 I386 I458 i53o 3602 52 7 7833 7909 7985 8o6i 8137 8212 52 8 I674 I746 i8i8 18901 96212035 15 8 8288 8364 8440 85I6 85928667 5I 9 2107 1279 225I 232312395 2468 50o 9 8743 889 8895 897I 9047 9I23 50 10 Io.492540 26I2 26841275712829 2901 4 IO I.5I9I9999275 9351 9427 9503 9579 49 II 2973 304613II8 3190o326313335148 II 9655 973I 9807 9883 9959.35 48 I2 3407 3480 35523624 3697 3769 47 12 10.5201 1 OI87 o263 0340 o4I6 o49247 I3 384i 39I4 3986 4059 4I 3I4204 46 I3 o568 o644 072070797 0873 o949 46 I4 4276 4348 442I 449314566 4638 45 14 I025 IOI1 1178I 154 i33o 1407 45 I 5 47I I 47831485614928 50ooI 5074 441 I5 I483 i5591I635 1712 I 78 i864 44 I6 5I46 52I9 529I 536415437 5509 43' i6 194I 20I7 2o94 2170 2246 2323 43 17 5582 5654 5727 58oo 5872 5945 42' 17 2399 2476 2552 2628 2705 278I 42 i8 60I8 6090~ 6I63 6236 6309638I 4I 8 2858 2934301 I 3087 3164 324 4 I9 6454 652716600 6672 6745 68I814o0 19 33I7 3394 34703547 3623 37004o 20o I.496891 6964 7036 7I09 7827255 39 20 IO.523777 385313930o400714083 4160o39 21 7328 740I 7474 7547 7691 6792 38[ 2I 4237 43I3 4390 4467 4544 4620 38 22 7765 783817911 7984|8057 8I30 37 | 22 4697 4774 485I 4927 5004 5081 37 23 8203 8276 834918422o8495 8568 36 23 5i58 5235 53I2 5388 5465 554236 24 864i 87I4 8787 8860o 8934 9007 35 24 5619 5696 5773 585o 5927 600oo4 35 25 90809 I 531 922G6 92999372 9445 34 25 608 I 6 I 58 6235 63 I 2 6389 6466 34 26 95i9|9592 966519738198II 9885133] 26 6543 6620o6697 6774 685I 6928 33 27 9958..3I I04. 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Degrees.: Co-tangent of 16 Degrees. 1" 2"' 3" 4" 5" 6" 7" 8" 9" i " 2" 3" 4/ 5" 6" 7/1 8" 9/ P. Part 7 15 22 29 37 44 51 59 66 P. Part 8 15 23 31 39 46 54 62 70 G Os 8 L 0 G A R LOGARITHMIC SINES. Sine of 74 Degrees. d Sine of 75 Degrees. o 0'/ 10"'20"' 30" 40'- 50' o 0" 10" 20" 30"1 40" 50' 0 9.982842 2848 2854 2860 2866 2872 59 0 9.984944 4949 4955 496 496614972 59 I 2878 2884 2890 2896 2902 2908 58 4978 4983 4989 4995 5ooo 5oo6 58 2 29I4 2920 2926 2932 2938 2944 57 2 50II 507 502 5028 5034 5040 57 3 2950 2956 2962 2968 2974 2980 56 3 5o45 505 50o56 5062 5o68 5073 56 4 2986 2992 2998 3oo4 30Io 3o0I6 55 4 5o79 5o84 5090 5096 5IOI 5107 J5 5 3022 3028 3034 3040 3046 3052 54 5 5II315II8 5I24 5I29 5I35 5I4154 6 3058 3064 3070 3076 3082 3088 53 6 5146 5I52 5I57 5163 569 5I74153 7 3094 3Ioo 3Io6 3I2 3II8 3I24 52 7 5I80 5I85 5I9I 5I97 5202 5208 52 8 3I30 3I36 3I42 3I483I5413I60o 5 8 52I3 52I9 5224 5230 5236 524I 5 9 3I66 3I72 3I78 3x84 3I90o 396 15 9 5247 5252 5258 5264 5269 5275 5o I1 P.983202 3208 3214 3220 3226 3232 49 I1 S.985280 5286 529I 5297 5303 5308 49 II 3238 3244 3250o3256 32625368148 i 534 5395325 5330 5336 5342 48 12 3273 3279 3285 329I 3297 330347 12 5347 5353 5358 536414 59 5375 47 13 3309 33i5 3321 3327 3333 3339 46 i3 538I 5386 5392 5397 54o3 54o8 46 I4 3345 335i 3357 3363 3369 3375 45 i4 54i4 5419 5425 5430o 5436 5442 45 i5 338I 3386 3392 3398 34o4 34io 44 I5 54471545315458 5464 5469 5475 44 I6 * 34i6 3422 3428 3434 344o 3446 43 I6 548o 5486 5491 5497 5502 55o8 43 17 3452 3458 3464 3469 3475 348I 42 11 7 55I4155I915525 553o 5536 554I 42 i8 3487 3493 3499 35o5 35I 5i 3517 411 I8 5547 5552 5558 5563 5569 5574141 I9 3523 3529 3535 354o 3546 3552 14 19 I 558o 5585 559i 5596 5602 5607 40 20 9.983558 3564 3570 357613582 3588 39 20 9.9856I3 56I8 5624 5629 5635 5640 39 21 3594 3599 36o5 36II 36I7 3623 38 | 21 5646 565i 5657 5662 58 68 5673 38 22 3629 3635 364I 3647 3653 3658 37 22 5679 5684 5690 5695 5701 5706 37 23 3664 3670 3676 3682 3688 3694 36 23 57I21571r75723 5728 573415739136 24 3700 3705 37II137I713723 3729 35 24 57451575 5756 5 756 1 765767 5772 35 25 3735 374I 3747 3752 3758 3764 34 1 25 5778 5783 5789 5794 58oo 58o5 34 26 377o 377613782 3788 3794 3799 33 |26 58i 58I615822 5827 5832 5 8358 3833 27 38o5 38II 38 7 3823 3829 3835 32 | 27 5843 5849 5854 586o 5865 5871| 32 28 384o 384613852 3858 3864 3870 31 28 5876 5882 5887 5893 5898 59o03 3 29 3875 388i 3887 3893 3899 3905 30 29 5909 5914 592045925 593I 593613o 30|9.9839II 39I6 3922 3928 3934 3940 29 30 9.985942 5947 5952 5958 5963 5969 29 3rl 3946 395I 3957 39633969 3975 28 | 3I 5974 598015985 599I 5996 60oi 28 321 3986139863992|3998|4004 40I127 321 6007160I2160I816023 6029o6034127 331 40o5 402I14027 403 40 9 4o4 26 33 6039 6o45 6050 6o56 606I 6067 26 34 4o5o 4o5614o6214o68 4074 4079 25 341 6072 6077 6o83 6o88 609416099 25 35 4o85140 409714Io34IO8 4II4 24 35 6Io4|6IIo16II5 6I21 6I2616i13224 36 4120 41264132 4i37 4I4314I49 23 | 36 6I3716I42|6I48 6I53 6I596164 23 37 4I55 4I6I 4I6614I72 4I78 4I84122 37 6I69|6I756I80o16I86|6I9I 6I96|22 38 4190 4I951420I 4207 42I3 42I8 21 381 6202|620716212 62I8 6223 6229 21 39 4224 423014236 4242 4247 4253 20 39 6234 6239 6245 6250o6256 6261 20 4019.98425914265 42701427614282.42881 I9 4o 9.986266 6272 6277 6282 6288 6293/ 19 41 4294 429914305o43II143I714322 I8 4I 6299 630416309163i 56320o63251I8 42 4328 4334 434o 4345 1435I4357 I7 42 633i 6336163421634716352|635817 43 4363 4368 4374 438o 4386 439I i6 43 6363 6368 6374 6379 6384 639o0 i6 44 4397444044 I44i4442014426 i5 44 6395 64o0I 64o06164i 64 64 6422 5 45 4432 4437144431444914455 446o I4 45 6427 6433 6438 6443 6449 645414 46 4466 4472 4477 4483 4489 4495 13 461 6459 6465 6470o6475 648i 6486lI3 3 47 4500145061451I245I8 4523 4529 I2 471 649I|649716502|6507|6531365I81I2 48 4535 4540o454614552.4558145631 II 481 652365291 6534 6539 6545 6551o i 49 4569 4575 458o 4586 4592 4598 io 49 6555 656 16566 6571 6577 6582 10 50o9.984603 4609146I5 462o04626|4632 9 5o 9.986587 6593 6598 66o3 66o8 66,4 9 5i 4638 4643 464914655 4660o4666 8 15I 66I9g66241663o06635/6640o66461 8 52 4672 4677 468368689 46944700 71 521 665i 6656 666i 6667 6672 6677] 7 53 4706 4712 4717 4723 47294734 61531 6683 6688 6693 6699 6704 67091 6 54 47740o4746 475144757 476314768 5 54 6714/672o06725 673o0673616741I 5 55 477414780I4785 479I 4797|4802 4 55 6746 6751 67571676216767|6773| 4 561 48084881448I41948251483I!4836 3 561 67767836783678867946799 68o04 3 571 4842!484848531485914865 4870 2 57 6809168I5 682o068251683i168361 2 58 487614882i 4887 4893 4899 49o4 I 58 684i 6846 6852 6857 686216867 I 591 490 49II6492I 4927|4932 4938 o 591 6873 6878_6883_6888689416899_ o 611" 50L, 40" 30" 20"1 1 10"1 60" WI50 40" 30"1 2 10"I'(Y Co0-sine of 15 Degrees.; t Co-sine of 14 Degrees. P. p rt 111 211 3/1 4// 5// 6" 7/' 8" 91 P. Part 1"/ 2]; 3"1 4/1 5" 61 7VI 81 9/1 1 2 2 3 3 4 55 5 Part1 1 2 I2 3 3 4 4 S LOGARITHMIC TANGENTS. 99 Tangent of 74 Degrees. || Tangent of 75 Degrees. C' 101' 20 30" 40"5 10" 20/1 30"1 40" 50" o 2o 542s54 2583 2663 2742 2822 2901 59 o0 10.571948 2032 212 I I6 2200 2285 2369 59 | I 298 3060o 3I4o 321 93299 3378 58 1 24531253712622 27o062790 2875 58! 2 3458 3538 3617 3697 3777 3856 57 2 2959 3o44 3I28 32121 397 338 I 57 3 3936 40I6 4095 4I75 4255 4335156 3 3466 355o 3635 37I9 3804 3888 56 4 44I4 4494 4574 4654 4733148 I3155 4 3973 4058 4I421 4227 43 I I 4396 55 5 489314973505 3 5 I 3315~ I 35292 54 5 448 [456514650o 4735 48 I 914904154 6 5372 5452 5532 5612 569215772 53 6 4989 5073 5I58 5243 5328 54I3 53 7 5852 5932 61o I2 6092 6I 72 6252 52 7 5497 5582 5667 5752 5837 5922 52 8 6332 6412 649216572 6653 6733 5I 8 60071609I 6I76 626I 6346 643I 51 9 68i3 6893 6973 7053 7I33 72I4 50 9 65I6 660I 6686 677I 6856 694I 5o ioIOI.54729417374 7454 7535 76I5 7695149 Io 1o.577026 7II2 7197 7282 7367 7452 49 I I 7775 7856 7936 8oi6 8097 8I77148 I I 7537 7622 7708 7793 7878 796348 12 8257 8338 84I8 8498 8579 8659147 I12 80488 34 82 I9 83483 908475 47 i3 8740 8820o890I 898I 9062 9142 46 i3 8560 8646 873I 88I6 8902 8987146 I4 922319303 93841946419545 9625 45 I4 9073 9I58 9243 9329 944 1950oo45 15 97o6 9787 9867 9948. 28. 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I29.217.3o4.39I 24 36 9964..46.28. 2I0.293.375 23 136 io 590479 o566 o654 0741 0829|09I623 37 I0.560457 o54o 0622 o704 0787 o869 22 37 1oo4 I 09 I I 79 I 266 354 44 22 38 o952 0o34 I I16 II99 I28I 364 2I | 38 1529 i6I6 I704 I792 I879 I967 21 39 I446 1529 i6ii I694 1776 185920o 39 2055 2I42 2230 23I8 2406 2493 20 4o I0.56,941 2024 2106o2I89 2272123541I91 40 I0.59258I 2669 275712845 2932|,3020 19 41 2437 2520 2602 2685 2768 2850o i8 4I 3io8 3I96 3284 3372 346o 3548 I8 42 2933 30I6 3099 3I8I 3264 3347171 42 3636 3724 382 39oo00 3988 4076 I7 43 3430o 35I2 3595 3678 3761 3844 I6J 43 4I64 4252 4340 4428 45I6 46o4 16 44 3927 4oI0o 4093 475 4258 434i i5 44 4692 4781 4869 4957 5o45 5I33I5 45 4424 4507 4590 4673 4756 4839 I4 145 5222 53io 53988 5486 557515663 I4 46 4922 5005 508815I72 5255 5338I31 46'575I 584015928160I7 6Io 5j6I93 13 47 5421 5504 5587 5670 5754 5837 1 247 6282 6370 6459 6547 66366724 1I2 48 5920 6003 6o86 6I 706253 63361 I 48 68I3 690 6990 7078171677256 11 49 6420 65o3 6586 6669 6753 6836j io 49 7344 7433 7522 76IO 7699 7788 iO 501 0.56692017003 7086 71 70 7253173371 9 50 Io. 597876 7965180548 I43 823I18320 9 5i 7420 7504 7587 767I 7754 7838j 8 5i 8409 8498 8587 8675 8764 8853 8 52 792I18005 8088 8I72 8255 83391 71 52 8942 903I 9I20 9209 929819387 7 53 8423 850618590o8674 875718841 61 53 9476 9565 9654 9743 983219921 6 54 8925 9oo8 90o92 9176 926o9343 5J 54 Io.6000Iooooioo o89 o0278o367 o456 5 55 9427 95 19595 9679 9763 9846 4 155 o545 o635 o724 0813 09020992 4 56 9930. I4..981. I82.266.3501 311 56 Io8 I I70 126o0 "349 I438 1528 3 57 Io.5704340o5I8 o602 o686 0770 o854 21 57 1617 1706 r796 I885 I975 2064 2 58 o9381o022 IIo 6 1190 I274 1358j I |58 2i54 2243 2333 2422 25I2 260I i 5; 1442I'527 16II I6 95 1779 Ij63J ol 59 269I 278I 2870 2960 3o5o 3139 _ 60" 150" 40" j30" 20" 10" - 60" 50" 40Y' 30"'0 1 d0" Co-tanger-t of 15 Degrees. Co-tangent of 14 Degrees. 1 " 2" 31' 4" 5// 6" 7/" 8" 9/" 1/1 2" 3/1 4" 5" 6' 7/1 8" 9"P.Part 8 16 25 33 41 49 57 65 74 P art l9 17 26 35 43 52 61 69 78 IL 0 LOGARITHMIC NIN ES. [-~ Sine of 76 Degrees. J i Sine of 77 Degrees. _ f O 10"' 20" 30" 401 50 W 0" 10' 20" 30"1 40' 50" 0 9.986904 69960 9 69269 6925 69359 0 9.988724 8729 8734 873918743 8748 59 I 6936 694I 6946 695I 6957 6962 58 I 8753 8758 8763 8768 8772 8777 58 2 6967 6972 6978 6983 6988 6993 57] 2 8782 8787 8792 8797 8802 88o6 57 3 6998 7004 7009 70oI470oI97025156 3 88 188I6 882I 8826 883I 8835156 4 7030 7035 7040 7045 705I 7056 55 4 884o 8845 885o 8855 8860 8864 5 5 5 7061 7066 7072 7077 7o82 7o87 54 5 8869 8874 8879 8884 8889 8893 54 6 7092 7098 7I03 7I0817II3 7II 853 6 8898 8038903 8908 893 8988922153 7 7124 7129 7I34 7I39 7144 7I50 52 7 8927 8932 8937 8942 8946 8951 52 8 7I55 7160 7165 7I170 776 7I81 5I 8 8956 896I 8966 897o 8975 8980o5I 9 7I86 719I 7196 7202 7207 7212 5o 9 8985 8990 8994 8999 9004 goog 50 I o9.9872 177222 7228172331723817243149 I09.9890I4 90I8 9023 9028 903o, 9038149 11 7248 7253 7259 7264 7269 7274 48 II 9042 o4790 52 905719062 9066 48 12 7279 7284 7290 7295 7300 73o5 47 12 907I 9076 908I 9085 909019095 47 I3 73I0 73I5 732I 73261733I 7336 46 I3 oo00 9I05 I09914 9II9 91I24 46'4 734 17346 7352 735717362 7367145 i4 9128 9I33 9I38 9143 914819152 45 i5 7372 7377 7383 7388 7393 7398 44 I5 9I57 9I62 9167 9I71 9I76 9I8I 44 i6 74037408174I3174I9 7424 7429 43 i6 9186 9190 9I95 9200 9205 920943 I7 7434 7439 7444 7449 7454746o42 I7 92I4 92I19 9224 9228 9233 9238 42 I8 7465 7470 74175 7480 7485 749o 4I I8 9243 9247 9252 9257 9262 9266 4I Ig 7496 750o 7506 75II 75i6 7521 4o I9 927I 9276 928I 9285 9290 9295 4o 20 9.987526 7531 7537 7542 7547 7552 39 20 9.989300 93041 93093 934 938 9323 39 21 7557 7562 7567 7572 757717583 38 2I 9328 9333 9337[9342 9347 9352 38 22 758817593 7598 760317608761337 22 9356 936I 936619370 9375 9380 37 23 7618 7623 7628 7634 7639 7644 36 23 9385 9389 9394 9399 9403 94o8 36 24 7649 7654 7659 7664 7669 7674 35 2 4 94I3 94I7 9422 9427 9432 9436 35 25 7679 7684 7690 7695 7700 77o5 34 25 9441 9446 9450 9455 9460 9464 34 26 7710 7715S7720 7725 7730 7735133 26.9469 947419479 9483 948819493 33 27 7740 7745 7750 7756 7761 7766 3132 127 949719502 9507 95195I95I6 952I 3 28 777I1 7776 7781 7786 779' 7796 3I 28 9525 9530 9535 9539 9544 9549J 3 29 780I 7806 78II 7816 7821 7826 30o 291 95531 9558 9563 9568 9572 9577i30 30 9.987832 7837 7842 7847 7852 7857 29 30 9.989582 9586 9591 9596 9600 9605 29 3I| 7862 786717872 7877 7882 7887 28 3I 96Io096I4 9619962319626 8 9633[28 32 7892 7897 790217907 79I2 79I7 27 32 9637 9642 9647 965I 9656 966I 27 33 7922 7927 7932 7937 7942 7947 26 33 9665 9670 9675 9679 9684l 96t89 34 795317958 796317968 7973 7978 25 34 9693 9698 9703 9707 9712 97xI625 351 7983 7988 7993i7998 8oo3 8oo8 24 35 9729726 97 973 19740 9744 14 36 8oI3J80I8 8023i8028 8033 8o38 23 36 974919753 9758 9763 9767 9772|23 37 8o43 8048 8o5380o58 8o63 8o68 22 37 977719 78786 97909795 9800 22 38 8073 8078 8o83 8088 8093180981 38 984 98098 984 98i1982339827 21 39 81o3|8IO8 8II3181I8I8I23 8I2820o 39 9832J9837 9841 9846)985019855,1o 4o 9.988133 8i38 8I438I48 8i53 8I58 19 4o 9. 989860 9864 9869 9873 9878 9883 19 41 8I63 8I6881I738178|8I838i88 I88 I 41 988719892 9896 990'9906 990o i8 42 8I938I98 803 820882I3 82I8II7 42 995[9999924992999339933993817 43 8223 8227 8232 8237 8242 82476 [ 43 9942 9947 9952 9956 996I 9965 i6 44 8252 8257 8262 8267 8272 8277 11 44 9970 9974 9979 9984 9988 9993 i5 45 8282 8287 8292 829718302 83071I4 45 9997...2...6..i z.. I6...20I14 46 83I2 831718322 8327 8332 8337 13 46 9. 99o25 oo0029oo340038 oo43 oo48 3 47 8342 8346 835i 8356 836i 8366 12 47 0052 0057 oo006I oo0066 0070 0075 2 48 837i183761838,18386 8391[83961 I 48 007900oo84o0088 0093o00o980102I 49 84o0I 84o0684I 184I 6428428425 1o 491 OI071I IOII60120012501291 IO 50o9.988430 8435 8440o844518450o8455 9 l 5o9.990o341i38 oI43 oI48|oI52 0157 9 5IR 846o08465 8470 847568480 8484 8 Si ooI6IoI66?70 o01751oI790oI84 8 52 8489 84948499 85o0485o0985i41 7 | 52 oI881I93 OI97 0202020211 I 7 53 85i198524 8529 8534 853818543 6 53 02I5 0220 022510229|023410238 6 541 854818553 8558 8563 8568 8573j 5 54 o243 o247 0252 0256o 026 0265 5 55 8578 8583 858718592 8597 8602 4 1155 o2700274 0o279 0283o2880292 4 56/ 8607186Ix|86171862218626J863I 3 56 o297 30o 30o6 o 03olo3I 5o0319 3 57 8636 864i 86461865i 8656 866I 2 57 0324 0328 o333 0337 0342 o346J 2 58 8666 8670o8675 868o 8685 8690o I 1 58 o35i o355 o36o o364 o369 o373 I 59 869587008704 8787098741,487I91 59 o378 o382Jo3861 039I 395o o04 o 60 " i 1_' 1 20" 10" 40 60" 5" 0 30 1 1' 10 Co-sine of 13 Degrees. i Co-sine of 12 Degrees. /P. Pat 1 2/ 31 4'.5. 6" 7/1 8" 9/ P art 1" 2" 3/ 4/ 511 6" 7' 8" 91 1 1 22 3 3 4 4 5 n 2 2 3 3 4 4 LOGARITHMIC TANGENTS. 1 Ii _. _ Tangent of 76 Degrees. 4 Tangent of 77 Degrees. 0"2 10" 210" 30" 40" 50" _ 0' 10"j 20" 30" 40" 50" o01I0.6032291331 3408 3498 358813678 590 I oo.63663616732 6828 6924, 4; zi6 59!1 3767 3857 3947 4037 4I27 4217 58 I 723 7309 7405 75o1 7597 7694 58 2 4306 14396 4486 4576 4666 4756 57 2 7790 7886 7983 8079 8175 8272 57 3] A84614936 5026 5 I I 6 5206529656 3 8368 8465 856i 8657 8754 885056 {4 5386 5477 5567 5657 5747 5837 55 4 8947 9043 9140 9237 9333 9430 55 }' ~l 5927 601I7 608 6I98 6288 6378 54 5 9526 9623 9720 986 99I3.. IO54 61 6469165591664916740~6830~6920~ 53 61~0.640I070 20~3 30300397 0494 059I 53 7 701 I 7IOI 7I927282 7372 7463 52 7 0687 0784 0881 0979 1075 1172 52 1 8. 7553 764417734 7825 79I5 8006 51 8 I269 I366 463I 5601I6571I754 5I 9 8097 8I87 8278 8368 8459 855o 5o 9 I85I I948 2046 2I43 2240 2337 5o io Io 6o8640 8731 8822 893 900qo3 9094 4O I0 O.6424234 2I 2629 2726 2823 2 2 49 I I 9I85 9276 9367 9457 9548 9639 48 I I 30I8 3II5 32I3 33I 3407 3505 48 12 9730 982I 9912.. 3..94. I 85 47 12 3602 3700 3797 3895 3992 4090 47 13 Io. 61o2761o367 o458 o549 o640 073I 46 13 4I87 42854383 4480 4578|4676|46 4 o822 ogJ3 Ioo4 io95 II 86 I278 45 I4 4773 4871 4969 5066 5i6415262345 I 51 369 I46o I555 I 642J 734 1825 44 i 5 536o 5458 555 55653 575i 15849 4 16I I 162008 2099 2I90 2282 2373143 i6 59476045 6143 6241 6339 6437 43 17 2464 2556 2647 2739 2830 2922|42 17 6535 6633 673I 6829 6927 7026|42 i8 3013 31 05 3I96 3288 3379 347I 4I I8 7124 7222 7320 7418 75 I 176 I 54 19 3562 365413746 3837 3929 402I140 19 773 78 I 7910 8008 8Io068205 40 20 Io. 6I4II2 420414296 4388 4479 4571 39 20 10.64830384028500 8599 8697 8796 39 21 4663 4755 4847 4938 So3o 5122|38 21 8894 8993 909I g19o1928819387138 22 52I4 53o6 5398 5490o5582 5674137 22 9486 9584|9683 9782 9880|9979|37 23 5766 5858 5950 6042 6I34 6226-36 23 I0O.650078 0177 02760 374 0473|0572|36 24 63I8 64Ii 6503 6595 6687 6779 35 24 0671 0o770o869 0968 I 067 I I 66|35 25 6871 6964 7056 7148 7241 73331341 25 I265 I3641 463 1i562 i66 1760o 34 26 7425 7518 76I 07702 7795 7887 33 26 I8591I958 2058 2I57 2256|2355|33 27 7980 8072 8I64 8257 8349 8442 32| 27 4455 2554 2653 2752 21295i132 28 8534 8627 8720 8812 8905 89973 1 28 3o05 35o 3249 3349 343448 3548 3I 29 9090 9I83 9275 9368 946I 9554130 29 3647 374713846 3946 4046 4145 30 30 I O. 6I 964619739 9832 19925.71. I I ol29 301 I o. 654245143441444145441464314743 29 3i 10.620203 O296 0389 0482 0575 o668 28 3I 4843 494315o43 51 42 5242 5342 28 321 076 0854109471 io4o1 33 I22627 32 5442 554215642 5742 5842 5942127 33 3I9 I1412 i55oI 1598 169I I 784126 1 33 6042 6I42 6242 63422 6442 6542 26 341 878 1971 2o64 2I57 2250 2344 25 34 6642 6742 6842 6943 704743 74325 351 243712530 2624127I7128IO 2904|24 351 7243 73441 7444 7544 7645 7745 24 36 2997 30903 384 3277 337 13464 23 36 7845 7946 8046 8I4718247 8348123 37 3558 365i 3745 3838 3932 4025 22 |37 8448 8549 8649 8750 885o 8951 22 38 4II9t42I31430614400o4494 4587 2i | 38 9o52 9152|19253 935419454|9555 21 39 468I 4775148691496215056 5I50o20o 39 9656 975719857 9958..59.160|20 4o I. 625244 5338 543 I 5525 56915 7I 3I 957 1 4o] I.66026 I03620463 05640665 0766| I 9 4I 5807 5901 5995|6089 6I83 6277 I81 4I 0867o09681Io691II 701I271 i 372 18 42 637I 6465 6559 6653 6747 684i I7I 42 I473 I574 I676 1777 1878 1979 17 43 6936 7030 7I24 72i8 73I2 74071 i6 43 208I 2I82 2283 2385 2486 2587 i6 441 7501 7595 7689 7784 7878 7972|I5 44 2689 2790 2892 2993 3095 3I96 i5 45 8067 8161 8256|8350 8444 85391I4| 45 3298 3399 35o0 3602 3704 38o6 i4 46 8633 8728 8822 89 7 go90I I IO61 3 46 3907 4009 4I I I 4212|43I,444 I 61 3 47 920I 9295 939019484 9579 967412 47 458 4620o472I 4823[492515027 12 48 9768 9863 99581. 53. I47.242 II 48 5I29 5231 5333 5435 5537 5639 II 491 I. 630337 0432 0527 0622 07I 6 o8 I I Ioo l 491 574I 584315945 6047 6I49 6252 IO 50IO.630906 IooI IO96 1 I9I 1286 I38I 9 50o io.666354 6456 6558 666o 6763 68651 9 51I I476 1571 i666 I76I I857 1952| 8|| 5i 6967 7070 1727 7274 7377 7479J 8 52 2047 2I42 2237 2332 2428 2523| 7| 52| 7582 7684 7787 7889 7992 18941 7 53 26I8 27I3 2809|2904 2999 3o95 6| 53 8197 8299 8402 85o51860o787ioj 6 54 3I90 3285 338i 3476 3572 3667 51154 88I3 89I6 9o8 9I2r1 9224 9327j 5 55 3763 3858 3954 1449 4I45 4240o 41155 9430 9532 9635 9738 9841 9944| 4 56 4336 4432 4527|4623 4718 4814| 3 5610O.670047|01o50o0253 0356o0459 o562| 3 57 49Io 5006 5IO1I5I97 5293 53891 2 57 066607691o872 0975o1078II 81| 2 58 5485 558o 5676 57725868 559864. I'4 58 I2851 388I49I i595 I6981 8o0I I 59 6o6o16I56 6252|634816444 6541 0 ol I90o52008|2II2 22I5 23I8124221 0 i 60" 50" 40" 30" 0 20 10" 60t 50 1 40" 1 301 20" 10 Co-tangnt of 13 Degrees. Co-tangent of 12 Degrees. 1Part,", 2," 3,, 4, 5,1 6," 7"1 8/1 9,11 1",, 2,1 3/ 4,/ 5," 6// 7/' 8' 9,, I 9 19 28 37 46 56 65 74 83 Pat 100 30 40 50 60 70 80 90 0 2 LcGARITHMIC N1IN;-. O Sine of 78 Degrees. d Sine of 79 Degrees. /0" 10" 20 1 301 40" 5(Y' 0/ _ 10 20" 30" 40. 50" o 9.99go44 0409104I3.4I8 422 0427 59 o9.99194795 I951 3955 I959 I963 1967 59 I o43i o436 o440o o0445 0449 o454 58 I I97I 975 I979 1 I983 1987 I 992 58 2 o458 o4621o467 047'I 0476 o48o 57 2 19962000 2004 20082I22012 216 57 3 o485 0489 o494 o498 0o503 0507 56 3 2020 2024 2o 28 2032 20362o040 56 4 o5ii o5I6 o520,o525 o52g o534 55 4 2044 2049 2o 53 205712o6I 2o65 55 51. o538 0543 o547 o552 o556 oo56o 54 5 2o69 207312077 2081 2o85 2089 54 6 o565 o569 o57410578 o583 o587153 6 2o9312097 2II 2Io 512IO9 20 I I3 53 7 o5gIlo596 0600o60o5 1o6oo 614 52 7 2Ii8212221iI26[2I30o2I3412I38 52 8 o6I8 o622o627 o63I o636 o64o05I 8 2I42 2I461250o 25412158 1I62 51 9 o645 o649 o653 o658 o6621o667 5o 9 26612I7021I74 2178 218212I86 50 10o9.99067 o6750 o68o0o684 o689o69313 49 I9.992I90 2I 9412I98 220212206 22 IO 49 i I o6970 702o0706 0711I 075 0719148 I 2234 22182222 222612230 2234 48 12 0724 0728 o733 o737 074I o 746 47 12 2239 2-43 2247 225I 2255 2259 47 I3 0750 0755 0759 o763 1o768 0o77246 3 2263 2267 227I 2275 2279 2283 46 14 0777 o78I 0785 0790 0o794 0798 45 i4 2287 2291 2295 2299 2303 2307 45 I5 o8031o807 o8o86 82 2544 i 5 23 I I23I5 23Ig9 2323 2327 233i 4Z4 i6 o829 o833 o838 o842 o847 o85i 43 16 2335 2339 234312347 2351 2355 43 3I7 o855 o860 o864o868 873 o8774o8 2 7 2359 2363 2366 2370 2374 237842 i8 o8821o886 o8go9o8951o899ogg9o3 4 i8 238223862390386 39423982402/4 I9 09080912o09160921 092510929 40 I9 2406 24I 024I4124I8 242212426 4o 20 9.990934 o0938 0942 0947 095 o0955 39 20 9.992430 2434 2438 2442 2446 2450 39 2I og60 og640o969 0973 09770982 38 21 2454 2458 2462_246612470o2474 38 22 o0986 0990o09950999 ioo31 ioo8 37 22 2478 2482 2485 2489 249312497 37 23 IOI2 ioi6 102I I025 Io 09 o33 36 23 250Io2505 2509 2531 25717252 I 36 24 Io38 Io42o461 io5i io55 1059 35 24 2525 2529 2533 2537 254I12545 35 25 io64 io6811072 10771 o8 1o85 34 25 2549 2553 2556 2560 2564 2568 34 26 9OgO Io94I Io98 I io3 II07 iiii 33 26 2572 2576 2580 258412588,2592 33 27 1tI5 3II20II24 II28II133 1137 32 27 2596J2600 2604 2607|26I3 26I5 32 28 11I4I 11461II50o ii54 I58 II63 31 28 2619 2623 2627 263 26I35 2639 3I 291 ii67 II71 II76 ii8oI i84 i I88 30 "29 2643 2647 2651 2654 2658[2662 30 30o9. 99I I93| 11971201 1206 1210 I 21429 3o 9.99266612670o26741267812682[2686129 3i I218 I223Ji227 I23II 235 I 240128 3I 2690 2693 2697 270I 2705 2709 28 32. I244 I24811253 1257 I261 I265 27 32 27I3 27I7 272,2725 2728 2732 27 33 I270 i274 I278 I282 I287 I29I 26 33 2736 2740 2744 2748 2752 2756126 34 I295I 299 I304 I 3081 312 i3i6 25 34 2759 27632767 277I 1277512779125 351 I32I I3251 329 1I 33 338 I 342 24 35 2783 2787 2790 2794|2798 2802 24 361 I346 i350 1355 I 359 I 363 367 23 36 2806 2810 28I4 28I8 2821 2825123 37 I372 I376 I38o I384 I389 I393 22 37 2829 2833 2837 284I32845 2848 22 38 13971 4o0II4o6 i4io 4I14 I4i8 2I 38 2852 2856 286860 2864 2868 2871 2 39 1422 I427 I43i i4351 439 I444 20 39 2875 291883288 712887 289I 2895 20 40 9.993448 I452 I456 i460o 465 I469 I9 40 9.992898 2902o2906o2910o2914J291819 4I i473 14773I482 i486 I4go I494 i8 4i 2921 2925 2929 2933 2937 294I8 8 42 14981i5031I5071I5II1I5I53I5Ig5I 7 | 42 294412948 2952 295612960~2963317 43 1524 I528 I532 i536 I54o0 545 I6 43 2967 297I32975 2979 2983 29861 6 441 I549 i553 i557 i56i I566 I570 I5 44 2990 2994 2998 30023oo005300915 451 I574 I578 I582 i586 1I359 595 1I4 45 30I3130I7 302I 3024 3028 3032 I4 46 15991I6031I60716I2 I6I61i6201 i3 46 30361304o0304313047 305I13055i3 471 I624 I628 I632 I6371I64i I645 12 47 305913062 3066 3070 3074 3078 1I2 48 I649 i653 I657 I662 i666 I670 I1 48 3o8i 3o85 308913093 30971 3iooI II 491 I674 1678 I682 I687 I69I I695 IO 49 30o4 3io8 3II23II51 3II193I231I I 50o9.99I699 I703 I707 17I2 I 7I 63I720 9 50o9.993I271 3i3i 334 3i38 3I42 3146 9 5I I3724 I728 I732I 736 I74I I745 8 5I 3149 3I53 3I571 3i6i 3651 368| 8 |52 I3749 17531I757 176I I76513770 7 52 3I72 3I76 0 138o 3831 387 3I91I 7 53 I7741I778 I782 3786 1:79o:I794 6 53 3I95 3198 3202 3206 32io 32I33 6 54 I799 I8031I8071 i8 IISI5I8 I9 5 54 32I7 322I33225 3228 323213236j 5 55 / 823 i827 I832 I836 i840oi844 4 55 324o03243 3247 325i 3255 3258 4 56 I848 i8521I856 860186565 IS69 3 56 3262|3266|3270|3273|3277|328I 3 57 I873 1877 188i i'885 I889 1893 21 57 3284 3288 3292 3296 3299 33o3/ 2 58 I897 I90I3I906 I9IOI 9I4I938 I 58 33071331i 33I4133I8 3322 3325 I 59 I9221I9261 930 1934 3938 I9421 59 3329133331333713340 3344 3348 0 60" 50 1 4C0' 30i 20" 10 I60" 50" I 40/" 30/"'20" I 10 Co-sine of 11 Degrees. t1 | Co-sine of 10 Degrees. P. Part t 211 3" 4" 5" 6" 7"1 8" 9" |.'prI 1" 2" 3"1 4"1 5" 6"' 7' 7 "9". 0 1 1 r2 2 3 3 3 4 0 1 1 2 2 2 3 3 4 LO G A R IT H M IC TANGENTS. 10 3 4..Tangent of 78 Degrees. Tangent of 79 Degrees. G" | 10" 20" 30" 401 50" 0" 10" 20" 30" 40" WI' o 0I.672525 2629 2733 2836 2940o3043 59 o 010.7II348 I46o 573 i685 1798 I9IO 59 I 3147 3251 3354 3458 3562 3666 58 I 2023 235 2248 236I 2473 2586 58 3769 3873 3977 408I 4I85 4289 57 2 2699 28 I 2924 3037 3I5o 3263 57 3 t4393 4497 460I 4705 4809 49: 3 56 3 3376 3488 36o0I 37I4 3827 394o 56 4 -5017 5121 5225 5329 5433 5537155 4 4053 4I6714280 4393 45o0646I9 55 5 5642 5746 585o 5954 6o591 663 54 5 4732 4846 4959 5072 5I851 599154 6 6267 6372 6476 658o 6685 6789 53 6 54I2 5526 5639 5752 5866 5979 53 7 6894 6998 7103 7207 73I2 74I7 52 7 6093 6207 6320 6434 6547 666I 52 8 752I 7626 7731 7835 794~ 8o45 5I 8 6775 6889 7002 7I16 7230 7344 5I 9 8I849 825418359 8464 8569 8674 5o 9 7458 7572 7686 7799 79I3 8027 5o io 8778 8883 89881909319I9819303149 Io 8142 82568370848488598 8712 49 II 9408 95 I 3 96I819723 982919934 48 I I 8826 894I 9055 9I69928319398 48 12 IO 68oo39 oI44 02490355 o460 o565 47 12 95I1 9627 974I 98569970..85 47 i3 0670 0776 o0881 o0987 1092 1I97 46 I 31 I720I9903I4 428 o543 o658 0772 46 I4 I303 i4o8 I5I4 16I9 I725 I83o045 I4 o8871 I002 I I16 123i I346 I46i 45 I5 I936 2042 2147 2253 2359 2464 44 i 5 I576 I69 i8o6 I921 2036 2151 44 i6 2570 2676 2782 2887 2993 3099143 I6 2266 2381 2496 2611 2726 284i 43 I 7 3205 33 I I 34I 7 3523 3629 3735142 1I7 2957 3072 3 I87 3302 34I8 3533 42 I8 384I 3947 4053 4I59142651437I 41 i8 3649 3764 3879 3995 4 I 4226 41 19 4477 4584 4690 4796 4902 5009 4o I9 4342 4457 4573 4688 4804414920 4o 20 5 I I 5 522 - 5328 5434 554o 5647 39 20 5036 55I 5267 5383 5499 56i5 39 21 5753 586o 5966 6073 6179 6286 38 2I 573I 5847 5963 6079 6195 63ii 38 22 6392 6499 66o6 6712 68I9 6926 37 22 6427 6543 6659 6775 689I 7oo008 37 23 7032 7 I 3 7246 735317460 7567 36 23 7I24 7240 7356 7473 7589 7706 36 24 7673 7780 7887 7994 8ioI 8208 35 24 7822 7938 8o55 8I7I 8288 84o5 35 25 83i5 8422 8529 8636 8744 885I 34 25 8521 8638 8754 8871 8988 9I05 34 26 8958 9065 9I72 9280 9387 9494 33 26 9221 9338 9455 9572 9689 9806 33 27 9601 97099 8I6 9924..3i. 38 32 27 9923.40. I.57.274.39I.5o8 32 28 0o.690246 o353 o46 o568 o676 0784 3I 28 10. 730625 0742 o86o 0977 I 94I2 I I 3 291 o89 I o999 IIo07 214 1 322 143o 30 29 I329 i446 I563 I68 I 798 1I96 3o 30o I537 i645 I753 i86i I969 2077 29 3o 2033 2I5I 2268 2386 250 262I 29 3I 2I84 2292 24002 508 26I6 2724128 3I 2739 2856 2974 3092 32I O3327 28 32' 2832 294I 3049 3I 57 3265 3373 27 32 3445 3563 368i 3799 39I7 4o35 27 33 3481 359o3698 38o6 39I4 4023126 33 4153 4271 4389 4507 4625 4744 26 34 43 I 4239 4348 4456 4565 4673 25 34 4862 4980 5098 52I7 5335 5453 25 35 4782 4890 4999 5I 0752I615325124 35 5572 5690 5809 5927 604616I64 24 36 5433 5542 565i 5759 5868 5977 23 36 6283 64oi 6520 6639 6757 6876 23 37 6o86 6I95 63o3 6412 6521 6630 22 37 6995 7II3 7232 7351 7470 7589 22 38 6739 6848 6957 7066 7175 7284 2 38 7708 7827 7946 8o65 8i84 83o3 21 39 7393 7503 76I2 7721I 7830 7939 20 39 8422 8541 866o 8780 8899 19go8 2o 4o 8049 8i 58 8267 8376 8486 8595 I91 4o 937 925719376 9496 96I5 9734jI9 41 87o05 88I4 8924 9033 9I43 9252 i8 4I 9854 9973..93.2I3.332.452 i8 42 9362 947I 958i 969I 9800oo991o I7 42 I. 74057I 06g o8II Og3I io5o II70 17 43 x1o. 700020 0 29 o023910349 o459 o569 [6 43 I290 I4Io I53o i 650 1770 890 i6 44 0678 10788 o898 ioo8 III8 I228 i5 44 2010 2130 2250 2370 2490 261I i5 45 I338 i448 i558 i668 1779 1889 I4 45 2731 285I 297i 3092 32I2 3332 I4 46 1999 2109 2219 2330 2440 2550 I3 46 3453 357313694 38 I43935 4o55 I3 47 2661 277I 288I 2992 3I02 32I3 12 47 4I 76 4297 44I17 4538 4659 4779 12 4j 3323 3434 3544 3655 3765 3876 II 48 4900 o502I 5I42 5263 5384 55o5 I 49I 3987 4097 4208 43I 94429 4540 I o 49 5626 5747 5868 5989 6iio 623 IIo 5o 465 1 47624873 4984 5095 520o5 9 50o 63526473 6595[6716 68376959 9 5i 53i6 5427 5538 5649 576I 5872 8 5I 7080 720I 7323 7444 7566 7687 8 52 5983 6094 62o5 63i6 6428 6539 7 52 7809 793 8052 8 I 74 8295 84 I7 7 53 6651 676I 6873 6984 7095 7207 6 53 8539 866i 8782 8904 9026 9148 6 54 7318 7430 754I 7653 7764 7876 5 54 9270 9392 9I14 9636 9758 9880 5 55 7987 8099 82 I I8322 8434 8546 4 55 I 0750002 0124 0247 o369g0491 o6i 3 4 56 8658 8769 888I 8993 90o5 9217 3 56 0736 0858 o980 1103I 225 1348 3 57 9329 9441 955,3 9665 9777 9889 2 57 I470 i593 715 I 838 196I 2083 2 581IO.7II0001I I3 o0225 0337 o449Jo562 I 58 2206 232912452 2574 269712820 I 59' o674907860o89851OIIII231I235 oj 59 2943 3066]3189q33I2 3435 3558 o 10" r50" 40" 3'0" ]Y 1(0" 60" 50" 401" 30" 20" 10" Co-tangent of 11 Degrees. 4 Co-tangent of 10 Degrees. * PPt1 //1 2" 3W' 4" 5" 6" 7" 8/" 9 Pp 1" 2" 3" 4" 59 6"1 7" 82 91! ar 11 22 32 43 54 65 75 86 97 art12 23 35 47 59 70 82 94 106 .04 LOGARITHMIC SINES. Sine of 80 Degfrees. Sine of 81 Degrees. [_ O 0" 10". 20'' 30"1 40" / 50" 0", 10"1 20" 30" 40" 5_6 09.99335I 335513359 3363 3366 337o 59 09. 994620 46234627 46346746334633 7 59 I 3374 3377 338i 3385 3389 3392 58 I 464o 4643 4647 465o 4653 4657 58 2 33961 3400 3403 3407 34 I 344 57 2 4660 466314667 4670 4673 4676157 3 34i8 3422 3426 3429 3433 3437 56 3 468014683 4686 4690 4693 4696 56 4 3440o3444 3448 345I 345513459 55 4 4700 4703 4706 47147 47I347I6 55 5. 3462 3466 3470 3473 3477 348i 54 5 4720 4723 4726 4729 4733 4736 54 (, 3484 3488 3492 3495 3499 3503 53 6 4739 4743 4746 4749 4 752 4756 53 7 35o6 35I0 35I4 35I7 352 3525 52 7 4759 4762 4766 4769 4772 4776 52 8 3528 3532 3536 3539 3543 3547 15i 81 477914782 4785 478914792 4795 5y 9 355o 3554 3558 356i 356 5 3569 50o 9 4798 4802 48o548 8o8 48I 2 48 I 5 I0 9.993572 3576 3580o3583 3587 359I49 9I o 9 *.948I81482 4825 4828 483I 4834 49 I I 359413598 360I13605 3609 3612148 I 4838 484Ij4844 48481485I 4854 48 I2 36I6 3620 3623 3627 363I 13634 47 12 4857 486I 48644867486 74874874 47 13 3638 364I 3645 3649 3652 365646 I3 4877 488o 48833 4887 4890 4893 46 14 3660 366313667 3670o3674i3678 45 i4 48961490oo4903 49o064909 49I3 45 i5 368i 3685 3689 3692 3696 3699 44 I5i 49-6 49'914922 4926 4929 4932 44 I6 3703j3707137o103714|37I7|3721 43 16 493514938|4942 4945 4948 495I 43 I7 3725 3728 3732 3735 3739 13743 42 7 4955 49558 496I 4964 4968 497I142 i8 3746 3750 3753 3757376I 13764 4I 8 4974 4977 4980 4984 4987 49904 4 19 3768 3771 3775 3779 37823786 4 I 91 4993 49971500 5oo3 15oo6 5500914c 20 9.993789 3793 3797 38oo0038043807 39 20 9.9950I3 5o06 50I19502215025 5029139 21 38II 38I4 38i8 3822 3825 3829 38 21 5032 5o35 5o38 50o4 5o45 5o48 38 22 3832 3836 384o 3843 3847 385o 37 J 22 5o5I 5o54 5057 5o6I 5o64 5o67 37 23 3854 3857 i 3864 38683872 36 23 507o 5073 5077 5o8o 5o83 5o86 36 24 3875 3879 3882 3886 3889 3893 35 24 508915092 5096 5099 5o I2 5Io5 35 25 389713900 3904j39071391 I 394 34 251 5Io8 5I215I5I 15 58 5I21 5I24134 26 39I8392139 25 3 928393239323936 33 26 5I27 55I3534 537 5I40 54333 27 393913943 3946|3950|395313957 32 | 271 5465I50o5153 5I56 559 5 6232 28 3960o3964 3967 39713 97974 3978 3I 28 5I65 5I6915I72 5175 5178 58I8 I3I 29 3982 13985 13989 3992 399613999 30 29 5I84 5I8815I9I 5I94 5197 5200oo30 3019.994003 4006401Io401I3440I714020 29 3o09.995203 520o652I152I3152I 65219 29 3i 4024 402 74o3i 4o34 4038 4o4i 28 3I 5:222 5225 5228 5232 5235 5238 28 32 4o45 4o048 405214055 4o59 4062 27 I 32 524I 5A44 5247 525o 5253 5257 27 33 406614069 407340Ao76 408o04083 26 33 5260o 526315266 5269 5272 5275 26 34 4087 4090o4094,4097 4IOI 4Io4 25 34 5278 5282 5285 5288 529I 5294 25 35 4Io8 41Ii4I I514I I814I214I125 24 35 5297153o005303 5307 53o10531324 36 4129 4132 41364I13914143 4146 23 | 36 53I6 53I9 5322 5325 5328 533I 23 37 4i5o 4I53 4I57,4I6o 4i64 4167 22 37 533453381534I 5344 534715350122 38 417I 4I744I1784I8I 41i844I188 2I 38 5353 535615359 5362 5365 5369 21 39 4191 4I95 4I9814202 42o054209 20 39 5372 5375 5378 538I 5384 5387 20 40o9.9942I2 42I6142I9 4223 42261423o I 1 40o9.995'3901539315396 5399 54o3 534o6 i 41 4233 4236 424o04243 4247 425o i8 54 41 5412 54I5 54i8 542I 5424 I8 42 4254 4257 426i 4264 4267 427I I7 42 5427 543o 15433 5436 5439 5442 I7 43 4274 4278 428I4285 4288 4292 i6 43 5446 5449 5452 5455 5458 546II 6 44 4295 4298 4302 4305 430914312 15 i 441 5464 54671547o 5473 5476 5479 i5 45 436 4319 43224326 43243 43294333 4 45 5482 5485 5488 5491 5494 5497 14 46 4336 434o 434314346 4350o 433 13 46 55oi 55o4 555507 55io 553 55I6 3 47 4357 436o 4363 4367 4370 4374 I2 47 55i915522 5525 5528 553i 5534 12 481 437714381 4384|43871439I 43941 I 481 553715540 554 3 5546 5549 55521 II 49 4398 44 44o4 44 44o8 44III44I5IO Io 49 5555 5558 556i 5564 5567 557o IO 50 9.9944I8 442I 4425 4428 4432 4435 9 S|o 509.995573 5576 5579 5582 5585 5588 9 5Ii 4438 442 4445 4454484452144551 8 5II 5595I94 5594 97 6oi 56 6o4 5671 8 52 4459144624465446944744724476 7 |j 52 56I0o56I315616 561915622 5625 7 53 447914482 4486 4489 4492 49496 6 53 5628 563[ 5634 5637 5640 5643 54 4499[450o345o645o945I34516 5 54 5646j564915052]5655 5658566 5 - 55 45i914523 4526 453o 4533 4536 4 551- 5664 56671567o 5672 15675 5678 4 56 454o 014543 4546 4550o 4553 4556 3 56 568i 5684 5687 569o 5693 5696 3 57 4560o456314566|457o04573S4576| 2 57 5699 570257o5 57o0857I1 57I4 2 58 458o14583145871459o14593J4597 I /I 58 57I7/5720 5723 5726 5729 5732 I 59 46004603460746o146131461 11 59 5735 573815741i5744:5747 5750 60"f 50" 40" 30" 20I 10" 60" 50" 40"- 30" 1 20" I 10 Co-sine of 9 Degrees. Co-sine of 8 Degrees. P Pa rtS 1" 21" 3"/ 4/!.51 6W 71 8 911 P. Pa.rS 1" 2" 3" 4" 5/1 (" 7" 1/ 8// sy I 0 1 1 -1 2. 2 2 3 3 0 1 1 1 - 2 2 2 3 3.....'.... 1~' j" 3:' 4" 5/' 6~'?J' 8] 9]~~ LOG AR IT EMIC T A JNTS. I. _ Tangent of 80 Degrees. ~ — Tangent of 81 Degrees. I I 10" 30 40 " 3 0" 10" 201 30" 40" 50" o 53681 3813928405 474429159 o0.800287 o0424 o56 0696o0833 969159 I 442I 4544 4667 479I 4914 5038 58 I IIo6 i242 I3 79 i5I6 I652 I 789 58 2 56 5285 54o8 5532 5655 5779 57 1926 2062 2199 2336 2473 26I0 57 5903 6026 6I5o 6274 6398 6522 56 3 2747 2884 302I 3i58 3295 3433 56 4 6646 6770 6894 7018 742 7266 55 4 357013707 3844139824I I9 4257 55 5 7390 7514 7638 7762 7886 8o0 I 54 5 439414532 4669 4807 4944 5082 54 g 8I35 8259 8384 8508 8633 8757 53 6 5220 5358 5495 5633 577'I 5909 53 7 8882 900oo6913 925519380 9'o5 52 7 6047 6i85 6323 646i 6599 6738 52 8 962919754 9879...4 128.253 5I 8 6876 70'4 7152 7291 7429 7568 15 9lo.760378 o503 o628 07530o878 Ioo3 50 9 7706 7845 7983 8122 8261 8399 50 I ~0 112811253 379 I 504 I 62975449 853818677 88I61895419093 9232 49 I I i880 2005 2I30 2256 2381 250748( II 937I 95Io09649 9788 9928..6748 12 2532 2758 2883 3009 3i35 3260 47 I2 I0.8I0206o0345 o485 0624o 64 o90o3 47 I3 338613512 363813763 3889 40I5 4611 I3 10421182 1322 I46i i6oI 74' 46 I4 4i4I14267.439314519 4645/477I 45 i4 I880|2020|2I60|2300|2440 2580 45 i5 4897 (5024 5rio 5276 5402/5529!44 i 2720 2860 3ooo 134o 13280 3421 44 | I61 5655 5781 5908 6o34 6I6I 16287 431 I6 356i 3701 3842 3982 4I2214263 43 64I4654o 6667 679416920 70474211 I7 4403 4544[4685[4825 4966 510~742 I8 7174 73oI 7427 7554 7681 780841i I8 5248 5388 552915670o5811 595214I 19 7935 806218I89 83I6 8443o8570 40 19 6093 6234{6375165I7{665816799 40 201 8698 8825 895219079 9207 9334 39 20 6941 7082 7223 7365 7506 7648 39 21 946Iz9589197I619844 9971i..99/38 21 7789o793I180737826148356{8498138 2210o.770227103541o4821o6iolo737 o086537 I 22 8640 8782[8924[9066 9208[9350|37 23 0993 1I2I1I249 I377 1504 I632136113 9492 9634 9776 99I8..6i.203 36 241 761 I889 20o17 2145 2273 24o035 24 Io.82o345 o488 o63o 0773 o9i5 io58 35 25 2529 2658 2786 294 30o43 317i 34 25 12oI I343 1486 1629 1772 19I 5 34 26 3300 342813557 3685138i413942 33 26 2o58 2200oo 234 248712630 2773133 27 A407, 4200oo4329 4457 458614715 321127 296 3059 3203 3346 3469 3633132 | S81 4844 4973 5102 5231 536o 5489 3I 128 3776 3920 4o63 42071435o 4494 31 29 56i8 5747 5876 6oo6 6i35 6264 30 29| 4638 4782 4925 5069 5,2I3 5357 3o 30 6393 6523 665216782 69II 7o41 29 13o 550I 5645 5789 5933 6078 6222 29 31 770 7300 7429.7559 7689 7818 28 31 636665 I I665 5679969447088|28 32 7948 8078 82o8 833818468 8598 27 132 7233 737717522 7667 78I217956127 33 8728 885818988[9Ii89248 9378126 33 8Iot 8246 8391 8536 868I18826126 34 9508 963919769 9899..29. i6o025 34 8971 9I I6 926i 9407 9552 9697 25 35 10.780290 42 o1055 10682 08I210943 24 35 9843 9988. I 34. 279 425.5724 36 10o74 20o41335 14661I5971 727 23 361 I0.8307I 6862 I8 I I 53 I 299 i44523 37 I858 1989 2120 2251 2382 2513 22 I37 1591 I737 I883 2029 2175 2322 2| 38 2644 2775 2907 3o38 3I69 33oo 2I 38 2468 26r4 2760 2907 3o53 3200~21 39 3432 3563 3695 3826 3957 4089 20 39 3346 3493 3639 3786 3933 4o8o 20 40 42201435214484 46I514747 4879 I9 4ol 4226 4373 4520 4667 48i4 4961 I9 4I o 50III4215274154o6 5538 56701 8 4I 5,b8 5255 5402 555o5697 5844 i8 421 5802o593416o66 169816330 64631 7 42 59926139628764346582672917 43 65951672716859 6992 7124 72571 6 143 6877 7025 7172 7320 7468 7616 6 44 7389 7522 76541778717919 8052I5 l44 7764179I2 8o6o 8208 8356 85o4 15 45 8i85 83I71845o 85831871618849 1145 8653 88oi 8949190981924619395 I4 46 8982}9II5 92481938195419647I3 46 9543 9692|9840o9989. I38.287 13 47 9780 9913..461.8o0.331.446I12 47I 0.840435 o584 0733 0882 1o3r ii8o 12 48IO. 79058007130847|09801 I I4 I247I I 1148 3291 4791I6281 7771i926/2076 II 49 i38i I515 I648 1782 IgI 620o50 Io l49 2225 2375 2524 26741282312973 io 50 2i83 23171245I 2585 2719,2853 91150 323 3272 3422 3572 372213872 9 5i 2987 3122132561339o 3524 3658 8115I 4022 4172 432214472 462314773 8 52 3793 3927 4062 4I96 433i 4465 7 52 4923 50745224 5374 5525 5675 7 53 46001473414869 50o4 5I38 5273 6153 5826 5977 6i27 6278 6429 658o1 6 54 54o08554315678 5812 5947 6082 51541 673i 6882 7033 7184 7335174861 5 | 551 62i8 6353 6488 6623 67586893 4155 7637 7789 7940o809I 8243 83941 4 56 7029 1764 7299 743517570 7706 3 56 854618697 884990oo 0 9152 9304 3! 57 7841 7977 8I12 824818384 85 19 2157 94569608 97609912..64.2i6| 2 1 581 8655 879I 892790o63H9I9919335 I 58 io.850368 o520o672 o825 0977 1129 1 59 947' 9607 9743 9879. 1.I5.I5I 059 1282| 434587I 17391 892~2045 0o 60" 1 50" 40" 30" 20 10 60/1 50" 40" 30" 20" 10 Co-tangent of 9 Degrees Co-tangent of 8 Degrees. P 1" 21 311 4'1 5I 61" 7"1 8'" Part 1" 2" 3" 4/" 511 6 7"1 8 911 P'Par4t 1.3 26 39 52 65 78 91 103 116 |P. Par 14 29 43 58 72 86 101 115 130 06 LOG A I T H M C SIN ES..'1 Sineof 82 Deg.rees. Sine of 823 Degrees. 1 0" 10" 20/" 30" 40/" 150", 0" 1,, 20" 30" 401 50, 0 9.995753 5756 57595762 57655768 59 0 g.99675 16753 6756 6758 676 676A 59 I 577I 5773 5776 5779 5782 5785 58 II 6766 6769 677I 6774 6777 6779 58 2 5788 5791 5794 5797 58oo 58o3 57 2 67826784 6 787 6789 6792 6795 57 3 5806 5809 58I2 58I5 58I8 582I 56 3 6797 6800o6802 6805 6807 68Io 56 4 5823 5826 58295832 5835 583855 4 6812 685 68 682 68236825 55 5 584I 5844 5847 585o 585853 5856 54 5 6828 6830o 68336835 6838 684i 54 6 5859 5862 5864 5867 5870 5873 53 6 6843 6846 6848 685i 6853 6856 53 7 5876 5879 5882 5885 5888 589I 52 7 6858 6861 6863 6866 6869 687i 52 8 5894 5897 5899 5902 5905 590851I 8 6874 6876 6879 6881 6884 6886 5I 9 59II 5914 5917 525923 5 926 50 9 6889 689I 6894 689616899 690I 50 I0 9.995928 593I 15934 5937 5940 5943 49 I 9.996904 6906699 69129 69I1469I7149 II 5946 5949 5952 5954 5957 5960 48 | I 69I9 6922 6924 6927 6929 6932 48 12 5963 5966 5969 5972 5975 5978 47 |I2 6934 6937 6939 6942 6944 6947 47 13 5980 5983 5986 5989 5992 5995 46 i3 6949 695216954 695716959 6962 46 14 5998 6oo0 60 03 6006 6009 6012 45 I4 6964 6967 6969 6972 6974 6977 45 i5 6o05 60o8 602I 6023 6026 6029 44 15 6979 6982 6984 6987 6989 6992 44 I6 6032 6o35 6038 604i 6043 6o46 43 i6 6994 6997 6999 7002 7004 7007443 7 6o49 60512 6o55 6o58 6o6I 6063 42 17 7009170II 7014 70I61 70I9 7021o42 18 6066 6069 6072,6075 6078 6o8I 4I I8 7024 7026 7029 7o03 703 417036|41 19 6083 6086 6089 6092 6095 6098 4o 1I 7039 704I 7044 7o46 7049 705IJ4o 209 996I00 6I03 6I06 6I09 6II2 6II5 39 20 9.997053 7056 7058 706I 7063J7066j39 21 6I7 I7 620 6I236I26 6I29 6i3i 38 21 7068 7075 7073 70776 778 7080o38 22 6I346137 6I4o06i43 6I46 6I48 37 22 708317o85 17170887 1709 709395 37 23.6151 6I5461576160o6I62 665136 23 70987I00O7I02 7I05 7I077II036 24 6I68617I|6174,6I77 6179 6I82 35 24 7II2 7II5'7II 7 7120 7122 7I24 35 25 6I85 6I886I9Ii6I93 3 6I966Ig99 34 25 7I27 7I297I32 734 7376713934 26 6202 6205 6207 62I0 62I3 6216 33 |26 7 7I4X17I44 7146 7I49 7151 7I54 33 27 6219 622I16224 6227623023232 32 27 7156|7158 7161 7163171667I68132 28 6235 6238 624I 6244 6246 6249 3I 28 717071731775 7I78 7i80 7182131 29 6252 6255|6257 6260 62636266 30 29 7I85|7187 719o0 792 71947197130 3o 9'996269 6271 6274 6277 6280 6282 29 30 9.997I99 7202 7204 7206 7209a 721I 29 3i 6285 6288 629Ir6293 6296 6299 28 31 72I4 72I6 7218 7221 7223 7226 28 32 6302 63o5 6307 63io63i3 63I6 27 32 7228 7230 7233 7235 7238 724o 27 33 63I8 632I 6324 6327 6329 6332 26 33 7242 7245 7247 7249 7252 7254 26 34 6335 6338 6340 6343 6346 6349 25 34 7257 7259 7261 7264 7266 7268 25 35 635i 6354 6357 6359 6362 6365 24 35 7271 7273 7276 7278 7280 7283 24 36 6368 6370 6373,6376 63791638I 23 36 728517287 7290 729292 7294 7297 23 37 6384 6387 6390 6392 6395 6398 22 37 7299 730I 7304 7306 7309 7311 22 38 6400 6403 6406 6409 64II 64I4 21 38 73I3173I617318 7320 7323 732512I 39 64I7 64I9 642216425 6428 6430 20 39 7327 7330 7337332 34 7337 7339 20 Ao 9 996433 6436 6438 644I 6444 6447 19 40o9 997341 7344 7346 7348 735I 735.3 19 4i 6449 6452 6455 6457 646o 6463 I8 4I 7355 7358 7360 736 2 7365 73671I8 42 6465 646867 6474 6476647917 42 7369 7372 7374 7376 7379 7381 I7 43 6482 6484 6487 6490o 6492 6495 i6 43 7383 7386 7388 7390 7393 7395 i6 44 6498 6500j6503 6506 65o8 65ii i5 44 7397 7399 7402 7404 740 6 40 7409 I5 451 654 65655 6514 65I7 6SI9 6522 6525 6527 14 45 74II 7413 746 7418 7420 7423 14 46 653o 6533 6535 6538 654i 6543 i3 46 7425 7427 7429 7432 7434 7436 13 47 6546 6549 655i 6554 6557 6559 I2 47 7439 744I 7443 7445 7448 7450 12 48 6562 6565 6567 6570 6573 6575 II 48 7452 7455 7457 7459 746I 7464J x 49 6578 6580o 6583 6586j6588 659I I0 49 7466 7468 747I 7473 7475 74771IO 5019. 996594 65966599 6602 66o4 6607 9 509.997480 7482 7484 7487 74891749I 9 5I 66Io 6612 66I5166I8 6620 6623 8 5I 7493749674987500750 0275o5 8 52 6625 6628 663i 6633 6636 6639 7 52 7507 7509 751175I4 756 175I8 1 7 53 664i 6644 6646 6649 6652 6654 6 53 7520 7523 7525 7527 7530 75321 6 54 6657 666016662 6665 6667 6670 5 54 753417536 7539 7541 7543 75451 5 55 6673 6675 6678 668i 6683 6686 4 55 754775507552 o 7557554 7556 75591 4 56 6688 6691 6694 6696 6699 6701 3 56 7561 7563 7565 7568 7570 7572[ 3 57 6704 6707 6709 6712 67I4 6717 2 57 7574 7577 7579 758I 7583 7585[ 2 58 6720 6722 6725 6727 6730 6733 I 58 7588 7590 7592 7594 7597 7599 I 59 6735 673816740o6743 6746 6748 0 S9 7601 760o3176o05760876I08 7612 o 60" 50" 40" 30 2" 10" 60 50" 40" 30" " 1 2 0" Co-sine of 7 Degrees. Co-sine of 6 Degrees. P. Fart 1" 2"' 1 4' "ll 6"' 7"' 8" 9" pptS 1" 2" 3" 4" 5" 6" 7" 8 9 " art 0 1 1 1 2 2. ar 0 0 1 1 1 1 2 2 2 L o A r I M I C T A N N T S. 1 07 Tangent of 82 Dearees. Tangent of 83 Degrees 0" -I10" 20" 30" 40' 50" t 0" 10" 20" 30" 40" 50" o Io0852,97 23502503 2656 28092962 59 1I.gio856 3 I3o 205 I379 553 [727 9 3:5 3268 3421 3575 3728 388i 58 I I902 20761225I 242612600 2775 58 2 4034 4I88 434i 4495 464914802 57 2 2950 3I25 33oo 3475 365o 3825 57 3 4956'51IO 5263 5417 557I 5725 56 3 4000 4I76 435I 4527 470o2s4878 56 4 5879 6033 6I87 634i 6496 665o 55 4 5053 5229154o5 558i 5757 5933 55 5 6804 6958 7 I 3 7267 74227576 54 5 6I09o6285 646I 6638168I4 6990o54 6 7731 7886 8o04 8195 8350 85o5 53 6 7I67 7343 7520 7697 7874 80o5 53 7 8660 883I 5 89709125 9280 9436 52 7 8227 84041858I 875918936 9 I 3 52 8 9591 9746 9902..57.2I2.368 5I 8 9290 9468 9645 98231.... 1 I78 51 9 Io.86o524 o679 o835 099 I I 46 I 302 501 I0. g20356|0534 07I o8go90 I o681246 50 Io I458 i6i4 1770 I 926 2082 2239149 Io0 I424 I602 178I I959 2I382316 49 11 2395 2551 2708 2864 3020|3177 48 I I 2495 2673 285213o3i 320 3389 48 12 333313490 3647 38o3 3960|4 17 47 12 356813747 3926 4I0 5 4285 4464 47 I3 4274 4433 4588 4745 4902 5059146 13} 464448231500oo3 583 5362 554246 34 5216 5374 5531 5688 5846 6003 45 i4 5722 59021608216262 6442 6623 45 I5 6i6x 63g 1647616634167921695o044 i5 6803698417I64 7345 7525 7706 4 I6 7I0~77265 7423 758I 7739 7898 43 i6 7887 8068 8249 8430 1 86 I 8792 43 I7 8056182i4 8372[853I 868918848142 I7 8973 9154|9336 95I 79699 9880 42 8 900oo69 65 932419482 964I 980014I I8 I. 930062 024410425 o607 0789 097 I4I 39 9959. I 8.2771.436.595. 754140 I9 II54 336 55I8 I 700 I883 206540 20o,.8709 3 I072 232 39 I I55II 7IO39 20 22481243026 3 2796 2979 3 62 39 21 I8701202921 892349 25082668 38 2I - 3345|3528137 I 389414078126 I 38 22 2828 2988 3I48 33o8 3468|3629|37 22 4444 4628148I 2499515I79 5363 37 23 3789 3949 4 o094270 4430o459I 36 23 5547 573I 595 60o99 6283 6467 36 24 475I 4912 5073 5234 5394 5555135 24 6652 6836 702I 72o5 7390 7575 35.25 57I615877 603816199 6360|6522 34 251 77607945 8I3o0 83I5 8500 8685 34 26 6683|6844 700617 67 732917490~33 26 8870 9056924I 9427196I2 9798133 27 7652178I 3797518i 37[8299|846I 32 27 99841. 69. 355. 54. 727. 9'i432 28 862318785 8947/9I309927 1943313 280I.94II00o I2861I472 I659 I 845 203231 29. 959619758992I..83.246.4o0830 29[ 22 I9240612592 2779 296613 53130 30o 1o.88057o 0734 0896 I059 I3222 I385 29 3o 334i 3528 37I5 3902 4090 4277 29 3i I548 17II I87420o38 2201 2364l28 3i 4465 46531484i 5028 52i6 5404 28 32 2528269I 2855130I8 3I82 3345127 32 5593 578I 596916657 6346 6534 27 33 3509o3673 38371400I 4i65 4329126 33 6723 6912 7I00 7289 747b1766726 34 449314657 482I 14985 5I50o53I4125 341 78568o45 8234 8424186I31880325 35 547915643 5808 5972 6I37 6302 24 35 8992 9I82]937I 9561 975I 994I 24 36 6467 6632 6796 6961 7127 7292 23 36 IO.950I3I 032I o5II 0702 0892 io8323| 37 7457 7622 7787 7953 8i i8 1828422 37 1273 i464 i654 I845 2036|2227t22 38 844918615 8781 89469II12|9278211 38 24I82509|2800 299I 3I8313374|2 39 9444|96IO 9776/9942..io8.27420o 39 3566 3757 3949 4I4I 4332452420 4o o.890o44I o0607 77310940oI o6 12731 9 4o 47I6 490815 oi 5293 548515678319 4I i440I60o6 I7731I94023Io7 2274 I81 4i 5870|6063|6255 64481664I 68341i8 42 3443I2608 277512942 3Iio032771I7 42 7027 7220o74I3 7606 7800o79931I7 43 3444|362 2377913947 4 I 5 4282] 6 43] 8I87838o08574 8768896I|9,i55i6' 44 4450o46I8 47861495451I22S5290oI5 44 9349|95441973819932.I261.32I3I5 45 5458|5626 579515963 6I 3I63oo004 45I o.96o05 507Ioo9o5 10991I2941I489i14 46 6468 6637 68o066974 7i43 73I2 I3 46 I684 I879 2074 2270 2465 2663 i3 47 7483 7650 78I9 7988 8I57 8326 12 47 2856 3052 3247 3443 3639 3835[I 481 8496|8665 8834900419I 731934313 I 48 403I 4227144241462o048 650o 31 I 49 95I319683 9852..'22,392.362Io 491 5209 540615603 58001599716I94I So 0o.qeo532o 702 o8731 o43 I2i3 i384} 91 50 639I 65886785 6983 7I80o7377 9 5 I 554 I 724 I895206622362407 8 I 5 I, 575177731797I 8i69|83678565| 8 521 2578|2749 2920o309I 3262134331 7 52 8763 896I 9I59 9358 9556 9755 7 53 36o053776 3947414 I I94290o4462| 6 53 9954. I52].35.i 55.o 749.948 6 154 4633 48o054977 5i491532015492 51 54IO, 97I,48 347 i546 746 I94521 45 5 55 5664 5837 6009 6I8I 6353 65261 411 55 2345 254527451294513I45 3345 4 56 6698 687I 7043 72I6 7388 756I 31156 345 3746 3946 4147 4347 4548 3 57 77347907 880808253 8426 8599 2 157 4749 4950o 55I 5352 5553 5755 2 58 87728946 9I I9192921946619639 I 1158 5956 6i571635916566 6762|6964 I 59 98319987.i6i.334-.5o8.682 o 59 7I6617368!7570 7773179758177 0 -60"I 50" 40'" 30" 20- 10" -60" 50" 40" 30" 20" 10" Co-tangent of 7 Degrees. x Co-tangent of 6 Degrees. 1" 2P " i 3" 4" 5" 6" 7" 8" 9/ P" 1" 2" 3" 4" 5" 6" 7" 8" 9I P.i art 16 33 49 65 81 98 114 130 146 P.*ar 19 37 5 75 94 112 131 1 0 168'atl19 37 5' 75 94 119J 131 l~O 1fir .0 8 L o C AR IT I' H I C;! r N E S. *fli Sine of 84 Degrees. d Sine of 85 Degrees. t 04: " i 10"/ 20" 1 30" 1 40 " 1 1' 0" 10" 2011" 30" 40/ 5iY' 0.9976I4 76I7, 76I9762I 7623 7625 59 O 9.998344 8346 8348 83508o 3528353 59 I 7628 7630 7632 7634 7636 7639 58 I 8355 8357 8359 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o85o9 85II85I3 85I5 44 16 7822 78247826 7828 7830 783343 i6 85I6185I818520 8522 8523 8525 43 I7 78351783783 839 784r17843 7845 42 17 8527 8529 853o 8532 8534 8535 42 i8 784717849 785217854 7856 78581 8 853718539 854i 8542 8544 8546 4I 19 7860 7862 7864 7866 7868 7870[40 i9 8548 8549 855i 8553 8554 8556 4o 20 9.997872 7875 7877 7879 7881 7883 39 20 9.998558 8560 8561 8563 8565 8566 39 21 7885 7887 7889 789I 7893 7895 38 21 8568 8570 8572 8573 8575 18577 38 22 7897 7900 7902 7904 7906 7908 37 22 8578 858o 8582185841858518587137 923 1 7912 i6 4 79798792 36 23 858918590g8592 8594 859518597 36 23 7910 791217914 7916 7918 3792036 241 7922 7924 7926 79291793I 7933 35 24 8599 860I 8602 86o4 86o6 8607 35 25 7935 7937 7939 794I 7943 7945 34 25 8609186II 86I2 86I4 86I6|86I7 34 26 7947 7949 795I 7953 79557957 33 26 86Ig 862I 8622 8624 8626 8627 33 27 7959 796 I 79637965 79671797~032 27 8629 863i 86331863418636 8638 32 28 7972 7974 7976 7978177979823 28 8639 864I 8643 8644 8646 8648 31 29 7984 7986 7988 7990 799299 9430 29 8649 8651 8653 8654 8656 8657 3o 3019.997996 79988000o 8002 80oo48oo6 29 30 9.998659|866I 8662 8664 8666 8667 29 3: 8oo8 08o080I2 8o0I4180618oi8 28 3I 8669|8671 86721867418676 8677128 32 8020 8022 8024 8026 8Q28 8o3o 27 32 8679 8681 8682 8684 8686 8687 27 33 8o32 8034 80368038 8o4o18042 26 33 868918690 8692 8694 8695 8697 26 34 8o4480o46 8o4818o050o 805218o54 25 34J 86991870018702 8704870o5 8707 25 35 8056 8058 806018062 8064 8066124 35 8708|87Io18712 87I3 87i5 8717 24 36 8068 8070 8072 8074 8076 8078 23 36 871818720o872I 8723 8725 8726 23 l 371 8080o 882!8o84 8o86 8o88 8090 22 37 8728 87291873I 8733 8734 8736 22 38 8092 80948096 8098 8Ioo 8oI02 2I 38 8738 8739 874I 8742 8744 8746 2I 39 8I04[8Io6 8o8 8IIo 811I2 8II4 20 39 8747 8749 8750 8752 8754 8755[20 4o 9.998116 8I 8 812o 812281I24 8126 9i 4o 9.998757 8758 8760 8762 8763 87651 I 42 8I283 83o I3I8I33 835 8I37 8 i 4i 8766 8768 8769 877I 8773 8774 i8 |42 8I39184I 8I438145 8I447 8I49 I7 42 8776|8777|8779 8785 8782 8784 17 431 8I5I 8I53!8I55 8I5718I59 8I6I I6 43 8785 8787 8788 8790 8 7987928793 i6 44 8I63 8165 8I67s8I688I70o 8172 i5 44 8795 8796 879818799 88o0 883i5 45 8I7418I76i8I788 8i8 I82:88I84 14 45 8804|8806|88078809 881 I o88 I 21 4 46 8I86 8188818I1o I92 8I94 8I95 I3 46 88i3 88i5 88I7 88i8 8820 8821 3 47 8I97 8I99 820I 8203 8205 8207 I2 47 8823 8824 8826 8827 8829 88311 2 48 8209/82IJ 82I3182I5182I7 821811 48 883218834[8835|8837 883818840oI | 49 8220 8222 8224 8226 82288230 IO 49 884i 8843 8844 8846 8848 88 49 I0 50o 9.998232 823418236 8238 8239 8241 9 50o9.99885I18852 8854 8855 885718858l 9 5i 8243 8245 8247 8249 825I 8253 8 51 886o 886i |8863 8864 8866 88671 8 52 8255 8257 8258 8260o8262.8264 7 52 886918870 8872 8873 8875188771 7 53 8266 8268 8270 8272 827318275 6 53 8878 888o 888i 8883 888 4 88 86 6 54 8277 8279 828I182838285 828758 5 54 8887 8889 8890 8o8892 8893 8895 5 55 8289 8298292 8294 829618298 4 551 8896 8898 8899 890I 8902 89o04 4 56 83oo 830283o383o5 830718309 3 56 89o5 8907 8989078 989Io 89 89I3| 3 57 83II|8313|83I5|83I6|83I8i8320 2 l 57 89,4189I6189I7|89I189218928922 2 58 8322 8324 8326832883 29'833 58 8923:8925 8926|8927 8929|893o I 59 8333 8335 8337 83391834I 8342 o 59 8932 8933!8935,8936 8938 8939 o0 60" 50' 40" 130"120"10" 60.50" 140" 1 30" 1 20" 1 10" Co-sine of 5 Degrees. J',q Co-sine of. 4 Degrees. I"i P. Part 1" 2" 3" 4" 5" 6" 7" 8" 9" P p 1" 2" 3" 4 5" 6" 7",'1 9, 0 0~ 1 1 1 1 1 2 2. *art 0 0 0 1 1 1 1 o,,jj~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ LOGARIT IMIC TANGENTF. I Tangent of 84 Degrees. d Toangent of 85 Degrees. | - (Y' 10" 20" 30" 401' 50" 1 - 0 10"1 20"/ 30/" 40/ 50" 010.978380 8582 8785 8988 9I91 9394 59 oII.o58048 8291 8533 8776 9019i9262159 I 9597 9800....3.206.4o.63 58 I 9506 9749 9993.236.48o.724 58 2 10.9808I7 I02I I224 1428 I632 i836 57 21II.060968 12I2 i456 I701 1945 2I90 57 3 204I 2245 2449 2654 2858 3063 56 3 2435 2680 2925 3I70 34i6 366i 56 4 3268 3472 3677 3882 4087 4293 55 4 390714I53 4399 4645 4891 5i38 55 5 4498 4703 4909 5 II4 5320 5526 54 5 5384 563i 5877 6124 6371 6619 54 6 573215938 6I44 635o 6556 6763 53 6 686617II3736I 7609 785718I05 53 7 6969 7I76 7382 7589 7796 8oo3 52 7 8353 860o 8850 9098 9347 9596 52 8 82I0 84I7 8624 883 I 9039 9246 5I 8 9845..94.343.593.842 1092 51 9 9454 9662 9869.. 77.285.493 50o 9 I.07o342 I592 I842 2092 2343 2593 5o Io0 0.990702 09I0 II8 i327' I535 1744 49 I 0 2844 3095 3346 3597 3848 4oo00 49 II I953 2I62 2371 2580 2789 2998148 II 435I 4603 4855 5I07 5359 56II 48 12 3208 34I7 3627 3836 4046 4256 47 I2 5864 6ii6 6369 6622 6875 7128 47 i3 4466 4676 4886 5096 5307 55I7 46 I3 7381 7635 7888 8142 8396 865o 4G I4 5728 5939 6149 6360o657I.6782 45 i4 8904 9159 94I3 9668 9922.I77 45 I5 6993 7205 74I6 7627 7839 805I 44 I5 I.0o80432 o688 0943 II99 I454 1710 44 I6 8262 8474 8686 8898 9I1I 9323 43 I6 I966 2222 2478 2735 299I 3248 43 17 9535 9748 9960.I73.386.59942 1I7 3505 3762 40I9 4276 4534 4791 42 8 I.000ooo812 1025 I238 I45i i665 1878 4I i8 5049 5307 5565 5823 6082 6340 o4 19 2092 2306 2519 2733 2947 3I6I 4o I9 659916858 71I 7 7376 7635 7894 4o 20o 3376 3590 38o4 4o09 4234 4448 39 20 8I54 84I4 8674 8934 9194 9454 39 21 4663 4878 50 93 530 8 5524 5739 38 21 97'I5 9975.236.497.758 1020 38 22 5955 617o06386 6602 68I8 7o34 37 22 II.09128I I543 1804 2066 2328 2590 37 23 7250 746617682 7899 85I5 8332 36 23 2853 31I5 3378 364i 3904 4I67 36 24 8549 8765 8982 9199 94I 79634 35 24 44301 4694 4957 522I 5485 5749 35 25 985I..69.286.5o4. 722.940 34 25 6oi3 6278 6542 6807 7072 7337 34 26 Ix.o II58 I376 1594 i8i3 203I 2250 33 26 7602 7868 8I33 8399 8665 893I 33 27 2468 2687 2906 3I25 3344 3563 32 27 9197 9463 9730 9996.263.530o32 28 3783 4002 4222 444I 466i 488I 3I 28 11. 100797 IO65 I332 I600I868,2I36 3[ 29 5ioi 532I 554i 5762 5982 6202 30 29 2404 2672 2940 3209 3478 3747 3o0 30 6423 6644 6865 7086 7307 7528 29 30 4oi6 4285|4555 4824 5094 5364|29 31 7749 797I 8I92 8414 8636 8857 28 3i 5634 590461I75 6445 67I6 6987!28 32 9079 y3oI 9524 9746 9968.I91 27 |32 725827529 78oI18072 8344 86I6 27 33 11.02o4I4 o636 o859 I082 3o05 1528 26 33 8888 9I60 9433 97o059978.25I.26 34 I1752 1975 2I99 2422 2646|2870 25 34 i.Io510240798 I07I1 I345 i;68 I892i, 35 3094 33I8 3542 3767 3991 42I6 24 35 2I67 2441 27I5 2990 3265 3540[24 36 4440o4665 4890oii 5 5340o5565 23 36 38i5 4090 4366 4642'49I7 5i93 23 37 579I 60o6 6242 6468 6693 6919 22 37 5470o 5746 6023 6299 6576 685322 38 7145 7372 7598 7824 8o05I8277 2I 38 7I3I174o087686 7963 8241 8520 2I 39 8504 873I 8958 9I85 9412 9640 20 39 8798 9076 9355 9634 99I3.I92 20 40 9867..95.322.55o.778 ioo6 I9 40 II.I20471 075 1 Io3i I3II 1591 1871 19 411.o3I234 1462 169I I9I9 2I48 2377 i8 41 25I1 2432 2713 2994 3275 3556 i8 42 2606 2835 3064 3293 3522 3752 17 42 3838 4II9 44oI 4683 4966 5248 I7 43 398 42I I 444I 467I 490oi 5i3I 6 43 553I 58i3 6096 6380o 6663 6946 I6 44 536i 5592 5822 6053 6284 65-4 i5 44 7230 75i4 7798 8o82 8367 865i I5 45 6745 6977 7208 7439 7671 7902 I4 45 893619221 9506 9792..77.363 i4 46 8134 8366 8598 8830o 9062 9295 3 46 1. I306490o9351I22I 50o8 I794208 11i3 47 9527 9760o 9992.225.458.69I 12 47 2368 265612943 3231 35i8 38o6 12 48|1I.040925 I 58I 39I I625 I859 2092 II 48 4094 4383 467 4960o 5249 5538 iIo 49 2326 2561 2795 3029 3264 3498 io 49 5827 6r17 6407 66971 6987 7277 10 50 3733 3968 4203 4438 4673 4908 9 50 7567 7858 8I49 844o 873I 9023 9 5I 5i44 5379 56I5 585i 6o87 6323 8 5I 93I4 9606 9898.190.483.775 8 52 6559]6795 7032 7268 75517742 7 52 II I4io68 I36I 654 I947 224I1 2535 7 53 7979182I6 8453 8691 8928 9166 6 53 2829 3I231 347 3712 4007 43oi 6 54 94031964I 9879 1II7.3561.5 94 5 54 4597 4892 5i87 5483 5779 6075 5 55 I.050832|I07I I3I0o 549 I788 2027 4~ 55 6372 6668 6965 7262 7559 7856 4 56 2266 25o6 2745 2985 3225 3465 3 56 81548452 875o 9048 9346 9645 3 571 37051394514I85 4426 4666 4907 2 57 9943.242.542.84 11I4III40o 2 58 5x48 5389 5630 587i 612 6354 x 58 II.I5740o 204I 234I 2642 2942 3s431 I 59, 6596 6837 7079 732I 7563 7806 o 59 3545 3846 4148 4449 4752 5054 o -- 60 50" 40" 30" 20" 10"o 60" 10"/ 40" 30" 201 10 iT Co-tangent of 5 Degrees. Co-tangent of 4 Degrees. I 1"2" 3"11 4" 51 6" 7/" 8" 911 pp 1" 2"3/ 411 5", 6" 7t 8" 9!, P. Part 22 44 66 88 110 132 154 177 199 P. Part) 27 54 81 108 135 162 188 215 2422 1 10 LOGARITHMIC aSINES..i Sine of 86 Degrees. Sine of 87 Degrees. 0" 10" 20" t 30" 4 4 5 0" l 10" 20 3'| 40" 50"Y 0o9.99894I 8942.8944.89451894718948 59 9994049406 9407 9408 9409 94o 59 8950 895I 895318954 8955 8957 58 I 94II 94I2 943 94I4 9415 94I6 58 2 8958 8960 896I 8963 4 84 8966 57 2 94i8 94I9 9420 942I 9422 9423 57 3 8967 8969 8970 897i 8973 8974 56 3 9424 9425 94226 9427 7 9428 943 56 4 8976 8977 8979 8980 8982 8983 55 4 943 9432 94339434 9435 9436 55 5 8984 8986 8987 8989 8990 8992 54 5 9437 9438 943994 40 9445 9442 54 6 8993 899518996 8997 8999 9000 53 6 9443 9445 9446 9447 9448 9449 53 7 9002 900319005 9006 9007 9009 52 7 9450 945I 9452 9453 9454 9455 52 8 901oi0o2 90 9 90oi5 901oi6 9oI7 5 8 9456 9457 9458 9459 9460 946I 55 9 90goI9 90209022 9023 9024 9026 5o 9 9463 9464 9465 9466 9467 9468 5o 109.999027 90299o030 9032 9033 9034 49 09 9999469 9470 947 947'2 9473 9474 49 II 9036 903719039 9040 904I 9043 48 II 9475 9476 9477 9478 9479 9480 48 12 9044 90469047I 9048 9050 9055 47 I2 948I 9482 9483 9484 9485 9486 47 i3 9053 905490505 9057 90o58 959 46 I3 9487 9488 9489 9490 949I 9492 46 I4 906I 9o6290o64 9065 9066 9068 45 I4 9493 9495 9496 9497 9498 9499 45 5 9069 9 90739071 9072 9073 9075 9076 44 5 9500950 9502 9503 9504 9505 44 i6 9077 907919080 9082 9o8 9o844384 4 6 9506 9507 9508 9509 95Io 95 II 43 17 9086 908790o88 909go 909I 9092 42 57 9512 9513 95I4 9515 9516 9517 42 x8 9094 9095S9097 9098 9099 9II0, 4 I8 95895I 1 9520 952I 9522 9523 4I I1 9oI02 9Io39Io 5 gIo6 9Io 7 9,09 4o I9 9524 952526 9527 9527 9528 4o 20 9.99910II 9I119I913 4 9I 5 9II7 39 20 9.999529 9530 953I 9532 9533 9534 39 21 9II8 912092 924 91229I I24 5 38 21 9535 9536 9537 9538 9539 9540 38 22 9I26 9528 9529 9i30 9I32 9I33 37 22 954I 9542 9543 9544 9545 ~546 37 23 9I34 9I36 9137 9I38 9I40 914I 36 23 9547 9548 9549 9550 955I 9552 36 24 9142 9I43 9I45 9I46 9I47 9I49 35 24 9553 9554 9555 9556 9557 9557 35 25 9I50o9I5I|9I5319I54 955 9I5734 25 9558 9559 9560 956I 9562 9563 34 26 9I58 9I5919I G 9I62 9I63!9I65 33 26 9564 9565 9566|9567 9568 9569 33 27 966 9I67 9I68i9I70 9I7I 9I 7232 27 9570 957I 957219573 9573 9574 32 28 9I74 917519I7691I78 9I799180o 31 28 9575 9576 9577 9578 9579 9580013 29{ 9I8I 9I83 9I84|9I85 9I87 9 I88 30 29 958I 9582 9583 9584 9585 9586 30 3o,..99989 1990 9I92993 91 94 9i9629 30o 9.999586 9587 9588|9589 959959 9529 3I 919791I9819I99 920I 9202 9203 28 31 9592 9593 9594i9595 959619597 28 32 92059206o9207|9208 92I0 92II 27 32 9597 9598 959919600oo960o 9602 27 33 9291292 2I5192I6 929179219 26 33 9603 9604 96059606 960696 69607 26 34 9220o922I19222 9224.922599226525 34 9608960996Io96II 9612 96I3 25 35 9227'922919230 9239232 9234 24 35 96I4 9614 961965 69667 9618 924 36 9235 9236 9237 9239 9240o924I 23 36 96I9 9620 962I 9622 9622 9623 23 37 9242 9244 9245 9246 9247 9249 22 37 9624 9625 962696 27 962819629 22 38 925o 925519252 9254 9255|92562I [ 38 9629 9630o 963i 9632 9633 9634 2I 39 9257 9258'9260o 926 9262 9263 20 39 9635 9635 9636 9637 9638.963920o 40 9.999265 9266 9267 9268 92709271 I9 4o 9.999640964 5 964I 964 2 9643:9644 I9 4:[ 9272 9273 9274 9276 9277 9278 I8 4I 9645 9646 9647 9647 9648 9649 I8'42. 9279 928o 9282 9283 9284 9285 I7 42 9650 965I 9652|9653 9653 9654 I7 43 - 9287 9288'9289 9290 929I 9293 i6 43 9655 9656 9657 9658 9658 9659 i6 44 9294 9295 9296 9297 9299 93001 5 441 9660 966I 5 2|9269663 9663 9664 15 45 9301 9302 9 03 930 9306 9307 i4 45 9665 9666 966719668 9668 9669 I4 46 9308 9309o93o193I2 93139341I3 3 46 9670967I 967296719672967319674i3 47 93I5 93I6 93I8 93I9 93200932I 52 47.9675 9676 96779677 9678 9679 12 48 9322 9323 9325 9326 9327 93285 5| 48 9680 9681 968i 9682 9683 9684 II 49 9329 933I 9332 93333 93349335 Io 49 9685 9685 9686 9687 9688 9689 |0 509.999336|9338 9339 9340 934l 9342 9 50 9.999689 9690 969I 9692 9693 9693 9 5; 9343 9344 934619347 9348 9349 8 5I 9694 9695|9696|96979769 98 8 52 9350o 1935i 9353 19354 9355 9356 7 52 9699 9700 9700970I9702 973 7 53 93579357 8 9359 936I 9362 9363 6 53 9704 9704 9705 9706 9707 9707 6 54 9364|9365|93669367 9369 9370 5 54 9708979197197I197I 97I2 5:55 937I19372 9373 9374 9375 9377 4 55 9753 97I497i4 97I5097I619717M 4 56 9378 9379 9380 938I 9382 9383| 3 56 971I7 97i8 97I919720 9 72O0.972 3 57 9384 9385 9387 9388 9389.939O 2 57 *9722 9723 9723 9724 9725 9726 2 58 9391 9392 9393 9394 939619397 I 58 9726 9727 9728 9729 9 729 973o 59 939893999 40040 942 94031 o 591 973 9732 9732 9733_9734197351 0''' 601"0'' 50" 1 " 50" 40" 30"" -- 2" 10" Co-sine of 3 Degrees. Co-sine of 2 Degrees.. t 1" 2/ 3/" 4/1 5/1 6/" 7"1 8/" 9" l " 2/" 3/1'*4 5"' 6" 711 8il 9 art 0 0 1 1 1 1 1 1 t 0 0 0 01 1 1 1 LOG A RIT H M1L, TANGENTS. 1 Tangent of 86 Degrees. Tangent of 87 Degrees. 0" i 10' 20"1 30"1 40" 50f"' 0" I 10" 1 20" 30" _ 40" 50"O'0 II. 55356 5659 5962 6265 6568 6872159 o i 1.28060o4 007 14I 1814 22I9 2623f59 I 7 75 7479 7784 80 88 8393 8697 58 I 3028 3433 3839 4245 4C52 5o58 58 I 2 9002 9308 963 99 I 9.224.53o0 57 2 5466 5873 628I 6689 7c98 7507 57 o1. I6o837 1143 i45o 1757 2064 237I 56 3 79178326 873791I479558i997056 4 2679 2987 3295 3603 39II 4220 55 4 11.29038207941 206 I6I9 2033 2446 55 5 4529 4838 5147 5457 5766 6076 54 5 2860 3275 3690 4I05 452I 4937 54 6 6387 6697 7008 7318 7629 794I 53 6 5354 5770 6i88 6605 7024 7442 53 7 8252 8564 887619188 9500 9813 52 7 7861 8280 870091I20 9541 9962 52 811 i I.70126 o4390752 I0661 379 i693 5I 8 I.300383 o805 1227 I649 2072 2496 51 9 2008 2322 2637 295I 3267 3582 50 9 2919 3344 3768 4193 46 91 5044 50 10 3897 4213 4529 4845 5i62j5478 9 Io 547I 5897 6325 6752 7180 7608 49 Ii 5795|6I I264306747 706517383 48 I I 80378466 8896 9326 9756.I87.48 12 7702~802o 8339 8658 8977 9'9747 12 II.3ro619 io5o 1483 1915 2348 2782 47 13 _ 96i6 9936.256.577.897 1218 46 I3 32I6 365o 4o85 452o04956 5392 46 1 I4 I. 18I 539 I860 218225042826 3148 45 i4 5828 6265 6702 7I401757880o 7145 15 347I 3793411 i 6 4440o4763 5087 44 5 8456 8896 9336 9776.217.659144 16 54 i 5735 6059 6384 670o97o34 43 i6 xI.3 tI oo I543 1985 2428 2872 3316i43 17I 7359 7685 8oiI 833718663 8990 42 7 3761 4206 465i 5097 5543 5990o42 I8 93 I 719644 9971.299.626.95414I 8 5437 6885 7333 7782|8231868o041 9I I. 191283 i 6I I940 2269 2598 2928 40 I9 I301 958I..32.483.935 1387140 2(1 3258 3588 39I8 4249 4579 491o 391 20 I I.32.I8,4o 2293 2747 320I 3656 4I 1 I|39 21 5242 5573 5905 6237|6569'6902 38 2 I 456750o23 548o 5937 6394 6852.38 22 7235 7568 790I 8235 8568 8902 37 22'7311 7770 8229 8689 950i5 96II137 23 9237 957I 9906.241.577.912 36 23 I.3400o721o534 o996 I459 I923|2387 36 24 i.201248 I 584 I921 2257 2594 293I 35 24 285I 33i6 378I 4247 47I4 5i8o 35 25 3269 3606 3944 4282 462I 4960.34 2,5 5648 6i16 6584 7o53 7522 7992 34 26 5299 5638 5977 6317 6657 6997 33 26 8463,8933 9405 9877.349.822 33 27 733817679 8020 836i 870319045 32 27 II..51296I 770 2244 2719 3195 3671 32 28 9387 9729..72.4I5.758 102 31 28 4147 4624 5I02 5580 6059 6538 3I 2911.211446 1790 2I34 2479'2823 3I69|30 29 70I8 7498 7979 8460|8942 9424 30 3o 35i41386o 4206 4552 4898 5245 29 3o 9907.39o.874 I359 i844 2329 29 31 5592 5939 6287 6635 6983 7331 28 3I II.3628I6 3302 3789 4277 4765 525428 32 7680 8029 837818728 9078 9428 27 32 5744 6234 6724 7215 7707 8I99 27 33 97781 I 29.48o.83I I I I83|I 534|26 33 86929I 85 9679. I 73. 668 I I 64|26 34 11.22i886 2239 2591 2944 3298 365I 25 34 1137I660 2I56 2654 3i5i 3650 4I49 25 351 400514359 4713 5068 5423 5778 24 35 4648 5I48 5649 650o 6652 715424 36 6134 6489 6845 7202 7558 79I5 23 1 36 7657 8I6I 8665 9170 9675.I8I 23 37 8273|86308988 9346 9705..63|22 37 II.3806871I 1941702 221027I193228|22 38II.230422 0782 I'I41 50o i86I 2222 2I 38 3738 4249 4760 5272 5785 6298 21 39 2583 2944 33o5 3667 4029 439I 20 39 68II 7325 7840 8356 8872 9388 20 4oI 475415ii6 548o 5843 6207 657I 19 4o 9906.424.942 I46i I98I 2501 19 41 6935 7300 7665 8030o8396 8762 I8 4I I I.393022 3544406645895 I I 3 5637 18 42 9I28 9495 986I.229.596.964 I7 42 6 6I687 72 I 13 7740 8267 8795 1 7 43 11.24I3321 I700 2069 2438 2807 3I7716 43 9323|9853.382.913 1I4441I976I16 44 3547139i7 4288 465915030 540o i5 44 II.402508o304I 3575 4IIo 4645 5I80o 5 45 5773 6I45 65I8 689I 7264 7637 I4 45 571I7 6254 6792 7330 7869 8400 9 4 46 8oI I 8385 8759 9I34 9509 9884|I 3 46 8949 9490..32. 574 I I 7 66 I 3 47 1.250260 o636 IOI2 1389|17662'1431I2 47 I I.41225275I 3296 3843|4390o4938|12 48 252112899 3277 3656 4034 4414II i48 5486 6036 6586 7136 7688 8240 I 49 4793 5173 5553 593463I5 6696o10 49 67921346 9900 1010 rI566 io 50 7078 7460 7842 8224 8607 899I 9 50o II.422123 2681 3240 3799 4358 4919 9 5i 937419758.1 42.527.9I2 1297 8 5i 548o06oA2 66o057168 7733 8298 8 52 I I.26I 683 2069 2455 2842 3229 3661 7 52 8863 943o0999791.5651 I33 I 702 7 53 400414392 4780 5I69 555585947 6 53 II.432273 2843 34151398714560o5134 6 54 6337 67277117 75o8178991829i 5 54 5709I6284 6860o7437 8oI5S8593! 5 55 8683 9075 9467 9860.254.647 41155 9172 9752.333.915 I497 20801 4 561II.270I41 i435 i830 2225 2620o30I6| 3 56 11.442664|3248|3834|4420o5007|5595 3 57 34I2 3809 4206 4603 500015398 2 57 6i83 6773 7363 7954 8546 9I38 2 58 5796 6i95 6594 6993 739317793 I 58 9732.326.921 I5I7 2II1132711 I 59 8194 8595 8996 9397 97991 2~02 o 1159 11.453309 390o845o8 5Io9 571163I3 o 60" 5" 140" 30" 20" 10". 60' 50" 40" 30" 20" 10"Co-tangent of 3 Degrees. 1 Co-tangent of 2 Degrees. R P. Part 1" 2" 3"/ 4"/ 5"1 6" 77 P8 9/' P (1/ 2" 3/ 4" 5 6/ 7// 8/1 9/P. Part.35 69 104 138 173 207 242 276 311 P. Part 48 97 145 193 242 290 338 387 435 t1i LOGARITHMIC SINES d Sine of 88 Degrees. Sine of 89 Degrees.'0" 10" 20" 301" 40" 50"'1 0" 10" 20"/ 30" 4(Y' 50"10o 9.9997359736 97379 739738 9739 59 0 9.999934 99993599359935993 59 I 9740 99740 974I 9742 9743 9743 58 I 9936 9936 9937 9937 9937 9938 58 2 9744 9745 9746 9746 9747 9748 57 2 9938 993919939 9939 9940 9940 57 3 9748 9749 9750 975I 975I 9752 56 3 9940 994I 994I 994I 9942 9942 56 4 9753 9753 9754 9755 9756 9756 55 4 9942 9943 9943 9943 9944 9944 55 5 9757 9758975897589759976 9760o54 5 9944 9945 9945 9945 9946 9946 54 6 976 97629763 9763 9764 9765 53 6 9946 9947 9947994 7 9949948 9948 53 7 9765 9766 9767 9767 9768 9769 52 7 9948 9949 9949 9949 9950 9950 52 8 9769 9770 9771977297729773 51 8 9950 995 I 995 I 995I 995219952 5I 9 9774 9774 9775 9776 9776 9777 50 9 9952 99539953 9953 9953 9954 50 109. 999778 9778 9779 9780 9780 9781 49 I09.999954 9954 9955 9955 9955 995649 I I 9782 9782 9783 9784 9784 9785 48 I I 5 9956 995695699579957 995 7 99548 I2 9786 9786 9787 9788 9788 9789 47 I 2 9958 9958 9958 9959 9959 9959 ~7 I3 9790 9790 979 9792 9792 9793146 I3 99599699609969960o996I 9961 46 I4 9794 9794 9795 9795 9796 9797 45 I4 996I 996I 9962 9962 9962 9963 45 i5 9797 9798 9799 9799 9800 980I 44 i 5 9963 9963 9963 9964 9964 9064 44 I6 9801 9802 9803 9803 980o49804 43 I6 9964 9965 9965 9965 9965 9966 43 1 7 9805 9806 9806 9807 9808 9do8 42 I7 9966 9966 9967 9967 9967 996799 42 x8 9809 9809 98I o 98 I I 981198 I 4I 8 996899689968 996899969996941 1I9 98I3 98138 1998I 98I4 9815 I9816 4o 19 9969 9969 99709970 99709 0 9970 4o 209.99986 981 I 798I781898 I 9819 39 20 9 9997I 9 97I 997I 997I 99729972 21 982098220982I 9822 9822 9823 38 21 997299729973 99739973938 22 982419824 9825 9825 9826 9827 37 22 9973 9974 974 9974 9974 9974 37 23 9827 982898289828 9829 98299830 36 23 9975 9975 9975 9976 9976 9976 36 24 983 I 983 I 9832 9832 9833 9834 35 24 9976 9976 9977 9977 9977 9977 35 25 9834 9835 9835 9836 9836 9837 34 25 9977 9978 9978 9978 9978 9979 34 26 9838 9838 9839 9839 9840 9840 33 26 9979 9979 9979 9979 9989 9980 33 27 984 9842 9842 9842 9843 9843 9844 32 27 9980 9980 9980 998 998I 998I 998 I 32 28 9844 9845 9846 9846 9847 9847 3i 28 9981 998 I 9982998299829982 9982 998 29 9848 9848 9849 9849 98598 985 30 29 9982 9983 9983 9983 9983 9983 30 30o 9.99985I 9852 9852|9853 9853 9854 29 309. 999983 9984 99849984 9984 9984 29 3I: 9854 9855 9856 9856 9857 9857 28 3i 9985 9985 9985 9985 9985 9985 28 32! 9858 9858 9859 9859 9860 9860 27 32 9986 9986 9986 9986 9986 9986 27 33 9861 9861 986219863 9863 9864 26 33 9987 9987 99879987 9987 9987126 34 9864 9865 986519866 9866 9867 25 34 9988 9988 9988 9988 9988 9988 25 35 9867 9868 8 9989 9869 9870 24 35 9989 9989 9989 9989 9989 9989 24 36 987 9870 9871 9879872 9872 9873 23 36 9989 9990 9990 9990~ 9990~ 99923 37 9873 9874 9874 9875 9875 9876 22 37 9990 9990~9991 9991 999I 999I 22 38 9876 9877 987719878 9878 9879 2i 38 999999 999 999I9992 9992 9992 21 39 9879 9880 9880 9881 988I 9882 20 39 9992 9992 9992 9992 9992 9993 20 40 9.999882 9883 9883 9884 9884 9885 19 4o 9 999993 9993 9993 9993 9993 9993 1I9 4I 9885 9886 9886 988719887 9887 988 8 4i 9993 9993 9994 9994 9994 9994 i8 42 9888 9889988989 989 98 989 989I7 42 9994 9994 9994 9994 9994 9995 I 7 43 9891 9892 9892 9892 9893 9893 i6 43 9995 9995 9995 9995 9995 9995 i6 44 9894 9894 98595 9895 9896 9896 i5 44 9995 9995 9995 9996 9996 9996 i5 45 9897 9897 9898 9898 9898 9899 I4 45 9996 9996 9996 9996 9996 9996 14 46 9899 9900 9900 990I 9901 9902 i3 46 9996 9996 9997 9997 9997 9997 I3 47 9902 9903 90319903 9904 9904 I2 47 9997 9997 9997 9997 9997 9997 I2 48 99059905 99 996 9906 99071 48 9997 9997 9997 9998 9998 9998 1 49 9907 998 9908 9908 9909 99 0999 o 49 9998 9998 9998 9998 9998 9998 Io 5019.999910o9910 99I I 99I I 99I2 99I2 9 50 9.999998 9 99 989998 9998 9998 9 5i 9939913 99I3 99I31 99I4 99995 8 5i 9999 9999 9999 9999 9999 9999 8 52 991I15 99I65 99I699 I799I7997 7 52 9999 99999999999999999 999 7 53 99I8 99I8 99I899199 19 9920 6 53 999999999999999999 99999 6 54 9920 9920 9921 9922 9922 9922 5 54 9999 9999 9999 9999999999.... 5 55 9922 9923 992399 24 992424 9924 4 55 o.oooooooooo oooo oooo0000 0000 0000 4 56 992599259926992699269927 3 56 oooo0oooo0oooo0oooo0oooo oooo 3 57 9927 9927992899289929 9929 2 57 oooo000000000000o 000ooo 00oo00oo0000 2oo 58 9929 993 9930993 993 993 I 58 0000 0000 0000 0oooooooo 00ooooooo 0000 0000oooooooo 59 9932 9932 993219933 9933 9933 o 59 oooooooooooooooo oooo oooo0000 0 -60" 50' 40" 30" 20" 10o" - 60" 50" 40" 30" 20" 10",g Co-sine of 1 Degree. Co-sine of 0_Degree. P. Parts 1" 2" 3"1 411 51 61" 7' 8" 91 pp 1" 2" 3" 4/" 5" 6/' 7 8" 9"'. 0 0 0 0 0 0 0 rt0 0 0 0 0 0 0 0 LO G A R I T H MIC TA N G E N T S.. 113.......... Tangent of 88 Degrees. P. P rt o,'0" o10"' 20" 30" 40"' 50' to 1". o I1.45696 1II.457520 11.458125 1 1.458731 11.459,338 i1.459945 59 6.6 I 460553 46ii63 461773 462383 462995 463608 58 6I. 2 464221 464836.46545I 466067, 466684 467302 57 6i.6 3 467920 468540 469I60 469782 4704o4 471027 56 62.2 4 47I65i 472276 472902 473528~ 474156 474785 55 62.7 5 4754I4 476044 476676 477308 47794I 478575 54 63.3 6 4792Io 479846 48o482 4811 20 481759 482398 53 63.8 7 483039 48368o 484323 484966 4856II 486256 52 64.4 8 486902 487549 488198 488847 489497 490o48 5i 65.o 9 490800 491453 492I07 492762 493418 494075 50 65.5 Io II.494733 II.495392 xI.496052 1t1.496713 II.497375 I1.498038 49 66. I 11 498702 499367 500033 500700 5oi368 502037 48 66.8 12 502707 503378 50405I 504724 505398 506073 47 67..4 13 506750 507427 5o8io6 508785 509466 5IoI48 46: 68.o I4 5Io83o 51I5I4 5I2199 5I2885 5I3572 514260 45 68.6 15 5I4950 515640 5i633i 517024 517717 5I84I2 44 69.3 i6 519io8 519805 520503, 521202 521903 522604 43 70.0 17 523307 5240oI 524715 52542I 526128 526837 42 70.7 i8 527546 52825771 528969, 529682 53o0396 531I1 41 71.3 19 531828 532545 533264 5339,84- 534705 535428 40 72.1 20 I i.536i5I I.536876 1t.537602. II,.53833o 1i.539058 I.539788 39 72.8 21 540519 54I251 541984 5427I9 543455 544192 38 73.5 22 544930 545670 5464ii 547,53 547896i 54864i 37' 74.3 23 549387 55o0I34 55088,3 55t632 552384 553I336. 36 75.0 24 553890 554645: 5554o0 556I59 5569I8 557678' 35 75. 8 25 558440 559203 559967; 560733 56I5oG 562268 34 76.6 26 563o038 563809 56458I, 565355 566i3o 566907 33 77.5 27 567685 568464 569245 570027 570811i 571596 32 78.3 28 572382 573170 573959 57475o 575542 576336 31 79. I 29 577131 577928' 578726; 579525 580326 581 28 30 80 o 30 II.58I932 I.582737 ii.583544 I.584353 Ix.585i63 I.585974 29 80.9 31 586787 587601 588417 58.9235 590054 590874 28 8i.8 32 591696 592520 593345 594I72 595000 595830 27 82.8 33 596662 597495 5983.30 599I66 6oooo4 600844 26 83.7 34 6oi685 602528' 603372 6042I8 6o05066 605915 25 84.7 35 606766 6076I9 608474 6o09330 6IOI87 611047 24 85.7 36 6I90o8 612771 6i3636 614502 615370 616240 23 86.7 37 617111 617985 6I886o 6 i9737 620615 621496 22 8q7.8 38 622378 623262 624147 625035 625924 6268I5 21 88.8 39 627708 628603 629500 63&399 63I299 632201 20 89.9 4o ii.633io5 1I.6340I2 11.6349i9 I I.635829 11.636741 xI.637655 19 9I.1 4I 638570 639488 640407 641329 642252 643177 i8 92.2 42 644o105 645034 645965 646899 647834 648771 I7 93.4 43 649711 650652 65 i595 652541 653488 654438 i6 94.6 44 655390 656343 657299 658257 659217 660I 79 I5 95.9 45 661144 66211- 663079 664050 665023 665998 I4 97.2 46 666975 667q55 668936 669920 670907 67I895 13 98.5 47 672886 673879 674874 675871 676871 677873 12 99.8 48 678878 679885 680894 6890o5 682919 683935 ii ioI.3 49 684954 685975 686998 688024 689052 690083 o I02.7 50 I1.69III6 I.692151 11-693I89 AI.694230 II.695273 11.696318 9 Io4.2 5; 697366 6984 17 699470 700526 70o584 702645 8 Io5.7 52' 703708 7o4774 705843 706914 707988 709065 7 107.2 53 7IOI44 7II2216 712311 713398 714488 71558I 6 Io8.9 54 716677 717775 718876 7I9980 72Io87 722196 5 iio.5 55 723309 724424 725542 726663 727787 728914 4 II2 2 56 730044 731176 7323I2 73345I 734592 735737 3 114.0 57 736885 738035 739I89 740346 74I506 742669 2 1I5.8 58 743835 745004 746I77 747352 74853i 749713 I I17.7 59 750898 752087 753279 754474 755672 756874 o I19.7 6"' 0W' I 40, 30"f 20" 0I d' Co-tangent of 1 Degree..,, ~.......,~~~~~1 114 AUXILIARY TA BLE FOR'INES, &C. 0 Degree. _ 1 Degree. 8 log.sin. A-log tan. A- log. cot. A+ o log. sin. A-log. tan. A- log. cot. A+ m log. A". log. A". log. A". _ log. A". log. A". log. A". l o 04.685575 4.6855751I5.3I4425 60 3600oo o4.685553 4.6856Ig I5.3I438I 6o 60 I 575 575 425 59 3660 I 552 620 380 59 I20 2 575 575 425 58 3720 2 55I 622 378 58 80o 3 575 575 425 57 3780 3 55i: 623 377 57 240 4 575 575 425 56 3840 4 550 625 375 56 30o 5' 575 575 4-25 55 3900 5 549 627 373 55 360 6 575 575 425 54 3960 6 548 628 372 54 420 7 5757 575 425 53 4020 7 547 630 370 53 480 8 574 576 424 52 40o8o 8 547 6321 368 52 54o 9 574 576 424 51 4140 9 546 633! 367 5I 600 10 4.685574 4.685576 i5.3I4424 50 4200 Io 4.685545 4.685635 i5.314365 50 660 ii 574 576 424 49 42601 I 544 637 363149 720 I2 574 577 423 48 432012 543 638 362 48 780 I3 574 577 423 47 4380oi3 542 640 360o47 84o0i4 574 577 423 46 4440oi4 54i 642 358 46 9001I5 573 578 422 45 4500oo i5 54o 644 356 45 9601I6 573 578 422 44 45601o6 539 646 354 44 I0201I7 573 578 422 43 46201I7 539 647 353 43 io8o0I8 573 579 42I142 468o0I8 538 649 35I 42 II401I9 573 579 421 41 474019 537 651 3494I I200 20 4.685572 4.6855801I5.314420 40 4800 20 4.685536 4.685653 I5.3I4347 40 1260 2I 572 580 420 39 4860 2I 535 655 345 39 I320 22 572 58I 4I9 38 4920 22 534 657 343 38 i380 23 572 58i 419 37 4980 23 533 659 34i 37 I440 24 57I 582 4I8 36 5040 24 532 66I 339 36 I500 25 57I 583 4I7 35 5I00 25 53I 663 337 35 1560 26 57I 583 417 34 5160o26 530 665 335 34 1620 27 570 584 4i6 33 5220 27 529 668 332 33 I1680 28 570 584 416 32 5280 28 527 670 330i32 I740 29 570 585 415131 5340 29 526 672 328 3I 8oo 30o 4.685569 4.685586 I5.3144I4 30 54oo 30 4.685525 4.685674 I5.314326 30 I86o 3I 569 587 413 29 546o3I 524 676 324 29 192o0 32 569 587 4I3 28 5520 32 5231 679 32I 28 980o 33 568 588 412 27 558o 33 522 68i 319 27 2040 34 568 589 4II 26 564o 34 52I 683 3I7 26 2I00o35 567 590 4I0O25 5700 35 520 685 315 25 2I60o36, 567 59I 409124 5'760 36 5i8 688 312 24.222037 566 592 408|23 582037 517 690 310 23 2280 38 566 593 407122 588o038 5i6 693 307 22 2340 39 566 593 407|2I 5940 39 515 695 305 2I 2400 40 4.685565 4.685594 I 5.314406 20 60004O 4.685514 4.685697 15.314303 20 2460 41 565 595 4051i9 6060 41 512 700 3001 9 2520 42 564 596 4o41I8 6I2o42 5Ii 7021 298 I8 2580 43 564 598 402|17 6I8o043 5io 705 295 I7 2640o44 563 599 4o0II6 6240 44 509 707 293 I6 2700o45 562 6oo 4oo00 5 6300o45 507 710 290 15 2760 46 562 6oI 399 14 6360o46 506 713 2871I4 2820 47 561 602 398 13 6420 47 5So5 715 285 13 2880 48 56I 6o3 39712 6480o48 503 718 282 I2 2940o49 560 6o4 3961ii1 6540 49 502 720 280 11 30o005o 4.685560 4.685605 5. 34395iao 6600 50 4.68550o14.6857231 5.314277I10 3060151 559 607 393 9 16660o 5, 4 99 726i 274 9 3120o52 558 608 392 8 6720o521 498 729 271 8 3i80o53 558 609 39|1 7 6780 53 497i 73I1 269 7 3240 54 557 6ii 389 6 6840o54 495 7344 266 6 3300o55 556 6I2 388 5 690o 55 494 737 2631 3360o56 556 6i3 387 4 6960 56 492 740 260 4 3420 57 555 615 385 3 7020 57 491' 7431 257 3 3480o58 554 6i6 384 2 7080 58 490 745 255 2 354059 554 6I8 382 I 714059 4881 748 252 I log. cos. A- log. ot. A- log. tan A+ log. cos. A- log. cot. A- log. tan. A+ ai log. c. A. log. c. A". log.a. f log c. A". I log. c. A". log. c. A". 89 Degrees. [ 88 Degrees. LOGARITHMIC TANGENTS, 11 5 Tangent of 89 Degrees. J Part 0" 10" 20" 30"/ 40" 50"'; t1". o0 I.758079 II.759287 II.760498 II.76I7I4 II. 762932' I.764154 59 I21.7 xI 765379 766608 767840 769076 770315 77I558 58 123.8 2 772805 774055 775308 776566 777826 779091 57 125.9 3 780359 781631 782907 784186 785470 786757 56 128.1 4 788047 789342 790641 79I943 793250 794560 55 i3o.4 5 795874 797I92 7985I5 799841 80117I 802506 54 I32.8 6 803844 805187 8o6534 807885 809240 8Io6oo 53 i35.3 7 8II964 8I3332 814704 8i608i 8I7462 818847 52 137.9 8 820237 821632 823031 824434 825842 827255 51 14o.6 9 828672 830094 831520 83295i 834387 835828 5o I43.4 Io II.837273 11.838724 I1.840179 11.841639 II.843Io4 II.844574 49 I46.2 II 846048 847528 849013 850503 851999 853499 48 149.3 12 855004 856515 85803i 859553 861079 862611 47 I52.4 13 864i49 865692 867240 868794 870354 87I9I9 46 I55.7 14 873490 875067 876649 878237 879831 88I43i 45 159.1 i5 883037 884648 886266 887890 889519 891155 44 I62.7 I6 892797 894446 896100oo 897761 899429 901103 43 i66.4 I7 902783 904470 906I63 907863 909569 9II283 42 I70.3 18 913003 914730 916464 918205 919953 92I707 4I I74.4 I9 923469 925239 927015 928799 930590 932388 4o 178.7 20 II.934194 11.936oo008 II.937829 1I.939658 II.941494 I1.943338 39 I83.3 21 945191 947051 948919 950795 952679 954572 38 i88.o 22 956473 958382 960299 962225 964I60 966Io3 37 I93.0 23 968055 970016 97I986 973965 975952 977949 36 i98.3 24 979956 981971 983996 986030 988074 990128 35 203.9 25 992191 994264 996347 998440 12.ooo543 I2.00oo2657 34 209.8 26 12.00478I 12.006915 12.oo0090o60 I2.II215 013382 015559 33 216.2 27 OI17747 OI9946 022156 024378 026611 028855 32 222.7 28 o3iii 033379 035659 037951 040255 042572 3I 229.8 29 044900 047242 o49596 051963 054342 056735 30 237,33 30 I2.o59I42 12.0o656I 12.o63994 12.066441 I2.o68902 I2.071377 29 245 4 31 073866 076369 078887 o81419 o83966 o86529 28 254.0 32 o8g106 og091699 094308 096932 099572 102228 27 263.2 33 I04901 107590 110296 113019 II5760 1185I7 26 2'73.2 34 121292 I24085 126896 I29726 I32574 i35440 25 283.9 35 138326 I4123I 144i56 I47Ioo 150065 53o050 24 295.4 36 i56o56 I59082 I62130 I65199 168290 171404 23 308.o 37 I74540 177698 i8o880 i84085 I87314 I90567 22 321.7 38 I93845 I97148 200476 203830 207210.210616 2I 336.7 39 214049 2175IO 220998 224515 228060 231635 20 353.x 40 12.235239 Ia2 238873 12.242538 1I2.246235 I2 249963 I2.253723 I9 371.2 41 257516 261342 265203 269098 273028 276995 i8 39I.3 42 280997 285037 28915 293232 297388 3oi584 I7 413.6 43 305821 33ioioo 314422 318787 323196 327650 i6 438.7 44 33215I 336698 34I294 345939 350634 35538i I5 467.0 45 360180 365032 369940 374903 379924 385004 I4 499.2 46 390r43 395345 400609 405938 4II333 416796 I3 536.2 47 422328 427932 4336o10 439362 445192 45iioo 12 579.1 48 45709I 463I65 469325 475574 481915 488349 II 629.4 49 494880 50I5Io 508244 5i5o83 522032 529094 io 689.4 50 I2.536273 12.543572 12.55o996 12.558549 I2 566236 I2.57406I 9 51 582030 590148 598421 606854 6I5454 624228 8 52 633i83 642327 651667 661212 670972 680956 7 53 69I175 701641 7I2365 723360 734641 746223 6 54 758122 770357 782946 795911 809275 823063 5 55' 837304 852027 867267 88306i 899452 9I6485 4 56 934214 952697 972002 992206 I3.OI3395 i3.o3567I 3 57 13.o59I53 i3.o83976 I3.iio3o5 I3.I38334 168297 200482 2 58 235244 273032 314425 360183 4ii335 469327 I 59 536274 6i5455 712365 837304 14.013395 14.314425 o 6MY' 5W' 401 30" 20Y' 10" Co.tangent o' 0 Degree. !1 6 NATURAL SINES. -d ~.0 10 20 30 40 50 60 70 8 9~ O 000ooo000 o7452 o34899 052336 069756 o871561 Io4528 12I869 139173I 156434 6o I 029I 7743 519o 2626 070047 7446 48i8 2158 9461 6722 59 2 0582 8034 548i' 29I7 0337 7735 5107 2447 9749 7009 58 3 o873 8325 5772 3207 o627 8025 5396 2735 I40037 7296 57 4 ii64 8616 6062 3498 0o9I7 83i5 5686 3024 0325 7584 56 5 i454 8907 6353 3788 1207 8605 5975 3313 o6i3 7871 55 6 I745 9197 6644 4079 I497 8894 6264 36oi 0901 8i58 54 7 ao36 9488 6934 4369 1788 9184 6553 3890 1189 8445 53 8 2327 9779 7225 4660 2078 9474 6843 41I79 I477 8732 52 9 2618 020070 7516 4950 2368 9763 7132 4467.7.......65 9020 5i 10 00oo2909 02o36 o37806 055241 072658 0900g53 I07421 124756 I42053 159307 50 II 3200 0652 8097 5531 2948 o343 7710 5045 2341 9594 49 12 3491 0942 8388 5822 3238 o633 7999 5333 2629 9881 48 13 3782 1233 8678 6112 3528 0922 8289 5622 29I7 160168 47 I4 4072 I524 8969 6402 38I8 1212 8578 59gIo 3205 o455 46 15 4363 I815 9260 6693 4Io8 I502 8867 6199 3493 0743 45 I6 4654 2106 9550 6983 4399 I79I 9156 6488 3780 Io3o044 I7 4945 2397 984I 7274 4689 2081 9445 6776 4o68 I317 43 I8 5236 2687 040132 7564 4979 237I 9734 7065 4356 1604142 19 5527 2978 0422 7854 5269 2660 110023 7353 4644 189gI1 4 20 oo58i8 023269 040713 o58145 075559 092950 1io3I 3 127642 I44932 162I78 4o 21 6o09 3560 Ioo4 8435 584.o 3239 0602 7930 5220 2465 39 22 6399 385I I294 8726 6135 3529 o89g 8219 5507 2752 38 23 6690 414I i585 goI6 6429 38I9 118o 8507 5795 303937 24 698I 4432 I876 9306 67I9 4I0o8 469 8796 6083 3326 36 25 7272 4723 2166 9597 7009 4398 1758 9084 637I 36i3 35 26 7563 5oi4 2457 9887 7299 4687 2047 9373 6659 3900 34 27 7854 5305 2748 060177 7589 4977 r 2336 9661 6946 4187 33 28 8i45 5595 3038 o468 7879 5267 2625 9949 7234 4474 32 29 8436 5886 3329 0758 8169 5556 294 I130238 7522 4761 13 30 0oo8727 026177 043619 061049 078459 095846 I13203 130526 147809 165048 30 3i 9017 6468 3910 1339 8749 6i35 3492 08i5 8097 5334 29 32 9308 6759 4201 I629 9039 6425 3781 Iio3 8385 5621 28 33 9599 7049 449I 1920 9329 6714 4070 1391 8672 5908 27 34 9890 7340 4782 2210 9619 7004 4359 i68o 8960 6I95 26 35 oioi8i 763i 5072 2500 9909 7293 4648 I968 9248 6482 25 36 0472 7922 5363 279I 080I99 7583 4937 2256 9535 6769 24 37 0763 8212 5654 3o8I o489 7872 5226 2545 9823 7056 23 38 io54 8503 5944 337I 0779 8162 5515 2833 I5oiII 7342 22 39 I344 8794 6235 366I Io69 845I 5804 312I 0398 762921I 4o oiI635 029o85 o46525 0o63952 08I359 098741 116093 i334Io 150686 I67916 20 41 1926 9375 68i6 4242 1649 9030 6382 3698 0973 8203 19 42 22I7 9666 7106 4532 I939 9320 6671 3986 I26I 8489 i8 43 2508 9957 7397 4823 2228 9609 6960 4274 I548 8776 17 44 2799 030248 7688 51i3 2518 9899 7249 4563 1836 9063 i6 45 3090 0539 7978 54e3 2808 IooI88 7537 4851 2123 93501 i5 46 3380 0829 8269 5,693 3098 0477 7826 5139 2411I 9636 14 47 367I I120 8559 5984 3388 0767 8ii5 5427 2698 9923 i3 48 3962 I41I 8850 6274 3678 Io56 8404 5716 2986 170209 12 49 4253 1702 9140 6564 3968 i346 8693 6004 3273 0496 1I 50 oi4544 03I992 049431 o66854 0o84258 Ioi635 II8982 136292 15356i 170783 io 5I 4835 2283 9721 7145 4547 I924 9270 6580 3848 Io69 9 52 5I26 2574 050012 7435 4837 22I4 9559 6868 4136 1356 8 53 54I6 2864 o302 7725 5127 2503 9848 7I56 4423 I643 7 54 5707 3155 0593 8o15 54I7 2793 I20137 7445 47IO I929 6 55 5998 3446 o883 8306 5707 3082 0426 7733'4998 2216 5 56 6289 3737 I174 8596 5997 3371 0o74 8021 5285 2502 4 57 6580 4027 i464 8886 6286 366i iOO3 8309 5572 27589 3 58 6871 43x8 I755 9176 6576 3950 I292 8597 5860 3075 2 59 7162 4609 2045 9466 6866 4239 I58I 8885 6147 3362 I 890 880 870 86~ 85~ 84~ 830 820 81 80. Natural Co-sines. t. P: 4. &5 4.85 4.84 4.84 4.83 4.83 4.82 4.8i 4.8o 4.78 NATURAL TANGENTS. 117 *1 00 oi 20 30 40 50 60 70- 80 90 0000000o017455 034921 o052408 069927 087489 io5io4 122785 i4o54i i 583846o I 0291 7746 5212 2699 070219 7782 5398 3o8o 0837 868359 2 0582 8037 5503 2991 o5ii 8075 5692 3375 i 134 898 i 58 3 0873 8328 5795 3283 0804 8368 5987 3670 i43i 9279 57 4 II64 86I9 6086 3575 iO96 866; 628I 3966 1728 9577 56 5 i454 89o10 6377 3866 1389 8954 6575 4261 2024 987655 6 1745 9201 6668 4i58 i68i' 9248 6869 4557 2321 i601o7454 7 2036 9492 6960 4450 1973 9541 7163 4852 2618 0472 53 8 2327 9783 7251 4742 2266 9834 7458 5147 2915 o77152 9 2618 020074 7542 5033 2558 090127 7752 5443 32I2 1069 5i 10 002909 020365 037834 o55325 072851 o09042I io80o46 125738 i435o8 i6i368 5o xI 3200 0656 8125 5617 3143 o714 8340 6034 3805 i66649 12 349I 0947 8416 5909 3435 1007 8635 6329 4Io2 1965148 13 3782 1238 8707 6200 3728 i3oo 8929, 6625 4399 2263 47 14 4072 1529 8999 6492 4020 1594 9223 6920 4696 256246 15 4363 1820 9290 6784 4313'I887 9518 7216 4993 286045 16 4654i 2111 958I 7076 4605 2180 9812 7512 5290 315944 17 4945 2402 9873 7368 4898 2474 II01o07. 7807 5587 3458'43 18 5236 2693 o4oxi64 7660 5190 2767 o4o i 8Io3 5884 3756!42 19 5527 2984 0456 7951 5483 3o6i 0695 8399 618i 4o554, 20 oo5818 023275 040747 058243 075775 093354 Io10990 128694 146478 io43544o 21 6o109 3566 io38 8535 6o68 3647 1284 8990 6776 4652 39 22 64oo00 3857 i33o 8827 636i 3941 1579 9286 7073 4951 38 23 6691 4148 1621 9119 6653 4234 1873 9582 7370 525037 24 6981 4439 1912 9411 6946 4528 2168 9877 7667 554936 25 7272 4731 2204 9703 7238 4821 2463 130173 7964 5848 35 26 7563 5022 2495 9995 7531 5115 2757 0469 8262 614734 27 7854 5313 2787 060287 7824 5408 3052 0765 8559 6446 33 28 8145 5604 3078 0579 8ii6 5702 3346 I06I 8856 6745 32 29 8436 5895 3370 0871 8409 5995 364i 1357 9154 7044131 3o 008727 026186 o4366i o6ii63 078702 096289 i13936 131652 149451 167343,30 3i 9018 6477 3952 1455 8994 6583 4230 1948 9748 7642 29 32 9309 6768 4244 1747 9287 6876 4525 2244 i 5oo46 7941 28 33 9600 7059 45351 2039 9580 7170 4820 2540 o343 8240 27 34 9891 7350 48271 2331 9873 7464 5114 2836 o64i 85392z! 35 oioi81 7641 5iI81 2623 o8o165 7757 5409 3132 0938 863825 36 0472 79331 54Io 2915 0458 8o5i 5704 3428 I236 913724 37 0763 82241 5701 3207 0751 8345 5999 3725 1533 943723 38 io54 85i5 5993 3499 io44 8638 6294 4021 i83i 973622 39 z345 8806 6284 3791 i336 8932 6588i 4317 2129 17oo003521 40o oi636 029097 046576 o64083 o081629 099226 ii6883 i346i3 152426 17033420 4. 1927 9388 6867 4375 1922 9519 7178 4909 2724 063419 42 2218 9679) 7159 4667 2215 9813 7473 5205 3022 0933 i8 43 2509 9970 7450 4959 2508 100o07 7768 5502 3319 1233 17 44 28o00o 030262 7742 5251 2801 o4oi 8063 5798 3617 15321i6 45 3091 0553 8033 5543 3094 0695 8358 6094 3915 i831I5 46 3382 o844 8325 5836 3386 0989 8653 6390 4213 2131 14 47 3673 i35 8617 6128 3679 1282 8948 6687 45io 243013 48 3964 1426 8908 6420 3972. 1576 9243 6983 4808 2730o12 49 4254 1717 9200 6712 4265 1870 9538 7279 5io6 3030oii o oio4545 032009 049491 067004 o84558 102164 ii9833 137576 i55404 173329 io 51 4836 23oo 00 97831 7296 4851 2458 120128 7872 5702 3629 9 52 5127 2591 050075 7589 5144 2752 04231 8169 6oo000 3929 8 53 5418 2882 o366i 7881 5437 3046 o718 8465 6298 4228 7 54 5709 3173 0658: 8173 5730 334o ioi3 8761 6596 4528 6 55 6oo000 3465 09491 8465 6023 3634 1308 9058 6894 4828 5 56 6291 3756 124Ii 8758 63i6 3928 16o4 9354 7192 5127 4 57 6582 4047 i533; 9050 6609 4222 1899 9651 7490 5427 3 58 6873 4338 1824 9342 6902 45i6 2194 9948 7788 5727 2 59 7164 4630o 2i16 9635 7196 481io 2489 140244 8086 6027 890 880 87' 860 850 840 83~ 820 81" 800 Natural Co-tangents. [to 4.8.5 4.85. 4.86 4.87 4.88 4.89 4.9' 4.93 4.96 4.98 a'11 S N ANATURAL SINES. a 100 110 120 130 140 15~ 16 170 18~ 190 o 173648 I90809 2079I2 22495I 241922 258819 275637 292372 309OI7 325568 60 I 3935 I095 8196 5234 2204 9100I 5917 2650 9294 5843 59 2 422I i380 848I 55i8 2486 938I 6197' 928 9570 6II8 58 3 4508 i666 8765 58o0 2769 9662 6476, 31c6 9847 6393 57 4 4794 g195 9050 6085 3o5I 9943 6756 3484 31OI23 6668 56 5 5o8o 2237 9334 6368 3333 260224 7035 3762 o4oo 6943 55 5367 2522 9619 665I 36i5 o505 7315I 4o4o 0676 7218 54 1 7 5653 2807 9903 6935 3897 o785 7594, 43i8 o0953 7493 53 8 5939 3093 210187 7218 4I79 io66 7874 4596 1229 7768 52 9 6226 3378 0472 7501 446I I347 8i53 4874 ISo6 8042 51 0 1I76512 193664 210756 227784 244743 261628 278432 295152 311782 328317 50 I 6798 3949 Io4o 8068 5025 1908 8712 5430 2059 8592 49 12 7o85 4234 I325 835i 5307 2189 899I 5708 2335 8867 48 13 737. 4520 1609 8634 5589 2470 9270 5986 26II 9141 47 i4 7657 4805 I893 89I7 587I 275I 9550 6264 2888 94I6 46 i5 7944 5090 2178 9200 6153 3o3i 9829 6542 3i64 9691 45 I6 8230 5376 2462 9484 6435 3312 280108 68I1 3440 9965 44 17 85i6 566I 2746 9767 6717 3592 o388 7097 37I6 330240 43 I8 8802 5946 3030 230050 6999 3873 o667 7375 399=2 o514[42 I9 9088 623I 3315 -o333 7281 4i54 0946 7653 4269' 078914i 20 I79375 1965I7 213599 2306:I6 247563 264434 281225 297930 3x4545 33io6314o 21 966I 6802 3883 0899 7845 4715 50o4 8208 4821 I338 39 22 9947 7087 4167 I I82. 8126 4995 1783 8486 5097 1612 38 23 180233 7372 445i I465 8408 5276 2062 8763 5373 I887137 24 0519 7657 4735 I748 8690 5556 234I 904I 5649 2161 36 25 o8o05 7942 5019 2031 8972 5837 2620 9318 5925 2435 35 26 Iog9 8228 5303 23I4 9253 6Ii-7 2900 9596 620I 2710o34 27 1377 8513 5588 2597 9535 6397' 3I79 9873 6477 2984 33 28 i663 8798 5872 2880 98I7 6678 3457 300oo151 6753 3258 32 29 1950 9083 6i56 3I63 250098 6958 3736 0428 7029 3533 31 30 182236 I99368 2I6440 233445 250380 267238 2840o5 300706 317305 3338o7 30 31 2522 9653 6724 3728 0662 75I9 4294 0983 7580 4o8I 29 32 2808 9938 7008, 4o0II 0943 7799 4573 126I 7856 4355 28 33 3094 2002231 7292 4294 I225 8079 4852 i538 8132 4629 27 34 3379 o5o8,5757 4577 50o6 8359 5131 1815 8408 490o3 26 35 3665 0793 7859 4859 I788 8640 5410 2093 8684 5178 25 36 395I 1078 8i43 5142 2069 8920' 5688 2370 8959 545224 37 4237 1363 8427 5425 2351 9200 5967 2647 9235 5726 23 38 4523 i648 87II 5708 2632 9480 6246 2924 9511 6000 22 39 4809 1933 8995 5990 2914 9760 6525 3202 9786 6274 21 4o 185095 2022I8 2I9279 236273 253195 270040 286803 303479 320062 336547 20 4.I 538i 2502 9562 6556 3477 0320 7082 3756 0337 6821 19 42 5667 2787 9846 6838 3758 o600 7361 4033 o6i3 7o95 I8 43 5952 3072 220130 7121 4039 o880 7639 43Io 0889 7369 17 44 6238 3357 o4I4 7403 432I ii60 79I8 4587 ii64 7643 16 45 6524 3642! 0697 7686 4602 144o 8I96, 4864 I439 79I71I5 46 68o 3927 098I 7968 4883 1720 8475 54I 1715 819go 14 47 7096 42II 1265 8251 57I65 2000 8753 54I8 I99o 8464 I 3 48 7381 4496 i548 8533 5446 2280 9032 5695 2266 8738 12 49 7667 478I I832 88i6 5727 256o 93Io 5972 254I 9012 II 50o I87953 2o050o65 222116 239098 256008 272840 289589 306249 322816 339285 io 51 8238 5350o 2399 9381 6289 3120 9867 6526 3092 9559 9 52 8524 56351 2683 9663 657I 34o00 290I45 68o3 3367 9832 8 53 88io 5920 2967 9946 6852 3679 0424! 78$o 3642 34oio6 7 54 90o95 6204 3250 240228 7133 3959 0702! 7357 3917 o380 6 55 938I 6489 3534 0510 74I4 4239 098I 7633 4193 o653 5 56 9667 6773 38I7 0793 7695 45I9 1259 7910 4468 0927 46 57 9952 7058 4Ioi I075 7976 4798 i537 8i87 4743 I20oo 3 58 190238 7343 4384 1357 8257 5078 i815 8464 5018 I473 2 59 0523 7627 4668 I64o 8538 5358 2094 8740 52931 I7471 I 790 780 1 770 76 7650 740 730 720~ 710. 700 Natural Co-sines. to 4 771 4.75 4.73 47I 4.69 4.67 4.65 4.62 4.60.57 NATURA L TANGENTS. j 19 A - 0 110 120 13" 141 150 160 170 180 190 o: -327 194380 212557 230868 249328 267949 286745 30573I 324920 344328 60 I 0,627 4682 286I 1175 9637 8261 7060 6049 524I 4653 59 2 6927 4984 3i65 I48i 9946 8573 7375 6367 5563 4978 58 3 7227 5286 3469 1788 250255 8885 7690 6685 5885 5304 57 4 7527 5588 3773 2094 o564 9197 8005 7oo3 6207 563o056 5 7827 5890 4077 2401 0873 9509 8320 7322 6528 5955 55 6 3I27 6192 438I 2707 ii83 9821 8635 7640 6850 6281 54 7 3427 6494 4686 3o0I4 I492 270133 8950 7959 7I72 6607 53 8 8727 6796 4990 332I I8oi o445 9266 8277 7494 6933 52 9 9028 7099 5294 3627 21iI 0757 9581 8596 78I7 7259 5I 1o. 9328 I954oI 215599 233934 252420 271069 289896 308914 328139 347585 50 1II 9628 7703 5903'424I 2729 1382 290211 9233'846I 7911 49 I2 9928 8005 6208 4548 3039 I694 0527 9552 8783 8237 48 13 o229z 8308 6512 4855 3348 2006 0842 9871 9106 8563 47 I4 0529 86io 6817 5162 3658 23I9 II58 3IOI89 94281 8889 46 I5 0829 8912 7121 5469 3968 263i 1473 o508 975I 9216 45 I6 II3o 9215' 7426 5776 4277 2944 I789 0827 330073 954244 17 430o 9517 7731 6o83 4587 3256 2.I 05 Ii46 0396 9868 43 IF 173I. 9820 8035 6390 4897 3569 2420 I465 0718 350195 42 19 2031 200122 8340 6697 5207 3882 2736 1784 i04i 0522 4I 2') I32332 200425 218645 237004 255516 274I94 293052 3I2104 33I364 350848 4o 2I 2632 0727 8950 7312 5826 4507 3368 2423 1687 II75 39 22 2933 Io3o 9254 76I9 6I36 4820 3684 2742 20IO 1502 38 23 3234 I333 9559 7926 6446 5133 4000 3062 2333 1829 37 24 3534 i635 9864 8234 6756 5446 43i6 338I 2656- 2I56 36 25 3835 1938 220169 854I 7066 5759 4632 3700 2979 2483 35 26 4i36 2241 0474 8848 7377 "6072 4948 4020 3302 2810 34 27 4437 2544 0779 9156 7687 6385 5265 434o 3625 3I37 33 28 4737 2847 io84 9464 7997 6698 558i 4659 3949 3464 32 29 5038 3I49: I389 977I 8307 7011 5897 4979 4272 3791I 3 30 185339 203452 22I695 240079 2586I8 277325 296213 315299 334595 354119 30 3I 5640 3755 2000 0386 8928 7638 6530 5619 4919 4446 29 32 594I 4058 2305 o694 9238 7951 6846 5939 5242 4773 28 33 6242 436I 2610 1002 9549 8265 7163 6258 5566 5101 27 34 6543 4664 29I6 I3Io 9859 8578 7480 6578 5890 5429 26 35 6844 4967 3221 16I8 260170 889I 7796 6899 6213' 5756 25 36 7145 5271 3526 1925 o480 9205 8II3 72I9 6537 6084 24 37 7446 5574 3832 2233 0791 9519 8430 7539 686i 6412 23 38 7747 5877 4137 254I II02 9832 8747 7859 7185 6740 22 39 8o48 6i8o 4443 2849 I4I3 280146 9063 8179 7509 706821 4o 188349 2o6483 224748 243I57 26I723 28o0460 299380 3I85oo 337833 35739620 4i 865I 6787 5o54 3466 2034 0773 9697 8820 8157 7724 19 42 8952 7090 5360 3774 2345 1087 3000o 4 9I41 848i 8052 i8 43 9253 7393 5665 4082 26561 I4o0 o33i 9461 8806 8380oI7 44 9555 7697 597I 4390 2967 17I5 0649 9782 913o 8708 i6 45 9856 8000 6277 4698 3278 2029 og966 320103 9454 9037 i5 46 I9oi57 8304 6583 5007 3589 2343 1283 0423 9779 9365 14 47 o459 8607 6889 53i5 3900 2657 I6oo 0744 34oio3 9694 I3 48 0760 8911 7I94 5624 4211 2971 19I8 io65 0428 360022 12 49 1062 92I4 7500 5932 4523 3286 2235 i386 0752 o35IiiI 5o 191363 209518 227806 24624I 264834 283600 302553 321707 34o077 360679 10 5I I665 9822 8112 6549 5145 3914 2870 2028 1402 IOO8 9 52 1966 210126 84i8 6858 5457 4229 3188 2349 1727 1337 8 53 2268 0429 8724 7I66 5768 4543 35o6 2670 2052 i666 7 54 2570 0733 9031 7475 6079 4857 3823 2991 2377 I995 6 55 2871 I137 9337 7784 6391 5172 414I 3312 2702 2324 5 56 3173 i34r 9643 8092 6702 5487 4459 3634 3027 2653 4 57 3475 1645 9949 84o0 7014 58o0 4777 3955 3352 2982 3 58 3777 1949 230255 8710 7326 6ii6 5095 4277 3677 3312 2 59 4078 2253 o0562 9019 7637 643I 54i3.4598 400o2 364I I 790 780 77 0 76 750 1 740 730 720 71~ 700 K Natural Co-tangents. P. P. to 1".5.~oi O5 5.9'.3 5.17 5.22 6.27 5.33.9 5.46' 120 NATURAL SINES. 200 210 220 230 240 250 2 270 280 290 0 342020o 358368 374607 390731 406737 422618 438371 45399o[ 469472 4848io 6o x 2293 864o 4876 0999 7002 2882 8633 4250 9728 5o64 59 2 2567 8911 5i46 1267 7268 3145 8894' 4509 9985 53i8 58 3 284o 9i83 54i6 i534 7534 3409 9i5517 4768470242 5573 57 4 3ii3 9454 5685 1802 7799 3673! 9417 5027! 0499 582756 5 3387 9725 5955 2070 8065 3936 9678 5286 0755 6o8i 55 6 3660 9997 6224 2337 8330 4199 9939 5545 IOI2 6335 54 7 3933 360268 6494 2605. 8596 4463 440200 5804 I268 6590 53 8 4206 o54o 6763 2872 886I 4726 0462 6063 1525 6844 52 9 4479 o8Ii 7033 3X4o 9I27 4990 0723 6322 1782 7098 Si 1o 344752 361o82 377302 393407 409392 425253 440984 45658o 472038 487352 50 II 5025 1353 7571 3675 9658 55i6! 1245 6839 2294 760649 12 5298 1625 784I 3942 9923 5779f i5o6 709& 2551 786048 13 5571 1896 8Iio 4209 4ioi88 6042i 1767 7357 2807 8II447 X4 5844 2167 8379 4477 0454 6306 2028 7615 3063 836746 i5 6117 2438 8649 4744 0719 6569 2289 7874 3320 8621 45 i6 6390 2709 8918 5oii 0984 6832 2550 8i33 3576 887544 17 6663 2980 9187 5278 1249 7095 2810 839I 3832 912943 z8 6936 3251 9456 5546 i5i4 7358 3071 8650 4088 938242 19 7208 3522 9725 58i3 I779 7621 3332 8908 4344 96361 20o 347481 363793 379994 396080o 42045 427884 443593 459166 474600oo 489890o40 21 7754 4064 380263 6347 23o10 8147 3853 9425 4856 49o0143 39 22 8027 4335 0532 66i4 2575' 84io 4114 9683 5112 039738 23 8299 4606 o8oi 688I 2840 8672 4375 9942 5368 065o 37 24 8572 4877 1o70 7148 3io4 8935 4635 4602o00 5624 0904 36 25 8845 5i48 I339 7415 3369 9198 4896 o458 588o 115735 -6 9II7 54i8 i6o8 7682 3634 9461 5i56 o716j 6i36'4 34 27 9390 5689 1877 7949 3899 9723 5417 0974].6392 x66433 28 9662, 59601 2146 8215 4i64 9986 5677 I2321 6647 191732 29 9935 6231 2415 8482 4429 430249 5937 1491i 6903 2I70 31 30 35020713665o01 382683 398749 414693 43o5ii 446198 46i749 47759 49214 3 31 0480o 6772 2952 90o6 4958 0774 6458 2007, 7414 267729 32 0752 7o42 3221 9283 5223 io36. 6718 22651 7670 293028 33 Io25 7313 3490 9549 5487 1299 6979 25231 79251 3i8327 34 I297 7584 3758 9816 5752 i56i 7239 2780o 8i8fl 343626 35 1569 7854 4027 4000oo82 6o6 1823 7499 3o38] 8436' 3689 25 36 1842 8125 4295 0349 6281 2086 7759 32961 86921 3942 24 37 214 8395 4564 o6i6 6545 2348 80o9! 3554] 8947w 495 23 38 2386 8665 4832 0882 68Io 26r0 82791 3812: 92o3 4448 22 39 2658 8936 5ioi I"49' 7074 2873 8539 4069 9458 470021 40 352931 369206 385369 4oI4i5 417338 433i351 448799 464327 479713 49495320 41 3203 9476 5638 i681 7603 3397 9059: 4584 9968 520619 42 3475 9747 59o6 1948 7867 3659 9319 4842 4802231 5459 i8 43 3747 370017 6174 2214 8i3i 3921 9579 5ioo 04791 57II 17 44 4og9 0287 6443 2480 8396 4i83 9839 5357 0734 5964 1G 45 4291 0557 6711 2747 8660 4445 45oo98 56i5 0989 6217 i5 46 4563 0828 6979 3oi3 8924 4707 0358 5872 1244 6469 i4 47 4835 0o98 7247 3279 9188 4969 o6i8 6129 1499 6722 I3 48 5Io7 i368 75i6 3545 9452 5231 0878 6387 1754 6974 12'49 5379 i638 7784 38ii 9716 5493 ii37 6644 2009 7226 iI So 35565i 3790o8 388052 404078 419980o 435755 45i397 46690o 482263 497479 IO S 59.3 2178 8320 4344 420244 6o017 i656 7158 2518 7731 9 52 6194 2448 8588 46io o5o8 6278 1916 7416 2773 7983 8 53 6466 2718 8856 4876 9772 6540 2175 7673 3028 8236 7 54 6738 2988 9124 5142 io36 6802 2435 7930 3282 8488 6 55 7o010o 3258 9392 54o8 i3oo 7063 2694 8187 3537 8740o 5 56 7281 35281 9660 5673 i563 7325 2953 8444 3792 8992 A 57 7553 3797 9928 5939 1827 7587 3213 8701 40461 9244 3 58 7825 4067 3901o96 6205 2091 7848 3472 8958 43oi 9496 z 59 8096 4337 0463 6471 2355 8iIo 3731 9215 4555 9748 I 69~ 680 67~ 66~ 656 646 63 60 6 61 60O Natural Co-sines. to I/.4.64 4.5i 4.48 4 45 4.4ij 4.38 4.34 4.3o 4.26 4.22 NATURAL TANGENTS. I121 200 210 220 230 24~ 25 26~ 27~ 280 290 0 363970 383864 404026 424475 445229466308 487733 509525 53I709 55430960 I 43oo00 4198 4365 48i8 5577 6662 80o93 9892 2083 4689159 2 4629 4532 4703 5162 5926 7016 8453 510258 2456 5070o58 3 4959 4866 5o042, 55o5 6275 7371 8813 0625 2829 545o]57 4 588 5200oo 5380o 5849 6624 7725 9174 0992 3203 5831i 56 5 56i8 5534 5719 6192 6973 8080o 9534 1359 3577 6212155 6 5948 5868 6058 6536 7322 8434 9895 I726 3950 6593 54 7 6278 6202 6397 6880 7671 8789 490256 2093 4324 6974 53 8 6608 6536 6736 7224 8020 9144 06I7 2460 4698 7355 52 9 6938 6871 7075 7568 8369 9499 0978 2828 5072 7736 51i 1o 367268 387205 407414 427912 4487191 469854 491339 513195 535446 558I18 50 11i 7598 7540 7753 8256 9068 470209 1700 3563 5821 8499[49 12 7928 7874 8092 86oi1 9418 0564 2061 3930 6195 888 148 13 8259 8209 8432 8945 9768 0920 2422 4298 6570 9263[47 14 8589 8544 8771 9289 45o0117 1275 2784 4666 6945 9645146 iS 8919 8879 9111 9634 0o467 163i 3145 5034 73191560027145 16 9250 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5502 76121 55oi34 3092 1!.5o 3805301o 4oo6461 42i046 44748 462771 484137 505867 527984 SSo5i31 573478 o 0 5i 0863 0984 1389 2095 3I24 4496 6232 8356 0892 3865 9 52 1196 1322 1731 2443 3478 4855 6598 8728 1271 4252 8 53 153o i66o 2074 2791 383i 5214 6963 g9oo i65o 4638 7 54 1863 I997 2417 3139 4185 5574 7329 9473 2030 5026 6 55 2196 2335 2759 3487 4538 5933 7695 9845 2409 5413 5 56 230 o 673 3o102 3835 4892 6293 8o6 11530218 2789 5800 4 57 2863 3oix'3445 4i83 5246 6653 8427 o95gI 3169 6187 3 58 3197 335o 3788 4532 56oo1 7o013 8793 0963 3549 65751 2 59 353o 3688 4132 488o 5954{ 7373 1599 i336 3929 6962 I 69: 680 670 66[ 650 640 630 620 610 60 Natural Co-tangenis. to 1,.535.6o 5.68 S.76 5.85 5.9S 6.o5 6 i6 6.28 6.4o 22 NATURAL SINES. 30 310 32 1 330 340 350 360 370 38~ 39 o 5000oo00 55o38 529919 544639 559193 573576 5877851 6o18i5 6I566 629320 6o I 0252 5287 530I66 4883 9434 38i5 8021 2047 589I 9546 59 2 o504 5537 o4I3 5127 9675 4053 8256 2280 6120 9772 58 3 0756 5786 0659 537I 9916 4291 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3791 8685 3419 7987 2384 6608 o653 45i5 8189 1673 5 56 4040 8932 3663 8228 2623 6844 o885 4744 84i6 1896 4 57 4290 9179 3907 8469 286I 7079 III8 4974 8642 211I 3 58 4539 9426 4151 8710 3oo00 7314 i35o 52o3 8868 2342 2 59 4789 9673 4395 8952 3338 7550o 583 5432 9094 2565 I 590 58~ I 570 566 55~ 54~ 530 52" 510 500 Natural Co-sines. to 1,.4.I8 4.i3 409og 4.04 4.00oo 3.95 3.90 3.85 3.80o 3.74 - NAT U R-A L TANGENTS..12 a 300 310 320 330 340 350 360 370 380 390 577350 6oo86i 624869 6494Q8 674509 700208 726543 753554 781286 809784 60 I 7738 I257 5274 982I 4932 o64I 6987 4oio I754 8I0266 59 2 8126 I653 5679 650235 5355 I075 7432 4467 2'223 0748 58 3 85I4 2049 6o83 0649 5779 I509 7877 4923 2692 1230 57 4 8903 2445 6488 Io63 6203 I943 8322 5380 316I I712 56 5 929I 2842 6894 I477 6627 2377 8767 5837 363I 2195 55 6 9680 3239 7299 1892 705 I 2812 9213 6294 4Ioo 2678 54 7 58oo0068 3635 7704 2306 7475 3246 9658 6751 4570 316i 53 8 0457 4032 8iIO 272I 7900 368I 730104 7209 5040 3644 52 9 o846 4429 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7479 I20I 5436 660214 5569 I539 8162 5480 3538 2384 34 27 7870 i6oi 5844, o63I 5997 1977 86ii 594I 4012 2872 33 28 8262 2001 6253 Io49 6425 2416 906I 6403 4486 3360 32 29 8653 2401 66.6I I467 6853 2854 9511 6865 4961 3848 3I 30 589045 61280I1' 637070 66i886 687281 713293 739961 767327 795436 824336 30 3I 9437 3201 7479 2304 7709 3732 7404I1 7789 5911 4825 29 32 9829 36oi 7888 2723 8i38 4171 o862 8252 6386 5314 28 33 590221 4002 8298 3I4I 8567 461 1312 8714 6862 5803 27 34 o6I3 4402 8707 3560 8995 5050 1763 9177 7337 6292 26 35 Ioo6 4803 9117 3979 9425 5490 2214 9640 7813 6782 25 36 I398 5204 9527 4398 9854 5930 2666 770104 8290 7272 24 37 I79I 5605 9937 48i8 690283 6370 3117 0567 8766 7762 23 38 2184 6oo06 640347 5237 0713 68io 3569 io3I 9242 8252 22 39 2577 6408 0757 5657 II43 7250 4020 1495 97I9 8743 2I 4o 592970 6I6809 641167 66607.7 69I572 717691 744472 77I959 800196 829234 20 4' 3363 72I1 I578 6497 2003 8132 4925 2423 0674 9725 19 42 3757 7613 I989 69I7 2433 8573 5377 2888 I I51 8302I6 I8 43 450o 8oi5 2399 7337 2863 9014 583o 3353 ~ I629 0707 I7 44 4544 8417 2810 7758 3294 9455 6282 38I8 2107 I 19916 45 4937 88I9 3222 8179 3725 9897 6735 4283 2585 I69I I5 46 533i 9221 3633 8599 4i56 720339 7189 4748 3063 2183 14 47 5725 9624 4044 9020 4587 0781 7642 5214 3542 2676 i3 48 6o20 620026 4456 9442 50o8 I223 8096 5680 402 I 31691I2 49 65i4 0429 4868 9863 5450 I665 8549 6i46 4500 3662 I I 50 596908 620832 645280 670284 69588I 722108 749003 776612 804979 834i55 Io 51 7303 1235 5692 o706 63i3 2550 9458 7078 5458 4648 9 52 7698 16381 6io4 II28 6745 2993 99I2 7545 5938 5142 8 53 8093 20421 65I6 i550 7177 3436 750366 8012 64i8 5636 7 54 8488 2445 6929 I972 7610 3879 o821 8479 6898 6i3o 6 55 8883 2849! 7342 2394 8042 4323 I276 8946 7379 6624 5 56 9278 3253 7755 2817 8475 4766 I731 9414 7859 7II9 4 57 9674 3657 8I68 3240 8908 52IO 2187 988I 8340 76141 3 58 600069 4o6I 858i 3662 934I 5654 2642 780349 882I 8109 2 59 o465 4465 8994 4085 9774 6098 3098 o817 9303 860o4 590 580 i 570. 56~ 550 540 530 520 51~ 500 Natural Co-tangents. P P, 6.53I 6.671 6.82 6.97 7.14 f 7.31 7. 50 7.70 7.92 8.4....... 77 ~7.92...... f 2 4 N A TU RAI JN ES..3 400 410 420 430 440 4o0 460 470 480 49"' o0 642788 656059 669131 68I998 694658 707I07 719340 731354 743I45 754710 60 I 3oio 6279 9347 2211 ~ 4868 7312 9542 1552 3339 4900 59 2 3233 6498 9563 2424 5077 7518 9744 1750 3534 5091 58 3 3456 6717 9779 2636 5286 7724 9946 1949 3728 5282 57 4 3679 6937 9995 2849 5495 7929 720148 2147 3923 5472 56 5 39OI 7156 670211 3o6i 5704 8i34 0349 2345 4117 5663 55 6 4I24 7375 0427 3274 59I3 834o o55I 2543 4312 5853 54 7 4346 7594 0642 3486 6122 8545 0753 274I 4506 6044 53 8 4569 7814 o858 3698 6330 8750 0954 2939 4700 6234 52 9 479I 8033 1074 3911 6539 8956 1156 3137 4894 6425 51 IO 6450I3 658252 67I289 684123 696748 709161 721357 733334 745088 7566I5 5o 11 5236 8471 i505 4335 6957 9366 1559 3532 5282 6805 49 12 5458 8689 172I 4547 7165 957I 1760 3730 5476 6995 48 13 568o 8908 1936 4759 7374 9776 1962 3927 5670 7185 47 I4 5902 91I27 2151 497I 7582 998I 2163 4125 5864 7375 46 15 6124 9346 2367 5i83 7790 7IOI85 2364 4323 6057 7565 45 I6 6346 9565 2582 5395 7999 0390 2565 4520 6251 7755 44 17 6568 9783 2797 5607 8207 0595 2766 47I7 6445 7945 43 i8 6790 660002 30i3 58i8 84i5 0799 2967 4915 6638 8134 42 19 70I12 0220 3228 6030 8623 Ioo4 3I68 5112 6832 8324 41 20 647233 660439 673443 686242 698832 7II209 723369 735309 747025 758514 40 21 7455 0657 3658 6453 9040 i4I3 3570 5506 7218 8703 39 22 7677 0875 3873 6665 9248 1617 3771 5703 7412 8893 38 23 7898 1094 4088 6876 9455 1822 397I 5900 7605 9082 37 24 8120 I312 4302 7088 9663 2026 4I72 6097 7798 9271 36 25 834I 1530 4517 7299 9871 2230 4372 6294 7991 9461 35 26 8563. I748 4732 7510 700079 2434 4573 649I 8i84 9650 34 27 8784 1966 4947 772I 0287 2639 4773 6687 8377 9839 33 28 9006 2184 5161 7932 0494 2843 4974 6884 8570 760028 32 29 9227 2402 5376 8I44 0702 3047 5174 7081 8763 0217 31 30 649448 662620 675590 688355 700909 713250 725374 737277 748956 760406 30 31 9669 2838 5805 8566 II1117 3454 5575 7474 9148 0595 29 32 9890 3o56 60I9 8776 I324 3658 5775 7670 934I 0784128 33 65o0Ir 3273 6233 8987 i531 3862 5975 7867 9534 0972127 34 0332 3491 6448 9198 1739 4066 6175 8063 9726 ii6i 26 35 o553 3709 6662. 9409 1946 4269 6375 8259.. 99g9g I350 25 36 0774 3926 6876 9620 2153 4473 6575 8455 750111 I538 24 37 0995 4144 7090 9830 2360 4676 6775 865I o303 1727 23 38 1216 436I 7304 690041 2567. 4880 6974 8848 0496 1915 22 39 1437 4579 7518 0251 2774 5083 7174 9043 o688 2I04 21 4oI I 657 664796 677732 690462 702981 715286 727374 739239 750880 762292 20 41 1878 5oi3 7946 o672, 3188 5490 7573 9435 1072 2480 19 42 2098 5230 8i6o 0882 3395 5693 7773 9631 1264 2668 i8 43 2319 5448 8373 1093 36oi 5896 7972 9827 i456 2856 I 7 44 2539 5665 8587 i3o3 3808 6099 8172 740023 i648 3044 i6 45 2760 5882 88oi 1513 40o5 6302 8371 0218 I840 3232 15 46 2980 6099 9014 1723 4221 6505 8570 o4I4 2032 3420 14 47 3200 63i6 9228 I933 4428 6708 8769 0609 2223 3608 13 48 3421 6532 9441 2143 4634 6911 8969 o805 2415 3796 12 49 364i 6749 9655 2353 484I 7113 9168 Iooo 2606 3984 I I 50 65386i 666966 679868 692563 705047 7r73I6 729367 74zI95 752798 764171 IO 5I 40o8 7183 68oo8I 2773 5253 7519 9566 139I 2989 4359 9 52 43oi 7399 0295 2983 5459 7721 9765 i586 318I 4547 8 53 4521 7616 o5o8 3192 5665 7924 9963 I781 3372 4734 7 54 474I 7833 0721 3402 5872 8I26 730162 1976 3563 4921 6 55 4961 8049 0934 3611 6078 8329 o36I 2171 3755 510o9 5 56 5i8o 8265 1I47 3821 6284 853I'o560 2366 3946 5296 4 57 5400 8482 i36o 4030 6489 8733 0758 2561 4137 5483 3 58 5620 8698 1573 4240 6695 8936 o957 2755 4328 5670 2. 59 5839 8914 1786 4449 6901 9138 1155 2950 4519 5857 I 490 480 470 460 450 1 440 430 42~ 410 40~. |' NNatural Co-sines. t.:P.3.691 3.63 1 3.57 1 3.521 3.46 3.4o0 3.34 | 3.27 1 3.21 3.x5 to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _. NATURAL TANGENTS. 1 2' 400 410 420 430 440 450 46~ 470 480 49o 839Ioo 869287 90o0404 93255 965689.ooooo00000.o3553 I7237 i.iio6i.I5037 60 I 9595 9798 o931! 3059 6251 00oo58 3613 7299 1126 5104 59 2 840092 870309 1458 3603 68I4 oii6 3674 7362 1 I91 5172 58 3 o588 0820 I985 4I48 7377 OI75. 3734' 7425 I256 5240 57 4 Io84 1332 2513 4693 79Zo 0233 3794 7487 1321 5308 56 5 i58i 1843 3o4I 5238 8504 029I 3855 7550 1387 5375 55 6 2078 2356 3569 5783 9067 o350 39I5 7613 I452 5443 54 7 2575 2868 4098 6329 9632 o408 3976 7676 I517 5511 53 8 3073 338I 4627 6875 970196 0467 4o36 7738 1582 5579 52 9 3571 3894 5i56 7422 0761 0525 4097 780I I648 5647 5I xo 844069 874407 905685 937968 971326 i.oo583 I.o4I58 1.07864 I.II713 I.I5715 5o II 4567 4920 6215 85i5 1892 0642 4218 7927 I778 5783 49 12 5066 5434 6745 9063 2458 0701 4279 7990 I844 5851 48 I3 5564 5948 7275 9610o 30214 0759 4340 8053 1909 5919 47 I4 6063 6462 7805 94oI58 3590 o818 44o0 8xi6 1975 5987 46 15 6562 6976 8336 0706 4I57 0876 446I 8179 2o4I 6056 45 I6 7062 749I 8867 I255 4724 0935 4522 8243 2106 61,24 44 17 7562 8006 9398 I8o3 5291 o994 4583 8306 2I72 6I9243 I8 8062 852I 9930 2352 5859 io53 4644 8369 2238 6261 42 19 8562 9037 910o462 2902 6427 III2 4705 8432 2303 6329'4I 20 849062 879553 9Io994 943451 976996I.o II701.04766 I.084961.12369x.16398 40 21I 9563 8800oo69 I526 4ooI 7564 I229 4827 8559 2435 6466 39 22 85oo64 o585 2059 4552 8133 I288 4888 8622 2501 6535 38 23 o565 IIo02 2592'5102 8703 1347 4949 8686 2567 6603 37 24 Io67 I619 3125 5653 9272 I4o6 5oio 8749 2633 667236 25 i568 2136 3659 5204 9842 i465 5072 8813 2699 674I 35 26 2070 2653 4193 6756 980413 i524 5I331 8876 2765 6809 34 27 2573 3171 4727 7307 0983 1583 5194 8940 2831 6878 33 28 3075 3689 526I 7859 I554 I642 5255 9003 2897 6947 32 29 3578 4207 5796 84I2 2126 1702 5317 9067 2963 7016 3I 3o 854o8I 884725 9I633I 948965 982697 1.I076I I.o5378 1.o913 1.I I3029 1.I7085 3o 31 4584 5244 6866 9518 3269 I820 5439 9195 3096 7154 29 32 5087 5763 7402 950071 3842 I879 550o 9258 3162 7223 38 33 559I 6282 7938 0624 44I4 I939 5562 9322 3228 7292 27 34 60o95 6802 8474 1178 4987 I998 5624 9386 3295 7361 26 35i 6599 732I 90Io I733 5560 2057 5685 9450 336I 7430 25 361 7I04 7842 9547 2287 6I134 2II7 5747 95I4 3428i8 7500 24 371 7608 8362 920084 2842 6708 2176 5809 9578 3494 756923 38 8I13 8882 0621 3397 7282 2236 5870 9642 356i 7638 22 39 86I9 94o3 1159 3953 7857 2295 5932 9706 3627 7708 2I 4o 859124 889924 92I697 954508 988432 1.02355.0o5994 I.o9770.I13694 I.17777 20 4I 9630 890446 2235 5064 9007 2414 6056 9834 376I 7846 19 42 860o36 09671 2773 562I 9582 2474 6II7 9899 3828 791618 43 o642 14891 33I2 6I77 99oI58 2533 6179 9963 3894 7986 I7 44 1I48 20I2! 385i 6734 0735 2593 624I.1o0027 396I 8055 i6 45 i655 25341 4390 7292 I311 2653 6303 oo009 4028 8125 i5 46 2162 30571 4930 7849 i888 2713 6365 o156 4095 81941I4 47 2669 3580 5470 8407 2465 2772 6427 0220 4162 8264 13 48 3177 4Io3i 6o010 8966 3043 2832 6489' 0285 4229 8334 12 49 3685 4627 655i 9524 362I 2892 655i 0349 4296 84o4 I I 50o 86493 895i51 927091 960083 994199 1.02952 I.o66I3 I.Io4I4 I.4363 I.18474 IO 5I 4701 56751 7632 0642 4778 3o12 6676 0478 4430 8544 9 52 5209 6I99 8174 I202 5357 3072 6738 0543 4498 86I4 8 53 57i8 6724 8715 176I 5936 3132 68oo o607 4565 8684 7 54 6227 7249 9257 2322 6515 3192 6862| 0672 4632 8754 6 55 6736 7774 9800 2882 7o95 3252 6925 0737 4699 8824 5 56 7246 8299 930342 3443 7676 3312 6987 0802 4767 8894 4 57 7756 8825 o885 4004 8256 3372 7049 0867 4834 8964 3 58 8266 935I I428 4565 8837 3433 7112 o931 4902 9035 2 59 8776 9877 I971. 5127 94I8 3493 7174 0996 4969 9o05 1 490 480 470 46 450 440 430~ 420 41~. 400~ 8.64 8. Natural Co-tangents. i Pto ~. p. I3 I~9 t2 I6 l. 21 —' I. f. |P,8.38| 8.64 8.92 9 2I 9.53 099 1.02 i.06 | II0 | Iil| 26 NATURAL I;INES.' 50~ C 510 520 530 540 550 560 570 5S~ 590 o 766044[ 777I46 7880I I 798636 8090I7 819Ig52 829038 838671 848o48 857I67 60 I 6231 7329 8190 88ii 9I88 931I9 9200 8829 8209 7317 59 2 64i8 75I2 8369 8985 9359 9486 9363 8987! 8356 7467 58 3 6605 7695 8548 gI60 9530 9652 9525 9146 85Io 76I6 57 4 6792 7878 8727 9335 9700 98I9 9688 9304~ 8664 7766 56 5 6979; 806o 8905 9510 987I1 9985 9850 9462 88I8 7915 55 6 7.655 8243 9084 9685 8Ioo42 820152 8300I2 9620 8972 8065 54 7 7352 8426 9263 9859 02I2 o03I8 0174 9778 9125 8214153 8 75381 8608' 944I 800034 o383 0485 0337 9936 9279 8364 52 9 7725 8791 9620 0208, o553 o65i. 0499 840094 9433 85i3 5 I0 7679II 778973 789798 800383 810723 8208I7 83o66I 840251 849586 858662 50 1I 8097 9I56 9977 0557 0894 0983 0823 0409 9739 88ii 49 I2 8284 9338 790155 073I Io64 II49 0984 0567 9893 8960148 131 8470 9520 0o333 0906 1234 i3i5 ii46 0724 85oo46 9I09o47 i4 8656 972 05I I0o8o i4o4 I48i I308 0882 0199 9258 46 i5 8842 9884 0690 I254 I5741 I647 1470 039 o352 9406 45 16 9028 780067| o868 1428 I744, i8i3 i63 IIg6 0505 9555 44 17 9214 0249 Io46 1602 I9I4 I978 1793 I354 o658 9704 43 x8 9400 o4301 I224 776 2084 2144 I954 1511 o8iI 9852 42 I9 9585 0612 I40I I949 22531 2310 2115 i668 o0964 86ooo000I 4 20 769771 780794 791579 802123 8I2423 8224751832277 84I825[ 85III7 860I49140 2I 9957 09761 1757 2297 2592 264I, 2438 I982 1269 0297 39 22 770142 1157 1935 2470 2762 2806 2599 2I391 1422 o446 38 23 0328 I339 2112 2644 2931 297I 2760 2296 1575 0594137 24 o5i3 1520 2290 28I7; 3OI 3i36 2921I 2452 I727 0742 36 25 o699 I702| 2467 2991 3270 3302 3082 2609 I879 0890g35 26 o884 i883 2644 3I64 3439 3467 3243 2766 2032'o38 34 27 I069 2065 2822 3337 3608 3632 3404 2922 2I84 I 86 33 28 I254: 22461 2999 35ii 3778 3797 3565 3079 2336 i334 32 29 I440o 2427 3I76 3684 3947 3961 3725 3235 2488 I48i 31 30 7716251 782608 793353 8o3857 8I4II6 824I26 833886 84339I 852640 86I629 3o 31 i8io 2789 3530 403o 4284 429I 4o46 35481 2792 1777 29 32 I995! 2970 3707 4203 4453 4456 4207 3704j 2944 I92428 33 2I79| 3i5i 3884 43761 422 4620 4367 3860 3096 2072 27 34 2364 33321 4o06 4548 479I 4785 4527 4oI6 3248 2219 26 35 2549 3513 4238 4721 49591 4949 4688 4172 3399 2366 25 36 2734 3693 44i5 4894 5I28 5II3 48484 4328 355I 2514~24 37 29I8 3874 459I 5o66 5296 5278 50081 4484 3702 2661 23 38 3Io3 4o55 4768| 52391 5465 5442 5i68 4640 3854 2808122 39 3287 4235 4944 54II 5633 5606 5328| 4795 4005 2955 21 40 7734721 7844i16 795I21 8o5584 8I580o 825770o 835488 84495I 854i561 86310220 4I 3656 4596 5297 5756 5969 5934 5648 5io6 4308 3249 19 42 3840 4776 5473 5928 6i38 6098 5807 5262 4459 3396 I8 43 4024 4957 565o 6Ioo 6306 6262 5967 54I7 46io 3542 17 44 42091 537 5826 6273 6474 6426 6I271 5573 476i 3689 i6 45 4393 5317 6002 6445 6642 6590 6286 5728 49I2 3836 I5 46 4577 5497 6178 6617 68091 6753 6446 5883 5063 3982 I4 47 476i 5677 6354 6788 6977 6971 6605 6038 5214 4128 I3 48 4944 58571 653o 6960 7I451 708I 6764 6I93 5364 4275 12 49 5I28 6037 6706 7132 73131 7244 6924 6348 55i5 442I II 50 775312 7862I7 796882 807304 8I748o 827407 837083 846503 855665 864567|:o 5i 5496 6396 7057 7475 7648 7571 7242 6658 58I6 47I3 9 52 5679 6576 7233 7647 78I5 7734 740I 68I3 5966 486o 8 53 5863 6756 7408 7818 7982 7897 7560 6967 6117 5oo6 7 54 6046 6935 7584 7990 8i1o 8o6o 77I9 7I22 6267 5I5i 6 55 6230 7114 7759 8I6I 8317 8223 7878 7277 64I7. 5297 5 56 64i3J 7294 7935 8333 8484 8386 8o36 7431 6567i 54431 4 57 6596. 7473 8iio 85o4 865I 8549 8i95 7585 6718 55891 3 58 6780; 7652 8285 8675 88i8 8712j 8354 7740 6868 57341 2 59 6963 7832 846o 8846 8985 88751 85I2 7894 7oI71 588o0 390 380'- 370 360 350 340 330 32 310 3Q0 Natural Co-sines. V p |o I 1 2 1. 2 2 I 24 to 3.o8 |3.o02 2.95.88 2.81 2.75 28.6o 2 5.3 2.46 NATURAL TANGENTS. 121 M 150o 510 520 530 540 550 560 570 580 L590 o0.19I75 1.23490 1.27994 I.32704 1.37638I.428 I5 1.48256 1.53986 i.6oo33 1 66428 6o I 9246 3563 8071 2785 7722 2903 8349 4o85 oi0137 6538 59 2 9316 3637 8 48 2865 7807 2992 8442 4i83 0241 6647 58 3 9387 3710 8225 2946 7891 3o80 8536 428I o345 6757 57 4 9657 3784 8302 3026 7976 3169 8629 4379 o449 6867 56 5 9528 3858 8379 3107 8o6o 3258 8722 4478 o553 6978 55 6 9599 3931 8456 3187 8i45 3347 8816 4576 0657 7088 54 7 9669 4005 8533 3268 8229 3436 8909 4675 0761 7198 53 8 9740 4079 86io 3349 8314 3525 9003 4774 o865 7309 52 9 9811 4i53 8687 3430 8399 36I4 9097 4873 0970 741, 5I lo 1.19882 1.24227 1.2876411.335 I1.38484I.43703 I.49190 I 54972 1.6 074 1.67530 5o II 9953 43oi 8842 3592 8568 3792 9284 5071 II79 7641 49 I211.20024 4375 8919 3673 8653 388I 9378 517o I283 7752 48 13 oo95 4449 8997 3754 8738 3970 9472 5269 i388 7863 47 I4 oi66 4523 9074 3835 8824 4060 9566 5368 1493 7974 46 I5 0237 4597 9152 3916 8909 4149 966I 5467 1598 8085 45 i6 o308 4672 9229 3998 8994 4239 9755 5567 I703 819644 17 0379 4746 9307 4079 9079 4329 9849 5666 i808 8308 43 i8 o45I 4820 9385 4i6o 9165 44I8 9944 5766 I9I4 84I1942 I9 0522 4895 9463 4242 9250 4508 X.5oo38 5866 2019 8531 4i 20 1.20593 1.24969 1.2954I I.34323 1.39336 1.44598 i.5oi33 I.55966 1.62125 i.68643 4o 21 o665 5044 9618 4405 9421 4688 0228 6065 2230 8754 39 22 0736 5118 9696 4487 9507 4778 0322 6165 2336 8866 38 23 o808 5193 9775 4568 9593 4868 0417 6265 2442 8979 37 24 0879 5268 9853 4650 9679 4958 0512 6366 2548 9gog 36 25 o951 5343 9931 4732 9764 5049 0607 6466 2654 9203 35 26 1023 54I7 1.30009 48I4 9850 5I39 0702 6566 2760 9316 34 27 1094 5492 0087 4896 9936 5229 0797 6667 2866 9428 33 28 ii66 5567 oi66 4978 1.40022 5320 o893 6767 2972 9541 32 29 1238 5642 0244 5060 0109 54io 0988 6868 3079 9653 3i 30 1.213Io 1.25717 1.30323 1.35142 1.40195 I.4550o i.51084 1.56969 i'.63I85.69766 30 31 1382 5792 040o 5224 028I 5592 1179 7069 3292.9879 29 32 I454 5867 o480 5307 o367 5682 1275 7170 3398 9992 28 33 I526 5943 o558 5389 o454 5773 1370 7271 3505 1.701o6 27 34 I598 6oi8 0637 5472 o540 5864 I466, 7372 36I2 0219 26 35 I670 6093 0716 5554 0627 5955 1562 7474 37I9 0332 25 36 1742 6I69 0795 5637 0714 6046 i658 7575 3826 0446 24 37 I814 6244 o873 5719 o80o 6137 1754 7676 3934 0560 23 38 I886 6319 o952 5802 0887 6229 1.850 7778 4o4I 0673 22 39 1959 6395 io03 5885 0974 6320 1946 7879 4i48 0787 21 4o 1.2203 I1.26471 i.3 I IO.35968 i.4io6i i.4641I 1. 52043 1.57981 I.64256 1.7090I 20 4I 2Io4 6546 1I90 6o05I 148 6503 2139 8083 43631 io5 I9 42 2176 6622 I269 6134 1235 6595 2235 8184 4471 1129 18 43 2249 6698 i348 6217 1322 6686 2332 8286 4579 1244 17 44 2321 6774 I427 6300 1409 6778 2429 8388 4687 i358 16 45 2394 6849 i507 6383 1497 6870 2525 8490 4795 I473 15 46 2467 6925 I586 6466 I584 6962 2622 8593 4903 I588 I4 47 2539 7001 1666 6549 1672 7053 2719 8695 50oI 1702 13 48 2612 7077 I745 6633 1759 7I46 2816 8797 5120 I817 12 49 2685 7153 1825 6716 1847 7238 2913 8900 5228 1932 II 50o.22758 1.27230 I.3I904 I.368oo001.41934 1.47330 I.53oo 1.59002..653371.72047 10 51 2831 7306 I984 6883 2022 7422 3107 91o5 5445 2163 9 52 2904 7382 2064 6967 2IIO 75I4 3205 9208 5554 2278 8 53 2977 7458 2144 7050 2198 7607 3302 9311 5663 2393 7 54 3o050 7535 2224 7134 2286 7699 34o00 94I4 5772 2509 6 55 3123 7611 2304 7218 2374 7792 3497 9517 588i 2625 5 56 3I96 7688 2384 7302 2462 7885 3595 9620 5990 2741 4 57 3270 7764 2464 7386 2550 7977 3693 97231 6099 2857 3 58 3343 7841 2544 7470 2638 8070 3791 98261 6209 2973 2 59 34I6 79I7 2624 7554 2726 8I63 3888 9930] 63I8 3089 I 390 380 37~0 360 350 340~ 330 32~ i 31.0 300 Natural Co-tangents. t P. P.t 1*20 I.25 I.3I I.37 I.44 I.5I I.59 | i.68 1.78] 1.88 i 28 NATURAL SINES, _ 600 61~ 62~ 63~ 64~ 65~ 66~ 67~ 68~ 69 o 866025 874620 882948 891007 898794 906308 913545 920505 927184 933580o60 I 6171 4761 3084 1I39 8922 643 3664 o6i8 7293 3685 59 2 63i6 4902 3221 I270 9049 6554 3782 0732 7402 3789 58 3 646 5042 3357 I402 9176 6676 3900 o846 7510 3893 57 4 6607 5i83 3493 I534 9304 6799 4oi8 0959 76I9 3997 56 5 6752 5324 3629 i666 9431 6922 4I36 1072 7728 4Ioi 55 6 6897 5465 3766 1798 9558 7044 4254 ii85 7836 4204 54 7 7042 5605 3902 1929 9685 7166 4372 I299 7945 4308 53 8 7187 5746 4038 2061 98I2 7289 4490 14I2 8053 44I2 52 9 733I 5886 4174 2I92 9939 74II 4607 1525 8i6I 4515 51 io 867476 876026 8843o09 892323 900065 907533 914725 921638 928270 934619 50 I1 762I 6167. 4445 2455 0192 7655 4842 1750 8378 4722 49 12 7765 6307 458I 2586 o319 7777 4960 i863 8486 4826 48 x3 79io 6447 47I7 27I7 0445 7899 5077 1976 8594 4929 47 x4 8054 6587 4852 2848 0572 8021 5194 2088 8702 503246 15 8199 6727 4988 2979 0698 8I43 53iI 220I 88o10 5i35 45 i6 8343 6867 5123 3Iio 0825 8265 5429 2313 8917 523844 17 8487 7006 5258 324I og5I 8387 5546 2426 9025 534I 43 i8 8632 7146 5394 3371 I077 8508 5663 2538 9133 5444142 19 8776 7286 5529 3502 1203 8630 5779 2650 9240 5547 41 20 868920 877425 885664 893633 g91329 908751 915896 922762 929348 935650 4o 21 9064 7565 5799 3763 1455 8872 6oi3 2875 9455 5752 39 22 9207 7704 5934 3894 i58I 8994 6I30 2986 9562 5855 38 23 9351 7844 6069 4024 1707 9115 6246 3098 9669 5957 37 24 9495 7983 6204 4i54 I833 9236 6363 3210 9776 6060o36 25 9639 8122 6338 4284 I958 9357 6479 3322 9884 6I62 35 26 9782 826I 6473 44i5 2084 9478 6595 3434 9990 6264 34 27 9926 8400 6608 4545 2209 9599 6712. 3545 930097 6366 33 28 870069 8539 6742 4675 2335 9720 6828 3657 0204 6468 32 29 0212 8678 6876 4805 2460 984I 6944 3768 o3ii 6570o31 3o 870356 878817 887011 894934 902585 90996I 917060 923880 930418 936672 30 3I 0499 8956 7145 5064 27I g910082 7I76 3991 0524 6774 29 32 0642 9095 7279 5194 2836 0202 7292 4102 o63I 687628 33 0785 9233 7413 5323 296I 0323 7408 4213 0737 697727 34 0928 9372 7548 5453 3086 0443 7523 4324 o843 7079 26 35 I071 95IO 768I1 5582 3210 o563 7639 4435 o0950 781 25 36 I214 9649 78I5 5712 3335 o684 7755 4546 o56 7282 24 37 1357 9787 7949 584I 3460 o8o4 7870 4657 1162 738323 38 I499 9925 8083 5970 3585 0924 7986 4768 1268 7485 22 39 I642 880063 8217 6099 3709 o44 8IOI 4878 1374 758621I 4o 87I784 880201 88835o 896229 903834 9I I 64 918216 924989 931480 93768720 4I 1927 0339 8484 6358 3958 1284 833i 5099 I586 7788 19 42 2069 o477 86I7 6486 4083 14o3 8446 52IO I691 7889 i8 43 2212 o615 875i 66i5 4207 1523 856I 5320 I797 7990 I7 44 2354 0753 8884 6744 433i I643 8676 5430 1902 8091 i6 45 2496 089I 90I7 6873 4/455 1762 8791 554I 2008 8191 i5 46 2638 1028 9150 7001 4579 I88I 8906 565! 2113 8292 I4 47 2780 1166 9283 7I30 4703 200I 902I 576I 22I9 8393 13 48 2922 I303 9416 7258 4827 2120 9135 5871 2324 8493 12 49 3064 I44I 9549 7387 495I 2239 9250 5980 2429 8593:I 50o 8732o6 88I578 889682 897515 9o5075 9I2358 9I9364 926090 932534 938694 ro 51 3347 1716 9815 7643 5198 2477 9479 6200 2639 8794 9 52 3489 I853 9948 7771 5322 2596 9593 63Io 2744 8894 8 53 363I g199 890080 7900 5445 2715 9707 64I9 2849 8994 7 54 3772 2127 0213 8028 5569 2834 9821 6529 2954 9094 6 55 3914 2264 o345 8i56 5692 2953 9936 6638 3o58 9194 5 56 4055 2401 0478 8283 58i5 3072 920050 6747 3i63 9294 4 57 4196 2538 o6io 84ii 5939 3g19 oi64 6857 3267 9394 3 58 4338 2674 0742 8539 6062 3309 0277 6966 3372 9493 2 59 4479 2811 0874 8666 6i85 3427 o3g9 7075 3476 9593 I 290 280 270 26~ 125~ | 240~ 23~ 2 22 210 2 20 Natural Co-sines. to I. f2.39 2.3I 2.24 2.I6 2.09 2.I I.93 I.86 1.78 1.70 L~~~~ NATURAL TANGENTS.. 00 610 620 630 640 6o5 660 670 680 69~ 0 I.73205 I.80405 1.88073 I.96261 2.o5030 2.I445I 2.246042.35585 2.47509 2.60509 6o I 3321 0529 8205 6402 5I82 46I4 4780 5776 77I6 0736159 2 3438 o653 8337 6544 5333 4777 4956 5967 7924 o096358 3 3555 0777 8469 6685 5485 4940 5I32 6I58 8132 II90 57 4 367 0o90g 8602 6827 5637 5I04 5309 6349 8340 1418 56 5 3788 I025 8734 6969 5790 5268 5486 654i 8549 I646 55 6 3905 ii5o 8867 7111 5942 5432 5663 6733 8758 I874 54 7 4022. I274 9000 7253 6094 5596 5840 6925 8967 2I03 53 8 4I4o I399 9133 7395 6247 5760 6oi8 7II8 9I77 2332 52 9 42571 I524 9266 7538 6400 5925 6196 73II 9386 256I5 r -0 1.74375 r.81649 1.89400 I.9768I 2.o6553 2.16090 2.26374 2.37504 2.495972.6279 o50 Ii 4492 I774 9533 7823 6706 6255 6552 7697 9807 3021 49 12 46io 1899 9667 7966 6860 6420 6730 789I 2.50018 3252 48 I3 4728 2025 980I 8IIo 70I4 6585 6909 8084 0229 3483 47 I4 4846 2150 9935 8253 7167 6751 7088 8279 o440 371446 I5 4964 2276 1.90069 8396 7321 6917 7267 8473 0652 3945.45, i6 5082 2402 0203 8540 7476 7083 7447 8668 o864 4I7744 I7 5200 2528 0337 8684 7630 7249 7626 8863 I076 440o43 Ii8 53I9 2654 0472 8828 7785 74I6 7806 9058 1289 464242 19 5437 2780 0607 8972 7939 7582 7987 9253 I502 48754I 20 1.75556 1.82906 I.9074I I.99gI6 2.o8094 2.I7749 2.28I67 2.39449 2.5 7I5 2.651O9 4o 21 I 5675 3033 0876 926I 8250 79I6 8348 9645 1929 5342 39 22 5794 3159 1oIz 9406 8405 8084 8528 984I 2142 557638 23 5913 3286 II47 9550 8560 825I 87I0 2.40038 2357 58ii 37 24 6032 34i3 1282 9695 8716 8419 8891 0235 257I 604636 25 6151 3540 I4I8 984I 8872 8587 9073 o432 2786 6281 35 26 6271 3667 I554 9986 9028 8755 9254 0629 3ooi 65I634 27 6390 3794 I690o2.0oo13 9184 8923 9437 0827 32I77 675233 28 65io 3922 I826 0277 934I 9092 9619 I025 3432 698932 29 6630 4049 1962 0423 9498 926I 980I 1223 3648 722531 3o I.76749 1.84I77 1.92098 2.oo.69 2.o9654 2.19430 2.299842.41421 2.53865 2.67462 3 31 6869 43o5 2235 o715 9811 9599 2.30167 1620 4082 770029 32 6990 4433 2371 0862 9969 9769 o35I I819 4299 793728 33 7I10 456I 2508 I00oo82.IOI26 9938 0534 2019 4516 8-75?7 34 7230 4689 2645 1S55 0284 2.20Io 8 0718.2218 4734 844 26 35 7351 48I8 2782 I302 0442 0278 0902 2418 4952 8653 25 36 747' 4946 2920 I449 o6oo 0449 Io86 26I8 5170 8892 24 37 7592 5075 3057 1596 0758 o619 I271 28I9 5389 9131 23 38 7713 5204 3I95 1743 o9gI6 0790 I456 3019 5608 937I 22 39 7834 5333 3332 189I 10o75 096I 164 3220 5827 961221 4o I.77955 I.85462 1.93470 2.02039 2.II233 2.2II32 2.3826 2.43422 2.56046 2.69853 20 41 8077 5591 3608 2187 I392 i3o4 2012 3623 6266 2.70094 I9 42 8I98 5720 3746 2335 1552 1475 2197 3825 6487 o335i8 43 83I9 5850 3885 2483 1711 1647 2383 4027 6707 o577I7 44 844i 5979 4023 2631 1871 81I9 2570 4230 6928 08I9 I6 45 8563 6og9 4162 2780 2030 1992 2756 4433 7I50 1062 i5 46 8685 6239 43oi 2929 219go 2164 2943 4636 7371 i3o05 4 47 8807 6369 444o 3078 2350 2337 3i3o 4839 7593 I548 i3 48 8929 6499 4579 3227 2511 2510 3317 5o43 7815 I792 12 49 9051 6630 4718 3376 2671 2683 3505 5246 8038 2036 II 50 1.79174 I.86760 1.94858 2.0o3526 2.12832 2.22857 2.33693 2.45451 2.5826I 2.7228 I1o 51 9296 6891 4997 3675 2993 3030 388i 5655 8484 2526 9 52 94I9 7021 5137 3825 3I54 3204 4069 5860 8708 277I 8 53 9542 7I52 5277 3975 3316 3378 4258 6065 8932 3017 7 54 9665 7283 5417 4I25 3477 3553 4447 6270 9156 3263 6 55 9788 7415 5557 4276 3639 3727 4636 6476 938I 3509 5 56 9911 7546 5698 4426 380o 3902 4825 6682 9606 3756 4 57 1.8oo34 7677 5838 4577 3963 4077 50I 5 6888 9831 4004 3 58 oi58 7809 5979 4728 4I25 4252 5205 7095 2.60057 4251 2 59 028i 794I 6120 4879 4288 4428 5395 7302 0283 4499 I 290~ 280 270 260 250 240 230 22~ 210 20~0 Natural Co-tangents. P: 200 2.3 2.27 2.44 2.62 2.82 3.05 3.3I 3.6i 3.95 tol'ao r 22. 130 NATURAL SINES. 4 700 710 720 730 740 750 760 770 780 790 0 939693 9455 19 95 Io57 956305 96I262 965926 970296 974370 978I48 98 I 62760 I 9792 56i3 1 46 6390 I342 6ooi o366 4435 8208 I683 59 2 9891 5708 1236 6475 I422 6076 o436 45oi 8268 I73 58 3 999I 5802 I326 6560 I502 615I o506 4566 8329 I 793 57 4 940090 5897 141i5 6644 I582 6226 0577 463I 8389 I849 56 5 OI89 5991 I5o50 6729 I662 63ox 0647 4696 8449 I904 55 6 0288 6o85 I594i 68i4. 1741. 6376 0716 476I 8509 I959 54 7 o387 6i8o 16841 6898 182I 645I 0786 4826 8569 20oi4 53 8 o486 6274 I773 6983 I9go 6526 o856 489I 8629 2069 52 9 o585 6368 I862 7067 I980 6600 0926 4956 8689 2123 51 Io 940684 946462 95g95I 957151 962059 966675 970995 975020 978748 982I78 5o I1 0782 6555 2040 7235 2139 6749 io65 5085 8808 2233 49 I12 o88I 6649 2129 7319 -228 6823 ii34 5149 8867 228748 I3 0979 6743 2218 7404 2297 6898 I 20o4 5214 89z- 2342 47 I4 1078 6837 2307, 7487 2376 6972 I273 5278 8986 2396 46 I5 II76 6930 23961 7571 2455 7046 I342 5342 9045 2450 45 16 1274 7024 24841 7655 2534 7I20 1411 5406 9105 2505. 4 17 I372 7II7 2573 7739 2613 7194 I48o 547I 9I64 2559 43 I8 I47I 7210 266I 7822 2692 7268 I549 5535 9223 2613 42 19 1569 7304 2750 7906 2770 7342 I618 5598 9282 2667 41 20 94I666 947397 952838 957990 962849 9674I5 97I687 975662 97934I 982721 4o 2I 1764 7490 2926 8073 2928 7489 I755 5726 9399 2774 39 22 1862 7583 30o5 8i56 3006 7562 1824 579o 9458 2.828 38 23 1960 7676 3io3 8239 3084 7636 I893 5853 9517 2882 37 24 2057 7768 3191 8323 3i63 7709 I96I 59I7 9575 2935 36 25 2I55 7861 3279 8406 324I 7782 2029 5980 9634 2989 35 26 2252 7954 3366 8489 3319 7856 2098 6044 9692 3042 34 27 2350 8046 3454 8572 3397 7929 2166 6Io7 9750 3096 33 28 2447 8I39 3542 8654 3475 8002 2234 6I70 9809 3149 32 29 2544 8231 3629 8737 3553 8075 2302 6233 9867 3902 3I 30 942641 948324 9537I7 958820 963630 968148 972370 976296 979925 983255 30 31 2739 84i6 3804 8902 3708 8220 2438 6359 9983 3308 29 32 2836 85o8 38921 8985 3786 8293 2506 6422 98004I 336I 28 33 2932 8600 3979 9067 3863 8366 2573 6485 0098 3414 27 34 3029 8692 4066 9150 394I 8438 2641 6547 oi56 3466 S6 35' 3I26 8784 4i53 9232 4oi8 85 I 2708 66io o024 3519 25 36 3223 8876 4240 9314 4095 8583 2776 6672 027I 3571 24 37 331'9 8968 4327 9396 4173'8656 2843 6735 0329 3624 23 38 34I6 9059 4414 9478 4250 8728 29II 6797 o386 3676 22 39 35I2 9151 45oi 9560 4327 8800 2978 6859 o443 3729 21 40 943609 949243 954588 959642 964404 968872 973045 97692I 98o05o0 98378I 20 4I 3705 9334. 4674 9724 448I 8944 3112 6984 o558 3833 i9 42 380o 9425 476i 9805 4557 9016 3I79 7046 o6i5 3885 i8 43 3897 95I7 4847 9887 4634 9088 3246 7108 0672 3937 17 44 3993 9608 4934 9968 47II 9g59 33I3 7I69 0728 3989i6 45 4089 9699 5020 960050 4787 923I.3379 723I 0785 4o4I 5 46 4I85 979o 5io6 oi3i 4864 9302 3446 7293 0842 4092 14 47' 4281 988I 5i92 02I2 4940 9374 35I2 7354 o899 4144I3 48 4376 9972 5278'0294 50o6 9445 3579 74I6 o0955 496 I2 49 4472 950063 5364 o375 5093 95I7 3645 7477 IOI2 4247 II 50 944568 950o54 9554501 96o456 96569g 969588 9737I2 977539 98i068 984298 Io 51 4663 0244 5536'o0537 5245 9659 3778 7600 II24 4350 9 52 4758 o335 5622 o6i8 5321 9730 3844 766I ii8i 44oi 8 53 4854 o425 5707 0698 5397 9801 3910 7722 1237 4452 7 54 4949 o5i6 5793 0779 5473 9872 3976 7783 1293 4503 6 55 5044 o606 5879 o860 5548 9943 4042 7844 1349 4554 5 56 5139 0696 5964 0940 5624 970014 4io8 7905 I4o5 4605 4 57 5234 0786 6049 102I 5700 0084 4I73 7966 i46o 4656 3 58 5329 0877 6i34 IoI 5775 oi55 4239 8026 I5I6 4707 2 59 542 0o967 6220 i18i 5850 0225 4305 8087 1572 4757 I — 190 18~ 170 1 6~ 15~1 140 13 2C 11 1l00 Natural Co-sines. I/..2 1, i,, 5,,,.6 i09 N ATURAL T C E N T S. 13 700 710 720 730 740 750 6' 770 780 790 0 2.74748 2.90421 3.07768 3.27085 3.48741 3.732054.010o78 4.33i48 4.70463 5.i4455 6o I 4997 0696 8073 7426 9125 364o 1576 3723 1137, 5256159 2 5246 0971 8379 7767 9509 4075 2074 43oo00 I813i 6058158 3 5496 I246 8685 8o109 9894 4512 2574 4879 24900 6863 57 4 5746 1523 8991 8452 3.50279 4950 3076 5459 31701 7671 56 5 5996 I799 9298 8795 o666 5388[ 3578 6o4oI 385i 848o 55 6 6247T 2076 9606 9139 Io53 5828 4o8i 6623! 4534 9293 54 7 6498! 2354 99I4! 9483 x44x 6268 4586 7207 5219 5.20107 53 8 6750 263213.I0223 9829! 1829 6709 5092 7793 5906 0925 52 9 7002 291o 0532 3.30o74 2219 7152 5599 838i 6595 1744 5I 10 2.77254 2.9318913. o842 3.3o52I 3 52609 3.7759514.0o6io07 4.38969 4.77286 5.22566 5o II 7507 3468i i1i53 0868 3ooi 8040 66i6 9560 7978 339149 12 7761 3748 I464 1216 3393 8485 7127{4.4o0152 8673 4218'48 [3 8oI4 4028 1775 i565 3785 8931 7639 0745 9370 5048 47 x4 8269 4309 2087 1914 4179 9378 8152 i34o4.8oo68 5880o46 i5 8523 4591 2400 2264 4573 9827i 8666 1936 0769 6715 45 i6 8778 4872 2713j 2614 4968 3.80276 9182 2534 I471 7553 44 17 9033 5155 3027 2965 5364 0726 9699 3I34 2175 8393 43 I8 9289 5437 334i 3317 5761 II7714.10216 3735 2882 923542 19 9545 5721 3656 3670 6159 i63o1 0736 4338 35905.3008041 20o2.79802 2.9600oo4 3.13972 3.34023 3.56557 3.82083 4.II25614.44942 4.843oo00 5.30928 4o 21 2.80059 6288 4288 4377 6957 2537 1778!' 5548 5oi3 1778 39 22 o3i6 6573 46o5 4732 7357 2992 23o01 6i55 5727 2631 38 23 0574 6858 4922 5087 7758 3449 2825 6764 6444 3487 37 24 o8336 744 5240 5443 816o 3906 3350 7374 7162 4345136 25:o091 7430 5558 58oo00 8562 4364 3877 7986 7882 520635 26 035o 7717 5877 6x58 8966, 4824 44o5 8600oo 8605 6070o34 27 i6io 8oo4 6I97, 65i61 9370' 5284 4934 9215 9330o 6936133 28 1870 8292 65I71' 6875! 9775 5745 5465 9832 4.9o0056 7805 32 91 2130 858o 6838 723413.6o0181 6208 59974.50451 0785 8677131 30o2.82391 2.98868 3.17159 3.37594'3.6o588 3.86671 4.I653o 4.51071 4.9i516 5.39552 30 3I 2653 9i581 7481j 79551 0996 7136 7064 1693 22495.4o42929 32 2914 9447 78041 83171 i4o5 76o01 7600 2316 2984 309 28 33 376 738j 8I71I 904 221 7600 23i6 29846 30712 33 3176 97381 8127 8679 i8i4 8068 8137 2941 3721 2192 27 34 3439 3.00028 8451I 9042 2224 8536 8675 3568 446o 307726 35 3702 0319i 87751 9406 2636 9004 9215 4196 5201 396625 36 3965 o611ii 9100oo 9771 3o48 9474 9756 4826 5945 485724 37 4229 0903 9426 3.4oi36 346i 9945 4.20298 5458 6690 5751 23 38 4494 ii96 9752 0502 38743.90417 0842 6o91 7438 664822 39 4758 1489 3.20079 0869 4289 0890o 387 6726 8i88 7548 21 4o 2.85023 3.01o783 3.20406 3.41236 3.64705 3.9i364 4.21933 4.57363 4.98940o 5.4845I 20 41 5289 2077 0734 i6o4 5121 1839 2481 8oor 9695 935619 42 5555 2372 io63 1973 5538 2316 3030 864i 5.oo45I 5.50264 i8 43 5822 2667 1392 2343 5957 2793 3580o 9283 I210o II7617 44 6089 2963 1722 2713 6376 3271 4132 9927 1971 20901I6 45 6356 3260 2053 3084 6796 3751 4685i4.60572 2734 300715 46 6624 3556 2384 3456 7217 4232 5239 1219 3499 3927I4 47 6892 3854 2715i 3829 7638 4713 5795 i868 4267 485i i3 48 716I 4152 3048 4202 8o6i 5196 6352 2518 5037 577712 49 7430 4450 338 4576 8485 568o 69ii 3171 5809 6706ii 50 2.877oo00 3.04749 3.23714 3.4495I 3.68909 3.96i65 4.27471 4.63825 5.o6584 5 57638 io 5I 7970 5049 4049 5327 9335 665i 8032 4480 7360 8573 9 52 8240 5349 4383 5703 9761 7139 8595 5i38 8039 9511 8 53 85ii 5649 47I91.60803.70o88 7627 9159 5797 8921 5.60452 7 54 8783 5950 50551 6458 o6i6 8117 9724 6458 9704 1397 6 55 9055 6252 5392 6837 io46 8607 4.30291 7121 5.o 490o 2344 5 56 9327 6554 5729 7216 1476 9099 0860 7786 I279 3295 4 57 9600 6857 6067 7596 I907 9592 i43o 8452 2o69i 4248 3 58 9873 7160 6406 7977 2338 4.oo00086; 2001 9121 2862. 5205 2 59 2.9o0147 7464 6745 8359 2771 0o5821 2573, 979I 3658 6i65 190 18i 170 160 150 14~ 130 120 1'00 Natural Co-tangents.,."P.45 4.82 5.3 16.0oi 6.79.73 8.90 io.35 2.20 i 4.6 to ill~~~~~~~~~. 79' [4 132 NATURAL SINES. 800 810 s~20 &30 840 85~ 860 1 870 880 89~ 0 984808 987688 990268 992546 994522 996I95 997564 998630 99939I 999848 60 x 4858 7734 0309 2582 4552 6220 7584 8645 9401 9853 59 2 4909 7779 o349 26I7 4583 6245 7604 866o 9411 985858 3 4959 7824 0389 2652 46i3 6270 7625 8675 942I 9863 57 4 5oo9 7870 o4291 2687 4643 6295 76451 8690 943o 9867 56 5 5059 7915 0469 2722 4673 6320 7664j 8705 944I 9872 55 6 5109 7960 0509 2757 4703 6345 76841 8719 9450 9877 54 7 5I59 8005 0549 2792 4733 6370 7704 8734 9460 9881 53 8 5209 8050o 0589 2827 4762 6395 7724 8749 9469 9886 52 9 5259 8094 o629 2862 4792 6419 7743 8763 9479 g890 51 I1 985309 988I39 990669 992896 994822 996444 997763 998778 999488 999894 50 II 5358 8i84 0708 2931 485i 6468 7782 8792 9497 9898 49:2 5408 8228 0748 2966 4881 - 6493 7801 8806 9507 990348 i3 5457 8273 0787 3000 4910 6517 782I 8820 9516 9907 47 14 5507 8317 0827 3034 4939 654I 7840 8834 9525 9910o46 i5 5556 8362 o866] 3068 4969 6566 7859 8848 9534 9914 45 I5 5605 840o6 0905 3io3 4998 6590 7878 8862 9542 9918 44 17 5654 8450 0944 3137 5027 66I4 7897 8876 9551 9922 43 I8 5703 8494 0983 3171 5o56 6637 7916 8890 9560 9925 42. I9 5752 8538 1022 3205 5084 666i 7934 8904 9568 9929 41 20 985801 988582 99o6I 993238 995II3 1 996685 997953 998917 999577 999932 40 21 585o 8626 II00 3272 5 r42 6709 7972 893I 9585 9936 39 22 5899 8669i II38i 3306 5170 6732 7990 8944 9594 993938 23 5947 873i II771 3339 5199 6756 8008 8957 9602 9942 37 24 5996 8756j I2I6! 3373 5227 6779 8027 897I 96I0 9945 36 25 6045 8800o I2541 3406 5256 6802 8o45 8984 96I8 9948 35 26 6093 8843 1292 3439 5284 6825 8o63 8997 9626 9951 34 27 6I4I 8886 i33I 3473 53I2 6848 808I 9010 9634 9954 33 28 6189 8930 I369 3506 5340 6872 8099 9023 9642 9957 32 29 6238 8973 I407 3539 5368 6894 8117 9035 9650 9959 3I 30 986286 9890o6 99I445 993572 995396 996917 998135 999048 999657 999962 30 31 6334 9059 1483 3605 5424 6940 8i53 9061 9665 9964 29 32 638i 9I02 152I 3638 5452 6963 8I70 9073 9672 9967 28 33 6429 9I45 iz558 3670 5479 6985 8i88 9086 9680 9969 27 34 6477 9187 1596 3703 5507 7008 8205 9098 9687 997I 26 35 6525 92301 I634 3735 5535 7030 8223 9111 9694 9974 25 36 6572 9272 1671 3768 5562 7053 8240 9I23 970I 9976 24 37 6620 93I5j I709 3800 5589 7075 8257 9135 9709 9978 23 38 6667 9357 1746 3833 56I7 7097 8274 947 97I6 9980 22 39 67:4 9399 1783 3865 5644 7II9 829I 9159 9722 9981 21 4o 986762 989442 99I820 993897 995671 997I41 998308 999I7I 999729 999983c-0 4.I 6809 9484 1857 3929 5698 7163 8325 9183 9736 9985 19 42 6856 9526 I894 3961 5725 7I85 8342 9I94 9743 9986 1i8 43 6903 9568 I931 3993 5752 7207 8359 9206 9749 9988 i7 44 695o 96I0; I968 4025 5778 7229 8375 92I8 9756 9989 i6 45 6996 9651 2005 4056 5805 7250 8392 9229 9762 9990 i5 46 7043 9693 2042 4o88 5832 7272 8408 9240 9768 9992 14 47 7090 9735 2078 4I20 5858 7293 8425 9252 9775 9993 I3 48 7136 9776 2115 4i5i 5884 7314 844i 9263 9781 9994|12 49 7183 9818 2151I 4182 591I 7336 8457 9274 9787 9995 ii 5o 987229 989859 992I87 9942I4 995937 997357 998473 999285 999793 979996 1i 51 7275 9900 2224 4245 5963 7378 8489 9296 9799 9997 9 52 7322 9942 2260 4276 5989 7399 85o05 9307 9804 9997 8 53 7368 9983 2296 4307 6oi5 7420 852I!, 9318 9810 9998 7 54 7414 990024 2332 4338 6o04 7441 85371 9328 9816 9998 6 55 7460 o0065 2368 4369 6067 7462 8552 9339 9821 93999 5 56 7506 oio5 2404 44o0 6093 7482 8568 9350 9827 99998 4 57 7551 oI46| 2439 443o 6ii8 7503 8583 936 9832.00000 3 i 58 7597 0187 2475 446I 6I44 7523 8599 9370 9837 09000 2 59 7643 0228 2511 449I 6169 7544 86i4 9381 9843 ooo I 90 8704 74 60 5 60 40 30 2 18 0 j Natural Co-sines. 5 o.8o 0.72 o.63j o.55 o.46 o.38 o.30 0.21 o.3 o.3 04 549 I82g995JrJ953 9 43998'J 997357J J999793J 9~9996/~~~~~ NATURAL T.ANGENTS. 1 3 1 800 ] 810 82' i 830 I 840 850 860 87~ 880 90 0 5.67128 6.3I375 7.I 53718.I4435.9.5I436 I I.430 I4.30071 9.08I 28.636357.2900 tC I 8094: 2566j 3042 6398 4io6 4685 3607 I879 877I 58.2612 59 2 90o641 376I 4553 8370 6-79I 5072 42I2 2959 29.I220 59.2659 58 3 5-700371 496I 607I 8.20352 9490 546, 4823 4o5fi 37 I 60.3o58 57 4 IOI31 6I65 7594 2344 9.622051 5853 5438 5I56 6245 6I.3829 56 5 I9921 7374 9I25 43451 49351 6248 6059 6273 8823 62.4992 55 6 29741 8587 7.2066I 6355 768ol0 6645 6685 7403 30o.446 63.6567 54 7 39601 9804 2204 8376}9.7o44xI 7045 73I7 8546 4 I6164.858o1 53 8 4949i6.41o26 3754 8.30406 32I7' 7448 7954 9702 6833166.1055 52 9 594I 2253 53IO1 2446 6009 7853 8596 20.0872 9599 67.40I9 5I Io 5-76937 6.43484 7.26873 8.34496 9.78817 I I.8262 I4.9244 20.2056 31.24I6 68.7501 50 II 7936 4720 8442 655519.864I 8673 9898 3253 5284 70.1533 /49 12 8938 596I 7.300I8 8625 4482 9087 15.0557 4465 8205 71.6r51 48 13 9944 7206 1600 8.40705 7338 9504 1222 5691 32.1181 73.I390 47 I415.80953 8456 3190 279519.902I 11 9923 1893 6932 42I3 74.7292146 x5 x966 97Io 4786 4896 3oI 12.o346 2571 8188 7303 76.39oo 00 45 i6 2982 6.50970 6389 7007 6007 0772 3254 9460o33.o452178. I263 44 17 4~OI 2234 7999 9128 893I 120I 3943 21.0747 3662 79.9434143 I8 5024 3503 96i6 8.5z1259 Io.o 0187 632 46381 2049 6935 8.8470 42 I9 6o05 4777 7.4I24o 3402 o483 2067 534o 3369 34.0273 83.8435 4I 20 5.87080 6.56o55 7.42871 8.55555 I0.0780oI2.2505 i5.6048 21I.4704 34.3678 85.9398 4o 2I 8rx4 7339 4509 77I8 io8o 2946 6762 6056 7I5I 88.i436 39 22 9151 8627 6I54 9893 038I 3390 7483 7426 35.o695 90o.4633 38 2315.90o191 992I 780618.62078 i683 3838 8211I 88I3 43I3192.9085 37 24 I236 6.6I2I91 9465 4275 i988 4288 8945 22.0217 8006195.4895 36 25 2283 2523 7.51132 6482 2294 4742 9687 1640o 36.I776 98.2179 35 26 3335 383I 2806 870I 2602 599 I 6.o435 308I 5627 I0OI.I107 34 27 4390 5z44 4487 8.7093I 2913) 566o IIgo90 454I 9560 1o4.171 33 28 5448 6463 6I76 3172 3224i 6124 I952 6020o37.3579I107.426132 29 651o 7787 7872 5425 35381 6591 27221 75I9 7686 1IIo.892 3 30o15.977 6.69Iz617.5957518.77689 I0.38541i2.7062 16.349.922.9038 38.i885 114.589 3o 3l 8646 6.7045017.6I287 9964 4172 7536 4283 23.o577 6I77 II8.540o29 32 9720 I789 3oo0058.82252 449I 8oi4l 50751 2I37 39.0568 I22.774128 33j6.oo7971 3i33, 4732 455i 48I3 8496 5874 3718 50591I27.321 27 341 I878 4483 6466 6862| 50361 8981 668i 532I 9655 132.2I9 26 35 2962 5838 8208 91851 54621 9469 7496 6945 4o.4358 I37.507 25 36 4o05 7199 995718.91520 5789 9962 8319 8593 9I74 143.237 24 37 5i43 8564 7.71715 3867 6I81 i3.o458 9150 24.0263 4I.4Io6 149.465 23 38 6240 9936 348o0 6227 6450 o958 999o 1957 9I58 I56.259 22 39 7340 6.8I3212 5254 8598 6783 146i 17.0837 3675 42.4335 163.7002I1 4o 6.o8444 6.82694 7.77035 9.00983 10.71I9x 13.I969 I7.I693 24.54I8 42.9641 171.885 20 41 9552 4082 8825 3379 7457 2480, 25581 7185 43.50o83I80.932 Ig 42 6.io664 5475 7.80622 5789 7797 2996 34321 8978 44.o66II90.984 IS 431 779 6874 2428 8211 8I39 3515 43I4 25.o0798 6386|202.219117 44 2899 8278 4242 9.IO646 8483 4039 5205 2644 45.226l1 24.858/6 6 45 40231 9688 6o64 3093 8829 4566 6Io6 4517 8294 229. 82 I5 46 5151i6.910o4 7895 5554 9178 5098 7051 64I8146.4489 245.552 I4 47 6283 2525 9734 8028 9529 5634 7934 8348|47.08531264.44 I31 48 74I9 3952 7.9I582 9.20516 9882 6I74 8863]26.o307 7395 286.478 I2 4o 8559 5385 3438 3016 1I.0237 6719| 9802 2296148.4I21|312.52I|II 506.I970316.9682317.953029.2955 30I I.o594 I3.7267 18.o75o 26.4316 49.Io391343.774 |o 5I 6.2085I 8268 7I76 8o58 o954 782I 17o81 6367 8I57 38I.97I 9q 52 2003 97I8 9o5819.3o599 i3i6 8378 2677 8450o5o.54851429.7i81 8 53 3 I 607.01 748.00948 3155 i68I 8940 3655127.o56651.3o32 49I.Io6[ 7 541 432a 2637 2848 5724 2048 9507 4645 27I5152.o8o71572.95 7[ 6 55 5486 4Io5 47561 8307 24I7 I4.0079 5645 4899 882I1687.5491 5 56 6655 5579 667419.40904 2789 o655 6656 7117 53.7086 859.436 4 57 7829 7o59 86oo 355 3I63j I235 7678 9372154.5613{xI45.99' 3 58 9007 8546[8.:o536 614iJ 354o0 I82 872II28.166455.44I5I7I8.87 2 S96 30I89 7I100381 248I1 878-i 39I9' 241 I 9755 3994 56.35o6 3437.75 1 90 80 7 6 50 40 30 20 10 00 Natural Co-tangents.,',, 80 22. g 2846 3783 528 88 I3,,.o,. 7.. | |...: I to~~~~~~~~~~~~~~~~~ I:.'7 2.46J-' ,34 NATURAL SECA N.T S. Deg. 0' 10' 201 30' 40' 50' P. Pt. I.oo000000 I.ooooo4. 000OI7 i.oooo38 1.oooo68 i.oooo106 89 2.5 I ooo000152 000207 000271 ooo343 000423 000512 88 7.6 2 oo000609 ooo715 ooo830 ooo000953 oo01084 0oo01224 87 127 3 I0372 OOI529 oo00695 001869 002051 002242 86 17.8 i ~ 002442 002650 002867 003092 003326 oo3569 85 22.9 5 oo003820. oo4080 00oo4348 004625 004911 005205 84 28,1 6 00oo5508.5820 oo6i4I 006470 oo6808 007154 83 33.3 7 007510 007874 008247 008629 009020 009419 82 38,6 8 009828 010245 0,o67I oiiio6 oii55o 012003 8i 43.9 9 012465 012936 o.34i6 013905 oI44o3 014910 8o 49.3 I o 1.o5427 1.OI5952 1.016487 1. 07030 1.017583 1.o18145 79 54.8 II I08717 019297 019887 020487 021095 02I713 78 60.4 12 022341 022977 023624 024280 o24945 025620 77 66.0 13 026304 026998 027702 028415 029138 029871 76 7I,8 I4 o3o064 o03366 032128 032900 033682 034474 75 77.7 I5 035276 o36088 o36gIO 037742 o38584 039437 74 83.7 i6 040299 041172 042055 042949 o43853 044767 73 89.8 17 045692 046627 047573 o48529 049496 050474 72 96.1 I8 051462 05246I o5347I 054492 055524 056567 71 I02.6 19 057621 o58686 o59762 060849 061947 063057 70 I09.2 20.o064178 i.o653Io I.o66454 1.o67609 1.o68776 1.o69955 69 116.i 21 071145' 072347 073561 074786 076024 077273 68 I23.2 22 078535 07980o8 809og4 o82392 o83703 o85025 67 i3o.4 23 o86360 087708 o8go68 09044I 091827 093225 66 I37.9 24 o094636 og6060 097498 098948 100411 ioi888 65 x45.6 25 10o3378 io488 1o6398 10o7929 109473 iiio3o 64 153.7 26 112602 114187 115787 I17400 II9028 120670 63 162.o0 27 I22326 123997 125682 127382 129096 130826 62 I70.7 28 I32570 134329 I361o4 137893 I39698 I41518 6i 179,7 29 I 43354 145205 147073 148956 i50854 152769 60 I89.1 30 1tx.547oi i.i56648 1.158612 I.160592 1.162589 1.164603 59 I98.9 31 i66633 i6868I I70746 172828 174927 I77044 58 209.1 32 179178 18I33' i8350o1 85689 187895 190120 57 219.7 33 192363 194625 196906 199205 201523 203861 56 230.9 34. 206218 208594 210991 213406 215842 218298 55 242.6 35 220775 223271 225789 228327 230886 233466 54 254.8 36 236068 238691 241336 244003 246691 249402 53 267.7 37 252136 254892 257671 260472 263298 266146 52 281.3 38 269018 271914 274834 277779 /280748 283741 51 295.6 39 286760 289803 292872 295967 299088 302234 50 3To.7 40 i.3o5407 1.308607 i.3ii833 1.315087 i.3i8368 1.321677 49 326.7 4I' 325013 328378 331771 335192 338643 342123 48 343.6 42 345633 349172 352742 356342 359972 363634 47 36i.5 43 367327 371052 374809 378598 382420 386275 46 38o.5 44 390164 394086 398042 402032 406057 4IOII8 45 400.7 45 4I42I4 4I8345 4225I3 426718 430960 435239 44 422.3 46 439557 4439I2 448306 452740 457213 461726 43 445.3 47 466279 470874 475509 480I87 484907 489670 42 469.8 48 494477 499327 504221 509160 514i45 519176 4I 496.2 49 524253 529377 534549 539769 545038 550356 40 524.4 50 I 555724 1.561142 1.566612 1.572134 1.577708 1.583335 39 554.7 5i 589016 594751 600542 606388 612291 61825I 38 587.4 52 624269 630346 636483 642680 648938 655258 37 622.7 53 66i640 668o86 674597 681173 687815 694524 36 660.9 54 701302 708148 7I 5o64 722o51 729110'73624z 35 702.2 55 743447 750727 758084 765517.773029 780620 34 747.2 56 788292 796045 80388i 8118oi 819806 827899 33 796.2 57 836078 844348 852707 861159 869704 878344 32 849.8 58 887080 895914 904847 913881 923017 932258 3I 908.5 59 941604 95io58 960621 970294 980081 989982 30 973.0 601 60' 40' 30' 20'. 0' Deg. Natural Co-secants. N ATURAL SECANT. 135 Deg.,0/ 10' 20' 30' 40' 50' to 11. 60 2.000000 2.00oI36 20 20393 2.030772 2.041276 2.051906 29 1044 06I 62665 073556 084579 095739 107036 118474 28 1123 62 I3oo54 I4I78I I53655 I6568I I77859 I90195 27 I2IO 63 202689 215346 228168 24I 58 254320 267657 26 I308 64 281172 294869 308750 322820 337083 35I542 25 1417 65 366202 38io65 396I37 4I142I 426922 442645 24 I539 66 458593 474773 491I87 507843 524744 541896 23 1678 67 5593O5 576975. 594914 613126 53I6I8 650396 22 1835 68 669467 688387 7085I4 728504 7488I4 769453 2I 2015 69 790428 8 I I 747 833419 85545i 877853 900635 20 2222 7C 2.923804 2 947372 2 971349 2.995744 3.020569 3.o45835 19 246 I 71 3.071553 3.097736 3.I24396 3. I 5545 I79198 207367 I8 2740 72 236068 265315 295123 325510 356490 388082 I1 3068 73 420304 453173 486711I 520937 555871 591536 i6 3458 74 627955 665152 703151 74I978 781660 822225 S5 3925 75 863703 906125 9A9522 993929 4. o39380 4.085913 4 4492 76 4.I33565 4.182378 4.232394 4.283658 3362I5 390116 13 5I90 77 4454it 502157 56o4o8 620226 681675 74482I 12 6062 78 809734 87649I 945I69 5.0I5852 5.o88628 5.163592 II 7I7I 79 5.240843 5.320486 5.402633 487404 574926 665333 Io 8612 80 5.758770 5.855392 5.955362 6.o58858 6.166067 6.277I93 9 8i 6.392453 6.5I208I 6.636329 6.765469 6.899794 7.039622 8 82 7.I85297 7.337I9I 7.4957II 7.66I298 7.834433 8.o0I5645 7 83 8.205509 8.404659 8.6I3790 8.83367I 9.065I5I 9.309170 6 84 9.566772 9.839123 1o.I2752 Io.43343 10.75849 I i.io455 5 85 11.47371 11.86837 I2.29I25 12.74549 i3.23472 I3.76311 4 86 I4.33559 I4.95788 15.63679 i6.38o4I I7. 9843 18.10262 3 87 19I10732 20.23028 2I.49368 22.92559 24.56212 26.4505I 2 88 28.65371 3I.25758 34.38232 38.20155 42.97571 49.11406 89 57.29869 68.75736 85.94561 114.5930 I71.8883 343.7752 0 60' 50' 40' 30' 20' 10' Deg. Natural Co-secants. LENGTHS OF CIRCULAR ARCS. Degrees. Minutes. [I Seconds. I.0174533 26.4537856 5i.8901179 I.0002909. ooooo48 2.0349066 27.47I2389 52.9075712 2.ooo58x8 2.0000097 3.0523599 28.4886922 53.9250245 3.0008ooo727 3 oooo.000045 4 -.698132 29.5o6I455 54.9424778 4.ooii636 4.0000194 5.0872665 30.5235988 55.9599311 5.Ioo4544 5.0000242 6.IO47I98 3I.54I052I 56.9773844 6.oo07453 6.0000291 7.122I730 32.5585o54 57.9948377 7.0020362 7.oooo339 8.I396263 33.5759587 58 1.0122910 8.002327I 8.oooo388 9 *I570796 34.5934119 59 I.0297443 9.0026180 9.ooo0436 10.1745329 35.6o08652 60 1 0471976 10.00oo29 I 10.oooo485 II.1919862 36.6283I85 65 I. 344640 I.o003998 II.ooo0533 12.2094395 37.6457718 70 1.22I7305 I2.0034907 I2.0000582 I3 2268928 38.6632251 75 I.3089969 i3 0oo37815 i3.oooo630 14.244346I 39.6806784 8o I.3962634 4.00oo40724 14.oooo679 i5.26I7994 40.6981317 85 1.4835299 5.00oo43633 5.0000727 I6.2792527 41.7i5585o 90~ I.5707963 i6.0046542 6.0000776 17.2967060 42.7330383 Ioo I.7453293, I7.0049451 1 7.0000824 i8.3141593 43.7504gI6 IIO I.9I98622 i8.00oo52360 8 0000873 I9.3316126 44.7579449 120 2.o094395I 19.oo55269 19.0000921 20.3490659 45.7853982 I30 2.2689280 20 20.0058178 20.0000970 21.3665191 46.80285I5 I40 2.44346IO 25.0072722 25.0001I212 22.3839724 47.8203047 I50 2.6179939 30.0087266 30.oooI454 a3.4014257 48.837758o0 60 2.7925268 40.oiI6355 4o.OOOI939 24.4188790 49.8552113 170 2.9670597 5o.oi45444 5o 0002424 25.4363323 50.8726646 8oI 3.I4I5927 6o.OI74533 60 0002909 ________.4o_____ - ___57_ 136 TRAVERSE TABLE. Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 6. Coure. Lat. Dep. Lat. Dep. Lat Dep. Lat. Dep. Lat. Dep. j Cours00. _ist o. O j I5 I.o000-44OO o007ooooo o.oo87o 33.oi 00I7oo ooo I7 o 028 89 45 30o 0000 0087 I, 9999 OI75 2.9999 0262 3.9998 o34911 4.9998 o436 30 45 o0.9999 oi31 9998 0262 9997 o393 9997 o524 9996 o654 15 1 O 9998 0175 9997 0349 9995 o524 9994 o698 9992 o873 89 0 I5 9998 0218 9995 o436 9993 o654 9990o 0873 9988 I o9 I 45 3o 9997 0262 9993 o524 9990 0785 9986 1047 9983 I309 30 45 9995 o305 999I o6ii 9986 o9I6 998I 1222 9977 I527 i5 2 0 9994 o349 9988 o698 9982 Io47 9976 I396 9970 1745 88 o 15 9992 0393 9985 0785 9977 II78 9969 I570 9961 1963 45 30 999o o436 9981 0872 997I I309 9962 I745 9952 2181 30 45 o.gg9988 o.o480 1.9977 o.960 2.9965 0.1439 3.9954 0.IgIg91 49942 0.2399gg I5 3 0 9986 0523 9973 1047 9959 I570 9945 2093 9931 2617 87 0 I5 9984 o567 9968 II34 9952 170I 9936 2268 9920 2835 45 3o 998I o6io 9963 1221 9944 i83i 9925 2442 9907 3052 30 45 9979 o654 9957 I3o8 9936 1962 99I4 26i6 9893 3270 i5 4 0 9976 o698 995I 1395 9927 2093 9903 2790 9878 3488 86 0 15 9973 0741 9945 1482 99I8 2223 9890 2964 9863 3705 45 30 9969 0785 9938 1569 9908 2354 9877 3i38 9846 3923 30 45 9966 o828 993I I656 9897 2484 9863 33 2 9828 4140 i5 5 o 9962 o872 9924 I1743 9886 2615 9848 3486 9810 4358 85 o I5 0.9958 o.O9I5 1.99I6 O.I830 2.98740.2745 3.9832 o.366o 4.9790 0.4575 45 30 9954 0958 9908 1917 9862 2875 98I6 3834 9770 4792 30 45 9950 I002 9899 2004 9849 3oo6 9799 4oo8 9748 5009 i5 6 o 9945 1045 9890 2091 9836 3i36 9781 4I8I 9726 522684 o i5 9941 1089 9881 2177 9822 3266 9762 4355 9703 5443 45 3o 9936 1132 987 I 2264 9807 3396 9743 4528 9679 566o 30 45 993I 1175 9861 2351 9792 3526 9723 470I 9653 5877 i5 -7 0 9925 12I9 985I 2437 9776 3656 9702 4875 9627 60o93 83 o i5 9920 I262 9840 2524 9760 3786 9680 5o48 9600 63io 45 30 9914 3o05 9829 2611 9743 3916 9658 5221 9572 6526 30 45 0o.99091o. O349 1:.98I 7 0.2697 2.9726 o.4046 3.9635 0.5394 4.9543 o0.6743 i5 8 0 9903 392 9805 2783 9708 4I75 9611 5567 95I3 6959 82 o i5 9897 i435 9793 2870 9690 43o5 9586 574011 9483 775 45 3o 9890 1478 9780 2956 9670 4434 956I 5912 9451 7390 30 45 9884 I152 9767 3042 965I 4564 9534 6o85 94I8 7606 IS 9 ol0 9877 I564 9754 3I29 9631 4693 9508 6257 9384 7822 8I 0 i5 9870 1607 9740 3215 96I0 4822 9480 643o 9350 8037 45 30 9863 i650 9726 330I 9589 4951 9451 6602 9314 8252 30 45 9856 I693 97I1 3387 9567 5o8o 9422 67741 9278 8467 15 Io o 9848 1736 9696 3473 9544 5209 9392 6946 9240 8682 80o 0 IS 0.9840 0.I779 1.9681I o.3559 2.952I o.5338 3.9362 0.7118 4.9202 o.8897! 45 30o 9833 1822 9665 36459 9498 5467 9330 7289 9163 9112 30 45 9825 I865 9649 3730 9474 5596 9298 7461 9 23 9326 i5 II O 98I6 1908 9633 38i6 9449 5724 9265 7632 9081 9540 79 0 i5 9808 1951 9616 3902 9424 5853 9231 7804 9039 9755 45 P 30 9799 994 9598 3987 9398 5981 9I97 7975 8996 99681 30 45 9790 2036 958I 4073 937I 6o109 962 8i46 8952 1.0I82 I a2 0 9781 2079 9563 4i58 9344 6237 9I26 83i6 8907 o396 78 0 I5 9772 2I22 9545 4244 9317 6365 9o89 8487 8862 0609 45 33 9763 2I64 9526 4329 9289 6493 9052 8658 88I5 0822 30 45 0o.g9753 0.22071i.9507 o444 2.9 260 0.662I 3.90 4 0.8828 4.8767 io35 5 13 9744 2250 9487 4499 923I 6749 897S 8998 87I9 12481 77 i5 9734 2292 9468 4584 9201 6876 8935 9I68 8669 r46o 45 30o 9724 2334 9447 4669 917I 7003 8895 9338 86i8 1672 3o 45 9713 2377 9427 4754 9'40 7131 8854 9507 8567 i884 -5 14 9703 2419 9406 4838 9109 7258 8812 9677 855 20o96176 o I5 9692 2462 9385 4923 9077 73851 8769 9846 8462 2308 s 5J 30 968I 2504 9363 5008 9044 75Ii 8726 I.ooi5 8407 2591 3o1 45 9670 2546 934I 5092 9011 7638 8682 oi84 8352 27301 5 i5 0 9659 2588 93I9 5I76 8978 7765 8637 03531 8296 2941 75 0 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. | - Dist Dist. 2. iDist. 3. Dist. 4. Dist. 5. L U11II - I. - TRAVERSE TABLE. 137 Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. lO. Course, Lat. Dep. Lat. Dep. Lat. Pep. Lat. Dep. Lat. Dep. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~io /0 / o 5 5. 9 0.0262 6.9999 o.0305 7.9999.349 8.99 9999o.o393 9.99.0436 5 930 19998 05249 999g7 O6II 9997 0698 9997 0785 9996 0873 30 45 9995 0785 99994 ogI6 9993 Io47].9992 1178 9991 3309 15 0 9991 1047 9989 1222 9988 1396 9986 1571 9985 1745 893 o 15] 9986 130o9 9983 1527[ 9981 1745 9979 1963 9976 2181 45 3o 9979 1571 9976 1832 9973 20o94 9969 2356 9966 2618 3S 45 9972 1832 9967 2138 9963 2443 9958 2748 9953 3054 O5 2 o 9963 2094 9957 2443 9951 2792 9945 3i41 9939 3490o1 88 o iS 9954 2356 9946 27481 9938 3z4i 9931 3533 9923 3926 45 30 9943 26I7 9933 3053 9924 3490o 9914 392611 9905 4362 30 45115.993I1 0.2879 6.9991 o.33581 7.9908 o.3838 8.9896. o.43i8 9.9885 0.4798 5 3 o 9918 3I4o0 9904 3664 9890 4187 9877 47IO 9863 52341 87 0 x5 9904 3402 9887 3968 9871 4535 9855 5IOz2 9839 5669 45 3o 9888 3663 9869 4273 9851 4884 9832 5494 9813 6io5 30 45 9872 3924 9850 4578 9829 5232 98071 5886 9786 654oi 15 4 o 9854 4185 9829 4883 9805 558i 9781 6278 9756 69761186 0 15 9835 4447 9808 588 9780 5929 9753 6670 9725 741 45 3o 9815 47081 9784 5492 9753 6277 9723 7061 9692 7846 30 45 9794 4968 9760 5797[ 9725 6625 9691 7453 9657 8281 I5 5 0 97721 5229 9734 6xoi 9696 6972 9658 7844 9619 8716 85 0 is 5.9748 0.5490o 6.9706 o.64ol 5 79664 0.7320 118.9622 0.8235 9.995809 150 45 30 9724 5751 9678 6709 9632 7668 9586 8626 9540 9585 30 45 9698 6oux 9648 7o013 9597 8oi5 9547 90171 94971I.OOI9 i5 6 oil 9671 6272 9617 7317 9562 8362 9507 9408 9452 o453 84 15 9643 6532 9584 762I 9525 8709 9465 9798 9406 0887 45 30 9614 6792 9550 7924 9486 9056 942 i.osi88 9357 1320 30 45 9584 7052 9515 8228 9445 9403 9376 0578 9307 1754 i5 7 0o 9553 73121 9478 853i 9404 9750 9329 0968 9255 21871183 0 i S 9520 7572 9440 88341 9360o1.oo96 9280 i358 920oo0 2620 l45 3o 9487 7832 9401 9137J 9316 0442 9230 I747 9144 Xo53 30 45j 5.9452 0.80o9gi 6.9361 0.9440 7.92691 1.o788118.9178, 1.237 9.90871,3485 x5 8 0 9416 835o 0119319 9742 9221 ii34 9124 2526 9027 3917 82 0:5 9379 86io 9276.oo0044 9172 1479 9069 2914 8965 4349 45 3o 9341 8869 9231 0347 9121 1825 90o1 33o3 8902 4781j 30 45 9302 9:27 9185l 0649 9069 2170 8953 3691 8836 5212 15 9 0 9261 9386 9138 0950 9o015 2515 8892 4079 8769 5643 18x 0 I5 9220 9645 9090 1252 8960 2859 883o 4467 8700 6074 45 30 9177 9903 9040 1553 8903 3204 8766 4854j 8629 65o5j 3o 45 9133 i.oi6i 8989 i854 8844 3548 8700 5241 8556 6935 15 I1 o0 9088 0419 8937 2155 8785 3892 8633 5628 8481 7365 8o0 I5 5.9042 i.o6771 6.88831 3.2456 7.87231.4235 18.8564 1.6oi 519.8404 I.77941 45 3o 8995 0934 8828 2756 866o 4579 8493 64o1 8325 822411 30 45'8947 II91 8772 3057J 8596 4922 8421 6787 8245 8652 15 ri 0 8898 1449 8714 3357 853o 5265 8346 7173 8i63 9081 79 0 15 8847 1705 8655 3656 8463 5607 8271 7558 8079 9509 45 30 8795 1962 8595 3956 8394 5949 8193 79431 7992 9937 30 45 8743 2219 8533 4255 8324 6291 8114 8328 7905 2.0364 i5 12 oil 8689 2475 8470 4554 8252 6633 8033 8712 7815 o791 78 0 is 8634 2731 84o6 4852 8178 6974 7951i 9096 7723 s118 45 30o 8578 2986 834i 5151 80o4 7315 7867 9480 7630'644 3o 45 5.8521 r.3242 16.82741.5449 17.8027 1.7656 8.7781 1.9863 19.7534 2.2070I i5 I3 oil 8462 3497 8206 5747 7950 7996 7693 2.0246 7437 24951 77 0 x5 84o3 3752 8137 6o44 7870 8336 7604 0628 7338 2920 45 30o 8342 4007 8066 634I 7790 8676 7513 1o0o 7237 3345J 30o 45 8281 4261 7994 6638'1707 905 7421 I392 7134 3769 15 4 0[ 8218 45i5 792I 6935 7624 9354 7327 1773 7P30 492 76 0 15i 8154 4769 7846 7231 7538 9692 7231 2154 6923 4i6S5 45 30 8089 5023 7770 7527 7452 2.oo0030 7133 2534 68i5 5o38 3o 45 8023 5276 7693 7822 7364 o368 7.),34 2914 6705 546o 15s I5 0o 7956 5529 7615 81171 7274 0706 6933 3294 6593 5882 75 o Pep. Lat. Dep. Lat. Pep. Lat. Pep. Lat. 7 Dp. Lat. Dist. 6. IDist. 7. Dist. 8.. Dist. 9. Dist. 10. ourse. ! i~ TRAVERSE TABLE. Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 5. Lat. Dep. 1at. Dep.- Lat. Dep. Lat Dep. Lat Dep. i 5 05o.9648 0o.2630 I.9296.526I 2.8944.789I 3.859I I.052I 4,8239 I.3I52 74 45 3o 9636 2672 9273 5345 8909 8017 8545 o6go l I82 3362 30 45 9625 2714 9249 5429 8874 8i43 8498 o858 8I23 3572 15 16 o 96I3 2756 9225 55I3 8838 8269 8450 I025 8063 3782 74 0 I5 96oo 2798 g20I 5597 88oi 8395 8402 II93 8002 3991 45 3o 9588 2840 9I76 5680 8765 8520 8353 I36I 794I 420I 3o 45 9576 2882 9151 5764 87271. 8646 83o3 I528 7879 44Io 15 17 o 9563 2924 9I26 5847 8689 877I 8252 I695 7815 46911 73 o I5 9550 2965 9Ioo 5931 865I 8896 8201 1862 7751 4827 45 30 9537 3007 9074 60o4 8612 902I 8149 2028 7686 5o35 - 3o 45 o.9524 o.3049g 1.948 o.6097 2.8572 o.9g46 3.8096 1.2 195 4.7620 I.5243 i5 0i oj 95ii 3090 902I 6i8o 8532 927I 8042 2361 7553 54511 72 0 i5 9497 3I32 8994 6263 849I 9395 7988 2527 7485 5658 45 3o 9483 3173 8966 6346 845o 9519 7933 2692 7416 5865 3o 45 9469 3214 8939 6429 8408 9643 7877 2858 7347 6072 i5 9 0o 9455 3256 8910 65ii 8366 9767 782I 3023 7276 6278 7I o i5 944I 3297 8882 6594 8323 9891 7764 3i88 7204 6485 45 30 9426 3338 8853 6676 8279 I.oo0014 7706 3352 7132 6690 30 45 94I2 3379 8824 6758 8235 oi38 7647 3517 7059 6896 15 20 0 9397 3420 8794 684o 8191 0261 7588 368i 6985 7101 70 o I 5o.9382.346 I.8764 0.6922 2.8 I46 i.o384 3.7528 I.3845 4.69 o I 7306 45 3o 9367 3502 8733 7004 8oo 0o506 7467 4008 6834 75IO 30 45 935I 3543 8703 7086 8o54 o629 7405 4172 6757 77I5 I.5 2I O 9336 3584 8672 7167 8007 075I 7343 4335 6679 79I8 69 o i5 9320 3624 8640 7249 7960 o873 7280 4498 66oo 8122 45 3o 9304 3665 8608 7330 7913 o995 7217 466o 6521 8325 30 45 9288 3706 8576 74II 7864 1117 7152 4822 644o 8528 i5 32 o 9272 3746 8544 7492 78I6 I238 7087 4984 6359 8730 68 o i5 9255 3786 85iI 7573 7766 I359 7022 5I46 6277 8932 45 30 9239 3827 8478 7654 77'6 I48I 6955 5307 6I94 9134 3o 45 o.9222 o.3867.8444 0o. 7734 2.7666 i.i6oi 3.6888 i.5468 4.6I Io I.93361 I5 23 o 9205 3907Ij 84Io 7851 76I5 I722 6820 5629 6025 95371l67 0 xI5 988 3947 8376 7895 7564 1842 6752.5790 5940 9737 45 3o 9I7I 3987 834I 7975 7512 I962 6682 5950 5853 9937 30 45 9153| 4027 |8306 8055 j459 2082 66I2 6IIO 5766 2.0I37 I5 24 O 9I35 4067 8271 8i35 7406 2202 6542 6269 5677 o337 66 o:5 9II8 4Io7 8235 82I4 7353 2322 6470 6429 5588 o536 45 30 910goo 4147 899 8294 7299 2441 6398 6588 5498 0735 3o 451 9081 4I87 8i63 8373 7244 2560 6326 6746 5407 o933 i5 25 o 9063 4226 8126 8452 7189 2679 6252 6905 53i5 1131l65 o I5s 0.9045 0.4266 1.8089 o.853I 2.7134 I.2797 3.6I78 I.7063 4.5223 21.I328 45 30o 9026 4305 8052 86Io 7078 29I5| 6io3 7220 5129 1526 30 45 9007 4344 8oI4 8689 7021 3o33 6028 7378 5o35 1722 i5 26 o 8988 4384 7976 8767 6964 3151 5952 7535 4940 i9i91164 o i5 8969 4423 7937 8846 6906 3269 5875 7692 4844 2114 45 30 8949 4462 7899 8924 6848 3386 5797 7848 4747 23I0 30 45 8930 450oI 7860 9002 6789 3503 579 8004 4649 2505 i5 27 o 89o1 4540 7820 9080o 6730 3620 564o 8I60 4550 27oo063 o i5 8890 4579 7780 9i57 6671 3736 556i 83i5 445i 2894 45 3o 8870 4617 7740 9235 66io 3852 548o 8470 435I 3087 30 4 5 c885oo.46561lI 77001.93 I 2 2.65501.3968 3.5400.8625 4.4249 2.3281 15 28 cl 8829 4695 7659 9389 6488 4084 53i8 8779 4147 3474162 o 5 8809 4733 7618 9466 6427 4200oo 5236 8933 4045 3666 45 30 8788 4772 7576 9543 6365 43i5 553 90o86 394I 3858 3o 45 8767t 48Io 7.535 9620 6302 443o 5069 9240 3836 4049 i5 99 o 8746 4848 7492 9696 6239 4544 4985 9392 373I 4240 61 o i5 8725 4886 745o 9772 6I75 4659 4900 9545 3625 4431 45 301o 8704 4924 7407 9848 6III 4773 484 9697! 35i8 462I 30 45 8682 4962 7364 9924 6o46 4886 4728 98491 34IO 48ii i5 30 ol 866 5000oo 732110000 598i 5ooo0 464I 2.0000 33oi 5000 60o 0 Dep. L|at. Dep. Lat.' Dep. Lat. Pep. Lat. Dep. Lat. Dist. 1. Dist. 2. 1 Dist. 3. Dist. 4. Dist. 5.Cuse TRAVERSE TABLE. 13.. -- Dist. 10. I~ist. 1 i)ist. 6. iDist. 7. Dist. 8. Dist. 9. Dist. 90. fCou~rse. - _ —.1 Lat. _Pep. Lat. Dep. Lat. Dep. - Lat. Dep. Lat. Dep. i 5.7887.5782'67535 1.842 7.783 2-.1042 8.683I 2.3673 9.6479 2-6303 74 45 30 7818 6o34 7454 8707 70901 379 6727 4o5i 6363 6724[ 30 45 7747 6286 7372 9o00o1 6996 1715 6621 443o 6246 7144 ID O 0 7676 6538 7288 9295 169o1 2051 65i4 4807 6126 7564 174 0 i 7603 6790 7203 9588 6804 2386 64o4 5i8511 6oo005 7983 45 30 7529 7041. 7117 9881 6706 2721 6294 556i 5882 8402l 30 45 7454 7292 7030 2.0174 66o6 3056 618i 5938 5757 8820 15 17 0 7378 7542 6941 o466 65o4 3390 6067 63i3 5630o 92371 73 15 7301 7792 6856 0758 6402 3723 5952 6689 5502 9654 45 30 7223 8042 6760 io49 6297 4o56 [ 5835 7o64 5372 3.0071 3o 45 5.7144 1.8292 16.666812. I341 7.6192 12.438911 8.57i 612.7438 9.524o0 3.o486 5 30 7063 854i 6574 1i63i1 6o85 4721 5595 7812 51io6 0902 72 0 i5 6982 8790 6479 192i 5976 5053 5473 8i851 4970 i3i6 45 30 6899 9038 6383 2211 5866 5384 5349 8557 4832 173o 30 45 68i6 9286 6285 25o01 5754 5715 5224 8930 4693 2144 15 9g o 6731 9534 6i86 2790 564i 6o45 5097 93o1 4552 255771 0 5 6645 9781 6o86 3078 5527 6375 4968 9672 4409 2969 45 3o 65581 2.02 8 5985 3366 54ii 6705 1483813.0043 4264 3381i 3o 45 6471 0275 5882 3654 5294 7033 4706 o4i3 4ii8 3792 i5 20 o 6382 0521 5778 3941 5175 7362 4572 0782 3969 4202 70 0 i 5.629 2.0767 6.5673 2..4228 87.5o55 2.7689 8.4437 3.ii5i 9.3819 3.462 45 30 6200 1o12 5567 45i5 4934 8o017 43oo i5911 3667 5021 3o 45 6io8 1257 5459 4800oo 48,i 8343 4162 i886 35I4 5429 i5 si o 6oiS 1502 5351 5o86 4686 8669 4022 2253 3358 5837 69 i5 5920 1746 5241 5371 456i 8995 388I 2619i 3201 6244 45 30o 585 gg99o 5I29 5655 4433 9320 3738 2985 3042 665o 30 45 5729 2233 5017 5939 43o5 9645 3593 3350o 2881 7056 15 22 0o 5631 2476 4903 6222 4175 9969 3447 3715 2718 7461 68 0 15 5532 2719 4788 65o5 4o43 3.o0292 3299 4078 2554 7865 45 3o 5433 2961 4672 6788 3910 o6iS 3149 4442 2388 8268 3o 45 5.5332 2.3203 6.4554 2.7070 7.3776 3.o937 8.2998 3.48o4 9.222o 3.8671 i 23 0 5230 3444o 4435 7351 3640 1258 2845 5i66 2050 907367 0 r5 5127 3685 43i5 76327 35o3 i58o 2691 5527 1879 9474 45 30 5024 3925 4194 79I2 3365 I90oo0 2535 5887 1706 9875 30 45 4919 4i65 4072 8192 3225 2220 2378 6247 I531 4.0275 i5 24 o 48,3 44o4 3948 8472 3084 2539 2219 66o6 i355 0674 66 0 iS 4706 4643 3823 8750 2941 2858 2059 6965 1176 1072 45 30 4598 4882 3697 9029 2797 3175 1897 7322 0996 1469 30 45 4489 5120 3570 9306l 2651 3493 1733 7679 o8i4 i866 i5 25 0 4378 5357 3442 9583 2505 3809 i568 8o36 o63. 2262 65 i5 5.4267 2.5594 6.3312 2.9860 17.2356 34 s25 8.i4oi 3.839 9.0446 4.2657 45 3o 4i55 583i 3i8i 3.oi36 2207 444j 1233 8746 0259 3oSi 30 45 4042 6067 3049 o4II 2056 4756 io63 gi9oo 0070 3445 i5 26 o 3928 6302 2916 o686 1904 5070 o89gi 9453 8.9879 3837 64 i5 3812 6537 2781 og6o 1750 5383 0719 9806 9687, 4229 45 30 3696 6772 2645 1234 i595 5696 o54414.oi58 9493 4620o 3 45 3579 7o006 2509 150o7 438 6oo8 o368 0509 9298 1Soo i5 27 0 346o 7239 2370 1779 1281 6319 0191 0859 9101 5399 63 0 i0 334I 7472 2231 2Q51 112 663o ooI2 12og9 8902 5787 45 30o 3221 7705 209I 2322 0961 6940 17.9831 1557 8701 6175 3o 45 5.30 99.7937 6.1949 3.2593 7.079913.7249 7.949 649 4.95 8.849914.656i i5 28 0 2977 8i68 i8o6 2863 o636 7558 9465 2252 8295 6947 62 0 i5 2853 8399 r662 3132 0471 7866 9280 2599 8089 7332 45 30 2729 863o 1517 340i o3oS 8173 9094 2944 7882 7716 30 45 2604 8859 1371 3669 0138 8479 89oS 3289 7673 8099 iS 29 0 2477 9089 I223 3937 6.9970 8785 8716 3633 7462 848i 6i 0 15 23S0 9317 175 4203 9800 9090 8525 3976 7250 8862 45 30 2221 9545 0925 4470 9628 9394 8332 43i8 7036 9242 3o 45 2092 9773 0774 473S 9456 9697 8i38 4659 6820 9622 i5 3o 0 196213.0000 0622 Sooo 928214.0000 7942 5ooo 66o3 5.0000 6o 0 Dep. La. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. 10. 4 0 TRAVERSE'rABLE. s Dist. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 5. Course., IDs.3 r.0Lat. IDep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. 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I 35.2. 95.4 55.6 i6.o 76..5 37.:2 98.0 59.0 20.2. 8.7 5 5 56 156.o i6.o 76. x 36.2. 96.4 56.6 i7.o 77.5 38.:2 99.o 60.0 2I.x. 8:2.7 56 57 57'~x7o 77' 379 97'4 57, x8. 78.5 39.29o.o600 2.0- 5 58580 78.o 78.x8,2 98.4 58.6 I9.o 79.5,4o.2 oi.o 62".1 23.3 84.75 59 59.0 I9.o1 79.Il 39.29. 99.4 59.71 20.0 80.5 Ix.., o2.ii 63.Ii 24.3 85.8 59j E R ID IONAL PARTS. 1 4 LATITUDE. 13' 140 150 160 170 18 - 190 200 210 220 230 24 0 786.8 848.5 9io.5 972.7 io35.3 o098.2 i I6.5 I225.I I289.21 353.7 I4I8.6 I484.I o I 87.8 49.5 II.5 73.8 36.3 99.3 62.5 26.2 90.3 54.8 I9.7 85.2 I 2 88.8 50.5 I2.6 74.8 37.4 I0oo.3 63.6 27.3 91.3 55.8 20.8 86.3 2 3 89.9 5i.6 I3.6 75.9 38.4 oi.4 64.7 28.3 92.4 56.9 2I.9 87.3 3 4 90.9 52.6 I4.6,76.9 39.5 02.4 65.7 29.4 93.5 58.o 23.O 88.4 4 5 9 I.9 53.6 15.7 78.0 4o0.5 o3.5 66.8 30.4 94.5 59.0 24.I 89.5 5 6 92.9 54.7 I6.7 79.0 4I.6 04.5 67.8 31.5 95.6 60.I 25.- 90.6 6 7 94.0 55.7 17.7 80.o 42.6 o5.6 68.9 32.6 96.7 6z.2 26.2 9I.7 7 8 95.0 56.7 I8.8 8I.I 43.7 o6.6 70.0 33.6 97.8 62.3 27.3 92.8 8 9 96.o 57.8 I9.8 82.I 44.7 07.7 71.0 34.7 98.8 63.4 28.4 93.9 9 Io0 797.o 858.8 92o.8 983.2 o45.8 I Io8.7 I I72.I I235.8I 299.9 I364.5 I429.5 I495.o0 o II 98.I 59.8 21.9 84.2 46.8 09g.8 73.I 36.8 i3oi.0 65.6 30.6 96. ii 12 99.I 60.9 22.9 85.2 47.9 IO.8 74.2 37.9 02.0 66.6 3I.7 97.2 I2 x38oo.i 61.9 23.9 86.3 48.9 II.9 75.2 39.0 03.i 67.7 32.8 98.3 I3 I4 OI.2 62.9 25.0 87.3 49.9 I2.9 76.3 4o.o 04.2 68.8 33.9 99.4 I4 i5 02.2 64.0 26.0 88.4 5i.o I4.o 77.4 4i.i o05.3 69.9 34.9 50oo.5 I5 I6 03.2 65.0 27.0 89.4 52.0 I5.o 78.4 42.2 o6.3 70.9 36.o oi.6 i6 I7 o4.2 66.0 28.I 9o0.4 53.1 I6.I 79.5 43.2 07.4 72. 37. 02.7 43.2 07~7 72'0 37'1 02'717 i8 05.3 67. 29.1 9I.5 54. 17.I 8o.5 44.3 o8.5 73. 38.2 03.8 i I9 o6.3 68. 3o.I 92.5 55.2 I8.2 8I.6 45.4 09.6 74.2 39.3 04.9 1 9 20 807.3 869. 93I.2 993.6 o56.2 I I I 9.2 82.7 I246.4 i3io.6 I375.3 i44o.4 I506.o 20 2I o8.4 70.1 32.2 94.6 57.3 20.3 83.7 47.5 11.7 76.4, 4i.5 07.I 2I 32 09.4 71.2 33.3 95.6 58.3 21.3 84.8 48.6 I12.8 77.4 42.6 08.2 22 33 Io.4 72.2 34.3 96.7 59.4 22.4 85.8 49.6 i3.8 78.5 43.7 09.3 23 24 I.4 5 3.2 35.3 97.7 6o.4 23.4 86.9 50.7 14.9 79.6 44.8 Io.4 24 25 I 2.5 71t4.3 36.3 98.8 6I.4 24.5 88.o 5i.8 i6.o 80.7 45.8 1I.5 25 26 i 3.5 75.3 37.4 99.8 62.5 25.5 89.o 52.8 17. 8I.8 46.9 I2.6 26 2 7 i4.5 76;.3 38.4 iooo.8 63.5 26.6 90o. 53.9 i8.I 82.8 48.o 13.7 27 28 15.5 77.4 39.5 oi.9 64.6 27.6 9.1I 55.0 19.2 83.9 49. I 4.8 28 19t i6.6 78.4 4o.5 02.9 65.6 28.7 92.2 56.0 20.3 85.o 50.2 I5.9 29 30 817.6 879.4 94I.6 xoo4.0o 166.7 II29.7 1x 93.2 I257.I I32.4 I 386.i i45I.3 5I 7.o 3o 31 I8.6 80.5 42.6 o5.o 67.7 3o.8 94.3 58.2 22.5 87.2 52.4 i8. 3I 32 I9.6 8.5 43.6 o6. I 68.8 3.8 95.4 59.2 23.5 88.3 53.5 19.2 32 33 20.7 82.5 44.7 07. 1 69.8 32.9 96.4 6o.3 24.6 89.4 54.6 20.3 33 34 21x.7 83.6 45.7 08. 70o.9 34.o 97.5 6i.4 25.7 9o.4 55.6 2I.4 34 35 22.j 84.6 46.7 09.2 72.o 35.I 98.5 62.4 26.7 91.5 56.7 22.5 35 36 23.8 85.6 47.8 Io.2 73.o 36.I 99.6 63.5 27.8 92.6 57.8 23.6 36 37 24.8 86.7 48.8 Ix.3 74. I 37.2 I200.7 64.6 28.9 93.7 58.9 24.7 37 38 25.8 87.7 499 I 2.3 75.I 38.2 OI.7 65.6 30.0 94.8 6o.o 25.8 38 39 26.9 88.7 50.9 i3.4 76.2 39.3 02.8 66.7 31.1 95.8 6i.I 26.9 39 4o 827.9 889.8 95.9 IoI4.4 1077.2 I 4o.I3 1203.9 1267.8 I332.I I396.9:462.2 1528.o 4o 4I1 28.9 90.8 53.0 5.4 78.3 4I.4 o4.9 68.8 33.2 98.0 63.3 29.1 4I 42 29.9 91.8 54.o i6.5 79.3 42.4 o6.o 69.9 34.3 99.I 64.4 30.2 42 43 3i.o 92.9 55.I I7.5 80.4 43.5 07.I 71.o 35.3 I400.2 65.5 3i.3 43 44 32.0 93.9 56.I i 8.6 8i.4 44.6 o8. I 72.1 36.4 oI.3 66.6 32.4 44 45 33.o 94.9 57.i z 9.6 8z.5 45.6 09.2 73.I 37.5 02.4 67.7 33.5 45 46 34. i 96.0 58.2 20.6 83.5 46.7 I0.2 74.2 38.6 o3.4 68.8 34.6 46 47 35.I 97.0 59.2 2I.7 84.6 47.7 i.3 75.3 39.7 o4.5 69.8 35.7 47 48 36.zI 98.o 60.2 22.7 85.6 48.8 I 2.4 76.3 40.7 o5.6 70.9 36.8 48 49 37.2 99.1I 6i.3 23.8 86.7 49.9 I3.4 77.4 4I.8 06.7 72.o 37.9 49 50 838.2 900.1 962.3 024.8 I087.7 I50o.9 2I4.5 I2785 I 342.9 1407.8 1473. I 539.o 5o 5i 39.2 oI.I 63.4 25.9 88.8 52.0 i5.5 79.5 44.o o8.8 74.2 4o0. 5i 52 40.2 02.2 64.4 26.9 89.8 53.o i6.6 8o.6 45. 09.9 75.3- 4.a2 5a 53 4i.3 o3.2 65.5 28.o 9~ 9 54.I I7.7 8I.7 46.I II.o 76.4 42.3 53 54 42.3 o4.3 66.5 29.0 9I.9 55.I I8.7 82.8 47.2 12.1 77.5 43.4 54 55 43.3 o5.3 67.5 3o.I 93.o 56.2 I9.8 83.8 48.3 13.2 78.6 44.5 55 56 44.4 o6.3 68.6 3i.x 94.o 57.2 2o.9 84.9 49.4 i4.3 79.7 45.6 56 57 45.4 07.4 69.6 32.2 95. I 58.3 21.9 86.o 5o0.4 5.4 80.8 46.7 57 58 46.4 o8.4 70o7 33.2 96.I 59.4 23.o 87.o 5I.5 I6.5 8I.9 47.8 58 59 47.5 09.4 71.7 34.3 97.2 60.4 24. I 88. 52.6 17.5 83.0o 48.9 59 4i M iftIDIONAL PARrS. LATITUDE. tr. 251 260 270 280 290 300 31~ 320 330 340 350.in. o I55o.o i6i6.5 i683.5 I75I.2 I8Ig.4 I888.4 1958.0 2028.4 2099.5 2I7I.5 2244.3 o I 5i.i I 7.6 84.6 52.3 20.6 89.5 59.2 29.6 2I00.7 72.7 45.5 I 2 52.2 18.7 85.8 53.4 21.7 90.7 60.4 30.7 OI.9 73.9 46.8 2 3 53.3 I9.8 86.9 54.6 22.9 91.9 6i.6 31.9 o3.1 75.1 48.0o 3 4 54.4 20.9 88.o 55.7 24.0 93.o 62.7 33.I o4.3 76.3 49.2 4 5 55.5 22.0 89.1 56.8 25.2 94. 63.9 34.3 05.5 77.5 5o.4 5 6 56.6 23.2 9o.3 58.0 26.3 95.3 65.o 35.5 06.7 78.7 5i.6 6 7 57.7 24.3 91.4 59.1 27.5 96.5 66.2 36.7 07.9 80.o 52.9 7 8 58.8 25.4 92.5 60.2 28.6 97.6 67.4 37.8 9. i 81.2 54.1 8 9 59.9 26.5 93.6 6i.4 29.7 98.8 68.5 39.0 io.3 82.4 55.3 9 0o I56I.o 1627.6 1694.8 1762.5 1830.9 1899.9 I969.7 2040.2 21II.5 2183.6 2256.5 Io 11 62.1 28.7 95.9 63.6 32.0 19IgOI.I 70.9 41.4 I2.7 84.8 57.8 I I2 63.2 29.8 97.0 64.8 33.2 02.3 72.0 42.6 I3.9 86.0 59.0 12 I3 64.4 3I.o 98.1 65.9 34.3 o3.4 73.2 43.8 I 5. 87.2 60.2 13 14 65.5 32.I 99.3 67.0 35.5 o4.6 74.4 44.9 I6.3 88.4 6i.4 14 15 66.6 33.2 I 700.4 68.2 36.6 05.7 75.6 46.I 17.5 89.6 62.7 5 i6 67.7 34.3 oi.5 69.3 37.8 o6.9 76.8 47.3 I8.7 90.8 63.9 16 17 68.8 35.4 02.6 70.5 38.9 o8. I 77.9 48.5 I9.8 92.o 65.I 17 I8 69.9 36.5 o3.8 71.6 4o.I 09.2 79.1I 49.7 2I.0 93.3 66.3 18 19 7I.o 37.6 04.9 72.7 41.2 Io.4 80.2 50.8 22.2 94.4 67.5 9 20 1I572.I I638.8I 706.o 1773.9 I 842.4 1911.5 1981.4 2052.0 2123.4 2I95.7 2268.8 20 2 1 73.2 39.9 07. 75.0 43.5 12.7 82.6 53.2 24.6 96.9 70.0 21 22 74.3 4I.o o8.3 76.I 44.6 I3.8 83.7 54.4 25.8 98.I 71.2 22 23 75.4' 42.1 o9.4 77.2 45.8i I5.o 84.9 55.6 27.o 99.3 72.5 23 24 76.5 43.2 io.5 78.4 46.9 16.2 86.I 56.8 28.2 2200.5 73.7 24 25 77.6 44.3 i.6 79.5 48.I I7.3 87.3 58.o 29.4 OI.7 74.9 25 26 78.7 45.5 12.8 80.6 49.2 i8.5 88.4 59.1 30.6 o3.0 76.1 26 27 79.8 46.6 I3.9 8i.8 5o.4 I9.6 89.6 60.3 31.8 o4.2 77.4 27 23 80.9 47.7 I5.o 83.0o 5.5 20.8 90.8 6i.5 33.0 o5.4 78.6 28 29 82.1 48.8 i6.I 84. I 52.7 2I.9 92.0 62.7 34.2 o6.6 79.8 29 3o 1583.2 1649.91 17I7.3 1785.2 i853.8 I923.1 1993.I 2063.9 2135.4 2207.8 228I.o 30 3I 84.3 5S.o i8.4 86.4 55.0 24.3 94.3 65.I 36.6 09.0 82.3 3I 32 85.4 52.2 19.5 87.5 56.1 25.4 95.5 66.2 37.8 Io.2 83.5 32 33 86.5 53.3 20.7 88.6 57.2 26.6 96.6 67.4 39.0 11.4 84.7 33 34 87.6 54.4 21.8 89.8 58.4 27.8 97.8 68.6 40.2 I2.7 86.o 34 35 88.7 55.5 22.9 90o.9 59.6 28.9 99.o 69.8 41.4 I3.9 87.2 35 36 89.8 56.6 24.0 92.I 60.7 3o.I 2000.2 7I.0 42.6 15.1 88.4 36 37 90.9 57.8 25.2 93.2 61.9 31.3 oi.3 72.2 43.8 i6.3 89.7 37 38 92.0 58.9 26.3 94.3 63.o 32.4 02.5 73.4 45.o 17.5 90.9 38 39 93. I 6o.o 27.4 95.5 64.2 33.6 03.7 74.5 46.2 I8.7 92.1 39 4o I 594.3I 66. I I I 728.6 1 796.6 I865.3 1934.7 2004.9 2075.7 2147.4 2219.9 2293.3 4o 41 95.4 62.2 29.7 97.8 66.5 35.9 06.0 76.9 48.6 21.2 94.6 41 42 96.5 63.4 30.8 98.9 67.6 37.I 07.2 78. I 49.8 22.4 95.8 42 43 97.6 64.5 31.9 i8oo.o 68.8 38.2 o8.4 79.3 5.o 23.6 97.o 43 44 98.7 65.6 33.I OI.2 69.9 39.4 09.6 80.5 52.2 24.8 98.3 44 45 99.8 66.7 34.2 02.3 71.1 4o.5 IO.7 81.7 53.4 26.0 99.5 45 46 I600.9 67.8 35.3 o3.5 72.2 4I.7 I1.9 82.9 54.6 27.2 2300.7 46 47 02.0 69.0 36.5 o4.6 73.4 42.9 13.1 84.o 55.8 28.5 02.0 47 48 o3.i 70o. 37.6 o5.7 74.5 44.0 I4.3 85.2 57'0 29.7 03.2 48 49 04.2 71.2 38.7 o6.9 75.7 45.2 5.4 86.4 58.2 30.9 o4.4 49 50 I6o5.4 I672.3 I739.9 I808.o I876.8 i946.4 2o0 6.6 2087.6 2I59.4 2232.I 2305.7 50 5i o6.5 73.4 4I.o 09.2 78.0 47.5 I7.8 88.8 60.7 33.3 06.9 51 52 07.6 74.5 42.I Io.3 79.2 48.7 I 9.0o.o 61.9 34.6 o8. 52 53 08.7 75.7 43.2 I1I.4 80.3 49.9 20.2 91.2 63.I 35.8 09.4 53 54 09.8 76.8 44.4 I2.6 8i.5 51.o 21.3 92.4 64.3 37.0 io.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 I2.0 79.0 46.6 I4.9 83.8 53.4 23.7 94.8 66.7 39.4 I 3. 1 56 57 3.i 80.2 47.8 i6.o 84.9 54.5 24.9 96.0 67.9 40.7 I4.3 57 58 14.2 8i.3 48.9 I7.2 86.I 55.7 26.0 97.1 69.1 41.9 15.5 58 59 15.4 82.4 50.0 i8.3 87.2 56.9 27.2 98.3 70.3 43.I I 6.8 59 MERIDIONAL PARTS. 14 LATITUDE. MIin. 360 370 38~ 390 400 410 420 430 ~ 440 450 460 min. 0 12318.o02392.6 2468.3 2545.o 2622,7270I.6 278.72863.1 2945.8 3030.0 3ii5.6 o I I9.2 93.9 69.5 46.2 24.o0 02.9 83.I 64.5 47.2 3i.4 I7.0o I 2 20.5 95.1 70.8 47.5 25.3 04.3 84.4 65.8 48.6 32.8 i8.5 2 3 21.7 96.4 72.1I 48.8 26.6 o5.6 85.8 67.2 5o.o 34.2 19.9 3 4 23 o 97.7 73.4 5o 1 27.9 o6.9 87. I68.5 51.4 35.6 21.4 4 5 24 2 98.9 74.6 5I.4 29.2 o8.3 88.5 70.0 52.8 37.0 22.8 5 6 25.4 2400.2 75.9 52.7 3o.5 og.6 89.8 7I.3 54.2 38.4 24.2 6 7 26.7 oI.4 77. I 54.0 3I.9I 0.I9 91.2 72.7 55.6 39.8 25.7 7 8 27.9 02.7 78.5 55.3 33.2 12.2 92.5 74.I 57.0 4I.3 27.I 8 9 29.1 03.9 79.7 56.6 34.5 i3.5 93.8 75.4 58.3 42.7 28.5 9 xo 12330.4 2405.2 248I.0 2557.8 2635.8 2714.9 2795.I 2876.8 2959.8 3o44.I 3 3o.o Io I I 3i.6 o6.4 82.2 59.I 37.I 16.2. 96.5 78.2 6I.I 45.5 3i.5 TI 12 32.9 07.7 83.5 60.4 38.4 I7.5 9,7.9 79.5 62.5 47.o 32.9 12 13 34. I o09.0o 84.8 61.7 39.7 I8.9 99.3 80.9 63.9 48.4 34.3 13 I4 35.3 10.2 86.i 63.o 4I.o 20.2 2800.6 82.3 65.3 49.8 35.8 I4 i5 36.6 I I.5 87.4 64.3 42.3 21.5 02.0 o3.7 66.7 5I.2 37,2 15 I6 37.8 I29.7 88.6 65.6 43.6 22.9 03.3 85.o 68. I 52.6 38.7 i 6 17 39.0 I4.o 89.9 66.9 44.9 24.2 04.7 86.4 69.5 54. I 40. I17 I8 40.3 I5.2 9I.2 68.2 46.3 25.5 o6.o 87.8 70.9 55.5 4I.6 I8 I 9 4i.5 i6.5 92.4 69.5 47.6 26.8 07.3 89.I 72.3 56.9 43.o I 9 20 92342.81 247.8 2493.7 2570.7 2648.9 2728.2 28o8.8 289o.5 2973.7 358.3'3i445 20 21I 44.o I9.o 95.0 72.0 50.2 29.5 Io.I 91.9 75. I 59.7 45.9 2 22 45.3 20.3 96.3 73.3 51.5 30.8 II.4 93.3 76.5 61.2 47.4 22 23 46.5 21.5 97.6 74.6 52.8 32.2 I 2.8 94.7 77.9 62.6 48.8 23 24 47'7 22.8 98.8 75.9 54.I 33.5 I4.I 96.0 79.3 64.o 50.3 24 25 49.0 24.0o 2500.I 77.2 55.5 34.8 15.5 97.4 80.7 65.4 5I.7 25 26 50.2 25.3 o.4 78.5 56.8 36.2 i6.8 98.8 82.I 66.9 53.2 26 27 5 1.5 26.5 02.7 79.8 58.I 37.5 I8.2 2900.2 83.5 68.3 54.6 27 28 52.7 27.8 03.9 8i.i 59.4 38.8 I9.5 oi.5 84.9 69-7 56., 28 ~9 54.o 29.I o5.2 82.4 60.7 40.2 20.9 02.9 86.3 7I.1 57.5 29 3e 2355.2 2430.3 2506.5 2583.7 2662.0 2741.5 2822.3 2904.3 2987.7 3072.6 3I59.0 30 3 56.5 3i.6 07.8 85.o 63.3 42.9 23.6 05.7 89. I 74.0 60.4 31 32 57.7 32.9 o9.o 86.3 64.6 44.2 25.o 07.1 90.5,75.4 6-f.9 32 33 58.9 34. I io.3 87.6 66.0o 45.5 26.3 o8.4 9I.9 76.9 63.3 33 34 60.2 35.4 ix.6 88.9 67-3 46.9 27.7 09.7 93.3 78.3 64.8 34 35 6I.4 36.7 I2.9 90.2 68.6 48.2 29.0 11.2 94.7 79.7 66.2 35 36 62.7 37.9 I4.2 9I.5 69.9[ 49.5 3o0.4 12.6 96.I 8I.I 67.7 36 37 63.9 39.2 15.4 92.8 7I.2 50.9 31.7 I4.o 97.5 82.6 69.I 37 38 65.2 4o04 I6.7 94. I 72.5 52.2 33.I 15.3 98.9 84.0 70.6 38 39 66.4 4I.7 I8.0o 95.4 73.9 53.5 34.5 16.7 3000.3 85.4 72.0 39 40 2367.6 2443.0 2519.3 2596.7 2675.2 2754.9 2835.8 298.I 3oo00I.8 3086.931I73.5 4o 4I 68.9 44.2 20.5 98.0 76.5 56.2 37.2 I9.5 o3.2 88.3 75.0 4I 42 70.2 45.5 21.8 99.3 77.8 57.6 38.6 20.9 o4.6 8971 76.4 42 43 7I.4 46.8 23.. 2600.5 79.1 58.9 39.9 22.3 o6.o 9I.2 77.9 43 44 72.6 48.0 24.4 oi.9 80.5 60.2 4r.3 23.6 07.4 92.6 79.3 44 45 73.9 49.3 25.7 o3.2 8i.8 6i.5 42.6 25.o o8.8 94.o 8o.8 45 46 75.I 50.6 27.0 o4.5 83.Il 62.9 44.0 26.4 IO.2 95.5 82.3 46 47 76.4 I51.8 28.3 o5.8 84.4 64.3 45.4 27.8 i.6 96.9 83.7 47 48 77.6 53. I 29.5 o7. I 8571 65.6 46.7 29.2 13.o 98.3 85.2 48 49 78.9 54.3 30.8 o8.4 87.I 66.9 48.I 30.6 I4.4 99.7 86.6 49 50o 2380.1 2455.6 2532.I 2609.7 2688.4 2768.3 2849.5 2932.o 3oi5.8 3IOI.2 3I88.I 5o 5i 8i.4 56.9 33.4 11.0 89.7 69.6 50.8 33.3 I7.2 02.6 89.6 5 i 52 82.6 58.I 34.7 12.3 91.0 7Io0 52.2 34.7 I8.7 0o4. 91.0 52 53 83.9 59.4 36.o i3.6 92.3 72.3 53.5 36.I 20.1 o5.6 92.5 53 54 85.I 60.7 37.2 I4.9 93.7 73.7 54.9 37.5 2I.5 07.o0 94.0 54 55 86.4 6I.9 38.5 16.2 95.o 75.o 56.3 38.9 22.9 o8.4 95.4 55 56 87.6 63.2 39.8 17.5 96.3 76.3 57.7 4o.3 24.3 og.8 96.9 56 57 88.9 64.5 4I.I I8.8 97.6i 77.7 59.0 4I.7 25.7 11.2 98.4 57 58 90.2 65.8 42.4 20.1 99.0 79.0 60.5 43.I 27.1 I2.7 99.8 58 59 91.4 67.0 43.6 21.4r2700.3 80.4 6J., 44.4 28.5 14.i13201.3 59 K 146 MERIDIONAL PARTS. LATITUDE. Min 470 480 490 500 51. 520 563- ~ 540 50 560 570 i 0 3202.7 329I.5 3382.i 3474.5 3568.8 3665.2 3763.8 3864.6 3968.o 4073.9 4I82 6 C I o4.2 93.o 83.6 76.0 70o.4 66.8 65.4 66.3 697 75.7 845 I 0o5.7 94.5 85.i 77.6 72.0 b8.4 67.1 68.o 71.5 77.5 86.3 2 3 o7.I 96.o 86.7 79.1 73.6 70.I 68.8 69,7 73.2 79.3 88. 3 4 o8.6 97.5 88.2 80.7 75.2 7I.7 70.4 7I.5 75.o 8I.T 90.0 4 5 10.I 99.0 89.7 82.3 76.8 73.3 72.1 73.2 76.7 82.9 91.8 5 6 11.5 3300.5 9I.3 83.8 78.4 75.0 73.7 74.9 78.4 84.7 93.7 6 7 I3.o 02.0 92.8 85.4 79.9 76.6 75.4 76.6 80.2 86.4 95.5 7 8 i4.5 o3.5 94.3 87.0 8i.5 78.2 77 I 78.3 82.0 88.2 97.3 8 9 I 5.9 o5.o 95.8 88.5 83.I 79.8 78.7 80.o 83.7 90.0 99.2 9 0 32 I 7.4 33o6.513397.4 3490. I 35847 368.5 3780.4 388 I.7 3985.4 409I.8 420 I.0 Io I I 18.9 o8.o 98.9 9I.6 86.3 83.I 82.I 83.4 87.2 93.6 02.9[ I 12 20.3 09.5 34oo.4 93.2 87.9 84.7 837 85.. I 88.9 95.4 04.7 I2 I3 21.8 I I.0 02.0 94.7 89.5 86.4 85.4 86.8 90.7 97.2 o6.6 13 I4 23.3 12.5 03.5 96.3 91.1 88.o 87. I 88.5 92.5 99.o o8.4 14 IS 24.8 I4o0 o5.o 97.9 92.7 89.6 88.8 90.2 94.2 4IOo.8 io.3 I5 i6 26.2 I5.5 o6.6 99.4 94.3 91.3 90.4 92.0 96.o 02.6 I2.I I6 17 27.7 17.0 o8.i 35oI.o 95.9 92.9 92.1 93.7 97.7 o4.4 14.0 17 I8 29.2 ix8.5 09.6 02.6 97.5 94.5 93.8 95.4 99.5 o6.2 I 5.8 I8 I 9 30.7 20.0 I I.r 04.1 99.I 96.2 95.5 97.1 4001.2 o8.o I7.7 I9 20 3232.1 3321.5 34127 35.73 360.7 3697.8 37971i 3898.8 4oo3.o 4109.8 4219.5 20 21 33.6 23.0 14.2 07.3 02.3 99.4 98.8 39o00.5 04.7 I.6 2.4 2 22 35., 24.5 15.7 o8.8- 3.9 370I.1 3800.5 02.2 o6.5 x3.4 23.2 22 23 36.6 26.o I7.3 Io.4 05.5 02.7 02.2 o4.0 o8.3 15.2 25, 23 24 38.0 27.5 i8.8 12.o0 07.1 o4.4 o3.8 05.7 Ioo0 17.I 27.0 24 25 39.5 29.0 20.4 13.5 08.7 o6.o o.5o 07.4 I.8 I8.9 28.8 25 26 4I.o 3o.6 21.9 1 5. I Io.3 07.6 07.2 09I 3.5 20.7 30.6 26 27 42.5 32.I 23.5 I6.7 I1.9 09.3 08.9 io.8 5.3 22.5 32.5 27 28 44; o 33.6 25.o 18.3 i 3.6 10 o.9 Io.5 I 2.5 I7.I 24.3 34.4 28 29 45.4 35I.1 26.5 I9.8 1 5.1 1 2.6 I 2.2 I4.3 I8.8 26.1 36.2 29 30 3246.9 3336.6 3428.0 352I.4 36I6.7 37I4.2 38I3.9 39 6.04020.6 4I27.9 4238.I 3o 31 I 48.4 38.I 29.6 23.o0 8.4 I5.8i I.6 1I77 22.4 29.7 40oo 31 32 49.9 39.6 3i.i 24.6 2o.o0 I7.5 I7.31 9.4 24.I 3I.5 4i.8 32 33 5I.4 4I.I 32.7 26.I 2I.6 I9.1 18.9 21.2 25.9 33.3i 43.7 33 34 52.8 42.6 34.21 27.7 23.2 20.8 20.6 22.9 27.7 35.21 45.5 34 35 54.3 44. I 35.8 29.3 24.8 22.4 22.3 24.6 29.4 37ol 47.4 35 36 55.8 4571 37.3 3o.8 26.4 24. I 24.0o 26.3 31.2 38.8 49.3 36 37 57.3 47.2 38.8 32.4 28.01 25.7 25.7 28.I 33.0o 40.6 5I.i 37 38 58.8 48.7 4o.4 34,0 29.6 27.4 27.4 29.81 34.8 42.4 53.0 38 39 6o.31 50.2 4i.9 35.6 3I.3 29.0 29.1 3i.5 36.5 44.2 54.9 39 40 326I.7 3351.7 3443.5 3537.i 3632.8 3730.7 383o.8 3933.2 4o38.3 4I46.I 4256.7 40 41 63.2 53.2 45.o 38.7 34.4 32.3 32.4 35.01 4o. I 47.9J 58.6 41 42 64.7 54.7 46.6 4o.3 36.I 34.o 34.I 36.7 4I.8 49.7 6o.5 42 43 66.2 56.2 48.I 4I.9 37.7 35.6 35.8 38.4 43.6 5i.5 62.3 43 44 67.7 57.8 49.7 43.5 39.3 37.3 37.5 40.2 45.4 53.4 64.2 44 45 69.2 59.3 5I.2 45.o 4o.9 38.9 39.2 41.9 47.2 55.2 66.I 45 46 70.7 6o.8 52.8 46.6 42.5 40.6 40.9 43.6 49-.o 57.0I 68.o 46 47 72.Ij 62.3 54.3 48.2 44.1 42.2 42.6 45.4 50.71 58.8 69.8 47 48 73.6[ 63.8 55.8 49.8 45.8 43.9 44.3 47 I 52.5 60.7 71.7 48 49 75.I 65.4 57.4 5I.4 47.4 45.5 46.o 48.8 54.3 62.5 73.6 49 50 3276.6 3366.9 3458.9 3553.o 3649.o 3747.2 3847.7j395o.6 4o56.I 4I64.3 4275.5 5o 5I 78.1 68.4 6o.5 54.6 5o.6 48.8 49.4 52.3 57.8 66.I 77.4 5I 52 79.6 69.9 62.o 56.i| 52.2 5o.5 51.1 54.o 59.6 68.0 79.2 52 53 8I.1 71.4 63.6 57.7 53.8 52.I 52.8 55.8 6i.4 69.8 8I.i 53 54 82.6 73.o 65.2 59.3 55.5 53.8 54.4 57.5 63,2 7I.6 83.o 54 55 84.I 74.5 66.7 60.9 57.1I 55.5 56.I 59.3 65.o0 73.5 84.9 55 56 85.6 76.o 68.3 62.5 58.7 57.x 57.8 6I.o 66.8i 75.3 86.8 56 1 57 87.I1 77.5 69.8 64.o 6o.3 58.8 59.5 62.7 68.5 77.I1 88.6 57 58 88.5 79.0 7I.4 65.6 6I.9 6o.l 6I.2 64.5 70.3 7Q.cg 9o.5 58 59 9o.o 8o.6 73.0 67.2 63.6 62.1 62.9 66.2 72.II 80.8 92.4 59 MERILIONAL PARTS. 147 I"LATITUDE. 580 590 600 61J 620 630 640 650'660 670 680 Min. 0 14294.3 449.I 4527.4 464-,.2 4775049049 5o39.4578.85323.55474.o0563o.8 o [1 96.2 40.I 29.4 5.3 771 07.1 41.7 8i.2 26.0 76.6 33.5 i 98. i 3.o 3i.4 53.4 79.3 09.4 44.o 83.5,,8.4 79.1 36.2 2 3 43oo.o i5.o 33.4 55.4 8.4.6 46.3 85.9 3o.9 81.7 38.8 3 4 o0.9 16.9 35,4 N15 83.5 {3.8 48.6 88.3 33.4 84.3 4x.5 4 5 03.7 I8.9 37.1 59.6 85.6 i6.o 5o.9 90.7 35.8 86.9 44.2 5 6 o5.6 20o.8 39.4 6b.6 87.8 18.2 53.i 93.1 38.3 89.4'46.9 6 7 07.5. 22.8 41.4 63.7 89.9 20.4 55.4 95.4 4o.8 92.o0 49.6 7 8 09.4 24.7 43.4 65.8 92.11 22.6 57.7 97.8 43.2 94.5 52.3 8 9 11.3 26.7 45.4 67.8 94.2 24.8 6o.o 5200.2 45.7 97.1 54.9 9 io 4313.2 4428.6 4547.4 4669.9 4796.3 4927.0 5062.3 5202.6 5348.2 5499.7 5657.6 io II 15.1 3o.6 49.4 72.0 98.5 29.2 64.6 04.9 50.7 5502.3 6o.3 ii 12 17.0 32.5 5i.4 74. 48oo.6 31.5 66.9 07.3 53.i o4.9 63.0 12 13 18.9 34.5 53.5 76.1 02.8 33.7 69.2 o09.7 55.6 07.4 65.7 13 I4 20.8 36.4 55.5 18.2 04.9 35.9 71.5 I 2. 58.i 0o.o 68.4 14 i5 22.7 38.4 57.5 8o.3 07.1 38.I 73.8 i4.5 6o.6 12.6 71. I 5 i6 24.6 4o.3 59.5 82.4 09.21 4.3 76. 16.9 63.i 15.2 73.8 i6 17 26.5 42.3 6i.5 84.5 11.4 42.6 78.4 19.3 65.6 17.8 76.5 I7 i8 28.4 44.2 63.5 86.5 43.5 44.8 80.7 2,,.6 68.0 20.4 79.2 18 19 30.3 46.2 65.6 88.6 I5.7 47.0 83.0 24.0 70.5 23.0 81.9 ig 20 4332.2 4448.2 4567.6 4690.7 4817.814949.2 5o85.3 5226.4 5373.o3 5525.6 5684.6 20 21 34.r 5o.i 69.6 92.8 20.0o 51.5 87.6 28.8 75.5 28.1 87.3 21 22 36.o 52.I 71.6 94.9 22.1 53.7 90o.0 31.2 78.0 30.7 90.0 22 23 37.9 54.I 73.6 97.0 24.3 55.9 92.3 33.6 8o.5 33.3 92.7 23 24 39.8 56.o 75.7 99.1 26.4 58.2 94.6 36.o 83.0o 35.9 95.5 24 25 4I.8 58.o 77.7 47o01.1 28.6 6o.4 96.9 38.4 85.5 38.6 98.2 25 26 43.7 60. 79.7 03.2 3o.8 62.6 99.2 4o.8 88.o 41.2 5700.9 9 26 27 45.6 61.9 81.7 o5.3 32.9 64.915ioi.5 43.2 go.5 43.8 o3.6 2'7 28 47.5 63.9 83.8 07.4 35.1 67.1 o3.8 45.7 93.0 46.4 o06.3 28 29 49.4 65.9 85.8 og.5 37.3 69.41 06.2 48.i 95.5 49.0 09.1 29 30 435i.314467.8 4587.8471.6.4839.4497I.6 5io8.5 525o.5 5398. 555i.6 5711.8 30 31 53.2 69.8 89.9 13.7 4I.6 73.8 io.8 52.9154oo.5 54.21 I4.5 31 32 55.1 71.8 91.9 i5.8 43.8 76.1 13.1 55.3 03.0o 56.8 17.3 32 33 57'1 73.7 93.9 17'9 45.9 78.3 15.5 57.7 o.5 59.4 20o.o0 33 34 59.0 75.7 96.0 20.0 48.i 8o.6 17.8 6o.I o8.i 62.1 22.7 34 35 60.9 77.7 98.0 22.1 5o.3 82.8 20. i 62.6 io.6 64.7 25.5 35 36 62.8 79.7 46oo.o 24.2 52.4 85. 22.4 65.o i3.o 67.3 28.2 36 37 64.7 8i.6 02.1 26.3 54.6 87.3 24.8 67.4 i5.6 69.9 30o.9 37 38 66.7 83.6 o4. i 8,4 56.8 89.6 27.1 69.8 I8.I 72.6 33.7 38 39 68.6 85.6 o6.i 30.5 59.o 91.8 29.4 72.2 20.6 75.2 36.4 39 40 4370.514487.6 60o8.2 4732.6 486i.i 4994.'iSi3i.8 52747 5423.2 5577.8 5739.2 40 41 72.4 89.6 10.2 34.7 63.3 96.3 34.I 77.1 25.7 8o.4 41.9 4' 42 74.3 91.5 12.3 36.8 65.51 98. 36.5 79.5 28.2 8 3.I 44.7 42 43 76.3 93.5 i4.3 39.0 67.715ooo.8 38.8 82.0 3o.8 85.7 47.51 43 44 78.2 95.5 i6.4 4I.i 69.9 o3.I 4i.i 84.4 33.3 88.4 50.21 44 45 8o.i 97.5 i8.4 43.2 72.0 o5.4 43.5 86.8 35.8 9gI.o 532.9 45 46 82.1 99.5 20.5 45.3 74.2 07.6 45.8 89.3 38.4 93.6 55.7146 47 84.o0 5oi.5 22.5 47.4 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 I2.2 5o.5 94.1 43.4 98.9 61.2 48 49 87.8 05.4 26.6 5i.6 8o.8 I4.4 52.9 96.6 46.o 56ox.6 64.o 49 50 4389.8 45o7.4 4628.7 J4753. 7 4883 i6.7 5i55.2 5299.o 448. 5604.2 766.8 o 51 9I.7 09.4 30.7 55.9 85.2 18.9 57.6 53oi.5 5i.o 06.9 69.5 5i 52 93.6 II.4 32.8 58.o 87.4 21.2 59.9 03.9 53.6 09.5 72.31 52 53 95.6 i3.4 34.8 60.i 89.5 23.5 62.3 o6.3 56.I 12.2 75.I 53 54 97.5 i5.4 36.9 62.2 91.7 25.8 64.6 o8.8 58.7 i4.8 77.9 54 55 99.4 I7.4 39.o0 64.4 93.9 28.0 67.0 11.2 6.2 i7.5 80.6 55 S6 440I.4 19.4 4x.o 66.5 96.1 3o.3 69.4 13.7 63.8 20.2 83.4 56 57 o3.3 21.4 43.o 68.6 98.3 32.6 71.7 i6. i 66.3 22.9 86.2 57 58 o5.3/ 23.4 45.I 70.7 49oo.5 34.9 74.'I 8.6 68.9 25.5 89.0 58 59 07.2 25.4 47.2 72.9 02.7 37.1 76.4 21.1 71.4 28.2 91.8 S9 148 M ERIDIONAL PARTS. LATITUDE. Min. 1 690 700 1 710 720 730 740'70 760 770 780 790 Mm. oj5794.65965.9 6145.716334.86534.4 6745.76970.372o.1 7467.2 7744.6 8457 0 I 97.4 68.8 48.8 38.I 37.8 49.4 74.2 14.2 71.7 49.4 5i.o 2 58oo. 71.8 5i.9 4i.3 41.3 53.0 78.1 i8.3 76.1 54.2 56.2 2 3 02.9 74.7 54.9 44.6 44.7 56.6 8I.9 22.5 80.6 59.0 61.5 3 4 o5.7 77.6 58.0 47.8 48.I 60.3 85.8 26.6 85.0 63.9 66.7 4 5 08.5 80.6 6i.i 51.1 51.6 63.9 89.7 3o.8 89.5 68.7 72.o 5 6 ii.3 83.5 64.2 54.3 55.o 67.6 93.6 35.o 94.0 73.5 77.3 6 7 14.2 86.4 67.3 57.6 58.5 71.2 97.5 39.1i 98.5 78.4 82.6 7 8 17.0 89.4 70.4' 6o.8 61.9 74.9 7001.4 43.3 7503.0 83.3 87.9 8 9 19.8 92.3 73.5 64. i 65.3 78.5 05.3 47.5 07.4 88.i 93.2 9 io 5822.6 5995.3 6176.6 6367.4 6568.8 6782.2 700o9.27251.77511.97793.o 8o98.5 io II 25.4 98.2 79'7 70.6 72.3 85.9 13.1 55.8 16.5 97.9 8io3.8 ii 12 28.2 6001oo.2 82.8 73.9 75.7 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 13,4 33.8 o7.1 89.0 80.4 82.6 96.9 24.8 68.4 30.0 12.6 19.9 14 i5 36.7 0o.o 92.I1 83.7 86.i 68oo.6 28.8 72.6 34.5 17.5 25.2 i5 i6 39.5 i3.o 95.2 87.0 89.6 04.3 32.7 76.8 39.1 22.4 3o.6 i6 17 42.3 i6.o 98.3 90.3 93.0 08.o 36.6 81.I 43.6 27.3 36.o 17 i8 45.i 18.9 6201.4 93.6 96.5 ii.6 4o.6 85.3 48.i 32.2 4I.3 i8 19 48.0 21.9 04.5 96.9 66oo00.0 15.4 44.5 89.5 52.7 37.2 46.7 19 20 585o.8 6024.9 6207.7 6400.2 660o3.5 6819.1 7048.5 7293.7 7557.3 7842. 8152.I 20 2I 5'3.6 27.8 io.8 o3.4 07.0 22.8 52.4 98.0 61.8 47.i 57.5 2 I 22 56.5 30.8 13.9 06.7 io.5 26.5 56.4 7302.2 66.4 52.0 62.9 22 23 59.3 33.8 17.0 1o.I 14.o 30.2 60.3 06.4 71.o 57.0 68.4 23 24 1 62.2 36.8 20.2 13.4 I7.5 33.9 64.3 10.7 75.5 61.9 73.8 24 25 65.0 39.8 23.3 16.7 2i.o 37.6 68.3 1i5.o 8o.i 66.9 79.2 25 26 67.8 42.7 26.5 20.0 24.5 41.3 72.2 19.2 84.7 71.9 84.7 26 27 70.7 45.7 29.6 23.3 28.0 45.I 76.2 23.5 89.3 76.9 90.I 27 28 73.5 48.7 32.7 26.6 31.5 48.8 80.2 27.7 93.9 81.9 95.6 28 29 76.4 5I.7 35.9 29.9 35.0 52.5 84.2 32.0 98.5 86.9 820I.1 29 3o 5879.2 6054.7 6239.0 16433.3 6638.5 6856.3 7088.2 7336.3 7603.2 7891.9 8206.6 30 31 82.1 57.7 42.2 36.6 42. 6o.o 92.2 4.6 07.8 96.9 12.1 31 32 85.0 60.7 45.3 39.9 45.6 63.8 96.2 44.9 12.4 7902.0 17.6 32 33 87.8 63.7 48.5 43.2 49.i 67.5 7100.2 49.2 17.0 07.0 23.1 33 34 90.7 66.7 51.7 46.6 52.6 71.3 04.2 53.5 21.7 12.0 28.6 34 35 93.6 69.7 54.8 49.9 56.2 75.0 08.2 57.8 26.3 17. 34.I 35 36 96.4 72.7 58.0 53.3 59.7 78.8 12.2 62.1 3i.o 22.1I 39.7 36 37 99.3 75.7 61.2 56.6 63.3 82.6 i6.3 66.4 35.6 27.2 45.2 37 38 5902.2 78.8 64.3 6o.o 66.8 86.3 20.3 70.7 4o.3 32.3 50o.8 38 39 oS.o 8i.8 67.5 63.3 70.4 90.1 24.3 75.1 45.0 37.3 56.3 39 4o 5907.9 6084.8 6270.7 6466.7 6673.9 6893.9 7128.4 7379.4 7649.7 7942.4 8261.9 4o 4' io.8 87.8 73.9 70.0 77.4 97.7 32.4 83.7 54.3 47.5 67.5 41 42 13.7 90.8 77.I 73.4 8i.o 69o0.5 36.4 88.i 59.0 52.6 73.1 42 43 i6.6 93.9 80.2 76.7 84.6 o5.3 40.5 92.4 63.7 57.7 78.6 43 944 9.;4 96.9 83.4 8o.i 88.2 o09. 44.5 96.8 68.4 62.8 84.3 44 45 22.3 99.9 86.6 83.5 97 12.9 48.6 7401.1 73.2 68.0 89.9 45.25.2~ ~ ~~ ~~~9- 713.2 68).o88.99' 46 25.2 6xo3.o 89.8 86.9 95.3 16.7 527 oS.5 77.9 73.1 95.5 46 47 28.1 o6.o 93.0 90.2 98.9 20.5 56.7 09.9 82.6 78.2183oi.i 47 48 3I.o og09.0 96.2 93.6 6702.5 24.3 60.8 14.3 87.3 83.4 06.7 48 49 33.9 12.1 99.4 97.0 o6.i.28.1 64.9 i8.6 92.1 88.5 12.4 49 5o 5936.8 6ii15.i 6302.6 6500.4 6709.7 6931.9 7169.0 7423.0 7696.8 7993.7/83i8.i 50 51 39.7 18.2 05.8 03.8 13.2 35.7 73.1 27.4 7701.5 98.9 23.8 51 52 42,6 21.2 09.1 07.2 i6.8 93.6 77.2 3i.8 o6.3 800oo4.o 29.4 52 53 45.5 24.3 12.3 io.6 20.4 43.4 81.2 36.2 II.I 09.2 35.1 53 54 48.4 27.3 15.5 14.o 24.0 47.2 85.3 40.6 15.8 i4.4 40o.8 54 55 51.3 30.4 18.7 I7.4 27.7 Si.i 89.5 45.I 20.6 I9.6 46.5 55 56 54.2 33.4 21.9 20.8 3i.3 54.9 93.6 49.5 25.4 24.8 52.2 56 57 57.2 36.5 25.I 24.2 34.9 58.8 97.7 53.9 30.2 30.0 58.o 57 58 6o.i 39.6 28.4 27.6 38.5 62.6 7201.8 58.3 35.0 35.2 63.7 58 59 63.o 42.6 3i.6 3i.o 42.1 66.5 o05.6 62.8 39.8 4o.5 69.4 59 I -~~~~~~~~___9. 5 CORRECTIONS TO MIDDLE LATITUDE. 14 14 Mid. DIFFERENCE OF LATITUDE. Mid. Lat. f1LalI Lat. 10 2"1 30 40 50 60 70 80 90 100j11 o120 130 140 150 160 170 180 190 20 0Lat. 0 I I I I I 1 01 2 3 5 7 9o2 i5 I8S222631 36419475259 65 72 I5 i6 0 1 2 3 46 6 9 I941 82 2I125 3 0[ 3439s3944 15056{ 62 69.1 o 17 0 I 2 3 4 6 8 II I4 7 20 24 28 33 38 43 48 54 6o 66 17 08 o I 1 3 4 6 8 Io 13 i6 20 23 27 32 36 41 46 52 58 64 18 9g o I I 3 4 6 8 Io 13 i6 19 22 26 30 35 4o 45 5o 56 6i i3 20 I 4 5 7 1 I 2 15 1822 25293438 43 48 54 60 20 21 o I I 2 4 5 7 9 I2 15 8 21 25 293 33 37 42 47 52 58 21 22 0 I I 2 4 5 7 9121 41 721 2 28 32 3614I46 o 5622 23 0 I 1 2 3 5 7 9 I, 14 17 20o23 27 3i 35 4o 45 So 55 23 24 0 I I 2 3 5 7 II 14 i6 20 23 27 31 35 39 44 49 54 24 25 o i I 2 3 5 7 9 I 131i6 I 923 26 30o34 39 43 48 53 25 26 0 I i 2 3 5 6 8 II 13 i6 19 22 26 30343842 47 52 26 27 0 I I 2 3 5 6 8 i i13 i6 19 22 25 29 33 37 42 47 52 27 28 0 I I 2 3 5 6 8 Io i3 i6 I8 22 25 29 33 37 4i 46 5I 28 290 I 1 2 3 5 6 8 1o 01315 182 1 25 28 32 37 41 46 51 129 3o 0 I I 2 3 5 6 8 Io B3 i5 I8 21 25 28132 36 4i 45 5o 3o 3i 0 I I 2 3 5 6 8 10 12 i5 I8 21 24 28 32 36 4o 45 So 3i 32 0 0 I 2 3 4 6 8 10 12 iS 1821 24283236 4o 45 So 32 33 0 0 I 2 3 4 6 81012152 1821 24 28 32 36 4o 45 491i33 34 0 0 I 2 3 4 6 8 10 12 15 i8 21 24 28 32 36 4o 45 49 34 35 0 0 i 2 3 4 6 8 10 I12 i i8 21 24 28 32 36 4o 45 49 35 36 0 I I 2 3 4 6 8 Io 1i2 i I8 21 24 28 32 36 4o 45 49 36 37 0 I I 2 3 4 6 8 io 12 i5 18 21 24 28 32 36 4o 45 49 37 38 0 I i 2 3 4 6 8 1o 12 i5 I8 21 24 28 32 36 4o 45 So 38 39 0 I I 2 3 4 6 8 IO 12 i5 I8 21 24 28 32 36 4o 45 So 39 4o 0 i I 2 3 5 6 8 Io i3 i5 I8 2I 25 28 32 36 4i 45 Sol 4o 41 0 Io i I — 3 5 6 8 Io0I3 I IS8 2I 25 28 32 3741I 46S 51 4I 42 0 1 I 2 3 56 81 ib o131i51 I8 22 25 29 33 37 41 46 511 42 43 o I I 2 3 5 6 8 Io0 i3 i6 iS 22 25 29 33 37 42 46 521 43 44 0 I I 2 3 5 6 81 Io013161 9g22 25 2933 38 42 47' 52 44 45 0 1 I 2 3 5 6 8 ii I3 i6 Ig 22 26 30 34 38 43 48 53 45 46 0 I I 2 3 5 6 8 Ii I3 i6 19 22 26 30 34 38 43 48 53 46 07 0 I 1 2 3 5 7 9 I I13 i6 19 23 26 30 35 39 44 49 54 47 48 0 1 I 2 3 5 7 9 II I4 17 20 23 27 31 35 4o 44 So 55 48 49 1o, I i 2 3 5 7 91II 14 I7 20 23 2 7 3i 36 40 45 So 56 49 5o o I I 2 4~ 5 7 9 1 I11I4'I 720 24 28 32 36[ 4i 46 5 I 57 5o Si 0.1 I 2 4 5 71 9 21241 I7 21 24 28 32 37 42 47 52 58 5x 52 O i I 2 4 5 -7. 9 12 iS I8 21 25 29 33 38 43 48 53 59 52 53 0 I I 2 4 5 7 10 12iS i8 21 25 29 34 38 43 49 54 Ch 53 54 o1I 1 2 4 5 77I2O 12I852 82 226 3o 34 39 44 5o 56 6'-. 54 55 0 I 1 2 4 6 810Io13 1 6 1922 26 3 135 40o45 51 57 63 55 36 0 I I 3.4 6 8 io 13 i6 19 23 27 31 36 4i 46 52 58 65 53 57 o 1 I 3 4 6 8 io 13 3i6 20 24 28 32 3.7 42 48 54 6o 66 57 58 0 I 2 3 4 6 8 I i4 17 20 24 28 33 38 43 49 55 6i 68 58 59 0 1 2 3 4 6 8 I I i4 17 2I125 29 34 39 45 So 57 63 70 59 6o 0 I 2 3 4 6 9 I I 14 i8 22 26 3o 35 4o 46 52 58 65 72 6o 6i 0 I 2 3 5 791 2 215 1 I8. 2 22 26 31 36 42 47 53 60o67 75 61 62 0 I12 3 5 7 9 I2 i5 I9g. -32.27 32 37 43 49 55 62 70 77 62 63 0 I 2 3 5 7 10 12 i6 20 24:-, 2 83 339 44 S I57 64 72 8o 63 64 0 I 2 3 5 7 10 i3 i6 20 24 29 344o 46 52 59 67 75 83 64 65 0 I 2 3 5 7 io B3 17 21 25 3o 36 4i 48 54 62 69 78 86 65 66 0 1 2 3 5 811141 8I 2 26 32 37 43 50o57 64 72 8i go 66 67 I 2 4 6 8 xii4 i8 23 28 33 39 45 52 59 67 76 85 94 67 68 0 1 2 4 6 8 12 iS 19 24 29 34 4o 47 54 62 70 79 89 99 68 69 o I 2 4 6 9 12 i6 20 25 3o 36 42 49 57 65 74 83 93 io4 69 70 0 1 2 4 6 9 13 16 21 26 32 3844 52 60o68 78 88 98110l 70 7 I I 0 4 7 Io 13 I7 22 27 33 4o 47 55 63 72 82 93 io4 ii6 71 72 o01_13_5 7Io I4_i8 23 29 35142 49158I671761871981ii.u124 72 01-~~~~~~~~~~~~~~~~3,45 1 5( LOGARITHMS FOR COMPUTING COMPOUND INIEREST, In computing compound intere st for long periods of time, it is necessary to have the following logarithms to more than six places. Number. Logarithm. Number. Logarithm. 1.0025.ooI08 438i3 1.0425 o1I807 60636. oo5o.002I6 60618 1.o45o.0 1911 62904 I.o075.oo0324 50548 1.0475.02015 40316 I.or25.00539 50319 I.0525.02222 21045 I.o5o.00646 60422 I.o55o.02325 24596 1.oI75.00753 44I79 1.o575-.02428 03760 1.0200.00860 0I7I8 i.o6oo.02530 58653 1.0225.00966 33I67 I.o625.02632 89387 I.0250.OI072 38654.0o650.0o2734 96078 1.0275.01178 18305 I.0675.02836 78837 I.o3oo.01283 72247 I.0700.0o7. 38 37777 1. o325.01389 006o3 I 1.0725.o3039 73009.o35o.OI494 o3498 I.o75o.o3I40 84643 1.o375.0I598 8I054.o775.03241 72788 x.o40oo.I0703 33393 I.o8oo.o3342 37555 NUMBERS OFTEN USED IN CALCULATIONS. Logarithmns. Circumference of a circle to diameter I Surface of a sphere to diameter.. - 3.145926 0.4975o Area of a circle to radius i.......... Area of a circle to diameter i -.7853982 9.895090 Capacity of a sphere to diameter I - -..=..._. ---.5235988 9.7i8999 Capacity of a sphere to radius I. -= 4.1 887902 0.622089.I 3.14I5926.-..................- 0.3183099 9.502850 Arc equal to radius expressed in degrees... -570.2957795 1.758123 Are equal to radius expressed in seconds.-..... = 206264".8 5.314425 Length of I degree in parts of radius. —--------- ---.0174533 8.24I877 Length of I minute in parts of radius - - =.000290gog 6.463726 Sine of I second. —.ooooo485 4.685575 Sine of 2 seconds -.... =.00000970 4.986605 Sine of 3 seconds.-.ooooI454 5.162696 Sine of 4 seconds................................=.0oooo939 5.287635 Sine of 5 seconds.-..-................. -=.00002424 5.384545 Sine of 6 seconds. --.000ooo2909 5.463726 Sine of 7 seconds.......oooo3394 5.530673 Sine of 8 seconds.=.0000.oooo3879 5.588665 Sine of 9 seconds..... =.oooo4363 5.639817 Base of Napier's system of logarithms.....= 2.7128I8 0.434294 Modulus of the common logarithms.. =.4342945 9.637784 360 degrees expressed in seconds -. = 1296000 6.II2605 24 hours expressed in seconds.-. _.:....... 864oo 4.936514 Number of feet in one mile.........=......... 5280 3.722634 THE END. 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