SECTION VIII. COMPUTATION OF THE CANON. ART. 223. THE trigonometrical canon is a set of tables containing the sines, cosines, tangents, &c. to every degree and minute of the quadrant. In the computation of these tables, it is common to find, in the first place, the sine and cosine of one minute; and then, by successive additions and multiplications, the sines, cosines, &c. of the larger arcs. For this purpose, it will be proper to begin with an arc whose sine or cosine is a known portion of the radius. The cosine of 60° is equal to half radius. (Art. 96. Cor.) A formula has been given, (Art. 210,) by which, when the cosine of an arc is known, the cosine of half that arc may be obtained. By successive bisections of 60°, we have the arcs If the radius be 1, and if a=60°, b=30°, c=15°, &c. then cos b=cos a= √1+x=0.8660254 cos e=cos &d= √1+cos d=0.9978589 Proceeding in this manner, by repeated extractions of the square root, we shall and the cosine of COMPUTATION OF THE CANON. 1 109 0° 0′ 52" 44" 3" 45"""" to be 0.99999996732 And the sine (Art. 94.)= √1-cos2=0.00025566346 This, however, does not give the sine of one minute exactly. The arc is a little less than a minute. But the ratio of very small arcs to each other, is so nearly equal to the ratio of their sines, that one may be taken for the other, without sensible errour. Now the circumference of a circle is divided into 21600 parts, for the arc of 1'; and into 24576, for the arc of 0° 0′ 52′′ 44′′ 3′′'" 45" Therefore 21600: 24576 :: 0.00025566346: 0.0002908882. which is the sine of 1 minute very nearly.* And the cosine = √1-sin2=0.9999999577. 224. Having computed the sine and cosine of one minute, we may proceed, in a contrary order, to find the sines and cosines of larger arcs. Making radius =1, and adding the two first equations in art. 208, we have sin(a+b)+sin(a-b)=2sin a cos b If we put b=1', and a=1', 2', 3', &c. successively, we shall have expressions for the sines and cosines of a series of arcs increasing regularly by one minute. Thus, sin(1'+1')=2sin 1'x cosl'-sin 0 0.0005817764, cos(1'+1')=2cos 1'x cos 1'-cos 0=0.9999998308 The constant multiplier here, cos 1' is 0.9999999577, which is equal to 1-0.0000000423. *See note L. 225. Calculating, in this manner, the sines and cosines from 1 minute up to 30 degrees, we shall have also the sines and cosines from 60° to 90°. For the sines of arcs between 0° and 30°, are the cosines of arcs between 60° and 90°. And the cosines of arcs between 0o and 30°, are the sines of arcs between 60° and 90°. (Art. 104.) 226. For the interval between 30° and 60°, the sines and cosines may be obtained by subtraction merely. As twice the sine of 30° is equal to radius; (Art. 96.) by making a=30°, the equation marked I, in article 224 will be come sin(30°+b)=cos b—sin(30° —b) And putting b=1', 2′, 3′, &c. successively, (30° 2′)=cos 2'-sin(29o 58') (30° 3')=cos 3'—sin(29° 57′) &c. &c. If the sines be calculated from 30° to 60°, the cosines will also be obtained. For the sines of arcs between 30° and 45°, are the cosines of arcs between 45° and 60°. And the sines of arcs between 45o and 60°, are the cosines of arcs between 30° and 45°.* (Art. 96.) 227. By the methods which have here been explained, the natural sines and cosines are found. The logarithms of these, 10 being in each instance added to the index, will be the artificial sines and cosines, by which trigonometrical calculations are commonly made. (Art. 102, 3.) 228. The tangents, cotangents, secants, and cosecants, are easily derived from the sines and cosines. By art. 93, NOTES. NOTE A. Page 1. THE name Logarithm is from eyes ratio, and disuas number. Considering the ratio of a to 1 as a simple ratio, that of a to 1 is a duplicate ratio, of a3 to 1 a triplicate ratio, &c. (Alg. 354.) Here the exponents or logarithms 2, 3, 4, &c. show how many times the simple ratio is repeated as a factor, to form the compound ratio. Thus the ratio of 100 to 1, is the square of the ratio of 10 to 1; the ratio of 1000 to 1, is the cube of the ratio of 10 to 1, &c. On this account, logarithms are called the measures of ratios; that is of the ratios which different numbers bear to unity. See the Introduction to Hutton's Tables, and Mercator's Logarithmo-Technia, in Maseres' Scriptores Logarithmici. NOTE B. p. 4. If 1 be added to-.09691, it becomes 1-.09691, which is equal to +.90309. The decimal is here rendered positive, by subtracting the figures from 1. But it is made 1 too great. This is compensated, by adding -1 to the integral part of the logarithm. So that -2-.09691-3+.90309. In the same manner, the decimal part of any logarithm which is wholly negative, may be rendered positive, by subtracting it from 1, and adding -1 to the index. The subtraction is most easily performed, by taking the right hand significant figure from 10, and each of the other figures from 9. (Art. 55.) On the other hand, if the index of a logarithm be negative, while the decimal part is positive; the whole may be rendered negative, by subtracting the decimal part from 1, and taking 1 from the index. NOTE C. p. 8. It is common to define logarithms to be a series of numbers in arithmetical progression, corresponding with another series in geometrical progression. This is calculated to perplex the learner, when, upon opening the tables, he finds that the natural numbers, as they stand there, instead of being in geometrical, are in arithmetical progression; and that the logarithms are not in arithmetical progression. It is true, that a geometrical series may be obtained, by taking out, here and there, a few of the natural numbers ; and that the logarithms of these will form an arithmetical series. But the definition is not applicable to the whole of the numbers and logarithms, as they stand in the tables. The supposition that positive and negative numbers have the same series of logarithms, (p. 7.) is attended with some theoretical difficulties. But these do not affect the practical rules for calculating by logarithms. NOTE D. p. 38. According to the scheme lately introduced into France, of dividing the denominations of weights, measures, &c. into tenths, hundredths, &c. the fourth part of a circle is divided into 100 degrees, a degree into 100 minutes, a minute into 100 seconds, &c. The whole circle contains 400 of these degrees; a plane triangle 200. If a right angle be taken for the measuring unit; degrees, minutes, and seconds, may be written as decimal fractions. Thus 36° 5′ 49′′ is 0.360549. According to the French division 10° 9° 100' =54' 1000"-324" English. NOTE E. p. 44. In Fig. 6th, let the arc AD-a, and ADB=2a. Draw BF perpendicular to AH. This will divide the right angled triangle ABH into two similar triangles. (Euc. 8. 6.) The angles ACD and AHB are equal. (Euc. 20. 3.) Therefore |