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 909. For the sines of arcs between 0° and 30°, are the cosines of arcs between 60o and 90°. And the cosines of arcs between 0° and 30°, are the sines of arcs between 60° and 90°. (Art. 104.)

226. For the interval between 30° and 60°, the sines and As twice cosines may be obtained by subtraction merely. 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,
sin(30° 1)=cos 1'-sin(29° 59')

(30° 2')=cos 2'-sin(29° 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 45° 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,

cos: R::R: sec
sin: R:: R: cosec

NOTES.

NOTE A. Page 1.

THE name Logarithm is from λόγος ratio, and αριθμός 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.

In Fig. 6th, let the arc AD-a, and ADB=2a. Draw BF perpendicular to AH. This will divide the right angled tri angle ABH into two similar triangles. (Euc. 8. 6.). The an gles ACD and AHB are equal. (Euc. 20. 3.) Therefore...

the four triangles ACG, AHB, FHB, and FAB are similar; and the line BH is twice CG, because BH: CG:: HA:CA.

The sides of the four triangles are,

A variety of proportions may be stated, between the homologous sides of these triangles: For instance,

By comparing the triangles ACG and ABF,

AC: AG:: AB: AF, that is, R: sin a:: 2sin a:vers 2a

4

R: cos a:: 2sin a: sin 2a

Sin a:cos a::vers 2a: sin 2a

AC:CG::AB: BF

AG:CG::AF: BF

By comparing the triangles ACG and BFH,

AC: CG:: BH: HF, that is, R:cos a:: 2cos a:vers. sup. 2a

AG:CG:: BF: HF

Sin a:cos a: sin 2a: vers. sup. 2a

That is, the product of radius into the versed sine of the supplement of twice a given arc, is equal to twice the square of the cosine of the arc.

And the product of the sine of an arc, into the versed sine of the supplement of twice the arc, is equal to the product of the cosine of the arc, into the sine of twice the arc, &c. &c.

NOTE F. p. 64.

The different methods of solving the same triangle, by making the different sides radius, may serve to verify each other. But they are not, in every instance, equally accurate. The differences in the sines of angles near 90°, and in the cosines of angles near 0°, are so small, as to leave an uncer tainty of several seconds in the result. Thus 9.9999998 is

P

given in Taylor's tables, as the cosine of every angle between 2' 52", and 3'41" and the sine of every angle between 89956! 19" and 89° 57' 8". But the differences in the tangents and cotangents are considerable, in every part of the quadrant. The method of solution should be varied, so as to avoid finding a very small angle by its cosine, or one near 90° by its sine. See the latter part of note H.

NOTE G. p. 69.

If the perpendicular be drawn from the angle opposite the longest side, it will always fall within the triangle; because the other two angles must, of course, be acute. But if one of the angles at the base be obtuse, the perpendicular will fall without the triangle, as CP; (Fig. 38.)

In this case, the side on which the perpendicular falls, is to the sum of the other two; as the difference of the latter, to the sum of the segments made by the perpendicular.

The demonstration is the same, as in the other case, except that AH-BP+PA, instead of BP-PA.

Thus in the circle BDHL (Fig. 38.) of which C is the centre,

{ ABXAH=ALX AD; therefore AB: AD:: AL: AH. But AD CD+CA=CB+CA

And AL-CL-CA=CB-CA
And AH-HP+PA=BP+PA

Therefore

AB: CB+CA :: CB-CA: BP+PA.

When the three sides are given, it may be known whether one of the angles is obtuse. For any angle of a triangle is obtuse or acute, according as the square of the side subtending the angle is greater, or less, than a right angle. (Euc. 12, 13. 2.)

NOTE H. p. 76.

SOLUTIONS OF TRIANGLES.

Any triangle whatever may be solved, by the theorems in section IV. But there are other methods, by which, in cer