But 2bc-a2-b2-c2 — a 2 —— (b —c)2=(a+b—c) (a−b+c)

(Alg. 235.)

Putting then h=(a+b+c), reducing, and extracting;

Ex. Given a, b, and c, as before, to find A and B.

By Art. 210, 2 Cos? ¿AR2+R×cos A.

Substituting and reducing, as in the demonstration of the

last theorem,

2bc+b+c2-α3 (b+c+a)(b+c—a)

2 Cos2A=R2x

2bc

2bc

Putting h=(a+b+c), reducing and extracting,

Ex. Given the sides 134, 108, SO; to find B and C.

The tangent is equal to the product of radius and the sine, divided by the cosine. (Art. 216.) By the last two theorems,

then,

Ex. Given the sides as before, to find A and C.

The three last theorems give the angle required, without ambiguity. For the half of any angle must be less than 90°. Of these different methods of solution, each has its advantages in particular cases. It is expedient to find an angle, sometimes by its sine, sometimes by its cosine, and soinetimes by its tangent.

By the first of the four preceding theorems, marked C, D, E, and F, the calculation is made for the sine of the whole

angle; by the others, for the sine, cosine, or tangent, of half the angle. For finding an angle near 90°, each of the three last theorems is preferable to the first. In the example above, A would have been uncertain to several seconds, by theorem C, if the other two angles had not been determined also.

But for a very small angle, the first method has an advantage over the others. The third, by which the calculation is made for the cosine of half the required angle, is in this case the most defective of the four. The second will not answer well for an angle which is almost 180°. For the half of this is almost 90°; and near 90°, the differences of the sines are very small.

NOTES.

NOTE A. Page 1.

THE name Logarithm is from λόγος, ratio, and ἀριθμὸς, 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. 7.

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. 43.

To revert a series, of the form

x=an+bn2+cn3+dn1+ en3 +, &c.

that is, to find the value of n, in terms of x, assume a series, with indeterminate co-efficients, (Alg. 490. b.)

=

Let n Ax+Bx2+Cx3+Dx4+Ex+, &c.

Finding the powers of this value of n, by multiplying the series into itself, and arranging the several terms according to the powers of x; we have