554. To test the error involved in the process of calculating the sines of multiples of 10".

Whatever error was originally involved in the value used for sin 10" is increased n. fold in calculating sin n. 10′′.

For the error arising from the term (2 – 2 cos a) sin na is less than 10-8 (the error in sin na) and may, therefore, be neglected. Assuming then that, if the error in sin a is 8, the errors up to sin na are multiples of 8, we have

sin (n + 1) a = 2 sin na – sin (n − 1) a

in which the error is clearly 2nd − (n − 1) 8, i.e. (n + 1) 8.

Thus, by induction, we have, universally, the error in sin na is n times the error in a.

Let us, then, use the value of sin 10" which is correct to within 10-13. Then the error in 60° would be less than 60 × 60 × 6 × 1. 10-13, i.e. less than 1.10-9. Hence the above value will enable us to calculate the sines up to 60° correctly to the first 8 decimal places.

555. When the sines of angles up to 60° have been calculated, the remaining sines may be found by simple addition from the formula

(1) sin (60° + 4) – sin (60° - A) = 2 cos 60° sin A = sin A. Or, stopping short at 45°, the remaining sines may be found by approximating to the value of /2 from the formula

(2) sin (45° + A) – sin (45° – A) = 2 cos 45° sin A = √2. sin A. Or, stopping short at 30°, the remaining sines up to 60° may be found by approximating to the value of 3 from the formula

(3) sin (30° + A) — sin (30° - A) = 2 cos 30° sin A = √3. sin A. After this we may use formula (1) to find the remaining sines.

556. Having found the sines, the other ratios may be found from the following:

cos A = sin (90° — A),

tan A = sin A÷cos A and tan (45° + A) – tan (45° – A) = 2 tan 24.

tan (90° – A),

557. The accuracy with which the calculations are performed may be tested by

(1) The known values of the ratios of certain angles (as proved in Chap. IV.), the surd expressions in which may be evaluated.

(2) The known values of the ratios of certain differences in the angles.

Thus since the ratios of 3° are known from those of 18° and 15°, the ratios of angles differing by 3° may be successively found independently.

(3) The following two so-called Formula of Verification, viz. :

sin (36° + A) -- sin (36° — A) — sin (72° + A) + sin (72° — A) = sin A. cos (36° + A) + cos (36° – A) – cos (72° + A) – cos (72° – 4) = cos A.

558. The method of constructing the tables of the logarithms of given numbers has been explained in Chapter XVIII. Having found the ratios of angles as above, we may, therefore, use the table of logarithms to find the logarithms of the ratios. But, by the method explained in the following article, we may find the logarithms of the ratios without using the table of the ratios.

559. To find the logarithm of the cosine of any angle. In order to calculate logarithms we require to use factor formulæ. Thus we have

The logarithms of the first one or two factors may thus be found from the table of logarithms; those of the remainder should be found from the formula

Thus after finding the logarithms of the first two factors by the tables, we have for the remainder to subtract

These series converge with sufficient rapidity for purposes of calculation.

560. After calculating the logarithms of any one ratio, to find the others requires no more complex operation than mere subtraction.

§ 2. THE THEORY OF INTERPOLATION.

561. We shall now examine the basis and limits of the theory of Proportional Differences, which is applied to interpolate values of quantities intermediate between two tabulated values. The general nature of the discussion may be thus indicated.

Let f(x) be some function of x, whose values are tabulated to an assigned degree of accuracy for successive values of a differing by an assigned amount. We shall show that to a certain determinable degree of accuracy

f(x+d)-f(x)= dx (some function of x)

when d does not exceed a certain amount.

The coefficient of d in the above equation is independent of d, but dependent upon x and upon the form of the function f. It is

called the Differential Coefficient of f(x) with respect to x; and is written f(x). Thus approximately

f(x + d) − f(x) = d⋅ ƒ1 (x).

The degree of accuracy of the above equation increases as d decreases. The general object of the discussion is to determine with what degree of accuracy it holds in any case.

562. After showing how far the equation

ƒ (x + d) − ƒ (x) = d. f1 (x)

holds true in any case, we may, by varying d and fixing x, write

If then d' is the amount by which the successive values of x in the tables differ, this equation gives us either

(1) f(x+d)-f(x) in terms of d,

or (2) d in terms of ƒ (x + d) − ƒ (x).

Though these two problems are theoretically identical, yet they require a different discussion when applying the theory to interpolate in the tables. For, firstly, the tabulated differences in x are constant so that those in ƒ(x) are variable; and, secondly, the accuracy with which f(x) is determined for any value of x is constant, but that with which x is determined for

of f(x) is variable.

any value

In consequence of this distinction, it will be found that ƒ (x + d) must be expanded in powers of d, not x + d in powers of ƒ (x + d) −ƒ (x).

563. To find the difference of the logarithms of two numbers approximately in terms of the difference of the numbers.

In this series the terms are continually decreasing and alternate in sign.

d

Hence, if is not greater than 10-", the equation

n

log (n + d) — log n = μ

holds as far as 2p decimal places.

d

n

564. To apply the logarithmic-difference formula to the interpolation of the required logarithm of a given number.

Tables are published of the logarithms of all integral numbers from 10000 to 100000. Hence, if we require the logarithm of a fraction lying between two such integers, the ratio of d to n is never greater than 10-4. Hence, if the logarithms were tabulated to 8 decimal places, we might interpolate in such a table, correctly to 8 decimal places, the logarithms of any fractional numbers by means of the formula

log (nd) - log n d

log (n+1)-log n ̄ Ï'

565. Conversely, to apply the logarithmic-difference formula to the interpolation of the required number corresponding to a given logarithm.

The logarithms are tabulated to 7 decimal places. In the difference formula, put d = 1. Thus