These are the expressions for each of the parts of a right angled triangle in terms of two others, because expressions for c' and c' would be exactly like (2) and (3). Substituting ▲, π —α, &c., for A', a', &c. in the above equations, they become

But these last equations are precisely what would be obtained by the application of Napier's rules, using the complements of b and c, and ▲- as the circular parts.

13. Napier's rules may be deduced as follows:

The following formulas have been derived in the foregoing pages for oblique angled triangles.

The above are but expressions of Napier's rules.

14. The case of solution treated at Art. 94, may be solved by Napier's Analogies. Thus if a, b and a be given, в may be calculated by the sin proportion, and c and c by the formulas

sin(a—b): sin } (a + b) : : tan † (A — B) : cot c
sin(A — B): sin } (A + B) : : tan} (a — b) : tan c

15. Napier's rules for the solution of right angled spherical triangles, though applicable to all cases, do not give results of that degree of accuracy which is some

* (2) and (3) are the same, and are derived from (1) by polar triangles.

+ Sine proportion, Art. 81.

+ Appendix II. See formula at top of p. 200.

times required, when the required part expressed by its sine is very small, or expressed by its cosine is very near 90°. The following formulas may in such cases be used.

changing p into a, and substituting the value of cos a, given by this last, we have

which is a formula to be employed, when в and c are given and a required.

II.

With the same data to find b use the formula

tanb =

✓ {tan [} (B — c) + 45°] tan [} (B +c) — 45°]}

derived from Napier's rules, which gives

from the formulas preceding (1) and (4) of Art. 12, App. I. and from

formula (42), Art. 15, App. I.

III.

The hypothenuse a and the side c being given to find the adjacent angle B, use the formula

V.

Finally, to obtain b when the opposite angle в and the hypothenuse a are given, we have, by Napier's rules,

16. The part of a spherical triangle determined by the proportion sin a sin b: sin A: sin в admits of a double value, since two arcs answer to the same sine; it becomes necessary, therefore, for us to inquire under what circumstances both these values are admissible, and how we may know which to choose when but one solution exists.

Referring to the fundamental formula (Art. 82), we have

in which expression we may remark that if cos b is numerically greater than either cos a or cos c, the second member must take the sign of cos b, consequently в and b must be of the same affection if sin b < sin a, or sin b < sin c, that is, an angl must be of the same species as its opposite side, if the sine of this side is less than the sine of either of the other sides.

But if cos b is numerically less than cos a, then whether the right hand member be + or - - will depend upon the magnitude of cos c, or cos c will have two values corresponding to + cos B, and cos B; hence an angle has two values, when the sine of its opposite side is greater than the sine of the other given side. In the proportion

sin A sin B:: sin a sin b

a being the required part, the nature of the arc b may be discussed, as in the preceding case.

By means of the polar triangles, we obtain from (2), in the same manner as at Art. 85, the formula

from which it follows, as in the foregoing case, that if cos B is numerically greater than cos A, B and b will be of the same affection. If cos B is numerically less than cos A, then both the values of b, given by the above proportion, will be admissible, for c may be determined so as to render cos b positive or negative. Hence any side will be of the same affection as its opposite angle, if the sine of this angle be less than the sine of either of the other angles; and the affection of the side b will be indeterminate if the sine of its opposite angle в be greater than the sine of the other given angle s.

17. In practice, the data for the solution of triangles are obtained by observation and measurement, and are liable to error from obvious and inevitable causes. It is true that from the great excellency of instruments, and the almost inconceivable accuracy of modern observation, these errors are extremely minute, yet in cases where precision is requisite, it becomes necessary to determine the effects which small errors in the data will produce upon the computed quantities, and to select the data and quæsita in such a manner that the given errors in the one shall entail the smallest possible on the other.

The principles of the Differential Calculus present an easy method for the purpose in question, and we shall here indicate the mode of proceeding, for the benefit of the student acquainted with that branch of mathematics.

Let us suppose that of the three data (for there are always three in the solution of a triangle), two have been obtained with sufficient accuracy, but the third x is liable to an error of a given amount, which we shall call h. Let u be the sought quantity. Two of the three data being considered constant, the sought quantity u may be considered as a function of the third x. The quantity z becoming x + h, let the quantity u become u', we have by Taylor's theorem,

u'-u is the error sought, and as his in practice very small, the higher powers of it may be neglected, and we may call

hence the following rule:

du

Multiply the given error by the differential coefficient of the sought quantity considered as a function of the given quantity liable to error, and the product will be the error in the sought quantity.

If two of the data be liable to given errors, the effect upon the sought quantity may be computed on similar principles, by considering the sought quantity as a function of the two data so liable to error, and differentiating it with respect to these two independent variables.

It is evident that the same method extends to the case where all the data are liable to given small errors. In this case the sought quantity is to be regarded as a fuuction of three independent variables, and its differential found as before.

EXAMPLE.

To determine the relation between the minute variations of the perpendicular side of a plane right angled triangle and the opposite angle, the remaining perpendicular side being considered constant.

Let c and c be the side and angle which are subject to variation, and the constant side. Then (Art. 41),

which is the multiplier of the given small variation in c to obtain that in c.

The theory of maxima and minima as explained in the Calculus, will here admit of an important and easy application, viz., to find under what circumstances u' — u will be least on the supposition of a given variation h in the variable datum, or in du dx

other words, under what circumstances the function will be a minimum.

17. The effects of small errors may be obtained, but with less brevity and elegance, without the aid of the differential calculus.

The following are specimens of the mode of proceeding.

PROBLEM I.

In a right-angled triangle one of the oblique angles being given, to determine the variation of the opposite side, arising from a small variation in the hypothenuse. Let c be the angle which does not change, c its opposite side, and a the hypothenuse; then dc denoting the variation of c, and da of a

sin e sin c sin a

sin (cdc)=sin c sin (a + da)

.. by subtraction,

sin (c + dc) — sin c=sin c {sin (a+da) — sin a} ;

that is (Art. 74, formula 4),

2 cos (cdc) sin dc 2 sin c cos (a + da) sin ✈ da
sin c cos (a+da) sin da

... sindc=

cos (c+dc

which variation will be the least possible when cot a is least, or when a=90°. If we restore theda which has been neglected, and write the above result thus:

dc tan c cot (a + } da) da ;

then, in the case of a 90°, the expression becomes

dc tan c tanda da;

or, considering the very small arc da to be equal to its tangent, we have in the case supposed