144 PHILOSOPHICAL TRANSACTIONS. [ANNO 1781. 



.i., ^ .»., *., &c. or some multiple of it, as twice, thrice, &c. Or x may be the 

 square, or the square root, or any other power of the sine, tangent, secant, or 

 versed sine, of an arc; in every one of which cases the formulae will put on dif- 

 ferent appearances, either with respect to the powers or co-efficients of the un- 

 known quantity, and yet admit ot the same kind of application. 



5. The tables may be rendered yet more extensively useful by inserting ex- 

 pressions for the sines, cosines, and tangents, of half the arc which has x for its 

 sine, tangent, secant, or versed sine; and also for the sines, cosines, and tan- 

 gents, of the odd multiples of this half arc, which expressions, together with 

 those already inserted, may be considered as the sines, cosines, and tangents, of 

 the multiples of an arc, the unknown quantity being the sine, tangent, &c. of 

 twice that arc. And this consideration may sometimes be applied to very useful 

 purposes. 



6. In order to render the formulae in the tables more general, I have put r for 

 the radius of the circle; whereas it will frequently happen, that the equation, 

 finally resulting from the resolution of a problem, especially those which relate 

 to the docrine of the sphere, will present itself in a form where the radius must 

 be taken equal to unity: what these forms are will readily appear by substituting 

 unity for r and its powers every where in the expression. 



Exam- 1 . — Required to find the value of x in an equation of the form x 3 — 

 r*x = a. 



If r 2 be expounded by 50, and a by 1 20, the equation may be reduced to 

 \/x X "/ (x~ — 50) = ^/ \20; and, consequently, by the tables, if x be consi- 

 dered as the secant of an arc, of which the radius is */50, then \/(x 2 — 50) 

 will be its tangent, and we shall have to find an arc, such that the tangent mul- 

 tiplied by the square root of the secant may be equal to \/l20; or, which 

 amounts to the same thing, such an arc that the log. tang, together with half 

 the log. secant may be equal to half the log. of 120. But because the tangent 

 and secant, here required, are to the radius of the >/ 50, the log. tangents and 

 secants in the tables must be increased by the logarithm of that number, and 

 therefore log. tang. + + log. 50 + x log. secant -f- J- log. 50 = 4- log. 120: or 

 log. tang. + i log. secant = ± log. 120 — f log. of 50. Hence, having taken 

 -J- the log. of 50 from i the log. of 120, run the eye along the tables of loga- 

 rithmic tangents and secants till an arc be found of which the sum of the log. 

 tangent and -i- the log. secant is equal to ig. 7053631, the remainder. In this 

 manner it will be readily found, that the sum of the log. tangent and 4. the log. 

 secant of 28° 37' is less than that difference by 2012, and that the sum of the 

 log. tangent and ~ the log. secant of 28° 38' is greater than it by 1337 : there- 

 fore 3349 (2012 + 1337) : 60" :: 2012 : 36". The exact arc therefore, of which 

 the sum of the log. tangent and 4 the log. secant is equal to 1 9.765363 1 is 28° 



