Tract 16.

WITHOUT TABLES.

265

A

n

a a a

- P - |Pf— =— — 4p_|_ —

J #')’ ')7. J 54 H

3/- 480r s

- &c; and

a = » x (*p- |U=mx(V( 2-2^/(1_^) ) - nearly.

4 1

Coro/. 7—And again, since — x (p - ? — i? 3 = ^? 4 5&c ; where q is = by subtracting this from — --—=

1 7Z

&c, and reducing, there will be obtained a = y^X

(144p - 39q - iq l ) = ~ X (14 V 2 - 2 V (1 - ? »)) - 39q -which will commonly give the angle exact to within a minuteof the truth. Where note, that the constant quantity is

= '54567409. And from the whole may be drawn the fol-lowing general problem.

PROBLEM.

To perform all the Cases of Trigonometry without any Tables .

Having any three parts of a triangle given, except the threeangles, the other three parts may be found, by some of thefollowing six general theorems.

1. a = \nx (V( 2 -V(-^) ) - -f) nearly. Or

A = 1^5 X ( 144v/ ( 2 _ VC 1 -^))- 39 -^-§p) more ^arly. i

a a? 3 a s 3.5a 7 3 5.7 a 9

2. A — nx (~+ 2>ar 3 + 2.4.5 p 5+ 2.4.6.7r 7 + 2.4.6.8.9?' 9 &C '^

3 ’ a VX( 'n 2.3m 3 + 2.3.4.5m s 2.3.4.5.6.7m 7

&c.)

4. r =

i +

2.3.M 3 2.3.4.5w s 2.3.4.5.6.7 m 7

&c.