Page:The Construction of the Wonderful Canon of Logarithms.djvu/101

NOTES ON TRIGONOMETRICAL PROPOSITIONS. 77

logarithm add, as before, 693147, and the sum, that is the number remaining since the sines are contrary, will be 638826, half of which, 319413, is the logarithm of 46° 36′ 0″. The arc of the given logarithm is therefore 93° 12′ 0″, the versed sine of which is 10558216, and since this is greater than radius it has a negative logarithm, namely —54321.

Demonstra- tion
c	b	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$	versed sine of arc	$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	c	d
a	b				a	d

x	c	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$	cont.	x	a	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$	cont.	½x c, sine of 30° 0′	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$	cont.
c	g		pro-	a	e		pro-	c g, sine of ½ arc c d		port.
c	h		port.	a	f		port.	c b, double of line c h		port.

Of the spherical triangle A B D ]

In finding the base we may pursue another method, namely:—

Add the logarithm of the versed sine of the given angle to the logarithms of the given sides, and the sum will be the logarithm of the difference between the versed sine of the difference of the sides and the versed sine of the base required. This difference being consequently known, add to it the versed sine of the difference of the sides, and the sum will be the versed sine of the base required.

For example, let the sides be 34° and 47’, their loga-

rithms