Page:The Construction of the Wonderful Canon of Logarithms.djvu/104

80 NOTES ON TRIGONOMETRICAL PROPOSITIONS.

						Logarithms.
Propor- tion	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$	Sine compl. ½ sum of angles	53°	11′	58″	2222368
		Sine compl. ½ diff. of angles	84°	18′	3″	49553
		Tangent ½ base	34°	30′	0″	3750122
		Tangent ½ sum of sides	40°	30′	0″	1577307

Another Example.

Let there be given

the angle 47°,
the sides containing it	$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	50°	35′	11″
		31°	6′	5″
‍		90°	41′	16″	sum.
		45°	20′	38″	half sum.
		44°	39′	22″	compl. of half sum.
		28°	29′	6″	difference.
		14°	14′	33″	half difference.
		28°	29′	6″	compl. of half diff.

						Logarithms.
Propor- tion 1.	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$	Sine compl. ½ sum of sides	44°	39′	22″	3526118
		Sine compl. ½ diff. of sides	75°	45′	27″	312192
		Tan. compl. ½ vert. angle	66°	30′	0″	−8328403
		Tan. ½ sum of angs. at base	72°	30′	0″	−11452329
Propor- tion 2.	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$	Sine ½ sum of sides	45°	20′	38″	3406418
		Sine ½ diff. of sides	14°	14′	33″	14023154
		Tan. compl. ½ vert. angle	66°	30′	0″	−8328403
		Tan. ½ diff. of angs. at base	38°	30′	0″	2288333

72°	30′
38°	30′	‍
111°	0′	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$	angles at the base.
34°	0′

And these relations are all uniformly maintained,

whether