The First Book of the Cross-Staff

CHAPT. VII. The use of the Line of Artifcial Sines.

This Line of Sines hath such use in finding a fourth Proportional, as the ordinary Canon of Sines; and the manner of finding it is alway such, as in this example.

• As the Sine of 90 gr. unto the Sine of 30 gr.
• So the Sine of 20 gr. unto the fourth Sine.

Extend the Compasses from the Sine of 90 gr. unto the Sine of 30 gr. the same extent will reach from the Sine of 20 gr. unto the Sine of 9 gr. 50 m.

Or you may extend them from the Sine of 90 gr. unto the Sine of 20 gr. the same extent will reach from the Sine of 30 gr. unto the Sine of 9 gr. 50 m. and such is the fourth proportional required.

In the like manner if the question proposed were,

• As the Sine of 30 gr. unto the Sine of 52 gr.;
• So the Sine of 38 gr. unto a fourth Sine.

Extend the Compasses in the Line of Sines from 30 gr. unto 52 gr. the same extend shall give the distance from 38 gr. unto 76 gr. Or, extend them from 30 gr. unto 38 gr. the same extent will reach from 52 gr. unto 76 gr. which is the fourth proportional Sine required.

And thus may the rest of all Sinical proportions be wrought two ways. The minutes which are wanting in the first degree of the Sines may be supplied by the Line of Numbers, as I shew in the next Chapter.