The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.

PROBL. 3.

To work proportions in Sines alone, by the Lines of natural Sines on the proportional side of the Instrument.

The general rule is this. Account the first term upon the Lines of Sines from the Cen∣ter, and enter the second term in the first so accounted,keeping the Sector at that Gage, account the third term on both lines from the Center, and taking the distance from the third term on one line to the third term on the other line, measure it upon the line of Sines from the beginning, and you have the fourth term. Ex. gr.

As the Radius is to the Sine 30, so is the Sine 40 to the Sine 18. 45.

There is but one exception in this Rule, and that is when the second term is greater than the first; yet the third lesser than the first, and in this case transpose the terms, by Chap 3. Probl. 3. Case 3.

But when the second term is not twice the length of the first, it may be wrought by the general Rule without any transposition of terms. Ex. gr.

As the Sine 30 is to the Sine 50, so is the Sine 20 to the Sine 31. 30. min.

And by consequent, when the third term is greater than the first, provided it be not upon the line, double the length thereof, it may be wrought by transposing the terms, although the second was twice the length of the first. Ex. gr.

As the Sine 20 is to the Sine 60, so is the Sine 42, to what Sine? which transposed is,

As the Sine 20 is to the Sine 42, so is the Sine 60 to the Sine 35. 30.

This case will remove the inconveniency mentioned, Chap. 3. Probl. 3. Case 4. of a double Radius. I intended there to have ad∣joyned the method of working proportions by natural Tangents alone, and by natural Sines, and Tangents, conjunctly: But consi∣dering

the multiplicity of proportions when the Tangents exceed 45. I suppose it too troublesome for beginners, and a needless variety for those that are already Mathema∣ticians. Sith, both may be eased by the ar∣tificial Sines and Tangents on the outward ledge, where I intend to treat of those Cases at large, and shall in this place only annex some proportions in Sines alone, for the ex∣ercise of young beginners.