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.

CHAP. IV.

Some uses of the Lines on the proportional side of the Instrument, viz. the Lines of natural Sines, Tangents, and Secants.

PROBL. 1.

To lay down any Sine, Tangent, or Secant to a Radius given. See Fig. 1.

IF you be to lay down a Sine, enter the Radius given in 90, and 90 upon the lines of Sines, keeping the Sector at that gage, set one point of your Compasses in the Sine re∣quired upon one line, and extend the other point to the same Sine upon the other Line: This distance is the length of the Sine requi∣red to the given Radius. Ex. gr. Suppose A. B. the Radius given, and I require the Sine 40. proportional to that Radius. Enter A. B. in 90, and 90 keeping the Sector at that gage, I take the distance, twixt 40 on one side, to 40 on the other, that is, C. D. the Sine required.

The work is the same, to lay down a Tan∣gent to any Radius given, provided you en∣ter

the given Radius in 45, and 45, on the line of Tangents. Only observe if the Tan∣gent required be less than 45. you must en∣ter the Radius in 45. and 45 next the end of the Rule. But when the Tangent required exceeds 45. enter the Radius given in 45, and 45 'twixt the center and end, and keeping the Sector at that Gage, take out the Tangent re∣quired. This is so plain, there needs no ex∣ample.

To lay down a Secant to any Radius given, is no more than to enter the Radius in the two pins at the beginning of the line of Secants; and keeping the Sector at that Gage, take the distance from the number of the Secant re∣quired on one side, to the same number on the other side, and that is the Secant sought at the Radius given.

The use of this Problem will be sufficient∣ly seen in delineating Dyals, and projecting the Sphere.

PROBL. 2.

To lay down any Angle required by the Lines of Sines, Tangents, and Secants. See Fig. 2.

There are two wayes of protracting an Angle by the Line of Sines, First if you use the Sines

in manner of Chords. Then having drawn the line A B at any distance of your Compass, set one point in B, and draw a mark to intersect the Line B A, as E F. Enter this distance B F in 30, and 30 upon the Lines of Sines, and keeping the Sector at that Gage, take out the Sine of half the Angle required, and setting one point where F intersects B A, turn the other toward E, and make the mark E, with a ruler draw B E and the Angle E B F is the Angle required, which here is 40. d.

A second method by the lines of Sines is thus, Enter B A Radius in the Lines of Sines, and keeping the Sector at that Gage, take out the Sine of your Angle required with that distance, setting one point of your Compas∣ses in A, sweep the ark D, a line drawn from B by the connexity of the Ark D, makes the Angle A B C 40 d. as before.

To protract an Angle by the Lines of Tan∣gents is easily done, draw B A the Radius upon A, erect a perpendicular, A C, enter B A in 45, and 45 on the Lines of Tangents, and taking out the Tangent required (as here 40) set it from A to C. Lastly, draw B C, and the Angle C B A is 40 d. as before.

In case you would protract an Angle by the Lines of Secants. Draw B A, and upon A erect the perpendicular A C, enter A B in

the beginning of the Lines of Secants, and take out the Secant of the Angle, with that distance, setting one point of your Compas∣ses in B, with the other cross the perpendi∣cular A C, as in C. This done, lay a Ruler to B, and the point of intersection, and draw the Line B C. So have you again the Angle C B A. 40. d. by another projection. These varieties are here inserted only to satisfie a friend, and recreate the young practitioner in trying the truth of his projection.

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.

PROBL. 4.

By the Lines of Natural Sines to lay down any Tangent, or Secant required to a Radius given.

In some Cases, especially for Dyalling, your Instrument may be defective of a Tan∣gent, or Secant for your purpose, Ex. gr. when the Tangent exceeds 76, or the Secant is more than 60. In these extremities use the following Remedies. First, for a Tangent.

As the cosine of the Ark is to the Radius given, so is the sine of the Ark to the length of the Tangent required. Secondly, for a Secant.

As the cosine of the Ark is to the Radius given, so is the Sine 90 to the length of the Secant required.

PROBL. 5.

The distance from the next Equinoctial Point gi∣ven to finde the Suns declination.

As the Radius to the sine of the Suns greatest declination, so is the sine of his di∣stance from the next Equinoctial Point to the sine of his present declination.

PROBL. 6.

The declination given to finde the Suns Equino∣ctial Distance.

As the sine of the greatest declination is to the sine of the present declination, so is the Radius to the sine of his Equinoctial Di∣stance.

PROBL. 7.

The Altitude, Declination, and Distance of the Sun from the Meridian given to finde his Azimuth.

As the cosine of the altitude, to the cosine of the hour from the Meridian, so is

the cosine of declination to the sine of the Azimuth.