3. To construct the line of rhumbs.*

Divide the arc B E into 8 equal parts, which mark with the figures 1, 2, 3, 4, 5, 6, 7, 8; and divide each of those parts into quarters; on B, as a centre, transfer the divisions of the arc to the chord BE, which, marked with the corresponding figures, will be a line of rhumbs.

4. To construct the line of sines.j

Through each of the divisions of the arc AD draw right lines parallel to the radius AC; and CD will be divided into a line of sines, which are to be numbered from C to D for the right sines; and from D to C for the versed sines. The versed sines may be continued to 180 degrees, by laying the divisions of the radius CD from C to E.

5. To construct the line of tangents.\

A rule on C, and the several divisions of the arc AD, will intersect the line DG, which will become a line of tangents, and is to be figured from D to G with 10, 20, 30, 40, &c.

6. To construct the line of secants.

The distances from the centre C to the divisions on the line of tangents, being transferred to the line CF from the

* Rhumbs here are chords, answering to the points of the Mariners' Compass, which are 32 in th^ circle.

†The sine of an arc is a right line, drawn from one end of an arc perpendicular to the radius, drawn to the other end. The versed sine is the part of the radius, included between the arc and its sine.

The tangent of an arc is a right line, touching that arc at one end, and terminated by a secant, drawn through the other end. § The secant of an arc is a right line drawn from the centre through one end of the arc, and terminated by the tangent, drawn from the other end.

centre C, will give the divisions of the line of secants; which must be numbered from A toward F with 10, 20, 30, &c.

7. To construct the line of semitangents, or the tangents of half the arcs.

A rule on E, and the several divisions of the arc AD, will intersect the radius CA, in the divisions of the semi or half tangents; mark these with the corresponding figures of the arc AD.

The semitangents on the plane scales are generally continued as far as the length of the rule, on which they are laid, will admit; the divisions beyond 90° are found by dividing the arc AE like the arc AD, then laying a rule by E and these divisions of the arc AE, the divisions of the semitangents above 90 degrees will be obtained on the line CA continued*

8. To construct the line of longitude.

Divide AH into 60 equal parts; through each of these divisions parallels to the radius AC will intersect the arc AE in as many points; from E, as a centre, the divisions of the arc EA, being transferred to the chord EA, will give the divisions of the line of longitude.

The points thus found on the quadrantal arc, taken from A to E, belong to the sines of the equally increasing sexagenary parts of the radius; and those arcs, reckoned from E, belong to the cosines of those sexagenary parts.

9. To construct the line of latitudes.

A rule on A, and the several divisions of the sines on CD» will intersect the arc BD, in as many points; on B, as a Centre, transfer the intersections of the arc BD, to the right line BD; number the divisions from B to D with 10, 20, SO, &c. to 90; and BD will be a line of latitudes.

10. To construct the line of hours.

Bisect the quadrantal arcs BD, BE, in a, b; divide the quadrantal arc ab into 6 equal parts, which gives IS degrees for each hour; and each of these into 4 others, which will give the quarters. A rule on C, and the several divisions of the arc ab, will intersect the line MN in the hour, &c. points, which are to be marked as in the figure.

11. To construct the line of inclination of meridians*

Bisect the arc EA in c; divide the quadrantal arc be into 90 equal parts; lay a rule on C and the several divisions of the arc be, and the intersections of the line HM will be the divisions of a line of inclination of meridians.

The use of these several lines will appear in the subse quent parts of the work.

PLANE TRIGONOMETRY.

PLANE Trigonometry teaches the relations and calcu

lations of the sides and angles of plane triangles.

The angles of triangles are measured by the number of degrees, contained in the arc cut off by the legs of the angle, and whose centre is the angular point. A right angle is therefore an angle of 90 degrees; and the sum of the three angles of every triangle, or two right angles, is equal to 180°. Wherefore, in a right-angled triangle, one acute angle being subtracted from 90°, the remainder will be the other; and the sum of any two angles of a triangle, being taken from 180°, will leave the third angle.

Degrees are marked at the top of the figure with a small ", minutes with ', seconds with ", and so on. Thus, 57° 30' 12", that is, 57 degrees, 30 minutes, and 12 secopds.

The complement of an arc is the difference between that arc and a quadrant. So BC=40° is the complement of AB =50°.

The supplement of an arc Is what it wants of a semicircle. So BCD=180° is the supplement of AB=50°.

Ала

The sine of an arc is the line, drawn from one end of the arc perpendicularly upon the diameter, drawn through the other end of the arc. So BE is the sine of AB or of BCD.

The versed sine of an arc is the part of the diameter between the sine and the beginning of the arc. So AE is the versed sine of AB, and DE the versed sine of BCD.

The tangent of an arc is the line, drawn perpendicularly from one end of the diameter passing through one end of the arc, and terminated by the line, drawn from the centre through the other end of the arc. So AG or DK is the tangent of AB, or of BCD.

The secant of an arc is the line, drawn from the centre through the end of the arc, and terminated by the tangent. So FG or FK is the secant of AB, or of BCD.

The cosine, cotangent, or cosecant, of an arc is the sine, tangent, or secant of the complement of that arc. So BH is the cosine, CI the cotangent, and FI the cosecant of AB.

From the definitions it is evident, that the sine, tangent, and secant, are common to two arcs> which are the supple