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 radi usCA, 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 continu ed 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, 30, &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 a b into 6 equal parts, which gives 15 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 bc into 90 equal parts; lay a rule on C and the several divisions of the arc bc, 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 subsequent 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 seconds.

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=130° 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

ments of each other. So the sine, tangent, or secant of 50° is also the sine, tangent, or secant of 130°.

The sine, tangent, or secant, of an angle is the sine, tangent, or secant, of the arc, or the degrees, by which the angle is measured.

The sine, tangent, and secant of every degree and minute in a quadrant are calculated to the radius 1, and ranged in ta bles for use. But because trigonometrical operations with these natural sines, tangents, and secants require tedious multiplications and divisions, the logarithms of them are taken, and ranged in tables also; and the logarithmic sines, tangents, and secants are commonly used, as they require only additions and subtractions, instead of the multiplications and divisions.

There are usually three methods of resolving triangles, or the cases of trigonometry; namely, Geometrical Construction, Arithmetical Computation, and Instrumental Operation.

In the first method; let the triangle be constructed by making the parts of the given magnitudes, namely, the sides from a scale of equal parts, and the angles from a scale of chords, or other instrument. Then measure the required parts by the same scale.

In the second method; having stated the terms of the proportion according to the rule, resolve it like all other proportions, in which a fourth term is to be found from three given terms, by multiplying the second and third together, and dividing the product by the first, in working with the natural numbers, whether they be sides, or sines, tangents, or secants, of angles. Or, in working with logarithms, add the logarithms of the second and third terms together, and from

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