which shall be of equal area to a polygon of any other number of sides.

PROP. XXXIX. fig. 19. To inscribe a square in a given circle, ACDB. Draw two diameters, AB and CD, intersecting each other at right angles: join the extremities of these diameters, and these lines will form a square, ACBD, within the given circle. For the angles AOC, COB, BOD, and DOA, being all equal, the chords subtending these angles must also be equal: but the angles at A, C, B, and D, being angles in a semicircle, are all right; consequently the figure ACBD is a square inscribed in the given circle, as was required.

The square may be placed in any position, as GEHF, whose extremities are equally bounded by the circle.

PROP. XL. fig. 20. To describe a regular hexagon in a given circle, whose centre is C. Take the radius of the circle AC, and set it off on the circumference to B and D; draw the diameters AG, BE, and DF; join all the extremities of these lines, and the figure ABFGED will be the hexagon required: for AB and AD being each made equal to AC, which is also equal to BC and CD, these twotwo triangles, CBA, CAD, are equilateral: and on account of the intersecting lines BE, AG, DF, the opposite triangles must also be equilateral; consequently, EG and GF each equal to the radius of the circle; that is, to ABand AD. In the same way, the remaining sides DE and BF may be proved to be each equal to the radius, or to AB and AD; consequently the six sides of the figure to be all equal; it is, therefore, a regular hexagon inscribed within the given circle, as was required.

If lines be drawn joining the alternate angles of the hexagon A, F, E, we obtain an equilateral triangle, inscribed within a given circle,

PROP.

PROP. XLI. fig. 21. To inscribe a regular octagon within a given circle. By the 39th Prop. construct a square whose angular points touch the circumference of the circle at ADBE; divide the sides AD and DB, into two equal parts; and through the sections and the centre C, draw diameters touching the circumference in the points F, G, H, I: then right lines joining these eight points will form the octagon AFDGBHEI, as was required to be done.

ON TRIGONOMETRY.

HAVING in the foregoing propositions explained the chief properties of certain plain geometrical figures, as the triangle, the square, the parallelogram, the circle, &c. it is now time to furnish the student with some observations on the application of the properties of one of these figures, namely, the triangle, to sundry very important purposes.

The branch of geometry which regards the properties of triangles, is termed Trigonometry, a name borrowed from the Greck language, signifying the art of measuring triangles.

Trigonometry is divided into two parts, Plane and Spherieal. Plane trigonometry is employed concerning such tri angles as are formed by three sides, or fines, all lying in the same plane, such as those drawn on a sheet of paper, on a smooth even table, or the like. Spherical trigonometry relates to those triangles whose sides are portions of circles, such as may be described on the surface of a celestial or terrestial globe, where the sides are all curves, and situated in different planes. This latter branch of trigonometry being founded on a knowledge of parts of geometry that have not been explained in the preceding propositions, it is not intended to enter upon it in this work; the attention

of

of the student will therefore be confined to plane trigonometry alone, and to some of its most common uses in ordinary life.

In every plane triangle there are three sides and three angles, which have all such proportions one to another, that if a certain number of these six parts be given, the remainder may be discovered: for instance, if the three sides be given, the three angles may be found; if two sides, and the angle formed by them, be given, the remaining side, and the other two angles, may be found; if a side, and the angles at each extremity be given, the remaining angle and sides may be found; but if the three angles alone be given, it will be impossible to ascertain any of the sides; their relative proportions, however, may be determined, as bearing a certain ratio to the opposite angles. See the concluding observations in page 353, on fig. 4 of Plate 1.

In speaking concerning fig. 22, Plate 1, (page 361), it was said, that if an angle were formed at the central point C by the radius CD, and the right line CHE, and the arch DH were described, then a line, such as DE, drawn perpendicular to CD, consequently touching the circle in D, and produced until it met CH produced in E, would become the tangent of the arch HD, or of the angle HCD; again, that if the right line HD was drawn joining the extremities of the arch, it would become the chord of that arch, or of the angle HCD; also, that if from H we let fall III perpendicularly upon CD, HI would become the sine of the same arch HD, or of the angle HCD; and lastly, that if the side CH was produced till it met the tangent at the point E, this line CE would become the secant of the same arch HD, or angle HCD. In this manner an infinite number of chords, sines, tangents, and secants, might be found, according to the boundless variety of angles which might be formed by the radius CD)

at the point C. Scales, therefore, may be constructed, containing the chords, sines, tangents, and secants of a convenient number of arches or angles, in the following manner. See Plate 3, fig. 1.

Upon the centre C, with any convenient radius, the larger the better, describe a quadrant, or a semicircle, as AFB; divide the arch, AB, into two equal parts in the point F, and join AF; then will AF be the chord of the angle ACF, or of the arch bounded by the points A and F. The whole circumference of every circle being supposed to be divided into 360 equal parts, called degrees, a quadrant, or fourth part of the circumference, must contain 90 degrees; if, therefore, we divide the arch AF into 9 equal parts, each part will contain 10 degrees. To do this, place one foot of the compasses in the point A, and opening the other to C, turn it round, and cut the arch in the point marked 60 in the figure, which will represent 60 degrees, for the radius of any circle is always equal to the chord of one sixth part of the circumference. Then placing the foot of the compasses in the point F, with the other, at the same opening, cut the arch in the point marked 30 degrees. The given quadrant is thus divided into three equal parts, cach of which is again to be divided, by repeated trials, into three other equal parts, when the several points on the arch will be obtained, viz. 10, 20, 30, 40, 50, 60, 70, 80, and 90, which will coincide with the point F. Then placing one. foot of the compasses in A, open up the other until it reach the point marked 10; sweep it round, agreeably to the dotted lines on the figure, and cut the right line, or chord, AF, in the point marked 10; again, with the foot fixed in A, and the opening up to 20 on the arch, draw a dotted arch cutting the chord AF in the point marked 20; and in this manner proceed from the fixed centre A, to ascertain the points on the chord, AF, corresponding to all the remaining divisions on the arch; then will the line AF

become

become a line of chords, and so may be transferred for use to a proper instrument or scale.

Having divided the quadrant BF into 9 equal parts, each containing 10 degrees, as was done with FA, numbered from B towards F, lay a ruler from 10 on BF to 10 on AF, and mark the point where it cuts CF to be marked also 10; then will C 10 be the sine of 10 degrees; again, by a ruler laid from 20 on AF to 20 on BF the point 20 will be obtained on CF, and C 20 will be the sine of 20 degrees: in the same way the points 30, 40, &c. will be obtained, and the whole radius or line, CF, will become a line of sines. Or, the same thing may be done by employing the quadrant FB alone: thus, from the point marked 80 on the arch FB, let fall a perpendicular in the radius CB, cutting it in the point marked 10; then from the points of the arch 70, 60, &c. draw other perpendiculars to ACB, marked 20, 30, &c. when the radius CB will also become a line. of sines.

Upon B, the extremity of the radius CB, erect the indefinite perpendicular BD; then from the centre C, through the several divisions 10, 20, 30, &c. of the arch FB, draw lines meeting BD in the corresponding points marked 10, 20, 30, &c.; then will the space, B 10, be the tangent of 10 degrees, B 20 the tangent of 20 degrees, and so on; B 80. the tangent of 80 degrees, and the line BD be the line of tangents.

Lastly, the line CF being produced indefinitely to E, with one foot of the compasses in C, open up the other to the point 10 upon the line of tangents BD, thus obtaining the length of the secant of 10 degrees; and turning it round to CFE, agreeably to the dotted arch, mark the point where it cuts FE, with the number 10, (omitted in the figure for want of room); next, from the same centre, C, opening up to the point marked 20 on BD, sweep the compasses round to FE, and mark 20; again, with the distance from

C to