PROB. 17. To make an angle of a proposed number of degrees, fuppofe 55°, by the line of chords. Fig. 20.

Draw the right line AB; about the center A, with a radius equal to the chord of 60°, defcribe the arc CD; then take the given number of degrees (55°) from the fame line of chords, and fet it from C to m, and through A and m draw the line AE; and EAB is the angle required.

PROB. 18. To measure a right lined angle by the line of chords.

With a radius equal to the chord of 60°, defcribe an arch about the angular point; take the arch in your compaffes, and apply it to the line of chords; and thus the quantity of the angle will be known.

When an obtufe angle is to be made or measured, its measure being greater than 90°, must be taken from the scale at twice: Thus, fuppose an angle of 160° was to be made; having described an arc with the chord of 60°, take 90° and 70°, or 80° and 80° from the line of chords, and fet them one after another on that arc.

DEFIN. 21. A regular polygon, is a figure whose fides and angles are all equal, and is denominated by the number of its fides. One of five fides is called a Pentagon, of fix fides, a Hexagon, &c.

22. The angle at the center of a polygon is contain. ed between two right lines drawn from its center to the extremities of any of its fides, and is found by dividing 360° by the number of fides.

23. The angle at the circumference of a polygon is contained between any two adjacent fides of the polygon, and is found by fubtracting the angle at the center from 180°.

A TABLE, fhewing the Names, number of Sides, Angles at the Center, and at the Circumference of regular Polygons, from three to twelve Sides, inclufive.

PROB. 19. To infcribe a regular polygon in a circle by the line of chords.

Make an angle at the center of the circle equal to the angle at the center of the polygon required to be drawn; this will give one fide of the polygon, which being applied round the circle, the lines joining the feveral points in the circumference will be the fides of the polygon.

Exam. Suppofe a pentagon were to be infcribed in a given circle ABD. Fig. 21.

At the center, make the angle ACB of 72°, and join AB; then applying AB round the circumference, will give the points D, E, F; join BD, DE, EF, FA, and ABDEF is the pentagon required.

In the fame manner, any other regular polygon may be infcribed in the circle.

V. B. The radius of the circle is equal to the fide of a hexagon infcribed in it.

PROB 20. To make any regular polygon on a given right line AB. Fig. 21.

At the points. A and B, make the angles CAB, ABC, each equal to half of the angle at the circumference of the polygon; and from the point C with the radius CA or CB, describe a circle; and AB properly applied in the circumference will give the fides of the polygon required.

Exam of a Pentagon Make the angles CAB and ABC each equal to 54°, the half of 108°; and from the center C, with the radius CA, describe a circle, in which apply the right lines BD, DE, EF, and FA, each equal to AB; and ABDEF is a pentagon defcribed upon the given line AB.

PROB. 21. To draw a tangent to the circle from any point A, which is not within the circle. Fig. 22. If the given point A is in the circumference, draw a line from the center C to A, and through A draw

I.

BAD

BAD at right angles to AC; and BD is the tangent required.

2. If the point A is without the circumference, join CA, and on it defcribe the femicircle AEC; then, through A and E, draw AD, which will touch the circle in the point E.

PROB. 22. To defcribe a circle about a given triangle ABC; or, to defcribe a circle through any three given points A, B, C, which do not lie in the fame right line. Fig. 23.

Bifect the fides of the triangle, or the distances of the points AB and AC, at right angles, by the right lines DF and EF; and from the point F, where they meet, with a radius equal to FA, FB, or FC, describe the circle ABC.

PROB. 23. To defcribe an oval figure reprefenting an ellipfe, whofe longest diameter AB is given. Fig. 24.

Divide AB into three equal parts in the points C and D; and about the centers C and D, with the radius CD, defcribe two circles, interfecting each other in the points E and F; through the points C and D, draw the lines EN, EM, FO, and FP; and about the centers F and E, with the radius EN, defcribe the arches NM and PO; and AHBG is the oval figure required.

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