40. Plane figures having more than four fides are, in general, called polygons: and they receive other particular names according to the number of their fides or angles.

41. A pentagon is a polygon of five fides; a hexagon has fix fides; a heptagon, feven; an octagon, eight; a nonagon, nine; a decagon, ten; an undecagon, eleven; and a dodecagon has twelve fides.

42. A regular polygon has all its fides and all its angles equal.—If they are not both equal, the polygon is irregular.

43. An equilateral triangle is alfo a regular figure of three fides, and the fquare is one of four: the former being alfo called a trigon, and the latter a tetragon.

44. A circle is a plane figure bounded by a curve line, called the circumference, which is every where equi-diftant from a certain point within, called its center.

N. B. The circumference itfelf is often called a circle.

53. The height or altitude of a figure, is a, perpendicular let fall from an angle, or its vertex, to the oppofite fide, called the bafe.

54. In a right-angled triangle, the fide oppofite the right angle, is called the hypotenuse; and the other two fides the legs, or fometimes the base and perpendicular.

55. When an angle is denoted by three letters, of which one stands at the angular point, and the other two on the two fides, that which stands at the angular point is read in the middle.

56. The circumference of every circle is fuppofed to be divided into 360 equal parts, called degrees; and each degree into 60 minutes, each minute into 60 feconds, and so on. Hence a femicircle contains 180 degrees, and a quadrant 90 degrees.

57. The measure of a right-lined angle, is an arc of any circle contained between the two lines which form that angle, the angular point being the center; and it is eftimated by the number of degrees contained in that arc. Hence a right-angle is an angle of 90 degrees.

The definition of folids, or bodies, will be given afterwards, when we come to treat of the menfuration of folids.

PROBLEM S.

PROBLEM I.

To divide a given Line A в into two Equal Parts.

From the centers A and B, with any radius greater than half AB, defcribe arcs cutting each other in Am and n. Draw the line mcn, and it will cut the given line into two equal parts in the middle point c.

PROBLEM II.

To divide a given Angle ABC into two Equal Parts.

From the center в, with any radius, defcribe the arc AC. From A and c, with one and the fame radius, defcribe arcs interfecting in m. Draw the line вm, and it will bifect the angle as required.

PROBLEM III.

B

m

To divide a Right Angle ABC into three Equal Parts.

From the center B, with any radius, defcribe the arc AC. From the center A, with the fame radius, crofs the arc AC in n. And with the center c, and the fame radius, cut the arc AC in m. Then through the points m and n draw вm and вn, and they will trifect the angle as required.

A

B

n

PRO

PROBLEM IV..

To draw a Line Parallel to a given Line A B. CASE I. When the Parallel Line is to be at a given Distance c.

From any two points m and n, in the line AB, with a radius equal to c, defcribe the arcs r and o: -Draw CD to touch thefe arcs, without cutting them, and it will be the parallel required.

A

r

D

B

m

C

n

CASE 2. When the Parallel Line is to pass through a given Point c.

From any point m, in the line AB, with the radius mc, defcribe the arc cn.-From the center c with the fame radius, defcribe the arc mr.-Take the arc cn in the compaffes, and apply it from m to r.-Through c and r draw DE, the parallel required.

D

A

C

r

-E

B

n

N. B. This problem is more eafily effected with a parallel ruler.

PROBLEM V.

To erect a Perpendicular from a given Point A in a

given Line Bc.

CASE I. When the Point is near the Middle of the Line.

On each fide of the point a take any two equal diftances Am, An. From the centers m, n, with any radius greater than am or an, describe two arcs interfecting in r.Through A and r draw the line Ar, and it will be the perpendicular as required.

B

L

r

C

CASE