31. All plane figures contained under more than four sides, are called polygons; of which those having five sides, are called pentagons; those having six sides, hexagons, and so on.

32. A regular polygon is one whose angles, as well as sides, are all equal.

33. A circle is a plane figure, bounded by one curve line A DEB, called the circumference or periphery, every part of which is equally distant from a certain point C within the circle, and this point is called the centre, Fig. 16.

34. The radius of a circle is a straight line drawn from the centre to the circumference, as C B, Fig. 17.

35. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the cir cumference, as A E, Fig. 17. It divides the circle into two equal parts, called semicircles.

36. A quadrant is one quarter of a circle, as ACB, Fig. 17.

Note. The fourth part of the circumference of a cir. cle is also called a quadrant.

37. A segment of a circle is the figure contained by a right line, and the part of the circumference it cuts off: thus AE BA and AEDA are segments of the circle ABED, Fig. 16.

38. An arc of a circle is any part of the circumference,

GEOMETRICAL PROBLEMS.

PROBLEM I.

To bisect a right line, AB, Fig. 18.

Open the dividers to any distance more than half the line AB, and with one foot in A, describe the arc CFD; with the same opening, and one foot in B, describe the arc CGD, meeting the first arc in C and D; from C to D draw the right line CD, cutting A B in E, which will be equally distant from A and B.

PROBLEM II.

At a given point A, in a right line E F, to erect a perpendicular, Fig. 19.

From the point A, lay off on each side, the equal distances A C, AD; from C and D, as centres, with any radius greater than A C or AD, describe two arcs intersecting each other in B; from A to B, draw the line AB, which will be the perpendicular required.

PROBLEM III.

To raise the perpendicnlar on the end of В of a right line AB' Fig. 20.

Take any point D not in the line AB, and with the

from E through D draw the right line EDC, cutting the periphery in C, and join CB, which will be perpendicular to AB.

PROBLEM IV.

To let fall a perpendicular upon a given line BC, from a given a point A, without it, Fig. 21.

In the line BC take any point D, and with it as a centre and distance DA describe an arc AGE, cutting BC in G, with G as a centre, and distance GA, describe an arc cutting AGE in E, and from A to E draw the line AFE; then AF will be perpendicular to AB.

PROBLEM V.

Through a given point A to draw a right line AB, parallel to a given right line CD, Fig. 22.

From the point A to any point F, in the line C D, draw the right line A F; with F as a centre and distance F A, describe the arc AE, and with the same distance and centre A describe the arc F G; make FB equal to AE, and through A and B draw the line AB, and it will be parallel to CD.

PROBLEM VI.

At a given point B, in a given right line LG, to make an angle equal to a given angle A, Fig. 23.

With the centre A and any distance AE, describe the arc DE, and with the same distance and centre B describe the arc FG; make HG equal to DE, and through Band H draw the line BH; then will the angle HBG be equal to the angle A.

PROBLEM VIL

To bisect any right lined angle BAC, Fig. 24.

In the lines AB and AC, from the point A, set off equal distances AD and AE; with the centres D and E and any distance more than half DE, describe two arcs cutting each other in F; from A through F draw the line AG, and it will bisect the angle BAC.

PROBLEM VIII.

To make a triangle of any three right lines D, E and F, of which any two together must be greater than the third, Fig. 25.

Make AB equal to D; with the centre A and distance equal to E, describe an arc, and with the centre B and distance equal to F describe another, arc, cutting the former in C; draw A C and BC, and ABC is the triangle required.

PROBLEM IX.

Upon a viven line AB to describe a square, Fig. 26.

At the end B of the line AB, by problem 3, erect the perpendicular BC, and make it equal to AB; with A and C as centres, and distance AB or BC describe two arcs cutting each other in D; draw A D and CD, then will ABCD be the square required.

PROBLEM. X.

To describe a circle that shall pass through the angu

By problem 1, bisect any two of the sides, as A C, B C, by the perpendiculars D E and F G; the point H where they intersect each other will be the centre of the circle; with this centre, and the distance from it to either of the points A, B, or C, describe the circle.

PROBLEM XI.

To divide a given right line AB into any number of equal parts, Fig. 28.

Draw the indefinite right line A P, making any angle with A B, also draw BQ parallel to A P, in each of which, take as many equal parts AM, MN, &c. Bo, on, &c. as the line A B is to be divided into; then draw Mm, Nn, &c. intersecting A B in E, F, &c. and it is done.

Fo make a plane diagonal scale, Fig. 29.

Draw eleven lines parallel to, and equidistant from each other; cut them at right angles by the equidistant lines BC; E F¿ 1, 9; 2, 7;-&c. then will B C, &c. be divided into ten equal parts; divide the lines E B, and F C, each into ten equal parts, and from the points of division on the line E B, draw diagonals to the points of division on the line FC: thus join E and the first division on F C, the first division on E B and the second on FC, &c.

Note.-Diagonal scales serve to take off dimensions or numbers of three figures. If the first large divisions be units, the second set of divisions, along E B, will be 10th parts; and the divisions in the altitude, along B C,