A

C

A

C

B

D

E

54. In a right-angled triangle, the side opposite to the right angle, is called the hypothenuse; and the other two sides are called the legs, and sometimes the base and perpendicular; thus, A, B is the base, B, C perpendicular, and A, C hypothenuse.

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

A and A, D is called the angle B, A, D, or D, A, B.

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

A

GEOMETRICAL PROBLEMS.

PROBLEM I.

To divide a line A, B into two equal parts.

Ж

B

Set one foot of the compasses in A, and opening them beyond the middle of the line, describe arches above and below the line; with the same extent of the compasses, set one foot in the point B, and describe arches crossing the former; draw a line from the intersection above the line to the intersection be

low the line, and it will divide the line A, B into two equal parts.

D

PROBLEM II.

To erect a perpendicular on the point C, in a given line.

C

F

Set one foot of the compasses in the given point C, extend the other foot to any distance at pleasure, as to D, and with that extent make the marks D and E. With the compasses, one foot in D, at any extent above half the distance D, E E, describe an arch above the line, and with the same extent, and one

foot in E, describe an arch crossing the former; draw a line from the intersection of the arches to the given point C, which will be perpendicular to the given line in the point C.

PROBLEM III.

To erect a perpendicular upon the end of a line.

Set one foot of the compasses in the given point B, open them to any convenient distance, and describe the arch C, D, E; set one foot in C, and with the same extent, cross the arch at D; with the same extent, cross the arch again from D to E; then with one foot of the compasses in D,

A

C

D

B

and with any extent above the half of D, E, describe an arch a; take the compasses from D, and keeping them at the same extent, with one foot in E, intersect the former arch a in a from thence draw a line to the point B, which will be a perpendicular to A, B.

PROBLEM IV.

From a given point a, to let fall a perpendicular to a given line A, B.

Set one foot of the compasses in the point a, extend the other so as to reach beyond the line A, B, and describe an arch to cut the line A, B in C and D; put one foot of the compasses in C, and with any extent above half C, D, describe an arch b, keeping the compasses at the same extent, put one foot in D, and intersect the arch bin b; through which intersection, and the point a, draw a, E, the perpendicular required.

PROBLEM V.

А С

D B

E

To draw a line parallel to a given line A, B.

Set one foot of the compasses in any part of the line, as at c; extend the compasses at pleasure, unless a distance be assigned, and describe an arch b; with the same extent in some other part

E

F

A

B

of the line A, B, as at e, describe the arch a; lay a rule to the extremities of the arches, and draw the line E, F, which will be parallel to the line A, B.

PROBLEM VI.

To make a triangle, whose sides shall be equal to three given lines, any two of which are longer than the third.

To make a square, whose sides shall be equal to a given line.

A

D

C

A

B

Let A be the given line; draw a line, A, B, equal to the given line; from B raise a perpendicu lar to C, equal to A, B; with the same extent, set one foot in C, and describe the arch D; also, with the same extent, set one foot in A, and intersect the arch D; lastly, draw the line A, D, and C, D, and the square will be completed.

In like manner, may a parallelogram be constructed, only attending to the difference between the length and breadth.

PROBLEM VIII.

To describe a circle, which shall pass through any three given points, not in a straight line.

Let the three given points be A, B, C, through which the cir

cle is to pass. Join the points A, B and B, C, with right lines, and bisect these lines; the point D, where the bisecting lines cross cach other, will be the centre of the circle required. Therefore, place one foot of the compasses in D, extending the other to either of the given points, and the circle, described by that radius, will pass through all the points.

B

C

A

D

Hence it will be easy to find the centre of any given circle; for, if any three points are taken in the circumference of the given circle, the centre will be found as above. The same may also be observed, when only a part of the circumference is given.

MENSURATION OF SOLIDS.

DEFINITIONS.

1. Solids are figures, having length, breadth, and thickness.

2. A prism is a solid, whose ends are any plane figures, which are equal and similar, and its sides are parallelograms.

NOTE. A prism is called a triangular prism, when its ends are triangles; a square prism, when its ends are squares; a pentagonal prism, when its ends are pentagons; and so on.

3. A cube is a square prism, having six sides, which are all squares.

4. A parallelopiped is a solid, having six rectangular sides, every opposite pair of which are equal, and parallel.

5. A cylinder is a round prism, having circles for its ends.

6. A pyramid is a solid, having any plane figure for a base, and its sides are triangles, whose vertices meet in a point at the top, called the vertex of the pyramid.

7. A cone is a round pyramid, having a circular base.

8. A sphere is a solid, bounded by one continued convex surface, every point of which is equally distant from a point within, called the centre. The sphere may be conceived to be formed by the revolution of a semicircle about its diameter, which remains fixed.

A hemisphere is half a sphere.

9. The segment of a pyramid, sphere, or any other solid, is a part cut off the top by a plane parallel to the base of that figure. 10. A frustum is the part that remains at the bottom, after the segment is cut off.

11. The sector of a sphere is composed of a segment less than a hemisphere, and of a cone, having the same base with the segment, and its vertex in the centre of the sphere.

12. The axis of a solid is a line drawn from the middle of one end to the middle of the opposite end; as between the opposite ends of a prism. The axis of a sphere is the same as a diameter, or a line passing through the centre, and terminating at the surface on both sides.

13. The height or altitude of a solid, is a line drawn from its vertex, or top, perpendicular to its base.