A: CB: D;

from which we obtain, by the preceding propositions, = A B: C - D=A: C = B : D

A+B:C+D

A+BA-BC+D: C — D.

Two proportions, as

and

A: BC: D

E: FG: H,

may evidently be multiplied together term by term, and the result

=

AXE:BX FCX G-DX H

is a new proportion.

Likewise, a proportion may be multiplied by itself any number of times in succession, and the squares, cubes, fourth powers, &c. of the terms form a new proportion. Thus, the proportion

GEOMETRY.

CHAPTER I.

GENERAL REMARKS AND DEFINITIONS.

1. Definition. Geometry is the Science of Position and Extension.

2. Definition. A Point has merely position, without any extension.

3. Definition.

Extension has three dimensions;

Length, Breadth, and Thickness.

4. Definition. A Line has only one dimension, namely, length.

5. Definition. A Surface has two dimensions; length and breadth.

6. Definition. A Solid has the three dimensions of extension; length, breadth, and thickness.

7. Scholium. The boundaries of solids are surfaces, the limits of surfaces are lines, and the extremities of lines are points.

The Point, then, on account of its simplicity, deserves our first consideration.

The Position of a Point; its Direction and Distance.

CHAPTER II.

THE POINT.

8. The Position of a Point is determined by its Direction and Distance from any known point; in other words, the Elements of its Position are Direction and Distance.

Remarks. The Direction of a Point is readily ascertained without any change in the position of the observer, whereas the determination of its distance is often more difficult, as it requires some change of place proportionate to the distance to be measured; thus, the direction of a star is seen at a glance, while the most profound science and the most accurate observations have not enabled the astonomer to ascertain its distance.

9. The Direction of a Point from the observer may be determined by a reference to some known direction, such as that of the zenith, the pole-star, &c.

The method by which one direction may thus be referred to another will be more definitely treated of in a succeeding article.

10. The Distance of a Point from the observer is the length of the shortest line drawn to the point; and it may be determined by a reference to some known length, such as an inch, a yard, a metre, a mile, &c.

The Direction of a line; the Straight and Curved Lines; the Plane.

CHAPTER III.

THE STRAIGHT LINE.

11. Definition. The Direction of a Line in any part is the direction of a point at that part from the next preceding point of the line.

a. Thus the direction of the line AB (fig. 1) at P is the same as the direction of P from 0.

b. In the same way, the direction of the line at P is the same as that of O from P, or the opposite direction to the preceding; and, consequently, a line has two different directions exactly opposed to each other, either of which may be assumed as the direction of the line.

12. Definition. A Straight line is one, the direction of which is the same throughout, as AB (fig. 2).

13. Definitions. A Broken or Polygonal Line is one, which is composed of straight lines, as ABCD (fig. 3).

A Curved Line is one, the direction of which is constantly changing, as AB (fig. 1).

14. Definition. A Plane is a surface in which any two points being taken, the straight line joining those points lies wholly in that plane.

15. Axiom. The direction of any point of a straight line from any preceding point, is the same as the direction of the line itself.

Thus the direction of P or B (fig. 2) from M or A is the same as that of the line AB.