If four quantities be proportional, and if the first and second be multiplied or divided by any quantity, and also the third and fourth, the resulting quantities will be proportional; or if the first and third be multiplied or divided by any quantity, and also the second and fourth, the resulting quantities will be propor

tional.

If there be several ranks of proportional quantities, the products of the corresponding terms will be proportional.

If four quantities be proportional, the like powers, and the like roots of those quantities will be proportional.

If three quantities be proportional, the first quantity will be to the third, as the square of the first is to the square of the second.

ELEMENTS OF GEOMETRY.

BOOK I.*

EXPLANATION OF TERMS.

ED.

A PROPOSITION is something which is proposed either to be done, or to be demonstrated; and is either a theorem or a problem.

The words in which a proposition is expressed are called the enunciation of the proposition.

A theorem is something which is proposed to be demonstrated; or, it is a truth which becomes evident by means of a train of reasoning called a demonstration.

A problem is something which is proposed to be done; or, it is a question proposed, which requires a solution.

A lemma is something which is premised, or previously demonstrated, with a design to facilitate the demonstration of a theorem, or the solution of a problem.

A corollary is a consequent truth, or proposition deduced immediately from some preceding truth, proposition, or demonstration.

A scholium is a remark made upon one or more preceding propositions, and tending to show their connection, or restriction, or extension, or utility.

An axiom is a self evident proposition.

A postulate is something required to be done, which is so easy and evident that its practicability cannot be denied.

* See Notes at the end of the volume,

.

DEFINITIONS.

A. GEOMETRY is that science which treats of the properties and relations of space; or, it is that science whose object is the measure of extension.

EDITOR.

B. Extension has three dimensions, length, breadth, and thickness.

ED.

C. Magnitude is that kind of quantity which we conceive to be extended, and divisible into parts. There are three sorts of magnitudes, a line, a surface, and a solid. ED.

1. A Point is that which has position, but not magnitude. 2. A Line is length without breadth.

Corollary. The extremities of a line are points; and the intersection of two lines is a point.

3. A straight line is that which every where tends the same way. See Note. ED.

4. Cor. Hence a straight line is the least distance between two points. ED. 5. A curve line is that which continually changes its direction between its extreme points.

ED.

6. A superficies, or surface, is that magnitude which has only length and breadth.

7. Cor. The extremities of a superficies are lines; and the intersection of one superficies with another is a line.

8. A plane superficies is that in which any two poins being taken, the straight line which joins them lies wholly in that. superficies.

9. If two straight lines diverge from the same point, the opening between them is called an angle.

ED.

10. An angle is formed by the meeting, or intersection of two lines. The point of concourse of the two lines is called the summit, vertex, or angular point.

ED.

The magnitude of an angle does not depend on the length of the two lines which form it, but on the wideness of their opening. Thus, the angle ABC is greater than the angle DBC. ED.

When several angles are at one point B, any one of them is expressed by three letters, of which the letter at the vertex of the angle is put between the other two letters, and one of these two is somewhere on one of those straight lines, and the other on the other line: thus, the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, BD, is named the angle ABD, or DBA; and that which is contained by BD, CB, is called the angle DBC, or CBD. If there be only one angle at a point, it may be expressed by a letter placed at that point, as the angle at E.

10. When a straight line standing on another straight line makes the adjacent angles equal to each other, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

Otherwise. When a straight line standing on another straight line does not incline to either side of it, the former line is said to be perpendicular to the latter; and the two angles which it forms with the latter are called right angles.

ED.

11. Cor. Hence all right angles are equal to one another. For they are formed by straight lines standing perpendicularly on other straight lines; and since they are formed exactly in the same manner they are necessarily equal to one another, oplast er mon pweth sail 12. An obtuse angle is that which is greater than a right angle. yg bd al ojimos A

13. An acute angle is that which is less than a right angle. Note. An angle which is either acute or obtuse is often called an oblique angle.

ED.

14. If two lines be in the same plane, and do not meet, though produced ever so far both ways, they are called parallel lines.

Otherwise. If two lines be in the same plane, and have no inclination to each other, they are called parallel lines. ED. 15. Cor. 1. Hence parallel lines are equidistant. For they neither accede to, nor recede from each other. ED.

16. Cor. 2. If two straight lines be perpendicular to the same straight line, they are parallel to each other. For they have no inclination to each other. ED.

17. A figure is that which is enclosed by one or more boundaries.

18. If a finite straight line be supposed to revolve in the same plane, about one of its extremities, which is fixed, until it arrive at the place from which it began to move, the surface described by the revolving line is called a circle.

ED.

19. The fixed point about which the line revolves is called the centre of the circle; the revolving line is called a radius; and the curve line described by the moveable extremity of the revolving line is called the circumference of the circle.

ED.

20. Cor. Hence a circle is a plane figure contained by a curve line, or circumference, and all straight lines drawn from the centre to the circumference are equal to one another.

Note. The circumference of a circle is often called a circle; and the arch of a semicircle is often called a semicircle. ED. 23. Rectilineal figures are those which are contained by straight lines.