If an angle stands by itself it may be designated by a single letter placed at the vertex; if two angles have a common vertex each is designated by three letters, of which the middle one denotes the vertex and the others the two sides: thus, the angle between QP and QR is called the angle Q; that between OA and OC is called the angle AOC; and that between OC and OD is called the angle COD.

Definition and Division of the Subject.

15. Lines, surfaces, volumes, and angles are called geometrical magnitudes.

16. Geometry is that branch of mathematics which treats of the properties and relations of geometrical magnitudes.

17. Geometry is divided into two parts: geometry of two dimensions, or plane geometry, in which the magnitudes considered lie in the same plane; and geometry of three dimensions in which the magnitudes considered do not lie in the same plane.

All the magnitudes considered in the first five books are supposed to lie in the same plane.

Definitions of Terms.

18. A theorem is an assertion whose truth is to be proved; the reasoning employed in making the proof is called a demonstration.

A problem is a question proposed, requiring an answer; the operation of finding the answer is called a solution.

Both theorems and problems are called propositions. 19. An axiom is an assertion whose truth is universally admitted.

20. A postulate is a problem whose solution is taken for granted.

21. A lemma is a proposition introduced out of the

regular course to aid in the demonstration of a theorem, or in the solution of a problem.

22. A scholium is an explanatory remark made with reference to one or more preceding propositions.

23. An hypothesis is a supposition made either in the enunciation of a proposition, or in the course of a demonstration, or of a solution.

Every proposition contains an hypothesis and a conclusion.

24. The following are some of the most important axioms and postulates.

Axioms.

1o. Things equal to the same thing are equal to each other. 2°. If equals are added to equals, the sums are equal. 3o. If equals are subtracted from equals, the remainders are equal.

4°. If equals are multiplied by equals, the products are equal.

5°. If equals are divided by equals, the quotients are equal. 6°. The whole is greater than any of its parts.

7. The whole is equal to the sum of all its parts.

8°. A straight line is the shortest path between two points.

Postulates.

1o. A straight line can be drawn through any two points. 2o. A straight line can be prolonged to any extent. 3o. A straight line can be constructed equal to a given straight line.

4°. A straight line can be bisected, that is, divided into two equal parts.

5°. An angle can be constructed equal to a given angle. 6°. An angle can be bisected.

Notation.

25. The signs and methods of notation used in Algebra are also employed in Geometry. The principal signs are the ordinary signs of addition, +; of subtraction, -; of multiplication, ; of division,÷; the radical sign, √; and the signs of proportion, :,::,:. The vinculum and the parenthesis are used to connect quantities that are to be operated on as a single quantity; coefficients are used to denote multiples; and exponents to denote powers. The symbols 1o, 2o, 3o, &c. are read first, second, third, &c. In addition to these methods of denotation, geometrical magnitudes are represented by pictorial symbols, called geometrical figures, in which points and lines are designated by letters placed so as to be convenient for reference.

NOTE.-In the references throughout this work, P. stands for proposition, B. for book, Cor. for corollary, Ax. for axiom, Prob. for problem, Post. for postulate, and Scho. for scholium. In referring to subjects in the same book the number of the book is not given.

BOOK I.

LINES, ANGLES, AND POLYGONS.

Generation and Classification of Angles.

26. An angle may be generated by a line revolving about one of its points in the same manner that one leg of a pair

of compasses turns about the hinge. Thus, if we suppose OC to start from the position OA and to revolve about O in the direction indicated by the arrow, it will generate an angle AOC, whose magnitude will obviously depend on the amount of turning, and not at all on the length of OC.

In this explanation we only consider that part of the revolving line which lies on one side of the vertex; if we suppose the line to be prolonged through the vertex, the prolongation may be regarded as a second line lying in an opposite direction. Thus, if we suppose QC to start from the position PA and to revolve about O in the direction indicated by the arrows, the part OC will generate an angle AOC and the opposite part, OQ, will generate another angle POQ.

P

27. If one line meets another making the two adjacent angles equal, each is called a right angle and the first ine is said to be perpendicular to A the second. Thus, if AOD and DOP

are equal, each is a right angle, and OD is perpendicular to PA, at O.

The point O is called the foot of the perpendicular OD.

28. An acute angle is one that is less than a right angle; an obtuse angle is one that is greater than a right angle: thus, AOC is acute, and COP is obtuse.

The angles AOC and COP are both called oblique, and the line OC is said to be oblique to PA.

29. If two lines intersect, that is, cross each other, they form four angles about the point of intersection. Those that

P

A

lie on the same side of one line and on opposite sides of the other are called adjacent, and those that lie on opposite sides of both are called. opposite, or vertical angles :

thus, AOC, COP, are adjacent, and AOC, POQ, are opposite angles.

Any one of the four angles about O has two adjacent angles and one opposite angle.

PROPOSITION I. THEOREM.

If two lines intersect, the opposite angles which they form are equal.

P

Let PA and QC be two lines that intersect, and suppose PA to remain fixed whilst QC revolves about their common point, O. If we suppose the moving line to start from the position PA and to revolve in the direction indicated by the arrows till it comes into some other position, QC, the part OC will generate the angle AOC and the opposite part OQ will generate the opposite angle POQ; but the amount