b

Let it be required to bisect angle AOB.

With O as a centre, and any radius, describe an arc intersecting OA at C and OB at D; with C and D as centres, and the same radius, describe arcs intersecting at E.

Then, the straight line OE will bisect angle AOB.

The reason for the construction will be found in § 58.

28. With a given vertex, and a given side, to construct an angle equal to a given angle.

B

E

-D

K

Let it be required to construct with E as the vertex, and ED as a side, an angle equal to angle ABC.

With B as a centre, and any radius, describe an arc intersecting AB at G and BC at H.

With E as a centre, and BG as a radius, describe an arc intersecting DE at K.

With K as a centre, and the distance from G to H as a radius, describe an arc intersecting the former arc at L.

Draw straight line ELF.

Then, angle DEF will equal angle ABC.

The reason for the construction will be found in § 53.

29. Given two sides and the included angle of a triangle, to construct the triangle.

Let it be required to construct the triangle having for two of its sides the straight lines m and n, and their included angle equal to angle E.

Draw line AB equal to m, and construct angle BAD equal to angle E (§ 28).

On AD take AC equal to n, and draw straight line BC.
Then, ABC is the required triangle.

30. Given a side and two adjacent angles of a triangle, to construct the triangle.

Let it be required to construct the triangle having for a side the straight line m, and its adjacent angles equal to angles F and G.

Draw line AB equal to m, and construct angle BAD equal to angle F (§ 28).

Draw line BE, making angle ABE equal to angle G, intersecting AD at C.

Then, ABC is the required triangle.

31. Given the three sides of a triangle, to construct the triangle.

Let it be required to construct the triangle having for its sides the straight lines m, n, and p.

Take the straight line AB equal to m.

With A as a centre, and n as a radius, describe an arc.

With B as a centre, and p as a radius, describe an arc intersecting the former arc at C.

Then, ABC is the required triangle.

32. We define an Axiom as a truth which is assumed without proof as being self-evident.

We define a Theorem as a truth requiring proof.

We define a Problem as a question proposed for solution. A Proposition is a general term for a theorem or a problem. A Postulate assumes that a certain problem can be solved. A Corollary is a truth which is an immediate consequence of the proposition which it follows.

An Hypothesis is a supposition, made either in the statement or the proof of a proposition.

33. Postulates.

We assume that the following problems can be solved:
1. A straight line can be drawn between any two points.

2. A straight line can be extended indefinitely in either direction.

1. Things which are equal to the same thing, or to equal things, are equal to each other.

2. If equals be added to equals, the sums will be equal.

3. If equals be subtracted from equals, the remainders will be equal.

4. If equals be multiplied by equals, the products will be equal. 5. If equals be divided by equals, the quotients will be equal. 6. But one straight line can be drawn between two points. 7. A straight line is the shortest line between two points. 8. The whole is equal to the sum of all its parts.

9. The whole is greater than any of its parts.

35. Since but one straight line can be drawn between two points, a straight line is said to be determined by any two of its points.

36. Symbols and Abbreviations.

The following symbols are used in the work:

+, plus.

-, minus.

A, triangle.

A, triangles.

X, multiplied by.

=, equals.

, perpendicular, is perpendicular to.

≈, equivalent, is equivalent Is, perpendiculars.

BOOK I

RECTILINEAR FIGURES

DEFINITIONS

37. We define an acute angle as an angle less than a right angle; as ABC.

We define an obtuse angle as an angle greater than a right angle; as DEF.

Acute and obtuse angles are called oblique angles; and intersecting lines which are not perpendicular, are said to be oblique to each other. We call two angles vertical when the sides of one are the prolongations of the sides of the other; as AEC and BED.

38. If angles AOB, BOC, COD, DOE, are all equal, we say that angle AOB is contained four times in angle AOE; and similarly for any number of equal parts of angle AOE.

39. We measure an angle by finding how many times it contains another angle taken as the unit of measure.

The usual unit of measure for angles is the degree, which is the ninetieth part of a right angle.

To express fractional parts of the unit, we divide the degree into sixty equal parts, called minutes, and the minute into sixty equal parts, called seconds.

We represent degrees, minutes, and seconds by the symbols ", ",", respectively.

40. If the sum of two angles is a right angle, or 90°, we call one the complement of the other; if their sum is two right angles, or 180°, we call one the supplement of the other.