This is the first example of contraposite reasoning in the Geometry. It can be done as follows:

If the two lines met in a second point, there would be two different straight lines through two points.

There cannot be two different straight lines through two points, by the straight-line axiom; therefore, by contraposite argument (i.e. the conclusion being untrue, the condition is also untrue), the two straight lines do not meet in two points.

In places where no ambiguity results, "line" will be used to mean straight line, since when any other kind of line is meant, the kind is always stated.

25. Sects.

A limited portion of a line is called a sect, or a line segment. That part of a line that lies between

the points A and B is a sect,

Α

B

is nothing to indicate otherwise, sect will mean straightline sect.

line, as ABCDE. A broken line is said to be closed if it is continuous; that is, if the line, when traced from any point through its entire length, is found to return to the starting point. ABCDE is not a closed line,

U

T

but RSTUV is closed.

The straight-line axiom

R

S

shows that a straight line is not continuous, and that it

does not intersect itself.

27. Closed-line Intersection Axiom. If a straight line of indefinite length passes through a point within the surface inclosed by any closed line, it intersects the closed line at least twice.

28. Curved Lines. A line, no part of which is straight, is called a curved line.

A curved line also is called A'

closed if it is continuous. (See § 26.)

B

29. Geometrical Figures. Any combination of points, lines, and surfaces, formed under given conditions, is a geometrical figure; as, a triangle might be defined as the figure formed by three lines meeting in pairs. If the figure is formed by straight lines, it is called a rectilinear figure.

30. Planes. A surface in which, any two points being taken, the straight line which joins them lies wholly in the surface is called a plane surface, or simply a plane.

A plane might be defined as the surface determined by any three of its points that are not in the same straight line.

The plane occupies the same position among surfaces that the straight line holds among lines. As indicated by the first definition given, it is straight through any two of its points. The surface of a blackboard is a familiar example of a plane.

31. Plane Geometry. Plane Geometry treats only of geometric figures that lie entirely in the same plane.

SECTION III. EQUALITY

32. There are three words used in Geometry to denote equality congruent, equivalent, and equal.

33. Congruence. Two figures are congruent when they can be made to coincide in every point.

34. Equivalence. Closed figures (or figures bounded by closed lines, see §§ 26, 28) are said to be equivalent when their boundaries inclose the same amount of surface.

35. Equality. The word equal is used somewhat in both senses, but in this syllabus it will be used only in those places where there can be no confusion between the ideas of congruence and equivalence. For example, sects will be said to be equal when they can be made to coincide, even though this fulfills the definition of congruence, for since a sect cannot inclose surface, there can be no confusion with equivalence.

36. Congruence includes equivalence, whereas equivalence does not imply congruence; a figure inclosed by a curved line might be equivalent to a figure inclosed by a broken line although it would be impossible to make them coincide.

37. Geometric Equality. In the strictest geometric sense, equality means that coincidence is possible, and in this the test for geometric equality differs from the test for arithmetic equality, for two arithmetic magnitudes are equal if they contain the same unit the same number of

times. Evidently, then, equivalence is to some extent an arithmetic property, but Geometry is applied so often to calculations of magnitudes in terms of a unit, that it is neither necessary nor desirable to attempt to distinguish too carefully between it and other mathematical subjects. In practical work, Geometry will be found to have many parts that involve Arithmetic and Algebra, and while the distinctions between the subjects may be kept in mind, their combined use is entirely legitimate.

38. Equality Axioms.

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

This axiom applies to both congruence and to equivalence; the first four following apply only to sects, angles (§ 44), and equivalent closed figures; they cannot be applied to the congruence of closed figures.

(2) If equals are added to the same thing or to equal things, the results are equal.

(3) If equals are taken from the same thing or from equal things, the results are equal.

(4) If equals are multiplied by the same thing or by equal things, the results are equal.

(5) If equals are divided by the same thing or by equal things, the results are equal.

(6) The whole equals the sum of all its parts.

39. Substitution Axiom. A magnitude may be put in place of an equal magnitude in any equation or statement of inequality.

This axiom is not independent of the equality axioms, but it is more convenient for many purposes. For example, if ak-b=r, and k=l; then, substituting 7 for k, al—b = r.

40. General and Geometric Axioms. Certain of the axioms apply not only to geometric magnitudes, but to all magnitudes, and are therefore called general axioms. The six equality axioms, the substitution axiom, and the axiom of inequality (§ 82), are general axioms. The straight-line axiom, and all others that refer to geometric conceptions only, are geometric axioms.

41. Axiom of Motion. Geometric figures can be moved about in space without altering them in any way. (Sometimes stated, “without altering their size or shape." This axiom is used in testing congruence, for one figure is sometimes supposed to have been placed upon (or superimposed upon) another figure, and whatever is known about the two figures is then used to determine whether or not they coincide.

42. Axiom of Division. Any magnitude can be divided into any number of equal parts. (The number must be a positive integer.)

If the magnitude is divided into two equal parts, it is said to be bisected; if into three equal parts, to be trisected, and the parts are called halves, and thirds, respectively.

* 43. A sect can be bisected by but one point.

If P and Q were both points of bisection (or midpoints) of AB, then AP and AQ would be equal, since each is AB÷2 (eq. eq.). Therefore AP and AQ coincide, and P falls on Q, making but one bisection point.

It might be thought at first that this fact was self-evi