each of its parts; also, that the indefinite expansion of that portion reproduces the whole line.

Let there now be imagined two points, then a third one not in a straight line with the first two, and a fourth one placed in the same position relatively to the two last ones as the third is placed relatively to the first two; then a fifth point placed in the same position relatively to the last two, as the fourth is placed relatively to the two points preceding it, and so on indefinitely; if the distance between these points be imagined to be infinitely small, the idea of a line may thus be conceived which will differ, not only in magnitude, but also in form, from the figure formed by the first two points, or any two of the points: this line is called a curved line. A still more complex line would be obtained in considering more than three points at first, and in varying the relative position of the succeeding ones. Whence

2. A curved line, or simply curve, is that which is not similar to each of its parts.

It is obvious that there can be but two kinds of lines,—the straight line and the curve; and that there can be but one kind of straight line, whilst the forms of curved lines may be varied to infinity.

It follows from what precedes, that two points suffice to produce in the mind the idea of a straight line, and that no less than three points are required to produce that of a curve. It follows equally from the same, that a curved line consists of infinitely small straight lines.

Let there be imagined a straight line moving indefinitely out of its length, so as to follow another straight line, and keeping constantly the same position relatively to it; the idea of a surface will thus be conceived, which will be similar to each of its parts, and is called a plane surface. Whence

3. A plane surface, or plane, is that which is similar to each of its parts.

Hence, the indefinite expansion of any portion of a plane reproduces the whole plane.

BOOK I.-PRELIMINARY DEFINITIONS.

35

Let there now be imagined, a straight line, which, in its motion out of its length, follows a curve, it will then be possible to conceive the idea of a surface which is no longer similar to each of its parts; the same would happen if a curve were conceived to move on a straight line or another curve: this surface is called a curved surface. Whence

4. A curved surface is that which is not similar to each of

its parts.

It is obvious that there can be but two kinds of surfaces, the plane and curved surfaces; and also that there can be but one kind of plane, whilst curved surfaces may be endlessly varied.

It follows from the preceding definitions that the idea of a plane is produced in the mind by two straight lines, whilst that of a curved surface is produced by curved lines only, or straight and curved lines together. It follows also from the same and from the definitions of the curved line, that a curved surface consists of infinitely small planes.

A plane is of its nature infinite, as is the straight line; it may contain points, straight lines, curved lines, and even finite portions of a plane; a curved surface may contain points, curved lines, finite portions of a curved surface, and sometimes straight lines, but no finite portion of a plane.

As a surface divides space into portions, these portions must be on different sides of the surface: the definition of a plane surface shows that it has only two sides.

5. The two sides of a plane are called its faces.

In the same manner a straight line contained in a surface has only two sides in that surface, and a portion of that surface on each side. Again, a point contained in a line has only two sides in that line and a portion of the line on each side.

6. A line, of which all points, and none else, have a common property, is called a locus.

III. DIVISIONS OF GEOMETRY.

Elementary Geometry is divided into two parts: Plane Geometry and Solid Geometry.

Plane Geometry has for its objects, lines contained in one and the same plane, and the portions of the plane limited by such lines.

The object of Solid Geometry are surfaces and geometrical bodies.

CHAPTER I.

ON STRAIGHT LINES.

SECTION I.

ON STRAIGHT LINES GENERALLY.

"Mathematical proofs, like diamonds, are hard as well as clear, and will be touched by nothing but strict reasoning." LOCKE.

DEFINITIONS.

7. When the length of a straight line is known, the line is said to be given in magnitude.

A straight line is of its nature infinite in magnitude and cannot be measured; but a portion of a straight line may be measured by another portion. It is evident that the unit for measuring a straight line must be a straight line.1

8. If two lines have one point in common, they are said to meet, or touch each other; if, when meeting, one of the two lines has some of its points on both sides of the other, the first is said to cut, or cross, the second, and their common point, or intersection, is called their point of section.

It follows from the definitions of the plane and of the straight line, that no line contained in a plane can have any of its points on both sides of a straight line without first cutting it.

9. A transversal to two or more straight lines is a straight line which cuts them in distinct points.

10. The meeting point of two lines which do not touch

1 A portion of an infinite straight line is generally called a straight line for the sake of brevity, but the mind must supply the rest of the expression when the sense of a sentence indicates that a portion only is meant,

each other, is the point in which they would meet if they were indefinitely produced.

II. The portion of a line intercepted between two portions of one or two other figures, is called an intercept.

12. The middle point of a line is that which divides it into two equal parts: the middle line of any two limited straight lines is that which joins1 their middle points.

POSTULATES.

IV. The elements of a straight line are all equal to another. V. There is one, and only one, line which is the shortest that can join two points.

VI. Through any point there may be any number of straight lines of any magnitude.

VII. If a plane be folded upon itself, the two parts coincide entirely with each other.

THEOREM 1.

All parts of a straight line are similar to each other.

For, if any two parts of a line had not the same form, the whole line could not have the same form as each of them, and therefore would not be a straight line [Def. 1].

COROLLARY. The direction of a straight line is the same throughout its extension.

1 Whenever in The Elements the joining of two points is spoken of, the meaning is that one point is to be joined to the other by a straight line.

2 Let the student demonstrate every corollary, and find out the converses and inverses of every proposition; also demonstrate that they are true or false as the case may be.

38

THEOREM 2.

A straight line turning upon two of its points, has none of its parts out of its first place.

For each part of a straight line being similar to the whole line [Def. 1], the figure formed by each part when turning must necessarily be the same as that formed by the whole line. Inversely

If a line, when turning upon two of its points, has any of its parts out of its first place, the line is not straight.

THEOREM 3.

From any point to another there cannot be more than one straight line.

A

N'
N

M

having their extremities at A and B.1

Let A M B and ANB B be two straight lines

Let the plane containing the two lines be folded upon the straight line A M B as an axis; then no part of A M B will be out of its first place [2]. Should any point N of the other line be out of A M B, that is, on one side thereof, then it would, after the folding, have a place N' on the other side, different from N, and A N B would not be a straight line [2]; but A N B is a straight line [Hyp.]; therefore no point of it is out of AM B; that is, the two lines A M B and A N B are one and the same straight line, which was to be demonstrated. Inversely

If one of two distinct lines joining two points be straight, the other line is not straight.

COROLLARY I. Two straight lines having the same extremities are not distinct from each other.

COROLLARY II. If a plane contain the two extremities of a straight line, it contains the whole line.

1 It is understood that the two lines are contained in the same plane, as all figures throughout Plane Geometry.