Def. 10.-If two straight lines in one plane have a common extremity, and either one of them be turned round this common extremity until it coincides with the other, the revolving line being always in the same plane, the smallest amount of turning required to effect this coincidence is called the angle between the two straight lines.

B

Thus the smallest amount of turning about A required to bring either of the straight lines AB or AC into coincidence with the other, the revolving line being always in the same plane, is called the angle between the lines AB and AC.

The angle in the last figure is sometimes denoted by the single letter A and sometimes, for greater distinctness, by the three letters BAC or CAB (A being always in the middle). The point A is called the vertex, and the straight lines AB and AC are called the arms of the angle, and these lines are said to contain, or include, or form the angle BAC.

Def. 11.-Angles are said to be equal when they can be placed one upon the other in such a way that the vertex and arms of the one can be made to coincide with the vertex and arms of the other.

Def. 12.-Angles are said to be adjacent when they have a common vertex and one common arm, their non-coincident arms lying on opposite sides of the common arm.

Thus the angles BAD and CAD are adjacent angles, of which AD is the common arm.

A

B

ے

C

Def. 13.-Angles are added together by placing them so as to be adjacent to each other: thus the angle BAC in the figure is called the sum of the two angles BAD and CAD.

Def. 14.-When two adjacent angles are equal to each other the common arm is said to bisect the angle between the two non-coincident arms.

Def. 15.-When two adjacent angles are equal to each other and the two non-coincident arms are in the same straight line, each of the equal adjacent angles is called a right angle, and the common arm is said to be perpendicular or at right angles to the straight line in which the non-coincident arms lie.

and OC is said to be perpendicular or at right angles to AB. Def. 16.-A triangle is a closed figure contained by three finite straight lines, which are called its sides.

Def. 17.-A triangle is called isosceles when two of its sides are equal, equilateral when all three sides are equal, and equiangular when all three angles are equal.

Def. 18.—When one of the angles of a triangle is a right angle, the triangle is said to be right angled, and in this case the side opposite to the right angle is called the hypothenuse.

Def. 19.-A triangle is sometimes regarded as standing upon one of its sides, which is then called its base; and the angle opposite to that side is then called the vertex.

When a truth is so obvious as to be manifest without demonstration it is called an axiom. The following statements are axioms.

Axiom 1.*-Two straight lines cannot intersect in more than one point.

* This axiom may be otherwise stated as follows:

D

If two straight lines have more than one point in common they lie in

one and the same straight line.

Thus in figure 1, the lines ACB and ADB cannot both of them be straight lines.

And in figure 2, the lines ABC and ABD cannot both of them be straight lines.

Axiom 2.-The length of the straight line joining any two points is less than the length of any broken line whatever joining the same two points.

Axiom 3.-All right angles are equal to one another.

A theorem is a geometrical truth capable of demonstration by reasoning from certain known truths.

A corollary to a theorem is a geometrical truth easily deducible from that theorem.

The sign = represents equality and is an abbreviation for the words is equal to, or are equal to, according as one or more things are referred to.

The sign+expresses addition and

subtraction.

Thus A+B is to be read as A together with B, or the sum of A and B, and A~B is to be read as the difference of A and B, or the remainder after subtracting the less of the two magnitudes A and B from the greater. The phrase A and B together is sometimes used for the sum of A and B.

We sometimes establish the proof of a theorem by assuming that it is not true, and proving that such assumption leads to an inconsistency or an impossibility, as for instance that one magnitude is both greater than and equal to another magnitude, and so on.

A proof of this kind is called an indirect proof, or a reductio ad absurdum. See for example Bk. I. Prop. 4.

The assumption, whether true or false, upon which any argument is based is called the hypothesis.

SECTION II.-NOTES TO THE PRECEDING DEFINITIONS AND AXIOMS.

1. The conceptions of a surface line and point may be assisted by the following illustrations :

Suppose a portion of space to be completely filled by two bodies, as a block of wood and a block of stone, of any shapes respectively, provided they exactly fit into each other, so that within the space in question there is no portion which is not completely filled up, then it is clear that there is a region within this space which belongs as much to the wood

as to the stone.

Such a region is called a surface or superficial space.

This surface can have no thickness, for if it had a thickness ever so small points might be found in it belonging entirely to the wood or to the stone, and such points could not, therefore, be situated on the surface.

Again, the block of wood might be situated alone in space, and then there would be a region belonging as much to the wood as to the surrounding space, and this region would be the bounding surface of the block of wood, or, to speak more correctly, of the portion of solid space occupied by the block of wood.

Again, we might suppose two surfaces in juxtaposition, the one coloured red and the other green, and it is clear that there would, in this case, be a region belonging as much to the red as to the green surface: such a region is called a line.

A line can have no thickness because it is a portion of a surface, and it may be made evident that a line can have no breadth, by reasoning similar to that employed to show that a surface can have no thickness.

If two lines intersect, any position in space common to both of them is called a point.

A point can have neither breadth nor thickness, inasmuch as it is situated on a line, and it may be shown to have no length, by reasoning similar to that employed to show that a surface has no thickness or a line no breadth.

2. All material objects must have length, breadth, and thickness, and therefore no body can be found which occupies such a region of space as a surface or a line; never

theless we can conceive such objects as existing, and we call them material surfaces, or lines.

A sheet of paper has length and breadth, each so much greater than its thickness that we come almost unconsciously to regard it as having length and breadth only, that is, as a material surface; if, however, it were viewed through a microscope the thickness would become apparent.

Gold may be hammered out into leaves of such exquisite tenuity that many thousands, laid one upon the other, would not, in the aggregate, have the thickness of an inch : each of these leaves suggests the idea of a material surface more forcibly than a sheet of paper does.

So a piece of fine string suggests the idea of a material line, a piece of thread does this yet more forcibly, and a spider's web more forcibly than either.

3. We may form the idea of a material straight line from a piece of thread or fine wire tightly stretched, and the portion of space occupied by such a body suggests the idea of a straight line.

It is important to observe the force of the word must in the definition of a straight line, for there is a certain curved line called a circle, of which it would be equally true that if one portion were applied to any other portion, so that the extremities of the one portion coincided with the extremities of the other portion, then every intermediate point of the one portion might coincide with an intermediate point of the other.

Thus, the portion AB of the circle ABC may be applied