to the portion CD so that the extremities and every intermediate point of AB coincide with the extremities and every intermediate point of CD.

But AB may also be applied to CD in such a way that A and B coinciding with C and D, the intermediate points of AB, do not coincide with the intermediate points of CD, as in the second of the two figures drawn above.

4. An idea of an angle may be gained by observing the hands of a watch, each hand being supposed to be indefinitely thin, or to be a material line. If these hands occupied certain given positions, and one of them were turned by means of the key until it came into the position immediately overlying the other, the smallest amount of turning required to effect this coincidence would be the angle between the two hands.

Suppose the long hand pointed to XII and the short hand to II, then we see that the amount of turning required to bring one hand into the position overlying the other is the same as it would be if the long hand were at XII and the short hand at X, and we should say that the angle between the hands in the first case was equal to the angle in the second case.

Again, if the long hand were at XII and the short hand at IV, we see that the amount of turning requisite in this case would be twice as great as that required in either of the last two cases, and therefore we should say that the angle between the hands in this case was twice as great as that in either of the former cases.

It is evident that the length of the hands has nothing to do with the magnitude of the angle between them.

5. The size and shape of a body may remain the same, when the body is transferred from one position to another in space.

By this means we frequently compare one portion of space with another.

For example, suppose that a material surface in the shape

of a triangle first occupied the space ABC, and was then

In such a case as this we often speak of transferring the triangle ABC to the triangle DEF; such language is really incorrect, inasmuch as a triangle, a line, &c., indicate fixed portions of space.

What is really transferred is the material surface which at one time exactly occupied the portion of superficial space indicated by the triangle ABC, and was afterwards made to occupy the space DEF.

This method has been employed to compare the magnitudes of lines and angles in Defs. 6 and 11 supra.

6. When we know that a certain line, or point, or surface must exist, we shall assume that we have found this line, or point, or surface, and reason upon such assumption accordingly.

for its extremities, and drawing such a line is called joining AB.

So, again, if AB be a finite straight line, we shall assume that straight lines, as BC or AE, may be drawn of any length, and making one straight line with AB.

The drawing such lines is called 'producing AB towards B or towards A respectively.'

Again, we assume that a point D may be found which is the middle point of the straight line AB, and the finding

such a point we call 'bisecting AB in D.' And so on in many similar cases.

7. Besides the axioms already explicitly referred to, there are many other geometrical truths assumed by us as too obvious to require express mention. E. g. :

If the indefinite straight line BD were drawn between the sides AB and

B

D

C

BC of the triangle ABC, as in the annexed figure, we should take it for granted

that the straight line BD must meet the side AC in some point.

A

D

And if C and D were two points situated on opposite

B

sides of the straight line AB of indefinite length, we should take it for granted that the straight line CD must meet the line AB.

Again, we assume it to be self-evident that if a point A within a closed figure DCE be joined to -B a point B without that figure, then the joining line must meet the boundary of the figure in some point; and similarly in

E

many other cases.

If we anticipate the definitions of the length of a curved line and of the equality of curved lines which are given hereafter in the course of Book II., and which have been postponed with the view of simplifying the commencement of the subject as much as possible, we may enunciate the second axiom of the text in the following more general language, namely:

The straight line joining any two points is the shortest distance between these points.

We may also prove that this is not really an independent statement, but that it follows from the first axiom and the assumption that one and only one line always exists which is shorter than any other line between two given points.

The proof will stand thus:

Let A and B be any two points, and AB the straight line which joins them, then AB shall be the shortest distance

between the points A and B.

Since there must be some line which is the A

shortest distance between A and B, if this

line be not AB let it, if possible, be ACB.

B

Let the line ACB be turned about AB until it coincides with some other line as ADB.

Because ACB may be made to coincide with ADB, the length of ACB must be equal to the length of ADB.

Because ACB is the shortest distance between the points A and B the length of ACB is less than that of ADB.

Therefore the length of ACB is both equal to and less than that of ADB, which is impossible.

Therefore ACB is not the shortest line between A and B, and in the same way it may be proved that no line except AB can be the shortest line between A and B. Therefore AB is the shortest line between A and B.

BOOK I.

SECTION I.-ON TRIANGLES.

PROPOSITION 1.

Any one side of a triangle is less than the sum and greater than the difference of the two remaining sides.

Let ABC be a triangle, then any one of its sides as BC shall be less than the sum, and greater than the difference, of the two remaining sides AB and AC.

B

A

Fig. 1.

C

Because the two points B and C are joined by the straight line BC and the broken line BAC,

therefore the length of BC is less than that of BAC

(Ax. 2),

that is, BC is less than BA+AC.

Again, suppose that AB is not greater than AC, Then AB+BC is greater than AC by what has been already proved,

If from the ends of a side of a triangle there be drawn two straight lines to a point within the triangle, these two lines shall be together less than the two remaining sides of the triangle.

From B and C the ends of the side BC of the triangle ABC, let the two straight lines BD and CD be drawn to