We now go forward :

Look at, and for the present confine your attention wholly to, the two straight lines BC and EF (do not think about the triangles). Observe that B is the beginning and C is the end of one of these straight lines, and that E

AA

been shown that the beginning B lies on the beginning E, and that the end C lies on the end F.

Now these two straight lines being together at their beginning and at their end, is it possible for them to separate at all between the beginning and the end? 'that is now the question.'

Here are a couple of pens; if the feather ends are brought together and the quill ends are brought together, they do separate between the be

ginning and the end, and by

thus separating they enclose

or shut up a space; but that is because the pens are not straight. Two straight lines cannot enclose a space (Ax. 10); it takes at least three straight lines to enclose or shut up a space. Therefore, the two straight lines BC and E F, being together at their beginning and together at their end, must keep together from beginning to end.

What is the end of all this? Why this: that the two straight lines BC and E F are together at their beginning, and are together at their end, and do not separate between the beginning and the end: and therefore they fill the same space.

In fact, if you attempt to draw the point of your pen along the line joining B and C, and again along the line. joining E and F, you find that you have in both cases drawn the point of your pen through the same space. But magni

tudes which fill the same space are equal. Therefore the base B C is equal to the base E F (Ax. 8).1

We have thus proved, in accordance with the eighth axiom, that the base B C is equal to the base E F. It will be a slight task compared with the foregoing to prove that the remaining angles, to which the equal sides are opposite, are equal, and that the triangles are equal.

These are also proved equal by axiom 8; we must therefore see if they fill the same space. First let us consider the angles A B C and DEF. Confine your attention wholly to these angles: they are respectively contained by the straight lines A B, B C, and D E, E F.

Now ask yourself the question, where is AB lying? The answer is, it is lying along D E, for we put it there. And where is B C now lying? It is lying along EF; we have just proved that it must come there. Whence it must follow that the angles ABC and DEF fill or occupy the same space. For if now you make (see fig. p. 46) the arc, or curved rim, described page 6, to indicate the angle contained by the straight lines A B, BC; that is, the angle ABC; and again, if you make a similar curved rim to indicate the angle contained by the straight lines D E, E F, that is, the angle DEF, you find that you draw the curved rims in both cases through the same space; that is, the angles A B C and D E F fill the same space, and therefore, in accordance with the eighth axiom, they are equal.

Again the angles ACB and D FE are proved to be equal in precisely the same manner. AC is now lying on DF [not because we put it there, as in the case of A B and D E, but] because, having put A B on D E, we proved that A C must lie along DF, and CB, as said above, has been proved to lie along FE, so that the two rims or arcs, if drawn to indicate the angles ACB and D FE, would both be drawn through the same space, thereby showing that the

If we show that two lines begin together, end together, and keep together from beginning to end, this is quite enough to prove that they fill the same space; for lines have no thickness; they can only fill or occupy space of one dimension—that is, length.

two angles ACB and DFE both fill the same space, and therefore again by the eighth axiom they are equal.

Once more. In order to prove that the triangles are equal, we follow the same line of reasoning, thus :—The straight line AB was placed on DE, and hence we proved that A C must lie along D F, also that BC must lie along EF; hence evidently if you put your finger first on the space enclosed or shut up within the three straight lines. A B, BC, and CA, and again on the space shut up or enclosed within the three straight lines D E, EF, and FD, you find you are putting your finger, in both cases, in the same space; that is, the triangles ABC and DEF both fill the same space,1 and therefore they are equal (Ax. 8).

It is all over now. You have thoroughly and rigorously proved, in accordance with the eighth axiom, that, if two triangles have two sides of the one equal to two sides of the other, and have likewise the angles contained by those sides equal, they are equal in every respect—that is, in respect of sides, in respect of angles, and in respect of area.

It will be useful to notice the three following remarks:-
:-

1. In order to prove that two lines fill the same space, it is necessary to show that they begin together, that they end together, and that they keep together from beginning to end.

2. In order to prove that two angles fill the same space, it is necessary to show that the two lines containing one angle lie, as far as they go, on the two lines which contain the other angle.

3. In order to prove that two areas fill the same space, it is necessary to show that the three boundary lines which enclose one of the areas, lie along the three boundary lines which enclose the other area.

It remains to write out the proposition for your guidance in the form in which it might be written in an examination.

It is only necessary to show that the boundaries (see Def. 13) of the triangles lie together, in order to prove that the triangles fill the same space, for triangles are surfaces which have only length and breadth and no thickness.

THE FOURTH PROPOSITION WRITTEN OUT.

PROPOSITION IV. THEOREM.

General Enunciation.

It is given that two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal:---

It is required to prove that the triangles are equal in every respect; that is, that their bases are equal, that their remaining angles are equal, each to each, viz., those to which the equal sides are opposite, and that the triangles are equal.

B

Particular Enunciation.

Let A B C and DEF be two triangles which

E

D

angle EDF:

have the two sides BA, AC, equal to the two sides ED, DF, each to each, viz., B A to E D, and A C to D F, and have likewise the included angle BAC equal to the included

It is required to prove that the triangles ABC and DEF are equal in every respect; that is, that the base B C is equal to the base E F, and that the remaining angles, to which the equal sides are opposite, are equal, each to each, viz., ABC to DEF, and ACB to DFE; also that the triangles ABC and DEF are equal.

Construction.

Apply the triangle A B C to the triangle D E F, so that the point A may be on the point D, and that the straight line A B may lie along the straight line DE.

Demonstration.

1. Then the point B will come on the point E, because the straight lines A B, DE are equal (hyp.). 2. And A B lying along D E, AC will lie along DF, because the included angles BAC and EDF are equal (hyp.).

3. And AC lying along D F, the point C will come on the point F, because the straight lines A C and DF are equal (hyp.).

4. And it has been proved that B will come on E.

5. And B lying on E, and C on F, the straight line BC must lie along the straight line EF from beginning to end, for if they separated, they would enclose a space, which is impossible (Ax. 10).

6. Wherefore the two straight lines B C, E F— beginning together, ending together, and keeping together from beginning to end-fill the same space, and are therefore equal (Ax. 8).

7. Also A B, B C lying along D E, E F, the angles ABC and DEF fill the same space, and are therefore equal (Ax. 8).

8. Likewise A C, C B lying along DF, FE, the angles ACB and DFE fill the same space, and therefore are equal (Ax. 8).

9. And the three boundary lines A B, BC, CA lying along D E, EF, FD, the triangles A B C and DEF fill the same space, and are therefore equal (Ax. 8).

Wherefore if two triangles have two sides of the one equal to two sides of the other, and have likewise the angles contained by those sides equal, they are equal in every respect, which was to be shownor quod erat demonstrandum, or Q.E.D.