we did not know that AC was equal to DF, the triangles might fall thus or otherwise;

DA

Thirdly it is because DF is equal to AC, that C and F coincide, but still the triangles might fall thus or otherwise;

AD

and, lastly, since two straight lines cannot enclose a space, they coincide thus:

E

AD

A close examination of the fourth proposition will shew the student that, besides assuming the right to superpose the triangles, Euclid assumes the converse of the eighth axiom, a definition of coincident angles, and the fact that a theorem is true if its contrary is false.

The converse of the eighth axiom is true of straight lines and of angles, but not of all magnitudes; for example, a square may be equal to a rhombus, but they could not be made to coincide.

The student must not, without consideration, admit that if one plane figure can be placed on another, so that the boundaries of the one, coincide with the boundaries of the other, each with each, the figures themselves must coincide. It is quite true that two plane figures so placed cannot enclose a space, and that therefore they do coincide and are equal, but still the question why remains unanswered. Euclid, whilst assuming this fact, considers it necessary to prove, in his eleventh book, that the common section of two planes which cut, is a straight line.

Euclid does not, prior to this proposition, give any definition of what is meant by the coincidence of angles; but we can, from the demonstration, infer what he means when he says: "angles coincide and are equal." The eighth axiom, translated from the Greek text, runs thus:—Things which can be fitted upon, or which coincide with each other, are equal. The student must clearly understand that an angle which is the inclination of two straight lines several miles in length, may be fitted on, and be said to coincide with, another angle which is contained by lines only a few inches in length.

The student should now write out or repeat the fourth proposition with each of the following diagrams. He will notice that he is not asked to deduce all the conclusions as enunciated by Euclid.

да

(79) In the triangles PQR and STV;

GIVEN PQ equal to SV, PR equal to ST, and the angle QPR equal to the angle VST.

IT IS REQUIRED TO PROVE that QR is equal to TV, the angle PQR equal to the angle SVT, and the angle PRQ equal to the angle STV.

(80) In the triangles AFC and AGB, (which overlap and partly fill the same space);

GIVEN AB equal to AC, and AG equal to AF, and the angle FAC equal to the angle GAB;

IT IS REQUIRED TO PROVE that FC is equal to GB, the angle ABG equal to the angle ACF, and the angle AFC equal to the angle AGB.

(81) In the triangles FBC and GCB, (which overlap and partly fill the same space);

GIVEN FB equal to GC, and FC equal to GB, and the angle BFC equal to the angle CGB;

IT IS REQUIRED TO PROVE that the angle FBC is equal to the angle GCB, and the angle BCF equal to the angle CBG.

A

(82) In the triangles ADC and BDC;

GIVEN that AC is equal to BC, CD common to the two triangles ACD and BCD, and the angle ACD equal to the angle BCD;

IT IS REQUIRED TO PROVE that AD is equal to BD.

I. 5. The above being the diagram for the fifth proposition, the triangles

ДА

which partly occupy the same space, thus:

are, by the fourth proposition, equal in all respects; but it is not necessary to show, as Euclid does, the triangles themselves equal in order to deduce the result.

which partly occupy the same space, thus:—

are, by the fourth proposition, equal in all respects.

A corollary to a proposition is an inference which may be deduced immediately from that proposition.

Nine corollaries are usually stated to the propositions of the first book; in the case of four only of these is it shewn how the inference can be be deduced. The student should write out rigid demonstrations

of the others.

I. 6. The sixth proposition is the converse of one part of the fifth. When Euclid has to establish the converse of a theorem already proved, he almost always uses the indirect method of proof; that is to say, he shows the proposition to be true by proving its contrary to be false. There are in the first book instances of a proposition and its converse being both proved directly, and there are also cases in which both proved indirectly.

are

This indirect method of proof is known as "Reductio ad absurdum.” It is perfectly rigid, but is sometimes regarded as being less satisfactory than a direct proof, inasmuch as it does not show why the theorem is true.

Here the student should refer back to page 9, and let him, in particular, distinguish clearly between the converse and the contrary of a proposition.

The converse of a proposition is not necessarily true, but, if the contrary is false, the proposition must be true.

In his sixth proposition Euclid proves that the sides of the triangle are equal, by showing that they cannot be unequal.

This proposition is not required by Euclid in his first book, and might have been proved directly by the twenty-sixth proposition.

I. 7. This proposition is demonstrated only in order to prove the eighth proposition. The student would find any attempt to convince himself of the truth of this theorem by the aid of instruments, impracticable, and the proposition affords a very good illustration of the unsatisfactory nature of such attempts. It is only the rigid accuracy of the demonstration which can convince us that the theorem is true.

The enunciation of this proposition would, if translated exactly from