(55)   The geometrical axioms are certain general propositions, the truth of which is taken to be self-evident, and incapable of being established by demonstration. According to the spirit of this science, the number of axioms should be as limited as possible. A proposition, however self-evident, has no title to be taken as an axiom, if its truth can be deduced from axioms already admitted. We have a remarkable instance of the rigid adherence to this principle in the twentieth proposition of the first book, where it is proved that ‘two sides of a triangle taken together are greater than the third;’ a proposition which is quite as self-evident as any of the received axioms, and much more self-evident than several of them.

On the other hand, if the truth of a proposition cannot be established by demonstration, we are compelled to take it as an axiom, even though it be not self-evident. Such is the case with the twelfth axiom. We shall postpone our observations on this axiom, however, for the present, and have to request that the student will omit it until he comes to read the commentary on the twenty-eighth proposition. See Appendix II.

Two magnitudes are said to be equal when they are capable of exactly covering one another, or filling the same space. In the most ordinary practical cases we use this test for determining equality; we apply the two things to be compared one to the other, and immediately infer their equality from their coincidence.

By the aid of this definition of equality we conceive that the second and third axioms might easily be deduced from the first. We shall not however pursue the discussion here.

^★★^★ The fourth and fifth axioms are not sufficiently definite. After the addition or subtraction of equal quantities, unequal quantities continue to be unequal. But it is also evident, that their difference, that is, the quantity by which the greater exceeds the less, will be the same after such addition or subtraction as before it.

The sixth and seventh axioms may very easily be inferred from the preceding ones.

The tenth axiom may be presented under various forms. It is equivalent to stating, that between any two points only one right line can be drawn. For if two different right lines could be drawn from one point to another, they would evidently enclose a space between them. It is also equivalent to stating, that two right lines being infinitely produced cannot intersect each other in more than one point; for if they intersected at two points, the parts of the lines between these points would enclose a space.

The eleventh axiom admits of demonstration. Let A B and E F be perpendicular to D C and H G. Take any equal parts E H, E G on H G measured from the point E, and on D C take parts from A equal to these (Prop. III. Book I.) Let the point H be conceived to be placed upon the point D. The points G and C must then be in the circumference of a circle described round the centre D, with the distance D C or H G as radius. Hence, if the line H G be conceived to be turned round this centre D, the point G must in some position coincide with C. In such a position every point of the line H G must coincide with C D (ax. 10.), and the middle points A and E must evidently coincide. Let the perpendiculars E F and A B be conceived to be placed at the same side of D C. They must then coincide, and therefore the right angle F E G will be equal to the right angle B A C. For if E F do not coincide with A B, let it take the position A K. The right angle K A C is equal to K A D (11), and therefore greater than B A D; but B A D is equal to B A C (11), and therefore K A C is greater than B A C. But K A C is a part of B A C, and therefore less than it, which is absurd; and therefore E F must coincide with A B, and the right angles B A C and F E G are equal.

The postulates may be considered as axioms. The first postulated, which declares the possibility of one right line joining two given points, is as much an axiom as the tenth axiom, which declares the impossibility of more than one right line joining them.

In like manner, the second postulate, which grants the power of producing a line, may be considered as an axiom, declaring that every finite straight line may have another placed at its extremity so to form with it one continued straight line. In fact, the straight line thus placed will be its production. This postulate is assumed as an axiom in the fourteenth proposition of the first book.

(56)   Those results which are obtained in geometry by a process of reasoning are called propositions. Geometrical propositions are of two species, problems and theorems.

A problem is a proposition in which something is proposed to be done; as a line to be drawn under some given conditions, some figure to be constructed, &c. The solution of the problem consists in showing how the thing required may be done by the aid of the rule and compass. The demonstration consists in proving that the process indicated in the solution really attains the required end.

A theorem is a proposition in which the truth of some principle is asserted. The object of the demonstration is to show how the truth of the proposed principle may be deduced from the axioms and definitions or other truths previously and independently established.

A problem is analogous to a postulate, and a theorem to an axiom.

A postulate is a problem, the solution of which is assumed.

An axiom is a theorem, the truth of which is granted without demonstration.

In order to effect the demonstration of a proposition, it frequently happens that other lines must be drawn besides those which are actually engaged in the enunciation of the proposition itself. The drawing of such lines is generally called the construction.

A corollary is an inference deduced immediately from a proposition.

A scholium is a note or observation on a proposition not containing any inference, or, at least, none of sufficient importance to entitle it to the name of a corollary.

A lemma is a proposition merely introduced for the purpose of establishing some more important proposition.