CHAPTER II.

A GEOMETRICAL proposition is a proposal to establish as true some new principle, or to show how to effect some new process; in the former case the proposition is called a theorem, and in the latter a problem.

The postulates are problems admitted to be solved, and the axioms are theorems admitted to be true without proof.

The enunciation states the nature of the proposal, first in general terms, and then sometimes with reference to some particular diagram; the part of the enunciation which is supposed to be true is called the hypothesis, and the inference to be deduced is the conclusion. For example, in the proposition "If two sides of a triangle are equal, the angles opposite to them shall be equal" the hypothesis is that the two sides are equal, and the conclusion to be proved is that the opposite angles are equal.

Starting with the hypothesis, processes are effected or assertions made, as allowed by the definitions, postulates, and axioms, until the conclusion is deduced; these intermediate steps constitute the construction and demonstration. When a fact has been thus established it may be asserted in proving other statements.

Taking for granted that, in the solution of a proposition, the construction is correct, and each step of the demonstration true, let us consider how the truth of the hypothesis and that of the conclusion depend on one another; bearing carefully in mind that the truth of the demonstration depends on the truth of the hypothesis, and not on the truth of the conclusion.

Suppose a magnitude called A to be equal to another magnitude B, and that from that fact a third magnitude C has been proved equal to B; then since magnitudes which are equal to the same magnitude are equal to one another, we conclude that is equal to C. Here the hypothesis is that A is equal to B, and the demonstration shows C to be equal to B; the conclusion is that A is equal to C.

We do not assert that A is equal to B, but prove that if A is equal to B, then A is equal to C. If the supposition that A is equal to B turned out to be false, we should not know that the conclusion that A is equal to C was also false, for if A is not equal to B, we might not be able to prove the intermediate step that B is equal to C. Again, if it is true that A is equal to C', it is not necessarily true that A is equal to B; for the demonstration that C is equal to B is dependent on the fact that A is equal to B,.and not on the fact that A is equal to C'; therefore, given A equal to C, we might not be able to show that B is equal to C, and hence should not know that A is equal to B. Lastly, if the conclusion turns out to be false, and every step in the demonstration is correct, that on which the demonstration depends, viz. the hypothesis, must also be false.

Generally then, it must be remembered that the hypothesis is not necessarily true, but that if it is true, the conclusion is true; but it does not follow that if the hypothesis is false the conclusion is false; establishing the conclusion does not establish the hypothesis, but a proof that the conclusion is false is a proof that the hypothesis is false.

One proposition is said to be the converse of another when the hypothesis of each is the conclusion of the other, and one proposition is said to be the contrary of another when, with the same hypothesis, the conclusion of the one denies that which the conclusion of the other asserts. Thus of the proposition: "If A is equal to B, then C is equal to D," the converse is: "If C is equal to D, then A is equal to B," and the contrary is: "If A is equal to B, then C is not equal to D."

If a proposition be true, it does not follow that its converse is true, but proving that the contrary of a proposition is false, establishes the truth of the proposition itself.

The obverse of a proposition is that if the hypothesis is false, the conclusion is also false; the obverse of a proposition is therefore not necessarily true.

The contrapositive of a proposition is, that if the conclusion is false, the hypothesis is also false, and this is always true if the proposition itself be true.

Of the proposition: "If A is B, then C is D," the obverse is: "If A is not B, then C is not D," and the contrapositive is: "If C is not D, then A is not B." The converse of the obverse of any proposition is the contrapositive of the same proposition.

If of a proposition, its converse, its contrapositive and its obverse, any two which are not contrapositive of one another, be true, all four are

true.

One of Euclid's books is supposed by some to have been lost, which took a comprehensive view of the subject, and guarded against the fallacies into which men were liable to fall. It is certainly astonishing what fallacious arguments beginners will accept as proofs; and the learner cannot be too strongly cautioned against being misled by false arguments.

In considering the different steps in the demonstration of a proposition, he must be very cautious in accepting as self-evident, any statements other than those assumed by Euclid; and, unless he is able to quote the definition, postulate, axiom, or formerly established fact on which the truth of a statement directly depends, he should consider it very carefully before admitting its truth.

The following are particular cases of some of the general principles stated in the axioms, and, as such, can be accepted as true, and referred to as axioms.

If the same magnitude be added to equals the wholes are equal. If equals be added to the same magnitude the wholes are equal. If the same magnitude be taken from equals, the remainders are equal.

If equals be taken from the same magnitude the remainders are equal.

If the same magnitude be added to unequals the wholes are unequal. If the same magnitude be taken from unequals the remainders are unequal.

Magnitudes which are double of equal magnitudes are equal to one

another.

Magnitudes which are halves of equal magnitudes are equal to one

another.

The following facts are assumed by Euclid in his first book to be self-evident but not mentioned in his definitions, postulates or axioms:

(a) A figure is said to be described on or applied to a straight line when that straight line becomes one of its boundaries.

(b) Any one figure may be placed on or adjacent to any other in any required position.

(c) If two equal straight lines be placed on one another so that an extremity of one coincides with an extremity of the other, the other extremities coincide.

(d) If the extremities of two lines which cannot enclose a space coincide, the lines coincide

(e) If the angular points of two angles coincide in position and the containing lines coincide in direction, each with each, the angles are equal.

(f) If two equal angles be placed on one another so that the angular points coincide in position, and one line containing one angle coincides in direction with one line containing the other, then the other containing lines are in the same direction.

(g) If one straight line is a boundary of each of two figures it is said to be common to both, and if two angles of two triangles coincide they are said to be common to both triangles.

(h) The circumference of a circle cuts any straight line drawn from its centre which is greater than the radius of the circle.

(i) If the centre of a circle is on one side of an unlimited straight line, and a point in the circumference is on the other side of it, the circumference of the circle will cut the straight line in two points.

(j) If the extremities of a given finite straight line be the centres of two circles, and the given straight line be the radius of each circle, the circumferences of these circles will cut.

(k) If the extremities of a given finite straight line be the centres of two circles, and if of the radii of these circles and the given line any two whereof are together greater than the third, the circumferences of the circles will cut.

(7) of any two magnitudes there is one that is not greater than the other.

(m) of any two unequal magnitudes one must be greater than the

other.

(n) One magnitude cannot be both equal to and greater than another.

(0) If one magnitude is not equal to nor less than a second magnitude, the first is greater than the second.

(p) The whole is equal to all its parts together.

(q) If a magnitude is double of one of two equal magnitudes, it is also double of the other.

(r) If two magnitudes are equal they are together double of one of them.

(8) If one magnitude is equal to a second, any multiple of the first is equal to the same multiple of the second.

(t) If one of two equal magnitudes be greater than a third then the other of the two equal magnitudes is also greater than the third.

(u) If a magnitude be greater than one of two equal magnitudes it is also greater than the other.

(v) If of three magnitudes the first is greater than the second and the second is greater than the third, much more then is the first greater than the third.

(w) If one magnitude is greater than a second and another magnitude be added to each, the first whole is greater than the second whole.

(x) Any one of two unlimited straight lines may be assumed to be greater than a given part of the other, or any other finite straight line.

(y) The straight line which divides the angle of a triangle into any two parts, will, if produced, cut the opposite side.

(*) A straight line drawn from a point without a triangle, parallel to one of its sides, will, if produced far enough, cut the other sides of the triangle or those sides produced.

(4) An unlimited straight line drawn parallel to one side of a parallelogram from a point without it will cut the adjacent sides produced.

(B) If a straight line cut one of two or more unlimited parallel straight lines, it will, if produced, cut the others.

(C) If through any two angular points of a triangle parallels be drawn to the opposite sides, these parallels will meet if produced.

(D) If two straight lines are respectively parallel to two other straight lines which meet at right angles to one another, the first two straight lines will meet if produced far enough.

(E) If a square be described on the hypotenuse of a right-angled triangle, a straight line drawn from the right angle parallel to one of those sides of the square which are adjacent to the hypotenuse will if produced far enough cut the hypotenuse and the side of the square which is opposite to the hypotenuse.

(F) The squares on equal straight lines are equal, and the sides of equal squares are equal.

(G) If the conclusion of a theorem is false the hypothesis is false, or, in other words, no false statement can be a necessary consequence of

a true statement.

Of the above statements, some are axioms, some definitions, and some postulates.