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Book I. remedied, if to Euclid's three postulates we be allowed to add
the following: If there be two equal straight lines, and if any figure whatever be constituted on one, a figure every way equal to it may be constituted on the other. Thus, if AB and DE be two equal straight lines, and ABC a triangle on the base AB, a triangle DEF, every way equal to ABC, may be supposed to be constituted on DE as a base. By this it is not meant to assert that the method of describing the triangle DEF is actually known, but merely that the triangle DEF may be conceived to exist, in all respects equal to the triangle ABC. Now there is no truth whatever that is better entitled than this to be ranked among the postulates or axioms of geometry; for the straight lines AB and DE being every way equal, there can be nothing belonging to one that may not also belong to the other.
On the strength of this postulate the fourth proposition is thus demonstrated.
If ABC, DEF be two triangles such that the two sides AB, AC of one are equal to the two ED, DF of the other, and the angle BAC contained by the sides AB, AC of one equal to the angle EDF contained by the sides ED, DF of the other; the triangles ABC and EDF are every way equal.
On AB let a triangle be constituted every way equal to the triangle DEF; then, if this triangle coincide with the triangle ABC, it is evident that the proposition is true, for it is equal to DEF by hypothesis, and to ABC because it coincides with it; wherefore ABC, DEF are equal to each other. But if it do not coincide with ABC, let it have the position ABG; and first suppose G not to fall on AC; then the angle BAG is not equal to the angle BAC. But the angle BAG is equal to the angle EDF; therefore EDF and BAC are
not equal, and they are also equal by hypothesis, which is im- Book I. possible. Therefore the point G must fall upon AC. Now, if it fall upon AC, but not at C, then AG is not equal to AC; but AG is equal to DF; therefore DF and AC are not equal, and they are also equal by supposition, which is impossible. Therefore G must coincide with C, and the triangle AGB with the triangle ACB. But AGB is every way equal to DEF, therefore ACB and DEF are also every way equal. Q. E. D.
By help of the same postulate the fifth proposition may be easily demonstrated.
Let ABC be an isosceles triangle, in which AB, AC are the equal sides; the angles ABC, ACB opposite to these sides are also equal.
Draw the straight line EF equal to BC, and suppose that on EF the triangle DEF is constituted every way equal to the triangle ABC, that is, having DE equal to AB, DF to
F AC, the angle EDF to the angle BAC, the angle DFE to the angle ACB, &c. Then, because DE is equal to AB, and AB is equal to AC, DE is equal to AC. For the same reason DF is equal to AB. Because DF is equal to AB, DE to AC, and the angle FDE to the angle BAC, the angle ABC is equal to the angle DFE (4. 1.). But the angle ACB is also, by hypothesis, equal to the angle DFE; therefore the angles ABC, ACB are equal to each other, Q. E. D.
Thus also the eighth proposition may be demonstrated independently of the seventh.
Let ABC, DEF be two triangles, of which the sides AB, AC are equal to the sides DE, DF, each to each, and the base BC to the base EF; the angle BAC is equal to the angle EDF,
On BC, and on the side of it opposite to the triangle ABC, let a triangle BGC be constituted every way equal to the triangle DEF, that is having GB equal to DĖ, GC to DF, the angle BGC to the angle EDF, &c. Join AG. Because GB and AB are each equal, by hypothesis, to DE, they
are equal to each other; therefore the triangle ABG is isosceles; wherefore the angle BAG is equal to the angle BGA (5. 1.). In the same way it may be shown that AC is equal to CG, and the angle CAG to the angle CGA. Therefore the two angles BAG, CAG together are equal to the two angles BGA, CGA together; that is, the whole angle BAC is equal to the whole angle BGC. But the angle BGC is, by hypothesis, equal to the angle EDF; therefore also the angle BAC is equal to the angle EDF. Q. E. D.
Such demonstrations, it must, however, be acknowledged, trespass against a rule which Euclid has uniformly adhered to throughout the Elements, except where he was forced by necessity to depart from it. This rule is, that nothing is ever supposed to be done, the manner of doing which has not been already taught, and the construction derived either directly from the three postulates laid down in the beginning, or from some problems already reduced to those postulates. Now this rule is not essential to geometrical demonstration, where, for the purpose of discovering the properties of figures, we are certainly at liberty to suppose any figure to be constructed, or
any line to be drawn, the existence of which does not involve Book I. an impossibility. The only use, therefore, of Euclid's rule is to guard against the introduction of impossible hypotheses, or the taking for granted that a thing may exist which in fact implies a contradiction. From such suppositions false conclusions might, no doubt, be deduced, and this rule is therefore useful as far as it answers the purpose of excluding them. But the foregoing postulatum could never lead to suppose the actual existence of any thing that is impossible ; for it only supposes the existence of a figure equal and similar to one already existing, but in a different part of space from it, or, to speak more precisely, having one of its sides in an assigned position. As there is no impossibility in the existence of one of these figures, it is evident that there can be none in the existence of the other.
Dr. Simson has very properly changed the enunciation of this proposition, which, as it stands in the original, is considerably embarrassed and obscure. His enunciation, with very little variation, is retained here.
It is essential to the truth of this proposition, that the straight lines drawn to the point within the triangle be drawn from the two extremities of the base; for, if they be drawn from other points of the base, their sum may exceed the sum of the sides of the triangle in any ratio that is less than that of two to one. This is demonstrated by Pappus Alexandrinus in the 3d Book of his Mathematical Collections, but the demonstration is of a kind that does not belong to this place.
Thomas Simpson, in his Elements of Geometry, has objected to Euclid's demonstration of this proposition, because it contains no proof that the two circles made use of in the construction of the problem must cut each other; and Dr.
Book I. Simson, on the other hand, always unwilling to acknowledge
the smallest blemish in the works of Euclid, contends that the demonstration is perfect. The truth, however, certainly is, that the demonstration admits some improvement; for the limitation that is made in the enunciation of any problem ought always to be shown to be necessarily connected with the construction of it, and this is what Euclid has neglected to do in the present instance. The defect may easily be supplied, and Dr. Simson himself has done it in effect in his note on this proposition, though he denies it to be necessary.
Because of the three straight lines DF, FG, GH, any two are greater than the third, by hypothesis, FD is less than FG and GH, that is, tha FH, and therefore the circle described from D the centre F, with the distance FD, must meet the line FE between F and H; and for the like reason the
A circle described from
B the centre G at the dis
Ctance GH, must meet DG between D and G; therefore one of these circles cannot be wholly within the other. Neither can one circle be wholly without the other, because DF and GH are greater than FG; therefore the two circles must intersect each other.
The subject of parallel lines is one of the most difficult in the Elements of Geometry. It has accordingly been treated of in a great variety of different ways, of which, perhaps, there is none that can be said to have given entire satisfaction. The difficulty consists in converting the 27th and 28th propositions of Euclid, or in demonstrating that two parallel straight lines, or such as do not meet each other, when they meet a third line, make the alternate angles with it equal, or which comes to the same, are equally inclined to it, and make the exterior angle equal to the interior and opposite. In order to