Book I.

If ABC, DEF be two triangles, fuch that the two fides AB and AC of the one are equal to the two ED, DF of the

A A

other, and the angle BAC, contained by the fides AB, AC of the one, equal to the angle EDF, contained by the fides ED, DF of the other; the triangles ABC and EDF are every way equal.

On AB let a triangle be conftituted every way equal to the triangle DEF; then, if this triangle coincide with the triangle ABC, it is evident that the propofition is true, for it is equal to DEF by hypothefis, and to ABC, because it coincides with it; wherefore ABC, DEF are equal to one another. But if it does not coincide with ABC, let it have the pofition ABG; and first, supposfe 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 ABC are not equal, and they are also equal by hypothefis, which is impoffible. Therefore the point G muft 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 fuppofition, which is impoffible. Therefore G muft coincide with C, and the triangle AGB with the triangle ACB. But AGB is every way equal to DEF, therefore ACB and DEF are alfo every way equal. Q. E. D.

By help of the fame poftulate, the 5th may also be very eafily demonftrated.

Let ABC be an ifofceles triangle, in which AB, AC are the equal fides; the angles ABC, ACB oppofite to these fides are alío equal.

Draw

Draw the ftraight line EF equal to BC, and fuppofe that Book I. on EF the triangle DEF is conftituted every way equal to the triangle ABC, that is, having DE equal to AB, DF to AC, the angle EDF to the angle BAC, the angle ACB to the angle DFE, &c.

AA

Then, because DE is equal to AB, and AB equal to AC, DE is equal to AC; and, for the fame reafon, DF is equal to AB. And 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 alfo, by hypothefis, equal to the angle DFE; therefore the angles ABC, ACB are equal to one another. Q.E. D.

Thus alfo, the 8th Propofition may be demonftrated independently of the 7th.

Let ABC, DEF be two triangles, of which the fides AB, AC are equal to the fides DE, DF, each to each, and also the bafe BC to the bafe EF; the angle BAC is equal to the angle EDF.

On BC, which is equal to EF, and on the fide of it oppofite to the triangle ABC, let a triangle BGC be conftituted every way equal to the triangle DEF, that is, having GB equal to DE, GC to DF, the angle BGC to the angle EDF, &c.: join AG.

Because GB and AB are each equal, by hypothefis, to DE, AB and GB are equal to one another, and the triangle ABG is ifofceles. Wherefore alfo (5. 1.) the angle BAG is equal

to

394

Book I.

to the angle BGA. In the fame way, it is fhewn that AC is equal to GC, and the angle CAG to the angle CGA. There

fore adding equals to equals, the two angles BAG, CAG together are equal to the two angles BGA, CGA together; that is, the whole angle BAC to the whole BGC. But the angle BGC is, by hypothefis, equal to the angle EDF, therefore alfo the angle BAC is equal to the angle EDF. Q.

E. D.

Such demonftrations, it must, however, be acknowledged, trespass against a rule which Euclid has uniformly adhered to throughout the Elements, except where he was forced by neceffity to depart from it. This rule is, that nothing is ever fuppofed to be done, the manner of doing which has not been already taught, and the contruction derived either directly from the three poftulates laid down in the beginning, or from fome problems already reduced to thofe poftulates. Now, this rule is not effential 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 an impoffibility. The only ufe, therefore, of Euclid's rule is to guard against the introduction of impoffible hypothefes, or the taking for granted that a thing may exift which in fact implies a contradiction; from fuch fuppofitions, false conclu

fions

1

395

fions might, no doubt, be deduced, and this rule is therefore Book I. ufeful as far as it answers the purpose of excluding them. But the foregoing poftulatum could never lead to suppose the actual existence of any thing that is impoffible; for it only fupposes the existence of a figure equal and fimilar to one already exifting, but in a different part of space from it, or, to fpeak more precifely, having one of its fides in an affigned pofition. As there is no impoffibility in the existence of one of these figures, it is evident that there can be none in the existence of the other.

PROP. VII.

Dr Simfon has very properly changed the enunciation of this propofition, which, as it ftands in the original, is confiderably embarraffed and obfcure. His enunciation, with very little variation, is retained here.

PROP. XXI.

It is effential to the truth of this propofition, that the ftraight lines drawn to the point within the triangle be drawn from the two extremities of the bafe; for, if they be drawn from other points of the bafe, their fum may exceeed the fum of the fides 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 demonftration is of a kind that does not belong to this place. If it be required fimply to fhew, that in certain cafes the fum of the two lines drawn to the point within the triangle may exceed the fum of the fides of the triangle, the demonftration is eafy, and is given nearly as follows by Pappus, and alfo by Proclus, in the 4th Book of his Commentary on Euclid.

Let ABC be a triangle, having the angle at A a right angle; let D be any point in AB; join CD, then CD will be greater than AC, because in the triangle ACD, the angle CAD is

greater

Book I.

greater than the angle ADC. From DC cut off DE equal
to AC; bifect CE in F, and

join BF; BF and FD are great-
er than BC and CA.

Because CF is equal to FE,
CF and FB are equal to EF
and FB, but CF and FB are
greater than BC, therefore EF
and FB are greater than BC. To
EF and FB add ED, and to BC

A

D

B

add AC, which is equal to ED by construction, and BF and FD will be greater than BC and CA. Q. E. D.

It is evident, that if the angle BAC be obtufe, the fame reafoning may be applied.

This propofition is a fufficient vindication of Euclid for having demonftrated the 21ft propofition, which fome affect to confider as felf-evident; for it proves, that the circumstance on which the truth of that propofition depends is not obvious, nor that which at firft fight it is fuppofed to be, viz. that of the one triangle being included within the other. For this reafon I cannot agree with M. Clairaut, that Euclid demonftrated this propofition only to avoid the cavils of the Sophifts. But I muft, at the fame time, obferve, that what the former has faid on the subject has certainly been misunderstood, and, in one refpect, unjustly cenfured by Dr Simfon. The exact tranflation of his words is as follows: "If Euclid has taken the trouble "to demonftrate, that a triangle included within another has "the fum of its fides lefs than the fum of the fides of the tri"angle in which it is included, we are not to be furprifed. "That geometer had to do with those obftinate Sophifts, who "made a point of refufing their affent to the most evident "truths," &c. (Elemens de Geometrie par M. Clairaut. Pref.)

Dr Simfon fuppofes M. Clairaut to mean, by the propofition which he enunciates here, that when one triangle is in cluded in another, the sum of the two fides of the included triangle is neceffarily lefs than the fum of the two fides of the triangle in which it is included, whether they be on the fame base or not. Now, this is not only not Euclid's propofition, as Dr Simfon remarks, but it is not true, and is direct

ly