at the point b, of the triangle a be, is bifected by the right line bf, and e f is greater than fa, it follows from what has been previously fhewn, that the fide be, is greater than the fide ba. But be has been fhewn to be equal to a c. The fide, therefore, a c, is greater than the fide a b; and the object of enquiry is exhibited. And it is manifeft that the inftitutor of the Elements, avoiding a variety of demonftration, refrains from this mode of demonftrating, and employs a method of proof, which leads from divifion to an impoffibility, because he was willing to fabricate the converse to the preceding, without any intervening medium. For the eighth theorem, indeed, which is the converfe of the fourth, brings great difturbance, because it makes converfion difficult to be known. For it is more excellent to exhibit converfe theorems, by preferving the continuity through an impoffible, than to deftroy the continuity by a principal demonftration. And hence, Euclid fhews almost all converse theorems by a deduction to an impoffibility.

PROPOSITION XX.

THEOREM XIII.

Two fides of every triangle, however taken, are greater than the remaining one.

The Epicureans oppofe the prefent theorem, afferting that it is manifest even to an afs; and that it requires no demonftration: and befides this, that it is alike the employment of the ignorant, to confider things manifeft as worthy of proof, and to affent to fuch as are of themselves immanifeft and unknown; for he who confounds thefe, feems to be ignorant of the difference between demonftrable and indemonftrable. But that the prefent theorem is known even to an afs, they evince from hence, that grafs being placed in one extremity of the fides, the ass feeking his food, wanders over one fide, and not over two, Against these we reply, that the present theorem is indeed manifest to sense, but not to reafon producing fcience: for this is the cafe in a variety of concerns. Thus for example, we are indubitably'

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certain

certain from fenfe, that fire warms, but it is the business of science to convince us how it warms; whether by an incorporeal power, or by corporeal sections; whether by fpherical, or pyramidal particles. Again, that we are moved is evident to fenfe, but it is difficult to affign a rational caufe how we are moved; whether over an impartible, or over an interval: but how can we run through infinite, fince every magnitude is divifible in infinitum? Let, therefore, the prefent theorem, that the two fides of a triangle are greater than the remainder, be manifeft to fenfe, yet it belongs to fcience to inform us how this is effected. And, thus much may fuffice against the Epicureans.

But it is requifite to relate the other demonftrations of the prefent theorem, fuch as Heron, and the familiars of Porphyry have fabricated, without producing the right line, after the manner of Euclid. Let there be a triangle a bc, it is requifite, therefore, to fhew, that that the fides ab, ac, are greater than the fide bc.

Bifect the

angle at a, by the right line a e.

Because, therefore, the angle aec, is external to the triangle a be, it is greater than the angle bae. But the angle ba e, was placed equal to the angle e a c. The angle, therefore, a ec, is greater than the angle e a c. Hence, the fide alfo ac, is greater than the fide c e. And for the fame reason the fide a b,

is

is greater than the fide be. For the angle a e bis external to the triangle a ec, and is greater than the angle ca e; that is than the angle eab. And on this account the fide a b, is greater than the fide be. The fides, therefore, a b, a c, are greater than the whole fide b c. And the like may be fhewn of the other fides. Let there again be a triangle a b c. If therefore the triangle a bc, be equilateral, two fides will be doubtless

greater than the remaining one: for when there are three equal quantities, any two are double of the remainder. But if it be ifofceles, it will have a base either lefs, or greater than each of the equal fides. If therefore the base be lefs, the two fides are given greater than the remainder. But if the base be greater, let it be bc, and cut off from it a part equal to either of the fides, which let be b e, and connect a e. Because, therefore, the angle a ec, is external to the triangle a e b, it is greater than the angle bae. On the fame account the angle ae b, is greater than the angle cae. Hence, the angles about the point e, are greater than the whole angle about the point a, of which be a is equal to ba e, fince a b is equal to be. The remainder, therefore, a ec, is greater than the remainder c ae. Hence, the fide a c, is greater than the fide ce. But the fide a b, was alfo equal to the fide be. The fides, therefore, a b, a c, are greater than the fide bc.

But

But if the triangle a bc, be fcalene, let the greateft fide be ab, the

middle a c, and the leaft c b. The greatest fide, therefore, affumed with either of the others, exceeds the remainder for by itself it is greater than either. But if we are defirous of fhewing that the fides a c, cb, are greater than the greatest fide a b, we muft employ the fame conftruction as in the ifofceles triangle, cutting off from the greater fide, a part equal to one of the other fides, and connecting the line ce, and ufing the external angles of the triangles.

Let there be again any triangle a b c, I fay that the fides a b, a c, are greater than the fide bc. For if they are not greater, they are

either equal or lefs. Let them be equal, and cut off b e, equal to a b. The remainder, therefore, e c, is equal to a c. Because then, ab, be, are equal, they fubtend equal angles; and this is likewife true of ac, ec, because they are equal. Hence, the angles at the point, are equal to the angles at the point a, which is impoffible. Again let

the

the fides a b, ac, be lefs than b c, and cut off b d, equal to a b, and

ec to a c. Because, therefore, a b is equal to b d, the angle bda, is

because a c is equal to c e, the Hence, the two angles b da,

not unequal to the angle ba d. And angle c e a, is equal to the angle e a c. ce a, are equal to the two b a d, and e a c. Again, because the angle bda is external to the triangle a dc, it is greater than the angle e ac: for it is greater than cad. By a fimilar reafon alfo, because the angle cea, is external to the triangle a be, it is greater than the angle bad: for it is greater than the angle ba e. Hence, the angles b da, ce a, are greater than the two bad, eac. But they were also equal to them, which is impoffible. The fides, therefore, ab, a c, are neither equal to, nor less than the fide bc, but greater. And the like may be exhibited in others.

PROPOSITION XXIV. THEOREM XIV. If upon one fide of a triangle, two right lines beginning from the extremities, are internally conftituted, the conftituted right lines will be lefs than the other fides of the triangle, but they will contain a greater angle. That which is expreffed by the propofition, is, indeed, manifeft; and the demonftration adopted by the elementary inftitutor, is evident; and the theorem is confequent to the first principles, fince it depends on two theorems, the one previously exhibited, and the fix

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