ticle upon the fame right line, is affumed, left we should exhibit upon another, two right lines equal each to each, and employ the propofition for the purpofe of circumvention. In the fecond place, he does not say upon what right line, to constitute two right lines fimply equal to two (for this is poffible) but each to each. For what wonderful thing is it, that he should take both equal to both, who extends one of the conftituted lines, and contracts the other? But each to each, (says he) is impoffible. In the third place, he adds the particle, to different points. For what, if fome one, when he has formed two lines equal to the first two, each to each, fhould connect these with thofe in the fame point, which joins the subject right lines in the vertex; and fhould conftitute thefe? For the extremes of equal right lines perfectly coincide. In the fourth place, he adds the particle to the fame parts t. For what if one fubject right line being given, we fhould place two of the right lines on one fide, and the other two on the oppofite fide, fo that this common right line should be the basis of the two triangles with oppofite vertexes? Left, therefore, we should form an erroneous figure, and charge our deception on the inftitutor of the Elements, he adds the particle to the fame parts. In the fifth place, he fubjoins, having the fame extremes with the two right lines first drawn. For it is poffible to conflitute upon the fame right line, two right lines equal to two, cach to each, drawn to different points, and to the fame parts, by employing the whole right line, and conftructing upon it, these two right lines; but then the lines laft drawn, will not have the fame extremes with thofe conftituted at firft.

For

if we conceive in a quadrangle two diagonals drawn on one of its fides, two lines shall be equal to two; a fide and diameter to its paralel fide, and the other diameter. But in this cafe the equal right lines will not have the fame extremes. For neither the parallel fides, nor the diameters, will mutually poffefs the fame extremes; and yet they will be equal. Thefe diftin&tions, therefore, being preferved, the truth of the propofition, and the certainty of the reafoning, is evinced.

See the Comment of Clavius on this propofition.

But

But perhaps, fome, notwithstanding all these terms producing fcience, will dare to object, that these hypotheses being admitted, it is poffible to effect what the geometrician affirms to be impoffible. For

let there be a right line a b, and upon this two lines a d, db, equal to two ac, cb, and let the former be external to the latter, being drawn to different points dc, and terminated in the fame extremes a and b. Let a c too, be equal to a d: but be to b d. This objection, then, we fhall confute, by connecting the line de, and producing the lines a c, and a d, to the points ef. For these being constructed, it is manifest that the triangle a c d is ifofceles, ad, being equal to a c, from hypothesis; and the angles under the bafe e c d, fdc are equal. The angle fd c, therefore, is greater than the angle bdc. Much more then is the angle b c d greater than the angle bd c. But again, because the line d b, is equal to the line bc, the angles are equal, i. e. the angle b c d, to the angle b d c. therefore, is both greater and equal, which is impoffible. And this

5

alfo at the base The fame angle,

is

is what we faid in our expofition of the fifth theorem, that though the equality of the angles under the base, was not useful to the demonftrations of the following theorems, yet it procured the greatest utility in the folution of objections. For in the prefent inftance we have confuted the objection, by inferring that, because a c, and a d, are equal, the angles e cd, and fd c, are alfo equal. In a fimilar manner in other theorems, it will appear to be peculiarly useful for the folution -of doubts*.

But if any one fhould fay that there may be conftituted upon the right line ab, right lines bd, bc, equal to the right lines a c, a d, of which be may be equal to a c, but b d to ad; and that in this cafe they will be drawn to different points a and b, to the fame parts, and will have the fame extremes with a c, and a d, viz, c, and d, what hall we reply to this affertion? Shall we fay that it is requifite to conftitute the firft lines, upon the right line a b, and their equals upon the fame right line? For this is what the inftitutor of the Elements affirms in the propofition. But here, a c, and a d, are not constituted upon the right line a b, but only on one of its points. Hence, the lines a c, c b, and a d, d b, which stand on the right line a b, are different from the right lines, which were placed in the beginning, and to which they ought to be conftituted equal. Though at the fame time it is neceffary that the right lines conftituted upon a b, fhould be equal to those conftituted upon a b. And thus much may fuffice for objections against the prefent queflion. But that the prefent theorem is exhibited by the inftitutor of the elements, by a deduction to an impoffibility, and that this impoffible oppofes the common conception, affirming that the whole is greater than its part; and that the fame thing cannot be both greater and equal, is fufficiently manifeft. But this theorem feems to have been affumed for the fake of the eighth theorem. For it confers to its demonftration, and is neither fimply an element, nor elementary: fince it does not extend its utility to a multitude. And hence, we find it very rarely em-> ployed by the geometrician.

And from hence, alfo appears the emptinefs and arrogance of Mr. Simfon's note to this propofition, which we have already exploded.

PROPO

PROPOSIT ON VIII. THEOREM V.

If two triangles have two fides equal to two, each to each, and have the base equal to the bafe: then the angles contained by the equal right lines, fhall be equal to

each other.

This eighth theorem is the converfe of the fourth but it is not. affumed according to a principal conversion. For it does not make the whole of its hypothesis a conclufion; and the whole conclufion an hypothefis. But connecting together fome part of the hypothesis of the fourth theorem, and fome part of the objects of enquiry, it exhibits one of the data which it contains. For the equality of two fides to two, is in each an hypothesis; but the equality of base to base, is, in the fourth, an object of investigation, but in the present a datum; and the equality of angle to angle, is, in the former, a datum, but in the latter, an object of enquiry. Hence, a change alone of data, and objects of investigation, produces converfion. But if any one defires to learn the cause why this theorem is placed in the order of the eighth propofition, and not immediately after the fourth, as its converse, in the fame manner as the fixth after the fifth, of which it is the converse, fince many converted propofitions follow their precedents, and are exhibited after them without any intervening medium, to this we must reply, that the eighth, indeed, is indigent of the feventh propofition. For its truth is evinced by a deduction to an im~ poffibility, but the nature of an impoffible becomes known from the feventh. And, this again, in its demonftration, is indigent of the fifth. Hence, the feventh and fifth theorems were neceffarily affumed, previous to the prefent. But because the converfe to the fifth obtained a demonftration eafy, and from things first, it was very properly placed after the fifth, on aecount of its alliance with that theorem; and beVOL. II. cause

K

caufe, fince it is fhewn by a deduction to an impoffibility, it confutes that which is impoffible from common conceptions, and not as the eighth from another theorem. For things oppofing common conceptions, are more evident for the purpose of confutation than fuch as contradict theorems: fince thefe are affumed by demonftration, but the knowlege of axioms is better than demonftration. But the institutor of the elements exhibits what is now propofed from the previoufly demonftrated feventh theorem.

But the familiars of Philo affert, that they can demonftrate this theorem, without being indigent of any other. For let there be conceived (fay they) two triangles, a b c d e f, having two fides equal

to two, and the bafe bc equal to the bafe e f. Likewife let the bafes

coincide