When it was propofed to exhibit equality to us, then it was requifite to make four theorems, receiving two in parallelograms, but the other two in triangles, fituated either upon the fame, or upon equal bases. But now by conversion, we neglect the theorems which are converse in parallelograms, and efteem fuch as are converfe in triangles worthy of relation. And the reason of this is, because the mode of demonftration in parallelograms, is the fame indifferently, by a deduction to an impoffibility, and the construction is fimilar. But we are content when we have exhibited the way in more fimple figures, I mean triangles, to leave to the more curious the fame mode of reason-` ing in the reft: fince it is eafy, at the fame time, to perceive that there is the fame method in these. For when we affume equal parallelograms, upon the fame bafe, or upon equal bases, we must say that they are also between the fame parallels. For if they are not, either one of them falls within, when the parallels which are in the other are produced; or without. But which ever cafe is affumed, when we receive it and its parallels, we may exhibit the fame confequences as in triangles, I mean that the whole will be equal to its part but this is impoffible. It is however manifeft, that the inftitutor of the Elements very properly adds the particle, and at the fame parts. For it is poffible that equal triangles, may be affumed upon the same base, one, indeed, at these parts, but the other at different parts, and yet thefe will not be entirely between the fame parallels : for neither will they be contained under the fame altitude. And on this account he added the particle.

But fince a parallel may be drawn in a two-fold refpect, according to an abfurd hypothefis, i. e. either within or without, Euclid draws it within: but we can exhibit the fame confequences, by drawing it without. For let the equal triangles ab c, dbc, be upon bafe, and at the fame parts, I fay that they are between the fame parallels, and that the right line connected at their vertices, is parallel to the bafe. Let the right line a d be connected. But if this is not parallel, let the line, external to this, i. e. a e be parallel, and let b d be produced to the point e, and connect e c. The triangle, therefore, abc,

abc, is equal to the triangle e b c, the whole to the part. But this is impoffible; and hence, the parallel line does not fall external to a d. But it is fhewn by the inftitutor of the Elements, that neither does it fall within: and hence a d, is parallel to b c. Hence too, equal triangles, which are at the fame parts, and upon the fame base are parallel to each other. And thus the remaining part of the deduction to an impoffibility is demonftrated. But it is worthy of obfervation, that fince the converfion of theorems is triple (for either the whole is converted to the whole, as we have noticed, in the eighteenth and nineteenth theorems; or the whole to the part, as the fixth and fifth; or the part to the part, as the eighth and the fourth for the whole is not a datum, in the one, and an object of investigation in the other : nor is the object of investigation, a datum, but a part) these triangular theorems appear to be of this kind. For, that the triangles are equal, is an object of inveftigation in the preceding; but this is not a datum alone in these, because it affumes, befides this, a part of that which was hypothefis in those. For to ftand upon the fame, or upon equal bases, is a datum in thefe, as well as in thofe, except that in these hypotheses he adds fomething which was neither an object of investigation, nor a datum in thefe; fince the particle at the fame parts, is over and above extrinfically affumed.

When it was propofed to exhibit equality to us, then it was requifite to make four theorems, receiving two in parallelograms, but the other two in triangles, fituated either upon the fame, or upon equai bafes. But now by conversion, we neglect the theorems which are converse in parallelograms, and efteem such as are converse in triangles worthy of relation. And the reafon of this is, because the mode of demonftration in parallelograms, is the fame indifferently, by a deduction to an impoffibility, and the construction is fimilar. But we are content when we have exhibited the way in more fimple figures, I mean triangles, to leave to the more curious the fame mode of reafoning in the reft: fince it is eafy, at the fame time, to perceive that there is the fame method in these. For when we affume equal parallelograms, upon the fame bafe, or upon equal bases, we must say that they are also between the fame parallels. For if they are not, either one of them falls within, when the parallels which are in the other are produced; or without But which ever cafe is affumed, when we receive it and its parallels, we may exhibit the fame consequences as in triangles, I mean that the whole will be equal to its part but this is impoffible. It is however manifeft, that the inftitutor of the Elements very properly adds the particle, and at the fame parts. For it is poffible that equal triangles, may be affumed upon the fame base, one, indeed, at these parts, but the other at different parts, and yet these will not be entirely between the fame parallels : for neither will they be contained under the fame altitude. And on this account he added the particle.

But fince a parallel may be drawn in a two-fold refpect, according to an abfurd hypothefis, i. e. either within or without, Euclid draws it within: but we can exhibit the fame confequences, by drawing it without. For let the equal triangles abc, dbc, be upon bafe, and at the fame parts, I fay that they are between the fame parallels, and that the right line connected at their vertices, is parallel to the bafe. Let the right line a d be connected. But if this is not parallel, let the line, external to this, i. e. a e be parallel, and let b d be produced to the point e, and connect e c. The triangle, therefore,

abc,

a b c, is equal to the triangle e bc, the whole to the part. But this is impoffible; and hence, the parallel line does not fall external to a d. But it is fhewn by the inftitutor of the Elements, that neither does it fall within and hence a d, is parallel to b c. Hence too, equal triangles, which are at the fame parts, and upon the fame base are parallel to each other. And thus the remaining part of the deduction to an impoffibility is demonftrated. But it is worthy of observation, that fince the conversion of theorems is triple (for either the whole is converted to the whole, as we have noticed, in the eighteenth and nineteenth theorems; or the whole to the part, as the fixth and fifth ; or the part to the part, as the eighth and the fourth for the whole is not a datum, in the one, and an object of investigation in the other : nor is the object of investigation, a datum, but a part) these triangular theorems appear to be of this kind. For, that the triangles are equal, is an object of investigation in the preceding; but this is not a datum alone in these, because it affumes, befides this, a part of that which was hypothefis in those. For to ftand upon the fame, or upon equal bases, is a datum in thefe, as well as in those, except that in these hypothefes he adds fomething which was neither an object of investigation, nor a datum in thefe; fince the particle at the fame parts, is over and above extrinfically affumed.

PROPOSITION XL. THEOREM XXX.

Equal triangles which are upon equal bafes, and at the fame parts, are between the fame parallels.

There is the fame mode of converfion too in the prefent theorem, and a fimilar demonftration; and that part of the deduction to an impoffibility, which is omitted by the inftitutor of the Elements, is demonftrated after the fame manner, and there is no occafion for repetition. But fince these three conditions are in the aforefaid propofitions, fituation upon equal, or on the fame bafes; pofition between the fame parallels; and equality of triangles and parallelograms, it is manifeft that we may variously convert, by. always connecting two, and leaving one. For we either fuppofed the bafes the fame, or equal, and triangles and parallelograms between the fame parallels, and thus we form four theorems; or we confider the triangles and parallelograms equal, and the bafes the fame, or equal, and thus we produce another four, two of which the elementary inftitutor omits, viz. those which respect parallelograms, but the other two relative to triangles, he exhibits; or laftly, when we have affumed them equal, and between the fame parallels, we prove the remainder, that they are either upon the fame, or upon equal bases, and produce another four, which the inftitutor of the Elements entirely neglects. For there is the fame demonstration in these, except that two of thefe four are not effentially true. Thus, equal parallelograms or triangles, between the fame parallels, are not neceffarily upon the fame base: but all this is true in these hypotheses, that they are upon the fame or equal bafes; but the other does not entirely follow the affumed hypotheses. Hence, as all these theorems are ten, the geometrician speaks of fix, and neglects four, left he should labour in vain, by repetition, fince the demonftration is the fame. For it may be fhewn in triangles, that if they are equal, and between the fame parallels, they will either be upon the fame, or upon equal bafes. For let it be denied, and if poffible, let the triangles a b c, def, have these conditions, upon unequal bafes bc, ef. Let too, bc, be the greater, and cut off bb, equal to

ef,