grams. For a parallelogram is formed, as well from those equal and parallel right lines, which are drawn in the beginning, as from those which conjoin them, and which are in like manner fhewn to be equal and parallel. Hence, the propofition which immediately follows the prefent, contemplates the properties effentially inherent in these spaces, in a parallelogram as it were already conftructed. And these things are indeed manifeft. But it is requifite to confider the diligence which this propofition contains. In the first place, indeed, that it is not fufficient, that the lines which are conjoined fhould be equal: for the lines which connect equals, are not entirely equal, unless they are alfo parallel. For a triangle being ifofceles, and a point being affumed in one of the equal fides, and through this a line being drawn parallel to the bafis, equal lines fhall indeed conjoin parallels to the bafis, and the bafis itfelf, yet these parallels fhall not also be equal; and the fides will not be parallel, because they coincide at the vertex of the triangle.

In the fecond place, he confiders that the subject right lines being parallel, is not fufficient to constitute the equality of the lines which conjoin them. For this is evident from the preceding conftruction of the ifofceles triangle; fince the drawn right line, and the basis, are parallel, and yet the lines which connect them are not parallel, becaufe they are parts of the fides of the ifofceles triangle. The parallel pofition, therefore, of the lines which are conjoined, is requifite to the equality of the connecting lines: but the equality of the latter is neceffary to the parallel pofition of the former. On this account the inftitutor of the Elements affumes each, in those which are conjoined, for the purpose of exhibiting, that the connecting lines are as well equal, as parallel to one another. But in the third place, he intimates, that right lines being fuppofed both equal and parallel, their connecting lines will not be univerfally equal and parallel. For unless we make the conjunctions at the fame parts as in this cafe, the connecting lines cannot be parallel (fince they will cut each other), fo they may be fometimes equal, and fometimes not. For if you affume a quadrangle, or oblong, as abcd, and connect

the

the right lines a d, b c, the diameters are indeed equal, but not parallel, and they conjoin the equal and parallel oppofite fides of the aforefaid spaces. But if the figure be a Rhombus, or a Rhomboides, the diameters of these, are not only non-parallels, but also unequal. For fince a b, is equal to cd, but a c is common, and the angle bac,

a

is unequal to the angle a cd, the bases also are unequal. The infti

tutor of the Elements, therefore, very properly confidered, that the lines which conjoin equal and parallel lines, ought to make the conjunction at the fame parts, left a c, b d, being fuppofed equal and parallel, we should affume a d, b c, as the connecting lines, and not a b, and c d. For he fhews that thefe latter are equal and parallel but that the former are, indeed, never parallel, but equal, as we have observed in a quadrangle and oblong, but never in a rhombus and rhomboides; as the oppofite to this has been proved to be true, because they are unequal, on account of the inequality of the angles internal, and fituated at the fame parts.

PROPOSITION XXXIV. THEOREM XXIV. The oppofite fides and angles of parallelogrammic spaces are equal to each other an they are bifected by the dia

meter.

As from the preceding theorem, he had affumed a parallelogram already conftructed, he now contemplates its primarily inherent properties, and fuch things as exprefs its peculiar conftitution. But these are the following: that the fides and angles which are oppofite, are equal, and that the spaces themselves are bifected by the diameter. For that part of the propofition relates to the spaces, which fays: and they are bifected by the diameter. So that the area itself, is that whole which is bifected, and not the angles through which the diameter paffes. These three properties then, are effentially inherent in paralrelograms, the equality of the oppofite fides and angles, and the bisection of the spaces by the diameter. And you may observe that the properties of parallelograms are investigated from all thefe, viz. from the fides, from the angles, and from the areas. But as there are foar kinds of parallelograms, which Euclid defines in the hypotheses *, viz. a quadrangle, oblong, rhombus, and rhomboides, it deferves to be re

* In the definitions which are with great propriety called by the Platonists hypothefes, be cause their evidence is admitted without proof, which at the fame time they are capable of reeciying form the first philofophy.

marked, that if we divide these four into rectangles, and non-rectangles, we shall find, that not only the diameters bisect these spaces, but that the diameters them felves, are, indeed, in rectangles equal, but in non-rectangles unequal, as was observed in the preceding theorem. But if we divide them into equilateral, and non-equilateral, we shall again find that in the equilateral figures, not only the spaces are bifected by the diameters, but likewife the angles through which they are drawn: but in non-equilaterals this is never the cafe. For in a quadrangle, and a rhombus, the diameters bifect the angles, and not the spaces only: but in an oblong, and a rhomboides, they alone bifect the spaces. For let there be a quadrangle, or a rhombus, gca b, g

C

a

and a diameter g b. Because, therefore, the fides g c, c b, are equal to the fides g a, a b (for they are equilateral), and the angles g cb, ga b, are equal (for they are opposite), and the bafis alfo is common, hence, all are equal to all; and on this account the angles cga, a b c, are bisected. Again, let there be an oblong, or rhomboides

given. If, therefore, the angle bac, and the angle c db, is bifected

by

by the diameter, but the angle ca d, is equal to the angle a a b, the angle alfo, ba d, will be equal to the angle adb. Hence, the fide alfo, a b, will be equal to the fide b d. But they are unequal; and confequently the angle ba c, is not bifected by the diameter, nor its equal the angle c d b. That I may therefore comprehend the whole in a few words, in a quadrangle the diameters are equal, on account of the rectitude of the angles, and the angles are bisected by the diameters, on account of the equality of the fides, and the areas are bifected by the diagonal, on account of the common property of parallelograms but in an oblong, the diameters are indeed equal, because it is a rectangle, but the angles are not bifected by the diameters, because it is not equilateral, though the divifion of spaces into equal parts, is alfo inherent in this figure, fo far as it is a parallelogram : but in a rhombus the diameters are unequal, because it is not a rectangle, but the spaces are not only bifected by thefe, because it is a parallelogram, but the angles alfo, because it is equilateral; and in the remaining figure, i. e. a rhomboides, the diameters are unequal, because it is not a rectangle, and the angles are cut by these into unequal parts, because it is not equilateral, and the spaces alone fituated at each part of the diagonals, are equal, becaufe it is a parallelogram. And thus much concerning obfervations of this kind, which exhibit the diverfity found in the four divifions of parallelograms.

But we must not pafs over in filence, the artificial confequence appearing in this theorem, that of theorems, fome are univerfals, but others non-univerfals. But we shall speak concerning each of these, when we divide the object of investigation, which has, indeed, one part univerfal, but the other non-univerfal. For though every theorem may seem to be univerfal, and every thing exhibited by the elementary inftitutor may appear to be of this kind (as in the present he may not only feem to affert, that in all parallelograms univerfally, the oppofite fides and angles are equal, but likewise that each is bifected by the diameter), yet we must say that fome things are univerfally exhibited, but others not univerfally. For it is cuftomary to call the univerfal which affirms the truth concerning every thing of which it is predicated, differently from that univerfal, comprehending all things in

which