Chapter VIII: Book IV (2)

In the second place, he considers that the subject right lines being parallel, is not sufficient to constitute the equality of the lines which conjoin them. For this is evident from the preceding construction of the isosceles triangle; since the drawn right line, and the basis, are parallel, and yet the lines which connect them are not parallel, because they are parts of the sides of the isosceles triangle. The parallel position, therefore, of the lines which are conjoined, is requisite to the equality of the connecting lines: but the equality of the latter is necessary to the parallel position of the former. On this account the institutor of the Elements assumes 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 supposed both equal and parallel, their connecting lines will not be universally equal and parallel. For unless we make the conjunctions at the same parts as in this case, the connecting lines cannot be parallel (since they will cut each other), so they may be sometimes equal, and sometimes not. For if you assume a quadrangle, or oblong, as _a b c d_, and connect the right lines _a d_, _b c_, the diameters are indeed equal, but not parallel, and they conjoin the equal and parallel opposite sides of the aforesaid spaces. But if the figure be a Rhombus, or a Rhomboides, the diameters of these, are not only non-parallels, but also unequal. For since _a b_, is equal to _c d_, but _a c_ is common, and the angle _b a c_, is unequal to the angle _a c d_, the bases also are unequal. The institutor of the Elements, therefore, very properly considered, that the lines which conjoin equal and parallel lines, ought to make the conjunction at the same parts, lest _a c_, _b d_, being supposed equal and parallel, we should assume _a d_, _b c_, as the connecting lines, and not _a b_, and _c d_. For he shews that these 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 opposite to this has been proved to be true, because they are unequal, on account of the inequality of the angles internal, and situated at the same parts.

PROPOSITION XXXIV. THEOREM XXIV.

The opposite sides and angles of parallelogrammic spaces are
equal to each other and they are bisected by the diameter.

As from the preceding theorem, he had assumed a parallelogram already constructed, he now contemplates its primarily inherent properties, and such things as express its peculiar constitution. But these are the following: _that the sides and angles which are opposite, are equal, and that the spaces themselves are bisected by the diameter_. For that part of the proposition relates to the spaces, which says: _and they are bisected by the diameter_. So that the area itself, is that whole which is bisected, and not the angles through which the diameter passes. These three properties then, are essentially inherent in parallelograms, the equality of the opposite sides and angles, and the bisection of the spaces by the diameter. And you may observe that the properties of parallelograms are investigated from all these, viz. from the sides, from the angles, and from the areas. But as there are four kinds of parallelograms, which Euclid defines in the hypotheses[34], viz. a quadrangle, oblong, rhombus, and rhomboides, it deserves to be remarked, 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 themselves, 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 bisected by the diameters, but likewise the angles through which they are drawn: but in non-equilaterals this is never the case. For in a quadrangle, and a rhombus, the diameters bisect the angles, and not the spaces only: but in an oblong, and a rhomboides, they alone bisect the spaces. For let there be a quadrangle, or a rhombus, _g c a b_, and a diameter _g b_. Because, therefore, the sides _g c_, _c b_, are equal to the sides _g a_, _a b_ (for they are equilateral), and the angles _g c b_, _g a b_, are equal (for they are opposite), and the basis also is common, hence, all are equal to all; and on this account the angles _c g a_, _a b c_, are bisected. Again, let there be an oblong, or rhomboides given. If, therefore, the angle _b a c_, and the angle _c d b_, is bisected by the diameter, but the angle _c a d_, is equal to the angle _a d b_, the angle also, _b a d_, will be equal to the angle _a d b_. Hence, the side also, _a b_, will be equal to the side _b d_. But they are unequal; and consequently the angle _b a c_, is not bisected 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 sides, and the areas are bisected 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 bisected by the diameters, because it is not equilateral, though the division of spaces into equal parts, is also inherent in this figure, so 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 bisected by these, because it is a parallelogram, but the angles also, 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 situated at each part of the diagonals, are equal, because it is a parallelogram. And thus much concerning observations of this kind, which exhibit the diversity found in the four divisions of parallelograms.

But we must not pass over in silence, the artificial consequence appearing in this theorem, that of theorems, some are universals, but others non-universals. But we shall speak concerning each of these, when we divide _the object of investigation_, which has, indeed, one part universal, but the other non-universal. For though every theorem may seem to be universal, and every thing exhibited by the elementary institutor may appear to be of this kind (as in the present he may not only seem to assert, that in all parallelograms universally, the opposite sides and angles are equal, but likewise that each is bisected by the diameter), yet we must say that some things are universally exhibited, but others not universally. For it is customary to call the _universal_ which affirms the truth concerning every thing of which it is predicated, differently from _that universal_, comprehending all things in which the same symptom is inherent. Thus it is universal, that every isosceles triangle has three angles equal to two right, because it is true of all isosceles triangles: and it is universal that every triangle has three angles equal to two right, because it comprehends all things, in which this is essentially inherent. On which account we affirm that the possession of three angles equal to two right, is to be primarily manifested of a triangle. According to this signification, therefore, of theorems, calling some universal, but others non-universal, we must affirm that the present theorem, has, indeed, one of its objects of investigation universal, but the other non-universal. For the possession of opposite sides and angles that are equal, is a universal, since it is alone inherent in parallelograms: but that the diameter bisects the space, is not universal, because it does not comprehend all things in which this symptom is beheld; for this is inherent in a circle and ellipsis. And it appears, indeed, that primary conceptions of such like concerns, are more particular, but that in their progress they comprehend the whole. For when the ancients had contemplated that a diameter bisects an ellipsis, circle, and parallelogram, they afterwards surveyed that which was common in these. But we are deceived (says Aristotle[35]) when a non-universal is exhibited as universal, because that common something in which the symptom is primarily inherent, is nameless. For we cannot say what that is, which is common to numbers and magnitudes, motions and sounds; and it is likewise difficult to express what is common to an ellipsis, circle, and parallelogram. For one of these figures is right-lined, but the other circular, and the third mixt; and on this account we conceive that he exhibits universally, who demonstrates that a diameter bisects every parallelogram, because we do not at the same time perceive that common something, on account of which, this is true. This then in parallelograms, is not an universal of this kind, on account of the aforesaid cause; but the proposition is universal, which asserts, that every parallelogram has its opposite sides and angles equal. For if any figure is supposed, having its opposite sides and angles equal, it may be shewn to be a parallelogram. Thus let such a figure be _a b c d_[36], and its diameter _a d_. Because, therefore, the sides _a b_, _b d_, are equal to the sides _a c_, _c d_, and the angles comprehended by them are equal, and the base common, all will be equal to all. The angle, therefore, _b a d_, is equal to the angle _a d c_, and the angle _a d b_, to the angle _c a d_. Hence, _a b_, is parallel to _c d_, and _a c_ to _b d_. And on this account the figure _a b c d_, is a parallelogram. And thus much may suffice for observations of this kind.

But the institutor of the Elements seems to have composed the name of parallelograms, by taking an occasion from the preceding theorem. For when he had shewn that right lines, which conjoin equal and parallel right lines at the same parts, were themselves also equal and parallel, it is evident that he pronounces as well the opposite sides which conjoin, as those which are conjoined, to be parallel: but that he very properly calls the figure which is contained by parallels, a parallelogram, in the same manner as he denominates that which is comprehended by right lines rectilineal. And it is evident that the institutor of the Elements places a parallelogram among quadrilateral figures. But it is worthy our observation and enquiry, whether every right-lined figure, which is composed from equal sides, since it is equilateral and equiangular, is to be called a parallelogram. For a figure of this kind also, has its opposite sides equal and parallel, as likewise the opposite angles equal. As for example, a sexangle, and an octangle, and a decangle. Thus, if you conceive a sexangle _a b c d e f_, and connect a right line _a c_, you may shew that _a f_ is parallel to _c d_. For the angle at the point _b_, is one right, and the third part of a right angle; and this is true of every angle of a sexangle, since it is equiangular. Besides the side _a b_, is equal to the side _b c_, for it is placed equilateral. Each of the angles, therefore, _b a c_, _b c a_, is a third part of a right angle. Hence, the angles _f a c_, _a c d_, are right angles. And on this account _a f_, is parallel to _c d_. In like manner we may shew that the other opposite sides are parallel, and the same may be evinced in an octangle, and in the remaining figures of this kind. If, therefore, that is a parallelogram which is comprehended by parallels oppositely situated, a parallelogram will likewise subsist among non-quadrilateral figures. But it appears that with the institutor of the Elements a parallelogram is quadrilateral. And this is particularly perspicuous in that theorem, in which he says, _that a parallelogram which has the same base with a triangle, and is between the same parallels, is double of the triangle_: for this is alone true in quadrilateral figures.

PROPOSITION XXXV. THEOREM XXV.

Parallelograms which are upon the same base, and between
the same parallels, are equal to each other.

As we have said that of theorems, some are universal, but others particular, and as dividing these we have subjoined, that some are also simple, but others composite, and have shewn the nature of each, so according to another distinction, we assert that some of these are local, but others non-local. But I call those local, to which the same symptom happens in a certain place; and I denominate the place of a line or a superficies, that situation, which produces one and the same symptom. For of local theorems some are constructed in lines, but others in superficies. And because of lines, some are plane, but others solid, the plane being those of which there is a simple conception in a plane, as of a right line: but the solid those whose origin appears from a certain section of a solid figure, as of cylindric, spiric, and conic lines, I should say, that of the local theorems which are constructed in lines, some have a plane, but others a solid place. The present theorem, therefore, is both local, and local in lines, and a plane. For the whole space which lies between the parallels, is the place of the parallelograms constructed upon the same base; and which the institutor of the Elements shews to be equal to each other. But of those local theorems which are called solid, let the following be an example[37]. _The parallelograms which are inscribed within the asymptotes and the hyperbola, are equal_: for it is evident that the hyperbola is a solid line.

But Chrysippus, as we are informed by Geminus, assimilates theorems of this kind to ideas. For as ideas comprehend the origin of infinites, in terminated limits, so in these also there is a comprehension of infinites, in terminated places, and by this boundary equality appears, since the altitude of the parallels remaining the same, if infinite parallelograms are conceived upon the same base, they may all be shewn to be equal to each other. The present, therefore, is with the institutor of the Elements, the first local theorem. And he appears, when, agreeable to an elementary mode, he had distinguished theorems by a variety, according to all possible divisions, with great propriety not to have omitted, considering their idea of this kind. Nevertheless, as his discourse, for the present, is concerning right lines, he delivers local plane theorems in right lines: but in the third book, as he treats concerning things which may be contemplated of circles, and their symptoms, he likewise teaches the particulars, which are constructed in circumferences belonging to local, and at the same time, plane theorems. And such, among these, is the theorem, which says, _that angles in the same segment, are equal to one another_. Also this which asserts, _that the angles in a semicircle are right_. For if infinite angles are constructed in a circumference, the same base remaining, they are all shewn to be equal; but if that which is comprehended by the base and the circumference, is a semicircle, they are all shewn to be right. And these, indeed, correspond in proportion to triangles and parallelograms upon the same base, and between the same parallels. And such is the species of theorems called local, by the ancient mathematicians.

But perhaps it may seem perfectly worthy of admiration, to such as are unskilled in contemplations of this kind, that parallelograms constructed upon the same base, and between the same parallels, should be equal to each other. For it may be asked, how is this possible, since the longitude of the spaces, constructed on the same base, increases in infinitum? Since as much as we produce the parallels, by so much we may also increase the longitudes of the parallelograms. But some one may not improperly enquire how, while this takes place, the equality of the spaces remains. For if the breadth is the same (since the base is one), but the length is greater, will not the space also be greater? The present theorem, therefore, and that which follows concerning triangles, are among the number of mathematical theorems, which are denominated admirable. For mathematicians in theorems, as the Stoics in arguments, have established a _place_, which is called admirable, and they place the present among theorems of this kind. The vulgar, therefore, are immediately astonished, when they hear that the multiplication of length does not destroy the equality of spaces on the same base. We must nevertheless assert, that equality and inequality possess the greatest power in increasing or diminishing the spaces of angles. For in proportion as we make angles unequal, in such proportion we diminish the space, if the length and breadth remain the same. Hence, the increase of length is necessary, that we may preserve equality. Thus, for example, let there be a parallelogram _a b c d_, and let the side _a c_ be produced in infinitum, and let it be a right-angled parallelogram; and lastly, on the base _b d_, construct another parallelogram _b e f d_. That the length, therefore, is increased is evident: for the side _b e_, is greater than the side _a b_, since the angle at the point _a_, is right. But this necessarily takes place, as the angles of the parallelogram _b e f d_, are unequal, and some of them are acute, but others obtuse: and this happens, because the side _b e_, approaches after a manner to the side _b d_, and contracts the space. For let _b g_ be taken equal to _a b_, and through _g_, draw _g h_ parallel to _b d_. The length, therefore, of the parallelogram _b d g h_, is equal to the length of the parallelogram _a b c d_, and the breadth is the same, and yet one space is less than the other; for it is less than _b e f d_. Hence, the inequality of angles diminishes the area, but the increment of length adding as much as the inequality of angles takes away, preserves the equality of the spaces. But the boundary of the increase of length, is the place of the parallel lines. For when both the parallelograms are rectangular, and have an equal ambit, the quadrangle is shewn to be greater than the oblong[38]: but when they are both equilateral, and have consequently an equal ambit, that which is rectangular, is shewn to be greater than that which is non-rectangular[39]. For the rectitude of angles, and the equality of sides, possesses universal power in the augmentation of spaces. It is on this account that a quadrangle is the greatest of all figures with an equal ambit, and a rhomboides the least. And these observations we shall demonstrate in another place[40]: for they more properly belong to the hypotheses of the second book.

But with respect to the present theorem, it is requisite to know, that when Euclid calls parallelograms equal, he means the spaces, and not the sides: for he now discourses of areas. And we must likewise observe, that he first mentions trapeziums in the demonstration of this theorem: from whence also it is manifest, that he does not improperly teach us concerning a trapezium, in the definitions, when he informs us that it is indeed of a quadrilateral species, but is not a parallelogram. For the figure which has not its opposite sides and angles equal, falls from the order of parallelograms. The institutor of the Elements, therefore, as he had chosen a more difficult case, demonstrates the thing proposed. But if any one should say, let the parallelograms _a b c d_, and _b d c e_, be upon the same base _d b_, so that the side _c d_ may be the diameter of the parallelogram _a b_, we can shew that according to this position they are equal. For the triangle _b c d_, is the half of each parallelogram: because _c d_ is the diameter of _a b_, but _c b_ of _d e_; and diameters bisect parallelograms. Hence, _a b_ is equal to the parallelogram _d e_. Again, if any one should suppose that the side _a c_, of the parallelogram _a b_, is cut by the side _d c_, and that the parallelograms are situated as _a d b e_, _b d c f_, we can shew that these also are equal. For since the side _a e_, is equal to the side _c f_ (each because opposite being equal to _d b_), let the common right line _c e_ be taken away. Hence, _a c_ is equal to _e f_. But _a d_, also, is equal to _e b_, and the angle _c a d_, to the angle _f e b_. For _a d_ is parallel to _e b_; and hence, the base _c d_, is equal to the base _f b_, and the whole triangle _a d c_, is equal to the whole triangle _e b f_. Let the common trapezium _c b_, be added. The whole, therefore, _a b_, is not unequal to the whole _d f_. And here you may observe that these are the only three cases. For the side _d c_, either cuts the side _e b_, according to the position of the elementary institutor; or it falls on the point _c_, as in the penultimate description: or it cuts the line _a e_, according to the present supposition. And thus the theorem is shewn to be true according to all its cases. Lastly, as there is a two-fold difference of trapeziums, and one kind has neither of its opposite sides parallel, but the other has one side parallel to one, this latter species of trapeziums is alone employed by the geometrician throughout the elements, and in the present description: for _c e_ is parallel to _d b_.

PROPOSITION XXXVI. THEOREM XXVI.

Parallelograms which are upon equal bases, and between
the same parallels, are equal to each other.

The preceding theorem assumed, indeed, the same bases, but this receives them equal, and different from each other. But it is common to both, to suppose the parallelograms between the same parallels. It is requisite, therefore, that they should neither fall within, nor without their subject parallel lines. For parallelograms are said to be between the same parallels, when their bases and opposite sides are adapted to the same parallels. As to the rest, the institutor of the Elements, as he had assumed the bases entirely separate, exhibits the theorem. But nothing hinders our receiving them with this hypothesis, so that they may have a common part. For let _a b_, _c d_, be parallelograms upon equal bases _e b_, _f d_, having a common part, and constructed between the same parallels, I say that they are equal. Let the lines _e c_, _b g_, be connected. Because, therefore, _e f_, is equal to _b d_ (for the base _e b_, was supposed equal to the base _f d_), but the side _c f_, is equal to the side _d g_, and the angle _c f e_, is equal to the angle _g d b_, and hence, _c e_ is equal to _b g_. But it is also parallel to it. Hence, _c b_ is a parallelogram, and has the same base with each of the parallelograms _a b_, _c d_, and is between the same parallels. The parallelogram, therefore, _a b_, is equal to the parallelogram _c d_.

But if any one should suppose that the bases of the parallelograms have neither a common part, nor are separate from each other, but (which is the only remaining hypothesis) that they touch each other in one point, as in the parallelograms _a e_, _e d_, we must say that the base _b e_ is equal to the base _e f_, and to the side _c d_. Hence, also, the right line _c b_, is equal to the right line _d e_, and is parallel to it. For the lines which join equal and parallel lines, are themselves also equal and parallel. Hence, _b d_ is a parallelogram, and is upon the same base, and between the same parallels, with the parallelograms _c b_, _d e_. The parallelograms, therefore, _c b_, _d e_, are equal. But according to the first conception of a theorem, we may divide the constructions by asserting that the bases have either a common part, or touch each other, or are distant from each other. It is however possible, that though they may touch each other, as _b e_, _e f_, yet the whole parallelogram _d e_ may be supposed external to the side _c e_; or one side of the parallelogram _c f_, may be the diameter of the parallelogram _a e_; or the side _c e_, may cut the side _a c_; or the side _a c_, being produced beyond _a_, the side _c e_, may fall as the diameter of the parallelogram increased towards _a_, when the side _d f_ becomes the same as a line drawn from _a_ to _f_, or the side _c e_, may cut the side _a c_, produced beyond _a_; or the side _a c_, may be still farther produced beyond _a_, so that the side _c e_ may fall beyond the point, to which _a c_ was extended in the preceding case, and the side _d f_, may cut the line produced beyond _a_[41]. * * *

PROPOSITION XXXVII. THEOREM XXVII.

Triangles which are upon the same base, and between
the same parallels, are equal to each other.

_The beginning of this Commentary is wanting._

* * * * *

* * for those being equal, the spaces are unequal; and when these are unequal, those are shewn to be equal. And this is the case with Chorographers, when they reason concerning the magnitudes of cities, from their ambits. But formerly, certain persons deceived their partners, in the distribution of their possessions, deluding them by an excess of ambit, so as to make them believe that they received a greater portion of land, when they received a greater ambit; and that they were gainers, by changing spaces into areas of less ambit. [42]Thus two isosceles triangles being proposed, one of which has each of its equal sides, containing five parts, but the base six: and the other has each of its equal sides five parts, but the base eight; and let these parts be, for instance, cubits, or digits, these triangles will very much deceive the ignorant in their choice. For the ambit of the one is eighteen, and of the other sixteen measures. But a geometrician is not ignorant that the spaces are equal, though the ambits are unequal; since the area of each is twelve measures. For if you draw a perpendicular from the vertex, you will bisect the bases, and cause the half of the one to be three but of the other four measures: but the perpendicular on the contrary, will be there equal to four, but here equal to three; since it is requisite that the square from the quinary, should be equal to the squares from the perpendicular, and the half of the base. But if the base of the one is equal to three, the perpendicular must be four; and if the base of the other is equal to four, its perpendicular must be three. When, therefore, you have multiplied the half of the base with the perpendicular, you will have a space equal to the triangle: but this is the same in each, whether you multiply the quaternary with the ternary, or the ternary with the quaternary. And we have made these observations for the purpose of shewing that the equality of spaces is not to be entirely received from the ambits. Nor should we wonder, that though triangles upon the same base, may be infinitely increased between the same parallels, according to the remaining sides, yet the equality of the spaces immutably remains. But those triangles are said to be between the same parallels, which have their bases upon one of the parallel lines, and fix their vertices on the remainder; and whose vertices being connected, form one right line, parallel to the bases on the same right line.

PROPOSITION XXXVIII. THEOREM XXVII.

Triangles which are upon equal bases, and between the
same parallels, are equal to each other.

The present theorem also is local, because it corresponds in proportion with parallelograms, and supposes the situation of triangles upon equal bases. But Euclid seems, to me, to have delivered one demonstration by the first proposition of the sixth book of these four theorems, two of which are exhibited in parallelograms, and two in triangles: and two of which are on the same base, and the other two on equal bases. But that Euclid has performed this is unknown to the vulgar. For after he had shewn that triangles and parallelograms, which are under the same altitude, have the same proportion to each other as their bases, nothing demonstrates all these four theorems more universally, from proportion, than this theorem: since to possess the same altitude, is nothing else than being constituted between the same parallels. For all figures between the same parallels, are under the same altitude, and the contrary: since the altitude is the perpendicular, which extends itself from one parallel to the rest. In that proposition, therefore, it is shewn by proportion, that triangles and parallelograms, under the same altitude, that is, situated between the same parallels, are to each other as their bases, and so when the bases are equal, the spaces are equal; and when those are double, these will be double; and when the bases have any other proportion, the spaces also will have to each other the same proportion. But for the present, because it is not proper that he should use proportion, who has not yet explained its nature, he is content with equality and identity alone: for the identity of bases is collected from equality. Hence, these four theorems are comprehended in that one; not only because he shews by one demonstration, whatever are contained in these four, but likewise, because he adds what was wanting to their perfection, viz. identity of proportion, though the bases are unequal. But that this theorem, also, has many cases, and that it is possible that the bases of the triangles may be assumed, either having the same part as in parallelograms; or possessing no common part, but touching each other according to one point; or entirely separate, so that a line may intervene between them, is manifest, even to such as are endued with slender capacities. And this too is evident, that according to all cases, however the bases or vertices may be situated, the same method of proceeding must be adopted as in parallelograms; viz. parallels to the sides must be drawn, and produced both ways, and the equality of the triangles exhibited.

PROPOSITION XXXIX. THEOREM XXXIX.

Equal triangles, which are upon the same base, and at the
same parts, are between the same parallels.

When it was proposed to exhibit equality to us, then it was requisite to make four theorems, receiving two in parallelograms, but the other two in triangles, situated either upon the same, or upon equal bases. But now by conversion, we neglect the theorems which are converse in parallelograms, and esteem such as are converse in triangles worthy of relation. And the reason of this is, because the mode of demonstration in parallelograms, is the same indifferently, by a deduction to an impossibility, and the construction is similar. But we are content when we have exhibited the way in more simple figures, I mean triangles, to leave to the more curious the same mode of reasoning in the rest: since it is easy, at the same time, to perceive that there is the same method in these. For when we assume equal parallelograms, upon the same base, or upon equal bases, we must say that they are also between the same 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 case is assumed, when we receive it and its parallels, we may exhibit the same consequences as in triangles, I mean that the whole will be equal to its part: but this is impossible. It is however manifest, that the institutor of the Elements very properly adds the particle, _and at the same parts_. For it is possible that equal triangles, may be assumed upon the same base, one, indeed, at these parts, but the other at different parts, and yet these will not be entirely between the same parallels: for neither will they be contained under the same altitude. And on this account he added the particle.

But since a parallel may be drawn in a two-fold respect, according to an absurd hypothesis, i.e. either within or without, Euclid draws it within: but we can exhibit the same consequences, by drawing it without. For let the equal triangles _a b c_, _d b c_, be upon base, and at the same parts, I say that they are between the same parallels, and that the right line connected at their vertices, is parallel to the base. 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, _a b c_, is equal to the triangle _e b c_, the whole to the part. But this is impossible; and hence, the parallel line does not fall external to _a d_. But it is shewn by the institutor 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 same parts, and upon the same base are parallel to each other. And thus the remaining part of the deduction to an impossibility is demonstrated. But it is worthy of observation, that since 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 sixth 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 assumes, besides this, a part of that which was hypothesis in those. For to stand upon the same, or upon equal bases, is a datum in these, as well as in those, except that in these hypotheses he adds something which was neither an object of investigation, nor a datum in these; since the particle _at the same parts_, is over and above extrinsically assumed.

PROPOSITION XL. THEOREM XXX.

Equal triangles which are upon equal bases, and at the
same parts, are between the same parallels.

There is the same mode of conversion too in the present theorem, and a similar demonstration; and that part of the deduction to an impossibility, which is omitted by the institutor of the Elements, is demonstrated after the same manner, and there is no occasion for repetition. But since these three conditions are in the aforesaid propositions, _situation upon equal, or on the same bases; position between the same parallels; and equality of triangles and parallelograms_, it is manifest that we may variously convert, by always connecting two, and leaving one. For we either supposed the bases the same, or equal, and triangles and parallelograms between the same parallels, and thus we form four theorems; or we consider the triangles and parallelograms equal, and the bases the same, or equal, and thus we produce another four, two of which the elementary institutor omits, viz. those which respect parallelograms, but the other two relative to triangles, he exhibits; or lastly, when we have assumed them equal, and between the same parallels, we prove the remainder, that they are either upon the same, or upon equal bases, and produce another four, which the institutor of the Elements entirely neglects. For there is the same demonstration in these, except that two of these four are not essentially true. Thus, equal parallelograms or triangles, between the same parallels, are not necessarily upon the same base: but all this is true in these hypotheses, that they are upon the same or equal bases; but the other does not entirely follow the assumed hypotheses. Hence, as all these theorems are ten, the geometrician speaks of six, and neglects four, lest he should labour in vain, by repetition, since the demonstration is the same. For it may be shewn in triangles, that if they are equal, and between the same parallels, they will either be upon the same, or upon equal bases. For let it be denied, and if possible, let the triangles _a b c_, _d e f_, have these conditions, upon unequal bases _b c_, _e f_. Let too, _b c_, be the greater, and cut off _b h_, equal to _e f_, and connect _a h_. Because, therefore, the triangles _a b h_, _d e f_, are upon equal bases, _b h_, _e f_, and between the same parallels, they are equal. But the triangles also, _a b c_, _d e f_, are supposed equal. Hence, the triangles _a b c_, _a b h_, are equal, which is impossible. The bases, therefore, of the triangles _a b c_, _d e f_, are not unequal. And the mode of demonstration will be the same in parallelograms. Since, therefore, the ostensive method is the same, and the impossibility the same, viz. that the whole is equal to its parts, it is not improperly omitted by the elementary institutor. And thus we have shewn, that there are necessarily ten theorems, and have enumerated what are omitted, and shewn the reason of their omission. But let us now pass to the following propositions.

PROPOSITION XLI. THEOREM XXXI.

If a parallelogram has the same base with a triangle, and is
between the same parallels, the parallelogram shall be double
the triangle.

The present theorem also is local, but it mingles the constructions of triangles and parallelograms, situated under the same altitude. As, therefore, we have separately surveyed parallelograms and triangles, so when we assume each of them in conjunction, and with the same condition, we contemplate their proportion to each other. In the former, therefore, an equality of proportion is apparent, since all upon the same bases, and between the same parallels, have a mutual equality, whether they are triangles, or parallelograms. But in these latter, the first of unequal proportions, I mean the duple, is exhibited: for he demonstrates that a parallelogram is double of a triangle, on the same base, and possessing the same altitude. But the elementary institutor shews the thing proposed, by supposing the vertex of the triangle external to the parallelogram. We can, however, demonstrate the consequence, by assuming the line which is parallel to their common base, in the other side of the parallelogram: for these are two cases of the theorem. Since in consequence of the two having the same base, it is necessary that the vertex of the triangle should either be within, or without the parallelogram. Let there be, therefore, a parallelogram _a b e d_, and a triangle _e c d_, and let a point _c_ be placed between the points _a_ and _b_, and connect the right line _a d_. Because, therefore, the parallelogram is double of the triangle _e c d_, but the triangle _a d c_, is equal to the triangle _e d c_, hence, the parallelogram is double of the triangle _e c d_. And hence it is evident that a parallelogram is double of a triangle on the same base. But if the bases are equal, we can shew the same by drawing the diameters of the parallelograms: for if the triangles are equal, the parallelogram which is double of the one, will also be double of the other. But triangles are equal, on account of the equality of bases, and the identity of altitude. The geometrician, therefore, very properly omits this, for the demonstration is the same: since they will either have the same part, or they will be conjoined in one point only, or they will be separate from each other. But in whatever manner they may receive this variety, there is one demonstration according to all the cases.

We can likewise demonstrate the converse propositions to this theorem, after the same manner. One of which is: _If a parallelogram is double of a triangle, and they have the same or equal bases, and are at the same parts, they shall be between the same parallels._ For if they are not the whole shall be equal to the part, and the same proportion shall prevail: since it is necessary that the vertex of the triangle should either fall within, or external to the parallels. But in either case, the same impossibility will be the result, by drawing a parallel to the base, through the vertex of the triangle. But the second converse theorem is: _If a parallelogram is double of a triangle, and both are between the same parallels, they will either be situated upon one base, or upon equal bases._ For if they are upon unequal bases, since we have assumed the figures to be equal, we may shew that the whole will be equal to its part. Hence, all these theorems end in this common impossible: and on this account, the institutor of the Elements leaves us to investigate the variety they contain, as he himself, has contracted his speculation to such as are more simple, and of a more primary nature. However, as we have recognized these observations, let us see for the sake of exercise, by not assuming a parallelogram, but a trapezium, two of whose sides only are parallel (because it has the same base with the triangle, while it is situated between the same parallels), let us, I say, consider what proportion it possesses to the triangle. That it has not, therefore, a duple proportion is evident: for if it had a duple ratio, it would be a parallelogram, since it is a quadrilateral figure. But I say that it is either greater than double or less; for since the two sides are parallel, one is greater, but the other less; because if equal, the sides conjoining them will be parallel. If, therefore, the triangle has its greater side for the base, the quadrilateral figure will be less than double of the triangle: but if the lesser side, it will be more than double. For let _a b c d_, be a quadrilateral figure, and let the side _a b_, be less than the side _c d_, and produce the side _a b_, in infinitum, and let the triangle _e c d_ have the same base with the quadrilateral figure, that is _c d_; and lastly, through _d_, draw _d f_, parallel to _a c_. Hence, the parallelogram _a c d f_, is double of the triangle _e c d_; and so the quadrilateral figure _a b c d_, is less than double of the triangle.

Again, let the triangle have the base _a b_, and draw _b f_, parallel to _a c_. The parallelogram, therefore, _a b f c_, is double of the triangle. And hence, the quadrilateral figure, _a b c d_, is more than double of the triangle. This being shewn; we affirm, that when there is a quadrilateral figure, whose two opposite sides only, are parallel, if one of the parallel sides is bisected, and right lines are drawn from it to the other side, the quadrilateral figure, is either more or less than double of the triangle resulting from such a construction. But if one of the sides by which the parallel lines are conjoined, is bisected, and certain right lines are drawn from it to the remaining side, the quadrilateral figure, will be perfectly double of the triangle which is produced. And this may be shewn as follows. Let there be a quadrilateral figure _a b c d_, and let the side _a d_, be parallel to the side _b c_, and bisect _d c_, in the point _e_, and connect the right lines _a e_, _e b_, and produce _b e_, till it coincides with _a d_, in some point, as _f_. Because, therefore, the angles at the point _e_, are equal, for they are vertical; likewise, because the angle _f d e_, is equal to the angle _b c e_, the side also _f e_, will be equal to the side _e b_, and the triangle _d e f_, will be equal to the triangle _b c e_. Let the common triangle _a d e_, be added. The whole triangle, therefore, _a e f_, is equal to the two triangles _a d e_, _b c e_. But the triangle _a e f_, is equal to the triangle _a e b_: for they are upon equal bases, _b e_, _e f_, and between the same parallels, if a line parallel to _b f_, is drawn[43]. Hence, the triangle _a e b_, is equal to the triangles _a d e_, _b c e_, and the quadrilateral figure _a b c d_, is double of the triangle _a e b_, which was to be shewn. After the same manner, we may shew, that if the side _a b_, is bisected, and certain right lines are drawn from it, to the side _e d_, the quadrilateral figure will be double of the triangle formed by such a construction. If, therefore, one of the sides by which the parallel lines are conjoined is bisected, and from it certain right lines are drawn to the remaining side, the quadrilateral figure shall be double of the triangle. And these things are demonstrated for the sake of geometrical exercise. Let us now proceed to the subsequent propositions.

PROPOSITION XLII. PROBLEM XI.

To construct a parallelogram equal to a given triangle, in a given
rectilineal angle[44].

* * * * *

PROPOSITION XLIII. THEOREM XXXII.

The complements of parallelograms, situated about the diameter
of every parallelogram, are equal to each other.

_The beginning of this commentary is wanting._

* * * * *

that parallelograms are not mutually conjoined according to one point, and that the complements are not quadrilateral; it is requisite that placing this also as a case, we should regard the same accident. For let there be a parallelogram _a b_, having the parallelograms _c k_, _d l_, about the same diameter, and let a certain right line _k l_, which is a part of the diameter intervene between them. Again, therefore, you may say the same, viz. that the triangle _a c d_, is equal to the triangle _b c d_, and the triangle _e c k_, to the triangle _k c f_; and likewise the triangle _d g l_, to the triangle _d h l_. The remaining figure, therefore, _a g l k e_, of five sides, is equal to the remaining five-sided figure _b f k l h_. But these were the complements. Again, if the parallelograms are neither conjoined according to a point, nor distant from each other, but mutually cut each other, on this hypothesis also, the demonstration will be the same. For let there be a parallelogram _a b_, and a diameter _c d_, and let parallelograms be constructed about it, one of which is _e c f l_, but the other, by which this also is intersected, _d g k h_. I say that the complements _f g_, _e h_, are equal. For since the whole triangle _d g k_, is equal to the whole triangle _d h k_, but a part of it also, the triangle _k l m_, is equal to the triangle _k l n_; (since _l k_ is a parallelogram); hence the remaining trapezium _d l n h_, is equal to the remaining trapezium _d l m g_. But the triangle _a d c_, is equal to the triangle _b c d_, and the triangle _f c l_, in the parallelogram _e f_, to the triangle _e c l_, and the trapezium _d g m l_, to the trapezium _d h n l_. The remaining quadrilateral figure, therefore _g f_, is not unequal to the remaining quadrilateral figure _e h_. And hence, the theorem is exhibited according to all its cases. But there are three only, and neither more nor less. For the parallelograms consisting about the same diameter, either cut each other or touch each other, according to a point, or are distant from each other by a certain part of the diameter.

But the institutor of the Elements assumes the appellation of _complements_, from the thing itself, so far as these also, besides two parallelograms, fill up the whole: and on this account, it was not of itself thought worthy of being remembered in the definitions. For, indeed, variety is requisite to its declaration, such as the knowledge of a parallelogram, and what those parallelograms are, which are about the diameter of the whole parallelogram; since, when these are explained, this likewise becomes known. But those parallelograms are about the same diameter, which have a part of the whole diameter for their own: and those which have not this condition, are by no means about the same diameter. For when the diameter of the whole parallelogram is cut by the sides of an internal parallelogram, then this parallelogram is not about the same diameter with the whole parallelogram. As for example, in the parallelogram _a b_, let the diameter _c d_, cut the side _e h_, of the parallelogram _c e_. The parallelogram, therefore, _e c_, is not about the same diameter with the parallelogram _c d_.

PROPOSITION XLIV. PROBLEM XII.

To a given right line, to apply a parallelogram equal to a given
triangle, in an angle which is equal to a given right lined
angle.