Commentary on Propositions I.33–I.48: Areas, Application & Pythagoras¶
Parallelograms & Locus Theorems (I.35), Pythagorean Application of Areas (Parabole, Hyperbole, Elleipsis, I.44), and Euclid’s Proof of I.47–I.48 — Friedlein (1873) pp. 384–436 · Book IV (Propositions XXXIII–XLVIII)
Note
Edition Track (English): Public-domain translation by Thomas Taylor (London, 1788–1789 / 1792), collated and synchronized with Gottfried Friedlein's 1873 Teubner Greek edition (pp. 384–436). Switch tracks above for the Greek (1873) text, Summary, or Notes.
Props. I.33–I.34: Parallelogrammic Areas & Diameter Bisection¶
Euclidean Reference: [I.34] · Friedlein pp. 384–393
PROPOSITION XXXIII. THEOREM XXIII.
The right lines which join equal and parallel right lines at the same parts, are themselves also equal and parallel.
The present theorem is, as it were, the confine of the consideration of parallels and parallelograms: for it seems to declare a certain symptom of parallel right lines, and delivers the latent origin of parallelograms. 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 shewn to be equal and parallel. Hence, the proposition which immediately follows the present, contemplates the properties essentially inherent in these spaces, in a parallelogram as it were already constructed. And these things are indeed manifest. But it is requisite to consider the diligence which this preposition contains. In the first place, indeed, that it is not sufficient, that the lines which are conjoined should be equal: for the lines which connect equals, are not entirely equal, unless they are also parallel. For a triangle being isosceles, and a point being assumed in one of the equal sides, and through this a line being drawn parallel to the basis, equal lines shall indeed conjoin parallels to the basis, and the basis itself, yet these parallels shall not also be equal; and the sides will not be parallel, because they coincide at the vertex of the triangle.
[Illustration]
[Illustration]
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.
[Illustration]
[Illustration]
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, 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) 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, 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.
[Illustration]
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.
Prop. I.35: Equal Parallelograms & Birth of Locus Theorems¶
Euclidean Reference: [I.35] · Friedlein pp. 394–400
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. 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.
[Illustration]
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: 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. 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: for they more properly belong to the hypotheses of the second book.
[Illustration]
[Illustration]
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.
Props. I.36–I.41: Equal Parallelograms & Triangles Between Parallels¶
Euclidean Reference: [I.41] · Friedlein pp. 400–415
PROPOSITION XXXVI. THEOREM XXVI.
Parallelograms which are upon equal bases, and between the same parallels, are equal to each other.
[Illustration]
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.
[Illustration]
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. * *
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. 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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
[Illustration]
[Illustration]
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. 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.
[Illustration]
Props. I.42–I.43: Area Construction & Complements About the Diameter¶
Euclidean Reference: [I.43] · Friedlein pp. 415–418
PROPOSITION XLII. PROBLEM XI.
To construct a parallelogram equal to a given triangle, in a given rectilineal angle.
PROPOSITION XLIII. THEOREM XXXII.
The complements of parallelograms, situated about the diameter of every parallelogram, are equal to each other.
[Illustration]
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.
[Illustration]
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.
[Illustration]
Props. I.44–I.45: Pythagorean Application of Areas (Parabole, Hyperbole, Elleipsis)¶
Euclidean Reference: [I.44] · Friedlein pp. 419–423
Editorial Key (Friedlein 419–420): On Prop. I.44, Eudemus testifies that parabole (exact application), hyperbole (exceeding), and elleipsis (falling-short) are ancient discoveries of the Pythagorean muse, later transferred by Apollonius to the conic sections.
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.
According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient, and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied; but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient. And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application, wishing to apply to a given right line a parallelogram equal to a given triangle; that we may not only have the construction of a parallelogram equal to a given triangle, but also an application to a determinate right line. As for example, a triangle being given, having an area of twelve feet, but a right line being proposed, whose length is four feet, we may apply to the right line a parallelogram equal to the triangle, if when we assume the whole length of four feet, we find how many feet the breadth ought to contain, that the parallelogram may become equal to the triangle. When, therefore, we have discovered that the breadth is three feet, and have multiplied the length with the breadth, the proposed angle being right, we shall obtain the desired space. And such is the verb to apply, formerly delivered by the Pythagoreans. But there are three things given in the present problem; one, the right line to which it is to be so applied, that it may become the whole side of that space; but the other is the triangle to which that which is applied ought to be equal; and the third is the angle to which it is requisite that the angle of the space should be equal. And here it is again perspicuous, that when the angle is right, the space which is applied, is either a quadrangle, or an oblong; but when it is either acute or obtuse, the space is either a rhombus, or rhomboides. Besides, this too is manifest, that the right line ought to be finite; since this cannot be accomplished on an infinite line. At the same time, therefore, as he says, to apply to a given right line, he indicates that the right line must be necessarily finite. But he uses in the construction of the present problem, the construction of a parallelogram equal to a given triangle; since, as we have observed, application is not the same with construction. For the latter, indeed, constructs both the whole space, and all the sides; but the former, when it has one side given, constitutes on this the space, because it is neither deficient, nor exceeds according to this extension, but uses this one side which comprehends the area. But you may perhaps say, why does he use theorems, when he shews triangles equal to triangles; but problems, when he shews triangles equal to parallelograms? We reply, it is because equality spontaneously arises in things of the same species; but requires origin, and fabrication, in things of a dissimilar species, on account of the mutation subsisting according to species, since it is by itself difficult of invention.
PROPOSITION XLV. PROBLEM XIII.
To construct a parallelogram equal to a given right-lined figure, in a given rectilineal angle.
The present is more universal than the two problems, in which he invented as well the construction, as the application of parallelograms equal to a given triangle. For whether a triangle, or a quadrangle, or any other quadrilateral figure is given, we may construct a parallelogram equal to it, by the present theorem; since every right-lined figure, as we have previously observed, may be essentially resolved into triangles, and we have delivered a method of discovering the multitude of triangles. When, therefore, we have resolved a given rectangle into triangles, and have constructed a parallelogram equal to one of them, and have applied to a given right line, parallelograms equal to the rest; then, by assuming that to which we have made the first application, we shall have a parallelogram composed from these parallelograms, equal to the right-lined figure composed from those triangles, and the thing desired, will be accomplished. Hence, though such a rectangle should be a figure of ten sides, yet, by resolving it into eight triangles, and constructing a parallelogram equal to one of them, and seven times applying parallelograms equal to the rest, we shall obtain the object of investigation. But, as it appears to me, the ancients being incited by this problem, sought how to describe a quadrangle equal to a circle. For if a parallelogram can be found equal to any right-lined figure, it deserves to be enquired whether right-lined figures also, can be shewn equal to such as are curve-lined. And Archimedes shews that every circle is equal to a right-angled triangle, one of whose radii is equal to one of the sides which are about the right angle of the triangle; but whose ambit is equal to the base. However, of this elsewhere: let us now proceed to the consequent propositions.
Prop. I.46: Constructing the Square & Posidonius’s Definition¶
Euclidean Reference: [I.46] · Friedlein pp. 423–425
PROPOSITION XLVI. PROBLEM XIV.
To describe a quadrangle from a given right line.
Euclid requires this problem, most particularly, in the construction of the following theorem. But he appears to have been desirous to deliver the origin of the two best rectilineal figures, viz. the equilateral triangle, and the quadrangle; because these right lined figures are required in the constitution of the mundane figures, and particularly of those four, to which origin and dissolution belong. For the icosaedron and the octaedron, and the pyramid, are composed from equilateral triangles; but the cube from quadrangles. And on this account, as it appears to me, he has principally constructed the former, but described the latter. For he has discovered appellations adapted to these figures: since the equilateral triangle, so far as its composition is various, requires construction; but the quadrangle, so far as it originates from one side, requires description. For we cannot produce a triangle in the same manner as a quadrangle, by multiplying the number of a given right line into itself; but when we have conjoined right lines produced by other means, with the extremities of the given right line, we construct from these one equilateral triangle: and the description of circles, profits in discovering that point from which it is requisite to connect right lines, to the extremes of the proposed right line. But these observations are indeed perspicuous.
[Illustration]
It may, however, be shewn, that the right lines, from which quadrangles are described, being equal, the quadrangles also shall be equal. For let the right lines a b, c d, be equal, and from a b, describe the quadrangle a b c g, but from c d, the quadrangle c d h f, and connect the right lines g b, h d. Because, therefore, the right lines a b, c d, are equal, a g, h c, are also equal; and they comprehend equal angles, and the base g b, is equal to the base h d, and the triangle a b g, to the triangle c d h; and the doubles of these are equal. Hence the quadrangle a c, is not unequal to the quadrangle c f. But the converse of this also is true. For if the quadrangles are equal, the right lines, also, from which they are described, will be equal. Thus let the quadrangles a f, c g, be equal, and let them be so placed, that the side a b, may be in a right line with the side b c. Since therefore, the angles are right, the right line also f b, will be in a direct position, with the right line b g. Let the right lines f c, a g, a f, c g, be connected. Because, therefore, the quadrangle a f, is equal to the quadrangle c g; the triangle, also, a f b, is equal to the triangle c b g. Let the common triangle b c f, be added. The whole triangle, therefore, a c f, is equal to the whole triangle c f g.
[Illustration]
Hence, a g is parallel to f c. Again, because, as well a f g, as the angle c g b, is the half of a right angle, a f, is parallel to c g. The right line, therefore, a f, is equal to the right line c g, since they are the opposite sides of a parallelogram. Because, therefore, there are two triangles, a b f, b c g, which have the alternate angles equal, since a f, c g, are parallel; likewise one side a f, equal to the side c g, the side, also, a b, shall be equal to the side b c, and the side b f, to the side b g. And thus it is shewn, that the quadrangles a f, c g, being equal, the sides, also, from which they are described are equal.
Prop. I.47: The Theorem of Pythagoras, Euclid’s Generalization & Triples¶
Euclidean Reference: [I.47] · Friedlein pp. 426–429
Editorial Key (Friedlein 426–429): On Prop. I.47, Proclus weighs the tradition of Pythagoras's discovery, celebrates Euclid's rigorous Book I proof and Book VI ([VI.31]) generalization, and records the Pythagorean and Platonic integer right-triangle rules.
PROPOSITION XLVII. THEOREM XXXIII.
In right angled triangles, the quadrangle, which is described from the side subtending the right angle, is equal to the quadrangles which are described from the sides comprehending the right angle.
If we attend to the historians of antiquity, we shall find them referring the present theorem to Pythagoras, and asserting that he sacrificed an ox or its invention. For my own part, I admire those who first investigated the truth of this theorem: but I possess a greater admiration for the elementary institutor, not only because he establishes its truth by evident demonstration, but likewise, because he persuades us by scientific reasons, which cannot be confuted of a theorem more universal than this in his sixth book. For in that he shews universally, that in right-angled triangles, the figure described from the side subtending the right angle, is equal to the figures described from the sides comprehending the right angle, when they are similar to the former figure, and are similarly described. For every quadrangle is similar to every quadrangle; but all right-lined figures similar to each other, are not quadrangles: since in triangles, and other multangles, similitude is inherent. Hence, the reason which demonstrates that the figure described from the side subtending the right angle, whether it is quadrangular, or of some other form, is equal to the figures subsisting about the right angle, similar to the former, and similarly described; exhibits something more universal, and which possesses a greater power of producing science, than the reason exhibits, affirming a quadrangle alone, equal to quadrangles. For in the former case, it becomes manifest by an universal ostension, that the rectitude of the angle affords to the figure described from its subtending side, equality, to all the figures, subsisting about its comprehending sides, similar to the former, and similarly described: just as obtuseness is the cause of excess; but acuteness of diminution. But how this theorem is evinced, will be perspicuous, when we comment on it in the sixth book.
But let us now consider the truth of the present theorem, only adding this, that universal ought not to be shewn here, by him who has taught nothing concerning the similitude of right-lined figures, and the doctrine of proportion: for many things which are here exhibited more particularly, are in that theorem shewn more universally by the same method. The institutor of the Elements, therefore, shews the thing proposed in the present, from the common contemplation of parallelograms. But since right-angled triangles are two-fold, i.e. either isosceles, or scalene; in isosceles triangles, we shall never find numbers corresponding with the sides: for there is no quadrangular number, exactly double of another quadrangular number; since the square from the septenary is double of the square from the quinary, by a deficience of unity. But in scalene triangles it is possible, that numbers may be assumed, so as evidently to evince, that the square from the side subtending the right angle, is equal to the squares from the sides subsisting about the right angle. And of this kind is the triangle in the republic, whose right angle is contained by the ternary, and quaternary, but is subtended by the quinary. The quadrangle, therefore, from the quinary, is equal to the quadrangles from the other numbers: for this is twenty-five; but the quadrangle from the ternary is nine, and from the quaternary sixteen. And thus what we have asserted is perspicuous in numbers.
But there are delivered certain methods of inventing triangles of this kind, one of which they refer to Plato, but the other to Pythagoras, as originating from odd numbers. For Pythagoras places a given odd number, as the least of the sides about the right angle, and when he has received the quadrangle produced from this number, and diminished it by unity, he places the half of the remainder, as the greatest of the sides about the right angle; and when he has added unity to this, he produces the remaining side which subtends the right angle. Thus for example, when he has assumed the ternary, and has produced from it a quadrangular number, and from this number nine, has taken unity, he assumes the half of eight, that is four, and to this again he adds unity, and makes five; and thus discovers a right-angled triangle, having one of its sides of three, but the other of four, and the other of five units. But the Platonic method originates from even numbers. For when he has assumed a given even number, he places it as one of the sides about the right angle, and when he has divided this into half, and has produced a quadrangular number from the half, when he has added unity to this quadrangle, he forms the subtending side, but when he has taken unity from the quadrangle, he forms the remaining side about the right angle. Thus for example, when he has assumed the number four, and has multiplied the half of this into itself, and produced four, when he takes away unity he forms the number three, but when he adds unity, he produces the number five; and thus he has the same triangle effected, as by the Pythagoric method. For the square from the number five, is equal to the squares from the numbers three, and four. And thus much for the digression of the present narration. But as the demonstration of the elementary institutor is perspicuous, I do not think, that any thing should be added, because it would be superfluous; but we should be content with what is written. For those who have added any thing more, as the familiars of Hero and Pappus, have been obliged to assume in an affair of no difficulty, some of the propositions of the sixth book; and the cause which regards this affair. We shall therefore pass on to the following theorem.
Prop. I.48: Converse of Pythagoras, Peroration & Manuscript Scholium¶
Euclidean Reference: [I.48] · Friedlein pp. 430–436
PROPOSITION XLVIII. THEOREM XXXIV.
If the quadrangle described from one side of a triangle, is equal to the quadrangles described from the other two sides of the triangle: then the angle comprehended by the remaining two sides of the triangle, is right.
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This theorem is the converse of the preceding, and the whole is converted to the whole. For if the triangle is rectangular, the quadrangle which is described, from the side subtending the right angle, is equal to the quadrangles described from the other sides: and if the square from this, is equal to the squares from the other sides, the triangle is rectangular, because it has the angle right, which is comprehended by the remaining sides. And the demonstration of the Elementary institutor is indeed conspicuous. But when there is a triangle a b c, having the quadrangle, which is described from the side a c, equal to the quadrangles from the sides a b, b c, since in the triangle, a right line from the point b, is raised at right angles to the side b c, if it should be said, that the right line must be raised at right angles, to other parts, and not at those to which the elementary institutor raises it, we assert that this is an impossibility. For it can neither fall within, nor without the triangle; and can be no other than a b. For if possible, let it fall as b e. Because, therefore, the angle e b c, is right, the angle c f b, is doubtless acute; and hence, the remaining angle a f b, will be obtuse. The side, therefore, a b, is greater than the side b f. Let a line b e, be placed equal to a b, and connect e c. Because, therefore, the angle e b c, is right, the quadrangle described from the side e c, is equal to the quadrangles from the sides e b, b c. But e b is equal to b a. The quadrangle, therefore, from the side e c, is equal to the quadrangles from the sides a b, b c. But the quadrangle from the side a c, was also equal to the same. Hence, the quadrangle from the side e c, is equal to that which is described from the side a c; and so e c is equal to a c. Two right lines, therefore, b e, e c, are equal to the two b a, a c, each to each, and are constructed upon the right line b c, which is impossible. And hence, the line raised at right angles, does not fall within the right line a b.
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But neither can it fall without, towards other parts of the right line a b. For if possible, let it fall as b g, and let b g be equal to a b, and connect c g. Because, therefore, the angle g b c, is right, the quadrangle described from the side g c, is equal to the quadrangles from the sides b g, b c. But the quadrangle also, from the side a c, was equal to the quadrangles from the sides a b, b c, but a b is equal to g b; and so g c is equal to a c. But the right line g b, also, is equal to the right line b a, upon one right line b c, which is impossible. Hence, the right line which is raised from the point b, at right angles to b c, neither falls within, nor without the side a b; and therefore falls upon it. And so the objection is dissolved. But the institutor of the Elements, thus far completes his first book, in which he has delivered many species of conversions; (for he often converts the whole of theorems to the whole, and wholes to parts, and parts to parts) and has invented a great variety of problems; (for he has delivered the sections, positions, constructions, and applications of lines and angles). He likewise touches upon that mathematical place which is called admirable; and sufficiently brings local theorems into our remembrance. Besides, he unfolds the elementary institution of universal and particular theorems, and indicates the difference of indeterminate, and determinate problems; all which, attending him in his progress, we have orderly explained. Lastly, he refers the whole book to one purpose, I mean the elementary institution, of the contemplation respecting the more simple rectilineal figures; and finally, he investigates their constructions, and considers their essential properties. But we, indeed, shall give thanks to the gods, should we be able to comment on the other books, in a similar manner. In the mean time, if other cares should prevent the execution of our design, it is my opinion, that such as are studious of these contemplations, ought to expound the other books, after the same mode; by investigating that which is every where difficult, and pertinent to the subject, and capable of an easy division. For, indeed, the commentaries which are circulated at the present period, are replete with great and various confusion, because, at the same time, they neither infer any assignation of cause, nor dialectic judgment, nor philosophic contemplation.