Chapter VII: Book IV (1)

Whatever can be said in an elementary institution, concerning the origin and equality of triangles, we may learn from the preceding discourse. But after this, the narration of Euclid is concerning quadrilateral figures, and he particularly teaches us concerning parallelograms, together with the contemplation of these delivering the doctrine of trapeziums. For a quadrilateral figure, (as we have formerly observed in our discourse on hypotheses,) is divided into parallelogram and trapezium; and a parallelogram into other certain species, and in like manner a trapezium. But because a parallelogram, on account of its participation of equality, possesses disposition and order, but a trapezium has neither the same, nor a similar order; Euclid’s principal discourse, is with propriety, concerning parallelograms, but he also contemplates together with these a trapezium. For from the section of parallelograms, the origin of trapeziums will appear, as will be manifest as we proceed. But because again, it is not possible that any thing can be said of the construction or equality of parallelograms, without the consideration of parallels, (for as it is manifest from the very name, that is, a parallelogram, which is circumscribed by parallel right lines in an opposite position,) hence, he necessarily assumes from parallels the beginning of his doctrine, but having advanced a little from these, he enters on the doctrine of parallelograms, employing one middle theorem, between the elementary institution of each, because he appears to contemplate a certain symptom inherent in parallels: but he delivers the first origin of a parallelogram. For such is the proposition, which says, that _right lines which join equal and parallel right lines towards the same parts, are themselves equal and parallel_. For in this theorem, indeed, a certain accident to equal and parallel right lines is considered: but from the connection a parallelogram appears, having its sides opposite and parallel. And from hence it is manifest that the discourse concerning parallels, was necessarily pre-assumed. But three things are to be assumed, essentially inherent in parallels, which they essentially express, and are converted with them, not only the three together, but every one separately assumed from the rest. Of these, one is, _that when a right line cuts parallel lines, the alternate angles are equal_; but the second, _that when a right line cuts parallel lines, the internal angles are equal to two right_; and the third, _that in consequence of a right line cutting parallel lines, the external is equal to the internal and opposite angle_. For when any one of these _symptoms_ is demonstrated, we have sufficient authority to affirm that the right lines are parallel. But other mathematicians, also, have been accustomed to discourse after this manner concerning lines, delivering the symptoms of every species. For Apollonius, in each of the conical lines, shews what a _symptom_ is, as also Nicomedes in his Treatise on Conchoids, and Hippias in his Quadratics, and Perseus in his Spirals. Since after their origin, that which is essentially inherent in these lines, and according to what it is inherent, being assumed, distinguishes a constructed form from all others. After the same manner, therefore, the institutor of the Elements, first of all investigates, the symptoms of parallels.

PROPOSITION XXVII. THEOREM XVIII.

If a right line falling upon two right lines, makes the
alternate angles equal to each other, those right lines shall be
parallel to each other.

In the present theorem it was not pre-assumed as evident that the right lines are in one plane, but this ought rather to be previously admitted in all theorems which are considered in a plane. This, however, is added, because it does not universally follow, that when the alternate angles are equal, the right lines will be parallel, unless they are in the same plane. For nothing hinders, but that a right line falling on right lines disposed in the shape of the letter X, one of which is situated in one plane, but the other in a different one, may make the alternate angles equal; and yet the right lines thus disposed will not be parallel. It was pre-assumed[27], therefore, that in a treatise on planes, we conceive every thing described in one and the same plane: and on this account, he does not require this addition in the present proposition. But it is requisite to know that the geometrician considers the particle _alternate_, in a two-fold respect, sometimes, indeed, according to a certain situation, but sometimes according to a certain consequence of proportions. And according to this last signification, the particle _alternate_ is used in the fifth book, and in such as are arithmetical: but agreeable to the former, both in this, and in all the other books concerning parallel right lines, and that which falls upon these. For he calls the angles alternate, which are not formed at the same parts, and are not successive to each other, which are distinct, indeed, from the incident line, but both of them exist within parallels, and differ in this, that the one has an upward, but the other a downward position. I say, for example, that when a right line _e f_, falls on the right lines _a b_, and _c d_, he calls the angles _a e f_, _d f e_, and also the angles _c f e_, _b e f_, _alternate_, or _altern_, because they have an alternate, or changed order, according to their position. But this too must be known, that from such a situation of right lines, all the symptoms become by division, six; three of which the geometrician alone receives; and three he omits. For we either assume the angles at the same parts, or not at the same. And if at the same parts, either both within the right lines, which shews them to be parallels; or both without, or one without, and the other within. And if not at the same parts, again, after the same manner, they are either both without the right lines, cutting the lines it is necessary to receive; or within; or one within, and the other without. But what we have said will become manifest by the same description as above. For let there be certain right lines _a b_, _c d_, and let a right line _e f_, fall upon them, and let it be produced to the points _h_ and _g_. If then you assume angles at the same parts, you will either place them both within, as _b e f_, and _e f d_, or as _a e f_, and _e f c_; or both without, as _h e b_, and _d f g_, or as _h e a_, and _c f g_; or one within, and the other without, as _h e b_, and _e f d_, or as _g f d_, and _f e b_, or as _h e a_, and _e f c_, or as _g f c_, and _a e f_: for these last are received in a quadruple respect. But if you assume the angles not at the same parts, you will either place both within, as _a e f_, and _e f d_, or as _c f e_, and _f e b_; or both without, as _a e h_, and _d f g_, or as _h e b_, and _c f g_; or one within, and the other without, and this again in a quadruple respect. For they will either be the angles _a e h_, and _e f d_; or _h e b_, and _e f c_; or _g f c_ and _f e b_; or _g f d_, and _f e a_. And besides these, there is no other assumption.

As, therefore, angles are assumed according to six modes, the geometrician combines three assumptions alone; and these consequent symptoms, are naturally adapted to express parallels. But of these three assumptions, one belongs to those angles which are not at the same parts, viz. to those which are only assumed within; and these he calls alternate, so that those, which are both external, and those, one of which is external, but the other internal, are omitted: but two of these assumptions belong to angles at the same parts, to those, indeed, which are both internal, which he says are equal to two right, and to those, one of which is internal, but the other external, which he says are equal, leaving indeed one assumption which supposes both the angles to be external. We therefore affirm that the same things will be consequent to the three omitted hypotheses. Thus, in the preceding figure, let both the external angles _h e b_, _d f g_, be at the same parts, I say that these are equal to two right angles. For the angle _d f e_, is equal to the angle _h e b_, and the angle _b e f_, to the angle _d f g_. But if the angles _b e f_, _e f d_, are equal to two right, the angles _d f g_, _h e b_, are equal to two right. Let again the angles _a e h_, _e f d_, not be towards the same parts, of which the one is within, but the other external, I say that these also are equal to two right angles. For if the angle _a e h_, is equal to the angle _b e f_, but the angles _b e f_, and _e f d_, are equal to two right, the angles, also, _a e h_, and _e f d_, are equal to two right. Again, let them not be at the same parts, but both without the right lines as _a e h_, _d f g_. I say that these are equal to one another. For if the angles _a e h_, and _b e f_, are equal to each other, but the angle _d f g_, is equal to the angle _b e f_, hence the angle _a e h_, is not unequal to the angle _d f g_. If, therefore, the things assumed by the geometrician, in three hypotheses are verified, all the same follow in the remaining three as indisputably true. Besides this too is to be observed, that in such as the geometrician receives these, according to two assumptions, the angles are supposed equal to each other, but when according to one assumption, equal to two right: but in these last on the contrary, according to two assumptions, they are supposed equal to two right angles, but according to one equal to each other. For since all the assumptions are six, it happens, indeed, from three, that the angles are equal to two right, but from the other three, that they are equal to each other. Hence, those which are omitted are not undeservedly contrary to the assumptions which are reckoned worthy of relation. But the geometrician appears to have chosen such hypotheses as either abound in affirmation, or are more simple, and, on this account of those angles which are not at the same parts, he assumed alone the internal, which he calls alternate: but of those at the same parts, he assumes as well the internal, as well as one internal and the other external, but he avoids the rest, either because they are more declared by negation, or because they are more various. However, whether this or some other be the cause, the number of the consequents to those hypotheses is from hence sufficiently manifest.

PROPOSITION XXVIII. THEOREM XIX.

If a right line falling upon two right lines, makes the external
equal to the internal angle, placed opposite, and at the same
parts, or makes the angles internally situated, and at the same
parts equal to two right, those right lines shall be parallel to
each other.

The preceding theorem receiving the angles, not at the same parts, but situated within right lines, shews that the right lines are parallel among themselves: but the present theorem proposes the two remaining hypotheses, of which one separates the angles according to the particles _without_ and _within_, but the other supposes them both within, and exhibits the same conclusion. But it may seem, perhaps, that the institutor of the Elements has inconveniently distributed the theorems. For it was necessary either to receive three hypotheses in a divided manner, and to make three theorems; or to collect all into one theorem, as Æneas Hierapolites does, who wrote a compendium of the Elements; or willing to divide them into two, to make an orderly division, and to assume the hypotheses separately, which contain equal angles, and separately that in which the angles are equal to two right. But in the present propositions, in one theorem he supposes the alternate angles equal, but in the other, the external to the internal, and the internal angles situated at the same parts equal to two right. What then is the cause of this division? Does he regard the equality of the angles to each other, or to two right, and on this account does not separate the proposed theorems from each other; or does he respect the angles being received at the same, or not at the same parts? For the preceding theorem does not respect angles at the same parts, since such as these are alternate: but the present regards such as are situated at the same parts, as is perspicuous from the proposition. But how the institutor of the Elements shews, that from the internal angles being equal to two right, the right lines are parallel, appears from his writings on this subject. Ptolemy, however, in the theorems in which he proposes to demonstrate that right lines produced from angles less than two right, coincide at the same parts, in which the angles less than two right are situated, shewing before all his theorems, that from the internal angles being equal to two right, the right lines are parallel, proves it in the following manner. Let there be two right lines _a b_, _c d_, and let a certain right line _e g f h_, so cut them, that it may make the angles _b f g_, and _f g d_, equal to two right, I say that those right lines are parallel, that is, will never coincide. For if it be possible, let them coincide while the right lines _b f_, _g d_, are produced in the point _k_. Because, therefore, the right line _e f_, stands upon the right line _a b_, it makes the angles _a f e_, _b f e_, equal to two right. In like manner because _f g_ stands upon _c d_, it makes the angles _c g f_, _d g f_, equal to two right. Hence, the four angles _b f e_, _a f e_, _c g f_, _d g f_, are equal to four right, two of which _b f g_, _f g d_, are supposed equal to two right. The remainders, therefore, _a f g_, _c g f_, are equal to two right. If then the right lines _f b_, _g d_, when produced, coincide, the internal angles being equal to two right, _f a_, and _g c_, also, shall coincide when produced: for the angles _a f g_, _c g f_, are also equal to two right. Either therefore the right lines shall coincide in both parts, or in neither, since these, as well as the former, are equal to two right. Let the right lines then _f a_, _g c_, coincide in the point _l_. But if this be admitted two right lines _l a f k_, _l c g k_, will comprehend space, which is impossible. It is not therefore possible, that the internal angles being equal to two right, the right lines can coincide. They are therefore parallel.

PROPOSITION XXIX. THEOREM XX.

A right line falling upon parallel right lines, makes the
alternate angles equal to each other; and the external equal to
the internal angle, oppositely situated, and at the same parts;
and the internal angles at the same parts equal to two right.

The present theorem is converted in both the preceding. For that which is _the object of investigation_, in each of them, forms the hypothesis: but what are _data_ in the preceding, he proposes to shew in the present. And this difference of converse theorems is not to be passed over in silence. I mean that every thing which is converted, is either converted as one to one, as the sixth proposition to the fifth; or as one to a many, as the present to the preceding; or as many to one, as will shortly be manifest[28]. But in the present theorem, the institutor of the Elements first employs the petition, which says: _If a right line falling upon two right lines, makes the angles situated internally, and at the same parts less than two right, those right lines whilst they are infinitely produced, will coincide at those parts in which the angles less than two right are situated_. But in our exposition of things prior to theorems[29], we have asserted, that this petition is not allowed by all to be indemonstrably evident. For how can this be the case when its converse is delivered among the theorems as demonstrable? For the theorem which says that the two internal angles of every triangle are less than two right, is the converse of this petition. Besides, the perpetual inclination of right lines, more and more, while they are produced, is not a certain sign of coincidence, because other lines are found perpetually inclining, and never coinciding, as we have already observed. Formerly, therefore, some, when they had pre-ordained this as a theorem, considered that which is assumed by the institutor of the Elements as a petition, to be worthy of demonstration. But this seems to be shewn by Ptolemy himself, in a book entitled: _That right lines which are produced from less than two right angles, coincide_. And this he proves by pre-assuming many things, which as far as to the present theorem, are already demonstrated by the elementary institutor; and he supposes that all are true (lest we should also superadd another confusion) and that this, as a small assumption, may be exhibited from the preceding. But this also is one of the things previously exhibited, which says, _that the right lines produced from two angles equal to two right, will never coincide_. I say, therefore, that the converse also is true, which says, _that right lines being parallel, if they are cut by one right line, the angles situated internally, and at the same parts, shall be equal to two right angles_. For it is necessary that a line cutting parallels, should either make the angles internally situated, and towards the same parts, equal to two right, or less, or greater than two right. Let the lines then, _a b_, _c d_, be parallel, and let the right line, _g f_, fall upon them, I say that it will not make the angles internal, and at the same parts greater than two right. For if the angles _a f g_, _c g f_, are greater than two right, the remainders _b f g_, _d g f_, are less than two right. But the same are also greater than two right. For _a f_, and _c g_, are not more parallel than _f b_, and _g d_. Hence, if the line which falls upon _a f_, _c g_, makes the internal angles greater than two right, that also which falls upon _f b_, _g d_, will make the internal greater than two right. But they are also less than two right (since the four, _a f g_, _c g f_, _b f g_, _d g f_, are equal to four right) which is impossible. In like manner we may plainly shew, that the right line which falls upon parallels, does not make the angles internal, and at the same parts, less than two right. But if it makes them neither greater nor less than two right, it remains that the incident line must make the angles internal, and at the same parts equal to two right. This then being previously shewn, the thing proposed, is doubtless demonstrated. For I say, that if a right line falling upon two right lines, makes the angles situated internally, and at the same parts, less than two right, if those right lines are produced they will coincide at those parts in which the angles less than two right are situated. For let them not coincide. But if they are non-coincident at those parts in which the angles less than two right are situated, much more will they be non-coincident at the other parts, in which the angles greater than two right are situated. Hence, the right lines will be non-coincident at both parts; and if this be true, they will be parallel. But it was shewn that the right line which falls on parallels, makes the angles internal and at the same parts equal to two right. The same, therefore, are both equal to, and less than two right, which is impossible.

Ptolemy having previously shewn this, and proceeding to the thing proposed, wishes to add something more accurate, and to shew that if a right line falling upon two right lines, makes the angles internal, and at the same parts, less than two right, the lines are not only coincident as has been shewn, but likewise that their coincidence takes place at those parts, in which the angles less than two right, and not at those in which the angles greater than two right are situated. For let there be two right lines _a b_, _c d_, and let a right line _e f g h_, falling upon them make the angles _a f g_, and _c g f_, less than two right. The remainders, therefore, are greater than two right; and thus it is shewn that the right lines coincide. But if they coincide, they will either coincide at the points _a_ and _c_, or at the points _b_ and _d_. Let them coincide at the points _b_ and _d_ in the point _k_. Because, therefore, the angles _a f g_, and _c g f_, are less than two right, but the angles _a f g_, _b f g_, are equal to two right, by taking away the common angle _a f g_, the angle _c g f_, will be less than the angle _b f g_. The external angle, therefore, of the triangle _g f k_, is less than the internal and opposite angle, which is impossible. Hence then, they do not coincide at these parts. But they do coincide; and consequently they will be coincident at the other parts, in which the angles less than two right are situated. And thus far Ptolemy.

But it is necessary to scrutinize this demonstration, lest perhaps there should be any perverse and captious reasoning in the assumed hypotheses, in those, I say, in which he affirms, that a right line cutting non-coincident right lines, by forming four internal angles, forms the angles at the same parts on each side, either equal to two right, or greater, or less than two right. For the division is not perfect; since nothing hinders our calling those lines non-coincident, which are produced from angles less than two right, denominating, indeed, the two angles at the same parts, greater than two right, but the two at the remaining parts less than two right and not admitting in these, one and the same proportion. But the division being imperfect, the thing proposed is by no means demonstrated. Besides this, also, is not to be passed over in silence against his demonstration, that he does not essentially shew that which is impossible. For it is not because a certain right line cutting parallels, makes the angles at the same parts on each side, greater or less than two right, that an absurdity on this account follows these hypotheses. Nevertheless, because the four angles within the lines which are cut, are equal to four right, on this account each of these hypotheses is impossible; since, if parallel right lines are not assumed, yet, when the same hypotheses are assumed, the same consequences will be the result. And such are our animadversions against the demonstration of Ptolemy: for the imbecility of his demonstration appears from what has been said.

Let us now consider those, who say it is impossible that lines produced from angles less than two right, should coincide. For when they have assumed two right lines _a b_, _c d_, and a right line _a c_, falling upon them, and making the two internal angles less than two right, they say it is possible that the right lines _a b_, _c d_, may be shewn to be non-coincident. For let _a c_ be bisected in _e_, and cut off from _a b_, a part _a f_, equal to _a e_: but from _c d_, a part _c g_, equal to _e c_. It is manifest, therefore, that the right lines _a f_, _c g_, will not coincide in the points _f_ and _g_. For if they coincide, these two in the triangle will be equal to _a c_, which is impossible. Let again _f g_ be connected, and bisected in _h_, and cut off equal parts. These, therefore, will not coincide on the same account, and this will be the case, in infinitum, by connecting the non-coincident points; and bisecting the connecting line, and by cutting from the right lines, lines equal to the halves of the connecting lines; for by this means they say, that the right lines _a b_, _c d_, will never coincide. To such as these we reply, that they indeed affirm that which is true, but not so much as they imagine. For it is not true that the point of coincidence is simply determined by this means, nor is it true that the lines by no means coincide. Thus, when the angles _b a c_, and _d c a_, are determined, the lines _a b_, and _c d_, will not coincide in the points _f_ and _g_, yet nothing hinders their coinciding in the points _k_ and _l_, though _f k_ and _g l_ should be equal to _f h_, and _h g_. For when _a k_ and _c l_ coincide, the angles _k f h_, _l g h_, will not remain the same, and a certain part of the right line _f g_, will be left external to the right lines _a k_ and _c l_; and so again the two lines _f k_, and _g l_, are so much greater than the base, as the interior parts of the right line _f g_, which they intercept. Besides this also is to be said to such as affirm the non-coincidence of lines extended from angles less than two right, that they destroy what they are unwilling to destroy. For let the same description be given. Whether, therefore, is it possible, or impossible to connect a right line from the point _a_, to the point _g_? For if it be impossible, besides destroying the fifth petition, they also destroy that which says, _that a right line may be drawn from every point to every point_: but if possible let it be connected. Because, therefore, the angles _f a c_, _g c a_, are less than two right, it is manifest that the angles also, _g a c_, _g c a_, are much less than two right. The right lines, therefore, _a g_, _c g_, will coincide in the point _g_, being produced from angles less than two right. Hence, it is not possible to affirm indeterminately, that lines produced from angles less than two right, will not coincide. It is however manifest, that some right lines produced from angles less than two right will coincide, though the present discourse seems to investigate this in all. For it may be said, that when the diminution of two right lines is indefinite, the lines will remain non-coincident according to such a diminution: but will coincide according to another less than this. But he who desires to behold a demonstration of this affair, must be informed that it is requisite for this purpose to pre-assume such an axiom as is employed by Aristotle[30] in proving the world to be finite, viz. _If from one point two right lines forming an angle are produced in infinitum, the distance of the lines infinitely produced will exceed every finite magnitude._ For he shews that when infinite right lines are produced from the centre to the circumference, the interval also contained between them will be infinite: since, if it be only finite, it is possible that the distance may be increased; and on this account the right lines will not be infinite. Right lines, therefore, infinitely produced, are distant from each other by an interval greater than every finite magnitude.

This being pre-supposed, I say that if any right line cuts the one of parallel right lines, it will also cut the other. For let _a b_ and _c d_ be parallels, and let the right line _e f g_ cut _a b_. I say that it will also cut _c d_. For since there are two right lines, which are produced infinitely from the point _f_, viz. _b f_, and _f g_, they shall have a distance greater than every magnitude. Hence, they shall exceed the quantity of the interval contained between the parallel lines. Since, therefore, their distance from each other is greater than that of the parallels, _f g_ shall cut _c d_. But this being demonstrated, we can exhibit the thing proposed in a consequent order. For let there be two right lines _a b_, _c d_, and let a right line _e f_, fall upon them, making the angles _b e f_, _d f e_, less than two right. I say that the right lines will coincide in those parts, in which the angles less than two right are situated. For since the angles _b e f_, _d f e_, are less than two right, let the angle _h e b_, be equal to the excess of two right angles above these angles, and produce _h e_, to the point _k_. Because, therefore, a right line _e f_, falls upon the right lines _h k_, _c d_, and makes the internal angles equal to two right, viz. the angles _h e f_, _d f e_, the right lines _h k_, _c d_, are parallel; and _a b_ cuts _k h_. It will therefore also cut _c d_, by the assumption previously exhibited. Hence, the right lines _a b_, _c d_, will coincide in those parts, in which the angles less than two right are situated. And on this account the thing proposed, is evinced[31].

PROPOSITION XXX. THEOREM XXI.

Right lines parallel to the same right line, are parallel to
each other.

The geometrician in these discourses which are conversant with relation, is accustomed to shew identity permeating through all quantities, having the same relation to the _same_. Thus among the axioms also he says, _things equal to the same, are equal to each other_: and in the following books he says, _things similar to the same, are similar to each other_, and _things having the same proportion to the same, have the same proportion to each other_. After this manner, therefore, he now also demonstrates, _that right lines parallel to the same, are parallel to each other_. But it happens that this is not true in all respects. For quantities double of the same, are not also double of each other: nor are those which are sesquialter of the same, sesquialter likewise to each other, but it appears to take place in those alone, which are univocally converted in equality, similitude, identity, and parallel position. For that which is parallel to a parallel, is itself also parallel. As that which is equal to an equal, is itself equal; and that which is similar to a similar, is itself similar. For the relation of parallels to each other, is similitude of position. He affirms, therefore, and shews, in the present theorem, that lines parallel to the same, are entirely so related, that they are also parallel to each other. And he also exhibits the parallels with an external position, and likewise a medium, to which these have a similar relation, that what he asserts may become manifest from a common conception. For if they coincide with each other on either side, and coincide with that which is situated in the middle, they will no longer be parallel to it.

But it is possible that he who changes the position may shew the same thing, and by the same methods which the geometrician employs in exhibiting his proposition. For instance, he assumes both _c d_, and _e f_, parallel to _a b_, both of them situated above, and _a b_ being beneath, and not in the middle. For a right line _h k l_, falling upon them, makes each of the angles _h k d_, _k l f_, equal to _a h k_, because they are alternate; and on this account it makes the angles _h k d_, _k l f_, equal to each other. The right lines, therefore, _c d_, _e f_, are parallel. But if any one should say that _a h_, _h b_, are parallel to _c d_, and are therefore parallel to each other, we reply that _a h_, _h b_, are parts of one parallel, and are not two parallel lines. For parallels are conceived to be infinitely produced, but _a h_, when produced, falls upon _h b_. It is therefore the same with _h b_, and not a different line. Hence, all the parts of a parallel, are parallel both to the right line, to which the whole was parallel, and to its parts. As for example, _a h_ is as well parallel to _k d_, as _h b_, to _c k_. For if they are infinitely produced, they will never coincide. And these remarks must be considered as not foreign from the purpose, both on account of sophistical importunities, and the juvenile habits of mathematical auditors. For the vulgar rejoice to find captious reasonings of this kind, and to procure vain molestation to the possessors of science. But it is not requisite to convert the present theorem, and to shew that lines parallel to each other, are also parallel to the same. For if we again suppose one line parallel to some other, the remainder also of these shall be parallel to it, and they will be parallel to the same, and we shall again return to the same proposition.

PROPOSITION XXXI. PROBLEM X.

Through a given point to draw a right line parallel to a
given right line.

It is requisite that we should not only learn the essential accidents of parallels, in the discourses of the elementary institutor, but also that we should relate their origin, and know how one right line becomes parallel to another: for origin every where renders the essence of subjects more known to us. And this the institutor of the Elements effects by the present problem. For having received a point and a right line, he draws through that point a line parallel to a right line. But we ought to pre-assume as necessary, that the point should entirely be placed external to the right line: for we must not place it in the right line, because it is said, _through a given point_; since no other, besides the given line, can be that which is drawn parallel through the point. Since, therefore, the point and the right line is divided, it indicates that the point is to be received external to the right line, which he manifests in a perpendicular by addition, commanding, _upon a given infinite right line, and from a given point which is not in it, to let fall a perpendicular_. One thing, therefore, which is common to both these problems, is, the external position of the point: but the other, that from the same point two perpendiculars cannot be let fall to the same right line, and that through the same point, two lines cannot be drawn parallel to the same right line. Hence, the institutor of the Elements commands in the singular number _to draw a right line_, in the former problem, _a perpendicular_, but in the present _a parallel_. And, _that_ indeed, has been shewn, but _this_ is manifest, from what is previously demonstrated. For if through the same point two parallels are drawn to the same right line, they would be parallel to each other, and coincide in the given point, which is impossible. But it is requisite to observe the differences of these two propositions, _from a given point_, and _through a given point_. For sometimes the point is the beginning of the right line which is drawn, and on this account the deduction is made from it: but sometimes the point is in the drawn right line, and on this account the drawing is made through the point. For the particle _through_, was not asserted, because the right line cuts a given point, but because it coincides with it, and terminates its own interval, in respect of that right line, by the distance of the point and the right line. Since as much as the given point is distant from the given right line, so much also is the interval of the parallel between itself and the right line.

PROPOSITION XXXII. THEOREM XXII.

One side of every triangle being produced, the external angle of
the triangle is equal to the two internal and opposite angles;
and the three internal angles of a triangle are equal to two
right angles.

As much as was deficient in the sixteenth and seventeenth theorem, so much Euclid adds in the present. For we not only learn by this theorem that the external angle of a triangle is greater than either of the internal and opposite angles, but likewise how much it is greater; since as it is equal to both, it is greater than either of the remaining angles. Nor do we alone know from this theorem, that any two angles of a triangle, are less than two right, but by how much they are less: for they are deficient by the remaining third. The former, therefore, were more indefinite theorems: but this brings with it, on both sides, a boundary to science. We must not, however, call them on his account superfluous: for they are of the greatest utility in many demonstrations; and the present is proved by their assistance. And besides this, it is necessary that our knowledge, proceeding from the imperfect to the perfect, should pass from indeterminate apprehensions, to determinate and certain propositions. But the institutor of the Elements, by drawing a parallel externally, exhibits each of the objects of investigation. It is, however, possible that the same thing may be shewn without drawing the parallel externally; and this, by only changing the order of the things exhibited. For Euclid first shews, that the external angle is equal to the internal and opposite, and from this he proves the remainder. But we shall demonstrate this by a contrary mode of proceeding. Let there be then a triangle _a b c_, and let the side _b c_ be produced to the point _e_. Then take a point _f_ in _b c_, and connect _a f_, and through the point _f_, let _f d_ be drawn parallel to _a b_. Because, therefore, _f d_ is parallel to _a b_, and a right line _a f_, falls upon these parallels, as also a right line _b c_, hence, the alternate angles are equal, and the external is equal to the internal angle. The whole, therefore, _a f c_, is equal to _f a b_, added to _a b f_. In like manner we may shew by drawing a parallel, that the angle _a f b_, is equal to the angles _f a c_, _a c f_. The two, therefore, _a f b_, _a f c_, are equal to the three angles of the triangle: and hence, the three angles of a triangle are equal to two right, viz. to _a f b_, added to _a f c_. But _a c f_, _a c e_, are also equal to two right angles. Let, therefore, the common angle _a c f_, be taken away; and then the remaining external angle will be equal to the internal and opposite angles. And after this manner may the present theorem be exhibited.

But Eudemus, the Peripatetic, ascribes the invention of this theorem to the Pythagoreans, I mean that every triangle has its internal angles equal to two right, and says that they demonstrate it in the following manner. Let there be a triangle _a b c_, and let there be drawn through the point _a_, a line _d e_, parallel to _b c_. Because, therefore, the right lines _d e_, _b c_, are parallel, the alternate angles are equal. Hence, the angle _d a b_, is equal to the angle _a b c_; and the angle _e a c_, to the angle _a c b_. Let the common angle _b a c_, be added. The angles, therefore, _d a b_, _b a c_, _c a e_, that is, the angles _d a b_, _b a e_, and that is two right, are equal to the three angles of the triangle. And such is the demonstration of the Pythagoreans.

But it is here requisite to deliver such theorems as are converse to the present theorem of the elementary institutor. For two are converted to one, since this is a composite, both, according to _the object of enquiry_, and the _datum_: for the _datum_ is two-fold, viz. the triangle, and one of its sides produced; and in like manner _the object of enquiry_. For one part says, that the external angle is equal to the internal and opposite angles: but the other, that the three internal angles are equal to two right. If therefore, we suppose that the external is equal to the internal and opposite angles, we may shew that one side is produced, and that the right line externally situated, is in a direct position with one of the sides of the triangle: but if the three internal angles are equal to two right, we may shew that the given figure is a triangle. And so _the whole object of enquiry_, is converse to _the whole datum_. Let there be then a triangle _a b c_, and let the external angle _a c d_, be equal to the internal and opposite angles, I say that the side _b c_, is produced to the point _d_, and that _b c d_, is one right line. For since the angle _a c d_, is equal to the internal and opposite angles, let the common angle _a c b_ be added. The angles, therefore, _a c d_, _a c b_, are equal to the three angles of the triangle _a b c_. But the three angles of the triangle _a b c_, are equal to two right. And hence, the angles _a c d_, _a c b_, are equal to two right. But if two right lines being consequently placed, and not at the same parts to any right line, and at a point in it, make the successive angles equal to two right, those right lines shall be in a direct position to each other. The right line, therefore, _b c_, is in a direct position to _c d_.

Let there be again, a certain right lined figure _a b c_, having three angles alone equal to two right, viz. _a_, _b_, and _c_, I say that the figure is a triangle, and that _a c_, is one right line. For let the right line _b d_ be connected. Because, therefore, the three angles of each of the triangles _a b d_, _b d c_, are equal to two right, of which the angles of the figure _a b c_, are equal to two right, the remainders _a d b_, _c d b_, are equal to two right, and they are placed about a right line _b d_. Hence _d c_, is in the same direction with _d a_; and so the side _a c_, is one right line. In like manner we may shew that the side _a b_, and the side _b c_, are each of them one right line. And consequently the figure _a b c_, is a triangle. If then a figure having internal angles equal to two right, is right-lined, it is perfectly a triangle: but it does not follow that a figure is a triangle merely because it has internal angles equal to two right. For you will find a figure constructed from circumferences, having its internal angles equal to two right. For let there be a quadrangle _a b c d_, and upon the side _a b_, let a semicircle _a e b_, be internally described: but upon the other sides, let the semicircles be externally described, as _f_, _g_, _h_. The figure, therefore, which is comprehended by the semicircles, has two angles _g a e_, _e b h_, equal to two right, viz. to _c a b_, _d b a_. For this was shewn in the petitions[32], and these angles alone are in this figure. There is, therefore, a certain figure not a triangle, which has its internal angles equal to two right. And thus much may suffice concerning converse theorems.

But as we have discovered that the three angles of every triangle are equal to two right, we ought to determine a certain method, by which we may find how many angles, of all other multangles, are equal to so many right angles; as for instance, of a quadrangle, quinquangle, and of all consequent multilateral figures. In the first place, therefore, it must be known, that every right-lined figure may be resolved into triangles, since a triangle is the principle of the constitution of all things, which Plato also asserts in the Timæus, when he teaches us that the rectitude of a plane basis is composed from triangles. But every figure is resolved into triangles less in number, by the binary, than its proper sides. If a quadrilateral figure, into two triangles: if a figure of five sides, into three: if of six sides, into four. For two triangles composed together, immediately form a quadrilateral figure. But the number of composite triangles by which the first constituted figure differs from its sides, is the measure of difference to the rest. Hence, every multilateral figure possesses more sides, by the binary, than the triangles into which it may be dissolved. But every triangle has been shewn to contain angles equal to two right. And hence, if the number of the angles be made double to that of the composite triangles, it will afford a multitude of right angles, to which the angles of every multangle will be equal. On this account every quadrilateral figure has angles equal to four right, since it is composed from two triangles: but every figure of five sides has angles equal to six right; and after the same manner of the rest in a consequent order. This one thing, therefore, is to be assumed from the present theorem, concerning all multangular and right-lined figures.

But there is another consequent to this, which is summarily as follows. In every right-lined figure, each of its sides being at the same time produced, the angles externally constituted are equal to four right. For it is requisite, indeed, that the successive right angles should be double of the multitude of the sides; because, in each they are constituted equal to two right. But the right angles equal to the internal angles being taken away, the remaining external angles are equal to four right. As for example, if the figure is a triangle, while every one of its sides is produced, at the same time internal and external angles are constructed equal to six right angles, of which the internal angles are equal to two right, but the remaining external angles to four right. But if the figure be quadrilateral, the angles are in all eight, since they are double of the sides, of which the internal are equal to four right, and the external to the four remaining angles, and the consequences will be similar in infinitum. But after these observations we may also collect, that by this theorem every angle of an equilateral triangle is two thirds of a right angle; but that an isosceles triangle, when the vertical angle is right, has each of its remaining angles the half of one right, as a semiquadrangle; and that a scalene triangle, when it is the half of an equilateral triangle, formed by a perpendicular drawn from any angle to its opposite side, has one angle right, but the other (which likewise belonged to the equilateral triangle) two thirds of a right angle, and the remainder by a necessary consequence, a third part of a right angle. For it is requisite that the three should be equal to two right. But I do not conceive that these remarks are foreign from our purpose, since they prepare us for the doctrine of Timæus. This also must be observed, that the possession of internal angles equal to two right is inherent essentially, and answering to the predication _according to what_, in a triangle. And on this account, Aristotle in his Treatise on Demonstration[33], employs this as an example, considering it _according to what_. As therefore _to be terminated_, is essentially and primarily inherent in every figure, so likewise the possession of internal angles equal to two right, is essentially and primarily inherent in a triangle, though not in every figure. And the truth of this theorem seems to present itself to us according to common conceptions. For if we conceive a right line, and two right lines standing on its extremities, and inclining to each other, so as to form a triangle, we shall find that in proportion to their inclination they diminish the right angles, which they form with the right line. Hence, obtaining as much angular quantity, by their inclination at the vertex, as they take away from the base, they necessarily form three angles equal to two right.

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.