Chapter V: Book III: Concerning Petitions and Axioms (4)

But if any one should say, that he who describes a circle will bisect _a b_ in _f_, we can again shew that this is impossible. For let all be described as before, and let the right line _f b_, be bisected in the point _h_. Because, therefore, _a f_, _f b_, are equal, but _c f_ common, and the base _c a_, is equal to the base _c b_, all are equal to all. Hence, the angles at the point _f_ are right. Again, because _f h_ is equal to _h b_, and _c h_ being connected, is common, and the base _c f_ is equal to the base _c b_, for they are from the centre, the angles at the point _h_, are right; for they are equal and successive. Because, therefore, each of the angles _c f h_, _c h f_, is right, _c f_ is equal to _c h_. But _c f_ is equal to _c e_, for they are from the centre, and hence _c h_ is not unequal to _c e_, which is impossible.

It now remains that we run over the third objection. For the circle which is described (say they) will cut the right line in the points _a_, _b_, and in the points _f_, _h_. We therefore bisecting the right line _a b_ in the point _k_, and connecting the lines _c a_, _c f_, _c k_, _c b_, can shew that this is impossible. For since _k a_, _k b_, are equal, and _c k_ is common, and the bases _c a_, _c b_, are equal, hence the angles at the points _a_ and _b_ are equal, and those at the point _k_ right. But each of the lines is equal to _c f_; and hence, the angles at the point _f_, are right; for they are equal, because successive. Therefore, _c f_ is equal to _c k_: for they subtend right angles. But _c f_ is equal to _c d_, since they are from the centre; and hence, _c d_ is equal to _c k_, which is impossible. Hence then, it is impossible that the circle which is described should cut the line _a b_ in one, two, or in more points than _a b_. And such are the objections against the present problem.

But there are also cases of the construction of this problem, which are to be distinguished from the objections. For case is not the same with objection; since the former shews the same thing differently, but the latter leads the objection to an inconvenience. But other expositors, not distinguishing these from one another, bring all into the same, so that it is uncertain, whether they enunciate to us in their writings, cases, or objections. We therefore distinguishing these, having enumerated the objections, shall now describe the cases of the problem. Let there be then an infinite right line _a b_, and a given point _c_. Now it may be said that there is no farther place in the other part of the perpendicular right line, but in that only where the point _c_ lies. Taking, therefore, in the right line _a b_ a point _d_, with the centre _c_, and interval _c d_, let us describe the circumference of a circle _d e f_, and bisecting _d f_ in _h_, let us connect the lines _c d_, _c h_, _c f_. Because, therefore, _d h_ is equal to _h f_, but _c h_ is common, and _c d_ is equal to _c f_, (for they are from the centre,) hence, the successive angles at the point _h_, are equal. They are, therefore, right. And hence, _c h_ is a perpendicular to _d f_. But if any one should also say that the described circle does not cut the right line _a b_, but touch it as the circle _d e_, by taking the point e externally, and using the centre _c_, and interval _c e_, as in the preceding, we shall obtain the object of our enquiry. And thus much we have said concerning the cases of the problem, for the sake of exercising the attention of the reader.

But if we are desirous of adding contemplation likewise to these two problems, a right line erected at right angles, seems to imitate a life tending on high from inferior concerns, ascending purely, and without contamination, and abiding inflexibly with regard to natures subordinate to its own. But a perpendicular is the image of a life perpendicularly descending, and the least of all replete with generative infinity. For a right angle is the symbol of an energy inflexible, and restrained in the comprehension of equality, bound, and finite. From whence, indeed, Timæus also calls the _other circle_ in the divine soul, possessing the reasons of sensible natures, _right_; for in our souls it is bent with flexions of every kind, and suffers various contortions and perturbations from the unceasing whirls of generation: but among _wholes_ it resides immaculate, uncontaminated, firm, and indeclinable, prior to sensible forms. But if likewise an infinite right line is the symbol of the whole of generation, which is moved infinitely and indeterminately, and besides this, of matter itself, which is deprived of bound and form: and if a point placed externally bears an image of an essence impartible, and separate from material natures, doubtless the deduced perpendicular will imitate that life which proceeds into generation with an undefiled progress from unity, and an impartible essence. But if a perpendicular cannot be shewn without circles, this also will be the symbol of an inflexibility inherent in life, through the medium of intellect. For life, indeed, since it subsists by itself as motion, is indeterminate: but it becomes terminated, and is filled with a pure and immaculate power, by participating and adhering to the circulations of intellect.

PROPOSITION XIII. THEOREM VI.

When a right line standing upon a right line forms angles, it either
forms two right, or angles equal to two right.

Euclid again passes on to theorems, consequent to things exhibited by problems. For after a perpendicular had been drawn to a right line, and a right line erected at right angles, it remained, to enquire if it should not be a perpendicular, what angles it would form, and how it would be affected to the line upon which it stands. This then he proves universally, _that every right line standing upon a certain line, and forming angles, either forms two right, if its state be indeclinable, firm, and never verging: or angles equal to two right, if it declines in one part, but is more distant from its subject line, in the other part_. For as much as it takes away from a right angle by its declination in one part, so much it adds by its distance in the other. But it is requisite to take notice, that in this proposition also, the Geometrician employs diligent care. For he does not simply say that every right line, standing upon a right line, forms either two right angles, or angles equal to two right, but he adds, _if it forms angles_. For what if standing on the extremity of a right line, it should form one angle, will it happen that this may be equal to two right? This certainly is impossible. Since every rectilineal angle is less than two right, as also every solid angle is less than four right. Hence, though you should receive that which appears to be the greatest of all obtuse angles, this also will increase, as that which does not yet receive the measure of two right angles. It is requisite, therefore, that the right line should stand in such a manner, that it may form angles. And these observations regard the productive diligence of science.

But what does he mean by adding the particle, _either two right, or equal to two right_? For when he has constituted two right, he forms angles equal to two right; since right angles are equal to each other. Shall we say that one of the equal angles is also common, but that the other of the equals is only proper? But we are accustomed when both _proper_ and _common_ is verified, to express every particular from that which is proper, but when we cannot effect this, we are content with that which is common for the explication of the subject concerns. This then, the equality of the successive angles, is common to right angles, but is not predicated of these alone: but this, that they are right, is peculiar to their equality. Hence, the assertion, _equal to two right_, alone signifies the inequality of the angles. For in these it is alone verified, but by no means in such as are equal. And this also the institutor of the Elements divides in opposition to two right. For since it is predicated by itself, it has a power of signifying that the angles on each side are unequal. But through these observations we may also perceive, that equality is the measure and bound of inequality. For though the increase and decrease of an obtuse and acute angle is indeterminate and infinite, yet it is said to receive limitation, and bound from a right angle; and each of them, indeed, separately, recedes from a similitude to the right; but both, according to one harmonizing union, are reduced to its bound. But as they can by no means perfectly equal the simplicity of a right angle, they receive an equality to it when doubled, the duad being the exemplar of their infinity, as of itself endued with an infinite nature. And this seems to procure a manifest image of the progression of primary causes; and of their abiding according to one boundary, in a manner perpetually the same, about the infinity of generation. For how could otherwise generation, which participates of the more and the less, and is carried in indefinite whirls, agree with intelligibles, and be equalled with them in a certain respect, unless by participating their natures, whilst they advance with prolific powers, and only multiply themselves in their progressions? For things which abide in their own simplicity and impartibility, are entirely separated from generable natures. And thus much is assumed from the present theorem, and applied to the knowledge of universals.

PROPOSITION XIV. THEOREM VI.

If to any right line, and at a point in it, two right lines
being placed in a consequent order, and not towards the same
parts, make the successive angles equal to two right, those
right lines shall be in a direct position to each other.

The present theorem is the converse of the foregoing: for such as are converse are always consequent to preceding theorems. Since, therefore, the former had constituted a right line upon a right line, and had shewn that it made the successive angles either two right, or equal to two right; in the present theorem he receives the equality of the angles to two right, which are formed at some right line, but he shews that it is one right line which produces their equality. Hence, that which was a _datum_ in the former, is in the present theorem an object of enquiry; and is shewn by a deduction to an impossibility. For after this manner the converse of theorems ought to be exhibited; but in problems they should receive principal demonstrations. But in this theorem we may also perceive the greatest and most admirable diligence of this proposition producing science. For in the first place, after he had said, _if to any right line_, he adds, _and at a point in it_; for what if the two extremes of the right line existing, one of the right lines should be drawn from the one extreme, but the other from the remaining one, and should form angles at the right line, equal to two right, would they on this account have a direct position? And how can this take place in lines drawn from different points of the right line? It is on this account also, that he adds, _and at a point in it_, since he is willing that both should be in the same point. But in the second place, because it is possible that the right lines which are drawn, may be at the same point, and not consequent (since we may receive infinite right lines placed at the same point) he adds the particle, _in a consequent order_. And in the third place, because the word _consequent_ may be considered as well at the same parts as on both sides: but because it is impossible that lines which are consequent at the same parts should be mutually in a direct position, this indeed he explains, but affords us an opportunity of considering that consequent right lines are to be received in position on both sides; since these also can be shewn to be in a right line. Let there be placed at the right line _a b_, and at a point in it _b_, towards the same parts, two right lines _b c_, _b d_, these, therefore, shall be consequent to each other. For no other right line is situated between them. But those things are _successive_, between which there is nothing similar. Thus we call the columns _consequent_, between which there is no other column: for though the air intervenes, yet nothing of the same kind is situated in the middle. Because, therefore, they lie towards the same parts, they are by no means in a direct position, although they form two angles equal to two right; I mean the angles at the point _b_. For nothing hinders but that the angle _a b d_, may contain in itself, one right, and a third part of a right angle: and that the angle _a b c_, may be two thirds of a right angle. And thus much concerning the proposition.

But one petition is employed in the construction, viz. the second, which begs _to produce a right line straight forwards_, as in the demonstration he uses the preceding theorem, and two axioms; i.e. the one which says, _things equal to the same, are equal to one another_; and also the one which affirms, _that if from equal things equals are taken away, the remainders shall be equal_. But at the collection of the impossibility, he employs the axiom, which says, _the whole is greater than its part_. For it is equal one common angle being taken away, which is impossible. But that it is possible to the same right line, and at a point in it, two right lines in a consequent position, and yet, towards the same parts, may form angles belonging to that one right line, equal to two right, we may shew with Porphyry, as follows. Let there be a certain right line _a b_, and any point in it _c_, and let _c d_ be raised at right angles to _a b_, and let the angle _d c b_ be bisected, by the line _c e_. Then from the point _e_, to the line _a b_, let there be drawn the perpendicular _e b_, and let _e b_ be produced, and place _f b_ equal to _e b_, and connect _c f_. Because, therefore, _e b_ is equal to _b f_, but _b c_ is _common_, and they contain equal angles (for they are right), hence, the base _e c_, is equal to the base _c f_. All, therefore, are equal to all. Hence, the angle _e c b_, is equal to the angle _f c b_. But the angle _e c b_ is the half of a right angle: because the right angle _d c b_ was bisected by the line _e c_. Hence, also the angle _f c b_, is the half of one right. The angle, therefore, _d c f_, is equal to one right, and the half of a right angle. But the angle _d c e_, also, is the half of a right angle. Hence to the right line _c d_, and to a point in it _c_, two right lines are consequently placed towards the same parts, viz. _c e_ and _c f_, forming angles equal to two right, _c e_ causing the half of a right angle, and _c f_ one and a half. Lest, therefore, we should enquire after things impossible to be effected, viz. how the right lines _c e_, _c f_, forming angles at the right line _d c_, equal to two right, can be in a direct position to one another, the Geometrician adds the particle _not towards the same parts_. It is requisite, therefore, that the right lines which form angles equal to two right, should be placed on both sides of the right line, being raised, indeed, from one point, but drawn to different parts of the right line.

PROPOSITION XV. THEOREM VIII.

If two right lines cut one another, they will form the angles at the
vertex equal.

We must call _successive_ angles different from such as are _vertical_. For these last originate from the section of two right lines: but the former from the mere dissection of the one by the other. Thus, if a right line remaining itself without section, but cutting another in its extremity, forms two angles, we denominate these _successive_ angles. But if the two right lines mutually cut each other, they form _vertical_ angles. And they are so called, because they have their vertices conjoined in the same point. But their vertices are the points, at which the planes, while they are contracted, form angles. This, therefore, is what the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented (according to Eudemus) by Thales: but was thought worthy of a demonstration producing science by the institutor of the Elements. But it is not exhibited from all the particulars requisite to a perfect proposition. For construction is wanting in the present theorem: but demonstration, which must be necessarily inherent, depends on the thirteenth theorem. But he uses two axioms, one of which is, _that things equal to the same, are equal among themselves_: and the other, _if from equal things equals are taken away, the remainders will be equal_. The theorem, indeed, of Euclid, is manifest, but another such is converted to the present theorem. _If to any right line, and at a point in it, two right lines, not assumed towards the same parts, make the vertical angles equal, those right lines shall be in a direct position to each other_. For let there be a certain right line _a b_, and any point in it _c_, and at the point _c_, let two right lines _c d_, _c e_, not towards the same parts be assumed, forming equal angles _a c d_, _b c e_. I say that _c d_, _c e_, are in a right line. For since the right line _c d_, insists upon the right line _a b_, it forms angles equal to two right, i.e. _d c a_, _d c b_. But the angle _d c a_, is equal to the angle _b c e_. Therefore, the angles _d c b_, _b c e_, are equal to two right. Because, therefore, to a certain right line _b c_, and at a point in it _c_, two consequent right lines _c d_, _c e_, not placed towards the same parts, form the successive angles equal to two right, those right lines _c d_, _c e_, are in a direct position to each other. The converse, therefore, to the present theorem, is exhibited. But the Geometrician seems to have neglected this, because it is easy to evince its truth, by the same method of deduction to an impossibility as we employed in exhibiting the fourteenth proposition. For the same things being supposed, I say that the right line _c d_, is in a direct position to _c e_. For if it be not, let _c f_ be taken in a right line with _c d_. Because, therefore, two right lines _a b_, _d f_, intersect each other, they will form the angles at the vertex equal. Hence, the angles _a c d_, _b c f_, are equal. But _a c d_, _b c e_, were also equal. The angle, therefore, _b c e_, is equal to the angle _b c f_, the greater to the less, which is impossible. Hence, no other right line, besides _c d_, is in a direct position to _c e_. The right lines, therefore, _c d_, _c e_, are in a direct position to each other, the angles at the vertex being supposed equal. Since then, there is the same demonstration which was pre-assumed in the fourteenth theorem, would it not have been superfluous to have produced this conversion? But for the sake of exercise, we have proved it as well by a deduction to an impossible, as by an ostensive method. However, this fifteenth theorem seems to rest upon the similitude of the parts of right lines, and their situation in their extremities. Because lines with these conditions, and mutually cutting each other, must necessarily possess similar inclinations on both sides to each other. Since circumferences, and universally non-right lines cutting one another, do not necessarily form the vertical angles equal, but sometimes equal, and sometimes unequal. For if two equal circles cut each other through the centres, or even not through the centres, they will form the lunular angles at the vertex equal: but not likewise the remaining angles, viz. those on both sides concave, and on both sides convex, but the one will be greater than the other. But in right lines, the situation in the extremities, causes the distance of one segment, to be equal to the distance of another.

COROLLARY.

From hence it is manifest that if two right lines cut each other, they
will make four angles equal to four right.

Corollary is one of the geometrical appellations, but it has a two-fold signification. For they denominate corollaries, whatever theorems are proved together with the demonstrations of others, becoming as it were the unexpected gain and emolument of the investigator: and likewise, whatever is the object of enquiry, but is indigent of invention, and is neither investigated for the sake of generation alone, nor of simple contemplation. For that the angles at the bases of isosceles triangles are equal, it is requisite to contemplate, and the knowledge of things in existence is of this kind. But to bisect an angle, or constitute a triangle, to cut off, or place an equal right line, all these demand that something may be performed. And again, to find the centre of a given circle, or two commensurable magnitudes being given to find their greatest common measure, with every thing of this kind, are, after a manner, situated between problems and theorems. For neither is the origin of _objects of enquiry_ inherent in these, nor contemplation alone, but invention. Since it is requisite to place the object of enquiry conspicuously and before our eyes. Such then are whatever corollaries Euclid wrote, for he constructed a book of corollaries. But we must now omit to speak of corollaries of this kind. However, such as occur in the elementary institution, appear at the same time with the demonstrations of other things, but they themselves are not principally investigated, as is evident in that which is proposed at present. For the design of the proposition is to enquire whether if two right lines mutually cutting each other, the angles at the vertex are equal. But whilst this is evinced, it is at the same time demonstrated, that the four angles which are formed, are equal to four right. For when we say let there be two right lines, _a b_, _c d_, cutting each other in the point _e_: because _a e_ stands upon _c d_, it makes the successive angles equal to two right. And again, because _b e_ stands upon _c d_, it also makes the successive angles equal to two right; then together with the object of enquiry we demonstrate, that the angles about the point _e_, are equal to four right. A corollary, therefore, is a theorem, unexpectedly emerging from the demonstration of another problem, or theorem. For we seem to fall upon corollaries, as it were, by a certain chance; and they offer themselves to our inspection, without being proposed, or investigated by us. Hence, we assimilate these also to gains. And perhaps those skilled in mathematical concerns, have imposed on them this appellation, shewing the vulgar, who rejoice in apparent gain, that these are the true gifts of divinity, and true gains, and not the objects of their sordid estimation. For this indeed produces that faculty resident in our nature, and adds the prolific power of science, to principal enquiries, manifesting the copious riches of theorems. And such is the property of corollaries.

But they are to be divided in the first place, according to sciences. For of corollaries, some are geometrical, but others arithmetical. Thus the present corollary is geometrical: but that which is added at the end of the second theorem of the seventh book of the arithmetical elements, is arithmetical. But afterwards they must be divided according to the principal objects of enquiry. For some things are consequent to problems, but others to theorems. Thus, the present is consequent to a theorem: but that which is placed in the second of the seventh book, is consequent to a problem. But in the third place, they must be divided according to their ostensions. For some are exhibited, together with ostensive methods, but others together with deductions to an impossible. Thus the present is shewn by a direct ostension: but that which is at the same time exhibited in the first of the third book, appears, together with a deduction to an impossible. But corollaries may also be divided in many other modes, but these may suffice our present purpose. The present corollary, however, teaching us that the place about one point is distributed into angles equal to four right, is subservient to that admirable theorem, which shews that the following three multangles about one point, can alone fill place, viz. the equilateral triangle, the quadrangle, and an equilateral, and equiangular sexangle. But the equilateral triangle must be six times assumed; since six two-thirds, form four right angles. But the sexangle must be three times formed; for every sexangular angle is equal to one right, and a third part of a right. And a quadrangle must be four times assumed: for every quadrangular angle is right. Hence, six equilateral triangles conjoined according to their angles, fill four right angles, as also three sexangles, and four quadrangles. But all other multangles, however composed, according to angles, are either deficient from four right, or exceed four right angles[21]; while these alone, according to the aforesaid numbers, are equal to four right. And this theorem is Pythagoric. But by the present corollary, if even more than two right lines should cut each other in one point, as for instance, three or four, or any other number, all the angles which they form, may be shewn to be equal to four right. For they will vindicate to themselves the place of four right angles. But it is manifest that the angles always become double to the number of right lines. And thus two right lines intersecting each other, there will be four angles equal to four right: but from the intersection of three lines, there will be six angles; and from four, eight, and so on, in infinitum. For the multitude of the right lines is always doubled: but the angles increase according to multitude, and are diminished according to magnitude, because it is the same four right angles, which is perpetually divided.

PROPOSITION XVI. THEOREM IX.

In every triangle having one side produced, the external angle
is greater than either of the internal and opposite angles.

Those who enunciated this proposition, and at the same time omitted the particle, _having one side produced_, perhaps afforded an occasion of objection to many others, as well as to Philip, (according to the narration of the mechanist Heron.) But such as were desirous of entirely removing this calumny, enunciated the theorem, with the proposed addition, corresponding with the general manner of the geometrician. For in the fifth theorem, being desirous to shew, that the angles under the base of an isosceles triangle are equal, he adds, that when the equal right lines are produced, the angles under the base are equal. Hence we infer, that though this proposition might be defective and imperfect it various copies, yet it was perfect and written entire, by the institutor of the Elements. What then does the proposition assert? _That in every triangle, if you produce one of its sides, you will find the angle constituted external to the triangle, greater than either of the internal and opposite angles._ For a little after, this angle will be shewn equal to both, but it is proved to be greater than either in the present; and he necessarily compares it with the opposite angles, and not with the _successive_ angle. For to this last it may be both equal and less: but it is greater than either of the former. Thus, if this triangle should be right angled, and you conceive one of the sides comprehending the right angle to be produced, the external will be equal to the successive angle. But if it should happen to be obtuse-angled, the internal angle may be greater than the external; and it is on this account that he does not compare the external with the successive angle, but with the opposite angles. For of the angles within a triangle, the successive angle borders on the external, but the two others are opposite. Hence, the external angle is greater than either of the successive, but may not exceed the successive angle to which it is proximate. But some conjoining these two theorems, I mean the present, and the following, enunciate the proposition thus. _In every triangle having one side produced, the external angle is greater than either of the internal and opposite angles; and any two of the internal angles, are less than two right._ But there is occasion for the connection of these theorems, because the geometrician himself, a little after, enunciates the proposition after this manner, in equal angles, for he says: _In any triangle having one of its sides produced, the external angle is equal to the two interior, and opposite angles; and the three internal angles of a triangle, are equal to two right_. Hence, they think it proper in the present similar case, to connect the objects of investigation, and to make the proposition a composite. But if the datum be enunciated with this addition, it also will be a composite, (since it is requisite to understand two things, viz. the subject triangle, and one side produced:) and if the datum be given without this, it will be a _composite_ in capacity, but _simple_ in energy; for this must be received at the same time as a datum; since while we suppose an external angle, we must pre-suppose the side as produced.

But we may assume from the present theorem, that it is impossible from the same point, for three equal right lines to fall on the same right line. Thus let there be drawn from one point _a_, three equal right lines, _a b_, _a c_, _a d_, to the right line _b d_. Because, therefore, _a b_ is equal to _a c_, the angles at the base are equal. Hence, the angle _a b c_, is equal to the angle _a c b_. Again, because _a b_ is equal to _a d_, the angle _a b d_, is equal to the angle _a d b_. But the angle _a b c_, was equal to the angle _a c b_. Hence, the angle _a c b_, is equal to the angle _a d b_, the external, to the internal and opposite, which is impossible. From the same point therefore, to the same right line, three equal right lines cannot be drawn. But by the present theorem, we can also demonstrate, that if a right line falling on two right lines, makes the external angle equal to the internal and opposite, those right lines will by no means make a triangle, nor coincide, because the same thing would be both greater and equal, which is impossible. Thus for example, let _a b_, _c d_, be right lines, and let the right line _e b_ falling on them make the equal angles _a b d_, _c d e_, the right lines _a b_, _c d_, will not coincide. For if they coincide, the equal angles remaining, the angle _c d e_, will be equal to the angle _a b d_. And since it is external, it will be greater than the internal and opposite angle. Hence, it is necessary, if they coincide, that the angles remain no longer equal, but that the angle at the point _d_, be augmented. For whether _a b_ remaining immoveable, we conceive that _c d_ is moved towards it, so as to coincide in the point _c_, we shall produce a greater distance in the angle _c d e_; since _c d_ approaches to _a b_, in the same proportion as it recedes from _d e_. Or whether _c d_, abiding, we conceive that _a b_ is moved towards it, in a similar manner, we shall by this means diminish the angle _a b d_; for it is at the same time carried towards _c d_, and to _b d_. Or whether we conceive both of them tending to each other, we shall find that _a b_ by tending to _c d_, contracts the angle _a b e_; and that _c d_, by receding from _d e_, on account of the motion to the line _a b_, increases the angle _c d e_. Hence, it is necessary, if it be a triangle, and if the right lines _a b_, _c d_, coincide, that the external angle must be also greater than the internal and opposite angle. For either the internal angle remaining, the external is increased, or the external abiding, the internal is diminished, or the internal is contracted, and the external is more dilated. But the cause of these consequences is the motion of the right lines, the one tending to those parts, where it diminishes the internal angle, but the other to the parts where it increases the external. And from this the reader should consider, how the origin of things produces the true causes of enquiries, which we have previously surveyed.

PROPOSITION XVII. THEOREM X.

The two angles of every triangle, taken all possible ways, are less
than two right.

In the present theorem he shews indeterminately, that any two angles of a triangle, are less than two right, but in the following theorems he determines how much they are less, and that they are deficient by the remaining angle of the triangle: for its three angles are equal to two right; and on this account the two remaining angles are less than two right. And, indeed, the demonstration of the elementary institutor proceeds in a manifest order; for it uses the preceding theorem. But if is necessary, as in the last proposition, by regarding the origin of triangles, to find the cause of the present symptom. Let then the right lines _a b_, _c d_, be at right angles to _b d_. If these lines then are to form a triangle, it is requisite they should incline to each other. But their inclination diminishes the internal angles, on which account they become less than two right: for they were right before their inclination. In like manner, if we conceive right lines standing at right angles, on the side _a b_, the same consequences will ensue respecting the inclination of the right lines; and the angles at the points _a_, _b_, will be less than two right; and so of the other side. This then is the cause of the proposition, and not the external angle being greater than either of the internal, and opposite angles: since it is not necessary that the side should be produced, nor that any angle should be constituted external to the triangle; but it is necessary that any two of the internal angles should be less than two right. Hence, it is necessary, as I have said, that the cause of this theorem should be the inclination of the right lines diminishing the angles at the base. But as the institutor of the elements exhibits the object of enquiry, by the external angle, we may accomplish this, without producing any one of the sides. Thus let there be a triangle _a b c_, and let there be taken in the side _b c_, any point _d_, and let _a d_ be connected. Because, therefore, one side of the triangle _a b d_, is produced, viz. _b d_, the external angle _a d c_, is greater than the internal _a b d_. Again, because one side of the triangle _a d c_ is produced, viz. _c d_, the external angle _a d b_, is greater than the internal _a c d_. But the angles about the right line _a d_, are equal to two right, by the thirteenth of this. Hence, the angles _a b c_, _a c b_, are less than two right. In like manner, we may shew, that the angles _b a c_, and _b c a_, are less than two right, by taking a point in the side _a c_, and by connecting the point _b_ with the assumed point. And again, we may affirm, that the angles _c a b_, _a b c_, are less than two right, by taking a point in the side _a b_, and by connecting a right line, from the point _c_, and the received point. And thus the thing proposed, is exhibited by the same theorem, without producing any side of the triangle. Hence, it is possible, that by this, the theorem may be proved, which asserts, _that from the same point, two perpendiculars cannot be drawn, to one right line_. For let there be drawn, if possible, from the point _a_, two perpendiculars _a b_, _a c_, to the right line _b c_. Then the angles _a b c_, _a c b_, are right. But because _a b c_ is a triangle, two of its angles are less than two right. The angles, therefore, _a b c_, _a c b_, are less than two right. But they are also equal to two right, because they are perpendiculars, which is impossible. Hence, from the same point, to the same right line, two perpendiculars cannot be drawn.

PROPOSITION XVIII. THEOREM XI.

The greater side of every triangle, subtends the greater angle.

That the equality of the sides in every triangle, forms the equality of the angles which they subtend, and that in like manner the equality of the angles shews the equality of their subtending sides, we learn from the fifth and sixth theorems. But that the equality of those angles, which are subtended by the sides, follows the inequality of the sides, and the contrary we now learn by the present eighteenth and nineteenth theorems. For the one shews that the greater angle is contained under the greater side, but the other, that the greater side subtends the greater angle; because these are mutually converted, but the same symptoms are contemplated in things contrary, as in the fifth and sixth theorems. But it is manifest, that we proportionally assume the greater and less side, in scalene triangles, that we distinguish the greatest, middle, and least, and the angles in a similar manner: but in isosceles triangles, the greater and less, simply assumed, are sufficient; for there is one side which is unequal to two, because it is either greater or less, as these theorems cannot take place in equilateral triangles. And here you may observe, that the theorems which exhibit the equality of angles or sides, agree with equilateral and isosceles triangles: but these which exhibit inequality to such as are isosceles and scalene. But the cause of this is, because of triangles, some are produced from equality alone, others from inequality alone, and others from the conjunction of both, which are partly constituted from equality, and partly through inequality. And some are allied to _bound_, others to _infinity_, and others are generated from the mixture of both. Hence the ternary permeates through all geometrical forms, as through lines, angles, and figures; and among figures, through such as are trilateral, quadrilateral, and all the rest in a consequent order. But _bound_, likewise, must be considered as inherent in geometrical forms, as well through similitude, as equality; and _infinite_, both by dissimilitude, and inequality; and that which is mixt, sometimes from the junction of similitudes, and dissimilitudes, and sometimes from the union of equalities, and inequalities. But the reason of this also, is because geometrical forms regard both quantity and quality. And we have assigned these, because, when we have determined these two, it will be manifest to us, that when the institutor of the Elements says, _of every triangle_, he does not also speak of the equilateral, but of that which has a greater and less side: for it is necessary to consider the object of enquiry, as consequent to the preceding datum; and that the triangle which has a greater and less side, contains a greater angle, under the greater side.

But because the geometrician, when in the construction he receives the triangle _a b c_, and the side _a c_, greater than the side _a b_, in order that he may shew, that the angle at the point _b_ is greater than at the point _c_, from the side _a c_, he cuts off a right line _a d_, equal to the side _a b_: on this account it may be said that it is necessary to make the ablation at the point _c_, let us therefore exhibit the thing proposed upon this hypothesis, according to Porphyry, as follows. Let _d c_ be equal to _a b_, and produce _a b_ to the point _e_, and place _b e_ equal to _d a_. The whole, therefore, _a e_, is equal to the whole _a c_. Connect _e c_. Because, therefore, _a e_ is equal to _a c_, the angle, also, _a e c_, is equal to the angle _a c e_, (by the fifth). Hence, the angle _a e c_ is greater than the angle _a c b_. But the angle, also, _a b c_ is greater than the angle _a e c_; because one side of the triangle _c b e_ is produced, viz. _b e_, and so the angle _a b c_, since it is external, is greater than the internal and opposite angle. Much more, therefore, is the angle _a b c_, greater than the angle _a c b_, which was to be shewn. And such are the geometrical exhibitions of the present theorem.

But it is manifest that the cause of this symptom is the amplification, or diminution according to magnitude, of the side subtending the angle. For when it is greater, it more amplifies the angle; but when less, at the same time it diminishes, and gives a greater contraction to the angle. And this takes place on account of the right line being situated in its extremities: for through its being placed in its extremities, it changes likewise the magnitudes of the angles, according to the increase and decrease which it receives. And this we affirm in one triangle, since it is possible that the same angle may be subtended by a greater or less right line; and that the same right line may subtend a greater and less angle. For let the triangle happen to be an isosceles one, _a b c_, and let there be taken in the side _a b_, a point _a_, and let _a e_ be taken equal to _a d_, and connect _d e_. The right lines therefore, _d e_, _b c_, subtend the angle at the point _a_, of which the one is greater, but the other less. And in the same manner infinite right lines, greater and less subtending the angle _a_. Again, let the triangle _a b c_, be isosceles, and let _b c_ be less than _b a_, _a c_, and construct upon _b c_, an equilateral triangle _b d c_, and connect _a d_, and produce it to _e_. Because, therefore, the angle _b d e_, of the triangle _a b d_, is external, it is greater than the angle _b a d_. In like manner the angle _c d e_, is greater than the angle _c a d_. The whole, therefore, _b d c_, is greater than the whole _b a c_, and the same right line subtends both, viz. the greater and the less angle. But it is shewn, that likewise greater and less right lines subtend the same angle. But in one and the same triangle, one right line subtends one angle, and the greater always the greater, and the less always the less, the cause of which we have contemplated.

PROPOSITION XIX. THEOREM XII.

The greater side of every triangle subtends the greater angle.

This is the converse of the preceding theorem; and the datum, as well as the object of enquiry, is simple in each. Add too, that what was conclusion there, is hypothesis here: and what was hypothesis there is conclusion in this. But the former precedes, because it has the inequality of the sides given; and this follows, because it supposes unequal angles. For _sides_, indeed, seem to contain right-lined figures, but the _angles_ appear to be contained; and the mode of demonstration in the former is ostensive, but in this it concludes the thing proposed by a deduction to an impossibility. The geometrician, therefore, by division, reasons concerning that which is impossible: for the angles being unequal, _I say_, (says he) _that the sides also subtending the unequal angles are unequal; and the greater subtends the greater given angle_. For if that which subtends the greater angle is not greater, it is either equal, or less. But if it be equal, the angles also which they subtend, are equal by the fifth. But if less, the angle also which it subtends, is less by the preceding: for it was shewn that the greater side subtends the greater angle, and the less the lesser. But the angles have a contrary position; and hence, the one side is greater than the other.

But it is possible that we may exhibit the thing proposed, without this division. For if the angle of a triangle be bisected, and the right line drawn to the base, cutting the angle, divides it into unequal parts, the sides containing that angle will be unequal, and the greater will be that which coincides with the greater segment of the base, but the less that which coincides with the lesser. Let there be a triangle _a b c_, and let the angle at _a_ be bisected, by the right line _a d_, and let _a d_ cut the base _b c_, into unequal parts, and let _c d_ be greater than _b d_. I say that the side _a c_ is greater than the side _a b_. Produce _a d_ to the point _e_, and place _d e_ equal to _a d_. And because _d c_ is greater than _d b_, place _d f_ equal to _b d_, and connect _e f_, and produce it to the point _g_. Because, therefore, _a d_ is equal to _d e_, and _b d_ to _d f_, the two are equal to the two, and they comprehend equal angles at the vertex. Hence, the base _b a_, is equal to the base _e f_, and all, therefore, are equal to all. On this account also the angle _d e f_, is equal to the angle _d a b_. But this is not unequal to _d a g_. Hence, the side _a g_, is equal to the side _e g_, by the sixth. The side, therefore, _a c_, is greater than the side _e f_. But the side _f e_, is equal to the side _a b_; and hence, the side _a c_, is greater than the side _a b_, which was to be demonstrated.

This being pre-assumed, we can shew that the greater side subtends the greater angle. Let there be a triangle _a b c_, having the angle at the point _b_, greater than the angle at the point _c_. I say that the side _a c_, is greater than the side _a b_. Let _b c_ be bisected in the point _d_, and connect _a d_, and draw _d e_, equal to _a d_, and connect _b e_. Because, therefore, _b d_, is equal to _d c_, and _a d_, to _d e_, the two are equal to the two, and they comprehend equal angles at the vertex. Hence, the base _b e_, is equal to the base _a c_, and all are equal to all. Hence too, the angle _d b e_, is equal to the angle at the point _c_, but less than the angle _a b d_. The angle, therefore _a b e_, is bisected by the right line _b f_. Hence, _e f_, is greater than _f a_. Because, therefore, the angle at the point _b_, of the triangle _a b e_, is bisected by the right line _b f_, and _e f_ is greater than _f a_, it follows from what has been previously shewn, that the side _b e_, is greater than the side _b a_. But _b e_ has been shewn to be equal to _a c_. The side, therefore, _a c_, is greater than the side _a b_; and the object of enquiry is exhibited. And it is manifest that the institutor of the Elements, avoiding a variety of demonstration, refrains from this mode of demonstrating, and employs a method of proof, which leads from division to an impossibility, because he was willing to fabricate the converse to the preceding, without any intervening medium. For the eighth theorem, indeed, which is the converse of the fourth, brings great disturbance, because it makes conversion difficult to be known. For it is more excellent to exhibit converse theorems, by preserving the continuity through an impossible, than to destroy the continuity by a principal demonstration. And hence, Euclid shews almost all converse theorems by a deduction to an impossibility.

PROPOSITION XX. THEOREM XIII.

Two sides of every triangle, however taken, are greater than the
remaining one.

The Epicureans oppose the present theorem, asserting that it is manifest even to an ass; and that it requires no demonstration: and besides this, that it is alike the employment of the ignorant, to consider things manifest as worthy of proof, and to assent to such as are of themselves manifest and unknown; for he who confounds these, seems to be ignorant of the difference between demonstrable and indemonstrable. But that the present theorem is known even to an ass, they evince from hence, that grass being placed in one extremity of the sides, the ass seeking his food, wanders over one side, and not over two. Against these we reply, that the present theorem is indeed manifest to sense, but not to reason producing science: for this is the case in a variety of concerns. Thus for example, we are indubitably certain from sense, that fire warms, but it is the business of science to convince us how it warms; whether by an incorporeal power, or by corporeal sections; whether by spherical, or pyramidal particles. Again, that we are moved is evident to sense, but it is difficult to assign a rational cause how we are moved; whether over an impartible, or over an interval: but how can we run through infinite, since every magnitude is divisible in infinitum? Let, therefore, the present theorem, that the two sides of a triangle are greater than the remainder, be manifest to sense, yet it belongs to science to inform us how this is effected. And thus much may suffice against the Epicureans.