Chapter IV: Book III: Concerning Petitions and Axioms (3)

The present theorem exhibits these two properties of theorems, _conversion_, and _a deduction to an impossibility_. For it is converted, indeed, in the preceding theorem, but its certainty is evinced by a deduction to an impossibility. It is requisite, therefore, to speak of each, whatever belongs to the present treatise. One kind of _conversion_ then, among geometricians, is denominated principally and properly, when the conclusions and hypotheses alternately receive theorems; so that the conclusion of the former becomes hypothesis in the latter; and hypothesis is inferred as the conclusion. As _that the angles at the base of an isosceles triangle are equal_. For here the isosceles triangle is the _hypothesis_: but the _conclusion_, the equality of the angles at the base. And _that where the angles at the base are equal, the triangles are isosceles_, which the present 6th theorem affirms. For here the equality of the angles at the base is the _hypothesis_; but the _conclusion_, the equality of the sides subtending the equal angles. But another kind of conversion, is alone according to a certain mutation of composites. For if the theorem be composite, beginning from many hypotheses, and ending in one conclusion, by receiving the conclusion, and one or more of the hypotheses, we infer some one of the other hypotheses as a conclusion. And after this manner the eighth theorem is the converse of the fourth. For the one says, that _equal bases subtend equal sides and angles_: but the other, that _equal sides being placed on equal bases, contain equal angles_. Of which the predication concerning _equal bases_ in the latter proposition, is the _conclusion_ of the former: but the predication concerning the position of _equal sides_, is one of the previously assumed hypotheses in the former theorem; and the _comprehension of equal angles_ is another hypothesis which this fourth proposition contains. In consequence therefore of these two _conversions_, the one which is called the principle, is uniform and determinate: but the other is various, advancing into a great number of theorems, and not converting in one, but in many, on account of the multitude of hypotheses, in composite theorems. But oftentimes in that which begins from two hypotheses, there is one which is converted, when the hypotheses are not all determinate, but some of them indeterminate.

It is here, however, requisite to observe, that many false and improper conversions take place. As that _every sexangular is a triangular number_[11]. For the converse is not also true, that _every triangular number is sexangular_. But the reason of this is, because the one is more common, but the other more particular. And one is alone predicated _totally_[12] of the other. But things in which, _that which is primary_, is inherent, and according to which it is received, in these, conversion also follows. And these observations, indeed, were not unknown to those mathematicians, the familiars of Menæchmus, and Amphinomus. But of theorems receiving conversion, some are usually called _precedents_, but others _converse_. For when supposing a certain genus, they demonstrate some symptom of its nature, they call this a _precedent_ theorem. But when on the contrary, they make the hypothesis a symptom, and the conclusion a genus, they denominate the theorem to which this happens _converse_. As for instance, the theorem which says, _every isosceles triangle has the angles at the base equal_, is a _precedent_. For that is subjoined which precedes by nature. I mean the genus itself, or the isosceles triangle. But that which says, _every triangle possessing two equal angles, has likewise the sides subtending those equal angles equal, and is isosceles_, is a _converse_ theorem. For it changes the subject, and its passion, supposing the latter, and from this exhibiting the former. And thus much concerning geometrical conversions.

But deductions to an impossibility, entirely end in an evident impossible, the contrary of which is confessed by all. It happens, however, that some of them end in such things as are opposed to Axioms, or Petitions, or Hypotheses; but others in things contradicting prior demonstrations. For the present sixth theorem shews that which happens to be impossible, because it destroys the common conception, affirming that _the whole is greater than its part_. But the eighth theorem falls, indeed, on an impossible, yet not on that endued with a power of destroying a common conception, but that exhibited by the seventh theorem. For what the seventh denies, this affirming exhibits to such as do not admit the object of investigation. But every deduction to an impossibility, which being received, opposes _the thing sought_, and on this hypothesis advances, until it falls upon the explored absurdity, and by this means destroys the hypothesis, corroborates that which was investigated from the first. But it is requisite to know, that all mathematical proofs are either _from principles_, or _to principles_, as Porphyry in a certain place affirms. And the proofs _from principles_, are two-fold. For they either emanate from common conceptions, and things self-evident: or from things previously exhibited. But proofs _to principles_ are endued with a power of either _establishing_ or _destroying principles_. And those, endued with a power of _establishing principles_, are called _resolutions_; and to these compositions are opposed. For it is possible that we may proceed in an orderly method from those principles to the object of investigation; and this is nothing else than composition. But those possessing a power of _destroying principles_, are called _deductions to an impossibility_. For it is the business of this mode to destroy some of the concessions, and objects of investigation. And in this, also, there is a certain ratiocination, though not the same as in resolution. For in deductions to an impossibility, _complexion_ is according to the second mode of hypothetical reasonings. As if _in triangles possessing equal angles, the sides subtending the equal angles are unequal; and the whole is equal to its part_: but this is impossible. _In triangles, therefore, possessing two equal angles, the sides subtending the equal angles are equal._ And thus much concerning what is called by geometricians, deduction to an impossibility.

But the institutor of the elements uses _conversion_ in the present proposition, for he receives the conclusion of the fifth as a datum, and adds its hypothesis as an object of enquiry: but he employs _deduction to an impossibility_, in the construction and demonstration. But if any should rise up, and assert that it is not necessary by taking a part from _a c_ equal to _a b_, to make the ablation at the point _c_, but at the point _a_, upon this hypothesis, we shall fall into the same impossibility. For let _a b_ be equal to _a d_, and having produced _b a_, let _a e_ be placed equal to _d c_. The whole _b e_, therefore, is equal to the whole _a c_.

Let _e c_ be connected. Because, therefore, _a c_ is equal to _b e_, but _b c_ is common, the two are equal to the two, and the angle at the point _b_, is equal to the angle _a c b_. For so it was established in the hypothesis. All, therefore, are equal to all, by the fourth theorem. Hence, the triangle _e b c_, is equal to the triangle _a b c_, the whole to the part, which is impossible. But because this also is manifest, it remains that we exhibit the rest of the conversion. For the institutor of the Elements converts the whole sixth theorem from a part of the fifth. But it is requisite to adjoin the remaining conversion. This, then, he receives as an hypothesis, _that the angles at the base of a certain triangle are equal_: but he shews that the triangle is _isosceles_. Let _a c b_, therefore, be a triangle, and let _a b_, _a c_, be produced to the points _d g_, and let the angles under the base be equal. I say that the triangle _a b c_, is isosceles. For let there be assumed in the line _a d_, the point _e_, and let _b e_ be taken equal to _c f_; and connect the lines _e c_, _b f_, _e f_. Because, therefore _b e_ is equal to _c f_, but _b c_ is common, the two will be equal to the two. And the angle _e b c_, is equal to the angle _f c b_; for they are under the base. All, therefore, are equal to all, by the fourth theorem. Hence the base _e c_, is equal to the base _f b_, and the angle _b e c_, to the angle _c f b_; and the angle _c b f_, to the angle _b c e_: for they subtend equal sides. But the whole angle _e b c_, was equal to the whole _f c b_, of which the angle _f b c_, is equal to the angle _e c b_. The remainder, therefore, _e b f_, is equal to the remainder _f c e_. But _b e_ is equal to _c f_, and _b f_ to _c e_, and they contain equal angles. All, therefore, are equal to all. Hence, also, the angle _b e f_, is equal to the angle _c f e_. Wherefore, the side _a e_, is equal to the side _a f_ (for it is shewn by the sixth) of which _b e_, is equal to _c f_. The remainder, therefore, _a b_, is equal to the remainder _a c_. And hence, the triangle _a b c_, is isosceles. It is, therefore, as well isosceles, if it possesses angles at the base equal: as if the sides being produced it has the angles under the base equal. Why then did not the institutor of the Elements convert the remaining part? Shall we say it was because the equality of the angles under the base in the fifth theorem, was exhibited for the sake of solving other doubts. But that proving the triangle to be isosceles, from the equality of the angles under the base, neither confers to a principal demonstration, nor to the solution of things investigated, the truth of which is confirmed in the following theorems, and that from the equality of the angles under the base, he is enabled to demonstrate that the triangle is isosceles? For if every right line, standing upon a right line, and forming two angles, makes them equal to two right; when the angles under the base are equal, those upon the base will be equal. And these being equal, the sides subtending them shall be equal. Euclid, therefore, having used this in the whole elementary institution, was enabled to conclude, that when the angles under the base are equal, the triangle is isosceles. Indeed he requires this also, for the demonstration of certain theorems: For shortly a theorem will appear, evincing, that if a right line standing on a right line, forms angles, it will either make two right, or angles equal to two right. And the theorems, indeed, preceding this, require no such conversion; but those which follow, are indigent of this, and establish their credibility from the present theorem.

PROPOSITION VII. THEOREM IV.

[13]Upon the same right line, two right lines cannot be
constituted equal to two other right lines each to each,
_drawn_ to different points, to the same parts, and having
the same extremes with the two right lines first drawn.

The present theorem possesses a rare property, which is not frequently found in propositions producing science. For to be formed by negation, and not by affirmation, is not their sufficiently distinguishing property. Indeed, the propositions, as well of geometrical as of arithmetical theorems, are for the most part affirmations. But the reason of this is, (as Aristotle says) because, an affirmative universal, especially agrees with sciences, as more proper, and not indigent of negation: but a universal negative requires affirmation, in order to produce evidence; for from negatives alone, there is neither demonstration nor reasoning. Hence, demonstrative sciences exhibit a multitude of affirmations, but rarely employ negative conclusions. However, the proposition of this theorem is full of admirable diligence, and is bound with every addition, by which it is rendered so certain and indubitable, that it cannot be confuted and overturned by the efforts of opposing calumniators. For in the first place, the particle _upon the same right line_, is assumed, lest we should exhibit upon _another_, two right lines equal each to each, and employ the proposition for the purpose of circumvention. In the second place, he does not say upon what right line, to constitute two right lines simply equal to two (for this is possible) but _each to each_. For what wonderful thing is it, that he should take both equal to both, who extends one of the constituted lines, and contracts the other? But each to each, (says he) is impossible. In the third place, he adds the particle, _to different points_. For what, if some one, when he has formed two lines equal to the first two, each to each, should connect these with those in the same point, which joins the subject right lines in the vertex; and should constitute these? For the extremes of equal right lines perfectly coincide. In the fourth place, he adds the particle _to the same parts_[14]. For what if one subject right line being given, we should place two of the right lines on one side, and the other two on the opposite side, so that this common right line should be the basis of the two triangles with opposite vertexes? Lest, therefore, we should form an erroneous figure, and charge our deception on the institutor of the Elements, he adds the particle _to the same parts_. In the fifth place, he subjoins, _having the same extremes with the two right lines first drawn_. For it is possible to constitute _upon the same right line, two right lines equal to two, each to each, drawn to different points, and to the same parts_, by employing the whole right line, and constructing upon it, these two right lines; but then the lines last drawn, will not have the same extremes with those constituted at first. For if we conceive in a quadrangle two diagonals drawn on one of its sides, two lines shall be equal to two; a side and diameter to its parallel side, and the other diameter. But in this case the equal right lines will not have the _same_ extremes. For neither the parallel sides, nor the diameters, will mutually possess the same extremes; and yet they will be equal. These distinctions, therefore, being preserved, the truth of the proposition, and the certainty of the reasoning, is evinced.

But perhaps, some, notwithstanding all these terms producing science, will dare to object, that these hypotheses being admitted, it is possible to effect what the geometrician affirms to be impossible. For let there be a right line _a b_, and upon this two lines _a d_, _d b_, equal to two _a c_, _c b_, and let the former be external to the latter, being drawn to different points _d c_, and terminated in the same extremes _a_ and _b_. Let _a c_ too, be equal to _a d_: but _b c_ to _b d_. This objection, then, we shall confute, by connecting the line _d c_, and producing the lines _a c_, and _a d_, to the points _e f_. For these being constructed, it is manifest that the triangle _a c d_ is isosceles, _a d_, being equal to _a c_, from hypothesis; and the angles under the base _e c d_, _f d c_ are equal. The angle _f d c_, therefore, is greater than the angle _b d c_. Much more then is the angle _b c d_ greater than the angle _b d c_. But again, because the line _d b_, is equal to the line _b c_, the angles also at the base are equal, i.e. the angle _b c d_, to the angle _b d c_. The same angle, therefore, is both greater and equal, which is impossible. And this is what we said in our exposition of the fifth theorem, that though the equality of the angles under the base, was not useful to the demonstrations of the following theorems, yet it procured the greatest utility in the solution of objections. For in the present instance we have confuted the objection, by inferring that, because _a c_, and _a d_, are equal, the angles _e c d_, and _f d c_, are also equal. In a similar manner in other theorems, it will appear to be peculiarly useful for the solution of doubts[15].

But if any one should say that there may be constituted upon the right line _a b_, right lines _b d_, _b c_, equal to the right lines _a c_, _a d_, of which _b c_ may be equal to _a c_, but _b d_ to _a d_; and that in this case they will be drawn to different points _a_ and _b_, to the same parts, and will have the same extremes with _a c_, and _a d_, viz. _c_, and _d_, what shall we reply to this assertion? Shall we say that it is requisite to constitute the first lines, upon the right line _a b_, and their equals upon the same right line? For this is what the institutor of the Elements affirms in the proposition. But here, _a c_, and _a d_, are not constituted upon the right line _a b_, but only on one of its points. Hence, the lines _a c_, _c b_, and _a d_, _d b_, which stand on the right line _a b_, are different from the right lines, which were placed in the beginning, and to which they ought to be constituted equal. Though at the same time it is necessary that the right lines constituted upon _a b_, should be equal to those constituted upon _a b_. And thus much may suffice for objections against the present question, But that the present theorem is exhibited by the institutor of the elements, by a deduction to an impossibility, and that this impossible opposes the common conception, affirming that _the whole is greater than its part_; and that _the same thing cannot be both greater and equal_, is sufficiently manifest. But this theorem seems to have been assumed for the sake of the eighth theorem. For it confers to its demonstration, and is neither simply an element, nor elementary: since it does not extend its utility to a multitude. And hence, we find it very rarely employed by the geometrician.

PROPOSITION VIII. THEOREM V.

If two triangles have two sides equal to two, each to each, and
have the base equal to the base: then the angles contained by
the equal right lines, shall be equal to each other.

This eighth theorem is the converse of the fourth: but it is not assumed according to a principal conversion. For it does not make the whole of its hypothesis a conclusion; and the whole conclusion an hypothesis. But connecting together some part of the hypothesis of the fourth theorem, and some part of the objects of enquiry, it exhibits one of the data which it contains. For the equality of two sides to two, is in each an hypothesis; but the equality of base to base, is, in the fourth, an object of investigation, but in the present a datum; and the equality of angle to angle, is, in the former, a datum, but in the latter, an object of enquiry. Hence, a change alone of data, and objects of investigation, produces conversion. But if any one desires to learn the cause why this theorem is placed in the order of the eighth proposition, and not immediately after the fourth, as its converse, in the same manner as the sixth after the fifth, of which it is the converse, since many converted propositions follow their precedents, and are exhibited after them without any intervening medium, to this we must reply, that the eighth, indeed, is indigent of the seventh proposition. For its truth is evinced by a deduction to an impossibility, but the nature of an impossible becomes known from the seventh. And, this again, in its demonstration, is indigent of the fifth. Hence, the seventh and fifth theorems were necessarily assumed, previous to the present. But because the _converse_ to the fifth obtained a demonstration easy, and from _things first_, it was very properly placed after the fifth, on account of its alliance with that theorem; and because, since it is shewn by a deduction to an impossibility, it confutes that which is impossible from common conceptions, and not as the eighth from another theorem. For things opposing common conceptions, are more evident for the purpose of confutation than such as contradict theorems: since these are assumed by demonstration, but the knowledge of axioms is better than demonstration. But the institutor of the elements exhibits what is now proposed from the previously demonstrated seventh theorem.

But the familiars of Philo assert, that they can demonstrate this theorem, without being indigent of any other. For let there be conceived (say they) two triangles, _a b c_, _d e f_, having two sides equal to two, and the base _b c_ equal to the base _e f_. Likewise let the bases coincide with each other; and let the two triangles _a b c_, _d e f_, be so placed in the same plane, that their vertices may be opposite, and so that _e f g_ may be the equal substitute of _a b c_. And let _e g_ be equal to _d e_, but _f g_ to _d f_. Hence, _f g_ will either be placed in a right line with _d f_, or not in a right line. And if not in a right line, it will either make with it an angle according to the internal part, or according to the external. Let it first be placed in a right line. Because, therefore, _d e_ is equal to _e g_, and _d f g_ is one line, the triangle _d e g_, is isosceles, and the angle at the point _d_, is equal to the angle at the point _g_. But if it does not lie in a right line, it will make an angle inward; and in this case let _d g_ be connected.

Because, therefore _e d_, _e g_, are equal, and the base is _d g_, the angle _e d g_ also, is equal to the angle _e g d_. Again, because _d f_ is equal to _f g_, and the base is _d g_, the angle, also, _f d g_, is equal to the angle _f g d_. But the angle _e d g_ was also equal to the angle _e g d_. Hence, the whole _e d f_, is equal to the whole _f g e_, which was required to be demonstrated. But in the third place, let _f g_ make an angle with _d f_, externally, and let the right line _d g_ be connected. Because, therefore _d e_, _e g_, are equal, and the base is _d g_, the angles _e d g_, _d g e_, are equal. Again, because _d f_, _f g_, are equal, and the base is _d g_, the angle _f d g_, is equal to the angle _f g d_. But the whole angles _e d g_, _d g e_, were mutually equal. Hence, the remaining angles _e d f_, _f g e_, will be equal to each other. And thus the thing proposed is invented according to any position of the right line _f g_, and we may demonstrate the theorem, without employing the seventh proposition.

Is, then (say they), the seventh proposition introduced in vain by the institutor of the elements? For if we only assume it on account of the eighth, but the eighth may be exhibited without it, does not the seventh appear entirely useless? To these enquiries we must reply in the words of our predecessors, that the seventh theorem, being demonstrated, is of the greatest utility to such as are skilled in astronomical concerns, when they discourse concerning the eclipses of the sun and moon. For, employing this theorem, they shew that three consequent eclipses, distant from each other by an equal space, cannot subsist. I say, in such a manner, that the second may be distant from the first by as great a space of time as the third from the second. For example, if the second is produced after the first, when six months and twenty days are elapsed; the third, will by no means be produced after the second, by the same, but by either a greater or less interval of time. But that this is the case may be demonstrated by the seventh theorem. And the institutor of the elements has not only exhibited the present as conferring to astronomy, but a multitude of other theorems and problems. For to what other end shall we say that the last problem of the fourth book was proposed, by which we are taught how to inscribe the side of a figure of fifteen angles in a circle, than for its relation to astronomy? For those who describe in a circle a quindecangle passing through the poles, will, by this means, obtain the distance of the poles of the equator from the poles of the zodiac. Since they are distant from each other by the side of a quindecangle. The institutor of the elements, therefore, appears by regarding astronomy, to have previously exhibited many things preparative to our advancement in that science. But when, at the same time, he saw that this seventh theorem is exhibited from the fifth, and proves the eighth without any variety, he assigned it the present place. The addition of Philo is, indeed, beautiful, but is not sufficiently adapted by its variety of cases to an elementary institution. And thus much in reply to the present question.

But if any one should doubt why he does not add so much in the eighth as in the fourth theorem, I mean, _that the triangles and the remaining angles are equal_; we must say, that because the equality of the vertical angle is demonstrated, it follows, that all are equal to all, by the fourth theorem. It was therefore alone necessary to demonstrate this by itself, but to assume all the rest as consequents. But it seems that the equality of the vertical angles causes the equality of the bases, and of the sides comprehending those angles. For when the bases are unequal, the same angles will not remain, though the containing equal sides are supposed, but while the base becomes less, the angle is at the same time diminished, and while that increases, the angle also receives a correspondent increase. Nor while the same bases remain, but the sides become unequal, will the angle remain; but while they are diminished, it will be increased; and while they are increased, it will be diminished: for angles, and their containing sides, suffer a contrary passion. Thus, if upon the same base, you conceive the sides descending to the lower part, you will diminish the sides, but increase the angle which they comprehend, and enlarge their distance from each other. But if you conceive the sides to be elevated, and to receive an addition as they rise, you will diminish the angle which they contain: for they will coincide the longer, when their vertex is more remote from the base. We may therefore certainly affirm that the identity of the basis and equality of the sides, in a triangle, determine the equality of its angle.

PROPOSITION IX. PROBLEM IV.

To bisect a given rectilineal angle.

Our author mingles theorems with problems, and connects problems with theorems, and through both completes the whole of his elementary institution, comparing as well subjects as the _symptoms_ subsisting about subjects themselves. Since, therefore, he had shewn in the preceding propositions, both in one triangle, from the equality of the sides, the consequent equality of the angles, and the contrary: and in a similar manner in two triangles, with this exception, that the mode of conversion in one and two triangles is different, he now passes to problems, and orders us to bisect a rectilineal angle. And it is manifest, that the angle here is given according to form: for it is called right-lined, and not of any kind whatever. Indeed, we cannot bisect every angle by the elementary institution; since it is doubtful whether every triangle can be bisected. For, perhaps, you may doubt whether it is possible to bisect a cornicular angle. But the ratio of the section is also distinguished in this problem, and this again not in vain. For to divide an angle in any given ratio, transcends the present construction: as, for example, into three, four, or five equal parts. Indeed, to trisect a right angle is possible, by employing a few of the propositions which are afterwards delivered[16]: but this cannot be effected in an acute angle, without passing on to other lines of a mixt species.[17] And this is manifested by the geometricians who propose to trisect a given rectilineal angle. For _Nichomedes_, indeed, from conchoidal lines, the origin, order, and symptoms of which, he delivers, as he was the inventor of their properties, trisects every right-lined angle. But others effect this from the quadrantal lines of _Hippias_ and _Nichomedes_, by employing mixt quadrantal lines. Others, again, being incited from the Helices of _Archimedes_, divide a given rectilineal angle, in a given ratio. But the consideration of these, because difficult to learners, we shall for the present omit; as it will, perhaps, be more convenient to examine this in the third book[18], where the institutor of the Elements bisects a given circumference. For there the same mode of enquiry presents itself with respect not only to bisection, but also trisection; and the ancients endeavoured, by employing the same lines, to divide every circumference into three equal parts. With great propriety, therefore, he who only mentions a right line and a circumference, alone bisects a right angle and a circumference. But conceiving that the species composed from these, through mixture, are difficult to explain and enumerate, without a curious examination, he omits all such enquiries as involve mixt lines in their consideration, and proposes to investigate in first and simple forms alone, such things as can either be produced or considered from these. And such, indeed, is the proposition of the present problem, _to bisect a given right lined angle_. For in the construction of this he uses one petition, and the first and third problem: but in the demonstration he employs the eighth theorem alone. Since problems entirely require demonstration (as we have already observed[19]) and through this they obtain a power of producing science. But perhaps, some may oppose the geometrician, by asserting that an equilateral triangle may be constituted by him, not having its vertex within the two right lines, but either upon, or external to each; and that this may be manifested by the elements. For let there be an angle _b a c_, which it is required to bisect. Then let _b a_ be taken equal to _a c_, and let _b c_ be connected, and upon it, let an equilateral triangle _b c d_ be constructed. This point _d_, therefore, is either within the right lines _a b_, _a c_, or upon _a b_, or _a c_, or external to both. Now the institutor of the Elements assumes them within; and hence, those who oppose the demonstration, will say the point is either placed on one of the right lines, or external to both. Let the point _d_ then be placed on the line _a b_, so that the triangle _b c d_ may be equilateral: _d b_, therefore, is equal to _d c_, and the angles at the base _c b d_, _b c d_, are equal. Hence, the whole, _b c e_, is greater than the angle _c b d_. Again, because _a b_, _c a_, are equal, the triangle _a b c_, is isosceles, and the angles under the base _b c_, will be equal. The angle, therefore, _b c e_, is equal to the angle _c b d_. But it was also greater, which is impossible. Hence, the vertex of the equilateral triangle cannot be in the right line _a b d_. In like manner we may shew that it cannot be in the right line _a c e_. Let it therefore, if possible be placed externally. Because, then _b d_ is equal to _c d_, the angles at the base are equal, viz. _b c d_, and _c b d_. Hence, the angle _b c d_, is greater than the angle _c b f_. Much more, therefore, is the angle _b c e_, greater than _c b f_: but it is also equal, because these angles are under the base _b c_, of an isosceles triangle _a b c_, and this is impossible. Hence, the point cannot fall in these parts external to the two right lines; and it may be similarly shewn that this is impossible in other parts. Here too you may again observe, that we destroy objections by using the second part of the fifth proposition, _that the angles under the base of an isosceles triangle are equal_. And this is what we have previously observed, that many things opposing science, are shewn to be debile, and easy of confutation, by the assistance of this theorem; and that such is the utility it affords to geometry.

But if any one should say that there is no place under the base, and yet that it is requisite to constitute the equilateral triangle at the same parts, in which the lines _b a_, _a c_, are situated; it will be necessary that the lines which are constituted should either coincide with _b a_, _a c_, if they also are equal to the base _c b_: or that they should fall external to them, if they are less than the base _b c_: or within, if _b a_, _a c_, are greater than _b c_. Let them, in the first place, coincide, and let _b a c_ be an equilateral triangle, and let there be taken in the side _a b_, the point _d_, and make _a e_ in the side _a c_, equal to _a d_, and connect the lines _d e_, _b e_, _c d_, _a f_. Because, therefore, _a b_ is equal to _a c_, and _a d_ to _a e_, the two _b a_, _a e_, are equal to the two _c a_, _a d_, and they comprehend the same angle. Hence, they are all equal to all, and the angle _d b e_, is equal to the angle _e c d_. But _d b_ is also equal to _e c_, and _b e_ to _c d_. All, therefore, are equal to all. Hence, the angle _d e b_, is equal to the angle _e d c_: for they subtend equal sides. And _d f_ is equal to _e f_, (by the sixth.) Because, therefore, _a e_ is equal to _a d_, and _a f_ is common, and the base _d f_, is equal to the base _e f_, the angle _d a e_ is bisected, which was required to be done.

But if the sides of the equilateral triangle fall external to the right lines _b a_, _a c_, let them be _b d_, _d c_, and having connected _d a_, let it be produced to the point _e_. Because, therefore _b d_, _d c_, are equal, but _d a_ is common, and the bases _b a_, _a c_, are equal, the angle, also, _b d a_, (by the eighth) is equal to the angle _c d a_. Again, _b d_, _d c_, are equal, and _d e_ is common, and they contain equal angles as we have shewn, the base also _b e_, is equal (by the fourth) to the base _e c_. Because, therefore, _a b_ is equal to _a c_, and _a e_ is common, the angle, also _b a e_, is equal to the angle _c a e_, which was to be shewn.

But if the sides of the equilateral triangle fall within the right lines _a b_, _a c_, as _b d_, _d c_, let again _a d_ be connected. Because, therefore, _b a_, is equal to _a c_, and _a d_ is common, but the base _b d_, is equal to the base _c d_, hence, the angle _b a d_ (by the eighth) is equal to _c a d_. The angle, therefore, at the point _a_, is bisected, in whatever manner the equilateral triangle may be constituted. And having thus summarily spoken concerning these, we shall now proceed to the following theorems, only adding, that the given angle may be given in a four-fold respect. _In position_, as when we say _to this right line, and to this point to place an angle_: for after this manner it is given. But _in form_, as when we call the angle right, or acute, obtuse, right-lined, or mixed. And _in proportion_, as when we call it double, or triple, greater, or less. And lastly, _in magnitude_, as when we call it the third part of a right angle. But the present angle is only given in form.

PROPOSITION X. PROBLEM V.

To bisect a given finite right line.

This, also, is a problem which supposes a finite right line, since we cannot terminate a line on both sides infinite. But the section of a line infinite on one side only, wherever the point is assumed, is made in unequal parts. For that part of the section which takes place on the infinite side, is necessarily greater than the remainder, because finite. Hence, the line required to be bisected, must be necessarily both ways finite. But perhaps, some excited by this problem, may think, that the doctrine of a line, not being composed from impartibles, is only previously received by geometricians as an hypothesis. For if it consists from impartibles, it either becomes finite, and receives its completion from _odd_, or from _even_ parts. But if from such as are _odd_, it will appear that an impartible also may be cut, while a right line is bisected. And if from such as are even, the section will be unequal, because, one part, as composed from more impartibles, will be greater than the remainder. It is therefore impossible to bisect a given right line, if magnitude consists from impartibles. But if it be not composed from impartibles, it may be divided in infinitum. It appears, therefore, (say they) to be received by common consent, and to be a geometrical principle, that magnitude is among the number of things infinitely divisible. Against these we reply in the words of Geminus, that geometricians previously receive according to a common conception, that continued quantity is divisible. For we call that continuous, which is composed from conjoined parts, and this it is in every respect possible to divide. But that continued quantity may be infinitely divided, they do not previously assume, but demonstrate from proper principles. For when they shew that incommensurability is found in magnitudes, and that all are not commensurable with each other, what else can we say they evince by this means, except this, that every magnitude may be divided into parts always divisible, and that we can never arrive at an impartible, by the most unwearied analysis, since this minimum would be the common measure of all magnitudes? This then is demonstrable, but that which says, _every thing continuous is divisible_, is an axiom. Hence, since a finite line also is continuous, it is divisible. And from this conception the institutor of the Elements cuts a finite right line into equal parts, but not as pre-assuming, that it is divisible in infinitum. For to be merely divisible, and to be infinitely divisible is not the same.

But the discourse of Xenocrates inferring indivisible lines, is confuted by this problem. For if it be a line, it is either right, and may be bisected; or circular, and it is greater than a certain right line; (since every circular has a certain right line less than itself); or it is mixt, and on this account is the more divisible, since composed from simple divisible lines. But this must be deferred to some posterior speculation. However, the geometrician bisects a finite right line, employing in the construction the first and ninth propositions; but using in the demonstration the fourth alone; for by the angles he shews the equality of the bases. But Apollonius Pergæus bisects a given finite right line after the following manner. Let there be (says he) a finite right line _a b_, which we are required to bisect, and with the centre _a_, but interval _a b_, let a circle be described. And again, with the centre _b_, but interval _b a_, let another circle be described, and let the right line _c d_, connect the common sections of the circles; this shall bisect the right line _a b_. For let the equal lines _d a_, _d b_, _c a_, _c b_, be connected; these being equal, because each is equal to _a b_. But _c d_ is common, and _d a_ is equal to _d b_ on the same account. Hence the angle _a c d_, is equal to the angle _b c d_; and so (by the fourth) _a b_ is bisected. Such then, according to Apollonius, is the demonstration of this problem, assumed, also, from an equilateral triangle; but instead of exhibiting the bisection of the line, from the bisection of the angle at the point _c_, it shews this from the equality of the bases. The demonstration, therefore, of the institutor of the Elements, is much better, since it is both more simple, and emanates from principles.

PROPOSITION XI. PROBLEM VI.

To raise a right line at right angles, to a given right line, from a
given point in that line.

Whether we receive a right line on both sides finite, or on both sides infinite, or on one side infinite but on the other finite, and a point in it, the construction of the present problem will conveniently succeed to the geometrician. For though the given point should be on the extremity of the right line, by producing it we can accomplish our purpose. But it is manifest that the point in the present problem is given in _position_, since it can only be placed in position in a right line. But the right line is given according to _form_; since its magnitude is not distinguished either by proportion or position. Hence, the institutor of the Elements, employing the first and third problem, together with the eighth proposition, and the tenth definition, exhibits the thing proposed. But if any placing the point on the extremity of the right line, should ask us without producing the line, to erect upon this a right line at right angles, we can likewise shew that this is possible to be effected. For let there be a right line _a b_, and a given point in it _a_, and let there be assumed in the line _a b_, any point _c_, and from this (as the present element teaches us) let a right line _c e_ be erected at right angles to _a b_. Then from _c e_, let _c d_ be taken equal to _a c_, and let the angle at the point _c_ be bisected by the line _c f_, and at the point _d_ let a right line be erected at right angles, coinciding with _f c_ in _f_; and lastly from the point _f_, to the point _a_, let _f a_ be connected. I say that the angle at the point _a_ is right. For since _d c_ is equal to _c a_, but _c f_ is common, and contains equal angles, (for the angle at the point _c_ was bisected) hence, _d f_ is equal to _f a_, and all in like manner (by the fourth) are equal to all. The angle, therefore, at the point _a_, is equal to the angle at _d_. But the angle at the point _d_ is right; and so consequently is the angle at _a_. And thus the thing required is effected. But the institutor of the Elements was not indigent of any such artifice: for he commands us to raise a line at right angles, but not at one right. It is requisite, therefore, not to receive the point in the extremity of the right line, because the perpendicular line forms angles with its subject right line, but not one angle alone.

But Apollonius raises a perpendicular as follows. Let the given right line, says he, be _a b_, and a given point in it _c_, but let there be assumed in _a c_ any point _d_, and from _c b_, take away _c e_, equal to _c d_. Then with the centre _d_, but interval _d e_, let a circle be described; and again with the centre _e_, but interval _e d_, let another circle be described, and let a right line be drawn from _f_ to _c_. I say that _f c_ is a perpendicular. For if _f d_, _f e_, are connected, they shall be equal. But _d c_, _c e_, are equal, and _f c_ is common. Hence, also, the angles at the point _c_ (by the eighth) are equal. They are therefore right. And here, is it not again obvious, that this demonstration is more various than that of Euclid, and requires the description of circles, that by this means an equilateral triangle may be described upon _d e_, and the problem exhibited? For all the rest are common to the demonstrations. But the demonstration by a semicircle is not worthy to be remembered, since it supposes many things which are afterwards exhibited, and entirely falls from the order of an elementary institution.

PROPOSITION XII. PROBLEM VII.

Upon a given infinite[20] right line, and from a given point which is
not in that line, to let fall a perpendicular.

Oenopides first investigated this problem, believing it useful for astrological purposes. But he calls a perpendicular, after the manner of the ancients, a gnomon, because a gnomon, also, is at right angles to the horizon, but the same line is at right angles with a perpendicular, from which it differs only in habitude, since, as he observes a gnomon has the same subject with a perpendicular. But again, a perpendicular is two-fold, that is, it is either plane or solid. Hence, when the point from which the perpendicular right line is drawn, is in the same plane, the perpendicular is called plane; but when the point is on high, and external to the subject plane, it is called solid. And the plane perpendicular, indeed, is drawn to a right line: but the solid to a plane. Hence, it is necessary, that this last should not only form right angles, with one right line, but with all right lines in the same plane. For the perpendicular is let fall on a plane. In the present problem, therefore, the institutor of the Elements proposes to let fall a plane perpendicular. For the deduction is proposed to a right line, and the discourse proceeds, so far as all are supposed to be in the same plane. Hence, in the line at right angles we do not require infinity, because the point is supposed to be in that right line. But in the present problem, respecting a perpendicular, he supposes the given right line infinite, because the point from which the perpendicular is to be drawn is placed external to the right line. For if it was not infinite, the point might be received externally, and yet in a direct position, so that the protracted right line would fall upon it, and the problem not succeed. Hence, he places the right line infinite, so that the point may be received at either of its parts; and that no place may be left, in which it can be in the same direction with the given right line, unless it is in the line, and has not an external position. And on this account the right line to which the perpendicular is to be drawn is considered as infinite.

But in what manner infinite can subsist, is a matter well worthy our contemplation. For it is manifest that a right line existing infinite, a plane also will be infinite, and this in energy, if the thing proposed by Euclid be true. That among sensible particulars, therefore, there can be no magnitude infinite, according to any distance, both the dæmoniacal Aristotle, and those who received their philosophy from him, have abundantly shewn. For neither that which is moved circularly, nor any other simple body can be infinite; since the place of each is limited. But neither in separate and impartible reasons is an infinite of this kind possible. For if they neither contain dimension, nor magnitude, much less can they contain infinite magnitude. It remains, therefore, that infinite can alone subsist in the phantasy, which at the same time the phantasy does not comprehend. For as soon as it understands, it induces form and bound to that which is understood, stops the transit of the phantasm by its intellection, pursues its progress, and infolds it in its shadowy embrace. The phantasy, therefore, is not infinite by intellection, but rather by advancing infinitely about that which is understood; and calling whatever it leaves innumerable, and incomprehensible by intelligence, infinite. For as the sight by not seeing understands darkness; so the phantasy by not understanding perceives infinite. Hence it pursues the progress of the infinite, because it is endued with an impartible power, capable of perpetually advancing: but it understands as if stopping in its progression, because infinite surpasses its comprehension. For it calls that infinite, which it leaves as unable to pass over in its pursuit. On this account when we place a given infinite line in the phantasy, in the same manner as we establish all other geometrical species, viz. triangles, circles, angles, lines, and all of this kind, we must not wonder how a line is infinite in energy, and how advancing infinitely, it applies itself to finite intellections. But cogitation, in which reasons and demonstrations reside, does not use infinite for the purpose of science, since infinite is by no means perceptible by science, but receiving it from hypothesis, it employs finite alone in its demonstrations, and assumes infinite not for the sake of infinite, but of that which is bounded and finite. For if we should grant to cogitation, that the given point, neither lies in a right line with the given finite right line, nor yet is so distant from it, that no part of the right line is subjected to the point, we shall no longer require an infinite line. That cogitation, therefore, when employing a right line, may use it without controversy and reproof, she supposes it to be infinite; and employs the infinity of the phantasy, as the foundation of infinite generation. And thus much may suffice for the present concerning the nature of infinite.

But it is now requisite that we should consider the objections which are urged against the construction of this problem. Let there be received, say they, an infinite right line _a b_, and let the given point be _c_, from which it is required to let fall a perpendicular, and let _d_ be a point on the other side, according to the geometrician. But the circle which cuts the right line _a b_, in the points _a_ and _b_, will cut it also in _f_, and will have a situation according to the figure. In answer to this, we must say, that it affirms an impossible case. For let the right line _a b_ be bisected in _h_, and let _c h_ be connected, and produced to the circumference, to the point _d_, and let _c a_, _c b_, _c f_, be connected. Because, therefore, these lines are from the centre, and _a h_, is equal to _h b_, but _c h_ is common, all are equal to all. Hence _c h_ forms right angles at the point _h_. Again, because _c a_, _c b_, are equal, they form equal angles at the points _a_ and _b_. But _c a_ also, is equal to _c f_, on which account the angle _c a f_, is equal to the angle _c f a_. In like manner the angle _c b f_ is equal to the angle _c f b_. Because, therefore, the angles at the points _a_ and _b_, are equal, the angle, also, _c f a_, is equal to the angle _c f b_, and they are successive, and consequently right. But each of the angles at the point _h_ is right. Hence, _c h_ is equal to _c f_. But _c f_ is also equal to _c d_, since they are from the centre. Therefore _c h_ is equal to _c d_, which is impossible. Hence, the circle does not cut the right line in any other points than _a_ and _b_.