Chapter III: Book III: Concerning Petitions and Axioms (2)

But let circles be described, and from their common section let the right lines _c e a_, _c e b_, be extended, having the common segment _c e_. It will therefore happen, that the lines extended from the common section, will be equal to the given line _a b_, and yet the sides of the triangle will not be also equal, but two will be less than the remainder, that is, than _a b_. And so this not being constituted, neither can the rest be constructed. Can then (says Zeno) the rest follow, though the principles are given, unless this also is previously received, that there are no common segments either of circles or of right lines? Against this objection then, we must affirm in the first place, that it was in a certain respect previously understood, that two right lines have no common segment. For the definition of a right line comprehends this property, since _that is a right line which is equally situated between its bounding points_; and the equality of the interval between the points to the right line, causes that which joins the points to be one, and the shortest line; so that if any one adapts it to another line, according to one of its parts, it must also agree with the line according to its remaining part; for since it is constituted in its extremities, because it is the shortest line, it is necessary that the whole should fall on the whole. But again, this was manifestly received in the Petitions: for the Petition which says, _that a terminated right line may be produced straight forwards_, perspicuously shews that the produced line ought to be one, and produced by one motion; but if any one is desirous to receive a demonstration of this assumption, let, if possible, _a b_ be the common segment of _a c_ and _a d_, and with the centre _b_, and interval _b d_, let the circle _a c d_ be described; because therefore the right line _a b c_, is drawn through the centre, _a f c_ is a semicircle; and because the right line _a b d_ likewise is drawn through the centre, _a e d_ is a semicircle. The semicircles, therefore, _a f c_, _a e d_, are equal to each other, which is impossible. But against this demonstration Zeno will perhaps say, that it is likewise requisite to demonstrate that the diameter bisects the circle, because we previously assume that there is not a common segment of two circumferences. Thus too we take for granted, that one circumference coincides with another, or if it does not coincide, that it either falls externally or internally. But nothing hinders (he will say) that the whole may not coincide with the whole, but according to some part. But to this Possidonius rightly answers, who laughs at the acute Epicurean, as if conscious that though the circumferences do not coincide according to a part, yet the demonstration will succeed; for according to that part in which they do not coincide, the one will fall within, and the other without, and the same absurdities will follow when right lines are extended from the centre to the external circumference; for those from the centre will be equal, as well the greater which is drawn to the external, as the less which is extended to the internal circle: either therefore the whole will coincide with the whole, and they will be equal; or coinciding according to a part, it will alternately vary according to the remainder, or no part will coincide with no part; and in this case it either falls within or without: but of this, enough. But Zeno also condemns the following demonstration of this particular: Let _a b_ be the common segment of two right lines _a c_, _a d_, and let be _b e_ erected at right angles to _a c_, the angle _e b c_, therefore, is a right one. Hence, if the angle _e b d_ is also right, they shall be equal, which is impossible; but if not, let _b f_ be erected at right angles to _a d_. The angle _f b a_, therefore, is right; but the angle _e b a_ was also right; and they are therefore mutually equal, which is impossible. This is the demonstration which Zeno opposes, as assuming that which is to be exhibited afterwards; I mean from a given point to raise a right line, at right angles, to a given right line. However, Possidonius observes, that indeed, a demonstration of this kind is never to be introduced into elementary institutions; but that Zeno calumniates Geometricians using their own as a flagitious demonstration; though there is some reason in their conduct. For there are right lines existing at right angles; since any two right lines are capable of forming a right angle; and this is previously assumed in our definition of a right angle. For we alone constitute a right angle from such an inclination; and it may perhaps be this which we have erected. Indeed, Epicurus himself, and all other philosophers admit, that not only many things possible may be supposed, but likewise many of an impossible matter, for the purpose of contemplating something consequent; and thus much concerning an equilateral triangle.

But it is requisite to construct other triangles, and in the first place an isosceles. Let _a b_, therefore, be a right line, upon which it is required to construct an isosceles triangle. Describe circles as in the construction of an equilateral triangle, and produce the line _a b_ on each side to the points _c d_; the line _c b_, therefore, is equal to _a d_. Again, with the centre _b_, and interval _c b_, let the circle _c e_ be described; and with the centre _a_, and the interval _d a_, the circle _d e_; and from the point _e_, in which the circles intersect each other, to the points _a_ and _b_, let the lines _e a_, _e b_, be extended. Because therefore, _e a_ is equal to _a d_; but _e b_ to _b c_, and _a d_ is equal to _b c_, _e a_ will also be equal to _e b_; but they are also greater than _a b_. The triangle _a b e_, therefore, is isosceles, which it was required to constitute. But let it be ordered to construct a scalene triangle upon the given right line _a b_. Describe circles with centres and intervals, as before, and let there be taken in the circumference of the circle, whose centre is _a_, the point _f_, and let the right line _a f_ be extended and produced to the point _g_; and likewise let the right line _g b_ be extended. Because, therefore, _a_ is a centre, _a f_ is equal to _a d_; and hence, _a g_ is greater than _a d_, that is, than _g b_. But _b_ also is a centre, _g b_, therefore, is equal to _c b_; and hence, _g b_ is greater than _b a_: but _g a_ is greater than _g b_; the three lines therefore _g b_, _b a_, _a g_, are unequal; and hence, the triangle _a b g_ is scalene. Hence too, three triangles are constructed; but these things are commonly known: however, this is beautiful in these triangles, that the equilateral existing on all sides equal, is constructed by one mode alone; but the isosceles, endued with equality in two sides only, has a two-fold construction: for the given right line is either less than both the equal ones (according to our present construction), or it is greater than both; but the scalene being unequal in all its sides, receives a triple construction; for the given right line is either the greatest of the three, or the least; or greater than the one, and less than the other; and indeed, it is proper to be exercised in each supposition, either by enlarging or contracting; but to us, what is already delivered, is sufficient. Let us now contemplate problems universally, some of which are produced simply, but others manifoldly, and others according to infinite modes. But (as Amphinomus observes) those which are simply constructed are _ordinate_: but those which receive a manifold composition, and are constructed according to number, are _middle_; and those which are varied in infinite ways, are _inordinate_. The manner, therefore, in which problems are constructed, simply or manifoldly, becomes manifest in the preceding triangles; for the equilateral is constituted simply; but of the other two, the one receives a two-fold, and the other a triple construction. But problems of the following kind, may take place in infinite modes; I mean _to divide a given right line in three proportional parts_; for if it be divided in a duple ratio, and the deficient quadrangular form, resulting from the less, be applied to the greater, it will be divided into three equal parts; but if the greater segment be more than double of the less, as for instance, triple, and a deficient quadrangular form, equal to that which results from the less, be applied to the greater, the line will be divided into three unequal parts. Because, therefore it may be divided into two parts, in infinite ways, the greater of which is either double or triple, (for multiplex proportion proceeds in infinitum), hence, it may be divided into three parts, according to infinite variations.

But it is requisite to know, that problem also is manifoldly predicated; for whatever is proposed may be called a problem, whether it is proposed for the sake of learning or operating. But in mathematical disciplines, that is properly called a problem, which is proposed for the purpose of contemplative energy. Since that which is performed in these, has contemplation for its end; and often, indeed, certain things, impossible to be executed, are called problems: but more properly that which is possible to be done, and neither exceeds, nor is deficient, is allotted an appellation of this kind; and the problem _exceeds_, which says, _to construct an equilateral triangle, having its vertical angle two thirds of one right_; for this is superfluous, and is added in vain: since it is a property inherent in every equilateral triangle. But of those which exceed, whatever are redundant with incongruous and non-existent symptoms, are called _impossibles_; but whatever are redundant with accidents, are called _greater problems_. But a _defective_ problem (which is also called a _less problem_) is that which requires some addition, that it may be reduced from inordination into order and scientific bound, as if any one should say, _to constitute an isosceles triangle_: for this is mutilated and indeterminate, and requires some one who may subjoin, what kind of an isosceles triangle, whether that which has its base greater than either of the equal sides; or that which has it less. Likewise, whether that which has the vertical angle double of each at the base, as a semiquadrangle; or that which has each of the angles at the base double of the vertical angle; or that which possesses these angles according to some other proportion, as triple or quadruple: for it is possible that it may be varied in infinite modes. From hence, therefore, it is manifest, that such things as are properly denominated problems, ought to avoid indetermination, and not to be of the number of things capable of infinite variation; though such as these are also called problems, through an equivocation of the word problem. The first problem, therefore, of these elements, excels the rest in the manner we have explained; for it neither _exceeds_, nor is _deficient_; it is neither constructed in a variety, nor according to infinite modes; and such ought to be the conditions of that which is to be the element of the rest.

PROPOSITION II. PROBLEM II.

To a given point to place a right line equal to a given
right line.

Of problems, as well as of theorems, some are without case, but others possess a multitude of cases. Whatever, therefore, have the same power acceding to many descriptions, and when their positions are changed, preserve the same mode of demonstration, these are said to have _case_; but such as proceed according to one position only, and one construction, are without _case_; for simply, _case_, appears about the construction both of theorems and problems. The second problem, therefore, has many cases; but a point is given in it _in position_, since it can only be given in this manner; but a right line, both in form and position, (for it is not simply _line_, but of such a kind.) For it is here enquired, _how to a given point to place a right line equal to a given right line_. But it is manifest that the point is entirely in the subject plane, in which the right line exists, and not in one more elevated. For in all problems and theorems respecting planes, we must conceive that one plane is subjected. But if any one should doubt how a line is to be placed equal to a given right line, for what if the given line be infinite? Since the present datum pertains both to finite and infinite: for every datum signifies that which is proposed and supposed by us for the sake of investigation. But this Euclid himself declares sometimes, saying, _upon a given terminated right line to construct an equilateral triangle_; but at other times, _upon a given infinite right line to let fall a perpendicular_. In answer then to this doubt, we must say, that when he orders us to place the line equal to a given right line, at a given point, he sufficiently evinces that the given line is finite; for every thing placed at a point, is terminated according to that point. Hence, the line equal to that which is given, must have a much prior termination. At the same time, therefore, in which he says _to a given point_, he terminates both the given right line, and its equal which is investigated.

But that the cases of the present problem are formed from the various position of a point, is manifest. For the given point is either placed external to, or in the given right line; and if in it, it will either be one of its extremities, or it will be situated within the extremes; and if external, it will either have a lateral position, so that a line drawn from it to the extremity of the given line will form an angle, or a direct position; so that if the line were produced, it would coincide with the external point. But the geometrician, indeed, considers the point as external, and receives it according to a lateral position; however, for the sake of exercise, all the positions are to be assumed, the more difficult of which we shall exhibit. For let there be given a right line _a b_, and a given point _c_, which lies between its extremes, and let there be constituted according to the doctrine of the elements, an equilateral triangle upon the right line _a c_, and let _d c_, _d a_, be produced; then, with the centre _a_, and the interval _a b_, let the circle _b e_ be described. And again, with the centre _d_, but with the interval _d e_, let the circle _d f_ be designed. Because, therefore, _a_ is the centre, _b a_ is equal to _a e_; and hence, _d e_ is equal to _d f_, the parts of which, _d a_, _d c_, are equal: for the triangle _d a c_ was established as equilateral. The remainder, therefore, _a e_, is equal to _c f_; but _a e_, as it was shewn, is equal to _a b_, and hence, _c f_ is equal to _a b_. To a given point, therefore, _c_, a right line _c f_ is placed equal to _a b_. With respect to the position of the point then, so many cases arise. But there are many more with respect to the constitution of the equilateral triangle, the extension of its sides, and the description of circles. For let there be assumed, as in this element, a point _a_, and a right line _b c_, but let _b a_ be extended. The equilateral triangle, therefore, will not be constituted on _b a_, with its vertex above (because there is no place for it), but beneath; let it, therefore, be _a d b_; _a d_, therefore, is either equal to _b c_, or greater or less. If then it be equal that which was required is performed. But if less with the centre _b_, and the interval _b c_, let a circle be described, and let _a d_, _d b_ be produced to the points _e_ and _g_; and with the centre _d_, but the interval _d g_, let a circle _g a_ be designed. Because, therefore, _d g_ is equal to _d e_, for they are drawn from the centre; and likewise because _a d_ is equal to _d b_, for the triangle is equilateral, the remainder _a e_ is equal to the remainder _b g_. But _b g_ is also equal to _b c_, for they proceed from the centre; and hence, _a e_ is equal to _b c_, which was required to be done. But if _a d_ is greater than _b c_ (for this is the last case), then with the centre _b_, and the interval _b c_, let a circle _e c_ be described. The line _d b_, therefore, shall cut the circle _e c_. Again, with the centre _d_, and interval _d e_, let the circle _e g_ be described. Because therefore, _d_ is the centre of the circle _g e_, _g d_ is equal to _d e_. But _d a_ was also equal to _d b_; the remainder, therefore, _a g_ is equal to the remainder _b e_. But _b e_ is equal to _b c_, for both proceed from the centre. Hence, _a g_ is equal to _b c_; and it is placed at the point _a_, as was required to be done. And though there are many other cases, the description of the above is sufficient for our present purpose. For from these it is possible for the more curious to exercise themselves in the rest. But formerly some destroying the construction and variety of this problem, reasoned thus. Let _a_ be a given point, but _b e_ a given right line, and with the centre _a_, but with an interval equal to _b e_, let a circle _d e_ be described. Then let a certain right line _a d_ be extended from the point _a_ to the circumference; and this shall be equal to _b e_: for the magnitude of the line from the centre, was equal to that of _b e_: and so that is done which was required. But he who thus reasons, _begs_, in the very beginning. For when he says with the centre _a_, but interval _b e_ describe a circle _e d_, he receives, in a certain manner, a line equal to _b e_, placed at the extremity _a_; and preserving the Petition, he makes one extremity of the interval a centre, but with the other describes a circle: however, in this case, the centre is in one place, but the interval in another. We by no means, therefore, approve this method of demonstration.

PROPOSITION III. PROBLEM III.

Two unequal right lines being given, from the greater
to cut off a part equal to the less.

This third problem, likewise, has a variety of cases. For the given unequal right lines are either mutually distant from each other, as with the institutor of the elements, or they are united according to one extreme; or the one cuts the other according to one of its extremities, and this in a two-fold manner. For either the greater cuts the less, or the less the greater. But if they are united according to one extreme, the demonstration is manifest. For employing the common extremity as a centre, and the lesser of the lines for an interval, you will describe a circle, and cut off from the greater, a part equal to the less; since as much as the circle intercepts within itself, will be equal to the less. But if the one cuts the other according to its extreme, either the greater will cut the greater, or the contrary. And if they mutually cut each other, they will either be mutually cut into equal parts, or into unequal; or the one will be cut into equal, and the other into unequal parts, and this in a two-fold respect. For all these present us with an admirable variety of exercise, some of which, out of a many, we shall exhibit. Let there be given the unequal right lines _a b_, _c d_, the greater of which is _c d_, and let it cut _a b_ in one of its extremities _c_; then with the centre _a_, but interval _a b_, let a circle _b f_ be described, and let an equilateral triangle _a e c_ be constructed upon _a c_, and produce _e a_, _e c_. Again, with the centre _e_, but interval _e f_, let the circle _g f_ be described; and with the centre _c_, and interval _c g_, the circle _g l_. Because therefore, _e f_ is equal to _e g_ (for the centre is _e_) of which _e a_ is equal to _e c_, the remainder _a f_, shall be equal to the remainder _c g_. But _a f_ is likewise equal to _a b_; for the centre is _a_. Hence, _c g_ will be equal to _a b_, and this is equal to _c l_, for the centre is the point _c_: _a b_, therefore, is equal to _c l_, which was required to be done.

But let _c d_ be less than _a b_, and let it cut _a b_ according to its extremity _c_; either, therefore, it will cut it in the middle, or not in the middle. Let it in the first place cut it in the middle; _c d_, therefore, is either the half of _a b_, and _a c_ is equal to _c d_, or it is less than half. And in this case with the centre _c_, and interval _c d_, describe a circle, and you will cut off from _a b_ a part equal to _c d_: Or it is greater than half; and then at the point _a_, placing _a f_, equal to _c d_, and describing a circle with the centre _a_, and interval _a f_, you will cut off from _a b_ a part equal to _a f_, that is to _c d_. But if _c d_ does not cut _a b_ in the middle, _c d_ shall either be its half, or greater than the half, or less. If therefore _c d_ is the half, or less than the half of _a b_, employing _c_ as a centre, and _c d_ as an interval, you will cut off from _a b_, a part equal to _e d_, as was required to be done. But if _c d_ is greater than the half, again at the point _a_[5] placing _a f_ equal to _c d_, you will accomplish the same. For with the centre _a_, but interval _a f_, you will describe a circle, cutting off from _a b_ a line equal to _a f_, that is, to _c d_. But if they mutually intersect, as _c d_, _a b_, then with the centre _b_, but interval _b a_, describe the circle _a f_, and let _b c_ be extended to the point _f_. Because therefore, _b f_, _c d_, are the two unequal right lines, and _c d_ cuts _b f_, according to its extremity, it is possible from _c d_ to make a line equal to _b f_; for this has been shewn in the first case of this problem. It is therefore possible, that a line equal to _a b_ may be cut off from _c d_; for _a b_ and _b f_ are mutually equal. Having, therefore, received these cases from division, we have endeavoured to exhibit their variety. But the demonstration of the elementary institutor is admirable, since it accords with all the preceding constructions. And it is possible, in every position, at the extremity of the greater, to place a line equal to the less, and using the same extreme as a centre, and placing the interval to describe a circle, which shall cut off from the greater, a line equal to the less, whether they mutually intersect, or one cuts the other, or they are constituted in a still different position.

PROPOSITION IV. THEOREM I.

If two triangles have two sides equal each to each; and have
likewise the angles equal; which are comprehended by the equal
sides; then they shall have their bases equal; and the two
triangles shall be equal; and the remaining angles opposite to
the equal sides shall be equal.

This is the first theorem in the institution of the elements, for all those which preceded were problems. The first, indeed, treating concerning the origin of triangles: but the second and third proposing to procure one right line equal to another. And of these the one produced an equal from an unequal line, but the other discovered an equal line by an ablation from one unequal. Since, therefore, equality, which is the first symptom in quantity, is to be constructed by us in a triangle and right line, it is delivered in the following theorem. For how can he who has not previously constructed triangles, and procured their origin, be learned in their essential accidents, and in the equality of angles and sides which they contain? How can he receive sides equal to sides, and right lines to other right lines, who has neither problematically investigated these, nor fabricated the invention of equal right lines? For if he should say it may happen before they are fabricated, that if two triangles have _this_ for a _symptom_, they shall likewise have _this particular symptom_; would it not, in this case, be easy to object to him, that we by no means know whether a triangle can be constructed? And should it be afterwards inferred, that if there are two triangles, they may have two sides equal to two sides, may we not also doubt this, whether it is possible that right lines may be mutually equal? And this particularly in geometrical forms, in which inequality not entirely existing, equality is likewise inherent. For we must learn that the cornicular is always unequal to an acute angle, and the same is true of the semicircular angle, and the transition from the greater to the less does not entirely take place through that which is equal. The institutor of the elements, therefore, first of all removing these objections, delivers also the construction of triangles (for it is common to three forms) and the origin of equal right lines, in a two-fold order. For he produces the one, not yet existing: but he acquires the other by an ablation from an unequal line. But after these he very properly subjoins the theorem, by which it is shewn how triangles having two sides equal to two, each to each, and the angles comprehended by the equal sides equal, have also the base equal to the base, the area equal to the area, and the remaining angles to the remaining angles. For there are three particulars exhibited in these triangles: but two data. Hence, the equality of the two sides is given, or two equal sides (and it is manifestly given in proportion) and the equality of the angle contained by the equal sides: but three particulars are investigated, the equality of _base_ to _base_, of _triangle_ to _triangle_, and of the _remaining angles_. But because it is possible that triangles may have two sides equal to two, and yet the theorem not be true, because the one is not equal to the other, but both together, on this account he adds in the data, that the sides are equal not simply, but one to the other. For if one of the triangles should have one of its sides of three units, but the other of four; and again, if the sides of the other triangle are respectively two, and five units, the angle comprehended by these being right, the two sides of the one triangle, will, indeed, taken together, be equal to the two sides of the other, or to seven units, yet the two triangles will not be equal. For the area of the one is six units[6], but of the other five. And the reason of this is, because the sides are not equal each to each. Hence, many, not observing this in the division of land, when they have received a greater, have thought just the same as if they had received an equal field; and this because both the sides containing one field, have been together equal to both the sides containing the other field. It is requisite, therefore, to receive the one equal to the other, and to mark wherever the institutor of the elements subjoins this, because he does not add it without occasion. For discoursing on the equality of equal angles, he adds the particle _comprehended by equal sides_, lest by speaking indeterminately we should assume some one of the angles at the bases. Besides, when in triangles no side is previously named, we must conceive the base to be the side opposite to our sight; but when two are previously received, the remaining side is necessarily the base. Hence, here too, the institutor of the elements having previously assumed two sides equal to two, calls the remainder the bases of the triangles. But a triangle is then said to be equal to a triangle, when their areas are equal. For it is possible, that though the ambits are equal, yet the areas may be unequal, on account of the inequality of angles. But I call the area, the space intercepted by the sides of the triangle: as also I denominate the ambit, the line composed from the three triangular sides. Each, therefore, is different, and it is requisite, indeed, that besides the equality of the ambits, according to each side, the angles should also be equal, if also area ought to be equal to area. But it happens in certain triangles, that though the areas are equal, yet the ambits are unequal; and that the ambits being equal, the areas are unequal. For if there be two isosceles triangles, each of whose equal sides contains five units, but the base of the one is eight, and of the other six units; he who is ignorant of geometry, will say that the greater triangle is that whose base contains eight units. For the whole ambit will be eighteen. But the geometrician will say, that the area of each triangle contains twelve units, and this he will demonstrate, by drawing in each triangle a perpendicular from the vertex, and multiplying this with either part of the segments of the base[7]. But it happens (as I have said) that though the ambits are equal, the spaces are unequal. Hence, certain persons formerly fraudulently deceived their partners in the division of fields, on account of the equality according to ambit, receiving a larger field. But one base is said to be equal to another, and one right line to another, when their extremes conjoined make the whole coincide with the whole. For every right line, indeed, agrees with every right line; but equal right lines mutually coincide according to their extremes. Again, one right-lined angle is said to be equal to another, when one of the comprehending sides of one angle being placed upon one of the other, the remaining side also coincides with the remainder: but when one of the remaining sides falls external to the other, the greater angle is that whose side falls externally; and the less whose side falls within. For there, indeed, the one contains, but in this case it is contained. But we must assume the equality of angles according to the convenience of sides in right lines, and in all of the same species, as in lunulars and systroides[8], and figures on both sides convex; because, it is possible that they may be equal, and yet the sides not mutually coincide. For a right angle is equal to a certain lunular angle, and yet it is not possible that right lines can coincide with circumferences. Besides, this also must be previously understood, that the angles are said to subtend the opposite sides. For every triangular angle is contained by two sides of the triangle, but is subtended by the remaining side. Hence, the geometrician, when he says that the angles are equal, adds, _which are opposite to the equal sides_, lest we should conceive it of no consequence whatever angle is received, and should think that he denominated any other two angles of the triangles equal, but we must call those equal which subtend equal sides. For equal sides mutually subtend equal angles. And such are the considerations necessary to the declaration of the present theorem.

But against the objection of our adversary[9], this must be previously assumed, that two right lines cannot comprehend space. For this the geometrician receives as evident. For if (says he) the extremes of the bases mutually coincide, the bases also shall coincide: but if not two right lines, will comprehend space. From whence, therefore, is the impossibility of this derived? Let there then be two right lines comprehending space _a c b_, _a d b_, and let them be infinitely produced. Then with the centre _b_, and interval _a b_, let a circle _a e f_ be described. Because, therefore, the line _a c b f_ is a diameter, _a c f_ is the half of the circumference. Again, because the line _a d b e_ is a diameter, _a e_, likewise, is one half of the circumference. Hence, _a e_, and _a c f_ are equal to the circumference, which is impossible. Two right lines therefore, cannot comprehend space; which the institutor of the elements knowing said, in the first Petition, _from every point, to every point, to draw a right line_, because one right line is always capable of uniting two points, but this is impossible for two right lines to effect. Many circumferences, indeed, may conjoin two points, both in the same, and in contrary parts: for by this means the extremities of a diameter conjoin two circumferences, but only one right line. But it is possible that both within and without semicircles, infinite circumferences conjoining given points may be described. And the reason of this is, because a right line is the least of lines, having the same extremes. But there is every where one _minimum_, and this always becomes the measure of the infinity of others. As therefore a right line; since it is one, becomes the measure of the infinity of right-lined angles (for by this we discover their quantity) so likewise a right line procures us the greatest utility in the mensuration of such as are non-rectilineal. And thus much may suffice concerning these.

But that the whole demonstration of the present theorem depends on common conceptions, rising as it were spontaneously, and emerging from the evidence of hypotheses, is manifest to every one. For since two sides are equal to two sides, each to each, they will mutually coincide. But since the angles contained by the equal sides are equal, they also shall mutually coincide. And when angle is placed on angle, and sides on sides, so as to touch, in every part, the extremities of the sides beneath shall also coincide. But if these, then _base_, shall agree with _base_. And if three with three, the whole triangle shall accord with the whole triangle, and all shall be equal to all. Hence, therefore, equality considered in things of the same species, appears to be the cause of the whole demonstration. For here are two axioms endued with a power of containing the whole method of the proposed theorem. One, indeed, affirming, that _things which mutually coincide, are equal_; and this is simply true, requiring no limitation, and is employed by the institutor of the elements both in the base, and in the space, and in the other angles. For these, says he, are equal, because they mutually coincide. But the other affirming that _things which are equal mutually coincide_. This, however, is not true in all, but in those of a similar species. But I call things similar in species, such as a right line when compared with a right line, one circumference with another of the same circle, and the angles comprehended by similar lines endued with a similar position. But of these, I say, that such as are equal, mutually coincide: so that in short, the whole demonstration is of this kind. These equals, therefore, are given, viz. two sides equal to two sides, and the angles which they comprehend, and these accord among themselves. But if these mutually coincide, the base also shall agree with the base, and all coincide with all. And if these accord, they are also equal. If then these are equal, it may at the same time be shewn that all are equal to all. And this appears to be the first mode of knowing triangles on all sides equal. And thus much concerning the whole demonstration.

But Carpus, the mechanist, who, in an astrological treatise, discourses of problems and theorems, says, “that they must not be passed over in silence, since they opportunely present themselves for investigation;” and lastly, entering on their distinction, he observes, “that the problematical genus precedes theorems in order. For in problems (says he) the invention of subjects is investigated prior to symptoms. Likewise, a problematical proposition is simple, and requires no artificial intelligence. For this commands us to accomplish something evident, as _to construct an equilateral triangle_, or _from two given unequal right lines, to cut off from the greater a part equal to the less_. For what is there in these difficult and obscure. But he affirms that the proposition of a theorem is difficult, and requires the most accurate power, and a judgment productive of science, that it may appear neither to exceed, nor to be deficient from truth; such, indeed, as the present, which is the first of theorems. Add too, that in problems, there is one common way invented by resolution, by proceeding according to which, we can happily accomplish our purpose. For after this manner the more easy kind of problems are investigated. But the treatise of theorems is so very difficult, that even to our time (says he) no one has been able to deliver any common method of their invention. Hence, on account of facility also, the problematical genus is more simple. But these being distinguished, it is on this account (says he) that in the elementary institution problems precede theorems, and from these the institution of the elements begins; and the first theorem is in order the fourth, not because the fourth is exhibited from the preceding, but because it is necessary they should precede as being problems, and this a theorem, though it should require none of the antecedent propositions for its demonstration. For the present theorem entirely employs common conceptions; and in a certain respect receives the same triangle in a different position. Since coincidence, and its consequent equality possess a sensible and manifest apprehension. But such being the demonstration of the first theorem, problems with great propriety precede, because they are universally allotted the primary place.” And perhaps, indeed, problems antecede theorems in order; and particularly among those who ascend to contemplation from the arts, which are conversant with sensible particulars: but theorems excel problems in dignity of nature. And it appears, that all geometry, so far as it conjoins itself with a variety of arts, energizes problematically: but so far as it coheres to the first science, it proceeds theorematically from problems to theorems, from things secondary to such as are first, and from things which more regard the arts, to such as are endued with a greater power of producing science. It is, therefore, vain to accuse Geminus, for affirming that theorems are prior to problems. For Carpus assigns a precedency to problems, according to order: but Geminus to theorems, according to a more perfect dignity. But of this fourth theorem, we have already observed, that in a certain respect it is indigent of the preceding problems, in which we learn the origin of triangles, and the invention of equality. But we now add, that since it is the most simple and principle of theorems (for it is naturally, as I may say, exhibited from primary conceptions alone), but demonstrates a certain symptom appearing about triangles, having two sides equal to two, each to each, and the two angles equal contained by the equal sides, it is with great propriety placed the first after problems, in which things subject to this symptom, and the data themselves are constructed.

PROPOSITION V. THEOREM II.

The angles at the base of an isosceles triangle are mutually equal; and the equal right lines being produced, the angles under the base shall be mutually equal.

Of theorems some are _simple_, but others _composite_. I call those _simple_, which, both according to hypotheses and conclusions, are indivisible, possessing one _datum_, and one object of investigation. Thus for example, if the institutor of the elements had said, _every isosceles triangle has the angles at the base equal_, it would have been a simple theorem. But theorems are composite, which are composed from many particulars, either having composite hypotheses, or conclusions from a simple hypothesis, or both. And of these, some are _complex_, but others _incomplex_. The _incomplex_ are such composites as cannot be divided into simple theorems, as the fourth proposition. For in this, both the _datum_ is a composite, and its consequent, yet it is impossible that the _datum_ can be divided into things simple, and become theorems. For if a triangle has its sides alone equal, or the angle at the vertex, the same consequences will not ensue. But the _complex_ are such as may be divided into things simple, as the theorem which says, _triangles and parallelograms of the same altitude, have the same proportion as their bases_. For it is possible to say by division, that _triangles of the same altitude, have the same proportion as their bases_, and in parallelograms after a similar manner. But of all composites, some are composed according to the conclusion, being excited from the same hypothesis: but others have their conclusion according to hypotheses, and infer the same conclusion in all: and others, lastly, are composed both according to the conclusion, and according to hypotheses. _Composition_, therefore, in the present case, is according to the _conclusion_, for there are three particulars concluded in this theorem, _that the bases are equal, that the triangles are equal, and that the remaining angles, under the base, are equal to the remaining angles_. But composition, according to _hypothesis_, is found in the common theorem _of triangles and parallelograms of the same altitude_. And _according to both_, in the theorem that _the diameters both of circles and ellipses, bisect as well the spaces as the lines containing the spaces_. But of _complex_ theorems, some are universal: but others conclude that which is universal from particulars. For if we should say that a diameter divides a circle, ellipsis, and parallelograms, we receive, indeed, every part of the complex, not universally, but we make that universal which is composed from all. But if we should say, that _in a circle, all lines passing through the centre, mutually bisect each other, and make equal angles of all the segments_, we should affirm a universal. For in an ellipsis all the angles of the segments are not equal, but those only which are formed by the diameter. But these compositions are entirely fabricated, for the sake of geometrical brevity and resolutions. For many things incomposite are not resolved, but composites alone afford convenience to a resolution tending to principles.

In consequence of these previous considerations then, we must call the fifth theorem a _composite_, and a composite, both with respect to the _datum_, and the _object of investigation_; and this the institutor of the elements exhibiting, divides this theorem, being one, and gives a separate position to the _data_, and the _things to be investigated_, for he says that _the angles at the base of an isosceles triangle are equal_; and again, that _the equal sides being produced, the angles under the base are equal_. For we must not think that there are two theorems, but one; and that this is a composite, both according to the _data_, and _thing sought_: and that each of these composites is perfect and true. Hence, conversion also is true in each. For if the angles at the base are equal, the triangle is isosceles: but if those under the base are equal, the equal right lines are produced, and the triangle is isosceles. But the institutor of the elements _converts_ the equality of the angles at the base; but not the equality of those under the base, though this is likewise true; the reason of which we shall shortly explain. But we shall now, in the first place, enquire on what account he demonstrates that the angles under the base are equal. For he never employs this in the construction or demonstration of other problems or theorems. It may be doubted, therefore, why, since it is useless, it was requisite to insert it in the present theorem? To this we must reply, that though it is never employed in the elements, yet it is most useful for the destruction of objections, and the solution of oppositions to theorems[10]. But it is artificial, and belongs to science to prepare solutions of things resisting its propositions, and to provide subsidies of answers; that not only true demonstrations may be fabricated from things previously demonstrated, but that from hence confutations of error may be produced. And from this geometrical order, you will likewise receive a rhetorical emolument. For he who can effect this in the discourses of rhetoric, who can foresee the oppositions to his following heads, and previous to their delivery, can first of all prepare solutions of them to others, he, indeed, will fabricate in a wonderful manner, a most excellent mode of disputation. The institutor of the elements, therefore, teaching us this in reality, previous to the theorems by which we solve opposing objections, employing such as are now exhibited, at the same time demonstrates, that the angles under the base of an isosceles triangle, are equal, and thus prepares a confutation of the falsehood such objections contain. But that from the present theorem we may solve the objections urged in the seventh and ninth propositions, will be perspicuous as we proceed. Hence, it appears, why Euclid does not convert the latter part of this theorem in the sixth, because it does not produce a principal utility, but confers to our advantage, accidentally, with respect to the whole of science.

But if any one should desire us without producing the equal right lines, to prove the angles at the base of an isosceles triangle equal, (for it is not requisite to demonstrate the equality of these, by those under the base) by transposing, in a manner, the construction, and fabricating those constructions within, which are made without the isosceles triangle, we may exhibit the thing proposed. Thus let _a b c_ be an isosceles triangle, and in the side _a b_, take any point _d_, and from _a c_, take _a e_, equal to _a d_, and draw the lines _b e_, _d c_, _d e_. Because, therefore, _a b_ is equal to _a c_, and _a d_ to _a e_, and the angle _a_ is common, _b e_ also shall be equal to _c d_, and the remaining angles to the remaining angles. Hence, the angle _a b e_, is equal to the angle _a c d_. Again, because _d b_ is equal to _e c_, and _b e_ to _d c_, and the angle _d b e_ to _e c d_; hence, the base, since it is common to both, is equal to itself, and all are equal to all. The angle, _e d b_, therefore, is equal to the angle _d e c_: and the angle _d e b_, is equal to the angle _e d c_. Hence, since the angle _e d b_, is equal to the angle _d e c_, from which the equal angles _d e b_, _e d c_, are taken, the remaining angles _b d c_, _c e b_ are equal. But the sides also _b d_, _d c_, are equal to the sides _c e_, _e b_, each to each, and the base _b c_ is common. All, therefore, are equal to all. Hence, the remaining angles also, subtending equal sides, are equal. The angle, therefore, _d b c_, is equal to the angle _e c b_. For the angle _d b c_, subtends the line _d c_: but the angle _e c b_, the line _e b_. The angles, therefore, at the base of an isosceles triangle, are equal, the equal right lines not being produced.

But Pappus demonstrates this yet shorter, without any addition in the following manner. Let _a b c_ be an isosceles triangle, having _a b_, equal to _a c_. We must conceive, therefore, this one triangle as if it was two, and reason thus. Because _a b_ is equal to _a c_, and _a c_ to _a b_, the two sides _a b_, _a c_, are equal to the two _a c_, _a b_, and the angle _b a c_, is equal to the angle _c a b_, (for it is the same.) All, therefore, are equal to all. The base _b c_, to the base _c b_. But the triangle _a b c_, to the triangle _a c b_; and the angle _a b c_, to the angle _a c b_, and the angle _a c b_, to the angle _a b c_. For they subtend equal sides, i.e. _a b_, _a c_. The angles, therefore, at the base of an isosceles triangle, are equal. And it seems that Pappus invented this mode of demonstration, when he considered that the institutor of the elements also, in the fourth theorem, when he had united two triangles, and had made them mutually coincide, thus forming one of two, by this means observed their equality throughout. In like manner it is possible, that we also, by an assumption contemplating two triangles in one, may demonstrate the equality of the angles at the base. Thanks, therefore are to be given to the ancient Thales for the invention of this theorem, as well as a multitude of others. For he, first, is said to have perceived and affirmed, that the angles at the base of every isosceles triangle are equal: and after the manner of the ancients, to have called them similar. But still more deserving of praise are those moderns, who have yet more universally demonstrated (among which number is Geminus) that equal right lines falling from one point, on a line of similar parts, form equal angles. For Geminus using this theorem, shews, that there are only three lines, and not more of similar parts, the _right_, the _circular_, and the _cylindric helix_; and this is properly universal, to which this symptom first agrees, just as the possession of two sides greater than the third, is shewn to be essentially inherent in every triangle. It is not, therefore, the property universally of every isosceles, though it belongs to every one, to possess angles at the base equal: but of equal right lines falling on a line of similar parts. For to subtend equal angles, is in these primarily inherent.

PROPOSITION VI. THEOREM III.

If two angles of a triangle be equal to each other, the sides
also which subtend the equal angles, shall be equal to one
another.