Commentary on Propositions I.16–I.32: Inequalities, Parallels & Angle-Sum

Exterior Angle, Epicurean Critique of I.20, Thales’ Ship-Distance Theorem (I.26), Ptolemy & Proclus on Postulate 5 (I.29), and the Pythagorean Angle-Sum Proof (I.32) — Friedlein (1873) pp. 305–384 · Book III (Props. XVI–XXVI) & Book IV (Props. XXVII–XXXII)

Note

Edition Track (English): Public-domain translation by Thomas Taylor (London, 1788–1789 / 1792), collated and synchronized with Gottfried Friedlein's 1873 Teubner Greek edition (pp. 305–384). Switch tracks above for the Greek (1873) text, Summary, or Notes.

Props. I.16–I.17: Exterior Angle & Sum of Two Interior Angles

[Folio Friedlein 305 | p. 305]

Euclidean Reference: [I.16] · Friedlein pp. 305–312

PROPOSITION XVI. THEOREM IX.

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

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

[Illustration]

[Illustration]

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

PROPOSITION XVII. THEOREM X.

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

[Illustration]

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

[Illustration]

Props. I.18–I.19: Greater Side & Greater Angle (Division Proof)

[Folio Friedlein 313 | p. 313]

Euclidean Reference: [I.18] · Friedlein pp. 313–321

PROPOSITION XVIII. THEOREM XI.

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

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

[Illustration]

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

[Illustration]

[Illustration]

[Illustration]

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

PROPOSITION XIX. THEOREM XII.

The greater side of every triangle subtends the greater angle.

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

[Illustration]

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

[Illustration]

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

Props. I.20–I.21: Triangle Inequality (Epicurean Critique, Heron, Porphyry)

[Folio Friedlein 322 | p. 322]

Euclidean Reference: [I.20] · Friedlein pp. 322–329

PROPOSITION XX. THEOREM XIII.

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

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

[Illustration]

[Illustration]

But it is requisite to relate the other demonstrations of the present theorem, such as Heron, and the familiars of Porphyry have fabricated, without producing the right line, after the manner of Euclid. Let there be a triangle a b c, it is requisite, therefore, to shew, that the sides a b, a c, are greater than the side b c. Bisect the angle at a, by the right line a e. Because, therefore, the angle a e c, is external to the triangle a b e, it is greater than the angle b a e. But the angle b a e, was placed equal to the angle e a c. The angle, therefore, a e c, is greater than the angle e a c. Hence, the side also a c, is greater than the side c e. And for the same reason the side a b, is greater than the side b e. For the angle a e b is external to the triangle a e c, and is greater than the angle c a e; that is than the angle e a b. And on this account the side a b, is greater than the side b e. The sides, therefore, a b, a c, are greater than the whole side b c. And the like may be shewn of the other sides. Let there again be a triangle a b c. If therefore the triangle a b c, be equilateral, two sides will be doubtless greater than the remaining one: for when there are three equal quantities, any two are double of the remainder. But if it be isosceles, it will have a base either less, or greater than each of the equal sides. If therefore the base be less, the two sides are given greater than the remainder. But if the base be greater, let it be b c, and cut off from it a part equal to either of the sides, which let be b e, and connect a e. Because, therefore, the angle a e c, is external to the triangle a e b, it is greater than the angle b a e. On the same account the angle a e b, is greater than the angle c a e. Hence, the angles about the point e, are greater than the whole angle about the point a, of which b e a is equal to b a e, since a b is equal to b e. The remainder, therefore, a e c, is greater than the remainder c a e. Hence, the side a c, is greater than the side c e. But the side a b, was also equal to the side b e. The sides, therefore, a b, a c, are greater than the side b c.

[Illustration]

But if the triangle a b c, be scalene, let the greatest side be a b, the middle a c, and the least c b. The greatest side, therefore, assumed with either of the others, exceeds the remainder: for by itself it is greater than either. But if we are desirous of shewing that the sides a c, c b, are greater than the greatest side a b, we must employ the same construction as in the isosceles triangle, cutting off from the greater side, a part equal to one of the other sides, and connecting the line c e, and using the external angles of the triangles.

[Illustration]

Let there be again any triangle a b c. I say that the sides a b, a c, are greater than the side b c. For if they are not greater, they are either equal or less. Let them be equal, and cut off b e, equal to a b. The remainder, therefore, e c, is equal to a c. Because then, a b, b e, are equal, they subtend equal angles; and this is likewise true of a c, e c, because they are equal. Hence, the angles at the point e, are equal to the angles at the point a, which is impossible. Again let the sides a b, a c, be less than b c, and cut off b d, equal to a b, and e c to a c. Because, therefore, a b is equal to b d, the angle b d a, is not unequal to the angle b a d. And because a c is equal to c e, the angle c e a, is equal to the angle e a c. Hence, the two angles b d a, c e a, are equal to the two b a d, and e a c. Again, because the angle b d a is external to the triangle a d c, it is greater than the angle e a c: for it is greater than c a d. By a similar reason also, because the angle c e a, is external to the triangle a b e, it is greater than the angle b a d: for it is greater than the angle b a e. Hence, the angles b d a, c e a, are greater than the two b a d, e a c. But they were also equal to them, which is impossible. The sides, therefore, a b, a c, are neither equal to, nor less than the side b c, but greater. And the like may be exhibited in others.

[Illustration]

PROPOSITION XXI. THEOREM XIV.

If upon one side of a triangle, two right lines beginning from the extremities, are internally constituted, the constituted right lines will be less than the other sides of the triangle, but they will contain a greater angle.

[Illustration]

That which is expressed by the proposition, is, indeed, manifest; and the demonstration adopted by the elementary institutor, is evident; and the theorem is consequent to the first principles, since it depends on two theorems, the one previously exhibited, and the sixteenth. For in order to shew, that the lines internally constituted, are less than the external, the theorem is required, which says, the two sides of every triangle, are greater than the remaining one: but for the purpose of confirming that the angle comprehended by them is greater than that comprehended by the external sides, that theorem procures the greatest utility, which says, the external angle of every triangle, is greater than the internal and opposite angle. But you will receive at the same time, conviction of geometrical diligence, and a commemoration of things admirable in the mathematical disciplines, if we shall shew that it is possible within a certain triangle, upon one of its sides, not upon the whole, but upon some one of its parts, to constitute two right lines greater than the external right lines; and again, others comprehending a less angle, and comprehended in the angle made by the external lines. For this being exhibited, it will at the same time be manifest, that the institutor of the Elements necessarily adds, that the internally constituted lines must begin from the extremities of the common basis; and must be constituted upon one whole side, and not upon any one of its parts: but likewise, as I have said, one of the admirable things which geometry contains, will be manifest. For is it not, indeed, admirable, that the lines constituted upon the whole side, should be less than the external sides: but that those constituted upon a part should be greater? Let there be then a right angled triangle a b c, having the angle at the point b right, and take in the side b c, any point d, and connect a d. Hence, a d is greater than a b. Take from a d, a part equal to a b, which let be d e, and bisect e a, in the point f, and connect f c. Because, therefore, a f c is a triangle, the lines a f, f c, are greater than a c. But a f is equal to f e. The right lines therefore, f e, f c, are greater than a c. But d e is equal to a b. Hence, the right lines f c, f d, are greater than the right lines a b, a c, and they are internally constituted.

[Illustration]

Let there be again an isosceles triangle a b c, having the base b c, greater than either of the equal sides. Then from b c, cut off b d, equal to a b, and connect a d, and take in a d, any point e, and connect e c. Because, therefore, a b, is equal to b d, the angle b a d, is also equal to the angle b d a. And because the angle b d a is external to the triangle e d c, it is greater than the internal and opposite d e c. Hence, the angle b a d, is greater than the angle d e c. Much more, therefore, is the angle b a c, greater than the angle d e c; and b a c is contained by the external lines, but d e c by internal lines. Within a triangle, therefore, right lines d e, e c, comprehending a lesser angle, are constituted within the angle comprehended by the external lines; and the thing proposed is shewn without employing the parallel lines of expositors. Hence, it is necessary that the constituted right lines should begin from the extremities of the basis: for those which are constituted upon any one of its parts, are shewn to be sometimes greater than the external lines, and to comprehend a lesser angle. But when they are constituted in this manner, beginning from the extremities, the species of triangles, called (ἀκιδοειδῆ) or, similar to the point of a spear, presents itself to our view; and is one of the admirable things contained in geometry, viz. to find a quadrilateral triangle. As for example, the triangle A B C. For it is contained by the four sides B A, A C, B D, D C; but it has three angles, one at B, the other at A, and the other at C. And hence, the present figure is a quadrilateral triangle.

[Illustration]

Props. I.22–I.23: Triangle Diorismos & Copying an Angle (Oenopides)

[Folio Friedlein 330 | p. 330]

Euclidean Reference: [I.23] · Friedlein pp. 330–335

PROPOSITION XXII. PROBLEM VIII.

To construct a triangle from three right lines, which are equal to three given right lines. But it is requisite that two of the lines must be greater than the remaining one, in whatever manner they may be taken.

[Illustration]

We again pass to problems, and Euclid commands us to construct a triangle from three proposed right lines, two of which are greater than the remaining one, equal to given right lines. Because he knew this in the first place, that it was impossible to construct a triangle from those same lines, which had already received the declared position: but that this was possible to be effected from their equals. In the next place, he knew it was necessary that two of the right lines about to complete the triangle, should be greater than the remaining one: for the two sides of every triangle are greater than the remaining one, however assumed, as we have shewn. On this account he adds, that it is necessary the first right lines remaining, to construct a triangle from three equal to them: but that it is requisite, any two, however taken, should be greater than the remainder, or there will not be a triangle from three lines equal to the given right lines. But by this means he also destroys all the objections which are urged against the construction, and which may be perfectly dissolved by this addition. Hence, the present problem ranks among things determined, and not among such as are indetermined: For of problems as well as of theorems, some are indeterminate, but others without termination. Thus if we should simply say, from three right lines which are equal to three given right lines, to construct a triangle, the problem is indeterminate and impossible. But if we add, two of which, however assumed, are greater than the remainder, the problem is determined and possible. For as the division of theorems takes place, according to true and false, so that of problems according to a possible and impossible enunciation. But that the objections which are urged against the construction, may be from hence dissolved, we shall learn from a little inspection: for we shall follow the words of the geometrician. Let there be three right lines, a b c, of which any two, however taken, are greater than the third, and let it be required to accomplish the thing proposed. Let there be placed a certain right line d e, on one part finite, as at the point d: and on the other part infinite. Then place d f equal to a, but f g to b: and g h to c. And from the centre f, but interval f d, let a circle k be described. Again, with the centre g, but interval g h, let the circle l be designed; and the circles will intersect each other. For this is assumed by the institutor of the Elements. But it may be asked how this takes place? For perhaps they either only touch each other, or they do not even touch. Since it is necessary that they should suffer some one of three cases, I mean that they should either intersect or touch, or be distant from each other. I say, therefore, that they necessarily intersect each other. For let them in the first place, touch each other. Because, therefore, the point f is the centre of the circle k, d f is equal to f n. And because the point g is the centre of the circle l, h g is equal to g m. The two, therefore, d f, g h, are equal to one, viz. to f g. But they were placed greater than one, as also a, together with c, is greater than b. They are therefore equal to it, and at the same time greater, which is impossible. Again, if it be possible, let the circles be distant from each other, as k and l. Because, therefore, the point f, is the centre of the circle k, d f, is equal to f n. And because the point g, is the centre of the circle l, h g is equal to g m. The whole, therefore, f g, is greater than the two, d f, h g: for f g, exceeds d f, g h, by n m. But it was supposed that d f, h g, were greater than f g, in the same manner as a and c are greater than b. For d f was placed equal to a, but f g, to b, and h g to c. It is necessary, therefore, that the circles k l, should intersect each other. Hence, the institutor of the Elements very properly receives them cutting one another: since of the three right lines, he supposes two greater than the third, however, they may be assumed, but neither equal to, nor less than one. But it is necessary that when the circles touch, two of the lines should be equal to the third; and that when they are distant from each other, two should be greater than the remainder.

[Illustration]

[Illustration]

PROPOSITION XXIII. PROBLEM IX.

On a given right line, and at a given point in it, to constitute an angle equal to a given right lined angle.

[Illustration]

This also is a problem, whose invention according to Eudemus is rather the gain of Oenopides than of Euclid: but it requires the construction of an angle, on a given right line; and at a given point in it equal to another right lined angle. This, then, Euclid necessarily adds, that the given angle must be rectilineal; because it is impossible that an angle can be construed on a right line equal to every angle. For it has been shewn that there are only two curve-lined angles equal to right-lined angles, viz. the angle of a lunular figure, which we have proved equal to every right-lined angle; and the angle of that figure similar to an axe, which is equal to two thirds of a right angle. But a lunular figure of this kind, which is called (πελεκοειδὲς) Pelecoides, is formed from two circles cutting each other through their centres. However, the construction of an angle on a certain right line, causes the constituted angle to become determinate, and not indifferent in species, but forms it either right-lined, or mixt. But since no mixt can be equal to a right-lined angle, it is manifest that this must be perfectly rectilineal. The institutor of the Elements, therefore, simply using the present problem, and constructing a triangle from three right lines, equal to three given lines, accomplishes the thing proposed. But you may receive a more exquisite construction of the triangle, by the following method. Let there be a given right line a b, and a given point in it a, and a given right-lined angle c d e. It is required, therefore, to accomplish the problem. Connect c e, and produce a b on both sides to the points f, g. Then place f a, equal to c d, and d e to a b, and b g to e c. And with the centre a, but interval a f, describe the circle k. And again, as in the preceding, with the centre b, but interval b g, describe the circle l. The circles, therefore, will cut each other, as we have shewn in the last proposition. Let them cut each other in the points m, n, and from these points draw right lines to the centres as in the figure. Because, therefore, f a, is equal to a m, and a n, but c d, is equal to f a; a m, and a n, will be each equal to c d. Again, because b g is equal to b m, and b n, but g b is not unequal to c e; b m, and b n, will be also equal to c e. But a b is equal to d e. The two therefore, a b, a m, are not unequal to the two d e, d c, and the base b m, is equal to the base c e. Hence, the angle m a b, is equal to the angle at the point d. And again, the two n a, a b, are equal to the two c d, d e, and the base n b, is equal to the base c e. The angle, therefore, n a b, is equal to the angle c d e, and the thing proposed is doubly accomplished: for we have not only constituted one, but two angles, equal to the given angle, on each side of the right line a b; so that in whatever part we may desire the construction to be made, it will be indubitable, and without contradiction. And this we have added to the construction of the elementary institutor.

[Illustration]

But we cannot praise the method of Apollonius, because it requires the assistance of the third book. For he receives any angle c d e, and a right line a b, and with the centre d, but interval c d, he describes a circumference c e. In like manner with the centre a, but interval a b, he describes a circumference b f; and intercepting a circumference c e, equal to b f, he connects the right line a f, and affirms that the angles a and d, insisting on equal circumferences, are equal. But it is necessary to pre-assume that a b is equal to c d, in order that the circles may be also equal. We therefore think that a demonstration of this kind requiring posterior propositions, is foreign from an elementary institution; and we give the preference to that of the geometrician, as consequent to principles.

Props. I.24–I.25: Hinge Theorems (Heron, Porphyry & Menelaus)

[Folio Friedlein 336 | p. 336]

Euclidean Reference: [I.24] · Friedlein pp. 336–347

PROPOSITION XXIV. THEOREM XV.

If two triangles have two sides equal to two, each to each, but the one angle contained by the equal right lines greater than the other: they shall also have the base of the one greater than the base of the other.

Euclid again passes on to theorems, and speaks concerning inequality in two triangles, in a manner similar to his discourse concerning equality. For supposing two triangles, having two sides equal to two, each to each, he sometimes places the vertical angle equal in each, and sometimes unequal; and he proceeds in a similar manner with respect to the base. Besides this, he demonstrates that the equality of the bases is consequent to the equality of the vertical angle, and that the equality of the vertical angles, is consequent to the equality of the bases: but he now shews that the inequality of the one, follows the inequality of the other. The present theorem, therefore, is opposite to the fourth: for that, indeed, supposes the vertical angles of the triangles equal, but this supposes them unequal. And that demonstrates the equality of their bases; but this proves them unequal, in the same manner as their angles. It precedes, however, the following theorem: for that deduces its proof of inequality from the bases to the angles subtending the bases: but this, on the contrary, reasons from the angles to the bases, which are under the angles. Hence it is, after this manner, the converse of its consequent proposition, but opposite to the eighth theorem. For the one from the equality of the bases, demonstrates the equality of the vertical angles, but the other from the inequality of the bases, shews that the vertical angles are unequal. It is, however, common to these four (two of which are conversant with equality, I mean the fourth, and the eighth, but two about inequality, the present and the following; and two begin from angles, viz. the fourth, and the object of investigation in the present, but two from bases, viz. the eighth, and the following proposition); it is common, I say, to all these four, as well to the fourth and the eighth, as to the twenty-fourth and twenty-fifth, to have two sides equal to two, each to each. For these being unequal, all enquiry is superfluous, and subject to deception. And thus much for a universal speculation concerning the present theorem.

[Illustration]

[Illustration]

But let us now consider the construction of the elementary institutor, and add to it where deficient. For Euclid receiving two triangles, a b c, d e f, having the sides a b, a c, equal to the sides d e, d f, each to each, and the angle at the point a, being greater than the angle at the point d, and willing to shew that the base b c, is greater than the base e f, on the right line e d, and at a point in it d, constitutes an angle e d h, equal to the angle at the point a. For the angle at the point a, is greater than the angle at the point d, and he connects d h, equal to a c. The right line, therefore, e h, produced to the point h, either falls above, or upon, or beneath the line e f. The institutor of the Elements, indeed, considers it as lying above the line. But let it be upon the right line. Again, therefore, we may exhibit the same. For the two a b, a c, are equal to the two d e, d h, and they contain equal angles. Hence, the base b c, is equal to the base e h. But e h is greater than e f; and on this account b c is greater than e f. Again, let it be placed beneath e f. Connecting, therefore, e h, we must say, that since a b, a c, are equal to d e, d h, and they comprehend equal angles, b c is also equal to e h. Because, therefore, within the triangle d e h, two right lines d f, f e, are constructed on the side d e, they are less than the external sides. But d h, is equal to d f: for it is equal to a c. Hence h e is greater than e f. But h e is equal to b c. And therefore, b c is greater than e f. The theorem, therefore, is exhibited according to every position. Why then, as is the fourth theorem, he at the same time demonstrated that the areas of triangles are equal, does he not add in the present, that besides the inequality of the bases, the areas also are unequal? Against this doubt we must say, that there is not the same proportion in equal, as in unequal angles and bases. For when the angles and bases are equal, the equality also of the triangles follows: but when they are unequal, it is not necessary that the inequality of the areas should be consequent; since the triangles may as well be equal, as unequal; and that may be greater, and likewise less, which contains the greater angle, and the greater base. On this account, therefore, the institutor of the Elements leaves the comparison of the triangles; to which we may add, that the contemplation of these, requires the doctrine of parallels.

[Illustration]

[Illustration]

But if it be requisite, that anticipating things which are afterwards exhibited, we at present make a comparison of areas, we must say, that if the angles a, d, are equal to two right, the triangles may be shewn to be equal: but when they are greater than two right, the lesser triangle will be that which contains the greater angle; and when they are less than two right, this will be the case with the greater triangle. For let the construction in the element be given, and produce e d, f d, to the points k, h; and let us suppose the angles b a c, e d f, equal to two right. Because, therefore, the angle b a c, is equal to the angle e d g, the angles e d g, e d f, are equal to two right. But the angles e d g, k d g, are also equal to two right. Let the common angle e d g be taken away, and the remainder e d f, will be equal to the remainder k d g. But e d f is equal to h d k; for they are vertical angles. Hence, the angle k d g, is equal to the angle h d k. And because the angle g d h, is external to the triangle g d f, it is equal to the two internal and opposite angles at the points g and f. But these angles are equal to each other, because d g is equal to d f. Hence, the angle g d h, is double of the angle at the point g, and of the angle at the point f. The angle, therefore, at the point g, is equal to the angle g d k, and they are alternate; and consequently d e is parallel to f g. The triangles, therefore, g d e, f d e, are upon the same base d e, and between the same parallels d e, g f; and are consequently equal. But the triangle g d e, is equal to the triangle a b c; and so the triangle d e f, is not unequal to the triangle a b c. And here you may observe, that we require three theorems belonging to the doctrine of parallels; one, indeed, affirming, that the external angle of every triangle is equal to the two internal and opposite angles: but the other, that if a right line falling upon two right lines, makes the alternate angles equal, the right lines are parallel; and the third, that triangles constituted upon the same base, and between the same parallels, are equal, which the institutor of the Elements also knowing, omits the comparison of triangles.

[Illustration]

But let the angles b a c, e d f, be greater than two right, and let the same things be constructed. Because, therefore, the angles b a c, e d f, i.e. the angles e d g, e d f, are greater than two right; but the angles e d g, g d k, are equal to two right, by taking away the common angle e d g, the angle e d f, is greater than the angle g d k. Hence, the angle g d h, is more than double of the angle g d k; and so the angle g d k, is less than the angle at the point g. Let g d k be placed equal to d g l, and let e l, and d l, be connected: g l, therefore, is parallel to d e; and hence, the triangles g d e, l d e, are equal. But the triangle l d e, is less than the triangle f d e. The triangle, therefore, g d e, is less than the triangle f d e. But the triangle g d e, is equal to the triangle a b c; and hence, the triangle a b c, is less than the triangle f d e, viz. is less than the triangle which contains the greater angle.

[Illustration]

In the third place, let the unequal angles be less than two right, and let the same things be constructed. Because, therefore, the angles e d g, g d k, are equal to two right, by taking away the common angle e d g, the whole g d h, is less than double of g d k. But it is double also of the angle at the point g. Hence, the angle g d k, is greater than the angle at the point g. Let the angle d g l, be placed equal to the angle g d k, and let g l coincide with e l, in the point l, and connect d l. Hence, g l is parallel to d e; and consequently the triangles g d e, l d e, are equal to each other. But the triangle l d e, is greater than the triangle f d e; and the triangle g d e, is equal to the triangle a b c. Hence, the triangle a b c, is greater than the triangle d f e. It is shewn, therefore, that the triangle a b c, is both equal to, and is also greater and less than the triangle d e f, the angles at the points a and d, being either equal to, or greater or less than two right. And thus, all the hypotheses may be accomplished. For what if the angle at the point a, should be one right, and the half of a right angle, but the angle at the point b, the half of one right, would not those two angles be equal to two right? But what if the angle at the point a, should be one right, and the half of a right, but the angle at the point b, two thirds of one right, would they not be greater than two right angles? And lastly, if the angle at the point a, should be one right, and the half of a right angle, but the angle at the point b, a third part of a right angle, would they not be less than two right, and the angle a be greater than the angle d? All these comparisons, therefore, are produced by the assistance of parallels; and hence, they are necessarily not found in the present elementary institution.

PROPOSITION XXV. THEOREM XVI.

If two triangles have two sides equal to two, each to each, but have the base of the one greater than the base of the other; they shall likewise have the angle contained by the equal sides in the one, greater than the angle contained by the equal sides in the other.

The present theorem is the opposite to the eighth, but the converse of the preceding. For the institutor of the Elements produces theorems concerning the equality and inequality of angles and bases, according to conjunction; in each of the conjunctions, receiving some as precedents, but others as converse. And in such as are precedent indeed, he employs direct ostensions: but in such as are converse, he uses deductions to an impossibility. After this manner he proceeds in some particular triangle, sometimes from the equality of the sides which it contains, shewing the consequent equality of the angles which they subtend: but sometimes from their inequality evincing inequality. And again, on the contrary, affirming that equality of sides is consequent to equality of angles, but inequality to inequality. However, that we may proceed to the thing proposed, we refer those who are desirous of learning how the geometrician shews when this is manifest, to his books on this subject. But we shall briefly relate the demonstrations which others produce of this proposition; and in the first place, that which Menelaus Alexandrinus invented and delivered. Let there be two triangles a b c, d e f, having the two sides a b, a c, equal to the two d e, d f, each to each, and the base b c, greater than the base e f, I say that the angle at the point a, is greater than the angle at the point d. For let there be cut from the base b c, a line b g, equal to the base e f, and construct at the point b, an angle g b h, equal to the angle d e f, and place b h equal to d e. Lastly, connect h g, and produce it to the point k, and connect a h. Because, therefore, b g is equal to e f, but b h to e d, the two are equal to the two, and they contain equal angles. Hence, g h is equal to d f, and the angle b h g, is not unequal to the angle e d f. And because g h is equal to d f, but d f to a c, g h, also, is equal to a c. Hence h k is greater than a c, and consequently is much greater than a k. The angle, therefore, k a h, is greater than the angle k h a. Again, because b h, is equal to a b, for it is equal to d e, the angle b h a, is equal to the angle b a h. Hence, the whole angle b h k, is less than the whole, b a c, but is shewn to be equal to the angle at the point d. The angle, therefore, b a c, is greater than the angle at the point d. And such is the demonstration of Menelaus.

[Illustration]

[Illustration]

But Heron, the mechanist, shews the same thing, in the following manner, without leading to an impossibility, as is the case with the demonstration of Euclid. Let there be two triangles a b c, d e f, with the same hypotheses as above. And because b c is greater than e f, let e f be produced, and place e g equal to b c; and in like manner extend d e, and place d h equal to d f. The circle, therefore, which is described with the centre d, and interval d f, will pass also through the point h. Let it be described as f k h. And because a c, a b, are together greater than b c, but these are equal to e h, and b c is equal to g e, hence the circle which is described with the centre e, but interval e g, will cut e h. Let it cut e h, as the circle g k, and connect from the common section of the circles to the centres, the right lines k d, k e. Because, therefore, the point d, is the centre of the circle h k f, d k is equal to d h, i.e. to a c. Again, because the point e, is the centre of the circle g k, the line e k, is equal to e g, i.e. to b c. Hence, since the two a b, a c, are equal to the two d e, d k, and the base b c, is equal to the base e k, the angle, also, b a c, is equal to the angle e d k. And thus the angle b a c, is greater than the angle f d e.

Prop. I.26: ASA/AAS Congruence & Thales’ Ship-Distance Theorem

[Folio Friedlein 347 | p. 347]

Euclidean Reference: [I.26] · Friedlein pp. 347–353

Editorial Key (Friedlein 352, 14–18): On Prop. I.26, Eudemus attributes ASA triangle congruence to Thales, noting that measuring the distance of ships at sea necessarily relies upon this theorem.

PROPOSITION XXVI. THEOREM XVII.

If two triangles have two angles equal to two, each to each, and one side equal to one side, either that which is adjacent to the equal angles, or that which subtends one of the equal angles: then they shall have the remaining sides equal to the remaining sides, each to each, and the remaining angle equal to the remaining angle.

[Illustration]

It is necessary, that he who wishes to compare triangles with each other, according to sides, angles, and areas, should either, by receiving the sides alone equal, enquire after the equality of angles; or by assuming the angles alone equal, investigate the equality of the sides; or by mingling the angles and sides, scrutinize the equality of angles and sides. Since, therefore, Euclid alone receives the angles equal, he could not likewise shew that the sides of the triangles are equal. For the least triangles are equiangular with the greatest, though at the same time they are excelled by them, both according to sides and comprehended space: but the angles of the former are separately equal to the angles of the latter. However, as he supposes the sides alone to be equal, he demonstrates that all are equal, by the eighth theorem, in which there are two triangles having two sides equal to two, each to each, and the base to the base, and these are shewn to be equiangular, and to possess a power of comprehending equal spaces. And the institutor of the Elements omits this addition, as necessarily following from the fourth, and requiring no demonstration. But when receiving sides and angles, he ought to receive either one side equal to one, and one angle equal to one; or one side, and two angles of the triangles, equal to two; or on the contrary, one angle and two sides; or one angle and three sides; or one side and three angles; or more than one side, and more than one angle. But when he had received one angle, and one side, he could by no means shew the thing proposed. I mean, the equality of the rest. For it is possible that two triangles which are equal, according to one side only, and one angle, may be entirely unequal as to the rest. Thus let there be a right line a b, perpendicularly erected upon the right line c d, but let b d be greater than b c, and connect a c, a d. In these triangles, therefore, there is one common side, and one angle equal to one, but all the rest are unequal. But it is lawful to receive one side, and two angles, and to prove the rest equal, and this he performs by the present theorem: though again, to suppose one side, and three equal angles, is superfluous; since from the equality of two alone, the equality of the rest is exhibited. Again, receiving one angle, and two equal sides, he demonstrates that the rest are equal in the fourth theorem. But it is superfluous to receive one angle, and three equal sides: for two equals being alone assumed, conclude the equality of the rest. Besides, it is superfluous to assume two sides, and two equal angles; or two sides, and three equal angles; or two angles and three sides; or three angles and three sides. For the consequents to fewer hypotheses attend likewise a greater multitude, while the hypotheses are received with proper conditions. Hence, three hypotheses requiring demonstration, present themselves to our view, one, which alone receives three sides; and another which assumes one side, and two angles, which the geometrician now proposes; and a third, the opposite to this. On this account, we have only these three theorems, concerning the equality of triangles, which are conversant in sides and angles; since all the other hypotheses are either invalid for the purpose of shewing the object of enquiry; or they are valid indeed, but superfluous, because the same things may be readily procured by fewer hypotheses. As, therefore, when he assumed two sides equal to two, and one angle equal to one, he did not, indeed, assume every angle, but (as it was proposed by him) that contained by equal right lines, in the same manner when he assumes two angles equal to two, and one side to one, he does not assume any side, but either that which is adjacent to the equal angles, or that which subtends one of the equal angles. For neither is it possible in the fourth theorem, by assuming any equal angle, nor in the present by assuming any side, to shew the equality of the rest.

[Illustration]

[Illustration]

Thus for example, an equilateral triangle a b c, being given, let the side b c be divided into unequal parts, by the line a d. Hence, there will be formed two triangles, having two sides a b, a d, equal to the two a c, a d, and one angle at the point b, equal to one angle at the point c, but the remaining sides will not also be equal, as for instance, the side b d, to the side d c: for they are unequal. But neither are the remaining angles equal: the reason of which is, because we receive an angle equal to an angle, but not the angle which is contained by equal sides. After the same manner, indeed, the present theorem also will appear dubious, unless we assume, according to the aforesaid condition, an equal side subtending one of the equal angles, or adjacent to the equal angles. For let there be a right angled triangle a b c, having the angle at the point b right, and the side b c, greater than the side b a, and let there be constructed on the right line b c, and at a point in it c, an angle b c d, equal to the angle b a c, and let b d, c d, produced, coincide in the point d. There are two triangles, therefore a b c, b c d, having one side b c common, and two angles equal to two, viz. a b c, to c b d (for they are right), and b a c to b c d, according to construction. Hence, as it appears the triangles are equal, and yet it may be shewn that the triangle b d c, is greater than the triangle a b c. But the reason of this is, because in the triangle a b c, we assume the common side b c, subtending one of the equal angles, viz. the angle at the point a: but in the triangle b c d, we assume the equal side, adjacent to the equal angles. It was requisite, therefore, in each, either to subtend one of the equal angles, or to be adjacent to the equal angles. But not observing this, we affirmed that triangle to be equal, which is necessarily greater: for is not the triangle b c d, greater than the triangle a b c? To be convinced of this, let there be constructed on the right line b c, and at a given point in it c, an angle f c b, equal to the angle a c b: for the angle b c d, as well as the angle at the point a, is greater than the angle a c b. Because, therefore, there are two triangles, a b c, b c f, having two angles a b c, b c a, equal to two c b f, b c f, each to each, and one side common, adjacent to the equal angles, viz. b c, the triangles are equal. But the triangle b c d is greater than the triangle b c f, and consequently it is also greater than the triangle a b c. But it was formerly shewn to be equal, on account of the assumption of any side: And thus much the diligence of Porphyry has supplied us on the present occasion. But Eudemus, in his Geometrical Narrations, refers the present theorem to Thales. For he says it is necessary to use this theorem in determining the distance of ships at sea, according to the method employed by Thales in this investigation. But from the preceding division we may briefly assume all the contemplation concerning the equality of triangles, and are enabled to relate the causes of things omitted, confuting those hypotheses, as either false, or superfluous. And thus far we determine the limits of the first section of the elementary institutor, because he forms the constructions and comparisons of triangles, according to equal and unequal. And by construction, indeed, he delivers their essence: but by comparison, their identity and diversity. For there are three things which are conversant about being, essence, same, and different, as well in quantities, as in qualities, according to the propriety of subjects. From these, therefore, as images it may be shewn, that every thing is the same with itself, and differs from itself, on account of the multitude which it contains; and that all things are the same with one another, and different from themselves. For both, in every triangle, and in more triangles than one, equality and inequality has been found to reside.

Props. I.27–I.28: Book IV Opening & Sufficient Conditions for Parallels

[Folio Friedlein 354 | p. 354]

Euclidean Reference: [I.27] · Friedlein pp. 354–364

BOOK IV.

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

PROPOSITION XXVII. THEOREM XVIII.

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

[Illustration]

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

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

PROPOSITION XXVIII. THEOREM XIX.

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

[Illustration]

The preceding theorem receiving the angles, not at the same parts, but situated within right lines, shews that the right lines are parallel among themselves: but the present theorem proposes the two remaining hypotheses, of which one separates the angles according to the particles without and within, but the other supposes them both within, and exhibits the same conclusion. But it may seem, perhaps, that the institutor of the Elements has inconveniently distributed the theorems. For it was necessary either to receive three hypotheses in a divided manner, and to make three theorems; or to collect all into one theorem, as Æneas Hierapolites does, who wrote a compendium of the Elements; or willing to divide them into two, to make an orderly division, and to assume the hypotheses separately, which contain equal angles, and separately that in which the angles are equal to two right. But in the present propositions, in one theorem he supposes the alternate angles equal, but in the other, the external to the internal, and the internal angles situated at the same parts equal to two right. What then is the cause of this division? Does he regard the equality of the angles to each other, or to two right, and on this account does not separate the proposed theorems from each other; or does he respect the angles being received at the same, or not at the same parts? For the preceding theorem does not respect angles at the same parts, since such as these are alternate: but the present regards such as are situated at the same parts, as is perspicuous from the proposition. But how the institutor of the Elements shews, that from the internal angles being equal to two right, the right lines are parallel, appears from his writings on this subject. Ptolemy, however, in the theorems in which he proposes to demonstrate that right lines produced from angles less than two right, coincide at the same parts, in which the angles less than two right are situated, shewing before all his theorems, that from the internal angles being equal to two right, the right lines are parallel, proves it in the following manner. Let there be two right lines a b, c d, and let a certain right line e g f h, so cut them, that it may make the angles b f g, and f g d, equal to two right, I say that those right lines are parallel, that is, will never coincide. For if it be possible, let them coincide while the right lines b f, g d, are produced in the point k. Because, therefore, the right line e f, stands upon the right line a b, it makes the angles a f e, b f e, equal to two right. In like manner because f g stands upon c d, it makes the angles c g f, d g f, equal to two right. Hence, the four angles b f e, a f e, c g f, d g f, are equal to four right, two of which b f g, f g d, are supposed equal to two right. The remainders, therefore, a f g, c g f, are equal to two right. If then the right lines f b, g d, when produced, coincide, the internal angles being equal to two right, f a, and g c, also, shall coincide when produced: for the angles a f g, c g f, are also equal to two right. Either therefore the right lines shall coincide in both parts, or in neither, since these, as well as the former, are equal to two right. Let the right lines then f a, g c, coincide in the point l. But if this be admitted two right lines l a f k, l c g k, will comprehend space, which is impossible. It is not therefore possible, that the internal angles being equal to two right, the right lines can coincide. They are therefore parallel.

Prop. I.29: Ptolemy’s Treatise & Proclus’s Attempted Proof of Postulate 5

[Folio Friedlein 364 | p. 364]

Euclidean Reference: [I.29] · Friedlein pp. 364–373

Editorial Key (Friedlein 365–373): On Prop. I.29, Proclus preserves Claudius Ptolemy's monograph on parallel lines, exposes its subtle circularity, and presents his own proof of Postulate 5 from Aristotle's bounded-divergence lemma (De Caelo I.5).

PROPOSITION XXIX. THEOREM XX.

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

[Illustration]

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

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

[Illustration]

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

[Illustration]

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

[Illustration]

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

[Illustration]

Props. I.30–I.31: Transitivity & Construction of Parallels

[Folio Friedlein 373 | p. 373]

Euclidean Reference: [I.30] · Friedlein pp. 373–376

PROPOSITION XXX. THEOREM XXI.

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

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

[Illustration]

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

PROPOSITION XXXI. PROBLEM X.

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

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

Prop. I.32: Triangle Angle-Sum, Pythagorean Proof & Polygon Formula

[Folio Friedlein 376 | p. 376]

Euclidean Reference: [I.32] · Friedlein pp. 376–384

Editorial Key (Friedlein 379, 2–15): On Prop. I.32, Eudemus records the ancient Pythagorean demonstration that the interior angles of every triangle equal two right angles by drawing a single parallel through the vertex.

PROPOSITION XXXII. THEOREM XXII.

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

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

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

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

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

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

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