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On BC, which is equal to EF, and on the side of it opposite to the Ériangle ABC, let a triangle BGC be constituted every way equal to the triangle DEF, that is, having GB equal to DE, GC to DF, the angle BGC to the angle EDF, &c. : join AG.
Because GB and AB are each equal, by hypothesis, to DE, AB and GB are equal to one another, and the triangle ABG is isosceles. Wherefore also (5. 1.) the angle BAG is equal to the angle BGA. In the same way, it is shewn that AC is equal to GC, and the angle CAG to the angle CGA. Therefore adding equals to equals, the two angles BAG, CAG together are equal to the two angles BGA, CGA together; that is, the whole angle BAC to the whole BGC. But the angle BGC is, by hypothesis, equal to the angle EDF, therefore also the angle BAC is equal to the angle EDF. Q. E. D.
Such demonstrations, it must, however, be acknowledged trespass against a rule which Euclid has uniformly adhered to throughout the Elements, except where he was forced by necessity to depart from it. This rule is, that nothing is ever supposed to be done, the manner of doing which has not been already taught, so that the construction is derived either directly from the three postulates laid down in the beginning, or from problems already reduced to those postulates. Now, this rule is not essential to geometrical demonstration, where, for the purpose of discovering the properties of figures, we are certainly at liberty to suppose any figure to be constructed, or any line to be drawn, the existence of which does not involve an impossibility. The only use, therefore, of Euclid's rule is to guard against the introduction of impossible hypotheses, or the taking for granted that a thing may exist which in fact implies a contradiction ; from such suppositions, false conclusions might, no doubt, be deduced, and the rule is therefore useful in as much as it answers the purpose of excluding them. But the foregoing postulatum could never lead to suppose the actual existence of any thing that is impossible ; for it only assumes the existence of a figure equal and similar to one already existing, but in a different part of space from it, or having one of its sides in an assigned position. As there is no impossibility in the existence of one of these figures, it is evident that there can be none in the existence of the other.
Dr. Simson has very properly changed the enunciation of this proposition, which, as it stands in the original, is considerably embarrassed and obscure. His enunciation, with very little variation, is retained here.
It is essential to the truth of this proposition, that the straight lines drawn to the point within the triangle be drawn from the two extremities of the base ; for, if they be drawn from other points of the base, their sum may exceed the sum of the sides of the triangle in any ratio that is less than that of two to one. This is demonstrated by Pappis
Alexandrinus in the 3d Book of his Mathematical Collections, but the demonstration is of a kind that does not belong to this place. If it be required simply to shew, that in certain cases the sum of the two lines drawn to the point within the triangle may exceed the sum of the sides of the triangle, the demonstration is easy, and is given nearly as follows by Pappus, and also by Proclus, in the 4th Book of bis Commentary on Euclid.
Let ABC be a triangle, having the angle at A a right angle ; let D be any point in AB ; join CD, then CD will be greater than AC, because in the triangle ACD the angle CAD is greaterthan the angle ADC. From DC cut off DE equal to AC; bi
С sect CE in F, and join BF ; BF and FD are greater than BC and CA.
Because CF is equal to FE, CF and FB are equal to EF and FB, but CF and FB are greater than BC, therefore EF and FB are greater than BC. To EF and FB add ED, and to BC add AC,
D which is equal to ED by construction, and BF and FD will be greater than BC and CA. Q. E. D.
It is evident, that if the angle BAC be obtuse, the same reasoning may be applied.
This proposition is a sufficient vindication of Euclid for having demonstrated the 21st proposition, which some affect to consider as selfevident; for it proves, that the circumstance on which the truth of that proposition depends is not obvious, nor that which at first sight it is supposed to be, viz. that of the one triangle being included within the other. For this reason I cannot agree with M. Clairaut, that Euclid demonstrated this proposition only to avoid the cavils of the Sophists. But I must, at the same time, observe, that what the French Geometer has said on the subject has certainly been misunderstood, and, in one respect, unjustly censured by Dr. Simson. The exact translation of his words is as follows : If Euclid has taken the trouble to demon“ strate, that a triangle included within another has the sum of its “ sides less than the sum of the sides of the triangle in which it is “ included, we are not to be surprised. That geometer had to do “ with those obstinate Sophists, who made a point of refusing their.as. “ sent to the most evident truths,” &c. (Elemens de Geometrie par M. Clairaut. Pref.)
Dr. Simson supposes M. Clairaut to mean, by the proposition which he enunciates here, that when one triangle is included in another, the sum of the two sides of the included triangle is necessarily less than the sum of the two sides of the triangle in which it is included, whether they be on the same base or not. Now, this is not only not Euclid's proposition, as Dr. Simson remarks, but it is not true, and is directly contrary to wbat has just been demonstrated from Proclus. But the fact seems to be, that M. Clairaut's meaning is entirely different, and that he intends to speak not of two of the sides of a triangle, but of all the three ; so that his proposition is, “ that when one
v: triangle is included within another, the sum of all the three sides of " the included triangle is less than the sum of all the three sides of the * other,” and this is without doubt true, though I think by no means self-evident. It must be acknowledged also, that it is not exactly Euclid's proposition, which, however, it comprehends under it, and is the general theorem, of which the other is only a particular case. Therefore, though M. Clairaut may be blamed for maintaining that to be an Axiom which requires demonstration, yet he is not to be accused of mistaking a false proposition for a true one.
Thomas Simpson in his Elements has objected to Euclid's demonstration of this proposition, because it contains no proof, that the two circles made use of in the construction of the Problem must cut one another; and Dr. Simson, on the other hand, always unwilling to acknowledge the smallest blemish in the works of Euclid, contends, that
demonstration is perfect. The truth, however, certainly is, that the demonstration admits of some improvement; for the limitation that is made in the enunciation of any Problem ought always to be shewn to be necessarily connected with the construction of it, and this is what Euclid has neglected to do in the present instance. The defect may easily be supplied, and Dr. Simson himself bas done it in effect in his note on this proposition, though he denies it to be necessary.
Because that of the three straight lines DF, FG, GH, any two are greater than the third, by hypothesis, FD is less than FG and GH, that is, than FH, and therefore the circle described from the centre F, with the distance FD must meet the line FE between F and H; and, for
C the like reason, the circle described from the centre G at the distance GH, must meet DG between D and G, and therefore, the one of these circles cannot be wholly within the other. Neither can the one be wholly without the other, because DF and GH are greater than FG ; the two circles must therefore intersect one another.
PROP. XXVII. and XXVIII.
Euclid has been guilty of a slight inaccuracy in the enunciations of these propositions, by omitting the condition, that the two straight lines on which the third line falls, making the alternate angles, &c. equal, must be in the same plane, without which they cannot be parallel, as is evident from the detinition of parallel lines. The only editor, I believe, who has remarked this omission, is M. de Foix, Duc DE CANDALLE, in his translation of the Elements published in 1566. How it has escaped the notice of subsequent commentators is not easily explained, unless because they thought it of little importance to correct an error by which nobody was likely to be misled.
The subject of parallel lines is one of the most difficult in the Elements of Geometry. It has accordingly been treated of in a great variety of different ways, of which, perhaps, there is none that can be said to have given entire satisfaction. The difficulty consists in converting the 27th and 28th of Euclid, or in demonstrating, that parallel straight lines, or such as do not meet one another, when they meet a third line, make the alternate angles with it equal, or, which comes to the same, are equally inclined to it, and make the exterior angle equal to the interior and opposite. In order to demonstrate this proposition, Euclid assumed it as an Axiom, that“ if a straight line meet two straight “ lines, so as to make the interior angles on the same side of it less than “two right angles, these straight lines being continually produced, “ will at length meet on the side on which the angles are that are less to than two right angles.” This proposition, however, is not self-evident, and ought the less to be received without proof, that, as Proclus has observed, the converse of it is a proposition that confessedly requires to be demonstrated. For the converse of it is, that two straight lines which meet one another make the interior angles, with any third line, less than two right angles ; or, in other words, that the two interior angles of any triangle are less than two right angles, which is the 17th of the First Book of the Elements : and it should seem, that a proposition can never rightly be taken for an Axiom, of which the converse requires a demonstration.
The methods by which Geometers have attempted to remove this, blemish from the Élements are of three kinds. 1. by a new definition of parallel lines. 2. by introducing a new Axiom concerning paraltel lines, more obvious than Euclid's. 3. By reasoning merely from the definition of parallels, and the properties of lines already demonstrated, without the assumption of any new Axiom.
1. One of the definitions that has been substituted for Euclid's is, that straight lines are parallel, which preserve always the same distance from one another, by the word distance being understood, a per
pendicular drawn to one of the lines from any point whatever in the other. If these perpendiculars be every where of the same length the straight lines are called parallel. This is the definition given by Wolfius, by Boscovich, and by Thomas Simpson, in the first edition of his Elements It is however a faulty definition, for it conceals an Axiom in it, and takes for granted a property of straight lines, that ought either to be laid down as self-evident, or demonstrated, if possible; as a Theorem. Thus, if from the three points A, B, and C of the straight line AC, perpendiculars AD, BE, CF be drawn all equal to one another, it is implied in the definition,
E that the points D, E and F are in the same straight line, which, though it be true, it was not the business of the definition to inform us of. Two perpendiculars, as AD
С and CF, are alone sufficient to determine the position of the straight line DF, and therefore the definition ought to be, “ that two straight lines are parallel, when there are two "points in the one, from which the perpendiculars drawn to the * other are equal, and on the same side of it.”
This is the definition of parallels which M. D'Alembert seems to prefer to all others ; but he acknowledges, and very justly, that it still remains a matter of difficulty to demonstrate, that all the perpendiculars drawn from the one of these lines to the other are equal. (Encyclopedie, Art. Parallele.)
Another definition that has been given of parallels is, that they are lines which make equal angles with a third line, toward the same parts, or such as make the exterior angle equal to the erior and opposite. Varignon, Bezout, and several other mathematicians, have adopted this definition, which, it must be acknowledged, is a perfectly good one, if
it be understood by it, that the two lines called parallel," are such as make equal angles with a certain third line, but not with any line that falls
upon them. It remains, therefore, to be demonstrated, That if AB and CD make equal angles with GH, they will do so also with any other line whatsoever. The definition, therefore, must be thus understood, That parallel lines are such as make equal angles with