On BC, which is equal to EF, and on the side of it opposite to the triangle 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. PROP. VII. 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. PROP. XXI. 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 Pappus Pp By help of the same postulate, the 5th may also be very easily demonstrated. Let ABC be an isosceles triangle, in which AB, AC are the equal sides; the angles ABC, ACB opposite to these sides are also equal. Draw the straight line EF equal to BC, and suppose that on EF the triangle DEF is constituted every way equal to the triangle ABC, that is, having DE equal to AB, DF to AC, the angle EDF to the angle BAC, the angle ACB to the angle DFE, &c. Then, becanse DE is equal to AB, and AB is equal to AC, DE is equal to AC; and for the same reason, DF is equal to AB. And because DF is equal to AB, DE to AC, and the angle FDE to the angle BAC, the angle ABC is equal to the angle DFE, (4. 1.). But the angle ACB is also, by hypothesis, equal to the angle DFE; therefore the angles ABC, ACB are equal to one another. Q. E. D. Thus also, the 8th proposition may be demonstrated independently of the 7th. Let ABC, DEF be two triangles, of which the sides AB, AC are equal to the sides DE, DF each to each, and also the base BC to the the base EF; the angle BAC is equal to the angle EDF. D B C F G On BC, which is equal to EF, and on the side of it opposite to the triangle 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. PROP. VII. PROP. XXI. Pp 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 his 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 greater than the angleADC. From DC cut off DE equal to AC; bi C F 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 E EF and FB add ED, and to BC add AC, A which is equal to ED by construction, 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 trou"ble to demonstrate, 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 refus" ing their assent 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 what 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 " 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. PROP. XXII. 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 the 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 has 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 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. |