this pons afinorum, upon a fuppofition that the fourth is perfectly understood: for except the circumstances of taking the equal angles CBG, BCF from the equal angles ABG, ACF; it has nothing peculiar to itself; every thing else consisting of particular prospects of the fourth propofition. But if any one is defirous to have some kind of reason for fixing his attention a little longer upon this propofition, by taking a fomewhat different view of the subject; let him fuppofe, instead of this fifth propofition, it is required to demonstrate that all the angles of an equilateral triangle are equal; for instance suppose the angles of the equilateral triangle ACB defcribed in the first propofition. Produce CA, and CB; and the fame conftruction and demonstration made use of in this fifth, will prove that the angle CAB is equal to CBA; and by producing AB and AC; in the same manner it may be proved that the angle ABC is equal to ACB; but it has been already proved that ABC is equal to CAB therefore, by the first common notion, the angle CAB is equal to ACB &c. I shall conclude this chapter by defiring the reader to take every opportunity of correcting a prejudice, which it will require all his art to remove. We cannot help drawing confequences from the very pofition which the lines accidentally happen to take in that particular figure which we reason upon, though this particular position make no part of the fuppofition. The ftudent may convince himself of this, if he read the fourth propofition by the afsistance of the figure which belongs to it; and then again making use of the two triangles ABG and ACF in the figure to propofition fifth; making the fame fuppofition in both cafes. If the confequences are fuggested to his mind more readily by one set of figures than another; this can arife from nothing but the stress laid upon the magnitude and fituation of the triangles, which, by the very fuppofition have no particular magnitude or fituation. But this prejudice will be leffered, and gradually removed by varying the position of the figures, turning them upside down; and by fuch other methods as will readily occur to the attentive reader; but above all by enlarging the figures, fetting the imagination to work until they cease to be the objects of our fenfes. The center of the moon, when in the equinoctial, joined by straight lines to the two poles of the Earth, would make a very proper isofceles triangle with which the ftudent might finish his fpeculations upon this propofition. INDIRECT demonftrations are generally ufed, when the propofition is the converfe of fome other which has been already demonftrated. A propofition is faid to be the converse of another, when the hypothefis or fuppofition in the one is the thing to be demonstrated in the other, and the contrary; thus the converfe of the fourth would be; fuppofing the base BC equal to the base EF; and the angle ABC equal to the angle DFE; as also, the angle ACB equal to the angle DFE; that then the two fides BA and AC will be equal to the two fides ED and DF, each to each; viz. those fides which are extended under the equal angles, that is AB equal to DE and AC to DF; as also the angle BAC contained by the two fides equal to EDF. Now this propofition admits of a direct demonstration; for if the point B be put upon E and BC upon EF, the point C will apply itself to F, because BC is equal to EF; and BC applying itself to EF; alfo BA will apply itself to ED, because the angle ABC is equal to DEF. Certainly for the fame reason CA will apply itself to FD, that is, because the angle ACB is equal to DFE: the point A will therefore apply itself to D; and fo AB will be equal to DE and AC to DF and the angle BAC will be equal to EDF. &c. But if we suppose the angles at B and C, not only equal to thofe at E and F, but alfo equal to one another; we may prove AB to be equal to DE as before; and then by putting B upon F and BC upon FE we can prove that G will apply itself to E; and that the line AB will be equal to DF: but it has been demonstrated that AB is equal to DE, therefore by the first common notion DE is equal to DF; and no doubt some authors would produce this as a demon a demonstration of the fixth propofition; but it is liable to the fame objections, as it converfe, which have been mentioned already. But instead of going any fuch way as this to work Euclid has given a very elegant demonstration of the fixth propofition, which is the converfe of the fifth; which the reader may please to turn to and examine; after which he may proceed with the following remarks. And first it will be worth his while to obferve the ingenuity which Euclid has displayed, in making the angles which are equal by the fuppofition, the angles between the equal fides of his two triangles; and BC being a fide common to both it remains only for him to make BD equal to AC; and to join DC. And here it may be asked, and it is also the most important question which can be put with regard to this propofition, how comes it to pafs that he cannot find by his construction that the points D and A are the fame? It is true fuch an inftrument as the compaffes would find out this if it discovered any thing; but this ufe of the inftrument is rejected for reasons given already. For the folution of this difficulty therefore the student may please to turn to the third propofition, and confider the apparatus made ufe of, for cutting off, a part from a line equal to another: and he will find that the fuppofing the problem poffible implies that the laft defcribed circle must cut AB that is that the point D must be between A and B. And when we find that this fuppofition leads to an abfurdity what are we then to conclude, but that the problem is really impoffible; which never can be if AB is greater than AC; therefore it is not greater. And I appeal to any one, whether the demonftration is either difficult, obfcure or inconclufive when it is confidered in this manner. Nothing can be so senseless as the objections usually made to indirect demonftrations. Every demonftration may be loaded with objections until it becomes indirect: the reader will find a fpecimen of that kind in my remarks upon the first propofition; where a caviller is introduced denying that the equilateral triangle there described is a fixt magnitude. But our author never carries his reafoning farther than good sense requires; so that it is only fuch propofitions as this fixth, which take an indirect form in his hands. And this kind of reasoning will not apply unless we know how how many ways the thing may be; as in the present instance AB must be equal to AC or longer or fhorter: or elfe we must get at fuch direct confequences from one part of the fuppofition, as will overturn the other as in the next propofition, which I must now beg the reader to peruse: after which I shall be ready to lend him my affiftance in removing such difficulties as ufually lie in the way of beginners. It is fuppofed here, time BC equal to BD. tion entirely upon this: that AC is equal to AD; and at the fame The learner will do well to fix his attenfirft tracing the confequences which follow from the fuppofition of AC's being equal to AD as far as they will go; which he may easily do; as it is only that the angle ACD с D B is equal to ADC. He is next to obferve the confequences which follow from the position of the lines; and these consequences are, that the angle ACD is greater than BCD, which follows from this common notion that the whole is greater than its part; and also that the angle BDC is greater than the angle ADC, which follows from the fame principle: and with a distinct impreffion of these things upon his mind, it is impoffible to miss the con- A clufion that the angle BCD is much less than BDC and this point being once gained; he is next to turn his thoughts to the other part of the fuppofition; in which it is pretended that BC is, alfo at the fame time, equal to BD: and the confequence which follows from this part of the supposition cannot fail to engage his attention; being no other than this, the angle BCD is therefore, by the fifth propofition, equal to BDC: When we fuppofe AC equal to AD we must conclude that the angle BCD is much less than BDC; but these very angles must be equal upon the fuppofition that BC is equal to BD; is not this faying in the strongest terms that the two fuppofitions are inconfiftent with one another; that is, that it is impoffible for AC to be equal to AD; and at the fame time, BC to be equal to BD. Which was to be demonstrated. It It will be a very neceffary and useful task for the reader to fet himself after he has finished every demonstration; to examine whether fome confequence has been drawn from every part of the suppofition; for though it may not always be convenient for the author to draw them, yet the reader should always do it for himself; for if the fuppofitions have no confequences to the purpose either expreffed or understood; they ought by all means to be omitted; and if the author trufted to the reader's ingenuity for finding them out he neglects his duty if this part of his business be overlooked. The neceffity of this practice being thus made evident; I shall explain my meaning by a particular example; it is faid that AC and AD are to have the fame extremity; but no use is made of this exprefly in the demonftration: and yet without this the demonstration could not proceed; because ACD could not be a triangle unless these lines had the fame extremity: neither could it be a triangle unless C and D were different points. It is moreover said that they are to be towards the fame parts: now if the points C and D were on different fides of the line AB; the angles ACD and BCD would either be two distinct angles and the one not a part of the other or if BCD be a part of ACD, then ADC would not be a part of BDC; which is abfolutely neceffary for bringing out the conclufion which we aim at. Lastly it is faid that the equal lines are to be terminated in the same extremity; and without this neither ACD nor BCD would be isosceles triangles; and then no inconsistency could follow, reason as long as we pleased. All this will be obvious by taking the points C and D on different fides of AB, and joining CA, CB; DA, DB and CD; fo that CD may cut AB or fall beyond the point B. And indeed it is very difficult to understand any general propofition, without fome representation of all the different pofitions which the lines can take and the reader cannot finish this propofition better than by supposing the point D within the triangle ACB, and to affift his imagination, by making AC and AD equal with a pair of compaffes; his demonftration will rest upon the fame principles; only producing AC and AD; he is to reafon upon the angles below the bafe; but every thing elfe will be the fame as in Euclid's demonstration. ; I fhall |