of the two fides; and as we must have an angle equal to the angle at the vertex of the ifofceles triangle, this angle must be made, which, in our prefent circumftances, will not be found fuch an eafy matter. And so much for hypotheses or suppositions. СНАР. V. Containing a critical examination of the fourth propofition. OBJECTIONS are brought against this propofition, as if the demonstration proceeded by a mechanical application of the two triangles to one another. But whoever starts such an objection as this, has not the most distant conception of the demonstration; or else he must be ignorant of the nature of a mechanical application. And a particular examination will put this beyond all doubt. Suppose two triangles of brass, made as accurately as poffible, all their different parts corresponding exactly. When we apply them to one another, all that we can fay is, that as far as our fenfes can judge, the parts feem to agree. Now what knowledge is got from this? Certainly by this application, no property of a triangle can be discovered; we may form a conclufion concerning the accuracy and neatness of the workmanship; but nothing farther. It is impoffible to say from this that any thing is equal, but what is made equal; no doubt a very curious discovery and tending greatly to the enlarging the boundaries of the science! The mistake here arifes from not confidering that it is impoffible to make the parts of the figure which are to reprefent the fuppofition unless you make at the fame time the parts expreffing the confequences which, it is faid, will follow. The parts reprefenting the fuppofition are to be tried and examined by the eye, as well as those which reprefent the confequences; fo that if any confequence follow it must follow from nothing. And the most attentive student will retire from this contemplation, with a very curious piece of information; namely, that he had seen two brafs figures fo contrived as to fit each other exactly. We We now proceed to confider the application made ufe of by Euclid in this fourth propofition. And firft it is to be observed that his triangles are required to have no particular pofition; therefore the point A may be fuppofed to be upon D as well as any where else; and likewife the line AB upon DE; and all this might happen to any two triangles not determined to any particular pofition. Then AB being to DE the point B must apply itself to E, it is impoffible to conceive it to have another fituation, which is very different from faying that as far as we can judge by our senses it seems to be fo; again if the angle BAC be equal to EDF, the line AC must take the direction of DF; and it is impoffible to conceive it to take any other; and fo likewife the point C must apply itself to F, whenever the line AC is equal to DF : and fo on to the coincidences of the different parts of the triangles; and all this is as different from a mechanical application, as light is from darknefs. For fuppofing what is taken for granted, it is impoffible to conceive the confequences not to follow ; and this is certainly science if there be any fuch thing in the world. But again it is alledged, why may not this method of applying be extended still farther by making it an inftrument for conftructing problems? as for inftance in the second propofition; why may we not suppose the line BC fo placed that B may be upon A, and the thing required is done? For a very good reason, because BC and the point A have each of them a fixt pofition already. Upon the whole then we may conclude, that this method of application is perfectly scientifical; and that whenever two triangles agree in the circumftances mentioned in this fuppofition; the confequences will always neceffarily follow; the bafes will be equal and the remaining angles, each to each, viz. thofe under which the equal fides are extended: Because no confequence is deduced from the lines having any particular length, but only from their being equal nor is it fuppofed that the angle contained by the fides is any particular angle, but only that the angles are equal. And this may fuffice for an answer to the objections commonly brought against the demonftration of this propofition. VOL. I. f CHAP. СНАР. VI. Containing an explanation of the fifth propofition. ONCE upon a time a certain father refolving not to be impofed upon by reports, determined to examine into his fon's progress in this fcience, produced the book and required him to demonftrate a propofition to which he referred: the young man though unacquainted with the subject, taking courage from his father's ignorance, began very impudently in fome such manner as follows; Because the angle ABC is equal to the angle CBA, therefore the angle DEF is equal to the angle CEF &c &c; ringing the changes upon fides and angles, until he had fpun out his demonftration to a decent length: and then kept filence in expectation of his fathers's opinion; who with a grave and important countenance remarked, "This is what we call demonstration." Every one is fenfible, that it is contrary to common sense to imagine that the letters of the alphabet thus repeated can have any meaning; but the indolent reader ought to be informed, that the repeating fuch phrases, in a regular order, mends the matter but very little, unless they convey to the mind their proper meaning: and unless the fourth propofition be well understood, the most of that which follows will be nothing but an infignificant jargon. As the whole fcience therefore depends fo much upon an accurate and comprehensive view of this propofition, it would be proper for the learner, before he proceeds farther to take the opinion of fome acquaintance skilled in these matters; who, by a particular examination might be able to determine, how far he can be properly faid to understand it; and this friend is authorized, upon his failure in any point, to admonish him, by faying, "this is "what we call demonstration.” I myself can trace every mistake concerning the following propofitions, or partial conception of their meaning, up to my ignorance of the full import and meaning of this propofition. For it is by no means to be understood as applicable only to fuch triangles, as one may make ufe of to affift the imagination in tracing the the steps of the demonftration; but as carrying with it this extenfive and general meaning, that all triangles which have two fides equal to two fides, each to each, and the angles, contained by those fides, equal; that all such triangles, I fay, have their bases equal; and the two remaining angles in every triangle, equal, each to each; viz. those angles under which the equal fides are extended. Or more properly that all fuch triangles, whatever be their number, are but one and the fame triangle. Or otherwise that the two fides, and the angle contained between them, fixes every part of the triangle, beyond a poffibility of change. But I shall now proceed to confider the fifth propofition. And the reader may recollect an attempt in the end of the fourth chapter, to derive this propofition from the fourth, or rather to make it only a particular cafe of that propofition, when the two fides of each triangle are equal to one another; and I there gave the reasons why it could not be confidered as fuch: nevertheless the artifice of this propofition will be the better understood by profecuting that scheme a little farther, producing the equal fides, and proving the angles below the bases of fuch triangles equal, in the fame manner as the equality of those above the base was there demonstrated. It was then obferved that the fuppofition was too complicated for this purpose; because not only DE and DF were to be equal to each other and to AB and AC, for this might have been allowed; but also the angles between the fides were to be equal, which could by no means be allowed; becaufe according to the fifth, only one angle being given, the other was to be made equal to it; which is not at prefent poflible. If the reader has profecuted this fpeculation far enough, he will certainly admire the ingenuity of our author for his contrivance to make these angles equal, in a very elegant manner indeed: and which, it is curious to observe, neceffarily requires that the fides of his two triangles fhould be unequal, For he makes this very angle which we are confidering common to both, by making the fides of the triangles take the fame direction, the shorter fide of the one being upon the longer fide of the other. But now it will be proper to turn to the demonstration itself; and after a careful examination of its different parts, I would recommend f 2 commend a particular attention to the ufe which might be made of it, in order to imprefs the last propofition more strongly upon the memory. The great advantage of problems above theorems, for fixing the attention of the learner, has been mentioned already: and here a little of that advantage may be gained, by the particular view of the fourth exhibited in this propofition; which will teach us how to represent by an actual construction, the fuppofitions in the last propofition: because, if we draw two undetermined straight lines, making any angle; we can cut off AB equal to AC; and AF to AG; and, by joining BG and FC form two triangles, with all the parts of the fuppofition in the last, accurately represented; and this not in imagination, but constructed by ourselves; and the confequences, not made, as was objected to the mechanical triangles, but left to follow from the construction ; the accuracy of which, the flow of imagination may examine by inftruments; and thus get a kind of palpable evidence for the truth of the propofition ; and by varying the inclination of the lines, and the length of the fides, climb up by degrees to fomething like a scientific conception of its meaning. The other two triangles FBC and CGB, though they represent a particular inftance of the fourth propofition, are not fo fit for this purpose, not being fo general as the first two triangles nor fa much in our power, or rather indeed they are not at all in our power: We may vary the angle and the two fides in the triangle ABG at pleasure; but we have no power over the fides, or angle BFC of the triangle FBC; for they become fuch as may happen from the construction of the firft triangle: therefore the epithet any could not be so properly applied to the triangle FBC as to ABG. I have recommended to thofe whofe imaginations are flow or inactive, these conftructions and inftrumental proofs of the feveral conclufions which follow from the fuppofitions in the fourth propofition. And I now, rather more earnestly, recommend the same to those whofe imaginations are difpofed to out run their judgement, left they should snatch the conclufions without the premises. I have only one remark more to make upon this propofition, which the judicious reader has already made for himself, namely, how little there is to attend to, in this very formidable proposition, |