as being the converse of it. For from this propofition we learn that whenever the two lines meet, the angles made by the cutting line, upon that fide, are lefs than two right angles; but we can by no means infer from this that the lines will meet whenever the angles are less than two right angles. But again he has demonstrated in the twenty eighth propofition; that when these angles are equal to two right angles the lines will never meet. But he cannot, from this fay, that this is the only cafe in which the lines will never meet: only thus far he may go, that the supposition, from which he infers that they will never meet, requires abfolutely that the angles fhould be exactly equal to two right angles; and the least deviation from this fuppofition will render his principles of no avail; as no confequence can follow from them; for their abfolute equality to two right angles and nothing lefs, is neceffary to prove that the lines will never meet. Now this is fufficient at leaft to ground a perfuafion, that in all other cafes they will meet; especially when it is confidered that such lines are inclined towards one another, as may be fhewn by drawing a perpendicular to one of the lines; which will prove that they are approaching towards one another upon the fide where the angles are less than two right angles and going farther from one another upon the fide where the angles are greater than two right angles; but the opinion thus acquired is not of the nature of a demonstration. Nor could he even say that a line intersecting one of two parallels, would meet the other; and indeed if this could be fhewn upon fcientific principles, it would be a demonftration of the thing in queftion: but this would be taking the thirtieth propofition for granted, the truth of which depends upon this very principle. We may therefore fuppofe that fuch things as thefe and numberless others of the fame kind must have occurred to a mind fo fruitful in expedients upon fuch occafions, and were all rejected by him as falling infinitely short of his idea of demonftration: wherefore without perplexing his reader with impotent attempts towards a demonstration; he judged it more proper to cut short this fruitlet's fearch by placing this principle among the common notions; and as it appears from this investigation that it was not placed here without reafon; fo it has a right to keep its place, until it can be shewn shewn that fome abfurdity can follow from the use of it. And upon this point I have only to add, that it is certain from the general tenour of his demonstrations, that if he could have exhibited it without any scientific defect, but in the clumfy dress in which some authors have arrayed it, he would have rejected the conclufion with difdain, and left it where we now find it. So that it is not a demonstration of any kind which was his object, but such an one as is both elegant and forcible. But further, I have mentioned before, what seems to be an exceeding good general rule of critifcifm in this science, that we ought always to examine if some consequence has been drawn from every part of the fuppofition or conftruction. Now there are several inftances where Euclid has laid no ftrefs, even, upon the most material part of the fuppofition; which fhews how little he minded the conviction of those obftinate fophifts, for whose fake according to Clairaut he gave his elements such a formal dress, to the great disgust of every Frenchman; because here would have been fo good a pretence for ftanding ftill, that no arguments could have perfwaded these fophifts to proceed while fuch blemishes remained. But to come to particulars. In the twenty fecond propofition, it is required to make a triangle of three given ftraight lines, but with this neceffary limitation; that any two of them must be greater than the third. However no use is made of this limitation in the demonftation of the propofition; and I am perfwaded that our author omitted to prove that the two circles will cut one another, which depends upon this limitation, because this proof does not admit of fuch a natural and elegant turn, as he had determined to give to every step of his demonstrations; there being no direct principle to refer to, by which the interfection of circles can be proved; that they have some space in common is the principle from which their interfection is to be inferred, which is indeed a common notion but not one of thofe, which he has felected to draw confequences from; concluding I fuppofe that all rational men, could make as ready and extenfive an use of such notions as thefe, as he could, and there→ fore, that it would be unfair in him, to appropriate to himself either the notions or their confequences; or according to Clairaut, thatt that such reasonings would only fall upon, what good sense had decided before hand, in a manner full as fatisfactory. Again, there is an inftance of the fame kind, in the twenty fourth propofition; where the most material part of the fuppofition is never once mentioned in the demonftration; viz. that the angle BAC is greater than the angle EDF; confequences are tacitly drawn from it, like the taking for granted that the circles will cut one another in the twenty fecond; for without this part of the fuppofition the angle EGF would not be a part of DGF, nor, if the point F was within the triangle DEG would this angle EGF be a part of the angle below the base of the isofceles triangle DFG; which is abfolutely neceffary in order to prove that the angle EFG is greater than EGF; or, in other words, it follows from this part of the fuppofition, that DG always falls upon the fame fide of DF which it is reprefented to be upon in the figure; but it would be difficult to give this reafon such an elegant turn as to entitle it to a place in one of Euclid's demonstrations. But the fophifts would stick at much less matters than these; they would require him to prove in the first propofition, that the two circles would cut one another; and that their intersection is a point and in the second that there would always be such points as G and L: alfo in the fifth that AF and AE were two fuch lines, that a part could be cut off from the one equal to the other, or that AE is greater than AF: they would likewife find several parts of the fuppofition in the seventh from which no confequences are drawn, which I have taken notice of before; in the twelfth propofition they would require him, to prove that a circle described with C for the center, and through any point on the other fide of AB, would cut the line AB; which would give occafion to his making use of the infinity of the line from which no consequence is drawn, as the demonftration now ftands: They might likewise ask him, upon the corollary to the fifteenth propofition, whether when two lines meet at a point they make angles at their meeting equal to four right angles: and in the thirtieth propofition, by what principle he makes a straight line pass through three points, that is, takes it for granted that it may pass through three points, viz. a point in each line; for he ought to have taken two, and to have proved by the eleventh common notion that there would always be a third: Nor would it be very impertinent of them to require him to prove that AH and FG will meet in the forty fourth propofition; and indeed in that inftance I cannot help being of opinion that the construction would have been more in Euclid's manner if he had made GH equal to BA and then joining HA had proved that HA was parallel to GB by the thirty third propofition. In fhort if it had been his intention to write for the conviction of the fophifts, his demonstrations would have had a very different form from that in which we now find them. And it will be eafy for any one who has read even the first book with attention to add a great many more to the above catalogue of omiffions. But I hope nobody will fo far mistake my meaning here as to fuppofe that I confider fuch questions as these, either as captious or improper. They are mentioned only to fhew that Euclid had some plan by which he directed himself in his demonstrations, very different from the fuppofition that it was to convince obftinate fophifts. For I am fo far from confidering fuch questions as improper; that I think they are fuch as every one must ask himself, and a great many to the fame purpose, otherwise I am certain he will have a very imperfect notion of the book; and whoever is in earnest to understand this fubject should take every opportunity of devifing fuch objections; the proper answers to which will be readily fuggested by what Euclid has faid. But this fhould be fet about upon fome regular plan; and I would therefore recommend it to the ftudent, after he has finished the first book to give it a fecond reading, examining every fuppofition and construction, and the confequences drawn from them; and their full importance in the propofition; which is not to be determined from the number of words in which they are expreffed, but by the dependence which the conclufion has upon them: for I have an instance just now in my view, in which I may fafely fay that the first reading hardly ever discovers the full importance of the fuppofition: what I mean is the fuppofition in the forty feventh propofition; from which the confequence that follows is; CA and AG as alfo BA and AH make but one ftraight line. Now whoever has read this propofition without perceiving that the conclusion depends entirely upon upon this, cannot be faid to have understood the demonftration; and that this is true appears; because the square BG is only double of the triangle BFC because GAC is a straight line. The reader is also carefully to observe whether the conclusions are general or particular; because it may so happen that a particular position of lines or points will lead to a conclufion, which will by no means follow if these are changed; and often if it do hold good it may require fome new steps to come at the conclufion. But it seems neceffary to illuftrate this by particular examples. In the eleventh propofition of this first book, it is required to draw a perpendicular to a line, from a point given in it. This may be divided into two cafes; for the point may be at some distance from the extremity, or it may be the extremity of the line. Euclid has confidered the first case only and there is but one false confequence which could be drawn, by taking the other for granted; and what is very remarkable Simfon in his edition has hit upon this very consequence, in attempting to prove that two straight lines cannot have a common fegment; because it must be taken for granted that two ftraight lines cannot have a common fegment before a perpendicular can be drawn from the extremity of the line; for as the line must be produced, without this limitation, it may take two directions. Again in the twenty fourth propofition, the point F may have three different pofitions; it may be without the triangle DEG or in the line EG or laftly within the triangle: For a particular instance the student may make such a triangle as ABC in the figureto this propofition; having AB longer than AC; and after the triangle DEG is made; with the center D and distance DG describe a circle; which will cut EG; now any straight line drawn to the circumference of this circle, from the point D, within the angle EDG, may represent the line DF; from which the three different pofitions will be obvious. When the point F falls within the triangle, the conclufion may be inferred from the twenty first propofition; but it would be more uniform to produce DF and DG; and to reafon upon the angles below the base, in the fame manner as Euclid has done upon those that are above it. It is true Simfon reduces this to one cafe, but I think not in the manner of Euclid; for |