"must be something farther. So again it cannot be conceived how "eternity should have flowed to the prefent time: and there is the "like fubtilty as to the infinite divifibility of lines, &c. all arifing "from the weakness of human thought." And whoever chuses to indulge this weakness of thought will be put into a fine train by reading what Locke has advanced upon the idea of infinity. But it is the end of all mathematical learning, inftead of lending any affistance to encourage this kind of dreaming, to root out fuch a difpofition from the human mind; and instead of fuch fooleries, to give it something to employ itself upon fuitable to the powers and faculties of a human foul. An understanding which has any force will always shake off such dreams; the indulging in which is the sign of a weak intellect. For the philofopher and the idiot agree perfectly in their notions of infinity; and the difference between them is only difcovered when they come to a comparison of determinate things and quantities.

The geometrical notion of infinity, as far as it regards extenfion, is derived from the fecond poftulate; let it be taken for granted that a straight line may be continued directly forward: The geometricians never trouble themselves with multiplying any affignable parts of extenfion, in order to be convinced that after they have advanced millions of miles, they are still as far from the end of an infinite line as they were at their firft setting out.

The two straight lines DE, DF in the second propofition, are infinite in the full and proper geometrical sense; and fo likewise are AD and AE in the fifth; and the third propofition may be applied to conftruct this, because AF is less than AE: but why is it lefs? Because AF is a finite and AE and infinite line and this inftance fully explains what is meant by an infinite line in this science; for it means only a line longer than any determinate line which they have occafion to take; and farther than this they give themselves no concern. In the ninth propofition AB and AC are infinite lines; and the point D is taken that we may have a fixt line AD and fo in other inftances.

I am now arrived at the conclufion of this differtation, in which I have endeavoured to point out fuch circumstances, as the thoughtlefs reader is apt to overlook. If it should seem strange to any one to

find such a cerimonious introduction to a fimple, well defined and demonstrative science, he will be pleafed to obferve fome peculiarities which distinguish this fubject remarkably from most others. It is true the language is plain and fignificant; but the truths to be communicated are conveyed in too few words to engage the attention of those who have been accustomed to the figurative language in which most other subjects are delivered; and where the reader, instead of having confequences deduced from every word, carries fomething with him if he attend to but one word of three. And for this reafon I have been a little diffuse and circumlocutory; that the tranfition, from the common form of speaking, to this very concife and accurate method of expreffion, might be made with greater ease by the reader, after fome part of the subject has been explained to him in his own way. But I must beg leave to introduce the student to Euclid himself, and defer all future intercourse with him until he has reached the end of the first book.

DISSERTATION III.

TH

HE first book of Euclid's elements will neceffarily fuggest a great variety of reflexions to a judicious reader; he will perceive a particular method of arrangement, and a particular manner of demonstration which makes that arrangement abfolutely neceffary; with fomething characteristic even in his way of conftructing problems. But an indolent reader requires to be put in mind of each of these particulars, otherwise he will be disposed to overlook them. Some remarks therefore upon each of these heads may be useful to put him into a proper train of thinking; and this shall be the subject of the present differtation.

CHA P. I.

In which Euclid's method of demonftration is proved to be necessary contrary to the opinion of Clairaut.

"Qu'Euclide se donne la peine de démontrer, que deux cercles qui fe coupent n'ont pas le même centre, qu'un triangle ren"fermé dans un autre, a la fomme de fes côtés plus petite que "celle des côtés du triangle dans lequel il eft renfermé; on n'en "fera pas surpris. Ce Géométre avoit à convaincre des Sophistes "obstinés, qui fe faifoient gloire de fe refufer aux vérités les plus “évidentes: il falloit donc qu'alors la Géométrie eut, comme la

Logique, le fecours des raifonnemens en forme, pour fermer la "bouche à la chicane. Mais les chofes ont changé de face. Tout "raisonnement qui tombe fur ce que le bon fens feul décide d' "avance, est aujourd'hui en pure perte, & n'eft propre qu'à ob"scurcir la vérité, & à dégoûter les Lecteurs."

VOL. I.

h

This

This is a very fingular opinion concerning the motives which led Euclid to that rigorous method of demonftration which he has adopted for we are here told that it was not choice, but the circumstances of the times in which he lived, which brought him to write as he has done: his living among fophifts, drove him beyond the bounds of good fenfe: for if he had been left to follow his own inclinations, we have here room to fuppofe that his demonstrations would have appeared in a very different form, or in other words, that had he been a Frenchman and lived in the fame polite and happy times as Clairaut he would have written just as he has done. The French would have taken his word for it, that two circles, which cut one another, could not have the fame center; and that polite nation furely never would have contradicted him if he had faid the fame thing of two Ellipfes: but one thing I am certain of, that, had they been acquainted with no other principles but fuch as this author would have us appeal to, they never could have contradicted fuch an affertion upon any good grounds.

Now what I mean to prove, in oppofition to this doctrine, is, that Euclid is remarkable for adhering to the very principles which Clairaut thinks he himself has gone upon: and I fhall now produce as many inftances as I think the reader's patience can well endure in proof of this affertion, namely, that Euclid every where diftinguishes himself by keeping clear of fuch things, as good fenfe would decide of itself before hand. Now as the method of these two authors is not fimply different, but directly oppofite the one to the other; Clairaut and I muft affign a very different office or employment to this fame good fenfe, the prefuming to teach which, he fays, answers no other end but to difguft the readers. But I am certain good fenfe will never be either affronted or difgufted to have any thing fet in a better or stronger light than what it appeared in before; and I challange any one to produce an inftance, where the thought is not improved and rendered more accurate by Euclid's reafoning; and it must be a strange kind of good sense, which could reckon fuch reasoning, as he fays en pure perte.

But as to his looking upon himself as bound to convince obftinate fophifts, it is very obvious that he never went in the leaft out of his way on their account; and guided himself by very different

maxims,

maxims, as will appear to every one who confiders his manner of reasoning, which is to the laft degree inconfiftent with any fuch fuppofition, as that he had the leaft intention to combat fuch prejudices as the fophifts raise, when they have a mind to sport with the credulity and ignorance of the multitude. It is true he furnishes us with the means of escaping out of their hands, if a proper use be made of his principles; but it never once entered his thoughts to combat their filly objections, which never could be obtruded upon any one, who is not entirely ignorant of the nature of the subject.

However, as a full proof of what I now affert, I would recommend it to those who may chufe to differ from me in opinion, to read Proclus's commentary, who fets himself in earnest about this hopeful task; and there, it is granted he will find reafonings, qui tombe fur ce que le bon fens feul décide d'avance; and I will allow that fuch reasonings are of no other ufe but to obfcure the truth, and disgust the readers. The dispute therefore is brought to this fhort iffue; let the reader peruse the first book of Euclid and then read Proclus's commentary, obferving the difference between the two kinds of compofition; and I am certain he will acquit Euclid of any design to convince obftinate fophifts; and moreover must be perfuaded that he has not only omitted every thing which good fense can intuitively determine; but that he has alfo left feveral things for the reader to fupply which would not come eafily under his own precise notion of demonstration; for Horace's rule was never better nor more properly applied than by this author;

et quæ

Defperat tractata nitefcere poffe, relinquit.

Several inftances of which, taken from his first book, I shall now proceed to enumerate, not confidering them as overfights, but as proofs of a moft refined and accurate judgement.

And first, to begin with that famous principle by which the meeting of two straight lines is determined: We may readily fuppofe that Euclid introduced the feventeenth propofition, with a view either to demonftrate or explain the eleventh common notion,

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