!

I fhall conclude this chapter with my opinion of indirect demonstrations, which fome very ignorantly object against. It would be abfurd to speak of the certainty of a demonftration: as to their ufe therefore, confidered as an exercise to the mind, I think they are preferable, for feveral reafons to direct demonstrations. Because in a direct demonstration, there is generally fome circumftance which catches the fenfes; and not being too difficult of perfuafion, where our own ease is so much concerned, we are disposed to rest very well fatisfied with that, though it may convey, but a very imperfect notion of the full extent of the demonftration. But the contrary happens in fuch demonftrations as are indirect; for the reasoning being generally at variance with the fenfible reprefentations of the magnitudes, it requires fome effort of the mind to get beyond the prejudice of fenfe, fo as to comprehend the force of the reafoning and when we cannot conquer our indolence or command our attention, it is a very decent excufe to lay the blame upon the nature of the demonftration. But I have been fo particular in my examination of these two propofitions, that the attentive reader can hardly want any farther information upon this head.

IT will now be proper for the reader to endeavour, from a careful examination of these firft feven propofitions, to collect as accurate a notion as poffible of a demonstration, according to the rules which Euclid has prescribed to himself.

We are apt to fatisfy ourselves of the truth of things, in the eafieft and shorteft way we can; and even when we chufe to confider the properties of magnitude, instead of having recourse to fome general principles, we truft to mechanical inftruments; and very often to vague conceptions collected at random from accidental obfervation. But our outhor's plan is very different from this: The definitions, poftulates and common notions, are the only foundation upon which his geometrical reasonings are founded;

and

and when he mentions any thing as a property of a figure, he confiders it as incumbent upon him to fhew that it follows from or agrees with the definition of that figure; and this he does by the affiftance of some common notion, poftulate or other definition. And whatever property is not reducible in this manner, he confiders as impoffible to be demonftrated; though it might be a very obvious truth from other principles. Truth therefore is not fo much his object as confiftency; it does not appear directly to be any part of his business to collect as many useful facts as poffible concerning the properties of figures; but only to be convinced that what he does produce have a folid foundation, making every part acquire additional ftrength by the confiftency of the whole. He will never allow it to be faid that fuch a property is fo fimple as not to require a demonstration; for to alledge this as a reafon for taking any thing for granted, would be, upon his plan the greatest abfurdity. For it may be said, if it be so very plain and obvious, whence does it derive this obviousness? If it be the property of a figure; that figure is defined to be fo and fo. How does it appear that it is confiftent with that definition? If you point out the confiftency, you have demonftrated the propofition; but if that cannot be done, it may be true according to your principles, but mine lead me abfolutely to reject it.

That this is Euclid's plan will appear from the whole of his work, even to the aftonishment of the attentive and judicious reader; when it is confidered how his invention must have been perplexed, by the obvious, but inaccurate experimental tests, which would for ever be presenting themselves to his imagination; and how his patience would be tired out by the number of steps, which he forsaw to be neceffary, before he could arrive regularly, at some conclusions seemingly very simple indeed. A man would feel him/ felf in a very aøkward fituation, who fhould fet about measuring magnitudes, without being able to judge when one straight line was longer than another; and yet it is wonderful, by how many fteps, this very teft is removed from the first principles; and which are all neceffary for giving it a fcientific ftability. We find this delivered in the nineteenth propofition of the first book ; but as this is rather foreign to my prefent purpose, and beyond the fup

pofed knowledge of the reader, it will be fufficient to have mentioned it by the bye, only as a fact to illuftrate my meaning. I fhall therefore now beg the attention of the ftudent, while I review the first feven propofitions, in fupport of my opinion concerning Euclid's plan of demonstation.

In the first propofition his principles are the third postulate; the first poftulate; the fifteenth definition; the first common notion; and lastly the twenty fourth definition. He is not satisfied to make an equilateral triangle any how; but he makes it by the affiftance of his own inftruments and principles. The fecond propofition is conftructed and demonftrated by the first poftulate, the first propofition, the third poftulate, the fifteenth definition and the third common notion; and we have alfo here a proof how tenacious our author is of his principles; for it is not abfolutely neceffary that in this fecond propofition, an equilateral triangle should be described upon AB; because an isofceles triangle would have answered the purpofe; for no confequence is drawn from AB's being equal to AD; as it would be fufficient if AD and DB were equal; but he could not make an isofceles triangle, confiftent with his own plan and principles. The third propofition refts upon the fame foundation; its demonftration being fupported by the fecond propofition, and the third poftulate.

In the fourth propofition the principles which we reason from, besides the fuppofition, are the eighth and twelfth common notions; to which every part of the demonstration may be reduced; for the application of the triangles to each other must be allowed to be a proper principle to reafon from, otherwise it is difficult to conceive what ufe can be made of this eighth common notion, which is the definition of equality; it will not matter much whether this putting of the triangles be reckoned a conftruction or not; I am rather inclinable to confider it as a thing of its own kind, and not to be understood either as an hypothesis or conftruction I know Euclid makes ufe of the expreffion the point being put, which feems to imply a kind of conftruction ; and yet would be doing no great violence to the language, to paraphrase it in this manner; the triangles have no particular position; that is they have no relation to any points, lines or angles, which can be

it

affected

affected by our giving them a particular pofition; let us fuppofe them then to have fuch a pofition, as it is neceffary magnitudes fhould have before that eighth common notion can be of any use; that is let us suppose their situation such, that the one is applied to the other; and the point A put upon D &c.

It is plain that Euclid himself found it impracticable to give this principle a more scientific form; and therefore instead of making a definition of it, left it among the common notions; but whatever opinion we adopt; it will answer our purpose, because every thing is inferred from the principles, which are laid down as fuch. If we were required to place triangles upon each other, which had a determinate pofition before, I would certainly confider it as a construction; but in the present case, I confider it only as a neceffary apparatus before any confequence can be drawn from the eighth common notion. And as to the objections, upon a conceit that this is a mechanical application, they have been very fully answered already; fo that upon the whole there is nothing in this propofition, but what agrees with the notion of a demonftration which has been delivered in the beginning of this chapter.

The fifth propofition feems by the references to depend upon the four first propofitions and the third common notion; but as the conftruction required is only the fimpleft cafe of the third, the first and second propofitions are not neceffary for cutting off AG equal to AF. But the fixth requires the four first propofitions; because the fimpleft cafe of the third, will not be fufficient for cutting off BD equal to AC; and, what I must beg the reader particularly to attend to, neither can it be done by a ruler and compaffes; because there is fomething fingular in this construction : for it is here required to do, what we afterwards find impoffible to be done; and the unlucky circumftance is, that whoever trufts to a ruler and compaffes will come to the knowledge of this too foon; a pair of compaffes lets him into the fecret immediately, before his mind is prepared for it by any scientific information; and hence will arise a gross mifconception of the author's meaning; because the reasoning muft appear ufelefs and unneceffary, being intended to discover to a man what he has found out already. In all fuch cafes as this therefore our ruler and compaffes are to be laid aside,

and

and fome method devised which may be more confiftent with the nature of the poftulates as here applied for by keeping strictly to the information wchih we get from them, we never can discover that we have been attempting to perform what is impoffible, before we are led to the abfurdity of concluding the triangle ABC to be equal to DCB. I would recommend the performance of such conftructions as this, by the hand unguided by any inftrument; as this will favour the condition of the propofition, because such a construction can make no discovery before the proper time. Not that I would fuppofe it performed; as that might fofter prejudices, ́ but rather reprefent every step of the conftruction; BC being joined already I would defcribc upon it an equilateral triangle producing the fides &c. as directed in the remarks on the third propofition, until the last described circle cut AB in fome fuch point as D; because the whole conftruction will be neceffary to convince the reader that he has no scientific notice of the impoffibility of the problem from the construction, but must wait for it until he come to the abfurdity. And hence it appears that Euclid requires every part of this propofition to be referred to the first principles; contrary to the fentiments of thofe fuperficial readers, who imagine that they have only to fuppofe a point D, and the thing is done.

The demonstration of the seventh is reducible to the fifth; and the ninth common notion. And thus it appears, that Euclid's intention is not to fhew that the propofition is true in any manner; but that it is immediately connected with, and depends upon his principles, or in other words that his aim is not to perfuade but to demonftrate.

CHA P. IX.

In which is explained the geometrical meaning of the words finite and infinite.

"THE human understanding, fays Bacon, fhoots itself out, "and cannot reft; but ftill goes on though to no purpose. Thus "'tis inconceivable there should be any bounds to the universe ; "yet it conftantly, and, as it were, neceffarily recurs, that there

"muft