five influence in fcience, and in common life; beginning with the fimpleft and clearest, and advancing gradually to thofe which are more complicated, or lefs perfpicuous. SECTI TION I. Of Mathematical Reafoning. THE HE evidence which takes place in pure mathematics, produceth the highest affurance and certainty in the mind of him who attends to it, and underftands it; for no principles are admitted into this fcience, but fuch as are either felf-evident, or fufceptible of demonftration. Should a man refuse to believe a demonstrated conclufion, the world would impute his obftinacy, either to want of understanding, or to want of honesty: for every person of understanding feels, that by mathematical demonstration he must be convinced whether he will or not. There are two kinds of mathematical demonftration. The firft is called direct; and takes place when a conclufion is inferred inferred from premifes which render it ne¬ as as to difprove thefe principles: and if you fay, that you do not believe them *, you 01 will be charged either with falfehood or with folly; you may as well hold your But who will pretend to prove a mathe- * Si quelque opiniaftre les nie de la voix, on ne l'en fçauroit empefcher; mais cela ne luy eft pas permis interieurement en fon efprit, parce que fa lumiere naturelle y repugne, qui est la partie où se rapporte la demonftration et le fyllogifme, et non aux paroles externes. Au moyen de quoy s'il fe trouve quelqu'un qui ne les puiffe entendre, cettuy-là eft incapable de difcipline. # Dialectique de Boujou, liv. 3. ch. 3. as as evident as any thing whatfoever can be*. You may bring the matter to the teft of the fenfes, by laying a few halfpence and farthings upon the table; but *Different opinions have prevailed concerning the na ture of thefe geometrical axioms. Some fuppofe, that an axiom is not felf-evident, except it imply an identical propofition; that therefore this axiom, It is impofible for the fame thing, at the fame time, to be and not to be, is the only axiom that can properly be called intuitive; and that all thofe other propofitions commonly called axioms, ought to be demonftrated by being refolved into this fundamental axiom. But if this could be done, which I fear is not poffible, mathematical truth would not be one whit more certain than it is. Thofe other axioms produce abfolute certainty, and produce it immediately, without any procefs of thought or reafoning that we can difcover. And if the truth of a propofition be clearly and certainly perceived by all men without proof, and if no proof whatsoever could make it more clear or more cer tain, it feems captious not to allow that propofition the name of Intuitive Axiom. Others fuppofe, that though the demonstration of mathematical axioms is not ablolutely neceffary, yet that thefe axioms are fufceptible of demonftration, and ought to be demonftrated to those who require it. Dr Barrow is of this opinion. So is Apollonius; who, agreeably to it, has attempted a demonftration of this axiom, That things equal to one and the fame thing are equal to one another. But whatever ac count we make of thefe opinions, they affect not our doctrine. However far the demonftration of axioms may be carried, it muft at laft terminate in one principle of common fenfe, if not in many; which principle we must take for granted whether we will or not. the the evidence of fenfe is not more unqueftionable, than that of abstract intuitive truth; and therefore the former evidence, though to one ignorant of the meaning of the terms, it might serve to explain and illustrate the latter, can never prove it. But not to rest any thing on the fignification we affix to the word proof, and to remove every poffibility of doubt as to this matter, let us fuppofe, that the evidence of external fenfe is more unquestionable than that of abftract intuitive truth, and that every intuitive principle in mathematics may. thus be brought to the teft of fenfe; and if we cannot call the evidence of fenfe a proof, let us call it a confirmation of the abstract principle: yet what do we gain by this method of illuftration? We only difcover, that the evidence of abstract intuitive truth is refolvible into, or may be illustrated by, the evidence of fenfe. And it will be feen in the next fection, that we believe in the evidence of external fenfe, not becaufe we can prove it to be true, but because the law of our nature determines us to believe in it without proof. So that in whatever way we view this fubject, the point we propose to illustrate J H |