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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.




Of Mathematical Reafoning.


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

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inferred from premifes which render it ne¬
ceffarily true: and this perhaps is a more
perfect, or at least a fimpler, kind of
proof, than the other; but both are e-
qually convincing. The other kind is call-
ed indirect, apagogical, or ducens ad absur-
dum; and takes place when, by fuppofing
a propofition falfe, we are neceffarily led
into an abfurdity, which there is no other
way to avoid, than by fuppofing the pro-
pofition true. In this manner it is proved,
that the propofition is not, and cannot be,
falfe; or, in other words, that it is cer-
tainly true. Every step in a mathemati-
cal proof either is felf-evident, or must
have been formerly demonftrated; and e-
very demonstration doth finally refolve
itself into intuitive or felf-evident princi-
ples, which it is impoffible to prove, and
equally impoffible to disbelieve. Thefe
first principles conftitute the foundation
of mathematical fcience: if you can dif-
prove them, you overturn the whole
feience; if you refufe to believe them,
you cannot, confiftently with this refufal,
acquiefce in
any mathematical truth what--
foever. But you may as well attempt to
blow out the fun with a pair of bellows,


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as to difprove thefe principles: and if you fay, that you do not believe them *, you 01 will be charged either with falfehood or

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with folly; you may as well hold your
hand in the fire, and say that you feel no
pain. By the law of our nature, we must
feel in the one cafe, and believe in the o-
ther; even as, by the fame law, we must
adhere to the earth, and cannot possibly
fall headlong to the clouds.

But who will pretend to prove a mathe-
matical axiom, That a whole is greater
than a part, or, That things equal to one
and the fame thing are equal to one ano-
ther? Every proof must be clearer and
more evident than the thing to be proved.
Can you then affume any more evident
principle, from which the truth of these
axioms may be confequentially inferred?
It is impoffible; because they are already

* 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.


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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.


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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



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