gate the origin of those which are the most important, and of the most extenfive influence in science, and common life *; beginning with the fimpleft and clearest, and advancing gradually to those which are more complicated, or less perfpicuous.

That the induction here given is fufficiently comprehenfive, will appear from the following analyfis.

All the objects of the human understanding may be reduced to two claffes, viz. Abstract Ideas, and Things really exifting.

Of Abstract Ideas, and their Relations, all our knowledge is certain, being founded on MATHEMATICAL EVIDENCE (a); which comprehends, 1. Intuitive Evidence, and, 2. the Evidence of strict demonftration.

We judge of Things really exifting; either, 1. from our own experience; or, 2. from the experience of other men.

1. Judging of Real Existences from our own experience, we attain either Certainty or Probability: Our knowledge is certain when supported by the evidence, 1. Of SENSE EXTERNAL (b) or INTERNAL (c); 2. Of MɛMORY (d); and, 3. Of LEGITIMATE INFERENCES OF THE CAUSE FROM THE EFFECT (e). Our knowledge is probable, when, from facts already experienced, we argue, 1. to facts OF THE SAME KIND (ƒ) not experienced; and, 2. to facts OF A SIMILAR KIND (g) not experienced.

2. Judging of Real Exiftences from the experience of other men, we have the EVIDENCE OF THEIR TESTIMONY (). The mode of understanding produced by that evidence is properly called Faith; and this faith fometimes amounts to probable opinion, and sometimes rifes even to abfolute certainty.

SECTION I.

Of Mathematical Reasoning.

THE evidence that takes place in pure mathematics, produces the highest affurance and certainty in the mind of him who attends to, and understands it; for no principles are admitted into this fcience, but fuch as are either felf-evident, or fufceptible of demonftration. Should a-man refufe to affent to a demonftrated conclufion, the world would impute the refufal, either to want of understanding, or to want of honesty: for every perfon of understanding feels, that by mathematical demonftration he must be convinced whether he will or not. There are two kinds of mathematical demonftration. The first is called direct; and takes place, when a conclufion is inferred from premifes that render it neceffarily true: and this perhaps is a more perfect, or at least a fimpler, kind of proof, than the other; but both are equally convincing. The other kind is called indirect, apagogical, or ducens ad abfurdum;" and takes place, when, by fuppofing a propofition falfe, we are led into an abfurdity, which there is no other way to avoid, than by fuppofing the propofition true. In this

2

manner

manner it is proved, that the propofition is not, and cannot be, falfe; in other words, that it is a certain truth. Every step in a mathematical proof, either is felf-evident, or must have been formerly demonstrated; and every demonstration does finally refolve itself into intuitive or felf-evident principles, which it is impoffible to prove, and equally impoffible. to difbelieve. These first principles

conftitute the foundation of mathematics : if you disprove them, you overturn the whole fcience; if you refufe to believe them, you cannot, confiftently with fuch refufal, acquiefce in any mathematical truth whatfoever. But you may as well attempt to blow out the fun, as to difprove thefe principles: and if you fay, that you do not believe them *, you will be charged either with falfehood or with folly; you may as well hand in the fire, and fay that you feel no pain. By the law of our nature, we muft feel in the one cafe, and believe in the other; even as, by the fame law, we must adhere to the earth, and cannot fall headlong to the clouds.

hold

your

* Si quelque opiniaftre les nie de la voix, on ne l'en fçauriot empefcher; mais cela ne luy eft pas permis interieurement en fon efprit, parce que fa lumiere naturelle y repugne, qui eft la partie où fe 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.

But

But who will pretend to prove a mathematical axiom, That a whole is greater than a part, or, That things equal to one and the fame thing are equal to one another? Every proof must be 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 as evident as any thing can be *. You2 may bring the matter to the teft of the fenfes, by laying a few halfpence and farthings upon the table; but the evidence of fenfe is not more unquestionable, than that of abftract intuitive truth; and therefore the for

mer

* Different opinions have prevailed concerning the nature of these geometrical axioms. Some fuppose, that an axiom is not felf-evident, except it imply an identical propofition; that therefore this axiom, It is impoffible 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, 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 whatever could make it more clear or more certain, it seems captious not to allow that propofition the name of Intuitive Axiom. Others fuppofe, that though the demonftration of mathematical axioms is not abfolutely neceffary, yet that these axioms are fufceptible of demonftration, and ought to be demonftrated to thofe who require it. Dr

Barrow

mer evidence, though to one ignorant of the meaning of the terms, it might ferve 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 abstract intuitive truth, and that every intuitive principle in mathematics may thus be brought to the test of sense; 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 illustration? We only discover, that the evidence of abstract intuitive truth is refolvable into, or may be illuftrated by, the evidence of fenfe. And it will be feen in the next fection, that we believe in the evidence of external fenfe, not because 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 mean to illustrate

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 account we make of thefe opinions, they affect not our doctrine. However far the demonftration of axioms may be carried, it must at last terminate in one principle of common fenfe, if not in many; which principle we must believe without proof, whether we will or no,

appears