mental knowledge, by infufing into it a diviner fpark from Ontology? Whatever be his purfuit, the ambition of that scholar must be very low in its aims, who should not aspire to catch one glimpse of the pure effences of things, as they are presented by the mirror of metaphyfics. But if it should appear, that the ontologists have mistaken the humble pofleriori for the high priori road; that they are juft as dependant upon fense and matter, as the mereft experimenter; that they have laboured to no better purpose than to cloak the fimpleft indications of fenfe in a fantaftic garb; and that even the claim of their fubtleties to ferve as a whetstone of a finer grain to the mind is groundless, fince the habit of difcrimination, as well as that of fixed attention, is to be perfectly acquired by studies, that are alfo capable of useful application; if for fuch reafons as thefe we may calmly be

hold

hold Ontology finking into that grave, in which Alchemy lies buried, ftill there will remain an abftract fcience, which has ftood the competition of the physical sciences, and indeed has grown with their growth. In mathematical reafoning, the mind grafps the conclufions with full affurance of their reality; we are fatisfied, that our advances in this fcience are actual acquifitions, and we find them as we go on continually capable of application.

It may therefore be interefting to enquire into those circumstances which conftitute the irrefiftable force of mathematical evidence. We fhall at the fame time, if we are fuccessful in this enquiry, difcover upon what depends the difference in the cogency of proof between demonftrative evidence, and fuch evidence as lefs powerfully commands our affent. Without this, I do not fee how we can ever

take

take a clear furvey of evidence in general, or enjoy the fatisfaction of accounting to ourfelves fully for our own conviction or belief.

It seems to me, in the present state of our knowledge, fo easy to point out the nature of this and the other forts of evidence, that I wonder how it can be miftaken. Yet frequently as the topic is expatiated upon, I know no book in which the true principles have been fully explained and applied; and in general, I have reafon to believe that very erroneous ideas prevail upon a subject, of unquestionable importance to the theory of the human understanding. I might recite the opinions of a confiderable number of writers, and offer arguments against them. But if I fucceed in establishing my own, I fhall at the fame time fufficiently refute what I imagine to be the mistakes of others; and the reader will

at

at once perceive how far each is wide of the truth, for all are not equally wide.

On examining a train of mathematical reafoning, we shall find, that at every step we proceed upon the evidence of the fenfes; or, to exprefs myself in different terms, I hope to be able to fhew that the mathema

1

tical Sciences are fciences of experiment and obfervation, founded folely upon the induction of particular facts, as much fo as mechanics, aftronomy, optics or chemistry. In the kind of evidence there is no difference; for it originates from perception in all these cafes alike, but mathematical experiments are more fimple and more perfectly within the grafp of our fenfes, and our perceptions of mathematical objects are clearer. So great indeed is the fimplicity of mathematical experiments, that at whatever moment we are called upon to reafon from them, we have the refult of many of them distinctly

in

in our memory; the obfervations cafually made in the courfe of life, leave a fufficient conviction of their truth upon the mind; and we are beforehand fo fully fatisfied as feldom to take the trouble of repeating them. The apparatus is fimple: no motion or change admonishes us, that we are engaged in an experimental enquiry; and this is, I fuppofe, the reafon why we are fo little aware of the nature of the intellectual procefs we are going through. Sometimes, however, notwithstanding we are fo well prepared, we do repeat some of these experiments; and there have probably been few teachers of geometry, who have not, at the beginning of their lectures, desired their pupils to repeat certain fundamental experiments, till they should have fatisfied themselves as to the refult.

No fooner do we look into an elementary treatise for the proofs of this opinion,

than