That the reasoning employed by Euclid in proof of the fourth proposition of his first book is completely demonstrative, will be readily granted by those who compare its different steps with the conclusions to which we were formerly led, when treating of the nature of mathematical demonstration. In none of these steps is any appeal made to facts resting on the evidence of sense, nor indeed to any facts whatever. The constant appeal is to the definition of equality."* "Let the triangle A B C (says Euclid) be applied to the triangle D E F; the point A to the point D, and the straight line A B to the straight line DE; the point B will coincide with the point E, be

matical reasoning.-"Unde meritò vir acutissimus Willebrordus Snellius luculentissimum appellat geometria supellectilis instrumentum hanc ipsam spapμcow. Eam igitur in demonstrationibus mathematicis qui fastidiunt et respuunt, ut mechanicæ crassitudinis ac avrozpysas aliquid redolentem, ipsissimam geometria basin labefactare student ; ast imprudenter et frustra. Nam apμriv geometræ suam non manu sed mente peragunt, non oculi sensu, sed animi judicio æstimant. Supponunt (id quod nulla manus præs tare, nullus sensus discernere valet) accuratam et perfectam congruentiam, ex eâque suppositâ justas et logicas eliciunt consequentias. Nullus hic regulæ, circini, vel normæ usus, nullus brachiorum labor, aut laterum contentio, rationis totum opus, artificium et machinatio est ; nil mechanicam sapiens av exigitur; nil, inquam, mechanicum, nisi quatenus omnis magnitudo sit aliquo modo materiæ involuta, sensibus exposita, visibilis et palpabilis, sic ut quod mens intelligi jubet, id manus qua dantenus exequi possit, et contemplationem praxis utcunque conetur æmulari. Quæ tamen imitatio geometricæ demonstrationis robur ac dignitatem nedum non infirmat aut deprimit, at validius constabilit, et atollit altius," &c. Lectiones Mathematica, Lect. III.

It was before observed (seep. 160) that Euclid's eighth (axiom, magnitudes which coincide with each other are equal) ought, in point of logical rigour, to have been stated in the form of a definition. In our present argument, however, it is not of material consequence whether this criticism be adopted or not. Whether we consider the proposition in question in the light of an axiom or of a definition, it is equally evident that it does not express a fact ascertained by observation or by experiment.

cause A B is equal to D E. And A B coinciding with DF, A C will coincide with DF, because the angle B A C is equal to the angle E D F." A similar remark will be found to apply to every remaining step of the reasoning; and, therefore, this reasoning possesses the peculiar characteristic which distinguishes mathematical evidence from that of all the other sciences,-that it rests wholly on hypotheses and definitions, and in no respect upon any statement of facts, true or false. The ideas, indeed, of extension of a triangle, and of equality, presuppose the exercise of our senses. Nay, the very idea of superposition involves that of motion, and consequently (as the parts of space are immoveable) of a material triangle. But where is there any thing analogous in all this, to those sensible facts, which are the principles of our reasoning in physics; and which, according as they have been accurately or inaccurately ascertained, determine the accuracy or inaccuracy of our conclusions? The material triangle itself, as conceived by the mathematician, is the object, not of sense, but of intellect. It is not an actual measure, liable to expansion or contraction, from the influence of heat or of cold; nor does it require, in the ideal use which is made of it by the student, the slightest address of hand or nicety of eye. Even in explaining the demonstration, for the first time, to a pupil, how slender soever his capacity might be, I do not believe that any teacher ever thought of illustrating its meaning by the actual application of the one triangle to the other. No teacher, at least, would do so, who had formed correct notions of the nature of mathematical science.

If the justness of these remarks be admitted, the demonsrtation in question must be allowed to be as well entitled

to the name, as any other which the mathematician can produce; for as our conclusions relative to the properties of the circle (considered in the light of hypothetical theorems) are not the less rigorously and necessarily true, that no material circle may any where exist corresponding exactly to the definition of that figure, so the proof given by Euclid of the fourth proposition, would not be the less demonstrative, although our senses were incomparably less acute than they are, and although no material triangle continued of the same magnitude for a single instant. Indeed, when we have once acquired the ideas of equality and of a common measure, our mathematical conclusions would not be in the least affected, if all the bodies in the universe should vanquish into nothing.

To many of my readers, I am perfectly aware, the foregoing remarks will be apt to appear tedious and superfluous. My only apology for the length to which they have extended is, my respect for the talents and learning of some of those writers who have lent the sanction of their authority to the logical errors which I have been endeavouring to correct; and the obvious inconsistency of these conclusions with the doctrine concerning the characteristics of mathematical or demonstrative evidence, which it was the chief object of this section to establish.*

This doctrine is concisely and clearly stated by a writer, whose acute and original, though very eccentric genius, seldom fails to redeem his wildest paradoxes by the new lights which he strikes out in defending them." "Demonstratio est syllogismus vel syllogismorum series à nominum definitionibus usque ad conclusionem ultimam derivata." Computatio sive Logica, cap. 6.

It will not, I trust, be inferred, from my having adopted, in the words of Hobbes, this detached proposition, that I am disposed to sanction any one of those conclusions which have been commonly supposed to be connected with

A a

SECTION IV.

Of our Reasonings concerning Probable or Contingent

Truths.

I.

Narrow Field of demonstrative Evidence.--Of demonstrative Evidence, when combined with that of SENSE, as in Practical Geometry; and with those of Sense and of INDUCTION, as in the Mechanical Philosophy.-Remarks on a Fundamental Law of Belief, involved in all our Reasonings concerning Contingent Truths.

Ir the account which has been given of the nature of demonstrative evidence be admitted, the province over which it extends must be limited almost entirely to the objects of pure mathematics. A science perfectly analo

it, in the mind of the author :-I say, supposed, because I am by no means satisfied, (notwithstanding the loose and unguarded manner in which he has stated some of his logical opinions) that justice has been done to his views and motives in this part of his works. My own notions on the subject of evidence in general, will be sufficiently unfolded in the progress of my speculations. In the meantime, to prevent the possibility of any misapprehension of my meaning, I think it proper once more to remark, that the definition of Hobbes, quoted above, is to be understood (according to my interpretation of it) as applying solely to the word demonstration in pure mathematics. The extension of the same term by Dr. Clarke and others, to reasonings which they have for their object, not conditional or hypothetical, but absolute truth, appears to me to have been attended with many serious inconveniencies, which these excellent authors did not foresee. Of the demonstrations with which Aristotle has attempted to fortify his syllogistic rules, 1 shall afterwards have occasion to examine the validity.

gous to this, in point of evidence, may indeed be conceived (as I have already remarked) to consist of a series of propositions relating to moral, to political, or to physical subjects, but as it could answer no other purpose than to display the ingenuity of the inventor, hardly any thing of the kind has been hitherto attempted. The only exception which I can think of, occurs in the speculations formerly mentioned under the title of theoretical mechanics.

But if the field of mathematical demonstration be limited entirely to hypothetical or conditional truths, whence (it may be asked) arises the extensive and the various utility of mathematical knowledge, in our physical researches, and in the arts of life? The answer, I apprehend, is to be found in certain peculiarities of those objects to which the suppositions of the mathematician are confined; in consequence of which peculiarities, real combinations of circumstances may fall under the examination of our senses, approximating far more nearly to what his definitions describe, than is to be expected in

The charge of unlimited scepticism brought against Hobbes, has, in my opinion, been occasioned, partly by his neglecting to draw the line between absolute and hypothetical truth, and partly by his applying the word demonstration to our reasonings in other sciences as well as in mathematics. To these causes may perhaps be added, the offence which his logical writings must have given to the Realists of his time.

It is not, however, to Realists alone, that the charge has been confined. Leibnitz himself has given some countenance to it, in a dissertation prefixed to a work of Marius Nizolius; and Brucker, in referring to this dissertation, has aggravated not a little the censure of Hobbes, which it seems to contain. "Quin si illustrem Leibnitzium audimus, Hobbesius quoque inter nominales referendus est, eam ob causam, quod ipso Occamo nominalior, rerum veritatem dicat in nominibus consistere, ac, quod, majus est, pendere ab arbitrio humano." Histor. Philosoph. de Ideis, p. 209. Augustæ Vindelicorum, 1723.