to his Elements of Geometry, a scholium concerning the final causes of circles and of straight lines, similar to that which, with such sublime effect, closes the Principia of Sir Isaac Newton.*

2. It yet remains for me to say a few words upon that superposition of triangles which is the ground-work of all our geometrical reasonings concerning the relations which different spaces bear to one another in respect of magnitude. And here I must take the liberty to remark, in the first place, that the fact in question has been stated in terms much too loose and incorrect for a logical argument. When it is said, that "all the fundamental theorems which relate to the comparison of triangles, derive their evidence from the mere superposition of the triangles themselves," it seems difficult, or rather impos

*In the course of my own experience, I have met with one person, of no common ingenuity, who seemed seriously disposed to consider the truths of geometry very nearly in this light. The person I allude to was James Ferguson, author of the justly popular works on Astronomy and Mechanics. In the year 1768, he paid a visit to Edinburgh, when I had not only an opportunity of attending his public course of lectures, but of frequently enjoying, in private, the pleasure of his very interesting conversation. I remember distinctly to have heard him say, that he had more than once attempted to study the Elements of Euclid; but found himself quite unable to enter into that species of reasoning. The second proposition of the first book, he mentioned particularly as one of his stumbling-blocks at the very outset ;-the circuitous process by which Euclid sets about an operation which never could puzzle, for a single moment, any man who had seen a pair of compasses, appearing to him altogether capricious and ludicrous. He added, at the same time, that as there were various geometrical theorems of which he had daily occasion to make use, he had satisfied himself of their truth, either by means of his compasses and scale, or by some mechanical contrivances of his own invention. Of one of these I have still a perfect recollection;-his mechanical or experimental demonstration of the 47th proposition of Euclid's first book, by cutting a card so as to afford an ocular proof, that the squares of the two sides actually filled the same space with the square of the hypothenuse.

To those who reflect on the disadvantages under which Mr. Ferguson had labored in point of education, and on the early and exclusive hold which experimental science had taken of his mind, it will not perhaps seem altogether unaccountable, that the refined and scrupulous logic of Euclid should have struck him as tedious, and even unsatisfactory, in comparison of that more summary and palpable evidence on which his judgment was accustomed to rest. Considering, however, the great number of years which have elapsed since this conversation took place, I should have hesitated about recording, solely on my own testimony, a fact so singular with respect to so distinguished a man, if I had not lately found, from Dr. Hutton's Mathematical Dictionary, that he also had heard from Mr. Ferguson's mouth, the most important of those particulars which I have now stated; and of which my own recollection is probably the more lively and circumstantial, in consequence of the very early period of my life when they fell under my notice.

"Mr. Ferguson's general mathematical knowledge," says Dr. Hutton, "was little or nothing. Of algebra, he understood little more than the notation; and he has often told me he could never demonstrate one proposition in Euclid's Elements; his conmethod being to satisfy himself, as to the truth of any problem, with a measurepale and compasses."-Hutton's Mathematical and Philosophical Dictione Ferguson.

sible, to annex to the adjective mere, an idea at all different from what would be conveyed, if the word actual were to be substituted in its place; more especially, when we attend to the assertion which immediately follows, that "this mode of proof is, in reality, nothing but an ultimate appeal, though of the easiest and most familiar kind, to external observation." But if this be, in truth, the sense in which we are to interpret the statement quoted above, (and I cannot conceive any other interpretation of which it admits,) it must appear obvious, upon the slightest reflection, that the statement proceeds upon a total misapprehension of the principle of superposition; inasmuch as it is not to an actual or mere superposition, but to an imaginary or ideal one, that any appeal is ever made by the geometer. Between these two modes of proof, the difference is not only wide, but radical and essential. The one would, indeed, level geometry with physics, in point of evidence, by building the whole of its reasonings on a fact ascertained by mechanical measurement: The other is addressed to the understanding, and to the understanding alone, and is as rigorously conclusive as it is possible for demonstration to be.*

* The same remark was, more than fifty years ago, made by D'Alembert, in reply to some mathematicians on the Continent, who, it would appear, had then adopted a paradox very nearly approaching to that which I am now combating, "Le principe de la superposition n'est point, comme l'ont prétendu plusieurs géomètres, une méthode de démontrer peu exacte et purement mécanique. La superposition, telle que les mathématiciens la conçoivent, ne consiste pas à appliquer grossièrement une figure sur une autre, pour juger par les yeux de leur égalité ou de leur différence, comme un ouvrier applique son pié sur une ligne pour la mesurer; elle consiste à imaginer une figure transportée sur une autre, et à conclure de l'égalité supposée de certaines parties de deux figures, la coincidence de ces parties entr'elles, et de leur coincidence la coincidence du reste: d'où résulte l'égalité et la similitude parfaites des figures entières."

About a century before the time when D'Alembert wrote these observations, a similar view of the subject was taken by Dr. Barrow, a writer who, like D'Alembert, added to the skill and originality of an inventive mathematician, the most refined, and, at the same time, the justest ideas concerning the theory of those intellectual processes which are subservient to mathematical reasoning.-"Unde meritò vir acutissimus Willebrordus Snellius luculentissimum appellat geometria supellectilis instrumentum hanc ipsam ipaguori. Eam igitur in demonstrationibus mathematicis qui fastidiunt et respuunt, ut mechanica crassitudinis ac avrovgyías aliquid redolentem, ipsissimam geometriæ basin labefactare student; ast imprudenter et frustra. Nam qáguori geometræ suam non manu sed mente peragunt, non oculi sensu, sed animi judicio æstimant. Supponunt (id quod nulla manus præstare, nullus sensus discernere valet) accuratam et perfectam congruentiam, ex eâque suppositâ justas et logicas eliciunt consequentias. Nullus hîc regula, circini, vel normæ usus, nullus brachiorum labor, aut laterum contentio, rationis totum opus, artificium et

That the resoning 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 ABC," says Euclid, "be applied to the triangle DEF; the point A to the point D, and the straight line A B to the straight line D E; the point B will coincide with the point E, because A B is equal to D E. And A B coinciding with D E, A C will coincide with D F, because the angle B A C is equal to the angle ED F." A similar remark will be found to apply to every remaining step of the reasoning; and, therefore, this resoning possesses the peculiar characteristic which distinguishes mathematical evidence from that af 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

machinatio est; nil mechanicam sapiens aurougyíav 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 quadantenus exequi possit, et contemplationem praxis utcunque conetur æmulari. Quæ tamen imitatio geometrica demonstrationis robur ac dignitatem nedum non infirmat aut deprimit, at validius constabilit, et attollit altius," &c.-Lectiones Mathematica, Lect. III.

It was before observed (see p. 119) that Euclid's eighth axiom (magnitudes which coincide with each other are equal) ought, in point of logical rigor, to have been stated in the form of a definition. In our present argument, however, it is not material consequence whether this criticism be adopted or not. Whether we

r the proposition in question in the light of an axiom or of a definition, it is evident that it does not express a fact ascertained by observation or by experi

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 this 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 demonstration 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 vanish into nothing.

To many of my readers, I am perfectly aware, the foregoing remarks will be apt to appear tedious and superfluous. My only apolygy 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 origin

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 analogous 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

al, 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 a 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 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 mean time 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 have for their object, not conditional or hypothetical, but absolute truth, appears to me to have been attended with many serious inconveniences, which these excellent authors did not foresee. Of the demonstrations with which Aristotle has attempted to fortify his syllogistic rules, I shall afterwards have occasion to examine the validity. 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. Philosph. de Ideis, p. 209. Augustæ Vindelicorum, 1723.