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 DEF; 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, 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 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 respeet upon any statement of facts, true or false. The ideas indeed

mentum hanc ipsam εφάρμοσιν. Eam igitur in demonstrationibus mathematicis qui fastidiunt et respuunt, ut mechanica crassitudinis ac aurougyías aliquid redolentem, ipsissimam geometria basin labefactare student; ast imprudenter et frustra. Nam ipúguor 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 hic regulæ, circini, vel normæ usus, nullus brachiorum labor, aut laterum contentio, rationis totum opus, artificium et 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 atollit altius," &c.-Lectiones Mathematica, lect. iii.

1 It was before observed, (see p. 126,) 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.

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 immovable,) of a material triangle. But where is there anything 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 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 anywhere 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 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.1

SECTION IV.-OF OUR REASONINGS CONCERNING PROBABLE OR CONTINGENT TRUTHS.

[SUBSECTION] 1.-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, [Expectation of the Constancy of Nature,] involved in all our Reasonings concerning Contingent Truths.

If 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 physical subjects; but as it could answer no other purpose than to display the ingenuity of the inventor, hardly anything 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 any other theoretical process of the human mind. Hence a corresponding coincidence between his abstract conclusions, and those facts in practical geometry and in physics which they help him to ascertain.

For the more complete illustration of this subject, it may be

observed, in the first place, that although the peculiar force of that reasoning which is properly called mathematical, depends on the circumstance of its principles being hypothetical, yet if, in any instance, the supposition could be ascertained as actually existing, the conclusion might, with the very same certainty, be applied. If I were satisfied, for example, that in a particular circle drawn on paper, all the radii were exactly equal, every property which Euclid has demonstrated of that curve. might be confidently affirmed to belong to this diagram. As the thing however here supposed is rendered impossible by the imperfection of our senses, the truths of geometry can never, in their practical applications, possess demonstrative evidence; but only that kind of evidence which our organs of perception enable us to obtain.

But although, in the practical applications of mathematics, the evidence of our conclusions differs essentially from that which belongs to the truths investigated in the theory, it does not therefore follow that these conclusions are the less important. In proportion to the accuracy of our data will be that of all our subsequent deductions; and it fortunately happens, that the same imperfections of sense which limit what is physically attainable in the former, limit also, to the very same extent, what is practically useful in the latter. The astonishing precision which the mechanical ingenuity of modern times has given to mathematical instruments has, in fact, communicated a nicety to the results of practical geometry, beyond the ordinary demands of human life, and far beyond the most sanguine anticipations of our forefathers.1

1 See a very interesting and able article, in the fifth volume of the Edin. burgh Review, on Colonel Mudge's account of the operations carried on for accomplishing a trigonometrical survey of England and Wales. I cannot deny myself the pleasure of quoting a few sentences.

"In two distances that were deduced from sets of triangles, the one measured by General Roy in 1787, the other by

Major Mudge in 1794, one of 24.133 miles, and the other of 38.688, the two measures agree within a foot as to the first distance, and sixteen inches as to the second. Such an agreement, where the observers and the instruments were both different, where the lines measured were of such extent, and deduced from such a variety of data, is probably without any other example, Coincidences of this sort are frequent in the trigonome