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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, 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 sci

posed, an inexact and purely mechanical mode of demonstration. Superposition in mathematics does not consist in applying one figure to the other, in order to judge by the eye whether they differ or coincide, just as a workman applies his foot-rule to a line in order to measure it; it consists in imagining one figure placed over the other, and concluding, from the supposed equality of certain parts of the two figures, the coincidence of these parts with each other, and from their coincidence inferring the coincidence of the other parts; whence results the perfect equality and similitude of the whole figures."]

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.

*It was before observed, 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 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.

ences, 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 immovable) 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 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 exists 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.

IV. OF OUR REASONINGS CONCERNING

PROBABLE OR

CONTINGENT TRUTHS.

1. Narrow field of demonstrative evidence. 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 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.

On the application of mathematics in practical geometry and physics. 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 èvidence 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.*

* See a very interesting and able article, in the fifth volume of the Edinburgh 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 agreed 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 trigonometrical survey, and prove how much more good instruments, used by skilful and attentive observers, are capable of performing, than the most sanguine theorist could have ever ventured to foretell.

"It is curious to compare the early essays of practical geometry with the perfections to which its operations have now reached, and to consider that, while the artist had made so little progress, the theorist had reached

This remarkable, and indeed singular coincidence of propositions purely hypothetical, with facts which fall under the examination of our senses, is owing, as I already hinted, to the peculiar nature of the objects about which mathematics is conversant; and to the opportunity which we have (in consequence of that mensurability,* which belongs to all of them) of adjusting, with a degree of accuracy approximating nearly to the truth, the data from which we are to reason in our practical operations, to those which are assumed in our theory. The only affections of matter which these objects comprehend are extension and figure; affections which matter possesses in common with space, and which may therefore be separated in fact, as well as abstracted in thought, from all its other sensible qualities. In examining, accordingly, the relations of quantity con

many of the sublimest heights of mathematical speculation; that the lat ter had found out the area of the circle, and calculated its circumference to more than a hundred places of decimals, when the former could hardly divide an arc into minutes of a degree; and that many excellent treatises had been written on the properties of curve lines, before a straight line of considerable length had ever been carefully drawn, or exactly measured on the surface of the earth."

* In an Essay on Quantity, by Dr. Reid, published in the transactions of the Royal Society of London, for the year 1748, mathematics is very correctly defined to be "the doctrine of measure." "The object of this science," the author observes, "is commonly said to be quantity; in which case, quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which has led some persons to apply mathematical reasoning to subjects that do not admit of it." The appropriate objects of this science are therefore such things alone as admit, not only of being increased and diminished, but of being multiplied and divided. In other words, the common quality which characterizes all of them is their mensurability.

In the same essay, Dr. Reid has illustrated, with much ingenuity, a distinction (hinted at by Aristotle) of quantity into proper and improper. "I call that," says he, “proper quantity, which is measured by its own kind; or which, of its own nature, is capable of being doubled or trebled, without taking in any quantity of a different kind as a measure of it. Thus a line is measured by known lines, as inches, feet, or miles; and the length of a foot being known, there can be no question about the length of two