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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.

If the acc int 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 of title 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.

mind of the author : 1 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 suffi. ciently 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 ab. solute 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. Philosoph. de Ideis, p. 209. Augustæ Vindelicorum, 1723.

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 I 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 subscquent 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 hy Major Mudge in 1794, one of 24,133 miles, and the other of 38,689, the two measures agreed within a foot as to the first distance, and 16 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 observe ers, are capable of performing, than the most sanguine theorist could have ever ventured to foretell.

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, *

“ It is curious to compare the early essays of practical geometry with the perfection to which its operations have now reached, and to consider that, while the artist had made so little progress, the theorist had reached many of the sublimest heights of mathematical speculation ; that the latter 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 arch 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 inay 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 Aristotlet) 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 feet, or of any part or multiple of a foot. This known length, by being multiplied or divided, is sufficient to give us a distinct idea of any length whatsoever.

“ Improper quantity is that which cannot be measured by its own kind, but to which we assign a measure in some proper quantity that is related to it. Thus velocity of motion, when we consider it by itself, cannot be measured. We may perceive one body to move faster, another slower, but we can perceive no proportion or ratio between their velocities, without taking in some quantity of another kind to measure them by. Having therefore observed, that by a greater velocity, a greater space is passed over in the same time, by a less velocity a less space, and by an equal velocity an equal space; we hence learn to measure velocity by the space passed over in a given time, and to reckon it to be in exact proportion to that; and having once assigned this measure to it, we can then, and not till then, conceive one velocity exactly double, or triple, or in any proportion to another. We can then introduce it into mathematical reasoning, without danger of error or confusion; and may use it as a measure of other improper quantities.

“ All the proper quantities we know may, I think, be reduced to these four : extension, duration, number, and proportion.

“ Velocity, the quantity of motion, density, elasticity, the vis insita and impressa, the various kinds of centripetal forces, and the different orders of Auxions, are all improper quantities; which, therefore, ought not to be admitted into mathematical reasoning, without having a measure of them assigned.

“ The measure of an improper quantity ought always to be included in the definition of it; for it is the giving it a measure that makes it a proper subject of mathematical reasoning. If all mathematicians had considered this, as carefully as Sir Isaac Newton has done, some trouble had been saved both to themselves and

* In this remark, Dr. Reid, as appears from the title of his paper, had an eye to the abuse of mathematical language by Dr. Hutcheson, who had recently carried it so far as to exhibit algebraical formulas for ascertaining the moral merit or demerit of particular actions.-See his inquiry into the Original of our Ideas of Beauty and Virtue.

+ Κυριως δε Ποσα ταυτα λεγεται μονα, τα δε αλλα παντα κατα συμβεβηκος εις ταιτα γαρ αποβλεποντες, και τα αλλα Ποσα λεγομεν.-Arist. Categ cap. vi. 17.

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 connected with these affections, we are not liable to be disturbed by those physical accidents, which, in the other applications of mathematical science, necessarily render the result, more or less, at variance with the theory. In measuring the height of a mountain, or in the survey of a country, if we are at due pains in ascertaining our data, and if we reason from them with mathematical strictness, the result may be depended on as accurate within very rrow limits; and as there is nothing but the incorrectness of our data by which the result can be vitiated, the limits of possible error may themselves be assigned. But, in the simplest applications of mathematics to mechanics or to physics, the abstractions which are necessary in the theory, must always leave out circumstances which are essentially connected with the effect. In demonstrating, for example, the property of the lever, we abstract entirely from its own weight, and consider it as an inflexible mathematical line :suppositions with which the fact cannot possibly correspond; and for which, of course, allowances (which nothing but physical experience can enable us to judge of,) must be made in practice.*

their readers. That great man, whose clear and comprehensive understanding appears even in his definitions, having frequent occasion to treat of such improper quantities, never fails to define them, so as to give a measure of them, either in proper quantities, or such as had known a measure. See the definitions prefixed to his Principia.'

With these important remarks I entirely agree, excepting only the enumeration here given of the different kinds of proper quantity, which is liable to obvious and insurmountable objections. It appears to me that, according to Reid's own definition, extension is the only proper quantity within the circle of our knowledge. Duration is manifestly not measured by duration, in the same manner as a line is measured by a line ; but by some regulated motion, as that of the hand of a clock, or of the shadow on a sun-dial. In this respect it is precisely on the same footing with velocities and forces, all of them being measured, in the last result, by extension. As to number and proportion, it might be easily shown, that neither of them falls under the definition of quantity, in any sense of that word. In proof of this assertion, which may at first sight seem somewhat paradoxical, I have only to refer to the inathematical lectures of Dr. Barrow, and to some very judicious observations introduced by Dr. Clarke in his controversy with Leibnitz. It is remarkable that, at the period when this essay was written, Dr. Reid should have been unacquainted with the speculations of these illustrious men on the same subject; but this detracts little from the merits of his memoir, which rest chiefly on the strictures it contains on the controversy between the Newtonians and Leibnitzians concerning the measure of forces.

* The following view of the relation between the theorems of pure geometry and their practical applications strikes me as singularly happy and luminous ; more especially the ingenious illustrations borrowed from the science of geometry itself.

“ Les vérités que la géométrie demontre sur l'étendue, sont des vérités purement hypothétiques. Ces vérités cependant n'en sont pas moins utiles, eu égard aux conséquences pratiques qui en risultent. Il est aisé de le faire sentir par une

Next to practical geometry, properly so called, one of the easiest applications of mathematical theory occurs in those branches of optics which are distinguished by the name of catoptrics and dioptrics. In these, the physical principles from which we reason are few and precisely definite, and the rest of the process is as purely geometrical as the Elements of Euclid.

In that part of astronomy, too, which relates solely to the phenomena, without any consideration of physical causes, our reasonings are purely geometrical. The data, indeed, on which we proceed must have been previously ascertained by observation; but the inferences we draw from these are connected with them by mathematical demonstration, and are accessible to all who are acquainted with the theory of spherics.

In physical astronomy, the law of gravitation becomes also a principle or datum in our reasonings; but as in the celestial phenomena it is disengaged from the effects of the various other causes which are combined with it near the surface of our planet, this branch of physics, as it is of all the most sublime and comprehensive in its objects, so it seems, in a greater degree than any other, to open a fair and advantageous field for mathematical ingenuity.

In the instances which have been last mentioned the evidence of our conclusions resolves ultimately not only in that of sense, but into another law of belief formerly mentioned ; that which leads us to expect the continuance, in future, of the established order of physical phenomena. A very striking illustration of this presents itself in the computations of the astronomer; on the faith of which he predicts, with the most perfect assurance, many centuries before they happen, the appearances which the heavenly bodies are to exhibit. The same fact is assumed in all our conclusions in natural comparaison tirée de la géométrie même. On connoit dans cette science des lignes courbes qui doivent s'approacher continuellement d'une ligne droite, sans la rencontrer jamais, et qui néanmoins, étant tracées sur le papier, se confondent sensiblement avec cette ligne droite au bout d'un assez petit espace. Il en est de même des propositions de géométrie ; elles sont la limite intelleciuelle des vérités physiques, le terme dont celles-ci peuvent approcher aussi près qu'on le desire, sans jainais y arriver exactement. Mais si les théorèms mathématiques n'ont pas rigoureusement lieu dans la nature, ils servent du moins à résoudre, avec une précision suffisante pour la pratique, les differentes questions qu'on peut se proposer sur l'étendue. Dans l'univers il n'y a point de cercle parfait ; mais plus un cercle approchera de l'etre, plus il approchera des propriétés rigoureuses du cercle parfait que la géométrie demontre; et il peut en approcher à un degré suffisant pour notre usage. Il en est de même des autres figures dont la geometrie détaille les propriétés. Pour démontrer en toute rigueur, les vérités relatives à la figure des corps, on est obligé de supposer dans cette figure une perfection arbitraire qui n'y sauroit être. En effet, si le cercle, par exemple, n'est pas supposé rigoureux, il faudra autant de théoremes differens sur le cercle qu'on imaginera de figures differentes plus ou moins approchantes du cercle parfait; et ces figures ellesinémes pourront encore ètre absolument hypothétiques, et n'avoir point de modèle existant dans la nature. Les lignes qu'on considère dans la geometrie usuelle, ne sont ni parfaitement droites, ni partaitement courbes ; les surfaces ne dont ni parfaitement planes, ni parfaitement curvilignes; mais il est nécessaire de les supposes telles, pour arriver à des vérités fixes et déterminées, dont on puisse faire ensuite l'application plus ou moins exacte aux lignes et aux surfaces physiques.” -D'Alembert, Elémens de Philosophie, Article Geométrie.

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