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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
son piéd 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." I
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 geometriae supellectilis instrumentum hanc ipsam sapucov. Eam igitur in demonstrationibus mathematicis qui fastidiunt et respuunt, ut mechanicae crassitudinis ac avrovpyias aliquid redolentem, ipsissimam geometriae basin labefactare student; ast imprudenter et frustra Nam μ geometrae suam non manu sed mente peragunt, non oculi sensu, sed animi judicio aestimant. Supponunt (id quod nulla manus praestare, nullus sensus discernere valet) accuratam et perfectam congruentiam, ex eâque suppositâ justas et logicas eliciunt consequentias Nullus hic regu lae, circini, vel normae usus, nullus brachiorum labor, aut laterum contentio, rationis to tum opus, artificium et machinatio est ; nil mechanicam sapiens autoupav exigitur; nil, inquam, mechanicum, nisi quatenus omnis magnitudo sit aliquo modo materiae involuta, sensibus exposita, visibilis et palpabilis, sic ut quod mens intelligi jubet, id manus quadantenus exequi possit, et contemplationem praxis utcunque conetur aemulari. Quae tamen imitatio geometricae demonstrationis robur ac dignitatem nedum non infirmat aut deprimit, at validius constabilit, et attollit altius," &c.—Lectiones Mathematicae, Lec. III.
[The principle of superposition is not, as many geometers suppose, a method of demonstration unsatisfactory and purely mechanical. Superposition, as understood by mathematicians, does not consist in applying coarsely one figure upon another, in order to judge by the eye of their equality or difference, as a labourer applies his rule to measure a line; it consists in imagining a figure transported and placed upon another, and in concluding from a supposed equality of certain parts of the two figures, the coincidence of the parts between them, and from their coincidence, the coincidence of the rest; whence results the perfect equality and resemblance of the two figures.]
[Hence the very acute W. Snellius calls this adaptation (superposition) the most beau tiful instrument in the laboratory of the geometer. They, therefore, who in the mathematical demonstrations reject and despise it, as partaking of mechanical coarseness and handicraft labour, strive, though foolishly and ineffectually, to subvert the very basis of geometry. For geometers perform this adaptation not by the hand, but the understanding, and measure not by the sense of the eye, but by the judgment of the intellect. They suppose, (what no hand can effect, no sense discern) an accurate and perfect coincidence, and from this supposition, they elicit consequences just and logical. Here are no scale, square and compasses, no labour of the hands or exercise of the muscles, the whole is the work, the artifice, and contrivance of reason. Nothing is required which has any resem blance to mechanical labour, nothing I say, mechanical, unless so far as all magnitude is in some sense involved in matter, exposed to the senses, visible and palpable; so that what the mind directs, that the hands in some sort perform, and action strives as far as possible to equal contemplation. Which imitation, so far from weakening or depressing the strength and dignity of geometrical demonstration, on the contrary, renders it more firm and elevated.]
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 "AB to the straight line DE; the point B will coincide with the "point E, because AB 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 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 mathe matical evidence from that of all the other sciences,-that it rests wholly on hypotheses and definitions, and in no respect upon any statements 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 slighest 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.
It was before observed (see p. 96) 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 ques tion 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.
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 errours 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.*
OF OUR REASONINGS CONCERNING PROBABLE OR CONTINGENT TRUTHS.
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.
Jemonstra. Ir the account which has been given of the nature of demonstraevid: limit. tive evidence be admitted, the province over which it extends must
ed to Math:
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 syllo gismorum 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 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 mathemat ics. 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 scien ces 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 him. self 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 cen
[Demonstration is a syllogism or a series of syllogisms, carried on from the definitions of words to the final conclusion.]
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.
But if the field of mathematical demonstration be limited entirely when 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.
7 utility of
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 demon- demonstra. strated of that curve might be confidently affirmed to belong to this tive evid diagram. As the thing, however, here supposed, is rendered impossible by the imperfection of our senses, the truths of geometry not found. can never, in their practical applications, possess demonstrative evi-in parti. dence; 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
sure 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 bumano." Histor. Philosoph. de Ideis, p. 209. Augustae Vindelicorum, 1723.
•[If we listen to the illustrious Leibnitz, Hobbes also is to be ranked among the nomin alists; for being more a nominalist than even Occam himself, he maintains that all truth consists in words, and still more, that it depends upon human judgment.]
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 un-modern times has given to mathematical instruments, has in fact, provement 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.*
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 mensurabilityľ which belongs to all of them) of adjusting, with a degree of accura Great ap. cy approximating nearly to the truth, the data from which we are "prosemato reason in our practical operations, to those which are assumed in tion to de our theory. The only affections of matter which these objects commonstrative prehend are extension and figure; affections which matter possesses certainty in common with space, and which may therefore be separated in 7 practi-fact, as well as abstracted in thought, from all its other sensible qualities. In examining, accordingly, the relations of quantity connected cal applica. with these affections, we are not liable to be disturbed by those tion of moth 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 narrow limits; and as there is nothing but the incorrectness of our data by which the result can be vitiated, the limits of possible errour may themselves be assigned. But, in the simplest applications of mathematics to me
physics. & mechanis
*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 sen
"In two distances that were deduced from sets of triangles, the one measured by ge neral 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 16 inches as to the second. Such an agreement, where the observers and the instruments were both dif ferent, 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 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 mathemati cal 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 bad 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."
+ See note (G.)