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sics. "The mathematical sciences (says Dr. Beddoes) are sciences "of experiment and observation, founded solely on the induction of "particular facts; as much so as mechanics, astronomy, optics, or "chemistry. In the kind of evidence there is no difference; for it "originates from perception in all these cases alike; but mathema"tical experiments are more simple, and more perfectly within the grasp of our senses, and our perceptions of mathematical objects "are clearer."*
A doctrine essentially the same, though expressed in terms not quite so revolting, has been lately sanctioned by Mr. Leslie; and it is to his view of the argument that I mean to confine my attention at present. "The whole structure of geometry (he remarks) is "grounded on the simple comparison of triangles; and all the fun"damental theorems which relate to this comparison, derive their "evidence from the mere superposition of the triangles themselves; "a mode of proof which, in reality, is nothing but an ultimate ap"peal, though of the easiest and most familiar kind, to external ob"servation." And, in another passage: "Geometry, like the other
Into this train of thinking, Dr Beddoes informs us, he was first led by Mr. Horne Tooke's speculations concerning language. "In whatever study you are engaged, to leave difficulties behind is distressing: and when these difficulties occur at your very entrance upon a science, professing to be so clear and certain as geometry, your feelings become still more uncomfortable; and you are dissatisfied with your own powers of comprehen sion. I therefore think it due to the author of EMEA ПITEPOENTA, to acknowledge my obligations to him for relieving me from this sort of distress For although I had of ten made the attempt, I could never solve certain difficulties in Euclid, till my reflections were revived and assisted by Mr. Tooke's discoveries." (See Observations on the Nature of Demonstrative Evidence, London, 1793, pp. 5, and 15.)
+ Elements of Geometry and of Geometrical Analysis, &c. By Mr. Leslie. Edinburgh, 1809.
The assertion that the whole structure of geometry is founded on the comparison of triangles, is expressed in terms too unqualified. D'Alembert has mentioned another principle as not less fundamental, the measurement of angles by circular arches. "Les propositions fondamentales de géométrie peuvent être réduites à deux; la mesure des angles par les arcs de cercle, et le principe de la superposition."‡ (Elémens des Philosophie, Art. Géométrie.) The same writer, however, justly observes, in another part of his works, that the measure of angles by circular arcs, is itself dependent on the principle of super position; and that, consequently, however extensive and important in its application, it is entitled only to rank with what he calls principles of a second order. "La mesure des angles par les arcs de cercle décrit de leur sommet, est elle-même dépendante du principe de la superposition Car quand on dit que la mesure d'un angle est l'arc circulaire décrit de sommet, on vent dire que si deux angles sont égaux, les angles décrits de leur sommet à même rayon seront égaux; vérité qui se démontre par le principe de la superposition, comme tout géometre tant soit peu initié dans cette science le sentira facilement."|| (Eclaircissemens sur les Elémens de Philosophie, § IV.)
Instead, therefore, of saying, that the whole structure of geometry is grounded on the comparison of triangles, it would be more correct to say, that it is grounded on the prin ciple of superposition.
[The fundamental propositions of Geometry may be reduced to two; the measure of angles by circular ares, and the principle of superposition.]
[The measure of angles by arcs of a circle described above them, is itself dependent upon the principle of superposition. For when we say that the measure of an angle is an arc of a circle described above, we mean to say, that if the two angles are equal, the arcs described above them with the same radius will be equal; a truth which is demonstrated by the principle of superposition, as every geometer ever so little initiated in the science, will readily perceive.]
"sciences which are not concerned about the operations of mind,
Before proceeding to make any remarks on this theory, it is proThis doctrine per to premise, that it involves two separate considerations, which nvolve 2 it is of material consequence to distinguish from each other. The first is, that extension and figure (the subjects of geometry) are quaConsidera-lities of body which are made known to us by our external senses alone, and which actually fall under the consideration of the natural philosopher, as well as of the mathematician. The second, that the whole fabric of geometrical science rests on the comparison of triangles, in forming which comparison, we are ultimately obliged to appeal (in the same manner as in establishing the first principles of physics) to a sensible and experimental proof.
1. In answer to the first of these allegations, it might perhaps be sufficient to observe, that in order to identify two sciences, it is not sume enough to state, that they are both conversant about the same objects; it is necessary farther to show, that, in both cases, these objects are considered in the same point of view, and give employment to the same faculties of the mind. er, and the botanist, are all occupied in various degrees and modes, The poet, the painter, the gardenwith the study of the vegetable kingdom: yet who has ever thought of confounding their several pursuits under one common name? The natural historian, the civil historian, the moralist, the logician, the dramatist, and the statesman, are all engaged in the study of man, and of the principles of human nature; yet how widely discriminated are these various departments of science and of art! how different are the kinds of evidence on which they respectively rest! how different the intellectual habits which they have a tendency to form! Indeed, if this mode of generalization were to be admitted as legitimate, it would lead us to blend all the objects of science into one and the same mass; in as much as it is by the same impressions on our external senses, that our intellectual faculties are, in the first instance, roused to action, and all the first elements of our knowledge unfolded.
In the instance, however, before us, there is a very remarkable marka. a-specialty, or rather singularity, which renders the attempt to idenspecial-tify the objects of geometrical and of physical science, incomparamore illogical than it would be to classify poetry with botany, or but the natural history of man with the political history of nations. This specialty arises from certain peculiarities in the metaphysical nature of those sensible qualities which fall under the consideration of the geometer; and which led me, in a different work, to distinguish them from other sensible qualities (both primary and seconda
*Elements of Geometry and of Geometrical Analysis, p. 453.
ry,) by bestowing on them the title of mathematical affections of =
Nor is this the only case in which the mind forms conclusions concerning space, to which those of the natural philosopher do not bear the remotest analogy. Is it from experience we learn that space is infinite? or (to express myself in more unexceptionable terms,) that no limits can be assigned to its immensity? Here is a fact, extending not only beyond the reach of our personal observation, but beyond the observation of all created beings; and a fact on which we pronounce with no less confidence, when in imagination we transport ourselves to the utmost verge of the material universe, than when we confine our thoughts to those regions of the globe which have been explored by travellers. How unlike those general laws which we investigate in physics, and which, how far soever we may find them to reach, may still, for any thing we are able to discover to the contrary, be only contingent, local, and temporary!
It must indeed be owned, with respect to the conclusions hitherto mentioned on the subject of space, that they are rather of a metaphysical than a mathematical nature; but are not, on that account the less applicable to our purpose; for if the theory of Beddoes had any foundation, it would lead us to identify with physics the former of these sciences as well as the latter; at least, all that part of the former which is employed about space, or extension,—a favourite object of metaphysical as well as of mathematical speculation. The truth, however, is, that some of our metaphysical conclusions concerning space are more nearly allied to geometrical theorems than we might be disposed at first to apprehend; being involved or implied in the most simple and fundamental propositions which occur in Euclid's Elements. When it is asserted, for example, that "if one straight line falls on two other straight lines, so as to make "the two interior angles on the same side together equal to two
Philosophical Essays, pp. 94, 95, 4to edit.
"right angles, these two straight lines, though indefinitely produc"ed, will never meet :"-is not the boundless immensity of space tacitly assumed as a thing unquestionable? And is not a universal affirmation made with respect to a fact which experience is equally incompetent to disprove or to confirm? In like manner, when it is said, that "triangles on the same base, and between the same par"allels are equal," do we feel ourselves the less ready to give our assent to the demonstration, if it should be supposed, that the one triangle is confined within the limits of the paper before us, and that the other, standing on the same base, has its vertex placed beyond the sphere of the fixed stars? In various instances, we are led, with a force equally imperious, to acquiesce in conclusions, which not only admit of no illustration or proof from the perceptions of sense, but which, at first sight, are apt to stagger and confound the faculty of imagination. It is sufficient to mention, as examples of this, the relation between the hyperbola and it asymptotes; and the still more obvious truth of the infinite divisibility of extension. What analogy is there between such propositions as these, and that which announces, that the mercury in the Torricellian tube will fall if carried up to the top of a mountain; or that the vibrations of a pendulum of a given length will be performed in the same time, while it remains in the same latitude? Were there, in reality, that analogy between mathematical and physical propositions, which Beddoes and his followers have fancied, the equality of the square of the hypothenuse of a right angled triangle to the squares described on the two other sides, and the proportion of 1, 2, 3, between the cone and its circumscribed hemisphere and cylinder, might, with fully as great propriety, be considered in the light of physical phenomena, as of geometrical theorems: Nor would it have been at all inconsistent with the logical unity of his work, if Mr. Leslie had annexed 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.*
In the course of my own experience, I have met with one person, of no common in. genuity, 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 member distinctly to have heard him say, that he had more than once attempted to stady 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 occular proof, that the squares of the two sides actually filled the same space with the square of the bypothenuse,
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 to 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 com"parison of triangles. derive their evidence from the mere superposi"tion of the triangles themselves," it seems difficult, or rather impossible, 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 of 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; in as much 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 ge●metry 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.*
To those who reflect on the disadvantages under which Mr. Ferguson had laboured 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 elaps ed 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 con sequence of the very early period of iny 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 constant method being to satisfy himself, as to the truth of any problem, with a measurement by scale and compasses." (Hutton's Mathematical and Philosophical Dictionary, Article, Ferguson.)
• 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 pa radox very nearly approaching to that which I am now combating. "Le principe de la superposition n'est point, comme l'ont prétendu plusieurs geometres, une méthode de démontrer pen exacté et purement méchanique. La superposition, telle que les mathé maticiens 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 15