to withdraw the attention from that unity of design which it is the noblest employment of philosophy to illustrate, by disguising it under the semblance of an eternal and necessary order, similar to what the mathematician delights to trace among the mutual relations of quantities and figures.

The consequence has been, (in too many physical systems,) to level the study of nature, in point of moral interest, with the investigations of the algebraist; an effect, too, which has taken place most remarkably where, from the sublimity of the subject, it was least to be expected-in the application of the mechanical philosophy to the phenomena of the heavens. But on this very extensive and important topic I must not enter at present.

In the opposite extreme to the error which I have now been endeavouring to correct, is a paradox which was broached, about twenty years ago, by the late ingenious Dr. Beddoes; and which has since been adopted by some writers, whose names are better entitled, on a question of this sort, to give weight to their opinions. By the partisans of this new doctrine it seems to be imagined that, so far from physics being a branch of mathematics, mathematics, and more particularly geometry, is in reality only a branch of physics. "The mathematical sciences,"

second (that the whole is greater than a part) is often used by Newton, and by all mathematicians, and many demonstrations rest upon it. In general, Newton, as well as all other mathematicians, grounds his demonstrations of mathematical propositions upon the axioms laid down by Euclid, or upon propositions which have been before demonstrated by help of these axioms.

"But it deserves to be particularly observed, that Newton, intending in the third book of his Principia to give a more scientific form to the physical part of astronomy, which he had at first composed in a popular form, thought proper to follow the example of Euclid, and to lay down first, in what he calls Regulæ Philosophandi, and in his Phænomena,

the first principles which he assumes in his reasoning.

"Nothing, therefore, could have been more unluckily adduced by Mr. Locke to support his aversion to first principles, than the example of Sir Isaac Newton."-Essays on the Intellectual Powers, pp. 647, 648, 4to edit.

1 I allude here more particularly to my learned friend, Mr. Leslie, whose high and justly merited reputation, both as a mathematician and an experimentalist, renders it indispensably necessary for me to take notice of some fundamental logical mistakes which he appears to me to have committed in the course of those ingenious excursions in which he occasionally indulges himself, beyond the strict limits of his favourite studies.

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 mathematical experiments are more simple, and more perfectly within the grasp of our senses, and our perceptions of mathematical objects are clearer."1

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 fundamental 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 appeal, though of the easiest and most familiar kind, to external observation."2 And, in another pas

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 comprehension. I therefore think it due to the author of EIIEA ПITEРOENTA, to acknowledge my obligations to him for relieving me from this sort of distress. For although I had often 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 de Philosophie, Art. Géométrie. The same writer, however, justly observes, in another part of his works, that the measure of angles by circular arches, is itself dependent on the principle of superposition; and that, consequently, however extensive and important in its applica tion, 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 son sommet, on veut dire que si deux angles sont égaux, les angles

sage: "Geometry, like the other sciences which are not concerned about the operations of mind, rests ultimately on external observations. But those ultimate facts are so few, so distinct and obvious, that the subsequent train of reasoning is safely pursued to unlimited extent, without ever appealing again to the evidence of the senses."1

Before proceeding to make any remarks on this theory, it is proper to premise, that it involves two separate considerations, which it is of material consequence to distinguish from each other. The first is, that extension and figure (the subjects of geometry) are qualities 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 enough to state that they are both conversant about the same objects; it is necessary farther to shew, that, in both cases, these objects are considered in the same point of view, and give employment to the same faculties of the mind. The poet, the painter, the gardener, and the botanist, are all occupied in various degrees and modes with 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

décrits de leur sommet à même rayon, seront égaux; vérité qui se démontre par le principe de la superposition, comme tout géomètre tant soit peu initié dans cette science le sentira facilement." -Eclaircissemens sur les Elémens de Philosophie, sect. 4.

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 principle of superposition.

1 Elements of Geometry and of Geometrical Analysis, p. 453.

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; inasmuch 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 specialty, or rather singularity, which renders the attempt to identify the objects of geometrical and of physical science, incomparably more illogical than it would be to classify poetry with botany, or 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 secondary,) by bestowing on them the title of mathematical affections of matter.1 Of these mathematical affections (magnitude and figure) our first notions are no doubt derived (as well as of hardness, softness, roughness, and smoothness) from the exercise of our external senses; but it is equally certain, that when the notions of magnitude and figure have once been acquired, the mind is immediately led to consider them as attributes of space no less than of body; and (abstracting them entirely from the other sensible qualities perceived in conjunction with them) becomes impressed with an irresistible conviction, that their existence is necessary and eternal, and that it would remain unchanged if all the bodies in the universe were annihilated. It is not our business here to inquire into the origin and grounds of this conviction. It is with the fact alone that we are concerned at present; and this I conceive to be one of the most obviously incontrovertible which the circle of our knowledge embraces. Let those explain it as they best can, who are of opinion, that 1 Philosophical Essays, pp. 94, 95, 4to edition; [infra, vol. v.] VOL. III.

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all the judgments of the human understanding rest ultimately on observation and experience.

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 anything 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 of a mathematical nature, but they 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 right angles, these two straight lines, though indefinitely produced, will never meet;" is not the boundless immensity of space tacitly assumed as a thing unquestionable? And is not