"In Mr. Spencer's subdivisions of mathematics, he has given a prominence to 'Descriptive Geometry' which might be regarded as arising from the partiality of the civil engineer for a branch of his own art, were it not that he says, 'I was ignorant of the existence of this as a separate division of mathematics, until it was described to me by Mr. Hirst, whom I have also to thank for pointing out the omission of the subdivision" Kinematics." It was only when seeking to affiliate and define "Descriptive Geometry," that I reached the conclusion, that there is a negatively quantitative mathematics, as well as a positively quantitative mathematics. In explanation of the term "negatively quantitative," it will suffice to instance the proposition that certain three lines will meet in a point as a negatively quantitative proposition, since it asserts the absence of any quantity of space between their intersections. Similarly, the assertion that certain three points will always fall in a straight line is negatively quantitative, since the conception of a straight line implies the negative of any lateral quantity or deviation.' The propositions selected by Mr. Spencer to illustrate what he calls Descriptive Geometry' are by no means peculiar to, or characteristic of, the art to which mathematicians have given this name. In the most elaborate and extensive treatises, no more is claimed for this art, than that it is an account, in a scientific order, of certain methods of geometrical construction useful in engineering and architecture, but inferior in scientific extension even to trigonometry, to which Mr. Spencer does not deign to descend. It is possible that Mr. Spencer has in mind certain propositions in the Higher Geometry' concerning relations of position and direction in points and lines: but these cannot be made to stand alone or independently of dimensional properties; and, if they could, they would be as appropriately named 'quantitative' mathematics as 'negatively quantitative.'"

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Must we then conclude that the writer who assumes to estimate a philosophical system like that of Mr. Spencer in the "North-American Review" is really unaware of the fundamental distinction between Science and Art? It would almost seem so. Ignoring the fact that Science is a statement of the relations among phenomena, and can include in its various divisions nothing more than the various classes of those relations, from which all practice based on knowledge

*North-American Review, April, 1865, p. 470.

of them is excluded, he actually supposes, that, by "Descriptive Geometry" Mr. Spencer means the art of plan-drawing! "Descriptive Geometry," in its scientific sense, no more means "certain methods of geometrical construction useful in engineering," &c., than "Geometry," in its scientific sense, means certain methods of earth-measuring. As from Geometry, which was originally a mathematical art, there has grown up a division of pure mathematics which has usurped the name of the art; so, beginning with the "Géométritric Descriptive" of Monge, in which theorems and their applications to drawing were mingled together, there has grown up a system of theorems which takes the name of " Descriptive Geometry," while omitting all mention of the practice which gave that name. If, because certain manufacturers and retailers are called "chemists," the reviewer had supposed that by "chemistry, as a branch of science, Mr. Spencer meant certain methods of preparing medicines and making dyes, he would have drawn an equally rational inference. He gains, however, by thus confounding science and art. It enables him to insinuate, by the quotation he makes, that Mr. Spencer, though educated as a civil engineer, was unacquainted with the branch of mathematical art which is especially familiar to engineers.* This insinuation it is unnecessary to meet: it disappears along with the reviewer's mistake on which it is based. It is needful only to point out what Mr. Spencer's admission really amounts to. Here, as in various places, Mr. Spencer has been careful to acknowledge aid derived from others; and, without stating that he was unacquainted with the propositions of "Descriptive Geometry," he candidly says he was not aware that they

* It is somewhat unfortunate for the writer's inference, that Mr. Spencer's first contribution to engineering literature (written before he was nineteen) is an account of a new and easier method of performing one of the most difficult problems of plan-drawing; namely, the delineation of the spiral courses of skew arches. See "The Civil Engineer and Architect's Journal" for May, 1839, p. 164.

† It happens, again unfortunate for the reviewer, that one division of “Descriptive Geometry" owes an original theorem to Mr. Spencer, which dates back to the time when he was seventeen; the theorem, namely, that the centres of the circles inscribed in all the triangles contained in any segment of a circle fall

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had been grouped into "a separate division of mathematics." Why he was not aware of this is easily explained. The title, "Descriptive Geometry," has never been adopted in England for the subject to which it was originally applied by the French: its modern restricted use is known only to professed mathematicians, and, as it now turns out, not even to all of these. This candor of Mr. Spencer, however, the reviewer takes advantage of, with what fairness we have seen. And then, showing the disingenuousness of his criticism, he seeks to ward off the charge of misrepresentation by saying "it is possible that Mr. Spencer has in mind certain propositions in the Higher Geometry' concerning relations of position and direction in points and lines." Indeed! it is possible that Mr. Spencer means that which, by his definition, he obviously does mean! Having first founded a charge of ignorance on a misrepresentation, the reviewer admits the possibility of another interpretation, which is, in truth, the only one Mr. Spencer's words will bear! There remains but to note the second clause of his last sentence: "But these [propositions in the Higher Geometry] cannot be made to stand alone, or independently of dimensional properties." To this the rejoinder is nothing else but a direct contradiction. If the reviewer asks for proof, we refer him to the recently published German work of Reye, entitled, "Geometrie der Lage." This will supply him with a whole volume full of propositions that wholly ignore "dimensional properties," are absolutely non-quantitative.

The readers of the "Christian Examiner" will remember an article which appeared in March of last year, entitled "Positivism in Theology." It is to this that most of the remarks we have to make will more especially apply. But first we offer a few words relating to the general plan of Mr. Spencer's system.

An early and thorough student of science in its various departments, and with a strongly philosophical turn of mind,

in the arc of a circle, which circle has its centre at the bisection of the arc of the complementary segment. This theorem he afterward published, with a demonstration, in The Civil Engineer and Architect's Journal for July, 1840,

p. 224.

it was but natural that Mr. Spencer's attention should have been drawn to the necessity and possibility of a more perfect organization than had hitherto been made of the general principles of knowledge, so as to form a connected and comprehensive philosophy of nature. This inclination was entirely coincident with the great tendency of modern inquiry, which is towards the disclosure of universal interdependence, harmony, and unity in nature. The problem of philosophy, as conceived by Mr. Spencer, was to represent this order and unity in thought. As the system was thus to be a mental reflex of the truth of nature, it was inevitable that he should take for its central and controlling idea the largest principle of connection and action which science has revealed in the universe; and this he discovered to be the Law of Evolution. The principle thus shadowed forth in so many directions, Mr. Spencer has worked out with more precision and completeness than any other thinker; and, holding it to be a universal law of nature, he has made it the organizing principle of his philosophical system. With it, that system, as such, must stand or fall. But to this scheme, which is to comprise some ten volumes in its development, he has prefixed an Introductory Essay of a hundred and twenty-two pages, discussing the question how far philosophy can go, and where she must stop, the bounds of legitimate inquiry, the limits of the knowable and the sphere of the unknowable; and he has here made an earnest and able attempt to fix the basis of a reconciliation between religion and science.

This introductory part, however, is by no means an essential portion of the philosophical system. Had Mr. Spencer not entered at all upon the question of the connection of the knowable and unknowable, his system of philosophy would still have been substantially what it is. For it is a perfectly possible thing, without expressing any opinion concerning the origin of things, to propound generalizations respecting the universal course of things, the order of phenomena, the connection and succession of events, as known to us in time. and space. The general doctrine of evolution may be enun

VOL. LXXXII.-NEW SERIES, VOL. III. NO. II.

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ciated and worked out in full detail, quite apart from all theological or ontological or metaphysical questions; and its truth or error is not in the least affected by the truth or error of Mr. Spencer's views respecting religion and science. Yet his critics have constantly committed the mistake of supposing, that, if they could throw doubt upon Mr. Spencer's doctrine regarding the relation of the Universe and its Cause, they thereby effectually disposed of his philosophy.

Now, so far as the philosophy proper is concerned, our reviewer has very little to say about it. He denies the adequacy of Mr. Spencer's method of inquiry to attain the result proposed, and carps at the law of Evolution. Mr. Spencer adopts the method by which modern knowledge has been created, first, the establishment of data; second, generalizing from these data; third, verification of the generalizations. His idea is, that, when we have thus reached the most general truth attainable, we have also arrived at the highest unity of knowledge; or, that the process by which knowledge is created is competent also to "unify" it. Not at all, says the reviewer. "Mere generalization is powerless to unify knowledge " (p. 240). Now, what is unifying knowl edge but reducing many facts to one fact? and what is this but generalizing? What is the highest unification of knowledge but the reduction of all facts to different forms of one fact? and what is this but generalization carried to its highest degree? To say that mere generalization is powerless to unify knowledge is to say that mere generalization is powerless to achieve generalization. Having thus, as he supposes, by a dash of his pen, discredited the grand tendency of modern intellect, what does the reader imagine he offers instead? He offers us the old file at which metaphysicians have been gnawing these thousands of years; and which will probably continue as sharp as at first, so long as this species of mental enterprise continues. "Its unity must be found in the equipoise and dynamic correlation of being and thought, which are welded into one in the act of knowledge itself." And, pray, what unification of fragmentary knowledges has ever been accomplished by that recipe?