sight the subject matter of the reasoning; on the right apprehension of which, with its peculiar character and attributes, all good reasoning, on all other subjects, must depend. 46 It is easy to show by examples, taken from the branches of mathematics which I have mentioned, that analytical modes of treating those subjects have, in fact, put out of sight the peculiarities of the conceptions which belong to each subject, and have merged all their special trains of reasoning in undistinguishing symbolical generalizations. In the Elements of Geometry, Ratio and Proportion are among the peculiar conceptions which belong to the subject; and of which the properties, as treated by Euclid, rest upon an especial Axiom, (Book v. Axiom v.) The mode in which, by means of this Axiom, the case of incommensurable quantities is reasoned upon, without any introduction of arbitrary assumptions or ungeometrical notions, has always been admired, by the cultivators of Geometry, as a beautiful and instructive example of mathematical subtilty and exactness of thought. In the analytical mode of treating the subject, a Ratio is identified with an Algebraical Fraction: and the reasonings about Ratios become operations upon Algebraical Fractions; in which operations, everything dependent upon the peculiar character of the conception disappears; and all the propositions of Geometry, in which Proportion is involved, are, on this scheme, made to depend upon Algebra. 47 Trigonometry was a science invented for the purpose of measuring Angles by means of Lines drawn in a certain manner, in a circle whose center is the angular point; and of using these measures for the solution of triangles. This subject has recently been modified so that Angles are measured by certain Algebraical Fractions, the original conception of the Circle being rejected. And in this manner, all the proposi tions of Trigonometry have been superseded by certain analytical formulæ involving those Algebraical Fractions. 48 Conic Sections were, till lately, treated as a geometrical subject; the Curves being defined, in some treatises, by the sections of a cone by a plane; in others, by certain simple relations of lines drawn in their own plane. But in either method, various conceptions were introduced, extensions of those of Elementary Geometry; as the conceptions of Tangents to such curves; of Properties analogous to those of the Circle; of properties of Conjugate Diameters; of properties of the Circle of Curvature; and the like. These properties were proved by Geometrical reasonings, built upon some simple fundamental properties, exhibiting at every step the evidence of the relation of the properties of the Conic Sections to those of the Circle; and supplying a transition to the properties of Curves in general. Of late, Conic Sections has been treated as a mere branch of analysis; the definitions of Tangents and Circles of Curvature have become Algebraical or Differential Formula; the analogies with the circle have also appeared only as interpretations of Algebraical Formulæ ; and the subject of Conic Sections has ceased to be of any meaning, as an introduction to the subject of curves in general; because the Conic Sections are treated only as curves in general; and any other class of curves might with equal propriety be made a separate branch; or rather, there is no propriety in so treating any class of curves; for all their mathematical interest, so treated, consists in their being examples of general methods. 49 Statics, the Mechanics of Equilibrium, depends upon certain fundamental truths, which were established by Archimedes, among the ancients, and by Stevinus, Galileo, and others, among the moderns. From these fundamental truths, by keeping steadily in view the conception of Statical Force, all ordinary pro blems may be solved by geometrical methods. But in modern times, the subject has been differently treated. The fundamental proposition, the Composition of Forces, or some equivalent one, has been proved (sometimes, even this, by analytical reasoning from assumed Axioms;) and then, all problems alike have been made to depend upon the equations which apply the fundamental properties, in the most general form, to every possible system of matter. In this manner, the conception of Force has been dismissed from the mind, as soon as the first steps of the science had been made. 50 The Doctrine of Bodies in Motion acted upon by Forces, was created by Galileo and his successors, and was applied by Newton to the System of the Universe, in such a manner as to draw to this doctrine the universal attention of the educated portion of mankind. Independently of the immense importance in the history of Inductive Astronomy which belongs to the Propositions of Newton's Principia, the work is a very beautiful series of examples of the application of the principles of Mechanics, combined with the properties of the Conic Sections, and other known geometrical propositions. In virtue of its combined merit and interest, this work is eminently fitted to be a part of the permanent mathematical studies of a Liberal Education, and specially of a Liberal English Education. But this subject also has been treated analytically. The forces by which bodies describe Conic Sections or any other orbits, the orbits which bodies will describe under the influence of any forces, the attraction of masses of attractive particles, and the like problems, have been investigated symbolically by means of the Differential Calculus, and other analytical processes; and hence the peculiar mechanical conceptions with which the speculators of Newton's time had to struggle, and which he followed out till they led to his remarkable discoveries, have been obliterated from the minds of most of our modern analytical mathematicians. 51 Even in Astronomy, though so much a science of observed phenomena, explanations of observed phenomena, and reasonings from them; mathematicians of analytical propensities have had a tendency to pass as briefly as possible over such observations, explanations, and reasonings; and to dwell mainly upon parts of the subject where principles, once established, gave a hold for their analytical instrument, the Calculus. They have thus assimilated this to other parts of their body of Mathematics and though Astronomy is an Inductive Science, explaining phenomena by its Theories, they have omitted out of it all that is Inductive, Explanatory, or especially Astronomical. : Remarks of the same kind might be made respecting other branches of Mathematics, but the subject has been pursued far enough, I trust, to justify the general opinion which I have delivered. 52 I assert, therefore, that these branches of Mathematics, thus analytically treated, do not possess that value as instruments of an exact and extensive discipline of the reason, which the geometrical branches of Mathematics do possess. Indeed it must appear, I think, from what has been said, not only that mere Analytical Mathematics does not possess so much value as Geometrical Mathematics for such purposes, but that, in truth, it possesses none at all, or at least very little. Analytical operations in Mathematics do not discipline the reason; they do not familiarize the student with a chain of syllogisms connected by a manifest necessity at every link: they do not show that many kinds of subjects may be held by such chains and at the same time, that the possibility of so reasoning on any subject must depend upon our conceiving the subject so distinctly as to be able to lay down axiomatic principles as the basis of our reasoning. : 52 With reference to analytical mathematics, the argument in favour of the use of Mathematics as a [PT. I.] 5 permanent educational study, loses all its force. If we can only have analytical mathematics in our system of education, we have little reason to wish to have in it any mathematics at all. Our education will be very imperfect without Mathematics, or some substitute for that element; but mere analytical mathematics does not remedy the imperfection. If we can only have analytical mathematics, it is well worth considering whether we may not find a much better educational study to supply its place in Logic, or Jurisprudence. The general belief, for undoubtedly it is a general belief, that Mathematics is a valuable element in education, has arisen through the use of Geometrical Mathematics. If Mathematics had only been presented to men in an analytical form, such a belief could not have arisen. If, in any place of education, Mathematics is studied only in an analytical form, such a belief must soon fade away. 53 I must request the Reader to observe that the consequences which I have spoken of, are spoken of as resulting from the use of mere Analysis, as the mathematical element of Education. If the geometrical modes of treating Mechanics, Conic Sections, and the Dynamics of the Universe, are carefully preserved and steadily employed, as Permanent Educational Studies, such analytical methods as I have mentioned may be brought before the Student with advantage, as further illustrations of the standard mathematical truths to which he is introduced; as examples of the various modes of arriving at mathematical truth; and as manifestations of the extent to which the solutions of problems may be simplified and extended. But this pursuit of simplicity and generality must not be allowed to interfere with the attention which is to be given to the standard modes of establishing such truths. Such a pursuit of mathematical variety, simplicity, and generality, if it take the place of the study of standard |