45 But again: Analytical reasoning is no sufficient discipline of the reason, on account of the way in which it puts out of sight the subject matter of the reasoning. In geometrical reasoning, we reason concerning things as they are; in the first place, in virtue of certain Axioms concerning them, which are selfevident from our conceptions of the things; and then, in virtue of Propositions deduced from those Axioms, which Propositions are considered as properties of the things. We reason concerning Straight Lines, Angles, Spaces, Curves, Forces, Inert Masses, conceiving them as Straight Lines, Angles, Spaces, Curves, Forces, Inert Masses. We are thus led to see that such reasoning as we employ in one case, is not applicable to such case only, but also to other classes of cases, in which the things reasoned of are of altogether a different kind. We pass from Geometry to Mechanics, not by identifying Forces with Lines, but by taking Axioms concerning Force, evident by its nature as Force, and by applying these Axioms, one after another; bearing in mind, at every step, the peculiar nature of Force. In this way, we are prepared to pass on, and to apply reasoning of the like rigour to other subjects, as different from these, as Force is from Space. In this way, Geometry, and Geometrical Mechanics, are a discipline for every kind of sound reasoning. We are prepared to bring the mental power to act by its syllogistic chain, upon all classes of conceptions. In Analysis, the contrary is the case. The Analyst does not retain in his mind, in virtue of his peculiar processes, any apprehension of the differences of the things about which he is supposed to be reasoning. All things alike, Lines, Angles, Forces, Masses, are represented by the letters of the alphabet. All curves, Conic Sections, Transcendental Curves, Curve Surfaces, are alike represented by the relations of co-ordinates; and for the sake of uniformity,

Straight Lines are represented in the same manner. All relations of Motion and Force are, in like manner, represented by the equations of co-ordinates of the points moving and acted on. When he has once placed before him these equations of co-ordinates, he no longer thinks at all about the special nature of the things originally spoken of. His reasonings are operations upon symbols; his results are equations. His final equation may give him an angle, or a radius of curvature, or an angular velocity, or a central force; but he has no separate processes of thought for these different cases. He obtains his result equally well, if he has forgotten, or does not know, which of these things his represents. I am quite willing to allow that this peculiarity arises from the perfection of analysis; from the entire generality of its symbols and its rules. What I am here saying is, that this is a kind of perfection which makes analysis of little value as a discipline of the reason for general purposes. For in reasoning for general purposes, it is quite necessary to bear in mind, at every step, the peculiar nature and attributes of things about which we reason. We cannot, in any subject, except analytical mathematics, express things by symbols once for all, and then go on with our reasoning, forgetting all their peculiarities. Any attempt to do this, (for such attempts have not been wanting,) leads to the most extravagant and inapplicable conclusions. Anything in common reasoning resembling such an attempt; as when men start with the definition of certain Technical Terms, and build systems by the combinations and supposed consequences of these,-belongs to a class of intellectual habits which it is the business of a good education to counteract, correct, and eradicate, not to confirm, aggravate, and extend. And therefore I say, that mere analytical reasoning is a bad discipline of the intellect, on account of the way in which it puts out of

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.