by moving bodies and the forces which act upon them, have been investigated by applying the differential calculus to the co-ordinates of the curves; instead of establishing those relations in the way in which Newton did. Such branches of mathematical science are often called Analytical, and Analysis is often opposed to Geometry. This opposition is not very exact; for Geometry has also its Analysis. We demonstrate a Theorem, or solve a Problem, by means of Geometrical Analysis, when we suppose the Theorem proved, or the Problem solved, and trace the consequences of this supposition into a known theorem, or a problem obviously soluble. In solving a question by means of Algebra we, in like manner, express the given supposition in algebraical language, using symbols of unknown, as well as of known quantities, and by tracing the consequences of this expression, we can often find the value of the unknown quantities. And this analytical process is so much more common in Algebra than in Geometry, that processes conducted by means of algebraical symbols are commonly called Analytical. I will adopt this phraseology, though, as I have said, it is not very exact, while I make a few remarks on the use of Analytical Mathematics as an instrument of Education*. I have already spoken on this subject, in my Thoughts on the Study of Mathematics as a part of a Liberal Education, published in 1835; and all subsequent experience appears to have confirmed the truth of the views there expressed.

*The reader may observe that the reasons here given for adopting geometrical rather than analytical modes of reasoning in our elementary university course of mathematics, apply rather against analytical generalities and special calculus, than against the introduction of simple algebraical processes. For example, these reasons would not exclude from Mechanics the method of finding the space fallen through in a given time, by means of assuming an arithmetical series and taking its limit: and the like processes.

41 The recommendation of the geometrical branches of mathematics, as parts of education, is, as we have seen, that they are an effectual discipline of the reason, and have always been familiar as such among educated men. On the other hand, the recommendations of analytical forms of mathematics are such as these; their supplying easier solutions of the problems with which the mathematician has to deal ;the symmetry and generality of their processes;-and their having, in consequence of these qualities, superseded geometrical methods in the mathematical literature of modern times. These merits of analytical processes have been shown, in a most striking manner, in the works of many great mathematicians of modern times; who have given to such processes great completeness and beauty, and have solved, by means of them, problems which had foiled the attempts of previous calculators. And these great works have been accompanied by many elementary works, which expound the like methods in a more limited form, accessible to common students, and applicable to simpler problems. We have to consider the advantages and disadvantages of employing in our Higher Education such Analytical Elementary Treatises, to the exclusion of geometrical modes of treating the same subjects.

42 The first reason which we have mentioned, why Mathematics, in the shape of Geometry, holds its place as an element of great and incomparable value among the permanent studies of a Liberal Education, is this: that it offers to us examples of solid and certain reasoning, by which the reasoning powers, and the apprehension of demonstrative proof, may be exercised, unfolded, and confirmed. This is eminently true of the Geometrical Forms of Elementary Geometry, Trigonometry, Conic Sections, Statics, and Dynamics. It is not true to the same extent, and hardly at all, of the Analytical Methods of treating

the same subjects. For, in the Geometrical Form of these sciences, we reason concerning subjects in virtue of the manner in which the subjects are conceived in the mind. In the Analytical Methods, on the other hand, we reason by means of symbols, by which symbols, quantities, and the relations of quantity, are represented; and by means of the general rules of combining and operating upon such symbols; without thinking of anything but these rules. When the supposed fundamental conditions are once translated into the language of Analysis, we dismiss from our minds altogether the conceptions of the things which the symbols represent; whether lines, angles, velocities, forces, or whatever else they may be. The mode of proceeding is the same, whichever of these be the matters in question; and the steps of the process are not acts of thought, in any other way than as the application of an assumed general rule to a particular case is an act of thought. We arrive at our conclusion, not by a necessary progress, in which we see the necessity at every step, but by a compulsory process, in which we accept the conclusion as necessary in virtue of the necessary truth of our rules of procedure, previously proved or supposed to be proved. In the one case, that of geometrical reasoning, we tread the ground ourselves, at every step feeling ourselves firm, and directing our steps to the end aimed at. In the other case, that of analytical calculation, we are carried along as in a rail-road carriage, entering it at one station, and coming out of it at another, without having any choice in our progress in the intermediate space.

43 It is plain that the latter is not a mode of exercising our own locomotive powers; and in the same manner analytical processes are not a mode of exercising our reasoning powers. It may be said that much thought and skill are required in the analyst, in

order that he may choose the best scheme of symbols, the best mode of combining them, the best analytical artifices for arriving at his result, and shortening the way to it. And in like manner, in travelling by a rail-road, thought and skill are requisite in order to select the line and the train, or the combination of lines and trains, which will lead us to the intended place. We must know the stations and times of the system, in order to use it. But still, this is not any exercise or discipline of the bodily frame. It may be the best way for men of business to travel, but it cannot fitly be made a part of the gymnastics of education. And just as little is mere analysis a discipline of the intellectual frame. It may be the best way for the professed mathematician to deal with the problems which he has to solve, but it cannot answer the purpose of that gymnastic of the reason, without which a liberal education cannot subsist.

44 Thus mere Analysis is not a suitable discipline of the reasoning powers, because analytical processes do not exhibit reasoning, in the common sense of the term, and in a form which resembles the common reasonings with which men are concerned: whereas geometry does exhibit reasoning in a form which resembles such common reasonings, except in so far as geometrical reasoning is more perfect and certain than most such reasonings. Geometry sets out from certain First Principles; namely, Axioms and Definitions; and at every step uses formulæ which, if they are really applicable, lead necessarily to the next step, by an evidence which the like forms of language express and convey, on all subjects, as well as Geometry. Its Because and Whereas, its But and For, its Wherefore and Therefore, are its connecting links, in the same sense in which they are the connecting links of all reasoning. If in other subjects we have First Principles equally certain, and Definitions equally precise,

we can reason in the same manner as in Geometry; and to reason conclusively, we must do so. All geometrical reasoning may be resolved into a series of syllogisms; and in its proper form, consists of a chain of enthymems, or implied syllogisms; and in like manner, all other sound reasoning on all subjects consists of a like chain of enthymems. In geometrical reasoning, each proposition, when once established, is used in establishing ulterior propositions, with as much confidence and promptitude as if it were itself a selfevident axiom. And in like manner, in all sound connected reasoning, a proposition, once established, is to be used with confidence and promptitude, in establishing ulterior propositions. And the habit of thus advancing, with clear conviction and active thought, from step to step of certain truth, is an intellectual habit of the greatest value; which a good education ought to form and render familiar; and which nothing but geometrical study can impart. In analytical reasoning, we have no such chains of syllogisms present to the mind. It may be said, indeed, that every step in analysis is a syllogism, in which the major is the Axiom, Things which are equal to the same are equal to one another; and the minor is a proposition that two certain forms of symbols have been proved to be equal to the same. But to this we shall reply, that the perpetual repetition of this elementary kind of syllogism, even if the process were so conceived, is no sufficient discipline in reasoning: and further, that the algebraical equality of two symbols does not exemplify a member of a syllogism, in any way which can make such reasoning an intellectual exercise. I repeat, therefore, that mere analytical processes are no proper discipline of the reason, on account of the difference of form between such reasoning and the reasoning with which men are mainly and commonly concerned.