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

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 formula 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.

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