calculation. As we have already said (21) calculation by means of symbols of number and quantity has, in many instances, taken the place of reasoning by means of the properties of space: and thus, in addition to the ancient Geometry, we have, in modern times, new branches of pure mathematics, Algebra, the Algebra of Curves, and the Differential Calculus. In these branches, new steps and modes of calculation, new advances in generalization and abstraction, new modes of dealing with symbols, such as may each be termed a New Calculus, have been constantly, in modern times, invented and published by mathematical writers. And these novelties, because they are novelties, and often because they render easy what before was difficult, are received with pleasure, and followed with interest by mathematical readers. And it is very fit that this should be so. But still, these Progressive portions of Mathematics cannot take the place of the Permanent portions, in our Higher Education, without destroying the value of our system. Wherever Mathematics has formed a part of a Liberal Education, as a discipline of the Reason, Geometry has been the branch of mathematics principally employed for this purpose. And for this purpose Geometry is especially fitted. For Geometry really consists entirely of manifest examples of perfect reasoning: the reasoning being expressed in words which convince the mind, in virtue of the special forms and relations to which they directly refer. But in Algebra, on the contrary, and in all the branches of Mathematics which have been derived from Algebra, we have, not so much examples of Reasoning, as of Applications of Rules; for the rules being at first proved by reasoning, once for all, the application of them no longer comes before us as an example of reasoning. And in the reasoning itself, quantities, and their relations, are not expressed in words and brought before the mind as objects of intuition, but are denoted

This

by symbols and rules of symbolic combination. mode of denoting the relations of space and number, and of obtaining their results, is in the highest degree ingenious and beautiful; but it can be an intellectual discipline to those only who fully master the higher steps of generalization and abstraction by a firm and connected mental progress from the lower of such steps upward; and this requires rather a professional mathematical education, than such a study of mathematics as must properly form part of a liberal education. Moreover, supposing these higher forms of Algebra to be thus completely mastered, the intellectual discipline which they afford is not a discipline in reasoning, but in the generalization of symbolical expression: and this appears to be a mental process of which a very small exercise is all that a liberal education requires. For instance, the general student may, especially if his mind has an aptitude for such studies, derive intellectual profit, from learning how the forms and properties of given curves are determined by symbolical expressions of co-ordinates, and of the relative rate of change of these co-ordinates, as in the Differential Calculus: but it can hardly be worth the while of such a student to bestow much time in ascending to that higher generalization in which these changes of the co-ordinate of a given curve are mixed with other changes, by which any one curve may be transformed into any other, as in the Calculus of Variations. Such steps of wide symbolical abstraction, however beautiful as subjects of contemplation to persons of congenial minds, are out of the range of any general system of Liberal Education.

32 All these branches of Algebra of which we have spoken may, as I have intimated, be considered rather as Progressive than as Permanent Studies; and therefore, not necessarily parts even of a Higher Education; since in order to the full cultivation of the

Reason they need not be possessed at all. They are fit matters of the study of the professed mathematician, when his general education is terminated. But of Geometry, on the other hand, it is not too much to say that it is a necessary part of a good education. There is no other study by which the Reason can be so exactly and rigorously exercised. In learning Geometry, as I have on a former occasion said*, the student is rendered familiar with the most perfect examples of strict inference; he is compelled habitually to fix his attention on those conditions on which the cogency of the demonstration depends; and in the mistakes and imperfect attempts at demonstration made by himself and others, he is presented with examples of the more natural fallacies, which he sees exposed and corrected. He is accustomed to a chain of deduction in which each link hangs from the preceding, yet without any insecurity in the whole; to an ascent, beginning from solid ground, in which each step, as soon as it is made, is a foundation for a further ascent, no less solid than the first self-evident truths. Hence he learns continuity of attention, coherency of thought, and confidence in the power of human Reason to arrive at the truth. These great advantages, resulting from the study of Geometry, have justly made it a part of every good system of Liberal Education, from the time of the Greeks to our own.

33 Arithmetic has usually been a portion of Education on somewhat different grounds; namely, not so much on account of its being an example of reasoning, as on account of its practical use in the business of life. To know and to be able familiarly to apply the rules of Arithmetic, is requisite on innumerable occasions of private and public business; and since this

University Education, p. 139: Thoughts on the Study of
Mathematics.

ability can never be so easily or completely acquired as in early youth, it ought to be a part of the business of the boy at school. For the like reasons, Mensuration ought to be learnt at an early period; that is, the Rules for determining the magnitude, in numbers, of lines, spaces, and solids, under given conditions; a branch of knowledge which differs from Geometry, as the practical from the speculative; and which, like other practical habits, may be most easily learnt in boyhood, leaving the theoretical aspect of the subject for the business of the Higher Education which comes at a later period. There is another reason for making Arithmetic a part of the school-learning of all who are to have a Liberal Education: namely, that without a very complete familiarity with actual arithmetical processes, none of the branches of Algebra can be at all understood. Algebra was, at first, a generalization and abstraction of Arithmetic; and whatever other shape it may take by successive steps in the minds of mathematicians, it will never be really understood by those students who do not go through this step. And, as we have already said, there is, in a general education, little or nothing gained by going beyond this. The successive generalizations of one or another New Calculus, may form subjects of progressive study for those whose Education is completed, but cannot enter into a general Education, without destroying the proportion of its parts.

34 I have spoken of Geometry as a necessary part of a Liberal Education. It may be asked, how far this Geometry extends? the Elements of Euclid, especially the first Six Books, are generally accepted as the essential portion of Geometry for this purpose. This portion of Mathematics is, however, insufficient fully to exercise the activity of the Reason, and to balance the influence of classical studies. If we consider what portions of mathematics may most properly

The

be added to Elementary Geometry, the parts that offer themselves are Solid Geometry, Conic Sections, Mechanics, Hydrostatics, Optics and Astronomy. Of Solid Geometry, we have an Elementary portion in the Eleventh and Twelfth Books of Euclid, and which has often, and very suitably been used for purposes of Education. Conic Sections are a very beautiful extension of Elementary Geometry; and would probably have been made a part of a general Education more commonly than it has been, if we had inherited from the Greeks any Treatise on the subject, as perfect as the Elements of Euclid are on their subjects. The properties of the Conic Sections are not merely so many propositions added to those of Elementary Geometry: there are introduced, in this branch of Geometry, new geometrical conceptions; for instance, that of the Curvature of a curve at any point. proofs of the properties of Conic Sections, discovered by the Greek geometers, have come down to us only in a fragmentary manner; and although there have appeared several modern treatises which are very good examples of geometry, no one of them has acquired a permanent and general place, as a part of a Liberal Education. This has arisen in part, at least in England, from the prevalence of a disposition among mathematical students in modern times to adopt the algebraical mode of treating these as well as other curves. But we may observe that the subject of Conic Sections, so treated, is of small comparative value as a portion of Education. If we make the Conic Sections merely examples of the application of Algebra to curves, they are of no more importance than Čissoids, Conchoids, or any other curves; and have little claim to be considered as a distinct part of our Educational Studies: while a geometrical system of Conic Sections is both a striking example of geometrical reasoning; a distinct member of an enlarged system of Geometry; [PT. I.]

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