various branches of Mathematics produce this effect, they destroy one of the main reasons why mathematical studies are accepted as parts of Education. For undoubtedly it has always been supposed, by those who have approved of such education, that the Mathematics so taught was to make men acquainted with those mental triumphs of past generations which have always occupied a conspicuous place in man's intellectual history. If our educational Mathematics does not do this, men in general, when they learn that the case is so, will be far less ready to assent to the value which we set upon the study. If our educational Mathematics give us no acquaintance with the works of Euclid and Archimedes, Galileo and Newton, men in general will look upon our Mathematical Education as illusory and worthless. If any one moderately acquainted with the general literature of the country knows more than our best mathematical students do, of the history of mathematical and physical discoveries, a praiser of our system will find, in general, averse and incredulous auditors. To know accurately those events in scientific history which other men know vaguely, is a most proper and characteristic superiority of a well-educated man; but to know certain general symbolical results, which are supposed to render all scientific history superfluous, is an accomplishment which can only be of little value in education: for a good education must connect us with the past, as well as with the future; even if such mere generalities did supply the best mode of dealing with all future problems; which, in fact, they are very far from doing. 65 For (to add one more to the points of advantage of geometrical over analytical forms of Mathematics for common educational purposes), it will generally be found that a person who has studied the branches of Mathematics in the more special and detached forms in which they were treated geometrically, before ana lytical generalities became so common, will be able to apply his knowledge to the calculation of practical results and the solution of problems, better than a person who has acquired his mathematical learning under general analytical forms. The geometrical student has a firmer hold of his principles than the analytical student has. The former holds his fundamental truths by means of his conceptions; the latter, by means of his symbols. In applying doctrines to particular cases, or in solving new problems, the former sees his way at every step, and shapes his course accordingly; the latter must commit himself to his equations; which, except he be a consummate analyst, he will not readily understand and interpret in their particular application. I have no doubt that in any application of geometrical, mechanical, or hydrostatical principles to a problem of moderate difficulty, supposing the problem new to both of two students; one, a geometer of the English school of forty or fifty years back, the other, a modern analyst, instructed in equal degrees; the former would much more accurately and certainly obtain a definite and correct solution. In the application of Mathematics to problems of engineering and the like, the generalities in which the analyst delights are a source of embarrassment and confusion, rather than of convenience and advantage. When particular problems are solved by particular considerations or particular artifices, the ingenuity thus exercised is a talent really more generally available than a knowledge of the general methods which express all problems alike, but actually solve none. 66 From the considerations which have thus been stated, I am led to the conclusion, that the geometrical modes of treating the various branches of Mathematics are those which are to be employed as Educational Studies. The geometrical forms of Trigonometry, Conic Sections, Statics, and Dynamics, and not any analytical substitutes for them, must be parts of a Liberal Education. This must be so, because thus alone can Mathematics be an intellectual discipline, strengthening the reasoning powers for other nonmathematical occupations; thus alone can the mathematical sciences be known in that historical shape with which a liberally educated person ought to be acquainted; and thus best is a person of moderate mathematical attainments able to apply to practical cases the knowledge which he possesses. I have hitherto spoken of that part of Education which consists of Permanent Mathematical Studies. But Progressive Mathematics may also advantageously enter into our Higher Education; and I proceed to speak of this portion of Educational Studies. SECT. 6. Of Progressive Mathematics as an 67 As I have already said, a liberal Education ought to include both Permanent Studies which connect men with the culture of past generations, and Progressive Studies which make them feel their community with the present generation, its businesses, interests and prospects. The Permanent Studies must necessarily precede, in order to form a foundation for the Progressive; for the present Progress has grown out of the past activity of men's minds; and cannot be intelligible, except to the student of past literature and established opinions. But the Progressive Studies must be added to the Permanent; for without this step, the meaning and tendencies of the past activity of men cannot be seen, nor our own business understood. And though Progressive Studies may form the business of life, as well as of the specially educational period of it, they may with advantage be begun in that period, before each man's course of study is, as in after life it generally is, disturbed and perplexed by the constant necessity of action. 68 This necessity of Progressive, as well as Permanent Studies, may be applied to Mathematical Studies in particular. For Mathematics, for the last three centuries, has been, and still is, a science in which a rapid progress is taking place. Old problems have been solved by new and simpler methods: new problems, formerly insoluble, have been successfully attacked; and the methods by which these successes have been attained have excited a strong admiration in men, on account of their ingenuity, generality, and symmetry. On these accounts, it is to be expected that those persons who cultivate mathematics will be drawn to give some of their attention to these modern and progressive portions of mathematics; and those who have to teach Mathematics as a part of Education, may naturally be led to introduce into their teaching a large share of this kind of Mathematics. 69 Almost all these modern portions of Mathematics are of the analytical kind. It may be useful to mention some of the most conspicuous of these newer mathematical trophies. The expression of the form and properties of Curves by means of their Co-ordinates was introduced by Descartes; and the like methods have since been extended to Curve Surfaces, giving rise to an extensive subject, the Application of Algebra to Geometry. The rates of change of variable quantities related in a given manner became the subject of the Differential Calculus, invented by Newton, (under the form of Fluxions) and by Leibnitz. The inverse process connected with this is the Integral Calculus; (the Method of Quadratures of the Newtonians.) Then come, as additional branches, Differential Equations, Finite Differences (originally termed the Method of Increments); Definite Integrals, and finally, the Calculus of Variations, which treats of the change in the forms of relation of related variable quantities. Moreover we have the application of all these modes of calculation to Mechanics (including Hydrostatics) and to the Mechanics of the Universe, cultivated to a wide extent by Euler, and carried to the highest pitch of generality and symmetry by Lagrange. To this may be added the application of analytical methods to other subjects, as Optics. All these portions of Mathematics lie open to us as parts of Progressive Science; and it is our business to select from them such a portion as may suit the purposes of a Liberal Education. 70 I have argued against the exclusive, or even the copious use of Analytical Mathematics among Permanent Educational Studies. It may be asked, whether the same objections apply to the study of such subjects as I have recommended, most of which are analytical in their form. To this we may reply, that the objections by no means apply in this case. When, by the pursuit of Permanent Mathematical Studies, the reasoning faculties have been educed and confirmed, the Student's powers of symbolical calculation, and his pleasure in symbolical symmetry and generality, need no longer be repressed or limited. He may go on following the reasonings and processes of the beautiful portions of mathematics above mentioned, as far as his taste and talents prompt him, or as the demands of other studies will allow. When the Student is once well disciplined in geometrical mathematics, he may pursue analysis safely and surely to any extent. be 71 But though modern mathematics may thus very fitly studied as a sequel to the older forms of mathematical science which must enter into a Liberal Education; these modern methods cannot supply the place of the ancient subjects as the Permanent Studies in our Educational course. This is sufficiently shown already, by what we have said of the unfitness of mere [Pr. I.] 6 |