ancient Geometry does. And the same is the case with regard to the other subjects. If our Trigonometry differs entirely from the Trigonometry which appears in the History of Mathematics; if our Conic Sections is a different science from the Conic Sections with which all ancient and most modern mathematicians are familiar; our new Mathematics cannot connect us with the ancient thinkers, as educational mathematics ought to do. If our Mechanics is a new Science, reducing all special cases, as the Mechanical Powers, to one universal case of equilibrium, and obliterating their peculiar characters, it cannot enable us to understand the familiar discussions of those whose Mechanics is of a more historical complexion. If our knowledge of the Mechanics of the Universe, though it implicitly includes the Newtonian truths, do not explicitly exhibit the Newtonian proofs and methods, it will not enable us to share in the interest with which those who know the history of science dwell upon the philosophical events and revolutions of the great Newtonian epoch. And even with regard to smaller problems, such as the Motion of Projectiles or the Oscillations of Pendulums, a person, in order to understand the principles which have been applied to their solution, must have them brought before him in the distinct and elementary form in which they were at first treated; and not merely as cases included in wide symbolical expressions of mechanical principles.

63 Thus the geometrical form of the mathematical subjects which have been mentioned are valuable parts of a Liberal Education, not only as being the best examples of rigorous reasoning, but as having been always regarded as the standard achievements of human reason; and thus possessing an historical as well as a disciplinal character. This historical character of the branches of Mathematics may be much obscured, and the consequent value of a mathematical education much

impaired, by treating the subjects in a merely analytical manner. Analysis presents each subject under its symbolical general forms, obtained by symbolical operations from fundamental principles; and thus puts out of sight, entirely or nearly, all the peculiar conceptions and terms which the original mathematical explorers of these subjects, proceeding in the geometrical way, had employed. A person may possess great knowledge of the analytical forms of Geometry, Trigonometry, Conic Sections, Mechanics, Dynamics, and yet know nothing at all of their history, or even of the principal terms in which their history is told. A person may be well acquainted with the formulæ of Analytical Trigonometry, and even able to combine them with skill; and may yet be ignorant of the meaning which the words Sine, Cosine, Tangent, Secant, Versed Sine, have had in all mathematical books till within these few years. With the same knowledge, he may be unable to solve a triangle in numbers, or to use a Table of Logarithms. A person may be well taught in Analytical Conic Sections, and may not know a single proposition of those which constituted the study of Conic Sections from the time of Apollonius to that of Newton, and which alone gave it its interest. A person who is thus ignorant of the propositions which belonged to Conic Sections in Newton's time, and of their demonstrations, will necessarily be unable to understand Newton's reasonings in the Principia; for these reasonings assume the ancient propositions, and follow a like mode of proof. And he may be entirely ignorant of every line of the Principia, and of every step of Newton's train of discovery and demonstration, and even unable to follow Newton's reasoning as presented by himself, though he is intimately acquainted with the modes in which the Mechanics of the Universe has been analytically treated.

64 Now if the analytical modes of treating the

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
Educational Study.

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