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analysis either to exercise the reasoning powers, or to render intelligible the history of human knowledge. Yet perhaps it may sometimes appear, both to teachers and to students, that it is a waste of time and a perverseness of judgment to adhere to the ancient kinds of mathematics, when we have, in the modern analysis, an instrument of greater power and range for the solution of problems, giving us the old results by more compendious methods; an instrument, too, in itself admirable for its beauty and generality. But to this we reply, that we require our Permanent Mathematical Studies, not as an instrument, but as an exercise of the intellectual powers: that it is not for their results, but for the intellectual habits which they generate, that such studies are pursued. To this we may add, as we have already stated, that in most minds, the significance of analytical methods is never fully understood, except when a foundation has been laid in geometrical studies. There is no more a waste of time in studying Geometry before we proceed to solve questions by the Differential Calculus, than there is a waste of time in making ourselves acquainted with the Grammar of a language before we try to read its philosophical or poetical literature. And a knowledge of the Sciences, as they have historically existed, is the best mode of enabling ourselves to understand their ultimate and most recent generalizations, as a knowledge of the etymology of words, and their transitions from one shade of meaning to another, is the best mode of learning to perceive all that is implied by words in their later use. There is therefore no waste of time, or perverseness of taste, when the mathematician retains and upholds, as an essential part of a general education, mathematical reasonings different from those which he would himself study or employ, if he had to deal with difficult and extensive problems. He does this, just as the most accomplished scholar

requires the student to study his grammar, though he himself has outgrown such study. Nor does the most careful regard to the maintenance of our geometrical studies, on this ground, imply any want of the intellectual taste which can perceive the beauty, and the intellectual power which can follow and continue the processes of the most refined modern analysis.

72 As I have already said, we have to select out of the whole range of modern mathematical literature those portions which are best suited to be admitted, as Progressive Studies, into a Liberal Education. In order to discuss this selection more conveniently, I will divide mathematical writings into three Classes, which I will call respectively Capital Works, Original Investigations, and Systematic Treatises.

By Capital Works in Mathematics, I mean works which hold a conspicuous place, both in mathematical history and in mathematical literature; this distinction being secured to them by their containing comprehensive and important truths connected by solid reasoning, and in them first presented as a connected whole. Such are the Principia of Newton, the Mechanica of Euler, the Théorie des Fonctions of Lagrange, the Mécanique Analytique of the same author, the Application de l'Algebre à la Géométrie of Monge, the Mécanique Céleste of Laplace.

By Original Investigations, I mean such publications as Memoirs in the Transactions of Scientific Bodies, and the like, in which mathematical investigations of detached problems are presented. Such investigations may afterwards be included in more complete Treatises; as many of Euler's and Laplace's investigations and original memoirs were afterwards included in those Capital Works of the same authors which have just been mentioned; or they may as yet not have taken this place in mathematical literature, as is the case with many highly important memoirs of

modern mathematicians; for instance, Poisson and Gauss among foreigners, and several of our own countrymen.

By Systematic Treatises I mean Treatises compiled out of the two preceding classes of works, of Elementary Treatises, intended as introductions to such works. Such are Lacroix's larger Traité des Calcul Differential et Integral, and his smaller Traité Elémentaire on the same subject. Such are the innumerable Elementary Treatises on the Differential Calculus, on Mechanics, and on other subjects, which have been published in recent times in France, Germany, and England.

I do not pretend that this division of modern mathematical literature can always be applied without difficulty; but, without attempting any rigour in the separation, the distinction will enable me, I hope, to speak intelligibly.

73 I remark, then, in the first place, that candidates for our highest mathematical honours ought to be encouraged to acquaint themselves with Capital Works, rather than with Systematic Treatises in which the same results are presented, more symmetrically and simply perhaps, but by mathematicians of inferior eminence. The historical interest belonging to every great work, added to its intellectual value, makes it fit that a man of liberal education, whose studies extend to the subject of the work, should be acquainted with the work itself, and not with any transcript of it. Moreover, it is always instructive and animating to study the works of men of genius. The mind appears to be elevated and ennobled by direct intercourse with the highest minds. On this account I recommend, for the highest students, Newton's Principia, rather than Maclaurin's account of Newton's Discoveries; Euler's Mechanica, rather than any modern Collection of Mechanical Problems; Lagrange's Mécanique Ana

lytique, rather than Poisson's or Francoeur's Mechanics; Laplace's Mechanics of the Universe, rather than Pontecoulant's. The derivative works which I have mentioned are excellent for their proper purposes; but the great original works are the proper study of a man who would pursue mathematics for the highest purposes of intellectual culture.

74 The works which I have enumerated in Article 72, as Capital Works, appear to me to suffice for the purposes of the highest Education; although others of equal, or almost equal, eminence might be added to the list, and will, of course, be read by a person who would be an accomplished mathematician. I think there are no works which have claims upon our notice superior to those above enumerated. Euler's other works, as for instance, the Treatise De Motu Corporum Rigidorum, and the solutions of Isoperimetrical Problems, have been so far superseded by more general methods, that they are not Capital Works in the same degree as the Mechanica. The Calculus Integralis of Euler is truly a Capital Work; but does not make so great a figure in mathematical history as the others which we have mentioned; and in which its substance, so far as most purposes are concerned, is included. Clairaut's Theorie de la Lune, Dalembert's Dynamique, and various other works, on the other hand, are of capital importance in mathematical history, but are included, as to their import, in the great works of Laplace and Lagrange. And since it is not possible to require or suppose, even in our highest students, an acquaintance with all the great works of mathematicians, I think it will be found that the list which I have given is ample, without being overwhelming as to its extent.

75 Whether any given mathematical work can properly be distinguished as one of the Capital Works of the subject, is a matter to be decided by the general

and permanent judgment of the mathematical world; and it must therefore be difficult to decide this question with regard to any new work. It is desirable, for the purposes of Education at least, that we should not be hasty in elevating the works of our contemporaries into this rank, and directing the attention of men to them as part of their educational studies. For the list of Capital Works which we already possess is sufficiently ample to occupy the time and thought, even of the most gifted student and to encourage a too ready pursuit of novelties, tends to promote the neglect or superficial study of the older works. For the same reason, it is not desirable, in general, to require or suppose in our students a knowledge of Original Investigations, which have not yet found their way into Capital Works, or Systematic Treatises. To urge young men, even of the greatest talents and industry, to rush into the vast field of mathematical memoirs which exists in the Transactions of Societies, and in detached Opuscules, would be to bewilder and overwhelm them. Such a course of reading is fit only for those who make mathematics a main business of life. And such a study could lead to no advantage comparable with the study of the great works of the best authors. I do not say that active-minded and industrious students should be prevented from pursuing such a line of reading, as far as their taste and their time will allow but this is not the course of reading which we ought to encourage by the turn which we give to the mathematical instructions and requirements which our education includes.

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76 Perhaps we may be a little more facile in encouraging the study of new investigations with regard to practical problems, than in recondite and speculative subjects. Problems of Engineering and Practical Mechanics naturally receive new solutions, as, in the progress of Art, they take new forms; and

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