it is desirable that our mathematical education should make our students acquainted with the best and most recent solutions of such problems; because, by this means their mathematical knowledge has solidity and permanence given it, by its verification in facts, and its coincidence with the experience of practical men. Hence, I would advise that we should introduce among the books of which we encourage the study among our best mathematical students, three excellent works of recent date: namely, Poncelet's Mécanique Industrielle, in which he has given modes of calculating the amount and expenditure of Labouring Force*; Professor Willis's Principles of Mechanism, in which he has classified all modes of communicating motion by machinery, and investigated their properties; and Count de Pambour's Theory of the Steam Engine, in which a sound mathematical theory is confirmed by judicious experiments. The study of these works would put our students in possession of the largest and most philosophical doctrines which apply to Engineering; and would thus give a tangible reality and practical value to their mathematical acquirements; while, at the same time they would find, in the works thus brought before them, excellent examples of mathematical rigour, ingenuity, and beauty.

77 Besides the great works of mathematical inventors, there have been produced many Systematic Treatises, in which the original inventions have been collected, methodized, and often simplified. I have said that our best mathematical students are to be directed to the great original works, rather than to these compilations. But in some measure, it will be necessary for all students to have recourse to such works.

*This term appears to me the best translation of Poncelet's term Travail; and I have accordingly used it in The Mechanics of Engineering, in which I have given some of Poncelet's results.

It is, for instance, more convenient for all students to acquire their knowledge of the Differential and Integral Calculus in a systematic work like Lacroix's Traité, rather than to gather up the various artifices of the Calculus out of the successive works of their inventors. And for the like reason, the Collection of Examples of the Applications of the Differential and Integral Calculus, by Dr Peacock*, and the Collection of Examples of the Application of the Calculus of Finite Differences, by Sir John Herschel, (both originally intended as a Supplement to the Translation of Lacroix's smaller Treatise) may be recommended as containing, within a convenient space, the substance of many investigations very instructive, but too numerous and extensive to be studied in their original form. To these works on Pure Mathematics, we may add Mr Airy's Tracts on certain subjects of Applied Mathematics, namely, the Lunar and Planetary Theories, the Figure of the Earth, Precession and Nutation, and the Undulatory Theory of Optics. These works are proper subjects of the labours of our best mathematical students.

78 With regard to students who have inferior ability and diligence, but who still are candidates for distinction in their mathematical studies, their course of reading must necessarily be different from that which I have described. Their acquaintance with analytical processes, and their habits of general reasoning will, in most cases, not allow them to go far in the reading of such Capital Works as we have spoken of; and they must, for the most part, acquire their knowledge from Systematic Treatises, and principally from Elementary Treatises. The selection of the Elementary Treatises

* This work may now be considered as replaced by the Examples of the Processes of the Differential and Integral Calculus, published in 1841, by the late lamented Mr Gregory; Dr Peacock not having had leisure to publish a second edition of his "Examples."

which are thus to be used is an important point in a scheme of education; and on this subject I shall venture to make one or two remarks.

79 In the first place, I remark that the Elementary Treatises which we use in our course of Education ought not to be too rapidly changed. For the change itself is an evil, inasmuch as it turns the student's attention to new proofs, instead of the application of known truths; and excites a craving for fresh novelties, in the place of a desire to overcome known difficulties. And moreover the new Elementary Treatises which are produced in modern times, for the most part, treat their subjects more entirely in an analytical manner than their predecessors had done; and are therefore more unfit for general educational purposes for the reasons already given in Section 5. If new Elementary Treatises of an analytical cast are easily introduced into our Educational System, the result will be very pernicious. If this be the case, the new parts of our course will naturally attract the most notice, and the greater part or the whole of the student's attention will be employed upon such novelties, without any real profit. For this change in the Elementary Treatises which are commonly read is not attended with any real progress of the sciences treated of. In every subject, treated analytically, the earliest steps of generalization admit of being variously presented; and every writer, and almost every teacher, thinks he can make some new advance in generality and symmetry. And all those devices, while new, please some readers, though they leave the science where they found it. Mere novelty appears like improvement, because it implies activity of mind in him who produces it. And thus, if new Elementary Treatises be readily admitted, the student will be perpetually carried by his teachers or his fellow-students from one elementary form of the subject to another. Instead of employing his mental labour in mastering

the difficulties of any one connected course of study, his thoughts are occupied in pursuing these detached novelties, and in considering which of them is most worthy of admiration, or in conjecturing which is most likely to receive honour. The result of such an occupation will probably be that he will know nothing well. He will ascribe an exaggerated value to those parts of his studies in which the new methods differ from the old. The conceit of a supposed knowledge of something which his predecessors did not know, will take the place of the satisfaction which he might feel from understanding what generations of thoughtful men before him have understood, or from following the intellectual processes by which real difficulties are overcome.

80 A too facile admission of new elementary works and new forms of old truths into our educational scheme, is likely to occasion a multiplication of such works, to the detriment rather than the advantage of our mathematical literature. For such works, produced on the first suggestion of some slight advance, fancied or real, in simplicity or generality or ingenuity, would not be likely to obtain any wide or permanent notice among the general mathematical public. Works so produced at a place of Education, might form a perpetual stream of transient local literature; and the students, in bestowing their attention upon such works, might be toiling on in paths held in no value in the rest of the mathematical world; and might be bestowing much labour on mathematical subjects, without approaching to any community of thought with the good mathematicians of their own and preceding times. In order to avoid such evils, I conceive that no book should be adopted into a course of education except by proper authority, and after mature deliberation. I shall afterwards venture to suggest the grounds on which such a choice should turn, and the nature of the authority by which the decision might be carried into effect.

81 It will of course be understood, from what has been said, that even when Elementary Analytical Treatises upon the various branches of mathematics have been selected and adopted in our educational course, they are not to supersede the Permanent Studies of which we have already spoken (17). Conic Sections, Mechanics, Hydrostatics, Optics, and Explanatory Astronomy, in their hitherto common form, should be mastered by every mathematical student; however he may afterwards study these subjects in the shape in which they have been presented by modern analysts. He will travel all the more securely in his analytical course in each subject, from having already gone over a part of the same ground, with the clear intuition which belongs to geometrical reasoning.

82 It may be remarked that the works which I have mentioned in Article 72 as "Capital Works," except Newton's Principia, are all by foreigners, and with the exception of Euler, by French writers of modern times. No English mathematician will be surprised at this; for the French mathematicians have undoubtedly of late been our masters and teachers. The pertinacity with which the English mathematicians clung to Newton's methods, and the mathematical controversies which soon after his time arose between Englishmen and foreigners, for a long time prevented his countrymen from adopting and following out the analytical generalizations introduced by his continental contemporaries. Yet it is not because we have no English works worthy of the mathematician's study, that I have mentioned none in my list, but because it appeared to me necessary to limit the list to few works of which the eminent place in mathematical literature is clear and undeniable. I might have recommended the beautiful geometrical investigations of Maclaurin; many ingenious solutions of problems by Emerson and Simpson; many labours of Ivory, not