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improvement is to be done at present, it must not only extend its effects to the Schools, but must begin at the Schools and I have the strongest conviction that those who wish to improve English education ought to direct their efforts to those quarters much rather than the Universities, as points on which their action, if successful, will produce a much wider and deeper effect.

364 Of course, when I speak here of the improvement of the system of the Great Schools, I do not mean any improvement in the mode of teaching Greek and Latin, but the improvement of combining, with Classical reading, other elements in the School education. I believe that in the Classical portion of education, and in the general tone and temper of the education of boys, great improvements have in recent years been introduced in most of our Great Schools: and I would beg the eminent scholars who direct the education of these Schools to believe that all which I say is said with the most entire respect for them, and with great admiration of the manner in which they discharge and elevate their office, so far as I am acquainted with the cases. They will, I am sure, see nothing but a fair discussion of an important subject about which we are all deeply interested, in any attempt to consider of what elements the education of a Great School ought to consist, and how these may be introduced into their places. I must treat the subject in a general manner; not only because such is the most proper mode of dealing with it, but because I really have not, in my mind, any reference to one particular place or mode of practice rather than another, amongst those which now exist.

365 After what I have already said, my readers will not be surprised at my again saying that the Mathematics ought to be taught at school, so far as to be a preparation for the Mathematics which are to be studied at the University; nor at my adding that [PT. II.]

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the present Mathematical teaching at several of our Great Schools fails of satisfying this condition with regard to a great number of their scholars, many of them very well instructed in Classics. Nor shall I here attempt further to illustrate these propositions. That Mathematics is a necessary portion of a Liberal Education, I have endeavoured to show in the first part. But Mathematics cannot be studied to any purpose at the University, except an effectual beginning is made at school. This is true, even of speculative portions of Mathematics, such as Geometry, in which the main point is to be able to understand and to state the proofs of the propositions which belong to the science. It is still more true of practical Sciences, such as Arithmetic, Algebra, and Practical Trigonometry, in which the learner has to apply rules and to perform operations, which it requires considerable time and application to learn to apply and to perform correctly, and still more, to perform both correctly and rapidly. If this is not learnt during the period of boyhood, at least with regard to Arithmetic, it is never learnt ; and when this is the case, all real progress in Mathematics is impossible. Yet how imperfectly Arithmetic is generally learnt at our Great Schools, is remarkable to the extent of being curious, besides being, as I conceive it is, a great misfortune to the boys. The sons of great merchants, bankers, and fundholders, when they leave school, are very generally incapable of calculating the discount upon a bill, and often not able to add up the sums of an account. And few indeed of the sons of our great landowners can calculate the area of a field of irregular, or even of regular form, and given dimensions. This appears to be a lamentable state of things on every account:-in its first and lowest bearing, because such ignorance is a great impediment in the practical business of life: in the next place, because Arithmetic is in itself a good discipline

of attention and application of mind, and when pursued into its applications, an admirable exercise of clearness of head and ingenuity:-in the next place, because, as the boys of the Middle Classes at Commercial Schools are commonly taught Arithmetic (and generally Mensuration also) effectively and well, the boys from the Great Schools have, in this respect, an education inferior to that which prevails in a lower stage of society-and, in the next place, again, because the want of Arithmetic makes it impossible that such young men should receive a good education at the University. On all these accounts, it appears to me in the highest degree desirable that Arithmetic, at least, should hold a fixed and prominent place in the system of our Great Schools. Such facts as I have above stated, show that at present it does not hold this place; and whatever means have hitherto been employed in teaching Arithmetic in the Great Schools, it is plain that they are insufficient with regard to the majority of the scholars: and I should say, plain, that new and better means should be devised and introduced.

366 I do not pretend to be sufficiently well acquainted with the system of Great Schools to be able to suggest how this may best be done: but it is plain, I conceive, that it must be done by making Arithmetic a part of the System of the School; not an appendage to the ordinary work of the school, enjoined it may be, and provided for, but not looked upon with any regard and respect by those who govern the school, and direct the minds of the boys. Arithmetic, and when that has been mastered, Geometry, Mensuration, Algebra, and Trigonometry, in succession, should form a part of the daily business of every school which is intended to prepare students for the University. I am aware that it has been said that any substantial attention to such subjects interferes with the Classical teaching; be

cause the Classes of boys framed according to their knowledge of Greek and Latin will differ from the Classes according to their knowledge of Mathematics. Of course this is a difficulty; but it is a difficulty which should be overcome. It has hitherto been in a great measure overcome in the University and in our Colleges. It is a difficulty which, if we yield to it, and allow it to deter us from the attempt to improve our education, will make it impossible for us to have a Liberal Education; because it will exclude all but one element. At this rate, we shall teach our boys Greek and Latin, and not teach them anything else, for fear it should interfere with Greek and Latin; and this, during the first eighteen or nineteen years of life, when they might learn the elements of all human knowledge, and acquire habits which would lead them into any part of literature or science, according to their intellectual tendencies.

367 It may be said that the University and the Colleges ought to compel the Schools to teach the Elements of Mathematics, by requiring a certain quantity of Mathematics of all their students;—that to a certain extent they do this; and may do it still further, if the present state of things in this respect is not satisfactory. This may be said: and it is true, and will I hope be acted on. I hope that the Colleges, or the University, will require of all students who come to them a real and practical acquaintance with Arithmetic and Geometry. But this will hardly continue to be the case, except something effectual is done to promote the learning of Arithmetic and Geometry at the Great Schools. We know that what is learnt merely in order to pass a single Examination, is always learnt in an imperfect and ineffective manner;-never forms the mental habits, or remains as a permanent possession. Arithmetic learnt for such a purpose, is not a familiar practical art, as, to be of any value, it

should be. Then consider again, what would be the effect, if a rule, that all students should bring with them to the University a good knowledge of Arithmetic and Geometry, were imposed with rigour, while the subject was taught at the schools, as now, reluctantly, and barely as far as was likely to be insisted on. Such consequences as this would ensue :-that some of the best scholars from the best schools, known already to be such by their performances at school, would, in consequence of their Mathematical deficiencies, be turned away from the gates of our Colleges when they presented themselves for admission. Is it likely that this would continue to be done rigorously and steadily?-Done too when the persons who hold the doors of those Colleges are themselves, perhaps, in a large proportion, men from Great Schools, sympathizing with the scholars of those schools:-sympathizing with their taste, their ambition, perhaps their ignorance. The unlikelihood that such a system of rigour will long be persisted in, except the Schools begin by really teaching Arithmetic and Geometry as a part of their system, makes me look to the latter course as the only one which is likely to prevent the further narrowing of our University Education.

368 Geometry, that is, a few Books of Euclid, has hitherto been commonly taught along with Arithmetic, at the Schools from which young men commonly proceed to the University. It is not quite so necessary that Geometry should be well studied at school as it is that Arithmetic should be well taught there because in Geometry, the learner has only to understand and to remember, whereas in Arithmetic he has to work in virtue of acquired habit. A student at the University, if he had very good mental talents, might perhaps go forwards and acquire a good knowledge of Mathematics, even if he had his Geometry to begin after his arrival. Still it is not very likely that