The arithmetic classes are the counterpart of the reading classes; and, according to the general character of the school, show all the intelligence and activity, or all the dulness and irregularity, which are to be seen in the latter. From the youngest upwards, however, the teaching of the several "rules," notwithstanding the hints of the "Calculator," is too little explanatory, and has too little reference to visible objects, except only in a few schools, as at New Windsor, Banbury, &c., where sensible objects are introduced among the younger children, and in others, as at Abbey-street and Louth, where an endeavour has been made to teach the rationale of numerical notation to this same class; but none of these attempts exhibit much success,-where sensible objects or the Pestalozzian Tables were used, through the monitors accepting simultaneous answers without an effort of attention from more than one or two leading children,—and in the other cases, from the monitors not themselves having sufficient power of explanation, though very apt and very intelligent in teaching and enforcing a dogmatic rule of operation. The arithmetic of the British schools is in effect, therefore, too entirely technical, like that of many schools of far higher pretensions, to be of that value to the mental culture of the children that it might be rendered, without any interruption to their progress in technical expertness; nor is there in most of the schools that occasional revertal to the early rules, and to those even of notation, which would awaken a child's perceptions to the value of the whole. It certainly appeared to me as though the full value and use of whole numbers was perceived by few who had not been compelled to look at them with eyes of discovery on encountering the new difficulties presented by fractions, and then only to a sufficient extent just to overcome them for the moment; for seldom, indeed, could any boy tell whence or why he borrowed 10 in simple subtraction, and few short of fractions had perfectly familiar command of the simple rules. Thus it always proves in a monitor's class, that three-fourths of the children who can do a sum in simple subtraction off-hand, correctly, and in a business-like style of figures, are boys far advanced in the "Calculator ;" and after them come a train of stumblers, successively worse, until, near the end of the class, would appear a total forgetfulness of the rule. The arithmetic of the girls schools is in fully as low a condition as the figures in the annexed Tables indicate; though here and there a mistress of remarkably vigorous mind will give it that clearness and vigour which characterizes every operation of her pupils.* Proper infantschools would supply what is chiefly wanted, as preliminary to the present course of arithmetic in the British schools; for the system in these schools assumes, in effect, as in the reading classes, that a considerable amount of elementary instruction has already been

* Gloucester.

conveyed (as it ought to be, wherever the population is sufficiently numerous) in schools and by teachers especially adapted to the purpose.

Nor is the course of mental arithmetic, contained in the excellent manual known as "Crossley's Calculator," though used in nearly all the British schools (Appendix XII., B.), made of that use to the intellectual progress of the children which the writer seems to have contemplated, and for which all its earlier exercises are peculiarly well adapted. The younger arithmeticians seldom share in these exercises, except when they are addressed simultaneously to the whole school, and the instruction is then virtually limited to a few; these few are chiefly children well advanced in the technical rules, who are thought not to need the early exercises of a disciplinal character, with constant reference to familiar objects, but are at once hurried on to the rules for giving expertness in the calculation of the value of dozens and grosses of articles at small prices. It is scarcely necessary to mention to those at all acquainted with British schools the extraordinary quickness exhibited by the most advanced children in these mental calculations; and the incredible rapidity with which, in some of the boys' schools, with the aid of extended Tables of multiplication of money, and of squares and cubes of numbers, the most intricate calculations are almost instantaneously solved. When carried at all to this extent, however, mental arithmetic is beyond the region of mental discipline for children whose education must terminate so early, whatever may be its value as a part of specific instruction for the occupations to which they are destined; the intellectual operation being limited to the application of the technical rules, without any necessary acquaintance with the properties of numbers on which those rules are founded.

The briefness of the time permitted for the education of the poor does, indeed, demand, not only that the principles presented to their minds shall be such as admit of ready application, but also that, to a certain extent, they shall have the technical use of numbers, as of language, beyond their conception of the rationale of the rules laid down for the use of them. Practically, the whole body of our elementary instruction, both public and private, goes upon this assumption, and leaves to a later period of a child's progress its instruction by a course of grammar and of mathematics in the rationale of the signs which it is habitually using. Art must still lead science in the elementary scholarship of all; but it is a great desideratum that art should not be allowed to leave behind altogether her graver but more faithful sister; and an incalculable service will be rendered to the children of the British schools when their system shall be such as to bring them to a clear conception of the rationale of the notation of whole numbers by the time that they have gone through the first four rules of arithmetic and their application to money; for with this

they will have acquired habits of attention and abstraction valuable through life; and without it they go very ill prepared to the rules of "proportion" and into "fractions," through which it is the usual practice to stumble blindfold.

The old method of writing down sums from the slate into an account-book is now little practised, except in the village schools; and certainly its main advantage, that of neatness, may be obtained by means more universally applicable, with less waste of time; means, however, which are not so universally in use as entirely to supply its place. There is a disadvantage, too, in the children commonly learning their tables only by ear; the aid of another sense, exercised upon a picture of the figures, would greatly facilitate the acquirement of their combinations, with a clear intelligence. But it is impossible now to dwell further upon details, beyond mentioning, as an instance of the power over the minds of mere boys which vigilance and system can give, that in the Launceston school, in which the whole of the arithmetic drafts are severally practised in mental operations by their respective monitors, I found, on coming round to the top draft from the lower, that its monitor had not a single boy to teach, while yet his grave countenance and audible whisper indicated intense and incessant application, which proved to be in putting to himself and answering hap-hazard questions in the multiplication of large numbers by the extended Table (from 13 to 20 times inclusively), in which my presence did not for a moment interrupt him.

In noticing the reading, writing, and arithmetic, I have now described the body of the instruction which is conveyed to the majority of the children who frequent the British schools; and in the girls' schools it may be said that the instruction of the majority of the children is not so much in reading, writing, and arithmetic, as in reading, writing, and needlework. All efficient instruction in the branches which yet remain to be noticed is restricted to the minority-for a minority they are, though an important. one in the aggregate-who attain to a position in the monitors' classes, or the upper sections of a simultaneous school, and are permitted to remain there, for some time, under the direct instruction of the master. It is in only a few of the most vigorous schools that there is any effective teaching of geography monitorially; and the limited number in which the great body of the school is thrown to drafts at all for the purpose of learning it, will be seen at once by a glance at the column of "Geography in Table X. How much fewer it is that employ monitorial agency in teaching grammar is shown by the next column, in which some, even of the rather larger numbers of children there entered, are taught collectively by the master, as all the smaller numbers are. In music, all the instruction is by the master collectively; in drawing and mensuration, by the master individually. The number here entered as learning music is the number learning by note.

PROGRESS in GEOGRAPHY, GRAMMAR, MUSIC, DRAWING, GEOMETRY, and MENSURATION, in the whole of the several Classes of Schools, and in the Monitors' Classes, only, of Fifty-two British Schools. (See Tables IX. and X.)

In the British schools, nearly one-fourth of the children in the boys' schools and nearly one-fifth in the girls' are returned as learning geography; while the proportion is only one-eighth in the British schools for both boys and girls in the same room, and onesixteenth in the village schools; the quality of the instruction declining in these with still greater rapidity. Of the top classes in

the British schools, however, 93 per cent. in boys' are learning geography, and 87 per cent. in those for both sexes in the same room, but only 51 per cent. in the girls' schools; and these figures fairly indicate the extent to which it is discountenanced in the girls' schools. The teachers generally are clear and intelligent in their teaching of geography; but it is less successful than it might easily be rendered by better appliances and a little modification of method. Until of late years the outfit of maps in the British schools appears to have consisted of a few outline maps, on too small a scale for efficient class teaching; and though these have been superseded, in nearly every school, by one good-sized map of Palestine; in a great number also by a large one of England, another of Europe, and a third of the two hemispheres; and in a few likewise by large ones of the different quarters of the globe, yet there is generally wanting a better supply of school-maps, which are denied to the anxious wishes of the teacher by the poverty of funds. Again, it appears to be the almost universal practice to begin by giving to the children abstract ideas of the rotundity and magnitude of the earth, and descending through its great divisions to the study of a map of England, instead of allowing the notion of a plane surface to continue for some time in teaching the children the relative distances and directions of the objects around them, with the cardinal points based on the direction of their own shadows when they leave school in the sunshine at noon; and gradually extending these ideas to the conception of a whole town, parish, district, country, island, continent, and globe. The effect of the contrary course is to make the learning of geography too truly what it is commonly called, "learning maps;" the different names being associated very readily, if not very permanently, with the spots of different colour on the surface of them, without any clear conception of the real position and dimensions of the regions pictured forth. It was in only the best schools that the top classes could decide truly, and by a considerable majority, the direction of the cardinal points; and not a few of the feebler in which it was almost unanimously voted that south was to the zenith and north. to the nadir, because the top and bottom of the map had been pointed out to them as the direction of north and south. This was particularly the case in the girls' schools, in one of which the children could tell the names of every known tribe of barbarians in Africa, and in another those of every petty island in the Pacific, without knowing the name or course of "the river” which ran through their respective towns; facts which made me entertain, with the ladies of the several committees, considerable doubt whether "geography" (so taught) were not rather an exercise breeding conceit than one either improving the character or informing the mind.

The manuals of geography employed by the teachers are various. They are merely the sources of their own information, and not