The idea of superficial measure, and the reason why we mul tiply two sides of a quadrangular figure in order to obtain the superficial content, may be illustrated as follows. Suppose a square table whose sides are 6 feet feet long, and another of the form of a parallelogram, 9 feet long, and 4 feet broad, the superficial feet contained in these dimensions may be represented as under6×6=36, and 9×4=36.

By such a representation it is at once seen what is meant by a square foot, and that the product of the length by the breadth of any dimension, or of the side of a square by itself, must neces sarily give the number of square feet, yards, inches, &c. in tne surface. It will also show that surfaces of very different shapes, or extent as to length or breadth, may contain the same superficial dimensions. In the same way we may illustrate the truth of such positions as the following:-That there are 144 inches in a square foot-9 square feet in a square yard-160 square poles in an acre-640 square acres in a square mile-27 cubical feet in a cubical yard, &c. For example, the number of square feet in a square yard, or in two square yards, &c. may be represented in either of the following modes.

1 Square Yard.

1 Square Yard.

2 Square Yards.

2 Square Yards.

When the dimensions of the mason work of a house are required, the different parts of the building, which require separate calculations, as the side-walls, the end-walls, the gables, the chimney-stalks, &c. should be separately delineated; and if such delineations are not found in the books where the questions are stated, the pupil, before proceeding to his calculations, should be desired to sketch a plan of the several dimensions which require his attention, in order that he may have a clear conception of the operations before him. Such questions as the following should likewise be illustrated by diagrams. "Glasgow is 44 miles west from Edinburgh; Peebles is exactly south from Edinburgh, and 49 miles in a straight line from Glasgow. What is the distance between Edinburgh and Peebles?" This question is taken from "Hamilton's Arithmetic,” and is inserted as one of the exercises connected with the extraction of the Square Root; but no figure or explanation is given, excepting the following foot-note. "The square of the hypothenuse of a right-angled triangle, is equal to the sum of the squares of the other two sides." It should be represented as under.

GLASGOW

44 Miles.

EDINBURGH

49 Miles.

In a similar manner should many other examples connected with the extraction of roots be illustrated. The following question can scarcely be understood or performed, without an illus. trative figure, and yet there is no figure given, nor hint suggested on the subject, in the book from which it is taken. "A ladder, 40 feet long, may be so placed as to reach a window 33 feet from the ground on one side of the street; and by only turning it over, without moving the foot out of its place, it will do the same by a window 21 feet high on the other side. Required the breadth of the street?" The following is the representation that should be given, which, with a knowledge of the geometrical proposition mentioned above, will enable an arithmetical tyro to perform the operation, and to perceive the reason of it.

By this figure, the pupil will see that his calculations must have a respect to two right-angled triangles, of which he has twosides of each given to find the other sides, the sum of which will be the breadth of the street. The nature of fractions may be illustrated in a similar manner. As fractions are parts of a unit, the denominator of any fraction may be considered as the number of parts into which the unit is supposed to be divided. The follow ing fractions,,,, may therefore be represented by a deline. ation, as follows:

By such delineations, the nature of a fraction, and the value of it, may be rendered obvious to the eye of a pupil.-A great many other questions and processes in arithmetic might, in this way, be rendered clear and interesting to the young practitioner in numbers; and where such sensible representations have a tendency to elucidate any process, they ought never to be omitted. In elementary books on arithmetic, such delineations and illustrations should frequently be given; and, where they are omitted, the pupil should be induced to exert his own judgment and imagination, in order to delineate whatever process is susceptible of such tangible representations.

I shall only remark further, on this head, that the questions given as exercises in the several rules of arithmetic, should be all of a practical nature, or such as will generally occur in the actual business of life-that the suppositions stated in any question should all be consistent with real facts and occurrences-that facts in relation to commerce, geography, astronomy, natural philoso phy, statistics, and other sciences, should be selected as exercises in the different rules, so that the pupil, while engaged in numeri

cal calculations, may at the same time be increasing his stock of general knowledge—and that questions of a trivial nature, which are only intended to puzzle and perplex, without having any practical tendency, be altogether discarded. In many of our arithmetical books for the use of schools, questions and exercises, in stead of being expressed in clear and definite terms, are frequently stated in such vague and indefinite language, that their object and meaning can scarcely be appreciated by the teacher, and far less by his pupils and exercises are given which have a tendency only to puzzle and confound the learner, without being capable of being applied to any one useful object or operation. Such questions as the following may be reckoned among this class. "Suppose £2 and of of a pound sterling will buy three yards and of of a yard of cloth, how much will of of a yard cost?" "The number of scholars in a school was 80; there were one-half more in the second form than in the first; the number in the third was of that in the second; and in the fourth, of the third. How many were there in each form?"

In some late publication, such as "Butler's Arithmetical Exercises," and "Chalmers' Introduction to Arithmetic," a considerable variety of biographical, historical, scientific, and miscellaneous information is interspersed and connected with the different questions and exercises. If the facts and processes alluded to in such publications, were sometimes represented by accurate pictures and delineations, it would tend to give the young an interest in the subject of their calculations, and to convey to their minds clear ideas of objects and operations, which cannot be so easily imparted by mere verbal descriptions; and consequently, would be adding to their store of genial information. The expense of books constructed on this plan, ought to be no obstacle in the way of their publication, when we consider the vast importance of conveying well-defined conceptions to juvenile minds, and of rendering every scholastic exercise in which they engage interesting and delightful.

SECTION V.-Grammar.

Grammar, considered in its most extensive sense, being a branch of the philosophy of mind, the study of it requires a considerable degree of mental exertion; and is, therefore, in its more abstract and minute details, beyond the comprehension of mere children. Few things are more absurd and preposterous than the practice, so generally prevalent, of attempting to teach grammar to children of five or six years of age, by making them commit to memory its definitions and technical rules, which to them