SECTION IV.-Arithmetic.

Arithmetic is the science which explains the properties and relations of numbers, and the method of computing by them. A knowledge of this subject should form a part of every system of education, as its principles and rules form the groundwork of all the computations connected with commerce, geometry, mensuration, geography, astronomy, navigation, and other departments of science.

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Having chalked such figures as the above, the children may be taught to say, "One line, one triangle, one circle, one square-two lines, two triangles, two circles, two squares-three lines, three triangles, three circles, three squares," &c. which may be continued to twelve or twenty, or any other moderate number. They may be likewise taught to repeat the numbers either backwards or forwards, thus: "One triangle, two triangles, three triangles, four triangles"-"Four circles, three circles, two circles, one circle." The nature of the four fundamental rules of arithmetic may be explained in a similar manner. Drawing five squares or lines on the board, and afterwards adding three, it would be seen that the sum of 5 and 3 is eight. Drawing twelve circles, and then rubbing out or crossing

Previous to engaging in the regular study of this science, and attempting its more complex operations, the general properties of numbers should be familiarly illustrated by sensible representations, in a manner similar to what is generally practised in infant schools. This may be done either in private by an intelligent parent, or in a public school, as an occasional amusement for those who have not entered on the regular study of arithmetic; which would prepare them for understanding its fundamental rules and computations. A variety of moveable objects, as peas, beans, beads, marbles, cubes, &c. may be provided, or perhaps small pieces of wood cut in the shape of cubes or parallelopipeds, as they do not roll, may be more convenient for this purpose and a method such as the following, corresponding to the spirit and plan of Pestalozzi, may be pursued. The teacher, placing one of the cubes before the children, says, "This is one cube;" the children at the same time repeat, "This is one cube." The teacher, adding another, says, "These are two cubes," which the children likewise repeat. This process may be continued till they advance to the number ten. Then, taking all the cubes from the table, and throwing down four, the question is put, How many cubes are on the table? which the children, after having been for some time familiarized to this mode of notation, will be able to answer. In like manner, other numbers may be successively placed on the table, and similar questions put. This process may be varied as follows: Plac-three of them, it will be seen that if 3 be taken from ing a parallelopiped or oblong figure before the children, the teacher may say, "Once one"-placing another at a little distance from the first, "Twice one"-adding another, "Three times one;" and so on, making the children repeat the numbers as the pieces are laid down. When the ten oblongs are thus arranged at equal distances and in a straight line, such questions as the following may be put. How many oblongs are there on the table? Do they lie close together? Is the first oblong placed nearer to the second than the second is to the third? Do their long sides lie in the direction of the window or of the door, &c.? Could they be placed differently without changing either their number or dis- When the arithmetical pupil proceeds to the comtance? When these questions are answered, they pound rules, as they are termed, care should be taken may then be desired either to shut their eyes or to to convey to his mind a well-defined idea of the return their backs to the table, when three oblongs lative value of money-the different measures of may be taken away, and the second moved nearer length, and their proportion to one another-the rethe first, and the question put, How many oblongs lative bulks or sizes of the measures of solidity and are there now? The children, having counted capacity-angular measures, or the divisions of the them, will say, "There are seven." How many circle-square measure-and the measure of time.were there before? "Ten." How many have I The value of money may be easily represented, by taken away? "Three." Did these oblongs under-placing six penny pieces or twelve half-pennies in a go any other change? "You have moved that row, and placing a sixpence opposite to them as the (pointing to it) nearer to the other." In order to value in silver; by laying five shillings in a similar vary these processes as much as possible, the chil-row, with a crown piece opposite; and twenty shildren should be desired to count the number of fin-lings, or four crowns, with a sovereign opposite as gers on one or both hands, the number of buttons on their jackets or waistcoats, the number of chairs or forms in the room, the number of books placed on a table or book-shelf, or any other object that may be near or around them. By such exercises, the idea of number and the relative positions of objects would soon be indelibly impressed on their minds, and their attention fixed on the subject of instruction.

12, nine will remain. In like manner the operations of multiplication and division might be illustrated. But it would be needless to dwell on such processes, as every intelligent parent and teacher can vary them to an indefinite extent, and render them subservient both to the amusement and the instruction of the young. From the want of such sensible representations of number, many young people have been left to the utmost confusion of thought in their first arithmetical processes, and even many expert calculators have remained through life ignorant of the rationale of the operations they were in the habit of performing.

the value in gold; and so on, with regard to other species of money. To convey a clear idea of mensures of length, in every school there should be accurate models or standards of an inch, a foot, a yard, and a pole. The relative proportions which these measures bear to each other should be familiarly illustrated, and certain objects fixed upon, either in the school or the adjacent premises, such as the length of a table. the breadth of a walk, the extent

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Height, 15.

such a question, or of the process by which the true | square mile-27 cubical feet in a cubical yard, &c. result may be obtained, without figures and accom-For example, the number of square feet in a square panying explanations. Yet no previous explanation yard, or in two square yards, &c. may be representis given in the book, of what is meant by the square ed in either of the following modes.

of any dimension, or of the method by which it may be obtained. Figures, such as the following, should accompany questions of this description.

Floor and Roof.

1 Square Yard.

When the dimensions of the mason work of a house are required, the different parts of the building, which require separate calculations, as the sidewalls, the end-walls, the gables, the chimney-stalks, delineations are not found in the books where the &c. should be separately delineated; and if such questions are stated, the pupil, before proceeding to of the several dimensions which require his athis calculations, should be desired to sketch a plan tention, 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 hypotenuse of a right-angled triangle, is equal to the sum of the squares of the other two sides." It should ha represented as under.

49 Miles.

PEEBLES

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 necessarily give the number of square feet, yards, inches, &c. in the sur- In a similar manner should many other examples face. It will also show that surfaces of very differ- connected with the extraction of roots be illustrated. ent shapes, or extent as to length or breadth, may The following question can scarcely be understood contain the same superficial dimensions. In the or performed, without an illustrative figure, and yet same way we may illustrate the truth of such posi- there is no figure given, nor hint suggested on the tions as the following:-That there are 144 inches subject, in the book from which it is taken. "A in a square foot-9 square feet in a square yard-ladder, 40 feet long, may be so placed as to reach 160 square poles in an acre-640 square acres in a a window 33 feet from the ground on one side of the

By this figure, the pupil will see that his calculations must have a respect to two right-angled triangles, of which he has two sides 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 following fractions, 2, ,, may therefore be represented by a delineation, as under.

| 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 occur rences-that facts in relation to commerce, geogra phy, astronomy, natural philosophy, statistics, and other sciences, should be selected as exercises in the different rules, so that the pupil, while engaged in numerical 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, instead 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 3 of of a pound sterling will buy 3 yards and of 3 of a yard of cloth, how much will of 3 of a yard cost?""The number of scholars in a school was 80; there Iwere 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 ?"

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By such delineations, the nature of a fraction, and the value of it, may be rendered obvious to the eye In some late publications, such as "Butler's of a pupil. A great many other questions and pro- Arithmetical Exercises," and "Chalmers' Introduccesses in arithmetic might, in this way, be render- tion to Arithmetic," a considerable variety of bioed clear and interesting to the young practitioner graphical, historical, scientific, and miscellaneous in numbers; and where such sensible representa-information is interspersed and connected with the tions have a tendency to elucidate any process, they different questions and exercises. If the facts and ought never to be omitted. In elementary books 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 general 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 welldefined conceptions to juvenile minds, and of rendering every scholastic exercise in which they engage interesting and delightful.

SECTION V.-Grammar.

the words beginning with J must be sought for under the letter I, and the words beginning with V, under the letter U, causing to every one a certain degree of trouble and perplexity, when searching for words beginning with any of these letters. Most of our school Dictionaries and Encyclopedias are still arranged on this absurd principle, which should now be universally discarded.

In the construction of our books of Grammar for the use of children,-instead of facilitating this study, we have done every thing to render it as dry and intricate as possible. We have definitions, general rules, exceptions to these rules, declensions and conjugations, profusely scattered throughout every part of these scholastic manuals, and a cart load of syntactical rules and examples, all of which must of course be crammed, like a mass of rubbish, into the memories of the little urchins, although they should not attach a single correct idea to any portion of such scholastic exercises. Nothing can be more simple than the English verb, which, unlike the Greek and Latin verb, has only two or three varieties in its termination; yet, we perplex the learner with no less than six different tenses-the present, the imperfect, the perfect, the pluperfect, the first future, and the future perfect, while nature and common sense point out only three distinctions of time in which an action may be performed; namely, the past, the present, and the future, which of course are subject to a few modifications. On the same principle on which we admit six tenses, we might introduce nearly double that number. Hence a celebrated grammarian, Mr. Harris, in a dissertation on this subject, enumerates no fewer than twelve tenses. It is quite easy to make a child understand that a man is now striking a piece of iron with a hammer, that he did the same thing yesterday, and will perform the same action to-morrow,

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 are nothing else than a collection of unmeaning sounds. In most instances they might as well be employed in repeating the names of the Greek characters, the jingles of the nursery, or a portion of the Turkish Alcoran. The following is the opinion of Lord Kaimes on this point:-" In teaching a language, it is the universal practice to begin with grammar, and to do every thing by rules. I affirm this to be a most preposterous method. Grammar is contrived for men, not for children. Its natural place is between language and logic: it ought to close lectures on the former, and to be the first lectures on the latter. It is a gross deception that a-in other words, that an action was performed at language cannot be taught without rules. A boy who is flogged into grammar rules, makes a shift to apply them; but he applies them by rote like a parrot. Boys, for the knowledge they acquire of a language, are not indebted to dry rules, but to practice an observation. To this day, I never think without shuddering, of Disputer's Grammar, which was my daily persecution during the most important period of my life. Deplorable it is that young creatures should be so punished, without being guilty of any fault, more than sufficient to produce a disgust at learning, instead of promoting it. Whence then this absurdity of persecuting boys with grammar rules?"

In most of our plans of education, instead of smoothing the path to knowledge, we have been careful to throw numerous difficulties and obstacles in the way. Not many years ago, we had two characters for the letter s, one of them so like the letter f, that, in many cases, the difference could not be perceived. We had likewise compound-letters, such as ct, sl, sh, &c. joined together in such an awkward manner, that the young could not distinguish them as the same letters they had previously recognised in their separate state; so that, in addition to the ungracious task of learning the letters of the alphabet in their insulated state, under the terror of the lash, they had to acquire the names and figures of a new set of characters, before they could peruse the simplest lessons in their primers. Such characters, it is to be hoped, are now for ever discarded. We have still, however, an absurd practice in our dictionaries and books of reference, which tends to perplex not only our tyros, but even our advanced students, when turning up such works-I mean the practice of confounding the letters I and J, and the letters U and V, which are as distinct from each other as a vowel is from a consonant; so that all

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some past time, is performing now, or will be performed at some future period; but it is almost impossible to convey to his mind a clear idea of twelve, or even of six, tenses, although a hundred distinctions and definitions should be crammed into his memory. A disposition to introduce quibbling and useless metaphysical distinctions has been the bane of theology, and one of the causes of the divisions of the Christian church. A similar disposition has rendered grammar perplexing and uninteresting to young minds, and prevented them from understanding or appreciating its nature and general principles. By attempting too much, in the first instance -by gorging their memories with all the distinctions, modifications, and rules, which grammarians have thought proper to inculcate, we have produced a disgust at the study, when, by attempting nothing more than they were able clearly to comprehend, we might have rendered it both delightful and instructive. There are, properly speaking no oblique cases in English nouns, excepting the possessive case, and yet, in some grammars, we have six cases specified, similar to those of Latin nouns; and in almost every book on grammar, three cases at least are considered as belonging to English nouns. On the same principle, we might affirm that there are as many cases as there are prepositions in the language; for every combination of a preposition with a noun forms a distinct relation, and consequently may be said to constitute a distinct case.Were it expedient in this place, many such remarks might be offered in reference to the absurdities and intricacies of our grammatical systems, and the perplexing and inefficient modes by which a knowledge of this subject is attempted to be communicated.

In communicating to the young a knowledge of grammar, or of any other subject, that plan which