afford the more useful, as well as the more interesting, examples. From whole numbers the pupil may proceed to the subject of Fractional Arithmetic; but here neither the ball-frame nor collections of counters, cubes, &c., will be available for explaining the principles upon which the rules are based. Some other mode of illustration must therefore be adopted; and the following is one which the writer has often employed with success : Let several lines of equal length be drawn upon the slate or black-board, and let the first be divided into two equal parts, the second into three, the third into five, &c., by compasses before the pupil's eyes. Let the lines be placed over each other, that their equality may be easily seen, and that all the lines may be supposed to represent the same unit or whole. Thus (1.) (2.) (3.) (4.) With the aid of such lines as these, it will be easy for the teacher to convey to the minds of his pupils clear ideas on most of the principles of Fractional Arithmetic. It will of course be necessary for him first to explain the simple definition of a fraction; although it will not perhaps be advisable at this early stage of the subject to show how fractions are expressed, or to use the terms nume rator, denominator, proper, or improper, compound, complex, &c. It is important that the lines should be equally divided; and for this reason a pair of the large wooden compasses used for the black-board should be at the teacher's command. Commencing with the line marked (1), the pupil should be told that each of the parts into which the line has been divided is called one second, or one-half, and he will of course easily perceive that the two parts together make up the whole line. In the second line each part will be called a third; and questions should be put in Addition and Subtraction, as the following:-How many thirds make up the whole line? How many are two-thirds and one-third? What part of the line will remain after cutting off one-third? What part, after cutting off two-thirds? Fourths will be taught from the first line, by subdividing the halves; and palpable proof will be offered to the pupil by the compasses. A new principle will now be derived, that one half is equal to two-fourths; and various exercises will follow, as in the former instances. When these divisions are thoroughly understood, the teacher may proceed to the other lines. If it is considered necessary to attach the idea of some particular unit to the line, it might be made exactly one foot in length, and the parts would be respectively called onehalf of a foot, one-third of a foot, &c. This phraseology may be further illustrated by such questions as the following:-A shilling may be divided into twelve equal parts, called pence: what fraction is one penny of a shilling? and other exercises of a similar nature. The next step will be to establish in the mind of the pupil the comparison of parts which are not in themselves commensurate, like fourths and seconds, sixths and thirds, &c. Fourths cannot be divided into an equal number of thirds, and the child will not at first be so familiar with the fact that one-fourth is less than one-third, as he is with regard to their reciprocals, that four is greater than three. Here then it will be necessary to adopt the subdivision of the line of thirds and fourths into twelfths, whereby it will be easily shown that one-fourth which is composed of three-twelfths, must be less than one-third which is made up of four-twelfths. When several examples of this kind have been investigated, and the child is able to perform readily any operations of addition and subtraction in connexion with the lines, he may be taught the arithmetical method of expressing fractions, or in other words that the denominator is written under the numerator, with a horizontal line between the two. The meaning of these terms may be illustrated from the lines, thus: calling upon the learner to name some fraction-for instance, twofifths he would be told by a reference to the line of fifths that the fraction belonged to the denomination of fifths, pointing out the same upon the line of fifths, and that the figure 5 would therefore represent its denominator (that which indicates the denomination). This being written down and a short line drawn over it, the fraction will be completed by placing the figure 2 above, which is called the numerator, because it numbers the value of the fraction. It is important that the pupil should not be confused with the terms numerator and denominator. Let him be told that the latter only represents the kind of fraction, whether fourths, fifths, &c., similarly to the signs £ s. d., which placed over (instead of under) numbers, denote that they are to be considered as pounds, shillings, and pence. To make him familiar with fractional expressions, easy exercises like the following may be first used 3/3 + 4 = 3, 4 + 4 − 4 = 4, &c. The pupil should then be required to point out, and also to write down, fractions having the same numerators but different denominators, and others having like denominators but unlike numerators, from which exercises the following results may be deduced : 1. Of two fractions having the same numerators, the greater is that which has the lesser denominator. 2. Of two fractions having the same denominator, the greater is that which has the greater numerator. The judgment of the instructor will be the best guide as to the length of time during which any of the exercises which have been mentioned should be continued, and also when the pupil should be required to consider fractions without reference to a line or any other particular unit. The rules for the conversion of Fractions by Reduction-those also for Addition, Subtraction, &c., of Fractions must all be carefully gone through, and the principles explained, if necessary, by reference to the lines. Whether the Rule of Three should be taught before or after Fractions, is a matter which it is scarcely necessary to discuss here. Writers on arithmetic generally place this rule immediately after Reduction and the compound rules, because its great usefulness in operations of a commercial nature makes it desirable that children should acquire some knowledge of it as early as possible. The principles of Ratio and Proportion may, no doubt, be more clearly explained by a reference to Fractions, and it may therefore be advisable to give the pupil some general ideas upon the subject. This may be done without entering into every particular connected with the rules for fractional quantities. The following lesson on the Rule of Three supposes a previous acquaintance with the nature of a fraction. LESSON ON THE RULE OF THREE. Preliminary. Ratio is the relation which one quantity bears to another of the same kind, in respect to magnitude. We can speak, for instance, of the ratio or relation between two sums of money, as 5 shillings and 50 shillings, 6 miles and 4 miles, 50 days and 9 days. But we cannot institute a comparison between things of a different kind, and therefore no relation or ratio can be said to exist between 5 days and 6 shillings, or 9 yards and 15 pence. One way of comparing two numbers, or of discovering their ratio, is to consider what part one is of the other: thus, What part is 4 shillings of 5 shillings? The part may be expressed by the fraction, and this fraction therefore expresses the ratio of 4 to 5. But the usual method of expressing ratio is by the sign () placed between the numbers. Thus, 4: 5 is the ratio of 4 to 5. What do you understand by the term proportion? The ratio of 4 to 5 was expressed by the fraction ‡. Mention another ratio which is equal to it; or, which is the same question, name a fraction equal to . Say. What ratio does express? How might the equality of these two ratios be expressed fractionally? . What sign here expresses the equality? Ans. The sign =. Now, what is the usual form of expression? Ans. 4: 5: 8:10. Point out the sign which expresses the equality? Ans. :: |