There is no principle in Multiplication that is not found in Addition; and Division is but a different kind of Subtraction.

The first examples in Addition should consist of such numbers that the sum of those under each denomination can not exceed nine. The first examples in Subtraction should consist of such numbers that each number of a certain denomination in the minuend should exceed the number of the same denomination in the subtrahend. The first examples in Multiplication should consist of such numbers that none of the products of numbers in the multiplicand by the multiplier can exceed nine. The first examples in Division should consist of such numbers that the divisor can be contained in each number of the dividend without a remainder. The first divisors used in what is called Long Division should be less than ten. In all cases the progress of the pupils should be gradual; but one point of difficulty should be presented at a time. Much practice should be allowed them in order to secure rapidity and accuracy in the performance of their work. Solutions should be neatly written upon blackboards and properly explained. Forms of explanation may be obtained from text-books; but teachers should be careful to have their pupils understand them and not merely cornmit them to memory. Teachers will find the construction of Addition, Subtraction, Multiplication, and Division tables, by their younger pupils, a very valuable auxiliary in familiarizing them with the processes involved. The terms applied to the numbers used in Subtraction are Minuend, Subtrahend, and Dif

ference. Any two of these being given, a third can be found. The same is true in Multiplication with reference to the Multiplicand, Multiplier, and Product; and in Division with reference to the Dividend, Divisor, and Quotient. I mention these facts here, in order to say that such problems present work of much value to learners.

7. Exercises in the Solution of practical Examples involving the four fundamental Rules. - Pupils not only need to know how to perform simple Arithmetical operations, but when they are required to be performed. For this purpose numerous practical problems must be presented. All text-books contain some such problems; but none of them within my knowledge contain one-fourth as many as are needed. The teacher must supply this deficiency. They are so well calculated to give interest to the study and to make pupils think, that I am disposed to consider them almost indispensable.

8. Exercises in imparting the Idea of a Fraction. The basis of all Arithmetical operations is the unit. The unit may be multiplied or divided, and these processes really constitute the whole of Pure Arithmetic. All Integers may be called multiplied units, and all Fractions, divided units. Particular whole numbers denote the extent of the multiplication, and particular fractions denote the nature of the division.

The idea of a fraction is formed upon seeing things broken up or divided. Pupils have the idea when they enter school, but the teacher must expand it by

exhibiting and naming the parts of objects. For this purpose, an apple may be cut into parts, a stick may be broken into pieces, or a line, a square, or a circle, drawn on a blackboard, may be divided into sections. Such instruction should be continued until the pupils can readily name the fraction upon seeing the object, or find an object which is represented by the fraction; or, in other words, until they learn to count fractionally.

9. Exerciscs in adding, subtracting, multiplying, and dividing fractions orally.-At this stage of their progress, pupils may perform orally with much advantage some of the simpler problems in Addition, Subtraction, Multiplication, and Division of Fractions. Such questions as the following may be asked: In Addition: What is the sum of one-half and one-half? one-third and one-third? one-fourth and two-fourths? one-half and one-fourth? one-half and one-third? &c.; in Subtraction: What is the difference between one and one-half? three-fourths and one-fourth? one-third and one-sixth? one-half and one-third? &c.; in Multiplication: What is the product of two times one-half? three times one-third? four times one-sixth? one-half times two? one-half times one-half? &c.; in Division: how many halves in one? in two? in five? how many times is two contained in one-half? in one-third? in two-fourths? how many times is one-fourth contained in one-fourth? in one-half? in one-eighth? &c. All this can be beautifully illustrated with squares drawn upon the blackboard and divided into the requisite numbei of parts. As soon as possible, however, pupils

should be taught to solve such problems without depending upon objects.

10. Exercises in teaching Fractional Expressions. When pupils have attained a clear idea of a fraction, it will not be difficult to teach them to express it. The simplest fractions are those in which the numerator is unity, and, therefore, pupils should first be taught to write,,,,, &c.; and afterwards fractions in which the numerator is greater than unity; as 3, 4, 3, 11, &c. Pupils may be required to write fractions representing the given parts of squares or circles drawn upon the blackboard, or they may divide such figures so that certain given fractions will represent them.

11. Exercises in the Addition, Subtraction, Multiplication, and Division of Fractions, and their Applications. -Pupils are now prepared to enter upon the work of adding, subtracting, multiplying, and dividing fractional numbers, and of making an application of them in the solution of practical problems. The work may be done orally or by writing. The simpler operations of fractions can be understood by inspection; but when pupils are prepared for it, the rules for finding the Greatest Common Divisor, the Least Common Multiple, and all other rules relating to Fractions must be rigidly demonstrated.

12. Exercises in Decimal Fractions.-With a knowledge of the Decimal Notation and of Common Fractions, it will be no difficult task for a pupil to learn Decimal Fractions, for there is no new prin

ciple involved. A Decimal Fraction is a fraction whose denominator is always 10 or some product of 10. Such fractions are written by placing a point, called the Decimal Point, before the numerator. This point indicates that the number of figures in the numerator to the right of it is equal to the number of cyphers in the denominator, and hence does away with the necessity of writing the denominator.

Instruction in Decimals must begin by making pupils thoroughly acquainted with the Decimal Notation. They must be taught both to read and to write Decimals with facility. The Decimal Notation may be taught in the same manner as the notation of integers; but this trouble need scarcely be taken, as pupils can almost as easily read or write tenths, hundredths, thousandths, as tens, hundreds, thousands.

All the rules in the Addition, Subtraction, Multiplication, or Division of Decimals may be shown to be true either by reducing the Decimals to Common Fractions, or from the nature of the Notation itself. Text-books exhibit both methods, and it is unnecessary to detail them here.

13. Exercises in Compound Numbers. In the Compound or Denominate Numbers, the units increase according to varying scales. These scales are fixed by some authority, and follow no regular law. Pupils must, therefore, commit them to memory; but when the tables of Weights and Measures are well understood, the Addition, Subtraction, Multiplication, and Division of Compound