65. ANALYSIS.

(a.) For convenience write the divisor at the left, and the quotient at the right of the dividend.

(b.) 483 is contained in 2630 tens of thousands, 5 tens of thousands times, with a remainder of 215 tens of thousands, equal to 2150 thousands; write the 5 tens of thousands in the place of tens of thousands in the quotient, and add the 2150 thousands to the 6 thousands of the dividend, which makes 2156 thousands.

(c.) 483 is contained in 2156 thousand 4 thousand times, with a remainder of 224 thousand equal to 2240 hundreds; write the 4 thousands in the place of thousands in the quotient, and add 2240 hundreds to the 2 hundreds of the dividend, which makes 2242 hundreds.

(d.) 483 is contained in 2242 hundreds 4 hundred times, with a remainder of 310 hundreds equal to 3100 tens; write the 4 hundred in hundreds' place in the quotient, and add the 3100 tens to the 4 tens of the dividend, which makes 3104 tens.

(e.) 483 is contained in 3104 tens 6 tens times, with a remainder of 206 tens equal to 2060 units; write the 6 tens in the tens' place in the quotient, and add the 2060 units to the 1 unit of the dividend, which makes 2061 units.

(f) 483 is contained in 2061 units 4 units times, with a remainder of 129 units; write the 4 units in units' place in the quotient, and divide the remainder of 129 units into 483 parts, giving 12 of a unit, which annex to the quotient.

(g.) Therefore 26,306,241 divided by 483 gives a quotient of 54,464123

*NOTE.-It is, perhaps, more convenient to place the divisor at the right of the dividend over the quotient.

(h.) RULE FOR LONG DIVISION.-Write the divisor at the left of the dividend as in short division.

Find how many times the divisor is contained in the LEAST number of the left hand figures of the dividend that will contain it, and place the quotient at the right of the dividend.

Multiply the divisor by this quotient figure, subtract the product from this partial dividend, and to the remainder annex the next figure of the dividend.

Divide as before, until all the figures of the dividend have been annexed and divided.

If any partial dividend will not contain the divisor, place a cipher in the quotient, annex the next figure of the dividend, and divide as before.

If there be a remainder after dividing all the figures of the dividend, it may be changed to a FRACTIONAL QUOTIENT by writing the DIVISOR underneath it.

(i.) PROOF.-The proof of long division is the same as that of short division.

NOTES.-1. The first step in Long Division is to find the quotient figure of the highest denomination.

2. Multiply the divisor by that figure and write the product under the first partial dividend, then observe that if the product is greater than the partial dividend, the quotient figure is TOO LARGE.

3. Subtract that product from the first partial dividend, and observe, that if the REMAINDER is equal to, or GREATER than the DIVISOR, the quotient figure is TOO

SMALL

4. Change the remainder to the next lower denomination, and add to it the corresponding denomination in the dividend.

LESSON V.

PRACTICAL EXAMPLES INVOLVING THE FUNDAMENTAL RULES.

Mr. Jones died leaving an estate worth $3,746, to be divided equally between 3 daughters and 2 sons after his wife had taken out her share amounting to $1,479; what was the share of each child?

ANALYTICAL STEP.-(a.) Find the amount to be divided after the wife has taken her share from the estate.

(b.) Find the number of children among whom the remainder is to be divided.

(c.) Find the amount which each child received.

EXERCISE.

Require the pupil to compose and analyze ten examples similar to the model.

QUESTIONS.-Repeat the analysis of Long Division. (65.) Repeat the rule. (65., h.) Give the proof. (65., 2.) What is addition? (31.) Illustrate the use of the sign of Addition. (32.) Subtraction. (43.) Multiplication. (51.) Division. (62.) What is the use of the cipher? (22.) What is the difference between the Roman and the Arabic notation? (18.,) (19.) What is the difference between the simple and the local value of a figure? (20.,) (21.) What is the difference between a simple and a complex problem? (8.,) (9.) What is the difference between a concrete and an abstract number? (4.,) (3.)

LESSON VI.

THE FUNDAMENTAL RULES-PRINCIPLES AND CONTRAC TIONS.

66. PRINCIPLES.

(a.) The greater of two numbers is equal to the less added to their difference.

*NOTE FOR THE TEACHER.-The teacher should in addition to the analytical steps require the Elementary Questions and Arithmetical Formulas, until the pupil can give them with promptness and accuracy, when they may be omitted.

Then 4186-3712+474 [Repeat prin. (a.)]

NOTE:-Let each principle be illustrated after the same manner on the blackboard.

(b.) The smaller of two numbers is equal to the remainder obtained by subtracting their difference from the greater. (c.) If the multiplier be a unit, the product will be equal to the multiplicand.

(d.) The multiplicand remaining the same, if the multiplier be increased or diminished any number of times, the product will be increased or diminished in the same ratio.

(e.) The multiplier remaining the same, if the multiplicand be increased or diminished any number of times, the product will be increased or diminished in the same ratio. (f) When the divisor is a unit, the quotient will be equal to the dividend.

(g.) When the divisor is equal to the dividend, the quo

tient will be a unit.

(h.) The dividend remaining the same, if the divisor is increased any number of times, the quotient will be decreased in the same ratio.

(i.) The dividend remaining the same, if the divisor is decreased any number of times, the quotient will be increased in the same ratio.

(j) The divisor remaining the same, if the dividend is 3

increased any number of times, the quotient will be increased in the same ratio.

(k.) The divisor remaining the same, if the dividend is decreased any number of times, the quotient will be decreased in the same ratio.

(1) The product of the divisor and quotient plus the remainder, equals the dividend.

(m.) The product divided by the multiplier is equal to the multiplicand.

(n.) From the sum of two numbers subtract their difference, and divide the remainder by 2; and the quotient will be the smaller number: to the smaller number add their difference, and the sum will be the larger number.

LESSON VII.

PROBLEMS FOUNDED ON THE PRECEDING PRINCIPLES.

NOTE FOR THE TEACHER.-The teacher should require the pupll to repeat the principle involved in each of these problems.

1. The less of two numbers is 398; their difference is 698; what is the greater? See prin. (a.)

2. The greater of two numbers is 7863; their difference is 6713; what is the smaller? See prin. (b.)

3. The multiplicand is 3679; the multiplier is 327; the product is 1203033; if the multiplier be increased 7 times, what will be the product?

4. The multiplier is 12; the product is 148,140; if the multiplier be increased 9 times, what will be the product? 5. The multiplier is 96; the product is 49164; if the multiplier be diminished 4 times, what will be the product?

6. The multiplier is 16; the multiplicand is 41862; if the multiplier be diminished 8 times, what will be the product?