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2. Multiply ‘008 by 34796, true to three decimal places, and prove as above.

3. Multiply ·5264 by `0428, true to three decimal places, and prove. 4. Multiply 1.729 by 7'218, true to four decimal places, and prove. 5. Multiply 26-45 by 39°46, true to two decimal places, and prove.

2. CONTRACTED DIVISION. When it is desirable in division to limit the number of decimal places in the quotient, it may be done as follows:

1. Take as many figures only, on the left hand side of the divisor, as the whole number of figures required to be in the quotient, and cut off the rest.

2. Let each remainder successively be a new dividend, without bringing down any figure from the original dividend, but, instead thereof, let another figure be continually cut off from the divisor for each quotient figure, till the whole is exhausted, observing, however, as in contracted multiplication, to increase each particular product by the nearest number of tens in the product of the quotient figure, into the figure last cut off in the divisor.

3. When the whole divisor does not contain as many figures as are re quired to be in the quotient, no figure should be cut off till the figures in the divisor shall be equal to the remaining figures required to be in the quotient, when the cutting off should commence as above directed.

Exemplifications for the Black-board. 1. Divide 74 33373 by 16346787, retaining three decimal places only, or five places in the quotient altogether.

74-33373(1,3,4,6,7187
67339 55-193 Ans.
6994
6734
260
135
125
121

4

Suggestive Questions.--How many tens are carried into the first partial product? How many into the second ? the third ? the fourth the fifth ?

The pupils should not write the partial product, but make the subtraction, as usual, mentally.

2. Divide *07567 by 2 32467, true to four decimal places ; or significant figures, the first being a cipher.

07567(2-3,2467

59 70326 Ans."

Suggestive Questions.—Are any tens carried into the first partial prod. uct? into the second ? the third ?

3. Divide 5'37341 by 3674, true to four decimal places.

5-37341(3,47,4
1633 1-4367
1374
252
28
2

Suggestive Questions.-Are any figures of the original dividend brought down to the partial dividends? If so, say how many, and why? Are any tens carried to the first partial product? to the second ? the third ? the fourth ? the fifth ?

4. Divide 1 by ‘3475, true to three decimal places.

1.0000(3,4,7,5
3050 2-878 Ans.
270
27

Suggestive Questions. How many tens are carried to the second partial product? to the third ? to the fourth ?

Exercises for the Slate or Black-board.

1. Divide 3-467 by 673367, true to four decimal places, and prove by dividing it in the usual manner.

2. Divide 1 by '462849, true to three decimal places, and prove. 3. Divide 16264531 by •92145, true to three decimal places, and prove. 4. Divide 36'4776 by «9834, true to the units' place, and prove. 5. Divide 2'12457 by “3268, true to three decimal places, and prove.

PROGRESSION.

Progression, in mathematics, signifies a regular or a proportional advance in increase or decrease in numbers. It is of two kinds :

1. Progression by differences, commonly, though improperly, called Arithmetical Progression.

2. Progression by quotients (or by ratios), quite as improperly called Geometrical Progression.

I. PROGRESSION BY DIFFERENCES.

Definitions. 1. If we take any number, and increase or diminish it continually by another number, we shall form a regular series of numbers, called a progression by differences. Thus :

2 4 6 8 10 12 12 10 8 6 4 2

form two series of progression by differences, the first ascending, the second descending, with the common difference 2.

2. The several numbers are called terms of the progression : the first and last terms are called the extremes, and the intermediate terms the

means.

3. Five things are to be considered in every progression by differences, any three of which being known, the remaining two can be found, namely

[blocks in formation]

CASE I. Where the first term, the common difference, and the number of terms, are given, to find the last, or any intermediate term (a, d, n, to find z).

Exemplification for the Black-board. 1. Form a progression of 6 terms, with 4 for the first term, and 3 for the common difference. [Let one of the class form it on the black-board, without copying it from the book.]

1st. 2d. 3d. 4th. 5th. 6th.

4 7 10 13 16 19 Suggestive Questions.—How often is d, the common difference, found in the 2d term ? What else does the 2d term contain ? Then write on your slate as follows:

Second=a+d. How often is d found in the 3d term ? Then write that term on your slate, under the 2d, and in a similar manner, namely, Third=a+2d, and so on with all the remaining terms. What does the 4th term contain besides a (the first term)? What the 5th ? the 3d ? Is the common difference, in any term, then, always repeated once less than its number? May not the following, then, be considered the first principle in Progression by Differences ? I. Every term in an increasing series of Progression by Differences con

sists of a (the first term), added to d (the common difference), taken once less than n, the number of that term.

Exercises for the Slate or Black-board.

1. When the first term is 6, and the common difference 2, what is the 4th term ? the 6th term ? the 3d term ?

2. A man bought 20 yards of cloth : he engaged to pay 6 cents for the first yard, 8 cents for the 2d, 10 for the 3d, and so on, increasing by the common difference 2 : how much did he pay for the last yard ?

Ans. 44 cents. 3. What is the 16th term of a progression by differences, whose first term is (), and the common difference 1 ?

4. What is the 18th term of a progression, whose first term is 3, and its common difference 4?

5. What is the 25th term of the progression 4, 9, 14, 19, 24, &c. ?

CASE II. Where a, z, and n are given, to find d (first, last, and number of terms

given, to find the common difference).

Exercises for the Slate or Black-board. 1. What is the common difference in a series whose first term is 4, last term 42, and number of terms 20? Prove by forming the series. If the 20th term proves to be 42, the process is correct.

Suggestive Questions. What does the 20th term contain besides a ? How, then, can d, the common difference, be found ?

2. What is the common difference in a series whose first term is 8, last term 15, and number of terms 8? Prove by forming the series. If the 8th term prove to be 15, the process is correct.

3. What is the common difference in a series whose first term is 3, last term 11, and number of terms 10? Prove as above.

4. What is the common difference in a series whose first term is 2, last term 14, and number of terms 8 ? Prove as above.

5. There are 21 persons whose ages are equally distant from each other. The youngest is 20 years old, and the eldest 60. What is the common difference of their ages ? Prove by finding the age of each.

CASE III.

Where a, z, and n, are given, to find s (first, last, and number of terms

given, to find the sum of all the terms).

Exemplification for the Black-board. 1. Find the sum of a series whose first term is 2, last term 22, and number of terms 11 ?

Ans. 264. Suggestive Questions.-What does the 11th term contain besides a ? How, then, can d, the common difference, be found ? [Let one of the class write the series on the black-board, and another write the same inverted immediately below, and also the sum of each pair of terms taken vertically.) What is the sum of the first and last terms? of the 2d and 10th ? of the 3d and 9th ? and so on to the end of the double series.

Are all these sums equal ? How many times does each sum contain the common difference? What else? Will this be the case in every series of the kind thus arranged? How many sums are there? How many terms? Will the number of sums and the number of terms always be the same in series thus arranged? What will be the product of one of these sums by the number of terms? To what, then, will half that product be equal? If the first, last, and number of terms be given, then, how can you find the sum of all the terms ? May not the following, then, be considered as the second Principle in Progression by Differences ? II. The sum of all the terms in a Progression by Differences, is equal to

half the product of the number of terms by the sum of the first and last terms.

Exercises for the Slate or Black-board.

ences.

1. What is the sum of a progression by differences, whose first term is 2, last term 100, and number of terms 50 ?

Ans. 2550. 2. A debt is to be discharged at 16 several payments, with equal differ

The first payment is to be $12, the last $100. What is the amount of the debt, neglecting all consideration of interest? Ans. $896.

3. A man engaged to travel to a certain place in 19 days, and to go but 6 miles the first day, incrcasing every day by an equal excess, so that the last day's journey may be 60 miles. What is the distance of the journey?

Ans. 627 miles. 4. How many strokes will the hammer of a common clock make on the bell during the space of 12 hours ?

Ans. 78. 5. Required the sum of all the numbers contained in a multiplication table, extending to 12 times 12.

Ans. 6084.

CASE IV.

Where z, n, and d are given, to find a (the last term, the number of

terms, and common difference given to find the first term).

Exercises for the Slate or Black-board.

1. The last term of a progression by differences is 119, the number of terms 24, and the common difference 5. What is the first term ? Ans. 4.

Suggestive Questions.-What does the last term consist of besides the first term? If that product, then, be taken away, what will remain ?

2. A note is to be paid in 10 annual instalments, the several payments being in a progression whose common difference is $30. The last payment is to be $400. What is to be the first payment ?

Ans. $130. 3. A man performs a journey in 19 days, travelling but a short distance the first day, and increasing his daily journey by 3 miles, until the last day, when he travelled 60 miles. How far did he travel the first day?

Ans. 6 miles. 4. The last term of a progression by differences is 363, the number of terms is 49, and the common difference . What is the first term ?

Ans. &.

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