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So in Division; which is the reverse of Multiplication, if a figure be removed one place to the right, it is diminished in a ten-fold proportion ; that is, it is divided by 10.

To divide 80 by 10, then, take away the cypher on the right, and the 8 will be one place lower; that is, it will be in the units' place instead of the tens': of course its value will be diminished ten-fold. The quotient, therefore, is 8.

Now if the dividend had been 89, instead of 80, taking away the 9 would have changed the 8 tens, to 8 units. as before ; that is, would have diminished them ten-fold, or divided them by 10, and as 9 is not great enough to make another ten, it would have been the remainder. Then,

REMOVING THE RIGHT HAND FIGURE OF ANY NUMBER DIVIDES IT BY 10, AND THE FIGURE SO REMOVED IS THE REMAINDER. The same reasoning will show that removing

two figures from the right of a number divides it by 100 ; and that the two figures so removed are the Remainder. Also that removing three, four, five figures, and so on, from the right of a number, divides it by 1,000, 10,000, and so on, and that the figures, so removed, are the Remainders. Hence, to divide by a number, consisting of 1, with cyphers annexed, REMOVE AS MANY FIGURES FROM THE RIGHT OF THE DIVIDEND, AS THE

THE FIGURES SO REMOVED, WILL BE THE REMAINDER; THOSE NOT REMOVED, THE QUOTIENT.

Note. It will not be necessary, actually to take the figures away ; but merely to place a point, to separate the quotient and remainder.

EXAMPLES FOR PRACTICE. 2. How many dollars in 7,963 cents ? A. $79.63. 3. How many dollars in 87,555 mills ? A. $87.555.

4. 100 men were to share equally a prize of $97,543, flow mucis was each man's share ? A. $975.43.

5. In an army of 100,000 men, an amount of pay of $2,775,000 was distributed, each man sharing an equal sum with the rest. What was each man's share ? A. $27.75.

6. Divide 33 by 10 | 10. Divide 81,960 by 100 7

10 | 11.

230,893 100 8.

DIVISOR CONTAINS CYPHERS.

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& XXXI. 1. At $40.00 a hogshead, how many hogsheads of molasses can I buy for $320.00 ?

40 is a composite number, made up of the factors 4 and 10. First divide by 10 according to the rule ; thus, 1,0)32,0 Then by 4; thus, 4)32

8 hogsheads. 2. Paid 80 laborers $570.00 distributing it equally. How many dollars had each?

80=8X10 Divide by 10, 1,0)57,0

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7+1 Rem. This remainder, of course, is one 10, or 10 dollars by the rule in $ xxix.

3. The paymaster of a garrison, distributed 5,845 dollars equally among 700 men. How many dollars did each receive ?

700=7X100 Divide by 100, 1,00)58,45

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8+2 Rem. The 2, remainder, is of course, 2 hundreds=200, to be added to the 45, first remainder. The true remainder, is therefore 245, and is found by writing the last remainder 2, before the first remainder 45; since this brings the 2 to the hundreds' place where it belongs.

4. In an army of 63,474 men, how many regiments of 4,800 ? 4,800=6X8X100. A. 13.1074.

Hence, when there are cyphers at the right of the divisor,

I. REMOVE THE CYPHERS, AND LIKEWISE AS MANY FIGURES FROM THE RIGHT OF THE DIVIDEND.

II. DIVIDE THE REMAINING FIGURES OF THE DIVIDEND, BY THE REMAINING FIGURES OF THE DIVISOR.

III. PREFIX THE REMAINDER FOUND BY THIS DIVISION, TO THE FIGURES REMOVED FROM THE DIVIDEND, FOR THE TRUE REMAINDER.

EXAMPLES FOR PRACTICE. 5. Divide 7,861 by 30. A. 2627. 6. Divide 21,564 by 20. A. 1,078747. 7. Divide 31,943 by 300. A. 106148.

8. Divide 1,151 by 20 ; 2,873 by 30; 9,999 by 90; 2,864 by 80 ; 71,843 by 200 ; 59,995 by 500; 77,724 by 9,000 ; 325,963 by 70,000 : 2,541,861 by 80,000.

XXXII. But there are many cases of Division, for which the preceding rules are insufficient. For instance,

1. If 23 yards of cloth cost $322, what cost 1 yard ? Our divisor, 23, consists of two figures. Of

23)322/14 course, it cannot be contained in the first fig

23 ure of the dividend.

We must therefore, take the first two figures. Thus, we must

92 find how often 23 is contained in 32. The

92 remainder must be prefixed to the next lower order, as in the preceding examples.

Then the only difference between dividing by a single figure, and by several, consists in this ; that in order to obtain the first quotient: figure, we must take as many figures of the dividend, as there are places in the divisor. Or, if the divisor be larger than the same number of figures in the dividend, we must take one more figure in the dividend.

2. If a man's income be 1,248 dollars a year what is that per week, allowing 52 weeks to the year? A. $24.

3. A privateer took a prize of $7,735. It was equally divided among 65 men. What amount had each ? A. $119.

4. Ă man bought 529 head of cattle for $15,341. What did lie give a head?

Ans. $29. 5. If a man's income be $49,640 a year, what is that á day allowing 365 days to the year ?

Ans. $136. 6. For $36.56 how many books can I buy at $4.57 each?

Ans. 8. It will be observed, that, we make a separate division for each quotient figure. The numbers, thus successively divided, are sometimes called the PARTIAL DIVIDENDS.

7. Divide 9,391 by 32. Ans. 293 .
8. Divide 28,609 by 51. Ans. 569 41.
Hence, we obtain a rule for division, in any case.
I. PLACE THE DIVISOR ON THE LEFT OF THE DIVIDEND.

II. FOR THE FIRST QUOTIENT FIGURE, DIVIDE AS MANY PLACES ON THE LEFT OF THE DIVIDEND AS THERE ARE PLACES IN THE DIVISOR; OR IF THESE BE NOT SUFFICIENT, TAKE ONE MORE.

III. MULTIPLY THE DIVISOR BY THIS QUOTIENT FIGURE, PLACE THE PRODUCT UNDER THE PARTIAL DIVIDEND, FRONT WHICH SUBTRACT IT, AND, TO THE REMAINDER, ANNEX THE. NEXT FIGURE OF THE DIVIDEND. This WILL BE THE SECOND PARTIAL DIVIDEND WHICH DIVIDE AS BEFORE.

IV. PROCEED IN THIS MANNER, TILL ALL THE FIGURES OF THE DIVIDEND ARE EMPLOYED, AND IF ANY PARTIAL DIVIDEND BE TOO SMALL TO CONTAIN THE DIVISOR, WRITE A CYPHER IN THE QUOTIENT, AND TREAT IT AS A REMAINDER.

EXAMPLES FOR PRACTICE. 9. Divide 251,104 by 472.

Ans. 532 10. Divide 863,256 by 736. Ans. 1,172 9.98 11. Divide 1,893,312 by 912.

Ans. 2,076. 12. Divide 47,254,149 by 4,674. Ans. 10,110 1677

12

13. Divide 761,858,465 by 8,465. Ans. 90,001. 14. Divide 119,181,693 by 38,473. A. 3,097 18. 15. Divide 230,293, 122,081 by 912,314.

Ans. 307,140

123 T4 16. Divide 7,328,946,264,418,232 by 814,313,515,623,303.

Ans. 9 1 3 4 6 3 3 8 0 8 5 0 5

81431351562330 3. As the quotient shows how many times the divisor is contained in the dividend, and the remainder what is left ; it is plain, that to prove Division, we must

MULTIPI.Y THE DIVISOR AND QUOTIENT TOGETHER, AND ADD THE REMAINDER TO THE PRODUCT. THE SUM OUGHT TO BE EQUAL TO THE DIVIDEND.

17. Divide 7,154 by 14 | 24. Divide 29,993 by 25
18.
29 | 25.

230,031 1

39 19. 13,243 37 | 26.

2,881,943 ,

46 20. 9,001 16 | 27.

4,960,002 201 21. 11,804 15 28.

13,801,804 » 315 22. 17,020 41 29.

175,990,806 " 7,864 23.

35,700 26 30. 27,341,020,003 "91,992

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♡ XXXIII. We have given (fXvi..) some of the tables of weights, measures, &c. The following are some that remain to be given. The first table contains the denominations of

WINE MEASURE.
gills, (gi.) make 1 pint, marked
2 pints

1 quart
quarts
1 gallon

gal.
10 gallons
1 anker

ank. 18 gallons

I runlet
314 gallons
1 barrel

bl.
42 gallons
1 tierce

tier 63 gallons I hogshead

hld. Si gallons 1 puncheon

pun. 2 hogsheads 1 pipe or butt 6 2 pipes

1 tun. Note. This measure is used for wine, brandy, spirits, mead, vinegar, foney, perry, cider oils, &c,

EAXMPLES POR PRACTICE. 1. Ilow many runlets in 925 gals. ? A. 51 run. 7 gals. How many tierces in 824 gals.? A. 19 tier. 26 gals.

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P. or b.

66

66

3. How many puncheons in 976 gallons ? In 1,823 ?

A. 11 pun. 52 gals. & 21 pun. 59 gals. 4. How many hhds. gals. qts. pts. and gi. in 11,934 gi. ?

A5 hhds. 57 gal. 3 qts. 1 pt. 2. gi. Note. Begin by dividing by 4, because 4 gi. make 1 pt. Then divide by 2, because 2 pts. make 1 qt., and so on. 5. How many tier. gal. qts. pts. and gi. in 38,254 gi. ?

A. 28 tier. 19 gal. 1 qt. 1 pt. 2 gi. 6. How many run. gal. qts. pts. and gi. in 38,254 gi. ?

A. 66 run. 7 gals. 1 qt. 1 pt. 2 gi. 7. How many gal. qts. and pts. in 218,363 pts. ?

A. 27,295 gal. 1. qt. 1 pt. 8. Change 3,834,579 gi. to hhds. gal: qts. &c. 9. How many bls. in 3,826 gal ? A. 1214 3. The pupil will probably be at a loss how to divide by 311. But he can find how many half.gallons there are in 314, and also how many half-gallons there are in 3,826. "Then he can divide the half-gallons in 3,826' by the half gallons. in 314.

10. How many bl. in 15,835 gal. ?
11. How many bl. gal. &c. in 17,936 gi. ?
12. How many bl. gal. &c. in 29,853,729 gi. ?

CLOTH MEASURE.
24 inches, (in.) make 1 nail, marked nl.
4 nails
1 quarter

qr. 4 quarters 1 yard

yd. 3 quarters

1 Ell Flemish E. FI: 43 quarters

1 Ell French

E. Fi. 5 quarters

1 Ell English

E. E. NOTE. This measure is used for cloths, and all goods sold by the card or ell. 13. How many yds. in 275 nls. ? A. 17 yds: 0 qr. 3 nls. 14. How many E. Fl. qrs. &c. in 2,753 in. ?

A. 101 E. Fl. 2 qrs. 3 nls. 11 in. NOTE. The pupil must reduce the 2, and the 2,753 to quarters of an inch, hefore dividing, as, in example 9, he reduced his numbers to halves. 15. How many E. E. qrs. &c. in 7,286 in. ?

A. 161 E. E. 4 qrs. 2 nls. in: 16. IIow many yds. qrs. &c. in 27,854 in. ? 17. How many aunes in 754 qrs. ? 18. How many aunes, qrs. &c. in 5,876 in. ? 19. How many aunes, qrs. &c. in 47,854 in.: 20. How many yds. qrs. &c. in 123,456,789 in. ?

CIRCULAR MEASURE, OR MOTION.
60 seconds, (") make 1 minute, marked
60 minutes

1 degree,
30 degrees

1 sign

S. 12 signs or 360°, The whole circle of the Zodiac.

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