5. An even number cannot divide, or measure, an odd number, nor a greater a less.

6. Half the sum of any two numbers, increased by half their differ. ence, will give the greater number; and half their sum diminished by half their difference, will give the less number.

7. The quotient, arising from the division of the sum of two, or more, numbers, is equal to the sum of the quotients arising from the division of the parts, separately, by the same divisor.

8. If any two numbers be separately divided by 9 or 3, and the two remainders multiplied together, and that product divided by 9 or 3, the last remainder will be the same as if you divided the product of the two first numbers by 9 or 3.

9. Any number divided by 9 or 3, will leave the same remainder as the sum of its digits divided by 9 or 3. Hence, if any number is divisible by 9 or 3, the sum of its digits is likewise divisible by 9 or 3, and vice versa. The method of proof by casting out the nines, in the precoding rules, depends upon this theorem.

(22.) Divide 714394756 by 1754.
(23.) Divide 47159407184 by 3574.
(24.) Divide 5719487194715 by 45705.
(25.) Divide 4715714937149387 by 17493.
(26.) Divide 671493471549375 by 47143.
(27.) Divide 571943007145 by 37149.
(28.) Divide 1714347149347 by 57143.
(29.) Divide 49371547149375 by 374567.
(30.) Divide 171493715947143 by 571007.
(31.) Divide 6754371495671594 by 678957.

division.

Examples to Prop. 4.

(32.) Divide 14715967899 by 145000. 145000)14715967899(101489 62899 Quotient.

145000

(33.) Divide 571436490075 by 36500.
(34.) Divide 194718490700 by 73000.
(35.) Divide 795498347594 by 47150.
(36.) Divide 1495070807149 by 371500.
(37.) Divide 6714934714934 by 754000.
(38.) Divide 1071491471430715 by 147500.
(39.) Divide 14714937493714957 by 157900.
(40.) Divide 7149374947194715 by 1749000.

CLASS II. Exercising all the Propositions.

(41.) Divide 714947349 by 90.

(42.) Divide 1714937148 by 14400.

(43.) Divide 7149371478 by 121.

(44.) Divide 71900715708 by 57149.

(45.) Divide 15241578750190521 by 123456789. (46.) Divide 121932631112635269 by 987654321. (47.) Divide 69616103498721931800 by 975005700. (48.) Divide 656458931996524171800 by 700489070. (49.) Divide 7149437149547 by 3714900. (50.) Divide 14714937148475 by 123456.

(51.) The sum of two numbers is 348, and their difference 194, required the numbers.

(52.) What number, multiplied by 365, will produce 315725?

(53.) What number, multiplied by 95, will give the same product as 157 by 570?

(54.) What number is that, from which if a twelfth part of 1728 be deducted, and the remainder increased by the ninety-fifth part of 82175, the sum will be 1185?

(55.) What number divided by 1185, will give 497 for the quotient, and leave just a fifth part of the divisor remaining?

(56.) Required the difference between six dozen dozen and half a dozen dozen.

(57.) Subtract 759 out of 171493745 as often as you can, and shew what the last remainder exceeds or falls short of 500.

TABLES OF ENGLISH COIN, WEIGHTS,

MEASURES, &c.

TABLE I. MONEY.

The lowest piece of money, used in England, is a farthing, and all accounts are kept in pounds, shillings, pence, and farthings. The pound sterling is an imaginary coin, value 20 shillings.

£. denotes pounds, s. shillings, and d. pence.
a farthing, or the quarter of any thing.
a halfpenny, or the half of any thing.

......

......

three farthings, or three-quarters of any thing.
Imaginary English Coin.

A mark value 13s. 4d.

An angel value 10s.

A Carolus.

.... 23s.

TABLE II. TROY WEIGHT.

By this weight are weighed gold, silver, jewels, amber,

and all liquors

Apothecaries, chemists, &c. use this weight in mixing medicines; but buy and sell their drugs by avoirdupois weight.