Comparing the last three examples with Example VI., we see that the whole four are worked in exactly the same way. In every case the partial dividends are 234, 565, and 316, respectively, and are regarded—each in its turn-as representing so many units. Because, in working the examples mechanically, we dispense with the words "tenths,' "hundredths," "thousandths," &c., as well as with "tens," "hundreds," "thousands," &c.—such words being employed only when we want to explain what we are doing.

67. Rule for Simple Division, when a Decimal occurs in the Dividend, but not in the Divisor: Divide as if there were no Decimal, and write the decimal point in the resulting quotient in such a position that each quotient figure shall stand in the same place as the most right-hand figure of the partial dividend which produced it.

3)12(4

6)24(4

9)36(4

12)48(4

30)120(4

300) 1200(4

Before we proceed further, it is to be observed that, instead of dividing a given dividend by a given divisor, we may, if we please, divide the double of the dividend by the double of the divisor, or 3 times the dividend by 3 times the divisor, or in a word any number of times the dividend by the same number of times the divisor: the quotient being the same in the one case as in the other. Thus, dividing 12 by 3 is the same- -so far as the quotient is concerned-as dividing the double of 12 by the double of 3, or 3 times 12 by 3 times 3, or any number of times 12 by the same number of times 3.

EXAMPLE X.-Divide 23.456 by 8.9.

Here we multiply both the divisor and the dividend by 10, in order to have a whole number for divisor: 8.9 X 10-89; 23456x10=234:56. We then divide 234:56 by 89, and find the required quotient to be 2·63. [See Ex. VII.]

EXAMPLE XI.-Divide 23456 by 89.

Here we take 89 for divisor, and remove the decimal point of the dividend two places to the right—that is, we multiply both of the given numbers by 100: 89X100=89; 23456×100= 23.456. We then divide 23.456 by 89, and find the required quotient to be 263. [See Ex. VIII.]

EXAMPLE XII.-Divide 0023456 by 089.

Here we take 89 for divisor, and remove the decimal point of the dividend three places to the right--that is, we multiply both

of the given numbers by 1,000: 089 × 1,000=89; •0023456x 1,000=2*3456. We then divide 2·3456 by 89, and find the required quotient to be 0263. [See Ex. IX.]

EXAMPLE XIII.-Divide 23456 by '0089.

Here we take 89 for divisor, and remove the decimal point of the dividend four places to the right—that is, we multiply both of the given numbers by 10,000: 0089X10,000=89; 23456X 10,000=234560000. We then obtain the required quotient

by dividing 234560000 by 89.

68. Rule for Simple Division, when a Decimal occurs in the Divisor: Regard the Divisor as a whole number, and remove the decimal point of the Dividend as many places to the right as there are decimal places in the Divisor; then, proceed as already directed. [§ 65, part 2; or § 67-as the case may be.]

NOTE. In applying this rule, the pupil should first set down the given divisor and dividend, and then make the necessary changes underneath the original pair of numbers being "cut off" by a horizontal line. Thus, in the case of the last two examples:

69. To prove an exercise in Division-whether it be Simple, Compound, or Fractional Division— we multiply the quotient by the divisor, and add the remainder should there be one-to the product. When the dividend is obtained for result, the work is presumed to have been correctly performed.

The reason of this is obvious. When, for instance, 789 pounds are divided equally between 4 persons [Ex. II.], each person receives 197 pounds, and 1 pound is left. The total amount distributed, therefore, is 4 times 197 pounds—that is, 788 pounds; and adding to this the amount (I pound) remaining undistributed, we necessarily obtain the original sum :- 197 X 4=788; 788+1=789.

The letters s. d. denoting pounds, shillings, and pence, respectively- -are the initials of the names of three old Roman Coins: "libra," "solidus," "denarius." A smaller amount than a penny is expressed as a fraction of a penny. Thus, a farthing is written d.-one-fourth of a penny; two farthings, or a halfpenny, d. one-half of a penny; and three-farthings, ¡d.— three-fourths of a penny.

66

Although the pound, the shilling, and the penny are the only coins of account "-that is, the only coins employed in the keeping of accounts, and in ordinary calculationsthere are, altogether, 12 coins in circulation: 2 gold coins-the sovereign and the half-sovereign; 7 silver coins the crown or five-shilling piece, the half-crown, the florin or two-shilling piece, the shilling, the sixpence, the fourpence, and the threepence; and three bronze coins-the penny, the half-penny, and the farthing.

Besides the twelve coins just mentioned, three silver ones -the two-penny, the three-half-penny, and the penny piece-are occasionally issued from the Mint, by special command of the Queen, for distribution amongst a number of poor persons on the Thursday before Easter-Maundy Thursday, as it is sometimes called in England; but those little coins, which are known as 66 Maundy* money," cannot be said to be in circulation, being, in most cases, retained as curiosities by the shopkeepers and other persons into whose hands they pass.

A new sovereign contains 5 dwts. 34 grs. of "standard" gold, in every 12 parts of which there are II parts of pure gold,

* The word Maundy is said to be derived from mand-Saxon for "basket:” because, formerly, it was customary for the English Sovereign to proceed to Whitehall on Maundy Thursday, and distribute alms from a basket.

F

and I part of an alloy consisting of a mixture of silver and copper; so that a Troy pound of standard gold can be coined into 4648 sovereigns (£46 14s. 6d.), and the Mint price of gold is thus fixed at £3 178. 10d. an ounce.

A new shilling contains 3 dwts. 15 grs. of standard silver, in every 40 parts of which there are 37 parts of pure silver, and 3 of copper; so that a Troy pound of standard silver can be coined into 66 shillings, and the Mint price of silver is thus fixed at 5s. 6d. an ounce.

The bronze coins, issued in 1860, are composed of a mixture of copper, zinc, and tin, in every 100 parts of which mixture there are 95 parts of copper, 4 of zinc, and I of tin. A bronze penny weighs 145.83 grs.; a bronze half-penny, 87.5 grs.; and a bronze farthing, 43'75 grs.: so that whilst two farthings contain exactly as much bronze as a half-penny, two half-pence contain more than a penny. The number of pence to the Avoirdupois pound is 48; of half-pence, (not 96, but) 80; and of farthings, 160. As a substitute for the ordinary Avoirdupois ounce, therefore, we might take 3 pence, or 5 half-pence, or 10 farthings. Moreover, as a substitute for a foot (12 inches), we might take 12 half-pence, placed in contact along a straight edge-a half-penny being exactly an inch in diameter, and 12 half-p -pence, consequently, representing a foot in length. A foot would also be represented by 4 pence, 4 half-pence, and 4 farthings, properly placed one after the other-a penny and a farthing being exactly 12 and 8 inches, respectively, in diameter.*

Pence are not a legal tender for more than a shilling; nor is silver a legal tender for more than 2 pounds.

The fact does not appear to be very generally known that, with a sufficient number of four-penny and three-penny pieces, a person could, except in three cases, dispense with the bronze coins in paying any amount containing no lower denomination than pence the exceptional amounts being Id., 2d., and 5d. Thus, Is. 5d. could be paid with 2 fourpences and 3 threepences; 28. Id. with 1 fourpence and 7 threepences; 38. 2d. with 2 fourpences and 10 threepences; &c. But of course such a sum as 5s. 93d., or 4s. 114d., or 6s. 10d. could not be paid entirely in silver, the lowest denomination not being pence.

Previously to the year 1825, when the Irish currency was assimilated to the British, the Irish penny was less in value than

* The "old" or "copper" coins, which are being withdrawn from circulation, are composed of pure copper. When new, a copper penny weighed 291 grs., and there were, consequently, 24 copper pence to the Avoirdupois pound. The Mint price of copper was thus fixed at about 28. a pound (Avoirdupois).

the British penny-13 of the former being worth only 12 of the latter; and although the people of Ireland kept their accounts in "pounds, shillings, and pence," there was no such thing, in the shape of a coin, as an Irish pound or shilling. The Irish shilling" was merely a name for "12 (Irish) pence," just as the Irish "pound" was a mere name for "20 (Irish) shillings." or "240 (Irish) pence." And as 13 Irish pence were represented by 12 British ones-so, 13 Irish shillings were represented by 12 British shillings, and 13 Irish pounds by 12 British pounds. We therefore diminish a given amount of "Irish" money by its thirteenth part when we want to find the equivalent amount in the present currency. Thus, the thirteenth part of £100 being £7 138. 10d., the subtraction of the latter amount from the former gives £92 68. 2d. as the equivalent, in British money, of £100 Irish.

Artisans divide the inch into eighths, whilst scientific people divide it into tenths, hundredths, &c. Formerly, the inch was divided into 12 equal parts, called lines; but this division is seldom-if ever-heard of now, except as a thing of the past.

An "Irish" perch is 7 yards long; and as there are 40 Irish perches in an Irish furlong, and 8 Irish furlongs in an Irish mile, there are (7X40×8=) 2,240 yards in an Irish mile, which, therefore, is 480 yards longer than an English milethe number of yards in an English mile being (512X40X8=) 1,760. A Scotch mile is 1,977 yards in length that is. 217 yards longer than an English mile. The relative lengths of the three miles might be thus expressed: a Scotch mile, about as long as an English mile and an eighth; an Irish mile. nearly as long as an English mile and a quarter. Eleven Irish are exactly equal in length to 14 English miles. A mile originally meant a thousand (mille) “paces”—that is, a length shorter than the present English mile by not quite 100 yards.

In the measurement of cloth, calico, linen, ribbons, &c.,

*The Greek foot was 12.14, the Roman foot 116, and the old French foot nearly 12.8 inches in length: yet, curiously enough, the average length of the human foot is only 10 3 inches.