parts added together, ought to be equal to the total of the whole column taken at once: 3rd, to cut off a line, the uppermost for instance of the account, and then add the remaining lines, the amount of which added to the line cut off, should be the same with the total first found, thus,

Total of this last sum and 1st line. 898229

Whatever be the quantity adopted for the unit, it may be supposed to be divided into a number of equal parts; and these parts may be of any determined magnitude: but if, for example we should say, that the pound of sterling money is divided into 960 farthings, it would be found extremely difficult either to reckon or to form a distinct conception of such a number of individual farthings, or of intermediate sums between 1 and 960. For this reason the pound is first divided into 20 equal parts called shillings; each shilling into 12 equal parts, called pence; and each penny into 4 equal parts called farthings: so that 1 pound will contain 20 shillings, or 240 pence, or 960 farthings.

When a sum is given consisting of one or more units, together with one or more of these subdivisional parts, it is said to be a complex sum; as 25 pounds, 14 shillings, 9 pence, 3 farthings; or written in this manner, £25.. 14s... 9d... 3qrs. where the mark £. stands for the Latin term libra, a pound in weight, such a quantity of silver having originally been the value of a pound sterling; sh. for shillings: d. being the first letter of the Latin word denarius, a denier or permy; and qrs. for quadrans meaning in Latin the fourth part of any thing.

A shop-keeper makes out a statement of the debts due to him in this manner. 20 12 4 S. D.

A's bill amounts to

£

qrs.

36 17 11 3

Here the several denominations must be carefully placed in the same columns, keeping units under units, tens under tens, hundreds under hundreds, &c. Then beginning with the column of farthings on the right hand, say 1 and 3 are 4, and 1 are 5, and 2 are 7, and 1 are 8, and (passing over the 0 nought or zero) 3 are 11 farthings. But as 4 farthings are one penny, and 11 will contain 4 twice and 3 over, the 11 farthings will be 2 pence and 3 farthings. In the column of farthings set down the remaining 3, and carry the 2 pence to the column of pence, and going first up the units say, 2 and 1 are 3, and 6 are 9, and 7 are 16, and 9 are 25, and (passing over the noughts) 1 are 26; to which adding the 3 tens in the column of tens or 30, you obtain 56 pence. But every 12 pence being 1 shilling, and 48 pence being 4 shillings, 56 pence will be 4 shillings and 8 pence over. This 8 pence must be written in the unit's place of the column of pence (filling up the ten's place, with a zero or a dot for regularity's sake) and the 4 shillings are to be carried to the unit's place of the column of shillings, which when added up will come to 41, and by adding the six tens in the place of tens, or 60, the whole will be 101 shillings. But every 20 shillings forming 1 pound, 101 shillings will form 5 pounds and 1 shilling over. This shilling is written in the column of shillings, with a zero or merely a dot before it, and the 5 pounds are carried to the unit's place of the column of pounds which summed up, as also the places of the tens and the hundreds, will give a total of 633 pounds. The whole amount of the seven bills will then stand £633 01 08 3.

The accuracy of addition of sums of money or any other complex numbers and quantities, is proved by the same methods with those pointed out for checking addition of simple integers.

Again, in adding together a number of quantities expressing weight, the mode of proceeding is exactly similar, as in the following example of several quantities of Avoirdupois weight, where the number of units of one denomination composing an unit of the next higher denomination is marked over the respective columns.

The total amount of the several quantities here set down will therefore be 1717 tons, 15 hundred weight, 2 quarters, 24 pounds, 7 ounces, and 14.drams or, as it ought to be written, drachms.

In calculating quantities of various denominations it is necessary to know the precise proportions of each of these to others among themselves: the reader will therefore here find a statement of measures and quantities of various sorts, the most frequently occurring in ordinary life. These measures and quantities are arranged according to the order in which they stand in the nature of things. 1st. Lineal or mere length and extent, 2nd, superficial or length and breadth of surfaces, 3rd, solid or cubic, consisting of length, breadth, and depth or thickness, 4th. capacity, divided into dry and liquid, and 5th, gravity or weight.

In Scotland the denominations of Long measure are the same as in England, but the respective measures are somewhat longer: thus the Scotch yard or ell, consisting of 3 feet, contains 37 Engush inches, instead of 36; and the Scotch chain contains 24 ells, equal to 74 English feet, instead of 22 yards or 66 feet. The Scotch furlong therefore consists of 246 2-thirds yards instead of 220, and the Scotch mile contains 5920 feet or 1973 yards, or very nearly 89,3 chains English.

2ud. Geographical or nautical measure.

10

It is not more common than erroneous to consider the geographical or nautical mile as the same with the English statute mile, consequently the marine league as equal to three, and the geographical

degree as equal to 60 English miles. The degree is the 360th part of the circumference of the earth, which, were it a perfectly regular sphere or globe, would contain about 24,913 English miles: consequently each degree would, instead of 60, contain 69 English miles, aud the geographical or nautical mile or minute of a degree would, instead of 5280 feet, consist of 6090 feet English; the league therefore, instead of three English miles, will contain very nearly 3. On the other hand the league employed all over France, in computing distances upon the road from place to place, is very little more than 2 English miles, and the poste by which distances on post roads are computed, consisting of 2 leagues, is only equal to five, and not to six English miles, as is generally supposed.

The geographical or nautical minute or mile and league, are the same in all parts of the world: but the local measures of distance are very unequal in different countries; a distinction but too little attended to, in modern accounts of travels, voyages, and systems of geography, particularly when translated or compiled from foreign languages. All maps and sea charts are laid down upon scales of geographical and nautical miles of 60 in the degree, unless where it is otherwise expressed. Maps of counties and small districts are frequently constructed with scales of English miles of 69 in the degree.

Scotch and Irish linens are bought and sold by the yard English: but Dutch linens are bought by the ell Flemish, and sold by the ell English.

II. Superficial measures, commonly termed Square measure, and Land measure.

In Scotland the acre consists of 4 roods, the rood of 40 falls, the fall of 36 square ells, the ell of 9 feet and 73 inches; so that 1 acre is equal to 4 roods, or 160 falls, or 5760 ells, or 54760 square feet Scotch: hence the Scotch acre contains 55056 English feet, whereas the English acre contains only 43560 feet, consequently 4 Scotch acres are a little more than 5 acres English. The regulated statute acre does not prevail over every part of England itself; for in the northern and western countries, as the length of the pole varies between 163 feet and 28, the acre must of course be greatly different in different quarters. These acres are styled customary measure. The customary acre in Wales is about 2 English acres; and the Irish acre is equal to one acre 2 roods and 19 poles English. The arpent or acre formerly used in France, was equal to 14 acre English, or 54450 feet, nearly agreeing with the Scotch acre, which was probably derived from the old French

measure.

Plastering, painting, paving, glazing, are computed by superficial or square measure.

A cube is a solid figure of which all the sides are of cqual length, and all the angles are right or square, each side or surface standing perpendicular upon another: 12 inches in the side multiplied by itself, give 144 square inches for the superficial foot, and this again multiplied by 12 will give 1728 solid inches for the cubic foot. Upon this measure is founded the whole system of guaging.