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added to the first row, will, if the operation be correct, give a total equal to that found by the addition of all the five rows of quantities.

In the practice of keeping accounts, it frequently happens that the whole of an account can not be contained in one page or folio of the book; when this is the case, the custom is to sum up the contents of that page or folio, and write it down at the bottom of the column, with the words Carried forward opposite to the sum; and the same sum or total is entered at the top of the money column of the subsequent page or folio, with the words Brought forward, leading to it. In this manner each preceding page is summed up, and the amount made the first article in the fol lowing page, until the account is closed, when the amount of the last page, as comprehending all the particular totals of the preceding pages, is to be considered as the total amount of the whole account.

OF SUBTRACTION.

Subtraction means that branch of calculation by which we find the difference between two given quantities; or agrecably to the meaning of the term, by which we draw a less sum from a greater, and so discover what quantity will be left as a remainder.

Although Subtraction be an operation, directly opposite to Addition, yet the numbers must be placed under one another as before; that is, the small number must be written under the great, and beginning at the units or first figures on the right hand, which may for instance be 8 and 5, we say if from 8 units, such as pounds, yards, &c, we take away 5, there will remain 3: or the difference between 5 and 8 is 3. Again, when one figure of the least number

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happens to be greater than the corresponding figure in the great number, as in subtracting 37 from 65; as it is impossible to take 7 units out of 5, we borrow, as it is called, 1 ten from the place of tens in the great number, and add it to the 5, thereby making 15; from which if we subtract or take away 7, the remainder will be 8. This 8 is accordingly written under the place of units, and the operation proceeds in this way. By borrowing the ten from the place of tens, the figure 6, may be supposed to be reduced to 5: we have then to subtract the 3 of the less number from the 5 of the greater, and the remainder or difference will be 2; that is in all 28. Or instead of taking 1 away from the 6, if we repay or add 1 to the 3, standing under it, we shall have 4, which again being subtracted from 6 will leave 2 as before.

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The Subtrahend or less number being written under the Minuend or greater, observing to place units under units, tens under tens, &c, we begin at the right hand, saying, if from 7 men 5 be taken away, 2 will remain behind this 2 therefore is written in the column of units. Again, if from 8 be taken 6, 2 will remain: Then from the third figure of the greater sum 1, we should take away 3; but this is impossible; we must therefore borrow 1 from the next figure on the left hand 2, which, if the 1 be considered as occupying the place of units, will become a ten, which 10 being added to the -1 will make 11. From this sum 11 we can subtract, as was first proposed, the 3, and the remainder will be 8, which is also written in its due place. Having borrowed 1 from the 2 in the fourth place of figures, this 2 comes to be reckoned as 1, and from it, as has just been done, we subtract the 5, by borrowing another ten from the

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fifth figure 6, making the 1'to be 11: then 5 taken from 11 will leave 6 for the remainder.-But instead of thus diminishing the number from which you have borrowed, in the upper row of figures, it is more customary to leave that number as it is in fact, and to add the ten which was borrowed, to the number to be subtracted: hence we will have, in the fourth place of figures 5 and 1 equal to 6, which being taken from the 2 in the first row, augmented by another ten borrowed from the 6 in the same row, and called 12, will leave a remainder of 6. Lastly, adding or carrying to the 1 in the Subtrahend, the ten that was borrowed, we have 2, which taken from the 6 in the first row or Minuend, will leave 4, to be written in the remainder; and the operation will be completed, showing the number 46,822, for the army remaining in the camp.

To prove Subtraction, you may either add the remainder and the less number together, which if no error have been committed, will give a total equal to the greater number; or from the greater number subtract the remainder, and the difference will be equal to the less number.

The following examples may be sufficient to shew the method of performing subtraction of complex numbers.

A trader retiring from business wishes to know how much clear property he will possess, when all his debts are paid. d. qrs.

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Beginning at the right hand, he subtracts 1 farthing from 3, and writes down the difference 2 in the remainder: then proceeding to the pence he finds he cannot take 8 pence from 5 pence; to this 5 therefore he borrows an unit from the next column, or 1 shilling, which is equal to 12 pence,

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and adding this 12 to the former 5 he has 17 pence, from which 8 being taken away, there will be a remainder of 9 pence, to be entered in the proper column. The unit or shilling thus borrowed he now repays or adds to the 15 shillings, thus made 16, which are to be subtracted from the 11 in the upper row: but this being impossible, an unit or pound, equal to 20 shillings must be borrowed and added to the 11 which will thus become 31, from which the 16 of the lower row being subtracted, 15 will remain. The pound borrowed is now added to the figure 7 in the units place of the column of pounds, and the amount subtracted from 15 (that is 5 and 10 borrowed from the adjoining figure) will leave 7 for a remainder :-and proceeding in this manner, as has been already shown, the total remainder will turn out to be .7687.. 15s... 09d. example.

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The accuracy of this subtraction will be proved by adding the remainder to the less number, which will give a total equal to the great number.

The following example is performed in the same way.

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To multiply one number by another, is to take the value of the one number as often as there are units in the other number: thus to multiply 8 by 5 is to take 8 five times; or in other words, as was before mentioned, it is to add 8 five

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Sum

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40

Multiplicand 8

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times together, as if the figures stood in a column as in the margin; where the amount of five times 8, is 40. But instead of this repeated addition of a number, which in many cases would be extremely inconvenient, we write down the number to be multiplied as in this example, calling it the Multiplicand, and under it, beginning at the right hand, that

units may stand under units, &c., we write the number denoting how often the value of the multiplicand is to be taken. This last number is called the multiplicator or Multiplier and observing that 8 repeated 5 times will amount to 40, we say 5 times 8 are 40; which sum is written down as in the margin, and is called the Product.

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The Multiplicand and Multiplicator are also termed Factors.

To assist in performing Multiplication, the following Table must be well gotten by heart.

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