As it is of great consequence in business to perform addition readily and exactly, the learner ought to practise it till it become quite familiar. If the learner can readily add any two digits, he will soon add a digit to a higher number with equal ease. It is only to add the unit place... Addition of that number to the digit; and if it exceed ten, it raises the amount accordingly. Thus, because 8 and 6 are 14, 48 and 6 are 54. It will be proper to mark down under the sums of each column, in a small hand, the figure that is carried to the next column. This prevents the trouble of going over the whole operation again, in case of interruption or mistake. If you wish to keep the account clean, mark down the sum and figure you carry on a separate paper, and after revising them, transcribe the sum only. After some practice, we ought to acquire the habit of adding two or more figures at one glance. This is particularly useful when two figures which amount to 10, as 6 and 4, or 7 and 3, stand together in the column.

Every operation in arithmetic ought to be revised, to prevent mistakes; and as one is apt to fall into the same mistake, if he revise it in the same manner he performed it, it is proper either to alter the order, or else to trace back the steps by which the operation advanced, which will lead us at last to the number we began with. Every method of proving accounts may be referred to one or other of these heads.

1st. Addition may be proved by any of the following methods: Repeat the operation, beginning at the top of the column, if you began at the foot when you wrought it.

2d. Divide the account into several parts; add these separately, and then add the sums together. If their amount correspond with the sum of the account when added at once, it may be presumed right. This method is particularly proper when you want to know the sums of the parts as well as that of the whole.

3d. Subtract the numbers successively from the sum; if the account be right, you will exhaust it exactly, and find no remainder.

When the given number consists of articles of different value, as pounds, shillings, and pence, or the like, which are called different denominations, the operations in arithmetic must be regulated by the value of the articles. We shall give here a few of the most useful tables.

I. Sterling Money. II. Avoirdupois Weight. 4 farthings = 1 penny, 16 drams = 1 ounce, oz. marked d., 16 ounces = 1 pound, lb. 12 pence = 1 shilling, s. 28 pounds = 1 quarter, qr. 20 shillings = 1 pound, £. 4 quart. = 1 hundredw, cwt. Also 6s. 8d. = 1 noble. 20 hundredw = 1 ton, T. 12s. = 1 angel. 13s. 4d. or two thirds of a pound = 1 merk.

Scots money is divided in the same manner as sterling, and has one twelfth of its value. A pound Scots is equal to 1s. 8d. sterling, a shilling Scots to a penny sterling, and a penny Scots to a twelfth part of a penny sterling; a merk Scots is two thirds of a pound Scots, or 13½d. sterling.

III. Troy Weight. IV. Apothecaries' Weight. 20 mites = 1 grain, gr. 24 grains = 1 pennyw, dwt. 20 pennyw = 1 ounce, oz. 12 ounces = 1 pound, lb. 20 grams = 1 scruple, 9 3 scruples = 1 dram, 3 8 drams = 1 ounce, 3 12 ounces = 1 pound, lb. 4 lippies = 1 peck 4 pecks = 1 firlot 4 firlots = 1 boll 16 bolls = 1 chaldar

V. English Dry Measure. VI. Scots Dry Measure. 2 pints = 1 quart 4 quarts = 1 gallon 2 gallons = 1 peck 4 pecks = 1 bushel 8 bushels = 1 quarter 4 lippies = 1 peck 4 pecks = 1 firlot 4 firlots = 1 boll 16 bolls = 1 chaldar

VII. English Land Measure. VIII. Scots Land Measure. 30½ square yards = 1 pole or perch 40 poles = 1 rood 4 roods = 1 acre 36 square ells = 1 fall 40 falls = 1 rood 4 roods = 1 acre

IX. Long Measure. X. Time. 12 inches = 1 foot 3 feet = 1 yard 5½ yards = 1 pole 40 poles = 1 furlong 8 furlongs = 1 mile 3 miles = 1 league 60 seconds = 1 minute 60 minutes = 1 hour 24 hours = 1 day 7 days = 1 week 365 days = 1 year 52 weeks and 1 day = 1 year

Rule for Compound Addition.—Arrange like quantities under like, and carry according to the value of the higher place.

Note 1. When you add a denomination which contains more columns than one, and from which you carry to the higher by 20, 30, or any even number of tens, first add the units of that column, and mark down their sum, carrying the tens to the next column; then add the tens, and carry to the higher denomination, by the number of tens that it contains of the lower. For example, in adding shillings, carry by 10 from the units to the tens, and by 2 from the tens to the pounds.

Note 2. If you do not carry by an even number of tens, first find the complete sum of the lower denomination, then inquire how many of the higher that sum contains, and carry accordingly, and mark the remainder, if any, under the column. For example, if the sum of a column of pence be 48, which is three shillings and sevenpence, mark 7 under the pence column, and carry 3 to that of the shillings.

Note 3. Some add the lower denominations after the following method; when they have reckoned as many as amounts to one of the higher denomination, or upwards, they mark a dot, and begin again with the excess of the number reckoned above the value of the denomination. The number of dots shows how many are carried, and the last reckoned number is placed under the column.

Examples.

| Sterling Money | Avoirdupois Weight | |---------------|-------------------| | £ 127 13 3 | 3 15 2 ¼ | | 43 5 10½ | 6 3 0 19 | | 806 18 7 | 5 7 3 26 | | 190 2 5½ | 3 2 2 0 | | 214 0 3 | 4 3 1 10 | | 85 15 4½ | 1 18 1 12 | | 1467 15 9¾ | 24 11 0 7 |

CHAP. III.—Subtraction.

Subtraction is the operation by which we take a lesser number from a greater, and find their difference. It is exactly opposite to addition, and is performed in a like manner, beginning at the greater, and reckoning downwards the units of the lesser. The greater is called the minuend, and the lesser the subtrahend.

If any figure of the subtrahend be greater than the corresponding figure of the minuend, we add ten to that of the minuend, and having found and marked the difference, we add one to the next place of the subtrahend. This is called borrowing ten. The reason will appear, if we consider that, when two numbers are equally increased by adding the same to both, their difference will not be altered. When we proceed as directed above, we add ten to the minuend, and we likewise add one to the higher place of the subtrahend, which is equal to ten of the lower place.

Rule.—Subtract units from units, tens from tens, and so on. If any figure of the subtrahend be greater than the corresponding one of the minuend, borrow ten.

Example. Minuend 173694 Subtrahend 21453 Remainder 152241 To prove subtraction, add the subtrahend and remainder together; if their sum be equal to the minuend, the account is right.

Or subtract the remainder from the minuend. If the difference be equal to the subtrahend, the account is right.

**Rule for Compound Subtraction.—Place like denominations under like; and borrow, when necessary, according to the value of the higher place.**

**Examples.**

| £ | s. | d. | Cut. | qr. | ℏ. | A. | R. | F. | E. | |---|----|----|------|-----|----|----|----|----|----| | 146 | 3 | 3 | 12 | 3 | 19 | 15 | 2 | 24 | 18 | | 58 | 7 | 6 | 4 | 3 | 24 | 12 | 2 | 36 | 7 | | 87 | 15 | 9 | 7 | 3 | 23 | 2 | 3 | 28 | 11 |

**Note 1.** The reason for borrowing is the same as in simple subtraction. Thus, in subtracting pence, we add 12 pence when necessary to the minuend, and at the next step we add one shilling to the subtrahend.

**Note 2.** When there are two places in the same denomination, if the next higher contain exactly so many tens, it is best to subtract the units first, borrowing ten when necessary; and then subtract the tens, borrowing, if there is occasion, according to the number of tens in the higher denomination.

**Note 3.** If the value of the higher denomination be not an even number of tens, subtract the units and tens at once, borrowing according to the value of the higher denomination.

**Note 4.** Some choose to subtract the place in the subtrahend, when it exceeds that of the minuend, from the value of the higher denomination, and add the minuend to the difference. This is only a different order of proceeding, and gives the same answer.

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**CHAP. IV.—Multiplication.**

In multiplication two numbers are given, and it is required to find how much the first amounts to when reckoned as many times as there are units in the second. Thus, 8 multiplied by 5, or 5 times 8, is 40. The given numbers (8 and 5) are called factors, the first (8) the multiplicand, the second (5) the multiplier, and the amount (40) the product.

This operation is nothing else than addition of the same number several times repeated. If we mark 8 five times under each other, and add them, the sum is 40. But as this kind of addition is of frequent and extensive use, in order to shorten the operation, we mark down the number only once, and conceive it to be repeated as often as there are units in the multiplier.

For this purpose the learner must be thoroughly acquainted with the following multiplication table, which is composed by adding each digit twelve times.

| Twice | Thrice | Four times | Five times | Six times | Seven times | Eight times | Nine times | Ten times | Eleven times | Twelve times | |-------|--------|------------|------------|-----------|-------------|-------------|------------|-----------|--------------|-------------| | 1 is 2 | 1 is 3 | 1 is 4 | 1 is 5 | 1 is 6 | 1 is 7 | 1 is 8 | 1 is 9 | 1 is 10 | 1 is 11 | 1 is 12 | | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 | | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 | | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 | | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 | | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 | | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |

If both factors be under 12, the table exhibits the product at once. If the multiplier only be under 12, we begin at the unit place and multiply the figures in their order, carrying the tens to the higher place, as in addition.

**Ex.** 76859 multiplied by 4, or 76859 added 4 times.

\[ \begin{array}{cccc} 4 & 76859 & 307436 \\ \end{array} \]

If the multiplier be 10, we annex a cipher to the multiplicand; if the multiplier be 100, we annex two ciphers, and so on. The reason is obvious, from the use of ciphers in notation.

If the multiplier be any digit, with one or more ciphers on the right hand, we multiply by the figure, and annex an equal number of ciphers to the product. Thus, if it be required to multiply by 50, we first multiply by 5, and then annex a cipher. It is the same thing as to add the multiplicand 50 times; and this might be done by writing the account at large, dividing the column into 10 parts of 5 lines, finding the sum of each part, and adding these ten sums together.

If the multiplier consists of several significant figures, we multiply separately by each, and add the products. It is the same as if we divided a long account of addition into parts corresponding to the figures of the multiplier.

**Example.** To multiply 7329 by 365.

\[ \begin{array}{cccc} 7329 & 7329 & 7329 & 36645 = 5 \text{ times.} \\ 5 & 60 & 300 & 439740 = 60 \text{ times.} \\ & & & 2198700 = 300 \text{ times.} \\ 36645 & 439740 & 2198700 & 2675085 = 365 \text{ times.} \\ \end{array} \]

It is obvious that 5 times the multiplicand added to 60 times, and to 300 times the same, must amount to the product required. In practice we place the products at once under each other; and as the ciphers arising from the higher places of the multiplier are lost in the addition, we omit them. Hence may be inferred the following