be shortened, by taking those proportional parts of the products found by the preceding numbers: for instance, in multiplying any sum by 639, when you have wrought with the nine, you may take one-third of the product for the product by 3; and doubling this last sum you will have the product by 6.

4th. In multiplying by 9, add 0 to the multiplicand, and beginning with the first figure of it, subtract it from 0, or 10, and so proceed subtracting each subsequent figure of the multiplicand, from the one preceding it, and setting down the remainder as a product: Thus in multiplying 436 by 9, you may write or suppose 0 on the right hand of 6, and say 6 from 10, and 4 remain, which is written down in the product; and then 3 and 1 carried are 4, which taken from 6 leave 2 for the product; 4 from 13 and 9 remain for the product; and carrying 1 for the ten just borrowed, say 1 from the 4 of the multiplicand, and S will remain to complete the product.

Multiplication may be performed not only with integral numbers, as of Pounds, Tons, &c. but of compound quantities such as Pounds, Shillings, Pence, &c. Tons. Hundreds, Quarters &c. in this manner: Suppose that in dividing the value of a prize at sea, between 6 Captains, each got. 266 .. 17 .. 8 .. 3. What was the value of the whole prize?

S. d. q.

£. 1601 .. 06

It was already observed that multiplication of Integers is only a short method of adding to itself the multiplicand, as often as there are units in the multiplier; the same is true of Compound Multiplication, as may by seen from this example; for if the share were written down 6 times, and these sums were added together, they would give a total equal to the above product, which is obtained

in this way-6 times 3 are 18 farthings, or 4 pence 2 farthings; set down the surplus 2, and carry the 4 to the pence, saying 6 times 8 are 48, and 4 carried are 52 pence, or 4 shillings and 4 pence; write the surplus 4, and carry 4 to the shillings: 6 times 17 are 102, and 4 are 106 shillings, equal to 5 pounds and 6 shillings, which 6 being written in the column of shillings, multiply the 266 pounds by 6, adding the 5 pounds carried, obtaining a product of £. 1601... 06 4. 2. for the total value of the prze. When the multiplier exceeds 12, or consists of 2 or more places of figures, find its component parts, and multiply the given quantity by one of these component parts, and the product by the other: for example,

Multiply
By 315

In this example, the multiplier 315 being a compound number, formed by the successive multiplication of 9, 7, and 5; multiply the quantity given by one of these component parts, as 9; then multiply the product by another compotent part, as 7, and the last product by 5, which will give the result, as if the whole multiplier 315 had been employed at once. It is of no consequence in what order these component parts are used, for 9 multiplied by 7, and the product by 5, will give the same result, as if they had been employed in this order, 5, 7, and 9; or 7, 9, and 5, &c.

It is however, convenient, when practicable, to employ these multipliers in such a way, as to remove some of the

lower divisons of the multiplicand; as in the preceding example, had 8 been the first component part, we would have had 8 times 12 equal to 96 ounces, or exactly 6 pounds, so that there would have been no surplus in the column of ounces, and the rest of the operation would have been so far abridged.

Had the multiplier been a prime number, instead of a composite; that is, for instance, had it in this example been 317, or 313, it would have been proper to have multiplied by the same component parts, and for 317,` which is 2 more than 315, to have added to the last product twice the amount of the first quantity given; and in the second case for 313, to have subtracted from the last product, twice the quantity given.

When the mutiplier is very large, you may multiply by 10, and that product again by 10, to obtain 100 times the number given; and if the multiplier is, or exceeds 1000, multiply again by 10; continuing still to do so, as often as may be necessary; then multiply the given number by the figure in the units' place of the multiplier; the first product by the 2 figure or tens' place; the second product by the 3d figure, or that in the hundreds' place, &c. All these products added together will give the total required.

For example. What is the amount of the Pay of 4325 Labourers at £. 36 15.. 8 each per annum.

L. 36

An application of multiplication which very frequently occurs, is to calculate the amount of any number of articles, in Value, Weight, Measure, or any other mode of reckoning. The multiplicand in such cases expresses the rate of value, &c. of one article, and the multiplier the number of articles to be estimated this multiplier is therefore always an abstract number, having no reference to any value or measure whatever. The only exception to this remark, and the objection is but apparent, is in performing operations in Mensuration, to find the superficial, or the solid quantity of any body whose dimensions are given in certain determinate measures, such as Inches, Feet, yards, &c. In these operations, yards, feet, and inches may be multiplied into yards, feet and inches, and the product will consist of quantities of similar, but not identical denominations; for feet multiplied by feet, will give not lineal but superficial feet, and this product again multiplied by feet will give solid feet. To assist in these operations, the following Table of superficial measures multiplied together, must be well understood.

Table for multiplying Yards, Feet, and Inches, by yards, feet, and inches.

The principles on which this Table is founded, will be explained when we come to treat of mensuration of Sur

faces;

faces; but in the mean time the following examples will show how this species of multiplication is performed.— How many yards of carpeting will cover a room 25 feet 8 inches long, and 17 feet 6 inches broad?

49 yards, 8 feet, 2 twelfth parts, equal to 1 sixth part of a foot, superficial measure.

The length and breadth being written in two lines, begin and multiply the 25 feet by the 17, setting down the product 425, which by the Table, must be square feet, as the result of multiplying lincal feet into one another. Then multiply the 17 feet by the 8 inches of the multiplicand, producing as above 136 figure sof a certain description, whose length is 1 foot, and whose breadth is 1 inch; to this add the product of 25 feet, multiplied by 6 inches or 150; and taking the 12th part, according to the table, of the sum 286, you will have 23 square feet, and 10 twelfth parts of 1 foot. Lastly, multiply the 8 inches by 6 inches, and the product 48 will be square inches equal to 4 of the above-mentioned figures, or 1 third part of 1 foot square. The 4 being set down, add the whole together, and you will have 449 square feet, and 2 twelfth parts: but as 9 square feet are 1 yard superficial, take the 9th part of 449, when you will have 49 yards 8 feet, and 1 sixth part, for the quantity of carpeting required.

Division.