DUODECIMALS, a term applied to an arithmetical me thod of ascertaining the number of square feet and square inches in a rectangular space, whose sides are given in feet and inches.

In this series of denominations (beginning with feet) every unit in the preceding denomination makes twelve in that which succeeds it : that is, every foot contains 12 inches, or firsts ; every first, 12 seconds ; and so on. There will be as many denominations in the product as in both factors taken together.

Feet are either marked with or without an f; inches are called firsts by the mark thus (') being placed above; seconds by the mark thus (") being placed above ; thirds thus ("'), and so on. In multiplying any two single denominations together, the value of the product will be known by adding the indices of the two factors. Thus, suppose 7 to be mul e tiplied by 11, then the product is 77, or 77 fifths, because adding the index " of the seven to the index "' of the eleven, produces v or fifths.

To multiply duodecimals together.—Write the multiplier under the multiplicand, so that the place of feet may stand under the last place of the multiplicand ; begin with the right-hand denomination of the multiplier, and multiply it by every denomination of the multiplicand, throwing the twelves out of every product, and carrying as many units to the next; place the remainders, if any, under the multiplier, so that like parts in the product may be under like parts of the multiplicand ; proceed with every successive figure of the multiplier, towards the left, in the same manner, always placing the first figure of the product under the multiplier; then the sum of the products will be the total.

In the first example, there is only one place of duodecimals in each factor; there are, therefore, two places in the product.

In the second example, there are two places of duodeci mals in the multiplicand, and one in the multiplier, which make three ; there are, therefore, three denomina tions in product.

Example 3.

1..9..4..1..7.. 11..2 ..3 1..0..9..8 ..'2 .. 4 ..3 9 1..0..11..5..6..8..2.. O.. 1..1..2..2.. 4..6 In this example, because there are no feet in the multiplier, the place is supplied by the cipher. The multiplicand has six places of duodecimals, and the multiplier seven ; there are, therefore, thirteen places of duodecimals in the product.

The first place of figures is feet, and the succeeding are the duodecimal places; the product is one foot, no inches, eleven seconds, five thirds, &c. But, independently of the consi deration of there being as many places of duodecimals in the product as in the multiplier and multiplicand, the method of placing the denominations of the fhetors gives the correct places of the product, since like parts of the product stand under like parts of the multiplicand ; it also shows the affinity, not only between duodecimals, but between decimals and every series of denominations, of which the same number in any place makes one of the next towards the left hand. The consideration is also useful, in discovering readily what kind of product arises by multiplying any two single denomina tions together.

When the number of feet runs very high in each factor, it will be much better to reduce all the denominat ions in both into the lowest, then multiply the factors, so reduced, and divide by 12 as often as there are duodeeimal places in the product.

Example 4.

ftft Multiply 6 .. , by 3 as in Example 1.

In this example, because there are seven places of duode cimals in the two factors, viz., four in the multiplicand, and three in the multiplier, the product 4613427 is divided seven times successively by 12.

There is another method of duodecimals, almost equally convenient. The rule is as follows : First.—Under the multiplicand, place the corresponding denominations of the multiplier.

Then multiply each denomination, from right to left, of the multiplicand, by each term of the multiplier successively from the left to the right, placing the first denomination of each row, or product, one place nearer to the right, and carry ing one from every twelve in the product of any denomi nation, to that which succeeds it towards the left, up to the place of feet ; then the sum of all the like products will be the total.

There is the same number of figures in each operation, but in the last the products are inverted. The first method, which is here used to prove the second, is similar to the com mon method of multiplication of integers, the first place of figures of every product being placed under the second deno mination towards the left.

When feet and inches only are concerned, the reader is referred to the articles CROSS MULTIPLICATION, and PRACTICE.