The duodecimal system of numerals is that in which the base is 12, instead of 10, as in the common decimal system. As a method of calculation it has fallen into almost entire disuse, and it is only in calculation of areas that duodecimals are now used. When the work becomes familiar, the work is practically as rapid as by using feet and decimals, and has the advantage of being absolutely accurate, which the decimal System is not, as inches cannot be expressed exactly in decimals of a foot. The principles upon which the work is based are as follows: When feet are multiplied by feet, or inches by inches, the product is, of course, square feet or square inches, respectively; but, when feet are multiplied by inches or vice versa, the results may, for want of a better name, be termed parts (although this name was given originally to the 144th part of a square foot). Suppose a square foot to be divided into 144 sq. in.; then a strip 12 in., or 1 ft., long, and

1 in. wide will correspond to 1 part. A strip 5 ft. long and 7 in. wide will contain 7 X 5, or 35 parts, which, divided by 12 (parts to a square foot), equals 2 sq.ft. and 11 parts, or

2 sq. ft. + (11 X 12) sq. in. Square inches may be reduced to parts by dividing by 12; thus, 54 sq. in. = 4 parts 6 sq. in. To illustrate these principles, let it be required to find the area of a room 18 ft. 10 in. long, and 16 ft. 7 in. wide. Place feet under feet, and inches under inches, thus:

Feet.

Inches.

18

10

16

7

301

4

10

11

10

312 sq. ft.

3 parts

10 sq. in.,

312 sq. ft. 46 sq. in.

or,

Begin by multiplying 10 in. by 16 ft.,which equals 160 parts, or 13 sq. ft. 4 parts. Place the 4 parts in the inches column, and carry 13 sq. ft. 18X 16 + 13 = 288 sq. ft.; 288 sq. ft. + 13 sq. ft. = 301 sq. ft. Next, multiply 10 in. by 7 in. = 70 sq. in., or 5 parts 10 sq. in. Set the 10 sq. in. to the right of the column of parts, as shown; 18 ft. X 7 in. + 5 parts = 126 parts + 5 parts carried = 131 parts; dividing by 12, 131 parts = 10 sq. ft. 11 parts. Set these down in the proper columns and add, beginning at the right; 4 parts +11 parts = 1 sq. ft. 3 parts. Expressed in square feet and square inches, the result is 312 sq. ft. 46 sq. in., which can be verified by reducing the given numbers to inches, multiplying them, and dividing by 144.