affords great advantage in calculation, seeing that we have only

1

to subtract-th. For example, if we would multiply 3848 by

7

m

, as the complement of is, we have only to subtract from 3848 its eighth part, thus: 3848 481= 3367, which is

easier than the following:

3848

7

8) 26936

3367

Again, if we would find the cost of 3848 yds. at 17 s. 6d. per yd., as 17 s. 6d. is £7, we subtract from 3848 its eighth, as above, and have £3367 for the required cost. is much easier than the following:

£

This

What is the cost of 97 yds. at 831 cts. per yd. ?

Then as 831 is of a

Here 97 yds. at $1, would be $97.

dollar, and its comp., we subtract from 97 its sixth part, and

have $80,831 for the required cost.

This is certainly more expeditious than the following:

As a mere introduction to the use which the intelligent stu dent may hereafter make of this principle, we here insert a few

nQ

dq

m

m = d.
mQ + do

m

be any quantity,

Then (77) n = m +d: consequently

=

:Q+ ; that is, Q is increased in a ratio

m

expressed by the fraction

or part by which

338. When n

of itself; that is,

по
m

n

m

=Q+ -. Thus 24 × =24+3=27.

Also, 24 yds. at $1,121 per yd. = $24 + $3

$27.

Here we have added to $24, the cost of 24 yds. at $1, the

cost of 24 yds. at $1, which is $3.

Proof.
24

1121 cts.

2688

12

$27,00

Required the cost of 291 yds. at $1,331 per yd.

Here the supplement is 331 cts. $1. We therefore add to 291, considered as $291, its third part, thus: 291 +93 $39, Änswer.

Here, as it is of no consequence which number we take for multiplicand, we add to $1,663 its fourth part, which is 413, and have 1,663 +413 $2081 or $208,331, for the answer.

Proof.

1,663 × 1000

8

1666,663

=208,331. See art. 273.

SECTION XVIII.

DUODECIMALS-BOARD MEASURE, ETC.

Duodecimals.

339. This rule is used in measuring cord-wood, artificer's work, &c. The foot is the principal measure, and is divided as follows:

1 foot, ft. is equal to 12 inches, or primes, marked thus,' (2)

The inch, or prime, is

of

7728

of a foot. of a foot, &c.

of a foot. The second is (2), or The third is (12)3, or 12 of 144

These parts, from their regular ratio of 12, are called duodecimals, which name is derived from the Latin duodecim, twelve.

Multiplication of Duodecimals.

340. From the nature of fractions, we shall easily perceive what parts will be produced in multiplying together any of the above denominations. For example, if we multiply by feet, as these are integers (whole numbers,) having a unit for denominator, it is evident that the denominator of the part multiplied will not be altered; thus, 3 feet multiplied by 2 feet, will give 6

6 =

feet; 3 in. multiplied by 2 ft., that is, 12, or 6 in. Again, 3 sec. multiplied by 2 ft., or 14X14, or 6 seconds, etc.

If we multiply by inches, as the denominator of these is 12, the denominator of the part multiplied will become the same as that of the next lower denomination; thus, 3 ft. multiplied by 2 in. will give 6 in.; that is, X6, or 6 inches; 3 in. multiplied by 2 in.; that is, 13 X 214, or 6 seconds; 3 sec. multiplied by 2 in., or X=1725, or 6 thirds, &c. From what has been said, the products of any other parts will easily be estimated by one who has learned fractions.

12

341. As the second is the square of the prime, the third, the cube, &c., if we consider the feet as having 0 for their index, the inches 1, the seconds 2, the thirds, 3, &c., if we multiply these together, the parts produced will have for their index the sum of the indices of the parts multiplied; thus, the index of the prime is 1; that of the second is 2, and the sum 3, of these indices, is the index of the parts produced when we multiply inches by seconds. It is the same for any other parts, seeing that their product contains the factor, as often as its index contains a unit.

For example, to multiply 7 ft. 3 in. 4 sec. by 2 ft. 4 in., we write the numbers thus:

and first multiply all the parts of the multiplicand by the 2 ft. of the multiplier. Beginning with the lowest denomination, which is here seconds, we say 2° × 4". 8", which we place under seconds; then 2° X 3' 6', which we write under inches: lastly, 2° 7° 14°, which we write under feet.

Then, to multiply by 4', we say 4' × 4′′ = 16′′" — 1′′+4′′"; we therefore write 4"" one place farther to the right, and retain the 1", to add to the next product. Then 4' X 3'12", and, adding the 1" retained, we have 13": =1'1"; we therefore write 1" under 8", and retain the l' to add to the next product. Lastly, 4' X 7°28', and 1' retained, is 29′ = 2° 5'; we therefore write 5' under the primes or inches, and the