RULE.

Reduce the factors to common fractions; multiply these fractions together, and reduce their product to a decimal.

2. What is the product of 3×·09×8·2×·9?

Ans. 249158.

76. Division of Infinite Decimals.

4

1. Divide 36 by 25. Here, 36-36; ·25=23. A÷÷ 33=23=389=1103=Ans. 1-4229249011857707509881.

Hence the

252

RULE.

Reduce both divisor and dividend to common fractions; find their quotient in a common fraction, and change it to a decimal.

2. Divide 13.5169533 by 4·297.

Ans. 3.145.

NOTE 1.-Infinite Decimals without the points limiting the repetends, are really Decimal Fractions, because they are entitled to the decimal denomi

nator.

But Infinite Decimals with the points, lose their character as decimals, because they drop the decimal denominator, and, universally, assume for a denominator as many 9s as there are figures in the repetend.

A repetend with the points always exactly expresses the value of the common fraction, from which it is produced." Without the points, it expresses the approximate value only of the common fraction.

The true name, therefore, of Infinite Decimals with the points, would be NOVIMALS, from the Latin novem, nine; as DECIMALS come from the Latin decem, ten.

77. DUODECIMALS.

Duodecimals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decimally into ten equal parts, is divided duodecimally into twelve equal parts, called primes, marked thus ('). Again, each of these parts is conceived to be divided into twelve other equal parts, called seconds ("). In like manner, each second is conceived to be divided into twelve equal parts, called thirds ('''); each third into twelve equal parts, called fourths; and so ou to any extent.

1

In this way of dividing a foot, it is obvious that

prime is

of1⁄2 of 11⁄2

of 11⁄2 of 11⁄2 of 11⁄2

1''''' fifth is 1⁄2

of 15 of 11⁄2 of 12 of 12 = 24532 of a foot, &c.

TABLE.

12" fourths make 1"" third,

12" thirds. . . 1" second,

12" seconds

12' primes.

Nor 1.-The marks.

1' prime,

1 foot.

****, &, which distinguish the different parts, are called the indices of the parts or denominations.

Nors 2-The divisions of a unit in duodecimals are uniform. just as in decimal fractions, with this difference: they decrease in a twelve fold proportion, 12 of a lower denomination making 1 of a higher. Operations in them are consequently the same as in whole numbers or decimals, except that 12 is the carrying number instead of 10.

78. Multiplication of Duodecimals.

Duodecimals are used in measuring surfaces and solids.

1. How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide?

Length x breadth = superficial contents (40, Prob. I.).

OPERATION.
feet.

Length, 16 7
Breadth, 1 3'

4 I' 16 7'

9"

a foot, and 3 inches = of a foot; consequently, the product of 7' x 3'=744 of a foot, that is, 21′′=1′ and 9"; wherefore we set down the 9", and reserve the 1' to be carried forward to its proper place. To multiply 16 feet by 3', is to take 31⁄2 of 1=1, that is, 48'; and the 1' which we reserved makes 49',= 4 feet 1'; we therefore set down the 1', and carry forward the 4 feet to its proper place. Then, multiplying the multiplicand by the 1 foot in the multiplier, and adding the two products together, we obtain the Answer, 20 feet 8′ and 9′′.

Ans. 20 8' 9′′

NOTE 1.-In all cases, the product of any two denominations will always be of the denomination denoted by the sum of their INDICES. Thus, in the above example, the sum of the indices of 7'x3' is ''; consequently, the product

is 21; and thus, primes multiplied by primes, produce seconds; primes multiplied by seconds, produce thirds; fourths multiplied by fifths, produce ninths, &c.

2. How many solid feet in a block 15 feet 8' long, 1 foot 5' wide, and 1 foot 4' thick?

I. Write the multiplier under the multiplicand, like denominations under like, and in multiplying, remember that the product of any two denominations will be of that denomination denoted by the sum of their indices.

II. Add the several products together, and the sum will be the product required.

EXAMPLES.

3. How many square feet in a stock of 15 boards, each of which is 12 feet 8' in length, and 13' wide? Ans. 205 feet 10'. 4. What is the product of 371 feet 2' 6'' multiplied by 181 feet 1' 9"? Ans. 67242 feet 10' 1" 4""' 6''''. 5. There is a room plastered, the compass of which is 47 feet 3', and the hight 7 feet 6'; what are the contents?

Ans. 39 yards 3 feet 4′ 6′′. 6. What will it cost to pave a court-yard, 26 feet 8' long by 24 feet 9' wide, at $90 per square yard? Ans. $66.

7. There is a house containing two rooms, each 16 feet by 15 feet 4'; a hall 24 feet by 10 feet 6'; three bedrooms, each

11 feet 4' by 8 feet; a pantry, 7 feet by 9 feet 6'; a kitchen, 14 feet 2' by 18 feet; and two chambers, each 16 feet by 20 feet 8': what did the work of flooring cost, at $:02 per square foot? Ans. $39.95.

79. Masons Work is estimated by the perch of 16 feet in length, 11⁄2 feet in width, and 1 foot in hight. A perch contains 24.75 cubic feet. If any wall be 14 feet thick, its contents in perches may be found by dividing its superficial contents by 161; but if it be any other thickness than 11⁄2 feet, its cubic contents must be divided by 24·75 (=243), to reduce it to perches.

Joiners, painters, plasterers, bricklayers, and masons, make no allowance for windows, doors, &c. Bricklayers and masons make no allowance for corners to the walls of houses, cellars, &c., but estimate their work by the girt, that is, the length of the wall on the outside.

1. The side walls of a cellar are each 32 feet 6′ long, the end walls 24 feet 6', and the whole are 7 feet high, and 11⁄2 feet thick; how many perches of stone are required, allowing nothing for waste, and for how many must the mason be paid?

Ans.

45 perches in the wall.
The mason must be paid for 48
{15 mason

48,

perches.

2. How many cord feet of wood in a load 7 feet long, 3 feet wide, and 3 feet 4 inches high, and what will it cost at $40 per cord foot? Ans. 43 cord feet, and it will cost $1·75.

3. How much wood in a load 10 feet in length, 3 feet 9′ in width, and 4 feet 8' in hight? and what will it cost at $1.92 per cord? Ans. 1 cord and 215 cord feet; and it will cost $2.621.

80. SCALE for Surveyors of Wood and Lumber.

By some surveyors of wood, dimensions are taken in feet and decimals of a foot. For this purpose, make a rule or scale 4 feet long, and divide it into feet, and each foot into ten equal parts. Such a rule will be found very convenient for surveyors of wood and of lumber, for painters, joiners, &c. ; for the dimensions taken by it being in feet and decimals of a foot, the casts will be no other than so many operations in decimal fractions.

1. How many square feet in a hearth-stone, which, by a rule, as above described, measures 4.5 feet in length, and 2·6 feet in width? and what will be its cost, at 75 cents per square foot? Ans. 117 feet; and it will cost $8.775.

2. How many cords in a load of wood, 7.5 feet in length, 3·6 feet in width, and 4-8 in hight? Ans. 1 cord 16 cubic feet. 3. How many cord feet in a load of wood 10 feet long, 3:4 feet wide, and 3.5 high? Ans. 76.

Questions.-77. Duodecimals? Division and subdivisions of a foot? Table? Note 1. Indices? Note 2. Divisions of a unit? Operations? 78. Multiplication of Duodecimals. Duodecimals used in what? Ex. 1. Operation and solution? Note 1. The product of two denominations will be of what denomination? Product of thirds multiplied by fourths? Rule? 79. Mason's work, how estimated? Perch? Joiner's, painter's, &c., work, no allowance for what? Bricklayers and masons, no allowance for what? Girt? 80. Scale for Surveyors of wood and lumber? How are dimensions taken by it?

81. COMPOUND NUMBERS.

When several abstract numbers, or several denominate numbers of the same unit value (4), are employed in an arithmetical calculation, they are called simple numbers, and operations with such numbers are called operations in simple numbers.

But when several numbers of different unit values are employed to express one quantity, the whole together is called a compound number. Thus, 12 rods, 9 yards, 2 feet, 6 inches, employed to express the length of a line, is a compound number. So also, 9 gallons, 2 quarts, 1 pint, employed to express a quantity of water, is a compound number.

NOTE 1.-The word denomination is used in compound numbers to denote the name of the unit considered. Thus, bushel and peck are names or denominations of measure; hour, minute, and second, are denominations of time.

English Money.

The Unit of English, called also Sterling Money, is the Pound Sterling. The denominations are pounds, shillings, pence, and farthings. These do not vary uniformly as the denominations U. S. currency do, but according to the following

TABLE.

NOTE 2-All the tables in Reduction of Compound Numbers must be carefully committed to memory by the pupil.

NOTE 3.-Farthings are often written as the fraction of a penny; thus

1 farthing = d., 2 farthings = d., 3 farthings = 3d.