by Duodecimal, or (as it is sometimes called,) Cross Multiplication, which we now proceed to explain. The examples in Exercise cxI. prove that

I. L in ft. with B in ft. gives

A. in sup. feet
A. in sup. primes
A. in sup. seconds
A. in sup. in. or sec.
A. in sup. thirds
A. in sup. fourths.

II. L in ft.
III. L in ft.

B in in.

B in lines,,

IV. Inches

Whence it is usual, erroneously to say that

I. Feet Feet give Feet.

Inches give Seconds.

II. Feet X Inches give Inches.*| V. Inches X Lines give Thirds.
III. Feet X Lines give Seconds. VI. Lines X Lines give Fourths.
The correct statement to be employed instead of the
foregoing, is

I. Units XUnits produce Units. IV.
II. Units X12ths produce 12ths.
III. Units X144ths pro. 144ths. VI.

V.

12ths X 12ths pro. 144ths. 12ths X144ths pro. 1728ths. 144ths X144ths pro. 20736ths.

GENERAL PRINCIPLE FOR MULTIPLICATION.

PRINCIPLE XXVII. Multiplying by a Number standing in any column of Duodecimals will remove the ProductFigures as many columns to the right as Multiplier is to the right of Units Column.

Compare this with Principle XII.

To Multiply Feet, Primes, Inches, &c. by any Duodecimal Multiplier.

First. Place Multiplier under Multiplicand; place for place, that is to say, Units under Units, 12ths under 12ths, &c., column for column.

Second. Multiply successively by each separate part of Multiplier; placing Product-Figures in their proper columns, according to Principle XXVII., and carrying one for every twelve from each column to next on the left.

FIRST EXAMPLE WORKED OUT. Find the area of a wall 14 ft. 11 in. by 9 ft. 6 in.

ft. in. li.

Length 14. 11. 3.

Example completed.
su. ft. pr. in.

14. 11. 3.

=

Ar. of strip 1 ft. wide.
No. of such strips.

11· 2· 5·3-"""=Ar. if 9·li. wide. 7.5.7.6.

=Ar. if 6 in. wide.

*Primes are here meant; but Artificers generally call the twelfth part of a sup. ft. an inch, at the same time aware that "144 sq. in. = 1.sq. ft."

a. 3.sup.in.X1=144 of 3. sup. in. (PR. IX.)=27. sup.4ths.=2. 3• Set down 3" and carry 2''''

b. 11. sup. pr.×14=144 of 11. sup. pr. (PR. Ix.) making, with 2 ́ ́ ́′ carried, 101· sup. 3rds., Set down 5" and carry 8"

=

=

99. sup. 3rds.,

8" 5"

c. 14 sup. ft.X14+=144 of 14′ sup. ft. (PR. IX.) = 126. sup. 2nds., or in., making, with 8-in. carried, 134′ sup. in. =11′′ 2′′′′ Set down 2" and carry 11"

d. Set down 11· primes.

Second Line.

Third Line.

=

a. 3'su. in. X6 of 3.su. in.
b. 11'su. pr. X6 of 11'su. pr.
c. 14'su. ft. X612 of 14'su. ft. 14
The work may be proved in a variety of ways.

18" a. 3.su. in. X9'— 27'su.in.
66. b. 11'su. pr.X999'su.pr.
c. 14'su. ft. X9'=126'su.ft.

I.

6.' | | 14.11.3.

134

8.9.2.3.

9"

133. 10. 6.

=A. 14. f. lg.

A. =142∙ 10∙ 0.114 if14′ft. 114′lg.

5.3.

A. 142. 100 114 if 14.ft. 114′lg.

V. By Reduction and Multiplication. See Second Example worked out, page 234.

CUSTOMARY AND STATUTE LINEAL MEASURES.
The old or customary lineal pole or landyard = 18.ft.
The statute lineal rod, pole, or perch

.. Ratio of Customary to Statute pole-18=3;=H

To Reduce Statute Lineal Units to Customary.
Multiply by 11 and Divide by 12.

To Reduce Customary Lineal Units to Statute.
Multiply by 12 and Divide by 11.

CUSTOMARY AND STATUTE SUPERFICIAL MEASURES.

An acre of old or customary measure contains the same number of roods and poles as the statute acre. But, Cust. Sup. po. 18 times 18. sup. ft. Whilst, Stat. Sup. po. 16 times 16 sup. ft. Hence, Ratio of Cust. Sup. po. to Stat. Sup. po.

And,

=

Ratio of Stat. Sup. po. to Cust. Sup. po.

=

324 sup. ft.

=

272 sup. ft.

324 144

2724

121

To Reduce Statute Area to Customary.
Multiply by 121 and Divide by 144.
To Reduce Customary Area to Statute.
Multiply by 144 and Divide by 121.

To find customary area of a small piece of land, take the dimensions in customary poles or landyards of 18. lin. ft. or 6 paces, each. The Area will then come out in customary superficial poles or landyards.

Or, take dimensions in feet. Find Area in sup.ft. Divide by 324.

EXERCISE CXII.

EXAMPLES IN THE MENSURATION OF PARALLELOGRAMS

AND TRIANGLES.

Explain every line, fully, in writing; specially noting the change of Unit. Work each Example in, at least, two ways; Vulgar Fractions to be one.

Find the Area

A.

1. Of a square field, measuring 948 links in the side.*

2. Of a parallelogram, 3450 links long, by 640 links broad.* 3. Of a rectangular estate, 19348 links long, by 8495 links wide.* 4. Of an oblong garden, 98 yards long, by 15 yards broad.* 5. Of a floor, a rectangle in shape, 18. ft. 11 in. by 14 ft. 3. in. 6. Of a quadrangular park, two adjacent sides of which measure, respectively, 1. mi. 5. fur. 25' po., and 45 chains."

Reduce both dimensions to miles, or to poles, or to chains. 7. Of a triangular plantation, which is 945 links in length, and 240 links in perpendicular breadth.*

*Examples marked thus to be worked out: I. In Statute Measure. II. In Old or Customary Measure.

B.

Find the respective Areas of the Rectangles having the lowing dimensions. Explain every line of the work. Solve ch in two ways, Vulgar Fractions to be one.

The cost of superficial measures may be found by Practice. EXERCISE CVII. F. and G., pages 261 and 262, but fractional methods are also frequently used. FIRST EXAMPLE WORKED OUT.

Find the cost of flooring a rectangular room, 32 ft. 8 in. long, by 18 ft. 6 in. broad, at 21's. per square of 100 superficial feet.

SECOND EXAMPLE WORKED OUT. What would be the cost of painting a wainscotted room, 12 ft. 3 in. high, 18. ft. 6 in. long, and 14 ft. 9 in. wide, at 74d. per square yard?

Example completed.

I.

Here, the breadth of the rectangle is the height of the room. And

Whole length of wall twice length and breadth of room.

.. Area painted

=(66×142) su. ft.=9803 su. ft.=1087 su.yd.

Now, 109 (or 9 doz. + 1.) at 73d. cost (7/6 × 9·)+71⁄2d.

.. Cost of painting:

=

£3. 8's. 1d. very nearly.

The error is only 2 of 74d., or 3% of a farthing.

II.

But, in computing sup. yds., much time and trouble will be saved by turning the lineal dimensions at once into yards, and multiplying, if necessary, by fractions.