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Cross Multiplication, a term used by artificers for thepeculiar arithmetical process employed in the mensuration oftheir work. Cross multiplication and duodecimals are gene-rally confounded together as being synonymous expressions;but the order of performing their operations is materiallydifferent. In cross multiplication, the parts are actually mul-tiplied crosswise, as well as in direct order, and the terms ofeach factor are confined to feet and inches j whereas, induodecimals, the terms of each of the factors are not confinedto number, and the parts are multiplied in the order of com-mon multiplication.
The rule in cross multiplication is as follows: Write thegiven numbers, as in addition. The sum of the products ofthe alternate parts, and the product of each pair of parts ineach denomination, will be the product of the whole. Itmust be remembered, that
Feet multiplied by feet give feet.
Feet multiplied by inches give inches.
Feet multiplied by seconds give seconds.
Inches multiplied by inches give seconds.
The method of performing the operation will be seen inthe following examples.—What is the area of a board, thelength of which is 25 feet 11 inches, and the breadth 23 feet7 inches ?
ft. in.
25 1123 7
0
6 5
= 7
X
11
14
7
= 7
X
25
21
1
= 23
X
11
575
= 23
X
25
611 2 5
Required the product of 58 feet 9 inches, by 36 feet 6 inches.
ft.
in.
58
9
36
6
0
4 6
= 6
X
9
29
= 58
X
6
27
= 36
X
9
2088
= 58
X
36
! I 2144 4 6
The following method is another form of cross multipli-cation, discovered by Mr. P. Nicholson, in which side oper-ations are unnecessary, and consequently, as a part of thework, must either be wrought on the margin, as above, or inthe mind: the work, by the method proposed, will be shorterthan the above, or less liable to mistake, if the side operationsbe not used. The rule is thus:
Multiply the inches together, and the product is seconds;divide the product by 12, if so divisible, the quotient isinches, and the remainder seconds; then, placing the feetand the inches in their respective order, one after the other,set the products of the two cross parts in inches, under theinches of the former quotient; add the three numbers toge-ther, and divide their sum by 12, the quotient will be feet,and the remainder inches ; multiply the feet of the multiplierby the feet of the multiplicand, and place the products underthe feet of the last division, then the sum of these productswill be the number of feet in the whole contents, and theremainders of inches and seconds, if any, the parts which
required to complete the whole product. foes 1
proof of the truth of the other.
First Example.
Second Example
ft. in.
ft. in.
25 11
58 9
23 7
36 6
6 5
4 6
175
348
253
324
12 ) 434
12 ) 676
36 2
56 4
75
348 |
50
174 1
611 2 5
2144
By this means, the whole is performed in ° n ® 0 ^ )gr efo rewithout laying any stress on the memory, and W” . . pr» c 'be particularly useful to those who have not sutn cl , gl . s oftice to fix the products and quotients of small n ’j' ^ v0 , i*their memory: and it is the shortest method of 1 w 01 *
the number of marginal figures be counted intoof the former method of operation. . j n „
In duodecimals the process is as follows : l> e g in . @ jjiih 1 'the last term, or the farthest from the left-hand in^ ^ o[Tl th®
ftP
c$'
plier, multiply by it each term of the multiplies 0 ^right-hand to the left, carrying one for every two v ^successive product, but the denominations must n ,| tj*ried farther than the place of feet; again, mterms of the multiplicand, by each successive te ^ [iri A e _the left of the multiplier; then place the products^ \e^the other, with the first term of every product f tfl lunder the second term of the product of the ho .^(jon 3immediately above ; then add the similar denom> ’
each product: the sum will be the whole produc ■ V ’in the first place, to put the highest den°m in ^piic* 11 ^the feet, under the first term on the left of the 10 g te rlll - t jjand the terms of the product under each respec t [ 0 n v/>the multiplicand will be of the same denomieach other. . , t0 juun 1 *
Example. —Suppose, again, it were require jg,feet 11 inches by 23 feet 7 inches, by duode
11 X 7 = 77
25
25 1123 7
6 5
25 X 7 = 175
611 2 5
11 X 23 = 253
iT'i - #
25 X 3 = 7525 X 20 = 500
596 t l„i
Cross multiplication is much better adapted to ^ art ) 'jperficial contents of a piece of work, where