CROSS-GRAINED wood its fibres in contrary directions to the surface, and which therefore cannot be made perfectly smooth by the operation of the plane, without either turning the plane or the stuff This most frequently arises from a twisted disposition of the fibres in the act of growing.
()loss Nirl.TIPLICATION, a term used by artificers for the peculiar arithmetical process employed in the mensuration of their work. Cross multiplication and duodecimals are gene rally MI1;11111(10;1 together as being synonymous expressions ; order pert inning their operations is materially different. In cross multiplication, the parts are actually mul tiplied crosswise, as \yell as in direct order. and the terms of each factorare confined to feet and inches; whereas, in duodecimals, the terms of each of the factors are not confined to number, and the parts are multiplied in the order of COM 111011 mu t pl lea tion.
The rule in cross multiplication is as follows : Write the given numbers, as in addition. The stun of the products of the alternate parts, and the product of each pair of parts in each denomination, will be the product of the whole. It must be remembered, that Feet multiplied by feet give feet.
Feet multiplied by inches give inches.
Feet multiplied by seconds give second.:. Inches multiplied by inches give seconds.
The method of performing the operation NVIII be seen in the following examples.—What is the area of a board. the length of which is 25 feet 11 inches, and the breadth 23 feet 7 inches ft. in.
23 11 23 7 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 the work, must either be wrought on the margin, as above, or in the mind : the work, by the method proposed, will be shorter than the above, or less liable to mistake, if the side operations be not used. The rule is thus : Multiply the inches together, and the product is seconds ; divide the product by t2, if so divisible, the quotient is inches, and the remainder seconds; then, placing the feet and the'inches in their respective order, one after the other, set the products of the two cross parts in inches, under the inches 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 multiplier by the feet of the multiplicand, and place the products under the feet of the last division, then the sum of these products will he the number of feet in the whole contents, and the remainders of inches and seconds, if any, the parts which are required to complete the whole product. We shall take the two Iiirmer examples, and the one method will be the best proof of the truth of the other.
By this means, the whole is performed in one operation, without laying any stress on the memory. and ++ ill therefore be particularly useful to those who have not sufficient prac tice to fix the products and quotients of small numbers on their memory : and it is the shortest method of the two, if the number of marginal figures be counted into the work of the former method of operation.
In duodecimals the process is as follows: beginning w ith the last term, or the furthest from the left-hand in the multi plier. multiply by it each term of the multiplicand from the right-hand to the left, carrying one for every tuu elve to each successive product, but the denominations must not be car ried farther than the place of ; again. multiply all the terms of the multiplicand, by each successive term towards the left of the multiplier ; then place the products one under the other, with the first term of every product on the left, under the second term of the product of the horizontal row immediately above ; then add the similar denominations of each product : the sum will be the whole product. Observe, in the. first place, to put the highest denomination, viz., the feet, under the first term on the left of the multiplicand, and the terms of the product under each respective term of the multiplicand will be of the same denomination with each other.
Example.—Suppose, again, it were required to multiply 25 feet 11 inches by 23 feet 7 inches, by duodecimals.
two terms in each factor, and where the feet of the multiplier rim to a high number, as well as the feet of the multipli cand, and more particularly the second mode of cross multi plication ; as in duodecimals, the calculator must either have a very good memory require no marginal work, or otherwise the quantity of marginal operations M ill exceed that of the principal work. Artificers and never take any account of I lenmninations less than inches, except in glazier's work, and hence cross multiplication is almost the only useful tnethod of finding the contents; but where more denominations are concerned, recourse must be had to duode cimals or decimals, as when the terms of each factor are more than two, cross multiplication cannot be applied.
When the terms of the multiplier are under 10, the operation will be exceedingly easy, as the following example will show.
Example.—Multiply S feet, 3 inches, 5 seconds, and 7 thirds, by 5 feet, 4 inches, 9 seconds, and 6 thirds.
placed alternately in a straight line, and the sides in conti guity with each other, the whole will Iiirm a rectangle, whose length is equal to the half' stun of the t w o •ircumfemices, and the breadth that of the ring; for all the middle breadths are in one straight line, equal to the length of the rectangle.
Example.—Suppose the greater circumference of a crown to be 24 feet S inches, and the lesser 21 tent 6 itches, and the breadth 2 feet 9 inches, required the superficial content? It will be perceived that there is a great difference between cross multiplication and duodecimals, and the purposes to which each may be most advantageously applied ; but the reader who wishes for thrther information on this subject, may have recourse to the articles DECIMALS, DUODECIMALS, and PRACTICE.