(Contributed by W. H. Brown, F.S.I.)

Having completed the "Taking-off," the dimensions should be "squared" and entered in the squaring column in black ink. The squaring should then be checked by another person, and ticked in red ink where found correct. Where the checker does not agree with any of the results he should have his own figures checked, and the requisite alteration should be made in red.

In squaring there are many short methods which are learned by experience. A few such will now be explained, in the hope that they may suggest the lines along which others may be looked for; and they are valuable insomuch that every expedient which shortens or simplifies the operation of squaring also lessens the liability to error. The expedient capable of the most universal application is perhaps that of varying the order of the factors. By this means it is often possible to get rid of fractions and sometimes also to eliminate the inches at the outset; for example:-

 3/4/3.6 2.3 .8 = 63'. o"

This means three times four times (or 12 times), a cube whose sides are 3 feet 6 inches, 2 feet 3 inches, and 8 inches respectively. By working in the following order the above result is arrived at without putting a figure to paper, namely:-

 3x8" = 2'. 0" and 2'. 0" x 3'. 6" = 7'. 0" 4 x 2'- 3" = 9'.o" 7'. 0" x 9'. 0" = 63'. 0"

In cubing certain scantlings it is sometimes possible to find a single divisor which will give the required result in one operation. For example, any length of 4 x 3-inch timber can be instantly cubed by dividing by 12, because 4 inches = 1/3 of a foot and 3 inches = 1/4, and 1/4 x 1/3 = 1/12.

 Thus: 126.3 .4 .3 = 10'. 5 1/4"

For similar reasons any length of 2 x 6-inch timber can be cubed by taking 12 as the divisor, and any length of 2 x 9-inch by taking 8 as the divisor.

Again, in multiplying by 4 1/2 feet x 3 inches it is often more convenient to add 1/8 and divide by 12.

 24. 0 .4 1/2 .3 = 2.3

which is worked mentally thus:- 24 feet 0 inches + 1/8 of 24 feet 0 inches = 24 feet 0 inches + 3 feet 0 inches = 27.0; which divided by 12 = 2 feet 3 inches.

It may be noted that this result, 2 3", though universally called 2 feet 3 inches by those engaged in building operations, means 2 cubic feet and three-twelfths of a cubic foot; for according to the practical man, a foot contains 12 inches, whether it be a lineal, super., or cubic foot.

## Duodecimals

Squaring dimensions is always performed by duodecimals, of which the above examples are modifications. The system consists in taking 1 foot as the unit and dividing it into twelfths; in the same way as in the decimal system the unit is divided into tenths. Thus 10 feet 6 1/2 inches would read as 10 + 6/12 + 6/144 feet, and would properly be written 10' 6. 6. Multiplication by duodecimals is performed as follows. When multiplying by feet the product will be in the same denomination as the figure multiplied.

When multiplying by 12ths (or "inches"), it will be moved one place to the right.

When multiplying by 144ths, it will be moved two places to the right; and by 1728ths, three places to the right, and so on.

For example, the process of multiplying 10 . 6 1/2 by 8 . 7 3/4 is performed as follows:-

 10.6. 6 8.7. 9 84 . 4 . 0 6.1. 9. 6 7 . 10 . 10 .6 91 . 1 . 8 . 4.6 super.

For all practical purposes this would be called 91 feet 2 inches super., ignoring the last two figures, calling each twelfth part an "inch," and the 8/144 of a super. foot 2/3 of an inch. It is usual, however, to call everything an "inch " which is more than 6/144 of a foot. In the case of a cubic dimension, say, 10 feet 6 1/2 inches x 8 feet 7 3/4 inches x 2 feet 6 1/2 inches, proceed as in the last example with the two first dimensions, arriving at the result of 91 feet 1.8.6. and multiply this by 2 feet 6\ inches, or 2 feet 6 . 6 as follows:-

 91 . 1 . 8. 4.6 2.6.6 182 . 3 . 4 . 9.0 45 . 6 . 10 . 2.3.0 3.9. 6.10.2.3.0 231 . 7. 9. 9.5.3.0 cube.

This result would be called 231 feet 8 inches.

Although in practice the system of duodecimals, or division into 12ths, is retained, the actual working can usually be more expeditiously performed by splitting the 12ths up into fractions of a foot having 1 for their numerators. For example, 6 nches = 1/2 a foot; 9 inches = 6 + 3 inches = 1/2 a foot + 1/2 of 6 inches; 8 inches = 4 + 4 inches = 1/3 + 1/3 of a foot; 7 inches = 4 + 3 inches = 1/3 + 1/4 of a foot; 4 1/2 inches = 3 + 1 1/2 inch = 1/4 of a foot + 1/2 of 3 inches; and the foregoing example would be worked as follows, varying the order of the factors as previously explained. The result is a species of "Practice" as shown below:-

 8.7 3/4 10.6 1/2 86.5 1/2 6" = 1/2 a foot, that is 1/2 the top line = 4 • 3 7/8 1/2" =1/12 of 6= 1/12 the last line = 4 1/23 nearly "A" 91 . 2 2.6 1/2 182 . 4 6" = 1/2 a foot, that is 1/2 of " A " = 45.7 1/2" =1/12 of 6",that is 1/12the last line = 3-9 7/12 231 . 8 cube

The example given is a very exceptional one, for the sake of illustrating the principle, as fractions of an inch rarely occur in the dimensions, except in the case of timber scantlings. They are, however, often unavoidably introduced in multiplying the first two factors of a cubic measurement together, but a little practice will soon teach the student to arrange his factors so as to avoid fractions whenever it is possible.

All dimensions, whether they have to be squared or not, should be carried into the squaring column, otherwise they are very liable to be overlooked when abstracting.

Where two or more consecutive dimensions of the same description occur they should be added up at the time of squaring. This saves the Abstractor's time, and space on the Abstract sheet. The person who checks the squaring should also check these totals, and remember to alter them when any of the dimensions comprising them have been found incorrect. All corrections made by the Checker should in turn be checked. The totals of similar consecutive items should be entered on the dimension sheets as shown in Fig. 32.

The Taker-off, when making collections for any of his dimensions, should do so "on waste," as it is called; that is, in the description column, so that the method by which the dimensions has been arrived at can be readily traced at any future time.