DUO

310

DUO

i I

“ This upper gallery, in which a man could barely makehis way, went quite round the tower, without any division orpartition ; but the lower gallery, underneath this, is partedoff into separate spaces, by great flag-stones placed upright;which several spaces, or little cells, were in general acces-sible only by means of holes in the floor or gallery above ;so that nothing can be more obvious, than that these cellswere intended for the keeping and preserving of stores ;whilst the upper gallery cannot but remind us somewhatof the little gallery within the wall of the round tower atBrunless.

“ Besides these galleries, there were, on the inside of thecircular wall, open to the circular enclosed apartment, fourperpendicular rows of small cavities, or, as they have beendescribed by others, four stages or nests of small square openholes, dividing the interior circular wall into four parts, andturning up from the lower part of the tower to the top; eachlittle hole, or nest, in the row, divided from that beneathonly by a sort of shelf, or flag-stone, and forming a littlecupboard.

“ The appearance of this sort of little cupboards, as wellas that of the sections of galleries, is similar to those in thistower, as they are seen in the wall of another dun, in thesame neighbourhood. And these square cavities seem obvi-ously to have been intended to hold the drinking horns, andother utensils for banqueting in these rude dens.”

DUODECIMALS, a term applied to an arithmetical me-thod of ascertaining the number of square feet and squareinches in a rectangular space, whose sides are given in feetand inches.

In this series of denominations (beginning with feet) everyunit in the preceding denomination makes twelve in thatwhich succeeds it: that is, every foot contains 12 inches, orfirsts ; every first, 12 seconds ; and so on. There will be asmany denominations in the product as- in both factors takentogether.

Feet are either marked with or without an f; inches arecalled firsts by the mark thus (') being placed above; secondsby the mark thus (") being placed above ; thirds thus ("'),and so on. In multiplying any two single denominationstogether, the value of the product will be known by adding

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the indices of the two factors. Thus, suppose 7 to be mul-

m V

tiplied by 11, then the product is 77, or 77 fifths, becauseadding the index " of the seven to the index of the eleven,produces v or fifths.

To multiply duodecimals together. —Write the multiplierunder the multiplicand, so that the place of feet may standunder the last place of the multiplicand ; begin with theright-hand denomination of the multiplier, and multiply itit by every denomination of the multiplicand, throwing thetwelves out of every product, and carrying as many units tothe next ; place the remainders, if any, under the multiplier,so that like parts in the product may be under like, parts ofthe multiplicand ; proceed with every successive figure ofthe multiplier, towards the left, in the same manner, alwaysplacing the first figure of the product under the multiplier ;then the sum of the products will be the total.

Example 1.

Multiply 6 •• 9' by 3 •• 6'n. / «

6-9

3..6

3-4

20-3

6 = 69X6= 6-9X3

Example 2.

Multiply 6 •• 5' - 4" by 3 ■a. , „

6-5-4

3-6

- ft. , /.

3-2-8-0 = 6-5-419-4-0 =6-5-4

Answer 22 - 6 - 8 - 0

In the first example, there is only one place of du j u ct.in each factor ; there are, therefore, two places in t *j u0 dect'In the second example, there are two places 0 ,vh' cmals in the multiplicand, and one in the multipmake together three ; there are, therefore, threetions in the product.

Example 3. ,

ft ' " "L n ‘

What is the product of 4'.3"2'.8"9" 1

by

2-8-9-5.

3-0-5

.8-7'

1 - 0 .

"l -o-ll •• 5 •• 6 - 8 - 2 - o - 1 - 1 2 - ^ “ \ ti iier,In this example, because there are no feet in the ilB sthe place is supplied by the cipher. The multip 1,c _ (lie resix places of duodecimals, and the multiplier se ve ”’juc‘-are, therefore, thirteen places of duodecimals in the P t)l eThe first place of figures is feet, and the succeeding e | eV -eflduodecimal places; the product is one foot, no inches. ^seconds, five thirds, &c. But, independently of tu . jn tl>«deration of there being as many places of duodecmw^d of

product as in the multiplier and multiplicand, the B rre ctplacing the denominations of the factors gives the ^places of the product, since like parts of the P r0 ? ‘finW’under like parts of the multiplicand ; it also shows the. fl „d

not only between duodecimals, but between dec '- be r i”every series of denominations, of which the same no® peany place makes one of the next towards the left ^consideration is also useful, in discovering readily * oa ,iP rof product arises by multiplying any two single detions together. J ° J to r,

When the number of feet runs very high in each t powill bemuch better toreduceall the denominations it“ djv id»the lowest then multiply the factors, so reduced, a^uct-by 12 as often as there are duodecimal places m tbe PExample 4.

6 as in Example 1*

Multiply 6-5-4

5-4

42

Answer 23 - 7 - 6

928

42

1856

3712

12)3897612)324812 )270 -8

22 - 6 - 8 the same as before.

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