MEN
180
MEN
to the same submultiple of the former difference ; nowbecause the equation n X — 5 —^ = a + b + c + d,
&c., to p + q, was obtained by dividing each side of the
a + q + 2b-l~2c + 2d, &c, to 2 p _l X -:--- — l
2 n — 2
by l, and multiply each side
equation
a + b + e + d, &c. top + qn
respectively by n, n — 1 , and then adding, or taking awaythe common parts; by making the divisor l a multiplier, andthe multipliers n, n — 1 , divisors ; then multiplying the
equation n X — — a + b + c-\-d, &c. to p + q by
l, and dividing it by n and n — 1, there will arise l Xa + q _ j a + 5 + c + <Z, &c. to p + q.
2 X (w—1) _ n X («—1)
Example. —Let a — 4, b = 12, c = 16, d z= 18, jo = 19,q = 18, and l = 10;
then l X
a + q
= 10 X
4+18
a
2 O — 1) 2 X (6 — 1)
+ b + c + d, &c. to p + q
= 22 ,
and / X , ,
n + (n —
4 + 12 + 16 + 18 + 19 +
= 10 x
6 X (6—1)
= 29; the difference is there-
( a + q v
—g— +6 + c+ d, &c. to p + q \
X l, and
+ b + c + d, 8 cc to p + q
X l.
1 I
Mensuration op Artificers’ Works. All such works,whether superficial or solid, are computed by the rules properfor the figure of them.
The most common instruments for taking the measuresare, a five-feet rod, divided into feet and quarters of a foot;and a rule, either divided into inches, or twelfth parts,and each twelfth part into twelve others ; a fractional partbeyond this division, measurers seldom, or never, take anyaccount of.
When the dimensions are taken by a rule divided in thismanner, the best methods to square the dimensions will thenbe by duodecimals, by the rule of practice, or by the multi-plication of vulgar fractions ; but, in the opinion of some,the best method of taking dimensions is with a rule, wheneach foot is divided into ten parts, and each part into tenother parts, or seconds, because the dimensions may be thensquared by the rules of multiplication of decimals, which isby far the shortest and readiest method. Those who con-tend that duodecimals, or cross multiplication, is the easiestmethod of squaring dimensions, as well as the most exact,are very much mistaken ; for if the dimensions are taken induodecimals, and reduced to decimals, and then squared, theoperation, in this case, will certainly be much longer than ifit had been done at once by duodecimals, and sometimes notso exact: but if the dimensions are taken in feet, tenths, &c.the operation will not only be easier and shorter, but in manycases will be much more exact than by duodecimals: the rea-son is obvious to those who consider that there are manycases in which it will be impossible to express, truly, a deci-mal scale equal to a duodecimal one; neither will it, in manycases, be possible to express accurately, a duodecimal scaleequal to a decimal one; duodecimals have the same property
with regard to twelfth parts, as decimals have to tenth P j a tetherefore, in many cases, duodecimals will sometimes c ,r ^ c j.and run on, ad infinitum, when reduced from decimals,mals will, when reduced from duodecimals; and farther, ^duodecimals are expressed by a series of twelfth p ar ^ ofdecimals by a series of tenth parts, in multiplying e ® e jvethe parts of the former, the trouble of dividing by 1will then be unavoidable, and more burdensome to tn ^ e rethan if the operation had been done by the latter,there is no such division to be made, but merely to ® ucommon multiplication, and point off the decima r
, 0O 1
in the product. ,
This last method is always to be preferred, as tnnatural, as well as the most easy of the two.
The mensuration of brickwork has already been tre» ^at considerable length under that head, but in order
to
ble ® 9
an 1
plete the article, we shall give a few more proiexamples.
Problem I.— To measure the vacuity of a .
Find the area of the outside of the window, and ® ^ fitthat by the number of half bricks thick, from the fa® 6 ^ th®sash-frame on the outside, to the face of the wall ^ fl( l Asame side; to the area so found, at half a brick tb* ^ o>the area of the inside vacuity multiplied by the n u ® tb ehalf bricks thick, from the face of the sash-frame ,^-pgioutside, to the face of the brick-work within the b°also add the area of the vacuity of the recess, the ^r,being taken from the bottom of the sash-frame to t ^jpl/and its width the same as the inside vacuity above ; ®this also by the number of half bricks thick, then t ji«of these will be the whole vacuity, or void space ^ \>e
whole window, at half a brick thick ; and if re Q u ' re ^reduced to the standard, divide the area so found by^,^_the area of the contents will be reduced to brick gExample .—Let the height of the outside vacuity b ^ tl ,eits breadth 4 feet, and half a brick thick ; the j
inside vacuity 8 feet, and its breadth 4 feet 9 inches, jpcbbricks thick; the recess is 2 feet 9 inches high, 4 f® e w’b 0wide, and half a brick thick ; required the area 01vacuity, at half a brick thick.
Ft. In.
8 0 height of the outside vacuity.
0 width of the outside vacuity.
32
0 area of the outside vacuity, half a brick tbic
9 width of the inside vacuity.0 height of the inside vacuity.
152
0
0 number of half bricks.
0 area of the inside vacuity, half a brick tbic
9 width of the inside vacuity. (
9 height of the recess from the floor to the— sash.
9 area at 1-J- brick thick.
0
3 area of the recess, half a brick thick.00
223 2 3 area of the whole vacuity, half
a brick
tb> cS