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For if the gauge point, then g2 = √ m2, or
g2 = m2 ==
Hence, m=g2a1, or g2 is the side of the polygon, whose content is m.
145. If the circular divisors are increased in the ratio of 2 to 3, the results are the spherical divisors; the spherical multipliers are the reciprocals of the divisors; and the spherical gauge points are the square roots of the divisors.
h, if h the height of the cylinder.
By 140, c = m1 Now, if d = the diameter of a sphere, and •5236 d3 2 7854 d3 2 d3 = = d3 ÷ 2m1 3m, =d3m. And if n is the reciprocal of m3, c = n,d3. Also the gauge point 93 = m is the diameter of a sphere, whose volume is = m multiplied by 93. 3
or m = .5236g32.
=m =2m1 =
For Conical Vessels.
146. The divisors are three times those for cylinders; the multipliers are their reciprocals; and the gauge points are the square roots of the divisors.
The reason why the divisors are three times as great as those for cylinders, is that the volume of a cylinder is three times that of a cone of the same base and height. It can also be proved, as is similarly done in the preceding articles, that the gauge point is the diameter of a cone, which at one inch of height is equal to the measure of capacity.
For Prismoidal Vessels.
147. If the divisors for rectilineal and cylindric figures are multiplied by 6, the products will be prismoidal divisors; their reciprocals the prismoidal multipliers; and the square roots of the prismoidal divisors the prismoidal gauge points.
TABLES OF MULTIPLIERS, DIVISORS, AND GAUGE POINTS. I. FOR PRISMATIC VESSELS WITH SQUARE BASES.
II. FOR CYLINDRIC VESSELS.
III. FOR REGULAR POLYGONAL PRISMATIC VESSELS.
Multipliers. Gauge Points.
IV. FOR CONICAL VESSELS.
WITH SQUARE ENDS.
WITH CIRCULAR ENDS.
V.-FOR SPHERICAL VESSELS.
VI. PRISMOIDAL VESSELS, FRUSTUMS, OR CYLINDROIDS.
Multipliers. Gauge Points.
THE SLIDING RULE FOR GAUGING.
148. Description of the rule.-On the first side of the rule the line marked A, and that marked B on the slider, are the same as those of the same names on the carpenter's sliding rule, already explained (48); but the latter has the series of numbers from 1 to 10 repeated twice. The third line on the first side of the gauging rule marked MD, or malt depth, is the same as the other two, except that it is inverted, and has the divisor 2218-19 for imperial bushels at each end opposite to 1 and 10 on A. On the line A the number 2218-19 is marked IMB.
On the second side of the rule are lines marked C and D, the former being on a slider. The line D is the continuation of the uppermost line, and these lines are the same as
those of the same name on the carpenter's rule explained in article 49. The numbers on the slider are the squares of those on the line D. The number 18-79 on D is marked IMG, this number being the circular gauge point for imperial gallons. At the number 47.1, the square gauge point for malt bushels, is marked MS; and for round vessels, at 53·14, the circular gauge point for malt bushels, is marked MR.
On the third side are lines for ullaging a standing cask marked Seg. St. or SS, denoting segments standing; and on the fourth side are lines for ullaging lying casks, marked Seg. Ly. or SL, for segments lying. The sliders in these two sides are the same as those in the first and second sides. The line SS is marked from 1 to 8 above the slider, and from 8 to 100 below it. The line SL is marked from 1 to 4 above, and from 4 to 100 below the slider.
On the inside of one of the sliders are marked the square and circular divisors and multipliers, and the gauge points for imperial bushels and gallons; square being denoted by S, and circular by C. On the inside of another slider is a line of inches, and other two lines for finding the mean diameters of casks of the first and second varieties.
The two sliders marked B, with the brass ends in contact, may be used in the same groove as one slider when necessary, and also the two sliders C.
149. Construction of the rule.-The lines on the first and second faces are of course constructed like the lines A, B, C, D, on the carpenter's rule, which are all logarithmic lines.
The lines of segments are constructed experimentally in this manner:-Construct the slider like those on the first and second sides, and assume a point on the rule, at which mark 100; then take a cask of the most common form, whose content is exactly 100 gallons; fill it with water; draw off one gallon, and measure the depth in inches of the remaining water; then, according as the cask is standing or lying, set the length or the bung diameter on the slider, which is previously constructed, opposite to the mark on the rule; then opposite to the number of wet inches on the slider mark the ullage 99 on the rule; next draw off another gallon, measure the depth of the water, and opposite to the number of inches of depth on the slider mark the ullage 98
on the rule; and proceed in the same manner till the cask is emptied, and the lines of segments will then be constructed.
150. The sliding rule referred to in the example and exercises of the following problem is that formerly described in article 49, or the first and second sides of the gauging rule.
151. PROBLEM I.-To gauge regular rectilineal and circular areas one inch deep.
Find the square of the side or the diameter in inches, and multiply or divide it by the proper multiplier or divisor for the regular figure.'
=na, or c= = ns2.
d2 ነቦ 1
By the Sliding Rule.
'Set the corresponding gauge point on D to 1 on C, against the given side or diameter on D is the answer on C.
For g is the side of a polygon or the diameter of a circle, whose area is = m, the measure of capacity, or = 1; and hence if s be the side or diameter,
hence, 2 Lg~2 Ls=L1~Lc (see 50 and 51.)
EXAMPLE. Find the area of a square cistern, whose side is 108 inches in imperial gallons.
c=ns2=·003606 × 108242.066.
Or set 16-65 on D to 1 on C, then opposite to 108 on D is 42.1 on C.
The number of integers in the fourth term on C can be known from the proportion 92:21:c, org:8=1:√c; for 8, divided by g, must give c, and as this number can he nearly found mentally, the integes in its square are