128. In making the calculations in gauging, according to the preceding rules, the multiplications and divisions are frequently performed by means of a Sliding Rule, on which are placed a number of logarithmic lines, similar to those on Gunter's Scale. See Trigonom. Sec. vi. and Note G. p. 141. Another instrument commonly used in gauging is the Diagonal Rod. By this, the capacity of a cask is very expeditiously found, from a single dimension, the distance from the bung to the intersection of the opposite stave with the head. The measure is taken by extending the rod through the cask, from the bung to the most distant part of the head. The number of gallons corresponding to the length of the line thus found, is marked on the rod. The logarithmic lines on the gauging rod are to be used in the same manner, as on the sliding rule.

ULLAGE OF CASKS.

129. When a cask is partly filled, the whole capacity is divided, by the surface of the liquor, into two portions; the least of which, whether full or empty, is called the ullage. In finding the ullage, the cask is supposed to be in one of two positions; either standing, with its axis perpendicular to the horizon; or lying, with its axis parallel to the horizon. The rules for ullage which are exact, particularly those for lying casks, are too complicated for common use. The following are considered as sufficiently near approximations. See Hutton's Mensuration.

PROBLEM VII.

To calculate the ullage of a STANDING cask.

130. Add together the squares of the diameter at the surface of the liquor, of the diameter of the nearest end, and of double the diameter in the middle between the other two; multiply the sum by of the distance between the surface and the nearest end, and the product by .0028 for ale gallons, or .0034 for wine gallons.

If D=the diameter of the surface of the liquor, d the diameter of the nearest end,

m=the middle diameter, and

7=the distance between the surface and the nearest end; The ullage in inches=(D2+d2+2m2) × ¿l×.7854.

Ex. If the diameter at the surface of the liquor, in a standing cask, be 32 inches, the diameter of the nearest end 24, the middle diameter 29, and the distance between the surface of the liquor and the nearest end 12; what is the ullage?

Ans. 27 ale gallons, or 333 wine gallons.

PROBLEM VIII.

To calculate the ullage of a LYING cask.

131. Divide the distance from the bung to the surface of the liquor, by the whole bung diameter, find the quotient in the column of heights or versed sines in a table of circular segments, take out the corresponding segment, and multiply it by the whole capacity of the cask, and the product by 11 for the part which is empty.

If the cask be not half full, divide the depth of the liquor by the whole bung diameter, take out the segment, multiply, &c. for the contents of the part which is full.

Ex. If the whole capacity of a lying cask be 41 ale gallons, or 49 wine gallons, the bung diameter 24 inches and the distance from the bung to the surface of the liquor 6 inches; what is the ullage?

Ans. 73 ale gallons, or 9 wine gallons.

86

NOTES.

NOTE A. p. 16.

ONE of the earliest approximations to the ratio of the circumference of a circle to its diameter, was that of Archimedes. He demonstrated that the ratio of the perimeter of a regular inscribed polygon of 96 sides, to the diameter of the circle, is greater than 3: 1; and that the ratio of the perimeter of a circumscribed polygon of 192 sides, to the diameter, is less than 34: 1, that is, than 22: 7.

Metius gave the ratio of 355: 113, which is more accurate than any other expressed in small numbers. This was confirmed by Vieta, who by inscribed and circumscribed polygons of 393216 sides, carried the approximation to ten places of figures, viz.

3.141592653.

Van Ceulen of Leyden afterwards extended it, by the laborious process of repeated bisections of an arc, to 36 places. This calculation was deemed of so much consequence at the time, that the numbers are said to have been put upon his tomb.

But since the invention of fluxions, methods much more expeditious have been devised, for approximating to the required ratio. These principally consist in finding the sum of a series, in which the length of an arc is expressed in terms of its tangent.

Ift the tangent of an arc, the radius being 1,

This series is in itself very simple. Nothing more is necessary to make it answer the purpose in practice, than that the arc be small, so as to render the series sufficiently converging, and that the tangent be expressed in such simple numbers, as can easily be raised to the several powers. The given series will be expressed in the most simple numbers, when the arc is 45°, whose tangent is equal to radius. If the radius be 1,

3

The arc of 45°=1-+-+-&c. And this multiplied by 8 gives the length of the whole circumference.

But a series in which the tangent is smaller, though it be less simple than this, is to be preferred, for the rapidity with which it converges. As the tangent of 30°, if the radius be 1,

And this multiplied into 12 will give the whole circumference. This was the series used by Dr. Halley. By this also, Mr. Abraham Sharp of Yorkshire computed the circumference to 72 places of figures, Mr. John Machin, Professor of Astronomy in Gresham college, to 100 places, and M. De Lagny to 128 places. Several expedients have been devised, by Machin, Euler, Dr. Hutton, and others, to reduce the labor of summing the terms of the series. See Euler's Analysis of Infinites, Hutton's Mensuration, Appendix to Maseres on the Negative Sign, and Lond. Phil. Trans. for 1776. For a demonstration that the diameter and the circumference of a circle are incommensurable, see Legendre's Geometry, Note IV.

The circumference of a circle whose diameter is 1, is

3.1415926535, 8979323846, 2643383279,
5028841971, 6939937510, 5826974944,
5923078164, 0628620899, 8628034825,
3121170679, 8214808651, 3272306647,
0938416+ or 7-.

NOTE B. p. 17.

The following multipliers may frequently be useful;

X.8862 the side of an equal square.

The diam'r of a circle

(x.866

The circumf.

X.707 the side of an ins'bed sq're.
the side of an inscribed
[equilateral triangle.

X.2821=the side of an equal square.
X.2251=the side of an inscribed square.
X.2756=the side of an ins❜bed eq'lat. triang.

X1.128 the diameter of an equal circle.

=

×3.545 the circumf. of an equal circle.

The side of a sq.X1.414 the diam. of the circums. circle.

X4.443 the cir. of the circumsc. circle.

NOTE C. p. 19.

The following approximating rules may be used for finding the arc of a circle.

1. The arc of a circle is nearly equal to of the difference between the chord of the whole arc, and 8 times the chord of half the arc.

2. If h the height of an arc, and d=the diameter of the circle;

5h

4. The arc=3(5d5d3h+4√dh) very nearly.

5. If s=the sine of an arc, and r=the radius of the circle;