The Length and Breadth of a Parallelogram being given, to find its Area in Malt Bushels by the Line M D.

RULE. Set either of the given dimenfions on B, to the other on MD, then against I on A, is the required

area on B.

EXAMPLE. How many malt bufhels can be contained on every inch of the depth of a çiftern, whose length is 180, and breadth 72 inches?

By fetting 72 on в, to 180 on MD, against 1 on A, will appear nearly 6 bufhels on B, which is the quantity required.

To find, by the Line MD, the Malt Bushels which may be contained in a Couch, Floor, or Cistern, whose

RULE.

length, breadth, and depth are given.

Set one of the dimenfions on в to another on MD, then against the third on A, will appear the content on B.

EXAMPLE. Required the number of bushels in the ciftern whofe length is 230, breadth 58.2, and depth 5.4 inches.

Having fet 230, on в to 5'4 on MD, against 58.2 on A, I find 33.6 bufhels on B, which is the content nearly.

The ufe of the other parts or marks on the rule will appear in the examples farther on.

CHAPTER II.

Of the Gauging or Diagonal Rod.

The diagonal rod is a fquare rule having four fides.

or faces, being generally four feet long, and folding together by means of joints.

It takes its name from its ufe in measuring the diagonals of casks, and computing the contents from the faid diagonal only. Where it may be noted, that by the diagonal of a cafk, is meant the line from the bung to the interfection of the head with the ftave oppofite to it, and is commonly the longest line that can be drawn from the middle of the bung to any part within the cask.

And, accordingly, upon one face of the rule is a scale of inches for taking the measure of the diagonal; to which is adapted the areas, in ale gallons, of circles to the correfponding diameters, like the lines on the under fides of the three flides in the fliding rule.

Upon the oppofite face are two fcales of ale and wine gallons, expreffing the contents of cafks having the correfponding diagonals; and thefe are the lines which chiefly conftitute the difference between this inftrument and the fliding rule; for all the other lines upon it are the fame with thofe on that inftrument, and are to be ufed in the fame manner.

EXAMPLE.

Let it be required to find the content of a cafk whofe diagonal measures 34.4 inches, which agrees with the cafk in the following chapter, whofe head and bung diameters are 32 and 24, and length, 40 inches; for if to the fquare of 20, half the length, be added the fquare of 28, half the fum of the diameters, the fquare root of the fum will be 344 nearly.

Now to this diagonal 34'4, correfponds, upon the rule, the content 90 ale or 111 wine gallons; which differs from all the contents, in the next chapter, obtained by confidering the cask as belonging

to

to each of the four propofed varieties; being indeed a kind of medium among them all, and falling in between the second and third variety; and fo anfwering to the most common form of cafks.

Of Cafks confidered as divided into feveral Varieties.

According to the cuftom of moft writers on this fubject, I fhall diftinguifh cafks into four forms or varieties, viz.

1. The middle fruftum of a spheroid,

2. The middle fruftum of a parabolic fpindle,
3. The two equal fruftums of a paraboloid,
4. The two equal fruftums of a cone.

I omit here the middle fruftums of circular, elliptic, and hyperbolic fpindles, on account of the difficulty of their rules, which renders them unfit for the purpose of practical gauging. And indeed fome of the above four forms are of very little real ufe; for very few, if any, cafks are to be met with which will hold fo much as the first form, or fo little as the third or fourth; fo that the fecond is the most generally, if not the only useful one, of the four varieties.

Note, 282 cubic inches make one ale gallon,

231
2150'42

wine gallon,

a malt bufhel.

It is also to be noted that the dimenfions are fup. posed to be inches, in the following rules.

PRO

PROBLEM I.

To find the Content of a Cafk of the First
or Spheroidal Variety.

*To the fquare of the head diameter add double the fquare of the bung diameter, and multiply the fum by the length of the cafk. Then let the product be mult. by 0009, or divided by 1077 for ale gallons, or mult. by '0011, or divided by 882 for wine gallons

BY THE SLIDING RULE.

Set the length on c to 32.82, for ale, or to 29.7, for wine, on D, and on D find the bung and head diameters, noting the numbers oppofite to them on c, then if the latter of these two numbers be added to the double of the former, the fum will be the measure in gallons.

EXAMPLE.

Required the content of a fpheroidal cask, whose bung and head diameters are 32 and 24, and length 40 inches.

By

* For, by prob. 12 fect. 5 part 3, the content in inches is (2 B2 + H2) × Ln, which being divided by 282 and 231, becomes 2B2 + H2 × L or (2B2 + H2) × 000928371 in the one cafe,

1077°157
2B2+H2

and

XL or (28+ H) X 00113333 in the other; 882.355 B being the bung and H the head diameter, and L the length of the cafk.- And in working by the fliding rule, it nced only be remarked that 32.82 and 297 are the roots of 1077.157 and S82.355.

By the Pen.

Here (2 × 32 +242) X 40 X gallons, the content required.

97.44 ale

{*0019} = {118-95 wine }

By the Sliding Rule. Having fet 40 on c to 32.82 on D, against 32 and 24 on D, stand 38 and 21.3, as near as can be judged, on c; then 2 × 38 + 21.3. 76+213973 ale gallons.

=

And having fet 40 on c to 297 on D, against 32 and 24 on D, ftand 46.5 and 26'1 on c; then 2 X 40·5 + 26′1 93 + 26·1 = 119.1 wine gallons.

PROBLEM II.

To find the Content of a Cask of the Second
or Parabolic Spindle form.

*To the fquare of the head diameter, add double that of the bung diameter, and from the fum take or of the fquare of the difference of the faid diameters; then multiply the remainder by the length, and the product multiplied or divided by the fame numbers as in the rule to the laft problem, will give the content.

BY THE SLIDING RULE.

As in the laft problem, fet the length one to 32.82 or 29.7 on D, and on D find both the bung and head diameters, and alfo their difference, taking out the three numbers oppofite to them on c; then if to twice the first be added the fecond, and of the

* For, by prob. 18 fect. 6 part 3, the content in inches is 8E+4Eн+ 3 H2 10 E2 + 4 BH + 5 H2 2 62 + 2 H2

15

Ln X (2 B2 + H2 fame numbers as in the last problem.