BY THE SLIDING RULE.

Find the difference between the bung and head diameters upon the fourth face of the rule, or infide of the third flider, and opposite thereto is, for each variety, a number to be added to the head diameter, for the mean diameter required.

So in the above example, against 8, the difference of the diameters, are found the numbers

5.60] which be- (29.60] for the refpective mean 5.10ing added 29.10 diameters: all of which 456 to 24,there 28 56 [are too great except the 4 12 j refult 28-12 2d, which is too little.

So that this method does not give the true mean diameter.

PROBLEM II.

To find the Content of a Cask by the Mean Diameter on the Sliding Rule.

Set the length on c to the

gauge point, 18.95 for ale, or 17.15 for wine, on D; then against the mean diameter on D, is the content on c.

EXAMPLE.

If the bung diameter be 32, the head 24; and the length 40 inches.

Having found the mean diameters as in the last problem, and fet 40 on c to 18.95 or 17.15 on D,

against

282

*The above gauge points are found thus, viz. √

*785398

Of the Gauging of all Cafks in general by means of Four Dimenfions, viz. the Length, the Bung and Head Diameters, and the Diameter taken in the Middle between the Bung and Head.

GENERAL PROBLEM.

To find the Content of Any Cask in Ale or Wine Gallons, by Four Dimensions.

*Add into one fum, the fquare of the bung diameter, the fquare of the head diameter, and the fquare

of

*This rule is taken from prob. 3 fect. 1 part 4; the num

*785398 6×231

785398

very nearly, and

6 x 282

0004641844 00043 nearly 000 more nearly.

And with regard to the operation by the fliding rule, it may be

=

6 × 231
x
•785398'

and 46.4 =

6 × 282
785398

obferved that 42 is The above rule, by the faid prob. 3 fect. 1 part 4, was proved to be accurately true, not only for all the four different varieties of casks, but alfo for all casks and folids generated from any conic fection whatever; and although the cafk be not precifely in the form of any fuch curve, the rule will give the content very near the truth; fo that whatever be the form of the cask, we may in all cafes be pretty fure of the content to within of a gallon, or perhaps lefs, fuppofing the dimenfions to be truly taken. So that more perfect than this, both with refpect to truth and expedition, nothing can be expected, or indeed wifhed for, in gauging

PP 4

whic

of double the middle diameter, and multiply that fum by the length of the cafk; then the product

0004 [ for ale I will give the
for wine

mult.
by

0005

}

content.

BY THE SLIDING RULE.

Set the length on c to {46.4 for ale

for wine

}

on D;

and find both the bung, head and middle diameters

on

22

20

18

16

14

12
10

which makes me hope that one day this method will come into general ufe with the practitioners in the excife; and till then, I am fully perfuaded that much of their practice must be mere guefs work. The pretended difficulty of taking the middle diameter may perhaps deter fome from ufing this method; but there cannot be any real difficulty in taking this diameter, except when a wooden hoop may happen to be at the part where the diameter ought to be taken; but that will very rarely happen. To and the fourth dimenfion or diameter in the middle be ween the bung and head; upon one fide of a fquare rule let be drawn a fcale of quarter inches, numbered both ways from nothing; and let the middle or o divifion be applied to the bung or middle of the cafk, as in the margin, parallel or nearly parallel to the axe, and in the direction of the staff; then whatever number of inches are in the whole length of the cask, it is evident that, from the nature of 20 the icale, the fame divifion, or number of quarter inches, will be oppofite to the where part the middle diameter must be taken, which here fuppofe to be 10; and at 10, on each part of the rule, measure the distance rs; then if the fum ofrs and rs be taken from the bung diameter, there will remain the required middle diameter, excepting the allow ance for the thickness of the staff, which must be fubtracted.

8

10

12

14

16

18

22

The

on D, noting the three numbers against them on c; then the fum of the firft and fecond with 4 times the third will be the content required.

EXAMPLE

Let the length of a cafk be 40 inches, the bung diameter 32, the head 24, and middle diameter 30.2 inches

The following collection of cafks happened in real practice; and their dimenfions were carefully taken; but their contents were computed by a fliding rule, and fo may not all be precisely

true.

From hence it appears that a fpheroidal cafk is a mere imaginary thing, the contents of real cafks being less than is affigned to them by that form; as indeed they ought, from the nature of the curves; for a fpheroidal cafk would be leaft curved in the middle, and the most at the ends; whereas a real cafk is the leaft curved at the ends, if it be any thing curved there at all; and indeed there is reafon to think it is not, as will appear in chap. 7.

Cafks gauged by four dimenfions.

Diff.

Diff.

Nu Len Head Bung Mid. Ale lefs Nu Len Head Bung Mid. Ale lefs mb gth diam. diam.Jiam.gallons than mb gth diam. diam.diam.gallons than

8

phr.

phr. 128 3 23 2 277 26°3 53.6101948 8 24°2 32 1 29.4 114.8 5.5 229.8 22.2 26 248 50°211 2051 223°331028°2111*3 5° 330 823 227.5 26.1 577 11 21 49°3 23.8 32.6 29°51170 6'1 4322245 301 28.4 70.61°3 22 48°C 28.2 33 831 4137°3 6.1 530 0 24 7 29 2 276 626182345*2 26 6 33 2 30°4 115.6 6.5 32.5 23.8 28.226.8 636 19 2451*6 36.6 +1.6 39.6 223°3 6.7 734 326 333 5311 90°42*92544 2 28 1 36 4 33 3134.6 6.8 34'5 26.. *433°C 30.7 89.03026 57°C 32°7 +2°C 39°1 236.6 6.8 +10 26 3 32° 30°2 102 230 2751°C 33°1 381 357 181°C 8.0 10 37°026*131*8:9′9 90*33*1 28 51.533°3400 37°2 197°C 8.7 144°5 34 440.838.8183.8 3.2 29 54°C 34°844 8415253.5 8.8 12470 26 3 53° 34°4126 335 3050°C 34°340°5377 1976 94 13 34 2272 338314 92.3 3.8 3149 0 29.5 36.0 33°C 148.1 9.4 1447025 332 0 29 7113° 14°3 3251033.5 39 2364 189.6 9.7 45 530 7 38 0 35 5157047 3351°C 33°4 39.8370 1940 9.8 44 6 24 7 32 2 29.6 106.647 3455 530 640 6370 207.210.2 1748624 232 1 29 4 114°44'93545.6 28.0 34 6 32 4134.8 12.0 1846 0 25 7 347 3**7125*35.5 3655035.848°C 43 2 282 2 17.8

inches nearly 912, which is taken upon the fuppofition that the cafk is fpheroidal.

Then (322+242 + 4 × 912) X 40 X 0004, T = 97'44 ale gallons; or multiplied by 00053, gives 118.95 wine gallons, for the content, the fame as at prob. 1 chap 3.

By the Sliding Rule. Having fet 40 on c to 46.4 on D, against 32, 24, and 30.2 on D, stand 19, 10.5, and 17 on c; then 19 + 10'5 + 4 × 17 = 97*50, ale gallons.

And, having fet 40 on c to 42 on D, against 32, 24, and 302 on D, ftand 23.2, 13, and 2017 on c; then 232 +13 + 4 X 207 119 wine gallons, for the content as before, nearly.

CHAPTER VI.

A New, Eafy, and Expeditious Method of computing the Content of a Cafk from Three Dimensions only, of Whatever Variety it may be.

*In the column of diameters, or middle column,

in

* The method of gauging here propofed, is no other than the rule for one form of folids, corrected fo as to make it agree, in content, with the most common or general form of cafks: and, for the more expedition, the fquares of all diameters, within certain limits, are divided by the proper conftant divifor, and the quotients arranged in the table oppofite to their correfponding diameters; that in practice nothing more may be required than to multiply thefe quotients by any affigned length of a cafk, for the content in gallons.

Now it hath been found that wine hogfheads generally contain about a gallon and a quarter lefs than the content affigned to them by the rule for fpheroidal cafks; to correct the rule for

that