To find the Ullage of a Standing Cafk by the Pen.

RULE 1.

*Add all together the fquare of the diameter at the furface of the liquor, the fquare of the diameter of the nearest end, and the fquare of double the diameter taken in the middle between the other two; then multiply the fum by the length between the furface and nearest end, and the product again by 0004 for ale gallons, or by 0005 for wine gallons, in the lefs part of the cafk, whether empty or filled.

EXAMPLE.

Taking the above diameters 24, 27, and 29 inches, and the wet inches 10; we shall have (242 + 542 + 292) X 10 X 0004 = 4333 004202 gallons for

the ullage required.

As the fquare of the length of the cafk, is to the fquare of the difference between the faid length and wet or dry inches, viz. the lefs of them; fo is the difference between the bung and head diameters, to a number, which being taken from the bung diameter, will leave the diameter of a cylinder of the fame length

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* The diameters at the furface of the liquor, and in the middle between it and the nearest head, will eafily be obtained, by fufpending a plummet by a rule laid over the middle of the head, fo as juft to touch the bulge of the cafk, and then measuring the distance of the ftring from the fide of the cafk at the surface of the liquor and in the middle between it and the nearest head; for then the doubles of these measures taken from the bung diameter, will leave the faid diameters required.

length with, and nearly equal to the part filled or empty, viz. the lefs of them*.

EXAMPLE.

Taking the fame example as in problem 1, we have 10) 3224: 4; and 32- 4/2/

as 402: (40

27 the mean diameter.

Then 27.5 X 10 X 002 = 21 ale gallons is the ullage required.

Note. The number 0027851 or 002 nearly, is the constant multiplier *785398 &c

282

To find the Ullage of a Lying Cask by the Pen.

Divide the wet inches by the bung diameter; find the quotient in the firft column of the table of circular fegments, at the end of the book, and take out the fegment oppofite to it; then multiply this fegment by the whole content of the cafk, and the product again by 1 for the ullage nearly.

EX

*This rule is founded on the reasonable fuppofition, that for a fmall part of cafk, the diameter in the middle may be confidered as a mean diameter; which is strictly true when the cafk is paraboloidal, and very near the truth when parabolic-fpindular; but the faid middle diameter, as given in the rule, is computed from the latter form, because the rule is thereby rendered not only eafier, but generally more exact.

Thus, by the property of the parabola, L2: (L1)2 :: B - H:

the bung, H the head, and м the mean diameter; alfo 1 is the length of the cafk, and I the wet or dry inches.

This method is evidently nothing elfe than taking the whole content, in fuch proportion to the ullage, as the whole bung circle bears to the fegment of it cut off by the furface of the liquor; and is nearer to the truth than any other practical rule that I can find.

EXAMPLE.

Taking the fame cafk as before, whofe length is 40, bung diameter 32, head diameter 24; and fuppofing

the wet inches to be 8.

8

32

By chapter 7, the whole content is nearly 92 ale gallons. Then ==25; oppofite to which, in the table of areas, is the fegment 15354621; hence 92 × 15354621 × 1 = 18 ale gallons, the ullage required.

SCHOLIU M.

Having delivered the neceffary rules for meafuring cafks, &c, I do not fuppofe that any thing more of the fubject of gauging is neceffary to be given in this book. For as to cifterns, couches, &c, tuns, coolers, &c, coppers, ftills, &c, which are firft fuppofed to be in the form of fome of the folids in the former parts of this work, and then meafured accordingly, no perfon can be at a lofs concerning them, who knows any thing of fuch folids in general; and to treat of them here, would induce me to a long and tedious repetition, only for the fake of pointing out the proper multipliers or divifors; which is, I think, a reafon very inadequate to fo cumbersome an increase of my book.

I fhall only juft obferve, that when tuns, &c, of oval bafes are to be gauged; as thofe bafes really measure to more than true ellipfes of the fame length and breadth, they ought to be meafured by the equi-diftant ordinate method, delivered in fection 2 of part 4.

And that when cafks are met with, which have different head diameters, they may be deemed incomplete cafks, and their contents confidered and measured as the ullage of a cafk.

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SEC

TH

SECTION III.

OF THE WORKS OF ARTIFICERS.

HE artificers whofe works are here to be treated of, are Bricklayers, 2 Carpenters and Joiners, 3 Glaziers, 4 Mafons, 5 Painters, 6 Pavers, 7 Plafterers, 8 Plumbers, and 9 Slaters and 1 ilers.

Artificers compute the contents of their works by feveral different measures.

As glazing and mafonry by the foot:

Painting, plaftering, paving, &c, by the yard of 9 fquare feet:

Flooring, partitioning, roofing, tiling, &c, by the fquare, of 100 fquare feet:

And brickwork either by the yard of 9 fquare feet, or by the perch, or fquare rod or pole, containing 2724 fquare feet, or 30 fquare yards, being the fquare of the rod or pole of 16 feet or 5 yards long.

As this number 272 is a troublesome number to divide by, the is often omitted in practice, and the content in feet divided only by the 272. But as this is not exact, it will be both better and easier to multiply the feet by 4, and then divide fucceffively by 9, 11, and 11. Alfo to divide fquare yards by 3c, first multiply them by 4, and then divide twice by 11.

All works, whether fuperficial or folid, are computed by the rules proper to the figure of them, whether it be a triangle, or rectangle, a parallelopiped, or any other figure.

For taking measures, the most common instrument is what is called the carpenters rule, of which it will be neceffary here to give a defcription.

СНАР

CHAPTER I.

Of the Common or Carpenters Rule.

This inftrument is otherwife called the fliding rule; and it is much used in timber measuring and artificers works, both for taking the dimenfions, and cafting up the contents.

The inftrument confifts of two equal pieces, each a foot in length, which are connected together by a folding joint.

One fide or face of the rule, is divided into inches, and half-quarters, or eighths. On the fame face alfo are feveral plane fcales, divided into twelfth parts by diagonal lines; which are used in planning dimenfions that are taken in feet and inches. The edge of the rule is commonly divided decimally, or into tenths; namely each foot into 10 equal parts, and each of these into 10 parts again: fo that by means of this laft fcale, dimenfions are taken in feet and tenths and hundredths, and multiplied as common decimal numbers, which is the best way.

On the one part of the other face are four lines, marked A, B, C, D, the two middle ones в and c being on a flider, which runs in a groove made in the ftock. The fame numbers ferve for both these two middle lines, the one being above the numbers, and the other below.

These four lines are logarithmic ones, and the three A, B, C, which are all equal to one another, are double lines, as they proceed twice over from 1 to 10. The other or loweft line D, is a fingle one, proceeding from 4 to 40 It is alfo called the girt line, from its ufe in cafting up the contents of trees and timber; and upon it are marked wo at 17.15, and AG at 18.95, the wine and ale gauge points, to make this inftrument ferve the purpofe of a gauging rule.

Upon the other part of this face there is a table of

the