are to consider each mean as the true diameter of the Copper throughout the space, regarded as cylindrical, to the middle of which the particular mean refers.

This being premised we shall proceed with an example in illustration, having no doubt whatever that the reader will readily comprehend our aim and method at every step.

EXAMPLE.

Let ABPQ be a section of a Copper with a rising crown, cut by a plane passing perpendicularly through the middle, and let PDQ represent the crown: it is required to gage and inch the Copper, of which ABPQ is a section.

Extend a cord, as AB, diametrically across the mouth of the Copper, and let CD be a plumb-line, touching AB, and hanging freely over D the centre of the crown; also let the length of CD be 78 inches.

Next move the plumb-line along AB, towards A, till it arrive at E, perpendicularly over P; and let the length of EP be 84 inches, the distance of E from C, being 39 inches. Then it will be,

8478= 6 inches altitude of the crown.

39 +39 78 inches diameter of the segment's base.

Here there are no odd tenths on the depth, and, consequently, we may, without inconvenience, begin our marks from the level of the crown-head, since every division of the Copper will consist of integral inches.

Descending now into the Copper, and with a pair of compasses opened to six inches full, striking a line all round, at the height of 6 inches from the angle formed by the crown and sides, we take two cross diameters of that circle, whereof the one, we shall suppose, is 81.5 inches, and the other 82.3 inches. these two diameters be added together, and the sum 'divided by 2, we shall have 81.9 inches for the mean diameter of the Copper at the top of the crown.

But,

If

81.9 x 70757·9 side of the inscribed square. With a sliding-cane or tape, therefore, extended to 57.9 inches, we inscribe the square, and, either with a chalk-line strike perpendiculars on the sides of the Copper at the four angles, or by the shadow of a plumbline from a candle in the centre, draw the lines with the hand.

We next begin at the level of the top of the crown, and proceed to set off ten-inch distances towards the

mouth on one of the chalk-lines; when, having made seven marks, we find the remaining portion above, to be eight inches over even tens, that is, 13 inches over the middle of the space included between the two marks made for the last two tens. Wherefore, regarding these 13 inches as one division, we make a mark at 6.5 inches from the mouth of the Copper, for the middle of this division; and now the lowermost six teninch marks will each be in the middle of a division; hence the other three perpendicular lines must be similarly marked.

This part of the operation being finished, we carefully measure the cross diameters, and enter them in a memorandum, with the mean diameter between each pair, and every other particular, such as the division to which they refer of the Copper, and the distance from the mouth at which they were taken. The form of the memorandum will, with reference to the figure, be as follows:

Memorandum of A. B.'s Copper, No. 1, Gaged, 10 May, 1820.

13

10

Cross Diameters Mean Diameter

6.5 inches 105·3...105.9 105.6 inches...fg 18.0 inches 105·1...105.5 | 105.3 inches...hi 28.0 inches 104·5...104·7 | 104.6 inches...kl 38.0 inches 103·9...103.5 103.7 inches...mn 980... 97.4 97.7 inches...ow

10

10

10

48 0 inches

CALCULATION.

The content of a circle, an inch deep, in ale gallons, for each of the mean diameters respectively, is now to be multiplied by the depth of the space to which the particular diameter applies, including the lowest portion of the Copper, as if the crown were flat: and the sum of these contents, after deducting the content of the spherical segment answering to the crown, will be the true content of the Copper.

The exact allowance to be made for the convexity of the crown, we deduce from calculation founded on PROB. XIX. SEC. III. of MENSURATION; hence the process will be as follows:

18.68 x 11

A. G.

51.23

A. G. 2201·40

CALCULATION OF THE SPHERICAL SEGMENT FORMING

THE CROWN.

[See page 212.]

36 square of the altitude

39 × 39 × 3 = 4563 thrice the sq. of half the base 4599 sum

Multiply by 6 the altitude

A.G.sp. div. 538-58)27594 (51.23 ale gallons.

But the content of the cylinder circumscribing the crown, for one inch deep, and mean diameter 81.9 as found by measurement at rs, is 60·85 ale gallons, obtained as follows:

18.68 ale gallons at an inch deep
Multiply by 6 the height of the crown

Product 112.08 content of the cylinder.
Subtract 51.23 for the crown

Remainder 60.85 ale gallons TO COVER THE CROWN.

It will now be proper to exhibit a Specimen of a Dimension Book, with the manner of entering the above Copper; where it will be seen that the content of each division of the Copper in gallons, is reduced in the last column, into Barrels, Firkins, and Gallons. This is done by dividing the gallons by 36 to integer places only, for barrels; and then dividing the remainder by 9 in the same way: for the first quotient being Barrels, the second will be Firkins, and the last remainder Gallons.