segment of a sphere, the depth 15 inches, the diameter of the base 60 inches, and the middle diameter 45 inches.

Ans. (602 +90 +0) × 15 × 0028326 = 82-85355 imperial gallons.

Set 46.02 on D to 15 on C; then at 60 on D is 25·5 on C. . . 46.02.

.

15

90

57.37 ..

82.87 imp.gal.

Or by the Rule in Prob. XV. Case 2, of MENSURATION OF SOLIDS.

Set 32.544 on D to 15 on C; then at 15 on D is 3.18 on C.

9. Required the content, in imperial bushels, of a hexagonal prism, of which the depth is 96 inches, and each side of the base 18 inches. Ans. 36-47255 imperial bushels. 10. Required the content, in imperial gallons, of a cylin drical vessel, of which the depth is 84 inches, and the diameter of the base 63 inches. Ans. 944-3775 imperial gallons.

11. Required the content, in pounds of hard soap, of a frustum of a pentagonal pyramid, the depth 60 inches, and the sides of the bases 18 and 6 inches. Ans. 593-355672 lbs. 12. Required the content of the frustum of a cone, in imperial gallons, the depth being 50 inches, and the diameters of the bases 24 and 30 inches. Ans. 103-67316 gallons.

PROB. XI. To gauge malt.

RULE. Take the depths at a great number of places, particularly where the malt is deepest, and where it is ebbest. Add all these depths, and divide the sum by the number of them for a mean depth. Find the content at one inch deep, as before, and multiply it by the mean depth.

1. Required the content of a rectangular floor of malt, of which the length is 72 inches, the breadth 48, and the depth, taken at five different places, 47, 54, 56, 4·9, and 4.4 inches. Ans. The sum of the depths, 25, divided by 5, gives 5 the mean; then 72 × 48 × 5 × 00045087.7898 imp. bushels.

By the Sliding-Rule.

Set the length 72 on B to the breadth 48 on MD; then gainst the depth 5 on A is 7:79 imperial bushels on B. 2. Suppose the length 270, the breadth 56-2, and the mean epth 5.2 inches. Required the quantity of malt.

Ans. 270 × 56.2 × 5·2 ×·000450835.570284 imp. bush. Set 270 on B to 56.2 on MD; and at 5.2 on A is 35:57 alt bushels on B.

Or find a mean proportional 123.2, between 270 and 56.2. Set 47-097 on D to 5-2 on C; then at 123-2 on D is 35.57 alt bushels on C.

3. Let the length be 140, the breadth 72, and the mean epth 18.2 inches. Required the quantity.

Ans. 82.7 imperial bushels. 4. Let the length be 1250, the breadth 360, and the mean epth 9 inches. Required the quantity.

Ans. 1825-74 imperial bushels. 5. How many imperial bushels of malt are in an octagonal istern, the length of the side being 10 feet, and the depth in ght different places 10.2, 96, 91, 98, 105, 107, 103, nd 10.4 inches? Ans. 24-83 imperial bushels. 6. There is an oval cistern of malt, of which the diameters re 72 and 48, and the depth 5 inches. Required its content. Ans. 6-118848 imperial bushels. Find a mean proportional 58.8, between 72 and 48. Set 53.144 on D to 5 on C; then at 58-8 on D is 6.12 nalt bushels on ̊C.

NOTE. Malt must be gauged several times. It is supposed o increase one-fifth in bulk in the cistern, and, after being 30 hours in the heap or floor, it is doubled by sprouting: herefore, to obtain the net measure, multiply it by 8 when auged in the couch, and by 5 when in the floor. When the neasure of the dry barley is given, multiply it by 12 to find vhat it should be in the couch, and that again by 1.6 to find what it should be in the floor.

7. What should be the couch and floor measure of 13.8 mperial bushels of dry barley?

Ans. 13.8 x 1.2

16.56 the couch measure.

16.56 × 1.6 26.496 the floor measure.

=

8. Suppose a floor to measure 100-8 imperial bushels, what should have been the couch measure? Ans. 63 bushels. 9. Suppose a couch to measure 56 bushels, what should have been the floor measure, and the quantity of dry barley? Ans. 89-6 the floor measure, and 463 bushels dry barley.

U

PROB. XII. To gauge open vessels.

B

These vessels being in the form of prisms, A cylinders, frustums, cylindroids, &c. their contents may be found by the preceding rules. But as they are often large and fixed vessels, their contents are generally required at every inch, or tenth part of an inch, of depth. These contents must therefore be found and placed in a table, so that, by taking the depth of the liquor, the content may be known at once from the table.

When the vessels are prisms or cylinders, find the content at one inch deep; and this doubled, tripled, &c. will give the contents at two, three, &c. inches. If the dimensions at the top and bottom be unequal, divide the difference of corre sponding sides or diameters at the bases by the depth, to get the difference at one inch deep; and this difference, added to the bottom diameter if it be less than that at the top, or subtracted from it if greater, will give the side or diameter at one inch deep; and the same difference, added to the side or diameter at one inch deep, or subtracted from it, will give it at two inches deep, and so on.

Having found the dimensions, find the content of each part; and, by adding them, the contents at all the depths will be found.

Generally the dimensions are found only in the middle of every six inches, and the content, being found from these dimensions for one inch deep, is added to itself six times, to get the contents for each of these six inches of depth.

1. Suppose an elliptical vessel to be 6 inches perpendicular depth, the axes at the top 65 and 60, and those at the bottom 110 and 100 inches, all taken parallel to the horizon, the vessel inclining so that it requires 15 gallons to reach to the upper part of the bottom where the axes were taken.

The difference of the two greater axes is 45, which, divided by 6, gives 7.5 inches, the difference for every inch of depth; and in the same manner the difference of the lesser axes for every inch of depth is 63 inches: consequently, at 1 inch from the bottom, the axes will be 72·5 and 66.7 inches; at 2 inches, 80 and 73.3 inches, and so on. These are placed in the second and third columns of the table; and the particular contents being found and added together regularly, both from the top and the bottom, are placed in the fourth and fifth columns.

In such vessels there is a place marked on the edge of the vessel for the dipping-place; and it is here supposed, that, at

he dipping-place, the wet inches are 2, when the 15 gallons re in the vessel to cover the bottom, and also that there is 1 ach dry at the top when the vessel is full.

Suppose the wet inches at the dipping-place to be 5; then gainst 5 wet inches in the column titled Content from Bottom, is found 90-84287 imperial gallons for the quantity f liquor in the vessel.

2. The depth of a circular mash-tun is 60, the top-diameter -8, and the bottom-diameter 36 inches, and supposing the ontent of the drip or fall to be 20 imperial gallons. Required the content of each 10 inches from the top, and also he whole content.

Ans. First 10 inches 62.57213, whole content 321.7852 mperial gallons.

3. Suppose the depth of a circular tun to be 80 inches, the op-diameter 50, and the bottom-diameter 30. Required the content of the tun, and also of every 10 inches from the bottom, allowing 10 gallons for the drip or fall.

Ans. Whole content 380-00836, content of first 10 inches 37-66211 imperial gallons.

Coolers, &c. are very wide and ebb, and their bottoms neven; therefore the depths must be taken at various parts, and their sum divided by the number of them, to get a mean Hepth. Tables are constructed for such vessels, exhibiting the content at every tenth part of an inch in depth. They are made and used in the same way as the last table.

It often happens that the depth taken at the dipping-place differs from the mean depth for which the table was calculated. The difference must be marked on the vessel and in the table, with the sign when the depth at the dipping-place is greater than the mean depth, or with the sign + if it be less ; and this difference must be subtracted or added to get the mean depth, before using the table.

--

Suppose the mean depth to be 4-89, and that at the dipping. place 5 inches; the difference, 0·11, must always be taken from the wet inches to reduce them to mean ones.

NOTE. When the wort is gauged hot, one-tenth part is deducted from the content, to find how much there will be when cold; as it has been found that 10 gallons of hot wort measure only about 9 gallons when the wort is cold.

1. The length of a cooler is 120, the breadth 84, and the depth at 10 equidistant places 4-6, 4-5, 4-7, 44, 4-2, 4, 39 37, 35, and 3 inches. How many gallons of hot wort will it contain, and how many gallons will there be when the wor is cold?

Ans. 115 6381 gallons hot, and 104-0743 gallons cold wort 2. Suppose the length to be 280, the breadth 200, and the mean depth 51 inches. Required the content in hot, and also in cold wort.

Ans. 808-99056 gallons hot, and 728-0915 gallons cold. PROB. XIII. To gauge a copper, still, &c.

If the greatest width be at the top, and the least at the bottom, or the contrary, take diameters perpendicular to one another at both ends, and also exactly in the middle, between the top and bottom. (By the bottom is meant the top of the crown in the bottom.) Then work by Prob. XII. of MEN SURATION OF SOLIDS: That is,

To 4 times the product of the middle diameters, add the products of those at the top and of those at the bottom. Multiply the sum by the depth from the top of the vessel të the top of the crown: the product, multiplied by 0004721 will give the imperial gallons in the content of all above the crown. Water must then be measured into the vessel, just to cover the crown; and this measure, added to that above, will give the whole content.

1. Let the depth to the top of the crown be 36 inches, the diameters at the top 116 and 115.5, at the top of the crown 111 and 110, and in the middle 114 and 113, and the liquor required to cover the crown 16-3 imperial gallons. Required the content.

Ans. (4x114x113+ 116 × 115.5+ 111 x 110)x36x ⚫0004721 = 1310-9726, and 1310-9726+16.3 the content of the crown 1327 2726 imperial gallons whole content.

If the broadest part be not at the top or bottom; suppose the vessel to be divided into two or more frustums, so that the broadest part of each frustum be at one end of it, and the least breadth at the other. Find the content of each frustum