The American Arithmetic

5. What is the content of a cask, representing an elliptic spindle, length 40; diameters 32 and 24? Ans. 117.2+ wine, 96.0f ale, &c.

In this example the number 67 is used.

6. What is the content of a cask representing, two frustums of a cone, diam. 24 and 32; length 40 Ans. 107.5 wine, 88·09 ale, &c.

In this example the number 515 is used.

To guage with calipers, or Gunter's scale.

On the calipers there is a brass pin fixed at the points 17.15 and at 18.95 ;* these are called guage points; 17.15 is used for guaging wine, and 18.95 is used for ale, or beer, &c.

RULE. For a cask representing the middle frustum of a spheroid, (see figure, case 13, solids) extend from 1 to 7; and with the same extent, set one foot of the dividers in the number expressing the difference between the bung and head diameter, and they will reach to a number, towards the left hand, which, when added to the head diameter will give the mean diameter of the cask; then set one foot of the dividers in the guage point, and extend to the mean diameter; and the same way set off twice that extent from the length of the cask, and they will reach to a number expressing the content of the cask.

NOTE 1. The same operation may be performed on the line of numbers, on Gunter's scale.

NOTE 2. If wine gallons are required use the guage point 17 -15; if ale gal. are required use the guage point 18-95; and if corn gal, are required you may use the point 185 respectively.

NOTE 3. If the cask represents the middle frustum of an elliptic spindle (see figure, case 14, solids) extend from 1 to 67, or if a little more straight, extend from 1 to 62; and for a cask formed by two frustums of a cone, extend from 1 to 51, or 52.

* These numbers are the square roots of the numbers 294.12 and 359.05 respectively.

Examples.

1. What is the content of a cask of rum, or wine whose length is 59.3 in.; and diameters 34.5, and 307; difference of diameters 3-8?

Operation.

Extend from 1 to 62; and that extent reaches from 3.8, (the difference of the diameters) towards the left hand to 2-4 nearly, which add to the head diameter: 30·7+24=33·1 mean diam.; then extend from the guage point 17.15 to 33.1 (the mean diameter), and set off twice that extent from 59.3 (the length of the cask), and they will reach to 220-8 nearly, which is the content of the cask.

2. What is the content of a cask of wine whose length is 38; diam..22, and 16 in.: difference of diameters 6? Ans. 52 wine, 42+ ale, &c.

Operation.

Extend from 1 to 67; and that extent will reach from 6, (the difference) to 37 nearly which being added to the head diameter 16, produces 19.7 for the mean diameter: then extend from the wine guage point to 197, (the mean diameter) and set off twice that extent from 38, the length; and the dividers will rest on 52 the answer.

NOTE.-Guagers make use of the guaging rod, to take the dimensions of the cask; and there are also tables on it by means of which the outs are taken; after the preceding rules are well committed to memory, the young practitioner would soon obtain a complete knowledge of guaging, by a little practice with the instruments

WATER LEVELLING,

Or to find whether water can be made to run from one place to another, or to find how much higher

one place is than another.

F W

AWB

F is the fountain, and E the place assigned to carry the water to

NOTE. To perform this business you must be provided with two staves divided into inches and tenths, which may be of any convenient length perhaps, 4 or 6 feet; and also a water level which may be made easily at any Joiners shop; the level may be of any convenient length dug out hollowing, so as to hold water; with two sights at the top to make the observations through having provided the staves and water level, go to the fountain with two assistants and observe the following rule.

RULE.-Order the first assistant to the fountain with one staff placed perpendicularly and the second assistant to any convenient place as at A, with his staff perpendicularly; then place the water level in the middle between them at w, and looking through the sights order the first assistant to move a piece of white paper up and down the staff till you can see it through the sights: then order him to notice the distance it rests from the ground; and going to the other end of the water level, order the second assistant to do likewise; and after he has marked down the distance the papers rest from the ground, order the first assistant to take the place of the second, and the second to take a new stand, as at B; place the water level between them, make observations as before, and order the assistants to mark down the distances the paper rests from the ground, and proceed to take another station and make new observations till you have arrived to the place assigned; and order the assistants to cast up their notes, and as much as the sec

ond assistant's notes exceed the first, so much does the ground descend; and so much as the second assistant's notes are less than the first, so much does the ground rise. In the preceding example the assistant's notes are as follows.

Second assistant's notes exceed the first by 6 inches of course the ground is 6 inches lower at E, than at F, and water will run.

MONIES OF THE KNOWN WORLD.

THE following tables exhibit to view the money of the known world, both real and imaginary, with the value of the same in the currency of the United States, estimated on the principle of allowing one hound sterling, to be equal to $4-44 cents in the United States.

Nors. The author is principally indebted to the Sarr MASTER'S DAILY ASSISTANT for these valuable extracts; and he has made no alteration from the tables in that book, excepting he has carried out the value of each piece of money in dollars, cents and mills, instead of carrying them out in pounds, ́ e, sterling.