1. Required the number which shall be to 36, as the square of 4 to that of 3.

Set 3 on D to 36 on C; then against 4 on D will be 64 on C, the answer.

2. What number is to 120, as the square of 3 to that of 2?

Ans. 270.

3. Increase the number 240 in the ratio of the square of 4 to that of 5. Ans. 375. 4. Diminish the number 392 in the ratio of the square of 7 to that of 6. Ans. 288. 5. Find the number to which 196 shall have the same ratio with the square of 7 to that of 9.

Ans. 324. PROB. VII. To find a number which shall be to a given one as the square roots of two given numbers.

Set the first term on C to the given number on D; then against the other term on C stands the answer on D.

1. To what number will 3 have the same ratio with the square root of 108 to that of 48 ?

Set 3 on D to 108 on C; then against 48 on C is 2 on D,

the answer.

2. To what number will 2 be as the square root of 120 to that of 270?

Ans. 3.

3. Required the number to which 256 shall be as the square root of 16 to that of 9.

4. Increase the number 433 in the ratio of the of 3 to that of 5.

Ans. 192.

square root

Ans. 559

5. Diminish the number 1414 in the ratio of the square root of 8 to that of 7.

Ans. 1393. PROB. VIII. Of multipliers, divisors, and gauge. points.

Instead of first finding the content of a vessel in inches, and afterwards reducing it to the measure of capacity required. which must often be done both by multiplying and dividing by known numbers, gaugers find the content in the measure required by means of a single multiplier or divisor.

These multipliers are got by dividing the multiplier used in finding the content by the divisor, which reduces the content to gallons, &c. Thus, to find the multiplier which, in circular vessels, will give the content in imperial gallons, vide 785398 by 277.274.

To find the divisor which will answer the same divide 277.274 by 785398.

di

purpose

Gauge-points are numbers made use of in working by the

ding-rule. The operation is made similar to that in rob. VI.; and for that purpose the square root of the divisor taken for the first term, and is called the Gauge-point.

TABLE I.

MULTIPLIERS, DIVISORS, AND GAUGE-POINTS, FOR
CYLINDRICAL VESSELS.

In this table, the first multiplier is that for finding the area of a circle, and its reciprocal is the first divisor. The other multipliers are got by dividing the first by the number of nches in a gallon, bushel, &c.; and the other divisors by multiplying the first divisor by the number of inches in a gallon, &c. The gauge-points are the square roots of the Hivisors.

If 1 be put instead of 7853982 at the top, tables may be Formed in the same way for square vessels. Thus, 1 divided by 27.14 gives '036846, the multiplier for hard soap.

TABLE II.

MULTIPLIERS, DIVISORS, AND GAUGE-POINTS, FOR
PRISMATIC VESSELS.

To find the multiplier, divisor, and gauge-point, for imperial

gallons in vessels of the form of a regular heptagon.

Divide the tabular multiplier 3-6339124 by 277-274: the uotient ⚫0131059 will be the multiplier. Divide 277.274 by 6339124: the quotient 76-3918 will be the divisor, and its quare root 8.7351 will be the gauge-point.

In the same manner the multipliers, divisors, and gaugeoints are found for any regular polygon.

TABLE III.

MULTIPLIERS, DIVISORS, AND GAUGE-POINTS, FOR

In pyramidal, conical, &c. vessels, where, in finding the content, we multiply by one-third of the length, the multiplier should be one-third of that in the table, the divisor must be three times as large as that in the table, and the gaugepoint must be the square root of three times the tabular divisor; and, in this case, use the whole length, instead of onethird of it. The same remarks are applicable to rules in which we multiply by any other part of the length.

PROB. IX. To gauge areas one inch deep,

I. When one side is given, set the gauge-point on D to 1 on C; and against the given side on D is the answer on C.

1. Suppose the side of a square to be 77 inches. Required its content, at 1 inch deep, in old wine and ale gallons.

Here the multipliers are 003546 and 004329, the divisors are 282 and 231, and the gauge-points 16-7929 for ale, and 15.1987 for wine gallons.

Set the gauge-point for ale, 16-7929 on D, to 1 on C; then against 77 on D will be found 21 ale gallons on C.

Set the gauge-point for wine, 15-1987 on D, to 1 on C; and against 77 on D is 25.7 on C, the wine gallons.

2. Required the content, in imperial gallons, of a square vessel at 1 inch deep, the side 98 inches. Ans. 34-6368 gals. 3. Required the content of a regular pentagon 1 inch deep, in hard soap and starch, the side 53 inches.

Ans. 178-07 lbs. hard soap, 138.874 lbs. starch. 4. Required the content of a regular octagon 1 inch deep, in tallow, the side 83 inches. Ans. 1098-5144 lbs.

II. When two dimensions are given, it is necessary, in working by the gauge-points, to find a mean proportional between the two factors, and to work with it by the preceding

rule.

1. Required the content, at 1 inch deep, of a rectangular vessel, of which the length is 1004 inches, and its breadth 20 inches, in imperial bushels and pounds of hard soap.

Set 100 on B to 1 on MD; then against 20 on A will

stand 906 of an imperial malt bushel on B.

Set 27.14 on A to 20 on B; then against 100-5 on A will stand 74.1 lbs. on B.