Since the area of a circle is 7854 when the diameter is 1, it is manifest, that if the number of cubic inches, in any measure whatever, be divided by 7854, the quotient will be the square of the diameter of a circle, which, being an inch deep, will contain that measure; for the square of the diameter of the circle multiplied by 7854 gives the area in square inches, and these, by conceiving them to be an inch in depth, become cubic inches. Hence the number 231 divided by 7854 will be the square of the diameter of the least circle, which, being an inch deep, will contain a wine gallon. So 282 divided by 7854 will quote the square of the least circle that will hold, at an inch deep, an ale gallon. In like manner 2150-42 divided by 7854 will be the square of the diameter of the least circle, which, being an inch deep, will contain a malt bushel; and so for any other measure whatever. The quotient arising from the division of the cubic inches in any proposed measure by 7854 is termed the Circular Divisor of that measure, being truly the square of the diameter of a circle, which being an inch deep, shall contain that measure ; and the square root of the quotient has the name of the Circular Gagepoint, being actually the DIAMETER of the circle, containing, for every inch in depth, the given measure. Hence the Circular Divisors and Circular Gagepoints for wine gallons, ale gallons, and malt bushels are obtained as follows: CIRCULAR DIVISORS AND CIRCULAR GAGE-POINTS. Divisor Dividend Quotient Square Roots 7854) 231-0000 (294-12 (17.15 wine gallons. 7854) 282-0000 ( 359-05 (18.95 ale gallons. 7854) 2150-4200 (2737.99 (52.32 malt bushels. Divisor Dividend Cir. Divisors Circular Gage-points Precisely in the same manner may the circular divisor and circular gage-point corresponding to any other measure or weight be ascertained, if the number of cubic inches in the proposed integer, be exactly known. The Circular Divisors and Circular Gage-points being now understood, we shall proceed with the Rule for calculating circular areas at an inch in depth. RULE. By the Pen. Divide the square of the diameter by the circular divisor for the proposed integer, and the quotient will be the content. By the Sliding Rule. Set the circular gage-point, for the proposed integer, on D, to 1 on C, and opposite to the given diameter on D, will be the content on C. EXAMPLE. Let KLMN be a circle, whereof the diameter LN is 120 inches; the content is required, for an inch deep, in wine gallons, ale gallons, and malt bushels. W.G. cir. divisor 294-12) 14400 (48.95 wine gallons. A. G. cir. divisor 359) 14400 (40·11 ale gallons. M. B. cir. divisor 2738) 14400 ( 5.25 malt bushels. BY THE SLIDING RULE. on D. on C. on D. on C. (48.95 W.G.) C. G. P. 18.95 to 1 against 120 are 40 11 A. G. 52.32 5.25 M. B. In treating of the Sliding Rule we defined, in page 42, the sense applied in Gaging to the word FACTOR, and shewed, in page 43, by an appropriate example, that a quotient in Division may sometimes be more expeditiously obtained by Multiplication. For this purpose it is necessary only to employ, as multiplier, the factor corresponding to the divisor. Now a factor is no more than the quotient that arises when 1 is divided by any divisor. Hence a Set of Factors for all the divisors in common use in Gaging is easily made, and should always be at hand, if not well fixed in the memory. The factors for the three circular divisors above, may be obtained either by dividing 1 by the proposed divisor, or by dividing 7854 (the area when the diameter is 1) by the number of cubic inches in the measure constituting the integer. Thus, Divisor Dividend Quotient 231) 7854 (00340 C. factor for wine gallons. 282) 7854 (00278 C. factor for ale gallons. 2150-42) 7854 (00036 C. factor for malt bushels. In like manner may the circular factor for any other solid measure be determined. The result will evidently be the same, if unity be divided by the circular divisor, as below: Divisor Dividend Quotient 294.12) 1 (00340 C. factor for wine gallons. 359.05) 1 (00278 C. factor for ale gallons. 2737-99) 1 (00036 C. factor for malt bushels. All this being clearly understood, and with reference to page 139, there will occur no difficulty in constructing the following useful Table. TABLE OF FACTORS, DIVISORS, AND GAGE-POINTS. By what has been said the reader will readily comprehend that the factors and divisors for squares are factors and divisors for areas of every shape reduced to square inches uniformly an inch in depth, that is, to cubic inches. The circular divisors and circular gage-points are used for circles only, and for ovals, as will be explained in the next problem. |