75. The manner of determining the area of a surface like the marginal figure, may be understood from the following explana nation. Let AB repre sent (on a reduced scale) a line 5 inches in length; then, evidently, if we pass from A to e, a distance of 1 inch, and draw the line ef, the figure ABfe will contain 5 square inches, i. e. 5 X 1 square inches. In like manner A Bhg will contain 10, or 5 × 2 square inches; and A B C D will contain 15, or 5 × 3 square inches; i. e. we multiply the numbers expressing the length and breadth together, and the product will be the number of square inches in the surface. (a) Likewise, the area divided by the length will give the breadth, and the area divided by the breadth will give the length; thus, 15 ÷ 5 = 3, and 153 = 5. NOTE. c. yd. } This table is used in measuring things which have length, breadth and thickness. 77. To determine the contents of a body in the form of the following figure, first find the area of the upper surface, ABCD, as in Art. 75; then going from A, B and C downward 1 inch to a, b and c, and passing a plane through a, b and c, we shall cut off 15 solid inches, i. e. 5 × 3 × 1 solid inches. D C So if a plane be passed through d, e and f, it will cut off 30, or 5 × 3 × 2 inches, etc.; i. e. the continued product of the numbers expressing the length, breadth and depth, will give the solid contents. (a) So, also, the solid contents divided by the area of the top face will give the depth. What are the solid contents of the above figure? NOTE 1. This table is used in measuring all liquids. The gallon is the unit. NOTE 2.-A gallon of oil, vinegar, cider, wine, brandy and other spirits, contains 231 cubic inches. NOTE 3. A gallon of milk, or of beer or other malt liquors, contains 282 cubic inches. NOTE 4.-Vessels of various capacities from 50 to 150 or more gallons, are indiscriminately called hogsheads, pipes, butts, tuns, etc. NOTE 1.. coal, etc. 1 4 1 8 32 16 64 220+ This table is used in measuring grain, fruit, potatoes, char NOTE 2. A bushel is equal to 2150 cubic inches, nearly. 1 Julian Year, J.yr. 12 Calendar Months (=365 or 366 Days), 1 Civil Year, c. yr. sec. 1. m. 1 J. yr. m. h. 1 60 (a) The twelve calendar months have the following number of days: January (Jan.) has 31 days; February (Feb.), 28; March (Mar.), 31; April (Apr.), 30; May, 31; June, 30; July, 31; August (Aug.), 31; September (Sept.), 30; October (Oct.), 31; November (Nov.), 30; December (Dec.), 31. (b) The number of days in each month may be easily remembered by committing the following lines: NOTE. This table is much used in trigonometrical and astronomical calculations. 83. A sheet folded in 2 leaves is called a folio. 84. REDUCTION is changing numbers of one denomination to those of another, without changing their value.* It is of two kinds, viz., Reduction Descending and Reduction Ascending. 85. Reduction Descending consists in changing numbers from a higher to a lower denomination. 86. Reduction Ascending is changing numbers from a lower to a higher denomination. 87. REDUCTION DESCENDING is performed by multiplication; thus, to reduce 15£ to shillings, we multiply 15 by 20, because there will be 20 times as many shillings as pounds. So to reduce 15£ and 12s. to shillings, we multiply 15 by 20, and to the product add the 12s. In a similar manner all such examples are reduced. Hence, To reduce the higher denominations of a compound number to a lower denomination, RULE.-Multiply the highest denomination given, by the num ber it takes of the next lower denomination to make one of this higher, and to the product add the number of the lower denomination; multiply this sum by the number it takes of the NEXT lower denomination to make one of THIS; add as before, and so proceed till the number is brought to the denomination required. Ex. 1. Reduce 11£ 17s. 9d. 3qr. to farthings. * Reduction, with merchants, means a diminution of the price of an article. |