1. What is the area of a 3. What is the area of a cir. circle whose diameter is 7, and cle whose diameter is 10 rods, circumference 22 feet? and circumference 31.416? 11=1 circumference. Ans. 78.54 rods. 3.5-1 diaineter. 4. How many square chains in a circular field, whose cir55 33 cumference is 44 chains, and diameter 14? Ans. 154 chains. 38.5 feet, Ans. 2. What is the area of a cir 5. How many square feet in cle whose diameter is 1, and a circle •whose circumference circumference 3.1416? is 63 feet? Ans. .7854. Ans. 315 feet. 311. The area of a circle given to find the diameter and circumference. RULE 1.-Divide the area by .7854, and the square root of the quotient will be the diameter. 2. Divide the area by .07958, and the square root of the quotient will be the circumference. 1. What is the diameter of a 3. I demand the length of a circle whose area is 154 rods? rope to be ticd to a horse's neck, that he may graze upon 7854 7854 ) 154.0000 ( 196 ( 14 rods. square feet of new feed every 7854 1 day, for 4 days, one end of the rope being each day fastened 75465 24 )96 to the same stake. 70686 96 1st circle contains 7854 feet :-.7854=10000, and „10000 47740 =100 diam. :-2–50 feet, the 47124 1st rope ; 2d circle contains 15708: 7854–20000, and ✓ 616 20000=141), or 704 feet, sec ond rope, &r.. 2. The area of a circle is Ist rope 50 feet. 701 feet. 78.5 feet; what is its circum Ans. 30 865 feet. ference ? Ans. 31.4 feet. 100 feel.) 4th " 312. To find the area of an oval, or eltipsis. RULE.—Multiply the longest and shortest diameters together, and the product by .7854; the last product will be the area 1. What is the area of an 2. What is the area of an oval, whose longest diameter oval whose longest diameter is is 5 feet, and shortest 4 feet? 21, and shortest 17 ? 5X4X.7854–15.7C8ft. Ans. Aps. 280.3878. 313. To find the area, or surface, of a globe or sphere. RULE.--Multiply the circumference by the diameter, and the product will be the area. 1. How many square feet in 3. What is the area of the the surface of a globe whose surface of a cannon shot, diameter is 14 inches, and cir- whose diameter is 1 inch? cumference 44 ? Ans. 3.1416 inches. 44X145616, Ans. 4. How many square inches 2. How many square miles in the surface of an 18 inch in the earth's surface, ils cir. artificial globe ? cumference being 25000, and Ans. 1017.8784. its diameter 7957} miles ? Ans, 198943750. 2. Mensuration of Solids. $14. Mensuration of Solids teaches to determine the spaces included by contiguous surfaces, and the sum of the measures of ihese including surfaces is the whole surface of the body. The measure of a solid is call ed its solidiiy, capacity, content, or volume. The content is estimated by the number of cubes, whose sides are inches, or feel, or yarus, &c. con. tained in the body. 315. To find the sclidity of a cube. (234) Rule.-Cube one of its sides, that is, multiply the side by itself, and that product by the side again, and the last product will be the answer. 1. If the length of the side 2. How many cubic inches of a cube be 22 feet, what is in a cube whose side is 24 ils solidity ? inches? 22x22x22-10648, Ans. Ans. 13824, 316. To find the solidity of a parallelopipedon. (69) RULE.--Multiply thc length by the breadth, and that product by the depth ; the last product will be the answer. 1. What is the content of a 2. How many feet in a stick parallelopipedon whose length of hewn timber 30 feet long, is 6 feet, its breadth 23 feet, / 9 inches broad, and 6 inches and its depth 1} feet? thick ? 6X2.5X1.75=26.25, or 26 Ans. 114 feet. feet. 317. To find the side of the largest stick of timber that can be hewn from a round log. RULE.—Extract the square root of twice the square of the semidiameter at the smallest end for the side of the stick when squared. 1. The diameter of a round 2. The diameter at the log at its smallest end is 16 | smallest end being 24 inches, inches; what will be the side | how large square will the stick of the largest squared stick of of timber hew? timber that can be hewed from Ans. 16.97 in. it? 78X8X2=11.31 in. Ans. 318. To find the solidity of a prism, or cylinder. RULE.—Multiply the area of the end by the length of the prism, for the content. 1. What is the content of a 2. What number of cubic riangular prism, the area of feet in a round stick of timber whose end is 2.7 feet, and whose diameter is 18 inches, hose length is 12 feet? and length 20 feet? 2.7 X 12–32.4 ft. Ans. Ans. 35.343. 319. To find the solidity of a pyramid. or cone. RULE.-Multiply the area of the base by tne height, and one third of the product will be the content. 1. What is the content of 2. What is the content of a a cone whose height is 12: triangular pyramid, its height: feet, and the diameter of the being 141 feet, and the sides bas. 2.3 feet? of its base being 5, 6 and 7 2x2={x}=*=64, feet? Ans. 71.035+ and 63x.7854 X 121:33 20.453125, Ans. 320. To find the solidity of a sphere.* RULE.—Multiply the cube of the diameter by.5236, or multiply the square of the diameter by one oth of the circumference. 1. What is the content of a 2. What is the solid content sphere whose diameter is 12 of the earth, its circumference inches ? 12X12X12X.5236 | being 25000 miles ? 104.7808, Ans. Ans. 26385814912 miles. Guaging. 321. Guaging teaches - to measure all kinds of vessels, as pipes, hogsheads, barrels, &c. ROLE.-To the square of the bung diameter add the square of the head diameter; multiply the sum by the length, and the product by .0014 for ale gallons, or by .0017 for wine gallons. 1 What is the content of a 2. What is the content of a cask, whose length is 40 inch- cask whose length is 20 inch€3, and its diameters 24 and 32 es, and diameters 12 and 16? inches? X 32+24x24X 40=64000, Ans. w gal. 64000X.2014=89.6 a. gal., Ans. 64000X.0017=108.8 w.gal., Ans. Ans. { 13.6 a, gali SECTION III. PHILOSOPHICAL MATTERS. Of the fall of Heavy Bodies. 322. Heary Bodies near the surface of the carth, fall one foot the first ..quarter of a second, threc feet the second quarter, five feet the third quar ter, and seven feet the fourth quarter, equal to 16 seet the first second. The velocities acquired by falling bodies, are in proportion to the squares of the times in which they fall; that is, if 3 hullets be dropped at the same time, and the first be stopped at the end of the first second, the second at the end of the second, and the third at the end of the third, the first will fiave fallen 16 feet, the second (2X2–4) four times 16, equal to 6+; and the third (3X359) nine times i6, equal to 144 feet, and so on. Or, if 16 The surface of a soliero is found by multiplying its diameter by its Cirrtunformace feet be multiplied by so many of the odd numbers, beginning at 1, as there are seconds in the given time, these several products will be the spaces passed through in each of the several seconds, and their sum will be the whole distance fallen. 323. The velocity given to find the space fallen through. RULE.—Divide the velocity in feet by 8, and the square of the quotient will be the space fallen through to acquire that velocity. 1. From what height must 2. From what height must a body fall to acquire the ve a budy fall to acquire a veloci. locity of a cannon ball, which ty of 1200 feet per second ? is about 660 feet per Ans 22500 feet. 660;-8=82.5, and 82.5X 82.5=806.25f=114 miles, Ans. 324. The time given to find the space fallen through. RULE.-Multiply the time in seconds by 4, and the square of the product will be the space fallen through in the given time. 1. How many feet will a 3. Ascending bodies are rebody fall in five seconds ? tarded in the same ratio that 5x4=20, and 20x20=400 descending bodies are accelerfeet, Ans. ated; therefore, if a ball, fired upwards, return to the earth 2. A stone, dropped into a in 16 seconds, how high did it well, reached the bottom in 3 | ascend ? The ball being half seconds; what was its depth ? | the time, or 8 seconds, in its 3X4=12, and 12X12–144 | ascent: therefore 8X4=32, feet, Ans. and 32X32=1024ft., Ans. 325. The velocily per second given to find the time. RULE.—Divide the given velocity by 8, and one fourth part of the quotient will be the answer. 1. How long must a body 2. How long must a body be be falling to acquire a velocity falling to acquire a velocity of of 160 feet per second ? 400 feet per second ? 160:8–20, and 20:45 Ans. 123 seconds. seconds, Ans. |