1. A trapezium is divided 2. What is the area of a into two triangles, by a diago- trapezium whose diagonal is Dal 42 rods long, and the per- 1084 feet, and the perpendicu. pendiculars let fall from the lars 564 and 604 feet? opposite angles of the two tri Ans. 63474 feel angles, are 18 rods and 16 rods; what is the area of the trape 3. How many square yards zium ? in a trapezium whose diagonal 42 42 336 is 65 feet, and the perpendicu. 9 8 378 lars let fallupon it 28 and 33.5 feet? 378 336 714 rods, Ans. Ans. 2227 ydse 309. To find the diameter and circumference of a cirebe, either from the other. (67) Rule 1.-As 7 is to 22, so is the diameter to the circumference, and as 22 is to 7, so is the circumference to the diameter. RULE 2.-As 113 is to 355, so is the diameter to the circum. ference, and as 355 is to 113 80 is the circumference to the diquieter. Rule 3.--As I is to 3.1416, so is the diameter to the circumference, and as 3.1416 is to 1, so is the circumference to the diameter. 1. What is the circumfer- 3. What is the diameter of once of a circle whose diaine- 2 circle whose circumference ter is 14 feet? is 50 rods? By Rule 1. As 22:7::50:15.9090, Ang. By Rule 2. Py Rule 2. As 113 : 355 :: 1+:43113, Ans. As 358 : 113 :: 50 : 15.9155, Ans. By Rule 3. By Rule 3, As 1 : 3.1416 :: 14: 13.9824, Aus. As 3.1416:17:50 : 15.9156, Aus. 2. Supposing the diameter 4. Supposing the circunferof the eurth to be 7958 miles, lence of the earth to be 25000 what is its circumference? miles, what is its diameter? Ans. 25000.8528 iniles. Ans. 7957* nearly. 310. To find the area of a circle. Rule.-Multiply half the circumference by half the diamo ter; or the square of the diameter by.7854, , -r the square of the circumference by .07958-the product will be the area 311, 312. MENSURATION OF SUPERFICIE.S. 135 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 rodom circumference 22 feet? and circumference 31.416? 11= circumference, Ans. 78.54 rods. 3.5$ diameter. 4. How many square chains 55 in a circular field, whose cir33 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 circumferenea circumference 3.1416? is 63 feet? Ans, .7854. Ans. 315 feet 311. The area of a circle given to find the diameter and cir. cumference. 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 tied to a horse's neck, that he may grazę 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 Ist circle contains 7854 feet --7854=10000, and, 10000 47740 =100 diam. - 250 feet, the 47124 1st rope ; 2d circle contains 15708 : 7854–20000, and ✓ 616 20000=141), or 700 feet, sec ond rope, &c, 2. The area of a circle is Ist rope 50 feet. 2d 78.5 feet; what is its circuma 704 feet. 3d Apa 864 feet, 1 ference ? Ans. 31.4 feet. 100 feet.) 41h 11 312. To find the area of an oval, or ellipsis. LULE.-Multiply the longest and shortest diameters together, god the product by .7854; the last product will be the arca 1. What is the area of an 2. What is the area of an oval, whose longest diameter oval whose longest diameter is 18 5 feet, and shortest 4 feet? 21, and shortest 17 ? 5X4X.785415.7081t. Ans. Ans. 280.3878 cannon 313, To find the area, or surface, of a globe or sphere. Role.--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 shot, diameter is 14 inches, and cir- | whose diameter is 1 inch? cumference 44 ? Ans. 3.1416 iaches. 44X14–616, 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 79572 miles ? Ans. 198943750. 2. Mensuration of Solids. 314, Mensuration of Solids teaches to delesmine the spaces included by contiguous surfaces, and the sum of the ineasures of ihese including surfaces is the whole surface of the body. The measure of a solid is call ed ils solidity, capacity, content, or volume. The content is estimated by the number of cubes, whose sides are inches, or feel, or yarıs, &c.com lained in the body. 315, To find the solidity of a cube. (254) 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? 22X22X 22=10648, Ang. Ans. 1:3824. 316. 7o find the solidity of a parallelopipedạn. (69) RULE.--Multiply the length by the breadth, and that product by the depth; the last product will be the answer. 317, 318, 319. YENSURATION OF SOLIDS. 137 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 25 feet, 9 inches broad, and. 6 inches and its depth 14 feet? thick ? 6X2.5X1.75–26.25, or 26} Ans. 11 feet. feet. 317. To find the side of the largest stick of timber that can be hewn from a round log. ROLE.--Extract the square root of twice the suare of the semidiameter at the smallest end for the side of the stick wheu equared. 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 show large square will the sticks of the largest squared stick of of timber hew? timber that can be hewed from Ang. 16.97 in it? V8X8X2-1131 in. Ans. 318. To find the solidity of a prism, or cylinder, ROLE.—Multiply the area of the end by the length of the prisin, for the content. 1. What is the content of a 2. What number of cubic triangular prism, the area of feet in a round stick of timber whose end is 2.7 feet, and whose diameter is 18 inches whose length is 12 feet? and length 20 fect? 2.7X12_2.4 ft. Ang. Ans, 33.343. 319. Tb firm the solidity of a pyramid, or cone. ROLE. --Multiply the area of the base by the height, and ove third of the product will be the content. 1. What is the content ofl 2. Wbat is the content of a #cone whose height is 12 triangular pyramid, its height fect, and the diameter of the being 145 feet, and the sides bag. 23 feet? of its brat being 5, 6 and 7 2X4={x}=*=64, feet? Ans. 71.035+ and 64.7854 X 12:33 20.453125, Ans. 320. To find the solidity of a spheri.* 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. 891. Graging teaches 10 measure all kinds of vessels, as piper, hogsheads, barrels, &c. Rule. To the square of the bung diameter add the square of the head diameter; multiply the sum by the length, and the prorlact by .0014 for ale gallons, or by .0017 for wine gallonse 1 What is the content of a 2. What is the content of a cark, whose length is 40 inch-cask whose length is 20 inches, and its diameters 24 and 32 es, and diameters 12 and 16? ir:ches? 11.2 a. gal, 32X32 +24X24X40=64000, Ans. 64000X.0014389.6 a. gal., Áns. 64000X.0017=108.8. w. gal., Ans. Ans. { 13.6 w.gal. SECTION III. PHILOSOPHICAL MATTERS. Of the Fall of Weavy Bodies. 322. Heary Bodies near the surface of the carth, fall one foot the first quarter of a second, three feel the second quarter, five feel the third quarier, and seven feet the fourth quarter, equal to 16 feet the first second. The velocities acquired by falling bodies, are in proportion to the squares of the times iu which they fall; that is, is 3 hullets be dropped at the same uime, and the first be stopped at the end of the first second, the second at the end of the seconıl, and the third at the end of the third, the first will have fallen 16 feet, the second (2X2=4) four times 16, cqual 10 64; and the third (3X3=9) nine times 16, equal to 144 feet, and so on. Or, if 16 • The surface of a sphere is found by mulüplying iis diameter by in pounfance |