THEOREM OF PAPPUS. If a plane curve lies wholly on one side of a line in its own plane, and revolving about that line as an axis, it generates thereby a surface of revolution, the area of which is equal to the product of the length of the revolving line into the path of its center of mass; and a solid the volume of which is equal to the revolving area into the length of the path described by its center of mass. XIX. MISCELLANEOUS MEASURE MENTS. 1. MASONS' AND BRICKLAYERS' WORK. Masons' work is sometimes measured by the cubic foot, and sometimes by the perch. A perch is 16 ft. long, 14 ft. wide, 1 ft. deep, and contains 16×14×1=244 cu. ft. Prob. CLIII. To find the number of perch in a piece of masonry. Rule. Find the solidity of the wall in cubic feet by the rules given for the mensuration of solids, and divide the product by 244. I. What is the cost of laying a wall 20 feet long, 7 ft. 9 in. high, and 2 feet thick, at 75 cts. a perch. II.< III. perch. ́1. 20 ft. the length of the wall, 2. 7 ft. 9 in. 7 ft.—the height of the wall, and 3. 2 ft. the thickness. 4... 20X7X2=310 cu. ft.=the solidity of the wall. 6. 310 cu. ft.=310-243-125 perches. 7. 75 cts. the cost of laying 1 perch. 8... 125X75cts.-$9.3913-the cost of laying 125 perches. .. It will cost $9.391 to lay1253 perches at 75 cts. a 2. GUAGING. Gauging is finding the contents of a vessel, in bushels, gallons, or barrels. Prob. CLIV. To gauge any vessel. Rule.-Find its solidity in cubic feet by rules already given; this multiplied by 1728÷2150.42 or .83, will give the contents in bushels; by 1728÷231. will give it in wine gallons, which divided by 31 will give the contents in barrels. Prob. CLV. To find the contents in gallons of a cask or barrel. Rule.-(1) When the staves are straight from the bung to each end; consider the cask two equal frustums of equal cones, and find its contents by the rule of Proh. XCIII. (2). When the staves are curved; Add to the head diameter (inside) two-tenths of the difference between the head and bung diameter; but if the staves are only slightly curved, add sixtenths of this difference; this gives the mean diameter; express it in inches, square it, multiply it by the length in inches, and this product by .0034; the product will be the contents in wine gallons. 3. LUMBER MEASURE. Prob. CLVI. To find the amount of square-edged inch boards that can be sawed from a round log. Doyle's Rule.-From the diameter in inches subtract four; the square of the remainder will be the number of square feet of inch boards yielded by a log 16 feet long. I. How much square-edged inch lumber can be cut from a log 32 in. in diameter, and 12 feet long? II. 3. 32 in.-4 in.=28 in.—the diameter less 4. 4. 844 ft. 282—the square of the diameter less 4, which by the rule, is the number of feet in a log 16 ft. long. 5. 12 ft. of 16 ft. 6... of 844 ft.=633 ft.=the number of feet of squareedged inch lumber that can be cut from the log. III. .. The number of square-edged inch lumber that can be cut from a round log 32 inches in diameter and 12 ft. long is 633 ft. 4. GRAIN AND HAY. Prob. CLVII. To find the quantity of grain in a wagon bed or in a bin. Rule.-Multiply the contents in cubic feet by 1728÷2150.42, or .83. I. How many bushels of shelled corn in a bin 40 feet long, 16 feet wide and 10 feet high? II. (1. 40 ft. the length of the bin. 2. 16 ft. the width of the bin, and 3. 10 ft. the height of the bin. 4. .. 40×16×10=6400 cu. ft. the contents of the bin in cu. ft.. 5. .. 6400×.83 bu.=5312 bu.=the contents of the bin in bu. III. ... The bin will hold 5312 bu. of shelled corn. Rule.-(1) For corn on the cob, deduct one-half for cob. (2) For corn not "shucked" deduct two-thirds for cob and shuck. I. How many bushels of corn on the cob will a wagon bed feet long, 3 feet wide, and 2 feet deep? hold that is 10 II. [in cu. ft 4. .. 10×3×2=734 cu. ft.—contents of the wagon bed 5... 734 X.8 bu=58.8 bu.=number of bushels of shelled corn the bed will hold. 6. .. of 58.8 bu=29.4 bu.=the number of bushels of corn on the cob that it will hold. III. .. The wagon bed will hold 29.4 bu, of corn on the cob. Prob. CLVIII. To find the quantity of hay in a stack,rick, or mow. Rule.-Divide the cubical contents in feet by 550 for clover or by 450 for timothy; the quotient will be the number of tons. Prob. CLXIX. To find the volume of any irregular solid. Rule.-Immerse the solid in a vessel of water and determine the quantity of water displaced. I A being curious to know the solid contents of a brush pile, put the brush into a vat 16 feet long, 10 feet wide, and 8 feet deep and containing 5 feet of water. He found, after putting in the brush, that the water rose 14 feet; what was the contents of the brush pile? II. III. 1. 16 ft. the length of the vat, 2. 10 ft. the width, and 3. 1 ft. the depth to which the water rose. 4. . 16×10×14=240 cu. ft. the volume of the brush pile. .. 240 cu. ft=the volume of the brush pile. XX. SOLUTIONS OF MISCELLANEOUS PROBLEMS. Prob. CLX. To find at what distance from either end, a trapezoid must be cut in two to have equal areas, the dividing line being parallel to the parallel sides. a Formula.―d=A÷[√}(b2+b2)+b]=1(b+b1) ÷[√√(b2+62)+b], where A is the area of the trapezoid, b the lower base, and b1, the upper base. √(b2+62) is the length of the dividing line. 19 Rule.-1. Extract the square root of half the sum of the squares of the parallel sides and the result will be the length of the dividing line. 2. Divide half the area of the whole trapezoid by half the sum of the dividing line and either end, and the quotient will be the distance of the dividing line from that end. I. I have an inch board 5 feet long, 17 inches wide at one end and 7 inches at the other; how far from the large end must it be cut straight across so that the two parts shall be equal? 1 By formula, d=4(b+b1)a÷÷÷[√+ (b 2+b2)+b] =(17+7)60÷÷[√}(172+72)+17]=720-30 -24 in. 2 ft. 1. Let ABCD be the board, [end, 2. AB 17 in.=b, the width of the large [board. 4. HK-5 ft. 60 in. a, the length of the 6. ABE:EGL:EDC::AB:LG2: DC2. But 8. 2EGL=2EDC+ABCD-EDC+EDC+ABCD -EDC+EAB. 9. . EGL (EDC+EAB), i. e., EGL is an arithme-tic mean between EAB and EDC. 10... GL2 (AB2+DC2)=(b2+b^2)=an arithmetic. mean between EAB and EDC, 11. GL (62+b′2)=√√2(b2+b ́2). II. 12. Draw CM perpendicular to AB. 13. FL=GL=4No2(b2+b′2). 14. IL-FL-FI(=KC=DC=1b)=√2(b2+b′2)—§6. 15. CM-HK-a. 16. MB (b—b′). and CIL, Then in the similar triangles CMB 17. MB:IL:: CM: CI, or (b-U′): (‡No2 (b2+b ́2)—1b)::a; CI. Whence 19. .. IM CM-CI-5 ft.-3 ft.-2 ft., the distance from the large end at which the board must be cut in two. to have equal areas. III. .. The board must be cut in two, at a distance of 2 feet from the large end, to have equal areas in both parts. (R. H. A., p. 407, prob. 101.), Prob. CLXI. To divide a trapezoid into n equal parts and find the length of each part. the width of the small end, b the width of the large end, and a the length of the trapezoid. h, is the length of the first part at the small end, h, the length of the second part, and so on. 2 I. A board ABCD whose length BC is 36 inches, width AB 8 inches and DC 4 inches, is divided into three equal pieces. Find the length of each piece. 42+82—4]=9[√32-4]-36(√2-1) = 14.911686 in. By formula, hi -11.442114 in. (n=3)b2+3b2 n α b-b1 n =36[2-3]-9.6462 in. 1. 4 in. the width DC of the small end, 2. 8 in. the width AB of the large end, and 3. 36 in. the length BC of the board. board. 5. of 216 sq. in. 72 sq. in. the area of 6. AK-AB-KB(=DC)=8 in.-4 in. 7. AK:DK::AB: BE, or 4 in.:36 in.::8 in.: BE. Whence, 8. BE (36X8)÷4-72 in. II. 9. [triangle ABE. (ABXBE)=(8×72)=288 sq. in. the area of the in.=72 10. ABE-ABCD 288 sq. in.-216 sq. in. 72 sq. in. area of the triangle DCE. |