As 282 A B 42: 30: 4,47, the area as before. PROB. II. To find a mean geometrical proportion between two given numbers. RULE. Multiply the two given numbers together, and extract the fquare root from their product, which will be the mean required. E. 1. What is the mean proportional between 36 and 64 ? 64 36 384 192 2304(48 the mean required. 88) 704 By the Rule. Set one of the given numbers upon C to the fame number upon D; then against the other given number upon C is the number fought upon D; thus: Note. D C D 36: 64: 48, mean as before. What is the mean between 42 and 30? 30 35,5 the mean required. This mean laft found is a mean proportional between the fum of the perpendiculars and half the diagonal of the trapezium, in Prob. 10. by which the area may be found by the lines C and D, as follows; thus: As 16,79 : 35,5 4,47, the area as before, found by the lines A and B. And in like manner may the area of the parallelogram, rhombus, rhomboides, and triangle be found. PROB. 12. To find the area of any regular polygon. RULE. Divide it into triangles, then find the area of one triangle, as in Prob. 9. and because there are as many triangles as there fides in the polygon, multiply the area of the triangle by the number of fides, and the product will be the area of the polygon. are EXAMPLE 351 EXAMPLE. What is the area of in ale gallons of a pentagon, whose fide is 50 inches, and the perpendicular 34,2 inches? 34,2 Perpendicular 25= one of the fides 1710 684 282)855,0(3,031 the area of the the triangle 5 Answer 15,155 Area of the polygon in ale gallons. PROB. 13. To find the area of a polygon, when the fide only is given. RULE. Multiply the fquare of the fide of any regular polygon, mentioned in the following table, by the common factor belonging to that polygon, and the product will be the area in inches, ale gallons, wine gallons, or malt bufhels respectively. EXAMPLE. Required the area of a pentagon in ale gallons, whofe fide is 50 inches? Answer 15,247500 Area in ale gallons, fame as before, nearly. In like manner may the area of any other regular polygon mentioned in the table be found, in any of the denominations there mentioned. PROB. 14. PROB. 14. To find the area of a circle. RULE. Square the diameter of the circle, and multiply or divide that fquare by the factors in the table for circles (page 346) and the product, or quotient, will be the area required. EXAMPLE. The diameter of a circle is 80 inches; what is its area in ale gallons? As 18,95 :: 80: 17,82, Area as before. The Rule being thus fet, it is like a table, for against any diameter on D, is the area in ale gallons on C. PROB. 15. To find the area of an ellipfis, or oval. RULE. Multiply the tranfverfe and conjugate diameters together, and that product multiply or divide by the factors in the table for circles (p. 346), then that product, or quotient, will be the area of the ellipfis. EXAMPLE. The tranfverfe diameter of an ellipfis is 72 inches, and the conjugate 50 inches; what is its area in ale gallons? 72 Tranfverfe diameter = 50=Conjugate diameter Note. If a mean proportion between the tranfverfe and conjugate diameters is found, the proportion will be the fame as a circle, and may be found on the lines C and D, C As 72 : D 72 thus: :: C 50 Tranf. Diam. Tranf. Diam. Conju. Diam. SOLIDS LXXI. OF SOLID BODIES. OLIDS are comprehended under length, breadth and depth: now by having the area given at one inch deep, it will be easy to find the content of any folid body, at any depth; for if the content at one inch deep be multiplied by the whole depth, the product will be the folid content of the body. PROB. I. To find the content of a folid, whofe bafes are either RULE 1. Multiply the length of the base by the breadth, and that product by the depth; and this laft product multiply or divide by the factors, &c. for fquares in the Table of factors (Page 346,) and the product, or quotient, will be the content required. 2. Or find the area of the base by Problems 5 and 6, of the last Section, and multiply that area by the depth, and the product will be the content. E. 1. There is a cube, each of whofe equal fides are 30 inches; what is the content in ale gallons? 30 30 900 30 282)27000(95,7 Content in ale gallons. 2538 1620 1410 2100 1974 126 The area of the base of this folid was found in PROBLEM 5. of the last Section to be 3,19 ale gallons. '.* 3,19× 30=95,7, the content in ale gallons, as before. As 16,79: 30 :: 30: 95,7 the content, as before. 2 Z E. 2. 354 E. 2. Required the content of a prifm in ale gallons, whofe length is 45,6 inches, breadth 27,5, and depth 21,5 inches. 45,6 = Length 27,5 2280 3192 912 Breadth 1254,00 21,5 = Depth 6270 1254 2508 282)26961,0(95,6 Content in ale gallons 2538 1581 1410 1710 1692 18 The proportion by the rule is the fame as that for a fquare bafe, when. there is a mean proportional found between the longest and shorteft fides of the bafe. In like manner the content of any right-lined folid may be found, either regular or irregular, if by the foregoing rules you find the area of its base, and multiply that area by its depth. PROB. 2. To find the content of a cylinder, by having the depth and diameter given. RULE. Multiply the fquare of the diameter by the depth, and divide by the circular divifors; the quotient will be the content required. EXAMPLE |