his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger: what was the eldest son's portion ? Ans. £25600. 4. A man bought 12 yards of cloth, giving 3 cents for the 1st yard, 6 for the 2nd, 12 for the 3rd, &c.: what did he pay for the last yard ? Ans. $61,44. CASE 11. § 195. Having given the ratio and the two extremes to find the sum of the series. RULE. Subtract the less extreme from the greater, divide the remainder by 1 less than the ratio, and to the quotient add the greater extreme : the sum will be the sum of the series. Q. How do you find the sum of the series ? EXAMPLES. 1. The first term is 3, the ratio 2, and last term 192 : what is the sum of the series ? 192—3=189 difference of the extremes, 2-1=1)189(189; then 189+192=381 Ans. 2. A gentleman married his daughter on New Year's day, and gave her husband ls. towards her portion, and was to double it on the first day of every month during the year: what was her portion? Ans. £204 15s. 3. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2nd, 9 for the 3rd, and so on to the last: what did he pay for the last bushel and for the 10 bushels ? Ans. last bushel $196,83, total cost $295,24. 4. A man has 6 children; to the 1st he gives $150, to the 2nd $300, to the 3rd $600, and so on, to each twice as much as the last : how much did the eldest receive and what was the amount received by them all ? Ans. Eldest $4800, total $9450 APPENDIX. MENSURATION. 8 196. A triangle is a figure bounded by three straight lines. Thus, BAC, is a triangle. The three lines BA, AC, BC, are called sides : and the three corners, B, A, and C, are called angles. The side BC is called the base. When a line like AD is drawn making the angle ADB equal to the angle ADC, then AD is said to be perpendicular to BC, and AD is called the altitude of the triangle. Each triangle BAD or DAC is called a right angled triangle. The side BA or the side AC, opposite the right angle, is called the hypothenuse. The area or content of a triangle is equal to half the product of its base by its altitude. EXAMPLES. 1. The base of a triangle is 40 yards and the perpendicular 20 yards : what is the area? We first multiply the 40 base by the altitude and the product is square yards, which we divide by 2 for the 2)800 area. Ans: 400 square yards. 2. In a triangular field the base is 40 chains and the perpendicular 15 chains : how much does it contain ? (see $ 64.) Ans. 30 acres 3. There is a triangular field of which the base is 35 rods and the perpendicular 26 rods: what is its content? Ans. 2A. 3R. 15P. OPERATION. 20 4. What is the area of a square field of which the sides are each 33,08 chains ? Ans. 109.A. IR. 28P+. 5. What is the area of a square piece of land of which the sides are 27 chains ? Ans. 6. What is the area of a square piece of land of which the sides are 25 rods each? Ans. 3A. 3R. 25 P. $ 197. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. The area or content of a rectangle is equal to the length multiplied by the breadth. EXAMPLES. 1. What is the content of a rectangular' field the length of which is 40 rods and the breath 20 rods ? Ans. 5 acres. 2. What is the content of a field 40 rods square ? Ans. 10 acres. 3. What is the content of a rectangular field 15 chains long and 5 chains broad. Ans. 4. What is the content of a field 25 chains long by 20 chains broad? Ans. 50 acres. 5. What is the content of a field 27 chains long and 9 rods broad. Ans. 6A. OR. 12P. § 198. A circle is a portion of a plane bounded by a curved line, every part of which is equally distant from a certain point within, called the centre. The curved line AEBD is called the A Ć circumference : the point C the centre ; the line AB passing through the centre, a diameter, and CB the radius. The circumference AEBD is 3,1416 times greater than the diameter AB. Hence, if the diameter is 1, the circumference will be 3,1416. Hence, also, if the diameter is known, the circumference is found by multiplying 3,1416 by the diameter. EXAMPLES. 1. The diameter of a circle is 4, what is the circumference ? OPERATION. The circumference is found by simply 3,1416 multiylying 3,1416 by the diameter. 4 Ans. 12,5664. 2. The diameter of a circle is 93, what is the circumference ? Ans. 3. The diameter of a circle is 20, what is the circumference ? Ans. 62,832. § 199. Since the circumference of a circle is 3,1416 times greater than the diameter, it follows that if the circumference is known we may find the diameter by dividing it by 3,1416. EXAMPLES. 1. What is the diameter of a circle whose circumference is 78,54. OPERATION. We divide the circumfer 3,1416)78,5400(25 62832 ence by 3,1416, the quotient 25 is the diameter. 157080 157080 2. What is the diameter of a circle whose circumference is 11652,1944 ? Ans. 37,09. 3. What is the diameter of a circle whose circumference is 6850 ? Ans. 2180,41 +. $ 200. The area or content of a circle is found by multiplying the square of the diameter by the decimal ,7854 EXAMPLES. OPERATION. 1. What is the area of a circle whose diameter is 6 ? We first square the diameter, giving 36, which we then 62=36 multiply by the decimal ,7854: ,7854x36=28,2744 the product is the area of the Ans. 28,2744 circle. 2. What is the area of a circle whose diameter is 10? Ans. 78,54. 3. What is the area of a circle whose diameter is 7? Ans. 4. How many square yards in a circle whose diameter is 3 feet. Ans. 1,069016+. $ 201. The surface of a sphere is formed by multiplying the square of the diameter by the decimal 3,1416. EXAMPLES. 1. What is the surface of a sphere whose diameter is 12 ? We simply multiply the deci- OPERATION. mal 3,1416 by the square of the 3,1416 diameter: the product is the 122=144 surface. | Ans. 452,3904 2. What is the surface of a sphere whose diameter is 7? Ans. 153,9384. 3. Required the number of square inches in the surface of a sphere whose diameter is 2 feet or 24 inches ? Ans. 4. Required the area of the surface of the earth, its mean diameter being 7918,7 miles ? Ans. 196996571,722104 sq. miles. $ 202. To find the solidity of a sphere-Multiply the surface by the diameter and divide the product by 6—the quotient will be the solidity. EXAMPLES. 1. What is the solidity of a sphere whose diameter is 12 ? |