RULE-As the given time, is to the square of the given diameter, is the required time, to the square of the required diameter. EXAMPLE. 1. Suppose a pipe 2 inches in diameter, will fill a cistern in 4 hours; what must be the diameter of another pipe which will fill the same cistern in 1 hour? 1 22 inches=2,5; and 2,5×2,5=6,25. ` Then, as 4 hours : 6,25 inches, : 1 hour: 25,00 inversely; and 25,00=5 inches. Ans. 5 inches. PROBLEM VI-When time is required, the diameters being given. RULE. As the square of one diameter, is to its time, so is the square of the other diameter inversely, to its time. EXAMPLE. 1. Suppose a pipe of 2in. bore, discharge a certain quantity of water in 4 hours; what time will a pipe of 4in. bore, re quire to discharge the same quantity? Ans. 1 hour. PROBLEM VII.-The diameter of a circle being given, to find the diameter of another circle, which shall be 2, 3, 4, &c., times greater or less than the given circle. RULE.-Square the given diameter, and if the required circle be greater, multiply the square of the diameter by the given proportion, and the square root of the product, will be the required diameter.But if the required circle be less, divide the square of the diameter by the given proportion, and the square root of the quotient will be the required diameter. EXAMPLES. 1. Let the diameter of a given circle be 12 inches; what is the diameter of one 4 times as large? Ans. 24 inches. NOTE.-The diameter of a circle, is a right line passing through the centre, and terminated both ways by the circumference. A, B, is the diameter of the circle A, f, B, e; and c, d, is the diameter, of the smaller circle, c, g, d, h. The circumference is the bounding line of a circle. A, f, B, e, is the circumference of the greater circle, A, f, B, e; and c, g, d, h, is the circumference of the circle c, g, d, h. The diameter of the lesser circle is half as long as the diameter of the greater circle, though the greater circle is 4 times as large. 2. The diameter of a circle is 24 inches; what is the diameter of one which is one fourth as large? Ans. 12in. 3. Seven men bought a grinding stone of 60 inches diameter, each paying part of the expense; what part of the diameter must each grind down for his share? Ans. The 1st, 4,4508-2d, 4,8400-3d, 5,3535-4th, 6,0765-5th, 7,2079-6th, 9,3935-7th, 22,6778 inches. PROBLEM VIII-To find the diameter of a circle equal" in area to an Ellipsis, whose longest and shortest diameters are given. RULE.-Multiply the two diameters of the ellipsis together, and the square root of the product will be the diameter of a circle equal in area to the ellipsis or oval. PROBLEM IX.-The sum of two numbers being given, and the difference of their squares; to find those numbers. RULE.-Divide the difference of their squares by the sum of the numbers, and the quotient will be their difference; then to half the sum of the given numbers, add half the difference for the greater number, and from half the sum subtract half the difference for the less. EXAMPLE. 1. The sum of two numbers is 40, and the difference of their squares is 320; what are the numbers? Ans. The greater is 24, the less 16. PROBLEM X-The difference of two numbers being given, and the difference of their squares; to find those numbers. RULE.-Divide the difference of the squares by the difference of the numbers, and the quotient will be their sum; then to half the sum add half the difference for the greater number, and from half the sum take half the difference for the less. EXAMPLE. 1. The difference of two numbers is 20, and the difference of their squares is 1200; what are the numbers? Ans. 40 the greater, 20 the less. PROBLÈM XI-Having the sum and product of two numbers given; "to find those numbers. RULE.-Square half the sum of the two numbers, and from the product, subtract the product of the two numbers; then the square root of the remainder added to half the sum of the two numbers, will give the greater number, and subtracted from half the sum, will give the less.. EXAMPLE. 1. The sum of two numbers is 52, and their product is 612; what are those two numbers? Ans. 34 the greater, and 18 the less. PROBLEM XII-Having the sum of the squares of two numbers given, and also the difference of their squares; to find those numbers. RULE. From the sum of the squares, take the difference of the squares, and the square root of half the remainder, will be the less number; and half the remainder taken from the sum of the squares, leaves the square of the greater; the square root of which will be the greater. EXAMPLE. 1. The sum of the squares of two numbers is 544, and the difference of their squares 256; what are those numbers? Ans. 12 the less, and 20 the greater. PROBLEM XIII.-The sum of two numbers and the sum of their squares being given; to find those numbers. RULE.-Subtract the sum of their squares from the square of their sum; then subtract this remainder from the sum of their squares, and the square root of the difference will be the difference between the two numbers; then to half their sum, add half their difference, and the sum will be the greater number; and half their difference taken from half their sum, will give the less.. EXAMPLE. 1. The sum of two numbers is 70, and the sum of their squares is 2900; what are the two numbers? Ans. 50 and 20. PROBLEM XIV-The difference of two numbers, and the sum of their squares being given; to find those numbers. RULE-Subtract the square of their difference from the sum of their squares; add the sum of the squares to the remainder, and the square root of this last sum, will be the sum of the numbers required; then to half the sum of the two numbers, add half the difference for the greater number, and from half the sum subtract half the difference for the less. EXAMPLE. 1. A number of crowns are to be divided between A and B in such a manner that A may have 60 crowns more than B, and that the sum of the square of the respective shares may be 11600; what number must each have? Ans. A must have 100 crowns, and B 40. QUESTIONS ON INVOLUTION, EVOLUTION, AND THE SQUARE ROOT. What is Involution? A. It is finding the powers of numbers. What is a power? A. It is the product arising from multiplying any number into itself continually a certain number of times. What is the power arising from multiplying any number into itself? A. the second power or square. How is the third power or cube found? A. By multiplying any number by its square. How is the power generally distinguished? A: By an index or exponent, placed at the right, and a little above the number, and this index or exponent is always one more than the number of multiplications to produce the power. When a simple fraction is raised to a power, does it increase or diminish its value? A. It diminishes its value in the same ratio, that numbers above unity are increased. When the exact root of a power cannot be found, what is the number called? A. It is called a surd number.— When the exact root can be found, what is the number called? A. It is called a rational number. What is Evolution? A. It is extracting or finding the roots of any given powers. What is the square root of any number? A. It is a number which multiplied into itself, will produce the given number. Why is the given number pointed off into periods of two figures each, or a period placed over units and one over every second figure, counting to the left, to obtain the square root ? A. Because any one figure raised to the second power, can never excead two figures. After the given number is pointed off into periods, how do you preceed to extract the root? A. I find the greatest square in the left hand period, placing it directly below that period, and its root in the quotient; then subtract, and to the remainder bring down the next period, placing double the root at the left for a divisor, and seek how often it is contained in the dividend, except the right hand figure, placing the result in the quotient, and likewise at the right hand of the divisor to complete the divisor; then multiply the divisor by this quotient figure and subtract as before, and to the remainder bring down the next period, and again place double the quotient to the left hand for a divisor, or bring down the old divisor without al teration except doubling the right hand figure. Why should the quotient be doubled for a divisor? A. Because the root, at all stages of the work, shows the side of a square, and the remaining part of the dividend must be so disposed of, as to preserve the Figure in its square form, or that the root may express one of its sides, and doubling the root gives the length of the two sides, and then inquire how often this length is contained in all except the right hand figure of the dividend, and place the result at the right hand of the divisor to make out the entire length of the addition, and then the quotient figure expresses the breadth of the addition. How is the square root of a vulgar fraction found? A. By reducing the fraction to its lowest terms, and then extracting the root of the numerator for a new numerator, and the root of the denominator for a denominator of the root required; but if the fraction be a surd, it should be reduced to a decimal, and its root extracted. What is the use of the square root? A. It is of great use to mechanicks of different kinds, in calculating the length of timbers, such as rafters, braces, &c.; to surveyors in calculating area, &c. &c. EXTRACTION OF THE CUBE ROOT, It-the finding of such a number, as being multiplied into its square, will produce the given number. A Cube is a square solid, having six equal sides, and each of the sides an exact square. The Root of the cube is the length of one of the sides of the square solid; for the length, breadth, and thickness of a cube or square solid are all alike; consequently, the root or length of one side raised to the third power, gives the solid contents; for raising the root or one side of the cube to the third power, is multiplying the length, breadth, and thickness of such a figure together. Therefore it follows, that extracting the cube root of any number of feet, yards, or rods, is finding the length of one side of a cube or solid square, of which the whole contents will be equal to the given number of feet, yards, or rods, &c. EXAMPLES. 1. How many solid feet are there in a cubick block, each -side of which measures 4 feet? Ans. 43-4x4x4=64ft. 2. How many solid feet in a cubick block, each side measuring 3 feet? Ans. 27 feet. 3. How many feet in length is the side of a cubick block which contains 8 solid feet? Ans. 2 feet. DEM.It is plain, in finding the cube root of 8, we have only to think of such a number for a root as will, when raised to the third power, equal, or, at least, not exceed 8; and the root must always be the greatest number possible that will not exceed the given number when raised to the third power, and 8 is the third power or cube of 2, which may be seen in the table of powers, or by raising 2, the root or one side of the cube, to the third power, thus 2×2×2=8, the given number. 4. What is the side of a cube or solid square containing 64 solid feet? Ans. 4 feet. T |