What is the product of 41 3ii Oiii Oiv 5v by 4ii 3iii 5iv? What is the product of 4f. 3i 2ii Siii 9iv 5v 3vi by 31 Oii 5iii Oiv Ov 6vi 2vii? Ans. 1 0 11 5 6 8 2 0 1 1 2 2 4 6 When there is a great number of feet in the multiplier, the best way is by the rule of practice, as follows: Multiply the feet together, as in common multiplication; and for the odd inches, &c. take the aliquot parts. Thus, for li take of a foot, or lii take of li, &c. If the feet in both the multiplicand and multiplier be large numbers, Multiply the feet only into each other: then, for the inches and seconds in the multiplier, take parts of the multiplicand; and for the inches and seconds of the multiplicand, take aliquot parts of the feet only in the multiplier; and the sum of all will be the product. DEFINITION.-A power is a number produced by multiplying any given number continually by itself a certain number of times. Any number is called the first power of itself. When multiplied by itself, the product is called the second power, and sometimes the square; if this be multiplied by the first power again, the product is called the third power, and sometimes the cube; and if this be multiplied by the first power again, the product is called the fourth power, and so on; that is, the power is denominated from the number which exceeds the multiplications by 1. 3d power 2 3 4 5 6 7 8 9 81 4th power NOTATION. The number which exceeds the multiplications by 1, is called the index or exponent of the power; so the index of the first power is 1, that of the second power is 2, that of the third is 3, and so on. Powers are commonly denoted by writing their indices above the first power: so the second power of three is denoted thus, 32; the third power thus, 33; the fourth power thus, 3*; and so on also the sixth power of 503 thus, 503°. Involution is the finding of powers; to do which, from their definition, there evidently comes the following problem : PROBLEM IX To raise a given number to any given power required. 1. Multiply the given number, or first power, continually by itself, till the number of multiplications be 1 less than the index of the power to be found, and the last product will be the power required. 2. But because fractions are multiplied by taking the products of their numerators and of their denominators, they will be involved by raising each of their terms to the power required and if a mixed number be proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule. The square of 3% or 7 is 7 × = = 11 =11.56. EVOLUTION, OR EXTRACTION OF ROOTS. DEFINITION. The root of any given number or power is such a number, as being multiplied by itself a certain`number of times, will produce the power; and it is denominated the first, second, third, fourth, &c. root respectively, as the number of multiplications made of it to produce the given power is 0, 1, 2, 3, &c. that is, the name of the root is taken from the number which exceeds the multiplications by 1, like the name of the power in involution. 3 : NOTATION.-Roots are sometimes denoted by writing ✔before the power, with the index of the root against it so the third root of 50 is 50, and the second root of it is 50, the index 2 being omitted; which index is always understood when a root is named or written without one. But if the power be expressed by several numbers with the sign + or &c. between them, then a line is drawn from the top of the sign of the root or radical sign, over all the parts of it; so the third root of 47-15 is 47-15. And sometimes roots are designed like powers, with the reciprocal of the index of the root above the given number. So the square root of 3 is 32; the square root of 50 is 50, and the third or cube root of it is 50; also the third or cube root of 47-15 is 47—15. And this method of notation is justly preserved in the modern algebra; because such roots, being considered as fractional powers, need no other directions for any operations to be made with them than those of integral powers. A number is called a complete power of any kind, when its root of the same kind can be accurately extracted; but if not, the number is called an imperfect power, and its root a surd or irrational quantity. So 4 is a complete power of the second kind, its root being 2; but it is an imperfect power of the third kind, its third or cube root being a surd quantity, which cannot be accurately extracted. Evolution is the finding of the roots of numbers, either accurately or in decimals to any proposed degree of accuracy. The power is first to be prepared for extraction, or evolution, by dividing it by means of points or commas, from the place of units to the left hand in integers, and to the right in decimal fractions, in periods containing each as many places of figures as are denoted by the index of the root, if the power contain a complete number of such periods; that is, each period to have two figures for the square root, three for the cube root, four for the fourth root, and so on. And when the last period in decimals is not complete, ciphers are added to complete it. Note. The root will contain just as many places of figures as there are periods or points in the given power; and they will be integers, or decimals respectively, as the periods are so from which they are found, or to which they correspond; that is, there will be as many integers or decimal figures in the root as there are periods of integers or decimals in the given number. PROBLEM X. To extract the square root. 1. Having divided the given number into periods of two figures each, find, from the table of powers in page 88, or otherwise, a square number, either equal to, or the next less than, the first period, which subtract from it, and place the root of the square on the right of the given number, after the manner of a quotient in division, for the first figure of the root required. 2. To the remainder annex the second period for a dividend; |