ELEMENTARY ALGEBRA, WITH BRIEF NOTICES OF ITS HISTORY. SECTION VI. INVOLUTION, EVOLUTION, BY ROBERT POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE, HON. LL.D. WILLIAM AND MARY COLLEGE, VA., U.S.. LONDON: INVOLUTION AND EVOLUTION, AND SURD NUMBERS. ART. I. The words Involution and Evolution are used in various senses, according to the subjects respecting which they are employed. In this part of mathematical science, the word Involution is employed to denote the process by which the powers of quantities are expressed or determined; and Evolution is the reverse process, by which the roots of quantities are expressed or determined. The word power is used to denote the product of two or more equal numbers or quantities; and the word root denotes the reverse of that denoted by the word power. The second power of any number or quantity is the product of two equal numbers or quantities; and conversely, the second root of any number or quantity is that number or quantity of which the second power will produce the given number or quantity. The second power of a number is also named the square* of that number, because the product of the two equal numbers which express the number of lineal units in the adjacent sides of a square, denotes the number of square units in the area of the square. In the same manner, the third power of any number or quantity is the product of three equal numbers or quantities; and the third root of any number or quantity is that number or quantity of which the third power is equal to the given number or quantity. The third power of any given number is also called the cube of that number, because the product of the three equal numbers which express the number of lineal units in the three adjacent edges of a cube, will denote the number of cubic units contained in the volume of the cube; and the third root of any given number is called the cube root, because if a given number denote the number of cubic units in the volume of a cube, the cube root of the given number will denote the number of lineal units in one of the edges of the cube. As the second and third powers of any quantity a have been assumed to be denoted by a2 and a3, so the second and third roots by a consistent analogy will be denoted by a1 and al. The words "square" and "cube" are the names of geometrical forms of a surface and a volume-modes of continued quantity, and cannot with strict propriety be applied to numbers. The proper analogous terms to the geometrical names square and cube are the second power and third power of numbers. But as the same words are applied both to the geometrical figures and the products of two and of three equal numbers, it is necessary that the distinction should be understood. 2. To find a rule for the formation of the square of any algebraical polynomial. Here (a+b)2= a2+2ab+b3, (a+b+c)2= a2+b2+c2+2ab+2ac+2bc, (a+b+c+d)2= a2+b2+c2+d2+2ab+2ac+2ad+2bc+2bd+2cd, (a+b+c+d+e)2 = a2+2a(b+c+d+e), +b2+2b(c+d+e), +c2+2c(d+e), +d2+2dc+e2, = a2+b2+c2+d2+e2+2ab+2ac+2ad+2ae+2bc+2bd +2be+2cd+2ce+2de. And so on, whatever may be the number of terms. Hence, it appears that the square of any polynomial consists of the sum of the squares of each term and twice the product of all the terms taken two and two. If there be n terms in the polynomial the rule may be thus expressed : Square the first term, and take the product of twice the first term and the sum of the remaining n-1 terms. Next square the second term, and take the product of twice the second term and the sum of the remaining n-2 terms, And so on to the square of the last two of the n terms. Since (a+b) a2+2ab+b2, it is obvious that 4x a2x b2= (+2ab)", or the square of every binomial consists of three terms, such that four times the product of the first and third is equal to the square of the second term. And, conversely, if four times the product of two terms of a trinomial be equal to the square of the remaining term, the trinomial is a complete square.* 3. Every positive quantity has two equal square roots, one positive and the other negative. Since the square of +a, or (+a)2 = +ax+a=+a2, and the square of -a, or (-a)2=—a ×—a=+a2. Hence, conversely, the square root of +a2, or {(+a)2}1 = (+a)? =+a, If in (a+b)2=a2+2ab+b2, 2a be written for a and 1 for b, then (2a+1)2=4a2+4a+1=4a(a+1)+1, which expresses the theorem. The square of the sum of any two consecutive numbers is equal to four times their product increased by unity. This theorem will be found useful in finding the squares of the odd natural numbers, by dividing the numbers into two parts whose difference is unity, as for instances : (19)2=4×10×9+1=360+1=361, (23)2=4.12.11+1=528+1=529, &c. And if in a − b2 = (a + b) (a−b), a+1 be put for a and a for b, then (a+1)2 - a2 = a2+2a+1−a2 = 2a+1, which expressed in words is, The difference of the squares of two consecutive numbers is equal to twice the smaller number increased by unity. and the square root of +a2, or {(-a)2 } * = ( —a)2 = —a, In the same manner, if there be more terms than one, and the square of b—a, or (b—a)2 = b2 — 2ab+b2. Hence, conversely, the square root of a2 - 2ab+b2 is a−b and b—a, or (a2-2ab+b2)* = ±(a—b). - It will be manifest from these conclusions that a negative quantity, as —a2, cannot have a square root. The square root, however, of a negative quantity admits of expression, but not of determination, the negative sign indicating the extraction impossible to be performed. The square root of a2 may be expressed in the forms (— a3)* and a(−1)*, or √ —a3 and a-1. The expression (—a2) has no intellegible meaning until it has been restored to a2 by the process of involution. When the data of a problem are incompatible with each other, the quæsita will always involve this expression; and, conversely, when this form of expression appears in the solution, the data of the problem are incompatible with each other. The square root of a negative quantity has been named an imaginary or impossible quantity, and when such an expression occurs in the solution of a problem, it indicates that the problem is impossible under the proposed conditions. As for instance if it be proposed to divide 12 into two parts such that their product shall be 40. If a denote one part, then 12-x will denote the other part, and x(12-x) = 40. The solution of this equation gives 6+/-4 and 6--4 for the two parts of 12; and these parts satisfy the equation, for their sum is 12, and their product is 40. But if the problem itself be considered, it will be seen that 12 can be divided six ways into two parts; namely, 1 and 11, 2 and 10, 3 and 9, 4 and 8, 5 and 7, 6 and 6; the respective products of these parts are 11, 20, 27, 32, 35, 36; and 36 is the greatest possible of all the products. It is therefore clear that 12 cannot be divided into two integral numbers whose product shall be 40, and this is indicated in the results obtained from the equation. It has also been judiciously remarked with respect to impossible expressions that "these symbols, although the parts of them are incapable of being numerically computed, yet, without the violation of perspicuity or logical accuracy, may be employed in calculation, since their origin and derivation, and consequently what they represent, are known.' In the second section of the first chapter of the Vija Ganita, the author states :—“The square of an affirmative or of a negative quantity is affirmative, and the root of an affirmative quantity is twofold, positive and negative. There is no square root of a negative quantity, for it is not a square." One of the commentators has given the following note on the text: "For, if it be maintained that a negative quantity may be a square, it must be shewn what it can be the square of. Now it |