CHAPTER III. APPLICATION OF ALGEBRA TO GEOMETRY. Of the Geometrical construction of Algebraic Quantities. 63. As lines, surfaces, and solids are quantities, each admits of the operations which are performed upon numbers and algebraic quantities. But the results of such operations may be estimated in two principal ways, either by numbers or by lines. The first of these, as it supposes that each of the given quantities is expressed by numbers, is at present attended with no difficulty; it is only necessary to substitute in the place of the letters the numerical quantities which they represent, and to perform the operations indicated by the disposition of the signs and letters. As to the manner of estimating by lines the results of solutions furnished by algebra, it is founded upon the import of certain fundamental expressions, to which all others are afterwards referred. We proceed to make known these expressions, and to explain how the others are referred to them. This is called constructing the algebraic quantities, or the problems which have led to these quantities. 64. Let it be proposed to construct such a quantity as ab, in C which a, b, c, stand for known lines. We draw two indefinite Fig. 30. lines AZ, AX (fig. 30), making any angle with each other; upon one of these lines AX, we take a part AB, equal to the line represented by c, and a part AD, equal to one or the other of the two lines a and b, a, for example; then upon the second AZ, we take a part AC, equal to the line b. Having joined the extremities B, C, of the first and third by the line BC, we draw, through the extremity D of the second, the line DE parallel to BC; this will determine upon AZ the part AE as the value of ab. For the parallels DE, BC, give this proportion, AB: AD: AC: AE (Geom. 197), C In other words, it is necessary to find a fourth proportional to three given lines c, a, b; and, as we have given the method of finding this fourth proportional, we can employ it for the con ab struction of the quantity (Geom. 237). a2 C It will be seen therefore, that, if it were proposed to construct C it might be done in the same manner, since in this case the line b is equal to a. If it were proposed to construct ab+bd it is to be observ c + d ; regarding there ed, that this quantity is the same as fore a + d as one line, represented by m, and c + d also as one line, represented by n, we shall have which refers itself to the preceding case. Let the quantity to be constructed be collected that a a-bb is equivalent to (a + b) (ab,) (Alg. 34), may be represented under the form (a+b) (a—b) ; C and we have only to find a fourth proportional to c, a + b, a — b. If the quantity to be constructed be form; and, having constructed in the manner just explained, we call m the line given by this construction; then x - becomes which is constructed as above shown. We see, therefore, that in order to construct we represent b it under the form X ; we then construct a2 C represented the value of this by m, we construct Thus the whole art consists in decomposing the quantity into ab a2 portions, each of which returns to the form or ; and, al though this process may appear difficult in some cases, yet we easily arrive at the object proposed, by employing transforma constructed, after what has been said, when m and n are known. Now to determine m and n, the equations b3 = a2 m, c2 = a n, and n = which are constructed according to b3 give m = a the method already laid down. Thus, while the quantity is rational, that is, without radical expressions, if the dimensions of the numerator do not exceed those of the denominator except by unity, we may always reduce the construction to the finding of a fourth proportional to three given lines. - It sometimes happens, that quantities present themselves under a form, that seems to render recourse to transformations of no use; it is when the quantity is not homogeneous, that is, when each of the terms of the numerator and denominator is not composed of the same number of factors; when the quantity, for a3 + b example, is such as c2 + ď But it should be observed, that we never arrive at a result of this kind, except when, in the course of an investigation, we suppose, with a view of simplifying the calculation, some one of the a3 + b2 c quantities equal to unity. If, for example, in a2 + c2 I 9 sup But, as we never un dertake to construct a quantity without knowing the elements which we are to use for this construction, we always know in each case what is the quantity which is supposed equal to unity. We can always therefore restore it, and the above difficulty cannot occur; because, as the number of dimensions must be the same in each term of the numerator, and also of the denominator, although the number of terms may be different in the one from what it is in the other, we restore in each term a power of the line, which is taken for unity, sufficiently raised to complete the a3 + b + c2 number of dimensions; thus, if I have to construct a+b2 d being supposed to be the line which is taken for unity, I write. a3 + b d2 + c2 d which I should construct by making b2 = d m, ad + b2 c2 = dn, and a3d2 p, which would change it into ed, when we have constructed the value of m, n, and p; namely, Hitherto we have supposed that the number of factors, or the dimensions of each term of the numerator exceeds the number of factors, or the dimensions of the denominator only by unity. It may exceed it by two or even three, but never by more than three, unless some line has been supposed equal to unity, or some of the factors do not represent numbers. 65. When the dimensions of the numerator of the proposed quantity exceed by two the dimensions of the denominator, the quantity expressed is a surface, the construction of which can always be reduced to that of a parallelogram, and consequently to that of a square. If, for example, the quantity to be constructed be a3 + a2 b a + c a x a2 + ab Now a + c a2 + ab Let us suppose therefore that m is the value of the a2 + ab will become a X m. Now I should consider it as a X constructed, after what has been laid down, by considering it as a + b a + c line thus obtained; then a X if we make a the altitude and m the base of a parallelogram, we shall have a X m for the surface of this parallelogram (Geom. 174,) therefore, reciprocally, this surface will represent a × m, a3 + a2 b or may be reduced to a3 + b c2 + d3 In like manner, the quantity a+ c a3 +amc+and or a a2+mc+n +nd). a similar construction by making b c = am, and d2 = an; for it the factor Now itself to the preceding construc will then become parallelopiped. If, for example, I had to construct I should consider this quantity as the same as a b x a2 + ab a3 b + a2b2 a + c a2 + ab ; and, a + c I represent by m, the line given by this construction, the question a + c 67. What has been said will suffice for constructing any rational quantity; we proceed now to radical quantities of the second degree. In order to construct ab, it is necessary to draw an indefinite Fig. 31. line AB (fig. 31), upon which we take the part CA, equal to the line a, and the part BC, equal to the line b; upon the whole AB as a diameter, we describe a semicircle, cutting in D, the perpendicular CD, raised upon AB at the point C; then CD will be the value of ab; that is, the value of ab is obtained by finding a mean proportional between the two quantities represented by a, b. Indeed, we have or whence AC: CD:: CD: CB, a: CD:: CD: b; -2 CD = a b, or CD = vab. |