shall express in all three cases the number of days that the sportsman brought the game. In the first case c=0, in the second case we must take the positive sign, in the third case the negative sign. (47) If one of two numbers be multiplied by m, and the other by n, the sum of the products is p; but if the first be multiplied by m', and the second by n', the sum of the products is p'. Required the two numbers. (48) An ingot of metal which weighs n pounds loses p pounds when weighed in water. This ingot is itself composed of two other metals, which we may call M and M'; now n pounds of M loses q pounds when weighed in water, and ʼn pounds of M' loses r pounds when weighed in water. much of each metal does the original ingot contain? How Ans. n(r-p) pounds of M, n(p-q) pounds of M'. REMARKS UPON EQUATIONS OF THE FIRST DEGREE. 152. Algebraic formulæ can offer no distinct ideas to the mind unless they represent a succession of numerical operations which can be actually performed. Thus, the quantity b-a, when considered by itself alone, can only signify an absurdity when a>b. It will be proper for us, therefore, to review the preceding calculations, since they sometimes present this difficulty. Every equation of the first degree may be reduced to one which has all its signs positive, such as ax+b=cx+d. Subtracting cr+b from each member, we then have ax-cx-d-b. (1)* This being premised, three different cases present themselves; 1o. d>b and a >c. 2o. One of these conditions only may hold good. 3o. b>d and c>a. In the first case the value of x in equation (2) resolves the problem without giving rise to any embarrassment; in the second and third cases it does not, at first, appear what signification we ought to attach to the value of x; and it is this that we propose to examine. In the second case one of the subtractions, d-b, a-c, is impossible; for example, let b>d and a>c; it is manifest that the proposed equation (1) is absurd, since the two terms ar and b of the first member are respectively greater than the two terms cr and d of the second. Hence, when we encounter a difficulty of this nature, we may be assured that the proposed prob * We can always change the negative terms of an equation into positive ones by transposing them from the member in which they are found to the other member. lem is absurd, since the equation is merely a faithful expression of its conditions in algebraic language. In the third case we suppose b>d and c>a; here both subtractions are impossible; but let us observe that, in order to solve equation (1), we subtracted from each member the quantity cr+b, an operation manifestly impossible, since each member <cr+b. This calculation being erroneous, let us subtract ar+d from each member; we then have This value of x, when compared with equation (2), differs from it in this only, that the signs of both terms of the fraction have been changed, and the solution is no longer obscure. We perceive that, when we meet with this third case, it points out to us that, instead of transposing all the terms involving the unknown quantity to the first member of the equation, we ought to place them in the second; and that it is unnecessary, in order to correct this error, to recommence the calculation; it is sufficient to change the signs of both numerator and denominator. When the equation is absurd, as in the second case, we may nevertheless make use of the negative solution obtained in this case; for if we substitute -x for x, the proposed equation becomes a value equal to that in (2), but positive. If, then, we modify the question in such a manner as to agree with this new equation, this second problem, which will bear a marked resemblance to the first, will no longer be absurd, and, with the exception of the sign, will have the same solution. Let us take, for example, the following problem: A father, aged 42 years, has a son aged 12; in how many years will the age of the son be one fourth of that of the father? Let the number of years required. Thus the problem is absurd. But if we substitute -x for, the equation becomes and the conditions corresponding to this equation change the problem to the following: A father, aged 42 years, has a son aged 12; how many years have elapsed since the age of the son was one fourth of that of the father?* * As a problem is translated into algebraic language by means of an equation, so an equation may be translated back into a problem, provided the general nature of the problem be known. Take another example. What number of dollars is that, the sum of the third and fifth parts of which, diminished by 7, is equal to the original number? or Here Whence -7-x. x=-15. The problem is absurd; but, substituting -x for +x, and the problem should read, What number of dollars is that, the third and fifth parts of which, when increased by 7, give the original number? 153. With regard to the interpretation of negative results in the solution of problems, then, we may, from what is seen above, establish the following general principle : When we find a negative value for the unknown quantity in problems of the first degree, it points out an absurdity in the conditions of the problem proposed; provided the equation be a faithful representation of the problem, and of the true meaning of all the conditions. The value so obtained, neglecting its sign, may be considered as the answer to a problem which differs from the one proposed in this only, that certain quantities which were additive in the first have become subtractive in the second, and reciprocally. 154. The equation (2) presents still two varieties. If a=c, we have whence b=d; if, therefore, b be not equal to d, the problem would seem absurd.* But the expression d-b m -, or, in general, where m may be any quantity, 0' m represents a number infinitely great. For, if we take a fraction the small n m er we make n, the greater will the number represented by finity, which corresponds to n=0. Or, we may say, to prove infinite, that 0 * The absurdity is removed by considering that finite quantities have no effect when added to infinite ones; that, in comparison with infinities, finite quantities are all equal to one another, and all equal to zero. a finite quantity evidently contains an infinite number of zeros. for the value of r in this case is x=∞.* The symbol which it appears that the product of zero by infinity is finite. So, also, or the quotient of a finite quantity by infinity, is zero. 155. If, in equation (2), a=c, and b=d, we have x= 0 in this case the original equation becomes ax+b=ax+b. m :0, Here the two members of the equation are equal, whatever may be the value of x, which is altogether arbitrary, and may have any value at pleasure. We perceive, then, that a problem is indeterminate, and is susceptible of an infinite number of solutions, when the value of the unknown quantity appears 0 It is, however, highly important to observe, that the expression does not always indicate that the problem is indeterminate, but merely the existence of a factor common to both terms of the fraction, which factor becomes 0 under a particular hypothesis. Suppose, for example, that the solution of a problem is exhibited under the a3-b3 form x= If, in this formula, we make a=b, then x=ō• *This infinite value of expressions like may be sometimes positive, sometimes nega m 0 tive, and sometimes indifferently positive or negative. m 1o. Let there be the formula z in which m and n are two invariable numbers, '(n—~) 2' which we suppose positive, and different from zero, while z can have all possible values. Making z=n, we have x= But as the denominator, (n—z)22, is always positive, whatever z may be, the infinity here should be regarded as designating the positive infinity. 20. By analogous reasoning, we see that if we have the formula x= and z=n, we should have the negative infinity z=-0. m 30. Let there be the formula x=- The hypothesis z=n gives still x= but here the infinity will have an ambiguous sign. Suppose, at first, ≈<, and cause z to increase, the formula will give increasing values, which will be all positive. On the contrary, taking z>n, then diminishing ≈ till it becomes equal to a, the formula gives increasing values, which are negative. Therefore, the hypothesis z=n ought to be considered as causing the formula to take two infinite values, the one positive and the other negative. This is indicated by writing I= 1. The ∞ is here the transition value between+ and -. Zero is also a transition value between + and z>n, the value of a in changing from+tosign must always pass through ◊ or ∞. out changing sign, as in x=( =(n-2)2, and For, let x=n-z: if z<n, and ≈ increase till passes through 0. Quantities in changing They may, however, pass through 0 or ∞ with But we must remark, that a3-b3 may be put under the form (a—b) (a2+ab+b2), and that a2-b is equivalent to (a−b) (a+b); hence the above value of x will be (a−b)(a2+ab+b2) x= (a-b)(a+b) Now if, before making the hypothesis a=b, we suppress the common factor a-b, the value of x becomes an expression which, under the hypothesis that a=b, is reduced to 0 0' making a b, the value of r becomes x=, in consequence of the existence of the common factor a -b; but if, in the first instance, we suppress the common factor a-b, the value of r becomes an expression which, under the hypothesis that a=b, is reduced to 2a 0 0 From this it appears that the symbol in algebra sometimes indicates the existence of a factor common to the two terms of the fraction which is reduced to that form. Hence, before we can pronounce with certainty upon the true value of such a fraction, we must ascertain whether its terms involve a common factor. If none such be found to exist, then we conclude that the equation in question is really indeterminate. If a common factor be found to exist, we must suppress it, and then make anew the particular hypothesis. This will now give us the true value of the fraction, which may present itself under A A O one of the three forms B'0' 0' In the first case, the equation is determinate; in the second, it is impossible in finite numbers; in the third, it is indeterminate. 0 There are other forms of indetermination besides ō; for, whatever be the |