Now how are we to interpret the sign, which affects the insulated quantity 14? We understand its import, when there are two quantities separated from each other by the sign, and when the quantity to be subtracted is less than that from which it is to be taken; but how can we subtract a quantity when it is not connected with another in the member where it is found? To clear up this difficulty, it is best to go back to the equations, which express the conditions of the question; for the nearer we approach to the enunciation, the closer shall we bring together the circumstances which have given rise to the present uncertainty.

I resume the equation

12x + 7y = 46;

I put in the place of x its value 5, and it becomes

60+ 7y = 46.

This equation, by mere inspection, presents an absurdity. It is impossible to make the number 46 by adding any thing to the number 60, which exceeds it already.

I take also the second equation,

8x+5y=30,

and putting 5 in the place of x, I find

40+5y= 30;

the same absurdity as before, since the number 30 is to be formed by adding something to the number 40.

Now the quantities 12 x or 60 in the first equation, 8 x or 40 in the second, represent what the laborer earned by his own work; the quantities 7y and 5 y stand for the earnings of his wife and son, while the numbers 46 and 30 express the sum given as the common wages of the three; we must see then at once in what consists the absurdity.

According to the question, the laborer earned more by himself, than he did by the assistance of his wife and son; it is impossible then to consider what is allowed to the woman and son, as augmenting the pay of the laborer.

But if, instead of counting the allowance made to the two latter persons as positive, we regard it as a charge placed to the account

of the laborer, then it would be necessary to deduct it from his wages; and the equations would no longer involve a contradiction, as they would become

we deduce from the one as well as from the other

y = 2;

and we conclude from it, that if the laborer earned 5 francs per day, his wife and son were the occasion of an expense of 2 francs, which may otherwise be proved thus.

It is very clear then, that in order to render the proposed problem with the first conditions possible, instead of the enunciation in article 56., we must substitute this;

A laborer worked for a person 12 days, having had with him the first 7 days, his wife and son at a certain expense, and he received 46 francs; he worked afterwards 8 days, during 5 of which he had with him his wife and son at an expense as before, and he received 30 francs. It is required to find how much he earned per day, and what was the sum charged him per day on account of his wife and son.

Calling the daily wages of the laborer, and y the daily expense of wife and son, the equations of the problem will evidently be

and being resolved after the manner of those in art. 56., they will give

x = 5 francs, y = 2 francs.

59. In every case, where we find, for the value of the unknown quantity, a number affected with the sign, we can rectify the

enunciation in a manner analogous to the preceding, by examining, with care, what that quantity is, among those which are additive in the first equation, which ought to be subtractive in the second; but algebra supersedes the use of every inquiry of this kind, when we have learnt to make a proper use of expressions affected with the sign; for these expressions, being deduced from the equations of the problem, must satisfy those equations; that is to say, by subjecting them to the operations indicated in the equation, we ought to find for the first member a value equal to that of the second. Thus the expression drawn from the equations

7

14

must, consistently with the value of x 5, as deduced from these

same equations, verify them both.

=

The substitution of the value of x gives, in the first place,

60+ 7y = 46,
405y30.

It remains to make the substitution of 14 in the place of y;

7

and for this purpose we must multiply by 7 and by 5, having regard to the sign with which the numerator of the fraction is affected.

If we apply the rule relative to the signs given in art. 42. for division, we have

besides, by the rule for the signs in multiplication, we find

and are verified, not by adding the two parts of the first member, but in reality by subtracting the second from the first, as was done above, after considering the proper import of the equations.

60. The problem in art. 58. does not admit of a solution in the sense in which it is first enunciated; that is to say, by addition, or regarding as an accession the sum considered with

reference to the wife and son of the laborer; neither does the second enunciation consist with the data of the problem in art. 56. If we were to consider in this case y, as expressing a deduction, the equations thus obtained

and the substitution of the value of x would immediately change the equations to

The absurdity of these results is precisely contrary to that of the results in art. 58., since it relates to remainders greater than the numbers 60 and 40, from which the quantities 7 y and 5 y are to be subtracted.

The sign minus, which belongs to the expression of y, implies an absurdity; but this is not all, it does it away also; for, according to the rule for the signs,

and are verified by addition; consequently, the quantities and 5y, transformed into +14, +10, instead of expressing expenses incurred by the laborer, are regarded as a real gain. We are brought back then in this case, also to the true enunciation of the question.

61. We perceive by the preceding examples, that there may be, in the enunciations of a problem of the first degree, certain contradictions, which algebra not only makes known, but points out also how they may be reconciled, by rendering subtractive certain quantities which had been regarded as additive, or additive certain quantities which had been regarded as subtractive, or by giving to the unknown quantities values affected with the sign

See then what is to be understood, when we speak of values affected by the sign —, and of what are called negative solutions, resolving, in a sense opposite to the enunciation, the question in which they occur.

It follows from this, that we may regard, as but one single question, those, the enunciations of which are connected together in such a manner, that the solutions, which satisfy one of the enunciations, will, by a mere change of sign, satisfy the other also.

62. Since negative quantities resolve in a certain sense the problems, which give rise to them, it is proper to inquire a little more particularly into the use of these quantities, and to settle once for all the manner of performing operations in which they are concerned.

We have already made use of the rule for the signs, which had been previously determined for each of the fundamental operations; but the rules have not been demonstrated with reference to insulated quantities. In the case of subtraction, for example, we suppose that there was to be taken from a, the expression b — c, in which the negative quantity c was preceded by a positive quantity b. Strictly speaking, the reasoning does not depend upon the value of b; it would still apply when b = 0, which reduces the expression bc to c. But the theory of negative quantities being at the same time one of the most important and most difficult in algebra, it should be established upon a sure basis. To effect this, it is necessary to go back to the origin of negative quantities.

The greatest subtraction, that can be made from a quantity, is to take away the quantity itself, and in this case we have zero for a remainder; thus a a = 0. But when the quantity to be subtracted exceeds that from which it is to be taken, we cannot subtract it entirely; we can only make a reduction of the quantity to be subtracted, equal to the quantity from which it was to be taken. When, for example, it is required to subtract 5 from 3, or when we have the quantity 3-5; to take, in the first place, 3 from 5, we decompose 5 into two parts 3 and 2, the successive subtraction of which will amount to that of 5, and thus, instead of 35, we have the equivalent expression 3-3-2, which is reduced to 2. The sign, which precedes 2, shows what is necessary to complete the subtraction; so that, if we had added 2 to the first of the quantities, we should have had 3 + 2 — 5, or