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MODERN MATHEMATICS ALGEBRA 27 every other class of objects to which they are legitimately applicable. Perceiving farther that, in order to understand these relations, I should sometimes have to consider them one by one, and sometimes only to bear in mind, or embrace them in the aggregate, I thought that, in order the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple, or capable of being more distinctly represented to my imagina- tion and senses ; and on the other hand, that in order to retain them in the memory, or embrace an aggregate of many, I should express them by certain characters the briefest possible. In this way I believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other." Let us, then, consider the characteristics of algebra and geometry. We have seen, when giving an account, in the first chapter, of the works of Ahmes and Diophantus, that mathematicians early saw the advantage of representing an unknown number by a letter or some other sign that may denote various numbers ambiguously, writing down much as in geometrical analysis the relations which they bear, by the conditions of the problem, to other numbers, and then considering these relations. If the problem is determinate that is to say, if there are one or more definite solutions which can be proved to involve only numbers already fixed upon this consideration leads, by the use of certain rules of calculation, to the determination actual or approximate of this solution or solutions. Under certain circumstances, even if there is a solution, depending on a vari- able, we can find it and express it in a quite general way, by rules, but that need not occupy us here. Thus, suppose that you know my age, but that I do not know yours but wish to. You might say to me : "I was eight years old when you were born." Then I should think like this. Let x be the (unknown) number of years in your age at this moment and, say, 33 the number of years in my age at this moment ; then in essentials your statement can be translated by the equation "a:-8 = 33." The meaning of the signs "-,"" = ," and " + " are supposed to be known ; as indeed they are by most people
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