simple as that before us; at any rate, if an analogous process were applied to a more complicated case, it would lead to far more embarrassing statements, and the mind would have to exert a great stretch, in order to comprehend the whole, and get accurately through the calculation. We have said that the method of proceeding which has been adopted is the direct one; what we mean by this term is, that no artifice has been introduced, no previous reflection on what is probable, or what must follow from the nature or class of the problem, has been made use of. The solution is, in fact, a synthetic solution effected by reasoning from the data directly up to the thing sought. It may appear unnecessary to adopt such a solution, and consequently the thought may arise that difficulties have been introduced merely for the purpose of exhibiting them. Let us, then, have recourse to an artifice for obtaining the solution of the same problem, and it will still appear, although perhaps less markedly than in the former case, that the mind has to be burdened with operations tacitly performed on things not known, nor susceptible of representation. We may proceed as follows: A's money is four times B's. Now, whilst B's is increased by £15, let A's be increased in such a manner that it shall still remain four times B's. We must then increase A's money by £60. Now, the hypothesis is, that it is increased by £15 only; hence we have added £45 too much to it. But, in its present state, it is four times B's increased money, whereas, according to the hypothesis, it should be only three times the same: the additional £45 by which we have increased it is consequently just enough to make it contain B's increased money once more than it ought:—that is, £45 and B's increased money are one and the same thing. And since B's money has been increased by £15, it follows that B's original money was £30;-the same result as that obtained by the previous method. The difficulty which attaches itself to our previous solution is not so apparent in this; but a little attention will show that it is the simplicity of the example itself, and not that of the mode of solution, which prevents our feeling embarrassed by it. From the first, B's money has been singled out, and certain operations have been performed on it, and conclusions deduced from them; and although these operations are simply those of multiplication and addition, yet the mind has nevertheless to fix itself on B's money, as though it were a thing actually represented, and to work on it as on a quantity known and expressed. 3. The first step in Algebra is to relieve the mind from the burthen thus imposed on it, by representing the quantity sought by a symbol, and thus calling in the aid of the eye, whereby memory is almost if not altogether dispensed with, through the register that is kept of the state of the operations as they proceed; whilst all the other processes are effectually facilitated by the use of means previously in our possession, such as those of addition, subtraction, etc., in the form with which we are familiar. 4. The nature of the assistance which the mind receives from algebraic language, is perhaps hardly made clear by this example. It may be desirable to direct farther attention to it. Were a student required to demonstrate the forty-seventh Proposition of the First Book of Euclid's Elements, he would without hesitation make use of the proposition already proved, viz., that if a parallelogram and a triangle have the same base, and lie between the same parallels, the parallelogram is double of the triangle. Nor would it strike him that he had lost the spirit of the reasoning, though he had never paused to reflect on the evidence on which that proposition rests. On the contrary, he would deem it to be his duty to concentrate his attention on the application of that fact. The relation between the parallelogram and the triangle would be simply kept before the eyes as a means of advancing to something beyond. It is the same with the results of algebra. The effect of notation is to bring conclusions before the eye as they are arrived at, and thus to relieve the mind from a repetition of operations, and to confine it, at each separate step, to the arguments which that step demands. 5. That, therefore, which constitutes the transition from Arithmetic to Algebra is nothing more than this, that the number required is represented in a visible form to the eye. This form is usually one of the letters of the alphabet; in some instances the Greek and other characters are introduced. The readiest representatives of number for mental calculation are, indeed, the fingers. The adoption of these as tallies would give us pictorial algebra, bearing the same relation to our proper science that the pictorial Roman numerical notation does to the system we employ in our ordinary arithmetic. To exhibit this in the example given above. Let one finger or | represent B's money; then I will represent A's money, After adding £15 to each we find them to be £15 for B's money (15), and III £15 for A's (III 15). But we are told that in this state A's money is only three times B's; and the eye at once satisfies us that I must be twice £15, or £30. No process could be more satisfactory than this. But the student will find that if he applies it to more complicated problems, those for instance in which the unknown quantity has to be repeatedly divided, the pictorial method is cumbrous and unmanageable. He will certainly be satisfied with the common method of representing an unknown quantity by a letter. 6. Not only do algebraists represent the things sought by letters, but very frequently they use a like notation for the things given; the utility of this will appear when some progress has been made in the science. Most modern writers make use of the earlier letters, a, b, c, etc., to represent the things given, and the later letters, such as x, y, z, to represent those sought. The term magnitude is sometimes used to direct the mind to the immediate contemplation of the thing treated of; but as all algebraical operations depend on comparison or combination, it is not a convenient term, nor shall we very frequently adopt it, at least in that sense. The word quantity is employed as an extension of the word number: in such phrases as numerical quantity, its meaning is obvious; thus, one-half, though not a number, is a numerical quantity. We understand the word quantity, then, to be a term which has its rise and use in the adoption of an unit of measure. Thus, if the unit of measure be one foot, the quantity which stands for the length of a room, is a thing of which the mind has a distinct conception, although it may not be an exact number of feet or inches. The expression is also used with reference to the results of operations which are not so readily expressible as the above, as in the phrase, an impossible quantity. The letters made use of to represent numerical quantities are sometimes called SYMBOLS OF QUANTITY. 7. Abbreviations are introduced to save the trouble of a repetition of words, such as the phrases increased by, diminished by, etc. These abbreviations are sometimes called SYMBOLS OF OPERATION. 8. The symbol +, which is read plus, signifies increased by, so that, when two quantities or numbers are connected by this symbol placed between them, we are to understand that the second is to be added to the first. It is also termed the positive sign. When no sign appears before a quantity, understood. must be The symbol, which is read minus, signifies diminished by, so that the second of the two quantities which it connects must be subtracted from the first. It is also termed the negative sign. When the signs of two quantities are both or both - the quantities are said to have like signs; and when one is + and the other unlike signs. The sign is sometimes used to denote the difference between two quantities, when it is uncertain which is the greater thus, ab denotes the difference between a and b ; and is equivalent to ab if a is greater than b, and to b if a is less than b. a 9. The symbol ×, which is read multiplied by, denotes that the former of the two quantities between which it is placed is to be multiplied by the latter. This sign is generally omitted when the quantities are simple, such as single letters, or numbers combined with letters; thus, 4a is the same thing as 4 x a. Sometimes a point is placed between the quantities, thus, 4. a. A quantity which multiplies another is called the coefficient of it; and as the multiplier and multiplicand are convertible, either of them may be said to be the coefficient of the other, but the term is commonly applied to the numerical multiplier. The word factor is also used to denote either of them; thus, 4 and a are the factors of 4a. When no numerical coefficient appears before a quantity, 1 must be understood; thus, a and la are the same thing. The symbol, which is read divided by, denotes that the former of the two quantities which it connects is divided by the latter. This sign is not so often used as the others, its place being supplied by representing the division as a fraction, placing the former number above the line, the latter below: thus, instead of 43, we generally write. That this latter expression is equivalent to four-thirds, with which its form coincides, is evident from the definition of a fraction. = 10. The symbol of equality, equal to, denotes that the two things which it connects are equal to one another. = An expression in which quantities are connected by the symbol is called an equation: thus, 4+5=8+1 is an equation; x+7=2x-3 is an equation; and is read x plus 7 equals 2x minus 3. |