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The critical student, who peruses this Section, will not find it stated, with sufficient explicitness, what it is, for which the symbols of Algebra immediately stand. Sometimes they appear to be spoken of as the representatives of quantities, and, at other times, as those of numbers. The determination of this point will be found to be of considera. ble importance, in examining the general relations of the different departments of mathematics. To us it appears manifest, that the symbols of Algebra are exclusively the representatives of numbers. Most of those algebraic characters which denote operations to be performed, deno e those operations which are exclusively appropriate to numb-r. Even when, in the application of Algebra to Geometry, letters are put for lines and spaces, they are never to be regarded as immediate representatives of those magnitudes; but of numbers proportional to them. The operations of involution, evolution, &c.-operations which belong exclusively to numberare resorted to in the reduction of equations, in this, no less than in the other departments of Algebra. This principle, if established, settles the limits within which the algebraic notation may be employed in demonstrating geometrical problems; and shows, that the characters denoting operations on quantities, when adopted in geometrical demonstrations, acquire a meaning in some respects different from that which they possess in pure Algebra.• The meaning of the sign of multiplication, for example, as used in the late editions of Euclid's second Book, is totally different from the algebraic signification of the same character.—To speak of algebraic characters, as denoting quantities, is common among mathematical writers; and, if properly understood, can lead to no mistake. But it ought always to be considered as an abbreviated way of saying, that they denote quantities through the medium of numbers.
Besides a general view of the algebraic notation, this Section contains a luminous exhibition of the uses of the negative sign. We cannot conceive the familiar illustrations given by our author of a subject so difficult to the beginner, to merit the censure of Simpson, that they derogate from the dignity of science.' It will be conceded by all, that, when a negative solution is obtained for a problem, if it has any meaning, it can be only in the application of it to objects having a real existence. Why, then, may not those existing objects be pointed out to the student, from which the symhols in question derive all their significancy? To expect him to form a correct notion of an abstract symbol, and yet to
keep from his view those particular cases by which it is exemplified, will appear preposterous to those who admit, that the natural progress of the human mind is from particulars to generals.*
There is one circumstance connected with this subject, which needs explanation; but which we have looked for in vain, either from Mr. Day, or from any other work which has fallen into our hands. Writers generally content themselves with stating, that the sign plus is prehxed to positive quantities, or those which are to be added, and the sign minus to negatives, or those which are to be subtracted. But the student who advances at all beyond the bounds of the simplest elements, will find nothing more common than expressions containing a series of quantities connected by the positive sign, and made equal to 0. This, indeed, is the form into which equations of all orders are usually thrown, in treating of the general theory of their reduction. To one who has derived all his knowledge of the use of signs from the common books of elements, we venture to say, that such an expression as this will be utterly unintelligible. If he is told, that though all the quantities are to be added, yet some of them are negative, the addition of which will be equivalent to an arithmetical subtraction, he will probably think it a sufficient reply, that no character denotes their being negative, and that the sign minus ought never to be omitted. The explanation which appears necessary is briefly the following. The sign plus is, in such cases, used for the ambiguous sign. In the solution of the equations, 23 + bx -C=0, and xu - bac +c=0, there can plainly be no other difference in the values of x than this; that the terms which contain any odd power of b, or c, alone, as a factor, will have a contrary sign in the first from the corresponding terms in the second. This being the case, a single formula may express both solutions, by prefixing to such terms the ambiguous sign. But if the sign plus be written throughout before such terms as have the ambiguous sign, and the sign minus before such as have
* We cannot avoid remarking here, the very different aspect which science presents, as treated by the best English writers, and by those on the Continent. The former, by pursuing a method rigidly synthetical, compel us, indeed, to admit the truth of their conclusions; but leave us to wonder how they came by them. The latter often take us as their companions in groping their way through the dusky regions of analysis. They show us the manner in which they use their tools; and are not ashamed even to acquaint us with their blunders and unsuccessful experiments. The former method is best calculated to inspire the learner with a profound reverence for the talents of the author; the latter, to give him confidence in his own talents.
the ambiguous sign inverted, it will be as easy (and indeed more convenient, in practice, as might be easily shown) to change the signs of all those terms which contain an odd power of a co-efficient, which, in a particular case, is negative, as to select from the two which compose the ambiguous sign, the one required by the supposed negative co-efficient. On this account, the minus of the ambiguous sign, particularly in the higher departments of analysis, is frequently omitted in the general investigation; and left to be substituted, whenever the peculiarities of an individual case, included in the general formula, may require it.
Passing over the four succeeding Sections, which contain, in nearly the usual form, the four simple rules for integral and fractional expressions, we come to the subject of simple equations. It is brought forward into an earlier part of the system, than has been done by must English writers; and, we think, with the utmost propriety; as it affords the student a respite from that long and tedious series of operations, with abstract symbols, which exclusively occupy the greater part of most elementary works; and furnishes him with examples of the real meaning and use of these symbols, at that period of his progress when he most needs them.-An equation is defined by our author to be a proposition expressing in algebraic characters, the equality between one quantity, or set of quantities, and another.' Mr. Stewart objects to this kind of definition; and proposes the following: 'A proposition asserting the equivalence of two expressions of the same quantity.' It is not our design to enter at large into the merits of these definitions; and we will only just remark, that the observations already made on the true import of algebraic symbols lead us to give the preference, in point of precision, to the latter. The former, however, (omitting the phrase "in algebraic characters') is the true definition of all mathematical equations except those of Arithmetic and of Algebra. The reason of this distinction, is, that two equal numbers are necessarily identical; or, in other words, are only different expressions for the same number; while two magnitudes may be equal, and yet in regard to place, and various other accidents, be far from being the same magnitude.
In the ninth Section, while treating of evolution and surds, our author just adverts to the subject of impossible quantities. Perhaps more could not have been said on this interesting topic, in a work which proiesses to be strictly elementary. We are anxious, however, to see it logically and sys
tematically examined. The principles generally assumed, in the management of impossible quantities, ought to be reduced to a system; and, if possible, established on demonstrative evidence. If this cannot be done, the nature of the evidence, which such investigations carry with them, ought, both in kind and in degree, to be precisely estimated. Till so much is effected, the use of imaginary expressions should be discarded from all science, which lays claim to demonstrative certainty. But we may apply to the most distinguished analysts, in reference to this subject, a remark somewhere made by Maclaurin, in reference to the analysis of infinites,-that mathematicians have been more anxious to push their discoveries, than to examine minutely the grounds on which they were proceeding. Several attempts have, indeed, been made to divest the logic of impossible quantities of the obscurity in which it is involved; particularly in the Philosophical Transactions of London, by Professor Playfair, Mr. Woodhouse, and M. Buee. The attempts of the latter, if we may confide in the statement given of them by Professor Playfair, in the Edinburgh Review, are nearly abortive; and we can scarcely regard the views of the Professor himself as entitled to any better character. His paper, which is referred to by Mr. Day, for farther information on this subject, contains many ingenious observations on the analogy of circular arcs to hyperbolic areas; but, considered as a statement of the evidence attending investigations with impossible quantities, it appears meagre and unsatisfactory. He is, in the first place, mistaken in supposing, that imaginary expressions are never used, except in investigations which concern circular arcs. The writings of Euler show that they have important applications to the indeterminate analysis. But to refer all the evidence attending the investigating of the properties of the circle, by means of imaginary symbols, to the analogy between these symbols and certain others free from imaginary expressions, which belong to the hyperbola appears to us to be annihilating it.
We will close these remarks with presenting, to our younger readers, a demonstration of a certain principle in reference to imaginary quantities, which they will often find taken for granted, * but which, so far as we know, has never yet been demonstrated. This is, that when real and impossible quantities occur in the same equation, those quantities which are real, and those which are impossible, on opposite sides of
* Euler's Alg. vol. ii. chap. 12. Vince's Flux. p. 20, &c. VOL. IX.
the equation, are respectively equal. All expressions involving imaginary terms, it has been demonstrated by D’A. lembert, may be reduced to the following form: A+ BV We have then only to show, that, in the equation A + B =a+6v-1, A = a, and consequently Bv1=bv-1, or B = b. Let the quantities a and B vi be transposed; and, by writing b-B=d, we shall have A-a=dvl. The second member of this equation is an impossible quantity; and, as the first is real, the equation is absurd in every case, except that in which Aa= 0, or A = a; and consequently B v1 = b V-1, or B = b. This is the principle, by means of which we obtain a very large part of the conclusions, derived from the introduction of imaginary expressions.
In delivering the doctrines of proportion, in Section twelfth, we have the same definition of ratio, as is given by Professor Adrain and others; who make it to consist in the quotient of the antecedent, divided by the consequent. This definition is perfectly adapted to the nature of Algebra, and leads to a much more concise and extended deduction of the properties of ratio and proportion, than that of Euclid. It would be unreasonable to infer from this, however, that the method of Euclid ought to be laid aside. The definition given above, as it necessarily involves fractions and surds, is applicable to magnitude only as denoted by number. The method of Euclid, on the contrary, avoids entirely the consideration of quotients; and considers magnitude as it exists independently o number; the only view of the subject, which, in our opinion, can consistently find a place in the elements of pure Geometry. The doctrine of ratio, as laid down and pursued by our author, supposes the practicability of multiplying one magnitude by another;—a supposition, which, when magnitude is considered independently of number, as in Geometry, is nothing less than a solecism. We have been led into these remarks, in consequence of the disposition, too common among late writers, to decry the ancient mode of treat. ing proportion, and to tarnish the character of pure Geometry, by blending it with the doctrines of number.
The succeeding Section, on Variation, will be found to supply an important chasm, which the student, who has used the common elementary books, must have found, in the transition from pure to mixed mathematics. The fundamental laws of general proportion are necessarily so often taken for granted, in Mechanical Philosophy, as well as in the higher