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numerical solution can never be demonstrated, by the principles of Algebra: And hence the fundamental operations of Arithmetic, if susceptible of demonstration at all, must be established on principles peculiar to the nature of numbers, and derived from the established scheme of notation. We cannot avoid thinking, that the logic of Mr. Day's whole system is somewhat impaired by the want of such demonstrations as could have found a place only in an introductory treatise of Arithmetic. The correctness of the common arithmetical operations with integers, fractions, and surds, he often takes for granted; but he has neither demonstrated those operations himself, --nor referred the student, for such a demonstration, to any other author.
For the two reasons implied in the foregoing remarks, we think, that the utility of the work would have been enhanced, if it had been introduced by a brief exhibition of the doctrines of number. Most of those details which fit a system of Arithmetic for the use of the artificer, or the merchant, might have been omitted; while the fundamental operations, both with numbers of the same and of different denominations, the theory of fractions, of ratio and proportion, of powers and roots, and of progressions and combinations, should have been presented to the learner, at least in the text, in the form of pure science.
The general Introduction, which contains a concise statement of the objects of mathematical science, of its great divisions, and of its practical applications, is sketched with the hand of a master, and will be read with interest, even by the advanced student. We are not confident, however, that the definition of quantity, which is intended to include all the objects of mathematical investigation, will be regarded as sufficiently precise. Any thing which can be increased or diminished, or which is capable
of being measured,' says our author, 'is called quantity. The former part of this definition appears to include too much; and the latter too little. Many things are capable, in the strictest sense of the modifications, of increase and diminution; such, for example, as pleasure and pain; which yet no one ever thought of ranking among the objects of mathematical inquiry. Although it is remarked, with truth, in illustrating the foregoing definition, that one colour cannot, with any propriety, be said to be greater or less than another;' yet it is because different colours are heterogeneous. The same colour, however, admits of different degrees of intensity, and is, therefore, susceptible of increase and diminution. It is not easy, we confess,
to draw the precise line where the objects of mathematical investigation cease. One body may be said, with propriety, to be more luminous than another; but shall luminousness be therefore considered as a quantity? Heat, considered, not as a substance, but as an agent, is capable of different degrees, and can even be measured with considerable exactness; but it may still be doubtful whether it ought to be ranked among mathematical quantities. On the other hand, the latter part of the definition seems to exclude number from the list of quantities; which, far from being capable of being measured,' appears to us to be, in the language of Mr. Locke, itself that which measures all measurables.' What do we mean, when we speak of any object as capable of being measured, but that it contains some other of the same kind a certain number of times, either exactly, or with a definite excess? To apply this definition to number itself would be absurd.
We find a difficulty, we own, in uniting number in a common definition with the other objects of scientific investigation. It is an abstract notion derived from those other objects, and alike common to them all. The latter would be most correctly defined, by considering them as susceptible, not only of the relations of greater and less, but of multiple and part. * The relations of greater and less are predicable of pain, colour, beauty, &c.; but we cannot conceive the possibility of applying to either a measure which shall determine it in one instance to be double or triple of what it is in another. To constitute our idea of mathematical quantity, however, it is not necessary that the object itself should be
* Since writing the above, we have unexpectedly found an almost ver. bal coincidence with these views, in a paper of Dr. Reid, which is published in the Philosophical Transactions for 1748. · The object of this science,' he observes, • is commonly said to be quantity; in which case, quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which has led some persons to apply mathematical reasoning to subjects that do not admit of it.' To this quotation Mr. Stewart subjoins: “The appropriate objects of this science are therefore such things alone as admit not only of being increased and diminished, but of being multiplied and divided.' În exact accordance with the views expressed above, he remarks of number, in the same note, that it might be easily shown that it does not fall under the definition of quantity, in any sense of that word.' Philosophy of the Human Mind, vol. ü, p. 489, 491, Edin. Ed. 418–20. Phil. Ed.
† We mean, in regard to the degree in which these modes exist at any assigned moment; for, in one respect, pleasure and pain, at least, have a singular analogy to some species of mathematical quantity. If a certain pain be supposed to continue uniform, it is as evident that the aggregate suffered varies as the time of its continuance, as that time is the measure of the distance described by uniform motion.
directly conceived capable of exact multiplication. We sometimes conceive of it through the medium of its attendants or effects; and it is then only necessary that the measure by which we represent it, should possess this property. Thus heat is measured by the expansion of mercury; accelerating force, by velocity and time jointly; velocity itself, by space and time; specific gravity, by weight and bulk. It may be thought that this view of the subject will indefinitely multiply the objects of mathematical investigation; since there is scarcely an object in nature which may not have some quantity assumed for its measure. Some of the best ethical writers, such as Wollaston, Hutcheson, &c. have carried it so far as even to apply numerical reasonings to the affections and operations of the mind. We have already aknowledged the difficulty of assigning the exact number of quantities, in the mathematical sense; but it is easy to limit them to a moderate number, and, at least, to exclude the objects of morals and metaphysics. Whatever comes under the denomination of quantity ought not only to be conceived capable, in theory, of exact measurement; but of being actually measured, with such a degree of accuracy, that the conclusions mathematically deduced from the assumed measure may be practically correct. It may, indeed, serve to fix our ideas, to consider, for example, the force of temptation, or the intensity of pain, as proportional to numbers; and the conclusions derived from such suppositions by the common arithmetical operations, will be hypothetically correct. But the hypotheses from which these conclusions are derived, are applicable in so vague a manner, at best, to the affections of mind, that they have little practical utility; and no one has ever thought of ranking these affections among mathematical quantities.-On the other hand, those collective objects in which the number of individuals is the only circumstance regarded, (abstracting from the magnitude and other peculiarities of each,) are susceptible of a perfect measurement. Numerical computations, concerning such collectives as an army, or a sum of money, may be mathematically exact, even in practice. In all other cases, as the nature of the quantity requires each unit to be in some respect equal to every other, although we conceive it in its own nature capable of exact measurements,—yet practice does not, as in the former case, perfectly coincide with theory. It is beyond human power, for example, to draw a line, which shall be exactly ten miles in length; or to describe a triangle, which shall exactly coincide with the triangles of Geometry. But lines may be measured in a manner so nearly exact, and figures may be found among material objects, or may be described, approaching so near to those of Geometry, that, in the former case, the numerical computations derived from the supposition that the measurement is perfect, and, in the latter, the application of the properties previously established concerning the figures of pure Geometry, is attended with no sensible error. So the expansion of mercury is not exactly proportioned to the increase of heat; and is, indeed, much farther from being an accurate measure, than those which have been just mentioned; yet it is so nearly the case, that the conclusions derived from the supposition of its being a perfect standard, are by no means destitute of practical utility. But should the measure assumed for any object, be applicable to it in so loose a manner, that a system of mathematical conclusions derived from it would have little practical use, such an object ought to be excluded from the number of mathematical quantities. Such, for example, in the present state of science, are magnetism, friction, and the resistance of fluids. But it is natural to conclude, that the number of quantities will increase with the advancing state of human knowledge. There is nothing incredible in the supposition, that accurate measures may be hereafter found for those qualities and operations of things around us, which are now regarded as incapable of measurement. It was not till the discoveries of Newton, let it be remembered, that gravitation could be regarded as a mathematical quantity.
We question the correctness of that part of our author's classification, which assigns to Fluxions a place among the higher branches of Algebra. Fluxionary processes, it is true, require the aid of Algebra; as those of Algebra often require the aid of Arithmetic and Geometry. The first principles of Fluxions appear, nevertheless, to be as distinct from those of Algebra, as the first principles of Algebra are from those of Ariihmetic. The latter, indeed, can scarcely be said to differ at all, except in regard to their universality. Fluxions might, we think, be with equal propriety ranked under the head of Geometry; and this classification would doubtless be preferred, by that large class of scientific writers who have treated of Algebra under the title of universal Arithmetic, and of Fluxions, under that of the Geometry of infinites. Although the algebraic notation may be conveniently employed in the fluxional calculus,—as it is by the best modern writers in some parts of pure Geometry; yet the principles of this calculus may be exhibited in a manner much
more analogous to Geometry than to Algebra. This method Maclaurin has actually adopted; and it appears to be the mode of investigation most appropriate to the nature of the subject. The first principles of Fluxions belong primarily to geometrical magnitude, which is considered as produced by motion; and only secondarily to abstract number, which may be considered as varying proportionally to variable magnitudes. Then at least are the views taken of the subject by the English writers, in accordance with those of the great inventor; and if some later writers on the Continent* have found means of arriving at the same results, without reference to motion, or even to infinitesimals, it can have no effect on the place which Fluxions, properly so termed, ought to hold, among the subjects of mathematical inquiry.
We cannot deny ourselves the pleasure of quoting our author's reply to an objection against the utility of scientific studies, which may have had weight in the minds of many of our readers. After enumerating the various practical applications of science to the arts of common life, he observes:
• It is true that in many of the branches which have been mentioned, the ordinary business is frequently transacted, and the mechanical operations performed, by persons who have not been regularly instructed in a course of mathematics. Machines are framed, lands are surveyed, and ships are steered, by men who have never thoroughly investigated the principles, which lie at the foundation of their respective arts. The reason of this is, that the methods of proceeding, in their several occupations, have been pointed out to them, by the genius and labour of others. The mechanic often works by rules, which men of science have provided for his use, and of which he knows nothing more than the practical application. The mariner calculates his longitude by tables, for which he is indebted to mathematicians and astronomers of no ordinary attainments. In this manner, even the abstruse parts of the mathematics are made to contribute their aid, to the common arts of life.' p. 7.
On the definition of Algebra, at the head of the first Section, which makes it a general method of investigating the relations of quantities, principally by letters,' we have nothing to remark, after the apology which the author makes for its necessary imperfection, except that it seems to make the use of letters essential to Algebra. It is only in the character of symbols that letters are of any consequence in Algebra: any other symbols whatever, if generally agreed on, would equally answer the purpose. In this particular, there seems to be no occasion for any deviation from the language of the definition usually given.
* Landen, Legrange, &c.