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branches of pure mathematics, that it is a matter of surprise, that a place has not been more generally assigned to them, in the treatises of elementary writers.
The Section on Mathematical Infinity deserves the attentive perusal of all, who are inclined to doubt the conclusive. ness of those investigations which involve infinites and infini. tesimals. Such persons, if we are not mistaken, will find several distinctions laid down, and several principles of operation with infinites explained, which, if earlier attended to, would have saved them much perplexity. It has been a favourite object with many who are no friends to mathematical science, to bring into discredit its pretensions to superior clearness of evidence, by appealing to the supposed absurdities deducible from the doctrine of infinites. It must be admitted, that some writers have been either so negligent in the choice of language--so iond of the marvellous and profound, or so imperfectly acquainted with their subject, as to present the consequences oí this doctrine in a very paradoxical, if not in a very questionable point o view.
We have certainly no more partiality than Mr. Hume, for the language of those who speak of orders of lines as actually existing, each of which is infinitely smaller than the preceding. In our view, as little is added to the extent, as to the logic of mathematical science, by such representations. But, that two magnitudes may be supposed to increas beyond any assignable limits, while one of them is continually greater, in any fin te ratio, than the other, is one of the most natural and familiar ideas possible. Likewise, that those magnitudes which are in some respects infinite, and in others finite, may differ in any assignable ratio, in those respects in which each is finite, involves no difficulty. Thus two parallelopipeds may be conceived to be extended indefinitely in length. Now, if one were disposed to maintain, that, in point of length, both, being infinite, must necessarily be equal, yet in regard to their solidity, which universally varies as the length, breadth, and thickness, two of which are supposed finite, they may differ in any assignable ratio, we see no good reason why he should be hindred from so thinking; and, indeed, to maintain that their length was the same, would be virtually conceding that their solidity was as their breadth and thickness. Here then, at least, we have two magnitudes, each greater than any assignable one of the same kind, which an objector must, according to his own principles, admit to be capable of differing in any finite ratio. Had the different modes in which objects may be supposed to become
infinite, and the fact that they may be infinite in some re. spects, and finite in others, been kept clearly in view, many - useless disputes would have been saved to the world, on metaphysical, as well as on mathematical, subjects.
The position of the sixteenth, seventeenth and eighteenth Sections will probably be regarded as a deviation from that judicious arrangement, which is generally so conspicuous throughout this work. Neither of them has any connexion with the sub;ects of the six preceding Sections; although each is intimately connected with those which precede the latter. We cannot feel entirely reconciled, we confess, to seeing one branch of algebraic division thrown off nearly two hundred pages from the rest. Division by compound divisions must undoubtedly be deferred, till the nature of powers is explained; but we see nothing to forbid inserting it in immediate connexion with the division of powers in Section eighth. Having explained the division of simple quantities affected with exponents, and, in an earlier part of the work, that of simple quantities not thus affected, this seems to be the proper place for combining them both, and of applying the simple rules just laid down, to compound divisions, as our author has actually done, to compound multipliers.—This Section, wherever placed, should have contained the method of finding the greatest common measure of algebraic quantities; and hence of reducing fractions to their lowest terms;an operation which is contained in no part of the work. We presume that it was an oversight in the author, which he will remedy, in a future edition,
We are also of opinion, that the Binomial Theorem, and the evolution of compound quantities, would have been better placed in connexion with the involution and evolution of simple quantities, in the eighth and ninth Sections. In treating of the latter, all the elements are introduced by the combination of which the rules for the former are framed. This arrangement would have been attended with another advantage,--that of avoiding an anticipation of part of the subject of the seventeenth Section, in explaining the reduction of affected quadratics. We can think of no objection to this arrangement, unless it be in regard to the Binomial Theorem; the investigation of which is somewhat more general and difficult of apprehension, than the other subjects which come into so early a part of the course. But the lucid manner in which its successive steps are unfolded by Mr. Day, and especially the circumstance that he has attempted no de
monstration of it, will remove this objection, at least in regard to those who have mastered the subject of surds.
We were going to propose some alterations in the Section on Infinite Series; but, where so much more matter must be excluded, than can possibly be admitted, in a work of so limited an extent, we are sensible that two individuals can scarcely be expected to agree perfectly, in the selection of what appears to possess the greatest practical importance. We shall not, therefore, trouble our readers, nor the author, with any proposed improvements,-any farther than to suggest, that, as the method of indeterminate coefficients must come up to view sooner or later in the course, this seems to be the proper place for illustrating the general principle on which that method depends.
The twentieth Section, which is the last on subjects algebraical, briefly treats of the composition and resolution of the higher orders of equations. The author has judiciously omitted noticing most of the topics connected with this subject. No branch of analysis has been pushed into longer and more laborious details than this of the general structure of equations; and none, in our view, has less practical importance. Even the rule of Cardan would probably be laid aside, for the common methods of approximation, by every practical mathematician, who had an equation of the third order to solve, and whose only objects were accuracy and expedition. Those who aim at a thorough knowledge of analysis will, of course, read the works of such writers as Lagrange, Euler, Bezout, and Waring, on the general theory of equations; but these topics are of too speculative a nature to find a place in an elementary treatise.
Next follows an explanation, somewhat in detail, of the application of algebraic symbols, to geometrical reasoning, and geometrical problems. The subject of this Section is one of so great practical importance that we could not wish it to have been shorter. The substitution of the short hand of Algebra, for the prolix method of the ancients, in analyzing the objects of Geometry, is one of those rare improvements of modern times, for which the memory of Des Cartes will long be cherished by the friends of science. But the logical correctness of this application, it has been reserved for later writers to evince; and we have never seen it done in so miasterly a manner as by Professor Day. The illustrations of this subject are followed by a few of the simplest applications of analysis to the solution of geometrical problems. To these might have been added, with advantage, a list of
theorems and problems to be demonstrated and solved by the student. We know of no exercise, in the whole circle of mathematical science, more advantageous than the solution of problems, which require the united aid of Algebra and Geometry. Besides giving scope to the powers of inven. tion, and requiring the continual application and revisal of principles already acquired, they lead the mind to contemplate the relations existing between the different branches of science, as well as to the formation of those enlarged views, which distinguish the profound mathematician from the mere sciolist.—The concluding Section, containing a brief view of the equations of curves and of geometrical loci, derives its importance, in a work of this kind, from its intimate connexion with the quadrature and rectification of curves, and the cubature of solids, in the last department of the Course. Leaving the first Number (for our limits oblige us to exclude many observations which we had intended to make) we proceed to the second; which treats of Logarithms and Plane Trigonometry.
An intimate acquaintance with the nature and uses of Logarithms is so essential a preparative for most of the subsequent branches of mathematics,-particularly for Plane and Spherical Trigonometry, Surveying, Navigation, and Astronomy,—that no reasonable objection can be made to an ac. count of them, which might otherwise appear too minute for an elementary work. There are several subjects usually included in books of Algebra--such as Compound Interest (to which Annuities might have been added with advantage) and Exponential Equations, which our author has very properly deferred, till the use of logarithms has been explained. It would be desirable, as many will probably pursue his course to this Number, who will stop short of Fluxions, and as there are many modes oi computing logarithms, which are independent of this calculus, that one of the simplest had been selected, and made the subject of a note. Probably more will have a curiosity to learn the manner of computing logarithms, than that of the trigonometrical canon; which our author has admitted into his text, and which equally be. longs, naturally, to the higher Geometry.
Mr. Day rejects a common definition of Logarithms, which makes them a series of numbers in arithmetical progression, corresponding to another series in geometrical progression, on the ground, that it is not the logarithms, but the natural numbers, which are in arithmetical progression. If it were added to this definition, by way of explanation, that the ratio
of the geometrical series is considered as extremely near to unity, and that by far the greater part of each series, towards the beginning, is omitted, as not being wanted in practice, the obscurity, which our author anticipates, would be, in a great measure, removed. On the other hand, it appears to us, that the definition which he has given the exponents of a series of powers and roots'--will be regarded by the learner as obscure, unless accompanied by considerable explanation. The student, who has come directly from Algebra, will perceive very little resemblance between the long decimal fractions, which he finds in logarithmic tables, and what he has been before taught to consider as exponents. To ourselves, we recollect, this appeared much the greater paradox of the two. Mr. Day's explanations, indeed, render the subject sufficiently clear; but his definition, considered by itself, is liable to the additional objections of not indicating whether logarithms are the exponents of powers and roots of the same, or of different numbers, and of not noticing the fact, that, in truth, by far the greater part of logarithms denote neither powers nor roots, but the powers of roots. Were we to hazard a definition, we would say, that the logarithm of any number is the exponent denoting such a root of a power of a given number (termed the radix of the system) as is equal to that number;' a definition, which, if it has not the merit of conciseness, at least distinguishes logarithms from every thing else.
If any of the Sections in the Trigonometry would admit of being condensed, it is that on right angled triangles, (as has been already hinted,) and those on the use of the plane and Gunter's scales. Little use, we believe, is made of these scales, in working proportions; and, even in Surveying, they are now in a great measure laid aside, for the more accurate method of arithmetical computation. Had half a dozen pages been taken from these Sections and devoted to Trigonometrical Analysis, it would have given room for the intro duction of those general formulas for the sines, cosines, and tangents of multiple arcs, which seem too important to be passed unnoticed, even in an elementary work.
We observe, that, under the second theorem of Obliqueangled Trigonometry, our author has adopted, with a little variation, the elegant demonstration of Simpson, instead of the clumsy one of Webber. The formal demonstration of the third theorem might have been superceded by a simple reference to Euclid, l. 6.; which contains the same proportion in the form of an equation.