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several kinds of mathematical knowledge connected with the subject. The information afforded will be found very valuable; and it is usually exhibited with great neatness and perspicuity. We find it, however, our duty to complain, that the work is not altogether what we could wish to have seen it. One great object of the author in composing the treatise, is lost it does not stand alone. References are frequently made to other books, for the supply of important steps. in the investigation; and these references, it would seem, are sometimes intended to display the author's extent of reading. His manner also, though neat, is too diffuse: had he aimed at compression, instead of an exhibition of acquirements, he would undoubtedly have produced a treatise, equally useful and equally perspicuous, in half the compass. He often dilates, where conciseness would be highly commendable; and sometimes, though less frequently, he passes slightly over topics, which, either from their real moment, or the importance formerly attached to them, demanded a more ample notice. Under the head Equation of Payments, for example, there is no mention of the controversy carried on at different times by Hutton, Burrow, Todd, Keith, and others, relative to the proper method of treating this rule: and, what is more extraordinary, Mr. Baily takes no notice of the inutility of the double sign (±) given in the algebraical formula for the equated time of two payments. His thea±√a2-4b;
orem for the equated time is of the form x= 2 1 the double sign being retained. He will find it proved, in a little book on Arithmetic, published more than twenty years ago by Mr. Bonny castle, that "the double sign made use of by Mr. Malcolm, and every author since, who has given his method, cannot obtain; and that there is no ambiguity in the problem."
To convince Mr. B. that our censure of his manner of reference to other authors is not the result of an ungenerous wish to depreciate an ingenious work, but proceeds from a desire to render it more worthy of public approbation, we beg him to reconsider the references in his notes at pp. 15, 40, 61, 140. We would ask him, how far it was necessary to direct his readers to four tables of logarithms would not Hutton's or Callet's do alone? but must all the old book shops in London be hunted over to find Sherwin's and Gardiner's? Again, (p. 61) after being referred to the common method of summing an infinite geometrical series', we are told that the sum of any infinite geometrical series decreasing is equal to the square of the first term divided by the difference be
tween the first and second,' and for a proof of this we are directed to Bernoulli de Seriebus infinitis, Cor. to prop. 8.' Now, when Malcolm and half the writers on Arithmetic, and more than half the writers on Algebra, exhibit this proposition either explicitly or by implication, where was the need of this reference? It could not be to convince the plain English accountant that Mr. B. had read a scarce Latin work on Infinite Series; because it would still remain a matter of doubt, though quite as important for the reader to know, whether Bernoulli's Works are in the library of our author, or in the library of the London Institution.—Önce
For the construction, &c. of the logarithmic curve, the reader may consult Robertson's Geo. Treat. on the Conic Sec. 8vo, 1802. Keill's Tract on Logarithms at the end of his edition of Euclid. Euler's Introd. in Anal. Inf. vol. ii. La Croix Calcul. Diff. et Integ. Emerson on Curve Lines: or Hugenii Opera Reliqua, vol. ii. in which latter work, the principal properties of this curve are pointed out.' p. 140.
Where is the utility of all these references, or where indeed was the necessity for any? All that Robertson says about the logarithmic curves may be comprized in a page; and this would have been amply sufficient for Mr. B.'s purpose when incorporated with what he has given himself. But we have probably said more than enough, to check the indulgence of this propensity in the treatise Mr. Baily promises on Life Annuities and Assurances. We will now make an extract of a different nature; an extract which, though it is founded on a remark made before by Woodhouse, La Croix, and others, is of sufficient importance to deserve a place here.
The terms of every art or science should be clear, definite, and explicit, and though they may not always be sufficiently precise, yet they should never tend to convey any false ideas on the subject. By using the term Hyperbolic logarithm, an idea is immediately entertained that this is the only system of logarithms that can be expressed by the hyperbola: whereas, not only the common system but every other system whatever may be expressed by means of that curve; and the only dif ference is that in the former, or hyperbolic logarithm, the asymptotes of the curve, are at right angles to each other; but in the latter or common logarithm, they form an angle of 25°. 44. 25". These are generally called Briggs's logarithms, after the name of their inventor; and the former, for the same reason, I have here called the Neperean logarithm,' p. 14..
Although there are several parts of the treatise before us which we have read with satisfaction; yet we confess we were most pleased with the note (E) in the Appendix, where our author explains the application of the Logarithmic curve to the doctrine of Interest and Annuities; because it shews. the advantages that result from giving scope to the powers of Vol. V.
imagination in mathematical pursuits. Some of our readers, no doubt, will be startled at our associating 'powers of imagination' with mathematical pursuits; having settled it in their minds as a sort of axiom, that to indulge in the latter is to ruin the former. On the same principle, differently applied, certain mathematical tutors and authors, of a new sect, have affirmed that science must not be degraded by metaphor.' On a former occasion*, we explained the reason of the mathematician's acquiescence in some of his deductions as certainty, though made from mysterious processes, while we noticed several remarkable instances in which mathematical reasoning is analogous, in its nature and results, to the reasoning employed on the most important religious topics. We will now endeavour to shew that mathematical inquiries, so far from being unfriendly to the play of imagination or the indulgence of fiction, frequently call in their assistance; and, instead of being degraded by metaphor, are often nothing else than a continued metaphor.
The province of imagination, as it is explained by our best metaphysicians, is to make a selection of qualities and of circumstances from a variety of different objects, and, by combining and disposing these, to form a new creation of its own.' Now, this describes accurately what is effected every day in the process of mathematical investigation. Instances of this mental magic will occur to every reader of competent information, in the application of algebra to geometry, in transferring the principles of motion to the ideal or fictious generation of surfaces and solids, whether of rotation, of translation, or of expansion, in the whole theory of fluxions and all its applications, in the ap propriation of pure analysis to the doctrine of chances, in the geometry of curves, in the application of that doctrine to political arithmetic, the duration of life, &c., in the appropriation of analysis to trigonometry, and of these conjointly to physical astronomy. Mathematical invention, in these and all its other varieties, is, in truth, the fruit of imagination and every new solution is, strictly speaking, a distinct invention. Mathematicians have in this way more than half created the wondrous world they see and their ideal creations are distinguished from most others in this, that they can at once be applied to realities, and turned to purposes of obvious and striking use. Let us take the example furnished by Mr. Baily, to shew in what manner these 'selections are made' and new creations formed.' From the various properties of arithmetical and geometrical progressions, this was selected; that if any arithmetical and geometrical progrés
* See Review of Bonnycastle's Trigonometry, Vol. IV. pp 52-59.
sion were arranged in parallel order, term by term, the several terms of the former would serve as exponents of the corresponding terms of the latter, in such manner, that, by the mere addition and subtraction of the former, what might be accom plished by the multiplication and division of the latter should not merely be indicated but ascertained and the same analogy was found to subsist between the multiplication and division of the arithmetical terms, and the involution and evolution of the geometrical terms:-from this selection, moulded by a rich imagination, arose logarithms. A farther selection. of one universal property of logarithms, and a grouping of this with another property selected from the doctrine of equations, led to the general formula, y = bax, from which new creation,' according to the manner of the algebraists, all the particular properties of logarithms could be made to flow with perfect ease and simplicity. It was another distinct effort of a fertile imagination, to draw together the remote analogies of equations and of curves, and make such a selection as should define curves by means of equations, and thus render them mutually illustrative and determinative of each other's properties: it was a farther effort still, to select from the multifarious properties of curve lines those which depicted, if we may so say, the nature of logarithms, and thus make a portrait of those remarkable numbers: and it was another effort of a like kind, a selection' terminating in a new creation,' that singled out, from the attributes of annuities and reversions (apparently so impregnable against the attacks of imagination), some so strictly connected with the nature of the logarithmic curve, as to make the latter a perfect representative of the former.
Poetry is doubtless the child of imagination: yet how has poetry been defined? The father of criticism has denominated poetry sexyn ubunioxn an imitative art ; while Baron Bielfield defines it as the art of expressing our thoughts by fiction. Neither of these definitions, is, in our estimation, sufficiently comprehensive; though Aristotle's was admitted for ages, and Bielfield's is often cited as the most appropriate that has yet been given. We mention them, merely to shew that great critics, in their definitions of that fine offspring of imagination, have selected qualities, that equally distinguish what in the opinion of many requires not the aid of imagination at all. For is not mathematics an imitative art?' an art by which we form to ourselves things not in being, exhibit things absent, and represent things past ?'* And that this department of science deals in fiction, every
one knows; with this remarkable advantage, that the fictions of the mathematicians are made to contribute to the discovery of truth. But we shall be told that the fictions of the poets tend to illustrate the effects of realities: true; and we enjoy the delights which this quality of poetry is adapted to impart. None can admire more warmly than ourselves, the admirable selection' and new creation' of the great dramatic poet, when he shews the power of gold, and the mischievous consequences of slander, in the passages set at the foot of our page*: or be more forcibly struck with the powers of Hogarth, when by a new creation' he depicts the miserable consequences of dram-drinking, in his singular picture called Gin: and we at once acknowledge these, as displaying considerable powers of imagination. All that we are now asserting is, that efforts of imagination, equally strong, lively, and illustrative of effects and consequences, result from the exercise of investigation among mathematicians: and we appeal to Mr. Baily's figure 4, where he exhibits to the eye, by means of logarithmic curves, the different accumulative powers of money lent out at 2, 5, and 10 per cent, and indeed all the leading theorems of interest and annuities, as a full confirmation of our position.
As we have never before indulged in a disquisition of this kind, we shall the more readily be favoured with the attention of our readers, while we now shew, as briefly as possible, that mathematical science is not degraded by metaphor.' What is a metaphor? An act of the imagination figuring one thing by another: thus, by a metaphor human life is figured to be a voyage at sea, There is a tide in the affairs of men,' &c. By a metaphor, the qualities of a conqueror are figured by those of a lion; and one of the
*Gold. "O thou sweet king-killer, and dear divorce
'Twixt natural son and sire! thou bright defiler
And mak'st them kiss! that speak'st with every tongue,
Whose edge is sharper than the sword; whose tongue
Rides on the posting winds, and doth belie