A question has arisen, whether impossible expressions ought to be employed in a science, that above all others boasts of the accuracy of its reasoning, and the certainty of its conclusions ? The question ought certainly to be an swered in the negative, if in any case the theory led to er ror, or if it were of more difficult application than the ana lytic theories, which involve only possible quantities. But neither of these is the case. The conclusions it brings out,_ are in perfect agreement with those obtained by other me thods ; and although it cannot be doubted, but that every truth that can be discovered by the imaginary analysis, may also be found by methods of reasoning which are quite un exceptionable ; yet, generally speaking, these last are less concise, and of more difficult application. Indeed such is the power of imaginary expression as an engine of analy sis, that we are not sure whether some theorems have been proved in any other way. It would be extremely valua ble, even if its conclusions were not stamped with the charac ter of mathematical truth, until they had been verified by the more strict methods. It cannot be doubted, but that the theory of infinities in its loosest form, has brought to light truths which would hardly have been acknowledged, unless they had been afterwards established upon more unexceptionable principles. The vast improvement which
that method of analysis has received between its first ap plications by Kepler, and the highly finished and almost perfect form under which it has been delivered by La Grange, gives us reason to hope, that the obscurity in which the doctrine of imaginary quantities has hitherto been involved, may in time be dispelled, and that it may be brought to the same state of perfection as the other more intelligible mathematical theories.
On the subject of impossible quantities, consult D'Alembeit, Alen:. de l'?cad. de Berlin An. 1716, page 182. Opuscules Math. t. 5.
Euler, same work, 1749, page 122 and 139.
Foncenex, liIiscellan. Taurinensia, t. 1 and 2.
John Bernoulli, Commercium Epistolicunz.
Schubert, Nova Acta Petrol:. t. 13, page 172.
Maclaurin, Fluxions, art. 669, 763.
Playfair, Lond. Phil. Trans. 1778.
Tu remain- NI issery, Theorie purenzent Algobraique des Quantites Imaginarics.
Edinburgh Review, vol. xii. July 1808.
Annales do Mathematiques, torn. iv. pages 20, 23, 61, 133, 222, 364.
Gompertz, The Principles and Application of Imaginary Quantities, book I. Lond 1817.