In I.:you:a-lox Assn EVOLUTION a method of performing the operations required in Sturm's theorem was proposed, which avoids useless writing. Mr. Young (' Math. Dissertations,' p. 143) pro posed another, of much the same degree of abbreviation. Sturm's theorem however reckuires so much operation, that there can be littlo doubt of that of }(airier being a more easy mode of working any particular case. It is not however as a key to the mere numerical solution of equations that either of these theorems must be viewed : the insight which they give into the nature of equations, and still more that which they are likely hereafter to give (for neither is more than a germ), will render them both important steps in the progress of algebra.
Since it is sufficient to the theorem that the last function v, should retain one sign, and not vanish, we may stop in the process when we arrive at any function of which all the roots are known, or can be discovered, to be impossible. And it is easily shown, as was done by Sturm himself, that even when there are equal roots, eo that the last, v,, is neither constant nor always of the same sign, the theorem still remains true, so far as to give the number of differene roots which lie between any two given limits, without any information as to tho number of times which each root should be repeated.
For instance, in the article cited we find v v,= +1 45 We need not go further, for v, has none but imaginary roots. Now when x= , the criterion is +—+ ; when x=0, it is — + + ; and when x= + co , + + +. Consequently there is one negative root, one positive root, and a pair of imaginary roots.
The following example is from Mr. Young (p. 101) : it is an instance given by Fourier in illukration of his own method, and Sturm's is applied to it by 3Ir. Young, to show the superior certainty of the latter. Of that certainty no one can doubt, but the process exhibited e excite impressions of the grand, the pathetic, or the humorous, whether they are addressed to the ear or merely to the eye. And great effects are also to be achieved by the arrangement of words, not only in the production of melody and cadence, but in a higher kind of gratification or excitement—as by the luminous disposition of all the parts of the sentence, by the presentment of every term at the place best fitted to bring out its whole import, by all the resources of what the gramma rians call inversion, ellipsis, and other figures of speech ; which, indeed, wherever they are'properly used, are no deviations from natural syntax at all, but, on the contrary, the most natural forms that can be employed. For, while writing is an art, it is nevertheless moat true that, like all the other arts the purpose of which is to give expression to mind, the guiding and controlling principle of its exercise, its life and being, as we may say, must ever be as exact and sympathetic a conformity as possible to the thoughts or emotions of the writer. Whatever more
style is than the mere expression of thought, that much it must be at the least. A powerful thinker may not always be a powerful writer, but no man can be a powerful writer who is not a powerful thinker.
Even the humblest quality of style, mere perspicuity, cannot be attained without a corresponding degree of clearness of thought. We sometimes meet with a perspicuity which is little more than gram matical, and hardly belongs to style at all ; but even that implies distinct conceptions so far as they go—a limpid stream of thought, however little depth or spaciousness of intellect. And as for all higher attributes, it is manifest that they cannot be found in the style, if they do not exist in the mind of the writer. The only fountains from which a man's words can derive the animation of true passion, or poetry, or wit, must be his own head and heart.
The lowest kind of writing that deserves the name of a style at all (unless it is to be called a bad style) ought, as we' have observed, to be perfectly perspicuous—that is to say, readily and completely intelli gible in so far as the understanding of it depends merely upon a knowledge of the language. The subject may be a difficult one ; but that is only a reason for more pains being bestowed to make the style clear and easy, by a lucid arrangement and the avoidance of all ambiguities of expression. But although this rule may be justly insisted upon where nothing beyond such perspicuity is desirable, it will not bear to he ao rigidly enforced in regard to the higher kinds of style. Here some sacrifice even of perspicuity is at times to be sub mitted to, for the sake of appropriate effects which could not be otherwise attained. sEschylus, no doubt, might have made his cho russes, Pindar his odes, Tacitus his historio pictures, more easily comprehensible, better fitted for the use of such readers as would always run while they read, by greater diffuseness and dilution of style; but much more, certainly, would have been lost than gained by the attempt. What is to be desired in the highest kinds of writing, as in the highest creations of all the fine arts, is not perfect compre hensibility at a glance, but rather that fulness and profundity of meaning which can never be wholly comprehended, but supplies inexhaustibly something new to be seen and felt every time we return to the work.