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That the Italic Greeks should have conceived that idea was perhaps not so much because they were astronomers as because they were practical geographers and geometers. Pythagoras, as we have noted, was born at Samos, and, therefore, made a relatively long sea voyage in passing to Italy.
The famous spiral of the geometers is the general plan followed by the Mollusc rolling its stone sheath. Where do these glairy creatures pick up this science? We are told that the Mollusc derives from the Worm. One day, the Worm, rendered frisky by the sun, emancipated itself, brandished its tail and twisted it into a corkscrew for sheer glee.
They are so delicate and multitudinous that it requires a very delicate and neat sense to appreciate them. . . . So it is as rare for geometers to be men of subtle wit as it is for the latter to be geometers, because geometers like to treat these nicer matters geometrically, and so make themselves ridiculous; they like to commence with definition, and then go on to principles—a mode which does not at all suit this sort of reasoning.
Such men would probably not be great geometers, because geometry involves a multitude of principles, and because the mind which may penetrate thoroughly a few principles to their depth may not be at all able to penetrate things which combine a multitude of principles. . . . There are two sorts of mind: the one fathoms rapidly and deeply the consequences of principles—this is the observant and accurate mind; the other embraces a great multitude of principles, without confounding them—and this is the mathematical mind.
An observant eye perceives already some traces of their efforts in the writings of the mathematicians of the Alexandrian School. These traces, it must be acknowledged, are so slight and so imperfect, that we should truly be justified in referring the origin of this branch of analysis only to the excellent labours of our countryman Vieta. Descartes, to whom we render very imperfect justice when we content ourselves with saying that he taught us much when he taught us to doubt, occupied his attention also for a short time with this problem, and left upon it the indelible impress of his powerful mind. Hudde gave for a particular but very important case rules to which nothing has since been added; Rolle, of the Academy of Sciences, devoted to this one subject his entire life. Among our neighbours on the other side of the channel, Harriot, Newton, Maclaurin, Stirling, Waring, I may say all the illustrious geometers which England produced in the last century, made it also the subject of their researches. Some years afterwards the names of Daniel Barnoulli, of Euler, and of Fontaine came to be added to so many great names. Finally, Lagrange in his turn embarked in the same career, and at the very commencement of his researches he succeeded in substituting for the imperfect, although very ingenious, essays of his predecessors, a complete method which was free from every objection. From that instant the dignity of science was satisfied; but in such a case it would not be permitted to say with the poet: "Le temps ne fait rien
A similar expedition had been despatched from France about the same time to Peru in South America, for the purpose of measuring an arc of the meridian under the equator, but the results had not been ascertained at the time to which the author alludes in the text. Translator. It may perhaps be asked why we place Lagrange among the French geometers?
This rank, which was lost for a moment, was brilliantly regained, an achievement for which we are indebted to four geometers. When Newton, giving to his discoveries a generality which the laws of Kepler did not imply, imagined that the different planets were not only attracted by the sun, but that they also attract each other, he introduced into the heavens a cause of universal disturbance.
It must be remembered that in these days we know of the physical necessity which requires that a planet shall revolve in an ellipse and not in any other curve. But Kepler had no such knowledge. Even to the last hour of his life he remained in ignorance of the existence of any natural cause which ordained that planets should follow those particular curves which geometers know so well.
His chief work was a treatise on Conic Sections. It is said that he was the first to introduce the words ellipse and hyperbola. So late as the eleventh century his complete works were extant in Arabic. Modern geometers describe him as handling his subjects with less power than his great predecessor Archimedes, but nevertheless displaying extreme precision and beauty in his methods.
The two kinds of speculation have been pursued, for the most part, by two different classes of persons, the geometers and the metaphysicians; for it has been far more the occupation of metaphysicians than of geometers to discuss such questions as I have stated, the nature of geometrical proofs, geometrical axioms, the geometrical faculty, and the like.