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The route was always, within the limits of precision of the calculations, that which Clairaut had indicated beforehand. The prediction of the illustrious geometer was verified in regard both to time and space: astronomy had just achieved a great and important triumph, and, as usual, had destroyed at one blow a disgraceful and inveterate prejudice.
According to the opinion of very competent judges the result of the indirect method might not impossibly merit the preference. The movements of the moon proved a fertile mine of research to our great geometer. His penetrating intellect discovered in them unknown treasures.
If this axiom of parallels does not follow from the other axioms, then from these latter we may construct a system of geometry in which the axiom of parallels shall not be true. This was done by Lobatchewsky and Bolyai, the one a Russian the other a Hungarian geometer, about 1830.
Accordingly I wrote to Professor Miller, of Cambridge, and this geometer has kindly read over the following statement, drawn up from his information, and tells me that it is strictly correct:
Pythagoras must have his pupils understand music and geometry; and by music he intended, all the arts, every department of life that came under the sway of the Nine Muses. Why? Because, as he taught, God is Poet and Geometer. Chaos is only on the outer rim of existence; as you get nearer the heart of thing, order and rhythm, geometry and poetry, are more and more found.
But, if we look at the subject in an intellectual light, we shall find a better explanation. Forms are the embodiment of thought or law. For the common eye they must be embodied in material shape; while to the geometer and the artist, they may be so distinctly shadowed forth in conception as to need no material figure to render their beauty appreciable.
Laplace, who never contented himself with a vague statement, sought to determine in numbers the rapid cooling of our globe which Buffon had so eloquently but so gratuitously announced. Nothing could be more simple, better connected, or more demonstrative, than the chain of deductions of the celebrated geometer. A body diminishes in volume when it cools.
It was impossible that the great geometer, who had succeeded so well in the study of the tides of the ocean, should not have occupied his attention with the tides of the atmosphere; that he should not have submitted to the delicate and definitive tests of a rigorous calculus, the generally diffused opinions respecting the influence of the moon upon the height of the barometer and other meteorological phenomena.
After more than twenty-two centuries it still survives, a model of accuracy, perspicuity, and a standard of exact demonstration. This great geometer not only wrote on other mathematical topics, such as Conic Sections and Porisms, but there are imputed to him treatises on Harmonics and Optics, the latter subject being discussed on the hypothesis of rays issuing from the eye to the object.
The geometer has made in his first reflection so clear and violent an abstraction from the sun's actual bulk and qualities that he will never imagine himself to be speaking of anything but a concretion in discourse.
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