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It was not so with our scientific societies, the members of which regarded with regret the approaching destruction of our planetary system. The Academy of Sciences called the attention of geometers of all countries to these menacing perturbations. Euler and Lagrange descended into the arena. Never did their mathematical genius shine with a brighter lustre. Still, the question remained undecided.
Under this new aspect, it will be requisite to submit the question to a new examination. Any geometers and astronomers who wish to know all the extent of Bailly's labours, must not content themselves with consulting the collections in the Academy of Sciences; for he published, at the beginning of 1766, a separate work under the modest title of Essay on the Theory of Jupiter's Satellites.
In the intellectual decline of Alexandria, indolent methods were preferred to laborious observation and severe mental exercise. The schools of Neo-Platonism were crowded with speculative mystics, such as Ammonius Saccas and Plotinus. These took the place of the severe geometers of the old Museum.
Other famous geometers could also be mentioned, but such men as Euclid, Archimedes, and Apollonius are enough to show that geometry was cultivated to a great extent by the philosophers of antiquity. It progressively advanced, like philosophy itself, from the time of Thales, until it had reached the perfection of which it was capable, when it became merged into astronomical science.
The varieties of physical phenomena with which those sciences are conversant, have been found to answer to determinable varieties in the quantity of some circumstance or other; or at least to varieties of form or position, for which corresponding equations of quantity had already been, or were susceptible of being, discovered by geometers.
By howsoever high authorities this doctrine may have been sanctioned, it appears to me psychologically incorrect. Our idea of a point, I apprehend to be simply our idea of the minimum visibile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable.
It is a widely diffused species, occurring probably throughout the Northern States. We have taken the moth in Northern Maine. We have received from Mr. W. V. Andrews the supposed larvæ of this moth. They are "loopers," that is, they walk with a looping gait, as if measuring off the ground they walk over, whence the name "Geometers," more usually applied to them.
These "Elements" soon became the universal study of geometers throughout the civilized world. They were translated into the Arabic, and through the Arabians were made known to mediaeval Europe. There can be no doubt that this work is one of the highest triumphs of human genius, and has been valued more than any single monument of antiquity.
Scientifically speaking, Helmholtz and Thomson are as well entitled to reason upon the assumption of a perfectly frictionless fluid as geometers in general are entitled to assume perfect lines without breadth and perfect surfaces without thickness.
Hence there arose a succession of great geometers, and their conceptions were applied to the construction of new mechanical contrivances on the one hand, and to the elaboration of theories of sidereal mechanics on the other.
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