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One of the most curious of these cases was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness.
The scientific studies inaugurated by Aristotle were here continued by a succession of great astronomers, geometers, chemists, and physicians, for whose use were furnished a botanical garden, a menagerie of animals, and facilities for human dissection, the first school of anatomy ever known.
The reason why some finer spirits are not geometers is, that they cannot turn their attention at all to the principles of geometry; but geometers fail in finer perception, because they do not see all that is before them, and being accustomed to the plain and palpable principles of geometry, and never reasoning until they have well ascertained and handled their principles, they lose themselves in matters of intellectual subtlety, where the principles are not so easily laid hold of.
The researches of geometers on the theory of gravitation had shown that an effect of this kind must be produced by the disturbing action of the planets on the earth. Laplace determined the secular displacement of the plane of the earth's orbit due to each of the planets, and in this way ascertained the whole effect of perturbation upon the obliquity of the ecliptic.
His hold upon the audiences of the giddy city was, however, much weakened by Hypatia, the daughter of Theon, the mathematician, who not only distinguished herself by her expositions of the doctrines of Plato and Aristotle, but also by her comments on the writings of Apollonius and other geometers.
These, however, are not inferences from experience; they are, properly speaking, conclusions deduced from the principle of the Sufficient Reason.” And to this day there are few geometers who would not think it far more scientific to establish these or any other premises in this way, than to rest their evidence on that familiar experience which in the case in question might have been so safely appealed to.
Further on we read respecting algebraic functions that "most functions were concrete in their origin even those which are at present the most purely abstract; and the ancients discovered only through geometrical definitions elementary algebraic properties of functions to which a numerical value was not attached till long afterwards, rendering abstract to us what was concrete to the old geometers."
For some time, however, there were capable geometers who kept up the tradition, filling in details, devising alternative solutions of problems, or discovering new curves for use or investigation. Nicomedes, probably intermediate in date between Eratosthenes and Apollonius, was the inventor of the conchoid or cochloid, of which, according to Pappus, there were three varieties.
Among that beautiful family of moths, the Phalænidæ, comprising the Geometers, Loopers, or Span-worms, are two formidable foes to fruit growers. The habits of the Canker worm should be well known. With proper care and well-directed energy, we believe their attacks can be in a great measure prevented.
The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another.
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