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Updated: May 26, 2025
When the geometrician defines for us the point, the line, the surface, and the solid, and when he sets before us an array of axioms, or self-evident truths, we follow him with confidence because he seems to be telling us things that we can directly see to be reasonable; indeed, to be telling us things that we have always known.
Perhaps the geometrician, or the accurate-minded reader, will doubt whether the primrose ring was made at all seeing that the wall of Ward C cut off nearly thirty degrees of it. But in the world of fancy geometrical accuracy does not hold; and the only important thing is believing that the ring has been made.
I may have occasion hereafter to speak of "roads," but I mean by the word nothing more than the geometrician means by a "line" simple longitudinal extension without any of the sensible qualities which are popularly associated with it.
We know that Pascal discovered the thirty-second proposition of Euclid all by himself. It is true that it has been concluded, wrongly perhaps, that he had also discovered all the earlier ones, the order followed by the Greek geometrician not being necessary, and not excluding other arrangements. However it be, reasoning alone was not enough for that discovery.
He is a believer in the correspondences of things, of the continuity in nature; a mystic as well as a geometrician. Yet such a realist is he that he quarrels with any artist who does not see "the latent heroic in every natural movement." Therefore he does not force the pose of his model, preferring attitudes or gestures voluntarily adopted.
XVII. However, Marcellus escaped unhurt, and sarcastically said to his own engineers: "Are we to give in to this Briareus of a geometrician, who sits at his ease by the seashore and plays at upsetting our ships, to our lasting disgrace, and surpasses the hundred-handed giant of fable by hurling so many weapons at us at once?"
The Chamber of Commerce proceeded at once to make trouble with regard to the paying of Feuerbach’s bill. An ugly quarrel arose in which Rüdiger, the geometrician, who had always been an ardent champion of Feuerbach, took the artist’s part. It finally reached the point where Rüdiger left the city, swearing he would never return.
One may make this still more evident in lines. A line may have twists and turns, ups and downs, points of reflexion and points of inflexion, interruptions and other variations, so that one sees neither rhyme nor reason therein, especially when taking into account only a portion of the line; and yet it may be that one can give its equation and construction, wherein a geometrician would find the reason and the fittingness of all these so-called irregularities.
A geometrician exactly knew the different energies acting in each case, with the properties of the particles moved, could demonstrate that after the causes given, each particle acted precisely as it ought to act, and that it could not have acted otherwise than it did.
If it be wished to substitute any other word for nature, the question will still remain as it did, as to the cause of her existence; the end she has in view. It is not by changing of terms that a geometrician can solve problems; one word will throw no more light on a subject than another, unless that word carries a certain degree of conviction in the ideas which it generates.
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