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It sometimes happens that a man has no answer to make, and yet is not persuaded. He is overthrown without the feeling of being convinced. He feels at the bottom of his heart a scruple, a repugnance, which hinders him from believing what has been proved to him. A geometrician demonstrates to you that between a circle and a tangent you may thread a number of curves, and yet cannot get one straight line to pass. Your eyes, your reason, tell you the contrary. The geometrician gravely answers you that it is an infinitesimal of the second order. You stare in stupid silence and quit the field all astonished, without having any clear idea, without comprehending anything, and without having any reply to make.

Consult but a geometrician of more candor, and he explains the mystery to you.

"We suppose," says he, "what cannot be in nature, lines which have length without breadth. Naturally and philosophically speaking, it is impossible for one real line to penetrate another. No curve, nor no right line, can pass between two real lines that touch one another. These theorems that puzzle you are but sports of the imagination, ideal chimeras, whereas true geometry is the art of measuring things actually existent."