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His family had been prominent in many walks of life since the fourteenth century, and, whether in the land of the Saxons or of the Slavs, represented the cream of the Jewries in which they lived. His father was a Maskil of great repute, who had written several treatises, in Hebrew, on algebra, geometry, optics, and kindred subjects.
There are so many pleasant things to do not always very easy things, much of my work in Algebra and Geometry is hard: but I love it all, especially Greek. Just think, I shall soon finish my grammar! Then comes the "Iliad." What an inexpressible joy it will be to read about Achilles, and Ulysses, and Andromache and Athene, and the rest of my old friends in their own glorious language!
How, being so contained and implied, it could be new truth, and how the theorems of geometry, so different in appearance from the definitions and axioms, could be all contained in these, was a difficulty which no, one, I thought, had sufficiently felt, and which, at all events, no one had succeeded in clearing up.
Neither natural ability without instruction nor instruction without natural ability can make the perfect artist. Let him be educated, skilful with the pencil, instructed in geometry, know much history, have followed the philosophers with attention, understand music, have some knowledge of medicine, know the opinions of the jurists, and be acquainted with astronomy and the theory of the heavens.
'Metageometries' have been invented by Riemann and Lobatschewski as rivals to the assumptions of Euclid, and the brilliant writings of Poincaré have explained the human devices on which mathematical concepts rest. Euclidean geometry is reduced to a useful interpretation of the data of experience; it is not theoretically the only one.
There is grave danger that the method will not be differentiated from the subject, the ideal from the context of the situation. To many children learning how to study in connection with history, or to be critical in geometry, or to be scientific in the laboratory, has never been separated from the particular situation.
And with all this was a self-discipline in the two great knowledges by which men have climbed from savages to gods language and mathematics. He was told one day that there was an English grammar in a house six miles from his home, and he at once walked off to borrow it. And he studied geometry and algebra alone.
But, besides that the existence of such things is not actual or possible consistently with the constitution of the earth, neither can they even be conceived as existing. The exact correspondence, then, between the facts and those first principles of geometry which are involved in the so-called definitions, is a fiction, and is merely supposed. Mechanics has the first law of motion.
"I've found out the trouble with John." "The trouble with John?" Her voice trembled. "Yes, the trouble with John is that he lacks blood at the brain. He is trying to make a living organism out of a skeleton to build the world over on a skull and cross-bones and it can't be done. I admire John as much as I ever did. He is as logical as a problem in geometry.
But when Berkeley goes further than this, and declares that there are no "ideas" common to the "ideas" of touch and those of sight, it appears to me that he has fallen into a great error, and one which is the chief source of his paradoxes about geometry. Berkeley in fact employs the word "idea" in this instance to denote two totally different classes of feelings, or states of consciousness.
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