„And thus far historians produce the perfection of this science. But Euclid was not much junior to these, who collected elements, and constructed many of those things which were invented by Eudoxus; and perfected many which were discovered by Theætetus. Besides, he reduced to invincible demonstrations, such things as were exhibited by others with a weaker arm. But he lived in the times of the first Ptolemy: for Archimedes mentions Euclid, in his first book, and also in others. Besides, they relate that Euclid was asked by Ptolomy, whether there was any shorter way to the attainment of geometry than by his elementary institution, and that he answered, there was no other royal path which led to geometry. Euclid, therefore, was junior to the familiars of Plato, but more ancient than Eratosthenes and Archimedes (for these lived at one and the same time, according to the tradition of Eratosthenes) but he was of the Platonic sect, and familiar with its philosophy: and from hence he appointed the constitution of those figures which are called Platonic, as the end of his elementary institutions.“

—  Proklos

Ch. IV. On the Origin of Geometry, and its Inventors.

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Proklos Foto
griechischer Philosoph 412 - 485

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Thomas Little Heath Foto

„After Pythagoras, Anaxagoras the Clazomenian succeeded, who undertook many things pertaining to geometry. And Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato mentions in his Rivals, as one who obtained mathematical glory. To these succeeded Hippocrates, the Chian, who invented the quadrature of the lunula, and Theodorus the Cyrenean, both of them eminent in geometrical knowledge. For the first of these, Hippocrates composed geometrical elements: but Plato, who was posterior to these, caused as well geometry itself, as the other mathematical disciplines, to receive a remarkable addition, on account of the great study he bestowed in their investigation. This he himself manifests, and his books, replete with mathematical discourses, evince: to which we may add, that he every where excites whatever in them is wonderful, and extends to philosophy. But in his time also lived Leodamas the Thasian, Architas the Tarentine, and Theætetus the Athenian; by whom theorems were increased, and advanced to a more skilful constitution. But Neoclides was junior to Leodamas, and his disciple was Leon; who added many things to those thought of by former geometricians. So that Leon also constructed elements more accurate, both on account of their multitude, and on account of the use which they exhibit: and besides this, he discovered a method of determining when a problem, whose investigation is sought for, is possible, and when it is impossible.“

—  Proclus Greek philosopher 412 - 485

Ch. IV.

Thomas Little Heath Foto
Thomas Little Heath Foto
Thomas Little Heath Foto
Florian Cajori Foto

„It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences.“

—  Florian Cajori, buch A History of Mathematics

Quelle: A History of Mathematics (1893), p. 30 Reported in Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914.

Pappus of Alexandria Foto
Freeman Dyson Foto

„The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry? …It cannot be answered by a definition.“

—  Freeman Dyson, buch Infinite in All Directions

Quelle: Infinite in All Directions (1988), Ch. 2 : Butterflies and Superstrings, p. 17
Kontext: Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." …A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry?... It cannot be answered by a definition.

Eugène Delacroix Foto
Swathi Thirunal Rama Varma Foto
James Whitbread Lee Glaisher Foto
Immanuel Kant Foto

„Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. … A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw hi the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts a priori, placed into that figure and represented in it, so that, in order to know anything with certainty a priori, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept.“

—  Immanuel Kant, buch Critique of Pure Reason

Preface to the Second Edition [Tr. F. Max Müller], (New York, 1900), p. 690; as cited in: Robert Edouard Moritz, Memorabilia mathematica or, The philomath's quotation-book https://openlibrary.org/books/OL14022383M/Memorabilia_mathematica, Published 1914. p. 10
Critique of Pure Reason (1781; 1787)

Abraham Lincoln Foto

„He studied and nearly mastered the six books of Euclid since he was a member of Congress. He regrets his want of education, and does what he can to supply the want. In his tenth year he was kicked by a horse, and apparently killed for a time.“

—  Abraham Lincoln 16th President of the United States 1809 - 1865

1860s, A Short Autobiography (1860)
Kontext: Abraham now thinks that the aggregate of all his schooling did not amount to one year. He was never in a college or academy as a student, and never inside of a college or academy building till since he had a law license. What he has in the way of education he has picked up. After he was twenty-three and had separated from his father, he studied English grammar — imperfectly of course, but so as to speak and write as well as he now does. He studied and nearly mastered the six books of Euclid since he was a member of Congress. He regrets his want of education, and does what he can to supply the want. In his tenth year he was kicked by a horse, and apparently killed for a time.<!--pp. 9-10

„Euclid's postulates came from the Pythagorean theorem, not the other way around.“

—  Richard Hamming American mathematician and information theorist 1915 - 1998

The Unreasonable Effectiveness of Mathematics (1980)
Kontext: The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around.

Thomas Little Heath Foto

„But now if any one hath a mind to come over to their sect, he is not immediately admitted, but he is prescribed the same method of living which they use for a year, while he continues excluded'; and they give him also a small hatchet, and the fore-mentioned girdle, and the white garment. And when he hath given evidence, during that time, that he can observe their continence, he approaches nearer to their way of living, and is made a partaker of the waters of purification; yet is he not even now admitted to live with them; for after this demonstration of his fortitude, his temper is tried two more years; and if he appear to be worthy, they then admit him into their society. And before he is allowed to touch their common food, he is obliged to take tremendous oaths, that, in the first place, he will exercise piety towards God, and then that he will observe justice towards men, and that he will do no harm to any one, either of his own accord, or by the command of others; that he will always hate the wicked, and be assistant to the righteous; that he will ever show fidelity to all men, and especially to those in authority, because no one obtains the government without God's assistance; and that if he be in authority, he will at no time whatever abuse his authority, nor endeavor to outshine his subjects either in his garments, or any other finery; that he will be perpetually a lover of truth, and propose to himself to reprove those that tell lies; that he will keep his hands clear from theft, and his soul from unlawful gains; and that he will neither conceal anything from those of his own sect, nor discover any of their doctrines to others, no, not though anyone should compel him so to do at the hazard of his life. Moreover, he swears to communicate their doctrines to no one any otherwise than as he received them himself; that he will abstain from robbery, and will equally preserve the books belonging to their sect, and the names of the angels [or messengers]. These are the oaths by which they secure their proselytes to themselves.“

—  Josephus on the Essenes

Jewish War

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