"The first important example solved by Descartes in his geometry is the "problem of Pappus"... Of this celebrated problem the Greeks solved only the special case... By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia."
René Descartes was a French philosopher, scientist, logician, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science during the Renaissance era. Mathematics was paramount to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry.
"Doubt is the origin of wisdom and Latin: Dubium sapientiae initium. This has been attributed to Descartes, including here previously, but no original attribution has been found. Descartes Meditationes de prima philosophia has been cited as the source of Dubium sapientiae initium, but this quote is not found in this work."
"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis..."
"An optimist may see a light where there is none, but why must the pessimist always run to blow it out?"
"He has given us only some beautiful beginnings, without getting to the bottom of things. ...he is still far from the true analysis and the general art of discovery. For I am convinced that his mechanics is full of errors, that his physics goes too fast, that his geometry is too narrow, and that his metaphysics is all these things."
"Modesty was not a condition from which Descartes suffered."
"René Descartes played his part in the world as a man in a mask. The phrase is his own. It implied no conscious duplicity but a certain apartness rendering him incapable of taking his share unreservedly in the game of life. His attitude was that of an unimpassioned spectator. He looked on and learned while others struggled for the stakes. A born recluse, he remained solitary even in the throng of social intercourse. ...The transport felt by Bacon at the glorious vision of a future in which the harvest he had sown would be reaped was not shared by Descartes. He indeed held himself to be the sower and the reaper in one. Calmly, and with settled conviction, he claimed to have virtually expounded all the phenomena of nature. He had crossed the frontier of the new world of knowledge; those who desired to follow him needed only to purge their eyes with the euphrasy of methodic doubt in order to obtain those crystal-clear intuitions from which, by sure reasoning, universal science could be deduced. ...He was a mathematician of first rate originality and power. ...Unfortunately, however, he was misled by false analogies of thought; he sought to universalise what was by its nature special and restricted; and thus pursued the flitting vision of a deceptive unity in knowledge. As the upshot, he gave to his philosophy a pseudo-mathematical character, and sacrificed the solid achievement of much by aiming at the illusory certainty of everything."
"The absolute requisites for the study of this work... are a knowledge of algebra to the binomial at least, plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry. ...A. De Morgan. London July 1, 1836"
"A free people will always refuse to put up with preventable poverty. If freedom is to be saved and enlarged, poverty must be ended. There is no other solution. The problem of how to prevent these three forces from coming into head-on collision is the principal study of the more politically conscious Conservative leaders. How can wealth persuade poverty to use its political freedom to keep wealth in power? Here lies the whole art of Conservative politics in the twentieth century."
"A clear prediction in an area undergoing vigorous study permits doctrines to be subject to disproof. The last posture a bureaucratic religion wishes to find itself in is vulnerability to disproof, where an experiment can be performed on which the religion stands or falls."
"Spencer argued in a very articulate way for the communality of these processes of self-organization, and used his ideas to make a theory of sociology. However, he was not able to put these ideas into mathematical form or argue them from first principles. And no one else has, either — doing so is perhaps the central problem in the study of complex systems."
"The commercial exploitation of the web has become a growing facet of the world economy, particularly in the last several years. In June 1999 NUA Internet Surveys estimated that 179 million people are connected to the Internet worldwide. A recent study by the sponsored by estimated that the "Internet Economy" generated $300 billion in revenue in the United States alone."
"In biology the Cartesian view of living organisms as machines, constructed from separate parts, still provides the dominant conceptual framework. Although Descartes simple mechanistic biology could not be carried very far and had to be modified considerably during the subsequent three hundred years, the belief that all aspects of living organisms can be understood by reducing them to their smallest constituents, and by studying the mechanisms through which these interact, lies at the very basis of most contemporary biological thinking. This passage from a current textbook on modern biology is a clear expression of the reductionist credo: One of the acid tests of understanding an object is the ability to put it together from its component parts. Ultimately, molecular biologists will attempt to subject their understanding of cell structure and function to this sort of test by trying to synthesize a cell."