"Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length—for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension."

“The relationship of point to line bothered the Greeks and led Aristotle to separate the two. Though he admits points are on lines, he says that a line is not made up of points and that the continuous cannot be made up of the discrete. This distinction contributed also to the presumed need for separating number from geometry, since to the Greeks numbers were discrete and geometry dealt with continuous magnitudes.”

“In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers.”

“Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.”

“Fermat knew that under reflection light takes the path requiring least time and, convinced that nature does indeed act simply and economically, affirmed in letters of 1657 and 1662 his Principle of Least Time, which states that light always takes the path requiring least time. He had doubted the correctness of the law of refraction of light but when he found in 1661 that he could deduce it from his Principle, he not only resolved his doubts about the law but felt all the more certain that his Principle was correct. ...Huygens, who had at first objected to Fermat's Principle, showed that it does hold for the propagation of light in media with variable indices of refraction. Even Newton's first law of motion, which states that the straight line or shortest distance is the natural motion of a body, showed nature's desire to economize. These examples suggested that there might be a more general principle. The search for such a principle was undertaken by Maupertuis.”

“The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by... Al Khowarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation... When the Moors reached Spain... algebrista... came to mean a bonesetter... and signs reading Algebrista y Sangrador were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts.”

“The famous sixteenth-century algebraist Jerome Cardan called negative roots fictitious, and the founder of modern symbolic algebra, François Viète, discarded negative roots entirely. Descartes, called them false on the ground that they represented numbers less than nothing and so were meaningless.”