"I was informed by the priests at Thebes, that king Sesostris made a distribution of the territory of Egypt among all his subjects, assigning to each an equal portion of land in the form of a quadrangle, and that from these allotments he used to derive his revenue by exacting every year a certain tax. In cases however where a part of the land was washed away by the annual inundations of the Nile, the proprietor was permitted to present himself before the king, and signify what had happened. The king then used to send proper officers to examine and ascertain, by admeasurement, how much of the land had been washed away, in order that the amount of tax to be paid for the future, might be proportional to the land which remained. From this circumstance I am of opinion, that Geometry derived its origin; and from hence it was transmitted into Greece."
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the no
"I would myself say that the purely imaginary objects are the only realities, the ὂντως ὂντα, in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line."
"[T]he system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of... practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their proper dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience... not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it from "purely axiomatic geometry."
"One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. ... for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such an association geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not."
"Each of five men—Lobachewsky, Bolyai, Plücker, Riemann, Lie—invented as part of his lifework as much (or more) new geometry as was created by all the Greek mathematicians in the two or three centuries of their greatest activity."
"The authors on may be divided into... theoretical and practical... [N]one... have combined the theory with the practice... to render the subject plain and intelligible... [T]he most valuable and scientifical are... abstruse, and the practical scarcely furnish... the rationale... The object of the ensuing treatise is to simplify the theory, yet to retain a methodical and accurate... investigation, and to exemplify this theory by... important... useful examples. ...[D]emonstrations are frequently founded on principles strictly Geometrical ...and sometimes ...by algebraical signs, particularly where the Geometrical ...would require a complicated figure, or a ...tedious process. ...[T]he algebraical mode of deduction tends greatly to simplify... yet... definitions and... elementary parts... must be acquired from Geometrical principles illustrated by diagrams; otherwise a student will never obtain a clear and satisfactory knowledge... Should any person attempt to teach the elementary principles of the science by... algebraic characters, and algebraic formulae alone, without the aid of Geometry, he would... deceive both himself and his pupils."
"Geometry can in no way be viewed... as a branch of mathematics, instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry."