"This close affinity of arithmetic and geometry comes about, rather, because geometry is, as it were, subordinate to arithmetic, and applies universal principles of arithmetic to its special objects." (John Wallis, "Mathesis Universalis", 1657)
"It hath been an old remark, that Geometry is an excellent Logic." (George Berkeley, "The Analyst; Or, A Discourse Addressed to an Infidel Mathematician", 1734)
"Geometry is a true natural science: - only more simple, and therefore more perfect than any other." (Auguste Comte, “Course of Positive Philosophy”, 1830)
"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur les Surfaces Élastiques", 1880)
"[…] geometry is the art of reasoning well from badly drawn figures [...]" (Henri Poincaré, 1895)
"We must [...] maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)
"Any mathematical science is a body of theorems deduced from a set of axioms. A geometry is a mathematical science." (John H C Whitehead, "The Foundation of Differential Geometry", 1932)
"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry." (Edward Kasner & James R Newman, "New Names for Old", 1956)
"Greek geometry is a wider gate, through which the eye might grasp an austere, yet essential landscape." (Dan Barbilian, 1967)
"Geometry is the study of form and shape."
"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension." (John L Casti, "Five Golden Rules", 1995)
"Traditional geometry is the study of the properties of spaces or objects that have integral dimensions."
"Practical geometry is an empirical undertaking, living and breathing and sweating in the real world where measurements are always approximate and things are fudged or smeared or jumbled up." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"Geometry is the science which restores the situation that existed before the creation of the world and tries to fill the 'gap', relinquishing the help of matter." (Lucian Blaga)
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