On Geometry (1800-1849)

"Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes ; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic." (Jacques Ozanam et al, "Recreations in Mathematics and Natural Philosophy", 1803)

"Whatever relates to extent and quantity may be represented by geometrical figures. Statistical projections which speak to the senses without fatiguing the mind, possess the advantage of fixing the attention on a great number of important facts." (Alexander von Humboldt, 1811)

"I am ever more convinced that the necessity of our geometry cannot be proved - at least not by human reason for human reason. It is possible that in another lifetime we will arrive at other conclusions on the nature of space that we now have no access to. In the meantime we must not put geometry on a par with arithmetic that exists purely a priori but rather with mechanics.” (Carl F Gauss, [letter to Heinrich W Olbers] 1817)

"Geometry is a true natural science: - only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Everybody studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone." (Auguste Comte,"Course of Positive Philosophy", 1830

"The process by which I propose to accomplish this is one essentially graphical; by which term I understand not a mere substitution of geometrical construction and measurement for numerical calculation, but one which has for its object to perform that which no system of calculation can possibly do, by bringing in the aid of the eye and hand to guide the judgment, in a case where judgment only, and not calculation, can be of any avail." (John F W Herschel, "On the investigation of the orbits of revolving double stars: Being a supplement to a paper entitled 'micrometrical measures of double stars'", Memoirs of the Royal Astronomical Society, 1833)

"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic." (William Chambers & Robert Chambers, "Chambers's Information for the People" Vol. 2, 1835)

"The doctrines of pure geometry often, and in many questions, give a simple and natural way to penetrate to the origin of truths, to lay bare the mysterious chain which unites them, and to make them known individually, luminously and completely.” (Michek Chasles, “Aperçu historique sur l'origine et le développement des méthodes en géométrie", 1837)

"Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "System of Logic", 1843)

"Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space." (James J Sylvester, "A Probationary Lecture on Geometry", 1844)

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept." (Hermann GGrassmann, "Ausdehnungslehre", 1844)

"The connected course of reasoning by which any Geometrical truth is established is called a demonstration." (Robert Potts, "Euclid's Elements of Geometry", 1845)

"In all cases, however, it must be kept in view that every geometrical truth is deduced by a comparison between two others, which agree, one in one particular part, and the other in another, with the conclusion so deduced." (?,"Miscallanea Mathematica", American Railroad Journal, No. X, 1846)