On Geometry (Unsourced)

"A child[’s …] first geometrical discoveries are topological [...] If you ask him to copy a square or a triangle, he draws a closed circle." (Jean Piaget)

"A geometrical theory in physical interpretation can never be validated with mathematical certainty […] like any other theory of empirical science, it can acquire only a more or less high degree of confirmation." (Carl G Hempel)

"An essential defect of previous presentations of geometry is that one usually returns to discrete numerical ratios in the treatment of similarity theory. This procedure, which at first seems simple, soon enough becomes entangled in complicated investigations concerning incommensurable magnitudes, as we have already hinted above; and the initial impression of simplicity is revenged upon problems of a purely geometrical procedure by the appearance of a set of difficult investigations of a completely heterogeneous type, which shed no light on the essence of spatial magnitudes. To be sure, one cannot eliminate the problem of measuring spatial magnitudes and expressing the results of these measurements numerically. But this problem cannot originate in geometry itself, but only arises when one, equipped on the one hand with the concept of number and on the other with spatial perceptions, applies them to that problem, and thus in a mixed branch that one can in a general sense call by the name 'theory of measurement' […] To relegate the theory of similarity, and even that of surface area, to this branch as has previously occurred" (not to the form but to the substance) is to steal the essential content from what is called" (pure) geometry." (Hermann G Grassmann)

"Architecture is the triumph of human imagination over materials, methods, and men, to put man into possession of his own Earth. It is at least the geometric pattern of things, of life, of the human and social world. It is at best that magic framework of reality that we sometimes touch upon when we use the word order." (Frank L Wright)

"Born at the same time with the Greek art, the mathematics kept in its canvas, in its intimate structure, a certain affinity with art. It comes to the same harmony in Euclid’s geometry as in the ancient temples. It is the same silence, the same balance in demonstrating a theorem as in the admirable columns of the Acropolis." (Gheorghe Ţiţeica)

"Geometrical truths are in a way asymptotes to physical truths, that is to say, the latter approach the former indefinitely near without ever reaching them exactly." (Jean le Rond D’Alembert)

"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)

"[It used to be that] geometry must, like logic, rely on formal reasoning in order to rebut the quibblers. But the tables have turned. All reasoning concerned with what common sense knows in advance, serves only to conceal the truth and to weary the reader and is today disregarded." (Alexis C Clairaut)

"It is very helpful to represent these things in this fashion since nothing enters the mind more readily than geometric figures." (René Descartes)

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin - for motion naturally comes after rest - nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities. So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness." (Nicomachus of Gerasa)

"Music is the arithmetic of sounds as optics is the geometry of light." (Claude Debussy)

"My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork, insofar as nearly all the manifold motions are caused by a most simple, magnetic, and material force, just as all motions of the clock are caused by a simple weight. And I also show how these physical causes are to be given numerical and geometrical expression." (Johannes Kepler)

"Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience." (Arthur Cayley)

"Poetry is as exact a science as geometry." (Gustave Flaubert)

"Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge." (Felix Klein)

"Show all these fanatics a little geometry, and they learn it quite easily. But, strangely enough, their minds are not thereby rectified. They perceive the truths of geometry, but it does not teach them to weigh probabilities. Their minds have set hard. They will reason in a topsy-turvy wall all their lives, and I am sorry for it." (Voltaire)

"The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems - the determination of the diagonal of a square and that of the circumference of a circle - revealed the existence of new mathematical beings for which no place could be found within the rational domain." (Tobias Dantzig)

"The axioms of geometry are - according to my way of thinking - not arbitrary, but sensible. statements, which are, in general, induced by space perception and are determined as to their precise content by expediency." (Felix Klein)

"[…] the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely esthetic feeling, which all mathematicians know. And this is sensitivity." (Henri Poincaré)

"The geometry of Algebraic Topology is so pretty, it would seem a pity to slight it and miss all the intuition which it provides. At deeper levels, algebra becomes increasingly important, so for the sake of balance it seems only fair to emphasize geometry at the beginning." (Allen Hatcher) 

"The knowledge of which geometry aims is the knowledge of the eternal." (Plato)

"The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation." (James J Sylvester)

"The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation." (J J Sylvester)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

Geometrical truths are in a way asymptotes to physical truths, that is to say, the latter approach the former indefinitely near without ever reaching them exactly." (Jean le Rond d’Alembert)"

"To follow a geometrical argument purely logically without having the figure on which the argument bears constantly before me is for me impossible." (Felix Klein)

"Topology is an elastic version of geometry that retains the idea of continuity but relaxes rigid metric notions of distance." (Samuel Eilenberg)

"Two other sciences in the same way will accurately treat of 'size': geometry, the part that abides and is at rest, [and] astronomy, that which moves and revolves." (Nicomachus of Gerasa)

"We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads." (Voltaire)

"We ought either to exclude all lines, beside the circle and right line, out of geometry, or admit them according to the simplicity of their descriptions, in which case the Conchoid yields to none except the circle. That is arithmetically more simple which is determined by the more simple equations, but that is geometrically more simple which is determined by the more simple drawing of lines." (Sir Isaac Newton)

"With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously." (Niels H Abel)