On Galileo Galilei - Historical Perspectives

"We pass with admiration along the great series of mathematicians, by whom the science of theoretical mechanics has been cultivated, from the time of Newton to our own. There is no group of men of science whose fame is higher or brighter. The great discoveries of Copernicus, Galileo, Newton, had fixed all eyes on those portions of human knowledge on which their successors employed their labors. The certainty belonging to this line of speculation seemed to elevate mathematicians above the students of other subjects; and the beauty of mathematical relations and the subtlety of intellect which may be shown in dealing with them, were fitted to win unbounded applause. The successors of Newton and the Bernoullis, as Euler, Clairaut, D’Alembert, Lagrange, Laplace, not to introduce living names, have been some of the most remarkable men of talent which the world has seen." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)

"Simple as the law of gravity now appears, and beautifully in accordance with all the observations of past and of present times, consider what it has cost of intellectual study. Copernicus, Galileo, Kepler, Euler, Lagrange, Laplace, all the great names which have exalted the character of man, by carrying out trains of reasoning unparalleled in every other science; these, and a host of others, each of whom might have been the Newton of another field, have all labored to work out, the consequences which resulted from that single law which he discovered. All that the human mind has produced - the brightest in genius, the most persevering in application, has been lavished on the details of the law of gravity." (Charles Babbage, "The Ninth Bridgewater Treatise: A Fragment", 1838)

"It is a vulgar belief that our astronomical knowledge dates only from the recent century when it was rescued from the monks who imprisoned Galileo; but Hipparchus [...] who among other achievements discovered the precession of the eqinoxes, ranks with the Newtons and the Keplers; and Copernicus, the modern father of our celestial science, avows himself, in his famous work, as only the champion of Pythagoras, whose system he enforces and illustrates. Even the most modish schemes of the day on the origin of things, which captivate as much by their novelty as their truth, may find their precursors in ancient sages, and after a careful analysis of the blended elements of imagination and induction which charaterise the new theories, they will be found mainly to rest on the atom of Epicurus and the monad of Thales. Scientific, like spiritual truth, has ever from the beginning been descending from heaven to man." (Benjamin Disraeli, "Lothair", 1879)

"In mechanics Descartes can hardly be said to have advanced beyond Galileo. [...] His statement of the first and second laws of motion was an improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens." (Florian Cajori, "A History of Mathematics", 1893)

"Roughly it amounts to this: mathematical analysis as it works today must make use of irrational numbers (such as the square root of two); the sense if any in which such numbers exist is hazy. Their reputed mathematical existence implies the disputed theories of the infi nite. The paradoxes remain. Without a satisfactory theory of irrational numbers, among other things, Achilles does not catch up with the tortoise, and the earth cannot turn on its axis. But as Galileo remarked, it does. It would seem to follow that something is wrong with our attempts to compass the infinite." (Eric T Bell, "Debunking Science", 1930)

"A pendulum is simply a small load suspended to a string or to a rod fixed at one end. If left alone it ends up hanging vertically, and if we push it away from the vertical, it starts beating. Galileo found that all beats last the same time, called the period, which depends on the length of the pendulum, but not on the amplitude of the beats or on the weight of the load. It also states that the period varies as the square root of the length: to double its period, one should make the pendulum four times as long. Making it heavier, or pushing it farther away from the vertical, has no effect. This property is known as isochrony, and it is the main reason why we are able to measure time with accuracy." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

On Pierre-Simon Laplace - Direct Perspectives

"The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered." (Augustus De Morgan, "Review of 'Théorie Analytique des Probabilites' par M. le Marquis de Laplace" 1820) 

"Whenever I meet in Laplace with the words “Thus it plainly appears”, I am sure that hours and perhaps days, of hard study will alone enable me to discover how it plainly appears." (Nathaniel Bowditch, "Mécanique céleste", 1829-39)

"The influence of electricity in producing decompositions, although of inestimable value as an instrument of discovery in chemical inquiries, can hardly be said to have been applied to the practical purposes of life, until the same powerful genius [Davy] which detected the principle, applied it, by a singular felicity of reasoning, to arrest the corrosion of the copper-sheathing of vessels. […] this was regarded as by Laplace as the greatest of Sir Humphry's discoveries." (Charles Babbage, "Reflections on the Decline of Science in England", 1830)

"The name of Sir Isaac Newton has by general consent been placed at the head of those great men who have been the ornaments of their species. […] The philosopher [Laplace], indeed, to whom posterity will probably assign a place next to Newton, has characterized the Principia as pre-eminent above all the productions of human intellect." (David Brewster, "Life of Sir Isaac Newton", 1831)

"The persons who have been employed on these problems of applying the properties of matter and the laws of motion to the explanation of the phenomena of the world, and who have brought to them the high and admirable qualities which such an office requires, have justly excited in a very eminent degree the admiration which mankind feels for great intellectual powers. Their names occupy a distinguished place in literary history; and probably there are no scientific reputations of the last century higher, and none more merited, than those earned by great mathematicians who have laboured with such wonderful success in unfolding the mechanism of the heavens; such for instance as D ’Alembert, Clairaut, Euler, Lagrange, Laplace." (William Whewell, "Astronomy and General Physics", 1833)

"We pass with admiration along the great series of mathematicians, by whom the science of theoretical mechanics has been cultivated, from the time of Newton to our own. There is no group of men of science whose fame is higher or brighter. The great discoveries of Copernicus, Galileo, Newton, had fixed all eyes on those portions of human knowledge on which their successors employed their labors. The certainty belonging to this line of speculation seemed to elevate mathematicians above the students of other subjects; and the beauty of mathematical relations and the subtlety of intellect which may be shown in dealing with them, were fitted to win unbounded applause. The successors of Newton and the Bernoullis, as Euler, Clairaut, D’Alembert, Lagrange, Laplace, not to introduce living names, have been some of the most remarkable men of talent which the world has seen." (William Whewell, "History of the Inductive Sciences" Vol. 1, 1837)

"Simple as the law of gravity now appears, and beautifully in accordance with all the observations of past and of present times, consider what it has cost of intellectual study. Copernicus, Galileo, Kepler, Euler, Lagrange, Laplace, all the great names which have exalted the character of man, by carrying out trains of reasoning unparalleled in every other science; these, and a host of others, each of whom might have been the Newton of another field, have all labored to work out, the consequences which resulted from that single law which he discovered. All that the human mind has produced - the brightest in genius, the most persevering in application, has been lavished on the details of the law of gravity." (Charles Babbage, "The Ninth Bridgewater Treatise: A Fragment", 1838)

"In general I would be cautious against […] plays of fancy and would not make way for their reception into scientific astronomy, which must have quite a different character. Laplace’s cosmogenic hypotheses belong in that class. Indeed, I do not deny that I sometimes amuse myself in a similar manner, only I would never publish the stuff. My thoughts about the inhabitants of celestial bodies, for example, belong in that category. For my part, I am (contrary to the usual opinion) convinced […] that the larger the cosmic body, the smaller are the inhabitants and other products. For example, on the sun trees, which in the same ratio would be larger than ours, as the sun exceeds the earth in magnitude, would not be able to exist, for on account of the much greater weight on the surface of the sun, all branches would break themselves off, in so far as the materials are not of a sort entirely heterogeneous with those on earth." (Carl F Gauss, (Letter to Heinrich Schumacher] 1847)

"No one can read the history of astronomy without perceiving that Copernicus, Newton, Laplace, are not new men, or a new kind of men, but that Thales, Anaximenes, Hipparchus, Empodocles, Aristorchus, Pythagorus, Oenipodes, had anticipated them." (Ralph W Emerson, "The Conduct of Life", 1860)

"Laplace considers astronomy a science of observation, because we can only observe the movements of the planets; we cannot reach them, indeed, to alter their course and to experiment with them. 'On earth', said Laplace, 'we make phenomena vary by experiments; in the sky, we carefully define all the phenomena presented to us by celestial motion.' Certain physicians call medicine a science of observations, because they wrongly think that experimentation is inapplicable to it." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible." (W W Rouse Ball, "A Short Account of the History of Mathematics", 1888) 

"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems." (Florian Cajori, "A History of Mathematics", 1893)

"From the infinitely great down to the infinitely small, all things are subject to [the laws of nature]. The sun and the planets follow the laws discovered by Newton and Laplace, just as the atoms in their combinations follow the laws of chemistry, as living creatures follow the laws of biology. It is only the imperfections of the human mind which multiply the divisions of the sciences, separating astronomy from physics or chemistry, the natural sciences from the social sciences. In essence, science is one. It is none other than the truth." (Vilfredo Pareto, "Cours d’Economie Politique", 1896-97)

"The first effect of the mind growing cultivated is that processes once multiple get to be performed in a single act. Lazarus has called this the progressive 'condensation' of thought. [...] Steps really sink from sight. An advanced thinker sees the relations of his topics is such masses and so instantaneously that when he comes to explain to younger minds it is often hard [...] Bowditch, who translated and annotated Laplace's Méchanique Céleste, said that whenever his author prefaced a proposition by the words 'it is evident', he knew that many hours of hard study lay before him." (William James, "The Principles of Psychology" Vol. II, 1918)

"The scientist who recognizes God knows only the God of Newton. To him the God imagined by Laplace and Comte is wholly inadequate. He feels that God is in nature, that the orderly ways in which nature works are themselves the manifestations of God's will and purpose. Its laws are his orderly way of working." (Arthur H Compton, "The Human Meaning of Science", 1940)

"As far as I see, such a theory [of the primeval atom] remains entirely outside any metaphysical or religious question. It leaves the materialist free to deny any transcendental Being. He may keep, for the bottom of space-time, the same attitude of mind he has been able to adopt for events occurring in non-singular places in space-time. For the believer, it removes any attempt to familiarity with God, as were Laplace’s chiquenaude or Jeans’ finger. It is consonant with the wording of Isaiah speaking of the 'Hidden God' hidden even in the beginning of the universe […] Science has not to surrender in face of the Universe and when Pascal tries to infer the existence of God from the supposed infinitude of Nature, we may think that he is looking in the wrong direction." (Monsignor G Lemaître, "The Primeval Atom Hypothesis and the Problem of Clusters of Galaxies", [in R. Stoops (ed.), "La Structure et l'Evolution de l'Univers"] 1958)

"Laplace's equation does not yield easily to straightforward treatment: fortunately in the development of conformal transformations there is no need to seek a formal solution of the equation. It is only necessary to note first that all fields and functions to be considered in this book are those that satisfy the inverse square equation when emanating from a point source and therefore they also satisfy Laplace's equation. The second point is that the equation is necessary for the development of other important equations that govern these particular fields. Such fields are called Laplacian. When the field is not Laplacian, more recondite methods are necessary for determining its distribution." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Why should a geometer, whose principal concern is in measurements of distance, desire to engage in analysis on a Riemannian manifold? For example, pondering the Laplacian, its eigenvalues and eigenfunctions? Here are some reasons, chosen from among many others. We note also here that the existence of a canonical elliptic differential operator on any Riemannian manifold, one which is moreover easy to define and manipulate, is one of the motivations to consider Riemannian geometry as a basic field of investigation. [...] Riemannian geometry is by its very essence differential, working on manifolds with a differentiable structure. This automatically leads to analysis. It is interesting to note here that, historically, many great contributions to the field of Riemannian geometry came from analysts." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

Jean-Baptiste-Joseph Fourier - Direct Perspectives

"It is true that Fourier had the opinion that the principal object of mathematics was public use and the explanation of natural phenomena; but a philosopher like him ought to know that the sole object of the science is the honor of the human spirit and that under this view a problem of [the theory of] numbers is worth as much as a problem on the system of the world." (Carl G J Jacobi [letter to Adrien-Marie Legendre], 1830)

"Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics." (Bernhard Riemann, 1854)

"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems." (Florian Cajori, "A History of Mathematics", 1893)

"It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment. - judgment, that is, in matters not admitting of certainty - balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)

"To understand, how noise is related to scale-freeness, we have to do some mathematics again. Noise is usually characterized by a mathematical trick. The seemingly random fluctuation of the signal is regarded as a sum of sinusoidal waves. The components of the million waves giving the final noise structure are characterized by their frequency. To describe noise, we plot the contribution (called spectral density) of the various waves we use to model the noise as a function of their frequency. This transformation is called a Fourier transformation [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"It has been said that the three most effective problem-solving devices in mathematics are calculus, complex variables, and the Fourier transform." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)

"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)

On Edward Beltrami - Historical Perspectives

"Early mathematicians had no difficulty accepting the first four of Euclid’s axioms. But the fifth was thought not to be so obvious as the other four, and mathematicians tried into the 19th century to derive the fifth postulate from the first four. All such attempts were doomed to failure, but in making the effort, investigators developed a great deal of modern mathematics. It was finally discovered (in the 19th century by Bolyai, Gauss, Lobachevsky, Beltrami, and others) that the fifth postulate is in fact independent of the first four and that one gets a perfectly legitimate geometry by discarding Euclid’s parallel axiom and replacing it with a different one: 'The hyperbolic parallel axiom: Given a line l and a point P not on l, there are infinitely many lines through P that are parallel to l.'" (Bruce Crauder et al, "Functions and Change: A Modeling Approach to College Algebra and Trigonometry", 2008)

"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"Beltrami started this train of thought in 1865, by asking which surfaces can be mapped onto the plane in such a way that their geodesics go to straight lines. He found that the answer is precisely the surfaces of constant Gaussian curvature. For example, great circles on the sphere can be mapped to lines on the plane, and the map that does the trick (to be precise, for the hemisphere) is central projection [...]. Rays from the center O to any great circle form a plane, which of course meets any other plane in a line. Thus projection from O to a plane sends great circles to lines, though only half of the sphere is mapped." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"When real numbers are used as coordinates, the number of coordinates is the dimension of the geometry. This is why we call the plane two-dimensional and space three-dimensional. However, one can also expect complex numbers to be useful [...]. What is remarkable is that complex numbers are if anything more appropriate for spherical and hyperbolic geometry than for Euclidean geometry. With hindsight, it is even possible to see hyperbolic geometry in properties of complex numbers that were studied as early as 1800, long before hyperbolic geometry was discussed by anyone. This was noticed by the third great contributor to non-Euclidean geometry after Beltrami and Klein - the French mathematician Henri Poincaré [...]" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"[...] quaternions give a nice approach to symmetric objects in three-dimensional space: the regular polyhedra. But this leads in turn to the regular polytopes, a family of four-dimensional symmetrical objects as remarkable as the regular polyhedra. One then becomes convinced that four-dimensional space is not just a set of quadruples; it is a world of genuine geometry." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

On Girard Desargues - Historical Perspectives

"We shall also demonstrate the following property, of which the original inventor is M. Desargues, of Lyon, one of the great minds of our time, and most versed in mathematics, amongst other topics, in conics, and whose writings on this subject, although small in number, have given ample testimony to those who have wished to receive of its knowledge. I am willing to confess that I owe the little I have found on this subject to his writings, and that I have endeavored, as far as possible, to imitate his method [...]" (Blaise Pascal, "Essais pour les coniques", 1640

"The famous geometer Desargues worked on the lines of Kepler and is now commonly credited with the authorship of some of the ideas of his predecessor. [...] the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. [...] Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity." (Charles Taylor, "The Geometry of Kepler and Newton", 1899)

"In general this geometry instead of dealing with definite triangles, polygons, circles, etc., in the Euclidean manner, is based on a consideration of all points of a straight line, of all lines through a common point and of the possible effects of setting up an orderly one-to-one correspondence between them. In particular, Desargues makes a comparative study of the different plane sections of a given cone, deducing from known properties of the circle analogous results for the other conic sections." (William T Sedgwick, "A Short History of Science", 1917)

"We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. He re-invented the epicycloid and showed its application to the construction of gear teeth, a subject elaborated more fully later by La Hire." (Florian Cajori, "A History of Mathematics", 1919)

"One of the first important steps to be taken in modern times... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the center is constant. He also treated the theory of poles and polars, although not using these terms." (David E Smith, "History of Mathematics" Vol. 1, 1923)

"Desargues the architect was doubtless influenced by what in his day was surrealism. In any event, he composed more like an artist than a geometer, inventing the most outrageous technical jargon in mathematics for the enlightenment of himself and the mystification of his disciples. Fortunately Desarguesian has long been a dead language." (Eric T Bell, "The Development of Mathematics", 1940

"[...] all the laws of algebra correspond to projective coincidences, and von Staudt showed that all the required coincidences follow from the theorems of Pappus and Desargues. Then in 1899 David Hilbert showed that all laws of algebra except the commutative law for multiplication follow from the Desargues theorem. And in 1932 Ruth Moufang showed that all except the commutative and associative laws follow from the little Desargues theorem. Thus the Pappus, Desargues, and little Desargues theorems are mysteriously aligned with the laws of multiplication!" (John Stillwell, "The Four Pillars of Geometry", 2000)

"Calculation with numbers is the obvious model for calculation with letters, but a geometric model is also conceivable, since numbers can be interpreted as lengths. Indeed, the coordinate geometry of Fermat and Descartes was based on algebra. They found that the curves studied by the Greeks can be represented by equations, and that algebra unlocks their secrets more easily and systematically than classical geometry. But to apply algebra in the first place, Fermat and Descartes assumed classical geometry. In particular, they used Euclid’s parallel axiom and the concept of length to derive the equation of a straight line,"(John Stillwell, "The Four Pillars of Geometry", 2000)

"The Pappus and Desargues theorems show that certain coincidences - three points lying on the same line - are in fact inevitable. In fact, all such coincidences can be explained as consequences of these two theorems [...] the Pappus and Desargues theorems do more than explain projective coincidences - they also explain where basic  algebra comes from!" (John Stillwell, "The Four Pillars of Geometry", 2000)

On Felix Klein - Historical Perspectives

"On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions; to determine whether on a given Riemann surface there always exists a function admitting the given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, say he, must pass, and the distribution of the current on the surface will define a function whose singularities will be precisely those called for by the enunciation." (Henri Poincaré, "La valeur de la science" ["The Value of Science"], 1904)

"It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment. - judgment, that is, in matters not admitting of certainty - balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"Symmetries of a geometric object are traditionally described by its automorphism group, which often is an object of the same geometric class (a topological space, an algebraic variety, etc.). Of course, such symmetries are only a particular type of morphisms, so that Klein’s Erlanger program is, in principle, subsumed by the general categorical approach." (Yuri I Manin, "Topics in Noncommutative Geometry", 1991)

"To summarize, affine torsion is about representing on affine space itself (i.e. on the corresponding Klein space) tiny closed curves of an affinely connected manifold and 'quantifying' its failure to close in the form of a vector, which one obtains through integration of the torsion. Let it not be forgotten that the representation of closed curves fails to close even if the torsion is zero but the affine curvature is not. But the reason for the failure to close is not the same one when the torsion is zero as when the affine curvature is zero. In general, both torsion and curvature will contribute to the failure to close." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"The Euler characteristic can be used to shorten the process, but for some cases a lengthy procedure is still necessary. Neither of these options provide a clear accounting for the ways in which the surfaces vary, e.g., which enclose cavities, which are non-orientable, etc., and neither can be completely generalized to higher-dimensional manifolds. Ideally, one would like some sort of algebraic invariant or computable quantity that would codify a lot of information: how many connected pieces a space has, how the gluing directions work, whether the surface is orientable or not, etc. The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have x = 0." (L Christine Kinsey. "Topology of Surfaces", 1993)

On David Hilbert

"Hilbert’s mathematics may be a pretty game with formulas, more amusing even than chess; but what does it have to do with knowledge, since its formulas should admittedly have no contentual significance by virtue of which they would express intuitive truths?" (Hermann Weyl, "Philosophie der Mathematik und Naturwissenschaften", 1927)

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined non-tentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920." (Eric T Bell, "The Development of Mathematics", 1940)

"Historically speaking, topology has followed two principal lines of development. In homology theory, dimension theory, and the study of manifolds, the basic motivation appears to have come from geometry. In these fields, topological spaces are looked upon as generalized geometric configurations, and the emphasis is placed on the structure of the spaces themselves. In the other direction, the main stimulus has been analysis. Continuous functions are the chief objects of interest here, and topological spaces are regarded primarily as carriers of such functions and as domains over which they can be integrated. These ideas lead naturally into the theory of Banach and Hilbert spaces and Banach algebras, the modern theory of integration, and abstract harmonic analysis on locally compact groups." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The theory of rings and ideals was put on a more systematic and axiomatic basis by Emmy Noether, one of the few great women mathematicians [...] Many results on rings and ideals were already known [...] but by properly formulating the abstract notions she was able to subsume these results under the abstract theory. Thus she reexpressed Hilbert's basic theorem [...] as follows: A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis. In this reforumulation she made the theory of invariants a part of abstract algebra." (Morris Kline, "Mathematical Thought From Ancient to Modern Times", 1972)

"The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. [...] Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa." (C West Churchman et al, "Thinking for Decisions Deduction Quantitative Methods", 1975)

"At the most elemental level, reality evanesces into something called Schröedinger's Wave Function: a mathematical abstraction which is best represented as a pattern in an infinite-dimensional space, Hilbert Space. Each point of the Hilbert Space represents a possible state of affairs. The wave function for some one physical or mental system takes the form of, let us say, a coloring in of Hilbert Space. The brightly colored parts represent likely states for the system, the dim parts represent less probable states of affairs." (Rudy Rucker, "The Sex Sphere", 1983) 

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"[...] all the laws of algebra correspond to projective coincidences, and von Staudt showed that all the required coincidences follow from the theorems of Pappus and Desargues. Then in 1899 David Hilbert showed that all laws of algebra except the commutative law for multiplication follow from the Desargues theorem. And in 1932 Ruth Moufang showed that all except the commutative and associative laws follow from the little Desargues theorem. Thus the Pappus, Desargues, and little Desargues theorems are mysteriously aligned with the laws of multiplication!" (John Stillwell, "The Four Pillars of Geometry", 2000)

On Michael Faraday - Historical Perspectives

"Taking him for all and all, I think it will be conceded that Michael Faraday was the greatest experimental philosopher the world has ever seen; and I will add the opinion, that the progress of future research will tend, not to dim or to diminish, but to enhance and glorify the labours of this mighty investigator." (John Tyndall, "Faraday as a Discoverer", 1868) 

"The contemplation of Nature, and his own relation to her, produced in Faraday, a kind of spiritual exaltation which makes itself manifest here. His religious feeling and his philosophy could not be kept apart; there was an habitual overflow of the one into the other." (John Tyndall, "Faraday as a Discoverer", 1868)

"When Faraday filled space with quivering lines of force, he was bringing mathematics into electricity. When Maxwell stated his famous laws about the electromagnetic field it was mathematics. The relativity theory of Einstein which makes gravity a fiction, and reduces the mechanics of the universe to geometry, is mathematical research." (James B Shaw, "The Spirit of Research", The Monist No. 4, 1922)

"Almost all great advances have sprung originally from disinterested motives. Scientific discoveries have been made for their own sake and not for their utilization, and a race of men without a disinterested love of knowledge would never have achieved our present scientific technique. […] Faraday, Maxwell, and Hertz, so far as can be discovered, never for a moment considered the possibility of any practical application of their investigations." (Bertrand Russell, "The Scientific Outlook", 1931)

"The concept of a field was actually introduced first in the context of electricity and magnetism by Michael Faraday as a means of 'visualizing' electromagnetic effects to replace certain mathematical ideas for which he had little talent. However, in the hands of James Maxwell, fields were molded into a physical entity having an existence of their own in the form of electromagnetic waves to be produced in 1887 by Hertz and used in 1901 by Marconi in the development of radio." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"There is in the chemist a form of thought by which all ideas become visible in the mind as strains of an imagined piece of music. This form of thought is developed in Faraday in the highest degree, whence it arises that to one who is not acquainted with this method of thinking, his scientific works seem barren and dry, and merely a series of researches strung together, while his oral discourse when he teaches or explains is intellectual, elegant, and of wonderful clearness." (Justus von Liebig) 

On Claudius Ptolemy - Historical Perspectives

"I have always read that the world, both land and water, was spherical, as the authority and researches of Ptolemy and all the others who have written on this subject demonstrate and prove, as do the eclipses of the moon and other experiments that are made from east to west, and the elevation of the North Star from north to south." (Christopher Columbus, [Letter to the Sovereigns on the Third Voyage], 1498)

"Yet the widespread planetary theories, advanced by Ptolemy and most other astronomers, although consistent with the numerical data, seemed likewise to present no small difficulty. For these theories were not adequate unless they also conceived certain equalizing circles, which made the planet appear to move at all times with uniform velocity neither on its deferent sphere nor about its own epicycle's center." (Nicolaus Copernicus, "De revolutionibus orbium coelestium" ["On the Revolutions of the Heavenly Spheres"], 1543)

"If Nicolaus Copernicus, the distinguished and incomparable master, in this work had not been deprived of exquisite and faultless instruments, he would have left us this science far more well-established. For he, if anybody, was outstanding and had the most perfect understanding of the geometrical and arithmetical requisites for building up this discipline. Nor was he in any respect inferior to Ptolemy; on the contrary, he surpassed him greatly in certain fields, particularly as far as the device of fi t ness and compendious harmony in hypotheses is concerned. And his apparently absurd opinion that the Earth revolves does not obstruct this estimate, because a circular motion designed to go on uniformly about another point than the very center of the circle, as actually found in the Ptolemaic hypotheses of all the planets except that of the Sun, offends against the very basic principles of our discipline in a far more absurd and intolerable way than does the attributing to the Earth one motion or another which, being a natural motion, turns out to be imperceptible. There does not at all arise from this assumption so many unsuitable consequences as most people think." (Tycho Brahe, [letter to Christopher Rothman] 1587)

"In astronomy the scenery is continually shifting, and the modes of language vary in proportion as this inexhaustible science makes progress in improvement, and supplies us with new theories. Ptolemy spake the language ot the people: to Copernicus we are indebted for the language of astronomy; which Tycho Brahe in some measure confounded: Kepler and Newton rectified his faults, and gave to astronomical language a superior degree of elegance and perfection. The discoveries of the present and future times will introduce in this respect farther changes. All these different modes of language will, nevertheless, continue to be always intelligible; and may always be preserved in a certain degree, and within certain limitations." (Johann H Lambert, "The System of the World", 1800)

"If the Greeks had not cultivated Conic Sections, Kepler could not have superseded Ptolemy; if the Greeks had cultivated Dynamics, Kepler might have anticipated Newton." (William Whewell, "History of the Inductive Sciences, from the Earliest to the Present Time", 1857)

"Newton's theory [of gravitation] is the circle of generalization which includes all the others [as Kepler's laws, Ptolemy's theory, etc.]; - the highest point of the inductive ascent; - the catastrophe of the philosophic drama to which Plato had prologized; - the point to which men's minds had been journeying for two thousand years." (William Whewell, "History of the Inductive Sciences, from the Earliest to the Present Time", 1857)

"All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors, and learned in what had been before them. There is not one exception. I do not say that every man has made direct acquaintance with the whole of his mental ancestry [...] But [...]it is remarkable how many of the greatest names in all departments of knowledge have been real antiquaries in their several subjects. [...] I may cite among those [...] in science, Aristotle, Plato, Ptolemy, Euclid, Archimedes, Roger Bacon, Copernicus, Francis Bacon, Ramus, Tycho Brahe, Galileo, Napier, Descartes, Leibnitz, Newton, Locke." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent." (Florian Cajori, "A History of Mathematics", 1893)

"Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates." (David E Smith, "History of Mathematics", 1923)

"Ptolemy [...] against the champions of this or that cosmology of the heavens [...] had dared to claim that it is legitimate to interpret the facts of astronomy by the simplest geometrical scheme which will 'save the phenomena', no matter whose metaphysics might be upset. His conception of the physical structure of the earth, however, prevented him from carrying through in earnest this principle of relativity, as his objections to the hypothesis that the earth moves amply show." (Edwin A Burtt, "The Metaphysical Foundations of Modern Physical Science", 1925)

"Great scientific minds, from Claudius Ptolemy of the second century to Isaac Newton of the seventeenth, invested their formidable intellects in attempts to deduce the nature of the universe from the statements and philosophies contained in religious writings. [...] Had any of these efforts worked, science and religion today might be one and the same. But they are not." (Neil deGrasse Tyson, "The Sky Is Not the Limit: Adventures of an Urban Astrophysicist", 2000)

"It is difficult to falsify the hypothesis of dark matter, because, as with Ptolemy's epicycles, true believers can always add additional arbitrary features and free parameters to overcome any conceivable difficulties that occur with the dark matter models." (John Moffat, "Reinventing Gravity", 2008)

On Archimedes of Syracuse - Historical Perspectives

"It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required." (Plutarch, cca. 1st century)

"[...] it is my opinion that this problem is what led the ancients to attempt the squaring of the circle. For if a parallelogram can be found equal to any rectilinear figure, it is worth inquiring whether it is not possible to prove that a rectilinear figure is equal to a circular area. Indeed Archimedes proved that a circle is equal to a right-angled triangle when its radius is equal to one of the sides about the right angle and its perimeter is equal to the base. But of this elsewhere." (Proclus Lycaeus, cca 5th century)

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent." (Florian Cajori, "A History of Mathematics", 1893)

"Eudoxes […] not only based the method [of exhaustion] on rigorous demonstration […] but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus, though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus […] another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, 'seems to have thought of everything' […] Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as equal or unequal [...] Democritus was already close on the track of infinitesimals." (Thomas L Heath, "Achimedes", 1920)

"Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates." (David E Smith, "History of Mathematics", 1923)

"Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub." (Richard J Trudeau, "Dots and Lines", 1976)

"The absolute scholar is in fact a rather uncanny being. He is instinct with Nietzsche's finding that to be interested in something, to be totally interested in it, is a libidinal thrust more powerful than love or hatred, more tenacious than faith or friendship - not infrequently, indeed, more compelling than personal life itself. Archimedes does not flee from his killers, he does not even turn his head to acknowledge their rush into his garden when he is immersed in the algebra of conic sections." (George Steiner, "The Cleric of Treason", 1980)

"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Once a mathematical result is proven to the satisfaction of the discipline, it doesn’t need to be re-evaluated in the light of new evidence or refuted, unless it contains a mistake. If it was true for Archimedes, then it is true today." (Peter Rowlett, "The Unplanned Impact of Mathematics", Nature, 2011)

On Apollonius of Perga - Historical Perspectives

"The so-called 'Treasury of Analysis' [also Heuristic] is, to put it shortly, a special body of doctrine for the use of those who, after having studied the ordinary Elements, are desirous of acquiring the ability to solve mathematical problems, and it is useful for this alone. It is the work of three men, Euclid, the author of the Elements, Apollonius of Perga, and Aristaeus the elder. It teaches the procedures of analysis and synthesis." (Pappus of Alexandria, cca. 4th century BC)

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent." (Florian Cajori, "A History of Mathematics", 1893)

“Any satisfactory reproduction of the Conics must fulfil certain essential conditions: (1) it should be Apollonius and nothing but Apollonius, and nothing should be altered either in the substance or in the order of his thought, (2) it should be complete, leaving out nothing of any significance or importance, (3) it should exhibit under different headings the successive divisions of the subject, so that the definite scheme followed by the author may be seen as a whole." (Thomas L Heath, "Apollonius of Perga", 1896)

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs some times used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree." (Florian Cajori, "A History of Mathematics", 1893)

"Perhaps nowhere does one find a better example of the value of historical knowledge for mathematicians than in the case of Fermat, for it is safe to say that, had he not been intimately acquainted with the geometry of Apollonius and Viéte, he would not have invented analytic geometry." (Carl B Boyer, "History of Analytic Geometry", 1956)

"We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies - such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later." (Arthur Koestler, "The Sleepwalkers: A History of Man's Changing Vision of the Universe", 1959)

On Adrien-Marie Legendre - Historical Perspectives

"It is Legendre’s eternal glory to have discovered the kernel of an important branch of analysis and by the work of half a lifetime to have erected on these foundations an independent theory. [...] Only with the continued determination that enabled the great mathematician to return again and again to the subject was he able to overcome difficulties that, with the means he had at his command, must have seemed scarcely subduable." (P G Lejeune Dirichlet, 1852)

"Legendre's Law of Quadratic Reciprocity […] is […] the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic', and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. […] we find in the 'Opuscula Analytica' of Euler […] a memoir […] which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler […] expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined […] that he had obtained a satisfactory demonstration." (Henry J S Smith, "Report on the Theory of Numbers", 1859)

"The fear that the mathematician most chiefly concerned with the determination of numerical values had of the imaginary, was the reason that Legendre was prevented from taking the most important step in modern analysis, the introduction of doubly periodic functions." (Jacobi, 1832)

"Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; [...]" (James J Sylvester,"A Plea for the Mathematician", 1869)

"It is well known that the reluctance of Gauss to publish his discoveries was due to the rejection of his Disquisitiones arithmeticae by the French Academy, the rejection being accompanied by a sneer which, as Rouse Ball has said, would have been unjustifiable even if the work had been as worthless as the referees believed. It is the irony of fate that, but for this sneer, the Traite des fonctions elliptiques, the work of a Frenchman, might have assumed a different and vastly more valuable form, and Legendre might have been spared the pain of realizing that many years of his life had been practically wasted, had the method of inversion come to be published when Legendre’s age was fifty instead of seventy-six." (George N Watson, "The Marquis and the Land-Agent; A Tale of the Eighteenth Century", Math. Gazette 17, 1933)

On William K Clifford - Historical Perspectives

 "It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment. - judgment, that is, in matters not admitting of certainty - balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this kind." (Shiing-Shen Chern, "On Riemannian manifolds of four dimensions". Bulletin of the American Mathematical Society 51 (12), 1945)

"Hermann Grassmann made substantial contributions to algebra and geometry during the 19th century. His ideas were so advanced that many of his colleagues failed to recognize their merit, but later generations quickly gravitated toward Grassmann’s highly abstract and beautiful work. [...] Grassmann also invented the concept of an exterior algebra - another algebra with a special product called the exterior product. This abstract structure was related to the quaternions of Sir William Rowan Hamilton, and was later developed by William Clifford into a tool that has been quite useful in quantum mechanics. The exterior algebra is an important object of study in modern differential geometry. Grassmann’s ideas were quite advanced for his time, and they were accepted slowly [...]" (Tucker McElroy, "A to Z of Mathematicians", 2005)

"The idea of Clifford algebra is basic for Dirac’s theory of the relativistic electron, and hence it is crucial for the fundamental fermions in the Standard Model in particle physics." (Eberhard Zeidler, "Quantum Field Theory I: Gauge Theory", 2006)

"Geometric algebra combines the algebraic structure of Clifford algebra with the explicit geometric meaning of its mathematical elements at its foundation. So, formally, it is Clifford algebra endowed with geometrical information of and physical interpretation to all mathematical elements of the algebra. This intrusion of geometric consideration into the abstract system of Clifford algebra has enriched geometric algebra as a powerful mathematical theory." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

On François Viète - Historical Perspectives

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." (Pierre de Fermat, [Letter to Frénicle] 1657)

"In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question." (Florian Cajori, "A History of Mathematics", 1893)

"The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa." (Florian Cajori, "A History of Mathematics", 1893)

"Letters had been used before Vieta to denote numbers, but he introduced the practice for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation." (Eric T Bell, "The Development of Mathematics", 1940)

"Vieta's principle advance in trigonometry was his systematic application of algebra. [...] he worked freely with all six of the usual functions, and [...] obtained many of the fundamental identities algebraically. With Vieta, elementary (non-analytic) trigonometry was practically completed except on the computational side. All computation was greatly simplified early in the seventeenth century by the invention of logarithms." (Eric T Bell, "The Development of Mathematics", 1940)

"Perhaps nowhere does one find a better example of the value of historical knowledge for mathematicians than in the case of Fermat, for it is safe to say that, had he not been intimately acquainted with the geometry of Apollonius and Viéte, he would not have invented analytic geometry." (Carl B Boyer, "History of Analytic Geometry", 1956)

"Vieta used letters to denote numbers, whereas Descartes introduced letters to denote line segments. Vieta studied rules for manipulating numbers in an abstract manner, and Descartes accepted the existence of similar rules for manipulating line segments and greatly improved symbolism and algebraic technique. Thus, it seemed that numbers might be put into one-to-one correspondence with points on a geometric line, leading to a significant step in the evolution of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

On Blaise Pascal - Historical Perspectives

"Pascal's Thoughts are not a book. We must bear this observation in mind, if we are to judge of them aright. They are not a book; perhaps they may be two, or even more. They are — if we must give them a name and define them - they are Pascal himself - whole Pascal, excepting in so far as he was a geometrician, properly so called, and a natural philosopher. The Thoughts are only the papers on which this great man threw out, from time to time, all that occupied his powerful mind, until the excess of physical malady reduced him to complete inaction, and put, so to speak, the seals upon his genius. Great pains have been taken, and not without success, to reduce these scattered materials, by means of art, into a kind of whole. Sometimes, perhaps, the secret of the writer has been guessed; possibly, in certain cases, his intention has been entirely misunderstood." (Alexandre R Vinet, "Studies on Pascal", 1859

"One striking peculiarity of mathematics is its unlimited power of evolving examples and problems. A student may read a book of Euclid, or a few chapters of Algebra, and within that limited range of knowledge it is possible to set him exercises as real and as interesting as the propositions themselves which he has studied; deductions which might have pleased the Greek geometers, and algebraic propositions which Pascal and Fermat would not have disdained to investigate." (Isaac Todhunter, "Private Study of Mathematics"  [in "Conflict of Studies and other Essays" 1873)

"Most important for the history of science is the fact that Liber de Ludo Aleae,'The Book of Games of Chance', contains the first study of the principles of probability. [...] it would seem much more just to date the beginnings of probability theory from Cardano's treatise rather than the customary reckoning from Pascal's discussions with his friend de Méré and the ensuing correspondence with Fermat [...] at least a century after Cardano [...]" (Oystein Ore [Ed.], "Cardano the Gambling Scholar", 1953)

"Pascal was no obscurantist, but he measured how far science (esprit de géometrie) may go and become irrelevant to the problems of human destiny. The achievements of science are not denied by this stand, nor are they dismissed as the ostrich with its head burrowed in sand dismisses reality. What is asserted is that our original endowments as they reflect on the data of experience, are, with all their limitations and imperfections, the only necessary and indispensable means by which we confront existence, the only weapon commensurate with our struggle." (Thomas Molnar, "The Decline of the Intellectual", 1961)

"To philosophize is always to rehabilitate the essential importance of the human dimension, and hence the dignity of man. This was the meaning of the Socratic quest, and also the meaning of Pascal's anguish at the threshold of the Cartesian revolution in science. In this respect, Pascal is an even more significant figure of philosophy than Socrates: his was the first brutal reaction to the claim of science to remove man from his position as a privileged being in creation; nobody has so far been able to improve upon his demonstration why science is unsatisfactory to give man meaning and to fill his heart with joy." (Thomas Molnar, "The Decline of the Intellectual", 1961)

"Pascal's Triangle and all the early work in probability answered only one question: what is the probability of such-and-such an outcome? The answer to that question has limited value in most cases, because it leaves us with no sense of generality." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996) 

On Girolamo Cardano - Historical Perspectives

"I had scarcely entered the room when I saw on my table an open book I had not put there. It was the works of Cardano. I did not intend to read it, but my gaze fell as though compelled on a story told by that philosopher. He writes that he was studying one night by candlelight when he saw two tall old men come in through the closed doors of his room. He asked them many questions, and they finally told him they were from the Moon; whereupon they disappeared. I was so surprised, both by the book that had put itself on my table and by the page it was open to, that I took this chain of events for an inspiration from God, who was urging me to tell people that the Moon is a world." (Cyrano de Bergerac, "The Other World", 1657)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

" It appears [...] from this short chapter [Ars Magna, lib x. ch. 1], that he had discovered most of the principal properties of the roots of equations, and could point out the number and nature of the roots, partly from the signs of the terms, and partly from the magnitude and relations of the co-efficients." (Charles Hutton, "Algebra" [in "A Mathematical and Philosophical Dictionary", 1795)

"The Ars Magna is a great advance on any algebra previously published. Hitherto algebraists had confined their attention to those roots of equations which were positive. Cardan discussed negative and even complex roots, and proved that the latter would always occur in pairs, though he declined to commit himself to any explanation as to the meaning of these 'sophistic' quantities which he said were ingenious though useless." (W W Rouse Ball, "A Short Account of the History of Mathematics", 1888)

"The application of the theory [of probability] to mortality tables in any large way may be said to have started with John Graunt [...] The first tables of great importance, however, were those of Edmund Halley [...] however [...] Cardan seems to have been the first to have been the first to consider the problem in a printed work, although his treatment is very fanciful. He gives a brief table in his proposition 'Spatium vitae naturalis per spatium vitae fortuitum declarare', this appearing in the De Proportionibus Libri V [...]" (David E Smith (Ed.), "History of Mathematics" Vol. 2, 1925)

"The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra." (David E Smith (Ed.), "History of Mathematics" Vol. 2, 1925)

"Most important for the history of science is the fact that Liber de Ludo Aleae,'The Book of Games of Chance', contains the first study of the principles of probability. [...] it would seem much more just to date the beginnings of probability theory from Cardano's treatise rather than the customary reckoning from Pascal's discussions with his friend de Méré and the ensuing correspondence with Fermat [...] at least a century after Cardano [...]" (Oystein Ore [Ed.], "Cardano the Gambling Scholar", 1953)

"Cardano reasoned that the end of man is to know God and to mediate between the divine and the mortal. The rational soul is immortal and when permeated with light is inseparable from God. True wisdom is gained from union with God and by mathematics, as God has subjected the world to mathematical law." (Albert E Avey, "Handbook in the History of Philosophy", 1954) 

"Insolving the cubic and quartic equations, Cardan and Ferrari implicitly assumed the existence of roots [...]. Girard, in the early seventeenth century, was one of the first to assert the existence of roots, real or imaginary, though 'imaginary root' did not have a clear meaning in his work. As people became more comfortable with complex numbers, the existence of roots evolved into the existence of complex roots, which come in complex conjugate pairs when the coefficients are real. Thus the eighteenth-century version of the Fundamental Theorem of Algebra asserts that every nonconstant polynomial in R[x] factors into linear and quadratic factors with coefficients in R." (David A Cox, "Galois Theory" 2nd Ed., 2012)

On Johannes Kepler - Historical Perspectives

"In astronomy the scenery is continually shifting, and the modes of language vary in proportion as this inexhaustible science makes progress in improvement, and supplies us with new theories. Ptolemy spake the language ot the people: to Copernicus we are indebted for the language of astronomy; which Tycho Brahe in some measure confounded: Kepler and Newton rectified his faults, and gave to astronomical language a superior degree of elegance and perfection. The discoveries of the present and future times will introduce in this respect farther changes. All these different modes of language will, nevertheless, continue to be always intelligible; and may always be preserved in a certain degree, and within certain limitations." (Johann H Lambert, "The System of the World", 1800)

"Simple as the law of gravity now appears, and beautifully in accordance with all the observations of past and of present times, consider what it has cost of intellectual study. Copernicus, Galileo, Kepler, Euler, Lagrange, Laplace, all the great names which have exalted the character of man, by carrying out trains of reasoning unparalleled in every other science; these, and a host of others, each of whom might have been the Newton of another field, have all labored to work out, the consequences which resulted from that single law which he discovered. All that the human mind has produced - the brightest in genius, the most persevering in application, has been lavished on the details of the law of gravity." (Charles Babbage, "The Ninth Bridgewater Treatise: A Fragment", 1838)

"If the Greeks had not cultivated Conic Sections, Kepler could not have superseded Ptolemy; if the Greeks had cultivated Dynamics, Kepler might have anticipated Newton." (William Whewell, "History of the Inductive Sciences, from the Earliest to the Present Time", 1857)

"It is a vulgar belief that our astronomical knowledge dates only from the recent century when it was rescued from the monks who imprisoned Galileo; but Hipparchus [...] who among other achievements discovered the precession of the eqinoxes, ranks with the Newtons and the Keplers; and Copernicus, the modern father of our celestial science, avows himself, in his famous work, as only the champion of Pythagoras, whose system he enforces and illustrates. Even the most modish schemes of the day on the origin of things, which captivate as much by their novelty as their truth, may find their precursors in ancient sages, and after a careful analysis of the blended elements of imagination and induction which charaterise the new theories, they will be found mainly to rest on the atom of Epicurus and the monad of Thales. Scientific, like spiritual truth, has ever from the beginning been descending from heaven to man." (Benjamin Disraeli, "Lothair", 1879)

"The famous geometer Desargues worked on the lines of Kepler and is now commonly credited with the authorship of some of the ideas of his predecessor. [...] the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. [...] Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity." (Charles Taylor, "The Geometry of Kepler and Newton", 1899)

"It has been said that no science is established on a firm basis unless its generalisations can be expressed in terms of number, and it is the special province of mathematics to assist the investigator in finding numerical relations between phenomena. After experiment, then mathematics. While a science is in the experimental or observational stage, there is little scope for discerning numerical relations. It is only after the different workers have 'collected data' that the mathematician is able to deduce the required generalisation. Thus a Maxwell followed Faraday and a Newton completed Kepler." (Joseph W Mellor, "Higher Mathematics for Students of Chemistry and Physics", 1902)

"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact." (Albert Einstein, 1930)

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

On Nicolaus Copernicus - Historical Perspectives

"If Nicolaus Copernicus, the distinguished and incomparable master, in this work had not been deprived of exquisite and faultless instruments, he would have left us this science far more well-established. For he, if anybody, was outstanding and had the most perfect understanding of the geometrical and arithmetical requisites for building up this discipline. Nor was he in any respect inferior to Ptolemy; on the contrary, he surpassed him greatly in certain fields, particularly as far as the device of fi t ness and compendious harmony in hypotheses is concerned. And his apparently absurd opinion that the Earth revolves does not obstruct this estimate, because a circular motion designed to go on uniformly about another point than the very center of the circle, as actually found in the Ptolemaic hypotheses of all the planets except that of the Sun, offends against the very basic principles of our discipline in a far more absurd and intolerable way than does the attributing to the Earth one motion or another which, being a natural motion, turns out to be imperceptible. There does not at all arise from this assumption so many unsuitable consequences as most people think." (Tycho Brahe, [letter to Christopher Rothman] 1587)

"In astronomy the scenery is continually shifting, and the modes of language vary in proportion as this inexhaustible science makes progress in improvement, and supplies us with new theories. Ptolemy spake the language ot the people: to Copernicus we are indebted for the language of astronomy; which Tycho Brahe in some measure confounded: Kepler and Newton rectified his faults, and gave to astronomical language a superior degree of elegance and perfection. The discoveries of the present and future times will introduce in this respect farther changes. All these different modes of language will, nevertheless, continue to be always intelligible; and may always be preserved in a certain degree, and within certain limitations." (Johann H Lambert, "The System of the World", 1800)

"The system of Copernicus is, in fact, only a theory; but we have seen that astronomy can come at the truth only by carefully bringing under a review every possible hypothesis.This science, however, has made astonishing progress; by converting, at different periods apparent arrangements into such as are more conformable to what actually exista, we have left behind various theories founded in appearances, and penetrated, if not fully and demonstrably, at least in the way ot fair conjecture, even to the real and genuine order of things." (Johann H Lambert, "The System of the World", 1800)

"It is a vulgar belief that our astronomical knowledge dates only from the recent century when it was rescued from the monks who imprisoned Galileo; but Hipparchus [...] who among other achievements discovered the precession of the eqinoxes, ranks with the Newtons and the Keplers; and Copernicus, the modern father of our celestial science, avows himself, in his famous work, as only the champion of Pythagoras, whose system he enforces and illustrates. Even the most modish schemes of the day on the origin of things, which captivate as much by their novelty as their truth, may find their precursors in ancient sages, and after a careful analysis of the blended elements of imagination and induction which charaterise the new theories, they will be found mainly to rest on the atom of Epicurus and the monad of Thales. Scientific, like spiritual truth, has ever from the beginning been descending from heaven to man." (Benjamin Disraeli, "Lothair", 1879)

"Copernicus and Lobatchewsky were both of Slavic origin. Each of them has brought about a revolution in scientific ideas so great that it can only be compared with that wrought by the other. And the reason of the transcendent importance of these two changes is that they are changes in the conception of the Cosmos. [...] Now the enormous effect of the Copernican system, and of the astronomical discoveries that have followed it, is […] the change effected by Copernicus in the idea of the universe. But there was left another to be made. For the laws of space and motion. […] So, you see, there is a real parallel between the work of Copernicus and […] the work of Lobatchewsky." (William K Clifford, "The Postulates of Time And Space", [in "Lectures and Essays" Vol. 1] 1901)

"The present revolution of scientific thought follows in natural sequence on the great revolutions at earlier epochs in the history of science. Einstein’s special theory of relativity, which explains the indeterminateness of the frame of space and time, crowns the work of Copernicus who first led us to give up our insistence on a geocentric outlook on nature; Einstein's general theory of relativity, which reveals the curvature or non-Euclidean geometry of space and time, carries forward the rudimentary thought of those earlier astronomers who first contemplated the possibility that their existence lay on something which was not flat. These earlier revolutions are still a source of perplexity in childhood, which we soon outgrow; and a time will come when Einstein’s amazing revelations have likewise sunk into the commonplaces of educated thought." (Arthur S Eddington, "The Theory of Relativity and its Influence on Scientific Thought", 1922)