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 Geometrical Figures: On Parabola

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient." (Proclus Lycaeus, cca 5th century)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations." (Isaac Newton, "Arithmetica Universalis", 1707)

"Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes - function theory or analysis - considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola. [...] If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates, - reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology […] bears to anatomy." (Christian H Dillmann," Die Mathematik die Fackelträgerin einer neuen Zeit", 1889)

"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)

On Probability (1825-1849)

"The calculus of probability is equally applicable to things of all kinds, moral and physical and, if only in each case observations provide the necessary numerical data, it does not at all depend on their nature." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The probability of an event is our reason to believe that it will occur or occurred. [...] Probability depends on our knowledge about an event; for the same event it can differ for different persons. Thus, if a person only knows that an urn contains white and black balls, whereas another person alsoknows that there are more white balls than black ones, the latter has more grounds to believe in the extraction of a white ball. In other words, for him, that event has a higher probability than for the former." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen)." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"I consider the world probability as meaning the state of mind with respect to an assertion, a coming event, or any other matter on which absolute knowledge does not exist." (Augustus De Morgan, "Essay on Probability", 1838)

"By degree of probability we really mean, or ought to mean, degree of belief [...] Probability then, refers to and implies belief, more or less, and belief is but another name for imperfect knowledge, or it may be, expresses the mind in a state of imperfect knowledge. (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

"Without doubt, matter is unlimited in extent, and, in this sense, infinite; and the forces of Nature mould it into an innumerable number of worlds. Would it be at all astonishing if, from the universal dice-box, out of an innumberable number of throws, there should be thrown out one world infinitely perfect? Nay, does not the calculus of probabilities prove to us that one such world out of an infinite number, must be produced of necessity? (Philippe Buchez & William B Greene, "Remarks on the Science of History: Followed by an a priori autobiography", 1849)

On Geometrical Figures: On Ellipses

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient."  (Proclus Lycaeus, cca 5th century)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations." (Isaac Newton, "Arithmetica Universalis", 1707)

"The perturbations which the motions of planets suffer from the influence other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion." (Carl F Gauss, "Theoria motus corporum coelestium in sectionibus conicis solem ambientium", 1809)

"Beauty cannot be defined by abscissas and ordinates; neither are circles and ellipses created by their geometrical formulas." (Carl von Clausewitz, "On War", 1832)

"But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is analysis situs. In this discipline, two figures are equivalent whenever one can pass from one to the other by a continuous deformation; whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight-line segment since this segment is not closed. A sphere is equivalent to any convex surface; it is not equivalent to a torus since there is a hole in a torus and in a sphere there is not. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight lines drawn by an inexperienced hand have suffered unfortunate deviations and contain awkward bends. From the point of view of metric geometry, and even of projective geometry, the two figures are not equivalent; on the contrary, from the point of view of analysis situs, they are.” (Henri Poincaré, “Dernières pensées”, 1913)

"A circle no doubt has a certain appealing simplicity at the first glance, but one look at a healthy ellipse should have convinced even the most mystical of astronomers that that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has nothing to say." (Eric T Bell, "The Handmaiden of the Sciences", 1937)

"Since the ellipse is a closed curve it has a total length, λ say, and therefore f(l + λ) = f(l). The elliptic function f is periodic, with 'period' λ, just as the sine function is periodic with period 2π. However, as Gauss discovered in 1797, elliptic functions are even more interesting than this: they have a second, complex period. This discovery completely changed the face of calculus, by showing that some functions should be viewed as functions on the plane of complex numbers. And just as periodic functions on the line can be regarded as functions on a periodic line - that is, on the circle - elliptic functions can be regarded as functions on a doubly periodic plane - that is, on a 2-torus." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

On Geometrical Figures: On Conics

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient."  (Proclus Lycaeus, cca 5th century)

"Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth." (René Descartes, "La Géométrie", 1637)

"The perturbations which the motions of planets suffer from the influence other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion." (Carl F Gauss, "Theoria motus corporum coelestium in sectionibus conicis solem ambientium", 1809)

"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)

"No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without regard to any utility other than to satisfy the craving for knowledge." (Alfred N Whitehead, " An Introduction to Mathematics", 1911)

"The binomial theorem, trigonometry, conic sections, and all the rest of the higher mathematics are fields of knowledge that can be acquired with dreary labour by anyone who persistently applies his mind to them." (Stephen Coleridge, "The Idolatry of Science, 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)

"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)

"Nature does not seem full of circles and triangles to the ungeometrical; rather, mastery of the theory of triangles and circles, and later of conic sections, has taught the theorist, the experimenter, the carpenter, and even the artist to find them everywhere, from the heavenly motions to the pose of a Venus." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"[...] if two conics have five points in common, then they have infinitely many points in common. This geometric theorem is somewhat subtle but translates into a property of solutions of polynomial equations that makes more natural sense to a modern mathematician." (David Ruelle, “The Mathematician's Brain”, 2007)

On Geometrical Figures: On Hyperbolas

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient." (Proclus Lycaeus, cca 5th century)

"Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many, - circle, parabola, hyperbola, ellipse, etc." (Pierre de Fermat, "Introduction aux Lieux Plans et Solides", 1679)

"The operations performed with imaginary characters, though destitute of meaning themselves, are yet notes of reference to others which are significant. They, point out indirectly a method of demonstrating a certain property of the hyperbola, and then leave us to conclude from analogy, that the same property belongs also to the circle. All that we are assured of by the imaginary investigation is, that its conclusion may, with all the strictness of mathematical reasoning, be proved of the hyperbola; but if from thence we would transfer that conclusion to the circle, it must be in consequence of the principle just now mentioned. The investigation therefore resolves itself ultimately into an argument from analogy; and, after the strictest examination, will be found without any other claim to the evidence of demonstration." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"It is well known that an initial value problem for a nonlinear ordinary differential equation may very well fail to have a solution for all time; the solution may blow up after a finite time. The same is true for quasi-linear hyperbolic partial differential equations: solutions may break down after a finite time when their first derivatives blow up." (Peter D Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves", 1974)

"A surface which can be regarded as the set of successive position of a curve moving in space is said to be generated by the curve. The utility of this notion in constructing a surface geometrically, in a picture or as a model is increased as the complexity of the generator and its motion is decreased. When the generator is a straight line, it is called a ruled surface. Since you can exchange X and Y in the above analysis, the hyperbolic paraboloid is generated by a line in two ways. It is a doubly ruled surface." (George K Francis, "A Topological Picturebook", 1987)

"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)

"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

On Geometry: On Hyperbolic Geometry

"Historically, hyperbolic geometry was first developed on an axiomatic basis. It arose as a result of efforts to prove the axiom of parallels from the other axioms. Doubt persisted for a long time as to whether this axiom could be deduced from the remaining axioms of Euclidean geometry. In their attempts to prove this axiom, mathematicians used the method of 'proof by contradiction' i.e., they assumed that the axiom of parallels was false and tried, on the basis of this assumption, to obtain a contradiction. All of these attempts were fruitless. True, the theorems obtained by negating the axiom of parallels appeared strange, but they did not contradict one another. The issue was resolved when C. F. Gauss, N. I. Lobachevski and J. Bolyai first stated explicitly that by negating the axiom of parallels one arrives at a new geometry, just as consistent as the usual (Euclidean) geometry." (Isaak Yaglom, "Geometric Transformations", 1973)

"Hyperbolic geometry is exceptional among non-Euclidean geometries because it satisfies all the axioms of Euclidean geometry except for the axiom of parallels. Other non-Euclidean geometries differ more radically from Euclidean geometry; in some of them a line segment cannot be produced indefinitely in both directions, and in others, two points cannot always be joined by a line." (Isaak Yaglom, "Geometric Transformations", 1973)

"We shall now compare the non-Euclidean geometry of Lobachevski-Bolyai with the geometry of Euclid studied in high school. One is immediately struck by how much the two geometries have in common. In both geometries two points determine a unique line, and two lines can have at most one point in common (this follows from the fact that lines of hyperbolic geometry are segments of lines in the plane). Further, in both geometries it is possible to carry a point and a ray issuing from it, by a motion, into any other point and a preassigned ray issuing from the latter point. The hyperbolic length of a segment and magnitude of an angle share many properties with their Euclidean counterparts; for example, in both geometries the length of the sum of two segments is the sum of their lengths, and the measure of the sum of two angles is equal to the sum of their measures." (Isaak Yaglom, "Geometric Transformations", 1973)

"Together with plane Euclidean geometry, spherical and hyperbolic geometry are 2-dimensional geometries with the following properties: (1) distance, lines and angles are defined and invariant under motions; (2) the motions act transitively on points and directions at a point; (3) locally, incidence properties are as in plane Euclidean geometry." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"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, knowing their geometric properties […] 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." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"[...] the use of complex numbers reveals a connection between the exponential, or power function and the seemingly unrelated trigonometric functions. Without passing through the portal offered by the square root of minus one, the connection may be glimpsed, but not understood. The so-called hyperbolic functions arise from taking what are known as the even and odd parts of the exponential function." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Visions of Infinity", 2013)

"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

On Number Theory (2000-)

"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, "Uncle Petros and Goldbach's Conjecture", 2000)

"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Transcendental numbers then are numerous but exceedingly slippery. As a rule of thumb, a number that arises in mathematics is almost always transcendental unless it is obvious that it is not. However, showing that a particular number is transcendental can be exceedingly difficult. Number theory throws up endless problems of this kind where everyone feels sure what the answer must be but at the same time no-one has any real idea how it could ever by proved." (Peter M. Higgins, "Number Story: From Counting to Cryptography", 2008)

"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"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)

"While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers [...]. The two merge when mathematicians look for number-theoretic patterns hidden in those sequences." (Kelsey Houston-Edwards, "Mathematicians Set Numbers in Motion to Unlock Their Secrets", Quanta Magazine, 2021)

On Number Theory (1975-1999)

"The theory of number is the epipoem of mathematics." (Scott Buchanan, "Poetry and Mathematics", 1975)

"[Number theory] produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers; and yet…number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!" (Barry Mazur, "Number Theory as Gadfly", The American Mathematical Monthly, Volume 98, 1991)

"[…] number theory […] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato’s shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 5 July, 1993)

"Number theory [...] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato's shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 1993)

"At first glance the theory of numbers is deprived of any geometricity. But this is actually not the case. At the contemporary stage of development of computers it has become possible to explain to a wide range of readers that visual geometry helps not only to illustrate some abstract situations from the number theory, but sometimes also to solve new problems." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Modem geometry and topology take a special place in mathematics because many of the objects they deal with are treated using visual methods. […] Each mathematician has his own system of concepts of the intrinsic geometry of his (specific) mathematical world and visual images which he associated with some or other abstract concepts of mathematics (including algebra, number theory, analysis, etc.). It is noteworthy that sometimes one and the same abstraction brings about the same visual picture in different mathematicians, but these pictures born by imagination are in most cases very difficult to represent graphically, so to say, to draw." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Number theory is so difficult, albeit so fascinating, because mathematicians try to examine additive creations under a multiplicative light." (William Dunham, 1994)

"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." (Andreas Knauf, "Number Theory, Dynamical Systems and Statistical Mechanics", 1998)

On Number Theory (1950-1974)

"On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory." (Kurt Gödel, 1951)

"The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. ‘Perfect numbers’ certainly never did any good, but then they never did any particular harm." (John E Littlewood, "A Mathematician’s Miscellany", 1953)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"The modern era has uncovered for combinatorics a wide range of fascinating new problems. These have arisen in abstract algebra, topology, the foundations of mathematics, graph theory, game theory, linear programming, and in many other areas. Combinatorics has always been diversified. During our day this diversification has increased manifold. Nor are its many and varied problems successfully attacked in terms of a unified theory. Much of what we have said up to now applies with equal force to the theory of numbers. In fact, combinatorics and number theory are sister disciplines. They share a certain intersection of common knowledge, and each genuinely enriches the other." (Herbert J Ryser, "Combinatorial Mathematics", 1963)

"No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integerst hat refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are ‘undecidable’. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)