Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2015)
Mathematics at infinity
Jeremy Gray1, 2
Department of Mathematics and Statistics, The Open University, Milton Keynes, UK
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Points just beyond the plane, or points that have been lost by a transformation, were often rationalised as points at infinity, but this term meant different things to different people at different times. It was a figure of speech, a purely formal claim, and a claim that there are new real objects in geometry, before it became assimilated into the modern language concerning mathematical existence.
This essay looks for points, lines, and circles at infinity, as presented by the likes of Desargues, Newton, Euler, Poncelet, Plücker, Cayley, and others.
The existence of mathematical objects
Naively, mathematical objects have been taken to exist because they are somehow out there in the world, or (which is not exactly the same thing) because what we say or know about them helps us do things successfully that we could otherwise not do at all. This is discussed much more fully in the essays by Avigad and Gillies in this volume, and can only be taken as background here. But it is worth noting historically that explicit attention to mathematical ontology has been much more the concern of philosophers (Aristotle, Plato) than of mathematicians, with some notable exceptions (Leibniz) until the late 19th and early 20th centuries, when the arrival of modern mathematics generated the interest of the likes of Poincaré, Hilbert, Brouwer, Weyl and Gödel. We have, for example, no words from Euclid himself on the relationship between his Elements and the natural world. Nor, on the other hand, do we have any reason to believe that the statements of mathematics were regarded as other than true and applicable to the world before, it would seem, the discovery of a plausible non-Euclidean geometry early in the 19th century.
We lack a clear set of historical opinions about mathematical existence, and must make do in the main with naive analyses. If it is not altogether clear what a point was taken to be, either in Euclid’s Elements or in the world, it will not be clear what was meant by the term a ‘point at infinity’; if the nature of the concept of a natural number is not much discussed by mathematicians it is hard to know what to make of their remarks about the negative integers, still harder to interpret their remarks about , the square root of negative one.
To give just one indication of the problem, it is quite possible that mathematical objects may have been used but said not to exist, if they were taken to be terms in a theory that could be replaced by other, more elaborate and cumbersome objects that do exist. The vexed question of infinitesimals in the foundations of the calculus is a case in point; we shall see that points at infinity and the ‘number’ are others. Indeed, it is quite likely that the whole concept of existence, loaded as it is with a bias towards medium-sized spatio-temporal objects, may be the wrong concept with which to discuss what mathematicians have been talking about.
Despite their obvious differences the various conic sections—the circle, ellipse, parabola, and hyperbola—have been regarded as forming a family of curves at least since their description by Apollonius . Typically, problems arose when a complicated construction applied, say, to study an ellipse did not apply automatically to a hyperbola, and the case at interest here is when the construction produces a tangent to an ellipse but an asymptote to a hyperbola. In these cases, Apollonius seems to have regarded asymptotes as analogous to tangents, but nonetheless different; for example, in Book III of his Conics he presented theorems about tangents (such as theorems 37–40) after getting their exceptional cases that deal with asymptotes (theorems 30–36) out of the way. At no stage did Apollonius, or any other Greek geometer, generalise a construction by speaking of points at infinity: lines and curves either meet or they do not meet, they were never said to meet ‘at infinity’.
A particular case of this problem concerns the foci of a conic. The foci of the ellipse and the hyperbola were studied by Apollonius in Book III of his Conics, theorems 45–52, but only his contemporary Diocles considered the focus of the parabola and it was he who proved that the sun’s rays are reflected to a point by a parabolic mirror (see  4.C1, 2). Thereafter the matter seems well-understood and in need of no further analysis until Kepler wrote his Ad Vitellionem paralipomena  in 1604, the year before he decided that Mars travelled in an elliptical orbit, as described his Astronomia nova  of 1609. Kepler was interested in geometrical and physiological optics in this work, and wanted to give a unified account of the conic sections. He began by briefly describing the curves, which he depicted as plane sections of a cone and presented in a single plane. Kepler seems to have considered that the hyperbola consisted of two separate curves, not branches of one curve, and that the limiting case of the hyperbola was a single straight line, to judge by the way he dealt with plane sections of the conic that pass through the vertex of the cone. He then introduced their foci. These are the points F, F′ with the property, established by Apollonius, that the lines FP and F′P joining F and F′ to a point P on the conic make equal angles with the tangent to the conic at P. Kepler noted that the circle has a single focus, its centre, and the ellipse and hyperbola each have two. As for the parabola:1
In the Parabola one focus, D, is inside the conic section, the other is to be imagined (fingendus) either inside or outside, lying on the axis [of the curve] at an infinite distance from the first (infinito interuallo a priore remotus), so that if we draw the straight line HG or IG from this blind focus (ex illo caeco foco) to any point G on the conic section, the line will be parallel to the axis DK.
Low key though this may be, it seems to be the first time that a point was said to lie at infinity, and it would be interesting to know how forcefully Kepler wished to press his approach. However, it is quite possible to read it as making no ontological commitment at all; indeed, what can one say about the focus of the parabola that lies ‘at infinity’ that cannot be said immediately in the language of parallel lines? The idea that a family of parallel lines has a lot in common with a family of lines passing through a point was, as Andersen has very thoroughly explained in , familiar to both renaissance painters and those who offered mathematical accounts of their work.
The man who introduced points at infinity as more than a figure of speech was the mathematician and architect Girard Desargues, who was born in 1591 in Lyons. The Brouillon project or Rough Draft on Conics of 1639 [13, 14] was his second publication, the first being his twelve-page Perspective that was devoted to perspective constructions. It opens with a clear statement that points at infinity are to be accepted into geometry and treated on a par with other points in the plane or in three dimensions [13, p. 70].
In this work every straight line is, if necessary, taken to be produced to infinity in both directions. …
To convey that several straight lines are either parallel to one another or are all directed towards the same point we say that these straight lines belong to the same ordinance, which will indicate that in the one case as well as in the other it is as if they all converged to the same place.
The place to which several lines are thus taken to converge, in the one case as well as in the other, we call the butt of the ordinance of the lines.
To convey that we are considering the case in which several lines are parallel to one another we often say that the lines belong to the same ordinance, whose butt is at an infinite distance along each of them in both directions.
To convey that we are considering the case in which all the lines are directed to the same point we say that all the lines belong to the same ordinance, whose butt is at a finite distance along each of them.
Thus any two lines in the same plane belong to the same ordinance, whose butt is at a finite or infinite distance.
In this work every Plane is similarly taken to extend to infinity in all directions.
But if Desargues was happy in principle that a family of parallel lines meet at a point at infinity, he nonetheless found some occurrences of it difficult to accept, and they came up in the situations that Apollonius had preferred to keep separate. One topic that had occupied both Apollonius and, later in Hellenistic times Pappus, was the construction of the so-called fourth harmonic point. Given three points A, B, C on a line their fourth harmonic point D is the point D on that line such that
Such a set of four points has been given various names, such as Pappus’s ‘harmonic division’ or, Desargues’s term, ‘four points in involution’. The construction of the fourth harmonic point is as follows: one takes an arbitrary point G not on the line ABC and joins it to A and B, then one draws an arbitrary line through C that meets the line GA at the point E and GB at the point F, then one draws the lines AF and BE which meet at H, and then one draws the line G—it meets the line ABC at the required point D. Remarkably, it does not depend on any of the arbitrary choices. In the case where C is the midpoint of AB the point D will lie ‘at infinity’ because the line EF is now parallel to the line ABC. The Greek geometers had treated this as a special case, and noted that what plays the role of the above ratio is the simpler one AC: CB.
When Desargues came to this issue he remarked that it was incomprehensible [13, p. 82] but a few pages later [13, p. 85] he explained that a set of four points such as A, B, C, D for which occur in “as it were, two species of the same genus”, those in which the four points all lie at a finite distance and those in which the fourth point is at an in finite distance.
So we shall take good note that a straight line divided into two equal parts by a point and understood to be extended to infinity is one of the forms of an involution of four points.
Desargues’s Rough Draft on Conics was published in an edition of fifty copies, and it quickly met with success. It inspired Blaise Pascal, then only 16, to produce his theorem on six points on a conic in 1642, and in 1648 the engraver Abraham Bosse published the result that we know as Desargues’s theorem.2 However, the limited number of copies and the almost unreadable style of the Rough Draft on Conics with its many needless and obscure neologisms then combined to make the work disappear, until by 1679 only one copy was known, that, happily, was copied out by the mathematician Philippe de la Hire.
The difficult, but ultimately much more successful use of algebra in geometry developed by Descartes in his La géométrie (1637) surely also diminished the impact of Desargues’s approach. Descartes’s opinion of the Rough Draft on Conics was that the new terminology was unnecessary if Desargues was writing for experts and would only make the bookmark difficult; he also suggested using “the terminology and style of calculation of Arithmetic, as I did in my Geometry”. As for parallel lines meeting in a butt at an infinite distance, Descartes’ commented that3
this is very good, provided you use it, as I am sure you do, as an aid to understanding what is difficult to see in one of the types, by comparing it with the other, where it is clear, and not conversely.”
As for Philippe de la Hire, it is very likely that he had received this copy from his father Laurent de la Hire, a painter and pupil of Desargues who was also a friend and colleague of Bosse. In any case, it was long thought that only de la Hire’s copy that survived until a sole copy of the original was found in the Bibliothèqual Nationale and published by René Taton in 1951.4
Philippe de la Hire gave several accounts of the geometry of conic sections in the course of his life, of which his  is the most important. It can be regarded as a projective treatment of the subject, including a theory of poles and polars of conics, organised around the harmonic division of four points, and it is held together by an appreciation that any two conic sections are projectively equivalent, and the harmonic division is projectively invariant. From there the subject passed to Isaac Newton, who published what became called his ‘organic construction’ of conic sections in his Principia Mathematica in connection with the determination of orbits, and went on to give a largely projective classification of conic sections in , published as an appendix to his Opticks in 1704.
It is well known that Newton gave an essentially projective treatment of conic and cubic curves in this work, and the details can not be given here. In particular he gave examples of what he called the five divergent parabolas; these are curves that have a branch going to infinity but have no asymptote, unlike the hyperbolic branches that are tangent at infinity to their asymptotes. We would say that the tangent to a parabolic branch is the line at infinity; Newton said rather [31, p. 10] that “the tangent of the parabolic branch being at an infinite distance, vanishes, and is not to be found”. He then indicated 72 species of cubic curve (forgetting the other six, which he had in fact known about) and that every cubic curve is the projection of precisely one of five types of cubic curve (curves of the second genus, as he called them), writing “so the five divergent parabolas, by their shadows, generate all other curves of the second genus”. It was the work of several mathematicians to make sense of Newton’s account.
It is relevant here to note that Newton may not have accepted a line at infinity, but he did accept points. He noted [31, p. 25] that the straight lines in one direction may all meet a cubic curve in just one point, and so
we must conceive that those straight lines pass through two other points in the curve at an infinite distance. Two intersections of this sort, when they coincide, whether at a finite or an infinite distance, we shall call a double point.
He then repeated what he had said in the Principia about how double points can arise by applying his organic construction (a form of birational transformation) to a conic section.
Newton’s contributions might have stimulated a century of work, but it was not to be. The projective approach to geometry was developed by de Gua de Malves in his , but not by Leonhard Euler or Gabriel Cramer, and it declined in significance as the 18th century proceeded, until it was revived partly by Gaspard Monge and much more influentially by Jean Victor Poncelet.
We have reached a convenient point to ask about the ontological implications of the 17th-century work, but there can be no simple answer. The authors did not address the issue explicitly, beyond making it clear that they spoke of points at infinity as if they were as good as finite points. They did not treat them as a mere manner of speech that could easily be replaced with statements about parallel lines, and they did not treat theorems involving them as special cases of other theorems. But they did not discuss what could be meant by talking about new points ‘at infinity’, however, much this might seem to strain the idea of a geometry based on a concept of distance to breaking point. Indeed, the whole question of the relation of mathematical points to points in the world was little discussed. David Hume did discuss how the foundations of geometry can be obtained from sense experience in his A Treatise on Human Nature (1739), Part II “On the ideas of space and time”, but he did not raise the issue of points ‘at infinity’; rather, he was concerned with infinite divisibility and the notion of ‘point’ in geometry.
Instead, an ontological debate about points at infinity was more apparent in issues to do with the use of algebraic methods in geometry, and here Euler did have things to say. Some concerned points ‘at infinity’, others the appearance of points with complex coordinates.5 Euler had nothing to add to the idea that some curves may have points ‘at infinity’, which he accepted, but he had interesting views about points with complex coordinates, to which we turn after briefly reviewing attitudes to complex numbers and complex roots of equations.
Complex numbers, complex points
For many centuries, questions such as ‘what is the square root of negative one?’ or ‘what are the roots of ?’ were answered in various ways by mathematicians, but until the 16th century it was always possible to dismiss them. One could reply that there is no such number, and the equation has no roots. What made the question unavoidable was the discovery in the 16th century of methods for solving cubic equations, and specifically the growing realisation that the formula for the solution of a cubic equation with three real roots necessarily involved the square roots of negative numbers. The ‘casus irreducibilis’ or irreducible case, as it came to be called, was known to Cardano, who could not find a way round it in , his Ars magna of 1545, and in his L’Algebra  Bombelli produced a formalism for handling complex quantities.
Thereafter many different explanations of the question of ‘what is a complex number?’ were proposed. The ontological context was unpropitious. It was generally agreed that there were numbers, that is, the counting numbers or non-negative integers, and there were magnitudes, such as lengths. In this sense, numbers were properties of objects: one could own three sheep or purchase five yards of cloth. Negative numbers, and still more negative magnitudes, posed further problems, but might be associated with debts or directions along a line. What united all these ideas was that numbers and magnitudes were attributes, and accordingly the ontological question became ‘what objects have complex numbers or magnitudes as attributes?’. No answer was to be given for many decades that satisfied its audience, but Euler’s proposal in  was the simplest.
Euler stepped outside the existing ontological framework, after showing that there was no answer within it, and offered a different definition of number, as a formal quantity that we can handle algebraically exactly as we handle ordinary numbers and that can, indeed, produce ordinary numbers. It is a concept that is free of contradiction and it embraces the usual numbers.
143. And, since all numbers which it is possible to conceive are either greater or less than 0, or are 0 itself, it is evident that we cannot rank the square root of a negative number amongst possible numbers, and we must therefore say that it is an impossible quantity. In this manner we are led to the idea of numbers, which from their nature are impossible; and therefore they are usually called imaginary quantities, because they exist merely in the imagination.
144. All such expressions as , , &c. are consequently impossible, or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing; which necessarily constitutes them imaginary, or impossible.
145. But notwithstanding this, these numbers present themselves to the mind; they exist in our imagination, and we still have a sufficient idea of them; since we know that by is meant a number which, multiplied by itself, produces − 4; for this reason also, nothing prevents us from making use of these imaginary numbers, and employing them in calculation.
The new numbers can be called ‘imaginary’, but they are not impossible per se—they are simply not measuring numbers. Indeed:
149. It remains for us to remove any doubt which may be entertained concerning the utility of the numbers of which we have been speaking; for those numbers being impossible, it would not be surprising if they were thought entirely useless, and the object only of an idle speculation. This, however, would be a mistake; for the calculation of imaginary quantities is of the greatest importance, as questions frequently arise, of which we cannot immediately say whether they include any thing real and possible, or not; but when the solution of such a question leads to imaginary numbers, we are certain that what is required is impossible.
This is another valuable insight. The price of declaring complex numbers to be logically impossible is high: the calculation of unknown quantities requires that complex numbers can be accepted as answers, even though in a problem concerning quantities the valid conclusion would be that ‘what is required is impossible’. A similar view was held by Johann Heinrich Lambert, who wrote to the philosopher Immanuel Kant in 1770 that “The sign represents an unthinkable non-thing. And yet it can be used very well in finding theorems”.6
These views should not be taken as typical of the period, and nor as they incisive as they may seem. Euler was given to many observations and remarks of a foundational kind that resolve nothing; his remarks about infinitesimals and the interpretations of 0∕0 in his Introductio in analysis infinitorum (1748) sit alongside the remark that there is no number smaller than one part in ten million and some inspired manipulation of infinite and infinitesimal numbers that cannot properly justify the correct results they are presented as supporting. In the case of imaginary numbers, as is well known, Euler promptly went on to make sign errors in his calculations with these ‘numbers’ that we apparently know how to use, treating as a single-valued expression. And, as we shall shortly see, there are graver problems with his attitude to the ‘numbers’ that have to do with their status as answers.
Algebra was, of course, a generator of complex numbers. Whereas mathematicians of the 17th century had usually defined a curve to be of order n if it meets a straight line in at most n points, mathematicians of the 18th century preferred to define the order of a curve as the degree of its defining equation, and to prove a theorem to the effect that a curve of degree n meets a straight line in at most n points. This includes the case where the curve is defined by a polynomial equation of the form
and the other curve is the straight line y = 0. These meet in the points whose x-coordinates are found by solving the equation that is found by eliminating y from these two equations:
What became called the fundamental theorem of algebra is the claim that the polynomial equation has n roots, where, as always in the 18th century, the coefficients are all real. Or rather, and more commonly, in a form that Lagrange endorsed, it was claimed that a polynomial of degree n can be factored into a certain number, k, of linear terms (corresponding to the real roots of the equation) and a number j of quadratic terms (corresponding to the 2j complex conjugate roots) where . This indicates a degree of reservation about the status of complex roots.
The more general claim was that two curves of degrees m and n meet in at most mn points. This is true, for example, of two ellipses meeting in four points, but it is much harder to prove. Euler published a short paper  in 1750 that illustrated some of the difficulties involved. As he pointed out, for the theorem to be true one must be able to say that parallel lines meet in a point, and that a parabola meets a line parallel to its axis in two points—which requires the invention of points at infinity. One must be able to explain why two circles meet in at most two points but two ellipses can meet in four by accounting for the ‘missing points’, which will have complex coordinates. As he put it:7
the meaning of our proposition is that the number of the intersections can never be greater than mn, although it is very often smaller; and thus we may consider either that some intersections extend towards infinity, or that they become imaginary. So that in counting the intersections to infinity, the imaginary ones as well as the real ones, one may say, that the number of intersections is always = mn.
In November 1751 Euler wrote to Cramer (who had also considered the problem) that it led to ‘such tangled formulas that one completely loses patience in pursuing the calculations’.8
The method of eliminating a variable defied every challenge, and the Académie des Marines, the scientific school attached to the French Navy, was provoked to remark in 1770 that:9
The elimination of unknowns is one of the most important parts [of mathematics] to perfect, both because the extreme length of ordinary methods makes it so repugnant and because the general resolution of equations depends on it.
On this occasion they dismissed attempts by Euler and another mathematician, Étienne Bezout, because they had not shown that elimination lead to an equation of the right degree, mn, but finally in 1779 Bezout announced a satisfactory solution. The result has been called Bezout’s theorem ever since, although proofs of increasing rigour continued to be produced until the 20th century.
A curious aspect of complex points of intersection of two curves should be noted that does not apply to the case of the fundamental theorem of algebra. Let us fix one curve, of degree m say, and suppose that it meets a curve of degree n in k points with real coordinates and mn − k points with complex coordinates. Now let us vary the curve of degree n, and consider the case where the varying curve continues to cut the fixed curve in k points with real coordinates that depend on the varying curve. We may, if we like, suppose that each of the k points moves around on the fixed curve. But what of the mn − k points with complex coordinates? In what sense does the fixed curve meet the varying curve in complex points, unless one is prepared to say that both curves have complex points and they meet in a set of mn − k complex points that vary with the varying curve? It is much easier to talk around the same question in the context of the fundamental theorem of algebra than in the context of curves, but this problem was to remain unaddressed for several decades, as we shall see shortly when we discuss Plücker’s work.
A further contradiction appears with the use of complex quantities in analysis. Famously, in , his Introductio in analysin infinitorum of 1748, Euler showed how to unify the theory of the exponential, logarithmic, and trigonometric functions, and went on to resolve the question of what the logarithm of a negative and a complex number could be that had confused both Johann Bernoulli and Jean le Rond d’Alembert. His answer was that the logarithm is an infinitely many valued expression, corresponding to the fact that its inverse, the exponential function, is periodic with period 2π i (an answer d’Alembert refused to accept). It is odd, therefore, to see that the idea that the logarithm of a number is anyone of a set of values of the form a + 2k π i, where k is an integer was something some people could accept, and yet to find that complex quantities were only admitted into geometry and analysis on condition that they do not appear in the answer—but such was the case. There are many occasions in the writings of d’Alembert, Euler, and Lagrange where such quantities are admitted only because at the end of the calculation they ‘destroy themselves’, as the phrase had it. Or, as we would say, they occur in complex conjugate pairs and the imaginary parts cancel each other out. This is true in d’Alembert’s work on flows in the plane, in the work of Euler and Lagrange on cartography, and in Euler’s work on partial differential equations.
For example, d’Alembert used formal complex methods in his  of 1752. In Chapter IV § 45 he considered a two-dimensional flow in the (x, z)-plane, and in § § 57–60 he looked for conditions on functions M and N of x and zunder which the differentials
are exact. He argued that when they are exact there are functions p and q such that
and so the differentials
are also exact. This gave him an answer to his question in only a few further lines of work, but he then gave a second, simpler argument to the same effect. The definitions of p and q imply that
so qdx + pdz and pdx − qdz are exact differentials and therefore q + ip is a function of x − iz and q − ip is a function of x + iz. He now remarked that if one wants p and q to be real then q must be a function of the form
where the functions ξ and ζ have real coefficients (with a similar result for p that he omitted). This gave him expressions for p and q in which, in his phrase, the imaginary quantities destroy themselves.
Lagrange made a similarly formal use of imaginary quantities in cartography in his memoir  of 1781. This led him to the equation
in which u and t are obtained from coordinates on the Earth (taken to be a spheroid) and x and y are coordinates on the plane. He factorised each side formally into factors of the form
by introducing a new variable ω, such that . This gave him the equations
and for these equations to be integrable, he said that he would follow d’Alembert’s method, and so these conditions must hold:
from which he deduced that
and α ± i β must be a function of u ± it.
He then defined the prime meridian (given by t = 0) and used it to obtain expressions for x and y that, he said, “have the advantage that the imaginaries destroy themselves”. Once again the formal complex quantities act as a catalyst and are made to vanish in the end.
These examples from the 18th century firmly suggest that complex numbers in whatever context were acceptable as means to an end, but much less acceptable in the end itself.
The arrival of projective space
We have seen that Euler, and for that matter, Cramer, were willing to regard curves as having points at infinity, but most likely regarded them as a special kind of point, not to be confused with finite points. Such a position became inadequate when transformations were introduced into geometry that map points at infinity to finite points and finite points to ones at infinity.
The decisive figure in this context is Poncelet, but his presentation was so entangled with other, less acceptable ideas that other mathematicians—Michel Chasles in France, August Ferdinand Möbius, Jakob Steiner, Julius Plücker and ultimately von Staudt in Germany—made equally important contributions in the 1820s and 1830s.
Poncelet’s paradoxical name in his  for his discovery—‘non-metrical geometry’—highlights its novelty. It is a study of the properties of plane figures that are preserved under central projection from one plane to another. Straightness of the straight line is such a property, so is the fact that conic sections are curves that meet a straight line in at most two points, but length and angle are not such properties, and Poncelet was able to show that non-metrical geometry was a rich subject with clear connections to the metrical geometry of conic sections. But he did so in a way that won little support, because he urged a very general interpretation of figures through what he called a principle of continuity.
He introduced the principle to give geometry the generality of algebra, and he used it to talk indifferently of configurations in which a line meets a circle and in which a line does not meet a circle. He argued that when the line is moved from the first position to the second the points of intersection become imaginary, but many things can be said about them and the segment they define, and so it was appropriate to treat real and imaginary intersections on a par. He extended this principle greatly, to include imaginary centres of projection. From an ontological standpoint, as is argued in , he may not have been introducing new objects so much as changing the meaning of terms such as ‘intersect’. However, in so doing he made few converts, and the enduring part of his legacy was the elucidation of projective properties and projective transformations.
But Poncelet’s ideas were the source of a peculiar debate in the 19th century on the nature of complex points on real curves. Various authors put some effort into giving a real interpretation for the pairs of conjugate imaginary points that satisfy an algebraic equation with real coefficients. This is the subject of , one of the better, but less well-known, books by the historian of mathematics Julian Lowell Coolidge. Typically, complex points on real curves were interpreted as corresponding to an elliptic involution, which is a map of period two having conjugate imaginary fixed points. This theory was extensively developed by Laguerre, as Coolidge described.
Julius Plücker is an interesting case, and a more relevant one here, because the subject of the first half of his  deals with the affine coordinate geometry of algebraic curves; the larger second half is a rich study of cubic curves.
In this treatment a point is determined by its coordinates and therefore by the intersection of two lines, and Plücker remarked (1835, 19) that the treatment would be incomplete without an account of imaginary coordinates,
that is, those values that, based on given points, receive given imaginary linear functions. Such coordinates, like the corresponding functions, arise in pairs. Each coordinate, taken singly, corresponds to an imaginary straight line, two connected coordinates correspond to two imaginary straight lines that meet in a real point. But we can also determine geometrically the coordinates through the use of two real straight lines that meet in the point.
A long discussion of the algebra led Plücker to system of four lines that he called harmonical; two real lines with equations and two lines with equation , where that are real or imaginary according as λ is either real or imaginary. He called this system an involution of line pairs, because there is a linear map that fixes the lines of one pair and exchanges the other two, and distinguished the cases when the involution has no imaginary lines and when it has two.
He included in his treatment of cubic curves their points at infinity, and extended this study in his  to the geometry of the infinite branches of algebraic curves, in particular curves defined by equations of degree four. He did not study this by using homogeneous coordinates and changing axes. Instead, he worked with the Cartesian equation, which he then sought to write in a helpful form. Given curve defined by an equation F n (x, y) = 0 of degree n he took the equation of line in the form and looked at the expression
where G n−1 is a polynomial in x and y of degree n − 1 and in general will also be of degree n. However, the points where the line meets the curve are given by the zeros of , and if any of them lie at infinity then the degree of will drop. In particular, if the line is an asymptote of the curve then it is a tangent at infinity and the degree of must drop by two (or more, if the contact at infinity is of higher order). So his method was to work with suitably chosen lines p = 0 that meet, and preferably touch, the curve F = 0 at infinity.
As an example, suppose
This can be written as
with and , showing that the curve F = 0 has the line x = y as an asymptote.
Plücker developed this theory for real asymptotes and then considered imaginary asymptotes, but the study was nonetheless about the shape of real curves in the real plane. One can therefore wonder when the idea came in that a complex curves has complex points in the same way that a real curve has real points, and in so doing one is in the company of Cayley, who wrote [8, p. 316] that “I was under the impression that the theory as a known one; but I have not found it anywhere set out in detail”.
The obvious, and correct answer, is that this was first set out by Riemann in his [40, 41], but his treatment of complex algebraic curves as branched coverings of the complex sphere (although not exactly what Cayley had in mind, as he observed) makes no substantial use of points at infinity and need not be described here.
Projective transformations offered a way of evading the full implications of non-metrical geometry, in that they could always be used to concentrate attention on a finite part of the plane. A projective transformation can always be found to reduce a given quadrilateral to another, any conic to a circle, to bring in almost all the points at infinity in a given figure to the finite part of the plane. Chasles’s view in  seems to have been that one was study the properties of figures in the finite plane that are invariant under projective transformations. These properties were largely Euclidean, so although he showed that given five points A, B, C, D, P on a conic any other point Q on the conic has the property that the cross-ratio of the lines PA, PB, PC, PD is equal to the cross-ratio of the four lines QA, QB, QC, QD he did not prove the converse, and so he came up with a projective property of a conic but not a projective characterisation of conics.
After Poncelet, mathematicians in France and Germany gradually took up non-metrical geometry, and slowly disentangled a number of complicated questions—such as the projective definition of the basic figures—until it was possible to give entirely projective definitions of the basic concepts, a process that is usually said to conclude with the work of von Staudt [44, 45] in the 1850s or, perhaps, of Felix Klein in the early 1870s. All of this work had to discuss what was meant by points ‘at infinity’ because now a transformation can map a finite point to one at infinity and vice versa. So to say what a point at infinity might be, it became necessary to say what projective space might be.
A word should be said here about Pasch’s approach to geometry in his  of 1882. It is well known that he aimed to start from Helmholtz’s ideas about the empirical origin of the geometrical axioms and arrive at a coherent system of geometry by specifying at each stage how the empirical ideas were codified into geometrical statements. Once a family of empirical claims about geometrical objects was stated in the form of Grundsätze, every statement about these objects had to be proved on the basis of the Grundsätze alone. The hope was that in this way a logically impeccable geometrical system would be established with a clear relationship to physical space.
Pasch began with bounded regions of the plane and space, and codified the behaviour of points and line segments; the eponymous Pasch axioms belongs here. To deal with anomalous intersection properties, such as lines in a plane that do not meet, Pasch then extended the meaning of the terms ‘point’, ‘line’ and ‘plane’ so that he could say, for example [32, p. 41] that “Two lines in a plane always have a point in common”. In so doing he was following the lead of von Staudt and Klein, and he compared what he was doing to the way that the concept of ‘number’ is stretched in going from meaning ‘positive integer’ to number in general.
Pasch left it unclear how the experience of a finite but arbitrary part of a plane could ever lead unambiguously to the Euclidean theory of parallels, but that was what he had in mind. So his geometry did not have points or lines ‘at infinity’, only families of lines no two of which had points in common and families of planes no two of which had a line in common.
Plane projective space was a novelty. It was not metrical, so it could not be said to be the space we inhabit. Also, because it was not metrical it was not at first very clear how considerations of continuity could apply to it. Also, it has an unexpected global property of being non-orientable that is invisible in any finite part of the plane. Two definitions of it co-existed. One added points to the Euclidean plane that were the ‘intersection points’ of families of parallel lines, and then allowed points to belong indifferently to families of either convergent or parallel lines. The other turned to coordinate methods, pioneered by Möbius, Plücker, and Hesse in Germany.
If the second approach proved more powerful this was because the synthetic methods associated with the first approach seldom dealt successfully with curves other than conic sections, but the algebraic approach was fertile. Plane projective space could be regarded as the collection of all homogeneous triples of numbers, the triple [0, 0, 0] excluded. Mathematicians were at liberty to let the entries in the triples be real or complex numbers, and the ontological question was now clear: in what sense does a seemingly consistent set of mathematical definitions and theorems constitute a description of something that exists but does not correspond to an object in the world?
This question joined with several others in the second half of the 19th century. In what sense do the three dimensions of space and the fourth dimension of time constitute a four-dimensional space? In what sense do a set of four variables, such as two of position and two of momentum, constitute a four-dimensional space? If, following Grassmann in , we work with extensive magnitudes of order n, can we go further and speak of n-dimensional space? Is there any good reason to do so? Is there any logical reason not to? At what point does a theory go from being figures of speech to being something like a naturalistic account of an object that just happens not to be in the world? And indeed, at what point does a mathematician start to lose confidence in the comfortable connection between Euclidean geometry and the world around him or her?
The existence of Non-Euclidean geometry
The discovery of non-Euclidean geometry has been told in many places and will not be told here.10 It is well known that Bolyai and Lobachevskii independently and around 1830 published accounts of a geometry that differed from Euclid’s in only one respect—the definition of parallel lines—and that was a plausible, and above all, metrical account of space that, however, failed to convince any important mathematician except Gauss. He, however, did essentially nothing to advance their cause. But it is worth noting that what Bolyai and Lobachevskii did, and Gauss had not done in the 1830s, was to describe a three-dimensional space, not a two-dimensional one. Intuition is a poor guide in such work; so accustomed are we to reading geometry from the world around us that we have to struggle to imagine ourselves in another world entirely. It is true that neither Bolyai nor Lobachevskii were able to provide an entirely convincing account, inasmuch as a remote possibility of a self-contradiction in their work could not be ruled out, but the deeper reason that their work met with such strong rejections when it was discussed at all was that it required a wholesale reassessment of our relationship with the world, and surely implied that there were two possible, but incompatible, geometrical theories of it. Even if one was correct the other would be wrong, and the only way to decide which was correct was to introduce an empirical element. Gone would be centuries of naive confidence in the truth of (Euclidean) geometry.
Non-Euclidean geometry was first accepted, and first advocated successfully, by mathematicians who were long past worrying about the Euclidean doctrine of parallel lines, and for whom the whole question of what geometry was about had deepened greatly. The inspirational figure here was Bernhard Riemann, and his view, in his Habilitation lecture of 1854 (published posthumously in 1867 as ) was that geometry was the study of any collection of points (most likely themselves described by coordinates) in which it was possible to talk about distance. This included, but was not limited to, sets of points that were n-tuples of numbers and a distance formula that resembled the Pythagorean distance formula of Euclidean differential geometry. Above all, it rendered Euclidean two- and three-dimensional geometries just two among infinitely many and denied them any foundational role.
Geometry was now talk involving distance and angle however they might be defined. In particular, the non-Euclidean geometry of Bolyai and Lobachevskii was, Riemann implied, the geometry of points in an n-dimensional ball of (Euclidean) radius r in which the distance between two points is obtained from the formula (see , III, 4)
where is the curvature.
Inspired by Riemann’s ideas Beltrami in  gave a similar account, but with a different metric in which (non-Euclidean) geodesics appeared Euclidean straight lines. This allowed Klein in  to find a way of including non-Euclidean geometry as a special case of projective geometry, and Poincaré in 1881 to give further interpretations of the subject that mark its full acceptance into the world of mathematics.
But it was also clear that the existence of these new geometries was the product of a novel ontology. What the various metrical descriptions provided was a map of non-Euclidean space in almost the same sense that a map of Europe in an atlas provides an account of Europe. The difference is that Europe exists, it is patiently surveyed, and the results of those surveys provide the data for the map; all that exists in the world of non-Euclidean space is the map. The claim that the space exists is based on an argument that the map is internally self-consistent, and that nothing else is required for mathematical existence other than consistency.
It is well known that it took time for these ideas to be accepted, especially by philosophers. Kantians had to reckon with the clear suggestion from Kant that Euclidean geometry was the inevitable consequence of a simple construction; Frege simply refused to accept it because there could only be one set of truths about the world. Their reluctance says a lot about how geometry was taken to apply to the world, and it hints at a possible reason for the ultimate success of Bolyai and Lobachevskii. They set up a complicated system of points, lines, curves, and surfaces and then moved quickly to describe them in formulae. They could not give the ultimately convincing reason for placing trust in their formulae—that was the specific contribution of Riemann and Beltrami—but by displaying a new set of consistent formulae they opened up a path to an argument that would run in precisely the opposite direction to theirs: from formulae to interpretation.
Points at infinity in non-Euclidean geometry have a particular significance in non-Euclidean geometry. They were introduced by Klein in  in order to show how to locate Beltrami’s account within projective geometry. Klein interpreted Beltrami’s work as being about the space of points interior to a fixed conic together with the group of projective transformations that maps this space to itself. He obtained the distance between two points A and B in this space as a suitable multiple of the logarithm of the cross-ratio of the four points A, B, C, D, where C and D are the points where the line AB meets the conic . On this definition, points on are infinitely far away.
Points at infinity were also used by Poincaré in his work on non-Euclidean geometry (see, for example, ), but in a different way. What he called a Kleinian group acts on three-dimensional non-Euclidean space and a fundamental domain for the action can cut out a region of the two-dimensional boundary, which of course consists of points at infinity in the geometry. When this happens the study of the group is intimately tied to the study of this boundary under the action of the group.
In these cases it can be fairly argued that the points at infinity are not mysterious at all; they are simply points on the boundary of a finite region drawn in the Euclidean plane or Euclidean three-dimensional space. They nonetheless have an interpretation in the geometry under study that would be difficult to understand if that geometry was taken to be real.
On the other hand, it was beginning to be clear to mathematicians that all they could mean by mathematical existence was a consistent theory: if a theory of a class of objects was free of self-contradiction that those objects existed, even if they had no counterparts in the physical world. This was the explicit view of Poincaré and Hilbert by 1900.
The freedom from self-contradiction was not necessarily easy to establish, and Euclidean geometry is a case in point. Once the naive identification of Euclidean geometry and physical space was broken, it had to be asked of Euclidean and non-Euclidean geometry alike: do they exist (are they each self-consistent)? Famously, as Poincaré explained in [36, 37], if one of these two geometries has a consistent mathematical account, then so does the other, and therefore, for as long as they were taken to be the only possible geometries of space, if one must exist then both do. As for which would correctly describe space, however, his position was that it was a matter of convention and we could never decide on logical or empirical grounds alone. But what actually establishes the mathematical existence of the objects described in Euclidean geometry?
The naive answer could not be the definitions in Euclid’s Elements, which by the standards of 1900 were no definitions at all. A better answer was the space of Cartesian geometry, which replaced Euclid’s objects with objects described by algebra and the real numbers. Hilbert’s Grundlagen der Geometrie  went further, and offered the use of models of various axiom systems to establish the consistency of those axioms systems, but this ultimately led back to the question of the existence of the real numbers.
As Hilbert saw clearly by about 1900 existence questions in geometry were coming down to existence questions in arithmetic, and they led directly to questions in the new theory of sets and ongoing investigations into logic. Hilbert seems never to have doubted that infinite sets can be satisfactorily handled in mathematics—he proclaimed in  that mathematicians would never be banished from the paradise that Cantor had created.
But influential voices spoke out against this view: those of Kronecker, Poincaré, Brouwer, and Weyl, and others. What concerned them was the loose way in which statements were being made about infinite sets of objects. They inclined in various degrees to the view that in principle a statement about all members of a set of objects could only be said to be true if every member of the set had been examined. To be sure, this was an idealisation of the human mind, but it seemed clear that few statements about uncountably infinite sets could be strictly defended. One could certainly say that such a set contained an element of such-and-such a kind if that element could be produced, but arguments by contradiction that could not even in principle produce such an element or, worse, that purported to show that no such element existed, seemed doubtful at best.
French mathematicians around Borel raised other objections. It is one thing, they said, to contemplate an infinite sequence of, say, numbers that is given by a rule for producing either each number from the one before or for specifying the nth number, for each n, but is another thing to speak of an arbitrary sequence given by no rule at all. They also objected, for example, to the axiom of choice.
All these objections were critiques of the human mind. The concern was that if the objects of mathematics were not physical objects but conceptual, mental objects, then the (idealised) mind that produced these objects had to be analysed. In what sense could it be legitimately said to produce objects of all kinds except those that fell into self-contradiction?
The contrary view, held by Hadamard in France and by Hilbert, was that all that was needed was a consistent way of handling these infinite objects. Understanding, in the sense of mental grasp, was not required; all that was needed was valid use.
At this stage, points at infinity no longer posed special problems. Their ontology had become part of a general question about the existence of mathematical objects, particularly those created from classes of objects whose existence was granted (for example, ideal numbers as sets of ordinary numbers).
This essay has concentrated on problematic entities for mathematicians in the period before mathematics was given its modern, largely set-theoretic, form, and merely hinted that these difficulties were dissolved in the larger questions about an ontology for modern mathematics. In this volume those questions are addressed most directly by Avigad and Gillies, and they both offer a somewhat Wittgensteinian perspective. They argue that coherent use of a term, in a language that governs its use, does the bulk of the work in establishing the existence of a mathematical object. Gillies, borrowing a distinction from Frege, suggests that consistent use of a term gives it is sense, and allows that some objects (transfinite cardinals) may lack a reference. Avigad, with a nod to a Quinean pragmatism about how we operate in the world, likewise sees mathematics primarily as a linguistic activity.
This linguistic standpoint is supported by the topics discussed in this essay. At one point I asked, rhetorically, “At what point does a theory go from being figures of speech to being something like a naturalistic account of an object that just happens not to be in the world?” One answer to this question is to say that it rests on a false antithesis. It suggests that there is loose talk about things we make up, and hard talk about things around us. But this is inadequate for discussing mathematics, and results in the contortions of some philosophies of mathematics. It is surely better to ask: what more could we want from talk about mathematical objects than that it be consistent with naive mathematics and self-consistent? To put the point another way, the supposed distinction rests on an unduly naive idea of how we know about the world, and a naive epistemology has resulted in an impoverished ontology.
One thing mathematicians want from any discursive analysis of mathematics is that it respects their sense that what they validly discover is true or correct in some straight-forward way. It may be that in this system a certain theorem holds that is false in some other system (compare Euclidean and non-Euclidean geometry), but in each system taken separately certain theorems hold and others do not. What they find disturbing in linguistic analyses, I believe, is the word ‘game’. Dickens could have made Oliver Twist throw in his lot with Fagin and become a master criminal and no linguistic rule or rule for writing fiction, however broadly defined, would be violated; a mathematician cannot similarly decide that a well-defined space shall, after all, have negative curvature. The mathematicians’ hands are tied.
This much the linguistic analysis of mathematics respects: it is the formalised rules of a system that tie the mathematicians’ hands. The remaining puzzle is why this suggests so strongly to many mathematicians that they are dealing with objects when trying to formulate their ideas. In this they are like physicists, who it seems have a tendency to attribute homely properties to the most arcane particles that must otherwise be dealt with using highly abstract, and not always rigorous, mathematics, and we can speculate as to why this is.
As Stillwell suggests in another essay in this volume, a great deal of mathematics can be written in the formal language of ZFC and its underlying logic, and in this system a certain hyperbolic seven-dimensional manifold (or any other mathematical object) has a definition, albeit one that would take up some space to write out fully. Mathematicians working with this manifold habitually use some of its defining features (let us suppose that it is compact and has negative curvature of such-and-such a kind), and these will generally be the ones that distinguish it from other manifolds, or that it has because it is a manifold rather than merely a metric space. They also have a sense of how the theory ought to go by analogy with related fields, and they have quite naive intuitions, often of a spatial kind (say, about two- dimensional manifolds). These factors combine in talk about the properties of the hyperbolic seven-dimensional manifold, rather than in talk about the permissible deductions within the system that defines it, and the manifold becomes an object with properties that determine what can be said about it.
This happens, it is productive, and one day psychologists and neuro-anatomists may have something valuable to say about it. The relevant question here is whether this feeling should drive a philosophy. At this point, the parallel between the issue as posed for the mathematics of the period before the mid-19th century and today is quite close. Whenever mathematicians discuss objects that they agree they cannot kick (in the fashion of Doctor Johnson kicking a stone to refute Bishop Berkeley’s idealism) they have only talk about it. If this talk is sufficiently constrained to be coherent it seems to be a matter of time before the object is said to exist, be it a point or a line at infinity, complex points, or a seven-dimensional manifold. Whether there is anything more to this talk by way of ontology may be to ask for a false view of ontology.
K. Andersen (2007). The Geometry of an Art: The history of the mathematical theory of perspective from Alberti to Monge, Springer.
Apollonius, Conics Apollonii Pergaei quae graece exstant cum commentariis antiquis. Edidit et latine interpretatus est I. L. Heiberg. Gr. & Lat. Leipzig, 1891, 1893, Stuttgart 1974. See also Treatise on Conic Sections. Edited in modern notation, with introductions, including an essay on the earlier history of the subject, by T. L. Heath. Eng. Cambridge: University Press, 1896.
E. Beltrami (1868). Saggio di interpretazione della Geometria non-euclidea. Giornale di Matematiche 6, 284–312. French trans. by J. Hoüel, “Essai d’interprétation de la géométrie non-euclidienne,” Annales ećole normale supérieure 6, 251–288 (1869).
R. Bombelli (1572, 1579). L’Algebra, Bologna.
H.J.M. Bos, C. Hers, F. Oort, D.W. Raven (1987). Poncelet’s closure theorem, Expositiones mathematicae 5, 289–354.MATHMathSciNet
A. Bosse (1648). Maniere universelle de M. Desargues, Paris.
G. Cardano (1968). Ars magna, T. R. Witmer (transl.), MIT Press, 1968.
A. Cayley (1878). On the geometrical representation of imaginary variables by a real correspondence of two planes. Proc. LMS 9, 31–39 in Collected Mathematial Papers 10, 316–323.
M. Chasles (1837). Aperçu historique sur l’origine et le développement des méthodes en géométrie, Hayez, Brussels.
J.L. Coolidge (1924). The geometry of the complex domain. Clarendon Press, Oxford.MATH
J. le Rond D’Alembert (1752). Essai d’une nouvelle théorie de la résistance des fluides. Chez David l’aîné, Paris. Rep. Culture et Civilisation, Bruxelles 1966.
de Gua de Malves, J.P. (1740). Usages de l?analyse de Descartes, pour découvrir, sans le secours du calcul differéntiel, les propriétés, ou affections principales des lignes géométriques de tous les ordres. Paris.
G. Desargues (1986). The geometrical work of Girard Desargues, J.V. Field, J.J. Gray (eds.), Springer.
G. Desargues (1988). L’oeuvre mathématique de G. Desargues: textes publiés et commentés avec une introduction biographique et historique, 2nd edn., avec une post-face inédité par René Taton, Institut interdisciplinaire d’études epistemologiques.
L. Euler (1748). Introductio in analysin infinitorum, two vols, Opera Omnia (1) vols 8 and 9. English transl. Introduction to Analysis of the Infinite, Book I, Springer, 1988, Book II, Springer, 1990 (E101, 102).
L. Euler (1750). Demonstration sur le nombre des points, ou deux lignes des ordres quelconques peuvent se couper, Mémoires de l’académie des sciences de Berlin 4, 34–248 in Opera Omnia (1), 26, 46–59 (E148).
L. Euler (1770). Vollständige Einleitung zur Algebra in Opera Omnia (1) 1, (E387). English transl. Elements of Algebra, Rev. J. Hewlett, London 1840, repr. Springer, 1972.
J.G. Fauvel and J.J. Gray (1987). The history of mathematics – a reader, Macmillan.
H.G. Grassmann (1844). Die Wissenschaft der extensiven Grösse, oder die Ausdehnungslehre, eine neue mathematische Disciplin, etc., Leipzig. English transl. A new branch of mathematics: the “Ausdehnungslehre” of 1844 and other works by Hermann Grassmann; translated by Lloyd C. Kannenberg. Chicago: Open Court, c1995.
J.J. Gray (1989). Ideas of space: Euclidean, non-Euclidean, and relativistic. Clarendon Press, Oxford.MATH
J.J. Gray (2010). Worlds Out of Nothing; A Course on the History of Geometry in the 19th Century, 2nd rev. ed. Springer.
M.J. Greenberg (2008). Euclidean and non-euclidean geometries: development and history, 3rd ed. New York: W. H. Freeman; Basingstoke: Palgrave.
D. Hilbert (1899). Festschrift zur Feier der Enthüllung des Gauss–Weber–Denkmals in Göttingen [etc.] Grundlagen der Geometrie. Leipzig.
D. Hilbert (1926). Über das Unendlich, Mathematische Annalen 95, 161–190.MathSciNetCrossRef
I. Kant (1967). Philosophical correspondence, 1759–1799, ed. and transl. A. Zweig.
J. Kepler (1604). Ad Vitellionem paralipomena, quibus Astronomiae pars optica traditur; potissimum de artificiosa observatione et aestimatione diametrorum deliquiorumque solis et lunae. Cum exemplis insignium eclipsium, etc. Frankfurt.
J. Kepler (1609). Astronomia Nova, seu physica coelestis, tradita commentariis de motibus stellae Martis ex observationibus G.V. Tychonis Brahe. Prague.
C.F. Klein (1871). Ueber die sogenannte nicht-Euklidische Geometrie I, Mathematische Annalen 4, 573–615, in Gesammelte Mathematische Abhandlungen I, 254–305.
P. de la Hire (1685) Sectiones Conicae, in novem libros distributae, Paris.
J.-L. Lagrange (1781). Sur la construction des cartes géographiques. Nouv. Mém. Acad. Berlin, 161–210, in Oeuvres 4, 637–691.
I. Newton (1704) Enumeration of lines of the third order, Appendix to Opticks, transl. C.R.M. Talbot, London, 1860.
M. Pasch (1882). Vorlesungen über neuere Geometrie. Teubner, Leipzig.
J. Plücker (1835). System der analytischen Geometrie, auf neue Betrachtungsweisen gegründet, und insbesondere eine ausführliche Theorie der Curven dritter Ordnung enthaltend. Berlin.
J. Plücker (1839). Theorie der algebraischen Curven gegründet auf eine neue Behandlungsweise der analytischen Geometrie. Bonn.
H. Poincaré (1883). Mémoire sur les groupes Kleinéens. Acta Mathematica 3, 49–92. In Oeuvres 2, 258–299.
H. Poincaré (1902a). L’espace et la géométrie. La Science et l’hypothése 77–94. Paris, Flammarion. English transl. W. J. Greenstreet, Space and geometry, in Poincaré Science and hypothesis 51–71. London, Walter Scott, 1968.
H. Poincaré (1902b). L’expérience et la géométrie. La Science et l’hypothése 95–110. Paris, Flammarion. English transl. W. J. Greenstreet, Experiment and geometry, in Poincaré Science and hypothesis 72–88. London, Walter Scott, 1968.
J.V. Poncelet (1822). Traité des Propriétés projectives des Figures, Gauthier-Villars.
R. Rider (1981). Poisson and algebra: against an eighteenth century background, Siméon-Denis Poisson et la Science de son Temps, M. Métivier, P. Costabel, P. Dugac (eds), École Polytechnique.
G.F.B. Riemann (1851). Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen. In Werke, 3rd ed. 35–80. English transl. in Riemann (2004), 1–42.
G.F.B. Riemann (1857). Theorie der Abel’schen Functionen. Journal für Mathematik 54. In Gesammelte mathematische Werke, 120–144. English transl. in Riemann (2004), 79–134.
G.F.B. Riemann (1867). Ueber die Hypothesen welche der Geometrie zu Grunde liegen. K. Ges. Wiss. Göttingen, 13, 1–20. In Gesammelte mathematische Werke, 304–319. English transl. in Riemann (2004), 257–272.
G.F.B. Riemann (2004). Collected Papers, R. Baker, C. Christenson, H. Orde transl. and ed., Kendrick Press.
von Staudt, C.G.C. (1847). Geometrie der Lage. Nürnberg.
von Staudt, C.G.C. (1856–1860). Beiträge zur Geometrie der Lage, 3 vols. Nürnberg.
See [13, p. 187]. Kepler added that the straight line has two coincident foci that lie on the line itself.
See . This theorem may have appeared in Desargues’s third publication, the Leçons de ténèbres (1640), but that book is lost.
See Descartes’s letter to Desargues in [18, 11.D4].
De la Hire’s copy was edited and published by Poudra in 1864.
There was also the issue of how to count multiple points of intersection, but that cannot be pursued here.
See [25, nr. 61] and [18, 16.A3].
Transl. Winifried Marshall, Euler Archive.
Euler–Cramer correspondence, OO474 not yet published, quoted in [39, p. 168]).
Quoted in [39, p. 168]).
See ,  or .