Dunnington did not comment critically on Gauss’s lofly remark that Abel’s work in the 1820s brought him about one-third of the way. Schlesinger suggested (Gauss Werke, X.2, 184) that by 1828 Gauss’s mostly unpublished theory came conveniently in three parts: ‘The first third was the general theory of functions arising from the [hypergeometric series], the second the theory of the arithmetico-geometric mean and the modular function, and finally the third, which Abel published before Gauss, was the theory of elliptic functions in the strict sense.’ Gauss would presumably have appreciated Abel’s theory of when the division equations are solvable algebraically, and the investigation of the divisibility of the lemniscate by ruler and compass would have struck him most forcefully, inspired as it was by the hint he had dropped in his Disquisitiones Arithmeticae. Gauss’s judgement was based on his observation that what Abel presented was only an account of elliptic functions with a real modulus. But had Gauss chosen to comment a year later on Jacobi’s Fundamenta Nova, he would have seen a general theory of the transformations of elliptic functions surpassing anything he had written down. He would also have found something more like his theory of theta functions, but again only in the context of elliptic functions with real modulus. He might also have become aware of Abel’s account of the same power series in his paper [1828]. In that paper Abel also allowed the modulus to become purely imaginary, but in all respects his theory of a complex variable was, like Jacobi’s, entirely formal.
A fairer comparison of the work of Gauss and Abel would be that they were proceeding in the same direction, although Abel’s theory (and indeed Jacobi’s) lacked a good way of writing the new functions (such as Jacobi’s theta series were to provide), a rigorous theory of convergence, and an explanation of double periodicity. Gauss, but not Abel, had a theory of the modular function (which expresses the modulus as a function of the periods, k = k(K/K')) and knew the connection with differential equations (such as Legendre’s). The theory of transformations is novel, and was better developed by Abel than Gauss (not in his Recherches but in his later Précis).
Dunnington briefly mentioned Gauss’s published paper of 1812 on the hypergeometric series173
which Gauss treated as a complex function of a complex variable x, but dependent on real parameters α, β, γ. Using his newly-developed theory of this function. Gauss gave an immediate proof of the formula that had intrigued him for so long, A ∙ B = π/4, but he revealed none of his theory of the arithmetico-geometric mean and elliptic functions. He also wrote, but did not publish, a study of the differential equation that the hypergeometric series satisfies, which is called the hypergeometric equation:
A special case of it, as Gauss knew, is Legendre’s differential equation (α = β = ½ and γ = 1).
Gauss’s analysis of this equation shows very clearly that he appreciated the distinction between F as a function and F as an infinite series. The former is defined for all finite values of the variable except 0 and 1, while the latter is only defined when the variable is less than 1 in absolute value. But the series, when defined, takes a unique value for each value of the variable, whereas the function does not. Another important point Gauss did not make was that he had a powerful reason for studying the hypergeometric series, for although the differential equation satisfied by the functions 1/M(1 + x, l – x) and l/M(1, x) is not the hypergeometric equation, on making the transformation x2 = z the differential equation becomes the equation
which is the hypergeometric equation with α = β = ½, γ = 1 (in fact, Legendre’s equation).174 This special case is mentioned in Gauss’s unpublished notes of 1809, (see Werke X.l, 343).
Gauss’s ideas about the meaning of a complex integral were connected by Schlesinger with his discussion of the hypergeometric series, because Gauss, although inspired by Euler’s work, did not follow Euler’s representation of the series as an integral. Schlesinger connected this with Gauss’s earlier refusal to invert the general elliptic integral with complex modulus, and ascribed it to an awareness that a complex integral may well define a many-valued function of its upper end-point. Indeed, in Gauss’s letter to Bessel already quoted. Gauss observed that the value of a complex integral depends on the path between its end points, and then he wrote:
The integral ∫ ϕ x ∙ dx along two different paths will always have the same value if it is never the case that φ x = ∞ in the space between the curves representing the paths. This is a beautiful theorem whose not-too-difficult proof I will give at a suitable opportunity. . . . In any case this makes it immediately clear why a function arising from an integral ∫ ϕ x ∙ dx can have many values for a single value of x, for one can go round a point where φ x = ∞ either not at all, or once, or several times. For example, if one defines log x by ∫ dx/x, starting from x = 1, one comes to log x either without enclosing the point x = 0 or by going around it once or several times; each time the constant +2πi or –2πi enters; so the multiplicity of logarithms of any number are quite clear.
This is generally regarded as the first crucial insight into the integration of functions of a complex variable.
Gauss and non-Euclidean geometry
The claim, made on Gauss’s behalf, that he was a, or even the, discoverer of non-Euclidean geometry is very hard to decide because the evidence is so slight. It is nonetheless implicit in the excellent commentaries of Stäckel and Dombrowski [1979], as it is in Reichardt’s book [1976] and the broader but slighter survey by Coxeter [1977]. Proponents of this view, with Dunnington, are happy to tie documents written in the late 1820s and 1830s to cryptic claims made by Gauss for early achievements, and to equally elusive passages from the 1810s. In fact, the evidence points in another direction. It suggests that Gauss was aware that much needed to be done to Euclid’s Elements to make them rigorous, and that the geometrical nature of physical space was regarded by Gauss as more and more likely to be an empirical matter, but in this his instincts and insights were those of a scientist, not a mathematician.
Gauss was 22 when he confided to Wolfgang Bolyai that he was doubtful of the truth of geometry. He had already found too many mistakes in other people’s arguments in defence of the parallel postulate to be so confident any longer in their conclusion. He had begun to consider the fundamental assumptions of geometry at least two years earlier, in July 27, 1797, when he wrote in his mathematical diary only too cryptically that he had ‘demonstrated the possibility of a plane’. It is tempting to connect this with fragments of arguments dating from 1828 to 1832 in which Gauss investigated whether the locus of a line perpendicular to a fixed line and rotating about that fixed line has all the properties of a plane, because, in a famous letter to Bessel of January 1829, where Gauss claims to have harboured these thoughts for almost 40 years, he wrote that “apart from the well-known gap in Euclid’s geometry, there is another that, to my knowledge no-one has noticed and which is in no way easy to alleviate (although possible). This is the definition of a plane as a surface that contains the line joining any two of its points. This definition contains more than is necessary for the determination of the surface, and tacitly involves a theorem which must first be proved. . .” One knows from the later history of geometry, most clearly from the remarks of Pasch (Pasch [1882]) that trying to spell out what exactly elementary Euclidean geometry is about is extremely difficult.
By 1808 Gauss was aware that in the hypothetical non-Euclidean geometry similar triangles are congruent, and therefore there is an absolute measure of length. But at this stage, according to Schumacher, he found this conclusion absurd, and therefore held that the matter was still unclear. As he put it: “In the theory of parallels we are no further than Euclid was. This is the shameful part of mathematics, that sooner or later must be put in quite another form”. Evidently he did not then feel confident in a non-Euclidean geometry. By 1816 he had shifted his opinion to accommodate an absolute measure of length as paradoxical but not self-contradictory (Gauss to Gerling, April 1816, in Werke, 8, pp. 168–169) and n
ow he thought it would be remarkable if Euclid’s geometry was not true, because then we would have an a priori measure of length, such as the length of the side of an equilateral triangle with angle 59°59'59,99999". As Dunnington correctly observed, being remarkable is consistent with being attractive. But still there is no evidence that Gauss deduced anything specific about the new geometry.
In 1816 we do get a glimpse of what Gauss knew as reported by his former student Wachter. On a certain (unspecified) hypothesis, Wachter wrote to Gauss, the opposite of Euclidean geometry would apparently be true, which would involve us with an undetermined constant, a sphere of infinite radius which nonetheless lacks some properties of the plane, and the use of a transcendent trigonometry that probably generalises or underpins spherical trigonometry. Gauss now, as he wrote to Olbers, was coming “ever more to the opinion that the necessity of our geometry cannot be proved, at least not with human understanding. Perhaps in another life. . .. but for now geometry must stand, not with arithmetic which is pure a priori, but with mechanics.” (Gauss to Gerling, April 1816, in Gauss’s Werke, 8, pp. 177).
This passage has been quoted more often than it has been understood. How would a proof beyond human understanding differ from a proof that does not surpass human understanding? Would it be some argument, compelling even to God, that made Euclidean geometry the right geometry for Space? Arithmetic, it seems, has an apodictic status, a truth of a higher kind than the truth of geometry, which is down there with mechanics. But the passage does not say that there are two geometries at some logical level and some experiment must choose between them. It says that knowledge is lacking. Gauss did not claim to possess knowledge of a new geometry, which surely means that even the ideas he was discussing with Wachter he considered to be hypothetical, and capable of turning out to be false. The ‘transcendent trigonometry’ is usually taken to be the hyperbolic trigonometry appropriate to non-Euclidean geometry, but there is very little evidence to support any interpretation. Accordingly, when Gauss replied to Schweikart in March 1819 that he could “do all of astral geometry once the constant is given” we cannot be sure what, precisely. Gauss had formulae for. The only one dated to this period is the one in his reply to Schweikart for the maximum area of a triangle in terms of Schweikart’s Constant (the maximum altitude of an isosceles right-angled triangle). And all that his correspondence with Taurinus reveals is that, by 1824, Gauss was more comfortable than ever with the idea of a new geometry.
Far from being an exhaustive exploration of the problem of parallels, most of Gauss’s work before 1831 on the fundamentals of geometry is firmly in a style one can call classical. Point, line, plane, distance, and angle are taken as undefinable, primitive terms with, perhaps, a set of obscure relationships between them, which is in need of elucidation. Gauss’s study of the problem of parallels is mostly in this vein, and Bessel’s letter to Gauss of 1829 was further encouragement to Gauss to state that geometry has a reality outside our minds whose laws we cannot completely prescribe a priori. This is entirely consistent with the classical formulation. The concepts of point, line, plane and so forth are formed as all scientific concepts are, and the task of the mathematician is to get them truly clear in the mind. This might involve teasing out tacit assumptions and fitting them up with proofs, or it might call for the elaboration of new ideas about a hitherto unsuspected species of geometry, which might nonetheless turn out to be (for some value of an unknown constant) the true geometry of space. This was also the approach of Gerling, Crelle and Deahna (see Zormbala [1996]).
Adherence to the classical formulation denies trigonometric methods a fundamental role. And in fact there is very little evidence of Gaussian contributions to trigonometry in non-Euclidean geometry before the letter to Schumacher of 12 July 1831, where he says that the circumference of a semi-circle is ½ π k (er/k – e-r/k) where k is a very large constant that is infinite in Euclidean geometry.
In particular, there is no evidence that Gauss derived the relevant trigonometric formulae from the profound study of differential geometry that occupied him in the 1820s. What he did say in the Disquisitiones generales circa superficies curvas is summed up in what he regarded as one of the most elegant theorems in the theory of curved surfaces: “The excess over 180° of the sum of the angles of a triangle formed by shortest lines on a concavo-concave surface, or the deficit from 180° of the sum of the angles of a triangle formed by shortest lines on a concavo-convex surface, is measured by the area of the part of the sphere which corresponds, through the direction of the normals, to that triangle, if the whole surface of the sphere is set equal to 720 degrees.”
It is tempting to suppose that Gauss connected this elegant theorem with the study of non-Euclidean geometry, by considering a concave-convex surface of constant negative Gaussian curvature. Even if we do, we must still note that Gauss did not develop the trigonometry of triangles on surfaces of constant (positive or negative) curvature until after 1840, when he had read Lobachevskii’s Geometrische Untersuchungen. Moreover, he did not have an example of a surface of constant negative curvature to hand; there is every reason to suppose that Minding was the first to discover one. Minding’s example, moreover has a number of topological properties that rule it out as a model of space, notably self-intersecting geodesies and pairs of geodesies that meet in more than one point.
But even if (in contradiction to Hilbert’s later theorem about surfaces of constant negative curvature) Gauss had found a surface of constant negative curvature in space, it would only establish that there is a surface in space whose intrinsic geometry is non-Euclidean. It would not establish that space could be non-Euclidean, because space is three dimensional. There is no sign that Gauss had any of the concepts needed to formulate a theory of differential geometry in three dimensions. Nor did he have any of the essential mathematical machinery for doing differential geometry in three or more dimensions, without which almost nothing useful can be said.
What weight can the two formulae be made to carry? They are not difficult to obtain and manipulate if, as for example Taurinus did, one assumes that non-Euclidean geometry is described by the formulae of hyperbolic trigonometry—a natural enough assumption. To introduce hyperbolic trigonometry into the study of non-Euclidean geometry properly is, as Bolyai and Lobachevskii found, a considerable labour of which no trace remains in Gauss’s work. It is more plausible to imagine that he made the assumption, but did not derive it from basic principles. So, perhaps by 1816, Gauss was convinced of ideas like these:
there could be a non-Euclidean geometry, in which the angle sum of triangles is less than π,
the area of triangles is proportional to their angular defect and is bounded by a finite amount,
the trigonometric formulae for this geometry are those of hyperbolic trigonometry, and the analogy with spherical geometry and trigonometry extends to formulae for the circumference and area of circles.
Such a position is unsatisfactory, to Gauss and to us, because it is purely and simply an analogy with spherical geometry. But whereas spherical geometry starts with a sphere in the non-Euclidean case there is no surface. And even if one found non-Euclidean geometry on a surface in three-dimensional Euclidean space, it would not follow that three-dimensional space was non-Euclidean, any more than the existence of spheres in three-dimensional space forces the conclusion that space is a three-dimensional sphere.
It becomes clear that a mathematician persuaded of the truth of non-Euclidean geometry and seeking to convince others is almost driven to start by looking for, or creating, non-Euclidean three-dimensional space, and to derive a rich theory of non-Euclidean two-dimensional space from it—as Bolyai and Lobachevskii did, but not Gauss. The only hint we have that he explored the non-Euclidean three-dimensional case is the remark by Wachter, but what Wachter said was not encouraging: “Now the inconvenience arises that the parts of this surface are merely symmetrical, not, as in the plane, congruent; or, that the radius on one side is infinit
e and on the other imaginary” and more of the same. This is a long way from saying, what enthusiasts for Gauss’s grasp of non-Euclidean geometry suggest, that this is the Lobachevskian horosphere, a surface in non-Euclidean three-dimensional space on which the induced geometry is Euclidean.
Gauss, by contrast, possessed a scientist’s conviction in the possibility of a non-Euclidean geometry which was no less, and no greater, than that of Schweikart or Bessel. The grounds for his conviction are greater, but still insubstantial, because he lacks almost entirely the substantial body of argument that gives Bolyai and Lobachevskii their genuine claim to be the discoverers of non-Euclidean geometry.
Did Gauss then, as a scientist, make an empirical test of the matter? This is one of the most discussed questions in the whole subject of Gauss and non-Euclidean geometry. Those who believe that he did quote Sartorius von Waltershausen’s reminiscence, where on p. 81, he states that Gauss did check the truth of Euclidean geometry on measurements of the triangle formed by the mountains Brocken, Hohenhagen, and Inselsberg (BHI), and found it to be approximately true. This claim was most recently advanced by Scholz [1992], on the basis of a number, quoted by von Waltershausen elsewhere in his reminiscence, relating to the very close agreement between the measurements of this triangle and the predictions of Euclidean geometry (once the mountain tops are treated as three points on a sphere). Scholz concludes that “there is no longer any reason to doubt that Gauss himself conducted such a test of the angle sum theorem.” (Scholz [1992], p. 644).
Those who dispute that Gauss made such a test argue that the problem that occupied Gauss, and figures so prominently at the end of the Disquisitiones generales circa superficies curvas, is the question of the spheroidal or spherical shape of the Earth, and that von Waltershausen was simply confused about the hypothesis that Gauss found to be approximately confirmed. This is the opinion of Miller [1972].
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 52