Theorie der Parallellinien.

Leipzig: 1786.

First edition, very rare, and a copy with excellent provenance, of one of the most important works on non-Euclidean geometry preceding those of Bolyai and Lobachevsky half a century later. Lambert derived several fundamental results in this subject, and “no one else came so close to the truth without actually discovering non-Euclidean geometry” (Boyer, History of Mathematics, p. 504). “The memoir Theorie der Parallellinien (Theory of parallel lines) by Johann Heinrich Lambert (1727-1777), written probably in 1766, is a masterpiece of mathematical literature, and its author is one of the most outstanding minds of all times” (Papadopoulos & Théret). “In the introductory part of his treatise Lambert wrote: ‘This work deals with the difficulty encountered in the very beginnings of geometry and which, from the time of Euclid, has been a source of discomfort for those who do not just blindly follow the teachings of others but look for a basis for their convictions and do not wish to give up the least bit of the rigor found in most proofs. This difficulty immediately confronts every reader of Euclid’s Elements, for it is concealed not in his propositions but in the axioms with which he prefaced the first book’” (Rosenfeld, A History of Non-Euclidean Geometry, p. 99). This difficulty was the question of whether Euclid’s ‘Parallel Postulate’ – that through any given point not on a given straight line one can draw exactly one straight line that is parallel to (i.e., which does not intersect) the given line – could be deduced from the other axioms of Euclidean geometry. Girolamo Saccheri, in his Euclides ab omni naevo vindicatus (1733), had deduced many interesting consequences of denying the parallel postulate, but had ultimately concluded, erroneously, that denying it led to a contradiction. Lambert was the first to realize “that Euclid’s Parallel Postulate cannot be proved from the other Euclidean postulates and that it is possible to build a logically consistent system satisfying the other postulates but explicitly rejecting the Parallel Postulate” (Parkinson, Breakthroughs, 1766 & 1786). OCLC lists no copies in US; no copies on ABPC/RBH.

Provenance: Max Steck (1907-71), German-Swiss mathematician and mathematical historian (bookplate on front paste-down). Steck was the editor of Johann Heinrich Lambert: Schriften zur Perspektive (Berlin, 1943), which contains a Bibliographia Lambertiana (reprinted separately, Hildesheim, 1970).

Lambert became interested in the parallel postulate after having heard of Georg Simon Klügel’s dissertation Conatuum praecipuorum theoriam parallelarum demonstrandi from 1763, in which he had shown the flaws of all proofs so far of the parallel postulate. This inspired Lambert to take up the subject himself. Like Saccheri’s Euclides vindicatus, “Lambert wrote his Theorie der Parallellinien in an attempt to prove, by contradiction, the parallel postulate. He deduced remarkable consequences from the negation of that postulate. These consequences make his memoir one of the closest (probably the closest) text to hyperbolic geometry, among those that preceded the writings of Lobachevsky, Bolyai and Gauss. We recall by the way that hyperbolic geometry was acknowledged by the mathematical community as a sound geometry only around the year 1866, that is, one hundred years after Lambert wrote his memoir.

“To give the reader a feeling of the wealth of ideas developed in Lambert’s memoir, let us review some of the statements of hyperbolic geometry that it contains. Under the negation of Euclid’s parallel postulate, and if all the other postulates are untouched, the following properties hold:

(1) The angle sum in an arbitrary triangle is less than 180°.

(2) The area of triangles is proportional to angle defect, that is, the difference between 180° and the angle sum.

(3) There exist two coplanar disjoint lines having a common perpendicular and which diverge from each other on both sides of the perpendicular.

(4) Given two coplanar lines d1 and d2 having a common perpendicular, if we elevate in the same plane a perpendicular d3 to d1 at a point which is far enough from the foot of the common perpendicular, then d3 does not meet d2.

(5) Suppose we start from a given point in a plane the construction of a regular polygon, putting side by side segments having the same length and making at the junctions equal angles having ascertain value between 0 and 180°. Then, the set of vertices of these polygons is not necessarily on a circle. Equivalently, the perpendicular bisectors of the segment do not necessarily intersect.

(6) There exist canonical measures for length and area.

“Property (6) may need some comments. There are several ways of seeing the existence of such a canonical measure. For instance, we know that in hyperbolic geometry, there exists a unique equilateral triangle which has a given angle which we can choose in advance (provided it is between 0 and 60°). This establishes a bijection [one-to-one correspondence] between the set of angles between 0 and 60° and the set of lengths. We know that there is a canonical measure for angles (we take the total angle at each point to be equal to four right angles.) From the above bijection, we deduce a canonical measure for length. This fact is discussed by Lambert in §80 of his memoir. Several years after Lambert, Gauss noticed the same fact. In a letter to his friend Gerling, dated April 11, 1816 (cf. C. F. Gauss, Werke, Vol. VIII, p. 168), he writes: “It would have been desirable that Euclidean geometry be not true, because we would have an a priori universal measure. We could use the side of an equilateral triangle with angles 59°59’59,9999” as a unit of length”. We note by the way that there is also a canonical measure of lengths in spherical geometry, and in fact, a natural distance in this geometry is the so-called ‘angular distance’.

“It also follows from Lambert’s memoir that in some precise sense there are exactly three geometries, and that these geometries correspond to the fact that in some (equivalently, in any) triangle the angle sum is respectively equal, greater than, or less than two right angles. This observation by Lambert is at the basis of the analysis that he made of the quadrilaterals that are known as Lambert quadrilaterals, or Ibn al-Haytham–Lambert quadrilaterals. These are the trirectangular quadrilaterals (that is, quadrilaterals having three right angles), and Lambert studied them systematically, considering successively the cases where the fourth angle is obtuse, right or acute. It is fair to note here that Lambert was not the first to make such an analysis in the investigation of the parallel problem, and we mention the works of Gerolamo Saccheri (1667-1773) and, before him, Abu ‘Ali al-Hasan ibn al-Haytham and Umar al-Khayyam. The three geometries suggested by Lambert’s and his predecessors’ analysis correspond to constant zero, positive or negative curvature respectively, but of course Lambert and his predecessors did not have this notion of curvature. The interpretation of the three geometries in terms of curvature was given one century after Lambert’s work, by Beltrami.

“Another important general property on which Lambert made several comments and which he used thoroughly in his memoir is the following: there exist strong analogies between statements in the three geometries (Euclidean, spherical and hyperbolic), with the consequence that some of the statements in the three geometries may be treated in a unified manner. More precisely, he noticed that there exist propositions that are formally identical in the three geometries up to inverting some in-equalities or making them equalities. A well-known example is the fact that in Euclidean (respectively spherical, hyperbolic) geometry, the angle sum of triangles is equal to (respectively greater than, smaller than) two right angles … There exist several statements of the same type, in which one passes from one geometry to the other by inverting certain inequalities. Euclidean geometry appears in this setting as the frontier geometry between spherical and Euclidean geometries. In relation with this, Lambert noticed that certain formulae of hyperbolic geometry can be obtained by replacing, in certain formulae of spherical geometry, distances by the same distances multiplied by the imaginary number √−1, and by keeping angles untouched. One well-known example is the following: Take, as a model of spherical geometry, the sphere of radius r (or curvature 1/r2). Recall that the area of a spherical triangle is equal to

r2 (α + β + γ − π),

where α, β, γ are the angles (in radians). This result is attributed to Albert Girard (1595-1632), who stated it in his Invention nouvelle en algebra (1629). If instead of r we take an imaginary radius √−1r, we obtain, as a formula for area,

r2 (α + β + γ − π) = r2 (π – α – β − γ),

which is precisely the area of a triangle of angles α, β, γ in the hyperbolic space of constant curvature −1/r2. Lambert declares at the occasion of a closely related idea that ‘we should almost conclude that the third hypothesis occurs on an imaginary radius’” (Papadopoulos & Théret).

Lambert’s memoir is divided into three parts: §1 to 11, §12 to 26, and §27 to 88. The central ideas are the following.

“In the first part, the author recalls the problem of parallels, pre-senting Euclid’s eleventh axiom, and the position it occupies among the propositions and the other axioms of the Elements. He mentions several difficulties presented by this axiom, quoting commentaries and attempts at proofs by his predecessors. Lambert, who was a fervent reader of classical literature, certainly knew the works of the Greek commentators and their successors on the parallel problem. Further- more, he was aware of Klügel’s dissertation, written in 1763, which contains a description of 28 attempts to prove the parallel axiom. In particular, Lambert knew about Saccheri’s work. It is also good to note that Lambert had probably no intention to publish his manuscript in the state it reached us, which explains the fact that certain historical references (in particular to Saccheri) are missing in that manuscript.

“In the second part, Lambert presents some propositions of neutral geometry, that is, the geometry based on the Euclidean axioms from which the parallel axiom has been deleted. One reason for which he works out these propositions is that he thinks that they may be used to prove the parallel axiom.

“The third part is the most important part of the memoir. Lambert presents his own approach to prove the parallel axiom. He develops a theory based on the negation of that axiom, hoping that it will lead to a contradiction” (ibid.).

After Lambert’s death in 1777, the Berlin Academy bought Lambert's Nachlass, his unpublished manuscripts, notes and correspondence, on the recommendation of the Swiss mathematician Johann Georg Sulzer (1720-79). Following Sulzer’s death two years later, the task of editing the Nachlass was taken over by Johann III Bernoulli (1744-1807). The Academy sold Bernoulli the Nachlass on the condition that he made a large portion of it available to the public. In 1782 Bernoulli inserted a note on the Nachlass in the widely read journals Allerneueste Mannigfaltigkeiten and Teutscher Merkur. He also published a posthumous manuscript of Lambert in the Mémoires of the Berlin Academy, he edited the first volume of Lambert’s Logische und Philosophische Abhandlungen, and published the first volume of Lambert’s Deutscher gelehrter Briefwechsel. In subsequent years, Bernoulli published a second volume of the Logische und Philosophische Abhandlungen and four more volumes of the Deutscher gelehrter Briefwechsel (1781-87). Bernoulli published Lambert’s manuscripts on mathematics and physics in the journals that his friend Carl Friedrich Hindenburg (1741-1808) edited, the short-lived Leipziger Magazin für reine und angewandte Mathematik (1786-89) and Archiv für reine und angewandte Mathematik (1795-99). Finally Bernoulli sold the Nachlass to the Duke of Gotha, in whose library it was rediscovered in the early 20th century by Karl Bopp.

“Johann Heinrich Lambert was born on 26 August 1728 in Mühlhausen (today Mulhouse, France). Mühlhausen was at that time associated to Switzerland. Lambert received six years of formal education from the municipality but had to leave school to help his father, a tailor, when he was 12 years old. However, Lambert never stopped learning though he did not attend any formal school afterwards. He studied French, Latin and Mathematics largely on his own. He became an assistant to the city clerk of Mühlhausen, J. H. Reber, then a bookkeeper to an industrialist and finally in 1746 a secretary to Prof. J. R. Iselin in Basel. In this position he gained access to the knowledge of physics and mathematics of his time. In 1748 he obtained a position in Chur as a private tutor to a grandson of Count Peter von Salis. At the court of von Salis, Lambert could finally pursue his research on physics and optics. He travelled with his pupil through Europe, meeting many eminent scientists and continuously pursuing his research. He became a member of the ‘physikalisch-mathematische Gesellschaft’ of Basel in 1754. From 1759 Lambert travelled on his own through Europe. During that time he published his early masterpiece, the Photometria.

“Lambert was living in rather poor conditions, though he received some support from the academies he was a member of. After long deliberations and in spite of Lambert’s eccentric character he became a member of the Royal Academy of Berlin in 1765. Finally Lambert had a secure post and he started researching and publishing on diverse topics of his interest. In this time of high productivity he proved that π and e are irrational, wrote about philosophy, studied non-additive probabilities and made contributions to hyperbolic functions and to cartography. Lambert died in Berlin on 25 September 1777” (Hulliger, pp. 2-3).

Somerville, Bibliography of non-Euclidean geometry, p. 11; Somerville, Elements of non-Euclidean Geometry, pp. 13-15; Gray, Worlds Out of Nothing, pp. 82-84; Klein, Mathematical Thought from Ancient to Modern Times, pp. 868-9. Papadopoulos & Théret, ‘Hyperbolic geometry in the work of J. H. Lambert,’ Bulletin of the Indian Society for History of Mathematics 36 (2014), pp. 129-155. Lambert’s treatise was reprinted in Engel and Stäckel’s Die Theorie der Parallellinien von Euklid bis auf Gauss, 1895. Hulliger, ‘Johann Heinrich Lambert,’ Bulletin of the Swiss Statistical Society 14 (2003), pp. 4-10.



8vo (203 x 116 mm). Contained in: Leipziger Magazin für reine und angewandte Mathematik, which was a relatively minor and short-lived (1786–1789) mathematical journal published by Johann III Bernoulli and Carl Friedrich Hindenburg. Lambert’s paper is pp. 137-164 and pp. 325-358 of the 1st volume (1786) and is accompanied by 2 engraved plates. Offered here is a very fine copy of the relevant volume ([2], 556 pp. and 8 plates) bound in contemporary German boards with richly gilt spine – a beautiful and unmarked copy.

Item #4779

Price: $12,500.00

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