Appendix. Scientiam Spatii Absolute Veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem: adjecta ad casum falsitatis, quadratura circuli geometrica. [in:]BÓLYAI, Farkas. Tentamen Juventutem Studiosam in Elementa Matheseos Purae. Tomus primus [-secundus].

Maros Vásárhely [now Târgu Mureș, Romania]: Joseph and Simon Kaili, at the press of the Reform College, 1832-1833.

First edition of ‘the most extraordinary two dozen pages in the history of thought’ (Halsted) and one of the few absolute rarities among the classics of science. This work contains the independent foundation (along with the work of Lobachevsky) of non-Euclidean geometry. Of the 25 copies located in Lemley’s census, the present copy represents the most complete state of the text for both volumes.


First edition of ‘the most extraordinary two dozen pages in the history of thought’ (Halsted) and one of the few absolute rarities among the classics of science. This work contains the independent foundation (along with the work of Lobachevsky) of non-Euclidean geometry. I have located some 23 other copies worldwide, all of them exhibiting variations in issue or completeness (the present copy represents the most complete state of the text for both volumes).

Lobachevsky and János Bolyai had independently created non-Euclidean systems by challenging the ‘parallel postulate’ of Euclid. János Bolyai’s work was conceived in 1823, when he wrote to his father ‘I have now resolved to publish a work on the theory of parallels ... I have created a new universe from nothing’. It was published as an appendix to his father’s mathematical treatise, the Tentamen, 1832–3. Lobachevsky's work appeared in a Kazan academic periodical between 1829–1830, and in fuller form as Geometrische Untersuchungen, Berlin 1840. Whereas Lobachevsky initially had only demonstrated the possibility of a geometry in which Euclid’s fifth postulate (or 11th axiom) was untrue, János developed a geometry completely independent of the fifth postulate and applicable to varieties of curved space. However, the epochal significance of the work of these two was to remain largely unappreciated until the beginning of the twentieth century when it provided the mathematical basis for the Theory of Relativity.

Provenance: stamp of the publisher, press of the Reformed College, Maros Vásárhely (now the Romanian city of Târgu Mures) on free flyleaf and title of first volume. 

János began working on his new geometry early in the 1820s. His father tried to discourage him from attempting to prove or refute Euclid’s fifth postulate:

‘You should not tempt the parallels in this way. I know this way until its end – I also have measured this bottomless night, I have lost in it every light, every joy of my life ... You should shy away from it as if from lewd intercourse, it can deprive you of all your leisure, your health, your peace of mind and your entire happiness. This infinite darkness might perhaps absorb a thousand giant Newtonian towers, it will never be light on earth, and the miserable human race will never have something absolutely pure, not even geometry’ (quoted from DSB). Farkas Bolyai, however diffidently he felt about his son’s researches, did send the manuscript of the Appendix to Gauss: the first letter went unanswered, and a second letter only elicited the reply that Gauss could not praise it, because he himself had reached the same conclusions some 30 years earlier although he had not published his discovery! This assertion so discouraged János that it effectively terminated his career in creative mathematics, but his father did publish his paper as an appendix to his own textbook. Published in this form, in a small edition (the two lists of subscribers give some 79 names accounting for 156 copies), by the Reform College in the Transylvanian city Maros Vásárhely (now Târgu Mureș, Romania), the work was guaranteed immediate oblivion. It remained a forgotten masterpiece until, 35 years later, Riemann’s paper on the hypotheses of geometry reawakened interest in this field, with profound consequences for the mathematical description of real space.

In the rediscovery of János’s masterpiece, the father’s work was largely neglected. Farkas Bolyai was a close friend of Gauss and regarded by the latter as the only man who fully understood Gauss’s metaphysics of mathematics. ‘He can be taken as a precursor of Gottlob Frege, Pasch, and Georg Cantor; but, as with many pioneers, he did not enjoy the credit that accrued to those that followed him’ (DSB). He had worked on the parallel postulate and the possibilities of a non-Euclidean geometry from his earliest days as a mathematician in Göttingen, and had corresponded with Gauss on the subject, even sending him a manuscript entitled Theoria parallelarum, but it was his son János who was to achieve the breakthrough. The Appendix appears at the end of volume one, and is separately paginated [ii] 26 [2, errata], and with one plate in the volume specifically pertaining to the Appendix. There are further substantial references to the Appendix in the main body of Farkas’s text, primarily in the section ‘Generalis conspectus geometriae’ (Vol I, pp 442–502) and an important supplement to the Appendix in the second volume (pp 380–383). Apart from the Appendix, hardly any two copies of the Tentamen agree in collation, and the great variation amongst them, including cancel leaves and gatherings, indicates that the publishing history of this work was confused, and remains confusing.

This copy is unusual in a further respect; the first volume is larger than any others known. The Norman copy was also uncut but measured only 219 x 137 mm, whereas its second volume was 225 x 143 mm, closer to the measurements of our first volume.

Bolyai illustrates his textbook with 14 folding plates, five of which are augmented with numerous small flaps. These plates contain as many as 10 slips, often concealed one behind the other; plate 10 also displays a single volvelle, which has gone unrecorded in most bibliographies to date; although not described in the printed or on-line catalogue entries, it is present in most copies. One point of bibliographic confusion has been clarified: the Horblit/Grolier Catalogue (based on the Smithsonian copy) lists an overslip on plate 6 that is not recorded in any other copy. Upon investigation, it appears that an integral part of the plate (the lower portion of the diagram labelled T.144) was inadvertently detached during rebinding and subsequently reattached on a stub, leading to the conclusion that this was a required flap.

Currently 24 copies of the Tentamen are known to exist, including the present copy, and one (Berlin) that was lost in WWII. Of these 24, one comprises Janos Bolyai’s Appendix only. A further three comprise volume one only. In addition, some copies are seriously defective, apart from the standard issue variations. There are numerous variations in collation, etc. amongst these copies. Samuel Lemley is compiling a detailed census and concordance which will be available shortly.

Collation note: both volumes are in the most complete state (of several) possible, with all of the various addenda issued.

Dibner 116; Evans 13; Horblit 69b; Norman 259; Parkinson pp 295 and 296; Nagy, Ferenc, Bolyai: Biographia, biblioteka, bibliografia pp 353–4 (incorrectly calling for an additional (4) ff of preliminaries in vol I, and in volume II requiring 584 pp instead of 402 pp, perhaps a typographical error for 384, as in the first issue?)

2 vols, 8vo (I: 228 x 145 mm; II: 214 x 125 mm), I: pp [iv] XCVIII; 502; [ii] 26 [2, errata] XVI [subscriber’s list and Latin-Hungarian lexicon of mathematical terms], with one large folding letterpress table, and 4 folded engraved plates (plate 3 with 7 small folding slips); II: pp [vi] xvi [Index Tom II] 402, with 10 folded engraved plates (plate 7 with 10 slips, plate 8 with 4 slips, plate 9 with 3 slips and plate 10 with 5 slips and 1 volvelle), manuscript corrections to line 6 of p 380 vol II; vol II with some worming to inner blank margins and text in several gatherings, affecting some letters but still legible, also affecting first four plates in same vol, just touching some of the figures, some paper flaws as often; first volume uncut, second with some outer edges uncut, together in uniform contemporary blue boards, paper labels on spines, spines and joints cracked but sound, preserved in a morocco box.


Item #4659

Price: $200,000.00

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