## 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, the finest and most complete copy ever likely to come to the market, of “the most extraordinary two dozen pages in the history of thought” (Halsted), containing the independent foundation (along with the work of Lobachevsky) of non-Euclidean geometry. This work is one of the few absolute rarities among the classics of science: in his recently published census, Lemley locates 25 surviving copies, including this one (no. 21 in the census). Our copy is in the most complete state (1H, 2C in Lemley’s classification). Only two other copies in this state are known, both in institutional collections (Universitätsbibliothek Leipzig and the Bibliothèque universitaire des Sciences et Techniques in Bordeaux). There are only two copies in auctions records: the Norman copy (no. 2, Sotheby’s NY, 1998, $98,000), in original binding but not the most complete state; and no. 5 (Sotheby’s Paris, 2011, €120,750), in contemporary marbled boards, again not in the most complete state. “Despite its brevity and relative obscurity, János Bolyai’s twenty-six-page *Appendix* was epoch-making in the history of mathematics. In it, Bolyai (1802–60) created a non-Euclidean system of geometry by challenging Euclid’s fifth postulate, otherwise known as the axiom of parallelism. While its reception was initially muted, János’s work would ultimately provide the mathematical basis for Einstein’s special theory of relativity, developed be- tween 1905 and 1915” (Lemley, pp. 196-197). Nikolai Lobachevsky (1792-1856) and János Bolyai had independently created non-Euclidean systems by challenging the ‘parallel postulate’ of Euclid. It has been a matter of debate for centuries whether this postulate – that through any point not on a given straight line exactly one line can be drawn that does not intersect the given line – can be deduced from the other postulates of Euclidean geometry. “János began work on his new geometry in 1820 at the age of eighteen, though his father, Farkas Bolyai, attempted to dissuade him from disproving Euclid’s fifth postulate … After his son’s unanticipated success, however, Farkas Bolyai published János’s treatise as an appendix to the first volume of his mathematics textbook titled, *Tentamen Juventutem Studiosam*. Published in this form—as a postscript to a small edition printed by an unknown Hungarian college in a provincial town in Transylvania—the work was almost guaranteed obscurity” (Lemley, p. 197). A copy of the *Appendix* was sent to Gauss shorty after publication. ‘Gauss was impressed, writing to Gerling on 14 February 1832 that “I consider this young geometer von Bolyai to be a genius of the first order”’ (Gray, p. 125). However, “Gauss responded in a letter [to Farkas] dated 6 March 1832: ‘If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself.’ Gauss’s dismissive assertion of priority disheartened János and effectively ended the young geometer’s career in mathematics. And while Farkas viewed Gauss’s letter as transparent praise from a master mathematician, János doubted Gauss’s claim of priority and suspected his father of sharing his work prior to its publication. In the years following its publication, Bolyai’s Appendix remained an obscure masterpiece. However, interest in non-Euclidean geometry was revived gradually in the decades before Einstein’s theory of special relativity, and since then the *Appendix* has been rightly viewed as transformative in the history of mathematics” (Lemley, pp. 197-198). 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.

*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.

In about 300 BC Euclid wrote the* Elements*, a book which “has exercised an influence upon the human mind greater than that of any other work except the Bible” (DSB). Euclid stated five postulates from which he deduced all of the theorems and propositions in his work:

- To draw a straight line from any point to any other.
- To produce a finite straight line continuously in a straight line.
- To describe a circle with any centre and distance.
- That all right angles are equal to each other.
- That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclid was dissatisfied with the fifth postulate and he tried to avoid its use as long as possible – in fact the first 28 propositions of *The Elements* are proved without using it. In his commentary on the* Elements*, Proclus (410-485) noted several incorrect attempts to deduce the fifth postulate from the other four, including one by Ptolemy, and gave a false proof of his own. However, his work was important for stating the following postulate, which is equivalent to the fifth postulate (in the presence of the other postulates):

Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

This became known as ‘Playfair’s Axiom’ after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid’s fifth postulate by this axiom.

Over the following centuries many attempts were made to prove the fifth postulate from the other four, all of which were found to contain errors or hidden assumptions which turned out to be equivalent to the fifth postulate. For example, John Wallis (1616-1703) showed in 1663 that the fifth postulate is a consequence of (in fact, equivalent to) the following statement:

To each triangle, there exists a similar triangle of arbitrary magnitude.

Since this statement is ‘obviously’ true, the fifth postulate follows. More than a century later, Adrien-Marie Legendre (1752-1833) ‘proved’ the fifth postulate, assuming the ‘obvious’ fact that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle; this statement is also equivalent to the fifth postulate. Legendre also showed that the fifth postulate is equivalent to the following classic theorem of Euclidean geometry:

The sum of the angles of a triangle is equal to two right angles.

The most important of the attempts to prove the fifth postulate was made by Girolamo Saccheri (1667-1733) in his *Euclides ab omni naevo vindicates* (*Euclid Freed of Every Flaw*, 1733). He attempted a ‘proof by contradiction’ – he assumed that the fifth postulate was false and attempted to derive an obviously false conclusion. Saccheri satisfied himself that he had reached a contradiction, but in fact he had derived from the assumption that the fifth postulate was false many of the theorems of non-Euclidean geometry which Bolyai and Lobachevsky would prove a century later. In 1786, Johann Heinrich Lambert (1728-77) again assumed that the fifth postulate was false and derived many formulas which turned out to be formulas of non-Euclidean trigonometry.

Carl Friedrich Gauss (1777-1855) began work on the fifth postulate in 1792 when he was only 15 years old. By 1817 Gauss had become convinced that the fifth postulate could not be deduced from the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. But Gauss never published this work, perhaps because his thinking at this time was dominated by Kant who had stated that Euclidean geometry is an inevitable necessity of thought. Gauss did, however, discuss the theory of parallels with his friend Farkas Bolyai (1775-1856), who himself had given several false proofs of the fifth postulate. Indeed, Farkas sent Gauss a manuscript entitled *Theoria parallelarum.*

“János Bolyai received his early education in Marosvásárhely, where his father was professor of mathematics, physics, and chemistry at Evangelical-Reformed College … Form 1815 to 1818, he studied at the college where his father taught. The elder Bolyai had hopes that the son would go on to Göttingen to study with his friend Gauss, but he did not. In 1818 János entered the imperial engineering academy in Vienna, where he received a military education; he remained there until 1822.

“From his father, János had inherited an interest in the theory of parallels: but in 1820 his father warned him against trying to prove the Euclidean axiom that there can be only one parallel to a line through a point outside of it:

‘You should not tempt the parallels in this way, I know this way until its end—I also have measured this bottomless night. 1 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.’

“In the same year, however, János began to think in a direction that led him ultimately to a non-Euclidean geometry. He profited by conversations with Karl Szász, governor in the house of Count Alexis Teleki, In 1823, after vain attempts to prove the Euclidean axiom, he found his way by assuming that a geometry can be constructed without the parallel axiom: and he began to construct such a geometry. ‘From nothing I have created another entirely new world,’ he jubilantly wrote his lather in a letter of 3 November 1823 …

“While visiting his father in February 1825, János had shown him a manuscript that contained his theory of absolute space, that is a space in which, in a plane through a point *P* and a line *I* not through *P* there exists a pencil of lines through *P* which does not intersect *I*. When this pencil reduces to one line, the space satisfies the Euclidean axiom. Farkas Bolyai could not accept this geometry, mainly because it depended on an arbitrary constant, but he finally decided to send his son’s manuscript to Gauss. The first letter (20 June 1831) went unanswered, but Gauss did answer a second letter (16 January 1832). In this famous reply, dated 6 March 1832 and directed to his ‘old, unforgettable friend,’ Gauss said:

‘Now something about the work of your son. You will probably be shocked for a moment when I begin by saying *that I cannot praise it*, but I cannot do anything else, since to praise it would be to praise myself. The whole content of the paper, the path that your son has taken, and the results to which he has been led, agree almost everywhere with my own meditations, which have occupied me in part already for 30–35 years. Indeed, I am extremely astonished.’

“Further on, after mentioning that there had been a time when he had been inclined to write such a paper himself, Gauss continued, ‘Hence I am quite amazed, that now I have been saved the trouble, and I am very glad indeed that it is exactly the son of my ancient friend who has preceded me in such a remarkable way.’ Gauss ended with some minor remarks, among them a challenge to János to determine, in his geometry, the volume of a tetrahedron, and a critique of Kant’s theory of space.

“It is now known from Gauss’s diaries and from some of his letters that he was not exaggerating; but for János the letter was a terrible blow, since it robbed him of the priority. Even after he became convinced that Gauss spoke the truth, he felt that Gauss had done wrong in remaining silent about his discovery. Nevertheless, he allowed his father to publish his manuscript, which appeared as an appendix to the elder Bolyai’s *Tentamen* (1832), under the title ‘Appendix scientiam spatii absolute veram exhibens’ (‘Appendix Explaining the Absolutely True Science of Space’). This classic essay of twenty-four pages, which contains János’ system of non-Euclidean geometry, is the only work of his published in his lifetime” (DSB).

“János Bolyai’s *Appendix* consists of twenty-six densely printed pages bound near the end of the first volume of his father’s two-volume mathematics textbook, *Tentamen Juventutem Studiosam (Marosvásárhely*, 1832– 33; hereafter, *Tentamen*). Both volumes of the *Tentamen* were printed in only one edition and one printing at the press of the Reformed College by Joseph and Simeon Kali in the Transylvanian city of Marosvásárhely (present-day Târgu Mureș, Romania)” (Lemley, p. 188). The Reformed College was the oldest Hungarian school in Transylvania, and the town still houses the Teleki Library, established in 1802 by Count Samuel Teleki and which subsequently acquired János Bolyai's papers and now contains the Bolyai Museum.

“The *Appendix* is more than an adjunct to the *Tentamen*; it is referred to repeatedly therein, and the two works accordingly should be viewed as integral” (*ibid*.). 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).

“The subscription list printed at the end of the first volume lists seventy names subscribing 128 copies; an additional list in the second volume lists nine more names subscribing twenty-eight copies. These numbers suggest that the edition was made up of at least 156 copies, although a slightly larger number, likely in the range of 175–200 copies, is more probable. This suggests a survival rate of about ten percent, pending the discovery of additional copies. The presence of a supplemental list of subscribers at the end of the second volume suggests that subscription likely continued during printing” (Lemley, p. 196).

Lemley’s article contains a detailed bibliographical analysis of surviving copies, as their make-up, including this copy, varies widely. Norman notes: “The *Tentamen* was very crudely or amateurishly printed at a school press; copies exhibit the earmarks of non-professional or inexperienced publishing, particularly in the clumsy typography and numerous errata and corrigenda leaves, which must have made the *Tentamen* extremely difficult to use. These leaves were printed on different paper stocks and were obviously added after the original printing.”

Both volumes of this copy are in the most complete state (of several) possible, with all of the various addenda issued. 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.

Of the 25 surviving copies in Lemley’s census, seven are in private hands. A copy formerly held by the Berlin Staatsbibliothek is recorded as having been lost in World War II.

Dibner 116; Evans 13; Horblit 69b; Norman 259; Parkinson pp. 295 and 296; Gray, *Worlds out of Nothing*, 2007; Lemley, ‘A bibliographical description of Farkas Bolyai's Tentamen Juventutem and János Bolyai's Appendix (1832/3) with a census of copies’, *PBSA *113 (2019), 187-203 (this copy is no. 21 in the census); Nagy*, 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?).

Two 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, first volume uncut, second with some outer edges uncut (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 volume, just touching some of the figures, some paper flaws as often). Uniform contemporary blue boards, paper labels on spines (spines and joints cracked but sound), preserved in a morocco box.

Item #4659

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Price:
$285,000.00
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