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:] Bolyai, Farkas: Tentamen Juventutem Studiosam in Elementa Matheseos Purae. Tomus primus [-secundus]

First edition of János Bolyai’s Appendix, his "explanation of the absolutely true science of space", "the most extraordinary two dozen pages in the whole history of thought" (Halsted, 1896), a complete and consistent system of non-Euclidean geometry. Published as an appendix to his father Farkas Bolyai’s book on geometry, Tentamen Juventutem Studiosam in Elementa Matheseos Purae Introducendi. Condition: Volume 1 Title page with old collection stamps of "Mathematicai Seminarium", appearing on the top and bottom of the page, and an illegible ownership inscription, dated 1840. A cluster of small brown spots at the lower right corner, ghosting to the verso. Another cluster of such spots at the edge of p. LXVII. Grayish stains appear at the lower corner of the pages of signature 3E, they grow and merge into each other at sign. 3M, this stain remains visible until the end of the book. Quire 26 with printing defects, with no effect on legibility. In this copy the two-sided broadsheet letterpress plate ("Explicatio Signorum / Arbor Arithmeticae Geometriaeque") bound after the title page, its paper has light stains on the lower half. Quire V is bound – probably erroneously – after VII2, and quire 26 after 2D. Two handwritten leaves, numbered as LXXVII, with the text of "Recensio per auctorem ipsum facta" (ie. Addendum 5, which only appears in state 1E – according to Lemley’s classification – and paginated as LXXVII–LXX[V]III) bound after p. LXIV (XVIII4v). Some pages are slightly overtrimmed at the top edge, which occasionally effects the page numbers, or headpieces, except for p. 2[28]v (2D2v), where the first line of the text is trimmed. Plate 1–3 slighly overtrimmed, effecting the frame and the plate titles, a diagram on plate 2 is also trimmed. Plate 3 includes the four demanded folding diagrams. Volume 2 Title page with old stamps and dated ownership inscription as in the first volume. The first quire and the first leaves of the second are stained (a light-brown stain at the upper half of the leaves). An inkblot at the lower gutter affects each second leaf of the first seven quires (I2, 12, 22, etc.; suggests that the contamination happened before binding). With occasional notes in Latin, signs and diagrams at the margins in pencil, a few correction of the text in ink and a collection of unintelligible curly brackets on the right margin of Tab. 2. The volume contains an extra copy of Tab. 7. Tab. 6. overtrimmed, effecting a digram. Both copies of Tab. 7. are overtrimmed at the lower edge, the lower frame is entirely trimmed causing small damages to two diagrams. Otherwise each plate is complete, with all the demanded folding diagrams and vervolle. Lemley in his thorough bibliographical study and catalogue of the 26 known copies of the Tentamen (of which only 25 exists today), classifies and describes eight different states of the first (1A–H) and four of the second volume (2A–D) (Lemley, 2019). Both volumes of this copy, are a hitherto unrecorded state of the book, they are differing significantly from those presented in the study. Lemley divides the first volume into 11 sections, of which 1 (?2), 2 (I–VI4 VII2), 9 (A–U4 X4 Z4 J4 V4 2A–2U4 2X4 2Z4 2J4 2V4 3A–3P4), 10 (2A–C4 D2) and 11 (x8) are constant. Section 3–8 are addenda (add. 1–6) and it varies which copy contains which of them (Lamley assumes that the addenda "were inserted into as yet unsold copies of the original and sole issue during binding and distribution, thus producing the variety of states"). The "constant sections" are certainly presented in our copy too, and of the addenda it contains add. 1 (26) and add. 3 (XVIII). Add. 1 bound to an irregular place, between section 10 and 11 – as mentioned above. It’s not clear if it was the publisher’s or the first binder’s intention or mistake, however the aforementioned grayish stain which effects this part of the book, also appears on the leaves of add. 1, which suggests that it’s not a recent alteration of the order of th.…