Astronomy in Paulerinus's Fifteenth-Century Encyclopaedia Liber Viginti Arcium

Cf. Mistra Pavla Žídka Spravovna, ed. by Tobolka Zdenêk (Prague, 1908).

Cf. Włodek Sophia Zathey Georgius Zwiercan Marianus , Catalogus codicum manuscriptorum medii aevi Latinorum qui in bibliotheca Jagellonica Cracoviae asservantur (Wratislava, Warsaw and Cracow, 1980), i, 289–93.

Actually, such a statement is common in medieval encyclopaedias and the ‘brevity’ means the use of extracts from much more extended sources (cf. Ribémont B. , Encyclopedias , in Medieval science, technology, and medicine: An encyclopedia, ed. by Glick T. Livesey S. J. Wallis F. (New York and London, 2005).

Cf. Paulerinus (Pavel Žídek): Liber viginti arcium (ff. 185ra–190rb), ed. by Hadravová Alena (Clavis monumentorum litterarum, Regnum Bohemiae 3, Fontes 2; Prague , 1997). This book provides an edition of Paulerinus's Latin description of artisans, richly glossed in Old Czech and sometimes in German. The English preface on pp. xvii–xxvii summarizes also the previous literature. Editions of some other parts of the encyclopaedia (on astronomy and on animals) are in preparation.

Lemmata [A]striferalium orbium … motus, f. 131rb; [T]abularis astronomia, f. 134ra; Martis verum, f. 135rb; Mercurii verum, f. 135rb; Stacionem, retrogradacionem aut direccionem planete repperire, f. 135va; Quo … colore colorabitur Sol vel Luna, f. 136rb.

Muczkowski J. , Pauli Paulirini olim Paulus de Praga vocitati Viginti artium manuscriptum librum, cuius codex membranaceus in bibliotheca universitatis Jagellonicae Cracoviae asservatus Twardovio vulgo tribuitur (Cracow, 1835), 16.

Cantimpratensis Thomas , Liber de natura rerum, ed. by Boese H. (Berlin and New York, 1973).

F. 171rb.

Cf. Anglicus Bartholomaeus , De genuinis rerum coelestium, terrestrium et inferarum proprietatibus libri XIX (Frankfurt, 1601), Liber xi, 501.

F. 131ra: “Nota diligenter: Astronomia est sciencia simpliciter speculativa, quia est de motibus celi, que sunt res non operabiles a nobis. Dicitur autem practica secundum artem calculandi, que sit a voluntate nostra et operacione nostra.” (“Remember well: Astronomy is a science simply theoretical as it deals with the motion of the sky, which is matter which we cannot influence. It is also denoted as practical science according to the art of calculation, which is dependent on our will and operation.”) This example reveals the understanding of astronomy in its two components of theory and practice. The latter, however, does not consist in observation but calculation (in computistics and compiling of a calendar).

F. 131va: “Feci rotam pretorianam, que potest pretoriis applicari tam pro utilitate, quam decore, in qua qualibet die omnis planetarum motus et aspectus potes considerare.” (“I have made a town-hall circle, which can be fixed on a town-hall for use as well as for decoration because it is possible to find on it all planetary motions and aspects for every day.”) F. 133va: “Pretoriana est figura per me edita, in qua omni die potes faciliter scire omnium planetarum motus et aspectus et coniuncciones et opposiciones et quadraturas et alia.” (“The town-hall instrument is a figure elaborated by me, in which you can every day easily find out the motions of all the planets as well as their aspects, conjunctions, oppositions, quadratures and other matters.”) The construction of this astronomical instrument is mentioned by Paulerinus in ten other places.

Cf. e.g., Aristotle, Meteorologicorum libri quatuor (Basel, 1553); Aristotle, Météorologiques, ed. and transl. by Louis Pierre (Paris, 1982).

Les auctoritates Aristotelis: Un florilège médiéval, ed. by Hamesse Jacqueline (Étude historique et édition critique, Philosophes Médiévaux, xvii; Louvain , 1974).

“Ego quoque careo Theorica planetarum et // astronomie libris, quia iam per hussitas a Praga quindecim annis expulsus pronunc maneo in Plzna, ubi et librorum est carencia, et a nemine edoccio illud ponam, non sicut libenter vellem, sed sicut possum”.

“Luna … habet maculam in modum faciei humane consignatam”.

Cf. De facie in orbe lunae, Plutarchi moralia, v, fasc. 3, ed. by Hubert C. Pohlenz M. Drexler H. (Leipzig, 1960).

Cf. Calendaria et prognostica, Astronomica minora, Somnium, ed. by Bialas Volker Grössing Helmuth (Keplers Gesammelte Werke, xi/2; Munich , 1993); Kepler's Somnium: The dream or Posthumous work on lunar astronomy, transl. by Rosen Edward (Madison, Milwaukee and London, 1967); Kepler Johannes , Sen neboli Měsíční astronomie, transl. by Alena and Petr Hadrava (Prague, 2004); and Kepler Johannes , Sen czyli wydane posmiertnie dzielo poswiecone astronomii ksiezycowej, transl. by Dorota Sutkowska and Jaroslaw Włodarczyk (Warsaw, 2004).

Cf. Křišt'an z Prachatic: Stavba a Užití astrolábu (Cristannus de Prachaticz, Composition and use of the Astrolabe), ed. and transl. by Hadravová Alena Hadrava Petr (Prague, 2001). E.g., Paulerinus's lemmata Hora inequalis, Crepusculum, Ocasus Solis, Hora equalis, Horas inequales … convertere, Horas equales … convertere on f. 139ra—rb of Liber viginti arcium are based on Cristannus's rules 3–11 of the Use of the astrolabe (cf. pp. 433–6 of our edition). On Křišt'an, see Dictionary of scientific biography, in press.

Cf. Le nombre d'or: Étude de chronologie technique suivie de texte de la Massa compoti d'Alexandre de Villedieu, ed. by Van Wijk W. E. (The Hague, 1936). E.g., Paulerinus's lemmata Mane, Kalendarium, Menses usuales, Egipciaci dies, Officia mensium on f. 139ra–141va of Liber viginti arcium are based on Alexander's computus (cf. versus 58–64, 98–100, 111–14, on pp. 54–55 of our edition).

Spunar Pavel , Repertorium auctorum Bohemorum provectum idearum post universitatem Pragensem conditam illustrans, i (Studia Copernicana xxv; Wratislava , 1985), 116–32. E.g., Cristannus's Computus is found in ms. Prague, NK I G 24, f. 18v–54v.

Cf. Bartholomaeus Anglicus, op. cit. (ref. 9). Paulerinus's lemma [G]alaxias (f. 165vb) corresponds to Bartholomaeus's text on p. 384, and [P]liades (f. 165vb) to the text on p. 424 (cf. the Appendix). Lemmata [A]ries — [P]isces (f. 166ra—rb) correspond to the text on pp. 389–95, [I]anuarius–[D]ecember (ff. 166vb–167rb) to the text on pp. 446–50, [A]urora (f. 167rb) to p. 453, [C]horuscacio and [F]ulmen (f. 170vb) to p. 504, [P]luvia (f. 171rb) to p. 498, [P]ruina (f. 171rb) to p. 499, and [V]entus (f. 171va) to pp. 485–93.

Poulle Emmanuel , Les Tables Alphonsines avec les Canons de Jean de Saxe (Paris, 1984). Paulerinus's lemma Radices argumentorum on f. 132ra and Alfoncius, cum loquitur de revolucione annorum mundi on f. 137vb of Liber viginti arcium are based on John of Saxony's Canons (op. cit., 46–48).

Cf., e.g., the edition by Sentinus Jacobus Lucilius Johannes (Venice, 1482); critical edition: Thorndike Lynn , The Sphere of Sacrobosco and its commentators (Chicago1949).

Cf. Libellus Ioannis de Sacro Busto De anni ratione seu, ut vocatur vulgo, Computus ecclesiasticus, with preface by Philip Melanchthon (Wittenberg, 1574).

On the sphere: Items Spera, Recta spera and Obliqua spera (f. 141vb); on climates: Clima, Dyaremes, Dyachirenes, Dyaallexandrios, Dyarodios, Dyameres, Dyacoristenes, Dyatiferos, Octavum clima, Summa miliarium climatum … est, Ambitus terre (f. 142rb—vb); on risings and settings: Ortus signorum, …, Occasus signorum (f. 142vb); on calendar e.g. items Embolismalis mensis (f.138vb), Aliter festa mobilia … quere (f. 140rb).

E.g., lemma Mundus sensibilis (f. 165ra) corresponds to Boëthius, De consolatione philosophiae, 3,9.

Paulerinus's comprehensive lemma [N]ox (f. 167rb) gives dividing of the night into seven parts according to (H)rabanus Maurus's De universo libri viginti duo, 10,7 (Patrologia Latina 111, ed. by Migne J. P. (Paris, 1864), cols 292–3).

Eberhardi Bethuniensis Graecismus, ed. by Wrobel Johannes (Corpus grammaticorum Medii Aevi, i (Vratislava, 1887; reprint Hildesheim, 1987)).

The works of Johannes de Lineriis remain mostly unedited in manuscripts.

F. 165vb: ” [G]alaxias est circulus celestis, pulcer et candidus, incedens per medium celi ab oriente per Cancrum et Capricornum usque ad septemtrionem in modum Lactei circuli, dirigens de nocte navigantes et ittinerantes. Proveniens ex relucencia fulgoris stellarum multarum parvarum, que in eo videntur, vel secundum vulgi oppinionem ex vestigio Solis illic meato, vel secundum Anaxagoram ex reflexione luminis ad aerem.” Cf. Bartholomaeus Anglicus, op. cit. (ref. 9), 384: “Galaxias est circulus coelestis, caeteris circulis coeli pulcrior et candidior, incendens per medium coeli ab oriente usque ad ad septentrionem, per Cancrum et Capricornum procedens iterum ad punctum suum. Dicitur autem Lacteus circulus … unde et de nocte navigantes et itinerantes dirigit et deducit…. Ideo Anaxagoras et Democritus dixerunt Galaxiam esse ex reflectione luminis ad aerem”.

F. 165vb: ” [P]liades est constellacio ex concursu septem stellarum inter genua Thauri, que cum oriuntur, significant pluvias temperantes caliditatem nimiam estatis cum magna aeris conturbacione. Et oritur in sedecimo gradu Tauri et quanto serenior est aura et frigidior, tanto melius ab aliis stellis discernuntur. Quarum una stella est multum lata inter alias.” Cf. Bartholomaeus Anglicus, op. cit. (ref. 9), 424: “quanto autem aura serenior est et frigidior, tanto melius ab aliis discernuntur”.

According to different calendars, the day of St Arnolf was either 18 July or 16 August. Cisioianus in the table on f. 147r and 147v shows that Paulerinus took into account the first variant. In 1460 the conjunction of Moon and Sun occurred on 18 July at 5h 32min UT. According to the Parisian Alfonsine Tables it had to happen at 4h 43min after midnight in Toledo, i.e. at about 5h 55min of Prague local time. The true conjunctions, tabulated on f. 147r, give for the day 18 July a conjunction in years with numerus aureus 17, which corresponds also to the year 1460, at 10h 32min (from sunset). The conjunction occurred then in the fourth degree of the sign of Leo close to the ascending node (caput draconis, dragon's head) with the Moon at ecliptical latitude 0.5°, so that the Sun was really eclipsed. In a good agreement with Paulerinus's calculations, the band of totality started in southeastern Europe and continued through Central Asia, China and Japan to the Pacific, cf. Hermann Mucke and Jean Meeus, Canon of solar eclipses −2003 to +2526, Canon der Sonnenfinsternisse −2003 bis +2526 (Vienna, 1983). A conjunction of the Moon with the Sun occurred also on 16 August at 13h 55min UT, but so far from the nodes so that the Sun could not be eclipsed.

The day of St Procopius is 4 July, which agrees with the cisioianus on f. 147r. The opposition of the Moon and the Sun preceding the conjunction on St Arnolf occurred on 3 July 1460 at 19h 53min UT at latitude 0.76°. The corresponding eclipse of the Moon predicted by Paulerinus, with its actual midpoint at 20h 03min UT and actual duration of 2h 4min, according to Mucke and Meeus, op. cit. (ref. 32), was observed and described in Vienna by Johannes Regiomontanus: “There was a partial eclipse of the Moon during the night which followed the 3rd day of July; its beginning was exactly 7h 16min after midday. Moreover, the middle was at 8h 13min and the end at 9h 10min; it was 2;56 ecliptic digits. This was according to the tables for the meridian of Vienna. However, I myself observed the middle of this eclipse in the sky, and it seemed to be eclipsed rather more than 4 digits. Moreover at the end I measured the altitude of the Moon as 15 degrees 18 minutes. Also present was George (von Peuerbach), my teacher.” Quoted from Steele John M. , Observations and predictions of eclipse times by early astronomers (Dordrecht, 2000), 141. The sun set in Prague at approximately 19h 10min UT that day, so that according to the old Czech time, counted from sunset, the opposition (and maximum of the eclipse of the Moon) happened around one o'clock of the new day, July 4.

F. 136rb: “Visibilitatem eclipsis cognoscere est scire distincte per arcum diei et noctis, an hora vere coniunccionis vel opposicionis sit de die vel de nocte, horas incipiendo a meridie diei precedentis. Quod si Sol de nocte eclipsabitur, non cura, quia non videbitur, nisi esset prope ortum, sicut continget anno Domini MoCCCCLXo, ubi in die Arnolffi Sol orietur eclipsatus et Luna non videtur de die, nisi de nocte aut prope, sicut eodem anno in die Procopii, hora prima, Luna eclipsabitur et prius de die incipiet eclipsari, quasi una hora cum media. Et ideo, quando taliter contingit, sis cautus”.

The Day of Innocents is 28 December. The opposition of the Moon and the Sun occurred on 28 December 1460 at 0h 10min UT in the ecliptical latitude −0.33° and the eclipse of the Moon with the middle at 0h 13min lasted 3h 20min (cf. Mucke and Meeus, op. cit. (ref. 32)). In Prague it was approximately one o'clock after midnight, what means — In agreement with Paulerinus's prediction — The end of the eighth hour after the sunset. This eclipse was also observed in Vienna by Peuerbach and Regiomontanus. In addition to these three eclipses there was also a total eclipse of the Sun visible on 23 January 1460 in Southern America (Mucke and Meeus, op. cit. (ref. 32)).

Should be 23 June. The eclipse of the Moon was 22 June 1461 at 20h 43min UT, i.e. 23 June according to the old Czech dating from the sunset. This eclipse was observed in Vienna by Peuerbach and Regiomontanus, who, however, give incorrectly its middle as after midnight. This eclipse was also observed and recorded in Cairo by Ibn Taghri Birdi ( Stephenson F. Richard , Historical eclipses and Earth's rotation (Cambridge, 1997), 453).

The opposition occurred on 17 December 1461 at 15h 02min UT and the middle of the eclipse of the Moon at 14h 58min, which means almost at the sunset in Prague. Paulerinus's prediction expected it probably a little bit later and hence, according to the old Czech dating, on 18 December. This eclipse was observed by Regiomontanus in Rome, where he moved after the death of Peuerbach in autumn 1461 with Cardinal Bessarion ( Steele , op. cit. (ref. 33), 142).

The opposition occurred on 12 June 1462 at 2h 5min UT and the eclipse of the Moon reached maximum at 1h 57min. Regiomontanus, who observed it from Viterbo, gives the time “on the night which followed the 11th of June … 15 hours 15 minutes after midday” ( Steele , op. cit. (ref. 33), 142).

Correctly should be 21 November, because the conjunction occurred on 21 November 1462 at 11h 44min UT and there was a total eclipse of the Sun, but the band of totality was in visible only in equatorial Africa. Regiomontanus observed it as partial from Viterbo ( Steele , op. cit. (ref. 33), 142).

F. 136rb: “Pronosticacio ex eclipsibus: Diligenter scito, quod eclipses visibiles et presertim Solis numquam {numquam} aliquid boni servant; leditur enim omne vitale ex radiorum retraccione et anno supradicto Sol eclipsabitur in Leone, illo vero regibus et hominibus, quoad febres et cordis passiones. Eodem autem anno erunt quatuor eclipses, quarum due nominate sunt visibiles et tercia erit Lune, visibilis in die Innocentum, hora octava noctis precedentis, que terribiliter stabit visibilis ad duas horas, et ideo ille annus erit multum periculosus, in quo necesse erit vigilare principibus, ut futurum malum impediatur, vel si non possit, ita temperetur, ut levius fiat. Sexagesimo autem primo anno iterum erunt due eclipses visibiles, quarum prima erit 23. Iulii et secunda Dece<m>bris 18. die, et ambe Lune. Sexagesimo secundo iterum erunt due, quarum prima erit duodecima die Iunii, altera vero Solis erit 25. Novembris et cetera”.

For details cf. van Dalen Benno , Ancient and medieval astronomical tables: Mathematical structure and parameters values (Utrecht, 1993), or Hadrava Petr , “Mathematical investigation of mediaeval tables of stars and other astronomical tables” (in Czech), Scripta astronomica, x (2003), 2003–81.

There are several reasons for a possible discrepancy between the true values of parameters used in calculation of the tables and the most probable values and their errors estimated by least-squares fits. Firstly, the probable errors of parameters are calculated assuming that the errors of individual data points are statistically independent, which is not true if, for instance, some tabulated values are interpolated from others. Moreover, truncation or rounding of resulting values, as well as of intermediate values during the calculation, or errors in auxiliary tables (e.g. of chords) used in the original calculation, usually lead to a systematic difference with respect to the exact geometric model. (Examples are the wave-like features in Figures 3 and 5.) A use for the fitting of a model which would repeat the original algorithm with its shortcomings can, in principle, reduce this discrepancy and, moreover, yield evidence about the procedures used (cf. Benno van Dalen, op. cit. (ref. 41)).

Cf. Chabás José , “Astronomy in Salamanca: The Tabulae resolutae”, Journal for the history of astronomy, xxix (1998), 167–75 and references therein.

The authors thank Richard L. Kremer for noting this consistency of the data with the 17° shift in longitude.

E.g., Prague NK III C 2, Cracow BJ mss. 550, 551, 560, 1852, 1857, Kalocza FK 326, Vienna ÖNB 5145.

E.g., Prague NK XIII F 25, Cracow BJ mss. 1915, 1927, 1865.

F. 135ra: “Latitudo civitatis dicitur arcus meridionalis ipsius civitatis, quo distat punctus verticis civitatis ab equinocciali. Longitudo civitatis est arcus equinoccialis, quo distat meridianus eiusdem civitatis ab oriente vel a meridionali Arim. Arim est civitas in India sub equinocciali, sine longitudine et latitudine. Longitudo Prage ab oriente sit 29 gradus 30 minuta, elevacio poli est 50 gradus 7 minuta. Noruberga est occidentalior Praga 7 minutis, Praga est orientalior Eufordia 10 minutis horariis, Wienna est orientalior Praga 6 minutis”.

Cf. Poulle, op. cit. (ref. 22), Tables 15, 16, 21, 22, 23, 19, 20, 17, 18, 24 and 25 resp. on pp. 134–44.

Cisioianus is a mnemonic aid to remember the dates of important feasts according to the order of their initial syllables in a verse corresponding to each month. Cf. Doskočil Karel , “Vývoj cisiojánu u nás”, Sborník historický, vi (1959), 97–170; Nováková Julie , “České cisiojány od 14. století”, Studie ČSAV, iii (Prague, 1971); Friedrich Gustav , Rukovět' křest'anské chronologie (Prague and Litomyšl, 1997); and Bláhová Marie , Historická chronologie (Prague, 2001).

Kremer Richard L. , 2007, personal communication.

Cf. Poulle , op. cit. (ref. 22), 58.

The puzzle of interpreting this table has been solved by Richard L. Kremer. We should note that this table is written in white ink on a dark background because of a defect in the parchment. Just this spot was considered to be a devil's sign and the whole codex was kept chained under a stone (in the eighteenth century!) as a book of black magic belonging to Twardowski, a Cracovian Faust.

Cf. Poulle , op. cit. (ref. 22), Table 26 on pp. 145–47 for the Sun and Table 27 for the Moon (both rounded to integer minutes), and subsequent tables for other planets.

Cf. Beatriz Porres de Mateo, Table R78 in Annexe XXI of “Les tables astronomiques de Jean de Gmunden: Édition et étude comparative”, Ph.D. thesis, École Pratique des Hautes Études, Paris, 2003, 617.

Pedersen Fritz S. , The Toledan Tables (Copenhagen, 2002), 1457.

As an example, let us mention, e.g., ms. Prague NK X B 3 (cf. Truhlář Josef , Catalogus codicum manu scriptorum latinorum (Prague, 1906), 42–43), where on ff. 83r–84v are tables of mean motions to four significant figures in months (for a common year as in the Paulerinus's table, as well as for a leap-year), and on ff. 85r–86v are the tables in days. This manuscript was attributed to Iohannes Schindel by Truhlář J. (cf. Věstník Č. A., 1900, 473); however, B. Porres found its identity with tables by Petrus Cruciferus in ms. Prague NK VIII G 24 and mss. Vienna ÖNB 5245 and Bodleian Library, Canon. Misc. 499 (cf. Porres B. , “The dissemination of mediaeval astronomical tables in Central Europe in the 15th century” (in Czech), Scripta astronomica, x (2003), 2003–51). The same text is also contained in ms. Cracow BJ 610, cf. Rosińska Grażyna , Scientific writings and astronomical tables in Cracow (XIVth—XVIth centuries) (Studia Copernicana xxii; Wrocław, 1984), 242. A list of other similar tables is presented by Rosińska on pp. 510–11.

Cf. van Brummelen Glen , “Lunar and planetary interpolation tables in Ptolemy's Almagest”, Journal for the history of astronomy, xxv (1994), 297–311.

Toomer G. J. , Ptolemy's Almagest (London 1984).

Pedersen , op. cit. (ref. 55), 1253–8.

Regarding the problems with systematic errors outlined in ref. 42, the values of parameters found by fitting the data can be used for classification of the tables (identification of their sources, determining of stemma of copies, and quantification of the imperfection of a copy), but they need not correspond to the values used by their authors. This is why we generally do not give the fitted results in the sexagesimal form commonly used in the manuscripts. An exception worth doing involves the epicycle radius r determined from equacio argumenti. For the standard choice R + e = 60 we get r = 5;14,47 by fitting the table in the Almagest, and r = 5;14,58,29 from the Toledan Tables, which agrees well with Ptolemy's value of r = 5;15. Our fit of the Parisian Alfonsine Tables gives r = 5;9,34,42.

As an example, cf. Hadravová Alena Černá Alena Homolková Milada Hadrava Petr , “The reply of M. Cristannus of Prachatice to the prophecy of M. Johannes Parisiensis”, Listy filologické, cxxiii (2000), 40–51.

F. 135va—vb: “Per pretorianam est levior modus querendi veram coniunccionem, similiter opposicionem, iuxta modum, quem tibi prius describsi [sic] et est multum precisus modus, numquam fallens in una hora. Maxima autem distancia inter mediam <et> veram coniunccionem sunt septem hore, sic scilicet: 1, 2, 3, 4, 5, 6, 7, et sex minuta, sic scilicet: 0, 1, 2, 3, 4, 5, 6, et ego levissimo modo traderem tibi modum querendi veram coniunccionem ex solo argumento. Sed ne alleviarem nimis hanc pergloriosam scienciam, ymo sic nimis alleviavi eam, quod per ydiotas eciam vilipendi poterit: Sed cum nimia fatiga capitis et oculorum meorum nulla enim sciencia tantum fatigavit caput meum, sicut ista!”.

For instance the Sacrobosco texts De sphaera mundi (c. 1220, cf. ref. 23) or his Computus ecclesiasticus (c. 1232, cf. ref. 24), included among Paulerinus's sources, were frequently reprinted in the sixteenth century and used as important elementary textbooks in European universities up to the first half of the seventeenth century. Similarly, some of the traditional ideas expressed in Paulerinus's other ‘arts’ can also be found in works of leading personalities of subsequent centuries. An example is the generally admired bestiary by Leonardo da Vinci, which reflects the same traditional claims, which can also be found in the section of Liber viginti arcium dealing with zoology and in other previous works. Even Johannes Kepler mentions as a matter of fact in his Somnium the birth of ducks from pitch.