¹ This paper is based on eight of the surviving manuscript versions of the De reprobatione. One of these, Vatican Barberini MS. 350, is of minor value since it breaks off after a few pages (see Thorndike, Lynn, A History of Magic and Experimental Science (8 vols. New York, 1923–1928), vol. 4, pp. 175–177Google Scholar. The other manuscripts are: Bibliothèque Nationale MS. 16,401, ff. 55r–67v; Stift Melk MS. 51, ff. 210r–218r; Vienna, Nationalbibliotek MS. 5203, ff. 100r–117v; Vatican Latin MS. 4082, ff. 87r–97r; Utrecht MS. 725, ff. 218r–246r; Princeton, Garret MS. 95, ff. 146–167v; Prague, Metropolitankapitel MS. 1272, ff. 45r–54r. For this report of some of Henry of Langenstein's ideas, I have relied on the Bibliothèque Nationale manuscript. Recently, Dr. J. D. North of the Museum of the History of Science, Oxford, has kindly drawn my attention to yet another version of the De reprobatione (Bodleian Library MS. Bodley 300, Item 5) which he discovered in an Oxford manuscript.

² Bibliothèque Nationale MS. 16,401, f. 62r.

³ Henry of Langenstein may have been born in 1325. He studied at Paris, was an M.A. by 1363, and had received a doctorate in theology by 1375. From 1383 until his death in 1397, he taught in the theological faculty at the University of Vienna. Denifle, Heinrich and Chatelain, Emil, Chartularium uniuersitatis Parisiensis (4 vols., Paris, 1889–1897), vol. 3, 132–133, 300Google Scholar; Hartwig, Otto, Henricus de Langenstein, dictus de Hassia (Marburg, 1857), 5–14, 64–86Google Scholar; Aschbach, Joseph, Geschichte der Wiener Universität im ersten Jahrhunderte ihres Bestehens (Vienna, 1865), 1, 366–367.Google Scholar

⁴ Pedersen, Olaf, “The Theorica-planetarum-Literature of the Middle Ages,” Classica et Mediaevalia, xxiii (1962), 225–232.Google Scholar

⁵ The word argumentum as used in medieval astronomy has two meanings. It can designate “anomaly” as in the present case, or it can also refer to the independent variable, as in “argument of latitude” (argumentum latitudinis).

⁶ There is no drawing in any surviving manuscript of the De reprobatione that I have examined to illustrate Henry's argument.

⁷ Physical signs (signa physica) of 60° each.

⁸ In this and subsequent translations the words in square brackets have been added by me. “Si essent epicicli, sicut communiter invenitur in theoricis. sequitur linea preter centrum transeuntem circulo esse perpendicularem, quod est impossible, cum solum sit talis linea perpendicularis per centrum progrediens. Antecedens probatur nam media aux ponitur invariabilis in circumferentia in qua mensuratur velocitas epicicli circa proprium centrum quia alias, cum ab ea computetur motus epicicli uniformis in tabulis mediorum argumentorum, nunquam invenietur in eisdem tabulis veraciter distantia planete ab auge vera epicicli, sicut patet cuilibet intuenti figuram. Sed cum epiciclus est in auge ecentrici, notum est quod dyameter epicicli inter eiusdem augem mediam et oppositum est perpendicularis super ecentricum, quia in continuum ducta per eius centrum transiret. Vocetur ergo illa media aux A, que necessario invariabilis esset in dicta circumferentia, scilicet in qua consideretur revolutio epicicli. Ergo necessario linea descendens ab ea ad centrum epicicli est super ecentricum perpendicularis, et ubicumque fuerit epiciclus quia ipse describit ecentricum. Veniat ergo epiciclus ad aliquam distantiam ab auge, scilicet ad puntum B, longitudinis medie. Et trahitur linea ostendens augem mediam quam omnes trahunt a centro equantis, preterquam in luna, per centrum epicicli, que linea vel terminatur in A. vel in punctum alium distantem ab A. Non secundum quia in A puncto dicte circumferentie signabatur aux media cum epiciclus esset in auge, et sic esset iam variata ad alium punctum, quod est contra eorum ponentem predictam. Igitur, dabitur primum. Igitur, cum talis linea recta incipiat a puncto preter centrum ecentrici, et cum talis linea inter A et centrum epicicli sit ecentrico perpendicularis … clarissime sequitur propositum… Et sequitur unam revolutionem astri ab auge media epicicli ad eandem non continere 360°, quod est manifeste contra tabularum Operationem mediorum argumentorum quia quando in eis invenitur 6 signa, dicitur esse pianeta in auge media epicicli … Et ponatur iam epiciclus ad augem ecentrici, tunc notum est quod semidyameter eius descendens ab auge media super ecentricum epicicli est perpendicularis super convexum ecentrici, que semidyameter sit CD; et cum CD interim revertitur ad perpendicularitatem super ecentricum est facta revolutio 360° per suppositionem; et tamen centrum epicicli interim non rediit ad augem ecentrici, sicut patet calculanti tempus revolutionis epicicli et centri eius in tabulis. Igitur necessario, illa revolutione facta, aux media erit post C vel ante per arcum qui intercipitur inter lineam augis medie et lineam exeuntem per centrum ecentrici et centrum epicicli. Igitur necessario, facta revolutione 360° ab auge media, non dum astrum venit ad augem mediam, vel iam earn transat… Ex quo sequitur… quod non precise intevitur distantie planete ab auge media. Et ergo, si epiciclos et ecentricos sustineret, dicerem augem mediam in omni epiciclo ostendi per lineam egredientem a centro ecentrici per centrum epicicli, quod est utique necessarium cum motus circuitionis epicicli habeatur in tabulis uniformis…” Bibliothèque Nationale MS. 16,401, ff. 59r–59v.

⁹ Pedersen, O., op. cit. (4), 216.Google Scholar

¹⁰ I do not intend to imply that Henry necessarily derived this information from the theorica communis since there is no specific reference to that work in this passage from the De reprobations.

¹¹ “Aux media in epyciclo dicitur punctus in superiori parte epycicli qua terminat linea exiens a centro equantis per centrum epycicli. Et hec aux non variatur. Aux vera dicitur punctus quem terminat linea exiens a centro terre per centrum epycicli, et hec aux variatur secundum quod crescat et decrescat equatio centri in epyciclo.” Erfurt, Wissenschaftliche Bibliothek. Amplonian MS. F178, f. 81v.

¹² Johannes de Lineriis (Jean de Linières) flourished at Paris, 1320–35. He was the author of a number of widely used astronomical works.

¹³ “Tertius motus ymaginatus est motus augis medie epicycli, ut lune, excepto in hiis 4 erraticorum, incipiente epicyclo moveri ab auge deferentis, incipit aux media declinare ad orientem ab auge vera. Et respicit et reflectitur aux media sive dyameter ymaginatus qui extenditur inter augem mediam et eius oppositum ad centrum equantis.” Bibliothèque Nationale MS. 7281, f. 168r.

¹⁴ Herz, Norbert, Geschichte der Bahnbestimmung von Planeten und Kometen (Leipzig, 1887), Part 1, 113Google Scholar; Dreyer, J. L. E., A History of Astronomy from Thales to Kepler (New York: Dover, 1953), 154–156.Google Scholar

¹⁵ Ptolemy, Claudius, Almagest, trans. Manitius, K. (Leipzig, 1963), Book X, Chapter 6, p. 172.Google Scholar

¹⁶ Ibid., Book XI, Chapter 10, 254–255; 261 ff.

¹⁷ The diagram illustrating Langenstein's argument has suffered at the hands of the scribes. It is omitted in the Utrecht and Princeton manuscripts and is incorrectly drawn in all the other versions. The Bibliothèque Nationale manuscript best approaches Henry's intent. I have added the dotted lines and the letters H, N as well as the obvious designation centrum to bring the diagram in line with the text. The three circles in Henry's drawing are the ecliptic, the equant, and Mercury's small circle. While the diagram apparently places the epicycle centre on the equant, this does not directly affect Langenstein's argument.

¹⁸ “Movetur autem, ut dictum est, centrum epycicli ad orientem, et aux ecentrici deferentis ad occidentem in contrarias partes eque cito inter se.” Erfurt, Wissenschaftliche Bibliothek, Amplonian MS. F178, f. 82r.

¹⁹ “Dicunt communiter in theorica communi quandoque centrum epicicli Mercurii est in altera linearum contingentium parvum circulum transeuntem per centrum ecentrici et equantis eius, quod tunc centrum ecentrici sit in altero contactuum illarum cum circulo parvo, quod est similiter falsum. Quod ostendo supponendo eorum dicta communissima: scilicet, quod centrum ecentrici uniformiter moveatur describendo parvum circulum versus occidentem et eque velociter sicut epiciclus movetur in equante versus orientem, quod idem auctor dicere videtur et omnes tanquam principium concedunt. Quod si concedatur, trahatur una linearum contingentium, scilicet occidentalis HDN, ita quod D cadit super sectionem eius cum equante, et sit punctus contactus G, et trahatur EG. Et B sit centrum terre. Et trahatur FD a centro equantis. et O sit centrum ecentrici in summitate parvi circuli. Quibus stantibus, arguo sic: arcus OG est minor respectu parvi circuli quam arcus CD equantis respectu totius equantis, igitur cum eque cito et proportionaliter O describat parvum circulum sicut epiciclus equantem, sequitur quod O. centrum ecentrici, citius Veniat ad G, punctum contactus, quam epiciclus ad lineam contingentium BN circa oppositum augis equantis. Ista consequentia nota est ex hoc quod linea FOC transiens a centro equantis per centrum epicicli eque velociter movetur in equante, ex suppositis, sicut centrum ecentrici in parva circumferentia, ex quo etiam patet centrum ecentrici in ipsa difformiter moveri. Probatur antecedens principale quia ex quo linea BG est maior linea EG, cum sit maior BF, que secundum eos est equalis FE, et EG est equalis FE. Erit ergo angulus GEB maior angulo GBE, igitur erit etiam maior angulo KBD, composito angulo GBE. Sed idem angulus KBD est maior angulo KFD, igitur de principio ad ultimam, anglus GEB est maior angulo KFD. Ergo manifestum est angulum GEO obtusum esse minorem angulo obtuso CFD, igitur arcus OG est minor respectu parvi circuli quam CD, arcus equantis, respectu equantis. quod est totaliter propositum.” Bibliothèque Nationale MS. 16,401, f. 64r.

²⁰ “Cum igitur centrum epycicli est in auge, statim incipit ire versus orientem in suo deferente, et similiter centrum deferentis incipit ire versus occidentem in suo parvo circulo. Et quando centrum epycicli est in capite linee contingentis, quod caput est prope oppositum augis equantis, tunc centrum deferentis est in puncto contactus parvuli circuli cum ipsa linea, et tunc aux deferentis est in maxima remotione ab auge equantis, et tunc est centrum epycicli in opposito augis deferentis quare tunc est in maiori appropinquatione ad terram … Quantum currit epyciclus in una parte tantum occurit centrum deferentis in alia parte, ergo semper erunt in eadem linea transeunte per centrum terre et centrum deferentis. Et tamen nunquam appropinquabit tantum centrum epycicli centro terre quantum approquinquat in capitibus linearum contigentium.” Erfurt, Wissenschaftliche Bibliothek, Amplonian MS. F178, f. 82v.

²¹ Taken from Oxford, Bodleian, Digby MS. 168, f. 49v.

²² Almagest, ed. cit. (15), Book IX, Chapter 8, pp. 137–139.Google Scholar

²³ Ibid., Book IX, Chapter 6, p. 123.

²⁴ The “they” here and elsewhere refer to the medieval proponents and interpreters of Ptolemaic theory who comprise Henry's sources of information, including the author of the theorica communis.

²⁵ (1) “Nam dicit auctor theorice communis lineam ostendentem augem veram semper pertransire centrum epicicli, igitur aux vera est punctus epicicli maxime a terra distans … Igitur, quando pianeta est in auge epicicli epiciclus est erectus in ecentrico quo ad circumferentiam transitus. Tenet consequentia quia linea ostendens veram augem semper transit in superficie plana ecentrici, igitur cum astrum se posuerit ad terminum eius in convexo epicicli, quod est cum veniat ad augem veram, sicut patet per eundem auctorem quia comput at verum argumentum a termino eiusdem linee. Sed secundum consequens est manifeste contra eundem auctorem et omnes epiciclistas dicentes tunc planetam maxime a superficie ecentrici declinare dum se ad augem posuerit.” Bibliothèque Nationale MS. 16,401, f. 59v-f. 60r. (2) “Si illa circumferentia transitus in epiciclo ita quandoque declinat ab ecentrico et quandoque est erecta, utique capto argumento vero 15°, scilicet propter illam declinationem corresponderet maior equatio argumenti quam cum epiciclus esset erectus, tunc enim illud argumentum plus distaret a centro mundi. Sed illud est contra eos quoniam eiusdem argumenti non dicunt diversificari equationes nisi ex sola ecentricitate.” Bibliothèque Nationale MS. 16,401, f. 60r. (3) “Circumferentiam ecentrici et equantis dicunt esse in eadem plana superficie in qua cadit centrum mundi, igitur cum ecentricus Martis vel alterius declinet ab ecliptica, similiter illa superficies declinabit. Et ultra dicunt lineam medii motus equedistare linee exeunti a centro equantis per centrum epicicli quod cadit in eadem dicta superficie, ergo linea medii motus Martis declinat ab ecliptica declinatione eccentrici… Ergo linea medii motus Martis movetur difformiter in ecliptica quod quilibet faciliter comprehendit ymaginando sicut de ascensionibus signorum in spera recta.” Bibliothèque Nationale MS. 16,401, f. 63v.

²⁶ Nallino, C. A., Al-Bottānī sive Albatenii Opus astronomicum, Part II (Milan, 1907), 242.Google Scholar

²⁷ “Dicunt quod si medius motus solis addatur super argumentum verum Veneris vel Mercurii, proveniet (!) distantiam inter caput draconis Mercurii et planetam que vocatur argumentum latitudinis; quod est falsum advertenti ymaginationem motus sperarum deferentium augem ecentrici Mercurii. Nam huiusmodi spere moventur uniformiter versus occidentem super polos zodiaci, et ecentricus (!) Mercurii movetur in se equaliter versus orientem, ambo velocitate solis. Et sic notum est quod argumentum latitudinis Mercurii semper crescit uniformi velocitate dupla ad velocitatem solis.” Bibliothèque Nationale MS. 16,401, f. 64r.

²⁸ Kennedy, E. S., “A Survey of Islamic Astronomical Tables,” Transactions of the American Philosophical Society, N.S., 46, Part II: 1956, 148–150Google Scholar; Neugebauer, O., “The Astronomical Tables of Al-Khwārizmī,” Kongelige Danske Videnskabemes Selskab, Historisk-filosofiske Skrifter, Bind 4, No. 2 (Copenhagen, 1962), 34–41Google Scholar. Also: Suter, H., Die astronomischen Tafeln des Mohammed ibn Mūsā al-Khwärizmi (Copenhagen, 1914), 60–64.Google Scholar

²⁹ Neugebauer, , “Al-Khwārizmī”, p. 38.Google Scholar

³⁰ Zinner, E., “Die Tafeln von Toledo,” Osiris, 1 (1936), 752–753.CrossRefGoogle Scholar

³¹ de Virduno, Bernardus, Tractatus super totam astrologiam, ed. Hartmann, P. (Werl, 1961), 134–140.Google Scholar

³² Practica canonum Alfoncii, Erfurt, Wissenschaftliche Bibliothek, Amplonian MS. F386, f. 44v.Google Scholar

³³ de Lineriis, J., Tabulae astronomicae, Erfurt, Wissenschaftliche Bibliothek, Amplonian MS. F384, ff. 42–43v.Google Scholar

³⁴ John of Gmunden, , Canones seu problemata astronomica, Vatican, Palatine MS. 1411, ff. 26r–26vGoogle Scholar; ff. 76v–79r.

³⁵ Bvlug, Rudolph, “Johannes von Gmunden, der Bergründer der Himmelskunde auf deutschen Boden,” Akademie der Wissenschaften in Wien, Philosophisch-historische Klasse, Sitzungsberichte 222, Band iv (1943), 18 ff.Google Scholar

³⁶ The binary table has two columns of argument for values of θ and 360° —θ the quaternary table has a column for each quadrant of the circle reflecting the sinusoidal nature of the underlying function of this component of the latitude. See Neugebauer, , op. cit. (28), 35.Google Scholar

³⁷ “Alii 5 autem habent duplicem latitudinem; unam qua epiciclus declinat ab ecentrico; aliam ex ecentrico qui declinat a via solis. Et per tabulam binarii habetur latitude secundum epiciclum; per tabulam quaternarii inventur latitudo secundum ecentricum. Et dicitur “binarii” quia habet duplos introitus; “quatenarii” quia habet quaternos introitus. Et tabula binarii facta est ad medietatem circuli; tabula quaternarii ad tria signa … Tabula binarii est ad epiciclum, ideo intra ad eam cum vero argumento. Et quia tabula quaternarii est ad ecentricum, ideo intra ad eam per distantiam a nodo capitis. Et ideo, dividitur una latitudo per aliam.” Erfurt, Wissenschaftliche Bibliothek, Amplonian MS. F178, f. 83r.

³⁸ Neugebauer, , op. cit. (28), 34–35.Google Scholar

³⁹ It is best not to pursue the logic of this argument concerning Mercury too far. For one thing, Henry's sources, either Campanus or the author of the theorica communis certainly could have informed him that the apsidal line (and thus the apogee of Mercury's deferent) does not circle the deferent, but oscillates between the limits delineated by the tangents drawn from the earth to Mercury's small circle and extended to the deferent.

⁴⁰ Denifle, and Chatelain, , Chartularium, IIIGoogle Scholar. See the many references to Henry of Langenstein in the section entitled De schismate, 552 ff.Google Scholar

⁴¹ Aschbach, , Geschichte der Wiener Universität, pp. 378–380Google Scholar; more recently, Lhotsky, Aphons, “Die Wiener Artistenfakultät, 1365–1497,” Akademie der Wissenschaften, Philosophisch-historische Klasse, Sitzungsberichte, vol. ccxlvii, Band a (Vienna, 1965), 37, 57, 79.Google Scholar

⁴² “Henricus Hassianus centesimo abhinc et octogesimo fere anno primo mathematicas artes Lutetia Viennam transtulit.” Ramus, Petrus, Mathematicarum scholarum, (21 vols. Paris, 1569), ii, 64.Google Scholar

⁴³ In the introduction to his Tabulae eclypsium Magistri Georgii Peurbachii. Tabula primi mobilis Joannis de Monteregio (Vienna, 1514). For Tanstatter's encomium, see note (48).

⁴⁴ Zinner, Ernst, Leben und Wirken des Johannes Müller von Königsberg genannt Regiomontanus (Munich, 1938), 220.Google Scholar

⁴⁵ Lhotsky, , op.cit. (41), 114.Google Scholar

⁴⁶ According to Apfaltrer, Ernst, Scriptores antiquissimae ac celeberrimae universitatis Viennensis (Vienna, 1740), Pars I, 57Google Scholar, Henry of Hesse wrote a Theorica planetarum which was known to Peurbach. There is a work with that title listed in Houzeau and Lancaster (No. 1787) attributed to Henry, but with no manuscript given. Houzeau, J. C. and Lancaster, A., General Bibliography of Astronomy to the Year 1880, 3 vols. (New Edition, London, 1964), vol. 1, 523Google Scholar. In his 1888 bibliography of Langenstein's works, F. W. E. Roth also lists a theorica but gives no manuscript. He adds the comment that the work is unknown other than its mention by Apfaltrer. Roth, Zur Bibliographie des Henricus Hembuche de Hassia, I. Zinner lists a Sternkunde composed by Henry (No. 6332) in a manuscript at Munich (Munich Universität Bibliothek MS. Q.738) which dates from the fifteenth century. Zinner, E., Verzeichnis der astronomischen Handschriften des deutschen Kulturgebietes (Munich, 1925), 200Google Scholar. Thorndike and Kibre report this manuscript as containing an Expositio terminorum astronomie with Henry of Hesse as the suggested author in a margin. Thorndike, L. and Kibre, P., A Catalogue of Incipits of Mediaeval Scientific Writings in Latin (Revised Edition, Cambridge, 1963), 156Google Scholar. According to Zinner, Henry abo composed a treatise on the astrolabe in German while at Vienna. Zinner, E., Astronomische Instrumente (Munich, 1956), 424.Google Scholar

⁴⁷ His Questio de cometa (1368/69) and the Tractatus contra coniunctionistas (1373). Both have been edited in Pruckner, Hubert, Studien zu den astrologischen Schriften des Heinrich von Langenstein (Berlin, 1933).Google Scholar

⁴⁸ “Quant à la profundeur et à subtilité de ses connaissances en Astronomie, elles sont clairement attestées par le premier livre de ses Commentaires sur la Genèse.” Quoted in Duhem, P., Études sur Léonard de Vinci, troisième serie (Paris, 1913), 15.Google Scholar

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