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John Holbroke, the Tables of Cambridge, and the “true length of the year”: a forgotten episode in fifteenth-century astronomy

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Abstract

This article examines an unstudied set of astronomical tables for the meridian of Cambridge, also known as the Opus secundum, which the English theologian and astronomer John Holbroke, Master of Peterhouse, composed in 1433. These tables stand out from other late medieval adaptations of the Alfonsine Tables in using a different set of parameters for planetary mean motions, which Holbroke can be shown to have derived from a tropical year of \(365\frac{1}{4} - \frac{1}{132}\) or \(365.\overline{24}\) days. Implicit in this year length was a 33-year cycle of repeating solar longitudes and equinox times, which has left traces in other astronomical tables from fifteenth-century England. An analysis of the manuscript evidence suggests that Holbroke owed his value for the “true length of the year” to a certain Richard Monke, capellanus de Anglia, who employed this parameter and the corresponding 33-year cycle in an attempt to construct a perfect and perpetual solar calendar, leading to his Kalendarium verum anni mundi of 1434.

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Notes

  1. See North (1977), Kremer (2007, 2008, 2011), Husson (2012, 2014), Chabás and Goldstein (2013), and the relevant chapters in Chabás and Goldstein (2015).

  2. This holds true, at any rate, for tables cast by Christian astronomers. A greater degree of diversity and independence in the area of parameters was exhibited by Jewish authors such as Levi ben Gerson. See Goldstein (1974, 1988, 1992), Mancha (1998a, b). For other examples, see Goldstein (2001, 2003a, 2004).

  3. The early history of these tables is fraught with uncertainty, but the most likely scenario remains that the Alfonsine Tables known in Paris after c.1320 originated on the Iberian peninsula. See the discussions in North (1996), Chabás and Goldstein (2003), Swerdlow (2004), and the opposing viewpoint taken by Poulle (1987, 2005).

  4. See the examples mentioned in Nothaft (2014; 2017, pp. 227–235).

  5. See Emden (1963, p. 309), Wordsworth (1904, p. 143), Lovatt (1993, pp. 130, 133–134), Clarke (2002, p. 700), Snedegar (2004).

  6. See in particular Gunther (1937, pp. 138–141), and Price (1955, p. 8). Further remarks on Holbroke appear in North (1977, pp. 275–276, 1986, pp. 142–149, 156), Carey (1987, pp. 54–55, 1992, pp. 138, 145–148, 150–153, 155–157, 159, 215–218, 256).

  7. Snedegar (2004). The sixteenth-century comment appears as a marginal note on fol. 164r of MS Gloucester, Cathedral Library, 21 (see n. 42 below).

  8. In analogy to the standard convention for recurring decimals, the vinculum placed above the last five numbers in 365 d \(5;\overline{49,5,27,16,21}\; \hbox {h}\) indicates an indefinitely repeating sequence of sexagesimals.

  9. MS Cambridge, University Library, Ee.3.61, fols. 56r–68v. The approximate date of this copy seems to follow from Lewis Caerleon’s calculations pertaining to the solar eclipses of 17 May 1482 and 28 May 1481 on fols. 1r and 12v–15r. On the owner of this manuscript, see Kibre (1952); North (1976, III, pp. 217–220), Carey (2012, pp. 694–696).

  10. MS London, British Library, Egerton 889 (hereafter cited as H). See Halliwell (1840), Gunther (1923, pp. 65–66), North (1976, II, p. 37), Pedersen (1986, pp. 230–231), Clarke (2002, pp. 499, 557–558), Juste (2017).

  11. H, fol. 6v: “Memorandum quod magister Iohannes Holbrook quondam alme Universitatis Cantebrigiensis cancellarius, sacre pagine professor ac in artibus liberalibus, precipuus in astronomia tamen, peritissimus, et magister Collegii sancti Petri Cantebrigiensis, contulit eidem collegio antedicto in festo sancti Valentini anno domini \(1426^{\mathrm{to}}\) hunc librum astronomicum.”

  12. MS Cambridge, University Library, Ee.3.61, fol. 56r.

  13. A separate copy of this preface appears in MS Oxford, Bodleian Library, Bodley 300, fol. 132vb, a codex from Clare College, Cambridge. Later parts of this codex once used to include Holbroke’s tables, but these have gone missing. See Clarke (2002, pp. 154–155). In passing, one should also note the existence of a stand-alone table of epoch values for the year 1460s that were derived from Holbroke’s Opus primum: MSS Oxford, Bodleian Library, Ashmole 340, fol. 78r; Oxford, Bodleian Library, Ashmole, 346, fol. 20v.

  14. See North (1977, pp. 274, 300), and the less than reliable account in Harper (1975). Rede’s tables and Holbroke’s commentary on them appear in H on fols. 111r–120v.

  15. See Pedersen (2002, III, pp. 968–976), Chabás and Goldstein (2012, pp. 27–28). Versions of this table appear to be present in most other copies of William Rede’s tables. See e.g. MSS London, British Library, Egerton 847, fols. 142v–143v; Oxford, Bodleian Library, Digby 48, fols. 173v–174r; Oxford, Bodleian Library, Digby 97, fols. 18r–19r; Oxford, Bodleian Library, Wood D.8, fols. 83r–84r; Oxford, Jesus College, 46, fols. 31r–32v. The latter copy once belonged to William Rede.

  16. See Pedersen (2002, I, p. 256; II, pp. 434–436; IV, p. 1223).

  17. H, fol. 133v: “Adhuc inter antistetis tabulas ponuntur ascensiones signorum in circulo recto quibus permiscuit Arzachel tabulam equacionis dierum que supponit augem solis in 17 gradu Geminorum esse. Quoniam vero aux solis pervenit modo usque ad primum gradum Cancri patet eam hiis nostris diebus nimis a vero remotam.”

  18. See H, fols. 31r–52v. The table for the equation of time appears on fol. 38v. Other copies of this table can for instance be found in MSS Bernkastel-Kues, Cusanusstiftsbibliothek, 212, fol. 83r; Paris, Bibliothèque nationale de France, lat. 7286C, fol. 54r.

  19. H, fol. 133v: “Quare relicta hac immisi tabulam magistri Iohannis de Lineriis astronomi eximii, que licet nunc precisa non sit, eo quod supponit augem solis in 28 gradu Geminorum existere, est tamen veritati propinquior quam tabula Arzachel. Qui vero precisiorem habere desiderat racionem composicionis ex fine tercii Almagesti et ex 29 capitulo Albategni capiat.”

  20. H, fol. 153r: “Composui autem ante hec tempora alias tabulas mediorum motuum supponens quantitatem anni Alfoncii supponensque eis singulas radices ad eram domini Ihesu Christi pro meridie Cantebrigie fecique tabulam ascensionum signorum in circulo obliquo ad veram latitudinem eiusdem oppidi, que ante nusquam facta fuerat. Feci insuper non est diu tabulam novam equationis dierum cum noctibus suis supponens augem solis in fine primi gradus Cancri, ut sic tabula diuturnior existeret. Hunc si quis desiderat eam ante in hoc eodem libro reperiet.” The table for oblique ascensions mentioned in this passage appears in Hon fols. 149v–150r. See the discussion in North (1986, pp. 130–131).

  21. See on this format Samsó (1987, pp. 31–32), Chabás and Goldstein (2012, pp. 55–57).

  22. An even more extreme case of entries going up to sexagesimal ninths or tenths can be found among the tables drawn up in 1392 by John Westwyk to accompany his English-language treatise on the equatorium in MS Cambridge, Peterhouse, 75.I. See Falk (2016, pp. 12–15).

  23. Poulle (1984, p. 130). The purpose of this type of table is discussed in Chabás and Goldstein (2012, pp. 218–223), Goldstein and Chabás (2013).

  24. This value contains one sexagesimal place more than the corresponding table found in most manuscripts and the 1483 editio princeps of the Alfonsine Tables, where the first entry is 0;59,8,19,37,19,13,56\({^{\circ }}\)/d. See Poulle (1984, p. 134). The expanded version with ten rather than nine sexagesimal places also appears in MSS Cambridge, Peterhouse, 75.I, fol. 7v (s. \(\hbox {XIV}^{\mathrm{ex}})\); London, British Library, Royal 12.D.VI, fol. 7r (s. \(\hbox {XV}^{\mathrm{in}})\); Oxford, Bodleian Library, Digby 97, fol. 8v (s. \(\hbox {XV}^{\mathrm{med}})\).

  25. An effort to spell out this value down to the 17th (!) sexagesimal-fractional place was made in MS London, British Library, Royal 12.D.VI, fol. 8r, where a table shows that it takes the Sun 1 d 0;20,58,12,39,48,17,38,59,11,43,25,32,57,37,2,29,20 h to traverse one degree of the zodiac, but 365 d 5;49,15,58,49,45,53,55,10,20,33,17,45,42,14,56,0,0 h to complete the whole 360\({^{\circ }}\).

  26. Holbroke addresses his intervention in the preface, for which see H, fol. 133v: “Et quia inter Alfonsi tabulas due communiter reperiuntur revoluciones, scilicet annorum ac revoluciones mensium, que nec secum neque cum medio solis motu concordiam servant, prima nempe primo suo versu inventa est minus habens in tertiis, secunda autem superhabundat, sic neutra veram anni solaris continet quantitatem, propterea has ipse ad concordiam duxi et in eis veram anni quantitatem ex motu unius diei secundum Ptholomei documentum inventam, in una quidem usque ad sexta, in alia vero usque ad octava inscripsi. Correxi quoque et tabulam revolucionis ascendentium, quam super tabulam revolucionis annorum fundari certum est.”

  27. H, fol. 152v: “Gloriosus atque sublimis Deus a rerum exordio luminaria in firmamento posuit ut diem noctem que divid erent et in tempora mortalibus dies scilicet fierent atque annos [Genesis 1:14], sicut scriptum est ‘Fecit lunam in tempora sol cognovit occasum suum’ [Psalm 103:19]. Sicut enim lunaris cursus menses distinguit, unde et Greco vocabulo ‘mene’, a quo ‘mensis’, dicitur, lunam signat, ita et sol suo cursu annum efficit, cuius quantitas a variis varia describitur.”

  28. Ibid.: “Iulius namque, cuius decretum usque adhuc servat ecclesia, ex 365 diebus et 6 horis annum suum effecit. Ptholomeus autem, qui omnes antiquiores et sibi coetaneos astronomie vicit magisterio, annum ex 365 diebus, 14 minutis et 48 secundis constare dicit. Alfonsus vero huic tempori 14 secunda ac 50 tertia ademit. Nullus tamen ad veram atque precisam anni quantitatem pervenit, quamvis hii duo veritati propinquius ceteris accesserint. Talis namque ac tanta esse debet ut equinoctia et solsticia in kalendario immota permaneant. Abiecto tamen bisextili die anno \(132^{\mathrm{o}}\), quod si ab ipsius tempore Iulii factum esset, non hec solsticiorum aut equinoctiorum anticipacio, non error qui in pasche ceterorumque mobilium festorum celebritate frequenter nimis accidit, videretur.”

  29. Ibid.: “Si ergo veram anni quantitatem scire cupis, accipe quantitatem aliquam estimatam quam precisiorem autumas, ut puta Ptholomei seu Alfonsi, eamque de 6 horis subtrahe et habes superfluum 6 horarum supra quantitatem estimatam. Quod per 132 multiplica et si hore 24 precise ex multiplicatione proveniant, scito quod superfluum est verum et quantitas estimata est vera anni quantitas. [\(\ldots \)] Si autem ex multiplicatione minus 24 horis proveniat ipsum de 24 horis subtrahe et remanens per 132 divide et quod exierit superfluo adde et resultans a 6 horis deme et habebis anni certissimam quantitatem, per quam si zodiacum diviseris exibit medius motus solis in uno die et, econtra, si per medium cursum diviseris zodiacum exibit vera anni quantitas.” I have omitted Holbroke’s description of the converse case, where the surplus-difference found at first approximation is greater than 1/132th of a day.

  30. See on this point Meeus (2002, pp. 359–366), Steel (2000, pp. 380–383).

  31. See n. 28 for the quoted passage.

  32. See Nothaft (2015) and ch. 7 in Nothaft (2018).

  33. See Swerdlow (1974, 1986).

  34. See Nothaft (2015, pp. 186–190).

  35. H, fol. 152v: “Reperto igitur medio solis cursu in uno die per ipsum multiplica medium motum Alfonsi cuiusvis planete et productum divide per medium motum solis quem in tabulis Alfonsi habes, et exibit vere medius cursus cuiuslibet planete in uno die. Isto modo calculare poteris capita omnium mediorum motuum, tam planetarum, quam augium et stellarum fixarum, argumentorum quoque Veneris et Mercurii accessusque et recessus octave spere.”

  36. In his preface, Holbroke acknowledges that an extension of the entry-value beyond sexagesimals of the fifth place would have caused it to differ from the Alfonsine counterpart and announces his intention eventually to extend the table to the eighth place. See H, fol. 153r: “Motus autem augium non variat a tabula Alfonsi nisi in septimis et octavis et quoniam accessus et recessus octave spere non ponitur in tabulis ultra quinta, ideo in motu octave spere non accidit variacio. Si quis autem motum octave spere usque ad octava calculare vellet, inveniret diversitatem aliquam in radice motus illius, sicut in radicibus que ponuntur in capitibus aliarum tabularum, quamvis id non oporteat. Intendo tamen cum mihi vacacerit illam tabulam etiam usque ad octava deducere, ut plane pateat quid superaddendum sit motibus per Alfonsum repertis.”

  37. H, fol. 153r: “Pro medio cursu lune in die est hic Ptholomei canon specialis. Si multiplicaveris cursum medium solis diei unius per tempus inter duas coniunctiones medias et resultanti unam revolucionem addideris, scilicet 360 gradus, proveniet motus lune in mense, quem si per tempus inter duas coniunctiones diviseris exibit cursus medius lune diei unius, a quo si medium solis cursum subtraxeris proveniet media elongacio lune a sole, ut antedictum est. Mensis vero Ptholomei excedit mensem Alfonsi in 12 tertiis et 45 quartis hore unius.”

  38. See n. 37 for the relevant quote. A note at the very end of the preface (H, fol. 153r), which was probably added at a later stage, gives the length of the mean lunation as 29;31,50,7,41,26 d = 29 d 12;44,3,0,4,34,24 h. This would lead one to expect an elongation of 12;11,26,41,36,13,14,19,20\({^{\circ }}\)/d. Neither of these values seems to be attested in the previous astronomical literature.

  39. On the origin of this value, see Mancha (2002–2003), Goldstein (2003b, pp. 67–68).

  40. See the corresponding remarks in H, fol. 153r: “Sed quoniam medius motus lune per proprium canonem inventus est, melius estimo sibi ceteros lune motus proportionare quam ad motum solis. Quapropter auxiliante Deo aptavi tabulas argumenti lune medii, per quas equationum tabulas ingrediens cito capere poteris verissimum lune motum. Motus autem medius argumenti latitudinis lune habetur per additionem medii motus capitis supra medium motum lune.”

  41. MS Cambridge, Peterhouse, 267, fols. 96r–100r. This is a large-size codex of approximately 395 \(\times \) 275 mm made up exclusively of astronomical tables. A subscription at the very end reads “Expliciunt tabule magistri doctoris Holbrook perpetui magistri Collegii Sancti Petri in Cantabrigia” (fol. 100r). According to Kibre (1952, p. 103 (n. 32)), and Snedegar (2004), this manuscript was presented to Peterhouse by Holbroke himself, but I am not aware of any evidence to support this.

  42. MS Gloucester, Cathedral Library, 21, fol. 164r: “Tabule magistri Iohanne Holbroke fundate super verissimam anni quantitatem et facte erant gravi diligentia Cantebrigie anno domini \(1433^{\mathrm{o}}\).” The tables of the Opus secundum follow on fols. 164r–171ra. See Rhodes (1954–1958, p. 211).

  43. The same material was copied separately in MS Oxford, Bodleian Library, Lyell 37, fol. 123r.

  44. MS Oxford, Bodleian Library, Digby 97, fols. 71v–72r. An abbreviated version of the same text appears in MS Oxford, Hertford College, 4, fols. 53v–54r.

  45. MS Oxford, Bodleian Library, Digby 97, fol. 72r: “Deinde multiplica unum per alium et illud productum divide per medium motum solis in die secundum Alfonsum resolutum eciam in minutas fractiones et proveniet tibi veraciter motus medius argumenti Veneris, qui est 0.0.36.59.27.26.39.25.23.58 secundum magistrum Iohannem Holbrok.”

  46. MS Oxford, Bodleian Library, Ashmole 369, fol. 8r. The epoch values are for the mean Sun and the four lunar parameters. There are also radices for planetary apogees, but these are not Holbroke’s.

  47. Compare MS Oxford, Bodleian Library, Ashmole 369, fol. 7v and H, fol. 133v.

  48. H, fol. 152v: “Divide 24 horas per 132 et exibit superfluum quod a 6 horis demens adde quod restat diebus anni et habes anni quantitatem. Secundum R.M.” Below comes another entry comparing the true length of the year with the Alfonsine year length. The superfluum is here given as 0;0,10,31,33,24,4,49,43 h, implying 365 d 5;49,5,27,16,21,49,5,27 h  +  0;0,10,31,33,24,4,49,43 h = 365 d 5;49,15,58,49,45,53,55,10 h for the Alfonsine year.

  49. H, fol. 152v: “Igitur, quia nostris temporibus vera anni quantitas, que a sapientibus et prudentibus abscondita est, auxiliante Deo parvulis revelatur, ad Dei honorem, a quo omnis sapientia est, eius inventionem adque posteritatis solatium demonstremus.” The parts in italics derive from the Vulgate text of Matthew 11:25.

  50. See Voigts (1995, pp. 258, 278–279).

  51. MS Oxford, Bodleian Library, Ashmole 369, fol. 8r: “Anni ab origine mundi usque ad Christum a [!] ad nativitatem Christi perfecti fluxerunt 4909 secundum R.M. inchoando annum ab equinoctio vernali.” A version of the same statement appears in John of Argentine’s codex containing the Opus secundum. MS Gloucester, Cathedral Library, 21, fol. 9r: “Sed secundum R.M. inchoando annum ab equinoctio vernali anni ab origine mundi usque ad annum nativitatis Christi perfecti fluxerunt 4909.”

  52. North (1983, p. 91, 1992, p. 134), Carey (1992, p. 48).

  53. Voigts (1995, p. 284).

  54. MS Oxford, Bodleian Library, Laud. Misc. 594, fol. 15r: “Vera loca solis in nona spera anno mundi primo et deinde omni anno \(33^{\mathrm{o}}\) completo usque ad finem istius seculi secundum Ricardum Monke capellanum de Anglia, videlicet super situm medium inter orientem et occidentem, meridiem et septemtrionem.” The only serious attempt in the previous literature to discuss the structure of Monke’s calendar is North (1983, pp. 91–94), who misunderstood and mischaracterized the material in a number of ways, for instance by claiming that Monke worked with a cycle of 33 “Egyptian years” of 365 days.

  55. MS Oxford, Bodleian Library, Laud. Misc. 594, fol. 21r.

  56. MS Oxford, Bodleian Library, Laud. Misc. 594, fol. 14v: “Tabula de veris litteris dominicalibus et primacionibus ab origine mundi, in qua scribuntur littere rubie pro omnibus annis bissextilibus huius tabule. Et pro fixione perpetua equinoxii vernalis in kalendario veri anni mundi nulla habentur littera rubia in ultima linea transversali, secundum Ricardum Monke, capellanum de Anglia, anno Christi 1434.”

  57. MS Oxford, Bodleian Library, Ashmole 789, fol. 374r: “Ista tabula equacionis kalendarii anno Christi 1432 composita per simplicem sacerdotem aliqualiter in arte astronomie informatum docet equare, stabilire et confirmare kalendarium nostrum ad cursum solarem, sic quod numquam de cetero aliqua anni tempora ab eisdem locis sive diebus in quibus iam sunt usque ad consummacionem seculi variabunt sive recedent. [...] Et sic Deo volente varietas et instabilitas nostri kalendarii solaris ex antiquis primo compositi et usque ad presens servati in posterum cessabunt, quod concedat dominus noster Ihesus Christi verus solis iusticie.”

  58. MS Oxford, Bodleian Library, Ashmole 789, fol. 374v. On the bottom of the page we find another reference to Monke’s trademark world era: “Incarnacio Christi anno mundi 4909 imperfecto.” See n. 51 above.

  59. See also MS Oxford, Bodleian Library, Ashmole 789, fols. 372v–373r, for a set of tables that spell out the excess of revolution and accumulating Julian surplus for every year in a 132-year cycle, again using multiples of 5;49,5,27,16,21,49 h and 0;10,54,32,43,38,11 h. For year 132, these tables show complementary values of 23;59,59,59,59,59,48 h and 24;0,0,0,0,0,12 h as opposed to the 24 straight hours an application of the precise year length would have yielded. The 33-year table in MS Oxford, Bodleian Library, Digby 97, fol. 41r generates the same imperfect results.

  60. The tables in MS Digby 97, fol. 33r, which reflect the same principle, are hence most likely based on the Opus secundum, whereas the imperfect table found on fol. 41r of the same manuscript, may go back directly to Richard Monke (see n. 59).

  61. That the Church would have been better served with a 33-year intercalation cycle has been argued in more recent times by authors such as Duncan Steel, who unfortunately mars some valid points by interweaving them with a ludicrous historical conspiracy theory. See Steel (2000, pp. 187–194).

  62. See MS Oxford, Bodleian Library, Laud. Misc. 594, fol. 15r, where these paschal termini are indicated to fall within a range of 13 March–10 April (traditionally: 21 March–18 April), in line with the updated location of the vernal equinox (12 March) and the new Golden Numbers inscribed into Monke’s Kalendarium.

  63. The possibility of such a link was raised by North (1983, pp. 93–94). See on the Jalālī calendar Ginzel (1906, pp. 300–305).

  64. Ziggelaar (1983, pp. 215–217), Levi Della Vida (1948, pp. 19–22), Takahashi (2014, pp. 326–328).

Manuscripts Cited

  • Bernkastel-Kues, Cusanusstiftsbibliothek, 212

  • Cambridge, Peterhouse, 75.I

  • Cambridge, Peterhouse, 267

  • Cambridge, University Library, Ee.3.61

  • Gloucester, Cathedral Library, 21

  • London, British Library, Egerton 847

  • London, British Library, Egerton 889

  • London, British Library, Royal 12.D.VI

  • Oxford, Bodleian Library, Ashmole 340

  • Oxford, Bodleian Library, Ashmole 346

  • Oxford, Bodleian Library, Ashmole 369

  • Oxford, Bodleian Library, Ashmole 789

  • Oxford, Bodleian Library, Bodley 300

  • Oxford, Bodleian Library, Digby 48

  • Oxford, Bodleian Library, Digby 97

  • Oxford, Bodleian Library, Laud. Misc. 594

  • Oxford, Bodleian Library, Lyell 37

  • Oxford, Bodleian Library, Wood D.8

  • Oxford, Jesus College, 46

  • Paris, Bibliothèque nationale de France, lat. 7286C

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Acknowledgements

Warm thanks to José Chabás, Matthieu Husson, David Juste, and Noel Swerdlow for their helpful comments on earlier drafts.

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    C. Philipp E. Nothaft

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Communicated by Noel Swerdlow.

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Nothaft, C.P.E. John Holbroke, the Tables of Cambridge, and the “true length of the year”: a forgotten episode in fifteenth-century astronomy. Arch. Hist. Exact Sci. 72, 63–88 (2018). https://doi.org/10.1007/s00407-017-0200-0

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