John of Lignéres, or Johannes de Lineriis

(fl.France, first half of fourteenth century)

astronomy, mathematics.

Originally from the diocese of Amiens, where any of several communes could account for his name, John of Lign′ res lived in Paris from about 1320 to 1335. There he published astronomical and mathematical works on the basis of which he is, with justice, credited with diffusion of the Alfonsine tables in the Latin West.¹

In astronomy the work of John of Lignéres includes tables and canons of tables, a theory of the planets, and treatises on instruments. The tables and the cannons of tables have often been confused among themselves or with the works of other contemporary Paris astronomers named John: John of Murs, John of Saxony, John of Sicily, and John of Montfort. There are three canons by John of Lignéres

(1)The canons beginning Multiplicis philosophie variis radiis. . . are sometimes designated as the Canones super tabulas magnas; they provide the daily and annual variations of the mean motions and mean arguments of the planets in a form which, although not the most common, is not exceptional. The tables of equations, on the other hand, are completely original: one enters them with both mean argument and mean center at the same time and reads off directly a single compound equation, the sum of the equation of center and the corrected equation of argument; it is sufficient to add this compound equation to the mean motion in order to obtain the true position. The tables also permit the calculation of the mean and true conjunctions and oppositions of the sun and the moon; but there is no provision for the determination of the eclipses and for planetary latitudes. The radix of the mean motions is of the time of Christ, but the longitude is not specified (in all probability that of Paris); nevertheless, since the list of apogees is established by reckoning from 1320, Tabule magne may be dated approximately to that year.

The canons are dedicated, as is the treatise on the saphea (see below), to Robert of Florence, dean of Glasgow in 1325.² It is not certain whether these tables were calculated on the basis of the Alfonsine tables, which John of Lignéres would therefore already have known.³ Although certain characteristics (the use of physical signs of thirty degrees and not of natural signs of sixty degrees) are not decisive, the tables of the equations of Jupiter and Venus appear to be calculated following neither those of the Alfonsine tables nor the Ptolemaic eccentricities:⁴ insofar as one can judge on the basis of tables from which it is difficult to derive the equation of center and the equation of argument, the eccentricities used are not the customary ones.

The text of the canons of John of Lignéres is very concise. It was the object of an explanatory effort by John of Speyer, Circa canonem de inventione augium. . . which many have been written in 1348, since it contains an example calculated for that year.⁵

(2)In 1322⁶ John of Lignéres composed a set of tables completely different from the preceding.(One cannot tell, however, if they are earlier or later than the Tabule magne.) These tables, and especially their canons, are in three parts often found separately, particularly the first. The canons of the primum mobile (Cujuslibet arcus propositi sinum rectum. . .), in forty-four chapters, correspond to the trigonometric part of the tables and consider the problems linked to the daily movement of the sun: trigonometric operations, the determination of the ascendant and of the celestial houses, of equal and unequal hours, and so on. Three of the canons describe the instruments used in astronomical observation: Ptolemaic parallactic rulers and a quadrant firmly fixed in the plane of the meridian. The corresponding part of the tables is thus made up of a table of sines, a table of declinations (the maximum declination is 23° 33é 30”), and tables of right and oblique ascensions for the latitude of Cremona and of Paris (seventh clima). This portion of the canons is not dated, and the contents do not provide any chronological information whatever; but it is reasonable to suppose that the canons were published at the same times as the tables and the canons of the movements of the planets, with Which they form a harmonious ensemble.

The canons of the primum mobile were the object of a commentary, accompanied by many worked- out examples, by John of Saxony (ca.1335): Quia plures astrologorum diversos libros fecerunt. . .. One of the canons of the primum mobile, no.37, concerning the equation of the celestial houses, was printed at the end of the conons by John of Saxony that were appended to the edition of the Alfonsine tables published by Erhard Ratdolt in 1483.

The canons of the movements of the planets form the second part of the treatise whose first part comprises the canons of the primum mobile: Priores astrologi motus corporum celestium. . .. They treat the conversion of eras (very briefly), the determination of the true positions of the planets and of their latitudes, the mean and true conjunctions and oppositions of the sun and the moon and their eclipses, the coordinates of the stars, and the revolution of the years. The corresponding tables give (a) chronological schemes which permit the conversion from one era to another with a sexagesimal computation revolution of the years. to another with a sexagesimal computation of the years; (b) the mean motions and mean arguments of the planets for groups of twenty years (anni collecti), single years (anni expansi), months, and days—both for the epoch of the Christian era and 31 December 1320 at the meridian of Paris; and (c) the tables of equations according to the usual presentation.

These canons, which are usually dated by their explicit references to 1322, allude to the Alfonsine tables, to which the tables of John of Lignéres are certainly related, if only by the adoption of the motion compounded from precession and accession and recession for the planetary auges. Yet the longitudes of the stars, established by adding a constant to Ptolemy’s longitudes, are not the same as those of Alfonso X. Nor are the tables of equations of Jupiter and Venus those ordinarily found in the Alfonsine tables or those of the Tabule magne but, rather, hand, equations of the Toledan tables. On the other, hand, John of Ligné res used the physical signs of thirty degrees and not the natural sings of sixty degrees. J. L. E. Dreyer observed that the Castilian canons of the Alfonsine tables in their original version, as they were published by Rico y Sinobas, do not correspond at all to the tables commonly designated as Alfonsine; he believed he had found them in a state closer to the original in the Oxford tables constructed by William Reed with the year 1340 as radix.⁷ It is reasonable to suppose that the tables of John of Ligné res represent, twenty years before Reed, an analogous effort to reduce the Castilian tables to the meridian of Paris; he preserved their general structure, and notably three characteristics: the compound motion of the auges (precession and accession and recession), physical sings of thirty degrees, and tables of periodic movements presented on the basis of twenty- year anni collecti.

The part of the canons of 1322 dealing with the determination of eclipses frequently appears separately (Diversitatem aspectus lune in longitudine et latitudine), despite the many references made to the preceding canons—references that thereby lose all significance.

In 1483 some of the canons on the planetary motions were included in the edition of the Alfonsine tables and with the canons of John of Saxony: canons 21- 23, concerning the determination of the latitudes of the three superior planets and of Venus and Mercury, and canons 38 and 40, on eclipses.

3. Finally, John of Ligné res wrote canons beginning Quia ad inveniendum loca planetarum . . . in order to treat the tabular material ordinarily designated as Alfonsine tables.⁸ The sings are the natural signs of sixty degrees, and the tables of the mean motions and mean arguments of the planets consist of the sexagesimal multiples of the motions during the day. The chronological portion has, obviously, had to be increased since with this system, in order to enter the tables of mean motions, it is necessary to transform a date expressed in any given calendar into a number of days in sexagesimal numeration. As it is certain that the sexagesimal form of the Alfonsine tables does not represent their original state, there is firm evidence for believing that the transformation which they underwent was carried out at Paris in the 1320’s, either by John of Ligné res himself, under his direction, or under the direction of John of Murs. The date of these very succinet new canons cannot be determined from the text; but it is certainly later than that of the canons Priores astrologi . . . (1322) and may perhaps be earlier than 1327, when John of Saxony produced a new version of the canons of the Alfonsine tables.

For those who are aware of the almost universal diffusion of the Alfonsine tables at the end of the Middle Ages, almost to the exclusion of any other tables, there is no need to emphasize John of Ligné res’s exceptional role in the history of astronomy. The magnitude of the work that he and his collaborators accomplished in so few years is admirable. Although there is no formal proof of the existence of a team of workers, the terms in which John of Saxony expressed his admiration for his “maître” bear witness to the enthusiasm that John of Ligné res evoked.

In order to complete the account of John of Ligné res’s work on astronomical tables, we must notice the execution of an almanac conceived, like that of Ibn Tibbon (Profacius), on the principle of the “revolutions” of each planet and therefore theoretically usable in perpetuity, provided a correction is applied based on the number of revolutions intervening since the starting date (1321) of the almanac. The work appears to be preserved in only one manuscript, unfortunately incomplete, with a short canon: Subtrahe ab annis Christi 1320 annos Christi. . . .

John of Ligné res’s theory of the planets, Spera concentrica vel circulus dicitur. . . represents the theoretical exposition of the principles of the astronomy of planetary motions, the application of which is furnished by the Alfonsine tables. In particular this theory provides a detailed justification of the compound motion of the eighth sphere; in it the author strives to demonstrate at length the inanity of the solution recommended by Thā bit ibn Qurra (a motion of simple accession and recession). Furthermore, he promises to return, in a work which it is not known whether he wrote, to certain difficulties remaining under the Alfonsine theory. John of Ligné res provided no indication of the values of the planetary eccentricities, of the lengths of the radii of the epicycles, or of the values of the various motions at any particular date. The only precise information, the reference to the position of the star Alchimech in 1335, allows the text to be dated about that year.

John of Ligné res’s astronomical work also included treatises on three instruments: the saphea, the equatorium, and the directorium. The saphea is an astrolabe with a peculiar system of stereographic projection: the pole of projection is one of the points of intersection of the equator and the ecliptic, and the plane of projection is that of the colure of the solstices. Following a rather clumsy effort by William the Englishman in 1231 to reconstruct the principle of an instrument attributed to al-Zarqā l he no doubt had never seen, the saphea was introduced in the West by the translation, done by Ibn Tibbon in 1263, of al-Zarqā l’s treatise. The saphea described by John of Lignéres (Descriptiones que sunt in facie instrumenti notificate . . .) presents technically several improvements over al-Zarqā l’s instruments.⁹ The most notable is the use of kind of rete, the circulus mobilis, consisting of a graduated circle of the same diameter as the face and an arc of circle bearing the stereographic projection of the northern half of the zodiac, as in the classic astrolabe. The diameter that subtends this projection of the zodiac carries a graduation similar to almucantars on the meridian line of the astrolabe’s tablet; a rule graduated in the same manner can be mounted on the circulus mobilis, forming a given angle with the diameter of the latter. On such an instrument one may consider either (1) one of the diameters of the face as a horizon, in the projection which characterizes the saphea: the diameter of the circulus mobilis then serves to refer to this horizon every position located in the unique system of the almucantars and of the azimuths traced on the instrument for the diameter of the horizon (this is the principle used in al-Zarqā s canons); or (2) the almucantars of the face as the horizons of a tablet of the horizons in a classic stereographic projection bounded by the equator: the half of the ecliptic traced on the circulus mobilis then plays the same role as the ordinary rete. The judicious alternate use of both systems allowed John of Lignē res to offer simpler and more rapid solutions to the problems dealt with in al-Zarqā l’s canons without losing any of the saphea’s advantages.

John of Lignē res wrote two treatises on the equatorium. The first, Quia nobilissima scientia astronomie non potest . . ., is an adaptation of Campanus’ instrument.¹⁰ In order to find the true positions of the planets, Campanus recommended a series of three disks, that is, six “instruments’ (one on each face of one disk), which reproduced fairly closely the schema of the geometric analysis of the planetary motions. John of Lignéres maintained this principle but simplified the construction by adopting a common disk, to bear the equants of all the planets (but not the moon). To avoid difficulties in reading, the equants are represented by a circle without graduations; and these ones, which begin at a different point for each planet, are replaced by a graduated ring which is superposed on the equant in the position suitable to the planet for which one is operating. The radii of the deferents are represented by a small rule bearing, on one side, a nail to be fixed in the center of the deferent of the planet, and on the other, an epicycle at the center of which is turning another small rule bearing, at appropriate distances, the “bodies” of the planets. Two threads represent the radius of the equant which measures the mean center and the radius of the zodiac which passes through the planet.

The other treatise on the equatorium (uses: Primo linea recta que est in medio regule...; construction: Fiat primo regula de auricalco seu cupro...¹¹) is fundamentally different from the first. The problem is no longer to reproduce the geometric construction of a planet’s true position but, rather, to calculate graphically, so to speak, the angular corrections (equation of center and equation of argument) that, added to the mean motion of a planet, determine its true position. The sole function of the instrument’s five parts (the so-called ruler of the center of the epicycle, the disk of the centers, the epicycle, the square carrying the “bodies” of the planets, and the rule for reading off) therefore is to furnish and to position in relation to each other the parameters of the planets (eccentricities, epicycle radii). Successively determined-exactly as in a calculation carried out with the Alfonsine tables-are the equation of center on the basis of the mean center, then the equation of argument on the basis of the true center and the true argument; the true position is obtained by adding the two equations to the mean motion.

The astrologer Simon de Phares, whose account of John of Lignéres is otherwise fairly correct, attributes to him a directorium the incipit of which (“Accipe tabulam planam rotundam cujus...”) corresponds very closely to that of a text on this instrument preserved in at least four anonymous manuscripts: “Accipe tabulam planam mundam super cujus extremitatem....”¹². The directorium was used only for astrology: it served to “direct” a planet or a point in the zodiac having a particular astrological value, that is, to lead it to another point in the zodiac by counting the degrees of the equator corresponding to this course. In fact, it is very similar to the astrolabe, except that the fixed celestial reference sphere, represented only by the horizon of the place and by the meridian line, is made to turn above the sphere of the stars and of the zodiac. Since, in a good calculation of “direction,” the latitude of the planets must be taken into account, the zodiac is represented by a wide band on which are traced its almucantars and its azimuths, as far as six degrees on either side of the ecliptic. John of Lignéres’s directorium presents no special features.

Finally, a Vatican manuscript attributes to John of Ligneres an “armillary instrument” that is difficult to define (Rescriptions [read Descriptiones] que sunt in facie instrumenti notificare. Trianguli equilateri ex tribus quartis arcus circuli magni. . .): in the absence of a section on its construction, the uses and the brief description that precede them give a very imperfect idea of the instrument, which appears to derive from the new quadrant. John of Lingé res’sidea seems to have been to replace the rotation of the margarites (which, in the new quadrant, compensated for the immobility in which this instrument held the rete of theastrolabe because of the reduction of the latter to one of its quarters)₁₃ by the rotation of another quarter-disk bearing the oblique horizon. This conception amounted to a return to what had constituted the justification of the stereographic projection characterizing the astrolabe, that is, to the rotation of the sphere of the stars and of the zodiac on the celestial sphere used for reference but with a reversal, as in the directorium, of the respective roles of the spheres ¹⁴.

None of John of Ligneres’s treatises on astronomical instruments is dated or contains information from which a date can be established. Nevertheless, the preface to the canons of the Tabule magne, addressed to Robert of Florence, notes the simultaneous sending, along with the tables, of an equatorium and a “universal astrolabe.” The latter should be identified with the saphea ; as for the equatorium, defined as suitable to furnish “easily and rapidly the equations of the planets,” it is more likely to be the second of the instruments described above.¹⁵

We have seen the development that John of Ligneres gave to sexagesimal numeration in the astronomical tables, since the tables of the regular movements of the planets in the Alfonsine tables have been modified so as to permit the systematic use of this type of numeration. He was so aware of the astronomer’s need for its use that he introduced, at the beginning of the canons of the Tabule magne, a long section on the technique of working on the “physical minutes.” He took up the question again and expounded it in the Algorismus ntinutiarum, in which he simultaneously treated physical fractions and vulgar fractions. Its great success is attested to by the number of manuscripts in which the Algorismus is preserved.

NOTES

1. P. Duhem, Le systême du ntonde, IV (Paris, 1916), 578-581, following G. Bigourdan, maintains that John of Ligneres was alive after 1350 ; he bases this on a letter from Wendelin to Gassendi that mentions the positions of the stars determined by John of Ligneres and reproduced by John of Speyer in his Rescriptum super canones J. de Lineriis. As long as John of Speyer’s work had not been found, one could—just barely-give credence to this tale. But the Rescriptumof John of Speyer, identified through MS Paris lat. 10263— see E. Poulle, La bibliotheque scientifique d’un imprimeur hmnaniste au XV siecle(Geneva, 1963), p. 49-and dating from about 1348, makes no reference to any table of stars .

2. G . Sarton, Introduction to the History of Science,III (Baltimore, 1947), 649n .

3. Despite the title of the MS Paris lat. 7281, fol. 201 v : Canones super tabulas magnas per J. de Lineriis computatas ex tabulis Alfonsii (in another hand : ad meridianum Parisiensem).

4. The equations of Jupiter and Venus given in the Alfonsine tables use simultaneously two values for the eccentricities of these planets. See E. Poulle and O. Gingerich, “Les positions des planetes au moyen age : application du calcul electronique aux tables alphonsines,” in Comptes rendus des seances de l’Academie des inscriptions et belles-lettres (1967), pp. 531-548, esp. 541 .

5. See note 1 .

6. Some MSS, notably MS Paris lat. 7281, fol. 201v, which Duhem used in constructing his account, give the date as 1320 : it corresponds to the epoch of the tables (31 Dec. 1320), that is, to the beginning of the first year following the closest leap year to the date of composition of the tables and canons .

7. J . L. E. Dreyer, “On the Original Form of the Alfonsine Tables,” in Monthly Notices of the Royal Astronomical Society, 80 (1919-1920), 243-262. M. Rico y Sinobas, Libros del saber del rey d. Alfonso X de Castilln, IV (Madrid, 1866), 111-183 ; the tables actually published by Rico y Sinobas, in fats . (ibid ., pp. 185 ff.), are spurious, as J .-M. Millas Vallicrosa has shown in Estudios sobre Azarquiel (Madrid-Granada, 1943-1950), pp. 407-408 .

8. The Alfonsine tables reorganized at the time of John of Ligneres are shorter than those published in 1483, which were completed by tables of ascensions, by tables of proportion and by tables for the calculation of eclipses. A portion of this supplement, but not the whole of it, is borrowed from John of Ligneres’s tables of the primun mobile and from the part of the tables of 1322 dealing with the calculation of eclipses ; but the canons Quia ad inveniendum. . . are silent on the use of this part of the tables and give no special attention to eclipses. In the medieval MSS the list of the tables forming the Alfonsine tables varies considerably from one MS to another, and it is very difficult to reconstruct the original core of the text ; one can rely on little more than the uses specified by the canons .

9. John of Ligneres’s treatise on the saphea contains only uses, preceded by a chapter of description. But the MS Paris lat. 7295, which preserves the text of the treatise (fols. 2-14), also included (fols. 18v-19) two incomplete and unidentified drawings ; these must be compared with John of Ligneres’s text, which they illustrate most pertinently .

On the saphea, see G. Garcia Franco, Catalogo critico de astrolabios existentes en Esparia (Madrid, 1945), pp. 64-65 ; M. Michel, Traite de l’astrolabe (Paris, 1947), pp. 95-97 ; and E. Poulle, “Un instrument astronomique dans l’occident latin, la saphea,” in A Giuseppe Ermini (Spoleto, 1970), pp. 491-510, esp. pp . 499-502 .

10. In MS Oxford, Digby 57, fols. 130-132v, the same incipit introduces another treatise on the equatorium, composed at Oxford with 31 December 1350 as the r adix. MS Paris fr. 2013, fols. 2-8v, preserves a text in French (“Pour composer I’equatoire des sept planetes . . .”) presented as the translation, in 1415, of a treatise on the equatorium by John of Ligneres written in 1360 ; besides the fact that the date cannot be accepted, the instrument, although similar to John of Ligneres’s first equatorium, is not identical .

11. The part dealing with its construction is found in only one of the two MSS of the texts, and there it is placed after the uses. The incipit “Descriptiones (eorum) que sunt in equatorio . . .” noted by L. Thorndike and P. Kibre in Catalogue of Incipits, 2nd ed . (Cambridge, Mass ., 1963), col. 402, is the title of the descriptive chapter that broaches the section on the uses.

12. E. Wickersheimer, ed., Recueil des plus cé lé bres astrologues et quelques hommes doctes faict par Symon de Phares (Paris, 1929), p. 214.

13. On the new quadrant, see E. Poulle, “Le quadrant nouveau mé dié val,” in Journal des savants (1964), pp. 148- 167, 182- 214.

14. MS Berlin F. 246, fol. 155, preserves extracts from a Tractatus de mensurationibus by John of Ligné res: they are actually several of the chapters from the section on geometric uses in the treatise on the armillary instrument, a section extremely similar, in terms of its contents, to the treatise on the ancient quadrant by Robert the Englishman.

15. Paris lat. 7281, fol. 202: “Post multas excogitatas vias, feci instrumentum modici sumptus, levis ponderis, quantitate parvum et continentia magnum quod planetarium equatorium nuncupavi, eo quod in eo faciliter et prompte eorum equationes habetur; . . . unum composui instrumentum omnium predictorum instrumentorum [astrolabe, saphea, solid sphere] vires et excellentias continens quod merito universale astrolabium nuncupatur, eo quod unica superficie tota celi machina continetur et illa eadem cunctis regionibus applicatur. . .. Suscipiatis, o domine decane, instrumenta et tabulas que vobis. . . offero.”

BIBLIOGRAPHY

I. Original Works. Of the canons and tables written by John of Ligné res, only the canons of the primum mobile (the first part of the canons of 1322) have been published in part: M. Curtze, “Urkunden zur Geschichte der Trigonometrie im christlichen Mittelalter,” in Bibliotheca mathematica, ser, 3, 1 (1900), pt, 7, 321- 416, pp. 390- 413: “Die canones tabularum primi mobilis des Johannes de Lineriis”: there are the first nineteen canons (pp. 391- 403), followed by the titles of the succeeding canons, as well as by the tables of sines and chords and of shadows and the tabula proportionis (pp. 411- 413); canon 9 of the canones super tabulas latitudinum planetarum et etiam eclipsium (the second part of the canons of 1322) is also included (pp. 403- 404). See also J. - M. Millá s Vallicrosa (note 7), p. 414. To study John of Ligné res’s work on astronomical tables recourse to the MSS is therefore necessary.

The almanac of 1321 is in MS Philadelphia Free Library 3, fols. 3- 10 (the beginning is incomplete).

The Tabule magne are very rare. The canons are in Erfurt 4° 366, fols. 28- 32v; Paris lat. 7281, fols. 201v- 205v; and Paris lat. 10263, fols. 70- 78. The tables are in Erfurt F.388, fols. 1- 42; and (tables of equations only) Lisbon Ajuda 52- VI- 25, fols. 67- 92v.

The tables and the canons of 1322, on the other hand, are fairly common; but the tables themselves are seldom complete, probably because those among them that duplicated the Alfonsine tables were not so well not so well accepted as the latter and hence only the tables for the primum mobile those for the latitudes, and those for the eclipses were preserved: Basel F. II. 7, fols. 38- 57v, 62- 77v (incomplete canons and tables in part); Catania 85, fols. 144- 173 (canons), 192- 201v (partial tables); Erfurt 4° 366, fols. 1- 25v (canons only); Paris lat. 7281, fols. 178v- 201v (canons only); Paris lat. 7286 C, fols. 9- 58v (tables and canons), etc. The canons of the primum mobile often appear alone: Paris lat. 7286, fols. 35- 42v (unfinished canons and partial tables); paris lat. 7290 A, fols. 66- 75v; Paris lat. 7292, fols. 1- 12v; Paris lat. 7378 A, fols. 46- 52. The canons of the planetary movements likewise are frequently found by themselves: Cusa 212, fols. 74- 108v (with tables); Paris lat. 7407, fols. 40- 63, etc. Yhose of John of Ligné res’s tables and canons that, for the latitudes and the eclipses, complete the Alfonsine tables are sometimes integrated into the latter, as in Paris lat. 7432, fols. 224- 358v. The portin of the canons that treats eclipses can be found separately: Paris lat. 7329, fols. 127- 131v.

See also Cracow 557, fols. 58- 96v (canons and partial tables).

John of Ligné re’s canons on the Alfonsine tables, while much less common than those of John of Saxony, are neverthele not rare: Cusa 212, fols. 65- 66v; Oxford fols. 148v- 155; Paris lat. 7281, fols. 175- 178; Paris lat. 7286, fols. 1- 3v; Paris lat. 7405, fols. 1- 4v. Moreover, they often duplicate those of John of Saxony.

John of Ligné res’s other astronomical works do not seem to have had as great a diffusion. The theory of the planets is preserved in Cambridge Mm. 3. 11, fols. 76- 80v; Paris lat.7281, fols. 165- 172. Another Cambridge MS, Gg. 6. 3, fols. 237v- 260, also preserves this text in a version that appears to be quite different, but this MSis very mutilated and practically unusable.

The saphea can be found (the incipit of which is very similar to the one in Ibn Tibbon’s translation of al-Zarqā l’s treatise) in Erfurt 4° 355, fols. 73-81v; Erfurt 4° 366, fols. 40-49; Paris lat. 7295, fols. 2-14. The first chapter (description) was published in L. A. Sé dillot, “Mé moire sur les instrument6s astronomiques des Arabes,” in Mé moires pré senté s par divers savants á I’ Acadé mie des inscriptions et belles-letters, ser. I, 1 (1844), 1-220, see 188-189n.

The first treatise on the equatorium was published by D. J. Price as an appendix to the treatise attributed to Chaucer: The Equatorie of the Planetis (Cambridge, 1955), pp. 188-196, but the text is very defective and it is still necessary to refer to the MSS: Cambridge Gg. 6. 3, fols. 11-12v; Oxford, Digby 57, fols. 11-12v; Cracow 557, fols. 11-12v; Oxford, Digby 168, fols. 65v-66; and Vatican Palat. 1375, fols 8v-10v. The treatise on the equatorium preserved in Oxford, Digby 57, fols. 130-132v, under the same incipit, is not the one by John of Ligné res. The second equatorium is unpublished: Vatican Urbin. lat. 1399, fols. 16-21 (uses and construction); Oxford, Digby 228, fols. 53v-54V (uses only).

The treatise on the directorium, Accipe tabulam planam . . ., is found only anonymously in Florence. Magl. XX. 53, fols. 35-37; Oxford, Digby 48, fols. 91v-94; Salamanca 2621, fols. 21v-23; Wolfenbuttel 2816, fols. 125-126v. The armillary instrument is attributed to John of Ligné res in Vatican Urbin. lat. 1399, fols. 2-15.

The Algorismus minuutiarum (Modum representationis minutiarum unlgarium. . .) was published very early: Padua, 1483 (Klebs 167. 1) and Venice, 1540. See A. Favaro, “Intorno alla vita ed alle opere di Prosdocimo de’ Beldomandi,” in Bullettino di bibliografia e di storia delle scienze matematiche e fisiche,12 (1879), 115-125; D. E. Smith, Rara arithmetica (Boston, 1908), pp. 13-15; and H. L. L. Busard, “Het rekenen met breuken in de middeleeuwen, in het bijzonder bij Johannes de Lineriis,” in Mededelingen van de K. academie voor wetenschappen, letteren en schoone kunsten van België (1968). There are a great many MSS of this work.

II. Secondary Literature. Pierre Duhem, Le systé me du monde, IV (Paris, 1916), 60-69, 578-581; and L. Thorndike, A History of Magic and Experimental Science, III (New York, 1934), 253-262, although they supersede most of the earlier works—see G. Sarton, Introduction to the History of Science, III (Baltimore, 1947), 649-652—do not really bring John of Ligné res’s work into clear focus; the canons of the tables, especially, have been confused with each other and with the treatises on the instruments. Moreover, Duhem’s hypothesis that the Algorismus minutiarum ought to be attributed to John of Sicity rather than to John of Ligné res is not based on any serious evidence: the medieval attribution is unanimously to John of Ligné res.

Emmanuel Poulle