Ways to Read a Table: Reading and Interpolation Techniques in Canons of Early Fourteenth-Century Double-Argument Tables

‘Syzygy’ refers to the opposition or conjunction of the Sun and Moon. Mean syzygies refer to their mean positions, true syzygies to their true positions. The computation of true syzygies is the first step in the determination of eclipses which, of course, are astronomical phenomena of first importance both astrologically and theoretically.

Chabás J. Goldstein B. R. , “Nicholaus de Heybech and his table for finding true syzygy”, Historia mathematica , xix (1992), 265–89; Chabás J. Goldstein B. R. , “Computational astronomy: Five centuries of finding true syzygy”, Journal for the history of astronomy , xxviii (1997), 93–105; Porres B. Chabás J. , “John of Murs's Tabulae permanentes for finding true syzygies”, Journal for the history of astronomy , xxxii (2001), 63–72; Kremer R. L. , “Thoughts on John of Saxony's method for finding times of true syzygy”, Historia mathematica , xxx (2003), 236–77; Kremer R. L. , “Wenzel Faber's table for finding true syzygy”, Centaurus , xlv (2003), 305–29; Kremer R. L. , “John of Murs, Wenzel Faber and the computation of true syzygy in the fourteenth and fifteenth centuries”, in Mathematics celestial and terrestrial: Festschrift für Menso Folkerts zum 65. Geburtstag , ed. by Dauben J. W. (Halle, 2008), 147–60; Kremer R. L. , “Experimenting with paper instruments in fifteenth- and sixteenth-century astronomy: Computing syzygies with isotemporal lines and salt dishes”, Journal for the history of astronomy , xlii (2011), 223–58.

Probably they were inspired by Arabic double-argument tables such as those of the twelfth-century al-Andalusian astronomer Ibn al-Kammâd for the computation of true syzygies. Cf. Chabás Goldstein , “Computational astronomy” (ref. 2).

Somes of these tables have been published or reviewed. See Chabás J. Goldstein B. R. , “John of Murs's tables of 1321”, Journal for the history of astronomy , xl (2009), 297–320; Chabás J. Goldstein B. R. , “Early Alfonsine astronomy in Paris: The tables of John Vimond (1320)”, Suhayl , iv (2004), 207–94.

Canons are generally prescriptive or procedural texts that show the way to use certain sets of tables and eventually other astronomical instruments, in order to solve astronomical questions.

Poulle E. , Les sources astronomiques: Textes, tables, instruments (Paris, 1981).

For a monumental study of the relation between these two types of tradition see Pedersen F. S. , The Toledan Tables: A review of the manuscripts and the textual versions with an edition (4 vols, Copenhagen, 2002).

During the ancient periods, computation of the position of a planet was composed of two distinct steps. First one computed the mean position of a planet, i.e., its position had it been moving at constant velocity. Then one computed different corrections, also called equations, which need to be added or subtracted to the mean position in order to take into account different types of irregularities (anomalies) in the motion of the planet. This second step is by far more complex since in the Ptolemaic theory for all planets except the Sun, one of the equations depends simultaneously on two variables. In the standard Ptolemaic tables two successive equations are computed for each planet in long and complex calculations.

Erfurt F 4° 388 ff. 1r–42v (first half of the 15th century); London, BL addit. 24070 ff. 1–57v; Lisbon Ajuda 52-XII-35 ff. 66v–93r (equations only).

The canons are found in ff. 28r–32v.

Saby M. M. , “Les canons de Jean de Lignières sur les tables astronomiques de 1321: Edition critique, traduction et étude”, unpublished thesis, École des Chartes, Paris, 1987, 487.

The canons are found in ff. 70r–78r.

Poulle E. , La bibliothèque scientifique d'un imprimeur humaniste au XV^e siècle (Geneva, 1963), 23–4.

The canons are found at ff. 201v–205v. For this manuscript, see Poulle E. , “Jean de Murs et les tables alphonsines”, Archives d'histoire doctrinale et littéraire du Moyen Âge , lv (1980), 241–71; Boudet J. P. , Le “Recueil des plus célèbres astrologues” de Simon de Phares (2 vols, Paris, 1997–99).

Curiously, nothing is said about cases when one entry is less than 180° and the other is greater than 180°.

“Quere argumenti mediem in laterem sinistro tabule si fuissent minus sex signis et centrem mediem in capite tabule et hoc si precise poteris utrumque invenire et in angulo comuni invenies equationem quam addis cum medio motu si superius ascendendo prius occurrat tibi hec differentia a vel ab eo minue si ocurrat ‘m’ primo. Si vero argumentem mediem et centrem mediem precise non invenies tunc in eisdem locis quere numerum minorem et (31r) propinquirem et equationem in angulo comuni invenies ex scribe et scribe differentiam suppositam que differentia argumenti et hoc si argumentum precise non invenis. Et similiter differentia versus dextram positam ex scirbe que differentia centri et hoc etiam si centri precise non invenies. Et similter ex sribe differentiam istius differentie ad differentiam sibi recte suppositam quam habebis per subtractione minoris a maiore que est differentie argumenti et vocetur differentiam differentie. Et in accipiendo illas tres differentias nota bene utrum numerus sequentes cuius sunt differentie quem scilicet ex scripsisti sit maior vel minor illo quem accepisti. Et si sit maior pone super differentiam sibi correspondente ‘a’ et si sit minor ‘m’.” Erfurt 4° 366, ff. 30v–31.

Note that for this last difference the given instructions for the determination of the ‘a’ or ‘m’ values of the difference do not apply in any obvious manner.

“Et tunc cuiuslibet differentie argumenti accipe partem proportionallem secundum proportionem residui argumenti ad numerum per quod crescit argumenti in illo loco. Et tunc accipe partem proportionalem differentiem centri secundum proportionem residui centri ad numerum per quod crescit centrum et hoc secundum doctrinam prius datam queres partes proportionnales adde cum equationem prius pervenenta si iuxtam differentiem fiunt ‘a’ scripti vel minue si ‘m’ et habebis tunc equationem argumenti ultimo equatam quoque adde cum medio motu vel minue ut prius dictum est.” Paris BnF Lat. 10263, f. 75r.

Procedures for determining the signs of the different values are ignored for the moment.

“… et tunc accipe partem proportionallem secundum proportionem residui argumenti ad numerum per quod crescit argumenti in illo loco et tunc partem proportionalem que proveniet ex differentiam differentie adde cum differentie centri vel minue secundum quod iuxtam differentiam differentie scriptum est ‘a’ vel ‘m’ et ergo quod proveniet accipe partem proportionalem secundum proportionalem residui centro ad numerum per quod crescit centrum et hoc per differentiam procetis quem partes proportionales scilicet differentie argumenti prius servanta et differentia adde cum equationibus prius servanta si iuxta differentiam scriptam fuissent ‘a’ vel minue si ‘m’ et habebis tamen equationum argumenti ultimo equata quod adde ad medio motu vel minue ut prius dictum est.” Erfurt 4° 366, f. 31r.

Again procedures for determining signs of the different values are ignored for the moment. Note that E4 is not used in this more complex procedure.

Although it is by no means a proof, this hypothesis is reinforced by two elements that will be exposed in the next section of this paper. The simple procedure is found in at least one of John of Murs's texts; the second, more complex procedure is set out, with a different text, in another part of the canons of John of Lignères's Tabule magne.

From our modern point of view, none of the procedures is numerically effective as neither uses E4 in the computation. Thus both procedures become more and more inaccurate as the interpolated intervals grow larger and as the interpolated value should be closer to E4. A closer examination of numerical issues will be addressed below.

See ref. 22 and next section.

See ref. 2.

Chabás Goldstein , “Computational astronomy” (ref. 3).

Again ignoring the signs of the different quantities.

“Si longitudo fuissent in gradus tunc erant hori et minuta. Si vero fuissent solum in minutis tunc erunt minutis et secundas” (Erfurt 4° 366, f. 32v). This reminds us of the particular mathematical setting of the mean motion table in the Parisian Alphonsine corpus where arguments range from 0° to 60° and may represent sexagesimal multiples or divisions of the day and where similar rules explain the units of the values found in the table according to the units with which the tables are entered.

Kremer , “John of Murs” (ref. 2).

Chabás Goldstein , “Computational astronomy” (ref. 3); Porres Chabás , “John of Murs's tabulae permanentes” (ref. 3).

Kremer , “Thoughts on John of Saxony's method” (ref. 2); Kremer , “Wenzel Faber's table” (ref. 2).

Porres Chabás , “John of Murs's tabulae permanentes” (ref. 3).

We will return to John of Murs's expression of these procedures in the next section.

In cases where the two quantities have the same sign, you subtract the lesser from the greater and give the result the common sign of the two quantities or the opposite according to the orientation of the subtraction. In cases where the two quantities have different signs you add them and give the result the sign of the greater or the opposite according to the orientation of the subtraction. We note that John of Lignères here does not tackle this difficulty in his algorithm so that, strictly applied, his interpolation procedures becomes absurd in those situations.

This point will not be emphasized here as it is a major concern of the last section of this paper.

Generally they appear as superscripts to certain numbers in the tables. They may be noted in ink of a different colour.

The computation of syzygies offers an interesting exception to this general rule since for one of the numbers (elongation) the ‘a’ and ‘m’ signs are replaced by the phrases “elongation of the Moon” or “elongation of the Sun”, while other quantities are marked by the ‘a’ and ‘m’ signs. This nicely illustrates the principle we suggested at the end of the second part of this paper. Even in what may seem to be the arithmetical core of these numerical practices, astronomical meanings are present.

As they appear in the table, the ‘a’ and ‘m’ signs naturally mark the near-zero values of the tabulated function. This makes a fourth possible meaning for these signs, but we do not consider it on the same level as the others because it does not appear in the canons. Hence it seems quite probable that this meaning was unnoticed by medieval readers.

“Si precise argument predicta non inveneris, intra cum minoribus propinquioribus a dextris vel a sinistris, prout debes, et quod in angulo comuni invenies scribe cum signo addicionis vel minucionis cum differencia solis, que in lineis descendentibus scribitur, vel cum differentia lune que in lineis transversalibus invenitur, et cum utraque differencia fac signum addicionis si numerous anguli sequentis fuerit maior, vel minucionis si minor. Et postea de differencia solis summe partem proporcionalem secundum proporcionem residui argument solis cum quo intrare non potuisti ad 6 gradus. Quam partem proporcionalem tempori in angulo communi invento adde si cum differencia solis fuerit signum addicionis, vel minue si minucionis. Similiter fac de differencia lune secundum proporcionem residui argument lune, et quod provenerit adde vel minue tempori preequato, et proveniet tempus inter mediam et veram coniunctionem vel opposicionem. Et hoc totum est verum si cum differenciis solis et lune non invenies ‘a’ vel ‘m’.” Porres and Chabás, “John of Murs's tabulae permanentes” (ref. 2), 69.

This is the case where, a subtraction being impossible, we have to invert the operation.

This was the case for the difference of differences in John of Lignères's procedure described in the first part of this paper.

Two examples here from John of Murs Tabule permanentes: “… cum quo fac signum addictionis vel minucionis prout ili tempori prosito debebatur”; “… et tunc fac signum addicionis ubi per per primam regulam vel secundam faceres signum minucionis, vel econtrario”. Porres Chabás , “John of Murs's tabulae permanentes” (ref. 2), 69.

For example, it is not possible to subtract 8 from 5. If asked to do that you must do the opposite, subtract 5 from 8 and eventually modify all the operations affected by this inversion.

“Cui adde vel ab eo subtrahe partem proporcionalem differencie lune per aliquam predictarum regularum, et quod provenerit est scundus numerus, cum quo fac signum addicionis vel subtractionis prout debes ut prius.” Porres Chabás , “John of Murs's tabulae permanentes” (ref. 2), 69.

The first tested entry is D1 = D2 = D3 = D4 = 0, the second is D1 = D2 = D3 = 0 and D4 = 20, the third is D1 = D2 = D3 = 0 and D4 = 40, and so on, until you stop when D1 = D2 = D3 = D4 = 180. This creates a space of 10⁴ entries. This space of entries and the one used to test John of Murs procedures (ref. 46) include most of the situations actually found in the tables the interpolation procedures were designed for and so give a good indication of the overall numerical performance of the medieval procedures. Yet these spaces also contains many unrealistic situations, as for example when D1 = D2 = D3 = D4, this is why some specific case are also presented in the study.

The first tested entry is D1 = D2 = D3 = D4 = −120, the second is D1 = D2 = D3 = −120 and D4 = −100, the third is D1 = D2 = D3 = −120 and D4 = −80, and so on, until you stop when D1 = D2 = D3 = D4 = 140. This creates a space of 14⁴ entries.

Ossendrijver Mathieu , Babylonian mathematical astronomy: Texts and procedures (forthcoming), 29–32.

Usually this history is especially focused on Arabic and European algebraic traditions. Sometimes earlier traditions are mentioned in China around the classical Nine chapters on the art of mathematics or in India around Aryabhata and Brahmagupta. See Boyer Carl B. Merzbach Uta C. , A history of mathematics, 3rd edn (Hoboken, 2011).