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One of the characteristics of an exact science is to explain theoretically the processes addressed by that discipline and to treat them precisely in a quantitative way. As an exact science, astronomy makes extensive use of numerical computations. In early astronomy, this is best exemplified by astronomical tables in the tradition of Ptolemy's Almagest that dates to the 2nd century, A.D. Throughout the Middle Ages, astronomers compiled a great variety of astronomical tables to help computers determine the positions and other circumstances of the celestial bodies and to help them solve astronomical problems related to the daily rotation, the determination of the times of eclipses, etc. The simplest astronomical table consists of two columns of numbers, such that for each value in the first column there corresponds one and only one entry, representing what nowadays is called a function. The first column gives the successive values of the argument, currently called the independent variable. The information contained in such a table can be represented as a two-dimensional graph, although it was not done so at the time. Astronomical tables in the Almagest were not all of the simplest kind, for many of them had more than two columns, such that the entries in each column depended on a single argument. In the Latin West a special form of astronomical tables, introduced in the 14th century, is called a 'double argument table', that is, a table with two arguments, one set at the head of each column and another set at the head of each row, corresponding (in modern terms) to a function of two variables. This ingenious type of table can be represented as a three-dimensional graph. The advantage of double argument tables is that they reduce the number of steps in the computation of a planetary position, etc. But this meant that the table maker had to produce many more entries in the tables, which required him to perform a large number of computations. As far as we know, the earliest sets of double argument tables in the West were due to John of Murs (c. 1321), John of Ligneres in Paris (c. 1325), and William Batecombe in Oxford (1348), all of whom based their tables on the Parisian Alfonsine Tables that began to circulate in the 1320s.

Astronomers addressing a variety of problems put together different tables and compiled 'sets of tables', that is, consistent collections of astronomical tables embracing all or some aspects of mathematical astronomy and usually accompanied by a text, called 'canons', explaining their use. Most sets of tables compiled in western Europe in the Middle Ages followed the structure of those composed in Arabic, that is, handbooks called zijes (from the Arabic ztj, plural zljat).1

In the Renaissance the mathematical sciences played an important role in humanistic culture, and they were highly appreciated at various social levels.2 Mathematical astronomy was regarded as especially valuable, for it was associated with cosmology and philosophy, as well as with astrology and astrological medicine. In other words, there was a considerable market for publications that included almanacs, ephemerides, and lunaria, several of them ranking among the best sellers in scientific publications,3 as well as sets of astronomical tables. The investment of time and money by the printer in producing these sets of tables was significant, and the fact that more than one edition of the same set appeared is an indication of the popularity of this genre. In particular, Bianchini's tables were printed three times between 1495 and 1553.

By the time Bianchini compiled his tables (c. 1442), European astronomers had access to several variants of the Parisian Alfonsine Tables, that is, a set of tables which was recast beginning in the 1320s by a group of notable scholars working in Paris, all of them sharing the name, John ( John of Murs, John of Ligneres, and John of Saxony), and based on the work done by the astronomers in the service of King Alfonso X of Castile in the second half of the 13 th century. Unfortunately, only the canons of the original Castilian Alfonsine Tables are extant, not the tables themselves.4

All these sets of astronomical tables are in the tradition of Arabic zijes: they contain a great many tables and, at their core, are those for the determination of the positions and motions of the five planets and the luminaries. The position of a planet in longitude (along the ecliptic),

2 See, e.g., Rose, Renaissance of Mathematics.

3 See Chabas, Granollachs.

4 Chabas and Goldstein, Alfonsine Tables of Toledo.

as well as the Sun and the Moon, is computed from tabulated values for mean motions and equations, that is

X = X + AX + c, where X0 is the initial mean longitude of a planet at some epoch, AX is the increment in mean motion from epoch, ta, to another time, t, and c is the equation, i.e., the deviation from mean motion to be computed from a model for planetary motion. In turn, AX is a linear function of time,

AX = |i ■ (t - to), where | is the mean motion per day (or some other unit of time). In the case of the five planets, c depends on two variables, X and a, the argument of anomaly, and a = a0 + Aa + c', where a0 is the argument of anomaly at epoch, Aa is the increment in the argument of anomaly from t0 to t, and c' is the equation of anomaly. To compute a position of one of the 5 planets using the tables, one first needs to find 3 quantities:

X = X0 + AX, k = XA - X, where X is the mean position of the planet and k, the mean argument of center, is the distance of X from the planet's apogee, XA, and a = a0 + Aa.

Then, with k and a as arguments in one or more tables, X = X + c( k, a).

Note the both c and c' can be positive or negative. These rules are common to all sets of astronomical tables in the Ptolemaic tradition, and they have generated a large number of tables based on various ingenious procedures. For worked examples of Bianchini's procedure for finding the true longitude of the Moon and of Mars, see Tables 17 and 41, below.

The Alfonsine tradition lies within the Ptolemaic tradition, having a set of parameters in common as well as the same underlying model. Nevertheless, we find a great variety of tables in this tradition, and among them are those of Bianchini, the subject of this monograph. As we shall see below, in addition to tables for planetary longitude, there are tables for eclipses, planetary latitudes (the distance of a planet north or south of the ecliptic at any given time), etc. But let us first consider some of the sets of tables produced in the 14th century.

The Parisian astronomers drew up tables and wrote their corresponding canons, based on the earlier Castilian material. One characteristic feature of the resulting Parisian Alfonsine Tables is the division of a circle into 6 physical signs of 60°, although in many cases the circle is divided into 12 zodiacal signs of 30°, as was the case in almost all previous sets of tables. These tables rapidly superseded the Toledan Tables, compiled at the end of the 11th century, also in Toledo, and extant in hundreds of manuscripts.5

Among the collections of tables ascribed to the Johns are (i) the tables of 1321, with a short canon, by John of Murs; (ii) the so-called Tabule magne, by John of Ligneres; and (iii) the tables of 1322, with canons, also by John of Ligneres. Unfortunately, a thorough study of these tables is still lacking.

(i) John of Murs compiled a set of tables to compute the positions of the planets and the luminaries which are extant in two MSS in Oxford and Lisbon.6 In both of them physical signs of 60°, rather than zodiacal signs of 30°, are used. The tables are accompanied by a short text beginning Si vera loca planetarum per presentes tabulas volueris invenire a tempore incarnationis domini dato perfecto deme 1320 . . . The presentation given to the tables of the planets is indeed original, because the organizational principle is a succession of mean conjunctions of each planet with the Sun. For each of the planets and the Moon we are given several tables: the first lists a number of mean conjunctions with the Sun, where the radix is 1321 (1320 completed); the second is a double argument table (30 x 30) to find the true positions of the planet, or the Moon, at times between two successive conjunctions; the third displays the mean longitude and the mean argument of center of the planet; the fourth lists the equation of center (the correction to be applied to the mean argument of center to obtain the true argument of center); and the fifth is a reduced

5 Pedersen, Toledan Tables.

6 Poulle, "John of Murs," 133.

double argument table (16 x 7) for the latitude of the planet. For Mercury and Venus, there is another column for the 3rd component of latitude (see Tables 73 and 74, below). All the parameters used here will later be found in the Parisian Alfonsine Tables (e.g., the maximum equations of center for Jupiter and Venus are 5;57° and 2;10°, respectively, thus departing from the parameters used in the Toledan Tables). It is most noteworthy that the titles of all tables other than those with double argument indicate that they were computed for Toledo, adding that it is 0;48h distant from Paris, and use the radices given by King Alfonso of Castile.

(ii) In 1322 (a date that appears in the colophon of the manuscripts) John of Ligneres compiled another set of tables with canons in 44 chapters for the prime mover (Cujuslibet arcus propositi sinum rectum . . .), i.e., problems of trigonometry, the daily rotation, etc., and canons in 46 chapters for the motions of the planets and the determination of eclipses (Priores astrologi motus corporum celes-tium . . .). These canons have been edited, but not published.7 Both canons describe one set of tables computed for the meridian of Paris which have December 31, 1320 as epoch; these canons and tables are extant in many manuscripts and the latter are often referred to as 'Alfonsine tables'.

(iii) The Tabule magne by John of Ligneres were compiled in about 1325 and depend on the tables for 1322 by the same author, according to Poulle.8 They are associated with the incipit, Multiplicis philosophie variis radiis . . ., and seem to be extant in only a few MSS. Curiously enough, the tables in Erfurt, MS F.388, use zodiacal signs of 30° whereas those in Lisbon, MS Ajuda 52-XII-35, ff. 67r-92v, use physical signs of 60°. There are four types of tables. The first gives the daily mean motions for a year for Saturn, Jupiter, and Mars (mean motions in longitude and mean arguments of anomaly); mean arguments of anomaly for Venus and Mercury; mean motions of the Sun, Venus, and Mercury, and the lunar nodes; mean motions in longitudes and mean argument of anomaly of the Moon. The Erfurt MS gives radices for the planets without indicating the place for which they are valid, but the values in the text indicate that the place is Paris and the epoch, the Incarnation (i.e., Jan. 1, 1 A.D.).

7 Saby, Jean de Ligneres.

8 Poulle, "Alfonsine Tables and Alfonso X," 103.

The second displays the yearly mean motions of the above quantities from 1 year to 20 years (at intervals of 1y), from 20 years to 100 years (at intervals of 20y), and from 100 to 1000 years (at intervals of 100y). The third is for the planetary equations, presented as large double argument tables (31 x 60), and for the solar equation (with a maximum of 2;10°), giving a combined correction to be added or subtracted to the mean longitude of the planet to obtain its true position. The fourth presents the hourly true motion of the Moon. In the Erfurt MS the set continues with a voluminous table for the computation of the time from mean to true syzygy, known as Tabulae Permanentes, attributed to John of Murs.9

To these sets one should add the tables of John Vimond, an astronomer also working in Paris, who compiled tables with 1320 as epoch, for the use of students at the University of Paris.10 These tables form a coherent set with all the elements needed to compute the positions of the celestial bodies, and are prior to, and independent of, the tables compiled in the early 14th century, which we call the Parisian Alfonsine Tables, also based on Castilian sources. Vimond's tables and the Parisian Alfonsine Tables have many parameters in common both for mean motions and equations.

These sets, and possibly those by other astronomers, gave rise to the Parisian Alfonsine Tables. This set underwent significant developments from the mid-14th century to the time of Bianchini, and at least three different adaptations were produced in three different countries.

(i) The Oxford Tables of 1348, often called Tabule anglicane, have been ascribed to William Batecombe and are associated with the canons beginning Vera locum omnium planetarum in longitu-

dine____11 The tables for mean motions give the mean argument of center rather than the mean longitude of the planet. There are also double argument tables for the planets, as was the case with the Tabule magne by John of Ligneres and, according to North, this set was indeed the source for the double argument tables compiled in Oxford. In this case the entries are also given as a

9 Chabas and Porres, "True Syzygies."

10 Chabas and Goldstein, "Tables of John Vimond."

11 North, "Alfonsine Tables in England."

function of the mean argument of center and the mean argument of anomaly, but the procedure to compute the true position of the planet is made easier than in the Tabule magne.12 To illustrate the size of the double argument tables for the planets, it is perhaps sufficient to say that in the manuscripts they fill about 66 pages crowded with entries of about 39,600 numbers of one or two digits.

(ii) Around 1424 the Paduan astronomer Prosdocimo de' Beldomandi wrote canons and tables, based on Jacopo de Dondi's tables which, in turn, depended on the Parisian Alfonsine Tables.13 The associated canons begin with Facta et ordinata sunt quam plura et varia paria tabularum ad celestes motus, and the tables contain all that is needed to compute positions of the celestial bodies. They also include a catalogue of more than 1,000 stars which served as the basis for the star catalogues published in the incunabula editions of the Parisian Alfonsine Tables. The presentation of Prosdocimo's mean motion tables follows the pattern in the Toledan Tables and the Tables of Novara (a version of the Toledan Tables adapted to the Christian calendar), using groupings of 28 years, rather than the presentation in the editio princeps of the Alfonsine Tables (1483), where one finds tables with multiples from 1 to 60 of the basic parameter (i.e., the daily mean motion) in each case (to be used with time intervals in days arranged in strict sexagesimal notation, rather than in years, months, days, and hours). On the other hand, Prosdocimo's tables for planetary latitudes with 22 columns are peculiar and unprecedented.

(iii) The Austrian astronomer, John of Gmunden (c. 1380-1442), compiled several versions of a set of astronomical tables for which he also wrote canons, and they were intended to accompany his lectures at the University of Vienna from 1419 onwards.14 He collected many tables from his predecessors in the Alfonsine tradition and his set of astronomical tables (90 tables for the planets and the luminaries) is probably the most complete until that time. John of Gmunden reproduced earlier tables, adapted others, and modified the format of still others, but in all of them he was faithful to Alfonsine astronomy and did not depart from its parameters and

12 North, "Alfonsine Tables in England," 278-82.

13 Chabas, "Prosdocimo de' Beldomandi."

14 Porres, Jean de Gmunden.

procedures. As is the case with many other sets of tables mentioned above, John of Gmunden's tables were never printed, but they were widely diffused in Central Europe for about a century.

Although these sets are not identical either in the number of tables they contain or in the format of some particular tables, they are organized in a way that is similar to the Parisian Alfonsine Tables. This was the state of the Alfonsine material in about 1442, when Giovanni Bianchini completed his voluminous set of tables. As we shall see, his set depends directly on the Alfonsine tradition, but differs from all previous sets in various crucial ways: the tables for the planets and the luminaries have a consistent format based on an internal organizing principle different from other sets of tables (see Tables 17, 27, 34, 41, 48, 56, 63, and 64, below), and his tables contain the largest number of entries ever computed in the Alfonsine tradition.

Bianchini did not use a single epoch in his tables, but a variety of 'intermediate' epochs from which all quantities involved in a computation are counted. So, for the Moon Bianchini chose to refer all quantities related to its motion to its mean anomaly, counted from the beginning of the most recent anomalistic month, whatever the date on which this occurs. Analogously for the planets, the intermediate epoch for each of them is the beginning of their respective most recent periods of anomaly. In the case of syzygies, all quantities are counted from the beginning of the most recent synodic month. This is a characteristic feature of Bianchini's tables, an arrangement which differs substantially from other sets of tables, even those sharing the same models and parameters, such as the Parisian Alfonsine Tables. In order to link the values obtained through the use of this scheme with dates expressed in years, months, and days, Bianchini introduced tables for computing a set of radices whose entries are associated with specific moments traditionally used by astronomers, or dates in his own time, that have to be understood as the time intervals between the beginning of an anomalistic month or the beginning of a period of planetary anomaly and the date in question. In the case of a mean synodic month, one has to compute the time interval between the beginning of a specific mean synodic month and the Incarnation. After finding these epochs, one then has the arguments for entering double argument tables (that require interpolation) from which one obtains the quantities to be added to the mean quantities at the intermediate epoch. The whole construction reflects a deep astronomical insight, but has the disadvantage of being unfamiliar and complex, which probably made some practitioners of astronomy avoid these tables, given the relative simplicity of the Parisian Alfonsine Tables and other adaptations of them.

Probably due to their substantial size and complexity, the tables of Bianchini were not copied very often in manuscript, but frequently enough to suggest to the printer that there was a market for them. And so they were published in 1495 in Venice for the first time. In the meantime, they coexisted with at least two other sets of tables that also addressed all problems related to the motions of the planets and the luminaries: the Tabulae resolutae, compiled and originally diffused in central Europe (first edited by A. Lacher, and published in Frankfurt in 1511 under the title Tabulae resolutae de motibus planetarum aliorumque super celestium mobilium), and the set of tables compiled by Abraham Zacut in Spain (first published in Leiria, Portugal, in 1496). Both of them presented the Alfonsine material in their own special way.

The first version of the Tabulae resolutae seems to be the work of Petrus Cruciferus, who compiled them for the meridian of Wroclaw with 1424 as epoch. Various Polish astronomers, including Marcin Krol, Andrzej Grzymala, Albert of Brudzewo, and John of Glogovia expanded them, resulting in a set of tables in the Alfonsine tradition that had a wide diffusion in manuscript in the 15th century and then in print in the 16th century.15 A typical set of the Tabulae resolutae contain tables for the radices and mean motions of 12 quantities: the apogees and the fixed stars; access and recess of the 8th sphere; Sun, Venus, and Mercury; Moon; argument of the Moon; argument of latitude of the Moon; lunar node; Mars; Jupiter; Saturn; argument of Venus; and argument of Mercury. The entries are arranged according to a system of cyclical radices ad annos collectos at 20-year intervals. There are also tables for mean syzygies, listing four variables: time of mean syzygy, mean motion of the Moon, mean argument of the Moon, and mean argument of latitude of the Moon. Then follow tables for the positions and motions of the apogees of the planets, also tabulated at intervals of 20 years; tables for the equations; interpolation tables; tables for the daily motions of the Sun and the Moon, the equation of time, and the planetary stations and retrogradations; and tables for the rising times. There are no tables with double arguments in any of these versions.

15 Dobrzycki, "The Tabulae Resolutae"; Chabas, "Astronomy in Salamanca"; Chabas, "Diffusion of the Alfonsine Tables."

The Tabulae resolutae do not present any innovation in the parameters underlying the tables, for all are strictly in the Alfonsine tradition, but the material is presented differently from that in previous tables, possibly making them easier to use. They were printed several times—even as late as 1588—thus competing for a few decades with the Prutenic Tables, first printed in 1551, based on Copernicus's theories.16

The Great Composition (ha-Hibbur ha-gadol) is the set of tables compiled in Hebrew by Abraham Zacut (1452-1515) in Salamanca.17 It consists of about 65 tables and lengthy canons explaining their use. The tables have 1473 as epoch, they are arranged for the Christian calendar, and their entries are computed for the meridian of Salamanca. The work was finished around 1478 and three years later it was translated from Hebrew to Castilian. In the Hibbur there are some double argument tables, in particular for the latitudes of the five planets and for the unequal motion of Mercury. Zacut adheres to the Alfonsine tradition, which came to Salamanca in the form of the Tabulae resolutae, as well as to the astronomical tradition in Hebrew, which included works by Levi ben Gerson, Jacob ben David Bonjorn, and Judah ben Asher II (all 14th century). Two versions of Zacut's Hibbur (one with canons in Latin and the other with canons in Castilian), edited by the Portuguese scholar Joseph Vizinus, were published in 1496 in Leiria, Portugal, under the title Tabulae tabularum coelestium motuum sive Almanach Perpetuum. Shortly thereafter, in 1498, this work was reprinted in Venice, with three more editions appearing during the 16th century.

Thus, in the early years after the invention of printing with movable type, the Alfonsine corpus was well established in at least seven European countries, with no other set of tables to compete seriously with it. Printing definitely accelerated the diffusion of the various presentations of the Alfonsine Tables throughout Europe, but the first set of tables to be published was a form of the Parisian Alfonsine Tables. Before 1500 there were two editions which differ in some important ways from one another, printed in the same city:

1483 - Tabule astronomice illustrissimi Alfontij regis castelle. Venice: Erhard Ratdolt;

16 Reinhold, Prutenicae tabulae coelestium motuum.

17 Chabas and Goldstein, Abraham Zacut.

1492 - Tabule Astronomice Alfonsi Regis. Venice: Johannes Lucilius San-tritter.

Then followed the first edition of Bianchini's tables, also printed in Venice:

1495 - Giovanni Bianchini. Tabulae astronomiae. Venice: Simon Bevi-

laqua.

Before the turn of the century, however, another set of tables, the Almanack Perpetuum, came out in two editions, as mentioned above:

1496 - Abraham Zacut. Almanack Perpetuum. Leiria: Samuel d'Ortas;

1498 - Abraham Zacut. Epkemerides sive Almanack Perpetuum. Venice:

The editio princeps (1483) of the Parisian Alfonsine Tables devotes 120 pages to numerical tables, excluding the star catalogue. In these pages we have counted more than 51,000 numbers of one or two digits, of which 41,000 have been computed, and about 10,000 are for the arguments in the tables (i.e., belonging to sets of consecutive numbers). The tables authored by Bianchini fill 633 pages in the edition of 1495 and contain about 315,000 numbers of one or two digits, that is, more than 6 times the amount in the Parisian Alfonsine Tables. Of these, 300,000 have been computed (more than 7 times the amount in the Parisian Alfonsine Tables), and about 15,000 are for the arguments. It should be noted that the second edition of Bianchini's tables (1526) enlarged the number of tables from 68 to 111, thus considerably increasing this huge undertaking of printing numbers.

Bianchini was not the only one in the 15th century to produce planetary tables with a vast quantity of entries to be published later in the century by courageous printers. In the Almanack Perpetuum, printed in Leiria in 1496, with 306 pages of numerical tables (excluding the star list), we have counted more than 201,000 numbers, of which 187,000 have been computed (4.5 times the amount in the Parisian Alfonsine Tables), and about 14,000 are for the arguments. It is thus clear that Bianchini and, to a lesser extent, Zacut made enormous efforts to provide their readers with a lot of precise numerical information presented as astronomical tables that could help them compute accurately the positions of the luminaries and the five planets. In rendering this task easier, they surely increased the number of practitioners of astronomy, a purpose which was greatly facilitated by the use of printing (or even inconceivable without it).

CHAPTER ONE GIOVANNI BIANCHINI: LIFE AND WORK

What little is known about the life of Giovanni Bianchini (Latin: Iohannes Blanchinus) was reported by G. Federici Vescovini (1968). Bianchini was probably born in the first decade of the 15th century, and he died sometime after 1469. His family came from Florence, but his father, Amerigo, was established in Bologna by 1400. Bianchini was a merchant in Venice until 1427 and then worked for Nicolo d'Este (1383-1441), Marquis d'Este, Signore of Ferrara, Modena, Parma, and Reggio. Bianchini spent most of his life in Ferrara where, for about three decades, he served as administrator of the estate of the prominent d'Este family, first for Nicolo, and then for Leonello (1407-1450) and Borso (1413-1471). Bianchini also taught at the University of Ferrara,1 and it has been established that he visited other Italian cities including Milan, Venice, Bologna, and Rome.2 The date of his death is uncertain. It is known, however, that he was buried at Saint Paul's Church in Ferrara, which was destroyed by an earthquake in 1570.

Bianchini's scientific writings, all of which deal with astronomy and mathematics, were composed between 1440 and 1460. In 1463-1464 he corresponded with Regiomontanus (1436-1476). The five extant letters (two from Bianchini and three from Regiomontanus) have been edited repeatedly and much has been written about them.3 The letters mainly concern astronomical and mathematical problems and their solutions. The first extant letter is dated July 27, 1463: it was written by Regiomontanus, then in Venice, in reply to a letter from Bianchini in late June 1463 that does not survive. At that time Regiomontanus was in Venice accompanying Cardinal Johannes Bessarion (1395-1472) and had intended to visit Bianchini in Ferrara shortly before his arrival there, but the rumor that plague was ravaging Ferrara prevented him

1 Thorndike, "Bianchini in Paris Manuscripts," 5.

2 Magrini, "Joannes de Blanchinis Ferrariensis," 8.

3 Murr, Memorabilia; Curtze, "Briefwechsel Regiomontan's"; Gerl, Briefwechsel Regiomontanus-Bianchini; Zinner, Regiomontanus, his Life and Work, 60-69; Swerdlow, "Regiomontanus on Critical Problems."

from meeting Bianchini. Most historians think it unlikely that Bianchini and Regiomontanus ever met.

In this monograph we focus on Bianchini's best known work, his Tabulae astronomiae (1495), consisting of astronomical tables and a set of canons explaining their use. We will demonstrate that these tables depend on the Alfonsine Tables that were later printed in Venice in 1483 and 1492 (with extant manuscripts as early as the 14th century), but Bianchini added many special features which gave his tables a completely different presentation from that of the standard Alfonsine tables. The work was compiled in Ferrara and addressed to his patron, Leonello d'Este, probably in 1442.4 There follows a list of the manuscripts, with the sigla we assign to those that we cite, that are known to include this set of tables (or parts of it):

Cracow, Biblioteka Jagiellonska, MSS 555, 557, 603, and 606

Ferrara, Biblioteca Comunale Ariostea, MS I.147

Florence, Biblioteca Laurenziana, MS Pl. 29,33

Milan, Biblioteca Ambrosiana, MSS C. 207 inf. and C. 278 inf.

Naples, Biblioteca Nazionale, MSS VIII.C.34 [Na] and VIII.C.36 (only a few tables) Nuremberg, Stadtbibliothek, Cent V 57 [Nu] Oxford, Bodleian Library, MS Canon. Misc. 454

Paris, Bibliothèque nationale de France, MSS 7269, 7270, 7271, and 16212

Rome, Biblioteca Casanatense, MS 1673 [ Rc] Vatican, Biblioteca Apostolica, MS Pal. lat. 1375 [Va] Venice, Museo Correr, MS Cicogna 3748

Vienna, Österreichischer Nationalbibliothek, MSS 2293 and 5299

Va, ff. 1ra-8va, contain the Tables of Bianchini, heavily annotated by Johannes Virdung of Hassurt, as indicated on f. 1r. The tables themselves occupy ff. 185r-262v, and on their first page we are told that they were copied in Cracow in 1488 from June 12 to July 20.

In a letter dated 1456 to Johann Nihil, court astrologer to Emperor Frederick III, Georg Peurbach (1423-1461) described the calculation of ephemerides he had computed, together with Regiomontanus, for which they used Bianchini's tables.5 While in Vienna in 1460 Regiomon-tanus made a copy of these tables for himself and wrote an abridged

4 For the contents of some of the manuscripts that contain this work, see Boffito, "Tavole astronomiche di Bianchini"; Thorndike, "Bianchini in Paris Manuscripts"; Thorndike, "Bianchini in Italian Manuscripts"; and Rosinska, Scientific Writings.

5 Hellman and Swerdlow, "Peurbach (or Peuerbach), Georg," 474.

set of canons for them entitled, Canones breviati in tabulas Ioannis de Blanchinis, beginning with Augem planetarum comunem invenire (Nu, ff 5r-19v).6

In addition to the edito princeps of 1495, noted above, two editions of the canons and the tables appeared in the 16th century:

1526 - Tabule Joa[nni] Blanchini Bononiensis. Venice: Luca Antonio Giunta;

1553 - Luminarium atqueplanetarum motuum tabulae octoginta quinque. Basel: Joannes Hervagium.

In the edition of 1553 the authors are given as Giovanni Bianchini, Nicholaus Prugner, and Georg Peurbach. Indeed this volume contains two different works: Peurbach's Tabulae eclipsium consisting of canons and tables, and the Tables of Bianchini, edited by Prugner, which include a considerable number of tables compiled by the editor himself. The two earlier editions only contain Bianchini's work.

The book has two dedications, one to Leonello d'Este (d. 1450), opening with Consideranti mihi, dive Leonelle, . . ., and a later one addressed to Frederick III, Holy Roman Emperor (reigned: 1452-1493), beginning with Cum nuper maiestas tua, Serenissime Cesar . . . According to Magrini,7 the dedication to Emperor Frederick was suggested by Bianchini's patron at the time, Borso, and was presented together with the tables to the Emperor on the occasion of his visit to Ferrara in January 1452. A splendid picture of that ceremony in the form of a miniature, ascribed to Giorgio d'Alemagna,8 in a manuscript currently in Ferrara shows Bianchini kneeling before the Emperor, handing him his set of tables, and receiving from him his coat of arms. He is accompanied by Borso d'Este (see Frontispiece, opposite the title page).

Prior to the dedications in the first two printed editions, we find an encomium in praise of Bianchini's book that was written by Augustinus Moravus in January 1495 in Padua. In these editions the tables are preceded by canons consisting of an Introduction and 51 chapters. In contrast, MSS Na, Rc, and Va only include the dedication to Leonello d'Este, after which Bianchini's canons begin, consisting of an Introduc

6 For the dependence of various tables printed by Regiomontanus on Bianchini's Tabulae astronomiae, see Rosinska, "'Fifteenth-Century Roots' of Modern Mathematics", 67, n. 16.

7 Magrini, "Joannes de Blanchinis Ferrariensis," 10.

8 Medica, "Miniato da Giorgio d'Alemagna."

tion and 39 chapters (numbered consecutively in Na, but unnumbered in Va) or 53 chapters in two parts of 41 and 12 chapters in Rc. The folios in Na were not properly bound, with the result that the introduction is found on f. 1r and f. 11r-v, for folios 2r-v and 11r-v have been interchanged.

The Introduction begins with references to Ptolemy and the Almagest: "Ptholomeus qui merito illuminator divine artis astrologie vocari potest, in suo libro Almagesti . . .". Several scholars other than Ptolemy are mentioned (Hipparchus, Thabit ibn Qurra, al-Battanl, the compilers of the Toledan Tables, and Alfonso X, among others), but we note that none of them is later than Alfonso (d. 1284). The Introduction focuses on two astronomical matters, precession/trepidation and the latitude of the planets. Bianchini praises the work of Alfonso and, as will be seen below, in the first table in his treatise he addresses the problem of precession/trepidation strictly in accordance with the approach taken in the Alfonsine corpus. As for the planetary latitudes, Bianchini indicates that he compiled tables following the instructions given by Ptolemy in Book XIII of the Almagest, which is indeed the case, to overcome the "significant discrepancy from the truth, especially for Venus and Mercury," found in other sets of tables.9 The Introduction closes with some basic information helpful to the reader when using his tables: years are 365;15 days; the beginning of the year is March 1; the epoch is the Incarnation; physical signs of 60° are used, as in the Parisian Alfonsine Tables; computed examples and some tables are for Ferrara whose geographical coordinates are longitude 32° and latitude 45°; and motions are referred to the 9th sphere (i.e., the coordinates are tropical).

The canons, that is, the Introduction followed by 51 chapters, are the same in both editions (1495 and 1526), but for minor variants: in ed. 1495, Chapters 1, 2, and 3 are not numbered, and in ed. 1526 the chapter following 50 is numbered 49. The chapters in ed. 1495 are the following:

[1.] De modo operandi per tabulas Joannis Blanchini generaliter ad quemcumque meridianum volueris. [2.] Ad sciendum numerum dierum a principio anni ad quemcumque diem cuiusque mensis sequentis [3.] Locum augium comunium octave spere in nona invenire

9 Bianchini, Tabulae astronomiae (1495), a4v.

4. Ad inveniendum locum augis cuiuslibet planete

5. Medium motum solis per tabulas invenire volueris

6. Applicationem solis ad eius augem invenire

7. Ad idem per alias tabulas

8. Per augem solis auges aliorum planetarum invenire

9. Verum locum solis invenire

10. Introitum solis in ariete seu in alio signo vel gradu examinare

11. De examinatione cursus lune

12. Latitudinem lune per tabulas invenire

13. Verum locum capitis et caude draconis invenire

14. Vera loca planetarum trium scilicet superiorum, Veneris etiam atque Mercurii per tabulas invenire

15. De latitudine planetarum

16. De elongatione planetarum a sole et e converse

17. Radices Christi cuiuslibet planete ad quodlibet aliud tempus extendere

18. Exemplum ad inveniendum verum locum planetarum in longitudine et latitudine atque eorum distantiam a sole

19. Utrum planeta fuerit stationarius directus sive retrogradus

20. Utrum planeta fuerit orientalis sue occidentalis a sole

21. Tempus vere coniunctionis et oppositionis luminarium per tabulas Jo. Blan. invenire

22. Duodecim coniunctiones immediate sequentes faciliter invenire

23. Verum locum solis per totum annum velociter extendere

24. Verum locum lune per totum annum invenire

25. Vera loca trium superiorum, Veneris etiam atque Mercurii continuando extendere

26. Ad reducendum tempus calculi facti diebus non equatis ad dies equatos et postmodum ad horas horologii initius sexti climatis ad meridianum Ferrarie

27. Gradum ascendentem et subsequenter figuram duodecim domorum celi per tabulas Joannis Blanchini erigere

28. Nota admirabilem operationem per locum solis loca aliorum planetarum et eorum latitudinem per tabulas Jo. Blanchini invenire

29. Planetarum calculum per tabulas Jo. Bla. ad Alphonsii regulas reducere

30. De tribus inferioribus

31. Medium motum et argumentum solis per tabulas cuiusvis planetarum superiorum, Veneris quoquem et Mercurii invenire

32. Latitudinem planetarum et eorum centra et argumenta equata per tabulas invenire

33. De latitudine trium superiorum

34. De latitudine Veneris et Mercurii

35. De reformatione tabularum radicum planetarum

36. De examinatione tabularum lune

37. De examinatione tabularum trium superiorum, Veneris quoque et Mercurii

38. De elongatione planete ad solem et e converso

39. Ad inveniendum precise dies et horas stationum planetarum

40. Tempus revolutionum annorum mundi seu nativitatum ac etiam cuiuscumque alius principii punctaliter reperire

41. Gradum ascendentem et ceterarum domorum cuspides tam in radice alicuius principii quam in revolutione suorum annorum perscrutari

42. Vera loca planetarum in quacumque revolutione velociter indagare

43. Verum locum capitis in revolutionibus annorum invenire

44. Feriam cuiuscumque mensis latinorum literam dominicalem aureum numerum et indictionem in quolibet anno a nativitate Christi indagare

45. De festis mobilius inveniendis

46. De generali doctrina operationis tabularum Jo. Blan. ad quem-cumque calculum volueris in motibus planetarum que dicitur corona tabule

47. Utrum planeta sit auctus vel diminutus numero aut per calculum

48. Utrum planeta sit ascendens vel descendens in circulo sue augis

49. Exemplum de universali atque oportuno calculo tam in electioni-bus quam in interrogationibus nativitatibus annorum revolutioni-bus ac etiam de applicatione alicuius planete in aliquod signum vel gradum signi, que omnia ex compositione tabularum aperte demonstrantur

50. Moram nati in utero materno atque verum gradum ascendentem cuiuscumque nativitatis per tabulas Jo. Blan. perscrutari

51. De regulis multiplicandi atque dividendi in calculo

In the edition of 1526 the editor, Luca Gaurico, added 8 short paragraphs after Chapter 51. The title of the first is "Verum ascendentis gradum per earum seminis rectificare secundum Jacobum Dundum patavinum", apparently based on a previous work by the astronomer

Jacopo de Dondi of Padua (1298-1359), and that of the eighth, "Lati-tudinem 5 errantum supputare". On the other hand, ed. 1553 has only the first 18 chapters.

The tables come after the canons and have the same general title in the editions of 1495 and 1526, and a similar title is found in ed. 1553: Tabule Ethereorum Motuum Secundi videlicet mobilis: Luminarum at Planetarum viri perspicacissimi Joannis Blanchini. Omnium ex his que Alfonsum Sequuntur quem facillime. Sidere felici Incipiunt. Bianchini uses physical signs of 60° throughout this work but, curiously enough, not for the lunar anomaly, which he systematically gives in degrees from 0° to 360°.

In addition to his Tabulae astronomiae, Bianchini wrote a few other treatises:

1. Compositio instrumenti. This short treatise, dated 1442, addresses the construction and use of an instrument, called biffa, to determine the altitudes of the stars, and in it Bianchini explains the meaning and use of the decimal point. The text begins Primo composui duas figuras (rigas) equalis longitudinis . . ., and has been published.10

2. Canones tabularum super primo mobile. This treatise includes both text and tables devoted to spherical trigonometry. The text begins with Non veni solvere legem sed illuminare his qui in tenebris sedent . . . and gives two basic parameters: 40;45,4° for the latitude of Ferrara and 23;30,30° for the obliquity of the ecliptic.

3. Flores Almagesti. Bianchini's largest work, composed between 1440 and 1455, consists of 8, 9, or 10 treatises (the number of treatises varies in different manuscript traditions). The first treatise, Arithmetica, beginning Arithmetica dico quod determinatur per numeros . . ., deals with theoretical arithmetic and the resolution of numerical problems, and the second, called Arithmetica algebrae or De algebra, is devoted to the resolution of quadratic equations.11 They were written around 1440 in Ferrara and, together with the third treatise, De proportioni-bus, serve as a mathematical introduction to Bianchini's astronomy.12 The other treatises are directly concerned with astronomical matters and follow Ptolemy's Almagest, but do not go beyond Almagest VI.

10 Thorndike and Kibre, Catalogue of incipits, col. 1099; Garuti, "Compositio instrumenti."

11 Rosinska, "Euclidean spatium," 29.

12 Rosinska, "Bianchini's De Algebra."

4. Canones tabularum de eclipsibus luminarium. In this treatise, written between 1456 and 1460, Bianchini reports observations that he made of four lunar eclipses that took place in 1440, 1448, 1451, and 1455,13 as well as the computation of a solar eclipse to be observed at Ferrara in July 1460.14 The incipit is: In libro florum Almagesti per me Ioannem blanchinum demostratum est. . . .

5. Tabulas magistrales. This is a set of 6 or 7 tables, depending on the manuscripts, arranged in two groups. Among them are high precision tables for tangents and cosecants, where Bianchini abandons sexagesimal notation, replacing it with decimal notation, that is, defining the radius of the circle as R = 10n (with n = 3 for tangents and n = 4 for cosecants). Bianchini was the first mathematician in the West to use purely decimal tables of trigonometric functions, soon followed by Regiomontanus.15

In spite of their unusual presentation and lack of user-friendliness, Bianchini's tables were appreciated and used by quite a number of astronomers. To the names of his contemporaries, Peurbach and Regiomontanus, who have already been mentioned, we may add that of Alessandro Borromei (c. 1463), doctor of arts and medicine, who lived in Venice.16 Moreover, the copy of Bianchini's tables owned and annotated by Johannes Virdung of Hasfurt is now preserved in MS Va. In Cracow, Bianchini's work seems to have been especially well regarded, for several manuscripts preserve copies of his tables while others give evidence that they were used. At least two of the manuscripts were copied by Ioannes Zmora de Lesnicz, one in Perugia in 1453 (Cracow, MS 555) and the other in Cracow in 1456 (Cracow, MS 557). In another manuscript at the Biblioteka Jagiellonska, MS 1841, we find a long text (ff. 9-65) beginning Volo invenire verus motus solis pro meridiano cracoviensi ex Blankini tabulas, in 20 chapters explaining the use of Bianchini's tables and giving examples of computations for July 1447.17 Moreover, Biblioteka Jagiellonska, MS 2480, f. 140, contains tables based on those of Bianchini and computed for 1459; and MS 2478, ff. 96-105, contains Proposiciones in Tabulas magistri Iohannis Blankijni.

13 Thorndike, "Bianchini in Paris Manuscripts," 170.

14 Magrini, "Joannes de Blanchinis Ferrariensis," 25.

15 Rosinska, "Universities in Copernicus' Time."

16 Magrini, "Joannes de Blanchinis Ferrariensis," 16.

17 See also Rosinska, Scientific Writings, 457.

As noted by Rosinska: "In Cracow, Bianchini's mathematical (trigonometric) and astronomical tables were systematically copied, rearranged, and adapted to Cracow meridian, to begin with Martinus Rex' students and to finish with Copernicus, and with his younger colleagues in Cra-cow."18 On the other hand, the edition of 1526 we have consulted at the Library of the University of Barcelona indicates on its front page that this copy belonged to the Franciscan monastery of Barcelona "for the use of Jo. Salom", most probably the Franciscan priest who published a proposal in 1576 addressed to Pope Gregory XIII for correcting the Roman calendar.19 There was also some diffusion of Bianchini's canons and tables in Hebrew.20

In the preceding paragraph, we have not attempted to provide a comprehensive study of the reception of Bianchini's tables; rather, our goal has only been to demonstrate that his work was not neglected by his contemporaries and immediate successors.

18 Rosinska, "Universities in Copernicus' Time," 10.

19 Ziggelaar, "Papal Bull of 1582," 205-6.

20 See, e.g., Steinschneider, Die hebraeischen Uebersetzungen, 626-28; Steinschneider, Die hebraeischen Handschriften in Muenchen, 17; and Chabas and Goldstein, Abraham Zacut, 22.

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