Dr. Kim H. Veltman

1. Introduction

2. Arithmetic

3. Geometry

4. Perspective and Surveying

5. Astrology

6. Astronomy

7. Tables

8. Instruments

9. Astronomy, Geography and Cosmology

10. Conclusions

Historians of science frequently look back at the Renaissance in terms of isolated events and books that changed the course of early modern science. In this approach the period 1520-1560 is notable mainly for three works published in the year 1543: On the revolutions of the heavens by Nicholas Copernicus, On the fabric of the human body by Andreas Vesalius and the first vernacular edition of Euclid's Elements by Niccolo Tartaglia. There is also the widespread assumption that after Gutenberg every important idea was immediately published and hence only printed texts are significant. The story is not so simple. In 1471 Johannes M�ller (Regiomontanus) established the first printing press devoted specifically to the publication of scientific books About 1472 he issued a sheet 30x23 cm listing which books he planned to publish. These included the works of Euclid and Archimedes, the Conics of Apollonius, Serenus On the Cylinder, Ptolemy's Almagest, Geography, Music and Optics, Hero of Alexandria's Hydraulics, the Spherics of Menelaus and Theodosius, Astronomy of Hyginus, Arithmetic of Jordanus, the Optics of Witelo, plus a series of commentaries and books written by himself. When Regiomontanus died unexpectedly in 1474, his work was inherited first by Werner (1468-1528), then by Hartmann (1489-1564). As a result a number of the classics on his list were first published in the sixteenth century: e.g. the Almagest (1515), Witelo (1535), Hyginus (1535), Archimedes (1544).

Publication of both ancient and modern scientific texts progressed slowly. The half century between 1470 and 1520 saw some basic mediaeval works such as the Sphere of Sacrobosco 1499) and the Optics of Peckham (1482,1504); as well as modern works on astronomy, Stoffler (1514) and Lefevre d'Etaples; perspective, P‚lerin (1505,1509) and practical arithmetic, including Cirvelo (1505), Lefevre (1510,1514), K"bel (1514), Bonini (1517) and Martini (1519). But it was not until the period 1520-1560 that a recognizable corpus of scientific literature emerged in published form. This included early both early mediaeval texts such as Proclus' Two books on motion (1542) and late mediaeval manuscripts such as Saint Hildegard of Bingen's Physics (1533). As we are concerned specifically with the rise of systematic quantitative literature only passing reference will be made to isolated contributions in music by Walther (1538) or navigation by Saa (1549), or even to important advances in the life sciences such as botany, e.g. Brunfels (1533) and anatomy, Vesalius (1543,1545,1555), Estienne (1546), Columbus and Valverde (1559). Some mention must, however, be made of literature on weights and measures such as Agricola's Five books on weights and measures (1533), a second edition of which (1550) also contained a work by the lawyer Alciati, of emblem fame, on the topic of weights and measures. Also important in this context are The judgment of medical weights by Asculanus appended to Brunfel's Onomastikon of medicine (1534), Pasi's Tariff of corresponding weights and measures from the east to the west, from one country and place to another (1540); Cenalis' On the measures of liquids and pulses. On the truth of measures and weights (1546); a Synopsis of weights and measures by Neander (1555), Marheld's work on the weight and price of silver (1556) and Rudolff's study of comparative weights, lengths and coinage in different towns and countries (1557). The incentives for these interests were various. One was obviously linked with trade and commerce. Medicine, and pharmacy in particular, which required accurate doses, provided another incentive. Agricola, for instance, was a town physician. In his spare time he also visited local mines and smelters which led to his classic study (1546,1557,1558) of weights and measures in connection with the emerging earth sciences, particularly metallurgy and mineralogy, a topic also treated by Biringuccio (1540,1556,1558) and Entzelt (1551,1557).

Our main purpose however is to focus on a phenomenon, whereby authors on traditionally abstract subjects such as arithmetic and geometry became increasingly concerned with practical topics involving measurement both on earth (notably perspective, surveying, geography) and in the heavens (astrology and astronomy). This created a new mathematical framework for a theoretical and practical treatment of nature amenable to verification by means of instruments. Without this the explosion of measuring devices and literature on quantification in the period 1560-1600 would have been unthinkable as would the subsequent contributions towards synthesis by Galileo, Descartes, Huygens and Newton in the seventeenth century frequently associated as the key moments in early modern science.

With respect to arithmetic the period 1520-1560 brought at least 65 publications. In 1494 Luca Pacioli had published his great Summa of arithmetic, geometry, proportion and proportionality. This was reprinted in 1525 and remained a basic source. Adam Riese's classic work on arithmetic (1525) appeared together with Erhard Helm's book on gauging, i.e. the problem of volumetric measurement of wine barrels using rods with square roots or cube roots. Riese went through various editions (e.g. 1531 Erfurt; 1544 and 1550 Leipzig, the latter again with a section on gauging). K"bel's Reckoning book (1544, 1549) also combined arithmetic, geometry and gauging. Meanwhile with authors such as Frey (1543) and Helmreich (1561) gauging also emerged as an independent topic of publication.

The authors of this literature on arithmetic were often also engaged in a wide range of practical applications. For instance, Peter Apian, who was based at Ingolstadt, and wrote A new...instruction in all merchants' reckoning (1532,1543); also published on geography (1529, 1553), surveying and astronomical instruments (1532, 1533), astronomy (1533,1539,1540) and sundials (1541). Gemma Frisius, who wrote an Easy method of practical arithmetic (1544), was the teacher of Mercator at Louvain, edited Apianus' work on cosmography (e.g.1539), published a basic text on surveying and cartography (1533) and also wrote on astronomy and geography (e.g. 1530, 1553) and new instruments (e.g. 1545). In Paris, Oronce Fin‚, who produced a treatise on practical arithmetic (1555) also published on geometry (1544), astronomy (1534, 1538,1553) as well as astronomy, geography and hydrography together (1555). A series of centres also emerged: chiefly Frankfurt, N�rnberg, Leipzig, Paris and Venice. Other centres, notably Basel, Strasbourg and Wittemberg (e.g Albert 1544, Ammon 1544 with a preface by Melanchthon; Gemma Frisius 1544 1548 1556 and Medler 1550), also played an important part in the early history of protestantism. Many of these publications on arithmetic also involved geometry.

In geometry the period 1520-1560 was partly concerned with recovery and spread of the classics. There were six editions of Euclid's Elements (1533 1537 1550 Basel; 1536 Wiiemberg; 1548 Frankfurt and 1555 Augsburg). There were also editions of Archimedes (1544 Basel, 1558 Venice) and Psellus (1558 Paris). Geometry also acquired a particular meaning. In Greek geometry literally means "measurement of the earth", but the Greeks themselves had tended to keep a sharp distinction between theoretical and practical geometry. From the time of Boethius onwards this literal meaning had gained increasing acceptance and by the sixteenth century practical geometry was frequently equated with geometry itself. Hence D�rer's basic textbook in geometry was entitled Instruction in measurement (1525). A decade later Jakob K"bel published Geometry. On artful measuring and recording of all heights, surfaces, planes, widths and breadths such as towers, churches, buildings, trees, fields and lands (1535). A slightly more abstract approach was taken by Wolfgang Schmid in The first book of geometry. A brief instruction of what and whereupon geometry is based and how one can with the aid of the same, using a ruler and compass divide all lines, surfaces and bodies in a given proportion (1539). Even so the emphasis was now clearly on three dimensional aspects of geometry. Oronce Fin‚, who founded a chair of mathematics at Paris pursued this in his Book of practical geometry or on the practices of lengths, planes and solids, that is, the measurement of lines, surfaces and bodies, and other mechanical matters which follow as a corollary from the demonstrations of Euclid's Elements (1544). This served as a starting point for Pierre de la Ram‚e (Ramus), which explains why Fin‚'s successor included surveying and problems of volumetric measurement in his treatise on geometry.

K"bel's work was reprinted in 1556, a year which also saw the publication of the first two parts of Niccolo Tartaglia's General treatise on numbers and measures (1556-1560). Parts three to six followed in 1560, including, Part four In which the great majority of figures, both superficial and corporeal are reduced to numbers; Part five In which is shown the means of executing with the compass and the ruler all the propositions of Euclid and other philosophers and part six which dealt with algebra. By this time other trends were evident. The Greeks had made a clear distinction between geometry (continuous quantity) and arithmetic (discrete quantity). The equation of geometry and measurement meant that geometrical lines, surfaces and solids could now be treated in terms of arithmetical numbers, which prepared the way for Descartes new synthesis in the form of analytic geometry some 70 years later. It also meant that instruments and models could be used both for recording and demonstrating geometrical and arithmetical problems. The increased interplay between practical and theoretical thus forged a new nexus between mathematics, instrumentation and model making. By implication, surveying, geography, astronomy and cosmology were no longer problems for intellectual verbal debate. They involved a challenge of visual, mechanical, instrumental demonstration. That challenge led Kepler to make his famous models and eventually led him to see discrepancies that would never have emerged with such clarity in a traditional discussion.

In addition to textbooks which treated geometry literally as measurement of the earth there were also books specifically devoted to the topic of surveying. One of the classics in this context was Gemma Frisius' Booklet on the means of describing places and finding their distances (1533), which described the use of timepieces in determining longitude (cf. Pogo, Isis, 1933). Johann Stoeffler adapted an astrolabe for the purposes of surveying in On artful measurement of all sizes, planes and declines in length, height, breadth and depth (1536). Kaspar Peucer was interested in the extension of these surveying principles to longitudes , latitudes and cartography in his book On the dimension of the earth (1550).

There was yet another context for these interests. From the early fifteenth century there had been significant connections between geometry, perspective and surveying, through Alberti, Filarete, Piero della Francesca, Francesco di Giorgio Martini and Leonardo da Vinci. The first of these to be published was Alberti (1540). Meanwhile, D�rer included a section on perspective in his Instruction of measurement (1525, 1532, 1535, 1538). Rodler who edited A beautiful useful booklet (1531,1546), set out to popularize D�rer and explicitly equated perspective and measurement. Hirschvogel (1543) who wrote on geometry and perspective also wrote a treatise on suveying. Ryff, wrote a large compendium (1547, 1558) of treatises on both perspective (Alberti, Serlio) and surveying. Such links had the consequence that surveying was more than a question of recording isolated points on a terrain: it was also a question of recording graphically the features of a terrain in a perspectival form. A number of artists were trained as surveyors and it was no coincidence therefore that such artists were frequently called in to settle land disputes in the latter sixteenth century, as PŠre De Dainville has shown. An understanding of these connections helps explain the close connections between geometry, surveying and perspective evidenced in the latter sixteenth century: e.g. Bartoli (1564), Barbaro (1568) or Danti in his edition of Vignola (1583), the interplay between surveying and perspective instruments from early examples such as Besson's cosmolabe to the sector in the seventeenth century.

In astrology there was also a movement to publish classic texts, particularly Ptolemy (1538,1541,1543,1550,1553,and 1556) and mediaeval manuscripts from both the Jewish ,e.g. Abraham ben Ezra (1537, 1545) and the Arabic tradition: Alchabitius (1521, 1560); Albohali (1546); and Ali-Ibn Abi al-Rajjal (1551), as well as the Latin west, notably, Pietro d'Abano (1552). Here too there was a rise of literature on quantitative quantitative measurement. A number of works simply attempted to predict the future on the basis of positions of the stars: e.g. Carion 1529, 1549, Gr�nbeck 1531, Torquatus 1535, Gauricus 1539. Frequently they predicted the coming year, as in Gasser's Prognosticum for the year 1545 (1544), Brotbeyel (1547) and Rheticus (1550) or they gave a forecast of the current year as in Heller's Practice books for 1548, 1549, 1551, 1553, 1554, 1557, 1560, 1561, 1580. Other books contained technical tables, lists and instruments. In 1532, for instance there appeared an anonymous: Geomancy...With accompanying tables, which hours of the day and night are governed by each planet. Two years later there was another anonymous: Geomancy...together with five tables with an appended technical instrument and rules which hour of the day and night is governed by each planet (1532). Another instrument was decribed by Fin‚, the mathematician, in his book On the twelve houses of the heavens and unequal hours (1536).

Besides superstition, one incentive for these studies was medicine, as in Dariot's Easy introduction to judgment of the stars. Fragment on knowing diseases and critical days from the motions of the stars (1557). A more pressing incentive was religion. At the time there was a widespread conviction that the end of the world was imminent. While knowing when it would happen might not save one's skin but could save one's soul. Hence those involved in astrology included the foremost astronomers: Regiomontanus' A new calendar of all kinds of medicine...with eclipses of the sun and moon until the year 1563 (1540) and Rheticus' Astronomical tables... On the ascensions of the signs in the level and oblique sphere for the latitude of 52 degrees (1545). This tradition explains why the Wolfenb�ttel copy of an anonymous German astronomy. On the nature, property and effect of the 7 signs of the heavens, ofthe 7 planets and the 36 celestial images and their stars...(1545) should be bound with technical works: M�nster's Outline and technical description of sundials and Schoner's Gnomonics. This also explains why comets, which were seen as barometers for disaster, emerged as an important topic: e.g. Brelochs (1531), Sch"ner (1531) Gasser (1538) or why, a generation later even the great Kepler was both an astrologer and an astronomer. Often, of course, the same term served to describe both astrology and astronomy, with the result that both fields remained interdependent.

The publication of Copernicus' work in 1543 was but one manifestation of a complex set of developments in astronomy. One aspect was the recovery, editing and publication of ancient texts. The most famous of these was Ptolemy's Almagest which went through at least seven editions in the period 1520-1560 ( Cologne 1537, Basel 1538, Rome 1539, basel 1541, Wittemberg 1549, Basel 1554 and Paris 1556). There were also editions of Proclus (T�bingen 1534, Rome 1539 and Basel 1549) and Hyginus (Basel 1535, Cologne 1539, Basel 1549). Another aspect was the publication of important mediaeval texts. The most famous of these was the Sphere of Sacrobosco (John of Holywood) which went through at least nine editions (Ingolstadt 1526, Venice 1531, Strasbourg 1533, Wittemberg 1549, Paris 1550 and Frankfurt 1549, 1552, 1552 and c.1560). Editions of mediaeval Arabic texts included those of Al-Petragius (Venice 1531), Al-Farghani in the translation of John of Seville (N�rnberg 1537), on whom Regiomontanus had lectured at Padua in 1464 and Al Battani in the edition of Plato of Tivoli (N�rnberg 1537). This had been the earliest statement of the cosine law for spherical triangles. Mediaeval Jewish texts included those of Abraham ben-Chijja (Basel 1546). Late mediaeval texts included hose of Peurbach such as his New Theory of the Planets with commentaries by Erasmus Reinhold (Wittemberg 1542, Paris 1553) and Oswald Schreckenfuchs (1556), his Epitome of the Almagest (Basel 1543 and N�rnberg 1550) and his Treatise on Ptolemy's propositions concerning sines and chords (N�rnberg 1541 and Basel 1561). Publications of Regiomontanus included his Introductory discourse on the mathematical disciplines (1537), plus various tables.

The principle of astronomical tables went back to Antiquity. The mediaeval period made its own contributions particularly through the Alfonsine tables, a new set of planetary tables based on Ptolemy's methods, produced in Toledo (1252). These were published in 1483 and superseded in 1551 when Erasmus Reinhold published his Prutenic Tables. Meanwhile in the latter fifteenth century Regiomontanus produced a Table of directions (1467) and Table of the first moveable sphere which were published in 1490 and 1514 respectively. It was, however, the period 1520-1560 that saw important developments in the context of these tables. The year 1535 brought a Perpetual lunar calendar until the year 1600; 1541 brought Peurbach's Treatise on the propositions of Ptolemy on sines and chords along with Regiomontanus' Construction of the sine tables and Tables of both major sine tables. In 1551 the same year that Reinhold produced his Prutenic tables, Leowitz published a Table of oblique ascensions and positions forthe latitude of 48 degrees 8 minutes at which...Augsburg is situated. More importantly, in 1551 Rheticus also published his Canon of the doctrine of triangles, which was the first table to give all six trigonometrical functions, the first extensive table of tangents and the first table of secants. This formed the basis for the great Opus Palatinus de triangulis published by his student Valentin Otto in 1596.

The great significance of these tables owed much to the new contexts that were emerging. The same Regiomontanus involved in the tables just mentioned was also the author of a treatise On triangles written in 1463 and first published in 1533. This work, initially designed specifically for astronomical purposes, was one of the starting points in the development of trigonometry. Given a lively interplay between astronomy and geography these trigonometric tables were soon applied equally to terrestrial and celestial problems. Age old analogies between earth and universe now acquired a new mathematical framework which was rendered the more precise by yet another factor: a dramatic development in scientific instruments. This meant that these mathematical functions were no longer simply a question of theoretical computation. They could be optically recorded and verified. Theory and practice were now interdependent.

In astronomy the use of instruments went back to Antiquity. The mediaeval period had seen the development and widespread use of instruments such as the astrolabe and quadrant. The fourteenth century had added the Jacob's staff. In the fifteenth century John of Gm�nden had made one of the earliest recorded collections of instruments. These were inherited by Peurbach and then by Regiomontanus, who developed new instruments, which in turn inspired further developments in the sixteenth century. In 1514, for instance, Werner improved upon the Jacob's Staff of Regiomontanus and also invented a meteoroscope to solve problems of spherical astronomy, inventions that were taken up by Peter Apian in his Introduction to geography (1533).

The period 1520-1560 saw a great rise in publications concerning astronomical instruments. Among the earliest of these was an Explanation of a new instrument of the sun with all its plates and circles (1528) by Sebastian M�nster, also important for his work on sundials In 1530 Gemma Frisius published his work On the principles of astronomy and cosmography and on the use of the globe by both. In 1532 Jacob K"bel published his Declaration of the astrolabe...useful and necessary not only to astrologers, doctors, geographers and other students of literature but also to mechanics in various trades (1532).That same year in Ingolstadt, Apian published his Astronomical quadrant (1532) which also contained other instruments for discerning both nocturnal and diurnal hours and learning about the sun, moon and the fixed and moving stars. A year later Apian published a description of a new sundial with the title Folio of the people and his Instrument book (1533) which contained a whole series of instruments both for astronomy and surveying.

Astrolabes had traditionally been constructed for a particular latitude. Travel to another latitude required changing the plate of the astrolabe. The sixteenth century saw the development of universal astrolabes. In Paris, Oronce Fin‚ published an Astrolabic quadrant serving for all the regions of Europe (1534) and Juan de Rojas included in his commentary an account of a universal astrolabe by Hugo Helt (1550). Gemma Frisius, adapting a method of the Toledan scholar, Ibn az-arquella (c.1029-c. 1087), published his Astrolabum Catholicum ( ) which became the most popular of these universal astrolabes. Meanwhile, Johann Stoeffler published an Elucidation on the construction and use of the astrolabe (1535) which contained a series of descriptions collected from different authors. A second edition more than twice the length of the original appeared in 1553.

Types of instruments continued to diversify. Mithob published a work On the use and structure of rings both spherical and mathematical (1535). Apianus produced his Astronomicum Caesareum (1540) dedicated to the emperor Charles V, which contained further instruments and elaborate paper models of the universe. (The Landgraf of Hesse who developed the first modern observatory in the west with a collection of scientific instruments owned a copy of this book and it probably served as a starting point for Jobst B�rgi's physical models of the universe in the 1580's and 1590's). A year later Apianus published an Instrument of sines or of the primum mobile (1541), which provided a mechanical aid in computing trigonometric sines. Seven years passed before Schoner issued his On the use of the astral globe (1548). In 1550 Engelhard published a Booklet on the composition of the astronomical and geometrical quadrant, which by its title confirmed a twin application to celestial and terrestrial realms.

There were also new links between instruments and projection methods. For instance, Stoeffler published Some descriptions of cosmography. On the cosmographic sphere. On the double projection of the earth on a plane (1537). Striking in this context is the development of complex sundial projections by those also active in mathematics and astronomy: e.g. Hartmann (1526), Apianus (1533), M�nster (1533, 1537); Dryander (1536, 1543) or the anonymous Summary booklet (1539, 1544). By 1550 M�nster was making claims for general principles in his Universal type of sundials of walls with fourfold hours: equal, unequal Bohemian and Italian.. The close connections between perspective projection and sundial projection were sometimes explicit in manuscripts of the period but only became published in the next generation with Barbaro (1568), Danti (1583) and Guidobaldo del Monte (e.g. 1600).

All these instruments and models were based on Ptolemaic conceptions of the universe and insomuch as they did not take into account the Copernican model are often simply dismissed as wrong. This overlooks a subtle and essential change which they helped to bring about. Astronomy and geography had traditionally been largely conceptual subjects. The new repertory of instruments brought both fields into the realm of measurable observation amenable at the same time to a new mathematical framework provided by trigonometry which evolved dramatically in the next generations. (The links between trigonometry and logarithms were explored by B�rgi from 1588 onwards and subsequently codified by Napier, Briggs and Vlacq in the decade 1618-1628). As a result maps and models of the earth and heavens were now verifiable in a new way, with an incentive to make an accurate fit between constructed model and physical original.

These developments were reinforced by trends to integrate astronomy and geography. This idea was already implicit in the second century when the same Ptolemy who wrote the Almagest on the structure of the heavens was also the author of the Geography. It became clearer in the late fifteenth century when Martin Behaim in N�rnberg made twin globes of both the earth and the heavens. The period 1520-1560 made brought this idea into the published literature. Gemma Frisius assumed it in his edition of Apianus' Cosmographical Book (1529) and made it explicit in his work On the principles of astronomy and cosmography and the use of the globe by each (1530, 1553), as did Walseem�ller in his Introduction to cosmography with some of the principles of geometry and astronomy necessary for each subject (1550). Sebastian M�nster, active as an astronomer, produced a Latin edition of Ptolemy's Geography (1540) and a Cosmography (1544) which went through at least 46 editions in six languages by 1650. These parallels between astronomy and geography also evolved in geographical literature proper: Glareanus (1528), the anonymous Canon (1536), Honter (1552) and Fidelis (1556) and Francesco Alunno's Ten books on the fabric of the world (1560) which included the views of Dante, Petrarch and Boccaccio, each of whose cosmologies had assumed a direct link between earth (with hell inside), purgatory and heaven. All these developments linking geography and astronomy, undermined ancient notions opposing imperfection of the earth with perfection of the heavens and helped to establish a new idea of universal laws, on earth as it is in heaven, basic to modern science.

Seen from this larger point of view the contribution of the period 1520-1560 is much more than the rejection of the past and establishment of a new paradigm as epitomized by the works of Copernicus and Vesalius. There was in fact a consolidation of earlier knowledge which included new editions of ancient works, texts from the Arabic and Jewish traditions, mediaeval contributions and even those from the fifteenth century, particularly of Peurbach and Regiomontanus. These publications brought into focus a cumulative dimension to knowledge; a new mathematical framework in the form of trigonometry and a whole range of new instruments which added quantitative dimensions to geography, astronomy and cosmology. This brought with it the emergence of centres where theoretical publications and practical instrument making went hand in hand: notably N�rnberg, Antwerp, Paris and by the latter sixteenth century Kassell, Florence and London. Communication between these centres provided a first step in the internationalization and institutionalization of science. By the mid seventeenth century when this role was taken up by the emerging scientific societies, this process took on a more systematic form.. Even so the frameworks established in the period 1520-1560 throw important light on the origins of the mania for measurement which Koyr‚ has identified as an essential feature of early modern science. And in that complex story these new links between mathematics, instruments and visualization may constitute a more fundamental a revolution than the obvious contributions of Copernicus and Galileo.