It’s that day of the year again. It seems to come around faster every time. On this day twelve years ago The Renaissance Mathematics first entered cyberspace. What does the word, twelve, actually mean? In the Germanic languages twelve and its equivalents means two left that is after counting to ten:
Old English twelf “twelve,” literally “two left” (over ten), from Proto-Germanic *twa-lif-, a compound of *twa- (from PIE root *dwo- “two”) + *lif- (from PIE root *leikw- “to leave”). Cognate with Old Saxon twelif, Old Norse tolf, Old Frisian twelef, Middle Dutch twalef, Dutch twaalf, Old High German zwelif, German zwölf, Gothic twalif.
Online Etymological Dictionary
Twelve features widely in culture, religion, science, and society in general. There were twelve apostles, twelve days of Christmas, the twelve Olympians (the major ancient Greek deities), The Twelve Tribes of Israel, jury of twelve good men and true, somehow twelve has always been a favoured number for humans. But this is a history of science blog and here we meet many instances of the number twelve.
The Romans used a base twelve or duodecimal number system, the only fraction that they used were twelfths. The remnants of this system are present in many countries that were once parts of the Roman Empire.
Table of units from a base of 12
French unit of length
English unit of length
English (Troy) unit of weight
Roman unit of weight
English unit of mass
Table taken from Wikipedia
Also, in English we still have the term dozen for twelve and gross for twelve squared, which reflect a twelve based number system.
There are modern societies in both the UK and the US that wish to replace our decimal system with a duodecimal one or as they prefer to call it Dozenal to avoid the decimal in duodecimal. They argue that because twelve has more factors than ten, a Dozenal system would be arithmetically preferable to a decimal one.
Our twelve-hour day has a different source. To tell the time at night Egyptian astronomers used the so-called decans, a set of thirty stars or groups of stars, which rise consecutively on the horizon throughout each earth rotation. In any given night twelve decans rose successively over the horizon dividing the night into twelve.
Gradually they developed the habit of also dividing the day into twelve units, our twelve-hour day. Originally, twelve seasonal hours, the length of which, varied throughout the year. In the early modern period these became our equinoctial hours of equal length. Hours are divided into sixty minutes and minutes into sixty seconds, sixty is a multiple of twelve.
Astronomy and astrology deliver up a two-significant-twelves. Twelve months in the year and twelve signs of the zodiac that are in fact related. The word month has the same etymological root and the word moon and originally referred to the lunar moon, which is about twenty-nine and a half days long. Early calendars were lunar calendars, but the solar year is about eleven days longer that twelve lunar months, so if you want to keep your calendar aligned with the solar year you have to add an extra lunar month about once every three years. The Greeks adopted the Metonic cycle, named after a Greek, but conceived by the Babylonians, in which seven extra months are added in nineteen solar years.
The Romans used a more random method in which an extra month was added by a political official when it was thought necessary. Because the dates for elections were determined by the calendar, this led to political corruptions with manipulation of the calendar. Julius Caesar solved the problem by introduction a solar calendar borrowed from the Egyptians with three hundred and sixty-five days divided up into twelve months. Nothing says there should be twelve months in a solar year, the French Revolutionary Calendar only had ten months, but by analogy to the lunar calendar twelve was chosen. Now, the solar year is closer to three hundred and sixty-five and a quarter days, which was known to the Egyptians and Caesar’s astronomical advisors, so you have to add an extra day approximately every four years. Caesar’s astronomical advisors got this slightly wrong leading to the whole Julian Calendar/Gregorian Calendar reform, which we won’t go into here.
That the ecliptic is divided into twelve thirty-degree signs of the zodiac, also goes back to the Egyptian solar calendar. The Egyptians divided the year into twelve thirty-day months, with five non-days between the beginning and the end of the year, making a total of three hundred and sixty-five days. These twelve thirty-day months became the twelve thirty-degree signs of the zodiac.
And so, the Renaissance Mathematicus enters its thirteenth year expectantly looking forward to what its Gemini horoscope will deliver. We wish all of our readers, commentators and supporters, both active and passive, all the best for our next circuit of the Sun and hope you enjoy the future blog posts.
There can’t be many Renaissance mathematici, whose existence was ennobled by a personal portrait by the master of the Renaissance portraits, Hans Holbein the younger. In fact, I only know of one, the German mathematicus, Nicolas Kratzer.
One might be excused for thinking that having received this singular honour that Kratzer had, in his lifetime, achieved something truly spectacular in the world of the Renaissance mathematical disciplines; however, almost the opposite is true. Kratzer appears to have produced nothing of any significance, was merely the designer and maker of sundials, and an elementary maths teacher, who was only portrayed by Holbein, because for a time they shared the same employers and were apparently mates.
So, who was Kratzer and how did he and Holbein become mates? Here we find a common problem with minor scientific figures in the Renaissance, there are no biographies, no handy archives giving extensive details of his life. All we have are a few, often vague, sometimes contradictory, traces in the proverbial sands of time from which historians have attempted to reconstruct at least a bare outline of his existence.
Kratzer was born in 1487 in Munich, the son of a saw-smith and it is probably that he learnt his metal working skills, as an instrument maker, from his father. He matriculated at the University of Köln 18 November 1506 and graduated BA 14 June 1509. He moved onto the University of Wittenberg, famous as the university of Martin Luther. However, this was before the Reformation and Wittenberg, a young university first founded in 1502, was then still Catholic. We now lose track of Kratzer, who is presumed to have then worked as an instrument maker. Sometime in the next years, probably in 1517, he copied some astronomical manuscripts at the Carthusian monastery of Maurbach, near Vienna.
In January 1517, Pieter Gillis (1486–1533) wrote to his erstwhile teacher Erasmus (1466–1536) that the skilled mathematician Kratzer was on his way with astrolabes and spheres, and a Greek book.
This firmly places Kratzer in the circle of humanist scholars, most famously Erasmus and Thomas More (1478–1535) author of Utopia, who founded the English Renaissance on the court of Henry VIII (1491–1547). Holbein was also a member of this circle. Erasmus and Holbein had earlier both worked for the printer/publisher collective of Petri-Froben-Amerbach in Basel. Erasmus as a copyeditor and Holbein as an illustrator. Holbein produced the illustrations for Erasmus’ In Praise of Folly (written 1509, published by Froben 1511)
Kratzer entered England either at the end of 1517 or the beginning of 1518. His first identifiable employment was in the household of Thomas More as maths teacher for a tutorial group that included More’s children. It can be assumed that it was here that he got to know Holbein, who was also employed by More.
For his portraits, Holbein produced very accurate complete sketches on paper first, which he then transferred geometrically to his prepared wooden panels to paint them. Around 1527, Holbein painted a group portrait of the More family that is no longer extant, but the sketch is. The figures in the sketch are identified in writing and the handwriting has been identified as Kratzer’s.
Like Holbein, Kratzer moved from More’s household to the court of Henry VIII, where he listed in the court accounts as the king’s astronomer with an income of £5 a quarter in 1529 and 1531. It is not very clear when he entered the King’s service but in 1520 Cuthbert Tunstall (1474–1559), later Prince-Bishop of Durham, wrote in a letter:
Met at Antwerp with [Nicolas Kratzer], an Almayn [German], devisor of the King’s horologes, who said the King had given him leave to be absent for a time.
Both Tunstall and Kratzer were probably in Antwerp for the coronation of Charles V (1500–1558) as King of Germany, which took place in Aachen. There are hints that Kratzer was there to negotiate with members of the German court on Henry’s behalf. Albrecht Dürer (1471–1528) was also in the Netherlands; he was hoping that Charles would continue the pension granted to him by Maximilian I, who had died in 1519. Dürer and Kratzer met in the house of Erasmus and Kratzer was present as Dürer sketched a portrait of Erasmus. He also drew a silver point portrait of Kratzer, which no longer exists.
Back in England Kratzer spent some time lecturing on mathematical topics at Oxford University during the 1520s. Here once again the reports are confused and contradictory. Some sources say he was there at the behest of the King, others that he was there in the service of Cardinal Wolsey. There are later claims that Kratzer was appointed a fellow of Corpus Christi College, but the college records do not confirm this. However, it is from the Oxford records that we know of Kratzer’s studies in Köln and Wittenberg, as he was incepted in Oxford as BA and MA, on the strength of his qualifications from the German institutions, in the spring of 1523.
During his time in Oxford, he is known to have erected two standing sundials in the college grounds, one of which bore an anti-Lutheran inscription.
Neither dial exists any longer and the only dial of his still there is a portable brass dial in the Oxford History of Science Museum, which is engraved with a cardinal’s hat on both side, which suggests it was made for Wolsey.
On 24 October 1524 Kratzer wrote the following to Dürer in Nürnberg
Dear Master Albert, I pray you to draw for me a model of the instrument that you saw at Herr Pirckheimer’s by which distances can be measured, and of which you spoke to me at Andarf [Antwerp], or that you will ask Herr Pirckheimer to send me a description of the said instrument… Also I desire to know what you ask for copies of all your prints, and if there is anything new at Nuremberg in my craft. I hear that our Hans, the astronomer, is dead. I wish you to write and tell me what he has left behind him, and about Stabius, what has become of his instruments and his blocks. Greet in my name Herr Pirckheimer. I hope shortly to make a map of England which is a great country, and was not known to Ptolemy; Herr Pirckheimer will be glad to see it. All who have written of it hitherto have only seen a small part of England, no more… I beg of you to send me the likeness of Stabius, fashioned to represent St. Kolman, and cut in wood…
Herr Pirckheimer is Willibald Pirckheimer (1470–1530), who was a lawyer, soldier, politician, and Renaissance humanist, who produced a new translation of Ptolemaeus’ Geographia from Greek into Latin.
He was Dürer’s life-long friend, (they were born in the same house), patron and probably lover. He was at the centre of the so-called Pirckheimer circle, a group of mostly mathematical humanists that included “Hans the astronomer, who was Johannes Werner (1468–1522), mathematician, astronomer, astrologer, geographer,
and cartographer and Johannes “Stabius” (c.1468–1522) mathematician, astronomer, astrologer, and cartographer.
Werner was almost certainly Dürer’s maths teacher and Stabius worked together with Dürer on various projects including his star maps. The two are perhaps best known for the Werner-Stabius heart shaped map projection.
Dürer replied to Kratzer 5 December 1524 saying that Pirckheimer was having the required instrument made for Kratzer and that the papers and instruments of Werner and Stabius had been dispersed.
Here it should be noted that Dürer, in his maths book, Underweysung der Messung mit dem Zirkel und Richtscheyt (Instruction in Measurement with Compass and Straightedge), published the first printed instructions in German on how to construct and orientate sundials. The drawing of one sundial in the book bears a very strong resemblance to the polyhedral sundial that Kratzer made for Cardinal Wolsey and presumably Kratzer was the original source of this illustration.
Kratzer is certainly the source of the mathematical instruments displayed on the top shelf of Holbein’s most famous painting the Ambassadors, as several of them are also to be seen in Holbein’s portrait of Kratzer.
Renaissance Mathematicus friend and guest blogger, Karl Galle, recently made me aware of a possible/probable indirect connection between Kratzer and Nicolas Copernicus (1473–1543). Georg Joachim Rheticus (1514–1574) relates that Copernicus’ best friend Tiedemann Giese (1480–1550) possessed his own astronomical instruments including a portable sundial sent to him from England. This was almost certainly sent to him by his brother Georg Giese (1497–1562) a prominent Hanseatic merchant trader, who lived in the Steelyard, the Hansa League depot in London, during the 1520s and 30s.
He was one of a number of Hanseatic merchants, whose portraits were painted by Holbein, so it is more than likely that the sundial was one made by Kratzer.
Sometime after 1530, Kratzer fades into the background with only occasional references to his activities. After 1550, even these ceased, so it is assumed that he had died around this time. In the first half of the sixteenth century England lagged behind mainland Europe in the mathematical disciplines including instrument making, so it is a natural assumption that Kratzer with his continental knowledge was a welcome guest in the Renaissance humanist circles of the English court, as was his younger contemporary, the Flemish engraver and instrument maker, Thomas Gemini (1510–1562). Lacking homegrown skilled instrument makers, the English welcomed foreign talent and Kratzer was one who benefited from this.
The Renaissance saw not only the introduction of new branches of mathematics, as I have outlined in the last three episodes in this series, but also over time major changes in the teaching of mathematics both inside and outside of the universities.
The undergraduate or arts faculty of the medieval university was nominally based on the so-called seven liberal arts, a concept that supposedly went back to the Pythagoreans. This consisted of the trivium – grammar, logic, and rhetoric – and the quadrivium – arithmetic, geometry, music, and astronomy – whereby the quadrivium was the mathematical disciplines. However, one needs to take a closer look at what the quadrivium actually entailed. The arithmetic was very low level, as was the music, actually in terms of mathematics the theory of proportions. Astronomy was almost entirely non-technical being based on John of Sacrobosco’s (c. 1195–c. 1256) Tractatus de Sphera (c. 1230). Because Sacrobosco’s Sphera was very basic it was complemented with a Theorica planetarum, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos. Only geometry had a serious mathematical core, being based on the first six books of The Elements of Euclid.
I said above, nominally, because in reality on most universities the quadrivium only had a niche existence. Maths lectures were often only offered on holidays, when normal lectures were not held. Also, the mathematical disciplines were not examination subjects. If a student didn’t have the necessary course credit for a mathematical discipline, they could often acquire it simply by paying the requisite tuition fees. Put another way, the mathematical disciplines were not taken particularly seriously in the early phase of the European universities. There were some exceptions to this, but they were that, exceptions.
Through out much of the Middle Ages there were no chairs for mathematics and so no professors. Very occasionally a special professor for mathematics would be appointed such as the chair created by Francois I at the Collège Royal in the 16th century for Oronce Fine (1494–1555) initially there were only chairs and professorships for the higher faculties, theology, law, and medicine. On the arts faculty the disciplines were taught by the postgraduate masters. The MA was a teaching licence. If somebody was particularly talented in a given discipline, they would be appointed to teach it, but otherwise the masters were appointed each year by drawing lots. To get the lot for mathematics was the equivalent of getting the short straw. This changed during the Renaissance, and we will return to when and why below but before we do, we need to first look at mathematics outside of the university.
During the medieval period preceding the Renaissance, trades people who had to do calculations used an abacus or counting board and almost certainly a master taught his apprentice, often his own son, how to use one. This first began to change during the so-called commercial revolution during which long distance trade increased significantly, banks were founded for the first time, double entry bookkeeping was introduced, and both the decimal place value number system and algebra were introduced to aid business and traded calculations. As I said earlier this led to the creation of the so-called abbacus, or in English reckoning schools with their abbacus or reckoning books.
The reckoning schools and books not only taught the new arithmetic and algebra but also elementary geometry and catered not only for the apprentice tradesmen but also for apprentice artists, engineers, and builder-architects. It is fairly certain, for example, that Albrecht Dürer, who would later go on to write an important maths textbook for apprentice artists, acquired his first knowledge of mathematics in a reckoning school. This was a fairly radical development in the formal teaching of mathematics at an elementary level, as the Latin schools, which prepared youths for a university education, taught no mathematics at all.
The first major change in the mathematic curriculum on the European universities was driven by astrology, or more precisely by astrological medicine or iatromathematics, as it was then called. As part of the humanist Renaissance, astro-medicine became the dominant form of medicine followed on the Renaissance universities; a development we will deal with later. In the early fifteenth century, in order to facilitate this change in the medical curriculum the humanist universities of Northern Italy and also the University of Cracow introduced chairs and professorships for mathematics, whose principal function was to teach astrology to medical students. Before they could practice astro-medicine the students had to learn how to cast a horoscope, which meant first acquiring the necessary mathematical and astronomical skills to do so. This was still the principal function of professors of mathematics in the early seventeenth century and Galileo, would have been expected to teach such courses both at Pisa and Padua.
As with other aspects of the humanist Renaissance this practice spread to northwards to the rest of Europe. The first chair for mathematics at a German university was established at the University of Ingolstadt, also to teach medical student astrology. Here interestingly, Conrad Celtis, know in Germany as the Arch Humanist, when he was appointed to teach poetics subverted the professors of mathematics slightly to include mathematical cartography in their remit. He took two of those professors, Johannes Stabius and Andreas Stiborius, when he moved to Vienna and set up his Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, this helped to advance the study and practice of mathematical cartography on the university.
Astrology also played a central role in the next major development in the status and teaching of mathematics on school and universities. Philipp Melanchthon (1497–1560) was a child prodigy. Having completed his master’s at the University of Heidelberg in 1512 but denied his degree because of his age, he transferred to the University of Tübingen, where he became enamoured with astrology under the influence of Johannes Stöffler (1452–1531), the recently appointed first professor of mathematics, a product of the mathematics department at Ingolstadt.
Melanchthon was appointed professor of Greek at Wittenberg in 1518, aged just twenty-one. Here he became Luther’s strongest supporter and was responsible for setting up the Lutheran Protestant education system during the early years of the reformation. Because of his passion for astrology, he established chairs for mathematics in all Protestant schools and university. Several of Melanchthon’s professors played important rolls in the emergence of the heliocentric astronomy.
The Lutheran Protestants thus adopted a full mathematical curriculum early in the sixteenth century, the Catholic education system had to wait until the end of the century for the same development. Founded in 1540, the Society of Jesus (the Jesuits) in their early years set up an education system to supply Catholics with the necessary arguments to combat the arguments of the Protestants. Initially this strongly Thomist education system did not include mathematics. Christoph Clavius (1538–1612), who joined the Jesuits in 1555, was a passionate mathematician, although it is not exactly clear where he acquired his mathematical education or from whom. By 1561 he was enrolled in the Collegio Romano, where he began teaching mathematics in 1563 and was appointed professor of mathematics in 1567. Clavius created an extensive and comprehensive mathematical curriculum that he wanted included in the Jesuit educational programme. Initially, this was rejected by conservative elements in the Society, but Clavius fought his corner and by the end of the century he had succeeded in making mathematics a central element in Jesuit education. He personally taught the first generation of teachers and wrote excellent modern textbooks for all the mathematical disciplines, including the new algebra. By 1626 there were 444 Jesuit colleges and 56 seminaries in Europe all of which taught mathematics in a modern form at a high level. Many leading Catholic mathematicians of the seventeenth century such as Descartes, Gassendi, and Cassini were products of this Jesuit education network.
By the beginning of the seventeenth century mathematics had become an established high-level subject in both Protestant and Catholic educational institutions throughout the European mainland, the one exception which lagged well behind the rest of Europe was England.
Well aware that the mathematical education in England was abysmal, a group of influential figures created a public lectureship for mathematics in London at the end of the seventeenth century. These lectures intended for soldiers, artisans and sailors were held from 1588 to 1592 by Thomas Hood (1556–1620), who also published books on practical mathematics in the same period. Other English practical mathematicians such as Robert Recorde, Leonard and Thomas Digges, Thomas Harriot and John Dee also gave private tuition and published books aimed at those such as cartographers and navigators, who needed mathematics.
In 1597, Gresham College was set up in London using money bequeathed by Sir Thomas Gresham (c. 1519–1597) to provide public lectures in both Latin and English in seven subjects, including geometry and astronomy. The professorships in these two mathematical disciplines have been occupied by many notable mathematical scholars over the centuries.
The two English universities, Oxford, and Cambridge, still lagged behind their continental colleagues, as far as the mathematical sciences were concerned. The first chairs at Oxford University for astronomy and geometry were the result of a private initiative. Henry Savile (1549–1622), an Oxford scholar, like many others in this period, travelled on the continent in order to acquire a mathematical education, primarily at the North German Universities, where several prominent Scottish mathematicians also acquired their mathematical education.
In 1619, he founded and endowed the Savilian Chairs for Astronomy and Geometry at Oxford. Many leading English mathematical scholars occupied these chairs throughout the seventeenth century, several of whom had previously been Gresham professors.
In the High Middle Ages the mathematical disciplines were treated as niche subjects on the medieval university. Throughout the Renaissance period this changed and with it the status and importance of mathematics. This change was also driven by the need for mathematics in the practical disciplines of cartography, navigation, surveying, astrology, and the emerging new astronomy; we will deal with these developments in future episodes. However, by the end of the Renaissance, mathematics had gained the high academic status that it still enjoys today.
Trying to write a comprehensive history of science up to the scientific revolution in a single volume is the historian of science’s equivalent to squaring the circle. It can’t actually be done, it must fall short in various areas, but doesn’t prevent them from trying. The latest to attempt squaring the history of science circle is Ofer Gal in his The Origins of Modern Science: From Antiquity to the Scientific Revolution.
Gal’s book has approximately 380 pages and given what I regard as the impossibility of his task, I decided, if possible, to cut him some slack in this review. To illustrate the problem, David Lindberg’s The Beginnings of Western Science, with which Gal is definitely competing, has approximately 370 pages and only goes up to 1450 and has been criticised for its omissions. The Cambridge History of Science requires three volumes with an approximate total of 2250 pages to cover the same period as Gal and its essays can best be regarded as introductions to further reading.
CUP are marketing Gal’s book as a textbook for schools and university students, which means, in my opinion, a higher commitment to historical factual accuracy, so where I might be prepared to cut some slack on possible omissions, I’m not prepared to forgive factual errors. If you are teaching beginners, which this book aims to do, then you have an obligation to get your facts right. The intended textbook nature is reflected in the academic apparatus. There is no central bibliography of sources, instead at the end of each section there is a brief list of primary and secondary sources for that section. This is preceded by a list of essay type questions on the section; questions that are more of a philosophical than historical nature. The book has neither foot nor endnotes but gives occasional sources for quotes within the main text in backets.
Gal’s book opens with a thirty-page section titled, Cathedrals, which left me wondering what to expect, when I began reading. Actually, I think it is possibly the best chapter in the whole book. What he does is to use the story of the origins and construction of the European medieval cathedrals to illustrate an important distinction, in epistemology, between knowing-how and knowing-that. It is also the first indication that in the world of the traditional history and philosophy of science Gal is more of a philosopher than a historian, an impression that is confirmed as the book progresses. At times throughout the book, I found myself missing something, actual science.
Chapter two takes the reader into the world of ancient Greek philosophy and give comparatively short and concise rundowns on the main schools of thought, which I have to admit I found rather opaque at times. However, it is clear that Gal thinks the Greeks invented science and that Aristotle is very much the main man. This sets the tone for the rest of the book, which follows a very conventional script that is, once again in my opinion, limited and dated.
The following section is the Birth of Astronomy, which Gal attributes entirely to the Greeks, no Egyptians, no Babylonians. He starts with Thomas Kuhn’s two sphere model that is the sphere of the Earth sitting at the centre of the sphere of the heavens and here we get a major factual error. He writes:
For the astronomers of ancient Mesopotamia and the Aegean region, that model was of two spheres: the image of our Earth, a sphere, nestled inside the bigger sphere of the heavens.
Unfortunately, for Gal, the astronomers of ancient Mesopotamia were flat earthers. Later in the section, Gal informs us that Babylonian astronomy was not science. I know an awful lot of historians of astronomy, who would be rather upset by this claim. Rather bizarrely in a section on ancient astronomy, the use of simple observation instruments is illustrated with woodcuts from a book from 1669 showing a cross-staff, first described in the 14th century by Levi den Gerson, and a backstaff, which was invented by John Davis in 1594. In the caption the backstaff is also falsely labelled a sextant. He could have included illustration of the armillary sphere and the dioptra, instruments that Hipparchus and Ptolemy actually used, instead.
Apart from these errors the section is a fairly standard rundown of Greek astronomical models and theories. As, apparently, the Greeks were the only people in antiquity who did science and the only science worth mentioning here is astronomy, we move on to the Middle Ages.
We get presented with a very scant description of the decline of science in late antiquity and then move on to the The Encyclopedic Tradition. Starting with the Romans, Cicero gets a positive nod and Pliny a much more substantial one. Under the medieval encyclopedist, we get Martianus Capella, who gets a couple of pages, whereas Isidore and Bede only manage a couple of lines each. We then get a more substantial take on the medieval Christian Church, although Seb Falk would be disappointed to note the lack of science here, the verge-and-foliot escapement and computus both get a very brief nod. Up next is the medieval university, which gets a comparatively long section, which however contains, in this context, a very strange attack on the university in the twenty first century. Gal also opinions:
They [medieval students] would study in two ways we still use and one which we have regrettably lost.
The three ways he describes are the lectura, the repititio, and the desputatio, so I must assume that Gal wishes to reintroduce the desputatio into the modern university! Following this are two whole pages on The Great Translation Project. This is somewhat naturally followed by Muslim Science. The section on the medieval university is slightly longer than that devoted here to the whole of Muslim science, with a strong emphasis on astronomy. In essence Gal has not written a book on the origins of modern science but one on the origins of modern astronomy with a couple of side notes nodding to other branches of the sciences. He devotes only a short paragraph to al-Haytham’s optics and the medieval scholars, who adopted it. Put another way, the same old same old.
The next section of the book bears the title The Seeds of Revolution and begins with a six-page philosophical, theological discourse featuring Ibn Rushd, Moshe ben Maimon and Thomas Aquinas. We now move on to the Renaissance. In this section the only nominal science that appears is Brunelleschi’s invention of linear perspective as an example of “the meeting of scholar and artisan.” A term in the title of the next subsection and throughout the section itself left me perplexed, The Movable Press and Its Cultural Impact. Can anybody help me? The history of printing is one of my areas of study and I have never ever come across the movable type printing press simply referred to as “the movable press.” I even spent half an hour searching the Internet and could not find the term anywhere. Does it exist or did Gal create it? The section itself is fairly standard. This is followed by a long section on Global Knowledge covering navigation and discovery, global commerce, practical mathematics driven by commerce, trade companies, and the Jesuits.
We then get a section, which is obviously a favourite area of Gal, given to space that he grants it, magic. Now I’m very much in favour of including what I would prefer to give the general title occult theories and practices rather than magic in a text on the history of science, so Gal wins a couple of plus points for this section. He starts with a philosophical presentation of the usual suspects, Neo-Platonism, Hermeticism, Kabbala et al. He then moves on to what he terms scientific magic, by which he means alchemy and astrology, which he admits are not really the same as magic, excusing himself by claiming that both are based on a form of magical thinking. He then attempts to explain each of them in less than three pages, producing a rather inadequate explanation in each case. In neither case does he address the impact that both alchemy and astrology actually had historically on the development of the sciences. Moving on we have Magic and the New Science. Here we get presented with cameos of the Bacons, both Roger and Francis, Pico della Mirandola, and Giambattista della Porta.
When dealing with Roger Bacon we get another example of Gal’s historical errors, he writes:
This enabled him to formulate great novelties, especially in optics. Theoretically, he turned Muslim optics into a theory of vision; practically, he is credited with the invention of the spectacles.
Here we have a classic double whammy. He didn’t turn Muslim optics into a theory of vision but rather took over and propagated the theory of vision of Ibn al-Haytham. I have no idea, who credits Roger Bacon with the invention of the spectacles, in all my extensive readings on the history of optics I have never come across such a claim, maybe just maybe, because it isn’t true.
Roughly two thirds of the way through we are now approaching modern science with a section titled, The Moving Earth. I’ll start right off by saying that it is somewhat symbolic of what I see as Gal’s dated approach that the book that he recommends for Copernicus’ ‘revolution’ is Thomas Kuhn’s The Copernican Revolution, a book that was factually false when it was first publish and hasn’t improved in the sixty years since. But I’m ahead of myself.
The section starts with a very brief sketch of Luther and the reformation, which function as a lead into a section titled, Counter-Reformation and the Calendar Reform. Here he briefly mentions the Jesuits, whom he dealt with earlier under Global Knowledge. He writes:
The Jesuits, as we’ve pointed out, turned from the strict logicism of traditional Church education to disciplines aimed at moving and persuading: rhetoric, theater, and dance. Even mathematics was taught (at least to missionaries-to-be) for its persuasive power.
Ignoring this rather strange presentation of the Jesuit strictly logical Thomist education programme, I will just address the last sentence. Clavius set up the most modern mathematical educational curriculum in Europe and probably the world, which was taught in all Jesuit schools and colleges throughout the world, describing it as “even mathematics was taught” really is historically highly inaccurate. Gal now delivers up something that I can only describe as historical bullshit, he writes: (I apologies for the scans but I couldn’t be arsed to type all of it.)
I could write a whole blog post trying to sort out this rubbish. The bit about pomp and circumstance is complete rubbish, as is, in this context, the section about knowing the exact time that had passed, since the birth of Christ. The only concern here is trying to determine the correct date on which to celebrate the movable feasts associated with Easter. The error in the length of the Julian year, which was eleven minutes not a quarter of an hour, also has nothing to do with the procession of the equinoxes but simply a false value for the length of the solar year. The Julian calendar was also originally Egyptian not Hellenistic. The Church decided vey early on to determine the date of Easter astronomically not by observation in order not to be seen following the Jewish practice. The calendar reform was not part of/inspired by the Reformation/Counter-Reformation but it had been on the Church’s books for centuries. There had been several reforms launched that were never completed, usually because the Pope, who had ordered it, had died and his successor had other things on his agenda when he mounted the Papal Throne. Famously, Regiomontanus died when called to Rome by the Pope to take on the calendar reform. The calendar reform that was authorized by the Council of Trent, had been set in motion several decades before the Council. Ptolemy’s Almagest had reached Europe twice in translations, both from the Greek and from Arabic, in the twelfth century and not first in the fifteenth century. What was published in the fifteenth century and had a major impact, Copernicus learnt his astronomy from it, was Peuerbach’s and Regiomontanus’ Epitoma in Almagestum Ptolemae
Just to close although it has nothing to do with the calendar reform, the name Commentariolus for Copernicus’ short manuscript from about 1514 on a heliocentric system, was coined much later by Tyco Brahe.
We now move on to Copernicus. His section on Copernicus and his astronomy is fairly good but we now meet another problem. For his Early Modern scientists, he includes brief biographical detail, which; as very much a biographical historian, I approve of, but they are unfortunately strewn with errors. He writes for example that Copernicus was “born in Northern Poland then under Prussian rule.” Copernicus was born in Toruń, at the time an autonomous, self-governing city under the protection of the Polish Crown. After briefly sketching Copernicus’ university studies he writes:
“Yet Copernicus had no interest in vita activa: throughout his life he made his living as a canon in Frombork (then Frauenburg), a medieval privilegium (a personally conferred status) with few obligations…”
The cathedral canons in Frombork were the government and civil service of the prince-bishopric of Warmia and Copernicus had very much a vita activa as physician to the bishop, as consultant on fiscal affairs, as diplomat, as governor of Allenstein, organizing its defences during a siege by the Teutonic Order, and much more. Copernicus’ life was anything but the quiet contemplative life of the scholar. Later he writes concerning Copernicus’ activities as astronomer, “his activities were supported by the patronage of his uncle, in whose Warmia house he set up his observatory.” Whilst Copernicus on completion of his studies initially lived in the bishop’s palace in Heilsberg from 1503 till 1510 as his uncle’s physician and secretary, following the death of his uncle he moved to Frombork, and it is here that he set up his putative observatory. Gal also writes, “It took him thirty years to turn his Commentariolus into a complete book – On the Revolutions – whose final proofs he reviewed on his death bed, never to see it actually in print.” The legend says the finished published book was laid in his hands on his death bed. He would hardly have been reviewing final proofs, as he was in a coma following a stroke.
This might all seem like nit picking on my part but if an author is going to include biographical details into, what is after all intended as a textbook, then they have an obligation to get the facts right, especially as they are well documented and readily accessible.
Rheticus gets a brief nod and then we get the standard slagging off of Osiander for his adlectorum. Here once again we get a couple of trivial biographical errors, Gal refers to Osiander as a Lutheran and as a Protestant priest. Osiander was not a Lutheran, he and Luther were rivals. Protestants are not priests but pastors and Osiander was never a pastor but a Protestant preacher. Of course, Gal has to waste space on Bruno, which is interesting as he largely ignores several seventeenth century scientists, who made major contributions to the development of modern science, such as Christiaan Huygens.
We are now well established on the big names rally towards the grand climax. Up next is Tycho Brahe, who, as usual, is falsely credited with being the first to determine that comets, nova et all were supralunar changing objects, thus contradicting Aristotle’s perfect heavens cosmology. History dictates that Kepler must follow Tycho, with a presentation of his Mysterium Cosmographicum. Gal says that Kepler’s mother “keen on his education” “sent him through the Protestants’ version of a Church education – grammar school, seminary and the University of Tübingen.” No mention of the fact that this education was only possible because Kepler won a scholarship. Gal also tells us:
By 1611, Rudolf’s colorful court brought about his demise, as Rudolf was forced off his throne by his brother Mathias, meaning that Kepler had to leave Prague. The last two decades of his life were sad: his financial and intellectual standing deteriorating, he moved back to the German-speaking lands – first to Linz, then Ulm, then Regensburg, and when his applications to university posts declined, he took increasingly lower positions as a provincial mathematician. … He died in poverty in Regensburg in 1630…
First off, Rudolph’s Prague was German speaking. Although Mathias required Kepler to leave Prague, he retained his position as Imperial Mathematicus (which Gal falsely names Imperial Astronomer), although actually getting paid for this post by the imperial treasury had always been a problem. He became district mathematicus in Linz in 1612 to ensure a regular income, a post he retained until 1626. He moved from Linz to Ulm in 1626 in order to get his Rudolphine Tables printed and published, which he then took to the Book Fair in Frankfurt, to sell in order to recuperate the costs of printing. From 1628 he was court advisor, read astrologer, to Wallenstein in Sagan. He travelled to the Reichstag in Regensburg in 1630, where he fell ill and died. He had never held a university post in his life and hadn’t attempted to get one since 1600.
Having messed up Kepler’s biography, Gal now messes up his science. Under the title, The New Physical Optics, Gal gets Kepler’s contribution to the science of optics horribly wrong. He writes:
Traditional optics was the mathematical theory of vision. It studied visual rays: straight lines which could only change direction: refracted by changing media or reflected by polished surfaces. Whether these visual rays were physical entities or just mathematical representations of the process of vision, and what this process consisted of, was much debated. (…) But there was no debate that vision is a direct, cognitive relation between the object and the mind, through the eye. Light, in all of these theories, had an important, but secondary role:
Kepler abolished this assumption. Nothing of the object, he claimed, comes to and through the eye. The subject matter of his optics was no longer vision but light:
This transformation in the history of optics was not consummated by Kepler at the beginning of the seventeenth century but by al-Kindi and al-Haytham more than seven hundred years earlier. This was the theory of vision of al-Haytham mentioned above and adopted by Roger Bacon.
We then get a reasonable account of Kepler’s Astronomia nova, except that he claims that Kepler’s difficulties in finally determining that the orbit of Mars was an ellipse was because he was trapped in the concept that the orbits must be circular, which is rubbish. Else where Gal goes as far as to claim that Kepler guessed that the orbit was an ellipse. I suggest that he reads Astronomia nova or at least James Voelkel’s excellent analysis of it, The Composition of Kepler’s Astronomia Nova (Princeton University Press, 2001) to learn how much solid mathematical analysis was invested in that determination.
As always Galileo must follow Kepler. We get a very brief introduction to the Sidereus Nuncius and then an account of Galileo as a social climber that carries on the series of biographical errors. Gal writes:
Galileo’s father Vincenzo (c. 1520–1591) (…) A lute player of humble origins, he taught himself musical theory and acquired a name and enough fortune to marry into minor (and penniless) nobility with a book on musical theory, in which he relentless and venomously assaulted the canonical theory as detached from real musical practices.
This is fascinatingly wrong, because Gal gives as his source for Galileo’s biography John Heilbron’s Galileo, where we can read on page 2 the following:
Although Galileo was born in Pisa, the hometown of his recalcitrant mother, he prided himself on being a noble of Florence through his father, Vincenzo Galilei, a musician and musical theorist. Vincenzo’s nobility did not imply wealth but the right to hold civic office and he lived in the straitened circumstances usual in his profession. His marriage to Giulia, whose family dealt in cloth, was a union of art and trade.
The errors continue:
…he returned to the University of Pisa to study medicine, but stayed in the lower faculties and taught mathematics there from 1589. Two years later, he moved to Padua, his salary rising slightly from 160 Scudi to 160 Ducats a year. In 1599, he invented a military compass and dedicated it to the Venetian Senate to have his salary doubled and his contract extended for six years. When Paolo Sarpi (1552–1623), Galileo’s friend and minor patron, arranged for the spyglass to be presented and dedicated to the Senate in 1609, Galileo’s salary was doubled again and he was tenured for life.
Galileo actually broke off his medical studies and left the university, took private lessons in mathematics and was then on the recommendation of Cardinal del Monte, the Medici Cardinal, appointed to the professorship for mathematics in Pisa. He didn’t invent the military or proportional compass and didn’t dedicate it to the Senate and his salary wasn’t doubled for doing so. Although he did manufacture and sell a superior model together with paid lessons in its use. His salary wasn’t doubled for presenting a telescope to the Senate but was increased to 1000 Scudi.
Of course, we have a section titled, The Galileo Affair: The Church Divorces Science, the title revealing everything we need to know about Gal’s opinion on the topic. No, the Church did not divorce science, as even a brief survey of seventeenth century science following Galileo’s trial clearly shows. Gal states that, “The investigation of the Galileo affair was charged to Cardinal Roberto Bellarmine…”, which simply isn’t true. He naturally points out that Bellarmine, “condemned Bruno to the stake some fifteen years earlier.” Nothing like a good smear campaign.
At one point Gal discuses Bellarmine’s letter to Foscarini and having quoted “…if there were a true demonstration that the sun is at the center of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary; and say rather that we do not understand them than that what is demonstrated is false.
makes the following interesting statement:
Bellarmine was no wide-eyed champion of humanist values. He was a powerful emissary of a domineering institution, and he wasn’t defending only human reason, but also the Church’s privilege to represent it. He wasn’t only stressing that the Church would abide by “a true demonstration,” but also that it retained the right to decide what the criteria for such a demonstration were, and when they’ are met. [my emphasis]
The emphasised statement is at very best highly questionable and at worst completely false. Bellarmine was a highly intelligent, highly educated scholar, who had earlier in his career taught university courses in astronomy. He was well aware what constituted a sound scientific demonstration and would almost certainly have acknowledged and accepted one if one was delivered, without question.
On Galileo’s questioning by the Roman Inquisition Gal writes:
After the first interrogation, he [Galileo] reached a deal which didn’t satisfy the pope and was interrogated again.
This is simply factually wrong; no deal was reached after the first interrogation.
This review is getting far too long, and I think I have already delivered enough evidence to justify what is going to be my conclusion so I will shorten the next sections.
Gal suddenly seems to discover that there were scientific areas other than astronomy and there follows a comparatively long section on the history of medicine that starts with William Harvey then back tracks to ancient Greece before summarising the history of medicine down to the seventeenth century. This is in general OK, but I don’t understand why he devotes four and a half pages to the Leechbook a relatively obscure medieval English medical text, whereas midwives warrant less than two pages.
We are on the home stretch and have reached The New Science, where we discoverer that Galileo originated the mechanical philosophy. Really? No, not really. First up we get told that Buridan originated impetus theory. There is no mention of Johann Philoponus, who actually originated it or the various Arabic scholars, who developed it further and from whom Buridan appropriated it, merely supplying the name. We then get Galileo on mechanics, once again with very little prehistory although both Tartaglia and Benedetti get a mention. Guidobaldo del Monte actually gets acknowledged for his share in the discovery of the parabola law. However, Gal suggests that the guessed it! It’s here that he states that Kepler guessed that the orbit of Mars is an ellipse.
Up next the usual suspects, Descartes and Bacon and I just can’t, although he does, surprisingly, acknowledge that Bacon didn’t really understand how science works. Whoever says Bacon must say scientific societies, with a long discourse on the air pump, which seems to imply that only Boyle and Hooke actually did air pump experiments.
We now reach the books conclusion Sciences Cathedral, remember that opening chapter? This is, naturally, Newton’s Principia. Bizarrely, this section is almost entirely devoted to the exchange of letters between Hooke and Newton on the concept of gravity. Or it appears somewhat bizarre until you realise that Gal has written a whole book about it and is just recycling.
Here we meet our last botched biographical sketch. Having presented Hooke’s biography with the early demise of his father and his resulting financial struggles to obtain an education, Gal turns his attention to Isaac and enlightens his readers with the following:
Isaac Newton: While Hooke was establishing his credentials as an experimenter and instrument builder in Oxford, Isaac Newton (1642–1726) was gaining a name as a mathematical wiz in Cambridge. Like, Hooke, he was an orphan of a provincial clergy man from a little town in Lincolnshire on the east coast of England, and like him he had to work as a servant-student until his talents shone through.
Hannah Newton-Smith née Ayscough, Newton’s mother, would be very surprised to learn that Isaac was an orphan, as she died in 1679, when Isaac was already 37 years old. She would be equally surprised to learn that Isaac’s father, also named Isaac, who died before he was born, was a provincial clergyman. In reality, he was a yeoman farmer. Hannah’s second husband, Newton’s stepfather, Barnabus Smith was the provincial clergyman. Woolsthorpe where Newton was born and grew up was a very little town indeed, in fact it was merely a hamlet. Unlike Hooke who had to work his way through university, Newton’s family were wealthy, when he inherited the family estate, they generated an annual income of £600, a very large sum in the seventeenth century. Why his mother insisted on him entering Cambridge as a subsizar, that is as a servant to other students is an unsolved puzzle. Gal continues:
Newton was a recluse, yet he seemed to have had an intellectual charisma that Hooke lacked. He became such a prodigy student of the great mathematician Isaac Barrow (1630–1677) that in 1669 Barrow resigned in his favour from Cambridge’ newly established, prestigious Lucasian Professorship pf Mathematics.
Here Gal is recycling old myths. Newton was never a student of Isaac Barrow. Barrow did not resign the Lucasian chair in Newton’s favour. He resigned to become a theologian. However, he did recommend Newton as his successor. Further on Gal informs us that:
Newton waited until Hooke’s death in 1703 to publish his Opticks – the subject of the earlier debate – and became the Secretary of the Royal Society, which he brought back from the disarray into which it had fallen after the death of Oldenburg and most of its early members.
I’m sure that the Royal Society will be mortified to learn that Gal has demoted its most famous President to the rank of mere Secretary. This chapter also includes a discussion of the historical development of the concept of force, which to put it mildly is defective, but I can’t be bothered to go into yet more detail. I will just close my analysis of the contents with what I hope was just a mental lapse. Gal writes:
Newton presents careful tables of the periods of the planets of the planets as well as those of the moons of Jupiter and Mercury [my emphasis].
I assume he meant to write Saturn.
To close I will return to the very beginning of the book the front cover. As one can see it is adorned with something that appears at first glance to be an astrolabe. However, all the astrolabe experts amongst my friends went “what the fuck is that?” on first viewing this image. It turns out that it is a souvenir keyring sold by the British Museum. Given that the Whipple Museum of the History of Science in Cambridge has some very beautiful astrolabe, I’m certain that the CUP could have done better than this. The publishers compound this monstrosity with the descriptive text:
Cover image: habaril, via Getty Images. Brass astrolabe, a medieval astronomical navigation instrument.
We have already established that it is in fact not an astrolabe. The astrolabe goes back at least to late antiquity if not earlier, the earliest known attribution is to Theon of Alexandria (C. 335–405 CE), and they continued to be manufactured and used well into the nineteenth century, so not just medieval. Finally, as David King, the greatest living expert on the astrolabe, says repeatably, the astrolabe is NOT a navigation instrument.
Gal’s The Origins of Modern Science has the potential to be a reasonable book, but it is not one that I would recommend as an introduction to the history of science for students. Large parts of it reflect an approach and a standard of knowledge that was still valid thirty or forty years ago, but the discipline has moved on since then. Even if this were not the case the long list of substantive errors that I have documented, and there are probably others that I missed, display a shoddy level of workmanship that should not exist in any history book, let alone in an introductory text for students.
 Ofer Gal, The Origins of Modern Science: From Antiquity to the Scientific Revolution, CUP, Cambridge 2021.
 David C. Lindberg, The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450, University of Chicago Press, Chicago and London, 2nd edition 2007.
In the High Middle Ages formal mathematics was totally dominated by Euclidian geometry, an inheritance from ancient Greece. In the Early Modern Period it would become slowly supplanted by algebra leading eventually to the creation of both analytical geometry and the calculus, the mathematics at the heart of the scientific revolution. However, in antiquity algebra predated Euclid’s geometry. Some form of algebra existed in China, India, Egypt, and Babylon before Euclid’s masterpiece was created.
The various states that occupied the so-called fertile crescent were states with all the power in the hands of a central government. Resources we collected in central warehouses and then distributed to the population. This required a numerate administration, who were responsible for recording, dividing up and accounting for those resources. To do this they first developed a sophisticated base sixty place value number system and then the arithmetic and algebra to manipulate that number system.
The Babylonian mathematical clerks developed their algebra to quite a high level. They could solve linear equation, including indeterminant ones. They had the general solution to the general quadratic equations but only considered positive solutions, as they had no concept of negative numbers. They could also solve various cubic equations but did no appear to have the general solution. Babylonian astronomers applied algebra to their work analysing centuries of observational data of the planetary positions and of solar and lunar eclipses. Their analysis led to accurate algebraic algorithms for predicting the positions of the planets. They also produced algebraic algorithms to predict lunar and solar eclipses. The algorithm for lunar eclipses was very accurate and reliable. For technical reasons the algorithm for solar eclipses could only predict when a solar eclipse might take place but not if it actually would take place. This correctly predicted the actual solar eclipses but produced more false predictions and no means in advance of deciding which was which.
Although the ancient Greeks inherited their astronomy and astrology from the Babylonians, they rejected the Babylonian algebraic approach to the mathematical problems substituting geometrical models for the Babylonian algebraic algorithms. This insistence on solving algebraic problems with geometry, not exclusive to the Greeks, led to much of our terminology for equations. X2 is the area of a square with a side length of X and so second order equations became quadratic, that is square, equations. The same argument leads to third order equations being named cubic equations.
Indian mathematicians, like the Babylonians, also developed a strong arithmetical/algebraic tradition, having like the Babylonians a place value number system. Of interest, there are algebraic problems, in the form of mathematic riddles, that turn up in very similar forms in several ancient cultures, India, Babylon and Egypt, which suggest some form of knowledge transfer in the past, but none has been found to date. Jens Høyrup hypothesised an earlier common source on which they all drew, rather than a direct knowledge transfer.
As with the Hindu-Arabic numerals and the rules for their use it was Brahmagupta (c. 598–c. 668), who in his Brāhmasphuṭasiddhānta provided the most developed presentation of the Indian algebra. He gave solutions to linear equations and the first presentation of the general solution to the quadratic equation in the form that it is still taught today with both positive and negative solutions. In general, his algebra was more advanced than the Babylonian.
As we saw in the last post the Brāhmasphuṭasiddhānta was translated into Arabic in about 770, where it became established. Once again it was a text from Muhammad ibn Musa al-Khwārizmī (c. 780–c. 850), his al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (the compendium on calculation by restoring and balancing”), which was translated into Latin by Robert of Chester in 1145. As also noted in the previous post al- Khwārizmī’s name gave us the word algorithm, although its meaning has changed down the years. The Arabic word al-Jabr, which in everyday language means “reunion of broken parts” becomes the European word algebra. In Spain, which was the major interface between Islamic and Christian culture, an algebrista was a bonesetter, a use that spread throughout Europe.
It should be noted that there were other Arabic algebra texts that were far more advanced mathematically than al-Khwārizmī’s al-Kitāb that were in the Middle Ages never translated into Latin.
As with the introduction of the Hindu-Arabic numerals Robert of Chester’s translation initially had comparatively little impact on the world of formal mathematics. Once again as with the Hindu-Arabic numerals, it was Leonardo of Pisa’s Liber Abbaci(1202, 2nd edition 1227), which also drew heavily on al-Khwārizmī, that established algebra, initially in Northern Italy as part of the commercial arithmetic taught in the abacus schools. Again, as with the Hindu-Arabic numerals the introduction of double entry bookkeeping along with other aspects of the commercial revolution accelerated the spread of the use of algebra. An acceleration increased by the publication of Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita in 1494.
Although algebra started life very much as a form of practical mathematics, throughout its history, its practitioners had developed it beyond their initial or basic needs. The clerks in Babylon and in the Islamic period doing mathematics for its own sake, pushing the boundaries of the discipline. The same took place in Europe during the Renaissance.
One such was the Frenchman Nicolas Chuquet (c. 1450– c. 1495), whose Triparty en la science des nombres was definitely more of an algebra book rather than a reckoning book.
It was never published in his lifetime but was heavily plagiarised. In Germany, in the sixteenth century a movement known as the Cossists developed, who wrote and published Coss books. These are algebra textbooks named after their use of the word Coss, derived from the Italian ‘cosa’ meaning thing, itself a translation of the Latin ‘res’, as a universal term for the unknown in an algebraic problem. The Cossists were generally reckoning masters, but their Coss books are different to their reckoning books.
The earliest Coss author was Christoff Rudolff (c.1500–before 1543), who published his Behend und hübsch Rechnung durch die kunstreichen regeln Algebre, so gemeinicklich die Coß genennt werden (Deft and nifty reckoning with the artful rules of Algebra, commonly called the Coss) in Straßburg in1525.
An extended improved edition under the title, Die Coss Christoffs Rudollfs, was published by Michael Stiffel (c. 1487–1567) in Königsberg in 1553. The last edition of Die Coss was published in Amsterdam in 1615. In the eighteenth century, Leonard Euler (1707–1783) used Rudollfs Coss as his algebra textbook. Michael Stiffel had published his own Coss, Arithmetica Integra, in Nürnberg in 1544.
Robert Recorde’s The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng the extraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers, published in London in 1557 is, as its title clearly states, in the Coss tradition.
Also, in the same tradition was the L’arithmétique by the Netherlander Simon Stevin (1548–1620) published in 1585.
The Coss authors were all outside the university system and the algebra remained outside too. The process that led to the acceptance of algebra in the university system was a slightly different parallel one. To some extent it begins with Pacioli’s claim in his Summa that, unlike the quadratic equation, there was no general solution to the cubic equation. However, at the beginning of the sixteenth century Scipione del Ferro (1465–1526), the professor of mathematics at the University of Bologna found a general solution to one form of the cubic equation. He died without revealing his discovery, which was then inherited by one of his students, Antonio Maria Fior. It was common in this period for mathematicians to challenge each other to public problem-solving competitions and Fior challenged Niccolo Fontana (1500–1557), known as Tartaglia, a leading mathematical exponent, letting it be known that he had a solution to the cubic equation. Tartaglia realised he was on a hiding to nothing and set about studying the problem and came up with a more general solution. On the day of the competition Tartaglia won hands down and achieved overnight fame. Polymath Gerolamo Cardano (1501–1576) seduced Tartaglia into revealing his solution in exchanged for introductions into higher social circles. The condition was however that Cardano was not allowed to publish the solution before Tartaglia had done so. Cardano, however, travelled to Bologna and discovered del Ferro’s solution. In the meantime, Cardano’s student Ludovico Ferrari (1522-1565) had discovered the general solution to the bi-quadratic or quartic equation.
Having expanded del Ferro’s solution to a general one for cubic equations, Cardano combined it with Ferrari’s general solution of the quartic and published them both in his major algebra book Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra), published by Johannes Petreius (c. 1497–1550) in Nürnberg in 1545.
He attributed the solution of the cubic to del Ferro much to the annoyance of Tartaglia, who still hadn’t published. Petreius had previously published Cardano’s abbacus book Practica arithmetice et mensurandi singularis in 1538.
Cardano, a physician, astrologer, philosopher, and mathematician, was a leading intellectual figure in Europe and Petreius was the leading scientific publisher, so the Ars Magna had a major impact helping to establish algebra as an accepted mathematical discipline. Although it’s a claim that I view sceptically, this impact in reflected in the general claim that Ars Magna was the first modern mathematics book.
Cardano had not only accepted positive and negative solutions to all equations but had also to a limited extent worked with complex numbers in the solution of equations as long as they cancelled out. Another Italian mathematician Rafael Bombelli (1526–1572) fully developed the use of complex numbers in his L’Algebra from 1572, an algebra textbook.
The last significant Renaissance algebra book published before the seventeenth century was probably the most significant, the In artem analyticem isagoge (1591) by the French mathematician François Viète (1540–1603). Viète pulled together, systemised, and provided a foundation for much of what had gone before. Most important he produced an algebra that was to a large extent symbolic in its presentation, which introduces a topic that I haven’t dealt with yet.
Historically there were three presentational forms of algebra. Rhetorical algebra in which everything is written out in full sentences with words and not symbols or numerals, for example X+1 = 2 becomes “the thing plus one equals two”. Babylonian and al-Khwārizmī’s algebra were both rhetorical algebras. The second form is syncopated algebra in which some of the words that repeated occur are reduced to abbreviations. This was the case with Brahmagupta’s Brāhmasphuṭasiddhānta. It was also the case with Diophantus’ Arithmetica (difficult to date but probably 3rd century CE). Diophantus’ Arithmetica is something that only became known again in Europe during the Renaissance. If Diophantus’ Arithmetica was originally syncopated is not known, as the oldest known manuscript dates from the 13th century CE. Viète drew heavily for his Isagoge from Diophantus. The third form is symbolic algebra in which both variables and operations are expressed with symbols. Some of the Arabic algebraists developed symbolic algebra but this was not transferred to Europe during the translation movement.
Most of the 16th century algebras are a mixture of rhetorical and syncopated algebra with occasional symbols for operations. The German university lecturer, Johannes Widmann (c. 1460–after 1498) introduced the symbols for plus and minus in his Mercantile Arithmetic oder Behende und hüpsche Rechenung auff allen Kauffmanschafft an abbacus book published in 1489.
Famously, Robert Recorde introduced the equals sign in his Whetstone of witte, although he didn’t invent it, as it had already been in use earlier in Italy.
Viète systematically used symbols for variables and constants, using vowels for the former and consonants for the latter. He, however, did not use symbols for operations. Although his system did not last long it had a massive influence on the symbolisation of algebra.
The introduction of symbols for operations was a rather haphazard process that had begun in the fifteenth century and wasn’t really completed until the eighteenth century. A significant publication was William Oughtred’s Clavis Mathematicae (1631), which was almost entirely symbolic, although most of the symbols he invented did not survive in the long run.
The Clavis Mathematicae went through numerous Latin editions during the seventeenth century and was used widely as a textbook both in Britain and on the continent. Many leading seventeenth century mathematicians, including John Wallis, Christopher Wren, Seth Ward, Isaac Newton, and Edmond Halley, learnt their algebra from it. The first English edition was produced in 1647 and Halley produced a new English translation in 1694.
Thomas Harriot (c. 1560–1621) and René Descartes (1596–1650) both made substantial contributions to the modern form of symbolic algebra.
In the seventeenth century both Pierre de Fermat (1601–1665) and Descartes combined algebra with geometry to create analytical geometry. Both of them, also contributed substantially to the conversion of calculus, which in its origins was geometrical, into an analytical algebraic discipline.
The acceptance and establishment of algebra in the first half of the 17th century was not uncontested and was certainly not a case of the modern scientists accepting and the last generation of traditionalists rejecting. Just to give one interesting example, Galileo, who is oft celebrated as the ‘father of modern science’, rejected it, whereas Christoph Clavius, the last of the great Ptolemaic astronomers not only accepted it but also wrote a textbook based on Viète’s algebra, which was then taught to and by all Jesuit mathematicians, and this although he was involved in a dispute with Viète about the calendar reform.
If you walk up Burgstraße in the city of Nürnberg in the direction of the castle, you will see in front of you the impressive Baroque Fembohaus, which from 1730 to 1852 was the seat of the cartographical publishing house Homännische Erben, that is “Homann’s Heirs” in English. But who was Homann and why was the business named after his heirs?
Fembohaus Source: Wikimedia Commons
Johann Baptist Homann (1664–1724) was born in Öberkammlach in the south of Bavaria. He was initial educated at a Jesuit school and at some point, entered the Dominican Cloister in Würzburg, where he undertook, according to his own account, his “studia humaniora et philosophica.”
Johann Baptists Homann (1664–1725) Portrait by Johann Wilhelm Windter (c. 1696– 1765) Source: Wikimedia Commons
In 1687 he left the cloister moved to Nürnberg and converted to Protestantism. Over the next ten years he vacillated between Catholicism and Protestantism, leaving Nürnberg during the Catholic phases, and returning during the Protestant phases. In 1691 in Nürnberg, he was registered for the first time as a notary public. Around the same time, he started his career as a map engraver. It is not known how or where he learnt this trade, although there are claims that he was entirely self-taught. A map of the district surrounding Nürnberg, produced in 1691/92, shows Homann already as a master in cartographic engraving. From 1693 to 1695 he was in Vienna, then he returned for a time to Nürnberg, leaving again for Erlangen in 1696. Around 1696 to 1697, he was engraving maps in Leipzig.
He appears to have final settled on life as a protestant and permanent residency in Nürnberg in 1698. In 1702 he established a dealership and publishing house for cartography in the city, producing and selling maps, globes, and atlases. His dealership also produced and sold scientific instruments. The field that Homann had chosen to enter was by the beginning of the eighteenth century well established and thriving, with a lot of very powerful competition, in particular from France and Holland. Homann entered the market from a mercantile standpoint rather than a scientific one. He set out to capture the market with high quality products sold more cheaply than the competition, marketing copies of maps rather than originals. In a relatively short time, he had established himself as the dominant cartographical publisher in Germany and also a European market leader.
Planiglobii Terrestris Cum Utroq[ue] Hemisphærio Cælesti Generalis Exhibitio, Nürnberg 1707 Source: Wikimedia Commons
His dealership offered single sheet maps for sale, but he became the first German cartographer to sell atlases on a large scale and is considered the second most important German cartographer after Mercator. His first atlas with forty maps appeared in 1707. This was expanded to the Großen Atlas über die ganze Welt (The Big Atlas of the Entire World), with one hundred and twenty-six maps in 1716.
A fine example of Homann’s 1716 map of Burgundy, one of France’s most important wine regions. Extends to include Lake Geneva in the southwest, Lorraine in the north, Champaigne (Champagne) and Angers to the northwest and Bourgogne to the west. Depicts mountains, forests, castles, and fortifications and features an elaborate title cartouche decorated with cherub winemakers in the bottom right. A fine example of this rare map. Produced by J. H. Homann for inclusion in the Grosser Atlas published in Nuremberg, 1716. Source: Wikimedia Commons
By 1729 it had around one hundred and fifty maps. Johann Baptist’s success was richly acknowledged in his own lifetime. In 1715 he was appointed a member of the Preußischen Akademie der Wissenschaften (The Prussian Academy of Science) and in 1716 he was appointed Imperial Geographer by the Holy Roman Emperor, Karl VI.
A detailed c. 1730 J. B. Homann map of Scandinavia. Depicts both Denmark, Norway, Sweden, Finland and the Baltic states of Livonia, Latvia and Curlandia. The map notes fortified cities, villages, roads, bridges, forests, castles and topography. The elaborate title cartouche in the upper left quadrant features angels supporting a title curtain and a medallion supporting an alternative title in French, Les Trois Covronnes du Nord . Printed in Nuremburg. This map must have been engraved before 1715 when Homann was appointed Geographer to the King. The map does not have the cum privilegio (with privilege; i.e. copyright authority given by the Emperor) as part of the title, however it was included in the c. 1750 Homann Heirs Maior Atlas Scholasticus ex Triginta Sex Generalibus et Specialibus…. as well as in Homann’s Grosser Atlas . Source: Wikimedia Commons
The publishing house continued to grow and prosper until Johann Baptist’s death in 1724, when it was inherited by his son Johann Christian Homann (1703–1730).
Johann Christian studied medicine and philosophy in Halle. He graduated doctor of medicine in 1725, following which he went on a study trip, first returning to Nürnberg in 1729. During his absence the publishing house was managed by Johann Georg Ebersberger (1695–1760) and later together with Johann Christian’s friend from university Johann Michael Franz (1700–1761).
Hand coloured copper engraving by J. Chr. Homann, showing noth west Africa with the Canary Islands and two large cityviews. Source: Wikimedia Commons
When Johann Christian died in 1730, he willed the business to Ebersberger and Franz, who would continue to run the business under the name Homännische Erben. The publishing house passed down through several generations until Georg Christoph Fembo (1781–1848) bought both halves of the business in 1804 and 1813. Fembo’s son closed the business in 1852 and in 1876 the entire collection of books, maps, engravings, and drawing were auctioned off, thus destroying a valuable source for the history of German cartography.
Today there is a big market for fictional maps based on fantasy literature such as Lord of the Rings. This is nothing new and Early Modern fiction also featured such fictional maps, for example Thomas More’s Utopia (1516). One very popular medieval myth concerns the Land of Cockaigne, a fictional paradise of pleasure and plenty also known as The Land of Milk and Honey. The German version is Schlaraffenland (literally the Land of the Lazy Apes). The most well-known version of the myth in the seventeenth century was written by Johann Andreas Schneblins (d. 1702) and based on Schneblins’ account of his travels in the utopia of Schlaraffenland Homann produced a map his very popular Accurata Utopiae Tabula.
“Accurata Utopiæ Tabula” (also named “Schlarraffenlandes”) designed by Johann Baptist Homann and printed in 1694 Source: Wikimedia Commons
From the very beginning one distinctive feature of the publishing house was Homann’s active cooperation with other scholars and craftsmen. From the beginning Johann Baptist worked closely with the engraver, art dealer, and publisher Christoph Weigel the Older (1665–1725).
Christoph Weigel, engraved by Bernhard Vogel of a portrait by Johann Kupetzky Source:Wikimedia Commons
Weigel’s most significant publication was his Ständebuch (1698) (difficult to translate but Book of the Trades and Guilds).
Gunpowder makers, engraving Regensburger Ständebuch, 1698, Christoph Weigel der Ältere (1654, 1725)
Weigel was very successful in his own right but he cooperated very closely with Homann on his map production.
Homann also cooperated closely with the scholar, author, schoolteacher, and textbook writer Johann Hübner (1668–1731).
Johann Hübner, engraving by Johann Kenckel Source: Wikimedia Commons
Together the two men produced school atlases according to Hübner’s pedagogical principles. In 1710 the Kleiner Atlas scholasticus von 18 Charten (Small School Atlas with 18 Maps) was published.
Kleiner Atlas scholasticus von 18 Charten
This was followed in 1719 by the Johann Baptist Homann / Johann Hübner: Atlas methodicus / explorandis juvenum profectibus in studio geographico ad methodum Hubnerianam accommodatus, a Johanne Baptista Homanno, Sacrae Caesareae Majestatis Geographo. Noribergae. Anno MDCCXIX.Methodischer Atlas / das ist, Art und Weise, wie die Jugend in Erlernung der Geographie füglich examiniret werden kann / nach Hübnerischer Lehr-Art eingerichtet von Johann Baptist Homann, Nürnberg, 1719. The title, given here in both Latin and German translates as Methodical Atlas in the manner in which the youth can be reasonably examined in the study of geography according to the pedagogic principles of Hübner, presented by Johann Baptist Homann.
Charte von Europa. Charte von Asia. Charte von Africa. Charte von America. Johanne Baptista Homanno, Norimbergae, 1719 Atlas methodicus / explorandis juvenum profectibus in studio geographico ad methodum Hubnerianam accommodatus
Johann Gottfried Gregorii (1685–1770) was a central figure in the intellectual life of eighteenth-century Germany. A geographer, cartographer, historian, genealogist, and political journalist, he put out a vast number of publications, mostly under the pseudonym Melissantes.
Johann Gottfried Gregorii Source: Wikimedia Commons
In his geographical, cartographical, and historical work he cooperated closely with both Johann Baptist Homann and Christoph Weigel.
One of the Homann publishing house’s most important cooperation’s was with the Nürnberg astronomer Johann Gabriel Doppelmayr (1677–1750).
Johann Gabriel Doppelmayr Source: Wikimedia Commons
Doppelmayr was professor for mathematics at the Aegidianum, Germany’s first modern high school, and is best known for two publication his Historische Nachricht Von den Nürnbergischen Mathematicis und Künstlern (1730), an invaluable source for historian of science and his celestial atlas, Atlas Novus Coelestis (1742). Doppelmayr had been supplying celestial charts for the Homann atlases but his Atlas Novus Coelestis, which was published by Homännische Erben, contained thirty spectacular colour plates and was a leading celestial atlas in the eighteenth century.
PHÆNOMENA circa quantitatem dierum artificialium et solarium perpetuo mutabilem, ex Hypothesi copernicana deducta, cum aliis tam Veterum quam recentiorum Philosophorum, Systematibus mundi notabilioribus, exhibita – Engraved between 1735 and 1742.
Doppelmayr’s successor as professor of mathematics at the Aegidianum was Georg Moritz Lowitz (1722–1774), who went on to become professor for practical mathematics at the University of Göttingen.
Georg Moriz Lowiz Source: Wikimedia Commons
He worked together with Johann Michael Franz and produced several astronomical publications for the Homännische Erben. Franz as well as being co-manager of the publishing house was also an active geographer, who became professor in Göttingen in 1755. He also published a series of his own books on geographical themes. He sold his share of the publishing house on his younger brother Jacob Heinrich Franz (1713–1769) in 1759.
Johann Michael Franz: Belgium, Luxemburg; Johann Michael Franz – Circulus Burgundicus – 1758
It was during his time in Nürnberg that he did his work on lunar astronomy. Like Lowitz, and Franz, Mayer also became a professor in Göttingen, in his case for economics and mathematics.
The three Göttingen professors–Lowitz, Franz, and Mayer–whilst still working for Homann in Nürnberg founded the Cosmographische Gesellschaft (Cosmographical Society), with the aim of improving the standards of cartography and astronomy. Due to lack of funding they never really got their plans of their grounds. Their only products being some propaganda publications for the society written by Franz and one publication from Mayer on his lunar research.
Each of the scholars, briefly sketched here was a leading figure in the intellectual landscape of eighteenth-century Germany and they were all to some extent rivals on the open knowledge market. However, they cooperated rather than competed with each other and in doing so increased the quality of their output.
The part of mathematics that we most use in our lives is numbers, the building blocks of arithmetic. Today, we mostly use the Hindu-Arabic numerals and the associated place value decimal system, but this was not always the case. In fact, although this number system first entered Europe during the 12th century translation movement, it didn’t become truly established until well into the Renaissance.
First, we will briefly track the Hindu-Arabic place value decimal system from its origins till its advent in Europe. The system emerged in India sometime late in the sixth century CE. Āryabhaṭa (476–550) a leading mathematician and astronomer doesn’t mention them in his Aryasiddhanta. The earliest known source being in the Āryabhaṭīyabhāṣya of Bhāskara I (c. 600–c. 680) another leading astronomer mathematician. The full system, as we know it today, was described in the Brāhmasphuṭasiddhānta of Brahmagupta (c. 598–c. 668 n. Chr.). The only difference is that he allows division by zero, which as we all learnt in the school is not on.
The Brāhmasphuṭasiddhānta was translated into Arabic in about 770 by Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (d. 777), Muhammad ibn Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (d. c. 800) and Yaʿqūb ibn Ṭāriq (d. c. 796). The first two are father and son. Having teams doing scientific translations in the middle ages was actually very common. I won’t go into detail, but it should be noted that it took several hundred years for this system to replace the existing number systems in Arabic culture, many mathematicians preferring their own systems, which they considered superior.
The system first came into Europe in the 12th century through the translation of a work by Muhammad ibn Musa al-Khwārizmī (c. 780–c. 850) by an unknown translator. No Arabic manuscript of this work is known to exist, and it is only known by its Latin title Algoritmi de Numero Indorum, where Algoritmi is a corruption of al-Khwārizmī.
This translation only had a very limited impact. The new number system was adopted by the scholars at the universities as part of computus in order to calculate the date of Easter and the other moveable Church feasts. Leading scholars such as Sacrobosco wrote textbooks to teach the new discipline, which was Algorimus, another corruption of al-Khwārizmī. The other mostly university-based scholars, who used mathematics extensively, the astronomers, continued to use a sexagesimal i.e., base sixty, number system that they had inherited from both the Greek and the Arabic astronomers. This system would stay in use by astronomers down to Copernicus’ De revolutionibus (1543) and beyond.
This is the opening page of a 1490 manuscript copy of Johannes de Sacro Bosco’s Tractatus de Arte Numerandi, also referred to as his Algorismus Source:
What about the world outside of the universities? In the outside world the new number system was simply ignored. Which raises the question why? People generally believe that the base ten place value number system is vastly superior to the Roman numeral system that existed in Europe in the Middle ages, so why didn’t the people immediately adopt it? After all you can’t do arithmetic with Roman numerals. The thing is people didn’t do arithmetic with Roman numerals, although it would have been possible using different algorithm to the ones we use for the decimal place-value system. People did the calculations using either finger reckoning
of counting boards, also known as reckoning boards or abacuses. They only used the roman numerals to record the results.
In the hands of a skilled operator the counting board is a powerful instrument. It can be used very simply for addition and subtraction and using the halving and doubling algorithms, almost as simply for multiplication and division. A skilled operator can even extract roots using a counting board. The counting board also offers the possibility in a business deal for the reckoning masters of both parties to observe and control the calculations on the counting board.
The widespread use of counting boards over many centuries is still reflected in modern word usage. The serving surface in a shop is called a counter because it was originally the counting board on which the shop owner did their calculations. The English finance ministry is called the Exchequer after a special kind of counting board on which they did they calculations in the past. Nobody pays much attention to the strange term bankrupt, which also has its origins in the use of counting boards. The original medieval banks in Northern Italy were simply tables, Italian banca, on the marketplace, on which a printed cloth counting board was spread out. If the bankers were caught cheating their customers, then the authorities came and symbolically destroyed their table, in Italian, banca rotta, broken table.
This was basically a book on commercial arithmetic, following its Arabic origins. The Arabic/Islamic culture used different number systems for different tasks and used the Hindu-Arabic numerals and the decimal place-value system extensively in commercial arithmetic, in general account keeping, to calculate rates of interest, shares in business deals and the division of inheritance according to the complex Islamic inheritance laws. Leonardo’s father was a customs officer in North Africa, and it was here that Leonard learnt of the Hindu-Arabic numerals and the decimal place-value system from Arab traders in its usage as commercial arithmetic.
This new introduction saw the gradual spread in Norther Italy of Scuole or Botteghe D’abbaco (reckoning schools) lead by a Maestri D’abbaco (reckoning master), who taught this new commercial arithmetic to apprentice traders from Abbaco Libro (reckoning books), which he usually wrote himself. Many leading Renaissance mathematici, including Peter Apian (1495–1552, Niccolò Fontana Tartaglia (c. 1500–1557), Gerolamo Cardano (1501–1576), Gemma Frisius (1508–1555) and Robert Recorde (c. 1512–1558), wrote a published abbacus books. The very first printed mathematics book the Arte dell’Abbaco also known as the Treviso Arithmetic (1478) was , as the title clearly states, an abacus book.
Un maestro d’abaco. Filippo Calandri, De arimetricha opusculum, Firenze 1491
This practice began to accelerate with the introduction of double entry bookkeeping. This was part of the more general so-called commercial revolution, which included the founding of the first banks and the introduction of bills of exchange to eliminate the necessity of traders carrying large amounts of gold or silver. Developments in Europe that lead to the Renaissance. The earliest known example of double entry bookkeeping is the Messari Report of the Republic of Genoa, 1340. The earliest account of double entry bookkeeping is the Libro dell’arte di mercatura by Benedetto Cotrugli (1416–1469), which circulated in manuscript but was never printed. The first printed account was in the highly successful Summa de arithmetica, geometria, proportioni et proportionalita of Fra. Luca Bartolemeo Pacioli (c.1447–1517) published in 1494, which contain the twenty-seven-page introduction to double entry bookkeeping, Particularis de computis et scripturis.
Particularis de computis et scripturis, about double-entry bookkeeping.
Beginning with the Southern German trading centres of Augsburg, Regensburg and Nürnberg, which all traded substantially with the Northern Italian commercial centres, the new commercial arithmetic and double entry bookkeeping began to expand throughout Europe. This saw the fairly rapid establishment of reckoning schools and the printing of reckoning books throughout the continent. We can see the partial establishment of the Hindu-Arabic numerals some four hundred years after their first introduction, although they were used principally for recording, the reckoning continuing to be done on a counting board, in many cases down to the eighteenth century.
By the end of the sixteenth century, the base ten positional value number system with Hindu-Arabic numerals had become well established across the whole spectrum of number use, throughout Europe. The Indian decimal system had no fractions and decimal fractions were first introduced into the Hindu-Arabic numerals by Abu’l Hasan Ahmad ibn Ibrahim Al-Uqlidisi in his Kitab al-Fusul fi al-Hisab al-Hindi around 952 and then again independently by Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (c. 1380–1429) in his Key to Arithmetic (1427). They first emerged in Europe in 1585 in Simon Stevin’s De Thiende also published in French as La Disme. The decimal point or comma was first used in Europe by Christoph Clavius (1538–1612) in the goniometric tables for his astrolabe in 1593. Its use became widespread through its adoption by John Napier in his Mirifici Logarithmorum Canonis Descriptio (1614).
However, at the end of the seventeenth century we still find both John Evelyn (1620–1706) and John Arbuthnot (1667–1735) discussing the transition from Roman to Hindu-Arabic numerals in their writings; the former somewhat wistfully, the later thankfully.
In the eighteenth century, Pierre-Simon Laplace reputedly said:
‘It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.’
A very positive judgement, with hindsight, of the base ten place value number system with Hindu-Arabic numerals but one that was obviously not shared in the Early Modern period when the system was initially on offer in Europe.
One of the defining aspects of the so-called scientific revolution was the massive increase in experimentation as a method to discover or confirm knowledge of the natural world, replacing the empirical observation or experience of Aristotelian scientia. Ignoring the trivial and fatuous, but unfortunately still widespread, claim that Galileo invented experimental science, it is an important area of the history of Early Modern science to trace and analyse how, when and where this methodological change took place. This transition, a very gradual one, actually took place in various areas of knowledge acquisition during the Renaissance and might well be regarded as one of the defining features of Renaissance science, separating it from its medieval predecessor.
One, perhaps surprising, area where this transition took place was in the testing of poisons and their antidotes, as brilliantly researched, described, analysed and reported by Alisha Rankin in her new book, The Poison Trials: Wonder Drugs, Experiment, and the Battle for Authority in Renaissance Science.
The starting point for Rankin’s fascinating story is that it was apparently considered acceptable for a large part of the sixteenth century to test poisons and above all their supposed antidotes on human beings. No, you didn’t misread that last sentence, during the sixteenth century test on poisons and their antidotes were carried out by physicians, with the active support of the ruling establishment, on condemned prisoners, truly shades of Mengele and Auschwitz.
This medical practice of testing didn’t, however, begin in the Renaissance but there are precedence cases throughout history beginning in antiquity and occurring intermittently all the way up to the Renaissance. Testing poisons and their antidotes mostly on animals, although tests on criminals existed as well. Rankin’s opening chapter is a detailed sketch and analysis of poison trials that preceded the Renaissance, as well as a general history of poisons and their antidotes.
Her second chapter then deals in detail with the trial ordered by Pope Clement VII in 1524 of the antidote Oleum Clementis created by the surgeon Gregorio Caravita. Here a new chapter in the history of testing was opened, as this antidote was tested on two condemned prisoners under the supervision of a physician. Both prisoners were given a dose of a known strong poison and one was given a dose of the antidote. The prisoner, who received the antidote survived and the trial and its results were publicised creating a medical sensation.
Rankin explains that there was an obsession with poison and poisoning amongst the rich and powerful during the Renaissance, so the interest in methods of both detecting poisons and combating their effects was very strong amongst those in power, the most likely victims of an attempted poisoning. This also meant that there was an interest amongst physicians, apothecaries, and empirics to find or create such potions, as a route to fame and fortune.
Having set the scene, in the rest of her book Rankin takes us through the sixteenth century through periods of testing on both humans and animals, into the seventeenth and the scientific revolution. Along route she introduces us to newly invented antidotes and their inventors/discoverers but also to an incredible amount of relevant contextual information.
We learn, for example, that poison antidotes were not considered poison specific but worked against all poisons, if they worked at all. We also learn that plagues were not, like other more common ailments, to be caused by an imbalance of the four humours, as taught by Galen, but were a poisoning of the body, so that a poison antidote, should or would function as a cure for plagues as well.
Alongside the purely medical descriptions, we also get the full spectrum of the social, political, cultural, ethical, and economic contexts in which the poison trials took place. A poison trial sanctioned by a head of state and carried out by a learned physician had, naturally, a completely different status to one carried out by an empiric on the town square during a local fair.
(I’m still hunting for a possible translation into modern English of the term empiric. This is, usually, simply translated as quack, and whilst it is true that many empirics were what we would now call quacks, the spectrum of their medical activities was not just confined to conning people. Quite a lot of them did offer genuine medical services, no more and no less effective than those of the university educated physicians.)
Rankin goes into great detail on how the physicians sought to present their trials, so that they were seen to be scholarly as opposed to the snake-oil salesman trials of the empirics. Writing detailed protocols of the progress of the victim’s condition following the administration of the poison dose and the antidote, noting times and nature of vomiting, sweating, diarrhoea etc, giving their trials at least the appearance of a controlled experiment. This is contrasted with the simple public presentations of the empirics.
Also, important, and highly relevant to the historical development of science, Rankin discusses and analyses the use of the terms, ‘experience’, ‘experiment’, and ‘proof’ in the descriptions of poison trials. The transition from Aristotelian experience to empirical experiment being one of the defining characteristics of the scientific revolution.
In the final section of her book Rankin expands her remit to cover the history of the universal cures on offer during the period, both the exotic imported kind as well as the locally discovered/invented ones.
An important element of the whole story that Rankin deals with extensively is how the various vendors of antidotes and universal cures advertised and promoted their wares. Hereby, the question whether reports of successful trials or testimonials from cured patient carried the greater weight is examined. We are of course well into the age of print and there was a flourishing market for books and pamphlets praising one’s own wonder products or damning those of one’s rivals.
Rankin tells a highly comprehensive tale of a fascinating piece of Renaissance medical history. It is thoroughly researched and presented in exhaustive detail. A true academic work, it has extensive endnotes (unfortunately not footnotes), a voluminous bibliography of both primary and secondary sources, and an excellent index. It is also pleasantly illustrated with the, in the meantime ubiquitous, grey scale illustrations. However, despite the academic rigour, Rankin has a light, literary style and her prose is truly a pleasure to read.
I really enjoyed reading this book and I would say that this volume is a must read for anybody involved in the history of Early Modern and/or Renaissance medicine but also more generally for those working on the history of Early Modern and/or Renaissance science, or simply Early Modern and/or Renaissance history. I would also recommend it, without reservations, for any general readers, who like to read well written accounts of interesting episodes in history.
 Alisha Rankin, The Poison Trials: Wonder Drugs, Experiment, and the Battle for Authority in Renaissance Science, The University of Chicago Press, Chicago & London, 2021
In the last two episodes we have looked at developments in printing and art that, as we will see later played an important role in the evolving Renaissance sciences. Today, we begin to look at another set of developments that were also important to various areas of the newly emerging practical sciences, those in mathematics. It is a standard cliché that mathematisation played a central roll in the scientific revolution but contrary to popular opinion the massive increase in the use of mathematics in the sciences didn’t begin in the seventeenth century and certainly not as the myth has it, with Galileo.
Medieval science was by no means completely devoid of mathematics despite the fact that it was predominantly Aristotelian, and Aristotle thought that mathematics was not scientia, that is, it did not deliver knowledge of the natural world. However, the mathematical sciences, most prominent astronomy and optics, had a fairly low status within medieval university culture.
One mathematical discipline that only really became re-established in Europe during the Renaissance was trigonometry. This might at first seem strange, as trigonometry had its birth in Greek spherical astronomy, a subject that was taught in the medieval university from the beginning as part of the quadrivium. However, the astronomy taught at the university was purely descriptive if not in fact even prescriptive. It consisted of very low-level descriptions of the geocentric cosmos based largely on John of Sacrobosco’s (c. 1195–c. 1256) Tractatus de Sphera (c. 1230). Sacrobosco taught at the university of Paris and also wrote a widely used Algorismus, De Arte Numerandi. Because Sacrobosco’s Sphera was very basic it was complimented with a Theorica planetarum, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos. Mathematical astronomy requiring trigonometry was not hardy taught and rarely practiced.
Both within and outside of the universities practical astronomy and astrology was largely conducted with the astrolabe, which is itself an analogue computing device and require no knowledge of trigonometry to operate.
Before we turn to the re-emergence of trigonometry in the medieval period and its re-establishment during the Renaissance, it pays to briefly retrace its path from its origins in ancient Greek astronomy to medieval Europe.
The earliest known use of trigonometry was in the astronomical work of Hipparchus, who reputedly had a table of chords in his astronomical work. This was spherical trigonometry, which uses the chords defining the arcs of circles to measure angles. Hipparchus’ work was lost and the earliest actual table of trigonometrical chords that we know of is in Ptolemaeus’ Mathēmatikē Syntaxis or Almagest, as it is usually called today.
The chord of an angle subtends the arc of the angle. Source: Wikimedia Commons
When Greek astronomy was appropriated in India, the Indian astronomers replaced Ptolemaeus’ chords with half chords thus creating the trigonometrical ratios now known to us, as the sine and the cosine.
It should be noted that the tangent and cotangent were also known in various ancient cultures. Because they were most often associated with the shadow cast by a gnomon (an upright pole or post used to track the course of the sun) they were most often known as the shadow functions but were not considered part of trigonometry, an astronomical discipline. So-called shadow boxes consisting of the tangent and cotangent used for determine heights and depths are often found on the backs of astrolabes.
Shadow box in the middle of a drawing of the reverse of Astrolabium Masha’Allah Public Library Bruges [nl] Ms. 522. Basically the tangent and cotangent functions when combined with the alidade
Islamic astronomers inherited their astronomy from both ancient Greece and India and chose to use the Indian trigonometrical half chord ratios rather than the Ptolemaic full cords. Various mathematicians and astronomers made improvements in the discipline both in better ways of calculating trigonometrical tables and producing new trigonometrical theorems. An important development was the integration of the tangent, cotangent, secant and cosecant into a unified trigonometry. This was first achieved by al-Battãnī (c.858–929) in his Exhaustive Treatise on Shadows, which as its title implies was a book on gnonomics (sundials) and not astronomy. The first to do so for astronomy was Abū al-Wafā (940–998) in his Almagest.
Image of Abū al-Wafā Source: Wikimedia Commons
It was this improved, advanced Arabic trigonometry that began to seep slowly into medieval Europe in the twelfth century during the translation movement, mostly through Spain. It’s reception in Europe was very slow.
The first medieval astronomers to seriously tackle trigonometry were the French Jewish astronomer, Levi ben Gershon (1288–1344), the English Abbot of St Albans, Richard of Wallingford (1292–1336) and the French monk, John of Murs (c. 1290–c. 1355) and a few others.
Johannes von Gmunden was instrumental in establishing the study of mathematics and astronomy at the University of Vienna, including trigonometry. His work in trigonometry was not especially original but displayed a working knowledge of the work of Levi ben Gershon, Richard of Wallingford, John of Murs as well as John of Lignères (died c. 1350) and Campanus of Novara (c. 200–1296). His Tractatus de sinibus, chordis et arcubus is most important for its probable influence on his successor Georg von Peuerbach.
Peuerbach produced an abridgement of Gmunden’s Tractatus and he also calculated a new sine table. This was not yet comparable with the sine table produced by Ulugh Beg (1394–1449) in Samarkand around the same time but set new standards for Europe at the time. It was Peuerbach’s student Johannes Regiomontanus, who made the biggest breakthrough in trigonometry in Europe with his De triangulis omnimodis (On triangles of every kind) in 1464. However, both Peuerbach’s sine table and Regiomontanus’ De triangulis omnimodis would have to wait until the next century before they were published. Regiomontanus’ On triangles did not include tangents, but he rectified this omission in his Tabulae Directionum. This is a guide to calculating Directions, a form of astrological prediction, which he wrote at the request for his patron, Archbishop Vitéz. This still exist in many manuscript copies, indicating its popularity. It was published posthumously in 1490 by Erhard Ratdolt and went through numerous editions, the last of which appeared in the early seventeenth century.
A 1584 edition of Regiomontanus’Tabulae Directionum Source
Peuerbach and Regiomontanus also produced their abridgement of Ptolemaeus’ Almagest, the Epitoma in Almagestum Ptolemae, published in 1496 in Venice by Johannes Hamman. This was an updated, modernised version of Ptolemaeus’ magnum opus and they also replaced his chord tables with modern sine tables. A typical Renaissance humanist project, initialled by Cardinal Basilios Bessarion (1403–1472), who was a major driving force in the Humanist Renaissance, who we will meet again later. The Epitoma became a standard astronomy textbook for the next century and was used extensively by Copernicus amongst others.
Title page Epitoma in Almagestum Ptolemae Source: Wikimedia Commons
Regiomontanus’ De triangulis omnimodis was edited by Johannes Schöner and finally published in Nürnberg in 1533 by Johannes Petreius, together with Peuerbach’s sine table, becoming a standard reference work for much of the next century. This was the first work published, in the European context, that treated trigonometry as an independent mathematical discipline and not just an aide to astronomy.
Copernicus (1473–1543,) naturally included modern trigonometrical tables in his De revolutionibus. When Georg Joachim Rheticus (1514–1574) travelled to Frombork in 1539 to visit Copernicus, one of the books he took with him as a present for Copernicus was Petreius’ edition of De triangulis omnimodis. Together they used the Regiomontanus text to improve the tables in De revolutionibus. When Rheticus took Copernicus’ manuscript to Nürnberg to be published, he took the trigonometrical section to Wittenberg and published it separately as De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in 1542, a year before De revolutionibus was published.
Rheticus’ action was the start of a career in trigonometry. Nine years later he published his Canon doctrinae triangvlorvmin in Leipzig. This was the first European publication to include all of the six standard trigonometrical ratios six hundred years after Islamic mathematics reached the same stage of development. Rheticus now dedicated his life to producing what would become the definitive work on trigonometrical tables his Opus palatinum de triangulis, however he died before he could complete and publish this work. It was finally completed by his student Valentin Otto (c. 1548–1603) and published in Neustadt and der Haardt in 1596.
Source: Wikimedia Commons
In the meantime, Bartholomäus Piticus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solution triangulorum tractatus brevis et perspicuous, one year earlier, in 1595.
Source:. Wikimedia Commons
This work was republished in expanded editions in 1600, 1608 and 1612. The tables contained in Pitiscus’ Trigonometria were calculated to five or six places, whereas those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. In comparison Peuerbach’s sine tables from the middle of the fifteenth century were only accurate to three places of decimals. However, on inspection, despite the years of effort that Rheticus and Otho had invested in the work, some of the calculations were found to be defective. Pitiscus recalculated them and republished the work as Magnus canon doctrinae triangulorum in 1607.
He published a second further improved version under the title Thesaurus mathematicus in 1613. These tables remained the definitive trigonometrical tables for three centuries only being replaced by Henri Andoyer’s tables in 1915–18.
In the seventeenth century a major change in trigonometry took place. Whereas throughout the Renaissance it had been handled as a branch of practical mathematics, used to solve spherical and plane triangles in astronomy, cartography, surveying and navigation, the various trigonometrical ratios now became mathematical functions in their own right, a branch of purely theoretical mathematics. This transition mirroring the general development in the sciences that occurred between the Renaissance and the scientific revolution, from practical to theoretical science.
As an Englishman brought up on tales, myths and legends of Francis Drake, Walter Raleigh, Admiral Lord Nelson, the invincible Royal Navy and Britannia rules the waves, I tend not to think about the fact that Britain was not always a great seafaring nation. As an island there were, of course, always fisher boats going about their business in the coastal waters and archaeology has shown us that people have been crossing the strip of water between Britain and the continent, as long as the island has been populated. However, British sailors only really began to set out onto the oceans for distant lands in competition to their Iberian brethren during the Early Modern Period. Before the start of these maritime endeavours there was a political movement in England to get those in power to take up the challenge and compete with the Spanish and the Portuguese in acquiring foreign colonies, gold, silver and exotic spices. One, today virtually unknown, man, whose writings played a not insignificant role in this political movement was the alchemist Ricard Eden (c. 1520–1576).
Richard Eden was born into an East Anglian family of cloth merchants and clerics, the son of George Eden a cloth merchant. He studied at Christ’s College Cambridge (1534–1537) and then Queen’s College, where he graduated BA in 1538 and MA in 1544. He studied under Sir Thomas Smith (1533–1577) a leading classicist of the period, who was also politically active and a major supporter of colonialism, which possibly influenced Eden’s own later involvement in the topic.
A c. 19th-century line engraving of Sir Thomas Smith. Source: Wikimedia Commons
Through Smith, Eden was introduced to John Cheke (1514–1557), Roger Ascham (c. 1515–1568) and William Cecil (1520–1598), all of whom were excellent classicists and statesmen. Cecil would go on under Elizabeth I to become the most powerful man in England. From the beginning Eden moved in the highest intellectual and political circles.
After leaving Cambridge Eden was appointed first to a position in the Treasury and then distiller of waters to the royal household, already indicating an interest in and a level of skill in alchemy. Eden probably acquired his interest in alchemy from his influential Cambridge friends, who were all eager advocates of the art. However, he lost the post, probably given to someone else by Somerset following Henry VIII’s death in 1547 and so was searching for a new employer or patron.
Through a chance meeting he became acquainted with the rich landowner Richard Whalley, who shared his interest in alchemy. Whalley provided him with a house for his family and an income, so that he could devote himself to both medicinal and transmutational alchemy. His activities as an alchemist are not of interest here but one aspect of his work for Whalley is relevant, as it marked the beginning of his career as a translator.
Whalley was obviously also interested in mining for metal ores, because he commissioned Eden to translate the whole of Biringuccio’s Pirotechnia into English. Although he denied processing any knowledge of metal ores, Eden accepted the commission and by 1552 he had completed twenty-two chapters, that is to the end of Book 2. Unfortunately, he lent the manuscript to somebody, who failed to return it and so the project was never finished. In fact, there was no English translation of the Pirotechnia before the twentieth century. Later he produced a new faithful translation of the first three chapters dealing with gold, silver and copper ores, only omitting Biringuccio’s attacks on alchemy, for inclusion, as we shall see, in one of his later works.
Title page, De la pirotechnia, 1540, Source: Science History Museum via Wikipedia Commons
In 1552, Eden fell out with Whalley and became a secretary to William Cecil. It is probable the Cecil employed him, as part of his scheme to launch a British challenge to the Iberian dominance in global trade. In his new position Eden now produced a translation of part of Book 5 of Sebastian Münster’s Cosmographia under the title A Treatyse of the New India in 1553. As I explained in an earlier blog post Münster’s Cosmographiawas highly influential and one of the biggest selling books of the sixteenth century.
This first cosmographical publication was followed in 1555 by his The Decades of the newe worlde or west India, containing the nauigations and conquests of the Spanyardes… This was a compendium of various translations including those three chapters of Biringuccio, probably figuring that most explorers of the Americas were there to find precious metals. The main parts of this compendium were taken from Pietro Martire d’Anghiera’s De orbe novo decades and Gonzalo Fernández de Oviedo y Valdés’ Natural hystoria de las Indias.
Source: The British Library
Pietro Martire d’Anghiera (1457–1526) was an Italian historian in the service of Spain, who wrote the first accounts of the explorations of Central and South America in a series of letters and reports, which were published together in Latin. His De orbe novo (1530) describes the first contacts between Europeans and Native Americans.
Source: Wikimedia Commons
Gonzalo Fernández de Oviedo y Valdés (1478–1557) was a Spanish colonist, who arrived in the West Indies a few years after Columbus. His Natural hystoria de las Indias (1526) was the first text to introduce Europeans to the hammock, the pineapple and tobacco.
MS page from Oviedo’s La Natural hystoria de las Indias. Written before 1535, this MS page is the earliest known representation of a pineapple Source: Wikimedia Commons
Important as these writings were as propaganda to further an English involvement in the new exploration movement in competition to the Iberian explorers, it was probably Eden’s next translation that was the most important.
As Margaret Schotte has excellently documented in her Sailing School (Johns Hopkins University Press, 2019) this new age of deep-sea exploration and discovery led the authorities in Spain and Portugal to the realisation that an active education and training of navigators was necessary. In 1552 the Spanish Casa de la Contratación established a formal school of navigation with a cátedra de cosmografia (chair of cosmography). This move to a formal instruction in navigation, of course, needed textbooks, which had not existed before. Martín Cortés de Albacar (1510–1582), who had been teaching navigation in Cádiz since 1530, published his Breve compendio de la sphere y de la arte de navegar in Seville in 1551.
Retrato de Martín Cortés, ilustración del Breve compendio de la sphera y de la arte de navegar, Sevilla, 1556. Biblioteca Nacional de España via Wikimedia Commons
In 1558, an English sea captain from Dover, Stephen Borough (1525–1584), who was an early Artic explorer, visited Seville and was admitted to the Casa de la Contratación as an honoured guest, where he learnt all about the latest instruments and the instruction for on going navigators. On his return to England, he took with him a copy of Cortés’ Breve compendio, which he had translated into English by Richard Eden, as The Arte of Navigation in 1561. This was the first English manual of navigation and was immensely popular going through at least six editions in the sixteenth century.
In 1562, Eden became a companion to Jean de Ferrières, Vidame of Chartres, a Huguenot aristocrat, who raised a Protestant army in England to fight in the French religious wars. Eden, who was acknowledged as an excellent linguist, stayed with de Ferrières until 1573 travelling extensively throughout France and Germany. Following the St. Batholomew’s Day massacre, which began in the night of 23–24 August 1572, Eden together with de Ferrières party fled from France arriving in England on 7 September 1573. At de Ferrières request, Elizabeth I admitted Eden to the Poor Knights of Windsor, a charitable organisation for retired soldiers, where he remained until his death in 1576.
After his return to England Eden translated the Dutch musician and astrologer, Jean Taisnier’s Opusculum perpetua memoria dignissimum, de natura magnetis et ejus effectibus, Item de motu continuio, which was a plagiarism of Petrus Peregrinus de Maricourt’s (fl. 1269) Epistola de magnete and a treatise on the fall of bodies by Giambattista Benedetti (1530–1590) into English.
This was published posthumously together with his Arte of Navigation in 1579. His final translation was of Ludovico de Varthema’s (c. 1470–1517) Intinerario a semi-fictional account of his travels in the east. This was published by Richard Willes in The History of Travayle an enlarged version of his Decades of the newe worlde in 1577.
Eden’s translations and publications played a significant role in the intellectual life of England in the sixteenth century and were republished by Richard Hakluyt (1553–1616) in his The Principal Navigations, Voiages, Traffiques and Discoueries of the English Nation (1589, 1598, 1600), another publication intended as propaganda to promote English colonies in America.
Unlike Sebastian Münster or Richard Hakluyt, Eden has been largely forgotten but he made important and significant contributions to the history of cosmography and deserves to be better known.
 I want to thank Jenny Rampling, whose book The Experimental Fire, which I reviewed here, made me aware of Richard Eden, although, I have to admit, he turns up, managing to slip by unnoticed in other books that I own and have read.
 The biographical details on Eden are mostly taken from the ODNB article. I would like to thank the three wonderful people, who provided me with a pdf of this article literally within seconds of me asking on Twitter
If your philosophy of [scientific] history claims that the sequence should have been A→B→C, and it is C→A→B, then your philosophy of history is wrong. You have to take the data of history seriously.
John S. Wilkins 30th August 2009
Culture is part of the unholy trinity—culture, chaos, and cock-up—which roam through our versions of history, substituting for traditional theories of causation. – Filipe Fernández–Armesto “Pathfinders: A Global History of Exploration”