Category Archives: Renaissance Science

Exposing Galileo’s strawmaning

There is a widespread, highly erroneous, popular perception in the world, much loved by gnu atheists and supporters of scientism, that as soon as Petreius published Copernicus’s De Revolutionibus in 1543 the question as to which was the correct astronomical/cosmological system for the cosmos was as good as settled and that when Galileo published his Dialogo[1] everything was finally done and dusted and anybody who still persisted in opposing the acceptance of the heliocentric world view, did so purely on grounds of ignorant, anti-science, religious prejudice. Readers of this blog will know that I have expended a certain amount of energy and several thousand words over the years countering this totally mistaken interpretation of the history of astronomy in the early modern period and today I’m going to add even more words to the struggle.

Galileo is held up by his numerous acolytes as a man of great scientific virtue, who preached a gospel of empirical scientific truth in the face of the superstitious, faith based errors of his numerous detractors; he was a true martyr for science. The fact that Galileo was capable of scientific skulduggery does not occur to them, but not only was he capable of such his work is littered with examples of it. One of his favourite tactics was not to present his opponents true views when criticising them but to create a strawman, claiming that this represents the views of his opponent and then to burn it down with his always-red-hot rhetorical flamethrower.

Towards the end of The First Day in the Dialogo, Galileo has Simplicio, the fall guy for geocentricity, introduce a “booklet of theses, which is full of novelties.” Salviati, who is the champion of heliocentricity and at the same time Galileo’s mouthpiece, ridicules this booklet as producing arguments full of “falsehoods and fallacies and contradictions” and as “thinking up, one by one, things that would be required to serve his purposes, instead of adjusting his purposes step by step to things as they are.” Galileo goes on to do a polemical hatchet job on what he claims are the main arguments in said “booklet of theses.” Amongst others he accuses the author of “setting himself up to refute another’s doctrine while remaining ignorant of the basic foundations upon which the whole structure are supported.”

The “booklet of theses”, which Galileo doesn’t name is in fact the splendidly titled:


English translation of the Latin title page Source: Notre Dame Press

Now most of the acolytes who fervently praise Galileo as the great defender of science against superstition probably have no idea who Johann Georg Locher was but they might well have heard of Christoph Scheiner, who was famously embroiled in a dispute with Galileo over the nature of sunspots and who first observed them with a telescope. In fact the authorship of the Mathematical Disquisitions has often falsely attributed to Scheiner and Galileo’s demolition of it seen as an extension of that dispute and it’s sequel in the pages of his Il Saggiatore.

Whereas Galileo’s Dialogo has been available to the general reader in a good English translation by Stillman Drake since 1953, anybody who wished to consult Locher’s Mathematical Disquisitions in order to check the veracity or lack thereof of Galileo’s account would have had to hunt down a 17th century Latin original in the rare books room of a specialist library. The playing field has now been levelled with the publication of an excellent modern English translation of Locher’s booklet by Renaissance Mathematicus friend, commentator and occasional guest contributor Chris Graney[2].


Graney’s translation (Christopher M. Graney, Mathematical Disquisitions: The Booklet of Theses Immortalised by Galileo, University of Notre Dame Press, Notre Dame, Indiana, 2017)  presents a more than somewhat different picture of Locher’s views on astronomy to that served up by Galileo in the Dialogo and in fact gives us a very clear picture of the definitely rational arguments presented by the opponents to heliocentricity in the early part of the seventeenth century. The translation contains an excellent explanatory introduction by Graney, extensive endnotes explaining various technical aspects of Locher’s text and also some of the specific translation decisions of difficult terms. (I should point out that another Renaissance Mathematicus friend, Darin Hayton acted as translation consultant on this volume). There is an extensive bibliography of the works consulted for the explanatory notes and an excellent index.

The book is very nicely presented by Notre Dame Press, with fine reproductions of Locher’s wonderful original illustrations.


Locher’s illustration to his discussion of diurnal rotation p. 32

Graney’s translation provides a great addition to his previous Setting Aside All Authority, which I reviewed here. Graney is doing sterling work in adjusting the very distorted view of the astronomical system discussion in the first half of the seventeenth century and anybody, who is seriously interested in learning the true facts of that discussion, should definitely read his latest contribution.




[1] By a strange cosmic coincidence the first printed copy of the Dialogo was presented to the dedicatee Ferdinando II d’Medici, Grand Duke of Tuscany 386 years ago today on 22 February 1632.

[2] At the end of my review of Setting Aside All Authority I wrote the following, which applies equally to this review; in this case I provided one of the cover blurbs for Chris’ latest book.

Disclosure; Chris Graney is not only a colleague, but he and his wife, Christina, are also personal friends of mine. Beyond that, Chris has written, at my request, several guest blogs here at the Renaissance Mathematicus, all of which were based on his research for the book. Even more relevant I was, purely by accident I hasten to add, one of those responsible for sending Chris off on the historical trail that led to him writing this book; a fact that is acknowledged on page xiv of the introduction. All of this, of course, disqualifies me as an impartial reviewer of this book but I’m going to review it anyway. Anybody who knows me, knows that I don’t pull punches and when the subject is history of science I don’t do favours for friends. If I thought Chris’ book was not up to par I might refrain from reviewing it and explain to him privately why. If I thought the book was truly bad I would warn him privately and still write a negative review to keep people from wasting their time with it. However, thankfully, none of this is the case, so I could with a clear conscience write the positive review you are reading. If you don’t trust my impartiality, fair enough, read somebody else’s review.




Filed under Book Reviews, Early Scientific Publishing, History of Astronomy, History of Mathematics, Myths of Science, Renaissance Science

Christmas Trilogy 2017: Bonus!

Yesterday was Johannes Kepler’s nominal birthday (as he was born before the calendar reform in a Protestant state his birthday on the Gregorian calendar would be 6 January!) and as in my wont, I posted a birthday post for the good Johannes. Of course I was far from being the only person to acknowledge his birthday and amongst many others somebody linked to the 2016 article on the website of the popular science magazine, Physics Today. Upon reading this brief tribute to my favourite seventeenth century polymath I cringed inwardly and didn’t know whether to let out a prolonged #histsigh or to turn loose the HistSci_Hulk; I have decided on the latter. Below the complete text of the offending document:

Born on 27 December 1571 in Weil der Stadt in the Holy Roman Empire, Johannes Kepler was an astronomer whose careful measurements led him to develop his three laws of planetary motion. He received a Lutheran education at the University of Tübingen and originally planned to be a theologian. Then one of his teachers gave him a copy of a book by Nicolaus Copernicus, sparking Kepler’s interest in astronomy. In 1600 Danish astronomer Tycho Brahe invited Kepler to Prague to help amass a precise set of astronomical measurements. Brahe died the following year, and Kepler inherited his mentor’s data and position as imperial mathematician to the Holy Roman emperor. In 1609 Kepler published Astronomia Nova, which included his first two laws of planetary motion; his third law was published in 1619. Kepler observed a supernova (though he called it a “new star”) and completed the detailed astronomical tables Brahe had been so determined to produce. Kepler also contributed research in optics and vision. Later in the century Isaac Newton would prove his law of universal gravitation by showing that it could produce Kepler’s orbits.

Born … in Weil der Stadt in the Holy Roman Empire… This contains something about which I have had bitter disputes on Wikipedia. There is a famous quip that the Holy Roman Empire was neither holy nor Roman nor an empire, it was also neither a country nor a state. The Holy Roman Empire was a loose feudal conglomeration of autonomous and semi-autonomous states. Weil der Stadt, Kepler’s birthplace was at the time of his birth in the autonomous Duchy of Württemberg.


Map of the Duchy of Württemberg 1619 by Pieter van den Keere. You can see Weyl (Weil der Stadt) roughly in the middle. Source: Wikimedia Commons

…Johannes Kepler was an astronomer whose careful measurements led him to develop his three laws of planetary motion. Kepler was a theorist, who didn’t on the whole take measurements careful or otherwise. The measurements that he used to derive his three laws were, of course, made very carefully by Tycho Brahe.

Kepler did not originally plan to be a theologian. He was on an educational tack designed to produce Lutheran Protestant pastors and schoolteachers. He would have become a pastor but was appointed to a position as a maths teacher instead.


Then one of his teachers gave him a copy of a book by Nicolaus Copernicus, sparking Kepler’s interest in astronomy. One of Kepler’s professors in Tübingen was Michael Maestlin, who in his courses taught Copernican heliocentric astronomy alongside the then dominant geocentric astronomy. Kepler took this course and developed an interest in heliocentrism. It was Maestlin who recognised Kepler’s aptitude for mathematics and recommended that he be appointed to a teaching post rather than a village church.

In 1600 Danish astronomer Tycho Brahe invited Kepler to Prague to help amass a precise set of astronomical measurements. Tycho Brahe invited Kepler to Prague not to help amass a precise set of astronomical measurements but to use his mathematical skills to turn the already amassed measurements into calculated orbits, ephemerides etc.

Brahe died the following year, and Kepler inherited his mentor’s data and position as imperial mathematician to the Holy Roman emperor. Kepler didn’t inherit his mentor’s data, Tycho’s daughter Elizabeth and her husband Frans Gansned Genaamd Tengnagel van de Camp did. This caused Kepler no end of problems, as he needed that data to realise his vision of a heliocentric astronomy. After tough negotiations, Tengnagel allowed Kepler to use the data but only if his name was included as co-author on any books that Kepler published based on it; a condition that Kepler duly fulfilled. Given my own inabilities to spell or write grammatically I’m not usually a grammar fetishist but, as I’m putting the boot in, Imperial Mathematician is a title and should be written with capital letters as in the emperor in Holy Roman Emperor.

Kepler observed a supernova (though he called it a “new star”). Well yes, as the term supernova was only coined in 1931 Kepler could hardly have used it. However, the nova part of the name, which simple means new, comes from Kepler’s term Stellar Nova, his being the most recent supernova observed with the naked eye.

…and completed the detailed astronomical tables Brahe had been so determined to produce. Kepler didn’t just complete them he produced them single-handedly, calculating, writing, typesetting, printing, publishing and selling them. This was the task assigned to him by Tycho and to which he was official appointed by the Emperor Rudolph II.

Physics Today is a fairly major popular science magazine but it would appear that they don’t really care enough about the history of science to indulge in a modicum of fact checking.





Filed under History of Astronomy, History of science, Myths of Science, Renaissance Science, Uncategorized

Christmas Trilogy 2017 Part 3: Kepler’s big book

Johannes Kepler was incredibly prolific, he published over eighty books and booklets over a very wide range of scientific and mathematical topics during his life. As far as he was concerned his magnum opus was his Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler’s The Harmony of the World) published in 1619 some twenty years after he first conceived it. Today in popular #histsci it is almost always only mentioned for the fact that it contains the third of his laws of planetary motion, the harmonic law. However it contains much, much more of interest and in what follows I will attempt to give a brief sketch of what is in fact an extraordinary book.


A brief glace at the description of the ‘five books’ thoughtfully provided by the author on the title page (1) would seem to present a mixed bag of topics apparently in some way connected by the word or concept harmonic. In order to understand what we are being presented with we have to go back to 1596 and Kepler’s first book Mysterium Cosmographicum (The Cosmographic Mystery). In this slim volume Kepler presents his answer to the question, why are there only six planets? His, to our eyes, surprising answer is that the spaces between the planets are defined by the regular so-called Platonic solids and as the are, and can only be, five of these there can only be six planets.

Using the data from the greatest and least distances between the planets in the Copernican system, Kepler’s theory produces an unexpectedly accurate fit. However the fit is not actually accurate enough and in 1598 Kepler began working on a subsidiary hypothesis to explain the inaccuracies. Unfortunately, the book that he had planned to bring out in 1599 got somewhat delayed by his other activities and obligations and didn’t appear until 1619 in the form of the Harmonice mundi.

The hypothesis that Kepler presents us with is a complex mix of ideas taken from Pythagoras, Plato, Euclid, Proclus and Ptolemaeus centred round the Pythagorean concept of the harmony of the spheres. Put very simply the theory developed by the Pythagoreans was that the seven planets (we are talking geocentric cosmology here) in their orbits form a musical scale than can, in some versions of the theory, only be heard by the enlightened members of the Pythagorean cult. This theory was developed out of the discovery that consonances (harmonious sounds) in music can be expressed in the ratio of simple whole numbers to each other (the octave for example is 1:2) and the Pythagorean belief that the integers are the building block of the cosmos.

This Pythagorean concept winds its way through European intellectual history, Ptolemaeus wrote a book on the subject, his Harmonice and it is the reason why music was one of the four disciplines of the mathematical quadrivium along with arithmetic, geometry and astronomy. Tycho Brahe designed his Uraniburg so that all the architectonic dimensions from the main walls to the window frames were in Pythagorean harmonic proportion to one another.


Tycho Brahe’s Uraniborg Blaeus Atlas Maior 1663 Source: Wikimedia Commons

It is also the reason why Isaac Newton decided that there should be seven colours in the rainbow, to match the seven notes of the musical scale. David Gregory tells us that Newton thought that gravity was the strings upon which the harmony of the spheres was played.

In his Harmony Kepler develops a whole new theory of harmony in order to rescue his geometrical vision of the cosmos. Unlike the Pythagoreans and Ptolemaeus who saw consonance as expressed by arithmetical ratios Kepler opted for a geometrical theory of consonance. He argued that consonances could only be constructed by ratios between the number of sides of regular polygons that can be constructed with a ruler and compass. The explication of this takes up the whole of the first book. I’m not going to go into details but interestingly, as part of his rejection of the number seven in his harmonic scheme Kepler goes to great lengths to show that the heptagon construction given by Dürer in his Underweysung der Messung mit dem Zirckel und Richtscheyt is only an approximation and not an exact construction. This shows that Dürer’s book was still being read nearly a hundred years after it was originally published.


In book two Kepler takes up Proclus’ theory that Euclid’s Elements builds systematically towards the construction of the five regular or Platonic solids, which are, in Plato’s philosophy, the elemental building blocks of the cosmos. Along the way in his investigation of the regular and semi-regular polyhedra Kepler delivers the first systematic study of the thirteen semi-regular Archimedean solids as well as discovering the first two star polyhedra. These important mathematical advances don’t seem to have interested Kepler, who is too involved in his revolutionary harmonic theory to notice. In the first two books Kepler displays an encyclopaedic knowledge of the mathematical literature.


The third book is devoted to music theory proper and is Kepler’s contribution to a debate that was raging under music theorist, including Galileo’s father Vincenzo Galilei, about the intervals on the musical scale at the beginning of the seventeenth century. Galilei supported the so-called traditional Pythagorean intonation, whereas Kepler sided with Gioseffo Zarlino who favoured the ‘modern’ just intonation. Although of course Kepler justification for his stance where based on his geometrical arguments. Another later participant in this debate was Marin Mersenne.


In the fourth book Kepler extends his new theory of harmony to, amongst other things, his astrology and his theory of the astrological aspects. Astrological aspects are when two or more planets are positioned on the zodiac or ecliptic at a significant angle to each other, for example 60° or 90°. In his Harmonice, Ptolemaeus, who in the Renaissance was regarded as the prime astrological authority, had already drawn a connection between musical theory and the astrological aspects; here Kepler replaces Ptolemaeus’ theory with his own, which sees the aspects are being derived directly from geometrical constructions. Interestingly Kepler, who had written and published quite extensively on astrology, rejected nearly the whole of traditional Greek astrology as humbug keeping only his theory of the astrological aspects as the only valid form of astrology. Kepler’s theory extended the number of influential aspects from the traditional five to twelve.

The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. He finally has success with the angular velocities of the planets in their orbits at perihelion and aphelion. He then examines the relationships between the tones thus generated by the different planets, constructing musical scales in the process. What he in missing in all of this is a grand unifying concept and this lacuna if filled by his harmonic law, his third law of planetary motion, P12/P22=R13/R23.


There is an appendix, which contains Kepler’s criticisms of part of Ptolemaeus’ Harmonice and Robert Fludd’s harmony theories. I blogged about the latter and the dispute that it triggered in an earlier post

With his book Kepler, who was a devoted Christian, was convinced that he had revealed the construction plan of his geometrical God’s cosmos. His grandiose theory became obsolete within less than fifty years of its publication, ironically pushed into obscurity by intellectual forces largely set into motion by Kepler in his Astronomia nova, his Epitome astronomiae Copernicanae and the Rudolphine Tables. All that has survived of his great project are his mathematical innovations in the first two books and the famous harmonic law. However if readers are prepared to put aside their modern perceptions and prejudices they can follow one of the great Renaissance minds on a fascinating intellectual journey into his vision of the cosmos.

(1) All of the illustration from the Harmonice mundi in this post are taken from the English translation The Harmy of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997


Filed under Early Scientific Publishing, History of Astrology, History of Astronomy, History of Mathematics, History of science, Renaissance Science, Uncategorized

The Albrecht Dürer or should that be the Bernhard Walther House?

On Saturday I did my history of astronomy tour of Nürnberg for some readers of this blog who were visiting the city[1]. As usually it ended at Nürnberg’s biggest tourist attraction the Albrecht Dürer House. There are of course good reasons for including Nürnberg’s most famous artist in such a tour, as readers of this blog should know. He wrote and published the very first printed maths book in German and was the artist involved in creating the first every printed European star maps. However this is another reason for including this building in a history of astronomy tour. Before it became the Albrecht Dürer House it had been the Bernhard Walther House and this was one of the reasons that motivated Dürer to purchase it. But who, I hear you say, was Bernhard Walther?

Bernhard Walther (Albrecht Dürer) House on Tiergärtentor Nürnberg
Photo: Monica Weidemann
Source: Wikipedia Commons

Bernhard Walther was born in Memmingen in Bavaria in 1430. The first really reliable fact we have about his life is when he became a citizen of Nürnberg in 1467; remember Nürnberg was an independent city-state in the fifteenth century. He was the general manager of the Nürnberg trading post of the Memmingen merchant traders the Vöhlin-Welser-Company. When Regiomontanus came to Nürnberg in 1471, he and Walther became friends and Walther became his astronomical assistant and companion. The accounts that claim that Walther was Regiomontanus’ patron are false, as are also the claims that the two of them built an observatory financed by Walther. They carried out their astronomical observations with portable instruments out in the streets. As well as astronomy Walther apparently learnt Greek from Regiomontanus, who had learnt the language whilst a member of Cardinal Bessarion’s household in Italy. We know of Walther’s abilities in the ancient language because they are mentioned in an ode that Conrad Celtis, the so-called arch humanist, wrote in his honour.

Regiomontanus had come to Nürnberg, according to his own account, to reform astronomy in two ways; firstly by starting a new programme of astronomical observations to replace those of Ptolemaeus corrupted by centuries of copying and recopying in manuscripts and secondly by printing and publishing new editions of the astronomical literature cleared of their errors through careful philological editing. Regiomontanus had chosen Nürnberg for his programme because the city made the best scientific instruments and because of its extensive communications network being aware of the fact that his programme was only achievable with the active assistance of other European astronomers. In an age without postal services, Nürnberg, as a major European trading city, had a private communications system second only to that of Venice.

Walther assisted Regiomontanus in both of his reform endeavours but they had only succeeded in publishing nine items, including the publishing house’s ambitious publication programme, when Regiomontanus again left Nürnberg in the direction of Rome to answer the Pope’s call to work on a calendar reform in 1475. Regiomontanus never returned from that journey, dying in Rome in 1476, presumable during some sort of epidemic. Walther did not continue the publishing endeavour, although he bought up Regiomontanus extensive collection of manuscripts, but he did carry on making a series of basic simple astronomical observations for the next almost thirty years. This was the first such series of astronomical observations carried out in Early Modern Europe, making Walther to an important if minor figure in the history of astronomy.

As the general manager of the trading company Walther occupied a house on the West side of the market place in Nürnberg, today Market Place No. 11. The original hose was destroyed in the Second World War.

Walther’s trading depot was on the west side of the Nürnberg market place, next door to the right of where the Körn & Berg bookshop now stands.

When he finally retired, seventy years old, he sold the house on the market place and bought the house on Tiergärtentor (The Zoo Gate) in 1501, which is now known as the Albrecht Dürer House. Walther substantially rebuilt the house adding the whole of what is now the top floor. He also had a small window let into the south gable with a stone window ledge; he used this window to make his astronomical observations resting his observing instruments on that stone ledge, this was his observatory. We know that Walther had this window constructed because in the document with which the city council gave permission for its construction, Walther had to give a guarantee that he wouldn’t empty his chamber pot out on to the roof of the neighbouring building.

Walther House with Observatory Window in the south gable
Photo: Nora Reim
Source: Astronomie in Nürnberg

Walther’s observation programme was comparatively simple and consisted largely of regularly determining the altitude of the Sun, observing eclipses and determining the positions of the planets during conjunctions etc. The latter set of observations leads to the assumption that the observations were principally for use by astrologers. This is not surprising as Regiomontanus was a practicing astrologer, with a very good reputation, whose stated intention in reforming astronomy was in order to improve astrological predictions. He claimed that such predictions were often wrong because the astronomical data on which they were based was inaccurate. Three of Walther’s observations found their way into Copernicus’ De revolutionibus, although we don’t know how they got there. Copernicus falsely attributes part of the used data to Johannes Schöner. In 1544 Schöner did publish Regiomontanus’ and Walther’s observations in his Scripta clarissimi Mathematici M. Joannis Regiomontani. Walther’s observation were, for their time, highly accurate only to be first superceded by those of Tycho Brahe at the end of the century.

Another little known Nürnberg astronomer, Conrad Heinfogel, referred to himself as a pupil of Bernard Walther and it was Heinfogel who provided the astronomical knowledge for Dürer’s star maps.

Largely forgotten today Walther was well known and highly regarded by his contemporaries and the astronomical community down to Tycho and Kepler, Tycho using Walther’s observations to check against his own. Walther died in 1504 and in 1509 Albrecht Dürer bought the house on the Tiergärtentor, partially because being himself a big fan of the mathematical sciences he desired to own Walther’s house. At the same time he also acquired ten manuscripts out of the Regiomontanus/Walther collection including an Elements of Euclid.

If you are ever in Nürnberg go round to the back of the Dürer house and you can see Walther’s observatory for yourself. However please be quite when doing so as the people who live next door get really pissed off with the tourists and the noise that they make.

[1] Any readers of the blog who visit Nürnberg are welcome to the same tour, you just need to arrange it in advance; all you have to do is buy me lunch at the end of it. A low price of a highly entertaining and educational tour that lasts between three and four hours!


Filed under History of Astrology, History of Astronomy, History of science, Renaissance Science, Uncategorized

The House of Blaeu vs.The House of Hondius – The Battle of the Globes and Atlases

There is a South to North trajectory in the evolution of the modern mathematical cartography in Europe over the two hundred years between fourteen hundred and sixteen hundred. Ptolemaic mathematical cartography re-entered Europe in Northern Italy with the first translation into Latin of his Geographia by Jacobus Angulus in 1406. Following this the first modern first modern cartographers, including Paolo dal Pozzo Toscanelli, were also situated in Northern Italy. By the middle of the fifteenth century the main centre of cartographical activity had moved north to Vienna and was centred around Kloster-Neuburg and the University with its First Viennese School of Mathematics, Georg von Peuerbach and Johannes Regiomontanus. Towards the end of the century printed editions of Ptolemaeus’ work began to appear both north and south of the Alps. The beginning of the sixteenth century saw the main centres of cartographic development in the Southern German sphere. Two principle schools are identifiable, the Nürnberg-Vienna school, whose main figures are Johannes Stabius, Peter Apian and Johannes Schöner, and the South-Western school with Waldseemüller and Ringmann in Saint-Dié-des-Vosges and Sebastian Münster in Basel. Again by the middle of the century the centre had once again moved northwards to Leuven and the Flemish school founded by Gemma Frisius and including the two great atlas makers Abraham Ortelius and Gerard Mercator. At the start of the seventeenth century the final step northwards had been taken and the new state of The United Provinces (The Netherlands) had taken the lead in modern cartography. This final step is the subject of this post.

Willem Janszoon Blaeu was born into a prosperous herring trading family in Alkmaar or Uitgeest in 1471. As would have been expected he was sent at an early age to Amsterdam to learn the family trade but it did not appeal to him and he worked instead as a carpenter and clerk in the office of his cousin. In late 1595 his life took a radical turn when he travelled to Hven to study astronomy under Tycho Brahe. It is not known what level of foreknowledge Blaeu took to Hven with him but he spent six months there studiously learning astronomy, instrument making, geodesy and cartography with Tycho and his staff. When he started his observing marathon Tycho had had a large brass globe constructed on which he, over the years, engraved the positions of all the stars that he had measured. Blaeu was given permission to transfer this data to a globe of his own. In 1596 he returned to Alkmaar and his wife Maertgen Cornilisdochter who bore his eldest son Joan on 21 September. On 21 February 1598 Blaeu in Alkmaar and Tycho in Hamburg both observed a lunar eclipse to determine the relative longitude of the two cities.

Portrait of Willem Janszoon Blaeu Artist unknown

Sometime in 1598/9 Blaeu took his family to Amsterdam and set up shop as a printer, instrument maker, globe maker and cartographer; making his first celestial globe, 34 cm diameter, for Adriaan Anthoniszoon, using Tycho’s data; this was the first publication of that data. However Blaeu’s new career was not going to be simple as he had an established competitor, Jocodus Hondius.

Jocodus Hondius was born Joost de Hondt in Wakken and grew up in Ghent, both now in Belgium, on 14 October 1563. He received an education in mathematics and learnt engraving, drawing and calligraphy. He had already established himself as a successful engraver when he was forced by the Spanish, as a Calvinist, to flee to London in 1584. In London he worked for and with Richard Hakluyt and Edward Wright and expanded his knowledge of geography and cartography through contact with the English explorers Francis Drake, Thomas Cavendish and Walter Raleigh. Around 1589 he published a wall map in London showing Drake’s voyage around the world. In 1593 he moved back to The Netherlands, establishing himself in Amsterdam.

Self-portrait of Jodocus Hondas taken from one of his maps

Portrait of Francis Drake by Jodocus Hondas from his Drake world map

He formed an alliance with the Plantin printing house in Leiden for who he made several globes. In 1602 he matriculated at the University of Leiden to study mathematics. In 1604 he made the most important decision of his career in that he bought the copper printing plates of the of both Mercator’s edition of Ptolemaeus’ Geographia and Mercator’s Atlas from his heirs.He published a new edition of Mercator’s Ptolemaeus, Claudïï Ptolemaeï Alexandrini geographicae libri octo graecog latini, in the same year. He set up his own publishing house in Amsterdam in December 1604. In the sixteenth century Mercator’s Atlas had failed to establish itself in a market dominated by Ortelius’ Theatum Orbis Terrarum, however Hondius republished it in 1606 with 36 new maps and it became a best seller.

Atlas sive Cosmographiae Meditationes de Fabrica Mundi et Frabicati Figura
Mercator (left) and Hondius (right) shown working together on tittle page of 1630 Atlas
Slightly ironical as they never met and both were dead by then.

Meanwhile Blaeu had established himself as a globe maker and astronomer. Following the tradition established by Johannes Schöner and continued by Mercator Blaeu issued a pair of 23.5 cm globes, terrestrial and celestial, in 1602. His rival Hondius introduced the southern constellation on a celestial globe produced in cooperation with the astronomer-cartographer Petrus Plancius in 1598. Blaeu followed suite in 1603. Hondius produced a pair of 53.5 cm globes in 1613; Blaeu countered with a pair of 68 cm globes in 1616, which remained the largest globes in production for over 70 years.

Hondas celestial globe 1600
Source: Linda Hall Library

A matching pair of Blaeu globes

As an astronomer Blaeu discovered the star P Cygni, only the third variable star to be discovered. In 1617 Willebrord Snellius published his Eratosthenes Batavus (The Dutch Eratosthenes) in which he described his measurement of a meridian arc between Alkmaar and Bergen op Zoom. This was done in consultation with Blaeu, who had learnt the art of triangulation from Tycho, using a quadrant, with a radius of more than 2 metres, constructed by Blaeu. Blaeu specialised in publishing books on navigation beginning in 1605 with his Nieuw graetbouck and established himself as the leading Dutch publisher of such literature.

Source: Wikimedia Commons

Title page
Source: Wikimedia Commons

Quadrant constructed by Blaeu for Snellius now in Museum Boerhaave in Leiden
Source: Wikimedia Commons

Jodocus Hondius died in 1612 and his sons Jodocus II and Henricus took over the publish house later going into partnership with Jan Janszoon their brother in law. They continued to publish new improved version of the Mercator-Hondius Atlas. Blaeu had already established himself as the successful publisher of wall maps when he began planning a major atlas to rival that of the house of Hondius. That rivalry is also reflected in a name change that Blaeu undertook in 1617. Up till then he had signed his work either Guilielmus Janssonius or Willem Janszoon, now he started add the name Blaeu to his signature probably to avoid confusion with Jan Janszoon (Janssonius), his rival.

Jan Janszoon Original copperplate from his Atlas Novus 1647

In 1630 the strangest episode in the battle of the globes and atlases took place when Jodocus II’s widow sold 37 of the copper plates of the Mercator-Hondius Atlas to Willem Blaeu. He published them together with maps of his own in his Atlantic Appendix in 1631. In 1636 Blaeu published the first two volumes of his own planned world atlas as Atlas Novus or Theatrum Orbis Terrarum, thus reviving the old Ortelius name.

In 1633 the States General (the government of the United Provinces) appointed Blaeu mapmaker of the Republic. In the same year he was appointed cartographer and hydrographer of the Vereenighde Oostindische Compagnie (VOC) – The Dutch East India Company. His son Joan inherited the VOC position, as did his grandson Joan II; The Blaeu family held this prestigious position from 1633 till 1712.

Willem Blaeu had great plans to publish several more volumes of his world atlas but he didn’t live to see them realised, dying 21 October 1638. The publishing house passed to his two sons Joan (1596-1673) and Cornelis (c.1610-1644). The last two volumes prepared by Willem appeared in 1640 and 1645. Joan completed his father’s atlas with a sixth volume in 1655.

Along with all his other achievements Willem Janszoon Blaeu made a substantial improvement to the mechanical printing press by adding a counter weight to the pressure bar in order to make the platen rise automatically. This ‘Blaeu’ or ‘Dutch’ press became standard throughout the low countries and was also introduced into England. The first printing press introduced into America in 1639 was a Blaeu press.

Although he held a doctorate in law, Joan devoted his life to the family cartographic publishing business. In 1662 he set the high point of the atlas battle with the House of Hondius with the publication of the Atlas Maior; containing 600 double page maps and 3,000 pages of text it was the most spectacular atlas of all time. Along with its lavish maps the Atlas Maior contained a map of Hven and pictures of the house and stellar observatory on the island where Willem Janszoon Blaeu first learnt his trade. Whereas Willem was careful not to take sides in the dispute between the different systems for the cosmos – geocentric, heliocentric, geo-heliocentric – in the Atlas Maior, Joan committed to heliocentricity.

Joan Blaeu. By J.van Rossum
Source: Wikimedia Commons

Blaeu Atlas Maior 1662-5, Volume 1
Nova Et Accvratissima Totius Terrarvm Orbis Tabvla
Source: National Library of Scotland

The rivalry between the Houses of Hondius and Blaeu, pushing each other to new heights of quality and accuracy in their maps and globes led to them totally dominating the European market in the first half of the sixteenth century, particularly in the production of globes where they almost had a monopoly. Globes in the period, which weren’t from one of the Amsterdam producers, were almost always pirated copies of their products.

As an interesting footnote, as with all things mathematical England lagged behind the continent in cartography and globe making. Although there had been earlier single globes made in on the island, England’s first commercial producer of terrestrial and celestial globes, Joseph Moxon, learnt his trade from Willem Janszoon Blaeu in Amsterdam. In 1634 Blaeu had published a manual on how to use globes, Tweevoudigh onderwijs van de Hemelsche en Aerdsche globen (Twofold instruction in the use of the celestial and terrestrial globes). In the 1660s, Moxon published his highly successful A Tutor to Astronomie and Geographie. Or an Easie and speedy way to know the Use of both the Globes, Cœlestial and Terrestrial : in six Books, which went through many editions, however the first edition was just an English translation of Blaeu’s earlier manual.

The Dutch painter Jan Vermeer often featured globes and maps in his paintings and it has been shown that these are all reproductions of products from the Blaeu publishing house.

Vermeer’s Art of Painting or The Allegory of Painting (c. 1666–68)
With Blaeu Wall Map
Google Art Project Source: Wikimedia Commons

Jan Vermeer The Astronomer with Blaeu celestial globe and right on the wall a Blaeu wall map
Source: Wikimedia Commons

Jan Vermeer The Geographer with Blaeu terrestrial globe and again right a Blaeu wall map
Source: Wikimedia Commons

The Blaeu wall map used in Vermeers’ The Astronomer and The Geographer

We tend to emphasise politicians, artists and big name scientists, as the people who shape culture in any given age but the cartographic publishing houses of Hondius and Blaeu made significant contributions to shaping the culture of The United Provinces in the so-called Dutch Golden Age and deserve to be much better known than they are.






Filed under Early Scientific Publishing, History of Astronomy, History of Cartography, History of Navigation, History of science, Renaissance Science

A Renaissance artist-engineer icon – Vitruvian Man

Leonardo da Vinci’s drawing of Vitruvian Man is one of the most well known graphic images in the world. Many people don’t even know the title I have used for the image and of those that do, many have no idea why it’s so called. Even less people are aware that the image is not unique or original to Leonardo, although his rendition is probably the most beautiful and most powerful, but is in fact an iconic concept in the work of Renaissance artist-engineers.

The origin of the Vitruvian Man is to be found in Vitruvius, De architectura (Ten Books of Architecture).[1] Vitruvius lived in ancient Rome in the first century BCE and his Ten Books of Architecture is the only known full treatise on architecture that we have from classical antiquity. Almost nothing is known about Vitruvius himself and even the full name that tradition has accredited him with, Marcus Vitruvius Pollio, is questionable the name Vitruvius being the only part that is certain. Although the book is nominally about architecture more than half of the text is about things we would not normally associate with a textbook on architecture such as astronomy, geography and natural philosophy to quote Tomas Noble Howe, himself quoting Frank Brown, “…the mission of Vitruvius is to present architecture as a liberal art, based on a Hellenistic belief of the unity of knowledge.”

It is against this background that we find the passages referencing the dimensions of the human body, the origins of the iconic diagram, in Book 3: Temples Chapter 1: First Principles of Symmetry.

  1. The composition of a temple is based on symmetry, whose principles architects should take the greatest care to master Symmetry derives from proportion, which is called analogia in Greek. Proportion is the mutual calibration of each element of the work and of the whole, from which the proportional system is achieved. No temple can have any compositional system without symmetry and proportion, unless, as it were, it has an exact system of correspondence to the likeness of a well-formed human being.


  1. For Nature composed the human body in such a way that the face, from the chin to the top of the forehead and the lowest roots of the hairline should be one-tenth [of the total height of the body]; the palm of the hand from the wrist to the tip of the middle finger should measure likewise; the head from the chin to the crown, one-eighth; from the top of the chest to the hairline including the base of the neck, one-sixth; from the centre of the chest to the crown of the had, one-fourth. Of the height of the face itself, one-third goes from the base of the chin to the lowermost part of the nostrils, another third from the base of the nostrils to the point between the eyebrows, from that point to the hairline, the forehead also measures one-third. The foot should be one-sixth the height, the cubit, one-fourth, the chest also one-fourth. The other limbs, as well, have their own commensurate proportions, which the famous ancient painters and sculptors employed to attain great and unending praise.


I have quoted theses passages in full to make it very clear that for Vitruvius the form of the human body is quite literally the mass of all things. Symmetry and proportion is everything and the human body is the model for this claim. In his next paragraph Vitruvius delivers up the construction plan for Vitruvian Man.

  1. Similarly, indeed, the elements of holy temples should have dimensions for each individual part that agree with the magnitude of the work. So, too, for example, the centre and midpoint of the human body is the navel. For if a person is imagined lying back with outstretched arms and feet within a circle whose centre is at the navel, the fingers and toes will trace the circumference of this circle as they move about. But to whatever extent a circular scheme may be present in the body, a square design may also be discerned there. For if we measure from the soles of the feet to the crown of the head, and this measurement is compared with that of the outstretched hands, one discovers that this breadth equals the height, just as in areas which have been squared off by use of the set square.


The illustrations are Thomas Noble Howe’s modern reconstructions but we have good reason to believe that manuscripts of Vitruvius’ work in antiquity would have had illustration.[2]

Given his unified approach to art, science, design, engineering and metaphysics it comes as no surprise that Vitruvius served as a major role model for the Renaissance artist-engineers and that his Ten Books of Architecture served them as a bible. We already find the Florentine artist Lorenzo Ghiberti (1378–1455), an acknowledged forerunner to the artist-engineers, quoting Vitruvius in his potted history of linear perspective; the humanist scholar Poggio Bracciolini having ‘rediscovered’ Vitruvius in 1406.

Vitruvian man emerged in the works of the so-called Sienese engineers. The first of these was Mariano di Jacopo (1382–1543) known as Taccola. Taccola an engineer produced two annotated manuscripts of drawings of machines De ingeneis (Concerning engines)

Machines, by Taccola, De ingeneis

and De machinis (Concerning machines).

Paddle boat system, by Taccola, De machinis (1449)

In his notes we find his rendition of Vitruvian Man, not an artistic one like Leonardo’s but the simple diagrammatic version of an engineer.

Taccola Vitruvian Man

Taccola was the major influence on a second Sienese engineer Francesco di Giorgio Martini (1493–1501), whose studies of machines are almost all based on those of Taccola.

Extract from a notebook of Francesco di Giorgio Martini, 1470


However unlike Taccola he was also a painter, a sculptor and a leading architect. His rendition of the Vitruvian Man is very simplistic

Trattato di architettura di Francesco di Giorgio Martini

He, however, went one stage further incorporating inscribed human bodies into the architectural drawings of his ‘temples’, the churches he designed.

Trattato di architettura di Francesco di Giorgio Martini

Trattato di architettura di Francesco di Giorgio Martini

Both Taccola and Francesco di Giorgio influenced Leonardo who processed manuscripts of the work of both men; his manuscript of di Giorgio being particularly heavily annotated.

Although Luca Pacioli (1445–1517) doesn’t include a version of the Vitruvian Man in his De divina proportioni (Venice, 1509), famously illustrated by Leonardo, the second part of the book Trattato dell’architettura (Treatise on Architecture) is a twenty chapter discussion of the theories of Vitruvius comparing the proportions of the human body to those of artificial structures.

Having considered the right arrangement of the human body, the ancients proportionedall their work, particularly the temples, in accordance with it. In the human body the discovered the two main figures without which it is impossible to achieve anything, namely the perfect circle and the square.

Luca Pacioli De divina proportione

Naturally the early printed editions of De architectura contain illustrations of the Vitruvian Man. The first printed and illustrated edition of De architecture edited by Italian architect and scholar, Fra. Giovanni Giocondo, in 1511 contained images for both square and circle:

The first Italian edition by Cesare Cesariano in 1521 also contains two images


Another edition from 1525 edited by Francesco Giorgi contains only one image of the circle.


The artist who spread the Italian concepts of linear perspective north of the Alps, Albrecht Dürer, was also obsessed with the idea of the perfect mathematical proportions of the human body and devoted a large part of his life to writing his magnum opus Vier Bücher von Menschlicher Proportion (Four Books on Human Proportion), published posthumously in 1528. Followed in 1532 by a Latin edition. Of interest is the fact that as he had almost completed his book he realised that the mathematics it contained was too difficult for the apprentice painters for whom he was writing so he wrote an introductory geometry book, Underweysung der Messung mit dem Zirkel und Richtscheyt (Instruction in Measurement with Compass and Straightedge). Dürer’s book does not contain a Vitruvian Man but contains many diagrams demonstrating the mathematical proportions of the human body.

Dürer Vier bücher von menschlicher Proportion

In the middle of the sixteenth century another Renaissance polymath, physician, astronomer, astrologer, mathematician and philosopher, Girolamo Cardano, wrote in his De subtilitate rerum (1552) that Vitruvius was one of the twelve persons who he supposes to have excelled all men in the force of genius and invention; and would not have scrupled to have given him the first place, if it could be imagined that he had delivered nothing but his own discoveries.

Since the ‘rediscovery’ of Leonardo in the eighteenth century his version of Vitruvian Man has been used, modified and parodied in a thousand different images, diagrams, adverts, poster and whatever. By a strange coincidence as I was preparing this post Monica Azzolini, Renaissance historian and Leonardo expert, posted two modern parodies of Leonardo’s Vitruvian Man on Facebook, which caught my fancy and I offer them for your amusement.


From the Uncyclopedia

 And of course a Ninja Turtle Leonardo Vitruvian Man

[1] All references to Vitruvius, Ten Books of Architecture are taken from the English translation edited by Ingrid D. Rowland (translator) and Thomas Noble Howe (illustrator), CUP, pb 2001

[2] On the subject of illustrations in scientific works in antiquity see: Alfred Stückelberger, Bild und Work: Das illustrierte Fachbuch in der antiken Naturwissenschaft, Medizin und Technik


Filed under Renaissance Science

Conrad Gesner Day 2017

Anyone who pokes around long enough here at the Renaissance Mathematicus will realise that I have a fondness for polymaths. It is in fact interesting how many of the leading researcher in history were in fact polymaths. One of my favourites is the Swiss Renaissance physician, classicist, Hebraist, natural historian, bibliographer and mountaineer, Conrad Gesner.

Conrad Gessner memorial at the Old Botanical Garden, Zürich Source: Wikimedia Commons

Conrad Gessner memorial at the Old Botanical Garden, Zürich
Source: Wikimedia Commons

Last year on the five hundredth anniversary of his birth I duly recycled my old Conrad Gesner post and discovered to my delight that I had a small but distinguished Gesner fan club on my Twitter stream. We spent a happy 24 plus hours tweeting and retweeting each other’s tributes to and admirations of the Swiss polymath. At some point in a flippant mood I suggested that we should celebrate an annual Conrad Gesner Day on, 26 March his birthday. The suggestion was taken up with enthusiasm by the others and so we parted.

A couple of months ago Gesner’s name came up again and I said I was serious about celebrating Conrad Gesner Day and all the others immediately responded that they were very much still up for it so it’s on. At the moment Biodiversity Heritage Library (BHL @BioDivLIbrary), Michelle Marshall (Historical SciArt (@HistSciArt), New York Academy of Medicine Center for History (@NYAMHistory), the rare book librarian at Smithsonian Libraries and I are committed to celebrating Conrad Gesner Day. What about you?

What is going to happen? That’s up to all those involved. You can post blog posts, post illustrations from Gesner’s works on Twitter, Facebook, Instagram, whatever, where ever. Post links to sites about Gesner. If you want to write something on Gesner but don’t have your own blog, contact me and I’ll post it here at the Renaissance Mathematicus. I will collect all the contributions and post a Whewell’s Gazette style links list here at RM on the Monday.

The aim is not to glorify Conrad Gesner but to raise peoples’ awareness of a fascinating and important figure in the history of Renaissance science. Join us! Make a contribution! We already have a hash tag .




Filed under History of science, Renaissance Science