Category Archives: History of Astronomy

The first calculating machine


Even in the world of polymath, Renaissance mathematici Wilhelm Schickard (1592–1635) sticks out for the sheer breadth of his activities. Professor of both Hebrew and mathematics at the University of Tübingen he was a multi-lingual philologist, mathematician, astronomer, optician, surveyor, geodesist, cartographer, graphic artist, woodblock cutter, copperplate engraver, printer and inventor. Born 22 April 1592 the son of the carpenter Lucas Schickard and the pastor’s daughter Margarete Gmelin he was probably destined for a life as a craftsman. However, his father died when he was only ten years old and his education was taken over by various pastor and schoolteacher uncles. Following the death of his father he was, like Kepler, from an impoverished background, like Kepler he received a stipend from the Duke of Württemburg from a scheme set up to provided pastors and teachers for the Protestant land. Like Kepler he was a student of the Tübinger Stift (hall of residence for protestant stipendiaries), where he graduated BA in 1609 and MA in 1611. He remained at the university studying theology until a suitable vacancy could be found for him. In 1613 he was considered for a church post together with another student but although he proved intellectually the superior was not chosen on grounds of his youth. In the following period he was appointed to two positions as a trainee priest. However in 1614 he returned to the Tübinger Stift as a Tutor for Hebrew.


Wilhelm Schickard, artist unknown Source: Wikimedia Commons

Here we come across the duality in Schickard’s personality and abilities. Like Kepler he had already found favour, as an undergraduate, with the professor for mathematics, Michael Maestlin, who obviously recognised his mathematical talent. However, another professor recognised his talent for Hebrew and encouraged him to follow this course of studies. On his return to Tübingen he became part of the circle of scholars who would start the whole Rosicrucian movement, most notably Johann Valentin Andreae, the author of the Chymical Wedding of Christian Rosenkreutz, who also shared Schickard’s interest in astronomy and mathematics.


Johann Valentin Andreae Source: Wikimedia Commons

Although Schickard appear not to have been involved in the Rosicrucian movement, the two stayed friends and correspondents for life. Another member of the group was the lawyer Christian Besold, who would later introduce Schickard to Kepler.


Christopher Besold etching by Schickard 1618

This group was made up of the brightest scholars in Tübingen and it says a lot that they took up Schickard into their company.

In late 1614 Schickard was appointed as a deacon to the parish of Nürtingen; in the Lutheran Church a deacon is a sort of second or assistant parish pastor. His church duties left him enough time to follow his other interests and he initially produced and printed with woodblocks a manuscript on optics. In the same period he began the study of Syriac. In 1617 Kepler came to Württtemburg to defend his mother against the charge of witchcraft, in which he was ably assisted by Christian Besold, who as already mentioned introduced Schickard to the Imperial Mathematicus. Kepler was much impressed and wrote, “I came again and again to Mästlin and discussed with him all aspects of the [Rudolphine] Tables. I also met an exceptional talent in Nürtingen, a young enthusiast for mathematics, Wilhelm Schickard, an extremely diligent mechanicus and also lover of the oriental languages.” Kepler was impressed with Schickard’s abilities as an artist and printer and employed him to provide illustrations for both the Epitome Astronomiae Copernicanae and the Harmonice Mundi. The two would remain friends and correspondents for life.


3D geometrical figures from Kepler’s Hamonice Mundi by Schickard

In 1608 Schickard was offered the professorship for Hebrew at the University of Tübingen; an offer he initially rejected because it paid less than his position as deacon and a university professor had a lower social status than an on going pastor. The university decided to appoint another candidate but the Duke, whose astronomical advisor Schickard had become, insisted that the university appoint Schickard at a higher salary and also appoint him to a position as student rector, to raise his income. On these conditions Schickard accepted and on 6 August 1619 he became a university professor. Schickard subsidised his income by offering private tuition in Chaldean, Rabbinic, mathematic, mechanic, perspective drawing, architecture, fortification construction, hydraulics and optics.


Page from a manuscript on the comets of 1618 written and illustrated by Schickard for the Duke of Württemberg

The Chaldean indicates his widening range of languages, which over the years would grow to include Ethiopian, Turkish, Arabic and Persian and he even took a stab at Malay and Chinese later in life. Schickard’s language acquisition was aimed at reading and translating text and not in acquiring the languages to communicate. Over the years Schickard acquired status and offices becoming a member of the university senate in 1628 and a school supervisor for the land of Württemberg a year later.  In 1631 he succeeded his teacher Michael Mästlin as professor of mathematics retaining his chair in Hebrew. He had been offered this succession in 1618 to make the chair of Hebrew chair more attractive but nobody had thought that Mästlin, then almost 70, would live for another 12 years after Schickard’s initial appointment.


Michael Mästlin portrait 1619 the year Schickard became professor for Hebrew (artist unknown)

In 1624 Schickard set himself the task of producing a new, more accurate map of the land of Württemberg. Well read, he used the latest methods as described by Willebrord Snell in his Eratosthenes Batavus (1617).


This project took Schickard many more years than he originally conceived. In 1629 he published a pamphlet in German describing how to carry out simple geodetic surveys in the hope that others would assist him in his work. Like Sebastian Münster’s similar appeal his overture fell on deaf ears. Later he used his annual school supervision trips to carry out the necessary work.


Part of Schickard’s map of Württemberg

Schickard established himself as a mathematician-astronomer and linguist with a Europe wide reputation. As well as Kepler and Andreae he stood in regular correspondence with such leading European scholars as Hugo Grotius, Pierre Gassendi, Élie Diodati, Ismaël Boulliau, Nicolas-Claude Fabri de Peiresc, Jean-Baptiste Morin, Willem Janszoon Blaeu and many others.

The last years of Schickard’s life were filled with tragedy. Following the death of Gustav Adolf in the Thirty Years War in 1632, the Protestant land of Württemberg was invaded by Catholic troops. Along with chaos and destruction, the invading army also brought the plague. Schickard’s wife had born nine children of which four, three girls and a boy, were still living in 1634. Within a sort time the plague claimed his wife and his three daughters leaving just Schickard and his son alive. The invading troops treated Schickard with respect because they wished to exploit his cartographical knowledge and abilities. In 1635 his sister became homeless and she and her three daughters moved into his home. Shortly thereafter they too became ill and one after another died. Initially Schickard fled with his son to escape the plague but unable to abandon his work he soon returned home and he also died on 23 October 1635, just 43 years old, followed one day later by his son.

One of the great ironies of history is that although Schickard was well known and successful throughout his life, today if he is known at all, it is for something that never became public in his own lifetime. Schickard is considered to be the inventor of the first mechanical calculator, an honour that for many years was accorded to Blaise Pascal. The supporters of Schickard and Pascal still dispute who should actually be accorded this honour, as Schickard’s calculator never really saw the light of day before the 20thcentury. The story of this invention is a fascinating one.

Inspired by Kepler’s construction of his logarithm tables to simplify his astronomical calculation Schickard conceived and constructed his Rechenuhr (calculating clock) for the same purpose in 1623.

The machine could add or subtract six figure numbers and included a set of Napier’s Bones on revolving cylinders to carry out multiplications and divisions. We know from a letter that a second machine he was constructing for Kepler was destroyed in a workshop fire in 1624 and here the project seems to have died. Knowledge of this fascinating invention disappeared with the deaths of Kepler and Schickard and Pascal became credited with having invented the earliest known mechanical calculator, the Pascaline.


A Pascaline signed by Pascal in 1652 Source: Wikimedia Commons

The first mention of the Rechenuhr was in Michael Gottlieb Hansch’s Kepler biography from 1718, which contained two letters from Schickard in Latin describing his invention. The first was just an announcement that he had made his calculating machine:

Further, I have therefore recently in a mechanical way done what you have done with calculation and constructed a machine out of eleven complete and six truncated wheels, which automatically reckons together given numbers instantly: adds, subtracts, multiplies and divides. You would laugh out loud if you were here and would experience, how the position to the left, if it goes past ten or a hundred, turns entirely by itself or by subtraction takes something away.

The second is a much more detailed description, which however obviously refers to an illustration or diagram and without which is difficult or even impossible to understand.

Schickard’s priority was also noted in the Stuttgarter Zeitschrift für Vernessungswesenin 1899. In the twentieth century Franz Hammer found a sketch amongst Kepler’s papers in the Pulkowo Observatory in St Petersburg that he realised was the missing diagram to the second Schickard letter.


The Rechenuhr sketch from Pulkowow from a letter to Kepler from 24 February 1624

Returning to Württemberg he found a second sketch with explanatory notes in German amongst Schickard’s papers in the Würtemmberger State Library in Stuttgart.


Hammer made his discoveries public at a maths conference in 1957 and said that Schickard’s drawings predated Pascal’s work by twenty years. In the following years Hammer and Bruno von Freytag-Löringhoff built a replica of Schickard’s Rechenuhr based on his diagrams and notes, proving that it could have functioned as Schickard had claimed.

Schickards Rechenmaschine

Schickard’s Rechenuhr. Reconstruction by Bruno Baron von Freytag-Löringhoff and Franz Hammer

Bruno von Freytag-Löringhoff travelled around over the years holding lectures on and demonstrations of his reconstructed Schickard Rechenuhr and thus with time Schickard became acknowledged as the first to invent a mechanical calculator, recognition only coming almost 450 years after his tragic plague death.




Filed under History of Astronomy, History of Computing, History of Mathematics, History of science, History of Technology, Renaissance Science, Uncategorized

Today in something is wrong on the Internet

When I was growing up one of the most widespread #histSTM myths, along with the claim that people in the Middle Ages believed the world was flat and Stone Age people lived in holes in the ground, was that Galileo Galilei invented the telescope. This myth actually has an interesting history that goes all the way back to the publication of the Sidereus Nuncius. Some of Galileo’s critics misinterpreting what he had written asserted that he was claiming to have invented the telescope, an assertion that Galileo strongly denied in a latter publication. Whatever, as I said when I was growing up it was common knowledge that Galileo had invented the telescope. During the 1960s and 1970s as history of science slowly crept out of its niche and became more public and more popular this myth was at some point put out of its misery and buried discretely, where, I thought, nobody would find it again. I was wrong.

When I wrote my essay on the origins of the reflecting telescope for the online journal AEON, my editor, Corey Powell, who is himself a first class science writer and an excellent editor, asked me to provide a list of reference books to help speed up the process of fact checking my essay. I was more than happy to oblige, as even more embarrassing than a fact checker finding a factual error in what I had written, and yes even I make mistakes, would be a reader finding a real clangour after my essay had been published. As it turned out I hadn’t made any mistakes or if I did nobody has noticed yet. Imagine my surprise when I read an essay published two days ago on AEON that stated Galileo had invented the telescope. Hadn’t it been fact checked? Or if so, didn’t the fact checker know that this was a myth?

The essay in question is titled Forging Islamic Science and was written by Nir Shafir and edited by Sally Davies. The offending claim was at the beginning of the second paragraph:

Besides the colours being a bit too vivid, and the brushstrokes a little too clean, what perturbed me were the telescopes. The telescope was known in the Middle East after Galileo invented it in the 17th century, but almost no illustrations or miniatures ever depicted such an object.

I tweeted the following to both the author’s and AEON’s Twitter accounts:

If the author is complaining about forgers getting historical details wrong he really shouldn’t write, “The telescope was known in the Middle East after Galileo invented it in the 17th century…”

The author obviously didn’t understand my criticism and tweeted back:

There are references to the use of telescopes for terrestrial observations, mainly military, in the Ottoman Empire, such as in evliya çelebi.

I replied:

Galileo did not invent the telescope! He wasn’t even the first astronomer to use one for astronomical observations!

Whereupon Sally Davies chimed in with the following:

Thank you for drawing this to our attention! A bit of ambiguity here; we have tweaked the wording to say he ‘developed’ the telescope.

Sorry but no ambiguity whatsoever, Galileo did not in anyway invent the telescope and as I will explain shortly ‘developed’ is just as bad.

Today the author re-entered the fray with the following:

Thank you for bringing this up. It’s always good to get the minor details right.

The invention of the telescope is one of the most significant moments in the whole history of science and technology, so attributing its invention to the completely false person is hardly a minor detail!

About that ‘developed’. A more recent myth, which has grown up around Galileo and his use of the telescope, is that he did something special in some sort of way to turn this relatively new invention into a scientific instrument usable for astronomical observations. He didn’t. The telescope that Galileo used to discover the Moons of Jupiter differed in no way either scientifically or technologically from the one that Hans Lipperhey demonstrated to the assembled prominence at the peace conference in Den Hague sometime between the 25thand 29thof September 1608. Lipperhey’s invention was even pointed at the night sky, “and even the stars which normally are not visible for us, because of the scanty proportion and feeble sight of our eyes, can be seen with this instrument.”[1]

Both instruments consisted of a tube with a biconvex or plano-convex objective lens at one end and a bi-concave or plano-concave eyepiece lens at the other end. The eyepiece lens also had a mask or stop to cut down the distortion caused around the edges of the lens. The only difference was in the focal lengths of the lenses used producing different magnitudes of magnification. Galileo’s use of other lenses to increase magnification was nothing special; it had been done earlier than Galileo by Thomas Harriot and at least contemporaneous if not earlier by Simon Marius. It was also done by numerous others, who constructed telescopes independently in those first few years of telescopic astronomical observation. The claims that Galileo had developed, improved, specialised, etc., etc., the telescope are merely mythological elements of the more general Galileo hagiography. Modern research has even revealed that contrary to his own claims Galileo probably did not (re)-construct the telescope purely from having heard reports about it but had almost certainly seen and handled one before he attempted to construct one himself.

Going back to the offending AEON essay, Sally Davies could have saved herself and Nir Sharfir if she had simply changed the sentence to:

The telescope was known in the Middle East after it was invented  in the late 16th early 17th century…(even I make mistakes)

What I intended to write before my brain threw a wobbly was:

The telescope was known in the Middle East after it was invented in late 1608…

 She doesn’t even need to mention Lipperhey’s name if she wants to avoid the on going debates about who really did invent the telescope.








[1]Embassies of the King of Siam Sent to His Excellency Prince Maurits, Arrived in The Hague on 10 September 1608


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

A sixteenth century bestseller by an amateur cosmographer

Sebastian Münster, who with his Cosmographia wrote and published what was probably the biggestselling book in the sixteenth century, was actually a professor for Hebrew by profession and only a passionate cosmographer in his free time. Born in Ingelheim am Rhein 20 January 1488 as the son of Endres Münster a churchwarden and master of the church hospital.


Sebastian Münster portrait by Christoph Amberger c. 1550 Source: Wikimedia Commons


Münster’s birthplace Ingelheim from the Cosmographia Source: Wikimedia Commons

He studied at a Franciscan school and entered the Order in 1505. In 1507 he was sent to Löwen and then Freiburg im Breisgau, where he studied under Gregor Reisch (c. 1467–1525), author of the well known encyclopaedic student textbook the Margarita Philosophica, in particular geography and Hebrew.


Ptolemeus and Astronomia from Gregor Reisch’s Margarita Philosophica Source: Wikimedia Commons

In 1509 he became a pupil of the humanist scholar Konrad Pelikan (1478–1556), who over the next five years taught him Hebrew, Greek, mathematics, and cosmography. In 1512 he was anointed a priest. Pelikan and Münster expanded their studies to include other Semitic languages, in particular Aramaic and Ethiopian.


Konrad Pelikan Source: Wikimedia Commons

From 1514 to 1518 he taught at the Franciscan high school in Tübingen. Parallel to his teaching he studied astrology, mathematics and cosmography under Johannes Stöffler. From 1518 he taught at the Franciscan high school in Basel and from 1521 to 1529 at the University of Heidelberg. In 1529 he left the Franciscan Order and became professor for Hebrew at the University of Basel as Pelikan’s successor, converting to Protestantism. In 1530 he married Anna Selber the widow of the printer/publisher Adam Petri, the cousin and printing teacher of Johannes Petreius. As a Hebraist he published extensively on language, theology and the Bible but it is his work as a cosmographer that interest us here. All of his books were published by his stepson Heinrich Petri.

In 1528 he published a pamphlet entitled Erklärung des neuen Instruments der Sunnen(Explanation of a new instrument of the Sun) in which he issued the following request, Let everyone lend a hand to complete a work in which shall be reflected…the entire land of Germany with all its territories, cities, towns, villages, distinguished castles and monasteries, its mountains, forests, rivers, lakes, and its products, as well as the characteristics and customs of its people, the noteworthy events that have happened and the antiquities which are still found in many places. He gave his readers instructions on how to record an area cartographically from a given point. This is the earliest indication of Münster’s intension to create a full geographical description of the German Empire. This first appeal proved in vain; it would be another sixteen years before he realised this high ambition. Münster satisfied himself with the publication of a small pamphlet Germaniae descriptioin 1530 based on a revised edition of a map of Middle Europe from Nicolaus Cusanus.

Turning his attention to ancient Greek geography Münster published Latin editions of Solinus’ Polyhistorand Pomponius Mela’s De situ orbis. In 1532 Münster drew a world map for Simon Grynaeus’ and Johann Huttich’s popular travel book Novus Orbis Regionum(“New World Regions”, which described the journeys of famous explorers. The map in not particular innovative and does not go much further in its information than the 1507 Waldseemüller world map. However it does contain a border of fascinating illustrations thought to have been created by Hans Holbein, who in his youth had worked for the Petri publishing house.


Münster’s 1532 World Map

In 1540 Münster issued his edition of Ptolemaeus’ Geographia, which was based on the Latin translation by Willibald Pirckheimer. His edition entitled, Geographia universalis, vetus et nova(“Universal Geography, Old and New”) was the first work to contain separate maps for each of the then four continents. In total the work contain forty-six maps drawn by Münster. The world map in this work differs substantially from the one from 1532.


Münster’s map of America Source: Wikimedia Commons

Münster’s  magnum opus his Cosmographiaor to give it its full title:

Cosmographia. Beschreibung aller Lender durch Sebastianum Münsterum: in welcher begriffen aller Voelker, Herrschaften, Stetten, und namhafftiger Flecken, herkommen: Sitten, Gebreüch, Ordnung, Glauben, Secten und Hantierung durch die gantze Welt und fürnemlich Teütscher Nation (Getruckt zu Basel: durch Henrichum Petri 1544)


Cosmographia title page

finally appeared in 1544 with contributions from over one hundred scholars from all over Europe, who provided maps and texts on various topics for inclusion in what was effectively an encyclopaedia. Over the next eighty years the work was published in thirty-seven editions, in German (21), Latin (5), French (6), Italian (3), Czech (1) and English (1) (although the English edition is an incomplete translation). The work was continually revised and expanded, the 1544 original had 600 pages and the final edition from 1628 1800. The work was published in six volumes, which in the 1598 edition were as follows:

Book I: Astronomy, Mathematics, Physical Geography, Cartography

Book II: England, Scotland, Ireland, Spain, France, Belgium, The Netherlands, Luxembourg, Savoy, Trier, Italy

Book III: Germany, Alsace, Switzerland, Austria, Carniola, Istria, Bohemia, Moravia, Silesia, Pomerania, Prussia, Livland

Book IV: Denmark, Norway, Sweden, Finland, Iceland, Hungary, Poland, Lithuania, Russia, Walachia, Bosnia, Bulgaria, Serbia, Greece, Turkey

Book V: Asia Minor, Cyprus, Armenia, Palestine, Arabia, Persia, Central Asia, Afghanistan, Scythia, Tartary India, Ceylon, Burma, China, East Indies, Madagascar, Zanzibar, America

Book VI: Mauritania, Tunisia, Libya, Egypt, Senegal, Gambia, Mali, South Africa, East Africa


Town plan of Bordeaux from the Cosmographia Source: Wikimedia Commons

As indicated in his original call for cooperation, Münster’s Cosmographia was much more than a simple atlas mapping the world but was an integrated description combining geography, cartography, history and ethnography to create an encyclopaedic depiction of the known world.


Chartre under attack from the Cosmographia Source: Wikimedia Commons

In total at least 50,000 German copies and 10,000 Latin ones left the Petri printing house in Basel over the eight-four years the book was in print, making it probably the biggest selling book, with the exception of the Bible, in the sixteenth century. The Cosmographiaset new standards in ‘modern’ geography and cartography and paved the way for the Civitates Orbis Terrarumof Georg Braun and Frans Hogenberg in 1572, the TheatrumOrbis Terrarum from Abraham Ortelius from 1570 and Mercator’s Atlas from 1595. Despite the competition from the superior atlases of Ortelius and Mercator, the Cosmographiasold well up to the final edition of 1628.

Münster’s Cosmographiais without a doubt a milestone in the evolution of modern cartography and geography and he deserves to be better known than he is. Bizarrely, although they mostly aren’t aware of it, Germans of a certain age are well aware of what Münster looks like, as his portrait was used for the 100 DM banknote from 1961 to 1995, when he was replaced by Clara Schumann.


Source: Wikimedia Commons



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

Renaissance mathematics and medicine

Anyone who read my last blog post might have noticed that the Renaissance mathematici Georg Tannstetter and Philipp Apian were both noted mathematicians and practicing physicians. In our day and age if someone was both a practicing doctor of medicine and a noted mathematician they would be viewed as something quite extraordinary but here we have not just one but two. In fact in the Renaissance the combination was quite common. Jakob Milich, who studied under Tannstetter in Vienna, was called to Wittenberg by Philipp Melanchthon in 1524, as professor for mathematics, where he taught both Erasmus Reinhold and Georg Joachim Rheticus. In 1536 he became professor for anatomy in Wittenberg and was succeeded by Rheticus as professor for mathematics. Rheticus in turn would later become a practicing physician in Krakow. The man, who Rheticus called his teacher, Nicolaus Copernicus, was another mathematical physician. My local Renaissance astronomer Simon Marius was another mathematician who studied and practiced medicine. That this was not a purely Germanic phenomenon is shown by the Welsh mathematicus and physician Robert Recorde and most notably by the Italian Gerolamo Cardano, who is credited with having written the first modern maths book, his Ars magna, and who was one of the most renowned physicians in Europe in his day.

These are only a few well-known examples but in fact it was very common for Renaissance mathematician to also be practicing physicians, so what was the connecting factor between these, for us, very distinct fields of study? There are in two interrelated factors that have to be taken into consideration, the first of which is astrology. The connection between medicine and astrology has a long history.

Greek legend says that Babylonian astrology was introduced into Greece by the Babylonian priest Berossus, who settled on the island of Kos in the third century BCE. Kos was the home of the Hippocratic School of medicine and astrology soon became an element in the Hippocratic Corpus. At the same time the same association between astrology and medicine came into Greek culture from Egypt in the form of the Greek-Egyptian god Hermes Trismegistos. Both the Egyptians and Babylonians had theories of lucky/unlucky, propitious/propitious days and these were integrated into the mix in the Greek lunar calendar. The Greeks developed the theory of the zodiac man, assigning the signs of the zodiac to the various part of the body. If a given part of the body was afflicted it would then be treated with the plants and minerals associated with its zodiac sign. The central role of astrology in medicine can be found in both the Hippocratic Corpus, in Airs, Waters, Placesit is stated that “astronomy is of the greatest assistance to medicine”and in Ptolemaeus’ Tetrabibloswe read, “The nature of the planets produce the forms and causes of the symptoms, since of the most important parts of man, Saturn is lord of the right ear, the spleen, the bladder, phlegm and the bones; Jupiter of touch, the lungs, the arteries and the seed; Mars of the left ear, the kidneys, the veins and the genitals; the sun of sight, the brain, the heart, the sinews and all on the right side; Venus of smell, the liver and muscles; Mercury of speech and thought, and the tongue, the bile and the buttocks; and the Moon of taste and of drinking, the mouth, the belly, the womb and all on the left side.” The connection between astrology was firmly established in Greek antiquity and was known as iatromathematica, health mathematics.

The theory of astrological medicine disappeared in Europe along with the rest of early science in the Early Medieval Period but was revived in the eighth century in the Islamic Empire when they took over the accumulated Greek Knowledge. The basic principles were fully accepted by the Islamic scholars and propagated down the centuries. When the translators moved into Spain and Sicily in the twelfth century they translated the Greek astrology and astrological medicine into Latin from Arabic along with rest of the Greek and Arabic sciences.

During the High Middle Ages, Christian scholars carried on an energetic debate about the legitimacy, or lack of it, of astrology. This debate centred on judicial astrology, this included natal astrology, mundane astrology, horary astrology, and electional astrology but excluded so called natural astrology, which included astrometeorology and astro-medicine both of which were regarded as scientific. To quote David Lindberg, “…no reputable physician of the later Middle Ages would have imagined that medicine could be successfully practiced without it.”


Woodcut of the Homo Signorum, or Zodiac Man, from a 1580 almanac. Source: Wikipedia Commons

Beginning in the fifteenth century during the humanist renaissance astrological medicine became the mainstream school medicine. It was believed that the cause, course and cure of an illness could be determined astrologically. In the humanist universities of Northern Italy and Poland dedicated chairs of mathematics were established, for the first time, which were actually chairs for astrology with the principle function of teaching astrology to medical students. Germany’s first dedicated chair for mathematics was founded at the University of Ingolstadt in about 1470 for the same reason.


Zodiac Man The Très Riches Heures du Duc de Berry c. 1412 Source: Wikimedia Commons

With the advent of moving type printing another role for mathematicians was producing astronomical/astrological calendars incorporating the phases of the moon, eclipses and other astronomical and astrological information needed by physicians to determine the correct days to administer blood lettings, purges and cuppings. These calendars were printed both as single sheet wall calendars and book form pocket calendars.


Renaissance Wall Calendar, 1544 Source: Ptak Science Books

These calendars were a major source of income for printer/publishers and for the mathematici who compiled them. Before he printed his legendary Bible, Johannes Guttenberg printed a wall calendar. Many civil authorities appointed an official calendar writer for their city or district; Johannes Schöner was official calendar writer for Nürnberg, Simon Marius for the court in Ansbach, Peter Apian for the city of Ingolstadt and Johannes Kepler for the city of Graz. Official calendar writers were still being employed in the eighteenth century. As I explained in an earlier post the pocket calendars led to the invention of the pocket diary.

Marius: [Alter und Newer Schreibcalender], [1602]  (Scan Nr. 150)

Simon Marius: Alter und Newer SchreibCalender auf das Jahr 1603 Title page Source: Deutsches Museum

With mainstream medicine based on astrology it was a short step for mathematicians to become physicians. Here we also meet the second factor. As a discipline, mathematics had a very low status in the Early Modern Period; in general mathematicians were regarded as craftsmen rather than academics. Those who worked in universities were at the very bottom of the academic hierarchy. At the medieval university it was only possible for graduates to advance to a doctorate in three disciplines, law, theology and medicine. It was not possible to do a doctorate in mathematics. With the dominance of iatromathematica, which depended on astrology, for which one in turn needed astronomy, for which one needed mathematics it was logical for mathematicians who wished to take a university doctorate, in order to gain a higher social status, to do so in medicine. The result of this is a fascinating period in European history from about 1400 to middle of the seventeenth century, where many of the leading mathematicians were also professional physicians. When astrology lost its status as a science this period came to an end.





Filed under History of Astrology, History of Astronomy, History of Mathematics, History of medicine, Renaissance Science, Uncategorized

The Bees of Ingolstadt

The tittle of this blog post is a play on the names of a father and son duo of influential sixteenth century Renaissance mathematici. The father was Peter Bienewitz born 16 April 1495 in Leisnig in Saxony just south of Leipzig. His father was a well off shoemaker and Peter was educated at the Latin school in Rochlitz and then from 1516 to 1519 at the University of Liepzig. It was here that he acquired the humanist name Apianus from Apis the Latin for a bee, a direct translation of the German Biene. From now on he became Petrus Apianus or simply Peter Apian.


Apianus on a 16th-century engraving by Theodor de Bry Source: Wikimedia Commons

In 1519 he went south to the University of Vienna to study under Georg Tannstetter a leading cosmographer of the period.


Georg Tannstetter Portrait ca. 1515, by Bernhard Strigel (1460 – 1528) Source: Wikimedia Commons

Tannstetter was a physician, mathematician astronomer and cartographer, who studied mathematics at the University of Ingolstadt under Andreas Stiborius and followed Conrad Celtis and Stiborius to Vienna in 1503 to teach at Celtis’ Collegium poetarum et mathematicorum. The relationship between teacher and student was a very close one. Tannstetter edited a map of Hungary that was later printed by Apian and the two of them produced the first printed edition of Witelo’s Perspectiva, which was printed and published by Petreius in Nürnberg in 1535. This was one of the books that Rheticus took with him to Frombork as a gift for Copernicus.

In 1520 Apian published a smaller updated version of the Waldseemüller/ Ringmann world map, which like the original from 1507 named the newly discovered fourth continent, America. Waldseemüller and Ringmann had realised their original error and on their 1513 Carte Marina dropped the name America, However, the use by Apian and by Johannes Schöner on his 1515 terrestrial globe meant that the name became established.


Apian’s copy of the Waldseemüller world map, naming the new fourth continent America Source: Wikimedia Commons

Apian graduated BA in 1521 and moved first to Regensburg then Landshut. In 1524 he printed and published his Cosmographicus liber, a book covering the full spectrum of cosmography – astronomy, cartography, navigation, surveying etc. The book became a sixteenth century best seller going through 30 expanded editions in 14 languages but after the first edition all subsequent editions were written by Gemma Frisius.


Title page of Apian’s Cosmpgraphia

In 1527 Apian was called to the University of Ingolstadt to set up a university printing shop and to become Lektor for mathematics. He maintained both positions until his death in 1552.

In 1528 he printed Tannstetter’s Tabula Hungariaethe earliest surviving printed map of Hungary. In the same year Apian dedicated his edition of Georg von Peuerbach’s New Planetary Theory to his “famous teacher and professor for mathematics” Tannstetter.


Tabula Hungarie ad quatuor latera Source: Wikimedia Commons

One year earlier he published a book on commercial arithmetic, Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen(A new and well-founded instruction in all Merchants Reckoning in three books, understood with fine rules and exercises). It was the first European book to include (on the cover), what is know as Pascal’s triangle, which was known earlier to both Chinese and Muslim mathematicians.


This is one of the volumes lying on the shelf in Holbein’s painting The Ambassadors. Like his Cosmographicusit was a bestseller.

In the 1530s Apian was one of a group of European astronomers, which included Schöner, Copernicus, Fracastoro and Pena, who closely observed the comets of that decade and began to question the Aristotelian theory that comets are sublunar meteorological phenomena. He was the first European to observe and publish that the comet’s tail always points away from the sun, a fact already known to Chinese astronomers. Fracastoro made the same observation, which led him and Pena to hypothesise that the comet’s tail was an optical phenomenon, sunlight focused through the lens like translucent body of the comet. These observations in the 1530s led to an increased interest in cometary observation and the determination in the 1570s by Mästlin, Tycho and others that comets are in fact supralunar objects.


Diagram by Peter Apian from his book Astronomicum Caesareum (1540) demonstrating that a comet’s tail points away from the Sun. The comet he depicted was that of 1531, which we now know as Halley’s Comet. Image courtesy Royal Astronomical Society.

Through the Cosmographicus he became a favourite of Karl V, the Holy Roman Emperor, and Apian became the Emperor’s astronomy tutor. Karl granted him the right to display a coat of arms in 1535 and knighted him in 1541. In 1544 Karl even appointed him Hofpfalzgraf (Imperial Count Palatine), a high ranking court official.

Apian’s association with Karl led to his most spectacular printing project, one of the most complicated and most beautiful books published in the sixteenth century, his Astronomicum Caesareum (1540). This extraordinary book is a presentation of the then Standard Ptolemaic astronomy in the form of a series of highly complex and beautifully designed volvelles. A vovelle or wheel chart is a form of paper analogue computer. A series of rotating paper discs mounted on a central axis or pin that can be used to calculate various mathematical functions such as the orbital positions of planets.


Astronomicum Caesareum title page

The Astronomicum Caesareumcontains two volvelles for each planet, one to calculate its longitude for a given time and one to calculate its latitude.


Astronomicum Caesareum volvelle for longitude for Saturn


Astronomicum Caesareum volvelle for the latitude for Saturn

There is also a calendar disc to determine the days of the week for a given year.


Astronomicum Caesareum calendar volvelle

Finally there are vovelles to determine the lunar phases  as well as lunar and solar eclipse.


Astronomicum Caesareum : Disc illustrating a total eclipse of the moon 6 Octobre 1530


Astronomicum Caesareum solar eclisse volvelle

Johannes Kepler was very rude about the Astronomicum Caesareum, calling it a thing of string and paper. Some have interpreted this as meaning that it had little impact. However, I think the reverse is true. Kepler was trying to diminish the status of a serious rival to his endeavours to promote the heliocentric system. Owen Gingerich carried out a census of 111 of the approximately 130 surviving copies of the book and thinks that these represent almost the whole print run. This book is so spectacular and so expensive that the copies rarely got seriously damaged of thrown away.

Like other contemporary mathematici Apian designed sundials and astronomical instruments as well as marketing diverse volvelles for calculation purposes. Apian died in 1552 and was succeeded on his chair for mathematics by his son Philipp, the second of the bees from Ingolstadt.

Philipp Apian was born 14 September 1531, as the fourth of fourteen children (nine sons and five daughters) to Peter Apian and his wife Katharina Mesner.


Philipp Apian painting by Hans Ulrich Alt Source: Wikimedia Commons

He started receiving tuition at the age of seven together with Prince Albrecht the future Duke of Bavaria, who would become his most important patron.


Duke Albrecht V of Bavaria Hans Muelich Source: Wikimedia Commons

He entered the University of Ingolstadt at the age of fourteen and studied under his father until he was eighteen. He completed his studies in Burgundy, Paris and Bourges. In 1552 aged just 21 he inherited his fathers printing business and his chair for mathematics on the University of Ingolstadt. As well as teaching mathematics at the university, which he had started before his father died, Philipp studied medicine. He graduated in medicine several years later during a journey to Italy, where he visited the universities of Padua, Ferrara and Bolgna.

In 1554 his former childhood friend Albrecht, now Duke of Bavaria, commissioned him to produce a new map of Bavaria. During the summers of the next seven years he surveyed the land and spent the following two years drawing the map. The 5 metres by 6 metres map at the scale of 1:45,000, hand coloured by Bartel Refinger was hung in the library of the Bavarian palace.


Philipp Apian’s map of Bavaria

In 1566 Jost Amman produced 24 woodblocks at the smaller scale of 1:144,000, which Apian printed in his own print shop. Editions of this smaller version of the map continued to be issued up to the nineteenth century.


Overview of the 24 woodblock prints of Apian’s map of Bavaria

In 1576 he also produced a terrestrial globe for Albrecht. Map, woodblocks, woodblock prints and globe are all still extant.


Apian’s terrestrial globe

In 1568 Phillip converted to Protestantism and in the following year was forced by the Jesuit, who controlled the University of Ingolstadt to resign his post. In the same year, he was appointed professor for mathematics at the Protestant University of Tübingen. In Tübingen his most famous pupil was Michael Mästlin, who succeeded him as professor for mathematics at the university and would become Johannes Kepler’s teacher. An irony of history is that Philipp was forced to resign in Tübingen in 1583 for refusing to sign the Formal of Concord, a commitment to Lutheran Protestantism against Calvinism. He continued to work as a cartographer until his death in 1589.

There is a genealogy of significant Southern German Renaissance mathematici: Andreas Stiborius (1464–1515) taught Georg Tannstetter (1482–1535), who taught Peter Apian (1495–1552), who taught Philipp Apian (1531–1589), who taught Michael Mästlin (1550–1631), who taught Johannes Kepler (1571–1630)













Filed under History of Astronomy, History of Cartography, History of Mathematics, Renaissance Science

Tycho’s last bastion

In the history of science, scholars who end up on the wrong side of history tend to get either forgotten and/or vilified. What do I mean by ‘end up on the wrong side of history’? This refers to scholars who defend a theory that in the end turns out to be wrong against one that in the end turns out to be right. My very first history of science post on this blog was about just such a figure, Christoph Clavius, who gets mocked by many as the last Ptolemaic dinosaur in the astronomy/cosmology debate at the beginning of the seventeenth century. In fact there is much to praise about Clavius, as I tried to make clear in my post and he made many positive contributions to the evolution of the mathematical sciences. Another man, who ended up on the wrong side of history in the same period is the Danish astronomer, Christen Sørensen, better known, if at all, by the name Longomontanus, the Latinised toponym based on Lomborg, the Jutland village where he was born on 4 October 1562 the son of a poor labourer, who died when he was only eight years old.


Longomontanus Source: Wikimedia Commons

Tycho Brahe backed the wrong astronomical theory in this period, a theory that is generally named after him although several people seem to have devised it independently of each other in the closing quarter of the sixteenth century. However, Tycho has not been forgotten because he delivered the new data with which Johannes Kepler created his elliptical model of the solar system. However, what people tend to ignore is that Tycho did not produce that data single-handedly, far from it.

The island of Hven, Tycho’s fiefdom, was a large-scale research institute with two observatories, an alchemy laboratory, a paper mill and a printing workshop.


Map of Hven from the Blaeu Atlas 1663, based on maps drawn by Tycho Brahe in the previous century Source: Wikimedia Commons

This enterprise was staffed by a veritable army of servants, technicians and research assistant with Tycho as the managing director and head of research.


Engraving of the mural quadrant from Brahe’s book Astronomiae instauratae mechanica (1598) Showing Tyco direction observations Source: Wikimedia Commons

Over the years the data that would prove so crucial to Kepler’s endeavours was collected, recorded and analysed by a long list of astronomical research assistants; by far and away the most important of those astronomical research assistants was Christen Sørensen called Longomontanus, who also inherited Tycho’s intellectual mantle and continued to defend his system into the seventeenth century until his death in 1647.

Christen Sørensen came from a very poor background so acquiring an education proved more than somewhat difficult. After the death of his father he was taken into care by an uncle who sent him to the village school in Lemvig. However, after three years his mother took him back to work on the farm; she only allowed him to study with the village pastor during the winter months. In 1577 he ran away to Viborg, where he studied at the cathedral school, supporting himself by working as a labourer. This arrangement meant that he only entered the university in Copenhagen in 1588, but with a good academic reputation. It was here at the university that he acquired his toponym, Longomontanus. In 1589 his professor recommended him to Tycho Brahe and he entered into service on the island of Hven.


Tycho Brahe’s Uraniborg main building from the 1663 Blaeu’s Atlas Major Centre of operations Source: Wikimedia Commons

He was probably instructed in Tycho’s methods by Elias Olsen Morsing, who served Tycho from 1583 to 1590, and Peter Jacobsen Flemløse, who served from 1577-1588 but stayed in working contact for several years more and became a good friend of Longomontanus. Longomontanus proved to be an excellent observer and spent his first three years working on Tycho’s star catalogue.


Stjerneborg Tycho Brahe’s second observatory on Hven: Johan Blaeu, Atlas Major, Amsterdam Source: Wikimedia Commons

Later he took on a wider range of responsibilities. In 1597, Tycho having clashed with the new king, the entire research institute prepared to leave Hven. Longomontanus was put in charge of the attempt to bring Tycho’s star catalogue up from 777 stars to 1,000. When Tycho left Copenhagen, destination unknown, Longomontanus asked for and received his discharge from Tycho’s service.

While Tycho wandered around Europe trying to find a new home for his observatory, Longomontanus also wandered around Europe attending various universities–Breslau, Leipzig and Rostock–and trying to find a new patron. He graduated MA in Rostock. During their respective wanderings, Tycho’s and Longomontanus’ paths crossed several times and the corresponded frequently, Tycho always urging Longomontanus to re-enter his service. In January 1600 Longomontanus finally succumbed and joined Tycho in his new quarters in Prague, where Johannes Kepler would soon join the party.

When Kepler became part of Tycho’s astronomical circus in Prague, Longomontanus the senior assistant was working on the reduction of the orbit of Mars. Tycho took him off this project putting him instead onto the orbit of the Moon and giving Mars to Kepler, a move that would prove history making. As should be well known, Kepler battled many years with the orbit of Mars finally determining that it was an ellipse thereby laying the foundation stone for his elliptical astronomy. The results of his battle were published in 1609, together with his first two laws of planetary motion, in his Astronomia nova.


Portrait of Johannes Kepler. Source: Wikimedia Commons

Meanwhile, Longomontanus having finished Tycho’s lunar theory and corrected his solar theory took his final departure from Tycho’s service, with letters of recommendation, on 4 August 1600.  When Tycho died 24 October 1601 it was thus Kepler, who became his successor as Imperial Mathematicus and inherited his data, if only after a long dispute with Tycho’s relatives, and not Longomontanus, which Tycho would certainly have preferred.

Longomontanus again wandered around Northern Europe finally becoming rector of his alma mater the cathedral school in Viborg in 1603. In 1605, supported by the Royal Chancellor, Christian Friis, he became extraordinary professor for mathematics at the University of Copenhagen, moving on to become professor for Latin literature in the same year. In 1607 he became professor for mathematics, and in 1621 his chair was transformed into an extraordinary chair for astronomy a post he held until his death.

As a professor in Copenhagen he was a member of an influential group of Hven alumni: Cort Aslakssøn (Hven 159-93) professor for theology, Christian Hansen Riber (Hven 1586-90) professor for Greek, as well as Johannes Stephanius (Hven 1582-84) professor for dialectic and Gellius Sascerides (Hven 1585-86) professor for medicine.

Kepler and Longomontanus corresponded for a time in the first decade of the seventeenth century but the exchange between the convinced supporter of heliocentricity and Tycho’s most loyal lieutenant was not a friendly one as can be seen from the following exchange:

Longomontanus wrote to Kepler 6th May 1604:

These and perhaps all other things that were discovered and worked out by Tycho during his restoration of astronomy for our eternal benefit, you, my dear Kepler, although submerged in shit in the Augean stable of old, do not scruple to equal. And you promise your labor in cleansing them anew and even triumph, as if we should recognise you as Hercules reborn. But certainly no one does, and prefers you to such a man, unless when all of it has been cleaned away, he understands that you have substituted more appropriate things in the heaven and in the celestial appearances. For in this is victory for the astronomer to be seen, in this, triumph. On the other hand, I seriously doubt that such things can ever be presented by you. However, I am concerned lest this sordid insolence of yours defile the excellent opinion of all good and intelligent men about the late Tycho, and become offensive.

Kepler responded early in 1605:

The tone of your reference to my Augean stable sticks in my mind. I entreat you to avoid chicanery, which is wont to be used frequently with regard to unpopular things. So that you might see that I have in mind how the Augean stable provided me with the certain conviction that I have not discredited astronomy – although you can gather from the present letter – I will use it with the greatest possible justification. But it is to be used as an analogy, not for those things that you or Tycho were responsible for constructing – which either blinded by rage or perverted by malice you quite wrongfully attributed to me – but rather in the comparison of the ancient hypotheses with my oval path2. You discredit my oval path. I hold up to you the hundred-times-more-absurd spirals of the ancients (which Tycho imitated by not setting up anything new but letting the old things remain). If you are angry that I cannot eliminate the oval path, how much more ought you to be angry with the spirals, which I abolished. It is as though I have sinned with the oval I have left, even though to you all the rest of the ancients do not sin with so many spirals. This is like being punished for leaving behind one barrow full of shit although I have cleaned the rest of the Augean stables. Or in your sense, you repudiate my oval as one wagon of manure while you tolerate the spirals which are the whole stable, to the extent that my oval is one wagon. But it is unpleasant to tarry in rebutting this most manifest slander.

 Whereas, as already mentioned above, Kepler presented his heliocentric theory to the world in 1609, Longomontanus first honoured Tycho’s memory with his Astronomia Danica in 1622. Using Tycho’s data Longomontanus provided planetary models and planetary tables for Tycho’s geo-heliocentric system. Longomontanus, however, differed from Tycho in that he adopted the diurnal rotation of Helisaeus Roeslin, Nicolaus Raimarus and David Origanus.


The Astronomia Danica saw two new editions in 1640 and 1663. For the five decades between 1620 and 1670 Kepler’s elliptical astronomy and the Tychonic geo-heliocentric system with diurnal rotation competed for supremacy in the European astronomical community with Kepler’s elliptical system finally triumphing.

 In 1625 Longomontanus suggested to the King, Christian IV, that he should build an observatory to replace Tycho’s Stjerneborg, which had been demolished in 1601. The observatory, the Rundetaarn (Round Tower), was conceived as part of the Trinitatis Complex: a university church, a library and the observatory. The foundation stone was laid on 7 July 1637 and the tower was finished in 1642. Longomontanus was appointed the first director of the observatory, after Leiden 1632 only the second national observatory in Europe.


Copenhagen – Rundetårn Source: Wikimedia Commons

Both Kepler and Longomontanus, who lost their fathers early, started life as paupers Both of them worked they way up to become leading European astronomers. Kepler has entered the pantheon of scientific gods, whereas Longomontanus has largely been assigned to the dustbin of history. Although Longomontanus cannot be considered Kepler’s equal, I think he deserves better, even if he did back the wrong theory.







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

Spicing up the evolution of the mathematical sciences

When we talk about the history of mathematics one thing that often gets forgotten is that from its beginnings right up to the latter part of the Early Modern Period almost all mathematics was developed to serve a particular practical function. For example, according to Greek legend geometry was first developed by the ancient Egyptians to measure (…metry) plots of land (geo…) following the annual Nile floods. Trade has always played a very central role in the development of mathematics, the weights and measures used to quantify the goods traded, the conversion rates of different currencies used by long distance traders, the calculation of final prices, taxes, surcharges etc. etc. A good historical example of this is the Islamic adoption of the Hindu place value decimal number system together with the associated arithmetic and algebra for use in trade, mirrored by the same adoption some time later by the Europeans through the trader Leonardo Pisano. In what follows I want to sketch the indirect impact that the spice trade had on the evolution of the mathematical sciences in Europe during the Renaissance.

The spice trade does not begin in the Renaissance and in fact had a long prehistory going back into antiquity. Both the ancient Egyptians and the Romans had extensive trade in spices from India and the Spice Islands, as indeed the ancient Chinese also did coming from the other direction.


The spice trade from India attracted the attention of the Ptolemaic dynasty, and subsequently the Roman empire. Source: Wikimedia Commons

Throughout history spice meant a much wider range of edible, medicinal, ritual and cosmetic products than our current usage and this trade was high volume and financially very rewarding. The Romans brought spices from India across the Indian Ocean themselves but by the Middle Ages that trade was dominated by the Arabs who brought the spices to the east coast of Africa and to the lands at the eastern end of the Mediterranean, known as the Levant; a second trade route existed overland from China to the Levant, the much fabled Silk Road. The Republic of Venice dominated the transfer of spices from the Levant into Europe, shipping them along the Mediterranean.


The economically important Silk Road (red) and spice trade routes (blue) blocked by the Ottoman Empire c. 1453 with the fall of the Byzantine Empire, spurring exploration motivated initially by the finding of a sea route around Africa and triggering the Age of Discovery. Source: Wikimedia Commons

Here I go local because it was Nürnberg, almost literally at the centre of Europe, whose traders collected the spices in Venice and distributed them throughout Europe. As Europe’s premier spice traders the Nürnberger Patrizier (from the Latin patrician), as they called themselves, grew very rich and looking for other investment possibilities bought up the metal ore mines in central Europe. In a short period of time they went from selling metal ore, to smelting the ore themselves and selling the metal, to working the metal and selling the finished products; each step producing more profit. They quite literally produced anything that could be made of metal from sewing needles to suits of armour. Scientific and mathematical instruments are also largely made of metal and so Nürnberg became Europe’s main centre for the manufacture of mathematical instruments in the Renaissance. The line from spice to mathematical instruments in Nürnberg is a straight one.


Torquetum designed by Johannes Praetorius and made in Nürnberg

By the middle of the fifteenth century the Levant had become a part of the Ottoman Empire, which now effectively controlled the flow of spices into Europe and put the screws on the prices. The Europeans needed to find an alternative way to acquire the much-desired products of India and the Spice Islands, cutting out the middlemen. This need led to the so-called age of discovery, which might more appropriately be called the age of international sea trade. The most desirable and profitable trade goods being those spices.

The Portuguese set out navigating their way down the west coast of Africa and in 1488 Bartolomeu Dias succeeded in rounding the southern most tip of Africa and entering the Indian Ocean.


Statue of Bartolomeu Dias at the High Commission of South Africa in London. Source: Wikimedia Commons

This showed that contrary to the Ptolemaic world maps the Indian Ocean was not an inland sea but that it could be entered from the south opening up a direct sea route to India and the Spice Islands.


A printed map from the 15th century depicting Ptolemy’s description of the Ecumene, (1482, Johannes Schnitzer, engraver). Showing the Indian Ocean bordered by land from the south Source: Wikimedia Commons

In 1497 Vasco da Gama took that advantage of this new knowledge and sailed around the Cape, up the east coast of Africa and then crossing the Indian Ocean to Goa; the final part of the journey only being made possible with the assistance of an Arab navigator.


The route followed in Vasco da Gama’s first voyage (1497–1499) Source: Wikimedia Commons

Famously, Christopher Columbus mistakenly believed that it would be simpler to sail west across, what he thought was, an open ocean to Japan and from there to the Spice Islands. So, as we all learn in school, he set out to do just that in 1492.

In fourteen hundred and ninety two

Columbus sailed the ocean blue.

The distance was of course much greater than he had calculated and when, what is now called, America had not been in the way he and his crews would almost certainly have all died of hunger somewhere out on the open seas.


Columbus’ voyage. Modern place names in black, Columbus’s place names in blue Source: Wikimedia Commons

The Portuguese would go on over the next two decades to conquer the Spice Islands setting up a period of extreme wealth for themselves. Meanwhile, the Spanish after the initial disappointment of realising that they had after all not reached Asia and the source of the spices began to exploit the gold and silver of South America, as well as the new, previously unknown spices, most famously chilli, that they found there. In the following centuries, eager also to cash in on the spice wealth, the English and French pushed out the Portuguese in India and the Dutch did the same in the Spice Islands themselves. The efforts to establish sea borne trading routes to Asia did not stop there. Much time, effort and money was expended by the Europeans in attempts to find the North West and North East Passages around the north of Canada and the north of Russia respectively; these efforts often failed spectacularly.

So, you might by now be asking, what does all this have to do with the evolution of the mathematical science as announced in the title? When those first Portuguese and Spanish expedition set out their knowledge of navigation and cartography was to say the least very rudimentary. These various attempts to reach Asia and the subsequent exploration of the Americas led to an increased effort to improve just those two areas of knowledge both of which are heavily based on mathematics. This had the knock on effect of attempts to improve astronomy on which both navigation and cartography depend. It is not chance or coincidence that the so-called age of discovery is also the period in which modern astronomy, navigation and cartography came of age. Long distance sea trading drove the developments in those mathematically based disciplines.

This is not something that happened overnight but there is a steady curve of improvement in this disciplines that can be observed over the two plus centuries that followed Dias’ first rounding of the Cape. New instruments to help determine latitude and later longitude such as mariners’ astrolabe (which is not really an astrolabe, around 1500) the backstaff (John Davis, 1594) and the Hadley quadrant (later sextant, 1731) were developed. The Gunter Scale or Gunter Rule, a straight edge with various logarithmic and trigonometrical scales, which together with a pair of compasses was used for cartographical calculations (Edmund Gunter, early seventeenth century). William Oughtred would go on to lay two Gunter Scales on each other and invent the slide rule, also used by navigators and cartographers to make calculations.

New surveying instruments such as the surveyor’s chain (also Edmund Gunter), the theodolite (Gregorius Reisch and Martin Waldseemüller independently of each other but both in 1512) and the plane table (various possible inventors, middle of the sixteenth century). Perhaps the most important development in both surveying and cartography being triangulation, first described in print by Gemma Frisius in 1533.

Cartography developed steadily throughout the sixteenth century with cartographers adding the new discoveries and new knowledge to their world maps (for example the legendary Waldseemüller world map naming America) and searching for new ways to project the three-dimensional earth globe onto two-dimensional maps. An early example being the Stabius-Werner cordiform projection used by Peter Apian, Oronce Fine and Mercator.


Cordiform projection in a map of the world by Apianus 1524 which is one of the earliest maps that shows America Source: Wikimedia Commons

This development eventually leading to the Mercator-Wright projection, a projection specifically designed for marine navigators based on Pedro Nunes discovery that a path of constant bearing is not a great circle but a spiral, known as a loxodrome or rhumb line. Nunes is just one example of a mathematical practitioner, who was appointed to an official position to develop and teach new methods of navigation and cartography to mariners, others were John Dee and Thomas Harriot.


Pedro Nunes was professor of mathematics at the University of Coimbra and Royal Cosmographer to the Portuguese Crown. Source: Wikimedia Commons

To outline all of the developments in astronomy, navigation and cartography that were driven by the demands the so-called age of discovery, itself triggered by the European demand for Asian spices would turn this blog post into a book but I will just mention one last thing. In his one volume history of mathematics, Ivor Grattan-Guinness calls this period the age of trigonometry. The period saw a strong development in the use of trigonometry because this is the mathematical discipline most necessary for astronomy, navigation and cartography. One could say a demand for spices led to a demand for geometrical angles.



Filed under History of Astronomy, History of Cartography, History of Navigation, Renaissance Science, Uncategorized