Category Archives: History of Mathematics

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.

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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.

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

 

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

Saxton and Speed two early Elizabethan cartographers and the Flemish influence

It is possible to date the start of the gradual emergence of modern cartography to the first decade of the fifteenth century when Jacopo d’Angelo produced the first Latin translation of Ptolemaeus’ Geographia(Geōgraphikḕ Hyphḗgēsis); this important Greek work had not been translated in the great wave of scientific translations in the High Middle Ages. This new knowledge of Ptolemaeus’ cartography with its projections and its longitude and latitude grids first took hold in Northern Italy, where its most famous early exponent was physician and mathematicus Paolo dal Pozzo Toscanelli (1379–1482) author of the so-called Columbus map.

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Paolo dal Pozzo Toscanelli Source: Wikimedia Commons

From Northern Italy the new cartographer spread fairly rapidly to Austria and by 1450 Vienna was a major centre for cartography. Not long after the invention of movable metal type printing editions of the Geographia began to appear, initially without maps but very soon with, and by 1500 various editions were making their way around Europe. From Vienna the knowledge of the new cartographer moved north into Germany, with two schools of cartography developing. The so-called historical school was centred on the St. Dié mapmakers in Lorraine and includes Sebastian Münster in Basel. Whereas the so-called mathematical school, also known as the Vienna-Nürnberg school, has Johannes Schöner and Peter Apian as its two most significant exponents. Both schools include both aspects of the Ptolemaic cartography, the historical and the mathematical, in their maps and the difference is rather more one of emphasis.

From Southern Germany the new cartography spread throughout Europe. Notable cartographers, for example, are Pedro Nunes in Portugal and Oronce Fine in France. However the major centre for the new cartography became the so-called Flemish school centred on Gemma Frisius at the University of Louvain. Its two most notable associates are Gerhard Mercator and Abraham Ortelius, the two most influential cartographers in the second half of the sixteenth century.

But what of England? As with the mathematical sciences in general, England lagged well behind the continent in terms of cartography. The first Britain to become acquainted with the new developments in cartography was probably John Dee, who following his graduation at Cambridge university travelled extensively on the continent and spent two years in Louvain studying and working together with both Gemma Frisius and Gerard Mercator.

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John Dee artist unknown Source: Wikimedia Commons

During his travels he also formed close friendships with both Abraham Ortelius and Pedro Nunes. When he returned to Britain Dee brought the latest developments in the Ptolemaic cartography, Frisius’ new methods of surveying through triangulation and the latest astronomical, cartographical and surveying instruments with him from Louvain. The Flemish/Dutch influence is clearly visible in the early English atlases.

It is probably no coincidence that the two ministers at Elizabeth’s court in London who pushed hard for the introduction of the new cartography into England were Sir Francis Walsingham, principle secretary to Elizabeth, and William Cecil, 1stBaron Burghley, Elizabeth’s chief advisor, both of whom were Dee’s, somewhat unreliable, patrons at court.

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Engraving of Queen Elizabeth I, William Cecil and Sir Francis Walsingham, by William Faithorne, 1655 Source: Wikimedia Commons

Their motivations for supporting the development of cartography were political and derived from military and commercial considerations and not academic or scientific ones. The same applies, of course to the general developments in cartography in the Early Modern Period throughout Europe. Burghley, the main driving force behind a new English cartography, possessed a self made atlas of manuscript maps from various sources that he is said to have always carried with him. Burghley’s atlas has survived and is now in the British Library.

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William Cecil, 1st Baron Burghley from NPG artist unkown Source: Wikimedia Commons

Burghley saw the necessity, on political grounds, of organising and financing a new, modern map of the British Isles. The first cartographer, who appears to have received a commission to map Britain from Burghley, was John Rudd (c. 1498–1597). In 1561 Rudd, a graduate of Clare College and a fellow of St. John’s Cambridge, was granted a two-year paid leave of absence from his various positions in the Church of England in order to travel the country with the objective of mapping England. It is not know if Rudd fulfilled his objective, as no map of England can with certainty be assigned to him. However, it has been speculated that his work was a source for the new map of Britain published by Mercator in his atlas. Several of the maps in Burghley’s handmade atlas are attributed to Rudd.

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This is a manuscript map of County Durham. It forms part of an atlas that belonged to William Cecil Lord Burghley, Elizabeth I’s Secretary of State. Burghley used this atlas to illustrate domestic matters. The map dates from 1569 and is by John Rudd, the man to whom Christopher Saxton was an apprentice to in 1570. Source: British Library

In his work Rudd employed Christopher Saxon (c. 1540–c.1610) as an assistant and taught him the art of surveying. Born in the West Riding of Yorkshire in 1542 or 1544 very little is known about Saxton’s childhood, although his employment by Rudd is confirmed by documents. On Burghley’s instructions Thomas Seckford, Master in Ordinary of the Court of Request at Elizabeth’s Court,

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Thomas Seckford Source: East Anglian Times

commissioned Christopher Saxton in 1573 to survey all the English counties and produce an atlas of the realm. It is not certain what motivated Burghley to act at this time but it might have been the publication of Ortelius’ Theatrum orbis terrarumin 1570, which was much admired in England, and/or Philipp Apian’s Bairischen Landtafeln from 1566

The survey of England began in 1574 and of Wales in 1577; it was completed in 1578. The individual maps were engraved as soon as finished and the proofs sent to Burghley.

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Saxton England and Wales proof map Source: British Library

In 1579 the maps were bound together and publish as a book for which Saxton received ten-year exclusive publication rights from the crown.

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Frontispiece Saxton The Counties of England and Wales Source: My facsimile copy

Although we now refer to Saxton’s book as an atlas at the time Mercator’s coining of the term still lay sixteen years in the future. It is not known how Saxton did his surveying work but the speed with which he completed the task and the general level of accuracy of his maps would suggest two things. Firstly that he had access to previous data, possibly from Rudd’s work amongst others, and secondly that he used triangulation in some form. The nationwide chain of beacons set up by the government to warn of a Spanish invasion from the Netherlands in 1567 would have provided him with a useful set of predetermined triangulation points. The pass issued by the Privey Council on 10 July 1576 for his survey of Wales also strongly suggests triangulation as his survey method.

An open Lettre to all Justices of peace mayours & others etc within the severall Shieres of Wales. That where the bearer hereof Christofer Saxton is appointed by her Maiestie vnder her signe and signet to set forth and describe Coates [Cartes] in particulerlie all the shieres in Wales. That the said Justices shalbe aiding and assisting vnto him to see him conducted vnto any towre Castle highe place or hill to view that country, and that he maybe accompanied with ij or iij honest men such as do best know the cuntrey for the best accomplishment of that service, and that at his departure from any towne or place that he hath taken the view of said towne do set forth a horseman that speak both welshe and englishe to safe conduct to the next market Towne, etc

“…any towre Castle highe place or hill to view that country…” strongly suggests triangulation.

There are 35 maps, each bearing the arms of the Queen and Thomas Seckford. Drawn by Saxton the maps were engraved by Augustine Ryther (5 maps) the only engraver who clearly identifies as English, Remigius Hogenberg (9 maps),Leonard Terwoort of Antwerp (5 maps), Cornelius Hogius (1 map), Johannes Rutlinger (1 map) all four of whom were Flemish, Francis Scatter (2 maps), Nicholas Reynold (1 maps) are both of uncertain origin. Of the remaining unsigned maps five are definitely engraved in the Flemish style. In general there are clearly Flemish elements in the style of all the maps.

It is not known for certain when John Dee and Christopher Saxton became acquainted and whether Dee had an influence on Saxton, but when Dee, as Warden of Christ’s College, Manchester (1595–1605), became involved in a boundary dispute he employed Saxton as his surveyor to settle the argument.

Although it sold well, Saxton’s atlas was not without its critics. Although some counties are presented on separate maps others are grouped together on one map. For example the 13 Welsh counties are presented on 7 maps or Kent, Sussex, Surry, Middlesex and London are all on one map. Because of the pages are uniform in size the map scales vary considerably. Also although the maps initially appear homogeneous the symbols used vary considerably from map to map, even between maps engraved by the same engraver.  None of Saxton’s maps have roads. All of these imperfections led to various attempts to improve on Saxton’s work.

The first of these was John Norden (c. 1547–1625), who started a series of county histories, each accompanied by a map, entitled Speculum Britanniae.

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John Norden Source

The first volume Speculum Britanniae: the First Parte: an Historicall, & Chorographicall Discription of Middlesexwas published in 1593. The manuscript is in the British Library with corrections in Burghley’s handwriting, which points to Burghley being Norden’s sponsor. 1595 he wrote a manuscript “Chorographical Description” of Middlesex, Essex,Surry,Sussex,Hampshire, Wight, Guernsey andJersey, dedicated to Queen Elizabeth. In 1596 he published his Preparative to the Speculum Britanniae, dedicated to Burghley. The only other volume published by Norden was Speculi Britaniae Pars: the Description of Hartfordshirein 1598. He completed accounts of five other counties in manuscript of which three were published posthumously in 1720, 1728 and 1840. It was probably Burghley’s death in 1598 that put an end to Norden’s project.

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John Norden’s map of Essex 1595 Source: British Library

The next attempt was undertaken by a friend and colleague of Norden’s, William Smith (c.1550–1618), antiquarian and Rouge Dragon at the College of Heralds. Unlike Norden, who had never left England, Smith spent five years living in Nürnberg, a major cartographical centre. In 1588 Smith completed The Particuler Description of England With The Portratures of Certaine the Chieffest Citties and Townes. Between 1602 and 1603 Smith anonymously published maps of Chester, Essex, Hertfordshire, Lancashire, Leicester, Norfolk Northamptonshire, Staffordshire, Suffolk, Surry, Warwickshire and Worcester probably engraved in Amsterdam and intended as sheets for a new atlas. Smith’s maps contain the roads missing from Saxton’s maps. It is thought that Smith abandoned his atlas project because of competition from John Speed (1551 or 52–1629)

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William Smith map of Lancashire 1598 Source: British Library

Speed born in Farndon, Cheshire followed his father into the tailoring business. Moving to London he became a Freeman of the Merchant Taylor’s Guild, who devoted his free time to cartography.

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John Speed Source: My facsimile copy

This activity attracted the attention of Sir Fulke Greville, 1stBaron Brooke (1554–1628) a leading Elizabethan statesman, who secured him a position at the Customs and with the support of the Queen, subsidized his map making.

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Fulke Greville, 1st Baron Brooke Portrait by Edmund Lodge Source: Wikimedia Commons

A member of the society of Antiquities he became friends with such people as the historian William Camden (1551–1623) and the cartographer William Smith, who assisted him with his research. Speed published his atlas, The Theatre of the Empire of Great Britaine, which was dated 1611 in 1612. He regarded it as a supplement to his Historie of Great Britainepublished in 1611. The use of the word theatre in the title reflects the influence of Ortelius, whose own The Theatre of the World was published in English in 1606. The Empire of Great Britain is a term introduced by John Dee.

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Speed title page Source: My facsimile copy

Whereas Christopher Saxton was a trained surveyor, who went out surveyed and drew his own maps, John Speed was a compiler who took his maps from various sources, including Saxton, and merely redrew them to a homogenous standard. Apart from Saxton (5 maps), Speed credits maps from John Norden (5), William Smith (2) and individual maps from Philip Symonson, John Harrington, William White, Thomas Durham and James Burrell. His maps of Wales are obviously based on Saxton’s although he doesn’t credit him. His of Ireland, which Saxton did not include in his atlas, are based on the work of Robert Lythe and Robert Jobson. However although not by him he doesn’t credit the majority of his maps. All of Speed’s maps were engraved by the Dutch engraver, Jodocus Hondius (1563–1612), who was himself one of the leading European cartographers and globe makers.

Obviously inspired by Saxton’s work Speed’s Theatre differs in that he includes Ireland and Scotland, both missing by Saxton. He gives each county a separate map and although he cannot reproduce them all to the same scale, due to page size, Like Saxton he includes a scale bar on each map. However, it is not known what length for the mile Speed used. His symbols are uniform through the entire book and on the back of each map sheet he includes topographical, administrative and historical comments. The margins of the maps often include the arms of the leading families or other informative historical drawings and following William Smith he includes plans and maps of the principle towns and cities. Speed’s Theatre is altogether a much more attractive and informative work than Saxton’s atlas even though it very clearly owes it existence to the earlier pioneering work, so it is fair to speak of a Saxon/Speed presentation of the counties of Britain.

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The Eastern half of Saxton’s map of Essex (above) Speed’s version of the same (below ) to illustrate the similarities and the differences. The place where I spent the first fifteen years of my life, Thorp (now Thorpe-le-Soken) is roughly in the middle of the Tendering Hundred in the North-East corner of the county. Going south from there Little Clacton and Great Clacton are both marked but my birthplace on the coast, Clacton-on-Sea, obviously didn’t exist yet when the maps were drawn. Speed’s map contains a plan of the town of Colchester, “Britain’s oldest town”, where I went to school.

Unlike the Netherlands, where the fierce competition between the houses of Blaeu and Hondius led to ever better, ever more spectacular atlases and globes throughout the seventeenth century, following the publication of Speed’s Theatre, cartography on that scale ceased almost entirely in Britain. This meant that Speed’s Theatreremained the standard cartographical work in Britain for more than one hundred years. The burst of cartographical activity between Rudd and Speed remained a bubble rather than the start of tradition, as Gemma Frisius’, Abraham Ortelius’ and Gerard Mercator’s work had become in the Netherlands.

 

 

 

 

 

 

 

 

 

 

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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.”

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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.

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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.

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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.

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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.

 

 

 

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Astronomicum Caesareum volvelle for longitude for Saturn

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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.

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Astronomicum Caesareum calendar volvelle

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

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

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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.

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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.

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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.

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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.

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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.

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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)

 

 

 

 

 

 

 

 

 

 

 

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

Does the world really need another Galileo hagiography?

When it was first advertised several people drew my attention to Michael E. Hobart’s The Great Rift: Literacy, Numeracy, and the Religion-Science Divide[1]and it had hardly appeared when others began to ask what I thought about it and whether one should read it? I find it kind of flattering but also kind of scary that people want to know my opinion of a book before committing but even I can’t read a more than 500 page, intellectually dense book at the drop of the proverbial hat. Curiosity peaked piqued I acquired a copy, for a thick bound volume it’s actually quite reasonably priced, and took it with me to America, as my travel book. I will now give my considered opinion of Hobart’s tome and I’m afraid that it’s largely negative.

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Hobart’s title says nearly everything about his book and to make sure you know where he is going he spells it out in detail in an 18-page introductory chapter The Rift between Religion and Science, which he attributes to the fact that in the seventeenth century science ceased to be verbal and became numerical. If this should awaken any suspicions in your mind, yes his whole thesis is centred round Galileo’s infamous two books diatribe in Il Saggiatore. As far as I can see the only new thing that Hobart introduces in his book is that he clothes his central thesis in the jargon of information technology, something that I found irritating.

The next 34 pages are devoted to explaining that in antiquity the world was described both philosophically and theologically in words. Moving on, we get a 124-page section dealing with numbers and mathematics entitled, From the “Imagination Mathematical” to the Threshold of Analysis. Here Hobart argues that in antiquity and the Middle Ages numbers were thing numbers, i.e. they were only used in connection with concrete objects and never in an abstract sense simply as numbers for themselves. His presentation suffers from selective confirmation bias of his theory, when talking about the use of numbers in the Middle ages he only examines and quotes the philosophers, ignoring the mathematicians, who very obviously used numbers differently.

He moves on to the High Middle Ages and the Renaissance and outlines what he sees as the liberation of numbers from their thing status through the introduction of the Hindu-Arabic numbers through Leonardo Pisano, the invention of music notation, the introduction of linear perspective in art and the introduction of both Scaliger’s chronology and the Gregorian calendar. Here once again his presentation definitely suffers from selective confirmation bias. He sees both Scaliger and the Gregorian calendar as the first uses of a universal time measuring system for years. Nowhere in his accounts of using numbers or the recording of time in years does he deal with astronomy in antiquity and down to the Early Modern Period. Astronomers used the Babylonian number system, just as abstract as the Hindu-Arabic system, and the Egyptian solar calendar in exactly the same way as Scaliger’s chronology. He also ignores, except somewhere in a brief not much later, the earlier use of the Hindu-Arabic number system in computos.

Here it is worth mentioning a criticism of others that Hobart brings later. In a chapter entitled, Towards the Mathematization of Matter, he briefly discusses Peter Harrison on science and religion and David Wootton on the introduction of a new terminology in the seventeenth century. He goes on to say, “…both of these fine scholars overlook just how the mathematical abstractions born of the new information technology and modern numeracy supplied an alternative to literacy as a means for discerning patterns in nature.” Two things occur to me here, firstly the mathematization of science as the principle driving force behind the so-called scientific revolution is one of the oldest and most discussed explanation of the emergence of modern science, so Hobart is only really offering old wine in new bottles and not the great revolutionary idea that he thinks he has discovered. The second is that in his book, The Invention of Science, David Wootton has a 47-page section entitled The Mathematization of the World, dealing with the changes in the use and perception of mathematics in the Renaissance that is, in my opinion, superior to Hobart’s account.

The third and final part of Hobart’s book is titled Galileo and the Analytical Temper and is a straight up hagiography. This starts with a gushing account of Galileo’s proportional compass or sector, prominent on the book’s cover. In all of his account of how fantastic and significant this instrument is Hobart neglects an important part of its history. He lets the reader assume that this is a Galileo invention, which is far from true. Although in other places Hobart mentions Galileo’s patron and mentor Guidobaldo del Monte he makes no mention of the fact that Galileo’s instrument was a modification and development of any earlier instrument of del Monte’s, which in turn was a modification of an instrument designed and constructed by Fabrizio Mordente.

This sets the tone for Hobart’s Galileo. He invents the scientific method, really? Then we get told, “Then in a dazzling stroke he pointed it [the telescope] skyward. He was not the first to do so, but he was certainly the first to exploit the new telescope, using it to expand beyond normal eyesight and peer into the vastness of space.” No he wasn’t!  Hobart gives us a long discourse on Galileo’s atomism explaining in detail his theory of floating bodies but neglects to point out that Galileo was simply wrong. He is even more crass when discussing Galileo’s theory of the tides in his Dialogo. After a long discourse on how brilliantly-scientific Galileo’s analysis leading to his theory is Hobart calmly informs us, “Galileo’s theory, of course was subsequently proved wrong by Newton…”! Yes, he really did write that! Galileo’s theory of the tides was contradicted by the empirical facts before he even published it and is the biggest example of blind hubris in all of Galileo’s works.

Hobart’s Galileo bias is also displayed in his treatment of Galileo’s conflicts with the Catholic Church and Catholic scientists. After a very good presentation of Galileo’s excellent proof, in his dispute with Scheiner, that the sunspots are on the surface of the sun and not satellites orbiting it. Hobart writes in an endnote, “A committed Aristotelian, Scheiner continued to advance fierce polemics against Galileo, but even he eventually accepted Galileo’s analysis.” In fact Scheiner accepted Galileo’s analysis fairly rapidly and went on to write the definitive work on sunspots. Hobart somehow neglects to mention that Galileo falsely accused Scheiner of plagiarism in his Il Saggiatore and then presented some of Scheiner’s results as his own in his Dialogo. Describing the dispute in 1615/16 Hobart quoting Bellarmino’s Foscarini letter, “I say that if there were a true demonstration that the sun is at the centre of the world and the earth in the third heaven, and that the sun does not circle the earth but the earth circles the sun, then one would have to proceed with great care in explaining the Scriptures that appear contrary, and say rather that we do not understand them, than that what is demonstrated is false”, goes on to say without justification that Bellarmino would not have accepted a scientific proof but only an Aristotelian one. This is, to put it mildly, pure crap. The behaviour of the Jesuit astronomers throughout the seventeenth century proves Hobart clearly wrong.

I’m not even going to bother with Hobart’s presentation of the circumstances surrounding the trial, it suffices to say that it doesn’t really confirm with the known facts.

I also have problems with Hobart’s central thesis, “The Great Rift.” At times he talks about it as if it was some sort of explosive event, as his title would suggest then admits on more than one occasion that it was a very long drawn out gradual process. Although he mentions it in asides he never really addresses the fact that long after Galileo many leading scientists were deeply religious and saw their scientific work as revealing God’s handy work; scientists such as Kepler and Newton who were just as analytical and even more mathematical than Galileo.

Throughout the book I kept getting the feeling that Hobart is simply out of touch with much of the more recent research in the history of science although he has obviously invested an incredible amount of work in his book, which boasts 144-pages of very extensive endnotes quoting a library full of literature. Yes, the mathematization of science played a significant role in the evolution of science. Yes, science and religion have been drifting slowly apart since the Early Modern Period but I don’t think that the mathematization of science is the all-encompassing reason for that separation that Hobart is trying to sell here. No, Galileo did not singlehandedly create modern science as Hobart seem to want us to believe, he was, as I pointed out in a somewhat notorious post several years ago, merely one amongst a crowd of researchers and scholars involved in that process at the end of the sixteenth and the beginning of the seventeenth centuries. Does Hobart’s book bring anything new to the table? No, I don’t think it does. Should one read it? That is up to the individual but if I had known what was in it before I read it, I wouldn’t have bothered.

 

 

 

[1]Michael E. Hobart, The Great Rift: Literacy, Numeracy, and the Religion-Science Divide, Harvard University Press, Cambridge & London, 2018

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Filed under Book Reviews, History of Astronomy, History of Mathematics, Uncategorized

Imre and me – a turning point

Today is once again the anniversary of the day I started this blog nine years ago. Nine years‽ I have difficulty believing that I have really churned out blog posts on a regular basis, with only minor breaks, for nine years now. It has become something of a tradition that on my blog anniversary I post something autobiographical and I have decided this year to maintain that tradition and explain why, when asked, I always name Imre Lakatos’ Proof and Refutations not just as my favourite book but as the most important/influential book in my life.

As regular readers might have gathered my life has been anything but the normal career path one might expect from a white, middle class, British man born and raised in Northeast Essex. It has taken many twists and turns, detoured down one or other dark alleyway, gone off the rails once or twice and generally not taken the trajectory that my parents and school teachers might have hoped or expected it to take.

In 1970 I went to university in Cardiff to study archaeology but after one year I decided that archaeology was not what I wanted to do and dropped out. I however continued to live in Cardiff apart from some time I spent living in Belgium and but that’s another story. During this period of my life I earned my living doing a myriad of different things whilst I was supposedly trying to work out what it was that I actually wanted to do. As I’ve said on several occasions I became addicted to the history of mathematics at the age of sixteen and during this phase of my life I continued to teach myself both the history of maths and more generally the history of science.

In 1976 my life took another left turn when I moved to Malmö in Sweden.

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Image of Malmö (Elbogen) in Scania, Southern Sweden from a German book (Civitates orbis terrarum, Vol. IV, by G. Braun & F. Hogenberg) .1580 Source: Wikimedia Commons

This was not my first attempt to move to Sweden there had been another abortive attempt a couple of years earlier but that is also another story. This time the move was not instituted by me but by my then partner K. K was a qualified nursery nurse and had applied for a job looking after the children of a pair of doctors in Malmö and her application had been successful. The couple agreed to my accompanying K on the condition that to pay my part of the rent of the flat (that went with the job) I would look after their garden until such time as I found work.

So after witnessing the rained out but brilliant Bob Marley open air in Cardiff football stadium in the summer of 76, we set of for a new life in Malmö. Not having employment my role was to do the cleaning, shopping, cooking and looking after the garden, all things I had been doing for years so no sweat. This left me with a lot of spare time and it wasn’t long before I discovered the Malmö public library. The Swedes are very pragmatic about languages; it is a country with a comparatively small population that lives from international trade so they start learning English in kindergarten. The result in that the public library has lots and lots of English books including a good section on the history and philosophy of mathematics and science, which soon became my happy hunting ground. Card catalogues sorted by subject are a great invention for finding new reading matter on the topic of your choice.

At that point in life I was purely a historian of mathematics with a bit of history of science on the side but in Malmö public library I discovered two books that would change that dramatically. The first was Stephan Körner’s The Philosophy of Mathematics–mathematics has a philosophy I didn’t know that–and the second was Karl Popper’s collection of papers, Conjectures and Refutations: The Growth of Scientific Knowledge. Both found their way back to our flat and were consumed with growing enthusiasm. From that point in my life I was no longer a historian of mathematics and science but had become that strange two-headed beast a historian and philosopher of mathematics and science.

Given the fundamental difference between empirical science and logically formal mathematics my next move might seem to some to be somewhat strange. However, I began to consider the question whether it would be possible to construct a Popperian philosophy of mathematics based on falsification. I gave this question much thought but made little progress. In 1977, for reasons I won’t expand upon here, we returned to the UK and Cardiff.

In Cardiff I continued to pursue my interest in both the histories and philosophies of mathematics and science. In those days I bought my books in a little bookshop in the Morgan Arcade in Cardiff.

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Morgan Arcade Source: Wikimedia Commons

One day the owner, whose name I can’t remember but who knew my taste in books said, “I’ve got something here that should interest you” and handed me a copy of Imre Lakatos’ Proofs and Refutations: The Logic of Mathematical Discovery[1]. I now for the first time held in my hands a Popperian philosophy of mathematics or as Lakatos puts it a philosophy of mathematics based on the theories of George Pólya, Karl Popper and Georg Hegel, a strange combination.

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This is still the copy that I bought on that fateful day in the small bookshop in the Morgan Arcade forty plus years ago

Lakatos was born Imre Lipschitz in Debrecen, Hungary in 1922. He studied mathematics, physics and philosophy graduating from the University of Debrecen in 1944. Following the German invasion in 1944 he formed a Marxist resistance group with his girlfriend and later wife. During the occupation he changed his Jewish name to Molnár to avoid persecution. After the War he changed it again to Lakatos in honour of his grandmother, who had died in Auschwitz. After the War he became a civil servant in the ministry of education and took a PhD from the University of Debrecen in 1948. He also studied as a post doc at the University of Moscow. Involved in political infighting he was imprisoned for revisionism from 1950 to 1953. One should point out that in the post War period Lakatos was a hard-line Stalinist and strong supporter of the communist government. His imprisonment however changed his political views and he began to oppose the government. Out of prison he returned to academic life and translated Georg Pólya’s How to Solve It[2] into Hungarian. When the Russians invaded in 1956, Lakatos fled to the UK via Vienna. He now took a second PhD at the University of Cambridge in 1961 under R.B. Braithwaite. In 1960 he was appointed to a position at the LSE where he remained until his comparatively early death at the age of 51 in 1974.

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Library of the London School of Economics and Political Science – Professor Imre Lakatos, c1960s Source: Wikimedia Commons

The book that I had acquired is a large part of Lakatos’ 1961 PhD thesis, published in book form posthumously[3], and extends Popper’s philosophy of logical discovery into the realm of mathematics. In his seminal work The Logic of Scientific Discovery, (which I had read shortly after discovering his Conjectures and Refutations) Karl Popper moved the discussion in the philosophy of science from justification to discovery. Most previous work in the philosophy of science had been devoted to attempting to justify the truth of accepted scientific theories; Popper’s work was concerned on a formal level at how we arrive at those theories. The same situation existed in the philosophy of mathematics. Philosophers of mathematics were concerned with the logical justification of proven mathematical theorems. Lakatos turned his attention instead to the historical evolution of mathematical theorem.

Proofs and Refutations is written in the form of a Socratic dialogue, although the discussion has more than two participants. A teacher and his class, the students all have Greek letters for names, who are trying to determine the relationship between the number of vertices, edges and faces in polyhedra, V-E+F = 2; a formula now known as the Euler characteristic or Euler’s Gem[4]. The discussion in the class follows and mirrors the evolution in spacial geometry that led to the discovery of this formula. Lakatos giving references to the historical origins of each step in the footnotes. The discussion takes the reader down many byways and cul de sacs and on many detours and around many corners where strange things are waiting to surprise the unwary reader.

The book is thoroughly researched and brilliantly written: erudite and witty, informative on a very high level but a delight to read. I don’t think I can express in words the effect that reading this book had on me. It inspired me to reach out to new heights in my intellectual endeavours, although I knew from the very beginning that I could never possibly reach the level on which Lakatos resided. Before reading Proofs and Refutations, history of mathematics had been a passionate hobby for me; afterwards it became the central aim in my life. I applied to go back to university in Cardiff to study philosophy, having already matriculated six years earlier to study archaeology this meant a one to one interview with a head of department. I completely blew the interview; I always do!

In 1980 I moved to Germany and in 1981 I applied to go to university in Erlangen to study mathematics, which I was able to do after having spent a year learning German. I wanted to choose philosophy as my subsidiary, which meant an interview with a professor. The man I met was Christian Thiel, a historian of logic and mathematics although I didn’t know that at the time, who was just starting his first year as professor, although he had earlier studied in Erlangen. We clicked immediately and although he no longer remembers on that day we discussed the theories of Imre Lakatos. As I documented here Christian Thiel became my mentor and is indirectly more than somewhat responsible for this blog

I have read a lot of books in my life and I continue to do so, although now much more slowly than in the past, but no book has ever had the same impact on me as Proofs and Refutations did the first time I read it. This is why I always name it when asked questions like, what was the most important book you have read or what is your all time favourite book.

 

[1] Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, eds. John Worrall and Elie Zahar, CUP, Cambridge etc., 1976

[2] How to Solve It is a wonderful little volume describing methods for solving mathematical problems; its methodology can also be used for a much wider range of problems and not just mathematical ones.

[3] Part of the thesis had been published as a series of four papers paper under the title Proofs and Refutations in The British Journal for the Philosophy of Science, 14 1963-64. The main part of the book is an expanded version of those original papers.

[4] I recommend David S. Richeson, Euler’s Gem, University Press Group Ltd., Reprint 2012

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Who cares about facts? – Make up your own, it’s much more fun!

Math Horizons is a magazine published by Taylor & Francis for the Mathematical Association of America aimed at undergraduates interested in mathematics: It publishes expository articles about “beautiful mathematics” as well as articles about the culture of mathematics covering mathematical people, institutions, humor, games, cartoons, and book reviews. (Description taken from Wikipedia, which attributes it to the Math Horizons instructions for authors January 3 2009). Apparently, however, authors are not expected to adhere to historical facts, they can, it seems, make up any old crap.

The latest edition of Math Horizons (Volume 25, Issue 3, February 2018) contains an article by a Stephen Luecking entitled Albrecht Dürer’s Celestial Geometry. As I am currently, for other reasons, refreshing my knowledge of Albrecht the mathematician I thought, oh that looks interesting I must read that. I wish I hadn’t.

Luecking’s sub-title seems innocent enough: Renaissance artist Albrecht Dürer designed a specialty compass for astronomical drawings, but when you read the article you discover that Luecking says an awful lot more and most of it is hogwash. What does he have to say?

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Albrecht Dürer Self-Portrait 1500 Source: Wikimedia Commons

Albrecht Dürer (1471–1528), noted Renaissance printer and painter, twice left his native Germany for sojourns to Italy, once from 1494 to 1495 and again from 1505 to 1507. During those years his wide-ranging intellect absorbed the culture and thinking of noted artists and mathematicians. Perhaps the most important
 outcome of these journeys was his
introduction to scientific methods. 
His embrace of these methods
 went on to condition his thinking 
for the rest of his life. 


So far so good. However what Dürer absorbed on those journeys to Italy was not scientific methods but linear perspective, the mathematical method, developed in Northern Italy in the fifteenth century, to enable artists to represent three dimensional reality realistically in a two dimensional picture. Dürer played a significant role in distributing these mathematical techniques in Europe north of the Alps. His obsession with mathematics in art led to him developing the theory that the secret of beauty lay in mathematical proportion to which de devoted a large part of the rest of his life. He published the results of his endeavours in his four-volume book on human proportions, Vier Bücher von Menschlicher Proportion, in the year of his death, 1528.

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Title page of Vier Bücher von menschlicher Proportion showing the monogram signature of artist Source: Wikimedia Commons

If Dürer wanted to learn scientific methods, by which, as we will see Luecking means astronomy, he could and probably did learn them at home in Nürnberg. Dürer was part of the humanist circle of Willibald Pirckheimer, he close friend and patron.

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Engraving of Willibald Pirckheimer at 53 by Albrecht Dürer, 1524. We live by the spirit. The rest belongs to death. Source: Wikimedia Commons

Franconian houses are built around a courtyard; Dürer was born in the rear building of the Pirckheimer house on the market square in Nürnberg. Although his parents bought their own house a few years later Albrecht and Willibald remained close friends and possibly even lovers all of their lives. Pirckheimer was a big supporter of the mathematical sciences—astronomy, mathematics, cartography and astrology—and his circle included, amongst others, Johannes Stabius, Johannes Werner, Erhard Etzlaub, Georg Hartmann, Konrad Heinfogel and Johannes Schöner all of whom were either astronomers, mathematicians, cartographers, instrument makers or globe makers some of them all five and all of them friends of Dürer.

Next up Luecking tells us:

One notable
consequence was Dürer’s abandonment of astrological subject
matter—a big seller for a printer
and publisher such as himself—in favor of astronomy.

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Albrecht Dürer Syphilis 1496 Syphilis was believed to have an astrological cause Source: Wikimedia Commons

Luecking offers no evidence or references for this claim, so I could offer none in saying that it is total rubbish, which it is. However I will give one example that shows that Albrecht Dürer was still interested in astrology in 1517. Lorenz Beheim (1457–1521) was a humanist, astrologer, physician and alchemist, who was a canon of the foundation of the St Stephan Church in Bamberg, he was a close friend of both Pirckheimer and Dürer and corresponded regularly with Pirckheimer. In a letter from 8 December 1517 he informed Pirckheimer that Johannes Schöner was coming to Nürnberg with printed celestial globes that could be used for astrology, which if his wished could be acquired by him and Albrecht Dürer. He would not have passed on the information if he thought that they wouldn’t be interested. Beheim also cast horoscopes for both Pirckheimer and Dürer.

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Gores for Johannes Schöner’s Celestial Globe 1517  Source: Hans Gaab, Die Sterne Über Nürnberg: Albrecht Dürer und seine Himmelskarten von 1515, Nürnberger Astronomische Gesellschaft, Michael Imhof Verlag, 2015 p. 115

 

Next up Luecking starts, as he means to go on, with pure poppycock. All of the above Nürnberger mathematician, who all played significant roles in Dürer’s life, were of course practicing astrologers.

Astronomy was not to be a casual interest. Just before his second trip to Italy, Dürer published De scientia motus orbis, a cosmological treatise by the Persian Jewish astronomer Masha’Allah ibn Atharī (ca. 740–815 CE). Since Masha’Allah wrote the treatise for laymen and included ample illustrations, it was a good choice for introducing Europeans to Arabic astronomy.

The claim that Dürer published Masha’Allah’s De scientia motus orbis is so mind bogglingly wrong anybody with any knowledge of the subject would immediately stop reading the article, as it is obviously a complete waste of time and effort. The book was actually edited and published by Johannes Stabius and printed by Weissenburger in Nürnberg in 1504.

The woodcut illustrations came from the workshop of Albrecht Dürer, but probably not from Dürer himself. There were traditionally attributed to Hans Süß von Kulmbach (1480–1522), one of Dürer’s assistants, who went on to become a successful painter in his own right, but modern research has shown that Süß didn’t move to Nürnberg until 1505, a year after the book was published.

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Hans Süß portrait  Source: Wikimedia Commons

Although Luecking wants Masha’Allah to be an astronomer he was in fact a very famous astrologer, who amongst other things cast the horoscope for the founding of Bagdad. De scientia motus orbis is indeed a book on Aristotelian cosmology and physics but it includes his theory that there are ten heavenly spheres not eight as claimed by Aristotle. His extra heavenly spheres play a significant role in his astrological theories. It is very common practice for astrologers, starting with Ptolemaeus, to publish their astronomy and astrology in separate books but they are seen as complimentary volumes. From their beginnings in ancient Babylon down to the middle of the seventeenth century astronomy and astrology were always seen as two sides of the same coin.

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Title page De scientia motus orbis Although this woodcut is usually titled The Astronomer I personally think the figure looks more like an astrologer Source: Wikimedia Commons

In 1509 Dürer purchased the entire library of Regiomontanus (1436–1476 CE) from the estate of Nuremberg businessman Bernhard Walther. Regiomontanus was Europe’s leading astronomer,
a noted mathematician, and a designer of astronomical instruments. Walther had sponsored Regiomontanus’s residency in Nuremberg between 1471 and 1475. Part of Walther’s largesse was to provide a print shop from which Regiomontanus published the world’s first scientific texts ever printed.

Regiomontanus was of course first and foremost an astrologer and most of those first scientific texts that he published in Nürnberg were astrological texts. Walther did not sponsor Regiomontanus’ residency in Nürnberg but was his colleague and student in his endeavours in the city. An analysis of Walther’s astronomical observation activities in Nürnberg after Regiomontanus’ death show that he too was an astrologer rather than an astronomer. When Regiomontanus came to Nürnberg he brought a very large number of manuscripts with him, intending to edit and publish them. When he died these passed into Walther’s possession, who added new books and manuscripts to the collection. The story of what happened to this scientific treasure when Walther died in 1504 is long and very complicated. In fact Dürer bought not “the entire library” but a mere ten manuscripts not when he bought Walther’s house, the famous Albrecht Dürer House, in 1509 but first in 1522.

In 1515, Dürer and Austrian cartographer and mathematician Johannes Stabius produced the first map of the world portraying the earth as a sphere.

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Johannes Statius portrait by Albrecht Dürer Source: Wikimedia Commons

The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The earliest know printed world map portraying the earth as a sphere is a woodcut in a Buchlein über die Kunst Corsmographia, (Booklet about the Art of Cosmographia) published in Nürnberg in about 1490. There are others that predate the Stabius-Dürer map most notably on the title page of Waldseemüller’s Die Welt Kugel (The Earth Sphere) published in Straßburg in 1509.

There are no surviving copies of the Stabius-Dürer world map from the sixteenth century so we don’t actually know what it was produced for. The woodblocks survived and were rediscovered in the 18th century.

It is however dedicated to both the Emperor Maximilian, Stabius’s employer who granted the printing licence, and Cardinal Matthäus Lang, so it might well have been commissioned by the latter. Lang commissioned the account of Magellan’s circumnavigation on which Schöner based his world map of that circumnavigation.

Afterward, Stabius proposed continuing their collaboration by publishing a star map—the first such map published in Europe. Their work relied heavily on data assembled by Regiomontanus, plus refinements from Walther.

It will probably not surprise you to discover that this was not “the first such map published in Europe. It’s the first printed one but there are earlier manuscript ones, two of which from 1435 in Vienna and 1503 in Nürnberg probably served as models for the Stabius–Dürer–Heinfogel one. Their work did not rely “heavily on data assembled by Regiomontanus, plus refinements from Walther” but was based on Ptolemaeus’ star catalogue from the Almagest. There is a historical problem in that there was not printed copy of that star catalogue available at the time so they probably work from one or more manuscripts and we don’t know which one(s). The star map contains the same dedications to Maximilian and Lang as the world map so one again might have been a commission from Lang, Stabius acting as the commissioning agent. Stabius and Lang studied together at the University of Ingolstadt.

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Stabs-Dürer-Heinfogel Star Map Northern Hemisphere Source: Ian Ridpath’s Star Tales

For more details on the star maps go here

The star map required imprinting the three- dimensional dome of the heavens onto a two- dimensional surface without extreme distortions, a task that fell to Stabius. He used a stereographic projection. In this method, rays originate at the pole in the opposite hemisphere, pass through a given point in the hemisphere, and yield a point on a circular surface.

You will note that I have included the name of Konrad Heinfogel to the producers of the map and it was actually he, and not Stabius, who was responsible for the projection of the map and the location of the individual stars. In fact in this project Johannes Stabius as commissioning agent was project leader, Konrad Heinfogel was the astronomical expert and Albrecht Dürer was the graphic artist hired to draw the illustration. Does one really have to point out that in the sixteenth century star maps were as much, if not more, for astrologers than for astronomers.

Luecking now goes off on an excurse about the history of stereographic projection, which ends with the following paragraph.

As the son of a goldsmith, Dürer’s exposure to stereographic projection would have been by way of the many astrolabes being fabricated in Nuremburg, then Europe’s major center for instrument makers. As the 16th century moved on, the market grew for such scientific objects as astrology slipped into astronomy. Handcrafted brass instruments, however, were affordable only to the wealthy, whereas printed items like the Dürer-Stabius maps reached a wider market.

Nürnberg was indeed the major European centre for the manufacture of scientific instruments during Dürer’s lifetime but scientific instrument makers and goldsmiths are two distinct professional groups, so Luecking’s argument falls rather flat, although of course Dürer would have well acquainted with the astrolabes made by his mathematical friends. Astrolabes are of course both astrological and astronomical instruments and astrology did not slip into astronomy during the 16th century. In fact the 16th century is regarded by historians as the golden age of astrology.

There now follows another excurse on the epicycle-deferent model of planetary orbits as a lead up to the articles thrilling conclusion.

In his 1525 book Die Messerung (On Measurement), Dürer presents an instrument of his own design used to draw these and other more general curves. This compass for drawing circles upon circles consisted of four telescoping arms and calibrated dials. An arm attached to the first dial could rotate in a full circle, a second arm fixed to another dial mounted on the end of this first arm could rotate around the end of the first arm, and so on.

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Dürer’s four arm compass

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Underweysung der Messung mit dem Zirkel und Richtscheyt Title Page

The title of Dürer’ 1525 book is actually Underweysung der Messung mit dem Zirckel und Richtscheyt (Instructions for Measuring with Compass and Straightedge). It is a basic introduction to geometry and its applications, which Dürer wrote when he realised that his Vier Bücher von Menschlicher Proportion was too advanced for the artist apprentices that he thought should read it. The idea was first read and digest the Underweysung then read the Vier Bücher von Menschlicher Proportion.

Luecking tells us that:

As a trained metalsmith, Dürer possessed the expertise to craft this complex tool. Precision calibration and adjustable arms allowed its user to plot an endless number of curves by setting the length of each telescoping arm and determining the rate at which the arms turned. This, in effect, constituted manual programming by setting the parameters of each curve plotted.

As a teenager Dürer did indeed serve an apprenticeship under his father as a goldsmith, but immediately on completing that apprenticeship he undertook a second apprenticeship as a painter with Michael Wolgemut from 1486 to 1490 and dedicated his life to painting and fine art printing. Luecking has already correctly stated that Nürnberg was the major European centre for scientific instrument making and Dürer almost certainly got one of those instrument makers to produce his multi-armed compass. Luecking describes the use to which Dürer put this instrument in drawing complex geometrical curves. He then goes on to claim that Dürer might actually have constructed it to draw the looping planetary orbits produced by the epicycle-deferent model. There is absolutely no evidence for this in the Underweysung and Luecking’s speculation is simple pulled out of thin air.

To summarise for those at the back who haven’t been paying attention. Dürer did not absorb scientific methods in Italy. He did not abandon astrology for astronomy. He didn’t publish Masha’Allah’s De scientia motus orbis, Johannes Stabius did. Dürer only bought ten of Regiomontanus’ manuscripts and not his entire library. The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The Stabius–Dürer–Heinfogel star charts were the first star-charts printed in Europe but by no means the first ones published. Star charts are as much astrological, as they are astronomical. Astrology did not slip into astronomy in the 16th century, which was rather the golden age of astrology. There is absolutely no evidence that Dürer’s multi-arm compass, as illustrated in his geometry book the Underweysung, was ever conceived for drawing the looping orbits of epicycle-deferent planetary models, let alone used for this purpose.

It comes as no surprise that Stephen Luecking is not a historian of mathematics or art for that matter. He is the aged (83), retired chairman of the art department of DePaul University in Chicago.

Whenever I come across an article as terrible as this one published by a leading scientific publisher in a journal from a major mathematical organisation such as the MAA I cringe. I ask myself if the commissioning editor even bothered to read the article; it was certainly not put out to peer review, as any knowledgeable Dürer expert would have projected it in an elegant geometrical curve into his trashcan. Above all I worry about the innocent undergraduates who are subjected to this absolute crap.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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