Category Archives: History of Mathematics

Why Mathematicus?

“The Renaissance Mathematiwot?”

“Mathematicus, it’s the Latin root of the word mathematician.”

“Then why can’t you just write The Renaissance Mathematician instead of showing off and confusing people?”

“Because a mathematicus is not the same as a mathematician.”

“But you just said…”

“Words evolve over time and change their meanings, what we now understand as the occupational profile of a mathematician has some things in common with the occupational profile of a Renaissance mathematicus but an awful lot more that isn’t. I will attempt to explain.”

The word mathematician actually has its origins in the Greek word mathema, which literally meant ‘that which is learnt’, and came to mean knowledge in general or more specifically scientific knowledge or mathematical knowledge. In the Hellenistic period, when Latin became the lingua franca, so to speak, the knowledge most associated with the word mathematica was astrological knowledge. In fact the terms for the professors[1] of such knowledge, mathematicus and astrologus, were synonymous. This led to the famous historical error that St. Augustine rejected mathematics, whereas his notorious attack on the mathematici[2] was launched not against mathematicians, as we understand the term, but against astrologers.

The earliest known portrait of Saint Augustine in a 6th-century fresco, Lateran, Rome Source: Wikimedia Commons

The earliest known portrait of Saint Augustine in a 6th-century fresco, Lateran, Rome
Source: Wikimedia Commons

However St. Augustine lived in North Africa in the fourth century CE and we are concerned with the European Renaissance, which, for the purposes of this post we will define as being from roughly 1400 to 1650 CE.

The Renaissance was a period of strong revival for Greek astrology and the two hundred and fifty years that I have bracketed have been called the golden age of astrology and the principle occupation of our mathematicus is still very much the casting and interpretation of horoscopes. Mathematics had played a very minor role at the medieval universities but the Renaissance humanist universities of Northern Italy and Krakow in Poland introduced dedicated chairs for mathematics in the early fifteenth century, which were in fact chairs for astrology, whose occupants were expected to teach astrology to the medical students for their astro-medicine or as it was known iatro-mathematics. All Renaissance professors of mathematics down to and including Galileo were expected to and did teach astrology.

A Renaissance Horoscope Kepler's Horoskop für Wallenstein Source: Wikimedia Commons

A Renaissance Horoscope
Kepler’s Horoskop für Wallenstein
Source: Wikimedia Commons

Of course, to teach astrology they also had to practice and teach astronomy, which in turn required the basics of mathematics – arithmetic, geometry and trigonometry – which is what our mathematicus has in common with the modern mathematician. Throughout this period the terms Astrologus, astronomus and mathematicus – astrologer, astronomer and mathematician ­– were synonymous.

A Renaissance mathematicus was not just required to be an astronomer but to quantify and describe the entire cosmos making him a cosmographer i.e. a geographer and cartographer as well as astronomer. A Renaissance geographer/cartographer also covered much that we would now consider to be history, rather than geography.

The Renaissance mathematicus was also in general expected to produce the tools of his trade meaning conceiving, designing and manufacturing or having manufactured the mathematical instruments needed for astronomer, surveying and cartography. Many were not just cartographers but also globe makers.

Many Renaissance mathematici earned their living outside of the universities. Most of these worked at courts both secular and clerical. Here once again their primary function was usually court astrologer but they were expected to fulfil any functions considered to fall within the scope of the mathematical science much of which we would see as assignments for architects and/or engineers rather than mathematicians. Like their university colleagues they were also instrument makers a principle function being horologist, i.e. clock maker, which mostly meant the design and construction of sundials.

If we pull all of this together our Renaissance mathematicus is an astrologer, astronomer, mathematician, geographer, cartographer, surveyor, architect, engineer, instrument designer and maker, and globe maker. This long list of functions with its strong emphasis on practical applications of knowledge means that it is common historical practice to refer to Renaissance mathematici as mathematical practitioners rather than mathematicians.

This very wide range of functions fulfilled by a Renaissance mathematicus leads to a common historiographical problem in the history of Renaissance mathematics, which I will explain with reference to one of my favourite Renaissance mathematici, Johannes Schöner.

Joan Schonerus Mathematicus Source: Wikimedia Commons

Joan Schonerus Mathematicus
Source: Wikimedia Commons

Schöner who was a school professor of mathematics for twenty years was an astrologer, astronomer, geographer, cartographer, instrument maker, globe maker, textbook author, and mathematical editor and like many other mathematici such as Peter Apian, Gemma Frisius, Oronce Fine and Gerard Mercator, he regarded all of his activities as different aspects or facets of one single discipline, mathematica. From the modern standpoint almost all of activities represent a separate discipline each of which has its own discipline historians, this means that our historical picture of Schöner is a very fragmented one.

Because he produced no original mathematics historians of mathematics tend to ignore him and although they should really be looking at how the discipline evolved in this period, many just spring over it. Historians of astronomy treat him as a minor figure, whilst ignoring his astrology although it was this that played the major role in his relationship to Rheticus and thus to the publication of Copernicus’ De revolutionibus. For historians of astrology, Schöner is a major figure in Renaissance astrology although a major study of his role and influence in the discipline still has to be written. Historians of geography tend to leave him to the historians of cartography, these whilst using the maps on his globes for their studies ignore his role in the history of globe making whilst doing so. For the historians of globe making, and yes it really is a separate discipline, Schöner is a central and highly significant figure as the founder of the long tradition of printed globe pairs but they don’t tend to look outside of their own discipline to see how his globe making fits together with his other activities. I’m still looking for a serious study of his activities as an instrument maker. There is also, as far as I know no real comprehensive study of his role as textbook author and editor, areas that tend to be the neglected stepchildren of the histories of science and technology. What is glaringly missing is a historiographical approach that treats the work of Schöner or of the Renaissance mathematici as an integrated coherent whole.

Western hemisphere of the Schöner globe from 1520. Source: Wikimedia Commons

Western hemisphere of the Schöner globe from 1520.
Source: Wikimedia Commons

The world of this blog is at its core the world of the Renaissance mathematici and thus we are the Renaissance Mathematicus and not the Renaissance Mathematician.

[1] That is professor in its original meaning donated somebody who claims to possessing a particular area of knowledge.

[2] Augustinus De Genesi ad Litteram,

Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant. II, xvii, 37

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Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, History of science, History of Technology, Renaissance Science

Two views of the celestial spheres

When the Bishop of Salisbury scanned the heavens in the 1670s it was difficult to know if he was contemplating the wonders of his God, or those of Kepler’s planetary laws. Seth Ward, the incumbent of the Salisbury bishopric, was both a successful Anglican churchman and an acknowledge astronomer, who did much to boost Kepler’s theories in the middle of the seventeenth century.

Greenhill, John; Seth Ward (1617-1689), Savilian Professor of Astronomy, Oxford (1649-1660) Source: Wikimedia Commons

Greenhill, John; Seth Ward (1617-1689), Savilian Professor of Astronomy, Oxford (1649-1660)
Source: Wikimedia Commons

Born in Aspenden in Hertfordshire on an unknown day in 1617, Seth Ward was the son of John Ward, an attorney, and his wife Mary Dalton. Having received a basic schooling he was admitted to Sidney-Sussex College, Cambridge on 1 December 1632, where he graduated B.A. in 1637 and M.A. on 27 July 1640, following which he was elected a fellow of the college. Ward was a keen mathematician, who, like many others in the Early Modern Period, was largely self-taught, studying William Oughtred’s Clavis Mathematicae together with fellow maths enthusiast Charles Scarburgh, a future physician to Charles II. Finding some passages difficult the two of them travelled to Albury in Surrey where Oughtred was rector. Here they took instruction from Oughtred and it was the start of a relationship between Ward and Oughtred that lasted until Oughtred’s death in 1660.

Sir Charles Scarborough Jean Demetrius (attributed to) Royal College of Physicians, London Source: Wikimedia Commons

Sir Charles Scarborough Jean Demetrius (attributed to)
Royal College of Physicians, London
Source: Wikimedia Commons

In 1643 Ward was appointed lecture for mathematics for the university but he did not exercise this post for very long. Some of the Cambridge colleges, and in particular Sidney-Sussex, Cromwell’s alma mater, became centres for the Puritan uprising and in 1644 Seth Ward, a devote Anglican, was expelled from his fellowship for refusing to sign the covenant. At first he took refuge with friends in and around London but then he went back to Albury where he received tuition in mathematics from Oughtred for several months. Afterwards he became private tutor in mathematics to the children of a friend, where he remained until 1649. Having used the Clavis Mathematicae, as a textbook whilst teaching at he university he made several suggestions for improving the book and persuaded Oughtred to publish a third edition in 1652

William Oughtred by Wenceslas Hollar 1646 Source: Wikimedia Commons

William Oughtred
by Wenceslas Hollar 1646
Source: Wikimedia Commons

In 1648 John Greaves, one of the first English translators of Arabic and Persian scientific texts into Latin, also became a victim of a Puritan purge and was evicted from the Savilian Chair for Astronomy at Oxford. Greaves recommended Ward as his successor and in 1649, having overcame his scruples, Ward took the oath to the English Commonwealth and was appointed Savilian Professor.

yooniqimages_102046418

These episodes, Wards expulsion from Sidney-Sussex and Greave’s from Oxford, serve to remind us that much of the scientific investigations that took place in the Early Modern Period, and which led to the creation of modern science, did so in the midst of the many bitter and very destructive religious wars that raged throughout Europe during this period. The scholars who carried out those investigations did not remain unscathed by these disturbances and careers were often deeply affected by them. The most notable example being, of course Johannes Kepler, who was tossed around by the Reformation and Counter-Reformation like a leaf in a storm. Anyone attempting to write a history of the science of this period has to, in my opinion, take these external vicissitudes into account; a history that does not do so is only a half history.

It was in his role as Savilian Professor that Ward made his greatest contribution to the development of the new heliocentric astronomy in an academic dispute with the French astronomer and mathematician Ismaël Boulliau (1605–1694).

Ismaël Boulliau  Source: Wikimedia Commons

Ismaël Boulliau
Source: Wikimedia Commons

Boulliau was an early supporter of the elliptical astronomy of Johannes Kepler, who however rejected much of Kepler’s ideas. In 1645 he published his own theories based on Kepler’s work in his Astronomia philolaïca. This was the first major work by another astronomer that incorporated Kepler’s elliptical astronomy. Ward another Keplerian wrote his own work In Ismaelis Bullialdi Astronomiæ Philolaicæ Fundamenta Inquisitio Brevis, which heavily criticised Boulliau’s theories and present his own, in his opinion superior, interpretations of Kepler’s ideas. He followed this with another more extensive presentation of his theories in 1656, Astronomia Geometrica; ubi Methodus proponitur qua Primariorum Planetarum Astronomia sive Elliptica sive Circularis possit Geometrice absolve. Boulliau responded in 1657 in his Ismaelis Bullialdi Astronomiæ Philolaicæ Fundamenta clarius explicata et asserta, printed in his Exercitationes Geometricæ tres in which he acknowledged errors in his own work but also pointing out inaccuracies in Ward’s. In final analysis both Boulliau and Ward were wrong, and we don’t need to go into detail her, but their dispute drew the attention of other mathematicians and astronomers to Kepler’s work and thus played a major role in its final acceptance as the preferred model for astronomy in the latter part of the seventeenth century.

The worst popular model of the emergence of modern astronomy in the Early Modern Period sees the inspiring creation of heliocentric astronomy by Copernicus in his De revolutionibus in the sixteenth century, the doting of a few ‘I’s and crossing of a few ‘T’s by Galileo and Kepler in the early seventeenth century followed by the triumphant completion of the whole by Newton in his Principia in 1687. Even those who acknowledge that Kepler created something new with his elliptical astronomy still spring directly to Newton and the Principia. In fact many scholars contributed to the development of the ideas of Kepler and Galileo in the decades between them and Isaac Newton and if we are going to correctly understand how science evolves it is important to give weight to the work of those supposedly minor figures. The scientific debate between Boulliau and Ward is a good example of an episode in the history of astronomy that we ignore at the peril of falsifying the evolution of a disciple that we are trying to understand.

Ward continued to make career as an astronomer mathematician. He was awarded an Oxford M.A. on 23 October 1649 and became a fellow of Wadham College in 1650. The mathematician John Wilkins was warden of Wadham and the centre of a group of likeminded enthusiasts for the emerging new sciences that at times included Robert Boyle, Robert Hooke, Christopher Wren, John Wallis and many others. This became known as the Philosophical Society of Oxford, and they would go on to become one of the founding groups of the Royal Society in the early 1660s.

During his time at Oxford Ward together with his friend John Wallis, the Savilian Professor of Geometry, became involved in a bitter dispute with the philosopher Thomas Hobbes on the teaching of geometry at Oxford and the latter’s claim to have squared the circle; he hadn’t it’s impossible but the proof of that impossibility came first a couple of hundred years later.

Thomas Hobbes Artist unknown

Thomas Hobbes Artist unknown

Ward however was able to expose the errors in Hobbes’ geometrical deductions. In some circles Ward is better known for this dispute than for his contributions to astronomy.

John Wallis by Godfrey Kneller Source: Wikimedia Commons

John Wallis by Godfrey Kneller
Source: Wikimedia Commons

When the alchemist and cleric John Webster launched an attack on the curriculum of the English universities in his Academiarum Examen (1654) Ward joined forces with John Wilkins to write a defence refuting Webster’s arguments, Viniciae Acadmiarum, which also included refutations of other prominent critics of Oxford and Cambridge.

Greenhill, John; John Wilkins (1614-1672), Warden (1648-1659); Wadham College, University of Oxford;

Greenhill, John; John Wilkins (1614-1672), Warden (1648-1659); Wadham College, University of Oxford;

Ward’s career as an astronomer and mathematician was very successful and his work was known and respected throughout Europe, where he stood in contact with many of the leading exponents of his discipline. However, his career in academic politics was not so successful. He received a doctorate in theology (D.D.) from Oxford in 1654 and one from Cambridge in 1659. He was elected principle of Jesus College, Oxford in 1657 but Cromwell appointed somebody else promising Ward compensation, which he never delivered. In 1659 he was appointed president of Trinity College, Oxford but because he was not qualified for the office he was compelled to resign in 1660. This appears to have been the final straw and in 1660 he left academia, resigning his professorship to take up a career in the Church of England, with the active support of the recently restored Charles II.

He proceeded through a series of clerical positions culminating in the bishopric in Salisbury in 1667. He was appointed chancellor of the Order of the Garter in 1671. Ward turned down the offer of the bishopric of Durham remaining in Salisbury until his death 6 January 1689. He was a very active churchman, just as he had been a very active university professor, and enjoyed as good a reputation as a bishop as he had enjoyed as an astronomer.

 

 

 

 

 

 

 

 

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Christmas Trilogy 2016 Part 3: The English Keplerians

For any scientific theory to succeed, no matter how good or true it is; it needs people who support and propagate it. Disciples, so to speak, who are prepared to spread the gospel. Kepler’s astronomical theories, his three laws of planetary motion and everything that went with them, were no different from every other theory in this aspect; they needed a fan club. On the continent of Europe the reception of Kepler’s theories was initially lukewarm to say the least and it was not only Galileo, who did his best to ignore them. Therefore it is somewhat surprising that they found a group of enthusiastic supporters right from the beginning in England. Surprising because in general in the first half of the seventeenth century England lagged well behind the continent in astronomy, as in all things mathematical.

The first Englishmen to pick up on Kepler’s theories was the small group around Thomas Harriot, who did so immediately after the publication of the Astronomia nova in 1609.

Portrait often claimed to be Thomas Harriot (1602), which hangs in Oriel College, Oxford. Source: Wikimedia Commons

Portrait often claimed to be Thomas Harriot (1602), which hangs in Oriel College, Oxford. Source: Wikimedia Commons

The group included not only Harriot but also his lens grinder Christopher Tooke, the Cornish MP Sir William Lower (c.1570–1615) and his Welsh neighbour John Prydderch (or Protheroe). Lower had long been an astronomical pupil of Harriot’s and had in turn introduced his neighbour Prydderch to the science.

The cartoon of Lower and Prydderch on page 265 of Seryddiaeth a Seryddwyr By J.S. Evans. Lower looks through a telescope while Prydderch holds a cross-staff. The cartoon had been used earlier by Arthur Mee in his book The Story of the Telescope in 1909. The artist was J. M. Staniforth, the artist-in-chief of the Western Mail newspaper.

The cartoon of Lower and Prydderch on page 265 of Seryddiaeth a Seryddwyr By J.S. Evans. Lower looks through a telescope while Prydderch holds a cross-staff. The cartoon had been used earlier by Arthur Mee in his book The Story of the Telescope in 1909. The artist was J. M. Staniforth, the artist-in-chief of the Western Mail newspaper.

This group was one of the very earliest astronomical telescopic observing teams, exchanging information and comparing observations already in 1609/10. In 1610 they were enthusiastically reading Astronomia nova and discussing the new elliptical astronomy. It was Lower, who had carefully observed Halley’s comet in 1607 (pre-telescope) together with Harriot, who first suggested that the orbits of comets would also be ellipses. Kepler still thought that comets move in straight lines. The Harriot group did not publish their active support of the Keplerian elliptical astronomy but Harriot was well networked within the mathematical communities of both England and the Continent. He had even earlier had a fairly substantial correspondence with Kepler on the topic of atmospheric refraction. It is a fairly safe assumption that Harriot’s and Lower’s support of Kepler’s theories was known to other contemporary English mathematical practitioners.

Our next group of English Keplerians is that initiated by the astronomical prodigy Jeremiah Horrocks (1618–1641). Horrocks was a self-taught astronomer who stumbled across Kepler’s theories, whilst on the search for reliable astronomical tables. He quickly established that Kepler’s Rudolphine Tables were superior to other available tables and soon became a disciple of Kepler’s elliptical astronomy. Horrocks passed on his enthusiasm for Kepler’s theories to his astronomical helpmate William Crabtree (1610–1644). In turn Crabtree seems to have been responsible for converting another young autodidactic astronomer William Gascoigne (1612–1644) to the Keplerian astronomical gospel. Crabtree referred to this little group as Nos Keplari. Horrocks contributed to the development of Keplerian astronomy with an elliptical model of the Moon’s orbit, something that Kepler had not achieved. This model was the one that would eventually make its way into Newton’s Principia. He also corrected and extended the Rudolphine Tables enabling Horrocks and Crabtree to become, famously, the first people ever to observe a transit of Venus.

opera_posthuma

Like Harriot’s group, Nos Keplari published little but they were collectively even better networked than Harriot. Horrocks had been at Oxford Emmanual College Cambridge with John Wallis and it was Wallis, a convinced nationalist, who propagated Horrocks’ posthumous astronomical reputation against foreign rivals, as he also did in the question of algebra for Harriot. Both Gascoigne and Crabtree had connections to the Towneley family, landed gentry who took a strong interest in the emerging modern science of the period. Later the Towneley’s who had connections to the Royal Society ensured that the work of Nos Keplari was not lost and forgotten, bringing it, amongst other things, to the attention of a young John Flamsteed, who would later become the first Astronomer Royal. . Gascoigne had connections to William Cavendish, the later Duke of Newcastle, under whose command he served at the battle of Marston Moor, where he died. William, his brother Charles and his wife Margaret were all enthusiastic supporters of the new sciences and important members of the English scientific and philosophical community. Gascoigne also corresponded with William Oughtred who served as private mathematics tutor to many leading members of the burgeoning English mathematical community. It is to two of Oughtred’s students that we now turn

William Oughtred by Wenceslas Hollar 1646

William Oughtred
by Wenceslas Hollar 1646

Seth Ward (1617–1689) studied at Oxford Cambridge University from 1636 to 1640 when he became a fellow of Sidney Sussex College.

Greenhill, John; Seth Ward (1617-1689), Savilian Professor of Astronomy, Oxford (1649-1660) Source: Wikimedia Commons

Greenhill, John; Seth Ward (1617-1689), Savilian Professor of Astronomy, Oxford (1649-1660)
Source: Wikimedia Commons

In the same year he took instruction in mathematics from William Oughtred. In 1649 he became Savilian Professor of Astronomy at Oxford University the same year that John Wallis was appointed Savilian Professor of Mathematics. Whilst serving as Savilian Professor, Ward became embroiled in a dispute about Keplerian astronomy with the French astronomer and mathematician Ismaël Boulliau.

Ismaël Boulliau  Source: Wikimedia Commons

Ismaël Boulliau
Source: Wikimedia Commons

Boulliau was an early and strong defender of Keplerian elliptical astronomy, who however rejected Kepler’s attempts to create a physical explanation of planetary orbits. Boulliau published his Keplerian theories in his Astronomia philoaïca in 1645. Ward attacked Boulliau’s model in his In Ismaelis Bullialdi astro-nomiae philolaicae fundamenta inquisitio brevis from 1653, presenting his own model for Kepler’s planetary laws. Boulliau responded to Ward’s attack in his De lineis spiralibus from 1657. Ward had amplified his own views in his Astronomia geometrica from 1656. This public exchange between two heavyweight champions of the elliptical astronomy did much to raise the general awareness of Kepler’s work in England. It has been suggested that the dispute was instrumental in bringing Newton’s attention to Kepler’s ideas, a claim that is however disputed by historians.

Ward went on to make a successful career in the Church of England, eventually becoming Bishop of Salisbury his successor, as Savilian Professor of Astronomy was another one of Oughtred’s student, Christopher Wren (1632–1723).

Christopher Wren by Godfrey Keller 1711  Source: Wikimedia Commons

Christopher Wren by Godfrey Keller 1711
Source: Wikimedia Commons

Wren is of course much better known as the foremost English architect of the seventeenth-century but started out as mathematician and astronomer. Wren studied at Wadham College Oxford from 1650 to 1653, where he was part of the circle of scientifically interested scholars centred on John Wilkins (1614–1672), the highly influential early supporter of heliocentric astronomy. The Wilkins group included at various times Seth Ward, John Wallis, Robert Boyle, William Petty and Robert Hooke amongst others and would go on to become one of the groups that founded the Royal Society. Wren was a protégé of Sir Charles Scarborough, a student of William Harvey who later became a famous physician in his own right; Scarborough had been a fellow student of Ward’s and was another student of Oughtred’s. Wren was appointed Gresham Professor of Astronomy and it was following his lectures at Gresham College that the meetings took place that would develop into the Royal Society. As already noted Wren then went on to succeed Ward as Savilian Professor for astronomy in 1661, a post that he resigned in 1673 when his work as Surveyor of the King’s Works (a post he took on in 1669), rebuilding London following the Great Fire of 1666, became too demanding. Wren enjoyed a good reputation as a mathematician and astronomer and like Ward was a convinced Keplerian.

Our final English Keplerian is Nicolaus Mercator (1620–1687), who was not English at all but German, but who lived in London from 1658 to 1682 teaching mathematics.

Nicolaus Mercator © 1996-2007 Eric W. Weisstein

Nicolaus Mercator
© 1996-2007 Eric W. Weisstein

In his first years in England Mercator corresponded with Boulliau on the subject of Horrock’s Transit of Venus observations. Mercator stood in contact with the leading English mathematicians, including Oughtred, John Pell and John Collins and in 1664 he published a defence of Keplerian astronomy Hypothesis astronomica nova. Mercator’s work contained an acceptable mathematical proof of Kepler’s second law, the area law, which had been a bone of contention ever since Kepler published it in 1609; Kepler’s own proof being highly debateable, to put it mildly. Mercator continued his defence of Kepler in his Institutiones astronomicae in 1676. It was probably through Mercator’s works, rather than Ward’s, that Newton became acquainted with Kepler’s astronomy. We still have Newton’s annotated copy of the latter work. Newton and Mercator were acquainted and corresponded with each other.

As I hope to have shown there was a strong continuing interest in England in Keplerian astronomy from its very beginnings in 1609 through to the 1660s when it had become de facto the astronomical model of choice in English scientific circles. As I stated at the outset, to become accepted a new scientific theory has to find supporters who are prepared to champion it against its critics. Kepler’s elliptical astronomy certainly found those supporters in England’s green and pleasant lands.

 

 

 

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

Christmas Trilogy 2016 Part 2: What a difference an engine makes

Charles Babbage is credited with having devised the first ever special-purpose mechanical computer as well as the first ever general-purpose mechanical computer. The first claim seems rather dubious in an age where there is general agreement that the Antikythera mechanism is some sort of analogue computer. However, Babbage did indeed conceive and design the Difference Engine, a special purpose mechanical computer, in the first half of the nineteenth century. But what is a Difference Engine and why “Difference”?

Both Babbage and John Herschel were deeply interested in mathematical tables – trigonometrical tables, logarithmic tables – when they were still students and Babbage started collecting as many different editions of such tables as he could find. His main object was to check them for mistakes. Such mathematical tables were essential for navigation and errors in the figures could lead to serious navigation error for the users. Today if I want to know the natural logarithm of a number, let’s take 23.483 for example, I just tip it into my pocket calculator, which cost me all of €18, and I instantly get an answer to nine decimal places, 3.156276755. In Babbage’s day one would have to look the answer up in a table each value of which had been arduously calculated by hand. The risk that those calculations contained errors was very high indeed.

Babbage reasoned that it should be possible to devise a machine that could carryout those arduous calculations free of error and if it included a printer, to print out the calculated answer avoiding printing errors as well. The result of this stream of thought was his Difference Engine but why Difference?

The London Science Museum's reconstruction of Difference Engine No. 2 Source: Wikimedia Commons

The London Science Museum’s reconstruction of Difference Engine No. 2
Source: Wikimedia Commons

Babbage needed to keep his machine as simple as possible, which meant that the simplest solution would be a machine that could calculate all the necessary tables with variations on one algorithm, where an algorithm is just a step-by-step recipe to solve a mathematical problem. However, he needed to calculate logarithms, sines, cosines and tangents, did such an algorithm exist. Yes it did and it had been discovered by Isaac Newton and known as the method of finite differences.

The method of finite differences describes a property shared by all polynomials. If it has been a while since you did any mathematics, polynomials are mathematical expressions of the type x2+5x-3 or 7x5-3x3+2x2-3x+6 or x2-2 etc, etc. If you tabulate the values of a given polynomial for x=0, x=1, x=2, x=3 and so on then subtract the first value from the second, the second from the third and so on you get a new column of numbers. Repeating the process with this column produces yet another column and so on. At some point in the process you end up with a column that is filled with a numerical constant. Confused? OK look at the table below!

 

x x3-3x2+6 xn+1-xn diff(1)n+1 –diff(1)n diff(2)n+1 – diff(2)n
0 6 -2 0 6
1 4 -2 6 6
2 2 4 12 6
3 6 16 18
4 22 34
5 56
6

 

As you can see this particular polynomial bottoms out, so to speak, with as constant of 6. If we now go back into the right hand column and enter a new 6 in the first free line then add this to its immediate left hand neighbour repeating this process across the table we arrive at the polynomial column with the next value for the polynomial. See below:

 

x x3-3x2+6 xn+1-xn diff(1)n+1 –diff(1)n diff(2)n+1 – diff(2)n
0 6 -2 0 6
1 4 -2 6 6
2 2 4 12 6
3 6 16 18 6
4 22 34
5 56
6

 

x x3-3x2+6 xn+1-xn diff(1)n+1 –diff(1)n diff(2)n+1 – diff(2)n
0 6 -2 0 6
1 4 -2 6 6
2 2 4 12 6
3 6 16 18 6
4 22 34 24
5 56
6

 

x x3-3x2+6 xn+1-xn diff(1)n+1 –diff(1)n diff(2)n+1 – diff(2)n
0 6 -2 0 6
1 4 -2 6 6
2 2 4 12 6
3 6 16 18 6
4 22 34 24
5 56 58
6

 

x x3-3x2+6 xn+1-xn diff(1)n+1 –diff(1)n diff(2)n+1 – diff(2)n
0 6 -2 0 6
1 4 -2 6 6
2 2 4 12 6
3 6 16 18 6
4 22 34 24
5 56 58
6 114

This means that if we set up our table and calculate enough values to determine the difference constant then we can by a process of simple addition calculate all further values of the polynomial. This is exactly what Babbage designed his difference engine to do.

If you’ve been paying attention you might notice that the method of finite differences applies to polynomials and Babbage wished to calculate were logarithmic and trigonometrical functions. This is however not a serious problem, through the use of other bits of higher mathematics, which we don’t need to go into here, it is possible to represent both logarithmic and trigonometrical functions as polynomials. There are some problems involved with using the method of finite differences with these polynomials but these are surmountable and Babbage was a good enough mathematician to cope with these difficulties.

Babbage now had a concept and a plan to realise it, all he now needed was the finances to put his plan into action. This was not a problem. Great Britain was a world power with a large empire and the British Government was more than ready to cough up the readies for a scheme to provide reliable mathematical tables for navigation for the Royal Navy and Merchant Marine that serviced, controlled and defended that empire. In total over a period of about ten years the Government provided Babbage with about £17, 000, literally a fortune in the early nineteen hundreds. What did they get for their money, in the end nothing!

Why didn’t Babbage deliver the Difference Engine? There is a widespread myth that Babbage’s computer couldn’t be built with the technology available in the first half of the nineteenth century. This is simply not true, as I said a myth. Several modules of the Difference Engine were built and functioned perfectly. Babbage himself had one, which he would demonstrate at his scientific soirées, amongst other things to demonstrate his theory of miracles.

The Difference Engine model used by Babbage for his demonstrations of his miracle theory Source: Wikimedia Commons

The Difference Engine model used by Babbage for his demonstrations of his miracle theory
Source: Wikimedia Commons

Other Difference Engines modules were exhibited and demonstrated at the Great Exhibition in Crystal Palace. So why didn’t Babbage finish building the Difference Engine and deliver it up to the British Government? Babbage was not an easy man, argumentative and prone to bitter disputes. He became embroiled in one such dispute with Joseph Clement, the engineer who was actually building the Difference Engine, about ownership of and rights to the tools developed to construct the engine and various already constructed elements. Joseph Clement won the dispute and decamped together with said tools and elements. By now Babbage was consumed with a passion for his new computing vision, the general purpose Analytical Engine. He now abandoned the Difference Engine and tried to convince the government to instead finance the, in his opinion, far superior Analytical Engine. Having sunk a fortune into the Difference Engine and receiving nothing in return, the government, not surprisingly, demurred. The much hyped Ada Lovelace Memoire on the Analytical Engine was just one of Babbage’s attempts to advertise his scheme and attract financing.

However, the story of the Difference Engine didn’t end there. Using knowledge that he had won through his work on the Analytical Engine, Babbage produced plans for an improved, simplified Difference Engine 2 at the beginning of the 1850s.

Per Georg Schutz Source: Wikimedia Commons

Per Georg Schutz
Source: Wikimedia Commons

The Swedish engineer Per Georg Scheutz, who had already been designing and building mechanical calculators, began to manufacture difference engines based on Babbage’s plans for the Difference Engine 2 in 1855. He even sold one to the British Government.

Scheutz Difference Engine No. 2. (1859) Maschine im Science Museum, London Source: Wikimedia Commons

Scheutz Difference Engine No. 2. (1859) Maschine im Science Museum, London
Source: Wikimedia Commons

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Filed under History of Computing, History of Mathematics, History of Technology, Myths of Science

Another public service announcement

Marius Book Launch

In September 2014 a conference was held in Nürnberg, as the climax of a year dedicated to celebrating the life and work of the Franconian astronomer, astrologer and mathematician Simon Marius, whose magnum opus Mundus Iovialis was published four hundred years earlier in 1614.

The papers held at that conference together with some other contributions from people who could not attend in person have now been collected together in the book Simon Marius und Seine Forschung, eds. Hans Gaab and Pierre Leich (= Acta Historica Astronomiae, Band 57) which will be official launched in the Thalia bookshop in Nürnberg on this coming Thursday, 13 October at 18:30 MET.

This volume contains papers by a wide range of scholars and could/should be of interest to anybody studying the histories of astronomy, astrology and/or mathematics in the Early Modern Period. It can be purchased online, after Thursday, directly from the publishers, Leipziger Universitätsverlag

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For those who would like to know more about the book including a table of contents (Inhaltsverzeichnis) they can inform themselves on the Marius Portal here.

For those who cannot read German, an English edition of the book is in planning for next year, for which further contributions on the life and work of Simon Marius would also be welcome. If anybody has any questions regarding this volume I would be happy to answer them.

 

P.S. For those waiting for blogging to resume here at the Renaissance Mathematicus I can report that there is light at the end of the tunnel!

 

 

 

 

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

A spirited defence

After I had, in my last blog post, mauled his Scientific American essay in my usual uncouth Rambo style, Michael Barany responded with great elegance and courtesy in a spirited defence of his historical claims to which I now intend to add some comments, thus extending this exchange by a fourth part.

On early practical mathematicians Michael Barany acknowledges that their work is for the public good but argues correctly that that doesn’t then a “public good”. I acknowledge that there is a difference and accept his point however I have a sneaky feeling that something is only referred to as a “public good” when somebody in power is trying to put one over on the great unwashed.

Barany thinks that the Liber Abbaci and per definition all the other abbacus books, only exist for a closed circle of insider and not for the general public. In fact abbacus books were used as textbooks in so-called abbacus schools, which were small private schools that taught the basics of arithmetic, algebra, geometry and bookkeeping open to all who could pay the fees demanded by the schoolteacher, who was very often the author of the abbacus book that he used for his teaching. It is true that the pupils were mostly the apprentices of tradesmen, builders and artists but they were at least in theory open to all and were not quite the closed shop that Michael Barany seems to be implying. In this context Michael Barany says that Recorde’s Pathway to Knowledge, a book on elementary Euclidean geometry, is eminently impractical. However elementary Euclidean geometry was part of the syllabus of all abbacus schools considered part of the necessary knowledge required by artist and builder/architect apprentices. In fact the first Italian vernacular translation of Euclid was made by Tartaglia, an abbacus schoolteacher.

Michael Barany makes some plausible but rather stretched argument to justify his couterpositioning of Recorde and Dee, which I don’t find totally convincing but slips into his argument the following gem. If you don’t like Dee as your English standard bearer for keeping mathematics close to one’s chest, try Thomas Harriot. Now I assume that this flippant comment was written tongue in cheek but just in case.

Michael Barany’s whole essay contrasts what he sees as two approaches to mathematics, those who see mathematics as a topic for everyone and those who view mathematics as a topic for an elitist clique. In the passage that I criticised in his original essay he presented Robert Recorde as an example of the former and John Dee as a representative of the latter. A contrast that he tries to defend in his reply, where this statement about Harriot turns up. Now his elitist argument is very much dependent on a clique or closed circle of trained experts or adepts who exchanged their arcane knowledge amongst themselves but not with outsiders. A good example of such behaviour in the history of science is alchemy and the alchemists. Harriot as an example of such behaviour is a complete flop. Thomas Harriot made significant discoveries in various fields of scientific endeavour, mathematics, dynamics, chemistry, optics, cartography and astronomy, however he never published any of his work and although he corresponded with other leading Renaissance scholars he also didn’t share his discoveries with these people. A good example of this is his correspondence with Kepler, where he discussed over several letters the problem of refraction but never once mentioned that he had already discovered what we now know as Snell’s Law. Harriot remained throughout his life a closed circle with exactly one member, not a very good example to illustrate Michael Barany’s thesis.

I claimed that there was no advance mathematics in Europe from late antiquity till the fifteenth century. Michael Barany counters this by saying: This cuts, for instance, the rich history of Islamic court mathematics out of the European history in which it emphatically belongs; it doesn’t cut it. Ignoring Islamic Andalusia, Islamic mathematics was developed outside of Europe and although it started to reappear in Europe during the twelfth and thirteen centuries during the translator period nobody within Europe was really capable of doing much with those advanced aspects of it before the fifteenth century, so I stand by my claim.

We now turn to Michael Barany’s defence of his original: In the seventeenth century’s Scientific Revolution, the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty. This he contrast with a, in his opinion, eighteenth century where mathematicians help sway over the scientific community. I basically implied that this claim was rubbish and I still stand by that to that, so what does Michael Barany produce in his defence.

In my original post I listed seven leading scholars of the seventeenth century who were mathematicians and whose very substantive contributions to the so-called scientific revolution was mathematical, on this Barany writes:

Thony pretends that naming some figures remembered today both for mathematics and for their contributions to the scientific revolution contradicts this well-established historical claim.

The, without any doubt, principle figures of the so-called scientific revolution are just some figures! Interesting? So what is Michael Barany’s well-established historical claim? We get offered the following:

Following Steven Shapin and many who have written since his classic 1988 article on Boyle’s relationship to mathematics, I chose to emphasize the conflicts between the experimental program associated with the scientific revolution and competing views on the role of mathematics in natural philosophy.

What we have here is an argument by authority, that of Steven Shapin, whose work and the conclusions that he draws are by no means undisputed, and one name Robert Boyle! Curiously a few days before I read this, science writer, John Gribbin, commentated on Facebook that Robert Hooke had to work out Boyle’s Law because Boyle was lousy at mathematics, might this explain his aversion to it? However Michael Barany does offer us a second argument:

But to take just his most famous example, Newton’s prestige in the Royal Society is generally seen today to have had at least as much to do with his Opticks and his other non-mathematical pursuits as with his calculus, which contemporaries almost uniformly found impenetrable.

Really? I seem to remember that twenty years before he published his Opticks, Old Isaac wrote another somewhat significant tome entitled Philosophiæ Naturalis Principia Mathematica [my emphasis], which was published by the Royal Society. It was this volume of mathematical physics that established Newton’s reputation, not only with the fellows of the Royal Society, but with the entire scientific community of Europe, even with those who rejected Newton’s central concept of gravity as action at a distance. This book led to Newton being elected President of the Royal Society, in 1704, the same year as the Opticks was published. The Opticks certainly enhanced Newton’s reputation but he was already considered almost universally by then to be the greatest living natural philosopher.

Is the Opticks truly non-mathematical? Well, actually no! When it was published it was the culmination of two thousand years of geometrical optics, a mathematical discipline that begins with Euclid, Hero and Ptolemaeus in antiquity and was developed by various Islamic scholars in the Middle Ages, most notably Ibn al-Haytham. One of the first mathematical sciences to re-enter Europe in the High Middle Ages it was propagated by Robert Grosseteste, Roger Bacon, John Peckham and Witelo. In the seventeenth-century it was one of the mainstream disciplines contributing to the so-called scientific revolution developed by Thomas Harriot, Johannes Kepler, Willebrord van Roijen Snell, Christoph Scheiner, René Descartes, Pierre Fermat, Christiaan Huygens, Robert Hooke, James Gregory and others. Newton built on and developed the work of all these people and published his results in his Opticks in 1706. Yes, some of his results are based on experiments but that does not make the results non-mathematical and if you bother to read the book you will find more than a smidgen of geometry there in.

In my opinion trying to recruit Newton as an example of non-mathematical experimental science is an act of desperation.

To be fair to Michael Barany the division between those who favoured non-mathematical experimental science and the mathematician really did exist in the seventeenth century, however it was largely confined to England and most prominently in the Royal Society. This is the conflict between the Baconians and the Newtonians that I have blogged about on several occasions in the past. Boyle, Hooke and Flamsteed, for example, were all Baconians who, following Francis Bacon, were not particularly fond of mathematical proofs. This conflict has an interesting history within the Royal Society, which led to disadvantages for the development of the mathematical sciences in England in the eighteenth century.

When the Royal Society was initially founded some mathematician did not become members because of the dominance of the Baconians and that despite the fact that the first President, William Brouncker, was a mathematician. Later under Newton’s presidency the mathematicians gained the ascendency, but first in 1712 after an eight-year guerrilla conflict between Newton and Hans Sloane, a Baconian and the society’s secretary. Following Newton’s death in 1727 (ns) the Baconians regained power and the result was that, whereas on the continent the mathematical sciences flourished and evolved throughout the eighteenth century, in England they withered and died, leading to a new power struggle in the nineteenth century featuring such figures as Charles Babbage and John Herschel.

To claim as Michael Barany does that this conflict within the English scientific community meant that mathematics played an inferior role in the seventeenth century is a bridge too far and contradicts the available historical facts. Yes, the mathematization of nature was not the only game in town and interestingly non-mathematical experimental science was not the only alternative. In fact the seventeenth century was a wonderful cuddle-muddle of conflicting meta-physical views on the sciences. However whatever Steven Shapin might or might not claim the seventeenth century was a very mathematical century and mathematics was the principle driving force behind the so-called scientific revolution. As a footnote I would point out that many of the leading experimental natural philosophers of the seventeenth century, such as Galileo, Pascal, Stevin and Newton, were mathematicians who interpreted and presented their results mathematically.

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Filed under History of Mathematics, History of science, Newton

Bertrand Russell did not write Principia Mathematica

Yesterday would have been Bertrand Russell’s 144th birthday and numerous people on the Internet took notice of the occasion. Unfortunately several of them, including some who should know better, included in their brief descriptions of his life and work the fact that he was the author of Principia Mathematica, he wasn’t. At this point some readers will probably be thinking that I have gone mad. Anybody who has an interest in the history of modern mathematics and logic knows that Bertrand Russell wrote Principia Mathematica. Sorry, he didn’t! The three volumes of Principia Mathematica were co-authored by Alfred North Whitehead and Bertrand Russell.

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Now you might think that I’m just splitting hairs but I’m not. If you note the order in which the authors are named you will observe that they are not listed alphabetically but that Whitehead is listed first, ahead of Russell. This is because Whitehead being senior to Russell, in both years and status within the Cambridge academic hierarchy, was considered to be the lead author. In fact Whitehead had been both Russell’s teacher, as an undergraduate, and his examiner in his viva voce, where he in his own account gave Russell a hard time because he knew that it was the last time that he would be his mathematical superior.

Alfred North Whitehead

Alfred North Whitehead

Both of them were interested in metamathematics and had published books on the subject: Whitehead’s A Treatise on Universal Algebra (1898) and Russell’s The Principles of Mathematics (1903). Both of them were working on second volumes of their respective works when they decided to combine forces on a joint work the result of the decision being the monumental three volumes of Principia Mathematica (Vol. I, 1910, Vol. II, 1912, Vol. III, 1913). According to Russell’s own account the first two volumes where a true collaborative effort, whilst volume three was almost entirely written by Whitehead.

Bertrand Russell 1907 Source: Wikimedia Commons

Bertrand Russell 1907
Source: Wikimedia Commons

People referring to Russell’s Principia Mathematica instead of Whitehead’s and Russell’s Principia Mathematica is not new but I have the feeling that it is becoming more common as the years progress. This is not a good thing because it is a gradual blending out, at least on a semi-popular level, of Alfred Whitehead’s important contributions to the history of logic and metamathematics. I think this is partially due to the paths that their lives took after the publication of Principia Mathematica.

The title page of the shortened version of the Principia Mathematica to *56 Source: Wikimedia Commons

The title page of the shortened version of the Principia Mathematica to *56
Source: Wikimedia Commons

Whilst Russell, amongst his many other activities, remained very active at the centre of the European logic and metamathematics community, Whitehead turned, after the First World War, comparatively late in life, to philosophy and in particular metaphysics going on to found what has become known as process philosophy and which became particularly influential in the USA.

In history, as in academia in general, getting your facts right is one of the basics, so if you have occasion to refer to Principia Mathematica then please remember that it was written by Whitehead and Russell and not just by Russell and if you are talking about Bertrand Russell then he was co-author of Principia Mathematica and not its author.

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Filed under History of Logic, History of Mathematics