He fought for his mother

There are not many books about the Renaissance mathematician and astronomer Johannes Kepler in which he only plays a supporting role but this is the case in Ulinka Rublack’s The Astronomer and the Witch: Johannes Kepler’s Fight for His Mother[1]. In fact in Rublack’s excellent book even Kepler’s mother, Katherina, the nominal subject of the book only really takes a supporting role; the lead role being taken by the context within which the whole tragic story unfolds and it is exactly this that makes this book so excellent.

book-cover

Regular readers of this blog will know that I champion the claims of Johannes Kepler to being the most significant natural philosopher of the Early Modern Period against the rival claims of Copernicus, Galileo, Descartes, Newton et al. So I am naturally interested in any new books that appear with Kepler as their subject. Having looked closely at one of the strangest events in Kepler’s unbelievably bizarre life, the arrest and trial of his mother, Katherina, on a charge of witchcraft – and having blogged about it twice – my interest was particularly piqued by an announcement of a new book on this topic. A decent, well-researched book in English devoted exclusively to the subject would be a very positive addition to the Kepler literature. Rublack’s book is just the bill.

Nearly all accounts of Katherina Kepler’s ordeal are merely chapters or sections in more general books about Kepler’s life and work and mostly deal chronologically with the original accusations of witchcraft, counter accusations, the attempted violent intimidation of Katherina, the frustrated strivings to bring charges against her tormentors, her arrest and finally the trial with its famous defence by Johannes. Except for thumbnail sketches of those involved very little attempt is ever made to place the occurrences into a wider or more general context and this is, as already said above, exactly the strength of Rublack’s book.

Rublack in having devoted an entire book to the whole affair draws back from the accusations, charges, counter charges and the trial itself to flesh out the story with the social, cultural, political and economic circumstances in which the whole sorry story took place. In doing so Rublack has created minor masterpiece of social history. Her research has obviously been deep and thorough and she displays a fine eye for detail, whilst maintaining a stirring narrative style that pulls the reader along at a steady pace.

One point in particular intrigued me having read all the prepublication advertising for the book, including several illuminating interviews on the subject with the author, as well as short essays by her. Rublack takes what might be seen as a strong feminist stand against the previous, exclusively male, characterisations of Katherina Kepler, all of which painted her as a mean spirited, crabby, old hag, who was, so to speak, largely to blame for the situation in which she found herself. Having over the years read almost all of these accounts I was curious how Rublack would justify her rejection of these portrayals of Katherina, which I knew were based on Kepler’s own accounts of his mother. Rublack does not disappoint. She points out quite correctly that Kepler’s description of his mother was written when he was still very young and is part of an almost psychopathic put down of himself and all those related or connected to him and calls rather his own mental state into question. Interestingly we have virtually no other accounts of Katherina from Johannes’ pen and to judge her purely on this one piece of strange juvenilia is probably, as Rublack makes very clear, a bridge too far. Piecing together all of the, admittedly scant, evidence Rublack paints a much more sympathetic picture of Katherina, a hard working, illiterate, sixteenth/seventeenth-century peasant woman, who had never had it easy in life but still managed to raise her children well and give them chances that she never had.

This book is not perfect, as Rublack relies in her accounts of Johannes on older standard biographies, whilst apparently not consulting some of the more recent scholarly studies of his life and work, and thus repeats several false claims concerning him. However I’m prepared to cut her some slack on this as none of the errors that she (unknowingly?) repeats have any direct bearing on the story of Katherina that she tells so skilfully.

The book is beautifully presented by the OUP. Printed in a pleasant, easy on the eyes typeface and charmingly illustrated with a large number of black and white pictures. The text is excellently annotated, but as always I would have preferred footnotes to endnotes, and there is an adequate index. I personally would have liked a separate bibliography but this might have been sacrificed on cost grounds, the hardback being available at a very civilised price for a serious academic volume. Although having called it that I should point out that the book is very accessible and readable for the non-expert or general reader.

I heartily recommend this book to anybody interested in seventeenth-century history, Johannes Kepler, the history of witchcraft or who just likes reading good informative, entertaining books, if one is allowed to call a book about the sufferings of an innocent woman entertaining. Put simply, it’s an excellent read that deserves to, and probably will, become the standard English text on the subject.

[1] Ulinka Rublack, The Astronomer and the Witch: Johannes Kepler’s Fight for His Mother. OUP, 2015

 

 

 

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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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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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Christmas Trilogy 2016 Part 1: Is Newtonian physics Newton’s physics?

Nature and nature’s laws lay hid in night;

God said “Let Newton be” and all was light.

Isaac Newton's Tomb in Westminster Abbey Photo: Klaus-Dieter Keller Source: Wikimedia Commons

Isaac Newton’s Tomb in Westminster Abbey
Photo: Klaus-Dieter Keller
Source: Wikimedia Commons

Alexander Pope’s epitaph sets the capstone on the myth of Newton’s achievements that had been under construction since the publication of the Principia in 1687. Newton had single-handedly delivered up the core of modern science – mechanics, astronomy/cosmology, optics with a side order of mathematics – all packed up and ready to go, just pay at the cash desk on your way out. We, of course, know (you do know don’t you?) that Pope’s claim is more than somewhat hyperbolic and that Newton’s achievements have, over the centuries since his death, been greatly exaggerated. But what about the mechanics? Surely that is something that Newton delivered up as a finished package in the Principia? We all learnt Newtonian physics at school, didn’t we, and that – the three laws of motion, the definition of force and the rest – is all straight out of the Principia, isn’t it? Newtonian physics is Newton’s physics, isn’t it? There is a rule in journalism/blogging that if the title of an article/post is in the form of a question then the answer is no. So Newtonian physics is not Newton’s physics, or is it? The answer is actually a qualified yes, Newtonian physics is Newton’s physics, but it’s very qualified.

Newton's own copy of his Principia, with hand-written corrections for the second edition Source: Wikimedia Commons

Newton’s own copy of his Principia, with hand-written corrections for the second edition
Source: Wikimedia Commons

The differences begin with the mathematics and this is important, after all Newton’s masterwork is The Mathematical Principles of Natural Philosophy with the emphasis very much on the mathematical. Newton wanted to differentiate his work, which he considered to be rigorously mathematical, from other versions of natural philosophy, in particular that of Descartes, which he saw as more speculatively philosophical. In this sense the Principia is a real change from much that went before and was rejected by some of a more philosophical and literary bent for exactly that reason. However Newton’s mathematics would prove a problem for any modern student learning Newtonian mechanics.

Our student would use calculus in his study of the mechanics writing his work either in Leibniz’s dx/dy notation or the more modern F’(x) = f(x) notation of the French mathematician, Lagrange (1736–1813). He won’t be using the dot notation developed by Newton and against which Babbage, Peacock, Herschel and the Analytical Society campaigned so hard at the beginning of the nineteenth century. In fact if our student turns to the Principia, he won’t find Newton’s dot notation calculus there either, as I explained in an earlier post Newton didn’t use calculus when writing the Principia, but did all of his mathematics with Euclidian geometry. This makes the Principia difficult to read for the modern reader and at times impenetrable. It should also be noted that although both Leibniz and Newton, independently of each other, codified a system of calculus – they didn’t invent it – at the end of the seventeenth century, they didn’t produce a completed system. A lot of the calculus that our student will be using was developed in the eighteenth century by such mathematicians as Pierre Varignon (1654–1722) in France and various Bernoullis as well as Leonard Euler (1707­1783) in Switzerland. The concept of limits that are so important to our modern student’s calculus proofs was first introduced by Bernard Bolzano (1781–1848), Augustin-Louis Cauchy (1789–1857) and above all Karl Theodor Wilhelm Weierstrass (1815–1897) in the nineteenth century.

Turning from the mathematics to the physics itself, although the core of what we now know as Newtonian mechanics can be found in the Principia, what we actually use/ teach today is actually an eighteenth-century synthesis of Newton’s work with elements taken from the works of Descartes and Leibniz; something our Isaac would definitely not have been very happy about, as he nursed a strong aversion to both of them.

A notable example of this synthesis concerns the relationship between mass, velocity and energy and was brought about one of the very few women to be involved in these developments in the eighteenth century, Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet, the French aristocrat, lover of Voltaire and translator of the first French edition of the Principia.

In the frontispiece to Voltaire's book on Newton's philosophy, du Châtelet appears as Voltaire's muse, reflecting Newton's heavenly insights down to Voltaire. Source: Wikimedia Commons

In the frontispiece to Voltaire’s book on Newton’s philosophy, du Châtelet appears as Voltaire’s muse, reflecting Newton’s heavenly insights down to Voltaire.
Source: Wikimedia Commons

One should remember that mechanics doesn’t begin with Newton; Simon Stevin, Galileo Galilei, Giovanni Alfonso Borelli, René Descartes, Christiaan Huygens and others all produced works on mechanics before Newton and a lot of their work flowed into the Principia. One of the problems of mechanics discussed in the seventeenth century was the physics of elastic and inelastic collisions, sounds horribly technical but it’s the physics of billiard and snooker for example, which Descartes famously got wrong. Part of the problem is the value of the energy[1] imparted upon impact by an object of mass m travelling at a velocity v upon impact.

Newton believed that the solution was simply mass times velocity, mv and belief is the right term his explanation being surprisingly non-mathematical and rather religious. Leibniz, however, thought that the solution was mass times velocity squared, again with very little scientific justification. The support for the two theories was divided largely along nationalist line, the Germans siding with Leibniz and the British with Newton and it was the French Newtonian Émilie du Châtelet who settled the dispute in favour of Leibniz. Drawing on experimental results produced by the Dutch Newtonian, Willem Jacob ‘s Gravesande (1688–1742), she was able to demonstrate the impact energy is indeed mv2.

Willem Jacob 's Gravesande (1688-1745) Portrait by Hendrik van Limborch (1681-1759) Source: Wikimedia Commons

Willem Jacob ‘s Gravesande (1688-1745) Portrait by Hendrik van Limborch (1681-1759)
Source: Wikimedia Commons

The purpose of this brief excurse into eighteenth-century physics is intended to show that contrary to Pope’s epitaph not even the great Isaac Newton can illuminate a whole branch of science in one sweep. He added a strong beam of light to many beacons already ignited by others throughout the seventeenth century but even he left many corners in the shadows for other researchers to find and illuminate in their turn.

 

 

 

 

[1] The use of the term energy here is of course anachronistic

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Filed under History of Physics, History of science, Myths of Science, Newton, Uncategorized

An anniversary

My mother died fifty years ago at midnight on the 24th December 1966. I had just turned fifteen, five days before, and was in many senses still fairly immature. At nine o’clock in the evening I was having my first ever adult conversation with my mother, on the subject of religion, enquiring what religious views she and my father held. I had recently come to the conclusion that I was an atheist and was curious what views my parents held. We were not a religious family and didn’t discus such things, so I was genuinely curious. She told me that my father was an atheist but that she was an agnostic. She added however that she categorically rejected all organised religions and having grown up in India in a Christian family she had personally experienced Christianity, Hinduism, Buddhism and Islam, so her rejection was well informed. In the middle of this, for me, fascinating conversation my mother suffered a massive heart attack and three hours later she was dead.

I didn’t go through five stages of grief; within twenty-four hours I went from a state of extreme shock, to boiling anger, to total shut down. This was not denial; I was more than aware that my mother was dead but was incapable of grieving or mourning. I refused to attend the funeral; I have no idea why and that was the state I remained in for a very, very long time. Looking back I now know that I desperately needed help, therapy, counselling or whatever but nobody was offering and I didn’t ask.

For the next nine months my two elder sisters and I rather grimly tried to maintain a semblance of family life. We were all fairly capable on a practical level because that was how we had been brought up but there was very little joy in existence at that time. At the end of summer in 67 my sisters both moved out to start their careers and my father dumped me in a boarding school. It was the school where I had been a dayboy for the previous four years so the rupture wasn’t total. For the next two years I was fairly miserable, mildly obstreperous and didn’t really give a shit about anything. The result was that I got expelled. I spent my A-level year living in London attending, the then notorious, Holland Park Comprehensive and consuming vast quantities of drugs. It was after all 69-70. Having scraped together an abysmal set of A-level results I now trundled off to Cardiff to study archaeology. Still not really giving a shit about very much I dropped out after one year.

I was now completely adrift with a head full of mental health problems and would basically remain so until 1993 when my father finally died after having the life slowly sucked out of him by emphysema over a period of about twenty year. As my father died the dams broke and I wept as I have never wept before or since in my life and I cry easily, often and copiously. I wasn’t weeping for my father, I did that later when I took my departure in the hospice and at his funeral, but for my mother. Twenty-seven years of grief, hurt, confusion and god only knows what poured out of me in the hours following the phone call telling me of my father’s death.

This is not a Hollywood movie, so I was not instantly ‘cured’ but took many years to finally come to terms with the circumstances of my mother’s death and find balm for my ravaged soul. Once many years later because of a chance remark about Christmas made by somebody in my presence I became haunted by my father’s voice on the phone dictating the telegram to my grandparents in Australia informing them that my mother had died. It took several weeks of professional psychiatric care and some fairly strong anti-depressants to once again banish that voice out of my head.

However, that night marks an important step in my long and weary fight to regain my mental health, which I talked about in my earlier post about my mental health problems, and now, as then, I’m not writing this to elicit sympathy or to self aggrandise, hey look how I’ve suffered, but in the vague hope that I might help somebody else in a similar situation.

If you have lost somebody you love under tragic circumstances or know somebody who has, in particular children, then please, please make sure that you or they grieve if necessary fetch professional help. Bottling up your grief will seriously damage you, gnawing at your soul like a bad tooth. You might not even be aware of the damage on a conscious level but believe me it’s there.

I don’t celebrate Christmas and never will, my bother and my sisters did and do because they have had children and grandchildren of their own, but I have never had children, which is good because I would have been a lousy parent, I was not even capable of coping with myself let alone being responsible for another vulnerable human being. However this post is my Christmas present for those who might be in need of it. It is given freely and if you can take anything positive from it then you are very welcome to do so.

 

 

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I was robbed (twice)! – Vague ramblings on rites of passage, anniversaries, calendrics and the human desire to control time – on the occasion of the winter solstice.

Sunrise at Stonehenge on the Winter Solstice Photo: Mark Grant Source: Wikimedia Commons

Sunrise at Stonehenge on the Winter Solstice
Photo: Mark Grant
Source: Wikimedia Commons

As I was growing up in a remote corner of North-East Essex there were two birthdays that were considered to mark important moments in a person’s life, the twenty-first and the sixty-fifth. The first marked the entry into adulthood and the second the exit out of the world of work. Both were celebrated as special occasions, the former with a lavish party and, in well off families, with a spectacular coming of age present, the later with a somewhat more sombre ceremony and traditional the presentation of a timepiece (quite why it is/was traditional to give people a timepiece when they retire I have absolutely no idea!) The celebration of such points in ones life are known as rites of passage because they mark the transition from one socio-cultural group to another – coming of age from the community of the children to that of the adults, retirement from the working community to the community of the retired. Humans find it necessary/comforting/important to mark these transitions in some significant way.

I was going on nineteen when the British government decided to reduce the age of majority from twenty-one to eighteen meaning that my transition into adulthood took place on some arbitrary date by act of parliament without any form of acknowledgement/ceremony or whatever. As the title of this post says, I was robbed! Two days ago I celebrated my sixty-fifth birthday or rather didn’t celebrate but I still turned sixty-five. The German government is in the process of incrementally raising the retirement age to sixty-seven so I would have been due to retire at sixty-five and six months. However that same government persuaded me to retire at the beginning of September, actually carried out retrospectively meaning once more I was robbed of my rite of passage. As, however, I am self employed in that work that I do, and continue to do, there would have been nobody to hand around the cucumber sandwiches and the plastic glasses of cheap bubbly or to hold a boring and embarrassing speech whilst presenting me with my timepiece anyway.

Being from a non-religious, middle class, English household, and not for example Jewish, I did not undergo a biological coming of age at a nominal puberty such as the Jewish Bar Mitzvah. That is unless you count the eleven plus exam and the transition from primary school to secondary school. Which, at my very elite and very posh, grammar school included the tradition of being dragged through a hedge backwards or having ones head stuck in a toilet bowl and flushed by members of the fifth form during the mid morning break on the first day of school, delights that I managed somehow to avoid. By the time I reached the fifth form the tradition had thankfully died out.

Human seem to have some sort of innate desire to mark time and to celebrate certain events on some sort of regular basis. On the secular side birthdays, wedding anniversaries, first meetings, for some final school exams and whatever. On the religious side, for all religions, a whole cartload of religious festivals of various types. As political communities independence days, armistice days and an assortment of other national holidays. These celebrations and the rites of passage discussed above have one thing in common they are almost all arbitrary, the one exception being anniversaries to which we will return to in a minute.

The only natural timekeepers that we have are the diurnal rotation of the earth, the phases of the moon and the apparent passage of the sun around the ecliptic, which give us respectively the day, the (lunar) month and the year. All other divisions of time are of our own devising and as such arbitrary. Calendars were invented to help us keep track of those days that we have chosen to mark out for special attention of some sort – a public holiday, a religious observance or whatever. They are crib sheets for rites and rituals, which as already remarked almost all take place on arbitrary days. Good examples of arbitrary ritual days are the rapidly approaching Christmas and New Years festivals, as I have pointed out for the latter in an earlier post, different cultures having different New Years celebrations on differing dates.

The only rituals that are in a sense not arbitrary are, because the solar year is periodic, anniversaries. These occur, with a little bit of fudging, once every three hundred and sixty-five days. The fudging is necessary because the solar year is, as should be well-known, a little bit longer than three hundred and sixty-five days. With the Gregorian calendar we have a tolerably good system of fudging, although other calendars, the Jewish and Islamic ones for example, do things differently.

Because the ecliptic is tilted at approximately twenty-three degrees with relation to the equator, known technically as the obliquity of the ecliptic, we have as a result the seasons and also four days in the solar year that are not arbitrary. These are the equinoxes and the solstices. The equinoxes are the days in spring, the vernal equinox around the twentieth of March, and autumn, the autumnal equinox around the twenty-second of September, when the sun appears to be over the equator and the day and night are equally long. The summer solstice (Northern hemisphere, winter for Southern hemisphere) takes place when the sun appears to be over the Tropic of Cancer (approximately 23° of latitude north of the equator), that is its Northern most point on its journey around the ecliptic, around the twenty-first of June, and marks the longest day and shortest night in the Northern hemisphere and vice versa in the Southern hemisphere. The winter solstice (Northern hemisphere, summer for Southern hemisphere) takes place when the sun appears to be over the Tropic of Capricorn (approximately 23° of latitude south of the equator), that is its Southern most point on its journey around the ecliptic, around the twenty-first of December, and marks the shortest day and longest night in the Northern hemisphere and vice versa in the Southern hemisphere.

Many of the folk customs that occur around these days are celebrations of these astronomical events, their origins often forgotten, as they are co-opted into other, oft religious, celebrations. This is certainly true for many of the Christmas customs, which have their origins in various winter solstice celebrations, now lost in the mists of history.

I celebrate neither Christmas nor New Years but am prepared to acknowledge the winter solstice as a fulcrum or turning point of the year, so I wish all of my readers all the best for their next three hundred and sixty-five and a bit days journey around the sun, it is of course we who orbit the sun and not the sun us, and may you enjoy in your own ways those arbitrary calendrical dates that you choose to celebrate.

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Filed under Autobiographical, History of Astronomy