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Some rather strange history of maths

Scientific American has a guest blog post with the title: Mathematicians Are Overselling the Idea That “Math Is Everywhere, which argues in its subtitle: The mathematics that is most important to society is the province of the exceptional few—and that’s always been true. Now I’m not really interested in the substantial argument of the article but the author, Michael J. Barany, opens his piece with some historical comments that I find to be substantially wrong; a situation made worse by the fact that the author is a historian of mathematics.

Barany’s third paragraph starts as follows:

In the first agricultural societies in the cradle of civilization, math connected the heavens and the earth. Priests used astronomical calculations to mark the seasons and interpret divine will, and their special command of mathematics gave them power and privilege in their societies.

We are taking about the area loosely known as Babylon, although the names and culture changed over the millennia, and it is largely a myth, not only for this culture, that astronomical calculations were used to mark the seasons. The Babylonian astrologers certainly interpreted the divine will but they were civil servants who whilst certainly belonging to the upper echelons of society did not have much in the way of power or privilege. They were trained experts who did a job for which they got paid. If they did it well they lived a peaceful life and if they did it badly they risked an awful lot, including their lives.

Barany continues as follows:

As early economies grew larger and more complex, merchants and craftsmen incorporated more and more basic mathematics into their work, but for them mathematics was a trick of the trade rather than a public good. For millennia, advanced math remained the concern of the well-off, as either a philosophical pastime or a means to assert special authority.

It is certainly true that merchants and craftsmen in advanced societies – Babylon, Greece, Rome – used basic mathematics in their work but as these people provide the bedrock of their societies ­– food, housing etc. – I think it is safe to say that their maths based activities were in general for the public good. As for advanced maths, and here I restrict myself to European history, it appeared no earlier than 1500 BCE in Babylon and had disappeared again by the fourth century CE with the collapse of the Roman Empire, so we are talking about two millennia at the most. Also for a large part of that time the Romans, who were the dominant power of the period, didn’t really have much interest in advance maths at all.

With the rebirth of European learned culture in the High Middle ages we have a society that founded the European universities but, like the Romans, didn’t really care for advanced maths, which only really began to reappear in the fifteenth century. Barany’s next paragraph contains an inherent contradiction:

The first relatively widespread suggestions that anything beyond simple practical math ought to have a wider reach date to what historians call the Early Modern period, beginning around five centuries ago, when many of our modern social structures and institutions started to take shape. Just as Martin Luther and other early Protestants began to insist that Scripture should be available to the masses in their own languages, scientific writers like Welsh polymath Robert Recorde used the relatively new technology of the printing press to promote math for the people. Recorde’s 1543 English arithmetic textbook began with an argument that “no man can do any thing alone, and much less talk or bargain with another, but he shall still have to do with number” and that numbers’ uses were “unnumerable” (pun intended).

Barany says, “that anything beyond simple practical math ought to have a wider reach…” and then goes on to suggest that this was typified by Robert Recorde with his The Grounde of Artes from 1543. Recorde’s book is very basic arithmetic; it is an abbacus or reckoning book for teaching basic arithmetic and book keeping to apprentices. In other words it is a book of simple practical maths. Historically what makes Recorde’s book interesting is that it is the first such book written in English, whereas on the continent such books had been being produced in the vernacular as manuscripts and then later as printed books since the thirteenth century when Leonardo of Pisa produced his Libre Abbaci, the book that gave the genre its name. Abbaci comes from the Italian verb to calculate or to reckon.

What however led me to write this post is the beginning of Barany’s next paragraph:

Far more influential and representative of this period, however, was Recorde’s contemporary John Dee, who used his mathematical reputation to gain a powerful position advising Queen Elizabeth I. Dee hewed so closely to the idea of math as a secret and privileged kind of knowledge that his detractors accused him of conjuring and other occult practices.

Barany is contrasting Recorde, man of the people bringing mathematic to the masses in his opinion with Dee an elitist defender of mathematics as secret and privileged knowledge. This would be quite funny if it wasn’t contained in an essay in Scientific American. Let us examine the two founders of the so-called English School of Mathematics a little more closely.

Robert Recorde who obtained a doctorate in medicine from Cambridge University was in fact personal physician to both Edward VI and Queen Mary. He served as comptroller of the Bristol Mint and supervisor of the Dublin Mint both important high level government appointments. Dee acquired a BA at St John’s College Cambridge and became a fellow of Trinity College. He then travelled extensively on the continent studying in Leuven under Gemma Frisius. Shortly after his return to England he was thrown into to prison on suspicion of sedition against Queen Mary; a charge of which he was eventually cleared. Although consulted oft by Queen Elizabeth he never, as opposed to Recorde, managed to obtain an official court appointment.

On the mathematical side Recorde did indeed write and publish, in English, a series of four introductory mathematics textbooks establishing the so-called English School of Mathematics. Following Recorde’s death it was Dee who edited and published further editions of Recorde’s mathematics books. Dee, having studied under Gemma Frisius and Gerard Mercator, introduced modern cartography and globe making into Britain. He also taught navigation and cartography to the captains of the Muscovy Trading Company. In his home in Mortlake, Dee assembled the largest mathematics library in Europe, which functioned as a sort of open university for all who wished to come and study with him. His most important pupil was his foster son Thomas Digges who went on to become the most important English mathematical practitioner of the next generation. Dee also wrote the preface to the first English translation of Euclid’s Elements by Henry Billingsley. The preface is a brilliant tour de force surveying, in English, all the existing branches of mathematics. Somehow this is not the picture of a man, who hewed so closely to the idea of math as a secret and privileged kind of knowledge. Dee was an evangelising populariser and propagator of mathematics for everyman.

It is however Barany’s next segment that should leave any historian of science or mathematics totally gobsmacked and gasping for words. He writes:

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.

What can I say? I hardly know where to begin. Let us just list the major seventeenth-century contributors to the so-called Scientific Revolution, which itself has been characterised as the mathematization of nature (my emphasis). Simon Stevin, Johannes Kepler, Galileo Galilei, René Descartes, Blaise Pascal, Christiaan Huygens and last but by no means least Isaac Newton. Every single one of them a mathematician, whose very substantial contributions to the so-called Scientific Revolution were all mathematical. I could also add an even longer list of not quite so well known mathematicians who contributed. The seventeenth century has also been characterised, by more than one historian of mathematics as the golden age of mathematics, producing as it did modern algebra, analytical geometry and calculus along with a whole raft full of other mathematical developments.

The only thing I can say in Barany’s defence is that he in apparently a history of modern, i.e. twentieth-century, mathematics. I would politely suggest that should he again venture somewhat deeper into the past that he first does a little more research.

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Uccello and the Problem of Space

Nice post on Renaissance master of linear perspective Uccello

In the Dark

The other night I was watching an old episode of the detective series Lewis and it reminded me of something I wanted to blog about but never found the time. The episode in question, The Point of Vanishing, involves a discussion of a painting which can be found in the Ashmolean Museum in Oxford:

Uccello_TheHunt

I won’t spoil the plot by explaining its role in the TV programme, but this work – called “The Hunt in the Forest” or “The Night Hunt” or some other variation on that title –  is by one of the leading figures of the Early Renaissance, Paolo Uccello, who was born in Florence and lived from about 1396 until 1475. He was most notable for his explorations of the use of perspective in painting, and specifically in “The Problem of Space”, i.e. how to convey the presence of three dimensions when the paint is…

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He died fighting for his King

On 2 June 1644 one of the biggest battles of the English Civil War took place on Marston Moor just outside of the city of York. The Parliamentary forces under Fairfax had, together with the Scottish Covenanters under the Earl of Leven had been besieging York, the principle Royalist stronghold in the North, the defence being led by the Marquess of Newcastle. Prince Rupert came to the aid of the beleaguered city with a substantial royalist army. Newcastle boke out of the city with his cavalry and joined Rupert and the two armies clashed on Marston Moor. The battle ended in a disastrous defeat for the royalist forces and marks a significant turning point in the war.

The Battle of Marston Moor 1644, by J. Barker Source: Wikimedia Commons

The Battle of Marston Moor 1644, by J. Barker
Source: Wikimedia Commons

This is all well and good but at first glance doesn’t appear to have a lot to do with the history of science. However if we zoom in a little closer Marston Moor actually has two connections with that history. William Cavendish, the Marquess of Newcastle, and his brother Charles were both actively engaged supporters of the new sciences developing at the time in Europe and having fled England following the royalist defeat, they eventually ended up in Paris as part of the court of Queen Henrietta Maria. Here the Cavendish brothers became part of a philosophical circle dedicated to the investigation of science that included Marin Mersenne, Kenelm Digby and Thomas Hobbes. In Paris William also met and married Margaret Lucas, one of Henrietta Maria’s chamber maids, who would later become notorious as Margaret ‘Mad Madge’ Cavendish, a female philosopher of science and extensively published author.

Our second history of science connection to the Battle of Marston Moor is a less happy one because amongst the 4 000 royalist soldiers who are estimated to have died there was the astronomer, inventor and instrument maker William Gascoigne (1612–1644). Gascoigne is today mostly only known to those interested in the fine details of the history of the telescope, something that hasn’t changed much since his own times when he only became widely known after his most important invention, the micrometer, was claimed by the Frenchman Adrien Auzout (1622–1691) in 1666, twenty two years after his untimely death.

Adrien Auzout's (1621-1692) Micrometer published in his book (1662) Source: Wikimedia Commons

Adrien Auzout’s (1621-1692) Micrometer published in his book (1662)
Source: Wikimedia Commons

Gascoigne was born into the landed gentry in the village of Thorpe-on-the Hill near Leeds. Little is known of his childhood or education, although he claimed to have studied at Oxford University, a claim that cannot be confirmed. Like many amateur astronomers Gascoigne was self taught and appears to have been a very skilled instrument maker as he made all of his telescopes himself, including grinding his own lenses. One of the problems of early telescopes was measuring the size of celestial objects viewed through them. There is no easy solution to this problem when using a Dutch or Galilean telescope, i.e. with a plano-convex objective and a plano-concave eyepiece, and Galileo soled the problem by attaching a metal grid to the side of his telescope and viewing the object under observation through the telescope with one eye whilst observing the grid with his other eye. A trick that is thought to have been possible for Galileo because of an optical peculiarity he seems to have been born with. This method could only produce rough approximate sizes.

The Keplerian or astronomical telescope, where both objective and eyepiece lenses are convex, provides a much simpler solution. The Focal plane is at the juncture of the two focal lengths of the lenses, which is inside the telescope tube, and here the Keplerian telescope produces a its image. It appears that Gascoigne was the first to utilize this fact. There is a story that Gascoigne was made aware of this phenomenon by a spider that had woven its web in his telescope tube in the crucial position allowing him to focus on what he was viewing and the spider’s web at the same time. The story is probably apocryphal bur astronomers continued to collect spider’s silk from the hedgerows to form the crosshairs in their astronomical telescope well into the nineteenth century. Whatever led Gascoigne to the discovery of the internal image, he soon went beyond the simple expedient of installing crosshairs into his telescopes.

Focal plane with image at (5) Source: Wikimedia Commons

Focal plane with image at (5)
Source: Wikimedia Commons

Gascoigne realised that this phenomenon would enable him to introduce a measuring device into the focal plane of his telescope and this is what he did. He produced a calliper the points of which could be moved towards or away from each other by means of turning a single screw. Along the base along which the calliper points moved was a measuring scale. Gascoigne could now make accurate measurements of the celestial objects he observed.

Being a self taught amateur astronomer and living as he did in a small northern village in an age when long distance communication was difficult and unreliable one might be forgiving for thinking that Gascoigne was isolated and to some extent he was but not completely. He communicated by mail, for example, with William Oughtred inventor of the slide rule and mathematics teacher of several notable seventeenth century mathematicians. This contact seems to have been initiated by Gascoigne who was surprisingly well informed about actual developments in mathematics and astronomy and is known to have owned all of the relevant literature. Through Oughtred Gascoigne was also introduced to Kenelm Digby with whom he also corresponded.

Perhaps more significantly Gascoigne was in contact with the Towneley family of Towneley Hall near Burnley, landed gentry who took an interest in the actual developments in mathematics and astronomy.

Towneley Hall Source: Wikimedia Commons

Towneley Hall
Source: Wikimedia Commons

Christopher Towneley (1604–1674) introduced a group of northern astronomers to each other including William Milbourne of Christ’s College Cambridge (M.A. 1623), William Crabtree a merchant from Salford, Jeremiah Horrocks curate from Much Hoole near Preston and Gascoigne. Crabtree and Horrocks, famously, were the first astronomers to observe a transit of Venus. Crabtree and Gascoigne became good friends with Crabtree visiting Gascoigne to view and inspect his instruments and the two of them corresponding extensively on both Gascoigne’s instrumental novelties and the contemporary developments in astronomy, in particular the theories of Johannes Kepler, which both Crabtree and Horrocks accepted and Gascoigne under Crabtree’s influence came to accept. It was through this correspondence that we have Crabtree’s account of Horrocks’ death.

"Crabtree watching the Transit of Venus A.D. 1639" by Ford Madox Brown, a mural at Manchester Town Hall. Source: Wikimedia Commons

“Crabtree watching the Transit of Venus A.D. 1639” by Ford Madox Brown, a mural at Manchester Town Hall.
Source: Wikimedia Commons

All of this might have been lost following the deaths of Horrocks (1641), Gascoigne (1644) and Crabtree (1644) if not for the Towneleys. When the Royal Society announced Auzout’s invention of the micrometre screw gauge in 1666 it was Richard Towneley (1629–1704), Christopher’s nephew and a mathematician and astronomer in his own right, who piped up and said I beg to differ. John Flamsteed (1646–1719) (another northerner, later to become the first Astronomer Royal, who was a protégée of Jonas Moore (1617–1679), yet another Lancastrian and a pupil of William Milbourne) travelled up north to investigate Towneley’s claims. Towneley demonstrated his micrometer screw gauge based on Gascoigne’s design to Flamsteed and the two of them travelled to Salford where Crabtree’s widow gave them the Crabtree Gascoigne correspondence. Flamsteed made notes from the correspondence but the originals remained in the possession of the Towneley family.

Robert Hooke drew diagrams of Towneley’s version of Gascoigne’s micrometer, which were published in the Philosophical Transactions of the Royal Society thus establishing Gascoigne’s priority and his right to be acknowledged the inventor of the micrometer screw gauge.

Robert Hooke - A Description of an Instrument for Dividing a Foot into Many Thousand Parts, and Thereby Measuring the Diameters of Planets to a Great Exactness, &c. as It Was Promised, Numb. 25. In: Philosophical Transactions. Band 2, Nummer 29, 11.  Source: Wikimedia Commons

Robert Hooke – A Description of an Instrument for Dividing a Foot into Many Thousand Parts, and Thereby Measuring the Diameters of Planets to a Great Exactness, &c. as It Was Promised, Numb. 25. In: Philosophical Transactions. Band 2, Nummer 29, 11.
Source: Wikimedia Commons

As a small side note it was Richard Towneley together with Henry Power (1623–1668) who first discovered what is now known as Boyle’s Law, which Power published in his Experimental Philosophy, in three Books in 1664, an important early work on microscopy and the corpuscular theory.

Henry Power, Experimental philosophy, in three books : containing new experiments microscopical, mercurial, magnetical ; with some deductions, and probable hypotheses, raised from them, in avouchment and illustration of the now famous atomical hypothesis. London, 1664 Source: NIH U.S:.National Library of Medicine

Henry Power, Experimental philosophy, in three books : containing new experiments microscopical, mercurial, magnetical ; with some deductions, and probable hypotheses, raised from them, in avouchment and illustration of the now famous atomical hypothesis. London, 1664
Source: NIH U.S:.National Library of Medicine

Much of Gascoigne’s original correspondence has become lost of time but enough has been recovered to give a vivid picture of this inventive and highly skilled astronomer and his contributions to the history of astronomy. More important the fragments of the Gascoigne story demonstrate very clearly that progress in science in not achieved through lone geniuses but through networks of researchers exchanging views and discoveries and encouraging each other to make further developments.

 

 

 

 

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The Goddess, her husband and his lovers

In recent days the science sections of the media have been full of the successful entering of orbit around Jupiter by the NASA probe Juno after its five-year, 2.8 billion kilometre journey from the Earth. Many of the reports also talk about the so-called Galilean moons, Jupiter’s four largest moons (there are currently 67 known moons of Jupiter), and Galileo’s discovery of them with the recently invented telescope in early 1610.

Jupiter_and_the_Galilean_Satellites

Montage of Jupiter’s four Galilean moons, in a composite image depicting part of Jupiter and their relative sizes (positions are illustrative, not actual). From top to bottom: Io, Europa, Ganymede, Callisto. Source: Wikimedia Commons

Juno was even carrying Lego models of the god Jupiter, the goddess Juno and Galileo holding a telescope.

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With the notable exception of the New York Times none of the reports mentioned that the Ansbach court mathematicus, Simon Marius, independently discovered the Galilean moons just one day later than Galileo. However, whereas Galileo rushed into print with his telescopic discoveries in his Sidereus nuncius in 1610, Marius waited until 1614 before publishing his discoveries in his Mundus Iovialis.

Houghton_IC6.G1333.610s_-_Sidereus_nuncius

Title page of Sidereus nuncius, 1610, by Galileo Galilei (1564-1642). *IC6.G1333.610s, Houghton Library, Harvard University

The four moons are named Io, Europa, Ganymede and Callisto after four of the lovers of Zeus, the Greek equivalent to Jupiter, and many people have made a joke about the fact that Juno, his wife, was on her way.

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Once again what none of the reports, with the exception of the New York Times, mention is that the names were not given to the moons by Galileo. Wishing to use his telescopic discoveries to leverage a position at the Medici court in Florence Galileo wrote a letter to Grand Duke Cosimo’s secretary on 13 February 1610 asking if the Grand Duke would prefer the moons to be called Cosmania after his name or, rather, since they are exactly four in number, dedicate them to all four brothers with the name Medicean Stars (All heavenly bodies were referred to as stars in the Renaissance). The secretary replied that Cosimo would prefer the latter and so the moons became the Medicean Stars in the Sidereus nuncius.

The New York Times report attributed the names Io, Europa, Ganymede and Callisto to Simon Marius and they did indeed first appear in print in his Mundus Iovialis. However the names were not thought up by Marius.

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Title page Mundus Iovialis Simon Marius 1614. Internet Archive

In the Mundus Iovialis Marius makes several naming suggestions. His first suggestion is to just number the moons I to IV, a system that was actually used by astronomers. His second suggestion follows Galileo in that he wishes to name them after his employer/patron the Margrave of Ansbach’s family and call them the Brandenberger Stars. Marius’ third suggestion is more than somewhat bizarre as he suggests naming them in analogy to the solar system planets, so the moon with the smallest orbit would be the Jupiter Mercury, the next the Jupiter Venus, the third the Jupiter Jupiter and the fourth the Jupiter Saturn. As I said bizarre. It is with Marius’ fourth suggestion that we finally arrive at Zeus’ lovers. After talking about Jupiter’s reputation for a bit on the side and describing his most notorious affairs Marius write the following:

 In Europa, Ganimedes puer, atque Calisto,

Lascivo nimium perplacuere Jovi.

Io, Europa, the young Ganymede and Calisto

appealed all too much to the lascivious Jove

In the next paragraph Marius goes on to explain that the idea for using these names for the moons was suggested to him by Johannes Kepler[1] when the two of them met at the Imperial Parliament in Regensburg in October 1613. He then names Kepler as co-godfather of these four stars. Marius closes his list of suggestions by saying that the whole thing should not be taken too seriously and everybody is free to adopt or reject his suggestions as they see fit.

So as we now know it was Kepler’s suggestion which finally won the naming contest for the four largest moons of Jupiter but it should be noted that the names were first adopted by the astronomical community in the nineteenth century but they first became the official names of the Galilean Moons in 1975 through a decision of the IAU (International Astronomical Union)

Anybody who wants to learn more about Simon Marius can do so at the Simon Marius Portal or become a member of the Simon Marius Society (Simon Marius Gesellschaft e.V.) via the portal, membership is free!

Addendum 7 July 2016: My attention has been drawn to a delightful pop song about the lascivious Jupiter, his dalliances with his satellites and the impending arrival of his wife, She’s Checking In (The NASA Juno Song) by Adam Sakellarides  h/t Daniel Fischer (@cosmos4u)

[1] It should be noted that Johannes Kepler loved coining names and terms for all things scientific. It is to him, for example, that we owe the term satellite, coined specifically for the Jupiter moons, and also the term camera obscura, which in shortened form is our modern camera.

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Retraction of a Retraction – Turns out I wasn’t wrong after all!

Yesterday I deleted a post about Caroline Herschel, the Google doodle that had been posted for her birthday and the type of telescope that she had used to make he comet discoveries only hours after I had posted it. I went on to claim that I was mistaken in my claims and to apologies widely on the Internet to anybody I might have misled. In the cold grey light of dawn I realised that I had indeed made an error, however not in my original post but in retracting it. How this came about can be attributed to one or more of the following:

1) Extreme tiredness

2) Mental and physical confusion brought about by my first day of remedial orthopaedic treatment

3) A brain fart

4) Early signs of dementia

5) Congenital stupidity

6) An inability to read

7) Late repercussion of my substantial drug abuse in my misspent youth

8) Aliens!

9) Add your own reason!

In my post I had said that the Google Doodle was wrong because it displayed Caroline Herschel with a ‘normal’ refractor, i.e. with lenses, telescope whereas the Herschel used reflectors, i.e. telescopes with mirror, which they designed and constructed themselves. So far, so good. Then in the comments Tony Angel drew my attention to an interesting article about Caroline Herschel and her comet hunting. I hastily, and here is where the problems begin, skimmed the article and read the following quote from Caroline:

I found I was to be trained for an assistant-astronomer, and by way of encouragement a telescope adapted for ‘sweeping,’ consisting of a tube with two glasses, such as was commonly used as a ‘finder,’ was given to me.

 I read “a tube with two glasses” as being the description of a refractor with two lenses, thought ‘Oh FUCK!’ and in a state of confused panic deleted my post.

In fact, if I had read a few lines further I would have seen the following:

Table 1. The Parameters of Caroline’s two alt-azimuth

                 Newtonian reflecting sweepers.

 And on the following page there is even a diagram of the two telescopes with a caption that clearly states that they are Newtonian reflectors. So now being once again of sound mind and body I shall restore my post with a little bit of added information i.e. the illustration of Caroline’s reflector telescopes.

 

 

 

 

 

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I WAS WRONG!

I have deleted my previous  post on the Caroline Herschel Google Doodle because I was wrong! I will explain tomorrow.

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Blogging Hiatus

There will be a four-week blogging hiatus both here and at Whewell’s Gazette for two different reasons. Firstly I am going into hospital for three weeks, but have no fear this is a positive development not a negative one. I suffer from scoliosis and I’m going to get three weeks of intensive remedial orthopaedic treatment to try and improve my condition. Directly following my last day of treatment, I shall then fly to Britain for a meeting of the Christie Clan in a twelfth-century manor house in the Welsh marches. I shall not be entirely incommunicado, as I will only be a day patient during my treatment. So I will still pop up on Twitter and Facebook, from time to time, but I don’t think I shall be doing any blogging in this period and Whewell’s Gazette is definitely out.

If you are at a loose end and looking for something to do during this period, Afton, the excessively charming three-year-old daughter of my very good #histsci friend and colleague Michael Barton (@darwinsbulldog), suffers from epilepsy and has recently undergone neurosurgery. As they live in America this means big medical bills. Michael and his wife, Catherine, have set up an appeal on gofundme to help pay those bills, so you could do me and Afton a favour and make a small donation to help pass the time until I’m up and blogging again.

Afton

Afton

 

 

 

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