Printing the Hindu-Arabic numbers

Arte dell’Abbaco, a book that many consider the first-ever printed mathematics book, was dated four hundred and forty years ago on 10 December 1478. I say many consider because the book, also known as the Treviso Arithmetic, is a commercial arithmetic textbook and some historians regard commercial arithmetic as a separate discipline and not really mathematics.

Calculation from the Arte dell’Abbaco

The unknown author explains his book thus:

I have often been asked by certain youths in whom I have much interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic, commonly called abbacus.

The Treviso Arithmetic is actually an abbacus book, those books on calculating with the Hindu-Arabic numerals that derive their existence from Leonardo Pisano’s Liber Abbaci. Like most abbacus books it is written in the vernacular, which in this case is the local Venetian dialect. If you don’t read 15^thcentury Venetian there is an English translation by Frank J. Swetz, Capitalism and Arithmetic: The New Math of the 15^thCentury – Including the Full Text of the Treviso Arithmetic of 1478, Open Court, 1987.

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Books about the book

Most readers are probably aware that I live not very far away from the Renaissance city of Nürnberg in Southern Germany. It is a city rich in the history of science particularly during the Renaissance and so it was only a mater of time, after I moved here, that I would get sucked into becoming a local historian. In the end it was the fact that Copernicus’ magnum opus was printed and published there that proved to be the bait. This, however, also took me down another path, the early history of scientific printing in which the city is particularly rich. Not only was it the home of Johannes Petreius, who printed and published the De revolutionibus, as well as many other important early scientific titles, but it was also where Johannes Müller, aka Regiomontanus, chose to set up the world’s first-ever scientific publishing house. Researching Regiomontanus as a printer publisher leads automatically to Erhard Ratdolt, who, whilst not a Nürnberger printer publisher, published several of those titles that Regiomontanus intended to publish but was unable to due to his untimely demise. Around 1500 CE, the world’s biggest printed publisher was the Nürnberger Anton Koberger, who printed, amongst many other volumes, the Liber Chronicarum. Better know as the Nuremberg Chronicle in English and Die Schedel’sche Weltchronik in German, the world’s first-ever printed encyclopaedia. As always when I develop an interest for a historical topic I try to view it not as isolated incidents but to develop knowledge of and a feeling for the complete historical context, as far as this is possible. This inevitably leads to the acquisition of books on the topic, preferably general, wide ranging, good quality reference books to which I can return as the situation demands. I now have a small, but I think, high-quality collection of books about the book. Last week saw a new addition to this collection Erik Kwakkel’s Books Before Print[1].

Having followed Erik on Twitter for a small eternity, at the same time reading his blog and also having had the pleasure of meeting him in person and hearing him lecture on the subject of the medieval book, I knew his book wouldn’t disappoint and it doesn’t. This is an introduction to the medieval book for people, who like me, have little or no knowledge of them. Basically a modified version of his blog on the subject it consists of short, clear simple chapters on each individual aspect of medieval manuscripts, divided into five sections: 1. Filling the Page: Script, Writing, and Page Design 2. Enhancing the Manuscript: Binding and Decoration 3. Reading in Context: Annotations, Bookmarks, and Libraries 4. The Margins of Manuscript Culture 5. Contextualizing the Medieval Manuscript.

Excellently structured, well written and beautifully illustrated this volume fulfils its intended purpose admirably; it really is everything you wanted to know about the medieval manuscript book and were too afraid to ask.

As I often get asked to recommend books on a given topic and so having started this post I decided to give a small overview of the books that I have and use on the history of the book. As a historian of science my main interest is in the invention of moving type printing, which according to conventional wisdom was one of the major driving forces of the so-called scientific revolution, thus most of the books I have deal primarily with the emergence of the printed book.

The Renaissance Mathematicus book-history-books bookshelf

However, the first book I would recommend is one for the general reader covering the entire history of the book from clay tablets to the modern printed book, Keith Houston’s The Book:A Cover-to-Cover Exploration of the Most Powerful Object of Our Time, which I reviewed here, so I won’t say anything more now. As a small bonus I also recommend Houston’s Shady Characters:The Secret Life of Punctuation, Symbol & Other Typographical Marks[2]. It’s eccentric, unique and a delight.

In his essay in TheCambridge Companion to the History of the Book(of which more later) Adrian Johns writes: “The introduction of Printing into western Europe has counted as the signature event of the history of the book ever since Lucien Febvre and Henri-Jean Martin’s l’Apparition du Livre launched the modern discipline in 1958. The purpose of l’Apparitionwas to demonstrate that Johann Gutenberg’ innovation was the most important turning point in human history, separating modernity from everything before”[3]The Febvre/Martin, The Coming of the Book[4]in English translation is a classic and was the book that introduced me to book history. Although now dated both in its historical facts and its historiography I still think it can be read with profit, although if wishing to quote anything from it one should check against more up to date works.

Next up is another absolute classic Elizabeth Eisenstein’s The printing press as an agent of change[5] probably the most famous and most influential volume on book history. Originally published in two volumes it is now available as a single volume paperback weighing in at just under 800 pages. Eisenstein introduced the concept of print culture, which she contrasts with the preceding age of the manuscript and to which she attributes massive influence (change) not only in the scientific revolution but also in the Reformation, claiming it as an unacknowledged revolution. It is a cornucopia of information, thoughts, ideas and theories that repays careful reading.

However Eisenstein’s central thesis does not go unchallenged. Our next book is Adrian Johns’ equally massive The Nature of the Book.[6] Johns’ sets out his stall thus, “The unifying concept of Eisenstein’s argument is that of “print culture.” This “culture” is characterized primarily in terms of certain traits that print is said to endow on texts. Specifically, those produced in such an environment are subject to conditions of standardization, dissemination, and fixity. The last of these is perhaps the most important.”[7] Johns’ then devotes his 700 plus pages to supposedly proving that Eisenstein’s “print culture” and above her fixity did not exist. Like Eisenstein’s tome it is also a cornucopia of information, thoughts, ideas and theories that repays careful reading. However, I personally don’t think he actually succeeds in proving his central thesis.

The American Historical Review staged a forum[8], introduced by Anthony Grafton, with a defence of her thesis by Eisenstein followed by a response from Johns and then a reply from Eisenstein in which the adversaries mostly argued past each other rather than with each other. However you can read both volumes and the forum and decide for yourself who is right! Happy reading.

If you wanted something shorter than the Eisenstein/Johns debate then you can turn to Andrew Pettegree’s The Book in the Renaissance.[9] Pettegree starts with the book before printing and follows with the invention of printing. He then introduces what he defines as the crisis in printing. This is the fact that there was not a large enough market for the Latin academic and theological texts that was the original fare of the earliest printing houses leading to an economic crisis. Out of this crisis emerged new forms of literature generated by the publishing houses to create new markets to finance their presses. This ‘creation of a European book market’, as he terms it is the central theme of Pettegree’s interesting and stimulating book.

Already mention above, The Cambridge Companion to the History of the Book (see footnote 3) is a collection of papers covering a wide-ranging series of book history topics from a very modern standpoint and is more than worth reading as a supplement to the volumes sketched above.

Another slightly dated but still useful volume is Colin Clair’s A History of European Printing.[10] This is basically an annotated chronology of the spread of the book printing business throughout Europe from its beginnings down to the end of the nineteenth century.

I close with a beautiful volume issued by the Gutenberg-Gesellschaft and Gutenberg-Museum, which is, unfortunately for those who don’t read the language, only available in German, Blockbücher des Mittelalters: Bilderfolgen als Lektüre.[11] Which is a collection of detailed essays on the books printed in Europe in the second half of the fifteenth century with woodblocks, issued as a guide to an exhibition of these books in the Gutenberg-Museum from 22 June to 1 September 1991. The book forms a complete history of this interesting anomaly in the European history of the printed book.

There has been, of course, since Levbre/Martin established the modern book history discipline with their tome in 1958 a vast flood of academic literature on the history of the book in Europe and indeed the world much of which the interested reader can find listed in the very extensive bibliographies of the volumes described above. As I also said above, happy reading!

 

 

[1]Erik Kwakkel, Books Before Print, ARC Humanities Press, Leeds, 2018

[2]Keith Houston, Shady Characters: The Secret Life of Punctuation, Symbol & Other Typographical Marks, W. W. Norton, New York & London, 2013.

[3]Adrian Johns, The coming of print to Europe, in The Cambridge Companion to the History of the Book, ed. Leslie Howsam, CUP, Cambridge, 2015

[4]Lucien Febvre and Henri-Jean Martin, The Coming of the Book, Verso, London & New York, ppb. 1997

[5]Elizabeth L. Eisenstein, The printing press as an agent of change: Communications and cultural transformations in early-modern Europe, CUP, Cambridge et al., ppb. 1980

[6]Adrian Johns, The Nature of the Book: Print and Knowledge in the Making, Chicago University Press, Chicago and London, ppb. 1998

[7]Johns, The Nature of the Book p. 10

[8]American Historical Review: Volume 107, Issue 1, 2002, pp. 84-128

[9]Andrew Pettegree, The Book in the Renaissance, Yale University Press, New Haven & London, ppb. 2011

[10]Colin Clair, A History of European Printing, Academic Press, London, New York, San Francisco, 1976

[11]Blockbücher des Mittelalters: Bilderfolgen als Lektüre, Herausgegeben von Gutenberg-Gesellschaft und Gutenberg-Museum, 1991.

Apples & Pears – comparing print technologies

 

On Facebook I recently stumbled across a link to a piece on 3 Quarks Daily, which in turn was only a lede for a short essay on the London Review of Books entitled, The Oldest Printed Book in the World. This is an article about the Chinese Dunhuang Diamond Sūtra

Frontispiece of the Chinese Diamond Sūtra, the oldest known dated printed book in the world. The colophon, at the inner end, reads: Reverently [caused to be] made for universal free distribution by Wang Jie on behalf of his two parents on the 13th of the 4th moon of the 9th year of Xiantong [i.e. 11th May, CE 868 ] Source: British Library via Wikimedia Commons

 from the ninth century explaining its origin and how it came to be housed in the British Library. The article contains the following sentence:

A colophon at the end of the Dunhuang Diamond Sūtra scroll dates it to 868, nearly six centuries before the first Gutenberg Bible.

Although not stated explicitly the intention of this sentence seems to be, the Chinese invented book printing six hundred years before the Europeans. Although on a very superficial level this is true it is actually a case of comparing apples with pears, as the two books in question are printed with very different reproduction technologies. The Dunhuang Diamond Sūtra is a woodblock print, whereas the Gutenberg Bible is printed with movable type.

First page of the first volume: The Epistle of St. Jerome from the University of Texas copy. Source: Ransom Center of the University of Texas at Austin via Wikimedia Commons

For woodblock printing the image to be printed is carved into a woodblock or rather the parts that are not to be printed are cut away with a knife or chisel. This is then inked and pressed onto the sheet of material, cloth or paper, to be printed. The used block produced by this difficult process can only be used to print this one page. With moveable type the individual pieces of type, or sorts, are composed into the image to be printed, inked and pressed into the sheet of material to be printed. When finished the sorts can be reused to compose a new page and so on. Once cut a set of woodblocks can only be used to print the same book over and over again. A full set of type can be continually reconfigured to print literally thousand of different books. This difference is important and the six hundred year gap throws up some very important and intriguing historical questions.

A case of cast metal type pieces and typeset matter in a composing stick Source: Wikimedia Commons

Central to these is the question of technological transfer. Woodblock printing was developed in East Asia sometime before the third century CE. The oldest fragments of printed cloth date to 220 CE. The oldest woodblock prints on paper date to the late seventh century CE. And as stated above to oldest extant woodblock printed book the Dunhuang Diamond Sūtra dates to 868 CE. Although the Chinese invention of paper arrived in Spain via the Islamic Empire in the late eleventh century CE and crossed the Alps into Northern Europe in the late fourteenth century CE, woodblock printing does not appear to have accompanied it. Strangely European books printed with woodblocks, block books, apparently only appeared after Gutenberg had introduced printing with movable type in the second half of the fifteenth century. There are a very limited number of such books mostly dating from the 1460s and 1470s and printed in the Netherlands of Southern Germany.

Block book – Biblia Pauperum (“Bible of the Poor”) Netherlands 1460s/70s Source: Wikimedia Commons

Gutenberg was by no means the first to use moveable type. Around 1040 CE a Chinese inventor, Bi Sheng (990–1051) invented a form of moveable type with the pieces of type made of ceramics. Beyond a short description of his invention nothing more is known about it and nothing he might have printed has survived. This was followed in East Asia by various other forms of moveable type carved from wood or made of various metals. The oldest book printed with wooden movable type was Records of Jingde County printed by Wang Zhen in 1298. In 1313 he published an account of his invention A method of making moveable wooden types for printing books.

A revolving typecase for wooden type in China, from Wang Zhen’s book published in 1313 Source: Wikimedia Commons

The oldest known book printed with metal moveable type is the two volume Jikji, a collection of excerpts from the analects of revered Buddhist monks, printed with metal type in Korea in 1377; that is at least seventy years before Gutenberg’s famous Bible. However, whereas 49 copies of Gutenberg’s Bible still exist, of which 21 are complete, only one copy of the second volume of the Jikji is still extant.

Korean movable type from 1377 used for the Jikji Source: Wikimedia Commons

Jikji or “Selected Teachings of Buddhist Sages and Seon Masters”, published in 1377, Korea during the Goryeo Dynasty. Source: Wikimedia Commons

Even within Europe Gutenberg was not the first to use moveable type, with several people experimenting with varying system. However Gutenberg was the first to produce anything functional and in reality his greatest inventions were not so much moveable type as the printing press (he converted a wine press) and printing ink or to put it another way he didn’t just invent moveable type but the whole printing process.

Replica of the Gutenberg press at the International Printing Museum in Carson, California Source: Wikimedia Commons

Although extensive effort has been invested into the research on the topic, no evidence has been found of a technology transfer from East Asia to Europe and it is thought that Gutenberg’s was an independent (re)invention.

Although my account is itself only a sketch of the development of printing, both woodblock and moveable type ( I don’t even touch upon book (re)production before woodblock printing or after moveable type), my main argument is that the London Review of Books article in just making its invalid comparison between the Dunhuang Diamond Sūtra and Gutenberg’s Bible creates an inadequate and distorted impression of a long and complex historical process; an impression that uninformed readers will take away with them. A mythical historical meme has been created “the first printed book is the Dunhuang Diamond Sūtra and not the Gutenberg Bible” to replace the Eurocentric myth that Gutenberg invented movable type printing and his Bible is the earliest printed book. If writing short popular historical pieces for the general public we should avoid simplistic descriptions and thereby the risk of creating myths rather than transmitting real knowledge.

 

Two Greek scholars butting heads in the Renaissance and the consequences for astronomy

The adversaries of the title were Georg of Trebizond (1395–1472) and Basilios Bessarion (1403–1472). There is an ironic twist to their names. George of Trebizond derived his name from his ancestors, who originated in the Empire of Trebizond but he was born in Crete. His later antagonist Basilios Bessarion, however, was born in Trebizond.

At sometime unknown point, whilst he was still relatively young, George of Trebizond moved to Italy, where he learnt Latin and acted as amanuensis to the politician Francesco Barbaro (1390–1454) in Venice. A brilliant Aristotelian scholar he entered the entourage of Pope Nicholas V (1397–1455) a convinced Aristotelian.

George of Trebizond Source: Wikimedia commons

Basilios Bessarion was educated in Constantinople then went in 1423 to study Plato under Georgius Gemistus (c.1355–c. 1452), known as Plethon, a highly influential revivalist and teacher of Neo-Platonism. He became an orthodox monk, advancing to abbot in 1436 and metropolitan of Nicaea in 1437. In 1439 he travelled with the Orthodox delegation to Italy to try to persuade the Catholic Church to join the Orthodox Church in a crusade against the Ottoman Turks. Bessarion’s political position led to him being heavily criticised in Byzantium and so he stayed in Italy where Pope Eugene IV (1383–1447) appointed him a cardinal of the Catholic Church. A convinced humanist he devoted his life to spreading support for humanism and to amassing a large private library, containing an extensive collection of Greek manuscripts. He presented his library to the Senate of Venice in 1468 and the 482 Greek manuscripts and 264 Latin manuscripts today still form the core of the St. Mark’s Biblioteca Marciana.

Basilios Bessarion Justus van Gent and Pedro Berruguete Source: Wikimedia Commons

Initially Bessarion and George of Trebizond were friends and Bessarion did much to support his colleague. However in the early 1450s their friendship began to unravel. In that year George undertook a translation from Greek into Latin of Ptolemaeus’ Mathēmatikē Syntaxis or as it is better known the Almagest, as a replacement for Gerard of Cremona’s twelfth-century translation from Arabic.  Bessarion lent him his best Greek manuscript for the purpose and suggested that he used Theon of Alexandria’s Commentary, as a guide. He duly produced his translation and an extensive commentary in nine months finishing in December 1451. His work was hurried, sloppy and strewn with errors and the Pope’s evaluator Jacopo di San Cassiano (ca.1400–ca.1454) judged the work deficient and the Pope, Nicholas V, rejected the dedication. Bessarion took issue with George’s treatment of Theon. The incident ruined George’s reputation and he was forced to flee from Rome.

The situation between the two Greek immigrants escalated when in 1458 George published a vicious attack on Plato in his Comparatio Aristotelis et Platonis, which historian James Hankins has described as “one of the most remarkable mixtures of learning and lunacy ever penned.” In this work he accused Plato of being a traitor to Athens, a besmircher of rhetoric, an advocate of paedophilia, and a pagan who lent aid and comfort to Greek Christians. Bessarion, a Platonist, could not let this stand and issued a powerful response, In calumnatorem Platonis, which was printed in 1469. The situation became even more heated when George offered to dedicate his Commentary on the Almagest to Mehmet II, the Ottoman Turk Sultan, who had conquered Constantinople and ended the Byzantine Empire. George entreated Mehmet to convert to Christianity, to conquer Rome and thus to unite Islam and Christianity under his sovereignty. Bessarion got hold of George’s correspondence with Mehmet and appealled to the Pope, Pius II (for whom George might have been working as an agent!), accusing George of treachery and George was imprisoned for four months in 1466-67. Released from prison, George now offered to dedicate both translation and commentary to Matthias Corvinus (1443–1490), the king of Hungary.

We now need to back peddle to 1460. In that year, Bessarion, who was a Papal legate, visited Vienna to negotiate with Frederick III and made the acquaintance of Georg von Peuerbach (1423–1461), who was at the time the leading astronomical scholar in Europe. Bessarion, still deeply upset by George’s abortive Almagest efforts, asked Peuerbach to produce a new commentary on Ptolemaeus’ work. Peuerbach acquiesced and began immediately to produce an epitome or digest of the Almagest. This was an updated, modernised, shortened, mathematically improved version of the Almagest. Peuerbach died in 1461, having only completed the first six of thirteen book of his epitome. He did, however, extract the deathbed promise from his star pupil, Regiomontanus, to finish the work. In the same year Regiomontanus left Vienna for Italy as a member Bessarion’s entourage, where he spent the next four years learning Greek, finishing the epitome and acting as Bessarion’s manuscript collector and librarian. The Epitome of the Almagestis a masterpiece:

The Epitome is neither a translation (an oft repeated error) nor a commentary but a detailed sometimes updated, overview of the Almagest. Swerdlow once called it “the finest textbook of Ptolemaic astronomy ever written.”[1]

I’ve already written an earlier blog post on Regiomontanus so we don’t need to outline the rest of his life but Shank does have an interesting hypothesis. He suggests that Regiomontanus went to Hungary at Bessarion’s behest in order to counter any influence that George might win at the Court of Corvinus through his second attempt to rededicate his Almagest and Commentary.

Johannes Regiomontanus, Woodcut Source: Wikimedia Commons

When he set up his printing business in Nürnberg, Regiomontanus published Peuerbach’s lectures on astronomy, Theoricae Novae Planetarum, as his first book.

Georg von Peuerbach: Theoricarum novarum planetarum testus, Paris 1515 Source: Wikimedia Commons

Peuerbach Theoricae novae planetarum 1473 Source: Wikimedia Commons

Although he included the Epitome in his publisher’s prospect he didn’t succeed in publishing it before his untimely death in 1476. The Epitoma in Almagestum Ptolemae was first published in 1496 in Venice by Johannes Hamman. Together with Peuerbach’s lectures the Epitome became the standard textbooks for teaching astronomy at the European universities for much of the next century. The influence of the Epitome goes much deeper than this in the history of astronomy.

Title page Epitoma in Almagestum Ptolemae Source: Wikimedia Commons

It is well known that Copernicus modelled his De revolutionibus on Ptolemaeus’ Almagest. In fact text analysis has shown that he actually modelled his magnum opus on the Peuerbach-Regiomontanus Epitome, for example taking most of his knowledge of Arabic astronomy from Regiomontanus’ work. This is, however, rather minor compared to what several expert think is the most important influence that Regiomontanus had on Copernicus.

Nicolaus Copernicus portrait from Town Hall in Toruń – 1580 Source: Wikimedia Commons

According to ancient Greek cosmology the planets orbit the earth with uniform circular motion. Any extended observation of the planets show that this is not the case and it was the job of the astronomers to construct geometrical model, which corrected the visible deviation from the cosmological norm; these deviations are known as the anomalies. Ptolemaeus had basically two geometrical tools to describe planetary orbits. With the eccentric deferent the centre of the circle that describes the orbit, the deferent, is not in the same position as the earth, i.e. the earth is not at the centre of the planets orbit. The alternative is the epicycle-deferent model in which the planet is carried around an epicycle, which is itself carried around the deferent. The mathematician Apollonius (late 3^rdcentury–early 2^ndcentury BCE) had shown that the two models were in fact mathematically equivalent; meaning any motion that could be described with the one model could equally well be described with the other.

Ptolemaeus, however, argued in the Almagest that whereas the retrograde motion (the so-called second anomaly, when the planet appears to reverse its orbital direction for a period of time) of the outer planets could be described with either model that of the inner planets (Venus and Mercury) could only be described with the epicycle-deferent model. In Book XII of the Epitome, Regiomontanus proved that the second anomaly of the inner planets could also be described with the eccentric deferent model. Without going into detail this seems to have led Copernicus directly to his heliocentric system for the inner planets, which he then extended to the outer ones.

Thinking hypothetically, if George had not written his translation of and commentary on the Almagest, then Bessarion would not has asked Peuerbach to write the Epitomeand Regiomontanus might never have provided Copernicus with that vital clue.

Regiomontanus wrote a second book inspired by George’s work. His Defensio Theonis contra Georgium Trapezuntium is a vast rambling mathematical work centred on a defence of Theon of Alexandria against what he saw as George’s unfair treatment of him. He accused George as having both misrepresenting Theon and plagiarising him. This work has never been published but Regiomontanus’ antagonism against George was known at the time. The Defensio was announced in Regiomontanus’ prospect and also in works published by Erhard Ratdolt. This situation led to a rather strange claim made by Pierre Gassendi. In the 1650s Gassendi published a collective biography of the great astronomers Brahe, Copernicus, Regiomontanus etc. in which he claimed that Regiomontanus was murdered in Rome by two of George’s sons in 1476. George had many vocal critics, none of whom were murdered and sensible historians think that Regiomontanus died in one of the epidemics that regularly swept Rome.

 

[1]Michael H. Shank, Regiomontanus and Astronomical Controversy in the Background of Copernicus, pp. 79-109 in Rivka Feldhay and F. Jamil Ragep eds., Before Copernicus: The Cultures and Contexts of Scientific Learning in the Fifteenth Century, McGill-Queen’s University Press, Montreal& Kingston, London, Chicago, 2017, p. 90

This blog post owes much to the above paper and to Michael H. Shank, The Almagest, Politics, and Apocalypticism in the Conflict between George of Trebizond and Cardinal Bessarion, in Almagest International Journal for the History of Scientific Ideas, Volume 8, Issue 2, 2017, pp. 49-83

Conversations in a sixteenth century prison cell

Science writer Michael Brooks has thought up a delightful conceit for his latest book.* The narrative takes place in a sixteenth century prison cell in Bologna in the form of a conversation between a twenty-first century quantum physicist (the author) and a Renaissance polymath. What makes this conversation particularly spicy is that the Renaissance polymath is physician, biologist, chemist, mathematician, astronomer, astrologer, philosopher, inventor, writer, auto-biographer, gambler and scoundrel Girolamo Cardano, although Brooks calls him by the English translation of his name Jerome. In case anybody is wondering why I listed autobiographer separately after writer, it is because Jerome was a pioneer in the field writing what is probably the first autobiography by a mathematician/astronomer/etc. in the Early Modern Period.

Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews. Source: Wikimedia Commons

So what do our unlikely pair talk about? We gets fragments of conversation about Jerome’s current situation; a broken old man rotting away the end of his more than extraordinary life in a prison cell with very little chance of reprieve. This leads to the visitor from the future, relating episodes out of that extraordinary life. The visitor also picks up some of Jerome’s seemingly more strange beliefs and relates them to some of the equally, seemingly strange phenomena of quantum mechanics. But why should anyone link the misadventures of an, albeit brilliant, Renaissance miscreant to quantum mechanics. Because our author sees Jerome the mathematician, and he was a brilliant one, as the great-great-great-great-great-great-great-great-great-great-great-great-great grandfather of quantum mechanics!

As most people know quantum mechanics is largely non-deterministic in the conventional sense and relies heavily on probability theory for its results. Jerome wrote the first mathematical tome on probability theory, a field he entered because of his professional gambling activities. He even included a section about how to cheat at cards. I said he was a scoundrel. The other thing turns up in his Ars Magna (printed and published by Johannes Petreius the publisher of Copernicus’ De revolutionibus in Nürnberg and often called, by maths historians, the first modern maths book); he was the first person to calculate with so-called imaginary numbers. That’s numbers using ‘i’ the square root of minus one. Jerome didn’t call it ‘i’ or the numbers imaginary, in fact he didn’t like them very much but realised one could use them when determining the roots of cubic equation, so, holding his nose, that is exactly what he did. Like probability theory ‘i’ plays a very major role in quantum mechanics.

What Michael Brooks offers up for his readers is a mixture of history of Renaissance science together with an explanation of many of the weird phenomena and explanations of those phenomena in quantum mechanics. A heady brew but it works; in fact it works wonderfully.

This is not really a history of science book or a modern physics science communications volume but it’s a bit of both served up as science entertainment for the science interested reader, lay or professional. Michael Brooks has a light touch, spiced with some irony and a twinkle in his eyes and he has produced a fine piece of science writing in a pocket-sized book just right for that long train journey, that boring intercontinental flight or for the week in hospital that this reviewer used to read it. If this was a five star reviewing system I would be tempted to give it six.

*  Michael Brooks, The Quantum Astrologer’s Handbook, Scribe, Melbourne & London, 2017

Exposing Galileo’s strawmanning

There is a widespread, highly erroneous, popular perception in the world, much loved by gnu atheists and supporters of scientism, that as soon as Petreius published Copernicus’s De Revolutionibus in 1543 the question as to which was the correct astronomical/cosmological system for the cosmos was as good as settled and that when Galileo published his Dialogo[1] everything was finally done and dusted and anybody who still persisted in opposing the acceptance of the heliocentric world view, did so purely on grounds of ignorant, anti-science, religious prejudice. Readers of this blog will know that I have expended a certain amount of energy and several thousand words over the years countering this totally mistaken interpretation of the history of astronomy in the early modern period and today I’m going to add even more words to the struggle.

Galileo is held up by his numerous acolytes as a man of great scientific virtue, who preached a gospel of empirical scientific truth in the face of the superstitious, faith based errors of his numerous detractors; he was a true martyr for science. The fact that Galileo was capable of scientific skulduggery does not occur to them, but not only was he capable of such, his work is littered with examples of it. One of his favourite tactics was not to present his opponents true views when criticising them but to create a strawman, claiming that this represents the views of his opponent and then to burn it down with his always-red-hot rhetorical flamethrower.

Towards the end of The First Day in the Dialogo, Galileo has Simplicio, the fall guy for geocentricity, introduce a “booklet of theses, which is full of novelties.” Salviati, who is the champion of heliocentricity and at the same time Galileo’s mouthpiece, ridicules this booklet as producing arguments full of “falsehoods and fallacies and contradictions” and as “thinking up, one by one, things that would be required to serve his purposes, instead of adjusting his purposes step by step to things as they are.” Galileo goes on to do a polemical hatchet job on what he claims are the main arguments in said “booklet of theses.” Amongst others he accuses the author of “setting himself up to refute another’s doctrine while remaining ignorant of the basic foundations upon which the whole structure are supported.”

The “booklet of theses”, which Galileo doesn’t name, is in fact the splendidly titled:

English translation of the Latin title page Source: Notre Dame Press

Now most of the acolytes who fervently praise Galileo as the great defender of science against superstition probably have no idea who Johann Georg Locher was but they might well have heard of Christoph Scheiner, who was famously embroiled in a dispute with Galileo over the nature of sunspots and who first observed them with a telescope. In fact the authorship of the Mathematical Disquisitions has often falsely attributed to Scheiner and Galileo’s demolition of it seen as an extension of that dispute and it’s sequel in the pages of his Il Saggiatore.

Whereas Galileo’s Dialogo has been available to the general reader in a good English translation by Stillman Drake since 1953, anybody who wished to consult Locher’s Mathematical Disquisitions in order to check the veracity or lack thereof of Galileo’s account would have had to hunt down a 17^th century Latin original in the rare books room of a specialist library. The playing field has now been levelled with the publication of an excellent modern English translation of Locher’s booklet by Renaissance Mathematicus friend, commentator and occasional guest contributor Chris Graney[2].

Graney’s translation (Christopher M. Graney, Mathematical Disquisitions: The Booklet of Theses Immortalised by Galileo, University of Notre Dame Press, Notre Dame, Indiana, 2017)  presents a more than somewhat different picture of Locher’s views on astronomy to that served up by Galileo in the Dialogo and in fact gives us a very clear picture of the definitely rational arguments presented by the opponents to heliocentricity in the early part of the seventeenth century. The translation contains an excellent explanatory introduction by Graney, extensive endnotes explaining various technical aspects of Locher’s text and also some of the specific translation decisions of difficult terms. (I should point out that another Renaissance Mathematicus friend, Darin Hayton acted as translation consultant on this volume). There is an extensive bibliography of the works consulted for the explanatory notes and an excellent index.

The book is very nicely presented by Notre Dame Press, with fine reproductions of Locher’s wonderful original illustrations.

Locher’s illustration to his discussion of diurnal rotation p. 32

Graney’s translation provides a great addition to his previous Setting Aside All Authority, which I reviewed here. Graney is doing sterling work in adjusting the very distorted view of the astronomical system discussion in the first half of the seventeenth century and anybody, who is seriously interested in learning the true facts of that discussion, should definitely read his latest contribution.

 

 

 

[1] By a strange cosmic coincidence the first printed copy of the Dialogo was presented to the dedicatee Ferdinando II d’Medici, Grand Duke of Tuscany 386 years ago today on 22 February 1632.

[2] At the end of my review of Setting Aside All Authority I wrote the following, which applies equally to this review; in this case I provided one of the cover blurbs for Chris’ latest book.

Disclosure; Chris Graney is not only a colleague, but he and his wife, Christina, are also personal friends of mine. Beyond that, Chris has written, at my request, several guest blogs here at the Renaissance Mathematicus, all of which were based on his research for the book. Even more relevant I was, purely by accident I hasten to add, one of those responsible for sending Chris off on the historical trail that led to him writing this book; a fact that is acknowledged on page xiv of the introduction. All of this, of course, disqualifies me as an impartial reviewer of this book but I’m going to review it anyway. Anybody who knows me, knows that I don’t pull punches and when the subject is history of science I don’t do favours for friends. If I thought Chris’ book was not up to par I might refrain from reviewing it and explain to him privately why. If I thought the book was truly bad I would warn him privately and still write a negative review to keep people from wasting their time with it. However, thankfully, none of this is the case, so I could with a clear conscience write the positive review you are reading. If you don’t trust my impartiality, fair enough, read somebody else’s review.

Addendum: The orthography of the neologism in the title was change—23,02,18— following a straw pole on Twitter

Christmas Trilogy 2017 Part 3: Kepler’s big book

Johannes Kepler was incredibly prolific, he published over eighty books and booklets over a very wide range of scientific and mathematical topics during his life. As far as he was concerned his magnum opus was his Ioannis Keppleri Harmonices mundi libri V (The Five Books of Johannes Kepler’s The Harmony of the World) published in 1619 some twenty years after he first conceived it. Today in popular #histsci it is almost always only mentioned for the fact that it contains the third of his laws of planetary motion, the harmonic law. However it contains much, much more of interest and in what follows I will attempt to give a brief sketch of what is in fact an extraordinary book.

A brief glace at the description of the ‘five books’ thoughtfully provided by the author on the title page (1) would seem to present a mixed bag of topics apparently in some way connected by the word or concept harmonic. In order to understand what we are being presented with we have to go back to 1596 and Kepler’s first book Mysterium Cosmographicum (The Cosmographic Mystery). In this slim volume Kepler presents his answer to the question, why are there only six planets? His, to our eyes, surprising answer is that the spaces between the planets are defined by the regular so-called Platonic solids and as the are, and can only be, five of these there can only be six planets.

Keplers model of the cosmos with the five Platonic solids

Using the data from the greatest and least distances between the planets in the Copernican system, Kepler’s theory produces an unexpectedly accurate fit. However the fit is not actually accurate enough and in 1598 Kepler began working on a subsidiary hypothesis to explain the inaccuracies. Unfortunately, the book that he had planned to bring out in 1599 got somewhat delayed by his other activities and obligations and didn’t appear until 1619 in the form of the Harmonice mundi.

The hypothesis that Kepler presents us with is a complex mix of ideas taken from Pythagoras, Plato, Euclid, Proclus and Ptolemaeus centred round the Pythagorean concept of the harmony of the spheres. Put very simply the theory developed by the Pythagoreans was that the seven planets (we are talking geocentric cosmology here) in their orbits form a musical scale than can, in some versions of the theory, only be heard by the enlightened members of the Pythagorean cult. This theory was developed out of the discovery that consonances (harmonious sounds) in music can be expressed in the ratio of simple whole numbers to each other (the octave for example is 1:2) and the Pythagorean belief that the integers are the building block of the cosmos.

This Pythagorean concept winds its way through European intellectual history, Ptolemaeus wrote a book on the subject, his Harmonice and it is the reason why music was one of the four disciplines of the mathematical quadrivium along with arithmetic, geometry and astronomy. Tycho Brahe designed his Uraniburg so that all the architectonic dimensions from the main walls to the window frames were in Pythagorean harmonic proportion to one another.

Tycho Brahe’s Uraniborg Blaeus Atlas Maior 1663 Source: Wikimedia Commons

It is also the reason why Isaac Newton decided that there should be seven colours in the rainbow, to match the seven notes of the musical scale. David Gregory tells us that Newton thought that gravity was the strings upon which the harmony of the spheres was played.

In his Harmony Kepler develops a whole new theory of harmony in order to rescue his geometrical vision of the cosmos. Unlike the Pythagoreans and Ptolemaeus who saw consonance as expressed by arithmetical ratios Kepler opted for a geometrical theory of consonance. He argued that consonances could only be constructed by ratios between the number of sides of regular polygons that can be constructed with a ruler and compass. The explication of this takes up the whole of the first book. I’m not going to go into details but interestingly, as part of his rejection of the number seven in his harmonic scheme Kepler goes to great lengths to show that the heptagon construction given by Dürer in his Underweysung der Messung mit dem Zirckel und Richtscheyt is only an approximation and not an exact construction. This shows that Dürer’s book was still being read nearly a hundred years after it was originally published.

In book two Kepler takes up Proclus’ theory that Euclid’s Elements builds systematically towards the construction of the five regular or Platonic solids, which are, in Plato’s philosophy, the elemental building blocks of the cosmos. Along the way in his investigation of the regular and semi-regular polyhedra Kepler delivers the first systematic study of the thirteen semi-regular Archimedean solids as well as discovering the first two star polyhedra. These important mathematical advances don’t seem to have interested Kepler, who is too involved in his revolutionary harmonic theory to notice. In the first two books Kepler displays an encyclopaedic knowledge of the mathematical literature.

The third book is devoted to music theory proper and is Kepler’s contribution to a debate that was raging under music theorist, including Galileo’s father Vincenzo Galilei, about the intervals on the musical scale at the beginning of the seventeenth century. Galilei supported the so-called traditional Pythagorean intonation, whereas Kepler sided with Gioseffo Zarlino who favoured the ‘modern’ just intonation. Although of course Kepler justification for his stance where based on his geometrical arguments. Another later participant in this debate was Marin Mersenne.

In the fourth book Kepler extends his new theory of harmony to, amongst other things, his astrology and his theory of the astrological aspects. Astrological aspects are when two or more planets are positioned on the zodiac or ecliptic at a significant angle to each other, for example 60° or 90°. In his Harmonice, Ptolemaeus, who in the Renaissance was regarded as the prime astrological authority, had already drawn a connection between musical theory and the astrological aspects; here Kepler replaces Ptolemaeus’ theory with his own, which sees the aspects are being derived directly from geometrical constructions. Interestingly Kepler, who had written and published quite extensively on astrology, rejected nearly the whole of traditional Greek astrology as humbug keeping only his theory of the astrological aspects as the only valid form of astrology. Kepler’s theory extended the number of influential aspects from the traditional five to twelve.

The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. He finally has success with the angular velocities of the planets in their orbits at perihelion and aphelion. He then examines the relationships between the tones thus generated by the different planets, constructing musical scales in the process. What he in missing in all of this is a grand unifying concept and this lacuna if filled by his harmonic law, his third law of planetary motion, P₁²/P₂²=R₁³/R₂³.

There is an appendix, which contains Kepler’s criticisms of part of Ptolemaeus’ Harmonice and Robert Fludd’s harmony theories. I blogged about the latter and the dispute that it triggered in an earlier post

With his book Kepler, who was a devoted Christian, was convinced that he had revealed the construction plan of his geometrical God’s cosmos. His grandiose theory became obsolete within less than fifty years of its publication, ironically pushed into obscurity by intellectual forces largely set into motion by Kepler in his Astronomia nova, his Epitome astronomiae Copernicanae and the Rudolphine Tables. All that has survived of his great project are his mathematical innovations in the first two books and the famous harmonic law. However if readers are prepared to put aside their modern perceptions and prejudices they can follow one of the great Renaissance minds on a fascinating intellectual journey into his vision of the cosmos.

(1) All of the illustration from the Harmonice mundi in this post are taken from the English translation The Harmy of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997