On the road to recovery?

Two weeks ago, I wrote a short piece explaining why I hadn’t posted my usual weekly blog post the day before and warning that I might not have recovered enough to write one for the following Wednesday, and so it came to pass. Much too late, I noticed through the general fog of a heavy virus infection of the head that I was deaf in my right ear, so last Tuesday it was off to the ear, nose, and throat specialist. It turns out that I have a heavy middle ear infection/inflammation, so a new course of treatment and a return to the doctor a week later. Well readers it didn’t work. So new course of treatment starting yesterday.

For the last ten days, or longer, I’ve had tinnitus, I’m deaf on my right ear and not hearing too good on the left. My head feels like it’s stuffed with cotton wool and my sense of balance is almost non-existent. Not the ideal situation to write intelligent #histSTM posts. I’m actually feeling better than I did a week ago and, with the hope that the new course of medication is going to work, I am slowly working towards resuming posting next Wednesday.

Till then you’ll have to content yourself with reading some old one!


Filed under Autobiographical

Shit Happens!

Some of you might have noticed that yesterday was Wednesday and there was no new substantive post here at the Renaissance Mathematicus. Did anybody notice, probably not. The explanation for this absence is quite simple, on the one hand pressure of work, on the other, I’m ill. I was to hold a public lecture yesterday, to which I will return in a minute, and was working hard to first decide what slides to make and secondly to make them and writing a blog post was definitely in second place on my schedule. Unfortunately, my health was deteriorating during the week, and I was running out of energy. By the time the weekend came round, when I might have written the planned post, I was definitely under the weather and had basically given up the idea of even attempting to write it. 

By Monday, I was desperately holding on and hoping I would still make it to my lecture on Wednesday. On Tuesday morning, I realised that my lecture was not going to happen, so I told the promoter and asked the local newspaper, who had published an interview with me to advertise the lecture, to publish a notice on Wednesday morning announcing that it had been cancelled. By Tuesday evening, my throat felt like somebody had cleaned it with a cheese grater, I could hardly talk and had difficulties negotiating the way from my living room into the kitchen, all of five metres. Wednesday morning, a friend drove me to my doctor, I couldn’t have made it under my own steam, where my very sweet lady doctor decided it was a virus and not a bacterium and so no antibiotics. Instead, a list of over-the-counter medicines to relieve the symptoms that cost me €33! Some symptoms have alleviated but I still feel like shit warmed up.

The missed lecture was a personal disaster in more than one way. For many years I have been part of a group that organises a series of history of science lectures every autumn for the local adult education classes. I suggested this year’s topic, the history of the involvement of Nürnberg’s printer/publishers in the very early phase of the printed scientific book. Together with my friend Pierre, we planned the programme and found experts to deliver each of the lectures which took a fair amount of time and effort. The lectures started four weeks ago, and the first three background lectures–the invention of paper, the Chinese invention of movable type printing, and Gutenberg’s reinvention of movable type printing–were all excellent and now it was my turn to introduce Regiomontanus and the world’s first scientific press. As the title says Shit Happens! 

I’m not sure if I will find the energy to write a blog post for next Wednesday, I’m struggling to finish this brief note, we will just have to wait and see. 


Filed under Autobiographical


Due to the impact of Isaac Newton and the mathematicians grouped around him, people often have a false impression of the role that England played in the history of the mathematical sciences during the Early Modern Period. As I have noted in the past, during the late medieval period and on down into the seventeenth century, England in fact lagged seriously behind continental Europe in the development of the mathematical sciences both on an institutional level, principally universities, and in terms of individual mathematical practitioners outside of the universities. Leading mathematical practitioners, working in England in the early sixteenth century, such as Thomas Gemini (1510–1562) and Nicolas Kratzer (1486/7–1550) were in fact immigrants, from the Netherlands and Germany respectively.

In the second half of the century the demand for mathematical practitioners in the fields of astrology, astronomy, navigation, cartography, surveying, and matters military was continually growing and England began to produce some home grown talent and take the mathematical disciplines more seriously, although the two universities, Oxford and Cambridge still remained aloof relying on enthusiastic informal teachers, such as Thomas Allen (1542–1632) rather than instituting proper chairs for the study and teaching of mathematics.

Outside of the universities ardent fans of the mathematical disciplines began to establish the so-called English school of mathematics, writing books in English, giving tuition, creating instruments, and carrying out mathematical tasks. Leading this group were the Welsh man, Robert Recorde (c. 1512–1558), who I shall return to in a later post, John Dee (1527–c. 1608), who I have dealt with in several post in the past, one of which outlines the English School, other important early members being, Dee’s friend Leonard Digges, and his son Thomas Digges (c. 1446–1595), who both deserve posts of their own, and Thomas Hood (1556–1620) the first officially appointed lecturer for mathematics in England.  I shall return to give all these worthy gentlemen, and others, the attention they deserve but today I shall outline the life and mathematical career of John Blagrave (d. 1611) a member of the landed gentry, who gained a strong reputation as a mathematical practitioner and in particular as a designer of mathematical instruments, the antiquary Anthony à Wood (1632–1695), author of Athenae Oxonienses. An Exact History of All the Writers and Bishops, who Have Had Their Education in the … University of Oxford from the Year 1500 to the End of the Year 1690, described him as “the flower of mathematicians of his age.”

John Blagrave was the second son of another John Blagrave of Bullmarsh, a district of Reading, and his wife Anne, the daughter of Sir Anthony Hungerford of Down-Ampney, an English soldier, sheriff, and courtier during the reign of Henry VIII, John junior was born into wealth in the town of Reading in Berkshire probably sometime in the 1560s. He was educated at Reading School, an old established grammar school, before going up to St John’s College Oxford, where he apparently acquired his love of mathematics. This raises the question as to whether he was another student, who benefitted from the tutoring skills of Thomas Allen (1542–1632). He left the university without graduating, not unusually for the sons of aristocrats and the gentry. He settled down in Southcot Lodge in Reading, an estate that he had inherited from his father and devoted himself to his mathematical studies and the design of mathematical instruments. He also worked as a surveyor and was amongst the first to draw estate maps to scale.

Harpsden a small parish near Henley-on-Thames Survey by John Blagrave 1589 Source

There are five known surviving works by Blagrave and one map, as opposed to a survey, of which the earliest his, The mathematical ievvel, from1585, which lends its name to the title of this post, is the most famous. The full title of this work is really quite extraordinary:


Shewing the making, and most excellent vse of a singuler Instrument So called: in that it performeth with wonderfull dexteritie, whatsoever is to be done, either by Quadrant, Ship, Circle, Cylinder, Ring, Dyall, Horoscope, Astrolabe, Sphere, Globe, or any such like heretofore deuised: yea or by most Tables commonly extant: and that generally to all places from Pole to Pole. 

The vse of which Ievvel, is so aboundant and ample, that it leadeth any man practising thereon, the direct pathway (from the first steppe to the last) through the whole Artes of Astronomy, Cosmography, Geography, Topography, Nauigation, Longitudes of Regions, Dyalling, Sphericall triangles, Setting figures, and briefely of whatsoeuer concerneth the Globe or Sphere: with great and incredible speede, plainenesse, facillitie, and pleasure:

The most part newly founde out by the Author, Compiled and published for the furtherance, aswell of Gentlemen and others desirous or Speculariue knowledge, and priuate practise: as also for the furnishing of such worthy mindes, Nauigators,and traueylers,that pretend long voyages or new discoueries: By John Blagave of Reading Gentleman and well willer to the Mathematickes; Who hath cut all the prints or pictures of the whole worke with his owne hands. 1585•

Dig the spelling!
Title Page Source Note the title page illustration is an  armillary sphere and not the Mathematical Jewel

Blagrave’s Mathematical Jewel is in fact a universal astrolabe, and by no means the first but probably the most extensively described. The astrolabe is indeed a multifunctional instrument, al-Sufi (903–983) describes over a thousand different uses for it, and Chaucer (c. 1340s–1400) in what is considered to be the first English language description of the astrolabe and its function, a pamphlet written for a child, describes at least forty different functions. However, the normal astrolabe has one drawback, the flat plates, called tympans of climata, that sit in the mater and are engraved with the stereographic projection of a portion of the celestial sphere are limited in their use to a fairly narrow band of latitude, meaning that if one wishes to use it at a different latitude you need a different climata. Most astrolabes have a set of plates each engraved on both side for a different band of latitude. This problem led to the invention of the universal astrolabe.

Full-page figure of the rete of Blagrave’s Jewel (Peterborough A.8.13) For more illustration from The Mathematical Jewel go here

The earliest known universal astrolabes are attributed to Abū Isḥāq Ibrāhīm ibn Yaḥyā al-Naqqāsh al-Zarqālī al-Tujibi (1029-1100), known simply as al-Zarqālī and in Latin as Arzachel, an Arabic astronomer, astrologer, and instrument maker from Al-Andalus, and another contemporary Arabic astronomer, instrument maker from Al-Andalus, Alī ibn Khalaf: Abū al‐Ḥasan ibn Aḥmar al‐Ṣaydalānī or simply Alī ibn Khalaf, about whom very little is known. In the Biographical Encyclopedia of Astronomers (Springer Reference, 2007, pp. 34-35) Roser Puig has this to say about the two Andalusian instrument makers: 

ʿAlī ibn Khalaf is the author of a treatise on the use of the lámina universal (universal plate) preserved only in a Spanish translation included in the Libros del Saber de Astronomía (III, 11–132), compiled by the Spanish King Alfonso X. To our knowledge, the Arabic original is lost. ʿAlī ibn Khalaf is also credited with the construction of a universal instrument called al‐asṭurlāb al‐maʾmūnī in the year 1071, dedicated to al‐Maʾmūn, ruler of Toledo. 

The universal plate and the ṣafīḥa (the plate) of Zarqalī (devised in 1048) are the first “universal instruments” (i.e., for all latitudes) developed in Andalus. Both are based on the stereographic meridian projection of each hemisphere, superimposing the projection of a half of the celestial sphere from the vernal point (and turning it) on to the projection of the other half from the autumnal point. However, their specific characteristics make them different instruments.

Al-Zarqālī’s universal astrolabe was known as the Azafea in Arabic and as the Saphaea in Europe.

A copy of al-Zarqālī’s astrolabe Source: Wikimedia Commons

Much closer to Blagrave’s time, Gemma Frisius (1508–1555) wrote about a universal astrolabe, published as the Medici ac Mathematici de astrolabio catholico liber quo latissime patientis instrumenti multiplex usus explicatur, in 1556. Better known than Frisius’ universal instrument was that of his one-time Spanish, student Juan de Rojas y Samiento (fl. 1540-1550) published in his Commentariorum in Astrolabium libri sex in 1551.


Although he never really left his home town of Reading and his work was in English, Blagrave, like the other members of the English School of Mathematics, was well aware of the developments in continental Europe and he quotes the work of leading European mathematical practitioners in his Mathematical Jewel, such as the Tübingen professor of mathematics, Johannes Stöffler (1452–1531), who wrote a highly influential volume on the construction of astrolabes, his Elucidatio fabricae ususque astrolabii originally published in 1513, which went through 16 editions up to 1620

or the works of Gemma Frisius, who was possibly the most influential mathematical practitioner of the sixteenth century. Blagrave’s Mathematical Jewel was based on Gemma Frisius astrolabio catholico.

Blagrave’s Mathematical Jewel was obviously popular because Joseph Moxon (1627–1691), England first specialist mathematical publisher, cartographer, instrument, and globe maker republished it under the title:

The catholique planisphaer which Mr. Blagrave calleth the mathematical jewel briefly and plainly discribed in five books : the first shewing the making of the instrument, the rest shewing the manifold vse of it, 1. for representing several projections of the sphere, 2. for resolving all problemes of the sphere, astronomical, astrological, and geographical, 4. for making all sorts of dials both without doors and within upon any walls, cielings, or floores, be they never so irregular, where-so-ever the direct or reflected beams of the sun may come : all which are to be done by this instrument with wonderous ease and delight : a treatise very usefull for marriners and for all ingenious men who love the arts mathematical / by John Palmer … ; hereunto is added a brief description of the cros-staf and a catalogue of eclipses observed by the same I.P.

Engraved frontispiece to John Palmer (ed.), ‘The Catholique Planispaer, which Mr Blagrave calleth the Mathematical Jewel’ (London, Joseph Moxon, 1658); woman, wearing necklace, bracelet, jewels in her hair, and a veil, and seated at a table, on which are a design of a mathematical sphere, a compass, and an open book; top left, portrait of John Blagrave, wearing a ruff; top right, portrait of John Palmer; top centre, an angel with trumpets.
Engraving David Loggan Source: British Museum

John Palmer (1612-1679), who was apparently rector of Ecton and archdeacon of Northampton, is variously described as the author or the editor of the volume, which was first published in 1658 and went through sixteen editions up to 1973.

Following The Mathematical Jewel, Blagrave published four further books on scientific instruments that we know of: 

Baculum Familliare, Catholicon sive Generale. A Booke of the making and use of a Staffe, newly invented by the Author, called the Familiar Staffe (London, 1590)

Astrolabium uranicum generale, a necessary and pleasaunt solace and recreation for navigators … compyled by John Blagrave (London, 1596)

An apollogie confirmation explanation and addition to the Vranicall astrolabe (London, 1597)

None of these survive in large numbers.

Blagrave also manufactured sundials and his fourth instrument book is about this: 

The art of dyalling in two parts (London, 1609)


Here there are considerably more surviving copies and even a modern reprint by Theatrum Orbis Terrarum Ltd., Da Capo Press, Amsterdam, New York, 1968.

People who don’t think about it tend to regard books on dialling, that is the mathematics of the construction and installation of sundials, as somehow odd. However, in this day and age, when almost everybody walks around with a mobile phone in their pocket with a highly accurate digital clock, we tend to forget that, for most of human history, time was not so instantly accessible. In the Early Modern period, mechanical clocks were few and far between and mostly unreliable. For time, people relied on sundials, which were common and widespread. From the invention of printing with movable type around 1450 up to about 1700, books on dialling constituted the largest genre of mathematical books printed and published. Designing and constructing sundials was a central part of the profession of mathematical practitioners. 

As well as the books there is one extant map:

Noua orbis terrarum descriptio opti[c]e proiecta secundu[m]q[ue] peritissimos Anglie geographos multis ni [sic] locis castigatissima et preceteris ipsiq[ue] globo nauigationi faciliter applcanda [sic] per Ioannem Blagrauum gen[er]osum Readingensem mathesibus beneuolentem Beniamin Wright Anglus Londinensis cµlator anno Domini 1596 

This is described as:

Two engraved maps, the first terrestrial, the second celestial (“Astrolabium uranicum generale …”). Evidently intended to illustrate Blagrave’s book “Astrolabium uranicum generale” but are not found in any copy of the latter.
The original is in the Bodleian Library.

When he died in 1611, Blagrave was buried in the St Laurence Church in Reading with a suitably mathematical monument. 

Blagrave is depicted surrounded by allegorical mathematical figures, with five women each holding the five platonic solids and Blagrave (in the center) depicted holding a globe and a quadrant.
The monument was the work of the sculptor Gerard Christmas (1576–1634), who later in life was appointed carver to the navy. It is not known who produced the drawing of the monument. 
Modern reconstruction of the armillary sphere from the cover of The Mathematical Jewel created by David Harber a descendent of John Blagrave

Blagrave was a minor, but not insignificant, participant in the mathematical community in England in the late sixteenth century. His work displays the typical Renaissance active interest in the practical mathematical disciplines, astronomy, navigation, surveying, and dialling. He seems to have enjoyed a good reputation and his Mathematical Jewel appears to have found a wide readership.  


Filed under Early Scientific Publishing, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, Renaissance Science

Renaissance science – XLVI

One area that is not usually counted among the sciences is cryptography, lying as it does, in this day and age, between, logic, mathematics, and informatics. In earlier times it is perhaps best viewed as a part of logic. Perhaps surprisingly, cryptography underwent a major development during the Renaissance provoked by an earlier development in the hands of Islamicate scholars.

Cryptography means literally hidden writing, coming from the Greek kryptos meaning hidden and graphiameaning write, express in written characters, so codes. We have very little evidence of the use of codes by the ancient Egyptians, Babylonians, or Greeks, although it can be assumed that they did so. It is known that the ancient Greeks sent secret messages by shaving the head of a slave, writing, or tattooing the message on their skull, and then waiting until the hair grew back. Not really encryption and anything but high-speed communication.

The most well-known system of encryption from antiquity is the Caesar cipher, used by Julius Caesar in his correspondence. This is a very simple substitution code in which each letter in the plain text is replaced by the letter so many places before or after it in the alphabet. Up till the Middle Ages, in Europe all codes were some form of simple substitution code.

The action of a Caesar cipher is to replace each plaintext letter with a different one a fixed number of places down the alphabet. The cipher illustrated here uses a left shift of three, so that (for example) each occurrence of E in the plaintext becomes B in the ciphertext: Wikipedia

The first significant work on cryptography was written the Arabic philologist Abu ‘Abd ar-Raḥmān al-Khalīl ibn Aḥmad ibn ‘Amr ibn Tammām al-Farāhīdī al-Azdī al-Yaḥmadī (718–786 CE) known as Al-Farāhīdī or Al-Khalīl. His book Kitab al-Muamma (Book of Cryptographic Messages), which has been lost, presents the use of permutations and combinations to list all possible Arabic words with and without vowel. 

Sculpture of al-Farahidi in Basra Source: Wikimedia Commons

Al-Khalīl’s book influenced the work on cryptography by the Arabic polymath Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (c. 801–873), whose theory of radiations played a significant role in the history of optics. In his book Risāla fī Istikhrāj al-Kutub al-Mu’ammāh (On Extracting Obscured Correspondence), al-Kindī introduced the concept of frequency analysis, which made all simple substitute codes accessible to solution. He wrote: 

One way to solve an encrypted message, if we know its language, is to find a different plaintext of the same language long enough to fill one sheet or so, and then we count the occurrences of each letter. We call the most frequently occurring letter the “first”, the next most occurring letter the “second”, the following most occurring letter the “third”, and so on, until we account for all the different letters in the plaintext sample. Then we look at the cipher text we want to solve and we also classify its symbols. We find the most occurring symbol and change it to the form of the “first” letter of the plaintext sample, the next most common symbol is changed to the form of the “second” letter, and the following most common symbol is changed to the form of the “third” letter, and so on, until we account for all symbols of the cryptogram we want to solve.

The first page of al-Kindi’s manuscript “On Deciphering Cryptographic Messages”, containing the oldest known description of cryptanalysis by frequency analysis. Source: Wikimedia Commons

Ai-Kindī’s explication of frequency analysis meant that a new more complex approach to encryption was necessary, and this was first delivered by the Renaissance polymath Leon Battista Alberti (1404–1472), whom we have already met in this series in the episodes on linear perspective and architecture. Alberti is one of those people, who deserves to be much better known than he is. 

Leon Battista Alberti Source: Wikimedia Commons

Alberti published the earliest known description of a polyalphabetic cipher in his De componendis cifris in 1467.

De componendis cifris frontispiece Source: Wikimedia Commons

In a polyalphabetic cipher not one but several substitution alphabets are used for different sections of the text, changing at random intervals, with a signal in the text to indicate for the reader deciphering the code of the alphabetical change. A polyalphabetic cipher is, in theory, not susceptible to frequency analysis; this, of course, is only true if one doesn’t know the method. In order to implement his method Alberti invented his cipher disc, which he explained in a letter was inspired by the recent invention of the movable type printing press. 

Alberti Cipher disc Source: Wikimedia Commons

How the whole thing is supposed to function is described by Alberti in his De componendis cifris

Chapter XIV. I will first describe the movable index. Suppose that we agreed to use the letter k as an index letter in the movable disk. At the moment of writing I will position the two disks of the formula as I wish, for example juxtaposing the index letter to capital B, with all other small letters corresponding to the capital letters above them. When writing to you, I will first write a capital B that corresponds to the index k in the formula. This means that if you want to read my message you must use the identical formula you have with you, turning the movable disk until the letter B corresponds to the index k. Thus all small letters in the ciphertext will receive the meaning and sound of those above them in the stationary disk. When I have written three or four words I will change the position of the index in our formula, turning the disk until, say, the index k is under capital R. Then I will write a capital R in my message and from this point onward the small k will no longer mean B but R, and the letters that follow in the text, will receive new meanings from the capital letters above them in the stationary disk. When you read the message you have received, you will be advised by the capital letter, which you know is only used as a signal, that from this moment the position of the movable disk and of the index has been changed. Hence, you will also place the index under that capital letter, and in this way you will be able to read and understand the text very easily. The four letters in the movable disk facing the four numbered cells of the outer ring will not have, so to speak, any meaning by themselves and may be inserted as nulls within the text. However, if used in groups or repeated, they will be of great advantage, as I will explain later on.

Chapter XV. We can also choose the index letter among the capital letters and agree between us which of them will be the index. Let us suppose we chose the letter B as an index. The first letter to appear in the message will be a small one at will, say q. Hence, turning the movable disk in the formula you will place this letter under the capital B that serves as an index. It follows that q will take the sound and meaning of B. For the other letters we will continue writing in the manner described earlier for the movable index. When it is necessary to change the set up of the disks in the formula, then I will insert one, and no more, of the numeral letters into the message, that is to say one of the letters of the small disk facing the numbers which corresponds to, let’s say, 3 or 4, etc. Turning the movable disk I will juxtapose this letter to the agreed upon index B and, successively, as required by the logic of writing, I will continue giving the value of the capitals to the small letters. To further confuse the scrutinizers you can also agree with your correspondent that the capital letters intermingled in the message have the function of nulls and must be disregarded, or you may resort to similar conventions, which are not worth recalling. Thus changing the position of the index by rotating the movable disk, one will be able to express the phonetic and semantic value of each capital letter by means of twenty-four different alphabetic characters, whereas each small letter can correspond to any capital letter or to any of the four numbers in the alphabet of the stationary disk. Now I come to the convenient use of the numbers, which is admirable.

I leave it to the reader to decipher Alberti’s instructions.

Alberti was not the only Renaissance scholar to suggest the adoption of a polyalphabetic cipher. The German, Benedictine monk, Johannes Trithemius (1462–1516), born Johann Heidenberg, was Abbot of the Abbey of Sponheim and from 1506 of the St, James’ Abbey in Würzburg.

Tomb relief of Johannes Trithemius by Tilman Riemenschneider Source: Wikimedia Commons

Trithemius was a polymath active as a lexicographer, chronicler, cryptographer, but above all he is known as an occultist. As I have noted in an earlier episode in this series the occult sciences played a significant part in Renaissance thought. Trithemius is considered to have had a major influence on both Paracelsus (c. 1493–1541) and Heinrich Cornelius Agrippa von Nettesheim (1486–1535).

Trithemius’ most famous work was his Steganographia (written c. 1499 but first published in 1606), which was initially thought to be about magic and was placed on the Index Librorum Prohibitorum (List of Prohibited Books) by the Catholic Church in 1609. In fact, the book was written in code and already in 1606, the first two volumes were shown to be about steganography (a word that Trithemius coined) and cryptography. Steganography is: 

The practice of hiding messages, so that the presence of the message itself is hidden, often by writing them in places where they may not be found until someone finds the secret message in whatever is being used to hide it. (def. Wiktionary). 

The third volume was, thought to be really about magic but has comparatively recently also shown to be about cryptography.

Source: Wikimedia Commons

Trithemius also wrote his Polygraphiae libri sex (Six books of polygraphia), the first printed book on cryptography, a further text on steganography, which was published posthumously in 1518.

Source: Wikimedia Commons

This work contains a progressive key polyalphabetic cipher now known as the Trithemius cipher. 

Unlike Alberti’s cipher, which switched alphabets at random intervals, Trithemius switched alphabets for each letter of the message. He started with a tabula recta, a square with 26 letters in it (although Trithemius, writing in Latin, used 24 letters). Each alphabet was shifted one letter to the left from the one above it, and started again with A after reaching Z (see table). Source: Wikimedia Commons

Our third Renaissance cryptographer was Blaise de Vigenère (1523–1596) a French, diplomat, cryptographer, translator, and alchemist.

Blaise de Vigenère Source: Wikimedia Commons

Although, he created a polyalphabetic cipher, the one that bears his name was actually first described by Giovan Battista Bellaso (1505–?) an Italian cryptographer in his La Cifra del Sig. Giovan Battista Belaso published in 1553. Bellaso went on to publish a second book, Novi et singolari modi di cifrare, in 1555 and a third one Il vero modo di scrivere in cifra, in 1565. Vigenère published his polyalphabetic cipher, first in 1586, in his Traicté des Chiffres ou Secrètes Manières d’Escrire. 1586. Both men’s ciphers were based on a so-called auto key but differed in detail. 

Source: Wikimedia Commons

An autokey cipher (also known as the autoclave cipher) is a cipher that incorporates the message (the plaintext) into the key. The key is generated from the message in some automated fashion, sometimes by selecting certain letters from the text or, more commonly, by adding a short primer key to the front of the message. (Def. Wikipedia). 

The Vigenère cipher was thought to be unbreakable and in fact Charles Lutwidge Dodgson (better known as Lewis Carroll), a very competent logician, said that it was unbreakable in 1868, unaware that Charles Babbage had already broken it earlier but had not published his results. The first to publish a general system to solve polyalphabetic ciphers, including the Vigenère cipher, was the German soldier, cryptographer, and archaeologist, Friedrich Wilhelm Kasiski (1805–1881) in his Die Geheimschriften und die Dechiffrir-Kunst (Secret Writing and the Art of Deciphering) in 1863, a publication that, at the time, went largely unnoticed. It was first in the nineteenth century that the art of cryptography evolved past the innovations of the Renaissance cryptographers, Alberti, Trithemius, Vigenère, and Bellaso. 


Filed under Renaissance Science

You Are History

Do you have children, grandchildren, nephews & nieces, grandnephews & grandnieces, the children of friends, age group 7–12, that you would like to give a book as a Christmas present. An entertaining but also educational book, a well written and beautifully illustrated book, a fascinating and intriguing book? Then look no further, I have the very thing for you. It’s Greg Jenner’s You Are History, illustrated by Jenny Taylor. 

That’s historian Greg Jenner star of radio and TV, he’s part of the crew that magic Horrible Histories onto your TV screens, and mega podcast star with his You’re Dead to Me. The book follows a child through its day, from being woken up by the alarm clock in the early morning to finally going to bed at night. At each station throughout the day the history of the everyday objects encountered is presented and explained is entertaining witty texts illustrated by wonderful pictures. Alarm clock, Toothpaste, letterbox, pencil case, chocolate, bicycle, cutlery, pyjamas, and, and, and, and, and … in total 137 objects that we encounter almost daily. This book is a winner. Greg is a serious historian, despite his love of bad jokes, and a little bird has told me that he has had everything fact checked by experts, so the child, who reads this delightful tome, is getting some solid history served up to them, as well as being entertained.

The book is available from 3 November, so get your orders in now and delight that child in your life on Christmas Day, or any other day for that matter.  

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Filed under Book Reviews

The Jesuits and science in the seventeenth century

One myth in the history of science that refuses to go away is that the Catholic Church was fundamental opposed to the modern science that emerged during the seventeenth century. They even according to one prevalent theory declared war on it. This myth is of course fuelled by the equally persistent false accounts of the conflicts between the Church and both Giordano Bruno and Galileo Galilei. English scholars, not necessarily historian of science, armed with their knowledge of the reputation of the Jesuits in Early Modern Protestant England, where they were denounced as spawn of the Devil, also polemicise against them, as particularly anti-progress including science. 

Long term readers of this blog will know that over the years I have posted numerous essays that try to correct this false perception both for the Catholic Church in general and for the Jesuits in particular. In fact, my very first substantive history of science post was an attempt to restore the reputation of Christoph Clavius, who introduced the mathematical sciences into the Jesuit’s educational programme, making them amongst the best educated mathematician of the period. 

Long ago I read The New Science and Jesuit Science: Seventeenth Century Perspectives, edited by Mordechai Feingold, (Springer, 2003), an excellent book, in which the first essay is by Michael John Gorman, and now I have been reading Gorman’s The Scientific Counter-RevolutionThe Jesuits and the Invention of Modern Science[1], of which the paperback was published this year, and which contains the essay mentioned above. This book is a must read for anybody interested in the Jesuit contribution to seventeenth century science.

I have included a scan of the back cover of the book, because the two ringing endorsements from Paula Findlen and Simon Schaffer, together with the publisher’s blurb should be enough to convince anybody interested in the topic that this is a must read, without further comment from me. However, I will, of course, make some comments.

This book is based on Gorman’s doctoral dissertation from 1999. It is not a continuous flowing narrative but, rather, a set of seven essays connected by a common theme, version of six of which have already been published in other contexts. Despite the previous publications, the book is more than worth reading, because the essays, as a whole, create a rounded picture of the Jesuits evolving role in mathematics and natural philosophy in the seventeenth century. 

The opening chapter, Establishing Mathematical AuthorityThe Politics of Christoph Clavius, the only one not previously published, deals with Clavius’ introduction of mathematics into the Jesuits’ educational curriculum with a strong emphasis on his motivation for doing so and the political arguments he used to justify this move. 

The second chapter, Mathematics and ModestyThe Problemata of Christoph Grienberger, illustrates the Jesuits personal requirement of humility and rejection of vanity in their work. This is illustrated with a description of the life and work of Christoph Grienberger, Clavius’ successor as professor for mathematics at the Collegio Romano, who almost completely disappeared behind his work. Perhaps the most extreme example of the abrogation of personality. 

The Jesuits provide a difficult subject for the historian of the scientific revolution because they are not the enemies of science that some try to paint them nor are they, despite their own substantial contributions, unrestricted supporters of the new developments in science. There is a tension between the Clavian application of modern mathematics and experimentation to the sciences and the fundamental Jesuit obligation to adhere to the Aristotelian science of Thomist philosophy. Gorman’s book expertly brings this paradox out into the open displaying and analysing it from all sides.

The third chapter of the book, Discipline, Authority and Jesuit CensorshipFrom the Galileo Trial to the Ordinatio Pro Studiis Superioribus, is an in-depth analysis of this paradox. Gorman actually argues, convincingly, that the Jesuits’ requirement that all scientific texts be submitted to the Oder’s authorities for censorship is an early form of peer review.

The fourth chapter, The Jesuits and the Vacuum Debate, was the one that I found most fascinating and from which I learnt the most. The invention of the Torricellian Tube, or barometer, unleashed a heated debate in the seventeenth century as to whether the empty space at the top of the tube was or was not a vacuum. Aristotle had argued that a vacuum could not exist, so the Jesuits were forced to take the side of denial, as we now know the wrong side. As is often the case in the history of science, their opposition actually stimulated the advancement of the science, as the supporters of the vacuum theory were required to counter their arguments. Interestingly the Jesuit opposition was not just philosophical. One of the principal supporters of the vacuum theory was another, non-Jesuit, Catholic scholar, and the Jesuits feared that if he won the debate that this would lead to a diminishing of their dominance in Catholic education. The whole chapter is an excellent example of the complexity of scientific evolution and its history.

There now follow two chapters on Athanasius Kircher, perhaps the most well-known scientific, Jesuit scholar of the century. The first, The Angel and the CompassAthanasius Kircher’s Geographical Project, deals with Kircher’s failed project to create a new, reformed  geography of the entire world by creating a world spanning network of researchers providing input to a central authority, himself. Although Kircher’s project, inspired by similar smaller collective research projects of Peiresc and Mersenne, in the end failed, Gorman sees the attempt as an important step in the evolving practice of scientific research. He also sees the Jesuits process of the central accumulation of scientific data in the Collegio Romano via world-wide correspondence, as an important model for others working in science.

The second chapter on Kircher, Between the Domonic and the MiraculousAthanasius Kircher and the Baroque Culture of Machines, takes a detailed look at Musaeum Kircherianum and its role as a public magnet to promote the Jesuit Order under the European civil and scientific aristocracies. He details the explanations of natural magic given by Kircher and above all his disciple Kasper Schott to explain the function of Kircher’s automatons, machines that initially appear to contradict the laws of nature but whose secret can be explained by those same laws.

Gorman’s final chapter, From ‘The Eyes Of All’ to ‘Usefull Quarries in Philosophy and Good Literature’: The Changing Reputation of the Jesuit Mathematicus, re-examines the trajectory of his book from Clavius to Kircher and how the emphasis and public presentation of the mathematicians of the Collegio Romano had changed over the century, not necessarily for the better. 

Gorman is an excellent writer and his prose is clear and a pleasure to red. The book is illustrated with a series of greyscale illustration. There are informative endnotes, no footnotes, at the end of each essay but unfortunately no general bibliography. There is a good index. 

At the end of the introduction, we can read:

A visit to the publisher’s website reveals the following

The site you are trying to reach has now been archived. Please contact onlineresources@bloomsbury.com for access.

Apart from this irritation, the book is, as already stated, excellent and definitely a must read for anybody interested in the Jesuits’ contribution to the evolution of science in the seventeenth century.

[1] Michael John Gorman, The Scientific Counter-RevolutionThe Jesuits and the Invention of Modern Science, Bloomsbury Academic, London, New York, Oxford, New Delhi, Sydney, 2020, ppb. 2022


Filed under Book Reviews

Renaissance science – XLV

70.8% of the earth’s surface is covered by the world ocean; we normally divide it up–Atlantic Ocean, Pacific Ocean, Indian Ocean, etc.– but they are all interconnected in one giant water mass.

The world ocean Source: Wikimedia Commons

Only 29.2% of the surface is land but, on that land, there are many enclosed seas, lakes, ponds, rivers, and streams so there is even more water. The human body is about 60% water, and humans are sometimes referred to as a water-based life form. The statistics are variable, but a healthy human can exist between one and two months without food but only two to four days without water. Brought to a simple formular, water is life.

When humans first began to settle, they did so on or near sources of water–lake shores, streams, rivers, natural springs. Where there was no obvious water supply people began to dig wells, there are wells dating back to 6500 BCE. As settlements grew the problem of water supply and sewage disposal became important and the profession of water manager or hydraulic engineer came into existence. Channelling of fresh water and sewage disposal, recycling of wastewater etc. Initial all of this was powered by gravity but over time other systems of moving water, such as the bucket water wheel or noria were developed for lifting water from one channel into another, appearing in Egypt around the fourth century BCE.

Close-up of the Noria do Mouchão Portugal Source: Wikimedia Commons

Probably the most spectacular surviving evidence of the water management in antiquity are the massive aqueducts built by Roman engineers to bring an adequate supply of drinking water to the Roman settlements. Alone the city of Rome had eleven aqueducts built between 312 BCE and 226 CE, the shortest of which the Aqua Appia from 312 BCE was 16.5 km long with a capacity of 73,000 m3 per day and the longest the Aqua Anio Novus from 52 CE was 87 km long with a capacity of 189,000 m3 per day. The Aqua Alexandrina from 226 CE was only 22 km long but had a capacity of 120,00 to 320,000 m3 per day.

Panorama view of the Roman Aqueduct of Segovia in 2014 Built first century CE originally 17 kilometres long Source Wikimedia Commons

The simplest water clock or clepsydra, a container with a hole in the bottom where the water was driven out by the force of gravity dates back to at least the sixteenth century BCE.

A reconstruction of the water clock used in ancient Greece (Museum of Ancient Agora/Athens) Figure 5: Water Clock/Clepsydra Source

It evolved over the centuries with complex feedback mechanism to keep the water level and thus the flow constant. Water clocks reach an extraordinary level of sophistication as illustrated by the Astronomical Clock Tower of Su Song (1020–1101 CE) in China

The original diagram of Su’s book showing the inner workings of his clocktower Source: Wikimedia Commons

and the Elephant Clock invented by the Islamic engineer al-Jazari (1136-1206). Al-Jazari invented many water powered devices.  

Al-Jazari’s elephant water clock (1206) Source: Wikimedia Commons

Much earlier the Greek engineer Hero of Alexandria (c. 10–c. 70 CE), as well as numerous devices driven by wind and steam, invented a stand-alone fountain that operates under self-contained hydro-static energy, known as Heron’s Fountain. 

Diagram of a functioning Heron’s fountain Source: Wikimedia Commons

All of the above is out of the realm of engineers. Another engineer Archimedes (c. 287–c. 212 BCE), is the subject of possibly the most well-known story in the history of science, one needs only utter the Greek word εὕρηκα (Eureka) to invoke visions of crowns of gold, bathtubs, and naked bearded man running through the streets shouting the word. In fact, you won’t find this story anywhere in Archimedes not insubstantial writings. The source of the story is in De architectura by Vitruvius (C. 80-70–after c. 15 BCE), so two hundred years after Archimedes lived. You can read the original in translation below:

Vitruvius “Ten Books on Architecture”, Ed. Ingrid D. Rowland & Thomas Noble Howard, (CUP, 1999) p. 108

However, Archimedes did write a book On Floating Bodies, which now only exists partially in Greek but in full in a medieval Latin translation. This book is the earliest known work of the branch of physics known as hydrostatics. It contains clear statement of two fundamental principles of hydrostatics, Firstly Archimedes’ principle:

Any body wholly or partially immersed in a fluid experiences an upward force (buoyancy) equal to the weight of the fluid displaced

Secondly the principle of floatation:

Any floating object displaces its own weight of fluid.

As well these two fundamental principles, he also discovered that a submerged object displaces a volume of water equal to its own volume. This is the discovery that led to the legendary of mythical Eureka incident. A crown of pure gold would have a different displacement volume to one of a gold and silver amalgam. The bath story was, as we will see later, highly implausible because it would be very, very difficult to measure the difference in the displaced volumes of water of the two crowns.

Whilst water management continued to develop through out the Middle Ages, with the invention of every better water mills etc., In the Renaissance the profession hydraulic engineer saw developments in two areas. Firstly, the increase in wealth and the development of residences saw the emergence of the Renaissance Garden. Large ornamental gardens the usually featured extensive and often spectacular water features.

Garden of Villa d’Este Tivoli (1550–1572) Source: Wikimedia Commons

The Renaissance mathematici employed by potentates and aristocrats were often expected to serve as hydraulic engineers alongside their other functions as instrument makers, astrologers etc. Secondly the major increase in mining for precious and semi-precious metals meant ever deeper mines, which brought with it the problem of pumping water out of the mines.

Archimedes’ On Floating Bodies was translated into Latin by William of Moerbeke (c. 1215–1286) in the thirteenth century and no complete Greek manuscript is known to exist. This translation was edited by Nicolò Tartaglia Fontana (c. 1506–1557) and published in print along with other works by Archimedes by Venturino Ruffinelli in Venice in 1543, as Opera Archimedis Syracvsani philosophi et mathematici ingeniosissimi

Opera Archimedis Syracvsani philosophi et mathematici ingeniosissimi1543 Source

The Nürnberger theologian and humanist Thomas Venatorius (1488–1551) edited the first printed edition of the Greek manuscripts of Archimedes, in a bilingual Greek/Latin edition, which was published in Basel by Johann Herwagen in 1544. The Greek manuscript had been brought to Nürnberg by the humanist scholar, Willibald Pirckheimer (1470–1530) from Rome and the Latin translation by Jacopo da Cremona (fl. 1450) was from the manuscript collection of Regiomontanus (1436-1476).


Venatorius claimed, in the foreword to the Archimedes edition to have studied mathematics under Johannes Schöner (1577–1547) but if then as a mature student in Nürnberg and not as a schoolboy. 

A reconstruction of On Floating Bodies was published by Federico Commandino (1509–1575) in Bologna in 1565. 


Tartaglia, who also produced an Italian edition of On floating Bodies, was the first Renaissance scholar to address Archimedes work on hydrostatics. It did not play a major role in his own work, but he was the first to draw attention to the relationship between the laws of fall and Archimedes’ thoughts on flotation. Tartaglia’s work was read by his one-time student, Giambattista Benedetti ((1530–1590), Galileo (1564–1642), and Simon Stevin (1548–1620), amongst other, and was almost certainly the introduction to Archimedes’ text for all three of them. 

Benedetti replaced Aristotle’s concepts of fall in a fluid directly with Archimedes’ ideas in his work on the laws of fall, equating resistance in the fluid with Archimedes’ upward force or buoyancy. This led him to his anticipations of Galileo’s work on the laws of fall. 

Moving onto Simon Stevin, who wrote a major work on hydrostatics, his De Beghinselen des Waterwichts (Principles on the weight of water) in 1586 and a never completed practical Preamble to the Practice of Hydrostatics.


One of Benedetti’s major works, Demonstratio propotionummotuum localiumcontra Aristotilem et omnes philosphos (1554) had been plagiarised by the French mathematician Jean Taisnier (1598–1562) Opusculum perpetua memoria dignissimum, de natura magnetis et ejus effectibus, Item de motu continuo (1562) and it was this that Stevin read rather than Benedetti’s original. Taisnier’s plagiarism was also translated into English by Richard Eden (c. 1520–1576) an alchemist and promotor of overseas exploration. Stevin a practical engineer ignored or rejected the equivalence between the laws of fall and the principle of buoyancy, concentrating instead on the relationship between flotation and the design of ship’s hulls. His major contribution was the so-called hydrostatic paradox often falsely attributed to Pascal. This states that the downward pressure exerted by a fluid in a vessel is only dependent on its depth and not on the width or length of the vessel. 

Of the three, Galileo is most well-known for his adherence to Archimedes. He clearly stated that in his natural philosophy he had replaced Aristotle with Archimedes as his ancient Greek authority, and this can be seen in his work. His very first work was an essay La Bilancetta (The Little Balance) written in 1586, but first published posthumously in 1644, which he presented to both Guidobaldo del Monte (1545–1607) and Christoph Clavius (1538–1612), both leading mathematical authorities, in the hope of winning their patronage. He was successful in both cases.

Galileo Galilei, La bilancetta, in Opere di Galileo Galilei (facsimile) Source:

Realising, that the famous bathtub story couldn’t actually have worked, Galileo tried to recreate how Archimedes might actually have done it. He devised a very accurate hydrostatic balance that would have made the discovery feasible. 

Later in life, when firmly established as court philosopher in Florence, Galileo was called upon by Cosimo II Medici to debate the principles of flotation with the Aristotelian physicist Lodovico delle Columbe (c. 1565–after 1623), as after dinner entertainment. As I have written before one of Galileo’s principal functions at the court in Florence was to provide such entertainment as a sort of intellectual court jester. Galileo was judged to have carried the day and his contribution to the debate was published in Italian, as Discorso intorno alle cose che stanno in su l’acqua, o che in quella si muovono, (Discourse on Bodies that Stay Atop Water, or Move in It) in 1612.


As was his wont, Galileo mocked his Aristotelian opponent is his brief essay, which brought him the enmity of the Northern Italian Aristotelians. Although Galileo’s approach to the topic was Archimedean, he couldn’t explain everything and not all that he said was correct. However, this little work enjoyed a widespread reception and was influential.

Our last Renaissance contribution to hydrostatics was made by Evangelista Torricelli (1608–1647), a student of Benedetto Castelli (1578–1643) himself a student of Galileo, and like Stevin’s work it came from the practical world rather than the world of science.

Evangelista Torricelli by Lorenzo Lippi Source: Wikimedia Commons

Torricelli was looking for a solution as to why a suction pump could only raise water to a hight of ten metres, as recounted in Galileo’s Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) (1638), a major problem for the expanding deep mining industry, which needed to pump water out of its mines. Torricelli in his investigations invented the Torricellian tube, later called the barometer, with which he demonstrated that there was a limit to the height of a column of liquid that the weight of the atmosphere, or air pressure, could support.

Torricelli’s experiment Source: Wikimedia Commons

He also incidentally demonstrated the existence of a vacuum, something Aristotle said could not exist. 

Torricelli’s work marks the transition from Renaissance science to what is called modern science. Building on the work of Benedetti, Stevin, Galileo, and Torricelli, Blaise Pascal (1623–1662) laid some of the modern foundation of hydrodynamics and hydrostatics, having a unit for pressure named after him and being sometimes falsely credited with discoveries that were actually made in the earlier phase by his predecessors. 

Painting of Pascal made by François II Quesnel for Gérard Edelinck in 1691. Source:Wikimedia Commons


Filed under History of Physics, History of Technology, Renaissance Science

Beinecke Library Redux: From Bad to Worse!

On Monday I wrote a quick blogpost on the not insubstantial errors in the description of one of Galileo’s lunar washes posted on the Beinecke Library blog. I was somewhat pleasantly surprised when within a day the description had been heavily edited, removing all the sections that I had criticised, even if no acknowledgement was made that changes had taken place or why.  In my elation over this turn of events I failed to properly read what now stood under Galileo’s image. One of my readers, Todd Timberlake author of Finding Our Place in the Solar System: The Scientific Story of the Solar System, was more observant than I and correctly stated that the modified version was now, if possible worse than the original. So, what had curator Richard Clemens done now?

Above: Detail, p. 18. Galileo, Siderevu nvncivs, QB41 G33 1610, copy 2. 

Left us examine what can only be described as a disaster, the text now reads:

Our mini-exhibits end with the vitrine holding several copies of Galileo’s first printed images of the moon made with the benefit of the telescope.  He shows the shadow the earth casts on the moon and the moon’s rocky surface. [my emphasis] A photograph at the back of the vitrine was taken in 1968, before humans landed on the moon. It shows Earth as seen from the moon—the first time we saw our own planet from another astronomical body. This rough black and white image eerily resembles Galileo’s lunar landscape.

The only time the Earth’s shadow is visible on the Moon is during a lunar eclipse when the Earth comes between the Sun and the Moon thus blocking off the Sun’s light. Galileo did not make drawings of any telescopic observations of the Moon during a lunar eclipse. What we actually have is an image of the Moon at third quarter put together by Galileo from his observations. The light side on the left is the half of the Moon that is visible at third quarter, the dark side on right is the half not visible. The jagged line down the middle is the so-called lunar terminator: the division between the illuminated and dark hemispheres of the Moon.

Photo of the Moon at third quarter Source:

Without being snarky, I think that Mr Clemens would do well to consult somebody who knows what they are talking about before writing his descriptions.


Filed under History of Astronomy, Myths of Science

Not with a bang but a whimper

Ethan Siegel is an astrophysicist, but he is better known as a highly successful science populariser, who even has his own Wikipedia page.  He first rose to fame as the author of the blog Starts With a Bang, which he launched in 2008. He expanded his brand, with the publication of popular books on physics. He expanded still further, making podcasts and writing posts under his brand name on MediumForbes, and Big Think. He is today one of the biggest names on the Internet in popularisation of physics. Here I’m going to look at his latest publication on Big ThinkBig Think is a multiplatform, multimedia Internet organisation who in their own words state: 

Our mission is to make you smater, faster.  At Big Think, we introduce you to the brightest minds and boldest ideas of our time, inviting viewers to explore new ways to work, live, and understand our ever-changing world. 

“Big Think challenges common sense assumptions and gives people permission to think in new ways.”

I’m sorry, but to my ears that sounds like those windy ads on the Internet that say, “Take our three-week course of our seminars once a week and you will be earning $100,000 a month within a year!”

So, what is the post of Dr Siegel on Big Think that has attracted the attention of The Renaissance Mathematicus and why? Our intrepid astrophysicist and physics populariser has decided to try his hand at history of science and has written a post about Johannes Kepler, Why Johannes Kepler is a scientist’s best role model. After all our author is a scientist and a successful science populariser, who has even won prestigious awards for his work, what could possibly go wrong, when he tries a bit of history of science? Unfortunately, as with other scientists and science populariser, who think they can do history of science, without investing serious time and effort in the discipline, almost everything.

Johannes Kepler unknown artist 1620

So why does Siegel think that the good Johannes should be every scientist’s role model? He tells us in his lede:

  • The annals of history are filled with scientists who had incredible, revolutionary ideas, sought out and found the evidence to support them, and initiated a scientific revolution. 
  • But much rarer is someone who has a brilliant idea, discovers that the evidence doesn’t quite fit, and instead of doggedly pursuing it, tosses it aside in favor of a newer, better, more successful idea. 
  • That’s exactly what separates Johannes Kepler from all of the other great scientists throughout history, and why, if we have to choose a scientific role model, we should admire him so thoroughly.

He then delivers four examples of famous scientists, who could not admit they were wrong:

  • Albert Einstein could never accept quantum indeterminism as a fundamental property of nature.
  • Arthur Eddington could never accept quantum degeneracy as a source for holding white dwarfs up against gravitational collapse.
  • Newton could never accept the experiments that demonstrated the wave nature of light, including interference and diffraction.
  • And Fred Hoyle could never accept the Big Bang as the correct story of our cosmic origins, even nearly 40 years after the critical evidence, in the form of the Cosmic Microwave Background, was discovered.

I already have a couple of comments here. Niels Bohr is on record as saying that Einstein through his intelligent, astute, and penetrating criticisms of quantum theory that demanded answers contributed more to the development of that theory than almost anybody else. Not least Bell’s theorem, one of the key developments in quantum theory, was based on his analysis of the Einstein–Podolsky–Rosen paradox. Opposition to theories based on knowledge are important to the evolution of those theories. 

Newton did in fact reject a wave theory of light in favour of a particle theory. However, he was able with his theory to explain all the known optical phenomenon. Moreover, when Hooke rejected his theory of colour saying that it wouldn’t work in a wave theory, Newton developed a wave theory, that was more advanced than those of Hooke and Huygens, to show that his theory of colour did work in a wave theory. Lastly, as I love to point out, Einstein won the Nobel Prize for physics, not for relativity, but for demonstrating that light consists of particles, so Newton wasn’t so wrong after all.   

More generally, there is a famous quote from Max Planck about the development of new theories in science:

A new scientific truth does not generally triumph by persuading its opponents and getting them to admit their errors, but rather by its opponents gradually dying out and giving way to a new generation that is raised on it.

He then goes on to tell us why Kepler was a spectacular exception. First, we get a popular rundown of the observable phenomena of the cosmos and why that led to a geocentric model. On the whole OK but littered with small errors. For example, he tells us:

The Earth was big, and its diameter had been measured precisely [my emphasis] by Eratosthenes in the 3rd century B.C.E.

This is, unfortunately, typical of Siegel’s hyperbolic style. Depending on which value for the stadium one takes, Eratosthenes’ estimate of the size of the earth was relatively close to the real value but by no means precise. Also, in antiquity no one knew how correct it was and most people actually accepted other values.

We then get a description of the deferent/epicycle model for the planets and Siegel tells us that Ptolemy made the best, most successful model of the Solar system that incorporated epicycles. Nothing to criticise here but there follows immediately a small misstep, he writes:

Going all the way back to ancient times, there was some evidence — from Archimedes and Aristarchus, among others — that a Sun-centered model for planetary motion was considered. 

First off you really shouldn’t use an expression like “ancient times.” We know that both Archimedes and Aristarchus lived and worked in the third century BCE, so we can say that. The expression “there was some evidence from Archimedes and Aristarchus, among others” is a load of waffle, which doesn’t actually tell the reader anything. According to a couple of secondary sources Aristarchus of Samos devised a heliocentric system. We don’t have anything about it from Aristarchus himself. Archimedes is one of the secondary sources but not in a work on astronomy or cosmology. Archimedes wrote a work on calculating and expressing large numbers, The Sand Reckoner, in which he calculated the number of grains of sand needed to fill the cosmos. He used Aristarchus’ heliocentric model, which he only mentions in passing, because the heliocentric cosmos is considered to be larger the than the geocentric one.

Siegel now moves onto Copernicus and once again delivers up historical rubbish:

Copernicus was frustrated to discover that his model gave less successful predictions when compared against Ptolemy’s. The only way Copernicus could devise to equal Ptolemy’s successes, in fact, relied on employing the same ad hoc fix: by adding epicycles, or small circles, atop his planetary orbits!

As stated, this is rubbish. From the very beginning Copernicus used deferent/epicycle models for the planetary orbits. He didn’t add epicycles as an ad hoc solution because his model gave less successful predictions when compared against Ptolemy’s. In fact, Copernicus didn’t produce any planetary tables before he died in the year that his De revolutionibus was published, so he couldn’t know about the comparative predictive powers of his and Ptolemy’s system. When Erasmus Reinhold (1511–1553) did produce his Prutenic Tables (1551), the first ones based on Copernicus’ model, it turned out that in some cases the predictions were better than in tables based on Ptolemy and in some cases worse. This was because Copernicus used the same, in the meantime corrupted through frequent copying, basic data for his models as Ptolemy. This problem was recognised by Tycho Brahe, which is why he set up his massive astronomical observation programme, on the island of Hven, in order to provide new basic data. It is to Tycho that Siegel now turns.

Tycho Brahe, for example, constructed the best naked eye astronomy setup in history, measuring the planets as precisely as human vision allows: to within one arc-minute (1/60th of a degree) during every night that planets were visible towards the end of the 1500s. His assistant, Johannes Kepler, attempted to make a glorious, beautiful model that fit the data precisely.

This is Siegel’s introduction to Kepler’s Mysterium Cosmographicum published in 1596, four years before he even met Tycho and began to work with him! Siegel now gives a brief description of the model presented in the Mysterium Cosmographicumand follows it up with a pile of absolute garbage.

Maybe our astrophysicist author has slipped into a parallel universe because what he presents here is hyperbollocks, an assorted collection of made-up “facts” thrown together in a narrative that bears absolutely no relation to what really happened in history. As a Kepler fan when I read this and the following paragraphs eight days ago, I began banging my head against the wall and haven’t stopped since. No pain can blot out the stupidity presented here. 

Kepler formulated this model in the 1590s, and Brahe boasted that only his observations could put such a model to the test. But no matter how Kepler did his calculations, not only did disagreements with observation remain, but Ptolemy’s geocentric model still made superior predictions.

Tycho made no such boast, that is simply made up and in fact he was not in any way interested in Kepler’s model. Kepler wanted to work with Tycho to get access to his data to fine tune his model, Tycho wanted to employ Kepler to do the mathematics necessary to turn his data into models for the planets orbits in his own geo-heliocentric model. When Kepler arrived in Prague, Tycho refused him access to the data he wanted out of fear of being plagiarised. Instead, he set Kepler to write a paper proving that Ursus had plagiarised him. The resulting essay is brilliant, was however first published in the nineteenth century, and has been described by Cambridge historian of science, Nicholas Jardine as The Birth of History and Philosophy of Science (CUP, 2nd rev. ed. 1988). Following this he was given the task of determining the orbit of Mars using Tycho’s data, to which I will return in a minute. 

At this point in his life Kepler made no attempt to improve his geometrical model. The phrase, Ptolemy’s geocentric model still made superior predictions is quite simple mind boggling for anybody who knows what they are talking about. The geometric model that Kepler presents in his Mysterium Cosmographicum is his answer to the question, why are there exactly six planets? Kepler argues that his completely rational God, who is a geometer, designed his cosmos rationally and geometrically and there are exactly six planets because there are only five regular Platonic solids to fill the spaces between them. Not our idea of rational but Kepler was mighty pleased with his “discovery.” This model makes no predictions of any kind!

Now we get to the crux of Siegel’s whole argument, Kepler admitting he was wrong:

In the face of this, what do you think Kepler did?

  • Did he tweak his model, attempting to save it?
  • Did he distrust the critical observations, demanding new, superior ones?
  • Did he make additional postulates that could explain what was truly occurring, even if it was unseen, in the context of his model?

No. Kepler did none of these. Instead, he did something revolutionary: he put his own ideas and his own favored model aside, and looked at the data to see if there was a better explanation that could be derived from demanding that any model needed to agree with the full suite of observational data.

Kepler didn’t tweak his model, at this time, attempting to save it, he certainly didn’t mistrust Tycho’s data, and he didn’t at this time add any postulates. He did put his model aside but not to look at the data to see if there was a better explanation that could be derived from demanding that any model needed to agree with the full suite of observational data. He was too busy doing other thing, things that served other purposes. 

If only we could all be so brave, so brilliant, and at the same time, so humble before the Universe itself! Kepler calculated that ellipses, not circles, would better fit the data that Brahe had so painstakingly acquired. Although it defied his intuition, his common sense, and even his personal preferences for how he felt the Universe ought to have behaved — indeed, he thought that the Mysterium Cosmographicum was a divine epiphany that had revealed God’s geometrical plan for the Universe to him — Kepler was successfully able to abandon his notion of “circles and spheres” and instead used what seemed to him to be an imperfect solution: ellipses.

Here without explicitly naming it, Siegel is referencing Kepler’s work on the orbit of Mars that he published in his Astronomia Nova in 1609. It was during the many years of his “War with Mars”, his own description, that he finally discovered his first two laws of planetary motion: 1: Planetary orbits are ellipses with the Sun at one focus of the ellipse 2: A line from the Sun to the planet sweeps out equal areas in equal periods of time. For a good description of the route to the Astronomia Nova, I recommend James R. Voelkel’s excellent The Composition of Kepler’s Astronomia nova (Princeton University Press, 2001). 

Siegel apparently thinks that this refutes Kepler’s Mysterium Cosmographicum, it doesn’t. The Mysterium Cosmographicum doesn’t deal with the shape of orbits at all. His model has the Platonic solids filling the spaces between the spheres. In the Ptolemaic deferent/epicycle system the orbits are not simple circle because of the epicycle. Ptolemy in his Planetary Hypothesis embedded the deferent/epicycle in a sphere but the book that got lost and was only rediscovered in the 1960s in a single Arabic copy. However, Peuerbach (1423–1461) revived this model in his Theoricae Novae Planetarum (written in 1454, published by Regiomontanus in 1472), which is almost certainly based on a now lost copy of the Planetary Hypothesis, with illustrations. 

Peuerbach’s illustration of a sphere containing a deferent/epicycle Source: Wikimedia Commons

Copernicus’ heliocentric system, which also uses the deferent/epicycle models would suffer from the same problem and it is between these spheres that Kepler places his Platonic solids, irrespective of the orbit inside the sphere. The system would work equally well for elliptical orbits, so Kepler’s discovery of them had no effect on his Mysterium Cosmographicum

Siegel gives a table of Tycho’s Mars observations with the following caption:

Tycho Brahe conducted some of the best observations of Mars prior to the invention of the telescope, and Kepler’s work largely leveraged that data. Here, Brahe’s observations of Mars’s orbit, particularly during retrograde episodes, provided an exquisite confirmation of Kepler’s elliptical orbit theory. [my emphasis]

Kepler used Tycho’s Mars data to derive his first two laws, so they can’t be used by him as confirmation. In fact, at the beginning he didn’t actual confirm his theory, simple assuming it applied to all the planets. It wasn’t until his Epitome Astronomiae Copernicanae published in three volumes from 1618 to 1621, after he had discovered his third law and done a substantial amount of the work reducing Tycho’s observational data to planetary tables, the Rudolphine Tables published in 1627 and on which he had begun to work as Tycho was still alive, that he demonstrated all three laws for all the known planets.

I will now return to that third law and the Harmonice mundi (1619) in which it first appeared. Kepler had already suggested the possibility of fine tuning the Mysterium Cosmographicum model with the Pythagorean concept of a harmony of the spheres and this is what his magnus opus Harmonice mundi was. He had already conceived it in the late 1590s but because of other commitments didn’t actually get round to writing it until the second decade of the seventeenth century. 

Having created his harmony of the spheres, in 1621 Kepler published an expanded second edition of Mysterium Cosmographicum, half as long again as the first, detailing in footnotes the corrections and improvements he had achieved in the 25 years since its first publication, so far from abandoning his first theory to produce his elliptical orbits as Seigel claims, Kepler spent his whole life working to improve it.

What is truly bizarre is that Siegel appears to be aware of this fact. He writes:

It cannot be emphasized enough what an achievement this is for science. Yes, there are many reasons to be critical of Kepler. He continued to promote his Mysterium Cosmographicum even though it was clear ellipses fit the data better. He continued to mix astronomy with astrology, becoming the most famous astrologer of his time.

As already explained in detail, he didn’t just promote his Mysterium Cosmographicum, he worked very hard for many years to improve it. The statement, becoming the most famous astrologer of his time is another example of hyperbollocks. Kepler was a well-known astrologer in Southern Germany and Austria but the most famous astrologer of his time I hardly think so. I would also note that the modern astro-scientists disdain for astrology, as displayed here by Seigel, displays their ignorance of the history of their own discipline. Astrology was the driving force behind the developments in astronomy for its first three thousand years of its existence. 

Siegel, like many scientists, who think they can write history of science without doing the detailed research, has taken a set of half facts, embroidered them with stuff that he simply made up and created a nice fairy tale that has very little to do with real history of science. A fairy tale that will be swallowed by his large fan base, who will believe it and make life difficult for real historians of science. 


Filed under History of Astronomy, Myths of Science

I expect better of you Beinecke

The Beinecke Rare Book & Manuscript Library is the rare book library and literary archive of the Yale University Library. Yesterday their Twitter account posted a tweet entitled GalileoSiderius Nunc, which linked to a blog post from July 11, 2022, by Raymond Clemens, Curator, Early Books & Manuscripts. 

It featured one of Galileo’s famous washes of the Moon from his Sidereus Nuncius (1610) followed by a short text.

Above: Detail, p. 18. Galileo, Siderevu nvncivs, QB41 G33 1610, copy 2. 

Our mini-exhibits end with the vitrine holding several copies of Galileo’s first printed images of the moon, the first ever made with the benefit of the telescope. For the first time, most Europeans were shown the dark side of the moon. Galileo’s sketches also emphasize its barren and rocky nature—well known to us today, but something of a revelation in the sixteenth century, when most people thought of the moon as another planet, thus generating its own light. Galileo was the first person to accurately depict the moons of Jupiter (which he called “Medicean stars,” after his patron, the Florentine Medici family). A photograph at the back of the vitrine was taken in 1968, before humans landed on the moon. It shows Earth as seen from the moon—the first time we saw our own planet from another astronomical body. This rough black and white image eerily resembles Galileo’s lunar landscape.

It is a mere 152 words long, not much room for errors, one might think, but one would be wrong.

We start with the heading. The title of Galileo’s book is Sidereus Nuncius and there one really shouldn’t shorten Nuncius to Nunc, as this actually changes the meaning from message or messenger to now! Also, it is Sidereus not Siderius!

Addendum: A reader on Twitter, more observant than I, has pointed out, correctly, that 1609 and 1610 are in the seventeenth century and not the sixteenth century as stated by Clemens.

In the first line Clemens writes: Galileo’s first printed images of the moon, the first ever made with the benefit of the telescope. I shall be generous and assume that with this ambiguous phrase he means first ever printed images made with the benefit of the telescope. If, however, he meant first ever images made with the benefit of the telescope, then he would be wrong as that honour goes Thomas Harriot.

The real hammer comes in the next sentence, where he writes:

For the first time, most Europeans were shown the dark side of the moon.

The first time I read this, I did a double take, could a curator of the Beinecke really have written something that mind bogglingly stupid? By definition the dark side of the moon is the side of the moon that can never be seen from the earth. The first images of it were made, not by Galileo in 1609, after all how could he, but by the Soviet Luna 3 space probe in 1959, 350 years later. 

The problems don’t end here, he writes:

Galileo’s sketches also emphasize its barren and rocky nature—well known to us today, but something of a revelation in the sixteenth century, when most people thought of the moon as another planet, thus generating its own light.

In the geocentric system the moon was indeed regarded as one of the seven planets, but in the heliocentric system, which Galileo promoted, it had become a satellite of the earth and was no longer considered a planet. There was a long and complicated discussion throughout the history of astronomy as to whether the planets generated their own light or not. However, within Western astronomy there was a fairy clear consensus that the moon reflected sunlight rather than generating its own light. A brief sketch of the history of this knowledge starts with Anaxagoras (d. 428 BCE). The great Islamic polymath Ibn al-Haytham (965–1039) clearly promoted that the moon reflected sunlight. In the century before Galileo, Leonardo (1452–1519) in his moon studies clearly stated that the moon was illuminated by reflected sunlight. However, he never published. 

Maybe, Clemens is confusing this with the first recognition of the true cause of earth shine, the faint light reflected from the earth that makes the whole moon visible during the first crescent, a recognition that is often falsely attributed to Galileo. However, here the laurels go to Leonardo but who, as always, didn’t publish. The first published correct account was made by Michael Mästlin (1550–1631)

 Clemens’ next statement appears to me to be simply bizarre:

Galileo was the first person to accurately depict the moons of Jupiter (which he called “Medicean stars,” after his patron, the Florentine Medici family).

Galileo was the first to discover the moons of Jupiter, just one day ahead of Simon Marius, but to state that he accurately depicted them is somewhat more than an exaggeration. For Galileo and Marius, the moons of Jupiter were small points of light in the sky, the positions of which they recorded as ink dots on a piece of paper. To call this accurate depiction is a joke.

Somehow, I expect a higher standard of public information from the Beinecke Library, one of the world’s leading rare book depositories. 

Addendum 18:30 CEST: The post on the Beinecke blog that this post refers to has now been heavily edited. Everything I criticised has been either removed or corrected but without acknowledgement anywhere!

Renaissance Mathematicus 1 Beinecke Library 0!


Filed under History of Astronomy, Myths of Science