Category Archives: History of Mathematics

There is no year zero!

I realise that in writing this post I am wasting my time, pissing against the wind, banging my head against a brick wall and all the other colourful expressions in the English language that describe embarking on a hopeless endeavour but I am renowned for being a pedantic curmudgeon and so I soldier on into the jaws of disappointment and defeat. I shall attempt to explain carefully and I hope clearly why the 31st of December of the year 2019 does not mark the end of the second decade of the 21st century. I know, I know but I must.

The core of the problem lies in the fact that we possess two basic sets of counting numbers, cardinals and ordinals. Now cardinals have nothing to do with the Holy Roman Catholic Church, a family of birds or a baseball team from St. Louis but are the numbers we use to say how many items there are in a group, a collection, a heap or as the mathematician prefer to call it a set. Let us look at a well-known example:

I’ll sing you twelve, O

Green grow the rushes, O

What are your twelve, O?

Twelve for the twelve Apostles

Eleven for the eleven who went to heaven,

Ten for the ten commandments,

Nine for the nine bright shiners,

Eight for the April Rainers.

Seven for the seven stars in the sky,

Six for the six proud walkers,

Five for the symbols at your door,

Four for the Gospel makers,

Three, three, the rivals,

Two, two, the lily-white boys,

Clothed all in green, O

One is one and all alone

And evermore shall be so.

This is the final round of an old English counting song the meaning of several lines of which remain intriguingly obscure. Starting with the fourth line from the top we have a set of 12 Apostles i.e. the original twelve follower of Jesus. One line further in, we have a set of 11, who went to heaven, presumably the Apostles minus Judas Iscariot. And so we proceed, each line refers to a group or set giving to number contained in it.

In everyday life we use cardinal numbers all the time. I bought 6 eggs today. There are 28 children in Johnny’s class. My car has 4 wheels and so on and so forth. The cardinal numbers also contain the number zero (0), which indicates that a particular group or set under discussion contain no items at all. There are currently zero kings of France. We can carry out all the usually simple arithmetical operations–addition, subtraction, multiplication and division–on the cardinal numbers including zero, with the exception that we can’t divide by zero; mathematicians say division by zero is not defined. So if Johnny’s class with its 28 members are joined by Jenny’s class with 27 members for the school trip there will be 55 children on the bus. I’m sure you can think up lots of other examples yourselves.

Ordinal numbers have a different function, there signify the position of items in a list, row, series etc. We also use different names for ordinal numbers to cardinal numbers, so instead of one, two three four…, we say first, second, third, fourth…etc. an example would be, Johnny was the fifth person in his class to get the flu this winter. Now, in the ordinal numbers there is no zero, it would be a contradiction in terms, as it can’t exist. Occasionally when there is an existing ordered list of principles or laws people will talk about the ‘zeroeth’ law, meaning one that wasn’t originally included but that they think should precede the existing ones.

When we talk about years we tend to use the words for cardinal numbers but in fact we are actually talking about ordinal numbers. What we call 2019 CE or AD i.e. two thousand and nineteen is in fact the two thousand and nineteenth year of the Common Era or the two thousand and nineteenth year of Our Lord. Whichever system of counting years one uses, Gregorian, Jewish, Muslim, Persian, Chinese, Hindu or whatever there is and never can be a year zero, it is, as stated abve, a contradiction in terms and cannot exist. Therefore the first decade, that is a group of ten year, in your calendrical system consists of the years one to ten or the first year to the tenth year, the second decade the years eleven to twenty or the eleventh year to the twentieth year and so on. The first century, that is a group of one hundred years, consists of the years one to one hundred or the first year to the one-hundredth year. First millennium, that is one thousand years, consists of the years one to one thousand or the first year to the one-thousandth year.

Going back to our starting point the first decade of the 21st century started on the 1st January 2001 and finished on the 31st December 2010. The second decade started on the 1st January 2011 and will end on the 31st December 2020 and not on 31st December 2019 as various innumerate people would have you believe.



Filed under Calendrics, History of Mathematics, Myths of Science

Finding your way on the Seven Seas in the Early Modern Period

I spend a lot of my time trying to unravel and understand the complex bundle that is Renaissance or Early Modern mathematics and the people who practiced it. Regular readers of this blog should by now be well aware that the Renaissance mathematici, or mathematical practitioners as they are generally known in English, did not work on mathematics as we would understand it today but on practical mathematics that we might be inclined, somewhat mistakenly, to label applied mathematics. One group of disciplines that we often find treated together by one and the same practitioner consists of astronomy, cartography, navigation and the design and construction of tables and instruments to aid the study of these. This being the case I was delighted to receive a review copy of Margaret E. Schotte’s Sailing School: Navigating Science and Skill, 1550–1800[1], which deals with exactly this group of practical mathematical skills as applied to the real world of deep-sea sailing.

Sailing School001.jpg

Schotte’s book takes the reader on a journey both through time and around the major sea going nations of Europe, explaining, as she goes, how each of these nations dealt with the problem of educating, or maybe that should rather be training, seamen to become navigators for their navel and merchant fleets, as the Europeans began to span the world in their sailing ships both for exploration and trade.

Having set the course for the reader in a detailed introduction, Schotte sets sail from the Iberian peninsular in the sixteenth century. It was from there that the first Europeans set out on deep-sea voyages and it was here that it was first realised that navigators for such voyages could and probably should be trained. Next we travel up the coast of the Atlantic to Holland in the seventeenth century, where the Dutch set out to conquer the oceans and establish themselves as the world’s leading maritime nation with a wide range of training possibilities for deep-sea navigators, extending the foundations laid by the Spanish and Portuguese. Towards the end of the century we seek harbour in France to see how the French are training their navigators. Next port of call is England, a land that would famously go on, in their own estimation, to rule the seven seas. In the eighteenth century we cross the Channel back to Holland and the advances made over the last hundred years. The final chapter takes us to the end of the eighteenth century and the extraordinary story of the English seaman Lieutenant Riou, whose ship the HMS Guardian hit an iceberg in the Southern Atlantic. Lacking enough boats to evacuate all of his crew and passengers, Riou made temporary repairs to his vessel and motivating his men to continuously pump out the waters leaking into the rump of his ship, he then by a process of masterful navigation, on a level with his contemporaries Cook and Bligh, brought the badly damaged frigate to safety in South Africa.

Sailing School004

In each of our ports of call Schotte outlines and explains the training conceived by the authorities for training navigators and examines how it was or was not put into practice. Methods of determining latitude and longitude, sailing speeds and distances covered are described and explained. The differences in approach to this training developed in each of the sea going European nations are carefully presented and contrasted. Of special interest is the breach in understanding of what is necessary for a trainee navigator between the mathematical practitioners, who were appointed to teach those trainees, and the seamen, who were being trained, a large yawning gap between theory and practice. When discussing the Dutch approach to training Schotte clearly describes why experienced coastal navigators do not, without retraining, make good deep-sea navigators. The methodologies of these two areas of the art of navigation are substantially different.

The reader gets introduced to the methodologies used by deep-sea navigators, the mathematics developed, the tables considered necessary and the instruments and charts that were put to use. Of particular interest are the rules of thumb utilised to make course corrections before accurate methods of determining longitude were developed. There are also detailed discussions about how one or other aspect of the art of navigation was emphasised in the training in one country but considered less important in another. One conclusion the Schotte draws is that there is not really a discernable gradient of progress in the methods taught and the methods of teaching them over the two hundred and fifty years covered by the book.

Sailing School003.jpg

As well as everything you wanted to know about navigating sailing ships but were too afraid to ask, Schotte also delivers interesting knowledge of other areas. Theories of education come to the fore but an aspect that I found particularly fascinating were her comments on the book trade. Throughout the period covered, the teachers of navigation wrote and marketed books on the art of navigation. These books were fairly diverse and written for differing readers. Some were conceived as textbooks for the apprentice navigators whilst others were obviously written for interested, educated laymen, who would never navigate a ship. Later, as written exams began to play a greater role in the education of the aspirant navigators, authors and publishers began to market books of specimen exam questions as preparation for the exams. These books also went through an interesting evolution. Schotte deals with this topic in quite a lot of detail discussing the authors, publishers and booksellers, who were engaged in this market of navigational literature. This is detailed enough to be of interest to book historians, who might not really be interested in the history of navigation per se.

Schotte is excellent writer and the book is truly a pleasure to read. On a physical level the book is beautifully presented with lots of fascinating and highly informative illustrations. The apparatus starts with a very useful glossary of technical terms. There is a very extensive bibliography and an equally extensive and useful index. My only complaint concerns the notes, which are endnotes and not footnotes. These are in fact very extensive and highly informative containing lots of additional information not contained in the main text. I found myself continually leafing back and forth between main text and endnotes, making continuous reading almost impossible. In the end I developed a method of reading so many pages of main text followed by reading the endnotes for that section of the main text, mentally noting the number of particular endnotes that I wished to especially consult. Not ideal by any means.

This book is an essential read for anybody directly or indirectly interested in the history of navigation and also the history of practical mathematics. If however you are generally interested in good, well researched, well written history then you will almost certainly get a great deal of pleasure from reading this book.

[1] Margaret E. Schotte, Sailing School: Navigating Science and Skill, 1550–1800, Johns Hopkins University Press, Baltimore, 2019.


Filed under Book Reviews, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, Renaissance Science, Uncategorized

Calculus for the curious

Some weeks ago I got involved in a discussion on Twitter about, which books to recommend on the history of calculus. Somebody chimed in that Steven Strogatz’s new book would tell you all that you needed to know about the history of calculus. I replied that I couldn’t comment on this, as I hadn’t read it. To my surprise Professor Strogatz popped up and asked me if I would like to have a copy of his book. Never one to turn down a freebee, I naturally said yes. Very soon after a copy of Infinite Powers: The Story of Calculus The Language of the Universe arrived in the post and landed on my to read pile. Having now read it I can comment on it and intend to do so.

For those, who don’t know Steven Strogatz, he is professor of applied mathematics at Cornell University and the successful author of best selling popular books on mathematics.


First off, Infinite Powers is not a history of calculus. It is a detailed introduction to what calculus is and how it works, with particular emphasis on its applications down the centuries, Strogatz is an applied mathematician, presented in a history-light frame story. Having said this, I’m definitely not knocking, what is an excellent book but I wouldn’t recommend it to anybody, who was really looking for a history of calculus, maybe, however, either as a prequel or as a follow up to reading a history of calculus.

The book is centred on what Strogatz calls The Infinity Principle, which lies at the heart of the whole of calculus:

To shed light on any continuous shape, object, motion, process, or phenomenon–no matter how wild and complicated it may appear–reimagine it as an infinite series of simpler parts, analyse those, and then add the results back together to make sense of the original whole.

Following the introduction of his infinity principle Strogatz gives a general discussion of its strengths and weakness before moving on in the first chapter proper to discuss infinity in all of its guises, familiar material and examples for anybody, who has read about the subject but a well done introduction for those who haven’t. Chapter 2 takes us  into the early days of calculus, although it didn’t yet have this name, and introduces us to The Man Who Harnessed Infinity, the legendary ancient Greek mathematician Archimedes and the method of exhaustion used to determine the value of π and the areas and volumes of various geometrical forms. Astute readers will have noticed that I wrote early days and not beginning and here is a good example of why I say that this is not a history of calculus. Although Archimedes put the method of exhaustion to good use he didn’t invent it, Eudoxus did. Strogatz does sort of mention this in passing but whereas Archimedes gets star billing, Eudoxus gets dismissed in half a sentence in brackets. The reader is left completely in the dark as to who, why, what Eudoxus is/was. OK here, but not OK in a real history of calculus. This criticism might seem petty but there are lots of similar examples throughout the book that I’m not going to list in this review and this is why the book is not a history of calculus and I don’t think Strogatz intended to write one; the book he has written is a different one and it is a very good one.

After Archimedes the book takes a big leap to the Early Modern Period and Galileo and Kepler with the justification that, “When Archimedes died, the mathematical study of nature nearly died along with him. […] In Renaissance Italy, a young mathematician named Galileo Galilei picked up where Archimedes had left off.” My inner historian of mathematics had an apoplectic fit on reading these statements. They ignore a vast amount of mathematics, in particular the work in the Middle Ages and the sixteenth century on which Galileo built the theories that Strogatz then presents here but I console myself with the thought that this is not a history of calculus let alone a history of mathematics. However, I’m being too negative, let us return to the book. The chapter deals with Galileo’s terrestrial laws of motion and Kepler’s astronomical laws of planetary motion. Following this brief introduction to the beginnings of modern science Strogatz moves into top gear with the beginnings of differential calculus. He guides the reader through the developments of seventeenth century mathematics, Fermat and Descartes and the birth of analytical geometry bringing together the recently introduced algebra and the, by then, traditional geometry. Moving on he deals with tangents, functions and derivatives. Strogatz is an excellent teacher he introduces a new concept carefully, explains it, and then shows how it can be applied to an everyday situation.

Having laid the foundations Strogatz move on naturally to the supposed founders of modern calculus, Leibnitz and Newton and their bringing together of the strands out of the past that make up calculus as we know it and how they fit together in the fundamental theorem of calculus. This is interwoven with the life stories of the two central figures. Again having introduced concepts and explained them Strogatz illustrates them with applications outside of pure mathematics.

Having established modern calculus the story moves on into the eighteenth century.  Here I have to point out that Strogatz perpetuates a couple of myths concerning Newton and the writing of his Principia. He writes that Newton took the concept of inertia from Galileo; he didn’t, he took it from Descartes, who in turn had it from Isaac Beeckman. A small point but as a historian I think an important one. Much more important he seems to be saying that Newton created the physics of Principia using calculus then translated it back into the language of Euclidian geometry, so as not to put off his readers. This is a widely believed myth but it is just that, a myth. To be fair it was a myth put into the world by Newton himself. All of the leading Newton experts have over the years very carefully scrutinised all of Newton’s writings and have found no evidence that Newton conceived and wrote Principia in any other form than the published one. Why he rejected the calculus, which he himself developed, as a working tool for his magnum opus is another complicated story that I won’t go into here but reject it he did[1].

After Principia, Strogatz finishes his book with a random selection of what might be termed calculus’ greatest hits, showing how it proved its power in solving a diverse series of problems. Interestingly he also addresses the future. There are those who think that calculus’ heyday is passed and that other, more modern mathematical tools will in future be used in the applied sciences to solve problems, Strogatz disagrees and sees a positive and active future for calculus as a central mathematical tool.

Despite all my negative comments, and I don’t think my readers would expect anything else from me, given my reputation, I genuinely think that this is on the whole an excellent book. Strogatz writes well and fluidly and despite the, sometimes, exacting content his book is a pleasure to read. He is also very obviously an excellent teacher, who is very good at clearly explaining oft, difficult concepts. I found it slightly disappointing that his story of calculus stops just when it begins to get philosophical and logically interesting i.e. when mathematicians began working on a safe foundation for the procedures that they had been using largely intuitively. See for example Euler, who made great strides in the development of calculus without any really defined concepts of convergence, divergence or limits, but who doesn’t appear here at all. However, Strogatz book is already 350-pages-long and if, using the same approach, he had continued the story down to and into the twentieth century it would probably have weighed in at a thousand plus pages!

Despite my historical criticisms, I would recommend Strogatz’s book, without reservations, to anybody and everybody, who wishes to achieve a clearer, deeper and better understanding of what calculus is, where it comes from, how it functions and above all, and this is Strogatz’s greatest strength, how it is applied to the solution of a wide range of very diverse problems in an equally wide and diverse range of topics.


[1] For a detailed analysis of Newton’s rejection of analytical methods in mathematics then I heartily recommend, Niccolò Guicciardini, Reading the Principia, CUP, 1999, but with the warning that it’s not an easy read!



Filed under Book Reviews, History of Mathematics

Mathematical aids for Early Modern astronomers.

Since its very beginnings in the Fertile Crescent, European astronomy has always involved a lot of complicated and tedious mathematical calculations. Those early astronomers described the orbits of planets, lunar eclipses and other astronomical phenomena using arithmetical or algebraic algorithms. In order to simplify the complex calculations needed for their algorithms the astronomers used pre-calculated tables of reciprocals, squares, cubes, square roots and cube roots.


Cuniform reciprocal table Source

The ancient Greeks, who inherited their astronomy from the Babylonians, based their astronomical models on geometry rather than algebra and so needed other calculation aids. They developed trigonometry for this work based on chords of a circle. The first chord tables are attributed to Hipparkhos (c. 190–c. 120 BCE) but they did not survive. The oldest surviving chord tables are in Ptolemaeus’ Mathēmatikē Syntaxis written in about 150 CE, which also contains a detailed explanation of how to calculate such a table in Chapter 10 of Book I.


Ptolemaeus’ Chord Table taken from Toomer’s Almagest translation. The 3rd and 6th columns are the interpolations necessary for angles between the given ones

Greek astronomy travelled to India, where the astronomers replaced Ptolemaeus’ chords with half chords, that is our sines. Islamic astronomers inherited their astronomy from the Indians with their sines and cosines and the Persian astronomer Abū al-Wafāʾ (940–998 CE) was using all six of the trigonometrical relations that we learnt at school (didn’t we!) in the tenth century.


Abū al-Wafāʾ Source: Wikimedia Commons

Astronomical trigonometry trickled slowly into medieval Europe and Regiomontanus (1536–1576)  (1436–1476) was the first European to produce a comprehensive work on trigonometry for astronomers, his De triangulis omnimodis, which was only edited by Johannes Schöner and published by Johannes Petreius in 1533.

Whilst trigonometry was a great aid to astronomers calculating trigonometrical tables was time consuming, tedious and difficult work.

A new calculating aid for astronomers emerged during the sixteenth century, prosthaphaeresis, by which, multiplications could be converted into additions using a series of trigonometrical identities:

Prosthaphaeresis appears to have first been used by Johannes Werner (1468–1522), who used the first two formulas with both sides multiplied by two.

However Werner never published his discovery and it first became known through the work of the itinerant mathematician Paul Wittich (c. 1546–1586), who taught it to both Tycho Brahe (1546–1601) on his island of Hven and to Jost Bürgi (1552–1632) in Kassel, who both developed it further. It is not known if Wittich learnt the method from Werner’s papers on one of his visits to Nürnberg or rediscovered it for himself. Bürgi in turn taught it to Nicolaus Reimers Baer (1551–1600) in in exchange translated Copernicus’ De revolutionibus into German for Bürgi, who couldn’t read Latin. This was the first German translation of De revolutionibus. As can be seen the method of prosthaphaeresis spread throughout Europe in the latter half of the sixteenth century but was soon to be superceded by a superior method of simplifying astronomical calculations by turning multiplications into additions, logarithms.

As is often the case in the histories of science and mathematics logarithms were not discovered by one person but almost simultaneously, independently by two, Jost Bürgi and John Napier (1550–1617) and both of them seem to have developed the idea through their acquaintance with prosthaphaeresis. I have already blogged about Jost Bürgi, so I will devote the rest of this post to John Napier.


John Napier, artist unknown Source: Wikimedia Commons

John Napier was the 8th Laird of Merchiston, an independently owned estate in the southwest of Edinburgh.


Merchiston Castle from an 1834 woodcut Source: Wikimedia Commons

His exact date of birth is not known and also very little is known about his childhood or education. It is assumed that he was home educated and he was enrolled at the University of St. Andrews at the age of thirteen. He appears not to have graduated at St. Andrews but is believed to have continued his education in Europe but where is not known. He returned to Scotland in 1571 fluent in Greek but where he had acquired it is not known. As a laird he was very active in the local politics. His intellectual reputation was established as a theologian rather than a mathematician.

It is not known how and when he became interested in mathematics but there is evidence that this interest was already established in the early 1570s, so he may have developed it during his foreign travels. It is thought that he learnt of prosthaphaeresis through John Craig (d. 1620) a Scottish mathematician and physician, who had studied and later taught at Frankfurt an der Oder, a pupil of Paul Wittich, who knew Tycho Brahe. Craig returned to Edinburgh in 1583 and is known to have had contact with Napier. The historian Anthony à Wood (1632–1695) wrote:

one Dr. Craig … coming out of Denmark into his own country called upon John Neper, baron of Murcheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus as ’tis said) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportionable numbers. [Neper is the Latin version of his family name]

Napier is thought to have begum work on the invention of logarithms about 1590. Logarithms exploit the relation ship between arithmetical and geometrical series. In modern terminology, as we all learnt at school, didn’t we:

Am x An = Am+n

Am/An = Am-n

These relationships were discussed by various mathematicians in the sixteenth century, without the modern notation, in particularly by Michael Stefil (1487–1567) in his Arithmetica integra (1544).


Michael Stifel Source: Wikimedia Commons


Michael Stifel’s Arithmetica Integra (1544) Source: Wikimedia Commons

What the rules for exponents show is that if one had tables to convert all numbers into powers of a given base then one could turn all multiplications and divisions into simple additions and subtractions of the exponents then using the tables to covert the result back into a number. This is what Napier did calling the result logarithms. The methodology Napier used to calculate his tables is too complex to deal with here but the work took him over twenty years and were published in his Mirifici logarithmorum canonis descriptio… (1614).


Napier coined the term logarithm from the Greek logos (ratio) and arithmos (number), meaning ratio-number. As well as the logarithm tables, the book contains seven pages of explanation on the nature of logarithms and their use. A secondary feature of Napier’s work is that he uses full decimal notation including the decimal point. He was not the first to do so but his doing so played an important role in the acceptance of this form of arithmetical notation. The book also contains important developments in spherical trigonometry.

Edward Wright  (baptised 1561–1615) produced an English translation of Napier’s Descriptio, which was approved by Napier, A Description of the Admirable Table of Logarithmes, which was published posthumously in 1616 by his son Samuel.


Gresham College was quick to take up Napier’s new invention and this resulted in Henry Briggs (1561–1630), the Gresham professor of geometry, travelling to Edinburgh from London to meet with Napier. As a result of this meeting Briggs, with Napier’s active support, developed tables of base ten logarithms, Logarithmorum chilias prima, which were publish in London sometime before Napier’s death in 1617.


He published a second extended set of base ten tables, Arithmetica logarithmica, in 1624.


Napier’s own tables are often said to be Natural Logarithms, that is with Euler’s number ‘e’ as base but this is not true. The base of Napierian logarithms is given by:

NapLog(x) = –107ln (x/107)

Natural logarithms have many fathers all of whom developed them before ‘e’ itself was discovered and defined; these include the Jesuit mathematicians Gregoire de Saint-Vincent (1584–1667) and Alphonse Antonio de Sarasa (1618–1667) around 1649, and Nicholas Mercator (c. 1620–1687) in his Logarithmotechnia (1688) but John Speidell (fl. 1600–1634), had already produced a table of not quite natural logarithms in 1619.


Napier’s son, Robert, published a second work by his father on logarithms, Mirifici logarithmorum canonis constructio; et eorum ad naturales ipsorum numeros habitudines, posthumously in 1619.


This was actually written earlier than the Descriptio, and describes the principle behind the logarithms and how they were calculated.

The English mathematician Edmund Gunter (1581–1626) developed a scale or rule containing trigonometrical and logarithmic scales, which could be used with a pair of compasses to solve navigational problems.


Table of Trigonometry, from the 1728 Cyclopaedia, Volume 2 featuring a Gunter’s scale Source: Wikimedia Commons

Out of two Gunter scales laid next to each other William Oughtred (1574–1660) developed the slide rule, basically a set of portable logarithm tables for carry out calculations.

Napier developed other aids to calculation, which he published in his Rabdologiae, seu numerationis per virgulas libri duo in 1617; the most interesting of which was his so called Napier’s Bones.


These are a set of multiplication tables embedded in rods. They can be used for multiplication, division and square root extraction.


An 18th century set of Napier’s bones Source: Wikimedia Commons

Wilhelm Schickard’s calculating machine incorporated a set of cylindrical Napier’s Bones to facilitate multiplication.

The Swiss mathematician Jost Bürgi (1552–1632) produced a set of logarithm tables independently of Napier at almost the same time, which were however first published at Kepler’s urging as, Arithmetische und Geometrische Progress Tabulen…, in 1620. However, unlike Napier, Bürgi delivered no explanation of the how his table were calculated.


Tables of logarithms became the standard calculation aid for all those making mathematical calculations down to the twentieth century. These were some of the mathematical tables that Babbage wanted to produce and print mechanically with his Difference Engine. When I was at secondary school in the 1960s I still carried out all my calculations with my trusty set of log tables, pocket calculators just beginning to appear as I transitioned from school to university but still too expensive for most people.


Not my copy but this is the set of log tables that accompanied me through my school years

Later in the late 1980s at university in Germany I had, in a lecture on the history of calculating, to explain to the listening students what log tables were, as they had never seen, let alone used, them. However for more than 350 years Napier’s invention served all those, who needed to make mathematical calculations well.














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

Mathematics or Physics–Mathematics vs. Physics–Mathematics and Physics

Graham Farmelo is a British physicist and science writer. He is the author of an excellent and highly praised biography of the British physicist P A M Dirac, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius(Faber and Faber, 2009), which won a couple of book awards. He is also the author of a book Winston Churchill role in British war time nuclear research, Churchill’s Bomb:A hidden history of Britain’s first nuclear weapon programme (Faber and Faber, 2014), which was also well received and highly praised. Now he has published a new book on the relationship between mathematics and modern physics, The Universe Speaks in Numbers: How Modern Maths Reveals Nature’s Deepest Secrets (Faber and Faber, 2019).


I must admit that when I first took Farmelo’s new book into my hands it was with somewhat trepidation. Although, I studied mathematics to about BSc level that was quite a few years ago and these days my active knowledge of maths doesn’t extend much beyond A-Level and I never studied physics beyond A-Level and don’t ask what my grade was. However, I did study a lot of the history of early twentieth century physics before I moved back to the Renaissance. Would I be able to cope with Farmelo’s book? I needn’t have worried there are no complex mathematical or physical expressions or formulas. Although I would point out that this is not a book for the beginner with no knowledge; if your mind baulks at terms like gauge theory, string theory or super symmetry then you should approach this text with caution.

The book is Farmelo’s contribution to the debate about the use of higher mathematics to create advanced theories in physics that are not based on experimental evidence or even worse confirmable through experiment. It might well be regarded as a counterpoint to Sabine Hossenfelder’s much discussed Lost in Math: How Beauty Leads Physics Astray(Basic Books, 2018), which Farmelo actually mentions on the flyleaf to his book; although he obviously started researching and writing his volume long before the Hossenfelder tome appeared on the market. The almost concurrent appearance of the two contradictory works on the same topic shows that the debate that has been simmering just below the surface for a number of years has now boiled over into the public sphere.

Farmelo’s book is a historical survey of the relationship between advanced mathematics and theoretical physics since the seventeenth century, with an emphasis on the developments in the twentieth century. He is basically asking the questions, is it better when mathematics and physics develop separately or together and If together should mathematics or physics take the lead in that development. He investigated this questions using the words of the physicists and mathematicians from their published papers, from public lectures and from interviews, many of which for the most recent developments he conducted himself. He starts in the early seventeenth century with Kepler and Galileo, who, although they used mathematics to express their theories, he doesn’t think really understand or appreciate the close relationship between mathematics and physics. I actually disagree with him to some extent on this, as he knows. Disclosure: I actually read and discussed the opening section of the book with him, at his request, when he was writing it but I don’t think my minuscule contribution disqualifies me from reviewing it.

For Farmelo the true interrelationship between higher mathematics and advanced theories in physics begins with Isaac Newton. A fairly conventional viewpoint, after all Newton did title his magnum opus The Mathematical Principles of Natural Philosophy. I’m not going to give a decade by decade account of the contents, for that you will have to read the book but he, quite correctly, devotes a lot of space to James Clerk Maxwell in the nineteenth century, who can, with justification, be described as having taken the relationship between mathematics and physics to a whole new level.

Maxwell naturally leads to Albert Einstein, a man, who with his search for a purely mathematical grand unification theory provoked the accusation of having left the realm of experiment based and experimentally verifiable physics; an accusation that led many to accuse him of having lost the plot. As the author of a biography of Paul Dirac, Farmelo naturally devote quite a lot of space to the man, who might be regarded as the mathematical theoretical physicist par excellence and who, as Farmelo emphasises, preached a gospel of the necessity of mathematically beautiful theories, as to some extent Einstein had also done.

Farmelo takes us through the creation of quantum mechanics and the attempts to combine it with the theories of relativity, which takes the reader up to the early decades following the Second World War, roughly the middle of the book. Here the book takes a sharp turn away from the historical retelling of the emergence of modern theoretical physics to the attempts to create a fundamental theory of existence using purely mathematical methods, read string theory, M theory, supersymmetry and everything associated with them. This is exactly the development in modern physics that Hossenfelder rejects in her book.

Farmelo is very sympathetic to the mathematicians and physicists, who have taken this path but he is in his account very even handed, letting the critics have their say and not just the supporters. His account is very thorough and documents both the advances and the disappointments in the field over the most recent decades. He gives much emphasis to the fruitful co-operations and exchanges that have taken place between mathematicians and theoretical physicists. I must say that as somebody who has followed the debate at a distance, having read Farmelo’s detailed account I came out of it more sympathetic to Hossenfelder’s standpoint than his.

As always with his books Farmelo’s account is excellently researched, much of the more recent material is based on interviews he conducted with the participants, and very elegantly written. Despite the density of the material he is dealing with, his prose is light and often witty, which makes it easier to grapple with the complex themes he is discussing. I would certainly recommend this book to anybody interested in the developments in modern theoretical physics; maybe to be read together with Hossenfelder’s volume. I would also make an excellent present for any young school leaver contemplating studying physics or one that had already started on down that path.


Filed under Book Reviews, History of Mathematics, History of Physics

Everything you wanted to know about Simon Marius and were too afraid to ask – now in English

Regular readers of this blog should by now be well aware of the fact that I belong to the Simon Marius Society a small group of scholars mostly from the area around Nürnberg, who dedicate some of their time and energy to re-establishing the reputation of the Franconian mathematicus Simon Marius (1573–1625), who infamously discovered the four largest moons of Jupiter literally one day later than Galileo Galilei and got accused of plagiarism for his troubles. Galileo may have discovered them first but Marius won, in the long term, the battle to name them.


Frontispiece of Mundus Iovialis Source:Wikimedia Commons

In 2014 the Simon Marius Society organised many activities to celebrate the four-hundredth anniversary of the publication of his opus magnum, Mundus Jovialis (The World of Jupiter). Amongst other things was an international conference held in Nürnberg, which covered all aspects of Marius’ life and work. The papers from this conference were published in German in 2016: Simon Marius und seine Forschung (Acta Historica Astronomiae), (AVA, Leipzig).


Now after much effort and some delays the expanded translation, now includes the full English text of Mundus Jovialis, has become available in English: Simon Marius and his Research, Springer, New York, 2019.


The ebook is already available and the hardback version will become available on 19 August. I apologize for the horrendous price but the problem of pricing by academic publishers is sadly well known. Having copyedited the entire volume, which means I have read the entire contents very carefully I can assure you that there is lots of good stuff to read not only about Simon Marius but also about astronomy, astrology, mathematics, court life in the seventeenth century and other topics of historical interest. If you can’t afford a copy yourself try to persuade you institutional library to buy one! If your university library buys a copy from Springer then students can order, through the library, a somewhat cheaper black and white copy of the book.


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

History of science on the Internet – the gift that keeps giving

Dear readers it is time once again for the Hist–Sci Hulk to flex his muscles and give a couple of Internet authors a good kicking for the crime of spreading history of science nonsense. First up we have a website called StarTeach Astronomy Education, who describe themselves as follows:


StarTeach Astronomy Education is a multimedia web-based astronomy education program designed especially for K-12 students and their teachers. The primary goal of the program is to aid in the classroom instruction of astronomy by providing a multimedia supplement that is available to schools with internet capability. In addition, it can also be used as research material for individual students.

The StarTeach Astronomy Education Program was created by Leslie Welser when she was a graduate student in the University of Nevada, Reno Physics Department. StarTeach has been recently updated to include topics for more advanced readers.

The section to which my attention was inadvertently directed was under the drop down menu Ancient Cultures with the title Arab and Islamic Astronomy.

Alone this title causes problem. Only yesterday I had an Internet exchange with Peter Adamson, the excellent creator of the History of Philosophy Without Any Gaps podcast and author of many books and articles based on that podcast, how exactly to describe the philosophy and science produced in the Middle Ages by the inhabitants of those areas of the world dominated by Islamic culture. The answer is certainly not Arab and Islamic, which would seem to suggest two separate areas of production or if only one that the astronomy was only produced by Islamic Arabs. The latter is definitively not the case as we have Arabs, Jews, Persians, Syriac Christians and various others. The subject is complex enough to warrant its own blog post, which I might or might not write sometime in the future.

The article opens with the following:

During the period when Western civilization was experiencing the dark ages [my emphasis], between 700-1200 A.D., an Islamic empire stretched from Central Asia to southern Europe.

The sound you can hear echoing around the world, is the sound of thousands of medievalist weeping and rending their clothes at the use of the term ‘the dark ages’. The preferred term is Early Middle ages. I should point out that the Arabic-Islamic dominance of a substantial part of the Eurasian landmass extended several centuries past 1200 CE and one of the things that Peter Adamson and I discussed is the fact that this empire actually split up fairly early into competing caliphates. Leaving out some minor quibbles we now come to this:

Another important religious use for astronomy was for the determination of latitude and longitude. Using the stars, particularly the pole star, as guides, several tables were compiled which calculated the latitude and longitude of important cities in the Islamic world.

Whereas you can determine latitude using altitude of the sun or the pole star and a bit of relatively simple trigonometry, you cannot determine longitude in this way. In fact the only way available in the Middle Ages to determine longitude astronomically is by simultaneous observation of eclipses, lunar or solar, and then comparing the timings. Next up with have:

Aside from religious uses, astronomy was used as a tool for navigation. The astrolabe, an instrument which calculated the positions of certain stars in order to determine direction, was invented by the Greeks and adopted and perfected by the Arabs (see picture below).

To quote David King, one of the world’s leading authorities on both the astrolabe and Islamic astronomy, the astrolabe was never used for navigation! It was actually the next sentences that finally provoked me into writing this blog post:

The sextant was developed by the Arabs to be a more sophisticated version of the astrolabe. This piece of technology ultimately became the cornerstone of navigation for European exploration.

When I first read this wonderful bullshit-double-whammy, I seriously started banging my head on my desk and praying for quick release from my pain. We are here confronted by a common problem in the history of science, the use of the same name for two or more completely different instruments. Our, by the way anonymous, I wouldn’t want to attach my name to this crap either, author apparently has no idea that the astronomical sextant and the navigational sextant are two completely different beast that apart from a scale representing one sixth of a circle, hence the name, have very little in common.

The astronomical sextant, to which the author intended to refer here, is anything but more sophisticated than the astrolabe and is in fact much, much simpler. It is a simple scale engraved on a segment, one sixth of a circle, with sights mounted on it used to determine the position of stars. I refer the reader to the Wikipedia article for more details. The navigational sextant, also not derived from the astrolabe, is a much more complex beast invented in the seventeenth century, the story of which you can read here. A change of topic:

Science was considered the ultimate scholarly pursuit in the Islamic world, and it was strongly supported by the nobility. Most scientists worked in the courts of regional leaders, and were financially rewarded for their achievements. In 830, the Khalifah, al-Ma’muun, founded Bayt-al-Hikman, the ‘House of Wisdom’, as a central gathering place for scholars to translate texts from Greek and Persian into Arabic. These texts formed the basis of Islamic scientific knowledge.

I think Qur’anic scholars might well dispute the claim that “Science was considered the ultimate scholarly pursuit in the Islamic world.” Another topic to which I might one day devote a whole blog post is the fact that the House of Wisdom, as it is usually presented, such as here, is a myth. For details see Dimitri Gutas, Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early ‘Abbāsid Society (2nd–4th/8th–10thcenturies).

I could go on and on but I will just consider one last gem:

One of the greatest Islamic astronomers was al-Khwarizmi (Abu Ja’far Muhammad ibn Musa Al-Khwarizmi), who lived in the 9th century and was the inventor of algebra[my emphasis].

The ancient Babylonians and the Indians would be amazed to discover that al-Khwarizmi invented algebra. His book, Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala, introduced algebra into medieval Europe and part of its title gave the discipline its modern name but algebra was being practice literally millennia before al-Khwarizmi was born.

Interestingly this dog’s dinner of an article, that is intended to educate school kids, isn’t based on academic literature on the subject but offers as its sources links to five other Internet articles, three of which no longer exist and one of which is about the navigational sextant and as such irrelevant.

My second candidate for a Hist_Sci Hulk kicking is from Quillette a self styled platform for free thought and is a review by Jared Marcel Pollen (whoever he is) of Ben Shapiro’s The Right Side of History. An extremely tedious review of an extremely tedious book by an extremely tedious person that wouldn’t normally occupy my thought stream for longer than thirty seconds were it not for the following totally fucked up piece of history of astronomy.

This also allows Shapiro to skirt the obvious hostility the church showed toward intellectual inquiry for centuries. Shapiro writes as if the church had never banned a book or burned a heretic:

Contrary to popular opinion, new discoveries weren’t invariably seen as heretical or dangerous to the dominion of the Church; in fact, the Church often supported scientific investigation.

The Church was indeed the only place where any kind of inquiry could be conducted at the time. It was the only place where people were literate and enjoyed steady funding and access to instruments. But the Church only encouraged inquiry to the extent that it could reinforce and expand its own doctrine, which is a bit like the state telling you that you are free to make whatever art you like, just as long as you don’t criticize the regime.

What we have here is the standard popular presentation of the medieval Catholic Church as some sort of all-powerful totalitarian state. The reality was actually somewhat different. The Church was not able to exercise the sort of total intellectual control that Pollen is here claiming and surprisingly diverse positions were possible and existed in reality .

Also Pollens claim that, “The Church was indeed the only place where any kind of inquiry could be conducted at the time. It was the only place where people were literate and enjoyed steady funding and access to instruments” is of course total rubbish. Just to take the example of the three leading writers on astronomy in the reception phase of Copernican heliocentricity–Tycho Brahe, Johannes Kepler and Galileo Galilei–all three of them were court mathematicians outside of the various churches. There are plenty of other examples of important and influential scholars from this period, who found employment outside of the churches.

The first major scientific challenge to the Church was heliocentrism. But Shapiro claims this was hardly an issue: “Nicolas Copernicus studied in parochial school and served the Church of Warmia as medical advisor; his publication of De revolutionibus… in March 1542, included a letter to Pope Paul III.”

Copernicus was in fact a canon of Frombork cathedral and as such part of the government of the prince-bishopric of Warmia. His role was much wider than that of cathedral physician. De revolutionibus was published in 1543 not 1542 but did in fact contain a dedicatory letter to Pope Paul II.

In fact, Copernicus had finished his treatise years earlier (there are records indicating that the manuscript had been completed as early as the 1530s), but he withheld it, aware that its publication could be life-threatening, and circulated only a few anonymous copies to his close friends.

The bulk of De revolutionibus was, as far as we can tell, finished by about 1530, however the reasons for Copernicus not publishing at that time are complex and contrary to popular opinion had very little to do with any fear of persecution by the Church and its publication would certainly not have been life-threatening; that claim is complete rubbish based on hindsight and fail interpretations of the cases of Giordano Bruno and Galileo Galilei. Despite censoring Copernicus for his relationship with his housekeeper, Danticus, the Prince-Bishop of Warmia, supported Copernicus work and even invited Gemma Frisius, who Danticus supported, to come to Frombork to work with Copernicus. Tiedmann Giese, Bishop of Kulm, an influential cleric and Copernicus’ best friend, had been urging him to publish for years. Lastly Nicholas Schönberg, Cardinal of Capua, wrote a letter to Copernicus urging him to make his work public. This letter was included in the front material of De revolutionibus. As can be seen Copernicus had solid support for his work within the Church hierarchy. After its publication there was no initial opposition to De revolutionibus from the side of the Church. Copernicus is considered to have held back with publication because he couldn’t provide the proof of the heliocentric hypothesis that he wished to. There were no anonymous copies of De revolutionibus circulated to close friends! I assume our author is confusing De revolutionibus with the Commentariolus, Copernicus unpublished manuscript, which first propagated his heliocentric hypothesis from around 1510.

The book was only published in its entirety on the eve of Copernicus’s death, and the letter to the pope, which was also anonymous, was not written by Copernicus, but by Andreas Osiander, a Lutheran preacher who had been given the job of overseeing the book’s publication. It was an attempt to soften the blow, and states, inter alia, that the author’s findings are only meant to aid the computation of the heavens, and do not even need to be considered true in order for the calculations to be useful.

The implication that Copernicus only gave De revolutionibus free for publication when he was dying is once again total rubbish. Rheticus took the finished manuscript away from Frombork in September 1541 more than eighteen months before Copernicus’ demise. The next sentence is mindboggling for anybody who knows anything about De revolutionibus and its publication history. Copernicus dedicated De revolutionibus to His Holiness Pope Paul III. The anonymous text added by Andreas Osiander during publication was the infamous ad lectorem a totally different text altogether.

The Church would continue to uphold the geocentric model for at least another 150 years, and wouldn’t get around to officially pardoning Galileo until 1992. However, Shapiro claims the persecution of Galileo was merely a PR move by the Church; an attempt to crack down on dissent in response to Protestant accusations of leniency and hypocrisy. The trial of Galileo also saw dozens of astronomical works, including De Revolutionibus, placed on the Church’s “List of Prohibited Books”

Not the geocentric but the geo-heliocentric model of Tycho Brahe, was upheld not only by the Church but also by a very large number astronomers for many decades because the heliocentric model proved extremely difficult to prove. I can’t really comment on Shapiro’s claim, not having read his book and not intending to do so, but whatever the Church’s reasons for the persecution Galileo, and they are very complex, they have absolutely nothing to do with Protestant accusations of leniency and hypocrisy. Galileo is said to have obtained permission from Urban to write his Dialogo, because he claimed that the Protestants were mocking the Catholic Church because of its ignorant stance in the astronomy/cosmology debate. It would appear that Shapiro got his fact confused to put it diplomatically.

“Dozens of astronomical works” is a crass exaggeration and those books that were placed on the Index were placed there in 1616, following Galileo’s first run in with the Church and not in 1633 following his trial. Interestingly, and I get weary of repeating this, De revolutionibus was only placed on the Index until corrected, which surprisingly was completed by 1620; the corrected version with only those parts corrected which claimed heliocentricity to be a fact rather than a hypothesis, was then given free by the Church.

What we have here in total is a collection of half remember facts and myths stirred up together with an added sauce of nonsense and then spewed onto the page without any consideration for truth, facts or accuracy. Kind of sums up Quillette in my opinion.






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