July 2, 2026 · 8:35 am

Just some guy with a blog

When somebody wants to dispute something that I say about the history of science or the history of mathematics on social media and can’t actually refute it with facts, they often resort to a comment like the one that is the title of this post. It is of course factually correct; I am just some guy with a blog and I’m even prepared to admit that the only real formal qualifications that I have are a very ancient set of very ropey A-levels. I’m quite happy to admit that I am to a large extent an autodidact but I do have quite a large amount of formal academic training acquired over the years. A lot of what I am now going to relate is already known to regular readers of this blog but I think it is worth repeating for any new readers who have joined since I last exposed the details of my somewhat unorthodox life journey.

I read my first book on the history of mathematics at about the age of five, Lancelot Hogben’s Man Must Measure, which I assume had belonged to my brother, who was six years older than me. I was a very precocious child and skipped not one but two grades in primary school and was still the most intelligent child in the class. I spent three years in the highest grade and at the end was teaching myself from the teacher’s books whilst the teacher taught the rest of the class. There was, however, a small problem, I couldn’t spell, with eleven I couldn’t even spell my own surname, and the proverbial drunken spider that had fallen into an ink well was a calligraphy master in comparison to me. Much value was attributed to spelling and good handwriting in my primary school and so I suffered, “with your intelligence!” I of course suffered from a learning disability , two in fact, but I didn’t know that at the time and in fact only discovered it when I was already over fifty years old but more of that later.

By now I was avidly reading of all the articles on numbers, number systems and mathematics in the Children’s Brittanica, a copy of which we had at home. But despite having sailed through my eleven plus and an emergency appendectomy all within twelve hours, I basically gave up on the school system because of my writing difficulties. As I have related in an earlier post, my mother died when I was fifteen and I ended up spending my fifth form and first year sixth as a boarder in the grammar school where I had previously been a day boy. At the end of my fifth year my boarding house master asked  me if I would be returning next year. I replied that depends on the O-level examiners. He replied that in that case he didn’t expect to see me. He was more than somewhat surprised when I got by far and away the best O-level results of any of the fifth form boarders. 

I now entered the first year sixth and was confronted with a new school headmaster, who played the role of the new broom that sweeps clean. I, shambling along in my usual couldn’t care less style, soon landed in his study. He asked me what I wished to study at university and I replied history. I had been deeply in love with history since I taught myself to read at three. He naturally then asked, why I was studying science A-levels to which I replied, because that’s what I’m good at. He then suggested that I could study archaeology with science A-levels. I wasn’t particularly keen on the idea, my father was an archaeologist, but I ended up going on my first excavation in the Easter holidays. I really enjoyed it and went on a much bigger and longer excavation in the summer holidays. The University of Cardiff excavation of the Roman fort at Usk in South Wales. I had in the meantime been expelled from my boarding school. 

I was now living in London, before being expelled during the school holidays and after full time. Sometime around then my father gave me his copy of Eric Temple Bell’s Men of Mathematics, a very inspiring book but historically highly inaccurate. I spent many years correcting in my head Bell’s historical errors. However, at the time I simply became addicted to the history of mathematics, lock, stock and barrel. Over the following years I would seek out and acquire books on the history of mathematics and later the history of science as well. Out of those early autodidact years I still have my copies of Carl B Boyer’s A History of Mathematics, Morris Kline’s Mathematical Thought from Ancient toModern Times, Arthur Koestler’s The Sleepwalkers, and Jacob Bronowski’s The Ascent of Man.

Back to the chronological narrative. I spent my A-level year at the then notorious Holland Park Comprehensive, the flagship of the Labour Government’s comprehensive education policy. I spent that school year smoking vast quantities of cannabis, going to amazing rock concerts in the nights and dropping acid about once a week. Come on, it was the schoolyear 69/70 and Holland Park-Ladbroke Grove was the centre of the British counterculture in the 1960s and on into the 1970s. It is not a surprise that my A-level results were, to put it mildly, bad. It’s a fucking miracle that I got any A–levels at all. 

I still managed to get a place to study archaeology at University College Cardiff, in those days one of the leading universities for the subject. I had spent another summer digging at Usk before the A-level results came out  and a lecturer recommended my selection for a place based on having got to know me on those excavation, I think it was more likely that he wanted to make an impression on my father, who was much further up the greasy pole than he was so, indirect nepotism. 

I studied archaeology, metallurgy and mathematics. I quite enjoyed the metallurgy but found the mathematics, which in that first year didn’t go beyond what I had already learnt at school, boring. I was stimulated by the archaeology but had immense problems with the essays because of my writing problems. However, all in all I decided it was not where I wanted to be and dropped out at the end of the year.

Over the next decade I continued to teach myself the histories of mathematics and science. In 1976 I lived for a time in Malmø in southern Sweden and in the well-stocked public library discovered both Stephen Körner’s The Philosophy of Mathematics and Karl Popper’s Conjectures and Refutations as a result of which I added the philosophies of mathematics and science to my private curriculum. Shortly after I returned to the UK an assistant in a small book shop, who knew my somewhat specialised taste in reading matter, drew my attention to Imre Lakatos’ Proofs and Refutations, which as I’ve documented elsewhere was possibly the most influential book in my life. 

Throughout the 1970s I did numerous things to pay the rent and buy groceries. For several years I spent the summers working as a field archaeologist and at one point for a whole year. Working on archaeology sites you learn an awful lot about the mechanisms of history. I also spent a lot of time time working as a sound and light technician in theatre, and as a set designer and constructer. I worked concerts as a stagehand and a ticket controller. I did a stint as an industrial painter in the docks. If somebody offers you work cleaning out the diesel tanks in the keel of a ship, so that welding repairs can be carried out, politely decline and walk away fast. I did the obligatory six months working in Cardiff’s nineteenth century East Moors steel works. I worked in the quality control for the ancient coke ovens, taking coal samples before they came into the ovens. The top of a nineteenth century coke ovens at three o’clock on a summer’s night is like something out of Dante’s Inferno, belching sulphurous fumes and sheets of flame. My final occupation before finally leaving Cardiff was renovating second-hand electric stoves for resale. It says something about the economic state of the UK in the 70s that it was a roaring trade.

In 1980 I moved to Germany. When people ask why I answer quite honestly, “I went on holiday and never went back again!” I’ve already documented the events leading up to this move so, I won’t repeat myself here. I couldn’t speak the language so, I went to evening classes to learn it. It was too slow for my taste and I asked if there was anything faster and got told there was a German as a foreign language course in Erlangen at the local university. I inquired and it turned out that I would have to register as a student to be able to do the German as a foreign language course. I had always intended to return to university so, I did. The German high school system closes with a high school certificate, (Das Abitur), which qualifies the holder to study anything at university. My science A-levels only counted as a restricted high school certificate so, I registered to study mathematics. 

After one year of the German as a foreign language course, I took and passed the final exam and could begin my mathematics degree. In those days the first degree at a German university was a master’s complete with thesis, which was expected to take four to five years but many students took longer. In mathematics it was called a diploma and I registered with philosophy as my subsidiary. As already related elsewhere, my philosophy professor was a historian of logic and mathematics and from the very beginning I followed an intensive history and philosophy of science course of studies with a set of excellent teachers. The maths department had absolutely no interest in the history of the discipline so, after two and a half years I changed from mathematics to a master’s in philosophy. In order to do so I required at least one humanities A-level so, I took German A-level courtesy of the British Institute in München. I took English philology and history as my subsidiaries.

By then, I had already been working for some time as one of the researchers in a major research project into the external history of formal logic. I now seriously devoted as much time as possible to the study of the history and philosophy of science. However, I was a mature student without a grant or parental support so, I was also working long hours to pay the rent and buy the groceries. I won’t list all the things I did to earn money but I did quite a lot of work as a stage hand for a large concert promoter. In the end I ended up working as the evening manager of a youth and cultural centre, with about  three thousand guests in its various activities on a Friday or Saturday night. I also worked there as a concert sound a lighting technician. In my “spare time” my hobby was running the jazz club in the centre. 

Not surprisingly, I was by now suffering from some fairly serious mental illnesses including a full blown alcoholism. After a total of ten years at the university, I had acquired all the course credits I needed to graduate MA and had written about two thirds of my master’s thesis and a very detailed annotated outline for my doctoral thesis! The planned doctoral thesis was originally going to be my master’s thesis  but had got too large and so I took one section out of it as a master’s thesis. As I have documented elsewhere, not surprisingly the wheels fell off and instead of graduating I took a time out in the local loony bin, going through withdrawal and trying to get my mental state back onto an even keel.

When I came out of the clinic it was obvious that I needed to reduce the stress in my life and as it was clear that as a forty year old graduate I would never get a job in the history of science, which are as rare as hen’s teeth for anybody, I dropped out of university for the second time in my life. It is often said that alcoholism is a symptom of an underlying phycological disturbance and that was certainly the problem in my case. In fact, within a year, I was back in the loony bin with clinical depression. I regularly attended the AA for several years and also had extensive out-patient psychotherapy but I still wasn’t really getting at the root problem.

However, after a time I started regularly attending my professor’s weekly research seminar in which graduate students and lecturers presented their actual research to public scrutiny. These weekly sittings had originally been part of the research project in which I had worked for years. In those days we often had guest lecturers and I got to meet and converse during the meals following the lectures with some of the world’s leading historian of mathematics and logic, including Ivor Grattan-Guiness, Martin Davis, and Joe Dauben. At the ICHS conference in Hamburg in 1989, I got to chaperone, the then ancient but still incredibly sharp, legendary historian of mathematics, Dirk Struik (1894–2000), who was the keynote speaker in our section of the conference. As well as reconnecting with my professor, if only as a guest, I also began to hold public lectures on the histories of science and mathematics, mostly in Nürnberg. I also became an active member of a group of historians of astronomy in Nürnberg. Later I became an active member of the Simon Marius Society. I also began my legendary history of astronomy tours of Nürnberg, which, in the meantime, have been enjoyed by quite a large number of professional historians of science.

I still had not come to terms with my inability to write. I could hold lectures on a myriad of topics without notes, which I know from feedback are very good but faced with a blank page I became hopelessly cramped. As I have noted elsewhere, I finally discovered at about the age of fifty that I have high grade AD(H)D and suffer from dysgraphia. I knew that some of my symptoms were indicative of dyslexia but I had no problems in reading. I had had my nose buried in a comic, magazine, newspaper, or book since I taught myself to read at the age of three. Dysgraphia is a disturbance of the part of the brain that controls writing! The solution turned out to be quite simple I don’t write, I hold a lecture in my head and write it down as I go along, hence my narrative style. 

Seventeen years ago, I started this blog to teach myself to write. It was an uphill struggle but the fact that I’m still hear seem to indicate that I succeeded. I have also over the years accumulated a small but growing number of real dead tree academic papers! My work has found favour with a surprisingly large number of the world’s leading historians of science. A fact that I find mildly embarrassing. Over the years I have been asked to fact check texts written by quite a few of those historians and surprisingly given my life long struggles with the written word, I am employed as a copy editor by historians writing in English, who are not native speakers. 

Yes, I’m just some guy with a blog, but I hope it is by now clear that that blog is written from a very solid and very real background as a historian of science, even though I never managed to get any formal qualifications along the way. I also research my blog posts carefully, mostly in the circa one thousand academic volumes of the personal research library that line the walls of my apartment. What I don’t already have at home I borrow from the university library.

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Filed under Autobiographical

June 26, 2026 · 7:23 am

Scientific and Mathematical Instruments in the Early Modern Period

Anybody who has been following this series will be aware that in England in the Early Modern Period, mathematical and scientific instruments played a central role in the world of practical mathematics. As we have seen from small beginnings in the middle of the sixteenth century the business of instrument making expanded to become a significant factor in the mathematical community by the middle of the seventeenth century. Of course, scientific and mathematical instruments had been in existence since antiquity but in the Early Modern Period the number of different types of instruments and the number of each type of instrument produced expanded immensely. One could argue that the production and use of such instruments was one of the defining aspects of the scientific and mathematical culture of the period.

However, instruments didn’t just define the period they also could be said to have defined the roles of the mathematical practitioners. Mathematical practitioner if not the name or definition of a career or profession but is a catch all term for all the diverse professions which were based on practical mathematics i.e. mathematics that is put to practical uses – surveyors, navigators, gunners, architects, shipwrights, sundial makers, military engineers, and book-keepers – many of whom we have met in the earlier posts in this series. These people often in their publications added the title mathematical practitioner, or a variant, to their job description in their publications, e.g.

For instance, Henry Bond’s A New Booke of Gauging is written by ‘Henry Bond, practitioner in the Mathematicks.’  Similar attributions appear in Seth Partridge’s Rabdologia, written by ‘Seth Partridge, Surveyor, and Practitioner in theMathematicks’ and William Purser’s Compound Interest and Annuities, in which Purser describes himself as ‘Mariner and Practitioner in the Mathematicks.[1]

Gauging is determining the volume of full barrels or ship’s holds for purposes of charging customs duty, which relies on some form of approximation based on the use of special instruments and mathematical calculation, as direct measurement and calculation is not possible because of the shapes. Literature on gauging and literature on dialling, the construction of sundials, were the most prolific form of mathematical literature published in the sixteenth and seventeenth centuries in England.  Johannes Kepler wrote a book on gauging in which he describes the determination of the volume of barrels using the method of exhaustion an early form of integration.  Rabdologia is instruction in the use of Napier’s Bones a simple calculating device invented by John Napier the inventor of logarithms. The fuller title of Partridges book is:

‘Rabdologia, or the Art of numbering by Rods … with many Examples for the practice of the same, first invented by Lord Napier, Baron of Merchiston, and since explained and made useful for all sorts of men. By Seth Partridge, Surveyor and Practitioner in the Mathematicks,’[2]

Source
A set of Napier’s Bones Source: Wikimedia Commons

This change in the role and status of practical mathematics can be clearly observed in the world of the surveyor. In the High Middle Ages, surveying was still carried out with the simplest of methods, ropes and measuring poles to determine distances or even simple pacing by trained walkers. The areas were then computed with simple geometrical methods to a very low level of accuracies. Boundaries were often vague and inaccurate. This would all change radically over the fifteenth and sixteenth centuries

First, came a demand for better more accurate surveying created by a series of factors. At the beginning of the fifteenth century the mathematical cartography of Ptolemaeus re-entered Europe with the translation of his Geographia into Latin by Giacomo or Jacopo d’Angelo of Scarperia better known in Latin as Jacobus Angelus from a Greek manuscript, found in Constantinople in about 1406.

Jacobus Angelus’ Latin translation of Ptolemaeus’ Geographia Early 15th century Source via Wikimedia Commons

By the beginning of the sixteenth century, Ptolemaic cartography had come to dominate throughout Europe. Secondly, changes in the legalities of land ownership demanded better and more accurate, estate maps with clearly defined, accurate legal boundaries. New modes of warfare also demanded better cartography of the territories that needed to be traversed by armies. 

All of this became possible first, through the invention and publication of triangulation in 1533 by Gemma Frisius (1508–1555), a mathematician, who was a university professor but who thought and worked like a mathematical practitioner.

Gemma Frisius’ presentation of triangulation Source: Wikimedia Commons

Taken up by cartographers throughout Europe his new surveying techniques spread quickly throughout Europe often through the offices of mathematicians such as Mercator (1512–1594) and John Dee (1527–1609) , who studied under Gemma Frisius in Louvain. The demand for new more accurate, mathematical methods of surveying brought with it a demand for new instruments to make the work possible. 

The plane table, the origins of which are not known for certain, emerged sometime in the sixteenth century. A portable drawing board on a tripod with plumb bob and spirit level to ensure it was level and in the correct position. With a mounted sighting device, it made it possible to survey small areas quickly and efficiently, transferring the sightings directly onto the paper on the table. For the larger areas needed for cartography angle measuring devices with sights were necessary. 

Philippe Danfrie (c.1532–1606) Surveying with a plane table 

Leonard Digges (c. 1515–1558) a prominent sixteenth century mathematical practitioner present new instruments in his guides to surveying, A Booke Named Tectonicon (1556), “for Surueryers, Landmeaters, Joyners, Carpenters, and Masons,”  all professions that required practical mathematics, and A Geometrical Treatise Named Pantometria completed and published posthumously by his son Thomas Digges (c. 1546–1595).

Source

In the Pantometria, Leonard introduced what he termed the theodolite and also his topographical instrument, We would now term his topographical instrument, which could measure both horizontal and vertical angles simultaneously a theodolite. Leonard’s theodolite could only measure angle in one dimension at a time. Later after the emergence of the telescope in 1608, theodolites would go on to acquire telescopic sights making them even better for triangulation over large areas. 

Theodolite 1590 Source:

Earlier than Digges’ instruments was the circumferentor which was used to measure horizontal angles, the origins of which are not known precisely but are thought to lie in the sixteenth century. These instruments made triangulation easier to carry out.

18th century circumferentor

Another innovation was the surveyor’s chain, which replaced the earlier ropes, which were subject to inaccuracies due to stretching. The most famous was the Gunter Chain, introduced by Edmond Gunter (1581–1626), which became the standard English Imperial chain. 100 links and 22 yards (66 feet) long, there are 10 chains in a furlong and 80 chains to a mile. A cricket pitch is still one chain long between the wickets. 

A Gunter chain photographed at Campus Martius Museum. Source: Wikimedia Commons

Gunter was also one of many instrument inventors who created a sector–a hinged rule with many different engraved scale used by surveyors, navigators, etc. to simplify calculations in the field. 

The Workes of Edmund Gunter 5th ed. Title page with diagrams of the sector on the fly leaf

Triangulation require trigonometrical calculations and these were made significantly easier when John Napier (1550–1617) published the first book on logarithms his Mirifici logarithmorum canonis descriptio… (1614). Unlike later tables of logarithms, which gave the logarithms of numbers, Napier’s tables were the logarithms of trigonometrical functions, providing an aid to calculation for both astronomers and surveyors.

Source: Wikimedia Commons

This brief account of some of the instruments that became tools of the trade for surveyors should serve and an example of how in the early modern instruments became a defining aspect of those careers that were dependent on practical mathematics. Another area where instruments played a defining role was in the mathematical education of gentlemen.

In the late sixteenth century it became increasingly obvious that gentlemen required a mathematical education alongside their more traditional training in such things as fencing, horse riding, and dancing. There were multiple reasons for this in a rapidly evolving society. Areas such as estate management, trade, sailing, and administration all became much more dependent on practical mathematics, for surveying, map reading, bookkeeping, astronomical navigation and even political arithmetic, an early form of statistical analysis. 

The necessity that Gentlemen are under, that would be Considerable in the Art of War or any great employment, (either in Church or State) which cannot well subsist without a considerable knowledge in the Mathematics; makes them to throw aside trifling Amusements and apply themselves to the Mathematical Sciences. [3]

However, already in the sixteenth century, Roger Ascham (1551–1568) wrote in his The Scholemaster, published posthumously in 1570, widely reprinted in many editions up to and into the twentieth century:

These sciences, as they sharpen men’s wits over-much, so the change men’s manners over-sore, if the be not moderately mingled, and wisely applied to some good use of life. Mark all mathematical heads, which be only and wholly bent to those sciences, how solitary they be them-selves, how unfit to live with others, and how unapt to serve in the world.[4]

Source

People were already dumping on nerds in the sixteenth century.

The conundrum, mathematical education but not too deep was solved by instructing gentlemen in the use of mathematical instruments. As Edward Stone noted in his translation from the French Nicholas Bion’s treatise on mathematical instruments, in 1723:

Mathematicks are now become a popular study, and make a part of the Education of almost every Gentleman… 

Mathematical Instruments are the means by which those Sciences are rendered useful in the Affairs of Life. By their assistance it is that subtile and abstract Speculation is reduced into Act. They connect as it were the Theory and Practice, and turn what was bare Contemplation, to the most substantial Uses. The Knowledge of these if the Knowledge of Practical Mathematicks: So that the Descriptions and Uses of Mathematical Instruments make, perhaps, one of the most serviceable Branches of Learning in the World.[5]

To some extent the gentleman was not required or expected to use all of the instruments himself. However, the instruction enabled him to discuss intelligently the work being carried out by his estate surveyors, or follow the bookkeeping accounts of his estate manager, perhaps checking the results of the calculations with one of the numerous multi-scaled calculating rules thrown onto the market by the professional mathematical instructors. When commanding a naval, exploration, or privateering vessel on the high seas to be able to discuss navigational instrument readings with his navigator. 

We saw that this type of mathematical instruction was a reality in the last episode of this series about the father and son mathematical instructors John and Euclid Speidel. We saw that that their cliental consisted largely amongst courtiers and gentry and both of them based part of their instruction on the mathematical instruments that they had designed themselves. 

In the Early modern Period mathematical instruments formed part of the definition of the mathematical practitioners who utilised them and were also the object of mathematical instruction of the members of the gentry who employed them.

[1] Hester Higton, Does using an instrument make you mathematical? Mathematical practitioners of the 17th century, Endeavour Vol. 25(1), 2001 pp. 18–22, p.18

[2] Partridge, Seth (1635–1703) Dictionary of National Biography, 1885–1900, Vol. 43

[3] John Jackson, Mathematical Lectures Being the First and Second That were read to the Mathematical Society at Manchester, Manchester, 1719 quote in A. J. Turner B.A., Mathematical instruments and the education of gentlemen, Annals of Science, 30(1), 1973, pp. 51–88, p. 54

[4] Turner, p. 54

[5] Turner, p. 51

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Filed under History of Mathematics, History of Navigation, Scientific Instrument Makers

June 11, 2026 · 7:40 am

17

The Renaissance Mathematicus has completed yet another orbit around the Sun and is, as of today, seventeen years old. Seventeen years of churning out history of science blog posts to keep myself occupied and hopefully entertain and maybe inform the one or other reader. Over the years I have either written something personal or something related to the number of the years on my blogiversaries and this year is no different. 

As regular readers should be aware I have had a rough year health wise. Back in October I managed to collapse a lung, not an experience I would recommend, which resulted in three very uncomfortable weeks in hospital and several weeks recovering from the aftereffects. I can, however, happily report that at the beginning of May I had a thorough lung examination and my lungs are currently in full working order. Let’s hope they stay that way! This year it was established that what I had thought over the last twenty or so years were degenerative orthopaedic problems are in fact largely neurological, caused by trapped nerves and compressed nerve roots. This has degenerated further and for distances more than about fifty metres, I am now reliant on my electric wheel chair, with which I terrorise the pedestrians of the village where I live. Actually, I have discovered that generally people are very kind and helpful, offering to get things off shelves in supermarkets that I can no longer reach. Getting old is to say the least interesting.

The Mathematicus Mobile

17 is the 7th prime number. It is a Fermat number and third of the five Fermat primes.

In number theory, it is also a Leyland number and a Leyland Prime of the format XY ± YX where X and Y are integers greater than 1 using 2 and 3 (23  + 32) and using 3 and 4 (34 – 43). It is also one of the six Euler lucky numbers which are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number.

However, it is in geometry that 17 takes a starring role.  

Although all of Theodorus’ work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus. 

Source: Wikimedia Commons

The Spiral of Theodorus with a maximum right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a hypotenuse of  √17

The strangest appearance of 17 in geometry is in the work Carl Frierich Gauss (1777–1855). The ancient Greeks developed the mathematical game of seeing what they could construct and which problems they could solve using nothing but an idealised straightedge and an idealised compass. 

The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to “collapse” when lifted from the page, so it may not be directly used to transfer distances. 

One can for example bisect a given angle but not trisect it. The three classic problems discussed by the Greeks were trisecting an angle, squaring a circle i.e. constructing a square with the same area as a given circle, and doubling a cube i.e. constructing a cube with double the volume of a given cube. None of this is solvable with a straightedge and compass. 

A more general problem is how many of the regular polygons can be constructed using only a straightedge and compass? All polygons with an even number of sides are constructable but what about those with an uneven number of side? In 1796, the then only nineteen years old, Gauss proved that the heptadecagon, the regular seventeen sided polygon, could be constructed with straightedge and compass.

Source: Wikimedia Commons

This achievement apparently convinced the young man to take up a career in mathematics rather than philology. In his Disquisitiones Arithmeticae (Arithmetical Investigations), a treatise on number theory published in 1801, he gave the general theory for determining whether a regular polygon could be constructed with straightedge and compass or not .

As always in my blogiversary post, I would like to that all pf the people who take the time to read my scribblings and especially those who comment and most of all those who correct my errors, it is always good to learn something new.

So off we go into the eighteenth year…

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Filed under Autobiographical, History of Mathematics

June 3, 2026 · 7:47 am

Take it to the limit one more time – a history of calculus 0

In the first half of the 1980s I studied mathematics, as a mature student, at the University of Erlangen. In those days a first degree at a German university was a master’s complete with dissertation. In mathematics this was called a diploma and the usual period of study was eight to nine semesters of four to four and a half years. A fair percentage of students took longer. The ground courses in mathematics, leading up to the first set of serious exams, was four semesters long and consisted of double lectures, totalling ninety minutes, four days a week–two day analysis and two days algebra. A double lecture generated eight to ten A4 pages of notes, it was a hard slog. There were exercise sheets each week for each course with afternoon sessions to analyse and correct them. After the first two semesters there were also seminars on more detailed topics, where we were required to prepare and hold a paper. I enjoyed the analysis not so much the algebra. 

I was required to have a subsidiary subject and had chosen philosophy. It was a lucky coincidence that my first semester was also the first semester of Christian Thiel as professor for philosophy in Erlangen. Alongside my mathematics studies I took a deep dive into the history and philosophy of science under Professor Thiel and several other excellent teachers. Because my real interest lay with the history of mathematics rather than mathematics itself, I changed after five semesters to a master’s degree in philosophy with history and English as my subsidiaries. I was now well along on the journey that would lead to me sitting at my computer writing this blog post.

Of course, entering university at the age of thirty was not my first encounter with analysis. I had as a teenager studied both O-level and A-level mathematics being one of those obnoxious kids who actually enjoyed doing mathematics. From the very beginning I sort of fell in love with analysis, here called calculus and really enjoyed derivatives and integrals. In the second year of my A-level studies I actually managed to get 102% in an exam! It was a test on derivatives in which there were fifty functions for which one had to write down the derivative. I banged it off at record speed and gave up my paper. When the exam was returned in the next lesson I was happy that I had got 98%, meaning I had only got one question wrong. Looking at the question I had got wrong, I had doubts and spoke to the teacher. He realised that my solution was, in fact, correct and it was his that was wrong. He corrected my score and gave me an extra two points for being right when he was wrong. This was unusual for me as I don’t usually do very well in exams, which is due to my AD(H)S. I normally know more about the topic than anybody else in the room but very often not the specific stuff being examined! 

Given my general love of history, I’d been a history junkie since I could read, and my love of mathematics, my father, a professional historian with a very wide range on interests, gave me his copy of Eric Temple Bell’s Men of Mathematics(Simon & Schuster, 1937). This an eminently readable and inspiring book but unfortunately, as I learnt over the succeeding years, historically very full of garbage. The historian of mathematics Ivor Grattan-Guiness (1941–2014) judged it so:

…perhaps the most widely read modern book on the history of mathematics. As it is also one of the worst, it can be said to have done a considerable disservice to the profession.[1]

However, at the time it inspired me immensely. Over the following years I began to acquire other books on the history of mathematics, then books on the history of science, then in the mid 1970s, I discovered both Stephen Korner’s Philosophy of Mathematics and Karl Poppers’s Conjectures and Refutations and became a full blown history and philosophy of science junkie. 

However, back to Eric Temple Bell, as already stated when I first read his seductive tome I was already deeply enamoured with the discipline of calculus and now learnt for the first time that calculus had been invented/discovered[2] at the same time by both Leibniz and Newton. This simultaneous discovery would lead to the most well know priority and plagiarism dispute in the history of mathematics. I was fascinated.

There are accounts of the ‘calculus wars’ in all the mainline histories of mathematics and the story has generated a boat load of academic papers. It is of course also dealt with in biographies of Newton and Leibniz. I own two books dedicated to the topic, A. Rupert Hall, Philosophers at War: The quarrel between Newton and Leibniz (CUP, 1980) and Jason Bardi, The Calculus Wars: Newton, Leibniz and the Greatest Mathematic Clash of All Time (High Stakes, 2006).

It makes for a good story but over the years I began to realise that Newton and Leibniz didn’t in fact invent calculus. Calculus evolved over a period of more than two thousand years with many mathematicians contributing to its growth and expansion. That growth accelerated throughout the seventeenth century and in the last decades of that century Leibniz and Newton pulled together the strands of the discipline and collated them into a single package, filling out some gaps but leaving others that would then be filled by other mathematicians over the next two hundred years. 

I have commented fairly often on what I see as a misrepresentation of the origins of calculus and even gave a brief sketch of parts of that evolution in a blog post on another blog as a guest blogger. Now I have decided to take a deep dive in the story and in an extended series of posts to present those two thousand years of evolution of the mathematical discipline for which Leibniz and Newton are, in my opinion given far too much credit. 

[1] Ivor Grattan-Guiness, Towards a Biography of Georg Cantor, Annals of Science, 27 (4), 345–391

[2] Choose the term that best fits your philosophy of mathematics. Having studied constructive logic at university, I tend towards invented.

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Filed under Autobiographical, History of Mathematics

May 28, 2026 · 8:52 am

The century and a half search by the British authorities–from the founding of the Royal Observatory at Greenwich to the dissolution of the Board of Longitude–to find a method to determine longitude at sea.

Back in 2010 the University of Cambridge and the National Maritime Museum at Greenwich launched a six year research programme, led by Simon Schaffer for Cambridge and Richard Dunn and Rebekah Higgitt for Greenwich: ‘The Board of Longitude 1714–1828: Science, Innovation and Empire in the Georgian World.’ “Its aim was to produce the first comprehensive history of the British Board of Longitude, examining its changing role as an influential player in Georgian culture.” Participants were the postdoctoral researchers Alexi Baker and Nicky Reeves, doctoral students Katy Barrett, Eóin Phillips and Sophie Waring, and Engagement Officer Katherine McAlpine.

During its run the team produced, in what might be termed the golden age of history of science blogs, a truly excellent blog reporting on all aspects of their work and progress. As well as three doctoral theses, two unpublished, and quite a few academic papers the research project generated several monographs. Richard Dunn, Navigational Instruments, (Oxford, 2016), Richard Dunn and Rebekah Higgitt eds., Navigational Enterprises in Europe and Its Empires, (Basingstoke, 2015), Rebekah Higgitt ed., Maskelyne Astronomer Royal, (London, 2014) and Katy Barrett, Looking for Longitude: A Cultural History, (Liverpool, 2022), her doctoral thesis, which I reviewed here. Dunn and Higgitt organised a major exhibition in the National Maritime Museum in Greenwich, which then went on world tour, for which the wrote the truly excellent Finding Longitude : How Ships, Clocks and Stars Helped Solve the Longitude Problem (Glasgow, 2014), which I reviewed here. Now the final report of this truly monumental research project has been published Alexi Baker, Richard Dunn, Rebekah Higgitt, Simon Schaffer and Sophie Waring, The Board of Longitude: Science Innovation and Empire, (CUP, 2025).

I will start by saying this book is everything one could wish on the historically important topic and more, but contains one of the most honest statements I have ever come across in such a project. Although their work has been staggeringly comprehensive the authors write in the conclusion to the introduction:

This volume represents a first attempt at more completely describing the work of the Commissioners of Longitude and the Board that they became. We must acknowledge, of course, that many questions remain. [my emphasis] In addition, there is much that can still be done to explore more fully the surviving papers of the Board of Longitude, not to mention the many additional archives and publications that shed light on this fascinating body.                                                         

They then go on to list the areas that could or should be further researched. 

Before the introduction the book opens with a six page, detailed time-line of the British search for longitude, a sort of annotated index to the subjects covered within. The introduction, itself gives sketches of the background and sources, the historiography of the Board, and the aims of the book  with brief outlines of the individual chapters. The introduction by itself is a highly readable paper on the topic that could with very little modification stand alone.

This stand-alone characteristic of the introduction is repeated throughout the book. Connected by the common theme of the book following the chronology of the Boards creation and existence as follows, the first chapter presents the situation before the Longitude Act then:

The first strand, comprising Chapters 2–5, 8, 11 and the Epilogue, offers a chronological account from the development of the first Longitude Act and the early work of the Commissioners through the development of the Board of Longitude as a standing body from the 1760s to its dissolution in 1828.m The second strand focuses on specific aspects of the Board’s work from the 1760s onwards, arranged to align with the book’s overall chronology. 

However, as noted above each of the chapters is written as a complete entity and could with only minor modifications be published as a separate papers. The combination of the encapsulated chapters and the timeline quasi index, means that anyone who wants a brief accurate introduction to a single episode or aspect of the British longitude story is well served with this volume.

It goes almost without saying that the level of historical research offered here is exemplary and each of the chapters is excellently written. The text is illustrated with grey scale pictures, tables and boxes for which there is a separate index at the front of the box. The volume closes with a very extensive glossary, an even more extensive bibliography and an excellent index.

Although the authors clearly state that the work on this important episode in maritime history is by no means complete, with this research project and the resulting volume, presented here, they had made a very major contribution, on a very high level. This volume is an absolute must for anybody and everybody, who has an interest in the topic and is an instant classic and a role model that sets very high standards for all undertaking major historical research projects.

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Filed under History of Navigation, History of Technology

May 20, 2026 · 7:31 am

Ibn al-Haytham did not invent the Scientific Method

It would appear that Ihtesham Ali, with his 29.5k followers on Twitter, is a repeat offender. After having totally ballsed up an account of al-Khwarizmi’s al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah, he then took on Ibn al-Haytham’s Kitāb al-Manāẓir; (Book of Optics) in 640 enthusiastic words. This one is not quite as bad a train wreck at his al-Khwarizmi post, but it too contains a lot of historically incorrect claims that it pays to examine and set straight, as they are widespread and uncritically accepted by those who don’t know better.

Ibn al–Haytham as imagined by Johannes Hevelius

Ihtesham Ali: An Arab scholar in 1011 was placed under house arrest in Cairo for 10 years. He used the time to invent the scientific method, prove how vision actually works, and write a 7-volume book that Newton studied 600 years later. I read about him last night and could not stop thinking about it. His name was Ibn al-Haytham. The book is called the “Book of Optics.” The textbook story names Bacon*, Galileo, and Descartes as the founders of modern science. All three of them came 600 years after Ibn al-Haytham. All three of them studied his work directly or through Latin translations. The man who actually invented the scientific method was working alone in a single room in Cairo while Europe was still in the Dark Ages. 

Our author starts with a bang, Ibn al-Haytham invented the scientific method, he proved how vision actually works and wrote a 7-volume book that Newton studied 600 years later. WOW! A true superman. I’ll deal with each of these three claims separately further on, here I’ll just address the 7-volume book, which seems to imply a monster work. The Book of Optics is not seven volumes but contains seven books, where books in earlier scientific publications would today be replaced with sections or chapters. A good comparative example is Kepler’s Harmonice Mundi, which has five books. The Latin manuscript of the Book of Optics is 130 folios so, about 260 pages. The seven books are:

  • Book I deals with al-Haytham’s theories on light, colours, and vision. 
  • Book II is where al-Haytham presents his theory of visual perception. 
  • Book III and Book IV present al-Haytham’s ideas on the errors in visual perception with Book VI focusing on errors related to reflection. 
  • Book V and Book VI provide experimental evidence for al-Haytham’s theories on reflection. 
  • Book VII deals with the concept of refraction. 

It is very clear that al-Haytham’s Book of Optics is modelled on the Optics of Ptolemaeus (fl.150 CE). Important to remember when considering his programme of experiments.

One should also, I think, when one is singing the praises of Ibn al-Haytham mention that he was a true polymath who produced scholarly works over a very wide range of topics. 

*Roger Bacon read al-Haytham, but as far as I know, Francis Bacon who is falsely credited with being the founder of modern science didn’t. I don’t think that Galileo read him either.

Ihtesham Ali: Here is the story almost nobody tells you. He was born in Basra around 965 CE. By his 40s he had a reputation across the Arab world as one of the most original minds alive. Then he made the mistake that almost killed him. He claimed publicly that he could regulate the flooding of the Nile. The mad caliph al-Hakim of Cairo summoned him to Egypt to do it. Ibn al-Haytham took one look at the river and realized the project was impossible with the technology of his era. The caliph had executed dozens of scholars for less. So he faked madness. The caliph believed him and put him under house arrest in his own home in Cairo for the next 10 years.

There are various version of the story of Ibn al-Haytham’s unfortunate interactions with al-Hakim bi-Amr Allah (985–1021) but I think the most important aspect is that following al-Hakim’s death in 1021, Ibn al-Haytham remained in Cairo for another twenty years until his own death. 

Ihtesham Ali: Most people would have lost their actual mind. He used the time to invent science. Before him, knowledge worked one way. You quoted authority. If Aristotle had said it, it was true. If Galen had written it, it was correct. The role of a scholar was to memorize and defend the ancient Greeks. I Ibn al-Haytham broke this completely. He wrote a sentence in the Book of Optics that quietly destroyed 1,400 years of intellectual culture. “The seeker after truth,” he said, “is not the one who follows his natural disposition to trust the writings of the ancients. The seeker after truth is the one who suspects them, questions them, and submits only to argument and experiment.” That single sentence is the foundation of modern science. He wrote it 600 years before the European Renaissance. The second thing he did was build the actual machinery of experimentation. He insisted that no claim about the physical world was acceptable until it had been verified by an experiment anyone could repeat. He gave detailed instructions for every experiment in his book. He told his readers, in writing, not to take his word for any of it. Build the equipment. Run the tests yourself. Verify or destroy my claims with your own eyes. 

The first five sentences here are, to put it mildly, total and utter rubbish. The Stoics, the Epicureans, and the Neo-Platonists, would all like to differ. Ibn a-Haytham did not destroy 1,400 years on intellectual culture because the intellectual culture our author is claiming for antiquity, late antiquity, and the middle ages simply didn’t exist. 

Ibn al-Haytham did indeed carry out an extensive programme of experimentation in his optics but that programme was an extended copy of the programme of experimentation that Ptolemaeus carried out in his Optics so, if that programme and the motivation behind it was the invention of the scientific method, then it was Ptolemaeus who was the inventor not Ibn al-Hytham. However, about four hundred years before Ptolemaeus, Archimedes also carried out extensive programmes of experimentation so, should he be regarded as the inventor?

There is of course the added problem that there is actually no such thing as “The scientific method” and attributions of its invention to Bacon, Galileo or Descartes are equally false. Nobody invented the scientific method!

He did indeed give detailed instructions for every experiment in his book, but as A Mark Smith, who probably knows more about al-Haytham’s optics than anybody else on the earth points out, it would have been literally impossible for al-Haytham to obtain the results he presents with his set ups and that many of the experiments were thought experiments and not real ones.

We now get down to the most common big al-Haytham myth. To start a little piece of terminology used to describe theories of vision: Theories of vision based on the concept of something entering the eye are termed intromission theories. Theories of vision based on the concept of something leaving the eye are termed extramission theories.

Ihtesham Ali: The third thing he did was use the method to overturn one of the most settled questions in physics. The Greeks had taught for centuries that vision worked because the eye emitted invisible rays. 

There was by no means a settled Greek theory of vision. In fact, there were four competing theories, which I have described in detail here. The Atomist had an intromission theory, Plato had an extramission theory with fire sent out by the eyes, but which had to be triggered by sunlight, Aristotle had an intromission theory, and the Stoics had a theory that is sort of a mixture. What is important in the context of al-Haytham’s work is that the presenters of geometrical optics, Euclid, Hero of Alexandria, and Ptolemaeus all based their theories on an extramission theory. All of these theories were known to the Islamicate philosophers.

Ihtesham Ali Ibn al-Haytham proved them wrong with a darkened room, a small hole, and a wall. The first camera obscura. He showed that light from the outside world enters the eye, the exact opposite of what every Greek thinker had taught. 

I would love for our author to explain just how, using a camera obscura, Ibn al-Haytham showed that light from the outside world enters the eye! He didn’t! 

We’ll start with the fact that although Ibn al-Haytham did do experiments with the camera obscura, it wasn’t the first camera obscura. The camera obscura effect had been known since the fourth century BCE when both Mozi (c. 470–c. 391 BCE) in China and Aristotle (384–322 BCE) in Greece wrote about it. Al-Kindi (c. 801–c. 873) wrote about pinhole images in his optical treatise De aspectibus to demonstrate that light travels in straight lines. Al–Haytham also used the camara obscura to demonstrate that light travels in straight lines and to study the theory of focal points. 

Al–Haytham did not prove the intromission theory of vision. Through a series of persuasive philosophical arguments, similar to, but not as extensive as, those presented by Ibn Sina (c. 980–1037) who supported Aristotle’s intromission theory, he demonstrated the implausibility of an extramission theory. 

His own intromission theory was based on the punctiform theory of reflection first propagated by al-Kindi, who argued for Euclid’s extramission theory. This stated that light is reflected from every point on a body in every direction and it was these reflections that created vision. Using the model of the eye presented by Galen but in which he hypothesised the eye lens was perfectly spherical. He argued that the reflected rays entered the eye, correct, and the image was formed in the lens, which is incorrect. The image is then transmitted to the brain along the optic nerve, which is correct. However, if the reflected rays are from every point on a body and in every direction, surely there would be a confusion of rays and images in the eye lens. Al–Haytham’s solution was to claim that only rays that met the lens perpendicularly actually entered the eye, all the others bounced off, so to speak. Although somewhat questionable his model did become influential.

Al–Haytham image of the eye Source: Wikimedia Commons

However, al–Haytham’s major contribution was to demonstrate mathematically that the geometric optics propagated by Euclid, Hero of Alexandria and Ptolemaeus function just as well with an intromission theory, as with the extramission theory that they had all used.

Ihtesham Ali: Two hundred years later his book was translated into Latin in Spain. Roger Bacon cited him. Kepler cited him. Galileo’s work on the telescope was built on his optics. Newton’s foundational work on light rested on his framework. 

The Kitāb al-Manāẓir was translated into Latin at the end of the twelfth century by an unknown scholar, I don’t think it is known where it was translated. The Latin De aspectibus was far more influential in Europe than the original was in Islamic culture. Roger Bacon (c. 1219–c. 1292) built his theory of vision around the work of Robert Grosseteste, who hadn’t read al–Haytham, and al–Haytham’s De aspectibus but also included the work of al–Kindi, Euclid and Ptolemaeus. John Peckham (c.1230–1274) in his Perspectiva cummunis wrote what is effectively an epitome of al–Haytham’s Deaspectibus, which became a university textbook. Witelo (c. 1230–1280) in his Perspectiva wrote an extended version of De aspectibus. However, also using a wide range of other sources. 

Thus, the optics of al–Haytham became established in medieval Europe but did not become totally dominant. In his De pictura (1450) the first published account of the geometric optics of linear perspective, Leon Batti Alberti (1404–1472) stated that in linear perspective it is irrelevant with you hold an extramission or intromission theory of vision.

Source: Wikimedia Commons

Al-Haytham’s De aspectibus was first printed and published by Friedrich Risner (c. 1533–1580), together with Witelo’s Perspectiva in his “Opticae thesaurus: Alhazeni Arabis libri septem, nuncprimum editi; Eiusdem liber De Crepusculis et nubium ascensionibus, Item Vitellonis Thuringopoloni libri X” (Optical Treasure: Seven books of Alhazen the Arab, published for the first time; His book On Twilight and the Rising of Clouds, Also of Vitello Thuringopoloni book X) (1572). This was the book that informed Kepler’s first major work on optics in 1604, but as the title, Ad Vitellionem Paralipomena, Quibus Astronomiae Pars Optica Traditur (Supplement to Witelo, in Which Is Expounded the Optical Part of Astronomy), indicates, it was Witelo’s derivative work rather that al–Haytham that influenced him. Kepler’s work, with its first correct account of vision, put an end to the perspectivist school of optics founded on al–Haytham’s work and introduced the new physical optics in place of optics as a theory of vision. 

Source: Wikimedia Commons

Do I really need to point out that Galileo’s work on the telescope was not built on the optics of al–Haytham? Telescopes are instruments utilising lenses, al–Haytham has virtually nothing on the optics of lenses. Al-Haytham has even less to do with the optics of Isaac Newton. 

Two of those who reposted Ihtesham Ali’s post, Lebanon John and Voorbij propaganda, both did so with the following comment:

Isaac newton’s book on optics is almost entirely plagiarised from this guy

The level of total fucking ignorance displayed in this statement wakes the HISTSCI_HULK in me. Would our two totally brainless re-posters care to point out where in De aspectibus I can find the correct explanation of the spectrum or failing that the detailed discussion of diffraction? I’m not going to hold my breath waiting for them to do so!

Ihtesham Ali: Walk into any physics department today. Ask who founded the scientific method. Almost nobody will say Ibn al-Haytham. The man who invented the way humanity actually knows things did the work under house arrest, with no funding, no laboratory, and a paranoid caliph next door waiting for an excuse to kill him. He did it anyway. Most of the world is still pretending it was someone else’s idea.

I can only repeat that Ibn al-Haytham did not found the scientific method! The literature in the history of science on Ibn al–Haytham is very extensive and his role in the histories of optics and astronomy is well known. His achievements, particularly in optics, are acknowledged and highly respected.  David Lindberg, one the greatest historians of optics wrote, “Alhazen [his Latin name] is undoubtedly the most significant figure in the history of optics between antiquity and the seventeenth century.” However, most scientists know very little about the history of their disciplines and quite a lot of them care even less. For example, if you ask physicists what Johannes Kepler achieved they will probably dish up the laws of planetary motion. Very few of them will know that he was one of those who laid the foundations of modern optics. 

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Filed under History of Optics, History of Physics, Myths of Science

May 15, 2026 · 7:41 am

A seventeenth-century teacher of geometry, who christened his son Euclid

In England from the middle of the sixteenth century the rapidly developing changes in navigation, cartography, surveying, and gunnery demanded the development of new systems and new skill in these areas.  Books by mathematical practitioners were rapidly followed by instrument makers, who developed, designed and produced the new mathematical instruments that were needed to make the observations, measurements and calculations required by the new systems. There also arose a demand for teachers to instruct those who needed it in the use of the new systems and the new instruments. 

The mathematicians John Dee (1527–1609) and Thomas Harriot (c. 1650–1621) were employed by the newly founded exploration companies to instruct their mariners in the latest developments in navigation and cartography needed on their voyages of discovery. Attempts were made to institutionalise such teaching, The appointment of Thomas Hood (1556–1620) as the first Mathematicall Lecturer to the Citie of London, in 1588, a post he only held for four years cannot be counted as a success, although his lectures were apparently popular during their brief existence. The establishment of Gresham College in 1597 was certainly longer lived than Hood’s lectureship but although the lectures were held in English as well as Latin, there is some doubt as to whether they reached enough of the ordinary people, who sought instruction. The still young East India Company, appointed Edward Wright (1561–1615) as their instructor in all things mathematical following the loss of his position as tutor to the Henry Frederick, Prince of Wales following Henry’s death in 1612. A post he only held for three years until his own death.

All of this meant there was a market for self-employed mathematical practitioners, who offered their services as teachers of all things mathematical to private customers. Two such freelance teachers of mathematics in seventeenth-century London were John Speidell (1577–1657) and his splendidly named, for a mathematics teacher, son Euclid (1631–1702). As with all mathematical practitioners we  know very little about the Speidells and would know almost nothing if it were not for the brief manuscript biography that Euclid drafted about himself and his father that was first discovered comparatively recently.

John Speidell gets a brief mention in the Brief Lives of John Aubrey (1626–1697):

Mr [John] Speidell […] taught Mathematiques in London and published a booke in 4to, named Speidell’s Geometrical Extraction. London 163[…] which made young men have a love to Geometrie. 

John Speidell was the son of the German Merchant, Sebastian Speydell from Bad Kreuznach, who had settled in London in about 1561. Sebastian was involved in the great Elizabeth recoinage from 1561. In 1564, Spydell, Thomas Thurland, Master of the Savoy Hospital and another German, Daniel Hochstetter, were granted a patent by Elizabeth I to mine and refine gold, copper, silver and mercury in England and Wales. According to his grandson, Euclid, Sebastian accumulated great wealth but at the time of his death in 1597 there had been great loses. It is not clear if Sebastian lost his fortune before his death or if John lost his inheritance. 

At about the age of twenty-eight, so around 1604, John married the widow of another mathematics teacher John Goodwyn (dates unknown). We know very little about Goodwyn, who is first mentioned in the book Navigator’s Supply by William Barlow (1544–1625), the Anglican cleric who invested much time and effort in the study of magnetism and navigation. Barlow writes of  Goodwyn:

A man unskillfull in the Latin tongue, yet having proper knowledge in Arithmetike, and Land-measuring, in the use of the Globe, and sundry other instruments: And having obteined, partly by his own industrie, and reading of English Writers (whereof there are many very good) and partly conference with learned men, (of which he is passing desirous) such ready knowledge and dexterite of teaching and practicing the groundes of those Artes, as (giving him but his due)I have not been acquainted with his like. 

The instrument maker and engraver Charles Whitwell (c. 1568–1611), who had provided the instrument illustration in the Navigator’s Supply, has written in an advertisement on the title page: 

If any man desire more ample instructions concerning the use of these instruments, hee may repayre unto Jhon [sic] Goodwin dwelling in Bucklerburye teacher of the growndes of these arts. 

John Goodwyn might have been tutor to the writer on surveying Arthur Hopton (1587/8–1614) author of, Speculum topographicum: or The topographicall glasse Containing the vse of the topographicall glasse. Theodelitus. Plaine table, and circumferentor. With many rules of geometry, astronomy, topography perspectiue, and hydrography. Newly set forth by Arthur Hopton Gentleman,

and the land surveyor Aaron Rathborne (1571/2, fl. 1605–1622) author of The Surveyor in Four Bookes (1616)

and was certainly the tutor of the mathematical practitioner John Reynolds, who apprenticed at the Gold Smiths Company in 1599, presumably fourteen years old, and went on to become a mathematician at the Royal Mint. It is not known if John Speidell, as well as marrying his widow, had been tutored by John Goodwyn. 

John Speidell was originally destined to follow his father into the life of a merchant but Euclid says that his father was only interested in music and mathematics. In the preface to his book A Geometricall Extraction (1617), John states that he began teaching mathematics in 1607.

Unusually, he offered his tuition in three languages English, French and Dutch. We also know from the same preface that he invented a calculating scale, about which we know very little as none have survived, he writes: 

not only these Problems contained in this booke […] but much more viz. in Arithmeticke, Geometrie, Astronomie, Navigation, Surveighing, fortification, sarchitecture, taking of heights & distances, and all other parts of the Mathematicks, &c. may be performed by a Mathematicall Scale [,] now newly (this present year) by me invented, farre beyond my former scale made in Anno. 1607 [,] the which with all other Mathematicall instruments, are made by my loving friends Mr. Elias Allen, over against St. Clements Church in the Strand, in Brasse; and Mr. John Tomson in Hosier lane by Smithfield (in Wood) and may also both in Wood and Brass be had with the instructions thereof by me at my house.

A veritable Swiss penknife of an instrument. 

John Speidel, Arthur Hopton and Aaron Rathborne all recommended Elias Allen (c. 1588–1653) and had their instruments made by him. Rathborne was friends with Speidell and with Henry Briggs (1561–1630). The firm of Tomson in Hosier Lane, run by John Tomson and later Anthony Tomson, presumably his son, was also used by Arthur Hopton to market his instruments along with other notable mathematical figures in the seventeenth century. Apparently the Tomsons only made instruments in wood and none of their instruments have survived. The close connections between Goodwyn, Speidel, Rathborne, Hopton, Tomson, and Allen demonstrate once again just how close knit the London mathematical community was. 

One major area of work undertaken by John Speidell involved the recent invention by John Napier (1550–1617) of logarithms. An area in which Henry Briggs was also very active. In 1619, he published a table entitled New Logarithms in which he calculated the natural logarithms of sines, tangents, and secants. A second edition was published in 1622 with an appendix containing  the natural logarithms of all numbers from 1–1000.

Along with Willian Oughtred (1574–1660) and Richard Norwood (c. 1590–1675) a mathematical practitioner from Bermuda, Speidell agitated for the abbreviations of the trigonometrical functions. In 1628, John added the book Arithmeticall Extraction to his publications.

In 1635, Sir Francis Kynaston, lawyer, courtier, poet and politician founded an academy of learning called the Musæum Minervæ offering a course of modern gentlemanly training. The full seven year course on offer was quite extraordinary, including fencing, music, languages, heraldry, the study of antiquities and coins, the procedures of the common law, husbandry, dancing and deportment, riding sculpture and writing. The sciences were represented by medicine (physiology and anatomy), astronomy, optics, fortification, and geometry. John Speidell was appointed Professor for Geometry. Despite a licence under the great seal, a grant of arms, a contribution from the Crown, the academy, beset by the plague in 1635 and 1637, never really got off the ground and had closed by 1639. John Speidel was back to surviving on teaching private pupils.

Fracis Kynaston was probably fairly typical of the type of men that John Speidel taught as Euclid noted in a comment:

…before the [civil] wars broke out his Practice was chiefly amongst the Courtiers. For in those days The Mathematical Arts were had in more Esteeme by the Gentry but when they left London he was forct to apply himself other ways which did not encourage him so much or prove so beneficial.

The 1640s were a hard time for John Speidel, a man closely associated with the royal court and in this time he only issued one work: A Briefe Treatise for the Measuring of glass, Board, Timber or Stone, Square or Round published by Thomas Harper (active 1614–1656) a London printer associated with many Catholic and Royalist publications. 

A year before John’s death, John Darker, about whom nothing is known, produced an elaborate illustrated manuscript combining John Speidel’s Geometricall Extraction with The Works of Edmund Gunter (1581–1626) creating a full course in both abstract and applied geometry, with the illustration of instruments all taken from Gunter’s book.

Euclid Speidel, born in 1631, was eighteen years old when his father died in 1649. We don’t actually know if his was christened Euclid, or whether he adopted the name later as appropriate for a geometry teacher. It is obvious that John gave Euclid a solid mathematical education. As a child he received an introduction to arithmetic, ‘my father did [teach?] me arithmetick as far as Trade might req[uire]’. By the age of thirteen Euclid was already giving arithmetic tuition to the children of the puritan parliamentarian Sir Robert Harley (1579–1665). Harley house was in the same area as Westminster school and one day Euclid met Dr. Richard Busby (1606–1695) the schools legendary head master, who held that post for fifty-five years and whose pupils included Robert Hooke, John Locke, and Christopher Wren amongst many other seventeenth-century luminaries:

…going through the abbey bare headed the then & now Master the Rever[end] Doctor Busby being in the abbey called me to him, and having after interrogated me of my Parents & Education, and if I then went to Latin School, and understanding me, I did not then, bidd me tell my father, if he would have me proceed further in Grammar learning and let me come to his school he would give me my Learning for nothing, which my Father accepted. 

After two years at Westminster school, Euclid returned to his mathematical education under his father: 

…when I desisted going to School my father sett me to study further in arithmetick, and to have ready hand in Practical Geometry, especially in his Geometrical Exercises, and to have some common uses of the Plane scale & sector and before I might leave my fathers house might to arithmetick as far as Doubl Position, and also beene in the Field with him several tymes to take the survey of several fields, and in anno 1648, we tooke the survey of Woods at Eitham in Kent. I might also learn some thing of instrumental Dyalling, but as to Triggonometry by Calculation either plane or Spherical I was a stranger. 

As can be seen from the earlier ‘arithmetick as far as Trade might req[uire]’ and the passage above the emphasis is very strongly on practical mathematics rather than the abstract mathematics of the university. Double position is a method of solving linear equations. It is interesting that John did not teach his son theoretical trigonometry as he had published a Breefe Treatise on Sphericall Triangles in 1627. 

His home education completed Euclid now took up a position as a lawyer’s clerk. However, this did not last long as in 1652 , now twenty-one years old, he went to sea, most probably as a captain’s clerk. He only stayed at sea for two years but saw service in the first Anglo-Dutch War (1652–54). Returning to London he followed in his his father’s footsteps, setting up as a teacher of practical mathematics, but also acquiring a position as an accounting clerk at the Excise Office. For his teaching he used his father’s Arithmeticall Extraction. However, this was already out of print and difficult to acquire  so, in 1686 he published an expanded second edition of the book. This was printed by H.C. for the cartographer, globe and instrument maker, and publisher Philip Lea (c. 1660?–1700). In 1688, he followed his father into the world of logarithms with his Logarithmotechnia: Or, The Making of Numbers Called Logarithms To Twenty Five Places, From A Geometrical Figure, With Speed, Ease and Certainty, which was printed by Henry Clark (active 1688s/90s), presumably the H. C. who printed the second edition of Arithmeticall Extraction.

Euclid also followed his father in designing a scale. A leading London instrument maker of the period Henry Sutton (c.1637–1665) made a dialling scale which is inscribed, In usum Euclides Speidell Angli. Euclid and Sutton both lived and worked in Threadneedle Street. Sutton’s most famous instrument were his quadrants. John Collins (1625–1683) is today best known as the seventeenth century’s mathematical groupie, who carried out correspondences with many leading European mathematician keeping them informed of all the latest developments. In 1658, he published a book, The Sector on a Quadrant, or, a Treatise Containing the Description and Use of Three Several Quadrants ‘With large Cuts of each Quadrant, printed from the original plates graved by Henry Sutton, either loose, or pasted upon Boards. Colins is described as ‘Accountant and  Student of Mathematiques’. Collins who had previously published very little went on to publish a series of books on mathematical instruments all connected to Henry Sutton’s work. 

Like Euclid, John Collins was a clerk, who had gone to sea (1642–1649), who on settling in London had set up as a mathematics teacher and had become an accountant to the Excise Office. The two men had a close relationship, which included Michael Dary (1613–1679) another mathematical practitioner, who held positions as exciseman and gunner of the Tower of London. The three men also maintained connections with Robert Hooke (1635–1703) and Edmund Halley (1656–1741). Euclid like his father was firmly embedded in a thriving mathematical community.

Euclid ceased teaching in the late 1670s or early 1680s but continued in his accountants position at the Excise Office. Euclid had both a son and a grandson named Euclid but neither of them became mathematicians. He died in 1702.

The biographies of the father and son, John and Euclid Speidell, gives us a window into the world of practical mathematics in the seventeenth century. A world that is very much centred on mathematics used in various activities such as surveying, dialling, cartography etc. and the instruments employed in them. People like the Speidells brought mathematics to a range of people outside of the schools and universities. 

Most of this essay, especially the quotes from Euclid Speidell’s biography, is extracted from Boris Jardine, The Life Mathematick: John and Euclid Speidell and the Centrality of Instrument’s in Seventeenth-Century Pedagogy, in Philip Beeley and Christopher D. Hollings eds. Beyond the Learned Academy: The Practice of Mathematics 1600–1850, (OUP, 2024), pp. 387–407.

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Filed under History of Mathematics, Scientific Instrument Makers

May 6, 2026 · 8:16 am

Al-Khwarizmi didn’t in any way originate, invent or create the algorithm

Somebody called Ihtesham Ali, with 29.5k followers on Twitter, posted an approximately nine hundred word hymn of praise for the Islamic mathematician, astronomer and geographer, Muhammad ibn Musa al-Khwarizmi, or simply al-Khwarizmi  (c.780–c. 850), in which he goes totally over the top in his description of his impact, in particular on computer science. Unfortunately, the core of what he writes is quite simply totally and horribly false. Although, his misconceptions are extreme, in core they are fairly widespread and I have decided to address them here. 

A stamp issued 6 September 1983 in the USSR, to celebrate roughly 1200 years since al-Khwārizmī’s birth Source: Wikimedia Commons

Ihtesham Ali: A Persian scholar finished a single math book in 9th century Baghdad that quietly became the foundation for every line of code running on Earth today. I started reading about him at midnight and could not believe how many things in my daily life trace back to one man. His name was Muhammad ibn Musa al-Khwarizmi. The book is called The Compendious Book on Calculation by Completion and Balancing. Every time you say the word algebra, you are saying his book title. Every time someone says the word algorithm, they are saying his name. Both English words come from him. Both are Latin transliterations of Arabic and of his own identity. The man did not just contribute to mathematics. He named it. Here is the part almost nobody tells you. Al-Khwarizmi was born around 780 CE in Khwarazm, in what is now Uzbekistan.

This opening block is largely correct except the claim that al-Khwarizmi’s book is the foundation for every line of code running on Earth today, which I will address later. However, there is one minor point that is common to all accounts of al-Khwarizmi that irritates me and that I would like to address, was he a Persian? Our author correctly states that Al-Khwarizmi was born around 780 CE in Khwarazm, in what is now Uzbekistan, a fact that is embedded in his name, al-Khwarizmi, which means literally from Khwarazm. Khwarazm had been a state in its own right since about five hundred BCE with its own language, an East Iranian language, Khwarezmian, and culture, until Turkic tribes invaded the territory in the ninth century CE. It became a vassal kingdom of the original Achaemenid Persian Empire and then again under the later Sasanian Empire, which itself was swallowed by the Abbasid Empire, hence the claim that al-Khwarizmi was Persian but he was obviously a Khwarezm native and not a Persian. Al-Khwarizmi literally means the native of Khwarezm. You might think I’m splitting hairs but Britain was a part of the Roman Empire for a couple of centuries and we refer to natives from this period as Britons and not as Romans!

Source:Wikimedia Commons

Ihtesham Ali: He moved to Baghdad and worked at a research institution called the House of Wisdom, which during the Islamic Golden Age was the single most important center of learning on the planet. The caliph al-Mamun hired the best mathematicians, astronomers, and philosophers from across three continents and put them in one building with one job. Translate, study, and produce new knowledge.

We have been here before. There is absolutely no evidence that the so-called House of Wisdom was an important centre of learning, let alone the single most important center of learning on the planet, it’s a myth. This is carefully spelled out by Dimitri Gutas in his Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early Abbasid Society (2nd–4th/8th–10th centuries),(Routledge, Oxford), ppb. 1998 pp. 53-60 and Lutz Richter-Bernburg, Potemkin in Baghdad: The Abbasid “House of Wisdom” as Constructed by 1001 inventions In Sonja Brentjes–Taner Edis­–Lutz Richter-Bernburg eds., 1001 Distortions: How (Not) to Narrate History of Science, Medicine, and Technology in Non-Western Science, Biblioteca Academica Orientalistik, Band 25, (Ergon Verlag) 2016 pp. 121–129 all of which I have documented here.

Ihtesham Ali: Al-Khwarizmi finished his book on algebra around 820 CE. The Arabic title contained the word al-jabr, which referred to one of the two operations he used to solve equations. When the book was translated into Latin in the 12th century, the Latin world did not have a word for what he had built. So they kept his Arabic word. Al-jabr became algebra. The discipline was named after a single Arabic word in the title of a single book by a single man.

Once again basically correct but one should, I think give the original Arabic title and the Latin translation title as provide by Robert of Chester (dates unknown) in 1145. The Arabic original is al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah and Robert of Chester’s Latin translation is Liber Algebræ et Almucabola. The first word of the title was borrowed into Medieval Latin as algebrāica to become the name for the discipline, eventually evolving into the modern algebra. 

Ihtesham Ali: The deeper insight is what he actually changed about how humans think. Before al-Khwarizmi, mathematical problems were solved geometrically. You drew shapes. You measured them. You compared areas. The Greeks had built an entire mathematical tradition on visual proofs and physical constructions. It was beautiful and limited. You could not solve a problem you could not draw. Al-Khwarizmi did something nobody had done before him at this scale. He said you could solve any problem using abstract symbols and rules. You did not need a shape. You needed a procedure. You moved terms across the equation. You cancelled like terms on both sides. You isolated the unknown. He invented the idea that mathematics is a manipulation of symbols according to rules, not a study of physical figures. That single shift made everything that came afterward possible. Calculus. Differential equations. Linear algebra. Quantum mechanics. None of it works if math is locked inside geometry. He pulled it out. 

Here, our author goes completely off the rails. I have to ask how someone can pontificate so authoritatively and enthusiastically about a subject, which they obviously know almost nothing about. Yes, before al-Khwarizmi the Greeks had solved problems geometrically. Euclid, one of al-Khwarizmi’s principle sources, had included much so-called geometrical algebra in his Elements. This is why first degree equations are linear, second degree equations quadratic, and third degree equations cubic. Linear equations reference a straight line, second degree equations a square and third degree equations a cube. However, well before the Greeks, ancient Egyptians, the ancient Chinese, the ancient Indians and the ancient Babylonians had all been solving equations algebraically. The Egyptians and the Chinese were well acquainted with the solution of linear equations, The Babylonians solved linear, quadratic and cubic equations algebraically. By about 1700 BCE , so more than two thousand years before al-Khwarizmi was born, the Babylonians had a version of the general solution for quadratic equations, although they only accepted positive solutions. Two centuries before al-Khwarizmi, the Indian astronomer and mathematician, Brahmagupta (c. 598–c. 688 CE), a major source and influence for al-Khwarizmi’s work, had the general solution for quadratic equations, basically, in the form still taught in schools today. He  accepted both positive and negative solutions, unlike his Babylonian predecessors. 

Traditional algebra, the theory of equations–the study of algebraic structures only emerged in the nineteenth century–was carried out in three different forms: Rhetorical algebra, in which equations are written in full sentences, Syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra, and Symbolic algebra, in which full symbolism is used. Early Egyptian and Chinese algebra was rhetorical. Babylonian algebra was largely rhetorical but was also partly syncopated. Greek Diophantine algebra, named after Diophantus, who lived in the third century CE, was syncopated.  Al-Khwarizmi’s algebra was one hundred percent rhetorical; he even used number words  not number symbols. There are absolutely no abstract symbols here! This is a typical passage from his book:

If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.

Al-Khwarizmi, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah, English translation, Frederic Rosen, The Algebra of Mohammed Ben Musa, (London: The Oriental Translation Fund, 1831) pp. 47–48.

Just as importantly, al-Khwarizmi offers both algebraic and geometric solutions. For example, when explaining the method of completing the square, he illustrates it with the same geometrical diagram that can be found in dozens of school maths textbooks today:

A page from al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah Arabic original Source: Wikimedia Commons
Pages from a 14th-century Arabic copy of the book, showing geometric solutions to two quadratic equations. Source: Wikimedia Commons

So, he in no way invented the idea that mathematics is a manipulation of symbols according to rules, not a study of physical figures. So, the follow on claim above is also nonsense. Al-Khwarizmi’s book is actually quite elemental, both the Babylonians and Brahmagupta produced algebra more advanced than him so, why is his book historically so important? Before al-Khwarizmi people were just problem solving; some of those problems were solved algebraically. Al-Khwarizmi’s al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah is the first book that handles algebra as a discipline.

Ihtesham Ali: The second thing he did is the one that changed how the world counted forever. He took the Hindu numeral system from Indian mathematics, refined it, and wrote a book introducing it to the Arab world. That system included the concept of zero as a placeholder, and a positional notation where the value of a digit depends on its location. Roman numerals could not do complex calculation. Hindu-Arabic numerals could. When his book on numerals was translated into Latin as Algoritmi de numero Indorum, the word Algoritmi was just the Latin spelling of his own name. Europeans started calling the new method “doing algorism,” then “running an algorithm.” The word for the most important concept in computer science is literally his name in Latin.

Al-Khwarizmi did indeed present the Hindu numeral system in a book, of which no original Arabic manuscript is known to have survived. He version is in no way refined in comparison to the presentation by Brahmagupta two hundred years earlier. That system contain zero not just as a placeholder but as a number in its own right. We have the usual irrelevant comment that Roman numerals could not do complex calculation. The calculation were carried out using an abacus and Roman numerals were merely used to record the results. The four surviving Latin manuscripts are titled:

  • Dixit Algorizmi 
  • Liber Alchoarismi de Practica Arismetice
  • Liber Ysagogarum Alchorismi
  • Liber Pulveris

The title Algoritmi de Numero Indorum was added to the first manuscript when it was published in 1857. 

Page from a Latin translation, beginning with “Dixit algorizmi” Source: Wikimedia Commons

Dixit Algorizmi means ‘Thus spake Al-Khwarizmi’ and a version of this Latin transliteration of his name algorism became the term used to describe calculating with the Hindu-Arabic place value decimal number system and also the name given to the text books used; famous and much used algorisms were written by Johannes de Sacrobosco (1195–c. 1256) and Robert Grosseteste (c. 1168–1253).

Ihtesham Ali: The third thing he did is the part that should haunt anyone who works in tech. His method of solving problems was systematic. Step one, do this. Step two, check that. Step three, if condition A, then do X, otherwise do Y. He wrote down procedures that could be followed by anyone, anywhere, who knew how to read. The procedure did not depend on intuition or genius. It worked because the steps worked. That is exactly what an algorithm is. A finite, deterministic procedure for solving a problem. He did not just give us the word. He gave us the entire concept of programming a thousand years before there was anything to program.

What our author is describing here is what is now termed the algorithmic method of problem solving and it didn’t originate with al-Khwarizmi; he didn’t invent or create it. Babylonian, Ancient Chinese, Ancient Egyptian, and Ancient Indian mathematics were all algorithmic. These cultures taught step by step recipes on how to solve a particular type of problem. They had no concept of a general proof; this was the Ancient Greek contribution to the evolution of mathematics. When al-Khwarizmi described his problem solving recipes he was merely continuing an established two thousand plus year old process. As noted the term algorism was first used for the art of calculating with the Hindu-Arabic number system. In the High Middle Ages, the term algorism first became algorithme in French and then algorithm because it was falsely believed to be derived from the Greek arithmos. Over the centuries the definition and meaning of the term evolved, an evolution that Jeffrey M Binder, who describes himself as a programmer, historian, and writer, sets out in his book, Language and the Rise of the Algorithm, (University of Chicago Press, 2022), which I reviewed here. 

Ihtesham Ali: When Alan Turing built the first abstract model of computation in 1936, when John von Neumann designed the first stored-program computer in 1945, when every engineer at Google, OpenAI, Anthropic, and DeepMind writes code in 2026, they are working in a paradigm that started with one man in Baghdad twelve centuries ago. The strangest part is what happens when you walk into any tech office in San Francisco or Bangalore or Lahore today. Engineers say the words algebra and algorithm hundreds of times a day. They do not know whose name they are saying. Almost nobody can spell al-Khwarizmi correctly on the first try. His original Arabic manuscript is preserved at Oxford. His book on Hindu numerals survives only in Latin translation. The Latin version was the textbook that taught medieval Europe how to count. The man who built the foundation of the AI revolution did not live to see a calculator. He died around 850 CE, a thousand years before the first electric current was sent through a wire. The civilization he built mathematics for collapsed. The library he wrote in burned. His own grave is unmarked. But every algorithm running on every machine on Earth right now still answers to his name.

All of this section is quite simply hogwash, because, as already explained above, although the term algorithm was originally derived from the Latin transliteration of his name he didn’t in any way originate, invent or create the algorithm. I would also note that the claim that Alan Turing built the first abstract model of computation in 1936 is also rubbish. In his legendary 1936 paper, On Computable Numbers, with an Application to the Entscheidungsproble, Alan Turing became one of three logicians or metamathematicians, who demonstrated that there are mathematical problems that cannot be finitely computed i.e. there is no single algorithm that infallibly gives a correct “yes” or “no” answer to each instance of the problem. I’m not really sure what an abstract model of computation is supposed to be, but I assume the first one was the oldest clay tablet from Mesopotamia on which a scribe explains how to carry out numerical addition. John von Neumann did not design the first stored-program computer in 1945! He wrote a detailed technical description of a computer designed by J. Presper Eckert and John Mauchly, which because it was published nullified the patent claims of Eckert and Mauchly. 

Personally, I would have thought that if one was going to pontificate with authority and great enthusiasm about a topic from the history of mathematics to one’s 29.5 k readers online then one should at least learn the real facts before committing oneself in public but what do I know. 

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Filed under History of Mathematics, Islamic science, Mediaeval Science

May 1, 2026 · 8:02 am

Joseph Moxon–a man of many talents

As I explained in a previous post Emery Molyneux (d. 1598) was the first English, printed-globe maker. After Molyneux departed from England to the Netherlands in 1597, where he died a year later, it would be almost sixty years before another Englishman began to print globes in England; that man was Joseph Moxon (1627–1691). With Moxon, as you will see, we have another practical mathematics link between England and the Netherlands in the Early Modern Period. We have already had Thomas Gemini (c.1510–1562), a Netherlander who became London’s first commercial instrument maker. John Dee (1527–c. 1608) one of England’s most important mathematical practitioners learnt his mathematics in Louvain from Gemma Frisius (1508–1555) and his pupil Gerard Mercator (1512–1564) and introduced Mercator’s printed globes into England when he returned from the continent.  Thomas Digges (c. 1546–1595), John Dee’s foster son, an early Copernican, and a leading mathematical practitioner had undertaken a tour of the Netherlands in 1578 inspecting fortifications and observing troops. In 1585, he joined Leicester’s expeditionary force to the Netherlands, as muster-master and trench-master. He became increasingly embroiled in disputes over various issues and in late 1588 he was official discharged. We saw that Molyneux’s globe gores were engraved by the Flemish engraver Jodocus Hondius (1563–1612), who would establish one of the seventeenth century’s cartographical publishing houses when he returned to the Netherlands.

Hondius had left the Netherlands in 1584 because of religious difficulties. The Moxon family went in the opposite direction, also because of religious difficulties. Joseph was born in Wakefield into an old Yorkshire family and educated at the Free Quenn Elizabeth Grammar School, where his uncle Peter Moxon was a governor. 

Joseph Moxon. Line engraving by F. H. van Hove, 1692. Source Welcome Collection via Wikimedia Commons

Joseph’s father, James Moxon, was a puritan, who was strongly opposed to the policies of William Laud (1573–1645), the Archbishop of Canterbury under Charles I from 1631–1645. In 1636, James left England for the Netherlands taking his sons James and Joseph with him. He set a printing shop, first in Delft then in Rotterdam, where he printed Puritan tracts. By 1638 he was printing English Bibles for the English Puritan community in Rotterdam. Laud was executed in 1645, followed by Charles I in 1649. The Moxons returned to London in 1646, where Joseph and his elder brother James continued to print and publish Puritan literature, with one exception, A Book of Drawing, Limning, Washing or Colouring of Mapps and Prints, printed for the engraver and publisher Thomas Jenner (died 1673), in 1647.

By 1650 Joseph had left the family busy, James carrying on alone. Joseph had taken up the study of globe and map making and developed a strong interest in practical mathematics. In the spring of 1652, he travelled to Amsterdam and acquired engraved copper globe-printing plates. Later in the same year in partnership with John Sugar, he began to sell 15 inch terrestrial and celestial globes. At the sign of the Atlas (shops didn’t have street numbers in those days) Joseph developed a business printing maps, charts, globes and paper mathematical instruments and publishing popular scientific books. Paper mathematical instruments is something that tend to get neglected by historians of mathematical instruments. Metal mathematical instruments were very expensive and there was a big trade in cheap printed paper instruments. These could be cut out and pasted onto wooden boards making a cheap functional alternative to the expensive metal instruments. Unfortunately, they are highly perishable and we only have very few examples that have survived the ravages of time.

Moxon’s original premises were in Cornhill and from 1665 to 1686 he was at Ludgate Hill except for six years when he was forced to move to Russell street following the Great Fire of London. 

In 1654 he published his first book, A Tutor to Astronomy and Geography, an unacknowledged translation from the Latin of Institutio Astronomica by Willem Janzoon Blaeu (1571–1638) Jodocus Hondius’ greatest rival. Over the next thirty years Moxon published more than thirty popular scientific expositions and technical handbooks, many of which he wrote himself. The volumes,  Vignola or The Complete Architect and A Tutor to Astronomy and Geography. Or an easie and speedy way to know the USE of both the GLOBES, Celestial and Terrestrial both of which he wrote himself went through several editions. 

Source

Moxon gained a reputation for printing mathematical texts (John Dansie’s Mathematical Manual, 1654, and Edward Wright’s Certain Errors in Navigation, (1657), and particularly of tabulated data (his tables of solar declination, Primum mobile, 1656, reprinted in John Newton’s A Help to Calculation, 1657, together with tables of logarithms); he also set the 230 pages of trigonometrical functions and logarithms in William Oughtred’s Trigonometria (1657).

Moxon sought to improve his status with a petition to Charles II, asking to be made Hydrographer to the King. By now Moxon was obviously well connected because his petition was supported by thirteen leading figures of the period–J. Newton D.D., L. Rooke, Walter Pope, John Collins, W. Chiffinch, E. Ashmole, G. Wharton , Henry Bonde, Jonas Moore, John Leeke, Thomas Harvie, Will Mar, Euclid Speidell–they stated that they were “Professors of Mathematics” meaning that the professed mathematics. Several of them had close connections at court.

Only Walter Pope and Lawrence Rooke were actually professors at Gresham College.

Moxon published Newton’s A Help to Calculation (1657), Leek’s Waterworks (1659), Moore’s Short Introduction Into the Art of Species (1660) and Bond’s Two Tables of Ranges, according to the Degree on Mounture (?)

On 10 January 1661/2 a warrant was issued by the King “To our Right Trusty and Right beloved Cousin and Councillor Edw: Earle of Manchester our Chamberlane of our household” as follows:

Our Will and Pleasure is that you give present order for the Swearing of Joseph Moxon our Servant in the quality of Hydrographer unto us for the making of Gloebes Mapps and Sea plats and he be admitted thereunto as out servant in ordinary and have and receive all Rights and Profitts Priviledges and advantages thereby in an ample manner as any our servants in that or like quality have doe or ought to enjoy and for do doeing  this shall be your warrant.

Joseph Moxon had come up in the world, from Puritan refugee in the Netherlands to official servant to the King of England, whose father the Puritans had executed. The appointment led to a boom in Moxon’s activities. In the following period he was involved in the production of almost forty volumes, as printer, published, translator, or author. 

Several of the supporters of Moxon’s petition were founding members of the Royal Society and over the next years he was in close contact with the society. He had close personal contact with Robert Hooke (1635–1703) the society’s demonstrator of experiments and the two men exchange information and undertook joint activities. 

In 1678 Moxon’s status rose once more when he was elected to the Royal Society. In its early years the society resemble more a gentleman’s club that a scientific society. The fellows were either high ranking scholars or aristocrats. Moxon was the first tradesman elected to the society and remained the only one to receive this honour in the seventeenth century. That his election was unusual is reflected in the vote on his membership. When somebody was proposed for fellowship the vote by show of hands was almost always unanimous. By Moxon’s election, in a secret vote, there were four votes against. 

Moxon was an active member for the first two years of his membership. In 1680, the possibility was raised of Moxon becoming the printer to the society. However, after a year of deliberation somebody else was appointed to this position. Following this Moxon took no further part in the activities of the society. It could have been the case that being a fellow of the society barred Moxon from receiving a paid position in the society. When Edmond Halley (1656–1742) was appointed secretary to the society he had to resign from his fellowship.

In his work as a printer, publisher, author Moxon was highly innovative. Moxon was the first to introduce pocket globes into England, sometime between 1659 and 1670.

A very rare Joseph Moxon 2 ¾-inch pocket globe, English, circa 1675 Source

As already noted he published a wide range of mathematical and technical literature but I will just highlight his three most significant publications. 

In 1678, he published the first issue of what was effectively a monthly subscription magazine with the main title Mechanick Exercises or, The Doctrine of Handy Works…, the series was completed by 1680 after a total of fourteens parts had been issued. Published later, as it book The full title was:  Mechanick exercises, or, The doctrine of handy-works : applied to the arts of smithing, joinery, carpentry, turning, bricklayery : to which is added Mechanick dyalling: shewing how to draw a true sun-dyal on any given plane …

Mechanick dyalling, or to give it its original full title, Mechanick dyalling teaching any man, though of an ordinary capacity and unlearned in the mathematicks, to draw a true sun-dyal on any given plane, however scituated : only with the help of a straight ruler and a pair of compasses, and without any arithmetical calculation had been published separately in 1668.

This was part of an interesting development, which Moxon didn’t start, in which the knowledge of the trades became increasingly public. Traditionally the hand-work trades were taught by a master to his apprentice(es), oft father and son, the knowledge imparted being kept secret, the classical “trade secrets.” In the early modern period as the barriers between practical knowledge and academic knowledge began to fall, one of the driving forces behind the so-called scientific revolution, various authors took to publishing those trade secrets to make them available to a wider public. Moxon took this to a new level.

The Mechanick Exercises might well have been instrumental in Moxon’s election to the Royal Society. In 1678, he had presented six issues of the work to the president of the society. He was much impressed, in particularly because the Royal Society had planned such a publication in 1660, which had never gotten off the ground.

Moxon’s Mechanick Exercises Volume II is regarded as his most important or significant publication. Originally published in twenty-four parts in 1683-84, the Mechanick Exercises: or the doctrine of handy-works applied to the art of printingwas the first comprehensive and detailed account of all aspects of printing written in English, or some sources say in any language. 

Moxon scored another first, also in any language, with his Mathematicks made Easie: Or, a Mathematical Dictionary, Explaining The Terma of Art, and Difficult Phrases used in Arithmetick, Geometry, Astronomy, Astrology, and other Mathematical Sciences first published in 1679. 

Note the presence of astrology in Moxon’s listing of the mathematical sciences. In 1682 he was heavily involved in an attempt to revive the London Society of Astrologers, originally created in 1649 by William Lilly and Elias Ashmole. Ashmole had been one of Moxon’s supporters in his petition to be appointed Hydrographer to the King. In general Moxon was deeply embedded in the scientific community in London in the second half of the seventeenth century. As well as those already mentioned at various point above, he is known to have made astronomical observations with Edmond Halley and he cut and cast the Irish characters commissioned by Robert Boyle (1627–1691) for the 1681-85 printing of the Bible in Irish.

Both volumes of the Mechanick Exercises and Mathematicks made Easie went through numerous editions right down to the present.

As an instrument maker, globe maker and printer Joseph Moxon was, like Elias Allen and Ralph Greatorex, part of the adhesive that turned a group of scientific researchers into scientific community but with the continued emphasis on big names and big discoveries in the history of science, people like Joseph Moxon don’t get the acknowledgement and recognition that they deserve. 

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April 24, 2026 · 7:36 am

From τὰ φυσικά (ta physika) to physics – LXIV

1023 days ago, I posted the first episode of this series tracking the development of physics from Aristotle’s τὰ φυσικά to the point where the term physics began to be used. Now in the sixty-fourth episode we have finally reached our destination. In that first episode I took a look at the term physics its origins in Aristotle’s Greek and how it changed down the centuries until it first emerged with its modern meaning in 1715. 

Although there is no link, the emergence of the term physics in its modern meaning is with certainty related to the publication of Newton’s Principia. Of course, Newton’s tome proudly contains the term Philosophiæ Naturalis (Natural Philosphy) in its title but it’s the other half of the title that is new Principia Mathematica  (Mathematical Principals). For Aristotle ta physika, the description of nature, could never be mathematical. Numbers are not natural object so, cannot be used to describe nature. Mathematics was confined to the so called mixed or subordinate sciences–astronomy, optics, statics–these are not natural philosophy. Newton’s description of nature is purely mathematical and this was one of the main points of criticism made by both Huygens and Leibniz. Newton’s gravity had no physical explanation. 

Despite these apparent failings Newton’s mechanics slowly but surely became dominant, the accepted norm. When we today refer to everyday physics, non-relativistic mechanics, the sort that’s taught in school we refer to it as classical of Newtonian mechanics or physics. However, the modern physics referred to as Newtonian physics is not Newton’s physics. 

The first thing that changed was that mathematicians on the continent replaced Newton’s extra created analytical Euclidian geometry with Leibniz’s calculus and then later the more modern F’(x) = f(x) notation of the French mathematician, Lagrange (1736–1813). Unfortunately, in England out of a sense of national pride, although the mathematicians replaced the analytical Euclidian geometry they did so with the much more unwieldly Newtonian analysis with its dot notation. This led to the infamous Analytical Society campaign in Cambridge, Newton’s own university, to promote “the principles of d-ism as opposed to the dot-age of the university” in Charles Babbage’s wonderful pun. 

Turning to the physics, Newton had woven together the astronomy concepts of Johannes Kepler (1571–1630) and Giovanni Alfonso Borelli (1608–1679), with the advances in mechanics made by Simon Stevin (1548–1620), Isaac Beeckman !588–1637), Galileo Galilei (1564–1642), Giovanni Alfonso Borelli, René Descartes (1596–1650), Christiaan Huygens (1629–1695) and others to create a unified terrestrial-celestial mechanics that explained mathematically all movement on the earth and in the heavens. However, despite the fact that he had modified and improved his masterpiece in the second (1713) and third (1726) editions, it was still by no means perfect. There were still grey areas that needed improvement. One was the theory of comets that as we have seen was significantly improved by the work Edmond Halley (1656–1742) in his 1706 publication.

Throughout the eighteenth century, people worked on improving, correcting, expanding the foundations that Newton had laid down in his Principia. Unfortunately, very little of that work took place in Britain, which became moribund in its reverence for Newton’s great achievement. In Switzerland, the Bernoullis and Leonard Euler (1707–1783) made significant progress, whilst in France the native-born Italian Joseph-Louis Lagrange (1736–1813) and the Frenchmen Pierre Simon Laplace (1749–1827), Adrien-Marie Legendre (1752–1833), Jean le Rond d’Alembert (1717–1783), Pierre Louis Maupertuis (1698–1759), and Émilie du Châtelet (1706–1749). What follows are very brief sketches of some of the major developments.

Daniel Bernoulli (1700–1782) incorporated the beginnings of the kinetic theory of gasses and hydrostatics into the more general mechanics. Perhaps most spectacular in celestial mechanics was Pierre Simon Laplace’s solution of the problem of the orbit of the Moon (one that Newton had failed to bring convincingly into his general theory of gravity) in his Exposition du système du monde (1796) without details, and more fully in his monumental five volume Traité de mécanique céleste (1798–1825), a work that can be regarded as the crowning glory of Newton’s celestial mechanics.

Two aspects of mechanics which were only beginning to emerge in the late seventeenth and early eighteenth centuries were energy and work. Newton had argued that kinetic energy, the energy released by a moving object on impact, was mv, where m was the mass and v the velocity of a moving object. Johann Bernoulli (1667–1748) and Leibniz had hypothesised that it was mv2 but without any real foundation for their claim. Willem s’ Gravesande (1688–1742) carried out a series of experiments in which he dropped steel balls into clay and measured the impact craters. Émilie du Châtelet took the results of his experiments and deduced theoretically that Leibniz was in fact right and E ≈ mv2. Work on the concept of energy continued throughout the eighteen and nineteenth centuries.

Work according to the modern definition is the energy transferred to or from an object via the application of force along a displacement. The term work was first used in 1826 but already in a letter to Huygens in 1637 Descartes wrote:

Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet. (Wikipedia)

In 1686 Leibniz wrote in his Brevis demonstratio:

The same force [“work” in modern terms] is necessary to raise body A of 1 pound (libra) to a height of 4 yards (ulnae), as is necessary to raise body B of 4 pounds to a height of 1 yard. (Wikipedia)

The English civil engineer John Smeaton (1724–1792), famous for building the third Eddystone Lighthouse (1755–59) did experiments relating power (his term for work) and kinetic energy, and supporting the conservation of energy, which he published in 1776 in the Philosophical Transactions of the Royal Society, of which he was a member. He supported Leibniz’s mv2, which made him unpopular with the other Royal Society members. His definition of power was  “the weight raised is multiplied by the height to which it can be raised in a given time,” which was very close to the definition for work introduced in the late 1820s by the French mathematician Gaspard-Gustave de Coriolis (1792–1843), who first used the term travail, the French for work in his Calcul de l’Effet des Machines (“Calculation of the Effect of Machines”)in 1829. He established the correct expression for kinetic energy, ⁠1/2⁠mv2, and its relation to mechanical work. The French engineer Jean-Victor Poncelet (1788–1867) independently introduced the term mechanical work and its relation to kinetic energy at around the same time. 

In 1773, the French chemist Antoine Lavoisier (1743–1794) stated the law of the conservation of mass based on his own experiment. By the beginning of the nineteenth century a large part of the body of classical physics erected on Newton’s foundations was in place. However, although Newton had written one of the most important books on optics, had discussed magnetism as another example of action at a distance and a series of electrical experiments were carried out at the Royal Society during his period as president, these three areas didn’t become integrated into the main body of physics until the discovery of the electromagnetic spectrum by Michael Faraday (1791–1867) and James Clerk Maxwell (1831–1879) in the middle of the nineteenth century. James Clerk Maxwell’s work developed by Oliver Heaviside into the famous four Maxwell’s Equations announced the beginnings of the fall of Newtonian physics and would lead the way to Einstein’s relativistic physics.

 

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