Category Archives: Book Reviews

History of science garbage dumps – two for the price of one

I acquired the book under review here back in 2024 shortly after it was first published and having read it was astounded at how bad it actually was. I started to write a review but after much effort and more than 8000 words, I was about half way through and couldn’t take anymore. I put it to one side aiming to take it up again later, I never did. Recently on an Internet forum somebody asked if anybody knew the book and what they thought of it? I replied with my honest opinion that I thought it was crap and was asked if I had written a review of it. I replied with the explanation given above and the enquirer asked if he could read my unfinished review, as did several others. Having dug it out and reread it, I sent it to those who wished to read it. I have now posted it here to warn any other potential readers  of those subpar tome. I will only point out that the chapters I haven’t reviewed here are just as bad as the ones I have reviewed.

Back in 2019, I decided to buy the highly praised debut book by Violet Moller, The Map of KnowledgeHow Classical Ideas Were Lost and FoundA History in Seven Cities (Picador, 2019). Typical of the promotion for this volumes was Peter Frankopan’s cover blurb:

A lovely debut from a gifted young author. Violet Moller brings to life the ways in which knowledge reached us from antiquity to the present day in a book that is as delightful as it is readable.

I have a lot of respect for Peter Frankopan so, I took the plunge. When I tried to read this highly praised volume, Frankopan was not alone in his gushing praise, I found it factually inaccurate and shoddy so I gave up. More recently an internet acquaintance asked me my opinion of the book and I said I wouldn’t recommend it. He then asked me why and I said, to be honest I have a negative view of the book  but I can’t remember why! One evening I took it down from the shelf and reread parts of it and came to the conclusion that it was even worse than I remembered.

Given these facts, it might seem surprising that, when Violet Moller’s new book was launched, with similar fanfares, although nothing from prominent historians, I bought a copy. Given the title, I simply couldn’t resist, Inside The Stargazers PalaceThe Transformation of Science in 16th-Century Northern Europe (One World, 2024). If I claim to have even a modicum of expertise in the history of science it is in The Transformation of Science in 16th-Century Northern Europe a topic to which a substantial percentage of this blog has been devoted over the last sixteen years. I wish I had resisted!

Moller sets out her intentions in a nineteen page PrologueBefore in which she explains, quite correctly, that substantial progress was made in advancing science in Europe in the sixteenth century before the seventeenth-century Scientific Revolution. She also correctly notes that these advances were still mixed up with the so-called occult sciences, astrology and alchemy. She writes:

The stars, or rather astronomy, will be our guide. This was the  most prestigious of the mathematical disciplines, one that long played a leading role in the development of science in part because it was often the starting point for investigation of the natural world. People have always built places to observe, to enhance their understanding of the night sky.

And then:

In my last book, The Map of Knowledge, I followed three major scientific texts as they were transmitted and transformed in the Middle Ages , following them on a thousand-year journey through seven cities that ended in 1500. This is where we will begin, taking up where we left off and travelling to seven places north of the Alps where people studied the stars and made instruments in their quest to deepen their understanding of the world around them.

If Moller had actually delivered that which she outlines in this prologue in an accurate factual manner then this would have been a good book, unfortunately, she doesn’t. The book is littered with errors inaccuracies and an incredible amount of waffle. A couple of inaccuracies from this prologue to give a taste before we dive into the morass. 

A central theme of the book is instruments and she writes:

By the second century CE, Ptolemy had an array of instruments at his fingertips, simple ones for measurement like quadrants but also more complex astrolabes and armillary spheres which could calculate and predict celestial activity.

Ptolemy did not have astrolabes. He uses the name for what we now call armillary spheres. Ptolemy’s armillary sphere was a large observational instrument, which was used to measure the positions of celestial object, as described in Book V of the Almagest,  and was not used for calculations. 

Throughout her book Moller seems to be obsessed with clocks and it starts in the prologue with the following:

Of all the astronomical instruments developed before the telescope, clocks were the most significant. Being able to accurately measure time had a profound influence on so many aspects of life, and a singular effect on the accuracy and potential use of astronomical observations. There had been clocks of various kinds for centuries; water clocks were popular in the Arab world and famously reached Europe when the caliph Harun al-Rashid sent one to the emperor Charlemagne – a classic example of one-upmanship masquerading as generosity. 

Let’s quote Wikipedia:

Water clocks are some of the oldest time-measuring instruments. The simplest form of water clock, with a bowl-shaped outflow, existed in Babylon, Egypt, and Persia around the 16th century BC. Other regions of the world, including Indiaand China, also provide early evidence of water clocks, but the earliest dates are less certain. Water clocks were used inancient Greece and in ancient Rome, as described by technical writers such as Ctesibius (died 222 BC) andVitruvius (died after 15 BC).

[…]

Some water clock designs were developed independently, and some knowledge was transferred through the spread of trade. These early water clocks were calibrated with a sundial. While never reaching a level of accuracy comparable to today’s standards of timekeeping, the water clock was a commonly used timekeeping device for millennia, until it was replaced by more accurate verge escapement mechanical clocks in Europe around 1300.

Put simple medieval Europeans didn’t need to be told by Harun al-Rashid what a water clock was.

We move to the first of Moller’s ‘seven places,’ Nuremberg! If only I knew an expert on the history of science in sixteenth century Nuremberg, who could point out Moller’s errors. Moller doesn’t actually engage with the extensive and varied mathematical and astronomical culture of sixteenth-century Nuremberg but only presents potted biographies of Regiomontanus, who lived and died in the fifteenth century, Albrecht Dürer, Wenzel Jamnitzer and a purple prose section on Augsburg and the Fugger. 

The chapter opens with a general account, which is OK, briefly discusses Hans Sachs’ The Book of Trades noting that it doesn’t contain an instrument maker, not surprising as it wasn’t a recognised trade, which she doesn’t mention. She then indulges her spleen for clocks by introducing Peter Henlein (1485–1542), she delivers a typical Moller nonsense:

In the early years of the sixteenth century Peter Henlein, a local artisan, made a small portable clock designed to be worn around the neck or fastened onto clothing—the first known watch, called a ‘living egg’ because of its shape and the miniscule steel cogs that turned inside it. Henlien’s workshop produced hundreds of these…

Henlein was indeed one of the first craftsmen to make small ornamental portable clocks which were often worn as pendants or attached to clothing and is credited with inventing the watch. He did not produce hundreds of these and he did not produce the egg shaped watches that became popular around 1580 twenty years after his death. 

Enter Regiomontanus who also featured in her earlier book and it is interesting to note that her account of his wanderings between leaving Vienna in 1461 and arriving in Nuremberg in 1471 is highly inaccurate in both books but the two accounts also contradict each other! I’ll just stick to the new book.

Moller quotes Regiomontanus’ letter in which he explains his move to Nuremberg in 1471 “on account of the availability of instruments, particularly the astronomical instruments on which the entire science of the heavens is based, but also on account of the very great ease of all sorts of communication with learned men everywhere.” 

She complains that Regiomontanus doesn’t receive the attention he deserves, “There is only one biography of him in English (translated from the German) and his presence in Nuremberg today is slight, although there is a small observatory named after him. I don’t quite know what Moller expects, although my friends at the highly active observatory will be pleased to have got a mention. I recently had a discussion on social media with other experts on the history of astronomy about producing a new Regiomontanus biography, which given his very broad pallet of activities would be a horrendous task. There are however numerous scientific papers on the various aspects of his life and work. 

We get a brief sketch of his life up to his work with Peuerbach in which Moller calls Regiomontanus a Latinized ‘nickname’–it’s a toponym! She also fails to note that he was never called that, the name was first used by Melanchthon sixty years after his death. We then get the following:

Then in 1461, disaster struck – Peuerbach died suddenly, aged just thirty-eight. Regiomontanus had lost his collaborator and friend, at a time when astronomers were thin on the ground. Fortunately, he had met someone the year before who could help. Cardinal Johannes Bessarion … Regiomontanus must have felt he had entered the very gates of heaven when he entered when he arrived at Bessarion’s elegant house and saw the library. 

Firstly, Bessarion’s name was not Johannes, it was Basilios. We get no mention of the fact that Bessarion had sought out Peuerbach to make new translation from the Greek of the Almagest, which he couldn’t do as he didn’t speak Greek but agreed to write an undated Epitome of the Almagest. Bessarion then invited him to return to Italy with him, Peuerbach accepted but only on the condition that Regiomontanus also went. So, Bessarion’s adoption of Regiomontanus into his familia was agreed before Peuerbach’s death. Moller tells us:

Regiomontanus taught Bessarion astronomy and mathematics , receiving tuition in Greek in return…

Bessarion did indeed teach Regiomontanus Greek but I know of no lessons in astronomy and maths for Bessarion.  We get told: 

He and Bessarion spent the next four years travelling around Italy together … In 1467 Regiomontanus was tempted back over the Alps by an offer from Matthias Corvinus, King of Hungary, whose recent victory against the Turks had left him in possession of several rare manuscripts. Unable to resist the prospect of new texts, Regiomontanus set off northwards to the Hungarian court in Buda, a rare beacon of humanism outside Italy. 

Regiomontanus left Italy in 1465 and it is not known where he was for the next two years, so if he headed northwards when he went to Hungary is not known. He received no offer from Matthias Corvinus but travelled to Esztergom (German Gran) to the court of the Archbishop János Vitéz (c. 1408–1472) as a potential member of staff for the newly established University of Bratislava. Vitéz had  earlier been a patron of Peuerbach’s and most probably wanted to engage Regiomontanus for his skills as an astrologer. Regiomontanus later transferred to Matthias Corvinus’ court in Buda. Corvinus did not have ‘several rare manuscripts’ but had established a royal library, the Bibliotheca Corviniana in 1465, one of the most renowned libraries in the Renaissance world, which had grown to about 3,000 codices, which included about four to five thousand various works, many of classical Greek and Latin authors. That is four times as large as the library of Bessarion that Moller waxes lyrical about over several pages.

Moller now tells us:

A few years later still in Matthias’ service, Regiomontanus wrote a letter to a fellow scholar at the University of Erfurt … This letter written in July 1471 is an invaluable source of information on his plans, … One of his priorities was to calculate new planetary tables based on his own improved observations; another was to set up a printing press to publish a selection of scientific works.  

Moller then tells how valuable Regiomontanus’ time in Italy was and can’t resist, “Italy was the main beneficiary of manuscripts brought from Constantinople after it was taken by the Ottomans in 1453…”  the flood of manuscripts out of Constantinople in 1453 is a myth. She adds, Occasionally he discovered manuscripts for himself…” Regiomontanus’ main occupation was to seek out and make copies of manuscripts for Bessarion. 

 We return to Matthias, “In 1471, King Matthias sent Regiomontanus to Nuremberg to work on a new set of astronomical tables based on new, improved observations.”

Actually, according to legend Regiomontanus was asked in Buda why astrological prognostication were so inaccurate, to which he replied because the astronomical data on which they are based is too inaccurate. He then requested permission from Matthias to leave Buda and travel to Nuremberg to carry out his programme of observations. The letter from July 1471, which Moller had actually quoted earlier in her narrative was actually written from Nuremberg, a fact that she quotes,  where he had been living since 2 June at the latest. 

Moller now rambles on extensively about Regiomontanus’ workshop, his famous Tradelist (1474), in which he announced the books he intended to publish, and his intention also to manufacture scientific instruments. In the early history of printing Regiomontanus’ Tradelist is a fascinating and interesting document and it is right to draw attention to it but Moller’s comments on his workshop are totally speculative as we know absolutely nothing about it. Although here Moller allows herself another blunder, she writes:

The list contains several works by Ptolemy, including his masterpiece on astronomy the Almagest, and Euclid’s foundational text on mathematics, the Elements (neither had been printed with their diagrams before [my emphasis])

Neither had even been printed! The first Latin edition of the Almagest printed in 1515 and the first Greek edition in 1538. The first edition of the Elements was that of Ratdolt in 1482.

Moller quotes from the Tradelist, “There shall be made also astronomical instruments for celestial observations,” leaving out the next sentence, “And also other things for common daily use, the names of which it would be tedious to relate.” She then goes on to say:

 “In this period, if you wanted an astrolabe, you either had to make on yourself using a manual, or specifically commission one from a goldsmith. There were no dedicated instrument shops, but as scholarship spread in Europe, more and more people became interested in measuring the stars, transmuting metals, and distilling tinctures. As the demand for astrolabes, glass vessels and other specialist equipment rose, people started making them to sell, setting up centres of production to cater for the new market. … Regiomontanus was a pioneer in this field…

Unfortunately for Moller, and her glorification of Regiomontanus the instrument maker, when he moved to Nuremberg there were already many workshop producing astronomical instruments, which is one of the two reasons he moved there, as he wrote in that letter from July 1471, which she quotes six pages earlier:

Quite recently I have made [observations] in the city of Nuremberg…for I have chosen it as my permanent home, not only on account of the availability of instrument, particularly the astronomical instrument on which the entire science of the heavens is based…

Apparently she doesn’t actually read what she writes! Nuremberg would continue to be the major centre for the production of scientific instruments until at least the middle of the sixteenth century. 

Moller tries to present the printer publisher, Regiomontanus, as some sort of highly influential role model in the history of scientific publishing but, although he was the first scientific publisher, there is little or no evidence that he influenced anybody apart from Erhard Ratdolt. Moller, of course, tries to push the story that Ratdolt learnt printing working for Regiomontanus in Nuremberg but there is absolutely no factual evidence for this theory. I argue that given the impact of Regiomontanus’ Ephemerides and Calendaria, and his reputation as an astrologer, that if Ratdolt had learnt his trade from him he would have loudly announced the fact, when he set up his own publishing house in Venice. 

Moller’s obsession with clocks come out with another quote from the Tradelist, “The Tradelist mentions of a planetarium or astronomical clock being made in the workshop, ‘a work clearly to be gazed upon as a marvel’…  The actual quote,  “In the workshop of the artisan a planetarium is under continuous development. A work clearly to be gazed upon as a marvel” makes no mention of an astronomical clock and a planetarium is not an astronomical clock.

Following Regiomontanus’ death, Moller gives out another piece of ahistorical garbage, she writes:

Mathematical printing continued, and in the following decades the city became a flourishing centre with Regiomontanus’ own  De Triangulis (1533), Copernicus’ De Revolutionibus (1543) and Cardano’s Ars Magna testament to ‘Regiomontanus’ importance, not only as a mathematician and astronomer, but also as a publicist a publicist and architect of the renaissance of mathematics.’ (The quote is from Paul Rose’s The Italian Renaissance of Mathematics p. 109).

All three books were published by Johannes Petreius, who had nothing to do with Regiomontanus’ efforts as a printer/publisher, but who had learnt the printing trade from his uncle Adam Petri in Basel before moving to Nuremberg in 1523 almost certainly to try and fill the gap left by the death of Anton Koberger. Moller never mentions Petreius  the most important scientific publisher in Europe in the first half of the sixteenth century or Koberger, who started printing in Nuremberg a year before Regiomontanus and was in the last decades of the fifteenth century and the first decades of the sixteenth, the biggest printer publisher in the whole of Europe. 

Moller now moves on to Regiomontanus’s partner in Nuremberg, Bernhard Walther and the house he purchased, when he retired in 1501. When I read what she now wrote I didn’t know whether to laugh or cry:

He had two windows and a balcony built onto the top floor of the southern gable  and installed his instruments there, creating a modest, yet ground-breaking observatory – the first identifiable one in northern Europe. 

Actually, Walther added the entire third floor to the building. I will show you a picture of his balcony to explain my reaction!

Walther House with Observatory Window in the south gable
Photo: Nora Reim
Source: Astronomie in Nürnberg

As you can see it is actually a stone window sill on which he supported his instruments when making observations. It is probably less that a metre long and maybe thirty centimetres wide at its widest point. Moller, who obviously not done the necessary research seriously thinks Walther built a balcony because  later when discussing the observatory of Wilhelm IV of Hesse-Kassell, she wonders whether Walther’s balcony served as a role model for Wilhelm’s observing balcony. 

She uses Walther’s house to introduce Albrecht Dürer, who bought the house in 1509, because it had been Walther’s house, and it is today a museum dedicated to Dürer. After a couple of introductory lines of biography Moller send Dürer off on his traditional journeyman years of travel and then writes:

When he returned to marry Agnes Frey, the daughter of a wealthy brass maker, he was a master of copperplate engraving, an almost unknown art in Nuremberg.

Copperplate engraving was an almost unknown art everywhere. Also, when Dürer returned to Nuremberg in 1494 he was anything but a master in the art but a shaky beginner. There are three small copperplate prints from that year that are very obviously the work of a beginner. Also, he didn’t learn the art during his journeyman years of travel. Copper plate engraving was invented by gold smiths and Dürer certainly learnt the art in his father’s workshop. Moller now tackles his first journey to the south, which she following the tradition went to Venice. Modern research doubts that on that first journey  Dürer ever left Germany. However, Moller writes:

There were many reasons for him to visit the magical city on the lagoon, but high on the list must have been visiting its printing presses and gathering expertise and contacts for the venture he was about to launch in Nuremberg.

The knowledgeable reader must ask himself at this point, why would Dürer visit Venice to look at printing presses, when by 1595 his godfather Anton Koberger was the biggest printer publisher in Europe. Koberger had printed the Nuremberg Chronicle in 1593, which is full of woodblock prints from the workshop of Michael Wolgemut, Dürer’s master! From Koberger and Wolgemut, Dürer could and did learn everything he needed to know about setting up a print workshop. 

We now get a piece of arrant bullshit:

The workshop gave Dürer control, just as it had Regiomontanus. Here he was able to oversee every stage of his cultural output, from initial design to finished painting or print. Dürer’s success in this endeavour, along with the house’s preservation, give us unprecedented access [my emphasis] to one of the most important and innovative workshops, there has ever been. It is a portal into the sixteenth century and the life of Albrecht Dürer. 

All leading Renaissance artists set up their own workshops giving them control. Whilst in detail different, Dürer’s workshop was no more innovative that those of Lorenzo Ghiberti (1378–1455), Andrea del Verrocchio (c. 1435–1488), Leonardo’s master, or Dürer’s own master, Michael  Wolgemut (1434–1519), who taught Dürer the art of woodblock printing and introduced him to the concept of selling prints individually, which Moller seems to think Dürer invented. He turned the concept into big business but he didn’t invent it. 

After a passage of purple prose about the workshop Moller delivers her next metaphorical history of art, pratfall: 

Dürer was awestruck by the natural world, obsessed with studying and capturing it. In 1503, he turned his forensic gaze upon a patch of weeds, dug from the surrounding countryside and carried back to the studio where, using pen, ink and watercolour he produced an image of ground-breaking naturalism and beauty. Every plant in the Great Piece of Turf is identifiable, each blade of grass perfectly rendered. This study of nature is scientific in detail and accuracy. Even the roots and soil are shown; it is the first image of its kind.

The Great Piece of Turf is not a parch of weeds, dug from the surrounding countryside and carried back to the studio, it is an artificial construct carefully put together to create an illusion of realism, which is in fact hyper-realistic.

Moller keeps trying to forge a link between Dürer and Regiomontanus that simply didn’t exist. For example, she write:

Inspired by what he had seen in Italy and by Regiomontanus’ enterprise in Nuremberg…

She seems to think that because Regiomontanus set up a printing works in Nuremberg to print books in 1471 that Dürer was copying him when he set up an artist’s workshop in 1495 specialising in woodcut prints. Dürer had served his apprenticeship in the workshop of Michael Wolgemut, who specialised in woodcut prints!  

The quote above has a bizarre footnote:

Dürer was a leading member of a circle of intellectuals who saw themselves as Regiomontanus’ successors, men like Walther, Willibald Pirckheimer, Johannes Werner and Johannes Schöner.

Apart from Walther, these men did not see themselves as Regiomontanus’ successors but had varied and complex backgrounds. Although Pirckheimer, Werner, and Schöner were all major scientific figures in Nuremberg during the period Moller covers , she makes no other mention of them or any attempt to describe their significant contributions to Renaissance science. Any non-expert reading her footnote would probably think, “who the fuck are they?”

After a couple of paragraphs of waffle about the importance of patronage, Moller now drifts off to write a five page gloss on the banking family the Fuggers of Augsburg in a chapter about Nuremberg. This ends in Antwerp where we then get the following:

Dürer, visiting in 1520 on his ill-fated mission to find a whale, noted that it was ‘constructed altogether new and at great expense, with a particular tower, wide and large, and with a beautiful garden’.

It would appear that Moller expects her readers to be fully informed about Dürer’s expedition to Zeeland to view a whale beached by a storm, because she gives no further explanation of this statement, except:

Dürer returned home to Nuremberg, weakened from an illness he had caught on his travels and disappointed he had neither secured Charles V’s patronage nor encountered a whale. 

Dürer didn’t travel to the Netherlands to see a whale, that was simply an accidental opportunity that occurred whilst he was there. He travelled because the Holy Roman Emperor Maximillian I had died in 1519 and with his death Dürer had lost his Imperial Pension. He travelled to the crowing of Charles V as emperor in Aachen to get his Imperial pension renewed , an endeavour in which he was successful. Apparently that news never reached Moller. 

Before leaving Dürer, it is interesting to note that in a book with the subtitle, The Transformation of Science in 16th-Century Northern Europe Moller completely ignores the three maths book Dürer authored, Various Lessons on the Fortification of Cities, Castles, and Localities (Etliche Underricht zu Befestigung der Stett, Schloss und Flecken) (1527), Four Books on Human Proportion (Vier Bücher von menschlicher Proportion) (1528) and Four Books on Measurement (Underweysung der Messung mit dem Zirckel und Richtscheyt or Instructions for Measuring with Compass and Ruler) (1525). The latter was the first mathematics book printed in German and was translated into Latin and several major European languages. He also, together with Johann Stabius produced a world map. Most telling in a book which the author says, The stars, or rather astronomy, will be our guide, she completely ignore the fact that Dürer provided the images for the first ever in Europe printed star maps produced by Johann Stabius and Conrad Heinfogel.

We now get a page and a half devoted to the goldsmith Wenzel Jamnitzer, who moved to Nuremberg in 1534, who as Moller points out was famous for his delicate gold flower but also for his book on the theory of perspective Perspectiva corporum regularium (Perspective of the Regular Solids), which was illustrated by Jost Amman (1539–1591). He was also an instrument maker. Moller tells us:

In 1562, Jamnitzer commissioned a portrait of himself. However, unlike Amman’s goldsmith in the Book of Trades, he is not painted holding the tools of his trade. In his left hand is a silver conversion rule he made himself, designed to compare the weights of different metals; in his right a variable proportional compass – precise mathematical instruments rather than pliers of hammers.

What Jamnitzer is holding in his hands are the tools of his trade! She then goes onto give other examples of Jamnitzer presented with mathematical instruments. Then she writes: 

In presenting himself as more than a craftsman, Wenzel was taking the mantle directly from Dürer, continuing his crusade to elevate the status of artists, scholars and artisans. His emphasis on the scientific aspects of his career shows how it was developing during the century, and with it, those who practiced it. 

Jamnitzer was possibly the best goldsmith who worked in Nuremberg during the Early Modern Period but he was by no means the only one who designed and made scientific or mathematical instruments and not even the first to do so. Moller is here trying to claim some sort of special status for Jamnitzer that he simply didn’t have. 

Moller closes this train wreck of a chapter with a quite frankly ludicrous claim.

Thanks to Regiomontanus, Dürer, and Jamnitzer, Nuremberg was the first place in northern Europe where the combination of commercial success and technological ambition came together to create a new world of knowledge, an inspiring example to others; the city remained a thriving centre of instrument making, but this example too was beginning to spread to other places.

Nuremberg was a major centre for the production of scientific instruments before Regiomontanus moved there; in fact, that’s one of the principle reasons he moved there. It is not known if Regiomontanus actually produced any instruments in Nuremberg. In terms of instrument made in Nuremberg, Jamnitzer was very much a late comer. Whilst Regiomontanus set standards for the quality of his scientific publishing, he general impact as a  printer/publisher was minimal compared to the contemporary publishing house of Anton Koberger or in scientific publishing compared to the slightly later Johannes Petreius. Although commercially more successful, Dürer’s workshop was no different to that of his master Michael Wolgemut, from whom he learnt the art of making and marketing woodcut prints. In general Moller completely ignores the people who actually made Nuremberg the centre of a new world of knowledge, Erhard Etzlaub, Willibald Pirckheimer, Johannes Werner and Johannes Schöner, Georg Hartman, Johannes Stabius (not a resident but a frequent visitor), Johannes Neudörffer and, Thomas Venatorius, and many other minor figures. 

Having right royally screwed the history of science of sixteenth century Nuremberg, Moller now takes us to the University of Louvain in the Spanish Netherlands. She opens with the arrival of a young John Dee in 1547 and tells us:

It’s hard to believe Dee would not have passed through Antwerp on his way to Louvain, which lies a few hours’ walk through the gently undulating countryside to the south-east.

Antwerp to Louvain is 43.5 kilometres as the crow flies so allowing for normal roads about fifty kilometres by road, it’s  not exactly what I would describe as a few hours walk. After a lot of waffle about Antwerp, Louvain and the Spanish Netherlands we arrive at the University of Louvain, and Moller informs us:

When Dee arrived, Louvain University had been educating young men for a little over a century. Known as the ‘Athens of Belgium,’ [Really? Belgium didn’t exist then!] it had grown quickly and was now only second to Paris in reputation.

“…now only second to Paris in reputation?” I known an awful lot of European universities who would seriously dispute that claim. Apart from anything else Louvain only acquired a university library in 1636. 

She continues:

Having completed the traditional BA degree, the three main MA subjects on offer were theology, philosophy and medicine. 

On the medieval university the MA was a teaching qualification, qualifying the holder to teach undergraduates. The advanced study was for a doctorate and the three subjects were theology, law and medicine.

We get a lot of background detail about the history of the university till we arrive at Andreas Vesalius, who we are told studied in the arts faculty as an undergraduate without a date, it was from 1528 to 1532. “Before long he became he became interested in the family business,” which was medicine. Moller then delivers up the story about Vesalius and Gemma Frisius stealing bits of a skeleton from a gibbet in 1536. Somehow she neglects to mention that Vesalius left Louvain to study medicine in Paris between 1533 and 1536, only returning to Louvain because of armed hostilities. 

We now get brief sketches of the life stories of Gemma Frisius and Gerhard Mercator. We are already eight pages into the chapter when finally on page nine we finally get something from the history of science:

In 1529, aged twenty-one and just one year after graduating his BA, he published a new edition of Peter Apian’s astronomical manual of 1524, Cosmographia, ‘carefully corrected and with all errors set to right, by Gemma Frisius’. 

So far so good but the title is Cosmographicus liber not Cosmographia and it is not an astronomical manual, it’s a cosmography manual as the title says, which means it covers astronomy, astrology, geography, cartography, navigation, surveying, instrument making etc. Moller continues:

Gemma Frisius had arrived, and from that moment on, the eyes of Europe looked to the Low Countries for progress in geography, cartography, and astronomy.

Correct would be, with the publication of the second edition of Apian’s Cosmographicus liber by Gemma Frisius, Louvain became a new additional centre for progress in geography, cartography, and astronomy, in northern Europe alongside Nuremberg, Ingolstadt, Vienna, Tübingen, Basel and Paris. Moller sinks deeper in the mire:

Apian’s text is a layman’s introduction to astronomy, geography and mathematical instruments, which Frisius adapted to make it more even more [sic] accessible. 

Written in Latin and highly technical, the Cosmographicus liber is hardly a layman’s introduction but a serious textbook for cosmography. Also, although Frisius expanded it, and would continue to do so over many new editions, he didn’t, in any real sense make it mor accessible.

Moller continues:

In a canny commercial move, he also began making instruments to sell alongside the text. There were very few workshops producing items like astrolabes and astronomer’s rings, while books like Cosmographia were introducing them to a wider audience, creating a new market. 

Nuremberg had a large number of workshops producing mathematical and astronomical instruments, which Moller simply chose to ignore in her highly inadequate account of the city. Georg Hartmann (1489–1564) for example produced sundials, astrolabes, armillary spheres and globes. He was probably the most prolific astrolabe maker in Europe, as he was the first to introduce the serial production of the instrument. We return to Moller:

His next move was to design ‘a geographical globe with the most important stars of the celestial sphere’ – a combined terrestrial and celestial globe. He worked in collaboration with his friend Gaspar van der Heyden, a local goldsmith who did the engraving work. 

[…]

He [Gaspar van der Heyden] had already made a globe in 1527 with the monk from Mechelen, Franciscus Monachus. The ‘gores’ (the petal-shaped segments on which the maps were printed before being pasted onto the globes) would have been printed at the publishers in Antwerp, but pasted and finished in the workshop where the spheres were made and inscribed, ‘Gaspar van der Heyden, from whom this work which cost much money and no less labour, may be acquire’.

Gemma published On the Principles of Astronomy and Cosmography, with Instructions for the Use of Globes, and information on the world and on Islands and Other Places Recently Discovered (like  his first book printed in Antwerp) to go with the globe. 

[…]

In the early sixteenth century, only a small number of workshops produced these marvellous objects [globes], usually engraved sphere of wood or metal made in commission for wealthy clients. The printing press made a new kind of globe possible, one that was made of two hollow hemispheres, usually of wood but sometimes papier mâché and plaster, glued together with the maps printed on gores and then pasted onto the surface. This type of globe was cheaper and easier to produce, enabling workshops to make theme in larger numbers for general sale rather than on commission, reducing the price and increasing their availability. Gemma saw the potential of this and ran with it. His combined globe, which was being produced in Louvain workshops by 1530, was the first of several that he designed, each one with improved geographical information which was constantly being updated by sailors and merchant returning to Antwerp from voyages.

There is an awful lot to unpack here. As far as we know the first cartographer to produce printed gores for a globe was Martin Waldseemüller (c.1470–1520), who made a very small globe, 12cm, of his famous world wall map, the first to use the name America, both in 1507. None of the globes have survived but four sets of gores are still extant. 

Unlike his map, Waldseemüller’s globe had little impact and it was Johannes Schöner (1477–1547), one of those mathematical practitioners from Nuremberg, who Moller ignored, who is credited with the first serial production of printed globes. Schöner produced a 27 cm terrestrial printed globe in 1515. This was followed by a matching celestial globe in 1517. He established the concept of matching pairs of terrestrial and celestial globes and the way that they were mounted that remained a standard down to the end of the nineteenth century. Standards also adopted by Gemma Frisius and his pupil Mercator. The cartography of the terrestrial globe was clearly based on the Waldseemüller wall map and the only surviving copy of the wall map, now in the Library of Congress, was that owned by Schöner. In 1533, Schöner produced a new pair of terrestrial and celestial globes with updated cartography. 

Although very few of Schöner’s globes have survived, they were made of  papier mâché and plaster, we now from correspondence that they were very much in demand and that he sold comparatively many of them, throughout Europe. The celestial globe in Hans Holbein’s painting The Ambassadors, painted in London in 1533, is one of Schöner’s and the small terrestrial globe is at least based on Schöner’s work. Schöner also printed books on how to use his globes, Luculentissima quaedam terrae tortius descriptio (A Very Clear Description of the Whole Earth) for his terrestrial globe and Solidi et sphaerici corporis sive globi astronomici canones usum (Manual for the Use of the Solid Spherical body and Astronomical Globe) for his celestial globe.

The demand for Schöner’s globes was very high and he could not fulfil it. In the 1520’s the Antwerp printer publisher, Roeland Bollaert had Schöner’s books but couldn’t get any of his globes. It was he who commissioned Franciscus Monachus (c. 1490–1565) together with Gaspar van der Heyden (c. 1496–c. 1549) to produce a terrestrial globe together with a descriptive book De Orbis Situ ac descriptione ad Reverendiss. D. archiepiscopum Panormitanum, Francisci, Monachi ordinis Franciscani, epistola sane qua luculenta. (A very exquisite letter from Francis, a monk of the Franciscan order, to the most reverend Archbishop of Palermo, touching the site and description of the globe), which he printed in  Antwerp, in 1524. None of the Monachus globes have survived.

In 1529, as Moller correctly pointed out Roeland Bollaert printed the second edition of Peter Apian’s Cosmographia as edited by the young Gemma Frisius. A year later he commissioned Frisius together with Gaspar van der Heyden to produce a new terrestrial globe and this is the globe that Moller describes as a combined terrestrial and celestial globe. For this Frisius wrote his De principiis astronomiae et cosmograpiae deque usu globi (Principles of Astronomy and Cosmography and the Use of the Globe), which was published by the Antwerp publisher Johannes Graheus. It is probably that Roeland Bollaert had died in the meantime. Monarchus had also acknowledged his debt to both Schöner and Peter Apian in his De Orbis Situ. None of these globes have survived. 

This globe “the first of several that he designed”! In 1536, Frisius produced, in imitation of Schöner, a matched pair of terrestrial and celestial globes. One of each has survived but the terrestrial globe has lost its stand. Interestingly, Frisius’ celestial globe uses for the constellations the images created by Dürer for the Stabius/Dürer/Heinfogel printed star maps that Moller didn’t think worth mentioning.  The globe from 1530 and the globe pair from 1536 were the only globes that Gemma Frisius produced. Moller claims the 1536 globe pair was commissioned by the Emperor Charles V, it wasn’t. Charles V granted him a patent which is something else altogether. 

Gerard Mercator, who was a pupil of Gemma Frisius, provided the italic inscriptions on the globe pair from 1536, Moller informs us:

Mercator had already made several maps by this point and had begun to use an Italian cursive script called cancellerescato mark up place names. 

Mercator’s earliest map, a wall map of the Holy land was produced in 1537 after he had finished work on Frisius’ globes. 

Having dealt with the history of Frisius’ globe production I’ll go back to Moller’s description of his publication in 1533 of the appendix to the third edition of the Cosmographia explaining triangulation. This she manages reasonably well although her explanation of triangulation is a bit terse. She then ruins it with the following:

Triangulation made it possible, for the first time, to correctly locate places on a map, to capture the vast tracts of the planet and plot them onto the page to scale. The whimsical maps of the Middle Ages like the Mappa Mundi in Hereford cathedral, which its absence of geographical knowledge, presented a vision of the cosmos based on imagination and faith, were gradually replaced by accurate charts and surveys.

I politely suggest that Moller takes a course of study in the histories of surveying and cartography. Whilst triangulation, as described by Frisius, improved the accuracy of surveying, map makers had been producing reasonably accurate maps long before Frisius was born, using other methods of surveying. Some of those methods were actually described in Peter Apian’s Cosmographia that Frisius took over. The Mappa Mundi in Hereford cathedral is one is termed a philosophical map and serves a different function, namely that of presenting a philosophical, in this case Christian, world view. She then goes off the rails with:

Maps enabled geography (the description of the world based on observation and measurement) to gradually eclipse cosmography (the conception of the universe based on philosophy and conjecture), changing the way humanity saw the world and how to approach it as an area of study.

I really don’t know where to begin in dismantling this wonderfully wrong pair of definitions. Perhaps we could start with the book that Ptolemy wrote in the second century CE, his Geōgraphik Hyphgēsis, lit. Geographical Guidance, which was titled in Latin in different edition both Geographia and Cosmographia. This was a collection of maps based as far as possible on observation and measurement, although it presupposed the philosophical assumption that the oecumene, i.e. Europe, Asia, Africa, constituted the entire world. Later the two words became distinguished, geography referring to what we now understand under the term, and cosmography referring to a description of the entire cosmos, which included geography as one of its constituents along with astronomy etc. Exactly that which Apian’s Cosmographia delivered. 

There are lots and lots of examples of maps based on observation and measurement, as far as it went, between Ptolemy’s Geographia and the invention of triangulation.

We now get a lot of filler about how the workshops in Louvain might have appeared, we don’t actually know, then Moller makes the following categorical claim: 

By the mid-1540s the workshops of Louvain were famous for producing the most accurate and most beautiful tools for studying astronomy that money could buy, eclipsing even the masters of Nuremberg.

This is hyperbolic hogwash. The instruments coming out of Louvain were indeed excellent quality but they did not eclipse the masters of Nuremberg. 

We get nothing almost nothing from Moller about Mercator’s cartographical work, although he is without doubt the most significant cartographer of the sixteenth century. We do get a longish account of his imprisonment on religious ground and then on his friendship with John Dee. Moller tells us that they spent their time discussing astrology. In this context she also claims: 

“Astrology had been under attack for several decades; Mercator and Dee were keen to ground it on a more scientific basis and place it within Copernicus’ new cosmographical framework.”  

Astrology was always under attack from somebody or other but I know of no  special state of attack in the first half of the sixteenth century: Steven Vanden Broecke has this to say about those discussions as related by John Dee:

Except for the present disc, Mercator has left no explicit record of his attitude towards astrology prior to his departure from Louvain to Duisburg in 1552. An important indirect source, however, is John Dee’s Propaedeumata Aphoristica (1558), which is dedicated to Mercator. After graduating from the University of Cambridge, the English polymath, John Dee (1527±1608) made two study tours to Louvain, one in the summer of 1547 and a second from June 1548 until at least July 1550. Apparently Dee spent much of his second stay at Louvain `learning and philosophizing ’ with Mercator. The precise content of these discussions is clarified in the preface: `Your next to last letter, in which you seemed to wish to refresh my memory of that noble debate formerly carried on between us, has given me an occasion to choose, in preference to all others, that subject which I am now to treat.’  

In other words, the topic of the Propaedeumata Aphoristica is the same as that of parts of Dee and Mercator’s debates at Louvain. Nicholas Clulee’s studies have established the Propaedeumata as Dee’s attempt to provide astrology with a firm physical and epistemological basis. In the common vein of Aristotelian natural philosophy, Dee explains that natural change is ultimately caused by celestial influence, adding the no less unexceptional conviction that such change is subject to a natural and predictable causality[1].

Traditionally Aristotelian, no mention of Copernicus!

Towards the end of the chapter Moller tells us:

Gemma died in 1555, and Mercator had left Louvain for the peace of Protestant Duisburg over the German border three years earlier, but the workshop continued to thrive under Gemma’s son Cornelius and his colleague Walter Arsenius. The number of instruments that survive suggest impressive production levels, and makers across the continent were influenced by the design and quality the city stood for, just as astronomers were enabled to make better, more accurate observations than ever before.

So much hogwash in one brief paragraph. Cornelius Gemma didn’t make instruments and in terms of the earlier comments on astrology in Louvain it is interesting to note that he shared in his father’s efforts to restore ancient Ptolemaic practice to astrology, drawing on the Tetrabiblos. What was that about Copernicus? Gualterus Arsenius, Gemma Frisius’ nephew, was the head of the family that actually produced the largest number of astronomical instruments in the Louvain workshop. The workshop was productive but no more or less so than other major European instrument workshops. The instruments from Louvain were no more accurate than those from earlier European workshops. 

Moller ramps up the stupidity a couple of lines further on:

The expertise in designing evermore accurate instruments enhanced the quality of observational data, its usefulness and status. This strengthened the role of instruments in the scientific enterprise; today, technology is so integral it is no longer possible to draw a line between the two. Modern astronomy is cutting-edge technology, and the complex telescopes that empower us to see into the darkest corners of the universe have their roots in the workshops of Louvain, and the standards and ideals that were generated there.

This is in the favourite expression of my friend the HISTSCI_HULK pure hyperbolics. The instruments makers in Louvain did not create any new or novel instruments and although their quality was high, their accuracy was not greater than other astronomical instrument makers in the sixteenth century. Lastly astronomy had been the cutting-edge technology for its time since at the latest Ptolemy. To suggest that modern astronomy has its roots in the workshops of Louvain any more than in the workshops of Nuremberg, of medieval Baghdad, ancient Alexandria or first millennium BCE Mesopotamia is quite simply bullshit.

Having entered Louvain with the young John Dee, Moller now takes us back with him to his house in Mortlake, in those days a small parish on the Thames about ten miles to the west of the City of London. Moller wishes to present Dee’s house and its library as one of her “Stargazer’s Palaces”. As with so many people who write about Dee she emphasises his occult activities whilst almost totally ignoring his scientific activities. She mentions, quoting Dee, that when he returned from Louvain he brought globes and scientific instruments with him, pointing out their scarcity in England at the time. Then she tells us:

Globes were not produced domestically until the 1590s, so the only way to get one, or two (from 1551 onwards the publication of Mercator’s celestial globe to go with the terrestrial one of 1541 set the fashion for them almost always being sold in pairs [my emphasis]), was to import them from abroad. 

Both Gemma Frisius and Mercator made matching pairs of terrestrial and celestial globes in imitation of Johannes Schöner, who “set the fashion for them almost always being sold in pairs.”

She mentions several times his financial problems and his difficulties in finding patrons/employment, whilst hardly mentioning his extensive, and historically very important, employment as an advisor and teacher of navigation, cartography etc. for the Muscovy Company amongst others. This is made even more bizarre, as she explains that Dee owned instruments designed by Richard Chancellor (c. 1521–1556). She writes:

Chancellor was a navigator who had been introduced to Dee by their mutual patron Sir Henry Sidney. He led voyages for the Muscovy Company which failed to find the Northeast Passage, but opened trade with Russia and took Chancellor to Ivan the Terrible’s court in Moscow.

She fails to mention that Dee, who worked as an advisor, teacher, and supplier of charts and instruments for ships masters and pilots of the Muscovy Company, was actually Chancellor teacher. He also wrote his The Astronomicall and Logisticall Rules and Canons to calculate the Ephemerides to be used on the first Northeast Passage voyage, by Willoughby and Chancellor.

We now get a classic, Moller writes:

Sailors needed instruments, especially astrolabes to help them navigate…

To quote David King, leading historian of scientific instruments and one of the greatest living experts on astrolabes, “astrolabes were never used for navigation.” In case you think she was referring to mariner’s astrolabe, she continues: 

…and one of these, now in a museum in Belgium, is engraved with Edward VI the Duke of Northumberland’s coats of arms. It was made in 1552 by Thomas Gemini, a founder of the instrument making trade in England, who was affected by the same religious persecution that pushed Mercator to flee Louvain and settle in the Protestant backwater of Duisburg. 

We then get the “life stories” of Thomas Gemini, Leonard Digges, and Thomas Digges all in one page of the book. …

At this point, as noted above I broke off in frustration!


[1] Steven Vanden Broecke, Dee, Mercator, and Louvain Instrument Making: An Undescribed Astrological Disc by Gerard Mercator (1551), Annals of Science, 58, 2001, 219-240 p. 226

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Filed under Book Reviews, History of Astronomy, History of Cartography, History of cosmography

Much Ado About Nothing

Regular readers will be well aware that a Renaissance Mathematicus book review is usually anything but short. I try as far as possible to give an accurate, informative,  outline sketch of the actual contents of the book under discussion. This leads automatically to a lengthy essay style review, the aim of which is to give potential readers a clear picture of what exactly they can expect if they decide to invest their time and money  in the volume in question. Given this approach to reviewing, how can I produce a Renaissance  Mathematicus style review of a book that is seven hundred and forty pages long and contains thirty nine academic papers covering a very wide array of different aspects of a single topic without it turning into a seemingly never ending essay? The simple answer is, I can’t so, what follows will be far less detailed and informative than is my want. 

So, what is the topic and what is the book that gives this topic so much attention? The topic is one that has fairly often put in an appearance here at the Renaissance Mathematicus, zero and the book is The Origin and Significance of ZeroAn interdisciplinary Perspective.[1] 

The book is the result of a cooperation between Closer to Truth, a broadcast and digital media not-for-profit organisation presenting a weekly half-hour television show which airs continuously since 2000 on over 200 PBS and public TV stations, and the Zero Project Foundation, which was set up in the Netherlands in 2015. Closer to Truth is the baby of the book’s one editor Robert Lawrence Kuhn and the Zero Project Foundation was set up by the book’s other editor Peter Gobets, who unfortunately passed away just before the book was published. You can view a Closer to Truth video on the Zero Project here.  and read about Closer to Truth here 

The book opens with a ten page preface in which Kuhn, a philosopher, talks about his life-long obsession with the concept of nothing and discusses a hierarchy of definition of nothing. The twelve page introduction from Gobets explains the motivation behind the Zero Project, its cooperation with Closer to Truth and the structure and intention of the book itself. 

The book is in four parts, whereby Part 0 consists of fifteen papers on Zero in Historical Perspective. Part 1 has sixteen paper on Zero in Religious, Philosophical and Linguistic Perspective, the papers  are as wide ranging as the title suggests. Part 2 Zero in the Arts is very short consisting of a very brief introduction by Gobets to eight art works by the artist British-Indian sculptor Sir Anish Mikhail Kapoor (b. 1924) devoted to Kapoor’s visualisation of the Buddhist concept of the void. Part 3 has seven papers under the title Zero in Science and Mathematics

The papers vary considerably, in length, in academic depth, some are fairly general and superficial, some are deeply researched, and writing quality i.e. readability but this is too be expected in a book that tries to pack so many different viewpoints into one volume. At times I got the feeling that some judicious editing would have improved it in general, less would have been more.

As somebody, who is primarily a historian of mathematics it is, of course, Part 0 Zero in Historical Perspectivethat most interested me. The section opens with two papers relating to the multiple appearances of zero as a concept, as a placeholder and as a number in different cultures and the historical problems of trying to establish if, when and how  influences or exchanges took place between those cultures and concepts. Neither paper is particularly helpful and the second Connecting Zeros by Mayank N. Vahia gives prominence to an ahistorical myth. He writes:

Indians were the first to work out the algebra of zero and opened the window to a completely new class of mathematics.

This was not true for the Europeans, to whom life without one was unimaginable. One was the natural smallest number for them. Zero made them uncomfortable. All cultures believed in one form or another, that there exists a Great God. This was the proverbial “One”. This Great Got then created the universe and the many variation in life. The one therefore pervades everything and remains even when all else is gone. 

In early Europe it was forbidden to study zero [my emphasis] as it was considered unnatural and against the working of the Great One who would always be present. 

I could write a whole blog post taking this heap of garbage apart. It comes as no surprise that it was written by a retired engineer who “has become interested in understanding the origin and growth of astronomy and science in India”. He should start by learning something about comparative religion about which he displays an unbelievable ignorance. Perhaps he could explain who the “Great God” is/was in pantheistic Hinduism? Although he doesn’t define what he means by early Europe, one has to assume he means the Middle Ages with its Christian culture, which I’m sorry to tell him, which, despite the widespread myth, never forbade the study of zero. 

Things improve when we get to the histories of zero in the individual cultures. There is an excellent paper, Babylonian Zero on the sexagesimal place-value number system in Mesopotamia and the introduction of a place holder zero and the separate concept of nothing as the result of an arithmetic operation.

There are two good papers on the Egyptian concepts of zero and nothing, Aspects of Zero in Ancient Egypt and The Zero Concept in Ancient Egypt, the latter includes a brief section on the Mayan concept of zero. Followed by an equally good one on zero in ancient Chinese mathematics, On the Placeholder in Numeration and the Numeral Zero in China.

As to be expected India features next with a short paper on the appearance of numerals in Reflection on Early Dated Inscriptions from South India followed by a longer one tracing the path from the religious term Śūnyameaning empty or void to the numeral zero, From Śūnya to Zero – an Enigmatic Journey, which includes section on the Egyptians, the Babylonians, the Incas, the Maya, China, Greece and India with reflection of the reception in Arabic and European culture. The two paragraphs here on the Incas and the Maya are the only mention of the development of zero in Middle America a serious lacuna in the book. This is followed by an essay on The Significance of Zero in Jaina Mathematics an interesting branch of Indian mathematics, somewhat outside the mainstream. 

Now we get the bizarre rantings of Jonathan J. Crabtree, Notes on the origin of the First Definition of Zero Consistent with Basic Physical Laws. Crabtree has been wittering on about his “great discovery” in elementary mathematical pedagogy to my knowledge for at least twenty years and an Internet search shows that it is closer to forty years. Crabtree thinks that English language elementary mathematics teaching is a disaster because it uses an at best ambiguous at worst false definition of multiplication. I write English language because the pesky British spread this abomination through the textbooks it distributed throughout the Empire. Crabtree attributes this pedagogical error to Henry Billingsley’s false translation of Euclid’s definition of multiplication. To this he has added that Europe didn’t understand the true nature of zero because the Arabs mistranslated Brahmagupta.  

Up next we next have a somewhat bizarre four page paper, Putting a Price on Zero about a historian of mathematics asking a class of mathematicians to explain how they would allocate royalties to the various cultures which are claimants for the invention of zero. A waste of printing ink in my opinion. 

Returning to more scholarly realms we now have an interesting article on a famous zero artifact, Revisiting Khmer Stele K-127. This stone stele discovered in1891 on the east bank of the Mekong River in Sambaur contains the date 604 of the śaka era, i.e. 682 CE, and is the oldest known inscription of the numeral zero.

Moving forward in time we get an essay on zero in Arabic arithmetic, The Medieval Arabic Zero. Comprehensive, detailed and highly informative this article meets to highest standards and one wished that it might have been used as a muster for the whole volume. This is followed by an excellent paper on Islamic numerals, Numeration in the Scientific Manuscripts of the Maghreb.

The final paper in Part 0, The Zero Triumphant is about the Tarot. This, however, is not the fortune telling Tarot but the original 15th century Italian card game, which was originally called ‘trionfi’ (i.e., ‘triumphs’ or ‘trumps’). This was played with an amalgamation of two packs of cards, the four-suited deck of playing cards brought into Europe via the Mamluk Empire from the Muslim Near East and a deck of 22 allegorical images originating in medieval Christian iconography. The Islamic deck was numbered with Hindu-Arabic numerals and the European Trumps cards had Roman numerals. The Fool or Crazy One (Il Mato or le Fol) is numbered 0.

A fascinating paper that is however flawed by repeating the myth served up in the second paper Connecting Zeros:

The concept of zero did not exist in the classical mathematics of the Greeks and Romans. And it was an abomination at first to the Christian West. What use did good Christians have for nothingness? God created something not nothing.

As noted above this is ahistorical bullshit. 

Each of the papers as footnotes and its own, oft very extensive, bibliography, and the book has a usable general index. Some but not all of the papers are illustrated. The book closes with an Epilogue by Peter Gobets with more thoughts about the Zero Project and the books role in it. 

Based on what I’ve read, and I admit to not having read the whole volume, I could have titled this review, The Good, The Bad and The Ugly. There are some excellent papers, some that are somewhat iffy and some that probably should not have made it into print. It is actually quite affordable given that it’s a Brill publication the hardback and the PDF both waying in at  €100 plus VAT on the publishers website but I’m not sure I would recommend buying it rather than borrowing it from a library to read the bits that interest the individual reader. I do have one last complaint, the book is so thick, so heavy, and so tightly bound that I literally found it impossible to find a way to read it comfortably. 


[1] The Origin and Significance of ZeroAn interdisciplinary Perspective, edited by Peter Gobels and Robert Lawrence Kuhn, Brill, 2024.

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A Christmas gift from the Gnomes of Ball Mansions.

Philip Ball is one of the best English science writers and with certainty one of the most if not the most prolific. He churns out books and article, with radio programs thrown in along the way, at a rate that is absolutely mindboggling. We here at the Renaissance Mathematicus exposed the secret of his production rate several years ago. Like Santa, who had gnomes in his workshops at the North Pole producing all those toys, Ball has a team of gnomes chained to writing desks in the cellars of Ball mansion busily scribbling away at his next publications.

A couple of years back the gnomes embarked on the production of a series of history of science coffee table books, richly illustrated volumes explaining the history of science for the non-expert. If you are looking for a last minute Christmas present, perhaps for a teenager fascinated by science, or just somebody who would like to delve into the history of science, without doing battle with an academic text, then these volumes are highly recommended.

The first volume to make its way out the gnomes production centre was The ElementsA Visual History of Their Discovery (Thames & Hudson, 2021) a beautifully illustrated book that takes the reader from the story of the four classical elements of Ancient Greece down to the artificially created atomic elements of the twentieth century.

Telling the story of the discovery of each element or group of elements along the way. Unfortunately, I feel obliged to point out that this, otherwise wonderful book, has a flaw. It seems that somewhere during the editing phase, the story of the discovery of mercury slipped through a gap and failed to make it into the published work. However, despite this highly regrettable lapsus the book is a delight to read and highly informative. 

In 2023, the gnomes turned their attention to the world of experimental science and delivered up Beautiful ExperimentsAn Illustrated History of Experimental Science (University of Chicago Press). This one truly delivers what the title promises.

The book has alternating chapters and interludes. The chapter looks at a set of historical experiments united by a common theme. For example, the theme of the first chapter is How Does the World Work and starts with Eratosthenes measuring the size of the world, followed by Foucault demonstrating diurnal rotation. Moving into modern physics we have Michelson and Morley attempting to detect the ether followed by Arthur Edington proving relativity. The first interlude asks the metaphysical questions, what is an experiment? and what makes a good experiment? We return to the world and the violation of parity, closing with the discovery of gravitational waves. 

This pattern is repeated in What Makes Things Happen?, with the interlude The Impact of New Techniques.  The third chapter asks What is The World Made From?, and its interlude questions the books title, What is a Beautiful Experiment? Chapter four is a theme from the history of science that is of particular interest to me, What is Light? and its interlude looks at The Art of Scientific Instrumentation. Moving on in chapter five we have the pregnant question, What is Life?, and an interlude about Thought Experiments. The book stays with the life sciences for the final chapter, How Do Organisms Behave, this time there is no interlude.

This book takes on a massive topic about which one could write a multi-volume encyclopaedia and masters it magnificently with a fine examples of classical experiments clearly explained and some intriguing metaphysical speculations about the nature of experimentation clearly expressed for the non-philosopher.

In 2025, the gnomes struck again with a truly magnificent volume, AlchemyAn Illustrated History of Elixirs, Experiments, and the Birth of Modern Science (Yale University Press).

All three books are beautifully illustrated but the alchemy volume takes the quality of the illustrations to a whole new level, which is due to the nature of the topic and the available pictures. On a general note, this is an excellent introduction to the history of alchemy. Despite the excellent work done by historians over the last half century explaining the rich and influential history of alchemy, there are still large numbers of people, who think that alchemy is just a bunch of crazies trying to turn lead into gold. This volume tells the real complex story of the discipline in non-academic terms for the lay reader. 

 There are chapters on the origins of alchemy in different period and cultures. Other chapters look at specific aspects of the topic such as chrysopoeia (the quest for gold) the uses of alchemy, the alchemical laboratory and others. In between are informative potted biographies of the leading figures in the history of alchemy. Towards the end the book handles the historically important transition from alchemy to chemistry, a topic that for far too long was swept under the carpet with the claim that the two had nothing to do with each other.

All three books have good indexes and a short but good list of suggestions for further reading. They are all excellently produced and are both pleasant to look at and to read. For the quality, all three are very reasonably priced and won’t require you to take out a second mortgage. Philip Ball is to be congratulated for having trained his gnomes to produce such desirable books. 

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Filed under Book Reviews, History of Alchemy, History of Chemistry, History of science

The history of science in medieval Islamicate culture.

This post is a first attempt to answer a question that came up when I was rude, not for the first time, on social media about Jim Al-Khalili’s book The House of Wisdom: How Arabic Science Saved Ancient Knowledge and Gave Us the Renaissance (Penguin, 2011). Al-Khalili is an excellent broadcaster, who is very good at presenting modern science, mostly in the form of interviews, to a lay public but he is not a historian of science. Two examples of things that for me disqualify Jim Al-Khalili, as an authority on the history of Arabic science. 

Firstly, during a radio panel discussion on the history of Arabic science he said, with reference to the House of Wisdom, “Professor Pormann will tell me that it didn’t exist, but I prefer to believe it did. Professor Peter Pormann is Professor of Classical and Greek-Arab Studies at the University of Manchester and a leading authority on the history of Arabic science. Secondly, Al-Khalili states categorically that because of his experimental programme in optics Ibn al-Haytham (c. 965–c. 1040) is the originator of the modern scientific method. Ibn al-Haytham’s experimental programme was a copy and extension of the very similar programme of Ptolemaeus (fl. 150 CE) so, if Ibn al-Haytham is the originator of the modern scientific method what does that make Ptolemaeus? Secondly, A. Mark Smith, historian of optics and leading authority on Ibn al-Haytham, thinks that most of al-Haytham’s experiments are only thought experiments because using the equipment he describes he would never have achieved the accurate results that he presents. 

So having dismissed Al-Khalili’ s book I naturally got asked what I would recommend instead. This is a problem as there is no really good general introduction to the science produced in medieval Islamicate culture, that is in the various areas dominated by Islam from the beginning of the seventh to the end of the sixteenth century. Please nobody recommend Jonathan Lyons’ The House of WisdomHow the Arabs Transformed Western Civilization (ppb. Bloomsbury, 2010) if anything it’s even worse that Al-Khalili’s book. The blurb from the book contains this little gem: “The Arabs could measure the earth’s circumference (a feat not matched in the West for eight hundred years).” Whoever wrote that has apparently never heard of Eratosthenes (c.276–c. 195 BCE) born in Cyrene, which last time I looked was in the West, about eight hundred years before Muhammad (C. 570–632 CE).  

I would recommend the three volume Encyclopedia of the History of Arabic Science, edited by Roshdi Rashed, (Routledge, 1996), which has its own Wikipedia article, but it was almost certainly expensive when it was first published and is now out of print. If, however, you have access to a university library, they might well have a copy or can get one for you via interlibrary loan. Written by an all star cast, it is excellent and reasonably accessible for the layman. 

Before I go further, why the circumlocution in my blog post title? There is a major problem about what to call the science under discussion. To simply call it Islamic science is problematic because not all the people who produced it were Muslims. There were also Jews, Christians, Sabians, and possibly Zoroastrians, who contributed to the science flowing out of the areas dominated by the Muslims. The alternative used in the titles of the books I have already mentioned is to call it Arabic science, because most of it was written in Arabic (most but not all), however, many of the authors were not Arabs. From now on I shall refer to it as alternatively as Islamic or Arabic science but with reservations.

One major problem is that the field of Arabic science and technology is very wide as can be seen from the three volumes of the Encyclopedia of the History of Arabic Science and neither AL-Khalili nor Lyons really cover the whole territory. Most of the books that I’m now going to recommend only cover a limited area but as already mentioned apart from the Encyclopedia of the History of Arabic Science there is no book that covers it all. 

The first book that I would recommend, but with some reservations, is Stephen P. Blake, Astronomy and Astrology in the Islamic World, (Edinburgh University Press, 2016). This is one of The New Edinburgh Islamic Surveys, series editor Carole Hillenbrand, about which I will say more later.

Blake opens his preface with the following sentence:

To compose a readable, nontechnical account of astronomy and astrology in the Muslim world is challenging. The topic is scientific (dependent on arcane mathematical theories and concepts), the period is long (covering nearly 1,000 years), the geography is extensive (stretching from India in the East to Spain in the West), and the context is crucial. To make sense of the Islamic era (from the middle of the eighth century CE until the middle of the sixteenth century), the narrative must begin three centuries before (with the Egyptians) and continue through the century following ( with Copernicus, Kepler, and Newton).

Having admirably sketched out here the problem that the author faces, one that applies to all the sciences in the Islamic World and not just astronomy and astrology, Blake goes on in his book to give a masterly attempt to meet the challenge and almost succeeds.

The first twenty pages are devoted to a brief, comprehensive but informative survey of the history of astronomy before Islam, covering Ancient Egypt, Mesopotamia, Ancient Greece with a concentration on the works of Ptolemaeus, Ancient India and Persia. Given its brevity a surprisingly good survey. The next sixteen pages covers the birth of Islam and in rapid succession all the usuall figures of early Arabic astronomy and astrology from Abu Ma’shar to al-Biruni giving details of their work and its influences from the sources mentioned in the first capital. 

The next ten pages opens with a very brief account of the Abbasids and astronomy in Baghdad, including the House of Wisdom, but emphasising the limited amount of large scale observation and leading up to the first large Islamic observatory in Isfahan founded around 1100 by Umar al-Khayyam (c. 1048–1131). In the early chapters on Islamic astronomy, Blake also goes into detail on the motivations of the Islamic astronomers, time keeping for the daily prayers, the Muslim lunar calendar, determining the direction of Mecca, the qibla, for prayer, and of course astrology.

Blake now goes geographically to the other end of the Islamic domination and devotes a chapter to Astronomy and Astrology in al-Andalus. He examines the similarities but above all the differences in the developments of the disciplines in the western end of the Islamic sphere of influence, to those in Asia. Important because it was al-Andalus that principally introduced Islamic astronomy and astrology into Europe. 

We now get chapters on the histories of the large scale observatories from Maragha, over Samarqand, and Istanbul ending in Shajahanabad in India. Each chapter deals with its founding, the astronomers who ran it and the results that they produced.

Despite the brevity of his book, the entire text is only one hundred and fifty pages long, Blakes tour of the history of Islamic astronomy and astrology is very comprehensive. At time I found it perhaps too condensed as a result, letting it read, at times, rather like a telephone book. I also found that Blake lets his own very obvious personal rejection of astrology gets in the way of his historical objectivity. Although he covers the astrology he does so grudgingly, referring to it constantly as a pseudo-science or a superstition.

Despite these minor quibbles I would whole heartedly recommend this book if it wasn’t for the final chapter, Medieval and early-Modern Europe. My friend the HISTSCI_HULK would probably call it a cluster fuck, as it contains errors that a book on this level should not contain. 

The book closes with a glossary which gives brief description of the instruments referred to throughout the book. Unfortunately, he repeats the totally erroneous claim that Hipparchus invented the astrolabe. 

Each chapter of Blake’s book has end notes that basically just refer the reader to the very extensive bibliography at the end of the book. There is also a good index after the bibliography. There are no illustration in the body of the text but there are eight pages of very nice colour illustration in the middle of the book.  

My second book, A Brief Introduction to Astronomy in the Middle East by John M. Steele (SAQI, 2008), I would whole heartedly recommend without reservation.

It is not just about Arabic or Islamic astronomy but delivers exactly what the title says. Steele is leading expert on the history of Mesopotamian science and it is here that his brief account begins, following an introduction that sets out the route that he intends to take.

The opening chapter, The Birth of Astronomy in the Middle East, gives a concise but informative cover of the evolution of astronomical activity in Mesopotamia from the invention of writing in the fourth millennium BCE down to the late Babylonian period in the first millennium BCE. Having followed that evolution Steele now devotes, for this short book, a long chapter to Late Babylonian Astronomy, which is the period in which the mathematical astronomy that is the very recognisable ancestor of our own modern astronomy came into being and evolved. Producing amongst other things the zodiac and mathematic models to predict astronomical phenomena. 

The third chapter, Astronomy in the Greek and Roman Middle East, opens with Alexander the Great’s conquest of Babylon in 331BCE, which signalled the start of the transfer of astronomy and its further development to Ancient Greece. A comparatively short capital, it documents the Greek adoption of geometrical models for astronomy to replace the arithmetical-algebraic models of the Babylonians. It also introduces the work of Hipparchus and naturally, above all Ptolemy, closing with his Almagest.

Nearly all of the rest of the book, almost the half, is dedicated to our actual topic Islamic astronomy. The first of three chapters, Astronomy in Medieval Islamic Culture, sketches the religious reasons why astronomy was important to Islamic culture during the medieval period, reasons that led them to adopt and develop the Babylonian and Greek astronomical heritage–the Islamic lunar calendar, the prescribed times of prayer, and the direction of prayer, towards Mecca. 

The second of the three chapters, Astronomical Observations and Instruments in the Medieval Islamic World, follows the evolution of the scientific methods that the astronomers undertook to fulfil those religious requirement and to go beyond them in developing a full blown, sophisticated astronomical science. 

The third chapter of the three, Medieval Planetary Theory, takes a look at the moves beyond Ptolemy that the Islamic astronomers undertook. New more accurate astronomical tables based on their own more accurate observations made with new improved instruments. Active criticisms of the more unsatisfactory aspects of Ptolemy’s Almagest and the development of new geometrical models with which to track the path of the planets, in particular the models of Nasīr al-Din al-Tūsī and Ibn al-Shatir, both of which, as is well known, later reemerged in the work of Copernicus.

The closing chapter of Steel’s  book, Legacies, deals very briefly with exactly this transmission of Islamic astronomical knowledge into Europe and into modern astronomy. The general theme of the transmission of knowledge and especial astronomical knowledge is the golden thread that winds its way through the whole of Steele, all too brief, book. 

Although Steele’s book actually has the same number of pages as Blake’s, those pages are noticeably smaller and although he covers in detail a much wider range of material, one doesn’t have Blakes mass of specific detail about individual aspects of Islamic astronomy, making his book an easier and pleasanter read. 

Steele’s book closes with a brief explanation of the sexagesimal number system. There are brief biographies for each chapter of the book and a very small number of endnotes giving the sources for the translation used. He also includes a short but usable index. There are black and white illustrations scattered throughout the text. 

My third book, Science and Islam:  A History by Ehsan Masood (Icon, 2009) is completely different, in that it is a popular book aimed at the general public, which gives an introduction to the whole spectrum of Islamic science.

Whilst by no means perfect, Masood’s book is less flawed than those of Al-Khalili and Lyons. Perhaps its greatest strength lies in that which is implied in the title, this is not a book about Islamic science but about the history of Islamic culture and the science that grew up within it; a subtle but importance difference.

In fact, the first seven chapters are devoted to a sketch of the history of Islam from the very beginning up to the destruction of the Eastern caliphate by the Mongols in the thirteenth and fourteenth centuries. The scientific centres and the leading scientific scholars get mentioned on the way but the emphasis is very much on the religion and the politics and how various aspects of both  led to an atmosphere in which science could emerge and grow.

Having first established this framework Masood now turns to a closer look at the science itself. In Part II of the book, he present the Branches of Learning. In The Best Gift From God, the title is part of a quote from Muhammad about good health, Masood presents a short but competent survey of Islamic medicine, covering the principle scholars, their fields and their books. Chapter nine, AstronomyThe Structured Heaven is a reasonably competent brief survey of the Islamic contributions to the study of the heavens and the astronomers, who made those contributions, which covers all the main bases. 

We get a similar competent survey of the Islamic world of mathematics in the tenth chapter, NumberThe Living Universe of Islam. Starting with the Arabic adoption of the Hindu number system, proceeding from there to algebra with the contributions of al-Khwarizmi and Omar Khayyam. A very brief look at Euclid’s Fifth Postulate and on into the world of geometry. We move on to a handful of pages in chapter eleven, At Home in the Elements in which Masood tries too hard to differentiate between alchemy and chemistry although he admits that at the time the division is not so clear. The final short chapter in this section, Ingenious Devices, is dedicated to technology but only briefly covers the automata and the water clocks. 

The third part of Masood’s book, Second Thoughts, starts with a chapter, An Endless Frontier, that looks at the significant Islamic contributions to the development of optics especially in Europe, and then moves on to explain how during the Renaissance, the mood in Europe turned against the Islamic contributions to medicine. It closes with a very brief look at medieval Islamic views on evolution. The penultimate chapter, One Chapter ClosesAnother Begins, examines the reasons for the decline of science in Islamicate cultures, finding part of the blame for a continuing decline in Western colonialism. The final chapter, Science and IslamLessons From History addresses the themes, Did science need Islam?, Did Islam need science?, and Islam and the new knowledge today.

The book has neither foot nor endnotes and no illustrations. At the end of the book there is a useful timeline and an extensive index. I think that the book, whilst not perfect, is a potentially good introduction for somebody approaching the history of science in the medieval Islamicate cultures for the first time.

When looking at texts on the history of mathematics, I recommended J.L. Berggren’s Episodes in the Mathematics of Medieval Islam, (Springer, 1986, ppb. 2003) as an excellent introduction to the topic, a recommendation that I, naturally, repeat here. 

I don’t know of any monographs dedicated to the history of alchemy in the medieval Islamicate cultures but the Encyclopedia of the History of Arabic Science, which I mention above, naturally, has an article on the topic. Lawrence M. Principe in his excellent The Secrets of Alchemy (The University of Chicago Press, 2013), a very readable general history of alchemy that I would recommend to anybody interested in the topic, also, naturally, has a chapter, DevelopmentArabic al-Kimiyā’

I stumbled across Stephen Blakes book, one of The New Edinburgh Islamic Surveys, series editor Carole Hillenbrand,  fairly recently and discovered that the series contains two further volumes relevant to this review, Donald R. Hill, Islamic Science and Engineering, (Edinburg University Press, 1993) and Peter E. Pormann & Emilie Savage-Smith, Medieval Islamic Medicine (Edinburg University Press, 2007), which I ordered and have arrived in the last few days and which I have started reading. Reviews will follow here in due course. There is also a new monster, 838 pages, Routledge Handbook on the Sciences in Islamicate SocietiesPractices from the 2nd/8th to the 13th/19th Centuries eds. Sonja Brentjes, Peter Barker, Rana Brentjes (Routledge , 2025), which I assume is the updated replacement for Encyclopedia of the History of Arabic Science. I have this on order but don’t know when it will turn up here. Will also review, when it does arrive.

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Filed under Book Reviews, History of Astrology, History of Astronomy, History of Islamic Science

A book you can count on 

Back in 2020, I wrote a very positive review of Benjamin Wardhaugh’s fascination volume, The Book of WonderThe Many Lives of Euclid’s Elements. This led me to also writing a positive review of Reading Mathematics in Early Modern Europe of which Wardhaugh was both a contributor and an editor. This was followed by a brief blog post on the research project, Reading Euclid, from which both books emerged.

I was recently very pleased to receive an email from Benjamin Wardhaugh asking if I would be interested in receiving a copy of his newest book, CountingHumans, History and the Infinite Lives of Numbers (William Collins, 2024). Given my more than passing interest in the history of numbers and the excellence of Wardhaugh’s Euclid book, I of course said yes. I have not been disappointed.

I mentioned above my interest in the history of numbers but as Wardhaugh points out that whilst numbers have been and are used in counting they are not necessarily required for the task and there is and has been plenty of counting without numbers. At one point I had to laugh at myself, skimming around the book and getting the general feel of it, I thought this is much more anthropology rather than history. That was when I first bothered to read the quote from The Times on the front cover:

An anthropological sweep through mathematical history from the Stone Age to the cyber age via six continents.

Histories of science or science related topics tend to follow a linear chronological narrative and within limits aim to be comprehensive. Do not expect this with Wardhaugh stimulating, fascinating and at time provocative dive into the history of counting. Why not? Wardhaugh provides the answer in the introduction which itself is not a continuous narrative but a series of fragmentary quotes that illustrate aspects of what counting is or even might be. In answer to the question, “But what is it?” Wardhaugh tells the reader:

‘Counting’ can seem like an unruly grab-bag of almost totally unrelated actions; a label covering a huge set of very different cultural practices. The range of different activities called counting seems too wide for comfort, and at least superficially, it is not clear what they all have in common; or even whether they have anything in common at all. 

Almost any definition of counting is problematic, but one of the best is attributed to the seventeenth century German philosopher Gottfried Leibniz. It says counting is repeated attention. Counting is what happens when you think ‘this…this…this…this’, and have some way of keeping track.

Wardhaugh goes on to tell us:

Counting does not have one single history.

[…]

The story of counting is shaped, instead, like a tree. It has several roots, many branches, and innumerable twigs and leaves. Counting has grown and travelled with the human species, ramifying into very nearly every culture past and present. Sometimes it is possible to follow a single branch for some distance: sometimes a branch turns out to cross, to touch (or nearly touch) other branches.

In this book, Wardhaugh takes his readers on a climb first, down into the root system,  then up and along some of the branches, across time and across the continents; pausing to examine a twig, pluck a leave, or sample some ripe fruit. The book is not written as a continuous narrative but is presented in eight sections, two of them explorations of the roots, the other six climbing amongst the diverse branches.

Each of the eight sections consists of a group of self-contained essays, which deal with an example or an aspect of the topic announced in the title of the section. In those section which deal with examples of counting, which is most but not all of them, Wardhaugh is very careful to give details of who is doing the counting, what they are counting, how they are counting and not least why they are counting. Through this process he makes it very clear that counting is as he wrote in his introduction, ‘an unruly grab-bag of almost totally unrelated actions’.

The first Roots section is not about counting per say but about Number sense before counting and consists of three essays on how people estimate the size of groups of objects without counting and the question whether this is unique to humans or whether animals possess the same innate ability. 

The second Roots section Counting before writing brings three essays on artifacts found in prehistoric Africa that might have been involved in the process of counting. Wardhaugh brings good arguments for such usage but also warns that in the end the claims remain speculative. A fourth essay deals speculatively but well argued with the possible origins of counting words.

Between the Roots sections and the Branches there is an interlude on numbers and their nature.

We now move into the realm of counting in a section titled, Counting with words and symbols in the Fertile Cresent.  The first essay here deals with the Sumerians, who developed the oldest number system of which we are aware.

We remain in Mesopotamia for the second essay and an example of a clever device that Wardhaugh uses often in his book. Instead of just dealing with abstract cultures as a whole, he uses the example of an individual within that culture who in some way or another demonstrates counting within that culture. Here we are in Arkadia and the individual is the monarch Tiglath-Pileser I, who is counting his plunder or spoils of war.  From Arkadia we move to Egypt and Teianti daughter of Djeho, who is counting coins buying a house. In both these examples Wardhaugh moves on from the specific to the general, looking at number systems in the cultures and what was counted and why. 

The second Branch is Counters and opens in Ancient Greece in Athens with Philokleon declaring his judgements in the court,  as a member of the jury, by casting counters into an urn. The essay expands into Greek number systems and the extensive use of counters in various contexts. The second essay takes us to Rome and Marcus Aurelius: Counting Years opening up a discussion of the Roman system of indicating numbers with hands and fingers and its survival down the centuries. The final essay in this section introduces the counting board, widespread in both these early cultures but for his example Wardhaugh takes us to the thirteenth century and Blanche of Castile overseeing the accounts. This leads into a general discussion of counting boards and jetons.

On the next Branch we climb into the history of the number symbols from India, an inevitable topic in such a book. For his entry, Wardhaugh choses, from the numerous possibilities, Bhaskara II and the Brahmi numerals, widening out to give a general sketch of the topic. Logic dictates that the next essay covers the use of India numerals in Islamicate culture and Wardhaugh here choses Ibn Mun’im to introduce us to Dust numerals. One of the longer essays, Wardhaugh covers a lot of ground sketching Arabic numerate culture. The third essay covers the early, very gradual transfer of Indian numerals to Europe. The European development continues with an in depth discussion of the seventeenth-century painting The Account Keeperby Nicolaes Maes. This branch closes in the nineteenth century with the weather records of Caroline Molesworth.

Wardhaugh takes a break for a second short but fascination interlude on Number symbols.

We have followed Wardhaugh from the fertile crescent, to the Mediterranean, and from there we followed the Indian numerals from India over the Islamic Empire to Europe. Now on the next Branch, we go to East Asia to look at Machine that count.

The first essay concerns the tax assessment of Hong Gongshou in China and introduced the Chinese use of counting rods, as a calculating aid. The next essay introduces the Chinese suanpan, Japanese soroban, Korean jupan, the wire and bead calculating device known as an abacus in English. This leads to the famous post WWII story of how the Japanese master of the soroban, Kiyoshi Matsuzaki, defeated the GI Thomas Wood using an electric calculating machine in a public calculation contest. This essay ends with the fascinating fact that suanpan and soroban operators can do mental arithmetic without their devices just by going through the moves in their imagination. 

The next essay spans Japan and America with an account of the nineteenth-century electric tabulating machines of Kawaguchi Ichitaro and Herman Hollerith used in both lands to tabulate census data. Hollerith would go on to found IBM which became the world’s biggest computer company. 

In the following essay we come right up to date, travel to Korea and meet Sia Yoon, who is counting likes. Here Wardhaugh makes his readers aware how much of the Internet world of social media is fixated on counting–like, followers, comments, reposts, …

A new Branch takes as out into the world’s biggest ocean and again back in time with Counting words and more in the Pacific World. In the first essay we count eggs in Ayankidarrba (Groote Eylandt), which lies about 50 kilometres off the Australian coast, leading to a more general discussion of counting in the various indigenous Australian languages. In the following essay we encounter the world of counting of the Oksapmin, one of the isolated peoples of Papua New Guinea. We then count leaves on the island of Tonga before moving into a general discussion of the complexities of the Tongan counting systems.

For the final Branch we leave Oceania to travel to the last continent, to look at Counting in the Americas. The first essay takes us into western Alaska and the tally sticks from the deserted village of Agaligmiut of the ancestors of the Yup’ik people. These found multiple uses. The next essay takes us to the Pacific coast in northern California, and the clam shell beads of the Pomo people. These developed into a trading currency amongst various peoples over a fairly widespread area. 

The final essay in this final Branch, takes the reader into the Mayan civilisation of Mesoamerica, where we meet the ruler of the city of Oxwitik, Waxalahun-Ubah-K’awil. Here Wardhaugh introduces his readers to what is probably the most complex system of counting ever developed by humans, the Mayan calendar systems. The Maya counted the days in a complex variety of ways often given dates simultaneously in more than one system. Wardhaugh skilfully guides his readers through the complexities. 

There are neither footnotes nor endnote but for each essay there is a brief but highly informative sketch of the sources on which it is based. These sketches refer to the ‘select’ forty-six page bibliography. There is also an extensive and very good index. Many of the essays are illustrated with greyscale images of the artifacts discussed in them.

In his conclusion Wardhaugh summarises his book much better than I could: 

The story of counting is a dense tree, with several roots and nearly an infinity of branches. Its end points – as the tree stands today – include elaborate routines with unmarked counters on marked surfaces, highly developed sets of number words, successful – and successfully exported – sets of number symbols, and electric machines whose internal states are, for some purposes, taken to represent numbers. They include smaller sets of words, sometimes accompanied by gestures, tally sticks, and written representations of number words. They include a world in which there is no counting whatever.

There is always a strange alchemy to counting which restlessly transforms one thing into another: days into tally marks; people into counters; books into magnetic tape.

This book has described a few of those processes. There have been thousands more, in all the thousands of languages living and dead, in all the thousands of cultures and hundreds of scripts. No two are alike. The real alchemy, perhaps, is in turning all f these processes into a single thing, and calling it counting.

Wardhaugh is an excellent expedition leader and it is a delight to follow him as he weaves his way through  the tree of the history of counting but who should read this book? At first glance it appears to be a very specialised, niche endeavour, something for a select group of historian of mathematics perhaps. But no, stop and think for a moment. Counting is an activity that is firmly embedded is our every day lives. One just needs to stop and think about the long list of everyday expression and clichés that revolve around counting:

One can or one cannot count on, counting down to, counting the days, count the ways, counting calories, count your blessings, count the pros and cons, take a deep beath and count to ten, that doesn’t count, count against, count out, count towards, count up, counting the votes…

Counting is an integral part of our social, cultural, and political existence and its history, presented in Wardhaugh’s excellent book, should appeal to anyone interested in human history. 

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Filed under Book Reviews, History of Mathematics

How high and above what?

Measurement lies at the heart of many areas of science, where there is almost a fetish for ever more exact measurement. However, one thing that most people are not aware of is how much of the measurement that we use everyday is based on convention or even arbitrary. Let us for a moment consider time, as I am typing this it is 8:35 am according to the clock on my computer. 

This is a momentary record of how far through the day I have progressed but almost everything about it is actually arbitrary convention adopted a couple of millennia in the past. The day itself is the only given part, as the name we give in English to the period that the Earth takes to revolve once about its axis. That we then divide this period into twenty four equal parts is merely an arbitrary convention introduced by the Ancient Egyptian astronomer. They divided the night into ten units determined by the rising of certain predetermined stars or groups of stars, the decans, over the horizon during the night. To this they added the periods of dusk and dawn to give twelve units by which they measured the night. Having divided the night inro twelve parts they then by analogy divided the day into twelve part making a total of twenty-four units. 

This system, however, has a major drawback the length of the day and night are not constant but vary throughout the year as the Sun appears to travel along its journey from the Tropic of Capricorn to the Tropic of Cancer and back again creating the solstices, longest and shortest periods of sunlight during the day, and the equinoxes, the two days in spring and autumn when the day and night are equally long. This meant that the length of those twelve units varies throughout the year lengthening and shortening as the Sun journeys between the Tropics. These varying units are called temporal or seasonal hours are where the accepted way of measuring time throughout the European Middle Ages. Our system of equal or equinoctial hours were introduced in antiquity by astronomers but for general public use only gradually adopted from the fourteenth century onwards in Europe. That we have twenty-four equal hours in one period of diurnal rotation is an arbitrary convention.

What about minutes and seconds, how come there are sixty of them? This we owe to the Ancient Babylonians who had a sexagesimal , that’s base sixty, number system. This meant that the fractions in the first position were one sixtieth of a whole number and not one tenth as in the decimal number system. The  fractions in the second position were one sixtieth of a sixtieth or one three-thousand-and-six-hundredth part of a whole number, our minutes and seconds. The names come from Latin pars minuta prima, little parts of the first type gives us the minute and pars minuta secunda, little parts of the second type gives us the second. Once again these divisions of the hour are an arbitrary convention. 

Hours, minutes and seconds as we use them are purely arbitrary convention there is nothing intrinsic about them. In fact, during the early phase of the French Revolution the revolutionaries tried to establish a decimal clock with twenty hours, ten and ten, instead of twenty-four and with one hundred minutes per hour and one hundred seconds per minute. It didn’t catch on. 

Horloge Republicaine Source: Wikimedia Commons

The last convention used in measuring time is when does the day begin? We now start our day in the middle of the night, at midnight, but this was not always the case. Different culture had different starting points for their daily time measurement, some used midnight, other dawn, so six in the morning, still others midday, twelve noon. As should be well-known in Judaism the day starts at dusk, so the Sabat starts at six in the evening on Friday and not on Saturday. All a matter of convention but in this case not quite so arbitrary as there is an intrinsic logic to each of the various starting points–dawn, dusk, midday, midnight. 

The village where I am typing this is situated at 49° 36´N, 11° 3´O. These are its coordinates in the rectangular grid system of latitude and longitude that we use to define the position of places on the surface of the Earth. This system was originally conceived by astronomers to define positions of celestial bodies on the celestial sphere. Later it was shrunk down to the surface of the globe to define positions on the surface of the Earth. 

Image from John George Hodgins, “Easy Lessons in General Geography, with maps and illustrations, etc” Source: British Library vis Wikimedia Commons

This measuring grid does have some intrinsic anchor points. Assuming the Earth to be a perfect sphere, in reality it’s more like a lumpy potato, the lines of longitude or meridians are the great circles that pass through the north and south poles. Another intrinsic fixed point in the great circle at right angles to the meridians around the middle of the sphere, the equator. The division of the circles into three hundred and sixty degree is another arbitrary convention that has its origins in a theoretical astronomical year of twelve thirty day months: a useful fiction for dividing up the ecliptic, the apparent path of the Sun around the Earth. Once more, thanks to the Babylonians we divide each degree into sixty minutes and each minute into sixty seconds. 

Determining a position north or south is fairly intrinsic in that one just takes the number of degrees/minute/seconds north or south of the equator, the poles being respectively at 90° north or south. Going east or west the problems start, as the equator is a circle, which has no natural beginning or end. There is a long and complicated history of where to set the beginning/end, 0°/180° points, which you can read in Charles Wither’s excellent Zero DegreesGeographies of the Prime Meridian (Harvard University Press, 2017) which I reviewed here

But wait the surface of the Earth is not flat, but is three dimensional, with mountains and valleys, plains and plateaus, so how do we measure this third dimension? The village where I am writing this is 312 metres above sea level, the peak of Mount Everest is 8,848.86 metres above sea level and the surface of the Dead Sea is 439.78 metres below sea level. Sea level is the point from which we measure to determine the height or depth of something on the surface of the Earth.

As far back as I can remember, I have always accepted that geographical altitude is expressed in so many feet or metres above or occasionally below sea level. However, I have never thought to ask what that actually means. How exactly is sea level even determined? Is the height of the sea the same over the entire surface of the globe. On the coast we know that the level of the sea is constantly changing between low tide and high tide and back again to low tide. So, when does one measure sea level, at low tide, at high tide, in the middle and how does one determine that? But the level of water on the coast doesn’t just change during the day, the high and low tides change continuously throughout the month so, when does one measure it? Historically, when and how did people start to try and determine sea level? When, if at all, was agreement reached throughout the  entire world as to what sea level is? If that agreement was never reached, does that mean that the sea level used in Paraguay is different to that used in the UK is different to that used in Borneo? 

The answers to these and lots more questions are delivered in great detail in the excellent new book from Wilko Graf von Hardenberg, Sea LevelA History, (The University of Chicago Press, 2024) part of the Oceans in Depth series edited by Katherine Anderson and Helen M. Rozwadowski. At the end of their forward to the book, Anderson and Rozwadowski write:

The history of sea level–as an idea and a practice–makes clear why historical ideas about oceans in three dimensions and on a planetary scale matter in the twenty-first century. 

My recommendation, if the topic interests you at all, don’t bother to waste time reading my review but go out and acquire a copy of this book and read it instead, you won’t regret doing so. 

In his introductory chapter, GvH opens with basically the same remark that I made above:

The idea of sea level as a benchmark for elevation has by now been around for so long as to go essentially unnoticed–we mention it without pausing to consider what it means. That the concept has a history is easily ignored. We tend to forget that sea level is–far from a natural index–a product of technically and culturally determined assumptions. In this book I tell the story of those assumptions.

In the introduction we learn that the systematic quantification of elevation in geographic and cartographic description doesn’t really begin till the late eighteenth century and then only very slowly. GvH outlines the few earlier historical exceptions and introduces the two principle methods of determining elevation, trigonometrically and barometrically, pointing out the difficulties met with before the latter became truly reliable. He goes on to the problem of making baselines, where exactly does one measures from. Then he takes the reader from individual base points to the concept of determining data set to try and establish sea level. The introductory material closes with a section Towards The Mean going from local data sets to some sort of universal baseline. The introduction closes with the following description of the rest of the next three chapters:

Chapter 1 traces the history of the idea of mean sea level as the most reliable reference point for elevations. Chapter 2 analyses how practical applications of this idea were connected to the infrastructural needs of national and colonial administrations. In chapter 3 I look at attempts to define mean sea level as a national and international standard.

In each of the three chapters GvH divides each topic into its various aspects and delivers up a detailed historical analysis of the debates, proposals and practical applications that took place. The reader gets presented with a master class of historical analysis delivered in a very accessible style.

Returning to the introduction:

In the two following chapters I turn my attention to the process by which a mathematical devised as a vertical datum for geodetical work became a crucial baseline for the assessment of anthropogenic change. Chapter 4 explores how modern ideas of global sea-level change induced by climate change are grounded in earlier research about ancient shorelines, submerged forests and glaciation. Chapter 5 provides an account of the further development, during the twentieth century, of theories of sea-level change induced by climate change and looks at the growing acceptance of the geological agency of humans.

As with the first three chapters the analysis of the historical developments, the debates pro and contra to proposed theories to explain the geological evidence and the growing consensus on the causes of long time sea-level changes are dealt with in clear and understandable detail. The master class continues. 

If GvH had finished his book with his fifth chapter, it would still be essential reading for historians of geography, cartography, geology and all with a general interest in first class history of science. However, his sixth chapter makes his whole book essential reading for all who take a serious interest in the rapid rise in sea-level being caused by the climate crisis and the resulting global warming. Once again as he writes in the introduction:

The final chapter is dedicated to present debates about sea level, including an account of the material limits of idealised averages and the environmental histories connected to a rising sea, from Miami to Bangladesh, Tuvala to England.

As with the previous chapters, GvH’s presentation and analysis of the current debate on sea-level rise due to global warming is clear, factual, and highly informative.

In just 130 pages of text in this slim volume, GvH tells the reader everything they might ever have wanted to know about the concept of sea level, its history and its high level relevance to the actual environmental debate on the climate crisis. His style is clear, simple, direct, and easy to understand with all technical details explained in a comprehensible manners. The analysis and explanations are backed up by detailed endnotes, a very extensive bibliography and an excellent index. The book is illustrated with black and white, and  grey scale pictures, maps, drawings and diagrams.

This is an excellent example of how to write a comprehensive introduction to a usually neglected area of the histories of science and technology that due to current circumstances has a strong relevance to environmental developments in the present time. Just read it!

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Filed under Book Reviews, History of Cartography, History of Geodesy

An episodic journey through the history of mathematics in forty short essays.

This is the next post in my serious on introductory books into the history of mathematics. In one sense the series started almost eight years ago when I wrote a short very negative put down of the book that introduced me to the history of mathematics, Eric Temple Bell’s Men of Mathematics. More recently some weeks back I wrote a very, very long two part total demolition of Kate Kitagawa & Timothy Revell, The Secret Lives of NumbersA Global History of Mathematics & Its Unsung Trailblazers. Here & here. On social media, I then got asked what would you then recommend? So, I followed up with a collective review of the books on the history of mathematics that I know and would recommend together with some books that are now market leaders and generally highly praised. In the comments to that post Fernando Q. Gouvêa recommended a book that he had coauthored with William P. Berlinghoff, Math Through the AgesA Gentle History for Teachers and Others, which came highly recommended by Glen Van Brummelen, whose wonderful books on the history of trigonometry I reviewed here. So, I acquired Gouvêa- Berlinghoff, Math Through the Ages and reviewed that as well. In my collective review I mentioned Snezana Lawrence, A Little History of Mathematics (Yale University Press, 2025), which at that point still hadn’t been published. It has now I have a copy which I have read, and what follows are my thoughts on it.

This is definitely not an academic book, it has no footnotes, endnotes or bibliography, and is obviously intended for the general interested reader. It has forty self-contained essays covering aspects of the history of mathematics in roughly chronological order. The book only has two hundred and eighty pages so, each essay averages seven pages. They are standalone vignettes with very occasional cross references in the form of see Chapter X. Lawrence covers most of the usual major themes but with a slight emphasis on the women in mathematics. 

Interestingly we open with a chapter where Lawrence inquires just what is mathematics and what is a mathematical object leading to a brief discussion of the Ishango bone. There follows a woefully inadequate presentation of Babylonian mathematics that concentrates on the tables of calculated values to aid arithmetic. We get nothing of the sophisticated and for its time advanced mathematical culture that the Babylonians developed. But what can you do in seven pages. Inevitably, Egypt follows Babylon and a reasonable brief account of the Papyrus Rhind. We move on to Greece and a pleasingly sensible account of the Pythagoreans leading into an account of Plato and the Platonic solids. We stay in Ancient Greece for mathematics all-time best seller, The Elements of Euclid. We move over to China and an account of The Nine Chapters on theMathematical Art, which emphasises the differences between the Chinese approach and that of Euclid. The chapter closes with the correct comment that Western and Chinese mathematics evolved largely independent of each other. 

We return to the West in late antiquity and the number theory of Diophantus. This allows our author to introduce her first female mathematician, Hypatia, who supposedly wrote a commentary on Diophantus. The account of Hypatia is fairly balanced and reasonable. We now go back to Asia and a fairly standard account of the introduction of zero as a number in early medieval India. 

We move to Baghdad and the usual hype about the House of Wisdom leading onto an account of the work of al-Khwārizmī. Interestingly when discussing his account of completing the square in quadratic equations, Lawrence now mentions that the Babylonians already knew how to do this. Before departing Baghdad, Lawence also briefly introduces the work of al-Kindi  and al-Karaji. We return to medieval Europe and a brief sketch of the translation movement reintroducing Greek and introducing Arabic mathematic into Europe. We get a brief account of the work of Adelard of Bath and a somewhat longer one of Fibonacci. 

The next chapter is unusual in an introductory text on the history of mathematics in that it features the work of both Levi ben Gerson, known as Gersonides, and Nicole Oresme during the European Middle Ages. Now firmly back in Europe we get introduced to the discovery of linear perspective with brief accounts of the work of Brunelleschi, Alberti, Uccello, della Francesca and closing with Luca Pacioli. Established in the Renaissance we get a whole chapter devoted to Tartaglia, Cardano, and the general solution of the cubic equation. Lawrence’s account of Cardano’s fascinating biography is a bit wonky. She says that he was denied membership of the College of Surgeons because of his unruly behaviour but doesn’t mention that the rejection was mainly because he was illegitimate. Later she says that in Milan he was a wealthy and well-respected mathematician, whereas in fact he had finally acquired his medical licence, given up teaching mathematics and became a wealthy and well-respected physician. Quibbles aside the account is quite reasonable. 

Cardano’s Ars Magna is an important algebra text and we stay in the world of algebra with the next chapter about Francoise Viète and the introduction of symbolic algebra. As is often the case, too much credit is given here to Viète, as the transition to symbolic algebra had begun before he wrote his book and continued in the work of others after him. Strangely the second half of the chapter is given over to the work on codes of Blaise de Vigenère.

Lawrence now takes a sharp left turn into the world of Johannes Kepler and presents a piece of nonsense, she write: 

“He studied at the University of Tübingen where he was surprised to learn that he was good at mathematics. He wrote his dissertation in defence od Copernicus’s 1543 heliocentric theory, which had established the Sun as being at the centre of the solar system.”

Where he was surprised to learn that he was good at mathematics? First I’ve ever heard of it but it’s the second sentence that’s just plain wrong. Kepler was studying to become a school teacher and never wrote a dissertation. What Lawrence is referring to, was a text for a student disputation, a standard practice at medieval universities. This is what later became, as Lawrence correctly points out, his proto-science fiction story Somnium. We then get Mysterium Cosmographicum dealt with in a paragraph, with, however, a false title Magisterium Cosmographicum. At break neck speed we get the first and second laws of planetary motion, although Lawrence apparently doesn’t know that he discovered the second law first. Pacing along we spring from snowflakes to the problems of living is a time of war, his mother witchcraft accusations, to music of the spheres all within eight lines! She slows down slightly for Harmonices Mundi and the discovery of his third law. Without going into great detail, she at least acknowledges that Harmonices Mundi is a treasure trove of mathematics. Having earlier introduced Kepler’s work on the Rudophine Tables, Lawrence brings the chapter to a close dedicating two pages to the invention of logarithm. She closes with Kepler’s accurate prediction of the transit of Mercury in 1631. The life and work of Johannes Kepler in seven and a half pages is simply a bridge too far. 

Having introduced logarithms, we now get a chapter on mathematical machines. Napier’s Bones lead onto Brigg’s improved logarithms. Next up are the mechanical calculators of Schickard and Pascal. The chapter closes with an account of Frans van Schooten’s device for drawing conic sections. 

Now well into the seventeenth century, we get introduced to the academy of Marin Mersenne. In this chapter Lawrence manages to dispose to both the analytical geometry of Descarte and the projective geometry of Desargues. On analytical geometry, Fermat doesn’t get a look in and Lawrence makes the classical mistake of attributing the rectangular coordinate system to Descartes. Whereas it was actually first introduced by Frans van Schooten in his expanded Latin translation of Descartes’ La Géométrie. Mid seventeenth century French mathematics leads almost automatically to the introduction of probability theory by Pascal and Fermat, this time getting a credit. Pascal’s triangle is introduced with a nod to his eleventh century Chinese predecessors but not to his European predecessors Jordanus de Nemore in the thirteenth century,  and Peter Apian in the sixteenth century. 

We return briefly to the far east, this time to Japan to learn about the parallel development of Bernoulli numbers by Seki Takakazu in Japan and Jacob Bernoulli in Switzerland at the beginning of the eighteenth century. This is followed by an account of the work of Takebe Katahiro, a follower of Seki, in particular his introduction of trigonometrical functions into Japan. This I found particular bizarre because trigonometry, a very major branch of mathematics with a long and complex history, doesn’t get mentioned anywhere else in Lawrence’s book. Having just met one Bernoulli the next chapter devotes nine whole pages to calculus and the Newton-Leibniz priority dispute. Along the way we get Johann Bernoulli’s brachistochrone problem, although Lawrence doesn’t give the curve its name, Newton’s solution and Bernoulli’s infamous “lion’s claw” quote. The chapter also gives Lawrence the chance to introduce he second female mathematician with Maria Agnesi, her calculus text book, and the witch of Agnesi. A very brief nod is also given to a third female mathematician Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet. 

Ever onwards into the eighteenth century, we now get introduced to, perhaps histories most prolific mathematician, Leonard Euler. From Euler’s vast output, Lawrence restricts herself to the beginnings of topology and graph theory and his summation of infinite series. The next chapter takes a look at Gaspard Monge and his invention of descriptive geometry. Truly a single theme chapter from which it benefits. I found it one of the best chapters in the book. We stay with geometry and the discovery of non-Euclidian geometry, covering Bolyai, Lobachevsky, and Gauss on hyperbolic geometry, with a very brief nod to Arthur Cayley for elliptical geometry at the end. 

We had a whole chapter on Tartaglia and Cardano and the story of the general solutions of the cubic and quartic equations. Lawrence now turns her attention to Abel and Galois, the failure to find a general solution to the quintic equation and the mathematics that grew out of it. 

The chapter on the nineteenth century development of algebraic logic opens with Lewis Carroll author of Alice in Wonderland, who Lawrence quite correctly points out was also a logician. This leads in to George Boole and his creation of algebraic logic. The chapter also includes Charles Babbage and his computers, which includes several false statements, which, unfortunately, seems to be the norm when people write about Babbage these days. “The birth of computers came not long after Boole…” Babbage began work on the Difference Engine in 1822, Boole published his first logic book, Mathematical Analysis of Logic in 1847. “His [Babbage’] partner in this pursuit was Ada Lovelace…” Lovelace was never Babbage’s partner in his work. After introducing Claude Shannon and his use of Boolean algebra to design electrical circuits we get, “from the early ideas of Babbage and Lovelace, Shannon and many other mathematicians and computer scientists following him…” Babbage’s work had absolutely no influence on the development of computers in the twentieth century. Lawrence now moves onto John Venn and diagrams writing, “But it was John Venn (1834–1923) who gave his name to diagrams of a particular sort that we use so often today.” As I love to point out, Venn actually called them Eulerian Circles.

The book is now moving into modern mathematics and I had the feeling which reading the later chapters that somehow Lawrence was more at home, more enthusiastic about the more modern developments than the earlier history. I can’t explain why or even give examples to back up this feeling. This should, however, not be seen in anyway as a criticism.

The following chapter opens with Henri Poincaré and his work on the three body problem in astronomy and the fact that there is no solution for it. Lawrence sees this as, her term, “the quiet birth of chaos.” We takes a leap to Edward Norton Lorenz and the butterfly effect before returning to the nineteenth century and Bernhard Reimann’s development in non-Euclidian geometry, which Poincaré took up and developed further. Moving on we get introduced to one of my favourite mathematicians, Georg Cantor and his investigations into the size of infinity. Where Cantor says must almost inevitably say Richard Dedekind, who also gets a look in here with his definition of real numbers. The chapter closes with, what is probably my favourite mathematical statement, “there are infinitely many infinities…” 

At the end of the nineteenth century mathematicians began to discover the fourth dimension, and thoughts of even higher dimensions, in geometry. This episode makes it possible for Lawrence to introduce two further mathematical women, Mary Everest Boole  the wife of George and  the author of mathematical textbooks in her own right. Mary was secretary to James Hinton, father of the mathematician Charles Hinton, who married her daughter Mary Ellen, and who wrote extensively about the fourth dimension. Mary Ellen’s sister Alicia Boole Stott, who had no formal mathematical studies, became famous for her ability to visualise four dimensional space. In the next chapter we leave the fourth dimension and enter the University of Göttingen of Felix Klein and David Hilbert. The emphasis here is the international mathematics congresses that they initiated and Hilbert’s legendary twenty-three unsolved problems presented at the 1900 Congress in Paris. Problems that would occupy leading mathematicians down to the present. 

Lawrence next takes us into the world of Russell, Whitehead, and Frege, and the rise of the philosophy of logicism followed by its fall at the hands of Gödel. Lawrence makes no comments on the competing systems of philosophy of mathematics in the search for secure foundations. We stay in the Cambridge of Russel and Whitehead and meet the extraordinary-brilliant philosopher, logician, economist , and mathematician Frank Ramsey, who despite dying at the age of twenty-eight left an amazing body of work for future generations. Lawrence takes a deeper look at Ramsey Theory in combinatorics. 

The next chapter is dedicated to another of Lawrence’s female mathematicians, Emmy Noether. After a slightly mangled biographical sketch covering Noethers journey to becoming a successful mathematician. Lawrence tells us that Klein and Hilbert invited her to Göttingen specifically to work on the theory of relativity. This is not true, she was invited there to teach mathematics and once there first turned her attention to the theory of relativity, producing the famous Noether’s Theorem in physics. Fortunately, because this is a book about the history of mathematics Lawrence doesn’t stop there but  gives emphasis to Noether’s massive contributions to the development of abstract algebra. 

The next chapter is dedicated to the twentieth century’s most famous non-existent mathematician Nicolas Bourbaki. Bourbaki was initially the pen name for the group publications of a group of young French mathematicians. The Bourbaki collective grew and developed and Lawrence traces the fascinating development of Bourbaki through the twentieth century and down to the present. From Bourbaki we move on to John von Neuman, Oskar Morgenstein and the evolution of game theory. Lawrence gives a detailed introduction to this work, although failing to note that it was built on Frank Ramsey’s work in probability, however she emphasises its significant impact on the world of modern mathematics. Lawrence continues with another new area of mathematics created in the twentieth century, that gets a lot of popular coverage, the fractals of Benoit Mandelbrot. From the very modern, Lawrence takes us back to the seventeenth century and Fermat’s last theorem. This is, of course, a lead in to the life and work of Andrew Wiles and his solution of that theorem.

Heading towards the end of her book, Lawrence gives her readers, what I found to be an especially good chapter, the life and work of the Iranian mathematician Maryam Mirzakhani, a female mathematician of extraordinary talent, who died at the comparatively early age of forty. She was the first of only two women to date to win the Fields Medal, mathematics most prestigious award. From a female mathematician, who displayed her extraordinary mathematical talent at a very early age, Lawrence now introduces us to an extraordinary male mathematician, who was definitely a child prodigy, Terence Tao, who won the Fields Medal at the age of thirty-one. 

In the penultimate chapter Lawrence, ties together several other chapters. She takes her readers back to the seventeenth century again for another problem that stumped mathematicians for almost  four hundred years, the Kepler conjecture. Sir Walter Raleigh had asked his house mathematician, Thomas Harriot, to find the most efficient way to stack cannon balls on a ship. Harriot shared the problem with Kepler during their correspondence and Kepler gave a conjectured solution to the problem, without a proof, in his wonderful pamphlet on snowflakes. Despite many efforts nobody came up with a proof over the centuries and finding one became one of Hilbert’s twenty-three problems in Paris in 1900. The proof was finally found by American mathematician, Thomas Hales, using a computer in a proof by exhaustions i.e. checking systematically all possible cases, in 2014. He was already too old to win the Fields Medal, as the recipients have to be under forty years old. However, as Lawrence relates the story doesn’t end here. As is often the case in modern mathematics, mathematicians asked, we have a solution for three dimensions what are the solutions for higher dimensions. Lawrence now introduces her final female mathematician the Ukrainian, Maryna Sergiivna Viazovska, who solved the problem for eight and twenty-four dimension earning her the Fields Medal in 2022.

Lawrence’s final chapter is entitled Dreams of New Mathematics and ruminates and speculates about the future of the discipline. 

As is my wont, I have sketched all forty of Lawrences chapters to give potential readers an idea of the width and depth of her endeavours. The short nature of each chapter presents a problem given the vast volume of potential material available on each theme with which one could fill them. In general Lawrence succeeds in producing an interesting and informative precis of each topic. In my opinion she fails is a few cases, her chapters on Kepler, and that on calculus,  for example. There are several obvious lacuna in her history, for example, as I noted above, nothing on trigonometry and although he gets mentioned several times in passing nothing on Archimedes, whose influence on the development of the mathematical sciences in the Early Modern Period, was immense.

She write fluidly and accessibly, making her book a good general introduction to the history of mathematics for interested beginners, older school pupils or students for example. As already noted the book has no footnotes, endnotes, or bibliography and I think it would have been improved by a couple of suggestions for further reading on each topic. There is, however, a good index. There are no illustrations in the text, as such, but there are simple black and white mathematical diagrams where needed. However, each chapter is headed by a black and white, artistic pseudo-woodcut hinting at the theme of the chapter, of which I have include a couple above. 

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Filed under Book Reviews, History of Mathematics

Mario Biagioli (1955–2025)

Over the years I have gained a reputation for being extremely negative about Galileo Galilei and his supposed achievements and contributions to the evolution of science. As I point out from time to time, much of what I say is not aimed at Galilei per say, but at those who continue to present him as some sort of one man scientific revolution, who single-handedly created modern science with the concomitant deification and hagiography. Not only that he became a martyr for science sacrificing his life on the altar of truth, Galileo can do no wrong, even when he is very obviously wrong, his theory of the tides for example, there is always an explanation why he was really on the right track. To deny this god like status is to commit blasphemy.

Naturally, over the years as somebody, whose main area of interest is the mathematical sciences in the Renaissance and the Early Modern Period, I have read a large number of papers and books about Signor Galileo and his contributions to astronomy and physics, as well as his problems with the Catholic Church.

Two of the very best books on Galileo, his times, his life and his work are both by Mario Biagioli,

Mario Biagioli

his Galileo Courtier: The Practice of Science in the Culture of Absolutism (University of Chicago Press, 1993)

and his Galileo’s Instruments of Credit: Telescopes, Images, Secrecy (University of Chicago Press, 2006)

Both books are, in my opinion, masterpieces in the history, sociology, and politics of science. There is no hagiography here. These are well researched, superbly written deep investigations into Galileo’s life and work focusing, amongst other things, on his personal motivations and how he used his science and technology to raise his social status. If you want to know about the real Galileo these books are a must read.

I found Biagioli’s comments on absolutism and favouritism as practiced by absolutist rulers of particular interest. Cardinal Maffeo Barberini had been a friend and supporter of Galileo’s since the publication of Sidereus Nuncius in 1610. In the following years he warned Galileo to tread softly and not to provoke the Church. He explained that the Church was capable of change, but was a large and complex organism that suffered from inertia like a sleeping bear (my analogy), change took time and Galileo should remain patient. Advice that Galileo famously ignored, instead poking the sleeping bear with a sharp stick, in 1615. Despite this when Barberini became Pope Urban VIII, he raised Galileo up to the status of court favourite. Biagioli’s hypothesis is that in 1632/33, at a time when his he was under immense political pressure, Urban cast Galileo down to demonstrate his political power. Those I raise up, I can also cast down. A widespread political strategy amongst absolutist rulers.

Mario Biagioli died seemingly unexpectedly in recent days, although it is now emerging that he had been ill for some time. A couple of months back he cancelled a planned lecture in Bavaria because of illness. His passing in a great loss not just to the history of science community. At the time of his death following a long and distinguish career, he was a Distinguished Professor of Law and Communication at UCLA, his research concentrating on the concept of intellectual property of which his books on Galileo were just one example.

I had the privilege and the pleasure of meeting him at the major conference in Middleburg in Holland in 2008, organised to celebrate the four-hundredth anniversary of the invention of the telescope. An open and friendly man, enthusiastic and with an intense and lively intellect. His books contributed much to my understanding of Galileo, and I was sad to learn of his death.

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Filed under Autobiographical, Book Reviews, History of science

The Land Between Two Rivers

I describe myself as a narrative historian of the contextual history of science or put simply I’m a story teller. One of the things that I firmly believe in as a historian of science is that context is supremely important. Science doesn’t arise and evolve in a vacuum but in a context that involves many different factors such as geography, environment, exigencies, politics, economics, beliefs, climate and so on and so forth. This is the reason why presentism as a form of historiography is so bad. It extracts elements of science out of their original contexts purely because they still exists in a similar form and are relevant in the modern context and try to make claims about the historical development of that particular piece of science, which is inevitably simply false. 

For myself, a wonderful example of the importance of context concerns astrology. For a long time, many historians of science dismissed astrology as irrelevant to their discipline because it was according to their definition not science, at best a mere superstition, at worst total nonsense. However, if you look closely at the context in which astrology arose and evolved, it becomes very obvious that astrology was a major driving force behind the origin and development of astronomy, a major branch of science. Ptolemaeus, who played a very major role in the development of both astronomy and astrology in Europe, actually viewed them as two aspects of one discipline. I sketched the parallel development of the two discipline here.

Another example concerns one of my central areas of interest the mathematical sciences in Renaissance Nürnberg. The emergence of Nürnberg as a major European centre for the mathematical disciplines during the Renaissance, initially had nothing to do with science per say but was driven by trade and economics, as I sketched here.

Context is, as I say, all important and requires the historian of science to climb out of their little scientific bubble and learn about the economic, political, social and cultural histories of the time and space in which the science they are studying originated and/or evolved. 

One early source of various aspects of human intellectual development that played an important role in the evolution of science is the area of Western Asia known as Mesopotamia, which lies within the Tigris–Euphrates river system. An area also know as the Fertile Cresent and The Cradle of Civilisation. I personally object to the last label as it’s Eurocentric. The Fertile Cresent civilisation, which includes Egypt on the Nile had a major influence on later developments in Europe, calling it “The” Cradle of Civilisation, however, ignores parallel developments of the Indus Valley Culture in Northwestern India and the Yangtze and Yellow River cultures in China during the same time period. It is interesting to note that these early major cultures all developed along large river systems an important environmental factor. Rivers provided the water so necessary for life and farming and also transport for travel and trade. 

Despite my objections to the labelling of Mesopotamia as The Cradle of Civilisation, it still plays a significant role in several aspects of the history of science. First and foremost, it was the first culture to develop a system of writing, a basic fundamental for the evolution of the sciences. Mesopotamia also produced the first systematic astrology and astronomy, which they quantified with the first place value number system. A sexagesimal system that continued in use in astronomy down to Copernicus. Using their advanced number system the cultures of Mesopotamia also produced some very sophisticated mathematics. With all of this in mind I was pleased to obtain a copy of the newly published Between Two Rivers: Ancient Mesopotamia and the Birth of History by Moudhy Al-Rashid[1] to be able to deepen my contextual knowledge of the cultures that produced these advances in science. 

Astute readers will have noted that that I wrote cultures, plural, rather than culture, singular, this is because although we talk of Ancient Mesopotamia, it existed for on going four thousand years and encompassed five different cultures–Sumerian, Akkadian, Assyrian, Babylonian, Persian–which although each had its own language were, however, united by a common written language, Sumerian.  To complicate matters those four cultures were not uniform but had well formed sub-cultures. On top of this the political structures varied over time. During some periods there existed diverse, independent city states cooperating and competing with each other. During others a single power ruled over large parts or even all of area encompassed by the two rivers. How do you write a history of this period of over three millennia with such diverse cultures and political systems? A detailed, chronological, systematic history would take up not just one book but a whole shelf full of books.

Moudhy Al-Rashid doesn’t even attempt to write such a history. Instead, she creates evocative, verbal sketches of central aspects of the whole location and period and shows how they play out in the various differing cultures and systems over the millennia. To begin with she explains that in writing the history of Mesopotamia, she is also writing the history of the beginnings of the discipline of history. History originally referred to the written record of human existence as opposed to prehistory which reconstructs the record of human existence before writing was invented. The Sumerians were the first culture to invent writing and by doing so unintentionally invented history. Because they wrote on baked or dried clay tablets, much of what they wrote has survived the ravages of time and Al-Rashid uses those myriad writings from all aspects of life in Mesopotamia to bring her sketches to life, but it is the strategy she uses to set those sketches in motion that is truly inventive.

She opens the book by taking her readers on a visit to a museum. During his excavations, in the 1920s, of the palace of the priestess-princess Enigaldi-Nanna in the city of Ur dating to about 600 BCE, Leonard Woolley (1880–1960) unearthed a room with an intact floor in which, amongst all the rubble he discovered artifacts from various epochs of Ur’s three and a half millennia existence. He labelled the room a museum. During an excavation in an intact layer, one normally only finds artifacts from a single period. This is in fact the beginning of archaeological, relative dating; artifacts from deeper layers are older. Al-Rashid explain that the room was not necessarily a museum, there could be other explanation for the assemblage of time spanning artifacts but says Woolley’s label has stuck. Throughout her book she gives brief insights into the world of archaeology and its practices.

Having introduced Woolley’s museum and its artifacts, each of Al-Rashid’s following chapters is dedicated to one or a group of archaeological artifacts. Having described the given artifact in its own context, she then goes on to spin out how that artifact plays out in the different cultures and time periods of the entire Mesopotamian history in those evocative, verbal sketches I mentioned earlier. The opening chapter features an inscribed clay drum, which here serves the function of an introduction to the invention, spread and use of the Sumerian cuneiform writing system. This chapter alone is worth the purchase price of the book.

The following chapter is centred around  an apparently very mundane object a building brick. Just as the Mesopotamian written culture is preserved on baked or dried clay tablets, the monumental architecture of those millennia was all constructed out of uniform, dried, clay bricks and, as Al-Rashid tells us, literally millions of them. This chapter is a tour of the Mesopotamian architecture and its building material. We get the mass production methods of the brick manufacturers explained to us together with another aspect that is missing from modern brick production, the bricks were inscribed. Each structure was built with bricks inscribed with a message relevant to the structure or its constructers. They even invented a method of stemple printing to mass produce those inscribed bricks. I never thought a book chapter on house bricks could be so fascinating. 

Up next is the statue of a king named Shulgi. After introducing him and his reign, Al-Rashid takes us on a journey through the concept of kingship and how it played out throughout Mesopotamian history. 

Having been to the apex of a hierarchical society, in the next chapter we go back down to the base and the school room, where children are taught the cuneiform system of writing. The large number of school writing tablets that have been found tell us a lot about the system of education that was employed in this first literate society. 

Al-Rashid now moves into my area and the widespread use of divination throughout Mesopotamian history including astrology, although strangely she never uses the word, which leads to astronomy or as she puts it in the chapter’s subtitle, The Birth of Science. The artifact around which this chapter is centred is a clay cone telling the story of Kudur-Mabuk who built a temple to the moon god Nanna in the nineteenth century BCE and whose daughter, Enanedu  became high-priestess of Ur just like Enigaldi-Nanna many centuries later.

The seventh chapter tells the story of a boundary stone, which tells of the donation of land by a king, leading us into a discussion of such stones and grants of land over the centuries. This discussion widens into a general one on the world of trade in the millennia of Mesopotamia’s existence and covers according to the subtitle, Slaves and Scribes, Weavers and Wives. We learn that woven fabrics were a major trade item, the men were usually the traders and their wives the weavers. Much of this information coming from letters exchanged between husbands in foreign parts trading and their wives at home running the business.

In any general history of a highly populated area that covers more that three thousand years war and its consequences has to be a central theme and Al-Rashid handles this in her penultimate chapter, revolving around a mace head, a symbol of power and a deadly weapon of war, as artifact. 

The book closes where it began in the palace of the priestess-princess Enigaldi-Nanna, this time examining the lady herself. Al-Rashid expands her investigation into the life and social, religious and cultural roles of Enigaldi-Nanna the priestess-princess into a wide-ranging survey of the role of women on all levels throughout the history of the region. 

Moudhy Al-Rashid is a professional academic, who has very obviously invested much time and effort into studying and researching the cultures which occupied Mesopotamia during the three plus millennia BCE, however her history of the area during this period is very much written for the general reader without sacrificing historical accuracy. She write in a bubbling, easy to read, narrative style interspersing her historical expositions with modern comparisons, such as comparing the beginnings of Sumerian writing that began with simple labels rather than words with our emojis. Other interjections tell of her experiences when first holding a historical artifact or attempting to get her brain around one or other of the languages. Sometimes she includes anecdotes  out of her personal life that parallel the historical events she is describing, such as her first experience of a comet compared to the records of comets in Mesopotamian astronomy.  All of this makes for a rich, entertaining narrative that pulls the reader along into the depths of ancient Mesopotamia. This mixture of ancient and present, historical and personal is emphasised in the six page recap, that closes the main text. 

Although this is a thoroughly popular book it has an impressive academic apparatus reflecting the authors scholarship. This begins with a list of selected artifacts cited ‘for further reading or research,’ ‘A Timeline of Ancient Mesopotamian History,’ a very extensive bibliography and a comprehensive set of end notes linking the text to that bibliography. End notes and bibliography show the reader interested in deepening their knowledge of Ancient Mesopotamia and its cultures how and where to begin their research. The book closes with a good index. There are sadly no illustrations.

If you want a fascinating introduction to Ancient Mesopotamian history or if you enjoy reading good popular history book then I can only recommend that you put Moudhy Al-Rashid’s excellent volume at the top of your reading list. I guaranty that you won’t regret it.  


[1] Moudhy Al-Rashid, Between Two Rivers: Ancient Mesopotamia and the Birth of History, hodder press, 2025.

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History of maths for beginners!

In the comments on my recent post on books on the history of maths Fernando Q. Gouvêa jumped in to draw attention to the book Math Through the AgesA Gentle History for Teachers and Others, which he coauthored with William P. Berlinghoff. I had not come across this book before, as I noted in my post I gave up reading general histories of maths long ago, so, I went looking and came across the following glowing recommendation:

Math Through the Ages is a treasure, one of the best history of math books at its level ever written. Somehow, it manages to stay true to a surprisingly sophisticated story, while respecting the needs of its audience. Its overview of the subject captures most of what one needs to know, and the 30 sketches are small gems of exposition that stimulate further exploration.” — Glen Van Brummelen, Quest University 

Glen Van Brummelen is an excellent historian of maths and regular readers will already know that I’m a very big fan of his books on the history of trigonometry, which I reviewed here. Intrigued I decided to acquire a copy and see if it lives up to Glen’s description. The book that I acquired is the paperback Dover reprint from 2019 of the second edition from Oxton House Publishers, Farmington, Maine, it has a recommended price in the USA of $18, which is certainly affordable for its intended audience, school, college and university students. The cover blurb says, “Designed for students just beginning their study of the discipline…”

The book distinguishes itself by not being a long continuous narrative covering the chronological progress of the historical development of mathematics, the usual format of one volume histories. Instead, it opens with a comparatively brief such survey, just fifty-eight pages entitled, The History of Mathematics in a Large Nutshell, before presenting the thirty sketches mentioned by Van Brummelen in his recommendation. These are brief expositions of single topics from the history of maths, each one conceived and presented as a complete, independent pedagogical unit. The sketches however contain references to other units where a term or concept used in the current unit is explained more fully. Normally when reviewing books, I usually leave the discussion of the bibliography till the end of my review but in this case the bibliography is an integral part of the unit presentation. The bibliography contains one hundred and eighty numbered  titles listed alphabetically by the authors’ names. At the end of each sketch there is a list of numbers of the titles where the reader can find more information on the topic of the sketch. Within the sketches themselves this recommendation system is also applied to single themes. The interaction between sketches and bibliography is a very central aspect of the books pedagogical structure.  

Given its very obvious limitations, my own far from complete survey of the history of mathematics now occupies more than two metres of bookshelf space and that is not counting the numerous books on the histories of astronomy, physics, navigation, cartography, technology and so forth, which I own, that contain elements on the history of mathematics, the Large Nutshell is actually reasonably good. It sketches the development of mathematics in Mesopotamia, Egypt, China, Ancient Greece, India, and the Islamic Empires, before leaping to Europe in the fifteenth and sixteenth centuries and from there is fairly large steps down to the present day. Naturally, a lot gets left out that other authors might have considered important but it still manages to give a general feeling of the depth and breadth of the history of mathematics. The real work begins with the Sketches.

Sketch No.1 is naturally Keeping CountWriting Whole Numbers. Our authors deliver competent but brief accounts of the Egyptian, Mesopotamian, Mayan, Roman and Hindu-Arabic number systems but don’t consider the Greek system worth mentioning. They of course emphasise the difficulty of calculating with Roman numerals, as everybody does, but at least mention the calculation was actually done with a counting board or abacus, without seeming to realise that abacus is the Greek name for a counting board, an error that leads them to a stupid statement in another sketch. They also don’t mention that everybody, Mesopotamians, Greeks, Egyptians etc used counting boards to do calculations. They of course mention this in order to emphasise the advantages of the Hindu-Arabic numerals for doing calculations on paper, which leads to another error later.

Sketch No.2 Reading and Writing ArithmeticThe Basic Symbols now makes a big leap to the introduction of symbolic arithmetic in the Renaissance to replace the previous rhetorical arithmetic. This precedes chronologically through the various innovations, including, unfortunately, the myth that Robert Recorde was the first to introduce the equal’s sign. Is OK, but they admit that their brief sketch skips over many, many symbols that were used from time to time. 

Almost predictably, Sketch No.3 is Nothing Becomes a NumberThe Story of Zero. We start with the Babylonians, who had a place value number system, and after a long time without one, introduced a place holder zero. We then spring to India and from them straight to the Arabs and the logistic origins of the word zero before being served up the first major error in the book. Our authors tell us:

By the 9th century A.D., the Indians had made a conceptual leap that ranks as one of the most important mathematical events of all time. They had begun to recognise sunya, the absence of quantity, as a quantity in its own right! That is, they had begun to treat zero as a number. For instance, the mathematician Mahāvīra wrote that a number multiplied by zero results  in zero, and that zero subtracted from a number leaves the number unchanged. He also claimed that a number divided by zero remain unchanged. A couple of centuries later. Bhāskara declared, a number divided by zero to be an infinite quantity.

This is quite frankly bizarre! Already in the 628 CE, Brahmagupta (c. 598–c. 668) had in Chater 12 of his Brāhma-sphua-siddhānta given the rules for addition, subtraction, multiplication, and division with positive and negative numbers and zero in the same way as they can be found in any elementary arithmetic textbook today, with the exception that he defined division by zero. His work was well known in India. In fact, Bhāskara II, mentioned by our authors, based his account on that of Brahmagupta. His work was also translated into Arabic in the eight century and provided the basis for al-Khwārimī’s account, which has not survived in Arabic but was translated into Latin in the twelfth century as Dixit Algorizmi, which was the first introduction of the Hindu-Arabic number system in Europe, as is noted by our authors. Our authors then over emphasise the reluctance of some European mathematicians to accept zero as a number, whilst introducing Thomas Harriot’s method for solving polynomials and zero as a defining property of rings and fields.

Sketch No.4 Broken Numbers: Writing Fractions is a quite good account of their history considering its brevity.

Sketch No.5 Less than NothingNegative Numbers is also a reasonably good account of the very problematic history of accepting the existence of negative numbers. Somewhat bizarrely, having totally ignored him on the topic of zero, our authors now state that:

A prominent Indian mathematician, Brahmagupta, recognised and worked with negative quantities to some extent as early as the 7th century. He treated positive numbers as possessions and negative numbers as debts, and also stated rules for adding, subtracting, multiplying, and dividing with negative numbers. 

It’s the same section of his book where he deals with zero as a number!

Sketch No. 6 By Ten and TenthsMetric Measurement is a good brief account of the French introduction of the metric system and its subsequent modernisation. I would have appreciated a brief nod for the English, polymath John Wilkins (1614–1672), who advocated for a metric system already in the seventeenth century. 

Sketch No. 7 Measuring the CircleThe Story of π a simple brief account. An interesting addition is a table showing the size of the error in calculating the circumference of a circular lake with a diameter of one kilometre using different historical values for π.

Sketch No. 8 The Cossic ArtWriting Algebra with Symbols is a competent sketch of the gradual but zig zag transition from rhetorical to symbolic algebra. 

Sketch No. 9 Linear ThinkingSolving First Degree Equations starts with the Egyptians mentions the Chinese but completely ignores the Babylonians. It then wanders off into an account of the false proposition method of solution.

Sketch No. 10 A Square and ThingsQuadratic Equations after one dimension we move onto two dimensions. This is devoted to al- Khwārimī’s discussion of methods of solving quadratic equations completely ignoring the fact that the Babylonians had the general solution of the quadratic equation a thousand years earlier, albeit in a different form to the one we teach schoolchildren and, of course ignoring negative solutions. Every worse ignoring  the fact that Brahmagupta had the general solution in the form we use today including negative solutions!

Sketch No. 11 Intrigue in Renaissance ItalySolving Cubic Equations and now onto three dimensions. We start with the Ancient Geek problem of trisecting an angle and then surprisingly, considering that this is supposed to be “for students just beginning their study of the subject,” we suddenly get a piece of fairly advanced trigonometry blasted into the text:

cos(3𝛂) = 4 cos3(𝛂) – 3 cos(𝛂)

to point out that the solution can be found with a cubic equation!

We move on to Omar Khayyám and his solution of cubic equations using conic sections. We then get a mangled version of the Antonio Fiore/Tartaglia challenge. Out authors claim that Tartaglia bragged that he could solve cubic equations and so Fiore, who had learnt the solution from his teacher Scipione del Ferro, challenged him to a mathematical contest. In fact, Tartaglia didn’t know how to solve them when challenged by Fiore and realising he was onto a hiding to nothing sat down and discovered how to solve them. End result victory for Tartaglia. They get the rest of the story right. How Cardano persuaded Tartaglia to reveal his solution after promising not to publish it before Tartaglia did. Then discovering that Scipione del Ferro had discovered the solution before Tartaglia, and having extended Tartaglia’s solution, went on and published anyway, although giving full credit to Tartaglia for his work. You can read the whole story here. Our authors, however, contradict themselves. In their Large Nutshell they gave a brief version of the story and wrote:

Once he knew Tartaglia’s method for solving some cubic equations. Cardano was able to generalise it to a way of solving any cubic equation. Felling he had made a contribution of his own, Cardano decided he was no longer bound by his promise of secrecy.

Now in Sketch 11 they write:

At this point, Cardano knew that he had made a real contribution to mathematics. But how could he publish it without breaking his promise? He found a way. He discovered that del Ferro had found the solution of a crucial case before Tartaglia had. Since he had the not promised to keep del Ferro’s solution secret, he felt he could publish it, even though it was identical to the one he had learnt from Tartaglia.

We then get Cardano’s discovery of conjugate pairs of complex numbers sometimes in the solutions of cubic equations and a somewhat confused explanation of his reaction to them. Our authors tell us that Cardano wrote to Tartaglia and asked him about the problem and Tartaglia simply suggest that Cardano had not understood how to solve such problems. This is true, quoting MacTutor:

One of the first problems that Cardan hit was that the formula sometimes involved square roots of negative numbers even though the answer was a ‘proper’ number. On 4 August 1539 Cardan wrote to Tartaglia:-

I have sent to enquire after the solution to various problems for which you have given me no answer, one of which concerns the cube equal to an unknown plus a number. I have certainly grasped this rule, but when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number, then, it appears, I cannot make it fit into the equation.

Indeed, Cardan gives precisely the conditions here for the formula to involve square roots of negative numbers. Tartaglia, by this time, greatly regretted telling Cardan the method and tried to confuse him with his reply (although in fact Tartaglia, like Cardan, would not have understood the complex numbers now entering into mathematics):-

… and thus, I say in reply that you have not mastered the true way of solving problems of this kind, and indeed I would say that your methods are totally false.

Our authors simply move on to state, It fell to Rafael Bombelli to resolve the issue.” 

This is selling Cardano short, firstly he was well aware that if one multiplies complex conjugate pairs together the imaginary term disappears as he demonstrated in Ars Magna, he simply thought that it didn’t make sense, mathematically. 

Dismissing mental tortures, and multiplying 5 + √-15 by 5 – √-15, we obtain 25 – (15). Therefore the product is 40. …. and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless. (MacTutor)

Secondly, Bombelli used Cardano’s work as the starting point for his own work. 

Sketch No. 12 A Cheerful FactThe Pythagorean Theorem This is a reasonable overview of some of the history of what is probably the most well-known of all mathematical theorems. The first half of the title is a Gilbert and Sullivan quote!

Sketch No. 13 A Marvellous ProofFermat’s Last Theorem Once again a reasonably synopsis of the history of another possible candidate for the most well-known of all mathematical theorems.

Sketch No. 14 On Beauty BareEuclid’s Plane Geometry Another reasonably synopsis of the world’s most famous and biggest selling maths textbook.

Sketch No. 15 In Perfect ShapeThe Platonic Solids  A brief, compact and largely accurate introduction to the history of the Platonic regular solids and the Archimedean semi-regular solids. I would have wished for more detail on the Renaissance rediscovery of the Archimedean solids, which was actually carried out by artists rather than mathematicians, a fact that our authors seem not to be aware of, as part of the explorations into the then new linear perspective. 

Sketch No. 16 Shapes by the NumbersCoordinate Geometry Another good synopsis covering both Fermat and Descartes and includes the important fact that the Cartesian system was actually popularised by the expanded Latin edition of La Géométrie put together by Frans van Schooten jr. Although our authors miss the junior. Having acknowledge his contribution they then miss one of his biggest contributions. Our authors write:

The two-axis rectangular coordinate system that we commonly attribute to Descarte seems instead to have evolved gradually during the century and a half after he published La Géométrie.

Rectangular Cartesian coordinates were introduced by Frans van Schooten jr. in his Latin edition.

Sketch No. 17 Impossible, Imaginary, UsefulComplex Numbers Having, in their discussion of cubic equations, failed to acknowledge the fact that Cardano actually showed how to eliminate conjugate pairs of complex numbers in his Ars Magna, even if doing so offended his feelings for mathematics, our authors now bring the relevant quote to this effect. Having opened with Cardano we then move onto Bombelli’s acceptance and Descartes’ rejection before our authors attribute an implicit anticipation of Euler’s Formula in the work of Abraham De Moivre, whilst ignoring the explicit formulation in the work of Roger Cotes. Dealing comparatively extensively with Euler our authors then move onto Argand and Gauss and the geometrical representation of complex numbers, whilst completely ignoring Casper Wessel.  

Sketch No. 18 Half is BetterSine and Cosine We now get introduced to the history of trigonometry. Once again a reasonable short presentation of the main points of the history down to the seventeenth century, with an all too brief comment about the development of the trigonometrical ratios as functions.

Sketch No. 19 Strange New WorldsThe Non-Euclidian Geometries Once again nothing here to criticise.

Sketch No. 20 In the Eye of the BeholderProjective Geometry Starts with some brief comments about the discovery of linear perspective during the Renaissance, then moves on to the development of projective geometry out of that and ends with Pascal’s Theorem. Once again no complaints on my part.

Sketch No. 21 What’s in a Game?The Start of Probability Theory Having had the teenage Pascal on conics we now have the mature Pascal on divvying up the stakes in an interrupted game of chance. This is followed by  standard story of the evolution of probability theory. 

Sketch No. 22 Making Sense of DataStatistics Becomes a Science Once again presents a standard account without any significant errors. 

Sketch No. 23 Machines that Think?Electronic Computers Having presented twenty-two sketches with only occasional historical error, our authors now come completely of the rails: To start: 

Some would say that the story begins 5000 years ago with the abacus, a calculating device of beads and rods that is still used today.

The Greek term abacus refers to a counting board. The calculating device of beads and rods, which is also referred to as an abacus, was developed in Asia and didn’t become known in Europe until the end of the seventeenth beginning of the eighteenth centuries, via Russia, well after the counting board had ceased to be used in Europe.

We get no mention of the astrolabe, which is universally described as an analogue computer,

We then get Napier’s Bones and Oughtred’s slide rule as calculating devices implying that they are somehow related. They aren’t the slide rule is based on logarithms and although Napier invented logarithms his Bones aren’t.

Next up is the Pascaline with, quotes correctly the information that they were difficult to manufacture and so were a flop, although they don’t put it that bluntly. No mention of Wilhelm Schickard, whose calculator was earlier that the Pascaline. Of course, if we have Pascal’s calculator then we must have Leibniz’s, and we get the following hammer statement:

Leibniz’s machine, the Stepped Reckoner, represented a major theoretical advance over the Pascaline in that its calculations were done in binary (base-two) arithmetic (my emphasis), the basis of all modern computer architecture.

When I read that I had a genuine what the fuck moment. There are three possibilities, either our authors are using a truly crappy source on the history of reckoning machines, or they are simply making shit up, or they created a truly bad syllogistic  argument.

1) Leibniz was one of the first European mathematicians to develop binary arithmetic

2) Leibniz created a reckoning machine

Therefore: Leibniz’s reckoning machine must have used binary arithmetic

Of course, the conclusion does not follow from the premises and Leibniz’s Stepped Reckoner used decimal not binary numbers. 

A fourth possibility is a truly cataclysmic typo but our authors repeat the claim at the beginning of the next sketch of that is definitively not the case.

We get a brief respite with the description of a successful stepped reckoner in the nineteenth century, then we move onto Charles Babbage and a total train wreck. Our authors tell us:

Early in the 19th century, Cambridge mathematics professor Charles Babbage began work on a machine for generating accurate logarithmic and astronomical tables.

[…]

By 1822, Babbage was literally cranking out tables with six-figure accuracy on a small machine he called a Difference Engine.

I think our authors live in an alternative universe, only this could explain how Babbage was “cranking out tables with six-figure accuracy” on a machine that was never built! They appear to be confusing the small working model he produced to demonstrate the principles on which the engine was to function with the full engine which was never constructed. They even include a picture of that small working model labelled, Babbage’s Difference Engine.

In 1832, Babbage and Joseph Clement produced a small working model (one-seventh of the plan),  which operated on 6-digit numbers by second-order differences. Lady Byron described seeing the working prototype in 1833: “We both went to see the thinking machine (or so it seems) last Monday. It raised several Nos. to the 2nd and 3rd powers, and extracted the root of a Quadratic equation.” Work on the larger engine was suspended in 1833. (Wikipedia)

1822 was the date that Babbage began work on the Difference Engine.  As to being small:

This first difference engine would have been composed of around 25,000 parts, weighed fifteen short tons (13,600 kg), and would have been 8 ft (2.4 m) tall. (Wikipedia)

Not exactly a desktop computer. 

Our authors continue with the ahistorical fantasies:

In 1801, Joseph-Marie Jacquard had designed a loom that wove complex patterns guided by a series of cards with holes punched in them. Its success in turning out “pre-programmed” patterns led Babbage to try to make a calculating machine that would accept instructions and data from punch cards.  He called the proposed device an Analytical Engine.

In 1801 Jacquard exhibited an earlier loom at the Exposition des produits de l’industrie française and was awarded a bronze medal. He started developing the punch card loom in 1804. The claim about Babbage is truly putting the cart before the horse. Following the death of his wife in 1827, Babbage undertook an extended journey throughout continental Europe, studying and researching all sorts of industrial plant to study and analyse their uses of mechanisation and automation. Something he had already done in the 1820s in Britain. He published the results of this research in his On the Economy of Machinery and Manufactures in 1832. To quote myself, “It would be safe to say that in 1832 Babbage knew more about mechanisation and automation that almost anybody else on the entire planet and what it was capable of doing and which activities could be mechanised and/or automated. It was in this situation that Babbage decided to transfer his main interest from the Difference Engine to developing the concept of the Analytical Engine conceived from the very beginning as a general-purpose computer capable of carrying out everything that could be accomplished by such a machine, far more than just a super number cruncher.” The Jacquard loom was just one of the machines he had studied on his odyssey and he incorporated its concept of punch card programming into the plans for his new computer.

Of course, our authors can’t resit repeating the Ada Lovelace myths and hagiography:

Babbage’s assistant in this undertaking was Augusta Ada Lovelace […] Lovelace translated, clarified and extended a French description of Babbage’s project, adding a large amount of original commentary. She expanded on the idea of “programming” the machine with punched-card instructions and wrote what is considered to be the first significant computer program…

Ada was in no way ever Babbage’s assistant. The notes added to the French description of Babbage’s project were cowritten with Babbage. They did not expand on the idea of “programming” the machine with punched-card instructions. The computer program included in Note G was written by Babbage and not Lovelace and wasn’t the first significant computer program. Babbage had already written several before the Lovelace translation.

Babbage in his autobiography Passages from the Life of a Philosopher (Longmans, 1864)

We now turn to George Boole and the advent of Boolean algebraic logic, which is dealt with extremely briefly but contains the following rather strange comment:

Although Boole reportedly thought his system would never have any practical applications…

I spent several years at university researching the life and work of George Boole and never came across any such claim. In fact, Boole himself applied his logical algebra to probability theory in Laws of Thought, as is stated quite clearly in its full title An Investigation of the Laws of Thought: on Which are Founded the Mathematical Theories of Logic and Probabilities.

After a very brief account of Herman Hollerith’s use of punch cards to accelerate the counting of 1880 US census, we take a big leap to Claude Shannon. 

The pieces started to come together in 1937, when Claude Shannon, in his master’s thesis at M.I.T., combined Boolean algebra with electrical relays and switching circuits to show how machines can “do” mathematical logic.

We get no mention of the fact that Shannon was working on the circuitry of Vannevar Bush’s Differential Analyser, an analogue computer. The Differential Analyser played a highly significant role in the history of the computer as all three, US, war time computers, the ABC, the Harvard Mark I, and the ENIAC, all mentioned by our authors,  were all projects set in motion to create an improved version of the Differential Analyser.

We move onto WWII:

Alan Turing, the mathematician who spearheaded the successful British attempt to break the German U-boat command’s so-called “Enigma code,” designed several electronic machines to help in the crypto analysis.

Turing designed one machine the Bombe based on the existing Polish Bomba, his design was improved by Gordon Welchman and the machine was actually designed and constructed by Harold Keen. We get a brief not totally accurate account of Max Newman, Tommy Flowers and the Colossus, before moving to Germany and Konrad Zuse. Our authors tell us:

Meanwhile in Germany, Konrad Zuse had also built a programmable electronic computer. His work begun in the late 1930s, resulted in a functioning electro-mechanical machine by sometime in the early 1940s, giving him some historical claim to the title of inventory of electronic computers. 

This contains a central contradiction an electro-mechanical computer is not an electronic computer. Also, we have a precise time line for the development of Zuse’s computers. His Z1, a purely mechanical computer was finished in 1938, his Z2, an electro-mechanical computer in 1939. The Z3, which our authors are talking about, another electro-mechanical computer, an improvement on the Z2, was finished in 1941. Our authors erroneously claimed that Leibniz’s steeped reckoner used binary arithmetic but didn’t think it necessary to point out that Zuse was the first to use binary rather than decimal in his computers beginning with the Z1. They also state:

However, wartime secrecy kept his work hidden as well.

Zuse already set up his computer company during the war and went straight into development and production after the war. Although he couldn’t complete construction of his Z4, begun in 1944, until 1949, delivering the finished product to the ETH in Zurich in 1950. 

The Z4 was arguably the world’s first commercial digital computer, and is the oldest surviving programmable computer. (Wikipedia).

Our authors now deliver surprisingly brief accounts of the ABC, the Harvard Mark I, and ENIAC, before delivering the John von Neumann myth.

John von Neuman is generally credited with devising a way to store programs inside a computer.

The stored program computer in question is the EDVAC devised by John Mauchly and J. Presper Eckert, the inventors of ENIAC. Von Neuman merely described their design, without attribution, in his First Draft of a Report on the EDVAC, thereby stealing the credit. 

The sketch ends with a very rapid gallop through the first stored program computers, EDSAC in Britain and UNIVAC I in the US, the introduction of first transistors and then integrated circuitry. 

There is so much good, detailed history of the development of computers that I simply don’t understand how our authors could get so much wrong.

Sketch No. 24 The Arithmetic of ReasoningBoolean Algebra A brief nod to Aristotelian logic and then the repeat of the false claim that Leibniz’s steeped reckoner used binary numbers leads us into Leibniz’s attempts to create a calculus of logic, which however our authors admit remained unpublished and unknown until the twentieth century. All of this is merely a lead in to Augustus De Morgan and George Boole. We get brief, pathetic biographies of both of them emphasising their struggles in life. This leads into a presentation of Boole’s logic, presented in the form of truth tables which didn’t exist till much later! We then get an extraordinary ahistorical statement:

De Morgan, too, was an influential, persuasive proponent of the algebraic treatment of logic. His publications helped to refine, extend, and popularise the system started by Boole.

De Morgan was an outspoken supporter of Boole’s work but , his publications did not help to refine, extend, and popularise the system started by Boole. We do however get De Morgan’s Laws and his logic of relations. The latter gives our authors the chance to wax lyrical about Charles Saunders Pierce. 

Towards the end of this very short sketch, we get the following, “But it was the work of Boole, De Morgan, C. S. Pierce and others…” (my emphasis) That “and others” covers a multitude of omissions, probably the most important being the work of William Stanley Jevons (1835–1882). An oft repeated truisms is that Boole’s algebraic logic is not Boolean algebra! It was first Jevons, who modifying Boole system turned it into the Boolean algebra used by computers today. 

Sketch No. 25 Beyond CountingInfinity and the Theory of Sets Enter stage right George Cantor. We get a reasonable introduction to Cantorian set theory, Kronecker’s objections and the problem of set theory paradoxes. We then get a long digression on nineteenth century neo-Thomism, metaphysics, infinite sets and the Mind of God. Sorry but this has no place in a very short, supposedly elementary introduction to the history of mathematics, “designed for students just beginning their study of the discipline.”

Sketch No. 26 Out of the ShadowsThe Tangent Function is a reasonable account of the gradual inclusion of the tangent function into the trigonometrical canon. My only criticism is although they correctly state that Regiomontanus introduced the tangent function in his Tabulae directionum profectionumque written in 1467. Our authors only give the first two words of the title and translate it as Table of Directions. This is not incorrect but left so probably leads to misconceptions. Directions here does not refer to finding ones way geographical but to a method used in astrology to determine from a birth horoscope the major events, including death, in the subject life. This require complex spherical trigonometrical calculation transferring points from one system of celestial coordinates to another system. Hence the need for tangents.

Sketch No. 27 Counting RatiosLogarithms A detailed introduction to the history of the invention of logarithms. In general, OK but a bit off the rails on the story of Jost Bürgi. 

As Napier and Briggs worked in Scotland, the Thirty Years’ War was spreading misery on the European continent. One of its side effects was the loss of most copies of a 1620 publication by Joost Bürgi, a Swiss clockmaker. Bürgi had discovered the basic principles of logarithms while assisting the astronomer Johannes Kepler in Prague in 1588, some years before Napier, but his book of tables was not published until six years after Napier’s Descriptio appeared. When most of the copies disappeared, Bürgi’s work faded into obscurity.

The failure of Bürgi’s work to make an impact almost certainly had to do with the facts that only a very small number were ever printed and it only consisted of a table of what we now call anti-logarithms with no explanation of how they were created or how to use them.  Also, in 1588 Kepler was still living and working in Graz and Bürgi was living and working in Kassel. Kepler first moved to Prague in 1600 and Bürgi in 1604. The reference to 1588 in Bürgi’s work is almost certainly to prosthaphaeresis, a method of using trigonometrical formulars to turn difficult multiplications and divisions into addition and subtractions which Bürgi had learnt from Paul Wittich, and not to logarithms. His work on logarithms almost certainly started later than that of Napier but was independent. Interestingly it is thought that Napier was led to logarithms after being taught  prosthaphaeresis by John Craig, who had also learnt it from Wittich

Sketch No. 28 Anyway You Slice ItConic Sections What is actually quite a good section on the history of conic sections is spoilt by one minor error and a cluster fuck concerning Kepler. The small error concerns Witelo (c.1230–after 1280), our authors write:

There is some evidence that Apollonius’s Conics was known in Europe as far back as the 113th century, when Erazmus Witelo used conics in his book on optics and perspective.

Witelo’s book is titled De Perspectiva and is purely a book on optics not perspective. Perspectivist optics is a school of optical theory derived from the optics of Ibn al-Haytham, whose book Kitāb al-Manāẓir is titled De Perspectiva in Latin translation. On to Kepler:

Sorry about the scans but I couldn’t be arsed to type out the whole section.

The camera obscura that Kepler assembled on the market place in Graz in the summer of 1600 was a small tent and not “a large wooden structure.” It was a pin hole camera and not a kind of large-size one. 

The pin hole camera problem, the image created is larger than it should be, had been known since the Middle Ages, was widely discussed in the literature, and Kepler had been expecting it before he began his observations.  His attention had been drawn to the problem by Tycho when he was in Prague at the beginning of 1600 to negotiate his move there to work with Tycho. Tycho had also drawn his attention to the problem of atmospheric refraction in the astronomy. These were the triggers that motivated his studies in optics. As noted he moved to Prague to work with Tycho but there was no Imperial Observatory in Prague. 

Kepler inherited Tycho’s position of Imperial Mathematicus but not his observation, which were inherited by his daughter. This led to several years of difficult negotiations before he obtained permission to publish his work based on those observations. Tycho had set Kepler on the problem of calculating the orbit of Mars, the work that led to his first two planetary laws, before his death.

The book that Kepler used for his optical studies and on which he based his own work was Friedrich Risner’s “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), which was published in 1572.

“Of course he thought of the conic sections!” Actually, when Kepler first realised that the orbit of Mars was some sort of oval, he didn’t think of the conic section at all. Something for which he criticised himself in his Astronomia Nova, when he finally realised, after much more wasted effort, that the orbit was actually an ellipse. A realisation not based on more measurements. The Astronomia Nova from 1609 only contains the first two planetary laws. The third was first published in his Harmonice Mundi in 1619.

 Sketch No. 29 Beyond the PaleIrrational Numbers Once again an acceptable historical account. 

Sketch No. 30 Barely TouchingFrom Tangents to Derivatives Not perfect but more than reasonable

The book ends with a What to Read Next which begins with The Reference Shelf. Here they recommend several one volume histories of mathematics starting with Victor J Katz and then moving onto Howard Eves’s An Introduction to the History of Mathematics and David M . Burton’s The History of Mathematics which they admit are both showing their age. They move on to lot more recommendations but studiously ignore, what was certainly the market leader before Katz, Carl B. Boyer, A History of Mathematics in the third edition edited by Uta Merzbach. I seriously wonder what they have against Boyer whose books on the history of mathematics are excellent. 

Having covered a fairly wide range of reference books, including a lot of Grattan-Guinness we now move onto Twelve Historical Books You Ought to Read. Of course, any such recommendation is subjective but some of their choices are more that questionable. They start, for example, with Tobias Dantzig’s Number the Language of Science. This was first published in 1930 and is definitively severely dated. 

Our authors also recommend Reviel Netz & William Noel, The Archimedes Codex (2007). This book was the object of an incredible media hype when it first appeared. Whilst the imaging techniques used to expose the Archimedean manuscript on the palimpsest are truly fascinating the claims made by the authors about the newly won knowledge about Archimedes works are extremely hyperbolic and contain more than a little bullshit. Should this really be on a list of twelve history of maths books one ought to read?

I did a double take when I saw that they recommend Eric Templer Bell’s Men of Mathematics. They themselves say:

The book has lost some of its original popularity, not (or at least not primarily) because of its politically incorrect, but rather because Bell takes too many liberties with his sources. (Some critics would say “because he makes things up.”) The book is fun to read, but don’t rely solely on Bell for facts.

My take don’t waste your time reading this crap book. I read it when I was sixteen and spent at least twenty years unlearning all the straight forward lies that Bell spews out!

Another disaster recommendation is Dava Sobel’s Longitude, which our authors say, “provides a good picture of the interactions among mathematics, astronomy, and navigation in the 18th century.” Sobel’s much hyped book give an extremely distorted “picture of the interactions among mathematics, astronomy, and navigation in the 18th century,” which at times is closer to a fantasy novel than a serios history book. If you really want to learn the true facts about  those interactions then read Richard Dunn & Rebekah Higgitt, Finding LongitudeHow ships, clocks and stars helped solve the longitude problem (Collins, 2014) and Katy Barrett, Looking for LongitudeA Cultural History (Liverpool University Press), both volumes are written by professional historians, who spent several years researching the topic in a major research project. 

The section closes with History Online, which can’t be any good because it doesn’t include The Renaissance Mathematicus! 🙃

The apparatus begins with a useful When They Lived, which simply lists lots of the most well-known mathematicians alphabetically with their birth and death dates. This is followed by the bibliography, already mentioned above, and a good index. There are numerous, small, black and white illustration scattered throughout the pages.

Despite my negative comment about some points in the book, I would actually recommend it as a reasonably priced, mostly accurate, introduction to the history of mathematics. A small criticism is it does at times display a US American bias, but it was written primarily for American students. A good jumping off point for somebody developing an interest in the discipline. In any case considerably better that my jumping off point, E. T. Bell!

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