God made all things by measure, number and weight
The first history of science, history of mathematics book I ever read was Lancelot Hogben’s Man Must Measure: The Wonderful World of Mathematics, when I was about six years old.
It almost certainly belonged to my older brother, who was six years older than I. This didn’t matter, everybody in our house had books and everybody could and did read everybody’s books. We were a household of readers. I got my first library card at three; there were weekly family excursions to the village library. But I digress.
It is seldom, when people discuss the history of mathematics for them to think about how or where it all begins. It begins with questions like how much? How many? How big? How small? How long? How short? How far? How near? All of these questions imply counting, comparison, and measurement. The need to quantify, to measure lies at the beginning of all systems of mathematics. The histories of mathematics, science, and technology all have a strong stream of mensuration, i.e., the act or process of measuring, running through them. Basically, without measurement they wouldn’t exist.
Throughout history measuring and measurement have also played a significant role in politics, often leading to political disputes. In modern history there have been at least three well known cases. The original introduction of the metric system during the French revolution, the battle of the systems, metric contra imperialism, during the nineteenth century, and most recently the bizarre wish of the supporters of Brexit to reintroduce the imperial system into the UK in their desire to distance themselves as far as possible from the evil EU.
It was with some anticipation that I greeted the news that James Vincent had written and published Beyond Measure: The Hidden History of Measurement. Vincent’s book is not actually a history of measurement on a nuts and bolts level i.e., systems of measurement, units of measurement and so on, but what I would call a social history of the uses of measurement. This is not a negative judgement; some parts of the book are excellent exactly because it is about the use and abuse of methods of measurement rather than the systems of measurement themselves.
Although roughly chronological, the book is not a systematic treatment of the use of measurement from the first group of hunter gatherers, who tried to work out an equitable method of dividing the spoils down to the recent redefinition of the kilogram in the metric system. The latter being apparently the episode that stimulated Vincent into writing his book. Such a volume would have to be encyclopaedic in scope, but is rather an episodic examination of various passages in the history of mensuration.
The first episode or chapter takes a rather sweeping look at what the author sees as the origins of measurement in the early civilisations of Egypt and Babylon. Whilst OK in and of itself, what about other cultures, civilisations, such as China or India just to mention the most obvious. This emphasises something that was already clear from the introduction this is the usual predominantly Eurocentric take on history.
The second chapter moves into the realm of politics and the role that measurement has always played in social order, with examples from all over the historical landscape. Measurement as a tool of political control. This demonstrates one of the strengths of Vincent’s socio-political approach. Particularly, his detailed analysis of how farmers, millers, and tax collectors all used different tricks to their advantage when measuring grain and the regulation that as a result were introduced is fascinating.
Vincent is, however, a journalist and not a historian and is working from secondary sources and in the introduction, we get the first of a series of really bad takes on the history of science that show Vincent relying on myths and clichés rather than doing proper research. He delivers up the following mess:
Consider, for example, the unlikely patron saint of patient measurement that is the sixteenth-century Danish nobleman Tycho Brahe. By most accounts Brahe was an eccentric, possessed of a huge fortune (his uncle Jørgen Brahe was one of the wealthiest men in the country), a metal nose (he lost the original in a duel), and a pet elk (which allegedly died after drinking too much beer and falling down the stairs of one of his castles). After witnessing the appearance of a new star in the night sky in 1572, one of the handful of supernovae ever seen in our galaxy, Brahe devoted himself to astronomy.
Tycho’s astronomical work was financed with his apanage from the Danish Crown, as a member of the aristocratical oligarchy that ruled Denmark. His uncle Jørgen, Vice-Admiral of the Danish navy, was not wealthier than Tycho’s father or his independently wealthy mother. Tycho had been actively interested in astronomy since 1560 and a serious astronomer since 1563, not first after observing the 1572 supernova.
After describing Tycho’s observational activities, Vincent writes:
It was the data collected here that would allow Brahe’s apprentice, the visionary German astronomer Johannes Kepler, to derive the first mathematical laws of planetary motion which correctly described the elliptical orbits of the planets…
I don’t know why people can’t get Kepler’s status in Prague right. He was not Tycho’s apprentice. He was thirty years old, a university graduate, who had studied under Michael Mästlin one of the leading astronomers in Europe. He was the author of a complex book on mathematical astronomy, which is why Tycho wanted to employ him. He was Tycho’s colleague, who succeeded him in his office as Imperial Mathematicus.
It might seem that I’m nit picking but if Vincent can’t get simple history of science facts right that he could look up on Wikipedia, then why should the reader place any faith in the rest of what he writes?
The third chapter launches its way into the so-called scientific revolution under the title, The Proper Subject of Measurement. Here Vincent selectively presents the Middle Ages in the worst possible anti-science light, although he does give a nod to the Oxford Calculatores but of course criticises them for being purely theoretical and not experimental. In Vincent’s version they have no predecessors, Philoponus or the Arabic scholars, and no successors, the Paris physicists. He then moves into the Renaissance in a section titled Measuring art, music, and time. First, we get a brief section on the introduction of linear perspective. Here Vincent, first, quoting Alberti, tells us:
I set this up between the eye and the object to be represented, so that the visual pyramid passes through the loose weave of the veil.
The ‘visual pyramid’ described by Alberti refers to medieval theories of optics. Prior to the thirteenth century, Western thinkers believed that vision worked via ‘extramission,’ with the eye emitting rays that interacted with the world like a ‘visual finger reaching out to palpated things’ (a mechanism captured by the Shakespearean imperative to ‘see feelingly’). Thanks largely to the work of the eleventh-century Arabic scholar Ibn al-Haytham, known in the West as Alhacen, this was succeeded by an ‘intromisionist’ explanation, which reverses the causality so that it is the eye that receives impressions from reality. It’s believed that these theories informed the work of artists like Alberti, encouraging the geometrical techniques of the perspective grids and creating a new incentive to divide the world into spatially abstract units.
Here, once again, we have Vincent perpetuating myths because he hasn’t done his homework. The visual pyramid is, of course, from Euclid and like the work of the other Greek promoters of geometrical optics was indeed based on an extramission theory of vision. As I have pointed out on numerous occasions the Greeks actually had both extramission and intromission theories of vision, as well as mixed models. Al-Haytham’s great achievement was not the introduction of an intromission theory, but was in showing that an intromission theory was compatible with the geometrical optics, inclusive visual pyramid, of Euclid et al. The geometrical optics of Alberti and other perspective theorists is pure Euclidian and does not reference al-Haytham. In fact, Alberti explicitly states that it is irrelevant whether the user of his system of linear perspective believes in an extramission or an intromission theory of vision.
Linear perspective is followed by a two page romp through the medieval invention of musical notation before turning to the invention of the mechanical clock. Here, Vincent makes the standard error of over emphasising the influence of the mechanical clock in the early centuries after its invention and introduction.
Without mentioning Thomas Kuhn, we now get a Kuhnian explanation of the so-called astronomical revolution, which is wonderfully or should that be horrifyingly anachronistic:
This model [the Aristotelian geocentric one] sustained its authority for centuries, but close observation of the night skies using increasingly accurate telescopes [my emphasis] revealed discrepancies. These were changes that belied its immutable status and movements that didn’t fit the predictions of a simple geocentric universe. A lot of work was done to make the older models account for such eccentricities, but as they accrued mathematical like sticky notes, [apparently sticky notes are the 21st century version of Kuhnian ‘circles upon circles’] doubts about their veracity became unavoidable.
Where to begin with what can only be described as a clusterfuck. The attempts to reform the Aristotelian-Ptolemaic geocentric model began at the latest with the first Viennese School of Mathematics in the middle of the fifteenth century, about one hundred and fifty years before the invention of the telescope. Those reform attempts began not because of any planetary problems with the model but because the data that it delivered was inaccurate. Major contributions to the development of a heliocentric model such as the work of Copernicus and Tycho Brahe, as well as Kepler’s first two laws of planetary motion also all predate the invention of the telescope. Kepler’s third law is also derived from pre-telescope data. The implication that the geocentric model collapsed under the weight of ad hoc explanation (the sticky notes) was Kuhn’s explanation for his infamous paradigm change and is quite simply wrong. I wrote 52 blog posts explaining what really happened, I’m not going to repeat myself here.
We now get the usual Galileo hagiography for example Vincent tells us:
It was Galileo who truly mathematised motion following the early attempts of the Oxford Calculators, using practical experiments to demonstrate flaws in Aristotelian wisdom.
Vincent ignores the fact that Aristotle’s concepts of motion had been thrown overboard long before and completely ignores the work of sixteenth century mathematicians, such as Tartaglia and Benedetti.
He then writes:
In one famous experiment he dropped cannonballs and musket balls from the Leaning Tower of Pisa (an exercise that likely never took place in the way Galileo claims [my emphasis]) and showed that, contra to Aristotle, objects accelerate at a uniform rate, not proportionally to their mass.
Galileo never claimed to have dropped anything from the Leaning Tower, somebody else said that he had and if it ‘never took place’, why fucking mention it?
Now the telescope:
From 1609, Galileo’s work moved to a new plane itself. Using home-made telescopes he’d constructed solely by reading descriptions of the device…
The myth, created by himself, that Galileo had never seen a telescope before he constructed one has been effectively debunked by Mario Biagioli. This is followed by the usually one man circus claims about the telescopic discoveries, completely ignoring the other early telescope observers. Copernicus and Kepler now each get a couple of lines before we head off to Isaac Newton. Vincent tells us that Newton devised the three laws of motion and the universal law of gravitation. He didn’t he took them from others and combined them to create his synthesis.
The fourth chapter of the book is concerned with the invention of the thermometer and the problems of creating accurate temperature scales. This chapter is OK, however, Vincent is a journalist and not a historian and relies on secondary sources written by historians. There is nothing wrong with this, it’s how I write my blog posts. In this chapter his source is the excellent work of Hasok Chang, which I’ve read myself and if any reader in really interested in this topic, I recommend that they read Chang rather than Chang filtered by Vincent. Once again, we have some very sloppy pieces of history of science, Vincent writes:
Writing in 1693, the English astronomer Edmond Halley, discoverer of the eponymous comet…”
Just for the record, Halley was much more than just an astronomer, he could for example have been featured along with Graunt in chapter seven, see below. It is wrong to credit Halley with the discovery of Comet Halley. The discoverer is the first person to observe a comet and record that observation. Comet Halley had been observed and recorded many times throughout history and Halley’s achievement was to recognise that those observations were all of one and the same comet.
A few pages further on Vincent writes:
Unlike caloric, phlogiston had mass, but Lavoisier disproved this theory, in part by showing how some substances gain weight when burned. (This would eventually lead to the discovery of oxygen as the key element in combustion.) [my emphasis]
I can hear both Carl Scheele and Joseph Priestley turning in their graves. Both of them discovered oxygen, independently of each other; Scheele discovered it first bur Priestly published first, and both were very much aware of its role in combustion and all of this well before Lavoisier became involved.
Chapter five is dedicated to the introduction of the metric system in France correctly giving more attention to the political aspects than the numerical ones. Once again, an excellent chapter.
Chapter six which deals primarily with land surveying had a grandiose title, A Grid Laid Across the World, but is in fact largely limited to the US. Having said that it is a very good and informative chapter, which explains how it came about that the majority of US towns and properties are laid out of a unified rectangular grid system. Most importantly it explains how the land grant systems with its mathematical surveying was utilised to deprive the indigenous population of their traditional territories. The chapter closes with a brief more general look at how mathematical surveying and mapping played a significant role in imperialist expansion, with a very trenchant quote from map historian, Matthew Edney, “The empire exists because it can be mapped; the meaning of empire is inscribed into each map.”
Unfortunately, this chapter also contains some more sloppy history of science, Vincent tells us:
In such a world, measurement of the land was of the utmost importance. As a result, sixteenth-century England gave rise to one of the most widely used measuring tools in the world: the surveyor’s chain, or Gunter’s chain, named after its inventor the seventeenth-century English priest and mathematician Edmund Gunter.
Sixteenth or seventeenth century? Which copy editor missed that one? It’s actually a bit of a problem because Gunter’s life starts in the one century and ends in the other, 1581–1626. However, we can safely say that he produced his chain in the seventeenth century. Vincent makes the classic error of attributing the invention of the surveyors’ chain to Gunter, to quote myself from my blog post on Renaissance surveying:
In English the surveyor’s chain is usually referred to as Gunter’s chain after the English practical mathematician Edmund Gunter (1581–1626) and he is also often referred to erroneously as the inventor of the surveyor’s chain but there are references to the use of the surveyor’s chain in 1579, before Gunter was born.
Even worse he writes:
Political theorist Hannah Arendt described the work of surveying and mapping that began with the colonisation of America as one of three great events that ‘stand at the threshold of the modern age and determine its character’ (the other two being the Reformation of the Catholic Church and the cosmological revolution begun by Galileo) [my emphasis]
I don’t know whether to attribute this arrant nonsense to Arendt or to Vincent. Whether he is quoting her or made this up himself he should know better, it’s complete bullshit. Whatever Galileo contributed to the ‘cosmological revolution,’ and it’s much, much less than is often claimed, he did not in anyway begin it. Never heard of Copernicus, Tycho, Kepler, Mr Vincent? Oh yes, you talk about them in chapter three!
Chapter seven turns to population statistics starting with the Royal Society and John Graunt’s Natural and Political Observations Made Upon the Bills of Mortality. Having dealt quite extensively with Graunt, with a nod to William Petty, but completely ignoring the work of Caspar Neumann and Edmond Halley, Vincent now gives a brief account of the origins of the new statistics. Strangely attributing this entirely to the astronomers, completely ignoring the work on probability in games of chance by Cardano, Fermat, Pascal, and Christian Huygens. He briefly mentions the work of Abraham de Moivre but ignores the equally important, if not more important work of Jacob Bernoulli. He now gives an extensive analysis of Quetelet’s application of statistics to the social sciences. Quetelet, being an astronomer, is Vincent’s reason d’être for claiming that it was astronomers, who initial developed statistics and not the gamblers. Quetelet’s the man who gave us the ubiquitous body mass index. The chapter then closes with a good section on the abuses of statistics in the social sciences, first in Galton’s eugenics and secondly in the misuse of IQ tests by Henry Goddard. All in all, one of the good essays in the book
Continuing the somewhat erratic course from theme to theme, the eighth chapter addresses what Vincent calls The Battle of the Standards: Metric vs Imperial and metrology’s culture war. A very thin chapter, more of a sketch that an in-depth analysis, which gives as much space to the post Brexit anti-metric loonies, as to the major debates of the nineteenth century. This is mainly so that Vincent can tell the tale of his excursion with said loonies to deface street signs as an act of rebellion.
In the ninth chapter, Vincent turns his attention to replacement of arbitrary definitions of units of measurement with definitions based on constants of nature, with an emphasis on the recent new definition of the kilogram. At various point in the book, Vincent steps out from his role of playing historian and presents himself in the first person as an investigative journalist, a device that I personally found irritating. In this chapter this is most pronounced. He opens with, “On a damp but cheerful Friday in November 2018, I travelled to the outskirts of Paris to witness the overthrow of a king.” He carries on in the same overblown style finally revealing that he, as a journalist was attending the conference officially launching the redefining of the kilogram, going on to explain in equally overblown terms how the kilogram was originally defined. The purple prose continues with the introduction of another attendee, his acquaintance, the German physicist, Stephan Schlamminger:
Schlamminger is something of a genius loci of metrology: an animating spirit full of cheer and knowledge, as comfortable in the weights and measures as a fire in a heath. He is also a key player in the American team that helped create the kilogram’s new definition. I’d spoken to him before, but always delighted in his enthusiasm and generosity. ‘James, James, James,’ he says in a rapid-fire German accent as he beckoned me to join his group. ‘Welcome to the party.’
We then get a long, overblown speech by Schlamminger about the history of the definitions in the metric system ending with an explanation, as to why the kilogram must be redefined.
This is followed by a long discourse over Charles Sanders Peirce and his attempts to define the metre using the speed of light, which failed. Vincent claims that Peirce was the first to attempt to attempt to define units of measurement using constants of nature, a claim that I find dubious, but it might be right. This leads on to Michelson and Morley defining the metre using the wavelength of sodium light, a definition that in modified form is still used today. The chapter closes with a long, very technical, and rather opaque explanation of the new definition of the kilogram based on Planck’s constant, h.
The final chapter of Vincent’s book is a sociological or anthropological mixed basket of wares under the title The Managed Life: Measurements place in modern society in our understanding of ourselves, which is far too short to in anyway fulfil its grandiose title.
The book closes with an epilogue that left me simply baffled. He tells a personal story about how he came to listen to Beethoven’s Ninth Symphony only when he had a personal success in his life and through this came to ruin his enjoyment of the piece. Despite his explanation I fail to see what the fuck this has to do with measurement.
The book has a rather small, random collection of colour prints, related to various bits of the text, in the middle. There are extensive endnotes relating bits of the text to there bibliographical sources, but no separate bibliography, and an extensive index.
I came away feeling that there is a good book contained in Vincent’s tome, struggling to get out. However, there is somehow too much in the way for it to emerge. Some of the individual essays are excellent and I particularly liked his strong emphasis on some of the negative results of applying systems of measurement. People reading this review might think that I, as a historian of science, have placed too much emphasis on his truly shoddy treatment of that discipline; ‘the cosmological revolution begun by Galileo,’ I ask you? However, as I have already stated if we can’t trust his research in this area, how much can we trust the rest of his work?
 Wisdom of Solomon 11:20
 James Vincent, Beyond Measure: The Hidden History of Measurement, Faber & Faber, London, 2022