One of my interests as a historian of practical mathematics is the history of the invention of triangulation and its applications in both cartography and geodesy, a subject on which I have, in the meantime, read a small library of academic books and papers. Now, at first glance, it would seem that the measuring of large triangles across the landscape is not the sort of exciting scientific endeavour that would inspire an author to put pen to paper or fingers to keyboard to produce a work of popular science history, however appearances can be deceptive. I possess five popular historical volumes on the application of triangulation to the solution of geodetic and cartographic problems, of which I have read the first three They are John Keay, *The Great Arc: The Dramatic Tale of How India was Mapped and Everest was Named*, HarperCollins, 2000, Ken Alder, *The Measure of All Things: The Seven Year Odyssey and Hidden Error That Transformed the World*, Free Press, 2002, Rachel Hewitt, *Map of a Nation: A Biography of the Ordnance Survey*, Granta Books, 2010, Larrie D. Ferreiro, *Measure of the Earth: The Enlightenment Expedition that Reshaped the World*, Basic Books, 2011 and Paul Murdin, *Full Meridian of Glory: Perilous Adventures in the Competition to Measure the Earth*, Copernicus Books, 2009. The first, as the title says, deals with the measurement of the meridian arc in India in the 19^{th} century and the resulting mapping of this sub-continent. The second deals with the re-measurement of the meridian arc in France to determine the basic unit of the metre for the metric system in the late 18^{th} century. The third is self-explanatory but also includes the measurement of a meridian arc in Britain. The last two deal with the history of the measurement, mostly by the French, of meridian arcs in France, Peru and Lapland to determine whether the earth is a prolate or an oblate spheroid, a story that I have already blogged about.

Given the fact that triangulation is a very major, or even the major, player in all of these books one could as reader expect the authors to have researched and in their respective books to explain the historical origins of this technique or methodology. This expectation is not really fulfilled. Let us examine what each of them has to say on the subject.

Paul Murdin writes the following:

[Willebrod] *Snell (sic) [1] had proposed the technique of triangulation as a way to measure the relative locations of points on the surface of the Earth (Smith 1986). The method, which is still the standard one, starts by the surveyors measuring a reference line between two stakes on a flat plain using standard sized sticks or chains (Fig. 7). A third stake is placed somewhere else on the plain at a significant location. From each end of the line, a surveyor sights the stake at the other end as well as the third stake, measuring the angle between with a theodolite.* [emphasis in original]

*The third stake is located relative to the other two by trigonometry of the triangle. From the reference points the surveyor can also sight natural vertical features in the landscape.*[…]

*All these features can be located relative to the others building a chain of triangles across the country – hence the term triangulation*(Murdin, p. 15)

*Snell’s techniques were demonstrated by relatively small-scale projects by him and other Dutch cartographers. The first survey by triangulation is regarded to be Snell’s survey prior to 1615 of the Netherlands from Alkmaar via Amsterdam, Leiden, Utrecht, Dordrecht and Breda to Bergen-op-Zoom, accumulating an overall distance of 120 km. *(Murdin p. 17)

Although he doesn’t actually directly say so Murdin implies that the technique of triangulation was invented by Snel; it wasn’t. Although Snel was almost certainly the first to measure a meridian arc using triangulation he certainly wasn’t the first to carry out a triangulation survey. I have included so much of the first quote because it saves me the bother of explaining the basic principles of triangulation. Let see whether Ferreiro does any better.

Writing about The Academy of Science’s proposal to extend Picard’s first triangulation survey in the late 17^{th} century he writes:

*The method for carrying out long-distance surveys had been around for several centuries and used a basic Euclidean premise: Given two angles of a triangle and the length of one side, the remaining sides and angles can be computed. This principle can be employed to measure over long distances by establishing a geodesic chain of triangles between two fixed points.*

* *Here nobody in credited with the invention and the couple of lines quoted contain a number of serious errors. The method had not been around for several centuries but was, as we will see, invented in the first half of the 16^{th} century. It does not use a basic Euclidean premise. Given two angles and one side of a triangle one can construct the triangle using Euclidean geometry but if you wish to compute the rest of the triangle, and the subsequent chain of triangles, which is indeed what is done in triangulation, you need trigonometry, which you will search for in vain in the *Elements* of Euclid. The oldest surviving source for trigonometry is Ptolemaeus’ *Syntaxis Mathematiké* from the middle of the 2^{nd} century CE more than 400 years later than Euclid. The trigonometry used in the Early Modern Period for triangulation came into Europe from India via the Islamic Empire during the High Middle Ages.

How do our other authors ~~fair~~ fare? Alder like Murdin plumps incorrectly for Snel, also misspelling his name:

*The modern technique for using triangles to measure earthly distances, however, was introduced in 1617 by Willebrod Snell, “The Dutch Eratosthenes,” on the frozen fields outside Leyden, and his *[my emphasis]

*method persisted for the next 360 years.*

To be fair to both Murdin and Alder many academic sources, that should know better, attribute the invention of triangulation to Snel.

Keay ignores the subject completely giving a brief description of the principles of the technique but wasting no thoughts on its origins. Of our five authors only Hewitt gets the attribution right, she writes:

*Triangulation had first emerged as a map-making method in the mid sixteenth century when the Flemish mathematician Gemma Frisius set out the idea in his Libellus de locorum describendum ratione (Booklet concerning a way of describing places),…*

So who was Gemma Frisius?

Gemma Frisius 17th C woodcut

E. de Boulonois

He was born Jemma Reinierzoon or Jemma son of Reinier to poor parents in Dokkum in Friesland on 9^{th} Dec 1508. His nom de plume Gemma Frisius is a Latinised onomatopoeic version of his birth name plus the toponym Frisius for Friesland. His parents died whilst he was still very young and he moved to relatives in Groningen, where he was educated in a cloister school. On 26^{th} February ~~1926~~ 1526 he matriculated as a poor scholar at the University of Louvain where he graduated Master of Arts in 1528. In 1529 there followed one of the strange unexplained episodes in the history of science. In ~~1924~~ 1524 the then young German graduate of the University of Vienna Peter Apian published his *Cosmographia*, a textbook on astronomy, astrology, surveying, cartography and other aspects of applied astronomy. Apian not only wrote this book but printed and published himself, as part of his efforts to establish himself as a scientific publisher. In 1529 a second improved edition of Apian’s *Cosmographia *was published but not by Apian, the second and all the subsequent highly successful extended and improved thirty-two editions of this book, in many different languages, were edited by Gemma Frisius. This was one of the most successful mathematical textbooks published in the sixteenth century and it is not known why Gemma and not Apian edited and published all but the original edition.

Following up on his edition of Apian’s *Cosmographia* Gemma Frisius began to make printed terrestrial and celestial globes moving the tradition of their manufacture from Nürnberg to the northern Germanic area and through his pupil Gerard Mercator into Holland, where the Dutch would dominate the European globe making industry for most of the seventeenth century. He also set up as an instrument maker and through his own efforts and those of his nephews, the Arsenius brothers, Louvain became a major centre for high quality mathematical instruments.

In ~~1934~~ 1534 he married and started to study medicine, graduating MA in 1536, which allowed him to practice medicine, going on to obtain his MD in 1541. At some point he became Professor of Medicine at the University of Louvain. During his medical studies he famously helped his fellow medical student Andreas Vesalius to steal a corpse from the gallows to conduct a bit of illicit dissection. Vesalius would of course go on to publish the most famous anatomy book of all times, his *De fabrica*, in 1543, in which he praises Gemma as a doctor and a mathematician.

Teaching medicine at the university, Gemma only taught mathematics privately producing in his time several famous pupils. I have already mentioned Gerard Mercator possibly the most well know of Gemma’s students but amongst other names known only to historians of mathematics and astronomy can be found the name of John Dee who came to Louvain after graduating at Cambridge to study at the feet of the master.

Gemma published nothing on medicine but lots of works on mathematics including, alongside the *Cosmographia*, the most successful arithmetic textbook of the sixteenth century. His *Radio astronomico*, a handbook on a new form of cross-staff of his own invention, published in 1545, contains the earliest printed, largely positive, discussion of Copernicus’ *De revolutionibus*.

Gemma was very successful and highly respected in his own lifetime and was one of the leading mathematical practitioners of Europe when he died of kidney stones on 25^{th} May 1555 not yet 47 years old.

Many of his innovations in the mathematical sciences were published as appendices to his various editions of the *Cosmographia* and it is here attached to the 1533 third edition (Gemma’s second) that we find the *Libellus de locorum describendum ratione*, his pamphlet outlining completely and in detail the technique of triangulation.

Gemma himself did not enjoy good health and was a thinker and not a doer so he almost certainly didn’t carry out any surveying work himself. We do know that Mercator used Gemma’s method when he surveyed the Duchy of Lorraine later in the century. Tycho Brahe who knew Gemma’s work well being a customer of his instrument workshop also conducted a survey of his island of Hven using Gemma’s triangulation, which was never actually finished. Willebrod Snel would also have been well acquainted with Gemma’s work and it is almost certainly the *Libellus* that was the source of his knowledge of triangulation.

Some sources claim rather vaguely that triangulation was acquired by the Europeans from the Arab mathematicians during the Renaissance but fail to give any source for these claims or to reference any Arabic works on the subject. More directly some sources claim that the great Islamic scholar al-Biruni, who wrote extensively on geography and geodesy, used triangulation. This claim is simply false. He used geometrical methods to determine the longitude and latitude of various cities but his calculations did not just use triangles and he had no measured base line and made no sightings. He merely constructed geometrical models of the positions of the towns respective to each other based on travellers’ tales of the scale of their separations.

Historically there is very little doubt that the technique of triangulation emerged once and once only in a pamphlet written and published by Gemma Frisius in 1533. It is a strange fact that relatively insignificant scientific discoveries and inventions proudly carry the names of their discoverers and inventors but most people, including the people who write books about it, never stop to consider who invented triangulation, which until the invention of GPS, was the only tool, and a very powerful one, capable of producing accurate maps with their incredible economic, political, military and scientific significance. Gemma Frisius belongs in the pantheon of great modern scholars for his invention and not forgotten and ignored even by those who earn money writing about the incredible applications that this invention made possible.

* *

[1] Snel is written with only one ‘l’, a common mistake in English texts. In his Latin publications Snel Latinised his name, as had his father also an academic mathematician, as Snellius. English authors translating back into the vernacular made the mistake of retaining the second ‘l’.

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Ah Gemma. Always felt a bit sorry that such a shameless self-publicist should be so little known today. A little ditty at the end of his arithmetic textbook (quoting from a 1540 Antwerp edition) reads “Gemma is my author. Who does not know Gemma? How strong is his wisdom and skill.”

Actually, his textbook isn’t great and I recommend Tunstall to any sixteenth century students I meet. Nor am I convinced he invented triangulation although he was doubtless the first into print. Which was typical of him really.

If he didn’t who did and what is your proof for that?

A handy summary, the misspelling of Snel possibly comes via the physicists who will invariably refer to “Snell’s” law. I’d heard of Frisius having read a large fraction of the books you mention at the top (and also Andrew Taylor’s biography of Mercator and one of Gauss) but I seem to recall finding out about Frisius via wikipedia rather than the books.

Crowdsource proofreading: I’m sure he didn’t get married in 1934 😉

I’m sure he didn’t get married in 1934What makes you think that?

There are several typos in the article where dates which should be 15XX are given as 19XX. Just search for 19 in the text

“How do our other authors fair?”

should be ‘fare’.

I enjoy reading your blog. thanks!

Thx! The collective spell checker is functioning well.

A bit off topic, but as you mentioned John Dee, I thought I’d share this quote from The Queen’s Agent: Francis Walsingham at the Court of Elizabeth I:

“Licensing her subjects to occupy America in her name could prove to be strategically useful; it might even be profitable. But was it legal? For an answer based on something more than patriotic enthusiasm, she turned to John Dee. Elizabeth had faith in Dee, who had studied the heavens to cast the best day for her coronation and kept her up to date with developments in natural philosophy. An eclectic education and a magpie mind had given Dee some understanding of Roman law, codified by the emperor Justinian during the sixth century and theoretically still regulating international relations in sixteenth-century Europe. If English settlement in America were to go ahead, it was vital that foreign powers-most obviously, Philip of Spain – couldn’t use it as a pretext for war.

“Dee’s solution, set out in a series of treatises and explained in personal audiences with Elizabeth, Burghley, and Walsingham, was a palimpsest of Roman law and Arthurian myth which only he could have come up with. When Francis Drake returned from his circumnavigation laden with treasure looted from South America, sparking a stern protest from the Spanish ambassador, Dee had his answer ready. As the descendant of King Arthur, Elizabeth had a prior claim to North America, which Dee called Atlantis, that predated Columbus’s discoveries by a thousand years. Perhaps aware that not everyone would share his belief in Arthur ‘s exploits, Dee buttressed his case by appealing to ancient legal precedent. Spain may have asserted her sovereignty over the northern parts of America, but she had done nothing to occupy the land; and physical occupation, under Roman law, was an essential part of establishing legal title.”

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Nice blog. Don’t know why you call him Flemish, when born in Dokkum. The period frisians were settled in Flanders was then long ended

I didn’t it’s a quote!

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Thank you for a very interesting article debunking a few myths. I feel however I must inform you that Snel’s first name is spelled “Willebrord” – not “Willebord” – on the memorial plaque where his house stood.

I fully share the previous gratitude and expand it also to the other Thony C’s pages linked to this one.

Vitali K.,

The St. Petersburg Society for Surveying & Mapping (NGO),

Russia

http://www.spbogik.ru

(containing a bilingual coverage of the Struve Geodetic Arc WH)

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I am surprised to read your saying that “Triangulation had first emerged as a map-making method in the mid-sixteenth century …” but perhaps you mean this is terms of mapping land only, or mean it in a more restricted sense than I’ve read it.

By the early fourteenth century the ‘rhumb-gridded cartes marine made in Majorca and Genoa rely on more than overlapping lines of the compass, or rendering the portolans’ straight-line description of winds an directions. It seems that ‘trig-points’ were also involved, and in fact anyone who has been out fishing and who finds a good ground will naturally look around the landscape and select three points to triangulate in order to find the place again. I won’t go on, since I realise you may mean only Europe, or only the dominant culture in Europe, or mean it is a technical mathematicians’ sense.

There is one fragment, commented on by Marston, which may be a record of a ship’s running ‘triangulations’ and using ‘trig-points’.

1. Thomas E. Marston, ‘An aid to Medieval Portolan-chart making?’, The Yale University Library Gazette , Vol. 46, No. 4 (April 1972), pp. 244-246.

If I’m using mathematical terms in a layman’s way, I apologise.

Trigonometry had been used to triangulate single points in astronomy since its inception for exactly that purpose in antiquity. However there is a significant difference between single point triangulation and the method of triangulation used in cartography to construct a network of triangles in order to measure a meridian or to map an expensive area of land. On term two different, if related, meanings.

It seems insufficient simply to find out and publish a correct point about a historical event (on Snel’s work in all details – 1968 Haasbroek N.D.). A complementation is also needed, a sort of happening: on-spot gathering of those who care, conference, anniversary celebration, etc. Public events of that kind promote wider and sooner publicity and adoption, help to reduce the number of doubters or questions like “who invented triangulation”. 2017 provided a very good reason to an event of the kind, as the year was the quatercentenary of Snel’s publication in which he described his using triangulation to measure a meridian arc and, by the way, to triangulate his home using three outdoor trigpoints. Either work prroved success and world priority.

(sorry for my English)