Triangulation ... it's not just "doing things with triangles"
/I am a bit of a purist on the matter!
Yesterday, I received, via the wonderful people of ILL, a chapter from a new book by Wolfgang Lefèvre on early modern textbooks and manuals in science and technology. In one of the inset accounts of ancillary technical information, I read
The triangulation method can be traced back to ancient Greek as well as medieval Arabic sources. In the early modern period, it was the Dutch physician Gemma Frisius (1508–1555) who proposed and described it as a key technique in connection with map making in his Libellus de locorum describendorum ratione (1533) — a book that was instrumental for the spread of this surveying method over many European countries in the following period. (Lefèvre 2021, 168)
The note for the paragraph reads:
59 Gemma Frisius‘ Libellus first appeared in Antwerp (apud Ioannes Grapheus, 1533). It became known to a broad readership when reissued as an appendix in Petrus Apianus’ Cosmographia of 1540. Frisius, together with a goldsmith, managed a workshop for the production of globes.
I was intrigued as to the nature of the Greek and Arabic sources that this poorly referenced statement might have referred to. [n1] What had I missed? So, as one does in this modern age, I consulted the twitters. The results were illuminating, and point to a persistent problem that just annoys me to my core.
The issue is simply this: “triangulation” as a method appears to be the most complex and fundamental method of surveying, but is little understood, even by map historians. People know that it involves triangles. So, any procedure involving triangles gets called “triangulation” whenever someone wants to elevate its status and give the procedure the status of “Science! (oooohh!)”
Early Modern pseudo-Triangulation
This is a problem in map history as a large part of surveying practice since the sixteenth century has relied on imaging triangles in the landscape and measuring the angles and some of the sides. When I did land surveying courses as an undergraduate, I remembered the simple mnemonic I had learned in school about the minimum magnitudes (interior angles, lengths of sides) are needed to define an entire triangle, and related the permutations to the types of surveying practice I was being taught:
AAA — all three angles are known — triangulation
ASA — one side of the triangle is known, together with the angles at either end — intersection
SAS — two sides are known, together with the angle between them — traverse
SSS — all three sides are known — trilateration
ASS — one angle and an adjacent and opposite side are known — ambiguous results, which make an ass of the student!
Intersection is often misleadingly called triangulation because the surveyor observes, or draws on a plane table, sight lines from two known places to a distant third location. I think this is a reflection of the colloquial use of “triangulate” to mean fixing one point from two others, a term misleading carried over into sociology (“social triangulation”).
The little classification is not perfect. Each surveying technique requires the combination of triangles and some other factors. In the case of triangulation, at least one side of a triangle has to be measured (the baseline) before that length is carried by trigonometrical calculations to all other sides. in effect, triangulation as a surveying technique is AAA in the field converted to SSS in the office. Here is a detail of the northernmost part of the chain of triangles measured by the French along the meridian of the Paris Observatory, from Jacques Cassini’s 1720 memoir:
The astronomer-surveyors sighted from towers to other towers. One baseline was measured along the beach at Dunkirk, between the fort and the “signal”; other baselines were used elsewhere along the chain. Other triangulations took the form of a network of triangles. In all cases, to be a triangulation requires the direct observation of many angles between imaginary lines to form a mass of triangles to be solved trigonometrically in the office. (Someone might try to do a triangulation graphically on a plane table, but usually, the triangulation was undertaken, the final points then plotted into paper to be fitted onto a plane table for use in filling in the topographical details.)
In editing Volume Four of The History of Cartography, Mary Pedley and I were careful about many things; one was that “triangulation” was explained well (Bendall 2019) and that no-one misused the term.
Ancient and Medieval pseudo-Triangulation
In the case of the quote that started me off, there is no evidence that anyone before the sixteenth century implemented the surveying practice of triangulation. From the answers I received on twitter, it seems that the medieval use of spherical trigonometry to resolve large triangles in the calculation of differences and latitude and longitude, by scholars such as al-Biruni, has been aggrandized as “triangulation.” But this is of course a process of trigonometry, not a careful and precise survey operation. The same method of turning itinerary distances into lat/long was probably used by Ptolemy in the second century CE, and probably by some earlier Greek scholars. It is a process that essentially turns itinerary distances and bearings (i.e., summary traverses) by means of a presumed size of the earth into differences of latitude and longitude.
More intriguing is a reference from a friend to a book arguing that the Greeks situated their religious sites according to some kind of geodetic triangulation (Manias 1969). Now, I don’t have the book, but I did find a relatively recent website on “The sacred triangles of ancient Greece and their well-hidden meaning” that seems to be based on Manias (1969). It’s an exercise in drawing triangles across Greece between certain locations, waving one’s hand, and going “Mystical science, oooohh!” Go look for yourself. But lots of triangles do not make triangulation!
Notes
n1. The book is typeset with the main text being about textbooks and manuals and then inset paragraphs of relevant, technical information. (At least I think so: I only requested the one chapter, so I haven’t seen any preliminary explanation by the author.) The inset paragraphs are not necessarily well sourced, seeming to rely on generally accepted work in the history of science and technology.
References
Bendall, A. Sarah. 2019. “Triangulation Surveying.” In Cartography in the European Enlightenment, edited by Matthew H. Edney and Mary S. Pedley, 1522–24. Vol. 4 of The History of Cartography. Chicago: University of Chicago Press.
Lefèvre, Wolfgang. 2021. “Practical Mathematics.” In his Minerva Meets Vulcan: Scientific and Technological Literature, 1450–1750, 147–73. Cham, Switz.: SpringerLink.
Manias, Theophanes. 1969. The Invisible Harmony of the Ancient Greek World and the Apocryphal Geometry of the Greeks. Athens. Greek edition, 1974.
