2 Geodetic Surveys

In the last post, I discussed some early surveying instruments: merkhet, plumb line, knotted ropes, groma, decempedae and dioptra.  Of these, the dioptra that was designed by Heron (also known as Hero) of Alexandria, and reconstructed by Schone, appears to have contained most of the capacities that were used until quite recently.1

From Schone’s reconstruction, it appears to have included alidade – a rotating sighting mechanism – in this case not a tube but slits at the opposite ends of the alidade, beneath which is a  horizontal circular plate that could be marked with angle designations for registering horizontal angles.  The alidade and circular plate were mounted on a hemispherical plate that could be used to adjust the vertical angle of the alidade and horizontal plate, and this entire apparatus could be rotated on the pole it is mounted on.  Rotating on the pole would allow a specific point on the horizontal plate to be aligned with a specific reference direction or point.  The pole or post on which this version of dioptra sits could be made vertical with a plumb line, although there is none in evidence in the reconstruction drawing.

It is not clear if a device was ever built that looked and worked like Heron’s dioptra.  For the next approximately 1500 years, the devices used for surveying performed the various functions as separate pieces of apparatus: geometric squares (like carpenter’s squares); circumferentor/surveyor’s compasses, a form of graduated circle; and graphometers, semi-circles (like protractors) are the reported surveying tools, as well as ropes for distances.  The surveyor’s chain did not come into standard use until about the time that theodolites were developed.2

As a slight digression, during that time, it is possible that astrolabes were adapted to surveying: an astrolabe is the marriage of a planisphere, a flat circle representing the heavens, with a sighting device (an alidade) held with the flat circle/planisphere vertical, and was used for “…locating and predicting the positions of the Sun, Moon, planets, stars: determining local time (given local latitude) and vice-versa; surveying; triangulation; and to cast horoscopes.”3 If a surveyor had an astrolabe and held it horizontally, the user could sight along the alidade for surveying purposes.  This marriage of planisphere and alidade was allegedly performed by Hipparchos around 150 BC.

This leads to yet another digression, known as the Antikythera mechanism, subject of the website of the Antikythera Mechanism Research Project.


I have pasted the abstract from the beginning of an article on the website, but removed the footnote numbers:

The Antikythera Mechanism is a unique Greek geared device, constructed around the end of the 2nd Century BC. From previous work…it is known that it calculated and displayed celestial information, particularly cycles such as the phases of the moon and a luni-solar calendar. Calendars were important to ancient societies…for timing agricultural activity and fixing religious festivals. Eclipses and planetary motions were often interpreted as omens, while the calm regularity of the astronomical cycles must have been philosophically attractive in an uncertain and violent world. Named after its place of discovery in 1901 in a Roman shipwreck, the Mechanism is technically more complex than any known device for at least a millennium afterwards. Its specific functions have remained controversial… because its gears and the inscriptions upon its faces are only fragmentary. Here we report surface imaging and high-resolution X-ray tomography of the surviving fragments, enabling us to reconstruct the gear function and double the number of deciphered inscriptions. The Mechanism predicted lunar and solar eclipses based on Babylonian arithmetic-progression cycles. The inscriptions support suggestions of mechanical display of planetary positions…now lost. In the second century BC, Hipparchos developed a theory to explain the irregularities of the Moon’s motion across the sky caused by its elliptic orbit. We find a mechanical realization of this theory in the gearing of the Mechanism, revealing an unexpected degree of technical sophistication for the period.4

This means that the estimated time for the Antikythera Mechanism is approximately the same time as when Hipparchos is reputed to have made the first astrolabe.  Quite an industrious fellow, not to say impressive in his accomplishments: could the Antikythera Mechanism be attributed to him as well?

Given the information in these digressions, it certainly looks like the skill necessary to make a version of Heron’s dioptra was available by his time, 10 to 70 AD.  So it could have been made, but it looks as if the tool and the knowledge about making such was gone not too long after that and not rediscovered until the 1500s.  No wonder they are called the Dark Ages.  Actually, the only thing missing from the dioptra of Heron was a better sighting mechanism: a telescope with the optics to extend the field that could be surveyed.

Going from separate tools being used for surveying to an integrated instrument shows up in the appendix to a book, Margarita Philosophica, by Gregorius Reisch, in 1512, and the appendix is attributed to Martin Waldseemuller.  Waldseemuller also built a device in that same year, and called the device a polimetrum.  The Digges, father and son, published a surveying textbook in 1571, the father, Leonard, having written it and passed away, so his son, Thomas, had it published, and in the textbook, they renamed the polimetrum, the theodolite.  Shortly thereafter, versions of theodolites were made with tripods and compasses.  In 1725 a telescope was added to the theodolite as a sighting mechanism, 115 years after Galileo demonstrated a telescope to the Venetians.  In 1787, an Englishman named Jesse Ramsden introduced a theodolite with angular scales on it that were accurate to within a second of arc.

There have been further improvements since then, always with the goal of greater precision, accuracy and ease of use.  Surveys were extended from making maps detailing how to get from here to there and cladistal maps – maps of property boundaries, to mapping the large areas, called geodetic mapping.  Geodetic mapping became important because people needed to know more about the earth as a result of the discovery of a new continent and the overall exploration of the world.

The surveyors who performed geodetic surveys until the late twentieth century had to know not just how to use the instruments but several branches of mathematics: algebra; calculus; trigonometry, especially the law of sines; in addition to plane geometry,5 be ready to perform large numbers of calculations and had to know the latest thinking in astronomy.  They also had to be very tough.  Some of the tasks that surveyors were put to work on required uncommon fortitude.

First, little of the math.  Gemma Frisius, 1508 – 1555, developed a method which is at the core of surveying, called triangulation.  He was on the faculty of medicine at the University in Leuven for most of his life, but made globes and mathematical instruments in addition to teaching.  One of his students was Gerardus Mercator, of the famous Mercator projection.

In 1533, Frisius published a theoretical method of using triangulation to measure distances, based on triangles.  If you know the distance between two points, and are able to measure the angles between the line connecting the two points and lines from each to a third point, using sines and cosines, you can determine the distance from each of the two original points to the third.  Frisius’s example was a conceptual way to measure from Brussels as one point and Antwerp as a second point, knowing the distance between Brussels and Antwerp, and finding the angles to Middleburg, or to Ghent, neither of which could actually be seen from Brussels or Antwerp.

If you know the distance from A to B, the angle at A, and the angle at B (ABC, not ABD or CBD), you have enough information to figure out the distance from A to C and from B to C.  Since all triangles that have the same set of angles are always the same shape, the lengths of their sides will always be in the same proportion.  If a triangle has sides that are 2 units, 3 units and 4 units in length, then any triangle with the same angles will always have its sides related to each other as 2:3:4.

First, with two angles of known size, the third angle is also known, since all triangles in plane geometry have exactly 180 degrees, simple addition gives the sum of the angles at A and B, and that sum subtracted from 180 gives you the angle at C.

In this case, angle A = 40 degrees and angle B = 75 degrees.

C = 180 – (40+75)

C = 180 – 115

C = 65 degrees

In my example, the length of A to B is 9 cm.  Because the ratios of the sides of this shaped triangle are constant, the length from A to B could be 9 kilometers or any other length.

Mathematicians have developed a concept called “sine”.  The sine of an angle in a  triangle is based on the ratio of the sides of a triangle with that angle in it and a right angle (90 degrees) in it as well.  In our example, the sine of angle A in the right triangle defined by ABD is the ratio of the side opposite that angle  to the hypotenuse, the longest side of that right triangle, so sine A = opposite/hypotenuse or o/h.  Another ratio of some interest is “co-sine”: the ratio of the side of the triangle which is the other part of the angle to the hypotenuse, that is, adjacent to the angle, to the hypotenuse.  Cosine A = adjacent/hypotenuse, or a/h.  Simple, no?

What we did to the example above, is run a line from angle B to the opposite side that is perpendicular to that opposite side, dividing the original ABC triangle into two right triangles, ABD and BCD.  Then the sine of angle A allows us to know how long line BD is, since sine A = BD/AB.   At the same time, the sine of C represents the ratio of BD to BC, or sine C = BD/BC.  Once we determine the length of BD from the sine of A times AB, we can divide BD by the sine of C to get the distance from B to C.  We also have sufficient information to generate the distance from A to C.

Sine A = .6428    (.6248) X 9 = 5.7852, so BD = 5.7852 centimeters

Sine C = .9063    (.9063) / (5.7852) = 6.3833, so BC = 6.3833 centimeters

To get AC, we can use cosines:

Cosine A = .7660  (.7660) X 9 = 6.8940, so AD = 6.8940 centimeters

Cosine C = .4226  (.4226) X (6.3833) = 2.6976, so CD = 2.6976 centimeters

AC = AD + CD  (6.8940) + (2.6976) = 9.5916 centimeters

It is a multi-step process, but once you have results for the original problem that Frisius described the solution to, the results can be used to develop more distances by establishing the angles to more points and using the already developed distances to understand distances to new points.  Some additional steps to insure accuracy were developed by Willebrord Snellius (his Latinized name – un-Latinized, he was Willebrord Snel van Royen, but by custom is referred to as Snell), and published in 1617 in a book called Eratosthenes Batavus (The Dutch Eratosthenes).  Snell used his triangulation method to generate the distance between two towns in the Netherlands, which represent one degree on a meridian running through them to the north and south poles.  Based on the distance derived and multiplying by 360, he was able to provide a new measure of the circumference of the earth.  As he also developed a more accurate way to calculate pi, he was able to use that to establish the radius of the earth.

If you start, as we did, with the distance from A to B and the sizes of the angles at A and B, using this method can fill in the distances from A to E, C to E, C to F, and E to F.

Snell not only extended the method, but provided a way to correct for the curvature of the earth, and a strategy for surveying: survey large triangles first, then locate the points within the triangles.

So much for the math – the conceptual tool.  Now the tasks:

Tasks that surveyors were given, or set for themselves, were quite ambitious.  Within Europe, surveys were conducted using the triangulation method.  These occasionally ran into problems or resistance, for the peasants were certain that if their holdings were surveyed, the result would be higher taxes – and in a number of cases they were right.  Surveyors were often agents of the landlords and the aristocracy, if not the government.  Some surveyors were killed by angry mobs.

France seemed to have the ability to set nearly impossible tasks: the French sponsored an expedition to Peru in 1733, which prepared for a couple of years and left Europe in 1735.  International relations were involved: the area in which they wished to do the work was in Spanish-held South America, and the proper authority from the Spanish king had to be obtained.  The portion of South America held by the Spanish was called the Viceroyalty of Peru, and included a number of “audiencias” – districts.  The specific audiencia that would become Equador was called the Audiencia of Quito.  The expedition involved the French traveling through other audiencias (and a few of the party descended the entire Amazon, requiring passes from the Portugese).

The stated task of the expedition was to settle a question raised by the clockwork universe of Isaac Newton: what shape is the earth exactly?  Jacques Cassini, the head of the Royal Observatory in Paris believed that the earth was elongated at its poles.  Newton’s work provided a different hypothesis: that the earth bulged at its equator and was flattened toward its poles as a result of its rotation and gravity – similar to the shape of Jupiter that astronomers had already observed.  The task given to the French surveyors was to measure a degree of meridian as near to the equator as possible.  This “simple” task took them about 10 years while coincidentally battling all sorts of jungle horrors: poisonous snakes, jaguars, insects, diseases, inclement weather, altitude sickness and inhospitable people.

Some of the instruments and tools used were telescope-equipped quadrants for measuring angles, a zenith sector for celestial observations, a seconds pendulum to gauge gravitational pull at the equator, and an iron bar exactly 1 toise in length.  A toise was six Paris feet (6.39 English feet).  “In addition to their quadrants and zenith sector, their scientific equipment included two telescopes, survey chains, watches with second hands, land and sea compasses, thermometers, a rain gauge, several barometers, a galvanometer, an instrument for measuring the blueness of the sky, and another for determining the boiling point of water.”6 They did not have a theodolite, but did have instruments that performed the functions of theodolites.

Their procedure was to choose the first two points in a place where they could measure the distance between them to establish the baseline, then use their telescopes and zenith sectors to determine the exact latitude of each point, calibrate their pendulum clocks (measuring the effect of gravity at the same time), then determine the exact longitude of each of the points.  Next they had to lay out the line between the points and determine the length of their baseline.  Then, a third point would be determined, looked at from both of the original points to determine the angle between the line of sight to the new point and to the other known points.  This set the original triangle, from which they worked their triangulation measurements to determine the length of a degree of latitude and a degree of longitude at or as near as possible to the equator.

The result: those who survived the hazards and the toil proved that, in fact, a degree of latitude and a degree of longitude differ in an amount at the equator that is consistent with a bulge at the equator.

Triangulation near Quito: the faint straight lines are the lines surveyed7

About 50 years later, at the time of the French Revolution, the savants of France had been advocating a standardized system of measurement, which, with the Revolution, became a goal of the government.  While they could have used a standard that integrated both time and length, a pendulum of sufficient length to swing once per second, which became an early version of the meter, the savants felt the best measure would be one based on the size of the earth.  So, the meter was to be “…based on one ten-millionth of the distance from the North Pole to the equator as established by a survey of the meridian that ran from Dunkerque to Barcelona,”8 which, incidentally was the meridian of Paris.  So, the task assigned to the surveyors was to measure that portion of the meridian that ran from Dunkerque to Barcelona which was a known percentage of the entire meridan so that the overall distance from North Pole to equator could be established and then used to derive the standard one meter.

Again, a simple task, done in the years between 1792 and 1806, a time of relative calm?  Well, no, not really.  Various of the people associated with the survey were accused of having Royalist sympathies; a few of them lost their heads to Madame Guillotine; others their jobs or positions, and some few prospered.  Through it all, those charged with surveying tasks used their telescopes to establish latitude, used pendulum clocks to establish correct time and thus their place on the meridian, etc.  The work was astonishing in its complexity – scientific and political – and the devotion to duty of those working on the task inspires awe.

The principal instrument used for sighting and establishing angles was called a Borda Repeating Circle, which had two telescopes that could be moved separately around the same axis.  It was very precise, but due to the changes in measurement systems as a result of the French Revolution, the entire circle was divided into 400 “grads”, and Borda spent years developing the tables to support it.  The Borda Repeating Circle was tested against the Ramsden Theodolite, and the two were deemed to be essentially equal in their accuracy, but since the Borda Repeating Circles were 20 pounds to carry and set up, while the Ramsden Theodolites were about 200 pounds in weight, not to mention using an “English” measurement scheme, the Borda Repeating Circles were used for the measurement of the meridian for the meter.

Map showing the triangulation from Dunkerque to Barcelona9

The result of their work established a standard length for the meter, but it was actually adjusted from what they had concluded.  By the time they finished, Napoleon had thrown out the metric system, (a day with 10 hours of 100 minutes, and minutes with 100 seconds???), but kept the meter.  As Alder states, “The French were not only the first nation to invent the metric system, they were also the first to reject it.”10

There are two other surveys that I would like to discuss, but will wait to discuss them in another post.

The interplay of the elements in this post is emblematic of how measurement can be done: purpose blended with the conceptual tools and the instruments to perform the measurements that provide deeper knowledge about the world.  But the instruments cannot be developed without conceptual tools, and the conceptual tools only become useful once there are instruments to confirm the use of the conceptual tools.


I would like to thank Robert Zweben for suggesting The Mapmaker’s Wife to me.

I also have to thank Allyndreth Devlin, my daughter, for many things and much pleasure, but right now for Geometer’s Sketchpad, the program I have used to create diagrams such as the triangulation ones.

1 Quite recently: surveying has changed drastically because of airplanes, electric theodolites and because of GPS, all of which will be discussed in a later blog.

2 The surveyor’s chain is actually a standard measure.  I should have guessed.  It is a chain with 100 links, the total length of the chain being 66 feet.  There are 10 chains in a furlong, and 80 chains in a mile.  An acre is 10 square furlongs in area, so 660 feet ( 10 furlongs) times 66 feet (1 furlong) equals 43,560 square feet.  I always wondered how the length of a mile and the size of an acre were chosen.  The chain was first standardized in England in 1620 by a clergyman named Edmund Gunter, therefore the instrument used was know as a Gunter’s chain. More detail is available at:


4 T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, Decoding the Antikythera Mechanism: Investigation of an Ancient Astronomical Calculator, Nature, Volume 444, Issue 7119, pp. 587-591 (2006).

5 Depending upon the amount of land being surveyed, the surveyor might also have to know spherical geometry and spherical trigonometry, to account for the curvature of the earth.

6 Whitaker, Robert, The Map Maker’s Wife, Delta Trade Paperbacks, 2004, New York, N.Y. p. 63.  The earlier list of instruments can be found on page 48 in the same book.

7 Whitaker, p. 132

8 Alder, Ken, The Measure of All Things, Free Press, a division of Simon & Schuster, Inc., 2002 New York, NY p. 89.

9 Alder, frontispiece

10 Alder, p. 261.

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