Surveying and Mapping

In the past few posts, I’ve been discussing maps and mapping.  I would like to build on those discussions by asking what would be needed in order to create a map.  To answer that question requires answering a few other questions:

What is the purpose of the map?

How is the map supposed to be used?

What information is required for the map to be useful for that purpose?

If we look at early maps for some elementary answers, we find that the earliest maps appear to have one of two purposes, either to provide directions on how to go from here to there; or to provide a way to understand some form of property rights or rights of land usage.

If the purpose is to insure capture of sufficient paths, directions and landmarks to remind ourselves or to insure that others are able to travel from here to there, then one set of information is important.

If the purpose is to delineate land divisions for ourselves and others to minimize conflict and insure that each member of the group knows which plots of land to work, then a slightly different set of information is necessary, though with some overlaps with the directions map.

The map for guiding travel would need:

  • Orientation – starting point, landmarks, easy way to identify geographical or geological elements encountered on the way.
  • A sense of scale – the path along the stream that is followed for three miles can’t be significantly shorter on the map than the three mile stretch between the two peaks to be traversed later in the journey.
  • Some labels or indications of important things to know, like “Ciudar – Region des Serpientes” (Be careful – area of snakes), an actual sign on the trail to Macchu Picchu that I encountered, indicating an area of bushmaster and coral snakes, or the name of the pass we had to cross a day or so earlier, Warmiwanusqa Pass, on the map, which translates to Pass of the Dead Woman – something else to know and exercise caution about?
  • Something to put the map onto – parchment, hide, clay tablet, paper, and a tool to use for inscribing the map.

Are there specific concepts that would be important for making the map?  Concepts such as distances, which could be time related, as in “two days travel” would probably be helpful.  Of course, if labels were to be used, language would have to be available, and a way to make symbolic marks indicating what would have been said by one experienced in the journey to a neophyte.

Are there measuring tools that would be needed?  I’m not sure, unless the intent was to make the scale absolutely reflect the actual distances, in which case most likely a ruler and possibly a dividers.  For orientation, a compass would be nice but compasses were developed fairly recently – within the last thousand years or so.

One possible tool to assist with orientation is a gnomon which is basically a straight shaft set vertically into the ground.  The word “gnomon” has come to mean the vertical portion of a sundial that throws a shadow, but as we are only interested in orientation, we don’t actually need the “hours”.  With a shaft acting as a gnomon set vertically into the ground, one could determine, from its shadow, the mid point of the day.  The direction of the shadow at the day’s middle would point in the direction we call “north” (at least for places above the Tropic of Cancer), and would line up approximately with the star at which the north axis pole points.  This could be used to orient the map, with an arrow indicating which direction the gnomon points at mid-day.

If that were to be used, another measurement tool would be required when setting up the gnomon, a plumb line to make sure the gnomon is as close to vertical as possible.

A map for establishing land use rights would need some of the above measuring tools and concepts, but in addition:

  • Geometry – the ability to make shapes on a map that reflects the shapes of the parcels of land, and the ability to make sense of those shapes.
  • Straight-edges – for marking the boundaries, on either the land or on the map.
  • Perhaps a way to deal with angles, whether 90 degrees or otherwise.
  • Some standards for lengths, and some mathematical understanding of how lengths relate the areas to be worked, as in length times width equals area (L x W = A).

There is little doubt that these requirements were met by both the Babylonians and the Egyptians at some point in their development.  The following is a double quote, since the book where I found this text content was written by George Gheverghese Joseph, The Crest of the Peacock, and in the book he includes a quote from Herodotus, The Histories (5th Century BC, Greece).  The next paragraph is from Herodotus, and the second one from Joseph:

Sesostris [Pharaoh Ramses II, c. 1300 BC] divided the land into lots and gave a square piece of equal size, from the produce of which he exacted an annual tax.  [If] any man’s holding was damaged by the encroachment of the river…The King…would send inspectors to measure the extent of the loss, in order that he might pay in future a fair proportion of the tax at which his property had been assessed.  Perhaps this was the way in which geometry was invented, and passed afterwards into Greece. (Herodotus, The Histories, p. 169)

He [Herodotus] also tells of the obliteration of the boundaries of these divisions by the overflowing Nile, regularly requiring the services of surveyors known as harpedonaptai (literally ‘rope-stretchers’).  Their skills must have impressed the Greeks, for Democritus (c. 410 BC) wrote that ‘no one surpasses me in the construction of lines with proof [?], not even the so-called rope-stretchers among the Egyptians’.  One can only suppose that ‘lines with proofs’ in this context refers to constructing lines with the help of a ruler and a compass.1

There is some further discussion of the Egyptian rope-stretchers in Dilke’s book, Mathematics and Measurement, in the British Museum series, Reading the Past.

When the floods receded, many landowners’ boundary marks had inevitably been washed away, so it is important that surveying should be carried out immediately.  There are Egyptian representations of surveyors employing knotted ropes (the knots indicating sub-divisions of linear measurement), the merkhet (a split centre-rib of a palm-leaf, used for sighting), and measuring rods.  The priests inaugurated this rapid re-survey of the land, which had to be ready for winter cultivation.  There is no evidence that land survey maps were used in dynastic Egypt for this operation.  But by means of exact area measurement and verbal descriptions, the status quo was re-established.2

The geometry required would be basic plane geometry concepts, such as squares, triangles, polygons, circles and areas.  Also required would be some understanding of equivalencies, such as a square that has sides that are twice that of another square, are equal in shape and angle, and the area is four times that of the smaller square.  Similarly shaped triangles have the same ratio in area, so two triangles with the same shape and angles, but one with sides twice the length of the smaller would be four times the area.  Additional understanding of ratios: if a gnomon casts a shadow at noon that is 2 feet long, and is 3 feet tall, then the tree nearby that casts a shadow at noon that is 20 feet long is 30 feet tall – a form of indirect measurement.  These concepts would require some measuring concepts and tools, with some standardized units of measurement.

The tools mentioned in the paragraph quoted from Dilke, above, knotted ropes, merkhet, and measuring rods, need a little explanation.  The knotted ropes, with the knots indicating sub-divisions, are most likely, ropes of a standard length, with the knots at specific spots along the standard length, such as at the half way point, the quarter point, and possibly other fractions.  The Egyptians are known to have used fractions in their mathematics.  Measuring rods would have similar use, except that a rigid rod is difficult to use for measuring the length across ground that is not level.  The merkhet mentioned in the paragraph is the center rib of a palm leaf, notched or split at one end.  It is not clear to me from my research how it was used, though I could make some guesses.  One source describes the rectangle with the plumb line as the merkhet, and the piece of palm leaf as the bay, and says that it was used for telling time.  The alternate source describes using it only at night, for viewing star positions.

Merkhet and Plumbline3

These tools and concepts were used for surveying, for laying out property lines.  They were not designed for making maps, but could easily have been adapted to map making.  They are among the earliest tools related to mapping that I have been able to find.  The next step in map making and surveying tools would probably be refinements on these.

By the time of the Romans, the need for tools for surveying was met by more sophisticated tools, although I have not been able to trace the path from Merkhet to groma (the Roman cross-staff with plumb line at each of the four ends of the crossed staves) and decempedae (10 foot measuring rod).  While I am not sure of the dates for the development of the groma, it appears to have preceded the early dioptras that are dated to the 3rd century BC.  The groma appears to have been used only for surveying, and the dioptra for astronomy.  The right angle under the dioptra sighting tube was eventually replaced with a hemispherical protractor, which permitted more precise angle measurement.  Further development of the dioptra was done by Heron of Alexandria, 10 AD to 70 AD.  A post had been added on which the dioptra sat, but Heron added the ability to use the dioptra in both horizontal and vertical directions, and screws for fine adjustment, making the dioptra much more precise.

Schematic version of groma: bottom pole is stuck into the ground, the bottom pole is adjusted so that it is vertical, by lining up the plumb lines, then the cross-poles are rotated to that the user can sight along them.

Schematic of early dioptra: the top is a tube for sighting through, there is a plumb line which allows the user to determine the angle of the sighting tube.

I remember being told about the Via Emilia while I was visiting in Florence, Italy.  What I was told is that the Romans had built a road that ran from Rimini on the eastern side of Italy straight through the countryside to where Milan is now 2000 years ago.  In checking, however, it turns out that the original road terminated in Piacenza, somewhat short of where Milan is now, but it did run straight for approximately 260 kilometers, and the Romans implanted colonies there that became Bologna, Parma, Reggio, Modena and others.  It is an impressive straight line, but there are other roads that the Romans built that are similar.  It is not known precisely what procedure was used, as there are no descriptions, nor is it clear which tool, groma or dioptra, was used.  But the Romans also set border lines, one of which is mentioned in Dilke to be 29 kilometers long with a variation from straight of no more that 2 meters along its whole length.

Some impressive milestones in surveying are tunnels started from both ends, using a method described by Heron of Alexandria, and the building of Roman aqueducts with the proper slope so that the water in them went where they wanted it to go.  Their sewers would have the same requirement.  The speculation mentioned in Dilke is that an advanced form of dioptra was used for the tunnels, aqueducts and roads.

Reconstruction by H. Schone of Hero’s dioptra, which could be used both for surveying and for astronomical observation.4

The next improvement had to wait until the 16th century, when the Digges, father and son, showed the theodolite to the world.


1 Joseph, George Gheverghese, The Crest of the Peacock, Non-European Roots of Mathematics, Princeton University Press, Princeton, N.J., 2000. p. 59.  The version of Herodotus that he used is Herodotus (1984), The Histories, London, Penguin. p. 59

2 Dilke, O.A.W., Mathematics and Measurement, Volume 2 in the Reading the Past series, 1987 by The Trustees of the British Museum, British Museum Publications Ltd, London. pp 7-8.

3 Ibid. p. 7

4 Ibid. p. 30

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