Considering Maps II – Virtual Lines

In the last post, I started discussing maps, mapping and the measurements required to construct maps.  I am going to continue, by looking at the coordinate system of longitude and latitude that Ptolemy refined and corrected, which is still in use today.  This system has a number of elements which he pulled together in a fashion that has provided a standard methodology for geographers, navigators and others to use, with the final element put into place by international convention in the 1880s.

The people who are important in the development of map making techniques often relied on or were astronomers.  Many of the results came from a combination of celestial and geographic measurement.  Celestial measurement adds an additional element to the geographic measurement: time and its measurement.

Watching the sky for patterns would have become very important for the agricultural communities that grew up after the end of the last ice age, around 12,000 years ago, and with several thousand years of doing so, the knowledge of the repeating patterns such as seasons was linked to the repeating patterns of the unchanging arrangements of the stars, and to the pattern of the sun moving against the backdrop of the stars.  The planets were special cases and had their own stories.

As observers of the sky, the people of the time would know about the sun’s position relative to the stars, and would have identified certain groupings of stars along the sun’s path as “constellations” that were discrete.  Approximately 12 were identified in the area of the sky that the sun moves through, which is called the ecliptic.  At some point, the approximately 30 day repeat pattern of the moon would be mapped against the sun’s movement along the ecliptic, and wow, 12 – 30s would be recognized as pretty close to a year (12 x 30 = 360).  Although the Egyptians knew that the year was 365 days long as early as 4241 BC by counting the days between the rising of Sirius, the Babylonians used 360 days for the length of their year.

Another observed pattern would be the length of daylight, its changes over the year and how that correlated with the height of the sun relative to what would become known as the southern horizon.  This would lead to a knowledge of solstices and equinoxes.  (It is a curious fact that all of the great civilizations that developed the early recorded astronomical observations -China, Babylon, Egypt, India (in large part), Greece, Maya – were all north of the equator.)

Another of the early elements was came from the Babylonian astronomers, among others, who saw that there seemed to be a place in the heavens that all of the stars moved around, but itself did not move much itself.  While we are familiar with Polaris (alpha ursa minor) being the pole star, due to the way that the earth wobbles on its axis, this would not always have been the case, evidently about 4500 years ago the pole star was alpha draconis, and from about 1500 BC to around 500 AD, beta ursa minor was recognized as the pole star.  Even so, this would have given an “unmoving” point to the nightly rotation of the stars, as if it were the center axle of a circular motion.

If these repeat patterns were characterized as occurring in a circle, it would not take much imagination to decide to divide a circle into 360 equal parts.  Evidently, that’s what the Babylonian astronomers and mathematicians did – they used 360 as the number of days in a year, and may have divided the circle into 360 parts.

The Babylonians were using a number system with base 60 on some of the earliest clay tablets, about 3000 BC.  There have been a number of proposals about why they used base 60, but since the proposals were developed well after the use began, and there is no documentation stating why the choice of 60, there is no way to know for sure.  But an extension of it is to the number of days in a year, though missing those pesky extra 5, possibly to make the calculations easier: 60 is divisible by 2. 3. 4. 5. 6. 10, 12, 15, 20, 30; and 360 shares those divisors along with a few more.  For whatever reason, the sexigismal is still active for us: 360 degrees in a circle, 60 minutes in an hour, 60 seconds in a minute.

In the 7th century BC, Babylonian astronomical information began to be acquired by Greek seekers of information, and continued through the time of Alexander the Great’s conquests (336 BC to 323 BC).  In the end, the Greeks possessed what is said to be a huge amount of accurate astronomical data compiled by the Babylonians.

Pythagoras of the famous theorem led a school of philosophy in the last quarter of the sixth century BC.  One of his hypotheses was that the earth was a sphere, though there is no record of how he reached that hypothesis.  The reason cited in several sources is that it was perhaps because “the sphere is the most beautiful of solid figures”.1 Aristotle (384 BC – 322 BC) built on this by providing three “proofs” based on observations.  He also tried to prove that the earth does not move, either around an axis or revolve around the sun.  Aristotle, among his other accomplishments, was Alexander’s tutor.

After Aristotle, there are three Greek mathematicians/scientists who have to be mentioned.  Euclid, about whom there is almost no personal information except an estimate that he must have been active around 300 BC, and may or may not have written before Archimedes, wrote a number of books, some of which have survived, and the most important of which is Elements, a treatise on geometry that has been the standard teaching text for the last 2300 years.  Elements is a comprehensive and logical treatment of geometry (and some other mathematical topics) that is a compilation of most everything that was known in plane geometry up to that point.

Archimedes (287 – 212 BC) is known for a number of scientific inventions and discoveries,  and is the prototype for the absent-minded professor who made an important realization about the displacement of water by solids when immersed in a bath and ran through the streets of Syracuse, where he lived, having forgotten to wrap even a towel around himself, shouting “Eureka!” because the displacement realization helped him to solve a knotty problem for the king of Syracuse.  He also systematized solid geometry in much the same way that Euclid systematized plane geometry.  Concepts in both plane and solid geometry were important for map makers to understand as they began their work.

Aristarchus of Samos (c. 310 to 250 BC) claimed that the earth moved around the sun and that it rotated about its axis.  This idea does not appear to become popular for another 1700 or so years, when Copernicus described it as a useful model, and attributed it to Aristarchus: giving proper credit.  But this model was not used for the making of maps of the world.  Aristarchus did create a more accurate sundial which became important in measuring the time of events that helped with map making.

And, it is at about this time, too, when making maps of the known world became an important activity.  Most of the concepts were in place for making maps that were more than local maps, though there had been some rather mythological attempts before this.  But with the great travels of Alexander as he conquered a version of the known world, trying create maps of the known world became more important.  The next actor in the list is Eratosthenes, who was introduced in the last post for his estimate of how large the world was.  Again, his dates are 276 to 194 BC.

By the time of Eratosthenes, the sages had figured out that if the earth was a sphere and the celestial sphere in which it is enclosed turned about the axis pointed at by the pole, there were some additional consequences.  One was that there was a place on the sphere of the earth that was half way between the pole at the top of the sphere and the pole at the bottom.  It is now called the equator, and presumably it was named that because it was an equal distance for each pole.  The sun was directly overhead of the equator two times per year: on the days of the spring and of the fall equinoxes.  The equal in this word refers to the fact that the sun was up the same number of hours as it was not up.

Also known was the fact that there was a northern-most point for the sun, occurring on the summer solstice, the day with the greatest number of hours of sun, and a southern-most point, on the day of the midwinter solstice, when the number of hours was the shortest.  Being clever, the sages realized that the days of the solstices also could be marked on the sphere as lines parallel to the equator, since the sun would be at its highest or lowest for the whole rotation of the celestial sphere on each of those days.  These lines are now known as the tropics, the northern one for the Tropic of Cancer, the constellation the sun moved into on the day of the solstice, and the southern one for the Tropic of Capricorn for the same reason.

Apparently this led to a number of additional lines parallel to the equator and the tropics, which were called “climates” to indicate climate regions of the earth.  The southern most line was the Tropic of Capricorn, below that, there had been no travelers who had returned with information so from there to the southern pole was treated as if humans did not exist there.  It is not clear from the literature that I’ve found if the earth south of the Tropic of Capricorn was considered to contain any land, habitable or not: there are some who mention a mythical southern land, but basically it appears to have been treated as if it were all hostile ocean.

The climates above the Tropic of Cancer, however, were sometimes treated as the line through a specific city, and sometimes as if to delineate a region.  There was a mythical island called Thule, which had scant reporting about it other than that it was cold there, and may have been Iceland, but also could have been a truncated version of the Scandinavian peninsula.  The important point, though is that there were those who treated various places as worthy of delineating regions with a line parallel to the equator and tropics.  These would ultimately become what we know as the lines of latitude.

Equally important, it was understood that a line that was perpendicular to the equator and ran to the north pole through a specific city, say Alexandria, this would not be parallel to a line running the same way through, say Athens, and that the lines would meet at the pole(s).  These would ultimately become the lines of longitude.

Eratosthenes created a map that had an early version of longitude and latitude laid out on it: the original has disappeared, but based on criticism of it by Hipparchus and Strabo, it has been reconstructed, a copy of which appears in the Brown book on page 51, and looks like this:

His parallels are wrong, his meridians (longitude) are wrong, his map is wrong.  But the beginnings of the concepts necessary to include latitude and longitude are present.

Hipparchus, (dates uncertain but after 127 BC) was so incensed by this map that he wrote an entire treatise which he called Against Eratosthenes.  Among other things, Hipparchus advocated that the parallels should be evenly spaced, and he, evidently, was the one who suggested basing their placement on the hours of the longest day, as in one at the latitude where the longest day is 13 hours, one at 14 hours, etc..  And he felt that the lines that were perpendicular to the equator should be evenly spaced at the equator, but joining at the poles.   One of my sources says that it was he who suggested dividing the sphere of the earth into 360 parts, later called degrees, though another source suggests that the Babylonians had divided the circle into 360 parts.

Hipparchus also has credited to him the invention of trigonometry, though knowing how my contemporaries and I struggled with trig, I fear the inventor has had more curses than thanks for doing so.  His trigonometric methods were based on isosceles triangles, which he used to calculate tables for chords for use in astronomy.

He also is credited with greater accuracy in his astronomical calculations, determining star positions, the lengths of the tropical year and sidereal years and one of his results was that the precession of the equinoxes was not less than 36 seconds and not more than 59 seconds per year, when the current figure is 50 seconds.2 I am not sure how the estimate was arrived at, since the tools he used or could have used were gnomons, sundials and potentially water clocks, none of which appear to me to have sufficient precision.  I will continue to research the question, and hope to be able to provide greater clarity in a future post.  But, since this estimate was made, in whatever form so that it could be translated into the above figures, there has to have been yet another set of concepts current and available for use: a day divided into 24 hours, with a way to judge fractions of a single hour.

Approximately 200 years after Hipparchus is when Claudius Ptolemy wrote The Almagest and Geographia, which are only two of his many works but are his best known.  His dates are about 100 AD to about 170 AD.  The work for which most people know him is the Almagest, which is his work on the universe that revolves around the earth.  In it are the mathematical explanations of the movements of the known planets, the sun and the moon.  Using the methods of the Almagest, astronomers and astrologers were able to make predictions about when planets would be where, and when eclipses of the sun and the moon could be expected.  Ptolemy states that having the sun be the center of movement of the universe would be a convenient explanation, but is ridiculous.  His reputation, after the Copernican revolution, was to suffer as a result.

In Geographia, Ptolemy was concerned with how the spherical world fits in the universe as he had mapped it.  As such, he tried to refine how maps of the known world should be made, and he followed Hipparchus’s suggestions of equal spacing of both parallels/latitude, and meridians/longitude.  As an additional aspect, he broke the degrees into smaller units, which have become known as minutes and seconds and are still used.  They come from the Latin translation of his divisions, “partes minutae primae” and “partes minutae secondae”.3

There are two reconstructions of map techniques that he is credited with which are visual demonstrations of one of the problems that he felt deserved thought.  That problem is that if the earth is a sphere, how can a map be accurately constructed on a flat piece of paper/parchment, whatever.

His first method was to use a simple conic projection which looks like the below figure, taken from the Brown book on page 70, which is based on a 16th century copy:

He also described a second method, which he called a superior method, but admitted that the easier method might be as useful for those who were somewhat lazy.  From Brown, again, on page 71:

It is important to notice a few things about his scheme: one is that he has included 180 degrees west to east of the possible 360, so only one half of the sphere, and he has further truncated the view by including starting at just above 60 degrees north to 15 degrees south.  This was considered the “habitable” portion of the earth.  The prime meridian in the west went through what was called “The Fortunate Isles”, probably the Canaries, and extended east to just beyond the southeastern corner of China, as it was postulated in Alexandria at the time of Ptolemy.

Here was the basis for creating maps of the world, and while the geography underneath the grid changed to more closely match the real geography of the earth, the grid has remained relatively constant, with the 15 degree divisions around the equator being retained roughly for the time zones.  The final step that I mentioned at the beginning of this post was to set the prime meridian as the one which runs through Greenwich, England, and the international dateline approximately 180 degrees from that, running through the Pacific Ocean.

So while Ptolemy thought that the earth couldn’t move, he did provide a reference scheme for mapping the earth.   While this reference has been used, now, for around 1700 years, it was not without its problems, and we can look at them in another post.


I have used a number of sources to pull this material together, and while I have not requested permission for the use of the three diagrams, I hope that I have sufficiently credited them.  The main sources that I have drawn this material from is:

Brown, Lloyd A., The Story of Maps, Dover Publications, New York, NY, 1979, by permission of Little, Brown and Company, Boston, MA, 1949, 1977.  Thorough, comprehensive, though at times, a little confusing.

Burton, David M., The History of Mathematics, An Introduction, Fourth Edition, WCB/McGraw-Hill, Boston, MA, 1999.  A marvelous history, the book was a gift from my daughter, Allyndreth Devlin, from the time when she worked for McGraw-Hill.

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.  Slim but authoritative.

Gullberg, Jan, Mathematics, From the Birth of Numbers, W.W.Norton & Company, New York, N.Y. 1997.  As stated on the copyright, the author prepared the entire camera ready book, all 1090 plus pages, on an Apple Macintosh Plus, using Microsoft Word.

Joseph, George Gheverghese, The Crest of the Peacock, Non-European Roots of Mathematics, Princeton University Press, Princeton, N.J., 2000.  An invaluable resource for those of us who have only been taught about a Western/European tradition.

Wilford, John Noble, The Mapmakers, Revised Edition, Vintage Books, New York, N.Y. 1981, 2000.  Well written and enjoyable.  The updated version predates MapQuest and Google Maps on the web, but is current up to that point.

And as always, periodic checks of the information on Wikipedia.

1Brown, Lloyd A., The Story of Maps, Dover Publications, New York, NY, 1979, by permission of Little, Brown and Company, Boston, MA, 1949, 1977. p.25.

2 Ibid p.53.

3 Wilford, John Noble, The Mapmakers, Revised Edition, Vintage Books, New York, N.Y. 1981, 2000. p. 30.

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