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(2006-10-19)     Obliquity of the Ecliptic
Latitude of the Tropic of Cancer.  Tilt of the Earth's axis of rotation.

Local  high noon  is the middle of the solar day.  It's when the Sun casts the shortest shadows.  On the summer solstice (June) and on the winter solstice (December)  the Sun's rays make two different angles with the local vertical.  The difference between these angles is always  twice  the  obliquity of the ecliptic.

Claudius Ptolemy (AD 85-165) reports that Eratosthenes of Cyrene (276-194 BC) had estimated the obliquity of the Ecliptic to be:

11/83 of a half circle (180°)   =   23.8554°   =   23°51'20". 

Eratosthenes, was merely 8' off the mark, which is typical of the uncertainty in good angular measurements from antiquity (0.2°). It turns out that the  obliquity of the ecliptic  changes slowly over time, but its value in the times of Eratosthenes (i.e., when he was in his late thirties) can be accurately estimated to be  23°43'30", by putting T = -22.4 in the following modern formula:

23°26'21.45" - 46.815" T - 0.0006" T² + 0.00181" T³  

The above is a standard approximation for the mean obliquity of the ecliptic, as a function of the time T counted from "January 1.5" of the year 2000 and expressed in "Julian centuries" of exactly 36525 days.

This means that, in the times of Eratosthenes, the Tropic of Cancer was about 17 nautical miles (30 km) to the north of its current (2006) latitude of 23°26'18". 

The above formula also says that the Tropic of Cancer was at the latitude quoted by Eratosthenes  (11p/83)  around 1347 BC.  Some have argued, backwards, that Eratosthenes did not measure the obliquity himself  (with a respectable accuracy for that period)  but had access to extremely accurate data from such earlier times...  This is either somewhat far-fetched or completely ludicrous.

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(2006-11-06)     The Ancient Wells of Syene
A vertical well in Syene is completely sunlit only once a year...

This ancient observation may have been part of the Egyptian folklore in the times of Eratosthenes.  Exactly  how ancient  an observation could that be?

The latitude of Syene (modern Aswan) is  about 24°06'N.  From the surface of the Earth, the radius of the Sun is seen at an angle of about 15'.

We're essentially told that the edge of the Sun was lighting up the entire bottom of a vertical well at Syene, just for a brief moment at noon on the summer solstice.  So, the center of the Sun must have been directly overhead at a point exactly 15 angular minutes (15 nautical miles) to the south.

Therefore, the latitude of the Tropic of Cancer must have been 23°51' at the time of the reports, if we assume they are perfectly accurate.  The above formula says that this happened about 33 centuries ago:  Around 1300 BC.

However, as the verticality of a well is certainly of limited precision,  that date doesn't mean much.  The legendary observations could be made even today with a well that's tilted by less than half a degree in the proper direction...  Any casual (or not-so-casual) observer will swear such a well to be "vertical".

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(2006-10-14)     252 000 stadia around   (700 stadia per degree)
The size of the Earth, according to Eratosthenes (276-194 BC).

Eratosthenes of Cyrene  became librarian of the  Great Library of Alexandria  around 240 BC, upon the death of his teacher Callimachus.

Eratosthenes knew the above story about the wells of Syene.  He took that to mean that the Sun was directly overhead at noon on the summer solstice in Syene (modern Aswan).  This is almost true, because Syene is  almost  on the  Tropic of Cancer.  Eratosthenes did not know about the slow evolution with time of the latitude of the Tropic of Cancer and he took the above at face value.  Let's do the same (slight) mistake by using the modern map at right, as if Eratosthenes were alive today...  From his own location in Alexandria, Eratosthenes could observe that, at noon on the summer solstice, the Sun's rays were tilted 1/50 of a full circle from the zenith  (i.e., 7.2° from the local vertical).  If we assume that Syene is due south from Alexandria, this means that the distance from Alexandria to Syene is 1/50 of the Earth circumference  (a posteriori, that's only 6% off).

The error from the difference in longitude between the two cities roughly compensates the error which places Syene on the Tropic of Cancer.  That's because, as the above map shows, the meridian of Alexandria  (about 30°E)  crosses the Tropic of Cancer at a point which is about the same distance from Alexandria as Syene (Aswan).

As the distance between Alexandria and Syene, was reputed to be 5000 stadia, Eratosthenes estimated the circumference of the Earth to be  250 000 stadia.  This estimate was then rounded up to  700 stadia  per degree, which corresponds actually to  252 000 stadia  for the whole circumference  (360°).

Unfortunately, we can't judge the absolute accuracy of that final result, because we don't know precisely what kind of  stadion  (or stadium)  was meant in the Alexandria-to-Syene distance quoted by Eratosthenes.

The traditional equivalences are 600 feet to a stadion and 8 stadia to a mile. 

The latter ratio justified the introduction of the current "statute" mile of 8 furlongs (1593) to replace the former "London mile" (itself based on the Roman ratio of 5000 feet to the mile).  The ratio of 600 feet to the furlong, which made the furlong a "modern" equivalent of the stadion, pertained to the deprecated "Saxon foot", which was worth 11/10 of the "modern" foot (henceforth, 1 ft = 0.3048 m).  A furlong is thus 660 ft.

However, the exact length of a Greek foot varied from one city to the next.  Arguably, Eratosthenes would have been likely to use the Attic stade of 185 m  (8 Attic stades to the Roman mile).  In any case, his estimate was certainly no worse than 20% off the mark and it may have been much better than that...

A circumference of  252 000 stadia  would be only 1% off  if  Eratosthenes, willingly or unwillingly, had been calling a "stade" an Egyptian surveying unit of  157 m,  which was sometimes identified with a Greek stadion.

That very low error figure of 1% is often quoted, but it's clearly misleading by itself, because intermediary steps do not attain the same accuracy.

The great achievement of Eratosthenes was to realize that the circumference of the Earth could be estimated with some accuracy from a single angular measurement and a few "well-known" facts, which happen to be  approximately  true.  By exaggerating the accuracy of the result, some commentators only cloud the issue.

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Archimedes  (287-212 BC)  quotes  300 000 stadia  as the figure "others have tried to prove" for the circumference of the Earth.  He does so in one of his most famous pieces, written or delivered shortly before his death:  De Arenae Numero  (The Sand Reckoner)  where his main concern with upper bounds led him to use a number ten times as large, just to be on the safe side.  There is very little doubt that Archimedes was thus referring to [a rounded up version of] the estimate of his younger contemporary.  Archimedes reportedly viewed Eratosthenes as his equal.  Arguably, this would be flattering to both scholars...

There may well have been some rivalry between the two men, which might be why Archimedes avoids mentionning the  name  of Eratosthenes in a text where he give meticulous credit to many others.

To Archimedes and Eratosthenes, the "traditional" estimate for the circumference of the Earth was most probably the one quoted by Aristotle (384-322 BC) in  On The Heavens, namely:  400 000 stadia.  This number was attributed by Aristotle himself to previous  mathematikoi  [the term usually applies to the elite followers of Phytagoras  (c.582-507 BC) but it has been argued that Aristotle could have meant to credit ancient Chaldean astronomers].  That tradition may help gauge the numerical breakthrough achieved by Eratosthenes.  It may also explain why Archimedes did not find it prudent to use the result of Eratosthenes in his famous  Sand Reckoner  essay.

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(2006-11-04)     Latitude and Longitude
Covering the Globle with a grid of parallels and meridians.

The idea of using a system of spherical coordinates to locate points on the Earth is credited to the Greek astronomer Hipparchus of Nicea (c.190-125 BC).  Hipparchus applied that idea to the heavens and produced a catalog of 1080 stars (Eratosthenes had previously listed 675 stars).

Latitude

There's no doubt that the notion of latitude is far more ancient than that.  Any bright shepherd who looks up at the sky several times during a single night, would notice that all star patterns revolve around a special point in the sky:  the celestial pole.  The angle from this direction to the plane of the horizon is the local latitude,  which can be measured to a precision of about 0.2° with elementary tools  (angular units need not be assumed; the result could be expressed as a fraction of a whole circle).  Even without such a formal measurement, this special angle could appear in local architecture as pointers to the celestial pole are erected and aligned by direct observation  (possibly for religious reasons).

By contrast, the next logical step was undoubtedly one of mankind's major prehistorical discovery:  "Latitude" (as defined above) changes from one place to the next!  The breakthrough was to have the idea that such a change might occur.  After that, observing it is comparatively easy...

The change is already noticeable after walking only 3 or 4 hours to the north or to the south (if you look carefully enough).  A major voyage would make it totally obvious...  We may thus guess that the modern notion of latitude is very old, since people have been navigating and observing changes in latitude for a very long time:

Sailing ships already traveled along the Nile river around 3100 BC.  Solid wooden boats existed before 6000 BC in Europe, skin and bark boats have been traced to 16 000 BC.  There's some evidence that people from Southeast Asia already had seagoing capabilities and sophisticated navigation skills as early as  60 000 BC  (some of them reached Australia and settled in Melanesia around  40 000 BC).  The Norwegian explorer Thor Heyerdahl (1914-2002) spent a lifetime proving that such prehistorical voyages where a practical possibility, starting with the celebrated  Voyage of the Kon-Tiki  in 1947.

Longitude

Longitude is a different story entirely.  Until reliable chronometers became available, longitude was mostly an intellectual construct based on the assumption that the Earth was spherical (or nearly so).  The difference in longitude between two points could only be estimated on firm land, by using surveying techniques after some fairly good knowledge of the size of the Earth had been gained to "calibrate" the whole process, like Eratosthenes did.  Hipparchus  (who was born as Eratosthenes died)  was thus in a position to make the notion of terrestrial longitude a practical proposition, by proposing the framework which has been underlying all efforts to map the Globe ever since.

However, more than 1600 years would pass before someone like Christopher Colombus would be willing to bet his life on the scholarly belief that the Ocean was small enough to sail through...

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(2006-10-17)     Itinerary Units: Land Leagues and Nautical Leagues
Matching land surveys and degrees of latitude at sea.

Perhaps the most interesting ancient itinerary unit is the  league.  It comes in two flavors, land league and nautical league.  Each of those has many definitions.

The Latin for "league" (leuga) comes from the Gallic leuca  [ not  the other way around ]  which was supposed to be equivalent to an hour of walking.  This land league was identified with 3 "miles" whenever and wherever some flavor of the "mile" was the dominant itinerary unit  (Roman mile, London mile, Statute mile).

The original "mile" was the military  Roman mile  of a thousand steps.  Each of those steps was properly a double-step (or  stride) which the Romans reckoned to be 5 (Roman) feet.

Land League(s) :

Officially, each flavor of the land league remained quite stable over time, although actual recorded measurements may show some lack of precision for both local land surveying and itinerary measurement.  Among the  many  "leagues" born in the Old World,  Roland Chardon singles out 5 which took hold in North America:

• French  lieue commune  of 3 Roman miles  (4444 m).
• French  grande lieue ordinaire  (3000 pas = 4872.609 m).
• French  lieue de poste  (2000 toises = 3898.0872 m).
• Mexican league,  legua legal
  (3000 pasos de Solomon = 5000 varas = 4191 m)
• Castilian legua común, legua regular antigua, modern legua
  (20000 pies de Burgos = 5572.7 m)

The Spanish system comes in different flavors whose basic units differ slightly, but all of them have 5 pies to the paso and 3 pies to the vara. The vara may also be subdivided into 4 cuartas or 8 ochavas.  The vara de Burgos was apparently first established in 1589, but was given its final metric definition (0.835905 m) only in 1852, as Spain was converting to the metric system.  It competes with the vara of California  (now identified with the ancient vara de Solomon)  which the Treaty of Guadalupe Hidalgo (1848) set to 33 inches (0.8382 m) to replace no fewer than 22 variants previously flourishing in California...  The so-called "vara of Texas" was defined in 1855  (3 of those are exactly 100 inches).

Nautical League(s) :

Each version of the nautical league was normally defined as a simple fraction of the (average) degree of latitude.  The nautical league which (barely) survives to this day is 1/20 of a degree (3 nautical miles) but another nautical league of 1/15 of a degree (4 nautical miles) used to be almost as common.  The ratio of the nautical units to the land units varied historically, as the accepted size of the Earth varied  (normally becoming more accurate with the passage of time). 

• Nautical league of 20 per degree  (one league is 3 modern nautical miles).
• Dutch or Spanish marine league of 15 per degree  (4 nautical miles).

In the early 1500s, these two were respectively equated to 3 and 4  Roman miles, which represents an underestimate of 20%, since a Roman mile is only 80% of a true  nautical mile.  That error was all but corrected by the mid 1600s.  The pre-metric value for the league "of 20 per degree" was  2850 toises  (5554.8 m).

The  conventional  modern value of the nautical league is 5556 m  (3 nautical miles of 1852 m).  The deprecated definition of the  nautical mile  as an "average minute of latitude"  is treacherous, because of the implied averaging over the surface of an oblate spheroid.  Also, "latitude" comes in two distinct flavors: geocentric and geodetic.

Still,  Livio C. Stecchini  argues that a "memory of the Roman calculation" of 75 Roman miles to the degree of latitude had been preserved trough medieval times.  This is so nearly perfect that it seems entirely too good to be true...

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(2008-03-10)     Amber, Compass and Lightning
The ancient mysteries of electricity and magnetism.

The word  electricity  comes from the greek word for  amber.  Amber is a translucent material which is actually hardened resin from pine trees.  It has one fascinating property:  If you rub it against wool, it attracts dust or dry leaves.  In modern terms, amber becomes negatively charged when rubbed.

The two kinds of electricity (negative and positive charges) were named by Benjamin Franklin (1707-1790).  They were first observed on various materials which, like amber, acquire a definite electric charge when rubbed.  For example, glass acquires a  positive  charge.  This phenomenon is known as  triboelectricity  (electricity produced by friction).

Electrostatic machines depend on it but the effect remains fairly difficult to quantify because it depends critically on a variety of factors which are difficult to control, including surface condition and humidity.  The following list, known as the  triboelectric series,  predicts fairly accurately (under typical conditions) which material will acquire a positive charge and which material will acquire a negative charge when they are separated after being rubbed on each other:  The earlier the material appears in the series, the more positive it will tend to be.

Magnetism, on the other hand, was first observed through the ability of a certain mineral to attract bits of iron.  The mineral was called  magnetite  because it was commonly found in a region named  Magnesia  (Central Greece).  The region gave its name to the rock  ( Fe₃O₄ )  the rock gave its name to the phenomenon.

Arguably, the  first scientific paper  ever written is a treatise on magnetism known as  Epistola de Magnete,  written in 1269 by the French scholar  Petrus Peregrinus  (Pierre Pèlerin de Maricourt ).  The notion of conservation of energy would emerge only  much  later, so we should forgive Peregrinus for believing magnetism could produce perpetual motion!  He was writing more than 3 centuries before the publication of  De Magnete (1600)  by  Sir William Gilbert (1544-1603)  [William Gylberde of Colchester].

One major contribution of Peregrinus was the observation that magnetic poles  (which he defined)  always come in opposite pairs.  A magnetic pole cannot be isolated from other poles of the opposite polarity; every piece of lodestone features both kinds of poles.  The modern statement  (for which no exceptions have been found to this day)  is that "there are no magnetic monopoles".  This is expressed mathematically by one of the four equations of Maxwell  (div B = 0).  Unlike the other three, that particular equation does not currently enjoy the privilege of a universally accepted specific name.  It's sometimes referred to as "Gauss's law for magnetism", which is dubious.  To be understood, I have been calling it the "Gauss-Weber law" myself, but it should really be called either the  Law of Peregrinus  or  Pèlerin's Law,  in honor of the scientific pioneer who first stated it  (in the language of his day).

Grosseteste

Petrus Peregrinus and the Dawn of Modern Science :

The  scientific method  [of comparing theories with observations]  was formally conceived by  Robert Grosseteste (1168-1253) at Oxford,  where he taught  Roger Bacon (1214-1294).  Bacon  and  Pierre Pèlerin de Maricourt  (Peregrinus)  belonged to the next generation, who would start practicing Science according to the rules laid down by  Grosseteste.

Roger Bacon's  own manuscripts  (c. 1267)  give high praise to  Peregrinus  (identified as  Magister Petrus de Maharn-Curia, Picardus)  whom Bacon had met in Paris  (apparently Bacon himself had no great interest in Science until he met Peregrinus).  Although most of the work of Peregrinus is now lost, we know that he was an outstanding mathematician, an astronomer, a physicist, a physician, an experimentalist and, above all, a pioneer of the  scientific method...  He may have been described as a recluse devoted to the study of Nature, but he was actually a military engineer who, in the words of Roger Bacon, was once able to assist Saint Louis  (Louis IX of France, 1214-1270)  "more than his whole army"  (Peregrinus apparently invented a new kind of armor).

De Maricourt's  Epistola de Magnete  was  "done in camp at the siege of Lucera, August 8, 1269".  Peregrinus was then serving the brother of  Saint-Louis,  Charles of Anjou, King of Sicily.  No other biographical details are known.

• Petrus Peregrinus de Maricourt and his Epistola de Magnete
  by  Silvanus P. Thompson, D.Sc., F.R.S.  (1906)
  Proceedings of the British Academy, Vol. II.  Oxford University Press.
• The Letter of Petrus Peregrinus on the Magnet, A.D. 1269
  translated by Brother Arnold, M.Sc.
  Introduction by Brother Potamian, D.Sc.  (1904).  Digitized in 2007.