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The above histogram starts with the birth of  Thales,  the earliest major scientist we can name  (c. 624 BC).  Scientific achievements of equal or greater magnitude predate Thales by at least  1200  years,  as the  cuneiform  inscription on a  famous clay tablet  goes to show:

(2018-09-14)     Mesopotamian Trigonometry   (Larsa,  c. 1800 BC)
Sexagesimal numbers recorded in cuneiform script on a famous tablet.

Apparently,  the tablet discussed below was first analyzed in:

Mathematical Cuneiform Texts   (177 pages).  American Oriental Series  # 29.
by  Otto E. Neugebauer (1899-1990)  and  Abraham Joseph Sachs (1914-1983).
Published jointly by the  American Oriental Society  and the
American Schools of Oriental Research  (New Haven, CT, 1945).

This is a burned clay tablet measuring  13 cm  by  9 cm.  Its style indicates it's from the kingdom of  Larsa;  dating it between  1822 BC and 1784 BC. 

The headings at the top of each column of  sexagesimal numbers  are written in  Akkadian  with some Sumerian  words  (as was the usual practice at the time).  We're happy to ignore them entirely.  Let the numbers speak for themselves...

This famous tablet is known far and wide as  Plimpton 322.  We'll simply call it the  Larsa tablet  in the following discussion.  It gives very special examples of what we now call  Pythagorean triples:  Integers which form the two sides and the  diagonal  of a rectangle:

h ²  +  w ²   =   d ²

The larger of the two sides  (h)  isn't listed in the  Larsa tablet,  but we've restored it in the following transcription,  as a new column  shaded in grey ,  after the  rightmost inscriptions  from the original tablet  (namely,  the line number from  1  to  15,  preceded by a word which translates as  row).  That  grey number ,  henceforth called  h,  is always a  regular sexagesimal number,  namely an integer whose reciprocal can be expressed exactly in the sexagesimal numeration system used by the Sumerians  (in this context,  the term  regular  seems due to  Neugebauer).  So,  the ratio  w/h  can be given  exactly  in sexagesimal and this is what appears in the first column  (there's no terminating decimal expression for this,  in any of the listed cases).  The second  (w)  and third  (d)  columns  are just  integers  which we chose to express in decimal,  rather than sexagesimal,  for the sake of modern readers  (all bases of numeration are equivalent for integers).

In two  highlighted cases ,  the ancient table forwent  coprime integers  in favor of something sexagesimally simpler:  As  60  is simpler than  4  in this context,  the celebrated 3,4,5  Pythagorean triple  appears here as 45,60,75.

Against clear photographic evidence  (consider especially rows 10 and 11)  some scholars think that a leading "1" sexagesimal digit once appeared to the left of all entries.  That would make the listed values correspond to:

(d/h)²   =   1  +  (w/h)²

This is utterly unimportant,  unless the table is extended to include very low values of  w/h  which would normally entail a leading zero sexagesimal digit  (something cuneiform notation would not handle properly until 300 BC or so).  We'll come back to that.

Putting it in a broader perspective :

All entries on the  Larsa tablet  are based on  even  values of  h,  with the exception of its last row  (#15)  which would correspond to an  odd value of  h  if simplest terms had been given  (as is the case for the equivalent entry  highlighted  in the table below).

Odd values are not only possible but some of them should have been particularly appealing to Mesopotamians  (e.g.,  the triple  8,17,15  which achieves ultimate sexagesimal simplicity in the form  32,68,60).

Such solutions are all of the following form,  where  a  is a power of  5  and  b  is a power of  3  (or vice-versa).

• w   =   ( a² - b² ) / 2
• d   =   ( a² + b² ) / 2
• h   =   a b

Those give positive solutions with  w < h   (in the spirit of the tablet)  iff :

Ö2 - 1   <   b / a   <   1

The next two  odd  values of  h  divide no power of  60  below the  8-th  power.  The  16  sexagesimal places required to express the associated squared ratios,  won't fit into the above table.  So,  we'll give those two in extenso,  which will also serve to explain how all entries are constructed...

If a ratio uses  n  sexagesimal digits  (after the  virtual  radix point,  which is never made explicit)  then its square requires  2n  digits,  unless  the trailing digit was a  "30"  (half of the exagesimal base)  in which case only  2n-1  digits suffice,  because squaring then creates a zero trailing digit which the Mesopotamians never recorded.  (Incidentally,  such trailing zeroes are optional in modern decimal,  except to indicate the  precision  of an approximative quantity.)  One example is given by the first line of the  Larsa tablet  where  .59.00.15  (only 3 digits)  is indeed the square of  .59.30  (2 sexagesimal digits).  Two more examples are found on lines  2  and  12.

• If  h   =   3⁸ 5⁵   =   20503125   then:
• w   =   (3¹⁶ - 5¹⁰) / 2   =   16640548
• d   =   (3¹⁶ + 5¹⁰) / 2   =   26406173
• (w/h)²   =   .39.31.21.38.45.33.27.10.11.46.35.23.27.24.26.40

• If  h   =   3⁸ 5⁶   =   102515625   then:
• w   =   (5¹² - 3¹⁶) / 2   =   100546952
• d   =   (5¹² + 3¹⁶) / 2   =   143593673
• (w/h)²   =   .57.43.03.41.04.34.12.58.25.59.47.10.25.11.06.40

The latter example yields the second-largest value of the  w/h  ratio seen so far  (just behind the top row of the  Larsa tablet).

The two families of solutions so illustrated are  infinite.  They start with:

In either case,  the power of  3  (denoted  n  in both parts)  gives the number of sexagesimal digits required to express the ratio  w/h  (not its square).  The two  highlighted values of  8  correspond to the two examples we gave in extenso and the seven smaller values are associated with the entriee in the previous table  (including  #15  from the  Larsa tablet).

---

Let's go back to the even values of  h  which provide the vast majority of the solutions.  There are now  three  possible prime factors for  h  (2,3,5).  So,  the classification is messier than what we just gave for the odd case  (where there are only two of those)  but the guiding principles are the same.

We must now use the following formulas,  where u and v are coprime divisors of  60ⁿ  not both odd,  for some parameter  n :

• w   =   u² - v²
• d   =   u² + v²
• h   =   2  u v

Positive solutions obeying   w < h   are obtained when the following holds:

Ö2 - 1   <   v / u   <   1

On the  Larsa tablet,  the two longest entries  (#2 and #10)  correspond to  n = 4.  Only entry #15 corresponds to an odd value of  h,  as already noted.  The remaining  12  entries correspond to  n = 1, 2 or 3, with an even  h.  There are  37  such possibilities  in toto.  Here are the  25  others:

As the  Larsa tablet  lists only the largest ratios in decreasing order,  most of the above wouldn't be expected in it,  with the notable exception of the top one  (the  highlighted entry   67319 : 98569)  which would rightly belong between the original entries  #4  and  #5  (let's call it  #4½).

Enumeration of the Solutions,  when  h  divides  60ⁿ

The enumeration is easy if we lift the restriction  w < h   and do not impose any positive lower-bound for the ratio  w/h.  Let's do that,  at first:

Positive solutions for odd values of  h  are obtained in this form:

h   =   a b       and       w   =   ½  | a²-b² |

In this,  the parameters  a  and  b  are  coprime  odd  divisors of  60ⁿ  (that's to say,  they are coprime divisors of  15ⁿ ).  To enumerate  positive  solutions,  we rule out  a = b = 1  and assume  WLG  that  b  is not divisible by  3.  Thus,  we find every solution uniquely in one of the two following way,  from an ordered pair  (p,q)  of parameters between  0  and  n :

• a = 3^p   and   b = 5^q   with  q  nonzero.
• a = 3^p 5^q   and   b = 1   with  p  nonzero.

Therefore,  there are   2 n (n+1)   solutions for odd values of  h.

On the other hand,  positive solutions for  even  h  are given in the form:

h   =   2 u v       and       w   =   | u²-v² |

In this,  u and v are coprime divisors of  60ⁿ  which are not both odd.  We may thus assume  WLG  that  u  is even  (nonzero)  and  v  is odd.

We may choose any value  m  between  1  and  2n-1  for the exponent of 2.

Let p and q be the respective exponents of 3 and 5  (both between 0 and n,  unless further restrictions are specified).  All solutions are obtained uniquely in one of the following four  mutually exclusive  forms:

• u = 2^m 3^p 5^q   and   v = 1.
• u = 2^m 5^q   and   v = 3^p   with  p  nonzero.
• u = 2^m 3^p   and   v = 5^q   with  q  nonzero.
• u = 2^m   and   v = 3^p 5^q   with  p   and  q   both nonzero.

Those four cases yield the following number of possibilities,  for even  h :

(2n-1) [ (n+1)² + 2 n (n+1) + n² ]   =   (2n-1) (2n+1)²

Adding this to our previous result,  we obtain the total number of solutions:

8 n³  +  6 n²  -  1

Only a small fraction of those satisfy the  Larsa constraint  (h > w):

What the decimal equivalent would be :

If  B  is the  base of numeration  (radix-B numeration)  a number whose expansion terminates  n  places after the radix point becomes an integer when multiplied into the  n-th  power of  B.  So,  the reciprocal of an integer terminates in radix  B  if and only if that integer divides some power of  B.  Such integers are precisely those whose  prime factors  are prime factors of  B.  That's 2, 3 and 5 in sexagesimal and just 2 and 5 in decimal.

In sexagesimal,  those are the integers whose prime factors are all less than or equal to  5.  They're now sometimes called  5-smooth.

Decimal :  1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1024, 1250, 1280, 1600, 2000 ... (A003592)
 
Sexagesimal :  1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150 ... (A051037)

Babylonian numbers (4:59)  by  Eleanor Robson  (Cambridge,  2011-03-18).
The Babylonian tablet Plimpton 322  by  Bill Casselman  (M446, Fall 2003).
 
News Releases and News Reports :  (perspective and/or conclusions are not entirely reliable)
Tablet Plimpton 322 (20:49)  by  Norman Wildberger  &  Daniel Mansfield  (2017-08-24).
Cuneiform Tablet and Babylonian Trigonometry (2:09)  Trinity Chavez  (RT, 2017-08-25).
Ancient Babylonian Tablet Deciphered after 70 Years (4:09)  ShantiUniverse  (2017-08-28).

(2009-12-11)     The [First] Theorem of Thales  (intercept theorem).
The  fundamental  theorem of (classical) geometry.

Quand l'ombre de l'homme sera égale à l'homme,
l'ombre de la Pyramide sera égale à la Pyramide.
 Bernard Lefèbvre,  lecturing on  Thales  (1973)

Thales of Miletus  was born in the seventh century BC.  An engineer by trade, he is the first of the  Seven Sages of Greece.  Thales is credited with the first rational speculations about Nature  (natural philosophy).

The advent of natural philosophy was a fundamental step on the way to a real understanding of Nature, compared to the primitive approach of "explaining" everything by divine intervention  (a viewpoint which is arguably still with us, unfortunately).  This became  modern physics  only with the revolutionary introduction of the  scientific method  of comparing speculations and observations!

Thales is also touted as the founder of  classical geometry,  although some of it predates him  (including the construction with straightedge and compass of the circle circumscribed to a triangle, by the Phrygian mathematician Euphorbus).

Legend has it that Thales was asked to tell the height of the  Pyramid  (possibly, the Pyramid of Cheops).  His answer came down to me  (via my high-school philosophy teacher)  in the eloquent form quoted at the beginning of this section.  Consider the  shadow  of the Pyramid and the shadow of a man  (or, rather, the shadow of a vertical pole whose height is easy to measure).  Here's the key:

[First]  Theorem of Thales :  If the corresponding sides of two triangles are parallel, the triangles are  similar  and the lengths of their sides are  proportional.

[ Pause ]

How does this help?  The two shadows may be proportional to the two heights and we can quickly measure the shadow and the height of the vertical pole  but  we know  neither  the height of the Pyramid nor the length of its shadow!  Think about it:  You are by yourself in this flat desert with your graduated yardstick next to a pole of known height.  How can you find the height of the Great Pyramid?

One solution is to look for triangles which do not involve the inaccessible center of the pyramid, as presented in the following picture  (courtesy of  Andrew Weimholt,  2013-11-21).

There's a  rudimentary  way to forgo any delicate sighting alignment or the measurement of long horizontal distances.  Can you guess what it is?

[ Answer ]

The geometry of Thales was formalized by Euclid three centuries later.  For over two millenia, it was thought to apply to our  physical  Universe.  The universe of classical geometry is postulated to be homogeneous  (Euclid's  fourth postulate  states that all right angles are equal)  and unaffected by scaling  (that's what Euclid's  fifth postulate  really means).

The scale invariance of the microscopic Universe was questioned by the ancient Greeks  (Democritus did speculate the existence of indivisible "atoms" with a definite size)  but it was thought to be an idiosyncracy of the  content  of the Universe...  Noneuclidean geometries were not even considered before the 19-th century.  However, we know now that the large-scale structure of our physical Universe is indeed noneuclidean  (this is a consequence of  General Relativity).  Yet, this conclusion could not have been reached without the likes of Thales and Euclid who set an ideal to compare against.

(2021-08-08)   Berlin Papyrus 6691 and Rhind Papyrus
A glimpse of what the ancient Egyptians knew.

Papyrus isn't nearly as robust as clay but we have more mathematical papyri than clay tablets from roughly the same historical period.  Partly because the ancoent Egyptian valued  recreational mathematics  more than contemporary societies did...

Moscow Papyrus  (c.1850 BC)   |   Vladimir Golenishchev  (1856-1947)
 
Berlin Papyrus 6691  (c.1800 BC)   |   Hans [Graf von] Schack-Schackenburg
 
Rhind Papyrus  by  Ahmes  (c.1550 BC, copied from c.2000 BC)   |   Alexander Henry Rhind (1833-1863)

(2016-06-02)   Anthyphairesis.  Pre-Eudoxian ratio theory.
The  Euclidean algorithm  predates  Euclid  by centuries.

Continually subtract the smaller from the larger.

Logically, what we now call  Euclid's algorithm  (formerly  anthyphairesis )  coprimality  and  Bézout's lemma  all come before the conceptualization of  prime numbers  and the  fundamental theorem of arithmetic  (whereby any positive integer has a unique factorization into primes).

The historian  David Fowler  has argued convincingly that this order of precedence was also a  chronological  one, during the early development of mathematical concepts in ancient Greece, centered on Plato's Academy...

Continued Fractions  by  Dr. Paul R. Hewitt,  University of Toledo  (2009-03-25).
 
"The Mathematics of Plato's Academy: A New Reconstruction"  by   Dr. David H. Fowler (1937-2004).

(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 87-165) reports that  Eratosthenes of Cyrene  (276 BC-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 this 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 time of Eratosthenes, the Tropic of Cancer was about 17 nautical miles (30 km)  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 used extremely accurate data from those earlier times...  This is either far-fetched or completely ludicrous.

(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".

Eratosthenes (6:41)  by  Carl Sagan  (Cosmos, Episode 1, 1980-02-28).
In this classic video, ignore Sagan's assertion that Eratosthenes had to "pay a man" to pace the distance between Alexandria and Syene  (this was already a well-known survey result at the time).  Do not let that detract you from the great moment of truth when Sagan bends the map and all falls into place.

(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, when his teacher Callimachus died.

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 saw 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 says 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 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, wittingly or unwittingly, 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.

---

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:  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 treated Eratosthenes as a peer.

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 didn't find it prudent to use the result of Eratosthenes in his own  Sand Reckoner  essay.

Details on the above work of Eratosthenes about the circumference of the Earth are only known to us through the astronomy textbook  Caelestia  (The Heavens)  by  Cleomedes,  which was written between AD 320  and  420,  according to  Otto Neugebauer (1899-1990).

(2019-01-03)     The Earth is Round
The ancient Greek scholars already knew that.  How can we be so sure?

This would have been a silly question a few short years ago,  since we now have pictures taken from distant outer space which leave absolutely no doubt concerning the  spherical  shape of the Earth  (careful analysis would reveal that it's closer to an  oblate spheroid).

As post-modern  pop culture  favors  bullshit receptivity  so much  (in a sense coined by the philosopher  Harry G. Frankfurt  in 2005)  in may now be useful to offer a quick justification for what shoukd be common knowledge:

The  above  angular measurements of Eratosthenes  involve  two  distant wells.  They enabled him to estimate the circumference of the Earth assuming it was round.  That would still be compatible with the hypothesis of a  flat Earth  if the Sun was not very distant.

With  three  wells or more that ambiguity would be lifted,  leaving only one possible explanation compatible with the observed angles:  All rays from the Sun are nearly parallel  (because the Sun must be very distant)  but they make different angles with the verticals at diferent locations with have different directions because of the  curvature  of the Earth surface.

Another  reliable clue that the Earth must be spherical is that it alwasys casts a circular shadow on the Moon during solar eclipses.

Observations within the atmosphere are nice too  (boats disappear hull first)  but they can be less reliable because air can easily curve light downward or upward,  depending on meteorological conditions:

The ever-present downward pressure-gradient is already enough to bend light-rays downward by about  half  of what's needed to make the surface of the Earth look flat.  When the temperature gradient is also downward  (e.g.,  warm air over cold water)  light rays can curve  more  than the surface of the Earth's surface  which makes the surface of the Ocean  look  like the inside surface of a shallow bowl.  In that case,  you can  see  boat hulls and islands well beyond the  geometric horizon.

The Absurdity of Flat Earth (8:59)  by  Neil deGrasse Tyson  (Star Talk,  2018-03-09).

(2010-07-04)     Computing the distance to the Moon
Aristarchus used lunar eclipses.  Hipparchus used solar eclipses.

Around 270 BC,  Aristarchus of Samos  remarked that the angular size of the shadow cast by the Earth on the Moon's orbit  (readily obtained by timing the maximum duration of a lunar eclipse)  gave the ratio of the size of the Earth to the Earth-Moon distance.  From this, he correctly deduced that the distance to the Moon was about  60  Earth radii.

The size of the Earth itself would be estimated later by Eratosthenes.

Hipparchus of Nicaea (c.190-126 BC)  confirmed that result independently by noting that a total solar eclipse over a known remote location (see below) was observed in Alexandria as a partal eclipse leaving  1/5  of the solar width  (30'  or 0.5°)  still visible.  So, the angular separation between those two earthly locations, seen from the Moon, was about  6'  (0.1°).

Assuming a knowledge of the two positions on Earth,  Hipparchus  (who invented trigonometry!)  could deduce the distance to the Moon  (as 573 times the distance separating two parallel sunrays through the two locations).  He  reportedly  deduced that the Moon's distance was no less than 59 times the radius of the Earth  (close to the modern number of 60.3).

---

Where and When ?

There is some debate concerning the date and location of the solar eclipse used by Hipparchus.

In the lifetime of Hipparchus, only one  total solar eclipse  occurred over  Syene  (technically, it was an annular eclipse).  It took place on August 17, 180 BC  at a time when Hipparchus was probably just a boy.  The previous total solar eclipse over Syene had occurred 80 years before, on September 16, 260 BC.  (Courtesy  Fred Espenak  of NASA, July 2003.)

However, it seems that Syene was not involved at all in this.  Reports to the contrary are probably simply due to a confusion with the related story about Eratosthenes estimating the size of the Earth.  Instead, Hipparchus reportedly used an eclipse over the  Hellespont  (the  Dardanelles strait).  It was most probably the solar eclipse of November 20, 129 BC  (astronomers assign the negative number -128 to the year 129 BC, just like they assign the number 0 to the yesr 1 BC, which came just before  AD 1).  Previous total or annular solar eclipses over the Dardanelles took place in 190 BC, 263 BC, 310 BC and 340 BC.

Earth-Moon distance by Hipparchus  (Cornell University)   Eclipse over Syene [?]
How distant is the Moon?  by David P. Stern  (NASA)   Eclipse over the Hellespont.

(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 Hipparchus of Nicaea (c.190-126 BC)  who first used it to map the heavens.

Latitude :

There's no doubt that the notion of latitude is extremely ancient.  Any smart shepherd who looks up several times in a single night, would notice that all star patterns revolve around a special point in the sky:  the celestial pole.

The celestial north pole is currently close to the position of the star we call  Polaris  or  North Star.  This wasn't always so, because of the precession of equinoxes  (discovered by Hipparchus in 130 BC):  The polar axis varies slowly, like the axis of a spinning top does.  In the main, over a period of about 26000 years,  it goes around a large circle of angular radius  e » 23.44°  (which is the  mean obliquity of the ecliptic ).  About 14000 years ago, the bright star  Vega  (a-Lyrae, magnitude 0.03)  was only  3°86'  from the celestial pole  (that angle is still more than 7 times the width of the Moon).
 
For completeness, note that the axis of the Earth oscillates around the position predicted by the above circular motion, just like the axis of a spinning top does  (nutation motion).  This translates into a periodic variation of the obliquity of the ecliptic, which astronomers approximate with a polynomial function of time, valid for a few centuries.

The angle between the celestial pole and 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 formal measuring, this special angle could be materialized by erecting pointers to the celestial pole, 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 the idea that such a change  might  occur.  After that,  actually observing it is relatively 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 were 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 when Eratosthenes died)  was thus in a position to make the notion of terrestrial longitude a practical proposition.

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...

(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 with 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  (equal to 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 was preserved trough medieval times.  This is so nearly perfect that it seems entirely too good to be true...

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

The word  electricity  comes from the greek word for  amber  (hlektron).  The  new latin  word  electricus  was coined by William Gilbert  in  De Magnete  (1600)  to denote the basic triboelectric properties of amber:

Amber is a transparent material consisting of hardened resin from conifers  (mostly of the family Sciadopityaceae that flourished in the Baltics 44 million years ago).  If you rub it against wool, polished amber attracts nearby dust or dry leaves.  Thales of Miletus (c. 624-546 BC) recorded that observation around 600 BC.

About  80%  of the World's amber  (100 000 metric tons)  was produced 44 million years ago in forests of the Baltic Region.  Natural Baltic amber  is sometimes known as  succinite  because it contains from 3% to 8% of succinic acid  (a solid also known as "spirit of amber").  Succinic acid is the third simplest dicarboxylic acid; its formula is  HOOC(CH₂ )₂ COOH  (butanedioic acid)  and it plays an essential metabolic rôle in the  Krebs cycle.

In 1620, Niccoló Cabeo (1586-1650)  observed that two electrified objects can either attract or repel.  An electrified object always attracts an unelectrified one.

In 1733, Charles François du Fay (1698-1739)  discovered that there are actually two  opposite  types of electrical charges, which he called  resinous  and  vitreous.  Unlike charges attract each other, like charges repel.

The attraction exerted by either kind of charges on any unelectrified object is due to the  influence  of a charged body on a neutral one.  Charges are redistributed within the latter:  Unlike charges are pulled closer and like charges are pushed away.  As distant charges have a lesser action than close ones, there's a net pull.

We now speak of negative charges  (resinous)  and positive charges  (vitreous)  according to the arbitrary algebraic sign convention which was introduced before 1746 by Benjamin Franklin (1706-1790)  to formulate the fundamental principle of  conservation of electric charge which is attributed jointly to him and to the British scientist William Watson (1715-1787).

Various materials acquire a definite electric charge when rubbed.  Amber becomes negatively charged.  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 precisely, because it depends critically on a variety of factors which are tough to control  (e.g., 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 against each other:  The earlier the material appears in the series, the more positive it will tend to be.

For many century, magnetism was perceived as a phenomenon unrelated to electricity.  Legend has it that it was first observed around 900 BC  (by a Greek shepherd called  Magnus)  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 Peregrinus should be forgiven for his misguided belief that magnetism might produce perpetual motion!  He was writing more than three centuries before  Sir William Gilbert (1544-1603)  [William Gylberde of Colchester]  published  De Magnete (1600).

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"  (the simplest magnetic distribution is a dipole).  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've 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 (1214-1292).  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  whom Bacon had met in Paris  (however, the object of that praise is only unambiguously  identified as  Magister Petrus de Maharn-Curia, Picardus  in a marginal gloss of a copy of Bacon's  Opus Tertium, which may have been added by someone else).  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 aforementioned words of Roger Bacon, was once able to help  Saint Louis  (Louis IX of France, 1214-1270)  "more than his whole army"  (as Peregrinus seems to have invented a new kind of armor).

One of the 39 extant copies of  De Maricourt's  Epistola de Magnete  attests that it 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.  The letter is adressed to a fellow soldier called  Sygerus of Foucaucourt  who was clearly a countryman/neighbor of  Pierre,  back in  Picardy  (the village of Foucaucourt is 12 km to the south of the village of Maricourt, across the  Somme  river).

• 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.

Wikipedia :   Petrus Peregrinus  de Maricourt.

(2017-05-17)     The Antikythera Mechanism
The earliest known  orrery.  (French:  La machine d'Anticythère.)

On 1902-05-17,  the Greek archaeologist Valerios Stais (1857-1923) spotted gears in a corroded chunk of metal recovered in 1901 from a wreck located  off Point Glyphadia on the island of Antikythera, West of the  Sea of Crete.

This was part of an ancient mechanical computer,  now known as the  Antikythera Mechanism.  Marking the  115^th  anniversary of this discovery,  Google  showed the following  Doodle  on their homepage, on May 17, 2017:

That intriguing artefact was found in the remains of a wooden box measuring appeoximately  34×18×9 cm.  It captured the imagination of generations of scholars.  It's the earliest known geocentric  orrery.  It comprised at least  37 gears.  The largest of these  (e3)  was about 13 cm in diameter,  with  223 teeth  (there are 223 synodic months in a  Saros). and has

In 1951,  Derek J. de Solla Price (1922-1983)  was the first major scholar to take a serious interest in the mechanism.  In 1974,  he concluded that it had been manufactured around  87 BC.

The Corinthian origin of the mechanism can be established by the names of some of the months engraved on it.  It was most likely manufactured in Corinth itself or one of its major colonies:  Corfu (Korkyra)  or  Syracuse,  home of  Archimedes (c. 287-212 BC)

Several concurring clues indicate that the shipwreck took place around  60 BC.  In 1976,  the team of  Jacques Coustau (1910-1997)  found coins in the wreck which were dated between 76 BC and 67 BC.  The amphorae are only slightly more recent.

The nature of the cargo seems to indicate that the ship was bringing Greek loot to Rome  (Ostia).  Few harbors could accomodate a commercial ship of that size.  This leaves only few possibilities for the origin of her last voyage:

• Kos  and/or  Rhodes.
• Athens  (Piraeus).
• Pergamon.
• Ephesus  (Efessos).

Tooth Counts in the Antikythera Mechanism :

The tooth-count of the  gears  of the  Antikythera Mechanism  allow the reproduction of heavenly motions using the best rational approximations known to the Ancients.  Some prime numbers are prominent  (A240136):

• 19: The approximate number of solar years in a Metonic cycle.
• 53: See below.
• 127: The approximate number of  sidereal  months in  half  a saros.
• 223:  The  exact  number of  synodic months  in a  saros.
• 235  =  5 . 47  : The number of lunar months in a Metonic cycle.

The  saros  (also called 18-year cycle)  is  defined  as exactly equal to 223 mean synodic months.  That's approximately  6585.321  mean solar days or 18.0296 Julian years.

The  eureka moment  of Tony Freeth can be summarized by the equation:

(9 . 53) / (19 . 223)   =   477 / 4237   =   0.11257965541656832664621...

It could have been discovered immediately by expanding an approximation of the right-hand-side as a  continued fraction,  but Tony's candid account of his own discovery of  53  (by trial and error)  makes for better video footage, albeit poorer mathematics.

Pin-and-Slot Couplings :

One remarkable device used many times in the  Antikythera Mechanism  is the  Pin & slot  coupling  (first uncovered by  Michael Wright)  whereby a pin parallel to the axis of a gear slides in a slot carved in another gear revolving around a slightly different center.

The average rate of rotation of two gears so coupled will be the same but the two rotations are not uniformly related to each other.

Bibliography   |   Michael T. Wright (1948-)   |   Allan G. Bromley (1947-2002)
 
The Antikythera Research Project   |   Mike Edmunds   |   Tony Freeth   |   Alexander Jones   |   Roger Hadland   |   Yanis Bitsakis   |   Xenophon Moussas
 
Wikipedia :   Orrery   |   Antikythera wreck   |   Antikythera mechanism
 
Return to Antikythera (2016).  Hellenic Ministry of Culture and Sports.
15 Intriguing Facts about the Antikythera Mechanism  by  Kristina Killgrove.
La fabuleuse machine d'Anticythère  by  Philippe Nicolet  (2015-07-03).
The Antikythera Mechanism: A Shocking Discovery from Ancient Greece  by  Tony Freeth  (2015-11-06).
 
"Making the Antikythera Mechanism" by Chris | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 (Clickspring, 2017-2018)

(2018-03-23)     Ancient Music and Musical Instruments
Monochord, lyre, kithara, barbiton.

A Brief History of Elementary Music Theory

• Pythagoras  of Samos (c.569-475 BC):  The Monochord.
• Fr. Marin Mersenne (1588-1648):  "Harmonie universelle" (1636).
• Hermann von Helmholtz (1821-1894).  Sensation of Tone (1863).

The Ancient Greek Kithara (9:03)  by  Peter Pringle  (2015-10-26).
c.1400 BC:  The Oldest Known Melody (5:42)  by  Michael Levy  (2009-05-13).
 
Hurrian Songs
 
Terpender (c.700 BC)   |   Archilochus (c.680-645 BC)   |   Arion (fl. 615 BC)

(2021-07-06)     The Pythagoreans
The cult of Pythagoras and the discovery of  irrational numbers.

Any religion consists of two distinct parts:

• Religious beliefs.  Mythology and theology.
• Religious practice.  Rules, rituals and festivals.

The Pythagoreans beliefs were derived from the Orphic form of the cult of Dionysos.  Their srict practices included an obsessive respect for beans,  believed to be a repository for some human souls.  Much of that was ridiculed by outsiders,  even in ancient times.

Pythagoras (c.569-c.495 BC)  received at least part of his mathematical education from  Anaximander of Miletus (610-546 BC)  head of the Milesian school founded by  Thales (c. 624-546 BC)  whom the young Pythagoras also met. 
 
On religious matters however,  Pythagoras followed the teachings of  Pherecydes of Syros  whose main tenet was the immortality of the soul through  metempsychosis.

Pythagoras was the first person in Greek history to call himself a  philosopher  (literally, a "lover of wisdom").  He was also the first to propose the momentous idea of a  spherical Earth,  which Plato accepted on aesthetical grounds before Aristotle gave three solid physical arguments for it:

• The celestial North is closer to the horizon after a long journey South.
• In any lunar eclipse, the shadow of the Earth is always perfectly round.
• Departing boats disappear behind the horizon  hull first.

The father of Pythagoras, called Mnesarchus, hailed from Tyre and was granted citizenship in Samos because he had brought corn to the city in a time of famine.  He settled in Samos as a gem engraver.  Mnesarchus was a religious man who consulted Pyhtia, priestess of Appollo  (the Oracle of Delphi)  who told him that his wife,  ,  was pregnant with a child destined to great fame.  Upon hearing this the mother changed her own name from Parthenas to Pythais and they chose to name their son Pythagoras after the Pythia.

Pythagoras was probably the main source for the first two books in  Euclid's Elements  but,  over time,  his achievements became considerably exaggerated,  casting much doubt on biographies which were written several centuries after his death.  Like Socrates,  Pythagoras didn't leave any written work of his own,  Unlike Socrates,  who was heralded by Plato and Xenophon,  Pythagoras wasn't written about by any of his followers,  who were not encouraged to do so  (to say the least).  The only extant contemporary account is a short excerpt from  Heraclitus of Ephesus (c.535-475 BC)  who was 35 years younger than Rythagoras.  Heraclitus wrote:

"Pythagoras, son of Mnesarchus, practiced inquiry more than any other man, and selecting from these writings he manufactured a wisdom for himself; much learning, artful knavery."

Samos  is an island off the coast of mainland  Anatolia  (Asia Minor)  were  Miletus  is located  (in modern-day Turkey).  Samos and Miletus waged a  war against each other in  440/439 BC.  Miletus was under the military protection of Athenian forces commanded by  Pericles.

A few notorious  Pythagoreans:

• Pythagoras of Samos  (c.569-c.495 BC).
• Hippasus of Metapontum  (c.530-c.450 BC).
• Philolaus of Croton  (c.475-c.385 BC)
• Archytas of Tarentum  (428-347 BC)
• Eudoxus of Cnidus  (408-355 BC)
• Nicomachus of Gerasa  (c. AD 60-c.120)

Cult of Hesiod   |   Cult of Dionysus (Bacchus)   |   Orphism nbsp; |   Pythagoreanism
 
The Dangerous Ratio  (c.520 BC)  by  Brian Clegg (1955-)  in NRICH  (2004, 2009).
 
What was up with Pythagoras? (8:49)  by  Vi Hart  (2012-06-12).
Pythagoras: Life's Music and Mathematics (1:10:44)  by  Michael Griffin  (2020-10-01).
The Pythagoreans and the Orphic mystery religion (29:58)   Leonard Peikoff  (Ayn Rand Institute, 2020-05-01).
10 Strange Facts About Pythagoras  by  Mark Oliver  (And My Quill, 2017-04-26).