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(2018-09-14)    
Sexagesimal numbers recorded in cuneiform script on a famous tablet.

Apparently,  the tablet discussed below was first analyzed in:

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. 

This famous tablet is known far and wide as Plimpton 322. We'll simply call it the Larsa tablet  in the followingdiscussion.  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 sexagesimalnumeration 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 theleft 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 includevery low values of  w/h  which would normally entail a leading zerosexagesimal 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 tablebelow).

Odd values are not only possible but some of them should have been particularlyappealing 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   =   (a²b²) / 2
• d   =   (a²b²) / 2
• h   =  a b

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

  <  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  h   =   3⁸5⁵   =   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   =   3⁸5⁶   =   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 areassociated 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 majorityof 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 thereare only two of those)  but the guiding principles are the same.

We must now use the following formulas,  where u and v are coprime divisorsof  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:

  <   v/ u   <   1

On the Larsa tablet,  the two longest entries  (#2 and #10)  correspondto  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 donot 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   =     |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^p5^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^m3^p5^q   and   v = 1.
• u = 2^m5^q   and   v = 3^p   with  p  nonzero.
• u = 2^m3^p   and   v = 5^q   with  q  nonzero.
• u = 2^m   and   v = 3^p5^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 pointbecomes 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.

(2009-12-11)    
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 theseventh 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).

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

Legend has it that Thales was asked to tell the heightof the Pyramid  (possibly, thePyramid of Cheops). His answer came down to me  (via my high-school philosophy teacher) in the eloquent form quoted at thebeginning 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]  : If the corresponding sidesof 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 andwe 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 nextto a pole of known height.  How can you find the height of theGreat Pyramid?

One solution is to look for triangles which do not involve the inaccessiblecenter 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 horizontaldistances.  Can you guess what it is?

[Answer ]

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

(2021-08-08)  
A glimpse of what the ancient Egyptians knew.

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

(2016-06-02)  
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 comebefore 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 precedencewas also a chronological  one, during the early developmentof mathematical concepts in ancient Greece, centered onPlato's Academy...

(2006-10-19)    
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 Suncasts the shortest shadows. On the summer solstice (June) and on the winter solstice (December)  the Sun's raysmake two different angles with the local vertical. The difference between these angles is always twice  the obliquity of the ecliptic.

ClaudiusPtolemy(AD 87-165) reports that Eratosthenes of Cyrene (276 BC-194 BC)  had estimated the obliquityof 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  

This means that, in the time of Eratosthenes, the Tropic of Cancer wasabout 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 byEratosthenes  (11/83)  around 1347 BC. Some have argued, backwards, that Eratosthenes did not measure theobliquity 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)    
A vertical well in Syene is completely sunlit only once a year...

This ancient observation may have been part of the Egyptian folklore in thetimes 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 theedgeof the Sun was lighting up the entire bottom of a vertical wellat Syene, just for a brief moment at noon on the summer solstice. So, thecenter of the Sun must have been directly overhead ata 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. Theabove formula says that this happened about33 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 tiltedby less than half a degree in the proper direction... Any casual (or not-so-casual) observer will swear such a well to be "vertical".

(2006-10-14)    
The size of the Earth, according toEratosthenes (276-194 BC).

Eratosthenes of Cyrene  became librarianof the Great Library of Alexandria around 240 BC,when his teacherCallimachus died.

Eratosthenes knew theabove story about the wells of Syene. He took that to mean that the Sun was directly overhead at noonon 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 timeof 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 is1/50 of the Earth circumference  (a posteriori, that's only 6% off).

As the distance between Alexandria and Syene,was reputed to be 5000 stadia,Eratosthenes estimated the circumference of the Earth tobe  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, becausewe don't know precisely what kind of stadion (or stadium)  wasmeant in the Alexandria-to-Syene distance quoted by Eratosthenes.

Thetraditional equivalences are600 feet to astadion and 8stadia to a mile. 

However, the exact length of a Greek foot varied from one city to the next. Arguably, Eratosthenes would have been likely to usethe Attic stade of 185 m (8 Attic stades to the Roman mile). In any case, his estimate was certainlyno worse than 20% off the mark and it may have been much better than that...

A circumference of  252 000stadia  would be only 1%off if  Eratosthenes, wittingly or unwittingly, hadbeen calling a "stade" an Egyptian surveying unit of  157 m, which was sometimes identified with a Greekstadion.

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

The great achievement of Eratostheneswas to realize that the circumference ofthe Earth could be estimated withsome accuracy from a single angular measurementand 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 usea numberten times as large, just to be on thesafe 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.

To Archimedes and Eratosthenes, the "traditional" estimate for thecircumference of the Earth was most probably the one quoted byAristotle (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 elitefollowers of Phytagoras  (c.582-507 BC) but it has been argued that Aristotlecould 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 usethe result of Eratosthenes in his own Sand Reckoner  essay.

(2019-01-03)    
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 doubtconcerning 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  (ina 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 ofEratosthenes  involve two distant wells.  They enabled him to estimate the circumference of the Earthassuming it was round.  That would still be compatible withthe hypothesis of a flat Earth  if the Sun wasnot 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 locationswith have different directions because of the curvature  of the Earth surface.

Another  reliable clue that the Earth must be spherical is thatit 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 reliablebecause 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 downwardby 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 surfaceof 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.

(2010-07-04)    
Aristarchus used lunar eclipses.  Hipparchus used solar eclipses.

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

Hipparchus of Nicaea (c.190-126 BC) confirmed that result independently by noting that a total solar eclipseover a known remote location (see below) was observed in Alexandriaas a partal eclipse leaving  1/5  of the solar width  (30' or 0.5°)  still visible.  So, the angular separationbetween 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 deducethe distance to the Moon  (as 573 times the distance separating two parallel sunraysthrough 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 byHipparchus.

In the lifetime ofHipparchus,only one total solar eclipse  occurred over Syene (technically, it was an annular eclipse). It took place onAugust 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,onSeptember16, 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 therelated 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 ofNovember 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 in190 BC,263 BC,310 BCand340 BC.

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

The idea of using a system of spherical coordinates to locate points onthe Earth is credited toHipparchus of Nicaea (c.190-126 BC) who first used it tomap 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 thesky:  thecelestial pole.

  23.44° (which is the mean obliquity of the ecliptic ). About 14000 years ago, the bright star Vega (-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 thatthe axis of the Earth oscillates aroundthe position predicted by the above circular motion, just likethe axis of a spinning top does (nutationmotion). This translates into a periodic variation of the obliquity of the ecliptic,which astronomers approximate with apolynomial function of time,valid for a few centuries.

The angle between the celestial pole and the plane of the horizon isthe locallatitude, which can be measured to a precision of about 0.2° with elementary tools (angular units need not be assumed; the resultcould be expressed as a fraction of a whole circle). Even without formal measuring,this special angle could be materialized by erectingpointers 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 prehistoricaldiscovery:  "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, sincepeople have been navigating and observing changes in latitude for a very long time:

Longitude :

Longitude is a different story entirely. Until reliable chronometers became available, longitude was mostly an intellectualconstruct based on the assumption that the Earth was spherical (or nearly so). The difference in longitude between two points could onlybe estimated on firm land, by using surveying techniquesaftersome fairly good knowledge of thesize of the Earth had been gainedto 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 thescholarly belief that the Ocean was small enough to sail through...

(2006-10-17)    
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 andnautical league (each with many definitions).

The Latin for "league" (leuga) comes from the Gallicleuca [ not  the other way around ]  which was supposedto 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).

Land League(s) :

Officially, each flavor of theland league remained quite stable over time,although actual recorded measurements may show some lack ofprecision 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  (2000toises =3898.0872 m).
• Mexican league, legua legal
  (3000pasos de Solomon = 5000 varas = 4191 m)
• Castilianlegua común,legua regular antigua, modernlegua
  (20000 piesde Burgos = 5572.7 m)

The Spanish system comes in different flavors whose basic units differ slightly,but all of them have 5pies to thepaso and 3pies to thevara. The vara may also be subdivided into 4cuartas or 8ochavas. Thevara de Burgos was apparently first established in 1589, but was givenits final metric definition (0.835905 m) only in 1852, as Spain was convertingto the metric system. It competes with thevara of California (now identified with the ancientvara de Solomon) which theTreaty 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 fractionof the (average) degree of latitude. The nautical league which (barely) survives to this day is 1/20 of a degree (3 nauticalmiles) but another nautical league of 1/15 of a degree (4 nautical miles) used to bealmost 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%, sincea 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).

Still, Livio C. Stecchini arguesthat a "memory of the Roman calculation" of 75 Roman miles to the degree of latitudewas preservedtrough medieval times. This is so nearly perfect that it seems entirely too good to be true...

(2008-03-10)    
The ancient mysteries of electricity and magnetism.

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

Amber is a transparent material consisting of hardened resin from conifers (mostly of the familySciadopityaceaethat flourished in theBaltics44 million years ago).  If you rub it against wool, polishedamber attracts nearby dust or dry leaves. Thales of Miletus (c. 624-546 BC)recorded that observation around 600 BC.

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

In 1733,CharlesFrançois 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.

We now speak of negative charges  (resinous) and positive charges  (vitreous)  according to the arbitraryalgebraic sign convention which was introduced before 1746byBenjamin Franklin(1706-1790)  to formulate the fundamental principle of conservation of electric chargewhich is attributed jointly to him and to the British scientistWilliamWatson (1715-1787).

Various materials acquire a definite electric charge when rubbed. Amber becomesnegatively 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 toquantify precisely, because it depends critically on a variety of factors which aretough 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 acquirea positive charge and which material will acquire a negative charge when they areseparated after being rubbed against each other:  The earlier the materialappears in the series, the more positive it will tend to be.

For many century,magnetism was perceived asa 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 mineralto attract bits of iron. The mineral was called magnetite  because it was commonlyfound in a region named Magnesia  (Central Greece). The region gave its name to the rock ( Fe₃O₄ )  the rock gave its name to thephenomenon.

Arguably, the first scientific paper  ever writtenis a treatise on magnetism known as Epistola de Magnete, written in 1269 by the French scholar PetrusPeregrinus  (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 magnetismmight produce perpetual motion! He was writing more than three centuries before Sir William Gilbert(1544-1603)  [] published De Magnete (1600).

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 accordingto 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 amarginal gloss of acopy 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 outstandingmathematician, an astronomer, a physicist, a physician, an experimentalistand, above all, a pioneer of the scientific method... He may have been described as a recluse devoted to the study of Nature, but he wasactually 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 ofLucera, 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 ofFoucaucourt 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 byBrother Potamian, D.Sc.  (1904). Digitized in 2007.

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

On 1902-05-17,  the Greek archaeologistValerios Stais (1857-1923)spotted gears in a corroded chunk of metal recovered in 1901 from a wrecklocated offPoint Glyphadiaon the island ofAntikythera,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.

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 approximationsknown 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-sideas a continued fraction,  but Tony's candid accountof his own discovery of  53  (by trial and error)  makes for bettervideo 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 revolvingaround a slightly different center.

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

(2018-03-23)    
Monochord, lyre, kithara, barbiton.

A Brief History of Elementary Music Theory

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

(2021-07-06)    
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 formof 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 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,  whichPlato accepted onaesthetical grounds beforeAristotlegave 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 citizenshipin 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 motherchanged her own name from Parthenas to Pythais and they chose to name their son Pythagorasafter 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,  castingmuch 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 offthe coast of mainland Anatolia  (Asia Minor) were Miletus is located  (in modern-day Turkey). Samos and Miletus waged a war against each otherin  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)