Plato Vindicated (1:04:20)
by Yuval Ne'eman (1999年08月26日).
How Greek Maths Changed the World (43:35)
Alan Davies
(RI, 2017年02月14日).
Panjagan
reinvented (17:22) by
Jörg Sprave (2020年09月02日).
History of Mathematics (1:45:15)
Marcus Du Sautoy (2023年02月03日).
The unreasonnable effectiveness of mathematics (53:50)
Nova (2015年05月02日).
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:
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.
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).
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.
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 :
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).
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 :
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:
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):
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年03月18日).
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年08月24日).
Cuneiform Tablet and Babylonian Trigonometry (2:09)
Trinity Chavez (RT, 2017年08月25日).
Ancient Babylonian Tablet Deciphered after 70 Years (4:09)
ShantiUniverse (2017年08月28日).
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年11月21日).
Measuring the height of the Great Pyramid
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.
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)
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年03月25日).
"The
Mathematics of Plato's Academy: A New Reconstruction" by
Dr.
David H. Fowler
(1937-2004).
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.
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年02月28日).
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.
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).
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年03月09日).
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).
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.
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.
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:
Thor Heyerdahl (1914-2002)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 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.
Christopher Columbus 1451-1506 (Colombo)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...
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.
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:
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).
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).
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...
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.
Direction of Magnetic Field (B) 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! William Gilbert (1544-1603) 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).
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).
Wikipedia : Petrus Peregrinus de Maricourt.
On 1902年05月17日, 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.
Location of the Antikythera shipwreck
This was part of an ancient mechanical computer, now known as the Antikythera Mechanism. Marking the 115th anniversary of this discovery, Google showed the following Doodle on their homepage, on May 17, 2017:
Antikythera Google Doodle 2017年05月17日
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:
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):
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.
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
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Michael T. Wright (1948-)
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Allan G. Bromley (1947-2002)
The Antikythera Research Project
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Mike Edmunds
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Tony Freeth
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Alexander Jones
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Roger Hadland
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Yanis Bitsakis
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Xenophon Moussas
Wikipedia :
Orrery
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Antikythera wreck
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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年07月03日).
The Antikythera Mechanism:
A Shocking Discovery from Ancient Greece by
Tony Freeth
(2015年11月06日).
"Making the Antikythera Mechanism" by Chris
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(Clickspring, 2017-2018)
The Ancient Greek Kithara (9:03)
by Peter Pringle (2015年10月26日).
c.1400 BC: The Oldest Known Melody (5:42)
by Michael Levy (2009年05月13日).
Hurrian Songs
Terpender (c.700 BC)
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Archilochus (c.680-645 BC)
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Arion (fl. 615 BC)
Any religion consists of two distinct parts:
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 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."
[画像: Come back later, we're still working on this one... ]
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:
Cult of Hesiod
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Cult of Dionysus (Bacchus)
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Orphism
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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年06月12日).
Pythagoras: Life's Music and Mathematics (1:10:44)
by Michael Griffin (2020年10月01日).
The Pythagoreans and the Orphic mystery religion (29:58)
Leonard Peikoff (Ayn Rand Institute, 2020年05月01日).
10
Strange Facts About Pythagoras
by Mark Oliver (And My Quill, 2017年04月26日).
