Numeral Systems

Following up on my “origins of the alphabet” chart, here is one for numerals. The chart was produced using R, and the pictures are purely illustrative – unlike the pictures in the alphabet chart, they do not relate to the origins of the symbols.


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Four worlds

The picture above shows four possible worlds: a (slightly oblate) sphere, a torus, a disc, and a Klein bottle (images © Anthony Dekker). The darkened end of the Klein bottle is shifted through the fourth dimension to connect with the other end, making it a one-sided surface (like the Möbius strip). How do we know which world we live in?


Measuring the Earth this (Southern) Christmas

In around 240 BC, Eratosthenes calculated the circumference of the Earth. The diagram above (from NOAA) shows how he did it. This Christmas, people in the Southern Hemisphere can repeat his work!

Eratosthenes knew that, at the summer solstice, the sun would be directly overhead at Syene (on the Tropic of Cancer) and would shine vertically down a well there. He also knew the distance to Syene.

On 21 December, the sun will be directly overhead on the Tropic of Capricorn at local noon. This table show the time of local noon on 21 December 2017, and the distance to the Tropic of Capricorn, for some Southern Hemisphere cities:

City Local Noon Distance to Tropic (km)
Adelaide 13:14 1270
Auckland 13:19 1490
Brisbane 11:46 450
Buenos Aires 12:52 1240
Darwin 12:45 1220
Hobart 13:09 2160
Johannesburg 12:06 310
Melbourne 13:18 1590
Perth 12:15 940
Santiago 13:41 1110
Sydney 12:53 1160

At exactly local noon, Eratosthenes measured the length (s) of the shadow of a tall column in his home town of Alexandria. He knew the height (h) of the column. He could then calculate the angle between the column and the sun’s rays using (in modern terms) the formula θ = arctan(s / h).

You can repeat Eratosthenes’ calculation by measuring the length of the shadow of a vertical stick (or anything else you know the height of), and using the arctan button on a calculator. Alternatively, the table below show the angles for various shadow lengths of a 1-metre stick. You could also attach a protractor to the top of the stick, run a thread from the to of the stick to the end of the shadow, and measure the angle directly.

The angle (θ) between the stick and the sun’s rays will also be the angle at the centre of the Earth (see the diagram at top). You can then calculate the circumference of the Earth using the distance to the Tropic of Capricorn and the fact that a full circle is 360° (the circumference of the Earth will be d × 360 / θ, where d is the distance to the Tropic of Capricorn).

Height (h) Shadow (s) Angle (θ)
1 0.02
1 0.03
1 0.05
1 0.07
1 0.09
1 0.11
1 0.12
1 0.14
1 0.16
1 0.18 10°
1 0.19 11°
1 0.21 12°
1 0.23 13°
1 0.25 14°
1 0.27 15°
1 0.29 16°
1 0.31 17°
1 0.32 18°
1 0.34 19°
1 0.36 20°
1 0.38 21°
1 0.4 22°
1 0.42 23°
1 0.45 24°
1 0.47 25°
1 0.49 26°
1 0.51 27°
1 0.53 28°
1 0.55 29°
1 0.58 30°
1 0.6 31°
1 0.62 32°
1 0.65 33°
1 0.67 34°
1 0.7 35°
1 0.73 36°
1 0.75 37°
1 0.78 38°
1 0.81 39°
1 0.84 40°
1 0.87 41°
1 0.9 42°
1 0.93 43°
1 0.97 44°
1 1 45°

The Plimpton 322 tablet re-examined

Several years back I blogged about the Plimpton 322 tablet – a Babylonian clay tablet from around 1,800 BC. It contains four columns of numbers, written in base 60 (with a small number of errors, as well as some numbers missing through damage – these are corrected below). For example, 1.59:00:15 = 1 + 59/60 + 0/3600 + 15/216000 = 1.983402777777778.

Column B of the tablet (with a label on the tablet containing the word “width”) is one of the sides of a Pythagorean triangle, and column C (with a label on the tablet containing the word “diagonal”) is the hypotenuse, such that C2 − B2 is always a perfect square (yellow in the diagram). Column A is exactly equal to C2 / (C2 − B2), the ratio of blue to yellow.

A B (“width”) C (“diagonal”) D
1.59:00:15 = 1.983402777777778 1:59 = 119 2:49 = 169 #1
1.56:56:58:14:50:06:15 = 1.949158552088692 56:07 = 3367 1:20:25 = 4825 #2
1.55:07:41:15:33:45 = 1.918802126736111 1:16:41 = 4601 1:50:49 = 6649 #3
1.53:10:29:32:52:16 = 1.886247906721536 3:31:49 = 12709 5:09:01 = 18541 #4
1.48:54:01:40 = 1.815007716049383 1:05 = 65 1:37 = 97 #5
1.47:06:41:40 = 1.785192901234568 5:19 = 319 8:01 = 481 #6
1.43:11:56:28:26:40 = 1.719983676268861 38:11 = 2291 59:01 = 3541 #7
1.41:33:45:14:03:45 = 1.692709418402778 13:19 = 799 20:49 = 1249 #8
1.38:33:36:36 = 1.642669444444444 8:01 = 481 12:49 = 769 #9
1.35:10:02:28:27:24:26:40 = 1.586122566110349 1:22:41 = 4961 2:16:01 = 8161 #10
1.33:45 = 1.5625 45 1:15 = 75 #11
1.29:21:54:02:15 = 1.489416840277778 27:59 = 1679 48:49 = 2929 #12
1.27:00:03:45 = 1.450017361111111 2:41 = 161 4:49 = 289 #13
1.25:48:51:35:06:40 = 1.430238820301783 29:31 = 1771 53:49 = 3229 #14
1.23:13:46:40 = 1.38716049382716 28 53 #15

What is this table all about? A good discussion is by Eleanor Robson [Words and pictures: new light on Plimpton 322,” American Mathematical Monthly, 109 (2): 105–120]. Robson sees Plimpton 322 as fitting into standard Babylonian mathematics, and interprets it as a teacher’s effort to produce a list of class problems.

Specifically, Robson believes that the table was generated by taking values of x (in descending order of x) from standard Babylonian reciprocal tables (specifically the values 2:24, 2:22:13:20, 2:20:37:30, 2:18:53:20, 2:15, 2:13:20, 2:09:36, 2:08, 2:05, 2:01:30, 2, 1:55:12, 1:52:30, 1:51:06:40, and 1:48) and then using the relationship (x − 1 / x)2 + 22 = (x + 1 / x)2 to generate Pythagorean triples. If we let y = (x − 1 / x) / 2 and z = (x + 1 / x) / 2, then B and C are multiples of y and z, and A = z2 / (z2 − y2).

Just recently, Daniel F. Mansfield and N. J. Wildberger [Plimpton 322 is Babylonian exact sexagesimal trigonometry,” Historia Mathematica, online 24 August 2017] interpret the table as proto-trigonometry. I find their explanation of the first column (“a related squared ratio which can be used as an index”) unconvincing, though. Why such a complex index? Robson calls such trigonometric interpretations “conceptually anachronistic,” and points out that there is no other evidence of the Babylonians doing trigonometry.

Mansfield and Wildberger also suggest that “the numbers on P322 are just too big to allow students to reasonably obtain the square roots of the quantities required.” However, I don’t think that’s true. The Babylonians loved to calculate. Using the standard square-root algorithm, even simplistic starting guesses for the square roots of the numbers in column A give convergence in 2 or 3 steps every time. For example, to get the square root of 1.59:00:15 (1.983402777777778), I start with 1.30:00:00 (1.5) as a guess. That gives 1.24:40:05 as the next iteration, then 1.24:30:01, and then 1.24:30:00 (1.408333333333333), which is the exact answer. That said, however, calculating those square roots was not actually necessary for the class problems envisaged by Robson.

Sadly, I do not think that Mansfield and Wildberger have made their case. Robson is, I believe, still correct on the meaning of this tablet.


A History of Science in 12 Books

Here are twelve influential books covering the history of science and mathematics. All of them have changed the world in some way:


1: Euclid’s Elements (c. 300 BC). Possibly the most influential mathematics book ever written, and used as a textbook for more than 2,000 years.


2: De rerum natura by Lucretius (c. 50 BC). An Epicurean, atomistic view of the universe, expressed as a lengthy poem.


3: The Vienna Dioscurides (c. 510 AD). Based on earlier Greek works, this illustrated guide to botany continued to have an influence for centuries after it was written.


4: De humani corporis fabrica by Andreas Vesalius (1543). The first modern anatomy book.


5: Galileo’s Dialogue Concerning the Two Chief World Systems (1632). The brilliant sales pitch for the idea that the Earth goes around the Sun.


6: Audubon’s The Birds of America (1827–1838). A classic work of ornithology.


7: Darwin’s On the Origin of Species (1859). The book which started the evolutionary ball rolling.


8: Beilstein’s Handbook of Organic Chemistry (1881). Still (revised, in digital form) the definitive reference work in organic chemistry.


9: Relativity: The Special and the General Theory by Albert Einstein (1916). An explanation of relativity by the man himself.


10: Éléments de mathématique by “Nicolas Bourbaki” (1935 onwards). A reworking of mathematics which gave us words like “injective.”


11: Algorithms + Data Structures = Programs by Niklaus Wirth (1976). One of the early influential books on structured programming.


12: Introduction to VLSI Systems by Carver Mead and Lynn Conway (1980). The book which revolutionised silicon chip design.

That’s four books of biology, four of other science, two of mathematics, and two of modern IT. I welcome any suggestions for other books I should have included.


350 years ago on Friday

On 26 May 1667, Abraham de Moivre was born. This French mathematician gave us, inter alia, the formula named after him:

De Moivre was born to French Protestant parents. When the Edict of Nantes was revoked, he was imprisoned for his beliefs for several years, after which he was allowed to leave for England. De Moivre made important contributions to probability theory, and was a pioneer of analytic geometry. Sadly, he was unable to get a university position in England, and he died in poverty.