The R100 and the R101

An instructive saga in the history of engineering is the story of the British airships R100 and R101. As part of a grand social experiment, the R100 was built by private industry (it was designed by Barnes Wallis), while the R101 was built by the British government (specifically, by the Air Ministry, under Lord Thomson). The R100 worked fine, and made a test flight to Canada in August 1930 (the trip took 78 hours). Here is the R100 over a Toronto building:

The R100 was huge. Here is a size comparison of the R100 (219 m long) and an Airbus A380 (73 m long):

While the government-built R101 used servo motors to control its gigantic rudder, the R100 team had worked out that the rudder could actually be operated quite easily by hand, using a steering wheel and cables. The government-built R101 was beset by poor choices, in fact. It contained overly heavy engines, a steel frame, and too much dead weight overall. After construction, the R101 had to be lengthened by inserting a new 14-metre section in the centre, in order to increase lift. This alteration caused a number of problems. Its design also allowed the internal hydrogen-filled gasbags to chafe against the frame, there were serious problems with the outer covering, and several “innovative” design ideas were never properly tested.

There was enormous political pressure for the R101 to fly before it was ready to do so. On the evening of 4 October 1930, it departed for India with a crowd of VIPs on board. It never arrived, crashing in bad weather over France, and bursting into flames. The disaster led to the R100 also being grounded, and the British government abandoned any thoughts of flying airships (as the rest of the world was to do after the Hindenburg disaster).

There are all kinds of lessons to be drawn from the saga of the R100 and the R101. One of them is that optimism is not a viable strategy for safety-critical engineering. Another is that engineers test things. As Kipling says, “They do not preach that their God will rouse them a little before the nuts work loose.” A third is that risky designs and fixed deadlines simply do not mix.


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.

The wash bottle

Washbottles, old (left, photo: Hannes Grobe) and new (right).

Wash bottles, in one form or another, have been a long-term feature of the chemistry lab. Once they were made of glass, and were operated by blowing. In more recent times, plastic squeeze bottles have been used.

See here for more posts on scientific equipment.

Joseph Dalton Hooker

The botanist Joseph Dalton Hooker was born 200 years ago, on 30 June 1817. Kew Gardens, of which he was the director, has a special event to commemorate him. Hooker travelled on expeditions to Antarctica, India, Palestine, Morocco, and the Western United States. The pictures below are from his The botany of the Antarctic voyage of H.M. discovery ships Erebus and Terror in the years 1839–1843, under the command of Captain Sir James Clark Ross. He also published several volumes on the botany of India.

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.

Harp History

After some feedback on my harp twins post, I thought I’d say something about the history of the harp. It’s one of the oldest musical instruments (following the flute and the drum). Harps are known to go back to 3500 BC, in Ur. Harp design has varied considerably over the 5500 years since then.

Harpist depicted on the Standard of Ur, c. 2500 BC

Later harps were of particular importance to the Celtic people, and the harp is still a symbol of Ireland today.

The medieval Queen Mary harp, c. 1400s (photo: David Monniaux)

A limitation of harps has been that the strings correspond only to the white keys on the piano. A significant improvement was the pedal harp – initially the single-action version, and from 1810 the double-action version. The double-action pedal harp is typically tuned to C major, the key of 7 flats. There are 7 pedals, with e.g. the C pedal connecting to all the C strings. Using the pedal can effectively shorten all the strings in this group to give either C or C (and the same for other groups of notes).

Child prodigy Alisa Sadikova playing the pedal harp (at age 9)

The pedal harp is the main concert instrument today. Garrison Keillor once described the instrument as “an instrument for a saint” because “it takes fourteen hours to tune a harp, which remains in tune for about twenty minutes, or until somebody opens the door.”

A modern electric lever harp (photo:Athy)

Smaller harps (including modern electric harps, like the one above) use levers to modify individual strings (which makes key changes much more difficult than with the pedal harp). Electric harps weighing up to 8 kg are described as “wearable,” which reminds me a little of this 11 kg grand-daddy of the laptop.

Camille and Kennerly Kitt playing “wearable” electric harps

The harp is often seen as a stereotypically feminine instrument – when I look at American harpists on Wikipedia, I count 10 men and 60 women. There are, however, exceptions.

Jakez François (president of French company Camac Harps) playing jazz