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.


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Universities and Wine

A few people have commented on my rather tongue-in-cheek post about solar car racing and beer. I don’t think the correlation there was actually spurious – there really is a tradition of excellent engineering education in the beer-producing areas of Europe, and both the beer production and the approach to engineering education have been exported around the world.

In the USA, for example, we have the influence of Stephen Timoshenko (1878–1972) at the University of Michigan and at Stanford. And we have the influence of Friedrich Müller / Frederick Miller (1824-1888) in the brewing industry.

But lest I be accused of some kind of pro-beer bias, the chart below shows national wine consumption (consumption this time, not production) compared to the date of the oldest university in the country (excluding universities less than a century old). Here we have universities (in the modern sense of the word) growing out of the wine-drinking areas of Europe, beginning with the University of Bologna. Once again, I think the data can be understood as a case of parallel exports:


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 Scientific Revolution and the Origins of Modern Science: a book review


The Scientific Revolution and the Origins of Modern Science by John Henry

I recently read the 3rd (2008) edition of John Henry’s The Scientific Revolution and the Origins of Modern Science – an excellent, though very brief, survey (it is 114 pages, not including the glossary and index).

Henry tends to see considerable continuity between the “natural magic” of medieval thought and the emerging scientific viewpoint, which was based on experiment and mathematical analysis. Personally, I think that he overstates the case a little. It is interesting that he never mentions Giordano Bruno, who was one of those who held on to the older magical view (then again, Bruno was not a scientist).


Replica of a van Leeuwenhoek microscope (photo: Jeroen Rouwkema)

Henry also puts emphasis on the emerging use of scientific instruments, such as the microscope and the telescope.


Galileo’s sketches of the moon, published in his Sidereus Nuncius of 1610

I was a little disappointed in the discussion of Galileo, which did not seem quite correct, but the main flaw in this book is its brevity. I’m giving it three stars.

* * *
The Scientific Revolution and the Origins of Modern Science by John Henry: 3 stars


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.