topics: |  math | |

A fascinating read.  Anecdotes from the history of science are used to
motivate the serious mathematics that underlies each of the problems
described.   While the mathematics is the goal, it is invariably the human
aspects that make the stories come alive - such as the opposition faced by
pioneers John Snow or Robert Koch in cholera bacillus identification, or
the very acrominous debates of the air defence committee when
F.A. Lindemann was added to it. (p.46)

Thomas Koerner, despite the umlauts in his name, is very British.  Although
a pure mathematician, he takes interest in very applied matters, and
here he takes us on a tour of some fascinating histories of technology.
the tales are also very British, ignoring all other points of view almost,
but nevertheless very entertaining, and covering a rather wide canvas.

These include identifying the causes for the spread of cholera, why U-boat
attacks against shipping is far less effective against convoys, and a
discovery of the history of Radar, more effective submarine hunting in WW2,
the metabolic rates of mammals, and the design of anchors.

a particularly elegant example is the applications of dimensionality
analysis, applied here for example, to infer that the pressurewave from an
atomic explosion will expand proportional to time^{2/5}.

mathematical modeling runs as a leitmotif in this fascinating history
of technology and operations research, unveiled often in military contexts.

Excerpts

Mathematics is, at least in part, the science of abstraction.
Mathematicians look at the rich complexity of the real world and replace it
with a simple system which, at best, palely reflects one or two aspects of
it.  Roads become lines, towns become points, weather becomes a series of
numbers (temperature, wind-speed, pressure, ... ) and human beings become
units. The object of the first part of this book is to show how useful such
abstraction can be. p.3

Dr. John Snow and the epidemiology of cholera

In 1818 Europe became aware of a terrifying epidemic raging in parts of
India. The disease, previously unknown to European science, struck suddenly,
manifesting itself in violent diarrhoea and vomiting followed by agonising
muscular cramps. An early description tells how

    The eyes surrounded by a dark circle are completely sunk in the sockets,
    the skin is livid ... the surface [of the skin] is now generally covered
    with cold sweat, the nails are blue, the skin of the hands and feet are
    corrugated as if they had been long steeped in water.

Often the skin turned blue or black and sometimes the convulsions
were so severe that the body was contracted into a ball which could
not be straightened out after death. The disease was named 'Cholera
Morbus' and killed perhaps half its victims. 3


the pump whose handle was removed.  Now a memorial
to John Snow in Broadwick St, Soho, London.
[source: wikimedia commons]

Search for the causes of cholera

Most probably cholera had always existed in India but the movement
of armies and the increase in long-distance trade brought about by the
expansion of the British and Russian empires now allowed it to spread.
In Russia infected villages were surrounded by troops with orders to
shoot anyone trying to leave. Spain made it a capital offence to leave
an infected town. In spite of all efforts to contain the disease, it swept
through Europe in the years 1830-32 killing one citizen in 20 in Russia,
one in 30 in Poland and Austria and many in every European country
before 'burning itself out'. In 1848 it returned and there were epidemic
outbreaks in Britain in most of the years up to 1855.

What was the cholera, how was it spread, how could it be prevented
and how cured? It was a disease of the poor which also killed the rich
— but that was and is true of most epidemic diseases.  Many people,
particularly reformers, felt sure that it was associated with dirt, poor
sanitation, bad water, bad air, bad diet and crowded living conditions.
Beyond that there was no agreement.  Was it contagious?  If so, how
could it evade all attempts at quarantine and why were doctors and
clergymen who attended the sick so often spared? Was it caused by
a poisonous miasma created by some process of fermentation in the
presence of bad drains or stagnant water? That would help explain
why it was a summer disease — except in Scotland where the outbreaks
occurred in winter.  Was the miasma, which some experts claimed to
have seen, electrical or was it something to do with ozone?

Snow's Water-borne disease theory

In 1849 Dr John Snow published yet another theory. Snow had
worked his way up from fairly humble beginnings to the top of the
medical profession, becoming one of the founding fathers of anaesthesia.
(In 1853 he was the doctor chosen to give chloroform to Queen Victoria
in childbirth.) He was a shy, diffident man, wholly immersed in his
work and devoted to the relief of suffering.
[Though, according to his biographer, 'in the last few years of his life he
so far threw off restraint as to visit the opera occasionally.'] p.4

Snow, as an expert on respiration, rejected the miasmatic theory. If the
disease were due to a 'miasma', surely the lungs would be affected
first. Since cholera was primarily a disease of the alimentary canal, the
cholera producing material must be swallowed 'and the increase of the
... cholera poison must take place in the interior of the stomach and the
bowels'. Unless strict cleanliness was observed, the cholera poison, once
excreted, would be transferred to hands and thence to food and drink ready to
infect further victims.
[but] he could only speculate about the nature of the cholera poison which
'having the property of reproducing its own kind, must necessarily have
some sort of structure, most likely that of a cell'. 5

[While the above sentence seems very reasonable in today's biology, in
another part - "the weakest part of his paper" Snow] suggests that the
'cholera poison' is particulate (like, say, tapeworm eggs) and so not
everybody who swallowed contaminated water would swallow the 'poison'. 17

However, rather than just collecting statistics, as many of his
contemporaries did, in the hope that something might emerge, he sought
statistical evidence for or against his particular theory of cholera. 5

[...]

A reasonable dose of aspirin reduces a headache, but may kill
one unfortunate individual in a million. 17

Statistical correlations

When cholera returned to London in 1854 Snow determined to visit
every house in these districts where a cholera death occurred and record
the name of the supplying company. As he remarks

    The enquiry was necessarily attended with a good deal of trouble' and in
    many cases it proved impossible to find anyone who knew the name of the
    water supplier. Fortunately a simple chemical test based on the high salt
    content of the Southwark and Vauxhall company's water ('part of that' he
    wrote 'which has passed through the kidneys ... of two millions and a
    quarter of the inhabitants of London') enabled him to deal with these
    cases as well. He communicated his initial findings to Farr 'who was much
    struck with the result' and arranged official assistance for the last
    three weeks of the investigation.

It is a maxim among practical statisticians that 'The data you need
are not the data you have, the data you have are not the data you
want and the data you want are not the data you need.' Although
the total number of houses supplied by each company was known, this
total was not broken down by district. The simplicity of Snow's 'grand
experiment' was thus marred by the fact that he could not directly
separate those districts supplied by both companies from those supplied
by only one or the other.  7

[Snow's ideas  were not widely accepted].  His book on cholera which cost
him £200 to prepare and publish sold only 56 copies. p.14

William Farr, of the Registrar-General's office, who was responsible for the
collection of the statistics on which the table was based had also analysed
them and found a strong correlation between height above sea level and deaths
from cholera. (Taking the epidemics of 1849 and 1853-4 together he found that
the mortality in the lowest parts of London was 15 times the mortality in
the highest.) 6

The 1854 Soho cholera outbreak and the Broad Street pump

In September 1854, there was a dreadful outbreak of cholera north of the
Thames in Soho.

	Within two hundred and fifty yards [roughly 250 metres] of the spot
	where Cambridge Street meets Broad Street there were upwards of five
	hundred fatal attacks of cholera in five days. The mortality in this
	limited area probably equalled] any that was ever caused in this
	country even by the plague; and was much more sudden, as the greater
	number of cases terminated in a few hours. ...

	As soon as I became acquainted with the situation and extent of this
	irruption of cholera, I suspected some contamination of the water of
	the much-frequented street-pump in Broad Street, near the end of
	Cambridge Street; but on examining the water, on the evening of the
	3rd September, I found so little impurity in it of an organic nature,
	that I hesitated to come to a conclusion.
		- John Snow, On the Mode of Communication of Cholera, 1855, p.38
		(source: http://www.ph.ucla.edu/epi/snow/snowbook_a2.html)


	the middle part of john snow's map showing the
	pump at Cambridge and Broad St, andthe addresses
	with cholera deaths (black bars).
	(no copyright; source: http://www.ph.ucla.edu/epi/snow/map1ea.htm)
	see larger map at wiki commons

The following is from Snow's letter to the editor of a medical journal:
(cited in: wiki 1854 Broad Street cholera outbreak)]

	On proceeding to the spot, I found that nearly all the deaths had
	taken place within a short distance of the [Broad Street] pump. There
	were only ten deaths in houses situated decidedly nearer to another
	street-pump. In five of these cases the families of the deceased
	persons informed me that they always sent to the pump in Broad
	Street, as they preferred the water to that of the pumps which were
	nearer. In three other cases, the deceased were children who went to
	school near the pump in Broad Street...

	With regard to the deaths occurring in the locality belonging to the
	pump, there were 61 instances in which I was informed that the
	deceased persons used to drink the pump water from Broad Street,
	either constantly or occasionally...

	The result of the inquiry, then, is, that there has been no
	particular outbreak or prevalence of cholera in this part of London
	except among the persons who were in the habit of drinking the water
	of the above-mentioned pump well.

	I had an interview with the Board of Guardians of St James's parish,
	on the evening of the 7th inst [September 7], and represented the
	above circumstances to them. In consequence of what I said, the
	handle of the pump was removed on the following day.

		—John Snow, letter to the editor of the Medical Times and Gazette

In his account of the outbreak Snow presents his evidence for believing
that the Broad Street pump was the cholera source in the form of a
map...

[However,] government inspectors appointed to report on the outbreak
entirely rejected Snow's theories.

    The extraordinary eruption of cholera in the Soho district which was
    carefully examined ... does not appear to afford any exception to
    generalisations respecting local states of uncleanliness, overcrowding,
    and imperfect ventilation. The suddenness of the outbreak, the immediate
    climax and short duration, all point to some atmospheric or other widely
    diffused agent still to be discovered, and forbid the assumption, in this
    instance, of any communication of the disease from person to person
    either by infection or contamination of water with the excretions of the
    sick.

[The Broad St pump handle was re-installed]

Although the vestrymen initially rejected Snow's condemnation of the pump,
they also set up their own committee of enquiry, which pressed ahead with a
door to door survey of the entire district.  The curate of a local church,
the Reverend Henry Whitehead, volunteered for the massive task of surveying
Broad Street itself and managed to interview well over half its original
inhabitants.  Initially opposed to Snow's theories, he was forced to change
his mind as the evidence accumulated and he found that of the 137 persons
who drank water from the pump 80 developed cholera whereas of the 297 who
did not drink the water only 20 did.

Finally he discovered the key to the mystery. Just before the outbreak a baby
girl had died of what could have been cholera in a house with a privy only a
few feet away from the Broad Street pump. Digging revealed that crude and
wrongly constructed drainage provided a nearly direct route for the
contamination of the pump water. Moreover, the day that the pump handle was
removed another occupant (the baby's father) contracted cholera and, in all
probability, it had only been Snow's timely intervention which had prevented
a second outbreak.

In 1883 Robert Koch isolated the bacillus for choleras, providing the
causative agent which Snow had been unable to give. 14
However, Koch's claim was also not accepted immediately. One of his
opponents drank a beaker full of Koch's bacilli to prove the falsity of the
theory and then, in a 'graphic demonstration of the power of a German
professor over his assistants' had the experiment repeated by his
assistant. Both became ill, the professor mildly so, the assistant more
seriously, but both survived.  16

The movement for clean water and proper sewerage, which was fuelled by many
other sources, reached a successful conclusion in Europe before the end of
the century. In 1883 Koch isolated the bacillus for choleras, providing the
causative agent which Snow had been unable to give. 14

The last major European outbreak of cholera occurred in Hamburg in 1892. The
merchant oligarchy which ruled the city had repeatedly postponed expensive
changes in its water supply system and the inhabitants drank untreated water
from the Elbe.  The neighbouring town of Altona had a water filtration
plant. A street divided the two cities. On one side the cholera raged
unchecked; the other side was spared almost completely. John Snow had won his
argument. 14

links: http://www.londonremembers.com/memorials/dr-john-snow-site-of-pump

Aspirin and Heart disease

Many cardiac deaths are caused by blood clots blocking the supply of blood
to the heart muscles.  It is natural to try the effect of administering
'clot dissolving' drugs but such drugs may cause bleeding and it is by no
means clear that the risks outweigh the benefits. 17

Testing must be done on a large enough sample.

Also, there is another subtle problem.  Suppose that an established procedure
A is being tested against a new procedure B. Doctors may tend to use A in
cases where they believe it will work and reserve B for the otherwise
hopeless cases where 'there is nothing to lose'. The new treatment may
therefore appear less effective simply because it is mainly given to the
iller patients.

In the ISIS-2 trial, 417 hospitals in 16 countries participated in the
experiment which ultimately involved 17187 patients.  Doctors called a phone
num which gave them a random assignment of patients to one of:
    - placebo
    - aspirin only
    - streptokinase only
    - aspirin plus streptokinase

The results from these four choices, in terms of mortality after 0-35 days,
are shown in this figure:


mortality after 35 days, with streptokinase + aspirin,
is 8% as opposed to 13% with aspirin.

The authors of the ISIS-2 report give an amusing illustration of
the difficulties attending relatively small scale trials. They analysed the
data using the patient's astrological birth signs (thus splitting the large
trial into twelve small trials) and discovered that, apparently, aspirin
had a slightly adverse effect on the mortality of those born under
Gemini and Libra but was strikingly beneficial for all the other signs!

We have no very clear idea why aspirin is so effective, but our figures show
that it is. Our dependence on statistics is galling. We can never be entirely
sure, even with the best conducted survey, that some subtle failure in design
may not vitiate our conclusions. We can never be comfortable when men play at
gods, giving one treatment to one patient and some other treatment (or no
treatment at all) to another simply on the throw of a die. But we have no
real choice.

Anti-submarine aircraft operations

It required about 170 man hours by maintenance staff and other personnel to
produce one hour's operational flying, and, on average, perhaps 200 hours of
flying to result in an attack. The few minutes of an attack thus represented
the result of at least 34 000 man hours work.  In 1941 about 2-3% of attacks
were successful — so about one and a half million man hours were required to
sink one submarine. What could be done? 62

Also, U-boats were usually better at spotting aircraft than the other way
around.
Given the estimated density of U-boats, the actual sightings were about 1/4
of what was expected.
These too were mostly as the U-boat was going down.
[Statistical analysis showed a normal distribution of the difference between
spotting time by U-boat vs by
aircraft - with about 66^ of boats not being sighted.

After re-painting the aircraft from black to white, this dropped from ~66% to
10-35%. 66

Further, the depth charges were set to explode at 100 ft - on the assumption
that it would have sighted the aircraft two minutes before the attack, and
that in this time it could dive to about 100 feet.

But accuracy was very low at 100ft.

E. J. Williams, an expert on the quantum theory of atomic collisions

	spotted a fallacy in the argument leading to the 100-foot depth
	setting. It might be true that on the average a U-boat might sight
	the aircraft a long way off and so manage to get to 100 feet before
	the attack. However, just in those cases the U-boat had disappeared
	out of sight of the aircraft for so long that the air crew could not
	know where to drop the depth charges, so that the effective accuracy
	in plan of the Aircraft versus submarine attack was very
	low.

Instead, the depth was set to 25ft, and this increased submarine kill
percentage by about 4x.  68

Further improvements were brought about by having the pilots bomb, not
"ahead" of where the U-boat was supposed to go (which was scattering the line
of depth charges quite randomly), but more or less directly on top.

The Japanese failed to introduce merchant convoys until the end of 1943,
failed to keep convoy radio silence and failed to develop effective
anti-submarine tactics. They lost seven-eighths of their merchant
shipping. 73



---blurb

What is the connection between the outbreak of cholera in Victorian Soho,
the Battle of the Atlantic, African Eve and the design of anchors? One
answer is that they are all examples chosen by Dr Tom Körner to show how a
little mathematics can shed light on the world around us, and deepen our
understanding of it. Dr Körner, an experienced author, describes a variety
of topics which continue to interest professional mathematicians, like
him. He does this using relatively simple terms and ideas, yet confronting
difficulties (which are often the starting point for new discoveries) and
avoiding condescension. If you have ever wondered what it is that
mathematicians do, and how they go about it, then read on. If you are a
mathematician wanting to explain to others how you spend your working days
(and nights), then seek inspiration here.