The Trivium and the Quadrivium

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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°

Guns, education, religion, and suicide

My earlier post indicated that gun laws in the US had little impact on the homicide rate, when demographic factors were taken into account. This makes sense – if I want to kill somebody, the lack of a gun will merely prompt me to choose another weapon. But what about suicide? The impulse to suicide is often brief, and easy access to a gun during a suicidal episode may increase the chance of dying.

To test this, I extended my previous dataset with data on educational attainment, data on religiosity, registered gun ownership data from the ATF, age-adjusted suicide rates from the CDC, poverty rates, unemployment rates, and other demographic data. I ran all that through a regression tree analysis, using R.

Suicide rates in the chart (click to zoom) are indicated by colour, ranging from 8 per 100,000 for New Jersey and New York (yellow) to 23.7 for Montana (black). Having a college degree seems to have a protective effect – states on the right of the chart, with more college degrees, had lower suicide rates. This may relate to the higher employability of college graduates. However, states at the top of the chart, with higher high school graduation rates, had higher suicide rates. I am not sure why this is the case.

Among the states with fewer college graduates, religion had a protective effect (this is consistent with other studies). States where 77% or more of the population said that religion was “somewhat important” or “very important” to them are indicated on the chart by triangles. For the states with fewer college graduates, the suicide rate was 13.6 per 100,000 for religious states, and 17.5 for less religious ones.

Finally, the highest-risk states (fewer college graduates and less religious) split according to gun ownership. States with more than 0.008 registered guns per capita are marked on the chart with an inner dot. Among the highest-risk states, the suicide rate increased from 13.9 per 100,000 to 18.6 when more guns were present. This group included Alaska (23.2 per 100,000), Arizona (17.5), Idaho (19.2), Maine (17), Montana (23.7), Nevada (18.6), North Dakota (17.3), Oregon (16.8), and Utah (21.4). Among the more religious states, registered gun ownership did not seem to have an effect (although, of course, registered gun ownership is a poor indicator of true gun ownership).

Thus the data does seem to suggest a link between gun ownership and suicide risk, but only when other risk factors are present (low religiosity and no college degree). This is exactly what we expected, and it means that suicidal (or potentially suicidal) people need to be kept away from guns.


Do gun laws save lives?

Somebody pointed me at this interesting data the other day. The chart above (click to zoom) combines the “Gun Law Score Card” from the Law Center to Prevent Gun Violence in the US with homicide rate data from Wikipedia and voting results from the last US election. Do gun laws reduce the chance of being murdered?

Obviously, “Blue” states tend to have stricter gun laws than “Red” states (an average of B− vs D−). “Blue” states also have lower homicide rates than “Red” states (4.5 vs 5.9), and this is statistically significant (p = 0.012). There is a weak (R2 = 6%) correlation between gun laws and homicide rates, but this relationship is not statistically significant.

Whatever it is that makes you less likely to be murdered in some states than others, it does not primarily seem to be the gun laws. Poverty may be one of the relevant factors, however – median household income explains 22% of the variance in homicide rates, and when this is taken into account, any effects due to gun laws or election results disappear. “Red” states are, on the whole, simply poorer (and, conversely, poor states are more likely to vote Republican and have weak gun laws). Other demographic factors, such as the number of people with college degrees, also seem to have explanatory value as far as the murder rate is concerned. However, the phenomenon of murder does not seem to be understood as well as it could be.


The Three-Body Problem trilogy: a book review


The Three-Body Problem trilogy by Liu Cixin

I recently read (in translation, of course) the popular The Three-Body Problem science fiction trilogy by Liu Cixin. These books explore the idea of first contact, and touch on several topics that I have posted about before (such as the Fermi paradox and the 3-body problem itself). I enjoyed reading them (the first two books more than the third). It was fascinating to read a Chinese view on some of the issues explored.

The novels are somewhat darker than classic Western science fiction, largely due to the shadow cast by the Cultural Revolution. But given the possibility of aliens like the Borg, the Daleks, and the Vang, perhaps interstellar optimism is just naive. And apparently, most contemporary Chinese science fiction is even more pessimistic than that of Liu (one of the characters in the first novel comments on this).

It seemed a little strange that Liu accepts the speed-of-light limit on space travel, but allows faster-than-light communication (which other authors have called an ansible). After all, both relativity and quantum mechanics forbid such a technology. Still, any depiction of truly advanced technology is going to read like fantasy, and the plot did require an ansible (although partway through the trilogy, the speed-of-light limit seemed to vanish even for ordinary communication).

These books (at least the first two) are well worth a read. Wired magazine also posted a review last year, and Nature had an interview with the author.


The Three-Body Problem trilogy by Liu Cixin: 3.5 stars


Logic in a box!

Having recently spent some time teaching a short course on logic and critical thinking, here is the core of the course reduced down to a box of 54 cards. These include:

  • 15 logic cards (summarising basic syllogistic and propositional logic rules),
  • 19 cards illustrating logical fallacies,
  • 5 cards for testing your ability to check validity, and
  • 15 logic-puzzle cards.

If you’re interested, more details can be downloaded from the game page (see the links in the “Downloads” section). The picture below shows some of the cards: