The Trivium and the Quadrivium



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.

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:

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:

Girl, flaunt those hoops!

American private liberal arts colleges constantly seem to be in the news (not, unfortunately, for their educational successes). Pitzer College, located in Claremont, California (and ranked 44th among liberal arts colleges in the United States by Times Higher Education), recently made the news when a college Resident Assistant announced that “white girls” should not wear hoop earrings.

Left: Greek hoop earring with lion head, 4th century BC; Right: Italian (Lombard) hoop earring with basket, 7th century AD

The strange thing is that hoop earrings are almost a cultural universal. They have been worn across the world, including in Europe and the Middle East. And not just by “girls” with skin of various colours – men have worn them too.

I must also say I’m grateful that the Greeks haven’t stopped the rest of the world from “cultural appropriation” of their philosophy, mathematics, and democratic ideals.

Left: Croatian hoop earring, 14th century AD; Right: Egyptian hoop earring, Roman Period or later

Education in the USA

The pie chart above shows the breakdown of elementary and secondary students in the USA. Data is for 2013, from the National Center for Education Statistics (with homeschooling numbers extrapolated from 2012 data). Educational options are a hot political issue right now, with changes to education policy likely under the Trump administration. What those changes will be is unclear.

Of course, the US educational system does need some improvement. In the 2015 PISA global education survey, the US ranked equal 23th in reading, 25th in science, and only equal 39th in mathematics. Canada did much better (2nd, 7th, and 10th), and Singapore came 1st in all three categories.