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°

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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:


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.


Chemical Compounds: the board game!

I have previously mentioned my strong interest in science / technology / engineering / mathematics education and in networks and in board games. This has prompted me to start designing educational games, such as the World Solar Challenge game. Joining the collection is my new Chemical Compounds game, which looks like this:

The online game store (faciliated by the wonderful people at The Game Crafter) has a free download link for the rules, should anyone wish to take a look. I also have a few other educational games there.


A scale model of the Solar System

The following images of the eight major planets and Pluto are to scale, with each image 500,000 km in width (the third image also includes the Moon). For comparison, the diameter of the Sun is 1,390,000 km, so you can team these planets up with a yellow circle 2.8 times the width of each image. Click on the images (which can also be found on the Homeschool Resources section of this blog) to zoom.

    
    
    

If you want distances from the Sun to each planet to be to scale as well (like this or this), they are shown below (in millions of km and multiples of the width of each image). If the images above are printed out 5 cm or 2 inches wide (i.e. at a scale of 1 cm = 100,000 km), then the sun would be 14 cm (5.5 inches) wide, and distances from the Sun to each planet would be as per the bottom two rows of the table.

58 108 150 228 779 1,434 2,872 4,495 5,906 million km
120 220 300 460 1,560 2,870 5,700 9,000 11,800 multiples
6 11 15 23 78 140 290 450 590 metres
19 35 49 75 260 470 940 1470 1940 feet

The image below (click to zoom) shows the complete solar system, but most planets are too small to see:


And So They Build by Bert Kitchen: a brief book review


And So They Build by Bert Kitchen

And So They Build by Bert Kitchen is a child’s introduction to construction by animals (also the subject of my previous review of a different book).


The story of the tailorbird

Kitchen provides 12 short descriptions of animal architects (mammals, birds, fish, and insects), along with 12 beautiful illustrations.


The harvest mouse

This short book is a good buy for parents of small children. I’m giving it the same rating as goodreads.


And So They Build by Bert Kitchen: 3.5 stars