Which of the following do you think the Greeks knew? (and how?) Π = ? The size of the Sun The height of the pyramids The Earth is a sphere The size of the Earth The distance to the Moon c2 = a2 + b2

Using History to Teach in the Present Beth Lilley @missblilley

Thales HT = HP ST SP (624 – 546 BC) Often considered the first Greek Mathematician & the father of Geometry Proved that two vertical angles where two lines intersect are equal One of his assignments was to measure the height of a pyramid: How would you measure the height of a tree outside? (no ladders allowed!) HT = HP ST SP

Myth Busting Not all Greeks believed we were at the centre of the Universe. Archimedes (287 – 212 BC) wrote that Aristarchus had a hypothesis that the sun was at the centre, and the world revolves around it.

Aristarchus (310 – 230 BC) Used proportions to work out the ratios of the distance to the Sun and Moon. Aristarchus made the realisation that when he saw a half moon, the sun and earth created a right-angled triangle with the centre of each celestial body being a vertex, the moon’s vertex being 90°. Measured the angle Sun-Earth-Moon at 29/30 of a right-angle = 87. Greeks had studied relationships in right-angled triangles, and used to find ratio of m to s. He found 1/18 < m < 1/20, ie. The distance to the Sun is between 18 and 20 times further than to the Moon. Actual distance ratio is 389, about 20 times bigger. Think error was in spotting when moon is exactly half full, the method works – the angle should be 89deg51min according to today’s measurements.

Aristarchus (310 – 230 BC) Earth Moon Sun When observing a solar eclipse, Aristarchus realised that the Moon fits over the Sun almost exactly. Can you use this and the information before to estimate a ratio for size? Using Similar Triangles, and his ratio of distance, he approximated that the Sun must be between 18 and 20 times bigger than the Moon. Using the same ratios as before meant that he was a factor of 20 out again, NASA calculates the Sun as 400 times larger than the Moon.

Aristarchus (310 – 230 BC) When observing a lunar eclipse, Aristarchus found the velocity of the moon: VM= 2rE 3 Aristarchus also knew that a full lunar cycle takes 720 hours, and so formulated the following equation: VM = 2πDM 720 Solving the equation you get DM = 60rE Sun Earth Used pi = 4, very inaccurate. Extremely accurate – today’s accepted values for the Distance to the Moon is between 57 and 63 earth radii Moon

Aristarchus (310 – 230 BC) Observing the moon setting takes 2 minutes, the equation: VM= 2rM = rM 2 Aristarchus also knew that the Moon’s orbit takes 24hrs = 1440minutes: VM = 2πDM 1440 Substitute in DM and solve the equations to get: rM = rE 3 Given that he uses pi = 4, this is inaccurate. Using pi=3.14 is much closer to today’s value of the moon being equal to 0.273 times earth’s size.

Myth Busting The Greek’s didn’t think the world was flat: They knew the shadow of the Earth was circular A ship’s mast appears on the horizon first The moon and the sun are both round Different stars can be seen at different latitudes Aristotle (384 - 322 BC) made the realisation that some stars could be seen in Egypt and not in Cyprus, and thereby showed that this means the earth must be spherical, as this phenomenon only occurs on a curved surface.

Eratosthenes (276 – 194 BC) Using the value for an Egyptian stadion, of 157.5m per stadion, this gives a circumference of 39375km – an error of less than 2%. However, there was another popular stadia measurement used in Eratosthenes’ time, the Attic stadion, with a value estimated to be 185m. Using the Attic stadion, his measurement was less accurate, at 46250km – a 16% error.

252,000 39,690,000 39,690 1.6%

1.09 x 1012 10913km 367% 109

Archimedes Is *quite* famous: (287 – 212 BC) Is *quite* famous: – he’s infamous for shouting Eureka when he found a way to find the volume of a crown; – he was very mechanically minded, and did a lot of work on discovering the advantages and effects of levers, “Give me a point to stand and I can move the Earth”; – he invented Archimedes’ Screw, which carries water upwards.

Archimedes In Maths, he found a relationship between the volume of a sphere and it’s encompassing cylinder, they are directly proportional. Test it out with a radius of 4.

Archimedes VC = πr2 x d VC = πr2 x 2r VC = 2πr3 If r is the radius of the sphere, the volume of the sphere is S r The volume of the cylinder must be area of the circular cross-section * height: VC = πr2 x d VC = πr2 x 2r VC = 2πr3 The volume of a sphere is two thirds the volume of the cylinder that encompasses it.

Archimedes SAS = 4πr2 SAC = 2πhr + 2πr2 SAC = 2π(2r)r + 2πr2 Interestingly, the same is true for the surface area. The surface area of the sphere is: SAS = 4πr2 r The surface area of the cylinder must be: SAC = 2πhr + 2πr2 SAC = 2π(2r)r + 2πr2 SAC = 4πr2 + 2πr2 SAC = 6πr2

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