1. State Euclids parallel postulate. Describe Saccheri's attempt to prove its truth in "Euclides ad omni naevo vindicatus"
In the description you could include some of the following in order to aid your description.
(a) An explanation of "reductio ad absurdum" and its relevance to the parallel postulate.
(b) A definition of Saccheri's quadrilateral. Also state the relevant names of all parts of Saccheri's quadrilateral.
(c) An explanation of the hypothesis of the acute angle and some of the Non-Euclidean theorems it led to. Use the following the illustrate your explanation.
"Let ABC be any right triangle and let M be the midpoint of the hypotenuse. At A construct angle BAD = angle ABC. From M draw MP perpendicular to CB. On AD mark off AQ = PB and MQ. Prove triangles AQM and BPM congruent thus showing that angle AQM is a right angle and points QMP collinear. Then ACPQ is a Lambert Quadrilateral with acute angle at A. Now show that under the hypothesis of the acute angle the sum of the angles of any triangle is less than 2 right angles.
2. One consequence of the parallel postulate was that Eratosthenes who lived in the third century BC determined the earth's circumference.
He found that the distance between Alexandria and Syrene (A and S) was 500 miles. He also found that angle b was 7.50 degrees. State the theorem concerned with parallel lines that he used and explain his estimate of the earth's circumference. Was it accurate?
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