By Robin Hartshorne
This e-book bargains a different chance to appreciate the essence of 1 of the good thinkers of western civilization. A guided studying of Euclid's parts results in a serious dialogue and rigorous smooth therapy of Euclid's geometry and its more moderen descendants, with whole proofs. subject matters contain the creation of coordinates, the idea of region, background of the parallel postulate, many of the non-Euclidean geometries, and the commonplace and semi-regular polyhedra.
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Extra info for Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)
Sample text
Join AD. Then AD is the angle bisector. 22 1. Euclid's Geometry Euclid's method is economical for him because it makes use of previously described constructions (I. l). If we count the number of steps to carry out this construction, we find seven: A Choose B at random on I (no step) 1. Circle center A radius AB, get C. 2. Draw BC. 3. Circle center B radius BC. 4. Circle center C radius CB, get D. 5. Draw BD. 6. Draw CD. 7. Draw AD, which is the angle bisector. If we are concerned only with making an independent construction for the angle bisector, there is no need to draw the lines BC, BD, CD.
11 (Campanus). Use the theory of content to show that the line DE joining the midpoint of two sides of a triangle is parallel to the third side. (Hint: Draw BE and DC. ll). The proof of this construction makes use of all the geometry he has developed so far, so that one could say that to understand fully this single result is tantamount to understanding all of the first four books of Euclid's geometry. It also raises questions of exposition that are central to our modern examination of Euclid's methods.
L). If we count the number of steps to carry out this construction, we find seven: A Choose B at random on I (no step) 1. Circle center A radius AB, get C. 2. Draw BC. 3. Circle center B radius BC. 4. Circle center C radius CB, get D. 5. Draw BD. 6. Draw CD. 7. Draw AD, which is the angle bisector. If we are concerned only with making an independent construction for the angle bisector, there is no need to draw the lines BC, BD, CD. Thus the construction reduces to four steps. In order to prove that this construction works, we might want to draw the lines BC, BD, CD and argue as Euclid did.