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Math 409/409G History of Mathematics

Math 409/409G History of Mathematics. Book I of the Elements Part III. In previous lessons you were introduced to Euclid’s axiomatic method, confronted with holes or problems with his method, and supplied with the modern day solution to these holes or problems.

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Math 409/409G History of Mathematics

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  1. Math 409/409GHistory of Mathematics Book I of the Elements Part III

  2. In previous lessons you were introduced to Euclid’s axiomatic method, confronted with holes or problems with his method, and supplied with the modern day solution to these holes or problems. You’re now ready to prove some of Euclid’s constructions.

  3. Bisecting an angle (P1.9) • Let BAD be given and let D be any point on side AB. (Ax. 2) • Construct E on AC so that . (P1.3)

  4. Construct segment DE. (Ax. 1) • On the side of DE not containing A, construct equilateral triangle DFE having side DE. (P1.1) • Then . (Def. equilateral )

  5. Construct segment DF. (Ax. 1) • Then ADF AEF. (SSS) • So BAF= CAF(Def. ), thus providing the bisection of BAC.

  6. Bisecting a segment (P1.10) Here’s an outline of the proof for a given segment AB. • Construct two circles having radius AB, one centered at A and the other at B. (Ax. 3) • Let C and D be the two points of intersection of these circles. (Ax. 7)

  7. Construct the segments displayed in the figure at the right. (Ax. 1) • All of these segments are equal in length. (Def. circle)

  8. Construct CD. (Ax. 1) • CAD CBD. (SSS) • 1  2. (Def. )

  9. ACM CBM where M is the intersection of segments CD and AB. (SAS) • So . (Def. ) This shows that a segment can be bisected.

  10. Constructing a perpendicular from a point on a line (P1.11) • Given a line and a point P on the line, construct two points A and B equidistant from P. (P1.3)

  11. Construct two circles having radius AB, one centered at A and the other at B. (Ax. 3) • Let the circles intersect at point C. (Ax. 7) • Construct line PC. (Ax. 1)

  12. Then PC perpendicular to the given line. SSS  ACP BCP  1 = 2  CP  AB (Def. )

  13. Constructing a perpendicular from a point not on a line (P1.12) • Given a line and a point P on the line, construct two points A and B equidistant from P. (P1.3)

  14. Construct the midpoint M of segment AB. (P1.10) • Construct PM. (Ax. 1) Then PM AB. (Def. )

  15. This ends the lesson on Book I of the Elements Part III

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