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Descartes's Mean Proportionals/Logarithm Machine

Descartes's Mean Proportionals/Logarithm Machine
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2018-09-27 00:02
VectorTrigonometryCoordinate GeometryOptimizationGeometric Construction
This example demonstrates the geometric construction of Descartes's mean proportionals and its connection to logarithms. Key math concepts include mean proportionals, geometric transformations, coordinate geometry, and logarithms. The construction utilizes GeoGebra commands such as Intersect, Point, Vector, Circle, Ray, OrthogonalLine, Semicircle, and Locus. Suitable for high school and above students to understand logarithmic principles and geometric constructions.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → Y
  2. Point(xAxis) → C_1
  3. Vector(Y, C_1) → u
  4. Circle(Y, 1) → d
  5. Point(d) → B
  6. Ray(Y, B) → e
  7. OrthogonalLine(B, e) → f
  8. Intersect(c, f) → C
  9. OrthogonalLine(C, c) → g
  10. Intersect(g, e) → D
  11. OrthogonalLine(D, e) → h
  12. Intersect(c, h) → E
  13. OrthogonalLine(E, c) → i
  14. Intersect(i, e) → F
  15. OrthogonalLine(F, e) → j
  16. Intersect(d, c) → I, A
  17. Intersect(c, j) → G
  18. Point(e) → X
  19. OrthogonalLine(G, c) → k
  20. Intersect(k, e) → H
  21. OrthogonalLine(H, e) → l
  22. Intersect(l, c) → N
  23. Point(c) → Z
  24. Semicircle(Y, E) → p
  25. Semicircle(Y, N) → r
  26. Semicircle(Y, G) → q
  27. Locus(D, B) → loc1
  28. Locus(F, B) → loc2
  29. Locus(H, B) → loc3
  30. Angle(Y, B, C) → α
  31. Angle(D, C, Y) → β
  32. Angle(Y, D, E) → γ
  33. Angle(F, E, Y) → δ
  34. Angle(Y, F, G) → ε
  35. Angle(H, G, Y) → ζ
  36. Angle(Y, H, N) → η

AI Drawing Prompt Reference

Draw the geometric construction of Descartes's mean proportionals and logarithm machine: start with coordinate axes and mark the origin; place a point on the x‑axis and draw a ray from the origin through that point; draw a unit circle centered at the origin; construct a series of perpendicular lines alternating between the ray and the circle to obtain intersection points; draw semicircles from the origin to selected points; trace the loci of moving points; and indicate the angles formed between the constructed lines.
The prompt above was compiled from the original drawing commands and is for reference only.