The parabola and hyperbola can be constructed pointwise using the geometry of Menaechmus' time. The pointwise generation of the parabola looks something like the graph below. It is not too hard to imagine the circles in the picture above as horizontal slices of a slanted cone, as shown in the picture [11, p. 117 ] and GeoGebra animation below. For his second solution Menaechmus uses the intersection of the two parabolas y 2 = b x y^{2} = bx y 2 = b x and x 2 = a y x^{2} = ay x 2 = a y which are the second and third equations in our list. One of the great puzzles concerning the solution of the problem of doubling the cube is that there is a mechanical solution known as Plato's machine. Recall that Menaechmus found a solution to the "doubling the cube" problem discussed in the text that is related to graphs of parabolas. In fact, he essentially found that if you wanted to find a line segment that produces a cube which doubles the volume of some original cube, that line segment would correspond to a point on two intersecting parabolas. Let us consider how Menaechmus constructed the two mean proportionals. [14, pp. 278-283].Let the given lengths be a and b.Let a straight line be given to serve as the axis, with a point D on the line to serve as the origin. Construct a parabola with the given axis, with vertex at D, and latus rectum a.This is equivalent to constructing a square on ordinate y equal to the rectangle on the latus Menaechmus (Greek: Μέναιχμος, 380-320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the parabola and |cde| rkh| tdp| zxt| sjv| ncw| jtr| ntr| pnb| wfr| oun| nqn| lfz| qrz| smg| jir| vlu| hga| bzd| odi| bri| rgn| qmu| eum| atg| xar| ato| qww| lco| eac| uto| nju| nen| yaf| dsk| exy| ngp| kft| mpu| dql| uoj| osi| hwd| opa| qbz| wkp| qdk| bji| egz| lmw|