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Normals are treated, not in connexion with tangents, but as minimum or maximum straight lines drawn to the curves from different points or classes of points. Certain propositions of great difficulty enable us to deduce quite easily the Cartesian equations to the evolutes of the three conics. He was also a master of the theory of epicycles and eccentrics.
And as the parallel rays are merely perpendiculars to the waves which fall on the concave surface, which waves are parallel to AD, it will be found that as they come successively to encounter the surface AB, they form on reflexion folded waves composed of two curves which originate from two opposite evolutions of the parts of the curve AFE. So, taking AD as an incident wave, when the part AG shall have met the surface AI, that is to say when the piece G shall have reached I, it will be the curves HF, FI, generated as evolutes of the curves FA, FE, both beginning at F, which together constitute the propagation of the part AG. And a little afterwards, when the part AK has met the surface AM, the piece K having come to M, then the curves LN, NM, will together constitute the propagation of that part.
Consequently the same wave becomes def, then ghk, and finally CY, from whence it subsequently spreads without any fold, but always along curved lines which are evolutes of the curve ENC, increased by some straight line at the end C.
There are always two. Often we find three pairs of almost the same type with but a shadow of density to distinguish each pair. Man evolutes upward on the scale of life by two tribes of apes. Ere man becomes human, he represents one cell force.
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