arcs GH, HA, etc., namely that in the quadrilaterals which enclose
them the side parallel to the axis is to the opposite side as 3 to 2.
Then also as 3 to 2 will the sum of the one set be to the sum of the
other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
supposing V to be the intersection of the curve EK and the ray FO.
But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
of BE to the semi-diameter of the spherical wave which emanated from
the point F while the light outside the transparent body traversed the
space BE. Then it appears that this wave will intersect the ray FM at
the same point V where it is intersected at right angles by the curve
EK, and consequently that the wave will touch this curve. In the same
way it can be proved that the same will apply to all the other waves
above mentioned, originating at the points G, H, etc.; to wit, that
they will touch the curve EK at the moment when the piece D of the
wave ED shall have reached E.
Now to say what these waves become after the rays have begun to cross
one another: it is that from thence they fold back and are composed of
two contiguous parts, one being a curve formed as evolute of the curve
ENC in one sense, and the other as evolute of the same curve in the
opposite sense. Thus the wave KE, while advancing toward the meeting
place becomes _abc_, whereof the part _ab_ is made by the evolute
_b_C, a portion of the curve ENC, while the end C remains attached;
and the part _bc_ by the evolute of the portion _b_E while the end E
remains attached. 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 is even, in this curve, a part EN which is straight, N being the
point where the perpendicular from the centre X of the sphere falls
upon the refraction of the ray DE, which I now suppose to touch the
sphere. The folding of the waves of light begins from the point N up
to the end of the curve C, which point is formed by taking AC to CX in
the proportion of the refraction, as here 3 to 2.
As many other points as may be desired in the curve NC are found by a
Theorem which Mr. Barrow has demonstrated in section 12 of his
_Lectiones Opticae_, though for another purpose. And it is to be noted
that a straight line equal in length to this curve can be given. For
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