olution of the curve ENC, with different lengths of thread, will cut
all the rays HL, GM, FO, etc., at right angles, and in such wise that
the parts of them intercepted between two such curves will all be
equal; for this follows from what has been demonstrated in our
treatise _de Motu Pendulorum_. Now imagining the incident rays as
being infinitely near to one another, if we consider two of them, as
RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
FS which intersects GM at P to have been described by evolution from
the curve NC, beginning at F, as far as which the thread is supposed
to extend, we may assume the small piece FP as a straight line
perpendicular to the ray GM, and similarly the arc GF as a straight
line. But GM being the refraction of the ray RG, and FP being
perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
proportion of the refraction; as was shown above in explaining the
discovery of Des Cartes. And the same thing occurs in all the small
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 e