nd 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
since it together with the line NE is equal to the line CK, which is
known, since DE is to AK in the proportion of the refraction, it
appears that by deducting EN from CK the remainder will be equal to
the curve NC.

Similarly the waves that are folded back in reflexion by a concave
spherical mirror can be found. Let ABC be the section, through the
axis, of a hollow hemisphere, the centre of which is D, its axis being
DB, parallel to which I suppose the rays of light to come. All the
reflexions of those rays which fall upon the quarter-circle AB will
touch a curved line AFE, of which line the end E is at the focus of
the hemisphere, that is to say, at the point which divides the
semi-diameter BD into two equal parts. The points through which this
curve ought to pass are found by taking, beyond A, some arc AO, and
making the arc OP double the length of it; then dividing the chord OP
at F in such wise that the part FP is three times the part FO; for
then F is one of the required points.

[Illustration]

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
surfac