be shown if L and K are other points in the curve CDE, that the times
along AL, LB, and along AK, KB, are always represented by the line AH,
and therefore equal to the said time along AD, DB.
In order to show further that the surfaces, which these curves will
generate by revolution, will direct all the rays which reach them from
the point A in such wise that they tend towards B, let there be
supposed a point K in the curve, farther from D than C is, but such
that the straight line AK falls from outside upon the curve which
serves for the refraction; and from the centre B let the arc KS be
described, cutting BD at S, and the straight line CB at R; and from
the centre A describe the arc DN meeting AK at N.
Since the sums of the times along AK, KB, and along AC, CB are equal,
if from the former sum one deducts the time along KB, and if from the
other one deducts the time along RB, there will remain the time along
AK as equal to the time along the two parts AC, CR. Consequently in
the time that the light has come along AK it will also have come along
AC and will in addition have made, in the medium from the centre C, a
partial spherical wave, having a semi-diameter equal to CR. And this
wave will necessarily touch the circumference KS at R, since CB cuts
this circumference at right angles. Similarly, having taken any other
point L in the curve, one can show that in the same time as the light
passes along AL it will also have come along AL and in addition will
have made a partial wave, from the centre L, which will touch the same
circumference KS. And so with all other points of the curve CDE. Then
at the moment that the light reaches K the arc KRS will be the
termination of the movement, which has spread from A through DCK. And
thus this same arc will constitute in the medium the propagation of
the wave emanating from A; which wave may be represented by the arc
DN, or by any other nearer the centre A. But all the pieces of the arc
KRS are propagated successively along straight lines which are
perpendicular to them, that is to say, which tend to the centre B (for
that can be demonstrated in the same way as we have proved above that
the pieces of spherical waves are propagated along the straight lines
coming from their centre), and these progressions of the pieces of the
waves constitute the rays themselves of light. It appears then that
all these rays tend here towards the point B.
One might also determine the point C, and all the others, in this
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