iven point D in the straight line AB. I say that, whether by
reflexion or by refraction, it is only necessary to make this surface
such that the path of the light from the point A to all points of the
curved line CDE, and from these to the point of concurrence (as here
the path along the straight lines AC, CB, along AL, LB, and along AD,
DB), shall be everywhere traversed in equal times: by which principle
the finding of these curves becomes very easy.

[Illustration]

So far as relates to the reflecting surface, since the sum of the
lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
ought to be an ellipse; and for refraction, the ratio of the
velocities of waves of light in the media A and B being supposed to be
known, for example that of 3 to 2 (which is the same, as we have
shown, as the ratio of the Sines in the refraction), it is only
necessary to make DH equal to 3/2 of DB; and having after that
described from the centre A some arc FC, cutting DB at F, then
describe another from centre B with its semi-diameter BX equal to 2/3
of FH; and the point of intersection of the two arcs will be one of
the points required, through which the curve should pass. For this
point, having been found in this fashion, it is easy forthwith to
demonstrate that the time along AC, CB, will be equal to the time
along AD, DB.

For assuming that the line AD represents the time which the light
takes to traverse this same distance AD in air, it is evident that DH,
equal to 3/2 of DB, will represent the time of the light along DB in
the medium, because it needs here more time in proportion as its speed
is slower. Therefore the whole line AH will represent the time along
AD, DB. Similarly the line AC or AF will represent the time along AC;
and FH being by construction equal to 3/2 of CB, it will represent the
time along CB in the medium; and in consequence the whole line AH will
represent also the time along AC, CB. Whence it appears that the time
along AC, CB, is equal to the time along AD, DB. And similarly it can
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 poin