Refrangibility do not converge to the same Focus; but if they flow from
a lucid Point, as far from the Lens on one side as their Foci are on the
other, the Focus of the most refrangible Rays shall be nearer to the
Lens than that of the least refrangible, by above the fourteenth Part of
the whole distance; and if they flow from a lucid Point, so very remote
from the Lens, that before their Incidence they may be accounted
parallel, the Focus of the most refrangible Rays shall be nearer to the
Lens than the Focus of the least refrangible, by about the 27th or 28th
Part of their whole distance from it. And the Diameter of the Circle in
the middle Space between those two Foci which they illuminate, when they
fall there on any Plane, perpendicular to the Axis (which Circle is the
least into which they can all be gathered) is about the 55th Part of the
Diameter of the Aperture of the Glass. So that 'tis a wonder, that
Telescopes represent Objects so distinct as they do. But were all the
Rays of Light equally refrangible, the Error arising only from the
Sphericalness of the Figures of Glasses would be many hundred times
less. For, if the Object-glass of a Telescope be Plano-convex, and the
Plane side be turned towards the Object, and the Diameter of the
Sphere, whereof this Glass is a Segment, be called D, and the
Semi-diameter of the Aperture of the Glass be called S, and the Sine of
Incidence out of Glass into Air, be to the Sine of Refraction as I to R;
the Rays which come parallel to the Axis of the Glass, shall in the
Place where the Image of the Object is most distinctly made, be
scattered all over a little Circle, whose Diameter is _(Rq/Iq) (S
cub./D quad.)_ very nearly,[H] as I gather by computing the Errors of
the Rays by the Method of infinite Series, and rejecting the Terms,
whose Quantities are inconsiderable. As for instance, if the Sine of
Incidence I, be to the Sine of Refraction R, as 20 to 31, and if D the
Diameter of the Sphere, to which the Convex-side of the Glass is ground,
be 100 Feet or 1200 Inches, and S the Semi-diameter of the Aperture be
two Inches, the Diameter of the little Circle, (that is (_Rq S
cub.)/(Iq D quad._)) will be (31 31 8)/(20 20 1200 1200) (or
961/72000000) Parts of an Inch. But the Diameter of the little Circle,
through which these Rays are scattered by unequal Refrangibility, will
be about the 55th Part of the Aperture of the Object-glass, which here
is four Inches. And therefore, the Error arising from the Spherical
Figure of the Glass, is to the Error arising from the different
Refrangibility of the Rays, as 961/72000000 to 4/55, that is as 1 to
5449; and therefore being in comparison so very little, deserves not to
be considered.
55