Lenses and Waves

1653 - TRACTATUS 23
object should leave the ocular parallel to the axis. 36 The foci of the lenses
should therefore coincide. For myopic people and those using a telescope to
project images things are different. In these cases the rays should be brought
to focus after they have passed the ocular and the foci of the lenses should
not coincide. In the second proposition, Huygens discussed the
configuration of two lenses required to project images and determined their
magnification. 37
Huygens aimed at providing a general and exact theory of the properties
of lenses and their configurations. The generality of Huygens’ theory reached
its high-point in a theorem that is inserted in part two of Tractatus as the sixth
proposition. It may be of a later date, as the manuscript is on a different kind
of paper and written with a different pen than the rest of this part. 38
Nevertheless, the theorem states that the magnification of an arbitrary
system of lenses remains the same when eye and object switch place. 39 This
theorem, so Huygens concluded his demonstration, would be useful in
determining the magnification and distinctness of images.
Figure 12 Analysis of Keplerian telescope with erector lens. See also Figure 13.
Huygens applied the theorem in the third and fourth proposition included by
the editors of Oeuvres Complètes in book three. The third proposition is
certainly of a later date, as it analyses the eyepiece Huygens invented in
1662. 40 The fourth proposition discusses a configuration of three convex
lenses proposed by Kepler in 1611 (Figure 12). 41 A telescope of two convex
lenses ordinarily produces a reversed image, but a third lens inserted between
the ocular and the objective may re-erect the image.
Huygens explained that an upright and sharp image is attained as follows
(Figure 13). 42 AC is the focal distance of the objective lens YAB, and HF the
focal distance of the ocular QHR. The third lens DET is identical with the
ocular with a focal distance EL = HF. It is placed so that EC = 2EL and EH =
3EL. In this case, point C on the axis is the ‘punctum correspondens’ for rays
through focus F of the ocular. Therefore a ray from S at a large distance is
refracted by the lenses in such a way that it leaves the ocular parallel to the
axis towards the eye PN. In order to determine the magnification by the
36 OC13, 244-247.
37 OC13, 246-253.
38 Hug29, 151-167.
39 OC13, 198-199.
40 OC13, 252n1. See below, section 3.1.2.
41 Dating this theorem is difficult. It may have been written in 1653, as the configuration was well-known.
Yet, Huygens also discussed the enlarged field of such a configuration, which may imply that it is of a
later date. See note 20 on page 16 above.
42 OC13, 258-261.