Euan N. K. Clarkson
University of Edinburgh, Department of Geology and Geophysics, Grant Institute, West Mains Road, Edinburgh EH9 3JW, Scotland, U. K.

Summary

Trilobites had to redevelop their corneal lenses after each molt, since the old lenses remained with the discarded exoskeleton. Earlier, Miller & Clarkson (1980) were able to reconstruct three main stages of the post-ecdysially developing lenses in the schizochroal compound eye of the Devonian trilobite Phacops rana milleri Stewart 1927. In this present work it is shown that the conical, then saucer-like and later wave-shaped proximal profile of the lens in these developmental stages is consistent with a Huygensian correction for spherical aberration as postulated for the adult eyes of some other trilobites by Clarkson & Levi-Setti (1975). The focal length of the developing lens is determined as a function of the lens thickness comparing and fitting the theoretically calculated Huygensian profiles to the experimentally reconstructed real lens surfaces. Using an empirical quadratic function fitted on to the variation of focal length versus lens thickness, a probable series of change of form of the developing lens in Phacops rana milleri is reconstructed computationally. On the basis of the geometric optical model presented, a further possible transitional stage between the shape of the last stage of the post-ecdysially developing lens and its mature form can be derived. Using geometric optical formulae for thick lenses and paraxial approximation, many features of image construction have been estimated for the post-ecdysial development of eye. The actual position of the retina below the lens, and whether this changed during post-ecdysial development, remains unknown from fossilized material. It has been possible, however, to calculate object position and magnification at all stages of post-ecdysial development, from the shape and thickness of the lens. Likewise, the positions of the retina for which Phacops rana milleri could take advantage of its spherically corrected Huygensian lenses are established here. It is probable that the retina was fixed or moved little during the post-ecdysial stages. If so the eye was myopic in the earliest developmental stage, but thereafter could see sharply at a distance of a few millimetres to a few centimetres from the visual surface. Depending on the receptor separation in the retina, the depth of focus estimated was several centimetres so the depth in object space could reach one decimetre, over which the image was in focus in the developing eye.

Figure Legends

Figure 1}: Three stages of the post-ecdysial development of the lens in {\em Phacops} {\em rana} {\em milleri} as reconstructed by Miller \& Clarkson (1980). (A) Initial small lens with conical, slightly incurved proximal surface; (B) thin biconvex lens with saucer-like proximal surface; (C) thick lens with wavy proximal profile, before the differentiation of the core and addition of the intralensar bowl. The values of the lens thickness $a$ of these developmental stages can be found in Table 1.

Figure 2}: Diagrammatic representation of the mature schizochroal compound eye of the Devonian trilobite {\em Phacops} {\em rana} {\em milleri} Stewart 1927, with lenses dissected to show internal structure. Details of corneal structure have been omitted for clarity (Miller \& Clarkson, 1980).

Figure 3}: Vertical (A) and horizontal (B) section of a lens in the eye of {\em Phacops} {\em rana} {\em milleri} (Clarkson, 1979).

Figure 4}: Structure of the developing eye in {\em Phacops} {\em rana} {\em milleri}. (A) Surface of the eye of the earliest post-ecdysial stage, after an etching process which has dissolved out the small conical lenses; $\times 60$ (cf. Figure 1A). (B) Detail of a single etched conical lens; $\times 315$. (C) Vertical section of lenses at the earliest post-ecdysial stage, showing thin cuticle and conical form of lenses; $\times 32$. (D) Vertical section through part of eye, a later post-ecdysial stage where the lens is now of biconvex form with saucer-like proximal surface; $\times 32$ (cf. Figure 1B). (E) Vertical section through lens at a still later stage, equivalent to Figure 1C; $\times 45$. (F) Vertical, etched section through mature lens, showing the intralensar bowl, core and etched-out cleavage planes in the calcite; $\times 60$ (cf. Figs 2 and 3A).

Figure 8}: Computationally reconstructed lenses in the post-ecdysially developing eye of {\em Phacops} {\em rana} {\em milleri} which fit best with the experimentally reconstructed real ones of Figure 1. (A) $a_1 = 150$ $\mu$m, $L_1 = 50$ $\mu$m; (B) $a_2 = 450$ $\mu$m, $L_2 = 950$ $\mu$m; (C) $a_3 = 650$ $\mu$m, $L_3 = 850$ $\mu$m. Ray tracing through the lens is represented in cases (B) and (C).

Figure 10}: Computational reconstruction of probable changes of form of the post-ecdysially developing lens in {\em Phacops} {\em rana} {\em milleri} as a function of the lens thickness $a$ using function $L(a)$ described by (19) and represented in Figure 9 with parameters $(q_1,q_2,q_3)$ of Table 1. $a_i = a_0 + (i - 1) \Delta a$, $i = 1,2,...,N$; $a_0 = 142$ $\mu$m. (A) $\Delta a = 100$ $\mu$m, $N = 8$; (B) $\Delta a = 50$ $\mu$m, $N = 13$.