The eye of the Insect tribe has been chosen for the present communication, not only from its great beauty and wonderful organization, but on account of its transparent portion (cornæ) presenting a multitude of well-defined planes, forming a reticulation which is especially calculated to excite our admiration. It is to this, therefore, and not to the interior, that our attention will be chiefly directed.
On examining the head of an insect we shall find a couple of protuberances more or less prominent, and situated symmetrically, one on each side. Their outline at the base is for the most part circular, elliptical, oval, or truncated; while their curved surfaces are spherical, spheroidal, pyriform, &c.
These horny, rounded, naked parts seem externally to represent the cornæ: of the eyes of Insects; at least they are appropriately so called from the analogy they bear to those transparent tunics in the higher classes of animals. They differ from these latter, however, in this respect, that, when viewed by the microscope, they display a number of hexagonal facets which constitute the media for the ingress of light to as many simple eyes. Under an ordinary lens, and by reflected light, the entire surface of one of these cornæ; presents a beautiful reticulation, like very fine wire gauze, with a minute papilla, or, at least, slight elevation in the centre of each mesh. These are resolved, however, by the aid of a compound microscope, and with a power of from 80 to 100 diameters, into an almost incredible number, when compared with the space they occupy, of minute, regular geometrical hexagons, well-defined and capable of being computed with comparative ease, their exceeding minuteness being taken into consideration. When viewed in this way the entire surface bears a resemblance to that which might easily and artificially be produced by straining a portion of Brussels lace with hexagonal meshes over a small hemisphere of ground glass. That this gives a tolerably fair idea of the intricate carving on the exterior may be further shown from the fact that delicate and beautiful casts in collodion* may be procured from the surface by giving this three or four coats with a camel’s-hair pencil. When dry it is peeled off in thin flales, upon which the impressions are left so distinct, that their hexagonal form can be discovered with a Coddington lens. This experiment will be found useful in examining the configuration of the facets of the hard and unyielding eyes of many of the Coleóptera, in which the reticulations become either distorted by corrugation or broken from the pressure required to flatten them. It will be observed, also, that by this method perfect casts of portions of the cornæ can be obtained without any dissection whatever, arid that these artificial exuviai, for such they really are, become available for microscopic investigations; obviating the necessity for a more lengthened or laborious preparation.
But to return. The dissection of the cornæ of an insect’s eye is by no means easy. I have generally used a small pair of scissors, with well-adjusted and pointed extremities, and a camel’s-hair pencil, having a portion of the hairs cut off at the end, which is thereby flattened. The extremity of the cedar handle, on the other hand, is shaved to a fine point, so that the brush may be the more easily revolved between the finger and thumb, and the coloured pigment on the interior may thus be scrubbed off by this simple process. A brush thus prepared and slightly moistened forms, as far as my experience goes, by far the best forceps for manipulating these objects preparatory to mounting; as, if only touched with any hard-pointed substance, they will often spring from the table from mere elasticity, and thus the labour of hours may be lost in one single moment. It does not appear to me desirable to attempt to flatten an entire cornæ by pressure and maceration, although I am aware this is generally recommended, but no useful purpose is really served either in developing the beauty or counting the number of its lenses. The rounded membrane, on the other hand, becomes, as might be anticipated if the margin remains intact, corrugated, and so one hexagon overlaps the other. It will be useful, therefore, to make two preparations of the eyes of one insect, the one entire, retaining its naturally curved form, not having been subjected to any pressure—the other nicked at its margin, or cut into small fragments and pressed flat between two slides.
Each of the hexagons above-mentioned is itself the slightly “ convex horny case of an eye. Their margins of separation are often thickly set with hair, as in the Bee; in other instances they are naked, as in the Dragon-fly, House-fly, &c. The number of these lenses has been calculated by various authors, and their almost incredible multitude has very justly excited astonishment. Hooke counted 7000 in the eye of a House-fly; Leeuwenhoek more than 12,000 in the eye of a Dragon-fly, and 4000 in the eye of a domestic fly; and Geoffroy cites a calculation, according to which there are 34,650 of such facets in the eye of a Butterfly.”
Having carefully examined with the microscope a small flattened portion of the eye of a Dragon-fly and a few analogous specimens, we are, I think, in a position to assume two things which will serve to form the basis in our calculations:—
1st. That the reticulations referred to are composed of perfect, regular, geometric hexagons; and 2ndly. That the hexagons are all of equal size.
Their number, in any individual specimen under investigation, might, of course, be ascertained by actual enumeration; the process however would be a very laborious one, and injurious to the sight. Leeuwenhoek computed them by assuming the prominent part of the eye to be hemispherical.* He then counted a single row of hexagons from the summit to the base, and this multiplied by four gave the great circle of a sphere, the area of which was then discovered by a simple arithmetical process. It will be observed, however, that those eyes only, the surface of whose common cornæ is hemispherical (and there is a large number in which it is not), can be treated in this way; and, if the facets could be thus computed, the results would be incorrect according to the method of Leeuwenhoek; inasmuch as in all his calculations the hexagons were reckoned as squares: thus many hundred were lost even in one single eye. Having pointed out this source of fallacy, we proceed to endeavour to correct it.
A mere inspection of the above square area of hexagons will show that such an outline, enclosing as many regular hexagons of a given size as it will contain, has a less number on the one side, A B, than on its adjacent side, AC. A closer examination will discover that these numbers bear a ratio of 8: 9.25, or of 1: 1.156; while, if the entire area is counted, not omitting the portions which are truncated by the sides of the square, it will be found about 74 (or 8 × 9.25). Those numbers are not, indeed, mathematically correct, but sufficiently so for our present purpose; for, doubtless, we have not failed to notice that if the side, A B, bad been squared in the ordinary way (8 × 8), and not treated as if it were composed of hexagons (8 × 9.25), we should have lost as many as ten planes even in a space containing so few hexagons; and these will vanish by hundreds instead of tens, as the area increases.
And, if we take a circle with a row of hexagons passing through its great diameter, A B, and calculate from this the entire number spread over its whole superficies, we shall soon discover how very far wide of the truth our results would be, supposing the hexagons were treated as squares. For, first, let it be required to find the area of a circle in squares with any number, say twenty, composing its diameter. Now, the square of the diameter × 7854 = the area: lienee 20“ × 7854 = 314.160 the area in squares.
Again, given a circle whose diameter = 20 regular hexagons, arranged with their sides in apposition (fig. 1), to find the area in hexagons. Now, as circles are to one another as the squares of their diameters, and as we have already seen that a square of hexagons = the product of 8: 9.25, or numbers in that ratio, we have:—8: 9.25:: 20: 23.125. Hence 20 × 23.125 = 462.5, the diameter squared, and 462.5 × 7854 = 363.247 the area in hexagons.
Or a circular area of hexagons may be thus found:— Given: a circle with twenty small hexagons (arranged side by side, fig. 1) passing through its great diameter, to find the area.
of the diameter. Now evidently 
= number of hexagons; we have therefore to find area of hexagon. 
From these and analogous calculations, tables might be constructed for all possible dimensions of the square and the circle, the side being given in the former case, and the diameter in the latter: —
A few only are necessary in this place; but even in these the columns of difference sufficiently indicate the loss likely to follow from miscalculation. I pass on to notice, however, that the only quadrilateral figure which will so contain a number of hexagons that its area may be discovered by squaring a side, is a rhomb of 60° and 120°; that is to say, two equilateral triangles placed base to base. When such a plane is occupied by regular hexagons, any side, A B, may be supposed to consist of small rhombs ranged side by side, each being exactly one-third (G) of one of the enclosed hexagons. All the sides are alike; hence it follows that, if one of them be multiplied into itself, and the product divided by three, the area of the rhomb, ABC I), in hexagons, is determined.
the number of hexagons in A B C D. But the sides themselves are deduced from a single row of hexagons E F extending across the rhomb perpendicularly with respect to A D and B C; and it is to be remarked that the number of rhombs in a side is always double of that of the hexagons composing this perpendicular senes. In the figure there are three such hexagons E F, and consequently six rhombs in a side. The hexagons can always be calculated therefore by the formula 
the area of ABCD.Allusion has been made to Leeuwenhoek’s calculations of the lenses of the silkworm’s eye. These may now be corrected.
We have seen how easily a surface of hexagons, whether it be circular or hemispherical, square or rhombic, may be computed from a single row; and we have now to procure sections of eyes, presenting such shapes for inspection under the microscope. To excise small fragments from such minute and fragile membranes, and those of regular and determinate figures, requires nice manipulation. The quadrilateral figures I have been in the habit of making, by enclosing the membrane between two pieces of gummed white paper, upon one side of which the parallelograms are drawn; they are then cut entirely through with a penknife, and soaked for a short time in cold water, which softens the gum, and thus separates the paper. Circular sections are made with a small punch, after having been enclosed between paper as above recommended. On the surface of a small circle of the eye of a Dragon-fly, excised with the smallest saddler’s punch, marked No. 1, I have counted about 800 facets; in another, a size or two larger, about 5000, and so on. I have not felt satisfied with many of these preparations, however, although several have come out very well. Their edges are often lacerated by the punch, while the parallelograms, when magnified, have presented considerable deviations from the true parallel. These inconveniences are obviated by making small apertures, of the required shape and size, in black paper, which are placed immediately over the specimens to be examined. The circular openings can be punched out, while the others can be removed with a sharp knife. A simple and not inelegant mode of procuring very small rhombic apertures, perfectly equilateral and equiangular, consists in excising two small equilateral triangles from two slips of black paper, and sliding one over the other until the small rhomb in the centre, produced by their mutual intersection, is of the required size. The cornæ is placed under this rhombic aperture, and the lenses are viewed and counted through it, by merely enumerating one row extending in a perpendicular direction, with respect to any two opposite or parallel sides, and joining them as in the dotted line of the annexed rhomb.
APPENDIX
This is the first time, I believe, that the collodion has been employed in the production of transparent membranes for microscopic purposes. There are reasons for supposing that it will enable us to construct a series of novel and highly interesting preparations, by its presenting the minute tracery observed on the surface of many opaque objects in a transparent form. In this way we can multiply impressions of specimens which are very beautiful or very rare. It bids fair, also, to put us into possession of the general configuration on the surface of certain minute fresh vegetable structures which become shrivelled, and their beauty obliterated in drying. It is best applied as follows:—A few chips of Red Sanderswood are shaken up in a drachm or two of good collodion; the surface of the object is then painted over four or five times, and in less than ten minutes the flake or cast of collodion can be peeled off, and mounted on a slide under a thin cover as a dry preparation.
The eye under examination was that of the Moth of the Silk-worm.
