PROBLEM XLIII.

To find the focus of any lens.

THE same reasoning, as that employed to determine the course of a ray passing through a glass sphere, might be employed in the present case. But for the sake of brevity, we shall only give a general rule, demonstrated by opticians, which includes all the cases possible in regard to lenses, whatever combinations may be formed of convexities and concavities. We shall then shew the application of it to a few of the principal cases. It is as follows.

As the sum of the semi-diameters of the two convexities, is to one of them; so is the diameter of the other, to the focal distance.

In the use of this rule, one thing in particular is to be observed. When one of the faces of the glass is plane, the radius of its sphericity must be considered as infinite; and when concave, the radius of the sphere, of which this concavity forms a part, must be considered as negative. This will be easily understood by those who are in the least familiar with algebra.

CASE I. When the lens is equally convex on both sides.

Let the radius of the convexity of each of the faces be, for example, equal to 12 inches. By the general rule we shall have this proportion: as the sum of the radii, or 24 inches, is to one of them, or 12 inches, so is the diameter of the other, or 24

inches, to a fourth term, which will be 12 inches, the focal distance. Hence it appears that a lens equally convex on both sides unites the solar rays, or in general rays parallel to its axis, at the distance of the radius of one of the two sphericities.

CASE II. When the lens is unequally convex on both sides.

If the radii of the convexities be 12 and 24, for instance, the following proportion must be employ ed: as 12+24, or 36, is to 12, the radius of one of the convexities, so is 48, the diameter of the other, to 16; or as 12+ 24, or 36, is to 24, the radius of one of the convexities, so is 24, the diameter of the other, to 16: the distance of the focus therefore will be 16 inches.

CASE III. When the lens has one side plane.

If the sphericity on the one side be as in the preceding case, we must say, by applying the general rule, as the sum of the radii of the two sphericities, viz, 12 and an infinite quantity, is to one of them, or the infinite quantity, so is 24, the diameter of the other convexity, to a fourth term, which will be 24; for the two first terms are equal, because an infinite quantity increased or diminished by a finite quantity, is always the same: the two last terms therefore are equal; and it hence follows, that a plano-convex glass has its focus at the distance of the diameter from its convexity.

CASE IV. When the lens is convex on the one side, and concave on the other.

Let the radius of the convexity be still 12, and that of the concavity 27. As a concavity is a ne

gative convexity, this number 27 must be taken with the sign prefixed. We shall therefore have this proportion.

As 12 inches - 27, or 15 inches, is to the radius of the concavity 27 (or as 15 is to 27), so is 24 inches, the diameter of the convexity, to 43. This is the focal distance of the lens, and is positive or real; that is to say, the rays falling parallel to the axis, will really be united beyond the glass. The concavity indeed having a greater diameter than the convexity, this must cause the rays to diverge less than the convexity causes them to converge. But if the concavity be of a less diameter than the convexity, the rays, instead of converging when they issue from the glass, will be divergent, and the focus will be before the glass: in this case it is called virtual. Thus, if the radius of the concavity be 12, and that of the convexity 27, we shall have, by the general rule: as 27-12, or 15, is to 27, so is 24 to 43 The last term being negative, it indicates that the focus is before the glass, and that the rays will issue from it divergent, as if they came from that point.

CASE V. When the lens is concave on both sides.

24 to

If the radii of the two concavities be 12 and 27 inches, we shall have this proportion as 1227 is to 27, or as 39 is to 27, so is 16 8 The last term being negative, it shews that the focus is only virtual, and that the rays, when they issue from the glass, will proceed diverging, as if they came from a point situated at the distance of 16 inches before the glass.

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CASE VI. When the lens is concave on one side, and plane on the other.

If the radius of the concavity be still 12, the above rule will give the following proportion: as — 12 + an infinite quantity, is to an infinite quantity, so is 24 to 24; for an infinite quantity, when it is diminished by a finite quantity, remains still the same. Thus it is seen that, in this case, the virtual focus of a plano-concave glass, or the point where the rays after their refraction seem to diverge, is at a distance equal to the diameter of the concavity, as the point to which they converge is in the case of the plano-convex glass.

These are all the cases that can occur in regard to lenses for that where the two concavities might be supposed equal is comprehended in the fifth.

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REMARK.

In all these calculations, we have supposed the thickness of the glass to be of no consequence in regard to the diameter of the sphericity, which is the most common case; but if the thickness of the glass were taken into consideration, the determinations would be different.

Of Burning Glasses.

Lenticular glasses furnish a third method of solving the problem, already solved by means of mirrors, viz, to unite the rays of the sun in such a manner, as to produce fire and inflammation: for a glass

of a few inches diameter will produce a heat sufficiently strong to set fire to tinder, linen, black or grey paper, &c.

The ancients were acquainted with this property in glass globes, and they even sometimes employed them for the above purpose. It was probably by means of a glass globe that the vestal fire was kindled. Some indeed have endeavoured to prove, that they produced this effect by lenses: but de la Hire has shewn, that this idea is entirely void of foundation, and that the burning glasses of the ancients were only glass globes, and consequently incapable of producing a very remarkable effect.

Baron von Tchirnhausen, who constructed the celebrated mirror already mentioned, made also a burning glass, the largest that had ever been seen. This mathematician, being near the Saxon glass manufactories, was enabled, about the year 1696, to procure plates of glass sufficiently thick and broad, to be converted into lenses several feet in diameter. One of them, of this size, inflamed combustible sub. stances at the distance of 12 feet. Its focus at this distance was about an inch and a half in diameter. But when it was required to make it produce its greatest effects, the focus was diminished by means of a second lens, placed parallel to the former, and at the distance of four feet. In this manner, the diameter of the focus was reduced to 8 lines, and it then fused metals, vitrified flint, tiles and slate, earthen ware, &c, in a word, it produced the same effects as the burning mirrors of which we have already spoken.

Some years ago a lens, which one might have taken for that of Tchirnhausen, was exhibited at