Surveying and Levelling Instruments, Theoretically and Practically Described. / For construction, qualities, selection, preservation, adjustments, and uses; with other apparatus and appliances used by civil engineers and surveyors in the field.

by William Ford Stanley

Fig. 5.—Diagram illustrating the principle of the lens.

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58.—Passage of a Ray of Light through a Prism or a Lens—Convex Refraction. If we comprehend the law of refraction exemplified above, art. 51, the path of a monochromatic ray through a prism or a lens is easily determined, taking into consideration the refraction index of the glass. In Fig. 5 let a″a‴ be the base of an equilateral prism, which base may also represent the axis of a lens linear or parallel with the direction from the centre of the eye to O. Now, if a ray of light pass from a small luminous object at O in the path a′ to the prism, we may assume all other parts of the prism covered, and the refraction of the glass be such that the ray will pass through it from this position in a horizontal direction, or that parallel to the assumed axis a″a‴, then the same ray will pass through the prism to equal distance from the centre of the prism,—that is, to the position of the eye shown by the ray continuing in the path a, the angles to or from the prism being equal; so that if we cover up all parts of this prism except a line parallel with its base joining the ends of the lines aa′, where it is shown passing through the prism, any ray of light from O, under the conditions given, will appear as a spot of light on the plane parallel to the base of the prism; or if we place our eye at the position shown, we shall see the image of the light O. If we take a prism of the same kind of glass, but of less angle, whose base is b″b‴, the refraction would then be less (that is in the ratio of the sines), that is if the ray pass through the prism at less distance from the base, so that the ray Ob′ would pass through horizontally as before, and emerge from the prism in the path b, also with equally less refraction, so that the ray would reach the eye at the same point as the more refracted ray. In like manner, if the prism were of still less angle with base c″c‴ and pass through the prism at a lower position, the refraction would be proportionally less, and therefore reach the eye at the same point.

59.—If we take the half lens shown in section in the figure, this may be considered to touch the surface of the prisms described tangentially in the lines a″a‴, b″b‴, and c″c‴, where the angles of contact of O, a, b, or c upon the prism would be equal to those upon the lens for an infinitely small extent of surface. Therefore, if we make the lens of such form that a ray of light may pass from any single point upon the line of its axis, and be refracted by every point of the surface of the lens to a single point or focus on the opposite side of the axis, such form would be a perfect lens. For simplicity of demonstration the refractions given above are made parallel with the axis of the lens. This parallelism could only occur with the object and the eye at equal distance from the centre of the lens, and with this distance also proportional to the amount of refraction of the glass used in the construction. If the rays were all parallel to each other upon incidence they would still be bent in the same ratio (to the sines of the angles of contact and departure), and this would bring the focus nearer to the glass; but it is evident the same principles would hold.

60.—As regards the action of the eye in this matter, it can only recognise the direction from which it receives the light, and not the processes the rays may have undergone before reaching it. Therefore the ray proceeding from O in the path b′, passing through the lens or prism and emerging in the path b, is recognised by the eye as the ray b only. So that the point of light O appears visually as proceeding from the direction bs, and this convergence or expansion of the point O, with its coincidence from the opposite side of the lens, produces the effect of magnification of the object represented by O.

61.—Concave Refraction.—In Fig. 6 a convex lens is shown in which the parallel rays L are drawn to a focus at F upon the principles just demonstrated. If the lens were made concave, as shown in section Fig. 7, by the same principles of refraction, it is evident that the rays would diverge, as the refraction bends the ray uniformly towards the thickest section of the glass. If two lenses are brought together, one with convex face, and one of the same radius of curvature, but with concave face, the rays in passing through would not be refracted. In this case the lens would be said to be corrected. A convex lens has a focus where the rays converge. A concave lens is said to have a negative focus equal to the focus of the convex lens, that will correct it, or make it equal, as regards refraction, to plane parallel glass.

Fig. 6.—Diagram convex lens.

Fig. 7.—Diagram concave lens.

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