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.

CHAPTER II.

THE TELESCOPE AS A PART OF A SURVEYING INSTRUMENT—GENERAL DESCRIPTION—QUALITIES—OPTICAL PRINCIPLES—REFRACTION OF GLASS—LIMIT OF REFRACTION—REFLECTION—PRISMS—LENSES, CONVEX AND CONCAVE—ABERRATION—FORMATION OF IMAGES—DISPERSION—ACHROMATISM—CURVATURE OF LENSES—TELESCOPES—EYE-PIECES—POWERS—DYNAMETER—CONSTRUCTION OF THE TELESCOPE, DIAPHRAGM—WEBS—LINES—POINTS—PARALLAX—EXAMINATION AND ADJUSTMENT.

45.—General Description of the Telescope.—This instrument forms part of the theodolite, level, some kinds of miner's dials, sextants, plane tables, and other surveying instruments. For this purpose it is made of similar construction to that of the refracting telescope used for astronomical purposes. The great object desirable in the telescope, when used as a part of a surveying instrument, is that it shall assist vision in obtaining the true direction, or pointing to the position of an object in such a manner that it can be employed to ascertain the angular position of two or more objects in relation to the position of the centre of the instrument upon which it is fixed; also to obtain relative altitude to this centre in relation to a distant station by the reading of a divided measure or staff placed thereon.

46.—The qualities desirable in a surveying telescope are, that sufficient rays of light may be collected from the object observed for it to be clearly seen as a whole, and in some cases that sufficient magnifying power should be available, in order that details or divisions painted upon a staff may be sharply defined. The amount of light received by the eye which is effective in producing distinct vision is in proportion to the extent of active surface of the object-glass converging the light rays. The magnifying power is regulated by the sum of the convexities of the lenses of the eye-piece upon principles to be explained. The surveying telescope is required to possess only a very limited field of view, but very great focal range, so that objects may be seen at any distance.

By the necessary optical arrangement of the telescope, which will be further described, the object observed is generally inverted. This inversion of the image as it appears, at first presents a little difficulty to the learner, but in practice this soon becomes so familiar as not to be even recognised mentally.

47.—Optical Principles involved in the Telescope.—To commence with the optical construction of the telescope, that this may be thoroughly understood, it is necessary to give brief details of some first principles upon which it is constructed, assuming that optics have not been made a special subject of study.

48.—Refraction of Glass.—The properties of a lens depend entirely upon the fact that a ray of light passing from air obliquely into the surface of a dense transparent medium (in this case of glass) and equally from the glass into air is bent, or, as it is termed, refracted, to a certain angle at the surface of contact of the air and glass. The ray of light entering the glass is termed the incident ray, that proceeding from it the emergent ray.

49.—There is no known medium, glass or other, which refracts a ray of white light at one uniform angle. The white ray is universally separated upon refraction, or dispersed, as it is termed, into rays of all colours of the rainbow. In considering refraction, therefore, in its simplest aspect we are compelled to take the refraction of one uniform ray which is distinguished by one colour, that forms a part of the white ray, as for instance the red, yellow, green, or blue, that is, a monochromatic ray, as it is termed, which gives a sharp refraction of its own coloured light only in its ray. Incandescent soda produces monochromatic rays, but in practice an intense flame behind a bright-coloured glass will answer the same purpose, as the coloured glass may be arranged to absorb all, or nearly all, parts of the white ray, except that of its own colour.

50.—Every transparent medium has a special quality of refraction. Therefore, different kinds of glass refract in different degrees within certain limited angles which will be hereafter considered. The refraction is uniformly in the plane containing the incident ray, and the perpendicular to the surface separating the two media. Every medium refracts monochromatic light equally according to the following law for any angle of refraction:—

Whatever the obliquity of the incident ray may be, when it passes from a rarer to a denser medium the ratio which the sine of the angle of incidence bears to the sine of the angle of refraction is constant for any two transparent media.

51.—The natural law by which the power of refraction of any medium may be shown, and consequently the magnifying power of a lens in the ratio of its curvative through this refraction may be exemplified, is illustrated by the diagram on the following page (Fig. 1).

PP', a line perpendicular to the surface of the plane of the medium (glass) with air above it, a ray of light would pass directly P to P' through the glass surface SS' without refraction, and so for all perpendicular incidences or emergences. By this perpendicular line PP', termed the normal, all refractions are measured. The incident ray I to C is refracted to R. Then if we call the angle ICP I, and the angle RCP' R, it is found by experiment that the perpendicular from I on PP' (or sin I) bears a certain proportion to the perpendicular from R on PP' (or sin R) according to the density of the glass. This proportion is generally expressed by the formula—sin I = µ sin R. Another incident ray I' to C would be refracted to R', and using similar notation to the above we have sin I' = µ sin R', and from this it follows that (sin I)/(sin R) = (sin I')/(sin R') = µ, which is called the index of refraction. Thus, if in a certain glass the sine of I measure 3 equal parts on any scale of length, and the sine R 2 parts on the same scale, the index of refraction of this glass would be 3 divided by 2 or 1·5.

Fig. 1.—Diagram of Refraction and Reflection.

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If the above process be reversed, and the ray of light R be refracted on passing from the glass to the air, it will be projected to I in the emergent ray, and follow the same law as that given above.

52.—Limit of Refraction—Reflection.—The sines to the angles ICP and I'CP' being constantly greater in proportion to the obliquity in the case of glass we are considering by 1/3 than the sine of the angles RCP' and R'CP' of the rays of incidence thrown upward upon the surface SS', it will be seen that at a certain angle or that in which the sine is 2/3 the radius, namely, 41° 48' 37, the equation given above makes sin I = 1 its maximum value; therefore, at any angle of incidence greater than this, the sine of refraction to continue in proportion would exceed the radius—an impossibility. The refraction, if possible, would carry the ray into the substance of the glass. This is therefore called the critical angle or angle of total reflection. At this point we may consider what must happen. By our rule, refraction must cease at the angle refraction becomes impossible by increase of sine, and as light cannot be extinguished in a transparent medium it must be reflected. Thus the ray r cannot be refracted in the proportion according to the rule given for sine I to sine R, as this would exceed the greatest sine, that is SC the radius, this ray will therefore be reflected at the surface from the point C, and pass in the direction r'. This property of refraction, continuing, as it were, into reflection, is made use of in many instruments.

53.—It may be worthy of repeating, as it is a mistake occasionally made by persons designing instruments for special purposes (as telemeters), that the refractions are not equal for varying angles of incidence, but only, as before stated, in the ratio of the sines. Thus there is no refraction P to P' a certain refraction I to R, and a greater refraction I' to R', the refraction constantly increasing with the angle of incidence.

54.—The Reflection of Light follows a very simple law, viz.:—The angle of reflection of a ray of light from a reflecting surface is equal and opposite to the angle of incidence upon it. Thus, in Fig. 2, let a ray of light IA fall upon the reflecting surface SS' at 30° of inclination to this surface, then this ray will be reflected from A to R at the angle RAS', which is also 30°. If an object be at O, and the eye at I, then the object will appear as though it were at O', as the eye only recognises the object in the direction from which it actually receives the light. The apparent angle S'AO' is equal to IAS, so that the point of a mirror from which an object reflected is received is in direct line between the eye and the apparent object. This observation will be found useful in placing mirrors.

Fig. 2.—Diagram reflections from a plane.

Fig. 3.—Reflection from a prism.

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55.—Prismatic Reflection. The same law as given above applies to internal reflection from glass. Let Fig. 3 represent the section of a prism ff', two plain surfaces of glass at right angles to each other, and the third side making an angle of 45° with each of the other two. The ray i will therefore pass perpendicularly through the plane f without refraction to meet the plane 45° and the angle of reflection, being equal to the angle of incidence, will leave this plane at 45°, and reach r. The angle of glass here given of 45° being greater than 41° 49', its extreme angle of refraction, the internal reflection will be therefore perfect.

56.—Prismatic Reflection, as this is termed, is largely used in optics in preference, where practicable, to open reflecting surfaces, from the certainty of keeping the reflecting surface clean; as dirt exterior to the reflecting surface of the prism does not affect the internal reflection in any degree.

57.—The reflection is shown for clearness from the plane (Fig. 2) as it actually occurs, or as it is measurable, independent of theory. In optics it is found much more convenient to take the reflection in relation to an imaginary line drawn perpendicular to the plane. In Fig. 4 NA is termed the normal. Taking the angles as before as 30° to the plane, the optical expression of this would be 60° to the normal, and the reflection of the incident ray IA to R would be in the angle IAR 60° + 60° = 120°, the amount the incident ray is deflected from its former course. This principle is important to be understood in the construction of the sextant and other reflecting instruments. In reflection the ray is found to follow the shortest path,—that is, the path I to R by reflection is shorter in the lines IAR, placed at equal angles to the normal, than it would be by any other possible path. As, for instance, it is shorter than IaR, shown by dotted lines.

Fig. 4.—Measurement of angle of reflection in optics.

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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 aa? 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 aa?, 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 bb?, 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 cc? 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 aa?, bb?, and cc?, 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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62.—Spherical Aberration.—If the surfaces of convex lenses are truly spherical, it is found, by an analysis too complex to be described in this work, that the rays which pass through at different distances from the axis converge to slightly different points of distance. This subject was at one time seriously discussed for the proper formation of objectives for telescopes; but at present it is entirely neglected by the optician, as it is found practically to be as difficult to make a lens truly spherical as one of the convergent or divergent form required under the special conditions present. The spherical form, as it is approximately produced from the grinding with spherical tools, being always nearly correct, the correct forms of object-glasses are made by figuring, which has been already referred to, art. 38. In eye-pieces the spherical aberration would cause some confusion were the glasses not adjusted in such a manner as largely to prevent this.

63.—The Formation of Images by Refraction from a Convex Lens.—If we take any double convex lens, as that shown in section Fig. 6, we find, if it is held towards the sun at a certain distance from a solid surface, we form a burning-glass,—that is, we produce an image of the sun where his rays of light and heat are refracted by the whole of the surfaces of the glass. The distance from the centre of the lens to the point of greatest light is called the solar focus of the lens,—that is, the point at which it concentrates or converges parallel rays, and forms the image of the sun. With parallel rays from the sun, the distance of focus is less than if these rays were divergent in any degree. Consequently the solar focus is less than that subtended by any object on the earth.

Fig. 8.—Diagram of the convergence of rays of light.

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64.—In the diagram, Fig. 8, a candle-flame at acb forms its focus at a?c?b?, where all rays converge to form an image in the following manner:—Every point of the candle throws its light upon every point of the surface of the lens, and, therefore, throws the image of each point to its focal position behind the lens, according to the direction of its refractions; so that, if we take all the separate points of light thrown from the candle, we then have a perfect image of it formed by an infinite number of separate focal points, and as the rays by their direction necessarily cross over the axis the image is in an inverted position.

65.—The whole of these lines would form a confusion if shown in a diagram. We may, therefore, take for illustration the exterior of a cone of rays proceeding from three points only. Thus the clear lines aa' and aa from the point of the flame would refract to the lower part of the image a?. The dotted lines bb' would proceed to the upper part of the image, as shown by the continuation of the dotted lines to b?, whereas the central dash lines c'c would form their images in the centre following the dash lines to c?, and thus, from the number of luminous points, the whole image of the candle would be produced at the foci b?c?a? in an inverted position.

66.—Dispersion of Light.—The conditions stated above for refraction of monochromatic light would not answer for perfect vision, which is only possible in clear white light. It therefore becomes necessary in practice to correct the quality of dispersion which light suffers in refraction through any dense medium. The evidence of dispersion by glass may be shown by a prism, as in the following diagram:—

Fig. 9.—Diagram showing chromatism of light by the prism.

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67.—In Fig. 9 let P represent the section of a prism of glass, covered except at the narrow opening a. Let a strong light, as shown, be covered, except from a narrow slit, then the ray from the light, refracted from a towards a' in the prism, will be dispersed or split up at a into the colours of the rainbow, shading from blue, green, and yellow, to red, within the prism. Upon emergent refraction at a' this dispersion will increase so that an image of the slot near the light, if thrown on a plane proceeding from the base of the prism to the right, will be represented at BGR by a prismatic or chromatic spectrum, as it is termed, shading off from blue to green, yellow, red.

68.—Achromatism of the Prism in the same Quality of Glass.—Taking the prism, Fig. 10, C as before, and applying a second exactly similar prism C' reversed upon the face of the first—then at every part of the process of dispersion from a point of white light under diffraction into the first prism, will by equal diffraction, in passing through the second prism, be brought to a point, where it will issue a white ray at the point a, as it entered at the point a; or, practically, the emergent ray will be achromatised. This principle must be followed in the manufacture of achromatic lenses, although under various indices of refraction and dispersion from differences in qualities of glasses. It is made use of in the achromatism of eye-pieces, and in combinations, and assures the achromatism of parallel glasses used for sextants under different angles of incidence.

Fig. 10.—Diagram perfect achromatism.

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69.—The Achromatic Lens.—The achromatism of a pair of lenses by which a large amount of refraction of pure white light is obtained, depends upon differences in the qualities of glasses which are due to their density and chemical composition, so that in one glass a less amount of dispersion is produced at an angle which gives an equal amount of refraction than in another. The combinations of glasses in use are crown and flint, as already described, art. 32, the crown being a light glass of soda and silica, the flint being a heavier glass containing silica, potash, and lead. In a certain kind of flint glass used for optical purposes, for a prism giving only slightly greater refraction than one of crown glass, the dispersion is about double. Therefore, we may combine a pair of glasses so as to obtain a desired amount of refraction from the combination if we make the crown glass refract something over double the amount we require for the perfected lens or prism, and diminish this quantity by the reverse refraction of the flint glass, thereby correcting the dispersion, as may be shown by the diagram on this page.

70.—In fig. 11 let C be a prism of crown glass giving over double the amount of refraction to a prism of flint glass F, but only of total dispersion equal to the thicker crown glass. The compound white ray of light a will then be dispersed upon refraction at the meeting faces of the two prisms, a certain quantity represented by the cone of rays shown, and again converge at a', an equal quantity on emergence from the exterior surface of the flint prism, so as to issue again a white ray, of which this system of prisms has refracted, but not dispersed, the light.

Fig. 11.—Showing principles of achromatism.

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71.—That the same principles given above for the prism will hold in the achromatic compound lens, is already demonstrated by the comparison of lenses and prisms shown in Fig. 5; but for the sake of clearness it may be again shown diagrammatically in Fig. 12 for an actual objective, wherein the parallel rays ab, proceeding from a distant object or star, are shown refracted to a'b', and coming to a focus at F, although dispersed at the meeting surfaces of the two glasses, as shown diagrammatically, by the internal cone of rays.

Fig. 12.—Showing achromatic objective.

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72.—Practically, the matter is not quite so simple as it would appear to be theoretically, by the above-described conditions, as we actually find the spectrum of a prism of flint glass of equal dispersion to one of crown glass does not give exactly similar extent of separate colours within its spectrum, the medium ray of the spectrum in the flint glass being nearer the blue than in the crown. Thus, this compound lens does not perfectly correct by inversion as it does in the perfect case discussed, and shown in Fig. 10. For this reason better definition is found by slight displacement and slight difference of total extent of dispersion of one of the spectra in coincidence on the meeting planes between the lenses, leaving in all cases a certain amount of residual colour, blue or red, uncorrected, by making the glass under- or over-corrected, as it is termed, which does not, however, seriously impair distinct vision. It is quite possible that, by some future improvements in the chemical constitution of the glass, this defect may be remedied. English glass-workers prefer to over-correct, German and French glasses are more often under-corrected.

73.—The measurements of refraction and dispersion being both in one direction, may be taken together within certain angular limits in one term in the construction of a lens as the ratio of dispersive powers, the indices being certain dark lines which are observed uniformly in the spectrum of the sun projected from a narrow slit. These lines or bands in the sun's spectrum are known to be due to metallic vapours which are present in his atmosphere, and can therefore be reproduced by the deflagration of like metals on a small scale. To certain of these lines letters of the alphabet have been applied. Of these letters, a pair of lines due to sodium vapour marked D, and three lines due to hydrogen, marked C, F and G, are commonly taken for reference of dispersion. Achromatism is generally considered duly corrected when the lines C and G are united. The middle of the spectrum between these lines is about E; and chromatic dispersion of optical flint and crown may be taken to be fairly corrected if the spectra are coincident in colour at this line.

74.—Curvatures in the Achromatic Lens.—A large amount of mathematical power has been expended upon this matter, but the perplexity of the subject is due to small differences of the material; and the impossibility of working absolutely true spherical curves has rendered this work of little practical value to the optician, who still resorts to the formulÆ of Dollond and Tully. Those who care to follow the subject beyond the scope of this work will find numerous papers in the Phil. Trans., and in the works of Herschel, Barlow, Coddington, Robinson, and Stokes, wherein what is known theoretically of the subject is fully investigated and discussed.

75.—For all small achromatics, such as are employed in surveying instruments with Chance's hard crown and dense flint, the following approximate formula is commonly employed, expressed in terms of the radius of the curved surface into f, the total focus of the finished objective, for first working before trial:—

1st.—Outside surface, f 2 convex,

2nd.—Inside " f 3 convex,

crown.

3rd.—Outside " f 3 concave,

4th.—Inside " 4f convex,

flint.

76.—By different makers the surfaces are changed as far as reversing the curvature of the front glass, and indeed very good glasses are made with the 1st, 2nd and 3rd = (f/2·5). In all cases true convergence of the white ray is only obtained by correction of the outer and inner surfaces, or by figuring, as it is technically termed, in which the curvature is not only made greater or less, but its character is altered generally in the direction from circular to elliptical section. The qualities of the object-glass cannot be over-estimated by the practical surveyor. A heavy instrument with inferior object-glass may be carried about for years, whereas a lighter instrument with good object-glass would perform better work. Excellent information upon this subject was given in a lecture before the Royal Institution by the eminent optician, Sir Howard Grubb, of Dublin.

77.—Optical Arrangements of the Telescope.—The earliest form of telescope is that of Kepler, Fig. 13. In this the rays from the object-glass cross in front of the eyeglass; consequently, the image is inverted. This form is at present little used except in combination with a separate eye-piece.

Fig. 13.—Kepler's telescope.

Fig. 14.—Galileo's telescope.

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78.—Galileo's Telescope, Fig. 14.—In this the eye-piece is a concave glass. This glass is placed inside the focal distance, so that the rays from the object-glass are bent to less convergence, that they may enter the pupil of the eye in a direction possible to reach the retina. The image in this telescope is maintained erect. This principle is used entirely for field and opera glasses, also for sextants and some other instruments where it is desirable to keep the image erect, and small power is required, sufficient only to obtain more distinct vision. The lines aa' in Figs. 13, 14 are termed the axis of the telescope.

79.—Optical Arrangement of the Huygenian Telescope.—In surveying instruments, where angles and directions are not taken by coincidence of direct and reflected images, it is necessary that the direction of the axis of the telescope should be clearly indicated. In this case the focus of a distant object—that is, its exact image—is projected upon a plane termed the diaphragm, Fig. 15, SS' upon which a visible object or index is placed, the position of which is picked up by a secondary telescopic arrangement, or eye-piece as it is technically termed.

Fig. 15.—Diagram of arrangement of lenses.

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80.—The arrangement of lenses in a surveying telescope is shown in the illustration above, where OG is the object-glass or objective, E the eye-glass, F the field-glass. The two lenses E and F, in their mountings, form the eye-piece EP. The dotted line a is the axis of the telescope, SS' is the focal plane of the object-glass, where a metal disc is placed with an opening in its centre—this is termed the diaphragm or technically, the index-stop. Across the opening in the disc, spider's webs or other fine visible objects are placed, to be described further on.

81.—Both the object-glass and the eye-piece are fitted in sliding tubes, which will be described presently, in such a manner that they may be made to approach or recede from the focal plane SS'. The nearest distance of the object-glass to this plane is the solar focus, or the distance at which a sharp image of the sun or a star placed in the axial line would be formed. The greatest distance of the object-glass from the focal-plane in most instruments is such that a clear image will be given on this plane SS' of an object placed at about twenty feet from it.

82.—The Ramsden Eye-piece, the optical arrangement of which is shown in Fig. 16, is also known as a positive eye-piece. It consists of two plano-convex lenses, the convex surfaces of which are turned towards each other. They are separated by a distance equal to two-thirds the focal length of either glass, and placed so that the diaphragm is one-fourth this focal length from the field-glass.

83.—This eye-piece is considered not to be quite so achromatic as another form known as the Huygenian eye-piece, but its spherical aberration is less than any other, and it gives what is necessary in all measuring instruments—a flat field of view, requiring no change of position to see the centre and border of the field with equal distinctness.

Fig. 16.—Ramsden eye-piece.

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84.—The Field of View should be as bright as possible. To ensure this, the field of the object-glass which is taken by the eye-piece at the position of the front of the eye should not be larger than the pupil. If the whole field of light enter the eye as it should do, the brightness will then vary directly as the square of the diameter of the object-glass, and inversely as the square of the magnifying power. The directions of the rays are shown by dotted lines as aa and a'a' for the Ramsden eye-piece in Fig. 16. This eye-piece is sometimes called an inverting eye-piece. It is not really so: the object-glass inverts its image and the eye-piece picks up the image in its inverted position. Two or three eye-pieces of this kind, of different magnifying powers, are sometimes supplied with one surveying instrument. The same form of eye-piece, being also a simple microscope, is used to read the divisions on the divided circles of theodolites, sextants, and other instruments, and for such purposes it is often desirable to ascertain its focal length.

85.—The Focal Length of the positive or Ramsden eye-piece is found by dividing the product of the focal lengths of the two lenses by their sum, diminished by the distance between them. Thus, if the focal length of each of the lenses be 1·5 inches, the distance between them 1 inch:—

1·5 × 1·5 3 - 1 = 1·125 inches.

86.—The Magnifying Power of the Telescope.—The focal length of the objective divided by that of the eye-piece gives the power of the telescope. Thus, a 14-inch telescope with the above eye-piece would have a power,

14 1·125 = 12·444, or 12½ nearly,

a very general lower power eye-piece with telescopes of this focus.

Fig. 17.—Dynameter.

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87.—Dynameter.—The magnifying power of a telescope may be ascertained, without any knowledge of the focus of the glasses used in its construction, by the use of a dynameter. This instrument, Fig. 17, consists of a compound microscope in which a finely divided transparent scale is placed in the mutual focus of its object-glass and of the eye-piece at a. The divisions of the scale may be ·01, ·02, or ·001 inches apart, adjusted so that a disc ·1 inch diameter at the exterior focus of the eye-piece may read a given quantity upon the scale. To use this apparatus, the flanged face is placed in front of the eye-piece of the telescope, previously set at solar focus. The telescope throws a circular image of its object-glass through the eye-piece, where it is picked up by the object-glass of the dynameter and brought to focus on the scale a, where it appears as a circular disc of light. If this image be measured by the scale, and the diameter of the object-glass be divided by this measure, the quotient will be the magnifying power of the telescope. There are several other forms of dynameter.

88.—The Erecting Eye-piece, sometimes supplied with theodolites and occasionally with other instruments, is the ordinary one of the common telescope, Fig. 18. The glasses are so arranged that the image brought to the focus of the telescope inverted is again erected, so that objects appear in their natural position. The complete eye-piece is of the same optical arrangement as that of a compound microscope. The arrangement of lenses is shown in the engraving on next page.

Fig. 18.—Optical arrangement of erecting eye-piece.

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89.—A object lens, B amplifying lens, C field lens, D eye lens. Stops are placed at d and d' to cut out extreme rays. The image is formed by the objective at O, and the light passes in the direction shown by fine lines, being thrown from side to side of the lenses. The ray is achromatised proportionally to its dispersion by the separate lenses, upon principles discussed art. 68 and shown Fig. 10, as independently of the small amount of opacity of the lenses, extreme rays are cut off, so that central portions only are used. This eye-piece suffers loss of light at each of the four lenses; therefore, a telescope with it, for equally distinct vision to that obtained by using the Ramsden eye-piece previously described, would require a larger objective.

This eye-piece is rarely used now, excepting with American instruments in which they are almost universal, as the very slight advantage of seeing the image erect is far outweighed by the loss of light it entails. The American manufacturers place them inside the telescope instead of outside, thus the telescopes look much the same as our ordinary ones, but the focal length of the object-glass is shortened by the length of the eye-piece, and as this takes up from three to four inches, a telescope which would appear to be say 10 inches solar focus is, in reality, only six or seven inches and consequently only about two-thirds the power.

Fig. 19.—Diagonal eye-piece, full size; S G sun-glass.

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It is astonishing that the Americans, who are usually so quick in adopting the most practical appliances, are so slow in seeing the advantage gained by the use of the now almost universal inverting eye-piece.

90.—The Diagonal Eye-piece, Fig. 19, is used upon transit instruments, theodolites, and occasionally upon mining-dials. It permits the telescope to be used by the observer looking at right angles to its axis. Thus, by the natural direction of the eye, stars or the sun may be observed to near the zenith, or the direction of a line cut by two lights at the bottom of a shaft may be observed from above by the telescope of a theodolite having a hollow centre on its ordinary stand, to check the magnetic bearing of the needle below ground, if this is assumed to be subject to local disturbance. The socket of this eye-piece screws upon the telescope and has a free inner tube for rotation, so that the 90° to the axis of the telescope may be placed at any angle to the axis of its cylindrical circumference; as, for instance, instead of being used vertically or for zenith stars, it may be used horizontally, where precipitous ground would not permit direct axial vision through the telescope. The reflecting arrangement of this eye-piece may be adapted either to the Ramsden or the erecting form. In either case the reflector is placed in the central portion of the eye-piece. In surveying instruments the reflector is generally a piece of polished speculum metal for portable instruments, but a prism of glass for larger fixed instruments. The general arrangement is shown in the section of a diagonal Ramsden eye-piece on page 42, full size. A object lens, D eye lens, adjustable for distance from the reflector R, S outer casing which permits adjustment for focusing, SG sun glass, the diaphragm being in front of A.

91.—When a rectangular prism is used for the reflector, it is worked with one plane 45°, as previously discussed, art. 55, Fig. 3. In place of one or both the 90° faces these surfaces are sometimes worked convex so as to form a magnifier, dispensing with one of the convex lenses of the eye-piece. A long diagonal eye-piece is necessary, where stars towards the zenith are to be observed, to prevent interference of the limb of a theodolite with the face of the observer.

92.—Reflecting Eye-piece is used to observe small stars, as for instance the circumpolar stars in the southern hemisphere, by illuminating the front of the webs or lines. A strong light thrown down the telescope from a reflector to illuminate the webs would tend to dim the effect of blackness of the sky and render these stars indistinct. In the eye-piece, Fig. 20, a piece of plain parallel glass is placed at an angle of 45° to the axis. This permits the webs to be clearly observed through the glass at the same time that it throws light from a lamp placed at a distance from the glazed aperture L by reflection of the surface of R sufficient for front illumination. The amount of light required is regulated by the distance of the lamp from L. This eye-piece is made to fit into the diagonal eye-piece casing, as S Fig. 19, E Fig. 20 being the position of the eye, F field-lens.

Fig. 20.—Reflector in eye-piece to illuminate the front of diaphragm.

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93.—Sun-glass.—Sextants and theodolites are supplied with a very dark glass or a combination of dark glasses fixed in a rim to form an eye-piece front, which screws or fits on in front of any eye-piece, to take observations of the sun for longitude or bearing, Fig. 19, SG. It needs no description, but is necessary to be mentioned to complete the optical arrangements of a telescope, as it is sometimes used for surveying purposes.

94.—The Body of a Telescope that forms part of a surveying instrument is constructed of a pair of triblet drawn tubes, Fig. 21, TT' T'. These tubes should be truly cylindrical and straight, so as to fit smoothly together, the one within the other, and slide in and out quite freely but without any play. The inner tube should be as long as practicable, so as to remain steady when racked out to the full extent required to focus near objects. The object-end R is generally enlarged so as to take the cell in which the objective O is placed, without cutting off any part of the light, or entailing the weight of larger tubes than is necessary to make use of the full field of the objective. The objective is generally held in its cell by an internally fitting screwed ring with milled edge, so that the glasses may be taken out and separated to be cleaned, and be easily replaced. Two notches or grooves are commonly made in the edges of the glasses, each of which is deep enough to take a small brass pin which is soldered to the inside of the cell. The second notch indicates relative position, so as to secure the glasses being replaced properly. In all cases the double convex crown glass is placed outwards from the telescope. A glass of large size should have a loose ring within the cell to act as a spring to save distortion of the objective from expansion or contraction of the metal; but this is not necessary in small surveying instruments. In some common telescopes the object-glass is burnished into its cell, in which case the glasses of the objective cannot be separated for cleaning.

Fig. 21.—Body of surveying telescope.

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Fig. 22.—Section Fig. 21, A to B.

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95.—Stops.—Within the inner tube two or more thin metal rings, termed technically stops SS and S'S', are placed to cut off any extraneous light that may enter the telescope obliquely, and which, if not stopped off, would produce a fogginess over the whole field of view. It is important that these stops should not cut out any part of the full aperture of the object-glass if it be a good one. In the manufacture of the telescope this is easily seen by looking in at the eye-piece of the unglazed telescope to see if the stops clear the objective cell. In the finished glazed telescope another method will be discussed further on.

96.—The inner or the outer tube of the body of the telescope slides towards or from the objective for focussing by means of a rack R and pinion P. The rack is soldered to the inner tube, and the pinion fitted in a cock-piece, as shown Fig. 22 C, on the outer tube. The pinion is moved by a large milled head M. This fitting should be made with care. The pinion should be very free, so that it does not lift the body at any tooth, and at the same time there should be no shake on the gearing. It needs considerable practice to rack a telescope properly.

97.—The outward part of the object end of the telescope is generally turned to fit the interior of a separate short tube, shown at R, which is placed over it. The outer end is closed by a ring to the size of the aperture of the objective. This is termed a ray-shade or sometimes a dew-cap. The ray-shade is extended when the telescope is directed to such an angle that the sun's rays would fall upon any part of the objective, and thereby cause internal reflections. A swivel shutter, Fig. 21, R', is placed upon the outward end of the ray-shade, which, when closed, as shown in the cut, forms a cap to the telescope. The eye-piece EP before described, art. 82, Fig. 16, is placed in a tube constructed upon the end of the telescope, in which it slides freely, to focus upon the diaphragm to be presently described. The telescope is mounted sometimes solidly upon a transverse axis, or it is mounted in turned bearings, or it has two collars placed round it which are turned quite equal and true, and are mounted on Y's to be hereafter described.

98.—Mechanical Adjustment of the Eye-piece.—In some large instruments the eye-pieces are racked for adjustment in the same manner as the object-glass already described. A better plan is to have an inner tube to the socket tube cut with a screw into this, and provided with a milled edge, so that the eye-piece may be screwed gently to focus upon the webs of the diaphragm.

Fig. 23.—Elevation of diaphragm.

Fig. 24.—Section of diaphragm.

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99.—The Diaphragm of the Telescope is so constructed as to permit the displacement of spiders' webs or other fine objects in any direction at right angles to the axis of the telescope, or in the vertical only in the dumpy level, to be described, the object in all cases being to adjust the crossing of the webs, lines, or points to the axis of the telescope. It will be convenient here to discuss a general form of diaphragm applicable to theodolites, mining-dials, and plane-tables only, which gives movement in two directions at right angles to each other.

100.—The diaphragm, Fig. 23, is formed of a stout disc of brass having a centre hole of about ·30 inch diameter. Upon the side which is placed next the eye-piece the hole is brought to a thin edge by an internal bevel or countersink, which leaves the hole much larger at its off surface, Fig. 24 P. The disc is held in its place and adjusted by four capstan-headed screws, termed collimating screws, two of which are shown in section as CC', the screws being tapped into the rim of the diaphragm frame P. The screws are placed through a stout collar. The theodolite diaphragm has generally three spiders' webs or lines crossed in the manner shown in the centre of Fig. 23. The eye-piece is screwed into the thick plate, Fig. 24, TT', and adjusts to the focus of the webs.

Fig. 25.—Webs wound off for use.

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101.—Webs.—It is a somewhat delicate process to web a diaphragm, but it is necessary that every surveyor abroad, out of the reach of an optician, should understand the method if his instrument were originally webbed. The webs are taken from a small or young garden spider. The best are taken when the spider has first commenced spinning. To wind off the web a fork is bent up out of a piece of thin brass wire. A long hairpin will answer for this purpose very well, or even a fork formed of a thin branching twig of a shrub; but if this last be used it should be thoroughly dry, or the webs will be broken or be baggy by its warping in drying.

102.—The web in connection with the spider is first attached to one prong of the fork by looping or by any sticky matter, if the web be not sufficiently sticky naturally. The spider is then suspended from the fork and jerked down a foot or so, and the web is wound off as shown in Fig. 25. The last length of web being attached by gum. A dozen or so of the forks may be taken from the same spider before she is exhausted. The webs are then gummed or varnished to the sides of the fork, and are ready for use at any future time. They are best preserved if placed in an air-tight box, which may have slots in an internal fitting to hold them. The small amount of spring given by the fork keeps the webs always taut. Where a living spider cannot be found, the open ties of an old web may be taken; but in this case, after the web is wound on the fork, it should be carefully washed by immersing it in clean water, and, if necessary, brushing it gently under water with a light camel-hair brush, examining it occasionally with a magnifier to see that it is sufficiently clean and free from knots for its purpose.

103.—To Fix the Webs, lines are drawn on the diaphragm, into which the webs are to fall. It is then varnished over the divided side with Canada balsam, laudanum, or other quick-drying, sticky varnish—at a pinch, sealing-wax dissolved in strong whisky will answer. The outer, or the unused web upon the fork, is lowered carefully over one of the most nearly vertical lines, and lightly pressed down to assure its perfect adhesion to the varnish. It is then either broken off or cut loose. The second nearly vertical line is then webbed in the same manner, and the horizontal line finally, being sure that this last cuts the intersection of the others. The diaphragm should then be put in a warm place to be allowed to thoroughly set without disturbance before it is fitted in the telescope.

104.—Platinum Wires are sometimes used in place of webs. These wires are made by drawing a piece of fine platinum wire, which has been previously soldered into a silver tube, to the greatest fineness possible with the draw-plate, and afterwards dissolving the silver off the platinum by nitric acid. The platinum wire is thus produced of less than ·001 inch diameter. For a time these wires were very popular, and it was thought that they would supersede the use of webs, but they do not appear entirely to answer expectation. The platinum drawn in this manner appears to lose some part of its elasticity. It is not easily attached, that is, it is liable to shift from its fixing, possibly from its contraction and expansion with change of temperature, not being of the same metal as the diaphragm. It also oxidises a little or becomes in some way corroded in use out of doors. It appears to answer better for astronomical telescopes, but the finest platinum wire obtainable is not so fine as a spider's web.

105.—Lines Ruled upon Glass.—A glass diaphragm is frequently used in a surveying instrument to replace the webs. Lines are ruled upon the glass in similar positions to the webs already described. They appear quite sharp in the eye-piece, and are more permanent than webs. Glass is also convenient for permitting space lines to be ruled for subtense measurements, a subject to be considered further on. The objections that have been found to glass are that it obstructs a little light, and is subject to dewing. The dewing is particularly annoying when temperature is lowering quickly, as a diaphragm may become bedewed many times in a few hours. In all cases where a glass diaphragm is used it should be placed in a ground metal fitting, so that it may be taken out in a minute to clean and be replaced with perfect certainty of its adjustment. It is a very convenient practice where webs are used to have a spare glass diaphragm to replace them should they become broken. This may be constructed by means of a ground metal fitting to be put in a webbed instrument in perfect adjustment in cases where it might be impossible to find a new web.

106.—Points.—The author for a large number of instruments employs very fine points in place of webs, which he highly recommends. These are fixed for support upon the margin of the diaphragm, and projected therefrom into the field of view. The points are formed of a special alloy, 75 platinum, 25 iridium, which has the hardness of steel, and is perfectly non-corrosive in air or moisture. They are made sufficiently stiff to be dusted with a camel-hair brush, supplied in the instrument case, without the slightest fear of disturbance of position in the instrument. They form a perfectly permanent index of sufficient stability to last in perfect adjustment as long as the instrument lasts in wear. One objection is that a point gives less field of observation for levelling than a line, but this does not hold if there is tangent adjustment to the instrument to bring the point up to its reading position. The value of the reading from these points will be discussed further on.

107.—Position of the Diaphragm in the Telescope.—If the objective be accurately centred, and its mounting true, the intersections of the webs, lines, or points should come exactly in the axis of the telescope; but it would never do to accept this without critical examination. Therefore the webs may be placed approximately in the centre, and adjusted true to the axis of the objective and the telescope by what is technically termed collimation. The first point, however, to be studied in this adjustment is to get the eye-piece and the objective accurately in focus with the webs. The same description of focussing which answers for collimation will answer also for ordinary use of the telescope.

108.—Adjustment of the Eye-piece to the Webs is effected by pushing in or drawing out the eye-piece in its tube with a slight screwing motion until the webs, lines, or points appear quite distinctly. To prevent confusion from the sighting of objects, it is better to take off the ray-shade, to point the telescope to the distance in opposition to the direction of the sun, and to keep the telescope rack fully extended, so that it is quite out of focus. When the light is not very bright a sheet of notepaper or an envelope may be placed obliquely in front of the object-glass to obtain a soft reflection from the sky. This method is always employed by some observers.

109.—Adjustment to Focus of the Objective.—Parallax.—The eye-piece remaining in focus, the telescope is racked out until the object desired to be brought into view, either for the collimation or for ordinary reading, is sighted. After this the milled head is moved as slowly as possible until what is thought to be the exact focus is obtained. The certainty of exact focus is not easily obtained by direct observation, but it may be obtained by what is termed observation for parallax, which must be taken in all cases when adjustment is required for collimation. Thus, having obtained the nearest possible adjustment by sighting a small object or a division upon the staff, bring the object to read exactly in a line above the horizontal web in the centre of the stop or the corner against a vertical web. If now the eye be moved up and down as far as the range of the eye-piece will permit vision of the centre of the webs, and the object sighted appears fixed at the same position to the webs, the focus is perfect. If, in moving the eye, the object sighted appears to follow its motion about the intersection of webs, the focus of the telescope lies beyond the webs; the objective must therefore be moved slightly nearer the webs by turning the milled head very gently. If, on the other hand, the object sighted moves in the opposite direction to the eye about the intersection of the webs, the focus of the telescope is towards the eye-piece, and the telescope requires slightly racking outwards by moving the milled head in the reverse direction. After a few trials the object and webs appear stationary, however obliquely observed.

110.—Collimation is the adjustment of the crossing of the webs of the diaphragm to the axis of the telescope and its object-glass. This is effected by adjustment of the opposite collimating screws, Fig. 24, CC', in two directions at right angles to each other. Where the telescope is placed in Y's or collars, this adjustment is made by placing the webs or lines in focus of the eye-piece and the object-glass of the telescope in focus upon a small distant object. Then if the telescope is rotated in all directions, and the small distant object cuts the crossing of the webs in all positions, it is said to be truly collimated. It is necessary to discuss the structure of various instruments to show the methods of collimating in special cases; therefore this subject will be again brought forward.

111.—The Qualities of a Telescope of a surveying instrument are best ascertained by its performance. The general method is to place a staff at the full range, 10 to 15 chains, and to see if the ·01 foot in fine bright weather is read clearly and sharply. This outdoor observation is not always possible, particularly in large towns, but it may very well be supplanted by reading at a short distance. The author made for the late Colonel Strange, F.R.S., whose knowledge of scientific instruments was of the highest order, a test-card for the Lambeth Observatory, to be placed at 25 feet from the instrument. This card had on one part fine lines ruled ·01 inch apart. A 14-inch telescope was considered sufficiently good if these lines could be clearly separated at this distance by the telescope when it was in correct focus. The dial of a watch, or an ivory scale, answers very well as a test object, as sharpness of outline is the point to be ascertained.

112.—A more refined technical method than that described above, which also tests the general accuracy of the optical arrangement of the telescope, is to fix a small disc of white writing-paper, say 1/8 inch diameter, cut out with the point of a pair of compasses with sharp outline, on a black surface of a board, paper, or cloth. If this be placed as before, 30 feet or more distant in a good light, and be correctly focussed in the telescope, a sharp image of it should be obtained. This focal position of the telescope may be temporarily marked upon the inner tube with a fine soft black-lead pencil. If now the object-glass be racked outwards or inwards from this line, say for about a twelfth of an inch, and the image appears to be surrounded with a uniform haze, the objective may be considered to be correctly formed, or to be free from spherical aberration, as it is termed, and the combination to be correctly centred. If the haze appears more on one side than the other the centring is defective. If the object remains fairly sharp when out of exact focus the curves of the lens are defective, as the shorter the range of focus the more perfect is the correction from spherical aberration.

113.—If the curves are not sufficiently correct to bring the image from all parts of the objective to a focus, such incorrect parts are useless, and a good glass of smaller size would be better. The fault is generally found in the marginal portion of the objective, which requires the greatest skill of the glass-worker. Therefore, a very good test to find whether the whole of the aperture of the objective is in effective use is to cut out a piece of card of the size of this aperture and to cut a second piece out of the centre of this, of half the diameter, so as to form a disc and a ring. If the objective be now covered by the ring and accurately focussed upon a test object, and this be then removed and replaced by the disc fixed over the centre of the objective, and the focus remains equally sharp, the curves may be said to be, practically, correctly worked.

114.—As the central part of an objective is more easily brought to correct curvature than the marginal parts it is not uncommon in inferior instruments to make the aperture of the central stop of the telescope cut off the margin of the objective. This renders it only equal to a smaller glass.

115.—Whether the full aperture of a telescope is used may be discovered by employing a second eye-piece—outside the regular eye-piece that is placed in the telescope—to pick up the image of the object glass formed through the eye-piece which is placed against the telescope in the manner of using a dynameter, art. 87. With the ordinary surveyor's level, two eye-pieces are commonly sold; one of these may be placed in the telescope and the other used to pick up the image of the object-glass. With a theodolite one eye-piece may be placed in the telescope, and one of the readers used to magnify the divisions of the limb may be used to pick up the image. The best manner of proceeding is to fix with water or thin gum two or three small pieces of paper, say 1/20, 1/10, and 1/7 inch square, close against the edge of the cell upon the face of the objective. Then focus the telescope on an object at some distance, say a chain or two. Now use the second eye-piece in front of the one in the telescope, and an image of the object-glass will be seen; and if the aperture is fully open all the pieces of paper in their places will be clearly distinguishable. If one or other piece is invisible, the margin of the glass is cut off to this extent. If the objects in front of the telescope tend to confuse, a piece of white paper may be placed obliquely to reflect the light of the sky into the telescope, which will at the same time fully illuminate the objective.

The discussion of the principle of the anallatic telescope, used only with the tacheometer, is deferred to another chapter, wherein subtense instruments are described.

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