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 XIV.

INSTRUMENTS OF REFLECTION—OCTANT OR QUADRANT—REFLECTING CIRCLE—SEXTANT—PRINCIPLE—PARALLAX—CONSTRUCTION—EXAMINATION—ADJUSTMENT—ARTIFICIAL HORIZON—SOUNDING SEXTANT—BOX SEXTANT—SUPPLEMENTARY ARC—IMPROVEMENTS UPON THIS—OPTICAL SQUARE—OPTICAL CROSS—APOMECOMETER.

615.—The Octant or Quadrant measures angles within 90° by an arc of 45°. This instrument is generally termed an octant on the Continent from the space of the divisions; a quadrant by English-speaking races, from the extent of angles it takes. The idea of bringing the reflection of an object from a mirror in line with the direct sight line from another object, to measure the angle at the position of the observer subtended by the two objects, was originally proposed and worked out in a manner by Hooke,[38] and also by Newton.[39] Newton's invention was the more simple and important. This was communicated to Dr. Halley, then Astronomer Royal, but it was left unpublished until after his death, when it was found in Newton's own handwriting among Dr. Halley's papers.

Newton employed two mirrors to obtain the reflection of an object placed at any angle of less than 90° to the axis of the telescope or sight tube, to throw an image directly through the tube. One of these mirrors was placed at an angle of 45° to the axis of the telescope and covered half its field aperture, so that a direct image of an object could be received by the eye from the open uncovered part of the telescope at the same time as the reflected image of another object from the mirror. The second mirror was placed so as to throw its reflection into the mirror on the end of the telescope without giving any obstruction to the open aperture. This side mirror was fixed with the centre of its plane over the axis of a movable arm which read upon an arc the amount of its angular displacement to 90°. The mirrors were so arranged that their faces should be parallel to each other when the movable arm was placed at the zero of the arc. The graduation of the arc was of double the closeness of the ordinary arc reading, so that the angular positions of the two mirrors in relation to each other was indicated according to the following law:—

That the angle between two reflections in the same plane is equal to twice the inclination of the reflecting surfaces to each other.

616.—Hadley's Quadrant.—In Newton's quadrant the arc was brought most inconveniently in front of the face. By Hadley's arrangement the telescope or sight line is brought in a direction about parallel with the chord of the arc, producing the very convenient form of instrument now in use. This instrument was exhibited at the Royal Society, 13th May, 1731.[40] It was tried experimentally by the Astronomer Royal, August, 1732, in a yacht excursion, when readings were taken satisfactorily within a minute of arc.[41] It afterwards came into general use.

The quadrant was at first held to be sufficient for measuring the sun's altitude for obtaining latitude, but Hadley, as early as 1731, saw the advantage of extending the arc so as to be able to observe the opposite horizon if the direct one was obscured. It was also found that measuring the moon's angular distance from a star beyond 90° was serviceable in determining longitude. He therefore proposed by a duplicate system of reflections to extend the arc by what is termed a back sight to 220°. The means he suggested, which were commonly carried out in instruments of the period, were found to be too complicated for practice.[42] In the meantime the construction of these instruments, originally framed of a combination of wood, ivory, and metal, was much improved by making the frame entirely of metal. There were also great improvements made in the optical parts, by which the arc of 90° was extended. In 1757 Captain Campbell had an instrument constructed of metal of 60° of arc which therefore read to 120°. This instrument, with details of improvement, principally by Ramsden,[43] became the modern sextant.

617.—Reflecting Circle.—As soon as the success of the sextant was assured there appeared to be a general desire to complete the circle by reflections, many inventors thinking this would possess great advantages over the arc of 120°, and we find therefore no lack of inventions to this end, even by eminent men. Reflecting circles, as they are termed, that were of sufficient merit to come into limited use, were designed by Mayer, 1770; Borda, 1787; Mendoza, 1801; Hassler, 1824; Fayrer, 1830. Troughton's circle of about this period was no doubt the best instrument of the class.[44] We have also meritorious reflecting circles by Pistor and Martins, and by Amici.[45] Although these instruments were used at sea to a limited extent, particularly on foreign ships, they were also used on land, where indeed they were more manageable. As no further reference to these reflecting circles will be given, anyone interested in the matter may refer to the books mentioned in the notes, where very full particulars of their structure are given. It was felt necessary to mention the subject here, as the same ideas are constantly cropping up as assumed advantages where previous experience is forgotten. Reflecting instruments at sea are tedious to use when the angle to be taken exceeds that taken in by the eye without movement of the whole body. On land, when the angle exceeds 120°, a theodolite is better; but supplementary angles may be taken with the sextant conveniently on land, where the portability of the instrument is of great consideration. This will be again brought forward in discussing box sextants with supplementary arc.

618.—The Sextant, of the invention of which some particulars have just been given, is only used as a surveying instrument for the exploration of new countries, for which employment—it may be used with or without a tripod or stand—it is found to be a most convenient, light, and portable instrument for the traveller for ascertaining longitude, latitude, and time with the aid only of an artificial horizon. Triangulation may also be taken with it of terrestrial objects, even for the complete circle, by repetitions from station to station in angles within 120°. The same principles which are followed in the construction of the nautical sextant are followed also in the manufacture of two modified forms of this sextant which are used for surveying only, the sounding sextant and the box sextant. As the nautical sextant is most open to observation of its parts it will be more convenient to discuss the construction and general arrangements of this instrument first.

619.—Optical Arrangements of the Sextant.—Newton in the description of his instrument placed the mirrors parallel to each other, that is, to zero of the arc, in his illustration for the demonstration of the principle. In this position he showed that the direction of the reflected ray is coincident with the direct ray entering the eye from the same object or star. This scheme the author has generally found the clearest for illustrating the principle to persons not well acquainted with optics, there being some difficulty in explaining the law just given, art. 615, from a more complicated scheme.

Fig. 283.—Reflection in direct line from two plain mirrors.

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620.—If two mirrors be placed with their faces parallel to each other in such a manner that a ray of light may continue after two reflections from them, the ray will continue its path parallel in its direction to its incidence upon the first mirror.

Let MM', Fig. 283, be two mirrors placed with their faces opposite and parallel to each other. Let the incident ray IM fall on the mirror M whose normal is a. Then, as the angles of incidence and reflection are equal, art. 54, it will be reflected at equal and opposite angle to the normal to M'. Let the normal of M' be a'. Then again, the incident line MM' will be reflected at equal angles to the normal to D', that is, as shown by the diagram, it will continue parallel with the incident ray and in such a position that an object at P would appear to the eye, placed at D', as though it were at P' in the direct line of sight.

621.—Parallax.—It will be seen by the figure that the point P does not appear to the eye at D' in its true position but at P' therefore with the mirrors MM' quite parallel, the points P and P' appear coincident, and would read as one point with the index of the sextant set at zero, that is, at the position when the mirrors are parallel to each other; whereas the points P and P' really subtend a small angle if direct lines be drawn from them to D'. It is therefore clear that the angle read by coincident reflection and direct or, as it is sometimes called, visual image is less than the true angle at about the position shown. This difference is called the error of parallax. When the object is distant this error is immeasurably small. The parallax error varies proportionately to the distance of the mirrors apart and with their angular position. If the mirrors are in such an angular position that the rays proceeding from an object impinging upon the centre of the first mirror would, if continued, reach the eye, there would be no error of parallax. This occurs in the nautical sextant at about 60°, and the parallax error increases on either side of this point.

622.—In the practice of surveying this small error is neglected. When the box sextant is used the mirrors are placed at a very small distance apart, and the parallax error therefore is extremely small even for near objects. Where two objects are to be triangulated, the one near and the other distant, the parallax error is much decreased or eliminated by taking the near object by direct vision, and the distant object by reflection. In this case, if the near object be towards the right hand, the sextant must be used in an inverted position. If the two objects be both near, a distant object may be sighted in the direction of one of them for the reflected image.

623.—It is readily seen that if the parallelism of the glasses shown in the figure be disturbed, say by a change in the relative angular position of M' so that the planes M and M' continued to subtend an angle to each other, then the normal of M' must also be changed in direction equal to this; but the ray MM' remaining constant, as there is no movement of M, this ray will therefore be displaced in its reflection from M' an amount equal to the angle of incidence on M' from its normal, plus the angle of reflection from the opposite side of the normal, that is, to double the amount of angular change of the position of the mirror or of its normal, which is the same thing. The sextant therefore reads, by change of position of one of its mirrors, half the angle of reflection upon its arc; and to make it read to the angular value of its reflection the divisions on the arc are made twice as close, that is, half degrees are made to read as degrees. This will be better explained by the following scheme.—

Fig. 284.—Principles of reflection of the sextant.

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624.—The above scheme, Fig. 284, is taken from Captain Magnaghi's admirable work before mentioned, which gives a very clear geometrical demonstration of the value of angular positions in compound reflection. A ray of light SR directed to a plane mirror R is reflected therefrom to a plane mirror R', following a plane of reflection perpendicular to the intersection of the two mirrors. The direction R'T of the ray reflected by the second mirror falls into the same plane of reflection, and makes with the direction SA of the incident ray an angle double that which is comprised between the two mirrors.

The two planes of reflection SAB and ABT unite in one because they both contain the line AB and the normal BP to the mirror R'.

In prolonging the normals of the mirrors to their point of intersection P we find that—

BTS = BAS - ABT;

but as ½ BAS - ½ ABT = BPA = BDA,

therefore BTS = 2 BDA.

625.—The mirrors being placed in the position shown in the figure, if we look through a telescope whose visual axis is placed in the line ET, with its objective to the mirror R', we see in the centre of the field of view the image of the object S reflected consecutively by the mirrors R and R'. We also see in the telescope whether the mirror R' is only a certain height above the plane of reflection, so as to permit half of the object-glass to receive the rays coming from the point E situated in the prolongation of the line TB, also the image of E which is necessarily coincident with that of S, because the rays by which each image is formed enter the telescope in the same direction BT. Therefore when the images of the two objects E and S appear superimposed or coincident in the middle of the field of view, we have an index given that the mirrors form an angle with each other which is half that which is made at the point T from the same objects, and when one is known the other is easily deduced.

626.—Nautical Sextant.—The ordinary construction of this instrument, Fig. 285, consists of a cast gun-metal frame, forming approximately in outline a segment of a circular disc AA including within its extreme radii about 155°.

Fig. 285.—Nautical or astronomical sextant.

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627.—The Limb G, which is made only about 1/12 inch in thickness, has generally a face of about ¾ inch in width, which is inlaid with silver or platinum, as Fig. 127, p. 186, to take the graduation to about 140°. The limb is stiffened by a deep, thin rib about ½ inch wide, supported by a corner hollow. The exterior radial arms and interspace framing, Fig. 285 MM, which vary very much in design according to the taste of the maker, is made generally of about 1/14 of an inch in width upon the face of the bars, with a depth of 3/8 inch. This arrangement of the bars placed edgewise gives great stiffness to the surface of the arc with little weight. A handle L, made generally of ebony, is supported on two standards or brace-pieces N, which are carried off to about 2 inches from the back of the frame to hold the handle parallel with the face. The handle has sometimes a hole bushed through it with metal, to support the sextant upon a corresponding pin forming part of a stand or tripod when the instrument is used for taking observations on land. Three feet are placed at the corners of the frame of the sextant, one shown at Q, to support it conveniently on a table to take the reading of an observation just made.

628.—At the centre of the arc a female axis of about 1½ inches in depth E is attached by three screws to the frame perpendicular to the plane of graduation. This carries the male axis, which centres the vernier on the vernier arm M. The axis is covered by a protecting tube which forms one of the three feet upon which the instrument rests when laid down. The vernier arm is made of gun-metal of about 1/16 inch in thickness and from 1 inch diminishing to ¾ inch in width. This is stiffened by a light rib on its upper side.

629.—The Vernier V reads upon an 8-inch sextant, that is, one of eight inches radius, to 10, the graduations being to 20' and the vernier taking 120 divisions. A description of the vernier reading was given, art. 318. The vernier falls upon the arc on the plan shown Fig. 127, p. 186. It is clamped near to position by the milled-headed screw H, and is adjusted by the tangent I. A magnifier J is placed on a jointed sling-piece K which traverses the vernier. This is sometimes provided with a ground glass shade to dull the silver for reading. The sling-piece moves the magnifier opposite to any division of the vernier.

630.—Over the axis of the vernier arm a large, oblong mirror, termed the index glass, A, is fixed with its face in a plane cutting the centre of the axis. The index glass is placed with its longest sides approximately in line with the vernier arm. This mirror is placed in a metal tray and is sometimes made adjustable by three screws; but it is better fixed by the maker by screwing the flange-piece, which forms one end of the tray, hard down. The index glass moves with the index arm and gives the first reflection of sun, moon, or star which falls thence upon the horizon glass B.

631.—The Horizon Glass B is placed upon a spur-piece formed in the same casting as the frame. This glass, which is worked perfectly parallel, has the lower half of its surface next the frame silvered. The silver is cut to a sharp line against the plain part. The horizon glass placed in its metal tray has adjustments given to it by means of capstan-headed screws in a manner that will be presently described.

632.—The Telescope screws into a ring fitted at R, which stands upon a bar erect from near the edge of the frame. The female screw by which the telescope is held is formed of two rings which adjust for the amount and direction of separation, so that the telescope may be directed coincident with the horizon glass. The bar or standard supports the ring fitting and is made of either square or triangular section, fitted accurately in a deep socket fitting, in which it slides to raise or lower the ring by means of a milled-headed screw placed on the end of the bar. This permits adjustment only sufficient to bring the axis of the telescope opposite the line of division between the plain and silvered parts of the horizon glass.

633.—Four Circular Shades, carried in square frames fitted with dark bluish-grey glasses, are jointed to the frame at C. These have nib-pieces at the upper corners, so that one or more of the shades may be turned up at a time by the finger-nail to intercept any surplus amount of light from a luminous body reflected from the index-glass; or the whole of the shades may be turned up when observation is made of the mid-day sun. Three other similar shades, but placed in circular frames are fixed at D, which hinge over and back, to be thrown in or out of interception, and are used to subdue the light from the horizon if required.

634.—The Telescopes used as a part of the sextant are generally two in number. One for direct vision is a short tube of about 3 inches in length, focussing at about 4 inches. The optical arrangement is the same as that of an opera glass, consisting of an achromatic object-glass of about 4 inches focus and a concave eye-glass of about 2 inches negative focus, Fig. 14. The second telescope is about 7 inches to 8 inches in length. This has two Huygenian eye-pieces, which have each a wired diaphragm at the mutual focus of the eye-piece and the object-glass. One of these has two fine wires placed parallel for use in adjusting the telescope, and the other has two pairs of crossed wires to indicate the centre of the field of view. There is also a plain pin-hole sight provided for open vision.

635.—The Case in which the instrument is packed is generally made of well-seasoned mahogany, dovetailed together at its corners. The fittings are made to put the instrument back in its case as it was last used within a wide range. A tommy-pin for adjustments and a hand magnifier are supplied with the instrument. The case is generally French polished inside as well as out to prevent absorption of moisture from sea air.

636.—Manufacture and Examination of the Nautical Sextant.—Besides the general good work that this instrument demands, the important points to be observed are, that the glasses should be of hard crown glass worked perfectly parallel from face to face; they should also be well polished. These observations apply to both the reflecting glasses and the shades. The silvering of the mirror should be protected with a good coating of copal varnish. The mirrors should be held by three points only, and be quite free from strain. The upper of the three points should detach, so as to be able to remove the glass at any time for resilvering. The axis should be fitted with all the care necessary for a theodolite, and be placed truly central to the arc. The extremity of the vernier arm when free of its clamp should traverse the arc at equal distance from its face and move with very light friction. The extreme lines of the vernier should cut equal divisions all along the arc 0° to 140°, observations being taken particularly at both ends and in the centre of the arc. The vernier should lie flat on the limb from end to end of the arc. The standard or stem-piece for elevating the telescope should move upwards and downwards stiffly but equally by the motion of its milled-headed screw. The division lines of the limb and vernier should be cut fine but very deep: they should be cut on the dividing engine from the axis of the sextant to ensure true centring of the arc, and not as in the usual plan of having the axis adjusted to the divisions.

Fig. 286.—Section of axis and index glass of sextant.

Fig. 287.—Section of limb and clamp and tangent.

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637.—Axis.—This is the most important part of the instrument, and requires the greatest care in construction. Fig. 286 represents this to a scale half size. a the axis, made of hard gun-metal, has a collar by which it is attached to the index arm. The axis is ground and burnished carefully into S the socket-piece, which is fitted into the frame and held down by three screws. At the end of the socket there is a collar-piece B attached upon an angular or tight conical fitting by the screw D, which prevents the axis rising out of its socket. The axis is covered by a cap L which protects it from injury, and this at the same time forms a leg to the instrument as before mentioned. The index glass I is mounted in a tray T shown in section. There are two points of contact at the lower part of the back of this glass, formed by pins, and one point of adjustment pressing against the clip G by a spring C in front, acting contra to a screw at the back E, which adjusts only a small distance to bring the index glass to perpendicularity. The flange-piece F is adjusted in the manufacture so as to leave very small separate adjusting to the index glass necessary.

638.—Section of the Limb and Clamp and Tangent.—The general arrangement is shown in Fig. 287. M arms of the frame; J section of the limb; C clamp attached to the tangent N for clamp and tangent motion, described art. 346; O milled head to clamp; N milled head to tangent. The vernier is shown at V, reading through an opening on the face of the index arm P. The rib to stiffen this arm is shown at R.

Fig. 288.—Vertical section of horizon glass.

Fig. 289.—Plan of section A to B.

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639.—The Adjustment Arrangement of the Horizon Glass.—This most important adjustment is constructed in various ways. The plan now generally thought to be the best is for the maker to fix the horizon glass frame firmly in its true position perfectly perpendicular to the surface of the frame, and to allow a small amount of adjustment to the glass only. A convenient plan of doing this is shown in the vertical section full size in Fig. 288. The frame FF is made in one casting, which has its base collar firmly fixed to the frame of the sextant. Fig. 289 is a cross section A to B. H the horizon glass is held upon its face by three points, one of which is shown at L, which is placed in the centre of the top. The lower front points are the exterior corners of a plate which is cut away between. This plate is held by the screw G. The screw G forms a kind of hinge which, together with the elasticity of the plate, gives a slight pressure directing the glass hard upon the points of the screws J and Q. The screw J resists this pressure lightly and permits adjustment of the horizon glass H to angular position in relation to the plane of the index glass to a small extent, by means of a pin placed in the capstan head J. The perpendicular position of the horizon glass, H, is secured by slight adjustment of the capstan head K, which moves against a spring L in the vertical centre of the top of H. This piece, with screw and spring, is attached to the horizon glass frame FF' by the screw M, so that it may be easily removed to replace or resilver the glass. The silver on the glass is cut to a sharp line at about the point H with a razor.

640.—Testing the Parallelism of the Surfaces of the Glasses.—The best method is to firmly fix a telescope provided with webbed or pointed index diaphragm so that the webs or points cut a distant, sharply defined object, or its edge only, quite clearly. If the glass to be tested be now placed in four directions agreeing with its four sides in front of the object-glass of the telescope, and it is worked perfectly parallel, and is free from striÆ, the distant object will not appear to be displaced by its presence in the slightest degree at any position. If the glass be not mounted and is quite square, should there be any very small error, the thickest or thinnest edge should be placed towards the frame; but in this case only a very small error is permissible. The coloured glasses require the same test as the white ones. Where the parallel glass to be tested is small, the object-glass of the telescope may be covered by a paper cap, with a small hole only left through its centre, sufficient to take the glass.

641.—The glasses, when fixed in the sextant, may be examined for parallelism approximately by setting them end up singly to the sun, with the sextant set at an angle that the direct and reflected images of the sun's limb appear just to touch, the eye-piece of the telescope being constantly covered by the sun-glass. If there be a want of parallelism, the image will be disturbed. One reason that the telescopic plan first proposed is better to be followed in the construction of the instrument, is that the telescope is fixed and that there is no indistinctness from unavoidable motion of the body, such as occurs when the sextant is held in the hand.

642.—The Quality of the Surfaces of the Glasses may be examined, both for flatness and brightness and for equality of density, by holding them so that the reflected image of a straight body, as for instance a stretched thin string placed at a distance, may be observed by reflection in glancing over the surfaces with the eye nearly parallel with its plane. If the glass be imperfect the image that reaches the eye will appear to be wavy. If the reflection appear misty, this is generally due to want of parallelism of the glass; but this mode of observation is altogether somewhat technical and difficult to attain without skill.

643.—To Silver the Index or Horizon Glass with Mercury.—Clean the glass thoroughly by boiling it in water containing an alkali (potash or soda), and then polish it off with whiting and water, using a clean piece of old linen or perfectly clean wash-leather. Do not touch the surface with the fingers. Take a piece of clean tin-foil freshly opened from the roll and cut out a piece slightly larger than the glass to be silvered. Lay this upon a smooth pad—an old leather book-cover answers. Place a single drop of clean mercury about the size of an ordinary shot upon the tin-foil and rub this gently over the surface until it is entirely silvered. Now pour very gently sufficient mercury upon the foil till the surface appears to be flooded. Take a sharply cut straight-edge formed of stiff writing-paper, and draw this over the surface of the mercury to clear it. Take a slip of clean smooth writing-paper very little wider than the foil and of about one and a half times its length: spring the paper to a slight curve and place one part of it over the silvered foil so that when it springs open it will cover it and exclude the air from the surface. Now give the glass a final polish and lay it upon the paper over the foil. Hold the glass down with slight pressure with the left hand, and slowly and steadily draw out the slip of paper in the linear direction of the surface of the glass with the right hand. This will take out the air between the foil and the glass, so as to bring the mercury in contact and leave a perfect mirror. It must now be set aside with the glass turned face downwards in an inclined position, so that the surplus mercury may drain off from the foil. Small slips of foil should be put at its lower edge, which, by their attraction for the mercury, will accelerate the draining. The mirror should not be touched after setting it up to drain for twelve hours at least, after which the surplus foil may be trimmed off. After another thirty hours or more it may have any varnish or other protection applied to the back of the silver.

644.—Where instruments are taken abroad mercury silvering may become spotted, so that a small store of mercury and tin-foil should be taken out with the sextant for resilvering. But it should be particularly observed that the mercury should never be placed in the same case with the instrument, as the smallest particle, if it touch the frame, will eat into the brass and destroy its strength. Sealing-wax dissolved in spirit answers for a varnish at the back of the foil fairly well after resilvering if proper varnish be not at hand. It is advisable before attempting to silver a sextant mirror to practise on a few slips of ordinary glass in order to get into the way of doing it. In modern practice base silver is deposited, and no mercury is used, but the process requires special skill.

645.—Adjustment of the Index Glass.—Hold the sextant clamped to about 60° in a horizontal position with the index glass near the eye. Look nearly along the plane of the glass in such a manner as to be able to see one part of the plane of the arc by direct vision, and another part by reflection of it at the same time. If the direct view and the reflected join in one line, and the arc appears as the continuity of a single plane, the index glass is perpendicular to the plane of the sextant. If this be not the case it can be adjusted by turning the set screw placed at the back of its upper centre, Fig. 286 E, very gently.

646.—Adjustment of the Horizon Glass to Perpendicularity.—Place the vernier at zero. Hold the plane of the sextant parallel to the horizon and observe if the image of the horizon seen by reflection at the edge of the silver line coincides exactly with the image received directly through the plain part of the glass. If it does so the horizon glass is perpendicular to the plane of the instrument, that is, assuming the index glass is also perpendicular. In this adjustment it is well to rock the plane of the instrument say 20°, to see that the horizon is cut as a clear line about its horizontal position for this amount of angle. If the mirror be not perpendicular adjust gently by the single screw at the top of the horizon glass frame. If the horizon be not water, the sharp outline of any distinct distant object will answer, or a piece of fine string placed at a distance and stretched straight.

647.—Adjustment for Index.—This is the adjustment for parallelism of the two mirrors at the zero of the arc. The sextant is clamped at zero as before: the arc of the instrument is turned in a vertical position and the horizon again observed. If this appears to cut a clear line through the plain glass and the mirror there is no index error, and the planes of the glasses are truly parallel to each other in this position. If the line is not continuous adjust gently by the lower screw, Fig. 288, at G.

648.—Adjustment of the Horizon Glass by the Sun.—This is a better adjustment than that given above, except that it introduces any error that may be due to the imperfection of the shades; and it is more difficult particularly for the first approximate adjustment. Arrange the telescope and shades so that a clear outline of the sun's limb may be observed without distressing the eye. Place the vernier at zero. Observe the sun, which will be most conveniently sighted at about 40° elevation, first with the plane of the frame vertical, and then horizontally perpendicular to this. If the sun presents a round disc in both these positions the sextant is in adjustment. If in the vertical position there appears to be a small notch at top and bottom of the sun's limb, the glass is not perpendicular to the plane of the instrument, and this requires adjustment by the screw at Fig. 288 K. If notches appear at the sides of the limb when it is held horizontally there is an index error, which may be adjusted at G if it be small.

649.—Index Error after Adjustment Allowance.—The limb of the sextant is graduated 5° beyond the zero position when the glasses are parallel to each other. This is called the arc of excess. The vernier is also divided three lines beyond its zero position, which is marked by an arrow. These extra divisions are placed on the instrument for correcting the index error by measurement of the angle subtended by the diameter of the sun's disc alternately on one side and the other of the zero line, in which observations, if the two readings agree, the sextant must be in perfect adjustment; when they do not agree half the error may be adjusted by the horizon glass. The same observations may also be made with a bright star by setting the index alternately on one and the other side of zero. When the sun is used the reflected and direct images are brought together, so that the two suns that appear in the instrument just touch limb to limb, first upon direct reading and then upon the arc of excess. When the division is adjusted very nearly, any small error, plus or minus, may be allowed as a constant for all readings. In observations of the sun care should be taken that the eye is protected, both by the sun-glass cover to the telescope and by sufficient use of the shades.

650.—Adjustment of the Telescope to Set its Axis Parallel to the Plane of the Sextant.—In fixing up the instrument after manufacture, the ring standard which carries the telescope is set at a measured distance from the plane of the frame, so that the centre of the ring coincides with the height of the silver line cut on the horizon glass. This is necessarily a primary adjustment. For final adjustment the long, inverting telescope is screwed home in the ring, and the eye-piece which has two parallel wires across its diaphragm placed in it. The telescope is brought to focus on any distant object, the eye-piece being turned at the same time to bring the wires parallel with the face of the instrument. Two objects are taken subtending an angle of 90° or over,—as the sun and moon, or the moon and a bright star,—and the index is moved so as to bring the objects, say the limbs of sun and moon, in contact with the wire nearest to the sextant, and fixed there. Then by changing the position of the instrument a little, the images are made to appear upon the wire furthest from the sextant. If the limbs of the sun and moon still remain in exact contact as they appeared before, the axis of the telescope is truly adjusted. If the limbs of the two objects appear to separate at the wire furthest from the sextant, the ring-adjusting screw furthest must be loosened a very little and the screw nearest the sextant tightened the same amount. If the reverse, and the images appear to overlap, adjust in the reverse direction. By repeating this operation a few times the contact will appear to be the same at both wires, and the axis of the telescope will be in collimation, that is parallel with the plane of the instrument. After the telescope is truly adjusted it may be raised or lowered a little to make the reflected and direct images appear equally clear.

651.—Final Examination of the Sextant.—It will be readily seen that an instrument, although correct in theory but depending upon perfection of workmanship in centring, division, surface and parallelism of glasses, and also in its adjustments, can scarcely be brought to perfection. The errors generally increase from the zero point, where adjustments are possible, and are greatest at 140°. In the ordinary commercial sextant of the dealers the errors of centring alone are commonly 3 minutes to 5 minutes, with like errors in other parts. It is therefore better, where the sextant has to be absolutely relied upon, to subject it to actual trial. The zero point can be readily fixed by rules already given; besides this the meridian altitude of several bright stars subtending angles of about 30°, 60°, 90°, and 120° should be measured either from a clear horizon or from a mercury artificial horizon, to be described presently, for angles under 60°, and the errors plus or minus tabulated. The data for the meridian altitudes of certain stars upon any night may be taken from the Nautical Almanac, which will require correction for the latitude and longitude of the observer. This subject is too complicated to be entered upon in detail here. At the present time the National Physical Laboratory undertakes the examination of sextants for a moderate fee. This is effected by means of fixed collimators, art. 229. For angles distributed over the arc the parallax error is eliminated by placing the collimators in pairs. The N.P.L. certificate may now be had with good instruments when purchased. It may be noted that an originally well-made instrument retains its qualities for all time, the wear of such instruments being inappreciable.

652.—To Use the Sextant the right foot should be placed nearly 2 feet in advance of the left and directed at right angles to it. In this position the body is firm. The instrument is supported by the right hand, the elbow being brought down firmly upon the body. The clamp screw first and then the tangent screw are moved by the thumb and finger of the left hand. Some practice is required to make a steady observation. To bring two objects into apparent juxtaposition, methods of observation for terrestrial objects will be reconsidered in discussing the box sextant further on. As regards celestial observations reference should be made to works on practical astronomy, as the subject would take too much space to be entered upon here. The whole subject, with many refinements of correction of parallax, etc., which fall beyond the limits of practical surveying with the sextant, is ably discussed in Chauvenet's Spherical and Practical Astronomy.

653.—Artificial Horizon.—For ascertaining the latitude of a place from the observation of a celestial body by means of a sextant, it is necessary to have some means of estimating the position of the horizon. A method of doing this, originally proposed by the elder George Adams, optician, 1748,[46] was to float a parallel disc of glass upon a basin of mercury, and to receive the reflected image of a star from the mercury by the sextant simultaneously with its direct image. The angle then given by the reading of the arc is double the angle at which the true horizon is placed relatively at the same time. This idea, carried out in a practical form in an instrument henceforth called the artificial horizon[47] is due to Wm. Jones, a well-known optician at the end of the 18th and beginning of the last centuries, who arranged convenient means of making the instrument portable, and to keep the mercury from disturbance of the air by covering it with a glass roof. The form of artificial horizon that he invented has been in common use ever since. He also invented another simpler form, which was that of taking the reflection from a piece of silvered, or of black, glass. The performance of the artificial horizon depends in any case entirely upon means of obtaining a reflection from a perfectly horizontal surface.

Fig. 290.—Diagram of artificial horizon.

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654.—Theory of the Artificial Horizon.—A ray M' Fig. 290, from a luminous body, at infinite distance will have its image reflected from a level reflecting surface SS' at an angle equal and opposite to the incident ray, the angles M'AS and EAS' being equal. Let E be the place of the eye or the sextant: this will receive a ray from the same distant body in direction ME, which is sensibly parallel with M'A. The angle MEA being double the angle of incidence M'AS, the half of this angle will therefore produce the horizontal line EH at the height of the observer's eye if the plane of reflection SS' be level. Therefore if we take half this angle MEA as it appears in the sextant, it will give an angular position of the object in relation to the horizon at the height of the eye, or be tangential to the surface of the earth. If M'AS be 30°, the angle AEM will be 60°, showing the elevation of object half this or 30°. The sextant takes 120° with certainty; therefore 60° will be the limit of meridian altitude the artificial horizon will measure.

Fig. 291.—Artificial horizon of black glass.

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655.—Artificial Horizon in Black Glass.—This instrument is the most portable, packing in a close pocket case. It is made of both circular and square form in plan. Fig. 291 is the Admiralty pattern. The black glass should have a truly plane surface. It is fixed over a brass tray by being floated on plaster of Paris to avoid strain. A light rim of brass is screwed down over the glass to keep it in position. There are three adjusting screws AA'A. It is adjusted to level by a loose level tube ground on its under face P. The level tube shown in detail Fig. 51, p. 94, is placed on the surface lineally with the two screws, Fig. 291 AA', and afterwards at a right angle to its first position with one end of the tube towards A. It is finally tested by traversing at the position shown in the Fig. 291, and at right angles to this direction. There is a great risk of getting a strain on the glass in fixing it in its frame. The author therefore prefers the circular form that leaves the glass quite free except at its fixings at three equidistant points only. In this kind of artificial horizon there is only one surface of glass to be worked true; therefore, there is perhaps less risk of error on this account than in other forms. On the other hand the mercury presents a more perfectly level plane. The circular artificial horizons are commonly made 3¼ inches diameter; weight, ¾ lb.; the oblong, Fig. 291, 4 inches by 3 inches; weight 2 lbs.

Fig. 292.—Artificial horizon, mercury.

Fig. 293.—Mercury bottle to the same.

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656.—Artificial Horizon of Mercury, Fig. 292. This instrument consists of an oblong tray of about 6 inches by 3 inches by ¾ inch in depth made of wrought iron. It is covered by a roof with two sloping sides at about 45° to the plane. The sides of the roof are glazed with worked parallel glass fixed by screws at three points. The mercury when out of use is contained in an iron screw-stoppered bottle, Fig. 293. It is poured into the open tray for use, and the tray is afterwards covered by the roof to prevent currents of air disturbing the level of the surface. After use the mercury is poured back into the bottle from the corner of the tray. It should be particularly observed that it is perfectly drained, as any free particles in the case in which all parts of the instrument are packed would be certain to attack the roof, which is made of brass and simply varnished. The instrument is packed in a mahogany case, size 7½ inches by 6 inches by 5 inches; weight, with 1 lb. of mercury, about 4¾ lbs.

657.—The Bottle, Fig. 293, is made of cast iron. It has a screwed plug stopper with a leather collar and a covering cap with a small hole through its apex. To pour out the mercury the cap and stopper are unscrewed, the plug is taken away, and the cover is screwed on again. The mercury then issues from the small hole in the cap. To return the mercury the cap is reversed and screwed upon the bottle. It then forms a funnel. The tray has a covered corner at which there is a small hole. This permits the mercury to be poured into the funnel without splashing. Both plug and cap are then screwed down firmly, and the bottle is placed in a secure fitting in the case.

Fig. 294.—Captain George's artificial horizon.

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658.—Captain George's Artificial Horizon,[48] Fig. 294. This is a great improvement on that last described. The instrument being made entirely of iron there is no risk of getting it injured by escape of the mercury. It is also much more portable and convenient. Two chambers E and M are connected together by a tube through a stem-piece in which there is a strong iron cock at a. The chamber E is cored out and form a bottle into which about 1 lb. of mercury is introduced by removing a screwed stopper at B. The chamber M is an open tray with a cover formed of a piece of parallel glass placed in an iron rim which screws down upon it. A milled-headed screw at C forms an air plug. The cock moves very stiffly by the leverage given by a tommy-pin, shown a', which is inserted in the hole at a. The chamber E is slightly elevated to cause the mercury to flow from it to M, the cock being turned on at the same time and the air screw C released a little. By the same arrangement, M being raised, the mercury flows back into the bottle for storage.

659.—For Using this Artificial Horizon, when the mercury is poured out in the tray M, it is levelled by the three screws AA'A so that it covers the bottom of the tray and presents a clear, level surface. A separate disc of parallel glass, which fits the tray M very loosely, is provided with the instrument. This floats on the surface of the mercury and keeps it quite still, even when the covering glass is removed. This arrangement is useful also in case of an accident to either of the glass covers. The disc is kept when out of use in a soft leather bag which fits in the tray M. This artificial horizon is generally carried in a solid leather case with sling to go over the shoulder. Its weight complete is about 4½ lbs.; size, 9½ inches by 4 inches by 1½ inches. The surface of mercury is a circle of 3 inches diameter.

660.—Improved Captain George's Artificial Horizon.—Mr. S. A. Ionides, C.E., has devised an improved form of the foregoing instrument shown at Fig. 295.

Fig. 295.—Ionides's artificial horizon.

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In this the container is formed beneath the horizon box with a plug tap fitted in the thickness of metal between the two; this form makes the whole much lighter and less than half the size of the usual Captain George's pattern.

661.—In Using the Artificial Horizon with the Sextant it is generally placed on the ground at such a distance in front of the observer that he can conveniently see the required reflection of the star or sun, the observer moving about until the reflection is obtained. This is a tedious process and requires some practice. It is much more easily effected if the sextant be mounted on a tripod or other stand. When a stand is used it has generally a universal joint, so as to be able to take surface angles also from the fixed position. When the altitude of objects on the earth is taken, the observation requires reduction for refraction, which becomes an important factor, although this is variable with atmospheric conditions; but upon the whole it always tends to make the object appear higher than it really is. Commonly one-seventh of curvature is used as an approximate correction. For solar and stellar refraction, works on astronomy should be consulted.

The index error of the sextant is corrected before refraction, when the natural horizon is employed. When the artificial horizon is used the index error is allowed before taking its half as a single measure. The artificial horizon is used also with the theodolite. It forms the most perfect means of adjusting the transverse axis by taking an observation of the pole star with the telescope, first directly and then by its reflection from the artificial horizon. If the images cut the centre of the webs in the two positions by the movement of the transverse axis only from the one to the other, this axis is proved perfectly level.

662.—Various schemes for obtaining the horizon by some system of levelling apparatus attached to the sextant have been devised, none of which are very practical, as they all depend upon a pendulum or a gravitation surface of a liquid or a gyroscope, and are all unstable as hand instruments. There have been numerous patents taken out with this object from that of Winter (1760) downwards, which anyone interested in the subject may consult.[49] The matter is mentioned here as the recurrence of the idea appears to be frequent.

Fig. 296.—Sounding sextant.

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663.—The Sounding Sextant.—This instrument is used for coast surveys. Angles are taken with it of objects, buoys, etc., from the land and also from a boat on the water for such objects or for others upon land. It is constructed upon the same principle as the ordinary nautical sextant; but as it is to be used as an all-day working instrument, and not for a few diurnal observations only, it is made much more solid, and its optical parts take a more extended field of view. The graduation is also stronger, such precision of reading only being required as may afterwards be plotted on a chart. This instrument is shown in perspective, Fig. 296. The index glass is large—about 2¼ inches by 1¼ inches. This is secured on all sides by a firm rim to the tray in which the glass is held at three points. The adjustment of the index glass is left under control, as it may occasionally be necessary to remove it from effects of spray upon and about it. The horizon glass is made about 1½ inches in width and ¾ inch in depth. This is entirely enclosed in a tray, the whole surface being a mirror without any plane part to the glass as with the ordinary sextant, so that it is entirely protected by the metal. By this arrangement the eye receives the direct ray from the object immediately before it, and the reflected ray from an object whose angular position is desired to be taken with it: but these images do not come exactly into contact, as the narrow frame interposes. It is, however, sufficiently near for terrestrial observations. The adjustment of the horizon glass to the perpendicular of the plane of the arc is the same as that shown in detail for the box sextant further on. The adjustment of the horizon glass to the index is by a stiff arm extended from the sole-plate projected into a loose opening, where it is held firmly by two opposing capstan-headed screws, as before described. The arc of the sextant is of 6 inches radius, graduated upon silver to 20', and reading by the vernier to single minutes only by the microscope. The clamp and tangent are the same as those described for the nautical sextant. The frame is straight braced. The telescope has a wide field, with achromatic object-glass of 4½ inches focus, the clear aperture being 1-1/8 inches. The supporting ring of the telescope has no rising stem or collimating adjustment, but is solidly fixed in its true position by the maker. The ring carries a plain disc pin-hole sight, which takes the place of the telescope for near observations. The instrument in use is held in the hand by a firm oblong handle. The instrument rests, if required for reading, upon three legs as the ordinary sextant. Its weight is about 2¾ lbs., or when packed in its case, 5 lbs. Its examination and adjustment are of the same kind as those just described for the nautical sextant.

664.—Box Sextant.—This very neat and portable instrument was invented by the late William Jones.[50] It is used for taking angles within 120° upon the surface of the land to within a single minute of arc. It has become deservedly popular with British surveyors as a land surveying instrument, and is equally so as a military one. It is the same in principle as the nautical sextant already described, but it possesses the great merit—as a surveying instrument constantly in hand—that all its glasses and delicate parts are securely protected from accidental injury by being covered; whereas the nautical sextant, made for one or two diurnal observations only, has all these parts exposed. And it is not only that all parts are protected when the instrument is in use, but they are all doubly protected by the covering box when carried about out of use; so that it is found that a well-made box sextant set originally in perfect adjustment will retain this adjustment in average use for very many years. The author has seen an instrument twenty years in use still in perfect adjustment. The box which covers the instrument out of use forms also a most convenient handle or support for it when in use by attaching it in a reversed position underneath, as it appears in Fig. 297. This attachment is made either by a screw cut entirely round the body of the instrument, or, what is much better, by a bayonet fitting, for the reason that large screws of this description are liable to cross thread. The general description of the outer parts is as follows:—

Fig. 297.—Perspective view of the box sextant ready for use.

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665.—C a covering box which inverts from the position shown in the figure and covers the instrument. This has a diameter of 3 inches and a depth of 1½ inches. B box containing the optical and moving parts of the sextant. A axis of index glass. This axis also carries a toothed segment fixed close under the front of the box, by which both the index glass and index are moved by means of a pinion to be described. The index carries a vernier divided into 30, which reads into the arc to single minutes; the arc is divided to half degrees on silver. The magnifier is centred by a swivel hinge joint over the axis, so as to permit it to be brought to focus upon the arc at any position. This magnifier is held down on the front of the box when out of use by a nib catch at a position of about 80° of the arc. O a milled head, the axis of which carries a pinion which works into the segment above described under the index glass. The pinion is about 1 to 9 of the segment, so that the index traverses the arc of 60° (reading 120°) by one-and-a-half turns. This gives a conveniently slow motion to the index glass, and enables this sextant, if it be well made, to be set rapidly with great precision. S two nibs, part of two levers for putting the shades in or out of action.

666.—In the closed form of sextant the shades block the reflecting position between the index and the horizon glass. For surface surveying they have therefore to be opened out, through an opening closed by a slide shutter which moves by a stud in a slot on the under side. The shades consist of one green and one dense red glass which must be worked parallel, as before described for the nautical sextant. These are used for taking altitudes of the sun, for adjustments only.

667.—The Key K is a milled head which screws out, and carries a watch-key pipe at the end of its stem by which adjustments may be made from three square-headed screws fitting its pipe, two of which are close to b, the axis of the horizon glass. These adjust perpendicularly to the plane of the arc. One screw at a adjusts the parallelism of the index and horizon glasses when the index is at zero.

668.—The Telescope is achromatic, with draw tube for focussing. It magnifies about 2½ diameters. It has a concave eye-glass, and therefore gives an erect image, Fig. 14. A sun-glass E screws over the eye-glass when it is required for sun observations. The telescope is attached to the sextant by means of a crank-piece upon the telescope which is fixed by the mill-headed screw T' and two steady pins. The crank-piece screws in reverse position upon the telescope for portability before putting it by in its case.

669.—By some makers the telescope is made to slide into the body of the sextant and thus become quite portable. This plan is a very neat one, but it requires care to see that the shades do not interfere before it is put by. The weight of the entire sextant with its solid leather case is about 18 oz. only. For close work the telescope is not generally used. A sliding shutter pierced with a small hole covers the telescope opening into the sextant, which is used as a sight hole.

Fig. 298.—Box sextant under the face.

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670.—The Interior or Optical and Mechanical part of the Sextant is shown Fig. 298. I index glass, fixed over the toothed segment on the same axis. The pinion is shown working into the segment moved by the milled head O of Fig. 297 on the face of the sextant. Fig. 298: horizon glass, cut by ED, adjusts to the vertical by screws CC', which have square fittings on the face of the instrument, shown Figs. 299 and 300 full size. The differential adjustment between horizon and index glasses is made by a screw with a square fitting at P. This adjustment acts by screwing against a helical spring, shown at Q. The reflected rays enter by a wide window in the side of the box, Fig. 298 d, the direct rays by a small window f. The path of a ray is shown by fine lines from R to E, for the positions in which the index and horizon glasses are placed. The pin-hole opposite which the eye is placed is shown white. S shades with their axis are shown cut off, to prevent confusion of other parts. They are simply round discs of parallel glass on arms which rise from the back of the face by pressure of the nibs at S.

Fig. 299.—Plan of horizon glass.

Fig. 300.—Section of the same.

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671.—The Construction of the Box Sextant may be fairly inferred from inspection of the engravings. The face-plate is made of a casing in brass 1/8 inch thick, which should be well hammered to harden and stiffen it. The axis, which has a wide collar, is fitted into a hole in the plate, first by turning it as exactly as possible, and then by burnishing it in by friction, the hole being broached slightly conical with a D-broach. The careful fitting of the axis is an important part. The horizon glass frame, Fig. 300, is held down by a central screw which fits tightly both in its fore hole and thread. The flange of the tray F is cut to an angle on its under side to permit adjusting to verticality by rocking over this angle, by tightening and loosening the adjusting screws cc' which protrude in square heads to the face of the instrument. The horizon glass, H, which is half silvered, is fixed in a tray-piece which has two narrow fillets turned to the face of the glass, and a spring-piece at the back brought up by a screw a. This glass is entirely open at its unsilvered part. The toothed segment should be cut upon its own axis, and although fitted to the pinion without any looseness, it should not press the index axis. The silver is inlaid in the arc on the plan shown Fig. 127. The vernier is soldered closely on the index and should read down to a fine clean edge.

672.—Examination of the Box Sextant.—The glasses should be cleanly silvered, with a sharp, clear cut between the silver and the clear glass of the horizon glass. The pinion should move softly and equally in causing the index arm to traverse the arc. If the pinion be moved in little jerks backwards and forwards there should be no shake, but the index should follow every slight motion. The magnifier rising joint should move rather stiffer than the traversing joint, so that the focus is not changed by traversing across the arc. The magnifier should have about 1 inch or less focus, and should stand square to the plane of the sextant when in focus. The graduation should be deep and fine, and the vernier should read 30 = 29 at the two ends and the centre of the arc. If there be a small excess or defect of vernier to arc, this should be noted and allowed for, either at the time of reading or as an index error. The sliding fittings of the pin-hole sight, shades, and under shutter should move firmly but not stiffly. The telescope should fit without shake. The covering box should fit well in both positions of cover or hand-hold.

673.—Adjustment.—The box sextant is best adjusted by the sun upon the plan described art. 648. The adjusting screws, as already stated, are moved by the key, which unscrews from the face of the sextant, Fig. 297 K. The adjustment is made permanently by the maker, except only that of the horizon glass, which is at the command of the user. The adjustment to perpendicularity of face is made by the two screws upon the face near b; adjustment to zero of arc by the screw at the side a. In defect of appearance of the sun, the sextant may be adjusted to any clear, sharp line, as that of a stretched piece of twine, for perpendicularity of plane, and to any object of clear outline sufficiently distant, say at half a mile, to avoid error of parallax for index zero, art. 621.

674.—Use of the Box Sextant.—The sextant has its under shutter opened by pressing the stud attached over in its slot. The nibs of the shade levers, Fig. 297 S, are then raised and the shades depressed. The cover is then screwed, or slid on if it fixes with bayonet notches, upon the under side of the sextant to form the hand-hold. The pin-hole sight is pressed over for use if not already in its position, unless it be intended to use the telescope. The box sextant is held in the left hand, with the right-hand thumb and forefinger constantly holding the milled head, and turning this so as to bring the two objects, of which it is desired to obtain the angular position, from the observer, exactly in apparent juxtaposition, the one over the other. In turning the milled head it is better to let all the other fingers of the right hand clutch and steady the instrument. To take angles objects should be observed that cut sharp, erect outlines, as buildings, posts, trees, etc., if possible. In open country it is necessary to use pickets, to be described further on. With pickets the reflected image of the upper half of one picket should form a continuous outline with the direct image of the lower half of the other picket in the eye, so that the pair of pickets appear as one. Where an angle greater than 120° is required an intermediate picket is set up, and angles taken to the right and left of this are added together.

675.—It must always be remembered that the sextant takes angular positions actually, whereas plans are made in azimuthal angles. There are some not very satisfactory means of approximate correction for this, for which books on surveying may be consulted; but altogether the sextant is not very useful for taking angles for plans on other than fairly level ground, wherein it has proved a most valuable and sufficiently exact instrument. Where ground is undulatory fairly good work may be done with it by taking stations for exterior triangles at equal heights on the hillsides, as ascertained by a hand level or clinometer to be described, or sometimes from hilltop to hilltop where these are of fairly equal heights. For sketch plans of very hilly or mountainous districts the prismatic compass, art. 148, is better, as this gives, although with less precision than the sextant, its angles in azimuth.

Fig. 301.—Interior construction of box sextant with supplementary arc.

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676.—Box Sextant with Supplementary Arc.—This sextant is preferred by many because of its more extended use. It is complete as an ordinary sextant for angles up to 120°; but if it be thought desirable to extend the angles to 220°—by a single observation this may be done. The ordinary arrangement of the box sextant just described is left intact and forms the upper part of the instrument. This arrangement, as in the box sextant, is attached entirely to the face or arc plate, the only difference being that the index glass is made of less depth. For the supplementary arc arrangement a mirror is fixed upon the lower or sole plate exactly under the position of the index glass. This mirror is termed the supplementary index glass. The position of the face of the index glass is at right angles to the face of the ordinary index glass when the index is at zero. The arrangement of glasses is shown Fig. 301: MM' index glasses. The supplementary angle is read through a separate pin-hole sight which is placed at about 90° from the pin-hole sight of the proper sextant and a little lower down on the rim. The arc of this sextant reads in the ordinary manner, left to right, to an inner circle of figures for angles from 0° to 130°. The supplementary arc reads by the same vernier, and is figured in the same manner at the tens; but it reads into an outer circle of figures which progress in the reverse direction, that is, right to left. The readings of the supplementary arc are from 90° to 220°, so that for a certain range, that is, for angles from 90° to 130°, these may be taken either by direct arc or by supplementary arc. The supplementary angle is taken by means of the coincident images of two reflections, one from the index glass and one from the supplementary index glass, and not by one direct and one reflected image as in the sextant proper.

Figs. 302, 303.—Diagram of supplementary arc sextant.

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677.—Theory of Supplementary Angles to the Sextant.—For the measurement of these angles we have to consider direct reflections only of two reflecting planes placed one above the other nearly in contact, so that the images projected from both planes may reach the eye superimposed. Let Fig. 302 II' be the surface of a mirror (index glass) which is movable to any angle in relation to the face of the mirror SS' (supplementary index). For demonstration of the principle these mirrors are shown in this diagram at 90° to each other; therefore coincident reflections will be at 90° + 90° = 180°. Let the lines FC and BC form a right line (180°); F fore sight and B back sight. An object at F would be reflected from the mirror II' to the eye at E, the angles FCI and ECI' being equal. Another object at B reflected from the face of the mirror SS' would also reach the eye at E, the angles BCS' and ECS' being equal. And as the angles FCI and BCI' are equal in crossing a right line, the line FCB must be also a right line (180°) which is indicated by the angle of coincidence of the two reflections to E. The positions of the reflections are shown as angular measurements upon the graduated arc.

678.—In Fig. 303 let SS' remain as before, angle BCE will remain as shown in both figures. Move the index glass from the position II' of Fig. 302 to the position JJ' of Fig. 303, so that after this movement the eye at E would receive the image of an object at a new position F' as reflected from the mirror JJ', F'CJ and ECJ' being equal. In this process, as the reflector JJ' in the angle ICJ would have moved half the angle JCF, the record of this movement upon the index, which moves with JJ', is at the same time double the true angular difference, as with the sextant proper fully described, the graduations being in both cases the same pro ratÂ. The increase of angle is taken supplementary to the angle given by the first reflection, by addition to this angle in a direction right to left from the right line of the former sight EC; consequently this increase is read backward on the sextant, that is, right to left, and is indicated by the outer line of numerals.

679.—Manufacture.—The general structure of this instrument is nearly the same as the ordinary box sextant, except the parts just referred to. The supplementary index glass is an ordinary mirror similar to the index glass but of only ¼ inch in depth: it is mounted in the same way. Its adjustments are similar to the horizon glass in kind, but there are no exterior screws, this glass being permanently fixed by the maker. Opposite the supplementary index glass a wide window is cut through the rim of the case near the sole plate to take sight of the object at angles exceeding 120°, so that in this sextant two large windows are cut out opposite to each other. The diameter of this sextant is 3 inches; the exterior depth about 1-5/8 inches, that is, 1/8 inch deeper than the ordinary box sextant. It weighs about 20 oz. It is carried in a solid leather case with strap to pass over the shoulder.

680.—Examination and Adjustment.—Examination will be nearly the same as for the common box sextant. The most important point is that the readings taken within both arcs should be alike, assuming, which is necessary, that the part comprising the sextant proper is perfectly adjusted. Thus there is a 90° on both direct and reverse arcs. The 90° may be measured by any pair of objects on the direct arc, and afterwards compared by shifting the index to the 90°, on the supplementary arc. If no object be found at 90°, then 95° 30' or any other quantity may be compared. It is also well to compare readings at or about 120° on both arcs. The 90° and 120° fall in the same position in the reading, and this checks any error in either. If the adjustment be not fairly perfect, the instrument should be returned to the maker. Indeed, this sextant would be better without any external means of adjustment, leaving these to be made by the optician in such a permanent form that they will not be liable to change. It is, as the plain box sextant, exceptionally protected from accident.

681.—In using this instrument the arc up to 120° is taken exactly as with the plain box sextant. Beyond 120° the sextant is shifted to take sight through the supplementary pin-hole, being particular to observe that the pinion is now turned the reverse way to increase the angle, and that the vernier reads for the supplementary arc right to left. It is in this reversing, if not carefully performed, that a little difficulty is experienced in using this instrument.

682.—Box Sextant, with Continuous Arc to 240°.—This instrument is an improvement by the author upon one originally designed by Mr. W. Franklin. The reading is taken continuously from the same sight-hole and by the same arc, and in a direct manner without any reversal for part of the arc. This sextant reads with certainty to 240°.

683.—In the construction of this sextant there are two horizon glasses superimposed one above the other and crossing each other, with faces which are adjustable for perpendicularity at an angle of 120°. The horizon glass is divided top from bottom by a clear band cut through it, as in the old form of back-sight nautical sextants. One of the wide glasses reflects into the upper, and the other into the lower mirror of the horizon glass. The pin-hole sight or the telescope is placed in the same position as in the plain box sextant described. The horizon glass is fixed and both the index mirrors adjust to angular positions, or one index glass only and the horizon glass is adjusted, this arrangement being optional. The arc is graduated as the plain box sextant, but it reads with two rows of figures from 0° to 120°, and from 120° to 240°, the 0° of the under line being under 120° of the upper. When the arc is set to zero the index glasses are in such a position that the direct vision and the reflection as seen in the upper mirror of the horizon glass are coincident for direct images, as at the zero of the plain sextant, but at this point the lower mirror of the horizon glass reflects to the eye an object at 120°. When the index is moved forward the angles continue onward, reflected from both glasses, so that the upper reads on 10°, 20°, 30°, etc., whereas the lower read 130°, 140°, 150°, etc.; so that if the objects desired to be triangulated are under 120° the coincidence is seen in the upper mirror, and if over this in the lower, the great distance of 120° apart of the angles preventing the risk of accidentally taking the one for the other. In the compact form of a box sextant this instrument embraces the uses of the ordinary reflecting circle of double the diameter, due to the entire circle graduation; and the range is sufficient, as beyond 240° the head materially interferes with observation. The size and weight of the instrument are generally but little over that of the plain box sextant. The adjustments are made permanently by the maker. The use of this instrument is fully inferred from the description given. The construction is shown in Fig. 304, E place of the eye with direct ray through the horizon glass H to O. The index glass I is that of the ordinary sextant, shown by dotted lines, throwing the image of an object at P to the upper horizon glass and thence to the eye at E. B is the fixed supplementary glass with its surface at 60° to the lower horizon glass at A. The sight lines from an object at Q are reflected from B to A and thence to E. A spring arrangement shown SS with a milled head underneath permits the lower glass A to be drawn down to convert the instrument into a simple box sextant.

Fig. 304.—Stanley's continuous arc box sextant.

Fig. 305.—Section of supplementary horizon arrangement.

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684.—Details of Spring Arrangement to the supplementary horizon glass are shown in Fig. 305 full size in section. The springs SS in Fig. 304 and S Fig. 305 form two points of support to the horizon glass, the silvered face of which is shown at A. A third point of contact is near D, placed in the centre of the end of the supporting plate for the horizon glass. When the screw R, which is placed in a loose fitting, is released, the springs bring the supporting plate tight up to D and hold the horizon glass firmly in an elevated position. When the screw R is tightened it brings this glass down. The horizon glass is adjusted over a rocking centre by the screws CC'. A screw and collar b prevent the loss of the screw R. By this arrangement the horizon glass is brought in or out of the field of view, in order to use the supplementary arc or for leaving it as a plain sextant.

Fig. 306.—Stanley's portable surveying sextant.

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685.—Open Surveying Sextants, similar to nautical sextants but generally smaller and of stronger construction, preceded the box sextant, and are still used to a limited extent upon the Continent, particularly with some form of supplementary arc, or arrangement to produce a large part of the reflecting circle. These forms are also occasionally revived by the opticians of our own country. The reason of this is easily seen. To the optician who lives in a town, moves on a level surface, and has comfortably warm hands, even in the winter, to hold and move the separate parts of an instrument, the open sextant appears the most perfect, as he can get at every part of it easily to clean and adjust. The surveyor takes another view of the subject. He is exposed in the open country to all weathers and all difficulties of movement over the land; therefore that form of instrument which is best protected and least liable to injury by a fall will be sure to be popular with him. It is upon these conditions the box sextant of some form is generally preferred.

A handy form of portable surveying sextant has been devised by the author and is shown at Fig. 306.

The arc is of 4 inches radius and is divided on silver to read 20, is complete with shades and telescope and packs into a case 7 × 6 × 2½ inches.

Fig. 307.—Optical square.

Fig. 308.—Double optical square.

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686.—Optical Square.—This extremely handy little instrument is invaluable for taking offsets in chaining for any irregularity or obliquity to the right line in the boundaries of fields, hedgerows, fences, streams, etc., giving as it does instantly at sight a right angle to any object that may be sighted on either hand. The instrument is optically constructed exactly as a box sextant; but the glasses are fixed with their faces permanently at the angle of 45° to each other, by which means the reflection of 90° is truly given on principles fully discussed at the commencement of this chapter. This instrument being made very small, that is, 2 inches or less in diameter, it is found most convenient for manipulation to place the adjustments to the larger glass, that is, the index glass. The horizon glass, Fig. 307, h is therefore fixed firmly, like the index glass of the box sextant, by two screws to the sole plate. The index glass i is held and adjusted in exactly the same manner as the horizon glass of the box sextant, as shown in detail, Figs. 299, 300, the only difference being that the frame which holds the glass is made of the entire height. The rim of the case of the optical square is formed of a short length, 3/8 inch to 5/8 inch, of a pair of telescope tubes which slide easily together. One of these is attached to the sole plate and the other to the cover, so that at first they close together as a box and lid. All the openings required for sight, as Fig. 307 at Q for horizon sight, o for index sight, and e for pin-hole or eye sight, are cut through the two tubes.

Fig. 309.—Optical square.

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687.—The inner case is cut in the plane of some part of the circumference of the instrument from a pin-hole into a bayonet notch, made with a horizontal slot for the two cases to revolve upon each other upon a pin, sufficiently to close and open the sight holes. This plan secures the instrument from any intrusion of dust when it is closed and out of use. An adjusting key is placed in the case, held by a tube or stud at the position k. The weight of the entire instrument is about 4 oz. if of ordinary make; but smaller ones are made in German-silver or silver, 1¼ inches diameter, 3/8 inch thick, weighing under 2 oz. These latter are very convenient for the waistcoat pocket, and are equally as exact as the larger instruments. Fig. 309 shows the general outward appearance of the optical square.

688.—Examination and Adjustment of the Optical Square.—Place two pickets in an open space at a distance apart, the further the better. Range an intermediate short picket in right line with these or the top of a stake the height of the eye, or what is better still, if at hand, the top of a tripod stand. Place the optical square over the intermediate station or tripod. Place another picket, which we will distinguish as the 90° picket, at a distance, and make this appear in the optical square coincident by reflection with the direct sight of one of the pickets in the right line from our station. Turn the optical square right over on its place, and looking in the opposite direction take a sight at the other right line picket and observe the 90° picket. If this still appears coincident with the direct line in reflection the optical square is in perfect adjustment. If it does not appear so, half the difference must be adjusted by means of the key taken from the interior of the case and placed on the square at k, Fig. 307, and this observation repeated until the 90° is correct.

689.—In Using the Optical Square it is customary to walk along the chain line at about the desired position for taking an offset, looking by direct vision through the plain part of the horizon glass h at a fore sight object until the required object is sighted by reflection at right angles to this, where it appears by coincidence of image with the fore sight. The heel of the forward foot in stepping indicates fairly the vertical position of the optical square; but some surveyors prefer the use of a drop arrow to fix the point. The offset is then chained in the line.

690.—Double Optical Square.—This instrument is exactly what its title indicates, that is two optical squares, the one placed exactly over the other, the one reflecting to the right hand and the other to the left. A simpler name, however, would be an optical cross. This arrangement of reflectors greatly extends it use. First, as regards the 90°, this need not depend in any way upon the position of the observer, as two objects may be observed, one to the right and one to the left, to appear to cut the direct forward line of sight, and therefore to cut the base line at the exact position of the instrument at right angles to it. Secondly, an intermediate station can be found in direct line between any two points, as the 90° + 90° forms this line.

691.—The arrangement of the optical part of the instrument is shown Fig. 308. The two index glasses CD are fixed at equal angles to the direct line of sight EO. The two horizon glasses AB are superimposed with the interval of a small space, 1/16 inch, between them. The horizon glasses are each separately adjusted so that their reflecting planes are respectively 45° to the index glass from which they receive the reflections. The diameter of the instrument as usually made is about 2¼ inches; its depth 7/8 inch. The weight is about 9 oz. It is generally carried in a light, solid leather, sling case. Total weight with instrument, 12 oz.

692.—Examination and Adjustment of the Double Optical Square.—1. Place the instrument, as already described for the optical square, at a station intermediate between two pickets. Examine the right angles, first looking towards one picket and then towards the other from the same position, as with the optical square, turning it over for this examination. 2. Turn the instrument half round and examine it this way also by turning it over again in like manner. Adjust either horizon glass if required. 3. Now take the position for the eye of the former 90° and see whether the extreme pickets appear in true position by the exact coincidence of their images at 180°. 4. Do this again, facing the opposite way and turning the instrument half round. If the extreme pickets still range in line from the central station the adjustment is perfect. If they do not do so half the error must be corrected by returning to the first and second adjustments to find out between which pair of mirrors it lies. For this adjustment the instrument is much better to be placed upon the top of a tripod, as the position of the axis should remain fixed after turning it over or changing the direction of the instrument. It is only from severe accident that the maker's adjustment will be disturbed.

693.—Apomecometer.—This little instrument, the invention of Mr. R. C. Millar, is intended to measure the height of buildings, trees, etc., by measuring the distance from the vertical upon the surface of the ground. It performs one of the functions of the box sextant in the same manner as the optical square, that is, to measure a single angle by reflection. The angle measured is 45°, consequently by measuring a space upon level ground up to a vertical, the vertical will be known, this being equal to the horizontal. Of course this will always be approximate, as the ground will seldom be truly level; but by taking a position, even on an incline, as nearly as possible level with the object, a very fair estimate may be made. Horizontal distances may be measured in the same manner from a perpendicular to any line.

694.—The instrument is constructed in exactly the same manner as the optical square just described as regards its mirrors and its adjustments, but the faces of the mirrors are fixed at the angle of 22° 30', so as to give a reflection of 45°, upon principles fully discussed. In Fig. 310, A is the index glass, B the horizon glass, E the pin-hole sight. There is a window opposite the index glass, and one behind the horizon glass, each sufficient to take in a wide field of view at about 45° and in the direct line E to H. These windows close by rotation of the casing of the box, which is made as the optical square. When closed the instrument is dust-tight and may be carried in the waistcoat pocket loose, or in a light snap leather case. Its size is 1¼ inches diameter, 3/8 inch in thickness, weight 2 oz. in German silver.

695.—The Use of the Apomecometer.—To measure the altitude of a building the open side nearest level is selected, and a station for observation is taken which is at a distance thought to be approximate to the height. The instrument is held edgewise with the pin-hole sight to the eye, and the reflection of a point of the building about level with the eye is observed by direct vision through the instrument. At the same time there will appear a reflection of the summit of the building. If we now walk backwards or forwards, as the case demands, keeping sight of a level object, as for instance in Fig. 311 the plinth of a building, then at a certain point the summit of the building will appear by coincident reflection. The height of the object will be the same as the distance plus the height of the observer's eye. This distance may be measured on the ground, or if a rough estimate is sufficient it may be stepped, the principle of which is shown by Fig. 310 in the line OH, being equal to FH. If a part of an object is required to be measured such part may be taken on the horizontal plane, as for instance the height of the figure in Fig. 311, by ab being = ed, as the base ab can easily be measured. An approximate may be found by dropping a small pebble at a and at b and then measuring the distance apart of these pebbles.

Fig. 310.—Optical details of the apomecometer.

Fig. 311.—Scheme for measuring heights.

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696.—The distance of an inaccessible object may be measured, as for instance a buoy at sea, by measuring in any straight line double the distance and taking equal angles thereto by the apomecometer on any direct line. An approximate idea may be formed by walking over measuring points. As for instance, b being a buoy at sea, Fig. 312, walk from e, at which a walking-stick may be set up, towards an object o. At E the buoy and object o will appear to be coincident. Then drop a stone or make a mark directly under the instrument. Walk on till beyond E' and turn to face e. Now in returning, the buoy and the object e will appear coincident at E'. The distance EE' is double that of the intermediate a to b.

Fig. 312.—Scheme for measuring distances.

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