The necessity of seeing the horizon, in order to find the latitude and longitude of a ship at sea, has generally precluded the taking of observations of altitude at night when the number of celestial bodies shining in the firmament is the greatest and would present the most numerous opportunities for determining geographical position if the altitude could be measured without the help of the sea horizon. And even during the daytime navigators are often sensible of this inconvenience on account of the obscuration of the horizon by haze or fog while the luminary continues to be visible in the sky.
In 1733, two years after he had invented the reflecting sextant now in universal use among navigators for taking astronomical observations at sea, John Hadley brought into notice the instrument, illustrated in Fig. 1 for supplying the deficiency of his sextant when the horizon is not visible, by mounting a sensitive spirit-level on a graduated quadrant fitted with a telescope, in such a manner that, when the level is nearly horizontal, the observer focuses the telescope on the celestial body and at the same time closes a stop-cock in the level-vial. Then the reading of the level-scale furnishes a correction to be added to or subtracted from the reading of the quadrant according to which side of the zero of the level-scale the center of the liquid column is displayed. In the Philosophical Transactions of the Royal Society (London), No. 430, for the months of November and December, 1733, Hadley describes the manner of observing with this instrument as follows:
Holding the quadrant in a vertical position, with that limb to which the level is fixed parallel to the horizon, raise the index to some division of the arc, as near as you can to the true altitude of the object; which is supposed to be near the meridian, and consequently to alter its altitude but slowly; then turning the key of the stop-cock so as to let the spirit of wine pass through the small hole in it, keep the image of the object as close to the thread on the vane as you can, endeavoring that the unavoidable vibrations of it above and below the thread, may be equal, both in respect of their length and the swiftness of their motions. Continue this until the spirit seems quite settled to some part of the scale, and some thing longer. This it will do slowly, but without any sensible vibrations; for the stop-cock allowing it no passage through the small hole in its key, will give such a check to its motions, as not only to stop those vibrations, but also to hinder its being thrown backwards and forwards in the tube by any shocks of the instrument; and yet, as far as I have observed, will not prevent its settling (with sufficient truth, though slowly) to the lowest part of the tube. About half a minute of time or more may be necessary for this, according as the aforesaid small hole is greater or less in proportion to the bore of the tube. When you judge the spirit quite settled, turn the stop-cock again: it is of no importance that the image of the object be exactly on the thread at the instant that this is done. Observe against what degree, and part of a degree, each end of the spirit in the tube stands. If your scale be numbered like the upper one in the figure, and the quantity of spirit be exact, both ends will agree, and the degree and parts marked must be added to, or subtracted from the altitude shown by the index, according to the directions. If the ends do not exactly agree, take the mean between them. If you use the under scale, subtract the less number from the greater and add or subtract the excess, the number resulting will show the mean elevation of the index during the latter part of the observation, and will differ from the true altitude of the object about half so much as the vibrations of its image above and below the aforementioned thread on the vane fail of compensating one another during that time. If either end of the spirit leave the scale, the index must be removed three or four degrees, and the observation repeated.
Instead of the curved tubes A and B, two straight ones might be used, set together so as to make a very obtuse angle in the middle; but then it will be convenient to have the quantity of spirit more exactly fitted to the scale, because the allowing for the difference will be something more troublesome.
If the observer has an assistant to attend to the level, while he himself observes the object, the whole apparatus of the brass tube and stop-cock may be omitted, substituting in its room only a plug with a small hole in it, which may be wrapped round with a very thin slice of cork, and so thrust down into the middle of the glass tube. The cutting of the glass tube in half in the middle may likewise be avoided, if, instead of the stopcock at G, there be one fixed in one or both of the pipes I and K, to open and stop the passage of the air, having a larger hole in their keys, there being also a plug with a small hole thrust down into the middle of the tube, as before.
The bore of the small pipes I and K, and the tube H, must not be so narrow as to make it difficult to reduce the spirit into its place, if by any accident either end of it should get into them.
I have been informed that an object may be kept in view without much difficulty, even in pretty rough weather, through a telescope magnifying about ten times. Now as such telescopes seldom comprehend an area of much more than one degree in diameter, or at most 1 degree and 20 minutes, it follows that the axis of the telescope is always kept within 40 minutes at the most of the object, and that is the greatest vibration of the image above and below the thread on the vane. If this be allowed, it seems reasonable to expect that the mean of the vibrations one way should not exceed the mean of those at the other more than by about 1/5 or 1/6 part of the greatest vibrations; i. e., about 7 or 8 minutes, the half of which will be the error of observation. In still weather it will probably be much less, if the instrument be in the hands of a person moderately skilful in observing.
The modern form of this zenith-distance quadrant or clinometer possesses the advantage of bringing the images of the observed body and a level-bubble into coincidence in the field of view. The manner in which this result is effected is illustrated in Fig. 2, in which the telescope tube mounted on one of the bounding radii of the quadrant is shown to be cut away in its under part to receive, upon a perforated mirror placed in the tube, the image of the level-bubble in the level-vial attached to the index-arm. The angle of inclination of the mirror is such that when the level-bubble is in mid-position in its vial it will be visible in the eye-piece of the telescope in an upright position, while through the perforation in the mirror the cross-hairs of telescope and the image of the observed object may be seen. It is advisable first to point the telescope toward the object to be observed, bringing its image into the middle of the square formed by the perpendicular pairs of parallel cross-hairs, and then to move the index-arm until the level-bubble appears in the field of view with its limits extending to equal distances beyond the upper and lower cross-hairs. The accuracy of the observation is found to be improved when the level-bubble is allowed to run from up to down and then to approach coincidence of images by a down to up movement. In aeronautics the instrument has been considerably used on account of its applicability when the surface of the earth is shut out by clouds or darkness from the observer's view. For night use, a small electric bulb is arranged to light the level-bubble from below.
When H. M. S. Victory was lost with all on board, near the middle of the eighteenth century, there was in the ship a dynamical, artificial horizon invented by a Mr. Serson who, after having submitted his invention to tests in the presence of naval officers, had been authorized by the Admiralty to proceed to sea in that frigate to make observations with his instrument during the voyage, which were to be compared with those taken in the usual way by the ship's officers. This instrument consisted simply of a revolving, horizontal, reflecting surface. In its final form, a disc of speculum metal mounted on a pivot at its center of gravity was set in motion by means of a cord wound on a drum. After being set in motion, the top tended to maintain its spindle in a vertical direction, that is to say, the disc remained spinning in a horizontal position, and all objects at rest and reflected by its surface to an observer also at rest appeared entirely without motion. Then, by means of Hadley's reflecting sextant, the angle between the celestial body to be observed and its reflected image in the revolving speculum could be measured and would indicate twice the apparent altitude. The following description, in which reference is made to Figs. 3, 4 and 5, is from the Gentlemen's Magazine (London), 1754:
THE "SPECULUM" DESCRIBED
It is made of the metal used for reflecting telescopes; something more than three inches diameter, but no thicker than is necessary for grinding and polishing it to an exquisitely true plane, that being essential to its just performance; as also is the perpendicularity of its axe, which must be of hardened steel, whose lower end, which extends but a small matter beneath the speculum's lower surface, terminates in a cone whose point is a little rounded off. Its other end, which rises half an inch above the polished surface, is filed square. The speculum is let into the upper edge of a brass hoop, half an inch deep, and thick enough to bear being turned away thinner and thinner, in a lathe, till the conical point of the axe, or the point of rotation, be found to be precisely in the common centre of gravity of the speculum, hoop, and axe, taken all together; this being the main intention of the hoop. For if the centre of gravity be higher than the point of rotation, the top will not spin so long, and will be more easily put out of its position, tho' it will recover it again; and if the centre of gravity be below the point of rotation, the speculum will never recover its once lost horizontality, but keep in a kind of vibratory rotation, till it ceases to move.
a, Fig. 5, represents the speculum with the square end of its axe b, the other end thereof, or the point of rotation, being supposed to rest on a small piece of agate, chrystal or hardened steel, wrought and polished to a shallow concavity, and set fast into the stand dd; which piece is not visible in this Fig.; but is shown at p, Fig. 4, m being the conical termination of the axe.
The apparatus for giving the whirling motion to the speculum consists of a strong handle 1, Fig. 3, with a shouldered tenon at its lower end, to be let into the mortise e, Fig. 5, and readily taken out again, and of the two arms o, o, Fig. 3, into which is fastened the hollow brass cylinder r, within which slides a solid steel one, about half an inch of whose upper end is filed square, to fit into the square hollow of another short brass cylinder, f; and its lower end is hollowed square half an inch up, to go upon the square upper part of the axe of the speculum, as a watch key upon the square arbour of the fusee. This sliding cylinder, by means of a small helical spring near its bottom is thrust up, so that its square end rises a full inch above the top of the hollow cylinder r, and the upper arm o. A groove is wrought lengthwise on the small cylinder f, to receive a little spring fastened by its upper end, but loose at the other, with a small catch h at its extremity.
When the speculum is to be set a going, the conical end of its axe is placed in the centre or middle of the polished concave, and the handle l fixed on. Then must the cylinder f be put on upon the square broach k, and about 3 quarters of a yard of strong ribband must be coiled round the said cylinder pretty tight, by turning round the handle of the broach, whereby all but the catch end of the spring will be buried in its groove. The ribband being secured on, by sticking a little pin in it, the small cylinder must be taken from the broach, and thrust on upon the square end of the solid cylinder, and both together pressed strongly downwards by the thumb on top, till the hollow square of the solid cylinder embraces the axe of the speculum, and the little catch lays hold of the top of the hollow cylinder, and keeps all confin'd. Lastly, the pin being taken out of the ribband, it must be enrolled a turn or two, and laid fast hold of by the operator's right hand; and the bottom of the handle, l, together with the stand d, grasped and firmly held down with his left; when giving a smart and continued tug, he whirls the top violently round and wholly uncoils and disengages the ribbands; at which instant the catch flies back, and suffers the helical spring to push up the sliding solid-cylinder, and so leaves the top to spin at freedom.
THE MANNER OF OBSERVING WITH THE "SPECULUM" AT SEA
The handle l being remov'd, and the top spinning briskly must, in case it blows hard, be defended by a covering, two of whose sides are glass planes well polish'd, perfectly parallel and joined together at right angles, like the ridge of a house; but if it be calm, such covering will be better omitted. The observer places himself so as to see the image of the sun in the speculum; then looking through the sight of his quadrant, he moves the index till the image of the sun, reflected by the speculum of the quadrant, is perfectly united with this image reflected by the whirling speculum; and then, as has been said before, the index Q, Fig. 5, shows on the limb, an angle equal to the angle Sas, which is the double of the sun's apparent altitude above the true horizon.
After Serson perished, it is not known that further investigations were made in the direction which he had proposed until the latter part of the nineteenth century when Admiral Fleuriais designed a gyroscopic horizon as an attachment to the frame of the sextant of reflection. As with Serson's horizon, the basic principle is the tendency of a top suspended at its center of gravity to maintain its axis of rotation in a vertical position, but the mechanical details and methods of observation are quite different. Fleuriais reduced the size of the rotating part to about 10 centimeters in diameter, and, in order to eliminate the effects of air resistance, he provided for spinning the horizon in a vacuum. Figs. 6 and 7 will serve to show the arrangement of parts. The Fleuriais method of observing the altitude and eliminating the effects of the precession of the top are ingenious and unique. On opposite ends of a diameter of the rotating disc are mounted two plano-convex lenses with the convex side facing away from the center. These two lenses have equal focal distances; and they are mounted so that their plane faces are at one focal length away from each other. On one lens, lines are ruled from the middle to the top, about to minutes apart and parallel to the surface of the disc; and on the opposite lens similar lines are ruled, but from the middle to the bottom. When the top is set in rapid motion, the retina of the eye perceives these lines as a grating with 21 lines; the center line being numbered zero. Now as the top precesses, the image of the celestial body as seen in the telescope appears to oscillate between positions on the grating. These positions can be read directly; and the zero point calculated. The angle at which the top spins is then known and the necessary correction made to the reading on the limb of the sextant. An example of the use of the grating is given herewith. Such observations must take into account the only case of practical importance; namely, that of a pivot that is rounded. In this case the spindle will describe an equiangular spiral, and, therefore, the readings on one side of the zero line will not equal those on the opposite side.
Example.—Observation of the altitude of the moon:
Let H4 = instrumental altitude.
Ho=the reading of the limb of the sextant.
l(x)=reading on the grating.
l1, l2, l3....etc.=limits of excursions of the image of the moon on the grating.
The theory of the gyroscopic horizon is based upon the equations of motion of a rigid body by applying them to the motion of a rotating body suspended at its center of gravity. The equations reduce down to the following:
Therefore, the time of one-half revolution of precession is independent of the mass of the top and of the inclination to the vertical of the axis of the top.
The theory must take into consideration two main cases: Firstly, where the top is spinning on a stationary point, and secondly, where the effects of the rotation of the earth must be taken into account. Again, under each of these heads, the nature of the pivot must be considered, and the velocity of the observed body must be taken into account.
When the support is motionless and the top is spinning on a sharp pivot, the altitude is given by
where oa' and oa are the excursions of the image on the grating.
The case applies when the observed body is motionless, as a star on the meridian.
When the star is in motion, by using the mean value we do not get exact results, but, provided that the minimum inclination of the axis of the top is 1°, the results will be close enough for practical purposes. This conclusion can be deduced from the equations given above.
In the case of a rounded pivot, the spindle describes an equi-angular spiral instead of a circle, as in the case of the sharp pivot.
Then, referring to the diagram,
In regard to the influence of the rotation of the earth, the gyro-scope in a vacuum turns from right to left of the observer when he stands with his head toward the pole of the trace of the spindle of the top. Furthermore, the center of gravity is below the point of support. The movement of precession would then take place from left to right of an observer having his head toward the zenith.
This motion around the axis of the earth combined with the precession of the top will give a certain absolute motion of the spindle, which may be illustrated by the accompanying diagram. The first precession is caused by gravity and may be drawn by OV. The second precession is caused by rotation of the earth and takes place in the direction of the axis of the earth. These two forces are proportional to the velocities of the corresponding precessions.
Let the first force be represented by OV and the second in the direction of the axis of the earth by OS, then the resultant will be OV’. The numerical values of the forces will be
The precession of the top will actually take place around OV’.
This correction must be applied according to the way the top turns. If it turns from left to right, the observed altitude facing northward, will be smaller; and, if to the southward, the observed altitude will be larger. This correction will also be greatest when the observation is made on the meridian and on the equator.
The motions of the observer also have effects upon the top, which in turn will affect the reading as taken on the grating.
If the top precesses around OV and if the axis is accelerated along OU there will be a precession around OU as an axis. This combination of precessions will make an actual precession around OT.
The practical effects of the motion of the observer may be classed under the following heads:
1. When the ship is stopped and the motion is irregular, the error will be very small.
2. When the ship is pitching and rolling not very violently, then the greatest error will be 3'.
3. In the condition of most violent pitching and rolling, the error will be between 5' and 6', but this condition is seldom met with.
The corrections to apply to the observed altitude may then be expressed by the following formula:
Ha=Hi+e+i+c
in which
Ha=apparent altitude.
Hi= instrumental altitude.
e =correction for instrumental error.
i= correction for rotation of earth. .
c=correction for error affecting all measured height.
In the Annales Hydrographiques, Paris, 1901, Captain Guyou has provided tables and diagrams which facilitate the reduction of observations made by the gyroscope-sextant; and an improved instrument, illustrated in Figs. 11, 12 and 13, was produced in 1904 by Ponthus and Therrode, in which the gyroscopic horizon is rotated on its axis by a current of air blown against the vanes on its periphery, and the grating, illuminated by an electric lamp, is formed by transparent lines ruled on a black surface with a lens placed opposite. A systematic and scientific examination of this instrument has been made, of which an account is given by M. Favé, hydrographic engineer-in-chief, French Navy, in the Annales Hydrographiques for 1904. After many trials, it has been shown that the greatest liability to error in a perfectly new instrument of this type is 2', although after much usage larger errors are found to creep in.
There may, of course, be other methods of approach toward the evolution of a gyroscopic horizon besides those of Serson and of Fleuriais; but in relation to these two—one employing a rotating speculum tending to maintain a horizontal reflecting surface in which the angle between the observed celestial body and its reflected image could be obtained, and the other employing a small top whose angle of precession could be measured for the purpose of correcting the instrumental reading of the altitude—practically nothing has been done to improve the former which appears to be free of limitations which have marked the development of the latter. It is a matter of great difficulty to read directly to minutes on a scale of small diameter, as of a top on a sextant, whatever combination of lenses might be used; and any method by which such a top can be set in motion in a vacuum precludes that control of its speed by the operator by which alone he may obtain the most stable position under varying circumstances.