In Chapter 16 of my book, The Nautical Sextant, I explain some of the various ways of calibrating a sextant. For the most part, they employ methods and equipment not accessible to the amateur, but sometimes I let my amateur enthusiasms run away with me. Over a period of years I have gradually accumulated for relatively small sums versatile precision measuring devices that, with some loving attention, could be given a second lease of life. For measuring small angles with great accuracy and precision, the autocollimator is almost indispensable. About ten years ago I spotted a “Job lot of five Hilger and Watts telescopes” and thought I could see in the picture two Hilger and Watts Microptic autocollimators, less their bases, the tube of another, and an Angle Dekkor (which need not concern us further). As the price was £100, I snapped them up. I made bases for the two autocollimators and overhauled them, while the third tube now forms part of a horizontal collimator for calibrating bubble sextants. Most people will perhaps need the terms “collimator” and “autocollimator” to be explained. It comes from the Latin colineare, to make straight, but a medieval scribe probably miscopied it as collimare and so it has stayed.
This is simply a good quality lens held rigidly in a tube with, at its focus, a set of cross wires. The wires are illuminated by a lamp behind them (Figure 1). Since the wires are at the focus of the lens, the rays that make up their image as they leave the lens are parallel, and so the wires appear as if they were at infinity. It is as if a very distant object had been imported into the workshop. If the wires are made to be adjustable so that the centre line of the tube coincides with the optical axis of the lens-crosswire combination, the addition of a sensitive level to the tube gives us a means of importing a horizontal line of sight.
Imagine now that the wires are illuminated indirectly by having the light reflected on to them by a semi-reflective mirror (or ” beam splitter”) set at 45 degrees and that the emerging parallel rays (full red in Figure 2) strike a mirror (full black) that is exactly at right angles to the axis of the instrument. It is obvious that the rays will be reflected back to where they came from and through the semi-reflector. Suppose now that the mirror is tilted through a small angle, α, shown in broken black in Figure 2. The rays will be deflected through twice this angle (broken red). If an eyepiece, with a measuring scale or other device at its focus is used to examine the displacement of the reflected image of the cross wires, it will be possible to deduce this angle if the focal length, f, of the lens is known. In fact, the shift of the image will be 2f tan α. Note that the distance of the mirror from the instrument does not appear in the equation.
The Hilger and Watts Microptic autocollimator has a pair of moveable wires for measuring the deflection. The wires are moved by means of a micrometer drum that is calibrated directly in minutes and seconds of an arc, with the least division being 0.2 seconds and the total range being 10 minutes. Figure 3 shows a view through the eyepiece with the parallel setting wires deliberately offset from the reflected image by 15 seconds. The parallel wires can be centred with great precision around the image. It is easy to see an error of as little as 2 seconds, and the experienced user is said to do rather better.
At the heart of the calibrator is a Soviet SNO-T sextant stripped of shades, mirrors and telescope mounting and mounted on three stout feet which have been ground and lapped so that the axis of the index arm is at right angles to the surface on which the sextant is placed. I did not of course sacrifice one of these fine sextants; I was able to obtain the part-instrument for US$45 from a source in India.
In place of the index mirror I mounted a 6 mm-thick steel sub-plate (Figure 4) ,which has a flat, a conical hole and a vee groove in which sit the ball-ended adjusting screws of the top plate (Figure 5). The random holes betray its source as a scrap heap.
The top plate in its turn carries an assortment of conical holes, a moveable hardened steel vee groove, and flats to accommodate the legs of the sextant being calibrated. This method of mounting prevents movement in any direction except upwards, while avoiding redundancy of constraints that might introduce stresses and strains; and if the mountings are accidentally disturbed it is easy to relocate them accurately.
Finally, a counterweight slides along an arm attached to the bottom plate in order to reduce asymmetry of loading of the index arm bearing. It counters the weights of the plates themselves together with the weight of the sextant being calibrated. There is a scale on the arm to cover sextants weighing between 0.8 and 1.6 kg. Before the calibrator can be put into service it must itself be calibrated.
Calibrating the calibrator
This uses the method described by Bouillon, Delisle and Pichard in the Canadian Journal of Physics for May 1, 1976 (Vol 54, No 9, pp 917 to 927). While it is an extremely tedious method in which it is very easy to displace instruments accidentally, it does have the merit of comparing the sextant with a fundamental standard of angle, the straight line. This of course always subtends exactly 180 degrees. It requires two autocollimators (though one plus a plain collimator would do), an accurately plane surface like a large surface table, a good quality mounted auxiliary mirror and an auxiliary base for the sextant. Though the authors do not mention it, the better the quality of the index mirror, the sharper is the reflected image as seen in the autocollimator and the easier it is to make accurate settings.
Setting the standard
Before embarking on this, it is as well to overhaul the index arm bearing and micrometer mechanism to ensure that everything operates with silky smoothness and without any trace of backlash between the micrometer screw and rack. The latter is adjusted by means of a screw whose head lies under a blob of yellow paint at the rear of the mechanism.
The axes of the two autocollimators have first to be set facing each other and parallel to the table surface, and for this a good quality mirror mounted on an adjustable stand is needed. Its face should be set approximately perpendicular to the table and then it is placed between the autocollimators (Figure 7). It is viewed first through one instrument and the angle a of the reflected image of the horizontal wire on the vertical scale is noted. It is then rotated through 180 degrees and the vertical scale reading b obtained on the second instrument. If this instrument is now adjusted by an amount (a – b)/2 its axis will then be horizontal. If the other autocollimator is also brought to the same reading, then their axes will be horizontal and coincide with each other in the horizontal plane. Note this reading. Figure 7 may help to make this clearer.
Once this has been done, the autocollimators are adjusted to align the images of the vertical wires with each other. In practice, the setting wires are offset somewhat from the midline and set to embrace the image of the other’s wire, so that the direct image does not obscure the reflected images. It is convenient to set the micrometer of one to, say, 4′ 30″ and the other to 5 ′ 30″. The axes of the autocollimators are then aligned in a straight line in both the vertical and horizontal plane and neither micrometer is changed until the end of the calibration process. Blobs of plasticine help to prevent accidental movement of the micrometers. These are visible in Figure 8, below.
Setting axis and mirror
The upper plate is used as an auxiliary base for this process in which the plane of the sextant is brought parallel to the reference surface or, more exactly, that the index arm axis is brought vertical to the surface. In an undamaged sextant of good quality the two should mean practically the same thing. A dial indicator is used to bring each end and the middle of the arc into the same plane. After this, when an autocollimator is directed at the index mirror with the index at each end and at the middle of the arc, the readings of the horizontal wire on the vertical scale should be closely the same and if they are not, the base adjusting screw at the index mirror corner is adjusted until they are. The index mirror, temporarily borrowed from another sextant, can then be adjusted to make the direct and reflected images of the horizontal wire coincide, so that the mirror is then aligned with the axis of rotation. Note also that sextant legs have been borrowed too.
Dividing the standard
The sextant and one autocollimator, designated the moving autocollimator are now used as an optical divider to divide the standard of 180 degrees into segments, which must be factors of 180. Seven and a half degrees is the smallest practicable factor for my instruments, representing 15 degrees on the sextant scale, as the bases interfere with each other at smaller values. The lamp of the fixed autocollimator can be temporarily switched off, to avoid confusion of images.
The starting position is with the sextant index and micrometer set to 15 degrees exactly, using a magnifying glass to ensure accurate setting of the micrometer (Figure 8). The sextant on its base is then moved so that the reflected image of the vertical crosswire is precisely between the setting wires. Taking great care not to move the sextant, it is then set to zero, which of course then resets the mirror at an angle of 7½ degrees to its previous position. The auto collimator this time is moved to intercept the beam and to bring the image of the wire back between the setting wires.
Then the sextant is re-set to 15 degrees and rotated to bring the vertical wire back again between the setting wires, after which it is again set to zero and the autocollimator moved. After 24 movements of the sextant and 24 of the autocollimator, the latter will be alongside the fixed autocollimator at a nominal angle of 7½ degrees (Figure 9). To avoid confusion, its lamp is then switched off and the fixed autocollimator switched on.
If there is no error of the sextant segment 0 to 15 degrees, when it is brought finally to zero the index mirror should be at right angles to the axis of the fixed autocollimator. If there is an error, Ε, in the segment, plus or minus, it will have been applied to each movement. The total can then be measured by bringing the vertical wire between the setting wires of the fixed autocollimator and noting the reading, ε. But this error applies to 24 movements (180/7.5) and the actual error Ε = ε/24.
There will of course have been setting errors of the sextant, which with care can be reduced to well under 3 seconds, thanks to vernier acuity aided by a magnifying glass and the superb workmanship of the SNO-T sextant (Figure 10), and there will have been reading errors of the autocollimators, which should also have been well under 3 seconds. However, these errors will have been averaged over 24 movements, and if the errors are random rather than systematic, will tend to cancel out, improving accuracy by a factor of √24, or very nearly five.
In a separate investigation of setting errors in which I tried to determine the errors involved in repeatedly re-setting the sextant micrometer to zero, always approaching from the same direction, I found the standard deviation of thirty readings to be 0.24″. This is a measure of dispersion of results around the mean and implies that 95 percent of results are likely to be within ± 0.47 seconds, say 0.5 seconds.
- E could then of course simply be applied to each reading 15 to 30°, 30 to 45°, and so on, leaving the autocollimators where they lie at a nominal 7½° while moving the sextant, but this would not average out setting errors. Instead, I elected to measure the error for each 15 degree segment by going through the full dividing procedure, to take advantage of the averaging effect. This was very tedious, but by adopting a zombie-like state and moving in an unhurried way, I believe I have avoided gross errors, and I had to do it only once. The results are given in Table 1. The last-but-one column gives the error, plus or minus, for each 15 degrees segment and the final column gives the cumulative error from zero. These are truly remarkably small errors and a great tribute to Russian workmanship, though the results for the two other SNO-T sextants I have calibrated showed errors of between 0.7 and 14 seconds at various points on the arc.
- Calibrating a sextant
- The borrowed index mirror and legs had now to be replaced by the sub-plate and short legs mentioned in the first paragraph, and the top plate replaced on top of the sub-plate with its ball-ended feet resting in their respective places. After checking that the legs of the instrument to be calibrated are tight, it is placed on the top plate, one leg in the vee groove, one in a conical hole and one on a flat. The position of the vee groove can be altered and, within reason, new conical holes drilled to accommodate different sizes of sextant. The autocollimators have to be raised higher off the surface table so that their axes are roughly at the same height as the index mirror of the sextant. Scrap pieces of large diameter steel water pipe with the ends faced in the lathe make excellent raising blocks for this purpose. The axes of the autocollimators are then brought parallel to the reference surface as described above under Setting the standard following which one autocollimator can be returned to its box. Bouillon et al.’s paper describes a method of setting the index error to zero using the two autocollimators, but as this should be checked each time the sextant is used, I have omitted it here.
- Next the sextant frame is set parallel to the plane and the index mirror adjusted as described above under Setting axis and mirror. The sextant is then set to zero and the calibrator to 120 degrees (Figure 11), using a magnifying glass to aid acuity and approaching the setting by rotating the micrometers always in the same direction, to avoid the effects of any backlash in either instrument. If you overshoot, it is necessary to go back half a turn or so and try again. The remaining autocollimator micrometer is set to zero and then directed at the index mirror to intercept the reflected image of the vertical wire between the setting wires. The autocollimator is not moved from this position and can be guarded against minor knocks by anchoring the base with plasticine.
- Calibration can now begin. The sextant is set to 15 degrees while the calibrator is set to 105 degrees. If there is no error, this should bring the vertical wire back between the setting wires, and if there is an error, this can be measured by re-setting the wires and reading the autocollimator micrometer, also applying the calibrator error with due regard to its sign, plus or minus. The sextant is next set to 30 degrees and the calibrator to 90 degrees, and so on. For a quintant, the calibrator is set at 135 degrees when the sextant is at zero. The calibration procedure is so rapid, once the sextant has been set up, that it is easy to do a set of three observations and average the results. the results for a SNO-T sextant are given in Table 2 .
For most purposes of course, such precision is not required and results could be given to the nearest 5 or 10 seconds, or, as C Plath, Cassens and Plath and Weems and Plath did, simply state that the instrument is free of error for practical use. Nowadays, probably only “Lunartics”, those insterested in making lunar distance observations, would wish to know the actual value of the errors. However, I hope that this illustration of what is possible will be of interest to a few people and should be grateful if they would point out (kindly of course) important errors or omissions.