Appendices
I. Details of the Theodolite Observations at Stonehenge
The instrument chiefly employed was a six-inch transit theodolite by Cooke with verniers reading to 20″ in altitude and azimuth. Most of the observations were made at two points very near the axis, which may be designated by a, b. Station a was at a distance of 61 feet to the south-west of the centre of the temple, and b 364 feet to the north-east. The distance from the centre of Stonehenge to Salisbury Spire being 41,981 feet, the calculated corrections for parallax at the points of observation with reference to Salisbury Spire are:—
| Station | a | + | 4 | ′ | 12 | ″. |
| „ | b | - | 25 | ′ | 20 | ″. |
(1) Relative Azimuths.—Theodolite at station a—
| Salisbury Spire | 0 | ° | 0 | ′ | 0 | ″ |
| N. side of opening in N.E. trilithon of the external ring | 237 | 27 | 40 | |||
| Tree in middle of clump on Sidbury Hill | 237 | 40 | 20 | |||
| Highest point of Friar’s Heel | 239 | 47 | 25 | |||
| S. side of opening in N.E. trilithon | 240 | 14 | 40 | |||
| Middle of opening in N.E. trilithon | 238 | 51 | 10 |
(2) Absolute Azimuths.—All the azimuths were referred to that of Salisbury Spire, the azimuth of which was determined by observations of the Sun and Polaris.
(a) Observation of Sun, June 23, 1901, 3.30-3.40 P.M.
| Mean of observed altitudes of Sun | 41 | ° | 26 | ′ | 35 | ″ | ||||||
| Refraction | - | 1 | ′ | 4 | ″ | } | 0 | 0 | 58 | |||
| Parallax | + | 6 | ||||||||||
| True altitude of Sun’s centre | 41 | 25 | 37 | |||||||||
Latitude = 51° 10′ 42″. Sun’s declination = 23° 26′ 43″. Using the formula cos2 1⁄2 A = sin 1⁄2(Δ + c - z) sin 1⁄2(Δ + z - c) sin c. sin z
where A = azimuth from south, Δ = polar distance,
c = co-latitude, and z = zenith distance,
we get
| Azimuth of Sun | S. | 75 | ° | 30 | ′ | 30 | ″ | W. |
| Mean circle reading on Sun | 84 | 38 | 35 | |||||
| Azimuth of Salisbury Spire | S. | 9 | 8 | 5 | E. | |||
(b) Observations of Polaris.—June 23, 1901. Time of greatest easterly elongation, calculated by formula cos h = tan φ cot δ, is G.M.T. 1.34 A.M.
Azimuth at greatest easterly elongation, calculated by the formula sin A = cos δ sec φ
is 181° 57′ 0″ from south.
| Observed maximum reading of circle | 256 | ° | 33 | ′ | 0 | ″ | ||
| True azimuth of star | 181 | 57 | 0 | |||||
| Meridian (S.) reading of circle | 74 | 36 | 0 | |||||
| Circle reading on Salisbury Spire | 65 | 28 | 0 | |||||
| Azimuth of Salisbury Spire | S. | 9 | 8 | 0 | E. |
The mean of the two determinations gives for the azimuth of Salisbury Spire S. 9° 8′ 2″ E. This result agrees well with the value of the azimuth communicated by the Ordnance Survey Office, namely, 9° 4′ 8″ from the centre of the circle, which being corrected by +4′ 12″ for the position of station a, is increased to 9° 8′ 20″.
Hence, from the point of observation a, 9° 8′ 20″ has been adopted as the azimuth of Salisbury Spire.
We thus get the following absolute values of the principal azimuths from the point a:
| Highest point of Friar’s Heel | 239 | ° | 47 | ′ | 25 | ″ | ||
| -9 | 8 | 20 | ||||||
| 230 | 39 | 5 | ||||||
| or | N. | 50 | 39 | 5 | E. | |||
| Middle of opening in N.E. trilithon | 238 | 51 | 10 | |||||
| -9 | 8 | 20 | ||||||
| 229 | 42 | 50 | ||||||
| or | N. | 49 | 42 | 50 | E. |
The difference of 81⁄2′ between this and the assumed axis 49° 34′ 18″ is so slight that considering the indirect method which has necessarily been employed in determining the axis of the temple from the position of the leaning stone, and the want of verticality, parallelism and straightness of the inner surfaces of the opening in the N.E. trilithon, we are justified in adopting the azimuth of the avenue as that of the temple.
Next, with regard to the determination of the azimuth of the avenue as indicated by the line of pegs to which reference is made on p. 65. The small angle between the nearest pegs A and B (which are supposed to be parallel to the axis of the avenue), observed from station a, was measured, and the corresponding calculated correction was applied to the ascertained true bearing of the more distant peg B.
Thus
| True bearing of peg B = | 238 | ° | 35 | ′ | 0 | ″ | ||
| Calculated correction to peg A = | 0 | 12 | 8 | |||||
| True bearing of line AB | 238 | 47 | 8 | |||||
| Bearing of Salisbury Spire | 189 | 8 | 20 | |||||
| True bearing of a line parallel to the axis of near part of avenue | N. | 49 | 38 | 48 | E. |
The mean of the three independent determinations by another observer was 49° 39′ 6″.
The calculated bearing of the more distant part of the axis of the avenue determined in the same manner by observations from station b is 49° 32′ 54″. The mean of the two, namely, 49° 35′ 51″, justifies the adoption of the value 49° 34′ 18″ as given by the Ordnance Survey for the straight line from Stonehenge to Sidbury Hill.
(3) Observation of Sunrise.—On the morning of June 25, 1901, sunrise was observed from station a, and a setting made as nearly as possible on the middle of the visible segment as soon as could be done after the Sun appeared.
The telescope was then set on the highest point of the Friar’s Heel, and the latter was found to be 8′ 40″ south of the Sun.
| Sun’s declination at time of observation | 23 | ° | 25 | ′ | 5 | ″ | |||||||
| Elevation of horizon at point of sunrise | 0 | 35 | 48 | ||||||||||
| Assuming 2′ vertical of Sun to have been visible at observation, we have apparent altitude of Sun’s upper limb | 0 | 37 | 48 | ||||||||||
| Refraction | - | 27 | ′ | 27 | ″ | } | -0 | 27 | 18 | ||||
| Parallax | + | 0 | 9 | ||||||||||
| True altitude of upper limb | 0 | 10 | 30 | ||||||||||
| Sun’s semi-diameter | 0 | 15 | 46 | ||||||||||
| True altitude of Sun’s centre | -0 | 5 | 16 | ||||||||||
| From this it results that the true azimuth of the Sun at the time of observation | = | N. | 50 | ° | 30 | ′ | 54 | ″ | E. | ||||
| And since azimuth of Friar’s Heel | = | 50 | 39 | 5 | |||||||||
| 2′ of sunrise should be N. of Friar’s Heel | 0 | 8 | 11 | ||||||||||
| Observed difference of azimuth | = | 0 | 8 | 40 | |||||||||
| Observed - calculated | = | 0 | 0 | 29 | |||||||||
The observation thus agrees with calculation, if we suppose about 2′ of the Sun’s limb to have been above the horizon when it was made, and therefore substantially confirms the azimuth above given of the Friar’s Heel and generally the data adopted.
II. Hints on Making, and Method of Reducing, the Field Observations
It will probably be found useful if I give here a few hints as to the precautions which must be taken in making the field observations and an example of their reduction to an astronomical basis.
For the azimuths of the sight-lines the investigator of these monuments cannot do better than use the 25-inch, or 6-inch, maps published by the Ordnance Survey. Their accuracy is of a very high order and is not likely to be exceeded, even if approached, by any casual observer having to make his own special arrangements for correct time before he can begin his surveying work.
In some cases, however, it may be found that the Survey has not included every outstanding stone which may be found by an investigator on making a careful search; many of the stones are covered by gorse, &c., and are not, therefore, easily found.
In such cases the azimuth of some object that is marked on the map should be taken as a reference line and the difference of azimuth between that and the unmarked objects determined. By this means the azimuths of all the sight-lines may be obtained.
When using the 25-inch maps for determining azimuths it must be borne in mind that the side-lines are not, necessarily, due north and south. The Director-General of the Ordnance Survey, Southampton, will probably on application state the correction to be applied to the azimuths on this account, and this should be applied, of course, to each of the values obtained.
If for any reason it is found necessary or desirable to make observations of the azimuths independently of the Ordnance Survey, full instructions as to the method of procedure may be found in an inexpensive instruction book133 issued by the Board of Education. The instructions given on p. 49, § 3, are most generally applicable, and the form on p. 76 will be found very handy for recording and reducing the observations.
In making observations of the angular elevation of the horizon a good theodolite is essential. Both verniers should be read, the mean taken, and then the telescope should be reversed in its Ys, reset, and both readings taken again. One setting and reading are of little use.
The Ordnance Survey maps may also be employed in a preliminary reconnaissance to obtain approximate values of the horizon elevations. This may be done by measuring the distances and contour-lines shown on the one-inch maps. This method, however, is only very roughly approximate owing to the fact that sharp but very local elevations close to the monuments may not appear on these maps and yet be of sufficient magnitude to cause large errors in the results.
Where trees, houses, &c., top the horizon, they should, of course, be neglected and the elevation of the ground level, at that spot, taken. Should the top of the azimuth mark (stone, &c.) show above the actual horizon, its elevation should be recorded and not that of the horizon.
Having measured the angular elevation of the horizon along the sight-line, it is necessary to convert this into actual zenith distance and to apply the refraction correction before the computations of declination can be made.
The process of doing this and of calculating the declination will be gathered from the examples given below:—
Data.
Monument:—E. circle Tregeseal, lat. 50° 8′ N. i.e. colat = 39° 52′.
Alignment. Centre of circle to Longstone.
Az. (from 25″ Ordnance Map). N. 66° 38′ E.
Elevation of horizon (measured) 2° 10′.
Reference to the May-Sun curve, given on p. 263, indicates that this is probably an alignment to the sunrise on May morning. Therefore, in determining the zenith distance, the correction for the sun’s semi-diameter (16′) must be taken into account, allowing that 2′ of the sun’s disc was above the horizon when the observation was made.
Zenith Distance:—
| Zenith | distance of | true | horizon | = | 90° | ||
| „ | „ | local | „ | = 90° - 2° 10′ | = | 87° | 50′ |
Bessel’s tables show that refraction, at altitude 2° 10′, raises sun 17′. If 2′ of sun’s limb is above horizon, sun’s centre is 14′ below.
∴ True zenith distance of sun’s centre = 87° 50′ + 17′ + 14′ = 88° 21′.
Declination:—
Having obtained the zenith distance, and the azimuth, the latitude being known, the N.P.D. (North Polar Distance) of the sun may be found by the following equations:—
(1) tan θ = tan z. cos A, where θ is the subsidiary angle which must be determined for the purpose of computation, z is the true zenith distance, and A is the distance from the North point.
(2) cos Δ = cos z. cos (c - θ)134 cos θ, where Δ is the N.P.D. of the celestial object, and c is the colatitude (90° - lat.) of the place of observation.
In the example taken this gives us—
(1) tan θ = tan 88° 21′. cos 66° 38′
θ = 85° 50′ 45″
(2) cos Δ = cos 88° 21′. cos (39° 52′ - 85° 50′ 45″) cos 85° 50′ 45″
Δ = 73° 57′ 50″
Declination, δ, = (90° - Δ) = 16° 2′ 10″ N.
Reference to the Nautical Almanac shows that this is the sun’s declination on May 5 and August 9. We may therefore conclude that the Long-stone was erected to mark the May sunrise, as seen from the Tregeseal Circle.
Had we been dealing with a star, instead of the sun, the only modification necessary in the process of calculating the declination would have been to omit the semi-diameter correction of 14′.
Having obtained a declination, we must refer to the curves given on pp. 115-6 in order to see if there is any star which fits it, and to find the date.
Take, for example, the case of the apex of Carn Kenidjack, as seen from the Tregeseal circle—
Az. = N. 12° 8′ E.; hill = 4° 0.′ lat. = 50° 8′.
This gives us a declination of 42° 33′ N., and a reference to the stellar-declination curves (p. 115-6) shows that Arcturus had that declination in 2330 B.C. From the table given on p. 117, we see that at that epoch Arcturus acted as warning-star for the August sun.
In cases where the elevation of the horizon is 30′, or in preliminary examinations, where it may be assumed as 30′, the refraction exactly counterbalances the hill, and therefore the true zenith distance at the moment of star-rise is 90°. Hence the N.P.D. of the star may be found from the following simple equation—
(3) cos Δ = cos A cos λ
where Δ and A have the same significance as before and λ is the latitude of the place of observation.
Notes
133 Demonstrations and Practical Work in Astronomical Physics at the Royal College of Science, South Kensington. Wyman and Sons, 1s.
134 cos (c - θ) = cos -(c - θ).