Measurement of Horizontal Angles.—Angles were measured with a 6-inch theodolite furnished with reading microscopes graduated to 10″ and permitting of reliable estimations to 1″. The angles between main points were read on four arcs to eliminate circle errors. Intersected points were observed on one arc only. The average error of closure of main triangles was 3″·3.
Field Computation and Plotting of Triangulated Points.—The triangles were computed by the ordinary method, but the angles were rounded off to 10″ to enable the sines to be taken direct from the logarithmic tables, and the logarithms were only taken to five places. The length of the sides having been thus found, the geographical positions were found by the ordinary L M Z computation, using, however, only two latitude terms and 5-place logarithms, while azimuths were only taken out to the nearest 10″. The abbreviated form of computation used will be best illustrated by an example:—[63]
Computation of Position of △ No. 260 from No. 275.
| l | = | 20042 m. | 275 | ⎰ ⎱ | φ | = | 23° 55′ 30″·6 | ||||
| Z | = | 64° 29′ 20″ | E of N. | λ | = | 34° 54′ 36″·9 | |||||
| Log l | = | 4·30194 | Log l2 | = | 8·604 | ||||||
| Log cos Z | = | 1·63416 | Log sin2 Z | = | 1·911 | ||||||
| B | 2·51194 | C | = | 9·053 | |||||||
| 2·44804 | = | 1·568 | |||||||||
| dφ, | 1st | term | = | + 280″·6 | Log l | = | 4·30194 | ||||
| 2nd | term | = | − 0″·4 | Log sin Z | = | 1·95545 | |||||
| dφ | = | 280″·2 | A′ | = | 2·50948 | ||||||
| = | 4′ 40″·2 | = | 2·76687 | ||||||||
| φ | = | 23° 55′ 30″·6 | Log cos φ′ | = | 1·96072 | ||||||
| φ′ | = | 24° 0′ 10″·8 | = | 2·80615 | |||||||
| dλ | = | 640″·0 | |||||||||
| Whence | = | 10′ 40″·0 | |||||||||
| 260 | ⎰ ⎱ | φ′ | = | 24° 0′ 10″·8 | λ | = | 34° 54′ 36″·9 | ||||
| λ′ | = | 35° 5′ 16″·9 | λ′ | = | 35° 5′ 16″·9 | ||||||
In the above, it will be noticed that the azimuth is always noted as so much east or west of north or south. If this convention be adopted, one may consider the first term of dφ as always +, and the second term will be + if the azimuth contains the word south,—if it is from the north, while the total dφ is to be added or subtracted according as one is going north or south. A somewhat similar convention is adopted in neglecting the sign of dλ till the actual addition or subtraction is made. It was found in the field that this method prevented any mistake of sign, while being much simpler to work than one involving angles greater than 90°.
The geographical coordinates thus found were plotted directly on to the plane-table sheets, on which the graticule at 10′ intervals was the first thing drawn. The odd minutes and seconds were first converted into minutes and decimals, and then into kilometres by multiplying by the factors appropriate to the latitude, so that the plotting could be done by the ordinary scale of kilometres. To avoid difficulties of paper-shrinkage, as many points as possible were plotted at the time of drawing the graticule, and in general the points had to be plotted as far ahead as possible for controlling the traversing and sketching.
Astronomical Observations.—Astronomical checks on the triangulation were obtained by observations of latitude at certain selected main stations 60-120 kilometres apart, and by azimuth observations for certain main lines.
The method used for latitude was that of observing the times of equal altitudes of three or more stars, selected as near to the meridian[64] as possible. This method presents great advantages over the usual Polaris and circummeridian altitudes, in that the observations are more easily made, and yield much more accurate results, because uncertainties in refraction are largely eliminated and the errors of circle graduation are not involved, the altitudes not being read at all.[65] The theodolite used was the same as was employed in triangulation, and the times were taken by a half chronometer watch, preferably one marking sidereal time with a rate which could be considered negligible during the hour or so occupied by the observation. The first star taken was usually Polaris, and the vertical circle was left clamped at its altitude. For the other stars, any dislevelment was corrected by touching up the levelling screws just before the instant of observation; this was found better, than taking bubble readings and correcting for slight difference of altitude.
The method which I found best in the field for reducing the observations differs somewhat from that described by Chauvenet. Assuming approximate values for the latitude and watch error, I first calculated the altitude of each star from the formula
sin h = sin φ sin ε + cos φ cos δ cos t