One of the most critical tasks carried out before construction began was the survey that determined exactly where each of the main bearings was to be located. This was carried out using surveying equipment and a process of triangulation from a set baseline. The survey, using a theodolite, was based on set survey marks at Mrs Macquarie’s point (G), in the grounds of Government House (A, B, E and F), on Kirribilli Head (H) and in Lavender Bay (J).
The triangulation process allowed the centreline of the proposed Bridge (C, D) to be located precisely, and from there the position of the main bearings was set.
In fact two different surveys were carried out, one by Dorman, Long & Co. Ltd. surveyors and the other by NSW Public Works Department surveyors. Each survey had its own baseline (AB and FE) and the resulting measurements were averaged to ensure a high level of accuracy.
The first task in the survey was to set out an accurate baseline. The two baselines were set out in the grounds of Government House using temperature-stable invar nickel steel tape:
- Baseline (AB) mean value adopted 1677.3492 ft (511.256 m)
- Baseline (EF) mean value adopted 1915.9577 ft (583.984 m)
- Baseline (JK) mean value adopted 1182.3915 ft (360.393 m)
The second task was to set up the theodolites and accurately measure the angles to the various survey marks. As described by JJ Bradfield, ‘the angles were read 24 times and the mean reading used in the calculation. With this instrument the error in closing the three angles of a triangle 180° averages ¼ of a second; an error of 1 second of arc in a mile [1.61 km] sight would amount to ¼" [6.35 mm].’ (Bradfield, cited in Mackaness, 2006, p 197).
The third task was to use trigonometry to calculate the accurate distance between ‘C’ and ‘D’. In Bradfield's words ’... the length of the centre line across the harbour ‘CD’ was determined by triangulation from each of these baselines’ to be 2268.447 ft [691.423 m]. The length of the arch span [K, L], 1650’ [502.92 m] was accurately fixed by measurement from ‘C’ and ‘D’ ’. (Bradfield, cited in Mackaness, 2006, p 197)
(The original surveys and calculations were all done long before the existence of computers, or even mechanical calculators. The surveyors and engineers used slide rules and 7-figure trigonometric and logarithm tables.)
Knowing the position of the ends of the arch, it was a simple matter to survey the centre of each main bearing on each side of the arch centreline.
Given the baseline AB = 100 m and the angles measured by surveyors.
Calculate the length of CD
The solution to this problem is rather complex and gives an insight into the mathematics that was involved — without a computer — in ensuring that the Bridge centreline was accurately defined.
AB is the baseline 100 m long. The steps in the calculation are:
Step 1 Extend the baseline (AB) upward.
Step 2 Draw a line perpendicular to AB to pass through D, intersecting AB at B'.
Step 3 Use triangles BAD and BDB' to calculate the length of BB' (60·6 m).
Step 4 Draw a line perpendicular to AB to pass through C, intersecting AB at A'.
Step 5 Use triangles BA'C and AA'C to solve for length AA' (61·93 m).
Step 6 Use trigonometry to find the lengths of A'C and DB' (118·96 m and 60·42 m).
Subtract these to find C' along A'C.
Step 7 Use Pythagoras’s theorem to find CD from the lengths of CC' and C'D.
Answer: CD = 119·4 m.
In the Bridge survey the differing heights of survey points had to be taken into account. Can you think why?