I understand that you can have a device with angular measurements for rotation and elevation, and use trigonometry to calculate the distances... but only if you have some distances to start with. How did they accurately measure the first straight-line distance over any distance significantly long enough to give usable angles? And wouldn't the error in other distances calculated from this very quickly accumulate out of control?


2 Answers 2


The device for taking horizontal and vertical angles that you mention is called a theodolite. Theodolites only started being phased out as the main surveying tool in the 1980s when total stations where introduced. Below is a Soviet theodolite from 1958, (ex Wikipedia).

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Theodolites were analogue devices and the angles measured had to be written in a notebook. Total stations were electronic devices, essentially electronic theodolites, with electronic distance measuring devices, based on infrared signals. These devices could be connected to a portable electronic memory unit with a keypad to store the measurements. The surveyor still had to manually enter a point identifier for each reading, but didn't have to enter the measured angles.

When starting a survey, a reference marker from the nation system of surveying markers closest to the surveying region was chosen as this had a known/established northing, easting and elevation. A picture of US survey marker follows (from Wikipedia).

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The theodolite would be set up and the first reading would be to the known marker peg to establish the baseline for the survey.

For very accurate surveys a surveying target, on tripod, was placed over the survey marker; either a plate with a cross on it or a short pointed rod with the point upwards. A similar target would then be placed on a temporary marker and the horizontal angle between the two targets measured. The vertical angle from theodolite's horizontal plane (in the eye piece) to the first target would be measured as would the vertical angle to the second target.

Each theodolite has specifically marker dot on it, at eye piece (telescope) height. This is the reference marker for the theodolite from which lateral distances are measured. A measuring tape was place against the dot on the theodolite and the other end of the tape was place at the centre of each target cross or the tips of each pointed target rod, to measure the slope distances. The measuring tape had to have a certain tension applied and the readings would be recorded. Later, in the office, the measured slope distances would be corrected for tape sag. Additionally, the heights of the theodolite and the two targets, above the ground, would be measured with a tape measure.

Having done all of that, another temporary marker would be established, the theodolite moved between the last two pegs and the process repeated.

For each set up, the heights of the theodolite and targets was needed as were the slope distances, vertical angles and horizontal angle. Using trigonometry on all this data one could determine the co-ordinates and elevation of each peg.

Another method used of measuring was called stadia. This used a theodolite but instead of a cross targets or pointed rod targets being used to sight to at each of the survey pegs, surveying rods were used. See the picture below from http://www.tigersupplies.com

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The surveying rod would be placed on each peg and three height reading were taken from the surveying rod: the top cross hair, the central (main) cross hairs and the bottom cross. See the picture below.

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The reading from the central cross hairs gives the height for the elevation. The difference between the upper and lower cross hair readings multiplied by an optical constant for the optics of the theodolite gave the distance between the surveying rod and the theodolite. Except for some Japanese theodolites, the optical constant was 100.

In the picture above, the cross hair readings are 1.500, 1.422 and 1.344.

Irrespective of which method was used. To make adjustments for surveying errors, a closed traverse was done whereby after everything that needed to be surveyed was measured, the last reading was back to the first peg surveyed. If the co-ordinates in 3D matched there were no errors. If they didn't each of the readings would need to be adjusted to close the traverse with "no errors"

To minimize errors, the shorter the lateral distances the better, as there was less tape sag. For measurements that required high levels of accuracy, such as when assembling large equipment in hot climates the work would be done during the early mornings to minimize or eliminate heat shimmer.

  • 1
    $\begingroup$ Wow, incredible... the multiple cross-hairs is genius! $\endgroup$
    – jhabbott
    Commented Jan 23, 2015 at 4:03
  • 3
    $\begingroup$ I forgot to mention, in the day before calculators, surveyors had to use 7 figure logarithm table to do all their calculations. I never had to use them but you can imagine the calc could be tedious & surveying was a lot more different than it is today $\endgroup$
    – Fred
    Commented Jan 23, 2015 at 4:12
  • $\begingroup$ And not just theodolites, you have tacometers, chains, ropes, levelling tables, astrolabe, compass... Theodolite is just the precursor of the total station. $\endgroup$ Commented Sep 22, 2019 at 5:10

The "first straight-line distance" in any survey is called the "baseline", and surveyors over the centuries have worked out many ways to minimize the error in the baseline to begin with, and to account for error accumulation as they survey additional points.

There are also many ways to cross-check measurements using independent methods that allows some errors to be mitigated.

The basic instruments of surveying are the transit for measuring horizontal and vertical angles and the rod and the chain (and the tape, in modern times) for measuring distances. But the real art in surveying is not how well you can take the raw measurements, but your ability to account for the errors and minimize them.

All of this was developed long before the Internet, and the best information on these techniques will be found in books, not online.

The key instrument for establishing a baseline distance is the steel tape. Under proper conditions — paying attention to details such as temperature and tension — a 100-foot tape can typically be read with an accuracy of 1/100 of a foot (about 1/8 inch), giving an overall accuracy of 1 part in 10,000.

A land survey will generally start with a measured baseline at one end of the area, and then after surveying the intermediate points, a second baseline between two points at the far end of the area will be measured as well. This allows most of the systematic errors for all of the intermediate points to be cancelled out.


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