Pythagoras was a larger than life pre-Socratic philosopher, born on the island of Samos. The Pythagorean Theorem relates the lengths of the three sides of a right triangle. Undoubtedly the most familiar theorem in mathematics, it had been known previously in Mesopotamian, Indian and Chinese cultures, but it was elaborated, made part of the over-all belief system and first proven by the Pythagoreans.

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse:

A^{2} + B^{2} = C^{2}

Where A and B are the legs and C is the hypotenuse.

As for triangles that do not have a right angle, it is necessary to resort to a trigonometric function, the cosine, which may be found by consulting the trigonometric tables or by using a scientific calculator. The applicable formula, the Law of Cosines, is:

A^{2} + B^{2} – 2AB cosθ = C^{2}

where θ is the angle between sides A and B.

If θ is 90°, then cosθ = 0, in which instance the Pythagorean theorem reappears.

Accordingly, it is a simple matter by extracting a square root to find the length of any side when the other two are known.

Little did Pythagoras know that his formula would make practical the invention of the surveyor’s transit. Invented in the late 1700s, the formal name for the transit is a theodolite. It consists of a moveable telescope mounted so it can rotate around horizontal and vertical axes and provide angular readouts. These readouts indicate the orientation of the telescope and are used to relate the first point sighted through the telescope to subsequent sightings of other points from the same theodolite position. These angles can be measured with accuracies down to microradians or seconds of arc.

The modern theodolite has evolved into what is known as a total station that measures angles and distances electronically and stores them in electronic memory. To make a measurement with a theodolite, the operator first centers it over what’s called the station mark (basically just a reference mark). The vertical axis of the theodolite is centered over the station mark using a centering plate known as a tribrach. Then the operator levels the base of the instrument to make the vertical axis vertical, usually via a in-built bubble-level. Then the operator removes parallax error by focusing the optics (objective) and eye-piece. The objective is re-focused for each subsequent sighting from the station because of the different distances to the targets.

The surveyor takes sightings by adjusting the telescope’s vertical and horizontal angular orientation so the cross-hairs align with the desired sighting point. Both angles are read from scales and recorded. The next object is then sighted and recorded without moving the position of the instrument and tripod.

The earliest angular readouts were from open vernier scales directly visible to the eye. Gradually these scales were enclosed for physical protection, and finally became an indirect optical readout, with convoluted light paths to bring them to a convenient place on the instrument for viewing. Of course, modern digital theodolites have electronic displays.

Surveyors use theodolites for triangulation, a process that was invented in the 1500s. Starting from a baseline, they divide an area to be mapped into a series of triangles. Once they have identified the points that mark each triangle, they use their theodolites to measure the length of each side and the height of the corners. From this information they can use the Pythagorean theorem to deduce lengths.

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