The modern surveyor blends skills from several disciplines, including those of mathematician, law scholar, expert measurer, and translator. By interpreting the legal description and applying the science of measurement, the surveyor translates the legal description into tangible positions on the ground. These positions on the ground can be used as a basis for construction or for perpetuation of a particular location, possibly to show delineation of ownership or to document change over time.
Of course, the surveyor must be familiar with mathematics, especially the application of trigonometry. Most traditional surveying is plane surveying, which does not take into account the curvature of the earth. For most surveying projects, the curvature of the earth is slight enough that the effects can be ignored, greatly simplifying the calculations involved. For projects involving greater distances, the curvature of the earth must be taken into account; this is geodetic surveying, an application of geodesy.
Assuming that plane surveying is acceptable for a particular scope of work, the surveyor's measurements are typically gathered with a theodolite, an instrument that is set up over a recoverable point. The theodolite is a sophisticated instrument combining the capabilites of objects with which we are all familiar: a telescope, a ruler, and a protractor. The telescope with crosshair is used for sighting greater distances than the unaided eye can see, and with greater precision. A laser, used in conjunction with a specialized prism, acts as the "ruler", and provides a method for measuring slope distances. On most contemporary survey instruments, a digital readout provides angular measurements (our "protractor") in both the horizontal and vertical planes. After the gathering of these raw measurements, trigonometry can be used to convert the data to a more usable form, usually in the form of XYZ coordinates. The vertical angle and slope distance are converted from polar measurements (angle and distance) to provide a difference in elevation (delta Z coordinate) and horizontal distance. The horizontal distance and horizontal angle are converted from polar measurements to rectangular coordinates (delta X and Y coordinates). Simpler forms of math are used to keep the coordinates on a common datum, most of which are just logical reasoning applied to spacial relationships.
It is important for a surveyor to be able to choose appropriate tools and methodology to perform a specific task with the required degree of precision and accuracy. One would not use a yardstick to measure a mile, or a vehicle odometer to measure ten feet, with any reasonable expectation of obtaining accurate results. Although this may not be thought of as mathematics, it is the understanding of many foundational elements of mathematics (significant figures, reasonable expectation, even the "mathematics" of common sense) that would lead the surveyor to intelligently choose one method over another for a particular task.
The surveyor must also be familiar with the laws governing the practice of land surveying and with laws regarding the ownership of real property (land). These laws may originate on the federal, state, or local level. The surveyor must be able to make judgements and decisions based upon preponderance of evidence (evidence that carries the most weight or is superior in some way. The surveyor must also understand the application of statutes, and be able to document and defend all decisions made.