J.J. with new equipment, its very easy to compute the horizontal distance. Almost all modern equpment will display the horizontal distance and the slope distance of everything we measure. Its much easier than it sounds. In the old days, like Ergon says, they used a steel tape (a chain in the real old days) and a hand level and plumb bobs. I can tell you very few surveyors used hand levels and just estimated how high or low the guy in front of them was. Picture going down a steep hill, the guy on the downhill side holding the tape horizontal, maybe as much as head high, and using a plumb bob to make a mark at his feet, then repeating this process until you measure the distance you want. It's pretty slow and not that accurate. Now we use a EDM and measure the distances electronically, and measure the angles to compute the horizontal distance between two points.
This is also a good example of how newer surveys can vary so much. Say a survey done in 1960 vs one done in 2007. The 1960 survey would have been done with a steel tape, where today modern equipment would be used. The new survey could vary by several feet over the 1960 survey. That is why property corners should hold and landmarks should hold instead of absoulute distances. 100 years from now they may laugh at how inaccurate my surveys are today.