# Measuring Heights, Depths, and Longitude of Planets

Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)

The title of Chapter 7 promises more than it delivers, “Incipit septima distinctio de inuentione altitudinum rerum eleuatarum et profunditatum atque longitudinum planitierum.” Regardless of promises, the chapter offers only trivial exercises in measuring heights of trees, to judge their suitability as masts for ships. Not a word about finding depths, say of wells. It is curious that some of the Italian translations do offer examples of measuring depths of wells. Could this be evidence of a defect in the exemplary text followed by the copyist of, say, Urbino 292?The focus of the chapter is on measuring heights, using similar triangles, as is done in most middle schools today, although in two different forms. The first is a quadrant, also called by him “an oroscope,” for which he presented detailed instructions for both construction and use. The second is “a wooden triangle.” It appears in [4] as a 3-4-5 right triangle for measuring heights, again by similar triangles. Despite the practicability of these instruments, the incompleteness of the chapter is bothersome: there is nothing about finding depths or longitude of planets. All the other chapters are quite complete; a few are even overflowing with information. Why does this chapter seem to have been clipped of some contents?The Latin manuscripts offer no suggestion. The Italian manuscripts, on the other hand, do offer problems seeking depths of wells. Yet none of the material in the corresponding chapter of the Riccardiana manuscript appears here. Furthermore, the figures do not always match the text, and the text was in need of severe correction, especially in [7]. New figures were drawn. Note the use of the word equiangular (corresponding angles are equal) that is so different from modern use (all angles are equal). Finally, there is a single example where the Pythagorean theorem is used.

## Keywords

Similar Triangle Ground Distance Exemplary Text Equal Circumference Seventh Part
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