A forgotten contrivance: a study of the diagonal scale and its appearance in mathematics texts from 1714 to the present

Abstract

The diagonal scale, whose origins are credited to Rabbi Levi Ben Gerson (1288–1344), is a two-dimensional scale, drawn or etched on a flat surface, used to measure short lengths to finer subdivisions of a unit. In modern times, the scale is used to measure up to a precision of three figures. The diagonal scale was a prevalent tool in mathematics for many years, up until about the late 1900s, when advances in mensuration, such as the vernier scale and the caliper, eradicated its use as a practical measuring instrument. Today, we see the scale mainly in architecture and drafting books, whereas in the past it appeared in scores of mathematics texts for school, practical, and recreational use. This paper is based on a Master’s thesis, and will cover the origin and history of the diagonal scale, a demonstration of its uses, and examples of its appearance in early mathematics texts.

Notes

¹ The nonius scale was invented by Pedro Nunes (1502–1578), a Portuguese mathematician, in attempts to improve the accuracy of the astrolabe. It consisted of 45 finely subdivided concentric scales on the instrument (Crato 2010, 61).

² Invented by Edmund Gunter (1581–1626), Gunter’s scale (or Gunter’s rule) was a two-foot long rule which had a number of different scales upon it, including a plane scale, diagonal scale, and logarithmic scales, and may have evolved into the more familiar slide rule.

³ A chain is a land measuring unit, common in the 1700s and 1800s when metal chains (called Gunter’s chain) were used to measure land. The chain was made of 100 metal links, each 7.92 inches long. The entire chain measures 22 yards long (Leifeste 2012, 242).