Astrolabes and quadrants were instruments first devised in antiquity and developed in the Middle Ages to chart the movements of stars, and they also became very useful tools for surveying.
One could use sightlines and angles to calculate unknown distances from known ones, as seen in this depiction of a surveying problem from a book by Leonard Digges (c. 1515-c. 1559) (Image 1). Digges is also credited with inventing the theodolite, a specialist surveying tool that used a refracting lens to help calculate angles.
Though helpful for observation, tools such as the astrolabe, quadrant, and theodolite required their user to have a good knowledge of arithmetic. Only a select few had the ability to carry out complex calculations, and especially in the context of war, it became clear that a tool dedicated to calculations was needed.
Reading the book of nature
Little known at the time, Galileo Galilei (1564-1642) was a lecturer at the University of Padua, where he taught courses on practical subjects such as military architecture. During this time, behind closed doors, he set an instrument maker to work on a device based on the proportional compass, a drawing tool, to aid in artillery calculations.
Galileo was probably not the first to 'invent' such a device, which became known as a sector. He may have gotten the idea from a friend, and London mathematician Thomas Hood was already familiar with a version of the instrument at the time. However, Galileo's writings did much to publicize the device. He became known for his view of mathematics as the core of natural philosophy, which was novel in the strength of his insistence:
"[The book of nature] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." Galileo Galilei, Il Saggiatore, 1626 (1)
We might say that Galileo saw the sector as a way to become better acquainted with nature through simplified calculations.
Sectors had different sets of rulings for different types of calculations, identical on each arm. A pair of dividers was used to match points along the rulings, and it could also be used as a protractor to measure angles. Some sectors even included a clamp, to suit their role as an artillery compass.
While it might seem to us like only a ruler, its design was revolutionary in that it portrayed measurement lines alongside 'artificial lines': sines, tangents, and logarithms. Combined with knowledge of proportions between triangles, the sector was a powerful instrument.
