The sector, also known as a proportional compass or military compass, was a major calculating instrument in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length joined by a hinge. A number of scales are inscribed upon the instrument which facilitate various mathematical calculations. It was used for solving problems in proportion, multiplication and division, geometry, and trigonometry, and for computing various mathematical functions, such as square roots and cube roots. Its several scales permitted easy and direct solutions of problems in gunnery, surveying and navigation. The sector derives its name from the fourth proposition of the sixth book of Euclid, where it is demonstrated that similar triangles have their like sides proportional. Some sectors also incorporated a quadrant, and sometimes a clamp at the end of one leg which allowed the device to be used as a gunner's quadrant.
History
The sector was invented, essentially simultaneously and independently, by a number of different people prior to the start of the 17th century.
Fabrizio Mordente (1532 – ca 1608) was an Italian mathematician who is best known for his invention of the "proportional eight-pointed compass" which has two arms with cursors that allow the solution of problems in measuring the circumference, area and angles of a circle. In 1567 he published a single sheet treatise in Venice showing illustrations of his device. In 1585 Giordano Bruno used Mordente's compass to refute Aristotle's hypothesis on the incommensurability of infinitesimals, thus confirming the existence of the "minimum" which laid the basis of his own atomic theory. Guidobaldo del Monte developed a "polymetric compass" c. 1670, including a scale for constructing regular polygons. The Italian astronomer Galileo Galilei added further scales in the 1590s, and published a book on the subject in 1606. Galileo's sector was first designed for military applications, but evolved into a general purpose calculating tool.
The two earliest known sectors in England were made by Robert Beckit and Charles Whitwell, respectively, both dated 1597. These have a strong resemblance to the description of the device given by English mathematician Thomas Hood's 1598 book. The sector Hood described was intended for use as a surveying instrument and included sights and a mounting socket for attaching the instrument to a pole or post, as well as an arc scale and an additional sliding leg. In the 1600s, the British mathematician Edmund Gunter dispensed with accessories but added additional scales, including a meridian line with divisions proportional to the spacing of latitudes along a meridian on the Mercator projection, privately distributing a Latin manuscript explaining its construction and use. Gunter published this in English as De Sectore et Radio in 1623.
Galileo's sector
Galileo first developed his sector in the early 1590s as a tool for artillerymen. By 1597 it had evolved into an instrument that had much broader utility. It could be used, for example, to calculate the area of any plane figure constructed from a combination of straight lines and semi-circles. Galileo was determined to improve his sector so that it could be used to calculate the area of any shape discussed in Euclid's Elements. To do this, he needed to add the capability to calculate the area of circular segments. It took him more than a year to solve this problem. The instrument we know today as Galileo's sector is the version with this added capability that he began to produce in 1599 with the help of the instrument maker Marc'Antonio Mazzoleni. Galileo provided Mazzoleni and his family with room and board, and paid him two-thirds of the 35 lire selling price; Galileo would charge 120 lire for a course teaching the use of the instrument, about half the annual wage of a skilled craftsmen. Most of his customers were wealthy noblemen, including Archduke Ferdinand, to whom Galileo sold a sector made of silver. More than a hundred were made in all, but only three are known to exist today: one in the Putnam Gallery at Harvard University, one in the Museum of Decorative Art in Milan's Castello Sforzesco, and one in the Galileo Museum in Florence.