The sector, also known as a sector rule, proportional compass, or military compass, is a major calculating instrument that was 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 is 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. Galileo described how to perform 32 different calculations with the sector in his 1606 manual. In the introduction, Galileo wrote that his intention in producing the sector was to enable people who had not studied mathematics to perform complex calculations without having to know the mathematical details involved. The sector was used in combination with a divider, also called a compass. Each arm of the sector was marked with four lines on the front, and three on the back, and the pivot had a dimple that would accept the point of a divider. The lines and scales on each arm were identical, and arranged in the same order as you moved from the inner edge to the outer edge, thus forming seven pairs of lines. All the calculations could be performed with some combination of five very simple steps: measuring some length, separation or object width with the divider; opening the arms of the sector and setting the crosswise distance between two corresponding points on a pair of lines to the divider separation; measuring the crosswise distance between two corresponding points on a pair of lines once the sector had been set to some separation; reading a value from one of the scales at a point where the crosswise distances matches a divider separation; and reading a value off a scale where the distance from the pivot matches a divider. Galileo did not describe how the scales were constructed, he considered that a trade secret, but the details can be inferred. Scale markings were placed with an accuracy of about 1%.
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