The sector was one of the most familiar of mathematical instruments between the 17th and 19th centuries. It was however devised just before 1600 and was first published in 1598 by the English mathematical practitioner Thomas Hood. An independent version developed by Galileo Galilei in the 1590s was published early in the 17th century, and many other designs subsequently followed.

The instrument can be used to graphically solve questions of proportion, and relies on the principle of similar triangles. Its vital feature is a pair of jointed legs, which carry paired geometrical scales. In use, problems are set up using a pair of dividers to determine the appropriate opening of the jointed legs and the answer is taken off directly as a dimension using the dividers. Specialised scales for area, volume and trigonometrical calculations, as well as simpler arithmetical problems were quickly added to the basic design.

Different versions of the instrument also took different forms and adopted additional features. The type publicised by Hood was intended for use as a surveying instrument, and included not only sights and a mounting socket for attaching the instrument to a pole or post, but also an arc scale and an additional sliding leg. Galileo's earliest examples were intended to be used as gunner's levels as well as calculating devices.

As a simple example of the use of the sector, consider the problem of dividing a line AB into a given number of equal parts, say 7 parts (see the accompanying figures). A pair of dividers is set to the length of the line AB. On the sector's scales of equal parts a number easily divisible by 7 is found, perhaps 140. The sector's legs are then opened until the points of the dividers can span the distance between the marks of 140 on the sectoral scale. Keeping the sector at this opening, the dividers are then closed until their points span the distance between the two points marked 20 on the sector's scale of equal parts. This distance is a seventh of the given line AB, and can be stepped out along its length.

Thomas Hood, The Making and Use of the Geometricall Instrument called a Sector (London, 1598); Galileo Galilei, Operations of the Geometric and Military Compass, translated and introduced by S. Drake (Washington, 1978)

Stephen Johnston
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