# Sector (instrument)

(Redirected from Proportional compass)

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, trigonometry, multiplication and division, and for various functions, such as squares 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. It has four parts, two legs with a pivot (the articulation), a quadrant and a clamp (the curved part at the end of the leg) that enables the compass to function 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.

Credit for the invention is often given to either Thomas Hood, a British mathematician, or to the Italian mathematician and astronomer Galileo Galilei. Galileo, with the help of his personal instrument maker Marc'Antonio Mazzoleni, created more than 100 copies of his military compass design and trained students in its use between 1595 and 1598. Of the credited inventors, Galileo is certainly the most famous, and earlier studies usually attributed its invention to him.

## The scales

The following is a description of the instrument as it was constructed by Galileo, and for which he wrote a popular manual. The terminating values are arbitrary and varied from manufacturer to manufacturer.

### The Arithmetic Lines

The innermost scales of the instrument are called the Arithmetic Lines from their division in arithmetical progression, that is, by equal additions which proceed out to the number 250. It is a linear scale generated by the function $f(n)=Ln/250$ , where n is an integer between 1 and 250, inclusive, and L is the length at mark 250.

### The Geometric Lines

The next scales are called the Geometric Lines and are divided in geometric progression out to 50. The lengths on the geometric lines vary as the square root of the labeled values. If L represents the length at 50, then the generating function is: $f(n)=L(n/50)^{1/2}$ , where n is a positive integer less than or equal to 50.

### The Stereometric Lines

The Stereometric Lines are so called because their divisions are according to the ratios of solid bodies, out to 148. One of this scale's applications is to calculate, when given one side of any solid body, the side of a similar one that has a given volume ratio to the first. If L is the scale length at 148, then the scale-generating function is: $f(n)=L(n/148)^{1/3}$ , where n is a positive integer less than or equal to 148.

### The Metallic Lines

These lines have divisions on which appeared these symbols: Au, Pb, Ag, Cu, Fe, Sn, Mar, Sto, (gold, lead, silver, copper, iron, tin, marble, and stone). From these you can get the ratios and differences of specific weight found between the materials. With the instrument set at any opening, the intervals between any correspondingly marked pair of points will give the diameters of balls (or sides of other solid bodies) similar to one another and equal in weight.

### The Polygraphic Lines

From the given information, the side length and the number of sides, the Polygraphic lines yield the radius of the circle that will contain the required regular polygon. If the polygon required has n sides, then the central angle opposite one side will be 360/n.

### The Tetragonic Lines

Tetragonic Lines are so called from their principal use, which is to square all regular areas and the circle as well. The divisions of this scale use the function: $f(n)=L(3^{1/2}\tan(180/n)/n)^{1/2}$ , between the values of 3 and 13.