Dimensional reduction

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Photo of a sector

The other side of the same sector

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

Galileo's geometrical and military compass, thought to have been made c. 1604 by Mazzoleni

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: $f(n)=L(31/2\tan (180/n)/n{)}^{1/2}$, between the values of 3 and 13.

The Added Lines

These Added Lines are marked with two series of numbers, of which the outer series begins at a certain mark called D followed by the numbers 1, 2, 3, 4, and so on out to 18. The inner series begins from this mark D, going on then to 1, 2, 3, 4, and so on, also out to 18. They were used in conjunction with the other scales for a number of complex calculations.

Use

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.

The sector was a very useful instrument at a time when artisans and military men were poorly educated in mathematics and, often, were unable to perform even elementary arithmetical operations. The inaccuracy induced by the analog scales of the sector were usually of no concern to those attempting to find a rapid solution to an approximate problem. It is striking, however, that the disciplines to which these instruments were applied, particularly perspective, music, architecture and fortification, traditionally classed as mechanical sciences, soon emerged as mathematical sciences in the seventeenth century. Indeed there is evidence that the universality of these practical applications helped to make possible the universality of science at a theoretical level. Hence this technology was not simply a consequence of advances in science. Rather, the technology helped make possible the mathematical sciences that led to modern science.

Bibliography

• Ralf Kern: Wissenschaftliche Instrumente in ihrer Zeit. Vom 15. – 19. Jahrhundert. Verlag der Buchhandlung Walther König 2010, ISBN 978-3-86560-772-0

References

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• A typical sector and how to use it
• The Scales of the Galilean Sector quotations from: "The Geometric and Military Compass” by G. Galilei
• Cole Military Sector at the IBM Archives