Sector (instrument)

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.

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.

[1] 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.

[2] 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.

[3] 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.

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,[4] privately distributing a Latin manuscript explaining its construction and use.

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.

[5] Most of his customers were wealthy noblemen, including Archduke Ferdinand, to whom Galileo sold a sector made of silver.

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.

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.

If the initial investment is P0, set the divider to the distance from the pivot to the point marked at P0 on the arithmetic lines.

Now set the crosswise distance at 100-100 again to the current divider separation and repeat the procedure for as many periods as needed.

Open the sector and set the crosswise distance at some intermediate value on the geometric lines to the divider separation, any number will do, say 20.

If this number is greater than 50 (the largest value on the geometric lines scale) then it must be reduced, in this example perhaps divided by 3 to make 29.

The metallic lines, the outermost pair on the front face, are marked with the symbols "ORO" (for oro, gold), PIO (for piombo, lead), "AR" (for argento, silver), "RA" (for rame, copper), "FE" (for ferro, iron), "ST" (for stagno, tin), "MA" (for marmo, marble), and "PIE" (for pietra, stone).

These symbols are arranged by decreasing specific weights or densities, with distance proportional to the inverse cube root.

These lines were of interest to artillerymen to solve the problem of “making the caliber”, that is how to figure out the correct powder charge to use for a cannonball of some size and material, when the correct charge is known for a cannonball of a different size and material.

The polygraphic lines, innermost scale on the back of the instrument, is labelled from 3 to 15, and the distance from the pivot is inversely proportional to the side length of a regular polygon of

The procedure for finding the radius of the enclosing circle is as follows: Open the sector and set the crosswise distance at the point 6–6 on the polygraphic lines to the desired side length.

The outer scale is linear and runs from 18 down to 0 as you move away from the pivot, and the zero point is marked with a ⌓, the symbol for a circular segment.

Set the sector crosswise on the added lines at the zero of the outer scale to the half-chord length,

The crosswise distance between the points n-n on the inner scale is the side length of the square equal in area to the circular segment.

, we set up a pair of similar triangles that share the angle made by the arms of the sector at the pivot, so that

A typical English sector, probably from the early 19th century, made of ivory with a brass hinge. This side has scales for lines of lines (L), secants (S), chords (C), and polygons (POL), along with a scale of 10ths of inches on the outer edges forming a straight 12-inch rule when the sector is fully opened, and a scale of 100ths of a foot marked along the side (only barely visible in this picture).
The other side of the same sector, with scales for a line of sines (S) and two lines of tangents (T), along with logarithmic Gunter's scales for numbers (N), sines (S), and tangents (T) on the outer edges.
De fabrica et usu menti ad omnia horarum genera describenda (1592), where Giovanni Paolo Gallucci is among the first to describe the sector
Clément Cyriaque de Mangin , Usage du compas de proportion , 1637
Brass sector with dividers, probably made in Dresden around 1630
Galileo's geometrical and military compass, thought to have been made c. 1604 by Mazzoleni
Figure showing the scales of Galileo's military compass, from his manual on the device.
The scale of polygons on a typical sector has lines proportional to the side length of a regular polygon of n sides inscribed in a given circle. In Galileo's sector design, he inverted this scale so that the numbers increase as they go away from the hinge and are more uniformly spaced. Later designs both in England and Continental Europe reverted to the original polygon scale.