The Sector

Galileo's geometric and military compass, and the dividers that read it.

Two flat brass legs, hinged at one end, carry the same scales running out from the hinge. Open them, and every pair of matching marks is joined by an invisible chord; all those chords belong to similar triangles, so each is in proportion to its mark. A pair of dividers lifts a length off the page, lays it across the legs, and carries back its proportional part. With one opening the sector divides a line, scales a plan, works the rule of three, doubles a square, and sizes a cannonball.

Plate I. The sector opened on its joint, the dividers, and a given line AB.

Mark on the line of lines:

Drag a leg's tip to open the sector; drag the dividers by their head, or by either point.

Open the sector until AB, in the dividers, fits across 100–100 of the line of lines.
Divide AB into seven equal parts: fit AB across 70–70, then take the span across 10–10.
Find x with 3 : 5 = 12 : x. Take 12 along the line, fit it across 30–30, take 50–50, and lay it along the line to read x.
Scale a line of 40 parts to seven tenths: fit it across 100–100, then take 70–70.

a.The legs turn on a round joint. Their inner edges, and every line engraved on them, run out from the centre of the joint. Drag the knob at either tip to open them.

b.The line of lines, Galileo's arithmetic line, is cut in equal parts: mark n stands n parts from the centre. One part is the unit of every length on this plate.

c.The dividers carry a length. Their points catch on any mark, on the centre, and on A and B. Carried by the head, they keep their span.

d.Laid across the legs at n–n, the dividers close a triangle with the centre. The triangle at 100–100 has the same angles, so the chord at n is n/100 of the chord at 100.

e.The given line AB. Drag B to change it. When the dividers hold less than AB, they are stepped along it to show how many times they go.

¶Commentary

Galileo's compass. Galileo worked the instrument out at Padua in the later 1590s. From about 1597 his instrument maker, Marcantonio Mazzoleni, made copies in Galileo's house, and Galileo taught their use to private pupils. In 1606 he printed Le operazioni del compasso geometrico et militare, a short manual of worked operations, in a small private edition dedicated to the young Cosimo de' Medici. When Baldassare Capra published a Latin account of the same instrument as his own in 1607, Galileo answered with his Difesa.

Similar triangles. Lines engraved from the centre of the joint make the same angle with their partners on the other leg, at every opening. The chord between mark n and mark n is therefore 2r·sin α, with r the mark's distance from the centre and α half the angle between the two lines, which the opening fixes. Set one chord to a given length, and every other chord on the same pair of lines is fixed in proportion: the sector has turned the rule of three into a single opening of a hinge. The chord at n–n can never exceed twice n parts, so small numbers are scaled up first: 3 : 5 is worked as 30 : 50.

Planes and solids. If the side of a figure is doubled, its area grows four times and its volume eight. So the line of planes puts its marks at distances in proportion to √n, and the line of solids at ∛n. A chord across planes 40–40 is then √2 times the chord across 20–20, whatever the opening, and across solids 40–40 it is ∛2 times. The line of planes also finds mean proportionals, and so square roots; the line of solids finds the side of a cube of twice the volume, the old Delian problem, by reading rather than construction.

For whom. The title says geometric and military. Galileo's instrument also carried a line of metals, giving the sizes of balls of equal weight in gold, lead, silver, copper, iron, and other materials, and a quadrant with a plumb line for laying a gun. A gunner could find the bore for a heavier shot; a surveyor could reduce a plan to scale or divide a field; a merchant could work the rule of three and exchange money, all without written arithmetic.

The English sector. Thomas Hood published an English instrument under the name of a sector in 1598, and Edmund Gunter's De sectore et radio (1623) made the sector a standard tool of English navigators and surveyors. Sectors were made in brass and ivory into the nineteenth century and are found in many cases of drawing instruments.

The proportional compass. Reduction compasses with a sliding pivot were in use by about 1600; one design is credited to Jost Bürgi. They do in one setting what the sector does with two: the pivot is set once for the ratio, and each length taken with the long points is given at once, reduced, by the short ones. The sector, by contrast, can be set afresh for any ratio by opening the hinge.

Further. Galileo Galilei, Operations of the Geometric and Military Compass, 1606, translated with an introduction by Stillman Drake (Smithsonian Institution Press, 1978); Stillman Drake, Galileo at Work: His Scientific Biography (University of Chicago Press, 1978); and Edmund Gunter, De sectore et radio (London, 1623).

Drawn for the Wending House, 2026. Drag anything that looks like brass.The cabinet of instruments