Galileo was born the same year as William Shakespeare. He was a builder or instruments and used them to measure earthly and extraterrestrial phenomena. He published his findings, only to become embroiled in disputes with the religious authorities. In a trial before the Inquisition, he was compelled to recant his views and he spent his final years under house arrest.

Galileo’s position was that Copernicus rather than Aristotle was correct in regard to the physical layout of all that exists. A lot of Galileo’s findings were based on observations he made using instruments he’d built himself, particularly his optical telescope. His first instrument was a 3x refractor, soon upgraded to over 30x. In 1610, he observed what he described as “three fixed stars” situated in a straight line intersecting Jupiter. Subsequent observations revealed they oscillated from side to side of the giant planet, and the obvious conclusion was that they were orbiting about it. This was further evidence refuting Aristotle’s geocentric notion of things.

Top, a sector rule. Below, a conventional mechanical divider.

Another Galileo invention was the sector, though some credit the invention to an Englishman named Thomas Hood. A sector is a pair of legs connected at one end by a rotating joint. In this regard, it resembles a simple mechanical divider except that the sector contains numerical scales inscribed upon each leg. In use, problems are set up using a pair of dividers to determine the appropriate opening of the jointed legs. The answer is taken as a dimension using the dividers.

There are numerous scales on the sector. The innermost scales are called the arithmetic lines because they are marked in an arithmetical progression. The next scales are geometric lines and are divided in lengths which vary as the square root of the labeled values. Next are stereometric lines are so called because their divisions are according to the ratios of solid bodies, so when given one edge of any solid body, the corresponding edge of a similar one has a given volume ratio to the first. Metallic lines provide a means of getting the ratios and differences of specific weight 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. Polygraphic lines yield the radius of the circle that will contain s specified regular polygon. Tetragonic Lines give the square of all regular areas and circles. Added Lines are marked with two series of numbers and used in conjunction with other scales for a number of complex calculations. Specialized scales for area, volume and trigonometrical calculations, as well as simpler arithmetical problems, eventually got added.

Sectors are still widely used woodworking tools. They were even used by early cannoneers to accurately aim their cannons.

The sector tool is based on rules of geometry, specifically that similar triangles have similar length ratios between their various sides. For example, when two triangles of different sizes each have the same three angles, the ratio of the length of the sides to the ratio of the length of the bases are the same.

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. For example, suppose the goal is to determine the circumference of a circle knowing its diameter. The first step is to open the jaws of a mechanical divider so the tips equal the diameter in question. The sector is then adjusted so the divider tips hit D (for diameter) on the line of circles on both jaws of the sector. Without changing the sector jaw angle, the mechanical divider tips are then moved to the C (circumference) spot on the line of circles and adjusted so they hit C on both scales. Finally, the mechanical divider is moved to the line of calculation where one tip goes on the zero point. The answer is read at the point on this line where the other mechanical divider tip touches.

Precision sector rules can be a bit pricey. We recently found one on eBay, made from ivory and measuring 6-in long, going for a mere $245. Fortunately, there are also DIY plans for making your own sector available, including a project covered in a recent issue of *Make:* magazine.