17th-century Englishman making calculations with a sector and dividers, from Edmund Gunter's 1623 De sector & radio.
A 17th-century Englishman making calculations with a sector and dividers, from Edmund Gunter's 1623 De sector & radio. Courtesy of Dibner Library, Smithsonian Institution Libraries.

Historians have used the term "mathematical practitioners" to refer to the Europeans who worked on mathematics in the 16th and 17th centuries. During the Renaissance and Scientific Revolution, mathematicians often held the equivalent of several modern-day jobs. For example, Thomas Blundeville (c. 1522–c. 1606), who is mentioned in the protractors object group, owned a country estate in England; tutored other gentlemen in mathematics; wrote about horsemanship, history and geography, logic, and astronomy; and may have practiced law. Besides indicating wide interests and employment, the term also tells us that these mathematicians were greatly concerned with the practical uses of mathematics. For instance, Niccoló Tartaglia pioneered a mathematical treatment of artillery shot in 1537 and 1546, inspiring Galileo's work on projectiles, among others. To carry out their activities, mathematical practitioners used, designed, and made instruments. Surviving early modern examples illustrate their concerns in mathematics as well as the quality of craftsmanship produced in workshops.

The sector was one of these objects that blended theory and practice to visually attractive effect. A sector generally has two arms that are connected by a hinge and covered on both sides with scales useful for problems in practical mathematics. The practitioner used a pair of dividers to measure distances and transfer the distances to the scales, thus opening up the sector to the initial distance and setting up proportions from which the solution distance was read by measuring again with the dividers. A sector may be made of brass, ivory, or wood. Most are 6 to 7 inches long when folded, although some are as short as 3 inches or as long as 13 inches. Instruments of this type were introduced in Italy, probably by Guidobaldo del Monte, and first described by G. P. Gallucci in a book about mathematical instruments published in Venice in 1598. Galileo added scales for military engineering to the device and publicized his version through Operations of the Geometric and Military Compass in 1606.

Meanwhile, in London, Thomas Hood independently wrote The Making and Use of the Geometricall Instrument Called a Sector in 1598. In fact, there were several other names for similar instruments developed in Europe between 1560 and 1620, including "proportional compass" and pantomètre, and ultimately three main styles evolved, associated with Italy, France, and England. As can be seen on the following pages, the styles can usually be distinguished by the sets of scales found on the sectors.

Nearly three-quarters of the 23 sectors in the mathematics collections came from collectors or dealers of rare instruments, but one of the 19th-century English-style sectors was associated with the Drapers, a prominent American scientific family that among other things established an observatory in New York City in 1868, while two others were used by a British officer who fought in the War of 1812 and then settled in Canada. One of the 18th-century French-style sectors arrived in the United States in the early 20th century with a German clockmaker, and it is possible that he was still using the instrument.

As calculating tools, sectors were supplanted by logarithmic tables and slide rules, although they continued to be manufactured in Great Britain into the late 19th century, generally as a decorative inclusion in a set of mathematical instruments. In part because the use of sectors was declining at the time that Americans were establishing firms to manufacture and sell mathematical instruments, the instrument was not widely used in this country. Eighteenth-century surveyors, such as George Washington, did purchase sectors overseas and carried the instrument among their equipment. James W. Queen of Philadelphia was one of the few firms to offer sectors for sale, advertising a 6" ivory instrument in 1867 for $2.25. Oddly, even though the company was founded in 1867, when others were occasionally selling sectors, Keuffel & Esser of New York only offered sectors in 1909 and 1913, by which time the instrument had fallen from common use.

Advertisement for sectors in Catalogue of Keuffel & Esser Co
Advertisement for sectors in Catalogue of Keuffel & Esser Co. (New York, 1909), 316. Courtesy of the NMAH mathematics collections.

The digitization of this group of artifacts was made possible through the generous support of Edward and Diane Straker.