Rectangular Protractor

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In the 18th and 19th centuries, draftsmen needed to prepare surveying and architectural drawings according to a variety of scales. Their work might call for them to reduce life-size conditions in a range from 1 inch to the foot to 1/8 inch to the foot. Draftsmen also needed a convenient way to keep all of the scales at hand without cluttering up their toolboxes with many drawing instruments. Instrument makers offered rectangular protractors as one tool to solve these problems.
A rectangular protractor has the angle markings around three edges of a rectangle instead of along the arc of a circle or half-circle. The object looks like a ruler and fits neatly inside a pocket or a case of instruments. There is space in the interior of the rectangle—and on both sides—for other scales. Early rectangular protractors were often made of metals such as brass, but in the 19th century, makers manufactured large quantities of rectangular protractors from ivory.
This ivory rectangular protractor typifies the general form of these instruments. It is graduated by single degrees and marked by tens from 10° to 170° in both directions, from left to right and from right to left. The front of the protractor also contains scales for 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, and 1 inch to the foot, as well as a scale of chords. Draftsmen used a scale of chords together with compasses to draw arcs on which they could construct angles, particularly when a protractor was not available. This scale is divided by single degrees and marked from 10° to 90° by tens.
The back of the protractor bears scales dividing the inch into 60, 50, 45, 40, 35, and 30 parts. These scales were also useful for creating and reading scale drawings. For instance, 1 inch on the 40 scale corresponded to 4 inches in real life, while 1 inch on the 60 scale corresponded to 6 inches in real life. Additionally, this protractor contains a diagonal scale, for reading off fractions of an inch, and a scale of cosines. John William Christopher Draper and James Christopher Draper of Pittsburgh donated this protractor to the Smithsonian in 1973.
Reference: Amy Ackerberg-Hastings, "Rectangular Protractors and the Mathematics Classroom," in Hands on History: A Resource for Teaching Mathematics, edited by Amy Shell-Gellasch, MAA Notes no. 72 (Washington, DC: The Mathematical Association of America, 2007), 35–40.
Currently not on view
date made
19th century
Physical Description
ivory (overall material)
overall: .2 cm x 15.2 cm x 4.4 cm; 3/32 in x 5 31/32 in x 1 23/32 in
ID Number
accession number
catalog number
Credit Line
Gift of John William Christopher Draper and James Christopher Draper
Drafting, Engineering
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Data Source
National Museum of American History


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