Smithsonian museums continue to be closed to support the effort to contain the spread of COVID-19. Read a message from our director, and check our website and social media for updates.

Arm Protractor and Goniometer Invented by Samuel L. Penfield

Arm Protractor and Goniometer Invented by Samuel L. Penfield

<< >>
Usage conditions apply
It may seem obvious that surveyors, navigators, and mathematics students would need to measure angles. These practitioners are not the only audiences for protractors, however, since angle measurement is necessary in a variety of other fields. Geologists examine the faces and edges of crystals and recreate the crystals' structures by drawing stereographic projections. In 1900, Samuel Lewis Penfield (1856–1906), a geology professor at Yale University who earned his Ph.B. from Yale in 1877, patented two forms of "contact-goniometer." (A goniometer measures plane angles.) These instruments, along with a stereographic protractor and beam compasses that Penfield patented in 1901, helped establish the technique of stereographic projections in crystallography. Penfield aimed to simplify the work involved in the technique and to produce an inexpensive instrument.
This semicircular paper protractor appears to be an example of the contact-goniometer awarded patent number 655,004 on July 31, 1900. It is printed on a white rectangular card. It is divided by half-angles and marked by tens in both directions (counterclockwise and clockwise) from 0° to 180°. A ruler, divided to millimeters and marked by ones from 0 cm to 14 cm, is printed along the top edge of the card. A diagonal scale and scales for dividing the inch into 10, 12, and 16 parts appear in the interior of the protractor. A celluloid arm is attached at the vertex of the protractor. Users were to set angles according to a horizontal line on the arm but then draw angles along the edge of the arm.
The lower left corner of the card reads: ARM PROTRACTOR AND GONIOMETER (/) Designed by S. L. Penfield. A mark at the lower right corner of the scales of equal parts affirms that the protractor is: ENGINE DIVIDED. The protractor is contained in a paper envelope, which also holds a sheet of instructions written by Penfield. The envelope is imprinted: ARM PROTRACTOR AND GONIOMETER. It is also stamped: Cenco 88210.
The Mineralogical Laboratory of Yale's Sheffield Scientific School offered early versions of this instrument. Central Scientific Company of Chicago (abbreviated Cenco) sold the Penfield arm protractor and goniometer in this form from as early as 1909 (as item 427) to as late as 1950 (as item 88210). In 1914, the instrument cost 67 cents. William C. Marshall (of Bridgeport Works and formerly at the Sheffield Scientific School at Yale) listed it as a required tool in Elementary Machine Drawing and Design (New York: McGraw-Hill, 1912), 8. Edward Salisbury Dana discussed Penfield's instrument in more detail in A Textbook of Mineralogy, With an Extended Treatise on Crystallography and Physical Mineralogy, 3rd ed. rev. William E. Ford (New York: John Wiley & Sons, 1922), 33–40.
References: Samuel Lewis Penfield, "Contact-Goniometer" (U.S. Patent 655,004 filed April 2, 1900), "Contact-Goniometer" (U.S. Patent 655,005 filed April 2, 1900), "Stereographic Protractor" (U.S. Patent 667,570 filed October 25, 1900), and "Beam-Compass" (U.S. Patent 673,687 filed December 31, 1900); S. L. Penfield, "The Stereographical Projection and its Possibilities, from a Graphical Standpoint," American Journal of Science 4th ser. 11 (1901): 1–24, 115–144; Central Scientific Company, Physical and Chemical Apparatus Catalogue M (May, 1914), 42; L. V. Pirsson, "Samuel Lewis Penfield," American Journal of Science, 4th ser. 22 (1906): 353–367; Horace L. Wells, "Samuel Lewis Penfield, 1856–1906," in Biographical Memoirs (Washington, D.C.: National Academy of Sciences, 1909), vi:119–146; Shellie Snell, "Central Scientific Company: A Brief History," Robert A. Paselk Scientific Instrument Museum, Humboldt State University,
Currently not on view
Object Name
date made
patent date
Central Scientific Company
Penfield, Samuel Lewis
place made
United States: Illinois, Chicago
Physical Description
metal (overall material)
plastic (overall material)
paper (overall material)
overall:.3 cm x 16.2 cm x 8.5 cm; 1/8 in x 6 3/8 in x 3 11/32 in
ID Number
accession number
catalog number
Credit Line
Gift of Kenyon College Department of Physics
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Data Source
National Museum of American History
Nominate this object for photography.   

Our collection database is a work in progress. We may update this record based on further research and review. Learn more about our approach to sharing our collection online.

If you would like to know how you can use content on this page, see the Smithsonian's Terms of Use. If you need to request an image for publication or other use, please visit Rights and Reproductions.

Note: Comment submission on our collection pages is temporarily unavailable. Please check back soon!

If you have a question or require a personal response, please visit our FAQ or contact page.