Trigonometry on the Sphere
Most trigonometry students look at triangles on a flat surface. However, people from ancient astronomers to modern navigators calculated the arc lengths and angles of triangles on a sphere. They used special globes and instruments to make measurements and teach. The Smithsonian collections are particularly rich in models for spherical trigonometry by Worcester, Massachusetts, high school teacher A. Harry Wheeler and his students. Examples of these models have dates ranging from 1915 to 1945, Wheeler used several schemes to identifying models  some are numbered, others lettered. Patterns, especially for models made in the 1940s, also survive. This object group attempts to separate models from such documentation, but not to arrange models by type or date.
Glen Van Brummelen, in his book Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry (Princeton: Princeton University Press, 2013), gives a history of early spherical trigonometry. For a general discussion of Wheeler’s models, see David L. Lindsay’s article “Albert Harry Wheeler (1873–1950): A Case Study in the Stratification of American Mathematical Activity,” published in Historia Mathematica in 1996 (vol. 23, pp. 269287).

Armillary Sphere
 Description
 This small instrument shows the relative positions of the equator, ecliptic, and other important astronomical circles. At the center, presumably representing the earth, is a small ivory ball. The “CASPAR VOPEL ARTE” inscription on the Tropic of Cancer refers to Caspar Vopel (15111561) of Cologne who taught mathematics and made mathematical instruments.
 Location
 Currently not on view
 date made
 ca 1550
 maker
 Vopel, Caspar
 ID Number
 PH.327302
 catalog number
 327302
 accession number
 272528
 Data Source
 National Museum of American History

Mechanical Navigator by F. E. Brandis, Sons and Company
 Description
 The mechanical navigator is an analog computing device designed to solve problems in spherical trigonometry arising in navigation. In this form, it was designed for instruction in navigation (another version was designed for use at sea). It allowed a student to compute a ship’s location from two sights in one operation.
 The instrument is a mechanical representation of the celestial sphere. A rotating ring mounted vertically on the right side represents the celestial equator. It is calibrated from 0 to 180 by quarterdegrees twice, representing celestial longitude. It also is graduated from 0 to 24 counterclockwise by one minute, and from 0 to XXIV clockwise by one minute. The iron housing inside the vertical circle is calibrated from 0 to 22 by one and labeled by constellation name. A vernier along the edge of this ring marks the meridian of the navigator.
 The instrument has two concentric rings which rotate in perpendicular planes. The outermost represents an hour circle. It is calibrated from 0 to 90 by quarterdegree, four times, and also bears hour lines. The inner ring represents the horizon circle. In addition to degree scales like those of the hour circle, it has is letters for eight cardinal points with sixteen subdivisions between each letter.
 A quadrant affixed perpendicular to the horizon ring, has scales calibrated scale along both sides that run from 0 to 90 degrees, divided to quarter degrees and marked every ten degrees. These represent degrees of latitude. All of these parts rotate on pivots. There are screws for setting the circles.
 The iron base, in the shape of a “T,” has handles at each end. A prior owner made a fitted wooden base for the navigator. The base has two boards with a space between them. Two removable wooden rods labeled in pencil “Left” and “Right” rest between the boards. A mark engraved on the vertical ring reads: F. E. BRANDIS, SONS & CO. (/) BROOKLYN, N.Y. (/) 2877.
 Frederick Ernest Brandis (18451916) was a German immigrant who began making and importing instruments in 1871. From the name of the firm, the instrument was made between 1890 and 1916. An eighteenpage typescript of the company’s instructions for using the mechanical navigator is stored in the accession file. According to an account of the instrument published in Engineering News in 1914, the mechanical navigator sold for $2400.
 Another example of the mechanical navigator has long been on loan to the physical sciences collection.
 References:
 Brandis & Sons Mfg. Co., Instruments of Precision . . . Catalogue No. 20 (Brooklyn, New York, n.d.), pp. 294297.
 "Instrument for Solving Problems of Navigation," Scientific American (July 16, 1910): 44,56,57.
 “An Instrument for Solving Spherical Triangles Mechanically,” Engineering News, vol. 71 #4, January 22, 1914, pp. 180181.
 Mimeographed instructions describing the instrument and its use in detail, are in the accession file.
 Location
 Currently not on view
 date made
 18901916
 maker
 F. E. Brandis, Sons and Company
 ID Number
 MA.314665
 accession number
 208323
 catalog number
 314665
 Data Source
 National Museum of American History

Slated Globe
 Description
 Erasable surfaces like slates and blackboards have been used in the United States since the late 18th century and became popular in the first half of the 19th century. A few teachers also acquired globes painted with “liquid slating” that could be marked with a slate pencil or chalk. These were used in teaching geography, astronomy, navigation, and spherical trigonometry. Commercial slated globes sold from the 1850s onward. This example, which comes with its own stand, is undated and unmarked. A small hour circle is near the North Pole. The meridian circle of the stand is graduated to degrees on both sides.
 The object was received at the museum from the National Bureau of Standards in the 1960s and transferred to the collections some years later.
 References:
 Accession file.
 P. A. Kidwell, A. AckerbergHastings, and D. L. Roberts, Tools of American Mathematics Teaching 18002000, Baltimore: Johns Hopkins University Press, 2008, esp. pp. 2930.
 D. J. Warner, “Geography of Heaven and Earth, Part 4,” Rittenhouse, 2, 1988, esp. 110112, 120, 127129.
 Location
 Currently not on view
 ID Number
 1989.0188.01
 catalog number
 1989.0188.01
 accession number
 1989.0188
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Two Intersecting Spherical Triangles
 Description
 This small tan paper model is cut, folded, and held together with tape. It shows two intersecting spherical triangles, one labeled “Triangle 1,” with vertices A, B, C; the other labeled “Triangle 2” with vertices A_{2}, B_{2}, and C_{2}. The model is undated and has no Wheeler number.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.509
 accession number
 304723
 catalog number
 304723.509
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Spherical Triangle
 Description
 This cut folded and taped object shows a spherical triangle with vertices labeled A, B, and C. It is probably a fragment of a larger model.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.697
 accession number
 304723
 catalog number
 304723.697
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Quadrantal Spherical Triangle
 Description
 This cut and glued plastic model shows a spherical triangle formed from three quadrantal (but not perpendicular) sectors of discs. It is unmarked and undated.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 1979.0102.214
 accession number
 1979.0102
 catalog number
 1979.0102.214
 Data Source
 National Museum of American History

Geometric Model by A . Harry Wheeler, Oblique Spherical Triangle
 Description
 During World War II, A. Harry Wheeler made several models relating to spherical trigonometry. This one shows four quadrants of the celestial globe. One represents the equator. The other three pass are perpendicular to the equator through the pole. One passes through point T (Tokyo), another through point H (Honolulu), and the third through point S (San Francisco). A trihedral angle made from pieces of white plastic creates a spherical triangle joining these three points.
 For the pattern for this model, see 1979.3002.084. The pattern is dated 1945.
 Location
 Currently not on view
 date made
 1945
 maker
 Wheeler, Albert Harry
 ID Number
 1979.0102.343
 accession number
 1979.0102
 catalog number
 1979.0102.343
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Polar Spherical Triangles
 Description
 This cut and glued clear plastic model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. A sticker on it reads: PT. One polar triangle is inside the other. The vertices are unlabeled. The model is undated and has no Wheeler number.
 For a discussion of polar triangles, see 304723.159. Compare 304723.491, 304723.516, 304723.521, and 304723.213.
 Location
 Currently not on view
 Maker
 A. Harry Wheeler
 ID Number
 1979.0102.212
 accession number
 1979.0102
 catalog number
 1979.0102.212
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Spherical Polar Triangles
 Description
 This cut and folded tan paper model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. A line perpendicular to the plane of a great circle of a sphere intersects the sphere in two points called poles (for example, on the earth, the great circle of the equator has poles the North Pole and South Pole). In the model, the outer spherical triangle has vertices labeled A, B, and C. Vertices of the inner spherical triangle are A_{2}, B_{2}, and C_{2}. A is the pole nearest A_{2} of the great circle of the sphere that includes the arc B_{2} C_{2}. B is the pole nearest B_{2} of the great circle that includes the arc A_{2}C_{2}. C is the pole nearest C_{2} of the great circle that includes arc A_{2}B_{2}. Also, spherical triangle A_{2}B_{2}C_{2} is the polar triangle of spherical triangle ABC (A_{2} is the pole nearest A of a great circle through BC and so forth).
 The model is among those Wheeler dubbed collapsible.
 Reference:
 G. van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton: Princeton University Press, 2013.
 Location
 Currently not on view
 date made
 1916
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.159
 accession number
 304723
 catalog number
 304723.159
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Spherical Polar Triangles
 Description
 This cut and folded tan paper model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. A line perpendicular to the plane of a great circle of a sphere intersects the sphere in two points called poles (for example, on the earth, the great circle of the equator has poles the North Pole and South Pole). In the model, the outer spherical triangle has vertices labeled A, B, and C. Sides a, b, and c are opposite the corresponding vertices. Vertices of the inner spherical triangle are A_{2}, B_{2}, and C_{2}, with sides a _{2}, b_{2}, and c_{2}. A is the pole nearest A_{2} of the great circle of the sphere that includes the arc B_{2} C_{2}. B is the pole nearest B_{2} of the great circle that includes the arc A_{2}C_{2}. C is the pole nearest C_{2} of the great circle that includes arc A_{2}B_{2}. Also, spherical triangle A_{2}B_{2}C_{2} is the polar triangle of spherical triangle ABC (A_{2} is the pole nearest A of a great circle through BC and so forth).
 In this model, the point C moves along the arc AC and the point B_{2} along the arc B_{2}C_{2}.
 The model is among those Wheeler dubbed collapsible.
 Reference:
 G. van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton: Princeton University Press, 2013.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.169
 accession number
 304723
 catalog number
 304723.169
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Symmetrical Spherical Triangles, Each Separated into Three Isoceles Spherical Triangles
 Description
 This cut and folded tan paper model shows two spherical triangles symmetrically located on opposite side of a sphere (only a great circle of the sphere is shown). Both of the spherical triangles are divided into three isoceles spherical triangles.
 The model is among those Wheeler dubbed collapsible.
 Compare MA.304723.174.
 Location
 Currently not on view
 date made
 1916 08 19
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.168
 accession number
 304723
 catalog number
 304723.168
 Data Source
 National Museum of American History

Geometric Model by a. Harry Wheeler, Trirectangular Spherical Triangle
 Description
 This small plastic mode consists of three quadrants of a disc with a common center glued together at right angles to form a spherical triangle with three right angles. Such spherical triangles are called trirectangular. A sticker on the model reads: TR.ST.
 Location
 Currently not on view
 date made
 ca 1945
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.709
 accession number
 304723
 catalog number
 304723.709
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Two Polar Spherical Triangles
 Description
 This cut and folded tan paper model shows two spherical triangles on the same section of a sphere, one inside the other. It is collapsible.
 Compare MA.304723.176.
 Location
 Currently not on view
 date made
 1916
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.599
 accession number
 304723
 catalog number
 304723.599
 Data Source
 National Museum of American History

Geometrical Model by A. Harry Wheeler, Isosceles Spherical Triangle
 Description
 Three segments of a white plastic disc glued together to form an isosceles spherical triangle. The model is marked "IST."
 For pattern, see 1979.3002.095.
 Location
 Currently not on view
 date made
 ca 1945
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.708
 accession number
 304723
 catalog number
 304723.708
 Data Source
 National Museum of American History

Geometric Models for Spherical Trigonometry by A. Harry Wheeler and his Students Kello Kern Bland and D. Parker
 Description
 These four paper and three plastic models represent principles of spherical trigonometry. One is signed Kello Kern Bland, another D. Parker.
 Location
 Currently not on view
 date made
 1943
 maker
 Wheeler, Albert Harry
 ID Number
 1979.3002.010
 catalog number
 1979.3002.010
 nonaccession number
 1979.3002
 Data Source
 National Museum of American History

Geometrical Model by A. Harry Wheeler, Oblique Spherical Triangle
 Description
 This small plastic model is of a scalene oblique spherical triangle. It consists of three discs of a sphere with a common center, whose straight edges are glued together to form the spherical triangle. None of the angles between the discs is a right angle (hence the triangle is oblique) and the three angles are unequal (hence the triangle is scalene).
 Models MA.304723.707 and MA.304723.710 are mirror images.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.710
 accession number
 304723
 catalog number
 304723.710
 Data Source
 National Museum of American History

Geometrical Model by A. Harry Wheeler, Oblique Spherical Triangle
 Description
 This small plastic model is of a scalene oblique spherical triangle. It consists of three discs of a sphere with a common center, whose straight edges are glued together to form the spherical triangle. None of the angles between the discs is a right angle (hence the triangle is oblique) and the three angles are unequal (hence the triangle is scalene).
 Models MA.304723.707 and MA.304723.710 are mirror images.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.707
 accession number
 304723
 catalog number
 304723.707
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Polar Spherical Triangles
 Description
 This cut and folded tan paper model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. A sticker on it reads: PTa^{*}, One polar triangle is inside the other. The vertices are unlabeled. The model is undated and has no Wheeler number.
 For a discussion of polar triangles, see MA.304723.159. Compare MA.304723.516. For a related drawing (undated) see 1979.3002.087.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.521
 accession number
 304723
 catalog number
 304723.521
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Symmetrical Spherical Triangles
 Description
 This cut and folded tan paper model shows two spherical triangles symmetrically located on opposite side of a sphere (only arcs of the sphere is shown). Both of the spherical triangles are subdivided, although the divisions have collapsed.
 The model is among those Wheeler dubbed collapsible.
 Compare MA.304723.168.
 Location
 Currently not on view
 date made
 1916 08 20
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.174
 accession number
 304723
 catalog number
 304723.174
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Collapsible Spherical Polar Triangle
 Description
 This cut and folded tan paper model ishows a spherical triangle inside a circle of the sphere.
 Location
 Currently not on view
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.597
 accession number
 304723
 catalog number
 304723.597
 Data Source
 National Museum of American History