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. 269-287).

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.
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 (1511-1561) 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
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).
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 quarter-degrees 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 quarter-degree, 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 (1845-1916) 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 eighteen-page 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. 294-297.
"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. 180-181.
Mimeographed instructions describing the instrument and its use in detail, are in the accession file.
Location
Currently not on view
date made
1890-1916
maker
F. E. Brandis, Sons and Company
ID Number
MA.314665
accession number
208323
catalog number
314665
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.
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. Ackerberg-Hastings, and D. L. Roberts, Tools of American Mathematics Teaching 1800-2000, Baltimore: Johns Hopkins University Press, 2008, esp. pp. 29-30.
D. J. Warner, “Geography of Heaven and Earth, Part 4,” Rittenhouse, 2, 1988, esp. 110-112, 120, 127-129.
Location
Currently not on view
ID Number
1989.0188.01
catalog number
1989.0188.01
accession number
1989.0188
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 A2, B2, and C2.
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 A2, B2, and C2. 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
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.Currently not on view
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
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.Currently not on view
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
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.
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
This cut and folded tan paper model is one of several in which A. Harry Wheeler illustrated properties of 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 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 A2, B2, and C2. A is the pole nearest A2 of the great circle of the sphere that includes the arc B2 C2. B is the pole nearest B2 of the great circle that includes the arc A2C2. C is the pole nearest C2 of the great circle that includes arc A2B2. Also, spherical triangle A2B2C2 is the polar triangle of spherical triangle ABC (A2 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
This cut and folded tan paper model is one of several in which A. Harry Wheeler illustrated properties of 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 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 A2, B2, and C2, with sides a 2, b2, and c2. A is the pole nearest A2 of the great circle of the sphere that includes the arc B2 C2. B is the pole nearest B2 of the great circle that includes the arc A2C2. C is the pole nearest C2 of the great circle that includes arc A2B2. Also, spherical triangle A2B2C2 is the polar triangle of spherical triangle ABC (A2 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 B2 along the arc B2C2.
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
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).
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
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.Currently not on view
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
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.Currently not on view
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
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.Currently not on view
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
These four paper and three plastic models represent principles of spherical trigonometry. One is signed Kello Kern Bland, another D. Parker.Currently not on view
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
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.
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
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.
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
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.
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
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).
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
This cut and folded tan paper model ishows a spherical triangle inside a circle of the sphere.Currently not on view
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
This cut and folded tan paper model displays one small and one larger spherical triangle on the same sphere.The model is what Wheeler dubbed collapsible.Compare MA.304723.599.Currently not on view
Description
This cut and folded tan paper model displays one small and one larger spherical triangle on the same sphere.
The model is what Wheeler dubbed collapsible.
Compare MA.304723.599.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.176
accession number
304723
catalog number
304723.176
This cut and folded tan paper model shows two symmetrical spherical triangles opposite one another on a sphere.Currently not on view
Description
This cut and folded tan paper model shows two symmetrical spherical triangles opposite one another on a sphere.
Location
Currently not on view
date made
1918
maker
Wheeler, Albert Harry
ID Number
MA.304723.173
accession number
304723
catalog number
304723.173
This flexible paper model consists of two intersecting spherical triangles. Various letters are written on the faces. A pencil mark on the object reads: August-20-1916. A tag attached to the model reads: 302.Currently not on view
Description
This flexible paper model consists of two intersecting spherical triangles. Various letters are written on the faces. A pencil mark on the object reads: August-20-1916. A tag attached to the model reads: 302.
Location
Currently not on view
date made
1916 08 30
maker
Wheeler, Albert Harry
ID Number
MA.304723.177
accession number
304723
catalog number
304723.177
This cut and folded tan paper model show two polar spherical triangles in opposite hemispheres. A mark reads: 308.This is of the style of models Wheeler called collapsible.Currently not on view
Description
This cut and folded tan paper model show two polar spherical triangles in opposite hemispheres. A mark reads: 308.
This is of the style of models Wheeler called collapsible.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.704
accession number
304723
catalog number
304723.704
This cut and folded paper model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. One polar triangle is inside the other. The vertices are unlabeled.
Description
This cut and folded paper model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. One polar triangle is inside the other. The vertices are unlabeled. The model is undated and has no Wheeler number.
A loose piece stored with the model has the label: QUADRANTAL ARC
For a discussion of polar triangles, see MA.304723.159.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.690
accession number
304723
catalog number
304723.690

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