This packet of materials was provided to the donor when he took a course on the use of tabulating equipment at IBM in Endicott, New York, in the 1950s. Included are a name tag, a punch card, three postcards, two leaflets, and an envelope.
The astrolabe is an astronomical calculating device used from ancient times into the eighteenth century. Measuring the height of a star using the back of the instrument, and knowing the latitude, one could find the time of night and the position of other stars. The openwork piece on the front, called the rete, is a star map of the northern sky. Pointers on the rete correspond to stars; the outermost circle is the Tropic of Capricorn, and the circle that is off-center represents the zodiac, the apparent annual motion of the sun. This brass astrolabe has a body and throne plate (there are no separate plates), a handle, ring, rete, alidade, pin, wedge, and index arm. It is signed d (/) q pnel in Gothic script – this may be an owner’s mark.
This isnstrument is sometimes referred to as "Parnel's astrolabe."
Reference:
For a detailed description of this object, see Sharon Gibbs with George Saliba, Planispheric Astrolabes from the National Museum of American History, Washington, D.C.: Smithsonian Institution Press, 1984, pp. 13, 150-151. The object is referred to in the catalog as CCA No. 304.
Robert T. Gunther, Astrolabes of the World, vol II, Oxford: Oxford University Press, 1932, p. 483.
John Davis, "Two Medieval English Astrolabes in the Smithsonian Museum," Bulletin of the Scientific Instrument Society, 2023, #157, pp. 2-17. Davis dates this instrument to second quarter of the fourteenth century.
This instrument has an L- shaped wooden base with six rubber feet. The base supports a wooden framework which holds a hollow metal drum. A piece of paper with various words typed on it is taped to the drum. The drum has a metal cover with a slit that reveals one row of words. Two slides in the slit can cover some of these words. The drum is moved by a handle (which is at one end) or a motor (which is on the on the part of the L, and joined to the other end of the drum). A mark on the motor reads: E/ dison/ konowatt.
Compare the memory drum with number 1979.3065.01. Both were acquired at the same time from the Psychology Department at the University of Arkansas.
Joel S. Freund, an emeritus faculty member at the University of Arkansas psychology department who helped to arrange the donation, suggests that these memory drums may have been associated with research of psychologist John A. McGeoch, who was at the University of Arkansas from 1928 to 1930.
References:
Personal Communication, Joel S. Freund, December 5, 2019.
John A. McGeoch, “The Influence of Degree of Learning upon Retroactive Inhibition,” American Journal of Psychology, vol. 41 #2 (April 1929), pp. 252-262. On p. 252, McGeoch refers to using a Chicago memory drum (a product of C. H. Stoelting Company of Chicago).
John A. McGeoch and William T. McDonald, “Meaningful Relation and Retroactive Inhibition,“ American Journal of Psychology, vo. 43 #4 (October 1931), pp. 579-588. By this time McGeoch had moved to the University of Missouri, but he reports on work done at the University of Arkansas. He mentions an “electrically driven memory drum: on p. 581.
This is one of a set of four models of ruled surfaces originally published by Ludwig Brill in 1891. The series was designed by C. Tesch, an assistant to Christian Wiener (1826-1896), a professor at the technical high school in Karlsruhe. Tesch also designed Series 20 of Brill’s models (also published in 1891) and, after graduation, Series 22 (published in 1894, by which time Tesch is listed as an engineer with no academic affiliation).
This, the third model in Brill's Series 18, has a black metal frame and should have red and green threads, with gold beads. The three sides are roughly straight, with complex curves on the top and bottom. One side is broken off. The threads in this example are presently missing.
All the models in the series are meant to show ruled surfaces of degree three. As in #2, there should be two lines of intersection of the ruled surfaces, marked by beads, but these lines would meet only infinitely far from the model.
This example of the model was exhibited at the Columbian Exposition, a World’s fair held in Chicago in 1893.
Compare 1985.0112.179.
References:
L. Brill, Catalog, 1892, p. 64-65, 44-45.
Richter, K., Online Collection of Mathematical Models, Martin-Luther-Universität Halle-Wittenberg. This collection has the web address: https://www2.mathematik.uni-halle.de/modellsammlung/. The example of this model shown there has number A3-032 in that collection.
Tesch. C., Modelle der Regelfaechen dritten Gerades. A copy of this essay, which gives a brief historical account of the development of the model at Karlsruhe, is online at http://modellsammlung.uni-goettingen.de/data/Texts/M1/Modelle72-75_Tesch.pdf. It was posted in connection with the copy of the model at the University of Goettingen (their model 74).
The astrolabe is an astronomical calculating device used from ancient times into the eighteenth century. Measuring the height of a star using the back of the instrument, and knowing the latitude, one could find the time of night and the position of other stars. The openwork piece on the front, called the rete, is a star map of the northern sky. Pointers on the rete correspond to stars; the outermost circle is the Tropic of Capricorn, and the circle that is off-center represents the zodiac, the apparent annual motion of the sun. Engraved plates that fit below the rete have scales of altitude and azimuth (arc of the horizon) for specific latitudes. This brass astrolabe has four plates; one may well be a replacement. It was made in Nuremberg by Georg Hartman in 1537. An inscription on the inside of the instrument states that it once belonged to the Italian mathematician and astronomer Galileo Galilei (1564-1642).
Reference:
For a detailed description of this object, see Sharon Gibbs with George Saliba, Planispheric Astrolabes from the National Museum of American History, Washington, D.C.: Smithsonian Institution Press, 1984, pp. 146-150. The object is referred to in the catalog as CCA No. 262.
The Belgian physicist Joseph Plateau (1801–1883) performed a sequence of experiments using soap bubbles. One investigation led him to show that when two soap bubbles join, the two exterior surfaces and the interface between the two bubbles will all be spherical segments. Furthermore, the angles between these surfaces will be 120 degrees.
Crockett Johnson's painting illustrates this phenomenon. It also displays Plateau's study of the situation that arises when three soap bubbles meet. Plateau discovered that when three bubbles join, the centers of curvature (marked by double circles in the figure) of the three overlapping surfaces are collinear.
This painting was most likely inspired by a figure located in an article by C. Vernon Boys entitled "The Soap-bubble." James R. Newman included this essay in his book entitled The World of Mathematics (p. 900). Crockett Johnson had this publication in his personal library, and the figure in his copy is annotated.
The artist chose several pastel shades to illustrate his painting. This created a wide range of shades and tints that allows the painting to appear three-dimensional. Crockett Johnson chose to depict each sphere in its entirety, rather than showing just the exterior surfaces as Boys did. This helps the viewer visualize Plateau's experiment.
This painting was executed in oil on masonite and has a wood and chrome frame. It is #23 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) GEOMETRY OF A TRIPLE BUBBLE (/) (PLATEAU).
Crockett Johnson's interest in regular polygons included the pentagram, or five-pointed star. The relation between the pentagon and the star is simple. If each side of a regular pentagon is extended, a regular five-pointed star results. Similarly, connecting each diagonal of a regular pentagon creates a regular five-pointed star. The star will have a pentagon in it, so the method is self-perpetuating.
A method for a pentagram's construction in described in Book IV, Proposition II of Euclid's Elements, but the construction illustrated in this painting is the artist's own creation. It builds on the relationship between the sides of a regular five-pointed star and the golden ratio. As Crockett Johnson may have recalled from his earlier paintings, the five rectangles that surround the central pentagon of the star are golden, that is to say the ratio of the length of the two equal sides of the triangle to the side of the enclosed pentagon is (1 + √5) / 2. Hence one can construct the star by finding a line segment divided in this ratio. No figure by Crockett Johnson showing his construction has been found.
The pentagram, executed appropriately enough in hues of gold, contrasts vividly with the purple background in Star Construction.
The painting is #103 in the series. It is in oil or acrylic on pressed wood and has a gold-colored metal frame. The painting is unsigned and undated. Compare #46 (1979.1093.33) and #64 (1979.1093.39).
This cut, folded and glued tan paper model has six faces, two triangles and four quadrilaterals. A tag reads: 81. A mark reads: Hexahedron III (/) (f) Faces 6, Faces 4,4,4,4,3,3 (/) (e) Vertices 7 (/) (k) Edges 11, No. 81 (/) e-k+f=2 (/) A. Harry Wheeler (/) Aug-22-1934.
This is one of a series of models of irregular hexagons that Wheeler made on August 22, 1934.
Compare 1979.0102.012, 1979.0102.014, 1979.0102.018, 1979.0102.020, 1979.0102.022, 1979.0102.025, and 1979.0102.026.
The astrolabe is an astronomical calculating device used from ancient times into the nineteenth century. Measuring the height of a star using the alidade on the back of the instrument, and knowing the latitude, one could find the time of night and the position of other stars. The openwork piece on the front, called the rete, is a star map of the northern sky. Pointers on the rete correspond to stars; the outermost circle is the Tropic of Capricorn, and the circle that is off-center represents the zodiac, the apparent annual motion of the sun. Engraved plates that fit below the rete have scales of altitude and azimuth (arc of the horizon) for specific latitudes. This brass Persian astrolabe has a rim with throne attached to a back plate with throne, handle, ring, an alidade with pin, a rete, two plates, a ringlet, and a wedge at the front that holds the instrument together. The instrument is signed: ‘amal [?] Khalil.
Reference:
For a detailed description of this object, see Sharon Gibbs with George Saliba, Planispheric Astrolabes from the National Museum of American History, Washington, D.C.: Smithsonian Institution Press, 1984, pp. 160-162. The object is referred to in the catalog as CCA No. 2567.
The astrolabe is an astronomical calculating device used from ancient times into the eighteenth century. Measuring the height of a star using the alidade on the back of the instrument, and knowing the latitude, one could find the time of night and the position of other stars. The openwork piece on the front, called the rete, is a star map of the northern sky. Pointers on the rete correspond to stars; the outermost circle is the Tropic of Capricorn, and the circle that is off-center represents the zodiac, the apparent annual motion of the sun. Engraved plates that fit below the rete have scales of altitude and azimuth (arc of the horizon) for specific latitudes. This brass Persian astrolabe has a mater or body with throne, a handle, a ring, an alidade with pin, a rete, four plates, and a wedge at the front that holds the instrument together.
The instrument is dated about 1715 A.D. It is signed with the mark of ‘Abd al-A’imma.
Compare 333589, 336114, and 316761, which are all by the same maker.
Those making mathematical instruments for surveying, navigation, or the classroom have long been interested in creating equal divisions of the circle. Ancient geometers knew how to divide a circle into 2, 3, or 5 parts, and as well as into multiples of these numbers. For them to draw polygons with other numbers of sides required more than a straightedge and compass.
In 1796, as an undergraduate at the University of Göttingen, Friedrich Gauss proposed a theorem severely limiting the number of regular polygons that could be constructed using ruler and compass alone. He also found a way of constructing the 17-gon.
Crockett Johnson, who himself would develop a great interest in constructing regular polygons, drew this painting to illustrate Gauss's discovery. His painting follows a somewhat later solution to the problem presented by Karl von Staudt in 1842, modified by Heinrich Schroeter in 1872, and then published by the eminent mathematician Felix Klein. Klein's detailed account was in Crockett Johnson's library, and a figure from it is heavily annotated.
This oil painting on masonite is #70 in the series. It is signed: CJ69. The back is marked: SEVENTEEN SIDES (GAUSS) (/) Crockett Johnson 1969. The painting has a black background and a wood and metal frame. There are two adjacent purple triangles in the center, with a white circle inscribed in them. The triangles have various dark gray regions, and the circle has various light gray regions and one dark gray segment. The length of the top edge of this segment is the chord of the circle corresponding to length of the side of an inscribed 17-sided regular polygon.
Reference: Felix Klein, Famous Problems of Elementary Geometry (1956), pp. 16–41, esp. 41.
The astrolabe is an astronomical calculating device used from ancient times into the nineteenth century. Measuring the height of a star using the alidade on the back of the instrument, and knowing the latitude, one could find the time of night and the position of other stars. The openwork piece on the front, called the rete, is a star map of the northern sky. Pointers on the rete correspond to stars; the outermost circle is the Tropic of Capricorn, and the circle that is off-center represents the zodiac, the apparent annual motion of the sun. Engraved plates that fit below the rete have scales of altitude and azimuth (arc of the horizon) for specific latitudes. This brass Persian astrolabe has a body with throne, a handle, a ring, an alidade with pin, a rete, seven plates, a ringlet, and a wedge at the front that holds the instrument together. The instrument has no signature.
For a detailed description of this object, see Sharon Gibbs with George Saliba, Planispheric Astrolabes from the National Museum of American History, Washington, D.C.: Smithsonian Institution Press, 1984, pp. 98-100. The object is referred to in the catalog as CCA No. 57.
In the 17th century, the natural philosophers Isaac Newton and Gottfried Liebniz developed much of the general theory of the relationship between variable mathematical quantities and their rates of change (differential calculus), as well as the connection between rates of change and variable quantities (integral calculus).
Newton called these rates of change "fluxions." This painting is based on a diagram from an article by H. W. Turnbull in Newman's The World of Mathematics. Here Turnbull described the change in the variable quantity y (OM) in terms of another variable quantity, x (ON). The resulting curve is represented by APT.
Crockett Johnson's painting is based loosely on these mathematical ideas. He inverted the figure from Turnbull. In his words: "The painting is an inversion of the usual textbook depiction of the method, which is one of bringing together a fixed part and a ‘moving’ part of a problem on a cartesian chart, upon which a curve then can be plotted toward ultimate solution."
The arc at the center of this painting is a circular, with a tangent line below it. The region between the arc and the tangent is painted white. Part of the tangent line is the hypotenuse of a right triangle which lies below it and is painted black. The rest of the lower part of the painting is dark purple. Above the arc is a dark purple area, above this a gray region. The painting has a wood and metal frame.
This oil painting on pressed wood is #20 in the series. It is unsigned, but inscribed on the back: Crockett Johnson 1966 (/) FLUXIONS (NEWTON).
References: James R. Newman, The World of Mathematics (1956), p. 143. This volume was in the library of Crockett Johnson. The figure on this page is annotated.
Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings," Leonardo, 5 (1972): pp. 97–8.
The astrolabe is an astronomical calculating device used from ancient times into the nineteenth century. Measuring the height of a star using the alidade on the back of the instrument, and knowing the latitude, one could find the time of night and the position of other stars. The openwork piece on the front, called the rete, is a star map of the northern sky. Pointers on the rete correspond to stars; the outermost circle is the Tropic of Capricorn, and the circle that is off-center represents the zodiac, the apparent annual motion of the sun. Engraved plates that fit below the rete have scales of altitude and azimuth (arc of the horizon) for specific latitudes. This brass astrolabe has a rim with throne attached to a back plate with throne, handle, ring, three plates, an alidade with pin, a rete, and a wedge at the front that holds the instrument together. The instrument is signed: ‘Ali ben Mu[. . .]ammad ben Abdallahben Faraji. The inscriptions are in the script of the western Arabic world.
Reference:
For a detailed description of this object, see Sharon Gibbs with George Saliba, Planispheric Astrolabes from the National Museum of American History, Washington, D.C.: Smithsonian Institution Press, 1984, pp. 17, 171-173. The object is referred to in the catalog as CCA No. 2571.
Mathematicians have explored the properties of polyhedra since ancient times, but it was the Swiss scholar Leonhard Euler (1707–1783) who proved the formula V-E+F = 2. That is, for a simple convex polyhedron (e.g. one with no holes, so that it can be deformed into a sphere) the number of vertices minus the number of edges plus the number of faces is two. An equivalent formula had been presented by Descartes in an unpublished treatise on polyhedra. However, this formula was first proved and published by Euler in 1751 and bears his name.
Crockett Johnson's painting echoes a figure from a presentation of Euler's formula found in Richard Courant and Herbert Robbins's article “Topology,” which is in James R. Newman's The World the Mathematics (1956), p. 584. This book was in the artist’s library, but the figure that relates to this painting is not annotated.
To understand the painting we must understand the mathematical argument. It starts with a hexahedron, a simple, six-sided, box-shaped object. First, one face of the hexahedron is removed, and the figure is stretched so that it lies flat (imagine that the hexahedron is made of a malleable substance so that it can be stretched). While stretching the figure can change the length of the edges and the area and shape of the faces, it will not change the number of vertices, edges, or faces.
For the "stretched" figure, V-E+F = 8 - 12 + 5 = 1, so that, if the removed face is counted, the result is V-E+F = 2 for the original polyhedron. The next step is to triangulate each face (this is indicated by the diagonal lines in the third figure). If, in triangle ABC [C is not shown in Newman, though it is referred to], edge AC is removed, the number of edges and the number of faces are both reduced by one, so V-E+F is unchanged. This is done for each outer triangle.
Next, if edges DF and EF are removed from triangle DEF, then one face, one vertex, and two edges are removed as well, and V-E+F is unchanged. Again, this is done for each outer triangle. This yields a rectangle from which a right triangle is removed. Again, this will leave V-E+F unchanged. This last step will also yield a figure for which V-E+F = 3-3+1. As previously stated, if we count the removed face from the initial step, then V-E+F = 2 for the given polyhedron.
The “triangulated” diagram was the one Crockett Johnson chose to paint. Each segment of the painting is given its own color so as to indicate each step of the proof. Crockett Johnson executed the two right triangles that form the center rectangle in the most contrasting hues. This draws the viewer’s eyes to this section and thus emphasizes the finale of Euler's proof. This approach to the proof of Euler's polyhedral formula was pioneered by the French mathematician Augustin Louis Cauchy in 1813.
This oil painting on masonite is #39 in the series. It was completed in 1966 and is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 (/) POLYHEDRON FORMULA (EULER). It has a wood and chrome frame.
Reference:
David Richeson, “The Polyhedral Formula,” in Leonhard Euler: Life, Work and Legacy, editors R. E. Bradley and C. E. Sandifer (2007), pp. 431–34.
Each of these three identical white plaster models has the dimensions given. These U-shaped solids have six quadrilaterals, two triangles and an octagon for faces. They are part of a series of models designed by A. Schoenflies in Göttingen to illustrate the regular partition of space. Schoenflies designed “stones” which could be arranged into larger blocks (sometimes with congruent stones and sometimes using stones that were mirror images of one another). The series was first published by Brill in 1891. The congruent plaster stones that comprise the object with museum number 1985.0112.177 could be arranged to form a block with museum number 1985.0112.166. These objects have no maker’s marks.
A model of this block and stones at the University of Göttingen (#329 in their collection) has two stones and a block. The Smithsonian collections include these three such stones and a block.
This example of the model was exhibited at the Columbian Exposition, a World’s Fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill, 1892, pp. 46-47, 90-91.
A Schoenflies, “Uber Reguläre Gebietstheilungen des Raumes,” Nachrichten von der Königl. Gesellschaft der Wissenschaften, #9, June 27, 1888, pp. 223-237.
Göttingen Collection of Mathematical Models, presently online at http://modellsammlung.uni-goettingen.de/, accessed September 6, 2019.