| National Museum of American History

The French lawyer and mathematician Pierre de Fermat (1601–1665) was one of the first to develop a systematic way to find the straight line which best approximates a curve at any point. This line is called the tangent line.

Description: The French lawyer and mathematician Pierre de Fermat (1601–1665) was one of the first to develop a systematic way to find the straight line which best approximates a curve at any point. This line is called the tangent line. This painting shows a curve with two horizontal tangent lines. Assuming that the curve is plotted against a horizontal axis, one line passes through a maximum of a curve, the other through a minimum. An article by H. W. Turnbull, "The Great Mathematicians," published in The World of Mathematics by James R. Newman, emphasized how Fermat's method might be applied to find maximum and minimum values of a curve plotted above a horizontal line (see his figures 14 and 16). Crockett Johnson owned and read the book, and annotated the first figure. The second figure more closely resembles the painting.; Computing the maximum and minimum value of functions by finding tangents became a standard technique of the differential calculus developed by Isaac Newton and Gottfried Leibniz later in the 17th century.; Curve Tangents is painting #12 in the Crockett Johnson series. It was executed in oil on masonite, completed in 1966, and is signed: CJ66. The painting has a wood and metal frame.
Location: Currently not on view

date made: 1966

referenced: Fermat, Pierre de
painter: Johnson, Crockett

ID Number: 1979.1093.07
catalog number: 1979.1093.07
accession number: 1979.1093

Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the problem of Delos.

Description: Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the problem of Delos. Crockett Johnson wrote of this problem: "Plutarch mentions it, crediting as his source a now lost version of the legend written by the third century BC Alexandrian Greek astronomer Eratosthenes, who first measured the size of the Earth. Suffering from plague, Athens sent a delegation to Delos, Apollo’s birthplace, to consult its oracle. The oracle’s instruction to the Athenians, to double the size of their cubical altar stone, presented an impossible problem. . . ."(p. 99). Hence the reference to the problem of Delos in the title of the painting.; Isaac Newton suggested a solution to the problem in his book Arithmetica Universalis, first published in 1707. His construction served as the basis of the painting. Newton’s figure, as redrawn by Crockett Johnson, begins with a base (OA), bisected at a point (B), with an equilateral triangle (OCB) constructed on one of the halves of the base. Newton then extended the sides of this triangle through one vertex. Placing a marked straightedge at one end of the base (O), he rotated the rule so that the distance between the two lines extended equaled the sides of the triangle (in the figure, DE = OB = BA = OC = BC). If these line segments are of length one, one can show that the line segment OD is of length equal to the cube root of two, as desired.; In Crockett Johnson’s painting, the line OA slants across the bottom and the line ODE is vertical on the left. The four squares drawn from the upper left corner (point E) have sides of length 1, the cube root of 2, the cube root of 4, and two. The distance DE (1) represents the edge of the side and the volume of a unit cube, while the sides of three larger squares represent the edge (the cube root of 2), the side (the square of the cube root of 2) and the volume (the cube of the cube root of two) of the doubled cube.; This oil painting on masonite is #56 in the series and dates from 1970. The work is signed: CJ70. It is inscribed on the back: PROBLEM OF DELOS (/) CONSTRUCTED FROM A SOLUTION BY (/) ISAAC NEWTON (ARITHMETICA UNIVERSALIS) (/) Crockett Johnson 1970. The painting has a wood and metal frame. For related documentation see 1979.3083.04.06. See also painting number 85 (1979.1093.55), with the references given there.; Reference: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,” Leonardo 5 (1972): pp. 98–9.

date made: 1970

referenced: Newton, Isaac
painter: Johnson, Crockett

ID Number: 1979.1093.36
catalog number: 1979.1093.36
accession number: 1979.1093

In 1966, Crockett Johnson carefully read Nathan A. Court's book College Geometry, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting.

Description: In 1966, Crockett Johnson carefully read Nathan A. Court's book College Geometry, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting. The work shows two orthoganol circles, that is to say two circles in which the square of the line of centers equals the sum of the squares of the radii. A right triangle formed by the line of centers and two radii that intersect is shown. The small right triangle in light purple in the painting is this triangle.; Crockett Johnson's painting combines a drawing of this triangle with a more complex figure used in a discussion of further properties of lines drawn in orthoganal circles. In particular, suppose that one draws a line segment from a point outside a circle that intersects it in two points, and selects a fourth point on the line that divides the segment harmonically. For a single exterior point, all these such points lie on a single line, perpendicular to the line of centers of the two circles, which is called the polar line.; The painting is #38 in the series. It has a background in two shades of cream, and a light tan wooden frame. It shows two circles that overlap slightly and have various sections. The circles are in shades of blue, purple and cream. The painting is signed: CJ66.; References: Nathan A. Court, College Geometry (1964 printing), p. 175–78. This volume was in Crockett Johnson's library.; T. L. Heath, ed., Apollonius of Perga: Treatise on Conic Sections (1961 reprint). This volume was not in Crockett Johnson's library.
Location: Currently not on view

date made: 1966

referenced: Apollonius of Perga
painter: Johnson, Crockett

ID Number: 1979.1093.26
catalog number: 1979.1093.26
accession number: 1979.1093

Two polygons are said to be homothetic if they are similar and their corresponding sides are parallel.

Description: Two polygons are said to be homothetic if they are similar and their corresponding sides are parallel. If two polygons are homothetic, then the lines joining their corresponding vertices meet at a point.; The diagram on which this painting is based is intended to illustrate the homothetic nature of two polygons ABCDE . . . and A'B'C'D'E' . . . From the title, it appears that Crockett Johnson wished to call attention of homothetic triangular pairs ABS and A'B'S, BCS and B'C'S, CDS and C'D'S, DES and D'E'S, etc. The painting follows a diagram that appears in Nathan A. Court's College Geometry (1964 printing). Court's diagram suggests how one constructs a polygon homothetic to a given polygon. Hippocrates of Chios, the foremost mathematician of the fifth century BC, knew of similarity properties, but there is no evidence that he dealt with the concept of homothecy.; To illustrate his figure, the artist chose four colors; red, yellow, teal, and purple. He used one tint and one shade of each of these four colors. The larger polygon is painted in tints while the smaller polygon is painted in shades. The progression of the colors follows the order of the color wheel, and the black background enhances the vibrancy of the painting.; Homothetic Triangles, painting #17 in the Crockett Johnson series, is painted in oil on masonite. The work was completed in 1966 and is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 (/) HOMOTHETIC TRIANGLES (/) (HIPPOCRATES OF CHIOS). It has a black wooden frame.; References: Court, Nathan A., College Geometry, (1964 printing), 38-9.; van der Waarden, B. L., Science Awakening (1954 printing), 131-136.
Location: Currently not on view

date made: 1966

referenced: Hippocrates of Chios
painter: Johnson, Crockett

ID Number: 1979.1093.11
catalog number: 1979.1093.11
accession number: 1979.1093

Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings.

Description: Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings. This painting is based on a method of constructing a rectangle equal in area to a given rectangle, given one side of the rectangle to be constructed.; In the painting, suppose that the cream-colored rectangle on the bottom left is given, as well as a line segment extending from the upper right corner of it. Construct the small triangle on the upper left. Draw the three horizontal lines shown, as well as the diagonal of the rectangle constructed. Extend this diagonal until it meets the bottom line, creating another triangle. The length of the base of this triangle will be the side of the rectangle desired. This rectangle is on the upper right in the painting.; This construction has been associated with the ancient Pythagoreans. Crockett Johnson may well have learned it from Evans G. Valens, The Number of Things. The drawing on page 121 of this book is annotated, although the annotations are faint.; The oil painting is #48 in the series. It has a black background and a black wooden frame, with the two equal triangles in light shades. The painting is signed on the front: CJ69. It is signed on the back: RECTANGLES OF EQUAL AREA (/) (PYTHAGORAS) (/) Crockett Johnson 1969.
Location: Currently not on view

date made: 1969

referenced: Pythagoras
painter: Johnson, Crockett

ID Number: 1979.1093.34
catalog number: 1979.1093.34
accession number: 1979.1093

The mathematician Euclid lived around 300 BC, probably in Alexandria in what is now Egypt. Like most western scholars of his day, he wrote in Greek. Euclid prepared an introduction to mathematics known as The Elements.

Description: The mathematician Euclid lived around 300 BC, probably in Alexandria in what is now Egypt. Like most western scholars of his day, he wrote in Greek. Euclid prepared an introduction to mathematics known as The Elements. It was an eminently successful text, to the extent that most of the works he drew on are now lost. Translations of parts of The Elements were used in geometry teaching well into the nineteenth century in both Europe and the United States.; Euclid and other Greek geometers sought to prove theorems from basic definitions, postulates, and previously proven theorems. The book examined properties of triangles, circles, and more complex geometric figures. Euclid's emphasis on axiomatic structure became characteristic of much later mathematics, even though some of his postulates and proofs proved inadequate.; To honor Euclid's work, Crockett Johnson presented not a single mathematical result, but what he called a bouquet of triangular theorems. He did not state precisely which theorems relating to triangles he intended to illustrate in his painting, and preliminary drawings apparently have not survived. At the time, he was studying and carefully annotating Nathan A. Court's book College Geometry (1964). Court presents several theorems relating to lines through the midpoints of the side of a triangle that are suggested in the painting. The midpoints of the sides of the large triangle in the painting are joined to form a smaller one. According to Euclid, a line through two midpoints of sides of a triangle is parallel to the third side. Thus the construction creates a triangle similar to the initial triangle, with one fourth the area (both the height and the base of the initial triangle are halved). In the painting, triangles of this smaller size tile the plane. All three of the lines joining midpoints create triangles of this small size, and the large triangle at the center has an area four times as great.; The painting also suggests properties of the medians of the large triangle, that is to say, the lines joining each midpoint to the opposite vertex. The three medians meet in a point (point G in the figure from Court). It is not difficult to show that point G divides each median into two line segments, one twice as long as the other.; To focus attention on the large triangle, Crockett Johnson executed it in shades of white against a background of smaller dark black and gray triangles.; Bouquet of Triangle Theorems apparently is the artist's own construction. It was painted in oil or acrylic and is #26 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) BOUQUET OF TRIANGLE THEOREMS (/) (EUCLID).; Reference: Nathan A. Court, College Geometry, (1964 printing), p. 65. The figure on this page is not annotated.
Location: Currently not on view

date made: 1966

referenced: Euclid
painter: Johnson, Crockett

ID Number: 1979.1093.19
catalog number: 1979.1093.19
accession number: 1979.1093

Toward the end of his life, Crockett Johnson took up the problem of constructing a regular seven-sided polygon or heptagon. This construction, as Gauss had demonstrated, requires more than a straight edge and compass.

Description: Toward the end of his life, Crockett Johnson took up the problem of constructing a regular seven-sided polygon or heptagon. This construction, as Gauss had demonstrated, requires more than a straight edge and compass. Crockett Johnson used compass and a straight edge with a unit length marked on it. Archimedes and Newton had suggested that constructions of this sort could be used to trisect the angle and to find a cube with twice the volume of a given cube, and Crockett Johnson followed their example.; One may construct a heptagon given an angle of pi divided by seven. If an isosceles triangle with this vertex angle is inscribed in a circle, the base of the triangle will have the length of one side of a regular heptagon inscribed in that circle. According to Crockett Johnson's later account, in the fall of 1973, while having lunch in the city of Syracuse on Sicily during a tour of the Mediterranean, he toyed with seven toothpicks, arranging them in various patterns. Eventually he created an angle with his menu and wine list and arranged the seven toothpicks within the angle in crisscross patterns until his arrangement appeared as is shown in the painting.; Crockett Johnson realized that the vertex angle of the large isosceles triangle shown is exactly π/7 radians, as desired. The argument suggested by his diagram is more complex than what he later published. The numerical results shown in the figure suggest his willingness to carry out detailed calculations.; Heptagon from its Seven Sides, painted in 1973 and #107 in the series, shows a triangle with purple and white sections on a navy blue background. This oil or acrylic painting on masonite is signed on its back : HEPTAGON FROM (/) ITS SEVEN SIDES (/) (Color sketch for larger painting) (/) Crockett Johnson 1973. No larger painting on this pattern is at the Smithsonian.; Reference: Crockett Johnson, "A Construction for a Regular Heptagon," Mathematical Gazette, 1975, vol. 59, pp. 17–21.
Location: Currently not on view

date made: 1973

painter: Johnson, Crockett

ID Number: 1979.1093.74
catalog number: 1979.1093.74
accession number: 1979.1093

This oil painting on pressed wood, #52 in the series, shows an original construction of Crockett Johnson. He executed this work in 1968, three years after he began creating mathematical paintings.

Description: This oil painting on pressed wood, #52 in the series, shows an original construction of Crockett Johnson. He executed this work in 1968, three years after he began creating mathematical paintings. It is evident that the artist was very proud of this construction because he drew four paintings dealing with the problem of squaring the circle. The construction was part of Crockett Johnson's first original mathematical work, published in The Mathematical Gazette in early 1970. A diagram relating to the painting was published there.; To "square a circle," mathematically speaking, is to construct a square whose area is equal to that of a given circle using only a straightedge (an unmarked ruler) and a compass. It is an ancient problem dating from the time of Euclid and is one of three problems that eluded Greek geometers and continued to elude mathematicians for 2,000 years. In 1880, the German mathematician Ferdinand von Lindermann showed that squaring a circle in this way is impossible - pi is a transcendental number. Because this proof is complicated and difficult to understand, the problem of squaring a circle continues to attract amateur mathematicians like Crockett Johnson. Although he ultimately understood that the circle cannot be squared with a straightedge and compass, he managed to construct an approximate squaring.; Crockett Johnson began his construction with a circle of radius one. In this circle he inscribed a square. Therefore, in the figure, AO=OB=1 and OC=BC=√(2) / 2. AC=AO+OC=1 + √(2) / 2 and AB=√(AC² + BC²) which equals the square root of the quantity (2+√(2)). Crockett Johnson let N be the midpoint of OT and constructed KN parallel to AC. K is thus the midpoint of AB, and KN=AO - (AC)/2=1/2 - √(2) / 4. Next, he let P be the midpoint of OG, and he drew KP, which intersects AO at X. Crockett Johnson then computed NP=NO+OP=(√(2))/4+(1/2). Triangle POX is similar to triangle PNK, so XO/OP=KN/NP. From this equality it follows that XO=(3-2√(2))/2.; Also, AX=AO-XO=(2√(2)-1)/2 and XC=XO+OC=(3-√(2))/2. Crockett Johnson continued his approximation by constructing XY parallel to AB. It is evident that triangle XYC is similar to triangle ABC, and so XY/XC=AB/AC. This implies that XY=[√((2+√(2)) × (8-5√(2))]/2. Finally he constructed XZ=XY and computed AZ=AX+XZ=[2√(2)-1+(√(2+√(2)) × (8-5√(2))]/2 which approximately equals 1.7724386. Crockett Johnson knew that the square root of pi approximately equals 1.772454, and thus AZ is approximately equal to √(Π) - 0.000019. Knowing this value, he constructed a square with each side equal to AZ. The area of this square is (AZ)² = 3.1415258. This differs from the area of the circle by less than 0.0001. Thus, Crockett Johnson approximately squared the circle.; The painting is signed: CJ68. It is marked on the back: SQUARED CIRCLE* (/) Crockett Johnson 1968 (/) FLAT OIL ON PRESSED WOOD) (/) MATHEMATICALLY (/) DEMONSTRATED (/) TO √π + 0.000000001. It has a white wooden frame. Compare to painting #91 (1979.1093.60).; References: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,” Leonardo 5 (1972): p. 98.; C. Johnson, “A Geometrical look at √π," Mathematical Gazette, 54 (1970): p. 59–60. the figure is from p. 59.
Location: Currently not on view

date made: 1968

painter: Johnson, Crockett

ID Number: 1979.1093.35
catalog number: 1979.1093.35
accession number: 1979.1093

A transversal is a line that intersects a system of other lines or line segments. Here Crockett Johnson explores the properties of certain transversals of the sides of a triangle.

Description: A transversal is a line that intersects a system of other lines or line segments. Here Crockett Johnson explores the properties of certain transversals of the sides of a triangle. The Italian mathematician Giovanni Ceva showed in 1678 that lines drawn from a point to the vertices of a triangle divide the edges of the triangle into six segments such that the product of the length of three nonconsecutive segments equals the product of the remaining three segments.; This painting shows a triangle (in white), lines drawn from a point inside the triangle to the three vertices, and a line drawn from a point outside the triangle (toward the bottom of the painting) to the three vertices. Segments of the sides of the triangle to be multiplied together are of like color. Crockett Johnson's painting combines two diagrams on page 159 of Nathan Court's College Geometry (1964 printing). These diagrams are annotated in his copy of the volume. Several of the triangles adjacent to the central triangle were used by Court in his proof of Ceva's theorem.; The painting is #31 in the series. It is signed: CJ66. There is a wooden frame painted off-white.
Location: Currently not on view

date made: 1966

referenced: Ceva, Giovanni
painter: Johnson, Crockett

ID Number: 1979.1093.22
catalog number: 1979.1093.22
accession number: 1979.1093

Brass screw barrel microscope with several lenses, ivory handle, six ivory slides, and wooden box covered with dark fish skin. James Wilson, an optical instrument maker in London, described the form in 1702.

Description: Brass screw barrel microscope with several lenses, ivory handle, six ivory slides, and wooden box covered with dark fish skin. James Wilson, an optical instrument maker in London, described the form in 1702. He did not make the first instruments of this sort, and never claimed to, but the form was often associated with him.; Ref: James Wilson, “The Description and Manner of Using a Late Invented Set of Small Pocket-Microscopes, Made by James Wilson,” Philosophical Transactions of the Royal Society of London 23 (1702): 1241-1247.; Reginald Clay and Thomas Court, The History of the Microscope (London, 1932), pp. 44-50.
Location: Currently not on view

date made: 1730-1760

ID Number: 1991.0664.0906
accession number: 1991.0664
catalog number: M-06302
collector/donor number: SAP 994
catalog number: 1991.0664.0906

This painting reflects Crockett Johnson's enduring fascination with square roots and squaring.

Description: This painting reflects Crockett Johnson's enduring fascination with square roots and squaring. As the title suggests, it includes four squares whose areas are 1, 2, 3, and 4 square units, and seven line segments whose lengths are the square roots of 2, 3, 4, 5, 6, 7, and 8.; One may construct these squares and square roots by alternate applications of the Pythagorean theorem to squares running along the diagonal of the painting, and to rectangles running across the top (not all the rectangles are shown). More specifically, assume that the light-colored square in the upper left corner of the painting has side of length 1 (which equals the square root of 1). Then the diagonal is the square root of two, and a quarter circle with this radius centered at upper left corner cuts the sides of the square extended to determine two sides of a second, larger square. The area of this square (shown in the painting) is the square of the square root of 2, or two.; One can then consider the rectangle with side one and base square root of two that is in the upper left of the painting. It will have sides one and the square root of 2, and hence diagonal of length equal to the square root of three. The diagonal is not shown, but an circular arc with this radius forms the second arc in the painting. It determines the sides of a square with side equal to the square root of three and area 3. It also forms a rectangle with sides of length one and the square root of 4 (or two). This gives the third arc and the largest square in the painting.; By continuing the construction (further squares and rectangles are not shown), Crockett Johnson arrived at portions of circular arcs that cut the diameter at distances of the square roots of 5, 6, 7, and 8. Only one point on the last arc is shown. It is at the lower right corner of the painting.; Crockett Johnson executed the work in various shades and tints from his starting point at the white and pale-blue triangle to darker blues at the opposite corner.; This oil painting on masonite is not signed and its date of completion is unknown. It is #97 in the series.
Location: Currently not on view

date made: 1970-1975

painter: Johnson, Crockett

ID Number: 1979.1093.65
catalog number: 1979.1093.65
accession number: 1979.1093

Thomas Sinclair (ca 1805-1881) of Philadelphia produced this chromolithographic print of "Chrysomitris marginalis [Bonaparte] male and female" (common name: Black-chinned Siskin) after an original illustration by William Dreser (b. 1820, fl. 1849-1860).

Description (Brief): Thomas Sinclair (ca 1805-1881) of Philadelphia produced this chromolithographic print of "Chrysomitris marginalis [Bonaparte] male and female" (common name: Black-chinned Siskin) after an original illustration by William Dreser (b. 1820, fl. 1849-1860). The image was published as Plate XVII in Volume 2, following page 180 of Appendix F (Zoology-Birds) by John Cassin (1813-1869) in the report describing "The U.S. Naval Astronomical Expedition to the Southern Hemisphere during the Years 1849, 1850, 1851, and 1852" by James M. Gillis (1811-1865). The volume was printed in 1855 by A. O. P. Nicholson (1808-1876) of Washington, D.C.
Description: Thomas Sinclair (c.1805–1881) of Philadelphia printed this lithograph of “Chrysomitris Marginalis [Bonaparte] male and female," now "Carduelis barbata" or Black-chinned siskin, from an original sketch by William Dreser (c.1820–after 1860) of Philadelphia (1847–1860) and New York (1860). The illustration was published in 1855 by A.O.P. Nicholson in Washington, D.C. as Plate XVII in the “Birds” section of volume II of The United States Naval Astronomical Survey to the Southern Hemisphere, written by John Cassin (1813–1869).
Location: Currently not on view

date made: 1855

graphic artist: Sinclair, Thomas; Dreser, William
printer: Nicholson, A. O. P.
publisher: United States Navy
author: Cassin, John; Gilliss, James Melville

ID Number: 2008.0175.03
accession number: 2008.0175
catalog number: 2008.0175.03

This is the third painting by Crockett Johnson to represent the motion of bodies released from rest from a common point and moving along different inclined planes.

Description: This is the third painting by Crockett Johnson to represent the motion of bodies released from rest from a common point and moving along different inclined planes. In the Dialogues Concerning Two New Sciences (1638), Galileo argued that the points reached by the balls at a given time would lie on a circle. Two such circles and three inclined planes, as well as a vertical line of direct fall, are indicated in the painting. One circle has half the diameter of the other. Crockett Johnson also joins the base of points on the inclined planes to the base of the diameters of the circles, forming two sets of right triangles.; This oil painting on masonite is #96 in the series. It has a black background and a wooden and metal frame. It is signed on the back: VELOCITIES AND RIGHT TRIANGLES (GALILEO) (/) Crockett Johnson 1972. Compare to paintings #42 (1979.1093.30) and #71 (1979.1093.46), as well as the figure from Valens, The Attractive Universe: Gravity and the Shape of Space (1969), p. 135.
Location: Currently not on view

date made: 1972

referenced: Galilei, Galileo
painter: Johnson, Crockett

ID Number: 1979.1093.64
catalog number: 1979.1093.64
accession number: 1979.1093

William Dougal (1822–1895) of Washington, D.C. engraved this print of “Crotalus molassus [B & G],” or Black–tailed rattlesnake, from an original sketch likely drawn by John H. Richard (c.1807–1881) of Philadelphia.

Description (Brief): William Dougal (1822–1895) of Washington, D.C. engraved this print of “Crotalus molassus [B & G],” or Black–tailed rattlesnake, from an original sketch likely drawn by John H. Richard (c.1807–1881) of Philadelphia. The illustration was printed as Plate II in the “Reptiles” section of the second part of volume II of the Report on the United States and Mexican Boundary Survey, which was written by Spencer F. Baird (1823–1887). The volume was printed in 1859 by Cornelius Wendell of Washington, D.C.
Description: William Dougal (1822–1895) of Washington, D.C. engraved this print of “Crotalus molassus [B & G],” or Black–tailed rattlesnake, from an original sketch likely drawn by John H. Richard (c.1807–1881) of Philadelphia. The illustration was printed as Plate 2 in the “Reptiles” section of the second part of volume II of the Report on the United States and Mexican Boundary Survey, which was written by Spencer F. Baird (1823–1887). The volume was printed in 1859 by Cornelius Wendell of Washington, D.C.
Location: Currently not on view

graphic artist: Dougal, William H.
printer: Nicholson, A.O.P.
author: Emory, William H.
printer: Wendell, Cornelius
publisher: U.S. Department of the Interior
original artist: Richard, John H.
author: Baird, Spencer Fullerton
publisher: U.S. Army

ID Number: GA.1367
accession number: 1888.20627
catalog number: 1367

This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting.

Description: This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting. Here Crockett Johnson argues that the triangle ABF in the figure is the one he sought, with angle FAB being one seventh of pi. Segment CD in the figure, which appears in the painting, is the length of the edge of a regular 14-sided figure inscribed in a portion of the larger circle shown.; The painting, of oil or acrylic on masonite, is number 105 in the series. It is drawn in shades of cream, blue, and purple on a light purple background. It has a metal frame and is unsigned.
Location: Currently not on view

date made: ca 1973

painter: Johnson, Crockett

ID Number: 1979.1093.72
catalog number: 1979.1093.72
accession number: 1979.1093

To "square" a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle.

Description: To "square" a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle. It was not until the 19th century that the impossibility of squaring a circle was demonstrated.; This painting is an original construction by Crockett Johnson. It begins with the assumprion that the circle has been squared. In this case, Crockett Johnson performed a sequence of constructions that produce several additional squares, rectangles, and circles whose areas are geometrically related to that of the original circle. These figures are produced using traditional Euclidean geometry, and require only straightedge and compass.; The painting on masonite is #102 in the series. It has a blue-black background and a metal frame. It shows various superimposed sections of circles, squares, and rectangles in shades of light blue, dark blue, purple, white and blue-black. It is unsigned. See 1979.3083.02.13.; References: Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (1991), Chapter 5.; Crockett Johnson, "A Geometrical Look at the Square Root of Pi," Mathematical Gazette 54 (February, 1970): pp. 59–60.
Location: Currently not on view

date made: ca 1970

painter: Johnson, Crockett

ID Number: 1979.1093.69
catalog number: 1979.1093.69
accession number: 1979.1093

This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to Kepler's first two laws of planetary motion.

Description: This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to Kepler's first two laws of planetary motion. The ellipse represents the path of a planet and the white sections represent equal areas swept out in equal times. This work is a silk screen on paper. It is number 99 in the series, and is signed in the right corner: Crockett Johnson (/) 67. It draws on a figure from The World of Mathematics by James R. Newman.
Location: Currently not on view

date made: 1967

referenced: Kepler, Johannes
painter: Johnson, Crockett

ID Number: 1979.1093.66
catalog number: 1979.1093.66
accession number: 1979.1093

This piece is a further example of Crockett Johnson's exploration of Kepler’s first two laws of planetary motion. The ellipse represents the path of a planet, and the white sections represent equal areas swept out in equal times.

Description: This piece is a further example of Crockett Johnson's exploration of Kepler’s first two laws of planetary motion. The ellipse represents the path of a planet, and the white sections represent equal areas swept out in equal times. This work, a silk screen inked on paper board, is signed: CJ66. It is #76 in the series, and it echoes painting #22 (1979.1093.16) and painting #99 (1979.1093.66).
Location: Currently not on view

date made: 1966

referenced: Kepler, Johannes
painter: Johnson, Crockett

ID Number: 1979.1093.50
catalog number: 1979.1093.50
accession number: 1979.1093

This colored lithograph of "Buteo calurus [Cassin]," now "Buteo jamaicensis calurus" or Red-tailed Hawk, is believed to have been drawn on stone by William E.

Description (Brief): This colored lithograph of "Buteo calurus [Cassin]," now "Buteo jamaicensis calurus" or Red-tailed Hawk, is believed to have been drawn on stone by William E. Hitchcock (ca 1822-ca 1906), lithographed by Bowen & Company of Philadelphia (ca 1840-1870), and likely hand colored by Bowen firm colorists or Lavinia Bowen (ca 1820- ca 1872).; The image was published as Plate XIV in the "Zoological Portion of the Reports by Lieutenant E. G. Beckwith, Third Artillery, upon the Route near the 38th and 39th Parallels, surveyed by Captain J. W. Gunnison, Corps of Topographical Engineers, and upon the route near the Forty–First Parallel, surveyed by Lieut. E. G. Beckwith, Third Artillery.” The report was published in volume X of the “Reports and Surveys to Ascertain the Most Practable and Economical Route for a Railroad from the Mississippi River to the Pacific Ocean ... 1853, 1856, Volume X," printed in 1859 by Beverley Tucker of Washington, D.C.
Location: Currently not on view

date on report: 1854
date printed in book: 1859

original artist: Cassin, John
publisher: U.S. War Department
author: Beckwith, Edward Griffin
publisher: U.S. Army Corps of Engineers Topographic Command
printer: Nicholson, A. O. P.

ID Number: GA.16332.017
accession number: 1930.110179
catalog number: 16332.017

William H. Dougal (1822-1895) of Washington, D.C. produced this engraving of "Dryophis Vittatus, Grd" from an original illustration by John H. Richard (1807- ca 1881).

Description (Brief): William H. Dougal (1822-1895) of Washington, D.C. produced this engraving of "Dryophis Vittatus, Grd" from an original illustration by John H. Richard (1807- ca 1881). The image was published as Plate XXXVI in Volume 2, following page 210 of Appendix F (Zoology Reptiles) by Charles Girard (1822-1895) in the report describing "The U.S. Naval Astronomical Expedition to the Southern Hemisphere during the Years 1849, 1850, 1851, and 1852" by James M. Gillis (1811-1865). The volume was printed in 1855 by A. O. P. Nicholson (1808-1876) of Washington, D.C.
Location: Currently not on view

date made: 1855

original artist: Richard, John H.
graphic artist: Dougal, William H.
printer: Nicholson, A. O. P.
publisher: United States Navy
author: Gilliss, James Melville

ID Number: 2008.0175.30
accession number: 2008.0175
catalog number: 2008.0175.30

William H. Dougal (1822-1895) of Washington, D.C. produced this pre-publication engraving proof of “Trichomycterus maculatus [Cuv.

Description (Brief): William H. Dougal (1822-1895) of Washington, D.C. produced this pre-publication engraving proof of “Trichomycterus maculatus [Cuv. -Val.], Cheiroden pisciculus [Grd], Cystignathus taeniatus [Grd], and Phyllobates auratus [Grd]” now "Trichomycterus maculatus," "Cheiroden pisciculus," "Batrachyla taeniata," (Banded tree frog), and "Dendrobates auratus" (Poison dart frog, Green poison frog, Green and black poison dart frog) from an original illustration by John H. Richard (1807- ca 1881). The image was published as Plate XXXIV in Volume 2, following page 208 of Appendix F (Zoology-Fishes) by Charles Girard (1822-1895) in the report describing "The U.S. Naval Astronomical Expedition to the Southern Hemisphere during the Years 1849, 1850, 1851, and 1852" by James M. Gillis (1811-1865). The volume was printed in 1855 by A. O. P. Nicholson (1808-1876) of Washington, D.C. The print is also signed in pen "Correct, C.Girard."
Location: Currently not on view

date of book publication: 1855

original artist: Richard, John H.
graphic artist: Dougal, William H.
printer: Nicholson, A. O. P.
publisher: United States Navy
author: Girard, Charles; Gilliss, James Melville

ID Number: 2008.0175.16
accession number: 2008.0175
catalog number: 2008.0175.16

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.

Description: 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).
Location: Currently not on view

date made: 1966

referenced: Plateau, Joseph
painter: Johnson, Crockett

ID Number: 1979.1093.17
catalog number: 1979.1093.17
accession number: 1979.1093

This engraved printing plate was prepared to print an image of "Ptilonopus Perousei" (now Many-colored Fruit Dove, Ptilinopus perousii Peale (S.

Description (Brief): This engraved printing plate was prepared to print an image of "Ptilonopus Perousei" (now Many-colored Fruit Dove, Ptilinopus perousii Peale (S. polynesia)) for the publication "United States Exploring Expedition, During the Years 1838, 1839, 1840, 1841, 1842," Volume 8, Mammalogy and Ornithology, plate 33, in the edition Philadelphia : J. B. Lippincott & Co., 1858. The engraving was produced by Robert Hinshelwood after T. R. Peale.
Description: Robert Hinshelwood (1812–after 1875) of New York City engraved this copper printing plate after a drawing by Expedition Naturalist Titian Ramsey Peale. The image depicts the Ptilonopus Perousei (now Many-colored Fruit Dove, Ptilinopus perousii Peale [S. polynesia]). The engraved illustration was published as Plate 33 in Volume VIII, Mammalogy and Ornithology, by John Cassin, 1858.
Location: Currently not on view

Date made: 1858

publisher: Wilkes, Charles
original artist: Peale, Titian Ramsay
graphic artist: Hinshelwood, Robert
printer: Sherman, Conger
author: Cassin, John

ID Number: 1999.0145.413
catalog number: 1999.0145.413
accession number: 1999.0145

William Dougal (1822–1895) of Washington, D.C.

Description (Brief): William Dougal (1822–1895) of Washington, D.C. engraved this print of “Phrynosoma regale [Girard] and Doliosaurus m’callii [Girard]”—now "Phrynosoma solare" (Regal horned lizard) and "Phrynosoma mcallii" (Flat–tail horned lizard); from one or more original illustrations by John H. Richard (c.1807–1881) of Philadelphia. The illustration was published as Plate 28 in the “Reptiles” section of the second part of volume II of the Report on the United States and Mexican Boundary Survey, written by S.F. Baird (1823–1887). The volume was printed in 1859 by Cornelius Wendell of Washington, D.C.
Description: William Dougal (1822–1895) of Washington, D.C. engraved this print of “Phrynosoma regale [Girard] and Doliosaurus m’callii [Girard]”—now "Phrynosoma solare" (Regal horned lizard) and "Phrynosoma mcallii" (Flat–tail horned lizard); from an original sketch by John H. Richard (c.1807–1881) of Philadelphia. The illustration was printed as Plate 28 in the “Reptiles” section of the second part of volume II of the Report on the United States and Mexican Boundary Survey, written by S.F. Baird (1823–1887). The volume was printed in 1859 by Cornelius Wendell of Washington, D.C.
Location: Currently not on view

date of book publication: 1859

author: Baird, Spencer Fullerton
original artist: Richard, John H.
graphic artist: Dougal, William H.
printer: Wendell, Cornelius
author: Emory, William H.
publisher: U.S. Department of the Interior; U.S. Army

ID Number: 2009.0115.062
catalog number: 2009.0115.062
accession number: 2009.0115

Science & Mathematics

Painting - Curve Tangents (Fermat)

Painting - Problem of Delos Constructed from a Solution by Isaac Newton (Arithmetica Universalis)

Painting - Polar Line of a Point and a Circle (Apollonius)

Painting - Homothetic Triangles (Hippocrates of Chios)

Painting - Rectangles of Equal Area (Pythagoras)

Painting - Bouquet of Triangle Theorems (Euclid)

Painting - Heptagon from Its Seven Sides

Painting - Squared Circle

Painting - Transversals (Ceva)

Microscope

Painting - Squares of 1, 2, 3, 4 and Square Roots to 8

Lithograph of bird species "Chrysomitris Marginalis"

Painting - Velocities and Right Triangles (Galileo)

Engraving of snake species "Crotalus molossus"

Painting - Heptagon 1:3:3 Triangle

Painting - Euclidian Values of a Squared Circle

Painting - Law of Orbiting Velocities

Painting -Law of Orbiting Velocity (Kepler)

Lithograph of bird species "Buteo calurus"

Engraving of snake species "Dryophis vittatus"

Engraving of fish and frog species "Trichomycterus Maculatus, Cheiroden Pisciculus, Cystignathus Taeniatus, and Phyllobates Auratus"

Painting - Geometry of a Triple Bubble (Plateau)

Engraved printing plate "Ptilouopus Perousei"

Engraving of lizard species "Phrynosoma regale, Doliosaurus m'callii"