# Mathematical Paintings of Crockett Johnson

• Contents

Inspired by the allure of the space age, many Americans of the 1960s took great interest in mathematics and science. One of them was the cartoonist, book illustrator, and children’s author David Crockett Johnson. From 1965 until his death in 1975 Crockett Johnson painted over 100 works relating to mathematics and mathematical physics. Of these paintings, eighty are found in the collections of the National Museum of American History. We present them here, with related diagrams from the artist’s library and papers.

### Painting - Mystic Hexagon (Pascal)

This painting is based on a theorem generalized by the French mathematician Blaise Pascal in 1640, when he was sixteen years old. When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points.
Description
This painting is based on a theorem generalized by the French mathematician Blaise Pascal in 1640, when he was sixteen years old. When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points. Pappus, writing in the 4th century AD, had shown in his Mathematical Collections that these three points lie on the same line. In the painting, the circle and cream-colored hexagon are at the center, with the sectors associated with different pairs of lines shown in green, blue and gray. The three points of intersection are along the top; the line that would join them is not shown. Pascal generalized the theorem to include hexagons inscribed in any conic section, not just a circle. Hence the figure came to be known as "Pascal’s hexagon" or, to use Pascal’s terminology, the "mystic hexagon." Pascal’s work in this area is known primarily from notes on his manuscripts taken by the German mathematician Gottfried Leibniz after his death.
There is a discussion of Pascal’s hexagon in an article by Morris Kline on projective geometry published in James R. Newman's World of Mathematics (1956). A figure shown on page 629 of this work may have been the basis of Crockett Johnson's painting, although it is not annotated in his copy of the book.
The oil or acrylic painting on masonite is signed on the bottom right: CJ65. It is marked on the back: Crockett Johnson (/) "Mystic" Hexagon (/) (Pascal). It is #10 in the series.
References: Carl Boyer and Uta Merzbach, A History of Mathematics (1991), pp. 359–62.
Florian Cajori, A History of Elementary Mathematics (1897), 255–56.
Morris Bishop, Pascal: The Life of a Genius (1964), pp. 11, 81–7.
Location
Currently not on view
1965
referenced
Pascal, Blaise
painter
Johnson, Crockett
ID Number
1979.1093.05
catalog number
1979.1093.05
accession number
1979.1093

### Painting - Proof of the Pythagorean Theorem (Euclid)

The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides.
Description
The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides. This painting depicts the “windmill” figure found in Proposition 47 of Book I of Euclid’s Elements. Although the method of the proof depicted was written about 300 BC and is credited to Euclid, the theorem is named for Pythagoras, who lived 250 years earlier. It was known to the Babylonians centuries before then. However, knowing a theorem is different from demonstrating it, and the first surviving demonstration of this theorem is found in Euclid’s Elements.
Crockett Johnson based his painting on a diagram in Ivor Thomas’s article on Greek mathematics in The World of Mathematics, edited by James R. Newman (1956), p. 191. The proof is based on a comparison of areas. Euclid constructed a square on the hypotenuse BΓ of the right triangle ABΓ. The altitude of this triangle originating at right angle A is extended across this square. Euclid also constructed squares on the two shorter sides of the right triangle. He showed that the square on side AB was of equal area to the rectangle of sides BΔ and Δ;Λ. Similarly, the area of the square on side AΓ was of equal area to the rectangle of sides EΓ and EΛ. But then the square of the hypotenuse of the right triangle equals the sum of the squares of the shorter sides, as desired.
Crockett Johnson executed the right triangle in the neutral, yet highly contrasting, hues of white and black. Each square area that rests on the sides of the triangle is painted with a combination of one primary color and black. This draws the viewer’s attention to the areas that complete Euclid’s proof of the Pythagorean theorem.
Proof of the Pythagorean Theorem, painting #2 in the series, is one of Crockett Johnson’s earliest geometric paintings. It was completed in 1965 and is marked: CJ65. It also is signed on the back: Crockett Johnson 1965 (/) PROOF OF THE PYTHAGOREAN THEOREM (/) (EUCLID).
Location
Currently not on view
1965
referenced
Euclid
painter
Johnson, Crockett
ID Number
1979.1093.01
catalog number
1979.1093.01
accession number
1979.1093

### Painting - Reciprocation

In this oil or acrylic painting on masonite, Crockett Johnson illustrates a theorem presented by the Greek mathematician Pappus of Alexandria (3rd century AD). Suppose that one chooses three points on each of two line straight segments that do not intersect.
Description
In this oil or acrylic painting on masonite, Crockett Johnson illustrates a theorem presented by the Greek mathematician Pappus of Alexandria (3rd century AD). Suppose that one chooses three points on each of two line straight segments that do not intersect. Join each point to the two more distant points on the other lines. These lines meet in three points, which, according to the theorem, are themselves on a straight line.
The inspiration for this painting probably came from a figure in the article "The Great Mathematicians" by Herbert W. Turnbull found in the artist's copy of James R. Newman's The World of Mathematics (p. 112). This figure is annotated. It shows points A, B, and C on one line segment and D, E, and F on another line segment. Line segments AE and DB, AF and DC, and BF and EC intersect at 3 points (X, Y, and Z respectively), which are collinear. Turnbull's figure and Johnson's painting include nine points and nine lines that are arranged such that three of the points lie on each line and three of the lines lie on each point. If the words "point" and "line" are interchanged in the preceding sentence, its meaning holds true. This is the "reciprocation," or principle of duality, to which the painting's title refers.
Crockett Johnson chose a brown and green color scheme for this painting. The main figure, which is executed in seven tints and shades of brown, contains twelve triangles and two quadrilaterals. The background, which is divided by the line that contains the points X, Y, and Z, is executed in two shades of green. This color choice highlights Pappus' s theorem by dramatizing the line created by the points of intersection of AE and DB, AF and DC, and BC and EC. There wooden frame painted black.
Reciprocation is painting #6 in this series of mathematical paintings. It was completed in 1965 and is signed: CJ65.
Location
Currently not on view
1965
referenced
Pappus
painter
Johnson, Crockett
ID Number
1979.1093.02
catalog number
1979.1093.02
accession number
1979.1093

### Painting - Perspective (Alberti)

Artists used methods of projecting lines developed by the Italian humanist Leon Battista Alberti and his successors to create a sense of perspective in their paintings. In contrast, Crockett Johnson made these methods the subject of his painting.
Description
Artists used methods of projecting lines developed by the Italian humanist Leon Battista Alberti and his successors to create a sense of perspective in their paintings. In contrast, Crockett Johnson made these methods the subject of his painting. He followed a diagram in William M. Ivins Jr., Art & Geometry: A Study in Space Intuitions (1964 edition), p. 76. The figure in Crockett Johnson’s copy of the book is annotated. This painting has a triangle in the center that is divided by a diagonal line, with the left half painted a darker shade than the right. Inside the triangle is one large quadrilateral that is divided into four rows of quadrilaterals that are painted various shades of red, purple, blue, and white.
To represent three-dimensional objects on a two-dimensional canvas, an artist must render forms and figures in proper linear perspective. In 1435 Alberti wrote a treatise entitled De Pictura (On Painting) in which he outlined a process for creating an effective painting through the use of one-point perspective. Investigation of the mathematical concepts underlying the rules of perspective led to the development of a branch of mathematics called projective geometry.
Alberti’s method (as modified by Pelerin in the early 17th century) and Crockett Johnson’s painting begin with the selection of a vanishing point (point C in the figure from Ivins). The eye of the viewer is assumed to be across from and on the same level as C. The eye looks through the vertical painting at a picture that appears to continue behind the canvas. To portray on the canvas what the eye sees, the artist locates point A on the horizon (the horizontal through C). The artist then draws the diagonal from A to the lower right-hand corner of the painting (point I). The separation of the angle ICH into smaller, equal angles creates lines that delineate parallel lines in the picture plane. The horizontal lines that create small quadrilaterals, and thus the checkerboard effect, are determined by the intersections of the lines from C with the diagonals FH and EI.
This painting, #7 in the series, dates from 1966. It is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) PERSPECTIVE (ALBERTI). It is of acrylic or oil paint on masonite, and has a wooden frame.
Location
Currently not on view
1966
referenced
Alberti, Leon Battista
painter
Johnson, Crockett
ID Number
1979.1093.03
catalog number
1979.1093.03
accession number
1979.1093

### Painting - Squares of a 3-4-5 Triangle in Scalene Perspective (Dürer)

This painting, while similar in subject to the painting entitled Perspective (Alberti), depicts three planes perpendicular to the canvas. These three planes provide a detailed, three-dimensional view of space through the use of perspective.
Description
This painting, while similar in subject to the painting entitled Perspective (Alberti), depicts three planes perpendicular to the canvas. These three planes provide a detailed, three-dimensional view of space through the use of perspective. Three vanishing points are implied (though not shown) in the painting, one in each of the three planes.
The painting shows a 3-4-5 triangle surrounded by squares proportional in number to the square of the side. That is, the horizontal plane contains nine squares, the vertical plane contains sixteen squares, and the oblique plane, which represents the hypotenuse of the 3-4-5 triangle, contains twenty-five squares. This explains the extension of the vertical and oblique planes and reminds the viewer of the Pythagorean theorem.
The title of this painting points to the role of the German artist Albrecht Dürer (1471–1528) in creating ways of representing three-dimensional figures in a plane. Dürer is particularly remembered for a posthumously published treatise on human proportion. In his book entitled The Life and Art of Albrecht Dürer, art historian Erwin Panofsky explains that the work of Dürer with perspective demonstrated that the field was not just an element of painting and architecture, but an important branch of mathematics.
This construction may well have originated with Crockett Johnson. However, he may have been influenced by Figure 1 (p. 604) and Figure 3 (p. 608) in Panofsky’s article on Dürer as a Mathematician in The World of Mathematics, edited by James R. Newman (1956). Johnson did not annotate either of these diagrams. The oil painting was completed in 1965 and is signed: CJ65. It is #8 in his series of mathematical paintings.
Location
Currently not on view
1965
referenced
Duerer, Albrecht
painter
Johnson, Crockett
ID Number
1979.1093.04
catalog number
1979.1093.04
accession number
1979.1093

### Painting - Conic Curve (Apollonius)

In ancient times, the Greek mathematician Apollonius of Perga (about 240–190 BC) made extensive studies of conic sections, the curves formed when a plane slices a cone.
Description
In ancient times, the Greek mathematician Apollonius of Perga (about 240–190 BC) made extensive studies of conic sections, the curves formed when a plane slices a cone. Many centuries later, the French mathematician and philosopher René Descartes (1596–1650) showed how the curves studied by Apollonius might be related to points on a straight line. In particular, he introduced an equation in two variables expressing points on the curve in terms of points on the line. An article by H. W. Turnbull entitled "The Great Mathematicians" found in The World of Mathematics by James R. Newman discussed the interconnections between Apollonius and Descartes, and apparently was the basis of this painting. The copy of this book in Crockett Johnson's library is very faintly annotated on this page. Turnbull shows variable length ON, with corresponding points P on the curve.
The analytic approach to geometry taken by Descartes would be greatly refined and extended in the course of the seventeenth century.
Johnson executed his painting in white, purple, and gray. Each section is painted its own shade. This not only dramatizes the coordinate plane but highlights the curve that extends from the middle of the left edge to the top right corner of the painting.
Conic Curve, an oil or acrylic painting on masonite, is #11 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) CONIC CURVE (APOLLONIUS). It has a wooden frame.
Location
Currently not on view
1966
referenced
Apollonius of Perga
painter
Johnson, Crockett
ID Number
1979.1093.06
catalog number
1979.1093.06
accession number
1979.1093

### Painting - Curve Tangents (Fermat)

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
1966
referenced
Fermat, Pierre de
painter
Johnson, Crockett
ID Number
1979.1093.07
catalog number
1979.1093.07
accession number
1979.1093

### Painting - Pendulum Momentum (Galileo)

The Greek mathematician Aristotle, who lived from about 384 BC through 322 BC, believed that heavy bodies moved naturally downward, while lighter substances such as air naturally ascended.
Description
The Greek mathematician Aristotle, who lived from about 384 BC through 322 BC, believed that heavy bodies moved naturally downward, while lighter substances such as air naturally ascended. Other forms of terrestrial motion required a sustaining force, which was not expressed mathematically. The Italian Galileo Galilei (1564–1642) challenged Aristotle. He held that motion was persistent and would continue until acted upon by an opposing, outside force.
In a book entitled Dialogues Concerning the Two Chief World Systems, Galileo presented his ideas in a dispute between three men: Salviati, Sagredo, and Simplicio. Salviati, a spokesman for Galileo, explained his revolutionary ideas, one of which is illustrated by a diagram that was the basis for this painting. This image can be found in Crockett Johnson's copy of The World of Mathematics, a book by James R. Newman. It is probable that this image served as inspiration for this painting, although Johnson did not annotate this diagram.
In Galileo's Dialogues, Salviati argued that if a lead weight is suspended by a thread from point A (see figure) and is released from point C, it will swing to point D, which is located at the same height as the initial point C. Furthermore, Salviati stated that if a nail is placed at point E so that the thread will snag on it, then the weight will swing from point C to point B and then up to point G, which is also located at the same height as the initial point C. The same occurs if a nail is placed at point F below the line segment CD.
The painting is executed in purple that progresses from light tints to darker shades right to left. This gives the figure a sense of motion akin to that of a pendulum. The background is washed in gray and black. The line created by the initial and final height of the weight divides the background.
Pendulum Momentum, a work in oil on masonite, is painting #13 in the Crockett Johnson series. It was executed in 1966 and is signed: CJ66. There is a wooden frame painted black.
Location
Currently not on view
1966
referenced
Galilei, Galileo
painter
Johnson, Crockett
ID Number
1979.1093.08
catalog number
1979.1093.08
accession number
1979.1093

### Painting -Centers of Similitude (La Hire)

Two circles or other similar figures can be placed such that a line drawn from some fixed point to a point of one of them passes through a point on the other, such that the ratio of the distances from the fixed point to the two points is always the same.
Description
Two circles or other similar figures can be placed such that a line drawn from some fixed point to a point of one of them passes through a point on the other, such that the ratio of the distances from the fixed point to the two points is always the same. The fixed point is called the center of similitude. The circles shown in this painting have two centers of similitude, one between the circles and one to the right (the center of similitude between the circles is shown). Crockett Johnson apparently based his painting on a diagram from the book College Geometry by Nathan Altshiller Court (1964 printing). This diagram is annotated in his copy of the book. In the figure, the larger circle has center A, the smaller circle has center B, and the centers of similitude are the points S and S'. S is called the external center of similitude and S' is the internal center of similitude. The painting suggests several properties of centers of similitude. For example, lines joining corresponding endpoints of parallel diameters of the two circles, such as TT' in the figure, would meet at the external center of similitude. Lines joining opposite endpoints meet at the internal center of similitude.
This painting emphasizes the presence of the two circles and line segments relating to centers of similitude, but not the centers themselves. Indeed, the painting is too narrow to include the external center of similitude.
Some properties of centers of similitude were known to the Greeks. More extensive theorems were developed by the mathematician Gaspard Monge (1746–1818). It is not entirely clear why Crockett Johnson associated the painting with the artist and mathematician Phillipe de la Hire (1640–1718). A bibliographic note in the relevant section of Court reads: LHr., p. 42, rem. 8. However, Court was referring to an 1809 book by Simon A. J. LHuiler on the elements of analytic geometry.
This oil painting on masonite is #14 in Crockett Johnson's series. It was completed in 1966 and is signed: CJ66.
References: R. J. Archibald, "Centers of Similitude of Circles," American Mathematical Monthly, 22, #1 (1915), pp. 6–12; unpublished notes of J. B. Stroud.
Location
Currently not on view
1966
referenced
de la Hire, Phillipe
painter
Johnson, Crockett
ID Number
1979.1093.09
catalog number
1979.1093.09
accession number
1979.1093

### Painting - Measurement of the Earth (Eratosthenes)

The determination of the size and shape of the Earth has occupied philosophers from antiquity. Eratosthenes, a mathematician in the city of Alexandria in Egypt who lived from about 275 through 194 BC, proposed an ingenious way to measure the circumference of the Earth.
Description
The determination of the size and shape of the Earth has occupied philosophers from antiquity. Eratosthenes, a mathematician in the city of Alexandria in Egypt who lived from about 275 through 194 BC, proposed an ingenious way to measure the circumference of the Earth. It is illustrated by this painting. Eratosthenes claimed that the town of Syene (now Aswan) was directly south of Alexandria, and that the distance between the cities was known. Moreover, he reported that on a day when the vertical rod of a sundial cast no shadow at noon in Syene, the shadow cast by a similar rod at Alexandria formed an angle of 1/50 of a complete circle.
In the Crockett Johnson painting, the circle represents the Earth and the two line segments drawn from the center display the direction of the two rods. The two parallel lines represent rays of sunlight striking the Earth, the dark-purple region the shadowed area. The angle of the shadow equals the angle subtended at the center of the Earth, hence the circumference of the entire Earth can be computed when the angle and the distance of the cities is known.
Crockett Johnson's painting may be after a diagram from the book by James R. Newman entitled The World of Mathematics (p. 206), although the figure is not annotated. Newman published a brief extract describing ideas of Eratosthenes, based on a first century BC account by Cleomedes.
The Crockett Johnson painting is #15 in the series. It is marked on the back : Crockett Johnson 1966 (/) MEASUREMENT OF THE EARTH (/) (ERATOSTHENES).
Reference: O. Pederson and M. Phil, Early Physics and Astronomy (1974), p. 53.
Location
Currently not on view
1966
referenced
Eratosthenes
painter
Johnson, Crockett
ID Number
1979.1093.10
catalog number
1979.1093.10
accession number
1979.1093

### Painting - Homothetic Triangles (Hippocrates of Chios)

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
1966
referenced
Hippocrates of Chios
painter
Johnson, Crockett
ID Number
1979.1093.11
catalog number
1979.1093.11
accession number
1979.1093

### Painting - Pencil of Ratios (Monge)

The history of projective geometry begins with the work of the French mathematician Gerard Desargues (1591–1661). During his lifetime his work was well known in some mathematical circles, but after his death, his contributions to the field were largely forgotten.
Description
The history of projective geometry begins with the work of the French mathematician Gerard Desargues (1591–1661). During his lifetime his work was well known in some mathematical circles, but after his death, his contributions to the field were largely forgotten. When Gaspard Monge (1746–1818) and his student, Jean-Victor Poncelet (1788–1867) began their studies of projective geometry, they were largely unaware of the work of Desargues. This may be why Crockett Johnson included Monge's name as opposed to Desargues' in this painting's title.
One of the fundamental concepts of projective geometry, which was touched upon, but not fully understood, by the Greeks, is that of a cross-ratio, or "ratio of ratios." It is the topic of Johnson's painting. If points A, B, C, and D on line l are projected from point O, and if the line l’ crosses the four projected line segments, then the ratio of ratios (A’C’/C’B’)/(A’D’/ D’B’) of the corresponding points A’,B’,C’, and D’ is the same as the ratio of ratios (AC/CB)/(AD/DB). Thus, a cross-ratio is a projective invariant for all line segments l’.
The artist may have received inspiration for this painting from his copy of James R. Newman's The World of Mathematics (1956), p. 632. The figure is found there in an article by Morris Kilne entitled "Projective Geometry." This figure is not annotated, and the painting flips Kline's image.
Crockett Johnson chose purple, white, black, and brown to color this work. He executed the projection in three tints of purple and one shade of white. The background, which is divided by line l’, was executed in black and brown.
Pencil of Ratios, an oil painting on masonite, is #18 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) PENCIL OF RATIOS (MONGE). The painting is unframed.
Location
Currently not on view
1966
referenced
Monge, Gaspard
painter
Johnson, Crockett
ID Number
1979.1093.12
catalog number
1979.1093.12
accession number
1979.1093

### Painting - Square Root of Two (Descartes)

La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots.
Description
La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots. This construction is the subject of Crockett Johnson's painting. Descartes explained: "If the square root of GH is desired, I add, along the same straight line, FG equal to unity, then bisecting FH at K, I describe the circle FIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root." (this is a translation of portion of La Géométrie, as published by J. R. Newman, The World of Mathematics (1956), p. 241)
To understand Descartes' description and the title of this painting, consider the diagram. An angle inscribed in a semicircle is a right angle, thus triangle FGI is similar to triangle IGH. Because this two triangles are similar, their corresponding sides are proportional. Thus, G/IFG = GH/GI. But FG is equal to one, so GH is the square of GI, and GI the square root of GH desired.
In his painting, Crockett Johnson has flipped the image from La Géométrie found in his copy of The World of Mathematics. This figure is not annotated. The artist divided his painting into squares of area one, suggesting what came to be called Cartesian coordinates. The division indicates that the GH chosen has length two.
Johnson chose white for the section of the semicircle that contains the edge of length equal to the square root of GH. This section provides a vivid contrast against the dull, surrounding colors. Crockett Johnson purposefully creates this area of interest to draw focus to the result of Descartes' construction.
Square Root of Two is painting #19 in the series. It was painted in oil or acrylic on masonite, completed in 1965, and is signed: CJ65. The wooden frame is painted black.
Location
Currently not on view
1965
referenced
Descartes, Rene
painter
Johnson, Crockett
ID Number
1979.1093.13
catalog number
1979.1093.13
accession number
1979.1093

### Painting - Fluxions (Newton)

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
Description
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.
Location
Currently not on view
1966
referenced
Newton, Isaac
painter
Johnson, Crockett
ID Number
1979.1093.14
catalog number
1979.1093.14
accession number
1979.1093

### Painting - Cross-Ratio in a Conic (Poncelet)

From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle.
Description
From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle.
Nineteenth-century mathematicians were much interested in the properties of conics that were preserved under projection. They knew from the work of the ancient mathematician Pappus that the cross ratio of line segments created by two straight lines cutting the same pencil of lines was a constant. In Figure 5, which is from an article by Morris Kline in James R. Newman's The World of Mathematics, if line segment l’ crosses lines emanating from the point O at points A’, B’, C’ and D’; and line segment l croses the same lines at points A, B, C, and D, the cross ratio:
(A’C’/C’B’) / (A’D’/D’B’) = (AC/BC) / (AD/DB). In other words, it is independent of the cutting line. (see the Crockett Johnson painting Pencil of Ratios (Monge) ).
The French mathematician Michel Chasles introduced a related result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio.This opened up an entirely different way of describing conic sections. Crockett Johnson associated this particular painting with another French advocate of projective geometry, Victor Poncelet.
This oil painting on masonite is #21 in the series. It has a dark gray background and a wood and metal frame. It shows a large black ellipse with two pencils of lines linked to the same four lines of the ellipse. The painting is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 ( /) CROSS-RATIO IN A CONIC (/) (PONCELET). Compare painting #69 (1979.1093.44).
Reference: This painting is based on a figure in James R. Newman, The World of Mathematics (1956), p. 634. This volume was in the Crockett Johnson library. The figure on this page is annotated. For a figure on cross-ratios, see p. 632.
Location
Currently not on view
1966
referenced
Poncelet, Jean-Victor
painter
Johnson, Crockett
ID Number
1979.1093.15
catalog number
1979.1093.15
accession number
1979.1093

### Painting - Law of Orbiting Velocity (Kepler)

This work illustrates two laws of planetary motion proposed by the German mathematician Johannes Kepler (1571–1630) in his book Astronomia Nova (New Astronomy) of 1609. Kepler argued that planets move about the sun in elliptical orbits, with the sun at one focus of the ellipse.
Description
This work illustrates two laws of planetary motion proposed by the German mathematician Johannes Kepler (1571–1630) in his book Astronomia Nova (New Astronomy) of 1609. Kepler argued that planets move about the sun in elliptical orbits, with the sun at one focus of the ellipse. He also claimed that a planet moves about the sun in such a way that a line drawn from the planet to the sun sweeps out equal areas in equal times. The ellipse in the work represents the path of a planet and the white sections equal areas. The extraordinary contrast between the deep blue and white colors dramatize this phenomenon.
This oil painting on masonite has a wooden frame. It is signed: CJ65. It also is marked on the back: Crockett Johnson 1965 (/) LAW OF ORBITING VELOCITY (/) (KEPLER). It is #22 in the series. The work follows an annotated diagram from Crockett Johnson’s copy of Newman's The World of Mathematics (1956), p. 231. Compare to paintings #76 (1979.1093.50) and #99 (1979.1093.66).
Reference: Arthur Koestler, The Watershed (1960).
Location
Currently not on view
1965
referenced
Kepler, Johannes
painter
Johnson, Crockett
ID Number
1979.1093.16
catalog number
1979.1093.16
accession number
1979.1093

### Painting - Geometry of a Triple Bubble (Plateau)

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
1966
referenced
Plateau, Joseph
painter
Johnson, Crockett
ID Number
1979.1093.17
catalog number
1979.1093.17
accession number
1979.1093

### Painting - Harmonic Series from a Quadrilateral (Pappus)

The concept of a harmonic set of points can be traced back through Girard Desargues (1591–1661) and Pappus of Alexandria (3rd century AD) to Apollonius of Perga (240–190 BC). Crockett Johnson's painting seems to be based upon a figure associated with Pappus.
Description
The concept of a harmonic set of points can be traced back through Girard Desargues (1591–1661) and Pappus of Alexandria (3rd century AD) to Apollonius of Perga (240–190 BC). Crockett Johnson's painting seems to be based upon a figure associated with Pappus. It is likely that Crockett Johnson was inspired by a figure found in H. W. Turnbull's article "The Great Mathematicians" found in his copy of James R. Newman's The World of Mathematics, p. 111. This figure is annotated.
The construction begins with a given set of collinear points (A, B, and Y). An additional point (W) is sought such that AW, AB, and AY are in harmonic progression. That is, the terms AW, AB, and AY represent a progression of terms whose reciprocals form an arithmetic progression. To do this, any point Z, not on line AB, is chosen, and line segments ZA and ZB are constructed. Next, any point D, on ZA, is chosen, and DY, which will intersect ZB at C, is constructed. AC and DB intersect each other at X, and ZX will intersect AB at W. The location of point W is entirely independent of the choice of points Z and D. It follows that AW, AB, and AY form a harmonic progression, and thus the points A, W, B, and Y form a harmonic set.
Crockett Johnson flipped the annotated image for his painting. The boldest portion of his painting, and thus the area with greatest interest, is the quadrilateral ABCD. In addition, the background of his painting is divided into three differently colored sections to illustrate the harmonic series constructed from the quadrilateral. This careful color choice reinforces the painting's title.
This painting was executed in oil on masonite and is painting #24 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) HARMONIC SERIES FROM A QUADRILATERAL (/) (PAPPUS). It has a gray wooden frame.
Location
Currently not on view
1966
referenced
Pappus
painter
Johnson, Crockett
ID Number
1979.1093.18
catalog number
1979.1093.18
accession number
1979.1093

### Painting - Bouquet of Triangle Theorems (Euclid)

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
1966
referenced
Euclid
painter
Johnson, Crockett
ID Number
1979.1093.19
catalog number
1979.1093.19
accession number
1979.1093

### Painting - Point Collineation in the Triangle (Euler)

Leonhard Euler (1707–1783) was the most prolific mathematician of the eighteenth century. He made significant contributions to geometry, calculus, mechanics, and number theory.
Description
Leonhard Euler (1707–1783) was the most prolific mathematician of the eighteenth century. He made significant contributions to geometry, calculus, mechanics, and number theory. He produced more than 800 publications during his lifetime, almost half of which were dictated after his eyesight failed in 1766. While Euler is best remembered for his contributions to analysis and mechanics, his interests included geometry. This figure illustrates a theorem about triangles associated with his name.
Euler showed that three points related to a triangle lie on a common line. The first is the circumcenter (point O in the figure), the intersection of the perpendicular bisectors of the three sides. This point is the center of the circle which passes through the vertices of the triangle. Johnson also constructed the three medians of the triangle and the three altitudes of the triangle. The medians intersect in a common point (point N in the figure) and the altitudes meet at a third point (H in the figure). These three points, Euler showed, lie on the same line. In the painting, Crockett Johnson also constructed the circle that circumscribes the triangle, as well as a circle of half the radius known as the nine-point circle. For a full description of this circle, see painting #75 (1979.1093.49).
In the painting, the circumcircle is centered exactly on the backing, and the Euler line extends from the lower right corner to the upper left corner. This divides the work into two triangles of equal area. The right half of the painting was executed in shades of red and purple, while the left half of the painting was executed in shades of gray and black. Crockett Johnson also joined the nine points of the nine-point circle to form an irregular polygon.
This oil painting on masonite is #28 in the series. There is a wooden frame painted black. The work was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) POINT COLLINEATION IN THE TRIANGLE (/) (EULER). For a related painting, see #75 (1979.1093.49).
Reference: Nathan A. Court, College Geometry (1964 printing), p. 103, cover. The figure on p. 103 is annotated.
Location
Currently not on view
1966
referenced
Euler, Leonhard
painter
Johnson, Crockett
ID Number
1979.1093.20
catalog number
1979.1093.20
accession number
1979.1093

### Painting - Every Positive Integer (Gauss)

This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old.
Description
This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old. The proposition, one of Gauss’s many contributions to the branch of mathematics called number theory, states that every positive integer is the sum of three triangular numbers. The concept of triangular numbers dates to antiquity. Suppose one arranges dots in rows, with one in the first row, two in the second, three in the first and so forth. Three dots form a triangle, as do 6 dots, 10 dots, and 15 dots. The numbers 3, 6, 10, 15, and so forth are called "triangular numbers." The integers 0 and 1 are thought of as special cases of triangular numbers.
Crockett Johnson derived his painting from an entry in Gauss's diary published in an article by Eric Temple Bell included by James R. Newman in his book The World of Mathematics (1956), p. 304. The entry includes the phrase EUREKA in Greek, and indicates that any positive integer is the sum of three triangular numbers.
Crockett Johnson’s painting abstractly represents this theorem through the juxtaposition of three triangles. The triangles are equal, but each figure is painted a different color. It is possible that the artist chose to illustrate each triangle in its own color to demonstrate that each triangle generally represents its own triangular number when computing a positive integer. However, the triangles are congruent, which reminds the viewer that the triangles are related because they all represent a triangular number.
This work was painted in oil on masonite, completed in 1966, and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) EVERY POSITIVE INTEGER (/) (GAUSS). It is painting #29 in the series, and has a wooden frame.
Reference: J. R. Newman, The World of Mathematics, 1956, p. 304.
Location
Currently not on view
1966
referenced
Gauss, Carl Friedrich
painter
Johnson, Crockett
ID Number
1979.1093.21
catalog number
1979.1093.21
accession number
1979.1093

### Painting - Transversals (Ceva)

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
1966
referenced
Ceva, Giovanni
painter
Johnson, Crockett
ID Number
1979.1093.22
catalog number
1979.1093.22
accession number
1979.1093

### Painting - One Surface and One Edge (Möbius)

Most geometric surfaces have a distinct inside and outside. This painting shows one that doesn’t. Take a strip of material, give it a half-twist, and attach the ends together. The result is a band with only one surface and one edge.
Description
Most geometric surfaces have a distinct inside and outside. This painting shows one that doesn’t. Take a strip of material, give it a half-twist, and attach the ends together. The result is a band with only one surface and one edge. Mathematicians began to explore such surfaces in the nineteenth century. In 1858 German astronomer and mathematician August Ferdinand Möbius (1790–1868), who had studied theoretical astronomy under Carl Friedrich Gauss at the University of Goettingen, discovered the one-sided surface shown in the painting. It has come to be known by his name. As often happens in the history of mathematics, another scholar, Johann Benedict Listing, had found the same result a few months earlier. Listing did not publish his work until 1861.
If one attaches the ends of a strip of paper without a half twist, the resulting figure is a cylinder. The cylinder has two sides such that one can paint the outside surface red and the inside surface green. If you try to paint the outside surface of a Möbius band red you will paint the entire band red without crossing an edge. Similarly, if you try to paint the inside surface of a Möbius band green you will paint the entire surface green. A cylinder has an upper edge and a lower edge. However, if you start at a point on the edge of a Möbius band you will trace out its entire edge and return to the point at which you began. Since Möbius's time, mathematicians have discovered and explored many other one-sided surfaces.
This painting, #34 in the series, was executed in oil on masonite and is signed: CJ65. The strip is shown in three shades of gray based on the figure’s position. The shades of gray, especially the lightest shade, are striking against the rose-colored background, and this contrast allows the viewer to focus on the properties of the Möbius band. The painting has a wooden frame.
Crockett Johnson's painting is similar to illustrations in James R. Newman's The World of Mathematics (1956), p. 596. However, the figures are not annotated in the artist's copy of the book.
Location
Currently not on view
1965
referenced
Moebius, August Ferdinand
painter
Johnson, Crockett
ID Number
1979.1093.23
catalog number
1979.1093.23
accession number
1979.1093

### Painting - Simple Equation (Descartes)

In a pathbreaking book La Géométrie, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting.
Description
In a pathbreaking book La Géométrie, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting. More specifically, Descartes described geometrical methods for finding the roots of simple polynomials. He wrote (as translated from the original French): "Finally, if I have z² = az -b², I make NL equal to (1/2)a and LM equal to b as before: then, instead of joining the points M and N, I draw MQR parallel to LN, and with N as center describe a circle through L cutting MQR in the points Q and R; then z, the line sought, is either MQ or MR, for in this way it can be expressed in two ways, namely: z = (1/2)a + √((1/4)a² - b²) and z = (1/2)a - √((1/4)a² - b²)."
To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.
Crockett Johnson's painting directly imitates Descartes's figure found in Book I of La Géométrie. A translation of part of Book I is found in the artist’s copy of James R. Newman's The World of Mathematics. The figure on page 250 is annotated.
This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.
Location
Currently not on view
1966
referenced
Descartes, Rene
painter
Johnson, Crockett
ID Number
1979.1093.24
catalog number
1979.1093.24
accession number
1979.1093

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