#
Science & Mathematics

The Museum's collections hold thousands of objects related to chemistry, biology, physics, astronomy, and other sciences. Instruments range from early American telescopes to lasers. Rare glassware and other artifacts from the laboratory of Joseph Priestley, the discoverer of oxygen, are among the scientific treasures here. A Gilbert chemistry set of about 1937 and other objects testify to the pleasures of amateur science. Artifacts also help illuminate the social and political history of biology and the roles of women and minorities in science.

The mathematics collection holds artifacts from slide rules and flash cards to code-breaking equipment. More than 1,000 models demonstrate some of the problems and principles of mathematics, and 80 abstract paintings by illustrator and cartoonist Crockett Johnson show his visual interpretations of mathematical theorems.

"Science & Mathematics - Overview" showing 2182 items.

Page 4 of 219

## Painting -

*Pencil of Ratios (Monge)*- 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’B’/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

- date made
- 1966

- referenced
- Monge, Gaspard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.12

- catalog number
- 1979.1093.12

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Fluxions (Newton)*- 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

- date made
- 1966

- referenced
- Newton, Isaac

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.14

- catalog number
- 1979.1093.14

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Cross-Ratio in a Conic (Poncelet)*- 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

- date made
- 1966

- referenced
- Poncelet, Jean-Victor

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.15

- catalog number
- 1979.1093.15

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Law of Orbiting Velocity (Kepler)*- 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

- date made
- 1965

- referenced
- Kepler, Johannes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.16

- catalog number
- 1979.1093.16

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Geometry of a Triple Bubble (Plateau)*- 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

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Harmonic Series from a Quadrilateral (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

- date made
- 1966

- referenced
- Pappus

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.18

- catalog number
- 1979.1093.18

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Bouquet of Triangle Theorems (Euclid)*- 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

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Point Collineation in the Triangle (Euler)*- 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

- date made
- 1966

- referenced
- Euler, Leonhard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.20

- catalog number
- 1979.1093.20

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Every Positive Integer (Gauss)*- 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

- date made
- 1966

- referenced
- Gauss, Carl Friedrich

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.21

- catalog number
- 1979.1093.21

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Transversals (Ceva)*- 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

- Data Source
- National Museum of American History, Kenneth E. Behring Center