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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 117 items.

Page 1 of 12

## Painting -

*Reciprocation*- 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

- date made
- 1965

- referenced
- Pappus

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.02

- catalog number
- 1979.1093.02

- accession number
- 1979.1093

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

## Painting -

*Perspective (Alberti)*- 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

- date made
- 1966

- referenced
- Alberti, Leon Battista

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.03

- catalog number
- 1979.1093.03

- accession number
- 1979.1093

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

## Painting -

*Curve Tangents (Fermat)*- 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

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

## Painting -

*Pendulum Momentum (Galileo)*- 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

- date made
- 1966

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.08

- catalog number
- 1979.1093.08

- accession number
- 1979.1093

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

## Painting -

*Centers of Similitude (La Hire)*- 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

- date made
- 1966

- referenced
- de la Hire, Phillipe

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.09

- catalog number
- 1979.1093.09

- accession number
- 1979.1093

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

## Painting -

*Measurement of the Earth (Eratosthenes)*- 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

- date made
- 1966

- referenced
- Eratosthenes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.10

- catalog number
- 1979.1093.10

- accession number
- 1979.1093

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

## Painting -

*Homothetic Triangles (Hippocrates of Chios)*- 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

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

## 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

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