#
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 2082 items.

Page 7 of 209

## Painting -

*Aligned Triangles (Desargues)*- Description
- In the 17th century, the French engineer and architect Girard Desargues (1591–1661) explored interconnections between extensions of the lines within a pencil of three line segments (a pencil of line segments consists of several line segments originating at a common point). His theorems, as published in his own extremely obscure work and also by his contemporary, Abraham Bosse, were extended in the 19th century, and proved of fundamental importance to projective geometry.

- Crockett Johnson's library contains discussions of Desargues' theorem by H. S. M. Coxeter, N. A. Court, Heinrich Dorrie, and William M. Ivins. This painting most resembles a figure from Coxeter, although the diagram is not annotated. Suppose that the vertices of two triangles (PQR and P'Q'R' in Figure 1.5B from Coxeter) lie on a pencil of three line segments emanating from the point O. Suppose that similarly situated sides of the two triangles can be extended to meet in the three points denoted by A, C and B in the figure. According to Desargues' theorem, A, C, and B are collinear.

- In the painting, the two concurrent triangles are shown in shades of gray and black, while the top of the pencil of three lines is in shades of gold. Extensions of the sides and their points of intersection are clearly shown. Both the figure and the background of the painting are divided by the line joining the points of intersection

- The painting is #63 in the series. It is painted in oil or acrylic on masonite, and has a brown wooden frame. The painting is signed: CJ70.

- References:

- Newman, J. R.,
*The World of Mathematics*, p. 133. Figure annotated.

- Court, N. A.,
*College Geometry*(1952), pp. 163–5. The figure is not annotated.

- Coxeter, H. S. M.,
*The Real Projective Plane*, (1955 edition), p. 7. The figure resembles the painting but is not annotated.

- Dorrie, Heinrich,
*100 Great Problems of Elementary Mathematics: Their History and Solution*(1965), p. 267. There is an annotated figure here for another theorem of Desargues, the theorem of involution.

- Field, J. V.,
*The Invention of Infinity: Mathematics and Art in the Renaissance*(1997), pp. 190–206.

- Ivins, William M. Jr.,
*Art & Geometry: A Study in Space Intuitions*(1946), pp. 87–94.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Desargues, Girard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.38

- accession number
- 1979.1093

- catalog number
- 1979.1093.38

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

## Painting -

*Golden Rectangle*- Description
- Crockett Johnson annotated several diagrams in his copy of Valens’s book
*The Number of Things*, and used a few of them as the basis of paintings. This is one example. It shows three golden rectangles, the curves from a compass used to construct the rectangles, and a section of a five-pointed Pythagorean star.

- Euclid showed in his
*Elements*that it is possible to divide a line segment into two smaller segments wherein the ratio of the whole length to the longer part equals the ratio of the longer part to the smaller. He used this theorem in his construction of a regular pentagon. This ratio came to be called the “golden ratio.”

- A golden rectangle is a rectangle whose sides adhere to the golden ratio (in modern terms, the ratio of its length to its width equals (1 + √(5) ) /2, or about 1.62). The golden rectangle is described as the rectangle whose proportions are most pleasing to the eye.

- This painting shows the relationship between a golden rectangle and a five-pointed Pythagorean star by constructing the star from the rectangle. It follows a diagram on the top of page 131 in Evans G. Valens,
*The Number of Things*. This diagram is annotated. Valens describes a geometrical solution to the two expressions f x f = e x c and f = e - c, and associates it with the Pythagoreans. The right triangle on the upper part of Valens's drawing, with the short side and part of the hypotenuse equal to f, is shown facing to the left in the painting. It can be constructed from a square with side equal to the shorter side of the rectangle. Two of the smaller rectangles in the painting are also golden rectangles. Crockett Johnson also includes in the background the star shown by Valens and related lines.

- This painting on masonite, #64 in the series, dates from 1970 and is signed: CJ70. It also is marked on the back: ”GOLDEN RECTANGLE (/) Crockett Johnson 1970. It is executed in two hues of gold to emphasize individual sections. While this method creates a detailed and organized contrast, it disguises the three rectangles and the star. Compare paintings 1979.1093.33 (#46) and 1979.1093.70 (#103).

- Reference: Evans G. Valens,
*The Number of Things*(1964), p. 131.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.39

- accession number
- 1979.1093

- catalog number
- 1979.1093.39

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

## Painting -

*Geometric Mean (Archytas)*- Description
- This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as radius. Erect a perpendicular at the point where the line segments meet (d in the figure), and consider this to be the altitude of a right triangle inscribed in the semicircle. By similar triangles, the length of the perpendicular of a triangle inscribed in a semicircle is the geometric mean of the two lengths into which it divides the diameter of the circle. Hence the length of d is the mean of the segments e and f.

- This painting an orange-red background, and shows a triangle inscribed in an orange semicircle. The perpendicular from the right angle of the triangle divides the triangle into triangles similar to it, painted in black and white.

- The painting, and the attribution of the theorem to Archytas, are based on a passage from Evans G. Valens,
*The Number of Things: Pythagoras, Geometry and Humming Strings*(1964), p. 118. The figure on this page of this book from Crockett Johnson's library is annotated.

- This oil painting on masonite is #65 in the series. It is inscribed on the back: GEOMETRIC MEAN (ARCHYTAS) (/) Crockett Johnson 1968. It has a wooden frame.

- Location
- Currently not on view

- date made
- 1968

- referenced
- Archytas

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.40

- accession number
- 1979.1093

- catalog number
- 1979.1093.40

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

## Painting -

*Square Roots of One, Two and Three*- Description
- Crockett Johnson much enjoyed constructing square roots of numbers geometrically. He offered the following account of this painting, as well as the figure shown: "Let AN and BN be 1. Then the diagonal AB is the square root of 2, because it is the hypotenuse of a right triangle with sides of length √1 and √1. The large right triangle √1 plus √2 adds up to a hypotenuse of √3. The compass traces pronounce a statement and also declare its proof. The square root of 2 is 1.4142 . . . and the square root of 3 is 1.7321 . . . Their decimals run on and on but as produced by the compass and blind straightedge both numbers are quite as finite as 1. The triangle embodies three dimensions of the cube. CB is any edge, AB is a face diagonal, and AC is an internal diagonal." Crockett-Johnson described the source of the painting as "Artist's Construction, or Anybody's."

- The triangle with three sides equal to the lengths of interest is painted white. Remaining segments of the construction are in dark gray and purple, with a black background. The painting has a brown wooden frame.

- The painting is #66 in the series and is signed: CJ69. For a related painting, see #45 (1979.1093.32).

- Reference: "Geometric Geometric [sic] Paintings by Crockett Johnson" NMAH Collections.

- Location
- Currently not on view

- date made
- 1969

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.41

- accession number
- 1979.1093

- catalog number
- 1979.1093.41

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

## Painting -

*Squared Lunes (Hippocrates Of Chios)*- Description
- Classical Greek mathematicians were able to square all convex polygons. That is, given any polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straight edge. Figures with curved sides proved more difficult. However, as this painting suggests, the mathematician Hippocrates of Chios (5th century BC) squared a lune, a figure bounded by arcs of two circle with different radii (lunes resemble quarter moons, hence the name). Finding the area of a lune in terms of a square might seem more difficult than squaring a circle, but the latter problem would prove intractable.

- The painting follows annotated figures in Evans G. Valens's
*The Number of Things*(1964), p.103, which was part of Crockett Johnson's mathematical library. It corresponds to an early diagram in Valens's discussion of squaring the circle. According to Valens, Hippocrates began by arguing that the areas of similar segments of different circles are in the same ratio as the squares of their bases. Suppose an isosceles right triangle is inscribed in a semicircle of diameter c. Construct smaller semicircles of diameter a and b on the sides of the inscribed triangle. As the square of a plus the square of b equals the square of c, the area of the two smaller semicircles equals that of the large one. The proof goes on to consider the area of the two crescents and the triangle.

- In this version of
*Squared Lunes*Crockett Johnson uses brown, black, red, and white against a gray background. This oil painting is #67 in the series, and the first in the series with the title "Squared Lunes." It was completed in 1968 and is signed: CJ68. It is inscribed on the back: SQUARED LUNES (/) (HIIPPOCRATES OF CHIOS) (/) Crockett Johnson 1968. A related painting is #68 (1979.1093.43).

- Location
- Currently not on view

- date made
- 1968

- referenced
- Hippocrates of Chios

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.42

- accession number
- 1979.1093

- catalog number
- 1979.1093.42

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

## Painting -

*Squared Lunes (Hippocrates Of Chios)*- Description
- The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons. That is, given a polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straightedge. They were unable to square a circle. This painting is based on the earliest known squaring of a figure bounded by curves rather than straight lines. The mathematician Hippocrates squared a lune, a figure bounded by arcs of two circles with different radii. This achievement might seem more difficult than squaring a circle.

- Crockett Johnson's painting follows two annotated figures in Evans G. Valens's
*The Number of Things*(1964), pp. 103–104, a book in the artist’s mathematical library. The finished piece shows isosceles triangles T, and a second congruent triangle connected to it base to base to form a square. Also present in the painting are three lunes, two small and one large. The area of triangle T is equal to the sum of the areas of lunes A and B (see figures). The area of triangle T is also equal to the area of a lune composed of X, Y, and the area T-C. Furthermore, because triangle T is congruent to the triangle below it, triangle T is equal to the area of this lune. Thus, the area of the square is equal to the sum of the areas of the three lunes. In summary, Johnson pictorially represented a "squared" curvilinear region; that is, he successfully constructed a square with the same area as that of the region of three lunes bounded by curves.

- Crockett Johnson executed this painting in 4 tints and darker shades of purple upon a black background. The center triangle is the darkest shade of purple. As one moves outward, the colors grow lighter. This allows a dramatic distinction to be seen between the figure and the background, and thus puts a greater emphasis on the lunes.

- This oil painting on masonite is #68 in Crockett Johnson's series. Its date of completion is unknown and the work is unsigned. It is closely related to painting #67 (1979.1093.42).

- Location
- Currently not on view

- date made
- ca 1965

- referenced
- Hippocrates of Chios

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.43

- accession number
- 1979.1093

- catalog number
- 1979.1093.43

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

## Painting -

*Cross Ratio in an Ellipse (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 (a pencil of lines is a set of lines emanating from a common point). In the drawing, which is Figure 5 from an article by Morris Kline in James R. Newman's
*The World of Mathematics*(1956), 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 Crockett Johnson's 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 painting with both Chasles and another French advocate of projective geometry, Victor Poncelet.

- The painting, in oil or acrylic on masonite, is #69 in the series. It has a dark gray or blue background and a black wooden frame. It shows a white ellipse, two points on the ellipse (on the left side of the painting), and two pencils of lines that produce the same cross ratio. The painting is not signed. It is inscribed on the back, in Crockett Johnson’s hand: CROSS RATIO IN AN ELLIPSE (PONCELET) (/) Crockett Johnson 1968. Compare #21 (1979.1093.15).

- Reference: Morris Kline in James R. Newman,
*The World of Mathematics*(1956), p. 634. This volume was in Crockett Johnson's library. The figure on this page is annotated.

- Location
- Currently not on view

- date made
- 1968

- referenced
- Poncelet, Jean-Victor

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.44

- accession number
- 1979.1093

- catalog number
- 1979.1093.44

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

## Painting -

*Seventeen Sides (Gauss)*- Description
- Those making mathematical instruments for surveying, navigation, or the classroom have long been interested in creating equal divisions of the circle. Ancient geometers knew how to divide a circle into 2, 3, or 5 parts, and as well as into multiples of these numbers. For them to draw polygons with other numbers of sides required more than a straightedge and compass.

- In 1796, as an undergraduate at the University of Göttingen, Friedrich Gauss proposed a theorem severely limiting the number of regular polygons that could be constructed using ruler and compass alone. He also found a way of constructing the 17-gon.

- Crockett Johnson, who himself would develop a great interest in constructing regular polygons, drew this painting to illustrate Gauss's discovery. His painting follows a somewhat later solution to the problem presented by Karl von Staudt in 1842, modified by Heinrich Schroeter in 1872, and then published by the eminent mathematician Felix Klein. Klein's detailed account was in Crockett Johnson's library, and a figure from it is heavily annotated.

- This oil painting on masonite is #70 in the series. It is signed: CJ69. The back is marked: SEVENTEEN SIDES (GAUSS) (/) Crockett Johnson 1969. The painting has a black background and a wood and metal frame. There are two adjacent purple triangles in the center, with a white circle inscribed in them. The triangles have various dark gray regions, and the circle has various light gray regions and one dark gray segment. The length of the top edge of this segment is the chord of the circle corresponding to length of the side of an inscribed 17-sided regular polygon.

- Reference: Felix Klein,
*Famous Problems of Elementary Geometry*(1956), pp. 16–41, esp. 41.

- Location
- Currently not on view

- date made
- 1969

- referenced
- Gauss, Carl Friedrich

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.45

- accession number
- 1979.1093

- catalog number
- 1979.1093.45

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

## Painting -

*Law of Motion (Galileo)*- Description
- Crockett Johnson based this painting on the discussion of motion along inclined planes by Galileo Galilee in his
*Dialogues Concerning Two New Sciences*(1638). Here Galileo showed that if from a fixed point straight lines be extended indefinitely downwards and a point be imagined to move along each line at a constant speed, all starting from the fixed point at the same time and moving with equal speeds, the locus of the moving points will be an expanding circle.

- This painting shows four superimposed circles in various shades of gray, white and black. These circles all have a common point at the center top, and differ in radius. They are shaded into several regions which are divided by lines originating at the common point. The work has an orange background and a black wooden frame. It is probably based on a drawing in E. G. Valens,
*The Attractive Universe*(1969). This volume is in Crockett Johnson's library, annotated on the page indicated.

- The painting is #71 in the series. It is signed: CJ70.

- References: Galileo Galilee,
*Dialog Concerning Two New Sciences, Third Day*(Figure 59 in the Dover edition).

- E. G. Valens,
*The Attractive Universe: Gravity and the Shape of Space*, Cleveland and New York: World Publishing Company, 1969, p. 135.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.46

- catalog number
- 1979.1093.46

- accession number
- 1979.1093

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

## Painting -

*Rotated Triangle and Reflections*- Description
- Plane figures of the same size and shape can be moved about in several ways and preserve their size and form. Such congruent transformations, as they are called, are combinations of rotations about a point or a line, reflections about a line, or translations in which the figure moves about the plane but the directions of the sides is unchanged.

- This painting, which closely follows a diagram from a book by H. S. M. Coxeter, illustrates two properties of congruent transformations. First, a transformation in which only one point remains unchanged is a rotation. In the figure, the triangle PQR passes through a congruent transformation into the triangle PQ'R'. Suppose that the transformation consisted of a reflection. Then triangle PQR could be rotated about the line m to another triangle, PRR[1]. However, these two triangles have a line, and not simply a point, in common. Coxeter went on to argue that any congruent transformation can be constructed as the product of reflections, the number of which can be reduced to three.

- In the painting, as in the diagram, there are three congruent triangles. One light blue and gray triangle rotates into another light blue triangle above it to the right (the axis of rotation is perpendicular to the painting). The blue and blue-gray triangle is a rotation of the first triangle about the axis m, and a reflection of the other. The background is in two shades of gray, divided by this line of rotation.

- The painting is #73 in the series and signed: CJ70. It has a metal frame.

- Reference: H. S. M. Coxeter,
*The Real Projective Plane*, p. 153.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.47

- catalog number
- 1979.1093.47

- accession number
- 1979.1093

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