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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 2033 items.

Page 8 of 204

## 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

## Painting -

*Nine-Point Circle*- Description
- Although 18th- and 19th-century mathematicians were much interested in analysis and algebra, they continued to explore geometrical constructions. In 1765, the eminent Swiss-born mathematician Leonhard Euler showed that nine points constructed from a triangle lie on a circle. This circle would come to be called the Feuerbach circle after Karl Wilhelm Feuerbach, a professor at the gymnasium in Erlangen, Germany. In 1822, he published a paper explaining and proving the theorem.

- It seems likely that the direct inspiration for this painting was a figure in H. S. M. Coxeter’s
*The Real Projective Plane*(1955). A diagram on p. 143 of this book shows a triangle with its respective nine points. In his copy of the book, Crockett Johnson connected the points himself, thereby completing the circle (see the annotated figure). In addition, Johnson also annotated a figure in Nathan A. Court’s*College Geometry*(1964 printing), p. 103. Crockett Johnson's painting does not directly imitate either drawing, but it is evident that he studied each figure in creating his own construction.

- The first three points of the nine-point circle are the midpoints of the sides of triangle QRP (points L, M, and N in the annotated drawing). The second three points are the bases of the altitudes of the triangle (points A, B, C). These altitudes meet at a point (S). The midpoints of the lines joining the vertices of the triangle to the intersection of the altitudes create the last three points that indicate the nine-point circle (L’, M’, N’).

- The segments of the triangle that are not part of the circle are colored in shades of blue and gray. Those segments that are part of the circle are white and various shades of pink and yellow. The painting has a background defined by two shades of gray.

- This oil painting on masonite, #75 in the series, dates from 1970, is signed in the upper left corner : CJ70. It is inscribed on the back: NINE-POINT CIRCLE (/) Crockett Johnson 1970. There is a metal frame.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.49

- catalog number
- 1979.1093.49

- accession number
- 1979.1093

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

## Painting -

*Law of Orbiting Velocity (Kepler)*- Description
- This piece is a further example of Crockett Johnson's exploration of Kepler’s first two laws of planetary motion. The ellipse represents the path of a planet, and the white sections represent equal areas swept out in equal times. This work, a silk screen inked on paper board, is signed: CJ66. It is #76 in the series, and it echoes painting #22 (1979.1093.16) and painting #99 (1979.1093.66).

- Location
- Currently not on view

- date made
- 1966

- referenced
- Kepler, Johannes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.50

- catalog number
- 1979.1093.50

- accession number
- 1979.1093

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

## Painting -

*Numbers in a Spiral*- Description
- Some of Crockett Johnson's paintings reflect relatively recent research. Mathematicians had long been interested in the distribution of prime numbers. At a meeting in the early 1960s, physicist Stanislaw Ulam of the Los Alamos Scientific Laboratory in New Mexico passed the time by jotting down numbers in grid. One was at the center, the digits from 2 to 9 around it to form a square, the digits from 10 to 25 around this, and the spiral continued outward.

- Circling the prime numbers, Ulam was surprised to discover that they tended to lie on lines. He and several colleagues programmed the MANIAC computer to compute and plot a much larger number spiral, and published the result in the
*American Mathematical Monthly*in 1964. News of the event also created sufficient stir for*Scientific American*to feature their image on its March 1964 cover. Martin Gardner wrote a related column in that issue entitled “The Remarkable Lore of the Prime Numbers.”

- The painting is #77 in the series. It is unsigned and undated, and has a wooden frame painted white.

- Location
- Currently not on view

- date made
- ca 1965

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.51

- catalog number
- 1979.1093.51

- accession number
- 1979.1093

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

## Painting -

*Parabolic Triangles (Archimedes)*- Description
- According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. But finding the area bounded by curved surfaces was not an easy task. The parabola and other conic sections had been known for almost a century before Archimedes wrote a short treatise called
*Quadrature of the Parabola*in about 240 BC. This was the first demonstration of the area bounded by a conic section. In his proof, Archimedes first constructed a triangle whose sides consisted of two tangents of a parabola and the chord connecting the points of tangency. He then showed that the area under the parabola (shown in gray and black in the painting) is two thirds of the area of the triangle which circumscribes it. Once the area bounded by the tangent could be expressed in terms of the area of a triangle, it was easy to construct the corresponding square. Crockett Johnson’s painting follows two diagrams illustrating a discussion of Archimedes’s proof given by Heinrich Dorrie (Figure 54).

- This oil or acrylic painting on masonite is #78 in the series and is signed “CJ67” in the bottom left corner. It has a gray wooden frame. For a related painting, see #43 (1979.1093.31).

- References: Heinrich Dorrie, trans. David Antin,
*100 Great Problems of Elementary Mathematics: Their History and Solution*(1965), p. 239. This volume was in Crockett Johnson's library and the diagram in his copy is annotated.

- James R. Newman,
*The World of Mathematics*(1956), p. 105. This volume was in Crockett Johnson's library. The figure on this page (Figure 9) is annotated.

- Location
- Currently not on view

- date made
- 1967

- referenced
- Archimedes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.52

- catalog number
- 1979.1093.52

- accession number
- 1979.1093

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

## Painting -

*Duality (Pascal-Brianchon)*- Description
- As a 21-year-old student, the Frenchman Charles Jules Brianchon (1785–1864) discovered that in any hexagon circumscribed about a conic section (such as a circle), the three lines that join opposite diagonals meet in a single point. He also pointed out connections between his result and Pascal's theorem concerning the points of intersection of opposite sides of a hexagon inscribed in a conic section.

- In the painting, a hexagon (only the vertices are shown) is inscribed in a circle. Three diagonal lines (edges of the gray and black polygon) are collinear. The line in question is the line joining the points of intersection, white on one side and purple on the other. Crockett Johnson's painting closely resembles a diagram of A. S. Smogorzhevskii in which Brianchon's theorem is applied to a proof of Pascal's theorem.

- The painting on masonite is #81 in the series. It has a purple background and a black wooden frame. It is signed: CJ66.

- References: A. S. Smogorzhevskii,
*The Ruler in Geometrical Constructions*(1961), p. 37. This volume was in Crockett Johnson's library. The figure is not annotated.

- Carl Boyer and Uta Merzbach,
*A History of Mathematics*(1991), p. 534.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Pascal, Blaise

- Brianchon, Charles Julien

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.53

- catalog number
- 1979.1093.53

- accession number
- 1979.1093

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

## Painting -

*Pi Squared and Its Square Root*- Description
- This painting is part of Crockett Johnson's exploration of constructions that might take place if one could draw squares equal in area to circles. It incorporates elements of a figure in his papers that includes two squares and a rectangle. The smaller square (ABDX in Crockett Johnson's figure) is defined as having the same area as a circle, CFXE,circumscribing the rectangle (the rectangle of sides CE and EX in the figure).

- The circle is assumed to have radius one. Hence the area of the square is supposed to be pi, and the length of its side (e.g. AB or CF) the square root of pi. The area of the rectangle is assumed to be the square root of pi. Hence one has a painting that includes a square of area equal to pi and a rectangle of area equal to its square root. From such assumptions, Crockett Johnson went on to construct a line segment of length pi, which is not shown in the painting but does appear in the figure.

- The painting is #83 in the series. It is in oil or acrylic on masonite. There is a black wooden frame. The work is unsigned and undated.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.54

- catalog number
- 1979.1093.54

- accession number
- 1979.1093

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

## Painting -

*Doubled Cube (Newton)*- Description
- Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the Problem of Delos. Crockett Johnson wrote of this problem: "Plutarch mentions it, crediting as his source a now lost version of the legend written by the third century BC Alexandrian Greek astronomer Eratosthenes, who first measured the size of the Earth. Suffering from plague, Athens sent a delegation to Delos, Apollo’s birthplace, to consult its oracle. The oracle’s instruction to the Athenians, to double the size of their cubical altar stone, presented an impossible problem . . . . It could not be done with the compass and an unmarked straightedge."

- (p. 99).

- Crockett Johnson's paintings follow a construction proposed by the eminent English mathematician Isaac Newton. As Lucasian professor of mathematics at Cambridge University, Newton was required to deposit copies of his lectures in the university library. In 1683, after he had taught a course in algebra for 11 years, he finally deposited the notes for it. After Newton left Cambridge in 1696, his successor, William Whiston, arranged to have the lectures published in a book with the short title
*Arithmetica Universalis*. Latin editions of the book appeared in 1707, 1722, 1732, and 1761; and English translations in 1720, 1728, and 1769.

- In an appendix to this book, Newton discussed ways of finding the roots of numbers through geometric constructions. One problem was that of finding two mean proportions between given numbers. One case of this problem gives the cube root of a number. [Suppose the numbers are a and b and the proportionals x and y. Then a / x = x / y = y /b). Squaring the first and last term, a² / x² = y² / b². But, from the first equation, one also has x = y² / b. By substitution, a² / x² = x / b, or x³ = a² b. If a is 1, x is the cube root of b, as desired.]

- Newton and Crockett Johnson represented the quantities involved as lengths of the sides of triangles. Newton’s figure is #99 in his
*Arithmetica Universalis*. Crockett Johnson's figure is differently lettered, and the mirror image of that of Newton.

- Following the artist's notation (figure 1979.3083.04.05), suppose AB = 1, bisect it at M, and construct an equilateral triangle MBX on MB. Draw AX and MX extended. Using a marked straightedge, construct line segment BZY, intersecting AX at Z and MX at Y in such a way that XY = AM = MB = 1/2. Then the distance BZ will have a length of one half the cube root of 2, that is to say the length of the side of a cube of side 1/2.

- A proof of Newton’s construction is given in Dorrie. Crockett Johnson's copy of a drawing in this volume is annotated. The duplication of the cube also was discussed in at least two other books in Crockett Johnson's library. One is a copy of the 1764 edition of an English translation of the
*Arithmetica Universalis*, which Crockett Johnson purchased in January of 1972. The second is W. W. Rouse Ball’s*Mathematical Recreations and Essays*, which also discusses Newton's solution.

- Crockett Johnson's painting emphasized doubled lines in the construction, building on the theme of the painting. His diagram for the painting is oriented differently from the painting itself.

- This oil painting on masonite is #85 in the series. It depicts overlapping blue, pink and gray circular segments in two adjacent rectangles. These rectangles are divided by various lines into gray and black sections. A lighter gray border goes around the edge. There is a metal and wooden frame. The painting is unsigned. For a mathematically related painting, see #56 (1979.1093.36).

- References: Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings,"
*Leonardo*5 (1972): pp. 98–100. This specific painting is not discussed in the article.

- Heinrich Dorrie, trans. David Antin,
*100 Great Problems of Elementary Mathematics: Their History and Solution*(1965) p. 171. The figure on this page, figure 27, is annotated.

- Isaac Newton,
*Universal Arithmetick*, (1769), esp. pp. 486–87, figure 99. This volume was in Crockett Johnson's library. It is not annotated.

- W. W. Rouse Ball, rev. H. S. M. Coxeter,
*Mathematical Essays and Recreations*, (1962 printing), pp. 327–33. This is a slightly different construction. The volume was in Crockett Johnson's library.

- Isaac Newton,
*The Mathematical Works of Isaac Newton*, assembled by Derek T. Whiteside, vol. 2, (1967). This includes a reprint of the 1728 English translation of the*Arithmetica Universalis*.

- Location
- Currently not on view

- date made
- ca 1970

- referenced
- Newton, Isaac

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.55

- catalog number
- 1979.1093.55

- accession number
- 1979.1093

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

## Painting -

*Equal Triangles*- Description
- In this painting, based on his own construction, Crockett Johnson continued his exploration of rectilinear figures that could be made with a ruler and compass, assuming that one could construct a square of area equal to pi (e.g. if one could square the circle). More specifically, he assumed that he could construct a square of that area (the square in blue and dark blue in the painting) and found four triangles, also shown in shades of blue, that would be of equal area. A sheet from his papers presents his argument (1979.3083.04.02)

- The oil painting on masonite is #86 in the series. It is signed on the back: EQUAL TRIANGLES (/) Crockett Johnson 1972. There is a wood and chrome frame.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.56

- catalog number
- 1979.1093.56

- accession number
- 1979.1093

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

## Painting -

*Squares of 2, 4, 16 from Square Root of x*- Description
- In this painting, Crockett Johnson supposed that one was given two lengths, one the square root of the second. Although no numerical values were given, he sought to construct three squares, one the square root of the second and the second the square root of the third, and to give their values numerically. His solution is represented in the painting, and described in his notes as work from 1972.

- The three squares are visible, one the entire surface of the painting and the two others within it. The vertical lines point to the starting point of the painting, a line segment along the base and its square root. From here, Crockett Johnson constructed the elaborate geometrical argument illustrated by the painting. He claimed that he had constructed squares of area 2, 4, and 16. The ratios of the areas are as he describes, but the absolute numerical values depend on the units of measure.

- This oil painting on masonite is #88 in the series. It is unsigned. There is an inset metal strip in the wooden frame.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.57

- catalog number
- 1979.1093.57

- accession number
- 1979.1093

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