##
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 6 of 204

## Painting -

*Polyhedron Formula (Euler)*- Description
- Mathematicians have explored the properties of polyhedra since ancient times, but it was the Swiss scholar Leonhard Euler (1707–1783) who proved the formula V-E+F = 2. That is, for a simple convex polyhedron (e.g. one with no holes, so that it can be deformed into a sphere) the number of vertices minus the number of edges plus the number of faces is two. An equivalent formula had been presented by Descartes in an unpublished treatise on polyhedra. However, this formula was first proved and published by Euler in 1751 and bears his name.

- Crockett Johnson's painting echoes a figure from a presentation of Euler's formula found in Richard Courant and Herbert Robbins's article “Topology,” which is in James R. Newman's
*The World the Mathematics*(1956), p. 584. This book was in the artist’s library, but the figure that relates to this painting is not annotated.

- To understand the painting we must understand the mathematical argument. It starts with a hexahedron, a simple, six-sided, box-shaped object. First, one face of the hexahedron is removed, and the figure is stretched so that it lies flat (imagine that the hexahedron is made of a malleable substance so that it can be stretched). While stretching the figure can change the length of the edges and the area and shape of the faces, it will not change the number of vertices, edges, or faces.

- For the "stretched" figure, V-E+F = 8 - 12 + 5 = 1, so that, if the removed face is counted, the result is V-E+F = 2 for the original polyhedron. The next step is to triangulate each face (this is indicated by the diagonal lines in the third figure). If, in triangle ABC [C is not shown in Newman, though it is referred to], edge AC is removed, the number of edges and the number of faces are both reduced by one, so V-E+F is unchanged. This is done for each outer triangle.

- Next, if edges DF and EF are removed from triangle DEF, then one face, one vertex, and two edges are removed as well, and V-E+F is unchanged. Again, this is done for each outer triangle. This yields a rectangle from which a right triangle is removed. Again, this will leave V-E+F unchanged. This last step will also yield a figure for which V-E+F = 3-3+1. As previously stated, if we count the removed face from the initial step, then V-E+F = 2 for the given polyhedron.

- The “triangulated” diagram was the one Crockett Johnson chose to paint. Each segment of the painting is given its own color so as to indicate each step of the proof. Crockett Johnson executed the two right triangles that form the center rectangle in the most contrasting hues. This draws the viewer’s eyes to this section and thus emphasizes the finale of Euler's proof. This approach to the proof of Euler's polyhedral formula was pioneered by the French mathematician Augustin Louis Cauchy in 1813.

- This oil painting on masonite is #39 in the series. It was completed in 1966 and is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 (/) POLYHEDRON FORMULA (EULER). It has a wood and chrome frame.

- Reference:

- David Richeson, “The Polyhedral Formula,” in
*Leonhard Euler: Life, Work and Legacy*, editors R. E. Bradley and C. E. Sandifer (2007), pp. 431–34.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Euler, Leonhard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.27

- catalog number
- 1979.1093.27

- accession number
- 1979.1093

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

## Painting -

*Multiplication through Imaginary Numbers (Gauss)*- Description
- This painting was inspired by ideas of Carl Friedrich Gauss (1777–1855). In his 1797 doctoral thesis, Gauss proved what is now called the fundamental theorem of algebra. He showed that every polynomial with real coefficients must have at least one real or complex root. A complex number has the form a+bi, where a and b are real numbers and i represents the square root of negative one. The French mathematician René Descartes (1596–1650) called such numbers "imaginary", which explains the reference in the title. Gauss demonstrated that, just as real numbers can be represented by points on a coordinate line, complex numbers can be represented by points in the coordinate plane.

- The construction of this painting echoes a figure in an article on Gauss by Eric Temple Bell in J. R. Newman's
*The World of Mathematics*that illustrates the representation of points on a plane. This book was in Crockett Johnson's library, and the figure is annotated.

- In Bell's figure, real numbers c and -c are plotted on the x axis, the imaginary numbers ci and -ci are plotted on the y axis, and the complex number a+bi is shown in the first quadrant. The figure is meant to show that if a complex number a+bi is multiplied by the imaginary number i then the product is a complex number on the same circle but rotated ninety degrees counterclockwise. That is, i(a+bi) = ai+bi² = -b+ai. Thus, this complex number lies in the second quadrant. If this process is repeated the next product is -a-bi, which lies in the third quadrant. It is unknown why Johnson did not illustrate the fourth product.

- The colors of opposite quadrants of the painting are related. Those in quadrant three echo those of quadrant one and those of quadrant four echo those of quadrant two.This oil painting is #40 in the series. It is signed: CJ67.

- References:

- James R. Newman,
*The World of Mathematics*(1956), p. 308. This volume was in Crockett Johnson's library. The figure on this page is annotated.

- Location
- Currently not on view

- date made
- 1967

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.28

- catalog number
- 1979.1093.28

- accession number
- 1979.1093

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

## Painting -

*Locus of Point on Chord (Plato)*- Description
- The locus of the midpoints of the chords of a given circle that pass through a fixed point is a circle when the point lies inside of or on the circle. The small circle painted white is the locus of the midpoints of chords drawn in the large circle that pass through a point toward the top left of the inside of the circle. Three chords of the large circle are suggested. These are the diameter, whose midpoint is the center of the circle, a vertical chord through the point, and a horizontal chord through the point (only a small part of this chord is indicated). The painting is based on a diagram from
*College Geometry*by Nathan Court. It is unclear why Crockett Johnson associated this painting with Plato.

- The oil painting on masonite is #41 in the series. It has a background of two purple and gray rectangles. It has a metal and wooden frame. It shows a circle with a smaller circle inside it. The smaller circle is in two shades of white, the larger one in orange, black, gray and light purple. The painting is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) LOCUS OF POINT ON CHORD (PLATO).

- Reference: Nathan Court,
*College Geometry*, (1964 printing), p. 13. This figure is annotated in Crockett Johnson's copy of this volume.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Plato

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.29

- catalog number
- 1979.1093.29

- accession number
- 1979.1093

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

## Painting -

*Velocity on Inclined Planes (Galileo)*- Description
- This oil painting is based on a figure from Galileo Galilee's
*Dialogues Concerning Two New Sciences*(1638), Book 3. Here Galileo discussed the time of descent of bodies rolling without friction along inclined planes. He argued that if from the highest point in a vertical circle there be drawn any inclined planes meeting the circumference of the circle, the times of descent along these chords are equal to one another. This painting shows two inclined planes drawn from the highest point of a vertical circle, with a ball moving along each chord. Crockett Johnson probably became familiar with Galileo's figure by examining the translation of part of his book published in James R. Newman,*The World of Mathematics*, vol. 2, New York: Simon and Schuster, 1956, p. 751–52. This volume was in Crockett Johnson's library. The figure on p. 752 is annotated.

- The painting has a gray background and a metal and wooden frame. It shows two superimposed triangles (inclined planes), one reddish purple, and the other smaller one blue. Both of these triangles are inscribed in the same white circular arc. A light purple circle is shown near the bottom of the purple triangle, and a light blue circle near the bottom of the blue triangle.

- The work is # 42 in the series. It is signed: CJ66. Compare to paintings #96 (1979.1093.64) and #71 (1979.1093.46).

- Location
- Currently not on view

- date made
- 1966

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.30

- catalog number
- 1979.1093.30

- 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. 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 white and light green in the painting) is two thirds of the area of the triangle that 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 is based on diagrams illustrating a discussion of Archimedes’s proof given by H. Dorrie (Figure 54) or J. R. Newman (Figure 9).

- This oil painting is #43 in the series, and is signed: CJ69. It has a gray background and a gray frame. It shows a triangle that circumscribes a portion of a parabola. The large triangle is divided into a triangle in shades of light green, which touches a triangle in shades of dark green. The region between the triangles is divided into black and white areas. A second painting in the series, #78 (1979.1093.52) illustrates the same theorem.

- 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 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 is annotated.

- Location
- Currently not on view

- date made
- 1969

- referenced
- Archimedes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.31

- catalog number
- 1979.1093.31

- accession number
- 1979.1093

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

## Painting -

*Square Roots to Sixteen (Theodorus of Cyrene)*- Description
- Greek mathematicians knew that numbers could not always be represented as simple ratios of whole numbers. They devised ways to describe them geometrically. The title of this painting refers to Theodorus of Cyrene (about 465–398 BC), a Greek geometer who, according to the Greek mathematician Theaetetus (about 417–369 BC), constructed the square roots of the numbers from 3 through 17. Crockett Johnson's painting follows a diagram in Evans G. Valens's
*The Number of Things*that stops with the square root of 16.

- The construction of this oil or acrylic painting, #45 in the series, begins with a vertical line segment of length one. Crockett Johnson then drew a right angle at the base of the segment and an adjacent line with length one. From the Pythagorean theorem, it follows that a line from the center of the spiral has length equal to the square root of 2. The construction was continued until the last hypotenuse displayed length equal to the square root of 16.

- The painting, which looks like a seashell, shows a specific color pattern. The three dark gray triangles have hypotenuses whose lengths are whole numbers (the square roots of 4, 9, and 16). The six white triangles have hypotenuses whose lengths are irrational and are square roots of even integers. Finally, the six tan triangles have hypotenuses whose lengths are irrational and the square roots of odd integers.

- The painting dates from 1967 and is signed: CJ67. It is marked on the back: Crockett Johnson (/) SQUARE ROOTS TO SIXTEEN (/) (THEODORUS OF CYRENE).

- Location
- Currently not on view

- date made
- 1967

- referenced
- Theodorus of Cyrene

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.32

- catalog number
- 1979.1093.32

- accession number
- 1979.1093

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

## Painting -

*Golden Rectangle (Pythagoras)*- Description
- The ancient Greek mathematician 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. The ratio came to be called the golden ratio. If the sides of a rectangle are in the golden ratio, it is called a golden rectangle. Several Crockett Johnson paintings explore the golden ratio and related geometric figures. This paintings suggest how a golden rectangle can be constructed, given the length of its shorter side. On the right in the painting is the golden rectangle that results. Lines in a triangle on the left indicate how the rectangle could have been constructed. Also included are the outlines of a hexagon and a five-pointed star constructed once the ratio had been found.

- This painting 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.

- The painting on masonite is #46 in the series. It has a black and purple background and a black wooden frame. It is unsigned. The inscription on the back reads: GOLDEN RECTANGLE (/) (PYTHAGORAS) (/) Crockett Johnson 1968. Compare #103 (1979.1093.70) and #64 (1979.1093.39).

- Location
- Currently not on view

- date made
- 1968

- referenced
- Pythagoras

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.33

- catalog number
- 1979.1093.33

- accession number
- 1979.1093

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

## Painting -

*Rectangles of Equal Area (Pythagoras)*- Description
- Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings. This painting is based on a method of constructing a rectangle equal in area to a given rectangle, given one side of the rectangle to be constructed.

- In the painting, suppose that the cream-colored rectangle on the bottom left is given, as well as a line segment extending from the upper right corner of it. Construct the small triangle on the upper left. Draw the three horizontal lines shown, as well as the diagonal of the rectangle constructed. Extend this diagonal until it meets the bottom line, creating another triangle. The length of the base of this triangle will be the side of the rectangle desired. This rectangle is on the upper right in the painting.

- This construction has been associated with the ancient Pythagoreans. Crockett Johnson may well have learned it from Evans G. Valens,
*The Number of Things*. The drawing on page 121 of this book is annotated, although the annotations are faint.

- The oil painting is #48 in the series. It has a black background and a black wooden frame, with the two equal triangles in light shades. The painting is signed on the front: CJ69. It is signed on the back: RECTANGLES OF EQUAL AREA (/) (PYTHAGORAS) (/) Crockett Johnson 1969.

- Location
- Currently not on view

- date made
- 1969

- referenced
- Pythagoras

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.34

- catalog number
- 1979.1093.34

- accession number
- 1979.1093

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

## Painting -

*Squared Circle*- Description
- This oil painting on pressed wood, #52 in the series, shows an original construction of Crockett Johnson. He executed this work in 1968, three years after he began creating mathematical paintings. It is evident that the artist was very proud of this construction because he drew four paintings dealing with the problem of squaring the circle. The construction was part of Crockett Johnson's first original mathematical work, published in
*The Mathematical Gazette*in early 1970. A diagram relating to the painting was published there.

- To "square a circle," mathematically speaking, is to construct a square whose area is equal to that of a given circle using only a straightedge (an unmarked ruler) and a compass. It is an ancient problem dating from the time of Euclid and is one of three problems that eluded Greek geometers and continued to elude mathematicians for 2,000 years. In 1880, the German mathematician Ferdinand von Lindermann showed that squaring a circle in this way is impossible - pi is a transcendental number. Because this proof is complicated and difficult to understand, the problem of squaring a circle continues to attract amateur mathematicians like Crockett Johnson. Although he ultimately understood that the circle cannot be squared with a straightedge and compass, he managed to construct an approximate squaring.

- Crockett Johnson began his construction with a circle of radius one. In this circle he inscribed a square. Therefore, in the figure, AO=OB=1 and OC=BC=√(2) / 2. AC=AO+OC=1 + √(2) / 2 and AB=√(AC² + BC²) = &#*&#);(2+√(2)). Crockett Johnson let N be the midpoint of OT and constructed KN parallel to AC. K is thus the midpoint of AB, and KN=AO - (AC)/2=(2-&#*&#);(2)) / 4. Next, he let P be the midpoint of OG, and he drew KP, which intersects AO at X. Crockett Johnson then computed NP=NO+OP=(√(2))/4+(1/2). Triangle POX is similar to triangle PNK, so XO/OP=KN/NP. From this equality it follows that XO=(3-2√(2))/2.

- Also, AX=AO-XO=(2√(2)-1)/2 and XC=XO+OC=(3-√(2))/2. Crockett Johnson continued his approximation by constructing XY parallel to AB. It is evident that triangle XYC is similar to triangle ABC, and so XY/XC=AB/AC. This implies that XY=[√((2+√(2)) × (8-5√(2))]/2. Finally he constructed XZ=XY and computed AZ=AX+XZ=[2√(2)-1+(√(2+√(2)) × (8-5√(2))]/2 which approximately equals 1.772435. Crockett Johnson knew that the square root of pi approximately equals 1.772454, and thus AZ is approximately equal to √(Π) - 0.000019. Knowing this value, he constructed a square with each side equal to AZ. The area of this square is (AZ)² = 3.1415258. This differs from the area of the circle by less than 0.0001. Thus, Crockett Johnson approximately squared the circle.

- The painting is signed: CJ68. It is marked on the back: SQUARED CIRCLE* (/) Crockett Johnson 1968 (/) FLAT OIL ON PRESSED WOOD) (/) MATHEMATICALLY (/) DEMONSTRATED (/) TO √π + 0.000000001. It has a white wooden frame. Compare to painting #91 (1979.1093.60).

- References: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,”
*Leonardo*5 (1972): p. 98.

- C. Johnson, “A Geometrical look at √π,"
*Mathematical Gazette*, 54 (1970): p. 59–60. the figure is from p. 59.

- Location
- Currently not on view

- date made
- 1968

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.35

- catalog number
- 1979.1093.35

- accession number
- 1979.1093

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

## Painting -

*Problem of Delos Constructed from a Solution by Isaac Newton (Arithmetica Universalis)*- 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. . . ."(p. 99). Hence the reference to the problem of Delos in the title of the painting.

- Isaac Newton suggested a solution to the problem in his book
*Arithmetica Universalis*, first published in 1707. His construction served as the basis of the painting. Newton’s figure, as redrawn by Crockett Johnson, begins with a base (OA), bisected at a point (B), with an equilateral triangle (OCB) constructed on one of the halves of the base. Newton then extended the sides of this triangle through one vertex. Placing a marked straightedge at one end of the base (O), he rotated the rule so that the distance between the two lines extended equaled the sides of the triangle (in the figure, DE = OB = BA = OC = BC). If these line segments are of length one, one can show that the line segment OD is of length equal to the cube root of two, as desired.

- In Crockett Johnson’s painting, the line OA slants across the bottom and the line ODE is vertical on the left. The four squares drawn from the upper left corner (point E) have sides of length 1, the cube root of 2, the cube root of 4, and two. The distance DE (1) represents the edge of the side and the volume of a unit cube, while the sides of three larger squares represent the edge (the cube root of 2), the side (the square of the cube root of 2) and the volume (the cube of the cube root of two) of the doubled cube.

- This oil painting on masonite is #56 in the series and dates from 1970. The work is signed: CJ70. It is inscribed on the back: PROBLEM OF DELOS (/) CONSTRUCTED FROM A SOLUTION BY (/) ISAAC NEWTON (ARITHMETICA UNIVERSALIS) (/) Crockett Johnson 1970. The painting has a wood and metal frame. For related documentation see 1979.3083.04.06. See also painting number 85 (1979.1093.55), with the references given there.

- Reference: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,”
*Leonardo*5 (1972): pp. 98–9.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Newton, Isaac

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.36

- catalog number
- 1979.1093.36

- accession number
- 1979.1093

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