#
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 7 of 219

## 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 -

*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

## Painting -

*Square Root of Pi*- Description
- There are two paintings in the collection with this title. The geometry and construction of the two are identical (see painting #100 - 1979.1093.67).

- The only differences between the two paintings are the color scheme and the dimensions. This painting has a black and purple background and a white rectangle that represents the square root of pi. The method of the color scheme is similar to painting #89 because, like the electric blue rectangle in the other painting, the white color of the rectangle against the purple background creates a dramatic contrast that highlights the crux of Crockett Johnson's construction.

- This painting is #89 in the series. It was executed in oil on masonite, and has a black wooden frame. It is unsigned and undated.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.58

- catalog number
- 1979.1093.58

- accession number
- 1979.1093

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

## Painting -

*Square Root Of Pi - 0.00001*- Description
- This oil painting on masonite, #91 in the series, uses the same construction as that of painting #52 (see 1979.1093.35). Crockett Johnson's construction leads to a square with side approximately equal to 1.772435, which differs from the square root of pi by less than 0.00001, as the title states. Thus, a square with this side would have an area approximately equal to 3.1415258.

- Unlike painting #52 (1979.1093.35), the circle of this work is divided into four quadrants. Crockett Johnson chose darker shades and lighter tints of pink to illustrate his figure, which appear bold juxtaposed against the black background. The triangle executed in the lightest tint of pink and the shape executed in white with a pink tip adjoin the horizontal line segment that has an approximate length of the square root of pi.

- This painting was completed in 1972, is unsigned, and has a wooden frame accented with chrome. On the back is an inscription, partly obscured, that reads: - 0.00001 (/) Crockett Johnson 1972.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.60

- catalog number
- 1979.1093.60

- accession number
- 1979.1093

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

## Painting -

*Biblical Squared Circles*- Description
- This painting, #92 in the series, relates to a verse in the Old Testament (I Kings, Chapter VII, Verse 23) which states, "Also he made a molten sea of ten cubits brim to brim, round in compass, . . . and a line of thirty cubits did compass it round about." This verse tells us that the circular sea had a circumference of 30 cubits and a diameter of 10 cubits. Because the value of pi is defined as the ratio of a circle’s circumference to its diameter (pi = c/d), the ancient Hebrew text uses 30/10 = 3 as the value for pi.

- To illustrate this value of pi, Crockett Johnson inscribes the six-pointed Star of David within a circle. The curve joining two opposite points of the star (point C and point F in his figure) serves as a reminder of how to construct a six-pointed figure inside a circle. Furthermore, he inscribes a second, smaller circle inside the hexagon created by the six-pointed star.

- In this painting, it is assumed that the value of pi is 3. There are several relationships in the painting that involve this number. The inner circle has radius 1/2 and the outer circle has radius 1. Thus, the smaller circle has circumference pi and the larger circle has area pi. Triangle ABC in Crockett Johnson's figure is a 30-60-90 triangle with AC = 1, AB = 2, and CB equals the square root of 3. It follows that CD, BD, EA, EF, and AF also equal the square root of 3. The Star of David is composed of two overlapping equilateral triangles (triangles AEF and BCD in the figure). Triangle AEF has altitude AH = 3/2 and triangle BCD has altitude BG = 3/2. Thus, the sum of their altitudes is AH + BG = 3. It is also interesting to note that, although the dotted lines in the accompanying figure are not present in the painting, the area of the square created by the dotted corners equals three.

- In reference to this painting, Crockett Johnson wrote, "Each of the six sides of the two equilateral triangles equaled the square root of the area of the outer circle and the square root of the circumference of the inner circle; together the altitudes of the male and female triangles equaled the area of the outer circle and the circumference of the inner circle. Of course both of these circular dimensions are pi, but ecclesiastically pi equaled 3."

- The artist chose several tints and shades of blue for this painting. The illustration is darker underneath the curve from C to F than it is above, and the transition between each tint and shade is subtle. The choice of this one, “cool” color evokes a feeling of tranquility.

- This work was painted in oil on masonite, and has a wood and metal frame. It is unsigned and its date of completion is unknown.

- Reference: Biblical Squared Circles, 1979.3083.02.09, Crockett Johnson Collection.

- Location
- Currently not on view

- date made
- ca 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.61

- catalog number
- 1979.1093.61

- accession number
- 1979.1093

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

## Painting -

*Squared Rectangle and Euler Line*- Description
- Crockett Johnson had a longstanding interest in squaring figures, that is to say, constructing squares equal in area to other plane figures. Euclid had shown in his
*Elements*(Book II, Proposition 14) how to construct a square equal in area to a given rectangle. Crockett Johnson developed his own construction, one case of which served as the basis of this painting. The rectangle, the square of equal area, and a circle used in the demonstration are shown in various shades of pink.

- Two drawings from Crockett Johnson’s papers illustrate his ideas. The one that relates most closely to this painting is labeled A in his figure. In it, the given rectangle is ABED. The angles at the corner A and D are bisected, and the bisectors extended to meet at point C. The line from corner B through C meets side DE at point X. Line segments CL and XS are constructed parallel to AD. By this construction, the segment DL is half the length of AD. From center X, one may draw a line segment of length DL that intersects CL at point O. The figure and painting then show a circle of radius OX and center O that intersected side AD at V (where OV equals DL and is perpendicular to AD), and side BE at F. The point Y on the circle is on OV extended. As Crockett Johnson states in his notes, XY squared equals the product of AB and AD.

- The Euler line of a triangle includes three points. These are the intersections of the altitudes, of the perpendicular bisectors (lines perpendicular to the sides at their midpoints), and of the medians (lines drawn from a vertex to the midpoint of the opposite side). For an inscribed right triangle, both the perpendicular bisectors and the medians intersect in the center of the inscribing circle, while the altitudes meet at the right angle of the triangle. In the painting there are three right triangles inscribed in the circle. These are triangles XEF, XYF, and VXY in the diagram. The Euler line for the first two triangles is XOF, the Euler line for the third is VOY. The colors of Crockett Johnson's painting draws special attention to XOF, and it is this line he mentions in his figure for the painting.

- The painting is on masonite, and is #94 in the series. It has a blue-black background and a black wooden frame. It is signed on the back: SQUARED RECTANGLE AND EULER LINE (/) Crockett Johnson 1972.

- Location
- Currently not on view

- date made
- 1972

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.62

- catalog number
- 1979.1093.62

- accession number
- 1979.1093

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