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.

Crockett Johnson's interest in regular polygons included the pentagram, or five-pointed star. The relation between the pentagon and the star is simple. If each side of a regular pentagon is extended, a regular five-pointed star results.
Description
Crockett Johnson's interest in regular polygons included the pentagram, or five-pointed star. The relation between the pentagon and the star is simple. If each side of a regular pentagon is extended, a regular five-pointed star results. Similarly, connecting each diagonal of a regular pentagon creates a regular five-pointed star. The star will have a pentagon in it, so the method is self-perpetuating.
A method for a pentagram's construction in described in Book IV, Proposition II of Euclid's Elements, but the construction illustrated in this painting is the artist's own creation. It builds on the relationship between the sides of a regular five-pointed star and the golden ratio. As Crockett Johnson may have recalled from his earlier paintings, the five rectangles that surround the central pentagon of the star are golden, that is to say the ratio of the length of the two equal sides of the triangle to the side of the enclosed pentagon is (1 + √5) / 2. Hence one can construct the star by finding a line segment divided in this ratio. No figure by Crockett Johnson showing his construction has been found.
The pentagram, executed appropriately enough in hues of gold, contrasts vividly with the purple background in Star Construction.
The painting is #103 in the series. It is in oil or acrylic on pressed wood and has a gold-colored metal frame. The painting is unsigned and undated. Compare #46 (1979.1093.33) and #64 (1979.1093.39).
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.70
catalog number
1979.1093.70
accession number
1979.1093
This painting reflects Crockett Johnson's enduring fascination with square roots and squaring.
Description
This painting reflects Crockett Johnson's enduring fascination with square roots and squaring. As the title suggests, it includes four squares whose areas are 1, 2, 3, and 4 square units, and seven line segments whose lengths are the square roots of 2, 3, 4, 5, 6, 7, and 8.
One may construct these squares and square roots by alternate applications of the Pythagorean theorem to squares running along the diagonal of the painting, and to rectangles running across the top (not all the rectangles are shown). More specifically, assume that the light-colored square in the upper left corner of the painting has side of length 1 (which equals the square root of 1). Then the diagonal is the square root of two, and a quarter circle with this radius centered at upper left corner cuts the sides of the square extended to determine two sides of a second, larger square. The area of this square (shown in the painting) is the square of the square root of 2, or two.
One can then consider the rectangle with side one and base square root of two that is in the upper left of the painting. It will have sides one and the square root of 2, and hence diagonal of length equal to the square root of three. The diagonal is not shown, but an circular arc with this radius forms the second arc in the painting. It determines the sides of a square with side equal to the square root of three and area 3. It also forms a rectangle with sides of length one and the square root of 4 (or two). This gives the third arc and the largest square in the painting.
By continuing the construction (further squares and rectangles are not shown), Crockett Johnson arrived at portions of circular arcs that cut the diameter at distances of the square roots of 5, 6, 7, and 8. Only one point on the last arc is shown. It is at the lower right corner of the painting.
Crockett Johnson executed the work in various shades and tints from his starting point at the white and pale-blue triangle to darker blues at the opposite corner.
This oil painting on masonite is not signed and its date of completion is unknown. It is #97 in the series.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.65
catalog number
1979.1093.65
accession number
1979.1093
This whimsical painting is part of Crockett Johnson's exploration of ways to represent the sides and angles of a regular heptagon using line segments of equal length.
Description
This whimsical painting is part of Crockett Johnson's exploration of ways to represent the sides and angles of a regular heptagon using line segments of equal length. In its mathematics, it follows closely the construction from isosceles triangles within a rhombus used in the painting Heptagon from Ten Equal Lines (#104 in the series - 1979.1093.71). However, both the line segments shown and the appearance of the paintings are quite different.
Here three pairs of carefully selected equal lines at appropriate equal angles combine with a seventh line of equal length to give a construction of three sides and two angles of a regular heptagon. All but one of the endpoints of the lines lie on a parallelogram (the rhombus mentioned previously), hence the title. The segment of the heptagon is on the right side of the painting. In Crockett Johnson's figure for the work, the segment is lettered BCPE.
The painting, in oil or acrylic on masonite, is #106 in the series. It has a dark purple background. The pairs of line segments are in turquoise, green, and lavender, with the vertical one in white. This increases the drama of the painting, but obscures the heptagon. There is a wooden frame. The painting is signed on the back: HEPTAGON STATED BY (/) SEVEN TOOTHPICKS (/) (BETWEEN PARALLELS) (/) Crockett Johnson 1973.
Location
Currently not on view
date made
1973
painter
Johnson, Crockett
ID Number
1979.1093.73
catalog number
1979.1093.73
accession number
1979.1093
To "square" a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle.
Description
To "square" a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle. It was not until the 19th century that the impossibility of squaring a circle was demonstrated.
This painting is an original construction by Crockett Johnson. It begins with the assumprion that the circle has been squared. In this case, Crockett Johnson performed a sequence of constructions that produce several additional squares, rectangles, and circles whose areas are geometrically related to that of the original circle. These figures are produced using traditional Euclidean geometry, and require only straightedge and compass.
The painting on masonite is #102 in the series. It has a blue-black background and a metal frame. It shows various superimposed sections of circles, squares, and rectangles in shades of light blue, dark blue, purple, white and blue-black. It is unsigned. See 1979.3083.02.13.
References: Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (1991), Chapter 5.
Crockett Johnson, "A Geometrical Look at the Square Root of Pi," Mathematical Gazette 54 (February, 1970): pp. 59–60.
Location
Currently not on view
date made
ca 1970
painter
Johnson, Crockett
ID Number
1979.1093.69
catalog number
1979.1093.69
accession number
1979.1093
The construction of regular polygons using straightedge and compass alone is a problem that has intrigued mathematicians from ancient times.
Description
The construction of regular polygons using straightedge and compass alone is a problem that has intrigued mathematicians from ancient times. Crockett Johnson was particularly interested in the construction of regular seven-sided figures or heptagons, which require not only a compass but a marked straight edge. The mathematician Archimedes reportedly proposed such a construction, which was included in a treatise now lost. Relying heavily on Thomas Heath's Manual of Greek Mathematics, Crockett Johnson prepared this painting.
Archimedes had reduced the problem of finding a regular hexagon to that of finding two points that divided a line segment into two mean proportionals. He then used a construction somewhat like that of the painting to find a line segment divided as desired. Crockett Johnson's papers include not only photocopies of the relevant portion of Heath, but his own diagrams.
The painting is #104 in the series. It is in acrylic or oil on masonite., and has purple, yellow, green and blue sections. There is a black wooden frame. The painting is unsigned and undated. Relevant correspondence in the Crockett Johnson papers dates from 1974.
References: Heath, Thomas L., A Manual of Greek Mathematics (1963 edition), pp. 340–2.
Crockett Johnson, "A construction for a regular heptagon," Mathematical Gazette, 59 (March 1975): pp. 17–18.
Location
Currently not on view
date made
ca 1974
referenced
Archimedes
painter
Johnson, Crockett
ID Number
1979.1093.71
catalog number
1979.1093.71
accession number
1979.1093
This inverted microscope was used at Genentech, a biotechnology company. In a traditional light microscope (the kind most often used in high school biology classes), the light source comes from below a slide-mounted specimen and the observer views it from above.
Description (Brief)
This inverted microscope was used at Genentech, a biotechnology company. In a traditional light microscope (the kind most often used in high school biology classes), the light source comes from below a slide-mounted specimen and the observer views it from above. By contrast, an inverted microscope’s light source comes from above and the sample is viewed from the bottom.
This configuration eliminates the need for slide-mounting the specimen for observation and allows the observer to view samples in flasks or petri dishes. For this reason the inverted microscope is particularly useful in work with living cells and tissue culture, allowing both observation and manipulation of the sample.
Sources:
Goldstein, David. “Inverted Microscope.” Microscopy-UK. 1998. http://microscopy-uk.org.uk/mag/indexmag.html?http://microscopy-uk.org.uk/mag/artjul98/invert.html
Olympus. “Inverted biological microscope.” http://www.olympus-global.com/en/corc/history/story/micro/headstand/
Location
Currently not on view
date made
before 1995
circa 1970
user
Genentech, Inc.
maker
Olympus
ID Number
2012.0198.60
accession number
2012.0198
catalog number
2012.0198.60
This painting, based on a construction of Crockett Johnson, shows a central brown circle, a blue square, and a pink rectangle of equal area. Assuming the radius ot the circle is one, this area equals pi.
Description
This painting, based on a construction of Crockett Johnson, shows a central brown circle, a blue square, and a pink rectangle of equal area. Assuming the radius ot the circle is one, this area equals pi. The blue triangle has an approximate area of square root of pi, presenting the "triangular square root" in the title.
The diagram is from Crockett Johnson's papers. It begins with construction of a circle of radius one (the smaller circle with center X in the figure) and assumes he could find the square root of pi and construct the line XC equal to this as a side of the square shown. Assuming he can do this, the area of the square is pi. He then draws a circle of radius 2 centered at X , which intersects the square at F and extensions of the line XC at A and at N. Bisecting FX at O, he can draw a second unit circle centered at O. He joined A to B and F to N to obtain triangles XAB and XNF. Next, the artist constructed the semicircle with that intersects circle O at point I and the larger circle at point K. He then drew diameter KP and extended FI to H with IH = 1. To complete the illustration, Crockett Johnson outlined rectangle with sides HI and IP.
To show that the construction is correct, note that XC = JF = √(pi) because the square with side XC and circle O both have area pi. Triangle XNF = (1/2)(XN)(JF) = (1/2)(2)(√(pi)) = √(pi). To show that the rectangle with sides PI and HI has area pi observe that right triangle PIF is congruent to right triangle PFK. Thus P/IPF = PF/PK and PI = (PF)²/(PK) = (2JF)²/PK = 4(JF)²/PK = 4(√((pi))²)/4 = pi. So, the rectangle has area (HI)(PI) = (1)(pi) = pi, and the demonstration is complete.
This painting is executed in oil on masonite and is #90 in the series. The figures of the painting that display the painting’s title are colored in bright, bold colors while those shapes that constitute the background are less drastically highlighted. Thus, Crockett Johnson uses color to distinguish the important features of his construction.
This painting is unsigned and its date of completion is unknown.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.59
catalog number
1979.1093.59
accession number
1979.1093
Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the problem of Delos.
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.
date made
1970
referenced
Newton, Isaac
painter
Johnson, Crockett
ID Number
1979.1093.36
catalog number
1979.1093.36
accession number
1979.1093
Toward the end of his life, Crockett Johnson took up the problem of constructing a regular seven-sided polygon or heptagon. This construction, as Gauss had demonstrated, requires more than a straight edge and compass.
Description
Toward the end of his life, Crockett Johnson took up the problem of constructing a regular seven-sided polygon or heptagon. This construction, as Gauss had demonstrated, requires more than a straight edge and compass. Crockett Johnson used compass and a straight edge with a unit length marked on it. Archimedes and Newton had suggested that constructions of this sort could be used to trisect the angle and to find a cube with twice the volume of a given cube, and Crockett Johnson followed their example.
One may construct a heptagon given an angle of pi divided by seven. If an isosceles triangle with this vertex angle is inscribed in a circle, the base of the triangle will have the length of one side of a regular heptagon inscribed in that circle. According to Crockett Johnson's later account, in the fall of 1973, while having lunch in the city of Syracuse on Sicily during a tour of the Mediterranean, he toyed with seven toothpicks, arranging them in various patterns. Eventually he created an angle with his menu and wine list and arranged the seven toothpicks within the angle in crisscross patterns until his arrangement appeared as is shown in the painting.
Crockett Johnson realized that the vertex angle of the large isosceles triangle shown is exactly π/7 radians, as desired. The argument suggested by his diagram is more complex than what he later published. The numerical results shown in the figure suggest his willingness to carry out detailed calculations.
Heptagon from its Seven Sides, painted in 1973 and #107 in the series, shows a triangle with purple and white sections on a navy blue background. This oil or acrylic painting on masonite is signed on its back : HEPTAGON FROM (/) ITS SEVEN SIDES (/) (Color sketch for larger painting) (/) Crockett Johnson 1973. No larger painting on this pattern is at the Smithsonian.
Reference: Crockett Johnson, "A Construction for a Regular Heptagon," Mathematical Gazette, 1975, vol. 59, pp. 17–21.
Location
Currently not on view
date made
1973
painter
Johnson, Crockett
ID Number
1979.1093.74
catalog number
1979.1093.74
accession number
1979.1093
In this painting, Crockett Johnson supposed that one was given two lengths, one the square root of the second.
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
Working at the Lamont Geological Observatory, a Columbia University facility in Palisades, N.Y., Frank Press and his mentor, Maurice Ewing, developed seismometers that responded to surface waves of long-period and small-amplitude, whether caused by explosions or by earthquakes.
Description
Working at the Lamont Geological Observatory, a Columbia University facility in Palisades, N.Y., Frank Press and his mentor, Maurice Ewing, developed seismometers that responded to surface waves of long-period and small-amplitude, whether caused by explosions or by earthquakes. The first long-period vertical seismometer at Lamont came to public attention in early 1953 with news that it had recorded waves from a large earthquake that had recently occurred at Kamchatka, in the Soviet Union. A painting of a subsequent but similar Lamont instrument appeared on the cover of Scientific American in March 1959.
This example was made for the World Wide Standard Seismological Network. Established in 1961, the WWSSN was designed to detect underground nuclear tests and generate valuable information about the earth’s interior and its dynamic processes. The WWSSN was a key component of VELA Uniform, a Cold War project that was funded by the Advanced Research Projects Agency (ARPA), a branch of the Department of Defense. It was managed by the U.S. Coast and Geodetic Survey and then by the U.S. Geological Survey. That agency transferred this instrument to the Smithsonian in 1999.
Each of the 120 WWSSN stations had an instrument of this sort. This example was used in Junction City, Tex. It would have been linked to a matched galvanometer (such as 1999.0275.09) and a photographic drum recorder (such as 1999.0275.10). The “Sprengnether Instrument Co.” signature refers to a small shop in St. Louis, Mo., that specialized in seismological apparatus.
Like other long-period vertical seismometers developed at Lamont, this one was built around a “zero-length spring” of the sort that had been proposed in 1934 by Lucien LaCoste, a graduate student in physics at the University of Texas, and later incorporated into the gravity meters manufactured by LaCoste & Romberg.
Ref: United States Coast and Geodetic Survey, Instrumentation of the World-Wide Seismograph System, Model 10700 (Washington, D.C., 1962)
Ta-Liang Teng, “Seismic Instrumentation,” in Methods of Experimental Physics, vol. 24 part B, Geophysics (1987), pp. 56-58.
Location
Currently not on view
date made
1961-1962
maker
W. F. Sprengnether Instrument Co.
ID Number
1999.0275.03
catalog number
1999.0275.03
accession number
1999.0275
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).
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
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, . . .
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.
date made
ca 1972
painter
Johnson, Crockett
ID Number
1979.1093.61
catalog number
1979.1093.61
accession number
1979.1093
This is one of a series of paintings in which Crockett Johnson explored ways of constructing the regular heptagon. The construction is his own, and a drawing for it is attached to the back of the painting.
Description
This is one of a series of paintings in which Crockett Johnson explored ways of constructing the regular heptagon. The construction is his own, and a drawing for it is attached to the back of the painting. By an arrangement of ten equal line segments, he produced three sides and two angles of a regular heptagon. Two sides and one angle are actually shown in the painting.
Crockett Johnson supposed that four equal isosceles triangles, constructed with six equal line segments, were arranged as shown in his figure to form sides of a rhombus and of a parallelogram within it. Two adjacent sides of the rhombus also served as the long sides of equal triangles oriented in the opposite direction. Finally, a line parallel to one of these sides passed through points of intersection of the sides of triangles.
More specifically, in the drawing triangles BAF, DAR, DKE, and HBE are arranged within rhombus ABED, and around a central parallelogram. Two other equal triangles DES and BAG are also included. AFand EJ intersect at a point C and EK and BH at a point P. The tenth line, UL parallel to BE, passes through points C and P. Crockett Johnson claimed that BCPE represents three sides of a regular heptagon. His argument appears in his papers. The painting shows only the ten equal lines described in the title.
The sections of the rhombus are in black, white, and rose, with a purple background. There is a wooden frame painted purple. This oil painting on masonite is #109 in the series. It is marked on the back: HEPTAGON FROM TEN EQUAL LINES (/) Crockett Johnson 1973. Taped to the back is a sheet of paper with an explanation that is entitled: HEPTAGON FROM TEN EQUAL LINES.
Location
Currently not on view
date made
1973
painter
Johnson, Crockett
ID Number
1979.1093.75
catalog number
1979.1093.75
accession number
1979.1093
In this painting, Crockett Johnson continued his exploration of ways to find rectilinear figures of area approximately equal to pi with another of his own constructions. He took advantage of the fact that the square root of two is 1.414214, while pi is approximately 3.141597.
Description
In this painting, Crockett Johnson continued his exploration of ways to find rectilinear figures of area approximately equal to pi with another of his own constructions. He took advantage of the fact that the square root of two is 1.414214, while pi is approximately 3.141597. By constructing a length of one tenth the √2 and adding it to length three, he had a length 3.1414214 which, in his language, is an approximation of pi to .0001.
Here he assumed that the two large overlapping circles both have diameter two, and the smaller circle diameter one. The three blue and white squares then have sides of length one and diagonals of length √2. Suppose (as Crockett Johnson does) that one marks off a length of 1/10 along the side of the rightmost square, and erects a perpendicular. It will cut the diagonal of the small square to form a right triangle that has hypotenuse of length equal to one tenth √2, as desired. This then serves as the radius of a small circular arc, and is added on to the length of the sides of the three unit squares to form an approximate value of pi.
A diagram from Crockett Johnson's papers presents the mathematics of his construction.
The painting is #101 in the series. It has a black border and is unframed. It shows two overlapping circles of the same size, a smaller of half the diameter, and the arc of a still smaller circle. The circles are divided by straight lines into turquoise and white sections on the right side, which form the area approximately equal in area to one of the large circles. The length approximately equal to pi is across the bottom. Sections at the tleft side are in dark purple and black.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.68
catalog number
1979.1093.68
accession number
1979.1093
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.
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
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 is based on a figure that includes two squares and a rectangle.
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 is based on a figure that includes two squares and a rectangle. The smaller square (ABDX in Crockett Johnson's figure) is defined as having the same area as the circle circle with center O and diameters the diagonals of the rectangle with sides CE and EX. This circle also appears in his other diagram, although it does not appear in the painting. Other assumptions concerning the upper diagram are that the rectangle has area the square root of the area of the circle and that the triangles with sides CX and PX are isosceles and congruent.
If the small circle has radius one.and the are of the rectangle is assumed to be the square root of the area of that circle and the small square, the area of the rectangle is the square root of pi.
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
In a series of experiments between 1972 and 1974 Stanley Cohen, Herbert Boyer, and their colleagues, at Stanford University and the University of California, San Francisco, developed techniques that formed the basis of recombinant DNA technology and helped spur the birth of the b
Description (Brief)
In a series of experiments between 1972 and 1974 Stanley Cohen, Herbert Boyer, and their colleagues, at Stanford University and the University of California, San Francisco, developed techniques that formed the basis of recombinant DNA technology and helped spur the birth of the biotechnology industry.
This notebook was used by Stanley Cohen in his lab at Stanford University from January of 1972 through 1978 in his study of plasmids—a specific form of DNA found in some organisms, especially of bacteria. It chronicles his research on creating recombinant plasmids, starting with his efforts to break plasmids through mechanical shearing and following through his ground-breaking experiments employing restriction enzymes with Herbert Boyer.
While not technically a lab notebook—one containing a log of daily experiments—the notebook contains extra information on experiments, many sketches and maps of recombinant plasmids, and outlines for papers to be published (including on p. 51 the “Outline for Recombination Paper” that would become the paper “Construction of Biologically Functional Bacterial Plasmids In Vitro” published in the Proceedings of the National Academy of the Sciences in 1973.)
Scientists knew since 1959 that bacteria contain extra loops of DNA called “plasmids” in addition to their chromosome. In nature, bacteria can swap these plasmids with one another, quickly transferring beneficial genes like those that code for antibiotic resistance. By the early 1970s, investigators had isolated several plasmids as well as special enzymes known as “restriction endonucleases” that worked like scissors to cut open the loops of plasmids. Boyer had expertise with restriction endonucleases, and Cohen studied plasmids. After meeting at a conference in 1972, the two decided to combine their research efforts. Following preliminary experiments in 1973, the Cohen-Boyer team was able to cut open a plasmid loop, insert a gene from different bacteria and close the plasmid. This created a recombinant DNA molecule—a plasmid containing recombined DNA from two different sources.
Next, they inserted the plasmid into bacteria and demonstrated that the bacteria could use the new genes. They had created the first genetically modified organisms. A year later, the team used this technique to insert a gene from a frog into bacteria, proving that it was possible to transfer genes between two very different organisms. The technology for creating these “molecular chimeras” was patented on December 2, 1980 (U.S. Patent 4,237,224.)
The concept that genes from one organism could be inserted into another and still work was the foundation for the biotechnology industry, which emerged a few years later. Biotech companies use recombinant DNA to insert genes coding for useful products into bacteria and other organisms, turning them into tiny factories for making things from medicine to industrial chemicals. The earliest application of this technology was in the pharmaceutical industry. Learn more about this by searching for “Recombinant Pharmaceuticals” in our collection.
Source: Accession File
Location
Currently not on view
date made
1972-1978
used date
1972-1974
referenced
Boyer, Herbert
user
Cohen, Stanley N.
maker
Cohen, Stanley
ID Number
1987.0757.01
catalog number
1987.0757.01
accession number
1987.0757
patent number
4,237,224
This microforge was used at Genentech, a biotechnology company.Laboratory technicians use microforges to heat and shape glass in order to create very small, delicate instruments for work with living cells under a microscope.
Description (Brief)
This microforge was used at Genentech, a biotechnology company.
Laboratory technicians use microforges to heat and shape glass in order to create very small, delicate instruments for work with living cells under a microscope. This model was designed to “provide seven different basic operations for transforming fine capillary tubing, solid glass rods, and various fusible materials into an endless variety of micro-tools.” These micro-tools typically come in the form of extremely fine needles, pipettes, or hooks. They are used to manipulate or inject living cells under a microscope.
The need for the microforge developed after the invention of the micromanipulator, a tool designed to hold and manipulate tools under a microscope with a precision greater than that of the human hand.
Sources:
Curtin Scientific Company. “mini-maker!” Southwest Retort 22:9, May 1970. http://digital.library.unt.edu/ark:/67531/metadc111167/m1/24/
Institut Pasteur “Pierre de Fonbrune.” http://www.pasteur.fr/infosci/archives/fnb0.html
Location
Currently not on view
date made
about 1970
user
Genentech, Inc.
maker
Curtin Matheson Scientific, Inc.
ID Number
2012.0198.61
accession number
2012.0198
catalog number
2012.0198.61
maker number
V58092
model number
MF-67
serial number
46590
This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting.
Description
This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting. Here Crockett Johnson argues that the triangle ABF in the figure is the one he sought, with angle FAB being one seventh of pi. Segment CD in the figure, which appears in the painting, is the length of the edge of a regular 14-sided figure inscribed in a portion of the larger circle shown.
The painting, of oil or acrylic on masonite, is number 105 in the series. It is drawn in shades of cream, blue, and purple on a light purple background. It has a metal frame and is unsigned.
Location
Currently not on view
date made
ca 1973
painter
Johnson, Crockett
ID Number
1979.1093.72
catalog number
1979.1093.72
accession number
1979.1093
Working at the Lamont Geological Observatory, a Columbia University facility in Palisades, N.Y., Frank Press and his mentor, Maurice Ewing, designed seismometers that responded to surface waves of long-period and small-amplitude whether caused by explosions or by earthquakes.
Description
Working at the Lamont Geological Observatory, a Columbia University facility in Palisades, N.Y., Frank Press and his mentor, Maurice Ewing, designed seismometers that responded to surface waves of long-period and small-amplitude whether caused by explosions or by earthquakes. Their horizontal seismometer was of the “garden-gate” form: here, the horizontal boom attaches to the lower end of a vertical post, and a diagonal wire extends from the upper end of the post to the outer end of the boom. The first example was installed in 1953.
This example was made for the World Wide Standard Seismological Network. Established in 1961, the WWSSN was designed to detect underground nuclear tests, and generate valuable information about the earth’s interior and its dynamic processes. The WWSSN was a key component of VELA Uniform, a Cold War project that was funded by the Advanced Research Projects Agency (ARPA), a branch of the Department of Defense. It was managed by the U.S. Coast and Geodetic Survey and then by the U.S. Geological Survey. That agency transferred this instrument to the Smithsonian in 1999.
Each of the 120 stations in the WWSSN had two horizontal seismometers of this sort (one to capture the east-west component of the earth’s motions, and one to capture the north-south component). This example was used Junction City, Tx. It would have been linked to a matched galvanometer (such as 1999.0275.09) and a photographic drum recorder (such as 1999.0275.10). The “Sprengnether Instrument Co.” signature refers to a firm in St. Louis, Mo., that specialized in seismological instruments.
Ref: United States Coast and Geodetic Survey, Instrumentation of the World-Wide Seismograph System, Model 10700 (Washington, D.C., 1962).
W.F. Sprengnether Instrument Co., Inc., General Discription (sic) Long Period Horizontal Seismometer ([St. Louis], n.d.).
W.F. Sprengnether Instrument Co., Inc., Sprengnether Horizontal Component Seismometer, Series H ([St. Louis], n.d.).
Ta-Liang Teng, “Seismic Instrumentation,” in Methods of Experimental Physics, vol. 24 part B, Geophysics (1987), pp. 56-58.
Location
Currently not on view
date made
1961-1962
maker
Geotechnical Corporation
W. F. Sprengnether Instrument Co.
ID Number
1999.0275.04
catalog number
1999.0275.04
accession number
1999.0275
This painting shows three rectangles of equal area, one in shades of blue, one in shades of purple, and one in shades of pink.
Description
This painting shows three rectangles of equal area, one in shades of blue, one in shades of purple, and one in shades of pink. The height of the middle rectangle equals the height of the first rectangle less its own width, while the height of the third rectangle equals the height of the first triangle less the width of the first triangle. Crockett Johnson associated these properties with conic curves. The construction is that of the artist. The coloring was suggested by a recently discovered French cave painting. The narrow rectangle on the left side and the dark, thin triangle at the base were also added to correspond to the cave painting.
The oil painting on masonite is #60 in the series. It is signed: CJ70, and inscribed on the back: DIVISION OF THE SQUARE BY CONIC RECTANGLES (/) (GNOMON ADDED AT THE SUGGESTION OF A CRO-MAGNON (/) ARTIST OF LASCAUX (/) Crockett Johnson 1970. The painting is in a black wooden frame. For related documentation see 1979.3083.02.05.
Reference: Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings," Leonardo 5 (1972): pp. 98–101.
Location
Currently not on view
date made
1970
painter
Johnson, Crockett
ID Number
1979.1093.37
catalog number
1979.1093.37
accession number
1979.1093
This oil painting is an original construction of Crockett Johnson. Since there are no notes on this construction, there is no way to tell what, if any, geometrical statement Johnson was representing in this painting.
Description
This oil painting is an original construction of Crockett Johnson. Since there are no notes on this construction, there is no way to tell what, if any, geometrical statement Johnson was representing in this painting. It may be the case that he merely thought of a more artistic way to portray the rectangles with area the square root of pi that appear in notes used for another painting, “Pi Squared and its Square Root” (#83 - 1979.1093.54).
This painting has at its center a circle with center O and area pi. Also in the painting there are two rectangles, each of area the square root of pi, that share a diagonal that is the diameter of the circle with one end at point E. The black rectangle in the painting has sides CE and EX and the blue rectangle has sides DE and EF. The square in the painting is congruent to the square BDXA so it also has area pi, but it has been translated so its center is the same as the center of the circle, i.e. at O.
This is one of two paintings in the collection with this same title referring to the area the rectangles shown in the paintings. The geometry of the two is identical but the dimensions and colors are different. For this painting, #100 in the series, Johnson illustrates the subject, vividly through the electric blue color of the rectangle. Its partner, #89 in the series (1979.1093.58), displays the same rectangle in white, which contrasts brilliantly with its black and purple surroundings.
The painting is unsigned and its precise date is unknown. It has a plain wooden frame.
This painting is unsigned and its precise date is unknown. It has a plain wooden frame.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.67
catalog number
1979.1093.67
accession number
1979.1093
This oil painting on masonite, #91 in the series, uses the same construction as that of painting #52 (see 1979.1093.35).
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.
Some sources refer to this painting as Circle Squared to 0.0001.
date made
1972
painter
Johnson, Crockett
ID Number
1979.1093.60
catalog number
1979.1093.60
accession number
1979.1093

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