Science & Mathematics

Chemical Balance

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.

Name: Johnson, Crockett

Painting - Conic Curve (Apollonius)

In ancient times, the Greek mathematician Apollonius of Perga (about 240–190 BC) made extensive studies of conic sections, the curves formed when a plane slices a cone.
Description
In ancient times, the Greek mathematician Apollonius of Perga (about 240–190 BC) made extensive studies of conic sections, the curves formed when a plane slices a cone. Many centuries later, the French mathematician and philosopher René Descartes (1596–1650) showed how the curves studied by Apollonius might be related to points on a straight line. In particular, he introduced an equation in two variables expressing points on the curve in terms of points on the line. An article by H. W. Turnbull entitled "The Great Mathematicians" found in The World of Mathematics by James R. Newman discussed the interconnections between Apollonius and Descartes, and apparently was the basis of this painting. The copy of this book in Crockett Johnson's library is very faintly annotated on this page. Turnbull shows variable length ON, with corresponding points P on the curve.
The analytic approach to geometry taken by Descartes would be greatly refined and extended in the course of the seventeenth century.
Johnson executed his painting in white, purple, and gray. Each section is painted its own shade. This not only dramatizes the coordinate plane but highlights the curve that extends from the middle of the left edge to the top right corner of the painting.
Conic Curve, an oil or acrylic painting on masonite, is #11 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) CONIC CURVE (APOLLONIUS). It has a wooden frame.
Location
Currently not on view
date made
1966
referenced
Apollonius of Perga
painter
Johnson, Crockett
ID Number
1979.1093.06
catalog number
1979.1093.06
accession number
1979.1093

Painting - Pendulum Momentum (Galileo)

The Greek mathematician Aristotle, who lived from about 384 BC through 322 BC, believed that heavy bodies moved naturally downward, while lighter substances such as air naturally ascended.
Description
The Greek mathematician Aristotle, who lived from about 384 BC through 322 BC, believed that heavy bodies moved naturally downward, while lighter substances such as air naturally ascended. Other forms of terrestrial motion required a sustaining force, which was not expressed mathematically. The Italian Galileo Galilei (1564–1642) challenged Aristotle. He held that motion was persistent and would continue until acted upon by an opposing, outside force.
In a book entitled Dialogues Concerning the Two Chief World Systems, Galileo presented his ideas in a dispute between three men: Salviati, Sagredo, and Simplicio. Salviati, a spokesman for Galileo, explained his revolutionary ideas, one of which is illustrated by a diagram that was the basis for this painting. This image can be found in Crockett Johnson's copy of The World of Mathematics, a book by James R. Newman. It is probable that this image served as inspiration for this painting, although Johnson did not annotate this diagram.
In Galileo's Dialogues, Salviati argued that if a lead weight is suspended by a thread from point A (see figure) and is released from point C, it will swing to point D, which is located at the same height as the initial point C. Furthermore, Salviati stated that if a nail is placed at point E so that the thread will snag on it, then the weight will swing from point C to point B and then up to point G, which is also located at the same height as the initial point C. The same occurs if a nail is placed at point F below the line segment CD.
The painting is executed in purple that progresses from light tints to darker shades right to left. This gives the figure a sense of motion akin to that of a pendulum. The background is washed in gray and black. The line created by the initial and final height of the weight divides the background.
Pendulum Momentum, a work in oil on masonite, is painting #13 in the Crockett Johnson series. It was executed in 1966 and is signed: CJ66. There is a wooden frame painted black.
Location
Currently not on view
date made
1966
referenced
Galilei, Galileo
painter
Johnson, Crockett
ID Number
1979.1093.08
catalog number
1979.1093.08
accession number
1979.1093

Painting - Law of Orbiting Velocity (Kepler)

This work illustrates two laws of planetary motion proposed by the German mathematician Johannes Kepler (1571–1630) in his book Astronomia Nova (New Astronomy) of 1609. Kepler argued that planets move about the sun in elliptical orbits, with the sun at one focus of the ellipse.
Description
This work illustrates two laws of planetary motion proposed by the German mathematician Johannes Kepler (1571–1630) in his book Astronomia Nova (New Astronomy) of 1609. Kepler argued that planets move about the sun in elliptical orbits, with the sun at one focus of the ellipse. He also claimed that a planet moves about the sun in such a way that a line drawn from the planet to the sun sweeps out equal areas in equal times. The ellipse in the work represents the path of a planet and the white sections equal areas. The extraordinary contrast between the deep blue and white colors dramatize this phenomenon.
This oil painting on masonite has a wooden frame. It is signed: CJ65. It also is marked on the back: Crockett Johnson 1965 (/) LAW OF ORBITING VELOCITY (/) (KEPLER). It is #22 in the series. The work follows an annotated diagram from Crockett Johnson’s copy of Newman's The World of Mathematics (1956), p. 231. Compare to paintings #76 (1979.1093.50) and #99 (1979.1093.66).
Reference: Arthur Koestler, The Watershed (1960).
Location
Currently not on view
date made
1965
referenced
Kepler, Johannes
painter
Johnson, Crockett
ID Number
1979.1093.16
catalog number
1979.1093.16
accession number
1979.1093

Painting -Approximation of Pi to .0001

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

Painting - Squared Rectangle and Euler Line

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

Painting - Harmonic Series from a Quadrilateral (Pappus)

The concept of a harmonic set of points can be traced back through Girard Desargues (1591–1661) and Pappus of Alexandria (3rd century AD) to Apollonius of Perga (240–190 BC). Crockett Johnson's painting seems to be based upon a figure associated with Pappus.
Description
The concept of a harmonic set of points can be traced back through Girard Desargues (1591–1661) and Pappus of Alexandria (3rd century AD) to Apollonius of Perga (240–190 BC). Crockett Johnson's painting seems to be based upon a figure associated with Pappus. It is likely that Crockett Johnson was inspired by a figure found in H. W. Turnbull's article "The Great Mathematicians" found in his copy of James R. Newman's The World of Mathematics, p. 111. This figure is annotated.
The construction begins with a given set of collinear points (A, B, and Y). An additional point (W) is sought such that AW, AB, and AY are in harmonic progression. That is, the terms AW, AB, and AY represent a progression of terms whose reciprocals form an arithmetic progression. To do this, any point Z, not on line AB, is chosen, and line segments ZA and ZB are constructed. Next, any point D, on ZA, is chosen, and DY, which will intersect ZB at C, is constructed. AC and DB intersect each other at X, and ZX will intersect AB at W. The location of point W is entirely independent of the choice of points Z and D. It follows that AW, AB, and AY form a harmonic progression, and thus the points A, W, B, and Y form a harmonic set.
Crockett Johnson flipped the annotated image for his painting. The boldest portion of his painting, and thus the area with greatest interest, is the quadrilateral ABCD. In addition, the background of his painting is divided into three differently colored sections to illustrate the harmonic series constructed from the quadrilateral. This careful color choice reinforces the painting's title.
This painting was executed in oil on masonite and is painting #24 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) HARMONIC SERIES FROM A QUADRILATERAL (/) (PAPPUS). It has a gray wooden frame.
Location
Currently not on view
date made
1966
referenced
Pappus
painter
Johnson, Crockett
ID Number
1979.1093.18
catalog number
1979.1093.18
accession number
1979.1093

Painting - Pi Squared and Its Square Root

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

Painting - Heptagon 1:3:3 Triangle

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

Painting - Multiplication through Imaginary Numbers (Gauss)

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.
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

Painting - Division of the Square by Conic Rectangles

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

Painting - Mystic Hexagon (Pascal)

This painting is based on a theorem generalized by the French mathematician Blaise Pascal in 1640, when he was sixteen years old. When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points.
Description
This painting is based on a theorem generalized by the French mathematician Blaise Pascal in 1640, when he was sixteen years old. When the opposite sides of a irregular hexagon inscribed in a circle are extended, they meet in three points. Pappus, writing in the 4th century AD, had shown in his Mathematical Collections that these three points lie on the same line. In the painting, the circle and cream-colored hexagon are at the center, with the sectors associated with different pairs of lines shown in green, blue and gray. The three points of intersection are along the top; the line that would join them is not shown. Pascal generalized the theorem to include hexagons inscribed in any conic section, not just a circle. Hence the figure came to be known as "Pascal’s hexagon" or, to use Pascal’s terminology, the "mystic hexagon." Pascal’s work in this area is known primarily from notes on his manuscripts taken by the German mathematician Gottfried Leibniz after his death.
There is a discussion of Pascal’s hexagon in an article by Morris Kline on projective geometry published in James R. Newman's World of Mathematics (1956). A figure shown on page 629 of this work may have been the basis of Crockett Johnson's painting, although it is not annotated in his copy of the book.
The oil or acrylic painting on masonite is signed on the bottom right: CJ65. It is marked on the back: Crockett Johnson (/) "Mystic" Hexagon (/) (Pascal). It is #10 in the series.
References: Carl Boyer and Uta Merzbach, A History of Mathematics (1991), pp. 359–62.
Florian Cajori, A History of Elementary Mathematics (1897), 255–56.
Morris Bishop, Pascal: The Life of a Genius (1964), pp. 11, 81–7.
Location
Currently not on view
date made
1965
referenced
Pascal, Blaise
painter
Johnson, Crockett
ID Number
1979.1093.05
catalog number
1979.1093.05
accession number
1979.1093

Painting - Square Root of Pi

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

Painting - Proof of the Pythagorean Theorem (Euclid)

The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides.
Description
The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides. This painting depicts the “windmill” figure found in Proposition 47 of Book I of Euclid’s Elements. Although the method of the proof depicted was written about 300 BC and is credited to Euclid, the theorem is named for Pythagoras, who lived 250 years earlier. It was known to the Babylonians centuries before then. However, knowing a theorem is different from demonstrating it, and the first surviving demonstration of this theorem is found in Euclid’s Elements.
Crockett Johnson based his painting on a diagram in Ivor Thomas’s article on Greek mathematics in The World of Mathematics, edited by James R. Newman (1956), p. 191. The proof is based on a comparison of areas. Euclid constructed a square on the hypotenuse BΓ of the right triangle ABΓ. The altitude of this triangle originating at right angle A is extended across this square. Euclid also constructed squares on the two shorter sides of the right triangle. He showed that the square on side AB was of equal area to the rectangle of sides BΔ and Δ;Λ. Similarly, the area of the square on side AΓ was of equal area to the rectangle of sides EΓ and EΛ. But then the square of the hypotenuse of the right triangle equals the sum of the squares of the shorter sides, as desired.
Crockett Johnson executed the right triangle in the neutral, yet highly contrasting, hues of white and black. Each square area that rests on the sides of the triangle is painted with a combination of one primary color and black. This draws the viewer’s attention to the areas that complete Euclid’s proof of the Pythagorean theorem.
Proof of the Pythagorean Theorem, painting #2 in the series, is one of Crockett Johnson’s earliest geometric paintings. It was completed in 1965 and is marked: CJ65. It also is signed on the back: Crockett Johnson 1965 (/) PROOF OF THE PYTHAGOREAN THEOREM (/) (EUCLID).
Location
Currently not on view
date made
1965
referenced
Euclid
painter
Johnson, Crockett
ID Number
1979.1093.01
catalog number
1979.1093.01
accession number
1979.1093

Painting - Law of Orbiting Velocities

This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to Kepler's first two laws of planetary motion.
Description
This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to 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 is a silk screen on paper. It is number 99 in the series, and is signed in the right corner: Crockett Johnson (/) 67. It draws on a figure from The World of Mathematics by James R. Newman.
Location
Currently not on view
date made
1967
referenced
Kepler, Johannes
painter
Johnson, Crockett
ID Number
1979.1093.66
catalog number
1979.1093.66
accession number
1979.1093

Painting - Square Root Of Pi - 0.00001

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

Painting - Area and Perimeter of a Squared Circle

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 nineteenth century that the impossibility of squaring a circle was demonstrated.
This painting is an original construction by Crockett Johnson. It begins with the assumption that the circle has been squared, the area of the larger square equals that of the circle. Crockett Johnson then constructed a smaller square so that it has perimeter equal to the circumference of the circle. His diagram for the painting is shown, with the large square having side AB and the small one side of length AC.
The painting is #95 in the series. It has a black background. There is a rose circle superimposed on two gray squares. The painting is unsigned and has a metal frame.
Reference: Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (1991), pp. 65-7, pp. 71–2.
Location
Currently not on view
date made
1970-1975
painter
Johnson, Crockett
ID Number
1979.1093.63
catalog number
1979.1093.63
accession number
1979.1093

Painting - Square Root of Pi

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” (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 purple rectangle in the painting has sides CE and EX and the white 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 (see painting #100 - 1979.1093.67) but the dimensions and colors are different. The method of the color scheme of this painting, #89 in the series, is similar to painting #100 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 a rectangle with area the title of the painting.
This painting 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

Painting - Velocity on Inclined Planes (Galileo)

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.
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

Painting -Construction of Heptagon

This painting represents one of Crockett Johnson's early constructions of a heptagon. It shows a large purple circle, a pink triangle superimposed, and two smaller circles. Crockett Johnson's diagram for the painting is shown.
Description
This painting represents one of Crockett Johnson's early constructions of a heptagon. It shows a large purple circle, a pink triangle superimposed, and two smaller circles. Crockett Johnson's diagram for the painting is shown. Two equal circles are constructed, with the center of the first on the second and conversely (circles with centers C and D in the diagram), and a line segment drawn that includes their points of intersection. Then, in Crockett Johnson's words, "Against a straight edge controlling their alignment the sought points B, U, and E, are determined by the adjustment of compass arcs BC from U and EC from B. Angles FBC, CBD, DBE, and BAF are π/ 7." Detailed examination of the triangles in the drawing shows that this is indeed the case.
The colors of the painting highlight the circles, lines, and arcs central to the construction, and the largest of the resulting isosceles triangles with vertex angle π/7 is shown in bold shades of pink. The short line called CF in the drawing (as well as line segments CD and DE, which are not shown), is the length of the side of a heptagon inscribed in a circle centered at B with radius BF.
The oil on masonite work is #116 in the series. It has a gray background and a wood and metal frame. It is inscribed on the back: CONSTRUCTION OF HEPTAGON (/) . . .(8) (/) Crockett Johnson 1973.
Location
Currently not on view
date made
1973
painter
Johnson, Crockett
ID Number
1979.1093.78
accession number
1979.1093
catalog number
1979.1093.78

Painting - Construction of a Heptagon

This is one of three very similar Crockett Johnson paintings closely related to the construction of a side of an inscribed regular heptagon which the artist published in The Mathematical Gazette in 1975.
Description
This is one of three very similar Crockett Johnson paintings closely related to the construction of a side of an inscribed regular heptagon which the artist published in The Mathematical Gazette in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π / 7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon.
Crockett Johnson described the construction of his isosceles triangle in the diagram shown. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the largest triangle in the painting is triangle is ABF in the figure, with vertex angle equal to one seventh of pi. This angle is inscribed in the large circular arc KDC. The side of the heptagon is the chord KC.
This version of Crockett Johnson's construction of a heptagon is #115 in the series. It has a dark blue background and a wood and metal frame. The painting is an oil or acrylic on masonite. The work is unsigned. See also #108 (335571) and #117 (1979.1093.79).
References: Crockett Johnson, “A Construction for a Regular Heptagon,” Mathematical Gazette, 1975, vol. 59, pp. 17–21.
Location
Currently not on view
date made
ca 1975
painter
Johnson, Crockett
ID Number
1979.1093.77
accession number
1979.1093
catalog number
1979.1093.77

Painting - Construction of Heptagon

Three very similar paintings in the Crockett Johnson collection are closely related to the the construction of a side of an inscribed regular a heptagon which he published in The Mathematical Gazette in 1975.
Description
Three very similar paintings in the Crockett Johnson collection are closely related to the the construction of a side of an inscribed regular a heptagon which he published in The Mathematical Gazette in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π/ 7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon. Crockett Johnson described the construction of his isosceles triangle in the diagram reproduced. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the triangle is ABF. Three of the four light-colored sections of the painting highlight important points in the construction. The critical steps are drawing a perpendicular bisector to the line segment BF, marking off an arc of radius equal to the √(2) with center F, and measuring the unit length AO along a marked straightedge that passes through B and intersects the perpendicular bisector at A. Finally, one finds the side of the regular inscribed heptagon.
Construction of Heptagon is #117 in the series. The oil painting on masonite is in shades of purple, cream, turquoise, and black. It has a black wood and metal frame. The work is unsigned. The surface appears damaged, perhaps from water. See also #115 (1979.1093.77) and #108 (335571).
Reference: Crockett Johnson, “A Construction for a Regular Heptagon,” Mathematical Gazette, 1975, vol. 59, pp. 17–21.
Location
Currently not on view
date made
ca 1975
painter
Johnson, Crockett
ID Number
1979.1093.79
accession number
1979.1093
catalog number
1979.1093.79

Painting - Construction of the Heptagon

Three very similar paintings in the Crockett Johnson collection are closely related to the construction of a side of an inscribed regular a heptagon which he published in The Mathematical Gazette in 1975.
Description
Three very similar paintings in the Crockett Johnson collection are closely related to the construction of a side of an inscribed regular a heptagon which he published in The Mathematical Gazette in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π/7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon.
Crockett Johnson described the construction of his isosceles triangle in the diagram shown in the image. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the triangle is ABF. The light colors of the painting highlight important points in the construction - marking off an arc of radius equal to the square root of 2 with center F, measuring the unit length AO along a marked straight edge that passes through B and ends at point A on the perpendicular bisector, and finding the side of the regular inscribed heptagon.
This version of the construction of a heptagon is #108 in the series. The oil painting on masonite with chrome frame was completed in 1975 and is unsigned. It is marked on the back: Construction of the Heptagon (/) Crockett Johnson 1975. See also paintings #115 (1979.1093.77) and #117 (1979.1093.79) in the series.
Reference: Crockett Johnson, "A Construction for a Regular Heptagon," Mathematical Gazette, 1975, vol. 59, pp.17–21.
Location
Currently not on view
date made
1975
painter
Johnson, Crockett
ID Number
MA.335571
accession number
322732
catalog number
335571

Painting -Centers of Similitude (La Hire)

Two circles or other similar figures can be placed such that a line drawn from some fixed point to a point of one of them passes through a point on the other, such that the ratio of the distances from the fixed point to the two points is always the same.
Description
Two circles or other similar figures can be placed such that a line drawn from some fixed point to a point of one of them passes through a point on the other, such that the ratio of the distances from the fixed point to the two points is always the same. The fixed point is called the center of similitude. The circles shown in this painting have two centers of similitude, one between the circles and one to the right (the center of similitude between the circles is shown). Crockett Johnson apparently based his painting on a diagram from the book College Geometry by Nathan Altshiller Court (1964 printing). This diagram is annotated in his copy of the book. In the figure, the larger circle has center A, the smaller circle has center B, and the centers of similitude are the points S and S'. S is called the external center of similitude and S' is the internal center of similitude. The painting suggests several properties of centers of similitude. For example, lines joining corresponding endpoints of parallel diameters of the two circles, such as TT' in the figure, would meet at the external center of similitude. Lines joining opposite endpoints meet at the internal center of similitude.
This painting emphasizes the presence of the two circles and line segments relating to centers of similitude, but not the centers themselves. Indeed, the painting is too narrow to include the external center of similitude.
Some properties of centers of similitude were known to the Greeks. More extensive theorems were developed by the mathematician Gaspard Monge (1746–1818). It is not entirely clear why Crockett Johnson associated the painting with the artist and mathematician Phillipe de la Hire (1640–1718). A bibliographic note in the relevant section of Court reads: LHr., p. 42, rem. 8. However, Court was referring to an 1809 book by Simon A. J. LHuiler on the elements of analytic geometry.
This oil painting on masonite is #14 in Crockett Johnson's series. It was completed in 1966 and is signed: CJ66.
References: R. J. Archibald, "Centers of Similitude of Circles," American Mathematical Monthly, 22, #1 (1915), pp. 6–12; unpublished notes of J. B. Stroud.
Location
Currently not on view
date made
1966
referenced
de la Hire, Phillipe
painter
Johnson, Crockett
ID Number
1979.1093.09
catalog number
1979.1093.09
accession number
1979.1093

Painting - Square Roots to Sixteen (Theodorus of Cyrene)

Greek mathematicians knew that numbers could not always be represented as simple ratios of whole numbers. They devised ways to describe them geometrically.
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

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