| National Museum of American History

This ten-key electric printing adding machine has a brown metal and plastic frame with brown keys. The block of nine number keys has a 0 bar below it, and a subtraction bar and a blank bar to its right..

Description: This ten-key electric printing adding machine has a brown metal and plastic frame with brown keys. The block of nine number keys has a 0 bar below it, and a subtraction bar and a blank bar to its right.. A lever is in the right front corner and a red button in the upper right corner. To the left of the number keys are a clearance lever, a N (/) R lever, and a lever for which one setting is x. The place indicator is over the keyboard, and the printing mechanism, carriage, paper tape, and motor behind it. The machine allows one to enter 11 (possibly 12)-digit numbers and print 12 (possibly 13)-digit totals. In addition to numbers, the printing mechanism prints decimal markers and labels on both the right and the left of the numbers. A serrated edge helps to tear off the paper tape. The brown rubber cord is separate.; The model number is 76 86 54. The serial number is 2930-987. Dated from Smithsonian tag number. Walter J. Smith, who was a supply technician in Exhibits Production at the Smithsonian’s National Museum of American History from February 1979 to September 1994, used the machine.; Reference:; NOMDA’s Blue Book: Approximate January 1st Ages Adding Machines and Calculator Retail Prices, November, 1980, p. 57.
Location: Currently not on view

date made: 1967

maker: Victor Comptometer Corporation

ID Number: 1995.3069.01
maker number: 2930987
nonaccession number: 1995.3069
catalog number: 1995.3069.01

In the late nineteenth and early twentieth centuries, practitioners of the relatively new discipline of psychology developed a variety of objects for use in testing the intellectual abilities, skills, and response times of individuals.

Description: In the late nineteenth and early twentieth centuries, practitioners of the relatively new discipline of psychology developed a variety of objects for use in testing the intellectual abilities, skills, and response times of individuals. Shortly before the outbreak of World War I, they began to use paper-and-pencil tests to evaluate such human characteristics as intelligence, manual dexterity, work skills, academic achievement, personality, and character. The new methodology was used by the U.S. Army during World War I to test the intelligence of recruits. After the war, it spread widely in American schools, offices, and industry.; Psychologist Samuel Kavruck (1915-2009) accumulated a collection of tests during his long career at the U.S. Civil Service Commission, the U.S. Office of Education, and George Washington University. Three of them involve physical manipulation of wooden puzzles, the others are paper-and-pencil. The materials date from 1916 to 1966, with the bulk from between 1920 and 1950. In addition to tests, the collection includes score sheets, test keys, manuals, and related publications.; References:; Accession file.; “Samuel Kavruck GWU Professor,” Washington Post, March 16, 2009.
Location: Currently not on view

date made: 1960

maker: McLaughlin, K. F.; U.S. Department of Health, Education, and Welfare

ID Number: 1990.0034.001
catalog number: 1990.0034.001
accession number: 1990.0034

This is a relatively late model of the calculating machine invented by the Swede W. T. Odhner in the 1870s and manufactured in Sweden from after World War I. The pinwheel lever-set non-printing machine has a metal frame painted gray with ten metal pinwheels and a metal base.

Description: This is a relatively late model of the calculating machine invented by the Swede W. T. Odhner in the 1870s and manufactured in Sweden from after World War I. The pinwheel lever-set non-printing machine has a metal frame painted gray with ten metal pinwheels and a metal base. Numbers are set by rotating the pinwheels forward using levers that extend from the wheels. Digits inscribed on the frame next to the rotating pinwheels show the number set, for there is no separate set of windows to show these digits.; The carriage is at the front of the machine, with eight windows for the revolution counter on the left. The ten windows of the result register are right of the revolution register. Cranks at opposite ends of the carriage zero the registers on it. Two buttons move the carriage a single place to the right or left. A crank with a wooden knob on the right side of the machine rotates clockwise for addition and multiplication and counterclockwise for subtraction and division. Sliding decimal markers are above the pinwheels, the revolution counting register, and the result register. The machine has a plastic cover.; A mark on the top of the machine reads: Original-Odhner. A paper label glued to the back reads: SKRIVEMASKINE-EXPERTEN (/) KOBMAGERGADE 2 (/) PA. 5291 KOBEBHAVN K. The serial number, stamped on the bottom, reads: 229-2484 (/) MADE IN SWEDEN. A plate under the zeroing mechanism on the right side reads: 5275-00. The inside of the cover has the mark: U. S. Naugahyde (/) 66.; According to the donor, he purchased the machine on a trip to Europe in about 1960 (the instruction manual indicates that the instructions were issued in August of 1967)..; For related documentation, see 1991.0183.02.; Reference:; Accession file.
Location: Currently not on view

date made: ca 1960; 1967

maker: Aktiebolaget Original Odhner

ID Number: 1991.0183.01
accession number: 1991.0183
catalog number: 1991.0183.01
maker number: 229-2484

This model represents the U.S. Lighthouse Tender Joseph Henry, a side-wheeled steamer built by Howard & Company in Jeffersonville, Indiana, in 1880. This 180-foot-long vessel was built for service along the nation’s inland waterways.

Description: This model represents the U.S. Lighthouse Tender Joseph Henry, a side-wheeled steamer built by Howard & Company in Jeffersonville, Indiana, in 1880. This 180-foot-long vessel was built for service along the nation’s inland waterways. Lighthouse tenders served both coastal and inland areas by delivering supplies, fuel, news, and relief and maintenance crew to lighthouses and lightships. They also maintained aids to navigation, including markers identifying channels, shoals, and obstructions. Based out of Memphis, the Joseph Henry worked along the Mississippi and Missouri Rivers until 1904.; The vessel’s namesake, Joseph Henry, was America’s foremost scientist in the 19th century. His expertise was in the field of electromagnetism. Henry was a professor at the College of New Jersey (Princeton) when he was named the first Secretary of the Smithsonian Institution, a position he held from 1846 until his death in 1878. He also served on the U.S. Lighthouse Board (1852-78), and implemented various improvements in lighting and signaling during his tenure. This lighthouse tender was named in his honor at its launching two years after his death.

Date made: 1880; 1962
used: late 19th century

ID Number: TR.321486
catalog number: 321486
accession number: 245714

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

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

In a pathbreaking book La Géométrie, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting.

Description: In a pathbreaking book La Géométrie, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting. More specifically, Descartes described geometrical methods for finding the roots of simple polynomials. He wrote (as translated from the original French): "Finally, if I have z² = az -b², I make NL equal to (1/2)a and LM equal to b as before: then, instead of joining the points M and N, I draw MQR parallel to LN, and with N as center describe a circle through L cutting MQR in the points Q and R; then z, the line sought, is either MQ or MR, for in this way it can be expressed in two ways, namely: z = (1/2)a + √((1/4)a² - b²) and z = (1/2)a - √((1/4)a² - b²)."; To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.; Crockett Johnson's painting directly imitates Descartes's figure found in Book I of La Géométrie. A translation of part of Book I is found in the artist’s copy of James R. Newman's The World of Mathematics. The figure on page 250 is annotated.; This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.
Location: Currently not on view

date made: 1966

referenced: Descartes, Rene
painter: Johnson, Crockett

ID Number: 1979.1093.24
catalog number: 1979.1093.24
accession number: 1979.1093

Two polygons are said to be homothetic if they are similar and their corresponding sides are parallel.

Description: Two polygons are said to be homothetic if they are similar and their corresponding sides are parallel. If two polygons are homothetic, then the lines joining their corresponding vertices meet at a point.; The diagram on which this painting is based is intended to illustrate the homothetic nature of two polygons ABCDE . . . and A'B'C'D'E' . . . From the title, it appears that Crockett Johnson wished to call attention of homothetic triangular pairs ABS and A'B'S, BCS and B'C'S, CDS and C'D'S, DES and D'E'S, etc. The painting follows a diagram that appears in Nathan A. Court's College Geometry (1964 printing). Court's diagram suggests how one constructs a polygon homothetic to a given polygon. Hippocrates of Chios, the foremost mathematician of the fifth century BC, knew of similarity properties, but there is no evidence that he dealt with the concept of homothecy.; To illustrate his figure, the artist chose four colors; red, yellow, teal, and purple. He used one tint and one shade of each of these four colors. The larger polygon is painted in tints while the smaller polygon is painted in shades. The progression of the colors follows the order of the color wheel, and the black background enhances the vibrancy of the painting.; Homothetic Triangles, painting #17 in the Crockett Johnson series, is painted in oil on masonite. The work was completed in 1966 and is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 (/) HOMOTHETIC TRIANGLES (/) (HIPPOCRATES OF CHIOS). It has a black wooden frame.; References: Court, Nathan A., College Geometry, (1964 printing), 38-9.; van der Waarden, B. L., Science Awakening (1954 printing), 131-136.
Location: Currently not on view

date made: 1966

referenced: Hippocrates of Chios
painter: Johnson, Crockett

ID Number: 1979.1093.11
catalog number: 1979.1093.11
accession number: 1979.1093

In 1966, Crockett Johnson carefully read Nathan A. Court's book College Geometry, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting.

Description: In 1966, Crockett Johnson carefully read Nathan A. Court's book College Geometry, selecting diagrams that he thought would be suitable for paintings. In the chapter on harmonic division, he annotated several figures that relate to this painting. The work shows two orthoganol circles, that is to say two circles in which the square of the line of centers equals the sum of the squares of the radii. A right triangle formed by the line of centers and two radii that intersect is shown. The small right triangle in light purple in the painting is this triangle.; Crockett Johnson's painting combines a drawing of this triangle with a more complex figure used in a discussion of further properties of lines drawn in orthoganal circles. In particular, suppose that one draws a line segment from a point outside a circle that intersects it in two points, and selects a fourth point on the line that divides the segment harmonically. For a single exterior point, all these such points lie on a single line, perpendicular to the line of centers of the two circles, which is called the polar line.; The painting is #38 in the series. It has a background in two shades of cream, and a light tan wooden frame. It shows two circles that overlap slightly and have various sections. The circles are in shades of blue, purple and cream. The painting is signed: CJ66.; References: Nathan A. Court, College Geometry (1964 printing), p. 175–78. This volume was in Crockett Johnson's library.; T. L. Heath, ed., Apollonius of Perga: Treatise on Conic Sections (1961 reprint). This volume was not in Crockett Johnson's library.
Location: Currently not on view

date made: 1966

referenced: Apollonius of Perga
painter: Johnson, Crockett

ID Number: 1979.1093.26
catalog number: 1979.1093.26
accession number: 1979.1093

The French lawyer and mathematician Pierre de Fermat (1601–1665) was one of the first to develop a systematic way to find the straight line which best approximates a curve at any point. This line is called the tangent line.

Description: The French lawyer and mathematician Pierre de Fermat (1601–1665) was one of the first to develop a systematic way to find the straight line which best approximates a curve at any point. This line is called the tangent line. This painting shows a curve with two horizontal tangent lines. Assuming that the curve is plotted against a horizontal axis, one line passes through a maximum of a curve, the other through a minimum. An article by H. W. Turnbull, "The Great Mathematicians," published in The World of Mathematics by James R. Newman, emphasized how Fermat's method might be applied to find maximum and minimum values of a curve plotted above a horizontal line (see his figures 14 and 16). Crockett Johnson owned and read the book, and annotated the first figure. The second figure more closely resembles the painting.; Computing the maximum and minimum value of functions by finding tangents became a standard technique of the differential calculus developed by Isaac Newton and Gottfried Leibniz later in the 17th century.; Curve Tangents is painting #12 in the Crockett Johnson series. It was executed in oil on masonite, completed in 1966, and is signed: CJ66. The painting has a wood and metal frame.
Location: Currently not on view

date made: 1966

referenced: Fermat, Pierre de
painter: Johnson, Crockett

ID Number: 1979.1093.07
catalog number: 1979.1093.07
accession number: 1979.1093

According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. But finding the area bounded by curved surfaces was not an easy task.

Description: According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. But finding the area bounded by curved surfaces was not an easy task. The parabola and other conic sections had been known for almost a century before Archimedes wrote a short treatise called Quadrature of the Parabola in about 240 BC. This was the first demonstration of the area bounded by a conic section. In his proof, Archimedes first constructed a triangle whose sides consisted of two tangents of a parabola and the chord connecting the points of tangency. He then showed that the area under the parabola (shown in gray and black in the painting) is two thirds of the area of the triangle which circumscribes it. Once the area bounded by the tangent could be expressed in terms of the area of a triangle, it was easy to construct the corresponding square. Crockett Johnson’s painting follows two diagrams illustrating a discussion of Archimedes’s proof given by Heinrich Dorrie (Figure 54).; This oil or acrylic painting on masonite is #78 in the series and is signed “CJ67” in the bottom left corner. It has a gray wooden frame. For a related painting, see #43 (1979.1093.31).; References: Heinrich Dorrie, trans. David Antin, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965), p. 239. This volume was in Crockett Johnson's library and the diagram in his copy is annotated.; James R. Newman, The World of Mathematics (1956), p. 105. This volume was in Crockett Johnson's library. The figure on this page (Figure 9) is annotated.
Location: Currently not on view

date made: 1967

referenced: Archimedes
painter: Johnson, Crockett

ID Number: 1979.1093.52
catalog number: 1979.1093.52
accession number: 1979.1093

The mathematician Euclid lived around 300 BC, probably in Alexandria in what is now Egypt. Like most western scholars of his day, he wrote in Greek. Euclid prepared an introduction to mathematics known as The Elements.

Description: The mathematician Euclid lived around 300 BC, probably in Alexandria in what is now Egypt. Like most western scholars of his day, he wrote in Greek. Euclid prepared an introduction to mathematics known as The Elements. It was an eminently successful text, to the extent that most of the works he drew on are now lost. Translations of parts of The Elements were used in geometry teaching well into the nineteenth century in both Europe and the United States.; Euclid and other Greek geometers sought to prove theorems from basic definitions, postulates, and previously proven theorems. The book examined properties of triangles, circles, and more complex geometric figures. Euclid's emphasis on axiomatic structure became characteristic of much later mathematics, even though some of his postulates and proofs proved inadequate.; To honor Euclid's work, Crockett Johnson presented not a single mathematical result, but what he called a bouquet of triangular theorems. He did not state precisely which theorems relating to triangles he intended to illustrate in his painting, and preliminary drawings apparently have not survived. At the time, he was studying and carefully annotating Nathan A. Court's book College Geometry (1964). Court presents several theorems relating to lines through the midpoints of the side of a triangle that are suggested in the painting. The midpoints of the sides of the large triangle in the painting are joined to form a smaller one. According to Euclid, a line through two midpoints of sides of a triangle is parallel to the third side. Thus the construction creates a triangle similar to the initial triangle, with one fourth the area (both the height and the base of the initial triangle are halved). In the painting, triangles of this smaller size tile the plane. All three of the lines joining midpoints create triangles of this small size, and the large triangle at the center has an area four times as great.; The painting also suggests properties of the medians of the large triangle, that is to say, the lines joining each midpoint to the opposite vertex. The three medians meet in a point (point G in the figure from Court). It is not difficult to show that point G divides each median into two line segments, one twice as long as the other.; To focus attention on the large triangle, Crockett Johnson executed it in shades of white against a background of smaller dark black and gray triangles.; Bouquet of Triangle Theorems apparently is the artist's own construction. It was painted in oil or acrylic and is #26 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) BOUQUET OF TRIANGLE THEOREMS (/) (EUCLID).; Reference: Nathan A. Court, College Geometry, (1964 printing), p. 65. The figure on this page is not annotated.
Location: Currently not on view

date made: 1966

referenced: Euclid
painter: Johnson, Crockett

ID Number: 1979.1093.19
catalog number: 1979.1093.19
accession number: 1979.1093

Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings.

Description: Crockett Johnson used a wide range of geometrical constructions as the basis for his paintings. This painting is based on a method of constructing a rectangle equal in area to a given rectangle, given one side of the rectangle to be constructed.; In the painting, suppose that the cream-colored rectangle on the bottom left is given, as well as a line segment extending from the upper right corner of it. Construct the small triangle on the upper left. Draw the three horizontal lines shown, as well as the diagonal of the rectangle constructed. Extend this diagonal until it meets the bottom line, creating another triangle. The length of the base of this triangle will be the side of the rectangle desired. This rectangle is on the upper right in the painting.; This construction has been associated with the ancient Pythagoreans. Crockett Johnson may well have learned it from Evans G. Valens, The Number of Things. The drawing on page 121 of this book is annotated, although the annotations are faint.; The oil painting is #48 in the series. It has a black background and a black wooden frame, with the two equal triangles in light shades. The painting is signed on the front: CJ69. It is signed on the back: RECTANGLES OF EQUAL AREA (/) (PYTHAGORAS) (/) Crockett Johnson 1969.
Location: Currently not on view

date made: 1969

referenced: Pythagoras
painter: Johnson, Crockett

ID Number: 1979.1093.34
catalog number: 1979.1093.34
accession number: 1979.1093

A transversal is a line that intersects a system of other lines or line segments. Here Crockett Johnson explores the properties of certain transversals of the sides of a triangle.

Description: A transversal is a line that intersects a system of other lines or line segments. Here Crockett Johnson explores the properties of certain transversals of the sides of a triangle. The Italian mathematician Giovanni Ceva showed in 1678 that lines drawn from a point to the vertices of a triangle divide the edges of the triangle into six segments such that the product of the length of three nonconsecutive segments equals the product of the remaining three segments.; This painting shows a triangle (in white), lines drawn from a point inside the triangle to the three vertices, and a line drawn from a point outside the triangle (toward the bottom of the painting) to the three vertices. Segments of the sides of the triangle to be multiplied together are of like color. Crockett Johnson's painting combines two diagrams on page 159 of Nathan Court's College Geometry (1964 printing). These diagrams are annotated in his copy of the volume. Several of the triangles adjacent to the central triangle were used by Court in his proof of Ceva's theorem.; The painting is #31 in the series. It is signed: CJ66. There is a wooden frame painted off-white.
Location: Currently not on view

date made: 1966

referenced: Ceva, Giovanni
painter: Johnson, Crockett

ID Number: 1979.1093.22
catalog number: 1979.1093.22
accession number: 1979.1093

From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle.

Description: From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle. Nineteenth-century mathematicians were much interested in the properties of conics that were preserved under projection. They knew from the work of the ancient mathematician Pappus that the cross ratio of line segments created by two straight lines cutting the same pencil of lines was a constant (a pencil of lines is a set of lines emanating from a common point). In the drawing, which is Figure 5 from an article by Morris Kline in James R. Newman's The World of Mathematics (1956), if line segment l’ crosses lines emanating from the point O at points A’, B’, C’, and D’; and line segment l croses the same lines at points A, B, C, and D, the cross ratio:; (A’C’/C’B’) / (A’D’/D’B’) = (AC/BC) / (AD/DB), in other words it is independent of the cutting line. (see Crockett Johnson's painting Pencil of Ratios (Monge)).; The French mathematician Michel Chasles introduced a related result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio. This opened up an entirely different way of describing conic sections. Crockett Johnson associated this painting with both Chasles and another French advocate of projective geometry, Victor Poncelet.; The painting, in oil or acrylic on masonite, is #69 in the series. It has a dark gray or blue background and a black wooden frame. It shows a white ellipse, two points on the ellipse (on the left side of the painting), and two pencils of lines that produce the same cross ratio. The painting is not signed. It is inscribed on the back, in Crockett Johnson’s hand: CROSS RATIO IN AN ELLIPSE (PONCELET) (/) Crockett Johnson 1968. Compare #21 (1979.1093.15).; Reference: Morris Kline in James R. Newman, The World of Mathematics (1956), p. 634. This volume was in Crockett Johnson's library. The figure on this page is annotated.
Location: Currently not on view

date made: 1968

referenced: Poncelet, Jean-Victor
painter: Johnson, Crockett

ID Number: 1979.1093.44
accession number: 1979.1093
catalog number: 1979.1093.44

The ancient Greek mathematician Euclid showed in his Elements that it is possible to divide a line segment into two smaller segments wherein the ratio of the whole length to the longer part equals the ratio of the longer part to the smaller.

Description: The ancient Greek mathematician Euclid showed in his Elements that it is possible to divide a line segment into two smaller segments wherein the ratio of the whole length to the longer part equals the ratio of the longer part to the smaller. He used this theorem in his construction of a regular pentagon. The ratio came to be called the golden ratio. If the sides of a rectangle are in the golden ratio, it is called a golden rectangle. Several Crockett Johnson paintings explore the golden ratio and related geometric figures. This paintings suggest how a golden rectangle can be constructed, given the length of its shorter side. On the right in the painting is the golden rectangle that results. Lines in a triangle on the left indicate how the rectangle could have been constructed. Also included are the outlines of a hexagon and a five-pointed star constructed once the ratio had been found.; This painting follows a diagram on the top of page 131 in Evans G. Valens, The Number of Things. This diagram is annotated. Valens describes a geometrical solution to the two expressions f x f = e x c and f = e - c, and associates it with the Pythagoreans. The right triangle on the upper part of Valens's drawing, with the short side and part of the hypotenuse equal to f, is shown facing to the left in the painting. It can be constructed from a square with side equal to the shorter side of the rectangle. Two of the smaller rectangles in the painting are also golden rectangles. Crockett Johnson also includes in the background the star shown by Valens and related lines.; The painting on masonite is #46 in the series. It has a black and purple background and a black wooden frame. It is unsigned. The inscription on the back reads: GOLDEN RECTANGLE (/) (PYTHAGORAS) (/) Crockett Johnson 1968. Compare #103 (1979.1093.70) and #64 (1979.1093.39).
Location: Currently not on view

date made: 1968

referenced: Pythagoras
painter: Johnson, Crockett

ID Number: 1979.1093.33
catalog number: 1979.1093.33
accession number: 1979.1093

This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old.

Description: This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old. The proposition, one of Gauss’s many contributions to the branch of mathematics called number theory, states that every positive integer is the sum of three triangular numbers. The concept of triangular numbers dates to antiquity. Suppose one arranges dots in rows, with one in the first row, two in the second, three in the first and so forth. Three dots form a triangle, as do 6 dots, 10 dots, and 15 dots. The numbers 3, 6, 10, 15, and so forth are called "triangular numbers." The integers 0 and 1 are thought of as special cases of triangular numbers.; Crockett Johnson derived his painting from an entry in Gauss's diary published in an article by Eric Temple Bell included by James R. Newman in his book The World of Mathematics (1956), p. 304. The entry includes the phrase EUREKA in Greek, and indicates that any positive integer is the sum of three triangular numbers.; Crockett Johnson’s painting abstractly represents this theorem through the juxtaposition of three triangles. The triangles are equal, but each figure is painted a different color. It is possible that the artist chose to illustrate each triangle in its own color to demonstrate that each triangle generally represents its own triangular number when computing a positive integer. However, the triangles are congruent, which reminds the viewer that the triangles are related because they all represent a triangular number.; This work was painted in oil on masonite, completed in 1966, and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) EVERY POSITIVE INTEGER (/) (GAUSS). It is painting #29 in the series, and has a wooden frame.; Reference: J. R. Newman, The World of Mathematics, 1956, p. 304.
Location: Currently not on view

date made: 1966

referenced: Gauss, Carl Friedrich
painter: Johnson, Crockett

ID Number: 1979.1093.21
catalog number: 1979.1093.21
accession number: 1979.1093

The history of projective geometry begins with the work of the French mathematician Gerard Desargues (1591–1661). During his lifetime his work was well known in some mathematical circles, but after his death, his contributions to the field were largely forgotten.

Description: The history of projective geometry begins with the work of the French mathematician Gerard Desargues (1591–1661). During his lifetime his work was well known in some mathematical circles, but after his death, his contributions to the field were largely forgotten. When Gaspard Monge (1746–1818) and his student, Jean-Victor Poncelet (1788–1867) began their studies of projective geometry, they were largely unaware of the work of Desargues. This may be why Crockett Johnson included Monge's name as opposed to Desargues' in this painting's title.; One of the fundamental concepts of projective geometry, which was touched upon, but not fully understood, by the Greeks, is that of a cross-ratio, or "ratio of ratios." It is the topic of Johnson's painting. If points A, B, C, and D on line l are projected from point O, and if the line l’ crosses the four projected line segments, then the ratio of ratios (A’C’/C’B’)/(A’D’/ D’B’) of the corresponding points A’,B’,C’, and D’ is the same as the ratio of ratios (AC/CB)/(AD/DB). Thus, a cross-ratio is a projective invariant for all line segments l’.; The artist may have received inspiration for this painting from his copy of James R. Newman's The World of Mathematics (1956), p. 632. The figure is found there in an article by Morris Kilne entitled "Projective Geometry." This figure is not annotated, and the painting flips Kline's image.; Crockett Johnson chose purple, white, black, and brown to color this work. He executed the projection in three tints of purple and one shade of white. The background, which is divided by line l’, was executed in black and brown.; Pencil of Ratios, an oil painting on masonite, is #18 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) PENCIL OF RATIOS (MONGE). The painting is unframed.
Location: Currently not on view

date made: 1966

referenced: Monge, Gaspard
painter: Johnson, Crockett

ID Number: 1979.1093.12
catalog number: 1979.1093.12
accession number: 1979.1093

Leonhard Euler (1707–1783) was the most prolific mathematician of the eighteenth century. He made significant contributions to geometry, calculus, mechanics, and number theory.

Description: Leonhard Euler (1707–1783) was the most prolific mathematician of the eighteenth century. He made significant contributions to geometry, calculus, mechanics, and number theory. He produced more than 800 publications during his lifetime, almost half of which were dictated after his eyesight failed in 1766. While Euler is best remembered for his contributions to analysis and mechanics, his interests included geometry. This figure illustrates a theorem about triangles associated with his name.; Euler showed that three points related to a triangle lie on a common line. The first is the circumcenter (point O in the figure), the intersection of the perpendicular bisectors of the three sides. This point is the center of the circle which passes through the vertices of the triangle. Johnson also constructed the three medians of the triangle and the three altitudes of the triangle. The medians intersect in a common point (point N in the figure) and the altitudes meet at a third point (H in the figure). These three points, Euler showed, lie on the same line. In the painting, Crockett Johnson also constructed the circle that circumscribes the triangle, as well as a circle of half the radius known as the nine-point circle. For a full description of this circle, see painting #75 (1979.1093.49).; In the painting, the circumcircle is centered exactly on the backing, and the Euler line extends from the lower right corner to the upper left corner. This divides the work into two triangles of equal area. The right half of the painting was executed in shades of red and purple, while the left half of the painting was executed in shades of gray and black. Crockett Johnson also joined the nine points of the nine-point circle to form an irregular polygon.; This oil painting on masonite is #28 in the series. There is a wooden frame painted black. The work was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) POINT COLLINEATION IN THE TRIANGLE (/) (EULER). For a related painting, see #75 (1979.1093.49).; Reference: Nathan A. Court, College Geometry (1964 printing), p. 103, cover. The figure on p. 103 is annotated.
Location: Currently not on view

date made: 1966

referenced: Euler, Leonhard
painter: Johnson, Crockett

ID Number: 1979.1093.20
catalog number: 1979.1093.20
accession number: 1979.1093

Artists used methods of projecting lines developed by the Italian humanist Leon Battista Alberti and his successors to create a sense of perspective in their paintings. In contrast, Crockett Johnson made these methods the subject of his painting.

Description: Artists used methods of projecting lines developed by the Italian humanist Leon Battista Alberti and his successors to create a sense of perspective in their paintings. In contrast, Crockett Johnson made these methods the subject of his painting. He followed a diagram in William M. Ivins Jr., Art & Geometry: A Study in Space Intuitions (1964 edition), p. 76. The figure in Crockett Johnson’s copy of the book is annotated. This painting has a triangle in the center that is divided by a diagonal line, with the left half painted a darker shade than the right. Inside the triangle is one large quadrilateral that is divided into four rows of quadrilaterals that are painted various shades of red, purple, blue, and white.; To represent three-dimensional objects on a two-dimensional canvas, an artist must render forms and figures in proper linear perspective. In 1435 Alberti wrote a treatise entitled De Pictura (On Painting) in which he outlined a process for creating an effective painting through the use of one-point perspective. Investigation of the mathematical concepts underlying the rules of perspective led to the development of a branch of mathematics called projective geometry.; Alberti’s method (as modified by Pelerin in the early 17th century) and Crockett Johnson’s painting begin with the selection of a vanishing point (point C in the figure from Ivins). The eye of the viewer is assumed to be across from and on the same level as C. The eye looks through the vertical painting at a picture that appears to continue behind the canvas. To portray on the canvas what the eye sees, the artist locates point A on the horizon (the horizontal through C). The artist then draws the diagonal from A to the lower right-hand corner of the painting (point I). The separation of the angle ICH into smaller, equal angles creates lines that delineate parallel lines in the picture plane. The horizontal lines that create small quadrilaterals, and thus the checkerboard effect, are determined by the intersections of the lines from C with the diagonals FH and EI.; This painting, #7 in the series, dates from 1966. It is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) PERSPECTIVE (ALBERTI). It is of acrylic or oil paint on masonite, and has a wooden frame.
Location: Currently not on view

date made: 1966

referenced: Alberti, Leon Battista
painter: Johnson, Crockett

ID Number: 1979.1093.03
catalog number: 1979.1093.03
accession number: 1979.1093

This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as diameter.

Description: This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as diameter. Erect a perpendicular at the point where the line segments meet (d in the figure), and consider this to be the altitude of a right triangle inscribed in the semicircle. By similar triangles, the length of the perpendicular of a triangle inscribed in a semicircle is the geometric mean of the two lengths into which it divides the diameter of the circle. Hence the length of d is the mean of the segments e and f.; This painting an orange-red background, and shows a triangle inscribed in an orange semicircle. The perpendicular from the right angle of the triangle divides the triangle into triangles similar to it, painted in black and white.; The painting, and the attribution of the theorem to Archytas, are based on a passage from Evans G. Valens, The Number of Things: Pythagoras, Geometry and Humming Strings (1964), p. 118. The figure on this page of this book from Crockett Johnson's library is annotated.; This oil painting on masonite is #65 in the series. It is inscribed on the back: GEOMETRIC MEAN (ARCHYTAS) (/) Crockett Johnson 1968. It has a wooden frame.
Location: Currently not on view

date made: 1968

referenced: Archytas
painter: Johnson, Crockett

ID Number: 1979.1093.40
accession number: 1979.1093
catalog number: 1979.1093.40

This painting illustrates two different kinds of mathematical progressions, the geometric (on the top) and the arithmetic (on the bottom). Going across the top from left to right each section is twice as wide as the previous one, as in a geometric progression.

Description: This painting illustrates two different kinds of mathematical progressions, the geometric (on the top) and the arithmetic (on the bottom). Going across the top from left to right each section is twice as wide as the previous one, as in a geometric progression. Going across the bottom from right to left, each section is 1 unit wider than the previous one, as in an arithmetic progression.; If the width of the top sections, considered going from left to right, represents the numbers a, 2a, 4a, and 8a in a geometric progression, then the width of the bottom sections, going right to left, can represent logarithms of these numbers, b = log a, 2b =2 log a, 3b = 3 log a, and 4b =4 log a. Crockett Johnson may have sought to illustrate an account of logarithms given in an article by H. W. Turnbull in Newman's Men of Mathematics. This painting does not represent the traditional divisions of either a slide rule or a ruler.; The Scottish nobleman John Napier published his discovery of logarithms in 1614. The painting suggests how logarithms allow one to reduce multiplication (as in the terms of a geometric progression) to addition (as in the terms of an arithmetic progression). As addition is far simpler than multiplication, logarithms were widely used by people carrying out calculations from the seventeenth century onward.; The painting is #37 in the series. It is in oil or acrylic on masonite, and is signed: CJ66. There is a gray wooden frame.; Reference: H. W. Turnbull, “The Great Mathematicians,” in James R. Newman, The World of Mathematics, (1956), p. 124. This volume was in Crockett Johnson's library, but the figure is not annotated.
Location: Currently not on view

date made: 1966

referenced: Napier, John
painter: Johnson, Crockett

ID Number: 1979.1093.25
catalog number: 1979.1093.25
accession number: 1979.1093

La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots.

Description: La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots. This construction is the subject of Crockett Johnson's painting. Descartes explained: "If the square root of GH is desired, I add, along the same straight line, FG equal to unity, then bisecting FH at K, I describe the circle FIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root." (this is a translation of portion of La Géométrie, as published by J. R. Newman, The World of Mathematics (1956), p. 241); To understand Descartes' description and the title of this painting, consider the diagram. An angle inscribed in a semicircle is a right angle, thus triangle FGI is similar to triangle IGH. Because this two triangles are similar, their corresponding sides are proportional. Thus, G/IFG = GH/GI. But FG is equal to one, so GH is the square of GI, and GI the square root of GH desired.; In his painting, Crockett Johnson has flipped the image from La Géométrie found in his copy of The World of Mathematics. This figure is not annotated. The artist divided his painting into squares of area one, suggesting what came to be called Cartesian coordinates. The division indicates that the GH chosen has length two.; Johnson chose white for the section of the semicircle that contains the edge of length equal to the square root of GH. This section provides a vivid contrast against the dull, surrounding colors. Crockett Johnson purposefully creates this area of interest to draw focus to the result of Descartes' construction.; Square Root of Two is painting #19 in the series. It was painted in oil or acrylic on masonite, completed in 1965, and is signed: CJ65. The wooden frame is painted black.
Location: Currently not on view

date made: 1965

referenced: Descartes, Rene
painter: Johnson, Crockett

ID Number: 1979.1093.13
catalog number: 1979.1093.13
accession number: 1979.1093

The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons.

Description: The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons. That is, given a polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straightedge. They were unable to square a circle. This painting is based on the earliest known squaring of a figure bounded by curves rather than straight lines. The mathematician Hippocrates squared a lune, a figure bounded by arcs of two circles with different radii. This achievement might seem more difficult than squaring a circle.; Crockett Johnson's painting follows two annotated figures in Evans G. Valens's The Number of Things (1964), pp. 103–104, a book in the artist’s mathematical library. The finished piece shows isosceles triangles T, and a second congruent triangle connected to it base to base to form a square. Also present in the painting are three lunes, two small and one large. The area of triangle T is equal to the sum of the areas of lunes A and B (see figures). The area of triangle T is also equal to the area of a lune composed of X, Y, and the area T-C. Furthermore, because triangle T is congruent to the triangle below it, triangle T is equal to the area of this lune. Thus, the area of the square is equal to the sum of the areas of the three lunes. In summary, Johnson pictorially represented a "squared" curvilinear region; that is, he successfully constructed a square with the same area as that of the region of three lunes bounded by curves.; Although Valens called the crescent moon shape a crescent, Crockett Johnson used the term lune. This probably indicates that he also read Herbert Westren Turnball “The Great Mathematicians” in The World of Mathematics, edited by James R. Newman (1956), where the term lune is used. Also, on page page 91 of Turnball’s article there is a diagram on which the painting could have been based.; Crockett Johnson executed this painting in 4 tints and darker shades of purple upon a black background. The center triangle is the darkest shade of purple. As one moves outward, the colors grow lighter. This allows a dramatic distinction to be seen between the figure and the background, and thus puts a greater emphasis on the lunes.; This oil painting on masonite is #68 in Crockett Johnson's series. Its date of completion is unknown and the work is unsigned. It is closely related to painting #67 (1979.1093.42).
Location: Currently not on view

date made: ca 1965; ca 1966

referenced: Hippocrates of Chios
painter: Johnson, Crockett

ID Number: 1979.1093.43
accession number: 1979.1093
catalog number: 1979.1093.43

Science & Mathematics

Victor Automatic Calculator

Pamphlet, Understanding Testing Purposes and Interpretations for Pupil Development

Original Odhner Model 229 Calculating Machine

Ship Model, Lighthouse Tender Joseph Henry

Painting -Centers of Similitude (La Hire)

Painting - Square Roots to Sixteen (Theodorus of Cyrene)

Painting - Simple Equation (Descartes)

Painting - Homothetic Triangles (Hippocrates of Chios)

Painting - Polar Line of a Point and a Circle (Apollonius)

Painting - Curve Tangents (Fermat)

Painting - Parabolic Triangles (Archimedes)

Painting - Bouquet of Triangle Theorems (Euclid)

Painting - Rectangles of Equal Area (Pythagoras)

Painting - Transversals (Ceva)

Painting - Cross Ratio in an Ellipse (Poncelet)

Painting - Golden Rectangle (Pythagoras)

Painting - Every Positive Integer (Gauss)

Painting - Pencil of Ratios (Monge)

Painting - Point Collineation in the Triangle (Euler)

Painting - Perspective (Alberti)

Painting - Geometric Mean (Archytas)

Painting - Logarithms

Painting - Square Root of Two (Descartes)

Painting - Squared Lunes (Hippocrates Of Chios)