#
Science & Mathematics

The Museum's collections hold thousands of objects related to chemistry, biology, physics, astronomy, and other sciences. Instruments range from early American telescopes to lasers. Rare glassware and other artifacts from the laboratory of Joseph Priestley, the discoverer of oxygen, are among the scientific treasures here. A Gilbert chemistry set of about 1937 and other objects testify to the pleasures of amateur science. Artifacts also help illuminate the social and political history of biology and the roles of women and minorities in science.

The mathematics collection holds artifacts from slide rules and flash cards to code-breaking equipment. More than 1,000 models demonstrate some of the problems and principles of mathematics, and 80 abstract paintings by illustrator and cartoonist Crockett Johnson show his visual interpretations of mathematical theorems.

"Science & Mathematics - Overview" showing 2033 items.

Page 131 of 204

## Involutes of Circles, Kinematic Model by Martin Schilling, series 24, model 6, number 334

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the sixth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- An involute of a circle is a curve that is produced by tracing the end of a string that is wrapped around a circle as it is unwound while being kept taut. It is the envelope of all points that are perpendicular to the tangents of a circle.

- As with the three trochoidal models, these curves were used in the shaping of gear teeth in the 18th century. Following that, it was discovered that shaping the teeth of gears using the curve formed by the involute of a circle also increases the efficiency of gearage. Surprisingly, there are many applications of noncircular gears, such as elliptical, triangular, and quadrilateral gears. (See model 1982.0795.06.)

- In this model a toothed circular gear of radius 13 mm is mounted on the baseplate and can be turned via a crank on the underside of the baseplate. A thick piece of beveled glass is mounted above the apparatus. A dark metal toothed bar 45 mm long is attached to the circular gear so that as the crank turns the circular gear, the toothed bar is forced past the circular gear and rotates round it.

- Perpendicular to the bar is a thin clip with three small colored balls. A blue ball is attached at the edge of the bar where the bar will touch the circle and traces the involute of the circle in blue on the glass. A red ball is placed 33mm in front of the toothed side of the bar and produces a “stretched” involute in red. A green ball is 45mm behind the toothed side of the bar traces another “stretched” involute in green. German title is: Erzeugung von Kreisevolventen.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group II, model 6.

- Online demo at Mathworld by Wolfram: http://mathworld.wolfram.com/Involute.html

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.04

- catalog number
- 1982.0795.04

- accession number
- 1982.0795

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

## Cycloids, Kinematic Model by Martin Schilling, series 24, model 7, number 335

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the seventh in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- The cycloid solves the 17th-century problem posed by Swiss mathematician Johann Bernoulli known as the brachistochrone problem. This problem asks for the shape of the curve of fastest decent: the path that a ball would travel the fastest along under the influence of gravity.

- The cycloids are drawn by tracing the location of a point on the radius of a circle or its extension as the circle rolls along a straight line. Cycloids are members of the family of curves known as trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. The curve generated by a point on the circumference of the rolling circle is called an epicycloid, and a ball rolling on this curve (inverted) would travel faster than on any other path (the brachistochrone problem). Points either inside or outside the rolling circle generate curves called epitrochoids. The cycloid also solves the tautochrone problem, a curve for which a ball placed anywhere on the curve will reach the bottom under gravity in the same amount of time.

- An example of the application of the cycloid as a solution of the tautochrone problem is the pendulum clock designed by Dutch physicist Christopher Huygens. As the width of the swing of the pendulum decreases over time due to friction and air resistance, the time of the swing remains constant. Also, cycloidal curves are used in the shaping of gear teeth to reduce torque and improve efficiency.

- This model consists of a toothed metal disc linked to a bar that is toothed along one edge. A radius of the circle extending away from the bar has a place for a pin inside the circumference, a pin on the circumference, and a pin outside the circle. Rotating a crank below the baseplate of the model moves the circle along the edge of the bar, generating a curve above each point. The curves are indicated on the glass overlay of the mechanism. The curve generated by the point on the circumference of the circle is an epicycloid, depicted in blue on the glass; that generated by the point outside the circle is a prolate (from the Latin to elongate) cycloid, depicted in orange; and that generated by the point inside the circle is a curtate (from the Latin to shorten) cycloid, depicted in green. The German title of this model it: Erzeugung von Cycloiden (to produce cycloids).

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group II, model 7.

- Online demo at Wofram Mathworld: http://mathworld.wolfram.com/Cycloid.html

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.05

- catalog number
- 1982.0795.05

- accession number
- 1982.0795

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

## Twin Elliptical Gears, Kinematic Model by Martin Schilling, series 24, model 8, number 347

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the eighth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- Many machines need to produce a back and forth motion, such as the back and forth motion of the rods of a locomotive that drives the wheels. This back and forth motion is achieved by converting circular motion (produced by the pistons of the steam engine) to linear motion (of the rods). One way of achieving this in a smooth way is through a
*quick return*mechanism. This model uses two ellipses that are held in constant contact, producing an “elliptical gear.”

- As one ellipse rotates around the other, the distance between the fixed focus of one ellipse and the free focus of the other remains constant. This can be seen in the model by the placement of the arm. As the ellipses rotate about each other, the speed of rotation increase as the ellipses move towards a side-by-side orientation, and slows as the ellipses move towards an end-to-end alignment. Thus the velocity increases and decreases periodically as the ellipses rotate. The velocity ratio of the rotating gear is the portion of the length of the top arm over one ellipse divided by the remaining length (over the other ellipse.) Mathematically this velocity ratio varies from
*e/(1-e)*to*(1-e)/e*where*e*is the eccentricity of the (congruent) ellipses. The cyclic nature of the velocity of this motion is known as a “quick-return” mechanism, which converts rotational motion into reciprocating or oscillating motion.

- This model employs two identical elliptical metal plates (major axis 8 cm, minor axis 5 cm). Both ellipses were fixed to the baseplate at their right foci (though one ellipse is now detached) while the other foci are free. This allows the two ellipses to rotate around each other while remaining in contact. An 8 cm rigid arm connects the fixed foci of one ellipse to the free foci of the other.

- Beneath the free foci of the left ellipse is a metal point. As the (now missing) crank below the baseplate is rotated, the point traces out a circle on the paper covering of the baseplate. Using the thumb hold at the midpoint of the arm, the two ellipses can be made to rotate around each other. A small ball-type joint at the ends of the major axis of each ellipse allows the two ellipses to join together when they are aligned end-to-end. The German title of the model is: Gleichläufiges Zwillingskubelgetriebe mit seinen Polbahnen (same shape transmitted by twin cranks with their poles).

- References:

- Cundy, H. M., Rollett, A. P.,
*Mathematical Models*, Oxford University Press, 1961, pp. 230-233.

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp. 56-57. Series 24, group III, model 8.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.06

- catalog number
- 1982.0795.06

- accession number
- 1982.0795

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

## Twin Hyperbolic Gears, Kinematic Model by Martin Schilling, series 24, model 9, number 348

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the ninth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- This model is an example of a Watt’s linkage. Linkages are joined rods that move freely about pivot points used to produce a certain type of motion. A pair of fireplace pincers is an example of a very simple linkage. Producing straight line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. Scottsman James Watt (1736-1819), devised linkages to create linear motion for use in early steam engines. A Watt’s linkage is a three-bar linkage in which two bars of equal length rotate to produce congruent circles. The ends of these two radii are joined by a longer crossbar. As the radii counter-rotate, the midpoint of the crossbar traces out a Watt’s Curve.

- Watt’s Curve is related to the lemniscate, or a figure-eight-shaped curve. However, Watt’s Curve resembles a figure eight that has been compressed vertically so that the two lobes appear as circles that are flattened where they meet. As the midpoint of the crossbar traces the region of the lemniscate where the curve crosses itself, the motion is approximately linear.

- This model consists of two identical components (“bowties”), each comprised of two rounded hyperbolic metal plates (13 cm base, 5.5 cm altitude) joined by an armature of 9 cm. One bowtie is mounted on top of and offset by 7 cm from the bottom bowtie. An armature attaches the vertex of one plate to the vertex of its corresponding plate below. A crank below the baseplate connects to one arm. When the crank is rotated, the two connecting arms rotate in opposing circular paths, causing the top bowties to follow a roughly figure eight path. As each arm rotates through 180 degrees, the bowties align first to the left, then to the right. The German title is of this model it: Gegenläufiges Zwillingskubelgetriebe mit seinen Polbahnen.

- References:

- Guillet, George,
*Kinematics of Machines*, John Wiley & Sons, N.Y., 1930, pp. 217, 218.

*Watt’s Curve*, Mathworld, http://mathworld.wolfram.com/WattsCurve.html

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group III, model 9.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.07

- catalog number
- 1982.0795.07

- accession number
- 1982.0795

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

## Hart Inversor, Kinematic Model by Martin Schilling, series 24, model 11, number 350

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the eleventh in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- Linkages are joined rods that move freely about pivot points. A pair of fireplace pincers is an example of a very simple linkage. Producing straight-line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. In this context, “inverse” is a geometric term that refers to the process of using algebra and trigonometry to convert or invert one geometric shape into another. In this case, the inverse of the circle will be a straight line. So an “inversor” is a device that finds the inverse of a geometrical object: the conversion of a circle to a straight line in the case of this model.

- The Hart’s Inversor, also known as Hart’s Cell or Hart’s Linkage, is a contraparallelogram of four pin-connected links. It is similar to the Peaucellier Inversor, but is a four-bar linkage as opposed to a seven-bar linkage. It was invented and published by Harry Hart (1848-1920) in 1874.

- This model is made from four metal armatures, two measuring 9.5 cm, two 16.5 cm, in an “hourglass” configuration (the two longer arms crossing to form the waist of the hourglass) with two congruent triangles meeting at a common vertex.

- When the top and bottom arms are parallel to the top and bottom of the baseplate, the triangles are isosceles. The top arm is fixed to the base slightly to the right of its midpoint. Below this fixed point, a fifth arm is attached to a crank below the baseplate and attached to the underside of the upper cross arm slightly above the midpoint. This attachment can be rotated in a circle either by turning the crank or by using the polished fingerhold on the top of the cross arm.

- A pin below the fingerhold (now inserted into a piece of cork to avoid tearing the paper covering of the baseplate) traces part of a circle as seen in the image. This causes a fingerhold and pin (also in a piece of cork) on the second cross arm, slightly below its midpoint, to move laterally right and left across the baseplate in a straight-line motion. As the attachment is rotated, the triangles become progressively more scalene.

- In addition, this linkage has the following linearity property. When the linkage is in its original (isosceles) configuration, mark four points on each of the four arms such that the four points lie on a vertical line. Fix the top point and allow the second point (below the top point) to trace a circle. This causes the third point to trace a straight line, and all four points will remain colinear regardless of the configuration of the linkage.

- The German title of this model is: Inversor von Hart. The name plate on the model gives a date of 1874 for this model, most likely indicating the date of Hart’s discovery.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterrich*t, Halle a.s., Germany, 1911, pp 56-57. Series 24, group IV, model 11.

- Online demonstrations for this model can be found at www.cut-the-knot.org

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.08

- catalog number
- 1982.0795.08

- accession number
- 1982.0795

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

## Sylvester-Kempe Inversor, Kinematic Model by Martin Schilling, series 24, model 12, number 351

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the twelfth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- Linkages are joined rods that move freely about pivot points. A pair of fireplace pincers is an example of a very simple linkage. Producing straight-line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. In this context, “Inverse” is a geometric term that refers to the process of using algebra and trigonometry to convert or invert one geometric shape into another. In this case, the inverse of the circle will be a straight line. So an “inversor” is a device that finds the inverse of a geometrical object: the conversion of a circle to a straight line in the case of this model.

- A generalization of Hart’s Inversor, the Sylvester-Kempe Inversor is also known as a Quadruplane inversor and creates linear motion from circular motion. English mathematicians James Sylvester (1814-1897) and Alfred Kempe (1849-1922) developed the geometric theory behind these linkages in the 1870s. Kempe proved that every algebraic curve can be generated by a linkage using a Watt’s curve, after Scottish engineer James Watt (1736-1819).

- Unlike the other Schilling linkages in the collection, this one is not made of armatures. It consists of linked triangular metal plates (two large and two small). The smallest triangle is attached to the baseplate at a stationary pivot point. The triangles are linked together at the vertices to form a chain of triangles (small-large-small-large). As with the other linkages, this model has an armature that is attached to a small hand crank on the underside of the baseplate and attached to the vertex of one of the larger triangles that allows the linkage to rotate. It can also be moved by using one of two fingerholds attached to the top of two of the triangles at a vertex.

- As the linkage is rotated, a pin where the armature attaches to the large triangle traces out a circle, visible in the image. At the same time, a pin under the fingerhold on the opposite large triangle traces a straight line from left to right across the baseplate, also seen in the image. The German title of this model is: Inversor von Sylvester und Kempe. The nameplate on the model gives a date of 1875 for this model, most likely the date of discovery by Sylvester and Kempe.

- Reference:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp. 56-57. Series 24, group IV, model 12.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.09

- catalog number
- 1982.0795.09

- accession number
- 1982.0795

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

## Model of a Hyperbolic Paraboloid

- Description
- In the late nineteenth century, a few Americans began to make geometric models like those previously imported from Europe. This string model, made by the firm of Eberbach in Ann Arbor, Michigan, is very similar to one made in Germany at about the same time. The model is adjustable. When the metal triangles lie flat, the surface formed by the strings is a rhombus. If the tips of the triangles are raised, the threads form a surface called a hyperbolic paraboloid. The model came to the Smithsonian from the Department of Mathematics at the University of Michigan.

- Location
- Currently not on view

- maker
- Eberbach

- ID Number
- 1982.0795.31

- accession number
- 1982.0795

- catalog number
- 1982.0795.31

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

## Lawrence Engineering Service 10-B Mannheim Simplex Slide Rule

- Description
- This inexpensive 9-1/2 inch one-sided wooden slide rule is painted white on the front face. A, D, and K scales are on the base, and B, CI, and C scales are on one side of the slide. A plastic indicator is in a metal frame. The top of the base is marked: MADE IN U.S.A. It is also marked LAWRENCE ENGINEERING SERVICE, PERU, INDIANA and PAT. PEND. The right end of the slide is marked: 10-B. Tables for equivalents and conversions appear on the back of the instrument, which is in a cardboard box covered with black synthetic leather.

- George Lee Lawrence (1901–1976) established a firm in Chicago to make slide rules for photography. In 1935 he moved to Wabash, Ind., renamed the company Lawrence Engineering Service, and began to manufacture general purpose slide rules. In 1938 he relocated once more to Peru, Ind., probably to enlarge the factory. Lawrence's second wife, Vivian Breyer, received the company in their 1947 divorce. Its name was changed to Engineering Instruments, Inc., and the company remained in business until its building burned down in 1967. Thus, this rule dates between 1938 and 1947. The model 10-B sold for 25 cents during this period. There is no record that Lawrence ever received a patent for any aspect of his design or manufacturing process. According to the donor, this rule belonged to her father, George L. Sterns. Compare to 1980.0097.02.

- References: Bruce Babcock, "Lawrence Engineering Service — A Tale from an American Small Town,"
*Journal of the Oughtred Society*5, no. 2 (1996): 55–61; David G. Rance, "The Unique Lawrence,"*Proceedings of the 17th International Meeting of Slide Rule Collectors*(September 2011), 87–107, http://www.sliderules.nl/index.php?p=papers; Peter M. Hopp,*Slide Rules: Their History, Models, and Makers*(Mendham, N.J.: Astragal Press, 1999), 195–196; accession file.

- Location
- Currently not on view

- date made
- 1938-1947

- maker
- Lawrence Engineering Service

- ID Number
- 1983.0042.01

- accession number
- 1983.0042

- catalog number
- 1983.0042.01

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

## Standardized Test, Schorling-Clark-Potter Arithmetic Test

- Description
- By the 1920s, mathematics educators increasingly turned to standardized tests as a way to measure what students knew, to predict what they could learn, and to determine where they had difficulties. This test had sections on addition, subtraction, multiplication, and division. The final two sections were on fractions, decimals, and percentages; and on a combination of problems. The authors were Raleigh Schorling (1887–1950), John R. Clark (1887–1986), and Mary A. Potter (1889–1993?). World Book Company published the four page leaflet in 1928. Versions of the test would be published for decades.

- By 1926, when the test was first published, Schorling and Clark had obtained their PhDs from Teacher’s College of Columbia University. After earning his doctorate, Clark headed the mathematics department of the Chicago State Teacher’s College, and then in 1920 returned to teach in the Department of Mathematics Education at Teacher’s College. He remained there until his retirement in 1952.

- Schorling taught at the Lincoln School of Teacher’s College. He left in 1923 to become the first principal of the University High School at the University of Michigan, and completed his Teacher’s College doctorate in 1924. He remained at Ann Arbor for the rest of his career, serving as well as a professor of education at the university. Mary Potter obtained her undergraduate degree from Lawrence College in Appleton, Wisconsin, in 1913. She taught in several Wisconsin school districts, settling in Racine by 1920 and living there the rest of her working life.

- At the Lincoln School, Schorling and Clark worked to reform arithmetic education by emphasizing the affairs of daily life. Their efforts led them to author new textbooks as well as new tests. Schorling, Clark, and Potter were all active in the establishment of the National Council of Teachers of Mathematics in 1920.

- Location
- Currently not on view

- date made
- 1928

- maker
- Schorling, Raleigh

- Clark, John R.

- Potter, Mary A.

- ID Number
- 1983.0168.01

- catalog number
- 1983.0168.01

- accession number
- 1983.0168

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

## Prototype for Willis Two-Wheeled Planimeter

- Description
- Even after the apparent commercial success of his "improved planimeter," Edward Jones Willis (1866–1941), a steam and electrical engineer from Richmond, Va., continued to experiment with planimeter designs. On January 17, 1922, he received a patent for a planimeter that had the measuring wheel on a spindle instead of a wheeled carriage, and had a magnifying glass attachment.

- Willis noted that date on this instrument, but it bears little resemblance to the patent drawings. A brass arm with metal tracers at both ends moves perpendicularly to a brass frame. The arm is evenly divided in increments marked 99, 88, 77, 66, 55, 44, and 33. A metal bar at the front of the frame has a brass slide. Next are two brass wheels on spindles fastened to a brass plate in the center. At the back is a wooden triangular ruler with six scales on white celluloid. These scales divide the inch into 100, 50, 60, 30, 80, and 40 parts. The ruler is marked: J. L. ROBERTSON & SONS, N.Y. Presumably, the ruler came from one of the Improved Willis Planimeters made by Robertson between 1896 and the 1910s. The back of the frame is marked: E. J. WILLIS (/) RICHMOND, VA. (/) PAT. APL'D FOR.

- A metal rod has rectangular brass ends. A triangular metal plate has brass bolts holding prickers and a brass post that holds one end of the rod. A small brass clamp is loose in the crudely made wooden case, which appears to be made from a shipping crate. Handwriting on the inside of the lid reads: Edward J. Willis (/) Room 119 Mutual Bldng (/) P.O. Box 416 Richmond V
__a__(/) Jany 17^{h}1922 (/) WILLIS TWO WHEEL PLANIMETER. There is no record of a patent that applies specifically to this apparent prototype.

- For information on Willis's earlier patents and planimeters, see 1994.0356.02, MA*324247, MA*323703, and MA*323704. At the same time that Willis worked on his later planimeters, he became interested in celestial navigation and published two textbooks on the subject. He invented a navigating machine and an altitude-azimuth instrument in the 1930s.

- References: Edward J. Willis, "Planimeter" (U.S. Patent 1,404,180 issued January 17, 1922); Hyman A. Schwartz, "The Willis Planimeter,"
*Rittenhouse*7, no. 2 (1993): 60–64; Willem F.J. Mörzer Bruyns, "The Willis Navigating Machine: A Forgotten Invention,"*Rittenhouse*14 (June 2000): 13–25.

- Location
- Currently not on view

- date made
- 1922-1941

- maker
- Willis, Edward Jones

- ID Number
- 1983.0173.01

- catalog number
- 1983.0173.01

- accession number
- 1983.0173

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

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