#
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 2915 items.

Page 120 of 292

## 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

## Dip Circle

- Description
- The Department of Terrestrial Magnetism of the Carnegie Institution of Washington purchased this Lloyd-Creak circle in 1904. It is marked "Dover, Charlton Kent. Circle 169." With gimbal stand it cost $527. The vertical circle is silvered, graduated to 30 minutes, and read by opposite verniers to single minutes. The horizontal circle is silvered, graduated to 30 minutes, and read by vernier to single minutes. An auxiliary needle on top is used to determine magnetic declination.
Ref: Carnegie Institution of Washington. Department of Terrestrial Magnetism,

*Land Magnetic Observations, 1905-1910*(Washington, D.C., 1912), p. 47.

- Location
- Currently not on view

- Date made
- ca 1904

- maker
- Dover

- ID Number
- 1983.0039.01

- accession number
- 1983.0039

- catalog number
- 1983.0039.01

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

## Dip Circle

- Description
- The Department of Terrestrial Magnetism of the Carnegie Institution of Washington bought this Kew pattern dip circle in 1919. It is marked "Dover, Charlton Kent, Circle 240." With four needles, tripod, case, Kew certificate of examination, and importation charges, it cost $184.70. The vertical circle is silvered, graduated to 30 minutes, and read by opposite verniers to single minutes. The horizontal circle is silvered, graduated to 30 minutes, and read by vernier to single minutes.

- Location
- Currently not on view

- Date made
- ca 1919

- maker
- Dover

- ID Number
- 1983.0039.02

- accession number
- 1983.0039

- catalog number
- 1983.0039.02

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

## Dip Circle

- Description
- This Kew pattern dip circle dates from the early decades of the twentieth century. It is marked "Dover, Charlton Kent, Circle 158." A paper note in the wooden carrying case states that the U.S. Navy lent it for observations during the second International Polar Year which ran from September 1932 to September 1933. The loan may have been to the Carnegie Institution of Washington which, fifty years later, donated it to the Smithsonian.

- Location
- Currently not on view

- maker
- Dover

- ID Number
- 1983.0039.03

- accession number
- 1983.0039

- catalog number
- 1983.0039.03

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

## Magnetometer

- Description
- This theodolite magnetometer is based on the design that the U.S. Coast and Geodetic Survey developed in 1892-1893. It is similar in many ways to the instrument that the Survey had been using since the early 1880s, but with several new features. One is the octagonal shape of the collimating magnets. Another is the black velvet screen that connects the telescope with the suspension box: this cuts off stray light and eliminates the problems that had been caused by the glass window in the earlier form. It is marked "FAUTH & CO. WASH
^{N}D.C. 941" and "T.M.C.I. 1." The serial number suggests that it was made around 1895.This instrument belonged to the Department of Terrestrial Magnetism of the Carnegie Institution of Washington. Internal records indicate that D.T.M. purchased it from Kolesch & Co. in New York in 1906 (for $175), sent it to Bausch, Lomb, Saegmuller Co. for repairs (another $120), and kept it in service until 1919.

Ref: Edwin Smith, "Notes on Some Instruments Recently Made in the Instrument Division of the Coast and Geodetic Survey Office,"

*Annual*

*Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1894*, Appendix No. 8.

- Location
- Currently not on view

- Date made
- ca 1895

- maker
- Fauth

- Fauth & Co.

- ID Number
- 1983.0039.04

- accession number
- 1983.0039

- catalog number
- 1983.0039.04

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

## Variometer

- Description
- This instrument belonged to the Carnegie Institution of Washington, and was probably made in the early 1900s by Max Thomas Edelmann of Munich.
Ref: M. Th. Edelmann, "On the Construction of Earth-Magnetic Instruments,"

*Weather Bureau Bulletin*11 (1893): 511-539.

- Location
- Currently not on view

- maker
- Edelmann, Max Thomas

- ID Number
- 1983.0039.05

- accession number
- 1983.0039

- catalog number
- 1983.0039.05

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