#
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 2805 items.

Page 194 of 281

## Burroughs Automatic Total Tax Machine, a Bookkeeping Machine

- Description
- This version of the Burroughs Class 7 bookkeeping machine was designed specifically for calculating taxes and payrolls. It has a gray metal case. At the front are two sets of digit keys, one gray and one white, as well as several function keys on the left. These keys are in two rows. To the left is a key mounted so that the stem moves horizontally.

- Behind and above the digit keys is a space bar and four rows of a QWERTY typewriter keyboard. The “uppercase” symbols are not conventional. Behind the typewriter keyboard is a lever that can be set for any of seven registers. At the center behind the typing keys is a lever that can be set at “INACTIVE” or at any number from 9 to 15. Behind this is a wide carriage and a complex mechanism. The motor is under the machine.

- The machine has no stand.

- Marks on the front and the back read: Burroughs. A red tag attached to the machine reads: PATENT DEPT. (/) #203. A metal tag at the front of the machine has the serial number: A3857. This suggests a 1934 date. A mark on the gray plastic cover reads: Burroughs B. A mark on a white paper tag attached to the machine reads: For information (/) regarding this machine (/) see (/) H. Brown Room 587.

- A card in the accession file indicates that this is a Style 72 07 22 machine.

- References:

- Accession file.

*Price and Style Guide for Class 7 Machines*, Burroughs Corporation Papers, Charles Babbage Institute, University of Minnesota, Minneapolis, Minnesota.

- Location
- Currently not on view

- date made
- ca 1934

- maker
- Burroughs

- ID Number
- 1982.0794.38

- catalog number
- 1982.0794.38

- accession number
- 1982.0794

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

## Burroughs Style 30 17 07 Bookkeeping Machine

- Description
- This full-keyboard, printing bookkeeping machine has a grayish tan metal case and round tan, light tan, and brown keys. It has 17 columns of round plastic keys. The thirteen columns on the right are digit keys, split into five columns and eight columns. The eight columns of digit keys on the right are color-coded in dark tan and light tan. The five columns of digit keys left of these are color-coded in light tan and brown. The machine also has a column of symbol and year keys (with the years “54” and “53”) and three columns of month and date keys. In back of the keyboard is a row of 17 brown column release buttons. Behind the keyboard is a wide carriage and a paper feed. A tan cord is at the back.

- A mark on the front reads: Burroughs. A metal tag screwed to the bottom of the front reads: 98815. A red tag attached to the object reads: PATENT DEPT. (/) #211.

- Compare 1982.0794.34 and 1982.0794.21. The Burroughs Class 30 was introduced 1935. This object was model 211 in the collection of the Patent Division of Burroughs Corporation.

- Location
- Currently not on view

- date made
- ca 1953

- maker
- Burroughs Adding Machine Company

- ID Number
- 1982.0794.40

- catalog number
- 1982.0794.40

- accession number
- 1982.0794

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

## Burroughs Series M Bookkeeping Machine

- Description
- This relatively late Burroughs bookkeeping machine has a gray metal cover. In the front are two sets of digit keys, one white and one black, as well as various functions keys. In front of these are space return, motor tab, and full cents bars, as well an an unlabeled bar. Behind the two rows of digit keys is a QWERTY keyboard with four rows of keys. Symbols indicated on some of these keys are unconventional. A lever at the right sets the machine for different registers. A wide carriage is behind the keyboard. The machine has an electric motor.

- A mark on the front reads: Burroughs. A mark on a red paper tag attached to the machine reads: PATENT DEPT. (/) #212. A metal tag screwed to the front of the machine reads: A3451.

- The Burroughs Series M was introduced in 1951. This example was in the collections of the Patent Division of Burroughs Corporation.

- Reference:

- Burroughs Corporation Papers, Charles Babbage Institute, University of Minnesota, Minneapolis, Minnesota.

- Location
- Currently not on view

- date made
- ca 1951

- maker
- Burroughs Adding Machine Company

- ID Number
- 1982.0794.41

- catalog number
- 1982.0794.41

- accession number
- 1982.0794

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

## Epitrochoid, Kinematic Model by Martin Schilling, series 24, model 1, number 329

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

- Circles rolling around the outside of other circles, known as epicycles to the ancient Greeks, were used to describe the motions of the planets in a geocentric cosmology. These curves,called epitrochoids, are formed by tracing a point on the radius or the extension of the radius of a circle as it rolls around the outside of a second stationary circle.

- Epitrochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and hypotrochoids (see items 1982.0795.02 and 1982.0795.03). In the 18th century, it was found that when shaping the sides of gear teeth as the valley between teeth, using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

- Depending on the distance of the tracing point from the center of the rolling circle, an infinite number of curves can be formed. The three curves depicted in this model are bicyclic, meaning the smaller circle needs to rotate around the larger circle twice before returning to is original configuration. The ratio of the radii of the two circles will determine the number of nodes in the curve and how many rotations are required before the tracing point returns to its starting configuration. The Spirograph toy produces various types of epitrochoids. (See item 2005.0055.02)

- In this model, a toothed metal disc of radius 30 mm links to a smaller toothed metal disc of radius 12mm. Rotating a crank beneath the baseplate rolls the smaller disc around the outer edge of the larger disk. The model illustrates three curves that may be generated by the motion of a point at a fixed distance along the radius of a circle when the circle rolls around the outer edge of a larger circle.

- The green point within the smaller circle (at radius 4mm) produces the green curve on the glass overlay of the model. The blue point on the circumference of the smaller circle (in this special case, the curve is known as an epicycloid) produces the blue curve. The third, represented by a red curve on the glass, is on the extension of a radius of the smaller circle (20mm). As the smaller circle rolls, the point moves inside the larger circle. The German title of this model is: Erzeugung der Epitrochoiden als solche mit freiem Centrum.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 1.

- Chironis, Nicholas,
*Gear Design and Applications*, 1967, p. 160.

- Davis, W.O.,
*Gears for Small Machines*, 1953, p. 9.

- Grant, George,
*Teeth of Gears*, 1891, p. 69.

- Material for educators can be found online at the Durango Bill website.

- Online demonstrations can be found at the Wolfram website Mathworld.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.01

- catalog number
- 1982.0795.01

- accession number
- 1982.0795

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

## Hypotrochoids, Kinematic Model by Martin Schilling, series 24, model 3, number 331

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

- The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Thus hypotrochoids are curves formed by tracing a point on the radius or extension of the radius of a circle rolling around the inside of another stationary circle.

- Hypotochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and epitrochoids (see item 1982.0795.01).

- An infinite number of hypotrochoids can be formed, depending on the distance of the tracing point from the center of the rolling circle. The ratio of the radius of the rolling disc to the radius of the outer ring will determine the number of nodes the hypotochoid will have. In this model, the curves each have five nodes. Hypotochoids, for which the tracing point is on the extension of the radius, form curves that resemble petalled flowers and are called roses. The Spirograph toy produces various types of hypotrochoids. (See item 2005.0055.2) In the 18th century, it was found that shaping the sides of gear teeth and the valley between teeth by using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

- This model consists of a stationary toothed metal ring (with teeth on the inner edge of the ring) of radius 80 mm. A toothed metal disc of radius 32 mm is attached to a brass arm of 7 cm that can be rotated by turning a crank below the baseplate. As the arm is rotated, the disc rolls around the inside of the ring. Three points lie along the radius of the disk and trace corresponding curves, or roulettes, on the glass overlay.

- The blue point on the circumference of the disc traces a blue five-pointed star shape referred to as a hypocycloid. The green point on the radius of the disc traces a green curve inside the ring, and the red point on the extension of the radius of the disc traces a curve that extends past the radius of the ring. The German title of this model is: Erzeugung der Hypotrochoiden als soche mit freiem Centrum.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 3.

- An online demonstration can be found at http://mathworld.wolfram.com/Hypotrochoid.html

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.02

- catalog number
- 1982.0795.02

- accession number
- 1982.0795

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

## Hypotrochoid with overlay, Kinematic Model by Martin Schilling, series 24, model 4, number 332

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

- The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Thus hypotrochoids are curves formed by tracing a point on the radius or extension of the radius of a circle rolling around the inside of another stationary circle.

- Hypotochoids are members of the family of curves called trochoids; curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve, and include the cycloids (see item 1982.0795.05) and epitrochoids (see item 1982.0795.01). An infinite number of hypotrochoids can be formed, depending on the distance of the tracing point from the center of the rolling circle. The ratio of the radius of the rolling disc to the radius of the outer ring will determine the number of nodes the hypotochoid will have.

- In this model, the curves each have five nodes. Hypotochoids, for which the tracing point is on the extension of the radius, form curves that resemble petalled flowers and are called roses. The Spirograph toy produces various types of hypotrochoids. (See item 2005.0055.2) In the 18th century, it was found that shaping the sides of gear teeth, and the valley between teeth, using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

- As with Schilling’s model number 3 of a hypotrochoid, this model has a stationary toothed metal ring of radius 60 mm. A toothed disc of radius 36 mm rolls around the inside of the ring by the use of a crank below the baseplate. However, this model has a semitransparent glass disc of the same radius as the ring attached to the rotating disc.

- Traced on this glass disc is a red epitrochoid that would be formed by an imaginary point on the extension of the radius of a circle rotating on the outside of the disc in the model. A green point on this curve traces a green star-shaped hypotrochoid on the stationary glass overlay of the model as the disc is rotated.

- The hypotrochoid can also be generated by imagining a point on the extension of the radius of the rotating disc. An orange point on the green hypotrochoid aligns with the epitrochoid and shows how the epitrochoid can be generated from a hypotrocoid and vice-versa. The German title of this model is: Erzeugung der Hypotrochoiden als soche mit bedecktem Centrum.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 4.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.03

- catalog number
- 1982.0795.03

- accession number
- 1982.0795

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

## 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

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