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Date: circa 1856

Inventory Number: 5468

Classification: Demonstration Model

Subject:

Maker: Fabre et Cie

Inventor: Théodore Olivier (1793 - 1853)

Cultural Region:

Place of Origin:

City of Use:

Dimensions:

55 x 27 x 27 cm (21 5/8 x 10 5/8 x 10 5/8 in.)

DescriptionThe model sits in a square mahogany box. Each of the four bottom corners is raised on a small round stand. There is a base board trim around the bottom of the box and the top panel extends slightly forming a lip around the top of the box. There is a brass plaque in the center of one edge on the top panel. Henceforth the edge on which the plaque is centered will be used as the 'front' of the box, e.g. the 'right side' of the box is on the right of someone facing center.

The model consists of two differently sized, intersecting cylinders. Each cylinder is 'on its side', meaning that the planes circumscribed by the circles at each end of each cylinder are perpendicular to the top panel of the box. Each cylinder is fixed in a brass frame. Each frame consists, in part, of two parallel thin, horizontal brass bars that run from end to end of the cylinder. The parallel bars are attached by a small brass bar at each end. From each such connecting bar, a small, rectangular sheet of brass with a width the length of the bar rises and attaches to the brass circumference of the circle at that end of the cylinder.

In each frame, a thin brass strip whose flat side is perpendicular to the plane of the brass sheet also rises from the small brass connecting bar. However, instead of ceasing at the edge of the circle, this sheet extends to the top of the circle; the topmost subsection of this strip of brass is the fixed vertical diameter of the circle. There is a similar strip of brass fixed across the horizontal diameter of the each circle.

The two sets of horizontal, parallel bars -- one to support each cylinder -- are each attached to a solid, cylindrical pivot fixed at the center of the top panel of the mahogany box. The parallel bars supporting the larger cylinder are closest to the bottom of the box and the parallel bars supporting the smaller cylinder are at the top of the pivot. The top edge of the smaller cylinder is flush with the top edge of the larger cylinder. As such it is held higher off of the box than the larger one. This is in part the case because the rectangular sheet of metal that rises at each end of the frame to support the smaller circles is longer (approximately double) than the height of those supporting the larger circles.

The circumference of each large circle has 120 equidistant holes around it. There are 60 long, golden strings, folded in half such that each end goes through one of two neighboring holes. Each end goes through two neighboring holes on the circumference of the circle at the other end. The strings are pulled taut between the two circles so that each string represents a ruled edge of the cylinder.

The circumference of each small circle is similarly divided with equally spaced holes and its strings are folded and pulled taut between them.

Crucially, the two sets of taut strings (one set for each cylinder) are intersecting: the strings of the smaller cylinder pass through the strings of the larger cylinder. At each point of intersection -- when one of the strings from the smaller cylinder crosses a string from the larger cylinder -- a small black ring is looped around both strings. That is to say that each point at which a ruled edge of each cylinder crosses the point is marked. The result is that the black markers provide a visual pattern of the intersection of the two spheres. The two cylinders can rotate relative to one another, such that the angle between their horizontal axes (i.e. the axes running through the center of the circular ends) changes. As they rotate, the shape of their intersecting ruled edges changes.

The model consists of two differently sized, intersecting cylinders. Each cylinder is 'on its side', meaning that the planes circumscribed by the circles at each end of each cylinder are perpendicular to the top panel of the box. Each cylinder is fixed in a brass frame. Each frame consists, in part, of two parallel thin, horizontal brass bars that run from end to end of the cylinder. The parallel bars are attached by a small brass bar at each end. From each such connecting bar, a small, rectangular sheet of brass with a width the length of the bar rises and attaches to the brass circumference of the circle at that end of the cylinder.

In each frame, a thin brass strip whose flat side is perpendicular to the plane of the brass sheet also rises from the small brass connecting bar. However, instead of ceasing at the edge of the circle, this sheet extends to the top of the circle; the topmost subsection of this strip of brass is the fixed vertical diameter of the circle. There is a similar strip of brass fixed across the horizontal diameter of the each circle.

The two sets of horizontal, parallel bars -- one to support each cylinder -- are each attached to a solid, cylindrical pivot fixed at the center of the top panel of the mahogany box. The parallel bars supporting the larger cylinder are closest to the bottom of the box and the parallel bars supporting the smaller cylinder are at the top of the pivot. The top edge of the smaller cylinder is flush with the top edge of the larger cylinder. As such it is held higher off of the box than the larger one. This is in part the case because the rectangular sheet of metal that rises at each end of the frame to support the smaller circles is longer (approximately double) than the height of those supporting the larger circles.

The circumference of each large circle has 120 equidistant holes around it. There are 60 long, golden strings, folded in half such that each end goes through one of two neighboring holes. Each end goes through two neighboring holes on the circumference of the circle at the other end. The strings are pulled taut between the two circles so that each string represents a ruled edge of the cylinder.

The circumference of each small circle is similarly divided with equally spaced holes and its strings are folded and pulled taut between them.

Crucially, the two sets of taut strings (one set for each cylinder) are intersecting: the strings of the smaller cylinder pass through the strings of the larger cylinder. At each point of intersection -- when one of the strings from the smaller cylinder crosses a string from the larger cylinder -- a small black ring is looped around both strings. That is to say that each point at which a ruled edge of each cylinder crosses the point is marked. The result is that the black markers provide a visual pattern of the intersection of the two spheres. The two cylinders can rotate relative to one another, such that the angle between their horizontal axes (i.e. the axes running through the center of the circular ends) changes. As they rotate, the shape of their intersecting ruled edges changes.

In Collection(s)

Signedengraved on a plaque centered on the edge of wooden base: INV(T) TH. OLIVIER, 1830 / FEC(T) FABRE ET C(IE) SUCC(R) DE PIXII / Paris

[all items in parentheses are superscript font in the engraving]

[all items in parentheses are superscript font in the engraving]

FunctionThéodore Olivier designed these string models as pedagogical tools for students and professors of Descriptive Geometry. Descriptive Geometry is the branch of mathematics devoted to the study of three dimensional objects in terms of two dimensional projections or representations of those objects. Of particular interest are the two-dimensional cross-sections of geometric objects, the surfaces of intersection-objects between two three-dimensional objects, the surfaces of three-dimensional objects, and the two-dimensional projections, or shadows, of three dimensional objects. In her article, "The Physicalist Tradition in Early Nineteenth Century French Geometry", Lorraine Daston indicates that Descriptive Geometry was deeply connected to the application-based pedagogical tradition of the newly founded (in 1784) École Polytechnique[1]. As such, physicalist, mechanics-based transformations like 'projection', 'rotation', and 'cross-section' were given geometrical interpretations and the resulting objects studied. In spite of being a rigorous and abstract field of mathematics, Descriptive Geometry was also extremely useful for stone-cutters, engineers, artillery specialists, military personnel, and other technically oriented students of the École.

Descriptive Geometry was developed by Gaspard Monge, a professor and the curriculum director at the École Polytechnique in the late 1700s and early 1800s. He proposed Descriptive (or Synthetic) Geometry as an alternative to the algebraic mathematical paradigm of the 18th century. Monge was in part responding to a general mathematical concern over the lack of rigorous grounding for the calculus. Most attempts to make the calculus rigorous were algebraic. However, there was also a growing concern that algebra was too un-tethered from experience and physical reality and that. According to Daston, part of the anxiety surrounding algebra was a result of the rise in popularity of Lockean empiricism which holds that "all valid knowledge is necessarily rooted in experience".[2] She elaborates as follows:

"The guarantee of a mathematical argument was for them ultimately a psychological one, which depended on the clarity and vivacity of the mental images before the mind's eye of the mathematician. Although this view has an (early) Cartesian ring to it, the emphasis upon "imagination or vision" betrays the influence of Locke in linking the essential clarity and distinctness to perception"[3].

Theodore Olivier studied Descriptive Geometry at the École Polytechnique and went on to be a professor himself at the École Centrale des Arts et Manufacture and a tutor at the École Polytechnique. In the spirit of the École Polytechnique and Monge's mathematical empiricism, Olivier was firmly convinced that in order for students to appreciate the complexities of geometric surfaces, they must be able to 'see' them:

"C'est ainsi que l'on commence à comprendre, que, lorsque l'on veut parler aux élèves des propriétés d'une surface, la première chose à faire est de mettre sous leurs yeux le relief de cette surface, pour qu'ils voient distinctement ce dont on veut leur parler"[4].

As such, he developed his models to enable students of Descriptive Geometry to see the surfaces of geometric objects and to observe the changing patterns of their intersections and certain features of their surfaces.

According to William C. Stone, a mathematics professor at Union College that has repaired and collected many Olivier models, there are 'about' forty-five models in the original set [5]. All but two models in the collection represent the three-dimensional geometric objects by virtue of ruled surfaces. A ruled surface is a surface that can be swept out by the motion of a straight line. The models represent such surfaces by virtue of being constituted by strings pulled taunt between two points to represent straight lines. Each string is like a moment in the line's sweeping path of the object in question. Olivier's string models are like a series of snap shots in the construction of a surface by the motion of a straight line.

According to the "Thirty-Second Annual report of the President of Harvard College" in the academic year of 1856 - 1857, Harvard acquired a complete set of Olivier models in that year for use in mathematics classes. The Report stated the following:

"A complete set of the celebrated models of Olivier has been procured for the use of the mathematical classes. These models, of which there are above fifty, are curiously contrived, by means of moveable parts connected with silken cords, so as to illustrate all the higher researches in Descriptive Geometry, presenting to the eye of the pupil obvious solutions of problems upon complicated surfaces, which might even exhaust the powers of the higher Analytic Geometry. The cost of the collection was about a thousand dollars, the whole amount having been contributed by a few friends of the College, at the instance of Professor Peirce"[6].

Only four of Harvard's Olivier models survive in the Collection of Scientific Instruments.

This model represents the variable intersections between two cylinders. The model consists of two cylinders, one larger (hereafter called cylinder L) and one smaller (hereafter called cylinder S). Let the horizontal axis of L be a line between the center of the two large brass discs and the horizontal axis of S be a line between the centers of the two smaller brass discs. The cylinders themselves are fixed in this model: all strings are parallel to the appropriate horizontal axis, which itself is also fixed. The user can manually change the angle between the horizontal axis of L and the horizontal axis of S by rotating both or either cylinder. The configuration of the surface marked out by the black loops at each point where a string from L and a string from S intersect shifts as the cylinders rotate. The model provides a useful visualization for students required to calculate the volume or the surface area of the intersection of two cylinders.

[1] Daston, Lorraine. "The Physicalist Tradition in Early Nineteenth Century French Geometry", in STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE, 17 (1986), pp. 269 - 295.

[2] Daston, p. 271.

[3] Daston, p. 274.

[4] Olivier, Theodore. COURS DE GEOMETRIE DESCRIPTIVE; PREMIERE PARTIE: DU POINT, DE LA DROITE ET DU PLAN. Paris: Carilian-Goeury et Vor Dalmost, Libraire des Corps des Ponts et Chaussées et des Mines, Quai des Augustins, no. 49, 1852, p. x

[5] Stone, William C. "Old Science, new art" in UNION COLLEGE, Volume 75, Number 3, (January-February 1983), pp. 10 - 11.

[6] Walker, James. "Thirty-Second Annual Report of the President of Harvard College to the Overseers, exhibiting the state of the institution for the academic year 1856 - 57", Cambridge: Metcalf and Company, Printers to the University, 1858.

Descriptive Geometry was developed by Gaspard Monge, a professor and the curriculum director at the École Polytechnique in the late 1700s and early 1800s. He proposed Descriptive (or Synthetic) Geometry as an alternative to the algebraic mathematical paradigm of the 18th century. Monge was in part responding to a general mathematical concern over the lack of rigorous grounding for the calculus. Most attempts to make the calculus rigorous were algebraic. However, there was also a growing concern that algebra was too un-tethered from experience and physical reality and that. According to Daston, part of the anxiety surrounding algebra was a result of the rise in popularity of Lockean empiricism which holds that "all valid knowledge is necessarily rooted in experience".[2] She elaborates as follows:

"The guarantee of a mathematical argument was for them ultimately a psychological one, which depended on the clarity and vivacity of the mental images before the mind's eye of the mathematician. Although this view has an (early) Cartesian ring to it, the emphasis upon "imagination or vision" betrays the influence of Locke in linking the essential clarity and distinctness to perception"[3].

Theodore Olivier studied Descriptive Geometry at the École Polytechnique and went on to be a professor himself at the École Centrale des Arts et Manufacture and a tutor at the École Polytechnique. In the spirit of the École Polytechnique and Monge's mathematical empiricism, Olivier was firmly convinced that in order for students to appreciate the complexities of geometric surfaces, they must be able to 'see' them:

"C'est ainsi que l'on commence à comprendre, que, lorsque l'on veut parler aux élèves des propriétés d'une surface, la première chose à faire est de mettre sous leurs yeux le relief de cette surface, pour qu'ils voient distinctement ce dont on veut leur parler"[4].

As such, he developed his models to enable students of Descriptive Geometry to see the surfaces of geometric objects and to observe the changing patterns of their intersections and certain features of their surfaces.

According to William C. Stone, a mathematics professor at Union College that has repaired and collected many Olivier models, there are 'about' forty-five models in the original set [5]. All but two models in the collection represent the three-dimensional geometric objects by virtue of ruled surfaces. A ruled surface is a surface that can be swept out by the motion of a straight line. The models represent such surfaces by virtue of being constituted by strings pulled taunt between two points to represent straight lines. Each string is like a moment in the line's sweeping path of the object in question. Olivier's string models are like a series of snap shots in the construction of a surface by the motion of a straight line.

According to the "Thirty-Second Annual report of the President of Harvard College" in the academic year of 1856 - 1857, Harvard acquired a complete set of Olivier models in that year for use in mathematics classes. The Report stated the following:

"A complete set of the celebrated models of Olivier has been procured for the use of the mathematical classes. These models, of which there are above fifty, are curiously contrived, by means of moveable parts connected with silken cords, so as to illustrate all the higher researches in Descriptive Geometry, presenting to the eye of the pupil obvious solutions of problems upon complicated surfaces, which might even exhaust the powers of the higher Analytic Geometry. The cost of the collection was about a thousand dollars, the whole amount having been contributed by a few friends of the College, at the instance of Professor Peirce"[6].

Only four of Harvard's Olivier models survive in the Collection of Scientific Instruments.

This model represents the variable intersections between two cylinders. The model consists of two cylinders, one larger (hereafter called cylinder L) and one smaller (hereafter called cylinder S). Let the horizontal axis of L be a line between the center of the two large brass discs and the horizontal axis of S be a line between the centers of the two smaller brass discs. The cylinders themselves are fixed in this model: all strings are parallel to the appropriate horizontal axis, which itself is also fixed. The user can manually change the angle between the horizontal axis of L and the horizontal axis of S by rotating both or either cylinder. The configuration of the surface marked out by the black loops at each point where a string from L and a string from S intersect shifts as the cylinders rotate. The model provides a useful visualization for students required to calculate the volume or the surface area of the intersection of two cylinders.

[1] Daston, Lorraine. "The Physicalist Tradition in Early Nineteenth Century French Geometry", in STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE, 17 (1986), pp. 269 - 295.

[2] Daston, p. 271.

[3] Daston, p. 274.

[4] Olivier, Theodore. COURS DE GEOMETRIE DESCRIPTIVE; PREMIERE PARTIE: DU POINT, DE LA DROITE ET DU PLAN. Paris: Carilian-Goeury et Vor Dalmost, Libraire des Corps des Ponts et Chaussées et des Mines, Quai des Augustins, no. 49, 1852, p. x

[5] Stone, William C. "Old Science, new art" in UNION COLLEGE, Volume 75, Number 3, (January-February 1983), pp. 10 - 11.

[6] Walker, James. "Thirty-Second Annual Report of the President of Harvard College to the Overseers, exhibiting the state of the institution for the academic year 1856 - 57", Cambridge: Metcalf and Company, Printers to the University, 1858.

Historical AttributesDesigned by Théodore Olivier, professor of Descriptive Geometry at the École Centrale des Arts et Manufacture.

According to the 32nd Annual Report of the President of Harvard College, in 1856 - 1857, this model is one of "above fifty" purchased for the pedagogical use of the Mathematics Department at Harvard University. The collection cost about $1,000.

According to the 32nd Annual Report of the President of Harvard College, in 1856 - 1857, this model is one of "above fifty" purchased for the pedagogical use of the Mathematics Department at Harvard University. The collection cost about $1,000.

Curatorial RemarksCleaned and repaired 1984 by E. Gay.

Curatorial analysis and history of damages and repairs to the Intersecting Cylinders Model in the hand of Ebenezer Gay (assistant curator at the Harvard University Collection of Scientific Instruments), in object file

Curatorial analysis and history of damages and repairs to the Intersecting Cylinders Model in the hand of Ebenezer Gay (assistant curator at the Harvard University Collection of Scientific Instruments), in object file

Primary SourcesComberousse, Ch. de. *Histoire de L'Ecole Centrale des Arts et Manufactures, Depuis sa Fondation jusqu'a ce Jour*. Paris: Gauthier-Villars, Imprimeur-Libraire du Bureau des Longitudes, de l'Ecole Polytechnique, Quai des Augustins, no. 55, 1879.

Olivier, Theodore.*Cours de Géométrie Descriptive; Première Partie: Du Point, de la Droite, et du Plan.* Paris: Carilian-Goeury et Vor Dalmost, Libraire des Corps des Ponts et Chaussées et des Mines, Quai des Augustins, no. 49, 1852.

Walker, James. "Thirty-Second Annual Report of the President of Harvard College to the Overseers, exhibiting the state of the institution for the academic year 1856 - 57", Cambridge: Metcalf and Company, Printers to the University, 1858.

Olivier, Theodore.

Walker, James. "Thirty-Second Annual Report of the President of Harvard College to the Overseers, exhibiting the state of the institution for the academic year 1856 - 57", Cambridge: Metcalf and Company, Printers to the University, 1858.

ProvenanceTaken from the attic of Pierce Hall, Department of Engineering, Harvard University, May 1961.

Related WorksBelhoste, Bruno. "The Ecole Polytechnique and Mathematics in Nineteenth Century France" in *Changing Images in Mathematics: From the French Revolution to the New Millenium*, ed. Umberto Bottazini, Amy Dahan Dalmedico (2001).

Brenni, Paolo. "Dumotiez and Pixii: The Transofrmation of French Philosophical Instruments" in*Bulletin of the Scientific Instrument Society*, (2006), pp. 10 - 16.

Daston, Lorraine. "The Physicalist Tradition in Early Nineteenth Century French Geometry", in*Studies in the History and Philosophy of Science* 17 (1986), pp. 269 - 295.

Shell-Gellasch, Amy. "The Olivier String Models at West Point" in*Rittenhouse* Vol. 17 (2003), pp. 71 - 84.

Brenni, Paolo. "Dumotiez and Pixii: The Transofrmation of French Philosophical Instruments" in

Daston, Lorraine. "The Physicalist Tradition in Early Nineteenth Century French Geometry", in

Shell-Gellasch, Amy. "The Olivier String Models at West Point" in

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