Olivier geometric string figure of intersecting cones
Date: circa 1856
Inventory Number: 5470
Classification: Demonstration Model
Subject:
Maker: Fabre et Cie
Inventor: Théodore Olivier (1793 - 1853)
Cultural Region:
Place of Origin:
City of Use:
Dimensions:
69 × 42.5 × 23.7 cm (27 3/16 × 16 3/4 × 9 5/16 in.)
DescriptionThe model sits on a rectangular mahogany box with two differently sized circular holes in the top panel. 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 of the long edges 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 four outer walls of the box can be removed revealing four unfinished table legs.
There is a teardrop-shaped brass plate screwed to the top panel of the box such that the pointed end is on the right hand side and the rounded end is on the left hand side. The brass teardrop has two holes it such that it frames the holes in the wood. The larger hole is through the round end of the teardrop and the small hole is through the smaller end on the right.
There is a small rectangular hole at the pointed rightmost end of the teardrop plate. There is a round brass gear with four notches fixed in the hole such that only the top half is above the top panel of the box. There is a long brass bar whose rightmost end is attached to the center of the gear by a pivot. The bar has a rectangular hole running almost its entire length that acts as a track for three sliding components. The bar has a latch such that it can be fixed in any of the four notches on the gear. The latch -- one of the three sliding components -- is attached to a handle. The handle can be loosened to move it away from the gear, permitting the brass bar to pivot freely, and it can be tightened again once the latch is slid into the desired notch. The minimum angle between the front edge of the box and the brass bar is achieved when the bar is in the leftmost position, and the maximum angle is achieved when the bar is in the rightmost position.
The other two sliding components are further up the bar to the left. Each component consists of two connected rectangles, one sitting on either side of the long brass bar holding it in place in the track. Each is equipped with a handle on the top of the bar that can be loosened, permitting the rectangles to slide side to side, and tightened to hold it in place.
There are two small holes at the leftmost end of each slider. There is a set of long hemp strings tied together attached to each of the left hand and right hand sliders. The strings are folded in half such that each end of each string goes through one of two neighboring holes in the brass frame of each figure-eight hole. There are small equally spaced holes around the circumference of each of the figure-eight holes respectively. Each end of each string descends from the slider and passes through one of two neighboring holes in the figure-eight circles beneath. The strings descending from the slider on the right move to the left in order to pass through the holes surrounding the smaller hole on the left. The strings descending from the slider on the left move to the right in order to pass through the holes surrounding the smaller hole on the right. There is a grey, elongated egg-shaped weight tied to the bottom of each string end pulling the strings taut. As such, each string acts as a straight line on the surface of the modeled object. The weights are hidden by the mahogany box but can be seen when the outer walls are removed. Because the strings start from a single point on the left of the slider and descend into a larger circle, the shapes created are cones. Crucially, because the strings pass from the right hand slider to the left hand circle and vice versa, the cones will intersect. The shape of the intersection is a function of the angle of the main brass bar and the positions of the brass sliders.
There is a teardrop-shaped brass plate screwed to the top panel of the box such that the pointed end is on the right hand side and the rounded end is on the left hand side. The brass teardrop has two holes it such that it frames the holes in the wood. The larger hole is through the round end of the teardrop and the small hole is through the smaller end on the right.
There is a small rectangular hole at the pointed rightmost end of the teardrop plate. There is a round brass gear with four notches fixed in the hole such that only the top half is above the top panel of the box. There is a long brass bar whose rightmost end is attached to the center of the gear by a pivot. The bar has a rectangular hole running almost its entire length that acts as a track for three sliding components. The bar has a latch such that it can be fixed in any of the four notches on the gear. The latch -- one of the three sliding components -- is attached to a handle. The handle can be loosened to move it away from the gear, permitting the brass bar to pivot freely, and it can be tightened again once the latch is slid into the desired notch. The minimum angle between the front edge of the box and the brass bar is achieved when the bar is in the leftmost position, and the maximum angle is achieved when the bar is in the rightmost position.
The other two sliding components are further up the bar to the left. Each component consists of two connected rectangles, one sitting on either side of the long brass bar holding it in place in the track. Each is equipped with a handle on the top of the bar that can be loosened, permitting the rectangles to slide side to side, and tightened to hold it in place.
There are two small holes at the leftmost end of each slider. There is a set of long hemp strings tied together attached to each of the left hand and right hand sliders. The strings are folded in half such that each end of each string goes through one of two neighboring holes in the brass frame of each figure-eight hole. There are small equally spaced holes around the circumference of each of the figure-eight holes respectively. Each end of each string descends from the slider and passes through one of two neighboring holes in the figure-eight circles beneath. The strings descending from the slider on the right move to the left in order to pass through the holes surrounding the smaller hole on the left. The strings descending from the slider on the left move to the right in order to pass through the holes surrounding the smaller hole on the right. There is a grey, elongated egg-shaped weight tied to the bottom of each string end pulling the strings taut. As such, each string acts as a straight line on the surface of the modeled object. The weights are hidden by the mahogany box but can be seen when the outer walls are removed. Because the strings start from a single point on the left of the slider and descend into a larger circle, the shapes created are cones. Crucially, because the strings pass from the right hand slider to the left hand circle and vice versa, the cones will intersect. The shape of the intersection is a function of the angle of the main brass bar and the positions of the brass sliders.
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 taught 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 cones. Each cone is attached to a brass bar, one end of which is attached to a gear on the top panel of the mahogany box. The user can manually alter the angle of the brass bar with the topmost panel of the box by loosening a handle near the gear and sliding the bar into various positions on the gear. Each cone is attached to the bar at the top by a handle. The user can also manually alter the position of each cone's topmost point on the bar by loosening the handle and sliding it left or right along the bar. The cone on the left is larger and the cone on the right smaller; let them be called L and S respectively. Let the axis of L be a line that passes from the top of the cone that intersects the bar and passes through the center of the leftmost circular opening on the box. Let the axis of S be similarly defined for the cone on the right. Both possible manual adjustments alter the angle between the axis of L and the axis of S. The shape and configuration of the intersection between the two cones changes accordingly.
[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 taught 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 cones. Each cone is attached to a brass bar, one end of which is attached to a gear on the top panel of the mahogany box. The user can manually alter the angle of the brass bar with the topmost panel of the box by loosening a handle near the gear and sliding the bar into various positions on the gear. Each cone is attached to the bar at the top by a handle. The user can also manually alter the position of each cone's topmost point on the bar by loosening the handle and sliding it left or right along the bar. The cone on the left is larger and the cone on the right smaller; let them be called L and S respectively. Let the axis of L be a line that passes from the top of the cone that intersects the bar and passes through the center of the leftmost circular opening on the box. Let the axis of S be similarly defined for the cone on the right. Both possible manual adjustments alter the angle between the axis of L and the axis of S. The shape and configuration of the intersection between the two cones changes accordingly.
[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 Theodore Olivier, graduate of L'Ecole Polytechnique and for time, a tutor there. He was appointed Professor of Descriptive Geometry at the École Centrale des Arts et Manufacture in 1839. He wrote several books, including "Development de Geometrie descriptive" (Paris: 1843) and "Cours de geometrie Descriptive" (Paris, 1845), that emphasized the importance of space intuition and visualization.
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 Remarksletter from William C. Stone at the Department of Mathematics at Union College to Mr. Ebenezer Gay, Assistant Curator at the Science Center, Harvard University, concerning aspects of the model's design, dated April 20, 1983; letter from William C. Stone at the Department of Mathematics at Union College to Mr. Ebenezer Gay, Assistant Curator at the Science Center, Harvard University, concerning aspects of the model's design dated May 30, 1983, in object file; pencil-drawn diagrams, measurements, and notes representing various components of the model's design from the letters and photographs of W.C. Stone, in object file; detailed three-page invoice and restoration report from Richard L. Ketchen (horologist) dated July 25, 1995, in object file; two original photos, and photocopy entitled "Detail for reconstruction of String Figure", in object file; seven photographs that accompanied Richard L. Ketchen's restoration report
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. 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.
ProvenanceTaken from the attic of Pierce Hall, Department of Engineering, Harvard University, in May 1961.
