Object Name: Olivier geometric string figure of intersecting cones

Dimensions: 68 x 62.5 x 25.3 cm (26 3/4 x 24 5/8 x 9 15/16 in.)

Description: The model sits on a rectangular mahogany box with two equally 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.

The frame of the model consists of three, squared brass bars: two vertical bars rise from either long end of the rectangular top panel and one horizontal bar that connects them at the top. At the top of each vertical brass bar there is an aesthetic embellish. Next to each vertical bar is another, shorter (less than half the height) vertical bar also with an aesthetic embellish at the top. There is a flat, brass, perfectly symmetrical figure-eight fixed in between the two shorter vertical bars.

There are two rectangular, brass sliding sleeves on the long horizontal top bar. Each one has a handle on the top (above the bar) and a disc on the bottom (beneath the bar). The discs are the same size and they are smaller than the circular holes carved out by the figure-eight brass strip below. The circles are positioned such that the plane circumscribed by the circumference is parallel to the top panel of the box.

The circumference of each disc is divided by small, equidistant holes. There are long golden strings folded in half such that each end passes through one of two neighboring holes on each disc. The circumference of each circle carved out by the brass figure eight is similarly divided into equidistant holes. Strings suspended from the left-hand disc are pulled to the side and pass through the holes of the right-hand figure-eight circle. Strings suspended from the right-hand disc are pulled to the side and pass through he holes of the left-hand figure-eight circle. There is a grey, elongated egg-shaped weight tied to each end of each string. The weights pull the strings taut such that each acts as a straight line on the surface of the cone. The model therefore represents the ruled surfaces of the modeled objects. The weights are concealed inside the mahogany box. They can be seen when the outer walls of the box are removed. Because the bottom figure-eight circles are bigger than the top disc circles, the strings form the ruled edges of two cones. Crucially, because the strings pass from the disc on one side to the figure-eight circle on the other side, the cones intersect in the middle.

A small black ring is looped around each intersection between a string from the left-to-right hand cone and a string from the right-to-left hand cone. Each black ring is a point on the circumference of the intersection shape (some form of ellipse). There are two axes of intersection: one horizontal (the circumscribed plane is parallel to the top panel of the box) and one vertical (the circumscribed plane is perpendicular to the top panel). As mentioned above, the two discs are attached to the top horizontal brass bar on sliding sleeves. The disc on the right can be slid all the way to the right end of the bar and as far to the left as the placement of the left disc permits (i.e. they cannot cross each other) and vice versa. As the discs are slid back and forth, the specificities of the modeled cones vary as do the intersection shapes.

Signed: engraved 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]

Function: Théodore 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 Lockeian 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 taut 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 ends at the top with a brass disc attached to a horizontal bar by a handle. The user can manually alter the angle between the two cones by loosening the handles and sliding one or both of the discs left or right along the horizontal brass bar, changing the position of the top of the cones. There is a black loop around each point at which a string from the cone on the left intersects a string from the cone on the right. The intersection consists of two perpendicular ellipses that extend and contract according to the relative positions of the cones.

[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 Attributes: Designed 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 "Developement 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.

Curatorial Remarks: detailed three-page invoice and restoration report from Richard L. Ketchen (horologist) to William Andrewes for the restoration of the Geometric String Figure Model # 5469, dated July 25, 1995, in object file

Primary Sources: Comberousse, 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.

Provenance: Taken from the attic of Pierce Hall, Department of Engineering, Harvard University, in May 1961.

Related Works: Belhoste, 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.