Geometric Models  Jullien Models for Descriptive Geometry
Known as the father of descriptive geometry, Gaspard Monge (17461818) was born in Beaune, Burgundy in France. Attending college at an early age, Monge showed mathematical talent early on. He drew a plan of his home town at age eighteen that brought him to the attention of the École Royale du Génie in Mézières where he became a draftsman in 1765. The next year he was asked to draw plans for a fortification. Putting his mathematical talent to use, Monge devised his own method of representing the vertical and horizontal components of an architectural drawing.
His new approach to geometry came to be known as descriptive geometry. Monge describes his new geometry as “representing with exactitude, within drawings that have but two dimensions, objects that have three.” In particular, by showing the vertical and horizontal projections of an object on one piece of paper (the paper is divided in half horizontally with the vertical projection on the top and the horizontal projection on the bottom) geometric properties can be employed to determine various elements of the surface depicted such as angles of intersection and lengths. All such properties of threedimensional objects are essential for the accurate design and construction of various buildings, as well as other design objectives of engineering. The term descriptive geometry is still used for this method of representing the vertical and horizontal projections of an object. However, the modern term for the method is orthographic projection.
Label on Jullien's Collection of Reliefs 
France found this new geometry so important to fortification design that it was held as a state secret for several years. An example is the design of star forts. Star forts were invented in Italy in the fifteenth century and became common in Europe and the New World the following century. With the advent of larger cannons, fortifications needed to be more thoughtfully designed to withstand and deflect cannon fire. The broad bases and angled walls of star forts helped deflect cannon fire. The straight, angled walls allowed defenders to launch enfilade or flanking fire: firing crossways at the enemy from the points of the stars so the attackers have no safe place to fire from and keeps attackers further from the walls. Previously, forts often had rounded walls which allowed for “dead zones” where defenders could not fire upon the attackers. Beautiful images of star forts can be found online.
Monge went on to become a teacher at Mézières as well as a member of the Académie des Sciences. He further developed descriptive geometry, teaching it and publishing text books. The teaching of descriptive geometry quickly spread throughout France and ultimately at the United States Military Academy at West Point, founded in 1802. Other practitioners took up the mantel and published texts on descriptive geometry. One such teacher and text book writer was French mathematician A. Jullien.
Jullien, taught at the Lycee SainteBartie in Paris. He wrote a descriptive geometry text book, Cours élémentaire de géométrie descriptive. The 3^{rd} edition published in 1881 is available online through Google Books. The reliefs or models in the Smithsonian collection are teaching aids made by Jullien to supplement this textbook, just as modern mathematical text books come with online applications that show the geometry of the mathematics being discussed. Each of the thirty reliefs show a construction of descriptive geometry. The reliefs start with the most simple of geometric ideas and progress to more sophisticated constructions. The models held by the Smithsonian were produced in the mid1870s, but after 1873. In that year a set of Jullien models won a certificate of merit at the Scientific Exposition in Vienna.
The thirty models are housed in a handcrafted wooden box lined in pink and cream stripped satin. A small pamphlet entitled Notice Explicative describing the assembly and concept of each model is also in the Smithsonian’s collections.
Box holding Jullien's Collection of Reliefs 
References:
Martínez, A.O., Kinematics: The Lost Origins of Einstein’s Relativity, Johns Hopkins Press, 2009, g. 45.
J. J. O'Connor and E F Robertson, Gaspard Monge, Mac Tutor History of Mathematics website, https://wwwhistory.mcs.stand.ac.uk/Biographies/Monge.html
Types of Castles and The History of Castles: Star Forts, https://www.castlesandmanorhouses.com/types_10_star.htm.

Models for Descriptive Geometry by A. Jullien
 Description
 This wooden box holds thirty models (called reliefs in French). Each is a folded card held in place by a metal mount that can rest in the palm of the hand. Each model depicts a concept of descriptive geometry. Wires and string show the geometric concepts in three dimensions and the two projections are depicted on the horizontal and vertical portion of the card. The strings are held in place by small washers on the back of the cards.
 Individual models are described in records 1986.0885.01.01 through 1986.0885.01.30. A related manual is 1986.0885.02.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 COLL.1986.0885
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Representation of a Point
 Description
 For the discussion that follows, the following conventions will be used to explain the location of points in each model. Any point in space can be denoted by a coordinate triple (x,y,z). This is the threedimensional version of the (x,y) plane from Euclidian geometry learned in high school. For our purposes, the xaxis will be the horizontal line stretching left and right at the fold of each relief. The yaxis will be on the horizontal plane (paper card) of each model, appearing to be coming out of the plane of each image. The zaxis will be the vertical axis of each relief and lies on the vertical plane (paper card) of each relief. In all of the reliefs, the xcoordinate is irrelevant since each projection will be with respect to the y and z planes. When needed, a point will be referred to in two coordinates only (y,z), leaving off the xcoordinate for brevity. Positive will be the forward or upward direction and negative values will be behind or below the cards of each relief.
 In each relief, a point of interest that is on the vertical or horizontal plane will be marked by a letter and a small hole or dot. Points in space are shown by the bend in a wire that pierces the cards at the y and z intercepts (a,0) and (0,b) respectively and will be denoted (a, b). Lines are shown by black or red strings threaded between points on the cards or by wires and will be denoted by the point on the horizontal plan followed by the point on the vertical plane, such as ab. The title of each relief is actually a construction. For example, relief seven is entitled “line perpendicular to a plane.” This is actually a task, “construct a line perpendicular to a given plane in space.” Following Jullien, we will assume the directions are to construct the item in question. There are many projections shown for the construction of each item. For simplicity, I have only described the relevant items in each model, leaving out all the mathematical details. It would take a whole textbook on the topic to rigorously go through each model. And that is the point of the models, to supplement the textbook Jullien wrote. The reliefs slowly progress from simple to complex, starting with the depiction of points and lines and ending with the construction of a pyramid, guiding the students through the constructions of descriptive geometry. The progression of the reliefs follows the textbook.
 In this particular model, nine points are shown for all the possible combinations of a positive, negative or zero value for y or z. For example, the first point on the left shows a point with both y and z coordinates positive.
 Reference:
 A. Jullien, Cours élémentaire de géométrie descriptive. . ., Paris: GauthierVillars, 1875. There were editions of this book as late as 1887.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.01
 catalog number
 1986.0885.01.01
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Representation of a Line (Trace, Angle of Projection with a Plane, Distance between Two Points)
 Description
 Two projections are shown. The left shows the vertical and horizontal projection of a line. The right shows a line rotated to the horizontal and vertical plane at a point of contact with each plane to show the angle of the line with each plane.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.02
 catalog number
 1986.0885.01.02
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Representation of a Line, Special Case
 Description
 This relief shows the special case of a line that passes through the origin (0, 0, 0) and another point (a, b, c). The line is represented by the wire.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.03
 catalog number
 1986.0885.01.03
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Representation of a Line (Special Cases)
 Description
 This relief shows four more special cases of lines. It is one of two reliefs that are about twice as long as the rest. The wires and strings are missing, so it is difficult to visualize the representations. (Note that the wire coming out of the horizontal plane on the far right is not correct, it is in fact one of the support pieces. This is how the relief was configured when it arrived at the Smithsonian.)
 Figure 1 on the left shows the vertical and horizontal projections of two lines as well as the intersection of the two lines projected onto the vertical plane.
 Figure 2 shows the angle of inclination of the line with the horizontal plane by rotating it down onto the horizontal plane.
 Figure 3 shows the angle of a line coming up through the horizontal plane with the vertical, again by rotating it onto the horizontal.
 Figure 4 shows the vertical projection of a line and the horizontal projection of its vertical component.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.04
 catalog number
 1986.0885.01.04
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Representation of a Plane (Projection of Lines on the Plane)
 Description
 Figure 1 on the left shows plane APP’ cutting obliquely through the space with the black string showing a line on the plane. The vertical and horizontal projections of the plane are also depicted.
 Figure 2 on the right shows the vertical and horizontal projection of a line in the plane BQB’.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.05
 catalog number
 1986.0885.01.05
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Parallel Lines (Given a Plane and a Line Parallel to the Plane, Construct the Parallel Plane Containing the Line)
 Description
 Plane APA’ (right construction) and line through (m, m’) parallel to the plane and to line cd’ on the plane are given (red strings). Then the plane BQB’ parallel to plane APA’ is constructed.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.06
 catalog number
 1986.0885.01.06
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Line Perpendicular to a Plane
 Description
 Plane APA’ is given. The line perpendicular to the plane is represented by the oblique wire passing through (m, m’). The point of intersection of the line and the plane would be about halfway along the wire between (m, m’) and (c,0).
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.07
 catalog number
 1986.0885.01.07
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Horizontal Line to a Plane (Construct a Line Parallel to a Given Plane through a Point)
 Description
 Plane APA’ is given. The line shown by the wire through (m, m’) and (c, c’) is constructed parallel to both the plane APA’ and the horizontal plane.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.08
 catalog number
 1986.0885.01.08
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Line of Greatest Slope
 Description
 Plane APA’ and point (m, m’) are given. Construct the line on the plane with largest slope or inclination from the horizontal. Theory says that this line would be perpendicular to the horizontal trace of the plane, line PA. So the result is the black string from point c on the horizontal plane to point A’ on the vertical plane. The red string from point A is in the plane; the red string from point d forms the plane perpendicular to plane APA’ that the line lies on. All three lines meet at point (m, m’).
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.09
 catalog number
 1986.0885.01.09
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Rotation of a Point about a Vertical or Horizontal Line
 Description
 Figure 1 shows the rotation and projections of point (m, m’) about a horizontal line perpendicular to the vertical plane through point a’. Figure 2 shows the rotation and projections of point (m, m’) about the vertical line perpendicular to the horizontal at point a.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.10
 catalog number
 1986.0885.01.10
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Rotation of a Point about an Axis in the Horizontal Plane
 Description
 Point (m, m’) is rotated about line ab. Point n on the horizontal plane is the foot of the perpendicular from the point to line ab. Point M_{1} is the result of rotation of the point about line nm; M_{2} is the result of rotation about line ab.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.11
 catalog number
 1986.0885.01.11
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Image (Rotation) of a Vertical Plane about a Line in the Horizontal Plane
 Description
 Plane APA’ is a plane perpendicular to the horizontal plane with point (m, m’) on APA’ (at corner of wire). Vertical and horizontal projections are shown, as well as rotation about line AP in the horizontal plane.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.12
 catalog number
 1986.0885.01.12
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Image (Rotation) of Any Plane
 Description
 Plane APA’ is any plane in space with point (m, m’). The two red strings, the black string and the horizontal wire are all lines on the plane. Vertical and horizontal projections are given, as well as the results of rotation of the plane about lines AP, nm, cn, and ce.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.13
 catalog number
 1986.0885.01.13
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Intersection of Two Planes
 Description
 Planes APA’ and BQB’ (where A=B=c and A’=B’=d’) are given. Then line d’c’ is the vertical projection of the intersection of the two planes while line dc is the horizontal projection of the intersection.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.14
 catalog number
 1986.0885.01.14
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Intersection of Two Planes with Parallel Horizontal Projections
 Description
 Planes APA’ and BQB’ are parallel in the horizontal plane (see their respective horizontal projections AP and BQ) and intersect along line (o, c’)(d, d’) (wire). This intersection is also parallel to the horizontal projections of the two planes (observe that cd is also parallel in the horizontal plane).
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.15
 catalog number
 1986.0885.01.15
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Intersection of Two Planes Parallel to a Line on the Ground (The xaxis)
 Description
 Planes CDD’C’ and ABB’A’ are both parallel to the xaxis (crease in the card). They intersect in line (e, e’)(f, f’) (wire) which is also parallel to the xaxis. The planes can be visualize by imagining both red strings extending left and right. Both projections of this intersection are shown as well as the rotation of it about the horizontal line perpendicular to the xaxis PA.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.16
 catalog number
 1986.0885.01.16
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  General Construction of the Intersection of a Line and a Plane
 Description
 The plane APA’ is intersected by line bc’ represented by the black string. The red string represents a line on the plane which bc’ intersects at point (m, m’). Horizontal and vertical projections of these lines are shown.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.17
 catalog number
 1986.0885.01.17
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Special Case of the Intersection of a Line and a Plane
 Description
 As with relief 17, APA’ is again the plane and the red string de’ is on the plane. In this relief, the line is represented by the wire coming out of the horizontal plane and away from the vertical plane (it intersects the vertical plane below the horizontal plane). The point of intersection is at (m, m’) where the wire, the string and the bent wire meet. The horizontal and vertical projections are shown.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.18
 catalog number
 1986.0885.01.18
 accession number
 1986.0885
 Data Source
 National Museum of American History

Model for Descriptive Geometry by A. Jullien  Three Points Determine a Plane
 Description
 The three points in space are represented at the bends in the three wires at points (a, a’), (b, b’) and (c, c’). The red lines connect the points in pairs showing the resulting triangle that lies on the plane that was to be constructed.
 For more details, see COLL.1986.0885 and 1986.0885.01.01.
 Location
 Currently not on view
 date made
 ca 1880
 maker
 Jullien, A.
 ID Number
 1986.0885.01.19
 catalog number
 1986.0885.01.19
 accession number
 1986.0885
 Data Source
 National Museum of American History