Abstract
Apart from the growing acceptance of the fold as a legitimate proof practice in geometry, two main mathematical traditions in the nineteenth century used, integrated and conceptualized folding, each showing its advantages and limitations.
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Notes
- 1.
I do not aim in this short survey to give a full account of the development of the mathematical tradition of modeling in the nineteenth century, but rather to give a background for the role folded models played in it. For an extensive investigation of this topic (see: Sattelmacher 2013, 2014). See also: Sattelmacher (2016), regarding the financial aspect that these mathematical models had.
- 2.
Cf. Sakarovitch (2010).
- 3.
- 4.
Daston (1986, p. 271). According to Daston, “[w]ithin a Lockean psychology, the clarity of a concept corresponds to its proximity to experience. The more remote an idea was from the vividness of direct sensation, the more it risked vagueness and imprecision. Conversely, the more closely a concept approximated the realm of experience, the sharper its mental outlines. Therefore, the purely intellectual ‘beings of reason’ defended by Euler appeared eminently untrustworthy to late-eighteenth century adherents of a Lockean philosophy of knowledge.” (ibid, p. 272). Therefore, Chasles described Viete’s algebraic symbols as leading us on a mysterious path to the desired results (ibid, p. 276) by means of indeterminate objects. See also: Glas (1986) and Glas (2002, p. 717): “The mathematics of Monge and Carnot was not of a lesser standard, but of a different type to that of the analyticians. It was aimed at the design of models for all kinds of technical devices, and at providing effective methods for investigating technical problems.”
- 5.
Lazare Carnot (1753–1823) was a French statesman, general, military engineer, and mathematician.
- 6.
See: Daston (1986, p. 273).
- 7.
See: Sattelmacher (2016, pp. 137–138), for a survey of the model tradition in France, also during the 18th century. As Sattelmacher remarks, it is with the establishment of the CNAM (Conservatoire national des arts et métiers) in 1794 that this tradition in France became more abundant. Before that, the physical instruments were to be found in “scientific cabinets” (ibid.)
- 8.
In the Catalogue des Collections issued by the CNAM (see: Conservatoire National des Arts et Metiers 1882, p. 31), it is mentioned that “Gaspard Monge had executed models made of silk threads for the teaching of descriptive geometry during the creation of the École polytechnique. In the collections of this school there existed in 1814 two models of great dimensions, the one represented the double rectilinear generation of the one-sheet hyperboloid […], the other the double rectilinear generation of a hyperbolic paraboloid.” [Gaspard Monge avait fait exécuter des modèles en fils de soie pour l’enseignement de la géométrie descriptive, lors de la création de l’ École polytechnique. Il existait en 1814, dans les collections de cette École, deux modèles, de grandes dimensions, représentant l’un la double génération rectiligne de l’hyperboloïde à une nappe […], et l’autre la double génération rectiligne d’un paraboloïde hyperbolique].
- 9.
Klein (1926, p. 78).
- 10.
See: Shell-Gellasch (2003).
- 11.
On the influence of the European on the American academy (see: Grabiner 1977).
- 12.
See, e.g., Daston (1986, pp. 294–295).
- 13.
For a survey on the epistemological role of models in Germany (see: Mehrtens 2004; see also Rowe 2013; Sattelmacher 2013). Models also played a role in mathematical education in the United States (see: Kidwell 1996) and in Italy (see: Giacardi 2015), but this would take us outside the scope of this short survey. See also Sect. 4.1.2.2 for a discussion on Eugenio Beltrami’s models in Italy.
- 14.
See: Klein (1921, p. 3): “Plücker himself once told me that he had been particularly encouraged by the communication with Faraday; he himself had used the construction of models as a means to clarify, for himself as a non-expert, the mathematical formulas necessary for him.” [Plücker selbst erzählte mir [Klein] einmal, dass er namentlich durch den Verkehr mit Faraday dazu angeregt worden sei; dieser selbst habe die Modellkonstruktion als Mittel benutzt, um sich als Nichtfachmann die ihm jeweils notwendigen mathematischen Formeln verständlich zu machen].
- 15.
The fact that there are 27 lines on a smooth cubic surface was discovered via a correspondence between Arthur Cayley and George Salmon in 1849. Cayley proved that the number of lines must be bounded (see: Cayley 1849), whereas Salmon (1849) proved that the expected number of such lines ought to be 27.
- 16.
For a study regarding the models of Kummer surfaces and the history regarding their production that emphasizes Klein’s role (see: Rowe 2013).
- 17.
- 18.
Lindemann (1927, p. 161).
- 19.
- 20.
Lindemann (1927, p. 159). See also: Dyck (1892, p. 258). See also Hashagen (2003, p. 421), concerning Henrici’s participation in Dyck’s exhibition. Cf. also Brill (1887, p. 75): “For me [Brill], the paraboloid model made of cardboard discs (constructed by Henrici in London) appears worthy of attention.” [Mir [Brill] selbst war namentlich das (von Henrici in London konstruierte) Paraboloid-Modell aus Kartonscheiben der Beachtung wert erschienen].
- 21.
- 22.
For a description of the models in these collections: Fischer (1986).
- 23.
For example, see: Betsch (2014), for a detailed description of a model constructed by one of Brill’s students, Christian Betsch.
- 24.
- 25.
- 26.
For a biography of Felix Klein, see: Tobies (1981).
- 27.
See: Sattelmacher (2014).
- 28.
Klein (1893 [1911], p. 32). I will return to the full citation below.
- 29.
- 30.
Wiener (1912).
- 31.
Fischer (1986, vol. 1, p. ix). To give another example, in 1926, Klein had already stated that Olivier’s models were in bad condition: Klein (1926, p. 78): “Unfortunately, Olivier’s models in the Conservatoire des Arts et Metiers are now completely destroyed due to the lack of durability of the silk threads.” [Leider sind Oliviers Modelle im Conservatoire des Arts et Metiers infolge der mangelnden Haltbarkeit der Seidenfäden jetzt ganz zerfallen].
- 32.
Mehrtens (2004, p. 293).
- 33.
Ibid., p. 294.
- 34.
Olivier (1852, p. x): “Thus we begin to understand that when one wishes to speak to the pupils on the properties of a surface, the first thing to be done is to place before their eyes the relief of this surface.” [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].
- 35.
Olivier (1845b, p. 64): “It is necessary to learn to represent the idea of surfaces and the curves, whose forms of a greater or lesser complication are varied to infinity. It is necessary to see them through the eyes of the mind, to cut, to touch, to wrap, according to the given conditions.” [Il faut apprendre à se représenter en idée des surfaces et des courbes dont les formes d’une complication plus ou moins grande sont variées à l’infini. Il faut les voir par les yeux de l’esprit, se couper, se toucher, s’envelopper, suivant des conditions données].—Olivier writes that he cites Charles Dupin from Considérations préliminaire sur le applications de la géométrie, but, in fact, he cites it from Dupin’s Considérations générales sur les applications de la géométrie. See: Dupin (1822, p. 1).
- 36.
Rowe (2013, p. 5, p. 11).
- 37.
Klein (1926, p. 78): “Like today, so it was back then that the purpose of the model was not to compensate for the weakness of the intuition [Anschauung], but to develop a vivid, clear intuition, a goal best achieved by the self-making of models.” [Wie heute, so war auch damals der Zweck des Modells, nicht etwa Schwäche der Anschauung auszugleichen, sondern eine lebendige, deutliche Anschauung zu entwickeln, ein Ziel, das vor allem durch das Selbstanfertigen von Modellen am besten erreicht wird].
- 38.
Ulf Hashagen notes regarding the Napier Tercentenary Exhibition in Edinburgh in July 1914 that “remarkably British scientists […] had made an exhibition concept with a pragmatic classification of the mathematical artefacts—similar to the 1876 exhibition in London—and did not present symbolically enhanced objects as their German colleagues.” (Hashagen 2015, p. 2840)
- 39.
Klein (1893 [1911], pp. 108–109).
- 40.
Ibid, p. 32.
- 41.
Klein (1872, p. 42): “Es gibt eine eigentliche Geometrie, die nicht, wie die im Texte besprochenen Untersuchungen, nur eine veranschaulichte Form abstrakterer Untersuchungen sein will. In ihr gilt es, die räumlichen Figuren nach ihrer vollen gestaltlichen Wirklichkeit aufzufassen und (was die mathematische Seite ist) die für sie geltenden Beziehungen als evidente Folgen der Grundsätze räumlicher Anschauung zu verstehen. Ein Modell—mag es nun ausgeführt und angeschaut oder nur lebhaft vorgestellt sein—ist für diese Geometrie nicht ein Mittel zum Zwecke sondern die Sache selbst.” (Translation taken from: Mehrtens 2004, p. 289)
- 42.
Klein (1872, p. 7): “Streifen wir jetzt das mathematisch unwesentliche sinnliche Bild ab, und erblicken im Raume nur eine mehrfach ausgedehnte Mannigfaltigkeit, also, indem wir an der gewohnten Vorstellung des Punctes als Raumelement festhalten, eine dreifach ausgedehnte. Nach Analogie mit den räumlichen Transformationen reden wir von Transformationen der Mannigfaltigkeit; auch sie bilden Gruppen.”
- 43.
For models as “epistemic things,” see: Mehrtens (2004, pp. 289–291).
- 44.
Brill (1887, 77): “The manufacturer of a model was free to write a treatise on it, the publication of which under his name stimulated him to undertake the often tedious calculations and drawings on which the practical work was based.” [Dem Verfertiger eines Modells stand es frei, eine Abhandlung zu demselben zu schreiben, deren Veröffentlichung unter seinem Namen nicht wenig dazu anreizte, die oft mühsamen Rechnungen und Zeichnungen, welche der praktischen Ausführung zu Grunde lagen, durchzuführen].
- 45.
Ibid.: “Öfter veranlasste umgekehrt das Modell nachträgliche Untersuchungen über Besonderheiten des dargestellten Gebildes.” Brill gives an example of a model of a surface with negative curvature, in which it was proved that it is an Enneper surface—a minimal surface (i.e., a surface that locally minimizes its own area; the strict definition is that the mean curvature would always be 0), discovered in 1864 by Alfred Enneper (1820–1885), which was not known until that time.
- 46.
- 47.
Though establishing a direct connection, if it even exists, is outside the scope of this section.
- 48.
See: Sohncke (1854, pp. 147–148), in which one can find a reference to Dupin models built in 1842. The models are mentioned in the “Construirende und analytische Geometrie” section.
- 49.
A similar box of polyhedrons is to be found in the Musée des arts et métiers in Paris (see Fig. 4.3).
- 50.
Olivier (1843).
- 51.
Ibid, p. 192: “En tranchant à mi-épaisseur le carton le long de l’arête droite, suivant laquelle deux faces planes du polyèdre s’assemblent, il a pu facilement plier le développement plan et reformer le polyèdre en relief.”
- 52.
See Sect. 2.2.2.1.
- 53.
Olivier (1845a).
- 54.
Ibid., p. 11.
- 55.
Vauvilliers (1847, p. 443). In the report, it is added that Dupin “entrusted the execution to Messrs. Molteni, who organized a method of manufacture that allows delivering these solid at prices that should facilitate the introduction in schools.” By “Messrs. Molteni,” Olivier refers to the company Molteni et Siegler. As a side note, the history of the firm also reflects another side of the decline of the fabrication of scientific models (see: Guérin 2015). In 1782, B. Molteni (probably Joseph Antoine Balthazar Molteni) opened a store for optical instruments in Sainte-Apolline Street in Paris. His son, Pierre François Antoine Molteni, continued this business, opening a second store at the Palais-Royal. In 1841, Ferdinand Siegler, one of the employees, became a partner. Jules Molteni, the second son of Francis, joined the leadership of the company in 1843 and the two brothers were the chiefs of the firm until 1864. François Marie Alfred Molteni, born in Paris in 1837, took over around 1865. Throughout the first half of the 19th century, the firm was devoted mainly to the fabrication of instruments for use in optical sciences, physics and mathematics, taking part in the 1851 Great Exhibition in London. And indeed, the mass production of Dupin’s models was given to this firm. During the second half of the 19th century, however, the domain of scientific apparatuses was abandoned under the leadership of Alfred Molteni, who is well known for his saying, “la projection faite homme” (Cf. Mannoni 2000, p. 286ff.). The firm became one of the most famous in France in the field of projection or the “magic” lantern, and the domain of projections in general (see, e.g.: Molteni 1881).
- 56.
For Monge’s treatment of projections of polyhedra, see: Belhoste and Taton (1992).
- 57.
This is in contrast to Marie’s 1835 descriptions of the same properties, which were written in a separate text (see Sect. 2.2.2.1).
- 58.
I follow here: Weitzenböck (1956) and especially ibid., pp. 11–18. On the development of the concept of the n-dimensional manifold beginning from the 1850s with Riemann, see, e.g.: Scholz (1999), esp. pp. 25–30. The introduction to and acceptance of the four-dimensional space by the general public was prompted in 1884 with the publication of the novel Flatland: A Romance of Many Dimensions by Edwin A. Abbott (under the pseudonym A Square, see: Abbott and Square 1884), which dealt with the perception of two-dimensional beings in a three-dimensional world. The novel served as an analogy, which helped in understanding how the fourth dimension should be conceived for beings living in a three-dimensional world.
- 59.
See: Schläfli (1901). However, Schläfli’s work Theorie der vielfachen Kontinuität was mainly ignored, and its importance was recognized only at the end of the 19th century, published fully only in 1901. It is important to note that the manuscript does not contain a single figure. Schläfli researched in his manuscript the higher dimensional analogues of polygons and polyhedra and introduced (what is today called) the Schläfli symbol. He proved that there are exactly six convex regular polytopes in four dimensions, but only three in dimension n where n ≥ 5.
- 60.
As mentioned above, there are six convex regular four-dimensional polytopes. They are the 5-cell (made of five tetrahedra), the 8-cell (made of eight cubes, also called a tesseract), the 16-cell (made of sixteen tetrahedra), the 24-cell (made of twenty-four octahedra), the 120-cell (made of one hundred and twenty dodecahedra), and the 600-cell (made of six hundred tetrahedra).
- 61.
See: Rowe (2010).
- 62.
Eneström (1905, p. 423).
- 63.
Cf. Rowe (2010, p. 54).
- 64.
Ibid., p. 45. However, Schlegel did use, in his 1882 paper “Quelques théorèmes de géométrie à n dimensions,” the same symbolism as Grassmann (Schlegel 1882, p. 173).
- 65.
Schlegel (1883) (The 1883 manuscript was already mentioned in the 1882 paper—see the former footnote—and was probably written before it).
- 66.
- 67.
- 68.
As was mentioned, in 1852, Schläfli discovered the six regular convex four-dimensional polytopes. Washington Irving Stringham rediscovered these polytopes in 1880 and published his results, together with two plates of drawings, in: Stringham (1880). Schlegel was aware of Stringham’s work and mentions him several times in his papers (e.g., Schlegel 1882, pp. 177, 203, 1883, pp. 447–8, 453–5). For a discussion on Stringham’s methods, see: Robbin (2006, pp. 4–11).
- 69.
- 70.
Schilling (1911, p. 32). It is indicated on p. 34 that the text was written in 1886.
- 71.
Dyck (1892, pp. 253–254).
- 72.
Schilling (1911, pp. 31–34).
- 73.
Schlegel (1885, p. 5): “Es muss uns eben genügen, in den beiden Gebieten der Ebene und des Raums, welche uns für unsere Construktion zur Verfügung stehen, genau analog und in analoger Weise deutungsfähige Abbildungen herzustellen, und, während unser Verstand sich durch diese Analogie befriedigt erklärt, auf die Analogie der durch das Auge vermittelten Anschauungen und der durch die Phantasie bewirkten Umdeutungen zu verzichten.”
- 74.
Dyck (1892, pp. 253–254).
- 75.
Stringham also drew his three-dimensional models in: Stringham (1880, plate I and II).
- 76.
See, e.g., Robbin (2006, p. 5): “Stringham’s approach, both in his drawings and in his mathematics, was to define the three-dimensional cells, or coverings, of four-dimensional figures, and then, in keeping with the mechanical drawing techniques of his time, to imagine the cells folded up to make a four-dimensional figure.” (emphasis by M.F.) However, the terminology that Stringham uses does not include the explicit folding of a three-dimensional net into a four-dimensional polytope; see, for example, Stringham (1880, p. 3): “In particular, the fourfold pentahedroid [the 5-cell] has 5 summits, 10 edges, 10 triangular and 5 tetrahedral boundaries. To construct this figure select any one summit of each of four tetrahedra and unite them. Bring the faces, which lie adjacent to each other, two and two into coincidence. There will remain four faces still free; take a fifth tetrahedron, and join each one of its faces to one of these four remaining ones. The resulting figure will be the complete fourfold pentahedroid.”
- 77.
See: Robbin (2006, p. 9).
- 78.
Schilling (1911, p. 31): “Nets for No. 6 [the 120-cell] for the purpose of cutting and gluing together the paper models” [Netze zu Nr. 6 (Hundertzwanzigzell), behufs Ausschneiden und zusammenkleben der Papiermodelle]. Again, this is in contrast to Stringham, who does not explicitly mention that folding is involved in the procedure.
- 79.
Ibid., p. 34: “Wird doch das Anschauungsvermögen bei Betrachtung der einzelnen Teile der Drahtkörper durch Vergleichung derselben mit den Cartonmodellen wesentlich unterstützt, da bei letzteren vermöge ihrer Darstellungsweise, entgegen den Drahtkörpern, alle Kanten weniger ins Auge fallen, die Flächen jedes einzelnen Körpers dagegen mehr hervortreten und eine klarere Vorstellung von der Gestaltung jedes einzelnen Polyeders ermöglichen.”
- 80.
- 81.
For a biography of Boole Scott, see, e.g., Coxeter (1987).
- 82.
- 83.
Hinton (1888, pp. i–vi) (immediately after p. 216)
- 84.
ibid, pp. vi–vii : “N. B. Models—It is unquestionably a most important part of the process of learning space to construct these, and the reader should do so, however roughly and hastily. […] Much of the work can be done with plain cubes by using names without colours, but further on the reader will find colours necessary to enable him to grasp and retain the complex series of observations. Coloured models can easily be made by covering kindergarten cubes with white paper and painting them with water-colour, and, if permanence be desired, dipping them in size and copal varnish.”
- 85.
Boole Stott (1900).
- 86.
- 87.
The only passage that describes the physical properties of her models is to be found in Boole Stott (1900, p. 14): “In practically constructing the sections I have found that their symmetry is made more obvious by colouring the faces. The letters on the faces of diagrams I–XIV denote colours, the plane and accented letters denote corresponding colours and the remaining sections are the same as those given with the plane and accented letters interchanged.”
- 88.
- 89.
Schlegel (1883, p. 345) (emphasis by M.F.): “[…] man [kann] durch Zusammenfaltung eines homogen zusammengesetzten (n − 1)-dimensionalen Gebildes im n-dimensionalen ebenen Raume und nachträgliche Hinzufügung des fehlenden Grenzgebildes ein homogenes n-dimensionales Gebilde erzeugen. Namentlich aber kann man in den Fällen, wo die wirkliche Construction sich der Anschauung entzieht, aus der Existenz des abbildenden Gebildes auf diejenige des abgebildeten schließen, da die wesentlichen Eigenschaften des letzteren nach der Definition nur in seiner Grenze hervortreten.”
- 90.
Ibid., p. 438, footnote 1: “der umgekehrte Prozess (Umstülpung der Ecken eines Tetraeders nach innen) [...]” (emphasis by M.F.).
- 91.
Emsmann (1880).
- 92.
Interestingly, Emsmann refers to van ’t Hoff’s “Die Lagerung der Atome im Raume” and his model of the tetrahedron as an example of a space with four axes: “Und der ganzen Arbeit van ’t Hoffs liegt das vierdimensionale Axensystem zu Grunde!” (ibid., p. 259), where Emsmann immediately mentions afterwards that he (Emsmann) constructed a four-dimensional tetrahedron out of cardboard (referring possibly to van ’t Hoff’s cardboard models), although it is not clear whether this model is a three-dimensional one with four axes (in which the axes are the lines connecting the center of the regular tetrahedron with its vertices) or a “real” four-dimensional tetrahedron, and what Emsmann means by that. For a discussion on van ’t Hoff’s models, see Sect. 4.1.3.1.
- 93.
Schlegel (1886, p. 133).
- 94.
Schlegel (1888, p. 21): “Faltet man [...].”
- 95.
Ibid. “[…] wenn uns ein vierdimensionaler Raum zur Verfügung stände, und die Möglichkeit, Gegenstände in denselben hinein zu versetzen, gegeben wäre, die eine von zwei symmetrischen Ecken erst aus unserem Welträume in diesen vierdimensionalen Raum bringen, dort umkehren (d. h. Innen- und Außenseite vertauschen) und dann in unseren Raum zurückbringen […]” (emphasis by M.F.).
- 96.
Ibid., p. 22.
- 97.
I follow here: Giacardi (2015).
- 98.
Ibid., p. 12.
- 99.
Note here that the folding in this section does not create creases, but rather concerns the bending of a surface. However, Beltrami himself regards the surfaces as “folded.”
- 100.
This is in contrast to elliptic geometry, in which, given a line L and a point p not on L, there exists no line parallel to L passing through p (i.e., all lines intersect in this geometry).
- 101.
A metric, roughly explained, gives the way to compute the distance between two points, and more precisely, indicates the infinitesimal distance on a manifold between two points. Thus, for example, for the two-dimensional Euclidean plane with coordinates (x, y), the Euclidean metric is ds 2 = dx 2 + dy 2, which is intuitively derived from the distance formula between two points in the two-dimensional plane, i.e., if x = (x 1, x 2 ) , y = (y 1, y 2), then the distance is\( s=\sqrt{{\left({x}_2-{x}_1\right)}^2+{\left({y}_2-{y}_1\right)}^2} \). (In modern terms, when the local coordinates on a surface are (u, v), the metric is expressed via the first fundamental form: ds 2 = Edu 2 + 2Fdudv + Gdv 2, when E, F, G are functions of u and v). However, looking on (smooth) surfaces in three-dimensional Euclidean space, the induced metric (induced from the Euclidean space) is not necessarily ds 2 = dx 2 + dy 2; for example, on a sphere, the distance between two points is measured along the great circle (on the sphere) connecting them, and not along the chord that also connects them (since this chord does not lie on the sphere). Using spherical coordinates (θ and ϕ, when θ is the angle measured from the z axis, and ϕ is the angle from the x axis in the xy plane), one obtains that a point on a unit radius sphere (x, y, z) is represented by the coordinates: x = sinθ cosϕ, y = sinθ sinϕ, z = cosθ and the metric is ds 2 = dθ 2 + sin 2 θdϕ 2. However, as Beltrami exemplifies, one does not necessarily have to use the induced metric from the embedded space.
- 102.
- 103.
- 104.
A geodesic is a generalization of the notion of a “straight line” for surfaces (or manifolds) with another metric, when the shortest segment is not necessarily a straight line. For example, on a sphere (where the metric is not the Euclidean one), the geodesics are the great circles.
- 105.
This axiom is verified in: Beltrami (1868, p. 383).
- 106.
Ibid., p. 381.
- 107.
In the year 1839, Ernst Minding (1805–1885) already proved that two surfaces having the same constant curvature could be mapped isometrically onto each other. Hilbert proved in 1901, however, that there exists no complete regular (i.e., continuous and smooth) surface S having a constant negative Gaussian curvature immersed in three-dimensional Euclidean space (Hilbert 1901). Beltrami hence proved that the surfaces of revolution—that he found with the induced metric from the ambient space—surfaces having a constant negative curvature, are isometric only to parts of the pseudosphere.
- 108.
Beltrami (1868, p. 375): “abbiamo tentato di trovare un substrato reale a quella dottrina, prima di ammettere per essa la necessità di un nuovo ordine di enti e di concetti.”
- 109.
Ibid., p. 390, equation (12).
- 110.
Ibid., p. 393, equation (14).
- 111.
Ibid., p. 394, equation (17).
- 112.
In: Boi et al. (1998, p. 80) (emphasis by M.F.): “J’ai eu […] une idée bizarre, que je vous communique à cause qu’il pourrait vous être plus facile qu’à moi de la mettre à exécution. J’ai voulu tenter de construire matériellement la surface pseudosphérique, sur laquelle se réalisent les théorèmes de la géométrie non-euclidienne. […] Si donc on considère la surface comprise entre deux // méridiens, assez rapprochés pour qu’on puisse la remplacer, sur une certaine longueur, par un plan, on peut, par des morceaux de papier convenablement découpés, reproduire les trapèzes curvilignes dont la surface véritable peut être censée se composer.”
- 113.
Beltrami also offered to construct models out of copper (ibid., p. 82).
- 114.
See the figure that Beltrami drew in this letter in: ibid., pp. 81, 82.
- 115.
See: ibid., p. 82, concerning a theorem that Beltrami proved only 3 years after its discovery using the model.
- 116.
Ibid., p. 86 (cursive by M.F.): “Ce matin, avec l’aide d’un de mes élèves, qui est bon dessinateur, j’ai découpé un modèle en carton qui est assez bien réussi, et qui me servira pour un nouvel essai de construction d’une surface pseudosphérique. Vous parlez de propositions empiriques qu’on pourrait trouver par ce moyen, et vous avez parfaitement raison, car ici il s’agit de surfaces dont on ne possède pas les équations générales. Voici précisément une proposition empirique que j’ai commencé à soupçonner: Une surface pseudosphérique peut toujours être pliée de manière qu’une quelconque de ses lignes géodésiques devienne une ligne droite.”
- 117.
ibid., pp. 91–92. See also: Capelo and Ferrari (1982, pp. 246– 247).
- 118.
Boi et al. (1998, p. 202): “La superficie di rotazione seconda cui esso è ripiegato è quella cui si riferisce l’equazione (14) del mio Saggio […], e il suo meridiano è una curva trascendente la cui equazione non può aversi in termini finiti.”
- 119.
For a survey of the constructions of Beltrami’s models and the correspondence between him and Cremona, see: Capelo and Ferrari (1982).
- 120.
Ibid., p. 242.
- 121.
- 122.
See: Beltrami (1872, pp. 394, 397).
- 123.
Beltrami (1867, p. 318): “Rappresentiamo con: ds 2 = Edu 2 + 2Fdudv + Gdv 2 il quadrato dell’elemento lineare della superficie S che dobbiamo considerare. Non sarà inutile il rammentare fin dal principio che quando si riguarda una superficie come definita dalla sola espressione del suo elemento lineare, bisogna prescindere da ogni concetto od imagine che implichi una concreta determinazione della sua forma in relazione ad oggetti esterni.”
- 124.
Beltrami (1868–1869, p. 427): “It should be observed […] that, while the concepts belonging to simple planimetry receive in this manner a true and proper interpretation, since they turn out to be constructive upon a real surface, those which embrace three dimensions will only admit an analytical representation, because the space in which such a representation would come to be concretized [concretarsi] is different from that to which we generally give that name.”
- 125.
Boi et al. (1998, p. 92): “Je vous avoue franchement que, lorsque le nombre des variables dans ces expressions est plus grand que 2, leur construction dépasse, en général, les bornes de l’expérience géométrique, mais cela arrive (si je ne me trompe) de la même manière que pour la théorie de Lobatschewsky, je veux dire sans l’intervention d’une impossibilité d’ordre purement logique, ou formel.”
- 126.
Ibid., p. 204. Beltrami means the model presented in Fig. 4.11.
- 127.
Ibid., p. 205.
- 128.
Ibid (emphasis by M.F.): “Si l’on fait abstraction de ces difficultés, qu’on pourrait appeler d’ordre pratique, il me semble que la surface, logiquement considérée, soit infinie, à la même manière du plan.”
- 129.
This is also to be seen in the letters Beltrami sent to Domenico Chelini on November 12, 1867, and August 7, 1868: Beltrami talks about his own “interpretation” of the theory of Lobatschewsky, but does not mention the possibility of a material “interpretation” or a visualization. See: Enea (2009, pp. 106, 124).
- 130.
Hilbert (1901, p. 87) (emphasis by M.F.): “die GANZE LOBATSCHEFSKIJsche Ebene [...] [kann] nicht durch eine analytische Fläche negativer konstanter Krümmung auf die BELTRAMIsche Weise zur Darstellung gebracht werden.” It should be mentioned that in the 1970s, the geometer William Thurston suggested another material model of the hyperbolic plane, constructed by gluing thin pieces of paper annuli one to another (Thurston 1997, pp. 49–50), though he was not aware of Beltrami’s constructions. This gave the inspiration to Daina Taimina in 1997 to weave her crochet models of the hyperbolic plane (see: Henderson and Taimina 2001). Note that none of these two models are the same as Beltrami’s model of the pseudosphere, but rather isometric to parts of them.
- 131.
- 132.
Riemann (1868, p. 133): “Ich habe mir daher zunächst die Aufgabe gestellt, den Begriff einer mehrfach ausgedehnten Größe aus allgemeinen Größenbegriffen zu construiren.”
- 133.
Ibid., p. 135.
- 134.
Scholz (1999, p. 26).
- 135.
Riemann (1868, p. 135).
- 136.
- 137.
- 138.
- 139.
See: Riemann (1851, pp. 28–29), where a fold (Falte) occurs when dt/dz is infinite. However, other remarks regarding Falte or Umfaltung are not given anymore.
- 140.
Serret (1868, p. 296), or Serret (1880, p. 296): “Soit une portion de surface courbe terminée par un contour C; nous nommerons aire de cette surface la limite S vers laquelle tend l’aire d’une surface polyèdrale inscrite formée de faces triangulaires et terminée par un contour polygonal Γ ayant pour limite le contour C.” (Translation from: Cesari 1956, pp. 24–25).
- 141.
See: Cassina (1950), for a complete description of the Schwarz-Peano discovery and the role Genocchi played in communicating it.
- 142.
Hermite (1883, pp. 35–36).
- 143.
Kennedy (1980, p. 10).
- 144.
- 145.
- 146.
Peano (1903, pp. 300–301).
- 147.
See: Cassina (1950, p. 317).
- 148.
- 149.
In: Cassina (1950, p. 318): “Pour vous donner une idée nette de mon exemple très-simple d’un polyèdre inscrit dans la surface d’un cylindre droit, de la sorte que la surface de ce polyèdre inscrit peut devenir infiniment grande, je vous ai fait un petit modèle de papier, ci-inclus. Après avoir coupé plusieurs fois le fil gris vous aurez le modèle.” Schwarz’s usage of models is not surprising, as he initiated the transformation of the Modellkammer at the University of Göttingen into a state-of-the-art collection of mathematical tools and models used in contemporary research. See: Burmann et al. (2001).
- 150.
Cassina (1950, p. 318).
- 151.
Ibid., p. 323
- 152.
See: Greco et al. (2016, sec. 7).
- 153.
Lebesgue (1902, p. 231), footnote (*) and also: Ibid., pp. 298–299.
- 154.
Dyck (1892, p. 56) (in Nachtragkatalog).
- 155.
Wiener (1887).
- 156.
See Sect. 4.1.2.4 for an extensive survey of Hermann Wiener’s work.
- 157.
Ibid., pp. 28–30.
- 158.
According to Aristotle in De Anima, “a line by its motion produces a surface” (Aristotle 1928–1952, vol. 3, 409a).
- 159.
Gaussian curvature is defined as the product of the two principal curvatures, which are the eigenvalues of the second fundamental form of the surface in question (the second fundamental form being a quadratic form defined on the tangent plane to a point on the surface). See, e.g.: Pressely (2001, p. 147).
- 160.
- 161.
In Euler’s words: “quorum superficiem itidem in planum explicare licet,” in: Euler (1772, p. 3).
- 162.
Ibid., pp. 7, 8, 11, 27, 31 and 34.
- 163.
Ibid., p. 7.
- 164.
- 165.
Ibid., pp. 489–490.
- 166.
For example in: Mémoire sur les développées, les rayons de courbure, et les différens genres d’inflexions des courbes a double courbure (Monge 1785 [1771], pp. 517–519, 521, 536, etc.).
- 167.
It is interesting to note that in 1750, Gabriel Cramer (1704–1752) described an inflection point as a point where the curve is folded: “Le Point A est apellé Point d’Inflexion parce qu’en ce point la Courbe est comme pliée & fléchie.” (Cramer 1750, p. 402). However, Cramer never repeats this explanation, and it seems that it was not common.
- 168.
Reich (1973, pp. 297–298).
- 169.
Cajori (1929, pp. 434–435).
- 170.
Reich (2007, p. 499): “Gauß’ new interpretation of surfaces, […] now became in some sense two-dimensional manifolds. Euler instead spoke of ‘Solids the surfaces of which can be developed on the plane’ […] Euler’s surfaces were still the boundaries of solids. Gauß’ surfaces, on the other hand, stood alone, without solids and even without the surrounding space. These ideas were totally new and Euler was no predecessor. Thus, in differential geometry Euler was not important for Gauß.”
- 171.
Wiener (1887, p. 29): “[wir] müssen […] auch die abwickelbare krumme Fläche, wenn wir Eigenschaften derselben aus dem Begriffe der Abwickelbarkeit herleiten wollen, als die Grenzgestalt eines ohne Faltung oder Bruch abwickelbaren Vielflaches ansehen, und dieser Grenzgestalt uns annähern, indem wir jede Seitenfläche sich der Grenze Null annähern lassen.”
- 172.
Ibid., p. 30.
- 173.
Ibid.: “In Fig. 16 ist ein solches [Vielflach] mit vierflächigen Ecken veranschaulicht, welches man durch dreimaliges Hin- und Herbiegen eines Blattes Papier [...] herstellen kann [...]. Das Vielflach selbst ist nicht geschlossen.”
- 174.
Ibid.
- 175.
Ibid., p. 31.
- 176.
Ibid.: “Nach Art dieses abwickelbaren Vielflachs mit geschlossenen endlichen Seitenflächen kann man auch abwickelbare Flächen mit unendlich kleinen ebenen Flächenelementen bilden.”
- 177.
See: Volkert (1986, pp. 110–133) for a detailed historical discussion on this function.
- 178.
Weierstrass (1895).
- 179.
Wiener (1887, p. 33): “die Summenkurve nähert sich mit zunehmenden n der Gestalt des geradlinigen Zickzacks, erstere des regelmäßigen, letztere eines nicht regelmäßigen.”
- 180.
Ibid., p. 35: “Es ist hiermit eine nicht geradlinige abwickelbare Fläche mit unendlich kleinen (geschlossenen) Flächenelementen durch ihr Entstehungsgesetz und ihre Gleichung gegeben, welche vorgestellt, aber nicht durch Zeichnung oder ein Modell dargestellt werden kann.”
- 181.
See: Volkert (1986, pp. 129–133) for the reaction of nineteenth century mathematicians concerning the Weierstrass “monster”. Note that Wiener considered this “monster” not as a sign for the need to abandon “intuition”, but rather as what points towards its possible change (ibid., p. 131).
- 182.
This, for example, is how Deleuze might have described it (see Appendix II).
- 183.
Peano (1890b, p. 159): “[…] les fractions X et X ′, bien que de forme différente, ont la même valeur.” See also: Volkert (1986, pp. 107–108).
- 184.
Dyck (1892, p. 56) (in Nachtragkatalog).
- 185.
Ibid.: “[…] die gewöhnlichen mit unendlich langen ebenen Flächenelementen.”
- 186.
Lebesgue (1899, p. 1504): “on peut passer de cette singularité unique à des singularités en nombre infini et l’on arrive ainsi à des surfaces applicables sur le plan et ne contenant aucun segment de droite.”
- 187.
Ibid., p. 1505: “Pour prévoir ce résultat, il suffisait d’ailleurs de remarquer combien la forme des surfaces physiquement applicables sur le plan, comme celles que l’on obtient en froissant une feuille de papier, diffère de la forme des surfaces développables.”
- 188.
- 189.
A shortened version of this section appeared in: Friedman (2016).
- 190.
Geometrical Theory of the Representation of Binary Forms by Groups of Points on the Line (Wiener 1885).
- 191.
- 192.
Wiener (1882): “Es wird Sie sicher interessieren, zu hören, dass wir beabsichtigen, von den zahlreichen Fadenmodellen der Sammlung im hiesigen Polytechnikum, eine Reihe herauszugeben. Mein Vater hat wegen dieser Sache schon im Sommer mit L. Brill in Darmstadt gesprochen.”
- 193.
See: Hashagen (2003), for an in-depth biography of Dyck.
- 194.
For more details on the Munich exhibition, see: ibid., pp. 431–436 and Joachim Fischer’s introduction in: Dyck (1892, pp. viii–xxii).
- 195.
Klein’s paper is called “Geometrisches zur Abzahlung der reellen Wurzeln algebraischer Gleichungen” (ibid., 3). At Klein’s request, Dyck organized another exhibition on mathematical education, which took place in Chicago in 1893 within the framework of the mathematical conference that was held there.
- 196.
Wiener (1893a).
- 197.
Ibid., p. 52: “Wie man in einem Blatt Papier durch Zusammenfalten eine Gerade herstellen kann, und durch Aufeinanderlegen der zwei Hälften dieser Geraden eine dazu senkrechte Gerade erhält, so lassen sich ohne Hilfe von Zirkel und Lineal, allein durch Zusammenfalten aus einem Papierstreifen [...] das Quadrat und das regelmäßige Dreieck und Fünfeck herstellen; und aus ihnen können durch richtiges Aneinanderfügen dieser Vielecke und durch Zusammenkleben die 5 Platonischen Körper gewonnen werden [...].”
- 198.
Ibid., p. 53: “Am überraschendsten ist die Anfertigung des regelmäßigen Fünfecks.”
- 199.
Ibid., pp. 52–53: “Um den Winkel von 60° [...] zu construieren, drittele man den gestreckten Winkel, indem, man an irgend einem Punkte des einen Streifenrandes sowohl die eine, wie die andere Hälfte dieses Randes umlegt. Dabei entstehen zwei Knicklinien; an die erste wird die zweite Hälfte des Randes, an die zweite die erste Hälfte des Randes angelegt. Dadurch sind aus dem gestreckten Winkel drei gleiche Winkel gemacht, nämlich der zwischen den Knicklinien liegende, gleich denen zwischen, je einer Knicklinie und einer Randhälfte liegenden Winkeln. Es bedarf nur einer kurzen Übung, um die zuerst vorläufig gewählten Knicklinien mit geringer Verschiebung in die verlangte Lage zu bringen [Nr. 3.].”
- 200.
A similar construction can be found in: Flachsmeyer (2008, p. 87).
- 201.
This follows from the following theorem: In a given right triangle, the hypotenuse is twice as long as the shorter leg if and only if the angles of the triangle are 30°, 60°, 90°, where the shorter leg is opposite to the 30° angle.
- 202.
Which is on the same line as the segment EB′.
- 203.
- 204.
Wiener (1893a, p. 53): “Man mache in den Papierstreifen einen Knoten, zieht ihn Langsam zu und drücke die übereinanderliegenden Teile des Streifens in die Ebene, indem man darauf achtet, dass die beiden Enden des Streifens an der Stelle, wo sie austreten, ohne Spielraum anliegen.”
- 205.
For a proof, see Sect. 5.1.3.3.
- 206.
See: ibid., p. 54. Unfortunately, without an accompanying model or figure (both are missing from the catalog), it is almost impossible to understand how Wiener constructed this shape.
- 207.
Wiener (1893b).
- 208.
Wiener (1905b, p. 9): “The six models were prepared by the editor initially for the Mathematical Institute of the University of Halle and were exhibited at the exhibitions of mathematical models in Munich and Chicago in 1892.” [Die 6 Modelle sind vom Herausgeber zuerst für das mathematische Institut der Universität Halle angefertigt worden und waren auf den Ausstellungen mathematischer Modelle in München und Chicago im Jahre 1892 ausgestellt].
- 209.
Wiener (1893b, p. 54): “Die geschlossenen Systeme von Spiegelaxen zur Erzeugung der Bewegungsgruppen der regelmäßigen Körper.”
- 210.
Ibid., p. 55: “Diese Linien bieten ein gruppentheoretisches Interesse dar: in beiden Fallen liegt ein System von Geraden vor, von denen jede an jeder anderen gespiegelt wieder eine Gerade des Systems ergibt; man kann ein solches System als ein geschlossenes System von Spiegelaxen und die Spiegelungen an ihnen als ein geschlossenes System von Spiegelungen bezeichnen.”
- 211.
Wiener’s models were, of course, not the only models of polyhedra and regular polyhedra that were presented in Dyck’s exhibition, and other models also connected these polyhedra with group theoretic considerations. See: Dyck (1892, pp. 246–254).
- 212.
Ibid., p. 246: “Die Modelle sollen, für die Zwecke des Unterrichts, die Herstellung mit den möglichst einfachen Mitteln illustrieren.”
- 213.
- 214.
Blumenthal (1935, pp. 402–403).
- 215.
Wiener (1894, p. 70).
- 216.
Ibid., pp. 74–75.
- 217.
See, e.g., Wiener (1890a).
- 218.
In light of the research on the foundations of geometry during the last years of the nineteenth century, it is clear that Wiener’s research on the Grundbegriffe was also a part of this tradition.
- 219.
Schönbeck (1985, pp. 5–17).
- 220.
Schönbeck does not emphasize Wiener’s occupation with mathematical physical models and comments only once on this subject: Ibid., p. 3: “In the succession of the work of Gaspard Monge in France, Christian Wiener was the first to systematically design, document, and implement geometric models for university teaching in Germany. Felix Klein, according to his own testimony, thanked him for his ‘valuable suggestions’ and ‘decisive impressions.’ Hermann Wiener later carried on his father’s work.”
- 221.
It should also be noted that Schönbeck posits Wiener as putting a great emphasis on finding a purely axiomatic basis for geometry, but as we will see, Wiener was mainly, indeed almost solely, interested in modeling later in his life, abandoning almost completely the theoretical approach.
- 222.
- 223.
See: Wiener (1890a), where every translation [Verschiebung] and rotation [Drehung] on the plane are decomposed into a composition of two reflections [Umwendungen], which I translated here as “half turn”; see an explanation in the following passage. See also: Wiener (1890b, p. 74), where every transformation that is an involution is replaceable by a composition of Umwendungen.
- 224.
See the above discussion on: Wiener (1894), or the theorem on the “isogonalen Punktverwandtschaft” (ibid., 75).
- 225.
As presented in: Schönbeck (1985, Sect. 3.3.3).
- 226.
- 227.
The term itself was not Wiener’s invention. See, for example: Henrici and Treutlein (1881, pp. 17–27).
- 228.
Wiener (1890a, p. 14): “die halbe Umdrehung um eine Axe.”
- 229.
Ibid., p. 16: “durch Spiegelung an der Geraden.”
- 230.
“Der Begriff ‘Spiegelung an einer Geraden’ gestattet nicht nur eine leichtere Ausdrucksweise, sondern ist, wie ich meine, auch anschaulicher als der der Umwendung. Doch glaubte ich auf diese Vortheile verzichten zu müssen, um selbst den Schein eines der Mechanik fremden Begriffs zu vermeiden.” (ibid., p. 16, footnote 1)
- 231.
Schönbeck (1985, Sect. 3.3.1).
- 232.
Wiener (1890b, p. 74), footnote 1: “Um diese wichtigen Verhältnisse in die Anschauung aufzunehmen, verfertige man sich ein Modell, bestehend aus einem Parallelogramm A 1 B 1 A 2 B 2, das man aus starkem Papier ausschneidet und längs der Diagonale A 1 A 2 umbiegt.”
- 233.
Wiener intended to show the fundamental relations between the basic elements (i.e., the reflections, see, e.g., Wiener (1891a), in which an algebraic group theoretic analysis of the relations [Verwandtschaften] between the generators of a group of involutions is performed).
- 234.
Wiener (1890a, p. 16): “[The] extension [of the geometry of reflections] is not without application to relationships found in nature; the recent studies about the crystal structure show the need to take the symmetrical systems also into consideration” [diese Erweiterung [ist] nicht ohne Anwendung auf Verhältnisse, die in der Natur vorkommen; die neueren Untersuchungen über die Kristallstruktur zeigen die Notwendigkeit, auch symmetrische Systeme mit in Betracht zu ziehen].
- 235.
Wiener (1890c, p. 248): “Durch die Auffassung der Bewegung als Verwandtschaft wird die Möglichkeit zu Erweiterungen gegeben. So kann man sich eine Verwandtschaft zwischen den Punkten des Raumes vorstellen, vermöge welcher, wie bei der vorhin geschilderten Congruenz, alle gegenseitigen Abstände der Punkte des Systems S2 denen der entsprechenden Punkte in S1 gleich geblieben sind, doch so, dass Jedem Tetraeder des einen Systems kein congruentes, sondern ein symmetrisches des anderen entspricht.” Note that here, symmetry is a relation between two bodies.
- 236.
With respect to folding-based geometry, this approach appears at the beginning of the twentieth century (see Sect. 5.1.1). Also, although Wiener does indicate that through two points A 1, A 2 there is only one half turn, and that one can find a unique half turn, switching between A 1 and A 2, that is perpendicular to line A 1 A 2 (see Wiener 1890b, p. 74, footnote 1), he does not formulate these facts in terms of axioms or basic operations.
- 237.
Wiener (1893a, p. 52): “One can produce a line in a piece of paper through folding.” [Man [kann] in einem Blatt Papier durch Zusammenfalten eine Gerade herstellen].
- 238.
Klein (1895, pp. 32–33).
- 239.
Klein (1897, p. 42).
- 240.
Sattelmacher (2013, p. 311), footnote 45.
- 241.
Wiener (1901).
- 242.
- 243.
Wiener (1885, p. 2): “I renounce completely the advantage of geometrical intuition, in order to allow the strictly systematic construction to be advanced all the more clearly.” [verzichte ich vollständig auf den Vortheil der geometrischen Anschauung, um den streng systematischen Aufbau um so klarer hervortreten zu lassen].
- 244.
Wiener (1890b, p. 87): “Die Methode, die wir im bisher Gesagten befolgt haben, war eine wesentlich anschauliche. Von dem sich selbst aufdrängenden Formalismus haben wir insofern Gebrauch gemacht, als es zur Abkürzung des Ausdrucks unumgänglich nöthig war, doch nur so, dass wir bei jedem Zeichen eine anschauliche Operation im Sinne hatten.”
- 245.
Wiener (1890c, p. 246): “Every step that we make in the process of computation can also be represented by the geometrical constructions [Gebilden], so that we will deal with an intuitive method.” [Jeder Schritt, den wir dabei rechnerisch machen, lässt sich auch an den geometrischen Gebilden anschaulich vorfolgen, so dass wir es also mit einem anschaulichen Verfahren zu thun haben].
- 246.
For example, Wiener (1891b).
- 247.
Wiener (1901, p. 1): “Am bequemsten gewinnt man eine Übersicht an der Hand von Fadenmodellen dieser Kegel. [...] Wenn so diese Modelle mancherlei unmittelbar abzulesen gestatten, so regen sie andrerseits Fragen an, deren Beantwortung ein tieferes Eindringen verlangt.”
- 248.
Wiener and the mathematical institute of the Technischen Hochschule in Darmstadt had already produced more than a 180 models in 1904 (Wiener 1905a, p. 746).
- 249.
Wiener (1913, pp. 295–296): “Sind aber wirklich jene Anschauungsformen, die aus Natur und Leben geschöpft sind, durch die Formeln überflüssig geworden? [...] Die Möglichkeit, durch Abstraktion Begriffe zu bilden, hängt von der Fähigkeit ab, den Stoff mit sich herumzutragen, und diese wird im Unterricht bis zu einem gewissen Grade beim Lösen von Aufgaben durch Zeichnung oder Rechnung erworben, aber weit mehr durch Anschauungsmittel aller Art [...].” Compare also Sattelmacher (2013, p. 308).
- 250.
Wiener (1905a, p. 739): “Daraus entspringt die Forderung, die man an geometrische Modelle stellen muß, wenn sie auf das Auge wirksam sein sollen: sie dürfen nicht aus Flächen, sondern sollen aus Linien bestehend dargestellt werden, d. h. das ganze Modell bestehe aus Draht-Stäben und -Kurven, vielleicht auch mit eingespannten Fäden.”
- 251.
Ibid., p. 746.
- 252.
Treutlein and Wiener (1912, p. 38) (Modelle 132, 133).
- 253.
Ibid.: “Durch Grenzübergang findet man daraus auch solche Flachen mit unendlich kleinen, aus geschlossenen ebenen Vielecken bestehenden Flächenelementen.”
- 254.
Ibid.
- 255.
Wiener (1906, p. 4).
- 256.
This following subsection is based on: Friedman (2017). I thank the editors of the journal Theory of Science for giving me permission to include materials from this paper in this book.
- 257.
- 258.
For an extensive analysis of the beginnings of stereochemistry, see: Ramberg (2003).
- 259.
Pasteur (1848).
- 260.
Today called l-(+)-Tartaric Acid (or dextrotartaric acid) and d-(-)-Tartaric Acid (or levotartaric acid).
- 261.
Hoff (1874a).
- 262.
Le Bel (1874). Note that Le Bel did not draw any models in his paper.
- 263.
Several models for representing molecules existed before and during the time of Van ’t Hoff’s models. Among these models, the most famous one is the Stick and Ball model of Kekulé, which definitely influenced Van ’t Hoff. Other models were the croquet ball models of August Wilhelm Hofmann, the brass strip models of James Dewar or Crum Brown’s Structural Diagrams. All of these models, except Kekulé’s, were two-dimensional and were not intended to represent a three-dimensional structure of a molecule. See: Meinel (2004).
- 264.
For example, see: Rocke (2010, pp. 228–260). “[A]t least four other chemists explicitly invoked the carbon tetrahedron during the 1860s: Pasteur in 1860 […], [Aleksandr] Butlerov in 1862, [Marc Antoine] Gaudin in 1865, and [Emmanuele] Paternò in 1869. [Johannes] Wislicenus stressed the need to consider three-dimensional spatial considerations for certain molecules at least three times before 1874.” (ibid., p. 252). Thus, for example, Paternò “had [already] used the concept of a tetrahedral carbon atom for the explanation of a case of isomerism.” (ibid., p. 251) See: Paternò (1869).
- 265.
Ramberg (2003, p. 53).
- 266.
Hoff (1874b).
- 267.
These “glyptic formulas” are, in fact, sculpted three-dimensional models of molecules made of croquet balls (representing atoms), connected by metal rods. However, these sculptures were actually flat. See: Meinel (2004, pp. 250–252).
- 268.
- 269.
Ramberg and Somsen (2001, p. 67).
- 270.
Ibid., pp. 67–68.
- 271.
Ibid., p. 71. Cf. the French 1874 translation, Hoff (1874b, p. 452): “la dissemblance de ces figures, dont le nombre se réduit à deux, annonce un cas d’isomérie, qui n’est pas impliqué dans le mode ordinaire de représentation.”
- 272.
Hoff (1875).
- 273.
Ibid., p. 7, footnote 1: “Il y aura peut-être quelque difficulté à suivre mon raisonnement; je l’ai senti moi-même, et je me suis servi de figures en carton pour faciliter la représentation.”
- 274.
V. M. (Anonymous) (1876, p. 1): “der Brochüre des Herrn van ’t Hoff ist eine grosse Anzahl aus Pappe gefertigter Modelle beigegeben, welche das Verständniss wesentlich erleichtern.” The article is kept in the Archives of the Museum Boerhaave, Leiden (arch 208).
- 275.
Ibid.
- 276.
Spek (2006).
- 277.
Rocke (2010, p. 254): “[V]an ’t Hoff proposed in 1874 that molecular asymmetry, thus nonsuperposability, is established only by the chemical distinguishability of the four geometrically indistinguishable vertices of the regular tetrahedron.” (emphasis in the original).
- 278.
Spek (2006, p. 163).
- 279.
- 280.
- 281.
See: Ramberg and Somsen (2001, p. 55).
- 282.
Ramberg (2003, p. 358). See also: ibid., p. 86, indicating that Hermann was also “mathematically inclined,” as Wislicenus described him.
- 283.
Hoff (1877, p. 46): “Zur Erleichterung der Vorstellung ist es erforderlich, die im ersten Abschnitt beschriebenen Figuren durch Modelle sich zur directen Anschauung zu bringen. In den folgenden Figuren sind die Netze der betreffenden sterischen Figuren entworfen. Dieselben werden am besten aus mässig dickem Cartonpapier ausgeschnitten. Die punktirten Linien werden mit einem scharfen Federmesser vorsichtig angeritzt. Durch Umknicken längs der geritzten Linien wird die Figur räumlich zusammengebracht und durch Festkleben der seitlich angebrachten Ausschnitte an die innere Seite der Flächen zusammengehalten” (cursive by M.F.).
- 284.
- 285.
Van ’t Hoff wrote the following to Bremer: “Stellen de drie driehoekjes in een der grootere, wit, ≡ en zwart, de groepen H, OH en CO 2 H voor, in ’t geval van Erythit H, OH en CH 2 OH, zoo komt men tot de 4 denkbare isomeren, als men de figuur langs AB dubbel omvouwt, zoodat de driehoekjes twee aan twee op elkaat vallen” (in: Jorissen 1924, p. 496). However, Bremer never folded the letter: no crease is apparent in the letter itself (the letter can be found in the Archives of the Museum Boerhaave, Leiden (arch 208)).
- 286.
Ibid., p. 497.
- 287.
Hoff (1877, pp. 48, 50).
- 288.
Ibid., p. 52: “Wenn man das Netz in der Weise zusammenfügt, dass einmal die obere Seite, das andere Mal die untere Seite zur Außenfläche der entstehenden Figur wird, so erhält man zwei enantiomorphe Tetraeder, welche die beiden Isomeren der oben erwähnten Combination darstellen.”
- 289.
Ibid., p. 50.
- 290.
Ramberg (2003, p. 85).
- 291.
Hoff (1887, pp. 26–27): “Pour bien saisir la différence des deux groupements dont il s’agit on peut faire usage de deux tétraèdres en carton, coupés et collés d’après les Fig. 6 et 7; les quatre groups différents supposés aux sommets des tétraèdres sont indiqués par des couleurs […].”
- 292.
Hoff (1894, p. 6): “Zur Erläuterung der Sachlage durch das Modell kann man sich der Tetraeder aus Pappe bedienen, bei denen die verschiedenen Gruppen durch angeklebte Käppchen aus farbigem Papier erläutert werden [...].”
- 293.
Ibid., pp. 37 and 38.
- 294.
See: Jorissen (1893). The lecture notes by Jorissen can be found in the Archives of the Museum Boerhaave, Leiden. The lecture notes to which I refer to have the title “Theoretische Chemie,” notebook num. IV (inside the third book), and contain thirty-three written pages on the right side of the notebook. The left side usually remains blank, but sometimes contains either drawings or remarks.
- 295.
Ibid., p. 2, left side.
- 296.
- 297.
Wislicenus (1887, pp. 12–13, 16).
- 298.
Jorissen (1893, pp. 30–31).
- 299.
Ibid., p. 30 (left side)
- 300.
Ramberg (2000, p. 39).
- 301.
Indeed, all three edges coming out of the nitrogen atom in Fig. 4.34 are on the same plane.
- 302.
Sachse (1888, p. 2531), footnote 2: “Da selbst die klarste Beschreibung dieser räumlichen Verhältnisse keinen genügenden Ersatz eines Modelles zu bieten vermag, Zeichnungen andererseits höchst complicirt ausfallen würden, so ist im Interesse möglichster Kürze alles folgende unter der Voraussetzung dargestellt, dass der geehrte Leser die weiteren Erörterungen durchweg am Modell verfolgt.”
- 303.
Ibid.
- 304.
Ibid., p. 2532. Sachse does not even supply a drawing in this paper, and gives the reader—in contrast to his own objection!—a mere description of how to build the model. However, in his paper from 1890, Sachse does give a drawing of his model (see Fig. 4.36).
- 305.
See: Ramsay (1975, p. 91).
- 306.
Baeyer (1885).
- 307.
Ibid., p. 2278: “Eine Vorstellung von der Bedeutung dieses Satzes [regarding the 109°,28′ normal angle] kann man sich leicht machen, wenn man von dem Kekulé’schen Kugelmodel ausgeht, und annimmt, dass die Drähte, einer elastischen Feder ähnlich, nach allen Richtungen hin beweglich sind.”
- 308.
The planar structure for the carbon atoms that Baeyer proposed for the molecules C 4 H 8 and C 5 H 10 was also wrong: i.e., the ring of the carbon atoms of both of these molecules is non-planar. For C 4 H 8, the ring of carbon atoms has a folded or “puckered” conformation, and for C 5 H 10, the ring has an unstable puckered shape that fluctuates. The ring of carbons for C 3 H 6 is planar, but this molecule is highly unstable. Note the molecule propene has the same chemical formula as C 3 H 6 but is non-planar, hence these two molecules are isomers.
- 309.
Sachse (1890, p. 1368): “Es ist also dabei stets ein gewisser Widerstand zu überwinden.”
- 310.
Ibid., p. 1368, footnote 1: “Für unser Modell wurde die am leichtesten darstellbare Phase gewählt.”
- 311.
Ibid., p. 1365: “Aus der Lösung dieser Aufgabe, welche zu einem System von 3 Gleichungen führt, ergeben sich zwei verschiedene Configurationen. Glücklicherweise haben dieselben gewisse geometrische Eigenschaften, welche eine Veranschaulichung dieser Systeme durch bequem herzustellende Modelle gestatten.”
- 312.
Sachse (1892, pp. 228–241).
- 313.
Ibid., p. 229: “Wir wissen also jetzt, dass für den drei-, vier-, fünfgliedrigen Methylenring eine solche Konfiguration, in welcher gar keine Abweichungen von der natürlichen Gleichgewichtslage vorkommen, nicht möglich ist. Für den sechsgliedrigen Ring dagegen bleibt vorläufig die Frage noch offen.”
- 314.
Ibid., p. 231.
- 315.
See: ibid., p. 238.
- 316.
Ibid., p. 241. It is not clear whether by “models,” Sachse refers to Kekulé’s examples, since on the one hand, Sachse (in this passage) also talked about Kekulé’s “Kugelmodelle” (i.e., the ball and stick models), and on the other, as we saw, Baeyer was familiar with Kekulé’s models. It may be that Sachse referred to his own models.
- 317.
See: Russel (1975, pp. 164–169).
- 318.
Hantzsch (1893, p. 98).
- 319.
Ibid., p. 97: “Diese Verhältnisse lassen sich auch durch die auf pag. 93 befindlichen Symbole anschaulich darstellen.”
- 320.
Jorissen (1893, p. 23) (right side).
- 321.
Mohr (1918).
- 322.
The crystal structure of diamond was the first crystal structure to be determined by X-ray diffraction. William Lawrence Bragg and his father William Henry Bragg published this in 1913.
- 323.
Ibid., p. 318.
- 324.
See: Russel (1975, p. 164).
- 325.
The 1892 paper of Sachse is mentioned only once; see: Mohr (1918, p. 316), footnote 1.
- 326.
Ibid., p. 351.
- 327.
Ibid.
- 328.
Russel (1975, pp. 170–175).
- 329.
Ibid., p. 175.
- 330.
See: Klein (2003, p. 246): “[T]he syntax of paper tools—their visual form, rules of construction and combination, maneuverability—shapes scientists’ production of chains of representation […] formula equations came into the fore as a means of justification.”
- 331.
The similarity to Dupin’s models (see Sect. 4.1.2.1) is noticeable, in the sense that in both models, the paper that was read was also to be folded. However, Sachse’s aim was to represent a molecule via the folded models, whereas for Dupin, the folded polyhedra were not a representation but rather the desired object.
- 332.
Cf. ibid., pp. 23–35.
- 333.
Cf. Francfouer (2000).
- 334.
Berzelius (1814, p. 52) (emphasis by M.F.) Jacob Berzelius (1779–1848) is one of the founders of modern chemistry. As Ursula Klein describes, “[c]hemical formulas, such as H 2 O for water or H 2 SO 4 for sulfuric acid, were introduced in 1813 by […] Berzelius.” (Klein 2003, p. 2). See Klein’s book (2003) for the epistemological consequences of the Berzelian formulas.
- 335.
Regarding the relation of chemists to mathematical arguments, see: Ramberg (2003, p. 112), where Wislicenus is described as considering “Van ’t Hoff’s theory of the asymmetric carbon atom a fully justified mathematical expansion of our chemical views”; and: ibid., pp. 189–191, concerning Victor Meyer; Meyer “oscillated between advocating the reduction of chemical theory to mathematical physics and then advocating its theoretical autonomy” (ibid., p. 189). Cf. also: Nye (1992).
- 336.
Sachse (1892, p. 203): “Lässt man, wie es bisher geschehen, die Resultate gelten, die man erhält, wenn man die bekannten Kekuléschen Kugelmodelle zu Ringen zu vereinigen sucht, so legt man damit—bewusst oder unbewusst—gewisse spezielle Eigenschaften, die den Modellen anhaften, den Atomen bei.”
- 337.
Ibid.: “Ein sicheres Vorwärtsschreiten in diesen Gebieten ist nur dann möglich, wenn man Schritt für Schritt sich auf dies notwendige Maß der Annahmen beschränkt, und demgemäß für ein jedes Theorem festzustellen sucht, welche Voraussetzungen es zu wirklich unumgänglichen Bedingungen hat.”
- 338.
“[…] [Die] Sprache der Mechanik […], in die sich ja die Sprache unserer Wissenschaft schliesslich auflösen soll.” In: Sachse (1893, p. 185).
- 339.
Cf. Francoeur (1997).
- 340.
Brill (1874, p. 2).
- 341.
See, e.g.: Mancosu (2005, pp. 14–17).
- 342.
- 343.
Fröbel (1826, p. 60): “Innerliches äußerlich, Äußerliches innerlich zu machen, für beydes die Einheit zu finden: dieß ist die allgemeine äußere Form, in welcher sich die Bestimmung des Menschen ausspricht; darum tritt auch jeder äußere Gegenstand dem Menschen mit der Anforderung entgegen, erkannt und in seinem Wesen, [...] anerkannt zu werden.”
- 344.
Ibid., p. 249: “Was ist Mathematik ihrem Wesen, ihrer Entstehung, ihrer Wirkung nach?—Sie ist als Erscheinung der Innen und der Außenwelt dem Menschen und der Natur gleich angehörig.”
- 345.
Ibid., p. 247: “[…] die Erscheinungen der Natur […] nicht im Traume siehst du sie; sie ist bleibend, überall umgibt sie dich; […] sie ist fest, Festgestalten bildet sie und auf einer Kristallwelt ruht sie.”
- 346.
Heiland (1993, p. 481).
- 347.
Ibid., p. 485: “Anstalt zur Pflege des Beschäftigungsbetriebes für Kinder und Jugend.”
- 348.
Cf. Brosterman (1997, p. 58): “The three-dimensional forms of the first six gifts inhabited the realm of objects. Modeled into simple expressions of other things, or organized as mathematical and artistic arrays, solidity was confirmation that their existence was actual. Transition to the seventh gift was considered a profound conceptual leap. Parquetry was the first of the gifts that Fröbel considered truly abstract, as it was used to create two-dimensional pictures of things rather than tangible things themselves.”
- 349.
See: Meinel (2004, pp. 267–268).
- 350.
Heiland (1993, p. 478): “In Frankfurt, he had already become familiar with Fichte’s writings, but Schelling’s speculative philosophy of identity and objective idealism appealed even more to him.”
- 351.
Hoffmann and Wächter (1986, p. 355): “Es herrscht nur ein Grundgesetz durch das ganze Universum. [...] Dieses Gesetz [ist] das Gesetz des + und—oder des Gegensatzes. Dieses Gesetz tritt aus der Mitte nach allen Seiten zugleich oder sphärisch heraus. Diesem sphärischen Gesetz ist unterworfen alles, was ist.”
- 352.
Fröbel (1826, p. 61): “Jedes Ding und Wesen, alles aber wird nur erkannt, [...], als die Verknüpfung mit dem Entgegengesetzten und die Auffindung des Einigenden geschieht.”
- 353.
For Haüy’s and Weiss’s conceptions of and work on the crystal and crystallography, see: Burckhardt (2013, pp. 16–47).
- 354.
Haüy (1801).
- 355.
See: Haüy (1804–1810).
- 356.
As Stapfer wrote to Pestalozzi, “H[er]r Weiss scheint Ihre Anstalten in Iferten mit nicht gemeinem Interesse und Scharfsinn beobachtet zu haben.” (Horlacher and Tröhler (2010, p. 423)
- 357.
A full survey would take us out of the framework of this book. See, however: Wagemann (1957, pp. 169–199).
- 358.
Several of Fröbel’s models of crystals can be seen in the Fröbel Museum in Bad Blankenburg, Germany.
- 359.
See: Weiss (1815, pp. 290–291).
- 360.
Ibid., p. 290: “Noch besser würde der Name kugliches oder Kugelsystem, griechisch sphäronomisches oder sphäroedrisches System gebraucht werden können.”
- 361.
Ibid., p. 289.
- 362.
Ibid. (emphasis by M.F.): “so wird zuförderst, das reguläre System den nicht-regulären entgegenzustellen, der schicklichste Ausgangspunkt für die Entwickelung der natürlichen Abtheilungen der Krystallisationssysteme seyn.”
- 363.
Hon and Goldstein (2008, p. 190).
- 364.
Ibid., p. 193: “While in 1790 Haüy speaks of symmetry as a geometrical expression of a relation of the parts of the whole, in 1791 he considers symmetry in the ‘disposition’ of the parts such that several axes can be identified.”
- 365.
Ibid.
- 366.
Wagemann (1957, p. 182).
- 367.
Ibid., pp. 182–185.
- 368.
Eduard Spranger describes two types of an apriori within Weiss’s approach: the first, a purely mathematical one, in which the solids would be categorized according to the possible axes systems; the second, a mechanical one, in which it is assumed that the crystals are created through the effect of a formative force. See: Spranger (1951, p. 45).
- 369.
In: Lange (1862, pp. 111–112): “[…] [die] Überzeugung [lag] in mir [...] daß nämlich selbst in diesen [...] sogenannten todten Steinen und Massen noch jetzt fortentwickelnde Thätigkeit und Wirksamkeit stattfindet. In der Mannigfaltigkeit der Form und Gestaltung erkannte ich ein auf das Verschiedenste modifiziertes Gesetz der Entwicklung und Gestaltung.”
- 370.
Fröbel (1826, p. 252): “Die Mathematik ist darum auch weder dem wirklichen Leben etwas Fremdes, noch aus demselben erst Abgezogenes; sie ist der Ausdruck des Lebens an sich, und darum ist ihr Wesen im Leben und durch sie das Leben erkennbar.”
- 371.
See: Spranger (1951, p. 47).
- 372.
See also: Heiland (2003, pp. 185–187).
- 373.
See, for example: Wirth-Steinbrück (1998).
- 374.
Similar to Louis Dupin’s models (see Sect. 4.1.2.1), Fröbel also developed as one of the gifts the ‘self-learning cube,’ a “mathematical” cube [Der “selbstlehrende” Würfel]. On the faces of the cube, information on its mathematical properties was written. However, in contrast to Dupin’s models, the children did not have to fold the cubes, and the cubes were not made of paper.
- 375.
Heiland (1993, p. 482).
- 376.
Fröbel and Bertha von Marenholtz-Bülow (1851, p. 730): “Andere werden sagen: das sind leere Formeln, mit welchen man keinen Hund aus dem Ofen lockt. Formeln sind es wohl; allein keine leeren; wer dieß sagt, weiß nicht,—daß die Erkenntniß unseres ganzen Weltalls, daß all unser positives mathematisches Wissen in allen Verzweigungen seiner Anwendung, zuletzt auf Formeln beruhet; daß ihr Wesen nur aus klar erkannter, in höchster Allgemeinheit aufgefaßter Gesetzmäßigkeit besteht. Und jeder Kundige weiß, was in der Weltallkunde von einer größeren oder geringeren Einfachheit einer Formel abhängt.”
- 377.
Fröbel and Leonhard Woepcke (1845, Abschrift, p. 19): “Eine weitere neue und große, so unterhaltende als belehrende und nützliche Abtheilung der Kinderbeschäftigungen ist das Umwandeln der Formen und zwar […] aus biegsamen Flächen, aus Papier; das Brechen und Falten verschiedener Formen und Gegenstände aus einer und eben derselben Geviertfläche, oder was das Gleiche ist: aus mehreren gleich großen Geviertflächen.” (cursive by M.F.)
- 378.
Fröbel and Berthold Auerbach (1847, Literaturliste and Spielmaterialien, p. 1).
- 379.
Fröbel (1874, pp. 371–388). The title in German is: “Anleitung zum Papierfalten. Ein Bruchstück. Eine entwickelnd-erziehende und unterhaltend-belehrende Kinderbeschäftigung für Kinder von 5 bis 7 Jahren und darüber, unter eingehender Mitwirkung von leitenden Erwachsenen.”
- 380.
“Leitender Faden bei den Beschäftigungsmitteln im Allgemeinen, als übersichtliche Einleitung in das Besondere.”
- 381.
“Das Papier-falten als Beschäftigungsmittel, von der Geviertfläche oder Geviertform ausgehend.”
- 382.
See: Heiland (1998, p. 155).
- 383.
Fröbel (1874, p. 388): “[…] ist nun die erste Grundform gegeben, aus welcher sich nun auch die ersten Hauptformen mit Nothwendigkeit entwickeln [...].”
- 384.
For a more elaborate study how napkin folding prompted the integration of paper folding within recreational mathematics, see: Friedman and Rougetet (2017).
- 385.
See: Sallas (2010).
- 386.
For more on Mattia Giegher, see: ibid., pp. 60–62, 70–72.
- 387.
- 388.
Harsdrffer already considered folding as a tactile practice, which transmits knowledge, which cannot be transmitted via symbols or visual means. See: Friedman and Rougetet (2017, p. 20).
- 389.
Sallas (2010, pp. 36, 44, 46).
- 390.
Ibid., pp. 116–117.
- 391.
Esveldt (1746).
- 392.
Ibid., pp. 127–137. For example, the author instructs as to the way in which to fold a square paper, where one of the resulting shapes is a rhombus (the resultant and the initial shape shown in Fig. 4.48): “Vouwt ’er dan een kruis in a, b, c, d. Neemt dan de hoek e. en vouwt ze na g. zo dat het punt e, tegen het midden van de vouw of boven verbeelde streep g, aankomt. Neemt dan de hoek h, en vouwt ze na i, zo dat het punt e. tegen het midden van de vouw of boven verbeelde streep i, aankomt, zo bekomt gy boven een punt a. Vouwt dan het punt a, tegens b, het middelste van de onderste linie dan is het Servet half toegeslagen. Neemt dan de hoek c, en vouwt die agter om tegens k. Neemt dan de hoek d, en vouwt die agter om tegens m, zo bekomt gy een vierkante ruit n, n, n, b. want de middelste n, moet men zig verbeelden dat in het middelpunt van het kruys staat, het geen alhier niet netter heeft konnen verbeeld worden, en de aan wederzyden van onderen uitstekende hoeken o, o, vouwt men mede agter om, gelyk in deeze Figuur verbeeld staat.” (ibid., p. 129).
- 393.
See: Sallas (2010, p. 49) for the geometrical (visual) instructions for folding various shapes in the 1833 Enzyklopädie der sämtlichen Frauenkünste of Caroline Leonhardt-Lyser and Cäcilie Seifer.
- 394.
As indicated in: Heerwart (1895, p. 4).
- 395.
From this series of folded figures from about 1810 to 1812, there are still 20 copies in the German Nationalmuseum in Nuremberg.
- 396.
- 397.
On the history of paper folding in Germany, surveying, among others, Giegher’s and Fröbel’s works, see: Lister (2003/2004).
- 398.
See, e.g.: Sallas (2010, pp. 94–95).
- 399.
Cf. Münkner (2008, pp. 151–155) and especially p. 155, on a discussion on folding of the baptismal letters as a bond between the concrete conceivability and the manual manipulation (cultural techniques from the order of techné) and between saying, showing and writing (as symbolic cultural techniques from the order of poiésis).
- 400.
Sallas (2010, p. 23).
- 401.
Indeed, Fröbel’s own baptismal letter was folded in this shape (the letter can be seen in the Fröbel Museum in Bad Blankenburg, Germany). The “Blintz” and the “Doppel-Blintz” shapes were also common basic forms in napkin folding, see: ibid., p. 49.
- 402.
See: Heiland (1998, pp. 171–173) for the complete analysis.
- 403.
Fröbel (1874, p. 372).
- 404.
It is important to note that Fröbel only numbered the first six “Spielgaben.” The numbering of the later occupations was given by his followers and was not uniform during the second half of the nineteenth century.
- 405.
Ibid., p. 373.
- 406.
Ibid., p. 374: “Cutting out is, therefore, the linking […] of a continuous with the parts and forms, so that a continuous remains, but in a relation to its division.” [Das Ausschneiden ist also das Verknüpfende [...] eines Stetigen mit dem Theilen und Formen, so, daß ein Stetiges zwar bleibt, aber in den Verhältnissen seiner Trennung].
- 407.
Fröbel’s Ausschneiden is, in fact, situated within a much larger tradition, which is known contemporarily as the fold and cut problem. The problem is as follows: folding a piece of paper flat and making one complete straight cut, can one, after unfolding it, obtain any shape whatsoever? The answer is affirmative, and was proved through two methods: the “straight skeleton” method (see: Demaine et al. 1999, Demaine et al. 2000a) and the “disk packing” method (see: Bern et al. 2001). For a detailed survey of the two methods, see: Demaine and O’Rourke (2007, Chapter 17). As Demaine and O’Rourke describe, whereas Martin Gardner first posed the problem explicitly in 1960 in his Mathematical Games series in Scientific American concerning the possible obtained shapes, it was also considered by the magicians Gerald Loe and Harry Houdini. The first published reference to the fold and cut problem from the eighteenth century is a Japanese book, Wakoku Chiyekurabe (Mathematical Contests), by Kan Chu Sen, published in 1721. See: ibid., p. 254. See also Sect. 5.1.3 on the methods of Ball and MacLoed, who, in 1892, used the fold and cut techniques for the education of geometry.
- 408.
Fröbel (1874, p. 375): “[…] von […] innerste[m] Entwickelungsgrund und Gesetz […].”
- 409.
Ibid., p. 376: “Das Merkwürdige dabei ist, daß aus dem stetig Ungeformten, oder aus dem ungeformt Stetigen durch drei Brüche und drei Schnitte die gesetzmäßigste und einfachste Form, das Geviert, entsteht. [...]. [Es] sollte blos bewiesen werden, daß aus dem Ungeformten durch gesetzmäßige Trennung das Geformte und hier, in diesem besonderen Falle, das Geviert hervorgeht.”
- 410.
Ibid., p. 379.
- 411.
Ibid., p. 383: “gleiche Form bedingt nicht gleiche Größe, oder bei gleicher Form kann die Größe sehr verschieden sein […].”
- 412.
Ibid., p. 386: “Viertel ist gleich Viertel.”
- 413.
The resulting shape of the folded paper is identical to the “Doppel Blintz” folding.
- 414.
Fröbel (1874, pp. 387–388).
- 415.
For the importance of the principle of transformation [Umwandlung] for Fröbel, as one of the basic principles that he adopts and adapts from crystallography, see: Wagemann (1957, pp. 186–188).
- 416.
Fröbel (1874, pp. 386–387).
- 417.
Ibid., pp. 382, 383.
- 418.
Ibid., p. 382.
- 419.
Wagemann (1957, p. 252).
- 420.
This also stands in contrast to napkin folding and the folding of the German baptismal certificates (Taufbriefe) in the shape of the “Blintz” or the “Doppel Blintz,” which always began with a square or a rectangle.
- 421.
Fröbel (1874, pp. 375, 378).
- 422.
See, for example: Soëtard (2003), and later in this section.
- 423.
In the United States, the kindergarten movement was much more influential. In 1856, Margarethe Schurz opened the first German-speaking kindergarten in Watertown, Wisconsin, and in 1860, Elizabeth Peabody opened an English-speaking kindergarten in Boston. In 1873, Susan Blow opened the first public kindergarten in the United States, and in 10 years time, every public school in St. Louis had a kindergarten. Maria Kraus-Boelté was also one of the pioneers of Fröbel-style education in the United States. However, concentrating on the developments in the United States and the place of folding within the reconceptualization of Fröbel’s ideas would take us outside the framework of this chapter and is deserving of a more elaborate study; see, however: Ball (1892a), MacLoed (1892); and Sect. 5.1.3.2 regarding the way in which these two American books handled the folding of the pentagon. Another important work that is outside the scope of this chapter is the work of the Dutch educator Elise van Calcar (1822–1904): De kleine papierwerkers (Calcar 1863), which presents, especially in the first volume (Wat men van een stukje papier al maken kan: het vouwen), a development of Fröbel’s ideas regarding folding and mathematics, especially with respect to other geometrical concepts and forms (such as parallel lines, equilateral and isosceles triangles and investigation of rectangles). See also: Drenth and Essen (2004) and Bakker (2017).
- 424.
As Helmut Heiland describes, during the second half of the nineteenth century, Fröbel’s ideas were conceptualized in a pragmatic way and were praxis-oriented. The gifts and the occupation were the materials with which one practiced in the pre-school (see: Heiland 2001). Obviously, other books promoting Fröbelian paper folding were published in Germany during the second half of the nineteenth century (see, for example: Barth and Niederley 1877), but reviewing all of them would take us beyond the scope of this chapter. See also: Rockstein (2006) and Schauwecker-Zimmer (2006) on Fröbel’s reception in Thüringen resp. in Sachsen and Bayern.
- 425.
Goldammer (1874).
- 426.
On Marenholz-Bülow’s work, see: Heiland (1990).
- 427.
Goldammer (1874, p. 7).
- 428.
Ibid., p. 121: “Fast Alles, was wir durch frühere Beschäftigungen an mathematischen Anschauungen gewonnen haben, findet sich im Faltblatt wieder. Es ist ein wahres Compendium elementarer Mathematik und deshalb mit Recht als zweckmäßigstes Hilfsmittel für den mathematischen Unterricht, so weit sich dieser auf die Gewinnung mathematischer Anschauungen beschränkt und von der Erwerbung mathematischer Begriffe fern hält, empfohlen worden.”
- 429.
Ibid., pp. 124–125.
- 430.
It should be remarked upon that Fröbel did not invent (or fold) what is now called the Fröbelstern (or the Advent star). One of the first appearances of this three-dimensional form appears in: Barth and Niederley (1877, pp. 22f). (Ibid., p. 22: “Wir bringen endlich an den Schluß dieses Abschnittes die Anleitung zur Verfertigung eines Sternes [...].”)
- 431.
See: Marenholtz-Bülow (1887, pp. 86–93).
- 432.
Ibid., p. 86.
- 433.
Ibid.
- 434.
Ibid., p. 89.
- 435.
Ibid., p. 90: “An der Hand des Gesetzes der Verknüpfung der Gegensätze lassen sich noch bedeutend mehr Formen finden [...]”.
- 436.
Heiland (1993, p. 489).
- 437.
Müller-Wunderlich (1900, pp. 3–4).
- 438.
Ibid., p. 4: “eine gute Vorübung für die späteren wirtschaftlichen Arbeiten der Mädchen im Hause [...].”
- 439.
Another notable educator was Minna Schellhorn (1829–1910), who founded kindergartens where the main activities also consisted of folding and its mathematical comprehension. See: Payne (1876, pp. 87–89), in which Payne describes a visit to Schellhorn’s kindergarten in Weimar, where mathematical concepts are exemplified and understood by means of folding.
- 440.
Elm (2015 [1882]).
- 441.
Hurwitz (1985, Diary 22, (18.12.1906–22.1.1908), p. 173). (underlined in the original)
- 442.
Oswald (2015, p. 129).
- 443.
- 444.
After Klein describes the “failed” methods, by his own opinion, of Pestalozzi and Herbart, he reveals why he favors Fröbel’s: “In order to reveal the real core of these educational monstrosities and to direct art education onto a rational path, Fröbel was required. He, and with him Harnisch, advanced the physical form, that is, the three-dimensional one, within child education. Both educators made their own course of education, namely, coming out of mineralogy and crystallography.” [Um den richtigen Kern aus diesen pädagogischen Monstrositäten herauszuschälen und die Erziehungskunst in vernünftigere Bahnen zu lenken, bedurfte es erst eines Fröbel. Er, und mit ihm Harnisch, stellte die körperliche Gestalt, also das Dreidimensionale, bei der Erziehung des Kindes voran. Bei beiden Pädagogen macht sich der eigene Bildungsgang, nämlich das Ausgehen von Mineralogie und Kristallographie, geltend] (Klein 1979 [1926], p. 128). However, Klein does indicate that Fröbel also preserved several of Pestalozzi’s erroneous methods (see: Klein 1925, pp. 251–252).
- 445.
Jacobs (1860, p. 147): “Le carré de papier que Fröbel donne à l’enfant renferme pour celui-ci toute une géométrie.”
- 446.
Ibid., p. 148: “découpez différentes feuilles de papier et vous aurez une provision de carrés avec lesquels vous ferez les formes fondamentales du pliage. Pour arriver à ces dernières on passe par une série de formes mathématiques.”
- 447.
Ibid., p. 149: “Deux moitiés d’une même chose sont égales; donc le triangle formé par l’oblique est égal au rectangle formé par la ligne horizontale [...].”
- 448.
Ibid., p. 154: “L’enfant doit sentir aussi comment une forme tire son développement d’une autre et comment on retrouve dans une figure transformée toutes les modifications qu’elle a subies.”
- 449.
Noël (1993, p. 433).
- 450.
- 451.
Kergomard (1886, p. 122).
- 452.
Ibid., p. 123: “L’invasion de la géométrie et de la philosophie, l’invasion de la synthèse et de l’analyse, l’invasion de la méthode qui, techniquement, part du concret pour arriver à l’abstrait, l’invasion de l’esprit allemand, en un mot, dans nos écoles maternelles, m’effraye et me désole.”
- 453.
Ibid., p. 124.
- 454.
Ibid., p. 126: “La seconde leçon consistera à le faire plier en deux parties égales.”
- 455.
An expression used regularly by Leblanc. See: d’Enfert (2003a, pp. 213–216).
- 456.
- 457.
Palmyre (1893). Palmyre, describing which manual labor should be used in the different classes, mentions Fröbel as an example of the “travail manuel” (ibid., p. 5) for the “classe enfantine,” indicating that the adults should fold several forms for the children, so that the children could unfold them (p. 13). When suggesting folding activities for the “cours élémentaire,” Palmyre suggests folding forms in shapes of boats, gondolas, boxes, windmills, etc., while failing to emphasize any mathematical aspect (pp. 36–54).
- 458.
Savineau (1897).
- 459.
Lebeaume (2007).
- 460.
Brehony (2000a, p. 60).
- 461.
- 462.
The writings of the three discussed persons are only a selection of manuscripts and manuals that were published in Great Britain during the second half of the nineteenth century dealing with Fröbel’s gifts and occupations in general, and folding in particular. For other manuals, see, for example: Gurney (1877, pp. 9–13), (in which paper folding is numbered as gift number 8, see: ibid., p. 9), Lyschinska (1880, pp. 64–88) and Kraus-Boelté and Kraus (1892, pp. 284–296).
- 463.
- 464.
Ronge and Ronge (1855, p. 48).
- 465.
E.g., ibid.: “By these foldings it will be clearly seen that the base of each triangle is longer than either of its sides, and that the two acute angles of each triangle are together equal to one right angle.”
- 466.
- 467.
See: Brehony (2000a, p. 70).
- 468.
Ibid., p. 73.
- 469.
- 470.
Heerwart (1897, p. 56).
- 471.
Ibid., p. 57.
- 472.
Ibid.
- 473.
Heerwart (1895). Heerwart published several more books (e.g., Heerwart 1889, 1894), explaining and expanding Fröbel’s gifts. Thus, for example, in the book Course on Paper-Cutting (Heerwart 1889), which used folding paper as a basic (though not mathematical) activity (by folding a square paper into a triangle and then again into a smaller one), Heerwart explains that Fröbel also considered the folding of a Hexagon (ibid., p. 6, Tafel VI), although Fröbel himself did not write these instructions specifically (see: Boldt 2003, pp. 114–115).
- 474.
Heerwart (1895, p. 4).
- 475.
Ibid., p. 5.
- 476.
Ibid. (emphasis by M.F.)
- 477.
Ibid., p. 7.
- 478.
Heerwart (1894, pp. 6–7).
- 479.
Ibid., p. 7: “In Paper-cutting the inner pieces are utilized outside the square. In Paper plaiting strips from without and from within are worked together to produce again a plane surface.”
- 480.
Heerwart (1895, p. 11).
- 481.
Ibid., Plate V. Heerwart remarks that, regarding this plate, all of the forms “were added after Fröbel’s time” (Ibid., p. 15).
- 482.
Ibid., p. 5.
- 483.
Ibid., p. 11. (bold in original)
- 484.
Ibid., p. 15.
- 485.
Ibid., p. 15 and plate IV (see Fig. 4.57). Recall also that Fröbel called “modelling” the change of three-dimensional forms.
- 486.
Although Heerwart did mention it several times (see, e.g.: Heerwart 1894, pp. 6–7).
- 487.
Ibid., p. 18.
- 488.
Ibid.
- 489.
Brehony (2000a, p. 75).
- 490.
Cf. Brehony (2000b).
- 491.
Woodham-Smith (1953, pp. 88–89).
- 492.
Douglas Wiggin and Archibald Smith, p. 225.
- 493.
National Froebel Union (1916, p. 53).
- 494.
- 495.
Murray (1903, p. 15).
- 496.
Ibid., p. 16.
- 497.
Ibid., p. 18.
- 498.
- 499.
The book Manuel pratique des jardins d’enfants de Frédéric Frœbel of Jacobs (Jacobs 1860) was translated into Italian in 1871, titled: Manuale pratico dei giardini d’infanzia ad uso delle educatrici e delle madri di famiglia (Jacobs 1871). The chapter called “La piegature” (Ibid., pp. 121–130) describes the same mathematical construction as in the French original.
- 500.
Schröder (1987, p. 70).
- 501.
“1. si rifiutava il geometrismo dei doni in nome di una diversa ‘cultura’; 2. si rifiutava la prospottiva ottimistica e romantica dei bambino […].” In: Grazzini (1973, p. 49).
- 502.
Albisetti (2009, p. 168).
- 503.
Agazzi and Pasquali (1973, pp. 61–86).
- 504.
Ibid., p. 75: “Fröbel did not say: geometrize the children’s brains; Fröbel, because of his long experience, has put in the hands of children a material base of geometry.” [Froebel non ha detto: geometrizzate il cervello dei bambini; Froebel, per via di lunghe esperienze ha messo nelle mani dei fanciulli un materiale a base di geometria].
- 505.
See: Jovine Bertoni (1976, pp. 21–22).
- 506.
- 507.
Agazzi and Pasquali (1973, p. 126).
- 508.
Pasquali (1892). (Intuitive geometry without instruments)
- 509.
Ibid., Prefazione: “Esercitar insieme l’occhio e la mano, e l’uno a l’altra far servire alla mente, è lo scopo del presente libretto. Fino a qual punto l’intuizione, applicata allo studio della geometria piana, possa condurre alla certezza sensibile lo giudicheranno gl’insegnanti ai quali sarò gratissimo dei loto benevoli suggerimenti.”
- 510.
Ibid., p. 14.
- 511.
Ibid., p. 15: “A forza di approssimazione, si eseguiscono due piegature concorrenti al vertice e si divide l’angolo in tre parti. Dividendo poi per metà ciascuna delle parti, si dividerà l’angolo in sei.”
- 512.
The other exercises deal with the constructions of triangles, quadrilaterals, pentagons, hexagons and octagons. Transformations of parts of shapes are also examined (i.e., e.g., the decomposing of a triangle in order to form the two parts of a rectangle), in order to show the equality of areas of certain shapes. See: ibid., pp. 36–54.
- 513.
For other instruction books in Italy for schools that involve folding, see: Giacardi (2015, Sect. 3).
- 514.
Rivelli (1897) (Stereometry applied to the development of solids and their construction in paper.)
- 515.
Ibid., p. 10.
- 516.
See, e.g.: ibid., pp. 15–16: “One folds the faces on one another following the lines of the drawing, which denote the edges: the solid ones are bent outside, the dotted to the inside […]” [Si pieghino le facce l’una sull’altra seguendo i tratti del disegno, che indicano spigoli: i tratti pieni si pieghino al di fuori, i punteggiati al di dentro […]].
- 517.
- 518.
Originally called “National Indian Association in Aid of Social Progress in India.”
- 519.
Allender (2016, pp. 223–227).
- 520.
Powell (2018).
- 521.
Fuller (1879).
- 522.
Ibid., pp. 365, 373.
- 523.
See: Brander (1885), in which a report by Brander is given, recommending that “kindergarten occupations should be introduced” (Ibid., p. 95). See also: Haur et al. (2014 , pp. 141–142), in which it is noted that the “Madras presidency took the lead in addressing the need for specially trained teachers for young children.” (ibid., p. 142). See also: Powell (2018).
- 524.
Haur et al. (2014, p. 146).
- 525.
Ibid., p. 144.
- 526.
Jayawardena (1995, pp. 57–59).
- 527.
Powell (2018).
- 528.
Aiyengar (1909, pp. 32–33).
- 529.
The name “Rao” is sometimes spelled as “Rau.”
- 530.
Rao (1915, p. 178).
- 531.
Hanumantha Rao is also known for helping Srinivasa Ramanujan get a scholarship of 75 rupees per month at the University of Madras: see Berndt and Rankin (1995, pp. 70, 73, 76).
- 532.
The first edition was published in 1885.
- 533.
Rao (1888, preface, p. i) (On the cover, the name “Rao” is spelled as “Rau”).
- 534.
Ibid., preface, p. ii.
- 535.
Ibid., preface, p. iii.
- 536.
Ibid. (emphasis by M.F.)
- 537.
Ibid., preface, p. vii: “Pestalozzi, the great educational reformer, recognized this and endeavored to bring about a change in the method of geometric instruction. After several adaptations and modification of Euclid’s Elements had been tried, it was found necessary to give it up altogether as a text-book for beginners […] If Euclid’s Elements is unsuited for beginners who study it in their own native tongue, how much more so should it be in this country, where it is taught in classes consisting generally of lads between 10 and 12, before they have had time to master the difficulties of a foreign language, and before too, I may add, they can benefit by its rigorous logic.”
- 538.
Ibid., p. ii.
- 539.
Ibid., p. iv. (cursive by M.F.).
- 540.
I extend warm thanks to Avril Powell for private communication regarding this subject. As Powell notes: “it was very common to refer only to ‘Kindergarten gifts’ or even ‘gifts’ alone […],” while reporting and teaching the Fröbelian methods in India, without mentioning his name (Powell 2016).
- 541.
Rao (1888, p. v).
- 542.
Ibid.
- 543.
Ibid., p. 3. See also: ibid., p. 5.
- 544.
Ibid., p. 28.
- 545.
These instruments replace the “costly Mathematical instruments.” Rao again mentions the financial aspect in Ibid., p. ix: “[…] waste paper costs nothing and country-made brown paper very little.”
- 546.
Ibid., p. x: under the title The young geometrician tools, Rao lists “Plenty of waste paper to be cut up or folded into geometrical forms.” It is important to note that Rao (in contrast to Sundara Row, as we will see later) does not reject the compass, the ruler or the straightedge as other tools, just because they belong to the Euclidean tradition.
- 547.
Ibid., p. ix. See also: ibid., p. 42.
- 548.
Ibid., p. xi.
- 549.
Ibid., p. x: “To obtain two [congruent] figures: take a sheet of paper, fold it in two and then cut it into any shape with a pair of scissors”, and in contrast, p. 47: “If the paper be folded along the crease […], the two triangles […] will be found to coincide.” See also pp. 85 and 93.
- 550.
Ibid., p. 11: “An easy mode of obtaining a straight line is to fold a sheet of paper.”
- 551.
Ibid., p. 52.
- 552.
Ibid., p. xi.
- 553.
This fact that Rao was aware of Henrici’s book raises the question of the transfer of knowledge from Britain to its colonies, and the other way around (this question is also relevant concerning how Sundara Row’s book was obtained by Felix Klein, see Sect. 5.1). As June Barrow-Green notes, “the caliber of mathematical emigrants to the colonies was generally rather high” but “it cannot be denied that the atmosphere in most of the colonial outposts was not conducive to either advanced level teaching or research” (Barrow-Green 2011, pp. 151, 152). See also: ibid., pp. 135–137 and pp. 144–151 regarding the setting up and the advancing of British mathematical departments and high mathematical education in India.
- 554.
Rao (1888, pp. 52, 70–71).
- 555.
Ibid., p. 93.
- 556.
Row (1893, pp. iii–iv).
- 557.
Ibid., p. iv.
- 558.
Rao (1915, p. 423).
- 559.
Bahadur (1914, p. 180).
- 560.
The indirect influence of Henrici will be discussed later.
- 561.
Row (1893, p. i). Recalling that Mary Gurney and Eleonore Heerwart have numbered paper folding as gift num. 8 (see Sect. 4.2.1.3), this points to another connection between the Fröbelian movement in England and its acceptance in India. Row mentions that these gifts can be bought in Messrs. Higginbotham and Co., probably referring to Madras’ known book store Higginbotham’s, established in the year 1844 by Abel Joshua Higginbotham, who came from Britain as a clandestine passenger. See: Muthiah (2003).
- 562.
Row (1893, p. iv).
- 563.
Ibid., p. ii.
- 564.
Ibid., p. vi.
- 565.
Ibid., p. 1.
- 566.
The passage from one square to the smaller one is done by folding along the four segments, which connect neighboring midpoints of the edges of the square.
- 567.
The Gauss-Wantzel theorem (proven in 1837) states that a regular n-gon is constructible with a straightedge and a compass if and only if n = 2k p 1 p 2 ∙ … ∙ p t where k and t are non-negative integers, and the p t ’s (when t > 0) are distinct Fermat primes (of the form: \( {2}^{\left({2}^m\right)}+1,m\ge 0 \)an integer). See: Wantzel (1837).
- 568.
(Row 1893, p. 32)
- 569.
The third one is the quadrature of the circle.
- 570.
Row mentions them in: ibid., p. v.
- 571.
Were this possible, then one could have constructed an angle of 40°, which is the angle bounded between the two rays connecting the center and the adjacent vertices of the nonagon.
- 572.
Ibid., p. 32.
- 573.
Gleason discusses the issue of which regular polygons would be constructible if angles could be trisected (Gleason 1988). This is a construction possible in paper folding (which was proved after Row’s time; in 1980, it was also proved by Abe that one can trisect an angle via folding; see: Fushimi (1980); see also Sect. 5.2.3). If this is the case, a regular n-gon is constructible if and only if n = 2k3s p 1 p 2 ∙ … ∙ p t where s, k and t are non-negative integers, and the p t ’s (when t > 0) are distinct Pierpont primes (prime of the form 2u3v + 1, u, v nonnegative integers). Hence, a heptagon is constructible using folding.
- 574.
Row (1893, p. 41). (cursive by M.F.)
- 575.
See: Dürer (1977 [1525], p. 360). Also, see: ibid., p. 359: “It is described in Eutokius’ commentary to Archimedes, and credited to Hippocrates in a letter by the mathematician Eratosthenes to King Ptolemy, which Eutokius quotes” (from the commentary of Walter L. Strauss to Dürer’s Underweysung der Messung). Note that Row was aware of Hippocrates’s method, see: Row (1893, p. 42).
- 576.
Ibid., pp. 65–67.
- 577.
An envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point.
- 578.
Ibid., p. 104.
- 579.
Ibid., p. 106.
- 580.
Ibid., p. 1. (cursive by M.F.)
- 581.
Ibid., p. 63.
- 582.
Ibid.
- 583.
Ibid., p. 64.
- 584.
Ibid., pp. 63–76.
- 585.
Ibid., p. 63.
- 586.
Note that folding a perpendicular crease was already mentioned at the beginning of the book (ibid., p. 2).
- 587.
Klein (1895, p. 33).
- 588.
It is registered in the old card catalog of the mathematics Library in Göttingen (where Klein was a professor at the time) that the earliest edition of Row’s book is from 1966. Earlier editions were not and still cannot be found in the library.
- 589.
Also, as one can see from the letters to be found in Klein’s Nachlass in Göttingen, addressed to Klein, either from Henrici or from Grace C. Young (who wrote a book on geometry and folding in 1905, see Sect. 5.1.2), neither of them mentions Row’s book.
- 590.
Klein (1897, p. 42).
- 591.
Row (1901, editor’s preface).
- 592.
Ibid.
- 593.
Row (1893, p. vi).
- 594.
Row (1901, p. xiv).
- 595.
E.g., the footnote in: ibid., p. 76 refers to Beman and Smith’s translation and work, which did not appear in the 1893 edition.
- 596.
Willson (1902, p. 465).
- 597.
Row (1901, p. 84, paragraph 153).
- 598.
Row (1906, preface). The citation also indicates that Row might have been aware of Klein’s recommendation of his book.
- 599.
Ibid., p. 1.
- 600.
Row mentions folding as a procedure for constructing or proving the equality of two segments only in: ibid., p. 17 and 26.
- 601.
Ibid., p. 31.
- 602.
Ibid.
- 603.
Ibid., p. 78.
- 604.
Ibid., p. 80
- 605.
Ibid., pp. 80, 83. However, Row does not give an explicit reference to the contribution by Wiener to Dyck’s catalog, and only mentions him by name: “Dr. Wiener has shown how the regular solids can be formed with a single strip of paper. For the tetrahedron, the octahedron and the icosahedron, he takes a strip folded into equilateral triangles […]” (Ibid., p. 80).
- 606.
Again, just as it is not clear how Klein obtained Row’s book, it is not clear how Row obtained Wiener’s paper or Dyck’s catalogue.
- 607.
Row (1893, pp. 6–7).
- 608.
Row (1907).
- 609.
Hanumantha Rao was probably the influence behind this, cf. Rao (1888), the introductory chapter, p. iv.
- 610.
E.g., Row (1909a, pp. 13, 69).
- 611.
- 612.
Row (1914, p. 222).
- 613.
- 614.
Cf. Boldt (2003, pp. 114–115).
- 615.
This distinction was made explicit, for example, in Descartes’s La Géométrie, in which Descartes distinguished between three classes of curves. The first class, known as “plane” curves, included Euclidean constructions involving straightedge and compass. The second class, known as the class of “solids,” was made up of conic sections. The third class included objects that could only be constructed by mechanical artifices or mechanisms (Descartes 1925, p. 43).
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Friedman, M. (2018). The Nineteenth Century: What Can and Cannot Be (Re)presented—On Models and Kindergartens. In: A History of Folding in Mathematics. Science Networks. Historical Studies, vol 59. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72487-4_4
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