Prolog to the Nineteenth Century: Accepting Folding as a Method of Inference

Friedman, Michael

doi:10.1007/978-3-319-72487-4_3

Prolog to the Nineteenth Century: Accepting Folding as a Method of Inference

Michael Friedman⁶

Chapter
First Online: 26 May 2018

1438 Accesses

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 59))

Abstract

Before going into the fourth chapter, which discusses two main traditions—firstly, the rise of the German tradition of folding in kindergarten, a tradition that spread throughout Europe starting from the mid-nineteenth century; secondly, the appearance of material mathematical models, including folded models, in the nineteenth century—another encounter between folding and mathematics occurred in the nineteenth century that deserves mention. The ramifications of this event are echoed in the attempts to prove parallel axioms (to be discussed in this chapter), as well as in Pacioli’s earlier forgotten constructions with folding (which I discussed in the second chapter). Hence, I will discuss this third tradition as one that appears alongside the two other subsequent traditions that I have just mentioned above.

This is a preview of subscription content, log in via an institution.

Chapter: USD 29.95; Price excludes VAT (USA)

eBook: USD 189.00; Price excludes VAT (USA)

Softcover Book: USD 249.99; Price excludes VAT (USA)

Hardcover Book: USD 249.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

1.
For two excellent acounts on the parallel postulated and non-Euclidean geometries, see: Greenberg (1974) and Rosenfeld (1988).
2.
See: Rashed and Vahabzadeh (1999). For a survey on Al-Khayyām’s work on parallels, see: Jaouiche (1986, pp. 75–98).
3.
Rosenfeld (1988, p. 64).
4.
Note that, according to Rosenfeld, Al-Khayyām’s constructions can be seen as the precursors of non-Euclidean geometries. A discussion regarding the epistemological status of these constructions is beyond the scope of this book.
5.
See: ibid., p. 67 and Rashed and Vahabzadeh (1999, p. 267).
6.
Rosenfeld (1988, p. 68).
7.
Rashed and Vahabzadeh (1999, p. 266).
8.
Ibid., p. 267.
9.
See, e.g., Jaouiche (1986, p. 49), for the texts of Thabit ibn Qurra (826–901), a Sabian scholar and mathematician, who also uses substantives derived from the same verb to describe superposition.
10.
Howard Eves (1969, pp. 123–124) claims that Blaise Pascal (1623–1662) proved that the sum of the angles in a triangle is 180°, using folding techniques. However, not only this claim is not supported by any other source (Eves mentions Pascal’s sister as a witness), but also the “proof” is eventually equivalent to “proving” the parallel axiom.
11.
See: Rosenfeld (1988, pp. 99–101).
12.
Lambert was not the first to consider these quadrilaterals: Saccheri, al-Haytham and al-Khayyām also considered similar constructions.
13.
For a survey of this manuscript, see: Papadopoulos and Theret (2014).
14.
See Sect. 2.1.1 for Dürer’s usage of this verb.
15.
E.g., Lambert (1786, § 24, § 29, § 40, § 49, § 50, § 53, § 55, § 57, § 76).
16.
Böhm (1771).
17.
For example, folding a segment along a perpendicular crease such that the two end points of the segment would be on each other bisects the segment (Böhm 1771, §2). It is also implied that if one folds a segment such that the two end points of the segment would be overlapping, then the obtained crease must be perpendicular.
18.
For example, Böhm writes: “Si super linea vectea CD quocunque puncto A erigatur perpetndicularis AB & planum, cui vtraque inscripta est, secundum istam perpendicularem plicetur; tum singula puncta partis AD cadunt super puncta partis AC. Etenim si AB perpendicularis ad CD, est BAD = BAC. Sed aequales anguli flbi invicem congruunt.” (ibid., p. 3)
19.
Ibid., §3.
20.
Suzanne (1809, p. 32).
21.
Ibid., p. 47.
22.
In fact, since an equivalent formulation of the parallel postulate is that “[t]here exists a quadrilateral in which all angles are right angles […],” then the problem deals with that which can be derived from the parallel postulate itself.
23.
This procedure is also presented at: ibid., p. 47.
24.
Ibid., p. 62.
25.
Ibid., p. 102, pp. 466–467.
26.
Francœur (1809, p. 189).
27.
E.g., ibid., pp. 194–195 (folding along the bisector of the main angle in an isosceles triangle would result in the congruence of the corresponding edges), p. 197 (folding one rectangle along another, where both share a common edge).
28.
Diderot and le d’Alembert (1751, p. 905): “Suivant les définitions précédentes l’axe d’une courbe est en général une ligne tirée dans le plan de cette courbe, & qui divise la courbe en deux parties égales, semblables, & semblablement posées de part & d’autre de cette ligne.” Translation taken from: Hon and Goldstein (2008, p. 194).
29.
See: Hon and Goldstein, pp. 193–194, regarding the usages of the term “axe” with respect to symmetry in the eighteenth century.
30.
“Si quelqu’un ignore l’effet maussade de la symétrie, il n’a qu’à regarder ce tableau. Tirez une ligne verticale du haut en bas; pliez la toile sur cette ligne, et vous verrez la moitié de l’enceinte tomber sur l’autre moitié.” (Diderot 1821, p. 281). Translation taken from: Goodman (1995, p. 117).
31.
Cf. also: Hon and Goldstein (2008, pp. 145–148), on symmetry in architecture and painting in Diderot’s thought.
32.
Francœur (1809, p. 189): “Le diamètre [DA] coupe le cercle en deux parties égales; car, en pliant la figure suivant DA les demi-cercle coincident”; a similar formulation appears in: ibid., p. 360.
33.
Ibid., p. 372: “l’axe des x coupe chacune de nos trois courbes en parties égales et qui se superposent, lorsqu’on plie la figure suivant cet axe.”
34.
Francœur (1828, p. 400).
35.
Ibid., p. 401: “[...] the ellipse is such that ABOD (fig. 203) is symmetrical with respect to the axis AO; So it is also symmetrical relative to BD, since +x and −x give the same value of y: Thus, when one folds the figure along AO or BD, the parts of the curve are superimposed and coincide.” [[…] l’ellipse est telle que ABOD (fig. 203) symétrique par rapport à l’axe AO; elle l’est aussi relativement a BD, puisque +x et −x donnent la même valeur de y: Ainsi, lorsqu’on plie la figure selon AO ou BD, les parties de la courbe se superposent et coïncident.]
36.
Ibid., p. 404, 417.
37.
Francœur refers at the margins of the text to figure 149.
38.
Francœur (1809, p. 280): “Lorsque deux polyèdres sont tels qu’on peut les placer l’un en dessus, l’autre en dessous d’un plan MN, de sorte que les sommets des angles polyèdres A, a, B, b… soient, deux à deux, à égale distance de ce plan, et sur une perpendiculaire Aa, Bb… à ce plan: ces deux polyèdres sont appelés Symétriques. […] Les polyèdres symétriques ont toutes leurs parties constituantes égales. Pour le prouver, plions le trapèze ABPQ suivant PQ, les lignes AP \( aP \, \acute{e} \)gales et perpendiculaires sur MN coïncideront […]”
39.
Hon and Goldstein (2008, p. 50).
40.
Ibid., p. 277 “A formal definition of bilateral symmetry requires a reference: a point, a line, or a plane.”—This plane was exactly what was introduced in 1809 in order to fold two bilateral symmetric bodies!—“A line that separates the two symmetrical sides is called an ‘axis’. As far as we can determine, this term was first linked to symmetry in a treatise on physics by Louis Benjamin Francœur […].”
41.
Ibid., p. 278. See also: Francœur (1804, pp. 49–50): “We shall say that a body is symmetrical with respect to a plane or an axis, when its molecules are arranged in pairs, at the same distance from this plane or axis.” [Nous dirons qu’un corps est symétrique par rapport à un plan ou à un axe, lorsque ses molécules—seront disposées deux à deux, à la même distance de ce plan ou de cet axe].
42.
See the former footnote.
43.
Hon and Goldstein (2008, p. 279).
44.
Euclid’s Elements was used as a textbook in Great Britain for learning Geometry, almost without change, until the end of the nineteenth century. Regarding the debate on Euclid’s method and the rival manuals that were proposed in Britain at that time, see: Moktefi (2011).
45.
On Lardner’s life, see: Martin (2015) and Hays (1981).
46.
Lardner (1828, p. ix): “Two thousand years have now rolled away since Euclid’s Elements were first used in the school of Alexandria and to this day they continue to be esteemed the best introduction to mathematical science. […] Some of the most eminent mathematicians have written, either original treatises, […]; but still the ‘Elements’ themselves have been invariably preferred […].”
47.
See the citation in the previous footnote. Regarding the success of Lardner’s Six Books of the Elements of Euclid, see: Martin (2015, pp. 82–83).
48.
See also: Jenkins (2007, pp. 158–161) regarding Lardner’s view of education of geometry.
49.
Interestingly, the detailed biography of Lardner (Martin 2015) does not even mention Lardner’s Treatise on Geometry (and therefore also neglects his ideas regarding folding). It seems that Giovanni Vacca (See: Vacca 1930, p. 46) was the first one to mention Lardner’s Treatise on Geometry in an historical survey from 1930, suggesting that Lardner was the first to use folding in plane geometry. See also Sect. 5.2.1 on Vacca’s paper.
50.
Lardner (1840, p. 3).
51.
Ibid., p. 4.
52.
Ibid., p. 5.
53.
Ibid., p. 6.
54.
Ibid., p. 18.
55.
E.g., ibid., pp. 25–26.
56.
Ibid., p. 32.
57.
Ibid., p. 77. The usage of folding appears in the process of proving the following proposition: “When two tangents are drawn from the same point P […] to the same circle they will be equal, and the line drawn to the centre will bisect the angle formed by them.”
58.
Ibid., p. 44.
59.
Ibid.
60.
Ibid., p. 73.
61.
Cf. also ibid., p. 249, 284 and 309 for similar examples.
62.
Ibid., p. 74.
63.
Ibid., p. 70.
64.
Hon and Goldstein (2008, p. 2).
65.
Ibid., pp. 50–51. See also: ibid., p. 58: “In 1794, in the context of solid geometry, Legendre applied the term, symmetrical, to a previously unrecognized relation of polyhedra: two polyhedra are symmetrical, that is, equal by symmetry, when their faces are respectively congruent, and the inclination of adjacent faces of one of these solids is equal to the inclination of the corresponding faces of the other, with the condition that the solid angles are equal but arranged in reverse order. Such solid bodies are similar and equal but not superposable.”
66.
Interestingly, in Lardner’s The First Six Books of the Elements of Euclid, the only folds that are mentioned are the physical ones (i.e., wrinkling); see: Lardner (1828, p. 285).
67.
Lardner (1840, p. 183).
68.
Ibid., p. 218. This is, of course, an implicit reference to developable surfaces. See Sect. 4.1.2.3.
69.
Ibid., p. 220.
70.
Ibid., p. 6.
71.
In: ibid., p. 7 (resp. p. 11) Lardner defines the mathematical line (resp. surface) as the result of movement of the mathematical point (resp. line).
72.
See: Lindemann (1927, pp. 159–160).
73.
For a more extensive survey of Henrici’s life and work, see: ibid and: Hill (1918).
74.
Henrici (1879, p. ix).
75.
Ibid., p. x. Regarding Henrici’s equation between physical and mathematical experience in particular and in the context of the changing research paradigms in geometry in Victorian England, see: Richards (1988, pp. 140–141,151–153).
76.
Henrici (1879, p. 56): “These properties of the bisectors of segments and angles may be used practically for finding them. Thus, if a segment is given on a straight edge of a piece of paper, we may bend the paper in such a manner that the two end points coincide, and then fold down the paper. The crease formed will bisect the edge. Similarly if a piece of paper on which we may suppose an angle given be cut along the limits of the angle, we may, by bending the paper, make the two limits coincide and fold the paper down. The crease hereby formed will be the bisector of the angle.”
77.
Ibid.
78.
Ibid., p. 59.
79.
Note that Humiaki Huzita later considered this construction to be a fundamental operation in the list of those operations for folding-based geometry (Huzita 1989a, p. 144). See also Sect. 6.1.
80.
Henrici (1879, p. 86).
81.
Ibid., p. 89 (cursive by M.F.)
82.
The following example is presented by Henrici in: ibid., p. 154: “Every diameter is an axis of symmetry. For if we fold over along a diameter d, every point on the part of the circle turned over must fall on some point on the other, as it is at the radius distance from the centre which remains fixed […].”
83.
Ibid., p. 100.
84.
Wright (1868, p. iii). The preface was written by the mathematician Thomas Archer Hirst (1830–1892); note that Hirst was the first president of the “Association for the Improvement of Geometrical Teaching” in 1871, aiming, along with others, to replace the Elements as the textbook for geometry.
85.
Ibid., p. vii.
86.
Ibid.
87.
Ibid., p. 10. Henrici’s proof is very similar to Wright’s proof.
88.
Ibid., p. 56.
89.
Ibid., pp. 75, 161.
90.
The first edition did not survey Henrici’s book.
91.
Dodgson (1885, pp. 94–95).
92.
Ibid., p. 95.
93.
Ibid., p. 179.
94.
Ibid., p. 180. This critique had already appeared in the first edition (from 1879) of Dodgson book.
95.
This can also be seen in the book by Giuseppe Ingrami: Elementi di geometria, from 1904. Ingrami mentions folding only once, as a “practical observation” to demonstrate congruence of triangles (Ingrami 1904, p. 66).
96.
Daston (1986, p. 272). This can also be seen with Diderot’s and Francœur’s explicit reference to folding as a (physical) operation.

References

Böhm A (1771) Rectis Parallelis Dissertatiuncula. Acta Philosophico-Medica Societatis Academicae Scientiarum Principalis Hassiacae 1–5
Google Scholar
Daston LJ (1986) The Physicalist tradition in early nineteenth century French geometry. Stud Hist Philos Sci Part A 17(3):269–295
Article MathSciNet Google Scholar
Diderot D (1821) Oeuvres de Denis Diderot, Salons, vol I. Brière, Paris
Google Scholar
Diderot D, d’Alembert J le R (1751) Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des métiers, vol 1. Briasson, Paris
Google Scholar
Dodgson CL (1885) Euclid and his modern rivals, 2nd edn. Macmillan, London
MATH Google Scholar
Francœur L-B (1804) Traite élémentaire de mécanique. Courcier, Paris
Google Scholar
Francœur L-B (1809) Cours complet de mathématiques pures, vol 1. Bernard, Paris
Google Scholar
Francœur L-B (1828) Cours complet de mathématiques pures, 3rd edn, vol 1. Bernard, Paris
Google Scholar
Goodman J (ed, trans) (1995) Diderot: on art, The Salon of 1765 and notes on painting, vol 1. Yale University Press, New Haven
Google Scholar
Greenberg MJ (1974) Euclidean and non-Euclidean geometries: development and history. W. H. Freeman, San Francisco
MATH Google Scholar
Hays JN (1981) The rise and fall of Dionysius Lardner. Ann Sci 38:527–542
Article Google Scholar
Henrici O (1879) On congruent figures. Longmans, Green, London
MATH Google Scholar
Hill MJM (1918) Obituary notice of Olaus Henrici. Proc Lond Math Soc 17(2):xlii–xlix
Google Scholar
Hon G, Goldstein BR (2008) From summetria to symmetry: the making of a revolutionary scientific concept. Springer, New York
Book Google Scholar
Huzita H (1989a) Axiomatic development of origami geometry. In: Huzita H (ed) Proceedings of the 1st international meeting of origami, science and technology. Comune di Ferrara and Centro Origami Diffusion, Ferrara, pp 143–158
Google Scholar
Ingrami G (1904) Elementi di geometria ad uso dei licei. P. Cuppini, Bologna
MATH Google Scholar
Jaouiche K (1986) La théorie des parallèles en pays d’Islam: Contribution à la préhistoire des geometries non-euclidiennes. Vrin, Paris
MATH Google Scholar
Jenkins A (2007) Space and the “March of Mind”. Literature and the physical sciences in Britain, 1815–1850. Oxford University Press, Oxford
Google Scholar
Lambert H (1786) Theorie der Parallellinien. Leipzig Mag Reine Angew Math 137–164, 325–358
Google Scholar
Lardner D (1828) The first six books of the elements of Euclid. John Taylor, London
Google Scholar
Lardner D (1840) A treatise on geometry and its application to the arts, the cabinet cyclopaedia. Longman, London
Google Scholar
Lindemann F (1927) Olaus Henrici. Jahresb Dtsch Math-Vereinigung 36:157–162
MATH Google Scholar
Martin AL (2015) Villain of steam: a life of Dionysius Lardner. Tyndall Scientific, Carlow
Google Scholar
Moktefi A (2011) Geometry. The Euclid debate. In: Flood R, Rice A, Wilson R (eds) Mathematics in Victorian Britain. Oxford University Press, Oxford, pp 321–336
Google Scholar
Papadopoulos A, Theret G (2014) La théorie des lignes parallèles de Johann Heinrich Lambert. Collection Sciences dans l’Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris
MATH Google Scholar
Rashed R, Vahabzadeh B (1999) Al-Khayyam Mathématicien [in Arabic]. Albert Blanchard, Paris
MATH Google Scholar
Richards JL (1988) Mathematical visions: the pursuit of geometry in Victorian England. Academic, Boston
MATH Google Scholar
Rosenfeld BA (1988) A history of non-Euclidean geometry. Evolution of the concept of a geometric space (trans: Shenitzer A). Springer, New York
Google Scholar
Suzanne P-H (1809) De la manière d’étudier les mathématiques: Ouvrage destiné à servir de guide auxgeunes gens, à ceux surtout qui aspirent à être admis à l’École normale ou à l’École impériale polytechnique. Bechet, Paris
Google Scholar
Vacca G (1930) Della piegatura della carta applicata alla geometria. Periodico Mat Ser IV 10:43–50
MATH Google Scholar
Wright RP (1868) The elements of plane geometry. Longman, London
Google Scholar

Download references

Author information

Authors and Affiliations

Humboldt-Universität zu Berlin, Berlin, Germany
Michael Friedman

Authors

Michael Friedman
View author publications
You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Cite this chapter

Friedman, M. (2018). Prolog to the Nineteenth Century: Accepting Folding as a Method of Inference. 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_3

Download citation

DOI: https://doi.org/10.1007/978-3-319-72487-4_3
Published: 26 May 2018
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-72486-7
Online ISBN: 978-3-319-72487-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)

Publish with us

Policies and ethics