• 1 On Jungius’s life and work, see: (Wohlwill, 1887; Kangro, 1968; Meinel, 1982, 1995; Wollgast, 1993, (...)

1Joachim Jungius (1587-1657) was a German mathematician, logician, and philosopher of science. He worked in Gießen, Lübeck, and Rostock; and from 1629 until 1657 he was a professor of natural sciences at the Akademisches Gymnasium in Hamburg. Mostly known today for his Logica Hamburgensis (1638), his understanding of mathematics was also influential during the 17th century. For Jungius — as he stated explicitly in his inaugural speech on March 19, 1629, in Hamburg (Jungius, 1929) — mathematics should not only be used to solve arithmetical and geometrical problems; it should also be at the basis of all scientific domains — which underlines the importance he attached in his work to observation and mathematical demonstration (Hübner, 1996). Jungius was also known for his Geometria empirica (1627), as well as for his studies on botany and on the corpuscular theory of chemistry. His scientific investigations were published in several books, including Disputationes (1642) and (posthumously) Doxoscopiae physicae minores (1662). His work was well known to Samuel Hartlib, Jan Amos Comenius, John Pell, and, later, Gottfried Wilhelm Leibniz, all of whom praised Jungius and his work.1

• 2 On how Jungius’s Nachlass was handled by Martin Fogel and his successors, see: (Meinel, 1984, p. ix (...)
• 3 The full title of the manuscript, probably given by Fogel, is “Texturæ Contemplatio. Auct. Joach. J (...)

2One of Jungius’s research themes between the 1620s and the 1640s was the investigation of textiles and textile practices (in particular, weaving, knitting, and plaiting). This interest culminated in a set of notes titled “Texturarum theoria modo sciendi physico inserviens,” which is mentioned by Martin Fogel in 1657 (shortly after Jungius’s death) in the inventory Fogel made of Jungius’s Nachlass (Meinel, 1984, p. xxiii).2 As with most parts of this Nachlass, it is unlikely that Jungius intended “Texturarum theoria” to become a text that was structured in any definite way; rather, it took the form of a collection of notes — one which was later copied and excerpted by Jungius’s students. The original collection of notes was subsequently lost, and what have survived are the copies and excerpts. These were then incorporated into Fogel’s Nachlass, where they were given the title Texturæ Contemplatio3 — henceforth TC. It is with TC that the present paper will mainly deal.

• 4 On the way in which Jungius wrote and arranged his notes, see: (Meinel, 1995).
• 5 In Gottfried Wilhelm Leibniz Bibliothek – Niedersächsische Landesbibliothek, Leibniz-Handschriften (...)

3TC was probably put together in Hamburg between 1660 and 1678, when former Jungius students, under the direction of Johannes Vagetius, arranged and transcribed Jungius’s numerous notes (Meinel, 1984, p. xii-xiv).4 Fogel took some of the copies with him to Hanover, where Leibniz managed to purchase Fogel’s library in 1678, shortly after Fogel’s death. Jungius’s original collection of notes was probably lost in 1691 during a fire in Vagetius’s house, where the notes were being stored. Neither TC nor “Texturarum theoria” were published in any form, even though Leibniz (2009, p. 147) expressed his wish for this to happen, and himself copied (and further edited) TC.5

• 6 To these one should add the inventions of the flying shuttle (in 1733) and the spinning jenny (arou (...)

6Textile practices such as spinning, knitting, plaiting, felting, basketry, and weaving are some of the oldest human practices, practices which have existed in various forms and which certainly do not have a single geographical origin, but were developed in different centers and in different ways. A proper outline of the history of European weaving and looms from the Middle Ages until their mechanization during the 18th century by Basile Bouchon, Jacques de Vaucanson, and Joseph Marie Jacquard (Broudy, 1993, p. 130-169) is beyond the scope of this paper. However, as Lewis Mumford notes, between the 15th and 18th century, “the [European] textile industries recorded the greatest number of improvements.” (Mumford, 1967, p. 144; cf. Bohnsack, 1981) Among these advancements and inventions, Mumford mentions the introduction of the spinning wheel from India, the fulling mills, Leonardo da Vinci’s invention of various textile machines at the end of the 15th century (da Vinci, 1979), the invention of a partly automatic spinning wheel in 1530, the invention of the ribbon loom (on which one could weave more than one ribbon at a time) in 1586 in Danzig, which spread throughout Europe during the 17th century, and last but not least the invention of the stocking frame (a mechanical knitting machine) in 1589 near Nottingham (Lewis, 1986; Plummer, 1972, p. 162-172).6 The late 17th century and the 18th century probably also saw a rise in the literacy of the weavers themselves, since it was during this period that the first books of weaving instructions, including weave patterns and notations, were printed and written by weavers – these books will be briefly discussed at section 2.1.

• 7 Saje is a fabric made of a mix of wool and linen, silk, or cotton.

7To concentrate only on Hamburg, where Jungius lived while composing his notes on weaving, it has been well documented that during the late 16th century this was one of the main destinations of Dutch religious refugees. These refugees were experts in, among other things, various techniques of weaving, especially silk weaving; an edict from 1586 describes some of these refugees as “saje makers” [“Sayenmacher”] (Rüdiger, 1874, p. 210).7 This edict permitted these refugees to both produce and sell their products. A similar edict issued in Hamburg in 1613 describes in more detail not only the refugees’ rights and what was prohibited to them, but also the composition and the dimensions of the fabrics produced (p. 211-232). Moreover, the establishment and mergers of guilds (“Brüderschaften”) underline the importance of the different branches of textile production in Hamburg. Various guilds — of knitters as well as of silk, velvet, and caffa weavers — were established in 1609, 1622, and 1628. In their records one finds that in ca. 1630 there were 183 “masters,” 18 former masters, and 30 widows registered working in these trades in Hamburg (Stieda, 1894, p. 422-424).

• 8 This question is also considered in the framework of the research on artisanal epistemology; see e. (...)
• 9 Wolzogen was “Swedish” in the sense that he later lived in Stockholm.

8In view of the mechanical developments and inventions in the domain of textile practices on the one hand, and the flourishing of these practices in the north of Germany during the 17th century on the other hand, the question that arises is how these practices were considered by scholars and mathematicians, especially during the 17th century.8 Jungius’s reflections and investigations on weaves and textiles have been mentioned above and will be examined thoroughly below; however, Jungius was not the only 17th-century scholar to have treated this subject. Samuel Hartlib, in a series of letters written in 1653 and 1657 to several scholars, also discusses innovations in weaving (Hartlib, 2002, 8/16/1A-5B). The title of this series is “Inventiones / Textoriæ / Wolzogens,” in reference to Johann Ludwig von Wolzogen (1600-1661), an Austrian philosopher, mathematician, and theologian. These letters can be seen in the context of Hartlib’s conversations with John Pell at the beginning of the 1650s. Hartlib writes that “[a] curious and accomodatious Weaving-Instrument mentioned by Mr Pell which the Swedish Baron Wolzogens wife did vse in Holland to get her subsistence by [is] much used in Bohemia wherby waiscots in all manner of colours are exactly woven [...]. Weaving / Woltzogen / in Sweden / Woltzogen being Mathematical can easily send a description with a delineation of it.”9 (Hartlib, 2002, 28/2/20A) Hartlib is referring to a loom found in Holland, hence probably to a manual ribbon loom, since these were widespread in Holland at the beginning of the 17th century. According to Hartlib, there is a close connection between this loom and mathematics or mathematical principles – although he does not explain what this might be.

• 10 A comparison between Jungius’s and Bacon’s use of terms derived from weaving practices and the infl (...)

9Bacon and Descartes expressed similar opinions concerning possible connections between mathematics and weaving. Bacon, in his 1620 Novum Organum, called for the gathering of knowledge via experiments, from which the universal laws of nature would be derived by means of inductive reasoning. Bacon underlines that these experiments should also take into account the knowledge of artisans: the “particular arts to be preferred” in such a project are those that “exhibit, alter and prepare natural bodies and the material of things such as agriculture, cookery, chemistry, [...] and the like,” as well as “weaving, carpentry, architecture [...].” (Bacon, 1620, p. 278) While Bacon, with his inductive method, might be expected to call for the inspection of manual and artisanal practices, one should also note his use of the term “textura,” which describes the order and arrangement of parts of bodies, and which has an affinity with weaving practices and terms (Gemelli, 1996, p. 186-188).10 But while Bacon delineated a connection between weaving practices and a theory of materials, an explicit connection to mathematics is absent.

• 11 Furthermore, Descartes emphasizes that, while these various activities “present us in the most dist (...)

10That being said, the fact that Descartes, who advocated a deductive method, called for a similar procedure of inspecting textile practices is more surprising. In the tenth rule of his Regulae ad directionem ingenii, Descartes notes that “in order to acquire discernment we should exercise our native intelligence by investigating what others have already discovered, and methodically survey even the most insignificant products of human skill, especially those which display or presuppose order.” He advises that “[one] must first tackle the simplest and least exalted arts, and especially those in which order prevails — such as weaving and carpet-making, or the more feminine arts of embroidery, in which threads are interwoven in an infinitely varied pattern.” (Descartes, 1701, p. 35) It should be recalled that, at that time, the textile industry was flourishing in France, and Descartes was certainly aware of this. Descartes’s emphasis on “display[ing] or presuppos[ing] order” may be interpreted in two ways: as referring either to the resulting “pattern” of the finished weave, or to the order of actions while using the loom — for example, the order in which one should step on the loom’s treadles, an aspect which is important, especially when weaving with a loom with more than two treadles. Indeed, the way threads are woven is to be decided before the realization of the fabric (i.e. before the production of the final result). Moreover, according to Descartes, the supposed mathematization of these textile practices must be done in a way that resembles “number games and [those] involving arithmetic.”11 (Descartes, 1701, p. 35)

• 12 In a list titled “Catalogue of Inventions (Catalogus inventionum),” one of the inventions that Hart (...)
• 13 This is not to imply that other scholars and mathematicians were not interested in the connections (...)

11Hence, the Baconian and the Cartesian conceptions of manual and haptic activities both reflect the need to discover (geometrical- mathematical) laws via manual practices, that is, by learning from and experimenting with crafts which, according to Bacon, “exhibit, alter and prepare [...] the material of things.” However, it is not clear whether either Descartes or Bacon was actually in contact with weavers or visited their ateliers. If Bacon’s term “textura” only referred to the spatial order of corpuscles, then one may claim that the transfer of knowledge enabled by the alleged trading zone between natural philosophers and weavers was only operating metaphorically. Moreover, Descartes, after comparing textile practices to arithmetic, adds that it “is surprising how much all these activities exercise our minds, provided of course we discovered them for ourselves and not from others” (Descartes, 1701, p. 35) — which points toward a separation between mathematical reasoning and artisanal knowledge. While Hartlib, on the contrary, referred to the novel inventions of textile machines — the ribbon loom or the stocking machine12 — it remains unclear how far he considered these mathematically. In this sense, Jungius’s TC serves as an explicit and unique meeting point between mathematics and weaving.13

• 14 One folio (fol. 41v) is even dated to 1654, though it might have been written by Fogel and not by J (...)

12Having presented the views of several thinkers on weaving and geometry, I now turn to Jungius’s TC. One aspect should be stressed at the outset: TC is not an autograph by Jungius but a posthumously reorganized copy of a collection of notes that Jungius wrote between 1621 and 1649.14 This set of notes, written in Latin and German, contains around 60 pages and is partially a description of a number of different weaving materials and techniques, partially a more scientific examination of fabrics, and partially an attempt to present weaves as geometrical, assigning to them “definitions” and “theorems.” TC, which is mentioned rarely in the secondary literature on Jungius (Kangro, 1968, p. 28, 243-245; Guhrauer, 1850, p. 292-294), has not been thoroughly examined, and none of the works that do mention it deals with how Jungius considered artisanal knowledge or, more specifically, textile practices. As we will see, Jungius wrote on textiles and weaving in other manuscripts too; however, these do not contain geometrical reflections.

• 15 This discussion comes in the framework of the changing meanings of scientia in the 17th century; se (...)
• 16 “[...] in Mathematicis praesertim meris sive abstractis earn principiorum evidentiam, eam praecepto (...)

14This section deals, among other topics, with Jungius’s mathematical and scientific reflections on textiles. But first one should clarify what these two adjectives meant for Jungius.15 As noted above, Jungius presented his view of mathematics in his inaugural speech in 1629. He saw pure mathematics as providing a possible structure for scientific inquiry, since in pure mathematics there exists not only an “evidence of principles” — principles here are considered in the sense of axioms — but also a constancy of definitions, theorems, and valid proofs (Jungius, 1929, p. 100).16 Moreover, the axioms of abstract mathematics can do almost without experience or observation — which is to say that, while the axioms of harmony, optics, and statics, for example, still draw to some extent on experience and observation, they do so to a much lesser degree than the axioms of physics (p. 102). Mathematics has the advantage of immediately obvious axioms, indisputable theorems, and conclusive proofs.

• 17 A detailed survey of Jungius’s scientific method is beyond the scope of this paper. See: (Clucas, 2 (...)
• 18 For an overview of the book, see: (Risse, 1964, p. 521-531).

15Turning to Jungius’s conception of his scientific method,17 this is presented, for example, in the fourth part of Jungius’s Logica Hamburgensis,18 which deals “with the application of logical induction to the natural sciences in a novel and controversial fashion.” (Clucas, 2010, p. 54) According to Stephen Clucas, Jungius belonged to a group of natural philosophers who formulated “a conception of observation as regulated experience, which could be legitimately appealed to in making truth-claims in natural philosophy.” (p. 57) Jungius examines two kinds of scientific method, inductio and demonstratio (Risse, 1964, p. 526-527), and presents the notion of an “empirical theorem” based on his own scientific method (Jungius, 1638, p. 207ff.). How proofs of such “theorems” are different from proofs in mathematics can be seen in Jungius’s discussion on proofs in chapter XIV of the fourth part of Logica Hamburgensis. Here, proofs are divided into two types, a priori and a posteriori, and a priori proofs are then divided further into a priori par excellence proofs and a priori proofs according to certain aspects (p. 238). While the a priori par excellence proofs deduce the conclusions from earlier knowledge (such as the proofs of arithmetic and geometry), a priori proofs according to certain aspects are proofs resulting from what occurs according to nature (prior to any observation), but becomes evident later through the demonstration. Examples of these types of proofs can be found in optics, astronomy, and physics (p. 238). And while in Logica Hamburgensis Jungius presents clear definitions of what constitutes a theorem and a proof in mathematics as opposed to physics and the other natural sciences, as we will see in what follows, the geometrical “theorems” presented in TC may not be as a priori (and detached from observation) as expected.

• 19 I will return to this discussion below. On Jungius, his axiomatic method, and its use with respect (...)

17A clear indication of TC’s mathematical orientation is to be seen in the first folios, which open with a set of definitions (“Definitiones”) and theorems (“Theorema”). There are at least 13 folios which can be associated with this set of pages dealing with a possible “geometrization” of weaving, which constitutes one of the largest groups in this set of notes. These folios may therefore be considered as Jungius’s most explicit attempt to provide a geometrical model for weaving, though this model should not be considered as axiomatic, since no axioms are presented.19 To analyze whether these notes could have led to the development of new mathematical theories, let us take a closer look at the definitions and theorems.

• 20 “p. 1 Texturæ Contemplatio ad scientiam situs.” In contrast to the titles, which were usually given (...)

18The most complete version of the set of definitions and theorems is to be found on folio 62r. The title of this folio is “p. 1[,] a contemplation of weaving[:] concerning knowledge of position,”20 and this is followed by a series of “definitions”:

• 22 “Textura quadrupla dicitur, si in una tela pagina sive facie unius seriei fila quaternis alterius s (...)

21The above series of definitions ends with a more elaborate explanation of the weave called textura quadrupla: “Quadruple weave is defined when in one fabric, whether on the side or on the surface, the threads of one row [the weft threads] go over a fifth [thread] with four threads of another row [the warp threads] having been sent under.”22 (TC, fol. 62r) But here one may notice that even if Jungius claims that in one row the weft thread passes below four warp threads, there is no way to induce whether in the next row there will be a shift in this pattern by one thread or by several threads. This is dealt with by Jungius’s “theorems,” which are in fact a collection of four statements about these quadruple weaves. These theorems are to be found on a folio titled “p. 2[,] a contemplation of weaving[:] concerning knowledge of position” (TC, fol. 63r). Here, Jungius makes explicit the relations between the rows of the weft — which may explain why he called these statements theorems. To give an example, the first theorem describes the characteristics of a specific “textura quadrupla” (today called 4/1 twill), when passing the weft thread from one side to the other:

• 24 (Emery, 2009, p. 108): “[...] in the [4/1] twill [...] each successive weft binds the next adjacent (...)

22While theorem I deals with what Jungius calls “continuous [continua] quadruple weave,” theorem II deals with the properties of “intermittent [intermittens] quadruple weave.” In modern terms, while the first weave is a 4/1 twill, the second is a 4/1 satin weave;24 both weaves are drawn in Fig. 2.

• 25 “p. 3[,] Texturæ Contemplatio ad scientiam situs.”

24In addition, it is important to note that in this set of folios Jungius is also aware that other twills exist, even drawing some of them. On a page titled “p. 3[,] a contemplation of weaving[:] concerning knowledge of position”25 (TC, fol. 8r), Jungius draws what are today called 2/2 and 3/3 twills (see Fig. 3; the 2/2 twill is characterized when two weft threads go below two warp threads, followed by a simple alternation in the next row).

• 26 To be more explicit: Folio 2r of TC contains the definitions, whereas folio 4r is a copy of them; t (...)
• 27 The postulates are: drawing a straight line between two points, continuing drawing a given line, an (...)
• 28 Concerning Geometria empirica, see: (Jungius, 1627), including the introduction by Bernd Elsner.
• 29 In the German adaptation of Geometria empirica, called Reiß-Kunst, published in 1629, the reliance (...)

25The attempt to introduce a geometrical structure of various weaves can be considered as one of the principal parts of TC. This is attested by the number of times the definitions and theorems were copied and recopied.26 How can one consider this geometrization in the context of Jungius’s work on geometry? The question arises whether in TC Jungius was aiming at developing a geometry based on a deductive, Euclidean style of reasoning, or at a more practical or empirical approach. Indeed, the structure of the definitions-theorems part of TC echoes the structure presented by Jungius in his book Geometria empirica (from 1627), although the latter book also contains postulates and problems as part of its argument. In Geometria empirica Jungius aims to solve the problems presented empirically with only three postulates,27 while not presenting any axioms (as in TC).28 Geometria empirica is not a revision of Euclid’s Elements or a selection of material from it, but rather an independent introduction to geometry; it presents an empirical approach to geometry, which is not based on purely logical deduction,29 and hence calls on students to practice and experiment.

• 30 See also: (Hall, 1979, p. 53) concerning terminology in artisanal practices in vernacular languages (...)
• 31 See: (TC, fol. 39r): “A biplex weave is when the thread of the weft lies over two [warp threads], [ (...)
• 32 Moreover, Ziegler’s and Lumscher’s manuscripts show that these works assume the literacy of the wea (...)
• 33 I will deal with knitting and knitters in section 3.

26Returning to TC, an indication of the experimental character of this manuscript is not only the development of theorems, but also Jungius’s hesitancy regarding the terms he was using to describe the various weaves.30 This terminological uncertainty and inconsistency raises the question of the character of the possible trading zone between Jungius and the weavers. For example, folio 39r, which is titled “Textoria,” begins with the “Definitio” of “textura biplexa,” being a 2/2 twill.31 On folio 8r, however, the very same textura biplexa is called “Textura simpla gemina intermittens.” Such inconsistencies prompt two questions concerning the actual or imagined trading zones and their influences: First, was Jungius sufficiently in contact with weavers to be aware of their terminology and to make use of it? And second, if he was in contact with them, would his geometrical reflections — had these been published or further transmitted — have been useful either to the weavers themselves or for the development of new domains of mathematics? Considering the first question, while it will be clear in the following that Jungius knew of or observed at least some of the stages of textile production, it is not clear whether the terms he applies were in use also among the weavers in Hamburg. Comparing Jungius’s notes to two of the first known printed manuscripts on pattern weaving — those by Marx Ziegler (from 1677) and Nathanael Lumscher (from 1708) — one does not find similar terminology, though these two manuscripts originated not from Hamburg but from the south of Germany (from Ulm and Kulmbach respectively).32 Concerning the second question, and concentrating first on the weavers,33 it is clear that these would have found the separation Jungius introduced between “definitions” and “theorems” to be an artificial one, since both described what every weaver would have already known. As noted above, besides the discussions on 4/1 twill vs. satin weave, which were presented as four theorems, the other definitions are partial, since they do not consider the relations between the consecutive rows of the weft. The definitions and theorems mostly reflect an analysis of the end product (the weave, the fabric) rather than of the practice itself. This may lead to the conclusion that the future audience for this set of notes, had it been published, was not the weavers but scholars and mathematicians (at least when considering the parts of the text written in Latin), and hence that the new knowledge prompted by this trading zone was not necessarily addressed to the artisans.

• 34 It is remarkable that a “proof” [“demonstration”] is mentioned only in theorem III (TC, fol. 63r). (...)
• 35 This certainly echoes Bacon’s reflections on “textura,” as noted above. On this notion in the thoug (...)
• 36 “Observavi per lentem vel et nudo visu, filorum situm multo rariorem esse situ filorum staminis.”

27This leads to the second part of the last question: how might the proposed geometrical structure have led to new mathematical knowledge? While trying to answer this question, one encounters another problem: the proposed separation between “definitions” and “theorems” is problematic for practical geometers too: the “theorems” do not follow from the “definitions,” and no logical structure is proposed for the deduction of new theorems.34 One can conjecture that the proposed title “ad scientiam situs” (concerning knowledge of position) relates to Jungius’s attempt to describe a geometry whose theorems are obtained by a consideration of relational position (e.g. the relations between the rows of the weft) and not of metrical relations;35 however, this attempt is never completed — although on several occasions Jungius does discuss the “position of the threads of the warp.”36 (TC, fol. 16r) This means that, even though Jungius’s theorems may be considered as descriptive (at least by weavers), they should still be considered as an attempt to reorganize artisanal knowledge and present it not only within a geometrical framework but also as an epistemic practice, that is, as something that may prompt new mathematical theorems.

• 37 The date indicated on all of these folios is “March [16]21”. The folios are 53r, 54r, 60r, and 61r.
• 38 The last two terms (weft and warp) are given both in German and in Latin: “Ein iedes geweb bestehet (...)

28While the introduction of the “definitions” and “theorems” for weaves would have been somewhat irrelevant to the weavers, if one takes a look at the German parts of the manuscript, one may find a clue to how, according to Jungius, mathematics was practically used in weaving. On a set of four folios, all written in German,37 Jungius gives a definition of weaves in general, noting the importance of order and position, and using geometrical terms such as “parallel” and “perpendicular”: “Every weave consists of two orders of threads presented in such a way that one order of threads is parallel or concurrent, and is in a perpendicular [position] to the threads of the other order. One order is [called] warp, the other series or order is [called] weft.”38 (TC, fol. 53r) This description, one may say, would have been obvious or even banal to the weaver. While this description uses basic geometrical terms, another folio presents a set of calculations:

• 40 Meinel is referring to the following manuscript by Jungius: Gottfried Wilhelm Leibniz Bibliothek - (...)
• 41 “Taffetum ita per anchiscopium apparebat.” Taffeta is a fabric made from silk and woven as a plain (...)
• 42 “Hac ex Anchiscopio deprehensa sunt.”
• 43 (TC, fol. 20r): “But then when I was looking through the microscope [...]” (“Verum cum deinde per a (...)
• 44 “fila [...] ita in situ [...] apparent per anchiscopium, at spatium aliquod sit inter fila haec sec (...)

32I will begin with Jungius’s investigation of fabrics with the aid of a magnifying lens: As Meinel notes, in 1633 Jungius used textiles to investigate “homogeneous substances” using a magnifying glass, observing that “they were in fact always heterogeneous if viewed through a microscope (anchiscopium).”40 (Meinel, 1988, p. 84) In TC, too, one finds descriptions of Jungius’s use of the microscope to investigate fabrics. On a set of two folios titled “taffeta as it appeared through a microscope,”41 Jungius notes that taffeta is “simple and uniform” [“simplex et uniformis”] (TC, fol. 15r). Following the investigation of this fabric, Jungius remarks that he “observed through a lens or by the naked eye [...] the position of the threads in the weft,” and that the order of the threads was detected by the microscope (TC, fol. 16r).42 He also emphasizes the use of the microscope to discover the order of woven threads, which he imagined to be simpler.43 Moreover, he points out that a diagrammatic representation of woven threads can be deceiving: “the threads of the row appear in this position [...] through the microscope, but there may be some space between the threads of the second row and the threads of the third row, [...] which has not been either observed or expressed in the previous diagrams.”44 (TC, fol. 25r) This indicates a certain mistrust in the knowledge that a diagram may convey, a mistrust that is at odds with the numerous diagrams and drawings which Jungius drew when composing the geometrical part in TC.

• 45 Sean Silver points out that “Hooke’s argument is to consider texture first as an artifact of the lo (...)
• 46 On how household items were used and reused in the observations and experiments of early modern nat (...)

33Jungius was not alone in examining textiles with the help of a magnifying glass; with this kind of investigation, he can be associated with other 17th century thinkers such as Robert Hooke and Henry Power, who also reflected on texture and textiles using the microscope (Silver, 2020, p. 185-189; Doherty, 2012, p. 223-224). Hooke’s Micrographia (1665), for example, presents research on textiles (e.g. linen or silk) carried out with the aid of a microscope, describing and depicting the texture and structure of these textiles.45 This may indicate that textiles, which were domestic objects, were available almost immediately to be examined;46 however, it also shows that Jungius’s (and Hooke’s) investigations went beyond a consideration of artisanal practices, since a microscope did not usually belong to the weaver’s set of instruments. Here again, in a somewhat similar way to Jungius’s geometrical reflections, one finds that the trading zone between weavers and Jungius also prompted investigations that lie outside the context of this zone. This leads me to the second aspect of Jungius’s work on textiles, which can be considered as departing from a mere description of artisanal practices: using textiles to clarify a theory of materials.

• 47 On Jungius’s reflections on textiles with respect to his theory of hypostatic and synhypostatic par (...)
• 48 “Ita filum est pars hypostatica panni; potest enima separari ex panno, ita ut non sit amplius pars (...)
• 49 “Exempli gratia si quis contactum, mutuum ordinem et situm filorum, secundum quem se invicem subeun (...)

34In the framework of his theory of materials, Jungius introduces two types of parts of a body: hypostatic parts, being parts which can exist outside a body after its decomposition; and synhypostatic parts, being parts which cannot exist or persist by themselves once a body has been decomposed (Jungius, 1982, p. 100). One of the examples given by Jungius to illustrate these two types of parts is a piece of fabric; this example is presented in his 1642 book Disputationum de principiis corporum (Jungius, 1642, p. 377).47 According to Jungius, the decomposition of a woven fabric gives rise to threads, which can exist independently of the fabric; furthermore, the threads themselves can be decomposed into fibers; thus, “the thread is a hypostatic part of the fabric; it can be separated from the linen in such a way that it is no longer part of a fabric. In the same way, the fiber is a part of the thread, which is twisted from several fibers.”48 (§34) However, if one considers the contact, arrangement, and position of the threads of the fabric, then while these are also parts of the fabric — since they give it its specificity and structure — in this case one can speak only of synhypostatic parts. For if the woven fabric is dissolved and ceases to be a fabric, then the order, position, and mutual contact of the threads no longer exist. Accordingly, a woven fabric consists of parts which are hypostatic and of parts which are synhypostatic, whereby the latter are destroyed during the fabric’s decomposition (§37).49

• 50 This note is found in the set of notes titled Doxoscopiae physicae minores (but it was not included (...)

35Moreover, in his Doxoscopiae physicae minores, which was published after his death in 1662, Jungius notes that the position and sequence of the threads are sufficient to distinguish different kinds of cloth (according to their color shimmer or their tautness). “Sinuous” woven fabrics, for example, can be stretched in any direction, whereas a rectangular fabric can be stretched only diagonally (Jungius, 1662, Part 2, Sect. 1, assert. 7, §95-99; see §85 regarding the flexibility of fabrics). In addition, in an undated note50 Jungius draws interwoven threads as an analogy to the synhypostatic parts (see Fig. 4), threads which in TC are described as “Textura Sinuosa” — and these threads are in fact knitted ones, as I will discuss below.

• 51 As indicated above (see section 2.2), Jungius reports that arithmetical calculations were recounted (...)

37Until now I have concentrated on Jungius’s mathematically and scientifically oriented reflections. The previous subsection made clear that Jungius’s considerations on fabrics went beyond their use in artisanal practices. Furthermore, section 2 as a whole raises the question, which was already partially answered positively, whether Jungius was in contact with the artisans themselves, and whether he saw with his own eyes how weaving or other textile practices (knitting or ribbon weaving, to give two examples) were done in practice.51 It also raises again the question of how Jungius reshaped and reorganized artisanal knowledge and practices. This reorganization could already be seen in the presentation of “theorems” for weaves or in the consideration of their synhypostatic parts, but one has to wonder whether it does not apply also to other textile practices. In the following, I will concentrate on these two questions, first, by concentrating on Jungius’s acquaintance with the treadle loom and with various practical stages of weaving, and, second, through a consideration of Jungius’s familiarity with knitting.

• 52 “Ungefehrl. 60 sajenmacher sind zu Hamburg.”

38In one of his notes, Jungius remarks that “[t]here are about 60 saje weavers in Hamburg”52 (TC, fol. 31r) — which indicates not only that he was aware of the existence of these weavers as a distinct group, but that he was also trying to approximate how many weavers there were in Hamburg. How Jungius obtained this information is unknown, but the question remains whether he visited the ateliers of the textile workers or observed their work? Did Jungius observe a loom? Answering this question is essential to establish the nature of the discussed trading zone between Jungius and the artisans.

• 53 As Meinel notes, Jungius used a single page for each topic he wrote about, which means that the cop (...)
• 54 The words are transcribed as they are written in the copy of Jungius’s notes on folio 55r of TC.
• 55 To explicate: by stepping on one treadle, one raises one set of heddles (and hence one set of threa (...)

40Folio 55r is divided into two parts.53 The upper part shows a two-treadle loom, for which Jungius has also drawn a foot pressing on one of the treadles. Some of the different parts of the loom are described in German: “kam” (“comb,” i.e. the reed), “einschlag” (the thread of the weft), “schlauff” (the loop of one of the threads of the warp around one of the heddles), and “pedahl” (treadle).54 However, this depiction of the loom is only partial, as can be seen when one compares it to another drawing of a two-treadle loom from 1586 (see Fig. 5b); for example, neither the shuttle nor the beater is drawn (or mentioned), nor is there any indication of the relations between the treadles and heddles either in the description or in the drawing. While the drawing in TC may make clear how a two-treadle loom operates,55 it does not indicate what happens when one works with looms having more than two treadles. By tying the treadles to several heddles, these looms can be used to obtain more complicated patterns. The process of determining which treadles of the loom are connected to which heddles is today called “tying up.” For example, to obtain a 2/2 twill pattern, a weave which Jungius certainly knew and which is shown in a drawing in TC (see Fig. 3), one would have to use a four-treadle loom, whereby each treadle is tied up to two heddles (or harnesses). Jungius was certainly aware that these looms existed: in another set of pages, Jungius mentions treadle looms with 3, 4, 5, 6, or even 8 treadles (TC, fols. 26r, 27r). Moreover, the relations determining the tying up — as well as, subsequently the connection between the tying up, the pressing of the treadles, and the resulting pattern – are not at all trivial; it has been shown (Brezine, 1993, 2009) that one can present these relations with the mathematical structure of matrix multiplication. This, however, could not have been known to Jungius, since matrix multiplication was only developed during the 19th century.

• 56 “Ein Zwele est surcula lignea, damit scheid sie die scheringe in obersten parallehl darnach scheide (...)
• 57 Compare also section 2.2. In (TC, fol. 51r) Jungius also describes the warping frame.

41The lower part of folio 55r contains a description of the laying out of the warp threads on a warping frame in an ordered manner: “A branch is a wooden twig; with this, one separates parallel-wise the warp on the top [branch]; then one separates according to the same shape below on the wooden nails of the frame. These two separations are done so that they [the threads] can then use the comb [reed] properly or can [be] put in it.”56 (TC, fol. 55r) As this is the only description Jungius gives of this process, one can only assume that the “wooden nails” he refers to are actually the wooden pegs around which the thread is wound up — as suggested by Jungius’s drawing in the lower part of the folio (see also Fig. 5c).57 But laying out the warp threads on a warping frame is only one step before starting to weave; how one threads the heddles, winds the warp, ties the warp on, or ties up the treadles to the heddles — all of these steps are not mentioned in TC.

• 58  This may be explained by the fact that Jungius was not present during such a preparation, or that (...)

42When considering the description of weaving as a practice, it is remarkable that there is no description of a loom’s preparation;58 nevertheless, Jungius does describe other textile practices: for example, the knitting of stockings (“gestrickte Strümpfe” (TC, fol. 56r)) and the weaving of baskets (fol. 58r). In the following, I would like briefly to discuss the presentation of knitting in TC, in order to examine again whether Jungius’s account would have been relevant to the practitioners themselves.

• 61 Compare also (Daston, 1991, p. 355) concerning the collection of Baconian facts during the 17th cen (...)
• 62 Recall that TC was composed over a period of almost 30 years.
• 63 See also: (Rabouin, 2020, p. 20). It should also be mentioned that the first published attempt to m (...)

47In sections 2 and 3, a survey of the main themes of Jungius’s TC was presented: from Jungius’s geometrical reflections and his arithmetical calculations to his investigations with the microscope and his theory of hypostatic and synhypostatic parts, finishing with his description of some artisanal practices, such as knitting or preparing the warp. As can be seen from the variety of the themes presented, TC, as a collection of notes, can be considered as having a non-hierarchical form made up of a scattered collection of slips of paper and pieces of knowledge (Krajewski, 2011, p. 17). Such a collection, whose final system of organization remains impossible to determine, can be described as a “Verzettelung” of knowledge, to cite Meinel (1995, p. 166), when it is not clear in advance how the various themes and notes relate to one another.61 Indeed, one may claim that these notes do not accumulate facts in order to create a coherent single body of knowledge on one single topic. Rather, they may be conceived as a non-centralized text to which further notes could (and would) be continuously added, and which never reached a final form or a final state of completion.62 That being said, and although this set of notes was never officially edited and published, Leibniz considered it important enough to make a partial copy, which he reorganized and edited himself, even copying the pattern of the knitted threads (see Fig. 8).63

• 64 See: (Long, 2011, p. 125-126), who notes that it is books and the printed medium which facilitated (...)

50The answer is more complex than a yes or a no and takes us back to the discussion on trading zones. As we have already seen in the discussion on Jungius’s weaving terminology above, it is possible that Jungius invented his own terms, which points to difficulties in establishing the existence of a “contact language.” While Jungius did not ignore either the practitioners or their working processes, weavers would have found his descriptions almost banal, and his philosophical and geometrical reflections on textiles irrelevant. This calls for a nuancing and relativizing of Pamela O. Long’s statement concerning trading zones in the early modern period, according to which these not only “consisted of arenas in which the learned taught the skilled, and the skilled taught the learned,” but also “in which the knowledge involved in each arena was valued by both kinds of ‘traders.’” (Long, 2011, p. 95) Which is to say that, while Jungius did meet and observe these artisans, the effects of these encounters on mathematicians on the one hand and on weavers on the other hand did not necessarily take place on both sides of this trading zone and were not necessarily reciprocal. But here we should relativize the claim just presented: due to the lack of written material or description from the weavers themselves (in Hamburg, in particular regarding their working methods), it is difficult to assess whether Jungius’s reflections exercised any influence on the weavers.64 Nevertheless, if TC had been edited and published, it could be considered as an attempt to combine theoretical and practical observations in a single work, sometimes presenting a more theoretical description of existing techniques alongside diagrams of the ideal, annotated shapes of threads, and alongside various definitions and theorems.