Abstract
This chapter investigates the status and designation of motion and genetic definitions in Henry Billingsley’s commentary on Euclid’s Elements. It also considers his discourse on the two definitions of the sphere by Euclid and Theodosius, comparing it with that held by Foix-Candale before him. In this framework, the different meanings Billingsley attributed to the notion of “description” are examined. The chapter concludes with Billingsley’s kinematic treatment of magnitudes in his commentary on Df. II.1 and its significance to the relation between arithmetic and geometry in the context of the Elements. This analysis shows in particular that Billingsley, in his consideration of genetic definitions, did not so much point to a metaphysical interpretation of the origin of geometrical objects, but rather emphasised the relationship between the abstract and concrete modes of production of geometrical objects. It also enables us to see that, although Billingsley admitted the distinction held by Foix-Candale between genetic definitions and definitions by property, he endowed genetic definitions with a stronger epistemic value than his French predecessor did. This chapter shows additionally how Billingsley saw in the motions of figures not only an object of geometry in its own right, but also a key to understanding certain fundamental principles of arithmetic and algebra.
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Notes
- 1.
This is assumed by his biographers on the basis that his father’s activity as haberdasher was settled in London.
- 2.
- 3.
Vermigli (transl. Billingsley 1568). Billingsley mentioned, in his commentary on the Elements, that he had completed an English translation of the Spherics of Theodosius, which he intended to publish, but I have not found any traces of this translation: Billingsley (1570, fol. 315v), Df. XI.12: “Theodosius in his booke De Sphericis (a booke very necessary for all those which will see the groundes and principles of Geometrie and Astronomie, which also I have translated into our vulgare tounge, ready to the presse)”. (My emphasis.)
- 4.
- 5.
Zamberti (1558). The fact that this volume belonged to and was annotated by Billingsley is indicated by the presence of the manuscript inscription Henricus Billingsley on the title page. This title page is reproduced in Archibald (1950, p. 447). This was previously noted in Halsted (1879), but R. Archibald was able to verify that the style of the inscription corresponds to Billingsley’s handwriting.
- 6.
- 7.
- 8.
- 9.
See supra, n. 14, p. 100.
- 10.
Billingsley (1570, sig. 2v).
- 11.
Billingsley notably took care to indicate the different parts of each proposition in the margin (e.g. construction, demonstration, things given, things required…).
- 12.
On practical mathematical culture in early modern England, see Harkness (2007, chap. 3).
- 13.
Proclus (Friedlein 1873, p. 57) and (Morrow 1992, p. 46): “Let us next speak of the science itself that investigates these forms. Magnitudes, figures and their boundaries, and the ratios that are found in them, as well as their properties, their various positions and motions—these are what geometry studies”. See supra, n. 55, p. 12.
- 14.
Billingsley (1570, fol. 311v).
- 15.
Although he wrote that the flow of the point is the principle of quantity in general at first (“from a point, which although it be indivisible, yet is it the beginning of all quantitie”), he afterwards limited it to continuous quantity (“of the motion and flowing therof is produced a line, and consequently all quantitie continuall”).
- 16.
Proclus (Friedlein 1873, p. 97). See supra, n. 107, p. 25.
- 17.
Billingsley (1570, fol. 1v), Df. I.2.
- 18.
Billingsley (1570, fol. 3r), Df. I.16.
- 19.
- 20.
Billingsley (1570, fol. 5v–6r), Post. 2: “And a line is a draught from one point to an other, therfore from the point b, which is the ende of the line ab, may be drawn a line to some other point, as to the point c, and from that to an other, and so infinitely.”
- 21.
Billingsley (1570, fol. 6r), Post. 3.
- 22.
Billingsley (1570, fol. 2r–v), Df. I.7.
- 23.
See supra, p. 10.
- 24.
This Pythagorean definition of the point as a unit which is endowed with a position is mentioned by Aristotle in De anima I.4, 409a6 (transl. Barnes 1995, I, p. 652): “for a point is a unit having position”, and, in Metaphysics M.8, 1084b26, by defining the unit as a point without position (Barnes 1995, vol. II, p. 1714): “for the unit is a point without position.” See also Proclus (Friedlein 1873, p. 95) and (Morrow 1992, p. 78): “Since the Pythagoreans, however, define the point as a unit that has position, we ought to inquire what they mean by saying this.”
- 25.
Billingsley (1570, fol. 1r), Df. I.1.
- 26.
See, for instance, Proclus (Friedlein 1873, pp. 54–55) and (Morrow 1992, pp. 43–44): “When, therefore, geometry says something about the circle or its diameter, or about its accidental characteristics, such as tangents to it or segments of it and the like, let us not say that it is instructing us either about the circles in the sense world, for it attempts to abstract from them, or about the form in the understanding. For the circle [in the understanding] is one, yet geometry speaks of many circles, setting them forth individually and studying the identical features in all of them; and that circle [in the understanding] is indivisible, yet the circle in geometry is divisible. Nevertheless we must grant the geometer that he is investigating the universal, only this universal is obviously the universal present in the imagined circles. Thus while he sees one circle [the circle in imagination], he is studying another, the circle in the understanding, yet he makes hid demonstrations about the former. For the understanding contains the ideas but, being unable to see them when they are wrapped up, unfolds and exposes them and presents them to the imagination sitting in the vestibule; and in imagination, or with its aid, it explicates its knowledge of them, happy in their separation from sensible things and finding in the matter of imagination a medium apt for receiving its forms.” (My emphasis.)
- 27.
Proclus (Friedlein 1873, pp. 95–96) and (Morrow 1992, p. 78): “Since the Pythagoreans, however, define the point as a unit that has position, we ought to inquire what they mean by saying this. That numbers are purer and more immaterial than magnitudes and that the starting-point of numbers is simpler than that of magnitudes are clear to everyone. But when they speak of the unit as not having position, I think they are indicating that unity and number—that is, abstract number—have their existence in thought; and that is why each number, such as five or seven, appears to every mind as one and not many, and as free of any extraneous figure or form. By contrast the point is projected in imagination and comes to be, as it were, in a place and embodied in intelligible matter. Hence the unit is without position, since it is immaterial and outside all extension and place; but the point has position because it occurs in the bosom of imagination and is therefore enmattered.”
- 28.
See supra, p. 147.
- 29.
Billingsley (1570, fol. 315r), Df. XI.12.
- 30.
The combined use of the verbs move, flow and glide to describe the generation of the line from the point tends to suggest that the flowing of the point is here understood in a quasi-physical manner, at least as relatable to concrete types of local motion.
- 31.
Billingsley (1570, fol. 315v), Df. XI.12.
- 32.
See, for instance, Billingsley (1570, fol. 57r), Prop. I.46: “[In marg. To describe a square mechanically] This is to be noted that if you will mechanically and redily, not regarding demonstration describe a square upon a line geven, as upon the line ab, after that you have erected the perpendiculer line ca upon the line ab, and put the line ae equall to the line ab: then open your compasse to the wydth of the line ab or ae, & set one foote thereof in the point e, and describe a peece of the circumference of a circle: and againe make the centre the point b, and describe also a piece of the circumference of a circle, namely, in such sort that the peece of the circuference of the one may cut the peece of the circumference of the other, as in the point d: and from the point of the intersection, draw unto the points e & b right lines: & so shalbe described a square.” (My emphasis.)
- 33.
Pacioli (1509b).
- 34.
Billingsley (1570, fol. 319r), Df. XI.24. (My emphasis.)
- 35.
Billingsley (1570, fol. 320r), Df. XI.25.
- 36.
According to H. Smith (Smith 2017), “pasted paper” (“such as paste-wives make womens pastes of”), would correspond to moulded paper that was used to support fashionable headgear.
- 37.
These were then not only to be made out of nets that were reproduced by the reader on pasted paper, as those shown in appendix of Df. XI.25 (fol. 320v–322v), but also out of printed paper slips provided in the book, which were to be cut out, folded and glued directly onto the printed two-dimensional diagram, as in the case of the diagrams of Df. XI.2–5, 9–11 (fol. 312r–313r and 314r–v).
- 38.
See supra, p. 151.
- 39.
Oosterhoff (2020). See also Lefèvre ’s words on this analogy in his commentary on Sacrobosco (Lefèvre 1495, sig. a4r): “Et haec profecto mire efficaciae descriptio est, quae aperte docet (quantum sensibilis materia recipere valet) artificialem constituere sphaeram, cuius utilem commodamque intelligentiam nostrae tempestatis artifices multis auri pondo comparare deberent: qui metallo, ligno, aut alia materia figuras torno exprimere volunt. Si itaque in levi calybe aut ferro, sumpto circino supra quancunque lineam semicirculus educatur qui ab arcu ad diametrum usque excavetur, quin immo et medium diametri interstitium, et mox ad arcum circumferentiamque excavatur ut ea ex parte ad scindendum secandumque fiat aptus, exurget instrumentum tornandis sphaeris (haud secus quam circinus circulis) aptissimum. Hanc utilitatem sua descriptione nobis attulit Euclides, illamque intendebat cum dicere sphaeram esse transitum dimidii circuli, quae (fixa diametro) quousque ad locum suum redeat circumducitur, abditam, occultamque tamen, ut solis studiosis pateret.”
- 40.
Billingsley did not himself write a commentary on Sacrobosco ’s Sphaera, but he did refer to it when dealing with Euclid’s definition of the sphere. Billingsley (1570, fol. 315v): “And it is fully round and solide, for that it is described of a semicircle which is perfectly round, as our countrey man Iohannes de Sacro Busco in his book of the Sphere, of this definition which he taketh out of Euclide, doth well collecte.” (My emphasis.) In this context, Billingsley attacked Sacrobosco ’s rendering of Euclid’s definition of the sphere, which states that only the circumference of the rotating semicircle, and not its whole surface, generates the sphere: “But it is to be noted and taken heede of, that none be deceived by the definition of a Sphere even by Iohannes de Sacro Busco: A Sphere (sayth he) is the passage or moving of the circumference of a semicircle, till it returne onto the place where it beganne, which agreeth not with Euclide. Euclide plainly sayth, that a Sphere is the passage or motion of a semicircle, and not the passage or motion of the circumference of a semicircle: neither can it be true that the circumference of a semicircle, which is a line should describe a body. It was before noted that every quantitie moved, describeth and produceth the quantities next unto it. Wherefore a line moved can not bring forth a body, but a superficies onely, which is the superficies and limite of the Sphere, and should not produce the body and solidity of the Sphere.”
- 41.
Billingsley (1570, fol. 315v), Df. XI.12: “This definition of Theodosius is more essentiall and naturall, then is the other geven by Euclide. The other did not so much declare the incard nature and substance of a Sphere, as it shewed the industry and knowledge of a producing of a Sphere, and therefore is a causall definition geven by the cause efficient, or rather a description then a definition. But this definition is very essentiall, declaring the nature and substance of a Sphere.” See also supra, p. 151.
- 42.
Aristotle , Physics II.3, 194b30–32 (Barnes 1995, I, fol. 333): “The primary source of the change or rest; e.g. the man who deliberated is a cause, the father is cause of the child, and generally what makes of what is made and what changes of what is changed.”
- 43.
On this debate, in which was challenged the notion that mathematical demonstrations represented the highest form of demonstrations (demonstrationes potissimae), i.e. demonstrations that display simultaneously the fact and its cause, see De Pace (1993, pp. 21–120), Mancosu (1996, pp. 10–33) and Higashi (2018). See also Feldhay (1998).
- 44.
Higashi (2018, pp. 123–276 and 364–393).
- 45.
See, for instance, Barrow (1734, V, p. 83): “it seems plain that Mathematical Demonstrations are eminently Causal, from whence, because they only fetch their Conclusions from Axioms which exhibit the principal and most universal Affections of all Quantities, and from Definitions which declare the constitutive Generations and essential Passions of particular Magnitudes. From whence the Propositions that arise from such Principles supposed, must needs flow from the intimate Essences and Causes of the Things.” (My emphasis.) On these authors and their attitude toward genetic definitions in geometry, see Mancosu (1996, pp. 94–100).
- 46.
This attitude is canonically expressed by Francis Bacon (1561–1626) in the New Organon II, Aphorism 9, in Bacon (transl. Silverthorne 2000, p. 109): “A true division of philosophy and the sciences arises from the two kinds of axioms which have been given above […]. The inquiry after forms, which are (at least by reason and their law) eternal and unmoving, would constitute metaphysics; the inquiry after the efficient and material causes, the latent process and latent structure (all of which are concerned with the common and ordinary course of nature, not the fundamental, eternal laws) would constitute physics; subordinate to these in the same manner are two practical arts: mechanics to physics; and magic to metaphysics (in its reformed sense), because of its broad ways and superior command over nature.” (Emphasis is proper to the quoted edition.) or by Descartes , in his Traité du monde (see infra, p. 266).
- 47.
Cf. Billingsley (1570, fol. 1v), Df. I.2: “A lyne is the movyng of a poynte, as the motion or draught of a pinne or a penne to your sense maketh a lyne”.
- 48.
On the relation between genetic definitions and the arithmetical understanding of magnitudes, notably within Billingsley’s commentary on Euclid, see also Malet (2006, pp. 73–74).
- 49.
Billingsley (1570, fol. 60r), Df. II.1.
- 50.
This axiom was a later addition to Euclid’s text (De Risi 2016b, p. 15).
- 51.
Vitrac (1990, I, p. 325).
- 52.
Billingsley (1570, fol. 60r–v), Df. II.1.
- 53.
See supra, p. 67.
- 54.
Billingsley (1570, fol. 1v), Df. I.3: “For a line hath his beginning from a point, and likewise endeth in a point: so that by this also it is manifest, that pointes, for their simplicitie and lacke of composition, are neither quantitie, nor partes of quantitie, but only the termes and endes of quantitie. […] And herein differeth a poynte in quantitie, from unitie in number for that although unitie be the beginning of nombers, and no number (as a point is the beginning of quantitie, and no quantitie) yet is unitie a part of number. For number is nothyng els but a collection of unities, and therfore may be devided into them, as into his partes. But a point is no part of quantitie, or of a lyne neither is a lyne composed of pointes, as number is of unities. For things indivisible being never so many added together, can never make a thing divisible, as an instant in time, is neither tyme, nor part of tyme, but only the beginning and end of time, and coupleth & joyneth partes of tyme together.” In this discourse, Billingsley is quite close to Foix-Candale in the passage quoted above, n. 32, p. 105.
- 55.
Billingsley (1570, fol. 62r), Prop. II.1: “Because that all the Propositions of this second booke for the most part are true both in lines and in numbers, and may be declared by both: therefore have I have added to every Proposition convenient numbers for the manifestation of the same. And to the end the studious and diligent reader may the more fully perceave and understand the agrement of this art of Geometry with the science of Arithmetique, and how nere & deare sisters they are together, so that the one cannot without great blemish be without the other, I have here also ioyned a little booke of Arithmetique written by one Barlaam, a Greeke authour a man of greate knowledge. In whiche booke are by the authour demonstrated many of the selfe same proprieties and passions in number, which Euclide in this his second boke hath demonstrated in magnitude, namely, the first ten propositions as they follow in order. Which is undoubtedly great pleasure to consider, also great increase & furniture of knowledge. Whose Propositions are set orderly after the propositions of Euclide, every one of Barlaam correspondent to the same of Euclide.” On Barlaam of Seminara ’s version of Euclid’s Book II, see Corry (2013).
- 56.
Billingsley (1570, fol. 60r), Book II, preface.
- 57.
However, there is no direct use here of the geometrical term ductus (or of an equivalent English term) to directly designate the arithmetical operation of multiplication, in spite of the fact that it was commonly used to speak of the generation of numbers by multiplication, especially in the case of square or cubic numbers. See supra, p. 68–69.
- 58.
See supra, n. 55, p. 163.
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Axworthy, A. (2021). Henry Billingsley. In: Motion and Genetic Definitions in the Sixteenth-Century Euclidean Tradition. Frontiers in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-95817-6_5
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