Abstract
Barrow’s second set of Lucasian lectures were delivered in 1665, and this set along with the other sets were published in 1683 as Lectiones mathematicæ; see Feingold, ‘Isaac Barrow: Divine, Scholar, Mathematician’, pp. 68, 79. The extracts of the lecture reproduced here are from the English translation in Barrow, The Usefulness of Mathematical Learning, pp. 163–85.
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Notes
- 1.
Barrow’s second set of Lucasian lectures were delivered in 1665, and this set along with the other sets were published in 1683 as Lectiones mathematicæ; see Feingold, ‘Isaac Barrow: Divine, Scholar, Mathematician’, pp. 68, 79. The extracts of the lecture reproduced here are from the English translation in Barrow, The Usefulness of Mathematical Learning, pp. 163–85.
- 2.
In the previous Lecture IX Barrow explored the logical foundations of infinitesimals. According to Malet, ‘Barrow, Wallis, and the Remaking of Seventeenth Century Indivisibles’, pp. 75–81, Barrow upheld the thesis that magnitudes are actually composed, or consist of, an infinite number of infinitely small parts; and by assuming that space itself is infinite, one can produce innumerable instances of ‘an infinite number within an infinite number’. Lecture IX, therefore, leads to the two ontological problems treated in Lecture X.
- 3.
See ibid., pp. 181 and 170.
- 4.
Although Gassendi postulated an infinite God, yet because he wished to avoid the radical natural-philosophical conclusions of Cusanus and Bruno, he tended to be wary of infinitude in any form; see Brush in Gassendi, Selected Works, p. 382.
- 5.
The name of Samuel Parker was added to this list by Roger North; see Kassler, Seeking Truth, p. 186 n.306.
- 6.
See Lee (tr.), Plato Timaeus, pp. 71–2.
- 7.
The Second Book of Moses called Exodus iii.14: ‘And God said unto Moses, I AM THAT I AM’.
- 8.
Colie, Paradoxia Epidemica, p. 25, also for some early modern solutions to the ontological paradoxes of being, becoming and creation, pp. 145–68, 219–51 and 300–28. For a more complete treatment of pp. 300–28, see Colie, ‘Some Paradoxes in the Language of Things’.
- 9.
In this mathematical emphasis, Barrow differed (as does Newton) from Gassendi, Helmont and More, although they all held that space is eternal and uncreated, as noted above.
- 10.
See Barrow, The Usefulness of Mathematical Learning, pp. 171–2. Note that Samuel Clarke may have reproduced this demonstration in a letter to Roger North which is no longer extant but which was paraphrased by North in his reply to Clarke; see Kassler, Seeking Truth, pp. 130–39, pp. 133–4.
- 11.
For his ‘significant characterisation of three-dimensionality of space’ in Principia mathematica, see Bochner, The Role of Mathematics in the Rise of Science, p. 249. For Barrow’s impact on Newton during the latter’s formative years, see Feingold, ‘Newton, Leibniz and Barrow too’, pp. 312–24.
- 12.
Barrow owned several works of Aristotle, for which see Feingold, ‘Isaac Barrow’s Library’, (Nos. 68–73), p. 342.
- 13.
Note that at some point in the next decade, Newton echoed Barrow in writing that space ‘has its own manner of existence which fits neither substances or accidents [of matter]’, so it cannot be said ‘to be nothing, since it is rather something ... and approaches more nearly to the nature of substance. There is no idea of nothing, nor has nothing any properties; but we have an exceptionally clear idea of [geometrical] extension, abstracting the dispositions and properties of a body so that there remains only the uniform and unlimited stretching out of space in length, breadth and depth’; see Newton, ‘De gravitatione’, p. 132 (additions in brackets mine).
- 14.
I.e., Zeno of Elea, whose ‘Device’ referred to is probably not one of his paradoxes but one or another of the extant antimonies that concern philosophical problems of infinity; see Kirk, Raven and Schofield, The Presocratic Philosophers, pp. 265–9, and for Plato’s indebtedness to these, p. 279.
- 15.
The Epistle of St. Paul, the Apostle, to the Ephesians iii.9: ‘...who created all things by Jesus Christ.’
- 16.
The Revelation of St. John the Divine: iv.11: ‘...for thou hast created all things, and for thy leisure they are and were created.’
- 17.
The Gospel according to St. John: i.3: ‘All things were made by him; and without him was not any thing made that was made.’
- 18.
The Book of the Prophet Isaiah: lxvi.2 (not, as above, 1): ‘For all those things hath mine hand made....’
- 19.
The Second Book of the Chronicles: ii.6: ‘... seeing the heaven and heaven of heavens cannot contain him? ’; vi.18: ‘...behold, heaven and the heaven of heavens cannot contain thee....’ For other similar passages, see The Fifth Book of Moses called Deuteronomy x.14, and The First Book of the Kings viii.27.
- 20.
I.e., Principia philosophiæ. See Descartes, Principles of Philosophy, Pt. II, §21, p. 49: ‘...we understand that this world, or the universe of material substance, has no limits to its extension. For wherever we may imagine those limits to be, we are always able, not merely to imagine other indefinitely extended spaces beyond them; but also to clearly perceive that these are as we conceive them to be, and, consequently, that they contain an indefinitely extended material substance. Because (as has now been shown at length), the idea of that extension which we conceive in any space whatever, is exactly the same as the idea of material substance.’ First published in Latin in 1644, then in French in 1647, Barrow owned a copy of the first Latin edition; see Feingold, ‘Isaac Barrow’s Library’, (No. 321, p. 349.
- 21.
OED: A Persian measure, usually reckoned as equal to between 3 and 3½ English miles (1555–1882).
- 22.
In a manner appropriate to an audience of mathematical students, the remainder of the paragraph constitutes a brief instance of generalising from the observable (known) to the unobservable (unknown), a move that is central to Newton’s more developed method of reasoning from experience.
- 23.
See [p. 168] above.
- 24.
I.e., the new worlds.
- 25.
Possibly a reference to Plato’s compressed description of the rotation of the ‘Same’ and ‘Different’; see Lee (tr.), Plato Timaeus, pp. 49–50 and n.1.
- 26.
See Miller and Miller (trs.), René Descartes Principles of Philosophy, Pt.II §25, p. 51, regarding ‘What movement is properly speaking’, in which Descartes distinguished between ‘a transference’ and ‘the force or action which transfers’.
- 27.
OED obs. rare: Capable of being placed (1734, Barrow the sole citation). See also Impone OED v. obs. 1. trans: To place or set upon something; to impose (1529–1729); 2. intrans.: To impose upon (1640).
- 28.
OED obs. rare: Capacity of being interposed (1734, Barrow the sole citation). See also Interpone OED v. obs. = Interpose v. I.1.b trans.: To place things with intervals or in alteration (1602–91).
- 29.
In his Lectiones geometricæ, published in 1670, Barrow also defined time (in its real or absolute sense) as a capacity thus: ‘Time ... does not denote an actual existence but merely the capacity or possibility of a continuity of existence, just as Space denotes the capacity for intervening magnitude.’ See Stone, Isaac Barrow, p. 136, who, pp. 135–8, reproduces from William Whewell’s 1860 edition of the mathematical works, the whole of Barrow’s lecture ‘with but slight excisions’. See also Burtt, The Metaphysical Foundations of Modern Science, pp. 155–61.
- 30.
OED a. B. Lying beyond or outside the world; of or belonging to things beyond the limits of the solar system.
- 31.
OED a. obs. rare: Capable of being traversed or passed through; traversible (1656).
- 32.
I.e., the opinions chiefly of Aristotle and Descartes. Note, however, that Barrow also included a critique of Hobbes’s definition of space that is not included here; see Barrow, The Usefulness of Mathematical Learning, pp. 179–80.
- 33.
According to Jones, The Epicurean Tradition, Epicurus imagined that ‘body’ and empty space are the sole existents of an infinite universe. But he distinguished between the terms ‘universe’ and ‘world’ (of which he imagined a plurality) by conceiving the former as unbounded and the latter as circumscribed within boundaries. Perhaps, therefore, Barrow’s cryptic statement above refers to the bounded worlds, which are compounds generated from an infinite number of indivisible minimæ partes that constitute the permanent substratum of matter.
- 34.
OED a. obs.: Happening or existing before something else; preceeding, antecedent, previous (1647–1794).
- 35.
See [p.177] above.
- 36.
Descartes’s, whose main goal in De géométrie was the application of algebra to geometry, applied this new method to the construction of curves in Bk. II; see Descartes, Discourse on Method, pp. 190–227. For a brief description of his method, see Mahoney in Crombie, Mahoney and Brown, ‘René du Perron Descartes’, Dictionary of Scientific Biography, vol. 4, p. 57. For Newton’s response to this Cartesian method, see Guicciardini, Isaac Newton, p. 6, et passim. According to the notice of Barrow’s Lectiones geometricæ in Philosophical Transactions of the Royal Society (1671), 6: 2260–2, five lectures were devoted to determining the ‘Tangents of Curve Lines, in the tenth of which he delivered ‘a general Analytical method of determining Tangents, extending to all sorts of Curve lines, both Geometrical and Mechanical (as Monsieur Des-Cartes distinguisheth)’. If Barrow did deliver these lectures (and there is some dispute about this), then it is probable they were read after the last set of his Lucasian Lectiones mathematicæ; see Feingold, ‘Isaac Barrow’, pp. 68–9.
- 37.
For this cryptic remark, see Barrow, A Treatise of the Pope’s Supremacy, p. 318: ‘the greatest tyranny that ever was invented in the world’ was ‘the pretence of infallibility’, for ‘the Pope, not content to make us do and say what he pleaseth, will have us also to think so’; and such authority ‘will inevitably produce depravation of Christian doctrine’. The doctrine of infallibility, which was formally defined in 1869–70 but defended earlier, was built on the Roman Catholic dogma of Petrine supremacy of the Pope; hence the title of Barrow’s tract.
- 38.
For Barrow’s lecture on time, see p. 147 n.29 above.
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Kassler, J.C. (2018). Appendix. In: Newton’s Sensorium: Anatomy of a Concept. Archimedes, vol 53. Springer, Cham. https://doi.org/10.1007/978-3-319-72053-1_6
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