# Euclid

‘A Work! Whose Propositions have such an admirable Connexion and Dependence, that take away but one, and the whole falls; whose Method is the most just, admitting nothing without a Demonstration, and no Demonstration but from what foregoes; and these so convincing, elegant and perspicuous, that it is beyond the Skill of Man to contrive better.’

Edmund Stone (ed.), Euclid’s Elements… Now first translated from Dr. Gregory’s edition (London, 1745), ii, Sig A3v.

Euclid’s famous Elements had first been printed at Venice in 1482 by Erhard Ratdolt. Worth owned a number of editions, primarily seventeenth and early eighteenth century printings. His earliest edition was by Federico Commandino, published in London in 1620; his second text was printed at the same venue some thirty-nine years later by the renowned Cambridge Fellow, Isaac Barrow; and, finally, his last edition of the Elements, by the Newtonian inspired David Gregory, was printed at Oxford in 1703. The predominance among these commentators of links with universities was no coincidence for Euclid was the geometrical text studied at universities by early modern mathematicians.

Euclid, Elementorum Euclidis libri tredecim. Secundum vetera exemplaria restituti. Ex versione Latina Federici Commandini aliquam multis in locis castigate (London, 1620), p. 5.

This illustration, from the very beginning of Commandino’s text, illustrates why Euclid’s text was so popular: it began with basic definitions, propositions, postulates and axioms and continued to explain geometry in an austere style which eschewed lengthy excursions into non mathematical material. The example above, of Euclid’s famous Parallel Postulate, concerning in this case non-parallelism, gives us some indication of his style:

‘That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.’

As Saito (2010) notes, the very austerity of Euclid’s style inevitably lead to problems of interpretation.

The editor of Worth’s earliest edition, Federico Commandino (1506-1575), had been trained as a physician and had a lucrative career as such. He was, however, considerably more interested in mathematics and his edition of Euclid was only one of a number of editions of works by ancient mathematicians he produced in the mid sixteenth century – indeed Worth’s copy of Commandino’s 1558 edition of the works of Archimedes is examined in this web exhibition also. Commandino’s edition of Archimedes had received financial support from his patron, Cardinal Ranuccio Farnese (1530-1565) and, on Farnese’s death, Farnese’s brother-in-law, Duke Guidobaldo II della Rovere (1514-1574), had taken over this role. It was at the request of the latter’s heir, Francesco Maria II della Rovere (1549-1631), that Commandino’s Latin edition of Euclid’s Elements saw the light in 1572. An Italian translation quickly followed in 1575 and another edition, printed at Pesaro in 1619. This in turn was followed by a bi-lingual 1620 edition in Latin and Greek, which was the edition owned by Worth. The latter had been printed at London and edited by the renowned English mathematician, Henry Briggs (1561-1630).

Euclidis Elementorum libri xv. breviter demonstrati, opera Is. Barrow, Cantabrigiensis, Coll. Trin. Soc… (London, 1659), provenance.

This provenance signature on Worth’s copy of Isaac Barrow’s 1659 edition of the Elements draws attention to the role of Barrow’s edition as the pre-eminent textbook of its kind. Peter Forth had been admitted to Trinity College Cambridge on 31 July 1672 and was awarded his B.A. in 1677 and his M.A. in 1680, before going on to Grays Inn to study law. He was, therefore, a student at the same college as the author of the work, the noted Cambridge mathematician, Isaac Barrow (1630-1677). Barrow had himself been a student at Trinity College Cambridge in the 1640s, graduating B.A. in March 1649 and shortly afterwards becoming a Fellow. During the 1650s he concentrated his energies on natural philosophy and natural history, producing a number of editions of mathematical works, including a compact edition of the Elements in 1656. Forced by his royalist sympathies to travel abroad between 1655 and 1659, he journeyed as far as Constantinople. At the Restoration, those same royalist sympathies paved the way for a host of honours, honours richly deserved by the learned Barrow. By 1662 he had been appointed Gresham Professor of Geometry and in the same year he became a Fellow of the Royal Society. In the following year he was appointed as the first Lucasian Professor of Mathematics at Cambridge, holding the position until 1669 before handing the chair to Isaac Newton. In 1673 he was appointed as Master of Trinity College Cambridge, a position he held until his death in May 1677.

Feingold (1993) has rightly pointed to Barrow’s importance as a mathematician, an importance all too often overshadowed by his more famous protégé, Newton, but contemporaries such as David Gregory were well aware of Barrow’s importance as a mathematician. As Guicciardini (2010) points out, Gregory in his ‘Calculus Differentialis Leibnizij et Methodus Fluxionum Newtoni tantum nomine tenus different […] et facile fluunt ex Methodo Tangentium Barrovij Lect: 10. Geom. Tradita’, a manuscript now in Christ Church Oxford, emphasised that Leibniz’s and Newton’s method ‘differ only in name’ and ‘flow easily from Barrow’s Method of Tangents’.

Barrow’s textbook was popular for a number of reasons: it was portable, designedly contained as many demonstrations as possible, and was succinct but at the same time left nothing out. Unlike Andreas Tacquet’s edition, which Barrow references in his introduction, Barrow’s own edition covered all the books of the Elements, using Pierre Hérigone’s 1644 Cursus mathematicus as his model. But Barrow not only wanted it to be both comprehensive and portable – he also advocated the use of symbolical over verbal demonstrations. All these aims were linked to the pedagogical nature of Barrow’s edition: as Feingold (1990) remarks, ‘With the possible exception of Barrow’s work on Archimedes, Apollonius, and Theodosius, all his mathematical publications were the outcome of his pedagogical work, and his first venture into print clearly demonstrates both his abilities as a teacher and his belief in the importance of teaching.’ The fact that the first edition of the book (in 1656) was dedicated to his students at the time of its composition, further emphasises the point. It quickly became a bestseller, earning from Barrow’s contemporary John Aubrey the encomium ‘Let them always have Barrow’s Euclid in their pockets.’

Euclid, Euclidis quæ supersunt omnia. Ex recensione Davidis Gregorii … (Oxford, 1703), p. 369.

In this illustration, from the beginning of Book XII of David Gregory’s 1703 dual language edition of the Elements, we see Euclid proving that ‘circles are to one another as the squares of their diameters’. The Newtonian David Gregory, elected to the Savilian Chair of Astronomy at Oxford in 1691, was as anxious as Barrow had been in Cambridge to extol the virtues of Euclid and in 1703 his seminal edition of the Elements was published by the University Press. As Feingold (1997) notes, in 1700 Gregory had outlined a plan for teaching mathematics in seven ‘schools’: ‘Euclid, trigonometry, algebra, mechanics, optics, the principles of astronomy and cosmology.’ As we can see, ‘Euclid’ and geometry were synonymous and as Feingold notes for the seventeenth-century curriculum at Oxford, study of geometry invariably began with at least the first six books of Euclid, often supplemented by Peter Ryff’s Quaestiones geometricae (the text of which was purchased by Worth). Gregory’s dual language Greek-Latin edition aimed to be as comprehensive as possible. Aware of the continuing appeal of Commandino’s edition, Gregory was at pains to emphasise that his new edition out-stripped all earlier editions. In doing so he included not only works genuinely by Euclid but also texts which had been falsely attributed to him for centuries. As Saito (2010) reminds us, all the Greek manuscripts which formed the basis of early modern editions of the Elements, and more generally Euclid’s work, derived from a recension which had been made by Theon of Alexandria in the fourth century AD, a recension which contained not only work by Euclid but propositions by Theon.

Composite image of Pythagorean theorem from all three editions in Worth.

Sources

Euclid, Euclidis Elementorum libri xv. breviter demonstrati, opera Is. Barrow, Cantabrigiensis, Coll. Trin. Soc... (London, 1659), edited by Barrow.
Euclid, Elementorum Euclidis libri tredecim. Secundum vetera exemplaria restituti. Ex versione Latina Federici Commandini aliquam multis in locis castigate (London, 1620), edited by Commandino.
Euclid, Euclidis quæ supersunt omnia. Ex recensione Davidis Gregorii … (Oxford, 1703), edited by Gregory.
Feingold, Mordechai (ed.), Before Newton: The Life and Times of Isaac Barrow (Cambridge University Press, 1990).
Feingold, Mordechai, ‘Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation’, Isis, 84, no.2 (1993), 310-338.
Feingold, Mordechai, ‘The Mathematical Sciences and New Philosophies’, in Nicholas Tyacke (ed.) The University of Oxford Vol IV: The Seventeenth Century (Oxford University Press, 1997).
Feingold, Mordechai, ‘Barrow, Isaac (1630–1677)’, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, May 2007 [http://www.oxforddnb.com/view/article/1541].
Grattan-Guinness, Ivor, The Fontana History of the Mathematical Sciences (London, 1997).
Guicciardini, Niccolò, ‘‘Gigantic implements of war’: images of Newton as a mathematician’, in Eleanor Robson and Jacqueline Stedall (eds.) The Oxford Handbook of The History of Mathematics (Oxford University Press, 2010), 712.
J J O’Connor and E F Robertson, Federico Commandino, MacTutor History of Mathematics, (University of St Andrews, Scotland, February 2000):
http://www-history.mcs.st-andrews.ac.uk/~history/Biographies/Commandino.html
Saito, Ken, ‘Reading ancient Greek mathematics’, in Eleanor Robson and Jacqueline Stedall (eds.) The Oxford Handbook of The History of Mathematics (Oxford University Press, 2010), 801-826.
J. E. Stephens (ed.), Aubrey on Education: A Hitherto unpublished manuscript by the author of Brief Lives (Abingdon, 2012).
Venn, John and J. A. Venn (eds.) Alumni Cantabrigiensis (Cambridge University Press, 1922), Part II volume I, 169.

Text: Elizabethanne Boran.