References
- A. C. Hilton, The Heathen Pass‐ee’, 1872 (in imitation of Bret Harte) in Michael Roberts (ed.), The Faber Book of Comic Verse (London, 1942), 240–2. I am grateful to Professor A. J. Meadows for this reference
- To avoid misunderstanding, by ‘rivals of Euclid’ (C. L. Dodgson's phrase) are to be understood those textbooks which tried to make Euclidean geometry clearer to tyros by the improvement of Euclid's definitions, the rearrangement of his theorems, and the use of new constructions and proofs not in Euclid. ‘Non‐Euclidean geometries’, on the other hand, refers to the new geometries of Lobatchevsky and Bolyai which used a different definition of parallelism from Euclid
- Hans , N. A. 1951 . New Trends in Education in the Eighteenth Century London
- Significant because it paved the way for the consideration of science and modern languages
- W. Whewell, On the Principles of English University Ėducation (London, 1837), 5, 42. The Appendix reprints an essay of 1835, Thoughts on the Study of Mathematics as a Part of Liberal Education’. Note also Baden Powell (Savilian Professor of Geometry at Oxford), The present state and future prospects of mathematical and physical studies in the University of Oxford (Oxford, 1832)
- Public Schools Commission [Clarendon] Report, Parliamentary Papers 1864 [3288] xx. My citations are to the Irish University reprint, British Parliamentary Papers (Education General), vol. 10, 456–86. Cited hereafter as PSC Report.
- Ibid., vol. 11, Q.6332 (Reverend S. T. Hawtrey). Cf. James M. Wilson; an autobiography, 1836–1931 (London, 1932), 57–9. The deficiencies in mathematics teaching were also to be found in newer foundations, like Marlborough College, which copied older foundations
- PSC Report, voL 10, 23
- The University of London matriculation examination (1838) demanded a knowledge of Euclid, as did the later established Oxford and Cambridge Local Examinations (1858)
- David Hilbert, as quoted by O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. (New York, 1962), 145
- For modern estimates of Euclid see T. L. Heath, The Thirteen Books of Euclid's Elements, 2 vols. (London, 1926), Dover (ed.), 3 vols. (New York, 1956); B. L. van der Waerdon, Science Awakening (Groningcn, 1954); J. Itard, Les livres arithmétiques d'Euclid (Paris, 1961); and Ivor Buliner‐Thomas and John Murdoch, ‘Euclid’ in Dictionary of Scientific Biography (New York, 1971), vol. 4, 414–59
- Whewell , W. 1845 . Of a Liberal Education in General and with a particular reference to the leading studies of the University of Cambridge 30 London
- Ibid., 65
- See, for example, Oliver Byrne's extraordinary The First Six Books of Euclid with coloured diagrams (W. Pickering, London, 1847) which used colours instead of letters (copy Maths. Association Library, Univ. of Leicester). The British Museum Catalogue of Printed Books, sub. Euclid, lists 238 editions of Euclid between 1801 and 1900; but this ignores the large number of editions by Simson and other editors, as well as the large number of other Geometries or Mathematical texts containing sections on geometry. Thus, P. J. Wallis, A check list of British Euclids up to 1850 (typescript, Dept. of Education, University of Newcastle, 1967) found 287 Euclids between 1801 and 1887
- Companion to the British Almanack … for 1849 (London, 1848). Of course, criticism of Euclid did not begin in the 1860's. See D. K. Wilson, The history of Mathematical teaching in Scotland to the end of the eighteenth century (London, 1935), 53–6
- Association for the Improvement of Geometrical Teaching. Nineteenth Report (Bedford, 1893), 15. These Reports will be cited as A1GT Report.
- J. Demogeot and H. Montucci, De L'Enseignment Secondaire en Angleterre et en Ecosse. Rapport addressi a son exc. M. Le Ministre de Vlnstruction Publique (Paris, 1868), 58 Iff. See R. D. Anderson, ‘French views of the English Public Schools: some nineteenth‐century episodes’, History of Education, 2 (1973), 159–72. The mathematician Legendre (1752–1833) produced his EUments de giometrie in 1794 (note the Engish trans, by the young Thomas Carlyle in 1824). It was revised continually by others during the nineteenth century and reached a 21st ed. in 1876
- I. Todhunter, The Conflict of Studies (London, 1873), 137–40. Demogeot taught literature at the Sorbonne, but Montucci, who was a modern languages teacher in a Parisian lycée, had studied mathematics. For Demogeot see Dictionnaire de Biographie Francaise, x, 991–2. Todhunter (1820–84), a Fellow of St John's College, Cambridge, produced one of the most successful of all the editions of Euclid used in the nineteenth century, The Elements of Euclid (London, 1862). Cf. Hilton's parody above
- Colenso , J. W. 1843 . Arithmetic designed for the use of schools London much reprinted; sales slumped after Colenso's notorious trials in the 1860s
- Schools Inquiry [Taunton] Commission, PP 1867–8 [3966], XXVIII, Pt. III. I have used the Irish University reprint Brit. Parl. Papers (Education General)‐ See vol. 19, Q.I664. Cited hereafter as SIC Report.
- Ibid., Q.2992. Key became a founder‐member of the AIGT. Cf. the evidence of the Astronomer Royal, G. B. Airy, PSC Report, vol. 12, 402, and further contrast the pro‐Euclid evidence of the Scots engineer, W. J. M. Rankine, SIC Report, vol. 19, Q.2325 and Thomas Selby of Taunton, ibid., vol. 20, Q.12444
- In 1867 Hirst succeeded De Morgan as Professor of Pure and Applied Mathematics at University College. At the time of the Taunton inquiry in 1868 he had become Professor of Pure Mathematics
- SIC Report, vol. 19, Q.2992
- See H. Hale Bellot . 1929 . University College London 1826–1926 321 – 2 . London
- Thomas Archer Hirst, Journal, 1 January 1854, ff. 1101–2 (Royal Institution Archives). See The Journals of Thomas Archer Hirst, FRS (1830–92), ed. W. H. Brock and R. M. MacLeod, in press. The episode is mentioned by John Tyndall (who had access to Hirst's Journal) in his lecture to the Royal Institution in 1854. See Lectures on Education Delivered in the Royal Institution (London, 1855), 181–2
- SIC Report, vol. 17, 30
- Intuitive proofs were any sound methods of reasoning—'if a direct proof can be found of any theorem, which naturally arises out of the data of the theorem, .it is to be preferred to a circuitous proof which depends on other [previously‐proved] theorems. (J. M. Wilson, Elementary Geometry, 3rd ed. (London, 1873), vi)
- Richard P. Wright, The Elements of Plane Geometry (London, 1868), vii. For Hirst's co‐authorship, see p. vi. Hirst himself published an anti‐Euclid syllabus of Lectures on Plane Geometry (London, 1870), which he had used in lecturing to the Ladies’ Educational Association in 1869
- Wilson, Autobiography (note 7), 66, and his ‘Early history of the Mathematical Association’, Mathematical Gazette, 10 (1921–2), 239–44
- J. M. Wilson, Elementary Geometry (Cambridge, 1868), 2nd ed. (Cambridge, 1869), 3rd ed. (London, 1873). Both Macmillan, and Longman (who published Wright's text), were in the vanguard of textbook publishing in the 1860s
- Educational Times, Sept 1868. Cf. Reverend Joshua Jones (headmaster of King William's College, Isle of Man), On the Unsuitableness of Euclid as a Textbook of Geometry (Liverpool, 1870)
- Athenaeum, 18 July 1868
- Todhunter, Conflict (note 18), Essay 5, 136–92; Charles L. Dodgson, Euclid and his Modern Rivals (London, 1879; reprint Dover Publications, 1974), 220–6 for a reproduction of De Morgan's review. Dodgson's Carrollean book went through an enlarged 2nd ed. in 1885. Note Dodgson's First Two Books of Euclid (Oxford, 1882)
- Wilson, Geometry (note 30), 3rd ed., v. Methodologically, his arguments for and against Euclid remind one of those used by him for and against science teaching. See F. W. Farrar, ed., Essays on a Liberal Education (London, 1867)
- Wright and Hirst also agreed (note 28, vii); but Dodgson, Rivals (note 33), 188, thought the syllogism had merits
- This comment led Todhunter, who slightly missed the point, into an interesting defence of British mathematics, which he thought as good as any in Europe (Conflict, note 18, 146). For Hirst's views on Continental geometry, see AIGT Report, I (1871), 9–11 and ibid., II (1872), 9–13
- Dodgson thought this a trivial objection since Euclid had survived 2000 years
- Rivals (note 33), 11–12, et passim.
- Wilson, Geometry (note 30), 3rd ed., ix; Todhunter, Conflict (note 18), 180, did not agree
- This was one reason why geometry was so strongly entrenched in the Scottish universities. See G. E. Davie, The Democratic Intellect (Edinburgh, 1961), chap. 7, and R. Olsen, ‘Scottish Philosophy and Mathematics 1750–1830’, J. History Ideas. 32 (1971), 29–44
- Conflict (note 18), 139–42. Perversely, Todhunter challenged Euclid's rivals to produce wranglers—which, of course, they were forbidden to do (173)
- Wilson gave sixpence to every Rugby boy who spotted an error. J. M. Wilson, ‘Early history …’ (note 29), 241
- Brit. Ass. Reports, XL (1871), Ixxii. The Committee's first report appeared in ibid., XLII (1873), 459–60; note also ibid., 45 (1876), 8–13
- On Tucker (1832–1905), see J. A. Venn, Alumni Cantabrigienses (Cambridge, 1954), PL II, vi, 240. For the LMS, which was founded by De Morgan, Hirst and others in 1866, see E. F. Collingwood, ‘A Century of the London Mathematical Society’, Journal LMS, XLI (1966), 577–94
- Nature 1 (1869–70), 534. The Headmaster's Conference, meeting at Sherborne in December 1870, expressed disquiet at the continued use of Euclid. See AIGT Report, I (1871), 13
- Nature II (1870), 141. For a sketch of Levett, see A. W. Siddons, ‘Progress’, Mathematical Gazette, 20 (1936), 7–26 (13–15)
- Richard Wormell, An Elementary Course of Plane Geometry (London, 1868), 2nd ed. 1870
- Nature III (1870–1), 169. Hirst was obviously elected Chairman because of his association with Tucker and because, apart from Cayley (a devoted Euclidean) he was the most distinguished geometer in England
- AIGT Report, II (1872) 23 (Wilson), 24 (Hirst, ‘no really valuable textbook would ever be produced by a committee or an association')
- AIGT Report, III (1973), 11. Only 13 Members replied! Only one of these was in favour of teaching ‘technical’ geometry
- The only AIGT members to stand out for technical geometry and, equally significantly, for the admixture of arithmetical and algebraic matter to textbooks of geometry, were the Reverend W. H. Laverty, a Fellow of Queen's College, Oxford, and J. F. Iselin, an Inspector of science schools for the Department of Science and Art See AIGT Report, II (1872), 21–2, 28–9
- AIGT Report, II (1872), 17 and ibid., Ill (1873), 12
- Note the assumption that the British Association possessed educational authority. Approval of the Headmaster's Conference was also sought
- See Brit. Ass. Reports, XLV (1876), 9. Of the 15 BA members, 7 were from the AIGT. Not unexpectedly the BA concluded that the AIGT syllabus was ‘drawn up with such care, and with regard to the essential conditions of the problem, as to render it highly desirable that it should be considered in detail by authorised representatives of the universities and the other great examining bodies of the United Kingdom’
- AIGT Report, V (1875), 11, quoting the circular of 21 November 1874, a copy of which is bound in J. S. Mackay"s copy of early AIGT Reports in the Mathematical Association Library, Leicester (MA 510.4/135)
- AIGT Report, VI (1878), 18–19
- I.e., the proof of a theorem appearing in Euclid Book II could be done in any way providing it did not involve using theorems that appeared after that theorem in Book II or later Books. The Cambridge Previous Examinations followed this rule after 1882
- Hirst was Assistant Registrar of the University of London between 1870 and 1872, but I have found no evidence to suggest that he influenced the Senate's decision of 1875–6
- For Hirst's anonymous review of Dodgson, see Nature XX (1879), 240–1. He thought most of Dodgson's points were ‘mere verbal quibbles’ over apparent inconsistencies which arose from the different standpoints of the rival authors
- On Hayward (1829–1903), see DNB. He and Hirst had corresponded on geometrical topics
- Circular to members from R. Levett dated 31 December 1880, bound in Mackay (note 55). There were no meetings of the AIGT in 1879 or 1880
- Wilson , J. M. 1878 . Elementary Geometry. Books I‐V. Containing the Subjects of Euclid's first six books: following the syllabus of geometry prepared by the Geometrical Association [sic], , 4th ed. Cambridge
- See AIGT Report. VII (1881), 28 and ibid., X (1884), 19
- Ibid., VII (1881), 5–10, list of members
- Ibid., 35
- On 12 December 1882 the Syndicate of the University of Cambridge recommended ‘some deviations from the old practice of merely asking questions directly from the propositions of Euclid as such, and that there should be questions on the subject matter of Euclid, as well as the propositions themselves’, AIGT Report. IX (1883), 16–17. This admission referred to examinations ‘previous’ to the tripos
- AIGT Report, XIII (1887), 21–2
- Cayley did not sign the Cambridge Board of Mathematics’ recommendation. See AIGT Report. XIV (1888), 28
- For the Cambridge attitude, see ibid., 23. For reminiscences concerning the AIGTs negotiations with examiners, see Siddons (note 46), 19–21
- Oxford was petitioned again in 1892—to J. J. Sylvester's embarrassment (R. C. Archibald, ‘Unpublished letters of J. J. Sylvester’, Osiris. I (1936)7 152–3); Oxford declined. See AIGT Report. XIX (1893), 7–8
- AIGT Report. VI (1878), 12
- For a summary of Perry's position, see Teaching Mathematics in Secondary Schools. Min. Educ. Pamphlet 36 (HMSO, London, 1958), 7–9
- For opportunities for women, see Evelyn Sharp, Hertha Ayrton 1854–1923 (London, 1926), 146–7. Hertha Ayrton was the wife of the electrical engineer W. E. Ayrton, a close friend of Perry
- J. Perry, ed., British Association Meeting at Glasgow, 1901. Discussion on the Teaching of Mathematics (London, 1901); Mathematical Gazette, II (1901–1904), 81–3
- See Mathematical Gazette, II (1901–1904), 105–6, where E. M. Langley also spelt out where the AIGT had gone wrong. The Perry campaigners also wanted the power of examiners and school inspectors reduced
- B. Russell, The Teaching of Euclid’, Mathematical Gazette. II (1901–1904), 165–7
- This letter to the BA Committee and the Mathematical Association from a ginger group of 23 mathematics school teachers demanded reforms of every aspect of school mathematics. It was first published in Nature, LXV (1901–2), 258–9 and rep. Mathematical Gazette, II (1901–1904), 143–6. For Forsyth's connection with the campaign, see Teaching Mathematics (note 72), 9
- For the capitulation, which included the Civil Service Commission, see Mathematical Gazette. II (1901–1904), 197–201, 349–51; ibid., Ill (1904–1906), 146–57; and Siddons (note 46), 19–21. On Forsyth, see DNB.
- Charles Godfrey and A. W. Siddons, Elementary geometry (Cambridge, 1903); for Godfrey's contributions to school geometry, see A. C. Howson, ‘Milestone or Millstone’, Mathematical Gazette, LVII (1973), 258–266. Clement Vavasor Durell, A Course of Plane Geometry for Advanced Students (London, 1909–10); A Concise Geometry (London, 1920). For later developments in mathematics teaching see the resume in The Teaching of Mathematics in Secondary Schools, Assistant Masters Association, 2nd ed. (Cambridge, 1973), 9–38; and A. G. Howson, ‘Milestone’
- My reflections are the result of reading D. Layton, Science for the People (London, 1973). For one link between the Perry movement and the PSSMA see W. D. Eggar, ‘On the teaching of science and mathematics’, Conference of Public School Science Masters, 19 January 1901 (priv. prin., London, 1902)
- ‘Science for All’, School Science Rev.. II (1920), 197–212. See W. H. Brock, ed., H. E. Armstrong and the Teaching of Science 1880–1930 (Cambridge, 1973), 53–5