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Nicholson had been led to the effort by a mathematician of the name of Theophilus Holdred, who showed him a method of his own, which to Nicholson appeared much confused. He then devised a plan on different lines, which the latter agreed to publish at the end of his own tract. Nicholson becoming dissatisfied with Holdred's proceedings, published his own plan in his ‘Rudiments of Algebra’ in 1819. On 1 July 1819 a paper on the same subject by Leonard Horner [q. v.] was read before the Royal Society. Nicholson considered that Horner's paper contained the substance of what he had just published, and wrote an account of the matter in the introduction to his ‘Essay on Involution and Evolution’ in 1820. The question of priority of invention is discussed in the ‘Companion to the British Almanack,’ 1839, pp. 43–6. He invented a new method of extracting the cube root, which is given in the ‘Civil Engineer,’ 1844 (p. 427). Nicholson never succeeded in turning his knowledge to pecuniary advantage. He was too apt to make use of his materials in more than one publication, and was involved in a chancery suit for some years, having violated his promise of making no further use of the plates in his ‘Architectural Dictionary.’ Towards the end of his life he entered into controversy with Sir Charles Fox [q. v.], engineer, as to his claim to having discovered a sure rule for the construction of the oblique arch. But Nicholson's mind was already enfeebled, and he proved unable to defend himself.

As an architect Nicholson did some useful work. The best of his executed designs are those for Castleton House and Corby Castle, both near Carlisle, a coffee-house at Paisley, additions to the university of Glasgow, and he laid out the town of Ardrossan in Ayrshire, intended as a fashionable bathing-place. Plans and elevations of all these are given in his ‘Architectural Dictionary,’ ii. 102–3, 774, 800. He also erected a timber bridge over the Clyde at Glasgow, and several dwelling-houses in the city.

His useful publications, most of which went through several editions both before and after his death, include: 1. ‘The Carpenter's New Guide,’ London, 1792, 1797, 1801, 1805, 1808, 1835; Philadelphia, 1848, 1854; London and Philadelphia, 1854, 1856; London, 1857. 2. ‘The Carpenter's and Joiner's Assistant,’ London, 1792, 1793, 1797, 1798, 1810. 3. ‘Principles of Architecture,’ London, 1795–8, 1809, 1836, 1841, 1848 (ed. Joseph Gwilt [q. v.]). 4. ‘The Student's Instructor,’ London, 1804, 1823, 1837, 1845. 5. ‘Mechanical Exercises,’ London, 1811, 1812, 1819, and under the title of ‘The Mechanic's Companion,’ London, 1824; Oxford, 1825; Philadelphia, 1856. 6. ‘Architectural Dictionary,’ London, 1812–19, 1835, 1852–4 (edited and largely rewritten by Lomax and Gunyon, 1855, 1857–62). The titles vary in the several editions; the last three contain portraits from a painting by W. Derby. 7. ‘A Treatise on Practical Perspective,’ London, 1815. 8. ‘An Introduction to the Method of Increments,’ London, 1817. 9. ‘Essays on the Combinatorial Analysis,’ London, 1818. 10. ‘The Rudiments of Algebra,’ London, 1819, 1824, 1837, 1839. 11. ‘Essay on Involution and Evolution,’ London, 1820 (for which Nicholson received the thanks of the Académie des Sciences at Paris). 12. ‘Treatise on the Construction of Staircases and Handrails,’ London, 1820, 1847. 13. ‘Analytical and Arithmetical Essays,’ London, 1820, 1821. 14. ‘Popular Course of Pure and Mixed Mathematics,’ London, 1822, 1823, 1825. 15. ‘Rudiments of Practical Perspective,’ London and Oxford, 1822. 16. ‘The New and Improved Practical Builder and Workman's Companion,’ London, 1823, 1837 (edited by T. Tredgold), 1847, 1848–50, 1853, 1861 (with a portrait by W. Derby). 17. ‘The Builder and Workman's New Director,’ London, 1824 (with portrait by T. Heaphy), 1827, 1834, 1836; Edinburgh, 1843; London, 1848. 18. ‘The Carpenter and Builder's Complete Measurer,’ London, 1827 (with portrait). 19. ‘Popular and Practical Treatise on Masonry and Stone-cutting,’ London, 1827, 1828, 1835, 1838. 20. ‘The School of Architecture and Engineering,’ five parts, London, 1828 (with portrait). 21. ‘Practical Masonry, Bricklaying, and Plastering’ (anon.), London, 1830 (revised by Tredgold. The portion on plastering was supplied by R. Robson, a journeyman plasterer). 22. ‘Treatise on Dialling,’ Newcastle, 1833, 1836. 23. ‘Treatise on Projection, with a Complete System of Isometrical Drawing,’ Newcastle, 1837; London, 1840. 24. ‘Guide to Railway Masonry,’ Newcastle, 1839; London, 1840, 1846; Carlisle, 1846; London, 1860 (with portrait by Edward Train). 25. ‘The Carpenter, Joiner, and Builder's Companion,’ London, 1846. 26. ‘Carpentry’ (anon.), London, 1849, 1857 (edited by Arthur Ashpitel; the book also contains works by other hands). 27. ‘Carpentry, Joining, and Building,’ London, 1851.

With John Rowbotham Nicholson published ‘A Practical System of Algebra,’ London, 1824, 1831, 1837, 1844, 1855, 1858, and a key to the same in 1825; and with his son, Michael Angelo Nicholson, ‘The Practical Cabinet Maker, Upholsterer, and Complete Decorator,’ London, 1826.

Nicholson also wrote articles on