An innovator in mathematics education, particularly in the teaching of geometry, Irish-born Oliver Byrne (1810-1880) lived in England and the United States at a time of prejudice against the Irish. Byrne faced physical and financial hardship and ridicule from his contemporaries for his mathematical and pedagogical innovations. His wife, Eleanor Rugg Byrne, published articles and books in meteorology at a time when women were discouraged from scientific and mathematical pursuits. This most comprehensive biography of Oliver Byrne to date discusses both Oliver's and Eleanor's accomplishments and highlights Oliver’s pedagogical philosophy, emphasizing his most innovative, famous, and visually stunning mathematical work on Euclidean geometry, *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners *(1847). We will conclude with a discussion of how Byrne's *Euclid,* and his pedagogical views, might be used in geometry, mathematics education, and mathematics history classes.

**Figure 1.** Lithograph of Oliver Byrne (from the original in the collection of Gerald L. Alexanderson, Michael and Elizabeth Valeriote Professor in Science at Santa Clara University.) The caption reads as follows:

OLIVER BYRNE.

LATE PROFESSOR OF MATHEMATICS, COLLEGE FOR CIVIL ENGINEERS.

Author of

*“The New and Improved System of Logarithms.”_“The Doctrine of Proportion.”_“The Practical,
Complete, and Correct Gager.”_“The Elements of Euclid by Colours.”_“A Practical,
Treatise on Spherical Trigonometry.”_“How to Measure the Length of a
Degree on the Earth's Surface by the assistance of Railroads."
&c., _&c.,_&c.*

Inventor of

*“The Patent Calculating Instruments.”~“The System of Facilitating the Acquirement of Geometry, &
of other Linear Arts and Sciences by Colours, &c.”~“Proposer of the New Theory of
the Earth, which accounts for many Astronomical, Geographical, and
Geological Phenomena, hitherto unaccounted for.”*

An innovator in mathematics education, particularly in the teaching of geometry, Irish-born Oliver Byrne (1810-1880) lived in England and the United States at a time of prejudice against the Irish. Byrne faced physical and financial hardship and ridicule from his contemporaries for his mathematical and pedagogical innovations. His wife, Eleanor Rugg Byrne, published articles and books in meteorology at a time when women were discouraged from scientific and mathematical pursuits. This most comprehensive biography of Oliver Byrne to date discusses both Oliver's and Eleanor's accomplishments and highlights Oliver’s pedagogical philosophy, emphasizing his most innovative, famous, and visually stunning mathematical work on Euclidean geometry,* The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners.* Finally, we will discuss how Byrne's *Euclid,* and his pedagogical views, might be used in geometry, mathematics education, and mathematics history classes.

**Figure 1.** Lithograph of Oliver Byrne (from the original in the collection of Gerald L. Alexanderson, Michael and Elizabeth Valeriote Professor in Science at Santa Clara University). The caption reads as follows:

OLIVER BYRNE.

LATE PROFESSOR OF MATHEMATICS, COLLEGE FOR CIVIL ENGINEERS.

Author of

*“The New and Improved System of Logarithms.”_“The Doctrine of Proportion.”_“The Practical,
Complete, and Correct Gager.”_“The Elements of Euclid by Colours.”_“A Practical,
Treatise on Spherical Trigonometry.”_“How to Measure the Length of a
Degree on the Earth's Surface by the assistance of Railroads."
&c., _&c.,_&c.*

Inventor of

*“The Patent Calculating Instruments.”~“The System of Facilitating the Acquirement of Geometry, &
of other Linear Arts and Sciences by Colours, &c.”~“Proposer of the New Theory of
the Earth, which accounts for many Astronomical, Geographical, and
Geological Phenomena, hitherto unaccounted for.”*

Mathematician, educator and civil engineer Oliver Byrne was born on 31 July 1810 in the copper mining village of Vale of Avoca in County Wicklow, Ireland, 40 miles south of Dublin.^{1} The son of Lawrence Oliver Byrne and Mary Byrne, Oliver Byrne had at least one brother and two sisters. His brother, John Byrne, two years younger than Oliver, edited a collection of his older brother’s mathematical works and co-authored at least one book with him.^{2} By 1839 Oliver Byrne was the “principal support of an aged mother and sisters in Ireland.”^{3} Married to Eleanor Rugg in England in 1845, Oliver Byrne died on 9 December 1880 and was buried at Maidstone, Kent, England.^{4} Eleanor died on 12 June 1897.^{5} The couple had no children.^{6}

Byrne's aptitude and passion for mathematics may have resulted from studying engineering. Although his obituary states that he “entered at Trinity College, Dublin, where he passed with distinction the various examinations,” it appears unlikely that Byrne graduated from or even matriculated at Trinity.^{7} He was not identified as “Oliver Byrne, B.A.” as were his degreed peers. Although some of his papers are in its archives and he was considered a Dublin native, Trinity College in Dublin has no record of Oliver Byrne.^{8} Some of his contemporaries, such as mathematician George Boole (1815-1864), were self-taught.^{9} Byrne may have been self-taught or apprenticed.

1 Oliver Byrne, application (6 January 1858), case number 987, vol. 31, Archives of Royal Literary Fund, London; Nineteenth Century Collections Online: "British Theatre, Music, and Literature: High and Popular Culture" Collection; Gale Digital Collections (http://www.galegroup.com : accessed 8 August 2014). Hereinafter, Byrne’s Royal Literary Fund applications will be referred to with “RLF application” followed by the date of application. All of Oliver Byrne’s RLF applications are case number 987. Newspaper clippings attached to applications are now out of order; for instance, an 1852 newspaper clipping may be found in an 1842 RLF application. Although Oliver Byrne’s obituary and more than one census records “Holland” or “Leyden” as his birthplace, when he self-reported in his Royal Literary Fund applications, Byrne gave the village of Avoca and County Wicklow, Ireland. Byrne complained of prejudice against the Irish and it is possible that his English wife, Eleanor Byrne, gave the Holland birthplace information. His *Kent Messenger and Maidstone Telegraph* obituary cited below describes the Byrne family as living in exile after the 1798 Irish Rebellion. Eleanor Byrne's 1881 Royal Literary Fund application lists her husband’s birthplace as “Leyden.” T. N. Schelhaas, Keeper of the Records of the City of Leiden [Netherlands], in a letter to Dr. Sid Kolpas, 24 November 1992, reported that he located no birth record for Oliver Byrne c. 1810. In Byrne’s United States naturalization records cited elsewhere, Byrne renounced "allegiance . . . particularly to the Queen of the United Kingdom of Great Britain and Ireland, of whom I am a subject."

2 Oliver Byrne and John Byrne, *The Fallacies of Our Own Time: Fallacy of Phrenology* (London: Sherwood, Gilbert and Piper, 1844).

3 For Oliver Byrne’s father’s name, see “London, England, Marriages and Banns, 1754-1921,” *Ancestry* (http://www.ancestry.com : accessed 25 July 2014); entry for marriage of Oliver Byrne and Eleanor Rugg (21 July 1845). For his mother’s given name, see Mary Byrne’s obituary in *Gentleman’s Magazine,* 194:213 (1853). For Byrne’s comment on his dependent mother and sisters, see RLF application (23 November 1839).

4 For marriage see “London, England, Marriages and Banns, 1754-1921,” *Ancestry* (http://www.ancestry.com : accessed 25 July 2014); entry for marriage of Oliver Byrne and Eleanor Rugg (21 July 1845). For death see General Register Office, Southport, England, Certified Copy Of An Entry Of Death, Oliver Byrne, December 9, 1880. For burial see “Register of burials in the burial ground of the parish of Maidstone, Kent, 1858-1881,” original records housed at the Canterbury Cathedral Archives, Canterbury, Kent; FHL microfilm 1,835,479.

5 General Register Office, Southport, England, Certified Copy Of An Entry Of Death, Eleanor Byrne, 12 June 1897. See also “England & Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1966,” Eleanor Byrne (12 June 1897); digital images, *Ancestry* (http://www.ancestry.com : accessed 30 July 2014). Entry, “Byrne, Eleanor of “Rosteague” Reginald-road Maidstone widow died 12 June 1897 Administration (with Will) London 19 July to George Philip Rugg M.D. Effects £89 19*s.* 1*d.*”

6 RLF application (6 January 1858). “Children None.” See also Principal Probate Registry [London, England], Will in the Estate of the Late Oliver Byrne, “Proved at London 28 February 1881 by the oath of Eleanor Bryne [*sic*] widow the relict daughter of John Rugg Esquire the sole executrix to whom admin was granted.”

7 “The Late Mr. Oliver Byrne,” *Kent Messenger and Maidstone Telegraph* (Kent, England), obituary, 18 December 1880 indicates Byrne “was entered at Trinity College, Dublin, where he passed with distinction the various examinations,” clipping, RLF application (3 January 1881).

8 Jane Maxwell (Dublin, Ireland) to Dr. Sid Kolpas, letter, 14 September 1992. Dr. Kolpas contacted Trinity College and Ms. Maxwell reported no record of Oliver Byrne ever attending the college. Nor is Byrne found in the online *Alumni Dublinenses,* a register of the students, graduates, professors and provosts of Trinity College in the University of Dublin (1593-1860) available at the Trinity College Dublin Library website. In his RLF application (28 May 1842), Byrne listed “Dublin” and the years 1832 and 1833 for his first two publications. For Dublin nativity, see “Repeal in Wicklow,” *Freeman’s Journal* (Dublin, Ireland), 22 May 1845, which reports “Mr. Oliver Byrne, the distinguished mathematician and engineer from London (he is a native of our town, to which he is an honour; but alas! our gifted countrymen must quit their native shores to seek those distinctions in a foreign clime denied to them at home).”

9 T. A. A. Broadbent, "George Boole (1815-1864)," *The Mathematical Gazette* (London: G. Bell and Sons, December 1964), 373.

At age 20 in Dublin, Oliver Byrne made his author’s debut with *A Treatise on Diophantine Algebra.* Although most often referenced as *Treatise on Algebra,* in his 1858 Royal Literary Fund (RLF) application, Byrne listed this volume specifically as *A Treatise on Diophantine Algebra,* published by Allen and Co., Dublin, 1830. Second, he published *A Pamphlet on the Teaching of Geometry by Coloured Diagrams, etc.; Applied to the First Book of Euclid* in 1831*.*^{10} By 1838, Byrne had expanded this pamphlet into a treatment of the first six books of Euclid's *Elements,* advertised in another of his publications as

"*In the Press,*

The second Edition of The Elements of Geometry, containing the first six Books of Euclid, in which the propositions are demonstrated by the substitution of *colours* for *letters,* the study being thus shortened and improved.

*Price Eight Shillings.*"

This book apparently never reached a seller; however, Byrne's famous *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners* was finally published in 1847 in London by William Pickering. An in depth discussion of Byrne’s lasting achievement, his *Euclid,* follows in a later section of this article, Byrne's *Euclid:* Geometry Understood via Color-coded Diagrams.

**Figure 2.** Title page of Oliver Byrne's *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners, *published in 1847. This book grew out of a pamphlet by Byrne containing just Book I of Euclid's *Elements* presented using colorful diagrams, published in 1831 when he was just 21 years old. (The image above is from the book in the collection of Sid Kolpas.)

Byrne’s other early publications included *How to Measure the Earth with the Assistance of Railroads* and *New and Improved System of Logarithms,* both in 1838, as well as a variety of mathematical pamphlets. A comprehensive list of Byrne’s known works is given in Appendix A of this article.

Not only a mathematician and author, Byrne also invented scientific instruments. On 10 March 1838, Abraham Parker, Surveyor, and Oliver Byrne, Professor of Mathematics, both of Gower Street, Bedford Square, London, filed for a patent covering an “Instrument for gauging malt and the fluid or solid content of casks and other vessels.”^{11} In 1843 Byrne wrote a “Description and use of an instrument to find the time by the sun, moon, or any of the visible fixed stars: as well as the names of those stars / invented and constructed for the Right Hon. Earl Fitzwilliam.”^{12} Later, in 1846, Byrne wrote of another invention, the Byrnegraph. The Byrnegraph was, according to Oliver Byrne, an improved proportional compass, designed on precise mathematical principles to remedy “the defects of the old proportional compasses” and extend

the application of the instrument of so much importance to the engineer, the artist, the architect, and the amateur, in every department of applied mathematics, when its results can be relied on with confidence.

Byrne claimed that traditional proportional compasses scarcely deserved the title “mathematical,” but that his version was constructed on purely mathematical principles, thus providing much greater accuracy.^{13}

Using the pseudonym E. B. Revilo (Oliver spelled backward), Byrne attempted to prove the truth of the Creed of Saint Athanasius through a mathematical analogy in a pamphlet titled *The Creed of Saint Athanasius proved by a Mathematical Parallel *and published in 1839.^{14} According to historian Daniel Cohen, Byrne sought to equate^{15}

each element of the Trinity (the Father, the Son, and the Holy Ghost) to infinity, since the Athanasian Creed considered all of them to be unlimited. Byrne then erected two vertical columns: the left containing the English Book of Common Prayer translation of the Quicunque Vult (the traditional description of the Athanasian Creed), the right containing parallel mathematical equations involving infinity that purported to establish the truth of the statements on the left.

The mathematician Augustus De Morgan (1806-1871), in *A Budget of Paradoxes,* judged this work to be nonsense and considered Byrne to be an eccentric mathematician.^{16} Janet Heine Barnett provided further analysis of DeMorgan’s writing on the works of Oliver Byrne in a recent article.^{17} Byrne also reviewed the work of his colleagues.^{18}

**Figure 3.** Title page of Augustus De Morgan's *A Budget of Paradoxes* (1872), a collection of the author's amusing and often critical writings about the attempted mathematical works of others. (This image is presented courtesy of the University of Pennsylvania Libraries and may be used in your classroom; for all other purposes, please seek permission from the University of Pennsylvania Libraries.)

Financial and legal problems plagued Oliver Byrne periodically throughout his entire life. His first of fifteen financial aid applications to the Royal Literary Fund occurred in 1839 while jailed as a debtor at White Cross Street Prison, London. On 23 November 1839, Byrne wrote,^{19}

I have been a resident in London for the last 8 years, during which time I have supported myself as a teacher of Mathematics, and have also published several works in connection with the mathematical sciences …. A native of the sister country I have had to struggle long and severly [sic] with adverse circumstances, and to encounter an almost overwhelming weight of prejudice, but my pupils have included some of the most distinguished in the ranks of society …. I have now unfortunately been arrested on account of a debt of thirty pounds which I have not been able to pay and now lay here at the suit of a merciless creditor whom I have not the means to satisfy.

Perhaps an example of Byrne’s claim of prejudice against the Irish occurred when a friend came to bail Byrne out in 1839 but it was refused by the plaintiff, even after the friend offered an even higher amount as security. The London newspaper *The Examiner *reported,^{20}

A person of the name of Oliver Byrne, a professor of mathematics, appeared in Court to be discharged on bail under the New Insolvent Debtors’ Act of last sessions.—His bail appeared in Court to justify, but one of them was opposed by Mr Woodruff, on behalf of the plaintiff, as to the bail’s sufficiency. The person opposed was pressed hard by counsel as to his means, and at last refused to answer any more questions, his feeling being much irritated at the interrogatories. The other bail, finding his co-bail in danger of passing as one of the sureties, then offered to pay the required sum into court as security for the insolvent’s appearance on the final day of hearing, rather than he should be inconvenienced by staying in prison for about five weeks.—This offer was refused by the Court. [This appears to us extremely unreasonable, vexatious, and unjust. The payment into Court of the sum required in security was obviously as good as or better than the best bail, and we are at a loss to conceive the pretext on which the offer can have been refused.]

Beginning with his first grant in 1839, the Royal Literary Fund’s support of Oliver Byrne amounted to £320 between 1839 and 1881 when his wife applied as his widow.^{21} David Williams founded The Royal Literary Fund in 1790 to “withdraw those apprehensions of extreme poverty, and those desponding views of futurity, which lead Genius and Talent from the path of Virtue.” Other Royal Literary Fund beneficiaries included novelists Bram Stoker, Charles Dickens, Jr., and James Joyce.^{22}

For a short time, Byrne taught mathematics in the College of Civil Engineers at Putney in South West London.^{23} Founded for the purpose of affording sound instruction in the theory and practical application of civil engineering and architecture, the College of Civil Engineers prepared students for a career in the field.^{24} Started in 1838 in Kentish Town, the school moved to Putney in August 1840.^{25} Putney House, one of two large mansions in the area, was converted into a College for Civil Engineers in 1839, founded by subscriptions among the nobility and others “for the purpose of conferring a superior education on the sons of respectable persons in the engineering, mathematical, and mechanical sciences.” The college closed in 1857 and the mansion was demolished.^{26}

Byrne adopted the specific title of “Professor of Mathematics at the College for Civil Engineers” (versus simply “Professor of Mathematics”) in August 1839.^{27} This was also the year of his first Royal Literary Fund (RLF) application, submitted from debtor’s prison due to a debt owed to an unspecified creditor. His brother, John Byrne, reported in *Miscellaneous Mathematical Papers of Oliver Byrne, Collected and edited by John Byrne,* that “lectures [were] delivered by Professor Oliver Byrne, at the Museum Lecture room, Philadelphia, in January 1842, . . ..”^{28} Thus, it appears that Byrne’s tenure at the College for Civil Engineers may have lasted just over two years, from August 1839 to December 1841. Concerning his resignation from his post at Putney College, Byrne himself later explained,^{29}

I resigned the professorship of Mathematics in the College for Civil Engineers, not without sufficient reason; the most persevering, in my position, would have done the same. Several of the Council resigned for the same reason. After my resignation, I went to the United States of America, hoping that I could stipulate with the government for a new plan of calculating the revenue. I travelled from City to City without success supporting myself by delivering lectures on popular mathematical subjects and by writing short articles for periodicals and newspapers.

10 Byrne assigned Dublin and 1832 and 1833 to these publications in their earliest known reference, Royal Literary Fund (RLF) application (28 May 1842). In RLF application (6 January 1858), Byrne entitled his first work *A Treatise on Diophantine Algebra* published by Allen & Company, Dublin, 1830, and he noted it “__very scarce__.” There is evidence of a Quaker printer, Richard Allen, active in Dublin in this time; see Bruce Nelson, *Irish Nationalists and the Making of the Irish Race* (Princeton: Princeton University Press, 2012), 99. The notice of Byrne's "second Edition of The Elements of Geometry, containing the first six Books of Euclid ..." appeared at the end of his 1838 publication, *How to Measure the Earth with the Assistance of Railroads**,* available at GoogleBooks.

11 David J. Bryden (Worcestershire, England) to Susan Hawes (Portland, Maine), "Oliver Byrne," email, 17 December 2014, Byrne Research Files; privately held by Susan Hawes, Portland, Maine. David Bryden spent most of his career as a museum curator. After his retirement, Bryden continues to research early scientific instruments. (“London meeting lecture Thursday 15 November 2012 Bankruptcy and insolvency in the English horological trade, 1720-1849 by David Bryden," newsletter; Antiquarian Horological Society (http://www.ahsoc.org : accessed 21 June 2015).

12 Oliver Byrne, "Description and use of an instrument to find the time by the sun, moon, or any of the visible fixed stars: as well as the names of those stars / invented and constructed for the Right Hon. Earl Fitzwilliam," manuscript, 1843; Harvard University Houghton Library.

13 Oliver Byrne, *Description and Use of the Byrnegraph: an instrument for multiplying, dividing, and comparing lines, angles, surfaces, and solids* (London: C. & J. Adlard, Bartholomew Close, 1846), 3.

14 Oliver Byrne, *The Creed of Saint Athanasius proved by a Mathematical Parallel* (London: William Day, 1839).

15 Daniel J. Cohen, *Equations from God: Pure Mathematics and Victorian Faith* (Baltimore: John Hopkins University Press, 2007), 73.

16 Augustus DeMorgan, *A Budget of Paradoxes* (London: Longmans Green & Co, 1872), 329.

17 Janet Heine Barnett, “Mathematics is a Plural Noun: The Case of Oliver Byrne,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-fifth Annual Meeting (Montreal),* 23:26-46 (2010).

18 Oliver Byrne, “Unwillingness of Man to Investigate,” *American Railroad Journal and Mechanics Magazine,* Vol VIII—New Series or Vol. XIV, 91-93 (New York: George C. Schaeffer, 1842).

19 RLF application (23 November 1839).

20 "Insolvent Debtors' Court, Monday," *The Examiner* (London), 29 December 1839.

21 MeasuringWorth.com: In 2013, the relative value of £320 in 1875 ranges from £26,300.00 to £407,100.00. Using Google to calculate British pounds into 2015 US dollars, this amount reflects a value of $41,000 - $639,000. Eleanor Byrne, widow of Oliver Byrne, RLF application (3 January 1881).

22 Archives of the Royal Literary Fund; Nineteenth Century Collections Online: "British Politics and Society” Collection; Gale Digital Collections (http://www.galegroup.com : 8 August 2014). For other beneficiaries, see “The RLF Archive,” Royal Literary Fund, (http://www.rlf.org.uk/home/the-rlf-archive : accessed 21 December 2014).

23 RLF application (23 November 1839). “1841 England Census,” [county] Surrey, [parish] Putney, enumeration district 5, folio 62, line 10, Oliver Byrne, professor; *Ancestry* (http://www.ancestry.com : accessed 30 July 2014).

24 Edward Walford, *Old and New London: A Narrative of Its History, Its People and Its Places. The Southern Suburbs Vol VI* (http://www.british-history.ac.uk : accessed 7 September 2014).

25 John Timbs, *Curiosities of London: Exhibiting the Most Rare and Remarkable Objects of Interest in the Metropolis* (London: D. Bogue, 1855).

26 Edward Walford, *Old and New London: A Narrative of Its History, Its People and Its Places. The Southern Suburbs Vol VI* (http://www.british-history.ac.uk : accessed 7 September 2014).

27 “British Association Fracas,” *The Mechanic's Magazine,* *Museum, Register, Journal, and Gazette,* April 6th - September 28th, 1939, Vol. XXXI (London: W.A. Robertson, 1839).

28 John Byrne, *The Miscellaneous Mathematical Papers of Oliver Byrne, Collected and edited by John Byrne* (London: Maynard, 1848), cited in Janet Heine Barnett, “The Dual Arithmetic of Oliver Byrne: A New Art which entirely supersedes the use of logarithms,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-sixth Annual Meeting (Dublin),* 24:12-30 (2011).

29 RLF application (28 May 1842). Example article Byrne wrote during this period, Oliver Byrne, “Unwillingness of Man to Investigate,” *American Railroad Journal and Mechanics Magazine,* Vol. VIII—New Series or Vol. XIV, 91-93 (New York: George C. Schaeffer, 1842).

In the fall of 1840, the Philanthropic Life Assurance Annuity & Endowment Society of London, listing Oliver Byrne, Consulting Actuary, sought incorporation by an Act of Parliament.^{30 }Soon after, Byrne defended the organization in a “Letter to the Editor” labeled as “(Advertisement)” by the newspaper.^{31} Little further mention is made of the company until years later when an inquiry to the Philanthropic Life Assurance Society had no record of Mrs. Mary Hodges. The Birmingham Police Court alleged Henry Lewis conned Mrs. Hodges by taking her believed Philanthropic Life Assurance Society entrance money two years prior.^{32}

About 1842, Oliver Byrne and Henry William Hull (B.A., C.E.) proposed a School of Mathematics, Engineering, Classics, and General Literature at Surrey Villa, near Lambeth Palace in London.^{33} The intent of the school was to provide practical lessons in Mathematics and Engineering using surveying equipment. Byrne's and Hull's prospectus states, “The chief object in the Institution will be, as much as possible, to combine practical illustrations with theoretical instruction.”^{34} There is no evidence that the school was ever established.

**Figure 4.** Title page of Byrne’s and Hull’s proposal for a school, c. 1842. © The British Library Board, 8365.b.48.(1.).

In 1844, Byrne and his brother John published *The Fallacies of Our Own Time: Fallacy of Phrenology*.^{35} Phrenology, based on measurements of the human skull in order to determine traits such as intelligence, is today considered a pseudoscience.^{36} In their book, the Byrne brothers espoused their own philosophy, writing^{37}

. . . the academic youth not only learn that the true equation of this life is, (Industry + Economy) x Virtue = Happiness: but, also, that ‘knowledge is the wing on which we fly to Heaven.’

During the same period, Byrne traveled to Ireland:^{38}

Professor Oliver Byrne, of London, passed through Mulingar [County Westmeath, Ireland] last week. He is examining the capabilities of the Royal Canal to construct a railway . . . between Dublin and Galway.

Between 1843 and 1847, articles by Byrne regularly appeared in London’s *Civil Engineer and Architect's Journal.* Only a portion of these articles are reprinted in the 1848 *Miscellaneous Mathematical Papers of Oliver Byrne, Part I,* edited by John Byrne. For a more complete list of articles, see Janet Heine Barnett’s “Mathematics is a Plural Noun: The Case of Oliver Byrne,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-fifth Annual Meeting (Montreal),* 23: 26-46 (2010).

Byrne married Eleanor Rugg (1822-1897), a native of Loose (near Maidstone), Kent, England in 1845.^{39} (Maidstone is 40 miles east-southeast of London.) According to David Bryden, a scholar of Georgian scientific instruments, Oliver Byrne “engaged in several railway projects,” including the South Midland Junction Railway and the Tring, Cambridge and Newmarket Railway.^{40} Bryden wrote, “During the early years of the railway boom [Oliver Byrne] was responsible for the forward planning of several proposed routes, and he undertook other practical consultancy work.”^{41} In one anecdote,

a raw Irishman, named Oliver Byrne, who was engineer of a proposed line that was opposed by the Duke of Buckingham, describing how, after ‘numerous affrays took place between his chainmen and assistant surveyors and the Dukes’

posse,

had the clever idea of using ladders to obtain the necessary survey data during the night without setting foot on the Duke’s estate.^{42}

Byrne acquired expensive engineering equipment and when one of the railways failed, he was sued and lost 75£.^{43} Later he lamented, “I attribute all of my present difficulties to my having been engaged as Engineer to several Railway companies in the year 1845.”^{44 }(To read this lament in Byrne's own hand, see the second sentence of the first letter in **Figure 5**, below.)

**Figure 5.** Oliver Byrne's letter of application to the Royal Literary Fund, dated 25 November 1848, followed by his letter of thanks, dated 15 December 1848, for the £20 granted him. (Archives of Royal Literary Fund, London, via Nineteenth Century Collections Online: "British Theatre, Music, and Literature: High and Popular Culture" Collection, Gale Digital Collections http://www.galegroup.com)

Soon after his marriage, Byrne received an appointment as “Her Majesty’s [Queen Victoria] Settlement Surveyor” in the Falkland Islands.^{45} To repay the money lost on the railway, Byrne reported in 1848 that he was^{46}

compelled to deliver up in court the copyrights of very valuable books, [including] that of my ‘Euclid by Colours.’ Everything I possessed being thus taken from me, I resigned the government appointment [as Falkland Islands surveyor], for want of means to procure an outfit and the passage money.

For this passage in Byrne's own hand, see the first letter in **Figure 5**, above, which indicates also that he was "prevented from leaving this Country" for the Falkland Islands. Nevertheless, the title page of his 1847 *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners* indicates “Surveyor Of Her Majesty's Settlements in the Falkland Islands.” Byrne may have worked in some civil service capacity, as his widow collected a civil service pension.^{47} Byrne's *Euclid in ... Coloured Diagrams* (1847) will be discussed in more detail in a later section of this article.

By 1849, Byrne and his wife Eleanor had left London and moved to New York City, presumably, like many other immigrants, to improve their lives. Or, perhaps career pressures or the failed 1848 Irish nationalist uprising demanded the move.^{48} Not long after his arrival, Byrne filed his intent to become a naturalized United States citizen.^{49} Over the next decade Byrne and his wife spent time in New York City, Philadelphia, and Jersey City, New Jersey.^{50}

The Irish newspaper, *Cork Examiner,* announced the May 1849 arrival in New York of brothers Oliver and John Byrne in an article titled, “The Exiled Patriots.” The article claimed the men “although not heralded by any special political reputations, have been welcomed by all who desire to see Irish talent and virtue enjoy a proper sphere.”^{51} John Byrne died two years later. According to Oliver Byrne’s 1853 dedication in *The American Engineer, Draftsman, and Machinist’s Assistant,* his brother, John O’Byrne “was born in Wicklow, Ireland, on the 27th of May, 1812, and died in New York on the 6th of April, 1851.”^{52}

Later that year, the Irish nationalist weekly newspaper, *The Nation,* also acknowledged Oliver Byrne’s move to the United States:^{53}

[Oliver Byrne] after a career of continued success for twenty years in Europe—after passing through every scientific ordeal, as author, inventor and professor; at one time laying down a railway through Duke of Buckingham's demesne [manor], while guarded night and day; at another, a ship building for Mehemet Ali in Egypt; exposing consistently the over-weening scientific pretensions of England, while teacher of the Prime Minister's family; an avowed chartist, yet ‘surveyor-general of the Falkland Islands’—after all these tests and trials, Mr. Byrne, in his vigorous middle age, has begun a fresh career of activity in New York. The Messrs. Appleton have already announced a great work under his editorship, viz. ‘A Dictionary of Machines, Mechanics, Engine-Work, and Engineering;’ to contain 2,000 ‘royal octavo pages,’ and ‘1,500 plates.’

30 "Philanthropic Life Assurance Annuity and Endowment Society . . . to be Incorporated by Act of Parliament,” *Northern Liberator and Champion* (Newcastle-Upon-Tyne, England), 26 September 1840, p. 1, col. 5. The company's address at the time was 432 West Strand, London.

31 “Letter to the Editor,” *The Morning Post *(London), Thursday, November 26, 1840, p. 6, col. 3.

32 *Aris's Birmingham Gazette* (Birmingham, England), 15 June 1857, p. 4, col. 6.

33 “Exemplary Institute for Mathematics, Engineering, Classics and General Literature, Surrey Villa, near Lambeth Palace,” advertisement, *The Times* (London), 19 November 1842, p. 1. col. 3.

34 Oliver Byrne and Henry William Hull, "Institute for Mathematics, Engineering, Classics and General Literature, Surrey Villa, near Lambeth Palace," school prospectus (London: Barnes, c. 1842), 8 pp.; photocopy privately held by Dr. Sid Kolpas, 2014.

35 Oliver Byrne and John Byrne, *The Fallacies of Our Own Time: Fallacy of Phrenology* (London: Sherwood, Gilbert and Piper, 1844).

36 Stephen Jay Gould, *The Mismeasure of Man* (New York: W.W. Norton, 1981).

37 Oliver Byrne and John Byrne, *The Fallacies of Our Own Time: Fallacy of Phrenology* (London: Sherwood, Gilbert and Piper, 1844), 79.

38 “Ireland,” *The Morning Post* (London), 25 November 1844, p. 1, col. 6.

39 “London, England, Marriages and Banns, 1754-1921,” *Ancestry* (http://www.ancestry.com : accessed 25 July 2014); entry for marriage of Oliver Byrne and Eleanor Rugg (21 July 1845). For Eleanor Rugg’s nativity see Eleanor Rugg (29 Oct 1822), daughter of John and Elizabeth Rugg, "England, Kent, Parish Registers, 1538-1911," index and images, *FamilySearch* (https://familysearch.org : accessed 18 March 2015); citing Christening, Loose, Kent, England, Kent Archives Office, Maidstone; FHL microfilm 1,473,735. See also “1891 England Census,” [county] Kent, [parish] Ashford St. Mary, enumeration district 5, folio 62, line 10, Eleanor Byrne; digital images, *Ancestry* (http://www.ancestry.com : accessed 30 July 2014).

40 “Court of Exchequer,” *The Times* (London), 16 January 1846, p. 7, col. 6. See also "Court of Exchequer," *The Times* (London), 3 June 1846, p. 8, col. 3. Byrne engaged with the South Midland Junction. Also acting engineer for Tring, Cambridge and Newmarket Railway Company, *The Times* (London), 8 October 1845, p. 1, col 3. For Midland see “Midland Junction Railway; extending from the town of Reading, Berkshire, to the station at Blisworth, Northamptonshire,” deposited November 1845; catalog description, “An exceptionally crude series of plans, some marked 'Proof', on which the basic survey lines are lithographed . . . each signed 'Oliver Byrne Engineer’"; The National Archives (http://discovery.nationalarchives.gov.uk : accessed 18 August 2015).

41 David J. Bryden (Worcestershire, England) to Sue Hawes (Portland, Maine), "Oliver Byrne," e-mail, 17 December 2014, Byrne Research Files; privately held by Sue Hawes, Portland, Maine.

42 John Kersley Fowler, *Echoes of Old County Life: Being Recollections of Sports, Politics, and Farming in the Good Old Times.* (London: E. Arnold, 1892), 153-154.

43 “Court of Exchequer,” The Times (London), 16 January 1846, p. 7, col. 6. See also "Court of Exchequer," The Times (London), 3 June 1846, p. 8, col. 3.

44 RLF application (25 November 1848). See **Figure 5**, above.

45 *The Times* (London), 29 Aug 1846, p. 3, col. 5. See also Sylvanus Urban, *Gentleman’s Magazine* (London: Nichols and Son, 1846).

46 RLF application (25 November 1848). See **Figure 5**, above.

47 "A Lady and Her Lost Money," *North-Eastern Gazette* (Middlesbrough, England), 11 Jan 1896; "19th Century British Newspapers," Gale Cengage Learning (http://www.galegroup.com : accessed 4 April 2015). The year before Eleanor Rugg Byrne died, "A Lady & Her Lost Money: The Maidstone Guardians [Kent, England] have been informed that a lady possessing £73 was in the workhouse. The Relieving Officer had found her destitute near Paddock Wood owing to losing large sums of money in the vicinity, but the police subsequently recovered £73. The lady was understood to be the widow of the well-known mathematician, Oliver Byrne, in receipt of a Civil Service pension of £50, and possessing small private means. It was decided to communicate with the lady's friends."

48 For discussion of career pressures see Janet Heine Barnett, “Mathematics is a Plural Noun: The Case of Oliver Byrne,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-fifth Annual Meeting (Montreal),* 23:26-46 (2010). Byrne moved to the United States soon after the failed 1848 Irish Rebellion and subsequently published Irish nationalist militaristic materials cited in this article.

49 “New York, Naturalization Petitions, 1794-1906,” Oliver Byrne, Court of Common Pleas for the City and County of New York, 7 April 1849; digital images, *Ancestry* (http://www.ancestry.com : accessed 7 August 2014); citing Petitions for Naturalization, 1793-1906, Record Group 85, National Archives at New York City, New York, U.S.A.

50 For Philadelphia see “The Collin’s as War Steamers,” *Trenton State Gazette,* 22 June 1854. For New York see RLF application (6 January 1858). 1850 U.S. Census, New York, New York, Ward 5, population schedule, p. 120 (written), dwelling 532, family 870, Oliver Byrne; *Ancestry* (http://www.ancestry.com : accessed 30 July 2014), citing National Archives Microfilm Publication M432. For Jersey City, see 1860 U.S. census, Hudson County, New Jersey, population schedule, Jersey City, dwelling 111, family 211 (Oliver Byrne); *Ancestry *(http://www.ancestry.com : accessed 28 July 2014) citing Family History Library microfilm 803,693.

51 “The Exiled Patriots,” *Cork Examiner *(Ireland), 7 May 1849; *FindMyPast* (http://www.findmypast.com : accessed 21 December 2014), “Last week arrived Messrs. George Duggan (of the Irish Board of works) and Oliver and John Byrne of Dublin; who, although not heralded by any special political reputations, have been welcomed by all who desire to see Irish talent and virtue enjoy a proper sphere. Mr. O. Byrne is already announced to lecture before one of our scientific associations.–New York Correspondent of the Freeman.”

52 Oliver Byrne, *The American Engineer, Draftsman, and Machinist’s Assistant* (Philadelphia: C.A. Brown & Co., 1853). John Byrne may have had a son. In RLF application (2 November 1872), Oliver Byrne writes, "Married, wife living, no family, reared and educated my brother's son, who is supposed to be lost in the war, as he is missing." Oliver Byrne’s brother John is most often listed as Byrne but occasionally, O’Byrne.

53 "Irish Science," newspaper clipping, *The Nation* (New York), 10 November 1849; Nineteenth Century Collections Online: "British Politics and Society," Gale Digital Collections (http://www.galegroup.com : 8 August 2014); Oliver Byrne, application, Royal Literary Fund case number 987, Vol. 31, 28 May 1842. For definition of “chartist,” see *Oxford English Dictionary;* online definition for chartist: “One of the body of political reformers (chiefly of the working classes) who arose in 1837–8, and whose principles were embodied in the document called the ‘People's Charter’ (charter n.1 1d). (The organization came to an end after 1848),” *Oxford English Dictionary* (http://www.oed.com : accessed 21 June 2015).

American publisher Appleton and Company of New York City employed Oliver Byrne to edit *A Dictionary of Machines, Mechanics, Engine Work, and Engineering.* This massive and popular work became known as *Appleton's Dictionary.* Published in 1852, the book went through multiple editions. Evidence indicates Byrne edited the dictionary only through the letter F.^{54} A positive review of *Appleton’s Dictionary* concluded,^{55}

We observe, however, with surprise, that the last number of the second volume is accompanied by new title-pages for both volumes, in which the name of the editor [Byrne] is altogether omitted. What is the meaning of this?

*Appleton's Mechanics' Magazine and Engineers' Journal* responded that a replacement editor was “invited by the Messrs. Appleton to take the entire charge of the editorial department of their great Dictionary, then in progress, and published as far as the letter F.”^{56}

While living in the United States, Byrne published three pamphlets advocating Irish rebellion against British rule.^{57} It was said at the time that, “[Oliver Byrne] did more than any man in America to infuse into his emigrant countrymen a military spirit.”^{58} Byrne dedicated *Freedom to Ireland* to the memory of William Byrne. “Billy Byrne” of County Wicklow, Ireland died a martyr of the Irish rebellion against British rule when he was executed in 1799. It is no surprise that Oliver Byrne made no mention of his American citizenship or his Irish nationalism efforts in any of his Royal Literary Fund applications.

Copyright issues arose for Byrne both in London and the United States. Byrne’s *Practical Metal Worker’s Assistant* was claimed “a barefaced piracy of great portions of two volumes on ‘Turnary and Mechanical Manipulation,’ published by Mr. Holtzappel, deceased, mechanist and writer.” Its sale by Chapman was prevented in England. A newspaper covering the case reported, "The plaintiff was the widow in the executrix of her ingenious husband [Mr. Holtzappel]."^{59} Present day further investigation uncovered no other accusations against Byrne for plagiarism. Before Byrne completed the five-year naturalization process and became a U.S. citizen in 1854, he himself reported that publisher Henry Carey Baird “put [Byrne] into law for selling a copyright, not being a citizen.”^{60} Byrne later published *The Young Geometrician* with Chapman and Hall in 1865.^{61}

Although it is not known if Byrne traveled to France, in 1856 he published a French guidebook on engineering and railroads entitled *Vade Mecum: De L’Ingénieur de Chemins de Fer Donnant*.^{62}

After a ten year stretch residing in the United States with no Royal Literary Fund financial aid applications, in 1858, Byrne requested RLF funds^{63}

to obtain means to bring myself, and family to England … I have no political patronage [in the United States], not having taken any part in the politics of that country … the only article of my own, in this room is a cooking stove, not yet paid for.

Before leaving the United States, Byrne made an “important metallic discovery.” He developed a process of smelting a new metal he called "Byrneore" or "Byrne metal." His metal looked like gold or silver but contained very little of either. Byrne claimed that his new alloy “would defy detection by more than nine-tenths of the dealers in wares manufactured of pure gold and silver.” Its weight and appearance were much like those of gold or silver, but at a considerably lower price. Byrneore was to be used “in the manufacture of watches, rings, pencils, spoons, forks, heads for canes, tooth-picks, pens, table sets, chandeliers, and in fact everything in which gold and silver has been used.”^{64} It is doubtful Byrneore ever went into commercial production.

To commemorate his new alloys, Byrne apparently commissioned medals with the profile of his wife, Eleanor Rugg Byrne, on the obverse.^{65} On the reverse of the medal was written “BYRNEORE GOLD 1859” with 13 stars. The medals were struck in copper plain edge, copper reeded edge (3 only), brass, and white metal.^{66}

**Figure 6a. **Eleanor Rugg Byrne medal made of Byrneore gold in 1859 (Images of a coin privately held by Dr. Sid Kolpas, who acquired it in 2015)

**Figure 6b.** It has been conjectured that Byrne's medals, made from his Byrneore, were manufactured by John D. Lovett, a seal engraver, or die sinker, in New York City. (See David Baldwin's Lovett Tokens & Medals website http://lovetttokensmedals.comĀ Image used courtesy of Baldwin)

54 Franklin Institute, *Journal of the Franklin Institute* (Philadelphia: Pergamon Press, 1851) 22:360.

55 Franklin Institute. *Journal of the Franklin Institute* (Philadelphia: Pergamon Press, 1851), 22:360.

56 Julius Walker Adams, *Appleton’s Mechanics' Magazine and Engineers' Journal,* Volume 1 (New York: D. Appleton & Co., 1852-1853), 767.

57 First, Oliver Byrne, *The first fifty lessons on military art and science* (New York: D. &. J. Sadlier, 1850). Second, Oliver Byrne, *Freedom to Ireland: The Art and Science of War for the People. The Pike Exercise, Foot Lancers, Light Infantry, and Rifle Drill. To which is Added a Short Practical Treatise on Small Arms, and Ammunition, Street and House Fighting, and Field Fortification* (Boston: Patrick Donahoe, 1853). Third, *Lectures on the Art and Science of War: addressed to Irish American citizen soldiers* (Boston: Patrick Donahoe, 1853).

58 John O’Hart, *The Irish and Anglo-Irish landed gentry: when Cromwell came to Ireland, or, a supplement to Irish pedigrees* (New York: Barnes & Noble, 1968). This comment is unattributed in the book.

59 "Rolls' Court, Chancery-Lane, Wednesday, July 14," *The Times *(London), 15 July 1852, p. 7, col. 1. See also "Holtzappel v. Chapman," newspaper clipping, c. 1852; RLF application (28 May 1842).

60 For H. C. Baird trouble, see RLF application (6 January 1858). For citizenship, see “New York, Naturalization Petitions, 1794-1906,” Oliver Byrne, Court of Common Pleas for the City and County of New York, date of naturalization 27 May 1854; *Ancestry* (http://www.ancestry.com : 7 August 2014; citing: Petitions for Naturalization, 1793-1906, Record Group 85, National Archives at New York City, New York, U.S.A.

61 “The Young Geometrician” advertisement, Chapman & Hall, Piccadilly; *The Times* (London), 12 February 1866, p. 13, col. 8.

62 *Vade Mecum De L’Ingénieur, de Chemins de Fer Donnant* (Paris: De Napoléon Chaix et Ce, 1856).

63 RLF application (6 January 1858).

64 "Important Metallic Discovery," *The Weekly Arizonian* (Tucson), 20 October 1859, p. 1, col. 3; *Arizona Digital Newspaper Program* (http://adnp.azlibrary.gov : accessed 25 January 2015).

65 “1424 Medal of Eleanor Rugg Byrne ; gilt and tin ; 3 each. 6 pieces,” *Catalog of J. S. Twining's Collection of Gold, Silver and Copper American Coins with a little collection of Bric-a-Brac Washington Pitchers and Japanese Curios to be sold at Auction, 27-29 April 1886, New York City,* (Boston: Marvin & Son, Numismatic Printers, 1886), 64.

66 Lovett Tokens & Medals (http://lovetttokensmedals.com : accessed 28 July 2015).

Byrne and his wife left the United States before the outbreak of the Civil War in 1861.^{67} Fifty years old and back in London, Byrne touted his^{68}

two new mathematical sciences, not yet known … the ‘Calculus of Form’ and ‘Dual Arithmetic.’ Since I arrived [in London] I have been very ill and am now quite penniless, after having laboured and studied so hard my entire life …. These two are my greatest works, but it may take me some time to get them published so as to make them profitable to myself.

Byrne first mentioned *The Calculus of Form* in a revised version of an article that first appeared in the *Civil Engineer and Architect's Journal* in 1847.^{69} In the revised version of this article later printed in *Miscellaneous Mathematical Papers,* Byrne touted his “new work on the Calculus, about to be published.”^{70} An advertisement in *Miscellaneous Mathematical Papers* announced *The Calculus of Form* already in press. In his 1849 work on logarithms, Byrne cited specific page references from *The Calculus of Form.* In a somewhat later publication, *Mechanics: Their Principles and Practical Applications,* Byrne stated concerning *The Calculus of Form:*^{71}

a very large work was printed on this new science in London, but was suppressed through conspiracy, the particulars of which we have not space to enter into here.

**Figure 7.** First page of *The Calculus of Form* © The British Library Board, C.40.i.5.

Byrne’s widow Eleanor donated a printed copy of *The Calculus of Form* to the Trinity College Library, Dublin. The Trinity College copy includes no use of color, although Byrne may have intended to use color if the book had been published. The British Library reports that the whole edition was cancelled except two copies.^{72}

In *The Calculus of Form,* Byrne viewed derivatives of geometric forms shrinking in size (going down a dimension), and integrals of geometric forms expanding (flowing) into one higher dimension. However, he also discussed the standard limit definition of the derivative. In the second author's 44 years of teaching and studying Calculus, he has used the same metaphors of going up and down one dimension for integration and differentiation. In fact, Isaac Newton called derivatives* fluxions* (changing by shrinking one dimension) and integrals *fluents* (flowing up one dimension). The second author does not think Byrne's work offers anything really new about teaching Calculus. However, it is helpful for many students to visually explain differentiation and integration using the fluxion and fluent approach, using movement (shrinking down a dimension and flowing up a dimension) to give a geometric and dynamic explanation of Calculus. The Trinity College collection includes a manuscript copy of Byrne’s *Trinal Calculus,* a work to which he sometimes referred during his later years and which appears to be the last text that he worked on.

Byrne contended that the new art of dual arithmetic “supersedes the use of all sorts of logarithms, and gives results with greater accuracy and precision than any other set of numbers and with less labour.”^{73} In this work, published in 1863, Byrne proposed a new method of performing faster calculations that involved breaking numbers down into the form “*a*(1.1)* ^{b}*(1.01)

**Figure 8.** Title page of Oliver Byrne's *Dual Arithmetic *(from the book in the collection of Sid Kolpas)

On 19 January 1866, Byrne, along with Rev. Walter Mitchell, gave a lecture at the Royal United Service Institution entitled, “The New Science of Dual Arithmetic Applied to Naval and Military Calculations.” After the lecture, the chairman commented,^{74}

Mr. Byrne has thrown down the gauntlet, and such immense consequences in every department of mathematical science are involved, that the system ought to be fully tested . . .. The system is easy to be comprehended by the lowest intelligence.

However, Augustus DeMorgan, in *A Budget of Paradoxes,* was derogatory in his judgment of dual arithmetic. DeMorgan implied that it showed the lengths to which some mathematicians would go in attempting to ease the labor of doing arithmetic. Unlike many of Byrne’s engineering and practical texts, *Dual Arithmetic: A New Art,* was not a bestseller. See Janet Barnett's article, “The Dual Arithmetic of Oliver Byrne: A New Art which entirely supersedes the use of logarithms” (2011), for further analysis of the “dual arithmetic” works of Oliver Byrne.^{75}

A gas explosion tragedy struck at the Byrne residence on Tollington Road, Holloway, London in 1864. “Terrific Explosion of Gas at Holloway,” a local newspaper reported,

Mr. Byrne on entering the front parlour with a light to see if all was safe before retiring was suddenly thrown down, the light having come in contact with the gas, which had been allowed to escape from the chandelier.

Both Byrne and his wife, Eleanor, were severely burned on the face and hands. On Byrne’s next Royal Literary Fund (RLF) application, his witness notes, “Mr. Byrne makes his mark in my presence, his hands having been severly [*sic*] burnt.”^{76} Years later Eleanor developed blindness attributed to the explosion.^{77}

Soon after the explosion, the Byrnes moved to Upper Sydenham in the Parish of Lewisham, England, near their latest publishers, the Spons.^{78} According to an 1891 obituary of the engineer and explosives expert Ernest Spon, Spon “was educated as a Civil Engineer for upwards of three years, from 1869 to 1872, in the office of the late Mr. Oliver Byrne.”^{79} Spon’s uncles, Edward Spon and Francis Nicolas Spon, published at least two of Byrne’s books in 1867 and 1868 and two publications for Byrne’s wife, Eleanor Rugg Byrne, in 1869 and 1870.^{80} Byrne edited *Spons’ Dictionary of Engineering, containing mechanical, military, and naval applications, with technical terms in French, German, Italian, and Spanish,* with the first volume published in 1869.^{81}

Despite the success of *Spons’ Dictionary,* Byrne did not enjoy the fruits of his labor. In his November 1872 RLF application, Byrne complained,

I have been engaged, during the last four years and a half, compiling and editing a Dictionary of Engineering … consisting of 1952 pages which contain upward of 4000 engravings…. The publishers E. & F. N. Spon, without the slightest notice, refused, on the 29th of July last, to pay me any more money and they furnished me with an erroneous account, to make it appear, that I was overpaid. … To avoid paying me the royalty agreed upon; and to get the Dictionary into their own hands … they have involved me in Law … may God protect me, I am in bad hands.

After receiving the RLF grant, Byrne wrote in his letter of thanks,^{82}

It would be impossible for me to overrate the value of aid at peculiar umbilicus points in my simple career. I do not like to use harsh words, although I have been often in the hands of robbers, but, never before this time, was I in the clutches of the thugs of trade and the smeddumtails [slime] of society.

The Spons themselves, in a Preface to Volume 8 of the *Dictionary,* reported only that, "Until August, 1872, the editorial department was conducted by Mr. Oliver Byrne, assisted by Mr. Ernest Spon; at that period Mr. Byrne ceased to be editor, and the work has been completed under the direction of Mr. E. Spon." The title page of each volume of the *Dictionary* recorded “edited by Byrne and Spon,” with publishers E. and F. N. Spon. E. & F. N. Spon, later known as Spon Press, persisted as a London publisher of building, construction, and civil engineering books until Taylor & Francis purchased the company in 1998.^{83}

**Figure 9.** Eleanor Rugg Byrne's balloon in her article, “Safety Concentric Balloon,” *The Civil Engineer and Architect’s Journal,* 25:154-155 (May 1862).

Byrne’s wife, Eleanor Rugg Byrne, was an innovator and author like her husband. Her 1862 article, “The Safety Concentric Balloon,” in the *Civil Engineer and Architect’s Journal,* discusses a design for an innovative balloon that removes “all dangers while in the air” and extends “the art [of recreational ballooning] by a new guiding principle.” Her balloon design used a series of concentric interior balloons, the innermost of which was made of a strong material and permanently charged with gas. The innermost balloon was not connected to the basket containing occupants and was used to support the steering and propelling of the balloon. The diagram shows a center metallic balloon, surrounded by four other balloons made of flexible material. The intent of the balloon was to increase the safety of recreational ballooning. It is unknown whether the balloon was ever manufactured.^{84} Her interest in ballooning may have resulted from her meteorological pursuits.

**Figure 10.** One of Eleanor Byrne’s meteorological publications. (This image is used by kind permission of the Syndics of Cambridge University Library.)

In 1869-1870 Eleanor published two books dealing with weather prediction based on observed patterns of historical seasonal variations, the wind, and the phases of the moon. Her weather prediction techniques were primitive by modern meteorological standards. She discussed how “the moon acts on the aqueous or vaporous qualities of the atmosphere in a manner similar to that in which she acts upon the sea.” Eleanor indicated that “the moon has mountains, atmosphere, seas, rivers, and lakes, like those we observe on the earth.” She estimated the earth's atmosphere at nearly 50 miles high.^{85} According to *Wikipedia,* the boundary of our atmosphere with outer space measures 62 miles. It is interesting, and not surprising, to note that Eleanor's books were published by Spon, a publisher for whom her husband, Oliver Byrne, worked.

67 RLF application (4 November 1861). Byrne letter dated 5 November 1861 uses 33 Argyle Street, Kings Cross, London address.

68 RLF application (4 November 1861).

69 Oliver Byrne, “A New Theory of the Earth, that Fully Accounts for Many Astronomical, Geographical, and Geological Phenomena, Hitherto Unaccounted For,” *Civil Engineer and Architect's Journal,* 10:99-101 (April 1847) and 10:133-134 (May 1847).

70 John Byrne, *The Miscellaneous Mathematical Papers of Oliver Byrne* (London: Maynard, 1848), 5.

71 Oliver Byrne, *Mechanics: their principles and practical applications* (New York: De Witt & Davenport, 1853), 95.

72 *Explore the British Library* (http://explore.bl.uk : accessed 28 July 2015); entry for “The Calculus of Form. [By Oliver Byrne.],” Unique Identification Number BLL01000569301.

73 RLF application (4 November 1861).

74 *Journal of the Royal United Service Institution* (London: W. Mitchell and Son, 1867), 10:61.

75 Janet Heine Barnett, “The Dual Arithmetic of Oliver Byrne: A New Art which entirely supersedes the use of logarithms,” *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-sixth Annual Meeting (Dublin),* 24:12-30 (2011).

76 RLF application (7 February 1864). Byrne’s 1864 Royal Literary Fund application contains a newspaper clipping entitled, “Terrific Explosion of Gas at Holloway.” Byrne was registered in 1865 to vote at this address (Byrne, Oliver, 21 Tollington Road) in “London, England, Electoral Registers, 1832-1965,” *Ancestry* (http://www.ancestry.com : accessed 30 July 2014).

77 RLF application (28 June 1880).

78 “1871 England Census,” Kent, Lewisham, Sydenham, p. 26, Oliver Byrne; *Ancestry* (http://www.ancestry.com : accessed 2 August 2014); citing Census Returns of England and Wales, 1871 (Kew, Surrey, England: The National Archives of the UK, Public Record Office, 1871).

79 Ernest Spon, death; *Minutes of proceedings of the Institution of Civil Engineers *(Great Britain: Institution of Civil Engineers, 1891), 315-316.

80 Oliver Byrne, *The Essential Elements of Practical Mechanics, based on the principle of work; designed for engineering students* (E. & F. N. Spon: London, 1867). Also Oliver Byrne, *General Method of Solving Equations of all degrees; applied particularly to equations of the second, third, fourth, and fifth degrees* (London: E. & F. N. Spon, 1868). For Eleanor Rugg Byrne, *A Short Philosophical Treatise on the Atmosphere, and the Influence of the Changes of the Moon on the Weather; and the Variations of the Weather Represented by Weather Indicators, from the First of July 1869, to the First of January, 1870, to Which is Subjoined Interesting Matter* (London: E. & F. N. Spon, 1869) and *The Useful Weather Guide, for the First Six Months of the Year 1870 *(London: E. & F. N. Spon, 1870).

81 Oliver Byrne ed., *Spons’ Dictionary of Engineering, containing mechanical, military, and naval applications, with technical terms in French, German, Italian, and Spanish* (London: E. & F. N. Spon, 1869).

82 RLF application (2 November 1872). For definition of “smeddumtails" as "ore sludge" or "ore slime,” as someone from a mining town would know, see http://www.mindat.org/glossary/smiddum_tails.

83 "Taylor & Francis Timeline" (http://www.ulib.niu.edu/publishers/TaylorFrancis.htm : accessed 28 July 2015).

84 Eleanor Rugg Byrne, “Safety Concentric Balloon,” *The Civil Engineer and Architect’s Journal,* 25:154-155 (May 1862).

85 Eleanor Rugg Byrne, *A Short Philosophical Treatise on the Atmosphere, and the Influence of the Changes of the Moon on the Weather; and the Variations of the Weather Represented by Weather Indicators, from the First of July 1869, to the First of January, 1870, to Which is Subjoined Interesting Matter* (London: E. & F. N. Spon, 1869). Secondly, Eleanor Rugg Byrne, *The Useful Weather Guide, for the First Six Months of the Year 1870* (London: E. & F. N. Spon, 1870).

About 1873 Oliver and Eleanor Byrne moved to Eleanor’s native Maidstone. As Oliver Byrne wrote that year,^{86}

I have met with nothing but difficulties and disappointments during the past year. I am sorry to add that I have received an injury to my right arm; and my right collar bone is dislocated from a cab-accident, which renders it painful for me to write. My wife too has suffered for the last four years from injury to her eyes, supposed to have been caused by the gas explosion which occurred when we lived at Holloway. She has had three operations performed on her eyes by Dr. Vernon at St. Bartholomew's Hospital. These circumstances oblige me to leave my present abode. We are going to Maidstone; where I hope my wife will get some attention and relief at the Kent Ophthalmic Hospital.

This move did not improve the couple’s situation. In 1875, Byrne wrote,^{87}

My wife's prolonged illness, and the coming winter, together with the non-settlement of my publishers as to the sale of my works make me look forward to nothing but distress. Therefore I implore you to excuse my asking once more, through your kindness and consideration to be helped on my way.

Additional applications for aid reflect ongoing illness. In 1878 Byrne wrote,^{88}

I am in great distress and obliged to apply again to the Royal Literary Fund for assistance. I have been ill during last winter and most of the time unable to leave my room. Messrs. Wilkins and Vernon of 9, Castle Street Holborn, printers and publishers failed with my last work in their hands, this rendered my position still worse. My illness and age, combined with the endurance of such disappointments have, of late, prevented me from writing much.

The following year he wrote,^{89}

I have been in great trouble, and distress since last winter, and my circumstances now are so serious that they compel me to apply to your Honorable Committee for kind consideration and help. I have lost friends through death, and some within the present year. I have tried to better my position, but, age with other influences that I was unable to restrain, have conspired in force against all my hard work of fifty years of authorship. I have not been paid according to agreements, which may be due to the pressure of the times. I have written much, and I am still writing, although in feeble health; but, publishers will not undertake anything of mine at present. I have been aided by friends but this help I cannot expect to continue.

Oliver Byrne died on 9 December 1880 of bronchial pneumonia at 46 Grecian Street in East Maidstone, Kent, at the age of 70 years.^{90} His wife Eleanor survived him by seventeen years.^{91} She received an 1881 RLF grant as Oliver Byrne’s widow and in 1891 headed her own household while “living on own means” at Ashford St. Mary, Kent.^{92} She died on 12 June 1897.^{93}

Many of Oliver Byrne’s works, primarily on the topics of Mathematics and Engineering, can be found on *Google Books,* some in their entirety. Many other of his works are available in reprint. There follows a list, not necessarily comprehensive, of Oliver Byrne’s known books and pamphlets.

Evidenced by his numerous appeals to the Royal Literary Fund in London for financial support between 1839 and 1880, Oliver Byrne’s tumultuous career as author and civil engineer resulted in some successes and many challenges. In his words,^{94}

All of my books, inventions, and important discoveries seem only to lead me into trouble.

In fact, his publications on civil engineering seem to have done quite well and perhaps sustained him during his adult life, while his innovative mathematical/educational publications did poorly.

One of these failed books is Byrne's best known publication today, his *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners.* In the next section, we examine this amazing text in more detail.

86 RLF application (27 November 1873).

87 RLF application (29 November 1875).

88 RLF application (30 June 1879).

89 RLF application (30 June 1879).

90 General Register Office, Southport, England, Certified Copy Of An Entry Of Death, Oliver Byrne, December 9, 1880. See also Oliver Byrne, obituary; *The Times* (London), 16 December 1880, p. 10, col. 3.

91 General Register Office, Southport, England, Certified Copy Of An Entry Of Death, Eleanor Byrne, 12 June 1897. See also “England & Wales, National Probate Calendar (Index of Wills and Administrations), 1858-1966,” Eleanor Byrne (12 June 1897); digital images, *Ancestry* (http://www.ancestry.com : accessed 30 July 2014).

92 “1891 England Census,” [county] Kent, [parish] Ashford St. Mary, enumeration district 5, folio 62, line 10, Eleanor Byrne; digital images, *Ancestry* (http://www.ancestry.com : accessed 30 July 2014). See also "A Lady and Her Lost Money," *North-Eastern Gazette* (Middlesbrough, England), 11 Jan 1896; transcription in earlier note.

93 General Register Office, Southport, England, Certified Copy Of An Entry Of Death, Eleanor Byrne, 12 June 1897.

94 RLF application (2 November 1872).

Oliver Byrne’s most celebrated work, *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners,* was published in London by William Pickering in 1847. Byrne claimed to have conducted experiments showing that Euclid’s *Elements* could be mastered using this color method “in less than one third of the time usually employed.”^{96} His expressed aim was “to teach people how to think and not what to think.”^{97} In his final Royal Literary Fund application in 1880, Byrne wrote,^{98}

These works … have a greater aim than mere illustration; I do not introduce colours for the purpose of entertainment, or to amuse by certain combinations of tint and form, but to assist the mind in its researches after truth, to increase the facilities of instruction, and to diffuse permanent knowledge.

**Figure 11.** Title page of Oliver Byrne's

Byrne’s 1847 *Euclid* was one of the first multicolor printed books, and is today the most renowned and valuable of his works. Many consider it the most attractive edition of *Euclid’s Elements* ever produced. Byrne's *Euclid* was extremely difficult and expensive to produce, requiring exact registration of the pages in order to print each color, the typeface, and the vignettes; therefore, only one thousand copies were originally published. According to retired University College Dublin meteorology professor and mathematics blogger, Peter Lynch, the book was regarded as a curiosity, and was largely ignored; it did not sell well at a price of 25 shillings, almost five times the typical book price at the time.^{99}

Historian of art and architecture Werner Oechslin, in his preface to the beautiful facsimile, *Oliver Byrne: The Elements of Euclid* (see **Figure 11**, below), wrote that^{100}

no one who holds it in his hands can resist the fascination of its illustrations. The pictures are more captivating because they simply suggest, concretely demonstrate ad oculus and thus assist in the comprehension of mathematical laws that initially seem most difficult and abstract.

**Figure 12.** Facsimile of Oliver Byrne's *The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners,* edited by Werner Oechslin and published as *Oliver Byrne: The Elements of Euclid* by Taschen in 2013. (Image of the copy owned by Sid Kolpas)

Byrne, according to Peter Lynch, “was not in the vanguard of mathematical thought.”^{101} In Byrne's time, Euclid's* Elements* was *the* geometry text used in British schools, but there was considerable debate about its suitability. Mathematicians such as Charles Dodgson (1832-1898), who wrote children's literature as Lewis Carroll, in his book *Euclid and His Modern Rivals* (1879), and Augustus De Morgan (1806-1871) debated how appropriate it was to teach Euclid's classical version of geometry versus a more modern approach.^{102} Byrne felt, as does Sid Kolpas, that geometry is the basis of all of mathematical science, and should provide a student's first formal experience with proof, the bedrock of mathematical thought.

Written and designed purportedly to simplify Euclid’s geometry, Byrne's *Euclid* was an extraordinary example of Victorian printing and was described by typographer and book designer Ruari McLean in *Victorian Book Design and Colour Printing* as “one of the oddest and most beautiful books of the whole century.”^{103} McLean described each page as

a unique riot of red, yellow and blue: on some pages letters and numbers only are printed in color, sprinkled over the pages like tiny wild flowers, demanding the most meticulous register; elsewhere, solid squares, triangles, and circles are printed in gaudy and theatrical colors, attaining a verve not seen again on book pages till the days of Dufy, Matisse and Derain.

McLean labeled Byrne’s work “… a decided complication of Euclid.” Often, a pedagogical prophet is not recognized by his or her peers.

**Figure 13.** This page from Oliver Byrne's *Elements of Euclid* displays “a unique riot of red, yellow, and blue, reminiscent of Matisse,” in the words of modern typographer and book designer Ruari McLean. (Image from the book in the collection of Sid Kolpas)

Historian of mathematics and mathematics educator Florian Cajori (1859-1930) had a mixed opinion of the pedagogical approach of Byrne's *Euclid.* After asserting that the use of colored diagrams and symbols was “a noteworthy device for aiding the young mind through sensuous stimulus,” he speculated that “the failure of the book is doubtless due to the want of moderation in the use of colors.”^{104} David Eugene Smith (1860-1944), another historian of mathematics and mathematics educator, was also guarded in his opinion of Byrne's approach, stating that “[t]here is some merit in speaking of the red triangle instead of the triangle ABC, but not enough to give the method any standing.”^{105} It is Sid Kolpas' opinion, in agreement with Cajori and Smith, that Byrne's *magnum opus* is not well suited as a stand-alone geometry text, but is best used as a supplement to a geometry course, along with labeled diagrams, traditional proof, and algebraic argument; that way, it is not "a decided complication of Euclid," but rather an aid to better understanding Euclid's arguments. Mathematician Bill Casselman believes that while Byrne's *Euclid'*s “failures are interesting, … for students it has proved to be a fruitful source of projects.”^{106} In fact, Kolpas has successfully used Byrne's work as a source of student projects, and as an inspiration for using color-coded diagrams within a traditional teaching approach.

Contrary to McLean, Cajori, and Smith, Edward R. Tufte, pioneer in the visualization of data, indicated in his *Envisioning Information* that Byrne’s design may greatly clarify Euclid’s *Elements* for students with a visual preference for learning.^{107} That is, Byrne created a book that would help right-brained students master the complexities of geometry! Modern pedagogy teaches us that it is best to appeal to the visual, auditory, and kinesthetic senses when teaching mathematics. In the preface to his *Euclid,* Byrne stated that the traditional oral and written demonstrations of Euclid are enhanced by color dissections. To quote Byrne’s preface:^{108}

Sounds which address the ear are lost and die

In one short hour, but these which strike the eye,

Live long upon the mind, the faithful sight

Engraves the knowledge with a beam of light.

Byrne further indicated in his preface that he subscribed to the pedagogy of the Swiss educator Johann Heinrich Pestalozzi (1746-1827), considered by many to be the Father of Modern Education. Pestalozzi felt that education should be interactive and should appeal to all of the senses; the use of colored dissections enhances that appeal. It should be noted that Byrne’s *Euclid* was not his only attempt at using color and strongly visual explanations of mathematical concepts, similar to “proofs without words.” In his *The Young Geometrician,* published in 1865, Byrne set out to teach geometric constructions with the use of color. Moreover, according to Ruari McLean in a letter to the second author, Byrne also intended to create a calculus text with the use of color dissections.^{109} The book was proofed, but never printed. Byrne indicated in the proof copy that the calculus text would be “uniform with The Coloured Euclid.”

**Figure 14.** Page from Oliver Byrne's *The Young Geometrician.* (Image from the book in the collection of Sid Kolpas)

Byrne’s * *was designed and printed by the acclaimed printer Charles Whittingham (1795-1876) of the Chiswick Press. The book's use of color was its most striking feature, with equal angles, lines, or polygonal regions assigned the same bright primary color, and its colored shapes surely must have presented the greatest challenge to the printer. However, its more traditional black print was intricate and beautiful, too. Each proposition was set in black Caslon italic, with a beautifully engraved four-line initial vignette, usually an “I” for “If” or “In” (see **Figure 15**). A four-line initial vignette therefore begins most pages in Byrne’s *Euclid.* Caslon is a group of beautiful serif typefaces designed by William Caslon (1692–1766). According to the Society of Printers, “It was at the Chiswick Press that the use of the old-face Caslon type was revived in 1843. . . [A] revival followed by printers throughout England.”^{110} At least three women assisted Whittingham: his daughters Charlotte and Elizabeth Whittingham studied art and calligraphy, and Mary Byfield turned their designs into beautiful wood engravings and made other contributions to the business as well. According to the *Dictionary of National Biography,*^{111}

Charlotte and Elizabeth were educated as artists, and from their designs came the greater part of the extensive collection of borders, monograms, head and tail pieces, and other embellishments still preserved and used. The engraver of most of the ornamental wood-blocks was Mary Byfield (d. 1871).

Byrne’s *Euclid* benefited from their beautiful work.

**Figure 15.** The "initials" above were used to begin first sentences of sections in Byrne's *Euclid.* (These images are provided courtesy of William A. Casselman, Department of Mathematics, University of British Columbia, and appear at his website www.math.ubc.ca. They were made from a copy of Byrne's 1847 *Euclid* held by the Thomas L. Fisher Rare Book Library, University of Toronto.)

According to Julie L. Mellby, graphic arts librarian at Princeton University, in her online article "Euclid in Color," Byrne's *Euclid* was exhibited in London at the Great Exhibition of 1851.^{112} Praise was given for its beauty and the artistry of the printing, which may have influenced future publications and artwork. However, the book was sold for an extravagant price by contemporary standards, placing it out of the reach of educators who were supposed to make use of this new way of teaching geometry. Given the royal stamp on the upper right hand corner of the title page indicating "Sample: Department Of Science And Art," the second author suspects that his copy of Byrne's *Euclid* might have been a sample copy at the Great Exhibition.

Today, due to its rarity and beauty, Byrne’s 1847 *Euclid* is an extremely valuable book, sold for an extravagant price by modern standards. At the time of this writing, if one could find a copy similar in condition to the one in the second author's rare book collection, it might cost as much as $22,500. Fortunately, one can purchase the beautiful full color facsimile shown in **Figure 12**, above, edited by art and architecture historian Werner Oechslin and published by Taschen America. Moreover, Bill Casselman has made available an image of each page of Byrne’s *Euclid *at the University of British Columbia’s web site: www.math.ubc.ca/~cass/Euclid/byrne.html. Teachers can make use of these images in their lessons.

Ideas for using Byrne’s *Euclid* in the mathematics classroom can be found on the next two pages.

96

97

98 RLF application (28 June 1880).

99 Peter Lynch, “That’s Maths: The rebel who brought Technicolour to Euclid,” *Irish Times,* February 20, 2014.

100 Werner Oechslin, ed., *Oliver Byrne: The Elements of Euclid* (Cologne, Germany: Taschen America LLC, 2013), 15.

101 Peter Lynch, “That’s Maths: The rebel who brought Technicolour to Euclid,” *Irish Times,* February 20, 2014.

102 Lewis Carroll (Charles Dodgson), *Euclid and his Modern Rivals* (New York: Dover Publications, 1973; originally published in 1879).

103 Ruari McLean, *Victorian Book Design and Colour Printing* (London: Faber and Faber, 1963), 50-51.

104 Florian Cajori, “Attempts Made During the Eighteenth and Nineteenth Centuries to Reform the Teaching of Geometry,” *The American Mathematical Monthly,* 17:10 (1910), 194. The opinions held by various historians about Byrne's *Euclid* are summarized in Janet Heine Barnett's article, “Mathematics is a Plural Noun: The Case of Oliver Byrne,”

105 Smith, David Eugene, ed., Augustus De Morgan, and Sophia Elizabeth De Morgan, *A Budget of Paradoxes,* by Augustus de Morgan, 2nd ed. (Chicago and London: The Open Court Publishing Company, 1915; originally published in 1872), Volume I, p. 329.

106 Bill Casselman, “Pictures and Proofs,” *AMS Notices,* 47:10 (2000), 1257-1266.

107 Edward R. Tufte, *Envisioning Information* (Cheshire, Connecticut: Graph Press, 1990).

108

109 Ruari McLean to Dr. Sid Kolpas, letter, June 29, 1993.

110 Society of Printers, *The Development of Printing as an Art: A Handbook of the Exhibition in Honor of the Bi-Centenary of Franklin's Birth Held at the Boston Public Library under the auspices of the Society of Printers* (Society of Printers: Boston, Massachusetts, 1906), 26.

111 Sidney Lee, ed., *Dictionary of National Biography,* 1885-1900 (London: Smith, Elder & Co., 1900), 61:148.

112 Mellby, Julie L., "Euclid in Color," Princeton University Library, Princeton, New Jersey, 2008. https://blogs.princeton.edu/graphicarts/2008/05/euclid_in_color.html

The following section of this article will discuss how Byrne's

Byrne’s *Euclid* uses little or no labeling of the diagrams and no algebraic arguments. It is primarily geometric, using appropriate bold primary colors to indicate the various parts of each diagram associated with a theorem to be proved. Therefore, Byrne’s proofs are the type of argument that Euclid used in his *Elements,* almost purely geometric in nature. Instructors teaching plane geometry at any level can use pages from Byrne’s *Euclid,* and require students to give the reasons for each step of the proof. In fact, students could also construct their own color-coded proofs using Byrne’s methods; in the 1970s, Sid Kolpas assigned each of his ninth grade Geometry students a different theorem, and had them create their own color dissection proofs similar to Byrne's. These proofs were displayed on the walls and windows of the classroom, and were used for periodic review throughout the semester. The Pythagorean Theorem and the Vertical Angles Theorem were chosen as the following two examples.

Oliver Byrne’s proof of the Pythagorean Theorem, Proposition XLVII of Book I of Euclid’s *Elements,* is a comprehensive color version of Euclid’s classic “Windmill” proof. Great care was taken in isolating various parts of the figure by choosing appropriate colors to aid in understanding the proof. According to Byrne's preface, careful choices of tint and form, and the isolation of parts of the diagram, help to better impress knowledge in the student’s mind.

Typically, today's geometry teachers use a simpler, similar triangles approach to prove the Pythagorean Theorem after the properties of similar triangles and proportions are established. Byrne’s color proof, a “proof without words” (or at least very few words!), could also be used to show students how Euclid proved it in his *Elements.* One should ask students to supply reasons, in words, for each step in Byrne’s color version of Euclid’s proof. Or they can devise their own color proof of whatever method they choose to prove the theorem.

**Figure 16.** Byrne's proof of the Pythagorean Theorem (Book I, Proposition 47 of Euclid’s *Elements*). (Images from the book in the collection of Sid Kolpas)

Referring to Byrne’s proof, students typically come up with reasons similar to the following for each step (adapted from Kolpas' *The Pythagorean Theorem: Eight Classic Proofs*^{113}):

- Construct blue and black squares on the legs, and a red square on the hypotenuse of the triangle.
- In a plane, through a point outside a given line, only one line can be drawn parallel to a given line; these are the dotted lines. Also, through two points exactly one line can be drawn. These are the solid black lines.
- All right angles are equal; these are the yellow angles. Therefore, by the addition property of equality, the overlapping obtuse angles (yellow and black) are equal.
- All sides of a square are equal. Therefore, the triangles isolated in the proof are congruent by side-angle-side.
- In a plane, if a line is perpendicular to each of two lines, then the two lines are parallel. Therefore, the solid yellow line is parallel to the dotted black line.
- The square on the longer leg and the triangle that overlaps it share a common base and altitude, so the area of the square is twice the area of the triangle. (If a parallelogram and a triangle share a common base and are contained in a common set of parallel lines, then the area of the parallelogram is twice the area of the triangle). The same relationship holds for the larger (blue) rectangular piece on the hypotenuse and its overlapping triangle.
- By substitution, the area of the square on the longer leg equals the area of the large blue rectangular piece on the hypotenuse.
- A similar argument can be used to demonstrate that the area of the square on the shorter leg equals the area of the small yellow rectangular piece on the hypotenuse. I ask students to supply this argument.
- By the addition property of equality, the sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse. Symbolically, if the legs are of lengths \(a\) and \(b,\) then the hypotenuse is of length \(c,\) where \(a^2+b^2=c^2.\)

Peter Lynch noted that "The Red and Blue Chair" designed in 1917 by Gerrit Rietveld has striking similarities to the color diagram used in Byrne’s proof of the Pythagorean Theorem.^{114}

**Figure 17.** Red and Blue Chair, by Gerrit Rietveld, 1917 (Creative Commons License)

Another example is Byrne’s color proof of the theorem that if two lines intersect, vertical angles are equal (Euclid’s *Elements,* Book I, Proposition XV).

**Figure 18.** Byrne's proof of the Vertical Angles Theorem (Book I, Proposition 15 of Euclid’s *Elements*). (Image from the book in the collection of Sid Kolpas)

One should ask students to supply reasons, in words, for each step in Byrne’s color version of Euclid’s proof. Or they can devise their own color proof of whatever method they choose to prove the theorem.

Students typically come up with reasons similar to the following for each step:

- We are given a black line intersecting a red line.
- The yellow acute angle and the red obtuse angle add to a line (they are supplementary, or they add to half of a circle or 180 degrees or, as Euclid would say, "two right angles").
- The black acute angle and the red obtuse angle add to a line (they are supplementary, or they add to half of a circle or 180 degrees).
- The yellow acute angle plus the red obtuse angle is equal to the black acute angle plus the red obtuse angle since they are both equal to 180 degrees.
- Subtracting the red obtuse angle results in the yellow acute angle being equal to the black acute angle.
- A similar argument can establish the fact that the red obtuse angle is equal to the blue obtuse angle. I ask students to supply this argument.

Pages from Byrne’s *Euclid *would also make for interesting pedagogical discussions in Mathematics Education classes of the use of color in teaching mathematics and of the strategy of isolating parts of a mathematical diagram. Oliver Byrne’s colorful work changed the second author’s teaching of high school Geometry when he was first introduced to it in a History of Mathematics class in 1970. Because of its influence on his teaching, he always used colored chalk and color coded isolations of the appropriate sections of the geometric argument, along with appropriate labels and algebra where necessary, to explain the proofs of theorems to his high school students. Student feedback regarding those methods was always highly positive. Students regularly used colored pencils and markers when doing their homework and writing their exams.

Oliver Byrne’s *Euclid* can certainly be used to supplement the teaching of geometry at all levels, visually enhancing students’ understanding of the geometric arguments through color and the isolation of parts of the diagram. The reader might want to use Byrne’s proof of the sum of the angles of a triangle equaling “two right angles” (or 180 degrees or \(\pi\) radians; see **Figure 19**) – or any other of Byrne’s color proofs – with their students supplying the reasons for each step, or creating their own color proofs.

**Figure 19.** Byrne’s proof that the sum of the angles of a triangle is equal to two right angles (Book I, Proposition 32 of Euclid’s *Elements*). (Image from the book in the collection of Sid Kolpas)

Instructors of history of mathematics classes might use Byrne's *Euclid* as a catalyst to discuss the history of mathematics textbook design, the debate over the teaching of geometry in Victorian England, and the rise of non-Euclidean geometry. A good source for the debate over geometry education in Victorian England can be found in *Euclid and His Modern Rivals* by the English mathematician Charles Lutwidge Dodgson, better known as Lewis Carroll.^{115} The book debates the pedagogical merit of thirteen contemporary geometry textbooks, demonstrating how each is either inferior to or equivalent to the approach of Euclid's *Elements.* As Oliver Byrne supported the use of Euclid's *Elements* in the mathematics curriculum, Dodgson supported *The Elements* as the geometry textbook that should be used in schools instead of the more modern geometry textbooks that were beginning to replace it.

Debate over the importance of proof as taught in the high school Geometry class, the correct way to teach Geometry, and how to integrate geometry into the mathematics curriculum continues to this day. Also, in the nineteenth century, non-Euclidean geometry was challenging Euclidean geometry's place as the only true system of geometry. History of mathematics students could discuss the development of such geometries, and how they would make their own color proofs of theorems in Lobachevsky's or Riemann's non-Euclidean geometries. For example, we know that the sum of the angles of a spherical triangle is greater than 180 degrees. An example of a color-aided presentation of a spherical triangle, whose angles sum to 270 degrees, can be found at http://www.math2earth.oriw.eu/ via Math2Earth > Outcomes > Spherical trigonometry: sum of angles of a triangle on a sphere.

113 Sidney J. Kolpas, *The Pythagorean Theorem: Eight Classic Proofs *(Addison-Wesley Publishing / Dale Seymour Publications, Palo Alto, California, 1992). Kolpas dedicated this book to Oliver Byrne since Byrne's *Euclid in Colours* inspired him to write it! The book includes other color dissection proofs of the Pythagorean Theorem that could be used in Geometry classes, by historical figures as diverse as the Indian astronomer and mathematician Bhaskara, the Italian artist and scientist Leonardo da Vinci, and the future U.S. President James A. Garfield.

114 Peter Lynch, “That’s Maths: The rebel who brought Technicolour to Euclid,” *Irish Times,* February 20, 2014. On the preceding page of this article, Peter Lynch was introduced as a meteorologist and mathematics blogger.

115 Lewis Carroll (Charles Dodgson), *Euclid and his Modern Rivals* (New York: Dover Publications, 1973; originally published in 1879).

Oliver Byrne’s life, prolific publication record, and pedagogical philosophy have intrigued the authors for decades. This article shares a little known Victorian mathematician with a wider audience and describes how his *Euclid in Colours* can be (and have been) used for lessons and projects in geometry classes, and also for discussions of history and pedagogy in courses in mathematics history and mathematics education. Software such as *Geometer’s Sketchpad,* often used to teach geometry classes today, is also in the spirit of Byrne’s pedagogical philosophy and use of color, which, in his own words, “engraves the knowledge with a beam of light.”^{116} During his lifetime, Byrne’s mathematical work was sometimes ignored or greeted with ridicule from his contemporaries. He also battled financial difficulties throughout his life, encountered nationalistic prejudice as an Irishman, and faced serious health issues in his later years. Despite the many challenges he encountered in his day, today Oliver Byrne can be recognized as a true pedagogical visionary, a true Matisse of Mathematics.

Susan M. Hawes is a genealogist based in Portland, Maine. She holds a Bachelor of Arts from the University of Colorado at Boulder. Her biographical research on Oliver Byrne stems from studying her great-grandfather Austin Thomas Byrne (1859-1934), surveyor, civil engineer, and author. Artifacts Austin Byrne left to the family include three of Oliver Byrne’s books.

Dr. Sid Kolpas has taught Mathematics for 44 years at the Elementary, Secondary, and College levels. He earned a BA (Magna Cum Laude) and MS in Mathematics from California State University, Northridge, and an EdD in Mathematics Curriculum and Instruction and Educational Psychology from the University of Southern California. In 1984 he received the McLuhan Distinguished Teachers Award. In 1991 he was selected as Burbank Teacher of the Year. In 2004 he received the Distinguished Faculty Award from Glendale Community College. In 2010, he received the prestigious Hayward Award for Excellence in Education given by the Board of Governors of the California Community Colleges. His interests include the use of technology in teaching, and Mathematics History. He has collected antiquarian mathematics books and ephemera for the past 44 years, incorporating his collection in his teaching. He is currently an Assistant Professor of Mathematics at Delaware County Community College in Media, Pennsylvania where he is the Custodian of the Liberal Arts Mathematics and Statistics curriculum.

116

Gale Digital Collections. "19th Century British Library Newspapers." http://galegroup.com

Adams, Julius Walker (editor). *Appleton’s Mechanics' Magazine and Engineers' Journal,* Volume 1. New York: D. Appleton & Co., 1852-1853.

“Alumni Dublinenses : a register of the students, graduates, professors and provosts of Trinity College in the University of Dublin (1593-1860),” Manuscripts & Archives Research Library, Trinity College Dublin http://digitalcollections.tcd.ie : accessed 21 August 2015).

Ancestry. 1841, 1871, and 1891 England Census, 1850, [London] Electoral Registers, [London] “Marriages and Banns, 1754-1921” and 1832-1965 and 1860 U.S. population schedules.

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Arizona. Tucson. *The Weekly Arizonan* [or *Arizonian*]. 20 October 1859.

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Barnett, Janet Heine. "Mathematics is a Plural Noun: The Case of Oliver Byrne." *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-fifth Annual Meeting (Montreal).* Volume 23, pp. 26 – 46 (2010).

Barnett, Janet Heine. "The Dual Arithmetic of Oliver Byrne: A New Art which entirely supersedes the use of logarithms." *Proceedings of the Canadian Society for the History and Philosophy of Mathematics: Thirty-sixth Annual Meeting (Dublin).* Volume 24, pp. 12 – 30 (2011).

Broadbent, T. A. A. "George Boole (1815-1864)." *The Mathematical Gazette*. London: G. Bell and Sons, December 1964.

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Bryden, David J., Worcestershire, England to Sue Hawes. “Oliver Byrne.” E-mail, 17 December 2014. Byrne Research Files. Privately held by Sue Hawes, Portland, Maine.

Byrne, Eleanor Rugg, “Safety Concentric Balloon,” *Civil Engineer and Architect’s Journal,* 25:154-155 (May 1862).

Byrne, Eleanor Rugg, *A Short Philosophical Treatise on the Atmosphere, and the Influence of the Changes of the Moon on the Weather; and the Variations of the Weather Represented by Weather Indicators, from the First of July 1869, to the First of January, 1870, to Which is Subjoined Interesting Matter* (London: E. & F. N. Spon, 1869).

Byrne, Eleanor Rugg, *The Useful Weather Guide, for the First Six Months of the Year 1870* (London: E. & F. N. Spon, 1870).

Byrne, John (editor). *Miscellaneous Mathematical Papers of Oliver Byrne.* London: Maynard, 1848.

Byrne, Oliver. *A Treatise on Diophantine Algebra.* Dublin: Allen & Company, 1830.

Byrne, Oliver. *A Pamphlet on the Teaching of Geometry by Coloured Diagrams, etc. Applied to the First Book of Euclid.* London: Printed for the Author, 1831.

Byrne, Oliver. *A Short Practical Treatise on Spherical Trigonometry, etc.* A. J. Valpy: London, 1835.

Byrne, Oliver, *New and Improved System of Logarithms.* London: Day & Son, 1838.

Byrne, Oliver. *How to Measure the Earth with the Assistance of Railroads*. Newcastle, England: Currie & Bowman, 1838. Available on Google Books.

Byrne, Oliver. *The Creed of Saint Athanasius proved by a Mathematical Parallel.* London: William Day, 1839.

Byrne, Oliver. *The Practical, Complete and Correct Gager, containing a description of Parker and Byrne’s patent calculating instruments; with their use and applications.* London: A. H. Bailey & Co., 1840.

Byrne, Oliver.* The Doctrine of Proportion clearly developed … or, the Fifth book of Euclid simplified*. London: J. Williams, 1841.

Byrne, Oliver, Applications to Royal Literary Fund, 1839-1881, case number 987, Archives of Royal Literary Fund, London. Nineteenth Century Collections Online: "British Theatre, Music, and Literature: High and Popular Culture" Collection. Gale Digital Collections http://galegroup.com

Byrne, Oliver and Henry William Hull. "Exemplary Institute for Mathematics, Engineering, Classics, and General Literature, Surrey Villa, near Lambeth Palace. Prospectus." London: W. Barnes, c. 1842.

Byrne, Oliver, “Unwillingness of Man to Investigate,” *American Railroad Journal and Mechanics Magazine, *Vol VIII—New Series or Vol. XIV, 91-93. New York: George C. Schaeffer, 1842.

Byrne, Oliver. "Description and use of an instrument to find the time by the sun, moon, or any of the visible fixed stars: as well as the names of those stars / invented and constructed for the Right Hon. Earl Fitzwilliam" (manuscript), 1843. Harvard University Houghton Library.

Byrne, Oliver and John Byrne. *Fallacies of Our Own Time: Fallacy of Phrenology.* London: Sherwood, Gilbert & Piper, 1844.

Byrne, Oliver. *Description and Use of the Byrnegraph: an instrument for multiplying, dividing, and comparing lines, angles, surfaces, and solids*. London: C. & J. Adlard, 1846.

Byrne, Oliver. *Oliver Byrne: The Elements of Euclid* (Werner Oechslin, editor), Taschen, 2013, and online via the University of British Columbia website: http://www.math.ubc.ca/~cass/Euclid/byrne.html (Bill Casselman, editor).

Byrne, Oliver. “A New Theory of the Earth, that Fully Accounts for Many Astronomical, Geographical, and Geological Phenomena, Hitherto Unaccounted For.” *Civil Engineer and Architect's Journal.* 10:99-101 (April 1847) and 10:133-134 (May 1847).

Byrne, Oliver. *The Calculus of Form.* Never published, 1848.

Byrne, Oliver. *Practical, Short, and Direct Method of Calculating the Logarithm of any given number corresponding to any given Logarithm.* New York: D. Appleton & Co., 1849.

Byrne, Oliver (editor). *Dictionary of Machines, Mechanics, Engine-work and Engineering*. Philadelphia: D. Appleton & Co., 1850.

Byrne, Oliver. *The first fifty lessons on military art and science*. New York: D. &. J. Sadlier, 1850.

Byrne, Oliver. *The Practical Metal-Worker’s Assistant: containing the arts of working all metals and alloys with the application of electro-metallurgy to manufacturing processes, etc.* Philadelphia: H.C. Baird, 1851.

Byrne, Oliver. *Pocketbook for Railroad and Civil Engineers.* New York: Shepherd, 1851.

Byrne, Oliver. *The Pocket Companion for Machinists, Mechanics and Engineers, etc*. New York: Dewitt & Davenport, 1851.

Byrne, Oliver. *The Practical Cotton Spinner, and Manufacturer.* Philadelphia: Henry Carey Baird, 1851.

Byrne, Oliver. *Practical Model Calculator, for the engineer, mechanic, machinist, manufacturer of engine-work, naval architect, miner, and millwright.* Philadelphia: H.C. Baird & Co, 1852, 1862, 1872.

Byrne, Oliver. *The American Engineer, Draftsman, and Machinist’s Assistant.* Philadelphia: C.A. Brown & Co., 1853.

Byrne, Oliver. *Mechanics: their principles and practical applications.* New York: De Witt & Davenport, 1853.

Byrne, Oliver. *Freedom to Ireland: The Art and Science of War for the People. The Pike Exercise, Foot Lancers, Light Infantry, and Rifle Drill. To which is Added a Short Practical Treatise on Small Arms, and Ammunition, Street and House Fighting, and Field Fortification.* Boston: Patrick Donahoe, 1853.

Byrne, Oliver. *Lectures On The Art And Science of War Addressed To Irish-American Citizen Soldiers.* Boston: Patrick Donahoe, 1853.

Byrne, Oliver. *The Handbook for the Artisan, Mechanic, and Engineer.* Philadelphia: T. K. Collins, Jr., 1853.

Byrne, Oliver. *The Calculator’s Constant Companion, for practical men, machinists, mechanics and engineers.* Philadelphia: J.W. Moore, 1854.

Byrne, Oliver. *The Evidence of Oliver Byrne in the Patent Case of Ross Winans' Eight-wheeled Car.* London: Murphy, 1855.

Byrne, Oliver.* Vade Mecum. De L’Ingénieur, de Chemins de Fer Donnant.* Paris: Imprimerie et Libraire Centrales des Chemins de Fer. De Napoléon Chaix et Ce, Rue Bergère, 1856.

Byrne, Oliver. *Pocket-Book for Railroad and Civil Engineers. Containing new, exact, and concise methods for laying out railroad curves, switches, etc.* New York: C. Shepard & Co., 1856.

Byrne, Oliver. *The Mechanics’ Manual: a pocket companion for working carpenters, joiners, etc.* New York : J.M. Fairchild & Co., 1856.

Byrne, Oliver. *Byrne's price book, ready reckoner and measurer: for merchants and traders; ship builders and lumber dealers; farmers and drovers; banks and stock companies.* New York: Philip J. Cozans, 1857.

Byrne, Oliver. *The apprentice, or First book for mechanics, machinists, and engineers.* New York: Philip J. Cozans, 1860.

Byrne, Oliver. “Calculations Respecting the Pressure of Steam on Cylinder covers and Other Disks,” *Civil Engineer and Architect's Journal* (December 1860), 24:353-354.

Byrne, Oliver. *Dual Arithmetic: A New Art.* London: Bell & Daldy, 1863.

Byrne, Oliver. *The Young Geometrician.* London: Chapman and Hall, 1865.

Byrne, Oliver. *The Young Dual Arithmetician; or, Dual arithmetic . . . Designed for elementary instruction, etc.* London: Bell & Daldy, 1866.

Byrne, Oliver. *Dual Arithmetic A New Art. Part II. The Descending Branch of the Art, and the Science of Dual Arithmetic.* London: Bell & Daldy, 1867.

Byrne, Oliver. *Tables of Dual Logarithms, Dual Numbers, and corresponding Natural Numbers, etc.* London: Bell & Daldy, 1867.

Byrne, Oliver.* The Essential Elements of Practical Mechanics, based on the principle of work; designed for engineering students.* E. & F. N. Spon: London, 1867.

Byrne, Oliver. *General Method of Solving Equations of all degrees; applied particularly to equations of the second, third, fourth, and fifth degrees.* London: E. & F. N. Spon, 1868.

Byrne, Oliver, ed., *Spons’ Dictionary of Engineering, containing mechanical, military, and naval applications, with technical terms in French, German, Italian, and Spanish.* London: E. & F. N. Spon, 1869.

Byrne, Oliver. *Byrne’s Treatise on Navigation & Nautical Astronomy.* London: Bentley & Son, 1877.

Byrne, Oliver. *The Geometry of Compasses; or, Problems resolved by the mere description of circles, and the use of coloured diagrams and symbols.* London: C. Lockwood & Co., 1877.

Byrne, Oliver. *Byrne’s Timber and Log Book: Ready Reckoner and Price Book ….* New York: The American News Company, 1878.

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Cajori, Florian. Attempts Made During the Eighteenth and Nineteenth Centuries to Reform the Teaching of Geometry. *The American Mathematical Monthly,* 17:10, 181-201, 1910.

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Casselman, Bill. Pictures and Proofs. *AMS Notices,* 47:10, 1257-1266, 2000. http://www.ams.org/notices/200010/fea-casselman.pdf

*Catalog of J. S. Twining's Collection of Gold, Silver and Copper American Coins with a little collection of Bric-a-Brac Washington Pitchers and Japanese Curios to be sold at Auction, 27-29 April 1886,* New York City. Marvin & Son: Boston, 1886.

Cohen, Daniel J. *Equations from God: Pure Mathematics and Victorian Faith.* Baltimore: Johns Hopkins University Press, 2007.

DeMorgan, Augustus. *A Budget of Paradoxes.* London: Longmans Green & Co, 1872. Available online via Project Gutenberg: Volume I: http://www.gutenberg.org/ebooks/23100, Volume II: http://www.gutenberg.org/ebooks/26408. Second edition: Smith, David Eugene (editor), Chicago and London: The Open Court Publishing Company, 1915. Available online via Internet Archive: https://archive.org/details/budgetofparadoxe02demorich

Dodgson, Charles Lutwidge [Lewis Carroll]. *Euclid and His Modern Rivals.* New York, N.Y.: Dover Publications, 1973; originally published in London: MacMillan & Co., 1879.

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England. Kent. *Kent Messenger and Maidstone Telegraph.* 18 December 1880.

England. Kent, Parish Registers, 1538-1911" index and images. FamilySearch. https://familysearch.org.

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Gould, Stephen Jay. *The Mismeasure of Man.* New York: W.W. Norton, 1981.

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Heath, Thomas L. *The Thirteen Books of Euclid’s Elements, translated from the text of Heiberg with Introduction and Commentary.* New York: Dover, 1956. Originally published as Euclid, Thomas L. Heath, and J. L. Heiberg. *The Thirteen Books of Euclid's Elements.* Cambridge: The University Press, 1908.

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Kolpas, Sidney J. *The Pythagorean Theorem: Eight Classic Proofs.* Addison-Wesley Publishing Company / Dale Seymour Publications. Palo Alto, California, 1992. Out of print. Copyright reverted to Sidney J. Kolpas.

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Oechslin, Werner (editor). *Oliver Byrne: The Elements of Euclid.* Taschen, 2013. (Facsimile of Byrne's 1847

O’Hart, John. *The Irish and Anglo-Irish landed gentry: when Cromwell came to Ireland, or, a supplement to Irish pedigrees.* New York: Barnes & Noble, 1968.

*Oxford English Dictionary.* http://www.oed.com.

Principal Probate Registry [London, England]. Will and Grant in the Estate of the Late Oliver Byrne, proved at London 28 February, 1881.

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Smith, David Eugene, editor, Augustus De Morgan, Sophia Elizabeth De Morgan, *A Budget of Paradoxes,* 2nd edition. Chicago and London: The Open Court Publishing Company, 1915. Available online via Internet Archive: https://archive.org/details/budgetofparadoxe02demorich

Society of Printers. *The Development of Printing as an Art: A Handbook of the Exhibition in Honor of the Bi-Centenary of Franklin's Birth Held at the Boston Public Library under the auspices of the Society of Printers.* Society of Printers. Boston, Massachusetts, 1906

*Spons’ Dictionary of Engineering, Civil, Mechanical, Military and Naval, with technical terms in French, German, Italian, and Spanish.* London: F. N. Spon, London and New York: 1869-74. (*Note:* Oliver Byrne contributed from 1869 to 1872.)

Volume 1: Available online via Internet Archive: https://archive.org/details/sponsdictionary00spongoog

Volume 6: Available online via Internet Archive: https://archive.org/details/sponsdictionary01spongoog

Spon, Ernest. Obituary. *Minutes of proceedings of the Institution of Civil Engineers.* Great Britain: Institution of Civil Engineers, 1891.

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Timbs, John. *Curiosities of London: Exhibiting the Most Rare and Remarkable Objects of Interest in the Metropolis.* London: D. Bogue, 1855.

Schelhaas, T. N. Keeper of the Records of the City of Leiden [Netherlands]. Letter to S. J. Kolpas. 24 November 1992.

Tomash, Erwin. The Erwin Tomash Library Catalogue. http://www.cbi.umn.edu/hostedpublications/Tomash/

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Tufte, Edward R. *Envisioning Information.* Cheshire, Connecticut: Graph Press, 1990.

Urban, Sylvanus (editor). *Gentleman’s Magazine.* London: Nichols and Son, October 1846.

Walford, Edward. *Old and New London: A Narrative of Its History, Its People and Its Places. The Southern Suburbs Vol VI.* London: Cassell & Company, 1879.

Wikipedia

Wikisource

*A Treatise on Diophantine Algebra.* Dublin: Allen and Co., 1830.

*A Pamphlet on the Teaching of Geometry by Coloured Diagrams, etc. Applied to the First Book of Euclid.* London: Printed for the Author. 1831.

*A Short Practical Treatise on Spherical Trigonometry, etc.* London: A. J. Valpy, 1835.

*How to Measure the Earth with the Assistance of Railroads.* Newcastle-upon-Tyne: Currie and Bowman, 1838.

*New and Improved System of Logarithms … Also, an appendix, containing tables of trigonometric formulae, etc.* William Day: London, 1838.

*The Navigator's Ready Calculator; designed for the practical Sailor, being a complete and easy introduction to navigation, and containing a newly-invented instrument, by means of which all of the cases in the different sailings are solved.* William Day: London, 1838 or 1839.

*The Creed of Saint Athanasius proved by a Mathematical Parallel.* London: William Day, 1839.

*The Practical, Complete and Correct Gager, containing a description of Parker and Byrne’s patent calculating instruments; with their use and applications.* London: A.H. Bailey & Co., 1840.

*The Doctrine of Proportion clearly developed … or, the Fifth book of Euclid simplified.* London: J. Williams, 1841.

"Exemplary Institute for Mathematics, Engineering, Classics and General Literature, Surrey Villa, near Lambeth Palace." Prospectus written with Henry William Hull. London: W. Barnes, c. 1842.

“Unwillingness of Man to Investigate,” *American Railroad Journal and Mechanics Magazine,* Vol VIII—New Series or Vol. XIV, 91-93. New York: George C. Schaeffer, 1842.

“Description and use of an instrument to find the time by the sun, moon, or any of the visible fixed stars:as well as the names of those stars / invented and constructed for the Right Hon. Earl Fitzwilliam.” Oliver Byrne. Manuscript. Houghton Library, Harvard University. Boston. 1843.

*The Fallacies of Our Own Time “Anti-Phrenology”* (with John Byrne). London: Sherwood, Gilbert & Piper, 1844.

*Description and Use of the Byrnegraph: an instrument for multiplying, dividing, and comparing lines, angles, surfaces, and solids.* London: C. & J. Adlard, 1846.

“A New Theory of the Earth, that Fully Accounts for Many Astronomical, Geographical, and Geological Phenomena, Hitherto Unaccounted For,” *Civil Engineer and Architect's Journal,* 10:99-101 (April 1847) and 10:133-134 (May 1847).

*The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners.* London: William Pickering, 1847.

*The Calculus of Form.* Never published, 1848.

*The Miscellaneous Mathematical Papers of Oliver Byrne,* Collected and edited by John Byrne. London: Maynard, 1848.

*Practical, Short, and Direct Method of Calculating the Logarithm of any given number corresponding to any given Logarithm.* New York: D. Appleton & Co., 1849.

*A Dictionary of Machines, Mechanics, Engine Work, and Engineering*. New York: D. Appleton & Co., 1851.

*Pocketbook for Railroad and Civil Engineers.* New York: Shepherd, 1851.

*The Practical Metal-Worker’s Assistant: containing the arts of working all metals and alloys with the application of electro-metallurgy to manufacturing processes, etc. A new, revised, and improved edition, with additions by John Scoffern…William Clay, William Fairbairn… and James Napier, etc.* Philadelphia: H.C. Baird, 1851 and 1864.

*The Pocket Companion for Machinists, Mechanics and Engineers, etc.* New York: Dewitt & Davenport, 1851.

*The Practical Cotton Spinner, and Manufacturer.* Philadelphia: Henry Carey Baird, 1851.

*The practical spinner, and manufacturer: the managers’, overlookers’, and mechanics’ companion. A comprehensive system of calculations of mill gearing and machinery … To which are added, compendious tables of yarns.* New York: R. Scott, 1851.

*Practical Model Calculator, for the engineer, mechanic, machinist, manufacturer of engine-work, naval architect, miner, and millwright.* Philadelphia: H.C. Baird & Co, 1852, 1862, 1872.

*Freedom to Ireland: The Art and Science of War for the People. The Pike Exercise, Foot Lancers, Light Infantry, and Rifle Drill. To which is Added a Short Practical Treatise on Small Arms, and Ammunition, Street and House Fighting, and Field Fortification.* Boston: Patrick Donahoe, 1853.

*Lectures on the Art and Science of War: addressed to Irish American citizen soldiers.* Boston: Patrick Donahoe, 1853.

*Mechanics: their principles and practical applications.* New York: De Witt & Davenport, 1853.

*The American Engineer, Draftsman, and Machinist’s Assistant.* Philadelphia: C.A. Brown & Co., 1853.

*The Handbook for the Artisan, Mechanic, and Engineer.* Philadelphia: T. K. Collins, Jr., 1853.

*The Calculator’s Constant Companion, for practical men, machinists, mechanics and engineers.* Philadelphia: J.W. Moore, 1854.

*The Evidence of Oliver Byrne in the Patent Case of Ross Winans' Eight-wheeled Car.* London: Murphy, 1855.

*Pocket-Book for Railroad and Civil Engineers. Containing new, exact, and concise methods for laying out railroad curves, switches, etc.* New York: C. Shepard & Co., 1856.

*The Mechanics’ Manual: a pocket companion for working carpenters, joiners, etc.* New York : J.M. Fairchild & Co., 1856.

*Vade Mecum. De L’Ingénieur, de Chemins de Fer Donnant*. Paris: Imprimerie et Libraire Centrales des Chemins de Fer. De Napoléon Chaix et Ce, Rue Bergère, 1856.

*Byrne's price book, ready reckoner and measurer: for merchants and traders; ship builders and lumber dealers; farmers and drovers; banks and stock companies.* New York: Philip J. Cozans, 1857.

*The apprentice, or First book for mechanics, machinists, and engineers.* New York: Philip J. Cozans, 1860. (Reprinted New York: Philip J. Cozans, 1863, 1864; New York: A. J. Fisher 1874.)

“Calculations Respecting the Pressure of Steam on Cylinder covers and Other Disks,” *Civil Engineer and Architect's Journal* (December 1860), 24:353-354.

*Dual Arithmetic: A New Art.* London: Bell & Daldy, 1863.

*The Young Geometrician; or, Practical geometry without compasses.* London: Chapman & Hall, 1865.

*The Young Dual Arithmetician; or, Dual arithmetic . . . Designed for elementary instruction, etc.* London: Bell & Daldy, 1866.

*Dual Arithmetic A New Art. Part II. The Descending Branch of the Art, and the Science of Dual Arithmetic.* London: Bell & Daldy, 1867.

*Tables of Dual Logarithms, Dual Numbers, and corresponding Natural Numbers, etc.* London: Bell & Daldy, 1867.

*The Essential Elements of Practical Mechanics, based on the principle of work; designed for engineering students*. London: E. & F. N. Spon, 1867.

G*eneral Method of Solving Equations of all degrees; applied particularly to equations of the second, third, fourth, and fifth degrees.* London: E. & F. N. Spon, 1868.

*Spons’ Dictionary of Engineering, Civil, Mechanical, Military and Naval, with technical terms in French, German, Italian, and Spanish.* London: F. N. Spon, London and New York: 1869-74. (*Note:* Oliver Byrne contributed from 1869 to 1872.)

Volume 1: Available online via Internet Archive: https://archive.org/details/sponsdictionary00spongoog

Volume 6: Available online via Internet Archive: https://archive.org/details/sponsdictionary01spongoog

*Byrne’s Treatise on Navigation and Nautical Astronomy.* London: Richard Bentley and Son, 1877.

*The Geometry of Compasses; or, Problems resolved by the mere description of circles, and the use of coloured diagrams and symbols.* London: C. Lockwood & Co., 1877.

*Byrne’s Timber and Log Book: Ready Reckoner and Price Book ….* New York: The American News Company, 1878

Spon, Edward. Oliver Byrne; Ernest Spon; Francis N. Spon. *Supplement to Spons’ dictionary of engineering, civil, mechanical, military, and naval.* London and New York: E. & F. N. Spon, 1879-81.

The following is a transcription of a photocopy of Oliver Byrne's Will (see below) privately held by Sid Kolpas.

Will in the Estate of the Late Oliver Byrne, proved at London 28 February, 1881.

On the 28^{th} day of February 1881 the will of Oliver Byrne formerly of No. 14 Fransfield Grove Upper Sydenham in the Parish of Lewisham in the County of Kent but late of Maidstone in the said County Civil Military and Mechanical Engineer deceased, who died on the 9^{th} day of December 1880 at Maidstone aforesaid was proved in the Principal Registry of the Probate Division of the High Court of Justice, by the oath of Eleanor Byrne of No. 46 Grecian Street Maidstone aforesaid Widow the Relict of the said Deceased (daughter of John Rugg Esquire) the sole executrix named in the said Will she having been first sworn duly to administer, ... Personal Estate under £100 / No Leaseholds

This is the last will and testament of me, Oliver Byrne of No. 14 Fransfield Grove Upper Sydenham in the parish of Lewisham in the county of Kent. Author of numerous works on Mathematics, Mechanics and Engineering Civil Military and Mechanical Engineer I direct that all my just debts funeral and testamentary expenses be paid and satisfied by my executrix hereinafter named as soon as conveniently may be after my decease I give devise and bequeath all my copyrights manuscripts and books to my dearly beloved wife Eleanor Byrne all money coming to me in any way whatever I give and bequeath to the said Eleanor Byrne I give and grant her power to will the copyrights of my works to whom she likes or to sell or dispose of them in any way that she said Eleanor Byrne may think proper I give devise and bequeath to my wife the said Eleanor Byrne all my personal property in furniture houses or lands that may be left to me by Will or otherwise to sell or dispose of as the said Eleanor Byrne may think right to and for her own use and benefit absolutely And I nominate constitute and appoint the said Eleanor Byrne the daughter of John Rugg Esquire of Maidstone Kent and sister of R. Rugg Esquire Surgeon of 65 Middle St., Brighton Sussex also sister of Dr. G P Rugg etc. etc. to be Executrix solely of this my Will and hereby revoking all former or other Wills and Testaments by me at any time heretofore made I declare this to be my last Will and Testament in witness whereof I the said Oliver Byrne Mathematician to have this my last Will and Testament set my hand the eighth day of March in the year of Our Lord one thousand eight hundred and seventy three [1873] – Oliver Byrne – signed and declared by the said Oliver Byrne the testator as and for his last Will and Testament in the presence of us who at his request in his presence and in the presence of each other all being present at the same time have hereunto subscribed our names as witnesses George Rivers 130 Patton Mill Rd. N. – James Walkley Lee 24 Essex Street, Islington N[orth].

Proved at London 28 February 1881 by the oath of Eleanor Byrne widow, the relict daughter of John Rugg Esq. the sole executrix to whom admin[istration] was granted.

**Figure 20.** Last Will and Final Testament of Oliver Byrne (photocopy privately held by Sid Kolpas)