THE AMERICAN MATHEMATICAL MONTHLY
VOL. ll. JANUARY. 1895. NO. 1.
BIOGRAPHY.
ALEXANDER MACFARLANE, M. A., D. Sc., LL. D.
by J. M. Colaw
Alexander Macfarlane was born at Blairgowrie, Scotland, April 21, 1851. He was educated at the public school, and at 13, became a regular pupil teacher in the employment of the Education Department. In 1869, having finished his apprenticeship as a teacher and saved a little money, Mr. Macfarlane went straight to the University of Edinburgh. At that time the curriculum for Master of Arts consisted of three departments–classical, mathematical, and philosophical; and it was customary for the more ambitious students to take the degree with honors in one of these departments.
Mr. Macfarlane first entered the Junior classes in Latin and in Greek, and at the end of the session, stood fourth in the former and fifth in the latter, in classes of 200, largely composed of High School graduates. He perceived that to carry himself through college it was necessary either to sacrifice a large part of his time to teaching, or else to study hard and pay his way by means of money prizes. He chose the bolder alternative. At the beginning of his second year, he won in open competition the Miller scholarship, worth $400. At the end of that year, he stood very high in Senior Latin and Greek and in Junior Mathematics. At the beginning of the third year, he won in open competition the Spence scholarship, worth $1,000. The financial difficulty was now solved; there remained a choice of a department for honors. He was urged by the professor of Latin to go forward in the Classics, but he felt that there was more scope for originality in philosophy. In his third year, he studied Senior Mathematics, Natural Philosophy and Logic. It was the custom of Professor Kelland to introduce Quaternions to his senior students. The addition of vectors was intelligible, but the product of vectors seemed to be a universal difficulty. The professor explained that in it the lefthand vector was to be considered as a sort of corkscrew turning the righthand vector through a right angle; but he did not explain how it ceased to be a corkscrew. To get light on the subject Mr. Macfarlane bought a copy of Tait’s Treatise on Quaternions, but found that it was addressed to mathematicians.
Before he entered the class of logic, Mr. Macfarlane was familiar with the works of Hamilton and Mill, and when a member of the class he read, at the invitation of the professor, a paper which criticized the statement of the Law of Excluded Middle given by Jevons in his Lessons on Logic. It was his intention to study for honors in logic and philosophy, but perceiving how much they depended on the principles of science, and especially of exact science, he took up the advanced classes in mathematics and physics as a secondary study. In Experimental and Mathematical Physics he gained the highest honors and the personal friendship of Professor Tait, then, as now, the greatest figure in the University. In 1874 he was appointed Neil Arnott Instructor in Physics, and in 1975 finished an unusually extensive course of undergraduate study by taking the degree of M. A. with honors in mathematics and physics.
The University record showed that he had passed each of the seven subjects of the pass examinations with high distinction. Having, after graduation, won in a competitive examination the MacLaren fellowship, worth $1,500, he proceeded to study for the recently instituted degree of Doctor of Science. After one year spent on chemistry, botany, and natural history, and two years on mathematics and physics, he obtained the doctorate in 1878. His thesis was an experimental research on the conditions governing the electric spark, and it was subsequently published in the Transactions of the Royal Society of Edinburg. It also brought him under the notice of the celebrated electrician and philosopher, Clark Maxwell, who made various suggestions for its extension.
In 1878, Dr. Macfarlane was elected a Fellow of the Royal Society of Edinburgh, and the first contribution which he read personally was a memoir on the Algebra of Logic. The memoir was referred by the Council to the professors of mathematics and of logic, and they reported that it was too mathematical for the one and too logical for the other to enable them to say what its value was. Dr. Macfarlane enlarged the memoir and published it as a small volume under the title of Principles of the Algebra of Logic (1879). The volume was received with favor, and brought the author into correspondence with Munro, Jevons, Venn, Caylev, Harley, Schroeder and Halsted who was then lecturing on the mathematical logicians at Johns Hopkins University. The main idea propounded is that of a limited and definite universe; also Euler`s diagrams were further developed. In 1879, he attended the meeting of the British Association at Sheffield, and there met many of the British savants.
During 1880, Dr. Macfarlane was interim Professor of Physics at the University of St. Andrews, and in 1881, he was appointed for the usual period of three years Examiner in Mathematics in the University of Edinburgh. During these years he contributed to the Royal Society of Edinburgh a series of experimental papers on electricity, and a series of mathematical papers on the Analysis of the Relationships of Consanguinuity and Affinity. A paper on this subject, which he read before the Anthropological Institute of London, contains as perfect a notation for relationship as is the Arabic notation for numbers. These papers, as well as those on the algebra of logic, now form part of the history of exact logic. He also contributed to the Royal Society of Edinburgh a note on plane algebra, which stated briefly the view he had arrived at concerning the imaginary algebra of the plane. It states that the fundamental quantity is versor rather than a vector, a view in advance of Argand’s, and indeed of much that has been written more recently. By means of this algebra of the plane, he deduced many series, some of which he propounded as problems in the Educational Times and the Mathematical Visitor. It was also during his tenure of office as examiner that he prepared the volume on Physical Arithmetic, a pioneer work, whose express object is to elucidate the logical processes involved in the application of arithmetic to physical problems.
In 1885, Dr. Macfarlane was called to the chair of physics at the University of Texas, where he became a colleague of his fellow logician, Dr. Halsted. That same year he met many of the American savants at the Ann Arbor meeting of the American Association. In 1887, he received the honorary degree of LL. D. from the University of Michigan on the occasion of their semi~centennial. His first years at the University of Texas were wholly taken up with organizing the department, but in 1889 he published as a sequel to Physical Arithmetic a volume of Elementary Mathematical Tables, distinguished for their comprehensiveness and uniformity. In 1889, he visited Paris at the time of the Exposition and met many of the continental savants at the meeting of the French Association.
On April 8, 1885, Macfarlane married Helen Mattie Swearingen. Her sister, Margaret, married math professor George Halsted, also a member of the University of Texas faculty.
On his return from Europe, he began to publish the results of his study of the algebra of space, which he approached as a logical generalization of the algebra of the plane. These papers are as follows: 1. Principles of the Algebra of Physics, read before the Washington meeting of the American Association in 1891, states the fundamental difficulties in the theory of Quaternions, lays stress on the distinction between vectors and versors, and deals mostly with the products of vectors. 2. On the Imaginary of Algebra, read at the Rochester meeting in 1892, gives an historical and critical account of the different interpretations of square root of (1), takes up the functions of versors, and shows that there are at least two distinct geometrical meanings of square root of (1). 3. The Fundamental Theorems of Analysis Generalized for Space, contributed to the New York Mathematical Society in 1892, investigates and proves the generalized form of the binomial and other theorems, and 4. On the Definitions of the Trigonometric Functions, read before the Mathematical Congress at Chicago in 1893, defines these functions so as to apply to the circle, hyperbole, ellipse, logarithmic spiral, and ncomplex curve partly circular, partly hyperbolic. 5. The Principles of Elliptic and Hyperbolic Analyses, read at the same place and time, extends spherical trigonometrical analysis to the other surface of the second order. 6. “The analytical treatment of alternating currents,” read before the International Electrical Congress at the same time, shows that plane algebra is the analysis needed for the problems of alternating currents 7. On Physical Addition or Composition, read before the Madison meeting of the American Association in 1893, treats in a uniform manner of the composition of various physical quantities located in space, ending with the composition of screwmotions. 8. On the Fundamental Principles of Exact Analysis, read before the Philosophical Society of Washington in 1894, discusses the fundamental laws of algebra, and the logical principle of generalization in analysis, 9. The Principles of Differentiation in Space Analysis, recently read before the American Mathematical Society at New York, investigates the differentiation of versors, and publishes the true generalization of Taylor`s theorem for space.
In 1891, Dr. Macfarlane took an active part in organizing the Texas Academy of Science, and for two years acted as its Honorary Secretary. He contributed many papers, among which may be mentioned "An Account of the Rainmaking Experiments in San Antonio," an article describing and criticizing the various modern methods of rainmaking, and a paper on "Exact Analysis as the Basis of Language," where his knowledge both of languages and of mathematics comes into play.
In 1894, Professor Macfarlane resigned from the University of Texas. Throughout the nine years he labored there, he gave the new University the full benefit of his varied experience as a teacher, his accurate knowledge of University affairs, and his widespread reputation as a savant. The course in mathematical physics was so well developed as to call forth a special article in the Rivista di Matematica, published at Turin, Italy.
Professor Macfarlane, in addition to being a member of numerous American and British societies, is a corresponding member of the Sociedad Cientifica Antonio Alzate, of Mexico, and the Circolo Matematico di Palermo, Italy. Personally he is a characteristic Scotsmansturdy, persevering, with a relish for hard work, thoughtful, courageous in his convictions, and endowed with more than the average share of the perfervidun ingenium Scotorum. He is unmarried, but it is announced that in this, as in other matters, good fortune awaits him. And as he is still a young man, it is not likely that we have seen the last of his contributions to mathematical analysis.
To the editors of the Electrical World we are indebted for the loan of the electrotype.
Alexander Macfarlane, MA, DSc, University of Edinburgh, FRSE June 15, 1885, regents approve an appointment of associate professor for Alexander Macfarlane. He received 5 votes vs 1 for Chancellor Garland of Vanderbilt. The next year at the January 29th meeting, Macfarlane proposed that the department enlarge by adding a room dedicated to laboratory and lab equipment. The regents approved $5000. In 1890, Macfarlane presented his record to the Board of Regents. He was competing with outside faculty for reappointment, he received a majority of the votes and was asked to make a proposal for improving the department. Over the next years, he pushed hard for additional staff and equipment to compete with schools such as Michigan and Cornell. He provided detailed comparative data with letters from faculty at the two school. He suggested that the department was not fulfilling its “first class” mission. In June 1894, in executive session, the Board requested that he tender his resignation from the University and that the position of chair be replaced with an associate professor, to be determined. The chair of biology was removed at the same time. The regents, as part of the same action, instructed both departments to do a major inventory report, casting a shadow on the two men. A group of alumni submitted a report of protest to the regents which was not accepted. (A December report found $900 to $1500 worth of materials, including some platinum, missing from the Chemical Laboratory. It was suggested a previous employee was responsible. No evidence of impropriety by either chair was presented.)
A 1951 report prepared by the Dean of Arts and Sciences, C. P Boner (physics), to determine if an instructorship should be named after Professor Halsted reported the following: "Both Professor Battle and Professor Porter are in agreement that Professor Halsted fully deserved the dismissal he got. According to Dr. Battle, Halsted was associated with Edward (biology), Everhardt (chemistry), and McFarlane (physics) in an effort to discredit the services of Messrs. Waggener, Wooldrige, and Wooten of the board of regents. Several of these men were discharged for their campaign, but Professor Halsted was continued on the faculty with his salary reduced by $500 for each of three years." Halsted was later dismissed from UT apparently for his "stuffing the ballot box" in connection with his candidacy for president of the Texas Academy of Science. Professor John W. Mallett, first chairman of the UT Faculty, once remarked,"Texas can send a man up higher, and let him down lower, than any other region on the face of the earth." Those dismissed were probably the source of Mallett's remark.
The regents appointed Dr. A. L. McRae of the Rolla School of Mines as associate professor and chair at a salary of $3000 for a term of one year. Like Macfarlane, he pushed for an electrical power generating plant for the campus or dedicated lines from the new Austin Power and Dam facility about to come on line. He requested $1500 for associated electrical equipment.
Petition by Professor of Law Clarence H. Miller and other alumni to have Professor MacFarlane reappointed was denied by the regents.
The biography of Macfarlane included below is from the Commemorative Biographical Record of the County of Kent, Ontario.
ALEXANDER MACFARLANE, one of the most distinguished citizens of Ontario, a leader in scientific thought and author of the highest merit, and a savant in whom both his native land and his adopted country take pride, was born at Blairgowrie, Scotland, April 21, 1851. His education was obtained in the public school and his selection as a pupilteacher as early as the age of thirteen years gives testimony to the quick ripening of his powers. His ambition was to reach the University of Edinburgh, and in 1869, he entered that great educational institution. Mr. Macfarlane first entered the junior classes in Latin and Greek, and at the end of the session, stood fourth in the former and fifth in the latter, in classes of 200, largely composed of high school graduates. At the beginning of his second year, he won the Miller scholarship, worth $400, in open competition, and at the beginning of the third year he won, in open competition, the Spence scholarship, worth $1,000. His third year of study was given to senior mathematics, natural philosophy and logic. It was the custom of Prof. Kelland to introduce quaternions to his Senior students. The addition of vectors was intelligible, but the product of vectors seemed to be a universal difficulty, and to assist in his understanding young Macfarlane purchased a copy of Tait’s Treatise on Quaternions. This was the beginning of his special work as a mathematician. Prior to entering the class of logic, Mr. Macfarlane had already become familiar with the works of Hamilton and Mill, and, while a member of the class, he read, at the invitation of the professor, a paper which criticized the statement of the law of excluded middle, given by Jevons in his Lessons on Logic, a paper which displayed unusual merit for so young a mind. It was his first intention to study for honors in logic and philosophy, but perceiving how much they depended upon the principles of science, he took up the advanced classes in mathematics and physics, and in mathematical physics he not only gained the highest honors but also the appreciation and the personal friendship of Prof. Tait, the head of the Physical Department of the University. In 1874, he was appointed Neil Arnott instructor in physics, and in 1875, finished an unusually extensive course of undergraduate study by taking the degree of MA with honors in mathematics and physics.
Having, after graduation, won in competitive examination the MacLaren fellowship, worth $1,500, he proceeded to study for the recentlyinstituted degree of doctor of science, and, after one year spent on chemistry, botany and natural history, and two years on mathematics and physics, he obtained the doctorate in 1878. His remarkable thesis was an experimental research on the conditions governing the electric spark, and it was subsequently published in the Transactions of the Royal Society of Edinburgh. It also brought him under the notice of the celebrated electrician and philosopher, Clerk Maxwell. In 1878, Dr. Macfarlane was elected a fellow of the Royal Society of Edinburgh, and the first contribution, which he read personally, was a memoir on the algebra of logic. In 1879, he enlarged its scope and published it under the title of Principles of the Algebra of Logic. This volume was received with favor and brought the author into correspondence with many of the leading scientists and savants of the world. In 1879, he was able to meet many of them at the meeting of the British Association at Sheffield. During 1880, Dr. MacFarlane was interim professor of physics at the University of St. Andrews, and in 1881, he was appointed, for the usual period of three years, Examiner in Mathematics in the University of Edinburgh. During these years, he contributed to the Royal Society a series of valuable papers on Analysis of the Relationships of Consanguinity and Affinity. A paper on this subject, read before the Anthropological Institute of London, contains as perfect a notation for relationship as is the Arabic notation for numbers. His notable paper on Plane Algebra and his Physical Arithmetic were prepared during his tenure of office as Examiner.
In 1885, Dr. Macfarlane was called to the Chair of Physics at the University of Texas, where he became a colleague of his fellow logician, Dr. Halsted, and during that same year, he met many American men of letters and science at the Ann Arbor meeting of the association. In 1887, he received the honorary degree of LL.D. from the University of Michigan. His first year at the University of Texas was wholly taken up with the organization of the department, but in 1889 appeared a sequel to Physical Arithmetic, namely, a volume of Elementary Mathematical Tables. During this year, he visited Paris, and at the meeting of the French Association became acquainted with many Continental savants. On his return from Europe, he began to publish the results of his study of the algebra of space, and a few of the notable papers read and prepared were the following, showing a mass of learning and an exactness of reasoning quite beyond the ordinary intelligence: Principles of the Algebra of Physics; On the Imaginary of Algebra; The Fundamental Theorems of Analysis Generalized for Space; On the Definitions of the Trigonometric Functions; The Principles of Elliptic and Hyperbolic Analysis; The Analytical Treatment of Alternating Currents; On the Fundamental Principles of Exact Analysis; and The Principles of Differentiation in Space Analysis. In 1891, Dr. Macfarlane took an active part in organizing the Texas Academy of Science, and for two years acted as its Honorary Secretary. He contributed many papers, among which may be mentioned: An Account of the Rainmaking Experiments in San Antonio and Exact Analysis as the Basis of Language. For nine years Prof. Macfarlane remained at the University of Texas, resigning in 1894. The benefits accruing to the institution through his connection with it placed it far ahead of competitors. The course in mathematical physics which he arranged called forth a special approving article from a mathematical journal published at Turin, Italy. Since 1885, he had been a member of the Canadian Institute, Toronto, and, in addition to belonging to a number of American and British societies, he also held membership with several of the leading ones of the European continent. He is prominently mentioned in the issue of Who’s Who in America.
Since coming to reside in Ontario, Prof. Macfarlane has continued to write many papers on the algebra of space and has carried on the work of secretary of an international society organized for promoting that branch of mathematics, and which includes in its membership many of the most active mathematicians of the several countries of the world. (After leaving Texas Dr. Macfarlane remaind friends with Texas Governor Francis R. Lubbock.)
Dr. Macfarlane is a grandson of Alexander and Jeanette (Steele) Macfarlane, honored old residents of Perthshire, Scotland. Their sons were: James, Peter, Alexander and Daniel, the last of whom was the Doctor’s father. The only member of this family who came to Canada was the late James MacFarlane.
Dr. Macfarlane married Miss Helen Swearingen*, daughter of Patrick and Mary E. (Toland) Swearingen, of Texas. The former, descended from one of the Dutch founders of New York, was an attorney of prominence and held the rank of lieutenant colonel in the Confederate army during the Civil war in the States. To Dr. and Mrs. Macfarlane have been born three sons, Alexander S., Robert H. K., and Henry S. In politics, Dr. Macfarlane favors the Liberal party; in religion he is a Presbyterian. He occupies his beautiful farm of 400 acres, on Lots 16 and 17, 6th Concession, during the summer season, his residence occupying its center. It is probably the most valuable, as it certainly is the most highly cultivated and improved, estate of the county, and everything is arranged in geometrical order. (From: Commemorative Biographical Record of the County of Kent, Ontario)
*(In 1886, UT mathematrics professor, George Bruce Halsted, wed Margaret Swearingen in Austin. She was the daughter of Patrick Swearingen of Brenham, TX and from one of the founding families of New Amsterdam later renamed New York City). Patrick Swearingen had another daughter, Helen Martha, who became Alexander Macfarlane's wife in 1895. Thus the university colleagues were, in fact, brothersinlaw. The wedding was announced in the American Mathematical Monthly (2:135). (Following the death of Dr. Macfarlane in 1914, his wife, Helen, returned to Austin and remained there until her death in 1927 Alexander and Helen had three sons: Clarence Alexander Swerington, Robert Harper K., and Henry S. In 1914, Clarence enrolled in UT. In the October 18, 1918 Austin americanStatesman it is reporter that Lieutenant Alexander Swearingen Macfarlane was wounded in action in Europe on September and has died. It is stated that he attended UT during 191415 taking engineering courses. Helen Macfarlane was born October 18, 1871 and died in Austin May 9, 1927. Harper Macfarlane became a major in the U. S. Army.—Mel Oakes)
Another biography:
Alexander Macfarlane FRSE (April 21, 1851–August 28, 1913) was a Scottish logician, physicist, and mathematician.
Macfarlane was born in Blairgowrie, Scotland and studied at the University of Edinburgh. His doctoral thesis, On the Conditions Governing the Electric Spark, was subsequently published in the Transactions of the Royal Society of Edinburgh. It brought him to the notice of James Clerk Maxwell, and, in 1878, Macfarlane was elected a fellow of the Royal Society of Edinburgh.
During his life, Macfarlane played a prominent role in research and education. He taught at the universities of Edinburgh and St. Andrew’s, was physics professor at the University of Texas (1885–1894), professor of advanced electricity, and later of mathematical physics at Lehigh University. Macfarlane was the secretary of the Quaternion Society and compiler of its publications.
Macfarlane was also the author of a popular 1916 collection of mathematical biographies (Ten British Mathematicians), a similar work on physicists (Lectures on Ten British Physicists of the Nineteenth Century, 1919), and a bibliography on quaternions in 1904. Significantly, by exploiting the concept of hyperbolic versor originating with James Cockle, he invented hyperbolic quaternions, which anticipated Minkowski space. (He was imbued with hyperbolic geometry through his brotherinlaw G. B. Halstead while they taught in Austin.)
Macfarlane actively participated in several International Congresses of Mathematicians including the primordial meeting in Chicago, 1893, and the Paris meeting of 1900 where he spoke on Application of Space Analysis to Curvilinear Coordinates.
Macfarlane retired to Chatham, Ontario, where he died in 1913. (From Absoluteastronomy.com)
Between 1901 and 1904, Macfarlane gave a series of lectures at LeHigh University to faculty, students and townspeople. Ten of these lectures on pure mathematicians are available for free on iTunes and are part of the University of South Florida "Lit2Go" project. Ten lectures on more physics related mathematicians are planned.
The recipient must, during the ensuing Summer and Winter Sessions assist the Professor of Natural Philosophy in the Laboratory.We note that Arnott made similar bequests to each of the four Scottish universities.
This was an account of a series of experiments made in the Natural Philosophy Laboratory of the University, to test the applicability of Angström's method of periodic variations of temperature to the determination of low conductivity. The wood was cut into discs of a standard thickness, and these were very tightly secured together, after the interposition of copperiron thermoelectric junctions (of very fine wire). One series of discs was cut parallel, the other perpendicular, to the fibre. The results were obtained very easily, and accorded satisfactorily with those obtained by more laborious methods.It was followed in the same year by a paper, jointly authored by Macfarlane, Knott and Charles Michie Smith (18541922) (known as Michie), entitled On the Electric Resistance of Iron at a High Temperature. Here is an extract from the Introduction:
The following paper is a continuation of a former brief one, communicated to the Society, and printed in the Proceedings, on the change of electric resistance of iron due to change of temperature. In a note appended to Professor Tait's paper on a "First Approximation to a Thermoelectric Diagram," attention was drawn to the curious phenomenon observed by Gore, that at a temperature about dull red heat, iron wire undergoes sudden changes in length, and also to the further discovery by Professor Barrett, that if the wire be cooling, a sudden reglow occurs simultaneously with these changes. These phenomena seemed to be connected with other known physical changes which take place in iron at this critical temperature, such as the loss of its magnetic properties, the remarkable bend of the iron line in the thermoelectric diagram, and the interesting alteration in the rate of change of electric resistance with respect to change of temperature, observable in iron at the same dull red heat.Macfarlane was awarded a D.Sc. on 23 April 1878 for his thesis On the Disruptive Discharge of Electricity which was published in the Transactions of the Royal Society of Edinburgh. He writes:
The experiments to which I shall refer were carried out in the physical laboratory of the University during the late summer session. I was ably assisted in conducting the experiments by three students of the laboratory,  Messrs H A Salvesen, G M Connor, and D E Stewart. The method which was used of measuring the difference of potential required to produce a disruptive discharge of electricity under given conditions, is that described in a paper communicated to the Royal Society of Edinburghin 1876 in the names of Mr J A Paton, M.A., and myself, and was suggested to me by Professor Tait as a means of attacking the experimental problems ...This was one of six papers Macfarlane published in 1878, all in the Proceedings or the Transactions of the Royal Society of Edinburgh. We note that by this time Macfarlane had both an M.A. and a B.Sc. In the same year, on 6 May, he was elected a fellow of the Royal Society of Edinburgh. He was proposed by Peter Guthrie Tait, Philip Kelland,Alexander Crum Brown and John Hutton Balfour. We note that Alexander Crum Brown was the Professor of Chemistry at Edinburgh, while John Hutton Balfour was the Professor of Botany.
I have made the following observations in the Natural Philosophy classroom of the United College, St Andrews, with the view of ascertaining whether the electromotive force required to cause a spark to pass between a small globe and a plate, or between a point and a plate, differs for the two kinds of electricity. Sir William Thomson suggested that I should apply to this question the method of measuring the electromotive force required to produce sparks, which I have described in papers already contributed to the Royal Society of Edinburgh. It is a problem to which Faraday attached great importance. He says in his 'Experimental Researches in Electricity': "The results connected with the different conditions of positive and negative discharge will have a far greater influence on the philosophy of electrical science than we at present imagine, especially if, as I believe, they depend on the peculiarity and degree of polarised condition which the molecules of the dielectrics concerned acquire."In 1881 he was appointed as an Examiner in Mathematics at the University of Edinburgh for three years. We give examples of four papers he set for the 188182 diet at THIS LINK.
In the preface to the new edition of the 'Treatise on Quaternions' Professor Taitsays, "It is disappointing to find how little progress has recently been made with the development of Quaternions, One cause, which has been specially active in France, is that workers at the subject have been more intent on modifying the notation, or the mode of presentation of the fundamental principles, than on extending the applications of the Calculus." At the end of the preface he quotes a few words from a letter which he received long ago from Hamilton  "Could anything be simpler or more satisfactory? Don't you feel, as well as think, that we are on the right track, and shall be thanked? Never mind when." I had the high privilege of studying under Professor Tait, and know well his singleminded devotion to exact science. I have always felt that Quaternions is on the right track, and that Hamilton and Tait deserve and will receive more and more as time goes on thanks of the highest order. But at the same time I am convinced that the notation can be improved; that the principles require to be corrected and extended; that there is a more complete algebra which unifies Quaternions, Grassmann's method and Determinants, and applies to physical quantities in space. The guiding idea in this paper is generalisation. What is sought for is an algebra which will apply directly to physical quantities, will include and unify the several branches of analysis, and when specialised will become ordinary algebra. That the time is opportune for a discussion of this problem is shown by recent discussion between Professors Tait and Gibbs in the columns of 'Nature' on the merits of Quaternions, vector Analysis, and Grassmann'smethod; and also by the discussion in the same journal of the meaning of algebraic symbols in applied mathematics.Macfarlane read the paper On the definitions of the trigonometric functions to the International Mathematical Congress in Chicago in August 1893. His paper begins:
In a paper on 'The Principles of the Algebra of Physics' I introduced a trigonometric notation for the partial products of two vectors, writing AB = cos AB + Sin AB, where cos AB denotes the positive scalar product, and Sin AB the directed vector product. To denote the magnitude of the vector product I used the notation sin AB without a capital: it is not the exact equivalent of the tensor, because the magnitude may be positive or negative. With the additional device of using the Greek letters to denote axes, it is possible to dispense with the peculiar symbols introduced into analysis by Hamilton and the spaceanalysis then assumes to a large extent the more familiar features of the ordinary analysis. The notation raises the question of the relation of spaceanalysis to trigonometry. If cos and sin are correct appellations of the products mentioned, are there products of two vectors which are correctly designated by tan, sec, cotan, cosec? At p. 87 of the Principles I give a brief answer to this question; but a complete answer called for a more thorough investigation than I had then time to make. This trigonometrical notation has been briefly discussed by Mr Heaviside ('The Electrician', 9 December 1892). He takes the position that vector algebra is far more simple and fundamental than trigonometry, and that it is a mistake to base vectorial notation upon that of a special application thereof of a more complicated nature. I believe that this paper will show that trigonometry is not an application of spaceanalysis, but an element of it; and that the ideas of this element are of the greatest importance in developing the higher elements of the analysis.In 1894, Macfarlane resigned his chair of physics at the University of Texas. On 8 April 1895 he married Helen Martha Swearingen (18701927) at Bexar, Texas. Helen had been born on 18 October 1870 in Washington County, Texas to Patrick Henry Swearingen (18341880), a lawyer and soldier, and Mary Eliza Toland (18431911). Helen's sister, Margaret Swearingen (born 1862 and known as Maggie), had married George Halsted in 1868. Alexander and Helen Macfarlane had five children: Alexander Swearingen Macfarlane (18961918); Robert Harper Kirby Macfarlane (19011980); Henry Swearingen Macfarlane (19031907), James Donald Macfarlane (19061929), and Margaretta Macfarlane (19091909). Notice the sad fact the only one of the five children lived beyond the age of 23; Alexander Swearingen Macfarlane was killed in action in France during World War I.
In several recent papers, I have investigated the vector expression for Lame'sfirst differential parameter in the case of orthogonal systems of curvilinear coordinates, and I have shown how to deduce the expression for Lame'ssecond differential parameter by means of direct operations of the calculus.In 1908 in Rome he gave the lecture On the Square of Hamilton's Delta and in 1912, in Cambridge, England, he gave the lecture On Vector Analysis as Generalised Algebra.
The results indicate that the method is not confined to orthogonal systems, but is applicable to what may be called conjugate systems. I shall first indicate the results for the spherical system of coordinates, then deduce the results for the complementary system of equilateralhyperboloidal coordinates, and finally show how the results are modified for an ellipsoidal system of coordinates.
Alexander Macfarlane Photo and Document Album 

