1.1. From the Preface.

This book may be described as a treatise on applied arithmetic, the applications being chiefly in physical science. Knowledge of the elements of pure arithmetic is assumed, but the more advanced methods are explained when their application happens to occur.

The progress of physical science has caused the idea of the unit to become more prominent in text-books on arithmetic; and the old form of rule of three has been replaced to a large extent by what is called the unitary method, or the method of reduction to the unit. That method, in my opinion, very imperfectly represents the reasoning process involved.

The method developed in this work may be called the equivalence method. Each quantity is analysed into unit, numerical value, and, when necessary, descriptive phrase. The rate, or law, or convention, according to which one quantity depends on one or more other quantities, is expressed by an equivalence. These equivalences are of two kinds, absolute and relative - the former expressing the equivalence of dependence, the latter the equivalence of substitution or replacement. Finally equivalences are combined according to a form which is a development of the Chain Rule.

The present work is a development of notes which I began to take when a student on the subject of units and the reasoning processes involved in elementary calculations. My experience as examiner has confirmed me in the opinion that an elementary text is necessary.

2.1. From the Preface.

These tables are designed to be useful not only in computing and in the graphic method, but also in the teaching of arithmetic and in the illustration of the theorems of algebra.

I have arranged the several tables on a uniform decimal plan, so that the entries for a particular number are generally found in the same position on the page. The arrangement is that of double entry, so that in general the order of reading is the same as for ordinary print. The argument and entry are so expressed that it is easy to find the entry corresponding to any other position of the decimal point in the argument. In most cases the whole of a table is seen at one opening of the pages, and the tenth compartment, when not requited for the main table, is filled with a short table which is in general auxiliary to the main table.

Special acknowledgments are due to Prof Hastings of Yale University, and Prof Halsted of this University. In the proof-reading and independent computation, I have received aid from D W Spence and J C Nagle, science students of this University.

3.1. From the Preface.

The International Catalogue of Scientific Literature places Quaternions as one division (83), and Ausdehnungslehre and Vector Analysis as another division (84) of Universal Algebra (8). The proximate divisions on either side are General Theory of Complex Numbers (82) and Matrices (85). These are the main and allied branches of Mathematics comprised in this Bibliography. I have also included publications in which vector ideas and methods are applied.

The contributions of each writer are arranged chronologically, the date taken being that of the volume in which the contribution appears. The mark † is printed before the first posthumous publication.

In the use of contractions for the names of serial publications, I have followed the system of the Jahrbuch über die Fortschritte der Mathematik; and I have likewise followed it in avoiding the use of Roman numerals. Where the number of the pages of a book is indicated by the sum of two numbers, the former number refers to the preface, and the latter to the text. The following contractions are used: M.P. for Collected Mathematical Papers; S.P. for Collected Scientific Papers; G.W. for Gesammelte Werke; Pr. for Programmabhandlung.

For facilities in the preparation of this Bibliography, I wish to thank the Librarian of the University of Michigan, and, for special assistance, the following members of the Association: Combebiac, Dickstein, van Elfrinkhof, Grassmann, Joly, and Schlegel. Almost every living writer has responded to the request to make the list of his own contributions complete.

Alexander Macfarlane.
17th November 1903.

4.1. From the Preface.

Since this Introduction to Vector Analysis and Quaternions was first published in 1896, the study of the subject has become much more general; and whereas some reviewers then regarded the analysis as a luxury, it is now recognised as a necessity for the exact student of physics or engineering. In America, Professor Hathaway has published a Primer of Quaternions (New York, 1896), and Dr Wilson has amplified and extended Professor Gibbs' lectures on vector analysis into a text-book for the use of students of mathematics and physics (New York, 1901). In Great Britain, Professor Henrici and Mr Turner have published a manual for students entitled Vectors and Rotors (London, 1903); Dr Knott has prepared a new edition of Kelland and Tait's Introduction to Quaternions (London, 1904); and Professor Joly has realised Hamilton's idea of a Manual of Quaternions (London, 1905). In Germany Dr Bucherer has published Elemente der Vektoranalysis (Leipzig, 1903) which has now reached a second edition.

Also the writings of the great masters have been rendered more accessible. A new edition of Hamilton's classic, the Elements of Quaternions, has been pre- pared by Professor Joly (London, 1899, 1901); Tait's Scientific Papers have been reprinted in collected form (Cambridge, 1898, 1900); and a complete edition of Grassmann's mathematical and physical works has been edited by Friedrich Engel with the assistance of several of the eminent mathematicians of Germany (Leipzig, 1894). In the same interval many papers, pamphlets, and discussions have appeared. For those who desire information on the literature of the subject a Bibliography has been published by the Association for the promotion of the study of Quaternions and Allied Mathematics (Dublin, 1904).

There is still much variety in the matter of notation, and the relation of Vector Analysis to Quaternions is still the subject of discussion (see Journal of the Deutsche Mathematiker-Vereinigung for 1904 and 1905).

Chatham, Ontario, Canada, December, 1905.

5.1. Review by: K.
The Monist 23 (2) (1913), 318-319.

This is Dr Macfarlane's "President's Address" as president of the International Association for Promoting Quaternions and Allied Mathematics. He says:

"We have before us the ordinary algebra founded on the straight line, or as Hamilton at one time preferred to say, on pure time. We have next the algebra of the complex quantity, founded on the plane; it is a portion only of plane algebra, for what is treated is the circular part only; the hyperbolic counterpart is almost wholly neglected. For instance, in the solution of the quadratic and the cubic equation, the roots are real and impossible so far as line algebra is concerned, but hyperbolic or circular so far as plane algebra is concerned. This plane algebra is a logical generalisation of line algebra, and every theorem in the latter has its generalised form in the former.

"There is a common belief that the algebra of the circular complex quantity rounds off and completes the domain of algebra, and we are furnished with a so-called reduction of every algebraic expression to the form of the circular complex quantity. But that argument is entirely fallacious; for in the plane there is a hyperbolic vector, and none of these can be reduced to the form mentioned. This matter was discussed before the American Institute of Electrical Engineers (Transactions, vol. 14, p. 163), the orthodox doctrine being championed by Mr Steinmetz, and the opposite by myself. My argument was derived from the investigation for the discharge of an electrical condenser; when the discharge is alternating the analysis leads to circular complex quantities and when it is not alternating to hyperbolic complex quantities which are analogous in every particular to the circular.

"Then we have before us three forms of space-analysis: the scalar, founded by Descartes, which makes use of axes, but provides no explicit notation for directed quantities whether line or angle ; the quaternionic, founded by Hamilton, which is characterised by a notation for versors or angles in space; the vectorial, founded by Grassmann, which is built on vector-units and compound units derived from them. For the past half century the masters of these several forms have been engaged in a triangular fight ; much has been written on vectors versus quaternions; and we have heard of a Thirty Years' War between one who could bend the bow of Hamilton and one equally skilled in the weapon of Descartes. It will surely be admitted that each branch contains part of the truth; it is therefore highly probable that none of them contains the whole truth, and that each has a part of the truth which the others have not. It has for long seemed to me that what is wanted is an analysis which will harmonise all three, and present itself as the space-generalisation of algebra. As to this conception of the oneness of the algebra of space, I may quote Sylvester's declaration that he would as soon acknowledge a plurality of gods as a plurality of algebras. Likewise, Gibbs at the close of his address to the Mathematics Section of the American Association, said we begin with multiple algebras and end with multiple algebra."

The pamphlet before us contains Professor Macfarlane's solution of the problem which consists mainly in a unification and generalisation of the principles of the algebra of space.

6.1. From the Preface.

During the years 1901-1904 Dr Alexander Macfarlane delivered, at Lehigh University, lectures on twenty-five British mathematicians of the nineteenth century. The manuscripts of twenty of these lectures have been found to be almost ready for the printer, although some marginal notes by the author indicate that he had certain additions in view. The editors have felt free to disregard such notes, and they here present ten lectures on ten pure mathematicians in essentially the same form as delivered. In a future volume it is hoped to issue lectures on ten mathematicians whose main work was in physics and astronomy.

These lectures were given to audiences composed of students, instructors and townspeople, and each occupied less than an hour in delivery. It should hence not be expected that a lecture can fully treat of all the activities of a mathematician, much less give critical analyses of his work and careful estimates of his influence. It is felt by the editors, however, that the lectures will prove interesting and inspiring to a wide circle of readers who have no acquaintance at first hand with the works of the men who are discussed, while they cannot fail to be of special interest to older readers who have such acquaintance.

It should be borne in mind that expressions such as "now," "recently," "ten years ago," etc., belong to the year when a lecture was delivered. On the first page of each lecture will be found the date of its delivery.

For six of the portraits given in the frontispiece the editors are indebted to the kindness of Dr David Eugene Smith, of Teachers College, Columbia University.

Alexander Macfarlane was born April 21, 1851, at Blairgowrie, Scotland. From 1871 to 1884 he was a student, instructor and examiner in physics at the University of Edinburgh, from 1885 to 1894 professor of physics in the University of Texas, and from 1895 to 1908 lecturer in electrical engineering and mathematical physics in Lehigh University. He was the author of papers on algebra of logic, vector analysis and quaternions, and of Monograph No. 8 of this series. He was twice secretary of the section of physics of the American Association for the Advancement of Science, and twice vice-president of the section of mathematics and astronomy. He was one of the founders of the International Association for Promoting the Study of Quaternions, and its president at the time of his death, which occurred at Chatham, Ontario, August 28, 1913. His personal acquaintance with British mathematicians of the nineteenth century imparts to many of these lectures a personal touch which greatly adds to their general interest.

6.2. Review by: Anon.
The Mathematics Teacher 9 (1) (1916), 66.

This is number 17 of the series of Mathematical Monographs issued by these publishers and contains an account of the leading British mathematicians of the last century. The names included are Peacock, De Morgan, William Rowan Hamilton, Boole, Cayley, Clifford, Smith, Sylvester, Kirkman, Todhunter. The sketches given are very interesting and instructive and the volume is in the usual attractive form of the series.

6.3. Review by: Thomas E Mason.
Bull. Amer. Math. Soc. 23 (4) (1917), 191-192.

This is somewhat of a departure from the preceding sixteen numbers of this series in subject matter, being an account of individual mathematicians and their contributions to mathematics, rather than a survey of some particular field of the subject. The ten men are George Peacock, Augustus De Morgan, Sir William Rowan Hamilton, George Boole, Arthur Cayley, William Kingdon Clifford, Henry John Stephen Smith, James Joseph Sylvester, Thomas Penyngton Kirkman, and Isaac Todhunter. The author had the advantage of personal acquaintance with a number of the men of whom he wrote, as well as the interest of having been born a fellow countryman. He has given us the life history of the man without too much detail, and yet with enough intimacy, so that we have a picture of the man as a man, as well as a scientist.

The book consists of ten lectures delivered to audiences composed of students, instructors, and townspeople at Lehigh University during the years 1901 to 1904. As each was delivered as a separate lecture, there is no direct connection between them. The critical analysis of the contributions of each mathematician is probably as well done as would be possible considering the character of the audience and the length of time at the disposal of the speaker. We get a very good general notion of what each contributed and its value to the science.

The ten men chosen are not equally well known to American mathematicians. Certainly we have a deeper interest in and knowledge of Sylvester than of the others. Todhunter would be best known for his many textbooks, and De Morgan for his Paradoxes. It is doubtful if American mathematicians know the English as well as the continental mathematicians, especially the German. American students of mathematics who have gone abroad for study have gone chiefly to the German universities. This has tended to bring American mathematicians into closer relations with the German mathematicians and their work than with the English. For that reason, this book on English mathematicians is valuable.

There is a tendency today to try to give some comprehension of the problems and methods and achievements of mathematics to those who are not specialists in mathematics. A recent book by Professor G A Miller, of the University of Illinois, is such an effort. These lectures by Dr Macfarlane serve that purpose, as they were not prepared for specialists in mathematics alone.

We have a partial promise in the preface that there will be a later volume dealing with ten mathematicians who worked chiefly in applied mathematics. We hope that this promise may be speedily fulfilled.

The reading of the page proofs does not seem to have been done carefully enough.

6.4. Review by: W J Greenstreet.
The Mathematical Gazette 9 (131) (1917), 146-152.

The present volume, we are delighted to hear, is the first instalment of twenty lectures given at Lehigh University some years before the author's lamented death. Macfarlane belongs to the little band of men who were proud to own allegiance to their teacher Tait, and he will be best known to a large number of mathematicians as a founder, as an industrious and active official, and ultimately as President of the International Association for the Promoting the Study of Quaternions.
...
The present instalment of these lectures is confined to studies of ten pure mathematicians: Peacock, De Morgan, William Rowan Hamilton, Boole, Cayley, Clifford, H J S Smith, Sylvester, Kirkman and Todhunter. Of De Morgan we have the Memoirs published by his wife Sophia in 1882. We have the definitive life of Hamilton by Robert Perceval Graves. Frederic Pollock contributed a charming prefatory notice to the collection of Clifford's Lectures and Essays published in 1879; and H J S Smith's appreciatory introduction to the Scientific Papers (1882) is well known. We might have expected a biography of Boole from the pen of his gifted wife, but a certain instability of temperament, which persisted after the guiding hand of her husband was withdrawn, unfortunately prevented this. For the rest, as far as we are aware, we must go in the main to short notices attached to editions of Collected Works, or to the obituaries in the Proceedings of such bodies as the Royal Society, the London Mathematical Society, or the Royal Astronomical Society. In the case of Kirkman, indeed, the dozen or so pages in this volume will be to a great extent a revelation, and we have met with many mathematicians who were barely aware of his existence. Of Todhunter we have three very characteristic papers in the Cambridge Review, vol. V. (1883-4), in which, however, we seem to see far more of the picturesque personality of the writer, dear old "Johnny Mayor," than of the subject of his discourse, the vivid touches it contains reminding us of Johnson's phrase on "reading Shakespeare by flashes of lightning." Of Todhunter, now that Besant is gone, there will he few who can say, as Professor Clifton writes to us, "I was introduced to Boole in his rooms." There are many still with us whose intimacy with some of the others would entitle them to write upon such attractive subjects. But we have said enough on the point to show that for the most part we are grateful for what the author has written upon these great figures in the mathematical history of early and mid-Victorian times.

6.5. Review by: W. J. Greenstreet
The Mathematical Gazette 32 (300) (1948), 146-152.

Note. This review is a reprinting of the 1917 one, an extract from which is presented as 4.4 above. We give here a different short extract.

A perusal of these pages will prove to the layman that mathematicians are not necessarily poor men of business, and that they are not as a rule devoid of humour. ... It was given to Macfarlane to lay these garlands on most of whom he had known, and all of whom he honoured. His hearers little thought that the volumes containing his lectures would in their turn serve so soon to remind us of one who has left behind him memories, fragrant and appreciative, on both sides of the great ocean which divided us from his manifold activities.

7.1. From the Preface.

During the years 1901-1904 Dr Alexander Macfarlane delivered, at Lehigh University, lectures on twenty-five British mathematicians of the nineteenth century. The manuscripts of twenty of these lectures were discovered in 1916, three years after the death of their author, to be almost ready for the printer, and ten of them, on ten pure mathematicians, were then published in Monograph No. 17 of this series. Lectures on ten mathematicians whose main work was in physics, astronomy, and engineering are given in this volume.

These lectures were given to audiences composed of students, instructors and townspeople, and each occupied less than an hour in delivery. It should hence not be expected that a lecture can fully treat of all the activities of a mathematician, much less give critical analyses of his work and careful estimates of his influence. It is felt by the editors, however, that the lectures will prove interesting and inspiring to a wide circle of readers who have no acquaintance at first hand with the works of the men who are discussed, while they cannot fail to be of special interest to older readers who have such acquaintance.

It should be borne in mind that expressions such as "now," "recently," "ten years ago," etc, belong to the year when a lecture was delivered. On the first page of each lecture will be found the date of its delivery.

7.2. Review by: George Sarton.
Isis 3 (2) (1920), 291.

During the years 1901-1904, Alex Macfarlane (Blairgowie, Scotland, 1851-Chatham, Ontario, 1913), delivered at Lehigh University, South Bethlehem, Pennsylvania, lectures on 25 British mathematicians of the nineteenth century. Ten of these lectures have already been published in Monograph, No. 17 "Ten British Mathematicians". They were devoted to: George Peacock; Aug De Morgan; Sir Will Rowan Hamilton; George Boole; Arthur Cayley; Will Kingdon Clifford; Henry John Stephen Smith; James Joseph Sylvester; Tho Penyngton Kirkman; Isaac Todhunter. Ten more, now published, are devoted to mathematicians whose main work was in physics, astronomy and engineering, to wit: James Clerk Maxwell; W J M Rankine; P G Tait; Kelvin; Charles Babbage; Will Whewell; Sir George G Stokes; Sir George B Airy; J C Adams; Sir John F W Herschel. There is an index (not carefully made!) and a plate with ten small portraits.

This book is very interesting, but much less than it could have been. Macfarlane's idea was to give in each lecture a sketch of a great scientist. We do not expect a sketch to replace a more complete biography, but we would expect the lecturer to lay more stress on the work upon which the fame of each man was based. We find nothing of the kind in these lectures. They are easy to read, but they are not really stimulating, because the essential of each life is not brought out more clearly; they are like paintings without focus and perspective. A few personal reminiscences add here and there a touch of colour, but the style is indifferent. The author must have been a kind and lovable man; he was also very impartial and one can but admire the candour with which he (president of the international association for promoting the study of quaternions!) reproduces Kelvin's sweeping indictment of the quaternions. The editors could have increased the value of this book by adding bibliographical notes to each sketch. As it is, the book is neither a work of reference, nor a collection of first hand documents. It must rather be considered as a series of literary essays, with but little literary value. The proofreading was not very carefully done. For instance I read on page 57, Lionville for Liouville, and on page 91, Fontenall for Fontenelle! Yet this book will be useful; I do not know any other of the same kind. It is not bad, but it could easily have been much better.

7.3. Review by: Anon.
The Mathematical Gazette 10 (148) (1920), 155-159.

The lectures published under this title are conceived on the same scale as the Ten British Mathematicians, already noticed in the columns of the Gazette (vol. ix, no. 131, pp. 146-152). Some readers may think that the line between "mathematician" and "physicist" has in some cases been oddly drawn, but it is idle to discuss the point in a posthumous work. Much more attention has been paid to the proofs of this volume than was received by its predecessor. "Lionville" is familiar enough, but "Fontenall" is a novel variant of Fontenelle: and in a Briton's lucubrations we do not expect to see "Brittanica." However, these appear to be the only slips, if "Menabréa" is correct, but we are afraid it is not.