MacTutor
William Wallace, LL.D., late Professor of Mathematics in the University of Edinburgh, was descended from a family in humble circumstances, which had been settled, for some generations, at the village of Kilconquhar in Fifeshire. His grandfather inherited a small property, the greater part of which he lost through injudicious management. His father established himself at Dysart, a sea-coast burgh in Fife, as a manufacturer of leather and shoes for exportation, and for some years carried on a considerable trade, which, however, was ruined by the breaking out of the American war. The subject of this memoir was born at Dysart on the 23d of September, 1768, and was the eldest of a numerous family.
In adverting to the circumstances of his early life, he used to relate that the first rudiments of his education were received from an aged widow in the town, who kept a school for children, and retailed small wares. About the age of seven, he was removed to a school of a better class, in which the principal branch of instruction was arithmetic. In this science, however, he had already been grounded by his father, and had made considerable proficiency in it before he was advanced to that department in the routine of his school progress. His attendance at school for instruction it can scarcely be called was discontinued when he had reached the age of ten or eleven years; and, according to his own statement, all he owed to the schoolmaster was the power of reading, and of forming, in a very indifferent way, characters by writing. His knowledge of arithmetic he owed to his father, and to his own strong liking for the subject.
In 1784, when in his sixteenth year, he was sent to Edinburgh to learn the trade of a bookbinder; and after a year or two of probation he entered upon a regular apprenticeship to this craft. But his passion for the acquisition of knowledge had been thoroughly roused by the perusal of some books which had fallen in his way; and, during the period of his apprenticeship, he devoted every spare moment to reading. These moments were, however, few. His master happened to be a person who had no sympathy with literary tastes, and no other concern about his apprentices than how to extract from them the greatest amount of labour. But his father, a man of considerable intelligence and strict religious principles, having removed with his family to Edinburgh, he had the comfort of residing, during this period, in the house of his parents, and the advantage of their society, encouragement, and moral superintendence, to which he professed himself to have been greatly indebted. His occupation, also, was in some respects favourable to the gratification of his tastes. Books of science were constantly passing through his hands, and his curiosity could not be restrained from occasionally casting a glance at their contents. He had also acquired a few mathematical books of his own; and such were his ardour and enthusiasm in their study, that it was his constant practice to take his meals with one of them in his hand, and to carry one in his pocket, to read on his way to and from the workshop. By this assiduous application, before he reached the age of twenty, he had read and made himself master of Cunn's Euclid, Ronayne's Algebra, Wright's Trigonometry, Wilson's Navigation, Emerson's Fluxions, Robertson's Translation of La Hire's Conic Sections, and Keill's Astronomy. Of these books he cherished the remembrance, as the means by which he had been enabled to grope his way into the region of the mathematics.
Hitherto, Mr. Wallace's efforts to acquire knowledge had been made under the most disadvantageous circumstances; without sympathy from any one but his father, and without a companion or friend to appreciate his exertions or applaud his success. But he was now approaching the turning point of his fortunes. He happened to become acquainted with an elderly person, a carpenter by occupation, who was employed by the celebrated Dr. John Robison, the Professor of Natural Philosophy, as an assistant in his class experiments. This man, though a great reader of books, was no mathematician; but he had sat too near the feet of Gamaliel not to have imbibed a respect for the science, and for the pursuits of his young friend. With an excusable vanity, he was in the habit of boasting of his intimacy with the professor, to whom he proposed to introduce Mr. Wallace. The latter, however, with great good sense, declined the kindly meant offer until the term of his apprenticeship had expired, when, though still with some diffidence and hesitation, he was prevailed upon to take advantage of it. Armed with a letter from his humble patron, he waited upon the professor, who received him with great kindness, examined him with respect to his proficiency in geometry and the conic sections, and inquired into the circumstances of his life, and the means by which he had made so much progress in the mathematics. In the course of the conversation Dr. Robison considerately took occasion to warn him that the study of mathematics was not likely to lead to any thing advantageous in the world: the reply was, that he was aware of the fact; but being, as it seemed, doomed to a life of labour, he hoped to sweeten the cup by the pleasure to be derived from the possession of knowledge. The interview ended with an invitation from the Professor to attend the course of lectures on Natural Philosophy then about to begin. Sensible as he was of the advantages which he could not fail to derive from such instruction, it required no small sacrifice on his part to accept the offer; for, being then employed as a journeyman, the time thus occupied could only be commanded by the abstraction of an equal portion from his hours of rest or sleep. Every difficulty, however, gave way before a determined will. The class was diligently attended: and he has been heard to say, that if he were asked which had been the happiest period of his existence, he would refer to that at which he attended the lectures on natural philosophy, when, for the first time in his life, he had the means of receiving sound instruction, and found himself in the company of young men devoted to the pursuit of knowledge.
Dr. Robison's next act of kindness was to introduce him to his colleague, Mr. Playfair, the Professor of Mathematics. Mr. Playfair was no less struck with the extent of his acquirements, and likewise offered him admission to the mathematical class. But attendance on two classes in one day being, in his circumstances, entirely out of the question, he was under the necessity of declining the offer, much, it may readily be believed, to his regret. Mr. Playfair, however, from this first interview, took a warm interest in his welfare, advised him with respect to the course of reading he should follow, supplied him with books from his own library, and continued his stedfast friend through life.
These details may appear trivial, or unnecessarily minute; but it can never be wholly uninteresting to trace the steps by which distinction in science or literature has been obtained when opposed by obstacles which might seem, and in ordinary cases prove to be, insurmountable. To the individual we are commemorating they were all-important some may receive encouragement from his example; and science itself is placed in an advantageous light when we see men so eminent as Professors Robison and Playfair taking trouble with, and giving help and encouragement to, a friendless young man, who had no claim on their good offices, and no other recommendation to them, than his successful struggles in acquiring the elements of those sciences which they themselves cultivated with such distinguished success. On the other hand, the merit must have been of no ordinary kind which, to persons of their experience, appeared so remarkable.
About the time he was attending Dr. Robison's lectures he was induced, by the prospect of having the command of a greater portion of time than had yet been at his disposal, to exchange his occupation for that of warehouseman in a printing-office. While in this occupation Dr. Robison paid him a visit, and proposed to him to give private lessons in geometry to one of his pupils. This proposal opened up a new prospect to him, and admitted the first gleam of hope of his being able to emancipate himself from the drudgery of manual labour. He now also began to acquire a knowledge of Latin, and in this, as in the study of mathematics, his manner of turning time and opportunity to account may afford encouragement to those who are in pursuit of knowledge under difficulties. A part of his duty in the printing-office was to collect the successive sheets of a work from a series of heaps arranged round a circuit of tables. While engaged in this monotonous occupation, he fixed up upon the wall a Latin vocabulary, from which he committed to memory a certain number of words every time he passed it in making his round. In his study of Latin, however, he received assistance from a student, to whom, in return, he gave instruction in mathematics.
After he had been engaged a few months in the printing-office, he entered into the employment of one of the principal book-sellers of Edinburgh in the capacity of shopman. This change was advantageous in several respects. His circumstances were now considerably improved, and he found leisure, not only to pursue his favourite studies, but to increase his stock of knowledge by general reading, and even to give private lessons in mathematics in the evenings. While in this situation he contrived to get a few lessons in French, and commenced his acquaintance with the works of the Continental mathematicians.
In 1793, while in his twenty-fifth year, he took the resolution to give up his employment, and support himself by teaching mathematics privately. This plan probably succeeded to the full extent of his moderate expectations. He now attended a course of lectures by Professor Playfair; and although, as the course was intended for an audience far behind him in mathematical acquirements, he had little to learn, the example of Playfair's manner – dignified, eloquent, and impressive, in a degree rarely equalled – was of great use to him in after-life. At the same time he also attended a course of chemistry, and by assiduous diligence endeavoured to repair, to the utmost of his power, the deficiencies of his early education.
In 1794, Mr. Wallace, on the recommendation of Professor Playfair, was appointed to the office of assistant teacher of mathematics in the academy at Perth. In respect of emolument the appointment was of no great value, but it gave him a settlement in life, with reasonable leisure to prosecute his mathematical studies, of which he did not fail to take advantage. In 1796, he presented his first memoir to the Royal Society of Edinburgh, entitled, "Some Geometrical Porisms, with Examples of their Application to the Solution of Problems." This paper, which contained some new and curious porismatic propositions, afforded ample proof of original and inventive powers; while his manner of conducting the investigation shewed how accurately he had imbibed the spirit and methods of the ancient geometrical analysis. About the same time, on the request of Dr. Robison, he contributed the article "Porism" to the third edition of the Encyclopedia Britannica; and, a few years later, when a new and greatly enlarged edition of that work was undertaken, he was enlisted as a regular contributor, and undertook to furnish the principal mathematical papers.
During the vacations of the Perth academy he paid regular visits to Edinburgh, where he continued to cultivate the friendship of Robison, Playfair, and other scientific men, to whom his now recognised talents and mathematical attainments procured him introductions. The first mark of literary distinction he received was that of Corresponding Member of the Edinburgh Academy of Physics; a society which, though not known by its published transactions, was at that time remarkable by reason of the cluster of talented persons of whom it was composed, several of whom have since attained the highest distinction in literature, philosophy, and public affairs. Such association could not fail to have a powerful effect in the developement of his mind, even though his residence at a distance from Edinburgh prevented him from attending many of the meetings.
In 1802, he presented a second paper to the Royal Society of Edinburgh, containing a new method of expressing the coefficients of the developement of the algebraic formula which represents the disturbing effect of the mutual action of two planets on each other. This was a contribution of great merit, and, immediately upon its publication, established his reputation as a mathematician of the first order. The volume of the Transactions in which it appeared was reviewed in the second number of the Edinburgh Review; and an able analysis of Mr. Wallace's paper was concluded with the following encomium: "We cannot conclude without expressing our sincere admiration of this excellent performance – excellent in every respect; and, trifling as it may appear to mathematicians, remarkable for a pure, perspicuous, and not inelegant style. It is a paper, equal, in our opinion, to whatever has been most admired of the greatest analysts. We remember nothing in the works of Euler or Lagrange which belongs to a higher order of excellence in the science." Mr. Wallace's method of developement depended ultimately upon the proportions which the perimeters of two ellipses bear to those of their circumscribing circles and in order to facilitate its application, he gave, in an appendix, a very beautiful and quickly converging series for the rectification of the ellipse, applicable to every case of eccentricity, and to every length of an are that can possibly occur in calculation. His merit with respect to this paper cannot be considered as having been diminished by the discovery he made some time after its publication, that in certain respects he had been anticipated by Legendre. The very little intercourse which at that time existed between this country and France, and the position of the author in a remote provincial town, are sufficient excuses for his not having been more accurately acquainted with the state of mathematical discovery on the Continent.
Mr. Wallace had been for several vears a contributor to some of the periodical publications in England in which mathematical questions were proposed, as Leybourn's Repository, the Gentleman's Mathematical Companion, and others of the same class. To this circumstance he attributed an incident which had an important influence on his future life. In 1803, he received a letter, under a feigned name, in which he was informed that an instructor in mathematics was wanted for the Royal Military College, then established at Great Marlow in Buckinghamshire, and recommended, if he thought of being a candidate for the office, to make an immediate application. Inquiry being made in the proper quarter, the information was found to be correct, but he ascertained also that it would be necessary to make his application in person. In matters affecting his own interests the disposition of his mind was not sanguine; and, as in the present case he had no influence to employ, and no other recommendation to carry with him than his skill in mathematics, his chances of success appeared so small that he would have been deterred by the length and inconveniences of the journey from thinking more of the subject, had he not been encouraged by his friend Professor Playfair. On his arrival at the Military College he found there were several competitors; but the persons who had to decide on the respective qualifications of the candidates gave their decision in his favour, and he was accordingly appointed to the office.
Mr. Wallace held this appointment upwards of sixteen years, during which period, the whole of his leisure time was unremittingly devoted to scientific study and literary labour, the fruits of which appear chiefly in his numerous contributions to the two great Encyclopedias then publishing in Edinburgh. This species of writing, which is not particularly well adapted to form the basis of a permanent reputation, was in a manner forced upon him by the circumstances of his position. On his appointment to the Perth Academy he had married, and after he joined the Military College his family began to increase rapidly. The inconveniences he had suffered from the defects of his own early education rendered him only more solicitous that his children should not labour under any disadvantages in this respect, and, as they grew up, he placed them at schools in Edinburgh. His official income being insufficient for this expense, he was led to engage in the works now referred to, rather with a view to add to his means, and to enable him to discharge a sacred duty, than for the sake of any distinction he was likely to get by them. No individual, perhaps, was ever less influenced by considerations of a worldly nature, or more ready to bestow time and labour upon objects from which he could neither receive nor expect any remuneration whatever.
In 1808, he contributed a paper to the Royal Society of Edinburgh, entitled "New Series for the Quadrature of the Conic Sections, and the Computation of Logarithms," and containing some very remarkable formule for the rectification of circular ares, with analogous expressions for the sectors of the equilateral hyperbola and the logarithms of numbers; all deduced from elementary principles, and without the use of the differential calculus or any equivalent method. At the time the paper was published, he believed the series to be entirely new, but he discovered afterwards that some of them had been previously given by Euler.
Mr. Wallace's services at the Military College were held in great estimation by the superior Officers, who frequently availed themselves of his practical sagacity in the adoption of regulations having respect not only to the course of instruction, but the general management of the establishment. One of the results of this deference to his recommendations (more particularly interesting to the Society), is the small observatory attached to the College, for the instruction of the officers of the senior department in practical astronomy. The plan of the building was originally furnished by Dr. Robertson of Oxford; but the superintendence and arrangement of all the details of construction were confided to Mr. Wallace, who visited most of the Observatories in the neighbourhood of London, for the purpose of acquiring hints and information. A transit-instrument, an astronomical circle by Ramsden, a reflecting circle, and a clock by Hardy, were procured, and some other instruments were ordered, but countermanded from an apprehension of opposition to the estimates in the House of Commons. Although an Observatory of this kind cannot be expected to produce results of any direct advantage to astronomy in the present state of the science, it must still be regarded as no unimportant appendage to a national establishment for the instruction of Officers for the public service.
In 1819 a vacancy occurred in the Mathematical Chair of the University of Edinburgh, through the death of Professor Playfair, and the appointment of Mr. Leslie to succeed him in that of Natural Philosophy, and Mr. Wallace resolved on presenting himself as a candidate. The patronage belongs to the magistrates of the city, who, having in general no pretensions to be capable of estimating degrees of merit in abstract science, necessarily form their opinions from the testimony of others, or notions of general fitness, and are liable to be acted upon by influences of various kinds. In the present case a very keen contest took place; for another competitor (a man of general talent and great respectability, though unknown as a mathematician) was strenously supported by a strong political party. The struggle terminated, however, in his election by a large majority of the voters. This was the crowning object of his ambition. Ever since his appointment to the Perth Academy, he had fixed his regards on a professorship in a Scottish university as the goal of all his exertions; but his elevation to the Chair of the Gregorys, of Maclaurin, Matthew Stewart, and Playfair, probably did not enter at that period into his most sanguine anticipations.
Mr. Wallace had reached the age of fifty-one when he was appointed to the mathematical professorship in Edinburgh; but he still retained both mentally and bodily all the energy and activity of his younger years. He held the office till 1838, when he resigned on account of ill-health, having been unable to perform his duties in person during the three previous sessions. Upon his resignation the honorary title of Doctor of Laws was conferred upon him by the University, and at the same time he received a pension from Government which he enjoyed during the few remaining years of his life, in consideration, as the warrant stated, of his attainments in science and literature, and his valuable services, up to a very advanced period of life, first in the Military College, and afterwards at the University of Edinburgh.
For some years after his establishment at Edinburgh, a considerable portion of his time was occupied in the preparation of his lectures, on which he bestowed great pains. When the new edition of the Encyclopedia Britannica was commenced, he undertook the revision of all the mathematical papers he had contributed, as well as some of those which had been written by Dr. Robison; and several of the more important treatises, particularly, Algebra, Conic Sections, and Fluxions, were remodelled and almost entirely rewritten. To the Transactions of the Royal Society of Edinburgh he contributed a paper, in 1823, on the Investigation of Formulæ for finding the logarithms of trigonometrical quantities from one another; one in 1831, entitled "Account of the Inven. tion of the Pantograph, and a Description of the Eidograph;" and one in 1839, on the Analogous Properties of Elliptic and Hyperbolic Sectors. His last contribution to that Society, published in Vol. XIV. of the Transactions, was entitled, "Solution of a Functional Equation, with its Application to the Parallelogram of Forces, and to Curves of Equilibration." This paper, in addition to the investigation of series adapted for calculation, contains a set of tables, to ten decimal places, of the corresponding values of the amplitude, ordinate, and are of a catenary, which are important in an engineering point of view, as they afford the data required for constructing arches having the forms of equilibrated curves. Similar tables, to eight places, had previously been given by Mr. Davies Gilbert in a paper on the mathematical theory of suspension bridges, in the Philosophical Transactions for 1826; but the numbers were found by Mr. Wallace to be erroneous, generally, in the three last decimal figures.
Mr. Wallace is the author of a paper in Vol. IX. of our Memoirs containing two elementary solutions of Kepler's problem by the angular calculus. In the Transactions of the Philosophical Society of Cambridge, Vol. VI., there is also a paper by him under the title of "Geometrical Theorems and Formule particularly applicable to some Geodetical Problems." For this subject he had a particular predilection; and in 1838, while confined to a sick-bed, he composed, and afterwards published at his own expense, a separate work entitled, "Geometrical Theorems and Analytical Formulæ, with their Application to the Solution of certain Geodetical Problems." This volume, which he appropriately dedicated to his friend Colonel Colby, contains the substance of his paper in the Cambridge Philosophical Transactions, with the addition of a considerable number of extremely elegant formulæ, most of them new, and some of them important in the practice of the higher geodesy.
Professor Wallace took great delight in all the practical applications of his science, and had a strong turn for mechanical invention. His attention having been directed to the imperfections of the Pantograph, he invented, in 1821, an instrument on a different principle to supply its place, to which he gave the name of Eidograph. This instrument answers the same purposes as the common pantograph, to which, however, it is greatly superior, both in the extent of its applications and the accuracy of its performance; for, while the similarity of the copy to the original, in all its parts, is preserved with geometrical accuracy, the copy may be reduced or enlarged in almost any proportion; or, by a particular modification of the instrument, it may even be reversed, and transferred immediately to metal or stone. This ingenious instrument, which would seem to be admirably adapted to the purposes of the engraver, was first described by him in Vol. XIII. of the Edinburgh Transactions to which reference has already been made. He has also described, in the Appendix to his Conic Sections, an Elliptograph, or instrument for describing an ellipse by continued motion, founded on a very beautiful property of the ellipse first pointed out, we believe, by him, namely, that the curve is organically described by any given point (not in the circumference) in the plane of a circle which rolls along the concave circumference of another fixed circle, the radius of which is twice that of the rolling circle. And in an Appendix to his Geometrical Theorems he has given the description of an instrument which he invented for the graphical solution of an important problem in surveying, viz. to determine the position of a station, having given the angles made by lines drawn from it to three other stations in the same plane, whose positions are known. This instrument, which he called a Chorograph (the problem which it solves having been proposed as a chorographical problem by Richard Townley in No. 69 of the Philosophical Transactions), is simple, compact, portable, and inexpensive; and in these respects has considerable advantages over the station-pointer, generally used for the same purpose.
Among the objects connected with the advancement of science to which Professor Wallace gave his aid, after his appointment to Edinburgh, there is one which it would be unpardonable to pass over without notice in this place, we allude to the Observatory now established there. Ever since the time of Maclaurin there had existed a small astronomical observatory in Edinburgh, but no provision was made for regular observation, nor, indeed, did it contain any instruments fit for the purpose. Through the exertions, chiefly of Professor Playfair, funds were at length raised, by private subscription, for the erection of an observatory adapted for observations of the most accurate kind. Mr. Playfair did not live to see the building completed, or means provided for obtaining instruments, or carrying on systematic observations; but Mr. Wallace, on becoming his successor, entered fully into his views, and, in concert with a few other individuals, used all his influence and exertions towards bringing the scheme to maturity. At length, after years of expectation and delay, the Government was prevailed upon to take the observatory under its protection, furnish it with instruments of the first class, appoint an astronomer and assistant, and provide for the regular publication of the observations. In bringing about this arrangement, Mr. Wallace's aid and recommendation were of essential service; and if any thing was wanting to complete the satisfaction which he felt at the result, it was to see the observatory placed under the care of his friend Professor Henderson, of whose distinguished merits as an astronomer it would be superfluous to speak to those who are in the habit of attending our meetings, or reading our Memoirs.
Although the works which Mr. Wallace has left behind him assure him a high place as an original and inventive mathematician, the talents with which he was endowed by nature were, doubtless, rendered less productive than they would have been by his want of early education, his residence during the best years of his life in the country at a distance from congenial society, and, perhaps, still more from the circumstance of so much of the time which his laborious public duties left at his disposal having been consumed in the preparation of his numerous treatises for the Encyclopedias. Thèse treatises being mostly of an elementary kind, and composed for the purpose of explaining the principles of the various branches of mathematical science, afforded little scope for originality. They possess, however, all the qualities which give value to the class of writings to which they belong; being remarkable for lucidity and precision of style, perspicuity of arrangement, elegance of demonstration, and admirable adaptation for self-instruction. The article "Conic Sections" in the last edition of the Encyclopedia Britannica has been translated into Russian, and used as a text-book in some of the schools for the instruction of naval Cadets in that empire. It has also been published as a separate work, and is one of the most elegant geometrical treatises on the subject in existence. Some of his other articles, besides their intrinsic value, had the accessory merit of being the first which were published in this country on the model of the French school, when the French mathematics were greatly superior to our own. His article "Fluxions," in Brewster's Encyclopedia, was the first systematic treatise in our language in which the differential notation was used. The date of the publication is 1815; but, as a point of history, it may be worth remarking, that this notation had been adopted several years previously, both by himself and his illustrious colleague, Mr. Ivory, in their contributions to the Mathematical Repository; and some instances of its use occur in an English work of much older date, Harris's Lexicon Technicum.
Mr. Wallace had made himself intimately acquainted with every department of mathematical knowledge, but the branch which he cultivated with the greatest affection was the ancient geometrical analysis. Of this subject he was a perfect master. His taste having been formed by the writings of Simson, Stewart, and Playfair, he had an unbounded admiration of the elegance and correctness of the Greek geometry; and he took credit to himself for having introduced the Elements of Euclid to the Military College, and restored them, as a class-book, to the University of Edinburgh. Another branch in which he excelled was the angular calculus, which he enriched with various new series and methods of considerable importance to the computer. All his memoirs exhibit ingenuity and fertility of invention, excellent taste, and an intimate acquaintance with those parts of analysis with which they are connected in its most improved state.
The perspicuity and methodical arrangement which distinguish his writings were equally conspicuous in his academical prelections. An intimate acquaintance with the history of scientific discovery, and the various applications of mathematical science, joined with a thorough knowledge of the particular subject under consideration, retentive memory, and a ready invention, rendered his lectures eminently instructive. They were delivered without the slightest attempt at ornament or effect; but they seldom failed to place the subject before the student in a strong, clear, and full light, and were animated with a genuine zeal for the progress of his pupils and the advancement of his science. His Chair had been raised to high degree of celebrity by a long line of illustrious predecessors, and it sustained, while occupied by him, no diminution either of efficiency or reputation.
Professor Wallace was not more distinguished by his mental endowments than for his moral virtues and private worth. In every relation of life his conduct was exemplary. In his family and domestic circle he was greatly beloved. In his general intercourse with the world he was upright, sincere, and independent. In society, his habitual cheerfulness and good humour, amiable manners, benevolent disposition, and a never-failing fund of anecdote, rendered him a delightful companion and a universal favourite. Generous and liberal in all his sentiments, he entertained no envy of the discoveries of his contemporaries; no jealousy of the reputation of younger men; but was ready at all times to applaud and encourage merit, wherever, and in whatever shape, it made its appearance. For such of his pupils as manifested any remarkable capacity or application he entertained an esteem almost amounting to affection; and he was always ready to use his influence, which was considerable, in order to forward their views in life or render them any service. In every measure affecting the public good, or the scientific renown of his country, he took a warm interest. He was the means of procuring a monument to be erected in Edinburgh to Napier, the celebrated inventor of logarithms; and the last occupation of his life was to investigate
the administration of some of the public charities of the city. Mr. Wallace was one of the original non-resident Fellows of this Society. He was also a Fellow of the Royal Society of Edinburgh a Corresponding Member of the Institution of Civil Engineers; an Honorary Member of the Cambridge Philosophical Society; and a few weeks before his death he was elected an Honorary Member of the Royal Irish Academy. After an illness which had for several years prevented him from mixing in society, he died at his residence in Edinburgh on the 28th of April, 1843, in the seventy-fifth year of his age, respected by all, and sincerely regretted by a wide circle of personal friends.
In adverting to the circumstances of his early life, he used to relate that the first rudiments of his education were received from an aged widow in the town, who kept a school for children, and retailed small wares. About the age of seven, he was removed to a school of a better class, in which the principal branch of instruction was arithmetic. In this science, however, he had already been grounded by his father, and had made considerable proficiency in it before he was advanced to that department in the routine of his school progress. His attendance at school for instruction it can scarcely be called was discontinued when he had reached the age of ten or eleven years; and, according to his own statement, all he owed to the schoolmaster was the power of reading, and of forming, in a very indifferent way, characters by writing. His knowledge of arithmetic he owed to his father, and to his own strong liking for the subject.
In 1784, when in his sixteenth year, he was sent to Edinburgh to learn the trade of a bookbinder; and after a year or two of probation he entered upon a regular apprenticeship to this craft. But his passion for the acquisition of knowledge had been thoroughly roused by the perusal of some books which had fallen in his way; and, during the period of his apprenticeship, he devoted every spare moment to reading. These moments were, however, few. His master happened to be a person who had no sympathy with literary tastes, and no other concern about his apprentices than how to extract from them the greatest amount of labour. But his father, a man of considerable intelligence and strict religious principles, having removed with his family to Edinburgh, he had the comfort of residing, during this period, in the house of his parents, and the advantage of their society, encouragement, and moral superintendence, to which he professed himself to have been greatly indebted. His occupation, also, was in some respects favourable to the gratification of his tastes. Books of science were constantly passing through his hands, and his curiosity could not be restrained from occasionally casting a glance at their contents. He had also acquired a few mathematical books of his own; and such were his ardour and enthusiasm in their study, that it was his constant practice to take his meals with one of them in his hand, and to carry one in his pocket, to read on his way to and from the workshop. By this assiduous application, before he reached the age of twenty, he had read and made himself master of Cunn's Euclid, Ronayne's Algebra, Wright's Trigonometry, Wilson's Navigation, Emerson's Fluxions, Robertson's Translation of La Hire's Conic Sections, and Keill's Astronomy. Of these books he cherished the remembrance, as the means by which he had been enabled to grope his way into the region of the mathematics.
Hitherto, Mr. Wallace's efforts to acquire knowledge had been made under the most disadvantageous circumstances; without sympathy from any one but his father, and without a companion or friend to appreciate his exertions or applaud his success. But he was now approaching the turning point of his fortunes. He happened to become acquainted with an elderly person, a carpenter by occupation, who was employed by the celebrated Dr. John Robison, the Professor of Natural Philosophy, as an assistant in his class experiments. This man, though a great reader of books, was no mathematician; but he had sat too near the feet of Gamaliel not to have imbibed a respect for the science, and for the pursuits of his young friend. With an excusable vanity, he was in the habit of boasting of his intimacy with the professor, to whom he proposed to introduce Mr. Wallace. The latter, however, with great good sense, declined the kindly meant offer until the term of his apprenticeship had expired, when, though still with some diffidence and hesitation, he was prevailed upon to take advantage of it. Armed with a letter from his humble patron, he waited upon the professor, who received him with great kindness, examined him with respect to his proficiency in geometry and the conic sections, and inquired into the circumstances of his life, and the means by which he had made so much progress in the mathematics. In the course of the conversation Dr. Robison considerately took occasion to warn him that the study of mathematics was not likely to lead to any thing advantageous in the world: the reply was, that he was aware of the fact; but being, as it seemed, doomed to a life of labour, he hoped to sweeten the cup by the pleasure to be derived from the possession of knowledge. The interview ended with an invitation from the Professor to attend the course of lectures on Natural Philosophy then about to begin. Sensible as he was of the advantages which he could not fail to derive from such instruction, it required no small sacrifice on his part to accept the offer; for, being then employed as a journeyman, the time thus occupied could only be commanded by the abstraction of an equal portion from his hours of rest or sleep. Every difficulty, however, gave way before a determined will. The class was diligently attended: and he has been heard to say, that if he were asked which had been the happiest period of his existence, he would refer to that at which he attended the lectures on natural philosophy, when, for the first time in his life, he had the means of receiving sound instruction, and found himself in the company of young men devoted to the pursuit of knowledge.
Dr. Robison's next act of kindness was to introduce him to his colleague, Mr. Playfair, the Professor of Mathematics. Mr. Playfair was no less struck with the extent of his acquirements, and likewise offered him admission to the mathematical class. But attendance on two classes in one day being, in his circumstances, entirely out of the question, he was under the necessity of declining the offer, much, it may readily be believed, to his regret. Mr. Playfair, however, from this first interview, took a warm interest in his welfare, advised him with respect to the course of reading he should follow, supplied him with books from his own library, and continued his stedfast friend through life.
These details may appear trivial, or unnecessarily minute; but it can never be wholly uninteresting to trace the steps by which distinction in science or literature has been obtained when opposed by obstacles which might seem, and in ordinary cases prove to be, insurmountable. To the individual we are commemorating they were all-important some may receive encouragement from his example; and science itself is placed in an advantageous light when we see men so eminent as Professors Robison and Playfair taking trouble with, and giving help and encouragement to, a friendless young man, who had no claim on their good offices, and no other recommendation to them, than his successful struggles in acquiring the elements of those sciences which they themselves cultivated with such distinguished success. On the other hand, the merit must have been of no ordinary kind which, to persons of their experience, appeared so remarkable.
About the time he was attending Dr. Robison's lectures he was induced, by the prospect of having the command of a greater portion of time than had yet been at his disposal, to exchange his occupation for that of warehouseman in a printing-office. While in this occupation Dr. Robison paid him a visit, and proposed to him to give private lessons in geometry to one of his pupils. This proposal opened up a new prospect to him, and admitted the first gleam of hope of his being able to emancipate himself from the drudgery of manual labour. He now also began to acquire a knowledge of Latin, and in this, as in the study of mathematics, his manner of turning time and opportunity to account may afford encouragement to those who are in pursuit of knowledge under difficulties. A part of his duty in the printing-office was to collect the successive sheets of a work from a series of heaps arranged round a circuit of tables. While engaged in this monotonous occupation, he fixed up upon the wall a Latin vocabulary, from which he committed to memory a certain number of words every time he passed it in making his round. In his study of Latin, however, he received assistance from a student, to whom, in return, he gave instruction in mathematics.
After he had been engaged a few months in the printing-office, he entered into the employment of one of the principal book-sellers of Edinburgh in the capacity of shopman. This change was advantageous in several respects. His circumstances were now considerably improved, and he found leisure, not only to pursue his favourite studies, but to increase his stock of knowledge by general reading, and even to give private lessons in mathematics in the evenings. While in this situation he contrived to get a few lessons in French, and commenced his acquaintance with the works of the Continental mathematicians.
In 1793, while in his twenty-fifth year, he took the resolution to give up his employment, and support himself by teaching mathematics privately. This plan probably succeeded to the full extent of his moderate expectations. He now attended a course of lectures by Professor Playfair; and although, as the course was intended for an audience far behind him in mathematical acquirements, he had little to learn, the example of Playfair's manner – dignified, eloquent, and impressive, in a degree rarely equalled – was of great use to him in after-life. At the same time he also attended a course of chemistry, and by assiduous diligence endeavoured to repair, to the utmost of his power, the deficiencies of his early education.
In 1794, Mr. Wallace, on the recommendation of Professor Playfair, was appointed to the office of assistant teacher of mathematics in the academy at Perth. In respect of emolument the appointment was of no great value, but it gave him a settlement in life, with reasonable leisure to prosecute his mathematical studies, of which he did not fail to take advantage. In 1796, he presented his first memoir to the Royal Society of Edinburgh, entitled, "Some Geometrical Porisms, with Examples of their Application to the Solution of Problems." This paper, which contained some new and curious porismatic propositions, afforded ample proof of original and inventive powers; while his manner of conducting the investigation shewed how accurately he had imbibed the spirit and methods of the ancient geometrical analysis. About the same time, on the request of Dr. Robison, he contributed the article "Porism" to the third edition of the Encyclopedia Britannica; and, a few years later, when a new and greatly enlarged edition of that work was undertaken, he was enlisted as a regular contributor, and undertook to furnish the principal mathematical papers.
During the vacations of the Perth academy he paid regular visits to Edinburgh, where he continued to cultivate the friendship of Robison, Playfair, and other scientific men, to whom his now recognised talents and mathematical attainments procured him introductions. The first mark of literary distinction he received was that of Corresponding Member of the Edinburgh Academy of Physics; a society which, though not known by its published transactions, was at that time remarkable by reason of the cluster of talented persons of whom it was composed, several of whom have since attained the highest distinction in literature, philosophy, and public affairs. Such association could not fail to have a powerful effect in the developement of his mind, even though his residence at a distance from Edinburgh prevented him from attending many of the meetings.
In 1802, he presented a second paper to the Royal Society of Edinburgh, containing a new method of expressing the coefficients of the developement of the algebraic formula which represents the disturbing effect of the mutual action of two planets on each other. This was a contribution of great merit, and, immediately upon its publication, established his reputation as a mathematician of the first order. The volume of the Transactions in which it appeared was reviewed in the second number of the Edinburgh Review; and an able analysis of Mr. Wallace's paper was concluded with the following encomium: "We cannot conclude without expressing our sincere admiration of this excellent performance – excellent in every respect; and, trifling as it may appear to mathematicians, remarkable for a pure, perspicuous, and not inelegant style. It is a paper, equal, in our opinion, to whatever has been most admired of the greatest analysts. We remember nothing in the works of Euler or Lagrange which belongs to a higher order of excellence in the science." Mr. Wallace's method of developement depended ultimately upon the proportions which the perimeters of two ellipses bear to those of their circumscribing circles and in order to facilitate its application, he gave, in an appendix, a very beautiful and quickly converging series for the rectification of the ellipse, applicable to every case of eccentricity, and to every length of an are that can possibly occur in calculation. His merit with respect to this paper cannot be considered as having been diminished by the discovery he made some time after its publication, that in certain respects he had been anticipated by Legendre. The very little intercourse which at that time existed between this country and France, and the position of the author in a remote provincial town, are sufficient excuses for his not having been more accurately acquainted with the state of mathematical discovery on the Continent.
Mr. Wallace had been for several vears a contributor to some of the periodical publications in England in which mathematical questions were proposed, as Leybourn's Repository, the Gentleman's Mathematical Companion, and others of the same class. To this circumstance he attributed an incident which had an important influence on his future life. In 1803, he received a letter, under a feigned name, in which he was informed that an instructor in mathematics was wanted for the Royal Military College, then established at Great Marlow in Buckinghamshire, and recommended, if he thought of being a candidate for the office, to make an immediate application. Inquiry being made in the proper quarter, the information was found to be correct, but he ascertained also that it would be necessary to make his application in person. In matters affecting his own interests the disposition of his mind was not sanguine; and, as in the present case he had no influence to employ, and no other recommendation to carry with him than his skill in mathematics, his chances of success appeared so small that he would have been deterred by the length and inconveniences of the journey from thinking more of the subject, had he not been encouraged by his friend Professor Playfair. On his arrival at the Military College he found there were several competitors; but the persons who had to decide on the respective qualifications of the candidates gave their decision in his favour, and he was accordingly appointed to the office.
Mr. Wallace held this appointment upwards of sixteen years, during which period, the whole of his leisure time was unremittingly devoted to scientific study and literary labour, the fruits of which appear chiefly in his numerous contributions to the two great Encyclopedias then publishing in Edinburgh. This species of writing, which is not particularly well adapted to form the basis of a permanent reputation, was in a manner forced upon him by the circumstances of his position. On his appointment to the Perth Academy he had married, and after he joined the Military College his family began to increase rapidly. The inconveniences he had suffered from the defects of his own early education rendered him only more solicitous that his children should not labour under any disadvantages in this respect, and, as they grew up, he placed them at schools in Edinburgh. His official income being insufficient for this expense, he was led to engage in the works now referred to, rather with a view to add to his means, and to enable him to discharge a sacred duty, than for the sake of any distinction he was likely to get by them. No individual, perhaps, was ever less influenced by considerations of a worldly nature, or more ready to bestow time and labour upon objects from which he could neither receive nor expect any remuneration whatever.
In 1808, he contributed a paper to the Royal Society of Edinburgh, entitled "New Series for the Quadrature of the Conic Sections, and the Computation of Logarithms," and containing some very remarkable formule for the rectification of circular ares, with analogous expressions for the sectors of the equilateral hyperbola and the logarithms of numbers; all deduced from elementary principles, and without the use of the differential calculus or any equivalent method. At the time the paper was published, he believed the series to be entirely new, but he discovered afterwards that some of them had been previously given by Euler.
Mr. Wallace's services at the Military College were held in great estimation by the superior Officers, who frequently availed themselves of his practical sagacity in the adoption of regulations having respect not only to the course of instruction, but the general management of the establishment. One of the results of this deference to his recommendations (more particularly interesting to the Society), is the small observatory attached to the College, for the instruction of the officers of the senior department in practical astronomy. The plan of the building was originally furnished by Dr. Robertson of Oxford; but the superintendence and arrangement of all the details of construction were confided to Mr. Wallace, who visited most of the Observatories in the neighbourhood of London, for the purpose of acquiring hints and information. A transit-instrument, an astronomical circle by Ramsden, a reflecting circle, and a clock by Hardy, were procured, and some other instruments were ordered, but countermanded from an apprehension of opposition to the estimates in the House of Commons. Although an Observatory of this kind cannot be expected to produce results of any direct advantage to astronomy in the present state of the science, it must still be regarded as no unimportant appendage to a national establishment for the instruction of Officers for the public service.
In 1819 a vacancy occurred in the Mathematical Chair of the University of Edinburgh, through the death of Professor Playfair, and the appointment of Mr. Leslie to succeed him in that of Natural Philosophy, and Mr. Wallace resolved on presenting himself as a candidate. The patronage belongs to the magistrates of the city, who, having in general no pretensions to be capable of estimating degrees of merit in abstract science, necessarily form their opinions from the testimony of others, or notions of general fitness, and are liable to be acted upon by influences of various kinds. In the present case a very keen contest took place; for another competitor (a man of general talent and great respectability, though unknown as a mathematician) was strenously supported by a strong political party. The struggle terminated, however, in his election by a large majority of the voters. This was the crowning object of his ambition. Ever since his appointment to the Perth Academy, he had fixed his regards on a professorship in a Scottish university as the goal of all his exertions; but his elevation to the Chair of the Gregorys, of Maclaurin, Matthew Stewart, and Playfair, probably did not enter at that period into his most sanguine anticipations.
Mr. Wallace had reached the age of fifty-one when he was appointed to the mathematical professorship in Edinburgh; but he still retained both mentally and bodily all the energy and activity of his younger years. He held the office till 1838, when he resigned on account of ill-health, having been unable to perform his duties in person during the three previous sessions. Upon his resignation the honorary title of Doctor of Laws was conferred upon him by the University, and at the same time he received a pension from Government which he enjoyed during the few remaining years of his life, in consideration, as the warrant stated, of his attainments in science and literature, and his valuable services, up to a very advanced period of life, first in the Military College, and afterwards at the University of Edinburgh.
For some years after his establishment at Edinburgh, a considerable portion of his time was occupied in the preparation of his lectures, on which he bestowed great pains. When the new edition of the Encyclopedia Britannica was commenced, he undertook the revision of all the mathematical papers he had contributed, as well as some of those which had been written by Dr. Robison; and several of the more important treatises, particularly, Algebra, Conic Sections, and Fluxions, were remodelled and almost entirely rewritten. To the Transactions of the Royal Society of Edinburgh he contributed a paper, in 1823, on the Investigation of Formulæ for finding the logarithms of trigonometrical quantities from one another; one in 1831, entitled "Account of the Inven. tion of the Pantograph, and a Description of the Eidograph;" and one in 1839, on the Analogous Properties of Elliptic and Hyperbolic Sectors. His last contribution to that Society, published in Vol. XIV. of the Transactions, was entitled, "Solution of a Functional Equation, with its Application to the Parallelogram of Forces, and to Curves of Equilibration." This paper, in addition to the investigation of series adapted for calculation, contains a set of tables, to ten decimal places, of the corresponding values of the amplitude, ordinate, and are of a catenary, which are important in an engineering point of view, as they afford the data required for constructing arches having the forms of equilibrated curves. Similar tables, to eight places, had previously been given by Mr. Davies Gilbert in a paper on the mathematical theory of suspension bridges, in the Philosophical Transactions for 1826; but the numbers were found by Mr. Wallace to be erroneous, generally, in the three last decimal figures.
Mr. Wallace is the author of a paper in Vol. IX. of our Memoirs containing two elementary solutions of Kepler's problem by the angular calculus. In the Transactions of the Philosophical Society of Cambridge, Vol. VI., there is also a paper by him under the title of "Geometrical Theorems and Formule particularly applicable to some Geodetical Problems." For this subject he had a particular predilection; and in 1838, while confined to a sick-bed, he composed, and afterwards published at his own expense, a separate work entitled, "Geometrical Theorems and Analytical Formulæ, with their Application to the Solution of certain Geodetical Problems." This volume, which he appropriately dedicated to his friend Colonel Colby, contains the substance of his paper in the Cambridge Philosophical Transactions, with the addition of a considerable number of extremely elegant formulæ, most of them new, and some of them important in the practice of the higher geodesy.
Professor Wallace took great delight in all the practical applications of his science, and had a strong turn for mechanical invention. His attention having been directed to the imperfections of the Pantograph, he invented, in 1821, an instrument on a different principle to supply its place, to which he gave the name of Eidograph. This instrument answers the same purposes as the common pantograph, to which, however, it is greatly superior, both in the extent of its applications and the accuracy of its performance; for, while the similarity of the copy to the original, in all its parts, is preserved with geometrical accuracy, the copy may be reduced or enlarged in almost any proportion; or, by a particular modification of the instrument, it may even be reversed, and transferred immediately to metal or stone. This ingenious instrument, which would seem to be admirably adapted to the purposes of the engraver, was first described by him in Vol. XIII. of the Edinburgh Transactions to which reference has already been made. He has also described, in the Appendix to his Conic Sections, an Elliptograph, or instrument for describing an ellipse by continued motion, founded on a very beautiful property of the ellipse first pointed out, we believe, by him, namely, that the curve is organically described by any given point (not in the circumference) in the plane of a circle which rolls along the concave circumference of another fixed circle, the radius of which is twice that of the rolling circle. And in an Appendix to his Geometrical Theorems he has given the description of an instrument which he invented for the graphical solution of an important problem in surveying, viz. to determine the position of a station, having given the angles made by lines drawn from it to three other stations in the same plane, whose positions are known. This instrument, which he called a Chorograph (the problem which it solves having been proposed as a chorographical problem by Richard Townley in No. 69 of the Philosophical Transactions), is simple, compact, portable, and inexpensive; and in these respects has considerable advantages over the station-pointer, generally used for the same purpose.
Among the objects connected with the advancement of science to which Professor Wallace gave his aid, after his appointment to Edinburgh, there is one which it would be unpardonable to pass over without notice in this place, we allude to the Observatory now established there. Ever since the time of Maclaurin there had existed a small astronomical observatory in Edinburgh, but no provision was made for regular observation, nor, indeed, did it contain any instruments fit for the purpose. Through the exertions, chiefly of Professor Playfair, funds were at length raised, by private subscription, for the erection of an observatory adapted for observations of the most accurate kind. Mr. Playfair did not live to see the building completed, or means provided for obtaining instruments, or carrying on systematic observations; but Mr. Wallace, on becoming his successor, entered fully into his views, and, in concert with a few other individuals, used all his influence and exertions towards bringing the scheme to maturity. At length, after years of expectation and delay, the Government was prevailed upon to take the observatory under its protection, furnish it with instruments of the first class, appoint an astronomer and assistant, and provide for the regular publication of the observations. In bringing about this arrangement, Mr. Wallace's aid and recommendation were of essential service; and if any thing was wanting to complete the satisfaction which he felt at the result, it was to see the observatory placed under the care of his friend Professor Henderson, of whose distinguished merits as an astronomer it would be superfluous to speak to those who are in the habit of attending our meetings, or reading our Memoirs.
Although the works which Mr. Wallace has left behind him assure him a high place as an original and inventive mathematician, the talents with which he was endowed by nature were, doubtless, rendered less productive than they would have been by his want of early education, his residence during the best years of his life in the country at a distance from congenial society, and, perhaps, still more from the circumstance of so much of the time which his laborious public duties left at his disposal having been consumed in the preparation of his numerous treatises for the Encyclopedias. Thèse treatises being mostly of an elementary kind, and composed for the purpose of explaining the principles of the various branches of mathematical science, afforded little scope for originality. They possess, however, all the qualities which give value to the class of writings to which they belong; being remarkable for lucidity and precision of style, perspicuity of arrangement, elegance of demonstration, and admirable adaptation for self-instruction. The article "Conic Sections" in the last edition of the Encyclopedia Britannica has been translated into Russian, and used as a text-book in some of the schools for the instruction of naval Cadets in that empire. It has also been published as a separate work, and is one of the most elegant geometrical treatises on the subject in existence. Some of his other articles, besides their intrinsic value, had the accessory merit of being the first which were published in this country on the model of the French school, when the French mathematics were greatly superior to our own. His article "Fluxions," in Brewster's Encyclopedia, was the first systematic treatise in our language in which the differential notation was used. The date of the publication is 1815; but, as a point of history, it may be worth remarking, that this notation had been adopted several years previously, both by himself and his illustrious colleague, Mr. Ivory, in their contributions to the Mathematical Repository; and some instances of its use occur in an English work of much older date, Harris's Lexicon Technicum.
Mr. Wallace had made himself intimately acquainted with every department of mathematical knowledge, but the branch which he cultivated with the greatest affection was the ancient geometrical analysis. Of this subject he was a perfect master. His taste having been formed by the writings of Simson, Stewart, and Playfair, he had an unbounded admiration of the elegance and correctness of the Greek geometry; and he took credit to himself for having introduced the Elements of Euclid to the Military College, and restored them, as a class-book, to the University of Edinburgh. Another branch in which he excelled was the angular calculus, which he enriched with various new series and methods of considerable importance to the computer. All his memoirs exhibit ingenuity and fertility of invention, excellent taste, and an intimate acquaintance with those parts of analysis with which they are connected in its most improved state.
The perspicuity and methodical arrangement which distinguish his writings were equally conspicuous in his academical prelections. An intimate acquaintance with the history of scientific discovery, and the various applications of mathematical science, joined with a thorough knowledge of the particular subject under consideration, retentive memory, and a ready invention, rendered his lectures eminently instructive. They were delivered without the slightest attempt at ornament or effect; but they seldom failed to place the subject before the student in a strong, clear, and full light, and were animated with a genuine zeal for the progress of his pupils and the advancement of his science. His Chair had been raised to high degree of celebrity by a long line of illustrious predecessors, and it sustained, while occupied by him, no diminution either of efficiency or reputation.
Professor Wallace was not more distinguished by his mental endowments than for his moral virtues and private worth. In every relation of life his conduct was exemplary. In his family and domestic circle he was greatly beloved. In his general intercourse with the world he was upright, sincere, and independent. In society, his habitual cheerfulness and good humour, amiable manners, benevolent disposition, and a never-failing fund of anecdote, rendered him a delightful companion and a universal favourite. Generous and liberal in all his sentiments, he entertained no envy of the discoveries of his contemporaries; no jealousy of the reputation of younger men; but was ready at all times to applaud and encourage merit, wherever, and in whatever shape, it made its appearance. For such of his pupils as manifested any remarkable capacity or application he entertained an esteem almost amounting to affection; and he was always ready to use his influence, which was considerable, in order to forward their views in life or render them any service. In every measure affecting the public good, or the scientific renown of his country, he took a warm interest. He was the means of procuring a monument to be erected in Edinburgh to Napier, the celebrated inventor of logarithms; and the last occupation of his life was to investigate
the administration of some of the public charities of the city. Mr. Wallace was one of the original non-resident Fellows of this Society. He was also a Fellow of the Royal Society of Edinburgh a Corresponding Member of the Institution of Civil Engineers; an Honorary Member of the Cambridge Philosophical Society; and a few weeks before his death he was elected an Honorary Member of the Royal Irish Academy. After an illness which had for several years prevented him from mixing in society, he died at his residence in Edinburgh on the 28th of April, 1843, in the seventy-fifth year of his age, respected by all, and sincerely regretted by a wide circle of personal friends.
William Wallace's obituary appeared in Journal of the Royal Astronomical Society 6:4 (1844), 31-41.