In my previous lectures I have tried to give you some idea of the history of mathematics and especially of the way in which its fundamental principles have been developed, with the purpose of showing what the early workers in the subject thought of it and what it is from our present point of view. I wish now to say something about the mathematician, the man who has made that history, as an individual. This topic lies outside the usual field of mathematical investigation; it is rather of a psychological character, for the most part, and has also a physiological and even a physical side.

It is impossible to define the mathematician otherwise than by describing the scope and method of his thought, and these are changing and extending so rapidly in their details that it is quite impossible to say what the mathematician of the future will be. Mathematics is commonly regarded by philosophers as a system of deductive reasoning; I am inclined to consider its sometime deductive character as the result of the way in which it has been developed, and it is certainly not merely a system of reasoning. Its operations include synthesis and construction as well as analysis and ratiocination; its more general results have been usually, perhaps always, obtained by induction and proved by deduction; in fact, the regular mode of discovery in mathematics is inductive, so that a somewhat extensive knowledge of mathematical facts is necessary for mathematical discovery. Induction seems to depend upon the association of ideas and to admit of no formal systematization; at all events such a systematization must be very complicated for us to be conscious of it. The acquisition of mathematical knowledge is, therefore, a very different thing from mathematical research; the former is possible to a considerable extant for every well-balanced and properly trained mind, while the latter can be achieved only by the mathematical genius. Professor Klein classes mathematicians as intuitive, formal, and critical; we may say that the first and highest class includes all those who are particularly characterized by great powers of induction, that the second class contains those that are particularly strong in deduction, and that the third class is made up of those in whom the power of mathematical thought is combined with a philosophical turn of mind that leads them to pay special attention to the mental processes involved in mathematical operations. Sylvester was a remarkable instance of the inductive mathematician. Speaking of his researches in the theory of numbers, he has repeatedly said to me something like this: "I have observed that the first hundred numbers of such and such a class have such and such a property, don’t you think I am justified in assuming that all the numbers of that class have that property?" Again, he would state a theorem in a lecture to his class and say, "I have not proved this theorem, but I am morally certain of it, yes, I would stake my life on the truth of it." Another instance is the so-called "fundamental theorem of invariants," which was stated by Cayley in 1853 and continually employed as true both by him and Sylvester, but which defied all attempts to prove it until Sylvester, in 1878 (that is twenty-five years after it was first enunciated) succeeded in finding a conclusive demonstration that it was valid. Steiner, before 1850, had stated hundreds of geometrical theorems without proof, of which many, I believe, have not yet been demonstrated. Anyone who has occasion to develop systematically any mathematical subject for himself will infallibly find that he knows or guesses to be true theorems for whose deductive proof he has not made suitable provision, so that he is compelled to go back and make additional assumptions, in the way of axioms and postulates, for the purpose of supplying the deficiency. There are good reasons to believe that the axioms and postulates of Euclid’s Elements were so added long after Euclid’s time.

As to the subject-matter of mathematics, there are as yet no indications that it has any, that is, that the field of its operations is in any way restricted excepting by the general limitations of human thought. As the science that takes cognizance of any class of phenomena become better known and more intimately associated as cause and effect that science becomes more capable of treatment by formal symbolic, that is mathematical, methods. Only in so far as such methods are applicable to a science it is possible for it to predict with certainty what phenomena will result from given conditions of any complexity.

Mathematics is, then, a system of methods with unlimited applications, its facts are assumptions and the results of applying those methods to the assumptions; I may define it briefly as formal symbolism or the art of stating facts symbolically and of deriving the necessary results of these statements by formal methods. The purely symbolic language (conventional notation) of mathematics is a necessary attribute of it and all attempts to popularize it by divesting it of this form must be destructive of its essential character. Its very usefulness depends upon its unequivocal and at the same time general, its brief and yet perfectly definite, statements, which enable us to carry out with absolute certainty mental processes of such a complicated nature as would be otherwise quite impossible. Mathematics is, then, par excellance an art. Sylvester used to say: "the fine arts are four: plastic, lyric, music, and mathematic."

I have already spoken of the difference between mathematical knowledge and the ability for mathematical research. The latter, which I shall designate as the mathematical talent, is the distinguishing characteristic of the mathematician, and I wish to protest against the application of this name to anyone simply because he is an excellent accountant, computer, surveyor, engineer, physicist, or even astronomer. Such an one may be a mathematician or he may not be. I do not mean to say that training in the higher branches of mathematics is necessary or even sufficient to entitle one to be called a mathematician, but that the application of mathematics to many purposes, and especially the ready and accurate use of numbers, may become so natural, by practice as to be mistaken for mathematical thought, the faculty for which is the birthright of the mathematician. A proof of this is to be found in the considerable number of children who succeed in performing satisfactorily their school tasks in arithmetic in comparison with the small number of those whoever accomplish anything more in mathematics; and I may add that a large proportion of those who are known as really good in arithmetic while in practice at school seem to loose all their facility with numbers shortly after this practice ceases. The real test, it seems to me, is the ability to do what one has not been taught to do, that is, the ability to devise means for one’s self. Arithmetical prodigies seem to do this unconsciously in a limited field, and to this extent they are mathematicians. The relation of the number-faculty to the mathematical talent in general is very uncertain. It seems to be impossible to say whether the mathematical talent is one or many; apparently, most of the great mathematicians have been "quick at figures," and certain physical characteristics seem to have been common to them all; still, there have been those who distinguished themselves as geometers or analysts without being skillful in calculation, and there have been arithmeticians and analysts who failed to grasp geometrical forms. Perhaps they have had in common one or more faculties that characterize the mathematician and have differed in other faculties that determine the form of mathematical thought. At all events, in the general study of the mathematical talent it is advisable to take account of all its phases, so that I am not inclined to exclude anyone from the Present consideration because he didn't do this or that thing.

However difficult it may be to give a satisfactory abstract definition of the mathematician, there is perhaps no type of man more definite, no type more readily distinguished from other types in the individual. The truly great mathematicians are known by their thoughts and their teachings, spoken and written, and their powers and modes of thought have been so peculiar as to justify the assumption that they constitute a special class among thinkers. The taste for mathematics and the ability to think mathematically, if I may so express it, are so uncommon that their existence in any individual, even to a moderate degree, is seldom doubtful. A taste for music, poetry, painting, sculpture, literature, or any of the sciences of observation, biology, chemistry, even physics and astronomy, may be the result of environment, apparently, but a taste for mathematics seems to be innate, if exists at all, and impossible to create where it does not exist. The mathematical talent seems, then, to be something peculiar to the individual and not a general characteristic of the race. "Nascitur non fit" is certainly as true of the mathematician as of the poet. Of course, I would not say that one could become a master in any art without an innate genius for is specialty; but it is noticeable that, while other men have sought recreation outside the line of their profession, the professional mathematician has seldom found his elsewhere than in pursuits more or less closely connected with his regular work. In fact, the mathematician is popularly known as a man with a hobby, which he rides continually and to the not infrequent annoyance of his non-mathematical associates. I do not mean, as you will presently see, that the possession of the mathematical talent is inconsistent with that of other talents; there have been not a few great mathematicians who attained high distinction in other fields; but, as a rule, the mathematical talent under favorable conditions so absorbs the attention of its possessor that he has neither time nor inclination for other occupations. On the other hand, circumstances, necessity, opportunity, policy, pecuniary remuneration, and popular consideration have, undoubtedly, turned into other professions many that might have achieved distinction in this line; from my experience as a teacher I am lead to think that few departments, if any, suffer so much from this kind of desertion as mathematics. I would suggest that the main objects of education ought to be not only to discover natural talents (not natural talent in general) and to develop them, but also to encourage by all possible means the choice of a profession that is likely to bring into play the strongest faculties of the individual. This would, to be sure, reassure the cooperation of the community in the active material encouragement of those professions that now offer but little prospect of personal gain to those who adopt them, but it would be a kind of "protection" that would be for the greatest advantage of the whole community. There seems to be a strong feeling on the part of very many that the teacher and the devotee of pure science are a burden to the community, that is, that then receive from the public a living (extending even to the luxuries of life and the means of accumulating property) without giving an equivalent in return. There is no doubt that the efficiency and reputation of the teaching profession have suffered greatly from its adoption as a means of livelihood, permanently or temporarily, by a large number of persons who have no particular talent for it, but who simply cannot find anything more agreeable to do or more profitable.

The production of mathematicians requires a certain amount and quality of culture in the community. We never find them among savages nor among what we call half-civilized peoples; they did not exist among the Saracens until they came under the influence of Greek culture. In ancient times, the Greeks, Hindus, and possibly the Chaldeans alone seemed to have developed mathematics to any great extent. So far as we can now say, only the crudest elements were known to the Egyptians and the Chinese; we must exclude the purely Greek school of Alexandria from consideration here, and what the Chinese tell us of their own developments is totally unreliable, all traces of such developments having disappeared at the present time. The Romans did nothing more than to apply the crude Egyptian methods of surveying, somewhat modified in later times by what they got from Greece. It may be remarked here that progress in the other arts has not generally been accompanied by a corresponding advance in mathematics. Even since the Renaissance the development of mathematics has progressed very unequally in different European countries. Spain served chiefly as a stepping-stone in the passage of Greek and Indian methods from the Saracens into Western Europe, Italy, Germany, and England laid the foundation of commercial arithmetic, and these with France perfected the notation of algebra, greatly improved its methods, and extended its results and its concepts. So far, the work in mathematics may be fairly regarded as simply a gradual improvement of ancient methods. But about 1630 began a new era; it seems as if the mathematical world then threw off the shackles of the past, and the history of modern mathematics began. For about fifty years France held easily the first place, and since has produced a continuous line of great mathematicians; probably no other country has exerted so great an influence on modern mathematics (excepting the inventors of new modes of thought, for example, the infinitesimal calculus, but France may be said to have produced their peers). With the exception of Kepler and Leibnitz, Germany had no great mathematicians before the end of the eighteenth century, but since that time she has held her own with the best. No British mathematician excepting, perhaps Newton and William Rowen Hamilton has exerted any great influence on the development of pure mathematics in foreign countries. The only Italian mathematician of the first rank was Cavalier, in the 1st half of the seventeenth century. Between the middle of the seventeenth and the latter part of the eighteenth centuries Switzerland had four great mathematicians. During the last century Norway produced two, Russia two, and Hungary one. I refer here only to those who have exerted an influence of so me consequence on the development of the subject. The chief cause of this diversity of development in different parts of Europe is, undoubtedly, the various measures of encouragement offered to mathematicians in the different countries. In this respect France has stood foremost in modern times, the French government having been continually on the lookout for talent, developing it with the greatest care, giving it the best opportunities for usefulness, and rewarding it liberally; and the same may of said of Germany during the last century. It is difficult to see how any such general encouragement can given, under the present conditions, in our own country, where educational institutions and scientific research are mainly supported by private generosity.

We know so little of the private lives of the ancient mathematicians that we are practically compelled to study the type as it is exhibited in the modern mathematician. Archimedes is the only one of the ancients about whose personal characteristics we can form any very definite idea; he seems to have been very simple in his habits, unassuming, although the relative and intimate friend of a king, often concentrated and therefore absent-minded even in the midst of the most exciting surroundings,–witness the story of his death.

Curiously enough, no bust or other portrait of any of the old mathematicians has come down to us, unless a bust in the Capitoline museum, commonly designated as that Aeschylos, but on entirely insufficient grounds, be really that of Archimedes. It will, therefore, be understood that what we shall say about the mathematical talent is drawn from and illustrated by the lives of modern mathematicians, and is to be applied only by inference to any others.

A. I do not claim the mental qualities to be named as the exclusive property of mathematicians, but I do say that the possession of such relatively uncommon faculties to so high a degree by such a large proportion of those who distinguished themselves in mathematical work is a sufficient basis for assuming that the mathematician is a special type of man and that these qualities are his characteristic marks. We must not expect to find all these qualities in every mathematician, for some of them may be obliterated or held in abeyance by unfavorable material conditions or even by the extraordinary development of other qualities.

Perhaps the most striking as well as the most usual characteristic of mathematicians is their precocity, although there are some exceptions to this rule. This precocity has not always shown itself in the line of mathematics, but there have been very few that attained distinction in this line who were not ardent students of something from their earliest childhood, mathematics, languages, or science. Many of them are said to have read fluently at the age of two or three years; it seems to have been impossible to keep them from books, when these were at all accessible. The exceptional cases appear to be of those whose worldly conditions were such as to make the early acquisition of knowledge out of the question. Thus, we read of a boy of 3 years of age discovering errors in his father's accounts, of another learning Latin verses at 4, of another reading 20 volumes of a large encyclopaedia before he was 14 and then learning Latin that he might read the works of the great mathematicians, and of still another who could write in nine different languages before he was 14 and who learned the infinitesimal calculus that he might know how to grind the lenses for a microscope he was making; and there are very many cases quite as remarkable as these. Somewhat related to this precocity, perhaps, is the power that some mathematicians have shown of acquiring languages or of prosecuting new sciences late in life.

While the mathematician's art is not essentially dependent upon the other arts and sciences, he has often found in them applications for his peculiar modes of thought and their unsolved problems have led to the invention of many new mathematical methods. This is particularly true of physics and astronomy. But the mathematician has often distinguished himself in subjects in which he could not be regarded as having any such objective interest. In most such cases it is probably impossible to say whether the different interests were coordinate or subordinate one to the other. At all events these cases show that the mathematical talent does not preclude other talents of a high order I believe that the mathematician is a kind of omnivorous animal endowed with a large amount of curiosity, so that, if the time ever comes when his attention is not absorbed by a mathematical object, he will turn it to any subject that presents itself, without thereby, be it said, abandoning his specialty. Mathematicians are often reproached with being one-sided, but the facts do not show that this criticism is just, in general. Any person with a decided talent is necessarily more or less one-sided, but that ought not to be a cause for reproach. In the present advanced state of science, the universal genius must be satisfied with attaining mediocrity, distinction is reserved for those who devote themselves almost entirely to one line of activity. The great mathematicians, or very many of them, however, have shown an enormous capacity for work and al together phenomenal faculty of acquiring knowledge, so that they have often accomplished what would appear to most people impossible tasks. Thus, Euler completed in three days computations for the Russian government for which three months had been allowed. This same Euler wrote over 800 books and articles, of which about 450 were produced after he had become blind in both eyes.

Abel wrote 48 original memoirs and shorter articles, which fill over 820 large quarto pages in the last edition of his works, and be died at the age of twenty-seven. The works of Lagrange have been published in 14 quarto volumes, those of Laplace in 13, and those of Cauchy are being published in 27 quarto volumes. There is hardly any profession, art, or science that has not been practiced with marked success by some great mathematician. Many mathematicians have been fond of music, some have been excellent performers, but it would probably be difficult to find a great musician who excelled as a mathematician. Albrecht Dürer and Leonardo da Vinci were mathematicians as well as painters. The late George Salmon had a chair of divinity and one of mathematics offered to him at one time, and Pope Sylvester II was a mathematician before he became the head of the Catholic hierarchy. Many men gifted with the mathematical talent have made a living and a reputation as lawyers, statesmen, or diplomats, and a few have practiced medicine. Descartes and Leibnitz were eminent mathematicians as well as philosophers, and Kant was somewhat of a mathematician; but it is remarkable that, while so many of the ancient philosophers were mathematicians, most of the modern philosophers seem to have absolutely no mathematical taste. Gauss, Young, William Rowan Hamilton, Clifford, and a host of others had a remarkable talent for languages as well as mathematics. Some mathematicians have written verses, but none, I think, could be called great poets. Monge won renown as a soldier as well as a geometer. Heredity has done very little for the great mathematicians. It is astonishing how many of them have been born in the lowest ranks of life, and there are very, very few cases where different generations of the same family have produced mathematicians of real distinction, although some taste for mathematics has run through a few generations of certain families. Curiously enough, there are several cases of brothers who had the mathematical talent, although their parents did not show it. Jacob and Johann Bernoulli furnish a case of this sort, and the latter had two sons of some mathematical ability, but not at all comparable to their father or their uncle. In a few families, father, son, and grandson have held professorships of mathematics; but I believe no case is known in which a son of a mathematician achieved distinction in this line equal to that of his father except that of Euler, (whose father wrote one mathematical paper under the auspices of Jacob Bernoulli, whatever that may mean), Claivant, and John Bolyai and also no case in which the mathematical talent after a first appearance in a family has passed over one or more generations to re-appear later. Of course, this apparent failure to inherit the talent may be due to one or other of the causes of desertion that I have mentioned already, but it is not likely that it should be so persistently the case; I believe, on the contrary, that the cases in which the talent appears to be inherited are largely the results of the peculiar opportunities that the son of a mathematician has to get a good training in the subject and to the personal influence of the father. Were it otherwise, how comes it that so many of the pupils of the great mathematicians have outstripped their sons?

It is difficult to say what effect nourishment has on the development of the mathematical talent. Certainly, many of the most eminent mathematicians have suffered great privations, even to the extent of reducing their food supply to the minimum, in order to pursue their favorite study, and many others have been educated by private or public charity, while but few have been reared in affluence. The mathematician Kästner said, "I do not fear a siege; I learned to starve at Leipzig." Some educators are of the opinion that students take greater advantage of their opportunities if they are obliged to do something towards their own support; but one of real talent needs no incentive, except the opportunity, to develop that talent; food, raiment, shelter, books and the other tools of his trade, and good advice are all that are necessary for him. While I am a firm believer in the advantages of a liberal education for all who would devote themselves to any kind of intellectual work, I am sure that children of real talent in some directions are seriously injured by being compelled to pursue studies that are distasteful to them. Alexander von Humboldt said, "Had I fallen into the hands of the present school training, I should have gone to ruin body and soul." Although I do not wish to deny in toto the proverb "mens sana in sano corpore," I will say that I believe it, like most proverbs, is generally falsely interpreted. Man, especially in the mass, is very prone to be carried away by a sentiment that would be fine if it could be realized, above all if it is expressed in a short sentence with an appropriate comparison, antithesis, or alliteration. But, in the present example, there is a great difference between a healthy body, that is one without disease, and an athlete, or even between a good digestion and a good appetite. For the average man, no doubt, the ability to endure severe and continued physical exertion is most important. I suppose also that a robust body and a phlegmatic temperament are most proof against the evil effects of care and worry, but the highest forms of intellectual activity seem to have generally found a more suitable arena in a somewhat frail body with a nervous temperament. If any one will read the lives of a large number of mathematicians, he will be surprised how often they are spoken of as lively, passionate, and even violent, "and also," as one writer says, "a fiery spirit seems to speak from their faces"; but they have usually possessed at the same time great powers of self-control.

The specific mental qualities that have usually characterized mathematicians seem to be those chiefly related to the will, if I may so describe them. The great mathematicians have certainly had strong wills, but it may be a question whether the development of the other faculties has been conditioned by this will-power, or whether the will is a habit acquired through the exercise of other faculties requiring control and a spur; I am inclined to the latter theory. First among these faculties I place concentration, the restriction of mental activity to a single object, a secondary faculty in general but of primary importance to the mathematician. The proverbial absentmindedness of mathematicians is but the sign of concentration. Closely related to the first is the second faculty of persistence, which recognizes nothing as impossible until it has been definitely proved to be so; it has overcome many difficulties and has been the cause of much wasted energy; also generally a secondary faculty. Most of the very eminent mathematicians have had what would usually be called wonderful memories for everything, almost without exception visual so that they remember better what they have seen or read than what they have heard. Recent experiments on college boys and girls also show that the taste for mathematics usually implies a visual memory. This kind of memory is generally accompanied by the habit of observation, by which I mean attention. There can be no doubt that the so-called "whist-memory" consists entirely in paying interested attention to each play and comparing it with other plays. Next to memory comes the group of analytic and synthetic faculties, which we may designate separately as penetration, judgement, and construction. The fact that the great mathematicians have possessed these three faculties is a substantial reputation of the popular opinion that they have often been stupid in matters not directly connected with their specialty. Indifference and ignorance are not stupidity, and the mathematician is less than any one else, perhaps, inclined to express an opinion on any subject to which he is indifferent or of which he is ignorant.

It is rather surprising that the chief and indeed regular mode of demonstration in all branches of mathematics to the close of the seventeenth century was the geometrical method, and it continued to be so in England until the first quarter of the nineteenth century had passed. Now the faculty most called into play in geometry is the imagination, not the power of thinking of the unreal, but that of forming definite mental images, keeping them in mind, breaking them up into their component parts, and recombining these elements to form new images. It is not wonderful, therefore, that mathematicians have been preeminent as possessors of this faculty. Finally, as a consequence of the possession of all these faculties to a high degree, mathematicians have exhibited an indefatigable industry. They seem to have found it practically impossible to stop the intellectual machinery. I have known Sylvester to get out of bed at three o'clock in the morning to develop an idea that had occurred to him, and even send out for some one to talk it over with him. This industry has often been unwisely directed, so that it produced no useful results, but I believe the natural impulse of the mathematician is to exercise his faculties, without regard to the utility of the results; and that is largely the reason why pure mathematics has advanced so far beyond its applications that it is regarded by many people as a useless species of mental gymnastics.

In spite of the enormous tax on the mind that mathematical work is commonly supposed to require, very few mathematicians are known to have been afflicted with mental disease. Cardano and Pascal were, undoubtedly, mentally unbalanced. Ampere, Fourier, John Bolyai, and a few others, were perhaps effected to such an extent as to make them not quite responsible for their actions. Monge, Möbius, and Huyghens were feeble-minded in their old age, and the latter is said to have recovered from a similar attack twenty years before. But there is no apparent evidence that their aberrations were due in any degree to mathematical activity, in fact, just in these oases we can trace the trouble to other causes. Newton is said not to have been quite in his right mind in his fifty-second year, six years after the publication of his last mathematical work, the Principia, to have suffered from nervousness and sleeplessness, and to have been temporarily out of his head; but he recovered and was, apparently, never so affected again; to be sure, he did no scientific work after this illness, unless e count the invention of the sextant, although he lived thirty-four years longer, wrote on prophesies and predictions, and held several important scientific and government offices. We may say that mathematicians, as a rule, have been long-lived. An enumeration of 300 mathematicians, physicists, and astronomers, taken at random showed the average length of life to be 65.6 years; but of a hundred of the most eminent, the average length of life was 72 years, 30 lived to be between 70 and 80, 27 between 80 and 90, and 6 between 90 and 100 years; but their reputation is not the result of longevity, depending, as it does, upon the quality and not the quantity of their work.

It is difficult to draw any satisfactory conclusion as to the frequency of the mathematical talent if we confine our attention to the great mathematicians; they, like those who have achieved distinction by any form of activity, constitute a very small part of the community. An estimate of the number of pupils in the higher classes of the several Gymnasien at Leipzig who were considered capable of pursuing mathematical studies further with advantage have 1 out of every 21 pupils. An enumeration of nearly 9000 articles in a biographical dictionary showed that about 2.3 percent of them had reference to mathematicians (including astronomers and mathematical physicists), that is, of those who attained sufficient distinction to be noticed in a biographical dictionary 2.8 per cent had a certain amount of mathematical ability; but it must be horn in mind, in drawing any conclusion from this, that many an article referred to several persons and, on the other hand, that many princely names occurred that would not have been cited if personal merit alone had been the basis of consideration.

Whether phrenology is of any scientific value in general or not, it is certain that a very large number of mathematicians show a physiognomic peculiarity that is striking. Gall, the originator of phrenology (Spurzheim was his friend and assistant), describes it thus: "the outer part of the roof of the orbita is depressed in such a manner that the upper edge of the hollow of the eve retains its natural arch only on the inner half, and that the outer half becomes a straight line, which runs sloping downward to its outer end. In consequence of this, the outer part of the upper lid falls and covers the eye, more than usually. Still more decided is the character when the lateral part of the orbita is pushed outward, so that the angular process of the frontal bone (the outer upper angle of the edge of the roof) projects sidewise over the front part of the temple." I know a great part of the living mathematicians personally I have studied the busts, and painted and engraved portraits of many others. In all, without exception, I have found the organ described. Consider the portrait of young Colburn; in him the outer part of the orbita is depressed and pushed out in such a way that this peculiarity did not escape the authors of the first notices about the young man in the American newspapers. Consider the portraits of Kepler, Newton, Leibnitz, Huyghens, Descartes, Euler, Roberval, Bernoulli, Lagrange, de la Place, Lalande, Herschell, Olbers, Bessel, Monge, Carnot, etc. P. J. Möbius, who has recently published a book on The Talent for Mathematics, says "Gall's statements about the mathematical organ are entirely correct, but I have three remarks to make: 1. Nature varies the forms more than might appear from Gall's description. 2. The mathematical organ is not equally developed on both sides, but as a rule more strongly on the left. 3. The mathematical organ consists in part of a thickening of the soft parts." Referring to Gall's description, Möbius continues "In fact, we find these modifications in almost all great mathematicians. However, sometimes the ridge running downward and backward from the lateral end of the eyebrows is more prominent and sometimes the slope of the outer edge of the orbita. Often the eye stands normal Iv open, often the outer half of the lid hangs down so that the eye is half closed and seems to slant. If we observe the face from in front, it also strikes us that the distance between the outer angle of the eye and the contour of the face is strikingly great. Sometimes the space between the outer angle of the eye and the end of the eyebrow is particularly broad, the distance between the eye and the edge of the forehead magnified. In such cases there is usually neither a distinct ridge nor a slope; the origin is then, so to say, built upward. In individual cases the angular process "is uniformly broadened with indefinite outlines, in others it projects with a sharp edge. The description is inaccurate in respect to the variety of nature. A definition that should be in accord with all variations is hard to be found. It would be, perhaps, best to say the mathematical organ consists in an abnormal formation of the angular process that amounts to magnification of the space inclosed by the process. We can say that on no two heads is the organ formed in exactly the same way." "For the important fact that the mathematical organ is to be found prevalently on the left, I need only refer to portraits. In the case of only moderate mathematical ability a distinct modification is usually to be found only on the left. But at times there is scarcely a perceptible difference between the left and the right, for example, in Leibnitz. At times, again, the right as well as the left is abnormally formed, but the one is entirely different from the other, for example, in Gauss. Here, also, the variations are many and it is neither practicable nor advisable to describe all particular cases in words. "Even in considering portraits one comes to the thought that the prominences are to be referred in part to an abnormally strong development of the skin. If we examine the living, we can convince ourselves by feeling that a considerable hyperplasia of the soft part exists. We feel distinctly that the angular process is indeed uncommonly strongly developed, but also that the skin, more definitely the hypodermal tissue, is thickened. The skin with an abundant cushion of fat forms a loose sack laid around the angular process. Very frequently we see also strikingly heavy eyebrows

Indeed, biographers, who otherwise do not say much of bodily peculiarities, not infrequently emphasize the heaviness of the brows and the length of the brow-hairs. Of Encke, for instance, C. Bruhns says that he had eyebrows with long thick hairs and that he often himself joked about it, because he could perceive all possible irradiations."

I should like now to give a few sketches, which will illustrate in individual cases what I have said in general. I ought to say that I have chosen these cases, not on account of their peculiarities, but on account of the prominence of the subjects as mathematicians and because the material for the sketches was at hand.

Leonhard Euler was the son of a clergyman and mathematician (a pupil of Jacob Bernoulli, at all events) and received his instruction from his father. He early showed an inclination for mathematics and determined, against his father's wishes, to devote himself to it. At sixteen he received the master's degree on the basis of a Latin comparison of the philosophies of Newton and Descartes. He then, for a short time, gave himself up to theological and oriental studies but soon returned to mathematics. At 19 he was called to the Academy at St. Petersburg, but on his arrival there the next year he was assigned to duty in the navy as a lieutenant. He became Professor of physics at 28 and at 26 took Daniel Bernoulli's place in the Academy. At 28 he carried out for the Russian government in three days some calculations for which three months had been assigned, in consequence of which exertion he was taken sick and recovered only with the loss of one eye. At 34 he was called to Berlin, but 25 years later returned to St. Petersburg. Soon afterwards he had another severe illness, which ended with the loss of the other eye. Six years later an operation restored his sight, but he very soon lost it again forever. He had a wonderful memory; he could repeat the whole of the Aeneid and saw in his mind so plainly the book from which he learned it that he could give the first and last lines of every single page. In the last year of his life, to pass the time, he gave his four grandchildren instruction in arithmetic and geometry. When he came to the extraction of roots, in order that he might have suitable examples, he calculated, during one sleepless night, the six lowest powers of all numbers under 20, and repeated the results several days afterward without hesitation. He had a great power of picturing geometrical figures in his mind, turning them into different positions and subjecting them to modifications; and was fond of chess. A son and several academicians helped him in his work. He had a large slate table, about which he walked guiding himself by the edge; and on this the blind man made rough sketches of his ideas; when his assistants came, he elaborated these sketches, they wrote down his words and read to him what they had written. At the age of 76 he had attacks of dizziness and, while joking with a grandson, fell into unconsciousness with the words "I am dying"; in a few hours he had ceased to live. His scientific activity was prodigious; his printed works include 32 quarto and 13 octavo volumes of independent books, beside more than 700 articles, of which some are very extensive. He was married at 26, and had 18 children, of whom, however, only three sons and two daughters were alive when he was 59. His sons, although capable men, had nothing like his ability.

Alexis Claude Clairaut was the second of 21 children of a mathematical teacher in Paris. At the age of 12 years he presented to the Academy a memoir that was highly praised by the referees, and at 16 wrote a long article of considerable importance. He died at the age of 52.

Johann Heinrich Lambert was the son of a poor tailor with many children. At 12 years of age he was a tailor, and later successively assistant city-clerk, book-keeper in an iron-foundry, private secretary, and tutor. His scientific activity apparently began when he was nearly 30, that is, very late, for a mathematician, and he died at 49.

Joseph Fourier was the son of a tailor. Up to the age of 13 he was, like other boys, fond of play; then suddenly, as he began the study of mathematics, he became an earnest student and collected candle-ends wherever he could find them to light his midnight studies. At first he wished to become an artillery-man but, being rejected, he joined the Benedictine monks as a novice. At 21, in consequence of the French revolution, he became a teacher of mathematics, first at Auxerres and then at Paris. He was a politician, went to Egypt with Napoleon, was in the diplomatic service, prefect at Grenoble, and Academician at Paris. In the latter part of his life, he kept his room very warm, had it heated even in summer, on account of rheumatism, and in the hottest weather dressed as even no traveler would who was condemned to spend the winter in the polar ice. He died at 62 of aneurism of the aorta.

Thomas Young was the son of Quakers. At 2 years of age he read fluently, at 4 he was acquainted with an enormous number of English writers and knew several Latin poems by heart. At 6 he first received regular instruction. At 8 he used to walk with a surveyor, heard him talk about his measurements, and read the subject up in a mathematical dictionary; from this time on he carried a quadrant with him in his walks, and calculated evenings the heights he had measured by day. From 9 to 14 he devoted himself to Latin, Greek, French, Italian, Hebrew, Persian, and Arabic. Once, when asked to give a specimen of his calligraphy by writing some sentences, he took a very long time to do it; but when urged to show what he had written, he produced the sentences in 9 languages. He studied also botany and made himself a microscope; that he might know how to grind the lenses properly, he studied the differential calculus. He was a physician by profession, but wrote on mathematics, physics, physiology, hieroglyphs, medicine, philosophy, grammar, technology, astronomy, music, painting, etc. His maxim was that every man could do everything; to prove it, he became rope-dancer, and circus rider. He was not successful as a physician; he gave up practice and became secretary of the Bureau of Longitudes. He died at the age of 56 of ossification of the aorta.

Gaspard Monge was the son of a small peddler. At school he always took first prizes in everything. At 14 he built a fire-engine; at 16 he drafted a plan of the city of Beaune, discovering methods of observation and constructing instruments for measuring angles, etc., and in the same year became Professor of Physics at Lyons. Afterward he attended the Artillery School at Mezieres, where he became Professor at 22. During this time he invented descriptive geometry. At 34 he was called to Paris and entered the Academy. The rest of his life is a romance; at 48 he was Minister of Marine, he helped to found the Ecole Polytecnique, was an intimate friend of Napoleon, whom he accompanied to Italy, Egypt, and Syria; and distinguished himself as a hero. On his return to France, he was made Senatos and Comte da Péluze. At 69 he was persecuted by the Restoration and driven out of the Academy. He became feeble-minded and "black darkness settled upon the noble and luminous intellect whose brilliancy all Europe bad admired." As a last resort to waken his sleeping mind they played the Marseillaise, but he remained perfectly indifferent and died at the age of 72. He was passionate, noble-minded, unselfish, and very warm-hearted. He loved tenderly his pupils, his family, the Emperor; he was very fond of children. His face was unusually broad, his eyes were deep-set and almost disappeared under bushy eyebrows.

André Maríe Ampère was the son of a merchant. Before he knew the numbers, he calculated with stones and beans. As a child he read 20 volumes of a large encyclopaedia, and even in the latter part of his life could repeat the long passages of it. At 14, having asked a book-seller for the works of Bernoulli and Euler and being told that they were in Latin, he learned Latin and read the works. As a boy he invented a universal language. When his father was beheaded, during the Revolution, he fell into a stupor which lasted a year. He was nervous, sentimental, and irascible, but kind and simple; was vary absent-minded and very near-sighted. Late in life he learned to enjoy music and had a passion for songs, but found classical music tedious. In his youth he wrote verses. He had a great inspiration for philosophy and was attracted by everything wonderful, particularly by animal-magnetism. He was always troubled by doubts, religious and otherwise. In later life he didn't want to read any more, and writing was always a burden to him. He could not work sitting, but must walk up and down. At last he busied himself only with the new classification of sciences. He died of lung-trouble at the age of 61. Just before his death, someone began to read to him from the Imitation of Christ; Ampère said he might stop, he knew the book by heart.

Karl Friedrich Gauss was the son of a brick-layer, afterwards gardener; his mother was a stone-cutter's daughter. He learned to read without instruction, asking the different members of the household the significance of the letters. He said himself that he learned to calculate before he could talk. At the age of three years he discovered errors in his father's accounts. He went to school at 7, began the study of arithmetic at 9, and entered the University at 18. He was at first undecided between philosophy and mathematics, but chose the latter. At 19 he discovered that a regular polygon of 17 sides can be inscribed in a circle geometrically, and at 24 published his Disquisitiones Arithmeticae, which made him famous. He was good at languages while in the Gymnasium, and remarkably so later. At 64 he studied Sanskrit and afterwards Russian. He died of heart disease at the age of 78.

Sir William Rowan Hamilton was born of Scotch parents in Dublin. His early education was carried on at home; under the influence of an uncle who was a good linguist he first devoted himself to linguistic studies; by the time he was 7 he could read Latin, Greek, French, and German with facility; and when 13 he was able to boast that be was familiar with as many languages as he had lived years, including Sanskrit and other Oriental languages. About this time he carne across a copy of Newton's Universal Arithmetic (a treatise on Algebra). This was his introduction to modern analysis; and be soon mastered the elements of analytic geometry and the calculus. He then read Newton's Principia and the four volumes then published of Laplace's Mécanique Celeste. In the latter he detected a mistake, and his paper on the subject written when he was 18 placed him at once in the front rank of mathematicians. In the following year be entered at Trinity College, Dublin. His university career is unique, for the chair of astronomy becoming vacant three years later, while he was yet an undergraduate, he was asked by the electors to stand for it, and was elected unanimously, it being understood that he should be left free to pursue his own line of study. He thus became Astronomer Royal of Ireland at the age of 22. He was extraordinarily industrious; he frequently spoke of working for twelve consecutive hours at mathematical research, and when immersed in these trances of discovery forgot his regular meal hours. His relaxations, such as they were, may be conjectured from a paragraph of a letter written the year before his death; he wrote: "The fact is that one of my early tastes was for metaphysics, and something has lately occurred to revive it. Another was for Eastern languages, and I chanced yesterday to light on the first sheet of a 'Persian grammar' written by myself forty years ago. These things, with others, may occasionally relax the bow–'non semper tendit' but 'many tastes one power', and my only power is mathematics." The study and writing of poetry were favorite recreations with him. What will probably be regarded as his greatest work is the invention of quaternions, which he first gave to the public in a volume published when he was 47. During the publication of this book he wrote to De Morgan, "I have just been correcting the slip which will bring it somewhat beyond 440 octavo pages. I first aimed at 200, but shall now congratulate myself if I get off under 500 pages." It was ultimately 888 pages. He was painfully fastidious on what he published, and left behind him an enormous bulk of manuscripts, of which not less than sixty great volumes have been deposited in the library of Trinity College, Dublin. There are also many other papers unpublished, as, for instance, a stupendous letter to Dr. Hart, which contained about 240 large folio pages and a postscript of 60 additional. He died at the age of 60 of a complication of ailments, which included gout and bronchitis. In one of his papers he predicted the phenomenon of conical retraction from purely theoretical investigations.

Of the women who have acquired reputations as mathematicians I find only four who are worthy of notice here. Hypatia, daughter of Theon of Alexandria, is said to have written on mathematical subjects, but none of her works have come down to us. Maria Gaetana Agnesi wrote a mathematical textbook at the age of 30 and was then appointed Professor of mathematics at the University of Bologna; she resigned this position, however, became a nun, and died at the age of 81. Sophie Germain, probably the greatest of the four, wrote several papers on mathematical physics and one on a pure mathematical topic, all, apparently, after she was 40. Her earliest published paper won the prize instituted, by Napoleon in consequence of Cladni's acoustical experiments. She was never married, was very peculiar, not leaving her room for yeas in succession, and died at the age of 55. Sonya Sophie Kavalevsky was the daughter of a General of artillery and granddaughter of an astronomer and mathematician. A somewhat peculiar uncle was fond of philosophizing and talking about mathematics although he knew nothing about it. This made an impression on the child and awakened in her the love of mathematics, as she said herself. The nursery was papered with the lithographed leaves of Ostrogradsky's lectures on the differential and integral calculus, which her father had heard as a young man. The child grubbed among the mysterious marks on the wall, learning by heart the formulae and fragments of the text. At 15 she received instruction in the differential calculus, and the teacher's explanations recalled to her mind the old formulae and words on the wall. Later she studied in Heidelberg and Berlin. For four years she pursued her studies under the direction of Weierstrass, and these studies had a decisive influence on her whole future scientific activity, which followed always the line indicated by him. "All her scientific work consisted in amplifications and developments of the theorems of the Master." At the age of 33 she was Privatdocent, and a year later Professor of analysis at Stockholm. One of her works secured a prize of the Paris Academy. For a long time she neglected mathematics altogether, and then could work on it only spasmodically. After the death of her husband she grew rapidly old, "her mind had lost its clearness," and she died at the age of 41.

I could multiply these sketches indefinitely, but I think I have said enough to give a general idea of the characteristics of the mathematical type.

I had hoped to show you the portraits of a number of mathematicians that illustrate what I have said about their physiognom peculiarities, but I have been unable to make the necessary arrangements. However, I have laid out in the Journal room of the University Library books containing a considerable number of such portraits, which you can examine at your leisure, if you are interested in such speculations, and I shall be very glad to give a demonstration of these portraits, if you will notify Mr. Wilson of your desire for it, stating the hour that would be most convenient for you.