“Divide et impera” is as true in algebra as in statecraft; but no less true and even more fertile is the maxim “auge et impera”.The more to do or to prove, the easier the doing or the proof.

As Crystallography was born of a chance observation by Haüy of the cleavage-planes of a single fortunately fragile specimen, … so out of the slender study of the Norwich Spiral has sprung the vast and interminable Calculus of Cyclodes, which strikes such far-spreading and tenacious roots into the profoundest strata of denumeration, and, by this and the multitudinous and multifarious dependent theories which cluster around it, reminds one of the Scriptural comparison of the Kingdom of Heaven “to a grain of mustard-seed which a man took and cast into his garden, and it grew and waxed a great tree, and the fowls of the air lodged in the branches of it.”

As lightning clears the air of impalpable vapours, so an incisive paradox frees the human intelligence from the lethargic influence of latent and unsuspected assumptions. Paradox is the slayer of Prejudice.

As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.

But for the persistence of a student of this university in urging upon me his desire to study with me the modern algebra I should never have been led into this investigation; and the new facts and principles which I have discovered in regard to it (important facts, I believe), would, so far as I am concerned, have remained still hidden in the womb of time. In vain I represented to this inquisitive student that he would do better to take up some other subject lying less off the beaten track of study, such as the higher parts of the calculus or elliptic functions, or the theory of substitutions, or I wot not what besides. He stuck with perfect respectfulness, but with invincible pertinacity, to his point. He would have the new algebra (Heaven knows where he had heard about it, for it is almost unknown in this continent), that or nothing. I was obliged to yield, and what was the consequence? In trying to throw light upon an obscure explanation in our text-book, my brain took fire, I plunged with re-quickened zeal into a subject which I had for years abandoned, and found food for thoughts which have engaged my attention for a considerable time past, and will probably occupy all my powers of contemplation advantageously for several months to come.

Chemistry has the same quickening and suggestive influence upon the algebraist as a visit to the Royal Academy, or the old masters may be supposed to have on a Browning or a Tennyson. Indeed it seems to me that an exact homology exists between painting and poetry on the one hand and modern chemistry and modern algebra on the other. In poetry and algebra we have the pure idea elaborated and expressed through the vehicle of language, in painting and chemistry the idea enveloped in matter, depending in part on manual processes and the resources of art for its due manifestation.

Does it not seem as if Algebra had attained to the dignity of a fine art, in which the workman has a free hand to develop his conceptions, as in a musical theme or a subject for a painting? It has reached a point where every properly developed algebraical composition, like a skillful landscape, is expected to suggest the notion of an infinite distance lying beyond the limits of the canvas.

Geometry may sometimes appear to take the lead of analysis, but in fact precedes it only as a servant goes before his master to clear the path and light him on his way. The interval between the two is as wide as between empiricism and science, as between the understanding and the reason, or as between the finite and the infinite.

He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer. I believe that it is impossible to do this, and to study geometry as it admits of being studied and am conscious it can be taught, without finding the reason invigorated, the invention quickened, the sentiment of the orderly and beautiful awakened and enhanced, and reverence for truth, the foundation of all integrity of character, converted into a fixed principle of the mental and moral constitution, according to the old and expressive adage “abeunt studia in mores”.

I know, indeed, and can conceive of no pursuit so antagonistic to the cultivation of the oratorical faculty … as the study of Mathematics. An eloquent mathematician must, from the nature of things, ever remain as rare a phenomenon as a talking fish, and it is certain that the more anyone gives himself up to the study of oratorical effect the less will he find himself in a fit state to mathematicize.

I should rejoice to see … Euclid honourably shelved or buried “deeper than did ever plummet sound” out of the schoolboys’ reach; morphology introduced into the elements of algebra; projection, correlation, and motion accepted as aids to geometry; the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity, continuity, infinity, and familiarization with the doctrines of the imaginary and inconceivable.

I should rejoice to see mathematics taught with that life and animation which the presence and example of her young and buoyant sister [natural and experimental science] could not fail to impart, short roads preferred to long ones.

I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.

Induction and analogy are the special characteristics of modern mathematics, in which theorems have given place to theories and no truth is regarded otherwise than as a link in an infinite chain. “Omne exit in infinitum” is their favorite motto and accepted axiom.

It is said of Jacobi, that he attracted the particular attention and friendship of Böckh, the director of the philological seminary at Berlin, by the great talent he displayed for philology, and only at the end of two years’ study at the University, and after a severe mental struggle, was able to make his final choice in favor of mathematics.

It is the constant aim of the mathematician to reduce all his expressions to their lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language.

Let him [the author] be permitted also in all humility to add … that in consequence of the large arrears of algebraical and arithmetical speculations waiting in his mind their turn to be called into outward existence, he is driven to the alternative of leaving the fruits of his meditations to perish (as has been the fate of too many foregone theories, the still-born progeny of his brain, now forever resolved back again into the primordial matter of thought), or venturing to produce from time to time such imperfect sketches as the present, calculated to evoke the mental co-operation of his readers, in whom the algebraical instinct has been to some extent developed, rather than to satisfy the strict demands of rigorously systematic exposition.

Little could Plato have imagined, when, indulging his instinctive love of the true and beautiful for their own sakes, he entered upon these refined speculations and revelled in a world of his own creation, that he was writing the grammar of the language in which it would be demonstrated in after ages that the pages of the universe are written.

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence.

May not Music be described as the Mathematic of sense, Mathematic as Music of the reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, Music the dream, Mathematic the working life each to receive its consummation from the other when the human intelligence, elevated to its perfect type, shall shine forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz!

Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence”.

Mr. Cayley, of whom it may be so truly said, whether the matter he takes in hand be great or small, "nihil tetigit quod non ornavit," …

No mathematician now-a-days sets any store on the discovery of isolated theorems, except as affording hints of an unsuspected new sphere of thought, like meteorites detached from some undiscovered planetary orb of speculation.

Number, place, and combination … the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.

Perhaps I may without immodesty lay claim to the appellation of Mathematical Adam, as I believe that I have given more names (passed into general circulation) of the creatures of the mathematical reason than all the other mathematicians of the age combined.

Professor Cayley has since informed me that the theorem about whose origin I was in doubt, will be found in Schläfli’s De Eliminatione. This is not the first unconscious plagiarism I have been guilty of towards this eminent man whose friendship I am proud to claim. A more glaring case occurs in a note by me in the Comptes Rendus, on the twenty-seven straight lines of cubic surfaces, where I believe I have followed (like one walking in his sleep), down to the very nomenclature and notation, the substance of a portion of a paper inserted by Schlafli in the Mathematical Journal, which bears my name as one of the editors upon the face.

So long as a man remains a gregarious and sociable being, he cannot cut himself off from the gratification of the instinct of imparting what he is learning, of propagating through others the ideas and impressions seething in his own brain, without stunting and atrophying his moral nature and drying up the surest sources of his future intellectual replenishment.

Surely the claim of mathematics to take a place among the liberal arts must now be admitted as fully made good. Whether we look at the advances made in modern geometry, in modern integral calculus, or in modern algebra, in each of these three a free handling of the material employed is now possible, and an almost unlimited scope is left to the regulated play of fancy. It seems to me that the whole of aesthetic (so far as at present revealed) may be regarded as a scheme having four centres, which may be treated as the four apices of a tetrahedron, namely Epic, Music, Plastic, and Mathematic. There will be found a common plane to every three of these, outside of which lies the fourth; and through every two may be drawn a common axis opposite to the axis passing through the other two. So far is certain and demonstrable. I think it also possible that there is a centre of gravity to each set of three, and that the line joining each such centre with the outside apex will intersect in a common point the centre of gravity of the whole body of aesthetic; but what that centre is or must be I have not had time to think out.

The discovery of the conic sections, attributed to Plato, first threw open the higher species of form to the contemplation of geometers. But for this discovery, which was probably regarded in Plato’s tune and long after him, as the unprofitable amusement of a speculative brain, the whole course of practical philosophy of the present day, of the science of astronomy, of the theory of projectiles, of the art of navigation, might have run in a different channel; and the greatest discovery that has ever been made in the history of the world, the law of universal gravitation, with its innumerable direct and indirect consequences and applications to every department of human research and industry, might never to this hour have been elicited.

The object of pure Physic is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence.

The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation.

The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.

There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all three of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order.
Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.

There is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], … or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being.

This [the fact that the pursuit of mathematics brings into harmonious action all the faculties of the human mind] accounts for the extraordinary longevity of all the greatest masters of the Analytic art, the Dii Majores of the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of the conic sections, who made mathematics his study and delight, who called them the handles or aids to philosophy, the medicine of the soul, and is said never to have let a day go by without inventing some new theorems, lived to 82; Newton, the crown and glory of his race, to 85; Archimedes, the nearest akin, probably, to Newton in genius, was 75, and might have lived on to be 100, for aught we can guess to the contrary, when he was slain by the impatient and ill mannered sergeant, sent to bring him before the Roman general, in the full vigour of his faculties, and in the very act of working out a problem; Pythagoras, in whose school, I believe, the word mathematician (used, however, in a somewhat wider than its present sense) originated, the second founder of geometry, the inventor of the matchless theorem which goes by his name, the pre-cognizer of the undoubtedly mis-called Copernican theory, the discoverer of the regular solids and the musical canon who stands at the very apex of this pyramid of fame, (if we may credit the tradition) after spending 22 years studying in Egypt, and 12 in Babylon, opened school when 56 or 57 years old in Magna Græcia, married a young wife when past 60, and died, carrying on his work with energy unspent to the last, at the age of 99. The mathematician lives long and lives young; the wings of his soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life.

Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul.

We are told that “Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.” I think no statement could have been made more opposite to the facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activities of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world (to which the inner one in each individual man may, I think, be conceived to stand somewhat in the same relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of the imagination and invention.

What a glorious title, Nature, a veritable stroke of genius to have hit upon. It is more than a cosmos, more than a universe. It includes the seen as well as the unseen, the possible as well as the actual, Nature and Nature's God, mind and matter. I am lost in admiration of the effulgent blaze of ideas it calls forth.
[Commenting on the title of the journal.]

Would it sound too presumptuous to speak of perception as a quintessence of sensation, language (that is, communicable thought) of perception, mathematics of language? We should then have four terms differentiating from inorganic matter and from each other the Vegetable, Animal, Rational, and Super-sensual modes of existence.

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.

If you enquire about him [J.J. Sylvester], you will hear his genius universally recognized but his power of teaching will probably be said to be quite deficient. Now there is no man living who is more luminary in his language, to those who have the capacity to comprehend him than Sylvester, provided the hearer is in a lucid interval. But as the barn yard fowl cannot understand the flight of the eagle, so it is the eaglet only who will be nourished by his instruction.

Among your pupils, sooner or later, there must be one. who has a genius for geometry. He will be Sylvester’s special pupil—the one pupil who will derive from his master, knowledge and enthusiasm—and that one pupil will give more reputation to your institution than the ten thousand, who will complain of the obscurity of Sylvester, and for whom you will provide another class of teachers.

J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid “deeper than e’er plummet sounded” out of the schoolboy’s reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson’s Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson’s additions and keeping strictly to the original treatise.

Professor Sylvester’s first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modem algebra. The attempt to lecture on this subject led him into new investigations in quantics.

Sylvester was incapable of reading mathematics in a purely receptive way. Apparently a subject either fired in his brain a train of active and restless thought, or it would not retain his attention at all. To a man of such a temperament, it would have been peculiarly helpful to live in an atmosphere in which his human associations would have supplied the stimulus which he could not find in mere reading. The great modern work in the theory of functions and in allied disciplines, he never became acquainted with …
What would have been the effect if, in the prime of his powers, he had been surrounded by the influences which prevail in Berlin or in Gottingen? It may be confidently taken for granted that he would have done splendid work in those domains of analysis, which have furnished the laurels of the great mathematicians of Germany and France in the second half of the present century.

The flights of the imagination which occur to the pure mathematician are in general so much better described in his formulas than in words, that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting— … the only successful attempt to invest mathematical reasoning with a halo of glory—that made in this section by Prof. Sylvester—is known to a comparative few, …

[J.J.] Sylvester’s methods! He had none. “Three lectures will be delivered on a New Universal Algebra,” he would say; then, “The course must be extended to twelve.” It did last all the rest of that year. The following year the course was to be Substitutions-Théorie, by Netto. We all got the text. He lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. “I must give one lecture a week on those,” he said. He could not confine himself to the hour, nor to the one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again. Statements like the following were not unfrequent in his lectures: “I haven’t proved this, but I am as sure as I can be of anything that it must be so. From this it will follow, etc.” At the next lecture it turned out that what he was so sure of was false. Never mind, he kept on forever guessing and trying, and presently a wonderful discovery followed, then another and another. Afterward he would go back and work it all over again, and surprise us with all sorts of side lights. He then made another leap in the dark, more treasures were discovered, and so on forever.

He [Sylvester] had one remarkable peculiarity. He seldom remembered theorems, propositions, etc., but had always to deduce them when he wished to use them. In this he was the very antithesis of Cayley, who was thoroughly conversant with everything that had been done in every branch of mathematics.
I remember once submitting to Sylvester some investigations that I had been engaged on, and he immediately denied my first statement, saying that such a proposition had never been heard of, let alone proved. To his astonishment, I showed him a paper of his own in which he had proved the proposition; in fact, I believe the object of his paper had been the very proof which was so strange to him.

His [J.J. Sylvester’s] lectures were generally the result of his thought for the preceding day or two, and often were suggested by ideas that came to him while talking. The one great advantage that this method had for his students was that everything was fresh, and we saw, as it were, the very genesis of his ideas. One could not help being inspired by such teaching.

When Cayley had reached his most advanced generalizations he proceeded to establish them directly by some method or other, though he seldom gave the clue by which they had first been obtained: a proceeding which does not tend to make his papers easy reading. …
His literary style is direct, simple and clear. His legal training had an influence, not merely upon his mode of arrangement but also upon his expression; the result is that his papers are severe and present a curious contrast to the luxuriant enthusiasm which pervades so many of Sylvester’s papers. He used to prepare his work for publication as soon as he carried his investigations in any subject far enough for his immediate purpose. … A paper once written out was promptly sent for publication; this practice he maintained throughout life. … The consequence is that he has left few arrears of unfinished or unpublished papers; his work has been given by himself to the world.

A short, broad man of tremendous vitality, the physical type of Hereward, the last of the English, and his brother-in-arms, Winter, Sylvester’s capacious head was ever lost in the highest cloud-lands of pure mathematics. Often in the dead of night he would get his favorite pupil, that he might communicate the very last product of his creative thought. Everything he saw suggested to him something new in the higher algebra. This transmutation of everything into new mathematics was a revelation to those who knew him intimately. They began to do it themselves. His ease and fertility of invention proved a constant encouragement, while his contempt for provincial stupidities, such as the American hieroglyphics for π and e, which have even found their way into Webster’s Dictionary, made each young worker apply to himself the strictest tests.

To know him [Sylvester] was to know one of the historic figures of all time, one of the immortals; and when he was really moved to speak, his eloquence equalled his genius.

I can see him [Sylvester] now, with his white beard and few locks of gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his figures and formulae on the board, sometimes explaining as he wrote, while we, his listeners, caught the reflected sounds from the board. But stop, something is not right, he pauses, his hand goes to his forehead to help his thought, he goes over the work again, emphasizes the leading points, and finally discovers his difficulty. Perhaps it is some error in his figures, perhaps an oversight in the reasoning. Sometimes, however, the difficulty is not elucidated, and then there is not much to the rest of the lecture. But at the next lecture we would hear of some new discovery that was the outcome of that difficulty, and of some article for the Journal, which he had begun. If a text-book had been taken up at the beginning, with the intention of following it, that text-book was most likely doomed to oblivion for the rest of the term, or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind, in consequence of that first difficulty. Other difficulties would soon appear, so that no text-book could last more than half of the term. In this way his class listened to almost all of the work that subsequently appeared in the Journal. It seemed to be the quality of his mind that he must adhere to one subject. He would think about it, talk about it to his class, and finally write about it for the Journal. The merest accident might start him, but once started, every moment, every thought was given to it, and, as much as possible, he read what others had done in the same direction; but this last seemed to be his real point; he could not read without finding difficulties in the way of understanding the author. Thus, often his own work reproduced what had been done by others, and he did not find it out until too late.
A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.
This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.

Sylvester’s writings are flowery and eloquent. He was able to make the dullest subject bright, fresh and interesting. His enthusiasm is evident in every line. He would get quite close up to his subject, so that everything else looked small in comparison, and for the time would think and make others think that the world contained no finer matter for contemplation. His handwriting was bad, and a trouble to his printers. His papers were finished with difficulty. No sooner was the manuscript in the editor’s hands than alterations, corrections, ameliorations and generalizations would suggest themselves to his mind, and every post would carry further directions to the editors and printers.

During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.

If we survey the mathematical works of Sylvester, we recognize indeed a considerable abundance, but in contradistinction to Cayley—not a versatility toward separate fields, but, with few exceptions—a confinement to arithmetic-algebraic branches. …
The concept of Function of a continuous variable, the fundamental concept of modern mathematics, plays no role, is indeed scarcely mentioned in the entire work of Sylvester—Sylvester was combinatorist [combinatoriker].

The enthusiasm of Sylvester for his own work, which manifests itself here as always, indicates one of his characteristic qualities: a high degree of subjectivity in his productions and publications. Sylvester was so fully possessed by the matter which for the time being engaged his attention, that it appeared to him and was designated by him as the summit of all that is important, remarkable and full of future promise. It would excite his phantasy and power of imagination in even a greater measure than his power of reflection, so much so that he could never marshal the ability to master his subject-matter, much less to present it in an orderly manner.
Considering that he was also somewhat of a poet, it will be easier to overlook the poetic flights which pervade his writing, often bombastic, sometimes furnishing apt illustrations; more damaging is the complete lack of form and orderliness of his publications and their sketchlike character, … which must be accredited at least as much to lack of objectivity as to a superfluity of ideas. Again, the text is permeated with associated emotional expressions, bizarre utterances and paradoxes and is everywhere accompanied by notes, which constitute an essential part of Sylvester’s method of presentation, embodying relations, whether proximate or remote, which momentarily suggested themselves. These notes, full of inspiration and occasional flashes of genius, are the more stimulating owing to their incompleteness. But none of his works manifest a desire to penetrate the subject from all sides and to allow it to mature; each mere surmise, conceptions which arose during publication, immature thoughts and even errors were ushered into publicity at the moment of their inception, with utmost carelessness, and always with complete unfamiliarity of the literature of the subject. Nowhere is there the least trace of self-criticism. No one can be expected to read the treatises entire, for in the form in which they are available they fail to give a clear view of the matter under contemplation.
Sylvester’s was not a harmoniously gifted or well-balanced mind, but rather an instinctively active and creative mind, free from egotism. His reasoning moved in generalizations, was frequently influenced by analysis and at times was guided even by mystical numerical relations. His reasoning consists less frequently of pure intelligible conclusions than of inductions, or rather conjectures incited by individual observations and verifications. In this he was guided by an algebraic sense, developed through long occupation with processes of forms, and this led him luckily to general fundamental truths which in some instances remain veiled. His lack of system is here offset by the advantage of freedom from purely mechanical logical activity.
The exponents of his essential characteristics are an intuitive talent and a faculty of invention to which we owe a series of ideas of lasting value and bearing the germs of fruitful methods. To no one more fittingly than to Sylvester can be applied one of the mottos of the Philosophic Magazine:
“Admiratio generat quaestionem, quaestio investigationem investigatio inventionem.”

Weierstrass related … that he followed Sylvester’s papers on the theory of algebraic forms very attentively until Sylvester began to employ Hebrew characters. That was more than he could stand and after that he quit him.