Mathematicians From Italy

"Cardan was the most distinguished astrologer of his time, and when he settled in Rome he received a pension in order to secure his services as astrologer to the papal court. This proved fatal to him for, having foretold that he should die on a particular date, he felt obliged to commit suicide in order to keep up his reputation—so at least the story runs."

"In 1570 he was imprisoned for heresy on account of his having published the horoscope of Christ, and when released he found himself... generally detested..."

"His career is an account of the most extraordinary and inconsistent acts. A gambler, if not a murderer, he was an ardent student of science, solving problems which had long baffled all investigation; at one time in his life he was devoted to intrigues which were a scandal even in the sixteenth century, at another he did nothing but rave on astrology, and yet at another he declared that philosophy was the only subject worthy of a man's attention. His was the genius that was closely allied to madness."

"Anticipations of Cardan are more truly wonderful when we consider that the symbolical language of algebra, that powerful instrument not only expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities besides the res or cosa, sometimes written shortly, for the assumed unknown number of an equation. But letters were not yet substituted for known quantities. Michael Stifel, in his Arithmetics Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers. It is very singular that discoveries of the greatest convenience, and apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so, that by dint of this acuteness they dispensed with the aid of these contrivances, in which we suppose that so much of the utility of algebraic expression consists."

"Jerome Cardan is... the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + px = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + px2 = q, and x3 - px2 = q. When the day of trial arrived, Tartaglia was able, not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and, four years afterwards, Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and, though he gave Tartaglia the credit of the discovery, revealed the process to the world."

"Every medieval and Renaissance court had a royal astrologer who advised the duke or prince he served. ...Men such as Roger Bacon, who even in the thirteenth century was a clear and outspoken champion of the experimental method in science, and Jerome Cardan, one of the foremost mathematicians and physicians of the sixteenth century, subscribed to astrology."

"It appears... from this short chapter [Ars Magna, lib x. ch. 1], that he had discovered most of the principal properties of the roots of equations, and could point out the number and nature of the roots, partly from the signs of the terms, and partly from the magnitude and relations of the co-efficients."

"You troubled mindes with tormentes loste that sighes and sobs consumes: (Who breathes and puffes from burning breast, both smothring smoke and fumes.) Come reade this booke that freelye bringes, a boxe of balme full swete, An oyle to noynt the brused partes, of everye heavye spirete. ...The lame whose lack of legges is death, unto a loftye mynde, Wyll kiss his crotche and creepe on knees,Cardanus workes to fynde."

"Cardano reasoned that the end of man is to know God and to mediate between the divine and the mortal. The is immortal and when permeated with is inseparable from God. True wisdom is gained from and by mathematics, as God has subjected the world to mathematical law."

"Although a long series of rules might be added and a long discourse given about them, we conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio refers to a line, quadratum to the surface, and cubum to a solid body, it would be very foolish for us to go beyond this point. Nature does no permit it."

"Since this art surpasses all human subtelty and the perspecuity of mortal talent and is truly a celestial gift and a very clear test of the capacity of man's minds, whoever applies himself to it will believe that there is nothing that he cannot understand."

"My father, in my earliest childhood, taught me the rudiments of arithmetic, and about that time made me acquainted with the arcana; whence he had come by this learning I know not. This was about my ninth year. Shortly after, he instructed me in the elements of the astronomy of Arabia, meanwhile trying to instill in me some system of theory for memorizing, for I had been poorly endowed with the ability to remember. After I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself. This is the knowledge I was able to acquire and learn without any elementary schooling..."

"I have not lost my faith; and this I must attribute more to a miracle than to my own wisdom; more to Divine Providence than to my own virtue. Steadfastly, in fact from my earliest childhood, I have made this my prayer, "Lord God... grant me long life, and wisdom, and health of mind and body.""

"What if one should address a word to the kings of the earth and say, "Not one of you but eats lice, flies, bugs, worms, fleas—nay the very filth of your servants! With what an attitude would they listen to such statements, though they be truths? What is this complacency then but an ignoring of conditions, a pretense of not being aware of what we know exists, or a will to set aside a fact by force? And so it is with everything else foul, vain, confused and untrue in our lives."

"This I recognize as unique and outstanding among my faults—the habit...of preferring to say above all things what I know to be displeasing to the ears of my hearers. ...I keep it up wilfully, in no way ignorant of how many enemies it makes for me. ...Yet I avoid this practice in the presence of my benefactors and of my superiors. It is enough not to fawn upon these, or at least not to flatter them."

"My personal affairs are not as highly esteemed as men commonly value their own interests—vain, empty affairs like those great clouds seen in the wake of the sunset which are meaningless and soon pass away."

"I have accustomed my features always to assume an expression quite contrary to my feelings; thus I am able to feign outwardly, yet within know nothing of dissumulation. This habit is easy if compared to the practice of hoping for nothing, which I have bent my efforts toward acquiring for fifteen successive years, and have at last succeeded."

"I am able to admit two distinct trains of thought to my mind at the same time."

"I am cold of heart, warm of brain, and given to never-ending meditation; I ponder over ideas, many and weighty, and even over things which can never come to pass."

"From these beginnings, as it were, have issued bitterness, contentious obstinancy, lack of amenity, hasty judgement, anger, and an intense desire for revenge—to say nothing of headstrong will; that which many damn, by word at least, was my delight."

"Among other myseries what I pray you tá be greater than whē a man riseth frō bed in the morning, to be incertaine of his returne to rest againe. or being in bed, whether his life shall continue tyll he ryse. besydes that, what labour, what hazard & care, are men constrained to abyde with these our brittle bodies, our feeble force, and incertayne lyfe: so as no nacion I thinke a man better or more fitlye named than the Spaniard, who in their language do terme a man shadow. And sure ther is nothing to be found of lesse assurance or soner passed than the lyfe of man, no... may more rightly be resembled to a shadow."

"So shall we voyd of all craft and sail, with true reason declare how much each man erreth in life, judgement, opinion, and will. Some things there are that so wel do prove themselves, as besides nature nede no profe at all."

"And wel we see ther is none alive that in every respect may be accompted happie, yea though mortall men were free from all calamities, yet the torments & feare of death should stil attend them But b:sides them, behold, what, and how manye evilles there bee, that unlesse the cloude of error bee removed, impossible it is to see the truth, or receive allay of our earthly woes."

"Better it is to have the worst, than none at all. for example we see, that houses are nedefull, such as can not possese & stately pallaces of stone, do persuade themselves to dwell in houses of timber and clap, and wanting them, are contented to inhabite the simple cotage, yea rather than not to be housed at all refuse not the pore cabbon, and most beggerly cave. So necessarie is this gift of consolacion, as there livith no man, but that hathe cause to embrace it. for in these things better is it to have any than none at al."

"The greatest advantage in gambling lies in not playing at all."

"Lumen aliquando per sui communicationem reddit obscuriorem superficiem corporis aliunde, ac prius illustratam."

"Lumen propagatur seu diffunditur non solum Directe, Refracte, ac Reflexe, sed etiam alio quodam quarto modo, Diffracte."

"The quest for our origin is the sweet fruit's juice which maintains satisfaction in the minds of the philosophers."

"The first published work describing the double entry accounting system and giving us insight into the logic behind the accounting entries is the Summa de Arithmetica, Geometria, Proportioni et Proportionalità... Repeatedly Pacioli stated that he described a system of bookkeeping that had been in use in Venice for more than 200 years, with the purpose of acquainting the merchants of his time with the method for keeping in good order their accounting books (chap. 1). Thus he did not mention things that were common practice a long time before 1494, as his treatise was a text for the untutored. Hence, Pacioli omitted most of the refinements common in practice of that day."

"Books should be closed each year, especially in partnership because frequent accounting makes for long friendship."

"The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square."

"1. Zero is a number. 2. The successor of any number is another number. 3. There are no two numbers with the same successor. 4. Zero is not the successor of a number. 5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers."

"1. 0 is a number. 2. The immediate successor of a number is also a number. 3. 0 is not the immediate successor of any number. 4. No two numbers have the same immediate successor. 5. Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers."

"These primitive propositions … suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. … All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions"

"Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define."

"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction."

"Certainly it is permitted to anyone to put forward whatever hypotheses he wishes, and to develop the logical consequences contained in those hypotheses. But in order that this work merit the name of Geometry, it is necessary that these hypotheses or postulates express the result of the more simple and elementary observations of physical figures."

"Quaestiones, quae ad mathematicae fundamenta pertinent, etsi hisce temporibus a multis tractatae, satisfacienti solutione et adhuc carent. Hic difficultas maxime en sermonis ambiguitate oritur. Quare summi interest verba ipsa, quibus utimur attente perpendere."

"He was a man I greatly admired from the moment I met him for the first time in 1900 at a Congress of Philosophy, which he dominated by the exactness of his mind."

"Bertrand Russell never wavered in acknowledging his intellectual debt to Giuseppe Peano. In many ways the contribution that Russell made to the foundations of mathematics, culminating in Principia Mathematica, strongly bears Peano's mark."

"I am fascinated by his gentle personality, his ability to attract lifelong disciples, his tolerance of human weakness, his perennial optimism. … Peano may not only be classified as a 19th century mathematician and logician, but because of his originality and influence, must be judged one of the great scientists of that century."

"Peano — whether in Logic or in Mathematics — never worked with pure symbolism — he always required that the primitive symbols introduced represent intuitive ideas to be explained with ordinary language."

"As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection."

"I cannot say whether I will still be doing geometry ten years from now. It also seems to me that the mine has maybe already become too deep and unless one finds new veins it might have to be abandoned. Physics and chemistry now offer a much more glowing richness and much easier exploitation. Also, the general taste has turned entirely in this direction, and it is not impossible that the place of Geometry in the Academies will someday become what the role of the Chairs of Arabic at the universities is now."

"Lagrange... was the first to draw sharply the line of demarcation between physics and metaphysics. The mechanical ideas of Descartes, Leibnitz, Maupertius, and even of Euler, had proved to be more or less hazy and unfruitful from a failure to separate those two distinct regions of thought. Lagrange put an end to this confusion, for no serious attempt has since been made to derive the laws of mechanics from a metaphysical basis."

"The value of his work [Mécanique Analytique] consists in the exposition of a general method by which every mechanical question may be stated in a single algebraic equation. The entire history of any mechanical system, as for example, the solar system, may thus be condensed into a single sentence; and its detailed interpretation becomes simply a question of algebra. No one who has not tried to cope with the difficulties presented by almost any mechanical problem can form a just appreciation of the great utility of such a labor-saving and thought-saving device. It has been well called 'a stupendous contribution to the economy of thought.'"

"The treatment of the kinetics of a material system by the method of generalised coordinates was first introduced by Lagrange, and has since his time been greatly developed by the investigations of different mathematicians. Independently of the highly interesting, although purely abstract science of theoretical dynamics which has resulted from these investigations, they have proved of great and continually increasing value in the application of mechanics to thermal, electrical and chemical theories, and the whole range of ."

"Full use of Lagrange's own made the unification of the varied principles of statistics and dynamics possible—in statistics by the use of the principle of virtual velocities, in dynamics by the use of . This led... to generalized coordinates and to the equation of motion in their "Lagrangian" form... Newton's geometrical approach was now fully discarded; Lagrange's book was a triumph of pure analysis."

"Analytic mechanics... was brought to the highest degree of perfection by Lagrange. Lagrange's aim is ('... 1788) to dispose, once and for all, of the reasoning necessary to resolve mechanical problems, by embodying as much as possible of it in a single formula. This he did. Every case... can now be dealt with by a very simple, highly symmetrical and perspicuous schema; and whatever reasoning is left is performed by purely mechanical methods. The mechanics of Lagrange is a stupendous contribution to the economy of thought."

"The questions here dealt with have occupied me since my earliest youth, when my interest for them was powerfully stimulated by the beautiful introductions of Lagrange to the chapters of his Analytic Mechanics..."