The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. The subsequent history may be divided into three periods... The first period is that... under Greek influence... the second is that of... the middle ages and the renaissance... the third is that of modern mathematics...
There are no absolutes... in mathematics or in its history.
In their lack of common mathematical curiosity, the algebraists of Islam and the European Renaissance were contemporaries of the ancient Egyptians. They wondered and were perplexed, of course; but there they stopped, because they lacked the Greek instinct for logical completeness and generality.
The fundamental definitions of the calculus, those of the derivative and integral, are now so clearly stated in textbooks on the subject... that it is easy to forget the difficulty with which these basic concepts have been developed.
The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition.
The derivative has throughout its development has been... precariously situated between the scientific phenomenon of velocity and the philosophical noumenon of motion.
Perhaps nowhere does one find a better example of the value of historical knowledge for mathematicians than in the case of Fermat, for it is safe to say that, had he not been intimately acquainted with the geometry of Apollonius and Viéte, he would not have invented analytic geometry.
Students enjoy... and gain in their understanding of today's mathematics through analyzing older and alternative approaches.
This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time.
The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science and that hardly anything ever done in mathematics has proved to be useless.
The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store.
The history of mathematics is important... as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development.
In mathematics the art of asking questions is more valuable than solving problems.
Chasles was... a contemporary of Steiner, Gauss, and Plücker, but he made the curious admission that he was unable to give proper weight to the contributions of German geometers owing to his ignorance of their language.
It has seemed to me for a number of years that a new book dealing with the history of the methods which men have employed in dealing with geometrical questions might be of use... My own inadequacy for the task has been abundantly evident to me, but it did not seem sufficient reason for not making the attempt.
In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy.
The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. ...it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics.
Pythagoras could not have been the discoverer of the relation, because... this property was known and used by scholars and artisans of Oriental lands thousands of years before Pythagoras... While deductive geometry is barely more than twenty-five hundred years old, empirical geometry is probably as old as civilization itself.
Mathematics, in an earlier view, is the science of space and quantity; in a later view, it is the science of pattern and deductive structure. Since the Greeks, mathematics is also the science of the infinite.
I would rather discover a single [geometrical] demonstration than become king of the Persians.
Like anything else, mathematics is created within the context of history, and it of interest to place Cardano's solution of the cubic two years after the publication of Copernicus's heliocentric theory and two years before the death of England's Henry VIII, or to emphasize the impact of the Restoration upon Cambridge University when a young scholar named Isaac Newton entered it in 1666.
Mathematics is the product of real, flesh-and-blood human beings whose lives may reflect the inspirational, the tragic, or the bizarre. ...Understanding something of the lives of these diverse individuals can only enhance an appreciation of their work.
In past centuries it was widely accepted that an understanding of, as well as a facility with numbers, is an essential part of an education. ...This book has been written with an intention of showing that numbers have been the centre of man's awareness of his surroundings since well before any times of which we have surviving records. It will show that numbers have provided an answer to man's cultural needs at least since any form of organized human society came into being.
The history of mathematics as an academic discipline is one of the youngest branches of the historical sciences. Even in Germany, where the history of science has a rather long tradition, only a few of the numerous universities have as yet established a chair—or even an institute—for the history of science, and positions expressly devoted to the history of mathematics are extremely rare.
Heinrich Wieleitner was one of the first professional historians of mathematics in Germany. ...He was particularly interested in Nicole Oresme and the latitude of forms, which he regarded as the forerunner of analytical geometry. Wieleitner was... according to Bortolotti, after the death of Tannery, Enestrom, and Zeuthen, ...the world's best historian of science.
The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries.
A student of history, who cares little for Greek or mathematics in particular, but who likes to watch how things grow, will be able to extract from these pages a notion of the whole history of mathematical science down to Newton's time...
Probably Greek logistic, or calculation, extended to more difficult operations... and... probably Greek arithmetic, or theory of numbers, owed much more to induction than is permitted to appear by its first and chief professors.
Mathematics is one of the most basic -- and most ancient -- types of knowledge. Yet the details of its historical development remain obscure to all but a few specialists.
Playfair... though he cannot condemn Cardan, seems to think Tartaglia rightly treated for concealing his discovery; and others have echoed this strain. Tartaglia himself says... that he meant to have divulged it ultimately; but, in that age, money as well as credit was to be got by keeping a secret: and those who censure him wholly forget that the solution of cubic equations was, in the actual state of algebra, perfectly devoid of any utility in the world.
Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since.
My main object has been to present a perfectly faithful reproduction of the treatises as they have come down to us, neither adding anything nor leaving out anything essential or important. The notes are for the most part intended to throw light on particular points in the text or to supply proofs of propositions assumed by Archimedes as known; sometimes I have thought it right to insert... notes designed to bring out the exact significance of those propositions...
The science of Germany and Italy and Spain and France; the science of the Arabs and the Hindus, and the beginnings of the science of the Egyptians and the Babylonians, all had a working part in the development of our modern science and, in particular, of arithmetic.
There is a familiar formula—perhaps the most compact and famous of all formulas—developed by Euler from a discovery of De Moivre: e + 1 0. ...It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making.
In the potentially democratic world which men of good will envision, the man in the street must be entitled to more mathematical stimulation than the puzzle column in the Sunday newspaper, an occasional profile of a Nobel prize winner, [or] an enigmatic summary of some recent discovery in applied mathematics...
The authors hope by publishing this work to demonstrate that the Arabs were not only transmitters of other cultures, but made their own significant contributions as well.
The mathematical genius can only carry on from the point which mathematical knowledge within his culture has already reached. Thus if Einstein had been born into a primitive tribe which was unable to count beyond three, life-long application to mathematics probably would not have carried him beyond the development of a decimal system based on fingers and toes.
Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations.
The solution of numerical cubic equations by intersecting conics was the greatest original contribution to algebra made by the Arabs. These solutions remained unknown to the Western world, and were rediscovered in the seventeenth century by Descartes, Thomas Baker, and Edmund Halley. The success of the Arab scholars in this field may have deterred them from trying methods of approximation
All things which can be known have number; for it is not possible that without number anything can either be conceived or known.
It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.
Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors.
For the circle is divisible into parts unlike in definition or notion, and so is each of the rectilineal figures; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike.
The field of mathematics is now so extensive that no one can [any] longer pretend to cover it, least of all the specialist in any one department. Furthermore it takes a century or more to weigh men and their discoveries, thus making the judgment of contemporaries often quite worthless.
The fact that arithmetic and geometry took such a notable step forward... was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B.C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales.
The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts.
I cannot satisfy myself that, when one is added to one, the one to which the addition is made becomes two, or that the two units added together make two by reason of the addition. I cannot understand how, when separated from the other, each of them was one and not two, and now, when they are brought together the mere juxtaposition or meeting of them should be the cause of their becoming two...
Some of the great treatises can be criticized for some startling omissions. Forsyth wrote six volumes on differential equations without a mention of Poincare's geometric theory of ordinary differential equations. Bourbaki has summarized the foundations of topology without a mention of Kuratowski, J. W. Alexander or Veblen.
The magnificent achievement of Bourbaki planned to present in an orderly sequence the whole of mathematics is already dated, and a new edition appears to need a complete revision based perhaps on category theory rather then on sets and logic.
The dull and pedestrian researches of Todhunter on the histories of probability, the calculus of variations and the theory of elasticity will always preserve their value for the historiographer.
The historical approach introduces us personally to the great mathematicians and infuses a humane and genial spirit into what can be the most arid and abstract study.
I cannot venture to suspect that in such a difficult subject I shall be quite free from error either in my exposition of the labours of others, or in my own contributions; but I hope that such errors will not be numerous nor important.
The relation between the jyā and the modern sine iswhere R is the radius of the base circle. ...we shall represent it with the now-standard Sin θ (the capital letter signifying that the function is R times the modern one.)
One of the most wholesome tendencies in the study of mathematics today is the desire to give increased attention to the history and genesis of the subject. This tendency has led to a more careful study of the works of the old Greek mathematicians Of these Pappus of Alexandria was among the last, and from the point of View of the historian one of the most important because it is in his works that we have the only authentic account of a large number of preceding mathematicians.
Mathematics as an Element in the History of Thought.
Both mathematics and the philosophy of mathematics stand to gain from an investigation of the evolution of mathematics. If it be true that 'In darkness dwells the people which knows its annals not,' it is equally true that the mathematician who ignores the evolutionary forces that have shaped his thinking, thereby loses a valuable perspective.
Much as the study of the evolution of a particular form of life can suggest patterns for more general forms, so can a study of a particular cultural item, such as mathematics, have significance for the general forms that cultural evolution takes.
Those people who do mathematics—the 'mathematicians'—are not only the possessors of the cultural element known as mathematics but, when taken as a group... can be considered as the bearers of a culture, in this case mathematics.
The evolution of number into the 'transfinite' was included only to emphasize the power of the forces acting within mathematics to compel this development—even against the philosophy of its most prominent creator, George Cantor (...numbers were extended, along with their arithmetic, to the non-finite, not as a mathematical whim, but for reasons of strong internal stresses.)
The evolution of number and geometry suffices to exhibit all the [cultural or anthropological] characteristics that are found in the development of more advanced mathematics.
The Authors who write near the beginnings of science, are, in general the most instructive: they take the reader more along with them, shew him the real difficulties, and, which is a main point, teach him the subject, the way by which they themselves learned it.