There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
Nikolai LobachevskyTag: science reality world mathematics abstract application scientist
It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.
Bernhard RiemannTag: science relationships darkness definitions philosophers space mathematics geometry euclid axioms scientist mathematician adrian-marie-legendre legendre renown
It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.
Adrien-Marie LegendreTag: science theories effort research regret proof scientist fermat lagrange leonhard-euler joseph-louis-lagrange number-theory pierre-de-fermat theorems
Et peut-être la posterité me saura gré de lui avoir fait connaître que les Anciens n’ont pas tout su. (And perhaps, posterity will thank me for having shown that the ancients did not know everything.)
Pierre de FermatTag: humor science knowledge funny posterity scientist ancients mathematician thank fermat-s-theorem
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 modem chemistry and modem 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.
James Joseph SylvesterTag: science art poetry language ideas scientists influence chemistry painting algebra tennyson matter masters browning toyal-academy
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.
Nicholas Murray ButlerTag: science reason philosophy philosophers scientists isaac-newton newton leibniz mathematics geometry descartes calculus sylvester supremacy gauss carl-friedrich-gauss gottfried-leibniz gottfried-wilhelm-leibniz analytical-geometry lobachevsky rene-descartes riemann
The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
W.W. Rouse BallTag: science style difficulty elegance scientists argument praise perfection proof interesting masters lagrange gauss laplace pierre-simon-laplace joseph-louis-lagrange carl-friedrich-gauss modern-analysis
Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Fermat, Wallis, and Barrow. It was Newton's good luck to come at a time when everything was ripe for the discovery, and his ability enabled him to construct almost at once a complete calculus.
W.W. Rouse BallTag: science history scientists discovery isaac-newton newton mathematics calculus fermat johannes-kepler kepler barrow pierre-de-fermat cavalieri infinitesimal napier wallis
Scientists still do not appear to understand sufficiently that all earth sciences must contribute evidence toward unveiling the state of our planet in earlier times, and that the truth of the matter can only be reached by combing all this evidence. ... It is only by combing the information furnished by all the earth sciences that we can hope to determine 'truth' here, that is to say, to find the picture that sets out all the known facts in the best arrangement and that therefore has the highest degree of probability. Further, we have to be prepared always for the possibility that each new discovery, no matter what science furnishes it, may modify the conclusions we draw.
Alfred WegenerTag: science truth understanding information scientists discovery evidence geology planet earth-sciences history-of-the-earth history-of-the-planet
The text-book is rare that stimulates its reader to ask, Why is this so? Or, How does this connect with what has been read elsewhere?
J. Norman CollieTag: science reading learning how questions why wonder curiosity discovery explorer scientist textbook mountaineer
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