Archimedes

4 Key excerpts on "Archimedes"

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  eBook - ePub

  Infinite Powers

  How Calculus Reveals the Secrets of the Universe

  • Steven Strogatz(Author)
  • 2019(Publication Date)
  • Mariner Books

    (Publisher)

  BCE , Archimedes—by then an old man, around seventy—helped defend the city by using his knowledge of pulleys and levers to make fantastical weapons, “war engines” such as grappling hooks and giant cranes that could lift the Roman ships out of the sea and shake the sailors from them like sand being shaken out of a shoe. As Plutarch described the terrifying scene, “A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.”

  In a more serious vein, all students of science and engineering remember Archimedes for his principle of buoyancy (a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced) and his law of the lever (heavy objects placed on opposite sides of a lever will balance if and only if their weights are in inverse proportion to their distances from the fulcrum). Both of these ideas have countless practical applications. Archimedes’s principle of buoyancy explains why some objects float and others do not. It also underlies all of naval architecture, the theory of ship stability, and the design of oil-drilling platforms at sea. And you rely on his law of the lever, even if unknowingly, every time you use a nail clipper or a crowbar.

  Archimedes might have been a formidable maker of war machines, and he undoubtedly was a brilliant scientist and engineer, but what really puts him in the pantheon is what he did for mathematics. He paved the way for integral calculus. Its deepest ideas are plainly visible in his work, but then they aren’t seen again for almost two millennia. To say he was ahead of his time would be putting it mildly. Has anyone ever been more ahead of his time?

  Two strategies appear again and again in his work. The first was his ardent use of the Infinity Principle. To probe the mysteries of circles, spheres, and other curved shapes, he always approximated them with rectilinear shapes made of lots of straight, flat pieces, faceted like jewels. By imagining more and more pieces and making them smaller and smaller, he pushed his approximations ever closer to the truth, approaching exactitude in the limit of infinitely many pieces. This strategy demanded that he be a wizard with sums and puzzles, since he ended up having to add many numbers or pieces back together to arrive at his conclusions.

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  Archimedes

  • Heath, Thomas Little, Sir(Authors)
  • 2011(Publication Date)
  • Perlego

    (Publisher)

  PIONEERS OF PROGRESS MEN OF SCIENCE Edited by S. CHAPMAN, M.A., D.Sc., F.R.S. Archimedes BY

  Sir THOMAS HEATH K.C.B., K.C.V.O., F.R.S.; Sc.D., Camb. Hon. D.Sc., Oxford

  Δός μοι ποῦ στῶ, καὶ κινῶ τὴν γῆν

  LONDON: SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE NEW YORK: THE MACMILLAN CO. 1920

    CONTENTS.

  CHAP.	PAGE
  I.	Archimedes	1
  II.	Greek Geometry to Archimedes	7
  III.	The Works of Archimedes	24
  IV.	Geometry in Archimedes	29
  V.	The Sandreckoner	45
  VI.	Mechanics	50
  VII.	Hydrostatics	53
  Bibliography	57
  Chronology	58

    CHAPTER I. Archimedes.

  If the ordinary person were asked to say off-hand what he knew of Archimedes, he would probably, at the most, be able to quote one or other of the well-known stories about him: how, after discovering the solution of some problem in the bath, he was so overjoyed that he ran naked to his house, shouting εὕρηκα, εὕρηκα (or, as we might say, “I’ve got it, I’ve got it”); or how he said “Give me a place to stand on and I will move the earth”; or again how he was killed, at the capture of Syracuse in the Second Punic War, by a Roman soldier who resented being told to get away from a diagram drawn on the ground which he was studying.

  And it is to be feared that few who are not experts in the history of mathematics have any acquaintance with the details of the original discoveries in mathematics of the greatest mathematician of antiquity, perhaps the greatest mathematical genius that the world has ever seen.

  History and tradition know Archimedes almost exclusively as the inventor of a number of ingenious mechanical appliances, things which naturally appeal more to the popular imagination than the subtleties of pure mathematics.

  Almost all that is told of Archimedes reaches us through the accounts by Polybius and Plutarch of the siege of Syracuse by Marcellus. He perished in the sack of that city in 212 B.C. , and, as he was then an old man (perhaps 75 years old), he must have been born about 287 B.C. He was the son of Phidias, an astronomer, and was a friend and kinsman of King Hieron of Syracuse and his son Gelon. He spent some time at Alexandria studying with the successors of Euclid (Euclid who flourished about 300 B.C.

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  The Handy Math Answer Book

  • Patricia Barnes-Svarney, Thomas E Svarney(Authors)
  • 2012(Publication Date)
  • Visible Ink Press

    (Publisher)

  Arithmetica, he solved equations in several variables for integral solutions, or what we called diophantine equations today. (For more about these equations, see “Algebra.”) He also calculated negative numbers as solutions to some equations, but he considered such answers absurd. What were Archimedes’s greatest contributions to mathematics?

  Historians consider Archimedes (c. 287-212 B.C.E. , Hellenic) to be one of the greatest Greek mathematicians of the classic era. Known for his discovery of the hydrostatic principle, he also excelled in the mechanics of simple machines; computed close limits on the value of “pi” by comparing polygons inscribed in and circumscribed about a circle; worked out the formula to calculate the volume of a sphere and cylinder; and expanded on Eudoxus’ method of exhaustion that would eventually lead to integral calculus. He also created a way of expressing any natural number, no matter how large; this was something that was not possible with Greek numerals. (For more information about Archimedes, see “Mathematical Analysis” and “Geometry and Trigonometry.”)

  The Greek mathematician Euclid contributed to the development of arithmetic and the geometric theory of quadratic equations. What Greek mathematician made major contributions to geometry?

  The Greek mathematician Euclid (c. 325-c. 270 B.C.E. ), contributed to the development of arithmetic and the geometric theory of quadratic equations. Although little is known about his life— except that he taught in Alexandria, Egypt—his greatest contributions to geometry are well understood. The elementary geometry many of us learn in high school is still largely based on Euclid. His thirteen books of geometry and other mathematics, titled Elements (or Stoicheion

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  Education

  How Old The New

  • James J. (James Joseph) Walsh(Author)
  • 2011(Publication Date)
  • Perlego

    (Publisher)

  As the Roman vessels were very small, propelled only by oars or at least with very small sail capacity, and as their means of offence was most crude and they had to approach surely within 100 feet of the wall to be effective, the old story therefore is probably entirely true. The other phase of history according to which Archimedes succeeded in constructing instruments by which the Roman vessels were lifted bodily out of the water, is probably also true, and certainly comes with great credibility of the man of whom it is told that, after having studied the lever, he declared that if he only had {76} some place to rest his lever, he could move the world. The well-known story of his discovery in hydrostatics, by which he was enabled to tell the King whether the royal goldsmiths had made his crown of solid gold or not, is very well authenticated. Archimedes realized the application of the principle of specific gravity in the solution of such problems while he was taking a bath. Quite forgetful of his state of nudity he ran through the streets, crying "Eureka! Eureka! I have found it! I have found it!" There are many other significant developments of hydrostatics and mechanics, besides specific gravity and the lever, the germs of which are at least attributed to Archimedes. He seems to have been one of the world's great eminent practical geniuses. That he should have been a product of Alexandria and should even have been a professor there would be a great surprise if we did not know Alexandria as a great scientific university. As it is, it is quite easy to understand how naturally he finds his place in the history of that university and how proud any modern university would be to have on the rolls of its students and professors a man who not only developed pure science but who made a series of practical applications that are of great value to mankind

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