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archimedes

The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.

105The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.

106Archimedes perished in the confusion of the sack while he was calmly pursuing his studies (Liv.

76Archimedes died at the capture of Syracuse by Marcellus, 212 B.C. In the general massacre which followed the fall of the city, Archimedes, while engaged in drawing a mathematical figure on the sand, was run through the body by a Roman soldier.

62As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its own volume of water; and he utilized it in the problem of the determination of the adulteration of the crown of Hiero.

32The theorem for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form I - cos 0 = 2 sin e 20.

33Although the heliocentric system is not mentioned in the treatise, a quotation in the Arenarius of Archimedes from a work of Aristarchus proves that he anticipated the great discovery of Copernicus.

21Since the area of a circle equals that of the rectilineal triangle whose base has the same length as the circumference and whose altitude equals the radius (Archimedes, KIKXou A ir, prop.i), it follows that, if a straight line could be drawn equal in length to the circumference, the required square could be found by an ordinary Euclidean construction; also, it is evident that, conversely, if a square equal in area to the circle could be obtained it would be possible to draw a straight line equal to the circumference.

21With Archimedes (287-212 B.C.) a notable advance was made.

21Further, in comparing the labours of Archimedes and Vieta, the effect of increased power of symbolical expression is very noticeable.

21To compare it on this score with the fundamental proposition of Archimedes, the latter must be put into a form similar to Snell's.

21II) whose centre is 0, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC AC. In modern trigonometrical notation the propositions to be compared stand as follows: 2 tan 20 >2 sin 28 (Archimedes); tan 10+2 sin 3B>0> 3 sin B (Snell).

21At Pavia in 1494 we find him taking up literary and grammatical studies, both in Latin and the vernacular; the former, no doubt, in order the more easily to read those among the ancients who had laboured in the fields that were his own, as Euclid, Galen, Celsus, Ptolemy, Pliny, Vitruvius and, above all, Archimedes; the latter with a growing hope of some day getting into proper form and order the mass of materials he was daily accumulating for treatises on all his manifold subjects of enquiry.

21Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them.

21Although the heliocentric system is not mentioned in the treatise, a quotation in the Arenarius of Archimedes from a work of Aristarchus proves that he anticipated the great discovery of Copernicus.

21At Pavia in 1494 we find him taking up literary and grammatical studies, both in Latin and the vernacular; the former, no doubt, in order the more easily to read those among the ancients who had laboured in the fields that were his own, as Euclid, Galen, Celsus, Ptolemy, Pliny, Vitruvius and, above all, Archimedes; the latter with a growing hope of some day getting into proper form and order the mass of materials he was daily accumulating for treatises on all his manifold subjects of enquiry.

21When Cicero was quaestor in Sicily (75 B.C.), he found the tomb of Archimedes, near the Agrigentine gate, overgrown with thorns and briers.

22Lessing' in 1773, which purports to have been sent by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes.

22If, therefore, the walls of the enclosure held the gas that is directly in contact with them, this equilibrium would be the actual state of affairs; and it would follow from the principle of Archimedes that, when extraneous forces such as gravity are not considered, the gas would exert no resultant force on any body immersed in it.

10The founder of the mathematical school was the celebrated Euclid (Eucleides); among its scholars were Archimedes; Apollonius of Perga, author of a treatise on Conic Sections; Eratosthenes, to whom we owe the first measurement of the earth; and Hipparchus, the founder of the epicyclical theory of the heavens, afterwards called the Ptolemaic system, from its most famous expositor, Claudius Ptolemaeus.

10This subject was investigated by Archimedes, who, by his "method of exhaustions," derived the principal results.

10The most important are :- Euclid's Elements; Euclid's Data; Optical Lectures, read in the public school of Cambridge; Thirteen Geometrical Lectures; The Works of Archimedes, the Four Books of Apollonius's Conic Sections, and Theodosius's Spherics, explained in a New Method; A Lecture, in which Archimedes' Theorems of the Sphere and Cylinder are investigated and briefly demonstrated; Mathematical Lectures, read in the public schools of the university of Cambridge.

10But to borrow Mr. Archimedes exclamation, Eureka!

11ARCHIMEDES (c. 287 -212 B.C.), Greek mathematician and inventor, was born at Syracuse, in Sicily.

11According to one story, Archimedes was puzzled till one day, as he was stepping into a bath and observed the water running over, it occurred to him that the excess of bulk occasioned by the introduction of alloy could be measured by putting the crown and an equal weight of gold separately into a vessel filled with water, and observing the difference of overflow.

11Archimedes maintained that each particle of a fluid mass, when in equilibrium, is equally pressed in every direction; and he inquired into the conditions according to which a solid body floating in a fluid should assume and preserve a position of equilibrium.

11As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR', or rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside FIG.

11Like another Archimedes, he requested that the logarithmic spiral should be engraven on his tombstone, with these words, Eadem mutata resurgo.

11Archimedes gave his results in the treatise IIepi Ti j c aOaipas Kai roD KUXLvbpov: he left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio.

11The third volume includes, however, some theological treatises, and the first part of it is occupied with editions of treatises on harmonics and other works of Greek geometers, some of them first editions from the MSS., and in general with Latin versions and notes (Ptolemy, Porphyrius, Briennius, Archimedes, Eutocius, Aristarchus and Pappus).

11pp. 313-339; Menge, Des Archimedes Kreismessung (Coblenz, 1874).

11quam Archimedes, we may direct our attention either to the infinite series of geometrical operations or to the corresponding infinite series of arithmetical operations.

11APOLLONIUS OF PERGA [PERGAEUS], Greek geometer of the Alexandrian school, was probably born some twenty-five years later than Archimedes, i.e.

11Then, by the principle of Archimedes, W = Vwo; or wo = W/V.

11Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.

11280-264 B.C.), was described by Archimedes in his Arenarius, only to be set aside Astronomisches aus Babylon (Freiburg im Breisgau, 1889).

11From the time of Archimedes there had existed a science of equilibrium, but the science of motion began with Galileo.

11Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces.

11He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential calculus.

11280-264 B.C.), was described by Archimedes in his Arenarius, only to be set aside Astronomisches aus Babylon (Freiburg im Breisgau, 1889).

11From the time of Archimedes there had existed a science of equilibrium, but the science of motion began with Galileo.

11Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces.

11At the Marchese's request he wrote, in 1588, a treatise on the centre of gravity in solids, which obtained for him, together with the title of "the Archimedes of his time," the honourable though not lucrative post of mathematical lecturer at the Pisan university.

12This has come down to us through a Latin version of an Arabic manuscript; it cannot, however, have been written by Archimedes in its present form, as his name is quoted in it more than once.

00Lastly, Archimedes is credited with the famous Cattle-Problem enunciated in the epigram edited by G.

00- The editio princeps of the works of Archimedes, with the commentary of Eutocius, is that printed at Basel, in 1544, in Greek and Latin, by Hervagius.

00Archimedes concluded from his measurements that the sun's diameter was greater than 27' and less than 32'; and even Tycho Brahe was so misled by his measures of the apparent diameters of the sun and moon as to conclude that a total eclipse of the sun was impossible.'

00Following Archimedes, Fagnano desired the curve to be engraved on his tombstone.

00He published the first Italian translation of Euclid (1543), and the earliest version from the Greek of some of the principal works of Archimedes (1543).

00Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated.

00Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio,was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin.

00His assault seawards was made mainly on Achradina,1 but the city was defended by a numerous soldiery and by what seems to have been still more formidable, the ingenious contrivances of Archimedes, whose engines dealt havoc among the Roman ships, and frustrated the attack on the fortifications on the northern slopes of Epipolae (Liv.

00This theorem is called generally the principle of Archimedes.

00Further, Copernicus could not have known of Aristarchus's doctrine, since Archimedes's work was not published till after Copernicus's death.

00These latter formulae are due to Archimedes.

00Archimedes >>

00Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a " rule" in words; in the 16th century it could be condensed into a " formula."

00Up to this point the credit of most that had been done may be set down to Archimedes.

00The problem he set himself was the exact converse of that of Archimedes.

00His first contribution 3 was a variation of the method of Archimedes.

0089, Citizen Eusebe Salverte calls attention to the poem "De Ponderibus et Mensuris" generally ascribed to Rhemnius Fannius Palaemon, and consequently 300 years older than Hypatia, in which the hydrometer is described and attributed to Archimedes.

00Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes (already mentioned in book i.

00The Bryozoa were also abundantinsomeregions (Polypora, Fenestella), including the remarkable form known as Archimedes.

00Considering his long life and reputation Aurispa produced little: Latin translations of the commentary of Hierocles on the golden verses of Pythagoras (1474) and of Philisci Consolatoria ad Ciceronem from Dio Cassius (not published till 1510); and, according to Gesner, a translation of the works of Archimedes.

00With Ricci's assistance, he rapidly mastered the elements of the science, and eventually extorted his father's reluctant permission to exchange Hippocrates and Galen for Euclid and Archimedes.

00on Archimedes).

00The first four books, of which the first three are dedicated to Eudemus, a pupil of Aristotle and author of the original Eudemian Summary, contain little that is original, and are principally based on the earlier works of Menaechmus, Aristaeus (probably a senior contemporary of Euclid, flourishing about a century later than Menaechmus), Euclid and Archimedes.

00But to borrow Mr. Archimedes exclamation, Eureka!

00For revenge, Archimedes devised a fiendish computational problem that involved truly immense numbers.

00Archimedes the numerical analyst Here we summarize the main points in the paper by Phillips with the above title.

00Stein has a sideline in history - his book " Archimedes: what did he do beside cry eureka?

00hit-and-miss approach to menu selection reminds me of the Acorn Archimedes.

00Archimedes (287-212 BC Greece) is reputed to have used powerful lodestones to pull the nails out of enemy ships thus sinking them.

00quadrature of the parabola Archimedes finds the area of a segment of a parabola cut off by any chord.

00ARCHIMEDES (c. 287 -212 B.C.), Greek mathematician and inventor, was born at Syracuse, in Sicily.

00This has been discredited because it is not mentioned by Polybius, Livy or Plutarch; but it is probable that Archimedes had constructed some such burning instrument, though the connexion of it with the destruction of the Roman fleet is more than doubtful.

00According to one story, Archimedes was puzzled till one day, as he was stepping into a bath and observed the water running over, it occurred to him that the excess of bulk occasioned by the introduction of alloy could be measured by putting the crown and an equal weight of gold separately into a vessel filled with water, and observing the difference of overflow.

00Archimedes died at the capture of Syracuse by Marcellus, 212 B.C. In the general massacre which followed the fall of the city, Archimedes, while engaged in drawing a mathematical figure on the sand, was run through the body by a Roman soldier.

00When Cicero was quaestor in Sicily (75 B.C.), he found the tomb of Archimedes, near the Agrigentine gate, overgrown with thorns and briers.

00- The range and importance of the scientific labours of Archimedes will be best understood from a brief account of those writings which have come down to us; and it need only be added that his greatest work was in geometry, where he so extended the method of exhaustion as originated by Eudoxus, and followed by Euclid, that it became in his hands, though purely geometrical in form, actually equivalent in several cases to integration, as expounded in the first chapters of our text-books on the integral calculus.

00Propositions I-II are preliminary, 13-20 contain tangential properties of the curve now known as the spiral of Archimedes, and 21-28 show how to express the area included between any portion of the curve and the radii vectores to its extremities.

00This has come down to us through a Latin version of an Arabic manuscript; it cannot, however, have been written by Archimedes in its present form, as his name is quoted in it more than once.

00Lastly, Archimedes is credited with the famous Cattle-Problem enunciated in the epigram edited by G.

00Lessing' in 1773, which purports to have been sent by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes.

00Of lost works by Archimedes we can identify the following: (I) investigations on polyhedra mentioned by Pappus; (2) Archai, Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) Peri zygon, On balances or levers; (4) Kentrobarika, On centres of gravity; (5) Katoptrika, an optical work from which Theon of Alexandria quotes a remark about refraction; (6) Ephodion, a Method, mentioned by Suidas; (7) Peri sphairopeoia, On Sphere-making, in which Archimedes explained the construction of the sphere which he made to imitate the motions of the sun, the moon and the five planets in the heavens.

00- The editio princeps of the works of Archimedes, with the commentary of Eutocius, is that printed at Basel, in 1544, in Greek and Latin, by Hervagius.

00Heath (The Works of Archimedes, Cambridge, 1897).

00On Archimedes himself, see Plutarch's Life of Marcellus.

00Screw Of Archimedes >>

00Archimedes concluded from his measurements that the sun's diameter was greater than 27' and less than 32'; and even Tycho Brahe was so misled by his measures of the apparent diameters of the sun and moon as to conclude that a total eclipse of the sun was impossible.'

00Following Archimedes, Fagnano desired the curve to be engraved on his tombstone.

00He published the first Italian translation of Euclid (1543), and the earliest version from the Greek of some of the principal works of Archimedes (1543).

00Archimedes, the famous mathematician, had a celestial globe of glass, in the centre of which was a small terrestrial globe.

00Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated.

00Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio,was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin.

00His assault seawards was made mainly on Achradina,1 but the city was defended by a numerous soldiery and by what seems to have been still more formidable, the ingenious contrivances of Archimedes, whose engines dealt havoc among the Roman ships, and frustrated the attack on the fortifications on the northern slopes of Epipolae (Liv.

00Archimedes perished in the confusion of the sack while he was calmly pursuing his studies (Liv.

00The fundamental principles of hydrostatics were first given by Archimedes in his work H€pi rwv o ovpEvwv, or De its quae vehuntur in humido, about 250 B.C., and were afterwards applied to experiments by Marino Ghetaldi (1566-1627) in his Promotus Archimedes (1603).

00Archimedes maintained that each particle of a fluid mass, when in equilibrium, is equally pressed in every direction; and he inquired into the conditions according to which a solid body floating in a fluid should assume and preserve a position of equilibrium.

00This theorem is called generally the principle of Archimedes.

00As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its own volume of water; and he utilized it in the problem of the determination of the adulteration of the crown of Hiero.

00As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR', or rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside FIG.

00- The principle of Archimedes in § 12 leads immediately to the conditions of equilibrium of a body supported freely in fluid, like a fish in water or a balloon in the air, or like a ship (fig.

00Like another Archimedes, he requested that the logarithmic spiral should be engraven on his tombstone, with these words, Eadem mutata resurgo.

00If, therefore, the walls of the enclosure held the gas that is directly in contact with them, this equilibrium would be the actual state of affairs; and it would follow from the principle of Archimedes that, when extraneous forces such as gravity are not considered, the gas would exert no resultant force on any body immersed in it.

00The founder of the mathematical school was the celebrated Euclid (Eucleides); among its scholars were Archimedes; Apollonius of Perga, author of a treatise on Conic Sections; Eratosthenes, to whom we owe the first measurement of the earth; and Hipparchus, the founder of the epicyclical theory of the heavens, afterwards called the Ptolemaic system, from its most famous expositor, Claudius Ptolemaeus.

00Further, Copernicus could not have known of Aristarchus's doctrine, since Archimedes's work was not published till after Copernicus's death.

00These latter formulae are due to Archimedes.

00This subject was investigated by Archimedes, who, by his "method of exhaustions," derived the principal results.

00Archimedes gave his results in the treatise IIepi Ti j c aOaipas Kai roD KUXLvbpov: he left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio.

00The third volume includes, however, some theological treatises, and the first part of it is occupied with editions of treatises on harmonics and other works of Greek geometers, some of them first editions from the MSS., and in general with Latin versions and notes (Ptolemy, Porphyrius, Briennius, Archimedes, Eutocius, Aristarchus and Pappus).

00Since the area of a circle equals that of the rectilineal triangle whose base has the same length as the circumference and whose altitude equals the radius (Archimedes, KIKXou A ir, prop.i), it follows that, if a straight line could be drawn equal in length to the circumference, the required square could be found by an ordinary Euclidean construction; also, it is evident that, conversely, if a square equal in area to the circle could be obtained it would be possible to draw a straight line equal to the circumference.

00Rectification and quadrature of the circle have thus been, since the time of Archimedes at least, practically identical problems. Again, since the circumferences of circles are proportional to their diameters - a proposition assumed to be true from the dawn almost of practical geometry - the rectification of the circle is seen to be transformable into finding the ratio of the circumference to the diameter.

00With Archimedes (287-212 B.C.) a notable advance was made.

00The conclusion from these therefore was that the ratio of circumference to diameter is 34 This is a most notable piece of work; the immature condition of arithmetic at the time was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever.5 No advance of any importance was made upon the achievement of Archimedes until after the revival of learning.

00pp. 313-339; Menge, Des Archimedes Kreismessung (Coblenz, 1874).

00The theorem for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form I - cos 0 = 2 sin e 20.

00Further, in comparing the labours of Archimedes and Vieta, the effect of increased power of symbolical expression is very noticeable.

00Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a " rule" in words; in the 16th century it could be condensed into a " formula."

00Up to this point the credit of most that had been done may be set down to Archimedes.

00To compare it on this score with the fundamental proposition of Archimedes, the latter must be put into a form similar to Snell's.

00II) whose centre is 0, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC AC. In modern trigonometrical notation the propositions to be compared stand as follows: 2 tan 20 >2 sin 28 (Archimedes); tan 10+2 sin 3B>0> 3 sin B (Snell).

00The problem he set himself was the exact converse of that of Archimedes.

00quam Archimedes, we may direct our attention either to the infinite series of geometrical operations or to the corresponding infinite series of arithmetical operations.

00His first contribution 3 was a variation of the method of Archimedes.

00APOLLONIUS OF PERGA [PERGAEUS], Greek geometer of the Alexandrian school, was probably born some twenty-five years later than Archimedes, i.e.

00(1) Ile /3 c Tou irvpiov, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Hepi On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) `H Ka06Xov lrpa-yµareta, perhaps a work on the general principles of mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) ' (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of 7r than the 37 and 3,4-A of Archimedes; (6) an arithmetical work (as to which see Pappus) on a system of expressing large numbers in language closer to that of common life than that of Archimedes' Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm.

0089, Citizen Eusebe Salverte calls attention to the poem "De Ponderibus et Mensuris" generally ascribed to Rhemnius Fannius Palaemon, and consequently 300 years older than Hypatia, in which the hydrometer is described and attributed to Archimedes.

00Then, by the principle of Archimedes, W = Vwo; or wo = W/V.

00Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes (already mentioned in book i.

00Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.

00The most important are :- Euclid's Elements; Euclid's Data; Optical Lectures, read in the public school of Cambridge; Thirteen Geometrical Lectures; The Works of Archimedes, the Four Books of Apollonius's Conic Sections, and Theodosius's Spherics, explained in a New Method; A Lecture, in which Archimedes' Theorems of the Sphere and Cylinder are investigated and briefly demonstrated; Mathematical Lectures, read in the public schools of the university of Cambridge.

00The Bryozoa were also abundantinsomeregions (Polypora, Fenestella), including the remarkable form known as Archimedes.

00The preface treats of Greek sciences, geometry, the discovery of specific gravity by Archimedes, and other discoveries of the Greeks, and of Romans of his time who have vied with the Greeks -- Lucretius in his poem De Rerum Natura, Cicero in rhetoric, and Varro in philology, as shown by his De Lingua Latina.

00Considering his long life and reputation Aurispa produced little: Latin translations of the commentary of Hierocles on the golden verses of Pythagoras (1474) and of Philisci Consolatoria ad Ciceronem from Dio Cassius (not published till 1510); and, according to Gesner, a translation of the works of Archimedes.

00Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them.

00With Ricci's assistance, he rapidly mastered the elements of the science, and eventually extorted his father's reluctant permission to exchange Hippocrates and Galen for Euclid and Archimedes.

00At the Marchese's request he wrote, in 1588, a treatise on the centre of gravity in solids, which obtained for him, together with the title of "the Archimedes of his time," the honourable though not lucrative post of mathematical lecturer at the Pisan university.

00on Archimedes).

00The first four books, of which the first three are dedicated to Eudemus, a pupil of Aristotle and author of the original Eudemian Summary, contain little that is original, and are principally based on the earlier works of Menaechmus, Aristaeus (probably a senior contemporary of Euclid, flourishing about a century later than Menaechmus), Euclid and Archimedes.

00He discovered a simpler method of quadrating parabolas than that of Archimedes, and a method of finding the greatest and the smallest ordinates of curved lines analogous to that of the then unknown differential calculus.

00In the Quadrature of the parabola Archimedes finds the area of a segment of a parabola cut off by any chord.

00

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