The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.
Archimedes perished in the confusion of the sack while he was calmly pursuing his studies (Liv.
At 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.
This 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.
According 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.
Archimedes 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.
When Cicero was quaestor in Sicily (75 B.C.), he found the tomb of Archimedes, near the Agrigentine gate, overgrown with thorns and briers.
- 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.
This 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.
Lastly, Archimedes is credited with the famous Cattle-Problem enunciated in the epigram edited by G.
Lessing' in 1773, which purports to have been sent by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes.
- 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.
Archimedes 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.'
Following Archimedes, Fagnano desired the curve to be engraved on his tombstone.
Archimedes, the famous mathematician, had a celestial globe of glass, in the centre of which was a small terrestrial globe.
Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated.
Archimedes' 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.
His 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.
As 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.
As 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.
Like another Archimedes, he requested that the logarithmic spiral should be engraven on his tombstone, with these words, Eadem mutata resurgo.
If, 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.
The 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.
Although 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.
This subject was investigated by Archimedes, who, by his "method of exhaustions," derived the principal results.
Archimedes 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.
With Archimedes (287-212 B.C.) a notable advance was made.
The 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.
Further, in comparing the labours of Archimedes and Vieta, the effect of increased power of symbolical expression is very noticeable.
To compare it on this score with the fundamental proposition of Archimedes, the latter must be put into a form similar to Snell's.
II) 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< HK, but > 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).
He 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.
Archimedes 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.
At 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.
From the time of Archimedes there had existed a science of equilibrium, but the science of motion began with Galileo.
Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them.
280-264 B.C.), was described by Archimedes in his Arenarius, only to be set aside Astronomisches aus Babylon (Freiburg im Breisgau, 1889).
The 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.
The 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.
Incidentally 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.
Then, by the principle of Archimedes, W = Vwo; or wo = W/V.
Since 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.