It was the oldest and most important of the five Greek cities in the region and it gave eastern Libya the classical name Cyrenaica that it has retained to modern times. Cyrene lies in a valley in the Jebel Akhdar uplands.
The following article is in two parts: Life and Works; Transmission of the Elements. Life and Works Although Euclid Latinized as Euclides is the most celebrated mathematician of all time, whose name became a synonym for geometry until the twentieth century, 1 only two facts of his life are known, and even these are not beyond dispute.
One is that he was intermediate in date between the pupils of Plato d. Until recently most scholars would have been content to say that Euclid was older than Archimedes on the ground that Euclid, Elements I.
Since Alexander ordered the foundation of the town in b. This agrees with what Proclus says about Euclid in his commentary on the first book of the Elements. This man lived 6 in the time of the first Ptolemy; 7 for Archimedes, who followed closely upon the first [Ptolemy], makes mention of Euclid, 8 and further they say that Ptolemy once asked him if there were a shorter way to the study of geometry than the Elements, to which he replied that there was no royal road to geometry.
It is clear, however, that Proclus, writing over six centuries later, had no independent knowledge, obviously relying upon Archimedes for his lower date. The story about the royal road is similar to a tale that Stobaeus tells about Menaechmus and Alexander.
The testimony of so zealous a Neoplatonist as Proclus is not, however, necessarily conclusive on this point. Autolycus was the teacher of Arcesilaus, who was born about b. It would be reasonable to suppose that Autolycus was at the height of his activities about b.
It is highly probable, however, quite apart from what Proclus says about his Platonism, that he attended the Academy, for Athens was the great center of mathematical studies at the time; and there he would have become acquainted with the highly original work of Eudoxus and Theaetetus.
He was probably invited to Alexandria when Demetrius of Phalerum, at the direction of Ptolemy Soter, was setting up the great library and museum. This was shortly after b. It is possible this had already been established by one or more books, but the only piece of internal or external evidence about the order in which Euclid wrote his works is that the Optics preceded the Phenomena because it is cited in the preface of the latter.
Euclid must be regarded as the founder of the great school of mathematics at Alexandria, which was unrivaled in antiquity. It was presumably at Alexandria, according to a story by Stobaeus, 17 that someone who had begun to learn geometry with Euclid asked him, after the first theorem, what he got out of such things.
A date about b. In the fourth century Theon of Alexandria reedited it, altering the language in some places with a view to greater clarity, interpolating intermediate steps, and supplying alternative proofs, separate cases, and corollaries.
Each book of the Elements is divided into propositions, which may be theorems, in which it is sought to prove something, or problems, in which it is sought to do something.
In many cases some of these divisions may be missing particularly the definition or the construction because they are not needed, but the general enunciation, proof, and conclusion are always found.
These formal divisions of a proposition in such detail are special to Euclid, for Autolycus before him—the only pre-Euclidean author to have any work survive entire—had normally given only a general enunciation and proof, although occasionally a conclusion is found; and Archimedes after him frequently omitted the general or particular enunciation.
The Greek mathematicians carefully distinguished between the analytic and the synthetic methods of proving a proposition. This collection of treatises included three by Euclid: But in the Elements the demonstrations proceed entirely by synthesis, that is, from the known to the unknown, and nowhere is appeal made to analysis, that is, the assumption of the thing to be proved or done and the deduction of the consequences until we reach something already accepted or proved true.
Euclid does, however, make frequent use of reductio ad absurdum or demonstratio per impossibile, showing that if the conclusion is not accepted, absurd or impossible results follow; and this may be regarded as a form of analysis.
There are also many pairs of converse propositions, and either one in a pair could be regarded as a piece of analysis for the solution of the other. No hint is given by Euclid about the way in which he first realized the truth of the propositions that he proves.
Majestically he proceeds by rigorous logical steps from one proved proposition to another, using them like steppingstones, until the final goal is reached. Aristotle had taught that to define an object is not to assert its existence; this must be either proved or assumed.
To describe a circle with any center and radius. In other words, he assumes the existence of points, straight lines, and circles as the basic elements of his geometry, and with these assumptions he is able to prove the existence of every other figure that he defines.
For example, the existence of a square, defined in Idefinition 22, is proved in 1. These three postulates do rather more, however, than assume the existence of the things defined.
The first postulate implies that between any two points only one straight line can be drawn; and this is equivalent to saying that if two straight lines have the same extremities, they coincide throughout their length, or that two straight lines cannot enclose a space.
The latter statement is interpolated in some of the manuscripts. The second postulate implies that a straight line can be produced in only one direction at either end, that is, the produced part in either direction is unique, and two straight lines cannot have a common segment.
The third postulate also implies the infinitude of space because no limit is placed upon the radius; it further implies that space is continuous, not discrete, because the radius may be indefinitely small. The fourth and fifth postulates are of a different order because they do not state that something can be done.
In the fourth the following is postulated: All right angles are equal to one another. This implies that a right angle is a determinate magnitude, so that it serves as a norm by which other angles can be measured, but it is also equivalent to an assumption of the homogeneity of space.
For if the assertion could be proved, it could be proved only by moving one right angle to another so as to make them coincide, which is an assumption of the invariability of figures or the homogeneity of space.Euclid (/ ˈ juː k l ɪ d /; Ancient Greek: A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre.
Although the apparent citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those. One of the first details we read about Archimedes ( BCE) in almost every account of his life is the famous scene where he runs wet and naked through the streets of Syracuse shouting “Eureka!, Eureka!” (“I have found it!”).
This nudist episode, however, fails to capture the respect that the life of the greatest Greek mathematician and . Archimedes died in the siege of Syracuse in b.c., according to Tzetzes, at the age of seventy-five; if so, he was born in Karl R.
Popper, Conjectures and Refutations, and Refutatuons, 3rd ed. (London, ), p. , has revived it (“Euclid the Geometrician you don’t mean the man “Eucleides,” in Dictionary of. Archimedes of Syracuse (Greek: Ἀρχιμήδης; – BCE) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
He is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.
Archimedes of Syracuse was an outstanding ancient Greek mathematician, inventor, physicist, engineer and also an astronomer. Although not much is known about his life, he is considered as one of the most eminent scientists and mathematicians of the classical era. Heracleides (Archimedes’ friend) had written a biography on him but Place Of Birth: Syracuse.
Invention of Archimedes. The architonnere is a machine of fine copper, which throws balls with a loud report and great force. It is used in the following manner:—One-third of the instrument contains a large quantity of charcoal fire.