# Greek Mathematical Works: Volume II, From Arist... [WORK]

The wonderful achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I (Loeb Classical Library no. 335) contains: The divisions of mathematics; mathematics in Greek education; calculation; arithmetical notation and operations, including square root and cube root; Pythagorean arithmetic, including properties of numbers; square root of 2; proportion and means; algebraic equations; Proclus; Thales; Pythagorean geometry; Democritus; Hippocrates of Chios; duplicating the cube and squaring the circle; trisecting angles; Theaetetus; Plato; Eudoxus of Cnidus (pyramid, cone); Aristotle (the infinite, the lever); Euclid.

## Greek Mathematical Works: Volume II, From Arist...

The wonderful achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I contains: The divisions of mathematics; mathematics in Greek education; calculation; arithmetical notation and operations, including square root and cube root; Pythagorean arithmetic, including properties of numbers; square root of 2; proportion and means; algebraic equations; Proclus; Thales; Pythagorean geometry; Democritus; Hippocrates of Chios; duplicating the cube and squaring the circle; trisecting angles; Theaetetus; Plato; Eudoxus of Cnidus (pyramid, cone, etc.); Aristotle (the infinite, the lever); Euclid.

Commentators on Aristotle from the 2nd century on tended tointerpret Aristotle's mathematical objects as mental objects, whichmade Aristotle more compatible with neo-Platonism. Later themechanistic movement in the late Renaissance treated Aristotle asdivorcing mathematics from physical sciences in order to drive a deeperwedge between their views and his. Because of this, it has been veryeasy to discount Aristotle as subscribing to a version of psychologismin mathematics. These tendencies contribute to the common view thatAristotle's views mathematics are marginal to his thought. Morerecently, however, some sympathetic readers have seen Aristotle asexpressing a fictionalist version of physicalism, the view that theobjects of mathematics are fictional entities grounded in physicalobjects. To the extent that this view is regarded as a plausible viewabout mathematics, Aristotle has regained his position.

Aristotle's discussions on the best format for a deductive science inthe Posterior Analytics reflect the practice of contemporarymathematics as taught and practiced in Plato's Academy, discussionsthere about the nature of mathematical sciences, and Aristotle's owndiscoveries in logic. Aristotle has two separate concerns. One evolvesfrom his argument that there must be first, unprovable principles forany science, in order to avoid both circularity and infinite regresses.The other evolves from his view that demonstrations must beexplanatory. (See subsections A, B, and C of 6, Demonstrationsand Demonstrative Sciences, of the entry Aristotle's logic.)

To solve the problems of separation and precision, contemporaryphilosophers such as Speusippus and possibly Plato posited a universeof mathematical entities which are perfect instances of mathematicalproperties, adequately multiple for any theorem we wish to prove, andseparate from the physical or perceptible world. Aristotle calls themmathematicals or intermediates,because they are intermediate between the Forms and physical objects,in as much as they are perfect, eternal, and unchanging like the Forms,but multiple like physical objects (cf., for example, Met. i.6987b14-18, iii 2, xiii.1-2). This solution is the ancestor of manyversions of platonism in mathematics.

Perceptible magnitudes have perceptible matter. A bronze sphere is aperceptible magnitude. For solving the plurality problem, Aristotleneeds to have many triangles with the same form. Since perceptiblematter is not part of the object considered (in abstraction orremoval), he needs to have a notion of matter which is the matter ofthe object: bronze sphere MINUS bronze (perceptible matter). Since thisobject must be a composite individual to distinguish it from otherindividuals with the same form, it will have matter. He calls suchmatter intelligible or mathematical matter. Aristotle has at least fourdifferent conceptions of intelligible matter in the middle books of theMetaphysics, Physics iv, and De anima i:

Most of the mathematical texts written in Greek survived through the copying of manuscripts over the centuries, though some fragments dating from antiquity have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.[27]

Netz has counted 144 ancient authors in the mathematical or exact sciences, from whom only 29 works are extant in Greek: Aristarchus, Autolycus, Philo of Byzantium, Biton, Apollonius, Archimedes, Euclid, Theodosius, Hypsicles, Athenaeus, Geminus, Hero, Apollodorus, Theon of Smyrna, Cleomedes, Nicomachus, Ptolemy, Gaudentius, Anatolius, Aristides Quintilian, Porphyry, Diophantus, Alypius, Damianus, Pappus, Serenus, Theon of Alexandria, Anthemius, and Eutocius.[64]

In his preface to his book On the Revolution of the Heavenly Spheres (1543), Nicolaus Copernicus cites various Pythagoreans as the most important influences on the development of his heliocentric model of the universe,[273][275] deliberately omitting mention of Aristarchus of Samos, a non-Pythagorean astronomer who had developed a fully heliocentric model in the fourth century BC, in effort to portray his model as fundamentally Pythagorean.[275] Johannes Kepler considered himself to be a Pythagorean.[273][276][277] He believed in the Pythagorean doctrine of musica universalis[278] and it was his search for the mathematical equations behind this doctrine that led to his discovery of the laws of planetary motion.[278] Kepler titled his book on the subject Harmonices Mundi (Harmonics of the World), after the Pythagorean teaching that had inspired him.[273][279] Near the conclusion of the book, Kepler describes himself falling asleep to the sound of the heavenly music, "warmed by having drunk a generous draught... from the cup of Pythagoras."[280] He also called Pythagoras the "grandfather" of all Copernicans.[281]

Archimedes was a great mathematician and was a master at visualising and manipulating space. He perfected the methods of integration and devised formulae to calculate the areas of many shapes and the volumes of many solids. He often used the method of exhaustion to uncover formulae. For example, he found a way to mathematically calculate the area underneath a parabolic curve; calculated a value for Pi more accurately than any previous mathematician; and proved that the area of a circle is equal to Pi multiplied by the square of its radius. He also showed that the volume of a sphere is two thirds the volume of a cylinder with the same height and radius. This last discovery was engraved into his tombstone.

HISTORY OF SCIENCE COLLECTION Claudius Ptolemy (circa 85-circa 165) Georg von Peurbach (1423-1461) Johann Müller of Königsberg, called Regiomontanus (1436-1476) Epitoma in Almagestum Ptolemaei ... Venice: Johannes Hamman for the editors, [31 August] 1496. Ptolemy's Almagest, a name derived from the medieval Latin form of its Arabic title, was the most important, encyclopedic, and complex astronomical and mathematical work of antiquity. Known in Greek as the "Mathematical Syntaxis" or the "Mathematical Collection," its thirteen books covered every aspect of mathematical astronomy. For over thirteen hundred years the Almagest remained the basis for all sophisticated astronomy. In 1460 Georg von Peurbach, professor of astronomy at the university of Vienna, was commissioned by Johann Cardinal Bessarion, Papal Legate to the Holy Roman Empire, to make a comprehensible Latin condensation of Ptolemy's work. Ignorant of Greek, he based his eptiome on a copy of Gerard of Cremona's 12th-century Latin translation of the "Syntaxis." Peurbach died just after finishing Book VI, and the remaining seven books were completed by his former student, Johann Müller of Königsberg, now known simply as Regiomontanus. The manuscript was completed sometime before April 28, 1463, but it was not until 20 years after Regiomontanus' death, that it was first printed under the joint editorship of Caspar Grosch and Stephan Römer. The Epitome provided easier access to Ptolemy's masterpiece, but it was more than a mere compressed translation. It added later observations, revised computations and offered critical commentary on obscure points and errors in the original text. Among the latter was the observation that Ptolemy's lunar theory required the moon's diameter to vary much more that it really did. This book is opened to that passage on lunar theory, Book V, proposition 22, which attracted the attention of Nicolaus Copernicus (1473-1543), then a young student at Bologna, who later overthrew the terra-centric Ptolemaic system with the helio-centric theory expounded his De revolutionibus orbium coelestium libri VI (Nuremberg: Johann Petreius, 1543-LOWNES). LOWNES COLLECTION OF SIGNIFICANT BOOKS IN THE HISTORY OF SCIENCE & TECHNOLOGYDiophantus of Alexandria (circa 200-circa 284) Arithmeticorum libri sex, et De numeris multangulis liber unus ... Paris: Sébastien Cramoisy for Ambrose Drouart, 1621.

The first Latin edition of the Arithmetica, translated from the Greek and edited by Wilhelm Xylander (also known as Wilhelm Holtzman, 1532-1576), was published in 1575. This first Greek and Latin edition incorporated Xylander's translation along with additional material and commentary supplied by Claude-Gaspard Bachet (1581-1638). It was in his copy of this edition of Diophantus that Pierre de Fermat (1601-1665) scribbled his famous "Last Theorem." It states that xn + yn = zn has no non-zero solutions for x, y, and z when n>2. He wrote: "To divide a cube into two cubes, a fourth power, or in general any power whatever into two powers of the same denomination above the second is impossible. I have discovered a truly remarkable proof which this margin is too small to hold." It was not until 1995 that Andrew Wiles proved this theorem, although using a method that Fermat would never have recognized. This volume is opened at Book II, proposition 8, "To divide a square number into two other square numbers," where Fermat wrote his "Last Theorem." HISTORY OF SCIENCE COLLECTION Pappus of Alexandria (circa 300-350) Mathematicarum collectionum libri V qui extant cum commentariis Federici Commandini ... Peasro: Hieronymous Concordia, 1588. 041b061a72