According to myth, the Pythagoreans were troubled to learn that the diagonal of a unit square is an irrational number because they thought natural numbers constituted the fundamental building blocks of the universe. This revelation purportedly resulted in the assassination of the individual who disclosed it. The Pythagoreans were not the only ones grappling with irrational numbers, referred to as “incommensurable numbers” in Greek geometry. For example, the diagonal of a square cannot be calculated using rational multiples of its side. Eudoxus of Cnidus (c. 390\u2013c. 340 BCE) is the mathematician recognized for tackling this issue.<\/p>\n
Eudoxus was born and passed away in Cnidus located on the southwestern coast of Anatolia. While none of his original texts have persisted, fragments are retained in Hipparchus\u2019 “Commentaries on the Phenomena of Aratus and Eudoxus.” Eudoxus was renowned as an astronomer, mathematician, physician, and legislator. The details of his life are gleaned from writings produced centuries later. He studied mathematics under the Pythagorean Archytas and medicine with Philiston of Locri in Magna Graecia (Southern Italy). At the age of twenty-three, he journeyed to Athens, attended lectures by Sophists, and spent sixteen months studying mathematics and astronomy in Egypt. Following his travels, he returned to Athens and is said to have led the Academy during Plato\u2019s absence before going back to Cnidus to pursue a political career.<\/p>\n
In the history of science, Eudoxus is famous for devising the homocentric sphere model to account for the erratic motions of planets. This model was refined by his pupil Callippus and later by Aristotle, serving as the standard Greek cosmology until the deferent-epicycle model introduced by Ptolemaeus. Nevertheless, Eudoxus’ mathematical contributions are what we focus on here.<\/p>\n
Eudoxus\u2019 mathematical influence is mainly evident in Euclid\u2019s “Elements,” particularly in Book V, which is largely attributed to him. He assisted mathematicians in circumventing the challenges posed by irrational numbers. The early Greek mathematicians were unable to perceive lengths, areas, or volumes in precise terms\u2014they simply compared them. Hence arose classical problems like squaring the circle and doubling the cube.<\/p>\n
Hippocrates of Chios (c. 470\u2013c. 421) made an attempt to square the circle (though he was unsuccessful) but achieved success in squaring the lune, highlighting the difficulties faced by Greek mathematicians with irrational numbers. The Pythagoreans encountered irrational numbers while trying to compare line segments. For example, dividing a square\u2019s side by its diagonal yields incommensurability. Eudoxus addressed this issue by substituting numbers with magnitudes\u2014lines, areas, volumes treated as non-numerical entities. Euclid\u2019s Elements, Book V, Definition 5 outlines Eudoxus\u2019 resolution.<\/p>\n
Magnitudes are said to possess the same ratio when, for any equimultiples of the first and third and of the second and fourth magnitudes, the results maintain proportional relationships. Utilizing modern notation: for quantities a, b, c, and d; a\/b = c\/d if certain conditions among positive integers m and n hold for equimultiples m*a, m*c, n*b, n*d.<\/p>\n
Eudoxus\u2019 technique avoided the challenge of irrational numbers by substituting arithmetical methods with geometric approaches rooted in magnitudes. The notorious irrational number from antiquity was Pi (\u03c0), which signifies the ratio of a circle’s circumference to its diameter. Early civilizations, such as the Babylonians and Egyptians, employed approximations for Pi in several practical scenarios.<\/p>\n
Similar to how he dealt with irrational numbers, Eudoxus circumvented the Pi dilemma by employing the method of exhaustion, a sophisticated evaluation method reflected in the Papyrus Rhind. This technique predates Eudoxus, having been utilized by Sophists like Antiphon and Bryson, who calculated the areas of circles through polygon approximations. Nonetheless, Eudoxus mathematically formalized these methods, as demonstrated in Euclid\u2019s Elements, Book XII, Proposition 2.<\/p>\n
The proof incorporates Archimedes\u2019 Axiom, which underlies the exhaustion method associated with Eudoxus, although it is sometimes ascribed to Hippocrates of Chios. Euclid further elucidates this in Book XII via propositions, showcasing volumetric relationships through the exhaustion technique.<\/p>\n
The phrase “method of exhaustion” wasn’t established until the 17th century by Jesuit mathematician Gr\u00e9goire de Saint-Vincent, following a resurgence by Nicolo Tartaglia in his edition of Archimedes, who frequently applied the method.<\/p>\n
Regarded as a precursor to integral calculus, the method of exhaustion shares a conceptual similarity with calculus: approximation using finite methods, steering clear of infinity and numerical values. In contrast, calculus employs the concept of limits, which are crucial for its higher computations. The geometrical magnitudes of the Greeks evolved into the real numbers of calculus, encompassing those that were once incommensurable.<\/p>\n","protected":false},"excerpt":{"rendered":"
According to myth, the Pythagoreans were troubled to learn that the diagonal of a unit square is an irrational number because they thought natural numbers constituted the fundamental building blocks of the universe. This revelation purportedly resulted in the assassination of the individual who disclosed it. The Pythagoreans were not the only ones grappling with […]<\/p>\n","protected":false},"author":1,"featured_media":376638,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"Default","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[175],"class_list":["post-376637","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-source-thonyc-wordpress-com"],"_links":{"self":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts\/376637","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=376637"}],"version-history":[{"count":0,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts\/376637\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/media\/376638"}],"wp:attachment":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=376637"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=376637"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=376637"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}