According to tradition, the Pythagoreans were heartbroken when they realized that the diagonal of a unit square represents an irrational number, as they held the belief that natural numbers were the fundamental elements of the universe. It is even claimed that they took the life of the individual who made this revelation. The struggle with the concept of irrational numbers, or incommensurable numbers as they were referred to by the Greeks in their geometric studies, was not exclusive to the Pythagoreans. In the case of the diagonal of a square, it cannot be measured against a specific fixed length. Therefore, if the square’s side measures 1 cm, the diagonal cannot be quantified by any rational multiple of 1 cm lengths. It becomes impossible to measure one against the other. The mathematician recognized for providing a solution to this dilemma was Eudoxus of Cnidus (circa 390\u2013circa 340 BCE).<\/p>\n
He was both born and passed away in Cnidus, located on the southwestern coast of Anatolia. None of his original texts have survived, but some fragments can be found in the Commentaries on the Phenomena of Aratus and Eudoxus authored by Hipparchus (circa 190\u2013circa 120 BCE), which remains the only surviving work of Hipparchus. As reported by Diogenes Laertius (active third century CE), Eudoxus was an astronomer, mathematician, physician, and legislator. It is essential to acknowledge that the scant information we possess about Eudoxus, similar to most early Greek mathematicians, comes from texts written centuries after his lifetime. He pursued mathematical studies under Archytas (435\/410\u2013360\/350 BCE), a Pythagorean, and medicine with Philiston of Locri (4th century BCE), both of whom resided in Magna Graecia, namely Southern Italy. At the age of twenty-three, he journeyed to Athens and devoted two months to attending the lectures of the Sophists. Subsequently, he spent sixteen months in Egypt delving into mathematics and astronomy. After further explorations, he returned to Athens and is believed to have taken over the leadership of the Academy during Plato\u2019s absence. Later, he went back to Cnidus and entered politics.<\/p>\n
In the realm of scientific history, Eudoxus is primarily recognized as the pioneer of the homocentric spheres model of the cosmos, designed to clarify the apparently erratic movements of the planets. This model was further refined by Callippus (circa 370\u2013circa 300 BCE), one of his pupils, and then by Aristotle (384\u2013322), who is also reported to have studied under Eudoxus at the Academy. It became, along with Aristotle\u2019s cosmology, the prevailing Greek model of the universe until it was eclipsed by the deferent-epicycle model of Ptolemaeus (active 2nd century CE). Nonetheless, our focus here is on Eudoxus\u2019 contributions to mathematics.<\/p>\n
To identify Eudoxus\u2019 mathematical contributions, we must look to Euclid\u2019s Elements, beginning with Book V, which is largely believed to have origins in Eudoxus\u2019 work. He assisted Greek mathematicians in navigating the challenges posed by irrational or incommensurable numbers. Early Greek mathematicians lacked foundational concepts of lengths, areas, or volumes and only engaged in comparisons. This led to two classic problems involving straightedge and compass: the squaring of the circle and the doubling of the cube. The first problem seeks a square with an area equivalent to that of a given circle, while the second seeks to determine which cube possesses double the volume of a specified cube.<\/p>\n
An excellent illustration of this challenge can be found in the sole surviving fragment from a mathematician who preceded Eudoxus by a century, Hippocrates of Chios (circa 470\u2013circa 421), who became the first of five authors before Euclid to compose an Elements. In his efforts to square the circle, a task in which he ultimately failed, Hippocrates was successful in squaring the lune.<\/p>\n
The Pythagoreans encountered irrational numbers through their comparisons of line segments. Given two line segments, a and b, where a > b, how many times can one segment b be divided into a? Given their fixation on natural numbers, the answer had to be articulated in numerical form. However, as previously mentioned, if one attempts to divide the side of a square by its diagonal, the answer becomes incommensurable. Eudoxus addressed this issue by substituting numbers with magnitudes. Lines, areas, and volumes are considered to possess a magnitude, an abstract size devoid of numerical value. In Definition 5 of Book V of The Elements, we then arrive at the following:<\/p>\n
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equim<\/p>\n","protected":false},"excerpt":{"rendered":"
According to tradition, the Pythagoreans were heartbroken when they realized that the diagonal of a unit square represents an irrational number, as they held the belief that natural numbers were the fundamental elements of the universe. It is even claimed that they took the life of the individual who made this revelation. The struggle with […]<\/p>\n","protected":false},"author":1,"featured_media":375380,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"Default","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[175],"class_list":["post-375379","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\/375379","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=375379"}],"version-history":[{"count":0,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts\/375379\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/media\/375380"}],"wp:attachment":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=375379"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=375379"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=375379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}