According to folklore, the Pythagoreans were appalled upon realizing that the diagonal of a unit square is an irrational number, as they held the belief that natural numbers were the fundamental elements of the universe. It is even claimed that they killed the individual who made this revelation. It wasn’t solely the Pythagoreans who struggled with the notion of irrational or incommensurable numbers, as referred to by the Greeks in their mathematical studies. Taking the diagonal of a square as an example, it cannot be measured with a fixed length. Therefore, if the side of the square measures 1 cm, the diagonal cannot be expressed as any rational multiple of 1 cm. There is no way to compare the two measurements directly. The mathematician credited with providing a resolution to this dilemma was Eudoxus of Cnidus (c. 390\u2013c. 340 BCE).<\/p>\n
He was born and passed away in Cnidus, located on Anatolia’s southwest coast. None of his original writings have endured, yet fragments are retained in the *Commentaries on the Phenomena of Aratus and Eudoxus* by Hipparchus (c. 190\u2013c. 120 BCE), which is the only surviving work of Hipparchus. Diogenes Laertius (fl. third century CE) reported that Eudoxus was an astronomer, mathematician, physician, and legislator. It should be acknowledged that the limited information available regarding Eudoxus, similar to most early Greek mathematicians, originates from texts written centuries after his lifetime. He pursued mathematical studies with Archytas (435\/410\u2013360\/350 BCE), a Pythagorean, and studied medicine under Philiston of Locri (4th century BCE), both residing in Magna Graecia, or Southern Italy. At the age of twenty-three, he traveled to Athens and spent two months attending the lectures of the Sophists. He then dedicated sixteen months in Egypt studying mathematics and astronomy. After additional travels, he returned to Athens and is said to have taken on leadership of the Academy during Plato\u2019s absence. Eventually, he returned to Cnidus and engaged in politics.<\/p>\n
In the narrative of science, Eudoxus is most recognized as the founder of the homocentric spheres model of the cosmos, which was developed to account for the apparently erratic motion of the planets. This model was further advanced by Callippus (c. 370\u2013c. 300 BCE), one of his students, and later by Aristotle (384\u2013322), who is also thought to have studied with Eudoxus in the Academy. Together with Aristotle’s cosmology, this model formed the standard Greek interpretation of the universe until it was supplanted by the deferent-epicycle model of Ptolemaeus (fl. 2nd century CE). However, our focus here remains on Eudoxus’ mathematical contributions.<\/p>\n
To explore Eudoxus\u2019 mathematical advancements, we must refer to Euclid\u2019s *Elements*, beginning with Book V, much of which is attributed to Eudoxus. In this book, he assists Greek mathematicians in circumventing the challenges presented by irrational or incommensurable numbers. Early Greek mathematicians lacked concepts for lines, areas, or volumes; they only compared them. This is exemplified in two classic problems involving straight edges and compasses\u2014squaring the circle and doubling the cube. The first problem asks for a square that has the same area as a given circle, while the second seeks to find which cube has double the volume of a specified cube.<\/p>\n
A fitting illustration of this is found in the sole surviving fragment of the work of Hippocrates of Chios (c. 470\u2013c. 421), a mathematician who lived a century before Eudoxus and was the first of five authors to write an *Elements* prior to Euclid. In his effort to square the circle, a task in which he ultimately failed, Hippocrates did succeed in squaring the lune.<\/p>\n
The Lune of Hippocrates. Partial solution of the \u201cSquaring the circle\u201d task, suggested by Hippocrates. The area of the shaded figure is equal to the area of the triangle ABC. Source: Wikimedia Commons<\/p>\n
The Pythagoreans encountered irrational numbers while comparing line segments. Given two segments, a and b, where a > b, how many times can one segment b be divided into segment a? Due to their fixation on natural numbers, the answer must, of course, be expressed numerically. However, as previously mentioned, when attempting to divide the side of a square by its diagonal, the outcome is incommensurable. Eudoxus addressed the issue by substituting numbers with magnitudes. Lines, areas, and volumes are characterized by a magnitude, an intrinsic size that is non-numerical.<\/p>\n","protected":false},"excerpt":{"rendered":"
According to folklore, the Pythagoreans were appalled upon realizing that the diagonal of a unit square is an irrational number, as they held the belief that natural numbers were the fundamental elements of the universe. It is even claimed that they killed the individual who made this revelation. It wasn’t solely the Pythagoreans who struggled […]<\/p>\n","protected":false},"author":2,"featured_media":374312,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"Default","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[175],"class_list":["post-374311","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\/374311","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=374311"}],"version-history":[{"count":0,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/posts\/374311\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=\/wp\/v2\/media\/374312"}],"wp:attachment":[{"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=374311"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=374311"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wolfscientific.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=374311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}