COUNTING ON EPIC: MATHEMATICAL POETRY AND HOMERIC EPIC IN ARCHIMEDES' <i>CATTLE PROBLEM</i>

In 1773, the celebrated enlightenment thinker G.E. Lessing discovered in Wolfenbüttel's Herzog August Library a manuscript which contained a previously unknown Ancient Greek poem. The manuscript identified the author as Archimedes (c.287-212 BCE), and the work became known as the Cattle Problem (henceforth CP). On the surface, its twenty-two couplets capitalise on Homer's depiction of the ‘Cattle of the Sun’ in Book 12 of the Odyssey and its numerical aspect. A description of the related proportions of black, white, brown and dappled herds of cattle, which are then configured geometrically on Sicily, creates a strikingly colourful image. The author's decision to encode a number into the figure of the Cattle of the Sun styles the poem as a response to, and expansion of, Homer's scene. Reading through the work, though, it becomes clear that the mathematics is more complex than that of Homer's Odyssey.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/rmu.2015.1

Archimedes\u27 Cattle Problem

Archimedes, born nearly 2,300 years ago, is a man of legacy within the mathematical and scientific world. Known for his work in the field of hydrostatics, mathematical calculus, and geometry, Archimedes’ mathematical advancements went unrivaled for 2,000 years until the time of Isaac Newton [5]. Archimedes’ Cattle Problem is equally a test of mathematical prowess in modern history as it was during its inception. Born in Syracuse, many scholars believe that Archimedes studied in Alexandria as a young man. It was there that he was taught by the followers of Euclid and grew to know many of his life-long contemporaries such as Eratosthenes of Cyrene and Conon of Samos [4]. It is Eratosthenes, alongside other Alexandrian mathematicians, to whom he posed his famous “Cattle Problem”: “If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions..

ARCHIMEDES IN THE CLASSROOM

Eureka! Eureka! What better way to excite and engage students than to bring Archimedes into the classroom? Archimedes is widely regarded as the greatest mathematician of antiquity. Very little is known about Archimedes’ personal life. Archimedes was born in Syracuse, Sicily around 287 B.C., and he was the son of an astronomer. He probably studied in Alexandria, Egypt under followers of Euclid. Today we know Archimedes as a brilliant mathematician and scientist, he spent much of his career inventing war machines. Despite orders not to harm Archimedes, his life was cut short in 212 B.C. when the Romans invaded Syracuse and killed Archimedes at age 75, more information on the history of Archimedes can be found in the article, Archimedes, by Chris Rorres of New York University. This paper will take a closer look at some of Archimedes’ most brilliant discoveries, and how they can be brought into the classroom. Unlike Plato and other great minds before him, Archimedes solved problems with anything and everything. That type of innovation and outside of the box thinking is exactly the type of skills students will need to succeed in today’s world

Pell's equation without irrational numbers

We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree.Comment: 10 pages, 3 figures added some references, fixed typos, added remarks on Speeding up the algorith

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

Poetry and Number in Graeco-Roman Antiquity

Ranges widely across Greek and Latin poetry to demonstrate the various roles played by number and how the treatment of counting and arithmetic was bound up with wider conceptions of the nature of poetry. Aimed at both classicists and those interested in the cultural history of mathematics

Famous numbers on a Chessboard

In this article it is shown how famous numbers like Pascal’s triangle, the Fibonaccinumbers, Catalan’s triangle, Delannoy’s square array, the Pell numbers and Schröder’striangle can be constructed on a chessboard with a rook, knight, bishop, king or queen.Furthermore, several new triangle sums, which are all named after chess pieces that areleapers and add up numbers according to the way they leap, are introduced. Finally anew theory of how Hipparchus, who lived around 150 BC, might have calculated histwo famous numbers with the aid of a ‘chessboard’ is presented

Applying Hallgren’s algorithm for solving Pell’s equation to finding the irrational slope of the launch of a billiard ball

This thesis is an exploration of Quantum Computing applied to Pell’s equation in an attempt to find solutions to the Billiard Ball Problem. Pell’s equation is a Diophantine equation in the form of x2 − ny2 = 1, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. We will be applying Hallgren’s algorithm for finding irrational periods in functions, in the context of billiard balls and their movement on a friction-less unit square billiard table. Our central research question has been the following: Given the cutting sequence of the billiard ball’s movement, can you find the irrational slope value in which the billiard ball was put in motion