Archimedes
Greek · Philosophy · Mathematics · Fragmentary Texts · Mathematics; Fragmentary Texts
6 works · 2,023 aligned sentences
Birth: 287 BC / Death: 212 BC
Syracuse · mathematician · physicist · astronomer
Archimedes's cattle problem
This work is an extremely challenging mathematical puzzle, attributed to Archimedes of Syracuse and sent to Eratosthenes of Alexandria, which tasks the reader with calculating the number of cattle belonging to the sun god, Helios. The text consists of a problem presented in poetic verse and an ancient commentary recording the numerical results of the calculation. In the poem, the relations between the numbers of bulls and cows of four different colors (white, black, dapple, and yellow) are presented as a series of simultaneous linear equations. The problem escalates in difficulty with additional conditions requiring the sums of certain cattle to form perfect square and triangular numbers, leading to an astronomically large integer solution. The subsequent commentary provides concrete numerical values for this complex calculation, illustrating the sophisticated level of ancient mathematics.
Philosophy1 chunks · §1-2108 aligned sentencesRead →Book of Lemmas
This book is a classical work of plane geometry comprising fifteen propositions and their proofs concerning circles, chords, tangents, and specific complex plane figures. Each chapter presents a geometric configuration, asserts a theorem, and provides a rigorous proof. The opening chapters deal with basic equality relations involving tangent circles, chords, and perpendiculars. In the middle section, the text focuses on the "arbelos" (shoemaker's knife), a figure formed by three semicircles, proving its area properties and the equality of its inscribed circles. The latter part explores the properties of orthogonal chords within a circle, relationships between tangents and secants, and the area of another unique figure, the "salinon" (salt-cellar). The final chapter concludes with the properties of a regular pentagon inscribed in a semicircle, deriving a corollary related to the golden ratio (extreme and mean ratio). The work showcases the exquisite logical progression of ancient geometry, resolving complex relations of areas and ratios through ingenious proofs.
Philosophy7 chunks · §1-3–§15751 aligned sentencesRead →Fragments
This work compiles scholarly fragments attributed to ancient Greek mathematicians and astronomers, such as Pappus, Theon, and Heron, focusing on geometry, optics, and astronomy. The first half centers on the theory of polyhedra, where Pappus presents Plato's five regular solids and Archimedes' thirteen semi-regular solids. It details the mathematical rules for calculating the number of faces, edges, and vertices for each, explaining the geometric process of constructing semi-regular solids by truncating the corners of regular ones. Additionally, a fragment from Heron discusses the fourteen-sided polyhedra known to Plato. The second half features Theon's explanation of Ptolemy's theory of atmospheric refraction, analyzing the physical and geometric principles behind why celestial bodies appear larger near the horizon, with reference to Archimedes.
Fragmentary Texts4 chunks · §1.1#1–§2.1204 aligned sentencesRead →Measurement of a Circle
This work, written by the ancient Greek mathematician Archimedes, presents rigorous mathematical proofs concerning the area and circumference of a circle. Consisting of three propositions, the treatise uses geometric methods to clarify the fundamental properties of a circle. In the first proposition, Archimedes proves that the area of any circle is equal to a right-angled triangle in which the height is equal to the radius and the base is equal to the circumference. The second proposition shows the approximate ratio of the area of a circle to that of its circumscribed square. In the final section, by using circumscribed and inscribed regular polygons of up to 96 sides, he systematically calculates the upper and lower bounds of the ratio of the circumference to the diameter. Ultimately, the work concludes by establishing that this ratio lies between 3 10/71 and 3 1/7.
Philosophy3 chunks · §1-2–§3#2233 aligned sentencesRead →Ostomachion
This work is a geometrical analysis by the ancient Greek mathematician Archimedes, focusing on the traditional puzzle known as the "Ostomachion." The Ostomachion is a puzzle in which a square is divided into 14 pieces of various shapes to be rearranged into diverse figures. Archimedes explores the theoretical background of how these pieces are divided and rearranged, conducting a rigorous mathematical inquiry. In the text, he presents proofs for preliminary geometrical theorems necessary to verify the angles and the straightness of the edges of each piece. Through these surviving fragments, the work demonstrates the author's mathematical approach, elevating a simple toy into a subject of rigorous geometric investigation.
Philosophy1 chunks · §195 aligned sentencesRead →The Sand Reckoner
Written as a letter addressed to King Gelo of Syracuse, this mathematical treatise proposes a unique system for representing and calculating arbitrarily large numbers, demonstrating it by estimating the number of grains of sand required to fill the universe. Archimedes begins by establishing astronomical premises, adopting the heliocentric model of Aristarchus of Samos and determining the sizes and distances of the Earth, Moon, and Sun through precise observational experiments and geometric proofs. To overcome the limitations of the traditional Greek numbering system, which only went up to a myriad of myriads, he defines a new system of counting based on periods and octads, while also introducing the law of exponents for geometric progressions. Ultimately, starting from the scale of a single poppy seed, he calculates the maximum number of sand grains that could fit into spheres of increasing size, culminating in the proof that even the vast universe of Aristarchus can be filled with a number of sand grains that easily fits within his newly established numerical system.
Philosophy9 chunks · §1#1–§4#3632 aligned sentencesRead →