Greece: Archimedes and Apollonius
Chapter 4 Archimedes
• “What we are told about Archimedes is a mix of a few hard facts and many legends. . . . Hard facts – the primary sources –are the axioms of history. Unfortunately, a scarcity of fact creates a vacuum that legends happily fill, and eventually fact and legend blur into each other. The legends resemble a computer virus that leaps from book to book, but are harder, even impossible, to eradicate.” – Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?, p. 1. Archimedes
• Facts: – Lived in Syracuse – Applied mathematics to practical problems as well as more theoretical problems – Died in 212 BCE at the hands of a Roman soldier during the attack on Syracuse by the forces of general Marcellus. Plutarch, in the first century A.D., gave three different stories told about the details of his death. Archimedes
• From sources written much later: – Died at the age of 75, which would put his birth at about 287 BCE (from The Book of Histories by Tzetzes, 12th century CE). – The “Eureka” story came from the Roman architect Vitruvius, about a century after Archimedes’ death. – Plutarch claimed Archimedes requested that a cylinder enclosing a sphere be put on his gravestone. Cicero claims to have found that gravestrone in about 75 CE. Archimedes
• From sources written much later: – From about a century after his death come tales of his prowess as a military engineer, creating catapults and grappling hooks connected to levers that lifted boats from the sea. – Another legend has it that he invented parabolic mirrors that set ships on fire. That is not likely. (See Mythbusters, episode 46) Archimedes’ Writings
• Planes in Equilibrium – An axiomatic development of The Law of the Lever: Two magnitudes balance at distances inversely proportional to the magnitudes. Archimedes’ Writings
• On Floating Bodies – The laws of hydrostatics, including Archimedes’ Principle: “Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.” Archimedes’ Writings
• Measurement of the • Archimedes used a Circle double “reductio ad – The area of any circle is absurdum” argument equal to that of a right involving the method of triangle in which one of exhaustion, showing the legs is equal to the that the area of the radius and the other to the circumference of the triangle could neither circle. be less than, or greater than, the area of the circle. Archimedes’ Writings
• Measurement of the • Archimedes did this by Circle using inscribing and – The ratio of the circumscribing regular circumference of any polygons of increasing circle to its diameter is number of sides, less than 3 but greater beginning with than 3 . hexagons and going up to 96‐gons. Each stage involved computation of ugly radicals. Archimedes’ Writings
• The Method – Discovered on a palimpsest in 1899, in Constantinople (now Istanbul, as all fans of TMBG know). – Disappeared during WWI, resurfacing in 1998. Archimedes’ Writings
• The Method –of • http://www.matematic discovery –involves asvisuales.com/english/ slicing areas and html/history/archimede volumes into s/parabola.html infinitesimal slices and “balancing” on lines with fulcrums, and employing the Law of the Lever to get ratios of those areas and volumes. Archimedes’ Writings
• On the Quadrature of • Along the way, proved the Parabola how to find the sum of – Proved, using a double a geometric series. reductio argument and exhaustion, that the area of a parabolic segment is of the area of the inscribed triangle. – A rigorous synthetic proof of a result from The Method. Archimedes’ Writings
• On the Sphere and the • Archimedes seemed to Cylinder be most proud of this – Showed that a sphere result, and asked that a has a volume two‐thirds sphere inscribed in a that of a circumscribed cylinder be placed on cylinder (i.e., of the his tomb. same height and diameter) – Showed that the sphere has an area two‐thirds that of the cylinder (including the bases). Archimedes’ Writings
• The Sand Reckoner • On Spirals – Archimedes calculates that – 28 propositions defining the number of grains of and exploring the sand required to fill the properties of what we call universe is 8×1063 (in an Archimedean spiral, modern notation). which is the set of all – Mentions that his father points corresponding to was an astronomer named the locations over time of a Phidius. point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. In polar coordinates, . Apollonius
• Born about 262 BCE, in Perga, on the Mediterranean coast of what is now Turkey. • Studied and probably worked much of his life in Alexandria. • Major contributions include: – Astronomy –the theory of deferent circles and epicycles – Mathematics –the most important work being Conics, a work in 8 volumes, of which 7 survive. Apollonius
• Of the Conics, T. L Heath, a major scholar of ancient Greek mathematics says, “... the treatise is a great classic which deserves to be more known than it is. What militates against its being read in its original form is the great extent of the exposition (it contains 387 separate propositions), due partly to the Greek habit of proving particular cases of a general proposition separately from the proposition itself, but more to the cumbersomeness of the enunciations of complicated propositions in general terms (without the help of letters to denote particular points) and to the elaborateness of the Euclidean form, to which Apollonius adheres throughout.” Apollonius
• In other words, it’s not much fun to read. • So, in the words of Inigo Montoya,
“Let me explain. No, there is too much. Let me sum up.” Apollonius
• Conics – 389 Propositions in 8 books, 7 of which we have (4 in Greek, 3 in Arabic). Before Apollonius
• Prior to Apollonius, conic sections were described in terms of the intersection of a cone and a plane, but: • The plane of intersection was always perpendicular to a side, and the vertex angle of the cone was either acute, right, or obtuse. Before Apollonius
mABC = 66 B mABC = 110
B
A C
A C
mABC = 90
B
A C Apollonius
• Used the “double” cone, and showed that the conics could be described by intersections with more arbitrary planes. Conics
• Developed methods very similar to those of analytic geometry, using for axes a diameter and a tangent: Conics
• Book I: Relations satisfied by the diameters and tangents of conics, and how to draw tangents to given conics. • Book II: How hyperbolas are related to their asymptotes • Book III: More tangents • Book IV: Intersections of conics These are considered “Basic” by Apollonius although he does prove new results, especially in Book III. • Books V – VII: – discuss normals to conics and in particular how they can be drawn from a point and propositions determining the center of curvature. – similarity of conics, – conjugate diameters Names of the Conics
• The parabola was considered to be the locus of points such that the square on the ordinate was equal to a rectangle on the abscissa and parameter ( ). • Translation: the square of y was equal to a multiple of x, or in other words , for a parabola with vertex at the origin. • These points could actually be “plotted” in a sense by using geometric algebra. Names of the Conics
• For ellipses and hyperbolas with one vertex at the origin, the equations can be written as: ∓ , or letting , as: , with the “+” for a hyperbola and the “–” for the ellipse. So, we have: Names of the Conics
• for the parabola, • for the hyperbola, and • for the ellipse.
• “Ellipsis” refers to a deficiency –leaving something out. • “Hyperbola” refers to an excess –a throwing beyond. • “Parabola” refers to placing beside, or a comparison. (Parable)