Greece

Chapters 6 and 7: Archimedes and Apollonius SOME ANCIENT GREEK DISTINCTIONS Arithmetic Versus Logistic

• Arithmetic referred to what we now call number theory –the study of properties of whole numbers, divisibility, primality, and such characteristics as perfect, amicable, abundant, and so forth. This use of the word lives on in the term higher arithmetic. • Logistic referred to what we now call arithmetic, that is, computation with whole numbers. Number Versus Magnitude

• Numbers are discrete, cannot be broken down indefinitely because you eventually came to a “1.” In this sense, any two numbers are commensurable because they could both be measured with a 1, if nothing bigger worked. • Magnitudes are continuous, and can be broken down indefinitely. You can always bisect a line segment, for example. Thus two magnitudes didn’t necessarily have to be commensurable (although of course they could be.) Analysis Versus Synthesis

• Synthesis refers to putting parts together to obtain a whole. • It is also used to describe the process of reasoning from the general to the particular, as in putting together axioms and theorems to prove a particular proposition. • Proofs of the kind Euclid wrote are referred to as synthetic. Analysis Versus Synthesis

• Analysis refers to taking things apart to see how they work, or so you can understand them. • It is also used to describe reasoning from the particular to the general, as in studying a particular problem to come up with a solution. • This is one general meaning of analysis: a way of solving problems, of finding the answers. Analysis Versus Synthesis

• A second meaning for analysis is specific to logic and theorem proving: beginning with what you wish to prove, and reasoning from that point in hopes you can arrive at the hypotheses, and then reversing the logical steps. • This doesn’t always work, of course, but in many cases it can yield a valid proof. Analysis Versus Synthesis

• Why does this matter? – First, much ancient Greek work is typified by synthesis –proving propositions based on postulates. One of Archimedes’ works is unique in that it gives us an insight into the analysis instead of just the synthesis that is more typical of ancient Greek works. – Second, since the Greek works were re‐discovered during the Renaissance, a lot of effort was spent trying to discover the “lost” analysis techniques of the Greeks. Analysis Versus Synthesis

• Why does this matter? – Third, it helps make sense of the label analytic geometry. We’ll talk about this more later. – Finally, even today we can recognize analysis and synthesis in two fundamental parts of doing mathematics –the discovery of reasonable conjectures, and the proof of conjectures as we turn them into theorems. More on this later, too. Syracuse and Perga 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: – He lived in Syracuse. – He 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 gravestone 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

• Planes in Equilibrium – The Law of the Lever was known in Aristotle’s time, but Archimedes developed it axiomatically from first principles, in a way that used ideas of static equilibrium. 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, nor 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. The Method

• Given parabolic H F segment ABC, cut off by segment . M • Construct a

perpendicular to at E K A. N

• Construct a tangent to B the parabola at C. P • These two lines meet at A F. O D C The Method

• Find the midpoint D of H F segment , and construct a perpendicular M there that meets the parabola at B, and at E E. K • Construct a ray that N intersects at K, and B extend to point H with KH P = KC. Segment will be A our lever, with fulcrum K. O D C The Method

• Finally, choose an H F

arbitrary point O on M and construct a perpendicular at O that intersects the parabola E K at P, at point N, and N at point M. B

• This is the set‐up. P

A O D C The Method

• We use two facts about H F

parabolas and their M tangents that are not part of our background knowledge from college E K algebra or pre‐calculus, N but were well‐known in B

Archimedes’ time: P

1. EB = BD A O D C The Method

• Examining right H F

triangles and M , we can see that they are similar. Thus their sides are in E K proportion, and we N have . B Combining this with P A we get O D C . The Method

• Finally since KC = HK, H F we arrive at . M • Cross multiply to get:

. E K • Using the Law of the N Lever, this means that B

the segment at P point H will balance the A segment at point N. O D C The Method

• Since this is true for any H F

point O along the M segment , then all the ’s added up, moved to H, will E K balance all the ’s left N exactly where they are. B

• We’re almost there….. P

A O D C The Method

Where “is” the big triangle H F

relative to the M segment ? Well, the center of gravity will be on E a median, 2/3 of the way K N from the vertex. In other Z words, at a point Z on the B segment , one third of P the way from K to C. A O D C The Method

• Thus the big triangle H F balances at 1/3 the distance from the M fulcrum that the

parabolic segment E would, and is therefore K N 3 times the area. Z B • Finally, we need to relate this to the small P

triangle subtended by A O D C the parabolic segment. The Method

• Now, how is the big F triangle compared to the little red triangle ? • First, drop a perpendicular from B to E with foot at point Q. K • Now medians split the area of a triangle in half, Q B and by similarity, K is a median of the big A triangle. Thus is D C half the area of . The Method

• Second, F by ASA, and by SAS. • So, the area of is E half of , or one K fourth of the big triangle , which is Q B 3 times the parabolic

A segment. D C The Method

• Putting this all together, F the area of is ¾ of the parabolic segment. E • Or, the parabolic K segment has 4/3 the area of the inscribed Q B triangle .

A D C Did Archimedes Consider this a Proof?

• No. • This was the “analysis” part of the problem, a way of discovering what the answer should be. • He did the “synthesis” part in On the Quadrature of the Parabola. Archimedes’ Writings

• On the Quadrature of • Along the way, he the Parabola proved how to find the – Proved, using a double sum of a geometric reductio argument and series. exhaustion, that the area of a parabolic segment is of the area of the inscribed triangle. – A rigorous synthetic proof of his 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 – Developed a way to discuss – 28 propositions defining verrrrrrry large numbers. and exploring the – Archimedes calculates that properties of what we call the number of grains of an Archimedean spiral, sand required to fill the which is the set of all universe is 8×1063 (in points corresponding to modern notation). the locations over time of a – Mentions that his father point moving away from a was an astronomer named fixed point with a constant Phidius. speed along a line which rotates with constant angular velocity. In polar coordinates, . Archimedes’ Spiral as a Trisectrix

• As the book points out, the Spiral of Archimedes can be used both to trisect an angle and to square a circle. Apollonius 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 (more on this later) – Mathematics –the most important work being Conics, a work in 8 volumes, of which 7 survive. Apollonius

• The lost works discussed on pp. 127 – 129 of your text are known largely from Pappus’ descriptions. • They were included in Pappus’ Treasury of Analysis (now lost) which was written for those who wished to be “capable of solving problems.” Note the meaning here of analysis. Apollonius

• The problems discussed

in your text are typical: AB = 2.99 cm AB = 1.50 Given two straight lines CD = 1.99 cm CD

A and a point on each B draw a third straight line thorough a third

given point that cuts off C D on the given line segments that are in a given ratio. Verging or Neusis Problems

• Insert a line segment of given length between L two given lines, such that it “verges to” a given point –that is, M goes through the point if extended. • Usually accomplished A with a marked straightedge. An Important Classic Problem

• Let three, four, or more lines be given in position; required the locus of the point from which the same number of lines may be drawn to meet them one to each, at given angles, such that, in the case of three lines, the rectangle of the first two lines may have a fixed relation to the square of the third; or, in the case of four lines, that the rectangle of the first and second may have a fixed relation to rectangle of the third and fourth, and so on. (Apollonius, via Pappus) • The Three‐ and Four‐line Locus Problems. More on this when we get to the 1600’s. 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). • The definitive treatment of conics up to the 1600’s, at least. 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. • Also, the cone was a single, right circular cone. Before Apollonius

mABC = 66 B mABC = 110

B

A

C

A C

mABC = 90

B

A C Apollonius

• Used the “double napped” cone, and showed that the conics could be described by intersections with more arbitrary planes. Conics

• As the book described, moved quickly from the three‐dimensional description of intersections of planes and cones to a “symptome” description that depended only on distances from given lines in the plane of the conic. • These given lines were something like what we could call axes. Conics

• Developed methods very similar to those of analytic geometry, often using diameters and tangents (or conjugate diameters) as axes because they had nice properties. Conjugate Diameters

• The midpoints of parallel 4 chords of an ellipse are all

collinear, lying on the 2 conjugate diameter. Of course, chords parallel to

that conjugate have -5 5 midpoints that define the

corresponding conjugate -2 diameter.

• These two diameters -4 meet at the center. Conjugate Diameters

• And, lines at the end of 4 one conjugate diameter that are parallel to the 2 other conjugate

diameter are tangent to -5 5 the ellipse.

-2

-4 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; more tangents and conjugate diameters • Book III: More tangents, the three‐ and four‐line locus problems • Book IV: Intersections of conics These are considered “Basic” by Apollonius although he does prove new results, especially in Books III and IV. • Books V – VII: – discuss normals to conics and in particular how they can be drawn from a point; 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)