Pitching a

Dr. Maurice Burke Montana State University- Bozeman October 19, 2017 What is a circle? Let’s zoom back 5000 years to 3000 BC.…… The time of the first pharaohs and humanity’s first writing in river cultures…… the birthday of the old tree on the right. Mammoths still walked the earth and Ötzi, the iceman, was hunting in the alps!

What did people then think a circle was????

Quiz: If C=3, is A >1 ?

Here are two “Biscuits” baked about 2000 BC by some Babylonian students. What do the numbers mean?

The circumference is 3 and the area is 45/60 or ¾. The Bab classic “What’s the area of a tree stump’s circular top” problem

“1 40 mu-uh-hi i-s:´ı-im ˇsu-ul-li-iˇs-ma = 5 ki-pa- at i-s:´ı-im i-l´ı = 5 ˇsu-ta-ki-il-ma 25 i-l´i = 25 a- na 5 i-gi-gub ´e-em i-ˇsi-ma = 2 05 A.ˇS`A i-l´I”

Translation: “Triple 1;40, (the diameter of) the top of the log, and 5, the circumference of the log, will come up. Square 5 and 25 will come up. Multiply 25 by 0;05, the coefficient, and 2;05, the area, will come up.

So it appears : • C = 3xD….. To us: C=πD • So their “π” was 3. But they had no theory of π - Just a pattern they noticed in circular regions. 퐶2 • To us, 퐴 = . To Babylonians, 4휋

퐶2 1 퐴 = = × 퐶2 푠표 4×3 12 5 A= × 퐶2 = 0; 05 × 퐶2) 60

• Hence, the coefficient 0;05 THERE IS NOTHING WRONG WITH THIS!!! I wonder if we asked students to come up with a formula for the area of an ellipse given its circumference and diameter…..

Here is a related Thought: Oldest Mathematics Exercises on Record, over 5000 years old!

• Tablet shows two exercises on calculating the area of

quadrilateral fields. Contrived

numbers reveal them as school exercises.

• What fascinates me is that the Surveyor Formula was used for Sumerian ca. 3200 BCE. Uruk over 2000 years in Egypt and Tablet W 19408 Mesopotamia!!!!

Surveyor Formula Eleanor Robson says: “The Old Babylonian circle was a figure—like all OB geometrical figures—conceptualized from the outside in. In such a situation, there could be no notion of measurable in the Old Babylonian period…. The radius was never conceptualized as a rotatable line.…The area of a circle is never calculated directly from its diameter… Both the circle and the circumference are called in Old Babylonian kippatum from the verb kapapum ‘to curve’. The diameter was necessary in order to conceive of a circumference or circle as a loop whose opposite points are all equidistant.” To use Linnaean terminology, their conceptual schema of circle, which includes its properties, major components and their relations, is a different species than our schema, but clearly part of the same genus. It is structurally different in fundamental way from our schema. Nonetheless, it has coherence and use. Fast Forward 2000 years 600 BC – 200 BC, The Greek Miracle 1. Definitions: Outgrowth of philosophical Arguing? 2. Theoretical discrimination of Number and Magnitude 3. Discovery, Exploration, and Development of “Circle”, and only a few (6) other , using synthetic geometry. No theory of “curves”. They are still tools. 4. Curves are physical, kinematic, stereometric and emphatically not numerical. They are “Continuous.” and that means unbroken, their parts touch; Locus is not a “set of points” but the place where points reside; points do not touch each other, they are not “parts” of a curve. So curve is more than a set of points. AHHHH ZENO! ’s Definitions (~300 BC) • Def. 2.A line is breadthless length. (Not a set of points! It’s a curve!) • Def. 4.A straight line is a line which lies evenly with the points on itself. (It is not the same as its points, they lie on it. Two points determine a line???) [Other defs: A plane figure is a surface with a boundary] • Def. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. (A circle is a region and the curve containing it!) [Evidence of evolution of schema: Refinement of structure] Greeks: Curves are initially used to solve problems that resist circles and lines. • Trisection Problem ( of - The oldest curve in mathematics next to the straight line and circle ~450 BC)

• Duplicating the Cube (Conics)

Quadratrix

• Kinematically defined

• Determined by the motions of two points (yellow dots in figure)

• Used for Trisecting Angle

• Constructed point by point with ruler and compass Your students can do these simulations Use GeoGebra….Now on Cell Phones! Let’s try it. The Delian Problem and Conics • : Construct the side of a cube whose volume is twice the volume of a given cube. So, if given cube has volume 1, then you must construct segment of length so that the cube with side length x would have a volume of 2. I. e. , 푥3 = 2, or 푥 = 3 2. (Perhaps the most consequential problem from antiquity.)

• Hippocrates (~450 BC) discovers that you can do this if you can construct two mean proportionals between segment a and a segment twice as long as a.

I.e. construct segments x and y such that 푎 푥 푦 a:x = x:y = y:2a or = = 푥 푦 2푎

• 푊푒 푤𝑖푙푙 푠푒푒 𝑖푛 푎 푚표푚푒푛푡 ℎ표푤 푡ℎ𝑖푠 푤표푟푘푠. The conic sections are discovered

Archytas (~400 BC), friend of Plato, teacher of Eudoxus (~380 BC), who is teacher of (~ 350 BC) who is tutor of Alexander the Great (~340 BC), shows how to find two mean proportionals between segments a and b by intersecting a cone with a cylinder with a torus, all surfaces defined by revolving lines and circles. Menaechmus simplifies the construction to the intersection of two planar sections of cones (Conics discovered!) Section of Right-Angled Cone Section of Acute-Angled Cone Exploration and Development of Conics

• Euclid book Conics (now lost) • (~250 BC) intersects “sections of right-angled cones” with “sections of obtuse-angled cones” to find where to section a sphere so that the volume of the two pieces have a some given ratio to each other. • E.g., where would you slice a sphere of radius 1 so that you cut off a third of its volume? Arch’s method reduces this to 1 finding h so that 4: ℎ2 = (3 − ℎ): . 4 • This, to us, is an equation ℎ3 + 1 = 3ℎ2. BUT, Arch was solving a proportion between magnitudes by a construction, ours is solving an equation of numbers by using algebraic/numeric operations. Apollonius: Theory of Conic Curves: Their formal baptism • Introduces language: absicssa, ordinate, parabola, hyperbola, ellipse. Considers the curves as a single general family rather than as three unrelated curves. • Defines conics stereometrically and uses the most general 3-D model, the oblique cone, and double-naped cone for his initial definitions and theorems. • His language is still geometric where square means a square region and not some number. He uses similar triangles and the Fundamental Property of Circles When two chords intersect inside a circle, the products of the segments of one equals the product of the segments of the other chord.

Corollary: If one chord is a diameter and is perpendicular to the other, then it bisects the other chord. Apollonius Parabola Apollonius Parabola

GeoGebra let’s us explore the conics and visualize the arguments of Apollonius.

In our algebraic way expressing things: 푥2= (4푝)푦 4p is the latus rectum of the parabola, so P is half the size of the gold segment. Your students can do these 3-D diagrams and study the ratios of components using GeoGebra….

So How To Solve 푥3 = 2 (Omar Khayyam’s Construction ~1100 AD) 1:x = x:y = y:2 1 푥 푦 i.e. = = 푥 푦 2 Find x and y

We have 푦 = 푥2 So, 푦2 = 푥4

But we have 푦2 = 2푥.

SO: 푥4 = 2푥 표푟 푥3 = 2

The Greek Legacy on Conics (i.e. The mindset for the next 2000 years!!) • There are no negative coordinates, let alone numerical coordinates. No numberline! • No equations of number and numerical operations, but proportions between magnitudes, given or constructed. Not easy to work with – you can’t even cross multiply! • The auxiliary lines are imposed on the curve after the curve is given and do not form a coordinate frame of reference independent of the curve. • The curve always passes through a point called origin. • Greek conceptual schema for curves is a different species, but of same genus as ours. Again, it is structurally different than ours.

Fast Forward 1500 Years (~1100 AD) Evolution never stops! • Arabian Scholars inherit the Greek Tradition. But gradually, focus on algebra becomes dominant part of their mathematics.

• Algebra is born as a subject of numbers and operations and equations. Equations are used and manipulated as mathematical objects. It is not symbolic, but rhetorical algebra. We find systems of equations and willingness to cross categorical lines in operating on quantity.

• The number concept evolves as Arab algebraists start to accept irrational numbers like square roots as solutions and as coefficients of equations. Negative numbers still frowned upon as false (but studied and operated on!!) and no consensus on the status of irrationals as numbers. Just what are these inexpressible things? Never coherently answered!

• Very few new curves get studied and coordinate geometry does not exist. Omar Khayyam (~ 1100 AD) • Bothered that some algebraists were operating on negative and irrationals as if they were numbers. Not Theoretically Justified! • First to systematically treat all cubic equations that had positive roots. Uses conic sections to solve them geometrically. Conjectures that there is no way to solve cubics purely algebraically in number, rational or irrational. • Solves geometry problems by representing them with equations, simplifying the equations using algebra until a standard form is obtained and then uses geometry to construct solution. • First to articulate a notion of number that sounds oddly like real numbers: “a ratio between magnitudes is conjoined with something numerical or in the potentiality of number.” 푥3 + 푎푥 = 푏 • “Cube plus sides (roots) equals a number.” (Note Rhetorical) We set 퐴퐵 to be a side of a square whose area equals the given number of roots. Then we construct a parallelepiped with a square base whose side is 퐴퐵 and its height is 퐵퐶, which we assume to equal the given number….We extend 퐴퐵 to Z, then construct the parabola MBD, with vertex B, axis 퐵푍 and its perpendicular side 퐴퐵…We construct a semicircle on 퐵퐶 which must intersect the section, say at D…. Khayyam’s Cubics Using Modern Notations: Evolution by Generalization Fast Forward 500 years – 16th Century Algebra Breaks its chains • Cardano (Italian): Crack the cubic and quartic. Geometry still the method for higher orders, but Algebra is Hot! • Viete (French): Quantity and operation abstracted by use of synchopated algebra with parameters. Generalized equations become objects of study! Theory of equations of a single unknown emerges. But he does not formulate the idea a numerical algebraic variable. • Stevin (Dutch): Number abstracted to linear continuum, including irrationals and maybe negatives. Put bluntly, measurement becomes so enhanced in the Renaissance that scientific theories need irrationals and common fractions to have status of numbers. And Scientists give them that status. “What can be done in geometry can also be done in arithmetic.” • Tartaglia (Italian): The study of motion, ballistics, curved trajectories, the scientific Renaissance driven by race for hegemony and wealth. • Galilleo (Italian): Use of curves to describe motions

Descartes and Fermat (~1630) Descartes and Fermat apply indeterminate equations in two unknowns to problems of geometry. [3x+1=5 is a determinate equation. 3y+2=x+6 is indeterminate] They independently come to the realization that “Whenever in a final equation two unknown quantities are found, we have a locus, and the extremity of one of those [unknowns] defining a line, straight or curved.” (Fermat)

(By 1650, number of known curves explodes!) According to Boyer:

“This brief sentence represents one of the most significant statements in the history of mathematics. It introduces not only analytic geometry, but also the immensely useful idea of an algebraic variable. …There appears to have been no appreciation before the time of Fermat and Descartes of the fact that, in general, a given equation in two unknown quantities determines per se a unique geometric curve.” What the Hell?!?! Why do we call it the “Cartesian Coordinate System”??? • Neither Fermat nor Descartes used the term “coordinate system” or the idea of two axes. Descartes gives us our symbols x and y to interchangably represent the abscissa and ordinate line segments. • Theirs is an ordinate and not a coordinate geometry! Please Welcome “Ordinate Geometry”

휽 Descartes establishes arithmetic of lengths: Defines a new algebra of lines

Both Fermat and Descartes are dealing with line segments not real numbers. But Descartes destroys homogeneity in algebra and provides geometric justification for Stevin’s insights about number, paving the way to the “ruler postulate” of modern mathematics, i.e. the existence of a number lines!

Hey, what is so special about the circle? Descartes’ compass Your students can do these linkages and study the resulting cubes using GeoGebra….

Let’s try it. Descartes Analyzes a Curve

His Ordinate Geometry: 1630-1800: Explosive Evolution From Ordinate to Coordinate Geometry Wallis (~1660) on Negative quantities: The Numberline is Born!

“…And though, as to bare algebraic notation, it [negative quantity] import a quantity less than nothing: Yet, when it comes to a physical application, it denotes as real a quantity as if the sign were +; but to be interpreted in a contrary sense….As for instance: supposing a man to have advanced or moved forward, from A toward B…and then retreat….So that +3 signifies 3 yards forward and -3 signifies 3 yards backwards from A, but still on the same line. And each defines (at least in the same infinite line) one single point.”

Evolution of Axes from lines with segments to numberlines • Wallis argues for arithmetization of curves and defines the conic curves by their equations with no reference to cones. • Leibnitz and Newton are first to systematically use two axes, usually oblique and sometimes at right . Neither uses the y-axis as a numberline but as a reference line to determine the angle of the ordinates. • Guisnee: 1718 book seems to be the first one in which both x and y coordinates in a rectangular Cartesian system are interpreted as the segments cut off on the two axes by perpendiculars to a given point. But he is still squeamish about negatives and linear equation like by = ax determine half-lines in the first quadrant only. • The distance formula was not systematically used in defining curves until mathematicians in the time of Lagrange (~1800) started routinely using rectangular coordinate systems in which plots of circle equations actually look like circles, with y and x axes both thought of as numberlines.

Descartes opened the gate for Newton (~1700) • Newton publishes essay enumerating curves given by cubic equations. It is the first instance of a work devoted solely to the theory of curves as such….Newton noted 72 species of cubic curves, and a curve of each species is carefully graphed! • There is no hesitation with respect to negative coordinates and the curves are plotted completely and correctly for all four quadrants. Y-axis still not on par with X-axis and not a numberline. Newton expands the notion of number: “By number we understand not so much a multitude of unities as the abstracted ratios of any quantity to another quantity of the same kind, which we take for unity. Number is threefold: integer, fracted, and surd, to which last, unity is incommensurable.”

His graph of 푥푦2 + 푒푦 = 푎푥3 + 푏푥2 + 푐푥 + 푑 Easy to Graph

Use sliders for the coefficients and generate family of curves of that type.

And then there was Euler (~1740) • The name “function” evolves from common usage as “the function of a house is to shelter…” As mathematicians focused on components of curves and noted dependency relations between the Y-values and the X-values, they gradually focused on curves where the ordinate Y was given by an analytic expression in X.….Y = Expression in X. • Euler was first great mathematician to make functions front and center in his work and initially defined functions as this analytic expression. But then struggle… • Later Euler (1755) “If some quantities so depend on other quantities that if the latter are changed the former undergoes change, then the former quantities are called functions of the latter. This definition applies rather widely and includes all ways in which one quantity could be determined by another. If, therefore, x denotes a variable quantity, then all quantities which depend upon x in any way, or are determined by it are called functions of x.”

19th Century

Euler avoids graphs and argues that analytic functions should be the focus of analysis and these are numerical, not geometric. The arithmetization of curves was almost complete but the numberline idea was still murky. Numbers were still thought of as abstract ratios of line segments and this was inadequate for settling disputes regarding continuity and infinitesimals. Dust clears with Boltzano, Dedekind, Weierstrass and Cantor (~ 1840 - 1870) defining the real numbers and making rigorous the definition of continuity we use today.

And then there was And then there was Today!

• National Curve Bank (Awesome) Check out the Mathematica collection. http://curvebank.calstatela.edu/home/ home.htm

• St. Andrews College Famous Curve Index http://www-history.mcs.st- and.ac.uk/history/Curves/Curves.html

Other Resources

There are free books….. There is free software I still use Geogebra Robert C. Yates A Handbook on Curves and Their Properties

Google it and download pdf My Math Ed Moment : A Tale as Old as Time (Sermon 39.6) • At first, math objects are as much physical (..ings) as conceptual (..ions). They are constructed and culturally used in many ways. Manifold conceptualizations. • The objects then get discovered as things in their own right (It’s everywhere, it’s everywhere!) Manifold names that implied contextual uses converge into the “essential form” or “informal definition” and the form is baptized with a common name. Number, curve and function are a few examples. • The essence of the (now ideal) mathematical object is explored and developed, with properties discovered and related. • The object is embedded in a theory of other such objects and finally defined within that theory, distinguishing it from other such objects in its class.

• Do we reverse this order in our curriculum? When we teach? If so, what implicit messages do our children hear? Learn it; It’s the Law!? Where is the personalizing when conventions are taken as truth about external, dour mathematical objects? Those earlier stages were the creative stages…We can still let our students experience them • Design and name their own curve and describe some of its properties • Make graphs with oblique axes • Construct conics in 3-D and experiment with the relations of ellipse and hyperbola segments • Solve quadratic equations geometrically • Use internet resources to study historical curves • Use Geogebra to animate curves and points on them. • Create their own linkages