1. History & Philosophy of Calculus, Session 1 ANCIENT GREEK GEOMETRY

2. CONIC SECTIONS

3.  Euclid’s Elements deals only with straight lines and circles, and figures that can be made with them.  In a sense these are the simplest of the “curves”.  The next step up from these objects are the conics.  Historically, these were motivated by the Delian problem and other puzzles in which they arise quite naturally. STRAIGHT LINES & CIRCLES

4.  What we now call “conic sections” were related to cones by Apollonius, at least a century after work on problems involving them began. CONICS

5. Page from a C9th Arabic translation of Apollonius’s treatise on conics.

6. ARCHIMEDES’ METHOD OF EXHAUSTION

7.  Quadrature is the art of comparing the areas of shapes. Which of these two shapes has the greater area?  The word means “squaring”. The way you classically compare two shapes’ areas is by constructing for each a square that has the same area. Then the two squares can be compared directly.  One ancient problem was the quadrature of the circle; this was never solved, and we now know it can’t be solved by the methods used by the Greeks. QUADRATURE

8. SQUARING THE CIRCLE

9.  The previous image suggests that the area of the circle is about half the circumference times the radius.  The more “slices” we take, the less curved the outer edge of the slice is; the closer it is to being a triangle. So the more accurate this estimate will be.  But the estimate is the same regardless of how many slices we take. So if we imagine taking an infinite number, we still get the same estimate.  Does this agree with what you learned at school about areas of circles?  C= 2πr  So our estimate says that A = ½ x 2πr x r = πr 2  Notice that Archimedes doesn’t need to tangle directly with the troublesome number π here! SQUARING THE CIRCLE

10.  A parabola is a type of conic section. Once you start working with these more complicated curves, it’s natural to ask how much of our existing geometry transfers over to them.  One natural question is whether we can find their areas. A parabola is an infinite curve but we can ask about the part cut off by an arbitrary line. SQUARING THE PARABOLA We want to find the areas of this region, bounded by the parabola (orange) and the straight line (black).

11.  Find the point on the parabola where the tangent is parallel to the cutting-off line.  Form a triangle as shown, using that point at the third vertex. FIRST APPROXIMATION

12. SECOND APPROXIMATION

13.  Notice that the second approximation uses exactly the same technique as the first, it just does it twice. We can repeat it as many times as we like.  Each time we repeat it, we’ll get a closer estimate to the area we want (though it’ll always be a little bit low).  Archimedes already had a proof that the areas of the two green triangles added together is a quarter that of the pink triangle.  So, the idea is to use this knowledge to ask what happens if we just keep on going, repeating the same procedure an infinite number of times. FROM FIRST TO SECOND

14.  STEP 1 gives a triangle of area A. This is easy to find.  STEP 2 gives a triangle of area A plus two more that add up to ¼A  STEP 3 will give us A + ¼A + ¼(¼A), since the next set of four triangles will have a quarter of the area of the two triangles from STEP 2.  STEP N will give us A + ¼A + ¼(¼A) + ¼(¼(¼A))+…  Notice that what we add each time is a LOT smaller than what we added last time. So we might not be surprised if this doesn’t “shoot off to infinity”.  And geometrically, we can see that the area will be constrained by the parabola; it can’t expand to infinity.  So the formula is: A + (¼)A + (¼) 2 A + (¼) 3 A + (¼) 4 A …  This goes on forever. But does that make any sense? What value could it possibly have? ARCHIMEDES’ APPROXIMATIONS

15. AN INFINITE SUM…

16. ARCHIMEDES’ FORMULA