Teaching Great Ideas of Mathematics with Mathematica

The authors use Mathematica to present some of the great ideas of mathematics to liberal arts students

by Alkiviadis G. Akritas and Zamir Bavel

Introduction.

In the spring semester of 1997 we introduced a new course at the University of Kansas.  The course was titled Journey through Genius, after the book with the same title by William Dunham [Dunham 1990]. We used  additional sources [Arnol'd 1990, Dunham 1994, and Polya 1954] and, at times, we presented additional  perspectives.

Since the majority of the students majored neither in mathematics nor in physics, it was obvious that the class would have difficulties comprehending the subtle points involved, carrying out some of the intricate and complicated computations requisite in certain sections of the  textbook, and completing the homework assignments.  Therefore, we decided not to emphasize hand calculations but, instead to focus on mastering the general concepts involved and to rely on computer-aided  symbolic manipulation with Mathematica.

The results were excellent and the students acknowledged that, after having taken this course, they had a better understanding of how mathematics was done.

Below we describe some of the topics covered and the ways we made use of Mathematica's functions and packages to enhance the students' understanding of these topics.

Eratosthenes, accuracy of computations and interval 
arithmetic.

Before the time of Galileo and Copernicus, there existed a man whose accomplishments formed the framework of modern geography, astronomy, and number theory. This man, Eratosthenes (ca. 284-192 BC), was one of the great men of ancient Greece.  He was born in Cyrene, the modern day Shahhat, in Libya.

In his youth, Eratosthenes studied at Alexandria and Athens. When he was forty, he began  tutoring the son of King Ptolemy III of Egypt.  After that he became the director of the Alexandria Library. It was then that  Eratosthenes’ greatest works were created.

Eratosthenes was a man of many facets—he knew at least a little of everything. To his  students he was known as "pentathlus", a champion in five athletic sports,  but "beta" (β) is the nickname he is best known for. Historians have  speculated that beta stood for being a second Plato, or second to his contemporary Archimedes, or just second best at everything.

By being knowledgeable in many areas, Eratosthenes made many contributions.  He included the fundamentals of astronomy in his long poem, Hermes. He also created a chronology and a calendar that included leap years. Most of the works written by Eratosthenes have been lost. The  only information about him and his work comes from commentators. Even some of  his most famous writings, such as On the Measurement of the Earth, have never been found.

Eratosthenes computed the circumference of the earth at 250,000 stadia.  His calculation was based on the formula

EarthCircumference  =  (360/7.2)* 5000 stadia

where 7.2 is the angle by which the sun missed being directly overhead in Alexandria on the first day of summer, and 5000 is the distance, in stadia, between Alexandria and Syene, where (on the same day) the sun was directly overhead (Figure 1).  (Alexandria and Syene are close to the meridian and their distance was known.)

According to the best estimate scholars have been able to determine, 1 stadium = 0.1575 km, with accuracy assumed to 0.5 meters [Zachary 1997].  Therefore, the circumference of the earth comes out to about

EarthCircumference*0.1575 km/stadia

which equals

Needs["Miscellaneous`Units`"]

Convert[39375 (10^3) Meter,  Mile]

or, about 24,466 miles.  We pointed out to the students that Eratosthenes' answer (given his  measurements) was as accurate as was possible to obtain—a statement the students  later verified.

In the process, the students learned that any result that involves measurements, cannot  be exact.  We called a measurement accurate if its true value lay (or, was believed to lie) within a specified range;  we also called a measurement precise if that range was "small".

We used the standard practice in science to describe the range in which a  measurement lies by giving only the significant digits of this measurement  (starting with the first nonzero integer).  Thus, rounding to the nearest  foot, when we say that a certain distance is 216.0 meters,  we imply that its true value is somewhere between 215.5 and 216.5 meters.  Saying that this distance is 216.732 meters is more precise (it has 6 significant digits) because now the range is much  narrower—the true value lies between 216.7315 and 216.7325.

Figure 1

There is insufficient information on the accuracy of the figure of 5000 stadia as the distance between Alexandria and Syene.  Since Eratosthenes used surveyors to obtain the figure, it is unlikely that the true value lies between 4999.5 and 5000.5.  The means of measurement of the times lead us to believe that we can do no better than to place the actual value between 4950 and 5050 stadia.

We then asked the students to verify our earlier statement.  Using Mathematica's Interval function (for doing interval arithmetic), the following expression yields  the circumference of the earth as an interval

EarthCircumference  = 
      ((360 )/Interval)[{7.15,  7.25}]  *
        (Interval[{4950,  5050}] * Interval[{0.15745,  0.15755}])

Map[Convert[# (10^3) Meter,  Mile]&, Last[EarthCircumference]]

within which lies Eratosthenes' answer.   (The currently accepted value for the circumference of the earth is 24,860 miles!)

Eratosthenes' estimation of the circumference of the earth implied a surface area much greater than the area of the earth known at the time. Thus the value of 250,000 stadia was not widely accepted.  About two centuries later an "improved" circumference of 180,000 stadia was computed which, combined with an overestimate of the size of Asia, led people to believe that it was only 70,000 stadia from the Atlantic ocean to India.

When Columbus was planning to sail westwards in 1492, hoping to reach India, the problem he was facing centered on the size, not the shape, of the earth.  It was the "improved" circumference of  the earth (180,000 stadia) that ultimately convinced Columbus to attempt his westward voyage to India.

The Bernoulli brothers, and exact versus approximate 
computations.

Jacob (1654-1705) and Johann (1667-1748) Bernoulli constitute mathematics'  most important fraternal success story—despite the fact that their  relationship was far from harmonious.  Born in Switzerland, they were  enthusiastic followers of Leibniz and were the major force in disseminating  and promoting the calculus throughout Europe.  Their efforts, perhaps as much  as those of Leibniz himself, gave the subject the flavor and appearance that  it retains today.  (It should be kept in mind that, de l'Hospital's book Analyse des infiniment petits, written in the vernacular rather than in Latin, was almost exclusively  Johann Bernoulli's in all but the name on the title page.)

Both Bernoulli brothers are associated with the harmonic series; Johann  proved that it is infinite, and Jacob published the result in his Treatise on Infinite Series (acknowledging his brother's priority).

The harmonic series is an example of a series that defies intuition.  One  must add its first 12,367 terms to get 10, and for the sum to climb to 20 it  takes about a quarter of a billion terms!  To think that the harmonic series could eventually surpass one  hundred, or one thousand seems completely out of the question.  But, of course it does.

Regarding the harmonic series, 1 + (1/2) + (1/3) + ⋯, one is tempted to try out several additions of ever increasing number of  successive terms in order to get a feeling for the overall sum.  These  additions can be done in (exact) rational arithmetic, in which case the  result is a fraction, or in (approximate) floating-point arithmetic.  For  example, the first few terms of the harmonic series can be computed as

1 + (1/2) + (1/3) + (1/4) + (1/5)

or as

1.0 + 0.5 + 0.3333 + 0.25 + 0.2

Given that (137/60) is approximated by the same number  2.2833, it is tempting to conclude  that there is no difference between the above two ways of adding terms.  We  demonstrated the opposite to the class as follows.

Floating-point arithmetic involves a finite set of numbers.  Therefore, an actual calculation may result in a number outside the floating-point set.  In this case, the result has to be approximated by its closest floating-point number —an act that generates round-off errors. (In case of ties, the conflict can be resolved by truncation, by rounding to the one even endpoint,  or by rounding away from zero.)  Therefore, when we add sufficiently many  floating-point numbers, each one of which is subjected to a round-off error,  these errors accumulate and may distort the final result.

To enhance the students' understanding of floating-point numbers and to  make clear that floating point arithmetic is not  always  accurate and results depend on the order of the arithmetic operations, we used Mathematica's standard add-on package.

Needs["\(NumericalMath`ComputerArithmetic` \*""]"\)

(To our knowledge, this package is the only tool available in the scientific community for simulating floating point arithmetic.)  With its help, we simulated a floating-point environment base 10, with 2 digits precision, and were able to show the main ideas using only 20 terms of the harmonic series.   Otherwise, using the full precision of the computer, we would have had to employ several thousand terms of the harmonic series in order to demonstrate the result below.

The expression

SetArithmetic[2, 10];

provides us with a base 10 environment with 2 digits precision (with our machine precision at 16).  Then, we generate in this environment the first 20 terms of the harmonic series

table1  =  Table[ComputerNumber[(1/i)],  {i,  1,  20}]

Many of these numbers are not in the simulated floating-point set and have to be approximated by the nearest floating-point equivalent, resulting in an abundance of round-off errors.  Therefore, the following expression computes their sum as

sum1 = Fold[Plus,  0,  table1]

while the true value is 3.59774 (to 5 decimals)

(∑_(i = 1)^20) (1. /i)

We can actually observe the running sum being formed in Table 1 below.

\(sum1 = FoldList[Plus, \ 0, \ table1]; 

    TableForm[\ SetAccuracy\ [Transpose[{table1, \ Drop[sum1, 1]}], 4], 
      TableHeadings -> {
          Automatic, {"Harmonic
 Series", \ "Running
 Sum "}}, 
      TableSpacing -> {1, 2}]\)

Table 1

Taking the sum of the first 40 terms, we again obtain the same value 3.80000, instead of the correct result (to five decimals) 4.27854.

\(table2\  = \ Table[ComputerNumber[1\/i], \ {i, \ 1, \ 40}]; 

    Fold[Plus, \ 0, \ table2]\)

Reversing the order of the addition, that is, starting the addition from the last of the twenty terms, yields quite different results:

sum2 = Fold[Plus, 0, Reverse[table1]]

In the case of the 40 terms, adding them in reverse order, results in 4.20000, instead of 4.27854.

Fold[Plus, 0, Reverse[table2]]

The students found these results impressive and learned to appreciate the dangers of floating-point arithmetic and of accumulating errors that do not cancel each other.

Euclid and the prime numbers

Euclid (330?-275? BC) is one of the most influential and best read  mathematicians of all time. His prize work, Elements, was the textbook of elementary geometry and logic up to the early  twentieth century. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians.

Very little is known about the life of Euclid. Both the dates and place of his birth and death are unknown. It is believed that he was educated at Plato's academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly-founded university in Alexandria. There, Euclid founded the school of mathematics and remained for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes.

Euclid's Elements is a thirteen-volume compilation of Greek mathematics and geometry. It is unknown how much, if any, of the work included in Elements is Euclid's original work; many of the theorems found can be traced to  previous thinkers including Euxodus, Thales, Hippocrates and Pythagoras.  However, the format of Elements—definitions and postulates followed by theorems whose proofs were based on those definitions and postulates—belongs to Euclid alone.. Every statement, no matter how  obvious, was proved. Euclid chose his postulates carefully, picking only the most basic and self-evident propositions as the basis of his work.  Before Euclid, rival  schools of geometry each had a different set of postulates, some of which were questionable.  Euclid's format helped standardize Greek mathematics.  As for the subject matter, it ran the gamut of ancient thought.  Especially noteworthy  subjects include the method of exhaustion, which would be used by Archimedes in the invention of integral calculus,  and the proof that the set of all prime numbers is infinite.

In preparation for Euclid's proof of the infinitude of primes, we  introduced the students to the historical worry of the Greeks that the number of primes may dwindle to nothing as their sizes increase.  This was demonstrated by such Mathematica computations as

PrimePi[100000]

(the number of primes less than or equal to 100,000) whereas

PrimePi[900000] - PrimePi[800000]

Archimedes

Archimedes is considered one of the greatest mathematicians of all time. In  his own time, he was known as "alpha" (α, the  first—compared with Eratosthenes' β), "the wise one,"  "the master" and "the great geometer" and his works and inventions  brought him fame that lasts to this very day.  He was one of the last great  Greek mathematicians.

Born in 287 B.C. in Syracuse, a Greek seaport colony in Sicily, Archimedes was the son of Phidias, an astronomer.  Except for his studies at Euclid's school in Alexandria, he spent his entire life in his birthplace.  Archimedes proved to be a master at mathematics and spent most of his time contemplating new problems to solve, becoming at times so involved in his work that he forgot to eat.  Lacking the blackboards and paper of modern times, he used any available surface, from the dust on the ground to ashes from an extinguished fire, to draw his geometric figures.  Never giving up an opportunity to ponder his work, after bathing and anointing himself with olive oil, he would trace figures in the oil on his own skin.

Much of Archimedes' fame comes from his relationship with Hiero, the king  of Syracuse, and Gelon, Hiero's son.  The great geometer had a close friendship with, and may have been related to, the monarch.  In any case, he  seemed to make a hobby out of solving the king's most complicated problems to  the utter amazement of the sovereign.  The story of the discovery of Archimedes' Principle ("Eureka!"—I  have found it!), which led to the field of hydrostatics, is well known. Another  time, Archimedes stated "Give me a place to stand on and I will move the  earth."  King Hiero, who was absolutely astonished by the statement, asked  him to prove it.  In the harbor was a ship that had proved impossible to  launch even by the combined efforts of all the men of Syracuse.  Archimedes,  who had been examining the properties of levers and pulleys, built a machine  that allowed him the single-handedly move the ship from a distance.  He  also had many other inventions including the Archimedes' watering screw and a  miniature planetarium.

Though he had many great practical inventions, Archimedes considered his purely  theoretical work to be his true calling.  His accomplishments are numerous.   His approximation of π between 3(10/71) and 3(1/7) was the most accurate of his time and he devised a new way to approximate  square roots.  Unhappy with the unwieldy Greek number system, he devised his  own that could accommodate larger numbers more easily.  However, as we will see below, his greatest invention was that of the integral calculus.   He also anticipated the invention of differential calculus as he devised  ways to approximate the slope of the tangent lines to his figures.  In  addition, he made many discoveries in geometry, mechanics and  other fields.

Archimedes was killed by a Roman soldier during the siege of Syracuse. The traditional story is that the mathematician was unaware of the taking of the city. While he was drawing figures in the dust, a Roman soldier stepped on them and demanded he come with him. Archimedes responded, "Don't disturb my circles!" The soldier was so enraged that he pulled out his sword and slew the great geometer.  When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

Archimedes' approximation of π

To compute the ratio of the circumference of any circle to its diameter, Archimedes inscribed and circumscribed regular polygons, and concentrated on their perimeters.  He started the process by inscribing a regular hexagon in a circle whose radius equaled the side of the hexagon; then he computed the perimeter of the hexagon.  He then doubled the number of sides of his inscribed polygon, to get a regular dodecagon whose perimeter he then calculated (Figure 2).

\(\(Needs["Graphics`MultipleListPlot`"]; 
    circle = Graphics[Circle[{0, 0}, 1]]; 

    Show[circle, Graphics[RegularPolygon[6]], Graphics[RegularPolygon[12]], 
      AspectRatio -> Automatic]; 
\)\)

Figure 2

Archimedes continued doubling the number of sides of the inscribed regular polygon to get a regular 24-gon, then a regular 48-gon, and finally a regular  96-gon, and calculated their respective perimeters.  He repeated the  comparable calculations for the circumscribed polygons.

Archimedes found that π was less than 3(1/7) but greater than 3(10/71).  Converting these rational numbers to floating-point arithmetic, we have  3.140845 < π < 3.142857.  What leaves modern mathematicians shaking their  heads in wonder is the fact that at each stage of his computations he had to  approximate sophisticated square roots, yet he never faltered.  (For example,  determining the perimeter of the dodecagon required a numerical value for the  square root of three.)  With our calculators and computers, this strikes us  as no real obstacle, but in Archimedes' time, not only were these devices  unknown, but there was not even a good number system to facilitate such  computations.

To demonstrate Archimedes' formidable task, and to show how slowly the  accuracy of the above calculations increases, we asked the students to repeat  Archimedes' calculations for the inscribed regular polygons using Mathematica.  They used the formula t = (Sqrt[2 - (Sqrt[4 - (s^2)])]) (from an earlier exercise)  to get the length t of the side of a regular polygon with twice as many sides as the polygon  with sides of length s.  They also used the functions Nest, and NestList, where, for example, Nest[f, x, 2] results in f[f[x]], whereas NestList[f, x, 2] results in the list {f[x], f[f[x]]}.

To show the square roots Archimedes had to approximate, we started with a  regular hexagon, s = 1, and obtained the length of the sides of a regular 12-gon, 24-gon,  48-gon, and 96-gon using exact arithmetic.  Based on these lengths, the  approximate value of π is as shown in Table 2.

\(lengthsOfSides = NestList[\@\(2 - \@\(4 - #\^2\)\)&, \ 1, 4]\);
TableForm[Prepend[
            Table[{6*(2^i), lengthsOfSides[[i + 1]], 
                lengthsOfSides[[i + 1]]*6. *(2^i), 
                lengthsOfSides[[i + 1]]*6. *(2^i)/2}, {i, 4}], {6, 1, 6, 
              3}],TableHeadings -> {
                None, {"No. of\nSides", "Length of each Side", 
                  "Peri-\n  meter", "   π\nApprox\n"}},TableSpacing -> {1, 2}]

Table 2

The last square root is the expression for the exact value of the length of the side of the 96-gon.  Based on this length we  obtain 3.14103 as a lower bound on the value of π , as compared to  Archimedes' 3.140845.  Archimedes' interval is slightly wider.

Table 2 verifies how slowly the accuracy increases.  As another example,  we started again with a regular hexagon, s = 1, and doubled the number of sides 7 times.  The following expression  approximates (with 20-digit precision) the length of the side of the  resulting 768-gon (768 = 6*2^7).

\(Nest[\@\(2 - \@\(4 - #\^2\)\)&, \ N[1, 20], 7]\)

When multiplied by 768 (the number of the sides), the perimeter of the  768-gon is 6.283167784296636817. Half of this last result is  3.141583892148318409, an approximation of π accurate to four decimals  3.1415.

Archimedes' discovery of the integral calculus

Archimedes computed the volume of a sphere with his now famous variable cross-sections method that is the forerunner to Riemann integration.  The presentation of this method below, is that of Polya [Polya 1954] interspersed with Mathematica diagrams for better understanding.

Archimedes regards the sphere as generated by a revolving circle, and he regards the circle as a locus, characterized by a relation between the distances of a variable point from two fixed rectangular axes of reference.  Written in modern notation, this relation is

(x^2)+ (y^2) = 2ax,

a circle with radius a that touches the y-axis at the origin, as shown in Figure 3 below for a circle of radius 1.   (Note that a = 1 throughout our discussion.)

\(a = 1; 

    sphere2D = 
      ParametricPlot[{{x, \@\(2*a*x - x\^2\)}, {x, 
            \(-\@\(2*a*x - x\^2\)\)}}, {x, 0, 2  a}, 
        AspectRatio -> Automatic, Axes -> True, 
        AxesLabel -> {"x", "y"}, Ticks -> {{0, 1, 2}, Automatic}]; 
    \)

Figure 3

The circle, revolving about the x-axis, generates a sphere.  As Polya indicates, using modern notation  does not distort Archimedes' idea.  On the contrary, it seems to be  suggestive.  It suggests motives which may lead us to Archimedes' idea today  and which are, perhaps, not too different from the motives that led  Archimedes himself to his discovery.

Consider the term (y^2) in the equation of the circle.  Observe that ((πy)^2) is the area of a variable cross-section of the sphere.  Yet Democritus found the volume of the cone by examining the variation  of its cross-section.  This leads us to rewrite the equation of the circle  in the form

π(x^2) + π(y^2)  =  π2ax

Now we can interpret ((πx)^2) as the variable cross-section of a cone, generated by the rotation of the  line y = x about the x-axis (Figure 4).

\(a = 1; 

    cone2D = ParametricPlot[{{x, x}, {x, \(-x\)}}, {x, 0, 2  a}, 
        AspectRatio -> Automatic, Axes -> True, 
        AxesLabel -> {"x", "y"}, Ticks -> {{0, 1, 2}, Automatic}, 

        		DisplayFunction -> Identity]; \)

\(\(Show[%, \ Graphics[Line[{{2  a, 2  a}, {2  a, \(-2\) a}}]], 
	
      DisplayFunction -> $DisplayFunction]; \)\)

Figure 4

This suggests that we seek an analogous interpretation of the remaining term  π2ax.  To see such an interpretation, we experiment with rewriting the equation, and so we hit upon the form

2a(((πy)^2)+((πx)^2)) = xπ(((2a))^2)

Now the term on the right-hand side of equation (3) is the cross-section  of a cylinder  generated by the rotation of the rectangle shown in Figure 5  about the x-axis.

\(a = 1; 

    cylinder2D = 
      Show[Graphics[
          Line[{{2  a, 0}, {2  a, 2  a}, {0, 2  a}, {0, \(-2\) a}, {2  a, 
                \(-2\) a}, {2  a, 0}}]], AspectRatio -> Automatic, 
        Axes -> True, AxesLabel -> {"x", "y"}, 
        Ticks -> {{0, 1, 2}, Automatic}]\ ; \)

Figure 5

We combine Figures 3, 4 and 5 to obtain Figure 6.

Show[sphere2D, cone2D, cylinder2D, AspectRatio -> Automatic, 
      Axes -> True, AxesLabel -> {"x", "y"}, 
      Ticks -> {{0, 1, 2}, Automatic}] ;

Figure 6

Revolving Figure 6 about the x-axis results in a sphere, a cone and a cylinder.  Notice that the cone  and the cylinder have the same base and height (the vertex of the cone is at the origin (0,0,0)).  The radius of the common base and the height have the same length 2a.  In Figure 7 below we generate these solids using Mathematica's add-on package Graphics`Shapes`.  (The function RotateShape below rotates a three-dimensional object by the specified Euler angles.)

Needs["Graphics`Shapes`"]

\(a = 1; 

    g1 = RotateShape[Graphics3D[Cylinder[2  a, 1, 20]], 0, 
		π\/2, 
          π\/2] // WireFrame; 
    g2 = RotateShape[Graphics3D[Cone[2  a, 1, 20]], 0, 
		π\/2, 
          π\/2] // WireFrame; 

    g3 = RotateShape[Graphics3D[Sphere[a, 20, 15]], 0, 
		π\/2, 
          π\/2] // WireFrame; 

    Show[RotateShape[{g1, g2, g3}, π, 0, 0], Boxed -> False, 
      ViewPoint -> {\(-1.010\), \ \(-5.546\), \ \(-0.484\)}]; \)

Figure 7

Several ideas are contained in the equation above.  We notice the areas of three circular disks, π(y^2), π(x^2), and ((π(2a))^2).  The three circles are the intersections of the same plane with three  solids of revolution.  The plane is perpendicular to the x-axis, and at the distance x from the origin (0 ≤ x ≤ 2a).  With Mathematica, we view this construction in Figure 8.

\(a = 1; 

    Show[Graphics3D[Cylinder[2  a, 1, 20]], Graphics3D[Cone[2  a, 1, 20]], 
      Graphics3D[Sphere[a, 20, 15]], PlotRange -> {\(-1\),  .6}, 
      Boxed -> False]; \)

Figure 8

Archimedes treats differently the disks, the areas of which appear on  different sides of the equation.  He leaves the disk with radius 2a, the cross-section of the cylinder, in its original position.  Yet he removes  the disks with radii y and x, cross-sections of the sphere and the cone, respectively, from their  original position and transports them to the point (-2a, 0) of the x-axis.  (We show the suspended disk of the sphere with radius y between two empty circles, and the disk of the cone with radius x between two filled squares.  The disk of the cylinder, at abscissa x, is shown between two filled circles.)  We let these disks with radii y and x hang with their center vertically under (-2a, 0), suspended by a string of negligible weight (Figure 9).  (This string  is an addition to Archimedes' original figure.)

\(a = 1; 

    Show[sphere2D, cone2D, cylinder2D, 
		
      Graphics[{{Dashing[{0.02, 0.02}], Circle[{\(-2\), \(-1\)}, 1]}, 
          Circle[{\(-2.91652\), \(- .6\)},  .1], 
          Circle[{\(-1.08348\), \(- .6\)},  .1], Circle[{ .59,  .9},  .1], 
          Circle[{ .59, \(- .9\)},  .1]}], 
	
      Graphics[{Disk[{ .6, 2},  .1], Disk[{ .6, \(-2\)},  .1]}], 
	
      Graphics[{Line[{{ .6, 2}, { .6, \(-2\)}}], 
          Line[{{\(-1.08348\), \(- .6\)}, {\(-2.91652\), \(- .6\)}}], 

          	{Dashing[{0.02, 0.02}], 
            Line[{{\(-2\) a, \(-2\) a}, {\(-4\) a, \(-4\) a}, {0, \(-4\) a}, {
                  \(-2\) a, \(-2\) a}}]}, 
	
          Line[{{\(-2.6\), \(-2.60\)}, {\(-1.4\), \(-2.60\)}}], 
	
          Line[{{\(-2\) a, 0}, {\(-2\) a, \(-2.60\)}}]}], 
	
      Graphics[{Point[{\(-2\) a, 0}], Point[{\(-2\) a, \(- .6\)}], 
          Point[{\(-2\) a, \(-2.60\)}]}], 
	
      Graphics[{Text["<---2--->", {\(-a\),  .2}], 
	
          Text["<---2--->", {a, 2  a +  .2}], 
	
          Text["-x-", { .3, \(-2\) a -  .1}]}], 
	
      Graphics[{
          Rectangle[{\(-2\) a -  .7, \(-2\) a -  .5}, {\(-2\) a -  .5, 
              \(-2\) a -  .7}], 
          Rectangle[{\(-2\) a +  .7, \(-2\) a -  .5}, {\(-2\) a +  .5, 
              \(-2\) a -  .7}], Rectangle[{ .7,  .5}, { .5,  .7}], 
          Rectangle[{ .7, \(- .5\)}, { .5, \(- .7\)}]}], 
	
      Prolog -> AbsolutePointSize[3], 
AspectRatio -> Automatic, 
      Axes -> True, 	AxesLabel -> {"x", "y"}, 
	
      PlotRange -> {{\(-4\), 4}, Automatic}]\ ; \)

Figure 9

Let us regard the x-axis as a lever, a rigid bar of negligible weight, and the origin o as its point of  suspension.  A moment is the product of the weight and the arm of the lever.   Our equation says that the moment of the two disks on the left-hand side  equals the moment of the one disk on the right-hand side and so, by the  mechanical law discovered by Archimedes, the lever is in equilibrium.

As x varies from 0 to 2a, we obtain all cross-sections of the cylinder; these cross-sections fill the cylinder.  To each cross-section of the cylinder there correspond two  cross-sections suspended from the point (-2a, 0) and these cross-sections fill the sphere and the cone respectively.  As their corresponding cross-sections, the sphere and the cone, hanging  from (-2a, 0), are in equilibrium with the cylinder.  Therefore, By Archimedes' mechanical law, the moments must be equal.   Let V denote the volume of the sphere, let us recall the expression for the volume of  the cone (due to Democritus) and also the volume of the cylinder and the obvious location of its center of gravity.  Passing from the moments of the cross-sections to the moments of the  corresponding solids, equation (3) leads to

2  a (V + ((π (((2  a))^2) 2  a)/3)) = 
      aπ (((2  a))^2) 2  a,

which yields

V = ((4  π (a^3))/3.)

Looking back at the foregoing, we see that the decisive step is that from the filling cross-sections to the full solids.  Yet this step is only heuristically assumed, not logically justified.  It is plausible but not demonstrative.  It is a guess, not a proof.  And Archimedes, representing the great tradition of Greek mathematical rigor, knows this full well: "The fact at which we arrived is not actually demonstrated by the argument used; but the argument has given a sort of indication that the conclusion is true."  This guess, however, is a guess with prospect.  The idea goes much beyond the requirements of the problem at hand, and has an immensely greater scope.

The passage from the cross-section to the whole solid, is in more modern language the transition from the infinitesimal part to the whole quantity, from the differential to the integral.  This transition is a great beginning, and Archimedes, who was a great enough man to see himself in historical perspective, knew it full well: "I am persuaded that this method will be of no little service to mathematics.  For I foresee that this method, once understood, will be used to discover other theorems that have not yet occurred to me, by other mathematicians, now living or yet unborn."

Introduction to modern integral calculus

We used the concepts and methods of the preceding section and the tools of Mathematica to make the students understand the "approach to the limit", the concept of area under the curve as an integral and the concept of successive approximation; without the formality of limits and integration.

Most of the students were unfamiliar with integration, either the concept or the practice.  Archimedes' variable cross-sections method, described above, served both as a motivation and an introduction to the subject.

We imitated Archimedes' method of the moving cross-section in the case of the (two-dimensional) ellipse defined by the equation ((x^2)/(a^2)) + ((y^2)/(b^2)) = 1, where we took a = 1, and b = 0.6, and a variable cross-section is a rectangle.

First we established an approximation of the area of the ellipse as the goal.  We took  the area of the ellipse as the area between the functions defined by f(x) and  g(x) below.

a  =  1; b  =  0.6;
f[x_] := b*(Sqrt[1 - ((x^2)/(a^2))]);g[x_]:=- b*(Sqrt[1 - ((x^2)/(a^2))]);

Using Mathematica we obtained the value 1.88496.

(∫_(-1)^1) 2*f[x] \[DifferentialD]x

We then used the Mathematica package AreaBetweenCurves.m [Finch and Lehmann 1992], modified to give upper and lower bounds, and computed the approximate area for 6 approximating rectangles (see Figure 10).   We then did the same for 40 and 80 rectangles (see Figures 11, and 12) at which point we abandoned the diagrammatic illustration as too dense, but continued computing successive approximations, as shown in Table 3 below.

<< AreaBetweenCurves.m

AreaBet[f, g, -1, 1, 6]

Figure 10

AreaBet[f, g, -1, 1, 40]

Figure 11

AreaBet[f, g, -1, 1, 80]

Figure 12

The approximation of the area by rectangles resulted from NAreaBetLower[f,g,-1,1,n], NAreaBet[f,g,-1,1,n], and NAreaBetUpper[f,g,-1,1,n] for n = 6, 40, 80, 250, 500, and 1000.

$RecursionLimit = 20000;
TableForm[
	Map[{#,NAreaBetLower[f, g, -1, 1, #], NAreaBet[f, g, -1, 1, #], NAreaBetUpper[f, g, -1, 1, #]}&,{6, 40, 80, 250, 500, 1000}]],TableHeadings -> {
              None, {"Recta\nngles", "Lower \n Bound", 
                "Mid\npoint", "Upper \n Bound\n"}},TableSpacing -> {1, 2}]

Table 3

Conclusion

It is not our intention to advocate that students do not study calculus or other means of calculating.  But even undergraduate students in mathematics, let alone the multitude who have not studied mathematics, would not be able to acquire most of the concepts presented in the Journey Through Genius course without the facilities of a symbolic computation system such as Mathematica.  We would be able to present only a small fraction of the present content of the course, and even then the students would have only a vague idea without the visual demonstrations that Mathematica makes possible.  In fact we champion the judicious use of Mathematica in such mathematics courses as calculus, not to deprive the students of learning computational techniques, but to enhance their understanding and perception of the principles and concepts involved.

Acknowledgements

Historical material was obtained from the following web sites: ¬http://www.shu.edu/~wachsmut/reals/history, (on Archimedes, and Euclid), and ¬http://www.thisweb.com/ThisMath/text/eratos.htm, (on Eratosthenes). Helpful discussions were held with Dr. Stelios Kapranidis of the University of South Carolina-Aiken.

References:

Arnol'd, V. I.. Huygens and Barrow, Newton and Hooke  (Translated from the Russian) Birkhäuser Verlag, Berlin, 1990.
Dunham, W.. Journey Through Genius-The Great Theorems of Mathematics.  J. Wiley, New York, 1990.
Dunham, W.. The Mathematical Universe.  J. Wiley, New York, 1994.
Finch, J.K. and M. Lehmann. Exploring Calculus with Mathematica. Addison-Wesley, New York, 1992.
Polya, G.. Induction and Analogy in Mathematics.  Princeton University Press, Princeton, New Jersey, 1954.
Zachary, J..  Introduction to Scientific Programming.  Telos, Santa Clara, California.  1997

ABOUT THE AUTHORS

Alkiviadis G. Akritas is (*? professor of....his interests include....*)

Zamir Bavel is (*? professor...interest...*)

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