Math (Несколько текстов для зачёта), страница 17

where the term on the right is seen to be 1/3 the area of the sector of the circle of radius R and angle Theta, yielding Proposition 2.

Any proof of this formula is equivalent to evaluating such integrals. Archimedes evaluated "[Greek text cannot be converted in ASCI text]"[sup R, sub 0]r[sup 2]dr by decomposing it into Riemann sums and obtaining a closed form for the sum 1[sup 2] + ... + n[sup 2]. In Proposition 2 this integral is computed by realizing it as the moment of a triangle and evaluating this as its weight multiplied by the distance of its center of gravity from the fulcrum.

The Way of Archimedes

The Calculus Reform movement has emphasized experimentation over rigor in calculus education and has been criticized as a result [53]. To defend its position that physical problems should be used to discover mathematical results, Harvard Calculus appeals to Archimedes and The Method [35, p. vii]:

The Way of Archimedes: Formal definitions and procedures evolve from the investigation of practical problems.

This principle accurately represents the works of Archimedes, but a disparity arises in that Harvard Calculus postpones mathematical rigor indefinitely; Archimedes's name should not be associated with such an endeavor. For example, the method of exhaustion used by Archimedes is essentially the epsilon-delta argument abandoned by Harvard Calculus, as B.L. van der Waerden writes [58, p. 220]:

... the estimations, which occur in the summing of infinite series and in limit operations, the `epsilontics', as the calculation with an arbitrary small epsilon is sometimes called, were for Archimedes an open book. In this respect, his thinking is entirely modern.

Moreover, Archimedes held in contempt those who did not furnish proofs of their results. In the introduction to On Spirals, Archimedes reveals that he intentionally announced false theorems in order to expose some of his contemporaries [6]:

... I wish now to put them in review one by one, particularly as it happens that there are two among them which [are wrong and which may serve as a warning to] those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible.

Harvard Calculus fails miserably when measured against this Way of Archimedes. Apart from the passage quoted above, the word "theorem" appears in [35] only in the name "Fundamental Theorem of Calculus." Compare this with a standard calculus text [22], which lists 130 theorems in its index. Even more revealing, the only instance of the word "proof" I located in [35] was in Archimedes's introduction to the method quoted above and used in [35] to justify "The Way of Archimedes." In fact, this quote emphasizes that discovery of the answer to a problem leads to a theorem whose proof is facilitated by knowledge of the answer. My interpretation is not Calculus Reform but

Problem-Solving: When faced with a problem, use any method that allows you to conjecture the answer, then find a rigorous proof.

A recent development: The second edition of [35] has taken a more moderate approach to Calculus Reform and now includes some complete proofs [35, 2nd Edition, p. 78] and the epsilon-delta definition of a limit [35, 2nd Edition, p. 128]. However, this new edition no longer includes "The Way of Archimedes."

Popular Misconceptions

It must be noted that the penultimate remark of the previous section paraphrases E.T. Bell [11, p. 31]: "In short he used mechanics to advance his mathematics. This is one of his titles to a modern mind: he used anything and everything that suggested itself as a weapon to attack his problems." However, strong opinions such as those expressed in [11] are fraught with danger, and it is instructive to include the continuation of this passage:

To a modern all is fair in war, love, and mathematics; to many of the ancients, mathematics was a stultified game to be played according to the prim rules imposed by the philosophically-minded Plato. According to Plato only a straightedge and a pair of compasses were to be permitted as the implements of construction in geometry. No wonder the classical geometers hammered their heads for centuries against `the three problems of antiquity': to trisect an angle; to construct a cube having double the volume of a given cube; to construct a square equal to a circle.

This has since been discredited, see [24] [41] (better yet, look at original sources, e.g., as collected in [54, Vol. 1, Chapter 9]); and van der Waerden writes [58, p. 263],

The idea, sometimes expressed, that the Greeks only permitted constructions by means of compasses and straight edge, is inadmissible. It is contradicted by the numerous constructions, which have been handed down, for the duplication of the cube and the trisection of the angle.

In particular, Archimedes trisected the angle with ruler and compass in Proposition 8 of The Book of Lemmas [6, p. 309], see [20] [31, Section 31]. The history of this misconception might prove an interesting subject for further study.

Unfortunately, it is only one of a number of popular misconceptions about the limitations of Greek science [56]. For example, Isaac Asimov (1920-1992) has written [5],

To the Greeks, experimentation seemed irrelevant. It interfered with and detracted from th e beauty of pure deduction ... To test a perfect theory with imperfect instruments did not impress the Greek philosophers as a valid way to gain knowledge ... The Greek rationalization for the "cult of uselessness" may similarly have been based on a feeling that to allow mundane knowledge (such as the distance from Athens to Corinth) to intrude on abstract thought was to allow imperfection to enter the Eden of true philosophy. Whatever the rationalization, the Greek thinkers were severely limited by their attitude. Greece was not barren of practical contributions to civilization, but even its great engineer, Archimedes of Syracuse, refused to write about his inventions and discoveries ... to maintain his amateur status, he broadcast only his achievements in pure mathematics.

This passage is contradicted by numerous examples of Greek scientific experiments, for example, Eratosthenes's measurement of the earth [4]. Asimov may be excused for paraphrasing Plutarch's account of Archimedes in his Life of Marcellus, written circa 75 AD [49] [54, Vol. 2, p. 31]:

Yet Archimedes possessed so lofty a spirit, so profound a soul, and such a wealth of scientific inquiry, that although he had acquired through his inventions a name and reputation for divine rather than human intelligence, he would not deign to leave behind a single writing on such subjects. Regarding the business of mechanics and every utilitarian art as ignoble or vulgar, he gave his zealous devotion only to those subjects whose elegance and subtlety are untrammeled by the necessities of life ...

Despite Plutarch's ancient credentials, he had no better insight into Archimedes's scientific contribution, which contradict his story. The reader is already aware that The Method shows that physical considerations played an important role in Greek mathematics. But Asimov and Plutarch are completely refuted by Archimedes in The Sand Reckoner [6] [18]:

While examining this question I have, for my part tried in the following manner, to show with the aid of instruments, the angle subtended by the sun, having its vertex at the eye. Clearly, the exact evaluation of this angle is not easy since neither vision, hands, nor the instruments required to measure this angle are reliable enough to measure it precisely. But this does not seem to me to be the place to discuss this question at length, especially because observations of this type ha ye often been reported. For the purposes of my proposition, it suffices to find an angle that is not greater than the angle subtended at the sun with vertex at the eye and to then find another angle which is not less than the angle subtended by the sun with vertex at the eye.

A long ruler having been placed on a vertical stand placed in the direction the rising sun is seen, a little cylinder was put vertically on the ruler immediately after sunrise. The sun, being at the horizon, can be looked at directly, and the ruler is oriented towards the sun and the eye placed at the end of the ruler. The cylinder being placed between the sun and the eye, occludes the sun. The cylinder is then moved further away from the eye and as soon as a small piece of the sun begins to show itself from each side of the cylinder, it is fixed.

If the eye were really to see from one point, tangents to the cylinder produced from the end of the ruler where the eye was placed would make an angle less than the angle subtended by the sun with vertex at the eye. But since the eyes do not see from a unique point, but from a certain size, one takes a certain size, of round shape, not smaller than the eye and one places it at the extremity of the ruler where the eye was placed ... the width of cylinders producing this effect is not smaller than the dimensions of the eye.

... It is therefore clear that the angle subtended by the sun with vertex at the eye is also smaller than the one hundred and sixty fourth part of a right angle, and greater than the two hundredth part of a right angle.

The correct value of the angular diameter of the sun is now known to average about 34' [26, p. 95], i.e., the 159th part of a right angle. It is important to note that this shows not only that ancient Greeks frequently performed experiments, but that Archimedes dealt with experimental error and also compensated for the fact that the human eye is part of the observational instrument, thus anticipating scientists such as Hermann von Helmoltz (1821-1894) [34]. A translation and analysis of The Sand Reckoner is given in [56].

Answers to Exercises

Exercise 1. A naive approach leads to incorrect results, evidence of the dangers of using infinitesimals, and indicating why Archimedes did not consider his method to be rigorous. For example, taking the radii of a circle of radius R, with respect to the circumference, and reordering them to form a rectangle, yields area 2piR[sup 2]. For a general figure, it's not even clear how to pick the radii. To make sense of what is going on, one regards radii as limits of sectors, i.e., infinitesimal triangles. In the case of the circle, this means that the weight of a radius, with respect to the circumference, is equal to one half its length. This can be loosely interpreted as the argument Archimedes used to compute the area of the circle [1]. In the general case, the following is justified:

Assumption 3. The weight of a radius is proportional to the square of its length.

In modern notation, this is simply

Multiple line equation(s) cannot be represented in ASCII text

where the radii have been chosen with respect to the unit circle. Given Assumption 3, one can compute the area of the spiral by using Pappus's argument [48, Book 4, Proposition 21], see also [32, p. 377] [41, p. 162].

To compute the weight of a spiral region, take each radius of the spiral, starting from the final radius, and place a disk with diameter equal to this radius at height the current angle so the resulting figure is a cone. Similarly, for each radius of the sector place a disk with diameter equal to this radius at height the current angle, resulting in a cylinder with the same base and height as the cone.

Since Euclid's Proposition 2 of Book 12 proves that "circles are to one another as the squares on the diameter, " Assumption 3 shows that the ratio of the weight of the spiral region to the weight of the sector is the same as the ratio of the volume of the cone to the volume of the cylinder. But Euclid's Proposition 10 of Book 12 proved that the volume of a cone is one third the cylinder with the same base and height, so the spiral weighs one third of the sector, which is the statement of Proposition 2. (Note that equilateral triangles could have been used instead of circles resulting in a pyramid whose volume is easier to compute.)

Knorr [40] comments that this appeal to three-dimensional figures might have been considered inelegant by Archimedes as it uses volumes to compute areas. On the other hand, reversing this argument and using the evaluation above shows that the volume of a cone can be computed by the mechanical method, a result which does not appear in The Method.

Exercise 2. In modern notation, Archimedes's formulation of Proposition 1 is Area of circle of radius R = Integral[sup R, sub 0] 2pirdr, for the integral represents the area of a right triangle with base R and height 2piR.

Exercise 3. This is equivalent to the fact that the length of an arc of fixed angle is proportional to its radius. In particular, pi exists, see [45] [56]. The proof is similar to [23, Book 12, Proposition 2] cited in Exercise 1, and is implicit in Archimedes's Measurement of the Circle. Similarly, the length of an arc of fixed radius is proportional to its angle.