This is chapter 11 of the [1]"The Phenomenon of Science" by [2]Valentin F.
   Turchin
     ____________________________________________________________________________

   Contents:
     * [3]NUMBER AND QUANTITY
     * [4]GEOMETRIC ALGEBRA
     * [5]ARCHIMEDES AND APOLLONIUS
     * [6]THE DECLINE OF GREEK MATHEMATICS
     * [7]ARITHMETIC ALGEBRA
     * [8]ITALY, SIXTEENTH CENTURY
     * [9]LETTER SYMBOLISM
     * [10]WHAT DID DESCARTES DO?
     * [11]THE RELATION AS AN OBJECT
     * [12]DESCARTES AND FERMAT
     * [13]THE PATH TO DISCOVERY
     ____________________________________________________________________________

                                     CHAPTER ELEVEN.
                                 From Euclid to Descartes

NUMBER AND QUANTITY

   DURING THE TIME of Pythagoras and the early Pythagoreans, the concept of number
   occupied the dominant place in Greek mathematics. The Pythagoreans believed that
   God had made numbers the basis of the world order. God is unity and the world is
   plurality. The divine harmony in the organization of the cosmos is seen in the
   form of numerical relationships. A substantial part in this conviction was played
   by the Pythagoreans discovery of the fact that combinations of sounds which are
   pleasant to hear are created in the cases where a string is shortened by the
   ratios formed by whole numbers such as 1:2 (octave), 2:3 (fifth), 3:4 (fourth),
   and so on. The numerical mysticism of the Pythagoreans reflected their belief in
   the fact that, in the last analysis, all the uniformities of natural phenomena
   derive from the properties of whole numbers.

   We see here an instance of the human inclination to overestimate new discoveries.
   The physicists of the late nineteenth century, like the Pythagoreans, believed
   that they had a universal key to all the phenomena of nature and with proper
   effort would be able to use this key to reveal the secret of any phenomenon. This
   key was the notion that space was filled by particles and fields governed by the
   equations of Newton and Maxwell. With the discovery of radioactivity and the
   diffraction of electrons, however, the physicists' arrogant posture crumbled.

   In the case of the Pythagoreans the same function was performed by discovery of
   the existence of incommensurable line segments, that is, segments such that the
   ratio of their lengths is not expressed by any ratio of whole numbers (rational
   number). The side of a square, and its diagonal are incommensurable, for example.
   It is easy to prove this statement using the Pythagorean theorem. In fact, let us
   suppose the opposite, namely that the diagonal of a square stands in some ratio
   m:n to its side. If the numbers m and n have common factors they can be reduced,
   so we shall consider that m and n do not have common factors. This means that in
   measuring length by some unitary segment, the length of the side is n and the
   length of the diagonal is m. It follows from the Pythagorean theorem that the
   equality m^2= 2n^2 must occur. Therefore, m^2 must be divisible by 2, an
   consequently 2 must be among the factors of m, that is, m = 2m[1]. Making this
   substitution we obtain 4m[1]^2 = 2n^2 , that is, 2m[1]^2 = n^2 . This means that
   n also must be divisible by 2, which contradicts the assumption that m and n do
   not have common factors. Aristotle often refers to this proof. It is believed
   that the proof had already been discovered by the Pythagoreans.

   If there are quantities which for a given scale are not expressed by numbers then
   the number can no longer be considered the foundation of foundations; it is
   removed from its pedestal. Mathematicians then must use the more general concept
   of geometric quantity and study the relations among quantities that may (although
   only occasionally) be expressed in a ratio of whole numbers. This approach lies
   at the foundation of all Greek mathematics beginning with the classical period.
   The relations we know as algebraic equalities were known to the Greeks in
   geometric formulation as relations among lengths, areas, and volumes of figures
   constructed in a definite manner.

GEOMETRIC ALGEBRA

   FIGURE 11.1 shows the well-known geometric interpretation of the relationship

   (a +b)^2 = a^2+2ab+b^2.

   [IMG.FIG11.1.GIF]

   Figure 11.1. Geometric interpretation of the identity (a +b)^2 = a^2+2ab+b^2.

   The equality (a+b)(a-b) = a^2 - b^2, which is equally commonplace from an
   algebraic point of view, requires more complex geometric consideration. The
   following theorem from the second book of Euclid's Elements corresponds to it.

   [IMG.FIG11.2.GIF]

   Figure 11.2. Geometric interpretation of the identity (a-b)(a+b) = a^2 - b^2

   "If a straight line be cut into equal and unequal segments, the rectangle
   contained by the unequal segments of the whole together with the square on the
   straight line between the points of the section is equal to the square on the
   half.''[14][1]

   The theorem is proved as follows. Rectangle ABFE is equal to rectangle BDHF.
   Rectangle BCGF is equal to rectangle GHKJ. If square FGJI is added to these two
   rectangles (which together form rectangle ACGE which is ''contained by the
   unequal segments of the whole'') what we end up with is precisely rectangle BDKI,
   which is constructed ''on the half.'' Thus we have the equality (a+b)(a--b) +
   b^2= a^2 which is equivalent to the equality above but does not contain the
   difficult-to-interpret subtraction of areas.

   Clearly, if these very simple algebraic relations require great effort to
   understand the formulation of the theorem--as well as inventiveness in
   constructing the proof--when they are expressed geometrically, then it is
   impossible to go far down this path. The Greeks proved themselves great masters
   in everything concerning geometry proper, but the line of mathematical
   development that began with algebra and later gave rise to the infinitesimal
   analysis and to modern axiomatic theories (that is to say, the line of
   development involving the use of the language of symbols rather than the language
   of figures) was completely inaccessible to them. Greek mathematics remained
   limited, confined to the narrow framework; of concepts having graphic geometric.

ARCHIMEDES AND APOLLONIUS

   DURING THE ALEXANDRIAN EPOCH (330 200 B.C.) two great learned men lived in whose
   work Greek mathematics reached its highest point. They were Archimedes (287-212
   B.C.) and Apollonius (265?-170? B.C.)

   In his works on geometry Archimedes goes far beyond the limits of the figures
   formed by straight lines and circles. He elaborates the theory of conic sections
   and studies spirals. Archimedes's main achievement in geometry is his many
   theories on the areas. volumes, and centers of gravity of figures and bodies
   formed by other than just straight lines and plane surfaces. He uses the "method
   of exhaustion.'' To illustrate the range of problems solved by Archimedes we
   shall list the problems included in his treatise entitled The Method whose
   purpose, as can be seen from the title, is not a full summary of results but
   rather an explanation of the method of his work. The Method contains solutions to
   the following 13 problems: area of a parabolic segment, volume of a spher, volume
   of a spheroid (ellipsoid of rotation), volume of a segment of a paraboloid of
   rotation, center of gravity of a segment of a paraboloid of rotation, center of
   gravity of a hemisphere, volume of a segment of a sphere, volume of a segment of
   a spheroid, center of gravity of a segment of a sphere, center of gravity of a
   segment of a spheroid, center of gravity of a segment of a hyperboloid of
   rotation, volume of a segment of a cylinder, and volume of the intersection of
   two cylinders (the last problem is without proof).

   Archimedes's investigations in the field of mechanics were just as important as
   his work on geometry. He discovered his famous ''law'' and studied the laws of
   equilibrium of bodies. He was extraordinarily skillful in making different
   mechanical devices and attachments. It was thanks to machines built under his
   direction that the inhabitants of Syracuse, his native city, repulsed the Romans'
   first attack on their city. Archimedes often used mechanical arguments as support
   in deriving geometric theorems. It would be a mistake to suppose, however, that
   Archimedes deviated at all from the traditional Greek way of thinking. He
   considered a problem solved only when he had found a logically flawless geometric
   proof. He viewed his mechanical inventions as amusements or as practical concerns
   of no scientific importance whatsoever. ''Although these inventions,'' Plutarch
   writes, ''made his superhuman wisdom famous, he nonetheless wrote nothing on
   these matters because he felt that the construction of all machines and all
   devices for practical use in general was a low and ignoble business. He himself
   strove only to remove himself, by his handsomeness and perfection, far from the
   kingdom of necessity".

   Of all his achievements Archimedes himself was proudest of his proof that the
   volume of a sphere inscribed in a cylinder is two thirds of the volume of the
   cylinder. In his will he asked that a cylinder with an inscribed sphere be shown
   on his gravestone. After the Romans took Syracuse and one of his soldiers
   (against orders, it is said) killed Archimedes, the Roman general Marcellus
   authorized Archimedes' relatives to carry out the wish of the deceased.

   Apollonius was primarily famous for his work on the theory of conic sections. His
   work is in fact a consistent algebraic investigation of second-order curves
   expressed in geometric language. In our day any college student can easily repeat
   Appolonius' results by employing the methods of analytic geometry. But Apollonius
   needed to show miraculous mathematical intuition and inventiveness to do the same
   thing within a purely geometric approach.

THE DECLINE OF GREEK MATHEMATICS

   ''AFTER APOLLONIUS,'' writes B. van der Waerden, ''Greek mathematics comes to a
   dead stop. It is true that there were some epigones, such as Diocles and
   Zenodorus, who, now and then, solved some small problem, which Archimedes and
   Apollonius had left for them, crumbs from the board of the great. It is also true
   that compendia were written, such as that of Pappus of Alexandria (300 A.D.); and
   ,geometry was applied to practical and to astronomical problems, which led to the
   development of plane and spherical trigonometry. But apart from trigonometry,
   nothing great nothing new appeared. The ,geometry of the conics remained in the
   form Apollonius gave it, until Descartes. Indeed the works of Apollonius were but
   little read and were even partly lost. The Method of Archimedes was lost sight of
   and the problem of integration remained where it was, until it was attacked new
   in the seventeenth century. . . "[15][2]

   The decline of Greek mathematics was in part caused by external factors--the
   political storms that engulfed Mediterranean civilization. Nonetheless, internal
   factors were decisive. In astronomy, van der Waerden notes, development continued
   steadily along an ascending line; there were short and long periods of
   stagnation, but after them the work was taken up again from the place where it
   had stopped. In geometry, however, regression plainly occurred. The reason is
   found, of course, in the lack of an algebraic language.

   "Equations of the first and second degree," we read in van der Waerden, ''can be
   expressed clearly in the language of geometric algebra and, if necessary, also
   those of the third degree. But to get beyond this point, one has to have recourse
   to the bothersome tool of proportions .

   "Hippocrates, for instance, reduced the cubic equation x^3 = V to the proportion

                                   a :x =x :y = y:b.

   and Archimedes wrote the cubic

                                   x^2 (a-x) = bc^2

   in the form

                                 (a-x) :b = c^2: x^2".

   "In this manner one can get to equations of the fourth degree; examples can be
       found in Apollonius.... But one cannot get any further; besides, one has to
       be a mathematician of genius, thoroughly versed in transforming proportions
       with the aid of geometric, to obtain results by this extremely cumbersome
       method. Anyone can use our algebraic notation, but only a gifted
       mathematician can deal with the Greek theory of proportions and with
       geometric algebra.
       Something has to be added, that is, the difficulty of the written tradition.
       Reading a proof in Apollonius requires extended and concentrated study.
       Instead of a concise algebraic formula, one finds a long sentence, in which
       each line segment is indicated by two letters which have to be located in the
       figure. To understand the line of thought, one is compelled to transcribe
       these sentences in modern concise formulas . . .
       An oral explanation makes it possible to indicate the line segments with the
       fingers; one can emphasize essentials and point out how the proof was found.
       All of this disappears in the written formulation of the strictly classical
       style. The proofs are logically sound, but they are not suggestive. One feels
       caught as in a logical mouse trap, but one fails to see the guiding line of
       thought.
       As long as there was no interruption. as long as each generation could hand
       over its method to the next, everything went well and the science flourished.
       But as soon as some external cause brought about an interruption in the oral
       tradition. and only books remained. it became extremely difficult to
       assimilate the work of the great precursors and next to impossible to pass
       beyond it.[16][3]

   But why, despite their high mathematical sophistication and abundance of talented
   mathematicians, did the Greeks fail to create an algebraic language? The usual
   answer is that their high mathematical sophistication was in fact what hindered
   them, more specifically their extremely rigorous requirements for logical
   strictness in a theory, for the Greeks could not consider irrational numbers, in
   which the values of geometric quantities are ordinarily expressed, as numbers: if
   line segments were incommensurate it was considered that a numerical relationship
   for them simply did not exist. Although this explanation is true in general,
   still it must be recognized as imprecise and superficial. Striving toward logical
   strictness cannot by itself be a negative factor in the development of
   mathematics. If it acts as a negative factor this will evidently be only in
   combination with certain other factors and the decisive role in this combination
   should certainly not be ascribed to the striving for strictness. Perfect logical
   strictness in his final formulations and proofs did not prevent Archimedes from
   using guiding considerations which were not strict. Then why did it obstruct the
   creation of an algebraic language? Of course, this is not simply a matter of a
   high standard of logical strictness, it concerns the whole way of thinking, the
   philosophy of mathematics. In creating the modern algebraic language Descartes
   went beyond the Greek canon, but this in no way means that he sinned against the
   laws of logic or that he neglected proof. He considered irrational numbers to be
   ''precise" also, not mere substitutions for their approximate values. Some
   problems with logic arose after the time of Descartes, during the age of swift
   development of the infinitesimal analysis. At that time mathematicians were so
   carried away by the rush of discoveries that they simply were not interested in
   logical subtleties. In the nineteenth century came time to pause and think. and
   then a solid logical basis was established for the analysis.

   We shall grasp the causes of the limitations of Greek mathematics after we review
   the substance of the revolution in mathematics made by Descartes.

ARITHMETIC ALGEBRA

   ADVANCES IN GEOMETRY forced the art of solving equations into the background. But
   this art continued to develop and gave rise to arithmetie algebra. The emergence
   of algebra from arithmetic was a typical metasystem transition. When an equation
   must be solved, whether it is formulated in everyday conversational language or
   in a specialized language, this is an arithmetic problem. And when the general
   method of solution is pointed out--by example, as is done in elementary school.
   or even in the form of a formula--we still do not go beyond arithmetic. Algebra
   begins when the equations themselves become the object of activity, when the
   properties of equations and rules for converting them are studied. Probably
   everyone who remembers his first acquaintance with algebra in school (if this was
   at the level of understanding, of course, not blind memorization) also remembers
   the happy feeling of surprise experienced when it turns out that various types of
   arithmetic problems whose solutions had seemed completely unrelated to one
   another are solved by the same conversions of equations according to a few simple
   and understandable rules. All the methods known previously fall into place in a
   harmonious system, new methods open up, new equations and whole classes of
   equations come under consideration (the law of branching growth of the
   penultimate level), and new concepts appear which have absolutely no meaning
   within arithmetic proper: negative, irrational, and imaginary numbers.

   In principle the creation of a specialized language is not essential for the
   development of algebra. In fact, however, only with the creation of a specialized
   language does the metasystem transition in people's minds conclude. The
   specialized language makes it possible to see with one's eyes that we are dealing
   with some new reality, in this case with equations, which can be viewed as an
   object of computations just as the objects (numbers) of the preceding level were.
   People typically do not notice the air they breathe and the language they speak.
   But a newly created specialized language goes outside the sphere of natural
   language and is in part nonlinguistic activity. This facilitates the metasystem
   transition. Of course, the practical advantages of using the specialized language
   also play an enormous part here; among them are making expressions visible,
   reducing time spent recopying, and so on.

   The Arab scholar Muhammed ibn Musa al-Khwarizmi (780-850) wrote several treatises
   on mathematics which were translated into Latin in the twelfth century and served
   as the most important textbooks in Europe for four centuries. One of them, the
   Arithmetic, gave Europeans the decimal system of numbers and the rules
   (algorithms--the name is based on al-Khwarizmi) for performing the four
   arithmetic operations on numbers written in this system. Another work was
   entitled Book of Al Jabr Wa'l Muqabala. The purpose of the book was to teach the
   art of solving equations, an art which is essential, as the author writes, ''in
   cases of inheritance, division of property, trade, in all business relationships,
   as well as when measuring land, laying canals, making geometric computations, and
   in other cases....'' Al Jabr and al Mugabala are two methods al-Khwarizmi uses to
   solve equations. He did not think up these methods himself; they were described
   and used in the Arithmetica of the Greek mathematician Diophantus (third century
   A.D.), who was famous for his methods of solving whole-number (''diophantine'')
   equations. In the same Arithmetica of Diophantus we find the rudiments of letter
   symbolism. Therefore, if anyone is to be considered the progenitor of arithmetic
   algebra it should obviously be Diophantus. But Europeans first heard of algebraic
   methods from al-Khwarizmi while the works of Diophantus became known much later.
   There is no special algebraic symbolism in al-Khwarizmi, not even in rudimentary
   form. The equations are written in natural language. But for brevity's sake, we
   shall describe these methods and give our examples using modern symbolic
   notation.

   Al Jabr involves moving elements being subtracted from one part of the equation
   to the other; al Muqabala involves subtracting the same element from both parts
   of the equation. Al-Khwarizmi considers these procedures different because he
   does not have the concept of a negative number. For example let us take the
   equation

                                   7x - 11 = 5x - 3.

   Applying the al Jabr method twice, for the 11 and 3, which are to be subtracted,
   we receive:

                                   7x + 3 = 5x + 11.

   Now we use the al Muqabala method twice, for 3 and 5x. We receive

                                        2x = 8.

   From this we see that

                                        x = 4.

   So although al-Khwarizmi does not use a special algebraic language, his book
   contains the first outlines of the algebraic approach. Europeans recognized the
   merits of this approach and developed it further. The very word algebra comes
   from the name of the first of al-Khawarizmi's methods.

ITALY, SIXTEENTH CENTURY

   IN THE FIRST HALF of the sixteenth century the efforts of Italian mathematicians
   led to major changes in algebra which were associated with very dramatic events.
   Scipione del Ferro (1465-1526). a professor at the University of Bologna, found a
   general solution to the cubic equation x^3 +px = q where p and q are positive.
   But del Ferro kept it secret, because it was very valuable in the problem-solving
   competitions which were held in Italy at that time. Before his death he revealed
   his secret to his student Fiore. In 1535 Fiore challenged the brilliant
   mathematician Niccolo Tartaglia (1499-1557) to a contest. Tartaglia knew that
   Fiore possessed a method of solving the cubic equation, so he made an all-out
   effort and found the solution himself. Tartaglia won the contest, but he also
   kept his discovery secret. Finally, Girolamo Cardano (1501-1576) tried in vain to
   find the algorithm for solving the cubic equation. In 1539 he finally appealed to
   Tartaglia to tell him the secret. Having received a "sacred oath'' of silence
   from Cardano, Tartaglia unveiled the secret, but only partially and in a rather
   unintelligible form. Cardano was not satisfied and made efforts to familiarize
   himself with the manuscript of the late del Ferro. In this he was successful, and
   in 1545 he published a book in which he reported his algorithm, which reduces the
   solution of a cubic equation to radicals (the "Cardano formula''). This same book
   contained one more discovery made by Cardano's student Luigi (Lodovico) Ferrari
   (1522-1565): the solution of a quartic equation in radicals. Tartaglia accused
   Cardano of violating his oath and began a bitter and lengthy polemic. It was
   under such conditions that modern mathematics made its first significant
   advances.

   Using a tool suggests ways to improve it. While striving toward a uniform
   solution to equations, mathematicians found that it was extremely useful in
   achieving this goal to introduce certain new objects and treat them as if these
   were numbers. And in fact they were called numbers although it was understood
   that they differed from ''real'' numbers: this was seen in the fact that they
   were given such epithets as "false'' "fictitious,'' ''incomprehensible,'' and
   "imaginary". What they correspond to in reality remained somewhat or entirely
   unclear. Whether their use was correct also remained debatable. Nonetheless, they
   began to be used increasingly widely, because with them it was possible to obtain
   finite results containing only "real" numbers which could not be obtained
   otherwise. A person consistently following the teachings of Plato could not use
   ''unreal'' numbers. But the Indian, Arabic, and Italian mathematicians were by no
   means consistent Platonists. For them a healthy curiosity and pragmatic
   considerations outweighed theoretical prohibitions. In this, however, they did
   make reservations and appeared to be apologizing for their ''incorrect''
   behavior.

   All "unreal'' numbers are products of the reverse movement in the arithmetic
   model: formally they are solutions to equations that cannot have solutions in the
   area of "real'' numbers. First of' all we must mention negative numbers. They are
   found in quite developed form in the Indian mathematician all Bahascara (twelfth
   century), who performed all four arithmetic operations on such numbers. The
   interpretation of the negative number as a debt was known to the Indians as early
   as the seventh century. In formulating the rules of operations on negative
   numbers. Bhascara calls them ''debts,'' and calls positive numbers ''property.''
   He does not choose to declare the negative number an abstract concept like the
   positive number. "People do not approve of abstract negative numbers,'' Bhascara
   writes. The attitude toward negative numbers in Europe in the fifteenth and
   sixteenth centuries was similar. In geometric interpretation negative roots are
   called ''false'' as distinguished from the 'true'' positive roots. The modern
   interpretation of negative numbers as points lying to the left of the zero point
   did not appear until Descartes' Géométrie (1637). Following tradition, Descartes
   called negative roots false.

   Formal operations on roots of numbers that cannot be extracted exactly go back to
   deep antiquity, when the concept of incommensurability of line segments had not
   yet appeared. In the fifteenth and sixteenth centuries people handled them
   cavalierly: they were helped here, of course, by the simple geometric
   interpretation. An understanding of the theoretical difficulty which arises from
   the incommensurability of line segments can be .seen by the fact that the numbers
   were called "irrational".

   The square of any number is positive: therefore the square root of a negative
   number does not exist among positive, negative, rational, or irrational numbers.
   But Cardano was daring enough to use (not w without reservations) the roots of
   negative numbers. ' Imaginary'' numbers thus appeared. The logic of using
   algebraic language drew mathematicians inexorably down an unknown path. It seemed
   wrong and mysterious. but intuition suggested that all these impossible numbers
   were profoundly meaningful and that the new path would prove useful. And it
   certainly did.

LETTER SYMBOLISM

   THE RUDIMENTS of algebraic letter symbolism are first encountered, as mentioned
   above, in Diophantus. Diophantus used a character resembling the Latins to
   designate an unknown. It is hypothesized that this designation originates from
   the last letter of the Greek word for number:
   a[rho][iota][theta]u[omicron][sigma] (arithmos). He also had abbreviated
   notations for the square cube, and other degrees of the unknown quantity. He did
   not have an addition sign: quantities being added were written in a series,
   something like an upside-down Greek letter [psi] was used as the subtraction
   sign, while the first letter of the Greek word [iota]d[omicron]d for ''equal" was
   used as the equal sign, everything else was expressed in words. Known quantities
   were always written in concrete numerical form while there were no designations
   for known, but arbitrary numbers.

   Diophantus' Arithmetica became known in Europe in 1463. In the late fifteenth and
   early sixteenth centuries European mathematicians first Italians and then others
   began to use abbreviated notations. These abbreviations gradually wandered from
   arithmetic algebra to geometric, and unknown geometric quantities also began to
   be designated by letters. In the late sixteenth century the Frenchman François
   Vieta (1540-1603) took the next important step. He introduced letter designations
   for known quantities and was thus able to write equations in general form. Vieta
   also introduced the term "coefficient.'' In external appearance Vieta's symbols
   are still rather far from modern ones. For example, Vieta writes

                                 [IMG.FIG11.2bis.GIF]

   instead of our notation

                                     D(2B^3 - D^3)
                                   ________________

                                       B^3 + D^3

   By the beginning of the seventeenth century the situation in European mathematics
   was as follows. There were two algebras. The first was arithmetic based on
   symbols created by the Europeans themselves and representing a substantial
   advance in comparison with the arithmetic of the ancients. The second algebra.
   geometric algebra. was part of geometry. It was taken. as was the whole of
   geometry, from the Greeks. The fundamentals were from Euclid's Elements and the
   further development came primarily from the works of Pappus of Alexandria and
   Apollonius, who had been thoroughly studied by that time. Nothing fundamentally
   new had been done in this field. We cannot say that there was no relationship at
   all between these two algebra: equation of degrees higher than the first could
   only receive geometric interpretation, for where else could squares, cubes, and
   higher degrees of an unknown number occur but in computing areas, volumes and
   manipulations of line segments related to complex systems of proportions? The
   very names of the second and third degrees, the square and the cube, illustrate
   this very eloquently. Nonetheless, the gap between the concepts of quantity (or
   magnitude) and number remained and in full conformity with the Greek canon only
   geometric proofs w ere considered real. When geometric objects--lengths, areas,
   and volumes--appeared in equations they operated either as geometric quantities
   or as concrete numbers. Geometric quantities were thought of as necessarily
   something spatial and, because of incommensurability not reducible to a number.

   This was the situation that René Descartes (1596-1650), one of the greatest
   thinkers who has ever lived, encountered.

WHAT DID DESCARTES DO?

   DESCARTES' ROLE as a philosopher is generally recognized. But when Descartes as a
   mathematician is discussed it is usually indicated that he "refined algebraic
   notations and created analytical geometry.'' Sometimes it is added that at
   approximately the same time the basic postulates of analytic geometry were
   proposed, independently of Descartes by his countryman Pierre de Fermat
   (1601-1665), while Vieta had already made full use of algebraic symbols. It comes
   out, thus, that there is no special cause to praise Descartes the mathematician,
   and in fact many authors writing about the history of mathematics do not give him
   his due. However, Descartes carried out a revolution in mathematics. He created
   something incomparably greater than analytic geometry (understood as the theory
   of curves on a plane). What he created was a new approach to describing the
   phenomena of reality: the modern mathematical language.

   It is sometimes said that Descartes ''reduced geometry to algebra" which means,
   of course numerical algebra, arithmetic algebra. This is a flagrant mistake. It
   is true that Descartes overcame the gap between quantity and number, between
   geometry and arithmetic. He did not achieve this by reducing one language to the
   other, however: he created a new language, the language of algebra. Not
   arithmetic algebra, not geometric algebra, simply algebra. In syntax the new
   language coincided with arithmetic algebra. but by semantics it coincided with
   geometric. In Descartes' language the symbols do not designate number or
   quantities, but relations of quantities. This is the essence of the revolution
   called out by Descartes.

   The modern reader will perhaps shrug his shoulders and think ''So what"? Could
   this logical nuance really have been very important?'' As it turns out, it was.
   It was precisely this ''nuance'' that had prevented the Greeks from taking the
   next step in their mathematics.

   We have become so accustomed to placing irrational numbers together with rational
   ones that we are no longer aware of the profound difference which exists between
   them. We write [SQRT2.GIF] as we write 4/5 and we call [SQRT2.GIF] number and,
   when necessary, substitute an approximate value for it. And there is no way we
   can understand why the ancient Greeks responded with such pain to the
   incommensurability of line segments. But if we think a little, we cannot help
   agreeing with the Greeks that [SQRT2.GIF] is not a number. It can be represented
   as an infinite process which generates the sequential characters of expansion of
   [SQRT2.GIF] in decimal fractions. It can also be pictured in the form of a
   boundary line in the field of rational numbers--one that divides rational numbers
   into two classes: those which are less than [SQRT2.GIF] and those which are
   greater than [SQRT2.GIF] . In this case the rule is very simple: the rational
   number a belongs to the first class if a^2 < 2 and to the second where this is
   not true. Finally, [SQRT2.GIF] can be pictured in the form of a relation between
   two line segments, between the diagonal of a square and its side in the
   particular case. These representations are equivalent to one another but they are
   not at all equivalent to the representation of the whole or fractional number.

   This by no means implies that we are making a mistake or not being sufficiently
   strict when we deal with [SQRT2.GIF] as a number. The goal of mathematics is to
   create linguistic models of reality, and all means which lead to this goal are
   good. Why shouldn't our language contain characters of the type [SQRT2.GIF] in
   addition to ones such as 4/5? It is my language and I will do what I want to with
   it.'' The only important thing is that we be able to interpret these characters
   and perform linguistic conversions on them. But we are able to interpret
   [SQRT2.GIF] . In practical computations the first of the three representations in
   the preceding paragraph may serve a. the basis of interpretation. while in
   geometry the third can be used. We can also carry out other computations with
   them. All that remains now is to refine the terminology. Let us stipulate that we
   shall use the term rational numbers for what were formerly called numbers, name
   the new objects irrational numbers, and use the term numbers for both (real
   numbers according to modern mathematical terminology). Thus, in the last
   analysis, there is no difference in principle between [SQRT2.GIF] and 4/5 and we
   have proved wiser than the Greeks. This wisdom was brought in as contraband by
   all those who operated with the symbol [SQRT2.GIF] as a number, while recognizing
   that it was "irrational". It was Descartes who substantiated this wisdom and
   established it as law.

THE RELATION AS AN OBJECT

   THE GREEKS' failure to create algebra is profoundly rooted in their philosophy.
   They did not even have arithmetic algebra. Arithmetic equations held little
   interest for them: after all, even quadratic equations do not, generally
   speaking, have exact numerical solutions. And approximate calculations and
   everything bound up with practical problems were uninteresting to them. On the
   other hand, the solution could have been found by geometric construction! But
   even if we assume that the Greek mathematicians of the Platonic school were
   familiar with arithmetic letter symbols it is difficult to imagine that they
   would have performed Descartes' scientific feat. To the Greeks relations were not
   ideas and therefore did not have real existence. Who would ever think of using a
   letter to designate something that does not exist? The Platonic idea is a
   generalized image. a form a characteristic: it can be pictured in the imagination
   as a more or less generalized object. All this is primary and has independent
   existence an existence even more real than that of things perceived by the
   senses. But what is a relation of line segments? Try to picture it and you will
   immediately see that what you are picturing is precisely two line segments, not
   any kind of relation. The concept of the relation of quantities reflects the
   process of measuring one by means of the other. But the process is not an idea in
   the Platonic sense: it is something secondary that does not really exist. Ideas
   are eternal and invariable, and by this alone have nothing in common with
   processes.

   Interestingly. the concept of the relation of quantities, which reflects
   characteristics of the measurement process, was introduced in strict mathematical
   form as early as Eudoxus and was included in the fifth book of Euclid's Elements.
   This was exactly the concept Descartes used. But the relation as an object is not
   found in Eudoxus or in later Greek mathematicians: after being introduced it
   slowly gave way to the proportion which it is easy to picture as a characteristic
   of four line segments formed by two parallel lines intersecting the sides of an
   angle.

   The concept of the relation of quantities is a linguistic construct, and quite a
   complex one. But Platonism did not permit the introduction of constructs in
   mathematics: it limited the basic concepts of mathematics to precisely
   representable static spatial images. Even fractions were considered somehow
   irregular by the Platonic school from the point of view of real mathematics. In
   The Republic we read: "If you want to divide a unit. learned mathematicians will
   laugh at you and will not permit you to do it: if you change a unit for small
   pieces of money they believe it has been turned into a set and are careful to
   avoid viewing the unit as consisting of parts rather than as a whole.'' With such
   an attitude toward rational numbers, why even talk about irrational ones!

   We can briefly summarize the influence of Platonic idealism on Greek mathematics
   as follows. By recognizing mathematical statements as objects to work with. the
   Greeks made a metasystem transition of enormous importance but then they
   immediately objectivized the basic elements of mathematical statements and began
   to view them as part of a nonlinguistic reality, "the world of ideas". In this
   way they closed off the path to further escalation of critical thinking to
   becoming aware of the basic elements (concepts) of mathematics as phenomema of
   language and to creating increasingly more complex mathematical constructs. The
   development of mathematics in Europe was a continuous liberation from the fetters
   of Platonism.

DESCARTES AND FERMAT

   IT IS VERY INSTRUCTIVE to compare the mathematical world of Descartes and Fermat.
   As a mathematician Fermat was as gifted as Descartes, perhaps even more so. This
   can be seen from his remarkable works on number theory. But he was an ardent
   disciple of the Greeks and continued the traditions. Fermat set forth his
   discoveries on number theory in remarks in the margins of Diophantus' Arithmetica
   His works on geometry originated as the result of efforts to prove certain
   statements referred to by Pappus as belonging to Apollonius, but presented
   without proofs. Reflecting on these problems, Fermat began to systematically
   represent the position of a point on a plane by the lengths of two line segments:
   the abscissa and the cardinal and represent the curve as an equation relating
   these segments. This idea was not at all new from a geometric point of view: it
   was a pivotal idea not only in Apollonius but even as far back as Archimedes, and
   it originates with even more ancient writers. Archimedes describes conic sections
   by their ''symptoms" that is, the proportions which connect the abscissas and
   ordinates of the points. As an example, let us take an ellipsis with the longer
   (major) axis AB.

   [IMG.FIG11.3.GIF]

   Figure 11.3.

   Perpendicular line PQ which is dropped from a certain point of the ellipsis P to
   axis AB is called the ordinate, and segments AQ and QB are the abscissas of this
   point (both terms are Latin translations of Archimedes' Greek terms). The ratio
   of the area of a square constructed on the ordinate to the area of the rectangle
   constructed on the two abscissas is the same for all points P lying on the
   ellipsis. This is the "symptom'' of the ellipsis, that is, in essence, its
   equation. It can be written as Y^2: X[1] X[2] = const. Analogous symptoms are
   established for the hyperbola and parabola. How is this not a system of
   coordinates ?

   Unlike the ancients, Fermat formulates the symptoms as equations in Vieta's
   language, not in the form of proportions described by words. This makes
   conversions easier: specifically, it can be seen immediately that it is more
   convenient to leave one abscissa than two. But the approach continues to be
   purely geometric, spatial.

   Fermat set forth his ideas in the treatise "Ad locos planos et solidos isagoge"
   (Introduction to Plane and Solid Loci). This work was published posthumously in
   1679, but it had been known to French mathematicians as early as the 1630s,
   somewhat earlier than Descartes' mathematical works.

   Descartes famous Géométrie came out in 1637. Descartes was not of course, at all
   influenced by Fermat (it is unknown whether he even read Fermat's treatise);
   Descartes' method took shape in the 1620's, long before the Géométrie was
   published. Nonetheless, the properly geometrical ideas of Descartes and Fermat
   are practically identical. But Descartes created a new algebra based on the
   concept of the relation of geometric quantities. In Vieta only similar quantities
   can be added and subtracted and coefficients must include an indication of their
   geometric nature. For example, the equation which we would write as

                                     A^3 + BA = D.

   Vieta wrote as follows:

                      A cubus + B planum in A aequatur D solido.

   This means the cube with edge A added to area B multiplied by A is equal to the
   volume D. Vieta and Fermat are intellectual prisoners of the Greek geometric
   algebra. Descartes breaks with it decisively. The relations Descartes' algebra
   deals with are not geometric, spatial objects, but theoretical concepts,
   "numbers". Descartes is not restricted by the requirement for uniformity of
   things being added or the general requirement of a spatial interpretation; he
   understands raising to a power as repeated multiplication and indicates the
   number of factors by a small digit above and to the right. Descartes' symbolism
   virtually coincides with our modern system.

THE PATH TO DISCOVERY

   FERMAT WAS ONLY a mathematician; Descartes was above all a philosopher. His
   reflections went far beyond mathematics and dealt with the problems of the
   essence of being and knowledge. Descartes was the founder of the philosophy of
   rationalism which affirms the human being's unlimited ability to understand the
   world on the basis of a small number of intuitively clear truths and proceeding
   forward step by step using definite rules or methods. These two words are key
   words for all Descartes' philosophy. The name of his first philosophical
   composition was Regulae ad directionem ingenii (Rules for the Direction of the
   Mind), and his second was Discours de la méthode (Discourse on the Method). The
   Discours de la méthode was published in 1637 in a single volume with three
   physico-mathematical treatises: "La Dioptrique" (the Dioptric), "les Meteores"
   (Meteors) and "la Gémétrie'' (Geometry). The Discours preceded them as a
   presentation of the philosophical principles on which the following parts were
   based. In this Discours Descartes proposes the following four principles of
   investigation:
   The first of these was to accept nothing as true which I did not clearly
       recognize to be so: that is to say, carefully to avoid precipitation and
       prejudice in judgements and to accept in them nothing more than what was
       presented to my mind so clearly and distinctly that I could have no occasion
       to doubt it.
       The second was to divide up each of the difficulties which I examined into as
       many parts as possible, and as seemed requisite in order that it might be
       resolved in the best manner possible.
       The third was to carry on my reflections into order, commencing with objects
       that were the more simple and easy to understand, in order to rise little by
       little, or by degrees, to knowledge of the most complex assuming order, even
       it a fictitious one, among those which do not follow a natural sequence
       relatively to one another.
       The last was in all cases to make enumerations so complete and reviews so
       general that I should be certain of having omitted nothing.[17][4]

   Descartes arrived at his mathematical ideas guided by these principles. Here is
   how he himself describes his path in Discours de la méthode:

   And I have not much trouble in discovering which objects it was necessary to
   begin with, for I already knew that it was with the most simple and those most
   easy to apprehend. Considering also that of all those who have hitherto sought
   for the truth in the Sciences, it has been the mathematicians alone who have been
   able to succeed in making any demonstrations, that is to say, producing reasons
   which are evident and certain. I did not doubt that it had been by means of a
   similar kind that they carried on their investigations.... But for all that I had
   no intention of trying to master all those particular sciences that receive in
   common the name of Mathematics; but observing that, although the objects are
   different, they do not fail to agree in this, that they take nothing under
   consideration but the various relationships or proportions which are present in
   these objects. I thought that it would be better if I only examined those
   proportions in their general aspect, and without viewing them otherwise than in
   the objects which would serve most to facilitate a knowledge of them. Not that I
   should in any way restrict them to these objects. for I might later on all the
   more easily apply them to all other objects to which they were applicable. Then,
   having carefully noted that in order to comprehend the proportions I should
   sometimes require to consider each one in particular. and sometimes merely keep
   them in min(l. or take them in groups. I thought that in order the better to
   consider them in detail, I should picture them in the form of lines, because I
   could find no method more simple nor mole capable of being distinctly represented
   to my imagination and senses. I considered, however, that in order to keep them
   in my memory or to embrace several at once, it would be essential that I should
   explain them by means of certain formulas, the shorter the better. And for this
   purpose it was requisite that I should borrow all that is best in Geometrical
   Analysis and Algebra, and correct the errors of the one by the other.[18][5]

   We can see from this extremely interesting testimony that Descartes was clearly
   aware of the semantic novelty of his language based on the abstract concept of
   the relation and applicable to all the phenomena of reality. Lines serve only to
   illustrate the concept of the relation, just as a collection of little sticks
   serves to illustrate the concept of number. In their mathematical works Descartes
   and subsequent mathematicians have followed tradition and used the term
   "quantity" for that which is designated by letters, but semantically these are
   not the spatial geometric quantities of the Greeks but rather their relations. In
   Descartes the concept of quantity is just as abstract as the concept of number.
   But of course, it cannot be reduced to the concept of number in the exact meaning
   of the word, that is, the rational number. Explaining his notations in the
   Géométrie, Descartes points out that they are similar (but not identical) to the
   notations of arithmetic algebra:
   Just as arithmetic consists of only four or five operations, namely, addition,
       subtraction, multiplication, division, and the extraction of roots, which may
       be considered a kind of division, so in geometry, to find required lines it
       is merely necessary to add or subtract other lines: or else, taking one line
       which I shall call unit) in order to relate it as closely) as possible to
       numbers, and which can in general be chosen arbitrarily and having given two
       other lines, to find a fourth line which shall be to one of the given lines
       as the other is to unity which is the same as multiplication: or, again, to
       find a fourth line which is to one of the given lines as unity is to the
       other (which is equivalent to division): or, finally, to find one, two, or
       several mean proportionals between units and some other line (which is the
       same as extracting the square root. cube root, etc., of the given line). And
       I shall not hesitate to introduce these arithmetical terms into geometry, for
       the sake of greater clearness".[19][6]

   The semantics of Descartes' algebraic language are much more complex than the
   semantics of the arithmetic and geometric languages which rely on graphic images.
   The use of such a language changes one's view of the relation between language
   and reality. It is discovered that the letters of mathematical language may
   signify not only numbers and figures. but also something much more abstract (to
   be more precise ''constructed''). This is where the invention of new mathematical
   languages and dialects and the introduction of new constructs began. The
   precedent was set by Descartes. Descartes in fact laid the foundation for
   describing the phenomena of reality by means of formalized symbolic languages.

   The immediate importance of Descartes' reform was that it untied the hands of
   mathematicians to create. in abstract symbolic form. the infinitesimal analysis
   whose basic ideas in geometric form were already known to the ancients. If we go
   just half a century from the publication date of the Géométrie we find ourselves
   in the age of Leibnitz and Newton, and 50 more years brings us to the age of
   Euler.

   The history of science shows that the greatest glory usually doesnot go to those
   who lay the foundations and, of course, not to those who work on the small
   finishing touches: rather it goes to those who are the first in a new line of
   thought to obtain major results which strike the imagination of their
   contemporaries or immediate descendants. In European physico-mathematical science
   this role was played by Newton. But as Newton said, ''If I have seen further than
   Descartes, it is by standing on the shoulders of giants.'' This is, of course,
   evidence of the modesty of a brilliant scientist but it is also a recognition of
   the debt of the first great successes to the pioneers who showed the way. The
   apple which made Newton famous grew on a tree planted by Descartes.
   _________________________________________________________________________________

   [20][1] The Thirteen Books of Euclid's Elements, translated and annotated by T.
   L. Heath, (Cambridge: Cambridge University Press, 1908), vol. 1. p. 382.

   [21][2] B. van der Waerden Science Awakening (New York: Oxford University Press,
   1969), p. 264.

   [22][3] B. van der Waerden Science Awakening, p. 266.

   [23][4] Descartes, Spinoza, Great Books of the Western World, Encyclopaedia
   Britanica Inc., Vol 31, 1952, p. 47.

   [24][5] Descartes, ibid., p. 47.

   [25][6] Descartes, ibid., p. 295
     ____________________________________________________________________________

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