Mathematics in the time of Descartes and Fermat

It is important to note that during the late 16th century, considerable improvement occurred in the matter of algebraic notation, the lack of which hindered elementary manipulation of formulae. The leading innovator in this respect was the French mathematician Fran¸cois Vi`ete (1540-1603). Vi`ete made several contributions to mathematical knowledge, as well as initiating scientific cryptology (during a French war against Spain). Vi`ete was greatly influenced by Greek mathematics and he sought to extend the methods that the Greeks had introduced. Building on the notion of analysis found in Greek texts of the late classical period, Vi`ete described a threefold method of problem-solving, as follows.

• zetetic analysis, in which a problem is transformed into an equation linking the unknown with known quantities.

•  poristic analysis, a procedure for exploring the truth of a theorem by symbolic manipulation.

•  exegetics, the means of transforming an equation found by zetetics to find the value of the unknown.

It is the third part of this procedure, the exegetics, which we would understand as the finding of the solution by algebraic manipulation, that provided the means of making algebra systematic. Vi`ete introduced the idea of antithesis, which involves the transfer of terms from one side of an equation to the other–a useful procedure to this day. This is the same idea as that involved in al-jabr, which we saw in the work of al-Khawarizmi.

For algebra, Vi`ete’s main achievements are found in his book In artem analyticem isagoge (Introduction to the Analytic Art) of 1590. In this work, he aimed to revive and extend the methods of Diophantus of Alexandria, and he used a few of Diophantus’s problems. His great innovation in this work is the use of letters to denote unknown or known quantities. Unknown quantities were denoted by capital letter vowels, such as A, E, I, O and U. Known or given quantities were denoted by capital consonants such as B, C, D, etc. Vi`ete did not entirely break away from the notation of his predecessors, such as Bombelli, as he wrote powers in terms of words or abbreviations. Thus A quadratum represented A2, B cubus represented B3, C quadrato-quadratum represented C4. These could also be shortened to A quad or C quad - quad. He also used the German symbols + and -.

For multiplication, he used the word in, and for fractions he used a fraction bar. Hence $\frac{A \ in \ B}{C \ quadratum} \ \text{represents} \ \frac{AB}{C^2}$

Similarly,

B in D quadratum 3 - B in A quadratum 3 represents 3BD2 - 3BA2.

Vi`ete’s partial use of symbolism enabled him to give general formulae for the solution of problems, rather than illustrations provided by specific examples. This helped in seeing the nature of general solutions, something not apparent in specific examples. He was also able to express many of the standard algebraic identities in symbolic form, as for example

(A+B)(A-B) = A2 -B2

or

(A + B)2 - (A - B)2 = 4AB.

Although known previously, these identities would have been expressed in words, often with geometric interpretations.

Vi`ete also showed how trigonometric functions can be used to investigate the solution of cubic equations in the irreducible case. Recall that if we try to find the roots of the cubic x3 + ax + b, the formula requires us to work with complex numbers if

$b^2 + \frac{4a^3}{27} < 0 .$

So let us assume that this inequality holds. Given a root б of this polynomial, we make a substitution б = r cosθ, where r is a real number and θ is an angle to be determined. Substituting,

r3 cos3θ + ar cosθ + b = 0.

Now recall the trigonometric identity

cos3θ = 4 cos3θ - 3 cosθ. We look for a number s so that

r3 cos3θ + ar cosθ = s cos3θ = 4s cos3θ - 3s cosθ. We can achieve this provided we take

r3 = 4s, ar = -3s.

Dividing we get,$r^2 = - \frac{4a}{3}$

Now our equation becomes s cos3θ = —b and we deduce that

$\cos 3\theta = - \frac{b}{s}$

We want to get real solutions for a and this requires that |cos 3θ |< 1, or cos2< 1. We therefore must have b2 — s2 < 0. But

$s^2 - \frac{a^2r^2}{9} - - \frac{4a^3}{27}$

Thus we require

$b^2 + \frac{4a^3}{27} \leq 0$

for this method to give real solutions, and this is precisely the condition that we have assumed, and is the one that occurs in the irreducible case of the cubic. We can solve for by taking inverse cosines and then divide by 3 to find θ. Then we evaluate cosθ and use a = rcosθ. At the time of Viete’s work, tables of cosines were quite common for astronomical and navigational purposes and a solution of this kind would have seemed perfectly calculable. We see now that Viete’s solution provides an alternative (non-algebraic) solution in terms of real numbers precisely in the case that the algebraic solution by Cardano’s method requires us to use complex numbers.

One of the major advances in mathematics of all time is the use of algebraic methods to solve problems of pure geometry. This innovation was due to the French mathematician and philosopher Rene Descartes (1596-1650). His work appeared in his publication La Geometrie, which was included as an appendix to his famous philosophical treatise Discours de la methode (1637). Interestingly, despite its scholarly nature, it was written in French, rather than the more erudite Latin. Descartes’s notation in this work is recognizably similar to our own, in marked distinction to that used by the Italians a century or even fifty years earlier. He employed the + and — signs, adopted from German arithmetic texts of the 16th centuries. He used exponential notation such as y3 for powers of unknowns, and surd signs for square roots. Variables are usually denoted by small roman letters from the end of the alphabet, and constant quantities by small roman letters from the beginning of the alphabet, such as a, b, c. Here again, this is the convention frequently adopted today. The only major difference is that he did not adopt the = sign for equality, but used instead a sign similar to one used to denote proportionality. He also, when multiplying a symbol such as y or y2 by an algebraic expression such as a2 + ab + b2, might write

the symbols as aa or bb and place them vertically above each other, with appropriate signs, bound together with a curly bracket. This may have been done to express formulae more compactly in horizontal form. Commentators have pointed out that Descartes’s purpose in La Geometrie was not purely practical, in the sense of aiding the location of points in space. The idea of Cartesian coordinates seems to be a misattribution to Descartes, as he himself did not use them. It is for his linking of algebra to geometry, an idea still vigorously pursued, that he most deserves attention. He did indeed make additional advances on properties of polynomials, for example, he stated a rule of signs, which estimates the number of real roots of a polynomial, and laid foundations for the beginning of the differential calculus, as it relates to the geometric problems of constructing tangents and normals to curves. Of special interest is his statement (not proved of course) that the number of roots of a polynomial equals its degree. This is expressed as:

Know then that in every equation there are as many distinct roots, that is, values of the unknown quantity, as is the number of dimensions of the unknown quantity.

Mathematicians see in this an anticipation of what is known as the ‘fundamental theorem of algebra’, in other words, the theorem that a polynomial of degree n has n roots, counted according to multiplicity, which we will talk about later. Descartes knew of the existence of imaginary roots and knew that these must be taken into account. For example, he wrote:

Thus, while we may conceive of the equation x3 — 6x2 + 13a; —10 = 0 as having three roots, yet there is only one real root, 2, while the other two, however we may increase, diminish, or multiply them in accordance with the rules just laid down, remain always imaginary.

Here, of course, the polynomial factorizes as (x — 2)(x2 — 4x + 5) and the quadratic factor x2 — 4x + 5 has non-real roots. Descartes also referred to negative roots as false roots, indicating the ambivalence still felt over negative numbers, despite the fact that negative numbers were clearly necessary, and useful in calculations.

At more or less the same time that Descartes was making great progress in the application of algebra to geometry, another French mathematician, Pierre de Fermat (1601-65), had also discovered this connection between algebra and geometry. Fermat may be considered to be an amateur mathematician, as his working life was spent in various aspects of the practice of law. He was influenced by his reading of Viete’s symbolic algebra and

theory of eqautions. He also thoroughly studied the Arithmetica of Diophantus, a scholarly edition of which was published in 1621. Fermat applied the newly available algebraic techniques to investigate difficult problems of number theory inspired by Diophantus, at least one of which has become notorious for its simplicity of statement and complexity of solution. As he was not a professional mathematician, Fermat did not publish or even make known many of his discoveries and his achievements were not fully appreciated until the 19th century.

As we have mentioned, Fermat, along with Descartes, is considered to be the principal founder of analytic geometry–the study of geometry by algebraic techniques. He was able to recover all of the classical results of the ancient Greek mathematician Apol-lonius concerning conic sections (ellipses, hyperbolas and parabolas) by the analysis of algebraic equations of degree 2. Fermat is also credited with the discovery of the method of determining maxima and minima of polynomials, something which is now routinely studied through the differential calculus. Fermat’s investigations began in the 1630’s, several decades before the definitive appearance of the differential calculus in the versions of Leibniz and Newton. Fermat also discovered some of the basic ideas of the integral calculus, which enabled him to find the areas under certain curves and the volumes enclosed by various surfaces. Thus, he is one of the anticipators of the great calculus revolution, that came to fruition in the 1670’s.

It is Fermat’s work in number theory that we will discuss in some detail in these notes. The Greeks had investigated properties of integers and Euclid’s Elements includes an account of what they had discovered. Greek mathematicians had a particular interest in perfect numbers. An integer n > 1 is said to be perfect if the sum of all its positive integer divisors, including 1 but excluding n itself, equals n. Thus, for example, 6 is perfect as its divisors less than 6 are 1, 2 and 3 and

1 + 2 + 3 = 6.

Similarly, 28 is perfect, as its divisors are 1, 2, 4, 7 and 14, whose sum is 28.

A means of producing even perfect numbers is described in Euclid’s Elements in the following way. Suppose that for some positive integer m, 2m - 1 is a prime. Then the integer 2m-1(2m - 1) is perfect. For, if we set 2m - 1 = p, where p is a prime, it is easy to see that the divisors of n = 2m-1p are

1, 2, 22, · · ·,2m-1, p, 2p,· · · ,2m-2p.

Now

1 + 2 + 2 + · · · + 2m-1 = 2m - 1 = p

and

p + 2p + · · · + 2m-2p = (2m~1 - 1)p

by the formula for summing a geometric progression. Thus it is clear that the sum of the divisors of n is n in this case. We obtain the perfect numbers 6 by taking m = 2 and 28 by taking m = 3. The next perfect number obtained is 24 · 31 = 496, when m = 5.

Primes of the form 2m - 1 have become known as Mersenne primes, after the 17th century French mathematician and priest Marin Mersenne. Mersenne primes play a special role in various parts of mathematics, especially group theory and number theory. In 1732, the Swiss mathematician Leonhard Euler proved that any even perfect number is necessarily one of the type described by Euclid (of the form 2m_1(2m - 1), with 2m - 1 a prime). To this day, no example of an odd perfect number has been found and they are conjectured not to exist. However, attempts to prove that there are no odd perfect numbers have been unsuccessful.

Fermat observed that, if 2m - 1 is a prime, then m must itself be a prime. For if m is composite, say m = rs, where r and s are integers with 1 < r < m, 1 < s < m, then 2r - 1 divides 2m - 1, since

2m - 1 = 2rs - 1 = (2r - 1)(2(r-1)s + 2(r - 2)s+ ... + 2s + 1).

Thus 2m - 1 can only be a prime if m is a prime, say m = p. We should note that not every number of the form 2P - 1, where p is a prime, is itself a prime. Nevertheless, we call any number of the form 2P - 1, where p is a prime, a Mersenne number. The smallest Mersenne number that is not a prime is 211 - 1 = 23 x 89. To investigate possible factors of Mersenne numbers, Fermat proved an important result, which has become known as Fermat’s Little Theorem. We can formulate this theorem as follows. Let r be a prime and let c be an integer not divisible by r. Then cr-l = 1 mod r, or in other words, r divides 2r-1 - 1. Now let I be the smallest positive integer such that r divides cl - 1. Then Fermat showed that I divides r - 1. We call I the order of c modulo r.

Suppose now that p is a prime and r is a prime divisor of the Mersenne number 2P - 1. It is straightforward to see that p is the order of 2 modulo r and thus p divides r - 1 by Fermat’s result. It follows that r - 1 = tp for some integer t and thus r = 1 + tp.

Using this result, it is easy to see that 23 = 1 + 2·11 might be a divisor of 211 - 1 and this proves to be the case. Fermat was led to discover the prime divisor 223 = 1 + 6 · 37 of the Mersenne number 237 - 1. Use of Fermat’s theorem makes it easier to search for possible prime divisors of Mersenne numbers. While many Mersenne numbers have proved not to be primes, several are primes. Indeed, many of the largest known primes are Mersenne primes. For example, 2127 - 1 was shown to be a prime in 1876 and it remained the largest known prime until 1951 (at the beginning of the era of high-speed computing machines). The Mersenne number 2216091 - 1, which is a 65, 050 digit integer, was shown to be a prime in 1985. It took 3 hours of computing time with a Cray computer to prove the primality of the number. Several special tests are available to determine whether Mersenne numbers are prime, and this accounts for the occurrence of many Mersenne primes in the lists of large prime numbers.

Fermat also investigated whether a number of the form 2m + 1 might be a prime. It is not hard to see that such a number cannot be a prime if m has an odd divisor bigger than 1. Therefore, such a number can only be a prime if m is a power of 2, say m = 2k. Fermat actually conjectured that when m = 2k, the number Fk defined by

$F_k = 2^{2^k} + 1$

is a prime. This is certainly true when k = 0, 1, 2, 3 or 4. However, Fermat’s conjecture seems to be spectacularly incorrect, as no value of k bigger than 4 has been found for which Fk is a prime. In 1732, Euler found the factorization

F5 = 232 + 1 = 641 x 6700417.

Since Euler’s factorization of F5, several other numbers of the form Fk have been factorized, while others have been proved to be composite without explicit factorizations being found. The numbers Fk are called Fermat numbers. So far, only five Fermat numbers have proved to be primes and it is even conjectured that there are no Fermat numbers Fk that are prime for k > 5. Of course, Fk is an enormously large number if k is at all large and so direct factorization is rarely feasible. Those Fermat numbers that are primes are called Fermat primes. Like the Mersenne primes, the Fermat primes still have an important position in mathematics.

Surprisingly, Fermat primes emerged in a piece of geometric research conducted more than one century after Fermat’s death. The German mathematician Carl Friedrich Gauss

(1777-1855) is known as one of the greatest mathematicians of all time. His early fame rests on his book Disquisitiones arithmeticae, published in 1801 when he was just 24. The book contains all the work on number theory that he had done up to that time. It includes the definitive result on the construction of regular polygons using ruler and compass alone. It had been known since the time of Euclid and earlier that we can construct a regular (equilateral) triangle, a regular pentagon, a regular hexagon, and any regular polygon having 2n sides, using a ruler and compass alone. No further constructions of such figures were known. Using his work on the roots of the polynomial xn1 (which are of course the complex roots of unity cos(2kπ/n)+i sin(22kπ/n)), Gauss proved the following fundamental result:

• Let pbea prime number. A regular polygon of p sides is constructible by ruler and compass alone if and only if p is a Fermat prime. A regular polygon of n sides is constructible by ruler and compass alone if and only if, in the prime factorization of n, we have n = 2tp1p2...pr, where p1, ..., pr are different Fermat primes.

It is said that Gauss’s discovery of the construction by ruler and compass of the regular 17-sided polygon in 1796, when he was only 18, was the crucial event that decided him on his career as a mathematician and not as a philologist. The actual construction is a little technical but Gauss’s work is considered to be the most original addition to the study of Euclidian geometry for over two thousand years.

Fermat’s interest in the problems described in the Arithmetica of Diophantus led him to invent a powerful tool in number-theoretic investigations. This is known as the method of descent. Roughly speaking, it is applied to certain integral Diophantine equations which we wish to show have no non-trivial integral solution. Starting with a presumed solution in positive integers, the method of descent tries to derive from this another solution involving strictly smaller positive integers. This produces a contradiction in the presumed solution, as we may descend to ever smaller positive solutions. In a sense, the method is like backwards induction. By this method, Fermat was able to prove that there are no integers x, y and z, all non-zero, that satisfy

x + y = z .

Fermat made use of the theeorem on Pythagorean triples that we previously mentioned. It is likely that he was also able to use the method of descent to prove that there are no

non-zero integers x, y and z that satisfy

x3 +y3 = z3,

as the method may also be applied here. His success in showing the non-existence of non-trivial integral solutions to these Diophantine equations must have led Fermat to believe that the method of descent could likewise prove that there are no non-zero integers x, y and z that satisfy

xn +yn = zn,

whenever n is an integer greater than 2. He certainly left a note in the margin of his copy of the Arithmetica stating that he had found such a solution. Nobody was able to make the method of descent apply in the cases where n > 4, and we must presume that Fermat was wrong in his claim. In fact, attempts to verify the truth of Fermat’s Last Theorem (as the claim became known) led to the development of a vast generalization of arithmetic known as algebraic number theory, in which the theory of primes and unique factorization is carried over to certain special types of complex number. Even this was not enough to prove Fermat’s Last Theorem in general, as the complete proof by Andrew Wiles in 1995 used techniques of algebraic geometry and modular form theory.

Classical Greek Geometry - 1

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