Chapter 5. Mathematics in the time of Descartes and Fermat

Chapter 5. Mathematics in the time of Descartes and FermatIt is important to note that during the late 16th century, considerable improvement occurredin the matter of algebraic notation, the lack of which hindered elementary manipulationof formulae. The leading innovator in this respect was the French mathematicianFrançois Viète (1540-1603). Viète made several contributions to mathematical knowledge,as well as initiating scientific cryptology (during a French war against Spain). Viète wasgreatly influenced by Greek mathematics and he sought to extend the methods that theGreeks had introduced. Building on the notion of analysis found in Greek texts of thelate classical period, Viète described a threefold method of problem-solving, as follows.• zetetic analysis, in which a problem is transformed into an equation linking theunknown with known quantities.• poristic analysis, a procedure for exploring the truth of a theorem by symbolicmanipulation.• exegetics, the means of transforming an equation found by zetetics to find thevalue of the unknown.It is the third part of this procedure, the exegetics, which we would understand as thefinding of the solution by algebraic manipulation, that provided the means of makingalgebra systematic. Viète introduced the idea of antithesis, which involves the transfer ofterms from one side of an equation to the other–a useful procedure to this day. This isthe same idea as that involved in al-jabr, which we saw in the work of al-Khawarizmi.For algebra, Viète’s main achievements are found in his book In artem analyticemisagoge (Introduction to the Analytic Art) of 1590. In this work, he aimed to reviveand extend the methods of Diophantus of Alexandria, and he used a few of Diophantus’sproblems. His great innovation in this work is the use of letters to denote unknown orknown 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. Viète did not entirely break away from the notation of his predecessors, suchas Bombelli, as he wrote powers in terms of words or abbreviations. Thus A quadratumrepresented A 2 , B cubus represented B 3 , C quadrato−quadratum represented C 4 . Thesecould also be shortened to A quad or C quad − quad. He also used the German symbols+ and −.1

Dividing we get,r 2 = − 4a 3 .Now our equation becomes s cos 3θ = −b and we deduce thatcos 3θ = − b s .We want to get real solutions for α and this requires that | cos 3θ| ≤ 1, or cos 2 3θ ≤ 1. Wetherefore must have b 2 − s 2 ≤ 0. ButThus we requires 2 = a2 r 29= − 4a327b 2 + 4a327 ≤ 0for this method to give real solutions, and this is precisely the condition that we haveassumed, and is the one that occurs in the irreducible case of the cubic. We can solve for3θ by taking inverse cosines and then divide by 3 to find θ. Then we evaluate cos θ and useα = r cos θ. At the time of Viète’s work, tables of cosines were quite common for astronomicaland navigational purposes and a solution of this kind would have seemed perfectlycalculable. We see now that Viète’s solution provides an alternative (non-algebraic) solutionin terms of real numbers precisely in the case that the algebraic solution by Cardano’smethod requires us to use complex numbers.One of the major advances in mathematics of all time is the use of algebraic methodsto solve problems of pure geometry. This innovation was due to the French mathematicianand philosopher René Descartes (1596-1650). His work appeared in his publicationLa Géométrie, which was included as an appendix to his famous philosophical treatiseDiscours de la méthode (1637). Interestingly, despite its scholarly nature, it was writtenin French, rather than the more erudite Latin. Descartes’s notation in this work is recognizablysimilar to our own, in marked distinction to that used by the Italians a centuryor even fifty years earlier. He employed the + and − signs, adopted from German arithmetictexts of the 16th centuries. He used exponential notation such as y 3 for powers ofunknowns, and surd signs for square roots. Variables are usually denoted by small romanletters from the end of the alphabet, and constant quantities by small roman letters fromthe beginning of the alphabet, such as a, b, c. Here again, this is the convention frequentlyadopted 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 multiplyinga symbol such as y or y 2 by an algebraic expression such as a 2 +ab+b 2 , might write3

served on various official bodies. His real interest was mathematics, which he studied inhis spare time.Unlike most noteworthy French mathematicians, Fermat never lived in Paris. Hecorresponded with leading French mathematicians, including Pascal and Mersenne. Fermatnever published any of his discoveries. For this reason, we do not usually know hisworking methods and cannot always be sure that facts he claimed as true were actuallyproved or merely conjectured. He certainly had numerous good ideas, and would drophints to his correspondents of how he had proceeded to prove things. His son publishedFermat’s manuscripts in 1679, and these provide us with most of our information abouthis methods.Fermat’s interests span a wide range of mathematical activity. He is known forhis work on the beginnings of probability theory, partly in response to questions raised byPascal. (It might be worth remarking that Cardano had made some earlier observations onelementary probability theory in the previous century.) He is also recognized for Fermat’sprinciple in physics. This principle states that light travels in such a way as to minimizethe time taken to pass between two given points. As light travels at different speeds indifferent media, for example, in air and in glass, it is possible to explain the observed lawsof reflection and refraction by this principle.Along with Descartes, Fermat is held to be one of the creators of modern analyticgeometry. He was influenced by Viète’s use of algebraic symbols and tried to apply algebrato study the work of Apollonius on geometric loci, such as conic sections. He was ableto relate the conic sections to what we now call equations of the second degree, such asx 2 − y 2 = 1 or x 2 = y. Before the development of the differential calculus later in the17th century, Fermat is credited with techniques for finding local maximum and minimumvalues on curves.It is for his work on number theory that Fermat is best remembered. Gauss held that,although the Arithmetica of Diophantus displayed great skill, it consisted of solutions toisolated problems which did not constitute a theory. It was only in the 17th century thataproper theory of numbers emerged, beginning with the work of Fermat, and continuedin the 18th century by Euler, Lagrange and Legendre. Gauss himself provided the mostcomplete work on number theory, his Disquisitiones arithmeticae, published in 1801 whenhe was only 24, in which he described the work of his predecessors and found numerous5

new lines of research.One popular theme in number theory has been that of sums of squares.integers can be written in the formx 2 + y 2 ,Whatwhere x and y are integers, or in the formx 2 + y 2 + z 2 ,where x, y and z are integers, and so on. Bachet (who translated the Arithmetica ofDiophantus into Latin in 1621, a translation much used by Fermat) conjectured thatevery integer n is a sum of four squares,n = x 2 + y 2 + z 2 + w 2 ,where x, y, z and w are integers. This was eventually rigorously proved by Lagrangein 1770’s. Fermat, in a letter to Mersenne in 1640, asserted that any prime p satisfyingp ≡ 1 mod 4 can be written asp = a 2 + b 2 ,for suitable integers a and b. If we assume that 0 < a < b, then a and b are unique. Exactdetails of his proof are not available but he claimed to use his method of infinite descent,which is a powerful method that may be applied to certain Diophantine equations. It is asort of induction in reverse. He probably used the technique in his study of the so-calledFermat problems, such as showing that there are no solutions to the equationx 3 + y 3 = z 3 ,where x, y and z are all non-zero integers. He certainly showed that there are no non-trivialinteger solutions to the equationx 4 + y 4 = z 4 ,using his method of infinite descent.His success in showing the non-existence of non-trivial integral solutions to theseDiophantine equations must have led Fermat to believe that the method of descent couldlikewise prove that there are no non-zero integers x, y and z that satisfyx n + y n = z n ,6

whenever n is an integer greater than 2. He certainly left a note in the margin of his copyof the Arithmetica stating that he had found such a solution. This assertion achievedinternational prominence with the publication of Fermat’s collected works in 1679, whichbrought Fermat’s claim out into the open. Nobody was able to make the method of descentapply 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 problem becameknown) led to the development of a vast generalization of arithmetic known as algebraicnumber theory, in which the theory of primes and unique factorization is carried over tocertain special types of complex number. Even this was not enough to prove Fermat’sLast Theorem in general, as the complete proof by Andrew Wiles in 1995 used techniquesof algebraic geometry and modular form theory.Fermat considered further problems concerning primes.that every prime p satisfying p ≡ 1 mod 3 is expressible asFor example, he claimedp = a 2 + 3b 2for suitable integers a and b (letter to Pascal, 1654). All of Fermat’s claims are true,and he may well have proved them. However, definitive proofs first appeared with Euler,and Gauss provided the most satisfactory approach to problems of the kind considered byFermat.We will now discuss some of Fermat’s work relating to primes. The Greeks hadinvestigated properties of integers and Euclid’s Elements includes an account of whatthey 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, including1 but excluding n itself, equals n. Thus, for example, 6 is perfect as its divisors less than6 are 1, 2 and 3 and1 + 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 thefollowing way. Suppose that for some positive integer m, 2 m − 1 is a prime. Then theinteger 2 m−1 (2 m − 1) is perfect. For, if we set 2 m − 1 = p, where p is a prime, it is easyto see that the divisors of n = 2 m−1 p are1, 2, 2 2 , · · · , 2 m−1 , p, 2p, · · · , 2 m−2 p.7

Now1 + 2 + 2 2 + · · · + 2 m−1 = 2 m − 1 = pandp + 2p + · · · + 2 m−2 p = (2 m−1 − 1)pby the formula for summing a geometric progression. Thus it is clear that the sum of thedivisors of n is n in this case. We obtain the perfect numbers 6 by taking m = 2 and 28by taking m = 3. The next perfect number obtained is 2 4 · 31 = 496, when m = 5.Primes of the form 2 m − 1 have become known as Mersenne primes, after the 17thcentury French mathematician and priest Marin Mersenne. Mersenne primes play a specialrole in various parts of mathematics, especially group theory and number theory. In1732, the Swiss mathematician Leonhard Euler proved that any even perfect number isnecessarily one of the type described by Euclid (of the form 2 m−1 (2 m − 1), with 2 m − 1a prime). To this day, no example of an odd perfect number has been found and theyare conjectured not to exist. However, attempts to prove that there are no odd perfectnumbers have been unsuccessful.Fermat observed that, if 2 m − 1 is a prime, then m must itself be a prime. For if mis composite, say m = rs, where r and s are integers with 1 < r < m, 1 < s < m, then2 r − 1 divides 2 m − 1, since2 m − 1 = 2 rs − 1 = (2 r − 1)(2 (r−1)s + 2 (r−2)s + · · · + 2 s + 1).Thus 2 m − 1 can only be a prime if m is a prime, say m = p. We should note that notevery number of the form 2 p − 1, where p is a prime, is itself a prime. Nevertheless, wecall any number of the form 2 p − 1, where p is a prime, a Mersenne number. The smallestMersenne number that is not a prime is 2 11 − 1 = 23 × 89. To investigate possible factorsof Mersenne numbers, Fermat proved an important result, which has become known asFermat’s Little Theorem. We can formulate this theorem as follows. Let r be a prime andlet c be an integer not divisible by r. Then c r−1 ≡ 1 mod r, or in other words, r divides2 r−1 −1. Now let l be the smallest positive integer such that r divides c l −1. Then Fermatshowed that l divides r − 1. We call l the order of c modulo r.Suppose now that p is a prime and r is a prime divisor of the Mersenne number2 p − 1. It is straightforward to see that p is the order of 2 modulo r and thus p dividesr − 1 by Fermat’s result. It follows that r − 1 = tp for some integer t and thus r = 1 + tp.8

Using this result, it is easy to see that 23 = 1 + 2 · 11 might be a divisor of 2 11 − 1 and thisproves to be the case. Fermat was led to discover the prime divisor 223 = 1 + 6 · 37 of theMersenne number 2 37 − 1. Use of Fermat’s theorem makes it easier to search for possibleprime divisors of Mersenne numbers. While many Mersenne numbers have proved not tobe primes, several are primes. Indeed, many of the largest known primes are Mersenneprimes. For example, 2 127 −1 was shown to be a prime in 1876 and it remained the largestknown prime until 1951 (at the beginning of the era of high-speed computing machines).The Mersenne number 2 216091 −1, which is a 65, 050 digit integer, was shown to be a primein 1985. It took 3 hours of computing time with a Cray computer to prove the primalityof the number. Several special tests are available to determine whether Mersenne numbersare prime, and this accounts for the occurrence of many Mersenne primes in the lists oflarge prime numbers.Fermat also investigated whether a number of the form 2 m + 1 might be a prime. Itis not hard to see that such a number cannot be a prime if m has an odd divisor biggerthan 1. Therefore, such a number can only be a prime if m is a power of 2, say m = 2 k .Fermat actually conjectured that when m = 2 k , the number F k defined byF k = 2 2k + 1is a prime. This is certainly true when k = 0, 1, 2, 3 or 4. However, Fermat’s conjectureseems to be spectacularly incorrect, as no value of k bigger than 4 has been found forwhich F k is a prime. In 1732, Euler found the factorizationF 5 = 2 32 + 1 = 641 × 6700417.Since Euler’s factorization of F 5 , several other numbers of the form F k have been factorized,while others have been proved to be composite without explicit factorizations being found.The numbers F k are called Fermat numbers. So far, only five Fermat numbers have provedto be primes and it is even conjectured that there are no Fermat numbers F k that areprime for k ≥ 5. Of course, F k is an enormously large number if k is at all large and sodirect factorization is rarely feasible. Those Fermat numbers that are primes are calledFermat primes. Like the Mersenne primes, the Fermat primes still have an importantposition in mathematics.Surprisingly, Fermat primes emerged in a piece of geometric research conducted morethan one century after Fermat’s death. The German mathematician Carl Friedrich Gauss9

(1777-1855) is known as one of the greatest mathematicians of all time. His early famerests on his book Disquisitiones arithmeticae, published in 1801 when he was just 24. Thebook contains all the work on number theory that he had done up to that time. It includesthe 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 polygonhaving 2 n sides, using a ruler and compass alone. No further constructions of such figureswere known. Using his work on the roots of the polynomial x n −1 (which are of course thecomplex roots of unity cos(2kπ/n)+i sin(2kπ/n)), Gauss proved the following fundamentalresult:• Let p be a prime number. A regular polygon of p sides is constructible by ruler andcompass alone if and only if p is a Fermat prime. A regular polygon of n sides isconstructible by ruler and compass alone if and only if, in the prime factorization ofn, we have n = 2 t p 1 p 2 · · · p r , where p 1 , . . . , p r are different Fermat primes.It is said that Gauss’s discovery of the construction by ruler and compass of the regular17-sided polygon in 1796, when he was only 18, was the crucial event that decided him onhis career as a mathematician and not as a philologist. The actual construction is a littletechnical but Gauss’s work is considered to be the most original addition to the study ofEuclidian geometry for over two thousand years.His success in showing the non-existence of non-trivial integral solutions to theseDiophantine equations must have led Fermat to believe that the method of descent couldlikewise prove that there are no non-zero integers x, y and z that satisfyx n + y n = z n ,whenever n is an integer greater than 2. He certainly left a note in the margin of his copyof the Arithmetica stating that he had found such a solution. Nobody was able to makethe method of descent apply in the cases where n > 4, and we must presume that Fermatwas 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 arithmeticknown as algebraic number theory, in which the theory of primes and unique factorizationis carried over to certain special types of complex number. Even this was not enough toprove Fermat’s Last Theorem in general, as the complete proof by Andrew Wiles in 1995used techniques of algebraic geometry and modular form theory.10