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April 2, 1997

The Transition Period

By the end of the 16 tex2html_wrap_inline450 century the European algebraists had achieved about as much as possible following the Islamic Tradition. They

were expert at algebraic manipulation,ß

could solve cubics and quartics,ß

had developed an effective syncopated notation, though they had yet no notation for arbitrary coefficients. This compelled them to explain methods by example. Formulas for the solution of equations did not yet exist.

Another trend that began during this time was the reconstruction of all of Greek mathematics. The basic library (Euclid, Ptolemy, etc.) had been translated earlier.

The Italian geometer Federigo Commandino (1509-1575) stands out here, as he prepared Latin translations of all known works of Archimedes, Apollonius, Pappus, Aristarchus, Autocyclus, Heron, and others. He was able to correct numerous errors that had crept in over the centuries of copying and translating.

The desire here, beside the link with the past, was to determine the Greek methods. Most of the Greek manuscripts were models of synthetic analysis (axiom, theorem, proof, tex2html_wrap_inline452 ) with little clue to methods of discovery. The work of Pappus (Book VII) was of some help here.

Francois Viéte [Vieta]
Born: Fontenay-le-Comte, Poitou (now Vendèe), 1540
Died: Paris, 23 February 1603

tex2html_wrap_inline454 Father's Occupation: Lawyer, Government Official

tex2html_wrap_inline454 Viète's father, Etienne Viète, was an attorney in Fontenay and a notary in Le Busseau. He was also a procureur du roi in Fontenay. Viète's grandfather was a merchant in the village of Foussay in Lower Poitou. Viète's mother was the first cousin to Barnabè Brisson, President of the Parlement de Paris under the League.

tex2html_wrap_inline454 All the evidence places the Viète family among the most distinguished in Fontenary. At least by the age of twenty, Viète was Sieur de la Bigotière. His two brothers both had distinguished positions.

tex2html_wrap_inline454 Viète's first scientific work was his set of lectures to Catherine Parthenay of which only Principes des cosmographie survives. This work introduced his student to the sphere, elements of geography, and elements of astronomy.

tex2html_wrap_inline454 His mathematical works are closely related to his cosmology and works in astronomy. In 1571 he published Canon mathematicus which was to serve as the trigonometric introduction to his Harmonicon coeleste which was never published.

Twenty years later he published In artem analyticum isagoge which was the earliest work on symbolic algebra.

In 1592 he began his dispute with Scaliger over his purported solutions to the classical problems with ruler and compass.

For all his achievement, however, mathematics was only a pastime for Viète, who was first and foremost a lawyer and public administrator.

Viète was involved in the calendar reform. He rejected Clavius's ideas and in 1602 published a vehement attack against the calendar reform and Clavius. The dispute ended only with Viète's death at the beginning of 1603.

tex2html_wrap_inline454 Means of Support: Government, Patronage Law, Personal Means

Francois Viète (1540-1603), a lawyer at the court of King Henry II in Tours and Paris; was one of the first men of talent who attempted to identify with the new Greek analysis.

Viète wrote several treatises, together called The Analytic Art. In them he reformulated the study of algebra for the solution of equations by focusing on their structure. Thus he developed the earliest articulated theory of equations.

He distinguished three types of analysis, after Pappus' two (theorematic, problematic). His trichotomy was:

Features of The Analytic Art:

Viète's symbolic operations

Viète's Theory of Equations

Viète also gave us our first closed form for tex2html_wrap_inline534 , from the formula


using trigonometry of inscribed figures. This can be derived starting with a inscribed square of area tex2html_wrap_inline538 , and proceeding to the circle by success doubling of the number of sides. Generate areas tex2html_wrap_inline540 . Now show tex2html_wrap_inline542 .

Algorithm: Let tex2html_wrap_inline544 . For tex2html_wrap_inline546 do tex2html_wrap_inline548 . Define tex2html_wrap_inline550 .

Large Simon Stevin

tex2html_wrap_inline454 Stevin was illegitimate. Though he was raised by his mother, his father's name was Antheunis Stevin, probably an artisan. ß

tex2html_wrap_inline454 By all evidence the family was poor.ß

tex2html_wrap_inline454 Nationality Birth: what is now Belgium Career: Netherlands Death: Netherlandsß

tex2html_wrap_inline454 Education Schooling: Leiden

tex2html_wrap_inline454 In 1581, after having been a bookkeeper in Antwerp and then a clerk in the tax office around Brugge, he moved to Leiden, enrolled in the Latin school, and in 1583 in the university. He remained enrolled until 1590. No evidence of a degree.

tex2html_wrap_inline454 Scientific Disciplines Mathematics, Engineering, Mechanics, Navigation, Astronomy, Hydraulics

tex2html_wrap_inline454 He published extensively on mathematics, engineering (both military and civil), and mechanics.

tex2html_wrap_inline454 His work on mathematics included practical surveying, with descriptions of instruments for that purpose.

Stevin's contributions to mathematics include:ß

Decimal Fractions. His work on decimal fractions was contained in his 1585 book, De Thiende (The Art of Tenths). It was widely read and translated. His goal in this book was as a teacher, to explain in full and elementary detail how to use the decimals. His idea was to indicate the power of ten associated with each decimal digit. We write for the approximation of tex2html_wrap_inline534






In the second part of his book he shows how to calculate with these numbers.

His notation was streamlined to the modern form only 30 years later in the 1616 English translation of Napier's Descriptio, complete with the decimal point.

Stevin also advocated the use of base 10 for units of all quantities. It would be more than 200 years before the French introduced the metric system.

In the history of science Stevins name is also prominent. He and a friend dropped two spheres of lead, one ten times the weight of the other, 30 feet onto a board. They noticed the sounds of striking to be almost simultaneous. (1586) Galileo's later similar experiment got more press.

Stevin wrote Hydrostatics in Dutch. (Translated into French by A. Girard, 1634 and into Latin by W. Snell, 1608. His methods influenced Kepler, Cavalieri and others.

In the Mechanics he makes center of gravity calculations. His methods modify and simplify the Archimedian' proof structure. He abandoned the routine of proofs by reductio ad absurdum. That is,

Stevin accepts that two quantities whose difference can be shown (arbitrarily) small are equal. Here's an example:

Theorem. The center of gravity of a triangle lies on its median.


To prove this he inscribes tex2html_wrap_inline576 with quadrilaterals as shown. Divide the median into tex2html_wrap_inline578 equal pieces and construct quadrilateral with sides parallel to the base and to the median. Let tex2html_wrap_inline580 denote the area of the triangle and let tex2html_wrap_inline582 denote the area of the quadrilaterals, then


By Euclid X-1, this can be made arbitrarily small, i.e.


Now let tex2html_wrap_inline590 and tex2html_wrap_inline592 denote the areas of tex2html_wrap_inline594 and tex2html_wrap_inline596 . Then




(Why?) Thus tex2html_wrap_inline602 are arbitrarily small, thus equal.

Here's lies logic:

  1. If two quantities differ they differ by a finite quantity.
  2. These quantities differ by less than any finite quantity.
  3. These quantities do not differ.

Yet he doesn't cite this as a general proposition, rather he repeats the argument in extenso each time.

Another example comes from Hydrostatics (1583).

Theorem. The total pressure on a vertical wall is 1/2 the pressure at the bottom.


Proof. He begins his proof by assuming the vertical wall is 1 foot square; area ACDE=1. He divides the wall into horizontal strips of width tex2html_wrap_inline608 feet. He argues that if such a strip were placed horizontally at a depth of h feet below the surface a weight of tex2html_wrap_inline612 of water would rest upon it. Placed vertically the pressure on the strip is


where tex2html_wrap_inline616 are the depths of the top and bottom of the strip. For the tex2html_wrap_inline618 strip,


Adding gives




and finally


His main intent, as always, was to provide enough understanding to make of mathematics a serviceable tool.

Large John Napier

Born: Edinburgh, 1550
Died: Edinburgh, 4 April 1617

tex2html_wrap_inline454 Sir Archibald Napier was the 7th Laird of Merchiston. The family had made its way up over the space of two centuries by service to the King. Sir Archibald, eventually became Master of the Mint.

tex2html_wrap_inline454 Education, St. Salvator's College, St. Andrews, 1563. He was apparently there only for a year and then went to the continent to study. He returned home by 1571 as a scholar competent in Greek.

tex2html_wrap_inline454 Rabdologiae, 1617, includes a number of calculating devices, including "Napier's bones," devices to aid multiplication (but not by logarithmic scales). Book II offers a practical treatment of mensuration rules.

tex2html_wrap_inline454 Napier was apparently reputed to be a magician, but the evidence is highly dubious.

tex2html_wrap_inline454 Napier did mathematics in his spare time.

In 1614, he wrote Mirifici Logarithmorum Canonis Descripto (Description of the Wonderful Canon of Logarithms), which contained a brief account showing how to use the tables.

It was in the second book, Mirifici Logarithmorum Canonis Constructio, (1619) he explains his idea of using geometry to improve arithmetical computations. The idea follows: Take two lines AB and CD


In modern terms: Let x=PB and y=CQ. Then


with tex2html_wrap_inline664 tex2html_wrap_inline666 . So,




when tex2html_wrap_inline672 so, tex2html_wrap_inline674 . This gives


Napier's definition lead directly to logarithms with base 1/e. Now Napier took effectively tex2html_wrap_inline678 . This gives: If




Now divide N and L by tex2html_wrap_inline688


The product rule is most important. Suppose




Napiers product role has an extra factor of tex2html_wrap_inline688 .

Napier even coined the word ``logarithm'' from the Greek: logos - ratio and arithmos - number.

Napier, of course, was aware of rules for product and quotient. He was unaware of a base.

The concept of logarithm function is implied throughout his work but this notion was not ready for formal definition.

Logarithms were a smash hit technology and their use immediately spread throughout Europe. Even Kepler used them.

Late in life, Napier thought it better to take the tex2html_wrap_inline698 and tex2html_wrap_inline700 . This he discussed with Henry Briggs (1561-1631), a professor of mathematics from Oxford.

Briggs returned to Oxford and began the new table. Starting from scratch he computed


54 times eventually approximating 1. All calculations were made at 30 decimal places. From these numbers he was able to build a table logarithms of closely spaced numbers.

In the meantime, actually earlier in 1588, the logarithm idea had occurred to the Swiss mathematician Jobst Bürgi (1552-1632). Bürgi published his Arithmetische und geometrische Progress-Tabulen in Prague, 1620.

Bürgi used tex2html_wrap_inline704 instead of tex2html_wrap_inline706 . He multiplied by tex2html_wrap_inline708 instead of tex2html_wrap_inline688 . If


Bürgi called 10L the ``red'' number corresponding to the ``black'' number N.

Large Galileo Galilei
Born: Pisa, 15 February 1564
Died: Arcetri, near Florence, 8 January 1642

tex2html_wrap_inline454 Father's Occupation: Musician, Merchantß

tex2html_wrap_inline454 Vincenzio Galilei was descended from a Florentine patrician family. He himself was a well known musician.

tex2html_wrap_inline454 He was not an economic success. He died leaving his oldest son (Galileo) with heavy financial responsibilities but no assets.

tex2html_wrap_inline454 He enrolled in Pisa in 1581 as a medical student, but left without a degree.

tex2html_wrap_inline454 Galileo was attracted to mathematics and studied it under Ostillio Ricci in 1583. After he left Pisa, he studied mathematics privately.

tex2html_wrap_inline454 Galileo was denounced to the Inquisition in 1615 and that he was tried and condemned by the Inquisition in 1633, living the rest of his life under house arrest. All of this was for Copernicanism, not for any heretical theological views.

tex2html_wrap_inline454 Scientific Disciplines Mechanics, Astronomy : Mathematics, Optics, Natural Philosophy

tex2html_wrap_inline454 He wrote De motu while at Pisa; Le mechaniche in the early 90's; work on motion during the first decade of the 17th century, with the composition of a treatise; the Discorsi in 1638.

tex2html_wrap_inline454 He began his telescopic observations, together with some thought on light and sight beginning in 1609.

tex2html_wrap_inline454 He wrote Il saggiatore, a work of many dimensions, including method and natural philosophy in general.

tex2html_wrap_inline454 He also wrote The Dialogo, far and away the leading polemic for the Copernican system, in 1632.

tex2html_wrap_inline454 During 1585-9 Galileo gave lessons in mathematics in Florence and Siena.

tex2html_wrap_inline454 1588, applied unsuccessfully for the chair in mathematics at Bologna. 1589, appointed to the chair in mathematics at Pisa. 1592, appointed to the chair in mathematics at Padua; there until 1610.

tex2html_wrap_inline454 While in Padua he produced his geometric and military compass and other instruments for sale.

tex2html_wrap_inline454 1610, appointed Mathematician and Philosopher to the Grand Duke Cosimo II, with a stipend of 1000 scudi. He was also professor of mathematics at Pisa, without obligation to teach or to reside in Pisa, and in fact his annual stipend came from the budget of the university.

tex2html_wrap_inline454 He improved the hydrostatic balance and the proportional compass. He described a crude clock to use with his method of determining longitude. He perfected the crude telescope into an astronomical instrument, and he developed a device, sort of a protomicrometer, to measure diameters of stars and planets. He developed a microscope. He developed a thermoscope.

tex2html_wrap_inline454 He developed a circle of young followers mostly in Florence-which included such as Viviani and Torricelli.

Galileo Galilei (1564-1642) was largely responsible for reformulating the laws of motion, considered earlier by the Greeks and medieval scholars.

He took a geometric approach, not algebraic. Nonetheless he applied mathematics in the service of motion on earth.

Galileo published Discourses and Mathematical Demonstrations Concerning Two New Sciences, 1638. Features:ß

Theorem. When a projectile is carried in motion compounded from equable horizontal and from naturally accelerated downward [motions], it describes a semi parabolic line in its movement.

The discovery of this result came from experiment of rolling balls off a table. This was his previous result about falling bodies tex2html_wrap_inline760 plus a knowledge of Apollonius.

He applied this to cannon fire, discovering that the maximum distance is achieved when the firing angle is tex2html_wrap_inline762 .

On factors such as air resistance he says:

Renè Descartes

His father was a counselor of the Parlement of Britany- noblesse de la robe.

Nationality Birth: French Career: French, Dutch, Swedish Death: Swedish

tex2html_wrap_inline454 In 1617 he set out for the Netherlands and the Dutch army. He wandered through Europe (at least he saw Germany and Italy, in addition to France) during the following eleven years before he settled in the Netherlands in 1628.

tex2html_wrap_inline454 Scientific Disciplines Mathematics, Natural Philosophy, Optics, Mechanics, Physiology, Music.

tex2html_wrap_inline454 Sometime during this period he inherited one-third of his mother's property, which he sold for about 27,000 livres. This produced enough income for him to live on.

tex2html_wrap_inline454 In 1628 Descartes left France for the Netherlands in order to isolate himself. It is clear that he lived quite comfortably; he did not aspire to live extravagantly. He himself asserted that he had received enough property from his family that he was free to choose where and how he would live. And he did.

tex2html_wrap_inline454 From an offer from Queen Christina of Sweden, Descartes moved to Sweden in 1649. (She planned to naturalize him and to incorporate him into the Swedish aristocracy with an estate on conquered German lands.)

tex2html_wrap_inline454 In Paris, he was in the circle of Mersenne, Mydorge, Morin, Hardy, Desargues, Villebressieu. In the Netherlands there was first Beeckman and then a network of followers that included Reneri, Regius, Constantijn Huygens, Herreboord, Heydanus, Golius, Schooten, Aemelius.

tex2html_wrap_inline454 He carried on a mathematical controversy with Fermat.

Renè Descartes (1596-1650) published his Discourse on the Method in 1637 in Leyden. In it he announced his program for philosophical research. In it, he

`` hoped, through systematic doubt, to reach clear and distinct ideas from which it would be possible to deduce innumerably valid conclusions''.

He believed everything was explainable in terms of matter and motion. The whole universe was just so. This Cartesian science enjoyed great popularity for almost a century - giving way only to the science of Newton.

The origin of analytic geometry came about while he was seeking to rediscover past truths from the Golden Age. Early on he discovered,

He must have come to analytic geometry sometime around 1628, the time he left France for Holland.

There he succeeded in solving the three and four line problems of Pappus (already known). This gave him confidence in his new methods.

La Geometrie was one of three appendices to Discourse. It is the earliest mathematical text that a present day student can follow without notational problems. Features:

Details of Book I:

tex2html_wrap_inline454 Here are things absent: rectangular coordinates, formulas for slope, distance, etc., no use of negative abscissas no new curves plotted from coordinates.

tex2html_wrap_inline454 Concludes Book I with the solution of the three and four line problems of Apollonius (Pappus).

Book II: Ovals of Descartes, given two foci tex2html_wrap_inline786 and tex2html_wrap_inline788



Book III - was a course on the elementary theory of equations. It included how to

Descartes was a professional mathematician, one of our first.

He is the ``father of modern philosophy."

He presented a changed scientific world view.

He established a branch of mathematics.

Oresme's latitude of forms more closely anticipates modern mathematical form or function but this played little role in the creation of analytic geometry.

Pierre de Fermat

His father had a prosperous leather business. He was also second consul of Beaumont. His mother brought the social status of the parliamentary noblesse de la robe to the family.

Education Schooling: Orleans, LD He received a solid classical secondary education. After studying with the Franciscans, he then studied with the Jesuits. He may have attended the University of Toulouse. He obtained the degree of Bachelor of Civil Laws from the University of Orleans in 1631.

In the Parlement of Toulouse, which was divided according to religion, he was a Catholic counselor.

Means of Support: Government, Personal Means

He corresponded with Carcavi, Brulart de Saint Martin, Mersenne, Roberval, Pascal, Huygens, Descartes, Frénicle, Gassendi, Lalouvere, Torricelli, Van Schooten, Digby, and Wallis.

Besides mathematical research and his profession Fermat's interests included:

Fermat had an enormous ego, and would communicate his theorems without proof and pose many difficult problems.

In 1636 he wrote his Introduction to Plane and Solid Loci, though it was not published in his lifetime. (At the same time Descartes was preparing his Discours de la méthode tex2html_wrap_inline452 )

Because he was familiar with Vièta's work, when he tried to reconstruct Apollonius' work, he replaced Apollonius' geometric analysis with algebraic analysis. This formed the beginning of Fermat's analytic geometry.

For example in Fermat's consideration of a theorem on an indeterminate number of points, he is led in the case of just two points to

  1. the correspondence between geometric loci and indeterminate algebraic equations in two or more variables, and
  2. a system of axes.

In modern terms, we suppose A = (-a,0) and B=(0,a). Let tex2html_wrap_inline816 be the circle (at the bisector of A and B), and let P = (x,y) be any point on the circle. Then


In the opening sentence, he says that if in solving algebraically a geometric problem one ends up with an equation in two unknowns, the resulting solution is a locus, straight line or curve.

After Vièta, Fermat sketched the simplest case of a linear equation: in Latin


(or Dx=By in modern notation).

He then shows a new method to prove the

Theorem: Given any number of fixed lines, in a plane, the locus of a point such that the sum of any multiples of the segments drawn at given angles from the point to the given lines is constant, is a straight line.

This follows from the fact that the segments are linear functions of the coordinates and Fermat's proposition that first degree equations represent straight lines.

Fermat then shows that tex2html_wrap_inline828 is a hyperbola and the form tex2html_wrap_inline830 can be reduced to tex2html_wrap_inline828 by a change of axes.

He establishes similar results for the parabola and for the ellipse.

His ``crowning'' result is this

Theorem. Given any number of fixed lines, the locus of a point such that the sum of the squares of the segments drawn at given right angles from the point to the lines is constant, is a solid locus (ellipse).

This would be nearly impossible to prove without analytic geometry.

We have, in modern terms


This gives an ellipse.

Summary. Fermat's exposition and clarity was better than Descartes'. His analytic geometry is closer to our own. He uses rectangular coordinates.

Fermat's Theory of Numbers.

Fermat's probability theory and analytic geometry were virtually ignored in his lifetime, and for almost 100 years after. He was given to secrecy of methods, and proofs are scarce.

His theory of numbers was widely circulated by Mersenne and others. It began with a study of perfect numbers.

He proved three propositions on this - communicated to Mersenne in 1640. The first result was

(1)  Theorem. If n is not prime tex2html_wrap_inline838 is not prime.



Thus the basic question is reduced to finding prime p for which tex2html_wrap_inline842 is prime. (These are called Mersenne primes.)

His next two theorems were

(2)  Theorem. If p is odd prime, then 2p divides tex2html_wrap_inline848 or p divides tex2html_wrap_inline852 .

(3)  Theorem. The only possible divisors of tex2html_wrap_inline842 have the form 2pk+1.

All without proofs! He gave a few examples.

Of course, Theorem 3 reduces the number of divisors one must check for primality of tex2html_wrap_inline842 .

There is a more general result he communicated to Bernard Frenicle de Bessy (1612-1676)

Theorem (Fermat's Little Theorem.) If p is any prime and a any positive integer then p divides tex2html_wrap_inline866 .



If a and p are relatively prime, then



Fermat gives no clue to discovery. However, this ``little theorem'' turns out to be very useful in number theory with many applications. The first published proof of this given by Euler (1732), in a much more general form. (Leibnitz left an earlier proof in manuscript form.)

Fermat also said that he believed all numbers of the form


were prime and announced a proof in 1659. (In 1732, Euler discovered that tex2html_wrap_inline878 is composite.) It is likely Fermat used his method of ``infinite descent'' using n=0,1,2,3,4 to arrive at the general conclusion.

Incidentally, using this method, which historically has been very important, Fermat showed that there is no integral right triangle whose area is a square.


Finally we note Fermat's Last Theorem:


has no integral solutions if tex2html_wrap_inline884 . It was resolved by Andrew Wiles in 1995.

Number theory faded after Fermat, re-emerging a century later.

Andriaan van Roomen

tex2html_wrap_inline454 Nationality Birth: Belgian Death: German

tex2html_wrap_inline454 Education Schooling: Louvain; M.A., M.D. He studied at the Jesuit College in Cologne.

tex2html_wrap_inline454 In Würzburg, where he was a professor of medicine and where he really created the medical faculty in a new university, he published a continuing series of medical theses defended by his students. A prolific author, Roomen wrote also on astronomy and natural philosophy. As with medicine, his opinions in these fields were traditional.

tex2html_wrap_inline454 As a mathematician he was especially concerned with trigonometry. He calculated the sides of the regular polygons, and from the polygon with 216 sides calculated the value of pi to sixteen places, achieving the same accuracy as Al Kashi. He also wrote a commentary on algebra.

tex2html_wrap_inline454 He corresponded with a considerable number of the mathematicians and scientists of his day, including Ludolph van Ceulen and Viete.

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Don Allen
Wed Apr 2 09:08:41 CST 1997