LECTURE

Euclid and the Mathematical Renaissance.I. Euclid and the Tradition of Classical Geometry. A. Euclid and 'The Elements'. B. 'The Elements' in Antiquity and the Middle Ages. C. 'The Elements' in the 16th and 17th Centuries.II. The Mathematical Renaissance in Italy.III. Mathematics in the 17th Century. A. Descartes et al.. B. Physical Intuition and the Growth of Mathematics. C. A Revolution in Mathematics?.IV. Mathematics, Mechanics, and Magic

[References: Bennet 1986; Boyer 1968; Cohen 1976; Crowe 1975; Dauben 1984; Debus 1975; Eamon 1983; Feingold 1984; Gjersten 1984; Hansen 1986; Keller 1985; Kline 1959; Kline 1972; Molland 1988; Rose 1975; Van Egmond 1988; Whitrow 1988]

.I. EUCLID AND THE TRADITION OF CLASSICAL GEOMETRY.A. EUCLID AND THE 'ELEMENTS'.TODAY I WOULD LIKE TO TO GIVE A VERY BRIEF SUMMARY OF

THE MAJOR TRENDS AND DEVELOPMENTS IN MATHEMATICS.

SO FAR WE HAVE LOOKED AT THE TRANSFORMATIONS IN THREE MAJOR SCIENTIFIC TRADITIONS: THE ONE FROM A GEOCENTRIC TO A HELIOCENTRIC ASTRONOMY; FROM ARISTOTELIAN TO GALILEAN MECHANICS; AND THE

COMPLEX TRANSITION FROM ARISTOTELIAN NATURAL PHILOSOPHY TO THE MECHANICAL PHILOSOPHY.

IRAONICALLY, MATHEMATICS IS NOT USUALLY TREATED AS A 'SCIENTIFIC TRADITION' IN COURSES ON THE SCIENTIFIC REVOLUTION.

AND I THINK THE TEXTS I HAVE ASKED YOU TO READ BEAR SILENT WITNESS TO THIS FACT.

THERE IS VERY LITTLE HISTORY OF MATHEMTICS TO BE FOUND IN ANY OF THEM EXCEPT AS IT RELATED DIRECTLY TO PROBLEMS IN ASTRONOMYOR MECHANICS.

YET IT SEEMS TO ME THAT THIS IS A MISTAKE.NOT ONLY DOES MATHEMATICS HAVE A LONG AND

DISTINGUISHED TRADITION QUITE SIMILAR TO THE THOSE OF THE PHYSICAL SCIENCES, IT IS ALSO VERY CLOSELY ASSOCIATED WITH THEM.

AND IT SEEMS TO ME THAT VARIOUS BRANCHES OF MATHEMTICS UNDERWENT SOMETHING LIKE REVOLUTIONARY TRANSFORMATIONS IN THE EARLY MODERN PERIOD.

HOWEVER, TO BEGIN THIS QUICK LOOK AT THE POSSIBILITY OF A 'MATHEMTICAL REVOLUTION' WE MUST START AT THE BEGINNING -- AND OF COURSE THAT MEANS WITH 'THE ELEMENTS' OF EUCLID.

FROM 300 B.C. TO 1900 AD, 'THE ELEMENTS' HAS HAD A LONGER ACTIVE PUBLICATION LIFE THAN ANY OTHER WORK OF NON-FICTIONAL.

NO OTHER WORK HAS BEEN AS FREQUENTLY EDITED, TRANSLATED, ABBREVIATED, AND COMMENTED UPON AS EUCLID'S 'THE ELEMENTS'.

YET WE KNOW ALMOST NOTHING ABOUT THE AUTHOR HIMSELF.

EUCLID IS BELIEVED TO HAVE LIVED DURING THE REIGN OF THE FIRST GREEK RULER OF EGYPT, PTOLEMY I, CA. 306-283 B.C..

HE IS CONSIDERED TO BE A MEMBER OF THAT GENERATION OF MATHEMATICIANS WHO LIVED BETWEEN THE TIME OF PLATO, WHO DIED IN 347 B.C AND ARCHIMEDES, WHO WAS BORN IN CA. 287 B.C..

AND IT SEEMS HE FOUNDED A SCHOOL OF MATHEMATICS IN ALEXANDRIA AND TAUGHT THERE.

LEGEND ATTRIBUTES TO EUCLID A RATHER DRY WIT.ALLEGEDLY HE WAS ASKED BY PTOLEMY I, HIS KING AND PUPIL,

IF THERE WAS AN EASIER WAY TO LEARN GEOMETRY THAN TO STUDY THE 'ELEMENTS'; HE RECEIVED THE CLASSIC REPLY, THAT NO, THERE IS NO ROYAL ROAD TO THE MASTERY OF GEOMETRY.

ANOTHER STORY HAS A YOUNG STUDENT ASKING EUCLID WHAT VALUE DOES THE STUDY OF GEOMETRY HAVE?; IN RESPONSE HIS TEACHER GAVE HIM A SMALL COIN SO THAT HE WOULD NOT FEEL HE WAS WASTING HIS TIME.

LIKE PTOLEMY - THE ASTRONOMER - EUCLID'S WORK REPRESENTS A CULMINATION OF A LONG TRADITION IN GREEK THOUGHT.

AND, LIKE PTOLEMY'S ALMAGEST, THE COHERENCE AND TREMENDOUS CONTEMPORARY SUCCESS OF EUCLID'S ELEMENTS RENDERED PREVIOUS WORKS IN THAT TRADITION OBSOLETE.

THUS WE HAVE ALMOST NO KNOWLEDGE OF HIS PREDECESSORS.

THOUGH HE CERTAINLY DREW HEAVILY ON THE WORKS OF EUDOXUS, ACONTEMPORARY OF PLATO, AS WELL AS ON THE PYTHAGOREAN SCHOOL.

AS WE ALL KNOWN -- OR VAGUELY REMEMBER -- 'THE ELEMENTS' CONSISTS OF 23 BASIC DEFINTIONS, 5 'COMMON NOTIONS' AND 5 POSTULATES OR AXIOMS.

THE "DEFINTIONS" ARE OF STRAIGHTFORWARD AND FREQUENTLY-USED TERMS; LIKE POINT, LINE, OBTUSE ANGLE, OR ISOCELES TRIANGLE.

THE "COMMON NOTIONS" ARE EQUALLY STRAIGHTFORWARD IDEAS CONCERNING COMPARISION AND COMMON-SENSE RULES OF INFERENCE:AS FOR EXAMPLE; 'THINGS WHICH ARE EQUAL TO THE SAME THING ARE EQUAL TO ONE ANOTHER'; OR 'THE WHOLE IS GREATER THAN THE PART'.

THE "POSTULATES" ARE UNPROVEN ASSERTIONS ABOUT THE NATURE OFGEOMETRY AND TOGETHER FORM THE FOUNDATION OF THE AXIOMATICSYSTEM.

THE MOTIVATION FOR ESTABLISHING A SMALL NUMBER OF SELF-EVIDENT AXIOMS IS TO GUARENTEE THE VALIDITY OF ALL SUBSEQUENT THEOREMS CONSTRUCTED FROM THESE AXIOMS.

EUCLIDEAN GEOMTERY HAS, OF COURSE, LONG BEEN VIEWED AS THE PARADIGM OF CERTAIN KNOWLEDGE BECAUSE OF ITS AXIOMATIC-DEDUCTIVE STRUCTURE.

IN FACT, WE HAVE ALREADY SEEN HOW DESCARTES SOUGHT TO EMPLOYTHIS STRATEGY IN HIS ATTEMPT TO CONSTRUCT A NEW AND CERTAIN NATURAL PHILOSOPHY.

B. THE 'ELEMENTS' IN ANTIQUITY & THE MIDDLE AGES.ALTHOUGH 'THE ELEMENTS' OF EUCLID CAN BE CONSIDERED

THE CULMINATION OF A LONG TRADITION IN GREEK MATHEMATICS, IT BY NO MEANS MARKED AN END TO THAT TRADITION.

'THE ELEMENTS' WAS WIDELY READ AND FREQUENTLY COMMENTED UPON IN ANTIQUITY AND WENT THROUGH THE HANDS OF SEVERAL EDITORS.

WE KNOW OF SIX EDITIONS WITH EXTENSIVE COMMENTARIES FROM GREEK ANTIQUITY.

HOWEVER, FOR OUR PURPOSES, THE MOST IMPORTANT EDITION WITH COMMENTARY CAME FROM THEON OF ALEXANDRIA, WHO LIVED DURING THE LATTER HALF OF THE 4TH CENTURY A.D..

FOR ALL EDITIONS BEFORE 19TH CENTURY DERIVE ULTIMATELY FROM THE TEXT PREPARED BY THEON.

AS WITH THE WRITINGS OF ARISTOTLE, EUCLID'S 'ELEMENTS' WAS TRANSLATED INTO ARABIC AND ENJOYED ENORMOUS ATTENTION FROM ISLAMIC SCHOLARS FOR MORE THAN 800 YEARS.

HOWEVER, KNOWLEDGE OF EULCID'S WORK WAS ALMOST COMPLETELYLOST IN THE LATIN WEST DURING THIS PERIOD.

PERHAPS NOTHING ILLUSTRATES THE DEPTHS TO WHICH KNOWLEDGE OF GEOMETRY HAD SUNK DURING THE MIDDLE AGES THAN THE FOLLOWING EPISODE:.

IN THE CORRESPONDENCE OF TWO LEARNED SCHOLARS FROM THE EARLY 11TH CENTURY, ONE FINDS THE FOLLOWING EXCHANGE CONCERNING A REFERENCE TO A PASSAGE IN EUCLID: THE ASSERTION WAS THAT THE INTERIOR ANGLES OF A TRIANGLE ARE EQUAL TO TWO RIGHT ANGLES:.

WHAT PUZZLED THE LEARNED CORRESPONDENTS WAS THE PHRASE 'INTERIOR ANGLES'.

CITING THE LEARNED BISHOP OF CHARTRE AS HIS AUTHORITY, ONE CORRESPONDENT SPECULATED THAT THEY WERE THE

ANGLES FORMED ON EITHER SIDE OF THE LINE DRAWN FROM APEX TO BASE.

THIS SORRY STATE OF AFFAIRS WAS NOT TO LAST FOR LONG.BEGINNING IN THE EARLY 12TH CENTURY, THERE WAS A GREAT

WAVE OFTRANSLATIONS OF GREEK SCIENTIFIC WORKS FROM ARABIC INTO LATIN.

THIS IS THE SAME WAVE OF TRANSLATIONS THAT BROUGHT THE WORKS OF PTOLEMY AND ARISTOTLE INTO THE LATIN WEST.

AFTER THE 12TH CENTURY, WE CAN BE CERTAIN THAT THERE WAS A GENERAL INCREASE IN THE LEVEL OF KNOWLEDGE OF EUCLIDEAN GEOMETRY.

GEOMETRY SOON BECAME AN ESTABLISHED PART OF THE UNIVERSITY, AND 'THE ELEMENTS' BECAME THE CENTRAL TEXT IN THE 'QUADRIVIUM'; THAT IS, THE PART OF THE UNIVERSITY CURRICULUM WHICH CONSISTED OF ARITHMETIC, ASTRONOMY, MUSIC (OR HARMONICS), AND GEOMETRY.

YET, DESPITE THE GROWTH IN KNOWLEDGE OF GEOMETRY AND ITS SECURE PLACE WITHIN THE UNIVERSITY STRUCTURE, EUCLID'S WORK DIDNOT BECOME CENTRAL TO SCHOLASTIC CONCERNS.

RATHER, IT SEEMS TO HAVE SERVED PRIMARILY PHILOSOPHICAL INTERESTS.

THAT IS, QUESTIONS WERE DIRECTED MORE TOWARD HOW GEOMETRIC KNOWLEDGE RELATES TO KNOWLEDGE IN NATURAL PHILOSOPHY OR MATHEMATICS GENERALLY RATHER THAN TO TECHNICAL QUESTIONS INTERNAL TO GEOMETRY ITSELF.

THERE WERE, FOR EXAMPLE, NO EXTENDED DEBATES ON THE PARALLELPOSTULATE IN THE LATIN WEST AS THERE HAD BEEN IN ISLAMIC CULTURE.

AND ALTHOUGH EUCLID RECEIVED CONSIDERABLY MORE ATTENTION FROM WESTERN SCHOLARS AFTER THE 12TH CENTURY THAN BEFORE, THENUMBER OF COMMENTARIES WRITTEN BY ARABIC SCHOLARS FAR OUTNUMBERS THOSE WRITTEN IN THE WEST.

UNLIKE THE WORKS OF ARISTOTLE, 'THE ELEMENTS' DID NOT BECOME PART OF THE COMMENTARY AND QUESTIONES TRADITION OF THE UNIVERSITIES.

C. THE 'ELEMENTS' IN THE 16TH & 17TH CENTURIES.ALL OF THIS CHANGED DRAMATICALLY IN THE 16TH CENTURY.THE FORCES DRIVING THIS CHANGE WERE CERTAINLY MANY

AND COMPLEX.YET IT SEEMS THAT TWO OF THE MOST IMPORTANT WERE THE

ADVENT OF PRINTING AND THE EXPANSION OF UNIVERSITY EDUCATION.

PERHAPS THE EASIEST WAY TO GET AN OVERVIEW OF THIS CHANGE IS TOTAKE A LOOK AT THE PUBLICATION HISTORY OF EUCLID'S 'ELEMENTS' FROM THE 15TH CENTURY ONWARD.

THE GRAPH I HANDED OUT EARLIER SHOWS THE NUMBER OF EDITIONS AND TRANSLATIONS OF ALL OR PART OF EUCLID'S WORK IN GEOMETRY FROM 1450 UNTIL 1700.

AND WHAT WE FIND WHEN WE ARRANGE THIS BIBLIOGRAPHICAL DATA GRAPHICALLY IS A REMARKABLE GROWTH IN THE EUCLIDEAN TRADITIONAFTER 1500 AND A SUSTAINED LEVEL OF INTEREST IN EUCLID THROUGHOUT THE EARLY MODERN PERIOD.

IF WE LOOK BEHIND THE BRUTE NUMBERS, WE FIND SEVERAL INTERESTING PATTERNS.

IN FACT, LET ME DIVIDE THESE PATTERNS INTO THREE PARTS: 1) RECOVERY AND ESTABLISHMENT OF THE TEXT; 2) DISSEMINATION; AND 3) IN THE 17TH CENTURY, TRANSFORMATION.

THE FIRST PRINTED EDITION OF EUCLID, PUBLISHED IN 1482, WAS BASEDON THE MEDEIVAL LATIN TRANSLATION OF CAMPANUS OF NOVARA -- AND THUS, FROM THE HUMANISTS' POINT OF VIEW, DEFECTIVE.

A NEW LATIN TRANSLATION FROM THE GREEK WAS PRINTED IN 1505 IN VENICE BY BARTOLOMEO ZAMBERTI.

ZAMBERTI RELIED ON A GREEK MANUSCRIPT DERIVING FROM THEON OFALEXANDRIA, EUCLID'S MOST CAPABLE AND INFLUENTIAL EDITOR IN ANTIQUITY.

AND HIS LATIN TRANSLATION SET THE STANDARD FOR THE NEXT 70 YEARS.

THE FIRST EDITION IN GREEK WAS PUBLISHED IN BASEL IN 1533 BY SIMON GRYNAEUS, A GERMAN THEOLOGIAN.

THIS WORK ALSO CONTAINED MANY GREEK COMMENTARIES AND IT REMAINED THE ONLY COMPLETE GREEK TEXT OF EUCLID UNTIL THE 18THCENTURY.

WITH TWO LATIN TRANSLATIONS AND A SCHOLARLY GREEK VERSION ALL IN PRINT, THE STAGE WAS SET FOR THE SECOND PHASE OF THE RENAISSANCE OF EUCLIDEAN GEOMETRY.

THIS SECOND PHASE -- WHAT I SHALL CALL THE PERIOD OF DISSEMINATION --OCCUPIES ROUGHLY THE SECOND THIRD OF THE 16TH CENTURY, AND IS CHARACTERIZED BY TWO DEVELOPMENTS:.

1) THE PRODUCTION OF A LARGE NUMBER OF EDITIONS IN LATIN FOR USEAS TEXTBOOKS IN THE SCHOOLS.

AND 2) THE TRANSLATION OF EUCLID INTO THE VERNACULARS.INDICATIVE OF THE FORMER TREND IS THE LATIN EDITION OF

EUCLID'S 'ELEMENTS' PUBLISHED IN 1537 IN BASEL.IT CONTAINED A PREFACE BY PHILIP MELACHTHON, THE

GERMAN REFORMER WHOM WE HAVE ENCOUNTERED BEFORE AS THE PATRON OF THE CIRCLE OF LUTHERAN ASTRONOMERS RECEPTIVE TO COPERNICAN ASTRONOMY.

THE PREFACE WAS ADDRESSED TO THE 'ZEALOUS AND DILIGENT ADOLESCENTS' FOR WHOM THE WORK WAS DESIGNED.

IN FACT MOST OF THE LARGE NUMBER OF LATIN EDITIONS ISSUED DURING THIS PERIOD WERE TAILORED TO THE NEEDS OF THE YOUNG UNIVERITY STUDENT.

HOWEVER, IN ADDITION TO EUCLID AS SCHOOL TEXTBOOK; IT SEEMS THERE WAS ALSO A EUCLID FOR THE LITERATE CRAFTSMAN.

THE COMMON BURGHER'S ACCESS TO EUCLID CAN BE ROUGHLY GAUGED BY THE APPEARANCE OF TRANSLATIONS OF EUCLID IN THE VERNACULARS.

THE FIRST TRANSLATION INTO ITALIAN WAS PUBLISHED IN 1543 IN VENICE; IT WAS DONE BY NICCOLO TARTAGLIA AND WENT THROUGH AT LEAST A HALF-DOZEN EDITIONS IN THE NEXT 40 YEARS.

ANOTHER ITALIAN TRANSLATION APPEARED TWO YEARS LATER IN ROME.

THE FIRST TRANSLATION INTO GERMAN APPEARED IN 1562 IN BASEL.

THE FIRST FRENCH EUCLID WAS PUBLISHED IN 1564 IN PARIS.A SECOND INDEPENDENT TRANSLATION INTO FRENCH

APPEARED THE FOLLOWING YEAR, ALSO IN PARIS.

IN 1570 THE FIRST ENGLISH TRANSLATION WAS PUBLISHED IN LONDON BY SIR HENRY BILLINGSLEY.

THIS IS THE EDITION THAT CARRIED THE FAMOUS PREFACE BY JOHN DEE,WHICH I HAVE ASKED YOU TO READ FOR TODAY.

AND FINALLY, EUCLID WAS TRANSLATED FOR THE FIRST TIME INTO SPANISH IN 1576.

NOW I HAVE ALREADY INDICATED THAT THE NUMEROUS LATIN EDITIONSOF EUCLID WERE INTENDED PRIMARILY FOR UNIVERSITY STUDENTS.

FOR WHOM WERE THE VERNACULAR EDITIONS INTENDED?.THE PREFACES OF MANY EDITIONS GIVE US VERY STRONG

EVIDENCE THAT THE AUDIENCE BEING TARGETED WAS THE BURGHER AND CRAFTSMAN.

INDEED, THE TRANSLATOR OF THE GERMAN EDITION OF 1562 STATES EXPLICITLY THAT THIS WORK IS INTENDED FOR THE LIKES OF PAINTERS, GOLDSMITHS, AND ARCHITECTS.

THE PRACTICAL ORIENTATION OF MANY OF THESE VERNACULAR EDITIONS CAN SCARCELY BE DENIED.

NOW YOU MIGHT ASK, OF WHAT USE CAN EUCLID BE TO CRAFTSMEN AND ARTISANS?.

WHAT PURPOSE CAN THE ABSTRACT, AXIOMATIC-DEDUCTIVE STRUCTUREOF EUCLIDEAN GEOMETRY SERVE IN THE DAY-TO-DAY WORLD OF THE LABORING BURGHER?.

I THINK NOTHING CAN ANSWER THESE QUESTIONS BETTER THAN THE PREFACE TO THE ENGLISH EDITION WRITTEN BY JOHN DEE.

AND SO I SHALL LEAVE THESE QUESTIONS FOR FRIDAY.LET ME INSTEAD TAKE A QUICK LOOK AT SOME OF THE EDITORS

AND TRANLSATORS OF EUCLID.WHAT WE FIND ARE A NUMBER OF THE MOST TALENTED

MATHEMATICIAN OF THE 16TH CENTURY.

PETER RAMUS, THE FRENCH PHILOSOPHER AND LOGICIAN, PUBLISHED BOTH A LATIN VERSION OF EUCLID IN 1545 AND AN EXTENSIVE LOGICAL ANALYSIS OF THE STRUCTURE OF EUCLIDEAN GEOMETRY IN 1559.

THIS, BY THE WAY, WAS THE SAME RAMUS WHO DISCUSSED ASTRONOMYWITH THE YOUNG TYCHO.

CHRISTOPER CLAVIUS, A PROFESSOR OF MATHEMATICS AT THE JESUIT COLLEGE IN ROME, PRODUCED A COMPENDIOUS AND VERY WIDELY USEDVERSION OF EUCLID.

IT CONTAINED AN ENORMOUS AMOUNT OF AUXILIARY MATERIAL DRAWN FROM MANY OF THE BEST AND MOST RECENT COMMENTATORS ON EUCLID.

IT QUICKLY BECAME THE DEFINITIVE SCHOLARLY EDITION AND HELPEDEARN FOR CLAVIUS THE EPITHET, 'THE EUCLID OF THE 16TH CENTURY'.

PERHAPS THE MOST COMPETENT 16TH-CENTURY MATHEMATICIAN TO TRY HIS HAND AT EDITING AND TRANSLATING EUCLID WAS FREDERICO COMMANDINO OF URBINO.

HIS LATIN EDITION OF 1572 SATISFIED THE HIGHEST STANDARDS OF HUMANIST SCHOLARSHIP.

AND HIS TRANSLATION INTO ITALIAN IN 1575 WAS VERY WIDELY USED AND OFTEN REPRINTED.

II. THE MATHEMATICAL RENAISSANCE IN ITALY.INDEED, THE WORKS OF COMMANDINO MARK THE HIGH POINT

OF WHATHAS BEEN CALLED THE "ITALIAN RENAISSANCE OF MATHEMATICS".

THIS RENAISSANCE IS MARKED NOT ONLY BY RECOVERY AND DISSEMINATION OF EUCLID BUT ALSO OF MANY OTHER CLASSICAL GREEKWORKS IN MATHEMATICS.

AS WE HAVE SEEN IN THE CASE OF THE REVIVAL AND REFORMATION OF PTOLEMAIC ASTRONOMY, THE RENAISSANCE OF MATHEMATICS ALSO HADITS ROOTS IN THE HUMANIST MOVEMENT OF THE 16TH CENTURY.

THE HUMANIST MOVEMENT ASSISTED IN THE REVIVAL OF GREEK MATHEMATICS IN AT LEAST THREE IMPORTANT WAYS.

1) FIRST, HUMANISTS COLLABORATED WITH MATHEMATICIANS IN THE RECOVERY OF GREEK MATHEMATICAL MANUSCRIPTS.

2) SECOND, HUMANISTS THEMSELVES TOOK AN ACTIVE PART IN THE TRANSLATION OF THESE TEXTS INTO LATIN.

3) AND THIRD, HUMANISTS AND MATHEMATICIANS WERE OFTEN IN PERSONAL CONTACT THROUGH COMMON CIRCLES OF FRIENDSHIP AND PATRONAGE.

HOWEVER, BEFORE GOING ANY FURTHER, PERHAPS I SHOULD TAKE A MOMENT TO CLARIFY WHAT I MEAN BY 'MATHEMATICS' AND 'MATHEMATICIAN'.

MATHEMATICS IN THE 16TH CENTURY CERTAINLY EMBRACED THE DISCIPLINES OF THE QUADRIVIUM; THAT IS, GEOMETRY, ARITHMETIC, MUSIC, AND ASTRONOMY.

BUT IT WOULD ALSO INCLUDE OPTICS, STATICS, AND MECHANICS SINCE THE PRINCIPLES UPON WHICH THESE FIELDS ARE BASED ESSENTIALLY GEOMETRIC.

BY THE SAME TOKEN THE LABEL 'MATHEMATICIAN' MAY BE APPLIED NOT ONLY TO THE PROFESSOR OF GEOMETRY BUT ALSO TO THOSE PROFESSIONS WHICH DREW HEAVILY UPON KNOWLEDGE OF GEOMETRY.

THUS NEO-PLATONIST PHILOSOPHERS, WHO BELIEVED THE COSMOS WASCONSTRUCTED ACCORDING TO GEOMETRIC PRINCIPLES, AS WELL AS ARTISTS AND ARCHITECTS, WHO

CONSTRUCTED THEIR CREATIONS ACCORDING TO LAWS OF PROPORTION, COULD BE NUMBERED AMONG THE MATHEMATICIANS.

AND BECAUSE OF THEIR NEED TO MASTER THE TECHNICAL DETAILS OF PLANETARY THEORY, BOTH ASTRONOMERS AND ASTROLOGERS WERE TYPICALLY CONSIDERED TO BE MATHEMATICIANS.

THUS, IN A WAY FEW OTHER FIELDS OF KNOWLEDGE COULD MATCH, MATHEMATICS CUT ACROSS LINES OF THEORY AND PRACTICE.

AND IT WAS EVEN ABLE TO MOVE ACROSS THE LINES SEPARATING ELITE FROM COMMON CULTURE.

JUST AS THE LOGICIAN AND PHILOSOPHER MIGHT CONTEMPLATE THE BEAUTY, SIMPLICITY, AND CERTAINTY OF GEOMETRY AS A SYSTEM OF AXIOMATIC KNOWLEDGE, THE CRAFTSMAN AND ARTIST COULD EMPLOY HIS KNOWLEDGE OF GEOMETRY TO IMPROVE HIS PRODUCT.

PERHAPS THIS EXPLAINS WHY MANY HUMANISTS FOUND MATHEMATICSAN ATTRACTIVE FIELD OF STUDY: IT COMBINED AESTHETICS WITH PRACTICALITY, CONTEMPLATION WITH ACTIVITY, AND CERTAINTY WITH EFFICACY.

WHATEVER THE REASONS, SOME HUMANIST EDUCATORS SAW IN GEOMETRY A DISCIPLINE WELL-SUITED TO THEIR NEW EDUTATIONAL PROGRAM.

IN ADDITION TO PEDAGOGICAL INTEREST, THE RECOVERY OF CLASSICALGREEK TREATISES IN MATHEMATICS ALSO BECAME PART OF ARISTOCRATIC FASHION.

AND MUCH OF THE COLLECTING AND CIRCULATING OF TREATISES IN MATHEMATICS WAS PERFORMED UNDER

THE AUSPICES OF ARISTOCRATICPATRONS OF THE HUMANITIES.

WEALTHY PRINCES AND POWERFUL ARISTOCRATS SOUGHT TO DISPLAY THEIR APPRECIATION OF CLASSICAL LEARNING BY ACQUIRING RARE AND VALUABLE GREEK MANUSCRIPTS -- THESE WERE THE MARKS OF THE WELL-EDUCATED, GENEROUS, AND REFINED PRINCE.

THE CREATION OF LARGE LIBRARIES OF GREEK TREATISES AND THE PATRONAGE OF HUMANIST SCHOLARS WHO COULD DECIPHER AND TRANSLATE THEM CAME TO BE PART OF THE COURT CULTURE OF THE NOBILITY.

INDEED, THIS FASHION SPREAD THROUGH THE HIERARCHY OF THE CATHOLIC CHURCH AND REACHED ITS GREATEST EXPRESSION IN THE SO-CALLED 'RENAISSANCE POPES'.

FROM 1450 TO ABOUT 1500, A NUMBEROF OF 'RENAISSANCE POPES' SUCCEEDED IN BUILDING UP ONE OF THE MOST IMPORTANT LIBRARIES OFGREEK MANUSCRIPTS IN THE EARLY MODERN PERIOD, THE 'BIBLIOTECA VATICANA'.

CONTEMPORARY INVENTORIES SHOW THAT IN 1443 THE VATICAN LIBRARY CONTAINED 340 MANUSCRIPTS, ONLY 2 OF WHICH WERE GREEK.

YET BY 1455, THE YEAR POPE NICHOLAS V DIED, THE FIRST AND GREATEST OF THE "RENAISSANCE POPES', THE COLLECTION HAD GROWN TO 1209 CODICES, 414 OF THEM IN GREEK.

AND BY 1484, THE LIBRARY CONTAINED MORE THAN 3700 MANUSCRIPTS, 853 OF WHICH WERE IN GREEK, MAKING IT THE LARGEST LIBRARY OF MANUSCRIPTS IN ALL OF EUROPE.

THE LIST OF MATHEMATICAL MANUSCRIPTS RECOVERED, CIRCULATED, AND TRANSLATED UNDER THE AUSPICES OF

RENAISSANCE PRINCES AND POPES IS TRULY IMPRESSIVE: PTOLEMY'S ALMAGEST AND HIS TREATISE ONGEOGRAPHY; PAPPUS' MATHEMATICAL COLLECTIONS; APOLLONIUS' TREATISE ON CONICS; THE MECHANICA FALSELY ATTRIBUTED TO ARISTOTLE (BUT ACTUALLY WRITTEN BY ONE OF HIS STUDENTS); DIAPHANTUS ON ALGEBRA, HERO ON MECHANICS AND PNEUMATICS, THEON ON GEOMETRY, AND -- MOST IMPORTANTLY -- SEVERAL WORKS BYARCHIMEDES.

THUS THE PUBLICATION HISTORY OF EUCLID IS SEEN TO BE PART OF A MUCH BROADER MOVEMENT.

A MOVEMENT THAT NOT ONLY EMBRACED A BROAD RANGE OF MATHEMATICAL DISCIPLINES BUT ONE THAT HAD BECOME LINKED TO COURT CULTURE, ARISTOCRATIC PATRONAGE, AND THE HUMANIST PROGRAM OF EDUCATIONAL REFORM.

III. MATHEMATICS IN THE 17TH CENTURY.AND OF COURSE THIS RENAISSANCE OF MATHEMATICAL

KNOWLEDGE WAS NOT WITHOUT ITS EFFECT IN OTHER AREAS OF EARLY MODERN SCIENCE.

YET FOR ALL THE INTEREST IN CLASSICAL MATHEMATICS DISPLAYED BYSCHOLARS OF THE 16TH CENTURY, IT WAS THE 17TH CENTURY THAT WITNESSED THE GREATEST ADVANCES.

INDEED, THIS BRINGS US TO THE THIRD PERIOD I REFERRED TO EARLIER; NAMELY, THE PERIOD OF TRANSFORMATION.

FOR IN THE 17TH CENTURY WE SEE NOT SIMPLY CONTINUED INTEREST INTHE EDITING OF CLASSICAL GREEK TREATISE IN MATHEMATICS, WE ALSO SEE A NUMBER OF VERY IMPORTANT ORIGINAL CONTRIBUTIONS.

THE LIST OF INNOVATIVE MATHEMATICIANS FROM THE 17TH CENTURY ISLONG ONE: NAPIER, DESCARTES, FERMAT, PASCAL, TORRICELLI, CAVALIERI, HUYGENS, NEWTON, LEIBNIZ.

AND THEIR ACCOMPLISHMENTS RANGE FROM THE INVENTION OF LOGARITHMS AND ANALYTIC GEOMETRY, TO THE DEVELOPMENT OF PROPABILITY THEORY AND THE CALCULUS.

ALTHOUGH I WILL NOT HAVE TIME TO PRESENT EVEN AN OVERVIEW OF 17TH-CENTURY MATHEMATICS, I DO WANT TO POINT TO WHAT SEEM TO BE THREE OF ITS MORE CHARACTERISTIC FEATURES.

1) THE FIRST AND MOST OBVIOUS CHARACTERISTIC IS THE VERY RICHNESS AND ORIGINALITY OF WORK DONE IN MATHEMATICS.

CLASSICAL GREEK MATHEMATICS HAD NOT ONLY BEEN THOROUGHLY MASTERED BY THE FIRST QUARTER OF THE 17TH CENTURY, ITS VARIOUS BRANCHES HAD BEEN VASTLY IMPROVED UPON BY A HOST OF HIGHLY TALENTED AND CREATIVE MATHEMATICIANS.

2) SECONDLY, THE CLASSICAL FIELDS OF PURE MATHEMMATICS, LIKE ALGEBRA, NUMBER THEORY, AND GEOMETRY EXPERIENCED RAPID ADVANCE PRIMARILY THROUGH AN INCREASE IN ABSTRACTION.

AREAS OF MIXED OR APPLIED MATHEMATICS -- LIKE OPTICS, HARMONICS,AND STATICS -- GAINED BOTH FROM INCREASED ABSTRACTION IN THEIR THEORETICAL TREATMENT AND FROM THE CONSTRAINTS IMPOSED BY CAREFUL EMPIRICAL DATA.

3) AND THIRD, IN ADDITION TO THE MODIFICATION OF OLD FIELDS OF CLASSICAL MATHEMATICS, THERE WERE A

NUMBER OF NEW FIELDS OF STUDY THAT BECAME MATHEMATIZIED FOR THE FIRST TIME: THE MOST NOTABLY EXAMPLES BEING PROBABILITY THEORY, KINEMATICS (RECALLED GALILEO'S PARABOLIC TRAJECTORIES), AND DYNAMICS (ESPECIALLY LAWS OF IMPACT AND MOTION).

A. PHYSICAL INTUITION AND THE GROWTH OF MATHEMATICS.

THE SHEER NUMBER OF FIRST-RATE MATHEMETICIANS WORKING IN THE 17TH CENTURY AND THE VARIOUS DIRECTIONS IN WHICH THEIR WORKS WENT MAKE IT IMPOSSIBLE TO GIVE 17TH-CENTURY MATHEMATICS A SIMPLE CHARACTERIZATION.

I WOULD NEVERTHELESS SUGGEST THAT TWO OF THE MORE IMPORTANTTHEMES CONCERN ABSTRACTION AND WHAT I WILL CALL "PHYSICAL INTUITION".

INCREASINGLY THE LEVEL OF ABSTRACTION IS SUCH A NATURAL PART MATHEMATICS THAT I DON'T THINK THERE IS MUCH NEED TO ELABORATE.

INDEED, THIS IS JUST A SLIGHTLY MORE PRECISE WAY OF SAYING THAT MATHEMATICS FLOURISHED AND CONQUERED NEW TERRITORY.

WHAT MAY NEED MORE EXPLAINING, HOWEVER, IS THE NOTION OF PHYSICAL INTUITION IN MATHEMATICS.

PERHAPS IT IS SIMPLEST AND MOST READILY IDENTIFIABLE FORM, PHYSICAL INTUITION MANIFESTED ITSELF IN THE WAY MANY 17TH-CENTURY MATHEMATICIANS APPROACHED PROBLEMS IN GEOMETRY.

THE GREEKS HAD DEVELOPED WHAT MIGHT BE CALLED 'KINETIC GEOMETRY' -- THAT IS, A GEOMETRY IN WHICH SIMPLE

FIGURES WERE FREE TO MOVE IN ORDER TO GENERATE NEW FIGURES.

A CIRCLE COULD BE MADE TO REVOLVE ABOUT A POINT OUTSIDE ITSELFTO GENERATE AN ANNULUS, OR TWO CROSSED LINES COULD REVOLVE ABOUT A VERTICAL AXIS TO GENERATE TWO CONES RESTING POINT TO POINT.

WHAT IS STRIKING IN THE 17TH-CENTURY IS THE EXTENT TO WHICH SUCH'KINETIC GEOMETRY' IS USED TO TREAT MATHEMATICAL PROBLEMS IN APHYSICAL OR SEMI-PHYISCAL MANNER.

THERE IS A LARGE CLASS OF MATHEMATICAL QUESTIONS TREATED IN THE 17TH CENTURY THAT CAN BE VISUALIZED KINETICALLY, OR EVEN CONSTRUCTED MECHANICALLY.

FOR EXAMPLE: WHAT IS THE SHAPE OF THE CURVE MADE BY A STRING SUSPENDED AT BOTH ENDS AND HANGING FREELY -- THAT IS, WITHOUT ANY WEIGHTS ATTACHED?.

DOES THE SHAPE OF THE CURVE CHANGE WHEN WEIGHTS ARE ATTACHED AT REGULAR INTERVALS?.

WHAT IS THE SHAPE OF THE CURVE TRACED OUT BY A STRING AS IT WRAPS ITSELF AROUND THE CIRCUMFERENCE OF A CIRCLE?.

WHAT ARE THE GEOMETRIC PROPERTIES OF THE CURVE GENERATED BY APOINT ON THE CIRCUMFERENCE OF A CIRCLE ROLLING ALONG A STRAIGHT LINE?.

ASSUMING UNIFORM ACCELERATION IN FREEFALL, WHAT IS THE SHAPE OF THE CURVE THAT ALLOWS THE FASTEST DESCENT OF AN OBJECT SLIDING ALONG IT?.

WHAT IS THE SHAPE OF THE CURVE TRACED OUT BY A POINT MOVING UNDER THE INFLUENCE OF AN INVERSE SQUARE LAW OF ATTRACTION?.

ALL OF THESE EXAMPLES, THOUGH ESSENTIALLY PROBLEMS IN GEOMETRY, ARE SOMEHOW DEPENDEND ON, OR CAN BE TRANSLATED INTO, A PHYSICAL PROBLEM.

NOT ONLY WERE THE PROBLEMS OF MATHEMATICS CAST IN TERMS OF APHYSICAL OR MECHANICAL SITUATION, THE SOLUTIONS OFTEN DEPENDED ON THE INSIGHTS PROVIDED BY FAMILIARITY WITH THE PHYSICAL WORLD.

NOW THIS SOMETIMES MEANT THAT THE PROOFS LACKED THE SORT OF RIGOR MATHEMATICIANS OF THE 19TH AND 20TH CENTURY HAVE COME TODEMAND.

IN THE OPINION OF ONE OF THE FOREMOST LIVING HISTORIANS OF MATHEMATICS, D.T. WHITESIDE, - QUOTE - "THE [FOUNDATIONS] WERE [NOT BY] SUITABLE ANALYTICAL JUSTIFICATION . . . [THEN] BY DIRECT APPEAL TO THE VISUAL PLAUSIBILITY OF A GEOMETRICAL MODEL." (WHITESIDE 1962, P. 384) .

IT IS THIS "DIRECT APPEAL TO VISUAL PLAUSIBILITY" THAT LIES AT THE HEART OF WHAT I MEAN BY PHYSICAL INTUITION IN 17TH-CENTURY MATHEMATICS.

THIS PHYSICAL INTUITION IS, I THINK, CLOSELY RELATED TO THE INTUITION THAT UNDERLAY THE MECHANICAL PHILOSOPHIES OF DESCARTES AND GASSENDI.

INDEED, ONE COULD ARGUE THAT WHAT DISTINGUISHES 17TH-CENTURYMATHEMATICS IS ITS HEAVY RELIANCE ON PROBLEMS DERIVED FROM THE PHYSICAL SCIENCES.

NOW HISTORIANS OF SCIENCE HAVE LONG NOTED THE TREMENDOUS IMPORTANCE OF MATHEMATICS IN THE DEVELOPMENT OF 17TH-CENTURYNATURAL PHILOSOPHY.

IMMANUEL KANT, WRITING IN THE LATTER HALF OF THE 18TH CENTURY,NOTED THAT "IN EVERY DEPARTMENT OF PHYSICAL SCIENCE THERE IS ONLY SO MUCH SCIENCE,

PROPERLY SO-CALLED, AS THERE IS MATHEMATICS" (KLINE 1959, P. VII).

AND ALEXANDER KOYRÉ NEVER TIRED OF THE MESSAGE THAT THE FUNDAMENTAL TRANSFORMATION RESULTING FROM THE SCIENTIFIC REVOLUTION WAS THE MATHEMATIZATION OF NATURE.

WHILE THIS IS UNDOUBTEDLY TRUE, IT SEEMS NO LESS TRUE THAT MATHEMATICS WAS JUST AS PROFOUNDLY ALTERED BY THE PROBLEMS AND METHODS OF THE NEW SCIENCE.

IN OTHER WORDS, ONE MUST ALSO ACKNOWLEDGE THE 'PHYSICALIZATION OF MATHEMATICS;' THAT IS, THE TRANSFORMATIONS INMATHEMATICS BROUGHT ABOUT BY THE EMPIRICAL, PHYSICAL, AND OBSERVATIONAL THRUST OF THE NEW EXPERIMENTAL AND MECHANICALPHILOSOPHIES.

INDEED, THERE IS CONSIDERABLE ADVANTAGE IN VIEWING THE RELATIONSHIP BETWEEM THE 'NEW SCIENCE' AND THE GROWTH OF MATHEMATICS IN THE 17TH-CENTURY AS BEING SYMBIOTIC.

AS AN ILLUSTRATION OF THIS POINT, WE NEED ONLY CONSIDER SOME OFTHE TECHNICAL PROBLEMS FACED BY NATURAL PHILOSOPHERS AND THE DEVELOPMENT OF MATHEMETICAL METHODS FOR TREATING CURVES.

IN ASTRONOMY KEPLER HAD TO KNOW HOW TO HANDLE THE GEOMETRIC PROPERTIES OF THE ELLIPSE.

AND BOTH HE AND TYCHO CONSIDERED HOW THE CURVED PATH OF LIGHT FROM SUN AND STARS AS IT PASSES THROUGH ATMOSPHERE AFFECTED THE ACCURACY OF THEIR OBSERVATIONAL DATA.

KEPLER AND GALILEO TRIED TO UNDERSTAND THEORETICALLY THE CURVATURE OF A LENS THAT WOULD MAKE THE BEST TELESCOPES.

AND IN MECHANICS, GALILEO NEEDED TO UNDERSTAND THE GEOMETRYOF THE PARABOLA FOR HIS WORK ON TRAJECTORIES.

IN FACT, IT IS NO EXAGGERATION TO SAY THAT ONE OF THE DEFINING PROBLEMS OF 17TH-CENTURY MATHEMATICS WAS THE GEOMETRIC TREATMENT OF CURVES.

AND WHEN WE CONSIDER THE CULMINATION OF 17TH-CENTURY MATHEMATICS -- BY THAT I MEAN THE INDEPENDENT INVENTION OF THE CALCULUS BY NEWTON AND LEIBNIZ TOWARD THE END OF THE CENTURY -- WE SEE THE EMERGENCE OF THE ULTIMATE METHOD FOR DEALING WITH CURVES.

WHAT IS MORE, THE INVENTION OF THE CALCULUS WAS, FOR BOTH NEWTON AND LEIBNIZ, CLOSELY TIED TO THEIR PURSUIT OF PHYSICAL PROBLEMS IN DYNAMICS AND KINEMATICS.

THUS THERE IS NO BETTER EXAMPLE THAN CALCULUS OF THE INTERRELATIONSHIP AMONG ABSTRACTION, 'KINETIC GEOMETRY', AND PHYSICAL INTUITION.

THE INTERRELATEDNESS OF MATHEMETICS AND THE NEW SCIENCE WASNOT LOST 17TH-CENTURY NATURAL PHILOSOPHERS.

FOR THEY THEMSELVES SAW THE NEED TO ESTABLISH NATURAL KNOWLEDGE ON A SECURE MATHEMATICAL FOUNDATION.

B. DESCARTES ET AL..AND NO ONE SAW THIS MORE CLEARLY THAN DESCARTES.

AND, OF COURSE, NO ONE CONTRIBUTED MORE TO THE UNDERSTANDINGOF GEOMETRIC CURVES EITHER.

I MENTIONED IN AN EARLIER LECTURE THE SERIES OF DREAMS DESCARTES HAD ON THE NIGHT OF NOVEMBER 10TH, 1619 AND HOW THE MEANING OF THE LAST DREAM CONVINCED HIM THAT MATHEMATICS WAS THE KEY TO ALL PHYSICAL TRUTH.

WRITING ALMOST TEN YEARS LATER, DESCARTES REFLECTS ON THE RELATIONSHIP BETWEEN GEOMETRY AND HIS NATURAL PHILOSOPHY: - QUOTE - "I HAVE RESOLVED TO QUIT ONLY ABSTRACT GEOMETRY, THAT ISTO SAY, THE CONSIDERATION OF QUESTIONS WHICH SERVE ONLY TO EXERCISE THE MIND, AND THIS, IN ORDER TO STUDY ANOTHER KIND OF GEOMETRY, WHICH HAS FOR ITS OBJECT THE EXPLANATION OF THE PHENOMENA OF NATURE" (KLINE 1959, P. 147).

DESCARTES COMPLAINED THAT GREEK GEOMETRY WAS SO MUCH TIED TO FIGURES THAT "IT CAN EXERCISE THE UNDERSTANDING ONLY ON CONDITION OF GREATLY FATIGUING THE IMAGINATION." (KLINE 1959, P. 150).

I TAKE THIS TO MEAN THE DESCARTES FELT THAT GREEK GEOMETERS RELIED TO HEAVILY ON GEOMETRIC FIGURES IN CONDUCTING THEIR MATHEMATICAL BUSINESS.

THAT IS, IN ORDER TO FOLLOW SOME OF THEIR MORE DIFFICULT PROOFS IN GEOMETRY, IT WAS NECESSARY TO VISUALIZE COMPLICATED FIGURES AND FOLLOW LONG AND INVOLVED DEDUCTIVE ARGUMENTS.

AND DESPITE HIS LOVE AND ESTEEM FOR THE AXIOMATIC-DEDUCTIVE CHARACTER OF GEOMETRY, DESCARTES REALIZED THAT, AT ONE LEVEL, GEOMETRY LACKED SYSTEMICITY.

IMPORTANT THEOREMS, ESPECIALLY THOSE CONCERNING THE CIRCLE, WERE COMPLICATED, AND EACH PROOF REQUIRED A NEW TYPE OF DEMONSTRATION SEEMINGLY INDEPENDENT OF OTHERS THOUGH ALL TREATED THE SAME CLASS OF GEOMETRIC FIGURES.

AND FINALLY, ALTHOUGH GEOMETRY CAN BE USED TO DESCRIBE THE ABSTRACT OR THEORETICAL PATH EXECUTED BY SOME OBJECT -- SAY A PARABOLIC TRAJECTORY, OR UNIFORM CIRCULAR MOTION -- IT CANNOT BE EMPLOYED TO GIVE CONCRETE QUANTITATIVE INFORMATION ABOUT THE ACTUAL PHYSICAL PATH OF A REAL OBJECT.

WHAT GEOMETRY LACKED WAS JUST THE SORT OF NUMERICAL INFORMATION THAT ALGEBRA PROVIDED.

INDEED, THE OLD DIVISION OF MATHEMATIC INTO TWO SCIENCES -- THE SCIENCE OF FIGURE AND THE SCIENCE OF NUMBER -- ONLY UNDERSCORES THE NON-NUMERICAL CHARACTER OF GEOMERTRY.

NOW CONSIDERABLE PROGRESS HAD BEEN ACHIEVED IN ALGEBRA IN THE LATE 16TH AND EARLY 17TH CENTURIES.

HOWEVER, IN REVIEWING THE CURRENT STATE OF AFFAIRS IN ALGEBRA,DESCARTES WAS AGAIN DISSATISFIED.

HE WROTE IN 1628,: "THERE HAVE BEEN SOME INGENIOUS MEN WHO HAVE TRIED IN THIS CENTURY TO REVIVE [ALGEBRA]; . . . IF ONLY IT COULDBE SO DISENTANGLED FROM THE MULTIPLE NUMBERS AND INEXPLICABLE FIGURES THAT OVERWHELM IT, THAT IT NO LONGER WOULD LACKED THE CLARITY AND SIMPLICITY THAT WE SUPPOSE SHOULD OBTAIN IN A TRUE MATHEMATICS." (DSB, P. 56).

THE END RESULT OF DESCARTES' DESIRE TO PURIFY ALGEBRA AND SYSTEMATIZE AND QUANTIFY GEOMETRY WAS, OF COURCE, HIS ANALYTICGEOMETRY.

NOW WE ARE ALL FAMILIAR -- OR WERE AT ONE TIME -- WITH THE CARTESIAN COORDINATE SYSTEM AND THE USE OF ALGEBRAIC EQUATIONS TO GENERATE CURVES IN TWO DIMENSIONS AND SURFACES IN THREE.

PRESUMING THIS FAMILIARITY IS STILL IN TACT AND GIVEN THE SHORTAGE OF TIME, I WILL SAY ONLY TWO THINGS ABOUT DESCARTES BRILLIANT SYNTHESIS OF ALGEBRA AND GEOMETRY.

FIRST, IN ANOTHER ONE OF THOSE IRONIES OF NOMENCLATURE, DESCARTES NEVER EMPLOYED THE CARTESIAN COORDINATE SYSTEM.

NOR DID HE EVER DEVELOPED HIS ANALYTIC GEOMETRY TO THE POINT WHERE HE COULD USE IT FOR SOLID GEOMETRY.

INSTEAD OF THE RIGID, THREE-DIMENSIONAL, MUTUALLY ORTHAGONALAXES WE HAVE COME TO IDENTIFY WITH DESCARTES, DESCARTES USED A SINGLE AXIS WITH A MOVING ORDINATE.

IMAGINE AN AXIS WITH A FIXED POINT "A" AND A VARIABLE DISTANCE "X" BEGINNING AT "A" AND ENDING AT "B".

AT "B," THERE IS A SECOND LINE "BC" WITH VARIABLE LENGTH "Y" SET ATA FIXED ANGLE "_" WITH RESPECT TO "AB".

NOW ACCORDING TO DESCARTES' SCHEME, THE LENGTH OF "BC" VARIESAS A FUNCTION OF "AB".

THAT IS, THE UNKNOWN LENGTH "Y" VARIES AS A FUNCTION OF "X," ANDSO THE END POINT "C" TRACES OUT THE PATH OF A GIVEN CURVE.

THUS DESCARTES' CURVES WERE NOT PLOTTED, AS WE ALL LEARNED TODO.

RATHER, THEY WERE GENERATED BY THE COMBINED 'MOTION' OF THE AXIS AND ORDINATE.

IF WE ATTACH COEFFICIENTS TO THE UNKNOWNS, WE OBTAIN FAMILIAR-LOOKING EQUATIONS.

TO TAKE THE SIMPLEST EXAMPLE: SUPPOSE DX = EY MEANS THAT RATE OF GROWTH OF "X" TO "Y" IS CONSTANT AND THE CURVE TRACED OUT BY "C" IS SIMPLY A STRAIGHT LINE, WITH SOMETHING LIKE THE SLOPE GIVENBY "E/D".

DESCARTES CHOSE FRESH PARAMETERS OF THE AXIS-ORDINATE SYSTEMFOR EACH NEW PROBLEM AND NEVER ADOPTED A RIGID ORTHAGONAL COORDINATE SYSTEM APPLICABLE TO ALL PROBLEM.

THE SECOND POINT I WANT TO MENTION ABOUT DESCARTES' ANALYTIC GEOMETRY IS THAT IT REPRESENTS YET ANOTHER EXAMPLE OF SIMULTANEOUS AND INDEPENDENT DISCOVERY.

DESCARTES PUBLISHED HIS GEOMETRY IN 1637.HOWEVER, ALREADY BY THE SPRING OF 1636, PIERRE DE

FERMAT HAD WRITTEN AND CIRCULATED IN MANUSCRIPT FORM A SYSTEM OF ANALYTIC GEOMETRY ALMOST IDENTICAL TO WHAT DESCARTES WOULDPUBLISH THE FOLLOWING YEAR.

FERMAT DID NOT USE A RIGID COORDINATE SYSTEM BUT, LIKE DESCARTES, A SINGLE AXIS AND MOVING ORDINATE.

AND HE COULD SOLVE APPROXIMATELY THE SAME RANGE OF PROBLEMS AS DESCARTES.

THUS THE TWO DISCOVERIES, DESPITE THEIR NEAR SIMULTANEITY AND THE NEAR IDENTITY OF THEIR BASIC CONSTRUCTIONS, SEEM TO BE FULLY INDEPENDENT.

INDEED, EACH DEVELOPED FROM DIFFERENT LINES OF RESEARCH.

A FINAL POINT WORTH MENTIONING IS THAT FERMAT, THOUGH CONSCIOUS OF THE IMPORTANCE OF HIS DISCOVERY, NEVER PUBLISHED AND NEVER RAISED A PRIORITY DISPUTE WITH DESCARTES OVER ANALYTICGEOMETRY.

IN FACT, FERMAT PUBLISHED NONE OF HIS MATHEMETICAL WORKS ON HIS OWN.

HOWEVER, AFTER HIS 'REDISCOVERY' IN THE 19TH CENTURY, HE IS NOW CONSIDERED ONE OF THE GREATEST MATHEMETICIANS OF THE 17TH CENTURY.

DESCARTES, ON THE OTHER HAND, DID PUBLISH, AND HIS FAME AS A MATHEMATICIAN WAS ALREADY WELL ESTABLISHED BY 1637 AND HAS CONTINUED UNABATED UNTIL TODAY.

YET ANOTHER EXAMPLE OF THE POWER OF PUBLICATION.

C. A REVOLUTION IN MATHEMATICS?IF WE STOP FOR A MOMENT AND LOOK AT THE

TRANSFORMATIONS THATMATHEMATICS WENT THROUGH BETWEEN 1550 AND 1650, THEY CERTAINLY SEEM NO LESS DRAMATIC THAN THOSE IN ASTRONOMY OR NATURAL PHILOSOPHY.

BUT SHOULD WE CHARACTERIZE THESE TRANSFORMATIONS AS REVOLUTIONS?.

AND SHOULD WE CALLI THEM KUHNIAN-TYPE DISCIPLINARY REVOLUTIONS?.

CERTAINLY BY THE 17TH CENTURY, GEOMETRY WAS A WELL-ESTABLISHED DISCOPLINE.

AND DESCARTES' ANALYTIC GEOMETRY AND CERTAINLY NEWTON'S CALCULUS ARE NICE EXAMPLES OF SUDDEN NEW PROBLEM-SOLVING TECHNIQUES; THAT IS, OF KUHNIAN-TYPE 'EXEMPLARS'.

AND IT WOULD SEEM THAT THE "NEW MATHEMATICS" HELPED CREATE ANEW WORLDVIEW.

IN LIGHT OF ALL THIS, CAN WE SPEAK OF A MATHEMATICAL REVOLUTION OF THE SORT KUHN MEANT?.

THE CONSENSUS AMONG HISTORIANS OF MATHEMATICS SEEMS TO BE NO, WE CANNOT.

JEAN-BAPTISTE FOURRIER, THE 19TH-CENTURY FRENCH MATHEMATICIAN,WAS ALSO A CAPABLE HISTORIAN OF MATHEMATICS.

HE WROTE THAT "THIS DIFFICULT SCIENCE [OF MATHEMATICS] IS FORMED SLOWLY, BUT IT PRESERVES EVERY PRINCIPLE IT HAS ONCE ACQUIRED; IT GROWS AND STRENGTHENS ITSELF IN THE MIDST OF MANY VARIATIONS AND ERRORS OF THE HUMAN MIND." (CROWE 1975, P. 165).

LATER IN THE 19TH CENTURY, A GERMAN MATHEMTICIAN-HISTORIAN WROTE - QUOTE - "IN MOST SCIENCES ONE GENERATION TEARS DOWN WHAT ANOTHER HAS BUILT . . . IN MATHEMATICS ALONE EACH GENERATION BUILDS A NEW STORY TO THE OLD STRUCTURE." (CROWE 1975, P. 165).

AND CLIFFORD TRUESDELL, ONE OF THE FOREMOST HISTORIANS OF MATHEMATICS PRESENTLY WORKING, BLUNTLY ASSERTS THAT - QUOTE - "WHILE 'IMAGINATION, FANCY, AND INVENTION' ARE THE SOUL OF MATHEMATICAL RESEARCH, IN MATHEMATICS THERE HAS NEVER YET BEEN A REVOLUTION." (CROWE 1975, P. 165).

NOW EACH OF THESE REJECTIONS OF THE NOTION OF A MATHEMATICALREVOLUTION RESTS ON THE ASSUMPTION IS THAT, FOR THERE TO BE A REVOLUTION "SOME PREVIOUSLY EXISTING ENTITY . . . MUST BE OVERTHROWN AND IRREVOCABLY DISCARDED." (CROWE 1975, P. 165).

IN ASTRONOMY THE COPERNICAN THEORY OVERTHREW THE PTOLEMAIC;AND IN PHYSICS GALILEO AND NEWTON OVERTHREW ARISTOTLE.

BUT, AS HISTORIANS OF MATHEMATICS LIKE TO POINT OUT, NEITHER DESCARTES' ANALYTIC GEOMETRY, NOR NEWTON'S CALCULUS, NOR EVENTHE NON-EUCLIDEAN GEOMETRIES OF THE 19TH CENTURY RENDERED EUCLIDEAN GEOMETRY INVALID.

THUS THERE IS NO PROFOUND OVERTHROW OF MATHEMATICAL THEORYAS THERE IS OF PHYSICAL THEORY.

AND THUS THERE CAN BE NO REVOLUTIONS IN MATHEMATICS.DESPITE THIS STRONG CONSENSUS BY EXPERTS IN THE

HISTORY OF MATHEMATICS -- AND WHILE CLAIMING NO EXPERTESE IN THE FIELD FORMYSELF -- I NEVERTHELESS DISAGREE.

IT SEEMS TO ME THAT THERE CAN BE REVOLUTIONS IN MATHEMATICS JUST AS IN THE NATURAL SCIENCES.

LETS LOOK AT THEIR EXAMPLE A BIT MORE CLOSELY.ALTHOUGH IN ONE SENSE PTOLEMY'S ASTRONOMY WAS INDEED

RENDERED OBSOLETE BY COPERNICUS, PTOLEMY'S CONSTRUCTIONS ANDTHEIR PREDICTIVE ACCURACY ARE JUST AS VALID AFTER COPERNICUS ASBEFORE.

ONE CAN, IN PRINCIPLE AND IN PRACTICE, GO BACK AND USE PTOLEMY'SCONSTRUCTIONS TO DO ASTRONOMY.

NOW CONSIDER THE SITUATION WITH GEOMETRY.IT IS QUITE TRUE THAT, IN ONE SENSE, DESCARTES' ANALYTICAL

GEOMETRY DID NOT OVERTHROW EUCLIDEAN GEOMETRY OR MAKE IT FALSE; ONE CAN STILL TRUST AND USE EUCLID'S PROOFS AFTER DESCARTES.

BUT, IN ANOTHER SENSE, DESCARTES DID MAKE EUCLID OBSOLETE; THAT IS, NO ONE REALLY USES THE

CUMBERSOME GEOMETRIC METHOD TO SOLVE PROBLEMS INVOLVING CURVES ONCE THERE IS ANALYTIC GEOMETRY AND CALCULUS.

THUS EUCLID HAS BEEN RENDERED JUST AS OBSOLETE AS PTOLEMY, THOUGH WITH NO GREATER LOSS OF CONCEPTUAL VALIDITY FOR THE LATTER THAN FOR THE FORMER.

THERE IS ANOTHER PIECE OF EVIDENCE THAT I THINK CAN BE USED AGAINST THOSE WHO WOULD DENY REVOLUTIONS IN MATHEMATICS.

ONE OF THE EARLIEST USAGES OF THE NOTION OF A REVOLUTION IN SCIENCE WAS APPLIED TO A TRANSFORMATION IN MATHEMATICS.

BERNARD DE FONTENELLE, THE FRENCH MAN OF LETTERS AND SCIENCE,RECOGNIZED THE PROFOUND CHANGE THAT CALCULUS EXERCISED OVER THE PRACTICE OF MATHEMATICS IN THE EARLY YEARS OF THE 18THCENTURY.

IN REFERRING TO THE WORK OF THE MARQUIS DE L'HOPITAL -- WHICH INTRODUCED THE SUBTLTIES OF INFINITESIMAL CALCULUS TO A FRENCHAUDIENCE -- FONTELLE WROTE, IN 1719, THAT - QUOTE - "IN THOSE DAYS [THAT IS IN THE LAST DECADE OF THE 17TH CENTURY] THE BOOK OF L'HOPITAL HAD APPEARED, AND ALMOST ALL THE MATHEMTICIANS BEGAN TO TURN TO THE . . . NEW GEOMETRY OF THE INFINITE . . . THE SURPASSING UNIVERSALITY OF ITS METHOD, THE ELEGANT BREVITY OF ITSDEMONSTRATIONS, THE FINESSE AND DIRECTNESS OF THE MOST DIFFICULT SOLUTIONS, ITS SINGULAR AND UNPRECEDENTED NOVELTY[;] IT ALL EMBELLISHES THE

SPIRIT AND HAS CREATED, IN THE WORLD OF GEOMETRY, AN UNMISTAKABLE REVOLUTION." (DAUBEN 1984, P. 83).

IN OTHER WORDS, TO A CONTEMPORARY OBSERVER -- AND I THINK WE SHOULD ALLOW CONTEMPORARIES THEIR SAY IN SUCH MATTERS -- THE CALCULUS UTTERLY SURPASSED IN ELEGANCE, UTILITY, AND POWER, THEGEOMETRY OF EUCLID.

THE WORLD OF GEOMETRY MAY NOT HAVE BEEN 'IRREVOCABLY DISCARDED' BUT IT NEVERTHELESS WAS NOW PURSUED UNDER VERY DIFFERENT TERMS.

AND IT IS THIS RADICAL AND SUDDEN TRANSFORMATION OF THE PROBLEM-SOLVING METHODS OF GEOMETRY WHICH DISTINGUISH A REVOLUTION.

JOSEPH DAUBEN, A MODERN HISTORIAN OF MATHEMAITCS SUMS UP THEMATTER IN THE FOLLOWING WORDS:.

"BECAUSE OF THE SPECIAL NATURE OF MATHEMETICS, IT IS NOT ALWAYSTHE CASE THAT AN OLDER ORDER IS REFUTED OR TURNED OUT. ALTHOUGH IT MAY PERSIST, THE OLD ORDER ... DOES SO UNDER VERY DIFFERENT TERMS, IN RADICALLY ALTERED OR EXPANDED CONTEXTS. ... THE CALCULUS [FOR EXAMPLE WAS] REVOLUTIONARY - [IT] CHANGED THECONTENT OF MATHEMATICS AND THE WAYS IN WHICH MATHEMATICS ISREGARDED. . . . [AND IT HAS] DONE MORE THAN SIMPLY ADD TO MATHEMATICS - [IT HAS] TRANSFORMED IT." (DAUBEN 1984, P. 95).

THUS IT WOULD SEEM THAT THERE ARE STRONG PARALLELS BETWEEN REVOLUTIONS IN THE PHYISCAL SCIENCES AND IN MATHEMATICS:.

IN BOTH THERE IS RESISTANCE TO CHANGE AND THE NEED FOR CONSESNUS IN SUPPORT OF NEW IDEAS.

IN BOTH THERE IS THE POSSIBILITY OF DRAMATIC SHIFTS FROM ONE PROBLEM-SOLVING PARADIGM TO A RADICALLY NEW PARADIGM.

AND THE PARTICIPANTS IN BOTH MAY HAVE A SENSE THAT THEIR FIELD HAS BEEN PROFOUNDLY ALTERED BY NEW PROBLEM-SOLVING TECHNIQUES.

THERE ARE, HOWEVER, IMPORTANT DIFFERENCES: 1) FIRST, MATHEMATICS POSSESSES A LOGICAL AND CONCEPTUAL STRUCTURE DIFFERENT FROM THAT OF SCIENCE.

PUT SIMPLY, AMONG THE MOST IMPORTANT GOALS OF MATHEMTICS AREINTERNAL CONSISTENCY AND LOGICAL RIGOR; WHEREAS FOR THE SCIENCES IT IS ACCURATE, TESTABLE PREDICTIONS AND/OR DESCRIPTION OF NATURAL PHENOMENA.

THUS IT IS MUCH MORE "NATURAL" FOR MATHEMATICS TO RETAIN OLDER THEORIES AND TO USE THEM AFTER THE ARRIVAL OF NEW PRARDIGMS.

2) SECONDLY, SINCE MATHEMATICS IS NOT GENUINELY EMPIRICAL IN NATURE; THAT IS, IT DOES NOT RELY ON EXTERNAL SENSORY EVIDENCE INTHE SAME WAY THE NATURAL SCIENCES DO, THERE CAN BE NO BUILD UP OF ANOMALIES TO A CRISIS STATE.

IN LIGHT OF THESE DIFFERENCES, ONE WOULD NOT EXPECT THAT MATHEMATICS SHOULD UNDERGO PRECISELY THE SAME SORT OF 'REVOLUTIONS' AS THE PHYSICAL SCIENCES