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]
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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.