"Annenberg Media" "¶" "CAPTIONING IS MADE POSSIBLE BY THE ANNENBERG/CPB PROJECT" "AFTERCENTURIES OFBRILLIANTATTEMPTS," "AGENERALMETHODFOR COMPUTING AREASOFCURVEDSHAPES" "WASACHIEVED INTHE17thCENTURY." "THATMETHODIS CALLED INTEGRATION." "IT'S OFTEN BEEN SAID" "ISAAC NEWTON WAS BORN THE SAME YEAR GALILEO DIED," "AS IF SOMEONE OF THAT CALIBER" "WERE ALWAYS NEEDED ON EARTH." "THE NEWTONS WERE A LINCOLNSHIRE FARMING FAMILY," "NEITHER PARTICULARY POOR NOR PARTICULARLY PROSPEROUS." "ISAAC WAS A STRANGE LAD." "HIS MOTHER REALIZED HE WAS TOO LAZY" "TO MAKE A DECENT FARMER." "THE ONLY THING SHE COULD DO WAS SEND HIM TO THE UNIVERSITY." "IF ISAAC'S FATHER HAD LIVED," "HE MIGHT HAVE GROWN UP TO BE AN EXTREMELY CLEVER FARMER." "INSTEAD, HE DID THINGS THAT STAGGER THE IMAGINATION." "ONE OF THEM HAS TO DO WITH INVENTING THE CALCULUS." "IN THE LATTER HALF OF THE THE 17th CENTURY," "AN ECCENTRIC ENGLISH SCIENTIST DISCOVERED THE CALCULUS." "HIS NAME WAS ISAAC NEWTON." "AND HIS DISCOVERY IS WITHOUT PARALLEL" "IN SCIENCE AND MATHEMATICS." "BUT CREDIT FOR DISCOVERING THE CALCULUS" "CAN'T BE GRANTED TO NEWTON ALONE." "IT'S RIGHTFULLY SHARED" "BY A SCHOLARLY DIPLOMAT FROM HANOVER," "AN ELECTORATE IN THE GERMAN STATE" "OF LOWER SAXONY," "IN THE 17th CENTURY." "HIS NAME--BARON GOTTFRIED WILHELM VON LEIBNIZ." "WORLDLY, OUTGOING, QUITE A FELLOW WITH THE LADIES." "IST SIE NOCH DA?" "ESISTNICHTGUT  UBELZUSPRECHEN." "LEIBNIZ WAS DIFFERENT FROM NEWTON" "IN ALMOST EVERY REALM," "EXCEPT THAT OF GENIUS." "LIKE NEWTON, LEIBNIZ WAS A MATHEMATICAL PRODIGY." "BUT UNLIKE NEWTON, WHOSE FORTE WAS THEORY," "LEIBNIZ FREQUENTLY DISPLAYED A PRACTICAL NATURE." "JA,NATURLICH." "DERMUSS SICHSELBSTUNTERSTUTZEN." "FOR EXAMPLE," "LEIBNIZ INVENTED A MECHANICAL CALCULATOR." "HIS CALCULATOR WAS PORTABLE" "AND COULD ACTUALLY EXTRACT SQUARE ROOTS." "HE WAS ACCLAIMED THROUGHOUT THE COURTS OF EUROPE" "FOR INVENTING IT" "AND WAS INVITED TO JOIN THE BRITISH ROYAL SOCIETY," "NO SMALL FEAT FOR A GERMAN." "HOWEVER, IT'S FOR THE CALCULUS," "NOT THE CALCULATOR," "THAT WILHELM LEIBNIZ IS BEST REMEMBERED." "ALTHOUGH LEIBNIZ AND NEWTON" "ARE RECALLED AS ITS CHIEF ARCHITECTS," "NEITHER ONE BUILT THE CALCULUS SINGLE-HANDEDLY." "BECAUSE LIKE ALL OTHER" "MAGNIFICENT MATHEMATICAL STRUCTURES," "THE CALCULUS IS A CULMINATION OF IDEAS." "ORESME..." "GALILEO..." "KEPLER..." "DESCARTES..." "FERMAT," "AND MANY OTHERS BEFORE NEWTON AND LEIBNIZ" "HAD ASKED QUESTIONS ABOUT SPEED, ACCELERATION," "AND THE BEHAVIOR OF CHANGING QUANTITIES." "QUESTIONS CONCERNING RATE OF CHANGE," "MANY OF WHICH ARE NOW ANSWERED ROUTINELY" "WITH DIFFERENTIAL CALCULUS," "CAN BE REDUCED TO A GEOMETRIC PROBLEM," "WHICH IS TO FIND THE TANGENT LINE" "AT AN ARBITRARY POINT OF A CURVE." "HOWEVER," "NOT ALL GEOMETRIC QUESTIONS INVOLVE MOTION." "SOME INVOLVE THINGS THAT ARE FIXED IN PLACE," "LIKE THE LAND ITSELF." "IN ANCIENT CIVILIZATIONS, SUCH AS EGYPT," "GEOMETRY HAD A NUMBER OF VERY PRACTICAL APPLICATIONS," "INCLUDING HOW TO MEASURE THE LAND." "THE VERY WORD GEOMETRY, IN FACT," "IS THE GREEK WORD FOR LAND MEASUREMENT." "IN THE GOLDEN AGE," "WHEN GREEK TRADERS ASKED QUESTIONS" "ABOUT SPECIFIC AREAS AND VOLUMES," "GREEK MATHEMATICIANS WERE ABLE TO OFFER SPECIFIC ANSWERS." "THEAREAOF ARECTANGLE" "ISTHEPRODUCT OF ITS BASE AND ITS ALTITUDE." "ATRIANGLEHASANAREA  1/2THATOF ARECTANGLE," "WITHTHESAMEBASE ANDALTITUDE." "ANYPLANEFIGURE BOUNDEDBYPOLYGONALLINES" "CANBEDIVIDED INTOTRIANGULARPIECES," "WHICHMAKESITSAREA" "THESUMOF THEAREAS OFALLTHEPIECES." "BUT HOW DOES ONE FIND THE AREA OF A CURVED FIGURE," "A PARABOLIC SEGMENT," "OR AN EVEN MORE EXOTIC SHAPE?" "THESE MATHEMATICAL QUESTIONS" "WERE CHALLENGES TO THE MIND AND SPIRIT OF GREECE." "IN AN INTELLECTUAL MARATHON WHICH MAY BE THOUGHT OF" "AS A RELAY RACE TOWARD THE CALCULUS," "GREEK MATHEMATICIANS COMPETED" "TO FIND A GENERAL METHOD OF QUADRATURES" "A PROCESS FOR CALCULATING AREAS OF CURVED FIGURES." "THERE WERE STRONG ENTRIES FROM OTHER CULTURES," "BUT THE FIRST LAP WAS A VICTORY FOR THE GREEKS." "THEIR IDEA WAS TO APPROXIMATE THE CURVED FIGURE" "WITH INSCRIBED POLYGONS," "ATTEMPTING TO EXHAUST THE WHOLE FIGURE." "BIT BY BIT," "THEY USED AN INCREASING NUMBER OF SIDES." "THIS ANCIENT GREEK APPROACH" "IS CALLED THE METHOD OF EXHAUSTION." "WITH THE METHOD OF EXHAUSTION" "AN APPLICATION OF THE CONCEPT" "OF THE LIMIT OF AN INFINITE SEQUENCE" "THE GREEKS SET THE MATHEMATICAL STANDARD" "FOR CENTURIES." "THEY ALMOST REACHED THE CALCULUS." "IF ARCHIMEDES HAD KNOWN THE SYMBOLS" "IN COMMON USE 18 CENTURIES LATER," "HE PROBABLY WOULD HAVE SUCCEEDED." "IN ANY CASE," "EVEN WITHOUT THE HELP OF ALGEBRA," "HE MADE A NUMBER OF SPECTACULAR DISCOVERIES." "ONE OF THE MORE REMARKABLE" "WAS THE QUADRATURE OF A PARABOLIC SEGMENT." "LETA OFT DENOTETHEAREA" "BETWEENTHEPARABOLA ANDTHEXAXIS" "FROMx = 0 TOx= t ." "ARCHIMEDES SHOWED," "BY AN INGENIOUS GEOMETRIC ARGUMENT," "THAT THIS AREA IS EQUAL TO 1/3 t-CUBED." "AS VALUABLE AS THE PARABOLIC SEGMENT WAS" "IN THE ADVANCE TOWARD THE CALCULUS," "IT'S A SPECIAL CASE RATHER THAN A GENERAL SHAPE." "AN EVEN MORE POWERFUL MATHEMATICS WAS NEEDED" "TO THRUST SCHOLARS FORWARD." "IT WOULD COME ALONG." "IT WOULD TAKE A VERY LONG TIME TO APPEAR." "NOT UNTIL THE EARLY 1600s" "DID A WANDERING MATHEMATICIAN AND ASTRONOMER," "JOHANNES KEPLER," "GET THE BALL ROLLING ONCE AGAIN." "JOHANNES KEPLER COMPUTED THE AREAS AND VOLUMES" "OF 92 NEW CURVED SHAPES." "KEPLER CAME CLOSE TO THE CALCULUS." "BUT LIKE THE GREEKS," "HE COULDN'T INVENT A GENERAL METHOD" "FOR ALL SHAPES AND FUNCTIONS." "BUT THIS TIME, THE RACE WASN'T DELAYED." "AND TWO FRENCHMEN PICKED UP THE TORCH" "PIERRE DE FERMAT" "AND RENE DESCARTES" "BLENDED ALGEBRA WITH GEOMETRY," "WHICH FINALLY MADE IT POSSIBLE" "TO DESCRIBE GEOMETRIC FIGURES WITH ALGEBRAIC EQUATIONS." "INA RECTANGULAR COORDINATESYSTEM," "ALINEOF SLOPEm" "CANBEDESCRIBEDBY THE SIMPLE ALGEBRAIC EQUATION y= m TIMESx." "ANDANALGEBRAICEQUATION FORA CIRCLE" "COMESDIRECTLYFROM THEPYTHAGOREANTHEOREM" "FORRIGHTTRIANGLES." "FERMAT CAME CLOSEST TO REACHING THE DERIVATIVE" "WITH HIS GENERAL METHOD" "FOR LOCATING THE HIGH AND LOW POINTS OF FUNCTIONS," "POINTS WHERE THE SLOPE IS ZERO." "THE RACE FOR THE CALCULUS WAS ENTERING ITS FINAL LAP." "THE SCIENTIFIC PROGRESS SPED ACROSS THE CHANNEL," "WHERE IT GENERATED EXCITEMENT AT TRINITY COLLEGE." "THE GREAT MATHEMATICAL ADVANCES," "MOST NOTABLY KEPLER'S AND FERMAT'S WORK," "ARRIVED IN CAMBRIDGE ON THE HEELS OF AN OBSCURE LAD" "FROM THE FARM COUNTRY OF LINCOLNSHIRE." "HE WAS ISAAC NEWTON." "AND AMONG OTHER ACCOMPLISHMENTS" "ANY ONE OF THEM ENOUGH TO EARN HIM A CHAPTER" "IN SCIENTIFIC HISTORY" "HE WOULD GO ON TO DISCOVER THE UNIVERSAL LAW OF GRAVITY" "AND TO REACH THE CALCULUS." "LATER, TO A MUCH OLDER NEWTON'S DISMAY," "IF NOT UTTER CONTEMPT," "SO WOULD WILHELM LEIBNIZ." "ALTHOUGH NEWTON CLAIMED," ""IF I HAVE SEEN FURTHER THAN OTHER MEN," "IT IS BECAUSE I STOOD ON THE SHOULDERS OF GIANTS,"" "IT WASN'T THIS MAN HE HAD IN MIND." "COPERNICUS, GALILEO, KEPLER" "IN NEWTON'S MIND," "THESE WERE THE SHOULDERS UPON WHICH HE'D STOOD." "LEIBNIZ AND NEWTON," "VICTORS IN THE RACE FOR THE CALCULUS." "WHILE THEIR DISCOVERY WAS MADE IN THE SAME FIELD," "IT WAS HARDLY MADE IN THE CONTEXT OR SPIRIT" "OF GOOD SPORTSMANSHIP." "YES." "I'LL GO SEE HIM AGAIN." "YES, WILL YOU?" "IT WAS A DRAMA WITH A CONFLICT" "THAT COULD HAVE BEEN WRITTEN BY SHAKESPEARE." "THIS WAS A DRAMA OF NO SMALL ACADEMIC CONSEQUENCE," "FOR AS THE REPUTATION OF ONE PLAYER WAS IN PERIL..." "YOU ARE A GUEST IN MY HOUSE." "SO, TOO, WAS THE VERY SANITY OF THE OTHER." "EARLIER, HOWEVER," "BEFORE THE SITUATION WENT BEYOND REASON," "NEWTON'S MIND WAS CAPABLE" "OF THE MOST RATIONAL THOUGHT IMAGINABLE." "AS A YOUNG MAN, INFLUENCED BY ISAAC BARROW," "THE CAMBRIDGE MATHEMATICIAN" "WHOM NEWTON HARDLY ACKNOWLEDGED AS HIS MENTOR," "NEWTON DISCOVERED HIS VERY OWN CALCULUS." "HE CALLED IT "THE METHOD OF FLUXIONS."" "NEWTON SAW THAT THE RATE OF CHANGE OF A FUNCTION" "GIVES THE SLOPE OF ITS GRAPH" "AND REALIZED THAT THE SLOPE ITSELF" "GENERATES A NEW QUANTITY, THE FLUXION." "TODAY, NEWTON'S FLUXION IS CALLED A DERIVATIVE." "THIS MATHEMATICS WAS STRONGER THAN THE GREEKS'," "AND MORE FLEXIBLE THAN FERMAT'S WORK." "WITH THIS MATHEMATICS," "NEWTON EXPLAINED THE MOTION OF OBJECTS IN SPACE" "AND WON THE RELAY RACE" "THAT HAD BEGUN TWO MILLENNIA BEFORE." "THIS MATHEMATICS WAS THE CALCULUS." "YET NEWTON KEPT HIS CALCULUS TO HIMSELF FOR 30 YEARS." "THE QUESTION IS--WHY?" "NO ONE WILL EVER KNOW FOR CERTAIN," "BUT ISAAC NEWTON, RATIONAL SCIENTIST," "LEFT LETTERS, HINTS, CONFESSIONS." "IN HIS OWN SECRET WORDS," "HE REVEALED FANTASIES OF BURNING HIS MOTHER." "HE JOTTED TWO MILLION WORDS" "ON ALCHEMY AND BIBLICAL PROPHECY" "AND DENIED CERTAIN TEACHINGS OF HIS CHURCH." "THAT WASN'T ALL HE DENIED HIMSELF." "ON HIS DEATHBED, AT THE AGE OF 85," "NEWTON CLAIMED HIS GREATEST TRIUMPH" "WAS TO DIE A VIRGIN." "ISAAC NEWTON WAS, NONETHELESS," "ONE OF THE MORE PRODUCTIVE MEN IN HISTORY" "DESPITE AN OBSESSION WITH SECRECY," "TERROR OF SCIENTIFIC CRITICISM AND OF THIEVERY." "FOR 30 YEARS, IN LETTERS TO HIS FRIENDS," "HE DROPPED CRYPTIC HINTS" "ABOUT HIS DISCOVERY OF THE CALCULUS," "PERHAPS TOO MANY HINTS." "SIE KENNEN IHN ZIEMLICH GUT, JA?" "OF COURSE, AS A DIPLOMAT, LEIBNIZ DEALT WITH SECRETS," "BUT IN A DIFFERENT WAY." "HE ALSO DEALT WITH NEWTON, THOUGH NEVER IN PERSON." "THEY CORRESPONDED FOR YEARS BUT NEVER MET." "NEWTON WROTE LEIBNIZ A CHALLENGE," "AN ANAGRAM HINTING AT FLUENTS AND FLUXIONS." "BUT THIS TIME," "NEWTON MIGHT HAVE BEEN TOOCLEVER." "INTRIGUED, LEIBNIZ MASTERED THE PREVIOUS WORK" "OF KEPLER AND FERMAT" "AND INVENTED HIS OWN CALCULUS." "IT MIGHT NOT HAVE BEEN THE FIRST," "BUT ONE FACT RISES ABOVE DEBATE" "IT WAS THE FIRST TO BE PUBLISHED." "LEIBNIZ'S WAS SLIGHTLY DIFFERENT FROM NEWTON'S," "AND, IN SOME WAYS, SLIGHTLY BETTER." "LEIBNIZ INCORPORATED THE IDEA OFCALCULATINGAREA" "BY IMAGINING A PICKET FENCE OFLITTLERECTANGLES." "IMAGINETHEM DIMINISHINGINSIZE," "ANDADDTHEIRAREAS." "EVENTUALLYTHEIRSUM APPROACHESA LIMIT" "WHICHISEQUAL TOTHEAREAOFTHEREGION" "UNDERTHEGRAPH OFA FUNCTION." "THISISTHEPROCESS OFINTEGRATION." "LEIBNIZSUMMARIZEDIT  WITHTHEINTEGRALSIGN," "ANELONGATEDS," "FROMTHELATINWORD FORSUM." "BOTH LEIBNIZ AND NEWTON, EACH A GENIUS WITH A NEW IDEA," "RECOGNIZED SOMETHING THE OTHERS HAD OVERLOOKED." "THEY SAW THE SURPRISING CONNECTION" "BETWEEN INTEGRATION AND DIFFERENTIATION." "THIS REMARKABLE CONNECTION CAN BE UNDERSTOOD" "BY CALCULATING THE AREA OF A PARABOLIC SEGMENT." "LET A OF t DENOTE THE AREA" "BETWEENTHEPARABOLA ANDTHExAXIS" "FROMx = 0 TOx= t ." "THISISAFUNCTIONOFt." "TRYTOFIND ITSDERIVATIVE,dA /dt." "IFTHEAREAFROMOTOt  IS A OF t," "THENTHEAREA FROM0 TOt+ DELTAt" "ISA OF(t +DELTAt)." "THEAREADELTAA ISTHESAMEASTHEAREA" "OFA RECTANGLE OFBASEDELTAt" "ANDHEIGHTx-SQUARED" "FORSOMEx." "ASDELTAtSHRINKSTO0,  x-SQUARED BECOMESt-SQUARED." "INOTHERWORDS," "THEDERIVATIVE" "OFTHEAREAFUNCTION" "ISt-SQUARED." "TRYINGVARIOUSFUNCTIONS" "TOSEEWHICHONE HAS  t-SQUARED AS ITS DERIVATIVE," "1/3t-CUBED" "EVENTUALLYCOMESTO MIND." "THATWORKS." "AND SO DOES 1/3 t-CUBED + A CONSTANT" "BECAUSE THE DERIVATIVE OF A CONSTANT" "IS ZERO." "THEAREAFUNCTIONAOF t" "IS 1/3 t-CUBED PLUS A CONSTANT." "SINCETHEAREAISZERO" "WHENt ISZERO," "THE CONSTANT IS ALSO ZERO." "SO THERE IT IS" "THE QUADRATURE OF THE PARABOLA," "USING THE CALCULUS OF NEWTON AND LEIBNIZ." "IN THE LANGUAGE OF CALCULUS," "THE INTEGRAL OF x-SQUARED FROM ZERO TO t" "IS 1/3 t-CUBED." "OF COURSE, ARCHIMEDES HAD ALREADY DONE THIS" "BY A GEOMETRIC METHOD." "BUT THE CALCULUS IS SUPERIOR." "CALCULUS APPLIES NOT ONLY TO THE PARABOLA," "BUT TO ANY SMOOTH CURVE." "IT SOLVES THE PROBLEM OF QUADRATURE IN GENERAL." "AND IT REVEALS THE LONG HIDDEN CONNECTION" "BETWEEN INTEGRATION AND DIFFERENTIATION." "DIFFERENTIATION AND INTEGRATION ARE INVERSE PROCESSES." "START WITH A FUNCTION" "AND INTEGRATE" "TO GET ITS AREA FUNCTION." "THENTAKETHEDERIVATIVE" "OFTHEAREAFUNCTION." "THERESULT" "ISTHEORIGINALFUNCTION." "THISPHENOMENONIS CALLED" "THE FIRST FUNDAMENTAL THEOREM OFCALCULUS." "ONTHEOTHERHAND," "THEPARABOLAILLUSTRATES SOMETHINGELSE." "FIRST,THEAREA OFTHEPARABOLICSEGMENT" "WASCALCULATED FROMx = 0 TOx= t ." "TOFINDTHEAREA BETWEEN ANY TWO VALUES OF x," "SAYFROMx= 1 TOx =2 ," "SIMPLYSUBTRACT." "TAKE THE AREA FROM ZERO TO 2" "MINUSTHEAREAFROMZERO TO1." "EXPRESSEDINITSGENERALFORM ," "THISIDEAIS CALLED" "THESECONDFUNDAMENTAL THEOREMOFCALCULUS." "ITSAYSTHATAFUNCTION" "ISEQUALTO THEINTEGRAL OF ITS DERIVATIVE + A CONSTANT." "NEWTON AND LEIBNIZ REALIZED" "THAT THE SECOND FUNDAMENTAL THEOREM," "WHICH WAS DISCOVERED BY SOLVING" "THE GEOMETRIC PROBLEM OF QUADRATURE," "ALSO HAS IMPORTANT CONSEQUENCES" "IN THE MECHANICS OF MOVING BODIES." "SINCE THE DERIVATIVE OF VELOCITY" "IS ACCELERATION," "VELOCITY CAN BE OBTAINED" "BY INTEGRATING THE ACCELERATION." "SINCE THE DERIVATIVE" "OF DISPLACEMENT IS VELOCITY," "DISPLACEMENT CAN BE OBTAINED" "BY INTEGRATING THE VELOCITY." "NEWTON WAS AWARE" "OF THE PROFOUND POWER" "OF THIS NEW MATHEMATICS." "BUT AS HE SAT POLISHING HIS CALCULUS" "AS IF IT WERE A MOST PRECIOUS APPLE," "ALL THE WHILE POSTING CRYPTIC HINTS" "OF ITS GOLDEN FLAVOR," "LEIBNIZ PUBLISHED HIS DIFFERENTIALS AND INTEGRALS." "THREE YEARS LATER," "NEWTON SOMEWHAT RELUCTANTLY PUBLISHED" "HIS MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY," "THEPRINCIPIA." "AND YET, EXCEPT FOR A FEW HINTS," "NEWTON KEPT HIS CALCULUS TO HIMSELF." "WHILE NEWTON REMAINED SECRETIVE," "THE SAME COULD NOT BE SAID OF HIS COUNTRYMEN," "WHO PRAISED HIM AT THE EXPENSE OF LEIBNIZ." "THEY CREATED CHAOS IN THE ROYAL SOCIETY" "OVER WHO FIRST INVENTED THE CALCULUS." "THE NEWTON-LEIBNIZ CONTROVERSY" "WAS BITTER IN TASTE AND MEAN IN SPIRIT." "ONE LOW POINT WAS THE CHARGE" "THAT LEIBNIZ HAD PLAGIARIZED IDEAS" "FROM NEWTON'S LETTERS." "LEIBNIZ WAS OUTRAGED AND DEMANDED A FORMAL HEARING." "SINCE ISAAC NEWTON HIMSELF" "WAS PRESIDENT OF THE ROYAL SOCIETY," "LEIBNIZ MIGHT HAVE MADE A WISER MOVE." "THIS HAS COME FOR YOU, SIR." "EVEN BEFORE PLEADING HIS CASE, LEIBNIZ WAS JUDGED GUILTY." "DO YOU WISH TO SEND A REPLY?" "NO REPLY." "THANK YOU, SIR." "AND SO LEIBNIZ, WHO'D BEGUN HIS CAREER IN MATHEMATICS" "WITH THE PHILOSOPHY OF OPTIMISM," "ENDED IT WITH AN EQUATION OF BITTERNESS." "HEARING OF THE GERMAN'S DEATH," "NEWTON BOASTED, "I HAD BROKEN LEIBNIZ'S HEART."" "IRONICALLY, THOUGH NEWTON'S FLUENTS AND FLUXIONS" "WERE TO RULE BRITANNIA FOR A CENTURY THEREAFTER," "LEIBNIZ'S NOTATION WAS BETTER," "AND IT BECAME THE UNIVERSAL LANGUAGE OF MATHEMATICS." "HIS SYMBOL FOR THE INTEGRAL" "CONVEYS A SOPHISTICATED IDEA." "THE INTEGRAL IS" "THE AREA OF THE REGION" "UNDER A CURVE," "AND AT THE SAME TIME, IT'S AN ANTIDERIVATIVE." "AND SO THE RACE FOR THE CALCULUS THAT BEGAN IN ANCIENT GREECE" "FINALLY ENDED IN A CONTROVERSIAL PHOTO FINISH." "THERE COULD HAVE BEEN ENOUGH GLORY FOR BOTH WINNERS." "BUT WHILE THE GLORY WASN'T SHARED," "THE PRIZE WAS." "NOT BETWEEN NEWTON AND LEIBNIZ," "BUT AMONG EVERY GENERATION SINCE." "THE INSIGHT OF LEIBNIZ AND NEWTON," "MOST NOTABLY THE CONNECTION" "BETWEEN DIFFERENTIATION AND INTEGRATION," "HAS EVOLVED INTO A TOOL" "OF INCOMPARABLE VALUE IN MATHEMATICS." "THANKS TO BOTH LEIBNIZ AND NEWTON," "DIFFERENTIAL AND INTEGRAL CALCULUS" "ARE THE COLLECTIVE LANGUAGE OF SCIENCE." "NEWTON'S CALCULUS WAS FIRST PUBLISHED IN THIS BOOK," "WHICH IS CALLED THEMETHODOF FLUXIONS." "IT'S ABOUT ISAAC NEWTON'S THEORY," "BUT IT'S NOT WRITTEN BYISAACNEWTON." "IT'S WRITTEN BY THIS MAN, JOHN COLSON." "FOR REASONS NOT ENTIRELY CLEAR," "NEWTON'S OWN MANUSCRIPT ABOUT HIS CALCULUS" "WAS NEVER PUBLISHED DURING HIS OWN LIFETIME." "IT MAY BE THAT THE PUBLISHER WAS A LITTLE SLOW" "IN GETTING IT OUT." "THAT SOMETIMES HAPPENS." "IT FINALLY CAME OUT IN 1969." "NEWTON'S FIGHT WITH LEIBNIZ WASN'T THE ONLY FIGHT" "HE HAD DURING HIS LIFETIME." "NEWTON WASN'T A VERY NICE MAN." "ONE OF HIS FAMOUS QUOTATIONS WAS" ""IF I SEE SO FAR," "IT'S BECAUSE I STAND ON THE SHOULDERS OF GIANTS."" "HE MAY HAVE," "BUT HE TRAMPLED ON OTHERS AS WELL." "I'LL SEE YOU ALL NEXT TIME." "CAPTIONING PERFORMED BY THE NATIONAL CAPTIONING INSTITUTE, INC." "CAPTIONS COPYRIGHT 1985 CALIFORNIA INSTITUTE OF TECHNOLOGY" "THE CORPORATION FOR COMMUNITY COLLEGE TELEVISION AND THE ANNENBERG/CPB PROJECT" "PUBLIC PERFORMANCE OF CAPTIONS PROHIBITED WITHOUT PERMISSION OF NATIONAL CAPTIONING INSTITUTE" "Annenberg Media" "¶" "For information about this and other Annenberg Media programs call 1-800-LEARNER and visit us at"