"Annenberg Media" "¶" "CAPTIONING IS MADE POSSIBLE BY THE ANNENBERG/CPB PROJECT" "ABODY IN UNIFORM CIRCULAR MOTION" "HASBOTHCONSTANTSPEED ANDCONSTANTACCELERATION." "UNIFORMCIRCULARMOTION" "CANDESCRIBE THEORBITOF THEMOON." "THE CIRCLE WAS INVENTED" "BY PRIMITIVE PEOPLES ALL OVER THE WORLD." "USUALLY IT HAD" "SOME MYSTICAL SIGNIFICANCE." "IT WAS ASSOCIATED" "WITH THE HORIZON" "OR WITH THE SKY OR THE GODS." "AFTER A WHILE, MYTHS TAKE ON" "COMPLICATED TRADITIONS AND FANCY LANGUAGE," "AND THEN THEY BECOME KNOWN AS PHILOSOPHY." "THE GREEKS, OF COURSE, WERE NOT MYSTICS." "THEY WERE PHILOSOPHERS." "THEY SAW CIRCLES IN THE SKY LIKE EVERYBODY ELSE," "BUT TO THEM, THEY REPRESENTED NOT SOME MYTH OF CREATION," "BUT GEOMETRY." "THEY IMAGINED THE UNIVERSE" "FILLED WITH SPHERES TURNING ON THEIR AXES" "SO THE MOTIONS WE SAW" "SEEMED TO US TO BE CIRCULAR MOTIONS." "AND SO THE PHILOSOPHER PLATO SAID," ""ALL MOTIONS IN THE HEAVENS" "MUST BE CIRCULAR MOTIONS AND UNIFORM."" "AND ASTRONOMERS SPENT THE NEXT 2,000 YEARS" "TRYING TO SHOW THAT HE WAS RIGHT." "ASTRONOMY WAS A PROFESSION." "ITS JOB WAS TO PREDICT" "WHERE EACH HEAVENLY BODY WOULD BE FOUND" "AT ANY GIVEN TIME ON ANY GIVEN NIGHT." "THESE PREDICTIONS WERE IMPORTANT" "BECAUSE THEY WERE USED FOR AGRICULTURE," "FOR NAVIGATION," "AND ABOVE ALL, IN THOSE DAYS, FOR CASTING HOROSCOPES." "ANYBODY COULD LOOK AT THE SKY AND SEE IF THEY WERE RIGHT." "REMEMBER, THE ANCIENTS SPENT MUCH MORE TIME" "LOOKING AT CLEAR NIGHT SKIES THAN WE DO." "USING CIRCULAR MOTIONS," "THEY HAD TO PREDICT WHERE THINGS WOULD BE," "AND BE RIGHT." "FOR THE STARS, IT WORKED WELL" "BECAUSE THE STARS SEEMED TO MOVE IN CIRCLES," "BUT THERE WERE OTHER BODIES UP THERE" "WHICH THE GREEKS CALLED PLANETS." "FOR THEM, IT DIDN'T SEEM TO WORK." "THEY DIDN'T MOVE ACROSS THE SKY AT CONSTANT SPEED." "THEY DIDN'T EVEN MOVE" "IN THE SAME DIRECTION ACROSS THE SKY." "THEY WOULD TURN AROUND AND GO BACKWARDS SOMETIMES." "TWO OF THEM, MERCURY AND VENUS," "ALWAYS SEEMED TO STAY FAIRLY CLOSE TO THE SUN," "WHEREAS THE OTHERS DIDN'T." "THEY DIDN'T ALWAYS HAVE THE SAME BRIGHTNESS," "MEANING THEY WEREN'T ALWAYS" "THE SAME DISTANCE FROM THE EARTH." "NEVERTHELESS, PLATO HAD SAID THAT ALL OF THOSE MOTIONS" "WERE DUE TO UNIFORM CIRCULAR MOTIONS," "AND THAT'S WHAT THE ASTRONOMERS HAD TO ACCOMPLISH." "THEY CONSTRUCTED A SYSTEM THAT WORKED REMARKABLY WELL." "IT WAS BASED NOT ON CIRCLES AROUND THE EARTH," "BUT ON CIRCLES AROUND OTHER CIRCLES." "THESE WERE CALLED EPICYCLES." "TO UNDERSTAND THAT," "WE HAVE TO MAKE A LITTLE CIRCLE OF OUR OWN" "AND GO BACK TO WHERE WE STARTED." "PROBABLY EVERY ANCIENT CULTURE DISCOVERED THE CIRCLE." "WALKING SYMBOLIC CIRCLES AROUND IT," "NATIVE AMERICANS WORSHIPPED THE SUN." "HINDUS NOTED WHAT APPEARED TO BE" "THE CIRCULAR MOTION OF THE HEAVENS," "AS DID THE QUECHUA INDIANS OF PERU" "AND EARLY JAPANESE ASTRONOMERS." "HEAVENLY FORMS AND FORMATIONS" "BECAME MODELS FOR EARTHLY STRUCTURES." "SOME OF THE EARLIEST BUILDINGS" "DATING FROM 5,000 B.C. IN MESOPOTAMIA," "HAVE A CIRCULAR GROUND PLAN." "EVEN A PRIMITIVE HAND CAN INSCRIBE CIRCLES." "WHILE THEY CAN BE EXPRESSED" "BY THE MORE SOPHISTICATED LANGUAGE OF MATHEMATICS," "THE CONCEPT REMAINS RELATIVELY SIMPLE." "THE VECTOR FROM THE CENTER TO EACH POINT ON A CIRCLE" "ALWAYS HAS THE SAME LENGTH." "THIS SIMPLICITY MIGHT HAVE BEEN PART OF ITS APPEAL." "FROM SIMPLE IDEAS, MAGNIFICENT STRUCTURES GREW." "IMAGES PROLIFERATED," "WERE EMBELLISHED UPON, AND FURTHER R ELABORATED." "THE CIRCULAR FORM OF ANCIENT STONEHENGE," "FOR EXAMPLE." "HUMANS IMITATED NATURE." "AND NATURE SEEMED TO IMITATE THEM IN RETURN." "DECORATION BECAME FUNCTION," "AND FUNCTION FOLLOWED FORM." "WITH THE HUMAN TOUCH," "NOT EVERYTHING WAS PERFECT IN THE NATURE OF THE CIRCLE." "SOMETIMES IT'S NECESSARY TO GO BACK TO THE DRAWING BOARD." "THE POSITION OF ANY POINT ON A CIRCLE" "CAN BE DESCRIBED BY ITS CARTESIAN COORDINATES," "OR BY ITS DISTANCE FROM THE CENTER" "AND THE ANGLE TO THE x AXIS." "THESE DESCRIPTIONS ARE RELATED BY TRIGONOMETRY..." "FOR JUST ABOUT ANYONE WITH A HEAD FOR FIGURES." "THE CIRCLE IS A POWERFUL SYMBOL," "ORGANIC TO THE UNIVERSE AND MOST OF ITS CIVILIZATIONS." "IN SOME CULTURES," "THE CIRCULAR FORM OF THE HEAVENS" "AND THE HEAVENLY CIRCULAR MOTIONS" "ARE AS MANIFEST TODAY AS THEY EVER WERE." "THE GREEK PHILOSOPHER PLATO DECLARED THE CIRCLE IDEAL" "AND ASSERTED THAT ALL OTHER FORMS WERE IMPERFECT." "HE DECIDED CIRCULAR MOTION DESCRIBES THE HEAVENS." "NOT MERELY CIRCULAR MOTION-- UNIFORM CIRCULAR MOTION." "EACH HEAVENLY BODY WAS TO BE A POINT" "MOVING ON A CIRCLE AT CONSTANT RATE." "THE VECTOR IS AT AN ANGLE THAT GROWS AT A CONSTANT RATE." "THAT RATE, OMEGA," "IS KNOWN AS THE ANGULAR SPEED OF THE OBJECT." "AT ANY TIME t," "THE ANGLE THETA = OMEGA t." "IN ONE REVOLUTION, THE OBJECT MOVES" "THROUGH 2 PI RADIANS IN TIME CAPITAL T." "THEREFORE, THE ANGULAR SPEED IS 2 PI/CAPITAL T." "PLATO SAID THAT GOD WAS IN THE HABIT OF GEOMETRIZING." "PERHAPS THAT'S WHY PLATO'S IDEALS" "OF THE UNIVERSE IN HARMONY" "STRUCK A RESPONSIVE CHORD" "WITH SCHOLARS AND MYSTICS FOR CENTURIES AFTERWARD." "FOR PLATO, THE PHYSICAL UNIVERSE WAS BOUNDED" "BY A ROTATING SPHERE OF FIXED STARS" "CENTERED ON THE STATIONARY FIXED EARTH." "INSIDE THE SPHERE," "TRAVELING IN CIRCULAR PATHS WITH INDIVIDUAL UNIFORM MOTION," "WERE SEVEN PLANETS" "THE MOON, VENUS, MERCURY," "THE SUN, MARS, JUPITER, AND SATURN" "IN THAT ORDER AND EXTENDING OUTWARDS." "GREEK ASTRONOMERS SAW THE PLANETS AS ERRATIC BODIES," "SO MUCH SO THAT THE VERY WORD PLANET" "MEANT TO WANDER, TO MISLEAD, OR TO DECEIVE." "PLANETS WERE DECEITFUL" "BECAUSE THEY REFUSED TO OBEY PLATO'S DICTUM OF PERFECTION," "UNIFORM CIRCULAR MOTION." "THIS PLATONIC IDEAL WAS EVENTUALLY ACCEPTED" "BY EUROPE'S NOBLE THINKERS, AND WHY NOT?" "THE RULING CLASS AND COMMON FOLK" "HAD MUCH IN COMMON WITH THE PLANETS." "IS NOT HUMAN BEHAVIOR, LIKE THAT OF THE PLANETS," "OFTEN ERRATIC?" "AND WHEN CERTAIN CONDITIONS EXIST," "DON'T HUMANS, LIKE THE PLANETS," "TEND TO WANDER OR APPEAR TO BE DECEITFUL?" "BUT IN THE ROYAL RENAISSANCE, OF COURSE," "NEITHER PLANET NOR NOBLEPERSON WAS SUBJECT TO CRITICISM" "FOR BEING LESS THAN PERFECT." "WHETHER IN THE HEAVENS OR ON EARTH," "ERRATIC BEHAVIOR DOESN'T NECESSARILY MEAN IMPERFECTION." "PERFECTION, AFTER ALL, IS IN THE EYE OF THE BEHOLDER," "AND SO IS REALITY ITSELF." "IN THE BEST OF ALL POSSIBLE WORLDS," "CAN THE NAKED EYE SEE REALITY PERFECTLY?" "NOT ALWAYS." "PERCEPTIONS OF REALITY MAY BE SHADOWS" "OF THE REAL THING." "SOMETHING THAT SEEMS TO HAVE ITS UPS AND DOWNS" "MAY BE GOING AROUND IN CIRCLES." "IN THE LANGUAGE OF VECTORS," "THE SHADOW OF A MOTION" "MAY BE ONLY ONE OF ITS COMPONENTS." "CIRCULAR MOTION ACTUALLY HAS TWO SEPARATE COMPONENTS." "THE ORIGINAL VECTOR IS THE SUM OF THESE." "AND THAT'S WHY CIRCULAR MOTION" "CAN BE DESCRIBED BY A VECTOR EQUATION." "THE MOTIONS OF THE PLANETS" "AS SEEN AGAINST THE BACKGROUND OF STARS" "WERE MUCH MORE COMPLICATED." "BECAUSE THEY SEEMED TO CONTRADICT THE IDEA" "OF UNIFORM CIRCULAR MOTION," "THEORETICAL MODELS OF PLANETARY MOTIONS" "WERE CREATED BY PLATO'S FOLLOWERS." "EACH ONE ATTEMPTED TO SAVE THE APPEARANCES," "IN OTHER WORDS, TO MAKE OBSERVED BEHAVIOR" "COMPATIBLE WITH THE PLATONIC IDEA." "ONE ATTEMPT," "AN INVENTION BY APOLLONIUS OF PERGA," "WAS CALLED THE EPICYCLE." "THE PLANETS MOVED ON SMALL CIRCLES" "ATTACHED TO A LARGER CIRCLE WHICH WAS ALSO TURNING." "THE LARGE CIRCLE WAS CALLED THE DEFERENT," "AND THE SMALLER ONE, THE EPICYCLE." "IN THE HANDS OF A MATHEMATICIAN," "THIS COMBINATION OF CIRCULAR MOTIONS" "CAN PRODUCE MANY FORMS." "VIEWED FROM THE CENTER OF THE DEFERENT," "THE PLANET SEEMS TO BEHAVE STRANGELY." "IF THE DEFERENT AND EPICYCLE ROTATE AT EQUAL SPEEDS," "THE FIGURE IS ANOTHER CIRCLE" "WHOSE CENTER IS OFFSET FROM THE EARTH." "IF THE EPICYCLE ROTATES" "EXACTLY TWICE AS FAST AS THE DEFERENT," "THE RESULT IS ANOTHER SMOOTH CURVE," "AN ELLIPSE." "THE ELLIPSE IS THE ACTUAL SHAPE" "OF A PLANET'S ORBIT AROUND THE SUN." "APOLLONIUS WROTE A CELEBRATED TREATISE ON CURVES," "ONE OF WHICH WAS THE ELLIPSE." "ANCIENT ASTRONOMY REACHED ITS HIGHEST POINT 400 YEARS LATER IN ALEXANDRIA," "THE QUEEN CITY OF EGYPT." "IN THAT FABLED CITY LIVED CLAUDIUS PTOLEMY." "AMONG MANY ACCOMPLISHMENTS, HE COMPILED THE ALMAGEST," "AN ENCYCLOPEDIA THAT SUMMARIZED FIVE CENTURIES OF GREEK SCIENCE" "INTO 13 EXTRAORDINARY VOLUMES." "PTOLEMY WAS NOT ABOVE ABANDONING" "PLATONIC THEORIES OF PERFECTION," "AND IN FACT, HE ADDED A FEW TWISTS OF HIS OWN." "THE DEFERENCE OF THE SUPERIOR PLANETS" "MARS, JUPITER, AND SATURN" "DID NOT ROTATE WITH UNIFORM RATES ABOUT THEIR CENTERS." "HE ADOPTED AN OVAL ORBIT FOR MERCURY." "AS STARTLING AS PTOLEMY'S VARIATIONS WERE," "THEY WERE LOST LIKE ESOTERIC FOOTNOTES" "IN THE SHEER MASS OF THE ALMAGEST." "THEY WERE ALSO OVERWHELMED" "BY THAT OVERRIDING PRINCIPLE OF THE GREEK ASTRONOMERS," "A SYSTEM BASED TOTALLY ON UNIFORM CIRCULAR MOTION." "PTOLEMY'S VARIATIONS ON PLATO'S THEME OF PERFECTION" "WERE COMPARATIVELY SUBTLE," "BUT THEY HAD A PROFOUND EFFECT ON NICHOLAS COPERNICUS." "WANTING TO REFORM THE ALMAGEST," "COPERNICUS PUT THE SUN" "AT THE CENTER OF THE SOLAR SYSTEM." "BY USING EPICYCLES LIBERALLY," "HE MANAGED TO SAVE APPEARANCES." "IN SPITE OF THE EPICYCLES," "COPERNICUS' UNIVERSE HAD PLANETS GOING AROUND THE SUN" "AND THE MOON GOING AROUND THE EARTH" "IN VERY NEARLY UNIFORM CIRCULAR MOTION." "TO MAINTAIN TRADITIONAL AESTHETIC PREFERENCES," "HE CREATED A MORE ACCURATE VISION" "OF THE SOLAR SYSTEM, THE COPERNICAN SYSTEM." "A CENTURY AND A HALF LATER," "ISAAC NEWTON REVEALED WHY IT WORKED," "STARTING WITH HIS THEORY OF THE MOON." "THE MOON IS ALWAYS FALLING, YET IT STAYS IN THE SKY." "THE GRAVITATIONAL FORCE FROM THE EARTH" "MAKES THE MOON ACCELERATE ACCORDING TO F = ma." "ALTHOUGH IT ACCELERATES ALWAYS," "IT MOVES AT NEARLY CONSTANT SPEED." "THIS SEEMS TO CONTRADICT" "THE VERY CONCEPT OF ACCELERATION," "BUT ACCELERATION IS THE RATE OF CHANGE OF VELOCITY," "AND VELOCITY IS A VECTOR." "THE SPEED IS ONLY ITS LENGTH." "A BODY CAN BE ACCELERATED EVEN IF ITS SPEED IS CONSTANT," "PROVIDED ITS DIRECTION CHANGES." "THIS POINT BECOMES CLEARER" "WITH THE INTRODUCTION OF ANOTHER CONCEPT," "THE DERIVATIVE OF A VECTOR." "THE DERIVATIVE OF A FUNCTION" "EXPLAINS HOW FAST IT'S CHANGING" "AT ANY GIVEN INSTANT." "THE DERIVATIVE OF A VECTOR" "EXPLAINS HOW FAST THE VECTOR IS CHANGING." "A VECTOR CAN CHANGE BY CHANGING ITS SIZE" "OR ITS DIRECTION OR BOTH." "THE CHANGE IN A VECTOR IS ITSELF A VECTOR." "THE VECTOR, v," "ALTHOUGH IT'S ALWAYS" "PERPENDICULAR TO r," "IS ITSELF UNDERGOING" "UNIFORM CIRCULAR MOTION." "IT'S DERIVATIVE IS THE ACCELERATION." "THAT SOLVES THE PUZZLE" "OF CONSTANT ACCELERATION AT CONSTANT SPEED." "AS A BODY UNDERGOES UNIFORM CIRCULAR MOTION," "THE ACCELERATION CONTINUALLY REDIRECTS" "THE VELOCITY VECTOR," "WHICH CONTINUALLY REDIRECTS THE RADIUS VECTOR." "THEY ALL KEEP IN STEP WITH EACH OTHER," "CONSTANT IN MAGNITUDE," "BUT CHANGING CONTINUALLY IN DIRECTION." "THE IDEA OF THE DERIVATIVE OF A VECTOR IS NOT NEW." "REMEMBER, F = ma IS ITSELF AN EQUATION" "ABOUT THE DERIVATIVES OF VECTORS." "THE ONLY DIFFERENCE IS THAT PREVIOUSLY," "BY TAKING DERIVATIVES ONE COMPONENT AT A TIME," "ONLY ORDINARY FUNCTIONS HAVE BEEN DIFFERENTIATED." "IN UNIFORM CIRCULAR MOTION," "THE VELOCITY IS PERPENDICULAR TOTHERADIUSVECTOR," "ANDTHEACCELERATION ISPERPENDICULAR" "TOTHEVELOCITYVECTOR." "SOITPOINTS ALONGTHERADIUSVECTOR," "BUTINTHEOPPOSITE DIRECTION." "BUT HOW BIG ARE THE VECTORS?" "vISEQUALTOOMEGAr..." "AND a IS EQUAL TO OMEGA v." "SO a IS EQUAL TO v-SQUARED/r." "THOSE ARE THE RELATIONSHIPS" "AMONG RADIUS, VELOCITY, AND ACCELERATION" "LOCKED TOGETHER" "IN THE ETERNAL DANCE OF UNIFORM CIRCULAR MOTION." "ISAAC NEWTON UNDERSTOOD THAT ANY BODY" "THAT MOVES IN UNIFORM CIRCULAR MOTION" "IS ALWAYS BEING ACCELERATED." "HE CALLED THIS CENTRIPETAL ACCELERATION." "ACCORDING TO NEWTON'S SECOND LAW," "A FORCE IS REQUIRED" "TO PRODUCE THAT CONTINUOUS ACCELERATION." "FOR A BODY IN ORBIT, THAT FORCE IS GRAVITY." "THE LENGTH OF THE RADIUS VECTOR" "DETERMINES THE GRAVITATIONAL FORCE." "THIS FORCE, THEN, DETERMINES THE ACCELERATION," "BUT CIRCULAR MOTION REQUIRES THIS RELATION" "AMONG RADIUS, SPEED," "AND ACCELERATION." "SO IF A BODY STARTS OUT WITH JUST THE RIGHT VELOCITY," "ITS SPEED STAYS CONSTANT," "AND UNIFORM CIRCULAR MOTION RESULTS." "NEWTON'S THEORIES ARE ESSENTIALLY MATHEMATICAL" "IN NATURE." "MUCH HAS BEEN MADE OF THEIR MATHEMATICAL CHARACTER." "WHAT'S TO BE MADE OF NEWTON'S NATURE AND CHARACTER?" "NEWTON SCHOLARS FREQUENTLY CHARACTERIZE HIM" "AS ACUTELY SUSPICIOUS AND JEALOUS," "OFTEN WITHOUT CAUSE, AND NEARLY ALWAYS TO A FAULT" "SOLITARY, ILL-TEMPERED, VINDICTIVE." "CONSIDERING THOSE TRAITS, ONLY THE MOST CREATIVE FICTION" "COULD PORTRAY SIR ISAAC NEWTON AS A JOLLY FELLOW." "HIS MERCURIAL DISPOSITION" "HARDLY MADE HIM A VICTOR IN CONTESTS OF PERSONALITY." "TOWARD HIS CONTEMPORARIES-- HE HAD NO PEERS" "HE WAS COOL, IF NOT ANTISOCIAL," "AND HISTORY IS ALMOST BLANK" "WITH ACCOUNTS OF SIR ISAAC'S LAUGHTER." "WILL YOU HAVE YOUR ROBE, SIR?" "YES, PLEASE." "WHAT'S FOR BREAKFAST?" "WELL, I GUESS YOUR FAVORITE, SIR." "NEWTON'S GENERAL DISDAIN" "FOR THE INTELLECTUAL ACHIEVEMENT OF HIS COLLEAGUES" "MAKES HIS ELECTION TO PARLIAMENT" "SEEM ALL THE MORE SURPRISING." "NEWTON WAS NOT WITHOUT THE GREAT FRIEND," "NOR THE ODD MOMENT OF LEVITY." "HIS PERSONALITY," "LIKE HIS EVALUATION OF COPERNICUS AND ARISTOTLE," "IS A STUDY IN CONTRASTS." "THE ESSENTIAL ISSUE IN THE MECHANIAL UNIVERSE" "IS NOT WHAT MOTIVATED NEWTON," "BUT WHAT NEWTON REVEALED AS A SCIENTIST." "REGARDING HIS REVELATIONS ABOUT THE MOON'S WORKINGS," "THE FACTS ARE CLEAR." "IT'S A LONG WAY FROM THE FARM" "THAT NOURISHED NEWTON'S IMAGINATION," "LONGER STILL FROM THE MYSTICISM" "THAT DROVE PRIMITIVES TO QUESTION THE HEAVENS," "A LONG WAY FROM THE PAST TO THE PRESENT" "AND INTO THE FUTURE." "NOW, IN THE INTEREST OF JUSTICE," "I WANT TO CHANGE THE SUBJECT COMPLETELY." "ONE OF THE MOST HEINOUS ACTIVITIES" "THAT GOVERNMENTS ENGAGE IN" "IS THE TORTURE OF PRISONERS." "TODAY, NO GOVERNMENT ANYWHERE IN THE WORLD" "OPENLY ADMITS TO TORTURING PRISONERS." "WE ALL KNOW THAT THEY DO IT SECRETLY AND ILLEGALLY." "IN EARLIER TIMES, HOWEVER," "IN MEDIEVAL TIMES RIGHT UP THROUGH THE RENAISSANCE," "TORTURE WAS AN OPEN, LEGAL PRACTICE." "THAT MEANT THAT THUMBSCREWS, RACKS, STAKES, AND FLAMES," "HORSES THAT PULL THE BODY IN DIFFERENT DIRECTIONS," "WERE INSTRUMENTS OF GOVERNMENT POLICY." "THE FACT THAT IT WAS A LEGAL INSTITUTION" "HAD COMPENSATIONS EVEN FOR THE VICTIMS." "THE EXECUTIONERS WERE SKILLED PROFESSIONALS." "FOR AN APPROPRIATE BRIBE," "THEY COULD PUT ON A GOOD SHOW WITHOUT CAUSING PAIN." "THAT WAS DONE BY MAKING SURE YOU WERE PAST CARING" "BEFORE THE EXCITING PART CAME." "THE FACT THAT IT WAS A LEGAL INSTITUTION" "MEANT IT WAS SUBJECT TO LAWS." "THEY DIDN'T JUST START BEATING A GUY UP." "BEFORE PHYSICAL TORTURE COULD TAKE PLACE," "JUDGES DELIBERATED ON THE MERITS OF THE CASE" "AND DECIDED HOW FAR THE PROCESS COULD GO." "THERE WERE ACTUALLY TWO STAGES THAT HAD TO BE GONE THROUGH" "BEFORE ACTUAL PHYSICAL TORTURE." "THE FIRST WAS THE VERBAL THREAT OF TORTURE." "IN THE TRIAL OF GALILEO," "THAT WAS ALL THAT WAS AUTHORIZED." "IN GALILEO'S CASE, THAT WAS ENOUGH." "THE NEXT STAGE WAS CALLED SHOWING THE INSTRUMENTS." "THE PRISONER WOULD BE SHOWN" "THE ACTUAL INSTRUMENTS OF TORTURE." "IT WAS A VERY IMPRESSIVE SIGHT," "AND IT ALMOST ALWAYS HAD THE DESIRED EFFECT." "YOU MAY BE WONDERING WHY I'M LECTURING TO YOU" "ON SUCH A BIZARRE SUBJECT." "WELL, THE PURPOSE IS TO INTRODUCE OUR NEXT TOPIC" "WHICH IS ABOUT THE USE OF FORCE." "I'LL SEE YOU THEN." "INUNIFORMCIRCULARMOTION," "THEVELOCITYVECTOR ISPERPENDICULAR" "TOTHERADIUSVECTOR," "ANDTHEACCELERATION IS INWARD ALONG THE RADIUS." "ACCORDINGTONEWTON'S SECONDLAW," "ANYBODYTHATEXECUTES UNIFORMCIRCULARMOTION" "ISBEINGACCELERATED BYSOMEEXTERNALFORCE." "FORA BODYIN ORBIT, THATFORCEIS GRAVITY." "CAPTIONING PERFORMED BY THE NATIONAL CAPTIONING INSTITUTE, INC." "CAPTIONS COPYRIGHT 1985 CALIFORNIA INSTITUTE OF TECHNOLOGY" "THE CORPORATION FOR COMMUNITY COLLEGE TELEVISION THE ANNENBERG/CPB PROJECT" "Annenberg Media" "¶" "For information about this and other Annenberg Media programs call 1-800-LEARNER and visit us at"