Galileo and Huygens on free fall: Mathematical and methodological differences

Galileo and Huygens on ee all: Ma hema ical and me hodological i e ences S e en Ducheyne Cen e o Logic and Philosophy o Science and he Cen e o His o y o Science, Ghen Uni e si y. S e [email p o ec ed] Dynamis Fecha de ecepción: 23 de mayo de 2007 [0211-9536] 2008; 28: 243-274 Fecha de acep ación: 15 de no iemb e de 2007 SUMMARY: 1.—In oduc ion. 2.—Galileo’s ea men o ee all. 3.—Huygens’s ea men o ee all. 4.—Compa ing Galileo and Huygens. ABSTRACT: In his essay, I will sc u inize he di e ences be ween Galileo’s and Huygens’s de- mons a ions o ee all, which can be ound espec i ely in he Disco si and he Ho ologium, om a ma hema ical, ep esen a ional and me hodological pe spec i e. I a gue ha mo e can be lea n om such an analysis han he hesis ha Huygens e-s yled Galilean mechanics which is a communis opinio. I shall a gue ha he di e ences in hei app oach on ee all highligh a signi ican ly di e en ma hema ical and me hodological ou look. PALABRAS CLAVE: Huygens, Galileo, caida lib e, mecánica, iloso ía na u al del siglo XVII, Ho o- logium Oscilla o ium, Disco si. KEYWORDS: Huygens, Galileo, ee all, mechanics, XVII h cen u y na u al philosophy, Ho olo- gium Oscilla o ium, Disco si. 1. In oduc ion In his essay, I shall explo e he main ma hema ical and me hodological di e ences be ween Galileo’s and Huygens’s ea men o ee all. I is my aim o cla i y and compa e he me hod(ology) employed by Galileo and Huygens in dealing wi h ee all. When I use «me hod(ology)» he e, I in end o e e o he ways in which scien i ic s a emen s a e demons a ed in a published ex —such s a egies will ypically include ma hema ical and ep esen a ional echniques. I do no ouch upon he me hodology ollowed du ing he p ocess o disco e y o scien i ic s a emen s. Needless o say, he con ex o jus i ica ion does no necessa ily ollow he con ex o disco e y. S e en Ducheyne Dynamis 2008; 28: 243-274 244 Co espondingly, I shall ocus on bo h Galileo’s and Huygens’s published esul s on ee all: Disco si e dimons azione ma ema iche in o no a duo nuo e scienze (1638) and Ho ologium oscilla o ium seu de mo u pendulo um ad ho ologia ap a o demons a iones geome icae (1673), espec i ely. The ollowing p oposi ions (demons anda) will be s udied —I indica e hei occu ence in bo h Galileo’s and Huygens’s p incipal wo k on ee all: Demons andum Galileo’s 3 d day o he Disco si Huygens’s 2nd pa o he Ho- ologium Accele a ed mo ion Galileo’s de ini ion o accele a ed mo ion P oposi ion I Mean-speed heo em P oposi ion I P oposi ion II + P oposi ion V Times-squa ed ule P oposi ion II P oposi ion III Odd-numbe ule Co olla y I o P oposi ion I P oposi ion IV Equal-heigh -equal Veloci y heo em Scholium P oposi ion VI Time-leng h p opo ion- Ali y o mo ion along Inclined planes P oposi ion III P oposi ion VII No e ha Galileo de ined na u ally accele a ed mo ion, bu demons- a ed i only indi ec ly by means o he imes-squa ed law 1. In he Dis- co si —con a y o he Ho ologium— he e is no di ec demons a ion o na u ally accele a ed mo ion —only i s indi ec empi ical consequences. On all o he occasions, we can s aigh o wa dly compa e Galileo’s and Huygens’s in e en ial s a egies (see he able). Galileo and Huygens p o ed hese p oposi ions each in a signi ican ly di e en way. Huygens concei ed o his demons a ions as being mo e clea («cla ius») o be e («op imè») han hose o iginally gi en by Galileo in he Disco si. Huygens howe e ully 1. As Huygens w i es: «Quod Galileus p incipij si e hypo hesis loco adsumsi , unde deinceps p o- po ionem spa io um quae aequalibus empo ibus à caden e anseun u demons a um dedi .», Huygens, Ch is iaan. Oeu es complè es de Ch is iaan Huygens. Vol. 17, Den Haag: M. Nijho ; 1888-1950, p. 127 (emphasis added). Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 245 acknowledged Galileo as his p edecesso 2. Huygens e en claimed o annul his in en ion o w i e a book-leng h s udy o simila con en like Galileo’s Disco si, since he did no wan o compose he Iliad a e Home 3. Huygens’s p oposi ions on ee all a e men ioned and p esen ed in some le el o de ail by his o ians o science, bu I hink he e is mo e we can lea n om hese p oposi ions —especially on he me hodological di e- ences be ween Galileo and Huygens. Taci ly —o e en explici ly 4— mos his o ians o science p esuppose ha Huygens’s p oposi ions we e only a ende ing explici o Galileo’s implici assump ions. This is ue o some ex en . Howe e , behind Huygens’s a emp o make Galileo’s doc ine mo e explici also lie p o ound me hodological conside a ions. This is my main message. Co espondingly, I shall sc u inize he in e en ial s eps made by Galileo and Huygens in hei p oo s conce ning na u ally accele a ed mo ion. Se e al au ho s ha e only b ie ly commen ed on he di e ence be ween Galileo’s and Huygens’s ma hema ical app oach on ee all —Ch is iane Vilain is a no able excep ion o his 5. F ançois De Gand , o ins ance, no es ha Huygens wished o demons a e Galileo’s law o ee all «wi hou explici ly 2. Snelde s, H.A.M. Ch is iaan Huygens’ and New on’s heo y o g a i a ion. No es and Reco ds o he Royal Socie y o London. 1989; 43 (2): 209-222, p. 219. Huygens explici ly e e s o Galileo a se e al occasions: Blackwell, Richa d J. Ch is iaan Huygens’s he pendulum clock o geome ical demons a ion conce ning he mo ion o pendula as applied o clocks. Ames: The Iowa S a e P ess; 1986. p. 12, 40 and 42. Fo a gene al s udy o Huygens’s in ellec ual biog aphy John Bell’s wo k: Bell, A.E. Ch is iaan Huygens and he de elopmen o science in he Se en een h Cen u y. London: Edwa d A nold; 1947 is s ill aluable —i con ains ele an algeb aic ansc ip ions o some esul s o Huygens. Rienk Ve mij’s book is also o in e es : Ve mij, Rienk. Huygens: De ma hema ise ing an de we kelijkheid. Diemen: Veen; 2004. Un o una ely, his wo k is only accessible o Du ch eade s. Galileo’s concep ion o ela i e mo ion is also ac able in Huygens’s wo k, see: Pièces conce nan la ques ion du «mo emen absolu». In: Huygens, n. 1, ol. 17, p. 213-233, 222 and 232. Fo a ca e ul analy- sis, see Mo mino, Gian anco. Pene alia mo us. La ondazione ela i is ica della meccanica in Ch is iaan Huygens, con l’edizione del Codex Hugenio um 7 A, La Nuo a I alia: Fi enze; 1993; Vilain, Ch is iane. Huygens e le mou emen ela i . Ph. D. disse a ion. Uni e si é Pa is 7; 1993. 3. Huygens, n. 1, ol. 11, p. 72-73. In an ea ly manusc ip (1659) on ee all, Huygens w o e down se e al p oposi ions con aining some o he ma e ial pe aining o he second pa o he Ho ologium. See: Pièces co espondan à quelques pa ies de la pa s secunda de «l’Ho olo- gium Oscilla o ium» de 1673, in i ulée «De descensu g a ium & mo u eo um in cycloïde». In: Huygens, n. 1, ol. 17, p. 125-137. 4. E.g., Yode , Joella G. Un olling Time. Ch is ian Huygens and he ma hema iza ion o na u e. New Yo k: Camb idge Uni e si y P ess; 1988, p. 47. 5. Vilain, Ca he ine. La loi galiléenne e la dynamique de Huygens. Re ue d’his oi e des ma hé- ma iques. 1996; 2: 95-117. S e en Ducheyne Dynamis 2008; 28: 243-274 246 supposing he dependence be ween ime and he a ia ion o eloci y —he e en belie ed i possible o de i e demons a i ely he undamen al p ope y o hea iness, ha a each equal in e al o ime he e comes o be added an equal eloci y» 6. Michel Blay no es ha Huygens’s app oach was «Euclidean in inspi a ion» and elied on «classical p ocedu es o geome y and a oiding, in pa icula , ecou se o in ini e sums» 7. Huygens aim was o p esen a « econs uc ion o Galilean mechanics consis en wi h he equi emen s o igo en o ced by Euclidean geome y» 8. His econs uc ion eschewed Galileo’s new bu a he unde eloped ma hema ical echniques 9. In simila ashion, Joella G. Yode s a es ha he axioma ic s uc u e o geome y was he model o logical igou o Huygens 10. Huygens seemed o ha e a p e e ence o classical-geome ical in e en ial s a egies 11. How can hese be ap ly cha ac e ized? H.J.M. Bos has b ie ly cha ac e ized Huygens’s ma- 6. De Gand , F ançois. Fo ce and Geome y in New on’s P incipia, ansla ed by Cu is Wilson. P ince on/New Je sey: P ince on Uni e si y P ess; 1995, p. 114. See also Vilain, n. 5, p. 117. 7. Blay, Michel. Reasoning wi h he in ini e. F om he closed wo ld o he ma hema ical uni e se, ansla ed by M.B. DeBe oise. Chicago: The Uni e si y o Chicago P ess; 1998, p. 27-28; see also, p. 37. This does no en ail, o cou se, ha Huygens ne e employed in ini esimals o in ini e sums («in ini a conside a a mul i udine») in his ma hema ical p oo s. Yode , n. 4, p. x. Fo Huygens’s usage o limi ing p ocedu es, see especially Bos, H.J.M. Huygens and ma hema - ics. In: Fle che , K., ed. P oceedings o he In e na ional Con e ence TITAN, F om disco e y o encoun e , 13-17 Ap il 2004. Noo dwijk: ESTEC; 2004, p. 67-80. In De Vi Cen i uga (1659), o ins ance, his ea men o cen i ugal o ce is ho oughly in ini esimal. Idem o Huygens’s de i a ion o he isoch ony o he cycloid. Yode , n. 4, p. 19-22 and 48-64. Aan Elzinga no es ha Huygens allowed in ini esimals in he con ex o disco e y. Elzinga, Aan . Re iew o S udies on Ch is ian Huygens. The B i ish Jou nal o he Philosophy o Science. 1983; 34 (3): 295-303 (35). 8. Blay, n. 7, p. 33; see also p. 36. Fo an o e iew o Huygens’s mechanics, see Gabbey, Alan. Huygens and Mechanics. In: Bos, H. J. M. e al., eds. S udies on Ch is iaan Huygens. In i ed Pape s om he Symposium on he li e and wo k o Ch is iaan Huygens. Ams e dam, 22-25 Augus 1979. Lisse: Swe s and Zei linge ; 1980, p. 166-199. 9. See Bos, H.J.M. Huygens and Ma hema ics. In: Bos e al. n. 8, p. 126-146, o a p esen a ion o he de elopmen o Huygens’s ma hema ics. 10. Yode , n. 4, p. 172. 11. Tha is no o say ha expe imen s we e o lesse impo ance o Huygens. In his a emp s o calcula e he s eng h o su ace g a i y (measu ed by he dis ance o all in one second), expe imen s we e o u e impo ance, Yode , n. 4, p. 9-43. In Huygens’s na u al philosophy, a ional p ocedu es we e combined wi h expe imen al ones. As Huygens himsel w o e: «Cum expe ien ia ac a ione dep ehendissem une penduli ib a iones na u a sua inaequales esse i a u la io es angus io ibus paulo plus empo is impendan , indeque e o is aliquid in ho logijs, p aese im quae ela e is i mo en u neccesa io acccide e, quaesi i quo pac o co ige e illam inaequali a em possem». (quo ed om a le e o Leopold de Medici, 28 h No embe 1660), Huygens, n. 1, ol. 3, p. 197; emphasis added. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 247 hema ical s yle as ollows 12. Fi s , Huygens’s classicism a ou ed s ic ly logical a gumen s based on educ io ad absu dum (as a means o a oid limi a gumen s, i.e. ma hema ical a gumen in ol ing in ini esimals (see 3 and 4)). Howe e , wha Bos does no men ion, one should ca e ully dis inguish be ween educ io ad absu dum1 used o show he alsi y o a hypo hesis and educ io o absu dum2 used o es ablish he alsi y o a claim’s nega ion 13 (and, hence, his me hod es ablishes he u h o a claim indi ec ly: om «no -no -A») we conclude: «A») 14. This indi ec usage o educ ion, which is a oided by Euclid, is based on he excluded middle. Huygens used his ype o educ ion in cases whe e i was clea ha he e a e only wo logical op ions a hand. Secondly, Huygens ac ually hough geome ically, i.e. he ocused on he ela ions in he igu es and did no use o mulas. Finally, Huygens also p e e ed axioma isa ion. Le me gi e an o e iew o his essay. In 2, I discuss Galileo’s p oposi- ions on ee all ha we e men ioned in ee able; in 3, we shall look a he co esponding p oposi ions in Huygens’s ea men o ee all. The eade will no ice ha I shall begin by unning h ough he p oo s and hen des- c ibe hem on a me a-le el. These analyses will be he inpu o ou cu en endea ou : o compa e he in e en ial s a egies o Galileo and Huygens (4). I shall also u he expand on Huygens’s ea ly ma hema ical classicism and poin o i s in ima e connec ion wi h his p e e ence o a mo e igid me hodology han hypo he ico-deduc i ism, which Huygens endo sed la e in his li e. I shall also a gue ha Huygens’s heo e ical ame-wo k is mo e uni ied in wo senses: (a) a b oade domain o applica ion is in ended and (b) some in e en ial s a egies a e ypically ecu en . 2. Galileo’s ea men o ee all My aim in his sec ion is o analyse he p oposi ions men ioned in he able in sec ion 1. In his and he ollowing sec ion I will s ay mo e desc ip i e. Theo em I, P oposi ion I is he mean-speed heo em o Me onian ule which s a es ha he « ime in which any space is a e sed by a body 12. Bos, n. 9, p. 131-132. 13. As P o esso Geo ge E. Smi h poin ed ou o me in p i a e co espondence. 14. This p ocedu e was, as is widely known, se e ely c i icised by he in ui ionis s in ma hema ics (e.g., L.E.J. B ouwe ). S e en Ducheyne Dynamis 2008; 28: 243-274 248 s a ing om es and uni o mly accele a ed is equal o he ime in which ha same space would be a e sed by he same body mo ing a a uni o m speed whose alue is he mean o he highes speed and he speed jus be o e accele a ion began» 15. AB ep esen s he ime in which he space CD is a e sed (hence, he dis ance is he independen a iable 16) by a body, which s a s o all a es om C («Rep aesen e u pe exis ensionem AB empus in quo a mobile la ione uni o mi e accele a a ex quie e in C con icia u spa ium CD» 17). See igu e 1. The ho izon al, pa allel lines ep esen wha we would oday call he ins an aneous eloci y (o mo e p ecisely, «c escen es eloci a is g adus pos ins ans A» 18). The iangle and 15. Galilei, Galileo. Dialogues conce ning wo new sciences, ansla ed by Hen y C ew and Al onso de Sal io. New Yo k: Do e ; 1954, p. 173. 16. Dijks e huis ema ks ha O esme used he a e sed ime as he independen a iable. Dijks- e huis, E.J. De mechanise ing an he we eldbeeld. Ams e dam: Meulenho ; 1950. p. 257. 17. Galilei, Galileo. Le ope e di Galileo Galilei. Nuo a Ris ampa della Edizione Nazionale. Edi ed by An onio Fa a o. Vol. 8, Flo ence: Ba bè a; 1968. p. 208. 18. This no ion was ne e explici ly de ined by Galileo. Michel Blay w i es on Galileo’s no ion o deg ee o eloci y: «While o a ce ain ex en i p e igu ed he concep o ins an aneous eloci y, i none heless emained subjec o he Galilean way o concei ing mo ion, which ega ded eloci y as an ‘in ensi e magni ude’ inc easing by successi e addi ions o deg ees». Blay, n. 7, p. 72. Figu e 1. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 249 he ec angle ep esen he o e all momen um acqui ed in a ime-in e al [ , ’] du ing uni o mly accele a ed mo ion (whe e he g adus eloci a is con- s an ly inc eases) and du ing uni o m mo ion (whe e he g adus eloci a is emains he same) espec i ely 19. The ex p oceeds as ollows: «Since each and e e y ins an o ime in he ime-in e al AB, om which poin s pa allels d awn in and limi ed by he iangle AEB ep esen he inc easing alues o g owing eloci y, and since pa allels con ained wi hin he ec angle ep esen he alues o a speed which is no inc easing, bu cons an , i appea s, in like manne , ha he momen a [momen a] assumed by he mo ing body may also be ep esen ed, in he case o he accele a ed mo ion, by he inc easing pa allels o he iangle AEB, and, in he case o he uni o m mo ion, by he pa allels o he ec angle GB. Fo , wha he momen a may lack in he i s pa o he accele a ed mo ion ( he de iciency o he momen a being ep esen ed by he pa allels o he iangle AGI) is made up by he momen a ep esen ed by he pa allels o he iangle IEF» 20. The pa allels o «ins an aneous» speed a e con ained («comp ehensae» o «con en ae») in he iangle. The «agg ega e» o all pa allels con ained in AEB equals he «agg ega e» o he pa allels con ained in AGFB 21. The deg ees o speed ha he uni o m accele a ed mo ion lack a e made up du ing he second hal 22. The ela ion be ween uni o m mo ion and uni o mly accele a ed mo ion is es ablished by he equali y be ween he su aces which ep esen hem. Galileo p esupposed ha he equali y o he wo in ini e se s o momen s o eloci y es ablishes he equali y o he co esponding o e all speeds 23. Galileo lacked adequa e ools o deal wi h his ho oughly 24. An impo an implici p emise is he ma hema ical assump ion ha an a ea is made up o inde ini ely many lines. Le me sum up how Galileo ep esen ed uni o mly accele a ed mo ion: 19. Galilei, n. 15, p. 173. 20. Galilei, n. 15, p. 173-174. 21. Blay, n. 7, p. 74. 22. Dijks e huis, E.J. Val en wo p: Een bijd age o de geschiedenis an de mechanica an A is o eles o New on. G oningen: P. Noo dho ; 1924, p. 257. 23. Dame ow, Pe e e al. Explo ing he Limi s o P eclassical Mechanics. New Yo k: Sp inge ; 1992. p. 230. 24. Cla elin, Mau ice. La Philosophie Na u elle de Galilée. Pa ís: A mand Colin; 1968, p. 316. S e en Ducheyne Dynamis 2008; 28: 243-274 250 (1) AB, a line consis ing o an in ini e se o poin s, ep esen s he ime needed o a e se a dis ance CD; e e y poin co esponds o an ins an o ime; A ep esen s he s a ing poin ( 0); B ep esen s he end poin ( n) (2) CD ep esen s an a bi a y dis ance (hence, i is he independ- en a iable) (3) in ini esimal ho izon al lines ep esen s he (ins an aneous) c escen es g adus eloci a is (4) AEB ep esen s he o ali y ( o idem eloci a is momen a) o he inc easing alues o g owing eloci y (hence, he agg ega e o he g adus eloci a is) (5) AGFB ep esen s he o ali y o he cons an alues o speed (hence he agg ega e o he cons an speeds) The aim is o show ha , in equal imes, a uni o m mo ion wi h ½ o e- all momen um o an accele a ed mo ion will a e se he same dis ance (neglec ing a ha poin he ques ion i such mo ions eally exis ). This p oposi ion will be used as an in e ence- icke o p oxy in he ollowing p oposi ion, i.e. uni o mly accele a ed mo ion will be educed o he al eady sol ed p oblem o uni o m mo ion. Theo em II, P oposi ion II is he squa ed- ime law which s a es ha he «spaces desc ibed by a body alling om es wi h a uni o mly ac- cele a ed mo ion a e o each o he as he squa es o he ime-in e als employed in a e sing hese dis ances» 25. The uni s o ime (« luxus empo is») a e ep esen ed on AB; he dis an- ces h ough which a body alls wi h a uni o m accele a ion s a ing om es a e ep esen ed by HI. See igu e 2. Time AD co esponds o leng h HL, AE o HM, AF o HN and AG o HI. AC is cons uc ed a an a bi a y angle on AB («quemcunque angulum»). OD and PE ep esen he maximum speed a D and E. 25. Galilei, n. 15, p. 175-176. Figu e 2. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 251 The p oo p oceeds as ollows 26. F om he mean-speed heo em i ollows ha he dis ances HM and HL a e he same as hose ha would be a e sed du ing AE and AD by a uni o m mo ion wi h hal he speeds o hose by which DO and EP a e ep esen ed. Since a io AE is o AD as ½ EP is o ½ DO o as EP o DO, he eloci ies a e o each o he as he ime-in e als ( ~ ). Galileo eplaced he accele a ed mo ions by uni o m mo ions. F om Theo em IV, P oposi ion IV (in he sec ion on uni o m mo- ion) which s a es ha «i wo pa icles a e ca ied wi h uni o m mo ion, bu each wi h a di e en speed, he dis ances co e ed by hem du ing unequal in e als o ime bea o each o he he compound a io o he speeds and ime in e als», Galileo concludes: x ~ ( × ) 27. Hence, he a io o he spaces a e sed is he same as he squa ed a io o he ime-in e als (hence: x ~ ²). Again, Galileo used in o ma ion abou a simple si ua ion (uni o m mo ion) o a less simple si ua ion (accele a ed mo ion). Galileo hen a gued om his amous inclined plane expe imen s ha he na u al phenomena ag ee o his p oposi ion. Galileo seems, a leas in he p esen a ional o exposi ional pa o his heo y, no o spend much a en ion on he de ails o he expe imen s. Le me sum up: (1) AB, a line consis ing o an in ini e se o poin s, ep esen s he ime needed o a e se a dis ance HI; e e y poin co esponds o an ins an o ime; A ep esen s he s a ing poin ( 0); B ep esen s he end poin ( n); ime-in e als AD, AE, AF and AG co espond o dis ances HL, HM, HN and HI (2) OD and PE ep esen he g adus eloci a is a ins an s o ime D and E (3) HL, HM, HN, HI ep esen he dis ances a e sed in ime-in e als AD, AE, AF, AG The p oo o he odd-numbe ule is s a ed as a co olla y o he i- mes-squa ed ule (see igu e 3). AO ep esen s he ime measu ed om he ini ial poin A. The ho izon al lines BC, IF, OP ep esen he eloci y a he co esponding poin s C, I, O. As Galileo assumed, he eloci y is p opo ional o he ime elapsed. By he mean speed heo em we know 26. See also Wisan, Wini ed L. The new science o mo ion. A s udy o Galileo’s De Mo u Locali. A chi e o His o y o Exac Sciences. 1974; 13 (2-3): 103-306 (286-288). 27. Gailei, n. 15, p. 157. S e en Ducheyne Dynamis 2008; 28: 243-274 258 will be a e sed. A G, he o al eloci y is ound by adding he uni o m componen , which is equal o wice he eloci y acqui ed a B, and he g a i a ional componen (« is g a i a is»), equal o he speed acqui ed a B 48. Hence, he eloci y acqui ed a he hi d uni o ime is h ee imes he eloci y acqui ed a he i s uni o ime. And so o h o all ollowing ( ini e) uni s o ime. Hence, in each amoun o ime equal inc emen s o speed a e made 49. The a gumen goes as ollows 50: 1: x1 = AB, 1 2: x2 = BE, 2 = 2. 1 (= uni o m componen 1 + accele a ed componen 1) 3: x3 = EG, 3 = 3. 1 (= uni o m componen 2 + accele a ed componen 1) [...] Huygens’s demons a ion is essen ially a s ep-by-s ep decomposi ion o downwa d mo ion. P oposi ion II s a es a p o isional e sion o he mean-dis ance heo- em: «The dis ance c ossed in a ce ain ime by a body beginning o all om es is one-hal he dis ance which i would c oss in an equal ime wi h a uni o m mo ion whose eloci y is equal o he eloci y acqui ed 51 a he las momen o he all» 52. Assuming he p e ious igu e, Huygens a gues ha dis ance BD is wice AB. In he i s ou uni s o ime he dis ances AB, BE, EG, and GK a e a e sed. Dis ances AE and EK a e o each o he as AB o BE. F om his i ollows ha KE/EA = EB/AB = DA/AB 53. F om P oposi ion I, i ollows ha 48. Hence, i is also implici ly supposed ha all occu s in an emp y and homogeneous space, whe e he ac ion o g a i y is cons an . See Vilain, Ch is iane. Espace e dynamique chez Ch is iaan Huygens. De Ze en iende Eeuw: Cul uu in de Nede landen in in e disciplinai pe spec ie . 1996; 12 (1): 235-243 (p. 241). The assump ion ha g a i y ac s cons an is alse, see sec ion 4. 49. Huygens w i es « eloci a es pe aequalia empo a aequali e auge i». Huygens, n. 1, ol. 18, p. 129. 50. x s ands o he x h uni o ime, xx o he dis ance a e sed a e he x h uni o ime, and x o he eloci y acqui ed a he x h uni o ime. The gene al o ma o Huygens solu ion is: xn = ½ n . ( n - 1). BD + n . AB. Vilain, n. 48, p. 113. 51. The La in ex s a es «cum eloci a e quam acquisi i ». Huygens, n. 1, ol.18, p. 129. 52. Blackwell, n. 37, p. 36. 53. Huygens, o cou se, o mula es hese geome ical ela ions e ba im. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 259 KE = 2.AB + 5.BD. We also know ha EA = 2.AB + BD. Hence: KE – EA = 4.BD. F om his: DB/BA = 4.DB/EA. The e o e, EA will be ou imes BA, which equals 2.AB + BD, BD = 2.AB. This p oposi ion p esupposes a p o- po ion be ween he dis ances a e sed by a alling body in equal imes, a supposi ion which Huygens la e shows how o a oid in P oposi ion V 54. Le us un again h ough he p oo 55: (1) AE/EK = AB/ BE (by cons uc ion) (2) KE/EA = EB/AB = DA/AB ( om (1)) (3) KE = 2.AB + 5.BD (by cons uc ion; see igu e 7) (4) EA = 2.AB + BD (by cons uc ion; see igu e 7) (5) KE – EA = 4.BD ((3) & (4)) (6) DB/BA = 4.DB/EA (by cons uc ion we know ha EA = 4.BA) (7) EA = 4.BA (6) (8) BD = 2.AB ((4) & (7)) 56 P oposi ion III con ains a o mula ion o he imes-squa ed law: «I wo dis ances a e c ossed by a alling body in any imes, each o which is measu ed om he beginning o he all, hese dis ances a e ela ed o each o he as he duplica e a io o hese imes, o as he squa es o he imes, o as he squa es o he eloci ies acqui ed a he end o hese imes» 57. F om P oposi ion II i ollows ha dis ance BD is wice AB, dis ance BE is iple AB, dis ance EG i e imes AB, dis ance GK se en imes AB, and so on o he emaining dis ances. Hence, he dis ances a e sed a ime uni s 1, 2, 3, 4, … e c. inc ease acco ding o he p og ession o odd numbe s s a ing ab uni a e: 1, 3, 5, 7, … e c. I « he imes a e assumed 54. Blackwell, n. 37, p. 40 55. René Dugas w o e: «Nous ci ons ces démons a ions, pa ce qu’elles di è en quan au ond de celles de Galilée. Elles on en e e in in e eni , à chaque ins an , la composi ion de la i esse acquise e de la chu e nou elle du g a e.» Dugas, René. His oi e de la mécanique. Neu châ el: Edi ions du G i on; 1950, p. 176. 56. Fo he eade ’s con enience: DB/BA = 4.DB/EA. Since DB/BA = 4.DB/(2.AB+BD), 2.DB.AB + DB² = 4.DB.BA. Thus: DB² = 4.DB.BA – 2.DB.AB = 2.DB.BA. F om his, we ob ain: 2.AB = DB²/DB = DB. 57. Blackwell, n. 37, p. 36. S e en Ducheyne Dynamis 2008; 28: 243-274 260 o be commensu able» 58, he dis ances a e ela ed o each o he as he squa ed a io o he co esponding imes 59. Nex , shows ha his esul «is easy o ex en o incommensu able imes» (ibid.): (1) Le us suppose: E/F > AB²/CD² – see igu e 8. In his case: AB²/ CG² = E/F, whe e CG is smalle han CD. F om CD sub ac DH, which is smalle han DG, he excess o CD o e CG (ibid., p. 37). Le his be done in such a way ha HC is commensu able o AB. Then ob iously: CH > CG. The squa es o he imes AB and CH will be as he dis ance E s ands o he dis ance i would a e se in he ime CH. The dis ance F a e sed in ime CD is la ge han his dis ance. F om his, we ha e: E/F < AB²/CH². Hence, AB²/CG² < AB²/CH². F om his i ollows ha CH² < CG² (and hus: CH < CG), which yields an inconsis ency. The e o e, we ejec he hypo hesis. (2) In a simila ashion we can de i e an inconsis ency om he hypo hesis ha E/F < AB²/CD². Huygens concludes his p oposi ion wi h he wo ds: «Finally, since he eloci ies acqui ed a he end o he imes AB and CD a e ela ed o each o he in he same way as hese imes, i is ob ious ha E is ela ed o F by he same a io as he squa es o he imes AB and CD in which hey a e c ossed» 60. The s uc u e o his p oo is: (1) BD = 2.AB (P oposi ion II) (2) BE = 3.AB (by idem) (3) EG = 5.AB (by idem) 58. Blackwell, n. 37, p. 37. The Encyclopaedia o Ma hema ics s a es ha wo magni udes o he same kind a e commensu able, i hey ha e a common measu e (i.e. a magni ude o he same kind con ained in an in eg al numbe s o imes in bo h o hem). I wo magni udes a e com- mensu able, hen hei a io is a a ional numbe (i no , hen i is an i a ional numbe ). See Hazewinkel, Michiel, ed. Encyclopaedia o Ma hema ics. Vol. 1, Do d ech /Bos on/London: Kluwe ; 1995, p. 714. 59. Huygens no es: «And since any sum o hese numbe s [i.e., 1, 3, 5, 7, … e c.], aken consecu- i ely, makes a squa e whose side equals he numbe o numbe s aken ( o example, i he i s h ee a e added, hey make nine; i ou six een), i ollows om his ha he dis ances c ossed by a alling body, each o which is aken om he beginning o he all, a e ela ed o each o he as he duplica e a io o he imes du ing which he all occu s, […]» Blackwell, n. 37, p. 37. 60. Blackwell, n. 37, p. 38. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 261 (4) GK = 7.AB (by idem) […] I we assume ha he imes a e commensu able, i ollows ha : x1/x2 = 1²/ 2² Tha he claim holds when he imes a e incom- mensu able can be by he ollowing educ io ad ab- su dum: (1) E/F > AB²/ CD² (ex hypo hesi) 61 (2) AB²/CG² = E/F, whe e CG < CD (by (1)) (3) DG = CD – CG, whe e HC is commensu able o AB (by (2)) (4) CH > CG (by (3)) (5) E/F < AB²/CH² (by (2) & (4)) (6) AB²/CG² < AB²/CH² (by (2) & (5)) (7) CH² < CG² ( om which i ollows: CH < CG) (by (6)) (8) Hence, we ejec E/F > AB²/ CD² (9) Finally: E/F = AB²/CD² (x1/x2 = 1²/ 2²) P oposi ion IV goes as ollows: «I a hea y body begins o mo e upwa ds wi h he same eloci y acqui ed a he end o a descen , hen in equal pa s o ime i will c oss he same dis ances upwa ds as i did downwa ds, and i will ise o he same heigh om which i descended. Also in equal pa s o ime i will lose equal amoun s o eloci y 62». This amoun s o p o ing ha in as many equal imes as he dis ances AB, BE, EG, and GK a e a e - sed by a body which alls om A, he same dis ances KG, GE, EB, and BA a e a e sed successi ely by he 61. The p oo can easily be cons uc ed o he hypo hesis: E/F < AB²/CD². 62. Blackwell, n. 37, p. 38. Figu e 8. Sou ce: Huy- gens, 1673, p. 26. S e en Ducheyne Dynamis 2008; 28: 243-274 262 same body when i mo es upwa ds beginning wi h he eloci y acqui ed a K (a e ee all om A) – see igu e 8. Huygens no es ha « o he sake o b e i y each eloci y 63 will be successi ely designa ed by he leng h o he dis ance c ossed by a body in uni o m mo ion wi h ha eloci y in one pa o ime» 64. When a body a i es a K, i has acqui ed eloci y KF (= GH + BD). I his eloci y is di ec ed upwa ds i will a e se he dis ance KF in one uni o ime. I we ake in o accoun he «ac ion o g a i y», his dis ance will be dec eased by FG (= AB) 65. The body ises only o G. A G he emaining eloci y is HG (= GD). In he second uni , o ime he body would a e se GD, om which we need o sub ac ED, which equals he ac ion o g a i y. A E, he emaining eloci y is FE (= GD – BD). I ha body mo es u he upwa ds (in he hi d uni o ime), by i s uni o m mo ion dis ance EA would no mally be a e sed in one uni o ime. F om EA we s ill need o sub ac he ac ion o g a i y, i.e. AB. The esul is ha he body will ise o B. In he ou h uni o ime, he body inally eaches A and no eloci y is le . The body does no mo e highe . F om his i ollows ha « he body ises o he same heigh om which i ell, and ha each dis ance c ossed in equal imes o descen is equally measu ed o in as many equal imes o ascen » 66. The s uc u e o P oposi ion IV is: Gi en: a K alling body’s eloci y is KF (= GH + BD) 1: when eloci y KF (= GH + BD) is di ec ed upwa ds: he body ises o G a G he emaining eloci y is HG (= GD) 2: when eloci y HG is di ec ed upwa ds: he body ises o E a E he emaining eloci y is FE (= GD – BD) 3: when eloci y FE is di ec ed upwa ds: he body ises o B a B he emaining eloci y is AB (= BD – AB) 4: 63. Wes all no es ha Huygens’s diag ams, con a y o Galileo’s, p esen ed he eloci ies and only inciden ally he pa hs; eloci y eme ged mo e clea ly han in Galileo’s mechanics as a physical quan i y. Wes all, Richa d. Fo ce in New on’s physics: The science o dynamics in he se en een h cen u y. Do d ech /Bos on/London: Else ie ; 1971, p. 153. 64. Blackwell, n. 37, p. 38. 65. Blackwell, n. 37, p. 38. 66. Blackwell, n. 37, p. 49. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 263 when eloci y AB is di ec ed upwa ds: he body ises o A a A emaining eloci y is ze o (AB – AB) No ice ha Huygens p o es his p oposi ion by illus a ing i wi h a case wi h ou uni s o ime. Ob iously, he demons a ion applies o any ini e se o subsequen uni s o ime. P oposi ion V con ains a new p oo o he mean-dis ance heo em, which Galileo ga e «in a less pe ec o m» 67: «The dis ance c ossed in a ce ain ime by a body which begins i s all om es is hal he dis ance which i would c oss in an equal ime wi h a uni o m mo ion ha ing he eloci y acqui ed a he las momen o he all 68». Le AH ep esen he o al ime o all and AC, CE, EG, … e c. he equal pa s o ime (see igu e 9). In AH a mo ing body a e ses a dis ance whose quan i y is ep esen ed («designe u ») by he plane P. HL ep esen s he e minal eloci y acqui ed a he end o he all («cele i a em in ine casus acquisi am»). AHLM ep esen s he dis ance c ossed in ime AH wi h eloci y HL. We need o show ha P is ½ AHML o ha P equals AHL. We p o e his by educ io ad absu dum 69. I P is no equal o ½ MH o AHL, hen i is ei he smalle o g ea e . Le us examine bo h cases. Keep in mind ha he dis ances a e ep esen ed by means o su aces. (1) Assume ha P is smalle han AHL. Le AH be di ided by a numbe o equal pa s AC, CE, EG, … e c. Then cons uc he ci cumsc ibed igu e ha is composed o ec angles whose al i udes equal each pa o he di ision o AH, namely he ec angles BC, DE, FG, … e c. Also cons uc wi hin he 67. Blackwell, n. 37, p. 40. Huygens no es ha he p oo o he mean-dis ance heo em in P opo- si ion II was based on he supposi ion ha he e is a p opo ion be ween he dis ances a e sed by alling bodies. Huygens ema ks: «This indeed mus be so because o he na u e o he way ha hings a e ela ed o each o he , and i his is denied, i mus be admi ed ha i is useless o sea ch o a p opo ion be ween hese dis ances». Blackwell, n. 37, p. 40. The mean-dis ance heo em can also be p o ed wi hou his supposi ion by using Galileo’s me hod («Galilei me hodum sequendo»). Huygens concludes: «Hence i will be a wo hwhile e o o w i e down he e mo e accu a ely he demons a ion which he ga e in a less pe ec o m». Blackwell, n. 37, p. 40. 68. Blackwell, n. 37, p. 40. 69. Michel Blay no es ha : «Huygens’ s a egy, hough i did in ol e he p opo ionali y o speed o ime, was easible only o he ex en ha i immedia ely subs i u ed dis ances o ime. Huygens’ easoning was, in a manne o speaking, s a ic». Blay, n. 7, p. 36. S e en Ducheyne Dynamis 2008; 28: 243-274 264 iangle an insc ibed igu e composed o ec angles o he same al i ude, namely he ec angles KE, OG, … e c. All his is done so ha he excess (equal o he lowes ec angle wi h base HL) o he ci cumsc ibed igu e o e he insc ibed igu e is less han he excess o AHL o e P. F om his, i ollows ha he excess o AHL o e he insc ibed igu e will be less han i s excess o e P. In his case, he insc ibed igu e is la ge han P. Since, by P oposi ion I, we know ha he eloci ies o alling bodies a e p opo ional o he imes o all, CK is he eloci y acqui ed a he end o he i s uni o ime, o AH/AC = HL/CK. Simila ly, EO is he eloci y acqui ed a he end o he second uni o ime. In he i s ins an o ime, a dis ance g ea e han ze o is a e sed. In he second uni o ime, a dis ance g ea e han KE is a e sed, since du ing CE dis ance KE would be a e sed by a uni o m mo ion wi h he eloci y CK, which is equal o he uni o m componen o which he ac ion o g a i y s ill needs o be added. Simila ly, du ing EG a dis ance g ea e han OG is a e sed. And so on o all successi e imes. Hence, he o al dis ance c ossed by an accele a ed mo ion will be g ea e han he insc ibed igu e. Tha dis ance was ab ini io assumed o be equal Figu e 9. Sou ce: Huygens, 1673, p. 29. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 265 o he plane P. Hence, he insc ibed igu e will be smalle han dis ance P. Thus, he plane P is no smalle han AHL. Ou ini ial hypo hesis leads o an inconsis ency and needs o be ejec ed. (2) Assume ha P is la ge han AHL. The excess o he ci cumsc ibed igu e o e he insc ibed igu e is less han he excess o P o e AHL. Hence, he ci cumsc ibed igu e will be less han plane P. In he i s uni o ime AC, he dis ance c ossed by an accele a ed mo ion is less han BC, because ha dis ance would be c ossed in he same ime wi h he uni o m eloci y CK which he body acqui es only a he end o ime CE. Simila ly, du ing CE a dis ance less han DE is a e sed (because i would be c ossed in he same ime CE wi h he uni o m eloci y EO which i acqui es only a he end o ime CE). And so on o all successi e imes. Hence, he whole dis- ance c ossed by an accele a ed mo ion will be less han he ci cumsc ibed igu e. Bu ha dis ance was ab ini io assumed o be equal o he plane P. Hence, he insc ibed igu e will be smalle han plane P. Thus, he plane P is no la ge han AHL. Ou ini ial hypo hesis leads o an inconsis ency and needs o be ejec ed. Since we ha e shown ha plane P is no la ge and no smalle han AHL, i ollows ha bo h mus be equal. The s uc u e o his p oo is he ollowing: Le us assume ha in (AH) a dis ance is a e sed ep esen ed by he plane P, ha HL ep esen s he e minal eloci y a he end o all along AH, and ha AHLM ep esen s he dis ance c ossed in ime AH wi h uni o m eloci y HL. We wan o p o e: P = ½ AHML = AHL. Suppose P ≠ ½ AHML ≠ AHL, hen wo op ions ((α) & (β)) a e open: (α) P < AHL (ex hypo hesi) (1) (a ea ci cumsc ibed igu e – a ea insc ibed igu e) < (AHL – P) (by cons uc ion) (2) (AHL – a ea insc ibed igu e) < (AHL – P) (by (1)) (3) a ea insc ibed igu e > P (by (2)) (4) 1: a dis ance g ea e han ze o is a e sed (by P oposi ion I) 2: a dis ance g ea e han KE is a e sed (by idem) 3: a dis ance g ea e han OG is a e sed (by idem) […] S e en Ducheyne Dynamis 2008; 28: 243-274 266 n: a dis ance g ea e han he g ea es ec angle o he insc ibed igu e is a e sed (by idem) 70 (5) Hence: whole dis ance c ossed by an accele a ed mo ion (= P) > insc ibed igu e (by [4]) (6) Hence: a ea insc ibed igu e < P (in con adic ion wi h (3)) (7) Finally, we ejec P < AHL (β) P > AHL (1) (a ea ci cumsc ibed igu e – a ea insc ibed igu e) < (P – AHL) (by cons uc ion) (2) a ea ci cumsc ibed igu e < P (by (1)) (4) 1: a dis ance smalle han BC is a e sed (by P oposi ion I) 2: a dis ance smalle han DE is a e sed (by idem) […] n: a dis ance smalle han he g ea es ec angle o he ci cums- c ibed igu e is a e sed (by idem) (5) Hence: whole dis ance c ossed by an accele a ed mo ion (= P) < ci cumsc ibed igu e (by (4)) (6) Hence: a ea ci cumsc ibed igu e > P (in con adic ion wi h (2)) (7) Finally, we ejec P > AHL Since bo h op ions a e un enable, we conclude P = ½ AHML = AHL. P oposi ion VI —o which «Galileo asked ha we accep is as in a sense being sel -e iden » 71 (ibid., p. 42)— can easily be de i ed: «The eloci ies acqui ed 72 by bodies alling h ough a iably inclined planes a e equal i he ele a ions o he planes a e equal» 73. 70. The e is no ma hema ical induc ion he e. Huygens cons uc ed his p oo wi h a ini e amoun o s eps p ecisely in o de o e ade Galileo’s p eca ious assump ion o in ini esimals. 71. The La in ex eads «u quodammodo pe se mani es am, Galileus pos ula i ». Huygens, n. 1, ol. 18, p. 141. E en Galileo’s la e addi ion o he scholium in he edi ion o 1654 could no con ince Huygens. Blackwell, n. 37, p. 42-43. 72. In a manusc ip om 1659 —Huygens’ annis mi abilis— Huygens used he Galilean e m «g adus eloci a is». Huygens, n. 1, ol. 17, p. 131. 73. Blackwell, n. 37, p. 43. Galileo and Huygens on ee all: Ma hema ical and me hodological di e ences Dynamis 2008; 28: 243-274 267 Le a body oll down om he inclined planes AB and CB, he heigh s o which AE and CD a e equal —see igu e 10. In bo h cases « he same deg ee o eloci y will be acqui ed» («eundem g adum eloci a is acqui- si u um») 74. I along an inclined plane CB, less eloci y han along AB we e o be acqui ed, he eloci y acqui ed along CB would be he same as on an a bi a y FB which has a heigh less han AE. F om P oposi ion IV, i ollows ha he eloci y acqui ed along CB is equi ed o make he body ascend h ough he whole o BC. I we hen suppose ha he all along FB is con inued h ough BC, «which i could do by e lec ion in he oblique di ec ion» 75, i would mo e up o C, i.e. up o a poin highe han he place om which i ell. This assump ion is absu d —since i iola es To icelli’s p inciple 76, which s a es ha he cen e o g a i y canno aise abo e i sel 77. Huygens inally no es ha : 74. Blackwell, n. 37, p. 43. 75. Blackwell, n. 37, p. 43. 76. See Lo ia, Gino; Vassu a, Giuseppe, eds. Ope e di E angelis a To icelli. Vol 2, Faenza: S abilimen o Tipo-li og a ico G. Mon ana i; 1919, p. 105, o To icelli’s own o mula ion. I am indeb ed o P o esso Geo ge E. Smi h o his e e ence. 77. See Huygens, n. 1, ol. 17, p. 132, 4n; Blackwell, n. 37, p. 108-109. Figu e 10. Sou ce: Huygens, 1673, p. 32. S e en Ducheyne Dynamis 2008; 28: 243-274 274 po ionali y o mo ion along e ical and inclined planes o mo ions along all cu es. Huygens’s p oposi ions, he e o e, applied o a g ea e domain, while Galileo’s p oposi ion had a mo e es ic ed scope 94. Acknowledgmen s The au ho wishes o hank Fabien Cha eix, Gian anco Mo mino. E ic Schliesse , Joella G. Yode , and Ch is iane Vilain o hei ad ice and guidance when wo king on his pape . He is indeb ed o Rienk Ve mij o se e al commen s and specially o Geo ge E. Smi h, o whom he had he absolu e honou o ecei e a co nucopia o use ul eedback, ema ks and c i icisms. This essay is dedica ed o he incompa able M . Smi h. ❚ 94. The ollowing s udies ha e helped he au ho a lo : Schliesse , E ic; Smi h, Geo ge E. Huygens’s 1688 Repo o he Di ec o s o he Du ch Eas Indian Company on he measu emen o longi ude a sea and he e idence i o e ed agains uni e sal g a i y. A chi e o His o y o Exac Sciences ( o hcoming). Cha eix, Fabien. La pésan eu dans l’uni e s méchanique de Ch is iaan Huygens. De Ze en iende Eeuw 1996; 12 (1): 244-252. Cha eix, Fabien. Expé ience e aison, la science chez Huygens (1629-1695). Re ue d’His oi e des Sciences. 2003; 56 (1): 79-112.