L’Hôpital’s Mmara no mfiase

Wɔwoo Guillaume-François-Antoine Marquis de l’Hôpital, Marquis de Sainte-Mesme, Comte d’Entremont et Seigneur d’Ouques-la-Chaise, a nkurɔfo frɛ no Guillaume L’Hôpital, wɔ 1661 mu wɔ Paris maa abusua bi a na wɔwɔ asraafo agyapade a tumi wom. Nanso, nea ɛne n’abusua apɛde ne adwene a na nnipa pii wɔ sɛ wɔyɛ adehye wɔ France bɔ abira no, na n’ani gye akontaabu ho fi ne mmofraase. Bere a na ɔrekɔ sraadi no, ɔyɛɛ ne ho sɛ ɔregye n’ahome wɔ ne ntamadan mu na osuaa geometry mmom. Bernard de Fontenelle kyerɛw ne ho asɛm wɔ n’ayeyi a ɔde maa L’Hôpital no mu sɛ:

Efisɛ ɛsɛ sɛ wogye tom sɛ ɛwom sɛ Franse man no wɔ suban pa te sɛ ɔman foforo biara de, nanso ɛda so ara wɔ saa atirimɔdensɛm a ɛnam so susuw ho sɛ ebia nyansahu ahorow a wɔde kɔ baabi pɔtee bi no ne anuonyamfo nhyia, ne sɛ ebia ɛnyɛ anuonyam kɛse sɛ wunnim hwee anaa. ... M’ankasa mahu wɔn a wɔsom wɔ bere koro no ara mu no bi, a wɔn ho dwiriw wɔn kɛse sɛ ɔbarima bi a ɔtraa ase te sɛ wɔn yɛ akontaabufo a wodi kan wɔ Europa no mu biako.

L’Hôpital gyaee Franse asraafo no esiane sɛ na onhu ade yiye nti, ɛwom sɛ na wɔka sɛ ɔpɛ sɛ ɔyɛ akontaabu bere nyinaa ara kwa de. Afei a wadi mfe aduonu anan no, ɔkɔɔ Congregation of the Oratory a ɛwɔ Nicolas Malebranche’s circle (kuw a wɔboaboa wɔn ho ano ma nkɔmmɔbɔ ne fekubɔ,) a na Paris akontaabufo ne nyansahufo atitiriw pii na wɔte mu no. Ɛhɔ no, ohyiaa Johann Bernoulli, Jakob Bernoulli a na ɔkyerɛkyerɛ Leibniz wɔ ne mmeranteberem na wobu no sɛ akontaabu mu ɔbenfo dedaw no nuabarima kumaa na ɔyɛ ɔkwasea kɛse. Ná L’Hôpital yɛ Bernoulli sukuuni a n’ani gye ho sen biara na ankyɛ na otuaa no ka sɛ ɔnkyerɛkyerɛ no kokoam, mmom.

L’Hôpital de ɔhaw ano aduru bi a efi adesua a Bernoulli de maa no no mu kɔmaa Christiaan Huygens a wanka sɛ ɛnyɛ n’ankasa de. Ntease wom sɛ esiane sɛ na Huygens nni adanse biara a ɛne eyi bɔ abira nti, ɔfaa no sɛ L’Hôpital na ɛyɛɛ saa. Bernoulli bo fuwii na ɔtwaa ne nkrataa a ɔtaa de kyerɛw ne L’Hôpital no mu asram asia–nanso ɔbubuu ne kommyɛ no bere a L’Hôpital srɛɛ no ​​sɛ ɔmma no “nneɛma a wɔahu” pii wɔ retainer a emu duru yɛ nkaribo ahasa (na ɛrenya nkɔanim) so no. Ɔsrɛɛ ne kyerɛkyerɛfo no nso sɛ ɔmma no hokwan ahorow a ɛyɛ soronko wɔ ne nkɔso ne ɔkasa ahorow ho. Bernoulli buae ntɛm ara sɛ sɛ L’Hôpital pɛ a, ɔrentintim biribiara bio wɔ n’asetra mu.

L’Hôpital de nea Bernoulli hui ne nsɛm a ɔkyerɛw fii ne kasa ahorow mu no tintim nea ɛbɛyɛ akontaabu nhoma a edi kan: Analyse de infiniment petits pour l’intelligence des lignes courbes (Analysis of Infinitely Small Quantities for the Understanding of Curves.) Wɔ mu no, ɔkyerɛ sɛnea wɔbɛsɔ anohyeto ahorow a anka wontumi nsi ho gyinae ahwɛ:

1. Ma kwan sɛ wobetumi afa (anaasɛ wɔde adi dwuma) dodow abien, a wɔn nsonsonoe yɛ dodow ketewaa bi a enni ano no anibiannaso so ama wɔn ho wɔn ho; anaasɛ (a ɛyɛ ade koro) sɛ wobetumi abu dodow bi a ɛkɔ soro anaasɛ ɛso tew dodow ketewaa bi a enni ano nkutoo sɛ ɛda so ara yɛ nea ɛte saa ara.

2. Ma kwan sɛ wobetumi abu curve sɛ nsensanee tẽẽ nketewa a enni ano dodow a enni ano a wɔaboaboa ano; anaasɛ (a ɛyɛ ade koro) sɛ polygon a ɛwɔ afã dodow a enni ano, a emu biara yɛ ketewaa a enni ano, a ɛnam anim a wɔne wɔn ho wɔn ho yɛ so kyerɛ sɛnea curve no bɛbɔ.

Ɛwom sɛ wɔamfa ankyerɛ wɔ ɔkwan a ɛfata so te sɛ nnɛyi calculus adesua nhoma ahorow mu de, nanso wɔ Stewart’s Calculus: Early Transcendentals Ɔfã 4.4 a ɛkyerɛkyerɛ mu no:

sɛnea L’Hôpital (a wɔafa aka wɔ nhoma no mu sɛ L’Hospital) mmara no, ne mfitiase asɛm no ne nnɛyi nsɛm a wɔasan aka no yɛ pɛ wɔ adwene mu. Sɛ L’Hôpital ka nsonsonoe nketenkete a enni ano ho asɛm a, eyi ne anohyeto ahorow a wɔde gyina hɔ ma no di nsɛ. Adwene a ɛne sɛ “ntrɛwmu nketewa a enni ano a ɛyɛ tẽẽ” no gyina hɔ ma geometric ntease a ɛfa nsonsonoe ho na ɛyɛ yɛn mprempren adwene a ɛfa nea efi mu ba ho no nana. Sɛ yɛka ne nyinaa bom a, sɛdeɛ ɛteɛ wɔ Ɔfa 4.4., L’Hôpital mfitiaseɛ nsusuiɛ no ka sɛ wɔbɛtumi adi nsusuiɛ a ɛnni ano ho dwuma denam dwumadie ahodoɔ no nsakraeɛ dodoɔ a wɔbɛhunu so.

Wɔn a wɔwɔ tema ma Johann Bernoulli ka sɛ wɔhyɛɛ no ​​ma ɔbrɛɛ ne ho ase maa anuonyamfo apɛde. Ɛmfa ho sɛ Bernoulli penee so mfiase no esiane sikasɛm mu abasamtu nti no, nhyehyɛe no kɔɔ so bere tenten kosii sɛ ɔyɛɛ ɔbenfo a odii yiye wɔ Groningen no. Bernoulli kae sɛ L’Hôpital nhoma no yɛ “ne de titiriw” bere a ne kan sukuuni no wu akyi nkutoo. Saa bere no, na Bernoulli din ayɛ kusuu bere a ɔne ne nuabarima panyin dii akasakasa mpɛn pii akyi. Saa bere no, na ɛyɛ gyinapɛn ma adehye sɛ wobetua nnwuma a efi adwumayɛfo a wɔwɔ tumi kɛse te sɛ amammuifo ne mmaranimfo hɔ ho ka, na na nnipa pii buu L’Hôpital sɛ akontaabufo a ne ho akokwaw wɔ n’ankasa mu.

Ade biako a wodii kan gyee akyinnye wɔ L’Hôpital adwuma no mudi mu kura ho ne brachistochrone haw no ano aduru a odii (a Johann Bernoulli de too gua wɔ 1696 mu, ɔhaw a ɛfa ɔkwan a wɔfa so sian ntɛmntɛm sen biara ho):

Ɔhaw Foforo a Wɔto nsa frɛ Nkontaabufo sɛ Wonni Ho Dwuma: Sɛ wɔde nsɛntitiriw abien A ne B ma wɔ wimhyɛn a ɛda fam mu a, sɛ wɔde ɔkwan AMB a, sɛ ɛsiane wɔ n’ankasa mu duru ase a, ɛtwam fi beae A kɔ beae B wɔ bere tiaa mu no mma abɔde ketewaa bi a ɛkɔ baabiara M.

Wɔkyerɛe sɛ mmuae a L’Hôpital de mae wɔ asɛmmisa no ho no nyɛ n’ankasa de, ebia na ne kyerɛkyerɛfo Bernoulli ankasa de.

Awiei koraa no, na L’Hôpital ho akokwaw wɔ Johann Bernoulli nkyerɛkyerɛ a wɔde bɛka abom mu na otintim opus a ɛho hia wɔ calculus a ɛrenya nkɔso ntɛmntɛm no mu, na ɛmaa atiefo pii tumi nyaa nkɔso ahorow. Nanso, n’adwuma no rentumi nkura mprempren gyinapɛn ahorow a ɛfa adesua mu nokwaredi ho no mu, na yebetumi aka sɛ ɔde ne sikasɛm mu gyinabea dii dwuma ɔkwammɔne so de bɛyɛɛ obi a wagye din wɔ nhomasua mu wɔ afeha a ɛto so dunwɔtwe France a na n’atipɛnfo no anni nneɛma foforo ankasa.

References

“Acta Eruditorum. 1696.” Internet Archive, Lipsiae : Apud J. Grossium et J.F. Gletitschium, 1 Jan. 1696, archive.org/details/s1id13206630.

Katz, Victor J. A History of Mathematics. 3rd ed., Pearson Education Limited, 2014.

L’Hospital, Guillaume François Antoine De, and M. Varignon. Analyse Des Infiniments Pettits, Pour l’intelligence Des Lignes Courbes. ALL-Éditions, 1988.

O’Connor, J J, and E F Robertson. “Guillaume François Antoine Marquis de L’Hôpital.” Maths History, University of St. Andrews School of Mathematics and Statistics, Dec. 2008, mathshistory.st-andrews.ac.uk/Biographies/De_LHopital/.

Stewart, James. Calculus: Early Transcendentals. Vol. 8.