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Thesis Chapters by Joaquim Berenguer
Our research concerns eighteenth century Spain and is focused on the development and diffusion of... more Our research concerns eighteenth century Spain and is focused on the development and diffusion of Differential and Integral Calculus or Calculus of Fluxions, as it was known in Great Britain. The analysis of Differential Calculus, as a new field of mathematics in eighteenth century Europe is not a new topic, but it has often been used to approach this topic from the “centre” and via the well-known scientists Isaac Newton or Gottfried Wilhelm Leibniz. Our thesis focus on the reception of this new calculus into Spain, through Tomàs Cerdà (1715-1791).
Cerdà was a Catalan Jesuit devoted to the teaching of mathematics in Barcelona and Madrid, in the mid-eighteenth century, and who published several mathematicals texts and prepared many others for future publication. One of these manuscripts is a treatise on Differential Calculus, the Tratado de Fluxiones, which is a translation and an adaptation of another book, The Doctrine and Application of Fluxions (1750) by an English mathematician, Thomas Simpson (1710-1761). Thus, the main aim of our thesis centers on the analysis of the role of Cerdà in the introduction of Differential and Integral Calculus in eighteenth Century Spain, by means his Tratado de Fluxiones.
Cerdà appropriated Simpson’s fluxional calculus and managed it into a new shape, basically by thinking of his pupils. The main topics of our thesis are how and why Cerdà decided who would be his guide for the introduction of Differential and Integral Calculus and what his own contributions were to this work. To that end, our research is aimed at a better understanding of the process of popularisation of scientific knowledge, in such a way that it becomes an active part of the building process of this knowledge.
The development of Differential and Integral Calculus in the eighteenth century in Europe as a sc... more The development of Differential and Integral Calculus in the eighteenth century in Europe as a scientific discipline is not a new topic. But it has often used to focus the view of this topic from the “centre” and from the well known scientists as Isaac Newton or Gottfried Wilhelm Leibniz. The main character in this project maybe, for many people, is not a well known figure; Tomàs Cerdà is a teacher in Barcelona and in Madrid during the last half of the eighteenth century, who “translates” English authors to Spanish, but, with his practice, he is really introducing the new Calculus in Spain and giving orientation on this new discipline to his disciples. The main topics of our paper will be how and why Cerdà decides who will be his guide on the introduction of Differential and Integral Calculus and which will be his own contributions in this work. Then, our task has focused on a better understanding of popularisation process of scientific knowledge, always in a way where this one is an active part of the building process of this knowledge.
Book Reviews by Joaquim Berenguer
Conference Presentations by Joaquim Berenguer
After the expulsion of Jesuits from Spain in 1767, Spanish political power tried to fill the void... more After the expulsion of Jesuits from Spain in 1767, Spanish political power tried to fill the void which had appeared in Jesuits schools. Francesc Subiràs, who had been Cerdà's pupil in his Mathematics classes and was well-connected with King's court in Madrid, was nominated Director of the College of Cordelles. In 1770, Subiràs tried to carry out an ambitious reform of Education in the old Jesuit College, where Mathematics and the new experimental sciences would have a preponderant role. This project did not obtain approval of the Monarchy and Subiràs resigned his position as director and moved to Madrid. Study of the so-called " Memorial de Subiràs " enables us to understand the educational reality of the College of Cordelles before 1767 and to learn about the educational model that the scientific community shaped around the Physical-mathematical Conference in Barcelona. Moreover this reformist initiative and its subsequent failure enables us to set out some hypotheses concerning the evident conflict between the different political and economical interests.
Books by Joaquim Berenguer
Papers by Joaquim Berenguer
El càlcul diferencial i integral va aparèixer a finals del segle XVII de la mà de Newton i Leibni... more El càlcul diferencial i integral va aparèixer a finals del segle XVII de la mà de Newton i Leibniz i durant el segle XVIII el nou càlcul es va desenvolupar a través d'aquestes dues escoles diferenciades. En particular, aquest nou càlcul es va introduir a Espanya a mitjans del segle XVIII, i un dels seus introductors més destacats va ser el matemàtic i ensenyant Tomàs Cerdà (1715-1791), que va prendre com a guia a un altre matemàtic i ensenyant anglès Thomas Simpson (1710-1761). El paper de la pràctica docent en la configuració d'una disciplina científica és un tema tractat darrerament per diversos autors, així com existeix una abundant literatura sobre les interconnexions entre els dos corrents del càlcul durant els segles XVII i XVIII 1. No pretenem repetir el que ja ha estat analitzat sota diferents enfocaments sinó simplement presentar el cas de Cerdà i Simpson com un exemple d'aquests dos temes més generals. Per altra banda, hi ha treballs molt rellevants 2 sobre el desenvolupament del càlcul a Espanya, sobre Cerdà i sobre les institucions per on aquest es va moure, que han facilitat la nostra recerca 3. La nostra aportació, en aquest article, es situa en el marc d'una reflexió general sobre el paper dels ensenyants, com Simpson i Cerdà, en la configuració d'una nova part de la matemàtica com era el càlcul diferencial i integral en el segle XVIII. En aquest sentit, pretenem analitzar els textos d'aquests autors així com la influència dels diferents corrents del càlcul durant aquest segle sobre aquests i més concretament mostrar la influència de determinades nocions leibnizianes, com són els infinitèsims, en l'obra bàsicament newtoni
El desenvolupament del càlcul diferencial i integral com a disciplina científica a Europa durant ... more El desenvolupament del càlcul diferencial i integral com a disciplina científica a Europa durant el segle XVIII no és un tema nou. Però s'ha acostumat a enfocar la visió d'aquesta formació molt sovint des del "centre" i a partir de les grans figures com Isaac Newton o Gottfried Wilhelm Leibniz. En el present treball el protagonista potser, per a molts, no és una figura de primera línia; Tomàs Cerdà, és un ensenyant a Barcelona i a Madrid durant la segona meitat del segle XVIII, que "tradueix" al castellà autors anglesos, però que amb la seva pràctica està realment introduint el nou càlcul a Espanya i donant, de fet, una orientació d'aquesta nova disciplina als seus deixebles. El com i per què Cerdà decideix quin serà el seu guia en la introducció del càlcul diferencial i integral i quines seran les seves pròpies aportacions en aquesta labor seran els temes centrals del nostre treball. La nostra tasca ha anat, així doncs, a entendre millor, el procés d...
Cerda (1715-1791) fue un jesuita catalan dedicado a la ensenanza de las matematicas en Barcelona ... more Cerda (1715-1791) fue un jesuita catalan dedicado a la ensenanza de las matematicas en Barcelona y en Madrid a mediados del siglo XVIII, que publico diversos textos matematicos y tenia preparados otros muchos para una futura publicacion. Uno de estos manuscritos es un tratado sobre calculo diferencial, el Tratado de Fluxiones, que es una adaptacion de otro libro, The Doctrine and Application of Fluxions (1750) de un matematico ingles, Thomas Simpson (1710-1761). Cerda dio una determinada orientacion al texto de Simpson, pensando fundamentalmente en sus discipulos.
Aquesta tesi doctoral ha analitzat el paper de Tomas Cerda (1715-1791) -un jesuita dedicat a l... more Aquesta tesi doctoral ha analitzat el paper de Tomas Cerda (1715-1791) -un jesuita dedicat a l'ensenyament- en la introduccio del calcul diferencial i integral a Espanya, a partir de l'estudi del seu “Tratado de fluxiones”. L'autor mostra en quina mesura el van influenciar els dos corrents paral·lels de teories i metodes d'aquest nou camp matematic, desenvolupats per Newton (1671) i Leibniz (1684), i quines van ser-hi les seves propies aportacions.
Differential calculus was introduced into eighteenth-century Spain through the teaching of severa... more Differential calculus was introduced into eighteenth-century Spain through the teaching of several authors in different scientific institutions. One of the more noteworthy of these Spanish authors was the Jesuit Tomàs Cerdà (1715–1791), who taught mathematics at the College of Cordelles in Barcelona and at the Imperial College in Madrid. This mathematician introduced differential calculus through the manuscript entitled ‘Tratado de Fluxiones’ (1757–1759), which had as a main source The Doctrine and Application of Fluxions (1750) by Thomas Simpson (1710–1761). Our aim in this paper is to analyse Cerdà’s special contribution to the introduction into Spain of the Newtonian theory of fluxions based on Simpson’s definition of a fluxion. Specifically, the paper shows that Cerdà deduced the fluxion of the product of two variables and the area under a curve by previously establishing the fluxion of a curvilinear surface, a particular and different approach to that employed by other contempo...
Cerdà (1715-1791) was a Catalan Jesuit devoted to the teaching of mathematics in Barcelona and Ma... more Cerdà (1715-1791) was a Catalan Jesuit devoted to the teaching of mathematics in Barcelona and Madrid, in the mid-eighteenth century, and who published several mathematicals texts and prepared many others for future publication. One of these manuscripts is a treatise on Differential Calculus, the Tratado de Fluxiones, which is an adaptation of another book, The Doctrine and Application of Fluxions (1750) by an English mathematician, Thomas Simpson (1710-1761). Cerdà managed Simpson’s text into a new shape, basically by thinking of his pupils.
Our research concerns eighteenth century Spain and is focused on the development and diffusion of... more Our research concerns eighteenth century Spain and is focused on the development and diffusion of Differential and Integral Calculus or Calculus of Fluxions, as it was known in Great Britain. The analysis of Differential Calculus, as a new field of mathematics in eighteenth century Europe is not a new topic, but it has often been used to approach this topic from the “centre” and via the well-known scientists Isaac Newton or Gottfried Wilhelm Leibniz. Our thesis focus on the reception of this new calculus into Spain, through Tomàs Cerdà (1715-1791).
Cerdà was a Catalan Jesuit devoted to the teaching of mathematics in Barcelona and Madrid, in the mid-eighteenth century, and who published several mathematicals texts and prepared many others for future publication. One of these manuscripts is a treatise on Differential Calculus, the Tratado de Fluxiones, which is a translation and an adaptation of another book, The Doctrine and Application of Fluxions (1750) by an English mathematician, Thomas Simpson (1710-1761). Thus, the main aim of our thesis centers on the analysis of the role of Cerdà in the introduction of Differential and Integral Calculus in eighteenth Century Spain, by means his Tratado de Fluxiones.
Cerdà appropriated Simpson’s fluxional calculus and managed it into a new shape, basically by thinking of his pupils. The main topics of our thesis are how and why Cerdà decided who would be his guide for the introduction of Differential and Integral Calculus and what his own contributions were to this work. To that end, our research is aimed at a better understanding of the process of popularisation of scientific knowledge, in such a way that it becomes an active part of the building process of this knowledge.
The development of Differential and Integral Calculus in the eighteenth century in Europe as a sc... more The development of Differential and Integral Calculus in the eighteenth century in Europe as a scientific discipline is not a new topic. But it has often used to focus the view of this topic from the “centre” and from the well known scientists as Isaac Newton or Gottfried Wilhelm Leibniz. The main character in this project maybe, for many people, is not a well known figure; Tomàs Cerdà is a teacher in Barcelona and in Madrid during the last half of the eighteenth century, who “translates” English authors to Spanish, but, with his practice, he is really introducing the new Calculus in Spain and giving orientation on this new discipline to his disciples. The main topics of our paper will be how and why Cerdà decides who will be his guide on the introduction of Differential and Integral Calculus and which will be his own contributions in this work. Then, our task has focused on a better understanding of popularisation process of scientific knowledge, always in a way where this one is an active part of the building process of this knowledge.
After the expulsion of Jesuits from Spain in 1767, Spanish political power tried to fill the void... more After the expulsion of Jesuits from Spain in 1767, Spanish political power tried to fill the void which had appeared in Jesuits schools. Francesc Subiràs, who had been Cerdà's pupil in his Mathematics classes and was well-connected with King's court in Madrid, was nominated Director of the College of Cordelles. In 1770, Subiràs tried to carry out an ambitious reform of Education in the old Jesuit College, where Mathematics and the new experimental sciences would have a preponderant role. This project did not obtain approval of the Monarchy and Subiràs resigned his position as director and moved to Madrid. Study of the so-called " Memorial de Subiràs " enables us to understand the educational reality of the College of Cordelles before 1767 and to learn about the educational model that the scientific community shaped around the Physical-mathematical Conference in Barcelona. Moreover this reformist initiative and its subsequent failure enables us to set out some hypotheses concerning the evident conflict between the different political and economical interests.
El càlcul diferencial i integral va aparèixer a finals del segle XVII de la mà de Newton i Leibni... more El càlcul diferencial i integral va aparèixer a finals del segle XVII de la mà de Newton i Leibniz i durant el segle XVIII el nou càlcul es va desenvolupar a través d'aquestes dues escoles diferenciades. En particular, aquest nou càlcul es va introduir a Espanya a mitjans del segle XVIII, i un dels seus introductors més destacats va ser el matemàtic i ensenyant Tomàs Cerdà (1715-1791), que va prendre com a guia a un altre matemàtic i ensenyant anglès Thomas Simpson (1710-1761). El paper de la pràctica docent en la configuració d'una disciplina científica és un tema tractat darrerament per diversos autors, així com existeix una abundant literatura sobre les interconnexions entre els dos corrents del càlcul durant els segles XVII i XVIII 1. No pretenem repetir el que ja ha estat analitzat sota diferents enfocaments sinó simplement presentar el cas de Cerdà i Simpson com un exemple d'aquests dos temes més generals. Per altra banda, hi ha treballs molt rellevants 2 sobre el desenvolupament del càlcul a Espanya, sobre Cerdà i sobre les institucions per on aquest es va moure, que han facilitat la nostra recerca 3. La nostra aportació, en aquest article, es situa en el marc d'una reflexió general sobre el paper dels ensenyants, com Simpson i Cerdà, en la configuració d'una nova part de la matemàtica com era el càlcul diferencial i integral en el segle XVIII. En aquest sentit, pretenem analitzar els textos d'aquests autors així com la influència dels diferents corrents del càlcul durant aquest segle sobre aquests i més concretament mostrar la influència de determinades nocions leibnizianes, com són els infinitèsims, en l'obra bàsicament newtoni
El desenvolupament del càlcul diferencial i integral com a disciplina científica a Europa durant ... more El desenvolupament del càlcul diferencial i integral com a disciplina científica a Europa durant el segle XVIII no és un tema nou. Però s'ha acostumat a enfocar la visió d'aquesta formació molt sovint des del "centre" i a partir de les grans figures com Isaac Newton o Gottfried Wilhelm Leibniz. En el present treball el protagonista potser, per a molts, no és una figura de primera línia; Tomàs Cerdà, és un ensenyant a Barcelona i a Madrid durant la segona meitat del segle XVIII, que "tradueix" al castellà autors anglesos, però que amb la seva pràctica està realment introduint el nou càlcul a Espanya i donant, de fet, una orientació d'aquesta nova disciplina als seus deixebles. El com i per què Cerdà decideix quin serà el seu guia en la introducció del càlcul diferencial i integral i quines seran les seves pròpies aportacions en aquesta labor seran els temes centrals del nostre treball. La nostra tasca ha anat, així doncs, a entendre millor, el procés d...
Cerda (1715-1791) fue un jesuita catalan dedicado a la ensenanza de las matematicas en Barcelona ... more Cerda (1715-1791) fue un jesuita catalan dedicado a la ensenanza de las matematicas en Barcelona y en Madrid a mediados del siglo XVIII, que publico diversos textos matematicos y tenia preparados otros muchos para una futura publicacion. Uno de estos manuscritos es un tratado sobre calculo diferencial, el Tratado de Fluxiones, que es una adaptacion de otro libro, The Doctrine and Application of Fluxions (1750) de un matematico ingles, Thomas Simpson (1710-1761). Cerda dio una determinada orientacion al texto de Simpson, pensando fundamentalmente en sus discipulos.
Aquesta tesi doctoral ha analitzat el paper de Tomas Cerda (1715-1791) -un jesuita dedicat a l... more Aquesta tesi doctoral ha analitzat el paper de Tomas Cerda (1715-1791) -un jesuita dedicat a l'ensenyament- en la introduccio del calcul diferencial i integral a Espanya, a partir de l'estudi del seu “Tratado de fluxiones”. L'autor mostra en quina mesura el van influenciar els dos corrents paral·lels de teories i metodes d'aquest nou camp matematic, desenvolupats per Newton (1671) i Leibniz (1684), i quines van ser-hi les seves propies aportacions.
Differential calculus was introduced into eighteenth-century Spain through the teaching of severa... more Differential calculus was introduced into eighteenth-century Spain through the teaching of several authors in different scientific institutions. One of the more noteworthy of these Spanish authors was the Jesuit Tomàs Cerdà (1715–1791), who taught mathematics at the College of Cordelles in Barcelona and at the Imperial College in Madrid. This mathematician introduced differential calculus through the manuscript entitled ‘Tratado de Fluxiones’ (1757–1759), which had as a main source The Doctrine and Application of Fluxions (1750) by Thomas Simpson (1710–1761). Our aim in this paper is to analyse Cerdà’s special contribution to the introduction into Spain of the Newtonian theory of fluxions based on Simpson’s definition of a fluxion. Specifically, the paper shows that Cerdà deduced the fluxion of the product of two variables and the area under a curve by previously establishing the fluxion of a curvilinear surface, a particular and different approach to that employed by other contempo...
Cerdà (1715-1791) was a Catalan Jesuit devoted to the teaching of mathematics in Barcelona and Ma... more Cerdà (1715-1791) was a Catalan Jesuit devoted to the teaching of mathematics in Barcelona and Madrid, in the mid-eighteenth century, and who published several mathematicals texts and prepared many others for future publication. One of these manuscripts is a treatise on Differential Calculus, the Tratado de Fluxiones, which is an adaptation of another book, The Doctrine and Application of Fluxions (1750) by an English mathematician, Thomas Simpson (1710-1761). Cerdà managed Simpson’s text into a new shape, basically by thinking of his pupils.