Irrationality and transcendence in the 18th and 19th centuries. A contextualized study of J. H. Lambert's Mémoire (1761/1768)

The object of this dissertation is to carry out a historical study on the development of irrational numbers and the concept of transcendence in mathematics during the 18th and 19th centuries. Although many authors are analyzed, the central character of the study is Johann Heinrich Lambert (1728--1777) and his Mémoires sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques (1761/1768). The main contributing portion --Part II of the dissertation consists of the rst annotated translation of this work, which is particularly known for including the rst demonstration of the irrationality of π, an analysis that is accompanied with a biography of its author. The research of this work and its author is not done separately. Rather, it is contextualized by analyzing the development of the so-called <<special irrationals>> in two markedly di erent centuries, the 18th and 19th, culminating in the complete theories of real numbers and the new conception of <<transcendental>> numbers. In Part I of the thesis, the situation prior to Lambert and mainly in the 18th century is analyzed an analysis in which Euler has a relevant role , but we also take a brief look at the last decades of the 16th and the 17th centuries, the time frame of the origin and development of analytical methods, key to progress towards a deeper understanding of irrational numbers. Lastly, Part III sheds light on the influence that Lambert's work had on later generations, and more generally, on the way in which irrational numbers went from being a mere quantities (classical image) to abstract objects de ned through in nite processes (modern image). This deeper knowledge of irrational numbers is particularly re ected in the rst demonstrations of transcendence, and in this sense, a key character is Joseph Liouville (1809-1882). This is not only because of his own contributions, but also because of his enormous influence through his Journal, his lectures, and the international network of contacts at his disposal, Dirichlet being a particularly distinguished case. To a large extent, the in uence of Liouville, which directly or indirectly was swayed by Lambert, is responsible for the famous results of Hermite and Lindemann on the transcendence of e and π --a concept (that of transcendence) which will not begin to be used in its modern meaning in a systematic way until the last decades of the 19th century, starting with the work of Dedekind.

Author dc:creator

• Dorrego López, Eduardo

Advisor dc:contributor.advisor

• Ferreirós Domínguez, José Manuel

Statement dc:rights

• Attribution-NonCommercial-NoDerivatives 4.0 Internacional

Licence dc:rights.uri

• http://creativecommons.org/licenses/by-nc-nd/4.0/

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/11441/125127

OAI identifier oai:identifier

oai:idus.us.es:11441/125127