@article {1019205, title = {Algebraic partitioning problems from {L}uca {P}acioli{\textquoteright}s {P}erugia manuscript (Vat. Lat. 3129)}, journal = {Sources and Commentaries in Exact Sciences}, volume = {11}, year = {2010}, pages = {3{\textendash}51}, issn = {1345-4617}, author = {Heeffer, Albrecht} } @article {666246, title = {The Abbaco Tradition (1300-1500): its role in the development of {E}uropean algebra}, journal = {Suuri Kaiseki Kenkyuujo koukyuuroku}, volume = {1625}, year = {2009}, pages = {23{\textendash}33}, abstract = {

Abbaco algebra is a coherent tradition of algebraic problem solving mostly based in the merchant cities of fourteenth and fifteenth-century Italy. This period is roughly situated between two important works dealing with algebra: the Liber Abbaci by Fibonacci (1202) and the Summa di Arithmetica et Geometria by Lucca Pacioli (1492). Such continuous tradition of mathematical practice was hardly known before the first transcriptions of extant manuscripts by Gino Arrighi from the 1960{\textquoteright}s and the ground-breaking work by Warren van Egmond (1980). After some decades of manuscript study and the recent assessment of Jens H{\o}yrup (2007) we now have a better understanding of this tradition. In this paper we provide an overview of the basic characteristics of the abbaco tradition and discuss the role it played towards the new symbolic algebra as it emerged in sixteenth-century Europe. We argue that its influence on the sixteenth century has largely been ignored and that the new ars analytica from the French algebraists should be understood as establishing new foundations for the general practice of abbaco problem solving.

}, issn = {1880-2818}, author = {Heeffer, Albrecht} } @article {heeffer2009abbacus, title = {The Abbacus Tradition: the Missing Link between Arabic and Early Symbolic Algebra?}, year = {2009}, abstract = {

Abbaco algebra is a coherent tradition of arithmetical and algebraic problem solving, mostly based in the merchant cities of fourteenth and fifteenth-century Italy. This period is roughly situated between two impor- tant works dealing with algebra: the Liber Abbaci by Fibonacci (1202) [5] and the Summa di Arithmetica et Geometria by Luca Pacioli (1492) [20]. Such continuous tradition of mathematical practice was hardly known be- fore the first transcriptions of extant manuscripts by Gino Arrighi from the 1960{\textquoteright}s (most notably [1], [2], [3] and [4]) and the ground-breaking work by Warren van Egmond [28]. After some decades of manuscript study and the recent assessment of Jens H{\o}yrup [10] we now have a bet- ter understanding of this tradition. Here we provide an overview of the basic characteristics of the abbaco tradition and discuss the role it played towards the new symbolic algebra as it emerged in sixteenth-century Europe.

}, author = {Heeffer, Albrecht} } @incollection {381663, title = {Abduction as a strategy for concept formation in mathematics: {C}ardano postulating a negative.}, booktitle = {Abduction and the process of scientific discovery}, year = {2007}, pages = {179{\textendash}194}, publisher = {Centro de Filosofia das Ci{\^e}ncias da Universidade de Lisboa}, abstract = {

When dealing with abductive reasoning in scientific discovery, historical case studies are focused mostly on the physical sciences, as with the discoveries of Kepler, Galilei and Newton. We will present a case study of abductive reasoning in early algebra. Two new concepts introduced by Cardano in his Ars Magna, imaginary numbers and a negative solution to a linear problem, can be explained as a result of a process of abduction. We will show that the first appearance of these new concepts fits very well Peirce’s original description of abductive reasoning. Abduction may be regarded as one important strategy for the formation of new concepts in mathematics.

}, author = {Heeffer, Albrecht}, editor = {Pombo, Olga and Gerner, Alexander} }