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**Works. IV. Part 1: Mathematical philosophy, pure natural theology. Part 2: The second special discipline of mathematical philosophy, the other part of pure mathematics: namely the archimetry, commonly used by the expression geometry. Edited and introduced by Thomas Behme.
(Werke IV. Teil 1: Philosophia mathematica, theologia naturalis solida. Teil 2: Philosophiae mathematicae secunda disciplina specialis, purae matheseos pars altera: videlicet archimetria, seu expressioris usus communissimi geometria.)**
*(Latin, German)*
Zbl 1284.01008

Clavis Pansophiae 3, 4. Stuttgart: Friedrich Frommann Verlag Günther Holzboog (ISBN 978-3-7728-2540-8/set). lxxvii, 372 p./pt.1; lvii, 536 p./pt.2. (2013).

The present edition of Erhard Weigel’s Philosophia mathematica, published originally in 1693, is part of a series of Weigel’s collected writings; for the first and third volume see [Zbl 1096.01017; Zbl 1143.01007]. Weigel’s treatise consists of two parts, entitled Theologia naturalis solida and Archimetria, respectively. Long (German) introductions of the editor to each part summarize the main ideas and place them within a broader context.

As Weigel’s earlier treatises mentioned above, also the Philosophia mathematica deals with a universal method of knowledge. In the Analysis Aristotelica ex Euclide restituta from 1658 the young Weigel had interpreted Aristotelian logic in the light of the Euclidean method in order to free it from the distortions of mediaeval scholasticism. In contrast, the present treatise critizises the restrictions of the Aristotelian method of knowledge and aims to overcome them by resorting to algebra. Weigel’s approach is rooted in the view that God’s creation follows mathematical laws and that men, by doing mathematics, take part in the divine wisdom. Thus the object of mathematics is universal; it may be applied to theology, to natural philosophy as well as to ethics. Weigel stresses the value of mathematics for moral education and argues for strengthening its role at school and university. His treatise is part of this pedagogical program and aims to introduce the epistemological and ontological foundations of his method to the youth. Therefore the mathematical parts are kept elementary. In the Archimetria Weigel illustrates how to produce new results via analysis (in the sense of ancient Greek philosophy) by reconsidering some theorems from Euclidean geometry and trigonometry. However, he does not believe that there is a general method of analysis; his derivations serve only as a guideline and source of inspiration. Another part of the Archimetria develops the metaphysics of motions since, for Weigel, by studying them, mathematics can be applied to natural philosophy and even to ethics.

Among the applications of mathematics presented by Weigel in his Theologia naturalis solida, the most important one is a proof of God’s existence. It has to be seen in the context of efforts, for example at the Royal Society, to which the treatise was dedicated, to counter atheism by a theology seconded by natural philosophy. The first publication of this proof in the 1670s had initiated a discussion with Gottfried Wilhelm Leibniz who critized its occasionalist parts. This is only one instance where the present treatise documents intersections and differences between the philosophies of Leibniz and his teacher Weigel. While their common interest in a universal method of knowledge based on mathematics was typical for the time, Leibniz took Cartesian analytical geometry and his own differential calculus, which first appeared in 1684, as a model. The editor notes that Weigel must have read Descartes, however his proofs do not use analytical geometry but are based on geometric and arithmetic arguments. Weigel’s concept of the human spirit as a mathematical point, which has no determinate direction, is close to ideas of Leibniz of a pointlike soul and his concept of monads. Here the influence might be mutual. In general, as the editor stresses, Weigel’s approach was less speculative and more didactic than Leibniz’s.

As the previous volumes, also the present one contains extensive annotations as well as name and subject indices (in German).

As Weigel’s earlier treatises mentioned above, also the Philosophia mathematica deals with a universal method of knowledge. In the Analysis Aristotelica ex Euclide restituta from 1658 the young Weigel had interpreted Aristotelian logic in the light of the Euclidean method in order to free it from the distortions of mediaeval scholasticism. In contrast, the present treatise critizises the restrictions of the Aristotelian method of knowledge and aims to overcome them by resorting to algebra. Weigel’s approach is rooted in the view that God’s creation follows mathematical laws and that men, by doing mathematics, take part in the divine wisdom. Thus the object of mathematics is universal; it may be applied to theology, to natural philosophy as well as to ethics. Weigel stresses the value of mathematics for moral education and argues for strengthening its role at school and university. His treatise is part of this pedagogical program and aims to introduce the epistemological and ontological foundations of his method to the youth. Therefore the mathematical parts are kept elementary. In the Archimetria Weigel illustrates how to produce new results via analysis (in the sense of ancient Greek philosophy) by reconsidering some theorems from Euclidean geometry and trigonometry. However, he does not believe that there is a general method of analysis; his derivations serve only as a guideline and source of inspiration. Another part of the Archimetria develops the metaphysics of motions since, for Weigel, by studying them, mathematics can be applied to natural philosophy and even to ethics.

Among the applications of mathematics presented by Weigel in his Theologia naturalis solida, the most important one is a proof of God’s existence. It has to be seen in the context of efforts, for example at the Royal Society, to which the treatise was dedicated, to counter atheism by a theology seconded by natural philosophy. The first publication of this proof in the 1670s had initiated a discussion with Gottfried Wilhelm Leibniz who critized its occasionalist parts. This is only one instance where the present treatise documents intersections and differences between the philosophies of Leibniz and his teacher Weigel. While their common interest in a universal method of knowledge based on mathematics was typical for the time, Leibniz took Cartesian analytical geometry and his own differential calculus, which first appeared in 1684, as a model. The editor notes that Weigel must have read Descartes, however his proofs do not use analytical geometry but are based on geometric and arithmetic arguments. Weigel’s concept of the human spirit as a mathematical point, which has no determinate direction, is close to ideas of Leibniz of a pointlike soul and his concept of monads. Here the influence might be mutual. In general, as the editor stresses, Weigel’s approach was less speculative and more didactic than Leibniz’s.

As the previous volumes, also the present one contains extensive annotations as well as name and subject indices (in German).

Reviewer: Charlotte Wahl (Hannover)

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

01A45 | History of mathematics in the 17th century |

00A30 | Philosophy of mathematics |

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\textit{E. Weigel} and \textit{T. Behme} (ed.), Werke IV. Teil 1: Philosophia mathematica, theologia naturalis solida. Teil 2: Philosophiae mathematicae secunda disciplina specialis, purae matheseos pars altera: videlicet archimetria, seu expressioris usus communissimi geometria (Latin). Stuttgart: Friedrich Frommann Verlag Günther Holzboog (2013; Zbl 1284.01008)