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[German version] [1] Philosophical Ἀναλογία, ἀνὰ λόγον ( Analogía, anà lógon) designate originally the mathematical proportion, i.e., the correspondence of numerical or volume relationships (definition: Euc. 5, Def. 5 and 6; 7, Def. 20; Aristot. Eth. Nic. 5,6; 1131 a 6-b9 Poet. 21,11-14; 1457 b 16-30). The teachings about proportions were used in the first instance as a tool to solve problems; it soon developed, however, into a general theoretical construction, which Euclid presented in the fifth book of …

[English version] [1] philosophisch Ἀναλογία, ἀνὰ λόγον bezeichnen urspr. die mathematische Proportion, d. h. die Gleichheit von Zahlen- oder Größenverhältnissen (Definition: Eukl. elem. 5, Def. 5 und 6; 7, Def. 20; Aristot. eth. Nic. 5,6; 1131 a 6-b9 poet. 21,11-14; 1457 b 16-30). Die Lehre von den Proportionen wurde zunächst als Problemlösungsinstrument benutzt, entwickelte sich jedoch sehr bald zu einer allg. theoretischen Konstruktion, die Euklid im fünften Buch der ›Elemente‹ darstellt. Platon …

I. Mesopotamia [German version] A. Numerical systems Before systems for representing numbers in writing were (further) developed, counting stones, known as calculi or tokens, were used in arithmetic. As first-order representations they enabled operations such as increasing, decreasing, combining, separating, and distributing. Their relationship to the numerical notations recorded in the oldest ‘texts’ ( c. 3300 BC; Uruk) is still discussed [2]. The numerical signs in these texts do not represent absolute numbers but context-dependent units of count…