De triangulis omnimodis libri quinque (Regiomontanus)
| Author | Regiomontanus, Johannes |
|---|---|
| Full title | Doctissmimi viri et mathematicarum disciplinarum eximii professoris Ioannis de Regio monte De triangulis omnimodis libri quinque. Quibus explicantur res necessariae cognitu, volentibus ad scientiarum astronomicarum perfectionem deuenire; quae cum nusquam alibi hoc tempore expositae habeantur, frustra sine harum instructione ad illam quisquam aspirarit.Accesserunt huc in calce pleraque Domini Nicolai Cusani de Quadratura circuli Deque recti ac curui commensuratione. Itemque Ioannis de monte Regio eadem de re ἐλεγκτικὰ [elenktika], hactenus a nemine publicata. Omnia recens in lucem edita fide et diligentia singulari. |
| Year | 1533 |
| Place | Nuremberg |
| Publisher/Printer | Petrejus, Johannes |
| Era | 16th century |
| Form/Genre | Monograph, Encyclopedic work |
| Discipline/Content | Mathematics |
| Digital copies | |
| Original | De triangulis omnimodis (e-rara.ch) |
| Digital sourcebook | 695816 |
| Description | De triangulis omnimodis is a 16th-century edition by Johannes Schöner of the original manuscript from 1464 by Regiomontanus. The latter's early death had prevented him to publish the work himself, as he had announced in his printing plan of 1474.
After a dedication letter of Schöner and a short preface to the reader by Regiomontanus, the first book starts with definitions (diffinitiones) of magnitudes, measurement and proportions as well as with "common conceptions" comparable to Eudlid's axioms. This is followed by 57 theoremata on problems in triangles. Mostly some quantities are given and the others have to be calculated. Regiomontanus uses the properties of right triangles and the sine function (referring to tables for numerical values). The second book starts with the law of sines and presents more complicated computations on triangles including its area. The third book is on spherical geometry and the fourth and fifth on spherical trigonometry with theorema XVII of book IV (p. 104) presenting the spherical law of sines. This is a significant improvement of spherical trigonometry compared to antique sources (like the Almagest of Ptolemy). In book V the cosine law for spherical triangles (theorema II) is formulated for the first time. Regiomontanus's knowledge of algebra is apparent in theorema 23 of book II, where he solves a trinonometric problem with the aid of algebra, for the first time in the Latin West. Regiomontanus refers to this technique as follows: "Per artem rei et census hoc problema absolvemus" and solves an exemplary problem in order to show the general method. The equation is formulated verbally using the common terms res for the unknown and census for its square as well as ducere in for multiplication and ducere in se for squaring a number. All propositions and reasonings are supported by helpful illustrations. Sometimes, the author demonstrates how to use a result carrying out the calculations for special examples. This is especially helpful in more complicated situations, as all relations of the respective quantities are expressed only verbally in the manner of the 15th and 16th century. Without doubt, De triangulis omnimodis is the starting point of trigonometry in Europe and represents a standard work for 16th century trigonometry. |
| References | Merzbach; Boyer 2011, 246–248
Complete Dictionary of Scientific Biography 2008, vol. 11, 350 |
| Cited in |
|
| How to cite this entry | Regiomontanus, Johannes: De triangulis omnimodis, in: Noscemus Wiki, URL: http://wiki.uibk.ac.at/noscemus/De_triangulis_omnimodis (last revision: 05.02.2026). |