# Geometry III: Theory of Surfaces (Encyclopaedia of Mathematical Sciences)

Its crown underlines its importance. An angel holds a laurel wreath that was meant for the author of the work. We will come back to this issue in the epilogue. On the ground-floor, a strange vehicle is drawn by the true constellations Bear and Lion. The armillary sphere that is a world model replaces the driver. The course is originally a running, a race track. The globes reveal 12 tilings of the ground-floor that is paved by squares.

Nine of them can be easily identified from the left to the right, from above to below : ballistics warfare , military architecture fortification , Tychonic world system, statics, gnomonics, practical geometry, algebra, geometry, and trigonometry. Obviously, the interests of an emperor being at war, shown in the armor of a knight, have influenced the selection of these nine represented mathematical disciplines out of the 22 dealt with in the volume. Utility is the leading aspect. It is not by chance that Schott will call practical geometry the noblest and most useful among all mathematical disciplines, trigonometry necessary and useful in the highest degree, gnomonics even divine.

The dedication to Leopold I. This is indeed an astonishing achievement because the volume comprehends folio pages and figures. This time Schott uses the popular metaphor employed for example by Francis Bacon before him and by Gottfried Wilhelm Leibniz after him: To pursue, science is something like a voyage of discovery. According to Schott, the emperor knows that the mathematical sciences, combined with the experience of philosophy, are jewels in the hands of princes, kings, and emperors. Kings and monarchs of the world always attached great interest to the mathematical sciences.

Without question, in the eyes of the princes of the seventeenth century mathematics was a discipline supporting the state, it was powerful. An astronomer is standing with commanding air on a sleigh bearing the garment of a sovereign and measures the celestial globe. He looks like God. And indeed, in the Middle Ages, God was painted in a similar pose as the highest geometer.

Schott is especially proud of his dedication. It is true that most of the eminent mathematicians of all times have dedicated their works to princes and sovereigns. But up to then nobody has dedicated a whole encyclopedia of the mathematical sciences to a sovereign. Schott had inherited his yearning for universality from his Lullistic teacher Athanasius Kircher.

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And indeed, the seventeenth century is the century of universal mathematics. While Schott was elaborating his encyclopedia, he was encouraged by many contemporaries. He received paraeneses , exhortations. He printed three of them at the beginning of his volume.

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One of these writings was the letter of his admired teacher Athanasius Kircher to mathematical beginners. He, Kircher, preferred to compare the studies of mathematics with a workshop officina.

### Theory of Surfaces

Kircher speaks about the comprehensive use of the divine mathematics sacra mathesis Schott , p. God himself was the first geometer who had created the world according to arithmetical and geometrical principles. Knowledge about the world presupposed the language of mathematics. Regularity and perfection of the unchangeable, geometrical structure of the world testified for its divine origin Romano — This theological view established for Clavius the importance of mathematics before all other disciplines, even before philosophy and theology.

God was the inventor of all mathematical disciplines. This importance compensates by far for the trouble of acquiring it because: ". Kircher cites Epicharmus, the best known poet of the Doric comedy.

Schott planned his textbook most carefully. He placed a preface in front of the whole work and in front of every of the 28 chapters as well. Therein he commented upon his aims and methods and defined the single mathematical, partial disciplines: a model of clarity and order! He addressed beginners and first candidates of mathematics. That he emphasized again and again. Hence, he inserted short introductions into areas or surveys of areas. He referred to further literature for additional studies. Schott did not want to be an innovator, any more than Kepler, nor did he want to treat everything exhaustively.

Here, if anywhere the wisdom proved to be true: Only when restricted does somebody prove to be a master.

## Geometry III

On the contrary, thanks to a wise self-restriction, he intended to explain the essentials in an easily comprehensible form for beginners. They were the only authors who had accomplished a similar work before him. The title page enumerates and illustrates theology, jurisprudence, medicine, philosophy, mechanics, and various subjects varia.

Mathematics is subsumed in this last group as it is indicated by the right lower picture and went short for that reason. Arithmetic, music, geometry, and astronomy are represented. Five chapters offered a medley of disciplines farragines disciplinarum. Yet, Schott remained a self-critical realist and remarked Schott , preface of the whole work p.

## Geometry III - Theory of Surfaces | Yu.D. Burago | Springer

We will come back to this point. Though he venerated his models, Schott deviated from their methods and opinions in special cases. It is indeed the science dealing with bounded quantity and this is either abstracted from sensible matter or immersed in it.

Indivisibles however are non-quantities by definition. In view of numerous classifications of the mathematical disciplines, Schott decides on a classification that is similar to that of Clavius Schott :3 :. Every subdivision is divided into further subunits. It might suffice to enumerate all 22 disciplines dealt with by Schott: Arithmetic, geometry, trigonometry, astronomy, astrology, chronology, geography, navigation, gnomonics, mechanics, statics, hydrostatics, hydraulic engineering, optics, catoptrics, dioptrics, fortification, warfare, tactics, musical theory, algebra, and theory of logarithms.

Alsted had excluded both types of architecture from the mathematical sciences. The five longest chapters by far 60—92 pages are geometry, geography, trigonometry, algebra, and astronomy. The shortest chapters four to six pages are dioptrics, statics, and tactics. Lopes, T. Lewiner, B. Medeiros, M.

## My Library

Lage, S. Arouca, F. Carmo, e L. Journal of the Brazilian Computer Society, Vol. Rebelo e A. Silva , On the Burnside problem in Diff M. Tomei, M. Carvalho, e J. Journal of Computational Physics, Vol.

go to link Volchan , Probability as typicality. Partially hyperbolic dynamics, laminations, and Teichmuller flow, Vol. Bordignon, R. Castro, H. Tavares, R.