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1. Historical development: overview From Greek antiquity onwards, the cultures of the Mediterranean considered geometry - the study of figures on the plane and in space - to be the epitome of an exact science, relying on strict proofs based on precisely formulated assumptions. Geometry moreover represented science’s most powerful language for expressing relations of size, trumping the arithmetica universalis. Its domain extended from direct measurement of spatial proportions through astronomy and optics to the art of construction.The early modern era witnessed a remar…

The term more geometrico (Latin: “in a geometric manner/style”) was introduced into European philosophy primarily through the writings of Baruch de Spinoza. In his  Ethica, ordine [or, often: more] geometrico demonstrata (1677; “Ethics, demonstrated in the geometric manner”), he presented a much-discussed system of philosophical ethics that he considered not only as categorically binding as the system of classical (Hellenistic) geometry, but also as treating its subjects just as literally and precisely as that geometry. …

The use of equipment to measure and calculate has accompanied the mathematical sciences and practices of calculation and draftsmanship since Grec-Roman antiquity, when drawing instruments – the ruler and the dividers – began to be used in astronomy along with various devices for measuring angles. The development of mathematical instruments for astronomy intensified greatly during the heyday of Arabian Islamic science (10th-13th centuries). The great Arabian observation centers like Maragha (13th…

1. Introduction The mathematical sciences (Latin disciplinae mathematicae) initially encompassed the fields of mathematical study that dated back to Hellenistic antiquity. Their status in the knowledge culture of the late Middle Ages was low and strictly limited to certain areas (cf. e.g. Mathematics, musical). Over the course of the 16th and 17th centuries, not only did the classical branches of the mathematical sciences enjoy a profound renewal, but new fields emerged alongside them, becoming establi…

A key aspect of the renewal of mathematics in the 17th century was the combination of geometric themes with symbolic/calculatory techniques related to the infinitesimal, that is, the infinitely small (Analysis, mathematical; Infinitesimals). This enabled a new way of conducting geometry, the most appropriate term for which in the 17th century is infinitesimal geometry and which from the 18th century onwards - the century of the elaboration of differential calculus - developed further as differential geometry.The main subjects of differential geometry in the 17th and…

At the end of the 17th century,  analysis situs (“analysis of the location”) or geometria situs (“geometry of the location”) came into use as an early term for the field of mathematics we today call topology, a new field initially developed very slowly during the 18th and 19th centuries (see Mathematical sciences). Alongside the mathematical theory of probability, its was the most important field of mathematics to be developed totally from the ground up in the Early Modern era, without any roots in antiquity.At the same time as modern analysis (Analysis, mathematical) was c…

1. Ancient and medieval traditionsOptics (from the Greek optikós, “of/for sight”) was already developing into a sophisticated mathematical science in Hellenistic Greece, incorporating three branches of optics in the stricter sense: (1) the theory of sight, the eye, and light, including philosophical, physiological, and mathematical aspects (e.g. central perspective); (2) catoptrics (the theory of reflection on mirrored surfaces); and (3) dioptrics (the theory of refraction at transitions between different optical media).Islamic authors, notably Ibn al-Haytham …

Ballistics (the science of the behavior of projectiles) was one of the disciplines that played a key role in the emergence of the new science of the 16th and 17th centuries. It provided both a motivation and one of the trickiest applications for mechanics as a mathematical science of motion. It was also related, not always straightforwardly, with early modern weaponry and artillery.Two concepts of the motion of a projectile were in competition. First, it was possible to see the phenomenon as the motion of a body through a continuous medium. Alternativel…

1. General The techniques of arithmetic and the media required are among the earliest cultural achievements of humankind. In many places (e.g. southern Mesopotamia at the end of the 4th millenium BCE), the introduction of writing was associated directly with the techniques of written arithmetical calculation and bookkeeping. Where arithmetical theory emerged, as in ancient Greece, practical calculation naturally remained in use as well; in many cases it had its own separate history, concealed in part by the history of academic mathematics [3]. The latter also includes the te…

1. BeginningsEven though the equality of quantities and figures or numbers and ratios is a relationship that had been in use in the mathematical sciences since their beginnings, solving equations, that is, the symbolically expressed equality of two algebraic expressions containing known and unknown quantities, did not become a focus of mathematical problem solving until around 1600 – with the advent of symbolic algebra and the analytic geometry dependent on it. Even earlier, though (building on s…

Mathematical knowledge and the application of mathematical processes have been with humanity since its beginnings and have been tangible from the earliest surviving textual evidence. By the Middle Ages at the latest, mathematics had also become an important area of Jewish scholarship. In modern Europe, and especially in the German-speaking areas, there was an extraordinarily high proportion of Jews in the scholarly culture of mathematics. Nearly all the Jewish mathematicians and those originally…

Mathematisches Wissen und die Verwendung mathematischer Verfahren begleiten die Menschheit seit ihren Anfängen und sind greifbar seit den frühesten erhaltenen schriftlichen Zeugnissen. Spätestens seit dem Mittelalter wurde die Mathematik auch ein wichtiges Feld jüdischer Gelehrsamkeit. Im modernen Europa, und besonders im deutschsprachigen Raum, ergab sich eine außergewöhnlich hohe Beteiligung von Juden an der Wissenschaftskultur der Mathematik. Nahezu alle jüdischen oder aus jüdischen Familien …

1. Schifffahrtspraxis Die europ. N. hatte in der Nz. weltweit regelmäßige Segelfahrten von einem Hafen zum anderen möglichst zuverlässig und sicher zu gewährleisten (Hochseeschifffahrt) [7]. Sie war Aufgabe des Kapitäns und des Steuermanns als seines Stellvertreters. Beide führten je eine Wache und meldeten dem anderen bei Wachwechsel den jeweiligen Verlauf. Auf dem achteren Aufbau hatten sie die nötige Rundsicht zur Kontrolle von Segelführung und Schiffszustand, Bahnführung, Schiffsstandort und Wetter. Von dort riefen…

1. Begriff und EntwicklungDas heutige Verständnis von Ph. als einer auf die Eigenschaften und Gesetze der unbelebten Materie gerichteten Wissenschaft, die aufgrund von Experiment und Messung math. formulierte Theorien entwickelt, entstand erst im späten 18. und frühen 19. Jh. aus der weiter gefassten Bestimmung als Naturlehre oder Naturphilosophie (Physikalische Wissenschaften). Zentral war die Zuspitzung auf einen spezifischen Gegenstandsbereich und die zunehmende Einbeziehung quantitativer und math. Verfahren (Quantifizierung und Messung) [6. Kap. 5]; [5].Ein we…

1. Concept and originsThe present-day understanding of physics as a science concerned with the properties and laws of inanimate matter and the development of mathematically-formulated theories on the basis of experiment and measurement arose only in the late 18th and early 19th centuries out of the wider sense of physics as natural philosophy (Physical sciences). The key development was the focus on a specific subject area and the increasing reliance on quantitative and mathematical procedures (Quantification and measurement) [6. chapter 5]; [5].One important impulse cam…

1. Seafaring practice European navigation in the early modern period was tasked with ensuring that regular passages under sail from one harbor to another worldwide were conducted as reliably and safely as possible (Deep sea navigation) [7]. It was the responsibility of the captain, with the helmsman as his deputy. The two took alternate watches, each informing the other of the prevailing course when changing the watch. The aft deck gave them the necessary clear all-round view to monitor the condition of the ship and sails, as we…