|
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates. According to Felix Klein,
A defining characteristic of synthetic geometry is the use of the axiomatic method to draw conclusions and solve problems, as opposed to analytic and algebraic geometries, where one would use analysis and algebraic techniques to obtain these geometric results. Euclidean geometry, as presented by Euclid, is the quintessential example of the use of the synthetic method. However, only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to the subject. As a field of study, synthetic geometry was most prominent during the 19th century when some geometers rejected coordinate methods in establishing the foundations of projective geometry and non-Euclidean geometries. It was the favourite method of Sir Isaac Newton for the solution of geometric problems. The geometer Jakob Steiner hated analytic geometry, and always gave preference to synthetic methods. ==Logical synthesis== The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives: * Primitives are the most basic ideas. Typically they include both objects and relationships. In geometry, the objects are things such as ''points'', ''lines'' and ''planes'', while a fundamental relationship is that of ''incidence'' – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs, the point being that the primitive terms are just empty placeholders and have no intrinsic properties. * Axioms are statements about these primitives; for example, ''any two points are together incident with just one line'' (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the ''building blocks'' of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「synthetic geometry」の詳細全文を読む スポンサード リンク
|