Search results
Results From The WOW.Com Content Network
However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. [4] Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): [5] Let the following be postulated:
Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions. [45] [k] These assumptions are intended to provide the logical basis for every subsequent theorem, i.e. serve as an axiomatic system. [46] [l] The common notions exclusively concern the comparison of magnitudes. [48]
This postulate does not specifically talk about parallel lines; [1] it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate.
Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, [7] Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. [8]
The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry). As such ...
Bernal coined the term biopoiesis in 1949 to refer to the origin of life. [33] In 1967, he suggested that it occurred in three "stages": the origin of biological monomers; the origin of biological polymers; the evolution from molecules to cells; Bernal suggested that evolution commenced between stages 1 and 2.
The earliest evidence for life on Earth includes: 3.8 billion-year-old biogenic hematite in a banded iron formation of the Nuvvuagittuq Greenstone Belt in Canada; [30] graphite in 3.7 billion-year-old metasedimentary rocks in western Greenland; [31] and microbial mat fossils in 3.48 billion-year-old sandstone in Western Australia.
Professor of biology Jerry Coyne sums up biological evolution succinctly: [3]. Life on Earth evolved gradually beginning with one primitive species – perhaps a self-replicating molecule – that lived more than 3.5 billion years ago; it then branched out over time, throwing off many new and diverse species; and the mechanism for most (but not all) of evolutionary change is natural selection.