#Common notion definition geometry series

The Postulates do not necessarily deductively follow from the Definitions, rather they are five rules offered by Euclid. They are a demonstration of the authority or authorship of Euclid.

#Common notion definition geometry series

While the Definitions are firm and unquestionable, the Postulates are a series of “requests” or “demands” placed upon the reader. Perhaps it is meant to be an exploration of the Platonic eidos. Perhaps Euclid’s Elements was not intended to be translated from the conceptual to the physical world (“earth-measurement”). creating a point and a line) takes precedence over the plane surface. It is worth noting that a plane surface does not appear first in the list of Definitions. We imagine an ancient geometer demonstrating Euclid’s Definitions in the dirt or on a chalk board.Īs the Definitions descend we begin with foundational elements like points and lines (Definitions 1-7), then with Definitions pertaining to proportions between foundational elements like angles (Definitions 8-13), and then Definitions concerning shapes or figures (A figure is defined in Definition 14, Definitions 13-18 concern circles, and Definitions 19-23 concern rectilinear figures). This is distinct from modern conceptions of rounded or spherical surfaces upon which to conduct geometric demonstrations. The assumption is that a) the straight could be produced indefinitely in a hypothetical situation and b) the straight lines are produced on an indefinitely flat plane/surface. This is evidenced by the final Definition of parallel lines (“straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”). A surface is presumed to be flat, unlike modern formulations of elliptical and non-linear geometry (i.e. Where do we draw these elements? On a surface (“that which has length and breadth only”). A point grants permission to draw a line (“breadthless length”) between two points.

A point gives us a sense place, perspective, and grounding.

The first Definition is of a point -an irreducible and indivisible element (“A point is that which has no part”). They are brief declarations that we can imagine as a response to Socratic questions, “what is…?” The Definitions do not permit a modern conception of the infinite. The Definitions proceed from small elements to constructions of shapes. The Definitions are 23 statements (they were later numbered by 16th century editors after the advent of the printing press). Things that are common occur last in order of importance. The Postulates follow the Definitions, and lastly we are offered a list of Common Notions. The Definitions appear first and a general descent occurs. There were other “Elements” books circulating in antiquity by Hippocrates, Leo, and Theudius, but Euclid superseded them all and none of the other books have fully survived into the modern day.Įuclid begins his Elements not with a series of “problems” or “equations” like many math modern textbooks but rather with a list of foundational metaphysical claims: Definitions, Postulates, and Common Notions. Take note of a common mistake: Euclid, the author of the Elements, is distinct from Euclid of Megara who appears in Plato’s Theaetetus.Įuclid appears briefly in Archimedes’s On the Sphere and the Cylinder and also in Apollonius’s Conics. Heath surmises that Euclid was originally schooled in Athens under the geometric pupils of Plato (in many ways we can see echoes of Plato found in Euclid’s Elements -recall the mathematical instruction of the boy in Plato’s Meno). He worked or perhaps founded a school in Alexandria, Egypt. The only two things we infer about his life, as referenced by ancient sources (primarily Diogenes Laërtius), is that he lived after Plato (died 347 BC) and before Archimedes (287 BC). The Elements has been cited by every major mathematical and scientific figure including Copernicus, Galileo, Kepler, Newton, Hobbes, Descartes, Spinoza, Whitehead, Russell, Einstein, and so on. Sadly, the Elements fell out of favor for students in the 20th century and very few, if any, students attempt to summit the extraordinary heights of Euclid in our modern era. The Elements was the essential geomtery textbook for nearly 2,000 years thanks to the preservation efforts of the Byzantines, Arabs, and English. The Elements is composed of thirteen books, each filled with propositions that beautifully unfold a theory of number, shape, proportion, and measurability. Euclid’s Elements ( “Stoikheîon”) is the foundational text of classical, axiomatic, and deductive geometry (“earth-measurement”).