![]() First note that Sextus wrote about 200 AD and it was believed until comparatively recently that Heron lived later than this. Is Definitions of terms in geometry based on Euclid's Elements or have the basic definitions from this work been inserted into later versions of The Elements? We have to consider what Sextus Empiricus says about definitions. In Knorr argues convincingly that this work is in fact due to Diophantus. which are very close to those given by Euclid. This contains 133 definitions of geometrical terms beginning with points, lines etc. The next point to note is that they are very similar to the work which is ascribed to Heron called Definitions of terms in geometry. If they were not in the text that Euclid wrote then of course he couldn't refer to them. The first comment would be that this would explain why Euclid never refers to the basic definitions. ![]() Some historians have suggested that the difference between the way that basic definitions occur at the beginning of Book I and of Book V is not because Euclid was less rigorous in Book V, rather they suggest that Euclid always left his basic concepts undefined and the definitions at the beginning of Book I are later additions. When Euclid introduces numbers in Book VII he does make a definition rather similar to the basic ones at the beginning of Book I:Ī unit is that by virtue of which each of the things that exist are called one.For example one might expect Euclid to postulate a + b = b + a, ( a + b ) + c = a + ( b + c ) a + b = b + a, (a + b) + c = a + (b + c) a + b = b + a, ( a + b ) + c = a + ( b + c ), etc., but he does not. When Euclid introduces magnitudes and numbers he gives some definitions but no postulates or common notions.However Euclid leaves the concept of magnitude undefined and this appears to modern readers as though Euclid has failed to set up magnitudes with the rigour for which he is famed. ![]() As we noted in The real numbers: Pythagoras to Stevin, Book V of The Elements considers magnitudes and the theory of proportion of magnitudes.For example there is no notion of ordering the points on a line, so the idea that one point is between two others is never defined, but of course it is used. Euclid never makes use of the definitions and never refers to them in the rest of the text.Euclid seems to define a point twice (definitions 1 and 3) and a line twice (definitions 2 and 4).Things equal to the same thing are also equal to one another. One can draw a straight line from any point to any point. The postulates are ones of construction such as: A straight line lies equally with respect to the points on itself. Then, before Euclid starts to prove theorems, he gives a list of common notions. Question 5: How does Euclid define a line?Īnswer: When Euclid first formalized the geometry, he defined a general line as “breathless length” as a straight line is a line “that lies evenly with the points on itself” in two dimensions.Book 1 of The Elements begins with numerous definitions followed by the famous five postulates. In addition, he also wrote works on conic sections, number theory, mathematical rigor, perspective, and spherical geometry. Question 4: How Euclid contribute to geometry?Īnswer: From a small set of axioms Euclid deduced the theorems of what is now known as Euclidean geometry. Question 3: What are the uses of Euclidean geometry?Īnswer: We use Euclidean geometry to design buildings, predict the location of moving objects, and survey land. Lastly, if the straight-line intersect two other straight-lines, and make the two interior angles on one side of it together less than two right angles, then the other straight line will meet at a point on the side with less than two right angles.It can be described as a circle with any given point as its center and any distance as its radius. ![]()
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