Introduction to Euclid's Geometry
|Philosophy of Mathematics|
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- Web resources:
- Syllabus documents
- Web resources
- Study.com - Shows an animated video explaining different types of angles.
- Eculid's axioms and postulates - Additional information on axioms and postulates
- Grade stack - This website gives slides explaining Euclidean Geometry.
- The following videos provide an introduction to axioms, postulates and lines
- Books and journals
- Karnataka Govt Text book – Class 8 : Part 1 ,
- Syllabus documents (CBSE, ICSE, IGCSE etc)
Concept 1 - Introduction to geometry
One interesting question about the assumptions for Euclid's system of geometry is the difference between the "axioms" and the "postulates." "Axiom" is from Greek axíôma, "worthy." An axiom is in some sense thought to be strongly self-evident. A "postulate," on the other hand, is simply postulated, e.g. "let" this be true. There need not even be a claim to truth, just the notion that we are going to do it this way and see what happens. Euclid's postulates, indeed, could be thought of as those assumptions that were necessary and sufficient to derive truths of geometry, of some of which we might otherwise already be intuitively persuaded. As first principles of geometry, however, both axioms and postulates, on Aristotle's understanding, would have to be self-evident. This never seemed entirely quite right, at least for the Fifth Postulate -- hence many centuries of trying to derive it as a Theorem. In the modern practice, as in Hilbert's geometry, the first principles of any formal deductive system are "axioms," regardless of what we think about their truth -- which in many cases has been a purely conventionalistic attitude. Given Kant's view of geometry, however, the Euclidean distinction could be restored: "axioms" would be analytic propositions, and "postulates" synthetic. Whether any of Euclid's original axioms are analytic is a good question.
It is useful to discuss with students about Euclid and his great contribution to Mathematics. The below two statements helps to understand and prove the theorems in geometry. Also, through a combination of activities, help the students understand results in the nature of axioms and postulates.
- Certain statements which are valid in all branches of mathematics whose validity is taken for granted without seeking mathematical proofs is called axioms
- Some statement which are taken for granted in a particular branches of mathematics is called postulates.
Concept 2 - Axioms and postulates
- First Axiom: Things which are equal to the same thing are also equal to one another.
- Second Axiom: If equals are added to equals, the whole are equal.
- Third Axiom: If equals be subtracted from equals, the remainders are equal.
- Fourth Axiom: Things which coincide with one another are equal to one another.
- Fifth Axiom: The whole is greater than the part.
- First Postulate: To draw a line from any point to any point.
- Second Postulate: To produce a finite straight line continuously in a straight line.
- Third Postulate: To describe a circle with any center and distance.
- Fourth Postulate: That all right angles are equal to one another.
- Fifth Postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side of which are the angles less than the two right angles.
Comparison of geometric parameters to check for their equality, for example if an area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.
Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of the “same kind.” Addition or elimination of equal parameters to equal quantities results in equal things, for which the relation of equality and the operation of subtraction make sense. In Euclid’s mathematics this relation and this operation apply not only to straight segments and numbers but also to geometrical objects.
From Euclid’s use of the word “equal” means “the same size”; two geometric figures are equal is justified by showing that one can be moved so that it coincides with the other.
Two individual things are connected together to form one entity then each of these two things that form a whole is smaller than the whole.
Solved problems/ key questions (earlier was hints for problems).
- What was the name of the book written by Euclid ? How many chapters did it have ?
- Given <P = <Q and <Q = <R, according to which axiom of Euclid, the relation between <P and <R is established ?
- If a + b = 8cm, Is it true to say that a + b + y = 8 + y ?
- If AB = 4cm, CD = 8cm and PQ = two times AB. Are CD and PQ equal ? Which axiom is used for proving this ?
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Assessments - question banks, formative assessment activities and summative assessment activities
Contributed by Rekha .D .R, Assistant Mistress, G.H.S , Jayanagar 9th Block, Bengaluru-69