Journey through genius chapter 2 study guide
JTG- Ch.2
Euclid’s Proof of the Pythagorean Theorem
Century and a half between Hippocrates and Euclid.
Plato esteemed geometry to be the entrance to his Academy. Let no man ignorant of geometry enter here.
“Logical scandal” Theorems were believed to be correct as stated but they lacked the material to prove them.
Euclid’s Elements was said to become the staple of mathematics or the standard. 13 books, 465 propositions (not all Euclid but rather a collection of great mathematicians work, started with 23 definitions, 5 postulates, 5 general axioms
Euclid defined a 90 degree angle as two equal adjacent angles along a straight line.
5 Postualates [It is possible] to draw a straight line from any point to any point. “ to produce a finite straight line continuously in a straight line. “ to describe a circle with any center and distance All right angles are equal to one another 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 on which the angles are less than two right angles.
Euclid’s first congruence was SAS came from proposition 4 of the first book
Veritical angle congruence → triangle exterior angle → AAS → transversal making parallel lines → alternate angles → three angles being equal to two right angles → PYTHAGOREAN THEOREM
Proposition I.47 In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Euclid’s diagram became known as the “Windmill” because of the resemblance.
Carl Friedrich Gauss thought that triangles had to be less or more than 180 degrees since he could not prove Euclid in saying that it is 180 exactly.
Johann Bolyai attempted to derive the parallel postulate. Also unsuccessful in trying to prove what Euclid did in the past, found himself in a whole new path of math.
Euclid’s Proof of the Pythagorean Theorem
Century and a half between Hippocrates and Euclid.
Plato esteemed geometry to be the entrance to his Academy. Let no man ignorant of geometry enter here.
“Logical scandal” Theorems were believed to be correct as stated but they lacked the material to prove them.
Euclid’s Elements was said to become the staple of mathematics or the standard. 13 books, 465 propositions (not all Euclid but rather a collection of great mathematicians work, started with 23 definitions, 5 postulates, 5 general axioms
Euclid defined a 90 degree angle as two equal adjacent angles along a straight line.
5 Postualates [It is possible] to draw a straight line from any point to any point. “ to produce a finite straight line continuously in a straight line. “ to describe a circle with any center and distance All right angles are equal to one another 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 on which the angles are less than two right angles.
Euclid’s first congruence was SAS came from proposition 4 of the first book
Veritical angle congruence → triangle exterior angle → AAS → transversal making parallel lines → alternate angles → three angles being equal to two right angles → PYTHAGOREAN THEOREM
Proposition I.47 In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Euclid’s diagram became known as the “Windmill” because of the resemblance.
Carl Friedrich Gauss thought that triangles had to be less or more than 180 degrees since he could not prove Euclid in saying that it is 180 exactly.
Johann Bolyai attempted to derive the parallel postulate. Also unsuccessful in trying to prove what Euclid did in the past, found himself in a whole new path of math.