Journey through Genius - The Great Theorems of Mathematics
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I love this. Love mathematics for the genius of their discovery. It's a mix of idea, imagination, inspiration, rigor, truth, beauty, simplicity.
It clicks, and when it does, you can see the picture. It's like it was always there but you couldn't see it.
This is the famous haha moment.
There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.
-- Bertrand Russel
In this book I shall explore a handful of the most important proofs - and the most ingenious logical arguments - from the history of mathematics.
In most sciences one generation tears down what another has built, and what one has established another undoes. In mathematics alone each gen eration adds a new story to the old structure.
-- Hermann Hankel
It has been said that talent is doing easily what others find difficult, but that genius is doing easily what others find impossible.
we are about to begin our journey through two millennia of mathematical landmarks. These results, old as they are, retain a freshness and display a sparkling virtuosity even after so many centuries.
Chap1ter 1 - Hippocrates' Quadrature of the Lune (ca. 440 B.C.)
The Appearance of Demonstrative Mathematics
Egyptian architects used a clever device for making right angles. They would tie 12 equally long segments of rope into a loop, as shown in Figure 1.1. Stretching five consecutive segments in a straight line from B to C and then pulling the rope taut at A, they thus formed a rigid triangle with a right angle BAG. This config uration, laid upon the ground, allowed the workers to construct a perfect right angle at the corner of a pyramid, temple, or other building.
Another great civilizations-flourished in Mesopotamia and produced mathematics signifi cantly more advanced than that of Egypt. The Babylonians.
they definitely understood the Pythagorean theorem in far more depth than their Egyptian counterparts;
it may seem a bit odd that the Babylonians chose a base-60 system.
their choice of base can still be seen in our measure ment of time (60 seconds per minute) and angles (6 X 60 · = 360 · in a circle).
Greece. Here there arose one of the most sig nificant civilizations of history, whose extraordinary achievements would forever influence the course of western culture.
Thales (ca. 640-ca. 546 B.C.), one of the so-called "Seven Wise Men" of antiqUity.
the father of demonstrative mathematics, the first scholar who supplied the "why".
As such, he is the earliest known mathematician.
Tradition holds that it was Thales who first proved the following geometric results:
- Vertical angles are equal.
- The angle sum of a triangle equals two right angles.
- The base angles of an isosceles triangle are equal.
- An angle inscribed in a semicircle is a right angle.
The proof given below is taken from Euclid's Elements, Book III:
THEOREM: An angle inscribed in a semicircle is a right angle.
Let a semicircle be drawn with center 0 and diameter BC, and choose any point A on the semicircle (Figure 1.4). We must prove that BAC is right.
Draw line OA and consider triangle AOB. Since OB and OA are of the semicircle, they have the same length, and so triangle AOB is isosceles.
Hence, as Thales had previously proved, angles ABO and BAO are equal (alpha).
Likewise, angles OAC and ACO are equal (beta).
But, from the large triangle BAC, we see that:
2 right angles = angles ABC + ACB + BAC = 2 (alpha + beta)
Hence, one right angle = alpha + beta = angle BAC.
This is exactly what we were to prove.
After Thales, the next major figure in Greek mathematics was Pythag oras. Born in Samos around 572 B.C.