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Question 1

Master Euclid

Answer 1

Euclid is the ancient Greek master of geometry.
The Elements is his geometrical masterpiece.

Question 2

The Structure of the Elements

Answer 2

Euclid’s Elements is composed of statements of
three types: definitions, postulates and
theorems.

Question 3

Definitions

Answer 3

A definition tells us precisely when a term or
phrase applies and precisely when it does not.

For instance, Euclid defines an isosceles triangle
as a triangle that has two and only two equal
sides. Thus if we know the side lengths of a
triangle, we know whether “isosceles” applies or
not.

Question 4

Postulates

Answer 4

The notion of a postulate is complex. We’ll have
much to say about it. The best we can do now is
this: a postulate is a basic assumption, where by
“basic” we mean “unproven”.

Here’s one of Euclid’s postulates: things equal to
the same are equal to one another.

Question 5

Theorems

Answer 5

A theorem (called by Euclid a “proposition”) is a
statement that’s been proven true. Proven how? It’s
been derived from definitions, postulates and
previously proven theorems. (What’s “derived
from” mean? We’re not ready for an answer.)

Here’s a famous theorem: in a right triangle, the
sum of the squares of the legs equals the square of
the hypotenuse.

Question 6

Mathematical System

Answer 6

It’s one in
which we prove every statement that is not a
definition or postulate. What binds the
statements within our system is proof.

Question 7

Provers

Answer 7

We build our system by
proof.