What Greek mathematicianwrote the book Elements?
The Greek mathematician and geometrician Euclid (c. 325–c. 270 BCE) made some of
the most significant improvements to geometry in his time. (For more about Euclid,
see “History of Mathematics” and “Foundations of Mathematics.”) One contribution
was his collection of 13 books on geometry and other mathematics, titled Elements
(or Stoicheionin Greek). This work has been called the world’s most definitive text on
geometry. The first six books offer elementary plane geometry, with sections on tri-
angles, rectangles, circles, polygons, proportions, and similarities; the rest of the
books present other mathematics of his day, including the theory of numbers (books 7
to 10), solid geometry, pyramids, and Platonic solids. These books were used for cen-
turies in western Europe; in fact, the elementary geometry many students learn in
high school today is still largely based on Euclid’s ideas on the subject. 167
GEOMETRY AND TRIGONOMETRY
What are the five postulates of Euclid?
E
uclid was also famous for his postulates, propositions (statements) that are
true without proof and deal with specific subject matter, such as the proper-
ties of geometric objects (for more information about postulates, see “Founda-
tions of Mathematics”). Along with definitions, Euclid began his text Elements
with five postulates. These postulates are as follows (some of which may seem
obvious to us now, but in Euclid’s time they had yet to be formally recorded):- It is possible to draw a straight line from any point to another point.
- It is possible to produce a finite straight line continuously in a straight line.
- It is possible to describe a circle with any center and radius.
- All right angles are equal to one another.
- Given any straight line and a point not on it, there “exists one and only
one straight line which passes” through that point and never intersects
the first line, no matter how far the lines are extended. Another way to say
this is: One and only one line can be drawn through a point parallel to a
given line. This is also called the parallel postulate.
Mathematicians first believed this last postulate could be derived from the first
four, but they now consider it to be independent of the others. In fact, this postu-
late leads to Euclidean geometry, and eventually to many non-Euclidean geome-
tries that are made possible by changing the assumption of this fifth postulate.
Like many early attempts at explaining mathematics, not all these postu-
lates tell the entire geometric story. There were still a large number of gaps,
many of which were gradually filled in over time.