of her younger brother and sister, Sandio and Dioshane,
so to elucidate Sandio is playing the part of Euclid, who in his
life wrote many books but the one called
'The Elements' (c300 BCE), is unequalled in the history of science and could
safely lay claim to being the most influential non-religious book of all
time, Euclid's great work consisted of thirteen books covering a vast body
of mathematical knowledge, spanning arithmetic, geometry and number theory, the
books are organised by subjects, covering every area of mathematics developed
by the Greeks:
·
Books
I - IV, and Book VI: Plane Geometry
·
Books
XI - XIII: Solid Geometry
·
Books
V and X: Magnitudes and Ratios
·
Books
VII - IX: Whole Numbers
the basic structure of the elements begins with
Euclid establishing axioms, the starting point from which he developed 465
propositions, progressing from his first established principles to the unknown
in a series of steps, a process that he called the 'Synthetic Approach.' He
looked at mathematics as a whole, but was concentrated on geometry and that
particular discipline formed the basis of his work,
Euclid's Axioms
Euclid
based his approach upon 10 axioms, statements that could be accepted as truths.
He called these axioms his 'postulates' and divided them into two groups of
five, the first set common to all mathematics, the second specific to geometry, some of these postulates seem to be self-explanatory to us, but Euclid operated
upon the principle that no axiom could be accepted without proof.
Euclid's First Group of
Postulates - the Common Notions:
1. Things which are equal to
the same thing are also equal to each other
2. If equals are added to
equals, the results are equal
3. If equals are subtracted
from equals, the remainders are equal
4. Things that coincide with
each other are equal to each other
5. The whole is greater than
the part
The
remaining five postulates were related specifically to geometry:
1. A straight line can be
drawn between any two points.
2. Any finite straight line
can be extended indefinitely in a straight line.
3. For any line segment, it is
possible to draw a circle using the segment as the radius and one end point as
the centre.
4. All right angles are
congruent (the same).
5. If
a straight line falling across two other straight lines results in the sum of
the angles on the same side less than two right angles, then the two straight
lines, if extended indefinitely, meet on the same side as the side where the
angle sums are less than two right angles.
(I should point out that
in 1795, John Playfair (1748-1819) offered an alternative version of the
Fifth Postulate, this alternative version gives rise to the identical geometry
as Euclid's, it is Playfair's version of the Fifth Postulate that often appears
in discussions of Euclidean Geometrys so,
5. might be read
as, Through a given point P not on a line L,
there is one and only one line in the plane of P and L which
does not meet L),
Euclid
felt that anybody who could read and understand words could understand his
notions and postulates but, to make sure, he included 23 definitions of common
words, such as 'point' and 'line', to ensure that there could be no semantic
errors, from this basis, he built his entire theory of plane geometry, which
has shaped mathematics, science and philosophy for centuries, he proved that it
is an impossibility to find the 'largest prime number,' because taking the
largest known prime number and adding one to the product of all previous primes
and the largest prime will give you another, larger prime number, all of that
from a single picture, way to go Sandio!
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