Diana's Family In The Philippines Sent Us This Picture,

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!