3. The Arrow. Does the arrow move when the archer shoots it at the target? If there is a reality of space, the arrow must at all times occupy a particular position in space on its way to the target. But for an arrow to occupy a position in space that is equal to its length is precisely what is meant when one says that the arrow is at rest. Since the arrow must always occupy such a position on its trajectory which is equal to its length, the arrow must be always at rest. Therefore motion is an illusion.
Oh, my Parmenides, do you not know that all things move in relative motion? Else no thing could move, no thought proceed from your noble mind! …
Euclid starts with the circle (Kuklos). This is a Schema( a figure) with an Oros(boundary) the extreme of a figure also called a periphery. This periphery we have given a latin derived name, a circumference. This extreme boundary Euclid calls a gramme, meaning that it can also be drawn as a gramme by a tool with a kentros(a sharp point). Such a tool we again latinise as a pair of compasses. Such a pair is an isos, that is a tool for measuring the orientation of two gramme specifically when they are ortes(orthogonal). Thus the relationships between the circle schema are manifold and complex, and it takes Euclid 3 books to arrive at a satisfactory arrangement of the most important ones.
In arranging the relations within a circle in a formal way that is based upon definition and use and observation and logic(comparisons and deductions from them} ,And according to the "Isis" System Where a vein is divided from a larger vein by deduction and definition, and also a larger vein is built up from a smaller vein by induction based on definition, he develops the combinatorial system for circles and figures within them. His 2 main combinatorial schema are the gnomon and the curved gnomon as schema in the plane formed by a circle. However, a plane is a more rudimentary idea as is a kentros, for a pair of compass cannot draw a periphery of a kuklos without a plane. It turns out that the idea of a plane requiresan idea called a Good gramme, and this in turn requires the idea of a seemeioon, a mysterious portal between worlds that draws subjective attention to itself and flings back information to the subjective processing centre about its region of space.
It turns out that the straight line can only be derived from the circle by an infinite iteration of intersecting arcs, but one cannot arrive at a flat surface without starting with a flat surface. Thus we can start with the notion of a point and a plne and definee a circle and a straight line can be constructed from circular arcs, from which we may define an identity between the plane and a system of such circles which construct a system of straight lines.
Such a system therefore contains every relationship between rectilinear schema and points, and moreover we may observe that in fact 3 points are all that is needed to "distinguish" a plane with 3 straight lines. As a consequence any circle is distinguished also by 3 such points , and thus we may distinguish a plane by these characteristics of a disc or 3 points that lie on such a discs periphery( indeed any 3 points within its schema!).
How does one therefore recognise a plane? This again is a consequence of an isos and a fixed kentros and our orthogonal sense. By rotating an isos tool so it is always ortes to us one arm of the tool remains in an orientation to the other tthat we define as planar. Again, nowadays we use a spirit level to determine this, and in fact we use the waer surface leve in a ppe for the most accurate artisan measurements.
We might think that these have been susperseded by lases, but in fact straight is not the same as planar, and in fact straight is derived from the more fundamentl planar. Do we not need straight to distinguish ortes? in fact we do not. Ortes is a subjective sensory measure we make which does not rely on straight. Thus in greek the line is never called straight, it is called "good". Such a good line is aesthetically pleasing and subject to iteration in definition toward some unachievable ideal. Since it is unachievable let it always be known as good enough!
Thus the seemeioon and the isos tool and rotation and ortes the sense of orthogonality are needed to develop the notion of plane along with the many examples of calm waters and flat grasslands. The isis system is then used to develop the combined picture of the formal realities vein upon vein, fibre upon fibre. …