A point is that which hath no part
A point is that which has no part.
Euclid and his school set out a developmental, systematic and thus logical exposition of the subject they were engaged in.
Why start at a point. I have discussed before the difficulties around the notion of point.
It is unlikely that Euclid started at a point, and more likely this is a redaction introduced over time as the system grew in influence and popularity.
If this observation is accurately translated then we have no problem, for it clearly states a point has no part. Thus the problem lies with the interpreters, and for me the most useful interpretation is that a point is an abstracted notion derived from the sharpened corners, hooks and vertices of objects, and serves as a reference marker to hook, position, or measure from.
The common, observational nature of these definitions is not to be overlooked, for Euclid wrote for those who were artisans, not just abstract thinkers. All his audience therefore knew how to abstract from a real thing and were never confused about there role in the elevation of everyday palpable thing to eternal verities.
In every definition Euclid appeals to their perceptions, there experiences , and their judgements of form, shape and surface to establish a common denotation, a common jargon that they will use to shorten communication, and to increase precision, but never to obscure or mystify for the purpose of aggrandizement. His 23RD DEFIFNITION is a very practical explanation of what he will denote by the term parallel.
Some have taken these definitions in the sense of properties, but i take them in the sense of attributional denotations, after all what is a property if it is not attributed?
Thus we may see Euclid's practical use of the sensory system to establish a common language, and to promote a common observation.
Without this kind of acceptance of common terms, no agreement can be reached .
Much is different between Euclid and modern terminology . Definition 8 for example defines angle as "angle using boundary lines", and the inclination of those boundary lines. This is very interesting, because an angle to euclid is a measure of inclination and is dynamic. Therefore if a boundary line is a circle the inclination to a straight line is constantly changing as one progresses along the boundary: inclination therefore can only be a measure of rotation relative to an orientation.
Euclid then denotes a special type of rotation one that is fixed and unmoving, in terms not of rotation, but motion along a direction traced out by a boundary line. Rectilinear motion produces fixed inclinations , non rectilinear motion produces variable inclinations/rotations.
Euclid uses this notion to demonstrate that a circle boundary is a greater rotation than any acute rectilinear rotation that is clockwise or less than any that is anticlockwise depending on which boundary you are rotating from as you move along the curve boundary.
The notion of angle therefore is precisely linked to rotational motion and comparison with articulated jointing.
`in exploring this in one of his theorems Euclid appeals to contradiction of established rules, as a explanation, by demonstration, why he asserts certain things. But the support in this manner is given by contradictions of simple real life situations, so for example a length cannot be both longer than it must be, when the length it must be is determined precisely.
The definition of plane angle, definition 8 is i suspect not couched using the modern notion of angle , because Euclid appears ti define form in terms of boundaries not angle. Thus we have trilateral, quadrilateral and multilateral rectilinear forms ,with these other curvilateral shapes.
The concept of angle is a very important concept for all of Greek geometry. Many of the propositions require angles even for their statements.
The two lines are meant to emanate from the same point; two intersecting lines will actually make four angles.
The concept is also a difficult one, and, surprisingly, broader than our modern concept of angle.
As can be seen from the next definition of rectilinear angle, angles do not have to have straight sides; they can have curves as sides. The size of the angle does not depend on the length of the sides, but is determined only by how the two sides meet. In the Elements nearly all the angles are rectilinear, but angles with curved sides appear in proposition III.16. In that proposition, a so-called horn angle CAE is described as the angle between a circle and a straight tangent line and is shown to be smaller than any rectilinear angle FAE. Even though the curved side of the horn angle extends beyond any rectilinear angle, it is considered to be smaller since near the vertex A of the angle, the curvilinear angle CAE is entirely contained in the rectilinear angle FAE. Thus, an angle doesn't have an extent.
As you may see , the interpreter has the problem of what is meant by the terms. I believe the notion of angle or hook is an foreign one to the greek notion of inclination/rotation as a dynamic constituent of motion along a "curve". For a rectilinear"curve": that is a straight line there is no rotation from a fixed line or orientation.
Greek geometry is dynamic and full of motion and actual sensate experience from which deductions and abstraction, definitions/fixed starting points are drawn. to not observe that that is the case is to create unusual problems of your own devising.
Euclid could easily consider a rotation relative to a curved reference trace, but it had been found that the most communicable reference frames were rectilinear trilateral forms with perpendiculars, and rotation relative to the boundaries of the space between the perpendicular could be most easily exposited.
To think that the greeks were not aware of relativity and the need to establish and use common reference frames is to be disingenuous
Note that there is no requirement that the angle be rectilinear, indeed, the horn angles mentioned before are not rectilinear, but they are less than right angles, and so are acute (notwithstanding Proclus' remarks to the contrary).
With these definition, we see another aspect of magnitudes, namely, two magnitudes of the same kind, such as two angles, can be compared for size. Euclid frequently uses what is known as the law of trichotomy: given two magnitudes F and G of the same kind, exactly one of the following three situations hold, F is less than G, F equals G, or F is greater than G. See the comments after the Common Notions for more discussion on magnitudes and the law of trichotomy.
Trichotomy is so evident in the very fabric ans structure of Hamilton's Exposition of conjugate functions or couples, that it is hard not to cry out Euclid/Eudoxus when reading this paper. Therefore the soundness of Hamiltons conceptions depend entirely on he soundness of the greek style of reasoning. Experiencing Hamilton's thorough exposition is a modern example of an understanding of greek deduction, unfettered by translation. Hamilton emulates the style and dependencies of the greek style and thus communicates an activity Also encoded in the greek, but not written in greek: Style.
In scripted form, in dance moves and in spoken or song words, the nonverbal communication is as important and relevant as the denotation of a grapheme or a phoneme. Thus, though Hamilton clearly limits his subject to "time", he , as he states, uses geometry analogically. Therefore, in reality, there is nothing that is not touched or informed or shaped by the greek science of geometry. I refer to it as spaciometry partly to free myself from modern misconceptions of dynamic greek geometry, and partly to emphasise that it is about the measurement of space by comparison of unit spaces in motion.
The revealing of the foundation of ars, that is motion sequents, though fundamental, and involving experiential,sensate perception, sensory mesh computational outputs, is more detailed rather than new , Greek dynamic geometry would always arrive at Newton' s Fluxions, Hamilton's couples and quaternions, my moment sequents, but only as the ideas of Brahmagupta, Bombelli, Descartes,and DE Fermat, Laplace,Lagrange all contributed to a common rhetoric and notation for referring to it.
Amongst the greek ideas, those who were students to the philosophers had understanding, those of us who have learned from translations of their work have conjecture and opinion. It is only as the utility of the ideas are vrified that we gain insight and understanding of what the philosophers undoubtedly knew and meant.
Do i mean by this that we have not advanced beyond our father? Indeed i do. And by what little we have gained in advancement, the more we have appreciated the state of their wisdom and knowledge and conjecture.However i am not saying that we should look back, or "go back", but rather we should go forward, bringing all wisdom forward with us so we might properly understand it by experience.
I am reminded that it is but 200 years since it was scientific dictum that life arose spontaneously from common everyday materials like wheat and rags! You may laugh, but it is still scientific dictum that life arose from common elements in chemical reactions under very violent conditions: the same idea but with more detail!