The definition of the spreading out product and the closing in product in the Vorrede was of particular difficulty for me to translate. Auseinandertreten and Annäherung are very general terms that have many meanings even in the limited context of just 2 Strecken!
I was driven to try to grasp how they related to the hands of a clock, because I had the parallelogram product , which is the construction of a parallelogram, clear in my mind. Bad as my German is , the clock hands analogy still seemed too definitive. In any case the clock hand was my explanatory interjection, Hermann does not use it , and I wondered why, if that was what he meant.
I now perceive that he was not thinking about the point rotation of the Strecken , the jostling or pushing against each other of the line segments was identifying their Contiguity! Auseinndertreten is agin identifying the Contiguity of 2 line segments particularly in a progression of line segments. Thus even if the line segments do not have the same orientation( nb not direction) travelling along them in the same direction is continuously " stepping one out of the other". The important properties of contiguity and being beyond the boundary of a preceding line segment is perhaps not usually dwelt on, but continuous and contiguous clearly are words that came from such sorts of meditations.
The contiguous property of any 2 objects is a concept of a common contact. It only really has validity in an ideal or formal setting when any 2 juxtaposed objects can be assigned one common interface or interconnect. In the real world 2 obJects pushed together retain there 2 distinct boundaries or they meld together and lose the distinct boundary altogether! Thus contiguity is an ideal interpose between these real states. One state we call discrete, the other we call continuous and the ideal in between state we call contiguous because we posit only 1 common boundary.
Auseinandertreten thus describes this contiguous progression of one distinguished entity to another.. In the case of a triangle AB has BC stepping out from it, but AC does not step out from either , even though it is contiguous with both at its endpoints.
In the case of a parallelogram AB , BC, CD and DA form a contiguous progression around the form. This I have identified in the cyclic interchange of labels as the root idea of Hetmanns anti commutative factorisation. AB.BC as a product clearly has the line segments stepping out of one another, whereas BC.AB does not. The second notation is related to the first by using a "– " sign. This simply means the progression is the reverse of each other.
However what Hetmann noticed was that meaning carries through from the fundamental " point" level where AB and BA are the reverse progression. This behaviour captured in the notation revealed to him the behaviour of the thinker, or the mathematician. The notation imports the observers or the constructors progressive sequencing. This is important in any synthesis process. Any manufacturing process relies on the proper sequence being followed.
Yet in arithmetic it was common to discount this progression. Here Hetmann recognised a fundamental dialectical contradictory statement of the same process. In one field it was ignored but in the geometrical construction arena it was vital. In this instance the resolution in going from geometry to arithmetic had created profound logical difficulties. It also had obscured an important natural product process. After trembling with shock for a few months Hemann embraced the distinction and moved forward, creating the concept of an anticommutative product and a whole new arithmetic based on it.
We call these differing arithmetics Algebras, chiefly because they screw with our brains, but etymologically because they relate to the Arabic word Al Jibr, which roughly translated still means screwing around with our brains!
So now Annäherung refers to the property of 2 things coming into contact. They must finally meet or collide and when they do they become contiguous. However, the directions are not the same in the objects or things so that one is not stepping out of the other. Rather both are colliding and opposing or prevented from entering the other by the interconnect or contiguous boundary.
In the triangle AC and BC are closing in on one mother nd meet at C. The orientations of the line segments means one cannot say the direction of travel is opposite. In the formal sense the direction of travel is the same for each orientation, because we define the orientation of a line segment relative to a principal orientation by rotation, and that principal orientation also is used to define its principal direction. Thus a rotation about a point does not change he principal direction of a line segment. However rotating a line segment by a half tun and then disconnecting it and translating its other end to the centre of rotation describes the concpt of opposite or opposing( gegenüber). In this case the Kline segments go from spreading out from each other by rotation to coming together or colliding by translation rotation and direction, providing there is a meet or point or interconnect, a contiguous boundary.
The ability to disconnect and translate line segments changes all these conceptual relations haphazardly, necessitating a re description of the positional relationships after each such change. Relativity is fundmental to our apprehension of everything.
In the context of the parallelogram we have line segments that progress continuously and thus can outré cyclic interchange. Or we can notate so that line segments collide with each other, or we can have paths of progression that eventually collide. Also within the parallrogram we can notate the triangle form, and so have progressions and collisions based on that level of analysi. Introducing the triangle into the parallelogram is natural for the geometrical system.
The natural introduction of the triangle is through a line from corner to corner diagonally. In geometrical texts one will find forms broken down into their " pieces" , each piece named and any important properties listed. What one did not normally see until Hermann was a listing of orientation and direction.. These were just " understood". So diagonal is an orientation, as is parallel. In the case of parallel it became fashionable to notate it with arrowheads pointing in the same direction. Direction was understood by the positioning of the point labels in the notation.
The introduction of the vertical projection, so as to utilise Pythagoras theorem or trigonometric ratio tables is a far more " mathematical" introduction of a triangle into a parallelogram. In this case the line segments, the vertical drop or perpendularity drop and the base line segment it is dropped onto are clearly not progressively contiguous. They are Closing in on one another and meeting at a point. This is such an important construction that it does deserve a name of its own, but the closing in product is not my first choice as a name!
The name of this construction has always been presented to me as dropping a perpendicular. However when you go back to Euclids construction it is not so neatly temed. The process is described as forming a line through a point not on a given line that meets the given line at a right angle, or similarly forming a line that bisects a given line segment. In relation to that label, "closing in product" is very good indeed
.
However, in the parallelogram we can use the diagonal in this way, or we can relabel the points so 2 adjacent sides of the parallelogram meet in this way. Given that we have such a choice it is no wonder that a formal convention is established. The formal convention that Hermann learned and knew was established by Justus , his father, in the school district of Stettin. However, Justus was not like any other primary educator of his time. He held very progressive views and philosophical positions. Thus his scheme and implementation was progressive and innovative, and gave the students of Stettin a fabulous advantage. Jakob Steiner is a former pupil of a Pestalozzi school like the one established by Justus in the Stettin school system. He, after Newton was one of the premier geometers of modern times. He eschewed using " algebra" preferring the synthetic approach. It is possible that Justus was heavily influenced by Jakobs curriculum ideas and implementation.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Steiner.htmlA nice twist is that Hermann was able to take up a gymnasium seat Jakob had just left when he was called to Berlin.
So the concept of line segments stepping out from one another is clearly ambiguous enough to incorporate the swinging out from each other around a common point, but each to its own. Hermann clearly distinguishes these swinging line segments in the Vorrede , but in the context of the closing in product, the trigonometric and hyperbolic trigonometric contexts. He retains the Auseinandertreten for the progressive contiguity of line segments, whether in a straight line or in a closed petimeter.
October 15, 2017, 08:00:59 AM
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