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NON-EUCLIDEAN GEOMETRY:
THE FALSITY OF EUCLID'S FIFTH POSTULATE.
(2004)
PAUL S. SIDLE.
Ā
János Bolyai(1802–1860);
Johann Friedrich Karl Gauss (1777–1855);
Nikolai Ivanovich Lobachevski (1793–1856);
Georg Friedrich Bernard Riemann (1826–1866).
NON-EUCLIDEAN GEOMETRY: THE FALSITY OF EUCLID'S FIFTH POSTULATE.
(2004)
PAUL S. SIDLE.
Geometry comes from the Greek geōmetria, representing the part of Mathematics
(Greek mathēmatikos: Science concerned with functions of values [Latin valēre,
of worth: estimate in exchange, equivalent of a thing, represents else
substituted for, etc., hence thing represented, etc.; by symbols {Greek
sumbolon, token: mark, character, letter, sign, etc.; interchangeable with sign
as token representing something}, signs {Latin signum: mark, token, 'picture',
motion, gesture, cue, symptom, indication, symbol, etc}, objects {object from
Latin objectāre: thing, 'figure', organized whole, gestalt of an external event:
involves as processes representing an event}, etc., for example by words,
mathematics, perceiving, visualizing, etc]; otherwise of relations in patterns
[structure]) concerned with functions (Latin, functiōnem: to do, perform,
operate; function of - how values relate; infinite-valued, differentialintegration, non-linear-asymmetry-non-additive) of curves, surfaces, shapes,
dimensions (Latin dīmensio from dīmetīrī, to measure: measurable extent, in
terms of length, breadth, height, thickness, etc), etc., hence not just
'spatial' but continuous magnitudes (Latin magnitūdo from magnus, great): size,
extent, amount, etc.; anything that can become measured; importance; etc. Where
arithmetic (Greek arthmētikē techne, act of counting), apart of Mathematics
instead concerned with numbers (computation of numbers, etc), hence discrete
magnitudes. Since it remains with the Greeks that any sophisticated geometry
has survived, who tended to view it as a 'deductively' (a priori, Latin for
prior to observation(s): 'deduction' [Latin dēdūcere, from Greek apagogic, to
lead: to 'deduce'; to infer necessary consequences from given premises] an
'identification' [Alfred Korzybski's (1933) term for treating an abstraction,
anything, etc., as the 'same', by the ignoring {'filtering' out} of facts] from
'universal' [Latin ūniversālis, of 'everything': 'always same, all true';
'common to all' cases – general] abstraction(s) to observation(s); part of
'logic', from Greek logikē: art of 'reason' [Latin ratiōnem: infer, 'think'
{Greek denkem, 'reason': infer, symbolize, formulating, etc., of higher orders
of abstracting}, 'thought', to theorize symbolically, etc]: inferring [Latin
inferre, to bring: higher order abstraction based on the 'facts' from
observation(s)], further the rules it operates by) organised (from Greek organon
[to organize, but meaning instrument else 'logic': an instrument for organizing
knowledge; a system of principles, rules of investigation, 'logic', validity,
etc.; equivalent to methodology, etc], but to make whole, equivalent to
abstraction, representation, gestalt, etc) discipline, from which a theoretical
knowledge of 'essence' (Latin essentia, 'to be': essential 'element' [Latin
elementum from Greek Stoichia, to 'analyse', 'atomize', etc {'el'}: divide to
the indivisible parts, the non-separable, contextually interchangeable whole(s)
from part(s)], equivalent to 'attribute' [Latin attribuere, to assign: typical
'element', equivalent to 'essence', 'property', etc], 'property' [Greek
proprium: that owned by; typical 'element', equivalent to 'attributes',
'essences', etc], etc), demonstrations (Latin demonstrātus: to show, indicate,
display, evidence of; describe-explain by means of specimens, experiments, etc.,
provide proof beyond doubt) can become evident. For example, the internal
angles of a triangle must equal two right angles (180o), given the apparently
undeniable truth of the kind of propositions (statements, etc) which Euclid (300
B.C.) adopts as axioms-or-postulates involving a definition of a triangle.
Where such a demonstration provides knowledge not merely that 'something is so',
but further an understanding of why it 'must be so', given the nature of things
concerned.
To Pythagoras (580-500 B.C.) along with Plato (428-347 B.C.) the example
of Mathematics suggested that knowable 'reality' (Latin realitātem: that
underlying appearances, 'existent', of actuality, etc) represented by shapes,
sizes, etc., as forms, 'must be' an unchanging realm behind that of changing
appearance. Aristotle (384-322 B.C.) instead insisted that shapes, sizes, etc.,
have no ‘existence’ except as aspects of physical, changeable things, which
represent the things, which have primary 'reality'. Such that from this point
of view the problems raised by geometry concerned the relation between forms
with items in the physical world, how knowledge of forms becomes possible,
further how this knowledge has uses in the physical world.
So, far from taking geometry as the Science (Latin scientia from scire,
to know: knowledge attained by experiment-observation-induction [Latin
inductiōnem, from Greek epagoge, to lead: to adduce; to infer premises from
given observations; hence a posteriori from Latin for after observation(s):
induction from observation(s) to abstraction(s)]) of physical 'space' (Latin
spatium: continuous extension, usually assumed as a 3-dimensional void; an
interval of 'time'; but not 'empty', hence interchangeable with 'time', via
fullness of changing 'matter'), the Greeks tended to see it as the Science of
continuous magnitudes, concerned with ratios, proportion, etc., further the
methods of construction enabling determination. The physical 'space' of the
Aristotelian (after Aristotle (350 B.C.)) universe is not the 'infinite (Latin,
infīnītus, not finite: greater than any assignable value, thus  > n),
homogeneous space' of Euclidean (E: geometry of 'straight lines' constructed by
Euclid (300 B.C.) from his fifth postulate) geometry, but the highly structured
(Latin structūra from struere, to build: emergent [Latin ēmergere: to appear,
become apparent, formed newly from dependent parts] from ordered [Latin,
ordinem: arrangement, disposition in space-time] relations [Latin relātus, to
refer: way in which a thing connects with; may become formulated in regard to
another as dependence-interdependence, similarity-difference, correspondencecontrast, reversible-interchangeable-equivalent-over-lapping, more-or-less,
etc]; equivalent to gestalt [German: form {shape hence 'figure'}, pattern,
configuration, unity, whole, organization, representation, etc., hence
emergence, thus structure; after Max Wertheimer (1912)], whole [combination of
parts], representation, etc) bounded series of nested spheres centred on the
earth.
Yet the acceptance of 'E' geometry came about from a cumulative number of
diverse factors operating from the beginning of the Renaissance, that led by the
18th Century to the connection of the physical with geometric 'space' to Isaac
Newton's (1687) 'absolute (Latin absolūtus, independent: self-existent, not
relative) space'. Perhaps the most important of these factors remain:
(1). René Descartes’ (1637) Cartesian co-ordinate representation of
extensions, shapes, etc.
(2). The geometrical treatment of problems of perspective:
(a). A tendency to treat geometry as descriptive of both perceptual with
physical 'space' as 3-dimensions, for example exercised in the work
of Italian artists (Latin ars artem: practical application, etc., of
Science) such as Leonardo De Vinci (1452-1519) who gave geometry a
role in accounts of the mechanics of 'visual' perception.
Figure 1.
Leonardo Di Ser Piero Da Vinci (April 15, 1452 – May 2, 1519).
(b). Development of the methods of projective geometry: 3-dimensional
'solids' transformed into 2-dimensional shapes in perspective; for
example used by Italian artists in compositions.
(3). The adoption of Nicolaus Copernicus' (1543) sun-centred view of the
universe, shattering Aristotle’s (350 B.C.) crystalline spheres, leaving
the earth spinning through an 'infinite, homogeneous 3-dimensional
E-space'.
Figure 2.
Nicolaus Copernicus (February 19, 1473 – May 24, 1543).
Thus geometry provided, further to some extent continues to provide a
paradigm (after Thomas Samuel Kuhn (1962), a framework, equivalent to overview,
orientation, system [Greek sustēma: methodology, orientation, organization, a
method, etc], methodology [principles concerning procedure for investigation,
verification, etc., along with those of evaluating; hence a system for
organizing knowledge], etc) for scientific knowledge along with understanding:
'apprehending' (Latin apprehensio, to seize: 'anxious', 'fear', etc; 'intuitive'
[Latin intuitiōnem, to look: anticipation, immediate 'apprehension', falsely
connected to 'innate' {Latin innātus, to be born: inherited; passed on, thus
known before birth} knowledge by 'rationalists' from Latin ratio, 'reason':
after Renē Descartes (1637), having origins with Plato's (381 B.C.) 'idealism'
from Greek idein, to see: epistemology asserting that our values come from
'innate' 'universal' idea], understanding, etc), comprehending (grasp as
intelligible, understanding, etc), assimilating (Latin assimilāre, liken to:
compare, make alike, absorb, etc); however mis-understood as only 'intellectual'
(Latin intelllectus, as 'intelligent': that involving 'reasoning', involving
understanding, etc.; distinguished from 'feelings', 'willing', etc), since
requiring insights, after Wolfgang Köhler (1925), gestalt term referring to
sudden re-organization (re-representation), realization, process of establishing
new organized wholes, etc. However the very feature of geometry which singles
it out as an orientation (world-view, methodology, paradigm, system, etc) for
other Sciences to aspire to, further became the source of epistemological (Greek
epistēmē, knowledge: theories of knowledge, ways of knowing, the justification
of values; part of 'Philosophy' from Greek philosophos: "Science of Sciences",
after Johann Gottlieb Fichte (1762-1814); pursuit of knowledge [know from Greek
gignōskein: to have experience of, hence experience], hence wisdom) problems
concerning the dynamic (Greek dunamikos: in motion, changing, etc) physical
world. Since geometry deals with a 'static (Greek statikos, to stand: 'without
motion, not active or changing'), unchanging timeless’ (Latin tempus:
interval(s) involving relations of a sequence of events, changes of something,
etc) world of the theorized physical 'space' consisting of pure shapes, sizes,
etc., (forms) a priori truths independent otherwise prior to experience,
contrary to the physical world of experience (Latin experientia from experīrī,
to go through): knowledge from observation (Latin observāre, to keep):
representation (stands for, corresponds to, depicts, shows, etc., events by
objects or-both symbols; equivalent to abstractions, organization, etc) of facts
(observations, observed actualities, what happened; of non-verbal events) by
Figure 3.
Plato (c. 428/427-348/347 B.C.);
Pythagoras of Samos (c. 580/572-500/490 B.C.);
Aristotle (c. 384–322 B.C.);
Proclus Lycaeus (February 8, 412 – April 17, 485);
Thales of Miletus (c. 640-546 B.C.);
Theaetetus of Athens (c. 417–369 B.C.);
Eudoxus of Cnidus (c. 410/408 – 355/347 B.C.).
perceiving, 'description(s)', etc.
An elaborate history of Greek geometry from the earliest beginnings,
became compiled by Eudemus (370-300 B.C.) of Rhodes, a pupil of Aristotle,
frequently referred to as the 'Eudemian Summary'. Further Proclus (412-85)
gives a brief review of the early history of geometry to the era of Euclid,
believed founded upon Eudemus. Thales (624-548 B.C.) of Miletus, had travelled
extensively, but during a visit to Egypt he became acquainted with the observed
rules of land surveying practised there. Referring to the Ionian school,
Proclus declared:
"Thales was the first to go into Egypt and bring back this learning
(geometry) into Greece. He discovered many propositions himself, and he
disclosed to his successors the underlying principles of many others, in some
cases his methods being more general in others more empirical".
Proclus further credits Thales with a knowledge of the following propositions:
(1). Any circle becomes bi-sected by their diameter.
(2). The angles at the base of an isosceles triangle 'equal' each other.
(3). If two 'straight lines' (extending 'uniformly' in one direction; not
bent, curved, etc) intersect, the vertically opposite angles 'equal' each
other.
(4). A triangle can become determined, if one side along with two adjacent
angles become known.
(5). The angle inscribed in a semi-circle involves a right angle.
Though Thales showed little interest in the practical aspect of
Mathematics, he supposedly has the credit of calculating the height of a pyramid
from the length of the shadow cast compared to that of a shadow cast from a
stick. Further the 'angle-sum rule' (the angles of a triangle come to 180o,
else two right angles) becomes ascribed to Thales, though the evidence for this
is not conclusive. However that Thales established the trend of geometry as a
'deductive' Science appears definitive. Ofcourse the Egyptians may well have
known the above facts, though if known they remained unrelated, since knowledge
for the Egyptians as for many pre-Greco civilizations become sought only as
necessity prompted it, usually involving mystic-magically superstitious beliefs
for the sake of power: capacity to do; whereas 'political' power entails,
exchange-power (material-organization) with co-ordination-power (leadership).
But for Thales such knowledge became the beginning of the Science of geometry.
Nevertheless from Eudemus, we gather that Euclid flourished about 300
B.C., (A.V. Howard (1961) suggests 330-275 B.C., though probably inaccurately)
further that the "Elements" ("Stoichia"), the work upon his reputation rests,
became compiled around 300 B.C. It appears probable from the style of his work,
that Euclid received his mathematical training in Athens from pupils of Plato,
if not at the Academy itself. However it appears definite that Euclid taught at
Alexandria, where he founded a school there. Further that Euclid wrote about a
dozen works other than the "Elements", of which only five of these have
survived: "Data", "Division Of Figures", "Phaenomena" along with "Optics".
However Euclid's (300 B.C.) "Elements" was not the first exposition of an
unification of geometry; we know of at least three earlier versions, including
one of Hippocrates (430 B.C.) of Chios. But Euclid's seems to have so surpassed
these that it alone has survived. Comprising of 13 books, comparatively few of
the propositions within the "Elements" appear as Euclid's own, since much of the
material derives from earlier sources. In the words of Eudemus:
"(Euclid)...put together the Elements, arranging in order many of
Eudemus' theorems, perfecting many of those due to Theaetetus, and also bringing
to irrefutable demonstrations the things which had only loosely been proved by
his predecessors".
Books I-IV concerned basic plane geometry, largely built upon the
knowledge of the Pythagoreans along with the sophists (sophism from Greek
sophos, wise: origins with Protagoras (490-420 B.C.), epistemology asserting
that our values come from personal [relative] belief(s)); Books V-VI develops a
theory of proportions, mostly credited to Eudoxus (408-355 B.C.) of Cnidus along
with Theaetetus (414-369 B.C.) of Athens, which overcame the problem of
incommensurable magnitudes, which later became explicitly tackled in Book X;
Books VII-IX deal with numbers along with ratios between numbers; while Books
XI-XIII chiefly became devoted to the geometry of 3-dimensions.
However Euclid's achievement involves a rigorous systematic organization
of geometry, one in which starting from axioms, definitions, postulates, etc.,
each proposition becomes proved directly from these, otherwise from these
together with propositions previously proved. Thus the "Elements" first printed
in a Latin translation from the Arabic in 1482, provided an early example of a
'deductive' organization of knowledge, which functioned as a paradigm for other
Sciences for over the next 2300 years.
Figure 4.
Representations of Euclid of Alexandria (c. 300 B.C.).
Yet Euclid's title to immortality has come to mean something other than
the supposed 'logical' perfection still falsely ascribed to him by many. This
involves that Euclid's (300 B.C) fifth postulate (axiom) as an assumption, is
false. The "Elements" begins with a list of definitions, postulates along with
'common notions'. Aristotle (350 B.C.) had made a sharp distinction between
axioms ('common notions') with postulates; the former having 'to be convincing
in themselves - truths common to all ('sum of parts', 'every' one of,
'everything', etc) considerations', while the latter appear less obvious not
pre-supposing assent, pertaining only to the subject under consideration. Some
later writers distinguished between the two kinds of assumptions by applying the
word axiom (Greek axios, worthy) to something known else accepted as obvious:
proposition assumed as self-evident truth, hence a priori; while the word
postulate (Latin postulāre, to demand) to something demanded: proposition though
provable, but assumed without proof. But we do not know whether Euclid
distinguished between the two kinds of assumptions (inference, etc., but a guess
if not backed by 'facts'), since surviving manuscripts are not in agreement
here. Nevertheless modern mathematicians view no essential difference between
an axiom with a postulate. Otherwise both termed premise (Latin proemissa): a
higher order abstraction (Latin abstractus, to draw: representing, organizing,
etc., of events by objects, symbols, etc., for example, perceiving, visualizing,
words, formulas, etc) from observation(s) (facts), hence inference, theory,
doctrine (Latin doctrīna, to teach: principle of scientific belief; see
premise), principle, analogy, equation, etc., any number of equivalent terms
linguistic or-both mathematical, dependent on context (Latin contextus, to
weave: 'meaning' related to circumstance; situation [background], relation of
parts, etc.; thus multi-ordinal [ordinal: defines a value’s order else position
in an 'aggregate'; multi-ordinality defines a value's order, relation to others,
etc., in a degree process, continuum, etc.; having multi-meaning, contextual to
level of abstracting] upon abstracting level), hence formulation. However in
most manuscripts of the "Elements" we find ten assumptions: five postulates of
which the first three involved postulates of construction, followed by five
'common notions', as quoted by Carl B. Boyer (1968):
"Postulates. Let the following be postulated:
(1). To draw a straight line from any point to any point.
(2). To produce a finite straight line continuously in a straight line.
(3). To describe a circle with any center and radius.
(4). That all right angles are equal.
(5). That, if a straight line falling on two straight lines makes the interior
angles on the same side less than two right angles, the two straight
lines, if produced indefinitely, meet on the side on which the angles are
less than the two right angles.
Common
(1).
(2).
(3).
(4).
(5).
notions:
Things which
If equals be
If equals be
Things which
The whole is
are equal to the same thing are also equal to one another.
added to equals, the wholes are equal.
subtracted from equals, the remainders are equal.
coincide with one another are equal to one another.
greater than the part".
However the fifth postulate entails that if a 'straight line' EH
intersects two other 'straight lines' AB along with CD at F with G, in such a
way that the angles BFG, DGF comes to less than two right angles. Refer to
Figure 5. Then the two 'straight lines' AB along with CD, will meet if extended
in the direction of B with D. Such that FBDG, will form a right-angled
triangle.
Nevertheless the fifth postulate alternatively known as the 'parallel'
postulate, since if the interior angles DGF, GFB, in Figure 5 come to two right
angles, then the two 'straight lines' AB with CD will not meet, hence continue
'parallel' to each other; can have many equivalent forms each of which becomes
'deducible' from any one of the others by means of the remaining first five
postulates of Euclid's geometry. Possibly the simplest of these equivalent
statements Eric Temple Bell (1937) phrases as:
"Given any straight line L and a point P not on L, then in the plane
determined by L and P it is possible to draw precisely one straight line L'
through P such that L' never meets L no matter how far L' and L are extended (in
either direction)".
Refer to Figure 6. As a nominal definition then, 'two straight lines lying in
one plane which never meet are parallel'. Thus the fifth postulate of Euclid
asserts that through P 'there is precisely one straight line parallel to L'.
E
A
B
F
D
G
C
H
Figure 5.
If the two interior angles F,G, come to less than two right angles, then the
line AB, CD, are not 'parallel'.
P
L'
L
Figure 6.
From Eric T. Bell's (1937) "Men Of Mathematics", volume 2.
Euclid's penetrating insight into geometrical knowledge convinced him
that this postulate could not be 'deduced' solely from the others (as many
previous failed attempts further served to establish), however wishing to use it
in proofs of many of his theorems, Euclid honestly included it with his other
postulates. Now Euclid's geometry of 'parallel straight lines' remains based on
his fifth postulate, so let us work through some theorem proofs to demonstrate
this basis of Euclidean geometry.
From Euclid's fifth postulate we can derive the following theorems (Greek
theorema, to behold: a proposition demonstrable by argument, proof, etc), each
following from the other, which have the following proofs.
Theorem 27: If a 'straight line' intersecting two other 'straight lines', make
the alternate angles equal each other, then these two 'straight lines shall be
parallel'. Refer to Figure 7.
E
A
G
B
K
C
H
D
F
Figure 7.
Let the 'straight line' EF intersect the two 'straight lines' AB, CD at G with
H, so as to make the alternate angles AGH, GHD equal to one another.
'So shall AB and CD be parallel'.
Proof.
For if AB with CD be not 'parallel', if produced they will meet towards B with
D, otherwise towards A with C.
If possible let AB with CD, when produced meet towards B with D at point K.
Such that KGH becomes a triangle, of which one side KG extends to A; therefore
the exterior angle AGH becomes greater than the interior opposite angle GHK.
But since the angle AGH remains equal to the angle GHK:
'Hypothesis'.
hence the angles AGH with GHK 'are both equal and unequal'; which is impossible.
Therefore AB with CD cannot meet when produced towards B with D.
Similarly by demonstrating A with C cannot meet:
'therefore AB and CD are parallel'.
Theorem 28: If a 'straight line' intersecting two other 'straight lines', make
an exterior angle equal to the interior opposite angle on the 'same' side of the
line; otherwise if it made the interior angles on the 'same' side together equal
to two right angles, then the two 'straight lines shall be parallel'. Refer to
Figure 8.
Figure 8.
Let the 'straight line' EF intersect the two 'straight lines' AB, CD in G with
H:
First, let the exterior angle EGB become equal to the interior opposite angle
GHD.
'Then shall AB and CD be parallel'.
Proof:
Because the angle EGB equals the angle GHD;
further because the angle EGB equals the vertically opposite angle AGH;
Therefore the angle AGH will equal the angle GHD;
but since as alternate angles;
'therefore AB and CD are parallel'.
Theorem 27.
Quad Erat Demonstrandum.
Secondly, let the two interior angles BGH, GHD together equal two right angles.
'Then shall AB and CD be parallel'.
Proof:
Because the angles BGH, GHD together equal two right angles;
further because the adjacent angles BGH, AGH together equal two right angles;
therefore the angles BGH, AGH will together equal the two angles BGH, GHD.
From these equals take the common angle BGH:
then the remaining angle AGH will equal the remaining angle GHD:
further as alternate angles;
'therefore AB and CD are parallel'.
Theroem 27.
Quad Erat Demonstrandum.
Theorem 29: If a 'straight line' intersects two 'parallel straight lines', then
it shall make the alternate angles equal to one another, further the exterior
angle equal to the interior opposite angle on the 'same' side; further the two
interior angles on the 'same' side equal to two right angles. Refer to Figure
9.
Figure 9.
Let the 'straight line' EF fall on the 'parallel straight lines' AB, CD.
Then (1). the angle AGH shall equal the alternate angle GHD;
(2). the exterior angle EGB shall equal the interior opposite angle GHD;
(3). the two interior angles BGH, GHD shall together equal two right
angles.
Proof:
(1). For if the angle AGH be not equal to the angle GHD, one of them must
become greater than the other.
If possible, let the angle AGH become greater than the angle GHD;
to each 'add' the angle BGH:
then the angles AGH, BGH together become greater than the angles BGH,
GHD.
But the adjacent angles AGH, BGH together equal two right angles;
therefore the angles BGH, GHD together become less than two right angles;
therefore, by postulate 5, AB with CD meets towards B with D.
But they never meet, since 'parallel'.
'Hypothesis'.
Therefore the angle AGH is not unequal to the angle GHD:
such that the angle AGH equals the alternate angle GHD.
(2). Again, because the angle AGH equals the vertically opposite angle EGB;
further because the angle AGH equals the angle GHD;
Proved.
therefore the exterior angle EGB equals interior opposite angle GHD.
(3). Lastly, the angle EGB equals the angle GHD;
Proved.
to each 'add' the angle BGH;
then the angles EGB, BGH together equal the angles BGH, GHD.
But the adjacent angles EGB, BGH together equal two right angles;
therefore the two interior angles BGH, GHD together equal two right
angles.
Quad Erat Demonstrandum.
These theorems based upon Euclid's (300 B.C.) fifth postulate intend to
prove beyond doubt that as H.S. Hall with F.H. Stevens (1902) had stated as
their axiom 10: "Two straight lines cannot enclose a space"; hence that
'straight lines exist'.
As pointed out, Euclid's (300 B.C.) object involved an erection of the
whole structure of his geometry upon a few definitions with postulates (axioms).
However not only Euclid's fifth postulate, but his fourth postulate became the
object of frequent attacks both in antiquity along with the modern era.
Allegedly the fourth postulate ("all right angles are equal") becomes capable of
proof by super-position. Yet it appears more than likely that Euclid was not
unaware of this, but had he accepted it, he would have had to postulate the
invariability (variable from Latin variāre: changeable; increase-or-decrease
proportionately-with-or-inversely to the increase-or-decrease of another value;
hence tending not to change) of figures both in size with shape on translation.
But Euclid had planned to use his fourth postulate to establish the congruence
of two triangles via his very first theorem. Actually, what Euclid intended as
postulate four appears tantamount to the invariability of figures on
translation.
However the infamous fifth postulate proved more critical. From
Clauchus Ptolemy (90-168) onwards attempts to prove it failed, but only
abandoned when finally realized that alternative geometries become devisable in
which this postulate plays no part. It is not that anyone questioned the truth
of it; the important point remains that it is not 'deducible', therefore cannot
be assumed without proof as expected by definition as a postulate else axiom.
One of the earliest attempts in the modern era to remove the uncertainty
(probability-uncertainty, interchangeable terms such that the uncertainty
[likelihood] of an event becomes measured by the ratio of the favourable chances
to the whole number of chances) surrounding the validity (to verify, test
[extent to which a test measures what it purports to measure, determined by
correlation between results], measure, correspond to actuality, etc.; concerning
truth-or-falsity; true as purports) of the axiom got made by Giovanni Girolamo
Saccheri (1667-1733) a Jesuit Priest who taught at colleges of his order in
Italy. Saccheri's (1733) "Euclides Ab Omni Naevo Vindicatus" ("Euclid Cleared Of
Every Flaw") intended to "vindicate Euclid from every blemish", arising out of
the doubtful status of the fifth postulate. Saccheri's developments appear to
have a starting point with the translations of Arabic mathematicians by John
Wallis (1616-1703) in the 17th Century.
Ibn-al-Haitham (965-1039) an Egyptian known in the West as Alhazen, had
begun with a tri-rectangular quadrilateral (often referred to as "Lambert's
quadrilateral"; refer to Figure 17), in order to prove that the fourth angle
must equal a right angle. Such that from this theorem on the quadrilateral, the
fifth postulate will become easily shown to follow. Where in his proof Alhazen
had assumed that the locus of a point that moves so as to remain equidistant
from a given line must necessarily 'be a parallel line to the given line' - an
assumption shown more recently as an equivalent to Euclid's postulate.
However Omar Khayyam (1050-1123) the Persian poet criticized Alhazen's
proof on the grounds that Aristotle had condemned the use of motion in geometry.
Instead Khayyam began with a quadrilateral, the two sides of which 'are equal
and perpendicular (at right angles from) to the base' (usually known as
"Saccheri quadrilateral"; refer to Figure 16), with which he investigated the
upper angles of the quadrilateral, which must necessarily equal each other. Of
the three angle possibilities:
(1). acute - angle < (less than) 90o;
(2). right - 90o angle;
(3). obtuse - angle > (greater than) 90o.
The first with third as possibilities, Khayyam ruled out on a principle of
Aristotle's, that two converging lines must intersect - an assumption equivalent
to Euclid's fifth postulate.
Nasir Eddin al-Tusi (else at-Tusi, 1201-1274) from Maragha, grandson of
Genghis Khan, brother of Kublai Khan, continued efforts to prove the fifth
postulate starting from the usual three 'hypotheses' (Greek hupothesis:
'deduced' particular, consequence, conclusion, etc) on a Saccheri quadrilateral.
Figure 10.
Claudius Ptolemaeus (c. 90-168), known as Ptolemy of Alexandria;
Johann Heinrich Lambert (1728-1777);
Adrien-Marie Legendre (September 18, 1752 – January 10, 1833);
Georg Simon Klügel (August 19, 1739 – August 4, 1812);
Alhazen (as known in the West) Ibn-al-Haitham (965-1039);
Omar Khayyam (1050-1123).
X
Y
?
A
?
B
Figure 11.
Girolamo Saccheri (1733) bi-rectangular isosceles quadrilateral
'Hypothesis' of the right angle.
X
Y
?
A
B
Figure 12.
Johann Heinrich Lambert's (1766) tri-rectangular quadrilateral.
Where his proof depended upon the following 'hypothesis', equivalent to Euclid's
axiom, given by Boyer (1966) as:
"If a line u is perpendicular to a line w at A, and if line v is
oblique to w at B, then the perpendiculars drawn from u upon v are less than AB
on the side on which v makes an acute angle with w and greater on the side on
which v makes an obtuse angle with w".
Nevertheless Saccheri had known of Nasir Eddin's efforts to prove the
fifth postulate half a millennium earlier, where upon Saccheri became determined
to establish postulate five by denying it, while seeking a consequent
contradiction. Refer to Figures 11, 13. Saccheri began with a bi-rectangular
isosceles quadrilateral, a quadrilateral which looks like a rectangle. Consider
a figure AXYB consisting of four 'straight lines' AB, BY, YX, XA, from which AB
the base, two equal perpendicular sides AX, BY, arise. Such that the base
angles XAB, YBA, must equal two right angles. Then if a line XY intersects the
two perpendicular lines AX, BY, then what might the angles AXY, BYX, become
equivalent to? Now without using the 'parallel' postulate, Saccheri easily
showed that these angles AXY, BYX, must equal each other; provable as a result
of the line XY intersecting two perpendicular lines AX, BY, both equal in
length.
Figure 13.
Nevertheless as Saccheri pointed out, this entails three possibilities for the
angle:
(1). 'hypothesis' of the acute angle;
(2). 'hypothesis' of the right angle;
(3). 'hypothesis' of the obtuse angle;
as Khayyam had considered. However Saccheri hoped to show by reductio ad
absurdum (disproving a false proposition by 'logically' 'deducing' an absurd
consequence) that 'hypothesis' 1 with 3 lead to absurdities, thus establishing
as a result 'hypothesis' 2 as a necessary consequence of Euclid's postulates
without recourse to using the fifth postulate. Such that if we assume these
angles as acute otherwise obtuse, then this led to conclusions at variance with
the rest of Euclid's axioms.
Saccheri had little trouble disposing of 'hypothesis' 3, for he
implicitly assumed a 'straight line' as an infinite extension: having 'same'
(Greek homos, 'identical': not other, different; exactly alike, undifferentiated
in 'all' respects) continuous extension. However from 'hypothesis' 1, he
derived theorem after theorem without encountering any difficulty. In fact
Saccheri had begun to build a wonderfully consistent
Non-Euclidean geometry, however so thoroughly inspired with the conviction that
only Euclid's geometry has validity that Saccheri permitted this 'prejudice'
(false assumptions, 'perceptions', etc., from 'filtering' out 'facts';
equivalent to 'identification(s)', etc) to interfere with his 'logic'. Where no
contradiction occurred, Saccheri twisted his 'reasoning' until 'hypothesis' 1
led to an absurdity.
Nevertheless Saccheri had only proved that without using Euclid's fifth
postulate, then it is impossible to prove that AXY, BYX, will despite looking at
them, equal two right angles. Further, if we assume that AXY, BYX, equal two
right angles, then we can prove Euclid's fifth postulate. Since the
'hypothesis' of the right angle has an equivalence to Euclid's axiom.
Indeed until about 1800, mathematicians who worked along these lines,
imagined that they operated with 'E' geometry so that they would eventually
'deduce' the fifth axiom as a theorem. Many believed that they had attained
this aim, though in fact they had merely replaced axiom five by other equivalent
assumptions, such as "two parallel lines are equidistant", otherwise "three noncollinear points always lie on a circle", etc.
Saccheri's conclusions became challenged by Johann Heinrich Lambert
(1728-1777), a Swiss-German theorist of a variety of interests not simply
constrained to Mathematics alone. In trying to complete what Saccheri had
attempted involving a proof that the denial of Euclid's 'parallel' postulate
would lead to a contradiction - Lambert wrote "Die Theorie Der Parallellinien"
in 1766, appearing posthumously in 1786. However instead of beginning with a
Saccheri quadrilateral, Lambert adopted as his starting point a quadrilateral
having three right angles. Refer to Figure 12. Lambert then considered for the
fourth angle the three possibilities: acute, right, along with obtuse. However
Lambert like Saccheri had a similar lack of success, as he wrote:
"Proofs of the Euclidean postulate can be developed to such an extent
that apparently a mere trifle remains. But a careful analysis shows that in
this seeming trifle lies the crux of the matter; usually it contains either the
proposition that is being proved or a postulate equivalent to it".
However in the case of the 'hypothesis' of the obtuse angle, Lambert
showed that this became realized on the surface of a sphere between great
circles, if the lines AB, BY, YX, XA, appeared as arcs of such circles. Refer
to Figure 25, discussed later. Further Lambert speculated that the 'hypothesis'
of the acute angle might correspond to a geometry on a novel surface, such as a
sphere of imaginary radius. Refer to Figure 19, discussed later.
Yet no one else other than Lambert, had ever come so close to the truth
without actually discovering Non-Euclidean (Ē) geometry.
But this was not the end, many mathematicians still tried to prove the
fifth postulate. For example, Georg Simon Klügel (1763) a German mathematician
listed nearly 30 attempts to prove postulate five, finally concluding that the
alleged proofs were 'all' unsound. Then Adrian Marie Legendre (1752-1833) a
French mathematician, made a determined attempt to prove the fifth postulate in
his "Elemens De Geometrie", by showing it as a consequence of the other
undisputed postulates. However Legendre's attempt though abortive, came to
inspire others to devise geometries without reference to Euclid's questionable
'parallel' axiom.
Non-Euclidean (Ē: Johann Friedrich Karl Gauss (1777-1855), Nikolai
Ivanovich Lobatchevski (1929-30), János Bolyai (1831), Georg Friedrich Bernhard
Riemann (1867), etc., revision of Euclid's (500 B.C) fifth postulate [false-tofacts assumption of 'straight lines'] to non-linearity) geometry concerns the
making of specific assumptions about points, lines, planes, along with 'space',
then drawing conclusions consistent with one's 'spatial' experience involving
objects of moderate size, yet rich in specific relationships that affront that
experience, particularly relationships concerning the 'concept' (Latin
conceptum, to conceive: 'generalized-universal' idea else notion) of
'parallelism' extended over a large distance. For example, that similar figures
'are necessarily' congruent (having the 'same shape and size'): since no plan,
model (diagrammatic-mathematical description), map, etc., can be truly accurate.
Such that, because it is impossible in practice to measure how far apart the
Figure 14.
From left to right.
Johann Friedrich Karl Gauss (April 30, 1777 – February 23, 1855);
János Bolyai (December 15, 1802 – January 27, 1860);
Nikolai Ivanovich Lobachevski (October 11, 1793 – February 24, 1856);
Georg Friedrich Bernard Riemann (September 17, 1826 – July 20, 1866).
lines may extend, it becomes quite possible that humanity lives within a Ē
universe. For example, in such a world, rail-road tracks can still appear
equidistant, but then they will not be perfectly 'straight'.
Karl (baptised Johann Friedrich Karl) Gauss (1777-1855) arose as an
infant prodigy of poor German parents, becoming perhaps the greatest
mathematical genius since Archimedes (287-212 B.C.). It appears that as early
as 1816, Gauss had considered the possibility of developing a geometry which
avoided the fifth postulate, yet would nevertheless appears as self-consistent
as that of Euclid's. However Gauss published nothing on the subject. This fact
along with many other prior discoveries unpublished, became known when the Royal
Society of Gottingen borrowed a diary, dated 1898 (though not 'all' of Gauss'
discoveries in the prolific period 1796-1814 appear), from a grandson of Gauss.
Later published as apart of Gauss' collected works (1863-1933 "Werke" compiled
by Ernst C.J. Schering; Felix Klein) in 1917. Such that Gauss in 1792, had
apparently began a much bolder-deeper study of the problem. Though he published
nothing on the subject, Gauss claimed in 1799, that he had discovered the
principles of a new geometry, based on the rejection of the 'parallel'
postulate. Which Felix Klein (1849-1925) gave as the first example of
hyperbolic geometry.
As to why Gauss held back on the great discoveries, well
some have claimed that he wished to leave only complete, convincing,
indisputable works. However Gauss appears to have engaged in his scientific
investigations, due more to personal curiosity, intending to publish perhaps
posthumously. Though in the case of publishing an account of E geometry, Gauss
may have 'feared' the criticisms of the traditionalists, mediocrity, etc.
More revealingly Gauss had committed some considerations in his
correspondence. Gauss wrote to his friend Farkas Bolyai in 1799:
"It might well be possible that, however far apart one took the
vertices of a triangle in space, its area was always under a given limit".
Whereas in a letter to Gauss, his pupil Friedrich Ludwig Wachter (1792-1817)
remarked that, if the fifth postulate is denied, a sphere the radius of which
tends to infinity (horosphere) approaches a limiting surface on which specific
curves appear just like the lines of the 'E' plane as geodesics on the surface.
Where a geodesic (geodesy from geōdaisia of geōdaiein, to divide: Science of
shape, area, etc., in terms of curvature upon earth's surface) of extrema the
'greatest' else 'least' arc of a circle, refers specifically to the shortest arc
between two points.
Nevertheless Stuart Hollingdale (1989) gives Gauss' alternative fifth
postulate as "there are at least two lines through a point parallel to a given
line".
Instead Gauss encouraged others to proceed with the construction of a
consistent Ē-system, such that in 1819, one of his correspondents, F.K.
Schweikart, asserted the ‘logical’ consistency of a geometry freed of the
‘parallel’ postulate. While another, F.A. Taurinus (1826) gave important
trigonometrical formulae for a Ē geometry by using the formulae of spherical
geometry with an imaginary radius.
Farkas Bolyai (1775-1856) a Hungarian mathematician had spent much of his
life trying to prove the fifth postulate, but when he found that his own son
János Bolyai (1802-60) had become absorbed in the problem of 'parallels', whom
in 1823 informed his father, that he had worked out a new theory of 'parallels'.
Farkas Bolyai the father Professor of Mathematics at Maras-Vasarhely, wrote to
the son, a dashing army officer:
"I entreat you, leave the Science of parallels alone...I have
travelled past all reefs of this infernal Dead Sea and have always come back
with a broken mast and torn soil...
For God's sake, I beseech you, give it up. Fear it no less than sensual
passions because it, too, may take all your time, and deprive you of your
health, peace of mind, and happiness in life".
The son not dissuaded, continued his efforts until about 1823 when he saw
the whole truth, declaring in his youthful enthusiasm, "I have created a new
universe from nothing!" János Bolyai understood that 'absolute' geometry (based
on the first four postulates of Euclid alone) branches out in two dimensions.
Where instead for his fifth postulate János Bolyai started from: given a point
not on a line, though which an infinite number of lines on the plane, each
'parallel' to the given line, can become drawn. Thus János Bolyai recognized
that two different but consistent geometries become possible (where 'E' ofcourse
remains one of them), whereupon sending his reflections to his father Farkas
Bolyai, had them published as an "Appendix Scientiam Spatii Absolute Veram
Exhibens" ("Appendix Explaining The Absolutely True Science Of Space") to his
treatise "Tentamen" which though bore the imprimatur date 1829, did not actually
appear until 1831-3. Of which George Bruce Halsted had wrote of it as: "the
most extraordinary two dozen pages in the whole history of thought".
Figure 15.
Farkas Bolyai (1775-1856).
In 1832 Farkas Bolyai proudly presented a copy to his friend Gauss
requesting an opinion on the unorthodox work of his son Janos, whose reply to
the father though contained a sincere approval had a devastating result on
János:
"I am unable to praise this work...To praise it would be to praise
myself. Indeed the whole contents of the work, the path taken by your son, the
results to which he is led, coincide almost entirely with my meditations which
occupied my mind partly for the last thirty-five years. So I remained quite
stupefied. So far as my own work is concerned...my intention was not to let it
be published during my lifetime...On the other hand, it was my idea to write
down all this later so that at least it should not perish with me. It is
therefore a pleasant surprise for me that I am spared this trouble, and I am
very glad that it is just the son of my friend who takes the precedence of me in
such a remarkable manner".
The temperamental János Bolyai became understandably disturbed, 'fearing'
that he would become deprived of priority. When Lobatchevski made a deeper
investigation, writing several books, Gauss sent him a letter of genuine praise,
arranging a recommendation that got Lobatchevski in 1842 elected to the
Gottingen scientific society. In letters to friends Gauss praised
Lobatchevski's work, though he refrained from supporting it in print, because it
appears that he 'feared' the jibes of the Boeotians. In marked contrast, the
unhappy János Bolyai received no recognition during his life-time. In 1848
János Bolyai read one of Lobatchevski's books (translated into German), praising
it warmly. However Janos remained too timid to introduce himself, further there
is no evidence that Lobatchevski appeared aware of Bolyai's work. Whereas Gauss
who knew them both, remained so preoccupied with his works, that he never took
the trouble to bring them together. Nevertheless, the continued lack of
recognition, further the publication of Lobatchevski's work in German in 1840,
so upset Janos Bolyai that he published nothing more.
However the earliest published Ē investigation came from the Russian
mathematician Nikolai Ivanovich Lobatchevski (1792-1856) entitled "O Nachalakh
Geometrii" ("On The Principles Of Geometry"), in the Kazanski Vestnik, a journal
published by the university, in 1829-30. Though Lobatchevski wrote his first
major work "Geometriya", in 1823, it was not published in the original form
until 1909. The basic geometrical studies that it embodies, however led
Lobatchevski to his chief discovery – Ē geometry – from which Lobatchevski
(1826) submitted a first account of hyperbolic geometry, which he termed
imaginary geometry, based on the rejection of Euclid’s (300 B.C.) fifth
postulate, in favour of the acute-angled ‘hypothesis’, on which as Professor at
the University of Kazan, he reported to the departments of Physics along with
Mathematics at a meeting held on 23rd February 1826, set out in "Exposition
Succincte Des Principes De La Geometrie Avec Une Demonstration Rigoureuse Du
Theoreme Des Paralleles". After which Lobatchevski (1828), published "New
Principles Of Geometry And A Number Applications", in a local mathematical
journal.
But then following Lobatchevski’s (1829-30) "On The Principles Of
Geometry", comprising the earlier "Exposition" (1826), he wrote out three full
accounts of the new geometry between 1835 to 1855.
Lobatchevski’s (1835)
"Voobrazhaemaya Geometriya" ("Imaginary Geometry") published in Uchenye Zapiski;
again published in French in 1837 to reach an international audience: "Geometrie
Imaginaire", in Crelle’s Journal Volume 17. Followed by Lobatchevski’s (183538) the "Novye Nachala Geometrii S Polnoi Teoriei Parallelnykh" ("New Principles
Of Geometry With A Complete Theory Of Parallels"). Whilst Lobatchevski’s (1840)
"Geometrische Untersuchungen Zur Theorie Der Parallellinien" ("Geometrical
Researches On The Theory Of Parallels"), appearing in German for an
international audience, with which both Gauss along with János Bolyai became
acquainted with. Finally Lobatchevski's (1855-56) last "Pangeometrie"
("Pangeometry") appearing both in French along with Russian, after his death.
Lobatchevski's revolutionary overview seems not to have come to him as
sudden inspiration. In an outline of geometry composed in 1823, presumably for
classroom use, Lobatchevski wrote of the fifth postulate: "no rigorous proof of
the truth of this had ever been discovered". However between 1825 to 1829
Lobatchevski had become thoroughly convinced that Euclid's fifth postulate
cannot be proved on the basis of the other four. Indeed in 1826 Lobatchevski
had read a French paper (now lost) "Une Demonstration Rigoreuse Du Theoreme Des
Paralleles", which may have proved pivotal. Nevertheless Lobatchevski with his
paper of 1829, "On The Principles Of Geometry", became the first mathematician
to take the revolutionary step of publishing a geometry specifically built on an
assumption in direct conflict with the 'parallel' postulate. As J.F. Scott
(1958) phrased it:
"Let there be in a plane, a straight line and a point lying outside
it. Of all the straight lines radiating from the given point, there are some
which cut the given line and some which do not. Separating these two classes of
lines is a boundary line; this boundary line is said to be parallel to the given
line. Consequently, through any given point there exist two straight lines
parallel to a given straight line, and each of these meets the given line at
infinity. This being so, a straight line has two distinct points at infinity".
Otherwise as Boyer (1968) more simply phrased it:
"Through a point C lying outside a line AB there can be drawn more
than one line in the plane and not meeting AB".
Refer to Figure 16. Thus Lobatchevski challenged the assumption that Euclid's
fifth postulate otherwise any of the equivalents, for example the 'hypothesis'
of the right angle, remains necessary to a consistent geometry, by producing a
system based on the 'hypothesis' of the acute angle in which there is not one
'parallel' through point C to a given 'straight line' but two.
The Ē geometry of which Gauss, Lobatchevski along with János Bolyai,
becomes termed hyperbolic (Greek huperballēin from hyperbola huperbolē, to throw
beyond: pertaining to otherwise of the nature of a hyperbola; a plane curve
formed by cutting a cone when the intersecting plane makes a greater angle with
the base than the side of the cone makes), because in order that the lines do
not meet, then they must diverge. To illustrate this we can use the following
imaginary construction, refer to Figure 18. Two lines in a plane can become
extended from a perpendicular, connecting points A with B. However instead of
remaining equidistant they become farther apart. Yet because the two curving
lines from the perpendicular diverge minutely, then as if with two 'straight
lines' the angles at A with B come to two right angles.
C
A
B
Figure 16.
Adapted from Nikolai Ivanovich Lobachevski's (1840) "Geometrical Researches On
The Theory Of Parallels".
Figure 17.
Nikolai Ivanovich Lobachevski's (1829) "On The Principles Of Geometry",
alternative fifth postulate.
A
B
Figure 18.
Adapted from Nikolai Ivanovich Lobachevski's (1829-30),
János Bolyai's (1831) hyperbolic geometry based on the 'hypothesis' of the
acute angle.
Similarly in Figure 16, neither of Lobatchevski's 'parallels' meet the
line to which both remain 'parallel', nor does any 'straight line' drawn through
point C, whilst lying within the angle formed by the two 'parallels'. This
apparently bizarre situation, becomes realized by the geodesics on a
pseudosphere. Refer to Figure 19. In 1868 Eugenio Beltrami (1835-1900) showed
that Lambert remained correct in his conjecture. However it was not a sphere
with an imaginary radius, but a surface of consistent negative curvature
generated by revolving the tractrix (extending spirally in curvature
indefinitely) above the axis - known as a pseudosphere. The surface (Figure 19)
looks like two infinitely long trumpets soldered together at their largest ends.
Now if on this surface we draw the four-sided figure AXYB of Figures 11-2
(Saccheri else Lambert's quadrilateral), with equal sides along with the right
angles as before, using geodesics we then find that the 'hypothesis' of the
acute angle will become realized.
Figure 19.
Construction above represents one half of a pseudo-sphere, which comprises of
two joined at the splayed end.
With this new postulate Lobatchevski 'deduced' a harmonious geometric
structure having no apparent 'logical' contradictions; which though in any sense
appeared a valid geometry, yet remained contrary to any 'common sense'. The
boldness of Lobatchevski's challenge with the successful outcome have inspired
mathematicians along with scientists to challenge other 'axioms', accepted
'truths', etc. As Bell (1937) puts it:
"The full impact of the Lobatchewskian method of challenging axioms
has probably yet to be felt. It is no exaggeration to call Lobatchewsky the
Copernicus of Geometry, for geometry is only a part of the vaster domain which
he renovated; it might even be just to designate him as a Copernicus of all
thought".
Ē geometry continued for several decades as fringe Mathematics (not
helped by Gauss, who refused to give his support in print) until it became
Figure 20.
Eugenio Beltrami (November 16, 1835 – February 18, 1900);
Augustin Louis Cauchy (August 21, 1789 – May 23, 1857);
Leonhard Euler (April 15, 1707 – September 18, 1783);
Jean le Rond d'Alembert (November 16, 1717 – October 29, 1783);
Friedrich W. Bessel (1784-1846);
Jean Baptiste Joseph Fourier (March 21, 1768 – May 16, 1830).
thoroughly integrated by Bernard (baptised Georg Friedrich Bernard) Riemann
(1826-66) a brilliant German mathematician, who studied at Gottingen under
Gauss. Riemann (1851) established his reputation as a foremost mathematician
with his doctoral dissertation, "Grundlagen Fur Eine Allgemeine Theorie Der
Funktionen Einer Veranderlichen Complexen Grosse" ("Foundations For A General
Theory Of Functions Of A Complex Variable") originally submitted for Gauss'
consideration. Of which Gauss reported to the Philosophical Faculty of the
University of Gottingen:
"The dissertation submitted by Herr Riemann offers convincing evidence
of the author's thorough and penetrating investigations in those parts of the
subject treated in the dissertation of a creative, active, truly mathematical
mind, and of a gloriously fertile originality. The presentation is perspicuous
and concise and, in places, beautiful. The majority of readers would have
preferred a greater clarity of arrangement. The whole is a substantial,
valuable work, which not only satisfies the standards demanded for doctoral
dissertations, but far exceeds them".
In Riemann's thesis in the theory of functions of a complex variable, we
find the so-called Cauchy(Augustin Louis Cauchy (1789-1857))-Riemann equations.
A single definite 'uniform' function e, depends upon the complex variable,
e = f (a + ib)
Equation 1.
where i, denotes  -1 , as known to Leonhard Euler (1707-83) along with Jean Le
Rond D'Alembert (1717-83). However as Gauss (1811) had informed a friend
Friedrich W. Bessel (1784-1846), the complex numbers (a, b) represented
geometrical points (x, y) on the plane of Cartesian (after René Descartes (15961650) 'analytic' geometry, such that:
e = f (x + iy)
Equation 2.
x + iy = z
Equation 3.
where for brevity,
because as x, y independently take on real values in any prescribed continuous
manner, the point z will wander about over the plane, though not at random but
determined by the values of x, y. Entailing that if in the complex (Gaussian)
plane one draws a simple closed curve, refer to Figure 26:
y
x + iy
z
-x
0
x
-y
Figure 21.
then if,
e = f (z) = f (x + iy)
Equation 4.
remains 'analytic' (has a derivative) at each point on the curve along with
within the curve then the line integral of f(z) taken along the curve will
become zero.
Then Cauchy (1831) announced the theorem that an 'analytic' function of a
complex variable:
w = f (z)
Equation 5.
can become expanded about a point,
z = zO
Equation 6.
in a power series that remains convergent for any values of z within a circle
having zO as centre, while passing through the singular point of f(z) nearest to
zO.
However Riemann then demonstrates what D'Alembert had arrived at in a
paper of 1732 on the resistance of fluids. If the 'analytic' function:
e = f (z) = f (x + iy)
Equation 7.
w = f (z) = u + iv
Equation 8.
then,
therefore,
du dv
 = 
dx dy
Equation 9.
with
du
dv
 = - 
dy
dx
Equation 10.
Ux = V y
Equation 11.
U y = - Vx
Equation 12.
otherwise,
with,
where a
Values of complex variables.
b ,
d, Differential.
e, w, 'Analytic' function.
f, Function of within ( ).
i, Value of  -1
u, Value representing x-axis co-ordinate.
v, Value representing y-axis co-ordinate.
x, Value co-ordinate along x-axis.
y, Value co-ordinate along y-axis.
z, Representing complex variable.
Nevertheless, Riemann's thesis led to the formulation of a Riemann surface
(2-dimensional manifold), anticipating how topology (mathematical concerns of
geometrical functions, under distortion) will ultimately function.
In 1854 Riemann became Privaldozent at the University of Gottingen, so as
per tradition had to give a Habilitationsschrift: a probationary essay, else
inaugural lecture. For the ordeal Riemann had prepared two topics (one
concerned a memoir on Fourier's [Jean Baptiste Joseph Fourier (1768-1830)]
trigonometric [Greek trigōnon, triangles; mētria, measurement: part of
Mathematics concerned with the measurement of the sides, angles, etc., of
triangles] series), but provided the faculty with a third on the foundations of
geometry. However contrary to Riemann's hoping-expectancy, Gauss intrigued,
chose the third topic as the one on which Riemann should prove his capability as
a lecturer before the critical faculty. The result became the most celebrated
probationary lecture in the history of Mathematics, presenting a deep-broad view
of the whole field of geometry. The thesis later published in 1867, had the
title: "Uber Die Hypothesen, Welche Der Geometrie Zu Grunde Liegen" ("On The
Hypotheses Which Lie At The Foundations Of Geometry"), within which he did not
present a specific example, instead urging a global view of geometry as a study
of manifolds (having many, various, forms, applications, component parts, etc.,
mathematically: an abstract surface of arbitrary dimension n) of any number of
dimensions in any kind of 'space'.
Riemann's Ē system differed more radically from that of Euclid than did
that of any of his predecessors. We have discussed that a perfectly 'logical'
system of geometry, can become devised in which any two lines may fail to meet,
even if the two interior angles which they make with a transversal, together
come to less than two right angles: the hyperbolic geometries of Lobatchevski
with Bolyai, involving the 'hypothesis' of the acute angle. However Riemannian
geometry has a more restricted formulation: the abandoning of infinitely
extending 'straight lines' incorporated in a plane geometry 'deduced' from
Saccheri's 'hypothesis' of the obtuse angle. Instead Riemann postulated that
though unbounded 'straight lines' remain 'finite' (Latin fīnītus, having limits:
'few-valued, allness') in length. Where Riemann's geometry instead becomes
termed elliptic (Greek elleipein from elleipsis, to fall short) because if
though unbounded 'finite', then the lines must converge, ultimately
intersecting; a geometry of an asymmetrical shphere. To illustrate this refer
to Figure 22, where the lines extending from a perpendicular connecting points A
with B, close together. Yet because the two curving lines from the
perpendicular converge minutely, then as if with two 'straight lines' the angle
at A with B come to two right angles. Such that a model for Riemann's elliptic
geometry involves the interpretation of a 'plane' as the surface
(2-dimensional case of the notion of a manifold) of a sphere, whereas a
'straight line' as a great circle on that sphere.
A
B
Figure 22.
Georg Friedrich Bernard Riemann (1854) elliptic geometry based on the
'hypothesis' of the obtuse angle.
Indeed as Lambert (1786) had demonstrated that the case of the
'hypothesis' of the obtuse angle, becomes realized on the surface of a sphere
between great circles if the lines AB, BY, YX, XA, of Figures 16-7, appeared as
arcs of such circles. Refer to Figure 25. Imagine on the sphere an equator
from which we may draw two geodesics on the surface (arcs of circles) through
the North pole perpendicular to the equator. In the northern hemisphere this
gives a triangle with curved sides two of which become equal in length. Such
that each side of this triangle, becomes an arc of a geodesic. Now if we draw
any other geodesic intersecting the two equal sides, so that the intercepted
parts between the equator with the intersecting line become equal, then we will
have on the sphere, the four-sided figure corresponding to the AXYB
quadrilateral previously discussed as Figures 11-2 on a plane. As before the
two angles at the base BAX, ABY, will equal two right angles, further the two
sides AX, BY, remain equal, however each of the equal angles AXY, BYX, will now
equal greater than a right angle.
However that curves appear over great distances as 'straight lines' on a
sphere remains a notion that Aristotle will have agreed with, for as expressed
by D.L. Hurd along with J.J. Kipling (1958) from Aristotle's (350 B.C.) "On The
Heavens":
"In a similar way there is doubt about the shape of the earth. To
some it seems to be spherical, but to others flat, in the form of a drum. To
support this opinion they urge that, when the sun rises and sets, he appears to
make a straight and not a circular occultation, as should be if the earth were
spherical. These men do not realize the distance of the sun from the earth and
the magnitude of the circumference, nor do they consider that, when seen cutting
a small circle, a part of the large circle appears at a distance as a straight
line. Because of this appearance, therefore, they ought not to deny that the
earth is round..."
Therefore a point moving along a 'straight line' will ultimately return
to the point from which it started out. A notion which Lobatchevski, had
previously found it quite impossible to accept. Such that as Riemann envisaged,
it would not be possible to draw a 'straight line' through a given point which
did not cut another 'straight line' as do arcs of great circles on a sphere; if
extended far enough. Leading Riemann to the formulation of spherical geometry
without 'parallels' as a general case, to that of the special case involving 'E'
geometry. As Bell (1937) interprets Euclid's geometry:
"Euclidean geometry is a limiting, or degenerate, case of geometry on
a sphere, being attained when the radius of the sphere becomes infinite".
Figure 24.
Eric Temple Bell (February 7, 1883 – December 21, 1960).
Riemann proposed that Saccheri's 'hypothesis' of the obtuse angle becomes
valid If Euclid's postulates 1, 2 with 5 become modified. In hyperbolic
geometry, postulate five is denied, because if the line from A in Figure 18
becomes replaced by one making a very slightly smaller angle with AB, the new
line from A along with the old one from B may converge at first attaining a
minimal distance, but then diverge. Refer to Figure 26. However in elliptic
geometry, not only is postulate five denied, but further postulates one
(interpreted as the shortest line between two points involves a 'straight line')
with two (interpreted as giving the line an infinite length) are denied, because
now a geodesic encloses, like a circle. As Scott (1958) reports, Riemann's
changes entailed:
(1). "Any two points determine a line".
N
X
Y
W
E
A
B
S
Figure 25.
Modified from Eric Temple Bell's (1937) "Men Of Mathematics", volume 2.
In Euclid's (c. 300 B.C.) geometry on a plane, any two geodesics intersect in
exactly one point unless 'parallel', when they do not intersect; but on a sphere
any two geodesics will intersect in precisely two points. Again on a plane, no
two geodesics can enclose a 'space' as Euclid (300 B.C.) assumed as a postulate;
on a sphere any two geodesics will enclose a 'space'. Therefore on a sphere it
is not Euclid's (300 B.C) fifth postulate which is true, nor the equivalent
'hypothesis' of the right angle, but the geometry which follows the 'hypothesis'
of the obtuse angle. Demonstrated in Figure 25, since though the four-sided
figure AXYB so created as on a plane, has equal sides with angles A,B, as equal
to right angles, yet on a sphere each of the angles X,Y, has angles greater than
right angles.
(2). "A line is unbounded".
(3). "Any two lines in a plane meet if they are produced far enough".
While in showing that E geometry with angles greater than two right angles
becomes realized on the surface of a sphere, Riemann simply verified the
consistency of the axioms deriving the geometry.
A
B
Figure 26.
Riemann's investigation led to the formulation of three distinct
geometries; these differ only in regard to the number of lines which might get
drawn through a given point 'parallel' to a given 'straight line'. Where only
one such line can become drawn (involving the 'hypothesis' of the right angle),
then we have the familiar geometry of Euclid; in as much as the 'plane' has an
equivalence to a surface with 'constant zero curvature'. If no such line can
become drawn (involving the 'hypothesis' of the obtuse angle), then we have
Riemann's elliptic geometry; where the 'plane' has an equivalence to a surface
of a sphere with 'constant' positive curvature. If however a pencil of lines of
'constant' angle can become drawn (involving the 'hypothesis' of the acute
angle), then we have the hyperbolic geometry of Lobatchevski-Bolyai; involving a
'plane' having an equivalence to a surface of a pseudosphere, with a 'constant'
negative curvature. In the others any two lines intersect, so long as they lie
within the surface.
Riemann (1854) had invented a spherical geometry (via the obtuse-angle
‘hypothesis’), where the second kind of Ē geometry, via the obtuse-angled
‘hypothesis’ appears realized. Where Riemann’s geometry though a priori,
appears far more disconcerting than the geometry of Gauss (1799), Lobatchevski
(1829-30), along with Bolyai (1831), which explains why mathematicians had
overlooked it even though it was one of the two possible cases considered by
Saccheri (1733).
Where however 'E' geometry can become regarded as an intermediary between
the two kinds of Ē geometries. But that in each case 'straight lines', are not
other than geodesics between two points. Such that the Ē geometries of
Lobatchevski, Bolyai along with Riemann can become distinguished from that of
Euclid's via the version of the fifth postulate involving 'parallels'. For
example, in following Cassius Jackson Keyser's (1922) exposition, while
referring to Figure 28.
"The given pathocircle is a; A is a point not on a; through A there is
evidently one and but one pathocircle b having no point in common with a; a and
b are, ofcourse, parallel to each other. This postulate, as you know, is the
Euclidean postulate par excellence - the one that mainly distinguishes Euclidean
geometry from the famous non-Euclidean geometries of Lobachevski and Riemann".
Yet Riemann's geometries appear Ē in a far more encompassing way than
Lobatchevskian geometry, where the question simply involves how many 'parallels'
appear possible though a point. Instead Riemann saw that geometry should not
even necessarily deal with points, lines, 'space', etc., in the ordinary sense,
but with ordered arrays of n-tuples combined according to specific rules.
In 1827 Gauss initiated a new branch of geometry known as differential
geometry, when his treatise "Disquisitiones Circa Superficies Curvas" appeared.
Figures 27.
Comparisons between Euclidean, Elliptical, Hyperbolic geometries.
A
a
b
Figure 28.
From Cassius Jackson Keyser's (1922) "Mathematical Philosophy".
Euclid's (c. 300 B.C.) 'parallel' (fifth) postulate entails that two 'straight
lines' will not meet if 'parallel' to each other. However since 'straight
lines' are nothing but parts of arcs of a circle, then two arcs can appear
'parallel' to each other. As Keyser (1922) demonstrated via Figure 28, given
pathocircle a, which has no point in common with pathocircle b, appears ofcourse
'parallel'.
From Keyser (1922), "Let O (point of origin) be a chosen point of . The
ensemble of all circles through O is called a bundle of circles. The bundle
includes, as infinite circles (i.e., circles of infinite radius), the straight
lines through O. Now, in thought, let us, once for all, remove the point O from
. Each circle of the bundle now lacks a point; we may call them pathocircles,
and speak of the O-bundle of pathocircles".
In other words, pathocircles, represent "Riemannian circles" ("Great circles"),
otherwise paths of arcs of circles.
Figure 29.
Archimedes of Syracuse (c. 287– 212 B.C.);
Gottfried Wilhelm Leibnitz (Leibniz otherwise von Leibniz; July 1, 1646
[21 June] – November 14, 1716);
Joseph Louis Comte Lagrange (January 25, 1736 – April 10, 1813);
Gaspard Monge, Comte de Péluse (May 9, 1746 – July 28, 1818).
The subject was not new since several of Gauss' predecessors, notably Leonard
Euler (1707-83), Joseph Louis Lagrange (1736-1813) along with Gaspard Mange
(1746-1818) had investigated geometry on particular kinds of curved surfaces,
using Gottfried Wilhelm Leibnitz's (1648-1716) with Isaac Newton's (1642-1727)
integral along with differential calculus (Latin, a pebble: after Isaac Newton
(1642-1727) with Gottfried Wilhelm Leibnitz (1646-1716), though pre-dated by
Archimedes' (287-212 B.C.) discovery; part of mathematics concerning functional
integrations ['summations'], otherwise differentials [changes]). Further Gauss
presumably drew upon insights while functioning as scientific adviser (1821-48)
to the Hanoverian along with Danish governments, in undertaking an extensive
geodetic survey of a portion of the earth's surface.
But importantly Riemann (1854) became inspired by this work, while
providing his Habilitationsschrift on the 'hypothesis' which lie at the
foundations of geometry. Which in turn led to the transformation of
differential geometry in Mathematical Physics, particularly in Albert Einstein's
(1916) theory of general relativity.
Three of the problems which Gauss considered in his work on surfaces
suggested theoretical extensions of a mathematical further scientific
importance: the measurement of curvature, the theory of conformal representation
(mapping) along with the applicability of surfaces; of which the first interests
us here. Roughly speaking 'ordinary' geometry remains interested in the
totality of a given diagram, figure, etc.; whereas differential geometry
concerns the characteristics of a curve, surface, etc., in the immediate
vicinity of a point on the curve, surface, etc. In relation to this Gauss
extended the work of Christian Huygens (1629-95) with Alexis Claude Clairaut
(1713-65) on the curvature of a plane (gauche curve) at a point by defining the
curvature of a surface at a point - the "Gaussian (total) curvature". The
problem involves devising some precise means for describing how the curvature of
a surface varies from point to point of the surface. Refer to Figure 33.
If on part of the surface S representing an unlooped closed curve C, one
erects normal lines N perpendicular to the plane which intersects the surface C
at points P, such that each point P on C will have a radius of curvature on our
supposed sphere. So that the directions of the curves with the maximum, along
with the minimum radii of curvature R with r (principal directions), represent
the principal radii of curvature of C at P. Where the Gaussian curvature of C
at points P, such that each point P on C will have a radius of curvature on our
supposed sphere. So that the directions of the curves with the maximum, along
with the minimum radii of curvature R with r (principal directions), represent
the principal radii of curvature of C at P. Where the Gaussian curvature of C at
P gets defined as:
K = 1/rR
Equation 13.
where K, Gaussian (total) curvature.
R, Maximum radii of curvature.
r, Minimum radius of curvature.
Another fundamental notion exploited by Gauss in the treatment of curved
surfaces involves parametric representation, using Euler's parametric equation.
In an n-dimensional manifold, n numbers becomes necessary both sufficient
to specify (individualize) each particular member of a 'class' of things, for
example points, sounds, colors, lines, etc. Using the language of geometry we
find it convenient to speak of any 2-dimensional manifold as a 'surface', hence
to apply to the manifold the geometrical 'reasoning'. Now it requires two
co-ordinates to specify a particular point on a plane. Similarly on the surface
of a sphere, otherwise on a spheroid like the Earth: the co-ordinates in this
case referred to as latitude along with longitude.
The notion of assigning an unique pair of numbers (co-ordinates) to the
position of any point whatever with respect to axes, originated with René
Descartes’ (1637) "La Geometrie"; who in so doing did not revise geometry, but
created it. Indeed one way say with considerable justification that 'E'
geometry reached the height of mathematical expression via Cartesian exposition.
Using Descartes arbitrary frames of reference, from a moment of insight, while
Figure 30.
René Descartes (March 31, 1596 – February 11, 1650).
pondering sick in bed seeing a branch of a tree framed by a window frame. For
example, for our axes we have x, y, over which a point may wander. Refer to
Figure 31. Such that the co-ordinates (x, y) of which an array of points on the
curve over which it wanders, representing a maximum to minimum range of values,
has a relation as an equation referred to as the equation of the curve.
y = f(x)
Equation 14.
where y, Unknown quantity.
f, Function of within ( ).
x, Known quantity.
Now suppose for simplicity that our curve appears apart of a circle.
Refer to Figure 34. Again for this circle, we have an equation describing the
co-ordinates of the system, equation 13. However what else can we do with this
equation? So instead of this particular equation we can derive a more
encompassing one (for example here, of the second degree with no cross-product
term, further with the coefficients of the highest powers of the co-ordinates
equal) by manipulating this equation algebraically.
From Pythagoras' theorem concerning right-angled triangles form Euclid's
"Elements" Book I theorem 47:
"In a right-angled triangle the square on the hypotenuse (side
opposite right angle) is equal to the sum of the squares on the other two
sides".
Refer to Figure 35.
From the figure in Figure 35, we can derive an equation in terms of the
'straight lines' composing the right-angle triangle ABC:
BC2 = AC2 + AB2
where BC, Hypotenuse side c in Figure 35.
AC, Side a in Figure 35.
AB, Side b in Figure 35.
Equation 15.
f(x): y
f(y): x
Figure 31.
After René Descartes’ (1637).
A graph using Cartesian co-ordinates.
Figure 32.
René Descartes’ (1637) Cartesian co-ordinate system for 3-dimensions.
Figure 33.
From Eric T. Bell's (1937) "Men Of Mathematics", volume 1.
Karl Johann Friedrich Gauss (1927) showed how to calculate the curvature, via
conformal mapping of curvature. For example in Figure 30, on a surface S, an
unlooped closed curve C, can have intersecting perpendicular to the plane,
lines N at points P.
-y
(x,y)
c
y
0
-x
+x
c
x
-y
Figure 34.
Otherwise in terms of triangle ABC's sides:
C2 = a 2 + b2
Equation 16.
Now using Pythagoras' equation we can calculate the distance from the O axis,
any point around the circle. Hence by substituting equation 16 with symbols in
Figure 34, we have:
C2 = x2 + y2
Equation 17.
Then in terms of y:
 x2 + c2 = y
Equation 18.
y = x+c
Equation 19.
y = f(x) + c
Equation 20.
Therefore,
otherwise,
where y,
f,
x,
c,
Unknown quantity.
Function of within ( ).
Known quantity.
Exponential 'constant'.
Figure 35.
"The Peacock's tail".
Finally we put back the results of our various algebraic manipulations
into their equivalents in terms of co-ordinates of points on the diagram, which
we had momentarily dispensed with. Algebra (Arab al-jebr, reunion of parts:
part of Mathematics concerning the investigation of relations, order, etc.,
[structure] of numbers via symbols, signs, etc) becomes easier to understand
than a cobweb of lines in the Greek method of geometric investigation. What we
have done, has involved the use of algebra for the discovery-investigations of
geometrical theorems concerning circles.
For 'straight lines' along with circles, this may not seem very exciting
since we knew how to do this long before in the Greek way. However the
innovative power of Descartes' method entails that we can start with equations
of any degree of complexity, while interpreting their algebraic functions
geometrically. Thus instead algebra becomes our instrument for investigating
the geometry of 'space'. Further what we have done can become external to a
'space' of any number of dimensions, for the plane we require two co-ordinates,
for ordinary 'solid' 'space' three, for the geometry of Quantum mechanics
(quantum from quantus, how much; mechanics from Greek mēchanikos from mēchanē,
machine: part of Physics concerned with functions of particle(s)-wave(s)(field(s)) as packets of mass[Greek maza: amount of 'matter' formed into a
coherent whole of indefinite shape, great proportion, etc]-energy [Greek
energeia, work: dynamic power, force, etc., capability of 'action']) along with
Relativity (after Albert Einstein (1905, 1916), part of Physics [Greek phusikē,
of nature: Science concerned with functions of matter-energy] concerned with
functions of mass-energy upon a space-time continuum) four co-ordinates, while
finally for 'space' as mathematicians view it as n-or-co-ordinates, otherwise as
many co-ordinates as appear points on a line.
However the fore-going considerations lead to the parametric
representation of surfaces. Gauss (1827) investigated the theory of surfaces,
which occur as curves embedded in a 3-dimensional 'space'. As a result Gauss
formulated an internal theory of surfaces without reference to the plenum (Latin
plēnus, full: fullness; space-time continuum, etc.; hence four-dimensional
space-time continuum), which they remain embedded, thus two-dimensional Gaussian
co-ordinates.
Let us imagine a task of mapping a thickly wooded region. But, that the
use of optical instruments is impossible. So, that there are no 'straight
lines' to deal with. This 'E' geometry will not in general be applicable to the
region as a whole, though perhaps 'E' geometry may appear applicable to small
regions which we may consider as 'flat'. Experience with differential along
with integral calculus, suggests that such approximations on a very small scale
appear reliable.
Now in order to conduct the survey Gauss devised a network of smoothly
curving lines (parameter curves) in two families u, v; of which each family of
curves will only intersect the other. Then to this network of curves,
consecutive numbers become consigned to each family. However that these numbers
(U, V numbers) do not represent lengths, angles, etc., as measurable
'quantities' (Latin quantitātem: measure, size, greatness, volume, number, etc.,
of a 'property'), but simply label's for the curves. Refer to Figure 36. Where
the procedure by which we can locate any point on the surface will appear as
simple as this. If our point P lies between the two curves x = 3 with x = 4,
then we can draw nine curves (in order to obtain a convenient decimal method of
labelling) between two curves labelling them 3, 1; 3, 2;…; 3, 9. Now if P lies
between curves 3, 1 with 3, 2, then we can draw nine similar curves between
these two curves labelling them 3, 11; 3, 12;...; 3, 19; etc. Performing a
similar procedure with the y curves. Such that we would succeed in assigning to
any point, as accurate a pair of numbered labels as we pleased, hence
establishing the Gaussian (parametric) co-ordinates of any point. Where the
Cartesian co-ordinates systems which we use in plane geometry simply represent
only special cases of Gaussian systems.
However to our network meshes of consecutively assigned numbers we must
introduce some measure relations. One way involves measuring the small meshes
one after another in order to plot them on our map. Such that when done we have
a complete map similar in structure to our region. Because of the smallness of
the meshes we can consider them as small parallelograms, hence such
parallelograms can become defined by the lengths of two adjacent sides with one
angle.
We may however proceed differently by measuring the distance directly.
Let us select one mesh, for example one bounded by the curves 2, 3, along with
the curves 6, 7. Now let us consider a point P within this mesh, further let us
denote this distance from the point 0 (u = 2, v = 6) by S. Let us draw from the
point P 'parallel' to our mesh lines, labelling the intersections with the mesh
lines A, B, respectively. Further let us draw PC perpendicular to the u
parameter curve. Refer to Figure 37.
The points A, B, will as a result now have numbers, labels, else Gaussian
co-ordinates in our network. The co-ordinates of A may become determined by
measuring the side of the parallelogram on which A lies, further the distance of
A from 0. We can regard the relation termed the ratio of these two lengths as
the increase of the u co-ordinates of A towards 0. Denoting this increases by
u, choosing 0 as the origin of the Gaussian co-ordinates. Similarly, we
determine the Gaussian co-ordinate of v of B as the ratio in which B cuts the
corresponding side. Therefore these two ratios, which we term as u, v, will
represent the co-ordinates of our point P.
Yet as ratios u, v, ofcourse do not give us the lengths of 0A, 0B, but
the lengths can become determined by further measurements for au, bv, where a,
b, represent definite numbers. If we move point P about, the Gaussian
co-ordinates change but the numbers a, b, which give the ratio of the Gaussian
co-ordinates to the true lengths remain unchanged.
Now we find the lengths S, the distance of the point P from 0, from the
right-angled triangle 0PC by Pythagoras' theorem equation 15 in substitution.
S2 = 0P2 = 0C2 + CP2
Equation 21.
where S,
0P,
0C,
CP,
Hypotenuse 0P in Figure 21.
Hypotenuse side S in Figure 21.
Side 0C, length on u parameter curve.
Side CP, length on v parameter curve perpendicular to u parameter
curve.
However from Figure 15 we can see that,
0C = 0A + AC
Equation 22.
where 0A, Length along u parameter curve for v parameter curve 'parallel' to P.
AC, Extended length along u parameter curve for v parameter curve as
perpendicular to P.
Therefore in terms of Pythagoras' theorem concerning obtuse angled triangles
from Euclid's "Elements" Book II theorem 12:
"In an obtuse-angled triangle, if a perpendicular is drawn from
either of the acute angles to the opposite side produced, the square on the side
subtending the obtuse angle is greater than the sum of the squares on the sides
containing the obtuse angle, by twice the rectangle contained by the side on
which, when produced, the perpendicular falls, and the line intercepted without
the triangle, between the perpendicular and the obtuse angle".
Otherwise more simply as:
"In an obtuse-angled triangle, the square on the side opposite of the
obtuse angle is equal to the sum of the squares on the other two sides plus
twice the product of one of these sides and the projection on it of the other".
Hence in terms of the Pythagorean rule formula 21, by substituting equation 22:
S2 = (0A +AC)2 + CP2
Equation 23.
Therefore,
S2 = A02 + 20A  AC + AC2 + CP2
Equation 24.
where for the right-angled triangle APC we have
AC2 + CP2 = AP2
Equation 25.
4
3
8
7
2
6
V
U
1
5
Figure 36.
Johann Friedrich Karl Gauss' (1827) internal theory of surfaces,
involving curves embedded in a 3-dimensional 'space', using Gaussian
(parametric) co-ordinates.
V=7
B
P
S
V=6
0
A
C
U=2
U=3
Figure 37.
Hence similarly in terms of equation 23:
S2 = 0A2 + 20A  AC + AP2
Equation 26.
However 0A = au, AP = 0B = bv, further as AC remains a projection (Latin projectum,
throw out: process of externalizing 'prejudices'; equivalent to
'identification(s)', etc.; geometrical representation; lines, rays, etc., from a
point, otherwise drawn through points of a given figure producing another
figure; otherwise Projective geometry: part of geometry concerning 'properties'
that which remain unchanged after projection; etc) of AP = bv, it further has a
fixed ratio to A, such that we may put AC = cv, from which we may obtain the
formula:
S2 = a2u2 + 2acuv + b2v2
Equation 27.
where a, b, c, Represent ratios given by fixed numbers.
Usually equation 26 has a different representation, a2 gets designated by g11, ac
by g12 while b2 by g22; where our equation becomes:
S2 = g11u2 + 2g12uv + g22v2
Equation 28.
where indexes 11, 12, 22, Refer to mere subscripts, labels, indices, etc.,
representing different g's have different values.
Finally in terms of variable (Latin variāre: changeable; increase-or-decrease
proportionately-with-or-inversely to the increase-or-decrease of another value)
values d, along with x = (v, u), from equations 11 with 12, as a quadratic
differential in 2-dimensions:
ds2 = g11dx12 + 2g12dx1dx2 + g22dx22
Equation 29.
where s, Length 0P.
d, Variable values of.
g11, Value au, (x, x).
g12, Value cv, (x, y).
g22, Value bv, (y, y).
x1, Gaussian (parameter) co-ordinate u.
x2, Gaussian (parameter) co-ordinate v.
We can derive the Pythagorean rule for right-angled triangles in terms of
parameter co-ordinates, from equation 28 by taking g11 = 1; g12 =0 along with g22
=1:
ds2 = dx12 + dx22
Equation 30.
Such that we can derive any formulae by equating some of the g's to zero's, to
one, etc. Hence for 3-dimensions we get:
ds2 = g11dx12 + g22dx22 + g33dx32 + 2g12dx1dx2
+ 2g13dx1dx3 + 2g23dx2dx3
Equation 31.
Therefore for 4-dimensions,
ds2 = g11dx12 + g22dx22 + g33dx32 + g44dx42 + 2g12dx1dx2
+ 2g13dx1dx3 + 2g14dx1dx4 + 2g23dx2dx3
+ 2g24dx2dx4 + 2g34dx3dx4
Equation 32.
Now formula 32 has become termed 'the generalized Pythagorean rule', of which
the ordinary form given previously remains a particular case. By comparing the
formulae 29, 32, with equation 34 for Pythagoras' theorem for an acute-angled
triangle:
"In any triangle, the square on the side opposite an acute angle is
equal to the sum of the squares on the other two sides minus twice the product
of one of these sides and the projection on it of the other".
Therefore with reference to Figure 38.
P
S

O
C
D
Figure 38.
Hence,
S2 = OP2 = OD2 – 2OC  CD + PD2
Equation 33.
Therefore in terms of parametric co-ordinates,
ds2 = dx12 - 2dx1dx2 + dx22
Equation 34.
That these g's are not equal for different systems of co-ordinates, further that
they remain factors in measure-determination which represent the geometry of the
surface considered. Therefore the above formulae can have an abbreviated
written form:
ds2 = gmndxmdxn
Equation 35.
where m , Dimensional Gaussian (parametric) co-ordinates 1, 2, 3, 4.
n
, Sum of.
Indeed Riemann observed that the theorem of Pythagoras can become further
generalized, such that it can become used to define a measure of length by
writing:
ds2
n
= 
ij = 1
gij dxi dxj
Equation 36.
where if n = 3, with gi = 1 (if i = j), O (if i  j), we have ‘E’ geometry described
in terms of Cartesian co-ordinates, while other choices for gij lead to a host of
new geometrical systems.
As factors of measure-determination, the g's with different indexes serve
as sides or-both angles (of trigonometric rules) for the determination of the
actual sizes of the quadrilaterals. They may have different values from mesh to
mesh, but if they become known for each mesh, then by formula 34, the true
distance of an arbitrary point P, within an arbitrary mesh from the origin can
become calculated.
As introduced our g's represent ratios, thereby representing numbers.
Such numbers may in turn become regarded as tensors (Latin tensus of tendere, to
stretch, tension, connectedness: to relate vectors [having direction-magnitude,
etc] involving transformations in terms of a co-ordinate system, as nonarbitrary relations; otherwise ratio of increase in length of a vector) of zero
rank for mathematical convenience; where the 'quantities' gxx, gxy, gyy, may
become treated as components of a tensor. Since this tensor determines the
measure relations in any particular region, it becomes termed the metrix
fundamental tensor ds2, a value representing geometric structure; mathematical
assignment of arbitrary inner products (multi-moments of dot products: inner
products having 'properties' comprising of points, vectors, etc., of a
manifold): having 'properties' of 'symmetry' (Greek summetria: having the 'same'
divisible proportions), 'linearity' (Latin, lineāris: 'uniform', 'straight', not
curving, etc., not extending in a continuum), etc., on each tangent plane ('set
sum' of 2-dimensional tangent [Latin tangens of tangere, touch: meeting at a
single point without intersecting it {even if produced}; ratio of the
perpendicular subtending the angle in a right-angled triangle to the base; etc]
vectors [a vector based on a 'linear' approximation from a point following that
surface] at each point P of manifold M) of manifold M. Such that with an
assignment values, a determination of the full geometry of the surface
(geometric surface, a two-dimensional Riemannian manifold), in a given region
can become achieved; further conversely, we can determine the fundamental tensor
in a given region from measurements made in that region, without any previous
knowledge of how our curved surface remains embedded in 'space' at the place in
question.
Therefore,
(pn) + ds2 = M(pn)
Equation 37.
ds2 = M(pn) - (pn)
Equation 38.
M = ds2
Equation 39.
it follows,
thus,
where
ds2, Metric fundamental tensor.
M, Manifold.
p, Patch of plane.
n
, Power of.
n
(p ), Surface.
M(pn), Geometric surface.
Where the metric tensor usually varies continuously from place to place such
that each geometric manifold may become regarded as the field of their metric
fundamental tensor. Therefore a manifold M of arbitrary dimensions furnished
with a (differential) inner product on each of their tangent 'spaces' (on which
tangent vectors operate) becomes termed a Riemannian manifold, hence the
resulting geometry a Riemannian geometry. Where 'E' geometry remains the
special case of Riemannian geometry obtained on the 'E space' En, with
assignments of dot product.
Nevertheless mathematicians have shown that the metric tensor defines a
number termed the Riemann scaler (Latin scāla, ladder: anything graduated; a
system of correspondence between different magnitudes, relative dimensions;
etc), which remains completely independent of the co-ordinate system, further
leads to the definition of the curvature tensor (Gaussian curvature), which can
become connected with the 'matter tensor', leading to a modelling of plenum as a
space-time (after Albert Einstein (1905, 1916), with Hermann Minkowski (1908),
'space' becomes interchangeable with 'time' because 'space' is not 'empty' but
filled with 'matter' which changes hence 'time') continuum (Latin continuus:
thing of continuous structure, etc): after Albert Einstein (1916), that 'space'
is not an 'empty nothingness' but filled with changing 'matter' (Latin mātēria,
stuff from Greek hulē: physical 'substance' [Latin substantia, that which a
thing consists: 'permanent', unchanging 'matter' having 'essences', variable
'properties' involving a doctrine of 'predicables’, that which 'can be
predicated {Latin proedicāre, to proclaim: assert as 'attribute'} or attributes,
essences predicated'], having mass, occupying 'space', etc., perceptible via
senses, distinguishing from 'thought', 'mind', etc.; subject, content [Latin
contentus, to contain: comprising of, abstracted as experience, etc], etc.,
opposed to form) in irregular amounts, hence a continuum of undulating
curvature, thus not of 'straight lines'.
Infact Isaac Newton's (1687) infinite extending 'space' in his
"Philosophiae Naturalis Principia Mathematica" ("Mathematical Principles Of
Natural Philosophy") is nothing more than the 'space' of 'E' geometry, involving
3-dimensions of longitude, latitude, radius, otherwise height, breadth, width.
For example, refer to Figure 40-41. Where 'time' can become 'added' (Latin
addere, to put: 'sum' of values) later.
Figure 39.
Isaac Newton, (January 4, 1643 – March 31 1727
[December 25, 1642 – March 20, 1727]);
However as Hermann Minkowski (1908) in a lecture on "Space And Time"
presented at the 80th Assembly of German Natural Scientists and Physicians at
Cologne, 21 September 1908, later published as a paper "Raum Und Zeit" ("Space
And Time") in Physikalische Zeitschrift 10, 1909, asserts:
"We will try to visualize the state of things by the graphic method.
Let x, y, z be rectangular co-ordinates for space and let t denote time. The
objects of our perception invariably include places and times in combination.
Nobody has ever noticed a place except at a time, or a time except at a place.
But I still respect the dogma that both space and time have independent
significance. A point of space at a point of time, that is, a system of values
x, y, z, t, I will call a world-point. The multiplicity of all thinkable x, y,
z, t systems of values we will christen the world".
Along with world (-point, -line, -postulate), Minkowski used point-event, along
with manifold. As Einstein (1920) in his popular exposition "Relativity: The
Special And The General Theory", elaborates:
"A four-dimensional continuum described by the 'co-ordinates' x1, x2,
x3, x4 was called 'world' by Minkowski, who also termed a point-event a 'worldpoint'.
...The four-dimensional 'world' bears a close similarity to the
three-dimensional 'space' of (Euclidean) analytical geometry.
...We can regard Minkowski's 'world' in a formal manner as a
four-dimensional Euclidean space (with imaginary time co-ordinate), the Lorentz
transformation corresponds to a 'rotation' of the co-ordinate system in the
four-dimensional 'world' ".
In order to solve this visualization problem Minkowski (1908) chose a
E3
Tp(M)
E2
E1
M
Where E 1-3,
T,
p,
M,
Figure 40.
Euclidean dimensions.
Tangent.
Point.
Manifold.
h
b
w
t
Figure 41.
Where h, Height, Lontitude, etc.
w, Width, Latitude, etc.
b, Breath, radius, etc.
t, 'Time', change, motion, etc.
Figure 42.
Lao Tse (c. 600 B.C.);
David Hume (April 26, 1711 – August 25, 1776);
Max Wertheimer (April 15, 1880 – October 12, 1943);
Hendrick Antoon Lorentz (July 18, 1853 – February 4, 1928);
Albert Einstein (March 14, 1879 – April 18, 1955);
Hermann Minkowski (June 22, 1864 – January 12, 1909).
h
t
s
Figure 43.
Hermann Minkowski's (1908) 2-dimensional geometry, relating 'space and time'
in a Pythagorean-like relationship.
Where s, 'Space'.
t, 'Time'.
h, Space-time interval.
Figure 44.
Equations for Euclidean, Non-Euclidean 'space'(-time).
two-dimensional graphic. For example, if we substitute 'space' for one of the
legs of a right-angled triangle, 'time' for the other leg, while space-time
interval for the hypotenuse. Refer to Figure 43. Such that we have a
relationship analogous to that between 'space', 'time', along with space-time
interval described in Einstein's (1905) Special theory of Relativity ("Zur
Elektrodynamik Bewegter Koper", Annalende Physik, 17. "On The Electrodynamics
Of Moving Bodies"). As Gary Zukov (1979) in "The Dancing Wu Li Masters",
explains:
"Actually this Pythagorean-like relationship was the discovery of
Einstein's mathematics teacher, Herman Minkowski, who was inspired by his most
famous student's special theory of relativity".
Such that where the Pythagoras’ theorem:
c2 = a2 + b 2
Equation 40.
While the equation for the space-time interval in the special theory of
relativity:
s2 = t2 – x2
Equation 41.
Where the Pythagorean theorem describes characteristics in 2-dimensional 'E'
'space'. While the equation for the space-time interval describes
characteristics in Minkowski's 2-dimensional space-time.
Yet Minkowski (1908) appears not to know of the work Riemann, nor of E
geometry, when he proclaimed:
"Three-dimensional geometry becomes a chapter in four-dimensional
physics".
Such that Minkowski (1908) had not made the Ē connection: 4-D results in
space-time curvature (if 'time' dilates, 'space' bends), hence continuum.
Instead Einstein (1916) in "The Foundation Of The General Theory Of Relativity"
("Die Grundlage Der Allgemeinen Relativitatstheorie", Annalen der Physik, 49),
made this explicitly, mathematically clear. Whereas Minkowski died in 1909.
However starting with with a system of values x, y, z, t, where,
x2 + y2 + z2 + t2
Equation 42.
represents an 'E' geometric (Pythagorean) equation for 4-D. We can characterize
Hendrik Antoon Lorentz's (1904) transformation by introducing the imaginary
-1ct in place of t, as time-variable; where we can denote i for  -1 . Such
that Minkowski's (1908) physico-mathematical (Greek phusikē, of nature;
mathēmatikos: extensional higher order abstracting, method(s) of formulating,
etc., combining both Physics with Mathematics) correction for an 'E' 'additive'
process gives:
x2 + y2 + z2 – ic2t2
Equation 43.
x2 + y2 + z2 = ic2t2
Equation 44.
in terms of c2t2,
If we introduce now the Minkowski-Sommerfeld U for the 'spatial' world-points,
U12 + U22 + U32 = ic2t2
Equation 45.
hence simplifying, we get a manifold of 4 world-points corresponding to a 'time'
variance (but invariant in 'linear' transfomations), otherwise world-time:
U42 = (ict)2
after Hermann Minkowski (1908).
where U, Minkowski world-point (manifold).
x, y, z, 'Space' co-ordinates.
Equation 46.
t, 'Time' co-ordinate.
c, Parameter.
i,  -1 .
Where what Minkowski must have understood, but found it difficult to
express via 'E' geometry, Alfred Korzybski (1933) in "Science And Sanity"
expresses clearly:
"The graphic picture of a moving point is a world-line. Rectilinear
uniform motion corresponds then to a straight world-line; accelerated motion, to
one that is curved".
Figure 45.
Alfred Habdank Skarbek Korzybski (July 3, 1879 – March 1, 1950).
Such that,
"Three dimensional kinematics becomes four-dimensional geometry,
three-dimensional dynamics can be considered as four-dimensional statics".
Indeed the problem involves one of visualizing, hampered by our
'Aristotelian ('A')-conditionality': education in 'Aristotelian methodology-orlogic' ('Aristotelian thinking'; methodology organizing knowledge as an
'intensional' [false-to-facts 'universalizing'] language 'habit(s)' based on the
laws of 'identity' ['everything is the same'], excluded third ['or not'], with
contradiction' ['but not both'], systematized by Aristotle (350 B.C.),
'dogmatised' [Greek dogma, to seem: doctrine, system of doctrines established
without question, ignoring 'facts', etc] by his followers), equivalent to
'identification', 'conditionality' (after Ivan Petrovich Pavlov (1906),
'associative' learning [Latin associāre: after Aristotle (350 B.C.), to form
connexion(s) accustomed in joining: 'associations' are false relations between
two events occurring {hence ordered} in spatio-temporal contiguity {Gottfried
Wilhelm Leibnitz's (1686), "action-by-contact"}; after David Hume (1739)], the
expectancy [likelihood of occurrence(s) from experience(s)] of a fixed
space-time contingency [likelihood of one event followed by another], such that
one event will follow another; involving for example 'Pavlovian conditioning',
after Ivan Petrovich Pavlov (1906), contingent contiguous 'associations' of
'stimuli' [Latin stimulus, goad: energy change, objects, events, originating a
'response' from Latin respondēre, respond, the consequence, both 'el'];
equivalent to 'identification(s)', etc), etc., hence 'Aristotelian-habitual
('fixed' disposition, way of 'acting' [Latin actiōnem, doing: exertion of
energy, influence, etc.; process of doing, performing, preventing, etc], etc.,
due to 'conditionality')-unconscious(conscious from Latin conscius, aware:
emergent self-awareness; 'mind' [Greek menos, rage; else psuchē, soul: emergent
'conscious' functioning], etc.; hence not conscious of abstracting, etc)-
conditionality', etc. Equivalent to Alfred Korzybski's (1933)
'identification(s)', the avoidance of which involves Korzybski's (1933)
consciousness abstracting: empirical order of evaluating, etc.
Therefore as Korzybski (1933) in "Science And Sanity" argues,
"Everything which happens must be structurally represented as
something, somewhere, at some 'time'...The four-dimensional language, which
describes happenings structurally more nearly as we experience them, is
precisely the language of 'event'".
that,
"As the 'space-time' continuum is the closest to our daily
experience".
then,
"The two-valued A, el, three-dimensional 'logic' does not apply to the
world of events, to the objective levels, etc., and, for reasons already
explained, does not apply to the study of the foundations of mathematics".
"If we approach the infinite-valued facts of life with one-, two-, or
even few-valued semantic attitudes, we must identify some of the indefinitely
many values into one, or a few values, and so approach the infinite-valued world
with an orientation which projects ignorantly or pathologically our restricted,
few-valued semantic evaluations on the infinite-valued individual facts of
experience".
Such that,
"The language of space-time is non-el".
Non-Elementalism (ēl, non-el): after Lao-Tse (600 B.C.), over-lapping,
interchangeable (equivalent, reversible, etc), functional (non-linear [not
'uniform', 'straight', etc., but curvature; such that a 'straight line' becomes
a point extending in a continuum; interchangeable with both non-additiveasymmetry]-asymmetry[not 'uniform' {irregular}, 'same' {incommensurable}, etc.,
in relations, for example left with right, up with down, etc., interchangeable
with both non-linear-non-additive]-non-additive[not sum of values, but
functional; interchangeable with both non-linear-asymmetry]) packets, etc.,
emergent, holism; rediscovered by Alfred Korzybski (1933).
Whereas,
"The language in the A-system represents, in principle, what maybe
called a three-dimensional and one-, two-, more generally few-valued linguistic
system structurally non-similar or to the infinite-valued, four-dimensional
event-process conditions. Let us analyse, for instance, the A-term 'apple'.
This represents, in principle, a name for a verbal, one-valued, and constant
intensional definition, in which space-time relations do not enter. What are
the structural facts of experience? The objects which we call 'apple'
represents a process which changes continually; besides, every single apple that
ever existed, or will exist, was an absolute individual, and different from any
other objective 'apple'. In applying such a three-dimensional, and one-valued
language to essentially infinite-valued processes, we only make proper
evaluation, and so adjustment and sanity, very difficult".
Further Korzybski (1937) in "General Semantics Seminar 1937", argues for more
dimensions:
"Dimensionality implies a number of factors. The number we need to
know before we know something. Motion involves four dimensions. Otherwise,
'space' and 'time' are indivisibly connected. Look at the smoke of my
cigarette. You see what is going on. How many factors - just fancy, for we can
do no better - do you have to know to follow up what is going on in this smoke.
At least four? No, a million would not cover it. I am trying to convey that
question of dimensionality over to you. Following that smoke how many factors
do you need to know, to get hold of that cloud of smoke? Otherwise, that cloud
of smoke is million-dimensional. Can we separate 'space' and 'time'? No".
Indeed as Korzybski argues, this multi-dimensional order (having many
dimensions, a manifold of at least four-dimensions, etc) involves an infinite
process as a variable finite, after Georg Ferdinand Ludwig Philipp Cantor
(1874), as changing combinations of variables; yet though the division of
variables in between two others remains a finite process, the potential of
variation (changes) becomes infinite. Where Korzybski (1933) continues in
"Science And Sanity":
"Structure represents the only extensional content of knowledge...as
given in terms of relations and multiordinal and multi-dimensional order".
"The only use-fullness of a map or a language depends on the
similarity of structure between the empirical world and the map-languages".
Where,
"...mathematics is the only language, at present, which in structure,
is similar to the structure of the world and the nervous system".
Such that instead of operating with an 'A'-system involving one-valued
terms, viewing 'pictures' as 2-D, 'models' as only 3-D, etc., introducing 'time'
later if required. We must as Korzybski argued, endeavour to perceivevisualize-describe-formulate in at least 4-D. Thus this can become achieved
with Korzybski's (1924) Structrual Differential (S.D.): model of the natural
order of abstracting, otherwise as a higher order abstracting, Time-binding:
capacity to improve on the accumulated abstractions of others, then transmitting
it for future generations. Such that Korzybski's SD intends to model changing
'reality', in terms of infinite-dimensionality, infinite-valued probabilities,
etc. As Korzybski (1933) explains:
"The use of the Structural Differential is necessary, because some
levels (Objective level, Event) are un-speakable".
Refer to Figure 45 (S.D.). Where from the "Event", the external happenings,
only abstractions as internal events become possible, involving non-verbal
(silent levels: equivalent to both the Event followed by Object [objective]
levels, the representations, dealing with, etc., of events via feeling(s)action(s) processes; after Alfred Korzybski (1933)) to verbal (pertaining to
higher orders of abstracting involving description(s) followed by inferences,
formulation(s), etc., of increasingly higher order; referring to the processes
of ‘thought’) levels. Entailing the feeling-thinking non-el connection via
perceiving-visualizing-insight-formulating. Otherwise Samuel J. Bois’ (1966)
evaluational (semantic) transacting: term for feeling(s)-thinking-action(s)with-an-environment (replacing Isaac Newton’s (1687) ‘el’ ‘reaction’, etc.,
further semantic to evaluating [due to Korzybski’s (1935) increasing concern
over folk-(mis)meanings of semantics], in Alfred Korzybski’s (1933) original
semantic reaction [s.r.], term for feeling(s)-thinking-action(s)-with-anenvironment), equivalent to evaluating. Whilst from Korzybski's (1950) paper
"The Role Of Language In The Perceptual Processes":
"Personally, I 'think' in terms of pictures, and how I speak about
those visualizations later is a different problem".
"There is a tremendous difference between 'thinking' in verbal terms,
and 'contemplating' inwardly silent, on nonverbal levels, and then searching for
the proper structure of language to fit the supposedly discovered structure of
the silent processes that modern science tries to find. If we 'think' verbally,
we act as biased observers and project onto the silent levels the structure of
the language we use, so remaining in our rut of orientations which make keen,
unbiased observations ('perceptions'?) and creative work well-nigh impossible.
In contrast, when we 'think' without words, or in pictures or visualizations
(which involve structure and, therefore, relations), we may discover new aspects
and relations on silent levels, and so may formulate important theoretical
results in the general search for a similarity of structure between the two
levels, silent and verbal. Practically all important advances are made in that
way".
"The points I have touched upon here: namely, the subject-predicate
type of structure, the 'is' of identity, two-valued 'either-or' orientations,
and elementalism, are perhaps the main factors of the Aristotelian language
structure that moulded our 'perceptions' and hindered the scientific
investigations which at this date have so greatly, in many instances freed us
from the older limitations and allowed us to 'see the world anew'. The
'discovery of the obvious' is well known to be the most difficult, simply
because the old habits of 'thinking' have blocked our capacity to 'see the old
anew' (Leibnitz)".
Where however ‘thinking’ in ‘pictures’ does not simply involve
2-dimensional constructs, nor that we visualize, perceive, etc., them in
2-dimensions, where any such related ‘motion’ emerges as structure from an order
of relations, involving dynamic 4-dimensional space-time. For example, Max
Wertheimer (1912) reported an experimental study in a famous paper,
Figure 46.
Structural Differential.
After Alfred Korzybski (1924).
"Experimentelle Studien Uber Das Sehen Von Bewegung" ("Experimental Study On The
Perception Of Movement") completed with Wolfgang Kohler along with Kurt Koffka
as subjects. It involved the "Phi-phenomenon" (dynamic-in-between-ness),
concerned with what the 'atomists' ('elementalists', origins with 'Physical
atomism', originated by Democritus (460-570 B.C.) with Leucippus (5th Century
B.C.): theory of the physical world, constituted by an infinite number of
indivisible corpuscles [atoms] moving randomly in an infinite void), for example
Wilhelm Max Wundt (1832-1920) who founded Structuralism (approach based on the
presumption that 'mental' [Latin mentālis, of 'mind'] experience becomes viewed
no matter how complex, as 'interactions of simple processes or elements', etc.;
otherwise that sensory experience remain 'el') considered as a visual illusion,
explained by eye-ball movements, after-images on the retina, etc. Involving
stationary objects, when shown in rapid succession appear to move. Showing that
we perceive 2-dimensional 'pictures' presented in an 'el' 'static' series as to
mimic 'motion' (such as arranging exposures via a tachistoscope) of the
'figures' within the 'pictures', instead as ēl 4-dimensional 'motion' (involving
a four-dimensional space-time). Wertheimer obtained the phenomenon (where
phi- represents whatever occurred between exposures, etc), when he arranged
his tachistoscopic exposures so that the total presentation period for the first
object, the in-between-interval, along with the exposure period for the second
object did not exceed one-tenth of a second; further since the minimal period
for eyeball 'reactions' remains somewhat greater (around 130 milliseconds), this
explanation must be excluded.
Perhaps the most definitive of Wertheimer's experiments, involved two
pairs of lines, presented in antagonistic apparent movements. Such that when
one line image of a pair disappears, the other appears above-or-below the first.
Negating any involvement of eye-ball movements, since obviously the eye-ball
cannot move in two opposed directions at the 'same time'. Whereby the
involvement of after-images got dispelled by gazing at a small bright cross
(else lamp filament) before producing the apparent movement. As discussed in
George Wilfred Hartmann's (1935) "Gestalt Psychology: A Survey Of Facts And
Principles".
As Kurt Koffka (1935) in "Principles Of Gestalt Psychology" observes:
"Wertheimer's paper and a number of publications which followed it
dealt chiefly or exclusively with stroboscopic motion, i.e., the case where
perceived motion is produced by stationary objects. Since it has been proved
beyond a doubt (Wertheimer, Cermak and Koffka, Dunker 1929, Brown 1931, Van der
Waals and Roelofs 1933) that as far as psychophysical dynamics are concerned
there is no difference between stroboscopic and 'real' motion, i.e., perceived
motion produced by actually moving objects".
Then as Korzybski (1933) continues in "Science And Sanity":
"...the Structural Differential...convey to the eye structural
differences between the world of the animal, the primitive man, and the infant,
which, no matter how complex, is extremely simple in comparison with the world
of 'civilized' adult. The first involves a one-valued orientation which, if
applied to the infinite-valued facts of life, gives extremely inadequate,
wasteful, and ultimately painful adjustment, where only the few strongest
survive. The second involves infinite-valued orientation, similar in structure
to the actual, empirical, infinite-valued facts of life, allowing a one-to-one
adjustment in evaluation with the facts in each individual case, and producing a
semantic flexibility, etc., necessary for adjustment".
"If we use a three-dimensional A language and apply such an 'all' to
such an infinite process then we simply produce a self-contradiction. If we
apply to such semantic process a four-dimensional 'all with a date', then we
have arrested, for the 'time' being, the process, or taken a static cross
section of the infinite process at that date; but then we deal with a finite.
Once we are constantly conscious of abstracting in different orders, these
subtle differences become quite clear. When treated as a variable finite it was
satisfactory and sufficient, and has proven to be a most creative notion in
mathematics".
Where Korzybski (1937) in "General Semantics Seminar 1937", has:
"For, the moment you index and date a passing event you have a
four-dimensional cross section that is static. This is four-dimensional,
remember.
Then,
A process can be made static in four dimensions".
"If you have a 'chair' as a definition, that is static, but when you
exhibit by extension, you exhibit things, chair , chair , chair , etc.; you have
any number of chairs. You have made 'chair' dynamic - changing. We can then
readjust the structure of the static language to the dynamic world".
Where Korzybski (1933) in "Science And Sanity", continues:
"The above considerations of order lead to a formulation of a
fundamental principle (a principle underlying the whole of the non-aristotelian
system); namely, that organisms which represent processes must develop in a
certain natural survival four-dimensional order, and that the reversal of that
order must lead to pathological (non-survival) developments".
"As the organism works as-a-whole, and as the training is
psycho-physiological in terms of order, reversing the reversed pathological
order, etc., (to consciousness of abstracting), organism-as-a-whole means must
be employed. For this purpose the Structural Differential has been developed".
"Our nervous system by its structure produces abstractions of
different orders, dynamic on some levels, static on others. The problems of
sanity and adjustment become problems of translation from one level to another".
"What is important for our s.r is that we realize the fact that the
gross macroscopic materials with which we are familiar are not simply what we
see, feel, etc., but consist of dynamic processes of some extremely fine
structure; and that we realize further that our 'senses' are not adapted to
register these processes without the help of extra-neural means and higher order
abstractions".
"If we introduce dynamic, shifting entities into static higher order
abstractions, rationality is impossible and we drift toward mysticism...The
first step of this translation has already been given in the notion of the
'variable'. The calculus carried it a step further".
Such that the SD becomes instrumental as an aid in the training of Korzybski's
(1933) Non-Aristotleian (Ā: Alfred Korzybski's (1933) extensional [orientating
to the non-verbal levels, factual evaluating] orientated revision of Aristotle's
(350 B.C.) methodology, based on General Semantic premises of non-identity,
non-allness with self-reflexiveness) systems along with General Semantics (G.S.:
ēl 'logic') Korzybski's (1933), Science of values, hence evaluation (French
evaluer: determine the value, hence representing): event(s)-insight-logic;
feeling ('sense of', perceiving, visualizing, 'emoting', etc)-thinking
(formulating)-about-an-environment, which may include the consideration of
possible 'action(s)'. Where Korzybski (1937) on GS, value, evaluation in "GS
Seminar 1937":
"...term semantics...Greek word meaning 'significance', 'value',
'meaning'...The science of GS is the science of values - evaluation".
Where Korzybski (1935, etc) elsewhere explains, that semantics (Greek semainein,
sign) introduced by Michel Breal (1897), to 'signify', value, 'mean', etc.,
dealing with 'meaning' in language; therefore should not be confused with
General Semantics; further where Korzybski (1935) became increasing concerned
with avoiding the folk-(mis)meanings involving semantics. While Korzybski
(1933) in "Science And Sanity" on infinite-valued, non-el 'logics':
"In a Ā-system, the 'logical' problems of freedom from contradiction
become also semantic problems of one-valued meanings made possible only under
infinite-valued, Ā, non-el general semantics, and the recognition of the Ā
multiordinality of terms, etc. A Ā-system introduces some fundamental
innovations, such as completely rejecting identity, elementalism, etc., and
becomes based on m.o structure and order, and so ultimately becomes non-el. The
A, (3+1)-dimensional el, (in the main) intensional system becomes a fourdimensional, non-el, (in the main) extensional system. In such a system we
cannot use the formulations of elementalistic 'logics' and 'psychologies', but
must have Ā, non-el general semantics, which when generalized become an entirely
general discipline applicable to all life, as well as to generalized
mathematics. For the above reasons I shall use the word 'logic', in its el
sense, with quotation marks; and use the term general semantics for a non-el, A
discipline corresponding to the el, A or Ā 'logics'".
Then Korzybski (1933) continues:
"We proceeded by similarities much too often considered as
identities, with the result that differences were neglected. But in actual
life, without some primitive metaphysics, we do not find identities, and
differences become as important as similarities.
...In building a Ā-system, we have to stress differences, build a
'non-system' on 'non-allness', and reject identity. The older semantic
inclinations and infantile or primitive tendencies were a necessary step in
human evolution. For sanity, we must outgrow these infantile semantic
fixations".
Where Korzybski (1935) "Neuro-Semantic And Neuro-Linguistic Mechanisms Of
Extensionalization: General Semantics As A Natural Experimental Science":
"General Semantics because of extensional (physico-mathematical)
methods accomplishes this (a Science of evaluating), requiring a complete
revision of existing doctrines".
Otherwise the above can become more abstractly represented by what James
Joseph Sylvester (1814-97) asserted, as quoted by Korzybski (1933) in "Science
And Sanity":
"In mathematics we look for similarities in differences and
differences in similarities".
So following from the "Event" each dimension follows a "logical destiny" (after
Alfred Korzybski (1924), based on Cassius Jackson Keyser's (1922) "Logical
fate"): expression for consequences follow from premises. Such that we can
'model' the differential evaluations of dimensionality, by modifying Korzybski's
SD to a Dimensional Differential (D.D.):
Event (Latin ēventus, to happen: occurrences, happenings, etc., in an
environment from French environ, round about: surrounding events): beyond
perceiving(Latin percipere, to take: sensing, representing [percepts] via
'sensory' processes: 'visual', 'auditory', 'taste', 'olfactory', with
'tactile')-visualizing(to replay the 'visual' part of the perceptual process, as
'mental images' ['picturing']; to imagine the possible both-others impossible,
as for example what might have happened from actual events transpired)describing(Latin dēscrībere, to write: an account of appearances,
interchangeable with verbal observation, along with 'model' as diagrammaticmathematical description)-formulating(formula, from Latin forma: verbalmathematical abstracting), from which we can evaluate in:
1-D : perceiving-visualizing-modelling-formulating:
e.g.: perceiving: walking a path, etc.
visualizing: 'linear': 'straight' line, etc.
describing: line on a sphere(description: longitude, etc), etc.
formulating: E1-manifold: x = x'; Euclid's fifth postulate, etc.
2-D : perceiving-visualizing-modelling-formulating:
e.g.: perceiving: framing a tree branch, etc.
visualizing: plane, etc.
describing: two perpendicular lines upon sphere (description:
longitude, latitude, etc); Edwin E. Abbot's (1991) "Flatland"; etc.
formulating: E2-manifold: x2 + y2; Descartes' algebraic translation;
Gaussian curvature K = 0; etc.
3-D : perceiving-visualizing-modelling-formulating:
e.g.: perceiving: ice-cubes, etc.
visualizing: cuboid, etc.
describing: two perpendicular lines upon sphere, radius of a sphere
(description: longitude, latitude, depth, etc), etc.
formulating: E3-manifold: x2 + y2 + z2; Euclid-Newton 'space';
Gaussian curvature K = determinant of shape operator on manifold; etc.
4-D : perceiving-visualizing-modelling-formulating:
e.g.: perceiving: Earth from orbit, etc.
visualizing: sphere, etc.
describing:two perpendicular lines upon sphere, radius of a sphere,
accounting for curvature (description: depth with curving longitude,
latitude, etc).
formulating: R4-manifold: u42 = (ict)2, Einstein-Minkowski
four-dimensional space-time continuum;
Gaussian curvature K > 1/rR; etc.
Etc.
Infinite-D : perceiving-visualizing-modelling-formulating:
e.g: perceiving: AK's cigarette smoke, etc.
visualizing: over-lapping folded spheres, etc.
describing: infinite number of infinite-overlapping-foldedspheres (2-D surfaces), on a sphere (description: depth of
infinite longitudes, latitudes, upon a curving longitudes,
latitudes, etc), etc.
formulating: R-manifold space-time: as yet unformulated, it
appears.
Such that 1-3 D as 'el' one-, two-, three-, dimensional-values (representations)
involve an 'additive ('and') linearity'; whilst 4-D as ēl four-, to -,
dimensional-value, involves functional non-linearity.
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