* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Ā - Non-Aristotelian Evaluating
History of trigonometry wikipedia , lookup
Algebraic geometry wikipedia , lookup
Surface (topology) wikipedia , lookup
Shape of the universe wikipedia , lookup
Anti-de Sitter space wikipedia , lookup
Four-dimensional space wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
Lie sphere geometry wikipedia , lookup
Cartan connection wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Analytic geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Riemannian connection on a surface wikipedia , lookup
Multilateration wikipedia , lookup
Differential geometry of surfaces wikipedia , lookup
Geometrization conjecture wikipedia , lookup
History of geometry wikipedia , lookup
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. REFERENCES. (1). Abbot, Edwin A. (1991) "Flatland: A Romance Of Many Dimensions". Princeton University Press. (2). Bell, Eric T. (1937) "Men Of Mathematics". Vol: 1-2. Pelican. (3). Bois, Samuel J. (1966) "The Art Of Awareness: A Handbook On General Semantics And Epistemics". 1978, fourth edition. Continuum Press. (4). Bolyai, János (1831) "Appendix Scientiam Spatii Absolute Veram Exhibens" ("The Science Of Absolute Space"). In (3). (5). Bonola, Roberto (1912) "La Geometria Non-Euclidean" ("Non-Euclidean Geometry"). 1955 edition, translated by H.S. Carslaw. Dover. (6). Boyer, Carl B. (1968) "A History Of Mathematics". Wiley. (7). Cantor, Georg Ferdinand Ludwig Philipp (1874) "Contributions To The Founding Of The Theory Of Transfinite Number". Translated by P.E.B. Jourdan, 1915. Dover. (8). Einstein, Albert (1920) "Relativity: The Special And The General Theory". Translated by R.W. Lawson. Methuen. (9). Einstein, Albert; Lorentz, Hendrik Antoon; Weyl, Hermann; Minkowski, Hermann (1923) "The Principle Of Relativity: A Collection Of Original Papers On The Special And General Theory Of Relativity". Translated by W. Perrett with G.B. Jeffery. 1952 edition. Dover. (10). Fauvel, John; Gray, Jeremy. Editors (1987) "The History Of Mathematics: A Reader". The Open University. (11). Gillispie, Charles Coulston (1973) "Dictionary Of Scientific Biography". Vol: I-XIV. Charles Scribner’s Sons. (12). Goetz, P.W. Editor-In-Chief (1985) "The New Encyclopaedia Britannica". Vol: 27. Encyclopaedia Britannica. (13). Greene, Brian (1999) "The Elegant Universe: Superstrings, Hidden Dimensions, And The Quest For The Ultimate Theory". Vintage. (14). Gregory, Richard L. (1987) "The Oxford Companion To The Mind". Oxford University Press. (15). Hall, H.S., Stevens, F.H. (1902) "A Text-Book Of Euclid's Elements: For The Use Of Schools". Books I-VI, XI. Macmillan. (16). Halsey, W.D. (1984) "Collier's Encyclopedia". Vol: 11. Macmillan. (17). Hartmann, George W. (1935) "Gestalt Psychology: A Survey Of Facts And Principles". Ronald Press Co. (18). Hollingdale, Stuart (1989) "Makers Of Mathematics". Penguin. (19). Honderich, Ted Editor (1995) "The Oxford Companion To Philosophy". Oxford University Press. (20). Howard, A.V. (1961) "Chambers Dictionary Of Scientists". Chambers. (21). Hume, David (1739) "A Treatise Of Human Nature". 1978 edition revised, edited by L.A. Selby-Bigge with P.H. Nidditch. Oxford University Press. (22). Hurd, D.L., Kipling, J.J. (1952) "The Origins And Growth Of Physical Science". Vol: 1-2. Pelican. (23). Hutchins, Robert M. Chief Editor (1952) "Encyclopaedia Britannica Great Books Of The Western World 8: Aristotle Volume I". Encyclopaedia Britannica Inc. (24). Hutchins, Robert M. Chief Editor (1952) "Encyclopaedia Britannica Great Books Of The Western World 9: Aristotle Volume II". Encyclopaedia Britannica Inc. (25). Hutchins, Robert M. Chief Editor (1952) "Encyclopaedia Britannica Great Books Of The Western World 11: Euclid, Archimedes, Apollonius Of Perga, Nicomachus". Encyclopaedia Britannica Inc. (26). Hutchins, Robert M. Chief Editor (1952) "Encyclopaedia Britannica Great Books Of The Western World 31: Descartes, Spinoza". Encyclopaedia Britannica Inc. (27). Katz, David (1950) "Gestalt Psychology: Its Nature And Significance". Ronald Press co. (28). Kendig, Marjorie (1990) "Alfred Korzybski Collected Writings 1920-1950". Institute of General Semantics. (29). Keyser, Cassius Jackson (1922) "Mathematical Philosophy: A Study Of Fate And Freedom, Lectures For Educated Layman". E.P. Dutton. (30). Klein, Felix; Schering, Ernst C.J. (1863-1933) "Werke" (Karl Gauss’ "Collected Works"). Vol: 1-2. Konigliche Gesellschaft Der Wessenshaften Zu Gottingen. (31). Koffka, Kurt (1935) "Principles Of Gestalt Psychology". Routledge and Kegan Paul Ltd. (32). Köhler, Wolfgang (1925) "The Mentality Of Apes". Harcourt Brace and Kegan Paul Ltd. (33). Korzybski, Alfred Habdank Skarbek (1921) "Manhood Of Humanity". Dutton. (34). Korzybski, Alfred Habdank Skarbek (1933) "Science And Sanity: An Introduction To Non-Aristotelian Systems And General Semantics". 1994, fifth edition. Institute of General Semantics. (35). Korzybski, Alfred Habdank Skarbek (1937) "General Semantics Seminar 1937: Transcription Of Notes From Lectures In General Semantics Given At Olivet College". Edited by Homer J. Moore. Institute of General Semantics. (36). Korzybski, Alfred Habdank Skarbek (1947) "Historical Note On The Structural Differential". Audio tape. Institute of General Semantics. (37). Korzybski, Alfred Habdank Skarbek (1948-9) "Alfred Korzybski: Intensive Seminar, December 27, 1948 - January 2, 1949". Audio tapes. Institute of General Semantics. (38). Kuhn, Thomas Samuel (1962) "The Structure Of Scientific Revolutions". University of Chicago Press. (39). Lobatchevski, Nikolai Ivanovich (1840) "Geometrische Undersuchungen Zur Theorie Der Parallelinen" ("Geometrical Researches On The Theory Of Parallels"). In (3). (40). Lobatchevski, Nikolai Ivanovich (1883-86) "Polnoe Sobranie Sochinenii Po Geometrii" ("Complete Geometrical Works". Vol: I-II. Vol: I, works in Russian; Vol: II, works in French and German. (41). Locke, John (1690) "An Essay Concerning Human Understanding". 1975 edition, edited by P.H. Nidditch. Claredon Press. (42). Minkowski, Hermann (1908) "Raum und Zeit" ("Space And Time"), in Physikalishe Zeitschrift, 10, 1909. From a lecture on 'space and time' at the 80th Assembly of German Natural Scientists and Physicians, 21 September, 1908. In (9). (43). Mordkowitz, Jeffrey A. (2001) "A Note On Evaluational Reactions". In James D. French. Editor-In-Chief "General Semantics Bulletin Numbers 65-68". Institute of General Semantics. (44). Newton, Isaac (1687) "Philosophiae Naturalis Principia Mathematica". Translated by Andrew Motte (1995) "The Principia". Prometheus. (45). O'Neill, Barrett (1966) "Elementary Differential Geometry". Academic Press. (46). Pavlov, Ivan Petrovich (1927) "Conditioned Reflexes: An Investigation Of The Physiological Activity Of The Cerebral Cortex". Translated by G.V. Anrep, 1960. Dover. (47). Plato (381 B.C.) "The Republic". Translated by Desmond Lee, 1955. Penguin. (48). Riemann, Georg Friedrich Bernhard (1854) "Uber Die Hypothesen Welche, Der Geometrie Zu Gunde Lingen". (49). Rox, R.W. (1963) "Certificate Mathematics". Book III. Edward Arnold Ltd. (50). Scott, J.F. (1960) "A History Of Mathematics: Form Antiquity To The Beginning Of The Nineteenth Century". Taylor and Francis Ltd. (51). Taton, Rene (1964) "A General History Of The Sciences". Vol: 1-4. Translated by A.J. Pomerons. Thames and Hudson. (52). Weinberg, Harry L. (1954) "Levels Of Knowing And Existence: Studies In General Semantics". Institute of General Semantics. (53). Wertheimer, Max (1912) "Experimentelle Studien Über Das Sehen Von Bewegung". Zeitschrift Für Psychologie, Vol: 61. (54). Wilson, Robert Anton (1993) "Quantum Psychology". New Falcon. (55). Zukov, Gary (1980) "The Dancing Wu Li Masters: An Overview Of The New Physics". Fontana/Collins.