· 5 years ago · Mar 01, 2020, 08:52 PM
1Pure
2mathematics
3is
4the
5study
6of
7mathematical
8concepts
9independently
10any
11application
12outside
13These
14may
15originate
16in
17real
18world
19concerns
20and
21results
22obtained
23later
24turn
25out
26to
27be
28useful
29for
30practical
31applications
32but
33pure
34mathematicians
35are
36not
37primarily
38motivated
39by
40such
41Instead
42appeal
43attributed
44intellectual
45challenge
46aesthetic
47beauty
48working
49logical
50consequences
51basic
52principles
53While
54has
55existed
56as
57an
58activity
59since
60at
61least
62Ancient
63Greece
64concept
65was
66elaborated
67upon
68around
69year
70[1
71after
72introduction
73theories
74with
75counter
76intuitive
77properties
78non
79Euclidean
80geometries
81Cantor
82theory
83infinite
84sets
85discovery
86apparent
87paradoxes
88continuous
89functions
90that
91nowhere
92differentiable
93Russell
94paradox
95This
96introduced
97need
98renewing
99rigor
100rewriting
101all
102accordingly
103systematic
104use
105axiomatic
106methods
107led
108many
109focus
110on
111its
112own
113sake
114Nevertheless
115almost
116remained
117problems
118coming
119from
120or
121less
122abstract
123Also
124which
125had
126seemed
127totally
128were
129eventually
130used
131applied
132areas
133mainly
134physics
135computer
136science
137famous
138early
139example
140Isaac
141Newton
142demonstration
143his
144law
145universal
146gravitation
147implied
148planets
149move
150orbits
151conic
152sections
153geometrical
154curves
155been
156studied
157antiquity
158Apollonius
159Another
160problem
161factoring
162large
163integers
164basis
165RSA
166cryptosystem
167widely
168secure
169internet
170communications
171[2
172It
173follows
174presently
175distinction
176between
177more
178philosophical
179point
180view
181mathematician
182preference
183than
184rigid
185subdivision
186In
187particular
188it
189uncommon
190some
191members
192department
193describe
194themselves
195geometry
196From
197Wikipedia
198free
199encyclopedia
200Jump
201navigationJump
202search
203Plane
204redirects
205here
206For
207other
208uses
209see
210disambiguation
211Detail
212Raphael
213The
214School
215Athens
216featuring
217Greek
218perhaps
219representing
220Euclid
221Archimedes
222using
223compass
224draw
225geometric
226construction
227Geometry
228Stereographic
229projection
230svg
231Projecting
232sphere
233plane
234OutlineHistory
235Branches
236[hide
237EuclideanNon
238Elliptic
239SphericalHyperbolicNon
240Archimedean
241geometryProjectiveAffineSyntheticAnalyticAlgebraic
242ArithmeticDiophantineDifferential
243RiemannianSymplecticDiscrete
244differentialComplexFiniteDiscrete
245Combinatorial
246DigitalConvexComputationalFractalIncidence
247ConceptsFeatures
248[show
249Zero
250dimensional
251One
252Two
253Three
254Four
255Geometers
256name
257period
258vte
259system
260Alexandrian
261he
262described
263textbook
264Elements
265method
266consists
267assuming
268small
269set
270intuitively
271appealing
272axioms
273deducing
274propositions
275theorems
276these
277Although
278stated
279earlier
280first
281show
282how
283could
284fit
285into
286comprehensive
287deductive
288begins
289still
290taught
291secondary
292school
293high
294examples
295formal
296proof
297goes
298solid
299three
300dimensions
301Much
302states
303what
304now
305called
306algebra
307number
308explained
309language
310two
311thousand
312years
313adjective
314unnecessary
315because
316no
317sort
318conceived
319so
320obvious
321possible
322exception
323parallel
324postulate
325theorem
326proved
327them
328deemed
329true
330absolute
331often
332metaphysical
333sense
334Today
335however
336self
337consistent
338known
339ones
340having
341discovered
342th
343century
344An
345implication
346Albert
347Einstein
348general
349relativity
350physical
351space
352itself
353good
354approximation
355only
356over
357short
358distances
359relative
360strength
361gravitational
362field
363[3
364synthetic
365proceeds
366logically
367describing
368objects
369points
370lines
371about
372those
373without
374coordinates
375specify
376contrast
377analytic
378translate
379algebraic
380formulas
381Contents
382Axioms
383Parallel
384Methods
385System
386measurement
387arithmetic
388Notation
389terminology
390Naming
391figures
392Complementary
393supplementary
394angles
395Modern
396versions
397notation
398Some
399important
400well
401Pons
402Asinorum
403Congruence
404triangles
405Triangle
406angle
407sum
408Pythagorean
409Thales
410Scaling
411area
412volume
413Applications
414As
415description
416structure
417Later
418work
419Descartes
420Treatment
421infinity
422Infinite
423processes
424Logical
425Classical
426logic
427standards
428Axiomatic
429formulations
430Constructive
431approaches
432pedagogy
433See
434also
435Notes
436References
437External
438links
439Main
440article
441systematization
442knowledge
443Its
444improvement
445treatments
446rapidly
447recognized
448result
449there
450little
451interest
452preserving
453they
454nearly
455lost
456There
457books
458Books
459IV
460VI
461discuss
462Many
463triangle
464taken
465together
466manner
467right
468Book
469proposition
470angled
471square
472side
473subtending
474equal
475squares
476sides
477containing
478VII
479deal
480numbers
481treated
482geometrically
483lengths
484line
485segments
486regions
487Notions
488prime
489rational
490irrational
491infinitely
492XI
493XIII
494concern
495typical
496ratio
497cone
498cylinder
499same
500height
501base
502platonic
503solids
504constructed
505Postulate
506If
507intersect
508third
509way
510inner
511one
512then
513inevitably
514must
515each
516if
517extended
518far
519enough
520statements
521derived
522simple
523Until
524advent
525considered
526obviously
527would
528equally
529However
530reasoning
531assumptions
532conclusions
533remains
534valid
535independent
536their
537reality
538[4
539Near
540beginning
541book
542gives
543five
544postulates
545terms
546constructions
547translated
548Thomas
549Heath
550[5
551Let
552following
553postulated
554To
555straight
556produce
557extend
558finite
559continuously
560circle
561centre
562distance
563radius
564That
565another
566[The
567falling
568make
569interior
570produced
571indefinitely
572meet
573explicitly
574asserts
575existence
576implicitly
577assumed
578unique
579include
580common
581notions
582Things
583thing
584Transitive
585property
586relation
587equals
588added
589wholes
590Addition
591equality
592subtracted
593differences
594Subtraction
595coincide
596Reflexive
597whole
598greater
599part
600scholars
601agree
602do
603provide
604complete
605foundation
606required
607presentation
608[6
609extensive
610ancients
611others
612They
613aspired
614create
615absolutely
616certain
617simpler
618impossible
619can
620construct
621systems
622obeying
623false
624[7
625himself
626seems
627have
628being
629qualitatively
630different
631evidenced
632organization
633alternative
634formulated
635equivalent
636context
637Playfair
638axiom
639through
640given
641most
642drawn
643never
644meets
645clause
646needed
647remaining
648exists
649segment
650equilateral
651includes
652made
653drawing
654circles
655centered
656taking
657intersection
658vertex
659constructive
660Postulates
661assert
662uniqueness
663assertions
664nature
665we
666told
667things
668exist
669creating
670unmarked
671straightedge
672[8
673this
674concrete
675modern
676saying
677even
678cannot
679within
680[9
681Strictly
682speaking
683paper
684models
685defined
686rather
687instances
688width
689will
690Though
691consider
692nonconstructive
693just
694sound
695proofs
696supplanted
697fallacious
698Pythagoreans
699involved
700usually
701statement
702Find
703greatest
704measure
705[10
706contradiction
707allows
708superposition
709figure
710transferred
711congruence
712moving
713coincides
714proving
715add
716sixth
717rigidity
718[11
719fundamental
720types
721measurements
722scale
723unit
724degree
725referred
726half
727arbitrarily
728picks
729nonzero
730length
731expressed
732represented
733copied
734onto
735end
736similarly
737subtraction
738Measurements
739rectangle
740represents
741product
742Because
743interpretation
744multiplication
745limited
746direct
747interpreting
748four
749avoided
750products
751although
752IX
753left
754congruent
755while
756similar
757last
758neither
759Congruences
760alter
761location
762orientation
763leave
764unchanged
765like
766latter
767invariants
768studying
769essence
770refers
771pair
772planar
773volumes
774respectively
775stronger
776term
777idea
778entire
779size
780shape
781Alternatively
782moved
783top
784matches
785up
786exactly
787Flipping
788allowed
789Thus
790x6
791x4
792letter
793mirror
794image
795Figures
796except
797differing
798sizes
799Corresponding
800shapes
801corresponding
802proportion
803Points
804customarily
805named
806capital
807letters
808alphabet
809Other
810listing
811sufficient
812pick
813unambiguously
814relevant
815ABC
816typically
817vertices
818Angles
819whose
820complementary
821formed
822when
823ray
824shares
825pointed
826direction
827original
828rays
829form
830Supplementary
831normally
832measured
833degrees
834radians
835textbooks
836define
837separate
838semi
839discussing
840object
841extends
842locutions
843occasionally
844either
845curved
846specific
847necessary
848pons
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850bridge
851asses
852isosceles
853case
854always
855legs
856hypotenuse
857AC
858diameter
859further
860under
861[12
862frequent
863role
864test
865intelligence
866reader
867harder
868followed
869might
870resemblance
871steep
872sure
873footed
874donkey
875cross
876[13
877determined
878specifying
879SAS
880ASA
881adjacent
882AAS
883Specifying
884SSA
885yield
886distinct
887unless
888specified
889Triangles
890SSS
891AAA
892necessarily
893[14
894causes
895every
896acute
897obtuse
898celebrated
899opposite
900Miletus
901where
902supposed
903means
904Prop
905III
906[15
907[16
908proportional
909linear
910displaystyle
911propto
912cube
913various
914special
915cases
916[17
917parallelepipedal
918[18
919constants
920proportionality
921successor
922who
923circumscribing
924[19
925[icon
926section
927needs
928expansion
929You
930help
931adding
932March
933status
934impractical
935give
936representative
937sampling
938surveyor
939level
940Sphere
941packing
942applies
943stack
944oranges
945parabolic
946brings
947light
948suggested
949etymology
950word
951earliest
952reasons
953surveying
954[20
955long
956before
957formally
958[21
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960directly
961Historically
962chains
963Gunter
964chain
965graduated
966theodolite
967determination
968arrangements
969finding
970efficient
971spheres
972error
973detection
974correction
975Geometric
976optics
977analyze
978focusing
979lenses
980mirrors
981art
982architecture
983water
984tower
985hemisphere
986calculated
987design
988origami
989extensively
990classical
991solved
992[22
993Quite
994lot
995CAD
996aided
997CAM
998manufacturing
999based
1000Design
1001bounded
1002planes
1003cylinders
1004cones
1005tori
1006etc
1007present
1008day
1009essential
1010everything
1011including
1012cars
1013airplanes
1014ships
1015smartphones
1016few
1017decades
1018ago
1019sophisticated
1020draftsmen
1021learned
1022fairly
1023advanced
1024Pascal
1025Brianchon
1026But
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1028done
1029programs
1030Motor
1031partsolutions
1032gif
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1035depend
1036[23
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1038change
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1040translations
1041reflections
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1049words
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1068commentators
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1070implications
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1074rigorous
1075reformulations
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1079separation
1080issues
1081Interpreting
1082spirit
1083approach
1084elliptic
1085variety
1086topologies
1087torus
1088surface
1089placed
1090tomb
1091request
1092BCE
1093colorful
1094whom
1095historical
1096anecdotes
1097recorded
1098remembered
1099along
1100ancient
1101foundations
1102put
1103place
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1108equations
1109enunciated
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1111investigation
1112Ren
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1114Frans
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1120[29
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1122equation
1123expressing
1124definition
1125PQ
1126sqrt
1127p_
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1129defining
1130px
1131py
1132qx
1133qy
1134metric
1135metrics
1136restriction
1137second
1138order
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1142Desargues
1143perspective
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1163[31
1164Leading
1165geometers
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1169naturally
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1174tools
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1177failed
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1190power
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1194generalization
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1199eliminates
1200whence
1201become
1202meaningless
1203retaining
1204parallelism
1205equivalence
1206continue
1207midpoint
1208Carnot
1209bius
1210systematically
1211signed
1212simplifying
1213unifying
1214[33
1215significant
1216development
1217occurred
1218nos
1219Bolyai
1220Nikolai
1221Ivanovich
1222Lobachevsky
1223separately
1224[34
1225Since
1226provably
1227relatively
1228realized
1229ten
1230suffice
1231contains
1232assumption
1233therefore
1234very
1235shown
1236above
1237constructs
1238usual
1239endpoints
1240His
1241guarantee
1242actually
1243continuity
1244completeness
1245Starting
1246Moritz
1247Pasch
1248improved
1249proposed
1250best
1251Hilbert
1252[35
1253George
1254Birkhoff
1255[36
1256Tarski
1257[37
1258disproof
1259stars
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1262photographed
1263during
1264solar
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1266starlight
1267bent
1268Sun
1269gravity
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1271interpreted
1272evidence
1273favor
1274prediction
1275cause
1276deviations
1277involves
1278time
1279Minkowski
1280shows
1281showing
1282[38
1283due
1284weak
1285Earth
1286sun
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1288technology
1289capable
1290detecting
1291predicted
1292verified
1293observations
1294slight
1295bending
1296considerations
1297software
1298runs
1299GPS
1300[39
1301sometimes
1302distinguished
1303did
1304distinctions
1305interpret
1306implying
1307notion
1308infinitesimal
1309quantities
1310previously
1311Eleatic
1312nobody
1313able
1314firm
1315Zeno
1316occurring
1317resolved
1318satisfaction
1319exhaustion
1320infinitesimals
1321[40
1322Proclus
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1324questions
1325demanding
1326claimed
1327divisibility
1328odd
1329constituting
1330[41
1331At
1332Otto
1333Stolz
1334Paul
1335du
1336Bois
1337Reymond
1338Giuseppe
1339Veronese
1340controversial
1341Leibniz
1342[42
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1344Abraham
1345Robinson
1346provided
1347[43
1348reason
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1350physically
1351require
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1353inspect
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1355intersected
1356distant
1357inspection
1358potentially
1359take
1360amount
1361[44
1362formulation
1363induction
1364implicit
1365infinitude
1366primes
1367[45
1368Supposed
1369involving
1370series
1371predated
1372discussions
1373giving
1374expression
1375partial
1376sums
1377commenting
1378possibility
1379letting
1380attention
1381expert
1382Please
1383talk
1384parameter
1385template
1386explain
1387issue
1388WikiProject
1389Mathematics
1390recruit
1391December
1392June
1393Real
1394closed
1395frequently
1396traditional
1397assumes
1398automatically
1399Placing
1400preoccupation
1401[46
1402primitive
1403undefined
1404clearly
1405forward
1406Alessandro
1407Padoa
1408Peano
1409delegation
1410Paris
1411conference
1412[47
1413begin
1414formulate
1415imagine
1416symbols
1417completely
1418devoid
1419meaning
1420unproved
1421simply
1422conditions
1423imposed
1424Then
1425ideas
1426initially
1427chosen
1428ignored
1429replace
1430mind
1431satisfies
1432thus
1433empirical
1434psychological
1435regarded
1436abstraction
1437specialized
1438successively
1439replaced
1440interpretations
1441Essai
1442une
1443orie
1444alg
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1446des
1447nombre
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1450Introduction
1451logique
1452ductive
1453quelconque
1454hierarchical
1455framework
1456said
1457Bertrand
1458[48
1459our
1460hypothesis
1461anything
1462deductions
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1464subject
1465know
1466talking
1467nor
1468metaphysicians
1469Such
1470foundational
1471range
1472foundationalism
1473formalism
1474correct
1475lya
1476How
1477Solve
1478dissertation
1479Trinity
1480College
1481Cambridge
1482summarized
1483changing
1484minds
1485philosophers
1486[49
1487conflict
1488experiment
1489empiricism
1490requiring
1491experimental
1492input
1493became
1494clear
1495applicability
1496matter
1497deciding
1498applicable
1499goal
1500identifying
1501deduced
1502outstanding
1503objectives
1504avoiding
1505hidden
1506ramifications
1507confirmed
1508experimentally
1509protractor
1510relies
1511heavily
1512[50
1513[51
1514[52
1515[53
1516Alfred
1517students
1518elementary
1519[54
1520[55
1521algorithm
1522doesn
1523violate
1524del
1525apply
1526[56
1527decidability
1528fields
1529model
1530process
1531axiomatization
1532exemplified
1533reduces
1534predicate
1535Greeks
1536emphasized
1537solving
1538[57
1539vice
1540versa
1541adequately
1542richer
1543[58
1544Proofs
1545derivations
1546premises
1547justify
1548Analytic
1549hypotheses
1550argue
1551arrived
1552thought
1553felt
1554forced
1555him
1556indicated
1557reluctance
1558[59
1559arrival
1560[60
1561Andrei
1562Nicholaevich
1563Kolmogorov
1564[61
1565[62
1566precursor
1567[63
1568[64
1569conviction
1570truth
1571contributing
1572experienced
1573something
1574already
1575likely
1576remain
1577purposeless
1578eyes
1579Celia
1580Hoyles
1581curricular
1582shaping
1583Absolute
1584coordinate
1585Incidence
1586List
1587interactive
1588Metric
1589Non
1590Ordered
1591Type
1592Angle
1593bisector
1594Butterfly
1595Ceva
1596Heron
1597formula
1598Menelaus
1599Nine
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2329negligible
2330More
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2353rectilinear
2354Spacetime
2355schematic
2356G_
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2364Phenomena
2365EquationsFormalisms
2366ascribed
2367starting
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2494Quantum
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2500bringing
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2508QFT
2509gravitons
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2516captured
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2518Every
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2520surrounded
2521conceptualized
2522exerting
2523Assuming
2524symmetrical
2525comparable
2526towards
2527observable
2528numerically
2529varies
2530slightly
2531depending
2532latitude
2533features
2534mountains
2535ridges
2536unusually
2537low
2538sub
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2540purposes
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2548denoted
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2560ignoring
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2591rotation
2592furthest
2593Equator
2594poles
2595Under
2596simplifies
2597mg
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2600stroboscopic
2601flash
2602During
2603ball
2604mm
2605total
2606units
2607potential
2608Ep
2609mgh
2610Wh
2611Similarly
2612tfrac
2613maximum
2614reached
2615vertically
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2631maintains
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2633Planets
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2638Hanford
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2641Washington
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2736Anomalous
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2740solely
2741losing
2742radiating
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2744Photons
2745travelling
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2747stop
2748returning
2749cosmic
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2751background
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2756hydrogen
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