1. QUADRATRIX FIG. i. \t\fi. !\ \. * . i /°. 7 7. !/ /. 'MM. FIG. 2. of this class arethose of dinostratus and EW Tschirnhausen, which are both related to the circle. http://99.1911encyclopedia.org/Q/QU/QUADRATRIX.htm

document.write(""); QUADRATRIX QUADRATRIX (from Lat. quadrator, squarer), in mathe matics, a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves FIG. i. ;/ i /° 'MM FIG. 2. of this class are those of Dinostratus and E. W. Tschirnhausen, which are both related to the circle. The cartesian equation to the curve is y = x cot —. which shows that the curve is symmetrical about the axis of y, and that it consists of a central portion flanked by infinite branches (fig. 2). The asymptotes are *= *=2na, n being an integer. The intercept on the axis of y is 2a/x; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected. The curve also permits the solution of the problems of duplicating a cube (q.v.) and trisecting an angle. The quadratrix of Tschirnhausen is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before. The mutual intersections of the lines drawn from the points of division of the arc parallel to AB, and the lines drawn parallel to BC through the points of division of AB, are points on the quadratrix (fig. 3). The cartesian equation is y — a cos Trx/2a. The curve is periodic, and cuts the axis of x at the points #= =*=(2n-i)a, n being an integer; the maximum values of y are =*=a. Its properties are similar to those of the quadratrix of Dinostratus.

2. History Of Mathematics: Chronology Of Mathematicians 350?). Thymaridas (c. 350). dinostratus (fl. c. 350) *SB http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

Chronological List of Mathematicians Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan Table of Contents 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below List of Mathematicians 1700 B.C.E. Ahmes (c. 1650 B.C.E.) *MT 700 B.C.E. Baudhayana (c. 700) 600 B.C.E. Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) 500 B.C.E. Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *MT Zeno of Elea (c. 490-c. 430) *MT Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *MT Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *MT Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB

3. History Of Mathematics: Greece Details development of mathematics in Greece. Includes maps, list of mathematicians, sources, and bibliography. 350?). Thymaridas (c. 350). dinostratus (c. 350). Speusippus (d. http://aleph0.clarku.edu/~djoyce/mathhist/greece.html

Greece Cities Abdera: Democritus Alexandria : Apollonius, Aristarchus, Diophantus, Eratosthenes, Euclid , Hypatia, Hypsicles, Heron, Menelaus, Pappus, Ptolemy, Theon Amisus: Dionysodorus Antinopolis: Serenus Apameia: Posidonius Athens: Aristotle, Plato, Ptolemy, Socrates, Theaetetus Byzantium (Constantinople): Philon, Proclus Chalcedon: Proclus, Xenocrates Chalcis: Iamblichus Chios: Hippocrates, Oenopides Clazomenae: Anaxagoras Cnidus: Eudoxus Croton: Philolaus, Pythagoras Cyrene: Eratosthenes, Nicoteles, Synesius, Theodorus Cyzicus: Callippus Elea: Parmenides, Zeno Elis: Hippias Gerasa: Nichmachus Larissa: Dominus Miletus: Anaximander, Anaximenes, Isidorus, Thales Nicaea: Hipparchus, Sporus, Theodosius Paros: Thymaridas Perga: Apollonius Pergamum: Apollonius Rhodes: Eudemus, Geminus, Posidonius Rome: Boethius Samos: Aristarchus, Conon, Pythagoras Smyrna: Theon Stagira: Aristotle Syene: Eratosthenes Syracuse: Archimedes Tarentum: Archytas, Pythagoras Thasos: Leodamas Tyre: Marinus, Porphyrius Mathematicians Thales of Miletus (c. 630-c 550)

4. Earliest Known Uses Of Some Of The Words Of Mathematics (Q) but it became known as a quadratrix when dinostratus used it for the quadrature of a circle (DSB, article "dinostratus"; http://members.aol.com/jeff570/q.html

Earliest Known Uses of Some of the Words of Mathematics (Q) Last revision: May 27, 2002 Q. E. D. Euclid (about 300 B. C.) concluded his proofs with hoper edei deiksai, which Medieval geometers translated as quod erat demonstrandum ("that which was to be proven"). In 1665 Benedictus de Spinoza (1632-1677) wrote a treatise on ethics, Ethica More Geometrico Demonstrata, in which he proved various moral propositions in a geometric manner. He wrote the abbreviation Q. E. D., as a seal upon his proof of each ethical proposition. The Q. E. D. abbreviation was also used by Isaac Newton in the Principia, by Galileo in a Latin text, and by Isaac Barrow, who additionally used quod erat faciendum (Q. E. F.), quod fieri nequit (Q. F. N.), and quod est absurdum (Q. E. A.). [Martin Ostwald, Sam Kutler, Robin Hartshorne, David Reed] QUADRANGLE is found in English in the fifteenth century. The word was later used later by Shakespeare. QUADRATIC is derived from the Latin quadratus, meaning "square." In English, quadratic was used in 1668 by John Wilkins (1614-1672) in An essay towards a real character, and a philosophical language

5. Chronology Of Mathematicians 350 MENAECHMUS CONIC SECTIONS. -350 dinostratus QUADRATRIX. -335 EUDEMUS HISTORY OF GEOMETRY http://www.erols.com/bram/timeline.html

CHRONOLOGY OF MATHEMATICIANS -1100 CHOU-PEI -585 THALES OF MILETUS: DEDUCTIVE GEOMETRY PYTHAGORAS : ARITHMETIC AND GEOMETRY -450 PARMENIDES: SPHERICAL EARTH -430 DEMOCRITUS -430 PHILOLAUS: ASTRONOMY -430 HIPPOCRATES OF CHIOS: ELEMENTS -428 ARCHYTAS -420 HIPPIAS: TRISECTRIX -360 EUDOXUS: PROPORTION AND EXHAUSTION -350 MENAECHMUS: CONIC SECTIONS -350 DINOSTRATUS: QUADRATRIX -335 EUDEMUS: HISTORY OF GEOMETRY -330 AUTOLYCUS: ON THE MOVING SPHERE -320 ARISTAEUS: CONICS EUCLID : THE ELEMENTS -260 ARISTARCHUS: HELIOCENTRIC ASTRONOMY -230 ERATOSTHENES: SIEVE -225 APOLLONIUS: CONICS -212 DEATH OF ARCHIMEDES -180 DIOCLES: CISSOID -180 NICOMEDES: CONCHOID -180 HYPSICLES: 360 DEGREE CIRCLE -150 PERSEUS: SPIRES -140 HIPPARCHUS: TRIGONOMETRY -60 GEMINUS: ON THE PARALLEL POSTULATE +75 HERON OF ALEXANDRIA 100 NICOMACHUS: ARITHMETICA 100 MENELAUS: SPHERICS 125 THEON OF SMYRNA: PLATONIC MATHEMATICS PTOLEMY : THE ALMAGEST 250 DIOPHANTUS: ARITHMETICA 320 PAPPUS: MATHEMATICAL COLLECTIONS 390 THEON OF ALEXANDRIA 415 DEATH OF HYPATIA 470 TSU CH'UNG-CHI: VALUE OF PI 476 ARYABHATA 485 DEATH OF PROCLUS 520 ANTHEMIUS OF TRALLES AND ISIDORE OF MILETUS 524 DEATH OF BOETHIUS 560 EUTOCIUS: COMMENTARIES ON ARCHIMEDES 628 BRAHMA-SPHUTA-SIDDHANTA 662 BISHOP SEBOKHT: HINDU NUMERALS 735 DEATH OF BEDE 775 HINDU WORKS TRANSLATED INTO ARABIC 830 AL-KHWARIZMI: ALGEBRA 901 DEATH OF THABIT IBN - QURRA 998 DEATH OF ABU'L - WEFA 1037 DEATH OF AVICENNA 1039 DEATH OF ALHAZEN

6. Lebensdaten Von Mathematikern Dini, Ulisse (14.11.1845 28.10.1918). dinostratus (um 390 - um 320 v. Chr.) http://www.mathe.tu-freiberg.de/~hebisch/cafe/lebensdaten.html

Diese Seite ist dem Andenken meines Vaters Otto Hebisch (1917 - 1998) gewidmet. By our fathers and their fathers in some old and distant town from places no one here remembers come the things we've handed down. Marc Cohn Dies ist eine Sammlung, die aus verschiedenen Quellen stammt, u. a. aus Jean Dieudonne, Geschichte der Mathematik, 1700 - 1900, VEB Deutscher Verlag der Wissenschaften, Berlin 1985. Helmut Gericke, Mathematik in Antike und Orient - Mathematik im Abendland, Fourier Verlag, Wiesbaden 1992. Otto Toeplitz, Die Entwicklung der Infinitesimalrechnung, Springer, Berlin 1949. MacTutor History of Mathematics archive A B C ... Z Abbe, Ernst (1840 - 1909) Abel, Niels Henrik (5.8.1802 - 6.4.1829) Abraham bar Hiyya (1070 - 1130) Abraham, Max (1875 - 1922) Abu Kamil, Shuja (um 850 - um 930) Abu'l-Wafa al'Buzjani (940 - 998) Ackermann, Wilhelm (1896 - 1962) Adams, John Couch (5.6.1819 - 21.1.1892) Adams, John Frank (5.11.1930 - 7.1.1989) Adelard von Bath (1075 - 1160) Adler, August (1863 - 1923) Adrain, Robert (1775 - 1843)

7. Dinostratus dinostratus. Born about 390 BC in Greece Died about 320 BC. dinostratusis mentioned by Proclus who says (see for example 1 or 3). http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Dinostratus.html

Dinostratus Born: about 390 BC in Greece Died: about 320 BC Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Dinostratus is mentioned by Proclus who says (see for example [1] or [3]):- Amyclas of Heraclea, one of the associates of Plato , and Menaechmus , a pupil of Eudoxus who had studied with Plato , and his brother Dinostratus made the whole of geometry still more perfect. It is usually claimed that Dinostratus used the quadratrix, discovered by Hippias , to solve the problem of squaring the circle Pappus tells us (see for example [1] or [3]):- For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming in other words the quadratrix It appears from this quote that Hippias discovered the curve but that it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Proclus , who claims to be quoting from Eudemus , writes (see [1]):- Nicomedes trisected any rectilinear angle by means of the conchoidal curves, of which he had handed down the origin, order, and properties, being himself the discoverer of their special characteristic. Others have done the same thing by means of the quadratrices of

8. Dinostratus Biography of dinostratus (390BC320BC) dinostratus. Born about 390 BC in Greece http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Dinostratus.html

Dinostratus Born: about 390 BC in Greece Died: about 320 BC Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Dinostratus is mentioned by Proclus who says (see for example [1] or [3]):- Amyclas of Heraclea, one of the associates of Plato , and Menaechmus , a pupil of Eudoxus who had studied with Plato , and his brother Dinostratus made the whole of geometry still more perfect. It is usually claimed that Dinostratus used the quadratrix, discovered by Hippias , to solve the problem of squaring the circle Pappus tells us (see for example [1] or [3]):- For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming in other words the quadratrix It appears from this quote that Hippias discovered the curve but that it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Proclus , who claims to be quoting from Eudemus , writes (see [1]):- Nicomedes trisected any rectilinear angle by means of the conchoidal curves, of which he had handed down the origin, order, and properties, being himself the discoverer of their special characteristic. Others have done the same thing by means of the quadratrices of

9. References For Dinostratus References for dinostratus. The URL of this page is http//wwwhistory.mcs.st-andrews.ac.uk/history/References/dinostratus.html. http://www-gap.dcs.st-and.ac.uk/~history/References/Dinostratus.html

References for Dinostratus Biography in Dictionary of Scientific Biography (New York 1970-1990). Books: G J Allman, Greek geometry from Thales to Euclid (Dublin, 1889). T L Heath, A History of Greek Mathematics I (Oxford, 1921). B L van der Waerden, Science awakening (Groningen, 1954). Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR April 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Dinostratus.html

10. References For Dinostratus References for the biography of dinostratus References for dinostratus. Biography in Dictionary of Scientific Biography (New York 19701990). http://www-groups.dcs.st-and.ac.uk/history/References/Dinostratus.html

References for Dinostratus Biography in Dictionary of Scientific Biography (New York 1970-1990). Books: G J Allman, Greek geometry from Thales to Euclid (Dublin, 1889). T L Heath, A History of Greek Mathematics I (Oxford, 1921). B L van der Waerden, Science awakening (Groningen, 1954). Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR April 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Dinostratus.html

11. Dinostratus dinostratus. Born about 390 BC in Greece Died about 320 BC. Showbirthplace location dinostratus is mentioned by Proclus who says. http://sfabel.tripod.com/mathematik/database/Dinostratus.html

Dinostratus Born: about 390 BC in Greece Died: about 320 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Dinostratus is mentioned by Proclus who says Amyclas of Heraclea, one of the associates of Plato , and Menaechmus , a pupil of Eudoxus who had studied with Plato , and his brother Dinostratus made the whole of geometry still more perfect. Dinostratus used the quadratrix, discovered by Hippias , to solve the problem of squaring the circle. Pappus tells us For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming in other words the quadratrix. It appears that Hippias discovered the curve but it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Dinostratus probably did much more work on geometry but nothing is known of it. References (2 books/articles) References elsewhere in this archive: Show me the quadratrix Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page History Topics Index Famous curves index ... Search Suggestions JOC/EFR December 1996

12. Dictionary: Dinostratus dinostratus, or of Tschirnhausen. http://www.hyperdictionary.com/dictionary/Dinostratus

Hyper Dictionary Dictionary Results About this site News Linkers Top Searches ... Webstats Dictionary Thesaurus Look up this word in the thesaurus: Dinostratus Dinostratus No definitions and no ... Dinostratus

13. Dinostratus dinostratus. It is usually claimed that dinostratus used the quadratrix,discovered by Hippias, to solve the problem of squaring the circle. http://www.math.hcmuns.edu.vn/~algebra/history/history/Mathematicians/Dinostratu

14. References For Dinostratus References for dinostratus. Biography in Dictionary of http//wwwhistory.mcs.st-andrews.ac.uk/history/References/dinostratus.html. http://www.math.hcmuns.edu.vn/~algebra/history/history/References/Dinostratus.ht

15. Footnotes . . . . .dinostratus dinostratus showed how to square the circleusing the trisectrix . . . . . http://www.math.tamu.edu/~don.allen/history/greekorg/footnode.html

Theodorus proved the incommensurability of , , , ...,. Archytas solved the duplication of the cube problem at the intersection of a cone, a torus, and a cylinder. ...histories Here the most remarkable fact must be that knowledge at that time must have been sufficiently broad and extensive to warrant histories ...Anaximander Anaximander further developed the air, water, fire theory as the original and primary form of the body, arguing that it was unnecessary to fix upon any one of them. He preferred the boundless as the source and destiny of all things. ...Anaximenes Anaximenes was actually a student of Anaximander. He regarded air as the origin and used the term 'air' as god ...proofs. It is doubtful that proofs provided by Thales match the rigor of logic based on the principles set out by Aristotle found in later periods. ...incommensurables. The discovery of incommensurables brought to a fore one of the principle difficulties in all of mathematics - the nature of infinity. ...discovered as attested by Archimedes. However, he did not rigorously prove these results. Recall that the formula for the volume pyramid was know to the Egyptians and the Babylonians. ...Persians. This was the time of Pericles. Athens became a rich trading center with a true democratic tradition. All citizens met annually to discuss the current affairs of state and to vote for leaders. Ionians and Pythagorean s were attracted to Athens. This was also the time of the conquest of Athens by Sparta.

16. Math Forum - Geometry.pre-college quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by dinostratus in 350 BC (MacTutor Archive). http://forum.swarthmore.edu/~sarah/HTMLthreads/geopre.descriptions.html

From the Geometry Forum Newsgroup Archives geometry.pre-college - brief descriptions About newsgroups Search newsgroups All geometry newsgroups [Switch to the Outline Version] ... Probability Problem: Broken Sticks (Archie Benton) 11/29/96 An 8-foot stick and a 22-foot stick are both randomly broken into two parts. What is the probability that the longer part of the 8-foot stick is longer than the shorter part of the 22-foot stick? Software and Hardware Advice (Carol Hattan) 11/07/96 Seeking advice on software and hardware for a high school math/science/technology lab. Equation of an Ellipse (Harold Stanley) 11/02/96 Can you always find the equation of an ellipse, knowing only two points on the ellipse and assuming that the ellipse is centered at the origin? If it is possible, how do you work it out mathematically? Geometry of Eggs (John A Benson) 10/18/96 Is it true that there is no equation, or system of equations, to describe the shape of an ordinary egg? [supereggs, Piet Hein] Lenart Sphere (Dennis Wallace) 10/16/96 Anyone interested in communicating with students about problems on the Lenart Sphere.? Anyone using it to help teach non-Euclidean geometry at the high school level?

17. Quadratrice De Dinostrate Translate this page QUADRATRICE DE DINOSTRATE dinostratus' (or Hippias') quadratrix,Quadratrix des dinostratus (oder des Hippias). Courbe étudiée http://www.mathcurve.com/courbes2d/dinostrate/dinostrate.shtml

courbe suivante courbes 2D courbes 3D surfaces ... fractals QUADRATRICE DE DINOSTRATE Dinostratus' (or Hippias') quadratrix, Quadratrix des Dinostratus (oder des Hippias) Autre nom : sectrice d'Hippias. La quadratrice de Dinostrate O ) et une composante de son vecteur vitesse constante (ici, la composante sur Oy O d'une Comme son nom l'indique, cette courbe est une quadratrice ; en effet : n -sectrice ; en effet courbe suivante courbes 2D courbes 3D surfaces ... Jacques MANDONNET

18. Academia THE ACADEMY 1, History. 1. dinostratus THE SQUARING OF THE CIRCLE.dinostratus proved that the trisectrix of Hippias could be used http://descartes.cnice.mecd.es/ingles/maths_workshop/A_history_of_Mathematics/Gr

THE ACADEMY 1 History DINOSTRATUS THE SQUARING OF THE CIRCLE Dinostratus proved that the trisectrix of Hippias could be used to solve this problem after discovering that the side of the square is the mean proportional between the arc of the quarter circle AC and the segment DQ. There are various stages to the reductio ad absurdum proof which are illustrated in the following windows: Let the circle with centre D and radius DR intersect the trisectrix at S and the side of the square at T. Draw the perpendicular SU to side DC from point S. As the arcs are proportional to the radii then AC/AB=TR/DR (2) From (1) and (2) it must follow that TR=AB (3) S is the point on the trisectrix which satisfies TR/SR=AB/SU (4) From (3) and (4) it follows that SR=SU However, this is absurd as the perpendicular is the shortest distance between a point and a line. Therefore, DR cannot be longer than DQ. 2.- We repeat this way of reasoning with the hypothesis

19. ACADEMIA_INDICE and the parabola. This discovery allowed the duplication of the cubeto be calculated. dinostratus (350 BC), He proved the squaring http://descartes.cnice.mecd.es/ingles/maths_workshop/A_history_of_Mathematics/Gr

PLATO: THE ACADEMY History 1. BACKGROUND TO THE PERIOD Aristotle and Plato in the centre of Raphael's painting "The School in Athens". The Vatican Museum. The Peloponnesian Wars took place in the IVth century B.C. Sparta fought against Athens and behind them other Greek towns followed them into warfare. Sparta called on Persia to help them keep control of the towns they had occupied. Athens and Thebes became allies and together managed to defeat Sparta. King Philip of Macedon took advantage of the situation and became ruler of Greece. His reign lasted from 360 B.C. to 336 B.C. when, upon his death, his son Alexander took the throne. Alexander the Great was responsible for the invasion of the Persian empire, which included Syria, Palestine, Egypt, Mesopotamia and Iran. This century began with the death of Socrates (399 B.C.) The two great philosophers Aristotle and Plato , one of Socrates students and admirers also belonged to this period along with Archytas. Aristotle was Alexander the Great's private tutor and instilled in him the superiority of the Hellenic culture and encouraged him to go East and extend his empire. Plato managed to bring the greatest thinkers of the time together at his Academy in Athens. His contributions to mathematics include his rigorous method of justifying solutions through logical reasoning, his

20. Squaring The Circle in this archive. Hippias and dinostratus are associated with themethod of squaring the circle using a quadratrix. The curve it http://www2.ittu.edu.tm/math/bosna.net/Geometry/papers/Squaring the circle/Squar

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