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 Dinostratus:     more detail

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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.

Extractions: 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

Extractions: 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 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 Ahmes (c. 1650 B.C.E.) *MT Baudhayana (c. 700) 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) 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

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

Extractions: 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

Extractions: 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

Extractions: 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)

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

Extractions: 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

Extractions: 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
http://www-gap.dcs.st-and.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

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

Extractions: 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

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

Extractions: 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://www.mathcurve.com/courbes2d/dinostrate/dinostrate.shtml

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

Extractions: 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