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1998 10 5 5 http://math.ntnu.edu.tw/horng

1970 International Study Group on the Relations beMathematics, HPM HPM

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tween the History and Pedagogy of

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1. PDFe-mail suhui_yu@yahoo.com.tw

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e-mailsuhui_yu@yahoo.com.tw4. http://math.ntnu.edu.tw/horng/letter/hpmletter.htm5.

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mailto:suui_hui_yuu@yahooyahoo..com.twmailto:Ze-mailsuhui_yu@yahoomail.topspeed.comailto:Ze-mailsuhui_yu@yahoomail.topspeed.cohttp://math.ntnu.edu.tw/~horng

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1987

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1980

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1966

Klein

biogenetic

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e1 e1 e e

( , ) ( , )d P F e d P L=

2 4y c= x2 2

2 2 1x ya b

+ = 2 2

2 2 1x ya b

=

A. Quetelet G. Dandelin 19

a x a 2a x, y

33 2ax =

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ay

yx

xa

2==

x

x y Menaechmus( 350B.C.)

yx

xa=

ay

xa

2= ayx =2 21 x

ay =

axy 22 = 221 ya

x =

y x x y ()(Conics)

(Apollonius of Perga, 262B. C. ~ 190B. C.)

(diameters)

Of any curved line which is in one plane I call that straight line the diameter which, drawn from the curved line, bisects all straight lines drawn to this curved line parallel to some straight line;

111213 ABCBC

ABCABC [I. 3]DEDEBCDFEFG [I. 7 and def. 4]ACFFHFGBC : BA, AC:: FH : FA K K KL DE KLHF, FL1

DFEAnd let such a section be called a parabola

K y KLx LF pHF (parameter)2y p= x

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(latus rectum) 12 ABCABC [I. 3]DEDEBCDFEFG [I. 7 and def. 4]FGACHAKAFGBCKFFGFL KA : BK, KC : : FH : FL M M MN DE N NOX FL HL X LOXP FN MN FX (applied to) FLFNLXHF, FL2

DFE And let such a section be call an hyperbola

M y MN

x FN 22py px xa

x= + pFL

(latus rectum)2a=HF 13 ABCABC

(subcontrariwise)DE

FGBCED [I. 7 and Def. 4]EEHEDAAKED AKBK, KC : : DEEH

L L LM FG LM (applied to) EHEMDE, EH3

And let such a section be call an ellipse L y LM

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x EM 22py px xa

x= pEH

(latus rectum)2a=ED 11 And let such a section be called a parabola, and let HF be called the straight line to which the straight lines drawn ordinatewise to the diameter FG are applied in square and let it also be called the upright side ().

parameter 11 FH 12 FL 13 EH the straight lines drawn ordinatewise to the diameter are applied in squarethe parameter of the ordinatewise to the diameter (parabola) KL FL HF HF (hyperbola)MN FN FL (ellipse)LM EM EH

parabola., hyperbola, ellipse

application of areas

(fall short) (exceed) (fit)elleipsis, defecthyperbole, excessparabole, a placing beside

11 upright side latus rectum

a : b :: c: d:: analogia M. Fried

(figure) upright side

22py px xa

=

22py px xa

=

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1. 11

(the straight line cut off by it on the diameter beginning from the sections vertex) (another straight line which has the ratio to the straight line between the angle of the cone and the vertex of the section that the square on the base of the axial triangle has to the rectangle contained by the remaining two sides of the triangle.) (And let such a section be called a parabola.)

2. 12

(applied to) (the straight line added along the diameter of the section and subtending the exterior angle of the triangle)

(the parameter) 2 (And let such a section be call an hyperbola)

3. 13

(subcontrariwise)

(applied to)

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(the parameter) (And let such a section be call an ellipse)

Apollonius (1952), Conics (tr. R. C. Taliaferro), in Great Books of the Western World, Encyclopaedia Britannica.

Bunt, L. N. H. et al (1988), The Historical Roots of Elementary Mathematics. New York: Dover. Eves, H. (1976), An Introduction to the History of Mathematics, New York: Holt, Rinehart and

Winston.

Fauvel, J and J. Gray ed.( 1987), The History of Mathematics: A Reader. London: The Open University.

Fried, M. (2003), The Use of Analogy in Book VII of Apollonius Conica, in Science in Context 16(3).

Isoda, Masami (2000), The Use of Technology in Teaching Mathematics with History--Teaching with modern technology inspired by the history of mathematics, in Proceedings of the HPM 2ooo Conference. Taipei: Department of Mathematics NTNU.

Lui, K.W. (2003), Study of Conic Sections and Prime Numbers in China: Cultural Influence on The Development, Application and Transmission of Mathematical Ideas, The University of Hong Kong.

Van Maanen, Jan A (1995), Alluvial Deposits, Conic Sections, and Improper Glasses, or History of Mathematics Applied in the Classroom, in F. Swetz et al eds., Learn From The Masters!. Washington, DC: The Mathematical Association of America.

Kline, M. (1983),

(2000) (1995)

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(The History of Mathematics) (Richard Mankiewicz) 2004 2 253 450 ISBN957-776-588-2

(Ian Stewart) =

(Mathophobia)

24

1

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35000 29

(1/21/3) ( 2/3 )

() 1400

(Vedic period) (Vedas)

(Timaeus)

()(Pythagoreanism) 2

(diameter) (radius)

3

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(al-Khwarizmi) (al-Karaji) (Omar Khayyam)

(Piero della Francesca)

(Albrecht Durer)

(John Napier) (Francis Bacon) (Girolamo Cardano) (Rene Descartes)(Nicolas Copernicus) (Johannes Kepler)

1202 (Leonardo de Pisa, c. 1180-c.1256, (Fibonacci) (Liber abbaci Book of the abbacus)abbacus b()(abacus)

Leonardo de Pisa Pisa Leonardode

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fromf

abbacus abacus () abbacus abacus bbbLiber abbaci

2r

2r

(Euclid)(Euler)

156 1929

1829

( Thomas L. Heath )

(infinity)

Thomas L. Heath

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

(indefinitely) (infinity)

HPM

1.

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2.

Morris Kline

2004

3.

2001